PHYSICAL REVIEW C 73, 035503 (2006)
Supe scaling and neu al cu en quasielas ic neu ino-nucleus sca e ing
J. E. Ama o,1M. B. Ba ba o,2J. A. Caballe o,3and T. W. Donnelly4
1Depa amen o de F´
ısica Mode na, Uni e sidad de G anada, E-18071 G anada, Spain
2Dipa imen o di Fisica Teo ica, Uni e si `
a di To ino and INFN, Sezione di To ino, Via P. Giu ia 1, I-10125 To ino, I aly
3Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo. 1065, E-41080 Se illa, Spain
4Cen e o Theo e ical Physics, Labo a o y o Nuclea Science and Depa men o Physics, Massachuse s Ins i u e o Technology,
Camb idge, Massachuse s 02139, USA
(Recei ed 10 Janua y 2006; published 29 Ma ch 2006)
The supe scaling app oach is applied o s udies o neu al cu en neu ino eac ions in he quasielas ic egime.
Using inpu om scaling analyses o elec on sca e ing da a, p edic ions o high-ene gy neu ino and an ineu ino
c oss sec ions a e gi en and compa ed wi h esul s ob ained using he ela i is ic Fe mi gas model. The in luence
o s angeness con en inside he nucleons in he nucleus is also explo ed.
DOI: 10.1103/PhysRe C.73.035503 PACS numbe (s): 25.30.P , 23.40.Bw, 24.10.J
I. INTRODUCTION
Inclusi e elec on sca e ing a in e media e o high ene gies
om nuclei is known o exhibi he phenomenon o scaling and
supe scaling [1–7]. A su icien ly high ene gies, ypically a
leas 500 MeV, one sees ha nea he quasielas ic peak he
c oss sec ion may be analyzed in e ms o a educed esponse
ob ained by di ision by a sui able N- and Z-weigh ed single-
nucleon elec omagne ic c oss sec ion and plo ed agains an
app op ia e kinema ic a iable o see he scaling beha iou .
Fi s , when he educed c oss sec ion is seen o depend only
on his kinema ic a iable— he scaling a iable—and no
on he momen um ans e one has scaling o he i s kind.
Second, i he educed c oss sec ion and scaling a iable ha e
been made dimensionless ia emo al o he momen um scale
cha ac e is ic o a gi en nucleus, and he esul s a e seen o be
independen o he pa icula nuclea species, one has scaling
o he second kind. When bo h ypes o scaling beha io occu
one says ha he c oss sec ions exhibi supe scaling. In he
abo e-ci ed s udies he app op ia e educed c oss sec ions and
scaling a iables ha e been discussed in dep h.
One inds ha in he ele an ene gy ange in he egion
below he quasielas ic (QE) peak, usually called he scaling
egion, scaling o he second kind is ound o be excellen
and scaling o he i s kind o be qui e good. Abo e he peak
scaling o he second kind is good; howe e , scaling o he i s
kind is clea ly iola ed. The las occu s o well-unde s ood
easons, namely in ha egion one has p ocesses o he han
quasi- ee knockou o nucleons playing an impo an ole.
Speci ically, he mos ob ious eac ion mechanism is ha o
exci ing a nucleon in he nucleus o a del a, which subsequen ly
decays in o a nucleon and a pion. Because he elemen a y c oss
sec ion o ha p ocess is no he elas ic eN c oss sec ion used
in de ining he scaling unc ion in oduced abo e, and because
he scaling a iable used in he usual analysis assumes he
kinema ics o he elas ic p ocess N→N, a he han o N→
which would now be app op ia e, i is no su p ising ha
scale b eaking occu s. Addi ionally, meson exchange cu en
e ec s a e known o iola e he scaling beha io , al hough
om modeling in his high-ene gy egime [8–13] hei e ec s
appea no o be he dominan ones.
Wha was app ecia ed o he i s ime in ecen wo k [14]
is ha i is possible o pu sue an app oach whe e bo h he
QE p ocess is ac i e (wi h i s educed esponse and scaling
a iable) and he inelas ic p ocess in he  egion is also
inco po a ed (wi h i s co esponding educed esponse and
scaling a iable). We oughly e e o he egion o exci a ion
o ming a peak ha lies abo e he maximum o he QE
esponse as he “peak,” al hough i should be unde s ood ha
he modeling ac ually includes he ull inelas ic esponse on
a nucleon ( esonan plus non esonan ) o kinema ics whe e
he (1232) is dominan .1In Re . [14] i was shown ha
an excellen ep esen a ion o he o al inclusi e elec on
sca e ing c oss sec ion om he scaling egion up o he
peak o he  egion is a ained by in e ing he p ocedu e.
Using he wo scaling unc ions, one o QE sca e ing and
one o he - egion, along wi h he co esponding N- and
Z-weigh ed elas ic (eN →eN) and inelas ic (eN →e)
elec on sca e ing c oss sec ions one inds excellen ag eemen
wi h exis ing high-quali y da a o e a wide ange o kinema ics
and o a ious nuclea species. O conside able impo ance
o wha was discussed in he es o Re . [14] and will be
discussed in he p esen wo k is he ac ha he quali y o he
analysis equi es he phenomenological scaling unc ions o
be qui e asymme ic, wi h ela i ely long ails ex ending in he
di ec ion o highe ene gy loss (posi i e alues o he scaling
a iables). Such is no ypically he case wi h mos models,
hese almos always being mo e nea ly symme ical abou he
peak in he scaling unc ion (see, howe e , Re . [15] whe e in
a leas one case he co ec beha io has been ob ained in a
model). This ac cas s conside able doub on mos exis ing
models o high-ene gy sca e ing in he QE and  egimes i
high-quali y esul s (say be e han 25%) a e desi ed.
Ha ing me wi h success in ex ending he scaling and
supe scaling analyses om he scaling egion, h ough he QE
peak egion and in o he  egion, in Re . [14] he scaling ideas
we e in e ed: gi en he scaling unc ions one can jus as well
mul iply by he elemen a y cha ge-changing (CC) neu ino
1Fo s ill highe -lying exci a ions and DIS a di e en app oach mus
be aken (see, o example, Re . [6]).
0556-2813/2006/73(3)/035503(12)/$23.00 035503-1 ©2006 The Ame ican Physical Socie y
AMARO, BARBARO, CABALLERO, AND DONNELLY PHYSICAL REVIEW C 73, 035503 (2006)
c oss sec ions now o ob ain he co esponding CC neu ino
and an ineu ino c oss sec ions on nuclei o in e media e o
high ene gies in he same egion o exci a ion. O he ela ed
wo k is p esen ed in Re s. [15,16,20]. Gi en he abili y o his
scaling app oach o ep oduce he elec on sca e ing c oss
sec ions, in con as o mos di ec modeling ha ails in de ail
o do so, we belie e ha such p edic ions o he analogous CC
neu ino eac ions should be e y obus . Clea ly such esul s
a e o ele ance o ongoing s udies o neu ino eac ions and
neu ino oscilla ions in his in e media e-ene gy egime.
In he p esen s udy hese scaling and supe scaling ideas
a e ca ied a s ep u he o include neu al-cu en (NC)
neu ino and an ineu ino sca e ing c oss sec ions, in his
case o sca e ing om 12C. Speci ically, he goal is o ob ain
esul s using he same analysis as discussed abo e (and in
de ail in Re . [14]) o he eac ions 12C(ν,p)νX, 12C(¯ν,p)¯νX
in ol ing p o on knockou , and 12C(ν, n)νX, 12C(¯ν,n)¯νX
in ol ing neu on knockou in he QE egime, he - egime
being le o a subsequen s udy.
A new ea u e eme ges wi h such a goal in mind, howe e ,
and ha a ises om he ac ha when one has an inciden
lep on, a sca e ing wi h exchange o a γ,W±o Z0, and
de ec s he sca e ed lep on (i.e., a cha ged lep on), he -
channel exchange o he app op ia e boson is con olled. In
con as , when he sca e ed lep on is a neu ino o an ineu ino,
and he e o e no de ec ed, bu ins ead a knocked-ou nucleon
is de ec ed, i is he u-channel whose kinema ics a e con olled
(see also Re . [17] o discussions o his case). Acco dingly,
in he scaling analysis i is no ob ious ha he wo ypes o
p ocesses a e simply ela ed, and he e o e o apply he scaling
ideas o NC neu ino and an ineu ino sca e ing, in pa icula
o discussions o di e en ial c oss sec ions as in he p esen
wo k, we i s ha e o add ess he issue o how he - and
u-channels a e ela ed.
The a icle is o ganized he ollowing way: in Sec. II we
begin wi h a basic discussion o he - and u-channel kinema ics
in ol ed in he semilep onic elec oweak p ocesses o in e es
(Sec. II A) ollowed by a b ie summa y in Sec. II B o he c oss
sec ion o malism and he ideas o scaling when in e ela ing
- and u-channel p ocesses. To keep he discussions ela i ely
b ie in his subsec ion, he de elopmen o he single-nucleon
NC neu ino and an ineu ino c oss sec ions is placed in an
Appendix. Fo o ien a ion in Sec. II C he ela i is ic Fe mi
gas (RFG) model is in oked and i s supe scaling p ope ies
summa ized. Then in Sec. III ou esul s a e p esen ed and ou
conclusions a e ga he ed in Sec. IV.
II. GENERAL FORMALISM FOR
U-CHANNEL SCATTERING
We begin he gene al discussion o how - and u-channel
semilep onic eac ions a e in e ela ed wi h a summa y o he
ele an kinema ic a iables in he p oblem.
A. Kinema ics
We conside gene al semilep onic quasi ee sca e ing om
nuclei in he Bo n app oxima ion.
FIG. 1. Kinema ics o semilep onic nucleon knockou eac ions
in he one-boson-exchange app oxima ion.
We s a wi h one basic assump ion ha is usually p esumed
o be a good app oxima ion in he kinema ic egion whe e
quasielas ic sca e ing is dominan , namely ha he inclusi e
c oss sec ions a e well ep esen ed by he sum o he in eg a ed
semi-inclusi e p o on and neu on emission c oss sec ions.
In doing so we a e neglec ing p ocesses ha occu o he
same kinema ics, bu ha e no emi ed nucleon in he inal
s a e (pho on emission, deu e on emission, alpha emission,
cohe en pion p oduc ion, e c., bu wi hou an emi ed nu-
cleon). The p ocess o in e es (see Fig. 1) has a lep on o
ou -momen um Kµ=(, k) sca e ed o ano he lep on o
ou -momen um Kµ=(,k), exchanging a ec o boson
wi h ou -momen um Qµ=Kµ−Kµ. The lep on ene gies
a e =√m2+k2and =√m2+k2, wi h m(m) he
mass o he ini ial ( inal) lep on. Fo NC neu ino sca e ing
m=m=0 (assuming ze o-mass neu inos). No e ha no
assump ion such as he plane-wa e impulse app oxima ion
is being in oked a his s age.
In he labo a o y sys em he ini ial nucleus is in i s g ound
s a e wi h ou -momen um Pµ
A=(M0
A,0). The inal had onic
s a e co esponds o a nucleon (N=po n) wi h ou -
momen um Pµ
N=(EN,pN) and ene gy EN=√m2
N+p2
N
plus an unobse ed daugh e nucleus wi h ou -momen um
Pµ
B=(EB,pB). As usual in semilep onic eac ions we in-
oduce he missing momen um p≡−pBand he exci a ion
ene gy E≡EB−E0
B, wi h E0
B=√(M0
B)2+p2,M0
Bbeing
he g ound-s a e mass o he daugh e sys em ( o de ails see
Re s. [2–4]).
Fo NC neu ino sca e ing we assume ha he neu ino
beam momen um is speci ied and he ou going nucleon is
de ec ed. Hence pNand he angle θkpN(be ween kand pN)
a e gi en. No e ha he sca e ed lep on’s ou -momen um is
no speci ied, as would be he case o -channel sca e ing.
In analogy wi h he -channel case, we can de ine a u-channel
exchanged ou -momen um
Qµ≡Kµ−Pµ
N=(ω,q).(1)
The abo e equa ion yields
q=|q|=k2+p2
N−2kpNcos θkpN.(2)
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SUPERSCALING AND NEUTRAL CURRENT . . . PHYSICAL REVIEW C 73, 035503 (2006)
pNq

k
q 
p 
k
z
x
FIG. 2. Vec o s ela ing -channel and u-channel kinema ic a i-
ables.
Fo con enience in looking a he kinema ics one can use a
coo dina e sys em ha ing he zaxis along q, wi h kand pN
lying in he xz plane. The ec o s kand p=k−qlie in a
plane o ming an angle φwi h he xz plane de ined abo e (see
Fig. 2).
The exclusi e p ocess illus a ed in Fig. 1 is ully de e -
mined by six kinema ic a iables, which can be chosen o be
(k,pN,θ
kpN,p,E,φ). The u-channel inclusi e c oss sec ion
o (k,pN,θ
kpN) ixed is ob ained by in eg a ing o e he
allowed egion in he (p, E) plane and o e he azimu hal
angle, 0 ⩽φ⩽2π. Again e e ing o Fig. 2, one sees ha
a ixed u-channel sca e ing kinema ics (i.e., he iangula
egion bounded by k,pN, and q ixed) and o a gi en poin in
he (p, E) plane, his φin eg a ion co esponds o ha ing he
iangle bounded by p,k, and q ixed in size and shape bu
o a ing abou he zaxis, namely h ough he ull ange o he
azimu hal angle φ. This clea ly implies ha he -channel
momen um ans e q a ies and ha he usual azimu hal
angle φ( o a ions abou q) does no co e he ull ange
(0,2π). This has consequences ha a e discussed in mo e
de ail below.
To de e mine he in eg a ion egion in he (p, E)
plane we use ene gy conse a ion, ob aining he ollowing
exp ession:
E=M0
A+ω−m2+q2+p2+2qpcos θqp
+M0
B2+p2.(3)
Following he usual y-scaling analysis we can now examine he
a ious cu es E=E(p) ha esul when a ious choices a e
made o cos θqp. Le us call he cu es E
±(p) when cos θqp=
±1:
E
±(p)=M0
A+ω−m2+(q±p)2+M0
B2+p2.
(4)
F om hese we can p oceed o ind he in e sec ions o he
cu es wi h he axis E=0. This leads o de ini ions o a
scaling a iable yand a maximum missing momen um Y:
y≡1
W2
XM0
A+ω2
X−M0
B2W2
X−qX(5)
Y≡1
W2
XM0
A+ω2
X−M0
B2W2
X+qX,(6)
whe e
WX=M0
A+ω2−q2(7)
X=1
2W2
X+M0
B2−m2.(8)
No e ha hese a e new a iables and no simply ela ed o
he a iables yand Y ha come om he amilia y-scaling
analysis [2–4,7]. The allowed egion is hen de e mined: o
y<0 one has −y⩽p⩽Ywi h 0 ⩽E⩽E
−(p), whe eas o
y>0 one has o 0 ⩽p⩽y he ange E
+(p)⩽E⩽E
−(p)
and o y⩽p⩽Y he ange 0 ⩽E⩽E
−(p). When y=0 one
co e s he la ges ange in missing momen um a he minimal
missing ene gy and acco dingly somewhe e nea his poin he
inclusi e in eg al is expec ed o be a a maximum; namely his
kinema ic poin co esponds app oxima ely o he QE peak.
Conce ning he azimu hal in eg a ion, no e ha kinema ic
a iables en e ing he usual -channel [such as he momen um
and ene gy ans e (q,ω), he lep on sca e ing angle θl
be ween kand k, and he solid angle de ining he ou going
nucleon momen um (θqpN,φ
N)] all depend on cos φ—see
he abo e discussions o Fig. 2. Thus he in eg a ion o e φ
implies an in eg a ion o e he azimu hal angle φN; howe e ,
as φ a ies, he in eg a ion implied o e φNis no being
done a cons an (q,ω). Fu he mo e, he ange o e which he
implied φNin eg a ion occu s is no in gene al he ull ange.
This implies ha he symme y p ope ies o he esponses RK
canno be used in he case o u-channel inclusi e sca e ing o
elimina e some o he esponses (e.g., he TL and TT e ms),
as is he case o -channel inclusi e sca e ing.
B. C oss sec ions and scaling
Nex we u n o a discussion o he basic c oss sec ions and
scaling a iables in ol ed in he p esen s udy. As discussed
abo e we conside only semi-inclusi e nucleon knockou
eac ions in building up he inclusi e c oss sec ions. The usual
p ocedu e [17] is o s a wi h he plane wa e impulse app oxi-
ma ion (PWIA) o he (l,lN) c oss sec ion and in eg a e o e
all uncons ained kinema ic a iables. Final-s a e in e ac ions
a e hen p esumed o occu a e he p ima y elec oweak
in e ac ion wi h a nucleon in he nucleus and so, o ins ance, a
succession o (N,2N) s eps occu ing du ing he ime e olu-
ion o he high-ene gy emi ed nucleon as i p oceeds h ough
he nuclea medium can cause a edis ibu ion o s eng h in
he missing-ene gy, missing-momen um plane (see Re . [25]
o ecen wo k along hese lines). Such p ocesses end o
mo e s eng h om lowe missing ene gies o highe ones
and he eby p oduce an asymme y in he scaling unc ion,
skewing i o la ge alues o ene gy loss ωo , equi alen ly, in
he posi i e scaling a iable di ec ion. O he app oaches [15]
also yield an asymme ic scaling unc ion—in ag eemen
wi h expe imen —when s ong inal-s a e in e ac ions a e
inco po a ed, again ia a shi o s eng h o highe missing
ene gies.
In con as , in he p esen wo k whe e ou emphasis is
placed on in e ela ing a ious inclusi e semilep onic p o-
cesses, and no on de ailed modeling o he eac ion chain, we
ake as gi en he ull semi-inclusi e nucleon knockou c oss
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AMARO, BARBARO, CABALLERO, AND DONNELLY PHYSICAL REVIEW C 73, 035503 (2006)
sec ion (i.e., gi en by na u e) and p oceed o in eg a e o ob ain
inclusi e c oss sec ions. Clea ly his does no imply ha we
ha e a ull unde s anding o he o me , only ha asymp o ic
s a es may be used o accoun o all open channels and ha
i is no necessa y o accoun o he en i e sequence o s eps
ha yields hese s a es.
Fo -inclusi e sca e ing, whe e Qµ≡(ω,q) is cons an
and he inal lep on is de ec ed (as in usual inclusi e elec-
on sca e ing o in cha ge-changing neu ino eac ions),
he inclusi e c oss sec ion is calcula ed by in eg a ing he
semi-inclusi e c oss sec ion dσ/dkdkdNdpNo e he
ejec ed nucleon (and summing o e p o ons and neu ons),
whe eas o he u-inclusi e sca e ing we a e conside ing
he e, whe e Qµ≡(ω,q) is cons an and he inal nucleon is
de ec ed, one has o in eg a e o e he inal lep on. Tha is we
ha e
dσ
dkdk=dNdpN
dσ
dkdkdNdpN
(9)
dσ
dNdpN=dkdkdσ
dkdkdNdpN
(10)
o - and u-channel eac ions, espec i ely. These in eg als
can be ans o med in o in eg als in he (p, E) plane using he
ela ions
dNdpN=EN
p2
N1
qpdpdEdφN(11)
dkdk=
k21
qpdpdEdφ.(12)
This leads o he ollowing exp essions o he inclusi e c oss
sec ions, in he -channel
dσ
dkdk=2π
qD
pdp dE
×2π
0
dφN
2πEN
p2
Ndσ
dkdkdNdpN
(13)
and in he u-channel
dσ
dNdpN=2π
qDu
pdp dE
×2π
0
dφ
2π
k2dσ
dkdkdNdpN
,(14)
espec i ely. The abo e exp essions a e simply connec ed o
one o he by in e changing he inal lep on a iables wi h he
inal nucleon a iables, bu o he ac ha he in eg a ion
egions D and Duin he (p, E) plane a e di e en in he
wo cases. The -channel case is discussed in Re . [17],
whe eas he u-channel case is ea ed in he ollowing
sec ion.
To his poin we ha e made only ela i ely weak app oxi-
ma ions by assuming ha he c oss sec ions in he quasielas ic
egion a e domina ed by in eg als o e he semi-inclusi e
nucleon knockou c oss sec ions. Following Re . [17] we w i e
he la e in e ms o p oduc s o single-nucleon elec oweak
c oss sec ions mul iplied by wha may be called he educed
c oss sec ion:
dσ
dkdkdNdpN=1
(2π)2
1
2
1
2Eg4DV(Q2)2
×lµνwµν k2
2p2
N
2EN
×(q,ω,θkk,φ,p,E),(15)
whe e Eis he ene gy o he s uck nucleon, gis he s eng h
o he e mion- ec o boson coupling, and DV(Q2)=(Q2−
M2
V)−1is he ec o boson p opaga o , whe eas lµν and wµν
a e he usual lep onic and (single-nucleon) had onic enso s,
espec i ely. Clea ly o he se s o independen a iables may
be used as a gumen s o he educed c oss sec ion (see below).
Nex we make wo s onge app oxima ions. Fi s , we
assume ha he single-nucleon c oss sec ion a ies only slowly
wi h (p, E) and may be emo ed om he in eg als o e
pand E. This has been e i ied o -channel eac ions as
long as he semi-inclusi e c oss sec ions a e peaked a low
missing-ene gy and missing-momen um (see, o example,
Re . [2]). In pa icula , o he -inclusi e case he ec o boson
p opaga o can be ex ac ed om he in eg al, and he same
applies o he single-nucleon o m ac o s appea ing in wµν,as
hey only depend on Q2. As a consequence, in -channel case
one can e i y ha he (p, E) dependence o he single-nucleon
c oss sec ion is weak a cons an (ω,q) and he e o e i s mean
alue (namely in eg a ed o e φNand di ided by 2π) can be
emo ed om he in eg a ions in Eq. (13). The u-channel case
is mo e complica ed and is deal wi h below.
I we make his app oxima ion we a e le wi h
dσ
dkdk≃σ( )
snF(y,q) (16)
whe e
F(y,q)≡D
pdp dE
E(q,ω,θkk,φ,p,E) (17)
depends on he scaling a iable yand he momen um ans e
q[1–4,7]. No e ha he educed c oss sec ion occu ing
abo e would be he spec al unc ion S(p, E), namely depen-
den only on (p, E) we e he PWIA o be assumed; howe e ,
no such assump ion is being made he e.
Second, we assume ac o iza ion in he sense ha he
educed c oss sec ion appea ing abo e depends only weakly
on he momen um ans e q, his dependence being con ained
mos ly in he single-nucleon c oss sec ion. No e ha , o
ins ance, esidual dependence in on he scaling a iable
yis no pa o he ac o iza ion assump ion. Such dependence
would no lead o any scaling iola ion. This means ha
ac o iza ion is no equi alen o assuming dependence only on
(p, E) as in he PWIA. Clea ly missing he e, o ins ance, a e
p ocesses in ol ing meson-exchange cu en s [8–13] ha in
his sense do no ac o , as hei dependences on qa e clea ly
no he same as hose con ained in he single-nucleon c oss
sec ion ha has been di ided ou o de ine he educed c oss
sec ion. Howe e , ou pas s udies o supe scaling show ha ,
o high-ene gy inclusi e sca e ing a quasielas ic kinema ics,
he scaling beha io is qui e well espec ed, wi h pe haps 10%
o so le o be explained by e ec s such as hose om MEC
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SUPERSCALING AND NEUTRAL CURRENT . . . PHYSICAL REVIEW C 73, 035503 (2006)
ha should b eak he scaling. Indeed, e en wi h ela i ely
s ong inal-s a e in e ac ions one inds in some modeling [15]
ha he scaling is main ained, sugges ing ha he abo e
assump ion is alid, a leas in he egion o he QE peak.
We no e in passing ha he iola ions o scaling o he i s
kind, namely some esidual dependence on he momen um
ans e q, e en a he le el o he abo e equa ion can s em
om wo di e en sou ces: (1) he egion o in eg a ion D
depends on qand only o asymp o ically high qdoes i
app oach a q-independen o m, and (2) he educed c oss
sec ion may con ain some weak dependence on q. Indeed,
om he obse a ion ha he app oach o i s -kind scaling is
om abo e, i.e., he measu ed educed c oss sec ion dec eases
wi h qbe o e eaching he scaling domain (see Re . [2], o
example), i appea s ha (2) mus occu , and no jus (1), which
would imply an app oach om below, because he in eg a ion
egion inc eases wi h q.
A high ene gies, whe e he scaling idea wo ks and scaling
o i s kind is easonably good, we ind ha F(y,q)≃F(y)≡
F(y,∞) and is no a unc ion o q, in e ec alida ing he
ac o iza ion assump ion and he quali y o he app oxima ion
whe e a mean alue o he single-nucleon c oss sec ion is
emo ed om he in eg als. This was used in Re . [14] o
p edic he cha ge-changing (CC) neu ino c oss sec ion: we
le na u e sol e o us he in eg al in Eq. (17) o ob ain an
empi ical unc ion F(y) om elec on sca e ing o be used in
CC neu ino s udies.
In he u-inclusi e case he abo e ac o iza ion is no i ial,
because Q2 a ies wi hin he in eg a ion egion. Howe e , one
can again assume ha
dσ
dNdpN≃σ(u)
sn F(y,q),(18)
whe e
F(y,q)≡Du
pdp dE
E≃F(y),(19)
p o ided he e ec i e NC single-nucleon c oss sec ion
σ(u)
sn =1
32π
1
qp2
N
ENg42π
0
dφ
2π
×lµν(k,k)wµν (p,pN)DV(Q2)2(20)
is almos independen o (p, E) o cons an (k,pN,θ
kpN).
This seems indeed o be he case, as shown om nume ical
s udies p esen ed below in Sec. III. Then, as in Re . [14], he
empi ically de e mined scaling unc ion F(y) can be used o
p edic ealis ic NC c oss sec ions.
To be able o use he scaling unc ion ob ained om
analyses o inclusi e elec on sca e ing da a o p edic ions o
neu ino eac ion c oss sec ions one u he assump ion mus
be made, namely he domains o in eg a ion in he in eg als
abo e mus be he same o a leas e y simila . In he case
o CC neu ino eac ions his is clea ly he case excep a
e y low ene gies o he muon case whe e he kinema ic
dependence on he muon mass is impo an in de e mining
D . Fo NC neu ino eac ions he in eg a ion domain Du
di e s o some deg ee om he one ha en e s in elec on
sca e ing, namely D . In pa icula , when de e mining he
scaling unc ion F(y,q) wi h inpu om elec on sca e ing
ha yields F(y,q), clea ly he i s s ep is o use he la e
e alua ed a y=yand o wo k in he scaling egime whe e
qand qa e bo h la ge enough o make he egions in he
(p, E) plane ex end o high pand high E(see he a gumen s
o elec on sca e ing scaling summa ized, o ins ance, in
Re . [2]). Unde hese ci cums ances he egions deno ed D
and Dudi e signi ican ly only a la ge E(also a la ge p,
bu he e one belie es he semi-inclusi e c oss sec ions a e
negligible). Acco dingly, gi en ha he semi-inclusi e c oss
sec ions a e domina ed by hei beha io s a low Eand low
p, one expec s he esul s o he in eg a ions in he wo cases,
-channel and u-channel, o be e y simila , and hus he scaling
unc ions will be essen ially he same. We e his no o be
he case, hen i would be likely ha i s -kind scaling o
inclusi e elec on sca e ing would no occu , in con adic ion
wi h obse a ion.
A u he di e ence be ween he - and u-sca e ing cases
should be s essed. In bo h cases he single-nucleon c oss
sec ion can be exp essed in e ms o esponse unc ions,
as shown in he appendix. Howe e , as men ioned abo e,
o -inclusi e p ocesses he special symme y abou he q
di ec ion can be exploi ed o emo e he TL, TT, and TL
esponses a e pe o ming he φN-in eg a ion, which simply
yields a ac o 2π.In heu-channel, ins ead, he un es ic ed
in eg a ion o e φyields an e ec i e in eg a ion o e φN
which is no uni o m and does no in gene al co e all o
he in e al (0,2π). As a consequence he TL, TT, and TL
esponses do con ibu e. As shown la e , hei con ibu ion is
supp essed and only he TL con ibu ion is ele an o he
kinema ics o in e es in he p esen s udy.
C. RFG and supe scaling
In his sec ion we discuss he NC neu ino c oss sec ion in
he RFG model, which co esponds o he ollowing exci a ion
ene gy
ERFG(p)=m2
N+k2
F−m2
N+p2(21)
and spec al unc ion
SRFG(p, E)=3kF
4TF
θ(kF−p)δ[E−ERFG(p)],(22)
whe e kFis he Fe mi momen um and TF=k2
F+m2
N−mN
he Fe mi kine ic ene gy.
Because o he del a unc ion in Eq. (22) he in eg a ion
egion in he (p, E) plane simply educes o a line and he
lowe limi in he in eg al o e pis gi en by he in e cep
o he cu e ERFG(p) wi h E
−(p) when y<0. When y>0
i is gi en by he in e cep o ERFG(p) wi h E
+(p)[E
−(p)]
when E
±(0) <T
F[E
±(0) >T
F]. By sol ing hese equa ions
i is easily shown ha he minimum momen um equi ed o a
nucleon o pa icipa e in he eac ion is
pmin =y(u)
RFG(23)
whe e
y(u)
RFG =smN
τ[λτ2ρ2+τ−κτρ] (24)
035503-5

AMARO, BARBARO, CABALLERO, AND DONNELLY PHYSICAL REVIEW C 73, 035503 (2006)
FIG. 3. The a ious egions co espond o
alues o he a io R≡σ(u)
sn (p, E)/σ(u)
sn (p=
|y|,E=0) di e ing om 1 by a mos 1 (lowes
egion), 1−2,2−5,5−10, and mo e han 10%
(highes egion). Fo his igu e p o on knockou
has been assumed; he neu on knockou case is
simila and no shown. Fo b e i y, in his igu e
we le θs and o he angle θkpN.
is he RFG y-scaling a iable associa ed wi h u-sca e ing
[hence he index (u) o dis inguish i om he usual -channel
a iable]. Mo eo e we ha e in oduced he dimensionless
kinema ic quan i ies κ≡q/2mN,λ
≡ω/2mN,τ=κ2−
λ2and de ined ρ≡1−1
4τ(1 −m2/m2
N). The sign sis
s≡sgn 1
τ[λτ2ρ2+τ−κτρ].(25)
As in elec on sca e ing, i is con enien o in oduce a
dimensionless scaling a iable
ψ(u)
RFG =smN
TF




1+y(u)
RFG
mN2
−1


1/2
,(26)
ep esen ing he minimum kine ic ene gy o he nucleons
pa icipa ing in he eac ion. By placing he spec al unc ion
o Eq. (22) in Eq. (19) one immedia ely inds he RFG scaling
unc ion
FRFGψ(u)
RFG=3kF
TFEF
Emin
dE dEδ(E−ERFG)
=3
4kF1−ψ(u)2
RFGθ1−ψ(u)2
RFG.(27)
P o iding he single-nucleon c oss sec ion is smoo hly
a ying wi hin he (p, E) in eg a ion egion, he di e en ial
RFG c oss sec ion can be ac o ized as shown in Eq. (18)
wi h he scaling unc ion gi en by Eq. (27). Mo e ealis ic
p edic ions can be gi en by using, ins ead o he RFG scaling
unc ion, he empi ical scaling unc ion as de e mined om QE
elec on sca e ing, as al eady done in Re . [14] o cha ged
cu en eac ions. These a e discussed in he nex sec ion.
III. RESULTS
Be o e p esen ing ou p edic ions o he c oss sec ion,
we es he alidi y o he scaling app oach in he u-channel.
To his end we analyze how he e ec i e NC single-nucleon
c oss sec ion σ(u)
sn gi en in Eq. (20) depends on he missing
momen um pand exci a ion ene gy E o selec ed alues o
he kinema ical a iables (k, EN,θ
kpN). To p oceed, we assume
he p o on knockou case and di ide σ(u)
sn e alua ed in he
whole (p, E)-plane by i s alue co esponding o p=|y|and
E=0. In wha ollows we use he cc2 o -shell p esc ip ion
o he nucleon cu en and he H¨
ohle pa ame iza ion o
he single-nucleon o m ac o s [18], igno ing he s angeness
con en o he nucleon, unless speci ied o he wise.
The esul s a e gi en in Fig. 3 in e ms o di e en shadings
ep esen ing he egions whe e his a io di e s om uni y by
a mos 1, 1–2, 2–5, 5–10% and mo e han 10%, espec i ely,
as indica ed in he op igh panel. The six g aphs co espond
o wo alues o he sca e ing angle θkpN:20
0( op panels) and
600(bo om panels). In each case, he ou going p o on kine ic
ene gies ha e been selec ed o co espond o he egions below,
abo e and close o he peak o he di e en ial c oss sec ion.
Al hough no shown he e, he esul s o neu on knockou a e
e y simila o hose o p o on knockou .
The esul s in Fig. 3 illus a e he alidi y o he scaling
app oach. Only o e y la ge alues o he exci a ion ene gy
does he e ec i e NC single-nucleon c oss sec ion depend sig-
ni ican ly on (p, E). In ac , es ic ing ou sel es o exci a ion
ene gies below wice he maximum alue o he RFG model,
E≃50 MeV, he dispe sion p esen ed by he a io is a mos
∼5–10%.
This ou come is also in acco dance wi h he esul s
p esen ed in Figs. 4 (p o on case) and 5 (neu on case). He e
we show he neu al cu en neu ino (uppe panels) and
an ineu ino (lowe panels) double di e en ial c oss sec ions
o sca e ing a 1 GeV om 12C as a unc ion o he ejec ed
p o on o neu on kine ic ene gy. The sca e ing angles ha e
been ixed as in he p e ious igu e.
Beginning wi h he RFG model, as in pas wo k he Fe mi
momen um o 12C is aken o be kF=228 MeV/cand esul s
a e gi en bo h using he ull RFG model (sho -dashed cu es)
and making use o he ac o iza ion app oach assumed in
Eq. (18) wi h he u-channel NC single-nucleon c oss sec ion
e alua ed a p=yRFG and E=ERFG (solid lines). One sees
ha he wo se s o esul s almos coincide in he whole TN
egion whe e he RFG c oss sec ion is de ined, indica ing ha
he scaling a gumen wo ks e y well.
Hence we may use he phenomenological scaling unc ion
ex ac ed om (e, e) da a, as was done in ou p e ious CC
neu ino eac ion analysis (Re . [14]), o p edic NC neu ino-
nucleus sca e ing c oss sec ions. These a e also plo ed in
Figs. 4 and 5 as long-dashed lines: hey a e seen o be lowe by
035503-6
SUPERSCALING AND NEUTRAL CURRENT . . . PHYSICAL REVIEW C 73, 035503 (2006)
FIG. 4. (Colo online) Quasielas ic di e en ial c oss sec ion o neu al cu en neu ino and an ineu ino sca e ing a 1 GeV om 12C
o p o on knockou ob ained using he RFG (sho -dashed), he ac o ized app oach wi h he RFG scaling unc ion (solid), and he empi ical
scaling unc ion (long-dashed). Fo b e i y, in his and in he ollowing igu es we le θNs and o he angle θkpN.
abou 25% a he peak han he RFG esul s, an e ec simila o
wha was ound in Re . [14] o he cha ge-changing p ocesses.
Mo eo e , he empi ical scaling unc ion leads o c oss sec ions
ex ending bo h below and abo e he kinema ical egion whe e
he RFG is de ined. In pa icula , he long ail displayed o
low TN alues (co esponding o posi i e alues o he scaling
a iable ψ) is no ewo hy. This ail a ises no only om he
asymme ic shape o he phenomenological scaling unc ion
bu also om he e ec i e NC single-nucleon c oss sec ion,
which inc eases signi ican ly o low TN alues.
On compa ing Figs. 4 and 5 we see ha he shapes o he
c oss sec ions o p o on and neu on knockou a e e y simila ,
al hough he magni udes a e somewha di e en : excep o
an ineu inos a o wa d angles, whe e he c oss sec ions a e
e y small, he neu on knockou esul s a e 30–50% highe
han o p o on knockou . This occu s because (in absence o
FIG. 5. (Colo online) As o he p e ious igu e, bu now showing he neu on knockou case.
035503-7
AMARO, BARBARO, CABALLERO, AND DONNELLY PHYSICAL REVIEW C 73, 035503 (2006)
FIG. 6. (Colo online) The sepa a e con ibu ions o he esponse unc ions o Eqs. (A15)–(A17) o he RFG neu ino c oss sec ion:
L(sho -dashed), T(solid), T(dashed), TT (do ed), TL(do -dashed), TL
(double-dashed). The uppe panels a e o p o on knockou and
he lowe o neu on knockou .
s angeness) bo h he ec o and he axial- ec o con ibu ions
a e la ge o neu ons han o p o ons, and hey sum up.
In pa icula , he AA piece is he same o pand n, because

GAp =−
GAn [see Eq. (A33)]. Howe e , om Eqs. (A28),
(A29), (A34), and (A35) one has ha 
GEp ≃−GEn and

GEn ≃−GEp . Hence, when compa ed wi h elec omagne ic
in e ac ions, he oles o p o ons and neu ons a e e e sed in
he weak neu al sec o , so ha |
GEn||
GEp |. Simila ly,
om Eqs. (A30), (A31), (A36), and (A37) one inds ha
|
GMn|>|
GMp|. Fo an ineu inos hings a e mo e delica e,
because he VA esponse has he opposi e sign. Fo ins ance,
o neu on knockou a θn=200 he sum VV +AA almos
exac ly cancels he in e e ence, explaining why he o wa d
angle ¯νneu on c oss sec ion is so small.
In Fig. 6 he con ibu ions o he sepa a e esponses o he
o al RFG c oss sec ion a e displayed. Clea ly he dominan
con ibu ions a ise om he RTand RT esponses, and in he
case o neu on knockou , om RLa low alues o he kine ic
ene gy. No e, howe e , ha al hough no dominan (see he
discussions in he appendix) he RTL esponse does p o ide
an impo an con ibu ion a backwa d angles. In pa icula ,
because i is nega i e a low kine ic ene gies and posi i e a
high, i skews he o e all c oss sec ion o highe alues o Tpo
Tn. Such an e ec is, as discussed abo e, absen o -channel
sca e ing whe e he TL,TL
, and TT esponses a e ze o.
Finally, in Figs. 7 (p o on knockou case) and 8 (neu on
knockou case) we explo e he dependence o he c oss sec ion
on he s angeness con en o he nucleon (Re s. [19,21,22]).
We compa e he esul s ob ained om he phenomenological
supe scaling unc ion in a si ua ion whe e no s angeness
is assumed (solid line) wi h he ones ob ained including
s angeness in he magne ic (long-dashed) and axial- ec o
(do ed) o m ac o s, using o µs=G(s)
M(0) a ep esen a i e
alue ex ac ed om he ecen wo ld s udies o PV elec on
sca e ing [23] and aking gs
A=G(s)
A(0) o be −0.2 [24]. The
e ec s om inclusion o elec ic s angeness a e no shown
he e, because G(s)
Ehas almos no in luence on he ull c oss
sec ions.
S a ing wi h he p o on knockou esul s in Fig. 7, we see
ha o he νcase magne ic s angeness ends o dec ease
he c oss sec ion, whe eas o ¯νi has he opposi e e ec
( he o wa d-angle ¯νc oss sec ions a e a he small and no
conside ed in his discussion). Fo bo h νand ¯ν he axial
s ange con ibu ion ends o inc ease he c oss sec ion, and
so he ne e ec o inco po a ing bo h ypes o s angeness
con en is ela i ely la ge in he ¯νcase han in he νcase.
Howe e , o he neu on knockou esul s shown in Fig. 8
he si ua ion is somewha di e en : o ν he oles o magne ic
and axial s angeness a e e e sed om wha is seen o p o on
knockou , an e ec ha is easily unde s ood by examining he
sign changes ha occu in going om p o ons o neu ons
(see appendix). Speci ically, GMp and GMn a e opposi e in
sign, whe eas, being isoscala , G(s)
Mis he same o p o ons
and neu ons. Simila ly, being isoscala G(s)
Adoes no change
sign in going om p o ons o neu ons, whe eas, being
iso ec o , G(3)
Adoes. The ¯νcase is anomalous: in his case
he in e e ence VA esponse ends o cancel he VV +AA
con ibu ions. Acco dingly, o neu on knockou including
magne ic s angeness, which inc eases bo h he VV and he VA
esponses, has li le ne e ec on he c oss sec ions, because he
wo e ec s cancel ou .
035503-8
SUPERSCALING AND NEUTRAL CURRENT . . . PHYSICAL REVIEW C 73, 035503 (2006)
FIG. 7. (Colo online) E ec s o s angeness and adia i e co ec ions in neu ino and an ineu ino c oss sec ions: no s angeness (solid),
µs=0.55 (dashed), gs
A=−0.2 (do ed), and all he abo e e ec s included (do -dashed). The case o p o on knockou is assumed.
FIG. 8. (Colo online) As o he p e ious igu e, bu now o neu on knockou .
035503-9