PHYSICAL REVIEW C 81, 055502 (2010)
Scaling unc ion and nucleon momen um dis ibu ion
J. A. Caballe o,1,*M. B. Ba ba o,2A. N. An ono ,3M. V. I ano ,3and T. W. Donnelly4
1Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Uni e sidad de Se illa, ES-41080 Se illa, 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
3Ins i u e o Nuclea Resea ch and Nuclea Ene gy, Bulga ian Academy o Sciences, BG-1784 So ia, Bulga ia
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 Ma ch 2010; published 18 May 2010)
Scaling s udies o inclusi e quasielas ic elec on sca e ing eac ions ha e been used in he pas as a basic ool
o ob ain in o ma ion on he nucleon momen um dis ibu ion in nuclei. Howe e , he connec ion be ween he
scaling unc ion, ex ac ed om he analysis o c oss-sec ion da a, and he spec al unc ion only exis s assuming
e y es ic ed app oxima ions. We e isi he basic exp essions in ol ed in scaling s udies and how hey can
be linked o he nucleon momen um dis ibu ion. In pa icula , he analysis applied in he pas o he so-called
scaling egion, ha is, nega i e alues o he scaling a iable y, is ex ended he e o posi i e y, as a “uni e sal”
supe scaling unc ion has been ex ac ed om he analysis o he sepa a ed longi udinal da a. This leads o esul s
ha clea ly di e om hose based solely on he nega i e-yscaling egion, p o iding new in o ma ion on how
he ene gy and momen um a e dis ibu ed in he spec al unc ion.
DOI: 10.1103/PhysRe C.81.055502 PACS numbe (s): 25.30.Fj, 21.60.Cs, 24.10.J , 21.10.F
I. INTRODUCTION: BASIC ASPECTS OF SCALING
Scaling s udies o inclusi e quasielas ic (QE) elec on-
nucleus sca e ing ha e la gely been conside ed o p o ide
a powe ul ool o ex ac ing he momen um dis ibu ion o
nucleons inside nuclei [1–7]. Such analyses ha e been applied
o ew-body sys ems, complex nuclei, and nuclea ma e , wi h
an impo an e o de o ed o es ima ing binding co ec ions
and, in pa icula , he high-momen um componen s o he
nucleon momen um dis ibu ion ha a e go e ned by sho -
ange co ela ions [8,9]. Howe e , cau ion should be bo ne
in mind o he conclusions eached, as a close ela ionship
be ween he momen um dis ibu ion and he scaling unc ion
only eme ges a e some app oxima ions a e made. These
a e linked no only o he gene al desc ip ion o he elec on
sca e ing eac ion mechanism, bu also o he in eg a ion limi s
in ol ed and he beha io o he spec al unc ion [1].
The phenomenon o yscaling eme ges om he analysis o
QE (e,e) eac ions. The scaling unc ion, de ined as he QE
(e,e) di e en ial c oss sec ion di ided by an app op ia e ac o
in ol ing he single-nucleon c oss sec ion [1,10–12], is shown
o depend only on a single a iable, y, gi en as a pa icula
combina ion o he wo independen a iables in he p ocess,
namely, he ene gy and momen um ans e s, ωand q.In he
QE domain and o alues o ωand qla ge enough, he basic
mechanism in (e,e) eac ions on nuclei co esponds o elas ic
sca e ing om indi idual nucleons in he nuclea medium
wi h “quasi ee” ejec ion o a nucleon om he nuclea sys em.
This implies ha he inclusi e (e,e) c oss sec ion is mainly
cons uc ed om he exclusi e (e,eN) p ocess, including he
con ibu ion o all nucleons in he a ge and in eg a ing o e
all (unobse ed) ejec ed nucleon a iables. In o he wo ds, QE
sca e ing o a nucleus is simply desc ibed as an incohe en
*[email p o ec ed]
sum o single-nucleon sca e ing p ocesses. This app oach,
which cons i u es he basis o he impulse app oxima ion
(IA), al hough being an o e simpli ied desc ip ion o (e,e)
eac ions, has demons a ed i s alidi y unde app op ia e kine-
ma ic condi ions. Mechanisms beyond he IA (co ela ions,
meson exchange cu en s, esca e ing p ocesses, e c.) may
play a signi ican ole in elec on sca e ing and, hence, may
lead o non-negligible scaling iola ions.
The IA p o ides an in ui i e explana ion o how he scaling
beha io eme ges om he analysis o da a. In his case he
QE (e,e) c oss sec ion is gi en by
dσ
dd(e,e)=
A

i=1
(ω,q)
pdpdEdφNiENi
qp2
Ni
×dσ
dddpNidNi(e,eNi)
,(1)
whe e he sum ex ends o all nucleons in he a ge and {,
}
e e o he sca e ed elec on a iables. The in eg a ion o e
he ejec ed (unobse ed) nucleon a iables {pNi,E
Ni,
Ni}
has been exp essed in e ms o he exci a ion ene gy Eo
he esidual nucleus and he missing momen um p.The
signi icance o hese a iables as well as he kinema ically
allowed in eg a ion egion deno ed (ω,q)isdiscussedin
de ail in nex sec ion.
Wi hin he IA, e alua ion o (e,eNi) c oss sec ions o
bo h p o on and neu on knockou de e mines he inclusi e
QE c oss sec ion. The s udy o exclusi e (e,eN) eac ions
has been p esen ed in p e ious wo k [13–18], ocusing on
di e en aspec s o he p oblem: inal-s a e in e ac ions (FSIs),
ela i i y, co ela ions, e c. Al hough such ing edien s ha e
been p o en o be essen ial o i expe imen al (e,eN) c oss
sec ions, in wha ollows we es ic ou a en ion o he
plane-wa e IA (PWIA), whe e he knocked-ou nucleon has
no in e ac ion wi h he esidual nucleus. Being he simples
0556-2813/2010/81(5)/055502(12) 055502-1 ©2010 The Ame ican Physical Socie y
CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010)
app oach o (e,eN) p ocesses, PWIA e ains impo an el-
a i is ic e ec s ha a e essen ial in desc ibing eac ions a
high qand ω. Mo eo e , he (e,eN) di e en ial c oss sec ion
in PWIA ac o izes in wo basic e ms: he elec on-nucleon
c oss sec ion o a mo ing, o -shell nucleon and he spec al
unc ion ha gi es he combined p obabili y o inding a
nucleon o ce ain momen um and ene gy in he nucleus
[16–18]. In gene al we can w i e
dσ
dddpNdNPWIA
(e,eN)=KσeN (q,ω;p, E,φ
N)S(p, E),
(2)
wi h Ka kinema ical ac o [19] and whe e pis he missing
momen um and E he exci a ion ene gy, essen ially he missing
ene gy minus he sepa a ion ene gy. I is impo an o poin
ou ha he ac o iza ion p ope y shown in Eq. (2) no longe
pe sis s i dynamical ela i is ic e ec s in he bound nucleons
a e inco po a ed, ha is, e ec s om he lowe componen s
in he ela i is ic wa e unc ions, e en in he plane-wa e
limi [20,21]. No e ha bo h he eN c oss sec ion and he
spec al unc ion depend on he wo in eg a ion a iables in
Eq. (1), pand E. To show how he scaling unc ion eme ges
om PWIA, u he assump ions a e needed. Fi s , he spec al
unc ion is assumed o be isospin independen , and second,
σeN is assumed o ha e a e y mild dependence on he missing
momen um and exci a ion ene gy, which is suppo ed by he
mos commonly used o -shell c oss sec ions [1]. Hence he
eN c oss sec ion can be e alua ed a ixed alues o pand E:
ypically he di e en ial c oss sec ion o inclusi e QE (e,e)
p ocesses is w i en in he o m
dσ
dd(e,e)∼
=σe(q,ω;p=|y|,E=0) ·F(q,ω),(3)
whe e he single-nucleon c oss sec ion is e alua ed a he
special kinema ics p=|y|(wi h y he scaling a iable; see
he nex sec ion) and E=0 ( he esidual nucleus in i s g ound
s a e). This co esponds o he lowes alue o he missing
momen um occu ing when E=0. The e m σe e e s o
he azimu hal-angle-a e aged single-nucleon c oss sec ion
and i also inco po a es he kinema ical ac o Kin Eq. (2)
and he con ibu ion o all nucleons in he a ge , ha is,
σe≡KA
i=1dφNiσeNi/2π.
The unc ion F(q,ω)inEq.(3) is known as he scaling
unc ion and i is gi en in PWIA in e ms o he spec al
unc ion:
F(q,ω)=2π
(q,ω)
pdpdES(p, E).(4)
A de ailed s udy o he scaling unc ion and i s connec ion wi h
he momen um dis ibu ion is p esen ed in he nex sec ion.
Howe e , le us s a by poin ing ou some gene al in e es ing
ea u es o his basic esul . Fi s , only in he case in which
i was possible o ex end he kinema ically allowed egion
(q,ω) o in ini y in he exci a ion ene gy plane, ha is,
Emax →∞, would he scaling unc ion be di ec ly linked o
he ue momen um dis ibu ion o he A-nuclea sys em:
n(p)≡∞
0
dES(p, E).(5)
Second, guided by he PWIA esul in Eq. (3), an expe i-
men al scaling unc ion can also be de ined by di iding he
expe imen al QE (e,e) c oss sec ion by he single-nucleon
unc ion, σe. A high enough alues o he momen um ans e
q, he unc ion Fexp(q,ω) has been shown o sa is y scaling
in he egion below he QE peak; ha is, Fexp becomes only a
unc ion o he scaling a iable y(see Re s. [1,11,12], and [22]
o de ails). No e ha Eq. (4) does no apply o Fexp(q,ω),
which inco po a es ing edien s no included in he simple
PWIA app oach: FSIs, meson exchange cu en s, esca e ing
p ocesses, e c. The con ibu ion o hese e ec s and hei
impac on he scaling phenomenon depend on he kinema ical
egion explo ed, leading, in pa icula , o a signi ican scaling
b eaking in he egion abo e he QE peak.
Fu he mo e, based on he analysis pe o med wi h he
ela i is ic Fe mi gas (RFG) model, and making use o
he sepa a e longi udinal (L) and ans e se (T)(e,e) da a,
expe imen al supe scaling unc ions ha e been in oduced:
exp(q,ω)≡kFFexp(q,ω),(6)
L(T)
exp (q,ω)≡kFFL(T)
exp (q,ω),(7)
whe e kFis he Fe mi momen um. In pa icula , he L
esponse is hough o ha e e y li le con ibu ion om meson
p oduc ion and om meson-exchange cu en s and hus should
be he place whe e he unde lying nuclea dynamics can
cleanly be esol ed. I has been shown o supe scale; ha
is, he unc ion L
exp shows only a e y mild dependence
on he momen um ans e q( i s -kind scaling) and he
nuclea sys em conside ed (second-kind scaling). This has led
o he in oduc ion o a uni e sal expe imen al supe scaling
unc ion ha cons i u es a s ong cons ain o any heo e ical
model desc ibing QE elec on sca e ing. No only should he
supe scaling beha io be ul illed, bu also he speci ic shape
o L
exp mus be ep oduced. This subjec has been s udied
in de ail in p e ious wo k showing he impo ance o FSI
and ela i i y [23–27], and hose s udies clea ly show ha
any conclusion abou he momen um dis ibu ion based on
Eq. (4) should be made wi h cau ion. Being awa e o his, i is
illus a i e, howe e , o analyze in de ail he basic app oaches
on which he “link” be ween he momen um dis ibu ion and
he scaling (supe scaling) unc ion is based. Mo eo e , he
usual analysis, es ic ed in he pas o he egion below he
QE peak, is now ex ended o he egion abo e he peak, as
he supe scaling unc ion L
exp is de ined o bo h nega i e and
posi i e alues o he scaling a iable (see discussion in he
nex sec ion).
II. THE SCALING FUNCTION
As al eady shown, in PWIA he scaling unc ion can be
exp essed as an in eg al o he spec al unc ion Sin he (p, E)
plane [Eq. (4)], wi h p he s uck nucleon’s momen um,
E(p)≡M∗2
B+p2−M02
B+p2⩾0(8)
he exci a ion ene gy o he ecoiling sys em B,M0
B he
g ound-s a e mass o he esidual nucleus, and M∗
B he gene al
in a ian mass o he daugh e inal s a e. The in eg a ion in
Eq. (4) is ex ended o he kinema ically allowed egion in
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SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010)
FIG. 1. (Colo online) Exci a ion ene gy co esponding o nega-
i e (le ) and posi i e ( igh ) alues o y.
he (p, E) plane a ixed alues o he momen um and ene gy
ans e , (q,ω). This is ep esen ed by (q,ω). The gene al
kinema ics co esponding o QE (e,e) p ocesses leads o he
E-in eg a ion ange [1,10]
max{0,E+}⩽E⩽E−,(9)
whe e
E±(p;q,ω)=M0
A+ω
−(q±p)2+m2
N+M02
B+p2(10)
and whe e M0
Ais he a ge nuclea mass and mN he nucleon
mass.
The in e cep s be ween he cu e E−and he paxis a e
deno ed −yand Y; ha is, E−(−y;q,ω)=E−(Y;q,ω)=0.
The in eg a ion egion (q,ω) is shown in Fig. 1 o ixed
alues o he ans e ed ene gy and momen um o ω<ω
QE
(le ) and ω>ω
QE ( igh ), wi h ωQE he ene gy a which he
quasielas ic peak (QEP) occu s. In he egion below he QEP,
yis nega i e and p=−y ep esen s he minimum alue o
he s uck nucleon’s momen um. Abo e he QEP yis posi i e
and he cu e E+cu s he in eg a ion egion when p<y.
In e ms o he independen a iables qand ω, he in e cep s
±yand Ya e gi en by
y(q,ω)=M0
A+ω2−M02
BW2−qW2,(11)
Y(q,ω)=M0
A+ω2−M02
BW2+qW2,(12)
wi h W≡√(M0
A+ω)2−q2 he cen e -o -mass ene gy and
≡(M02
B−m2
N+W2)/2. Then he scaling unc ion in
Eq. (4) can be ecas as ollows:
1
2πF(q,y)=Y(q,y)
−y
pdp E−(p;q,y)
0
dES(p, E)i y<0,
(13)
1
2πF(q,y)=y
0
pdp E−(p;q,y)
E+(p;q,y)
dES(p, E)+Y(q,y)
y
pdp
×E−(p;q,y)
0
dES(p, E)i y>0,(14)
o nega i e and posi i e alues o y, espec i ely. The analysis
p esen ed in he p e ious wo k has been es ic ed o he
nega i e-y egion, ha is, below he QEP, as his is he egion
whe e c oss-sec ion da a ul ill y-scaling p ope ies. The
unc ion Fexp does no scale o posi i e alues o ybecause
o he signi ican scaling iola ions in oduced by e ec s
beyond he IA, namely, inelas ic p ocesses and con ibu ions
om meson-exchange cu en s. Howe e , hese con ibu ions
mos ly eside in he pu ely ans e se esponse and a e
negligible in he Lchannel. The “uni e sal” supe scaling
unc ion ex ac ed om he analysis o he sepa a ed Lda a,
and de ined o bo h nega i e and posi i e alues o he
scaling a iable, explains ou in e es in ex ending he s udy
o he egion abo e he QEP. This s a egy, which o ces
us o employ he supe scaling unc ion L
exp o de e mine
FL
exp = L
exp/kFins ead o he usual y-scaling unc ion Fexp,
can lead o signi ican e ec s conce ning he momen um and
ene gy dis ibu ion in he spec al unc ion, as discussed he e.
In he p eceding exp essions we ha e chosen (p, E;q,y)as
independen a iables. In e ms o hese we can also exp ess
he ene gy ans e ,
ω(q,y)=(q+y)2+m2
N+M02
B+y2−M0
A,(15)
he limi s o he exci a ion ene gy,
E±(p;q,y)=(q+y)2+m2
N−(q±p)2+m2
N
+M02
B+y2−M02
B+p2,(16)
and he uppe limi o p,
Y(q,y)=
M02
B(2q+y)+2(q+y)M02
B+y2(q+y)2+m2
N+y2(q+y)2+m2
N
M02
B+2M02
B+y2(q+y)2+m2
N+2y(q+y)+m2
N
.(17)
In he he modynamic limi M0
B→∞, we ge
E±(p;q,y)→(q+y)2+m2
N−(q±p)2+m2
N
≡Eq+y−Eq±p,(18)
Y(q,y)→2q+y, (19)
whe e we ha e in oduced he nucleon ene gies Ek≡
√k2+m2
N. Mo eo e , no e ha in he limi o a e y
high momen um ans e , ha is, q|y|and qmN, he
p eceding limi ing alues educe o Y→2qand E±→y∓p.
Following p e ious a gumen s p esen ed in Re s. [1] and
[4], i is ins uc i e o spli he spec al unc ion in o wo
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CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010)
e ms, co esponding o ze o and ini e exci a ion ene gy,
espec i ely:
S(p, E)=n0(p)δ(E)+S1(p, E),(20)
wi h S1(p, E=0) =0, which, inse ed in o Eqs. (13) and (14),
yields
1
2πF(q,y < 0) =Y(q,y)
−y
pdp n0(p)+Y(q,y)
−y
pdp
×E−(p;q,y)
0
dES1(p, E),(21)
1
2πF(q,y > 0) =Y(q,y)
y
pdpn0(p)
+y
0
pdp E−(p;q,y)
E+(p;q,y)
dE
+Y(q,y)
y
pdp E−(p;q,y)
0
dES1(p, E).
(22)
To analyze how he scaling unc ion and he nucleon mo-
men um dis ibu ion a e connec ed, we p oceed by e alua ing
he de i a i es o he scaling unc ion Fwi h espec o yand
q. Making use o Leibniz’s o mula and choosing (p;q,y)as
he h ee emaining independen a iables, a e some algeb a
we inally ge he ollowing esul s.
A. Nega i e- y egion
1
2π
∂F
∂y =Yn
0(Y)∂Y
∂y −yn
0(−y)
+Y
−y
pdp ∂E−
∂y S1(p, E−),(23)
1
2π
∂F
∂q =Yn
0(Y)∂Y
∂q +Y
−y
pdp ∂E−
∂q S1(p, E−).
(24)
Making use o he limi s in Eq. (16) and assuming he
esidual mass M0
B o be much la ge han he momen a,
|y|,p,q,wesimplyha e
∂E−
∂y ≃q+y
Eq+y
,∂E−
∂q ≃q+y
Eq+y−q−p
Eq−p
.(25)
Likewise, he de i a i es o Y educe o ∂Y/∂y ≃1 and
∂Y/∂q ≃2.
In oducing hese esul s in he gene al exp essions in
Eqs. (23) and (24), we ge
1
2π
∂F
∂y =Yn
0(Y)−yn
0(−y)
+q+y
Eq+yY
−y
pdpS
1(p, E−),(26)
1
2π
∂F
∂q =2Yn
0(Y)+Y
−y
pdp
×q+y
Eq+y−q−p
Eq−pS1(p, E−),(27)
wi h E−and Ygi en in he he modynamic limi by Eqs. (18)
and (19). No e ha he exci ed-s a e con ibu ion in he
spec al unc ion, ha is, S1, is e alua ed a ene gies along he
cu e E−.
Fo qsu icien ly la ge, q−y, he uppe limi Ycan be
sa ely aken o ∞, and as limY→∞ Yn
0(Y)=0, he exp essions
o he de i a i es simpli y o
1
2π
∂F
∂y =−yn
0(−y)+q+y
Eq+y∞
−y
pdpS
1(p, E−),(28)
1
2π
∂F
∂q =∞
−y
pdpq+y
Eq+y−q−p
Eq−pS1(p, E−).(29)
I we u he assume ha S1is small o la ge alues o p,
so ha he main con ibu ion o he in eg al Eq. (29) comes
om p≃−y, hen we ge
lim
q→∞
∂F
∂q =0,(30)
namely, scaling o he i s kind ( he scaling unc ion Floses
i s dependence on q).
We also obse e ha , because a a ixed alue o y
he in eg a ion egion in Eq. (27) inc eases wi h qand he
in eg and is a posi i e unc ion, he asymp o ic alue F(y)
is eached om below (i.e., mono onically inc easing as a
unc ion o q) in any PWIA app oach, in con as wi h wha
expe imen al da a seem o indica e [11,12,22]. This is clea ly
illus a ed in Fig. 2, whe e he in eg a ion egion is shown o
di e en alues o he momen um ans e a ixed y, and i
is also consis en wi h esul s shown in Figs. 3(a) and 4(a).
In Fig. 3we p esen he supe scaling unc ion (ψ) e alua ed
wi hin he amewo k o he ela i is ic PWIA (RPWIA) (see
Re s. [24] and [25] o de ails) o di e en q alues and plo ed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
p [GeV/c]
0.1
0.2
0.3
0.4
ε [GeV]
q=0.5 GeV/c
q→∞
q=0.7 GeV/c
q=1.0 GeV/c
-y
FIG. 2. (Colo online) In eg a ion egion in he (E,p)plane
o y=−0.1 GeV/c and 12C as he a ge selec ed. Each cu e
co esponds o E− o a di e en momen um ans e .
055502-4
SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010)
-0.7-0.6 -0.5-0.4 -0.3-0.2 -0.1 0
ψ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(ψ)
q=0.5 GeV/c
q=0.6
q=0.7
q=0.8
q=0.9
q=1.0
0 0.2 0.4 0.6 0.8 11.2
ψ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) (b)
FIG. 3. (Colo online) Supe scaling unc ion (ψ) o nega i e
(a) and posi i e (b) alues o he scaling a iable ψ. Resul s
co espond o 12C(e,e) e alua ed in RPWIA o di e en momen um
ans e s.
agains he supe scaling a iable ψin he nega i e-ψ egion
(below he QEP). This a iable is gi en by [10,12]
ψ=1
√ξF
λ−τ
(1 +λ)τ+κ√τ(1 +τ)
,(31)
whe e λ≡ω/2mN,κ≡q/2mN, and τ≡|Q2|/4m2
N=κ2−
λ2. The scaling a iables yand ψa e closely connec ed [12]:
ψ=y
kF⎡
⎣1+1+m2
N
q2
1
2ηFy
kF+Oη2
F⎤
⎦≃y
kF
,
(32)
whe e ηF=kF/mN, and, as no ed abo e, he supe scaling
unc ion is connec ed o F ia ≡kF×F, wi h kF he
-0.7-0.6 -0.5-0.4 -0.3-0.2 -0.1 0
y/kF
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(y/kF)
q=0.5 GeV/c
q=0.6
q=0.7
q=0.8
q=0.9
q=1.0
0 0.2 0.4 0.6 0.8 1
y/kF
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) (b)
FIG. 4. (Colo online) Supe scaling unc ion o nega i e (a) and
posi i e (b) alues o he dimensionless scaling a iable y/kF. Resul s
co espond o 12C(e,e) e alua ed in RPWIA o di e en momen um
ans e s.
Fe mi momen um. The cu es in Fig. 3may be compa ed
wi h he RPWIA esul s o he supe scaling unc ion, now
o nega i e and posi i e alues o he dimensionless scaling
a iable y/kFob ained using he quad a ic o m o Eq. (32);
see Fig. 4.Asshown,a ixedψ(o y/kF) he unc ion (ψ)
inc eases wi h qin acco dance wi h he p e ious discussion.
The basic esul s shown in Figs. 3and 4demons a e ha ψ
and y/kFcan be used in e changeably as long as one does
no ocus on he ew pe cen di e ences seen in he igu es,
namely, o la ge magni udes o he scaling a iables.
In showing he esul s we choose 12C as an illus a i e
example. Indeed his nucleus is ele an o many neu ino
oscilla ion expe imen s, whe e supe scaling ideas can be used
o make eliable p edic ions o neu ino-nucleus c oss sec ions
[28]. Mo eo e , he analysis o he wo ld da a pe o med in
Re . [11] poin s o an excellen supe scaling in he so-called
scaling egion (ψ<0) o nuclei wi h A⩾12. No e, howe e ,
ha e en he 4He da a display a e y good supe scaling
beha io o la ge nega i e alues o he scaling a iable
(ψ<−0.2), while a he QEP he e is a 10% iola ion owing
o he e y di e en spec al unc ion o he ligh es nuclei.
B. Posi i e-y egion
In his case, as shown in Fig. 1( igh ), he in eg a ion
egion in he (p, E) plane is limi ed by he wo cu es, E+
and E−, in he missing momen um egion [0,y]. This makes
he de i a i e analysis somewha mo e complica ed. Mo eo e ,
he expe imen al da a show ha scaling a gumen s o he i s
kind do no apply o he unc ion F(q,ω) in his egion; ha is,
Fdoes no become a unc ion dependen only on he scaling
a iable y. On he con a y, i shows a s ong dependence
on he momen um ans e q. As al eady men ioned, his
is because o he impo an con ibu ions beyond he IA
con ained in he ans e se channel. The e o e, al hough he
analysis ha ollows is applied o F(q,y), i should be
clea ly s a ed ha only he use o he “uni e sal” (namely,
longi udinal) supe scaling unc ion L, in pa icula , he s udy
o i s de i a i e wi h espec o he scaling a iable in he
posi i e-y egion, can e eal impo an e ec s no accoun ed
o by he esul s ob ained in he nega i e-yscaling egion.
A e some algeb a, he de i a i es o he scaling unc ion
F(q,y)a egi enby
1
2π
∂F
∂y =Yn
0(Y)∂Y
∂y −yn
0(y)
+Y(q,y)
0
pdpS
1(p, E−)∂E−
∂y 
−y
0
pdpS
1(p, E+)∂E+
∂y ,(33)
1
2π
∂F
∂q =Yn
0(Y)∂Y
∂q 
+Y(q,y)
0
pdpS
1(p, E−)∂E−
∂q 
−y
0
pdpS
1(p, E+)∂E+
∂q .(34)
055502-5

CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010)
As in he p e ious case, om he gene al exp essions o
E±gi en in Eq. (16) and assuming he he modynamic limi ,
we ge
∂E±
∂y ≃q+y
Eq+y
,∂E±
∂q ≃q+y
Eq+y−q±p
Eq±p
,(35)
and he de i a i es educe o
1
2π
∂F
∂y =Yn
0(Y)−yn
0(y)+q+y
Eq+y
×Y
0
pdpS
1(p, E−)−y
0
pdpS
1(p, E+),
(36)
1
2π
∂F
∂q =2Yn
0(Y)+q+y
Eq+yY
0
pdpS
1(p, E−)
−y
0
pdpS
1(p, E+)
+y
0
pdpq+p
Eq+p
S1(p, E+)
−Y
0
pdpq−p
Eq+p
S1(p, E−).(37)
Mo eo e , in he limi o he momen um ans e la ge
enough, qy, so ha he condi ion limY→∞ Yn
0(Y)=0
holds, he exp essions o he de i a i es esul :
1
2π
∂F
∂y =−yn
0(y)+q+y
Eq+y∞
0
pdpS
1(p, E−)
−y
0
pdpS
1(p, E+),(38)
1
2π
∂F
∂q =∞
0
pdp q+y
Eq+y−q−p
Eq−pS1(p, E−)
−y
0
pdp q+y
Eq+y−q+p
Eq+pS1(p, E+).
(39)
No e ha in he limi in which ycan be neglec ed compa ed
wi h q, ha is, (q+y)/Eq+y→q/Eq, he same commen
applies o he a io (q+p)/Eq+pin ol ed in he second
in eg al in Eq. (39), as pis limi ed wi hin he ange [0,y].
Thus, in such a limi ing case,
y
0
pdp q+y
Eq+y−q+p
Eq+pS1(p, E+)≃0 o qy,
(40)
and only he i s in eg al in Eq. (39) su i es. Fu he mo e, i
he spec al unc ion is such ha we can neglec pcompa ed
wi h qinside he in eg al, we again ge scaling o he
i s kind: limq→∞(∂F/∂q)=0. This is s ic ly alid only
o e y la ge alues o qand i is en i ely based on he
app oxima ions leading o he exp ession in Eq. (4) ha
connec s he scaling unc ion o he spec al unc ion. As
shown in Fig. 4(b) (posi i e-y egion), he RPWIA scaling
unc ion shows a negligible dependence on he momen um
ans e o 0.3y/kF0.8(qy), whe eas o la ge
y/kF, scaling o he i s kind begins o be sligh ly iola ed.
The expe imen al scaling unc ion ex ac ed om he analysis
o da a a in e media e q alues (less han o o he o de o
he nucleon mass) shows e y impo an scaling iola ions in
he egion abo e he QEP (posi i e alues o y).
Wi h ega d o he dependence o he scaling unc ion F
wi h qa ixed y, we ge di e en beha io s o small and la ge
alues o y. Indeed om Eq. (39) we obse e ha in he case
o ybeing e y small (in he icini y o 0), he second in eg al
in Eq. (39) can be neglec ed. As he in eg and in he emaining
in eg al is posi i e, we ge ∂F/∂q > 0; ha is, he scaling
unc ion g ows wi h q. This beha io is in acco dance wi h ha
al eady shown in he nega i e-y egion. On he con a y, o
inc easing alues o y he i s in eg al in Eq. (39) is expec ed
o diminish signi ican ly, as he exci a ion ene gy cu e E−
along which S1is e alua ed lies much highe han E+(see
Fig. 5), and i is easonable o expec ha S1(p, E) ge s i s main
con ibu ion o alues o he momen um and ene gy ha a e
no oo la ge. Fo yla ge enough, only he second in eg al in
Eq. (39) su i es, and because i s in eg and is also posi i e, he
minus sign in on o i leads o ∂F/∂q < 0; ha is, he scaling
unc ion Fdec eases wi h q, changing i s beha io wi h espec
o he p e ious cases. I is in e es ing o poin ou ha his esul
is consis en wi h he in eg a ion egions shown in Fig. 5whe e,
o inc easing momen um ans e , he cu e E+mo es o
highe exci a ion ene gies in he (E,p) plane. This means ha
as qgoes up, egions a low (E,p) alues, whe e he spec al
unc ion mos ly esides, a e no kinema ically accessible
anymo e. A simila a gumen can be applied o he case o e y
small alues o y[see Fig. 5(a)]. Howe e , he e he in eg a ion
egion los as E+goes up wi h inc easing qis less impo an
han he e ec s in oduced by he g owing in eg a ion egion
a ached o E−. This gene al beha io is also in acco dance
wi h he RPWIA esul s o he supe scaling unc ion
shown in Fig. 4(b) (posi i e alues o y), o , al e na i ely,
Fig. 3(b). One sees ha inc eases wi h qup o ψ
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
ε [GeV]
00.20.4
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
p [GeV/c]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ε [GeV]
q=0.5 GeV/c
q→∞
q=0.7 GeV/c
q=1.0 GeV/c
q=0.5 GeV/c
q→∞
q=0.7 GeV/c
q=1.0 GeV/c
(a)
(b)
FIG. 5. (Colo online) As Fig. 2, bu now o posi i e alues o
y:(a)y=0.1 GeV/c;(b)y=0.5 GeV/c.
055502-6
SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010)
0.4; ha is, y/kF∼0.364 (q=0.5GeV/c), y/kF∼0.375
(q=1.0GeV/c), and y/kF∼0.382 (q=∞GeV/c), wi h
kF=1.2 m
−1 he Fe mi momen um. This co esponds o y∼
0.1GeV/c, which is he si ua ion ep esen ed in Fig. 5(a).Also
no e ha he qdependence o in he egion whe e y/kF>
0.3 shown in Fig. 4is e y weak. Examina ion o Figs. 5(a)
and 5(b) shows ha o la ge y alues, he ene gy cu es E±
lie e y high, and hence, as qinc eases, he in eg als in ol ed
inco po a e only addi ional con ibu ions ha a e e y small,
leading o a e y weak a ia ion wi h momen um ans e .
III. NUCLEON MOMENTUM DISTRIBUTION AND
THE SCALING FUNCTION
In he p e ious sec ion we ha e de i ed gene al in eg od-
i e en ial equa ions connec ing he de i a i es o he scaling
unc ion, ∂F/∂y and ∂F/∂q, wi h he spec al unc ion. Based
on hese esul s applied o bo h nega i e and posi i e alues o
y, in wha ollows we e isi he “usual” p ocedu e o ob ain
he nucleon momen um dis ibu ion unc ion om he analysis
o QE (e,e) da a. Because he kinema ics o elec on sca e ing
lead o ini e in eg a ion limi s, we may no a p io i d aw any
s ong conclusions abou he “ ue” momen um dis ibu ion
gi en as n(p)≡∞
0dES(p, E), namely, he in eg al o he
spec al unc ion up o in ini e exci a ion ene gy. Howe e ,
assuming he spec al unc ion o eside mos ly in he (p, E)
plane a alues o pand E ha a e no oo la ge, he p e ious
analyses applied o nega i e- and posi i e-y egions lead o
di e en esul s, hus p o iding impo an and complemen a y
in o ma ion on how he ene gy and momen um a e dis ibu ed
wi hin he spec al unc ion.
The usual p ocedu e conside ed in p e ious wo k [3,4] o
gene a e he nuclea momen um dis ibu ion om he scaling
unc ion has been based on he exp ession
n(k)=−1
2πy ∂F
∂y |y|=k
,(41)
which has been widely applied in he nega i e-y egion. In
wha ollows we ex end his s udy o he posi i e-y egion
based on he uni e sal supe scaling unc ion in oduced om
he analysis o he sepa a ed longi udinal da a.
Making use o he gene al exp essions gi en by Eqs. (26)
and (36) and assuming he limi ing case limY→∞ Yn
0(Y)=0,
which is alid i he momen um ans e qis su icien ly la ge,
he momen um dis ibu ion unc ions can be w i en as ollows:
ny<0(q,k)=n0(−y)−q+y
yEq+y∞
−y
pdpS
1(p, E−)−y=k
=n0(k)+q−k
kEq−k∞
k
pdpS
1(p, E−),(42)
ny>0(q,k)=n0(y)−q+y
yEq+y∞
0
pdpS
1(p, E−)
−y
0
pdpS
1(p, E+)y=k=n0(k)−q+k
kEq+k
×∞
0
pdpS
1(p, E−)−k
0
pdpS
1(p, E+).
(43)
As obse ed, bo h exp essions ecei e con ibu ions om
he A−1 sys em g ound s a e, n0(k), as well as om he
exci ed s a es desc ibed h ough S1(p, E). Al hough using
he same no a ion o he exci a ion ene gy E−, no e ha he
Ecu es ha en e in he spec al unc ion S1in Eqs. (42) and
(43) a e e y di e en (see Figs. 2and 5).
Conclusions abou he pa icula beha io o he p e ious
exp essions can only be d awn based on a speci ic model
o he spec al unc ion; howe e , i is illus a i e o discuss
some gene al, “model-independen ” p ope ies. Fo nega i e
y he unc ion in Eq. (42) exceeds he pu ely g ound-s a e
con ibu ion, ha is, ny<0(q,k)>n
0(k) o all q,k alues.
This means ha he con ibu ion om he exci ed s a es adds
o he g ound-s a e momen um dis ibu ion. Conce ning he
speci ic ole played by each one o he wo e ms in Eq. (42), i
is di icul o d aws ingen conclusions wi hou ha ing con ol
o e S1. As he momen um kg ows, he con ibu ion o he
in eg al in Eq. (42) is expec ed o diminish signi ican ly (S1
mos ly esiding a momen a and exci a ion ene gies ha a e no
oo la ge). A simila commen also applies o he g ound-s a e
con ibu ion, which dec eases as kge s la ge . The analysis o
Eq. (43) in he posi i e-y egion di e s because o he ela i e
con ibu ions p o ided by he wo in eg als linked o he
exci ed s a es. In his case he global esponse ny>0(q,k) can be
smalle and/o la ge han he pu ely g ound-s a e con ibu ion,
n0(k), depending on he speci ic missing momen um alue.
In wha ollows we discuss some pa icula si ua ions
in de ail, he eby d awing some p elimina y conclusions on
he gene al beha io shown by ny≶0(q,k). Le us s a by
conside ing he alue o he nucleon momen um k o be in he
icini y o 0. Thus, neglec ing kcompa ed wi h he momen-
um ans e q(kq) and assuming ∞
0pdpS
1(p, E−)
k
0pdpS
1(p, E+)→0, we can w i e
ny<0(q,k)≃n0(k)+q
kEq∞
k
pdpS
1(p, E−)>n
0(k),
(44)
ny>0(q,k)≃n0(k)−q
kEq∞
0
pdpS
1(p, E−)<n
0(k).
(45)
F om hese esul s he ollowing ela ion ( alid o ksmall
enough) occu s:
ny>0(q,k)⩽n0(k)⩽ny<0(q,k).(46)
Mo eo e , om Eqs. (44) and (45) he g ound-s a e con ibu-
ion is oughly gi en as n0(k)≃[ny<0+ny>0]/2.
As he nucleon momen um kg ows, he wo unc ions
ny<0(q,k) and ny>0(q,k)inEqs.(42) and (43) ge close ,
c ossing each o he a some speci ic k, such ha ny>0(q,k)>
ny<0(q,k) o la ge k. F om he in eg a ion egion in he
(E-p) plane shown in Fig. 5, and assuming mos o he
s eng h in he spec al unc ion o be loca ed a no oo
high pand E, we can conclude ha o in e media e o
high missing momen um alues he main con ibu ion in
ny>0(q,k) comes om he second in eg al in Eq. (43); ha
is, ny>0(q,k)≃[(q+k)/(kEq+k)] k
0pdpS
1(p, E+).
055502-7
CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010)
FIG. 6. (Colo online) A e age exp
L(ψ) compa ed wi h he
Gumbel dis ibu ion in Eq. (47) (solid cu e) and a i o he
expe imen al da a (dashed cu e).
To p o e hese gene al p ope ies, in wha ollows we
p esen esul s based on he de i a i e analysis making use o
he supe scaling unc ion (ψ). To simpli y he calcula ions
we ep esen (ψ) by means o he Gumbel p obabili y densi y
unc ion (i.e., he de i a i e o he Gumbel dis ibu ion):
G(ψ)=1
σexp −(ψ−µ)
σexp −exp −(ψ−µ)
σ.
(47)
In ou case he alues o he pa ame e s a e µ=0 and
σ=0.67 [ max
G= G(0) =0.55]. In Fig. 6we compa e he
Gumbel dis ibu ion [Eq. (47)] wi h L
exp(ψ) and a i o he
expe imen al da a [22]. As shown, he Gumbel dis ibu ion
nicely i s he da a. Mo eo e , i ul ills he uni a i y condi ion
+∞
−∞ (ψ)dψ =1. The nucleon momen um dis ibu ion is
e alua ed h ough he de i a i e o he scaling unc ion by
using Eq. (41) and ecalling ha =kFF, hus ge ing
n(k)=−1
2πy
1
kF
d (ψ(y))
dy |y|=k
,(48)
which, using he app oxima e ela ion ψ≃y/kF, can be
p esen ed in he o m
n(k)=− 1
2πk
1
kFd (ψ)
d(kF|ψ|)kF|ψ|=k
.(49)
No e ha i he supe scaling unc ion is no symme ic
wi h espec o ψ, as is he case o he expe imen al da a,
Eq. (49) yields di e en momen um dis ibu ions o nega i e
and posi i e alues o ψ, which a e deno ed n<and n>,
espec i ely. On he con a y, symme ic scaling unc ions,
like he RFG one, lead o n<=n>.
In he case o he Gumbel dis ibu ion, we ge (se ing
µ=0)
d G(ψ)
dψ =1
σ(e−ψ/σ −1) G(ψ),(50)
FIG. 7. (Colo online) Nucleon momen um dis ibu ion ex ac ed
h ough he de i a i e o he supe scaling unc ion gi en by he
Gumbel p obabili y densi y in Eq. (47). Resul s co esponding o
nega i e (solid line) and posi i e (dashed line) alues o he scaling
a iable a e compa ed.
which leads o
n<
G(k)=1
2πσk2
Fk[ek/(σkF)−1] G(−k/kF),(51)
n>
G(k)=1
2πσk2
Fk[1 −e−k/(σkF)] G(k/kF).(52)
In Fig. 7we p esen he esul s o nψ<0(k)=n<
G(k)/2
(solid line) and nψ>0(k)=n>
G(k)/2 (dashed line), wi h n<
G(k)
and n>
G(k)gi eninEqs.(51) and (52)(a kF=1.2 m
−1). As
expec ed, n<
G(k) and n>
G(k) (and nψ<0and nψ>0, espec i ely)
coincide in he limi ing case k=0:
n>
G(0) =n<
G(0) =1
2πσ3k3
Fe.(53)
Fo missing momen a up o k∼1 m
−1 he main con ibu ion
esides in n<, which is in acco dance wi h Eq. (46) and he
gene al discussion p esen ed abo e. A k≃1.3–1.4 m−1, ha
is, kclose o he Fe mi momen um, n<and n>c oss each
o he , wi h n>being much highe o la ge k alues. In ac ,
whe eas n<shows a s eep slope when kinc eases, which is in
acco dance wi h esul s based on independen -pa icle model
desc ip ions, n>p esen s a high momen um ail e y a om
n<and, hence, om shell-model esul s (see nex sec ion). As
al eady explained, his ail a in e media e o high kis linked
o he much la ge con ibu ion gi en by he spec al unc ion
S1when e alua ed along he cu e E+ins ead o E−.This
gene al beha io is illus a ed in Fig. 8, whe e he con ou
cu es E±co esponding o posi i e and nega i e y alues a e
p esen ed. The p esence o he ail a high momen um alues
in he nucleon momen um dis ibu ion is a clea signa u e o
he impo ance o nucleon-nucleon co ela ions. Because he
spec al unc ion maps e y di e en egions in he (E−k)
plane o nega i e and posi i e y(Fig. 8), he join analysis o
he wo kinema ical egions can p o ide impo an clues in he
knowledge o NN co ela ions. I should be poin ed ou ha
he unc ions nψ<0(k) and nψ>0(k), e alua ed h ough Eq. (49)
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SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
p [GeV/c]
0
0.1
0.2
0.3
0.4
0.5
ε [GeV]
y=-0.2 GeV/c
y=+0.2 GeV/c
y=-0.5 GeV/c
y=+0.5 GeV/c
y=0
ε+
ε-
FIG. 8. (Colo online) In eg a ion egion in he (E,p)plane
o q=1 GeV/c and di e en , nega i e and posi i e, alues o
he scaling a iable y. The con ou cu es E±in bo h egions a e
ep esen ed.
and p esen ed in Fig. 7, a e no malized o di e en alues
connec ed wi h he di e en a eas sub ended by he Gumbel
dis ibu ion unc ion G(ψ) a nega i e and posi i e ψ, ha is,
0.37 ( o ψ<0) and 0.63 (ψ>0).
In pa icula , i has been shown in Re s. [24] and [26]in
he amewo k o ela i is ic nuclea models ha he la ge
posi i e-ψ ail o he scaling unc ion is closely ela ed o FSIs,
while he nega i e-ψ egion is mo e a ec ed by ini ial-s a e
co ela ions, as also shown in he nex sec ion using he CDFM
model. The possibili y o connec ing di e en aspec s o he
momen um dis ibu ion o ini ial- and inal-s a e physics will
be explo ed u he in u u e wo k.
IV. NUCLEON MOMENTUM DISTRIBUTION WITHIN
THE COHERENT DENSITY FLUCTUATION MODEL
In his sec ion we gi e, as an example, he esul s o he
nucleon momen um dis ibu ion ex ac ed om he scaling
unc ion, ob ained wi hin he amewo k o a pa icula
nuclea model, namely, he cohe en densi y luc ua ion model
(CDFM) [29,30]. The la e is a na u al ex ension o ini e nu-
clei o he RFG model wi hin which he scaling a iable ψwas
in oduced.1The CDFM is based on he gene a o coo dina e
me hod [31] and includes long- ange NN co ela ions (LRC)
o collec i e ype. In [32,33] he scaling unc ion was de ined
wi hin he CDFM using he RFG scaling unc ion [10,34–36]
and applied i o a ious p ocesses [32,33,37–40].
In he CDFM model [29,30], he one-body densi y ma ix
ρ( , ) is an in ini e supe posi ion o one-body densi y ma i-
ces ρx( , ) co esponding o single Sla e de e minan wa e
unc ions o sys ems o A ee nucleons homogeneously dis-
ibu ed in a sphe e wi h adius x, densi y ρ0(x)≡3A/(4πx3),
and Fe mi momen um kF(x)≡[3π2
2ρ0(x)]1/3≡α
x[wi h α≡
1The scaling a iable ψdi e s om ψby a phenomenological
ene gy shi Es≃20 MeV ( o 12C) in oduced o ep oduce he
expe imen al posi ion o he QEP: ψ(q,ω)=ψ(q,ω −Es).
(9π
8A)1/3∼
=1.52A1/3]:
ρ( , )=∞
0|F(x)|2ρx( , )dx. (54)
The weigh unc ion |F(x)|2can be exp essed in an equi alen
way ei he by means o he densi y dis ibu ion [29,30,33],
|F(x)|2=− 1
ρ0(x)
dρ( )
d  =x
a dρ( )
d ⩽0,(55)
o by means o he nucleon momen um dis ibu ion [33],
|F(x)|2=−3π2
2
α
x5
dn(k)
dk k=α/x
a dn(k)
dk ⩽0.(56)
In Eqs. (55) and (56)
ρ( )d =A, n(k)dk=A, ∞
0|F(x)|2dx =1.
(57)
In he e sion o he CDFM app oach sugges ed in Re s. [32]
and [33], he scaling unc ion has he o m
(ψ)=α/(kF|ψ|)
0|F(x)|2 RFG(x,ψ)dx, (58)
whe e he RFG scaling unc ion is
RFG(x,ψ)=3
41−kFx|ψ|
α2
×⎧
⎨
⎩
1+!xmN
α"2kFx|ψ|
α2
×⎡
⎣2+α
xmN2
−21+α
xmN2⎤
⎦⎫
⎬
⎭
.
(59)
In he CDFM he Fe mi momen um kFis calcula ed o
each nucleus by
kF=∞
0
kF(x)|F(x)|2dx =∞
0
α
x|F(x)|2dx (60)
and is no a i ing pa ame e , as i is in he RFG model.
By using Eqs. (55) and (56)inEqs.(58) and (60), he CDFM
scaling unc ion (ψ) and kFcan be exp essed equi alen ly
by he densi y and momen um dis ibu ions [33]:
(ψ)=4π
Aα/(kF|ψ|)
0
ρ(x)x2 RFG(ψ,x)
+x3
3
d RFG(ψ,x)
dx dx, (61)
whe e RFG(ψ,x) is gi en by Eq. (59), and
(ψ)=4π
A∞
kF|ψ|
n(k)k2 RFG(ψ,k)
+k3
3
d RFG(ψ,k)
dk ,(62)
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