Charged-current neutrino interactions with nucleons and nuclei at intermediate energies

Cha ged-cu en neu ino in e ac ions wi h nucleons and nuclei a in e media e ene gies TESIS DOCTORAL Guille mo Daniel Megías Vázquez Depa amen o de Física A ómica, Molecula y Nuclea Facul ad de Física UNIVERSIDAD DE SEVILLA Mayo 2017 Cha ged-cu en neu ino in e ac ions wi h nucleons and nuclei a in e media e ene gies Memo ia de Tesis Doc o al ealizada po Guille mo Daniel Megías Vázquez pa a op a al g ado de Doc o en Física Nuclea Di igida po los Doc o es Ma ia B. Ba ba o Juan A. Caballe o Ca e e o Depa amen o de Física A ómica, Molecula y Nuclea Facul ad de Física UNIVERSIDAD DE SEVILLA Mayo 2017 Ag adecimien os Más de cua o años de abajo decaen a su es ado undamen al en es as páginas, las cuales cons i- uyen una en ana abie a hacia un mundo, el de la in es igación, donde muchas eces lo úl imo que uno sabe es po donde empeza . Sob e odo, cuando uno se en en a a la di ícil a ea de sin e- iza en unas b e es líneas la g a i ud hacia odas las pe sonas que, de una u o a o ma, han hecho posible es e ilusionan e p oyec o. En p ime luga , quie o ene unas palab as de ag adecimien o con mi di ec o de esis, Juan An onio Caballe o, po la con ianza deposi ada en mí pa a el desa ollo de es e abajo y su de- dicación e in e és cons an e en lle a es e p oyec o adelan e. Un excelen e in es igado , an o a ni el p o esional como pe sonal, que además ha sido p o agonis a en mi o mación académica desde que u ie a el place de dis u a de sus clases ha á ya unos años. Recue do cuando en la asigna u a de Física de Pa ículas, p opuso como abajo de in es igación “adop a una pa ícula”. En esos momen os mi duda osciló en e acoge al neu ino o al muón. Lo que yo no imaginaba po aquel en onces e a que pasados a ios años es a ía iendo en su madu ez a esas pa ículas lige as a las que un día i nace . Del mismo modo, ag adezco especialmen e a mi codi ec o a de esis, Ma ia Ba ba o, su apoyo y dedicación since a du an e es os años así como su hospi alidad du an e mis es ancias en Tu ín. A odo ello, hay que añadi el innegable es ue zo de habe co egido oda es a esis desde la dis ancia y su ayuda y apidez siemp e que he necesi ado algún in o me pa a los dis in os en edos bu oc á i- cos. Mención especial ambién me ecen Quique Ama o y Nacho Ruiz Simó de la Uni e sidad de G anada con quienes he colabo ado en los úl imos meses, así como la gene osidad y hospi alidad de Ca lo a Gius i du an e mi b e e es ancia en la Uni e sidad de Pa ia. De odos ellos gua do g andes ecue dos. Si bien no apa ece implíci amen e como di ec o de es a esis, g an pa e del conocimien o adqui ido du an e es a e apa y plasmado en es as páginas se lo debo a Bill Donnelly, de quien admi o p o undamen e su g an sabidu ía en el campo de la Física y a quien ag adezco sus ideas y su since o in e és du an e mi es ancia en el MIT. La hospi alidad de Bill así como la de su muje , Ba ba a, hicie on de mi expe iencia en Bos on un ecue do imbo able. En el plano económico, caben menciona las ayudas ecibidas po los dis in os p oyec os de in es igación de la Uni e sidad de Se illa y del Minis e io de Economía y Compe i i idad (an e- io men e de Ciencia), así como a los con a os p edoc o ales de la Fundación Cáma a y de los P oyec os de Excelencia de la Jun a de Andalucía, pe o cabe más aún ag adece hones amen e el es ue zo de los dis in os unciona ios y, en gene al, el de los abajado es, que con sus impues os con ibuyen a man ene la cada ez más complicada labo cien í ica en es e país. Tampoco puedo ob ia en es as líneas el indudable alo humano y p o esional de odos los in- es igado es que he conocido en dis in os cong esos a lo la go del mundo, donde he adqui ido una isión más amplia de la in es igación en sus dis in os ámbi os. Asimismo, en el plano más local, engo que ag adece al Clus e po aguan a con ida y a odos los compañe os del depa amen o el buen ambien e en el abajo y ue a de él, po los almue zos y los ca és jun os y po los cada ez más ol idados ie nes de nachos y u bolín que me hicie on e de una o ma más “ ealis a” los p ocesos de dispe sión. Pe o no sólo el mundo de la Física ha con ibuido a la ealización de es e p oyec o. Es a memo ia no hubie a sido posible sin esas pe sonas que dan un alo medible al obse able que es la ida. A los que es án incluso cuando no he enido iempo pa a queda , y a los que en cie o modo me end é que ol e a p esen a . Po ello, no puedo e mina es as páginas sin ag adece a quienes es án siemp e ahí pa a lee es ás líneas (aunque sólo in e accionen débilmen e con ellas) y a los que ya no es án pa a e las desde an ce ca. En especial a mi mad e y a mi pad e po ayuda me y enseña me a llega has a el día de hoy; y a Bibi po su ca iño, sus ánimos y po aguan a odos mis muones. Abs ac Nowadays, he in e es in neu inos ex ends o a la ge a ie y o ields in As ophysics, Nuclea Physics and Pa icle Physics. One o he open ques ions in heo e ical physics is he desc ip ion o neu ino oscilla ions o which an accu a e in e p e a ion o neu ino-nucleus eac ions is c ucial. In his con ex , ecen yea s ha e wi nessed an in ense expe imen al and heo e ical ac i i y o de- e mine he p ope ies o neu inos and hei in e ac ion wi h ma e . This PhD Thesis is hus ocused on he analysis o cha ged-cu en neu ino-nucleus eac ions a kinema ics o in e es o neu ino oscilla ion expe imen s, whe e he neu ino ene gy is ypically in he GeV egion. Addi ionally, weak in e ac ions in he nuclea medium a in e media e ene gies a e an ex ao dina y oppo uni y o s udy he dynamics o he nuclea many-body sys em, beyond he in o ma ion accesible om elec on and had on p obes, and o gain a deepe knowledge o he axial s uc u e and he s angeness con en o he nucleons. In any accele a o -based neu ino oscilla ion expe imen , neu inos a e p oduced as he decay p oduc s o successi e eac ions, hus implying a wide- anged ene gy beam. Hence, when in- e ac ing wi h he nuclea ma e , a la ge a ie y o nuclea e ec s come in o play, going om quasielas ic sca e ing o deep inelas ic p ocesses, mul i-nucleon exci a ions o meson p oduc ion ia nucleon esonances. Acco dingly, obus models ha p ope ly desc ibe neu ino-nucleus in- e ac ions o e he whole expe imen al ange (o he o de o 10s o MeV up o 10s o GeV) a e equi ed o he expe imen al analyses. No ice also ha he kinema ics in ol ed demand a ela- i is ic desc ip ion o he mic oscopic nuclea s uc u e. In his hesis, he analysis o hese p ocesses a e add essed by using ealis ic models ha p o- ide, wi hin a ully ela i is ic amewo k, an accu a e desc ip ion o he di e en eac ion mecha- nisms o ele ance o neu ino oscilla ion measu emen s. We begin analyzing neu ino sca e ing o ee nucleons and desc ibing he weak had onic esponses oge he wi h he inne s uc u e o he nucleons. Wi h he aim o achie eing a consis en analysis o cha ged-cu en quasielas ic (CCQE) neu ino in e ac ions wi h nuclei, we p esen he so-called SuSA 2 model, which is based on he supe scaling beha io exhibi ed by elec on sca e ing da a and makes use o he ela i is ic mean ield (RMF) heo y o desc ibe he nuclea e ec s a ising in neu ino-nucleus in e ac ions. This p esc ip ion accoun s o he inal-s a e in e ac ions (FSI) be ween he ou going nucleon and he esidual nucleus and allows o a desc ip ion in e ms o he di e en iso ec o /isoscala and axial/ ec o eac ion channels ha play a ole in weak in e ac ions. A e y high kinema ics, whe e FSI a e negligible, we app oach ou model o he ela i is ic plane wa e impulse app oxima ion (RPWIA) whe e no FSI a ec he ou going nucleon. Fu hemo e, a basic ea u e in his hesis also conce ns he e alua ion o mul i-nucleon exci- a ions, in pa icula wo-body meson exchange cu en s (2p-2h MEC) con ibu ions, which a e p o ed o be an essen ial ing edien o in e p e neu ino c oss sec ion measu emen s a in e me- 2 1. INTRODUCTION deciphe able by using elec on o pho on sca e ing. In his sense, he cha ged-cu en neu ino- nucleon sca e ing (CC ν-N) is he mo e e icien way o analyze he nucleon axial o m ac o . This subjec is add essed in de ail in his hesis. Mo eo e , he s udy o neu al-cu en neu ino- nucleon p ocesses (NC ν-N) shed ligh on he s ange sea qua k-gluon e ec s on he nucleon. Despi e i s ele ance in se e al ields o Physics, neu inos a e s ill elusi e pa icles. They only in e ac h ough weak o ces and a e ha dly de ec able, so hey can only be clea ly obse ed h ough he de ec ion o he seconda y pa icles p oduced in he p ocess. Fo his eason, hea y a ge s a e o en employed in neu ino expe imen s in o de o inc ease neu ino-nucleus c oss sec ions. A he same ime, a p ope heo e ical unde s anding o he weak nuclea esponse is a p e equisi e o he analysis o cu en and u u e neu ino oscilla ion expe imen s. The e o e, a p ecise s udy o neu ino oscilla ions equi es an adequa e desc ip ion o hei in- e ac ions wi h nuclei and nucleons. The neu ino-nucleus eac ion mechanisms a e di e se, going om he quasielas ic sca e ing o deep inelas ic sca e ing, mul i-nucleon p ocesses o he exci a- ion o nucleon esonances (see Chap e s 3-5 o de ails). Mos o hese egimes equi e bo h be e heo e ical and expe imen al unde s anding. Acco dingly, i would be ad an ageous o employ he knowledge ex ac ed om elec on-nucleus sca e ing as a solid benchma k o assess he alidi y o he heo e ical neu ino in e ac ion app oaches in he di e en nuclea egimes (see Chap e 6) as well as o imp o e he unde s anding o cu en neu ino oscilla ion expe imen s. Theo e ical models on (e,e′) eac ions can s ill be imp o ed by aking ca e o he ecen de- elopmen s in he inelas ic s uc u e unc ions along wi h he p og ess on pa on densi y unc ions (PDFs) h ough QCD calcula ions. Fu he mo e, he employmen o heo e ical models including mul inucleon exci a ions and ealis ic mean ield heo ies, in addi ion o he la ge amoun o (e,e′) da a o di e en nuclei, will likely help us o deepen ou unde s anding o he eac ion mech- anisms in elec omagne ic eac ions and i s ex ension o he weak sec o , i.e., o he analysis o neu ino in e ac ions. This, makes esea ch on neu ino in e ac ions an a ac i e ing edien o he u u e o Pa icle and Nuclea Physics. 1.2 Neu ino P ope ies and His o ical Con ex As p e iously men ioned, he “bi h” o neu ino da es back o 1930, when Pauli pos ula ed i s exis ence om he analysis o he con inuum ene gy spec um obse ed in he be a decay p ocess. This was inconsis en wi h he image o only elec ons being emi ed in he nuclea p ocess. The "possible" exis ence o an ex a pa icle ( he neu ino) wi h negligible mass and elec ically neu- al, ha is, a "silen " pa ne , was needed in o de o p ese e ene gy and momen um conse a ion. Thus he be a decay p ocess would imply he emission o no only elec ons bu also o new, un- known, pa icles ha would ga he he ene gy loss in he p ocess. Hence, he desc ip ion o he β-decay p ocess was de ined as: n→p+e−+ν. In he 30’s and 40’s new pa icles we e disco e ed, speci ically, he muon, e y simila o he elec on bu wi h a much la ge mass, and he pion, he pa icle p oposed by Yukawa o explain he s ong in e ac ion be ween nucleons inside he nuclei. A ca e ul s udy o he pion p o ed ha i decayed in o a muon which eme ged wi h an almos pe pendicula ack in ela ion o he pion, ensu ing he exis ence o an ex ao dina y ligh pa icle in he p ocess. I was sugges ed o be a neu ino. A ew mon hs la e , he expe imen al p oo o he muon decay in o an elec on led o he conclusion ha he la e mus be accompanied by wo neu inos. 1.2. NEUTRINO PROPERTIES AND HISTORICAL CONTEXT 3 In he ea ly 1950’s and despi e he di e en hypo heses abou he exis ence o he neu ino, he e had no expe imen al e idence ye . I was in 1956, when Cowan and Reines [4] de eloped a me hod o obse e he in e se βdecay (ν+p→n+e+), in which his elusi e pa icle was clea ly, unambiguously de ec ed. In hese expe imen s, he ac ual pa icle aken in o conside a ion was he an ineu ino (ν). Bu how could one dis inguish be ween neu ino and an ineu ino? This dis inc ion was explained by Da is and Ha me [5] in he la e 1950s, showing ha whe eas he p ocess ν+n→p+e−was commonly de ec ed, on he con a y, he ν+n→p+e−one ne e ook place. This was he beginning o he lep on numbe assignmen o neu inos and he es o lep ons as well as i s conse a ion law. The e o e, neu inos in muon decay should be pa icle and an ipa icle, whose di e ence comes om i s own helici y. This subjec will be add essed in de ail la e . The hypo hesis o neu inos wi h di e en la o s came up a e ealizing ha , unlike he p o- cess µ→e+ν+ν, he eac ion µ→e+γwas no obse ed. Consequen ly, i was p oposed he exis ence o wo di e en kinds o neu inos, one ela ed o he elec on and ano he one o he muon. The e o e, some o he p ocesses men ioned abo e a e eally gi en as: n→p+e−+νe π+→µ++νµ π−→µ−+νµ µ+→e++νe+νµ µ−→e−+νµ+νe Neu ino la o s The S anda d Model o Pa icle Physics con ains h ee di e en neu ino la o s (νe,νµand ντ), which espec i ely o m a double wi h he co esponding cha ged lep ons (e,µand τ) as shown in Table. 1.1. In he 1990s, he LSND expe imen [6] sugges ed he possibili y o u he neu inos o explain neu ino masses and in oduced a new ype o neu inos, called “s e ile” neu inos, whose exis ence is s ill unde in es iga ion. This pa icle would no in e ac wi h any o he (excep h ough g a i y). The sea ch o s e ile neu inos is an ac i e a ea o pa icle physics and i is expec ed ha u u e neu ino oscilla ion acili ies could shed mo e ligh on his issue. Helici y Wi hin he amewo k o he SM, neu inos a e massless lep ons and exhibi pu ely nega i e he- lici y (le -handed), i.e., hei spin is an ipa allel o hei momen um; whe eas he e e se occu s o an ineu inos,i.e., hei spin p ojec ion poin s ou pa allel (posi i e helici y) o he momen um. This e lec s he pa i y iola ion in elec oweak p ocesses. In he SM, only le -handed neu inos and igh -handed an ineu inos exis . Ne e heless, he disco e y ha neu inos ha e mass could modi y his desc ip ion. Mo eo e , i is no clea ye i neu inos a e Majo ana pa icles, ha is, neu ino and an ineu ino a e he same pa icles. Neu inoless double be a decay, which can be iewed as wo be a decay e en s wi h he p oduced an ineu inos immedia ely annihila ing wi h one ano he , is a possible way o de ec i neu inos a e hei own an ipa icles. Expe imen s a e unde way o sea ch o his ype o decay. 4 1. INTRODUCTION Cha ged lep ons Neu inos Name Symbol Cha ge Mass (MeV) Name Symbol Cha ge Mass (MeV) 1s gen. Elec on e−-1 0.511 Elec on neu ino νe0<3·10−6 Posi on e++1 Elec on an ineu ino νe0 2nd gen. Muon µ−-1 105.658 Muon neu ino νµ0<0.19 An imuon µ++1 Muon an ineu ino νµ0 3 d gen. Tauon τ−-1 1766.99 Tauon neu ino ντ0<18.2 An i auon τ++1 Tauon an ineu ino ντ0 Table 1.1: The P o oypical Family o Lep ons. Neu ino mass The expe imen al e idence o sola and a mosphe ic neu ino oscilla ions [7–9] led o he conclu- sion ha neu inos a e no massless, e en hough, a p esen , we do no know he absolu e alues o hei masses bu only he uppe limi s (see Table 1.1). In addi ion o he h ee di e en la o s, he e a e also h ee mass s a es ν1, ν2and ν3wi h masses m1,m2and m3 espec i ely, which a e associa ed o he la o s a es h ough he Pon eco o- Maki-Nakagawa-Saka a (PMNS) ma ix [10,11], *., νe νµ ντ+/-=*., Ue1Ue2Ue3 Uµ1Uµ2Uµ3 Uτ1Uτ2Uτ3+/-*., ν1 ν2 ν3+/-.(1.1) The PMNS ma ix can be exp essed as he p oduc o ou sub-ma ices [2], which con ain h ee mixing angles (θ12,θ13 and θ23). These pa ame e s de e mine he deg ee o which he mass and la ou s a es a e mixed, as well as di e se CP- iola ion e ms. 1.3 Neu ino Oscilla ions In 1998 he Supe -Kamiokande acili y [7] e ealed he la o change in a mosphe ic neu inos. A ew yea s la e , he Sudbu y Neu ino Obse a o y expe imen (SNO) [8] concluded ha he loss obse ed in he incoming elec on sola neu ino lux was due o neu ino oscilla ions. These ob- se a ions we e con i med in subsequen yea s in di e en neu ino expe imen s based on he use o eac o s and accele a o s [9]. The neu ino oscilla ion phenomenon, p edic ed by B uno Pon eco o [10], indica es he pos- sibili y ha a speci ic neu ino (elec onic, muonic o auonic) ans o ms in o a one wi h di e en la o . This e ec is o high expe imen al and heo e ical ele ance as i implies neu inos a e no massless. In his con ex , a p ope unde s anding o c oss sec ions in neu ino-nucleus eac ions is essen ial. In he simple hypo hesis ha only wo amilies o neu ino exis , he p obabili y ha a neu ino o ene gy Eνchanges i s la o om νi o ν a e a eling a dis ance Lis gi en by P(νi→ν )=sin22θsin2 ∆m2L 4Eν!,(1.2) whe e θis he mixing angle ela ed o he combina ion o he di e en la o s a es in o he mass s a es. The mass squa e di e ence pa ame e ∆m2=m2 1−m2 2is he di e ence be ween he wo 1.3. NEUTRINO OSCILLATIONS 5 mass s a es and is he only mass pa ame e ha can be measu ed in neu ino oscilla ion expe i- men s, which canno gi e in o ma ion on he absolu e masses. The a io be ween L/Eνand ∆m2 is op imized in neu ino acili ies o maximize he sensi i i y o neu ino oscilla ions. In he h ee- la o case he o mula o he oscilla ion p obabili ies is mo e complica ed and depends upon h ee mass squa es di e ences (∆m2 12 =m2 1−m2 2,∆m2 13 =m2 1−m2 3,∆m2 23 =m2 2−m2 3), h ee oscilla ion angles (θ12,θ13,θ23) and a phase δCP co esponding o he possible iola ion o CP symme y in he lep onic sec o . The measu emen s o he la e is one o he mos impo an challenges o u u e neu ino oscilla ion expe imen s. Thus he oscilla ion p obabili y is gi en by Pνα→νβ(L,E)=δα β −4X i>j Re(U∗ αiUβiUαjU∗ βj) sin2*, ∆m2 ij 4EL+- +2X i>j Im(U∗ αiUβiUαjU∗ βj) sin *, ∆m2 ij 2EL+-,∆m2 ij ≡m2 i−m2 j.(1.3) The Pa icle Da a G oup [2] p o ides a e iew o he up- o-da e alues o he oscilla ion pa- ame e s, shown in Table 1.2. Pa ame e Value θ12 33.9◦±1.0◦ θ13 9.1◦±0.6◦ θ23 39◦< θ23 <51◦ ∆m2 12 (7.50 ±0.20)10−5eV2 ∆m2 23 (2.32+0.12 −0.08)10−3eV2 Table 1.2: Cu en wo ld knowledge o neu ino oscilla ion pa ame e s. ∆m2 13 is de e mined h ough ∆m2 13 =∆m2 12 +∆m2 23. The e a e wo me hods o de e mine neu ino oscilla ions: •Appea ance mode: These expe imen s a e ocused on he sea ch o a new neu ino la o , absen in he o iginal beam, o an enhancemen o neu inos o a gi en la o in he ini ial beam. These e ec s a e de ec ed h ough he co esponding cha ged lep on p oduced ia cha ged-cu en weak in e ac ion: νl+N→l−+X(1.4) wi h l=e, µ,τ and X he inal had on s a e. These p ocesses will be add essed in ollowing chap e s. •Disappea ance mode: I is based on he educ ion o he expec ed numbe o a pa icula neu ino la o a he de ec o , as he case o he SNO expe imen [8]. This me hod equi es an accu a e unde s anding o he neu ino beam a he sou ce. Neu ino sou ces employed in oscilla ion expe imen s a e di e se, going om nuclea eac o s (νe), a mosphe ic (νe, νe, νµ, νµ), sola (νe) and accele a o s (νe, νe, νµ, νµ). In he case o sola and a mosphe ic neu inos, he e a e some cons ain s on Land Eνwhich p e en om an op i- mum combina ion o hese pa ame e s o he oscilla ion measu emen s. The e o e, he cu en 6 1. INTRODUCTION e o s ocus on accele a o -based neu ino expe imen s whe e a be e sensi i i y o oscilla ions can be achie ed. A de ailed analysis o cu en neu ino expe imen al acili ies will be p esen ed in Sec . 1.6. 1.4 In e ac ion wi h Ma e The neu ino in e ac ion wi h ma e is pu ely weak, ac ing a e y sho dis ances wi h a low in en- si y, and being able o modi y he la o o he pa icles in ol ed. Such in e ac ion is desc ibed by he elec oweak heo y, de eloped by Weinbe g [12], Salam [13] and Glashow [14] in he 1970s, by he exchange o he weak bosons W±and Z. The W±boson is connec ed wi h cha ged-cu en (CC) sca e ing p ocesses whe e he e is a cha ge exchange in he in e ac ion e ex, whe eas he Z boson is associa ed o neu al-cu en sca e ing whe e no cha ge exchange occu s. Mo eo e , he ac ha he elec oweak in e ac ion is 5 o 6 o de s o magni ude lowe han he elec omagne ic one, leads o ex emely low c oss sec ions which makes expe imen al p ocedu e highly demand- ing. In addi ion o his, he low in ensi y o weak in e ac ions allows o a pe u ba i e analysis, so ha he Bo n app oxima ion, ha is, he 1-boson exchange be ween he lep onic and had onic e ices, ep esen s an accu a e app oach. Unlike elec omagne ic sca e ing, he weak p ocesses do no conse e pa i y, which p o ides speci ic in o ma ion abou he axial s uc u e o he nucleon. In pa icula , he cha ged-cu en neu ino-nucleon (ν-N) sca e ing con ains he mos p ecise in o ma ion abou he nucleon axial o m ac o s (add essed in Chap e 2). On he o he side, neu al cu en p ocesses gi e essen ial de ails abou he s ange sea qua ks in he nucleon s uc u e. In Fig. 1.1, he Feynman diag ams ela ed o he elas ic lep on-had on eac ions a e shown. In weak in e ac ions, a dis inc ion be ween cha ged cu en p ocesses, whe e a cha ged lep on is emi - ed, and neu al cu en ones, whe e he neu ino does no change in he inal s a e, is conside ed. Figu e 1.1: Lep on-had on elas ic sca e ing p ocesses: a) Elec omagne ic in e ac ion. b) Cha ged-cu en weak in e ac ion. c) Neu al-cu en weak in e ac ion. Compa ed o CC neu ino eac ions, he expe imen al s udy o NC p ocesses is a ai ly de- manding ask due o he conside able di icul ies o ob ain in o ma ion in a p ocess wi h a educed c oss sec ion and wi hou cha ged lep ons in he inal s a e. Thus, he e en ecogni ion lies on he had on de ec ion. 1.4. INTERACTION WITH MATTER 7 1.4.1 Neu ino-nucleus sca e ing A he lowes neu ino ene gies, he mos p obable in e ac ion is he elas ic sca e ing whe e he ene gy ans e o he nucleus is no enough o elease an unbound nucleon so he nucleus ecoils in ac . The quasielas ic (QE) sca e ing appea s once he ene gy ans e is la ge enough o sca - e o a nucleon om he nucleus. When his p ocess occu s ia neu al-cu en sca e ing, all neu inos and an i-neu inos can sca e o bo h neu ons and p o ons in wha is e e ed o as neu al-cu en quasielas ic (NCQE) sca e ing: νl(νl)+N→νl(νl)+N. Once neu inos acqui e su icien ene gy o c ea e he cha ged lep on’s mass hey can also unde go he analogous cha ged- cu en quasielas ic (CCQE) in e ac ion: νl+n→p+l−and νl+p→n+l+. Fo Eνµ≈1 GeV, CCQE is he dominan in e ac ion. The in e ac ion o neu inos wi h nuclei a in e media e ene gies, o he o de o 100s o MeV up o 10s o GeV, plays an impo an ole in he p ecise de e mina ion o neu ino oscilla ion pa ame e s. A hese ene gies, nuclea e ec s a e e y signi ican , p o iding also ele an in o - ma ion on he axial had onic cu en s. F om he expe imen al poin o iew, he da a analysis needs o conside a la ge numbe o nuclea e ec s ha dis o he signals and p oduce new sou ces o backg ound ha a e absen in he elemen a y neu ino-nucleon p ocesses. In his sense, a comple e heo e ical model should include, a leas , h ee kinds o con ibu ions: (i) quasielas ic (QE) o low-in e media e ene gy ans e s, (ii) nucleon co ela ions and wo-body con ibu ions and (iii) he comple e inelas ic spec um, con aining pion p oduc ion a ising om he ∆(1232) esonance peak, non- esonan con ibu ions, o he meson p oduc ion as well as highe -ene gy nucleonic es- onances and deep inelas ic sca e ing p ocesses. In his wo k, we ocus on in e media e and high ene gies co e ing om QE eac ions o deep inelas ic sca e ing p ocesses. The di e en eac ion mechanisms a e b ie ly desc ibed in he ol- lowing lines: •Quasielas ic sca e ing: The mos ele an con ibu ion a in e media e ene gies (Eν∼1 GeV) in which a neu ino sca e s o a single bound nucleon being he la e ejec ed om he nuclea a ge . We can dis inguish be ween NCQE neu ino in e ac ions, whe e a neu ino is emi ed, and CCQE neu ino in e ac ions wi h a cha ged lep on in he inal s a e. Rega ding he no a ion employed in he expe imen al li e a u e, CCQE-like sca e ing (o CC0π) is de ined as he p ocess whe e one lep on and no pions a e de ec ed in he inal s a e. In his sense, QE-like p ocesses a e con amina ed wi h o he con ibu ions such as sho - ange co ela ions (SRC), mul inucleon emissions (np-nh) induced by meson-exchange cu en s (MEC) and by esca e ing p ocesses, pion abso p ion in he nuclea medium and pionless esonance decay. •2p-2h MEC con ibu ions: This co esponds o a weak boson being exchanged by a pai o nucleons (2-body cu en ) leading o he emission o wo nucleons om he p ima y e ex. This con ibu ion is essen ial o in e p e p ope ly he “dip” egion be ween he QE and ∆ peaks o (e,e′) eac ions as well as o ep oduce he neu ino QE-like expe imen al da a. •Resonance p oduc ion: A highe ene gies, which implies la ge Q2 alues, neu inos gain access o inelas ic sca e ing p ocesses. He e he a ge nucleon is “knocked” in o a ba y- onic esonance, a ∆o a hea ie esonance N∗depending on he ene gy ans e . These esonances decay in o a a ie y o inal s a es wi h combina ions o nucleons and mesons. Neu ino-induced pion p oduc ion is he dominan p ocess in he egion 0.5 GeV < Eν< 10 GeV. 8 1. INTRODUCTION •Deep inelas ic sca e ing (DIS): A highe ene gies he neu ino is able o ans e su icien momen um so he inne s uc u e o he nucleon can be esol ed. The neu ino can sca e di ec ly o any o he qua ks inside he nucleon, including hose which o m he “sea” o qua ks and an i-qua ks ha a e cons an ly popping in and ou o exis ence. A lowe ans e momen um he nucleons con ain mos ly up, down and s ange qua ks, bu highe ans e momen um alues ge access o highe -mass and sho e -li ed qua ks oo. DIS is he dominan p ocess o Eν> 10 GeV and i s mos isible consequence is he b eak up o he nucleon con aining he s uck qua k. The di e en eac ion mechanisms con ibu ing o (an i)neu ino-nucleus in e ac ions a e shown in Fig. 1.2 as unc ion o he incoming (an i)neu ino ene gy. In o de o gi e a mo e comple e de- sc ip ion o he sca e ing mechanisms, he nuclea esponse in e ms o he ene gy ans e ed o he nucleus is schema ically ep esen ed in Fig. 1.3 o a ixed momen um ans e q. In he la e , we can obse e how small ene gy ans e s o he o de o ens o MeV esul in elas ic sca e ing o he nucleus ollowed by a collec i e exci a ion o he nucleus. In e media e ene gies a e ela ed o he p ocess whe e he lep on sca e s o a bound nucleon gi ing a b oad QE peak cen e ed a ound ∼Q2/2M(being M he nucleon mass). A highe ene gies, mul inucleon exci a ions and nucleonic esonances s a o be ele an as well as DIS p ocesses when he inciden ene gy is enough o he nucleon b eak-up. No ice also he o e lap o he di e en eac ion mechanisms in he in e media e kinema ical egion (1-10 GeV). This e ec should be conside ed ca e ully when analyzing neu ino-nucleus expe imen al da a. 1.4.2 Elas ic neu ino-nucleon in e ac ions and nucleon o m ac o s Doing calcula ions o he in e ac ion o neu inos wi h nuclei p esen s addi ional complexi y o e elec ons wi h nuclei since, in addi ion o he pu ely ec o elec omagne ic esponses, new axial and in e e ence ec o -axial con ibu ions a ising om weak lep onic and had onic cu en s come in o play. Al hough essen ially composed o only h ee qua ks, nucleons a e cons an ly in e ac ing ia exchange o gluons which in u n can p oduce o he empo a y qua k/an i-qua k pai s. The ou - momen um o he weak boson (Q2) de e mines how much o he nucleon’s in e nal s uc u e is esol ed by a weak in e ac ion. The desc ip ion o he inne s uc u e o he nucleon can be p o- ided by he use o nucleon o m ac o s, ha a e gi en as phenomenological unc ions exp essed in e ms o Q2. The ype o in e ac ion de e mines which o m ac o s en e in he p ocess. A mo e de ailed desc ip ion o had onic o m ac o s will be add essed in Chap e 2.3. Wi h espec o he nucleon s uc u e, elas ic neu ino-nucleon sca e ing p ocesses a e pa icu- la ly impo an o neu ino physics o wo easons. Fi s , hey p o ide essen ial in o ma ion on he weak nucleon o m ac o s which a e di icul o inaccessible o o he sca e ing p obes. Second, he p e ious p ocesses enable he kinema ics o be comple ely econs uc ed, and hence he ini ial neu ino ene gy can be de e mined. This is c i ical o measu emen s o he oscilla ion pa ame e s. In spi e o his, mos o he expe imen s employ hea y a ge s in o de o ge la ge c oss sec ions. This makes he analysis o he p ocess much mo e complex, i.e., a g ea cau ion should be d awn on how o econs uc he kinema ics, and how o ge in o ma ion on he weak nucleon o m ac- o s. Rega ding he weak nucleon o m ac o s in CCQE neu ino sca e ing we can dis inguish ec o , F1(Q2) and F2(Q2), axial, FA(Q2), and pseudoscala , FP(Q2), ones, ela ed o he co e- 1.4. INTERACTION WITH MATTER 9 Figu e 1.2: To al neu ino (le ) and an ineu ino ( igh ) c oss sec ions pe nucleon in e ms o he inciden neu ino ene gy (Eν). Di e en eac ion channels a e shown sepa a ely. The igu es a e aken om [15]. 10 1. INTRODUCTION Figu e 1.3: Regimes o he nuclea esponse in e ms o he ene gy ans e ed (ω) o he nucleus. sponding componen s o he had onic cu en . The ec o o m ac o s, as a consequence o he conse ed ec o cu en hypo hesis (CVC) [16], can be de ined in e ms o he elec omagne ic ones ela ed o elec on-nucleon sca e ing. Conce ning he axial o m- ac o , FA(Q2), i is com- mon o assume a “dipole” o m, whe e wo con ol pa ame e s mus be de e mined expe imen ally: he “axial cha ge” gA=1.26 om be a-decay, and he “axial mass” MA=1.014±0.014 GeV om CCQE neu ino sca e ing on deu e ium and pion p oduc ion by elec ons [17]. I should be no ed hough ha he dipole o m is an assump ion mo i a ed by he analysis o elec omagne ic o m ac o s [18]. A s udy o he di e ences be ween dipole and monopole axial o m ac o is shown in Appendix D as well as in [19]. Al hough, o da e, neu ino expe imen s do no ha e he p ecision equi ed o es explici ly he Q2-dependence o he weak o m ac o s, a dipole o m wo ks nicely in mos si ua ions [18]. Finally, he pseudoscala o m ac o , FP, is ela ed o he axial one ia he “Goldbe ge -T eiman” ela ion. The basic ea u es o hese unc ions a e analyzed in Chap e 2.3. 1.5 Theo e ical app oaches o neu ino-nucleus sca e ing Wi h he aim o achie ing a comple e heo e ical desc ip ion o neu ino-nucleus in e ac ions, se - e al heo e ical g oups ha e de eloped di e en models o accoun o he di e en ing edien s needed o a deep unde s anding o expe imen al da a. The i s MiniBooNE CCQE measu emen s o neu ino in e ac ions on 12C mo i a ed a la ge heo e ical e o o explain he appa en dis- c epancy be ween he expe imen al c oss sec ions and he heo e ical p edic ions, which seem o unde es ima e he da a [20,21]. A i s app oach o disen angle his appa en disc epancy was o inc ease he alue o he nucleon axial mass (MA) om he wo ld’s a e aged alue (1.032 GeV) eme ging om deu e ium bubble chambe expe imen s [22, 23] o ∼1.35 GeV. Howe e , while imp o ing he heo e ical ag eemen wi h he MiniBooNE da a, such a high alue o MAdoes no ma ch wi h o he expe imen al da a, as NOMAD [24], no wi h o me elas ic neu ino c oss sec- ions on ligh nuclei [25–29]. The key o his issue lays on he de ini ion o he CCQE e en in he expe imen . Whils heo is de ine CCQE as he e en whe e one nucleon is knocked ou in he inal s a e along wi h a cha ged lep on, MiniBoone de ined CCQE as an e en wi h one lep on and no pion obse ed in he inal s a e, which has been “ eno malized” o CCQE-like. The e o e, he de ini ion o CCQE-like e en also includes e ec s beyond he Impulse App oxima ion, i.e., 1.6. EXPERIMENTAL STATUS 11 he neu ino only in e ac s wi h a single-bound nucleon. Thus, nucleon-nucleon co ela ions and mul inucleon emission p ocesses, such as 2p-2h MEC con ibu ions, should be accoun ed o in he analysis o expe imen al da a. F om his baseline, a la ge numbe o heo e ical g oups began o in es iga e on his opic. In pa icula , Ma ini e al. employed a local Fe mi gas model [30], aking in o accoun RPA co ela- ions, cohe en and incohe en pion p oduc ion as well as quasielas ic exci a ions. Besides his, he inclusion o con ibu ions a ising om np-nh exci a ions [30] led hem o a easonable ag eemen wi h MiniBooNE da a wi hou eso ing any e ec i e pa ame e o inc easing he MA alue. A simila app oach was aken by Nie es e al. [31], achie ing a good ag eemen wi h da a a e a sub ac ion o ∼10%. Mo eo e , he Pa ia g oup made use o a ela i is ic G een’s unc ion ap- p oach [32] including he imagina y pa o he ela i is ic op ical po en ial o desc ibe inal-s a e in e ac ions (FSI). Thei p edic ions a e in acco dance wi h da a o some pa icula choices o he op ical po en ials. The con ibu ions o one-body and wo-body cu en s ha e also been ana- lyzed in [33] using he spec al unc ion o malism. A semiphenomelogical model ha conside s he quan um-kine ic anspo heo y (GiBUU) has been also applied o he analysis o elec o- magne ic and weak in e ac ions [34], using a pa ame iza ion o he ans e se elec omagne ic con ibu ions as a basis o 2p-2h neu ino in e ac ions. In his hesis, he analysis o he elec on and neu ino-nucleus in e ac ions in he QE and in- elas ic egimes will be ca ied ou wi hin he amewo k o he Supe Scaling App oach (SuSA), which assumes he exis ence o uni e sal scaling unc ions o bo h elec omagne ic and weak in- e ac ions. The analysis o inclusi e (e,e′) expe imen al da a [35–38] has p o ed ha scaling is ul illed wi h e y good accu acy. This implies ha he educed c oss sec ion exhibi s indepen- dence o he momen um ans e ( i s -kind scaling) and o he nuclea a ge (second-kind scaling) when exp essed as a unc ion o he app op ia e scaling a iable, i sel a unc ion o he ene gy and momen um ans e . In a ecen ly imp o ed e sion, called SuSA 2 model, we ha e employed he Rela i is ic Mean Field Theo y (RMF) and he Rela i is ic Plane Wa e Impulse App oxima- ion (RPWIA) o ob ain a comple e se o scaling unc ions ha embody all he nuclea depen- dence o he in e ac ion, being alid o all nuclei. The desc ip ion o he he many-body physics o he in e ac ing nucleons wi hin he SuSA 2 model, akes in o accoun he di e en con ibu- ions o bo h longi udinal and ans e se nuclea esponses, as well as he iso ec o and isoscala channels. This is o g ea in e es o cha ged-cu en (CC) neu ino eac ions, which a e pu ely iso ec o . Rega ding he 2p-2h MEC con ibu ions and con a y o o he wo ks on his opic, we employ a mic oscopic calcula ion in a ully ela i is ic amewo k wi hou u he app oxima ions. The SuSA 2-MEC p edic ions, based on he use o he RMF/RPWIA model plus he 2p-2h MEC ully mic oscopic calcula ions, ha e been succes ully applied o he analysis o (e,e′) and CCQE neu ino-nucleus eac ions co e ing om low o e y high ene gies [38,39]. 1.6 Expe imen al s a us Once neu ino oscilla ions we e con i med in sola and a mosphe ic neu inos [7,8], an ample ex- pe imen al p og am was de eloped wi h he aim o s udying hese oscilla ions and de e mining p ecisely hei masses h ough he oscilla ion pa ame e s. This deep esea ch has enewed he in- e es on neu ino c oss sec ions as an essen ial ing edien on he analysis o neu ino oscilla ion expe imen s as well as on he s udy o he weak nucleon s uc u e. Ne e heless, due o he ex- emely educed c oss sec ions in weak eac ions, neu ino expe imen s a e highly complica ed. In he wo ds o Haim Ha a i [40], “Neu ino physics is la gely an a o lea ning a g ea deal by 18 1. INTRODUCTION collec ed by MINOS o iden i y and measu e hese ene ge ic pa icles. Figu e 1.9: Side- iew o he MINERνA expe imen . The neu ino beam a els le o igh h ough he a ious de ec o componen s [46]. MINERνA has p oduced CCQE-like c oss sec ions on CH [46, 47] which a e in ag eemen wi h heo e ical es ima ions including also mul inucleon exci a ions, wi hou u he ing edien s o la ge axial mass alues. I s ecen measu emen s on DIS c oss sec ions o di e en nuclei [54] could be aluable o assess he ele an ing edien s o he neu ino eac ion models a e y high ene gies. K2K The K2K expe imen [9] was designed in Japan o con i m a mosphe ic neu ino oscilla ions using Supe -Kamiokande as de ec o . A schema ic diag am o his de ec o is shown in Fig. 1.10. This consis s o a wa e Che enko de ec o , a wa e -based liquid scin illa o (SciFi, oxygen a ge ) and a muon ange de ec o (MRD). Thus, he impac o 12-GeV p o ons on an aluminium a ge p oduces neu inos wi h an ene gy sp ead om 1 o 1.5 GeV. Finally, in he SciFi de ec o he neu ino c oss sec ion is measu ed. Figu e 1.10: Schema ic iew o K2K de ec o [9]. 1.6. EXPERIMENTAL STATUS 19 T2K The T2K expe imen (Tokai o Kamioka) is a long-baseline neu ino expe imen in Japan which is ocused on neu ino oscilla ions and can be conside ed he successo o K2K. I employs a high- esolu ion de ec o wi h he aim o de e mining he neu ino ene gy spec um as well as i s la o and he esul ing c oss sec ions. T2K has made a sea ch o oscilla ions om muon neu inos o elec on neu inos, al eady published [55] as well as i s wo k on he measu emen o oscilla ions om muon neu inos o au ones. A la ge se o expe imen al da a om o wa d o backwa d angles has also been published o he analysis o CC0πp oduc ion p ocesses as well o he inclusi e ones [44,45,56]. Figu e 1.11: Schema ic iew o T2K de ec o [44]. The T2K expe imen sends an in ense beam o muon neu inos om Tokai, which is on he eas coas o Japan, o Kamioka a a dis ance o 295 km in wes e n Japan. The neu ino beam is p oduced om collisions be ween a p o on beam and a g aphi e a ge ; hese collisions p oduce pions, which quickly decay o muons and muon neu inos. The muons and any emaining p o ons and pions a e s opped by a second laye o g aphi e, bu he neu inos pass h ough i . T2K s udies neu ino oscilla ions wi h wo sepa a e de ec o s, bo h o which a e 2.5 deg ees away om he cen e o he neu ino beam. The ND280 nea de ec o is loca ed a 280 me es dis ance om he a ge , and measu es he numbe o muon neu inos in he beam be o e any os- cilla ion occu s. T2K neu inos ha e much highe ene gies han sola neu inos, and high-ene gy neu inos a e mo e likely o in e ac . A small numbe o muon neu inos in e ac wi h scin illa o o wa e in he ND280, and many o hese in e ac ions p oduce a muon. The muon can be de- ec ed since i ionises gas which is placed immedia ely a e he in e ac ion poin s. These ND280 measu emen s a e used o p edic he numbe o muon neu inos ha would be seen in he “ a de ec o ” Supe Kamiokande i he e we e no oscilla ions. The walls o Supe Kamiokande a e lined wi h mo e han 10,000 sensi i e pho o-mul iplie s, which de ec he cone o Ce enko ligh as a ing (see Figu e 1.12). This de ec o sys em can dis inguish muons o igina ed om muon neu inos (which p oduce a sha p ing) om elec ons a ising om elec on neu inos (which p oduce a mo e di use ing). A nex s ep in o he Kamiokande in es iga ions is he Hype -Kamiokande expe imen [57], which will ope a e in he same beam line as T2K. This de ec o consis s o a mega on scale wa e ank and ul a high sensi i i y pho osenso s. The Hype -Kamiokande de ec o is bo h a “mic o- scope”, used o obse e elemen a y pa icles, and also a “ elescope” o obse ing he Sun and su- 20 1. INTRODUCTION Figu e 1.12: Expe imen al iew o muon ings in he T2K de ec o [44]. pe no as, using neu inos. The Hype -Kamiokande p ojec has an ex emely ich physics po olio ha spans om he s udy o he CP iola ion in he lep onic sec o and neu ino mixing pa ame- e s using accele a o neu ino and an i-neu ino beams o p o on decay, a mosphe ic neu inos and neu inos om as onomical o igin. NOMAD The NOMAD expe imen (Neu ino Oscilla ion MAgne ic De ec o ) was designed o sea ch o ντ appea ance om neu ino oscilla ions in he CERN wide-band neu ino beam p oduced by he 450 GeV p o on synch o on. The single-pa icle econs uc ion and lep on iden i ica ion capabili y o he NOMAD de ec o allowed he sea ch o ντappea ance in mos o he lep onic and had onic τ decay channels and also o look o νµ→νeoscilla ions. A second phase o he NOMAD analysis s a ed a e he comple ion o he oscilla ion sea ches, wi h he aim o exploi ing he high quali y o he a ailable neu ino da a samples o p ecise measu emen s o c oss sec ions and pa icle p o- duc ion. This ac i i y could be also o in e es o oscilla ion s udies. The NOMAD de ec o consis ed o a high esolu ion magne ic de ec o . In 2000, NOMAD comple ed i s sea ch o νµ→ντ h ough hei cha ged-cu en in e ac ions ollowed by he τ decay in o one o wo neu inos and by he co esponding τdecay daugh e s. No oscilla ion signal was ound bu he limi se was in acco dance wi h he expec a ions o he p oposal, hus demon- s a ing he alidi y o he kinema ic me hod used by NOMAD o sea ch o ντcha ged-cu en in e ac ions. This collabo a ion p oduced in e es ing esul s abou he o al CCQE muon neu ino c oss sec- ion on 12C [24] which, unlike MiniBooNE, do no claim o a la ge axial mass alue o he nucleon. A goNeuT The A goNeuT (A gon Neu ino Tes ) Expe imen an on he NuMI beam line a he Fe miLab, om Sep embe 2009 o Feb ua y 2010. I is he i s s age o a p ojec ha makes use o Liquid A gon Time P ojec ion Chambe s (LA TPCs) as neu ino de ec o s. A goNeuT has collec ed hou- 1.6. EXPERIMENTAL STATUS 21 sands o beam neu ino e en s in he 0.1 -10 GeV ene gy ange du ing i s un and some esul s ha e been ecen ly published, including he i s measu emen s o he inclusi e muon neu ino cha ged cu en di e en ial c oss sec ions on a gon [48,49]. In a LA TPC expe imen , he ee elec ons a e d i ed owa ds he anode in a uni o m elec ic ield whe e hey a e egis e ed on wi es in mul iple (a leas wo) planes, due o elec omagne ic induc ion and collec ion o he elec ons on he las plane o wi es. Because he wi es in di e en planes a e a an angle wi h espec o each o he , i is possible o ge he ull 3D econs uc ion o an e en . The LA TPC, gi en i s abili y o simul aneous p ecise 3D and calo ime ic econs uc ion o pa icle in e ac ions, is an ex emely in e es ing de ec o echnology. I connec s physics goals, like s e ile neu ino sea ch, wi h echnology de elopmen miles ones ha will lead o a mul i- kilo on long-baseline neu ino expe imen . This has p o ided he i s e e da a o low ene gy neu ino in e ac ions wi hin a LA TPC, pa ing he way o cons uc ion o la ge de ec o s. I s goals included measu emen s o he CC inclusi e c oss sec ions in he 1-5 GeV ange, examining he e ec s o Final S a e In e ac ions (FSI) and es ing he Pa icle ID capabili ies o he LA TPC, especially he e/γ sepa a ion c ucial o u u e neu ino expe imen s. νSTORM νSTORM (Neu inos om STORed Muons) is a p oposed s o age ing acili y [58,59] o deli e beams o muon an ineu inos and elec on neu inos om posi i e muon decays (muon neu inos and elec on an ineu inos om nega i e muon decays), wi h a cen al muon momen um o 3.8 GeV/c and a momen um accep ance o 10%. The acili y will allow sea ches o eV-scale s e ile neu inos a be e han 10 sigma sensi i i y. I will be able o p o ide measu emen s o neu ino and an ineu ino-nucleus sca e ing c oss sec ions wi h pe cen -le el p ecision and will se e as a i s s ep owa ds de eloping muon accele a o s o pa icle physics. The lux o he neu ino beam can be de e mined wi h pe cen -le el accu acy o pe o m c oss-sec ion measu emen s o u u e neu ino oscilla ion expe imen s and o esol e he hin s o eV-scale s e ile neu inos. νSTORM may be conside ed as a i s s ep owa ds a Neu ino Fac o y and a Muon Collide . In his sense, ou s udies abou he di e ence be ween elec on neu ino eac ions and muon neu ino ones could be aluable o he analysis o i s expe imen al esul s. DUNE The Deep Unde g ound Neu ino Expe imen (DUNE), conduc ed wi h he de ec o s ins alled in he Long-Baseline Neu ino Facili y (LBNF) a Fe miLab, is expec ed o achie e impo an disco - e ies in o hcoming yea s, making de ini i e de e mina ions o neu ino p ope ies, he dynamics o he supe no ae ha p oduced he hea y elemen s necessa y o li e, and he possibili y o p o on decay. Wi h he LBNF acili ies and he de ec o s p o ided by DUNE [60], he DUNE Collabo a ion p oposes o disen angle he puzzle o neu inos wi h b oad sensi i i y o neu ino oscilla ion pa am- e e s in a single expe imen . The ocus o he scien i ic p og am is he de e mina ion o he neu ino mass hie a chy and he analysis o lep onic CP iola ion by p ecisely measu ing di e ences be- ween he oscilla ions o muon- ype neu inos and an ineu inos in o elec on- ype neu inos and an ineu inos, espec i ely. Fu he mo e, he DUNE expe imen also ocuses on de e mining he o de ing o he neu ino masses as well as on sea ching o neu inos beyond he cu en ly known h ee. 22 1. INTRODUCTION 1.7 S uc u e and mo i a ion o his hesis As al eady men ioned in p e ious sec ions, mos e en s analyzed by he di e en collabo a ions co espond o CCQE p ocesses whe e a muonic neu ino is sca e ed by a bound nucleon (νµ+n→ µ−+p). The employmen o ealis ic nuclea models is c ucial o he analysis o hese and o h- coming expe imen s, such as NOνA, MINERνA, MiniBooNE o T2K, ocused on he sea ch o neu ino oscilla ions νµ→νe. A p ope in e p e a ion o neu ino oscilla ions implies an accu a e desc ip ion o he CCQE p ocess in a wide ange o neu ino ene gies. I should be no ed ha he beam neu ino ene gy is no p ecisely de e mined and only an ene gy dis ibu ion o he incoming lux is p edic ed. The e- o e, a con ol o e he unce ain ies associa ed o he nuclea model is indispensable. Likewise, backg ound p ocesses a e also impo an and ha d o disen angle om he CCQE signal as he case o he pion p oduc ion and i s subsequen abso p ion by he nuclea a ge . O he e ec s, such as nucleon co ela ions, meson-exchange cu en s ela ed o mul i-nucleon knock-ou o inal-s a e in e ac ions (FSI), can also con ibu e signi ican ly o he inal esul and ha e o be app op ia ely accoun ed o in he nuclea models. Mo eo e , he oscilla ion p obabili y di ec ly depends on he neu ino ene gy, which mus be econs uc ed om he inal-s a e pa icles. This ein o ces he need o dispose o ealis ic models ha p o ide an accu a e desc ip ion o he mechanisms in ol ed in neu ino-nucleus in e ac ions. These models a e in u n used in he simula ions wi h which expe imen s de ine hei analyses and in e p e hei da a so i is essen ial o imp o e he unde s anding o neu ino in e ac ions in o de o imp o e neu ino oscilla ion measu emen s. The e o e, he ocus o he ollowing chap e s is o s udy elec oweak sca e ing p ocesses, deepening in neu ino-nucleus eac ions along he di e en nuclea egimes o in e es o neu ino oscilla ions. This will be add essed by using ealis ic models ha ea he p ocess in a ully el- a i is ic way, desc ibing bo h had onic and nuclea s uc u e. This is di ec ly connec ed wi h he g owing in e es in neu ino-oscilla ion expe imen s whe e an accu a e heo e ical desc ip ion o weak in e ac ions is essen ial o in e p e p ope ly he expe imen al esul s. The s uc u e o his hesis will be he ollowing: in Chap e 2 we analyze he case o cha ged- cu en neu ino-nucleon elas ic in e ac ions whe e he weak esponse unc ions and he nucleon o m ac o s a e desc ibed; in Chap e 3 we p esen he o malism o quasielas ic neu ino-nucleus in e ac ions wi hin he SuSA 2 model, which is based on he supe scaling beha io and he ela- i is ic mean ield heo y. The Chap e 4 ocuses on he 2p-2h MEC con ibu ions as an essen ial ing edien o he analysis o expe imen al da a, whe e we make use o an accu a e pa ame iza ion o ully ela i is ic mic oscopic calcula ions. The ex ension o he SuSA 2 model o he inelas- ic egime is de ailed in Chap e 5. A e wa d, in Chap e 6 we compa e he p e ious heo e ical desc ip ion wi h elec on-nucleus sca e ing da a o all kinema ics. This cons i u es a solid bench- ma k o assess he alidi y o ou model o he analysis o he exis ing cha ged-cu en neu ino expe imen al da a, as de ailed in Chap e 7. Finally, in Chap e 8 we p esen a summa y and he main conclusions o his PhD hesis. Chap e 2 Cha ged-cu en elas ic neu ino-nucleon sca e ing In his chap e we in oduce he o malism employed o desc ibe elas ic cha ged-cu en neu ino- nucleon (CC ν-N) sca e ing p ocesses, which will be o ele ance o unde s and p ope ly he CCQE neu ino-nucleus eac ions. Acco dingly, we also analyze he ele ance o he di e se elec- omagne ic and weak nucleon o m ac o s and he di e en single-nucleon esponses. This anal- ysis is ca ied ou wi hin he Bo n App oxima ion, whe e a single Wboson is exchanged, hus esul ing an accu a e app oach due o he educed alue o he weak coupling cons an and he la ge W-boson mass. 2.1 Gene al o malism The kinema ics o he CC ν-Nsca e ing p ocess (νl+n→l−+p), whe e l ep esen s e, µ o τ, and he ame o e e ence employed a e schema ically shown in Fig. 2.1. In his igu e, he elas ic neu ino-neu on in e ac ion ia he exchange o a W−is displayed, leading o a inal s a e composed o an ou going lep on and a p o on. The an ineu ino-nucleon p ocess (νl+p→l++n) is analogous o he p e ious one. Figu e 2.1: F ame o e e ence and kinema ics in ol ed in he CC neu ino-nucleon p ocess whe e he ou going lep on is a muon. 23 24 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING Acco dingly, he cha ged-cu en neu ino (an ineu ino) in e ac ion, which is pu ely iso ec o , is associa ed o isospin T=1 as he e is a cha ge exchange whe e he ini ial neu on (p o on) u ns in o a p o on (neu on) in he inal s a e. This implies an isospin change in he Z-componen as |∆Tz|=1. In neu al cu en in e ac ions (ν+N→ν+N), his Z-componen is ze o in such a way ha isoscala con ibu ions would also be pe mi ed. The ame o e e ence selec ed o hese p ocesses is he labo a o y ame in which he ini- ial nucleon is a es . I is aken o simplici y he momen um ans e o he nucleon along he Zaxis and, he e o e, he inal momen um o he nucleon. Thus, he ini ial and inal momen a o he lep ons a e con ained in he X Z plane. Fo comple eness, he ollowing analysis has also been add essed b ie ly o he elec omagne ic elec on-nucleon in e ac ion (e−+N→e−+N) in Appendix A. The no a ion employed o he kinema ics in CC weak p ocesses is summa ized as ollows: ➲Inciden neu ino νl: –4-momen um: kν=(Eν,~ kν) –Mass: mν=0→Eν=|~ kν| ➲Final lep on l−: –4-momen um: kl=(El,~ kl) –Mass: ml –Sca e ing angle: θl ➲Ini ial nucleon n: –4-momen um: Pi=(Mn,0) –Mass: Mn ➲Final nucleon p: –4-momen um: P =(E ,~q) –Mass: Mp ➲Exchanged boson W−: –4-momen um: Qν=(ω, ~qν≡qν) The ene gy-momen um conse a ion law in he lep onic and had onic e exes implies he ollowing ela ions: Lep onic e ex ➦Ene gy conse a ion: Eν−El=ω ➦Momen um conse a ion: ~ kν−~ kl=~q⇒q2=E2 ν+|~ kl|2−2Eν|~ kl|cos θl Had onic e ex ➦Ene gy conse a ion1:E =ω+MN⇒ω2+2MNω=q2⇒−Q2≡ |Q2|=2MNω ➦Momen um conse a ion: ~q=~ Pi+~ P ⇒~q=~ P As shown in he p e ious equa ions, he kinema ical a iables Eν,Eland θla e no independen so he kinema ics o he p ocess is comple ely de ined by de e mining wo o hem. 1No ice ha a gene ic nucleon mass MNis employed. This can be ei he he neu on o p o on mass depending on he neu ino o an ineu ino case, espec i ely. 2.1. GENERAL FORMALISM 25 In e ac ion hamil onian The ansi ion ampli ude S i o he neu ino-nucleon sca e ing p ocess is ob ained om he asso- cia ed in e ac ion hamil onian (HW), S i =−iZd4X HW(X).(2.1) wi h he hamil onian exp essed as HW(X)="g 2√2#2 J(l)† µ(X)Aµ (N)(X)="g 2√2#2 J(l)† µ(X)Dµν W(Q)J(N) ν(X),(2.2) whe e gis he dimensionless weak coupling cons an , ela ed wi h he Fe mi cons an , GF, as GF √2 =g2 8M2 W ;MW=80.401(38) GeV .(2.3) The 4-po en ial Aµ (N)(X), i.e., he ield gene a ed by he nucleon in he in e ac ion wi h he co e- sponding lep on, can be desc ibed in e ms o he had onic cu en , J(N) ν, and he weak p opaga o , Dµν W, Aµ (N)(X)=Zd4Y DW z }| { Zd4Q (2π)4*,−gµν +QµQν/M2 W Q2−M2 W+iε+-eiQ·(X−Y)J(N) ν(Y) ≈Zd4YZd4Q (2π)4eiQ·(X−Y)*, 1 M2 W+-Jµ (N)(Y),(2.4) whe e he app oxima ion |Q2| ≪ M2 W(MW=80.401 GeV/c2), alid o ene gies co esponding o he elas ic and QE egimes, has been conside ed. Nex we in oduce he lep onic and had onic cu en s, used o ob ain he double di e en ial c oss sec ion. Lep onic and had onic cu en s The lep onic j(l) µcu en is de ined as a sum o ec o (γµ) and axial (γµγ5) e ms associa ed o he ec o and axial s uc u e o he nucleon. The axial e m makes possible he pa i y iola ion in weak in e ac ions. The p e ious de ini ion is gene al o all weak p ocesses and, in pa icula , o he cha ged cu en ones. On he con a y, he analysis o he lep onic cu en (j(e) µ) in elec o- magne ic (e,e′) p ocesses only in ol es he ec o componen . Thus, he weak lep onic cu en is de ined as j(l) µ=ψl(kl,sl)γµ(1 ∓γ5)ψνl(kνl,sνl),(2.5) whe e he -(+) sign e e o neu ino (an ineu ino) p ocesses. Likewise, he had onic cu en s o cha ged-cu en weak p ocesses, Jµ (N), a e also composed o ec o and axial e ms associa ed o he ec o and axial s uc u e o he nucleon. On he con a y, only he ec o pa emains o elec omagne ic in e ac ions (see Appendix A o de ails), Jµ (N)=ψP (P ,S )H ΓµψPi(Pi,Si),(2.6) 26 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING whe e H Γµ=FV 1γµ+ iFV 2 2MN σµνQν | {z } H Γµ V +GAγµγ5+FPQµγ5 | {z } H Γµ A .(2.7) A i s o de in pe u ba ion heo y and conside ing plane wa es, he wa e unc ions ela ed o he had onic and lep onic pa icles can be de ined as ψα(X)= mα EαVuα(pα,sα)e−ipα·X,(2.8) whe e u(pα,sα) a e he Di ac spino s and α he co esponding single pa icle s a e. In he p e ious exp esion o he had onic cu en (2.6), he inne s uc u e o he nucleon is de- sc ibed by means o he pu ely-iso ec o nucleon o m ac o s (FV 1,FV 2,GAand FP), a ising om he CC weak in e ac ion and dependen on Q2. The isospin symme y allows o a desc ip ion o he iso ec o o m ac o s FV 1and FV 2in e ms o he elec omagne ic Di ac, Fp,n 1and Pauli, Fp,n 2, o m ac o s o p o ons and neu ons. Conse ed ec o cu en (CVC) When compa ing bo h elec on-nucleon and cha ged-cu en neu ino-nucleon sca e ing, i can be obse ed ha he elec omagne ic cu en (see Appendix A) and he ec o componen o he weak one a e ela ed h ough he conse ed la o cu en . This is knwon as he CVC (conse ed ec o cu en ) hypo hesis and implies ha he weak ec o componen is conse ed. In he wo ds o J.D. Walecka [16], he CVC implies ha he ec o pa o he single nucleon ma ix elemen o he cha ge changing weak cu en , wha e e he de ailed dynamic s uc u e o he nucleon, can be ob ained om elec on sca e ing h ough he elec omagne ic in e ac ion. The CVC hypo hesis assumes ha he ec o pa o he had onic cu en and he elec omag- ne ic cu en a e componen s o he same conse ed-cu en isospin mul iple and hus hei o m ac o s a e in e ela ed, FV 1(Q2)= Fp 1(Q2)−Fn 1(Q2) 2,FV 2(Q2)= µpFp 2(Q2)−µnFn 2(Q2) 2,(2.9) whe e µp=2.793 and µn=−1.913 a e he co esponding p o on and neu on magne ic momen s. Fo comple eness, he isoscala o m ac o s a e de ined as FS 1(Q2)= Fp 1(Q2)+Fn 1(Q2) 2,FS 2(Q2)= µpFp 2(Q2)+µnFn 2(Q2) 2.(2.10) Finally, he axial o m ac o , GA, desc ibes he axial-pseudo ec o s uc u e o he nucleon whe eas he FPone englobes he pseudoscala s uc u e. All hese o m ac o s ha e been s udied in de ail in he li e a u e and will be ca e ully analyzed in Sec ion 2.3. Addi ional in o ma ion on he elec oweak o m ac o s can also be ound in [61–63]. 2.1. GENERAL FORMALISM 27 Neu ino-nucleon elas ic c oss sec ion The double di e en ial c oss sec ion o he elas ic neu ino-nucleon in e ac ion can be exp essed, in e ms o he ou going lep on ene gy (El) and he solid sca e ing angle (Ωl), as: d2σ dEldΩl =|~ kl| |~ kνl| G2 F 4π2δ ω−|Q2| 2MN!Hηµν H Wµν ,(2.11) whe e Hηµν and H Wµν a e he lep onic and had onic enso s, espec i ely, which a ise om he con- ac ion o he lep onic and had onic cu en s. The lep onic enso eads, Hηµν =j(l)† µj(l) ν=k ,µki,ν −k ·kigµν +k ,ν ki,µ ±iεµναλ kα kλ i,(2.12) being he axial con ibu ion nega i e o neu inos and posi i e o an ineu inos as a consequence o hei di e en helici y. Mo eo e , he had onic enso H Wµν is gi en by: H Wµν =Jµ† (N)Jν (N)=H Wµν V+H Wµν A+H Wµν V A =−W1(Q2)gµν +W2(Q2)Pµ iPν i M2 N +iW3(Q2) 2M2 N εµν ρσPi,ρQσ +W4(Q2) M2 N QµQν+W5(Q2)Pµ iQν+QµPν i 2M2 N ,(2.13) whe e Wi(Q2) a e he had onic s uc u e unc ions de ined in e ms o he nucleon o m ac- o s [63], W1=|Q2| 4M2 NFV 1+FV 22 +(GA)2+G2 A(2.14) W2=(FV 1)2+|Q2| 4M2 N (FV 2)2+(GA)2(2.15) W3=2GA(FV 1+FV 2) (2.16) W4=|Q2| −4M2 N (4M2 N)2(FV 2)2−FPGA MN +|Q2| 4M2 N (FP)2(2.17) W5=W2.(2.18) In he p e ious exp ession (2.13), he an isymme ic e m co esponds o he axial con ibu ions and, in pa icula , o he in e e ence ec o -axial e m associa ed o W3. The con ac ion o he lep onic and had onic enso s Hηµν H Wµν esul s in he ollowing exp ession, d2σ dEldΩ =|~ kl| Eνl G2 F 4π2δ ω−|Q2| 2MN!Hηµν H Wµν =|~ kl|El G2 F 4π2δ ω−|Q2| 2MN!(2W1sin2θ 2+W2Elcos2θ 2±W3 Eνl+El MN Elsin2θ 2 + m2 l El(El+|~ kl|)"W1cos θ−W2 2cos θ±W3 2*, Eνl+El MN cos θ−El+|~ kl| MN+- +W4 2*, m2 l M2 N cos θ+2El(El+|~ kl|) M2 N sin2θ 2+-−W5El+|~ kl| 2MN#).(2.19) 34 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING Gals e dipola pa ame iza ion This pa ame iza ion, i s ly in oduced in 1971 [18], is widely used by he heo e ical and expe i- men al communi y and gi es a easonable desc ip ion o he p o on expe imen al da a o |Q2| ≤ 1 GeV2(∼5%). In he neu on case, he desc ip ion is no e y accu a e due o he da a unce ain y. I p esen s a high simplici y based on a dipola unc ional o m, Gp E(Q2)=GV D(Q2) (2.78) Gn E(Q2)=−µnτGV Dξn(2.79) Gp M(Q2)=µpGV D(Q2) (2.80) Gn M(Q2)=µnGV D(Q2),(2.81) whe e ξn=(1 +λnτ)−1wi h λn=5.6 and τ=|Q2|/4M2 N, and µp=2.793 and µn=−1.913 a e he p o on and neu on magne ic momen s. The dipola o m ac o , GV D, is desc ibed as: GV D(Q2)=1 1+|Q2| M2 V2≡1 1+λV Dτ2;GV D(0) =1 ; GV D(|Q2| → ∞)=0,(2.82) whe e he pa ame e alues a e λV D=4M2 N/M2 V=4.97 and MV=0.843 GeV. Fo Q2→0, he o m ac o s a e ixed by he elec ic cha ge and he nucleon magne ic momen s. Typically his pa ame iza ion is cons ained o be consis en wi h expe imen al da a o he neu on cha ge adius. Howe e , ecen and mo e p ecise expe imen al esul s [67] ha e shown he limi a ion o his pa ame iza ion o i all hese da a. This has mo i a ed he analysis o o he app oaches. Kelly pa ame iza ion Recen ly, a new pa ame iza ion o he elec omagne ic o m ac o s has been de eloped by J.J. Kelly [68]. This is basically an ex ension o he Gals e one, p o iding a easonable desc ip ion o all expe imen al da a. In his app oach, he elec ic and magne ic o m ac o s a e de ined as, G(Q2)∝ n X k=0 akτk 1+ n+2 X k=1 bkτk ,(2.83) whe e bo h nume a o and denomina o a e polynomials in τ. Wi h n=1 and a0=1, his pa ame iza ion p o ides excellen i s o Gp E,Gp M/µpand Gn M/µnusing only ou pa ame e s. Howe e , his app oach is less success ul o Gn E. Mo e speci ic de ails can be ound in [68]. VMD models: GKeX pa ame iza ion A mo e accu a e desc ip ion o he nucleon o m ac o s can be ob ained om he VMD ( ec o meson dominance) models. Wi hin his phenomenological desc ip ion, he o m ac o s a e ex- p essed in e ms o mesonic p opaga o s and meson-nucleon o m ac o s. The mos ep esen a i e VMD models a e he Ga i-K umpelmann pa ame iza ion [69], which inco po a es he desc ip ion 2.3. HADRONIC STRUCTURE 35 o high |Q2|in oduced by he pQCD; and he ecen ly-de eloped GKeX pa ame iza ion, also known as Lomon p esc ip ion [70]. The alidi y o he GKeX pa ame iza ion ex ends o e all ange o Q2whe e elec on da a exis (0.1-10 GeV). This desc ip ion ep esen s an ex ension o he Ga i-K umpelmann one, including also he e ec s a ising om he ec o mesons: ρ,ρ′,ω,ω′ and φ. 0.2 0.4 0.6 0.8 1 1.2 GE p/GD Gals e GKeX Da a 0.95 1 1.05 1.1 GM p/(µpGD) 0.001 0.01 0.1 1 10 |Q2| (GeV2) -0.2 0 0.2 0.4 0.6 GE n/GD 0.01 0.1 1 10 |Q2| (GeV2) 0.6 0.7 0.8 0.9 1 1.1 GM n/(µnGD) Figu e 2.2: Elec omagne ic Sachs o m ac o s as a unc ion o |Q2| o he Gals e and GKeX pa ame iza ions and di ided by he app op ia e dipole ac o . Expe imen al da a aken om [71]. In Fig. 2.2 we compa e he elec omagne ic o m ac o s, Gn,p EyGn,p M, di ided by he app o- p ia e dipole e m o he Gals e and GKeX pa ame iza ions. Bo h a e e y simila o low |Q2| alues, whe eas hei di e ences inc ease subs an ially wi h |Q2|. Fu he mo e, he GKeX model ep oduces he expe imen al esul s accu a ely, due o i s mo e sophis ica ed desc ip ion o he nu- cleon s uc u e. Fo comple eness, we also show in Fig. 2.3 he iso ec o o m ac o s, G(1) EyG(1) M, associa ed o he weak cu en o bo h pa ame iza ions. This analysis ein o ces he idea o using he GKex pa ame iza ion in he nucleon o m ac o s employed in he s udy o QE p ocesses. Ad- di ionally, a de ailed analysis o a ious nucleon o m ac o s’ pa ame iza ions and hei ele ance on he cha ged-cu en quasielas ic neu ino-nucleus c oss sec ion will be analyzed in Chap e 7. 36 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING 0.001 0.01 0.1 1 10 |Q2| (GeV2) -0.5 0 0.5 1 GE V/GD Gals e GKeX 0.001 0.01 0.1 1 10 |Q2| (GeV2) 0.85 0.9 0.95 1 1.05 1.1 GM V/(µV GD) Figu e 2.3: Iso ec o o m ac o s as a unc ion o |Q2| o he Gals e and GKeX pa am e iza ions whe e µVmakes e e ence o he iso ec o magne ic momen . 2.3.3 Axial s uc u e The weak had onic cu en shown in Eq. (2.6) also depends on he axial GA(Q2) and pseudoscala GP(Q2) o m ac o s. The axial one, GA, has been widely s udied in las yea s [71, 72] in o de o de e mine i s unc ional s uc u e. On he con a y, he pseudoscala GP≡FP/2MN, is ha de o analyze, being i s con ibu ion e y small in mos o he kinema ical si ua ions add essed and negligible o he momen um ans e ed in ol ed in β-decay p ocesses. Mo eo e , cu en da a a e no p ecise enough o de e mine he unc ional o m o GA, which is usually pa ame ized using a dipole o m, analogously o he ec o o m ac o s, GA(Q2)=gA 1+Q2 M2 A2,(2.84) whe e gA=−1.267 is he axial- ec o coupling cons an and MA=1.032(36) GeV is he nucleon axial mass. The gApa ame e is de e mined h ough β-decay p ocesses wi h neu ons in he elas ic limi Q2→0 [73], and he axial mass MA alue has been ex ac ed om deu e ium- illed bubble chambe expe imen s [17]. Mo e in o ma ion ega ding he unc ional o m o GAand he possible quenching o gAcan be ound, espec i ely, in Appendices D and E. The axial and pseudoscala o m ac o s can be connec ed making use o he PCAC (pa ially conse ed axial cu en ) hypo hesis (see [74] o de ails), h ough he Goldbe ge -T eiman ela ion, GP(Q2)= 4M2 N Q2+m2 π GA(Q2) ; mπ: pion mass .(2.85) In cu en neu ino-nucleus in es iga ions, one o he main sou ces o unce ain y comes om he axial o m ac o , and speci ically, i s dependence wi h Q2and MA. In his sense, ecen CCQE neu ino expe imen s on 12C, such as MiniBooNE o MINERνA, ha e es ima ed highe MA alues (MA≈1.35 GeV) in disag eemen wi h he s anda d es ima ions. I mus be aken in o accoun ha he s anda d alue, MA=1.03 GeV, is consis en wi h deu e ium expe imen al da a whe e nuclea e ec s a e negligible as well as wi h weak pion p oduc ion da a a low |Q2|. Hence he inc ease o 2.3. HADRONIC STRUCTURE 37 he axial mass alue in hese expe imen s mus be in e p e ed as he lack o some ing edien s in he model employed o he da a analysis. In pa icula , he ole played by mul inucleon e ec s, such as 2p-2h MEC in CCQE neu ino-nucleus expe imen s will be explo ed in Chap e 7 as a possible explana ion o he “appa en ” inc ease o MA. In Fig. 2.4, we analyze he axial o m ac o s, GAand GP, using he wo ld a e age axial mass alue as well as an inc ease alue o 1.35 GeV. Whils small di e ences eme ge a e y low |Q2|, we can obse e how GAinc eases wi h MA o la ge alues o |Q2|. On he con a y, his e ec is no isible o GPdue o he ac o 1 1+|Q2|/m2 π whe e mπ≪MA. Mo eo e , he pseudoscala o m ac o alls o ze o as e han he axial one which u ns in o a less ele an con ibu ion o pseudoscala e ec s in he egion o elas ic, QE sca e ing and beyond. ✵✵✵✁ ✵✵✁ ✵✁ ✁ ✁✵ ⑤✂ ✷ ⑤ ✄☎✆✝ ✷ ✮ ✵ ✞ ✵✟ ✵✠ ✵ ✡ ✁ ✁ ✞ ● ❆ ✴ ☛ ❆ ▼ ☞ ❂ ✁✌✵✍☎✆✝ ▼ ☞ ❂ ✁✌✍ ✎☎✆✝ ✵✵✵✵✁ ✵✵✵✁ ✵✵✁ ✵✁ ✁ ✁✵ ⑤✂ ✷ ⑤ ✄☎✆✝ ✷ ✮ ✵ ✵ ✞ ✵✟ ✵✠ ✵✡ ✁ ✁ ✞ ● P ✯ ✏ ♣ ✴ ✑ ☛ ❆ ✭ ✒ ✓ ✔ ✕ ❪ ▼ ☞ ❂ ✁✌✵✍☎✆✝ ▼ ☞ ❂ ✁✌✍✎ ☎✆✝ Figu e 2.4: Axial (GA) and pseudoscala (GP) o m ac o s o wo MA alues as a unc ion o |Q2|. Finally, i is wo h men ioning ha he di e ences be ween neu ino and an ineu ino c oss sec ions, which a ise basically om he V-A in e e ence e m (2.21), s a o disappea as |Q2| inc eases. No ice ha he RV A T′ esponse (see Eq. 2.68) depends on GA ha app oaches ze o o la ge |Q2|. Chap e 3 Cha ged-cu en quasielas ic neu ino-nucleus sca e ing A e p esen ing he o malism o neu ino-nucleon elas ic sca e ing, in his chap e we ex end ou heo e ical desc ip ion o cha ged-cu en neu ino-nucleus in e ac ion in he quasielas ic egime and he s udy o nuclea e ec s. This is o in e es o mos ecen neu ino expe imen s ha em- ploy di e en nuclea a ge s o measu e he oscilla ion pa ame e s a ene gies whe e he quasielas- ic egime domina es. Unlike he elas ic sca e ing whe e we only conside he inne s uc u e o he nucleon, quasielas- ic eac ions equi e a desc ip ion o he nuclea s uc u e. In his PhD hesis, he desc ip ion o he lep on-nucleus in e ac ion and, in pa icula , he nuclea dynamics, is add essed wi hin he con ex o he Supe Scaling App oach (SuSA), which assumes he exis ence o uni e sal scaling unc ions o bo h elec omagne ic and weak in e ac ions. Fo a p ope unde s anding o his app oach, we gi e a b ie desc ip ion o he ela i is ic Fe mi gas model ha will help o in oduce he basis o he SuSA model as a semiphenomelogical app oach based on he analysis o inclusi e (e,e′) da a. Nex , we ex end ou desc ip ion o he ela i is ic mean ield (RMF) heo y and he ela i is ic plane wa e impulse app oxima ion (RPWIA) models as a mo e sophis ica ed p ocedu e o include inal-s a e in e ac ion (FSI) e ec s as well as he mean ield gene a ed by he nuclea cons i uen s in he neu ino-nucleus in e ac ion. The RMF has he me i o ea ing inal-s a e in e ac ions in a ela i is ic amewo k, which is o ele ance o he analysis o neu ino expe imen s. In neu ino- nucleus in e ac ions, a e he inal-s a e pa icles ha e been c ea ed, hey p opaga e ou h ough he nucleus, unde going s ong in e ac ions wi h he o he nucleons inside he nucleus. These “ inal- s a e in e ac ions” can signi ican ly al e he momen um and di ec ion o he inal-s a e pa icles as well as he ype and numbe o pa icles. Acco dingly, pions and nucleons can be abso bed wi hin he nuclea medium o hei collisions wi h o he nucleons can gene a e addi ional pa icles. Thus a consis en and ela i is ic ea men o hese in e ac ions is essen ial o de e mine he con ibu ion o he di e en eac ion mechanisms o neu ino-nucleus c oss sec ion. The desc ip ion o he he many-body physics o he in e ac ing nucleons wi hin he RMF and RPWIA models is he e o e included in ou amewo k, in he so-called SuSA 2 (Supe Scaling App oach e sion 2) model. This app oach has been ecen ly applied o he analysis o QE elec on sca e ing da a [38] o se e al nuclea a ge s as well as o cha ged-cu en quasielas ic (CCQE) neu ino-nucleus expe imen s [39], yielding an accu a e desc ip ion o he expe imen al da a in bo h cases. 39 40 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 3.1 Gene al o malism o he CCQE p ocess In his sec ion we show schema ically he kinema ics in ol ed in s udies o lep on sca e ing om nuclei, ocusing on cha ge-changing neu ino eac ions. Mos kinema ic a iables ha e been p e- iously de ined in Chap e 2 o elas ic neu ino-nucleon sca e ing. He e we employ ha no a ion o (νl,l−) and (νl,l+) neu ino-nucleus eac ions. We speci ically analyze he CCQE neu ino sca e ing p ocess in which an inciden beam o neu inos wi h 4-momen um kµ=(Eν,kν) in e ac s wi h a nucleus. In he inal s a e, a cha ged lep on wi h 4-momen um k′µ=(E′ l,k′ l) eme ges as well as an ou going nucleon. This p ocess is media ed by a weak boson (W) and can be desc ibed as, νµ(νµ)+A→µ−(µ+)+p(n)+(A−1) (3.1) wi h A he nuclea a ge and (A−1) he esidual nucleus a e he in e ac ion. Compa ed o he elas ic case, he complexi y o he nuclea dynamics in oduces some unce ain ies in he de- sc ip ion, ela ed o he inne s uc u e o he nucleus o inal-s a e in e ac ions desc ibed abo e. Many heo e ical app oaches e alua e hese p ocesses in he Impulse App oxima ion (IA), which assumes he inciden lep on o only in e ac wi h a single bound nucleon as shown in Fig. 3.1. The in luence o he emaining nucleons o e he en i e p ocess is aken in o accoun in di e en ways depending on he nuclea model employed. Hence, he IA desc ibes he nuclea many-body ma ix elemen as a sum o single-nucleon cu en ma ix elemen s. Figu e 3.1: Schema ic iew o he cha ged-cu en neu ino-nucleus sca e ing p ocess in he Im- pulse App oxima ion (IA). 3.1. GENERAL FORMALISM OF THE CCQE PROCESS 41 3.1.1 Lep on sca e ing kinema ics Going in o de ail o he desc ip ion o lep on kinema ics, he neu ino and inal-lep on ene gies a e gi en as Eνl=qm2 ν+k2 ν(3.2) El=qm2 l+k′2 l,(3.3) whe e mνand mla e he masses o he inciden neu ino and ou going lep on, espec i ely. The 4-momen um ans e o he nucleus Qµ=(ω,q) is gi en by ω=Eν−El(3.4) q=kν−k′ l,(3.5) being ωand q he ene gy ans e and 3-momen um ans e , espec i ely. Gi en an exci a ion om a ge es mass Mi o some inal es mass M ≥Mi( ha is, he inal had onic es ame o al ene gy is W=M ), we de ine a so o exci a ion ene gy ω0≡1 2MiM2 −M2 i≥0,(3.6) ela ed o he mass excess o he esidual nucleus. This alue se s a minimum neu ino ene gy o he QE neu ino in e ac ion o ake place. In he case o NCQE sca e ing, his would be he minimum ene gy equi ed o he p ocess whe eas in he CCQE one we mus also conside he ene gy needed o c ea e he ou going-lep on mass. Then om ene gy-momen um conse a ion one has ω=ω0+|Q2| 2Mi .(3.7) Sol ing Eqs. (3.4) and (3.7), one ge s exp essions o he sca e ed lep on’s ene gy and 3-momen um. De ining ǫ1≡qM2 i+2MiEν+m2 ν+k2 νsin2θl(3.8) ǫ2≡qMi(Eν−ω0)+H M2,(3.9) whe e θlis he lep on sca e ing angle ( he angle be ween kνand k′ l) and H M=q(m2 l+m2 ν)/2, i can be shown ha k′ l=1 ǫ2 1ǫ2 2(kνcos θl)+(Mi+Eν)qǫ4 2−m2 lǫ2 1(3.10) El=1 ǫ2 1ǫ2 2(Mi+Eν)+(kνcos θl)qǫ4 2−m2 lǫ2 1,(3.11) whe e o he esul s o be eal o all sca e ing angles, he beam ene gy mus be g ea e han Eν,min, Eν,min =ml+ω0+ mlω0+(m2 l−m2)/2 Mi−ml .(3.12) This esul ep esen s he minimum ene gy equi ed o he CCQE p ocess in e ms o he lep on mass and he mass excess o he esidual nucleus. In addi ion, he con ibu ion o he ω0pa ame e 42 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING is gene ally smalle han he lep on mass. In he pa icula case o (ν, µ−) eac ion on 12C, we ha e mµ≫ω0as he mass excess o he esidual ni ogen, ∼15 MeV, is much smalle han he muon mass, mµ≈105 MeV. Taking in o accoun he mass di e ence be ween lep ons and nucleus, he hi d e m in Eq. (3.12) gi es a negligible con ibu ion in mos o he cases. Hence, o gi en alues o he exci a ion ene gy ω0, beam ene gy Eνand he sca e ing angle θl, he quan i ies ǫ1,2can be compu ed and om hem he inal lep on’s ene gy and 3-momen um a e ixed. The 4-momen um ans e is hen gi en as well. 3.2 Rela i is ic Fe mi Gas Model and Supe Scaling App oach Nex , in o de o desc ibe he s uc u e o he nuclea esponses in ou heo e ical p esc ip ion, we in oduce he concep o scaling and he main ea u es o he ela i is ic Fe mi gas model [75,76] ha has been used as a basis o he s udy o scaling and supe scaling beha io s. 3.2.1 The concep o scaling and supe scaling Scaling is a phenomenon obse ed in se e al a eas o Physics and is a undamen al pa o he scien i ic me hodology. This is no new; we can hink o Galileo’s obse a ions o he oscilla ions o a pendulum, Keple ’s disco e y o he equal a ea law o plane a y mo ion and New on’s in e se squa e law o g a i a ion. The es ablishmen o a scaling ela ionship be ween physical quan i ies e eals an unde lying d i ing mechanism, and i is he ask o Physics o unde s and and o p o ide a o malism o ha mechanism. The concep o yscaling in lep on-nucleus eac ions was i s ly in oduced by Wes [77] as an analog o xscaling in high-ene gy physics. The e i was shown ha a collec ion o non- ela i is ic poin -like cha ged “nucleons” wi h negligible inal-s a e in e ac ions leads o an inclu- si e quasielas ic elec on sca e ing c oss sec ion ha can be w i en in e ms o a ac o con aining a single-nucleon c oss sec ion imes a speci ic unc ion. In he limi o la ge momen um ans e s, his unc ion scales; ha is, i becomes a unc ion o only one a iable, usually ep esen ed by y, and la gely independen on he momen um ans e . Fu he s udies ocused on he analysis o his scaling beha io on inclusi e elec on-nucleus sca e ing da a in a ela i is ic amewo k [77] show ha his beha iou can also be obse ed o di e en nuclei [36,78]. Conce ning he di icul ies o his analysis, i should be no iced ha he explici ene gy dependence o he c oss sec ions complica es he a emp o ac o ize he elec on- nucleus c oss sec ion by ex ac ion o a unique single-nucleon c oss sec ion. In addi ion o his, he e also exis s he ques ion o he o -mass-shell ex apola ion o he single-nucleon cu en as well as he ac ha FSI and mul i-nucleon in e ac ions can also a ec he scaling analysis. In he pa icula case o QE elec on-nucleus sca e ing p ocesses, in mos o he models based on IA, he inclusi e (e,e′) c oss sec ion can be app oxima ed by a single-nucleon c oss sec ion imes a speci ic unc ion o (q,ω). In his case, he lep on in e ac s wi h a many-body sys em in such a way ha he ene gy ωand momen um qa e ans e ed only o indi idual cons i uen s o he complex sys em. Scaling occu s a some speci ic kinema ics, whe e he speci ic unc ion scales, ha is, i becomes dependen on only a single quan i y, namely, he scaling a iable ψ. This quan- i y, whose de ini ion is discussed la e , is in u n a unc ion o qand ω:ψ=ψ(q,ω). The unc ion ha esul s once he single-nucleon c oss sec ion has been di ided ou is called he scaling unc ion 3.2. RELATIVISTIC FERMI GAS MODEL AND SUPERSCALING APPROACH 43 = (q,ψ). In o he wo ds, o he ex en ha a some kinema ics his unc ion depends on ψ, bu no on q, one says ha ψ-scaling occu s. The s udy o he scaling unc ion can shed ligh on he dynamics o he nuclea sys em. Indeed, wi hin some speci ic app oaches, he scaling unc ion is ela ed o he momen um dis ibu ion o he nucleons in he nucleus (o , mo e gene ally, wi h he spec al unc ion) [79,80]. When s udying (e,e′) p ocesses i is use ul o in oduce he ollowing concep s: •Scaling o i s kind. This is ela ed o he concep o y-scaling: i is sa is ied when he scaling unc ion does no explici ly depend on he ans e ed momen um, bu only on ψ including i s implici dependence on qand ω. •Scaling o second kind. I is obse ed when he scaling unc ion is independen o he nuclea species. •Scaling o ze o h kind. I occu s when he scaling unc ions linked o he di e en chan- nels ha make up he c oss sec ion, longi udinal (L) and ans e se (T), a e equal. Hence in inclusi e elec on sca e ing, ze o h-kind scaling means ha he elec omagne ic (EM) scal- ing unc ions sa is y = L= T, whe e ep esen s he scaling unc ion associa ed o he o al c oss sec ion whe eas L,T e e o he scaling unc ions ob ained om he sepa a e longi udinal and ans e se esponses. •Supe scaling. Finally, when scaling o bo h he i s and second kinds occu s simul aneously one has supe scaling [35,36]. Wi h hese ing edien s i is clea ha he na u al s a ing poin o such an examina ion is he ela i is ic Fe mi gas (RFG) model in which he nucleus is desc ibed as a nonin e ac ing gas o nucleons. Indeed, his model ul ills exac ly all o he kinds o scaling p e iously de ined. The RFG has he appeal o simplici y while main aining impo an aspec s in he p oblem such as Lo en z co a iance and gauge in a iance in a ully ela i is ic way. Na u ally i igno es po en- ially impo an e ec s such as hose s emming om s ong inal-s a e in e ac ions o wo-body MEC and employs an o e simpli ied ini ial-s a e spec al unc ion; ne e heless, such ing edien s can be added o he basic model and appea no o in alida e i as a basic s a ing poin o analyses o scaling. The gene al p ocedu e used o de ine scaling unc ions consis s o cons uc ing he inclusi e c oss sec ion, o nuclea esponse unc ions, wi hin a pa icula heo e ical model (o expe imen al da a) and di ide hem by he app op ia e single-nucleon quan i y compu ed wi hin he RFG model. He e he wo d “app op ia e” en ails wo aspec s o be conside ed. Fi s , he usual analysis in he egion o he quasielas ic (QE) peak assumes ha he dominan p ocess is elas ic sca e ing om nucleons in he nuclea g ound s a e ollowed by quasi ee ejec ion o he nucleons om he nucleus, and hence he app op ia e single-nucleon o m ac o s a e he elas ic ones. Second, he nucleons in he nuclea g ound s a e a e mo ing (Fe mi mo ion) and acco dingly he single-nucleon c oss sec ion used mus ake his in o accoun . 3.2.2 Nuclea e ec s and dynamical pa ame e s in he RFG model The RFG model can be aken as a guide o inco po a ing ela i is ic ing edien s o mo e sophis i- ca ed models and, pa icula ly, as a i s app oxima ion o he nuclea dynamics in ol ed in he QE 50 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING Figu e 3.2: Scaling unc ion (ψ′) as a unc ion o ψ′ o di e en nuclei (A≥12) and se e al kinema ics co e ing om o wa d o e y backwa d angles as well as om low o e y high inci- den ene gies. The alues o Aco esponding o di e en symbols a e shown in he igu e. Da a aken om [36]. Ne e heless, he scaling beha io becomes pa icula ly clea i one s udies he expe imen al c oss sec ion sepa a ed in o i s longi udinal L(ψ′) and ans e se T(ψ′) con ibu ions, as shown in Fig. 3.3. The sepa a e longi udinal and ans e se con ibu ions lead o he conclusion ha he longi udinal expe imen al da a supe scale h oughou he whole egion o he QE peak, whe eas he ans e se da a do no scale, being scaling iola ions mo e p ominen in he egion abo e he QE peak (ω > ωQEP,i.e.,ψ′>0). Scaling iola ions a high ωoccu because o o he non-QE p o- cesses, such as meson p oduc ion and esonance exci a ions, which a e p edominan ly ans e se, come in o play. A e y high ans e ene gies (i.,e. high ψ′- alues) deep inelas ic sca e ing s a s o be ele an . Likewise, 2p-2h s a es induced by meson-exchange cu en s a e known o ha e a ele an con ibu ion in he “dip” egion be ween he QE and he ∆peaks. As we de ail in Chap- e 4, 2p-2h MEC con ibu ions a e p edominan ly ans e se o elec omagne ic in e ac ions. All hese con ibu ions beyond he IA a e esponsible o scaling iola ions, mainly obse able in he ans e se channel. O he mechanisms, no included in he p esen wo k, can in oduce some e ec s in he gene al discussion. This is he case o RPA nucleon-nucleon co ela ion e ec s [30] ha can modi y he longi udinal and ans e se esponses because o he e y di e en isospin cha ac e o he wo channels. Howe e , RPA is only ele an o e y low ene gy/momen um ans e s and, in o e all, hey a e expec ed o be e y small. The p esc ip ion adop ed in he i s e sion o he Supe Scaling App oach has been o employ he expe imen al longi udinal esponses o de ine a gene al scaling unc ion o bo h elec omag- ne ic longi udinal and ans e se channels. This implies ha bo h Land Tscaling unc ions a e oughly he same a e sub ac ing he non-scaling con ibu ions ela ed o p ocesses beyond he QE egime, ha is, ze o h kind scaling is ul illed. As we shall illus a e, he mos mode n e sion 3.3. SUPERSCALING APPROACH:A SEMIPHENOMELOGICAL MODEL 51 o he model (SuSA 2) con ains co ec ions o his assump ion based on Rela i is ic Mean Field heo y. Figu e 3.3: Scaling unc ion, L(ψ′) and T(ψ′), om he longi udinal and ans e se esponse, espec i ely, as a unc ion o ψ′ o di e en nuclei (A≥12) and o di e en alues o q(in MeV/c). Da a aken om [82]. F om he analysis o he sepa a e longi udinal da a shown in Fig. 3.3 (le panel), a “uni e sal” phenomenological Lsupe scaling unc ion Lhas been ex ac ed. In Fig. 3.4, we show he da a analysis. The solid cu e e e s o a i o he longi udinal da a gi en by he ollowing pa ame iza- ion, which co esponds o he SuSA scaling unc ion, SuSA(ψ′)≡ L(ψ′)=p1 [1 +p2 2(ψ′−p3)2](1 +ep4ψ′)(3.69) whe e p1=2.9883, p2=1.9438, p3=0.67310 and p4=−3.8538. The RFG scaling unc ion is also shown o e e ence (dashed line). No e ha SuSA(ψ′) p esen s an asymme ic shape and a ail ha ex ends owa ds posi i e al- ues o ψ′,i.e.,ω > ωQEP. I s maximum eaches abou ∼0.6. In con as , he RFG scaling unc ion is symme ic in he scaling a iable ψ′, is limi ed s ic ly o he egion −1≤ψ′≤+1 and has a maximum alue o 3/4. This beha io clea ly di e s om da a and, hence, he RFG does no ep oduce he scaling beha io shown by he incusi e elec on sca e ing da a. The p e ious supe scaling beha io shown in Fig. 3.4 has been e alua ed wi h o he models based on he ha monic oscilla ion shell model (HO+FSI) and he ela i is ic Fe mi gas (RFG+FSI) ha inco po a e di e en desc ip ions o FSI (see [83] o de ails). Some asymme y in he co e- sponding scaling unc ions eme ges om hese calcula ions, mainly due o he FSI, bu he heo e - ical p edic ions s ill di e signi ican ly om he da a. In subsequen sec ions we p o ide a de ailed s udy o he longi udinal and ans e se scaling unc ion wi hin he amewo k o he RMF heo y, analyzind he speci ic ole played by FSI and he ela i is ic nuclea dynamics. 52 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING -1 0 1 2 3 ψ/ 0 0,2 0,4 0,6 0,8 L (ψ/) RFG SuSA Figu e 3.4: A e aged expe imen al SuSA(ψ) e sus ψ′in he quasielas ic egion oge he wi h a phenomenological pa ame e iza ion o he selec ed (e,e′) longi udinal scaling da a ob ained om he RLda a on [82]. The in eg al o he cu e has been no malized o uni y. The RFG scaling unc ion is also shown as e e ence. . Mo eo e , in spi e o he di icul y in analyzing he ans e se scaling unc ion, p e ious s ud- ies [84] based on he modeling o he QE longi udinal esponse and con ibu ions om non-QE channels ha e p o ided some e idence ha he scaling o ze o h kind is no ully sa is ied by da a. In pa icula , hese s udies ha e ound ee′ T,exp > ee′ L,exp, a esul ha has mo i a ed a deep s udy o he scaling beha io as add essed in he nex sec ion. 3.4 Ex ension o he Supe scaling App oach om Rela i is ic Mean Field Theo y: he SuSA 2 Model As de ined be o e, he Supe Scaling App oach (SuSA) is based on he scaling p ope ies o he longi udinal esponse ex ac ed om (e,e′) da a o p edic cha ged-cu en quasielas ic (CCQE) neu ino- and an ineu ino-nucleus c oss sec ions [85]. Thus, SuSA is based on he hypo hesis ha he neu ino c oss sec ion scales as does he elec on sca e ing c oss sec ion. This ea u e is obse ed in mos o he models based on IA (see, o ins ance, [85–87]). Acco dingly, he SuSA model uses he expe imen al scaling unc ion ee′ L,exp as a uni e sal scaling unc ion and hen builds he di e en nuclea esponses by mul iplying i by he co esponding single-nucleon esponses. In mos IA app oaches, one inds ha once he single-nucleon c oss sec ion is emo ed in de ining he scaling unc ions, he longi udinal and ans e se esponses a e basically he same. Howe e , wi hin he amewo k o he RMF heo y, one inds ha 0 h-kind scaling is mildly b oken o momen um ans e s in he 1 GeV egion being T(ψ′)> L(ψ′). Mo eo e , no ice ha he ex ac ion o ee′ L,exp en ails he analysis o he pu ely- ec o longi udinal (e,e′) nuclea esponse, which combines isoscala +iso ec o con ibu ions. In con as , CC neu ino-nucleus e- 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 53 ac ions in ol e only iso ec o couplings and a e mainly domina ed by pu ely ans e se esponses (TVV +TAA and T′ V A). Thus, one could ques ion he alidi y o he Supe Scaling App oach. This subjec was s udied in [88] by analyzing he scaling unc ions e alua ed wi h he RMF model. The e, i was ound ha , con a y o wha one migh expec , he (e,e′) longi udinal scaling unc- ion ag ees wi h he o al (νl,l−) one (which is mainly ans e se) be e han does he ans e se scaling unc ion om (e,e′). This esul is explained by he di e en oles played by he iso ec o and isoscala nucleon o m ac o s in each p ocess (see [88] o de ails). These p e ious s udies ha e mo i a ed a supe scaling analysis based on he RMF heo y in o de o imp o e he analysis o neu ino eac ions. Wi hin he RMF model [89] he bound and sca e ed nucleon wa e unc ions a e solu ions o he Di ac-Ha ee equa ion in he p esence o ene gy-independen eal scala (a ac i e) and ec o ( epulsi e) po en ials. Since he same el- a i is ic po en ial is used o desc ibe he ini ial and inal nucleon s a es, he model is shown o p ese e he con inui y equa ion [90] ( his is s ic ly ue o he CC2 cu en ope a o [86]); hence he esul s a e almos independen o he pa icula gauge selec ed [86, 87]. In he RMF model he nucleons a e dynamically and s ongly o -shell and, as a consequence, he c oss sec ion is no ac o ized in o a spec al unc ion and an elemen a y lep on-nucleus c oss sec ion. The RMF has achie ed signi ican success in desc ibing QE elec on sca e ing da a. On he one hand, i s alidi y has been widely p o ed h ough compa isons wi h QE (e,e′) da a (see [86] and Sec . 3.5.2). In his connec ion, an impo an esul is ha he model ep oduces su p isingly well he magni ude and shape o ee′ L,exp,i.e., i yields an asymme ic longi udinal scaling unc ion, wi h mo e s eng h in he high-ω ail, and wi h a maximum alue (∼0.6) e y close o he expe - imen al one. On he o he hand, he model p edic s ee′ T> ee′ L. Fo ins ance, a q=500 MeV/c (1000 MeV/c) he ans e se RMF scaling unc ion a he maximum is 13% (20%) la ge han he longi udinal one. This iola ion o ze o h-kind scaling was analyzed in [88], whe e i was shown ha he o igin o such an e ec lies in he dis o ion o he lowe componen s o he ou going nu- cleon Di ac wa e unc ion by he FSI. Howe e , he RMF model also p esen s some d awbacks. Fi s , i p edic s a mode a e depen- dence o he scaling unc ion on he ans e ed momen um q. Fo inc easing alues o q he RMF model p esen s: i) a s ong shi o he scaling unc ions o highe ω alues, ii) oo much enhance- men o he a ea unde he ail o he unc ions, and iii) co espondingly oo se e e a dec ease in he maximum o he scaling unc ions. Despi e hese sho comings, he scaling unc ions ob ained wi hin he RMF model ep oduce easonably well he expe imen al longi udinal scaling unc ion in a wide kinema ical egion (see Sec ion 3.5). Second, ge ing esul s wi h he RMF model is compu a ionally e y expensi e, especially when he model is employed o p edic neu ino c oss sec ions whe e one has o old in he lux dis ibu ion o he inciden neu ino o o compu e o ally in eg a ed c oss sec ions. Hence in wha ollows, a e co ec ing o he oo s ong q-dependence o he RMF model, we shall implemen he main ea u es o he model in a new e sion o he SuSA app oach, called “SuSA 2”, ha makes i possible o ob ain nume ical p edic ions o com- pa e wi h da a using as codes, ye e aining he basic physics o he RMF. In summa y, in his wo k we ex end he o iginal SuSA model by inco po a ing in i s o malism in o ma ion om he RMF model. Thus we build he new model in such a way ha i ep oduces he expe imen al longi udinal scaling unc ion, p oduces ee′ T> ee′ L, akes in o accoun he di - e ences in he isoscala /iso ec o scaling unc ions and a oids he p oblems o he RMF model in he egion o high momen um ans e . 54 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 3.4.1 RMF and RPWIA scaling beha io In his sec ion we p esen a sys ema ic analysis o he scaling unc ions compu ed wi h he Rel- a i is ic Mean Field (RMF) and he Rela i is ic Plane Wa e Impulse App oxima ion (RPWIA). Bo h models a e based on he ela i is ic impulse app oxima ion (RIA) and p o ide a ully ela- i is ic desc ip ion o he sca e ing p ocess. The bound s a e Di ac-spino s a e he same in bo h models and co espond o he solu ions o he Di ac equa ion wi h scala and ec o po en ials. The wo p esc ip ions di e in he ea men o he inal s a e: he RPWIA desc ibes he ou going nucleon as a ela i is ic plane wa e while he RMF model accoun s o he FSI be ween he ou go- ing nucleon and he esidual nucleus using he same mean ield as used o he bound nucleon. In wha ollows, we analyze he scaling unc ions in ol ed in (e,e′), (ν, µ−) and (ν, µ+) eac ions as unc ions o he momen um ans e q. Due o he la ge numbe o exis ing elec on and neu ino expe imen al da a o 12C, mos o he calcula ions p esen ed in his hesis co espond o his nu- cleus. The ex ension o o he nuclea sys ems will be add essed in ollowing chap e s. We i s spli all di e en esponse unc ions by isola ing he isoscala (T=0) and iso ec o (T=1) con ibu ions in elec on sca e ing, and he ec o and axial con ibu ions o neu ino and an ineu ino induced eac ions: VV ( ec o - ec o ), AA (axial-axial), VA ( ec o -axial). This s a egy will allow us o ex ac clea in o ma ion on how he FSI a ec he di e en sec o s o he nuclea cu en . Fu he mo e, i will make i easie o explo e he ela ionships be ween he di e en esponses linked o (e,e′), (ν, µ−) and (ν, µ+) eac ions. Assuming cha ge symme y, he wo channels, Land T, accesible in elec on sca e ing (3.57) can be decomposed as a sum o isoscala (T=0) and iso ec o (T=1) con ibu ions. In e ms o he scaling unc ions, he nuclea esponses a e de ined as, Ree′ L,T(q,ω)=1 kF T=1,ee′ L,T(ψ′)GT=1 L,T(q,ω) + T=0,ee′ L,T(ψ′)GT=0 L,T(q,ω)g.(3.70) Simila ly, he cha ge-changing muon-neu ino (an ineu ino) esponses (3.54) can be gi en as, RVV,ν(ν) L(q,ω)=1 kF VV,ν(ν) L(ψ′)GVV L(q,ω) (3.71) RAA,ν(ν) CC (q,ω)=1 kF AA,ν(ν) CC (ψ′)GAA CC (q,ω) (3.72) RAA,ν(ν) CL (q,ω)=1 kF AA,ν(ν) CL (ψ′)GAA CL (q,ω) (3.73) RAA,ν(ν) LL (q,ω)=1 kF AA,ν(ν) LL (ψ′)GAA LL (q,ω) (3.74) Rν(ν) T(q,ω)=1 kF VV,ν(ν) T(ψ′)GVV T(q,ω) + AA,ν(ν) T(ψ′)GAA T(q,ω)g(3.75) Rν(ν) T′(q,ω)=1 kF V A,ν(ν) T′(ψ′)GV A T′(q,ω).(3.76) The GK’s e ms in (3.70) and (3.71–3.76) a e he single-nucleon esponses de ined in (3.55). No- ice ha he p e ious scaling unc ions K o neu ino induced eac ions a e pu ely iso ec o . 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 55 In he ollowing we examine h ee basic ea u es o he scaling unc ions in he RPWIA and RMF models: shape, posi ion and heigh o he peak, and he in eg als o he scaling unc ions o e ψ′. 3.4.2 Shape o he scaling unc ions The goal he e is o s udy he shape o all scaling unc ions. In Fig. 3.5 (Fig. 3.6), o di e en alues o q, we p esen he ans e se (longi udinal) RMF scaling unc ions no malized o he maximum alue co esponding o a e e ence unc ion, in his case VV,ν T, and eloca ed so ha he maximum is a ψ′=0. As al eady men ioned, he scaling a iable ψ′depends on q,ωand Eshi . Thus, o each scaling unc ion, Eshi is aken so ha he maximum is loca ed a ψ′=0. The esul s wi hin he RPWIA model a e p esen ed in Fig. 3.7. We do no p esen esul s o AA CC , AA CL , AA LL o neu ino and an ineu ino sca e ing, and T=0 T o elec on sca e ing because hey a e e y sensi i e o small e ec s due o cancella ions and/o o he smallness o he denomina o (G unc ion) which appea s in he de ini ion o he scaling unc- ion (3.55). The i s h ee a e seen o be insigni ican o neu ino eac ions as i will be shown in Chap e 7, whe eas he ou h does no en e in ha case and is known o be a mino co ec ion in he QE egime o elec on sca e ing. The esul s om he RPWIA model show ha all scaling unc ions ha e he same shape (see Fig. 3.7). This commen also applies o models based on non ela i is ic and semi ela i is ic de- sc ip ions (see [85,91]). -1 012 3 0 0.2 0.4 0.6 -1 012 3 0 0.2 0.4 0.6 T - ν T’ VA - ν T AA - ν T - aν T’ VA aν T AA - aν T (T=1) - e -1 012 3 ψ’ 0 0.2 0.4 0.6 -1 012 3 ψ’ 0 0.2 0.4 0.6 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figu e 3.5: T ans e se RMF scaling unc ions no malized o he maximum alue co esponding o an a bi a y e e ence unc ion and eloca ed a ψ′=0 (see ex o de ails). The con en ion used o label he di e en cu es is as ollows: “e” o elec on-induced eac ions and “ν” (“aν”) o neu ino- (an ineu ino-) induced eac ions. 56 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING -1 012 3 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 012 3 0 0.1 0.2 0.3 0.4 0.5 0.6 L - ν L - aν L T=1 - e L T=0 - e -1 012 3 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 012 3 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figu e 3.6: As in Fig. 3.5, bu now o he longi udinal RMF scaling unc ions. -1 012 3 0 0.2 0.4 0.6 0.8 -1 012 3 0 0.2 0.4 0.6 0.8 L - ν T - ν T’ VA - ν T AA - ν L - aν T - aν T’ VA - aν T AA - aν L T=1 - e T T=1 - e L T=0 - e -1 012 3 ψ’ 0 0.2 0.4 0.6 0.8 -1 012 3 ψ’ 0 0.2 0.4 0.6 0.8 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figu e 3.7: As in Fig. 3.5, bu in his case he esul s co espond o RPWIA. T ans e se and longi udinal se s a e p esen ed oge he . Wi hin he RMF model, all ans e se scaling unc ions app oxima ely collapse in a single one. On he con a y, he longi udinal esponses a e g ouped in wo se s: one co esponding o he pu e elec on iso ec o and neu ino (an ineu ino) VV- esponses, i.e., T=1,ee′ Land VV,ν(ν) L, and he o he o he isoscala con ibu ion o elec ons, namely, T=0,ee′ L. This esul eme ges o all q- alues and ends o be a he gene al. I is also no iceable ha he ail is highe and mo e ex ended o he ans e se esponses, whe eas o he longi udinal ones i ends o go down as e . 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 57 I is wo h obse ing ha in all cases he RMF scaling unc ions display a much mo e p o- nounced asymme ic shape han he RPWIA ones, an e ec ela ed o he speci ic ea men o inal s a e in e ac ions. 3.4.3 Heigh and posi ion o he peak o he scaling unc ion In he op (bo om) panel in Fig. 3.8 he peak-heigh o he ans e se (longi udinal) se o scaling unc ions is p esen ed as unc ion o q. The esul s co espond o RMF and RPWIA p edic ions. We obse e ha he peak-heigh s o he scaling unc ions wi hin RPWIA a e almos q-independen (and e y close o RFG alue o 3/4), while he RMF ones p esen a mild q-dependence in he ans e se se and a somewha s onge one o he longi udinal se . I is well known ha FSI end o dec ease he peak-heigh o he esponses pu ing he s eng h in he ails, especially a high ene gy loss. This is pa icula ly ue o he RMF app oach [86, 92] and models based on he Rela i is ic G een Func ion (RGF) [93, 94]. Simila e ec s ha e also been obse ed wi hin semi ela i is ic app oaches [85,91]. 400 600 800 1000 1200 1400 1600 q (MeV/c) 0.6 0.65 0.7 0.75 0.8 peak heigh T’ VA - ν T - ν T AA - ν T’ VA - aν T - aν T AA - aν T T=1 - e RPWIA RMF 400 600 800 1000 1200 1400 1600 q (MeV/c) 0.45 0.5 0.55 0.6 0.65 0.7 0.75 peak heigh L - ν L - aν L T=1 - e L T=0 - e RPWIA RMF Figu e 3.8: Top panel: Peak heigh o he ans e se se o scaling unc ions as a unc ion o he ans e ed momen um q. The uppe se o lines co esponds o he p edic ion wi hin RPWIA ( hin lines), while he lowe se o lines has been ob ained wi h he RMF model. Bo om panel: As o he op panel, bu now o he longi udinal se o scaling unc ions. 58 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING Mo e speci ically, in Fig. 3.8, we see ha he disc epancies be ween he RMF and RPWIA peak-heigh esul s a e age o ∼25% in he ans e se se . On he o he hand, hose disc epan- cies a e mo e s ongly q-dependen in he longi udinal sec o , eaching ∼30% (∼70%) in he lowe (highe ) q- egion o he longi udinal iso ec o esponses (blue lines). Finally, he di e ence be- ween he isoscala longi udinal (e,e′) scaling unc ion p oduced by RMF and RPWIA (magen a dashed-do ed lines) is somewha smalle : ∼20% (∼30%) o lowe (highe ) q. 400 600 800 1000 1200 1400 1600 q (MeV/c) 10 20 30 40 50 60 70 80 Eshi (MeV) T’ VA - ν T - ν T AA - ν T’ VA - aν T - aν T AA - aν T T=1 - e RMF RPWIA 400 600 800 1000 1200 1400 1600 q (MeV/c) 10 20 30 40 50 60 70 80 Eshi (MeV) L - ν L - aν L T=1 - e L T=0 - e RMF RPWIA Figu e 3.9: Top panel: Shi ene gy, Eshi , needed in o de o ha e he co esponding scaling unc ion peak loca ed a ψ′=0, as unc ion o q. Resul s o he ans e se se o scaling unc ions. Bo om panel: As o he op panel, bu now o he longi udinal se o scaling unc ions. In Fig. 3.9 we s udy he posi ion o he peak o he ans e se and longi udinal se s. To his scope we display he ene gy shi , Eshi , needed o place he peak o he scaling unc ion a ψ′=0 as a unc ion o q. In he op panel o Fig. 3.9 we see ha o he RPWIA ans e se scaling unc ion, Eshi is almos q-independen , while he co esponding RMF shi inc eases almos linea ly wi h he momen um ans e . This q-linea dependence o Eshi was al eady obse ed and discussed wi hin he amewo k o a semi ela i is ic model based on he use o he Di ac-equa ion- based po en ial [91]. App oxima ely he same beha io is obse ed o he longi udinal se (bo om panel in Fig. 3.9), al hough in his case he RPWIA esul s a e so ly linea ly dependen on q. I is also wo h men ioning ha he h ee ans e se scaling unc ions linked o he same neu ino o an ineu ino p ocess, VV T, AA Tand V A T′, collapse in a single line o RMF as o RPWIA. 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 59 F om he analysis o Figs. 3.8 and 3.9 one may conclude ha T=1,ee′ Lp esen s he same beha - io (heigh and posi ion) as VV,ν(ν) L(blue lines). The di e ences be ween hese h ee cu es a e app oxima ely cons an and a ise om he di e ences in he bound s a es in ol ed in he eac ion: p o on+neu on in (e,e′), neu on in (ν, µ−) and p o on in (ν, µ+). The Coulomb-FSI, namely, he elec omagne ic in e ac ion be ween he s uck nucleon and he esidual nucleus, which plays a ole when he ou going nucleon is a p o on, could also in oduce a di e ence; howe e , we ind ha i s e ec s a e negligible and ha he di e ences be ween, o ins ance, VV,ν Land VV,ν Lin RP- WIA (whe e no Coulomb-FSI a e in ol ed) a e almos he same as in RMF (see Figs. 3.8 and 3.9). The s ong q-dependence o he RMF peak posi ion, which keeps g owing wi h he momen um ans e , is a sho coming o he model, whose alidi y is ques ionable a e y high q. Indeed o high q he ou going nucleon ca ies a la ge kine ic ene gy so he e ec s o FSI should be sup- p essed o such kinema ics. In ac , i would be desi able ha he RMF esul s end o app oach he RPWIA ones o inc easing momen um ans e , i.e., he scaling unc ions should become mo e symme ic, and a sa u a ion o he peak-heigh educ ion and o he ene gy shi should be obse ed. Tha end is consis en wi h he scaling a gumen s [36, 37, 86], i.e., he expe imen al e idence o a uni e sal scaling unc ion o inc easing q. This is one o he mo i a ions o use an al e na i e model i one aims o ep oduce he expe imen al (e,e′) da a a medium- o-high mo- men um ans e s. A possible al e na i e o he beha io o he peak heigh , peak posi ion and shape o he scaling unc ions is o implemen he RMF model a low o in e media e-qand he RPWIA one o highe q- alues, as i is p oposed in ollowing sec ions. 3.4.4 Sum ules In Fig. 3.10, he alues o he in eg als o e ψ′o he di e en scaling unc ions wi hin RMF model a e p esen ed e sus q. These a e gi en by Si(q)=Z∞ −∞ i(ψ, q)dψ . (3.77) The in eg a ion limi s, deno ed by (−∞,+∞), ex end in eali y o he ange allowed by he kine- ma ics. The abo e in eg al in he case o he longi udinal (e,e′) scaling unc ion was shown o coincide, apa om some mino disc epancies asc ibed o he pa icula single-nucleon exp es- sions conside ed and he in luence o he nuclea scale in oduced, wi h he esul s ob ained using he s anda d exp ession o he Coulomb Sum Rule (see [95] o de ails)1. Hence in wha ollows we deno e he unc ions Si(q) simply as sum ules. We see ha all in eg als o he ans e se se a e abo e uni y and inc ease almos linea ly wi h q. On he con a y, he in eg als o VV,ν(ν) Land T=1,ee′ L(blue lines) a e below uni y and dec ease wi h qup o q=1100 MeV/c. F om q=900 MeV/c hey begin o be s able a ound he alue 0.7. Then, om q=1200 MeV/c o highe q- alues he in eg als s a g owing again. Howe e , no ice ha in ha q- egion he esul o he in eg als is e y sensi i e o he beha io o he ail o hese pa icula scaling unc ions (see Fig. 3.6). Finally, he alues o he in eg al o he longi udinal isoscala unc ion, T=0,ee′ L, is app oxima ely cons an and close o uni y. The beha io o he in eg als o he wo longi udinal scaling unc ions o (e,e′) is consis en wi h he analysis o he Coulomb sum ule o hese wo models (see [95]). 1Coulomb sum ule (CSR) in inclusi e elec on sca e ing s a es ha by in eg a ing he longi udinal s eng h o e he ull ange o ωa la ge q, one should ge he o al cha ge (numbe o p o ons) o he nucleus. 66 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING When displayed e sus he scaling a iable ψ′(Fig. 3.14), we no ice ha he non-blocked scaling unc ion is exac ly he same o all kinema ics and is only limi ed o he ange o ω- alues compa ible wi h he ixed q alue. As also obse ed, he PB e ec s only ha e an impac in he egion o low ψ′which is ela ed o he lowe alues o ω. 00.5 11.5 22.5 3 0 0.1 0.2 0.3 0.4 0.5 (ψ’) SuSA SuSAwo PB 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 -1 0 1 2 3 4 5 ψ’ 0 0.1 0.2 0.3 0.4 0.5 (ψ’) -1 0 1 2 3 4 5 ψ’ 0 0.1 0.2 0.3 0.4 0.5 q=50 MeV q=100 MeV q=250 MeV q=500 MeV Figu e 3.14: Supe scaling unc ion e sus ψ′a di e en q- ixed alues and e alua ed o he SuSA model wi h (SuSA) and wi hou (SuSAwoPB) Pauli blocking. . 0 0,1 0,2 0,3 0,4 ω (GeV) 0,1 0,2 0,3 0,4 0,5 blocked = (ψ) - (Φ) (Φ) = (-ω,q) (ψ) = (ω,q) q = 570 MeV/c 0 0,1 0,2 0,3 0,4 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 400 MeV/c 0 0,1 0,2 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 250 MeV/c -0,05 00,05 0,1 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 150 MeV/c 00,05 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 0,6 q = 100 MeV/c -0,025 00,025 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 0,6 q = 50 MeV/c Figu e 3.15: Supe scaling unc ion e sus ωa di e en q- ixed alues and e alua ed o he SuSA model wi h ( blocked) and wi hou ( (ψ)) Pauli blocking. The mi o unc ion ( (Φ)) is also shown as e e ence. 3.5. ANALYSIS OF THE SUSAV2MODEL 67 Nex we show he e ec s o PB in he SuSA and SuSA 2 models when compa ing wi h (e,e′) da a on 12C. Fo his pu pose, we display in Fig. 3.16 he SuSA esul s wi h and wi hou PB and compa e hem wi h a ew se s o da a a he kinema ics in which PB e ec s a e signi ican , i.e., e y low q. In he egion o low-ω, he PB e ec s can be obse ed on he wid h and peak heigh o he c oss sec ions. In gene al we conclude ha he ag eemen be ween SuSA and da a imp o es when PB is in oduced. SuSA wi hou PB (g een-dashed) p oduces c oss sec ions oo wide, while SuSA wi h PB (b own) p o ides na owe c oss sec ions in be e ag eemen wi h da a. This is pa icula ly ue in panels (1) and (2) in Fig. 3.16. The same commen s apply o Fig. 3.17 whe e SuSA 2 wi h and wi hou PB is compa ed wi h he same se o low-qda a. The lowes ene gy ans e da a, co esponding o he exci a ion o esonan and collec i e s a es, canno be desc ibed by any o he p esen models. 00.05 0.1 0.15 0 50000 100000 150000 200000 SuSA w PB SuSA w/o PB εi=400, θe=36º, q~238 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0.1 0.2 0.3 0.4 0 500000 1000000 1500000 dσ/dΩ/dω (nb/s /GeV) εi=1300, θe=11.9º, q~270 0 0.1 0.2 0 50000 100000 εi=480, θe=36º, q~285 00.05 0.1 0.15 ω (GeV/c) 0 10000 20000 30000 εi=320, θe=60º, q~296 0.1 0.2 ω (GeV/c) 0 200000 400000 600000 800000 εi=1300, θe=13.5º, q~305 (1) (2) (3) (4) (5) (6) Figu e 3.16: SuSA wi h and wi hou Pauli Blocking is compa ed wi h (e,e′) da a. Eshi =10 MeV has been employed. Da a aken om [100,101]. 68 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 00.05 0.1 0.15 0 50000 100000 150000 200000 SuSA 2 w PB SuSA 2 w/o PB εi=400, θe=36º, q~238 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0.1 0.2 0.3 0.4 0 500000 1000000 1500000 dσ/dΩ/dω (nb/s /GeV) εi=1300, θe=11.9º, q~270 0 0.1 0.2 0 50000 100000 εi=480, θe=36º, q~285 00.05 0.1 0.15 ω (GeV/c) 0 10000 20000 30000 εi=320, θe=60º, q~296 0.1 0.2 ω (GeV/c) 0 200000 400000 600000 800000 εi=1300, θe=13.5º, q~305 (1) (2) (3) (4) (5) (6) Figu e 3.17: SuSA 2 wi h and wi hou Pauli Blocking is compa ed wi h (e,e′) da a. Da a aken om [100,101]. A clea di e ence be ween SuSA and SuSA 2 (Figs. 3.16 and 3.17) is ha he la e clea ly o e es ima es he da a in he egion below and close o he peak whe eas he SuSA model un- de es ima es all da a o medium and high q- alues (Figs. 3.19-3.20). Howe e , in all cases he maximum is placed a ω.50−60 MeV whe e, as discussed in Sec . 3.5.2, he alidi y o he mod- els based on IA is ques ionable and no de ini i e conclusions can be d awn based on compa ison o model and da a in his ω- egion. Ne e heless, he ag eemen o he SuSA 2 model wi h da a in he low-ene gy ange can be imp o ed by de e mining mo e accu a ely he q0 ansi ion pa ame e be ween RMF and RPWIA con ibu ions, as shown in Sec . 6.1, in such a way ha he e ec s o FSI ia he RMF model be mo e signi ican o lowe kinema ics, hus educing he esul ing c oss sec ion. The e ec o Pauli Blocking is also o ele ance o he analysis o CCQE neu ino c oss sec ions, pa icula ly a low kinema ics, as will be shown in Chap e 7. 3.5. ANALYSIS OF THE SUSAV2MODEL 69 3.5.2 Analysis o he SuSA 2 model o (e,e′) eac ions wi hin he QE egime In his sec ion we p esen a sys ema ic compa ison o inclusi e 12C(e,e′) expe imen al c oss sec- ions and he p edic ions o he QE p ocess wi hin RMF, SuSA and SuSA 2 models. As men- ioned, da a co espond o he o al inclusi e c oss sec ion which includes con ibu ions om se - e al channels, mainly: QE sca e ing, inelas ic sca e ing, many-nucleon emission, e c. He e we only ocus on he QE p ocess, whe eas a mo e de ailed analysis o he inclusi e c oss sec ion including inelas ic p ocesses and 2p-2h MEC con ibu ions will be add essed in Chap e 6. The e- o e, one expec s ha he models do no ep oduce he o al inclusi e expe imen al da a co e- sponding o kinema ical si ua ions in which non-QE con ibu ions play some ole. Thus, he main in e es o he sys ema ic analysis p esen ed in his sec ion is he compa ison be ween SuSA 2 p edic ions and hose om he SuSA and RMF models. Full analyses o he inclusi e (e,e′) c oss sec ion (including desc ip ions o QE and non-QE con ibu ions) ha e been p esen ed wi h some success in he pas [84, 85] wi hin he con ex o he semiphenomelogical SuSA. Ou aim in he ollowing is o comple e he desc ip ion o he inclusi e p ocess wi hin he con ex o SuSA 2 model, ia he inclusion o he inelas ic spec um and 2p-2h con ibu ions. This will be de ailed in he nex chap e s. In Figs. 3.18-3.20 we p esen he compa ison o he (e,e′) expe imen al da a and models. Due o he la ge amoun o a ailable da a on 12C(e,e′) a di e en kinema ics (see [100,101]) in hese h ee igu es we only show some ep esen a i e examples. A mo e de ailed analysis will be d awn in Chap e 7. Each igu e is labeled by he inciden elec on ene gy, εi(in MeV), he sca e ing angle, θe, and he ans e ed momen um co esponding o he cen e o he quasielas ic peak, q (in MeV/c). Pauli Blocking has been included in he SuSA and SuSA 2 models ollowing he p ocedu e desc ibed in [96,97] and de ailed in Sec ion 3.5.1. 00.05 0.1 0.15 0 50000 100000 150000 200000 dσ/dΩ/dω (nb/s /GeV) εi=400, θe=36º, q~239 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0 0.1 0.2 ω (GeV) 0 50000 100000 150000 dσ/dΩ/dω (nb/s /GeV) εi=480, θe=36º, q~285 0 0.1 0.2 ω (GeV) 0 10000 20000 εi=361, θe=60º, q~333 (a) (b) (c) (d) Figu e 3.18: Compa ison o inclusi e 12C(e,e′) c oss sec ions and p edic ions o he RMF ( ed), SuSA (g een-dashed) and SuSA 2 (b own) models (see ex o de ails). Se o panels co espond- ing o low-q alues. Da a a e aken om [100,101]. 70 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 00.1 0.2 0 5000 10000 dσ/dΩ/dω (nb/s /GeV) εi=440, θe=60º, q~401 0.2 0.4 0.6 0 2000 4000 6000 εi=961, θe=37.5º, q~585 0 0.2 0.4 0.6 0.8 ω (GeV) 0 10000 20000 30000 dσ/dΩ/dω (nb/s /GeV) εi=2500, θe=15º, q~660 0.2 0.4 0.6 0.8 ω (GeV) 0 500 1000 1500 2000 εi=1299, θe=37.5º, q~792 (e) ( ) (g) (h) Figu e 3.19: Con inua ion o Fig. 3.18. Se o panels co esponding o medium-q alues. Da a a e aken om [100,101]. 0.2 0.4 0.6 0.8 200 400 600 800 1000 dσ/dΩ/dω (nb/s /GeV) εi=1501, θe=37.5º, q~917 0.2 0.4 0.6 0.8 0 200 400 600 εi=3595, θe=20º, q~1316 11.5 ω (GeV) 0 2 4 6 8 10 dσ/dΩ/dω (nb/s /GeV) εi=4045, θe=30º, q~2247 2.2 2.4 2.6 2.8 3 ω (GeV) 0 0.1 0.2 0.3 εi=4045, θe=55º, q~3457 (i) (j) (k) (l) Figu e 3.20: Con inua ion o Fig. 3.18. Se o panels co esponding o high-q alues. Da a a e aken om [100,101]. 3.5. ANALYSIS OF THE SUSAV2MODEL 71 The panels in Figs. 3.18-3.20 a e o ganized acco ding o he alue o he ans e ed momen- um (a he cen e o he QE peak) in h ee se s: low-q( om q=238 o q=333 MeV/c) in Fig. 3.18, medium-q( om q=401 o q=792 MeV/c) in Fig. 3.19 and, high-q( om q=917 o q=3457 MeV/c) in Fig. 3.20. The only phenomenological pa ame e s en e ing in he calcula- ion a e he Fe mi momen um kFand he ene gy shi Eshi . Fo hese we use kF=228 MeV/c (see [37]) in bo h SuSA and SuSA 2 models. A cons an ene gy shi o 20 MeV is employed in SuSA [37] while a q-dependen unc ion, he one desc ibed in Sec . 3.5, is used o Eshi in he SuSA 2 model. P elimina y alues o he SuSA 2 ansi ion pa ame e s o q0=800 MeV/c and ω0=200 MeV ha e been applied. We begin commen ing on he low-qpanels p esen ed in Fig. 3.18. The main con ibu ions o he c oss sec ion om non-QE p ocesses such as inelas ic p ocesses con ibu ions (∆- esonance) and MEC, a e e y small, e en negligible, in his low-q egion. In spi e o ha , when he ans- e ed ene gy is small (ω.50 −60 MeV) o he p ocesses such as collec i e e ec s con ibu e o he c oss sec ion making ques ionable he ea men o he sca e ing p ocess in e ms o IA-based models. This could explain, in pa , he gene al disag eemen be ween models and da a in ha ω egion in (a), (b) and (c) panels. Mo eo e , we can obse e ha he SuSA model seems o be shi ed o high ω alues wi h ega d o he expe imen al da a whe eas he SuSA 2 model ma ches wi h he expe imen al QE peak posi ion. A he same ime, he SuSA 2 esul s a e highe han he SuSA ones, which is mainly due o he na u al enhancemen on he RMF ans e se scaling unc ion. Some cla i ica ions a e called o ega ding he RMF esul s in Fig. 3.18, whe e sha p eso- nances appea a e y low ω alues. These co espond o 1p-1h exci a ions wi h he phase shi o a gi en pa ial wa e going h ough 90 deg ees. Wi h mo e complica ed many-body desc ip ions hese sha p ea u es a e smea ed ou . In summa y, in o de o es he goodness o he models in he kinema ical si ua ion o Fig. 3.18, one should ocus on he s udy o he ails o he c oss sec ions whe e la ge enough ω- alues (ω&50 −60 MeV) a e in ol ed. The e, one obse es ha SuSA p edic ions a e clea ly o e - shi ed o high ω- alues while RMF and SuSA 2 models i he da a easonably well. In addi ion, as expec ed, SuSA esul s a e sys ema ically below SuSA 2 and RMF ones a he QEP. We now discuss he esul s o medium-q alues p esen ed in Fig. 3.19. Fi s o all, one should men ion ha o he kinema ics o his igu e, in addi ion o he QE p ocess, non-QE con ibu ions a e essen ial o desc ibe he expe imen al c oss sec ions. Fo ins ance, in panels ( ), (g) and (h) he ∆-peak appea s clea ly de ined a ω alues abo e he QE peak. In panel (e) one sees ha in he egion a ound he cen e o he QE-peak, he RMF p edic ion is sligh ly abo e he SuSA 2 one, being close o he expe imen al da a. This is consis en wi h he beha io o he RMF scaling unc ion s udied in Sec . 3.4.3 (see Fig. 3.8), namely, he peak-heigh o he RMF scaling unc ions inc eases o dec easing q- alues. I he main non-QE con ibu ions a e no included in he modeling i is ha d o conclude which model is be e o ep oduce he pu ely QE c oss sec ion. Howe e , i seems easonable o conclude ha SuSA 2 imp o es he ag eemen wi h da a compa ed o SuSA. Fo ins ance, in he si ua ion o panel (e), i would be needed ha non-QE p ocesses would con ibu e mo e han 20% a ound he QE-peak in o de o SuSA i s he heigh o he da a. A 20% ac ion o he c oss sec ion linked o ∆- esonance and MEC con ibu ions is p obably oo much o ha kinema ics. Simila commen s and conclusions apply o he esul s in panel (d) o Fig. 3.18. 72 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING Fo q- alues close o 650 MeV/c (panels ( ) and (g)) RMF and SuSA 2 p oduce e y simila esul s because o he way in which SuSA 2 has been de ined (see Sec . 3.5). Fo highe q- alues, q&792 MeV/c ((h) panel), SuSA 2 and RMF p edic ions begin o depa om each o he . In pa icula , RMF esul s end o shi he peak o highe ω alues and o place mo e s eng h in he ail while SuSA 2 c oss sec ions end o be mo e symme ical due o he inc easing dominance o he RPWIA scaling beha io (see Sec . 3.5). This di e ence is mo e e iden o highe q- alues, as obse ed in panels (j)-(l) o Fig. 3.20. I is impo an o poin ou ha o he kinema ics p esen ed in Fig. 3.20 he non-QE con ibu ions a e no only impo an bu hey become dominan in he c oss sec ions. This is he case p esen ed in panels (k) and (l) whe e he QE-peak is no e en isible in he da a. Nex , we summa ize he main conclusions om he p esen compa ison o he SuSA and SuSA 2 models in he QE egime wi h he inclusi e (e,e′) da a: •Rega ding he enhancemen o he ans e se esponse, RT, in SuSA 2 compa ed wi h SuSA: p io o he addi ion o non-QE con ibu ions (which will be add essed in Chap e 6), he mos clea indica ions ha suppo he SuSA 2 assump ions a ise om he compa ison wi h da a a kinema ical si ua ions in which non-QE e ec s a e supposed o be small (panels (e) and (d) in Figs. 3.18 and 3.19, espec i ely). •Rega ding he ene gy-shi analysis: wi hin he SuSA model we ha e used a cons an ene gy shi o 20 MeV/c. F om he compa ison wi h he low-qse o expe imen al da a, Fig. 3.18, one concludes ha 20 MeV is a oo la ge shi . On he con a y, he compa ison wi h he high-qse o da a, Fig. 3.20, sugges s ha 20 MeV is p obably oo small. Then, one is led o conclude ha a cons an ene gy shi is no he bes op ion o ep oduce (e,e′) da a. These esul s suppo he idea o in oducing a q-dependen ene gy shi such as we made in he SuSA 2 model. The heo e ical jus i ica ion o his assump ion was al eady discussed in Sec . 3.5. In Chap e 7 we will show ha he SuSA 2 model desc ibes mo e accu a ely he neu ino induced eac ions han he semiphenomelogical SuSA ap oach. This esul mainly comes om he q-dependence on he ene gy shi , he enhancemen on he ans e se esponse ia RMF p e- sc ip ions as well as he possibili y o sepa a ing scaling unc ions in o isoscala and iso ec o con ibu ions. Finally, we should add a ema k conce ning he RMF/RPWIA ansi ion pa ame e s, q0and ω0, appea ing in Eq. 3.91. In he p e ious esul s (Figs. 3.18-3.20), we ha e applied a ixed alue o hese pa ame e s which ha e shown a easonable ag eemen wi h some expe imen al da a a he QE peak. Howe e , he ansi ion be ween he RMF and RPWIA models depends on he pa icula kinema ics in ol ed, namely on he momen um ans e q. Acco dingly, he ansi ion pa ame e , q0, is expec ed o inc ease wi h qin such a way ha he RMF con ibu ion will be dominan a low kinema ics whe e FSI e ec s a e o high ele ance, whe eas he RPWIA desc ip ion is needed a highe kinema ics. The e o e we in oduce a dependence on he momen um ans e q ha de e mines explici ly he ela i e RMF and RPWIA con ibu ions a di e en kinema ics. This analysis is ca ied ou oge he wi h he ex ension o he SuSA 2 o malism o he inelas ic spec um and applied o he s udy o he (e,e′) da a in Chap e 6. Chap e 4 2p-2h MEC con ibu ions o elec oweak eac ions In his Chap e , we e alua e and discuss he impac o wo-pa icle wo-hole meson-exchange cu - en s (2p-2h MEC) on elec on- and neu ino-nucleus c oss sec ions. The 2p-2h MEC esponses a e calcula ed wi hin he RFG model in which a ully Lo en z and ansi ionally in a ian calcu- la ion can be de eloped. In o de o educe he compu a ional ime, we make use o an accu a e pa ame iza ion o hese esponses. 4.1 In oduc ion As shown in Fig. 4.1, he 2p-2h MEC p ocess akes place when a weak o elec omagne ic bo- son om he lep onic cu en is exchanged by a pai o nucleons (2-body cu en ) leading o he emission o wo nucleons om he p ima y e ex. These s a es, whe e wo nucleons a e p omo ed abo e he Fe mi le el lea ing wo holes inside he Fe mi sea, a e known o gi e a la ge con ibu ion in he so-called “dip egion”, co esponding o exci a ion ene gies lying be ween he quasielas ic (QE) and ∆(1232) exci a ion peaks. These con ibu ions a e essen ial o a co ec in e p e a ion o Figu e 4.1: Schema ic iew o he 2p-2h MEC p ocess o lep on-nucleus in e ac ions. cu en and o hcoming neu ino oscilla ion expe imen s, which s ongly elies on ou unde s and- ing o neu ino-nucleus sca e ing a in e media e ene gies ( om 0.5 o 10 GeV) and in pa icula 73 74 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS o he nuclea -s uc u e e ec s in ol ed. A hese kinema ics, i has been p o ed ha p ocesses be- yond he IA, in which MEC play a majo ole, gi e a signi ican posi i e con ibu ion o he c oss sec ion which helps o accoun o he disc epancy obse ed in (e,e′) p ocesses be ween heo y and expe imen in he “dip” egion as well as o he disc epancies be ween some ecen neu ino CCQE measu emen s (e.g., MiniBooNE, NOMAD, MINERνA, T2K) [20, 21,24,46,47,56]. In- deed, he inclusion o 2p-2h MEC con ibu ions has allowed o explain hese da a wi hou modi y- ing any e ec i e pa ame e (such as he axial mass MA) [30,31,39,102,103]. All his suppo s he need o conside mechanisms such as inal-s a e in e ac ions (included in ou QE desc ip ion ia RMF heo y), nuclea co ela ions o MEC, in pa icula h ough hei con ibu ion o mul inucleon knock-ou a ound and beyond he QE peak as sugges ed by explici modeling [30,31,104,105]. As commen ed in Chap e 3, he 2p-2h MEC p ocesses oge he wi h inelas ic con ibu ions and, in pa icula , meson p oduc ion ia ba yon esonances such as he ∆, a e esponsible o scaling iola ions in (e,e′) eac ions, which a e mo e p ominen in he ans e se [84,85]. Howe e , e en below he meson p oduc ion h eshold he e a e scaling iola ions in he ans e se esponse [36], one sou ce o which is clea ly he MEC con ibu ions, again p edominan ly ans e se. These wo-body cu en s can exci e bo h one-pa icle one-hole (1p-1h) and wo-pa icle wo-hole (2p- 2h) s a es. Mos s udies o elec omagne ic (e,e′) p ocesses pe o med o low- o-in e media e momen um ans e s wi h MEC in he 1p-1h sec o (see, e.g., [106–109]) ha e shown a small e- duc ion o he o al esponse a he QE peak, mainly due o diag ams in ol ing he elec oexci a ion o he ∆ esonance. Ne e heless, hey a e oughly compensa ed by he posi i e con ibu ions o co ela ion diag ams, whe e he i ual boson couples o a co ela ed pai o nucleons. In his wo k we shall he e o e neglec hem and es ic ou a en ion o 2p-2h inal s a es, compu ed in a ully ela i is ic way. As discussed in p e ious wo ks [105, 110–113], ela i i y is an essen ial ing e- dien in he analysis o 2p-2h p ocesses a momen um ans e s abo e 400-500 MeV/c. A hese kinema ics, he non- ela i is ic educ ion can lowe he esul ing 2p-2h MEC c oss sec ion a ound a∼40% o mo e [105]. This s a emen is in connec ion wi h he kinema ical egions o in e es o neu ino oscilla ion expe imen s which ex end o ela i is ic domains. A hese q- alues, he s a ic app oxima ion used o he ∆p opaga o in he non- ela i is ic calcula ions o 2p-2h ans e se esponse unc ion [114] ails o explain he “dip” egion. Mo eo e , he p esence o nucleon-nucleon co ela ion in e ac ions in ol ing he one-nucleon cu en may lead o he exci a ion o 2p-2h inal s a es, and in e e ence be ween hese p ocesses and hose in ol ing MEC should be analyzed o assess hei ele ance. These e ec s, aken in o ac- coun in he RFG-based desc ip ions o 2p-2h p o ided by Nie es e al. [31] and Ma ini [30], a e no included explici ly in ou RFG MEC model, ha elies on a hyb id desc ip ion whe e he one- pa icle emission al eady con ains con ibu ions o nuclea ejec ions due o nuclea co ela ions — ia scaling unc ions om he SuSA 2 model. Explici calcula ions o he co ela ion-MEC in e e ence e ms a e s ill in p og ess and hei con ibu ions will be p esen ed in u he wo ks. The con ibu ions conside ed in his hesis englobe he 2p-2h s a es exci ed by he ac ion o meson-exchange cu en s wi hin a ully ela i is ic amewo k (see [110,112,113,115] o de ails), in ol ing i ual ∆ esonances as well as he seagull (con ac ) and pion-in- ligh cu en s ob ained in p e ious wo ks [110, 111]. De ia ions om he Fe mi gas model 2p-2h esponses p oduced by ing edien s such as inal-s a e in e ac ions, ini e nuclea e ec s o nuclea co ela ions a e expec ed o be mode a e, which would esul in small co ec ions in he impulsi e c oss sec ion as he MEC con ibu ions a e also mode a e. The p e ious assump ion is based on p e ious wo ks a low- o-in e media e momen um ans e s o 12C and 40Ca wi hin he amewo k o he con inuum shell model [114,116] as well as in o he analyses [117–119]. 4.2. GENERAL FORMALISM 75 4.2 Gene al o malism The e alua ion o he 2p-2h MEC con ibu ions is pe o med wi hin an exac mic oscopic cal- cula ion, whe e he wo-body cu en is he sum o seagull, pion-in- ligh , pion-pole and ∆-pole ope a o s and he basis wa e unc ions a e non-in e ac ing Di ac spino s. The ea u es o he RFG model allows o a ully Lo en z co a ian calcula ion o he MEC. In Re s. [110, 113, 115] i has been calcula ed o he i s ime he ully ela i is ic weak (wi h ec o and axial compo- nen s) cha ged meson-exchange cu en s o neu ino-nucleus in e ac ion in bo h longi udinal and ans e se channels as well as a comple e analysis o elec omagne ic eac ions. The nume ical in eg a ion me hod is desc ibed in [105,113,120] whe e he 2p-2h MEC had onic enso is calcu- la ed conside ing wo pa icles p′ 1and p′ 2abo e he Fe mi momen um in he inal s a e, p′ i>kF, and wo holes h1and h2below he Fe mi momen um, hi<kF, Wµν 2p−2h=V (2π)9Zd3p′ 1d3h1d3h2M4 E1E2E′ 1E′ 2 Θ(p′ 1,p′ 2,h1,h2) µν(p′ 1,p′ 2,h1,h2)δ(E′ 1+E′ 2−E1−E2−ω),(4.1) whe e by momen um conse a ion, p′ 2=h1+h2+q−p′ 1and Eiand E′ ia e he on-shell ene gies o he holes and pa icles. The only wo pa ame e s in he desc ip ion a e he Fe mi momen um kF associa ed o he nuclea species and he sepa a ion ene gy, i.e.,Eshi . Pauli-blocking e ec s a e also conside ed h ough he s ep unc ion Θ(p′ 1,p′ 2,h1,h2)=θ(p′ 2−kF)θ(p′ 1−kF)θ(kF−h1)θ(kF−h2).(4.2) The elemen a y 2p-2h had onic enso µν is de ined in e ms o he wo-body MEC an isym- me ized ma ix elemen jµ(1′,2′,1,2)A, µν (p′ 1,p′ 2,h1,h2)=1 4X s1s2s′ 1s′ 2X 1 2 ′ 1 ′ 2 jµ(1′,2′,1,2)∗ Ajν(1′,2′,1,2)A.(4.3) The MEC ope a o is w i en as he sum o ou con ibu ions, seagull (a,b), pion-in- ligh (c), pion-pole (d,e), and ∆pole ( –i), as shown in Fig. 4.2, jµ MEC =jµ sea +jµ π+jµ pole +jµ ∆.(4.4) The di e en con ibu ions a e cha ac e ized in e ms o how he i ual boson is a ached o he had onic e ex. The seagull o con ac e ms a e associa ed o he a achmen o he i ual boson o he N Nπ e ex whe eas he pion-in- ligh ope a o is e e ed o he di ec in e ac ion o he bo- son wi h he i ual pion. A a iance wi h he pion-in- ligh cu en , he pion-pole e ms has only he axial componen and he e o e i is absen in he elec omagne ic case. This con ibu ion could be conside ed as he “axial coun e pa ” o he pion-in- ligh e m, in he sense ha i con ains wo pion p opaga o s. MEC con ibu ions in ol ing he i ual ∆ esonance (∆-pole e ms) a e also o ele ance o he desc ip ion. Fo comple eness, i is wo h poin ing ou ha hose diag ams ha co espond o he exci a ion o a 2p2h+πs a e, hence o pion p oduc ion [121], a e implici ly included in he phenomenological inelas ic scaling unc ion desc ibed in Chap e ??. The exac e alua ion o he 2p-2h had onic enso (4.1) in a ully ela i is ic way in ol es nume ical se en-dimensional in eg a ions o a huge numbe o e ms. This makes he compu a ion, exac ly pe o med in [105,110,113,120], highly non- i ial. In o de o educe he compu a ional ime as well as o ease he implemen a ion o he esul s in Mon e Ca lo gene a o s used in he 82 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS 050 100 150 200 0 0.01 0.02 0.03 0.04 R2p-2h MEC (MeV-1) CC CL LL T T’ LEM TEM TAA TVV 0 100 200 300 400 500 600 ω (MeV) -0.01 0 0.01 0.02 0.03 R2p-2h MEC (MeV-1) 0 200 400 600 800 1000 ω (MeV) -0.01 -0.005 0 0.005 0.01 R2p-2h MEC (MeV-1) Figu e 4.9: Compa ison be ween 2p-2h MEC ans e se (T=TVV +TAA and T′=T′ V A) esponse unc ions and he longi udinal ones (CC,CL and LL) a q=200 MeV/c ( op panel), q=600 MeV/c (mid panel) and q=1000 MeV/c (bo om panel). 4.4. 2P-2HMEC RESPONSES FOR (νl,l)REACTIONS 83 We can also no ice a la ge ele ance o he weak longi udinal con ibu ions wi h espec o he elec omagne ic case. Ne e heless, when compu ing he o al MEC neu ino c oss sec ion, he con ibu ion o he CC and LL channels is oughly compensa ed by ha o he nega i e CL esponse, so ha o neu ino ene gies below ∼1 GeV he ne longi udinal con ibu ion plays a mino ole in he o al MEC esponse. This is illus a ed in Fig. 4.10, whe e he L,Tand T′con ibu ions o he 2p-2h MEC c oss sec ion a e displayed e sus he neu ino ene gy. A highe ene gies he Land T′con ibu ions become compa able, bo h being much smalle ha he dominan Tone. The balance be ween he longi udinal and ans e se 2p-2h channel discussed abo e is somehow di e en om he one eme ging in he elec omagne ic case. As desc ibed in Sec ion 4.3, he longi udinal elec omagne ic MEC esponse is indeed negligible wi h ega d o he ans e se one. Howe e , as illus a ed in Fig. 4.10, we no ice ha when compu ing he o al 2p-2h MEC weak c oss sec ion he longi udinal con ibu ion is domina ed by he axial channel and hus i plays a mo e ele an ole compa ed wi h he pu ely- ec o elec omagne ic case. Conce ning he ans e se esponses, i is no iceable ha he magni ude o he pu e axial and ec o channels o he c oss sec ion is e y simila . Mo eo e , he ec o -axial in e e ence con ibu ion eaches i s maximum a ound Eν∼1 GeV and dec eases a highe ene gies as a consequence o he beha io o he lep onic ac o VT′and he axial o m ac o GA(Q2), which anish as |Q2|inc eases (see Chap e 2 o de ails). 00.5 11.5 22.5 Eν (GeV) 0 1 2 3 4 σMEC (10-39cm2) L T T’ L+T+T’ 00.5 11.5 22.5 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 σMEC (10-39cm2) LVV LAA TVV TAA T’VA Figu e 4.10: Sepa a ion in o componen s o he o al 2p-2h MEC νµc oss sec ion displayed e sus neu ino ene gy Eν. The o al longi udinal (L), ans e se (T) and ans e se in e e ence (T′) con ibu ions a e shown (le panel) as well as he ne co n ibu ion (L+T+T′). Longi udinal and ans e se channels a e decomposed in o ec o and axial con ibu ions ( igh panel). 0500 1000 1500 2000 ω (MeV) 0 0.01 0.02 0.03 RT (MeV-1) q: 200-2000 MeV/c (s eps: 200 MeV/c) TAA TVV 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 RT’ VA (MeV-1) q: 200-2000 MeV/c (s eps: 200 MeV/c) Figu e 4.11: Compa ison be ween 2p-2h MEC axial ans e se (TAA) esponse unc ions and he ec o ones (TVV ) e sus ω(le panel). The ans e se ec o -axial in e e ence e m (T′ V A) is also shown ( igh panel). The cu es a e displayed om le o igh in s eps o q=200 MeV/c. 84 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS The analysis o he e olu ion wi h qo he indi idual ans e se componen s (see Fig. 4.11) shows ha he axial e m is la ge han he ec o one a low-in e media e kinema ics (q<800 MeV/c) whe eas he opposi e occu s a highe kinema ics. The in e e ence ec o -axial esponse unc ions yield an in e media e esul be ween he axial and ec o esponses. The p esen e alua ion o he 2p-2h MEC esponses and hei co esponding i s also ha e he me i o co e ing a e y wide q ange, including he ail o he esponses a high ψ′and ω alues. In gene al, (e,e′) da a a e a ely a ailable when ω→qand hence he high-ω egion was igno ed in some p esc ip ions. In con as , o CCQE eac ions one mus in eg a e o e a b oad neu ino spec um and hence, po en ially, he high-ω egion may be ele an . This has mo i a ed us o make an accu a e desc ip ion o he MEC con ibu ions also o hese kinema ics. In Fig. 4.5, he analysis o he RM EC T,VV esul s e sus ωshows a small con ibu ion below q<300 MeV/c as well as he ele ance o he ail in he esponse a q>800 MeV/c. On he o he hand, he ail o he MEC esponses a high q(q>1000 MeV/c) which appea s a ω&1000 MeV does no con ibu e signi ican ly o he c oss sec ion a lowe kinema ics, as can be deduced om Fig. 4.12. In ac , i we neglec he ail o he MEC esponse in ou pa ame iza ion, no signi ican di e ences eme ge excep a neu ino ene gies abo e 1 GeV whe e he comple e app oach yields somewha la ge con ibu ions, as seen in Fig. 4.13. I can also be deduced om Fig. 4.12 ha no signi ican MEC con ibu ions appea o q>2000 MeV/c, and he same is ue o la ge ω > 1000 MeV, showing ha he kinema ics whe e he MEC gi e he la ges con ibu ion o he c oss sec ion co esponds o q∼500 −1500 MeV/c and ω∼250 −1000 MeV, as also s a ed in [115]. A mo e de ailed analysis o he ele an kinema ic egions o he en i e 2p-2h MEC con ibu ion will be shown in Chap e 7. Mo eo e , he compa ison be ween he elec omagne ic 2p-2h MEC calcula ions and he (e,e′) da a will be add essed in Chap e 6 oge he wi h he QE and inelas ic con ibu ions. 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC MEC, q>100 MeV/c MEC, q>250 MeV/c MEC, q>500 MeV/c MEC, q>1000 MeV/c MEC, q>2000MeV/c 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC MEC, ω>50 MeV MEC, ω>100 MeV MEC, ω>250 MeV MEC, ω>500 MeV MEC, ω>1000 MeV MEC, ω>2000MeV Figu e 4.12: To al 2p-2h MEC νµc oss sec ion o he TVV channel pe a ge nucleon e alua ed excluding all con ibu ions coming om ans e ed momen um (uppe panel) and ene gy (lowe panel) below some selec ed alues, as indica ed in he igu e. The main me i o he pa ame iza ion p o ided he e o 2p-2h MEC elec omagne ic and weak eac ions on 12C is ha i ansla es a sophis ica ed and compu a ionally demanding mic oscopic calcula ion in o a smoo h pa ame iza ion which depends on he alues o he ans e a iables o he p ocess, easing i s implemen a ion in o Mon e Ca lo neu ino e en simula ions used in he analysis o expe imen s. In he ollowing sec ion, we s udy he ex ension o his pa ame iza ion o o he nuclei. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 85 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC (TVV) no ail MEC (TVV) Figu e 4.13: Compa ison be ween he o al 2p-2h MEC νµc oss sec ion pe a ge nucleon o he TVV channel wi h and wi hou conside ing he ail con ibu ion in he esponses. 4.5 Densi y dependence o 2p-2h meson-exchange cu en s In his sec ion, we analyze he densi y dependence o he 2p-2h MEC con ibu ions in lep on- nucleus in e ac ions in o de o ex end he abo e pa ame iza ion on 12C o o he nuclei. This is connec ed o he g owing in e es in he ex ension o hea ie nuclei, such as 16O, 40A , 56Fe and 208Pb, used in ongoing and u u e neu ino expe imen s. The e o e, an es ima ion o he densi y dependence o he 2p-2h MEC esponses would be ex emely use ul o ex apola e he esul s om he cu en pa ame iza ion on 12C o o he nuclei. In Chap e 3, he “supe scaling” beha io o he inclusi e elec on sca e ing da a was analyzed o a ious nuclei (see Figs. 3.2 and 3.3), showing ha , below he quasielas ic peak, hese da a we e independen on he momen um ans e (scaling o i s kind) and on he mass numbe (scaling o second kind). On he con a y, some scaling iola ion eme ged o ene gy ans e a ound and abo e he QE peak, mainly in he ans e se channel and asc ibed o eac ion mechanisms di e - en om one-nucleon knockou , such as 2p-2h MEC con ibu ions. This supe scaling beha io was also ep oduced h ough he SuSA 2 scaling unc ions a ising om he RMF heo y. Mo e speci ically, i is wo h men ioning ha he educed QE c oss sec ion, i.e. he QE c oss sec ion di ided by he app op ia e single-nucleon one, scales as ∼A/kF,kFbeing he Fe mi momen um. This can be deduced by simple inspec ion o he equa ions desc ibed in Sec ions 3.2-3.4. The Fe mi momen um o mos nuclei belongs o he ange 200-300 MeV/c (see Table 3.1 o a de ailed compa ison). In wha ollows we explo e he densi y dependence o he 2p-2h nuclea esponses. Since he beha io wi h densi y o he nuclea esponse is no expec ed o depend e y much on he speci ic channel o on he na u e o he p obe, o simplici y we ocus on he elec omagne ic 2p-2h ans e se esponse, which la gely domina es o e he longi udinal one. Ou s a ing poin is he e o e he elec omagne ic ans e se esponse o Z=Nnuclei, RMEC T, associa ed wi h meson-exchange cu en s ca ied by he pion and by he ∆- esonance, e alua ed wi hin he model o [110]. In a p e ious wo k [117] based on he non- ela i is ic Fe mi gas, i has been sugges ed ha he nuclea dependence o he 2p-2h MEC esponses scales as Ak2 F. Thus, in o de o es his scaling 86 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS ule wi hin ou ela i is ic app oach, we emo e he single-nucleon physics om he p oblem by de ining he ollowing educed esponse (pe nucleon) H FMEC T(q,ω)≡1 η2 F RMEC T(q,ω) Gee′ T(τ),(4.9) whe e he single-nucleon e m Gee′ Twas p e iously de ined in Sec ion 3.2.3 and depends on he p o on (GMp) and neu on (GMn) magne ic o m ac o s. Fo simplici y he e we neglec in he single-nucleon di iding ac o small con ibu ions coming om he mo ion o he nucleons (Wee′ 2∆ e m), whe e he elec ic o m ac o con ibu es, which depend on he Fe mi momen um (see Sec ion 3.2.3). No e ha he educed esponse H FMEC Tis di ided by η2 F=(kF/mN)2, oge he wi h he single-nucleon ac o Gee′ Tin he sea ch o he Ak2 Fscaling. We can also in oduce a 2p-2h MEC scaling unc ion, MEC T, de ined analogously o he ans e se scaling unc ion coming om he one-body esponse, MEC T(q,ω)= RMEC T(q,ω) Gee′ T(τ)=η2 FkFH FMEC T.(4.10) The pa icula case o asymme ic nuclei, Z,N equi es mo e in ol ed o malism and will be add essed in u u e wo k, al hough p elimina y s udies indica e ha he quali a i e beha io wi h kFdoes no change signi ican ly unless N−Zis e y la ge. 4.5.1 Analysis o esul s In Fig. 4.14 we show he dependence on qo he abo e desc ibed 2p-2h MEC scaling unc ion, MEC T, o 12C. Unlike he QE case, whe e he scaling unc ions o he RFG and RMF models “collapse” in o a single one o all q alues, he MEC scaling unc ions depend la gely on qand sp ead a beyond he QE egion (−1≤ψ′≤+1 wi hin he RFG model). This beha io is mo e p onounced o q: 400−600 MeV/c. As qinc eases, he MEC scaling unc ions a e mo e simila and hei maxima end o lowe ψ′ alues, ge ing close o he QE peak (ψ′=0). Fu he mo e, we can also no ice ha a e y high momen um ans e s he 2p-2h MEC con ibu ions a e e y signi ican in he deep scaling egion (la ge nega i e ψ′ alues), o he ex en ha hey may e en p o ide a con ibu ion simila o he QE one a e y low ω. Ne e heless, he QE and 2p-2h MEC con ibu ions o low ωand high qa e e y small. In gene al, he obse ed end wi h qo he 2p-2h MEC esponses is consis en wi h he iola ion o i s -kind scaling exhibi ed by he MEC in [111]. In Fig. 4.15 we display RMEC Tas a unc ion o he ene gy ans e ω o momen um ans e s q anging om 200 o 2000 MeV/c and h ee alues o he Fe mi momen um kF om 200 o 300 MeV/c. Unlike he 1-body quasielas ic case, i clea ly appea s ha he 2p-2h esponse unc ions inc ease wi h kF,i.e. wi h he mass numbe o he nuclea species. This mo i a es he sea ch o a second-kind scaling beha io in he 2p-2h MEC egime. In o de o s udy he kF-dependence o he esponses, we ix he momen um ans e o a speci ic alue in Fig.4.16, whe e we show he esponse RMEC T o q=800 MeV/c (uppe panels) and he same h ee alues o kFused abo e. In he lowe panels o Fig. 4.16 we display he scaled 2p-2h MEC esponse, H FMEC T, as a unc ion o he MEC scaling a iable ψ′ MEC (q,ω, kF) and o he quasiela ic one ψ′≡ψ′ QE (q,ω, kF). The MEC scaling a iable is de ined in he Appendix C, in analogy wi h he usual QE scaling a iable, o adjus he maximum o he 2p-2h esul s a ψ′ MEC ≈0. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 87 The esul s show ha he educed 2p-2h esponse oughly scales as k2 Fwhen ep esen ed as a unc ion o ψ′ MEC,i.e., he scaled 2p-2h MEC esponses shown blend a he peak in o a uni e sal esul . This scaling law is e y accu a e a he peak o he 2p-2h esponse, while i is iola ed o some ex en a la ge nega i e alues o he scaling a iable. These e ec s in he “deep scal- ing” egion a e educed when conside ing he usual scaling a iable ψ′ QE de ised o quasielas ic sca e ing. -1 012 3 4 5678 9 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 T MEC (ψ’) 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 Figu e 4.14: T ans e se 2p-2h MEC scaling unc ions MEC T o 12C e sus he usual QE scaling a iable ψ′ QE om q=400 MeV/c o 1500 MeV/c. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 ω (GeV) 0 2 4 6 8 10 12 14 RT MEC (GeV-1) kF=200 MeV/c kF=250 MeV/c kF=300 MeV/c q: 200-2000 MeV/c Figu e 4.15: The 2p-2h MEC esponse plo ed e sus ω o h ee alues o he Fe mi momen um kFand o di e en alues o he momen um ans e . The cu es a e displayed om le o igh in s eps o q=200 MeV/c. 88 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ω (GeV) 0 1 2 3 4 5 6 7 RT MEC (GeV-1) kF=200 MeV/c kF=250 MeV/c kF=300 MeV/c -2 0 2 4 6 8 10 ψ’QE 0 1 2 3 4 5 6 7 RT MEC (GeV-1) -2 -1 012 3 4 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ -2 0 2 4 6 8 10 ψ’QE 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ Figu e 4.16: Uppe panels: he 2p-2h MEC esponse plo ed e sus ω(le panel) and ψ′ QE ( igh panel) o q=800 MeV/c and Fe mi momen um kF a ying be ween 200 and 300 MeV/c. Lowe panels: he co esponding scaled 2p-2h MEC esponse H FMEC Tplo ed e sus he scaling a iables ψ′ MEC (le panel) and ψ′ QE ( igh panel). In Fig. 4.17 he scaled 2p-2h MEC esponse is now plo ed e sus ψ′ MEC o ou alues o q and kF. In pa icula , we ha e conside ed he cases o 12C (kF=228 MeV/c) and 40Ca (kF=241 MeV/c). The i s is clea ly ele an o ongoing neu ino oscilla ion s udies whe eas he second is a symme ic nucleus lying close o he impo an case o 40A o upcoming neu ino expe imen s. He e we see ha he same kF-dependence is alid o di e en alues o qas long as Pauli blocking is no ac i e, namely q>2kF. A lowe qand in he deep scaling egion his ype o scaling is mildly b oken. In pa icula , his co esponds o q.500 MeV/c and he lowes ω alues. Ne - e heless, his is no a ele an kinema ic egion o he analysis o 2p-2h MEC c oss sec ions on neu ino eac ions as shown in Fig. 4.12 ensu ing an accu a e ex ension o he 12C-pa ame iza ion o o he nuclei. Fo comple eness, we compa e in Fig. 4.18 he scaled esponses H FMEC T o di e en powe s o ηF o q=1000 MeV/c, which ein o ces he idea o he Ak2 Fscaling beha io obse ed in Fig. 4.17 also o he same alue o q. In pa icula , no scaling beha io is obse ed in e ms o AkF(le panel) e en hough some ag eemen be ween di e en kF alues appea s in he deep scaling egion. On he con a y, he analysis o he Ak3 Fdependence ( igh panel) only p oduces simila esul s o high ψ′ MEC alues which co esponds o high ene gy ans e s. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 89 -2 -1.5 -1 -0.5 00.5 11.5 ψ’MEC 0 1 2 3 4 FT MEC (GeV-1) ∫ q=500 MeV/c -2 -1 012 3 4 5 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ q=1000 MeV/c -2 -1 012 3 4 5678 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ q=1500 MeV/c -2 -1 012 3 4 5678 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) kF=200 MeV/c 12C (kF=228 MeV/c) 40Ca (kF=241 MeV/c) kF=300 MeV/c ∫ q=2000 MeV/c Figu e 4.17: The scaled 2p-2h MEC esponse H FMEC Tplo ed e sus he scaling a iable ψ′ MEC o di e en alues o q a ying om 500 o 2000 MeV/c and Fe mi momen um kFbe ween 200 and 300 MeV/c. The pa icula cases o 12C and 40Ca a e shown. -2 -1 012 3 4 5 ψ’MEC 0 0.5 1 1.5 2 2.5 RT MEC/GT/ηF (GeV-1) kF=200 MeV/c 12C (kF=228 MeV/c) 40Ca (kF=241 MeV/c) kF=300 MeV/c -2 -1 012 3 4 5 ψ’MEC 0 5 10 15 20 25 30 35 RT MEC/GT/ηF 3 (GeV-1) Figu e 4.18: The scaled 2p-2h MEC esponse H FMEC T o di e en powe s o ηn F≡(kF/mN)n,n=1 (le panel) and n=3 ( igh panel) plo ed e sus he scaling a iable ψ′ MEC o q=1000 MeV/c and Fe mi momen um kF a ying be ween 200 and 300 MeV/c. 90 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS A e analyzing he k2 Fdependence o he MEC esponses as well as he 1/kFdependence o he QE egime in e ms o he supe scaling beha io , we can d aw ou some conclusions abou he ela i e impo ance o bo h con ibu ions. Rega ding he kFdependence o di e en nuclei shown in Table 3.1, we can deduce ha o ligh e nuclei, whe e kFis changing mo e apidly wi h inc easing A, he size o he MEC ela i e o he QE peak changes no iceably as Abecomes la ge . As Ainc eases owa d hea ie nuclei, he nuclea densi y sa u a es, causing kF o slowly app oach he nuclea ma e alue o kF∼250 −260 MeV/c. This implies ha o hea ie nuclei all con- ibu ions will scale app oxima ely as A. The e o e, while he ela i e MEC con ibu ion will be la ges o hea y nuclei, i changes mos apidly when compa ing c oss sec ions o ligh nuclei. This is also obse ed in he analysis o (e,e′) inclusi e da a shown in Chap e 6 o di e en nuclei. 4.5.2 Neu ino-Oxygen 2p-2h MEC esponses Finally, we ocus on he 2p-2h MEC esponses o 16O, which is ele an o ecen neu ino ex- pe imen s [122]. In pa icula , in he T2K expe imen he a de ec o may ha e di e en nuclea a ge s, mine al oil and wa e , and i is hen c ucial o unde s and how o ex apola e he esul s om one a ge o ano he . In Fig. 4.19 we analyze he 2p-2h MEC esponses o neu ino in e ac ions on 16O compa ed wi h 12C o wo o he mos ele an con ibu ions (TVV and T′ V A), based on mic oscopic calcu- la ions by Ama o e al. [105]. As p e iously men ioned, he 2p-2h MEC esponses a e e alua ed wi hin he RFG model in a ully ela i is ic amewo k, o which he alues o he Fe mi momen- um and ene gy shi used a e, kF=228 MeV/c, Eshi =20 MeV o 12C and kF=230 MeV/c, Eshi =16 MeV o 16O. The kFand Eshi alues co esponding o 16O a e consis en wi h he analysis o elec on sca e ing da a, as de ailed in Chap e 6. We obse e in Fig. 4.19 (uppe panels) ha he esul s o 16O a e la ge han he 12C ones, which is mainly due o he di e en Anumbe s as he kF alues a e e y simila . A he same ime, he 16O esponses a e sligh ly shi ed o lowe ω alues wi h espec o he 12C ones as a consequence o he smalle Eshi o oxygen. Using now he p e ious esul ha he 2p-2h MEC con ibu ions scale as Ak2 Fwe can mul iply he 12C esul s (which a e pa ame ized) by he a io be ween he Anumbe s and he k2 F alues o bo h nuclei, A(16O)k2 F(16O) A(12C)k2 F(12C)≈1.35 ,(4.11) in o de o ep oduce he 16O ones. This is p esen ed in he lowe panels o Fig. 4.19 o he TVV and T′ V A channels, showing he high deg ee o accu acy o he scaling law o ex apola ing he 2p- 2h esponses om ca bon o oxygen o all alues o qand leading o he conclusion ha he Ak2 F dependence is also widely ul illed by he 2p-2h weak eac ions. No e also ha he di e en ene gy shi s ha e been co ec ed by shi ing he ca bon esul s 8 MeV o he le .3. Simila commen s also apply o all he emaining weak 2p-2h MEC esponses. The compa ison o ou heo e ical p esc ip ion wi h 16O (e,e′) and (νµ, µ−) da a will be add esed in Chap e s 6 and 7. 3Al hough he shi di e ence be ween 12C and 16O is, ∆Eshi =Eshi (12C) −Eshi (16O) =4 MeV, we ha e o double his alue when conside ing wo-nucleon emission. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 91 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 RTVV (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 0.03 RT’ (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 RTVV (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 0.03 RT’ (MeV-1) Figu e 4.19: 2p-2h MEC ec o - ec o ans e se (TVV ) esponse and he axial- ec o in e e ence (T′ V A) one. Uppe panels: Compa ison be ween he esul s o 12C (do s) and 16O (solid). Bo om panels: compa ison by e-scaling he 12C esul s wi h a ac o 1.35 (see ex ). The cu es a e displayed om le o igh in s eps o q=200 MeV/c om q=200 MeV/c up o 2000 MeV/c. Summa izing, we ha e shown ha he 2p-2h elec omagne ic and weak MEC esponse unc- ions oughly g ow as Ak2 F o Fe mi momen um a ying om 200 o 300 MeV/c. This scaling law is excellen a ound he MEC peak o in e media e and high alues o qwhe eas i s a s o b eak down a ound he Pauli-blocking egion (q=2kF), whe e 2p-2h MEC do no con ibu e sig- ni icanly o he neu ino c oss sec ions. Compa ed o he beha io o he 1-body esponse, which scales as A/kF, he ela i e impo ance o he 2p-2h con ibu ion g ows as k3 F. This esul allows one o ge an es ima e o he ele ance o hese con ibu ions o a a ie y o nuclei, o in e es in ongoing and u u e neu ino sca e ing expe imen s, and should acili a e he implemen a ion o 2p-2h e ec s in Mon e Ca lo gene a o s. 98 5. DEEP INELASTIC SCATTERING FORMALISM whe e, simila ly o he QE egime, we can de ine: UL=UCC =U00 =κ2 τ 1+τρ2W2(τ, ρ)−W1(τ, ρ)+W2(τ, ρ)D(κ,τ, ρ)g(5.37) ULL =U33 =λ2 κ2U00 (5.38) UCL =−1 2U03 +U30=−λκ κ2U00 (5.39) UT=U11 +U22 =2W1(τ, ρ)+W2(τ, ρ)D(κ, τ, ρ) (5.40) UT′=−i 2U12 −U21=±τ κ"1 2(ǫF+ǫ0)+λ ρ#W3(τ, ρ),(5.41) whe e Dis de ined as: D(κ, τ, ρ)=1 ǫF−ǫ0(ρ)ZǫF ǫ0(ρ) dǫZ2π 0 dΦ 2πη×D κ2 =τ κ2(1 3 ǫ2 F+ǫFǫ0(ρ)+ǫ0(ρ)2g+λǫF+ǫ0(ρ)+λ2)−(1 +τ) +(ρ−1) τ κ2λǫF+ǫ0(ρ)−τ(ρ+1) =ξF1−ψ2 X"1+ξFψ2 X−λ κψXqξF2+ξFψ2 X+τ 3κ2ξF1−ψ2 X#. (5.42) As p e iously discussed, and o a ixed alue o he in a ian mass µX, he RFG yields a scaling unc ion RFG(ψ′ X)= RFG L(ψ′ X)= RFG T(ψ′ X)=3 4(1 −ψ′2 X)θ(1 −ψ′2 X).(5.43) whe e ψ′ Xalso con ains he co esponding ene gy shi , Eshi , as o he QE egime, ψ′ X≡1 √ξF λ′−τ′ρ′ q(1 +λ′ρ′)τ′+κpτ′(τ′ρ′2+1) ,(5.44) wi h ρ′de ined as ρ′≡2H·Q Q2=1+1 4τ′µ′ X 2−1;µ′ X= W′ X MN =1 MNω′+Eh2−p2 X.(5.45) Then, we can inden i y in Eq. (5.36), he e m which englobes all he nuclea dependence o he in e ac ion, RFG, and eplace i by he one a ising om he SuSA o SuSA 2 models ( model), in a simila way as done o he QE egime in Chap e 3. The e o e, we calcula e he inelas ic nuclea esponses o each speci ic model (RFG, SuSA o SuSA 2) as RK inel (κ, τ)=N η3 FκξFZµmax X µmin X dµXµX model (ψ′ X)UK.(5.46) The p e ious o malism allows o a comple e desc ip ion o he inelas ic c oss sec ions on lep on-nucleus in e ac ions whe e he nuclea e ec s a e included by means o he SuSA 2 model. 5.3. INELASTIC STRUCTURE FUNCTIONS 99 The p ocedu e is simila o ha applied o he QE egime (see Chap e 3), keeping he same unc- ional o m o he RMF and RPWIA scaling unc ions (3.89) bu using a di e en scaling a iable (ψ′ X) and a di e en q0 ansi ion pa ame e o he blending unc ion (3.91). The speci ic de ails abou he applica ion o he SuSA 2 model o he inelas ic spec um as well as he es ima ion o he q0 ansi ion pa ame e be ween he RMF and RPWIA p esc ip ions o bo h quasielas ic and inelas ic egimes will be p esen ed in Chap e 6, whe e a de ailed analysis o he exis ing (e,e′) da a and he 2p-2h MEC con ibu ions is also p o ided. 5.3 Inelas ic s uc u e unc ions In his Sec ion, he p ope ies o single-nucleon inelas ic s uc u e unc ions o p o ons and neu- ons a e analyzed. The speci ic de ails abou he exis ing pa ame iza ions and empi ical models a e de ailed in Sec ion 5.4. As de ined in Sec ion 5.2, he inelas ic s uc u e unc ions F1,F2and F3depend on wo a i- ables Q2and x, o , equi alen ly, τand ρ. To de e mine hese inelas ic unc ions o gi en Q2and x, measu emen s o he di e en ial c oss sec ion a di e en sca e ing angles and incoming elec on beam ene gies a e needed. This will help in ob aining phenomenological i s o he single-nucleon inelas ic s uc u e unc ions o p o ons and neu ons. Speci ic in o ma ion on he p o on inelas ic o m ac o s can be ob ained om he analysis o elec on-p o on (ep) and elec on-deu e on (eD) sca e ing p ocesses. On he con a y, he neu on case is much ha de as no spedi ic elec on- neu on (en) p ocesses a e a ailable. Thus, in o ma ion on neu on inelas ic o m ac o s elies di ec ly on he join analysis o he wo sca e ing p ocesses men ioned abo e, namely, ep and eD. Expe imen ally i is obse ed ha bo h F1and F2a e almos independen o Q2in he limi o high Q2and ν. This is known as Bjo ken x-scaling [123] F1,2(x,Q2)→F1,2(x).(5.47) Unde he p e ious kinema ical condi ions, a ela ionship be ween F1and F2is gi en by he Callan- G oss ela ion [124] F2(x)=2xF1(x),(5.48) o iginally based on he pa on model. Some iola ion e ec s on he Callan-G oss ela ion a e e- la ed o second o de QCD co ec ions [125,126], which appea a xapp oaching ze o, i.e. in he highly deep-inelas ic egion, as well as o mesonic con ibu ions in he nuclea medium. The io- la ion due o mesonic and o he nuclea e ec s a e shown o be no iceable only in he egion o low xand Q2[127] whe e he inelas ic s uc u e unc ions a e e y educed. Viola ion o Callan-G oss ela ion in nuclei is o cu en in e es in ongoing expe imen s a JLab [128,129], whe e he mea- su emen s o F1and F2on nuclea a ge s will p o ide impo an in o ma ion on his subjec [130]. Bo h he Bjo ken Scaling and he Callan-G oss ela ionship can be explained assuming ha DIS is domina ed by he sca e ing o a single i ual pho on (boson) om a poin -like qua k wi hin he p o on (see Fig. 5.2). A e y high kinema ics whe e nucleon esonance s uc u es a e no subs an ially ele an , he pa on dis ibu ion unc ions a ising om pQCD (pe u ba i e QCD) p o ide a p ope ep esen a- ion o he inelas ic s uc u e unc ions. The ange o alidi y o his app oach will be discussed in Sec ion 5.4.4. 100 5. DEEP INELASTIC SCATTERING FORMALISM Figu e 5.2: DIS p ocess o elec omagne ic e-p eac ions desc ibed in e ms o inelas ic s uc u e unc ions (le panel) and o he qua k-pa on model ( igh panel). 5.3.1 Ex ension o he weak sec o The desc ip ion o he deep-inelas ic egime o weak in e ac ions implies he analysis o an ad- di ional s uc u e unc ion, F3(W3), ela ed o he pa i y iola ing con ibu ion associa ed o he V−Ain e e ence. An accu a e de e mina ion o his weak unc ion is ha d o achie e om neu- ino expe imen s as well as om pa i y- iola ing elec on sca e ing [131, 132] due o he la ge unce ain ies associa ed o he c oss sec ion measu emen s. Ne e heless, wi hin he qua k-pa on model, we can es ablish a ela ionship among he elec omagne ic and weak s uc u e unc ions and be ween F2and F3[74, 133, 134]. This is based on he assump ion ha he co esponding s uc u e unc ions Wican be w i en in e ms o qua k Qand an iqua k Qdis ibu ions [135,136] F2=νW2=Q+Q(5.49) F3=xνW3=Q − Q (5.50) and, hence, xνW3=νW2−2Q.(5.51) Fo elec on sca e ing, he isoscala F2s uc u e unc ion o he nucleon, de ined as he a e age o he p o on and neu on s uc u e unc ions, is gi en (a leading o de in αsand o h ee la o s) by FeN 2=1 2Fep 2+Fen 2=5x 18 u+u+d+d+x 9(s+s),(5.52) whe e u(u),d(d) and s(s) a e he dis ibu ions o he up, down and s ange qua ks (an iqua ks), espec i ely. The qua k dis ibu ions a e de ined o be hose in he p o on and he ac o s 5/18 and 1/9 a ise om he squa es o he qua k cha ges. Fo neu ino sca e ing, he co esponding F2 s uc u e unc ion is gi en by FνN 2=x(u+u+d+d+s+s),(5.53) whe e qua k cha ges a e no conside ed. In he mode a e and la ge-x egion, whe e s ange qua ks a e supp essed, he weak and elec omagne ic F2s uc u e unc ions app oxima ely sa is y, FeN 2≈5x 18 u+u+d+d≈5 18 FνN 2.(5.54) Unde his assump ion, which has been analyzed in connec ion wi h expe imen al esul s [135, 137–139], one can eadily ob ain he weak s uc u e unc ions om he exis ing pa ame iza ion o he elec omagne ic s uc u e unc ions and he an iqua k dis ibu ion.1 1In his wo k, he inelas ic c oss sec ions a e only calcula ed and compa ed wi h da a o elec omagne ic eac ions. Thei ex ension o he weak sec o and he cons uc ion o he app op ia e isoscala and iso ec o con ibu ions needed o CC and NC neu ino eac ions will be accoun ed o in u he wo ks. 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 101 5.4 Pa ame iza ion o he inelas ic s uc u e unc ions In his sec ion, we analyze wo di e en pa ame iza ions o he inelas ic s uc u e unc ions and a Pa on Dis ibu ion Func ion (PDF) model, showing hei capabili y o ep oduce (e,e′) da a a in e media e kinema ics o he inelas ic egime. 5.4.1 Bodek-Ri chie pa ame iza ion This app oach is based on phenomenological i s o he single-nucleon inelas ic s uc u e unc- ions W1and W2 o ep and en, being he la e ex ac ed om deu e on da a. The Bodek-Ri chie pa ame iza ion is a gene al i o Re s. [133, 134, 140, 141], which desc ibes bo h he deep in- elas ic and esonance egions, co e ing he en i e inelas ic spec um. The pa ame iza ion i s he SLAC da a published in [140], co e ing a |Q2| ange om 0.1 o 30 GeV2, and including scaling iola ions in e ms o a modi ied scaling a iable ωw. The s uc u e unc ion W2is desc ibed by νWep(en) 2(ν,Q2)=B(WX,Q2)g(ωw)ωw/ω , (5.55) g(ωw)= 7 X n=3 Cn(1 −1/ωw)n,(5.56) ωw=2MNν+a2 Q2+b2.(5.57) The modula ing unc ion B(WX,Q) con ains 12 pa ame e s ep esen ing he masses, wid hs, and ampli udes o he c oss sec ion o elec op oduc ion o he ou mos p ominen nucleon eso- nances, and eigh pa ame e s ep esen ing he WXdependence o he low-WXnon esonan con- ibu ion and single-pion p oduc ion h eshold. The modula ing unc ion is close o uni y in he deep-inelas ic egion (WX>2 GeV). The es o pa ame e s a e de ined in [133] whe e aand b a e he same o p o on (Wep 2) and neu on (Wen 2) and he Cncoe icien s a e di e en o p o ons and neu ons. The Wep 1and Wen 1s uc u e unc ions can be deduced om he Callan-G oss ela- ion (5.48). The W3 unc ions, ela ed o V A in e e ence, a e ob ained using he ela ion (5.51) wi h he an iqua k dis ibu ion, Q, gi en by Q(x,Q2)=1 2 1−1 ωw!7 B(WX,Q2)g(0)ωw ω.(5.58) The di e en Wi unc ions o p o ons and neu ons wi hin he Bodek-Ri chie pa ame iza ion will be shown in 5.4.4 in compa ison wi h o he pa ame iza ions. 5.4.2 Bos ed-Ch is y pa ame iza ion Al hough he Bodek-Ri chie [133] pa ame iza ion has been widely used o he analysis o he highly-inelas ic sca e ing egion [142], in ecen yea s new s udies, bo h heo e ical and expe - imen al, o he nucleon s uc u e unc ions in he esonance egion ha e been pe o med, indi- ca ing he need o mo e sophis ica ed pa ame iza ions. The kinema ical egion whe e nucleon esonances con ibu e is essen ial o he analysis o (e,e′) da a and pion p oduc ion esul s om inclusi e neu ino measu emen s. The e o e, i is mo e con enien o employ mo e ac ual exp es- sions o Wep(en) 1,2. In his sense, he Bos ed-Ch is y (B-C) pa ame iza ion o he p o on [143] and neu on [144] s uc u e unc ions seems o ep oduce be e he elec omagne ic beha io in he esonance egion. 102 5. DEEP INELASTIC SCATTERING FORMALISM This app oach is based on an empi ical i o desc ibe he measu emen s o inclusi e inelas ic elec on-p o on and elec on-deu e on c oss sec ions in he kinema ic ange o ou -momen um ans e 0 ≤ |Q2|<8 GeV2and inal s a e in a ian mass 1.1<WX<3.1 GeV, hus s a ing oughly om he pion p oduc ion egion o he highly-inelas ic egion. The i is cons ained by he high p ecision longi udinal and ans e se (L/T) sepa a ed c oss sec ion measu emen s om JLab Hall C [145]. Compa ed o p e ious i s, i co e s a wide kinema ic ange, i s bo h ans e se and longi u- dinal c oss sec ions, and ea u es smoo h ansi ions o he pho op oduc ion da a a Q2=0 and DIS da a a high |Q2|and WX. A he same ime, i p o ides an excellen desc ip ion o he eso- nan s uc u es seen in inclusi e (e,e′) c oss sec ions. As i will be shown in 5.4.4, i s ag eemen wi h da a in he egion o he ∆-peak wi hin he SuSA 2 model is be e han using he p e ious Bodek-Ri chie pa ame iza ion. 5.4.3 PDF model: GRV98 The Glück-Reya-Vog GRV98 model [146] employs e ec i e leading o de (LO) Pa on Dis ibu- ion Func ions o qua ks and an iqua ks o ge he inelas ic s uc u e unc ions F1,F2and F3. In his hesis, we use a ecen upda e [147] designed o bo h inelas ic neu ino- and elec on-nucleon sca e ing c oss sec ions. The model desc ibes exis ing inelas ic neu ino-nucleon sca e ing mea- su emen s and has been de eloped o analyze neu ino oscilla ion expe imen s in he ew GeV egion. The PDFs a e ex ac ed om global i s o a ious se s o deep inelas ic sca e ing da a a high ene gies and high |Q2|, whe e non-pe u ba i e QCD e ec s a e negligible. These e ec s a e ele an o lowe kinema ic egions. In his sense, an scaling a iable (ξw(x,Q2)) is employed o cons uc e ec i e LO PDFs ha accoun o he con ibu ions om a ge mass co ec ions, non-pe u ba i e QCD e ec s, and highe o de QCD e ms. The non-pe u ba i e e ec s om spec a o qua ks a e equi ed o be conside ed o |Q2|<1 GeV2. These co ec ions oge he wi h e ec s om low ene gy sca e ing da a a e pa ame ized and included in he desc ip ion. This model also accoun s o nucleonic esonances and includes pho op oducion da a abo e he ∆(1232). In gene al, he model gi es a easonable a e age c oss sec ion in he esonance egion and beyond as shown in [147] bu i seems o be inadequa e o desc ibe inelas ic (e,e′) da a a low- in e media e kinema ics, below |Q2|<1 GeV2. Rega ding his, we show in Sec ion 5.4.4 ha he GRV98 p esc ip ion only ma ches he empi ical Bos ed-Ch is y and Bodek-Ri chie s uc u e unc ions, based on i s o elec on sca e ing da a, a |Q2|&5 GeV2. 5.4.4 Compa ison o he di e en pa ame iza ions Nex , we p esen a compa ison o he di e en pa ame iza ions o he single-nucleon inelas ic s uc u e unc ions desc ibed abo e, also con on ing hem wi h (e,e′) expe imen al da a in he inelas ic egime. The elec omagne ic inelas ic s uc u e unc ions o p o ons and neu ons Wep(en) 1,2a e dis- played e sus he Bjo ken scaling a iable xin Figu e 5.3. The compa ison is ca ied ou o he Bodek-Ri chie and Bos ed-Ch is y pa ame iza ions as well as o he GRV98 PDFs p esc ip ion om low o high-Q2 alues. As obse ed, he GRV98 PDFs p oduces an a e age o he inelas- ic s uc u e unc ions h ough he pa on dis ibu ion unc ions whe eas he Bodek-Ri chie and 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 103 Bos ed-Ch is y pa ame iza ions, based on phenomenological i s o he s uc u e unc ions, ep o- duce in a mo e ealis ic way he esonance s uc u es obse ed in ep and ed eac ions. The h ee models appea o be simila a |Q2|abo e 5 GeV2 o low x alues whe e DIS p o- cesses domina e whe eas some mino di e ences appea as xge s close o 1, i.e., whe e esonance e ec s a e mo e signi ican . On he con a y, he di e ences among hese pa ame iza ions inc ease when mo ing o lowe |Q2| alues whe e esonan con ibu ions a e mo e ele an in he en i e x ange, e en a low x alues. Indeed, a |Q2|=1 GeV2, he GRV98 is oughly an a e age o he empi ical Bodek-Ri chie and Bos ed-Ch is y i s bu does no ep oduce he esonan s uc u es a all. The compa ison a he lowes |Q2| alues p oduces he mos di e ging pic u e be ween he empi ical i s and he GRV98 PDFs model. The e o e, i s applicabili y a ene gies o ele ance o cu en neu ino expe imen s is a he ques ionable. Focusing on he empi ical i s, we can obse e small di e ences be ween he p o on s uc u e unc ions Wep 1,2 o he Bodek-Ri chie and Bos ed-Ch is y pa ame iza ions (see Fig. 5.3), being he la e a bi la ge a low |Q2|and high x. Conce ning he neu on s uc u e unc ions, Wen 1,2, hey a e signi ican ly g ea e a lowe |Q2|and la ge x o he Bos ed-Ch is y i . Al hough, in gene al, he neu on s uc u e unc ions emain below he p o on ones, he opposi e occu s o he Bos ed- Ch is y pa ame iza ion a he lowes |Q2| alues. All hese di e ences be ween bo h pa ame iza- ions may be due o he mo e accu a e analysis o he esonances included in he Bos ed-Ch is y i as well as o he di e en p ocedu es o disen angle he neu on s uc u e unc ions om he p o on and deu e on ones. Simila commen s also apply o Fig. 5.4, whe e he sum o p o on and neu on con ibu ions a e shown o FeN 2and FeN 3wi hin he Bos ed-Ch is y and Bodek-Ri chie i s. The an iqua k dis- ibu ion (Q) ob ained h ough he p ocedu e desc ibed in Sec ion 5.4.1 (see also Re . [133]), is also displayed. As can be no iced, he an iqua k dis ibu ion is mo e ele an a |Q2|<1 GeV2and low x alues, hus implying ha he main di e ences be ween he F3and F2 unc ions appea a hese kinema ics. In Fig. 5.4, he ωdependence wi h xis also displayed, showing how ωinc eases as x ends o lowe alues. Rega ding he Bos ed-Ch is y app oach, some di e gences appea a e y low x o highe |Q2| alues. Ne e heless, hey co espond o he highly-inelas ic egime and ex emely high ω, which, in gene al, co esponds o kinema ically o bidden egions. The compa ison o he h ee pa ame iza ions wi h (e,e′) da a is shown in Fig. 5.5 o wo ep- esen a i e cases a in e media e kinema ics whe e he QE and ∆peaks a e easily ecognizable. A he beginning o he highly-inelas ic egion, i.e. la ge ω alues, whe e inelas ic s uc u e unc ions a e mo e simila (see Fig. 5.3), he h ee models end o be close . I can be also no iced ha he GRV98 PDFs o e s ima e he expe imen al da a in he ∆- esonance egion. A he same ime, i s con ibu ion in he “dip” egion be ween he QE and he ∆peaks is oo la ge in such a way ha he o al esul , a e conside ing QE and 2p-2h MEC con ibu ions, would clea ly o e es ima e he da a. On he con a y, he Bodek-Ri chie pa ame iza ion unde es ima es he ∆ egion as well as he beginning o he highly-inelas ic egime which may be in connec ion wi h i s poo e desc ip ion o he esonance s uc u es wi h espec o he Bos ed-Ch is y one. As commen ed o Figs. 5.3 and 5.4, he la ge Bos ed-Ch is y s uc u e unc ions esul he e in an inc eased inelas ic c oss sec ion. A he same ime, he accu a e inclusion o esonances in his pa ame iza ion allows o a be e desc ip ion o he (e,e′) da a in he ∆-peak egion wi hou con ibu ing ema kably o he “dip” egion. As can be deduced om Fig. 5.4, he di e ences be ween he empi ical i s dec ease as going deepe in o he inelas ic egime, i.e., la ge ω alues. 104 5. DEEP INELASTIC SCATTERING FORMALISM The e o e, on he basis o he p e ious analysis, he Bos ed-Ch is y pa ame iza ion eme ges as he mos app op ia e app oach o he analysis o he inelas ic egime. In his sense, he e e ence inelas ic s uc u e unc ions Fi o be employed in ou heo e ical desc ip ion a e shown in Fig. 5.6. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 0 0.1 0.2 0.3 0.4 0.5 0.6 Q2=1.0 GeV2/c2 0.2 0.4 0.6 0.8 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q2=5.0 GeV2/c2 Figu e 5.3: Inelas ic s uc u e unc ions Wep 1(black lines), Wen 1( ed lines), Wep 2(g een lines), Wen 2(blue lines) in e ms o x o he Bodek-Ri chie pa ame iza ion (solid lines), Bos ed-Ch is y pa ame iza ion (do -dashed lines) and GRV98 PDFs (dashed lines) a Q2=0.14,1.0 and 5.0 GeV2. 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 105 0 0,1 0,2 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 o Q xνW3 (B-R) νW2 (B-R) Q νW2 (B-C) xνW3 (B-C) 0 0,2 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) ω s x Q2=0.1 GeV2/c2, Ei= 2.0 GeV, q=0.6 GeV/c, ω=0.51 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0 0,2 0,4 0,6 0,8 νW2, xνW3 o Q 0 0,2 0,4 0,6 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) Q2=0.5 GeV2/c2, Ei= 2.0 GeV, q=1.0 GeV/c, ω=0.71 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 o Q 0 0,2 0,4 0,6 0,8 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) Q2=1.0 GeV2/c2, Ei= 2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 o Q 0 0,2 0,4 0,6 0,8 1 x 0 5 10 15 20 ω (GeV) Q2=5.0 GeV2/c2, Ei=10.0 GeV, q=5.0 GeV/c, ω=4.47 GeV 0 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 νW2, xνW3 o Q 0 0,2 0,4 0,6 0,8 1 x 0 20 40 60 80 100 120 140 ω (GeV) Q2=10.0 GeV2/c2, Ei=30.0 GeV, q=9.0 GeV/c, ω=8.43 GeV Figu e 5.4: Compa ison o he inelas ic s uc u e unc ions FeN 2=νWeN 2=νWen 2+νWep 2 and FeN 3=xνWeN 3=xνWen 3+xνWep 3 o he Bodek-Ri chie (B-R) and Bos ed-Chis y (B-C) pa ame iza ions. The an iqua k dis ibu ion, QeN =Qen +Qep, is also shown. The ωdependence wi h xis also displayed. 106 5. DEEP INELASTIC SCATTERING FORMALISM 0 0,1 0,2 0,3 0,4 0,5 ω (GeV) 0 500 1000 1500 2000 2500 Bos ed-Ch is y Bodek GRV98 E=680 MeV, θ=60o, qQE=610 MeV/c 0,2 0,4 0,6 ω (GeV) 0 1000 2000 3000 4000 E=1108 MeV, θ=37.5o, qQE=674.6 MeV/c Figu e 5.5: Compa ison o inclusi e 12C(e,e′) double di e en ial c oss sec ions and p edic- ions o he inelas ic egime o he Bodek-Ri chie pa ame iza ion (solid lines), Bos ed-Ch is y pa ame iza ion (do -dashed lines) and GRV98 PDFs (dashed lines) a di e en kinema ics (inci- den elec on beam and sca e ing angle) in e ms o he ene gy ans e ed o he nucleus (ω). Expe imen al da a aken om [100, 101]. The y-axis ep esen s d2σ/dΩ/dωin nb/GeV/s . The alue o qa he QE peak (qQE) is shown as e e ence. 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,2 0,4 0,6 0,8 2Mx(W1 p+W1 n) 2MxW1 n 2MxW1 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,1 0,2 0,3 0,4 0,5 ν(W2 p+W2 n) νW2 n νW2 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,1 0,2 0,3 0,4 0,5 xν(W3 p+W3 n) xνW3 n xνW3 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV Figu e 5.6: Re e ence inelas ic s uc u e unc ions F1=2M xW1( op panel), F2=νW2(mid panel) and F3=xνW3(bo om panel) o p o ons and neu ons and he sum o bo h con ibu ions a Q2=1 GeV2. The calcula ions a e pe o med wi hin he Bos ed-Ch is y pa ame iza ion. Chap e 6 Analysis o inclusi e elec on sca e ing wi hin he SuSA 2-MEC model Al hough he ocus in his PhD hesis conce ns he s udy o CC neu no-nucleus sca e ing p o- cesses, i is essen ial i s o es ou heo e ical p edic ions agains he a ailable inclusi e elec on sca e ing da a. This was al eady shown in Chap e 3 bu es ic ing ou sel es o he pu e QE egime. In his chap e he SuSA 2 model is ex ended o include he comple e inelas ic spec um — esonan , non esonan and deep inelas ic sca e ing — desc ibed in Chap e 5. We also conside he impac o 2p-2h meson-exchange cu en s ollowing he p ocedu e de ailed in Chap e 4. The p edic ions o he ull SuSA 2-MEC model a e i s compa ed wi h he exis ing inclusi e 12C(e,e′) da a and la e also applied o o he nuclei. The capabili y o he model o desc ibe elec on sca e - ing da a wi h accu acy gi es us con idence in i s subsequen ex ension, and alidi y, when applied o ecen neu ino oscilla ion expe imen s. This subjec will be add essed in Chap e ??. 6.1 SuSA 2 model o quasielas ic and inelas ic egimes In Chap e 3 a e y de ailed desc ip ion o he SuSA 2 model was gi en, ha , as known, inco - po a es he p edic ions om he RMF heo y and a ansi ion o he RPWIA model a high alues o he momen um ans e . This ansi ion be ween bo h RMF and RPWIA egimes is go e ned by a blending unc ion whose explici exp ession is also gi en in Chap e 3 (see Eqs. 3.89—3.91 o de ails). He e ou in e es is o ex end he SuSA 2 model o he inelas ic egime so i s p e- dic ions can be compa ed wi h da a co e ing he en i e ene gy spec um. This equi es o ha e a good con ol o he ansi ion pa ame e s (q0,ω0) ha de e mine he ela i e s eng h o he RMF and RPWIA esponses, and how he ansi ion be ween hem e ol es as he ans e momen um a ies. Acco dingly, he ansi ion pa ame e , q0, is expec ed o inc ease wi h qin such a way ha he RMF con ibu ion will be dominan a low kinema ics whe eas he RPWIA one s a s o be ele an a highe ene gies. The e o e we in oduce a dependence o he pa ame e q0on he momen um ans e q ha de e mines he ela i e RMF and RPWIA con ibu ions a di e en kine- ma ics. Mo eo e , his ansi ion occu s in a egion o wid h ω0, which is ixed a 200 MeV. The pa icula p ocedu e o de e mine he q0-beha io wi h qis in acco dance o he bes i o a la ge amoun o (e,e′) expe imen al da a on 12C in a wide kinema ical egion, co e ing om low o high q- alues (q: 239 −3432 MeV/c). Fo his analysis, 12C is employed as a ge e e ence due o he ample a ie y o exis ing da a o elec on sca e ing as well as i s ele ance o neu- ino oscilla ion expe imen s. The me hod applied o de e mine he RMF/RPWIA ansi ion in he SuSA 2 model in bo h QE and inelas ic egimes is based on a educed-χ2analysis o he da a se s. 107 1146. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL Some commen s conce ning he “dip” egion be ween he QE and he ∆peaks a e also in o de . This is he egion whe e he QE and he inelas ic con ibu ions o e lap he mos and whe e FSI e ec s ha modi y in a signi ican way he ail o he QE cu e a la ge ω- alues can in oduce an impo an impac . Mo eo e , he ole o he 2p-2h MEC e ec s is essen ial because i s maximum con ibu ion occu s in his egion. Thus, only a ealis ic calcula ion o hese ing edien s beyond he IA can desc ibe success ully he beha io o he c oss sec ion. To conclude, he acco dance be ween heo y and da a in he inelas ic egime, whe e a wide a ie y o e ec s a e aken in o accoun , also gi es us a g ea con idence in he eliabili y o ou calcula ions. The inelas ic pa o he c oss sec ion is domina ed by he ∆-peak ha mainly con- ibu es o he ans e se esponse unc ion. A low elec on sca e ing angles he longi udinal QE esponse unc ion domina es he c oss sec ion and he inelas ic con ibu ion is smalle (as will be shown in Sec ion 6.2.4). The opposi e holds a la ge sca e ing angles, whe e he ∆-peak con ibu- ion is impo an . On he o he hand, o inc easing alues o he ans e ed momen um he peaks co esponding o he ∆and QE domains become close , and hei o e lap inc eases signi ican ly. This gene al beha iou is clea ly shown by ou p edic ions compa ed wi h da a. In hose kine- ma ical si ua ions whe e inelas ic p ocesses a e expec ed o be impo an , ou esul s o he QE peak a e clea ly below he da a which is compensa ed by he la ge inelas ic con ibu ion. On he con a y, when he inelas ic con ibu ions a e expec ed o be small, he QE heo e ical p edic ions ge close o da a. No e also he excellen ag eemen in some si ua ions (bo om panels on Fig. 6.4) e en being awa e o he limi a ions and pa icula di icul ies in o de o ob ain phenomenological i s o he inelas ic s uc u e unc ions, and he educed c oss sec ions a hese kinema ics. 6.2.2 Sensi i i y o he model I is impo an o poin ou he no el ies in oduced in his PhD hesis compa ed wi h some p e i- ous s udies. Wi h ega ds o he o iginal “SuSA” esul s shown in [84], ha we e based only on he supe scaling unc ion ex ac ed om he analysis o he longi udinal (e,e′) da a and assuming he ans e se unc ion o be equal (scaling o ze o h kind), in he SuSA 2 app oach he enhancemen in he ans e se channel in oduced by he RMF model is inco po a ed. Mo eo e , he ole o FSI is ca e ully examined by making use o he e olu ion o he scaling unc ion om he RMF esponses o he RPWIA ones as he momen um ans e goes up. This explains why he p esen analysis p o ides a much mo e accu a e desc ip ion o he da a. No ice ha he SuSA 2 model makes bo h QE and inelas ic esul s highe han he SuSA ones. A simila ou come can be also ob- se ed in Sec ions 3.5.2 and 3.5.1 (see also [153]) whe e he s udy was es ic ed o he QE egion and a ixed alue o q0 ha can be app op ia e o he speci ic kinema ics conside ed was used. On he con a y, he e he aim is o p o ide a model capable o ep oducing (e,e′) c oss sec ions o a e y wide selec ion o kinema ics and including in each case he whole ene gy spec um. This is consis en wi h he q-dependence shown by q0in bo h egimes, QE and inelas ic. We ha e also es ed he sensi i i y o ou esul s o di e en choices in he alues o ω0,q0and Eshi o wo ep esen a i e kinema ical si ua ions (see Fig. 6.5). Conce ning he ω0pa ame e , a a ia ion o ±100 MeV leads o negligible e ec s, hence he alue o χ2is basically he same (uppe panels in Fig. 6.5). In he case o q0and Eshi , a ia ions o he o de o ±100 MeV/c (in q0) and ±5 MeV (Eshi ) lead o di e ences wi hin ∼20% on χ2, bu s ill p o iding a e y good ep esen a ion o he da a (see esul s p esen ed in he middle and bo om panels o Fig. 6.5). No e howe e ha q0is a dynamical pa ame e unning wi h q, whe eas he alue o Eshi is de e mined by he igh loca ion o he maxima in he scaling unc ions. Hence a signi ican a ia ion o hese h ee alues does no imply a wo sening in he ag eemen wi h da a. 6.2. ANALYSIS OF (e,e′)EXPERIMENTAL DATA 115 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 ω0+100 MeV, χ2 ed=4.15 ω0-100 MeV, χ2 ed=4.33 χ2 ed=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 50000 60000 70000 ω0+100 MeV, χ2 ed=12.22 ω0-100 MeV, χ2 ed=12.23 χ2 ed=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 q0+100 MeV/c, χ2 ed=4.59 q0-100 MeV/c, χ2 ed=4.25 χ2 ed=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 50000 60000 70000 q0+100 MeV/c, χ2 ed=10.98 q0-100 MeV/c, χ2 ed=14.30 χ2 ed=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 500 1000 1500 2000 2500 3000 Eshi -5 MeV, χ2 ed=7.37 Eshi +5 MeV, χ2 ed=3.44 χ2 ed=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 10000 20000 30000 40000 50000 60000 70000 Eshi -5 MeV, χ2 ed=16.89 Eshi +5 MeV, χ2 ed=8.60 χ2 ed=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c Figu e 6.5: Compa ison o inclusi e 12C(e,e′) c oss sec ions and p edic ions o he QE-SuSA 2 model (long-dashed ed line), 2p-2h MEC model (do -dashed b own line) and inelas ic-SuSA 2 model (long do -dashed o ange line). The sum o he h ee con ibu ions is ep esen ed wi h a solid blue line. I is also shown he o al con ibu ion by shi ing ω0( op panels), q0(middle panels) and Eshi (bo om panels). The y-axis ep esen s d2σ/dΩ/dωin nb/GeV/s , whe eas he x-axis ep esen s ωin GeV. 1166. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 6.2.3 Rele ance o he RMF/RPWIA e ec s Nex , we discuss he ele ance o he RMF and RPWIA app oaches in he SuSA 2 model. Whe eas he RMF p o ides an excellen desc ip ion o he expe imen al longi udinal scaling unc ion ex- ac ed om da a aken a in e media e q- alues, p oducing he equi ed asymme y and he en- hancemen o he ans e se esponse, he RPWIA app oach yields much mo e sui able esul s a highe alues o he momen um ans e whe e FSI e ec s a e signi ican ly educed. In Fig. 6.6 we p esen he c oss sec ions o a se o kinema ical si ua ions showing he isola ed con ibu ions eme ging om he wo models in he case o he QE egime. No ice ha we conside he e ec s in oduced by he blending unc ion and he q0pa ame e s in he RMF and RPWIA esul s. The pe cen age o he wo con ibu ions is gi en in each panel. As shown, o hose kinema ics ha co espond o he lowe alues o qQE ( op panels) he RMF esponse con ibu es he mos . As qQE inc eases, he RPWIA con ibu ion becomes ela i ely mo e impo an , app oaching he RMF one (see panels in he middle). Finally, o he highe qQE- alues (bo om panels) he beha io e e ses wi h he RPWIA esul being he main one esponsible o he QE esponse. 00.05 0.1 0 10000 20000 30000 40000 50000 RMF (86.3 %) RPWIA (13.7 %) 00.05 0.1 E=280 MeV, θ=60o, qQE=263.739 MeV/c 0 0.1 0.2 0.3 0.4 0 10000 20000 30000 40000 50000 RMF (76.7 %) RPWIA (23.3 %) 0 0.1 0.2 0.3 0.4 E=620 MeV, θ=36o, qQE=367 MeV/c 0 0.1 0.2 0.3 0 2000 4000 6000 8000 RMF (68.8 %) RPWIA (32.2 %) 0 0.1 0.2 0.3 E=500 MeV, θ=60o, qQE=456.6 MeV/c 00.05 0.1 0.15 0.2 0.25 0 500 1000 1500 2000 2500 3000 RMF (66.3 %) RPWIA (33.7 %) 00.05 0.1 0.15 0.2 0.25 E=400 MeV, θ=90o, qQE=489 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 RMF (52.6 %) RPWIA (47.4 %) 0 0.1 0.2 0.3 0.4 0.5 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.2 0.4 0.6 0.8 0 5000 10000 15000 20000 25000 30000 RMF (45.7 %) RPWIA (54.3 %) 0 0.2 0.4 0.6 0.8 E=2500 MeV, θ=15o, qQE=658.6 MeV/c 0.2 0.3 0.4 0.5 ω (GeV) 0 100 200 300 400 500 RMF (44.2 %) RPWIA (55.8 %) E=560 MeV, θ=145o, qQE=795 MeV/c 0.2 0.4 0.6 0.8 ω (GeV) 0 100 200 300 400 500 600 RMF (34.8 %) RPWIA (65.2 %) 0.2 0.4 0.6 0.8 E=3595 MeV, θ=20o, qQE=1316 MeV/c 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω (GeV) 0 10 20 30 40 50 RMF (30.7 %) RPWIA (69.3 %) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E=4045 MeV, θ=30o, qQE=2247 MeV/c Figu e 6.6: Compa ison o RMF and RPWIA con ibu ions in he QE egime. Also shown o e - e ence he p edic ions o he o al QE-SuSA 2 model (long-dashed ed line) and he o al inclusi e con ibu ion (solid blue line). The y-axis ep esen s d2σ/dΩ/dωin nb/GeV/s . To make clea e how bo h RMF and RPWIA app oaches con ibu e wi hin he SuSA 2 model, in Fig. 6.7 we p esen he speci ic pe cen ages asc ibed o he wo con ibu ions and how hey a y wi h qQE. The main a ia ion in he wo cases is p oduced in he egion o in e media e 6.2. ANALYSIS OF (e,e′)EXPERIMENTAL DATA 117 qQE- alues, namely, 250 .qQE .700 MeV/c. He e, he ela i e RMF con ibu ion quickly diminishes as qQE inc eases whe eas he opposi e occu s o he RPWIA. No e ha a qQE ∼700 MeV/c bo h models p oduce basically he same answe (∼50%) c ossing each o he , whe eas o qQE .500 MeV/c RPWIA gi es a e y mino con ibu ion, ha is, FSI a e essen ial o desc ibe da a a hese kinema ics. Finally, a highe qQE he RPWIA inc eases slowly, whe eas he RMF dec eases, al hough in bo h cases some kind o sa u a ion seems o eme ge app oaching he RPWIA pe cen age o ∼60 −70% (∼30 −40% o he RMF). Al hough no p esen ed he e o simplici y, simila conclusions a ise o he RMF and RPWIA con ibu ions in he inelas ic egime. 0500 1000 1500 2000 2500 3000 qQE (MeV/c) 0 20 40 60 80 100 % RPWIA % RMF Figu e 6.7: Compa ison o pe cen ages co esponding o he RMF and RPWIA con ibu ions in he QE egime as a unc ion o qQE. 6.2.4 Sepa a e L/T analysis The sepa a e analysis o he longi udinal and ans e se esponse unc ions o 12C is p esen ed in Fig. 6.8. We compa e ou p edic ions wi h da a aken om Jou dan [82] based on a Rosenblu h sepa a ion o he (e,e′) wo ld da a. In each case we isola e he con ibu ions co esponding o he QE, inelas ic and 2p-2h MEC sec o s. Th ee kinema ical si ua ions co esponding o ixed alues o he momen um ans e ha e been conside ed in Fig. 6.8: q=300 MeV/c ( op panel), 380 MeV/c (middle) and 570 MeV/c (bo om). As obse ed, he longi udinal channel is o ally domi- na ed by he QE con ibu ion. Only a e y la ge alues o ωdoes he inelas ic p ocess en e gi ing ise o a mino esponse, whe eas he e ec s due o 2p-2h MEC a e negligible. This esul is in ac- co dance wi h p e ious wo k [82,93,151,154], and i clea ly shows ha he longi udinal esponse is basically due o he IA. On he con a y, he ans e se sec o shows an impo an sensi i i y o MEC and inelas ic p ocesses. No e ha he inelas ic ans e se esponse gi es ise o he high ail shown by da a a la ge ω- alues, whe eas he 2p-2h MEC can modi y signi ican ly he ans e se esponse in he dip egion as well as in he maximum o he QE peak. I is also wo h men ioning ha he na u al enhancemen in he ans e se esponse a ising om he RMF model is necessa y in o de o ep oduce he sepa a e L/T da a, ejec ing he idea o 0- h kind scaling. 1186. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 050 100 150 200 250 300 0 0.01 0.02 0.03 0.04 R (MeV-1) L T 050 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 R (MeV-1) 050 100 150 200 250 300 350 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 R (MeV-1) Figu e 6.8: Analysis o he longi udinal (solid lines) and ans e se esponses (dashed lines) in (e,e′) sca e ing a q=300 MeV/c ( op panel), q=380 MeV/c (middle panel) and q=570 MeV/c (bo om panel). QE, MEC and inelas ic con ibu ions a e shown, espec i ely, as g een, blue and o ange lines. The o al esponse is shown by he black lines. Da a aken om [82]. 6.3. EXTENSION OF THE SUSAV2-MEC MODEL TO OTHER NUCLEI 119 F om esul s in Fig. 6.8 we obse e ha he model leads o a easonable ag eemen wi h da a in bo h channels, al hough some disc epancies also eme ge. No ice ha he longi udinal p edic ion a q=300 MeV/c (q=380 MeV/c) o e s ima es da a by ∼12% (∼15%). I is impo an o poin ou ha SuSA 2 is based on he exis ence o he scaling phenomenon o (e,e′) da a, and his is comple ely ul illed when he alue o qis la ge enough (q≥400 MeV/c). The e o e, he ex ension o he supe scaling app oach o low q- alues is no well es ablished e en hough a good ag eemen a low kinema ics has been achie ed in he p e ious sec ion. Fu he mo e, he mino disc epancies obse ed may also be due o he speci ic Rosenblu h sepa a ion me hod used in [82], which in oduces some le el o model dependence h ough y-scaling assump ions and he ea men o adia i e co ec ions. 6.3 Ex ension o he SuSA 2-MEC model o o he nuclei Once analyzed he capabili y o he SuSA 2 model o ep oduce he 12C(e,e′) da a, we ex end he p e ious o malism o he analysis o elec on sca e ing da a on o he nuclei. Fo his pu pose, no di e ences in he scaling unc ions a e assumed o he di e en nuclei excep o he alues used o he Fe mi momen um and ene gy shi (see Table 3.1 o de ails). The use o he same scaling unc ions o di e en nuclea sys ems is consis en wi h he p ope y o scaling o second ype, i.e, independence o he scaling unc ion wi h he nucleus, and i also ollows om he heo e ical p edic ions p o ided by he RMF and RPWIA models on which SuSA 2 elies. This has been s udied in de ail in p e ious wo ks (see [32, 86–88, 155]) whe e he elec omagne ic and weak scaling unc ions e alua ed wi h he RMF and RPWIA app oaches ha e been compa ed o 12C, 16O and 40Ca. In Figu e 6.9 we compa e he gene al RMF scaling unc ions o hese nuclei, which exhibi no ema kable di e ences in e ms o he nuclea species. In his sense, we apply he e e ence scaling unc ions o 12C o he analysis o QE and inelas ic egimes in o he nuclei. The ex ension o he 2p-2h MEC con ibu ions o o he a ge s was p e iously desc ibed in Sec ion 4.5. -2 -1 012 3 4 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (ψ’) 40Ca 16O 12C Figu e 6.9: Analysis o second kind scaling wi hin he RMF model o 12C, 16O and 40Ca. In he case o 16O, he kFand Eshi - alues selec ed (kF=230 MeV/c, Eshi =16 MeV/c) a e also consis en wi h he gene al end obse ed in [37], i.e., an inc ease o he Fe mi momen um 1206. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL wi h he nuclea densi y. This is a a iance wi h some p e ious wo ks [85–87,91] whe e 16O was desc ibed by using kF=216 MeV/c and Eshi =25 MeV. Al hough bo h se s o alues lead o small di e ences in he c oss sec ions, he p esen choice does p o ide a mo e consis en analysis o he supe scaling beha io in he deep scaling egion, and mo e impo an ly, i also imp o es he compa ison wi h elec on sca e ing da a. In pa icula , he analysis o he scaling beha io o he 16O(e,e′) da a in he deep scaling egion below he QE peak (ψ′=0), whe e no scaling iola ions a e expec ed, leads o he conclusion ha he p esen choice o kFand Eshi - alues wo ks be e when compa ing wi h o he nuclei (see Figu e 6.10). -2 -1.5 -1 -0.5 00.5 11.5 2 ψ’ 0.01 0.1 1 (ψ’) 4He, Ei=3595 MeV, θe=16o 12C, Ei=3595 MeV, θe=16o 27Al, Ei=3595 MeV, θe=16o 56Fe, Ei=3595 MeV, θe=16o 4He, Ei=730 MeV, θe=37.1o 12C, Ei=730 MeV, θe=37.1o 16O (kF=216 MeV/c), Ei=730 MeV, θe=37.1o 16O (kF=230 MeV/c), Ei=730 MeV, θe=37.1o Figu e 6.10: Expe imen al scaling da a o a ious nuclei and o di e en alues o he inciden ene gy (Ei) and sca e ing angle (θe). The 16O da a a e show o wo di e en alues o kF. The kF- alues o o he nuclei a e desc ibed in Table 3.1. In acco dance wi h he p e ious analysis, we show in Fig. 6.11 he p edic ions o he SuSA 2- MEC model o six di e en kinema ical si ua ions, co esponding o he a ailable (e,e′) da a on 16O. In all he cases we p esen he sepa a e con ibu ions o he QE, 2p-2h MEC and inelas ic egimes. The 2p-2h MEC esponses a e ex apola ed om he exac calcula ion pe o med o 12C assuming he scaling law R2p2h∼Ak2 Fdeduced in Re . [156] as well as in Sec ion 4.5. The inclusi e c oss sec ions a e gi en e sus he ans e ed ene gy (ω), and each panel co esponds o ixed alues o he inciden elec on ene gy (Ei) and he sca e ing angle (θ). Whe eas he la e is ixed o 320[157] excep o one case (cen e panel on he op, i.e, θ=37.10) [158], he elec on ene gy alues un om 700 MeV (le - op panel), whe e he QE peak domina es, o 1500 MeV ( igh -bo om) wi h he inelas ic channel gi ing a e y signi ican con ibu ion. This is due o he alues o he ans e ed momen um qin ol ed in each si ua ion. Al hough qis no ixed in each o he panels, i.e., i a ies as ωalso a ies, he ange o q- alues allowed by he kinema ics inc eases e y signi ican ly as he elec on ene gy g ows up ( o ixed sca e ing angles). Thus, o highe Ei he wo egimes, QE and inelas ic, o e lap s ongly, he inelas ic p ocesses being esponsible o he la ge c oss sec ions a inc easing alues o ω. This di e en ange o q- alues spanned in each panel also explains he ela i e ole played by he RMF e sus he RPWIA app oaches. 6.3. EXTENSION OF THE SUSAV2-MEC MODEL TO OTHER NUCLEI 121 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 10000 20000 30000 40000 50000 60000 70000 80000 QE 2p-2h MEC Inelas ic To al 16O, Ei=700MeV, θ=32o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 5000 10000 15000 20000 25000 30000 16O, Ei=737MeV, θ=37.1o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 10000 20000 30000 40000 16O, Ei=880MeV, θ=32o 0 0.2 0.4 0.6 0.8 ω (GeV) 0 5000 10000 15000 16O, Ei=1080MeV, θ=32o 0 0.2 0.4 0.6 0.8 ω (GeV) 0 2000 4000 6000 8000 10000 16O, Ei=1200MeV, θ=32o 0 0.2 0.4 0.6 0.8 1 ω (GeV) 0 1000 2000 3000 4000 16O, Ei=1500MeV, θ=32o Figu e 6.11: Compa ison o inclusi e 16O(e,e′) c oss sec ions and p edic ions o he SuSA 2- MEC model. The sepa a e con ibu ions o he pu e QE esponse (dashed line), he 2p-2h MEC (do -dashed), inelas ic (double-do dashed) a e displayed. The sum o he h ee con ibu ions is ep esen ed wi h a solid blue line. The yaxis ep esen s d2σ/dΩ/dωin nb/GeV/s . Da a om Re s. [157] and [158]. Al hough no shown in he igu e o simplici y, whe eas he RMF esponse domina es a lowe Ei- alues (panels om le o igh on he op), he e e se occu s, ha is, he scaling unc ion is essen ially gi en by he RPWIA p edic ion, as Eiinc eases oge he wi h q(panels on he bo om). As obse ed, he SuSA 2-MEC p edic ions a e in e y good acco dance wi h da a o all kine- ma ical si ua ions. Al hough he ela i e ole o he 2p2h-MEC e ec s is a he modes compa ed wi h he QE and inelas ic con ibu ions, a he peak o he 2p2h esponse he h ee con ibu ions a e compa able in size, as also obse ed o 12C. Fo comple eness, we also p esen in Figu e 6.12 he calcula ions o he hea ie a ge 40Ca (kF=241 MeV/c, Eshi =28 MeV) whe e he compa ison wi h da a is again e y p ecise om o wa d o e y backwa d angles. The analysis o hese esul s is ele an because o he simila i y wi h 40A , a a ge o in e es o ecen and o hcoming neu ino oscilla ion expe imen s. No e also ha he 2p-2h MEC con ibu ions a e mo e p ominen o 40Ca han o 12C and 16O wi h espec o he QE egime due o he di e en kFdependence o he QE and 2p-2h MEC con ibu- ions, A/kFand Ak2 F, espec i ely [156]. To conclude, he compa ison wi h a e y ligh nucleus as 4He (kF=200 MeV/c, Eshi =15 MeV) is displayed in Fig. 6.13. In his si ua ion, he SuSA 2 model unde es ima es he QE peak which may be a consequence o he s ong ec o and scala po en ials a ising om he RMF p esc ip ion. The e o e, he applicabili y o he SuSA 2 model and, speci ically, he 12C RMF scaling unc ions and hei associa ed FSI e ec s may be ques ionable o e y ligh nuclei, i.e. o e y low Fe mi momen um. We can also obse e, in acco dance wi h he densi y dependence shown in Sec ion 4.5, ha he 2p-2h MEC ela i e con ibu ion is signi ican ly smalle han he one obse ed o hea ie nuclei. 1226. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 5000 10000 15000 20000 25000 30000 40Ca, Ei=681MeV, θ=45.5o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 5000 10000 15000 40Ca, Ei=841MeV, θ=45.5o 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 5000 10000 15000 20000 QE Inelas ic 2p-2h To al 40Ca, Ei=560MeV, θ=60o 0 0.1 0.2 0.3 ω (GeV) 0 1000 2000 3000 40Ca, Ei=400MeV, θ=140o Figu e 6.12: Compa ison o inclusi e 40Ca(e,e′) c oss sec ions and p edic ions o he SuSA 2- MEC model. The sepa a e con ibu ions o he pu e QE esponse (dashed line), he 2p-2h MEC (do -dashed), inelas ic (double-do dashed) a e displayed. The sum o he h ee con ibu ions is ep esen ed wi h a solid blue line. The yaxis ep esen s d2σ/dΩ/dωin nb/GeV/s . Da a om Re s. [159] and [160]. 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 2000 4000 6000 8000 10000 12000 To al QE Inelas ic 2p-2h MEC 4He, Ei=737MeV, θ=37.1o 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 ω (GeV) 0 400 800 1200 4He, Ei=3595MeV, θ=16o Figu e 6.13: Compa ison o inclusi e 4He(e,e′) c oss sec ions and p edic ions o he SuSA 2- MEC model (solid blue line). The sepa a e con ibu ions o he pu e QE esponse (dashed line), he 2p-2h MEC (do -dashed), inelas ic (double-do dashed) a e displayed. The yaxis ep esen s d2σ/dΩ/dωin nb/GeV/s . Da a om Re s. [158] and [161]. 6.4. CONCLUSIONS 123 6.4 Conclusions In his Chap e he SuSA 2-MEC model applied o elec on sca e ing is ex ended o he whole ene gy spec um, inco po a ing he con ibu ions coming om he QE, inelas ic and wo-body me- son exchange cu en s. Wi hin his amewo k a gene al “blending” unc ion is in oduced o make he ansi ion be ween he RMF and RPWIA esponses. This unc ion is cons uc ed in e ms o a pa ame iza ion o he op imized blending egion gi en by a ansi ion pa ame e , q0, and i has been applied consis en ly o he QE as well as o he inelas ic egimes, using he same scaling unc- ions wi h an analogous RMF-RPWIA ansi ion. Al hough he use o mo e ee pa ame e s, as ω0 and/o he shi ene gy, leads o an e en be e ag eemen wi h da a in some pa icula cases, he speci ic pa ame iza ion assumed is no c i ical, and indeed, he p esen model is capable o ep o- ducing e y success ully he whole ene gy spec um o 12C(e,e′) da a a e y di e en kinema ics. In pa icula , he SuSA 2 model ep oduces accu a ely he posi ion, wid h and maximum o he QE peak o all kinema ics, whe eas he use o he single-nucleon inelas ic s uc u e unc ions oge he wi h he SuSA 2 scaling unc ions has led o a p ecise desc ip ion o he ∆- esonance egion and he en i e inelas ic egime. Acco dingly, he applica ion o he phenomenological SuSA app oach o he inclusi e (e,e′) eac ions would clea ly esul in an unde p edic ion o he da a, as i was p e iously sugges ed in Sec ion 3.5.2. Rema kably, we ha e also shown ha he SuSA 2-MEC model gi es a easonably good de- sc ip ion no only o he c oss sec ions bu also o he sepa a e longi udinal and ans e se esponse unc ions. This is a e y impo an es o models used in neu ino sca e ing s udies, since in his case he balance be ween he L and T channels is di e en om he (e,e′) case. All his gi es us a g ea con idence in he eliabili y o he model, p o iding a solid benchma k o assess i s alidi y when ex ended o he desc ip ion o neu ino- nucleus sca e ing. In his case, no only new e- sponses con ibu e, bu also he wide neu ino ene gy band implied by he ypical accele a o -based neu ino luxes makes i di icul o econs uc he neu ino ene gy. Thus, ing edien s beyond he ones usually assumed wi hin he IA can ha e a signi ican impac on he analysis o da a. A basic ea u e o ou p esen s udy, apa om he appliance o he SuSA 2 model o he QE and inelas ic egions, conce ns he e alua ion o he wo-body meson exchange cu en s in bo h longi udinal and ans e se channels. This ully ela i is ic calcula ion has allowed o a consis en e alua ion o high ene gies/momen a expe imen al da a. Fu he mo e, he pa ame iza ion o he exac esul s o he 2p-2h MEC esponses has allowed us o a oid he compu a ionally demanding mic oscopic calcula ion o he en i e se o kinema ics equi ed o he expe imen al da a p e- sen ed he e. Addi ionally, his analysis has been ex ended o o he nuclea species leading o an accu a e desc ip ion o he expe imen al da a o nuclei o ele ance in o hcoming neu ino expe imen s. Despi e he smalle amoun o a ailable (e,e′) expe imen al da a o some nuclei wi h espec o 12C, he g owing in e es o expe imen al collabo a ions on (e,e′) measu emen s, oge he wi h he ex ension o neu ino expe imen s o o he nuclea a ge s, is expec ed o shed ligh on hei nuclea e ec s and he associa ed pa ame e s ha play a ole in mos heo e ical desc ip ion o bo h elec on and neu ino eac ions. To conclude, we emphasize he impo ance o scaling a gumen s ha p o ide a p ope de- sc ip ion o elec on sca e ing da a be o e he analysis be ex ended o neu ino eac ions. In his sense, he ully ela i is ic analysis p esen ed in his chap e o he en i e ene gy spec um and he di e en nuclea esponses is o c ucial impo ance o he analysis o neu ino eac ions. 130 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS an ineu ino ene gy luxes, he quali y o he ag eemen wi h da a is a he simila in he wo cases. 0 0.4 0.8 1.2 1.6 2 0 5 10 15 20 d2σ/dTµ/dcosθµ 0.9 < cosθµ < 1.0 0 0.4 0.8 1.2 1.6 0 5 10 15 20 25 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 0.7 < cosθµ < 0.8 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 MiniBooNE QE+MEC QE MEC 0.6 < cosθµ < 0.7 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 d2σ/dTµ/dcosθµ 0.5 < cosθµ < 0.6 0 0.2 0.4 0.6 0.8 1 0 5 10 15 0.4 < cosθµ < 0.5 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 0.3 < cosθµ < 0.4 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 0.2 < cosθµ < 0.3 0 0.2 0.4 0.6 0.8 Tµ (GeV) 0 2 4 6 8 10 12 d2σ/dTµ/dcosθµ 0.1 < cosθµ < 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 0.0 < cosθµ < 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.1 < cosθµ < 0.0 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 2 4 6 8 10 -0.2 < cosθµ < -0.1 Figu e 7.3: MiniBoone lux- olded double di e en ial c oss sec ion pe a ge nucleon o he νµ CCQE p ocess on 12C. Resul s a e gi en in 10−39 cm2/GeV and displayed e sus he µ−kine ic ene gy Tµ o a ious bins o cos θµob ained wi hin he SuSA 2-MEC app oach. QE and 2p-2h MEC esul s a e also shown sepa a ely. Da a a e om [20]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 d2σ/dTµ/dcosθµ -0.3 < cosθµ < -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.4 < cosθµ < -0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.5 < cosθµ < -0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.6 < cosθµ < -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 d2σ/dTµ/dcosθµ -0.7 < cosθµ < -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.8 < cosθµ < -0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.9 < cosθµ < -0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -1.0 < cosθµ < -0.9 Figu e 7.4: As o Fig. 7.3 bu conside ing mo e backwa d kinema ics. Da a a e om [20]. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 131 0 0.4 0.8 1.2 1.6 0 2 4 6 8 10 12 d2σ/dTµ/dcosθµ 0.9 < cosθµ < 1.0 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 0.7 < cosθµ < 0.8 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 0.6 < cosθµ < 0.7 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 d2σ/dTµ/dcosθµ 0.5 < cosθµ < 0.6 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 30.4 < cosθµ < 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5 0.3 < cosθµ < 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 20.2 < cosθµ < 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 d2σ/dTµ/dcosθµ 0.1 < cosθµ < 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 0.0 < cosθµ < 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 -0.1 < cosθµ < 0.0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 d2σ/dTµ/dcosθµ -0.3 < cosθµ < -0.2 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 < cosθµ < -0.3 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 < cosθµ < -0.4 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.6 < cosθµ < -0.5 Figu e 7.5: As o Fig. 7.3 bu conside ing now he νµCCQE p ocess on 12C. Da a a e om [21]. A u he gene al commen on he p e ious esul s om Figs. 7.3—7.5 is in o de : he RMF p edic ions (shown in [163]) p oduce almos iden ical esul s o he ones om he SuSA 2 model a MiniBooNE kinema ics. A simila compa ison was also shown in Chap e 3 o he analysis o (e,e′) da a a in e media e q- alues. To comple e he p e ious discussion on he double di e en ial c oss sec ions, we p esen in Figs. 7.6 and 7.7 he esul s a e aged o e he muon kine ic ene gy bins as unc ions o he muon sca e ing angle o neu inos and an ineu inos, espec i ely. These g aphs complemen he p e- ious ones, and p o e he capabili y o he model o ep oduce he da a o a la ge a ie y o kinema ic si ua ions. The 2p-2h MEC con ibu ions inc ease he pu e QE esponse by ∼25 −35% (depending on he pa icula egion explo ed) and a e shown o be essen ial in o de o desc ibe he da a. As obse ed, he o al model ends o o e p edic he da a measu ed a angles close o ze o and Tµin he icini y o ∼0.8−1 GeV. This is consis en wi h esul s in p e ious igu es and he inabili y o he model o desc ibe p ope ly da a a e y small angles. Howe e , he la ges disc epancy be ween heo y and da a occu s a he smalles muon kine ic ene gy bins conside ed, i.e., 0.2<Tµ<0.4, in pa icula , o neu inos (Fig. 7.6) and angles bigge han 900(cos θµ<0). As seen, he da a a e highe by ∼25−30% han heo e ical p edic ions. This ou come is consis en wi h he esul s shown in he panels on he bo om in Figs. 7.3-7.5. 132 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 1.0 < Tµ < 1.1 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 0.9 < Tµ < 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 0.8 < Tµ < 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.7 < Tµ < 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 0.6 < Tµ < 0.7 -0.2 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 20 0.5 < Tµ < 0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 14 16 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.4 < Tµ < 0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 14 0.3 < Tµ < 0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 0.2 < Tµ < 0.3 Figu e 7.6: MiniBoone lux- olded double di e en ial c oss sec ion pe a ge nucleon o he νµ-12C CCQE p ocess displayed e sus cos θµ o a ious bins o Tµob ained wi hin he SuSA 2- MEC app oach. QE and 2p-2h MEC esul s a e also shown sepa a ely. Da a a e om [20]. 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 1.0 < Tµ < 1.1 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 0.9 < Tµ < 1.0 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 0.8 < Tµ < 0.9 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.7 < Tµ < 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 0.6 < Tµ < 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 0.5 < Tµ < 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 1 2 3 4 5 6 7 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.4 < Tµ < 0.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 1 2 3 4 5 0.3 < Tµ < 0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 0.5 1 1.5 2 2.5 3 3.5 40.2 < Tµ < 0.3 Figu e 7.7: As o Fig. 7.6, bu now o he νµCCQE p ocess on 12C. Da a a e om [21]. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 133 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 5 10 15 20 25 dσ/dcosθµ (10-39cm2) MiniBooNE QE+MEC MEC QE -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 2 4 6 8 10 12 dσ/dcosθµ (10-39cm2) 00.5 11.5 2 Tµ (GeV) 0 2 4 6 8 10 12 14 16 dσ/dTµ (10-39cm2/GeV) 00.5 11.5 2 Tµ (GeV) 0 0.5 1 1.5 2 2.5 3 dσ/dTµ (10-39cm2/GeV) Figu e 7.8: MiniBooNE lux-a e aged CCQE νµ-12C (νµ-12C) di e en ial c oss sec ion pe nu- cleon as a unc ion o he muon sca e ing angle ( op panels) and o he muon kine ic ene gy (bo - om panels). The le panels co espond o neu ino c oss sec ions and he igh ones o an ineu ino eac ions. Da a a e om [20,21]. In Fig. 7.8 esul s a e p esen ed o he MiniBooNE lux a e aged CCQE νµ(νµ)−12C di e en- ial c oss sec ion pe nucleon as a unc ion o he muon sca e ing angle ( op panels) and he muon kine ic ene gy (bo om panels). The in eg a ion o e he muon kine ic ene gy has been pe o med in he ange 0.2 GeV <Tµ<2.0 GeV. Panels on he le ( igh ) co espond o neu inos (an ineu- inos). As shown, and in consis ency wi h p e ious esul s, he SuSA 2-MEC model is capable o ep oducing he magni ude as well as he shape o he expe imen al c oss sec ion in all o he cases. 7.1.2 Analysis o longi udinal/ ans e se channels o neu ino eac ions In his sec ion we s udy in de ail he ele ance o he di e en longi udinal and ans e se channels ha con ibu e o he QE and 2p-2h MEC MiniBooNE c oss sec ions, also accoun ing o he co - esponding axial and ec o con ibu ions which a ise om he had onic cu en s. An analysis on he di e en channels o he 2p-2h MEC nuclea esponses and he o al c oss sec ion was add essed in Chap e 4 (see Figs. 4.9, 4.10 and 4.11), showing a p edominance o he ans e se o e he longi udinal ones, whe eas o he la e he con ibu ions a ising om ec o cu en s we e negligible in compa ison wi h he axial ones. Mo eo e , he sepa a e ans e se channels, TVV ,TAA and T′ V A, whe eas showing some ema kable di e ences o di e en q alues, con ibu e in a simila way o he o al c oss sec ion. This is due o he ele an kinema ic egions (see Sec ion 7.3 o de ails). He e we explo e he ele ance o he di e en channels in he speci ic kinema ics o he MiniBooNE expe imen . In Fig. 7.9 he sepa a e 2p-2h MEC con ibu ions o he di e en channels (L,TVV ,TAA and T′ V A) co esponding o he MiniBooNE double di e en ial c oss sec ion a di e en bins o he muon sca e ing angle. 134 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 00.5 11.5 2 0 1 2 3 4 5 d2σ/dTµ/dcosθµ (10-39cm2/GeV) 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 TOTAL L TVV TAA T’VA 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 d2σ/dTµ/dcosθµ (10-39cm2/GeV) 0.3 < cosθµ < 0.4 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 10.3 < cosθµ < 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 d2σ/dTµ/dcosθµ (10-39cm2/GeV) -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.2 0.4 0.6 0.8 1 d2σ/dTµ/dcosθµ (10-39cm2/GeV) -0.6 < cosθµ < -0.5 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 -0.6 < cosθµ < -0.5 Figu e 7.9: Compa ison o he di e en 2p-2h MEC channels o he νµ(le panels) and νµ( igh panels) MiniBooNE double di e en ial c oss sec ion. Resul s in Fig. 7.9 show he di e ences be ween he TAA and TVV con ibu ions, he la e being shi ed o highe Tµ alues by abou 50 MeV o all angula bins. A e y o wa d angles, i.e., lowe q- alues, he global magni ude o he AA channel is g ea e han he VV one, in acco dance wi h he esul s obse ed in Fig. 4.11. This igu e also shows clea ly ha a qo he o de o 400 MeV/c he TVV and TAA esponses di e oughly by a ac o 2 a he maximum. This di e ence dec eases o highe q- alues. Conce ning he in e e ence T′ V A componen , i s magni ude is no so di e en om he VV and AA ones a e y o wa d angles, being on he con a y he mos ele an con ibu ion a la ge angles. Finally, al hough he longi udinal channel gi es he smalles global con ibu ion, i s ole is essen ial in o de o in e p e an ineu ino sca e ing a backwa d angles. This is a consequence o he nega i e T′ V A e m o an ineu ino eac ions ha almos cancels ou he TVV +TAA con ibu ion. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 135 The conclusions ex ac ed om he p e ious analysis on he 2p-2h MEC c oss sec ion also apply o he sepa a e QE con ibu ions o neu ino and an ineu ino c oss sec ions. The di e en QE channels a e analyzed o he MiniBooNE double di e en ial c oss sec ions in Fig. 7.10, whe e he ans e se con ibu ion p edomina e a all kinema ics whils he ne longi udinal channel, e en being a e y small con ibu ion, is essen ial o desc ibe an ineu ino da a a backwa d kinema ics oge he wi h he 2p-2h longi udinal one. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 22 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) MiniBooNE SuSA 2 SuSA 2, L SuSA 2, T SuSA 2, TVV SuSA 2, TAA SuSA 2, T’VA 0.8 < cosθµ < 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 0.1 < cosθµ < 0.2 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 Tµ (GeV) 0 1 2 3 4 5 6 7 8 9 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 0.5 1 1.5 2 2.5 3 3.5 Figu e 7.10: Sepa a ion in o componen s o he MiniBooNE CCQE νµ( op panel) and νµ(bo oom panel) double-di e en ial c oss sec ion pe nucleon displayed e sus Tµ o a ious bins o cos θµ wi hin he SuSA 2 app oach. The MiniBooNE da a [20,21] a e also shown o e e ence. In Fig. 7.11 we show he b eakdown o he o al neu ino c oss sec ions in o indi idual L(= LVV +LAA), T(=TVV +TAA), TVV ,TAA and T′ V A con ibu ions, wi h he las occu ing as a posi i e (cons uc i e) e m in he neu ino c oss sec ion and a nega i e (des uc i e) e m in he an ineu- ino one. The sign o he T′ V A channel ep esen s he main di e ence be ween he o al neu ino and an ineu ino c oss sec ions (see also Fig. 7.1). Apa om he opposi e sign in he V A esponse, some mino di e ences be ween neu ino and an ineu ino c oss sec ions a ise om he di e en Coulomb dis o ions o he emi ed lep on (see Sec ion 7.3.2 o de ails) and he inal nuclei in- ol ed in he CC neu ino (ni ogen) and an ineu ino (bo on) sca e ing p ocesses on ca bon. We also no ice ha below 1 GeV he T′ V A esponse is highe han he TVV one and o he same o de as he TAA one. No e ha he maximum o he V A con ibu ion is a ound he peak o he MiniBooNE neu ino lux. On he con a y, he e ec s o V A con ibu ions o he c oss sec ions a e negligible a ene gies abo e 10 GeV as a consequence o he small axial o m ac o GAand lep onic ac o VT′ a high Eνand Q2 alues (see Chap e 2 o de ails). This is also in ag eemen wi h some p e ious QE esul s [19]. As a consequence, o e y high νµ(νµ) ene gies (abo e ∼10 GeV) he o al c oss sec ion o neu inos and an ineu inos is e y simila . Only he Land Tchannels con ibu e o he highe alues explo ed by NOMAD expe imen . On he con a y, in he egion explo ed by he MiniBooNE collabo a ion, he main con ibu ions come om he wo ans e se T,T′channels. 136 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 0.1 110 100 Eν (GeV) 0 2 4 6 8 10 12 14 16 σν (10-39cm2) SuSA 2 SuSA 2, TVV SuSA 2, TAA 0.1 110 100 0 2 4 6 8 10 12 14 16 SuSA 2, T’VA SuSA 2, T SuSA 2, L Figu e 7.11: Sepa a ion in o componen s o he CCQE νµc oss sec ion pe nucleon on 12C dis- played e sus neu ino ene gy Eνwi hin he SuSA 2 app oach. The MiniBooNE [20] and NO- MAD [24] da a a e also shown o e e ence. The asymme y be ween neu ino and an ineu ino c oss sec ions in oduced by he V A in- e e ence is analyzed in Fig. 7.12 whe e we show he expe imen al di e ence (σνµ−σνµ)exp om MiniBooNE, oge he wi h he co esponding heo e ical p edic ion om he SuSA 2-MEC model. This di e ence is app oxima ely equal o 2 (σνµ)SuSA 2 T′ V A +2 (σνµ)M EC T′ V A , apa om he mi- no di e ences be ween neu ino and an ineu ino eac ions desc ibed abo e. The esul om he pu ely QE esponses is also shown, ein o cing he impo ance o he 2p-2h MEC con ibu ions o he analysis o he MiniBooNE expe imen and hei ele ance in he in e e ence channel. 0.1 110 100 Eν (GeV) 0 2 4 6 8 10 σν-σν (10-39cm2) MiniBooNE (νµ-νµ) QE+MEC (νµ-νµ) QE (νµ-νµ) Figu e 7.12: Expe imen al di e ence be ween neu ino and an ineu ino c oss sec ions (σνµ−σνµ) om MiniBooNE, oge he wi h he co esponding heo e ical p edic ion om SuSA 2+MEC. The pu ely QE esponse is also shown. MiniBooNE da a a e om [20,21]. 7.2. CCQE-LIKE SCATTERING IN THE MINERνAEXPERIMENT 137 The asymme y o he nuclea e ec s o neu ino and an ineu ino is impo an o CP io- la ion s udies and i has been analyzed in p e ious wo ks [3, 115, 164]. Howe e , he inhe en di icul ies ela ed o he di e en neu ino and an ineu ino luxes esul in a po en ial obs acle o he in e p e a ion o expe imen s aimed a he measu emen o he CP iola ion angle. In his con ex , some p elimina y s udies on he CP asymme y om he MINERνA Collabo a ion ha e been ecen ly published [165]. Mo eo e , CP iola ion e ec s in he lep on sec o ela ed o he νµ and νec oss sec ions will be add essed in ollowing sec ions. 7.2 CCQE-like sca e ing in he MINERνA expe imen In his sec ion we apply he SuSA 2-MEC model o he analysis o he MINERνA expe imen . The MINERνA Collabo a ion has ecen ly measu ed di e en ial c oss sec ions o muonic and elec on neu ino and an ineu ino cha ged-cu en quasielas ic sca e ing on a hyd oca bon a - ge [46, 47]. The “quasielas ic” e en s a e de ined, in his case, as con aining no mesons in he inal s a e (CCQE-like) hus also including con ibu ions om mul i-nucleon exci a ions. The en- e gy lux ex ends up o 10 GeV and is peaked a Eν∼3 GeV o bo h neu inos and an ineu inos, i.e., in be ween MiniBooNE and NOMAD ene gy anges. The e o e, i s analysis can p o ide aluable in o ma ion on he ole played by 2p-2h meson-exchange cu en s in he nuclea dynam- ics [110,166,167]. 7.2.1 Analysis o cha ged-cu en muonic neu ino esul s In p e ious s udies o he MINERνA Collabo a ion [46, 47], he RMF, SuSA and SuSA 2 mod- els [97,153] as well as o he heo e ical app oaches [168–170] we e able o ep oduce he MINERνA da a wi hou he inclusion o np-nh exci a ions, unlike he MiniBooNE esul s. I is impo an o poin ou ha he MINERνA kinema ics, la ge han he MiniBooNE one, ex ends in o he pion p oduc ion egion and hence he expe imen al sub ac ion o hese e ec s o isola e CCQE-like e en s is mo e c i ical. No e ha in hese expe imen s only he in o ma ion ela ed o muon a i- ables is conside ed. Mo eo e , he ecen ly imp o ed analysis o he MINERνA lux esul s [171] leads o an inc ease o he MINERνA CCQE-like expe imen al c oss sec ions. Conside ing he ee alua ion o he MINERνA lux [171,172], we p esen in Fig. 7.13 he lux a e aged CCQE νµ(νµ) di e en ial c oss sec ion pe nucleon as a unc ion o he econs uc ed ou -momen um Q2 QE. This econs uc ed magni ude is ob ained ollowing he p ocedu e in o- duced in Sec ion 7.1 and also de ailed in [46,47]. The op panel in Fig. 7.13 e e s o νµ−12C whe eas he bo om panel con ains p edic ions and da a o νµ−CH. The mean ene gy o he MINERνA muonic lux is much highe han he Mini- BooNE one, abou 3.5 GeV o bo h νµand νµ. As obse ed, signi ican con ibu ions o he 2p-2h MEC, o he o de o ∼35 −40% (∼25%) a he maxima o νµ(νµ), a e needed in o de o ep oduce he expe imen al da a ha co espond o a new analysis pe o med by he MINERνA collabo a ion [171,172]. These da a exceed by ∼20% he ones al eady p esen ed in p e ious pub- lica ions [46,47] (using he unco ec ed MINERνA lux) ha , on he o he hand, we e consis en wi h calcula ions based exclusi ely on he impulse app oxima ion (see [97]). Thus, he new MINERνA analysis shows majo consis ency wi h he MiniBooNE da a and in alida es he p e ious conclusion [97] ha he MINERνA esul s can be ep oduced wi hou he 138 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS inclusion o 2p-2h con ibu ions. In spi e o he e y di e en muon neu ino (an ineu ino) ene gy luxes be ween MiniBooNE and MINERνA, 2p-2h MEC e ec s emain e y signi ican (on a e - age, 25 −35%) being hei con ibu ion essen ial in o de o ep oduce he da a. Fo comple eness, i is wo h men ioning ha in he speci ic condi ions o MINERνA, i clea ly appea s ha , e en i he neu ino ene gy is as la ge as 3 GeV, he p ocess is la gely domina ed by ela i ely small ene gy and momen um ans e , namely, ω < 500 MeV, q<1000 MeV, whe eas con ibu ions below ω < 50 MeV, q<200 MeV go e n he lowes Q2 QE egion. Mo e speci ic de ails can be ound in Sec ion 7.3 and, addi ionally, in Re . [97]. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 Q2 QE (GeV2) 0 5 10 15 20 dσ/dQ2 QE (10-39cm2/GeV2/neu on) Mine a QE 2p-2h MEC QE + 2p-2h MEC νµ - 12C 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 Q2 QE (GeV2) 0 2 4 6 8 10 12 14 16 dσ/dQ2 QE (10-39cm2/GeV2/p o on) νµ - CH Figu e 7.13: Flux- olded νµ−12C CCQE (uppe panel) and νµ−CH (lowe panel) sca e ing c oss sec ion pe a ge nucleon as a unc ion o Q2 QE and e alua ed in he SuSA 2 and SuSA 2-MEC models. MINERνA da a a e om [172]. 7.2. CCQE-LIKE SCATTERING IN THE MINERνAEXPERIMENT 139 7.2.2 Analysis o cha ged-cu en elec on neu ino esul s The p e ious conclusion holds also o he νeCCQE-like di e en ial c oss sec ions on hyd oca - bon published by MINERνA in [53]. These esul s a e p esen ed in Fig. 7.14 as a unc ion o he elec on ene gy ( op-le panel), elec on angle ( op- igh ) and econs uc ed ou -momen um (bo om-le ). Compa ed o he νµ(νµ) luxes, he νeand νeones ha e oughly he same shape in he egion o he peak bu he ail egion a la ge ene gies is signi ican ly highe in he elec onic case which so ly inc eases he a e age νeene gy up o 3.6 GeV. A de ailed compa ison o he νe e sus νµc oss sec ions is add essed in Sec ion 7.4. 12 3 4 5678 9 10 Ee (GeV) 0 0.5 1 1.5 2 2.5 dσ/dEe (10-39cm2/GeV/nucleon) 0510 15 20 25 30 35 θe (deg) 0 0.1 0.2 0.3 0.4 0.5 dσ/dθe (10-39cm2/deg ee/nucleon) MINERνA MEC QE QE+MEC 00.5 11.5 2 Q2 QE (GeV2) 0 2 4 6 8 10 12 dσ/dQ2 QE (10-39cm2/GeV2/nucleon) 00.5 11.5 2 Q2 QE (GeV2) 0.5 1 1.5 (dσνe+νe/dQ2 QE)/(dσνµ/dQ2 QE) Figu e 7.14: MINERνA lux-in eg a ed di e en ial νe-12C CCQE-like c oss sec ion pe nucleon s. elec on ene gy ( op le ) and elec on angle ( op igh ). The bo om panels show he νedi e - en ial c oss sec ion s. Q2 QE (bo om le ) and he a io be ween he lux a e aged CCQE νe+νeand νµc oss sec ions e sus Q2 QE (bo om igh ) compa ed wi h he SuSA 2-MEC p edic ion (solid ed line). Da a a e om [53]. In Fig. 7.14, esul s a e shown o he pu e QE esponse based on he IA, he 2p-2h MEC con ibu ion and he o al esponse. In all he cases he con ibu ion a he maximum coming om he 2p-2h MEC is oughly 30 −35% compa ed wi h he pu e QE esponse. These esul s a e simila o he ones al eady p esen ed o muon neu inos (an ineu inos), and hey show he impo ance o 2p-2h e ec s in o de o explain he beha io o da a. As obse ed, he model is capable o ep oducing success ully he da a. Fo comple eness, we p esen in he igh -bo om panel he esul s co esponding o he a io be ween he lux a e aged CCQE νe+νeand νµc oss sec ions e sus he econs uc ed ou -momen um. We compa e he p edic ions o he model ( ed cu e) wi h he da a, leading o a simila esul as when compa ed wi h GENIE p edic ions [53], i.e., he SuSA 2-model p edic s a a he la a io while he expe imen al da a seem o g ow up o Q2 QE ≈1 GeV2. Taking in o accoun ha he νe lux is negligible in compa ison wi h he νe