Physics Le e s B 711 (2012) 178–183
Con en s lis s a ailable a SciVe se ScienceDi ec
Physics Le e s B
www.else ie .com/loca e/physle b
Supe scaling p edic ions o neu ino-induced cha ged-cu en cha ged pion
p oduc ion a MiniBooNE
M.V. I ano a,b,∗,J.M.Udias
a,A.N.An ono
b, J.A. Caballe o c,M.B.Ba ba o
d, E. Moya de Gue aa
aG upo de Física Nuclea , Depa amen o de Física A ómica, Molecula y Nuclea , Facul ad de Ciencias Físicas, Uni e sidad Complu ense de Mad id, CEI Moncloa, Mad id E-28040, Spain
bIns i u e o Nuclea Resea ch and Nuclea Ene gy, Bulga ian Academy o Sciences, Sofia 1784, Bulga ia
cDepa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad de Se illa, 41080 Se illa, Spain
dDipa imen o di Fisica, Uni e si à di To ino and INFN, Sezione di To ino, Via P. Giu ia 1, 10125 To ino, I aly
a icle in o abs ac
A icle his o y:
Recei ed 9 Ma ch 2012
Recei ed in e ised o m 23 Ma ch 2012
Accep ed 23 Ma ch 2012
A ailable online 28 Ma ch 2012
Edi o : W. Hax on
Keywo ds:
Neu ino eac ions
Nuclea e ec s
Pion p oduc ion
Supe scaling app oxima ion (SuSA) p edic ions o neu ino-induced cha ged-cu en cha ged pion
p oduc ion in he - esonance egion a e explo ed unde MiniBooNE expe imen al condi ions. The esul s
ob ained wi hin SuSA o he flux-a e aged double-di e en ial c oss sec ions o he π+p oduc ion o
he νμ+CH2 eac ion as a unc ion o he muon kine ic ene gy and o he sca e ing angle, he c oss
sec ions a e aged o e he angle, he o al c oss sec ion o he π+p oduc ion, as well as CC1π+ o
CCQE c oss sec ion a io a e compa ed wi h he co esponding MiniBooNE expe imen al da a. The SuSA
p edic ions a e in good ag eemen wi h da a on neu ino flux a e age c oss sec ions, bu a somewha
di e en dependence on he neu ino ene gy is p edic ed han he one esul ing om he expe imen al
analysis.
©2012 Else ie B.V. All igh s ese ed.
1. In oduc ion
The p ope ies o neu inos, pa icula ly he pa ame e s o hei
oscilla ions, a e being s udied wi h inc easing in e es as hese
may ca y impo an in o ma ion abou he limi s o he S an-
da d Model. In mos neu ino expe imen s, he in e ac ions o he
neu inos occu wi h nucleons bound in nuclei. The influence o
nucleon–nucleon in e ac ions on he esponse o nuclei o neu ino
p obes mus hen be conside ed, ideally in a model independen
way. Model p edic ions o hese eac ions in ol e many di e -
en e ec s such as nuclea co ela ions, in e ac ions in he final
s a e, possible modifica ion o he nucleon p ope ies inside he
nuclea medium, ha p esen ly canno be compu ed in an unam-
biguous and p ecise way. This is pa icula ly ue o he chan-
nels whe e neu ino in e ac ions ake place by means o exci a ion
o a nucleon esonance and ul e io p oduc ion o mesons. The
da a on neu ino-induced cha ged-cu en (CC) cha ged and neu-
al pion p oduc ion c oss sec ions on mine al oil ecen ly eleased
by he MiniBooNE Collabo a ion [1] p o ides an unp eceden ed op-
po uni y o ca y ou a sys ema ic s udy o double di e en ial
c oss sec ion o he p ocesses, νμp→μ−pπ+,νμn→μ−nπ+,
νμn→μ−pπ0, a e aged o e he neu ino flux.
*Co esponding au ho a : G upo de Física Nuclea , Depa amen o de Física
A ómica, Molecula y Nuclea , Facul ad de Ciencias Físicas, Uni e sidad Complu ense
de Mad id, CEI Moncloa, Mad id E-28040, Spain.
E-mail add ess: [email p o ec ed] (M.V. I ano ).
One way o a oiding model-dependencies is o use he nuclea
esponse o o he lep onic p obes, such as elec ons, unde sim-
ila condi ions o he neu ino expe imen s. Thus, in his Le e
we compa e he SuSA p edic ions o neu ino-induced CC cha ged
pion p oduc ion c oss sec ions wi h MiniBooNE da a [1].Theex-
ensi e analyses o scaling [2–4] and supe scaling [5–10] phenom-
ena obse ed in elec on–nucleus sca e ing lead o he use o he
scaling unc ion di ec ly ex ac ed om (e,e)da a o p edic neu-
ino (an ineu ino)–nucleus c oss sec ions [11],no elyingona
pa icula nuclea s uc u e model. Wi hin SuSA a “supe scaling
unc ion” (ψ) is buil by ac o ing-ou he single-nucleon con en
o he double-di e en ial c oss sec ion and plo ing he emaining
nuclea esponse e sus a scaling a iable ψ(q,ω). App oxima e
scaling o he fi s kind, i.e., no explici dependence o (ψ) on
he momen um ans e q, can be seen a ans e ene gies below
he quasielas ic (QE) peak. Scaling o second kind, i.e., no depen-
dence o (ψ) on he mass numbe , u ns ou o be excellen in
he same egion. When scaling o bo h fi s and second ypes oc-
cu , one says ha supe scaling akes place.
The analyses o he wo ld da a on inclusi e elec on–nucleus
sca e ing [7] confi med he obse a ion o supe scaling and hus
jus ified he ex ac ion o a uni e sal nuclea esponse o be also
used o weak in e ac ing p obes. Howe e , while he e is a num-
be o heo e ical models ha exhibi supe scaling, such as o
ins ance he ela i is ic Fe mi gas (RFG) [5,6], he nuclea e-
sponse depa s om he one de i ed om he expe imen al da a.
This showed he necessi y o conside mo e complex dynamical
pic u es o fini e nuclea sys ems – beyond he RFG – in o de
0370-2693/$ – see on ma e ©2012 Else ie B.V. All igh s ese ed.
h p://dx.doi.o g/10.1016/j.physle b.2012.03.072
M.V. I ano e al. / Physics Le e s B 711 (2012) 178–183 179
o desc ibe he nuclea esponse a in e media e ene gies. SuSA
p edic ions a e based on he phenomenological supe scaling unc-
ion ex ac ed om he wo ld da a on quasielas ic elec on sca -
e ing [12]. The model has been applied o neu al cu en sca -
e ing [13] and i has also been ex ended o he - esonance
egion [11] whe e he esponse o he nuclea sys em p oceeds
h ough exci a ion o in e nal nucleonic deg ees o eedom. In-
deed, a non-quasielas ic c oss sec ion o he exci a ion egion in
which nucleon exci a ions, pa icula ly he , play a majo ole
was ob ained by sub ac ing om he da a QE-equi alen c oss sec-
ions gi en by SuSA [9,14]. This p ocedu e has been possible due
o he la ge amoun o a ailable high-quali y da a o inelas ic elec-
on sca e ing c oss sec ions on 12C, including also sepa a e in-
o ma ion on he longi udinal and ans e se esponses, he la e
con aining impo an con ibu ions in oduced by e ec s beyond
he impulse app oxima ion (non-nucleonic).
The SuSA p ocedu e has been al eady employed o desc ibe he
non-pionic (QE) c oss sec ion o he MiniBooNE ν- and ν-nucleus
c oss sec ion [15–17]. He e we ex end he analysis o CC pion
p oduc ion c oss sec ion measu ed a MiniBooNe, ha om he
heo e ical poin o iew can be seen as mo e challenging. Fo
ins ance, p ope ies in he nuclea medium, as well as bo h
cohe en and incohe en pion p oduc ion o he nucleus should
be conside ed in any heo e ical app oach, while in he SuSA p o-
cedu e hey a e included phenomenologically ex ac ed om he
elec on sca e ing da a. All wha is assumed wi hin SuSA ap-
p oach is an in e nal ac o iza ion o he nuclea esponse o a
weakly in e ac ing p obe in o a single-nucleon pa and a ‘nu-
clea unc ion’ accoun ing o he o e all in e ac ion among nucle-
ons. As men ioned be o e, he SuSA assump ions ha e been es ed
agains a g ea deal o elec on–nucleus sca e ing da a wi h ai
success. The ac o iza ion assump ion allows o apply he same
nuclea esponses de i ed om elec on sca e ing o neu ino-
induced eac ions, wi h a me e use o he adequa e single-nucleon
e ms o his case. To show he impo ance o nuclea in e ac-
ion e ec s as p edic ed wi hin SuSA, as a e e ence, we also show
esul s ob ained wi hin he RFG, wi h no in e ac ions among nu-
cleons, o which he scaling unc ion in he -domain is sim-
ply gi en as 
RFG(ψ)=3
4(1−ψ2)θ(1−ψ2)wi h ψ he di-
mensionless scaling a iable ex ac ed om he RFG analysis ha
inco po a es he ypical momen um scale o he selec ed nu-
cleus [8,11].
In Fig. 1 we compa e he - egion SuSA [11] and RFG scal-
ing unc ions, ha we use in ou s udy. He e he da a e e o 12C
and 16O and span a la ge ange o ene gies ( om 0.3 o4GeV)
and sca e ing angles ( om 12 o 145 deg ees). The expe imen al
poin s in Fig. 1 a e ex ac ed by sub ac ing om he o al c oss
sec ions he quasielas ic con ibu ion calcula ed using he uni e -
sal QE scaling unc ion QE(ψ). This analysis does no include he
con ibu ion associa ed o meson-exchange cu en s (MEC), which
a e impo an in he egion be ween he QE and peaks and a e
esponsible o he disag eemen be ween he da a and he fi ( ed
cu e) a la ge nega i e alues o ψ. These cu en s a e medi-
a ed by i ual pions and do no co espond o he p oduc ion o
eal pions; hence hey should be included in he “quasielas ic” e-
sponse, which o he MiniBooNE expe imen co esponds o he
absence o eal pions in he final s a e bu no o he pionic da a
discussed in his wo k. The con ibu ion o he MEC o he QE neu-
ino and an ineu ino sca e ing has been e alua ed in he SuSA
amewo k in wo ecen pape s [15,17].Wealsono e ha he
fi shown in Fig. 1 is es ic ed only o exci a ion ene gies a and
below he  esonance peak, whe e he esponse is domina ed
by he ; a highe ene gies o he esonances and e en ually he
ail o deep inelas ic sca e ing (DIS) con ibu e. This explains he
di e ence be ween he phenomenological supe scaling unc ion,
Fig. 1. (Colo online.) The SuSA scaling unc ion in he - egion (ψ)(solid line)
ex ac ed om he wo ld da a on elec on sca e ing [11]. The do ed line shows
he scaling unc ions (ψ)in he RFG model.
which aims o desc ibe he  esonance peak, and he da a ob-
se ed in Fig. 1 a posi i e ψ- alues.
2. Fo malism
In wha ollows we p esen he esul s o applying he SuSA
and RFG -scaling unc ion o neu ino-induced CC cha ged pion
p oduc ion. We ollow he o malism gi en in [11]. The cha ged
cu en neu ino c oss sec ion in he a ge labo a o y ame is
gi enin he o m
d2σ
dΩdk=(Gcos θck)2
2π21−|Q2|
4F2(1)
whe e Ω,kand a e he sca e ing angle, momen um and en-
e gy o he ou going muon, Gis he Fe mi cons an and θcis he
Cabibbo angle. The unc ion F2depends on he nuclea s uc u e
h ough he R esponses and can be w i en as [11,18]:
F2=
VCC RCC +2
VCL RCL +
VLL RLL +
VTRT+2
VTRT
ha is, as a gene alized Rosenblu h decomposi ion ha ing cha ge-
cha ge (CC), cha ge-longi udinal (CL), longi udinal-longi udinal (LL)
and wo ypes o ans e se (T,T) esponses (R’s) wi h he co e-
sponding lep onic kinema ical ac o s (V’s). The nuclea esponse
unc ions in - egion a e exp essed in e ms o he nuclea en-
so Wμν in he co esponding egion. The basic exp essions used
o calcula e he single-nucleon c oss sec ions a e gi en in [11].
These in ol e he lep onic and had onic enso s as well as he e-
sponse and s uc u e unc ions o single nucleons. A con enien
pa ame iza ion o he single-nucleon W+n→+ e ex is gi en
in e ms o eigh o m- ac o s: ou ec o (CV
3,4,5,6) and ou ax-
ial (CA
3,4,5,6) ones. Vec o o m ac o s ha e been de e mined om
he analysis o pho o and elec o-p oduc ion da a, mos ly on a
deu e on a ge . Among he axial o m ac o s, he mos impo an
con ibu ion comes om CA
5. The ac o CA
6, whose con ibu ion o
he di e en ial c oss sec ion anishes o massless lep ons, can be
ela ed o CA
5by PCAC. Since he e a e no o he heo e ical con-
s ain s o CA
3,4,5(q2), heyha e obefi ed oda a.Weuse wo
di e en pa ame e iza ions: he one gi en in [19] whe e deu e on
e ec s we e e alua ed (au ho s es ima ed ha he la e educe
he c oss sec ion by 10%), deno ed as “PR1”, and he one om [20],
called “PR2”.
Wi h hese ing edien s, we e alua e he c oss sec ion o CC
++ and +p oduc ion on p o on and neu on, espec i ely.
Once p oduced, he decays in o πNpai s. Fo he ampli udes
Ao pion p oduc ion he ollowing isospin decomposi ion ap-
plies: A(νlp→l−pπ+)=A3,A(νln→l−nπ+)=1
3A3+2√2
3A1,
A(νln→l−pπ0)=−√2
3A3+2
3A1,wi hA3being he ampli ude
o he isospin 3/2s a eo heπNsys em, p edominan ly , and
180 M.V. I ano e al. / Physics Le e s B 711 (2012) 178–183
Fig. 2. (Colo online.) The double-di e en ial c oss sec ion a e aged o e he neu ino ene gy flux as a unc ion o he muon kine ic ene gy Tμob ained by SuSA (solid line)
and RFG (do ed line) - egion scaling unc ions. In each subfigu e he esul s ha e been a e aged o e he co esponding angula bin o cos θ. “PR2” pa ame iza ion [20] is
used. The esul s a e compa ed wi h he MiniBooNE da a [1].
A1 he ampli ude o he isospin 1/2 s a e ha is no conside ed
he e.
3. Resul s and discussion
Fi s we p esen RFG and SuSA p edic ions o he double-
di e en ial c oss sec ion o CC neu ino-induced π+p oduc ion
on CH2a e aged o e he neu ino flux Φ(ν), namely
d2σ
dTμdcos θ=1
Φ o d2σ
dTμdcosθν
Φ(ν)dν,(2)
whe e Tμand θa e co espondingly he kine ic ene gy and sca -
e ing angle o he ou going muon, νis he neu ino ene gy and
Φ o is he o al in eg a ed νμflux ac o o he MiniBooNE expe -
imen (Φ o =5.19 ×10−10 [νμ/cm2/POT]). The double-di e en ial
c oss sec ion a e aged o e he neu ino ene gy flux as a unc-
ion o he muon kine ic ene gy Tμis p esen ed in Fig. 2.Each
panel co esponds o a bin o cos θ. The PR2 pa ame iza ion has
been conside ed. Resul s wi h he PR1 pa ame e iza ion a e abou
5% highe , ha is a measu e o he deg ee o unce ain y ha we
expec om he choice o he single-nucleon esponse o his e-
ac ion. We compa e he p edic ions o SuSA and RFG wi h he
MiniBooNE da a [1]. The nuclea a ge has been conside ed as ca -
bon and hyd ogen in he mine al oil a ge . Fig. 3 shows SuSA and
RFG p edic ions o he double-di e en ial c oss sec ion a e aged
o e he neu ino ene gy flux as a unc ion o he sca e ing angle
a fixed Tμcompa ed wi h da a [1].
Figs. 2 and 3 show a good ag eemen be ween da a and he
SuSA p edic ions o he flux-a e aged double-di e en ial c oss
sec ions. This applies o bo h pa ame e iza ions o he ec o
and axial o m ac o s. RFG esul s ha e simila shape as SuSA
ones bu , as expec ed, hey o e es ima e he da a o a la ge ex-
en .
In Fig. 4 a e shown he esul s ob ained by in eg a ing he flux-
a e aged double-di e en ial c oss sec ions o e angle:
M.V. I ano e al. / Physics Le e s B 711 (2012) 178–183 181
Fig. 3. (Colo online.) Same as Fig. 2 bu he double-di e en ial c oss sec ion a e aged o e he neu ino ene gy flux as a unc ion o he sca e ing angle, a e p esen ed.
dσ
dTμ=1
Φ o Φ(ν)d2σ
dTμdcos θν
d(cos θ)dν.(3)
The o al c oss sec ion o π+p oduc ion as a unc ion o
he neu ino ene gy along wi h he MiniBooNE da a a e displayed
in Fig. 5. Poo e ag eemen wi h da a han o he flux-a e aged
c oss sec ions p esen ed in Figs. 2–4 is clea ly obse ed. The da a
seem o ollow a mo e linea dependence wi h he ene gy up o
2 GeV han he heo y. Howe e , be o e d awing defini e conclu-
sions, one has o conside on one side ha he nuclea esponse
ex ac ed om he phenomenological elec on da a includes he
whole ( eal) pion p oduc ion s eng h ( i ual pion con ibu ion
ia MEC has al eady been emo ed) while in MiniBoone he da a
a e sensi i e only o he cases whe e a eal pion is seen away
om he nucleus. This means ha i he eal pion is p oduced
and hen abso bed du ing he final s a e, i will no be seen in
he obse ed MiniBoone pionic c oss sec ions. On he o he hand,
he un olding p ocedu e used o ex ac he da a o Fig. 5 is o
Fig. 4. (Colo online.) The dσ/dTμ esul s ob ained by in eg a ing he flux-a e aged
double-di e en ial c oss sec ions o e cosθ[Eqs. (3)] a e compa ed wi h he Mini-
BooNE da a [1]. Fo ec o and axial o m- ac o s wo pa ame e iza ions, “PR1” [19]
and “PR2” [20],a eused.
182 M.V. I ano e al. / Physics Le e s B 711 (2012) 178–183
Fig. 5. (Colo online.) The o al c oss sec ion o π+p oduc ion a e compa ed wi h
he MiniBooNE da a [1]. Fo ec o and axial o m- ac o s wo pa ame e iza ions,
“PR1” [19] and “PR2” [20],a eused.
Fig. 6. (Colo online.) The esul s o CC1π+ o CCQE c oss sec ion a io a e com-
pa ed wi h MiniBooNE da a (co ec ed o final s a e in e ac ions and escaled o
an isoscala a ge ) [25].
some ex en model dependen . Thus hese da a a e less di ec
and we conside he compa ison wi h he da a o Figs. 2–4 o be
o mo e significance. I is wo h men ioning some ecen publi-
ca ions whe e he p oblems wi h he econs uc ion o neu ino
ene gy Eνa e discussed in mo e de ails [21–23] and also he
e iew by Gallaghe , Ga ey, and Zelle [24], whe e he au ho s
conside in dep h neu ino–nucleus in e ac ions (in he medium-
ene gy egime, O(1GeV)) in espec o he mode n neu ino oscil-
la ion expe imen s.
Fig. 6 shows he a io o CC1π+(CC single-pion p oduc ion)
o CCQE (CC quasielas ic sca e ing) c oss sec ions om SuSA,
SuSA +MEC (2p–2hmeson-exchange cu en ) [16], and RFG ap-
p oaches in compa ison wi h he MiniBooNE da a co ec ed o
final s a e in e ac ions. All hese a ios ha e been escaled o an
isoscala a ge [25]. The esul s a e ob ained on he basis o o-
al c oss sec ions o CC1π+(gi en in Fig. 5) and CCQE [16].
A simila conclusion as he one in he p e ious figu e could
be d awn he e. I seems ha he e is oo much πp oduc ion
s eng h below 1.2 GeV, and oo li le beyond ha , compa ed o
da a.
Be o e concluding we would like o emind ha he scaling
unc ion used in ou app oach ep esen s a good fi o elec on
sca e ing da a only in he egion below he -peak, as shown in
Fig. 1. As an icipa ed, highe esonances and he ail o DIS come
in o play a high momen um and ene gy ans e . Howe e , in he
kinema ical condi ions o he MiniBooNE expe imen he dominan
con ibu ion is associa ed o -exci a ion and he e o e we only
include he la e in ou analysis. This is suppo ed by he ac ha
inclusi e elec on sca e ing da a in he same kinema ical domain
a e easonably well ep oduced by using he pu e -supe scaling
unc ion (see Re . [11]).
4. Conclusion
Summa izing, in his Le e we p esen esul s o he c oss sec-
ions o neu ino-induced CC π+p oduc ion ob ained wi h he
SuSA and RFG (shown as e e ence) models. The SuSA app oach
p o ides neu ino–nucleus c oss sec ion p edic ions, based on he
obse ed nuclea esponse o elec on p ojec ile and he uni e sal
cha ac e o he scaling unc ion. No ice ha SuSA p edic ions in-
co po a e e ec s o final s a e in e ac ion (FSI), he p ope ies o
he  esonance in he nuclea medium, bo h he con ibu ion o
cohe en and incohe en p oduc ion, e c. The ole o he FSI on
he one-pion p oduc ion has been conside ed o ins ance wi hin
he GIBUU anspo model [26], whe e i was shown ha in o -
de o ep oduce he da a, he o al c oss sec ion ob ained wi h
FSI included has o be mul iplied by a ac o o 1.5. He e we show
ha SuSA p edic ions a e in good ag eemen wi h he MiniBooNE
expe imen al da a o pionic c oss sec ion in he case o he flux
a e aged da a, while some disag eemen emains in he compa i-
son o un olded neu ino ene gy da a. No ice ha he acco dance
be ween SuSA and da a he e is be e han he one o he non-
pionic case, whe e he model was ound o unde p edic he da a
unless meson exchange cu en s we e explici ly included [16].We
conclude ha he SuSA scaling unc ion o he - egion (ex ac ed
om elec on sca e ing expe imen s) and i s ex ension o neu-
ino p ocesses may be e y use ul o p edic c oss sec ions o
neu ino-induced CC π+p oduc ion, no elying on specific mod-
els.
Acknowledgemen s
This wo k was pa ially suppo ed by DGI (Spain) and FEDER
unds unde con ac s FIS2011-28738-C02-01, FIS2008-01301, and
FIS2011-23565, as well as he Bulga ian Na ional Science Fund
unde con ac s No. DO-02-285 and DID-02/16-17.12.2009, by
Uni e sidad Complu ense de Mad id (UCM, 910059) and by he
INFN-MICINN Collabo a ion ag eemen . M.V.I. is g a e ul o he
wa m hospi ali y gi en by he UCM and o financial suppo
du ing his s ay he e om he Cen o Nacional de Física de Pa ícu-
las, As opa ículas y Nuclea (CPAN) o Spain (Re .: CSD2007-
00042) and FPA2010-17142. The au ho s would like o hank
Rex Tayloe om he MiniBooNE Collabo a ion o help ul discus-
sions.
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