Relativistic description of final-state interactions in neutral-current neutrino and antineutrino cross sections

PHYSICAL REVIEW C 88, 025502 (2013) Rela i is ic desc ip ion o inal-s a e in e ac ions in neu al-cu en neu ino and an ineu ino c oss sec ions R. Gonz´ alez-Jim´ enez and J. A. Caballe o Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Uni e sidad de Se illa, 41080 Se illa, Spain And ea Meucci and Ca lo a Gius i Dipa imen o di Fisica, Uni e si ` a degli S udi di Pa ia, and INFN, Sezione di Pa ia, Via A. Bassi 6, I-27100 Pa ia, I aly M. B. Ba ba o Dipa imen o di Fisica, Uni e si ` a di To ino and INFN, Sezione di To ino, Via P. Giu ia 1, 10125 To ino, I aly M. V. I ano G upo de F´ ısica Nuclea , Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Uni e sidad Complu ense de Mad id, CEI Moncloa, 28040 Mad id, Spain and Ins i u e o Nuclea Resea ch and Nuclea Ene gy, Bulga ian Academy o Sciences, So ia 1784, Bulga ia J. M. Ud´ ıas G upo de F´ ısica Nuclea , Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Uni e sidad Complu ense de Mad id, CEI Moncloa, 28040 Mad id, Spain (Recei ed 10 July 2013; published 21 Augus 2013) We e alua e semi-inclusi e neu al-cu en quasielas ic di e en ial neu ino and an ineu ino c oss sec ions wi hin he amewo k o he ela i is ic impulse app oxima ion. The esul s o he ela i is ic mean- ield and o he ela i is ic G een’s unc ion models a e compa ed. The sensi i i y o he s ange-qua k con en o he nucleon o m ac o is also discussed. The esul s o he models a e compa ed wi h he MiniBooNE expe imen al da a o neu ino sca e ing. Nume ical p edic ions o lux-a e aged an ineu ino sca e ing c oss sec ions a e also p esen ed. DOI: 10.1103/PhysRe C.88.025502 PACS numbe (s): 25.30.P , 13.15.+g, 24.10.J I. INTRODUCTION The esul s on neu ino oscilla ions published by di e en collabo a ions [1–15] ha e aised a la ge deba e o e he p ope ies o neu inos ha could lead o a mo e comple e unde s anding o neu ino physics. Because o he in e es in oscilla ion measu emen s, a ious expe imen al neu ino- nucleus di e en ial c oss sec ions ha e been p esen ed [16–21] and a e planned in he nea u u e [22–24]. A clea unde s and- ing o neu ino-nucleus eac ions wi h a p ecise de e mina ion o di e en ial c oss sec ions is c ucial o a p ope analysis o he expe imen al da a. The MiniBooNE Collabo a ion has ecen ly epo ed [18] a measu emen o he neu al-cu en elas ic (NCE) lux- a e aged di e en ial neu ino c oss sec ion on CH2as a unc ion o he ou -momen um ans e ed squa ed, Q2.The ene gy egion conside ed in he MiniBooNE expe imen s, wi h a e age neu ino ene gy o ≈0.8 GeV, equi es he use o a ela i is ic model wi h an adequa e desc ip ion o nuclea dynamics and cu en ope a o s. The ela i is ic Fe mi gas (RFG) model canno ep oduce he da a unless calcula ions a e pe o med wi h a alue o he axial mass MAsigni ican ly la ge (MA=1.39 ±0.11 GeV/c2) han he wo ld a e age alue om he deu e ium da a o MA≃1.03 GeV/c2[25,26]. I is easonable o assume he la ge axial mass equi ed by he RFG as an e ec i e alue o inco po a e in o he calcula ions o nuclea e ec s which a e no included in he RFG. A p ecise knowledge o lep on-nucleus c oss sec ions, whe e unce ain ies on nuclea e ec s a e educed as much as possible, is manda o y and a compa ison be ween he esul s o di e en models can be help ul o disen angle di e en physics aspec s in ol ed in he sca e ing p ocess. I would be a sound s a egy o equi e ha any nuclea model used o desc ibe neu ino-nucleus sca e ing succeed in he desc ip ion o a ailable elec on sca e ing da a in simila kinema ic egions [27]. A in e media e ene gy, quasielas ic (QE) elec on sca e ing calcula ions [28,29], which a e able o success ully desc ibe a wide numbe o expe imen al da a, can p o ide a use ul ool o s udy neu ino-induced p ocesses. Howe e , some o hese models based on he impulse app oxima ion (IA) ha e been shown o be unable o desc ibe he MiniBooNE da a o bo h cha ged-cu en (CC) and neu al-cu en (NC) eac ions [30–33]. This has been iewed as an indica ion ha he eac ion can ha e signi ican con ibu ions om e ec s beyond he IA. The con ibu ion o mul inucleon exci a ions o neu ino-nucleus sca e ing [34–40] has been ound sizable and able o b ing he heo y in ag eemen wi h he MiniBooNE c oss sec ions wi hou he need o inc ease he axial mass MA. On he o he hand, a ela i is ic calcula ion o wo-pa icle– wo-hole exci a ions, pe o med o bo h elec on and neu ino sca e ing [41–44], has shown ha wo body cu en s gi e a mo e modes con ibu ion a MiniBooNE kinema ics and a e unable o ully accoun o he da a. O he models in oke an enhancemen o he magne ic esponse a he han a modi ica ion on he axial mass o ge ag eemen wi h he MiniBooNE da a [45,46]. 025502-1 0556-2813/2013/88(2)/025502(10) ©2013 Ame ican Physical Socie y R. GONZ ´ ALEZ-JIM ´ ENEZ e al. PHYSICAL REVIEW C 88, 025502 (2013) A deepe unde s anding o he eac ion dynamics would equi e a ca e ul e alua ion o all nuclea e ec s and o he el- e ance o mul inucleon emission and o some non-nucleonic con ibu ions [47–51]. P e ious s udies ha e clea ly s a ed he ele ance o inal-s a e in e ac ions (FSI) o ep oduce he exclusi e (e, ep) c oss sec ion wi hin he dis o ed-wa e impulse app oxima ion (DWIA) [28,29,52–57] and he use o a complex op ical po en ial (OP). The imagina y pa o he OP p oduces an abso p ion ha educes he c oss sec ion and accoun s pa ly o he loss o he inciden lux o he open inelas ic channels. Fo he case o inclusi e sca e ing, whe e only he emi ed lep on is de ec ed, all elas ic and inelas ic channels con ibu e, and a di e en ea men o FSI is equi ed: since all inal-s a e channels a e e ained, he lux los in a channel is edis ibu ed in he o he channels, and in he sum o e all he channels he o al lux mus be conse ed. FSI ha e been conside ed in ela i is ic calcula ions o he inclusi e QE elec on- and neu ino-nucleus sca e ing unde di e en app oaches [58–70]. The simples one co esponds o he ela i is ic plane-wa e impulse app oxima ion (RPWIA), whe e FSI a e neglec ed. In some DWIA calcula ions FSI e ec s a e inco po a ed in he inal nucleon s a e by using eal po en ials, ei he e aining only he eal pa o he ela i is ic ene gy-dependen complex op ical po en ial (deno ed as ROP) o using he same ela i is ic mean- ield po en ial conside ed in desc ibing he ini ial nucleon s a e (RMF) [58,71]. No e ha he RMF, because o he use o he same s ong ene gy- independen eal po en ial o bo h bound and sca e ing s a es, ul ills he dispe sion ela ion [72] and main ains he con inui y equa ion. A di e en desc ip ion o FSI in ol es he use o ela i is ic G een’s unc ion (RGF) echniques [61,62,68,69,73–78]. In he RGF model he componen s o he nuclea esponse a e w i en in e ms o he single-pa icle op ical model G een’s unc ion; i s spec al ep esen a ion, which is based on a bio hogonal expansion in e ms o a non-He mi ian OP H and o i s He mi ian conjuga e H†, can be exploi ed o a oid he explici calcula ion o he single-pa icle G een’s unc ion and ob ain he componen s o he had on enso [61,62]. Calcula ions equi e ma ix elemen s o he same ype as he DWIA ones o he exclusi e (e, ep) p ocess in [53] bu in ol e eigen unc ions o bo h Hand H†, whe e he imagina y pa has an opposi e sign and gi es in one case a loss and in he o he case a gain o s eng h. The RGF o malism makes i possible o econs uc he lux los in o nonelas ic channels in he case o he inclusi e esponse s a ing om he complex OP which desc ibes elas ic nucleon-nucleus sca e ing da a. Mo eo e , a consis en ea men o FSI in bo h exclusi e and inclusi e sca e ing is p o ided, and, because o he analy ici y p ope ies o he OP, he Coulomb sum ule is ul illed [62,72,73]. A compa ison among hese di e en desc ip ions o FSI has been p esen ed in [68] o inclusi e QE elec on sca e ing, in [69] o cha ged-cu en quasielas ic (CCQE) neu ino sca e ing, and in [79] wi h he CCQE MiniBooNE da a. The beha io o elec on sca e ing da a and hei ela ed scaling and supe scaling unc ions a e success ully desc ibed by bo h RMF and RGF models. In he case o neu inos, he shape o he expe imen al CCQE c oss sec ions is well ep oduced by bo h models, al hough he RMF gene ally unde p edic s he CCQE MiniBooNE da a, while he RGF can ep oduce i s magni ude o some pa icula choices o he ela i is ic po en ial wi hou he need o inc ease he s anda d alue o he axial mass. In his wo k we ex end he compa ison be ween he esul s o he RGF and RMF models o NCE sca e ing. We no e ha he RGF is app op ia e o an inclusi e p ocess whe e only he emi ed lep on is de ec ed, whe eas in NCE sca e ing he inal lep on is usually no de ec ed and i is he nucleon in he inal s a e ha igge s he e en de ec ions. Thus NCE c oss sec ions a e usually semi-inclusi e in he had onic sec o , whe e e en s o which a leas one nucleon in he inal s a e is de ec ed a e expe imen ally selec ed. The desc ip ion o semi-inclusi e NCE sca e ing wi h he RGF app oach can eco e impo an con ibu ions ha a e no p esen in he RDWIA, o which he semi-inclusi e c oss sec ion is ob ained om he sum o all he in eg a ed single-nucleon knockou channels plus he abso p ion p oduced in each channel by he imagina y pa o he op ical po en ial. This is app op ia e o exclusi e sca e ing, bu i neglec s some inal-s a e channels which can con ibu e o he semi-inclusi e eac ion. The RGF, howe e , desc ibes he inclusi e p ocess and, as such, may include channels which a e no p esen in he semi-inclusi e NCE measu emen s. F om his poin o iew, he RDWIA can ep esen a lowe limi and he RGF an uppe limi o he semi-inclusi e NCE c oss sec ions. In compa ison wi h he MiniBooNE NCE da a, he RDWIA gene ally unde p edic s he expe imen al c oss sec ion, while he RGF esul s a e in easonable ag eemen wi h he NCE da a [80]. I is no easy o disen angle he ole o speci ic con ibu ions which may be neglec ed in he RDWIA o spu iously added in he RGF, in pa icula i we conside ha bo h RDWIA and RGF calcula ions make use o phenomenological op ical po en ials, ob ained h ough a i o elas ic p o on-nucleus sca e ing da a. In o de o cla i y he con en o he en- hancemen o he RGF c oss sec ions compa ed o hose o he IA models, a ca e ul e alua ion o all nuclea e ec s and o he ele ance o mul inucleon emission and o some non-nucleonic con ibu ions [48] is equi ed. The compa ison wi h he esul s o he RMF model, whe e only he pu ely nucleonic con ibu ion is included, can be help ul o a deepe unde s anding o nuclea e ec s, pa icula ly FSI, which may play a c ucial ole in he analysis o upcoming sca e ing da a, and o hei in luence in s udies o neu ino oscilla ions a in e media e o high ene gies. II. RESULTS In his sec ion he nume ical esul s o he RGF and RMF models a e compa ed o NCE neu ino and an ineu ino sca e ing on 12C. As a i s s ep, we ha e p o ed ha RPWIA c oss sec ions e alua ed wi h wo independen compu e p og ams (de eloped by he Pa ia and Mad id-Se illa g oups) a e almos iden ical. This gi es us enough con idence on he eliabili y o bo h calcula ions, and i ag ees wi h p e ious esul s ound in [68] o he inclusi e QE elec on sca e ing and in [69] o CCQE neu ino-nucleus sca e ing. Then he 025502-2 RELATIVISTIC DESCRIPTION OF FINAL-STATE ... PHYSICAL REVIEW C 88, 025502 (2013) 100 200 300 0 2 4 6 8 10 RGF-EDAI RGF-EDAD1 RMF RPWIA 100 200 300 0 5 10 15 200 400 600 0 2 4 6 dσ/dTN (10-41 cm2/MeV) 200 400 600 0 2 4 6 8 400 800 TN (MeV) 0 2 4 6 400 800 TN (MeV) 0 2 4 6 8 500 MeV 1000 MeV 2000 MeV 500 MeV 1000 MeV 2000 MeV (a) (b) (c) (d) (e) ( ) FIG. 1. (Colo online) Di e en ial c oss sec ions o NCE neu ino sca e ing on 12C as a unc ion o he kine ic ene gy o he emi ed p o on [panels (a), (c), and (e)] o neu on [panels (b), (d), and ( )] a εν=500,1000,and 2000 MeV calcula ed in he RPWIA ( hin solid lines), RMF (dashed lines), RGF-EDAD1 ( hick solid lines), and RGF-EDAI (dash-do ed lines). The ec o and axial- ec o s ange o m ac o s ha e been ixed o ze o. compa ison be ween he esul s co esponding o he RMF and RGF models is pe o med o he NCE neu ino- and an ineu ino-induced c oss sec ions and also o he a io be ween p o on- and neu on-knockou c oss sec ions. In all he calcula ions p esen ed in his wo k he bound nucleon s a es a e aken as sel -consis en Di ac-Ha ee solu ions de i ed wi hin a ela i is ic mean- ield app oach using a Lag angian con aining σ,ω, and ρmesons [81]. The di e en ial c oss sec ions o he NCE neu ino and an ineu ino sca e ing, e alua ed in he RPWIA, RMF, and RGF, a e p esen ed in Figs. 1and 2as a unc ion o he kine ic ene gy o he emi ed p o on o neu on o h ee di e en (an i)neu ino ene gies εν(¯ν)=500,1000,and 2000 MeV. The con ibu ion om s ange qua ks o he ec o and axial- ec o o m ac o s has been ixed o ze o. In addi ion, we no e ha in all he calcula ions p esen ed in his wo k we ha e used he s anda d alue o he axial mass, MA=1.03 GeV. A di e en alue o MAwould change he c oss sec ions bu no he compa ison be ween he esul s o he di e en models. In he RGF calcula ions we ha e used wo pa ame iza ions o he ela i is ic OP o 12C: he ene gy-dependen and A-dependen EDAD1 (whe e Ais he a omic numbe ) and he ene gy-dependen and A-independen EDAI phenomenologi- calOPso [82]. The EDAD1 pa ame iza ion is a global one, because i is ob ained h ough a i o elas ic p o on-sca e ing da a on a wide ange o nuclei and, as such, i depends on he a omic numbe A, whe eas he EDAI OP is cons uc ed only om elas ic p o on-12C phenomenology [82]. I leads o 100 200 300 0 1 2 3 4 5 RGF-EDAI RGF-EDAD1 RMF RPWIA 100 200 300 0 2 4 6 200 400 600 0 1 2 3 4 5 dσ/dTN (10-41 cm2/MeV) 200 400 600 0 2 4 6 400 800 TN (MeV) 0 1 2 3 4 400 800 TN (MeV) 0 1 2 3 4 5 500 MeV 1000 MeV 2000 MeV 500 MeV 1000 MeV 2000 MeV (a) (b) (c) (d) (e) ( ) FIG. 2. (Colo online) The same as in Fig. 1, bu o an ineu ino sca e ing. a be e desc ip ion o he inclusi e QE 12C(e, e) expe imen al c oss sec ion, as well as o CCQE and NCE esul s ha a e in be e ag eemen wi h he MiniBooNE da a wi hin he RGF app oach [68,79,80,83]. The RMF gi es c oss sec ions ha a e gene ally 30% lowe han he RPWIA ones a small ou going nucleon kine ic ene gy TN, bu wi h a longe ail ex ending owa d la ge alues o TN, i.e., highe alues o he ans e ed ene gy, ha is a ibu able o he s ong ene gy-independen scala and ec o po en ials adop ed in he RMF app oach. The RGF c oss sec ions a e gene ally la ge han he RPWIA and he RMF ones. In he RGF he imagina y pa o he op ical po en ial edis ibu es he lux in all he inal-s a e channels and, in each channel, he lux los owa d o he channels is eco e ed by he lux gained om he o he channels. The la ge c oss sec ions in he RGF a ise om he ansla ion o he s eng h o he o e all e ec s o inelas ic channels which a e no included in he o he models, such as, o ins ance, esca e ing p ocesses o he nucleon in i s way ou o he nucleus, non-nucleonic exci a ions which may a ise du ing nucleon p opaga ion, o also some mul inucleon p ocesses. These con ibu ions a e no included explici ly in he RGF, bu hey all buil phenomenologically on he abso p i e imagina y pa o he OP. Dispe sion ela ions wi hin he RGF would ansla e his s eng h in o he inclusi e RGF c oss sec ion. Howe e , he RGF is app op ia e o an inclusi e p ocess whe e only he emi ed lep on is de ec ed and can include con ibu ions o channels which a e p esen in an inclusi e bu no in a semi-inclusi e eac ion. F om his poin o iew, he RGF can be conside ed as an uppe limi o he NCE c oss sec ions. 025502-3 R. GONZ ´ ALEZ-JIM ´ ENEZ e al. PHYSICAL REVIEW C 88, 025502 (2013) The compa ison be ween he RGF esul s ob ained wi h he EDAD1 and EDAI po en ials can gi e an idea o how he p edic ions o he model a e a ec ed by unce ain ies in he de e mina ion o he phenomenological OP. The di e ences depend on he ene gy and momen um ans e and a e essen ially a ibu able o he di e en imagina y pa o he wo po en ials, which accoun s o he o e all e ec s o inelas ic channels and is no uni ocally de e mined only om elas ic phenomenology. In con as , he eal e m is simila o di e en pa ame iza ions and gi es simila esul s. The NCE expe imen s can also be used o look o possible s ange-qua k con ibu ions in he nucleon. The ole o s angeness con ibu ion o he elec ic and magne ic nucleon o m ac o s has been ecen ly analyzed o pa i y- iola ing elas ic elec on sca e ing [84]. Speci ic alues o he elec ic and magne ic s angeness we e p o ided making use o all a ailable da a a di e en ans e ed momen a Q2.The analysis o 1σand 2σcon idence ellipses showed ha ze o elec ic and magne ic s angeness we e excluded by mos o he i s. Howe e , he alues o he s angeness in he elec ic and magne ic sec o s compa ible wi h he p e ious s udy lead o e y mino e ec s in he sepa a e p o on and neu on con ibu ions o he c oss sec ion o neu ino and an ineu ino sca e ing. Mo eo e , hese “small” e ec s end o cancel, being negligible o he o al di e en ial c oss sec ions. Al hough his cancella ion also wo ks o he axial- ec o s angeness, i s ela i e con ibu ion o he sepa a e p o on and neu on c oss sec ions is much la ge han he one associa ed wi h he elec ic and magne ic channels. The e o e, in his pape we es ic ou sel es o he in luence o he axial- ec o s angeness and conside how he NCE an ineu ino c oss sec ions change when he desc ip ion o he axial- ec o o m ac o o he nucleon is modi ied. I is a common p esc ip ion o apply he dipole pa ame iza ion o he s ange axial o m ac o and o use he same alue o he axial mass used o he nons ange o m ac o as a cu o ; he s ange axial coupling cons an a Q2=0iss. A measu emen o ν(¯ν)-p o on elas ic sca e ing a he B ookha en Na ional Labo a o y a low Q2sugges ed a nonze o alue o s [16,85]. The MiniBooNE Collabo a ion used he a io o p o on- o-nucleon NCE c oss sec ions o ex ac s =0.08 ±0.26 [18] based on he RFG wi h MA=1.35 GeV/c2. The analysis pe o med in [86] wi h he RMF model led o s =0.04 ±0.28, while he COMPASS Collabo a ion epo ed a nega i e s = −0.08 ±0.01(s a .) ±0.02(sys .) as a esul o a measu emen o he deu e on spin asymme y [87], in ag eemen wi h he HERMES esul s [88]. The (an i)neu ino c oss sec ion can be unde s ood essen- ially by analyzing he beha io o he longi udinal esponse L, he pu e ec o ans e se esponse T, and he axial- ec o ans e se esponse T.InFig.3 he ela i e impo ance o hese h ee con ibu ions o he NCE an ineu ino di e en ial c oss sec ion is p esen ed o ε¯ν=500 MeV. Fo neu ino sca e ing he same sepa a ion holds bu he T esponse has opposi e sign. The in luence o s on each esponse, L,T, and T, and on sepa a e p o on and neu on e en s, is also explo ed. In o de o a oid complica ions ela ed o he desc ip ion o he FSI and/o o unce ain ies due o he pa icula model, calcula ions ha e been pe o med in he RPWIA. In he case 0 100 200 300 TN (MeV) -1 0 1 2 3 4 dσ/dTN (10-41cm2/MeV) T; Δs=-0.15 T; Δs=0.0 T; Δs=0.15 L; Δs=-0.15 L; Δs=0.0 L; Δs=0.15 T’; Δs=-0.15 T’; Δs=0.0 T’; Δs=0.15 0 100 200 300 TN (MeV) -1 0 1 2 3 4 ε ν = 500 MeV _ (a) (b) FIG. 3. (Colo online) Sepa a ed longi udinal, L(cen al se o lines), ans e se (symme ic), T( op se o lines), and ans e se axial- ec o (an isymme ic) T(bo om se o lines) o he NCE an ineu ino c oss sec ion a ε¯ν=500 MeV as a unc ion o he emi ed p o on (a) o neu on (b) kine ic ene gy. Calcula ions a e pe o med in he RPWIA. Solid, dashed, and do ed lines a e he esul s wi h s =0.0, −0.15, and +0.15, espec i ely. o p o on knockou , he ans e se esponse Tis la ge by a ac o o ≈2 han he ans e se axial- ec o esponse T, and he longi udinal esponse Lis e y small. In he case o neu on knockou , he T esponse is s ill la ge han he Tone bu he Lcon ibu ion is signi ican . No e ha he longi udinal esponse is o a la ge ex en insensi i e o s angeness. The NCE di e en ial c oss sec ions a e displayed in Fig. 4. The p o on c oss sec ion dec eases when inc easing s, while he neu on c oss sec ion has he opposi e beha io . Thus, he o al p o on+neu on c oss sec ion is almos independen o s in he ange −0.15 o 0.15. This esul is ob ained o bo h neu ino and an ineu ino sca e ing and is a he independen o he inciden (an i)neu ino ene gy. De e mining he s angeness con ibu ion o he axial o m ac o om measu emen s o NCE c oss sec ions is no easy. Theo e ical unce ain ies on he app oxima ions and on he ing edien s o he models a e usually la ge han he unce ain y ela ed o he s angeness con en o he nucleon. F om he expe imen al poin o iew, p ecise c oss sec ion measu emen s a e no easy o make due o di icul ies in he de e mina ion o he neu ino lux ela ed o he nuclea model dependence. The e o e, a ios o c oss sec ions ha e 100 200 300 TN (MeV) 0 1 2 3 4 5 dσ/dTN (10-41cm2/MeV) To al; Δs=-0.15 To al; Δs=0.0 To al; Δs=0.15 100 200 300 TN (MeV) 0 1 2 3 4 5 (a) (b) FIG. 4. (Colo online) NCE an ineu ino c oss sec ion a ε¯ν=500 MeV as a unc ion o he emi ed p o on (a) o neu on (b) kine ic ene gy. Calcula ions a e pe o med in he RPWIA. Solid, dashed, and do ed lines a e he esul s wi h s =0.0, −0.15, and +0.15, espec i ely. 025502-4 RELATIVISTIC DESCRIPTION OF FINAL-STATE ... PHYSICAL REVIEW C 88, 025502 (2013) 100 200 1 1.5 2 R [p/n] 100 200 1 1.5 2 R [p/(p+n)] 200 400 600 1 1.5 2 R [p/n] 200 400 600 1 1.5 2 R [p/(p+n)] 200 400 600 TN (GeV) 0.6 0.7 0.8 0.9 1 R [p/n] 200 400 600 TN (GeV) 0.6 0.7 0.8 0.9 1 R [p/(p+n)] εν = 500 MeV ε ν = 500 MeV εν = 1000 MeV ε ν = 1000 MeV _ _ εν = 2000 MeV ε ν = 2000 MeV _ (a) (b) (d) (c) (e) ( ) FIG. 5. (Colo online) Ra io o p o on- o-neu on c oss sec ions as a unc ion o he kine ic ene gy o he emi ed nucleon o neu ino [panels (a), (c), and (e)] and an ineu ino [panels (b), (d), and ( )]. Resul s o di e en desc ip ions o FSI a e compa ed. Line con en ions a e as in Fig. 1. All he esul s a e ob ained wi h s =0. been p oposed as al e na i e and use ul ools o sea ch o s angeness e ec s. The a io o p o on- o-neu on c oss sec ions was p oposed and discussed in [89–95]. This a io is e y sensi i e o s ange-qua k e ec s because he axial s angeness s in e e es wi h he iso ec o con ibu ion o he axial o m ac o gA≈1.27 wi h one sign in he nume a o and wi h he opposi e sign in he denomina o . In Fig. 5we display ou esul s o he p/n a io o h ee di e en neu ino and an ineu ino ene gies. In he case o a ios o c oss sec ions he dis o ion e ec s a e la gely educed and di e en models o desc ibe FSI a e expec ed o p oduce simila esul s. To make easie he compa ison be ween neu inos and an ineu inos we ha e chosen he same scale in bo h cases. This allows us o isualize clea ly he di e en e ec s in oduced by he models in bo h sca e ing eac ions. In he case o neu ino sca e ing he p/n a io is almos cons an and he RPWIA, RMF, and RGF esul s coincide up o a ew pe cen . As obse ed, in he egion o small nucleon kine ic ene gy he main di e ence in he neu ino case comes om he RGF-EDAI model wi h a small bump ( o εν=1 and 2 GeV) ha is no p esen in he o he app oaches. Fo la ge TN he a io s abilizes, wi h he disc epancy among he di e en models being a mos o he o de o ∼4%–5%. Finally, he di e ences inc ease a he la ges TN alues. No e ha in his egion he c oss sec ions a e e y small and show a signi ican sensi i i y o FSI and/o he h esholds used. The maximum unce ain y in he p o on/neu on a io linked o he di e en models is o he o de o ∼15% (εν=500 MeV) and ∼8% (εν=1and2GeV). 100 200 0.5 1 1.5 2 R [p/n] 100 200 0.5 1 1.5 2 200 400 600 0.5 1 1.5 2 R [p/n] Δs=-0.15 Δs=0 Δs=0.15 200 400 600 0.5 1 1.5 2 200 400 600 TN (GeV) 0.5 1 1.5 2 R [p/n] Δs=-0.080 ± 0.022 200 400 600 TN (GeV) 0.5 1 1.5 2 εν = 500 MeV ε ν = 500 MeV εν = 1000 MeV ε ν = 1000 MeV _ _ (a) (b) εν = 2000 MeV ε ν = 2000 MeV _ (c) (d) (e) ( ) FIG. 6. (Colo online) Ra io o p o on- o-neu on c oss sec ions as a unc ion o he kine ic ene gy o he emi ed nucleon o neu ino [panels (a), (c), and (e)] and an ineu ino [panels (b), (d), and ( )]. Calcula ions a e pe o med in he RPWIA and wi h di e en alues o s. The shadowed band e e s o esul s co esponding o he COMPASS-HERMES measu emen o he axial s angeness. La ge di e ences a e ob ained in he case o an ineu ino sca e ing, in pa icula o he RMF model, whose esul s a e signi ican ly enhanced wi h espec o he RGF ones o la ge alues o TN. Con a y o he case o neu inos, whe e he a io changes e y smoo hly wi h TN, o an ineu inos he slope o he a io goes up e y as wi h he nucleon ene gy. This e lec s he di e en beha io shown by he p o on and neu on c oss sec ions agains TN. A in e media e nucleon ene gies he unce ain y among he a ious models is o he o de o ∼12%–14%,wi h much la ge disc epancies o inc easing TN alues. Howe e , in his ene gy egion he c oss sec ion becomes signi ican ly lowe han i s maximum and a e y p ecise measu emen is equi ed o ob ain a clea esul . I is in e es ing o poin ou he simila i y among he esul s co esponding o RGF-EDAI, RFG-EDAD1, and RPWIA a ε¯ν=500 and 1000 MeV. In Fig. 6 he dependence o he RPWIA p/n a io on he s ange-qua k con ibu ion is p esen ed. The a io is enhanced when calcula ions a e pe o med wi h a nega i e s and supp essed when a posi i e s is conside ed. In he case o an ineu ino sca e ing he ole o s angeness con ibu ion is pa icula ly signi ican when a nega i e s is assumed wi h a la ge peak a TN≈0.7ε¯ν. The sensi i i y o he p/n a io o s, as well as o he s ange-qua k con ibu ion in he ec o o m ac o s, was analyzed in [65]. In pa icula , i was ob ained ha a mode a ely la ge and nega i e s angeness con ibu ion o he magne ic momen o he nucleon can cancel he peak in he p/n a io. Al hough a la ge s angeness 025502-5 R. GONZ ´ ALEZ-JIM ´ ENEZ e al. PHYSICAL REVIEW C 88, 025502 (2013) con ibu ion o he ec o o m ac o s is no suppo ed by any a ailable expe imen al e idence [84], i would be anyhow in iguing o look o possible s angeness e ec s wi h a di ec measu emen o his quan i y. We a e awa e ha a p ecise measu emen o he p/n a io is a ha d expe imen al ask, bu he i s measu emen o he MiniBooNE Collabo a ion [18] has p o en he alidi y o his expe imen al echnique and, hope ully, new da a will be a ailable in he u u e. In he esul s o Fig. 6, he unce ain y in he p o on/neu on a io associa ed wi h he axial s angeness is qui e la ge: in he case o neu inos he a io changes by a ac o o 2 when going om posi i e (s =0.15) o nega i e (s =−0.15) s angeness. This la ge ange o a iabili y o s is in acco dance wi h ν(¯ν) B ookha en da a [16,85] and also wi h he MiniBooNE esul s [18], bu he COMPASS measu emen s sugges a na owe in e al o he axial s angeness [87], which esul s in a educed ange o a ia ion o he p o- on/neu on a io. This is ep esen ed in Fig. 6by he shadowed band ha , as obse ed, is o he same o de o magni ude as he unce ain ies ela ed o he dis o ion e ec s. This sensi i i y o s ge s much la ge o an ineu inos, whe e he a io goes up e y as wi h inc easing TN alues. Howe e , as in he p e ious case o neu inos, he ange o a ia ion in R[p/n] associa ed o he COMPASS measu emen is simila o he unce ain y in oduced by nuclea model and/o dis o ion e ec s. Al hough his s udy is consis en wi h p e ious analyses, and i shows he capabili y o he a io R[p/n] as an use ul obse able in o de o ge p ecise in o ma ion on he axial- ec o s angeness con en in he nucleon, he esul s in Fig. 6indica e ha , owing o he ac ual p ecision in he axial s angeness gi en by he COMPASS expe imen , a deep and ca e ul analysis o he unce ain ies linked o ing edien s o he calcula ion such as nuclea models o FSI is equi ed. III. RESULTS AT MINIBOONE KINEMATICS The neu ino-nucleus NCE eac ion a MiniBooNE can be conside ed as sca e ing o an inciden neu ino o an ineu ino wi h a single nucleon bound in ca bon o ee in hyd ogen. Each con ibu ion is weigh ed by an e iciency co ec ion unc ion and a e aged o e he expe imen al (an i)neu ino lux [96]. Di e en ela i is ic desc ip ions o FSI we e p esen ed and compa ed wi h he NCE MiniBooNE da a in [80,86]. In Fig. 7we p esen ou RMF and RGF c oss sec ions o NCE (νN →νN) sca e ing and compa e hem wi h he expe imen al da a, whe e he a iable Q2 QE =2mNT is de ined by assuming ha he a ge nucleon is a es , mN being he nucleon mass and T he o al kine ic ene gy o he ou going nucleons. The RMF esul has a oo-so Q2beha io o ep oduce he expe imen al da a a small Q2, while he RGF p oduces la ge c oss sec ions, in be e ag eemen wi h he da a. The di e ence be ween he RGF esul s calcula ed wi h he wo op ical po en ials is signi ican , pa icula ly o small TN(Q2 QE) alues. This is consis en wi h he la ge disc epancies shown by he c oss sec ions e alua ed a ixed neu ino and an ineu ino ene gies (see Fig. 1). The RGF-EDAI c oss sec ion is in acco dance wi h he shape and he magni ude 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 TN (GeV) 0 1 2 3 4 dσ/dQ2 (10-39cm2/GeV2) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Q2 QE (GeV2) 0 1 2 3 4 dσ/dQ2 (10-39cm2/GeV2) MiniBooNE RMF RPWIA RGF EDAI RGF EDAD1 FIG. 7. (Colo online) NCE lux-a e aged (νN →νN)c oss sec ion as a unc ion o Q2. Line con en ions a e as in Fig. 1.The da a a e om [18]. o he da a. In con as he RGF-EDAD1 esul lies below he da a a he smalles alues o Q2conside ed in he igu e. The RMF app oach leads o he lowes c oss sec ion o low- o-in e media e alues o he ans e ed ou -momen um. Only o Q2 QE ⩾0.9GeV 2is he RMF ail highe han he RPWIA esul , bu i s ill lies below he wo RGF models. Howe e , in his kinema ical egime all he models a e able o ep oduce he da a wi hin he e o ba s. The MiniBooNE Collabo a ion has collec ed also an ex- ensi e da a se o neu al-cu en an ineu ino e en s whose analysis is cu en ly ongoing and some p elimina ies esul s a e a ailable [97,98]. In Fig. 8we show ou p edic ions o he NCE MiniBooNE (¯νN →¯νN) c oss sec ion. In hese calcula ions we use he se o e iciency coe icien s gi en in [18] o neu ino sca e ing. The selec ion o he an ineu ino NCE sample is sligh ly di e en om he neu ino sample, and he e o e he e iciencies a e simila only as a i s app oxima ion, since hey a e expec ed o be a li le bi di e en . Howe e , e en i i is no igo ous, he use o neu ino e iciencies o he an ineu inos is app oxima ely co ec . Simila ly o he neu ino case, he RMF gi es c oss sec ions ha a e lowe han he RPWIA ones whe eas he RGF p oduces la ge c oss sec ions. This is consis en wi h he esul s shown in Fig. 2 o ixed an ineu ino ene gies, whe e a signi ican 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 TN (GeV) 0 1 2 3 4 dσ/dQ2 (10-39cm2/GeV2) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Q2 QE (GeV2) 0 1 2 3 4 RGF-EDAI RGF-EDAD1 RMF RPWIA FIG. 8. (Colo online) The same as in Fig. 7, bu o he (¯νN →¯νN) c oss sec ion. 025502-6 RELATIVISTIC DESCRIPTION OF FINAL-STATE ... PHYSICAL REVIEW C 88, 025502 (2013) 12 ε ν (MeV) 0 2 4 6 12 ε ν (MeV) 0 0.5 1 12 ε ν (GeV) 0 0.05 0.1 12 ε ν (GeV) 0 0.02 0.04 (a) (b) __ dσ/TN(10-41cm2 MeV-1)xΦ[(1011 cm2 POT 50MeV)-1] (c) (d) FIG. 9. (Colo online) P oduc o he p o on+neu on NCE an ineu ino c oss sec ion and he an ineu ino MiniBooNE lux [96] as a unc ion o he an ineu ino ene gy ε¯νa ou ixed alues o he ou going nucleon kine ic ene gy TN: 108 (a), 252 (b), 540 (c), and 756 MeV (d). Line con en ions a e as in Fig. 1. disc epancy among he c oss sec ions ob ained wi h he a ious models is obse ed, wi h he smalles con ibu ion being o he RMF and he la ges one o RGF-EDAI. Fu he mo e, he RGF wi h he EDAD1 op ical po en ial gi es esul s which a e e y simila o he RPWIA calcula ion. The p edic ions o hese wo models, RPWIA and RGF-EDAD1, ag ee e y well wi h he p elimina y an ineu ino NCE MiniBooNE da a [97,98]. The cu es displayed in Figs. 7and 8in ol e a con- olu ion o e he expe imen al (an i)neu ino lux. In o de o be e unde s and hese esul s, in Fig. 9we p esen he p o on+neu on NCE an ineu ino c oss sec ion mul iplied by he an ineu ino MiniBooNE lux o [96]asa unc iono he an ineu ino ene gy o ou di e en alues o he kine ic ene gy o he emi ed nucleon. The calcula ions equi ed o he analysis in Fig. 9conside he en i e ene gy ange which is ele an o he MiniBooNE lux. I has been poin ed ou in [32,99] ha he lux-a e age p ocedu e in oduces addi ional unce ain ies and, he e o e, he MiniBooNE c oss sec ions can include con ibu ions om di e en kinema ic egions, whe e eac ion mechanisms o he han one-nucleon knockou can be dominan . Pa o hese con ibu ions, which a e no included in usual calcula ions based on he IA, can be eco e ed in he RGF by he imagina y pa o he phenomenological OP. The RMF gi es c oss sec ions ha a e lowe han he RPWIA ones a TN=108 and 252 MeV bu la ge a highe alues o TN. As al eady men ioned, his e ec is due o he s ong ene gy-independen po en ial adop ed in he RMF model. The la ge c oss sec ion in he RGF can be asc ibed o he con ibu ion o eac ion channels which a e no included in o he models based on he IA. The MiniBooNE Collabo a ion has also epo ed he (νp →νp)/(νN →νN) a io [18]. The denomina o o his 350 400 450 500 550 600 650 700 750 800 T ec (MeV) 0.1 0.2 0.3 0.4 0.5 (νp ⎯⎯> νp) / (νN ⎯⎯> νN) RGF-EDAD1 RGF-EDAI RMF RPWIA MA = 1.03 GeV, Δs = 0.0 FIG. 10. (Colo online) Ra io (νp →νp)/(νN →νN)asa unc ion o he econs uc ed ene gy compu ed wi hin RGF, RMF, and RPWIA models. Line con en ions a e as in Fig. 1. The da a a e om [18]. a io includes e en s wi h s anda d NCE selec ion cu s bu wi h he ene gy cu eplaced wi h 350 <T N<800 MeV and an addi ional “p o on/muon” cu in o de o educe muonlike backg ounds ha domina e he high- isible-ene gy egion. In he nume a o a e e en s om he so-called NCE p o on- en iched e en sample whe e wo addi ional cu s a e applied o supp ess neu on NCE e en s. The Mon e Ca lo simula ion shows ha only 10% o neu on NCE e en s gi e a con ibu ion o he νp →νp sample. Mo e de ails on he olding p ocedu e o calcula e his a io a e gi en in Appendix B o [100]. In Fig. 10 we p esen ou esul s o he (νp →νp)/ (νN →νN) a io wi h RGF, RMF, and RPWIA models as a unc ion o econs uc ed ene gy T ec. In ou calcula ions he axial s angeness s has been ixed o 0. All he models gi e e y close esul s which a e in ag eemen wi h expe imen al da a wi hin e o ba s; his is in acco dance wi h he ac ha in his kinema ical egime wi h TN>350 MeV all he models a e able o ep oduce he c oss-sec ion da a. IV. CONCLUSIONS This wo k ex ends p e ious compa a i e s udies o include he analysis o neu al-cu en neu ino-nucleus sca e ing eac ions. In p e ious wo ks we applied ou models o inclusi e elec on and cha ged-cu en neu ino sca e ing, p o iding also a compa ison wi h da a measu ed by he MiniBooNE collabo a ion. Ou main objec i e in his pape is o examine how capable ou heo e ical models a e a explaining he ecen da a on NC eac ions measu ed by MiniBooNE. In bo h cases, CC and NC p ocesses, he kinema ics in ol ed implies he use o ully ela i is ic models. This is he case o he ela i is ic mean- ield and he ela i is ic G een’s unc ion app oaches conside ed in his wo k. No only is ela i is ic kinema ics conside ed, bu also nuclea dynamics and cu - en ope a o s a e desc ibed wi hin a ela i is ic o malism. Mo eo e , inal s a e in e ac ions, an essen ial ing edien in he eac ion mechanism, a e also aken in o accoun by in oducing ela i is ic po en ials in he inal s a e and sol ing he Di ac equa ion. Whe eas in he RMF case he po en ial consis s o eal s ong ene gy-independen scala and ec o e ms 025502-7 R. GONZ ´ ALEZ-JIM ´ ENEZ e al. PHYSICAL REVIEW C 88, 025502 (2013) ( he same used o he bound nucleon s a es), he RGF makes use o phenomenological ene gy-dependen complex op ical po en ials. In his wo k esul s a e shown o wo choices o he op ical po en ial: EDAI and EDAD1. We ha e compa ed he p edic ions o he di e en ial c oss sec ions and he p o on/neu on a io. The o me shows an impo an dependence wi h he model, pa icula ly a small alues o he ou going nucleon kine ic ene gy. The RMF p o ides he lowes esul while he RGF ge s much mo e s eng h, al hough a signi ican dependence on he po en ials conside ed is also seen o he RGF case. This gene al esul applies o bo h neu ino and an ineu ino eac ions and occu s o e y di e en alues o he lep on (νμ/νμ) ene gy. This explains he signi ican di e ences obse ed o he NC lux-a e aged c oss sec ions, which a e also compa ed wi h MiniBooNE da a. F om ou analysis we conclude ha he la ges con ibu ion co esponding o RGF-EDAI is in acco dance wi h da a o neu inos, whe eas he o he models, in pa icula he RMF, lie clea ly below da a a small nucleon kine ic ene gies (TN). In con as , all models ep oduce he beha io o da a a la ge TN alues. Howe e , we ha e o keep in mind he la ge da a e o bands in his kinema ical egime. In addi ion o he unce ain ies associa ed wi h nuclea model and/o FSI desc ip ions, which a e pa icula ly ele an o he c oss sec ions, ano he ing edien o be ca e ully conside ed is he ole o s angeness in he nucleon. While s angeness in he elec ic and magne ic sec o s leads o e y mino e ec s, which a e almos negligible o he o al c oss sec ion, he dependence upon he axial- ec o s angeness is much mo e impo an . This is pa icula ly ue in he case o he sepa a e p o on and neu on con ibu ions o he c oss sec ions. The ole o he axial s angeness is opposi e in p o ons and neu ons, and i ends o cancel in he o al c oss sec ion. This jus i ies he use o o al c oss sec ions o analyze nuclea models and FSI dependencies, since hey a e almos independen o s (axial s angeness). Mo eo e , i also jus i ies he use o he p/n a io as a use ul obse able o ge in o ma ion on he axial s angeness. In his wo k we ha e analyzed in de ail he p o on/neu on a io by compa ing he p edic ions gi en by he RMF and RGF models. We ha e p o ed ha he a io only p esen s a weak dependence on he model, in pa icula , in he case o neu inos: he unce ain y is on a e age o he o de o ∼4%–5%. This disc epancy ge s signi ican ly highe o an ineu inos a inc easing alues o nucleon ene gy. In any case, hese di e ences a e usually smalle han he ones asc ibed o he use o di e en axial s angeness con en in he nucleon. In his case he p/n a io can change by mo e han a ac o o 2 when he a ia ion in s is in acco dance wi h he B ookha en and MiniBooNE da a. Howe e , he highly p ecise measu emen s gi en by COMPASS lead o an unce ain y in R[p/n] simila o he one asc ibed o dis o ion and nuclea model e ec s. Summa izing, we ha e applied wo di e en ela i is ic models ha inco po a e inal-s a e in e ac ions o he s udy o NCE neu ino- and an ineu ino-nucleus sca e ing p ocesses. We ha e p esen ed a de ailed analysis o he di e en ial c oss sec ions (wi h he sepa a e p o on and neu on con ibu ions) and he p/n a io. We ha e compa ed ou p edic ions wi h he ecen expe imen al da a aken by he MiniBooNE Collabo a- ion o neu inos and ha e gi en p edic ions o an ineu inos which can be also compa ed wi h da a when a ailable. We ha e p o ed he signi ican di e ences in oduced by he a ious models ha may indica e impo an e ec s asc ibed o co ela ion and meson exchange cu en s, which a e no ye inco po a ed in he models. Al hough he compa ison be ween RMF and RGF models may help us in disen angling di e en aspec s in ol ed in he physics o he p oblem, we should be cau ious in es ablishing inal conclusions be o e o he ing e- dien s beyond he impulse app oxima ion can be implemen ed in mo e e ined calcula ions and hei con ibu ions can be ca e ully examined. 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