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
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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.
ACKNOWLEDGMENTS
This wo k was pa ially suppo ed by he I alian MIUR
h ough he PRIN 2009 esea ch p ojec , by he Is i u o
Nazionale di Fisica Nuclea e unde Con ac No. MB31,
by Spanish DGI and FEDER unds (FIS2011-28738-C02-01
and FPA2010-17142), by he Jun a de Andalucia, by he
Spanish Consolide -Ingenio 2000 p og am CPAN (CSD2007-
00042), by he Campus o Excellence In e na ional (CEI) o
he Moncloa p ojec (Mad id) and Andalucia Tech, by he
INFN-MICINN collabo a ion ag eemen (AIC-D-2011-0704),
as well as by he Bulga ian Na ional Science Fund unde
Con ac s No. DO-02-285 and No. DID-02/16-17.12.2009.
MVI is g a e ul o he wa m hospi ali y gi en by he UCM
and o inancial suppo du ing his s ay he e om he
SiNuRSE ac ion wi hin he ENSAR Eu opean p ojec . RGJ
acknowledges suppo om he Minis e io de Educaci´
on
(Spain).
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