Inelas ic elec on-nucleus sca e ing and scaling a high inelas ici y
M. B. Ba ba o,1J. A. Caballe o,2T. W. Donnelly,3and C. Maie on2,4
1Dipa imen o di Fisica Teo ica, Uni e si à di To ino and INFN, Sezione di To ino, Via P. Giu ia 1, 10125 To ino, I aly
2Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad de Se illa, Apa ado Pos al 1065, E-41080 Se illa, Spain
3Cen 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
4INFN, Sezione di Ca ania, Via S. So ia 64, 95123 Ca ania, I aly
(Recei ed 24 No embe 2003; published 24 Ma ch 2004)
Highly inelas ic elec on sca e ing is analyzed wi hin he con ex o he uni ied ela i is ic app oach p e i-
ously conside ed in he case o quasielas ic kinema ics. Inelas ic ela i is ic Fe mi gas modeling ha includes
he comple e inelas ic spec um— esonan , non esonan , and deep inelas ic sca e ing—is elabo a ed and com-
pa ed wi h expe imen al da a. A phenomenological ex ension o he model based on di ec i s o da a is also
in oduced. Wi hin bo h models, c oss sec ions and esponse unc ions a e e alua ed and binding ene gy e ec s
a e analyzed. Finally, an in es iga ion o he second-kind scaling beha io is also p esen ed.
DOI: 10.1103/PhysRe C.69.035502 PACS numbe (s): 25.30.Fj, 24.10.J , 13.60.Hb
I. INTRODUCTION
In his wo k we conside highly inelas ic elec on sca e -
ing and compa e i s analysis wi h he case o quasielas ic
(QE)elec on sca e ing. The la e is domina ed by he p o-
cess whe e he exchanged i ual pho on in e ac s wi h a
nucleon in he nuclea g ound s a e and ejec s ha nucleon,
he eby o ming a nuclea pa icle-hole exci a ion. Co ec-
ions o his dominan p ocess in ol e going beyond he im-
pulse app oxima ion o accoun o wo-body cu en s, inal-
s a e in e ac ions, and nuclea co ela ions. Al hough hese
con ibu ions a e known no o be en i ely negligible [1–7]
his simple p ocess accoun s o he basic ea u e seen in he
icini y o elas ic sca e ing om a nucleon a es , namely,
he QE peak. Models such as hose discussed below ake in o
accoun he ac ha he nucleons in he nucleus a e mo ing
and a e bound and he eby p oduce a b oad peak in he in-
elas ic spec um. In he p esen wo k ou goal is o ex end
he analysis, s ill main aining he same basic ea u es o he
ela i is ic modeling used o he QE egion, and now ocus
on wha we call highly inelas ic sca e ing, o o b e i y,
simply he inelas ic egion. This includes e e y hing ha
goes beyond he QE p ocess: ha is, whe eas he QE p ocess
assumes elas ic sca e ing om he nucleons, he inelas ic
p ocess will assume inelas ic e-Nsca e ing. Fo ela i ely
low inal-s a e in a ian masses one lies in he egion o eso-
nance exci a ion and wo cases o his so ha e been ex-
plo ed in ecen wo k [8,9]. In he p esen s udy hese ideas
a e gene alized o include he comple e inelas ic spec um,
bo h esonan and non esonan , including deep inelas ic sca -
e ing (DIS), wi hin he con ex o he uni ied ela i is ic
app oach used in ou p e ious wo k.
Thus, in he p esen wo k ou goal is o begin by explo -
ing ex ensions o he ela i is ic Fe mi gas (RFG)model
[6,10,11] o an inelas ic e sion o his app oach. While his
bea s some connec ion wi h adi ional con olu ion models
o he high-ene gy esponse o nuclei (see, o example,
Re s. [12–15]) i is no he same in ha , albei wi hin a
model, i co ec ly inco po a es a speci ic ela i is ic nuclea
spec al unc ion in o he p oblem, whe eas some o he ap-
p oaches make addi ional assump ions and use only he in e-
g al o he spec al unc ion, namely, he nuclea momen um
dis ibu ion o make non ela i is ic app oxima ions when
dealing wi h he spec al unc ion.
This dis inc ion can be seen qui e clea ly in s udies o
i s - and second-kind scaling [11,16–20]and will no be
elabo a ed he e. Once he inelas ic RFG modeling is in hand,
i becomes clea ha i migh be use ul o explo e a phenom-
enological ex ension o his model, namely, wha we call he
ex ended ela i is ic Fe mi gas (ERFG). In his app oach we
ake he esul o doing he co ec in eg al o e he nuclea
spec al unc ion (i.e., no he ull in eg al, which is he mo-
men um dis ibu ion, as alluded o abo e)di ec ly om i s
made p e iously o he da a [21]. We shall see ha his has a
signi ican impac on he nuclea esponses a high inelas ic-
i y.An issue which will also become clea la e is ha he
s o y is no ye comple e: in addi ion o he modeling done in
he p esen wo k, whe e he ocus is placed on inco po a ing
inelas ic e ec s a high ene gies, he e a e s ill o he con i-
bu ions ha mus be added. Speci ically, in ecen wo k [22]
on 2p-2hmeson-exchange cu en e ec s i is seen ha a
signi ican incohe en con ibu ion mus be added o hose
explo ed he e. Gi en ha he wo k on 2p-2he ec s is, as
ye , incomple e—co ela ion con ibu ions a e p esen ly be-
ing included—i is p ema u e o make oo much o compa i-
sons wi h expe imen al da a, and, as we ema k la e in he
app op ia e places, he inal unde s anding o how all o he
a ious eac ion mechanisms en e , while becoming clea e
is no ye achie ed.
The pape is o ganized as ollows: in Sec. II we ecall he
gene al o malism o inelas ic elec on-nucleus sca e ing;
in Sec. III we de i e he exp essions o he inelas ic had-
onic enso in h ee di e en models: he pu e RFG model
[Sec. III A], he RFG including he e ec s o binding ene gy
[Sec. III B]and he ERFG [Sec. III C]; in Sec. IV we p esen
nume ical esul s o c oss sec ions [Sec. IV A], esponse
unc ions [Sec. IV B], and scaling unc ions [Sec. IV C]and
inally, in Sec. V we d aw ou conclusions.
PHYSICAL REVIEW C 69, 035502 (2004)
0556-2813/2004/69(3)/035502(16)/$22.50 ©2004 The Ame ican Physical Socie y69 035502-1
II. INELASTIC ELECTRON-NUCLEUS SCATTERING:
GENERAL FORMALISM
Wi h he goal ou lined abo e in mind, we s a by ew i -
ing he gene al exp essions ha apply in bo h elas ic and
inelas ic egimes. The gene al o malism desc ibing inclu-
si e elec on-nucleus sca e ing p ocesses is widely a ailable
[23–25]; he e we simply ocus on hose aspec s ha a e o
special ele ance o he discussion ha ollows. We ollow
he con en ions and me ic o Re . [26]and use capi al le -
e s o e e o ou - ec o s. The inciden and sca e ed elec-
on ou -momen a a e deno ed by Ki
␮
=共␧i,ki兲and K
␮
=共␧ ,k 兲. The had onic a iables, PA
␮
=共MA,0兲and PB
␮
=共EB,pB兲 ep esen he ou momen a o he a ge and e-
sidual nucleus, espec i ely. The ou -momen um ans e is
gi en by Q
␮
=共
␻
,q兲(we assume he Bo n app oxima ion,
i.e., only one i ual pho on exchanged in he p ocess).
Following s anda d p ocedu es he di e en ial c oss sec-
ion may be w i en
d
␴
d⍀ d␧ =2
␣
2
Q4
␧
␧i
␩
␮
␯
W
␮
␯
,共1兲
whe e
␣
is he ine s uc u e cons an ,
␩
␮
␯
is he lep onic
enso ha can be e alua ed di ec ly using ace echniques
关27兴, and W
␮
␯
is he had onic enso con aining all o he
nuclea s uc u e and dynamics in o ma ion. Assuming ha
he inal s a e can be desc ibed in e ms o a ecoiling nuclea
s a e 兩
␺
B典plus a 共highly兲inelas ic s a e 兩⌽X典, i s gene al ex-
p ession is gi en by
W
␮
␯
=兺
A兺
B兺
X具
␺
B,⌽X兩J
ˆ
␮
共q兲兩
␺
A典*具
␺
B,⌽X兩J
ˆ
␯
共q兲兩
␺
A典
⫻
␳
共EB兲dEB
␳
共EX兲dEX
␦
共␧i−␧ +EA−EB−EX兲,
共2兲
whe e 兺
¯
A共兺B兺X兲indica es he app op ia e a e age 共sum兲
o e ini ial 共 inal兲s a es. He e J
ˆ
␮
共q兲is he Fou ie ans o m
o he nuclea cu en ope a o e alua ed, 兩
␺
A典and 兩
␺
B,⌽X典
ep esen he ini ial and inal s a es, espec i ely, and he
dis ibu ion unc ions
␳
共EB兲and
␳
共EX兲a e in oduced o ac-
coun o he ene gy-momen um dispe sion ela ion o he
inal nuclea 共B兲and had onic 共X兲sys ems. In his wo k we
assume ha he inelas ici y o he p ocess is o ally ac-
coun ed o by he inal s a e ⌽X; hence o he ene gy dis-
ibu ion unc ion o he esidual nuclea sys em we use
␳
共EB兲=
␦
共EB−E
¯
B兲, whe e E
¯
B=冑pB
2+共MB
*兲2. No e ha W
␮
␯
in
Eq. 共2兲is mean o be e alua ed a pB+pX=q=ki−k .
The nuclea enso can equi alen ly be exp essed as an
in eg al in he 共E,p兲plane, wi h −p=pB he h ee-momen um
o he ecoiling daugh e nucleus and E⬅冑p2+共MB
*兲2
−冑p2+共MB
0兲2 he exci a ion ene gy o he esidual nucleus
(see Re . [19]). The domain o in eg a ion is he kinema i-
cally allowed egion
max关E共0兲,0兴艋E艋E共
␲
兲,共3兲
whe e
E共
␪
兲=MA
0+
␻
−冑共MB
0兲2+p2−冑WX
2+q2+p2+2pq cos
␪
共4兲
wi h
␪
he angle be ween pand q, and whe e WXis he
in a ian mass o he inal s a e. In he MB
0→⬁limi he
abo e exp ession becomes
E⬁共
␪
兲=mN+
␻
˜
−冑WX
2+q2+p2+2pq cos
␪
,共5兲
whe e mNis he nucleon mass,
␻
˜
⬅
␻
−ESand ES=MB
0+mN
−MA
0is he sepa a ion ene gy.
The uppe cu e E共
␲
兲c osses he paxis a p−=−yXand
p+=YX, whe e
yX=1
2W2关共MA
0+
␻
兲冑共W−MB
0兲2−WX
2
⫻冑共W+MB
0兲2−WX
2−2q⌳X兴共6兲
and
YX=1
2W2关共MA
0+
␻
兲冑共W−MB
0兲2−WX
2
⫻冑共W+MB
0兲2−WX
2+2q⌳X兴,共7兲
and whe e
W=冑共MA
0+
␻
兲2−q2and ⌳X=1
2关W2+共MB
0兲2−WX
2兴.
共8兲
The a iable yXis he gene aliza ion o he usual y-scaling
a iable o he inelas ic p ocess whe e a esonance Xis p o-
duced. In he limi MB
0→⬁i eads
yX,⬁=冑共
␻
˜
+mN兲2−WX
2−q.共9兲
No e ha he allowed egion dec eases wi h WXand col-
lapses o a poin when −yX=YX, which implies W=MB
0+WX
o , in he MB
0→⬁limi , 共yX,⬁兲min=−q, co esponding o
共WX兲max=
␻
˜
+mN. Summa izing, o ixed ou -momen um
ans e , he esonan mass is limi ed o he ange
mN+m
␲
艋WX艋mN+
␻
−ES.共10兲
III. THE RELATIVISTIC FERMI GAS MODEL
In his sec ion we p oceed by e alua ing he had onic
nuclea enso assuming he impulse app oxima ion and by
wo king wi hin he amewo k o he RFG model. In his
case, he i ual pho on is abso bed by an on-shell nucleon
desc ibed by a Di ac spino u共h,sh兲, wi h ene gy E
¯
h
=冑h2+mN
2. In eg a ing o e he momen a in he Fe mi sea,
he ollowing exp ession o he inelas ic had onic enso e-
sul s:
BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004)
035502-2
W
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
FdhmN
E
¯
h
冕
dEX
␦
共
␻
+E
¯
h−EX兲
⫻1
2兺
sh兺
Xi
␳
共EXi兲关⌽
¯
XiJ
ˆ
␮
u共h,sh兲兴*关⌽
¯
XiJ
ˆ
␯
u共h,sh兲兴,
共11兲
whe e Nis he numbe o nucleons 共p o ons o neu ons兲and
兰Fdh⬅兰dh
␪
共pF−h兲,pFbeing he Fe mi momen um. The
symbol 兰dEXs ands o he in eg al o e he ene gy o he
inelas ic inal s a e, while 兺Xiindica es in gene al he sum/
in eg al o e all he in e nal quan um numbe s o all possible
inelas ic inal s a es ⌽Xi, ha ing o al ene gy EXand o al
momen um pX, ixed by momen um conse a ion o be pX
=h+q.
The had onic enso in Eq. (11) o inelas ic p ocesses can
be also w i en in he o m
Winel
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
dEX
冕
FdhmN
E
¯
h
winel
␮
␯
共H,Q,EX兲
⫻
␦
共
␻
+E
¯
h−EX兲,共12兲
whe e H
␮
=共E
¯
h,h兲and we ha e in oduced he inelas ic
single-nucleon enso
winel
␮
␯
共H,Q,EX兲=1
2兺
sh兺
Xi
␳
共EXi兲关⌽
¯
XiJ
ˆ
␮
u共h,sh兲兴*
⫻关⌽
¯
XiJ
ˆ
␯
u共h,sh兲兴.共13兲
No e ha he abo e single-nucleon enso has dimensions o
E−1. As will be shown la e , his is in con as wi h ou pas
wo k on QE and N→⌬sca e ing whe e he single-nucleon
enso s we e de ined o be dimensionless.
Nex we choose o exp ess he inelas ic had onic enso in
Eq. (12)in e ms o he in a ian mass WX,
Winel
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
dWX
冕
FdhmNWX
E
¯
hEX
winel
␮
␯
共H,Q,EX兲
⫻
␦
共
␻
+E
¯
h−EX兲共14兲
wi h EX=冑pX
2+WX
2. The ene gy in eg al can be pe o med
by exploi ing he
␦
unc ion, yielding
Winel
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
FdhmN
E
¯
h
winel
␮
␯
共H,Q,
␻
+E
¯
h兲.共15兲
In he case o DIS on a single nucleon, he inelas ic enso
simply educes o he single-nucleon enso winel
␮
␯
.
Be o e en e ing in o a de ailed analysis o he inelas ic
nuclea enso , i is in e es ing o no ice how he usual ex-
p essions o he QE and N→⌬had onic enso s a e eco -
e ed om he gene al esul gi en in Eq. (11). Fi s , in he
case o QE sca e ing, he nuclea inal s a e is simply a
pa icle-hole exci a ion, hence, in he RFG model, ⌽Xde-
sc ibes an on-shell nucleon, namely, ⌽X=冑mN/E
¯
pu共p,sp兲.
The ene gy dis ibu ion unc ion is simply
␳
共EX兲=
␦
共EX
−E
¯
p兲and he sum o e he inal s a es educes o a sum o e
spin p ojec ions, 兺Xi=兺sp. The QE had onic enso hen
eads
WQE
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
FdhmN
2
E
¯
hE
¯
p
wQE
␮
␯
共H,Q兲
␦
共
␻
+E
¯
h−E
¯
p兲,
共16兲
whe e wQE
␮
␯
is he usual dimensionless QE single-nucleon en-
so
wQE
␮
␯
=1
2兺
sh兺
sp
关u
¯
共p,sp兲J
ˆ
␮
u共h,sh兲兴*关u
¯
共p,sp兲J
ˆ
␯
u共h,sh兲兴.
共17兲
In he case o he ansi ion N→⌬, he inal s a e ⌽X, wi hin
he con ex o he RFG model, is an on-shell ⌬, namely,
⌽X=冑m⌬/E
¯
⌬u⌬共p,s⌬兲, wi h on-shell ene gy E
¯
⌬=冑p2+m⌬
2.
The ene gy dis ibu ion unc ion in his case is
␳
共EX兲
=
␦
共EX−E
¯
⌬兲and 兺Xi=兺s⌬. The N→⌬had onic enso ha
esul s is
W⌬
␮
␯
共q,
␻
兲=3N
4
␲
pF
3
冕
FdhmN
2
E
¯
hE
¯
⌬
w⌬
␮
␯
共H,Q兲
␦
共
␻
+E
¯
h−E
¯
⌬兲
共18兲
wi h w⌬
␮
␯
he dimensionless nucleon-⌬ enso
w⌬
␮
␯
=m⌬
2mN兺
sh兺
s⌬
关u
¯
⌬共p,sp兲J
ˆ
␮
u共h,sh兲兴*关u
¯
⌬共p,sp兲J
ˆ
␯
u共h,sh兲兴.
共19兲
As expec ed, hese exp essions o he dimensionless single-
nucleon enso s coincide wi h he ones in oduced in Re .
关8,28兴. Likewise o he Rope esonance he exp essions ob-
ained in Re . 关9兴a e eco e ed.
A. The RFG inelas ic nuclea enso and esponse unc ions
In his sec ion we e alua e he inelas ic nuclea enso in
he RFG amewo k. Fo con enience, as usual we i s de-
ine he dimensionless a iables
␬
␮
=共␭,
␬
兲=
冉
␻
2mN,q
2mN
冊
,
␶
=
␬
2−␭2,
␩
F=pF
mN,
⑀
F=冑1+
␩
F
2,
共20兲
␩
␮
=共
⑀
¯
,
␩
兲=
冉
E
¯
h
mN,h
mN
冊
,
␮
X=WX
mN,
⑀
X=冑
␮
X
2+共
␩
+2
␬
兲2,
in e ms o which he had onic enso in Eq. (14) eads
INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004)
035502-3
Winel
␮
␯
共
␬
,␭兲=3N
4
␲
␩
F
3
冕
d
␮
X
冕
d
␩
␮
X
⑀
¯
⑀
Xwinel
␮
␯
共
␩
,
␮
X;
␬
,␭兲
⫻
␦
共2␭+
⑀
¯
−
⑀
X兲
␪
共
␩
F−
␩
兲.共21兲
Be o e p esen ing he explici esul s o he RFG e-
sponse unc ions, le us discuss an impo an ing edien o
he calcula ion, he single-nucleon inelas ic had onic enso
winel
␮
␯
. Fo unpola ized sca e ing, he la e can be pa am-
e ized in e ms o wo s uc u e unc ions, w1and w2, ac-
co ding o
winel
␮
␯
=−w1
冉
g
␮
␯
+
␬
␮
␬
␯
␶
冊
+w2共
␩
␮
+
␬
␮
␳
兲共
␩
␯
+
␬
␯
␳
兲.
共22兲
Fo on-shell nucleons, he s uc u e unc ions w1and w2de-
pend on wo a iables, he ou -momen um ans e Q2and
he in a ian mass WXo he inal s a e eached by he
nucleon, o , equi alen ly, he single-nucleon Bjo ken a i-
able
x=兩Q2兩
2H·Q=兩Q2兩
WX
2−mN
2−Q2=
␶
␩
·
␬
.共23兲
In ou o malism i is con enien o in oduce he inelas-
ici y pa ame e [8,29]
␳
⬅1+ 1
4
␶
共
␮
X
2−1兲,共24兲
he alue uni y co esponding o elas ic sca e ing. No e ha
␳
is simply linked o he Bjo ken scaling a iable o he
on-shell nucleon mo ing inside he a ge nucleus by he
ela ion
␳
=1/x, hus in he ollowing we will use
␳
as a gu-
men o he s uc u e unc ions w1,w2.
In p esen ing ou esul s we will also use he “labo a o y”
Bjo ken a iable
xL=兩Q2兩
2mN
␻
=
␶
␭,共25兲
co esponding o a single nucleon a es in he labo a o y
ame.
Le us now e u n o he inelas ic nuclea enso o Eq.
(21): a e pe o ming he pola angula in eg a ion by means
o he ene gy-conse ing
␦
unc ion one ge s
Winel
␮
␯
共
␬
,␭兲=3N
␶
2
␩
F
3
␬
冕
0
2
␲
d⌽
2
␲
冕
␳
1共
␬
,␭兲
␳
2共
␬
,␭兲d
␳
冕
⑀
0共
␳
兲
⑀
Fd
⑀
¯
⫻winel
␮
␯
共
⑀
¯
,
␪
0,
␳
;
␬
,␭兲,共26兲
whe e
cos
␪
0=1
␬
␩
共␭
⑀
¯
−
␶
␳
兲.共27兲
The condi ion 兩cos
␪
0兩艋1 ixes he in eg a ion limi s o e
⑀
¯
:
⑀
¯
艌
⑀
0共
␳
兲⬅
␬
冑1
␶
+
␳
2−␭
␳
.共28兲
Mo eo e , by equi ing ha
⑀
0共
␳
兲艋
⑀
Fand ha he eso-
nance mass is abo e he pion-p oduc ion h eshold (i.e.,
␮
X
艌
␮
h esh⬅1+
␮
␲
) he ollowing egion is ob ained o he
in eg a ion o e
␳
:
关
␳
1共
␬
,␭兲,
␳
2共
␬
,␭兲兴 =
冋
max
再
␭
⑀
F−
␬
␩
F
␶
,
␳
h esh
冎
,␭
⑀
F+
␬
␩
F
␶
册
共29兲
wi h
␳
h esh =1+
␮
␲
共
␮
␲
+2兲
4
␶
.共30兲
No e ha he uppe in eg a ion limi
␳
2共
␬
,␭兲always lies be-
low he cu o co esponding o Eq. 共10兲. Indeed using Eqs.
共10兲and 共24兲and keeping in mind ha he 共nega i e兲sepa-
a ion ene gy o he RFG is ES
RFG=−TF⬅mN共1−
⑀
F兲, he
maximum
␳
allowed by Eq. 共10兲 eads
␳
max
RFG =1+ 1
4
␶
关共2␭+
⑀
F兲2−1兴=
␳
2共
␬
,␭兲+1
4
␶
共2
␬
−
␩
F兲2,
共31兲
␳
2 he e o e esul ing in he mo e s ingen in eg a ion limi .
Now by w i ing he single-nucleon inelas ic enso in
e ms o s uc u e unc ions w1and w2as in Eq. (22)and
choosing he zdi ec ion along q, he in eg a ion o e ⌽and
⑀
¯
can be pe o med analy ically (see Appendix A)and he
had onic inelas ic enso can be exp essed in he gene al
o m
Winel
␮
␯
共
␬
,␭兲=3N
␶
2
␩
F
3
␬
␰
F
冕
␳
1共
␬
,␭兲
␳
2共
␬
,␭兲d
␳
共1−
␺
X
2兲
␪
共1−
␺
X
2兲
⫻U
␮
␯
共
␬
,
␶
,
␳
兲,共32兲
whe e
␰
F=
⑀
F−1 is he Fe mi kine ic ene gy and he inelas ic
scaling a iable
␺
X⬅sgn共␭−
␶
␳
兲冑
⑀
0共
␳
兲−1
⑀
F−1 共33兲
has been de ined. Fo each alue o
␳
共and hence
␮
X兲a
“peak” can hus be iden i ied, co esponding o he egion
−1艋
␺
X艋1, cen e ed a
␺
X=0, ␭P=
␶
P
␳
=1
2
␳
共冑1+4
␬
P
2
␳
2−1兲,
␬
P=冑
␶
P共1+
␶
P
␳
2兲,共34兲
whose wid h
⌬␭ =1
2关冑共2
␬
+
␩
F兲2+
␮
X
2−冑共2
␬
−
␩
F兲2+
␮
X
2兴⯝ 2
␬
␩
F
冑4
␬
2+
␮
X
2
共35兲
is a unc ion ha g ows wi h
␬
and dec eases wi h
␮
X.
BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004)
035502-4
The gene al exp ession o he enso U
␮
␯
is de i ed in
Appendix A. He e we only epo he longi udinal and ans-
e se componen s
UL=U00 =
␬
2
␶
关共1+
␶
␳
2兲w2共
␶
,
␳
兲−w1共
␶
,
␳
兲
+w2共
␶
,
␳
兲D共
␬
,
␶
,
␳
兲兴,共36兲
UT=U11 +U22 =2w1共
␶
,
␳
兲+w2共
␶
,
␳
兲D共
␬
,
␶
,
␳
兲,共37兲
which a e linked o he longi udinal and ans e se esponse
unc ions by he ollowing ela ions:
Rinel
L,T共
␬
,
␶
兲=3N
␶
2
␩
F
3
␬
␰
F
冕
−1
␺
X
max共
␬
,␭兲d
␺
X冏
⳵␳
⳵
␺
X冏共1−
␺
X
2兲
⫻UL,T„
␬
,
␶
,
␳
共
␺
X兲…,共38兲
wi h
␺
X
max共
␬
,␭兲= min
冦
1,
冢
␬
冑1
␶
+
␳
h esh
2−␭
␳
h esh −1
␰
F
冣
1/2
冧
共39兲
and
⳵␳
⳵
␺
X=−冑2
␰
F
␬
共1+
␰
F
␺
X
2兲−␭
␺
X冑2
␰
F
冉
1+1
2
␰
F
␺
X
2
冊
␶
冑1+1
2
␰
F
␺
X
2
.
共40兲
In Eqs. 共36兲and 共37兲 he unc ion
D共
␬
,
␶
,
␳
兲=1
⑀
F−
⑀
0共
␳
兲
冕
⑀
0共
␳
兲
⑀
Fd
⑀
¯
冕
0
2
␲
d⌽
2
␲
共
␩
⫻
␬
ˆ兲2
=
␶
␬
2
再
1
3关
⑀
F
2+
⑀
F
⑀
0共
␳
兲+
⑀
0共
␳
兲2兴
+␭关
⑀
F+
⑀
0共
␳
兲兴 +␭2
冎
−共1+
␶
兲+共
␳
−1兲
⫻
␶
␬
2兵␭关
⑀
F+
⑀
0共
␳
兲兴 −
␶
共
␳
+1兲其
=
␰
F共1−
␺
X
2兲
冋
1+
␰
F
␺
X
2−␭
␬␺
X冑
␰
F共2+
␰
F
␺
X
2兲
+
␶
3
␬
2
␰
F共1−
␺
X
2兲
册
共41兲
a ises om he Fe mi mo ion and goes o ze o as
␰
F→0;
being p opo ional o
␰
F⬵
␩
F
2/2Ⰶ1, his p o ides ela i ely
mode a e co ec ions o he es o he con ibu ions in Eqs.
共36兲and 共37兲.
The alue
␳
=1 co esponds o QE kinema ics: in his case
he well-known exp essions o he QE esponses a e eco -
e ed. The “ o al” obse ables a e hen ob ained by adding he
usual RFG QE esponse o he inelas ic esul s:
R o
L,T=RQE
L,T+Rinel
L,T.共42兲
In he deep inelas ic egime i is cus oma y o deal wi h
nuclea s uc u e unc ions W1,2
Aand/o F1,2
A. These can be
exp essed in e ms o he longi udinal and ans e se e-
sponse unc ions h ough he ollowing ela ions:
W1
A=1
2RT,共43兲
W2
A=
冉
␶
␬
2
冊
2RL+1
2
␶
␬
2RT共44兲
and
F1
A=mNW1
A,共45兲
F2
A=2mN␭W2
A.共46兲
B. E ec s o binding ene gy
In he s udy o supe scaling o inclusi e QE elec on
sca e ing om nuclei, an app op ia e scaling a iable
␺
⬘
was in oduced by including a small ene gy shi o ha e he
QE peak occu a he place whe e he scaling a iable is ze o.
A de ailed s udy o he sensi i i y o he scaling unc ion o
a ia ions o he Fe mi momen um and ene gy shi was p e-
sen ed in Re s. [16,18,20]. He e we ex end his analysis o
he inelas ic egion. In p inciple, he in oduc ion o an en-
e gy shi
␻
shi in he o malism is s aigh o wa d and he
calcula ion o he inelas ic esponses p oceeds as in he
␻
shi =0 case. Howe e , as will be made clea in he ollow-
ing, some complica ions a ise. Fi s , due o he gene al o m
assumed o he single-nucleon inelas ic had onic enso , a
ce ain asymme y appea s be ween he ene gy shi e ec s
in he longi udinal and ans e se esponses. These shi e -
ec s a e la ge in he longi udinal esponse. No ice ha his
asymme y al eady en e s a he le el o he QE nuclea e-
sponses. Second, he e exis s an ambigui y in he de ini ion
o he a iable which should be used as he Bjo ken x-scaling
a iable co esponding o he mo ing nucleon.
The e ec s o he inclusion o an ene gy shi on he in-
elas ic nuclea had onic enso ha e been s udied in he li -
e a u e, wi h pa icula emphasis on he s uc u e unc ion F2
and he Eu opean Muon Collabo a ion (EMC)e ec a la ge
alues o x, in he con ex o so called “binding models” (see,
o example, Re s. [30,31]and he gene al e iews [12,13]).
The app oach we ollow he e is he sel -consis en gene ali-
za ion o p e ious wo ks on he RFG. I is o mally simila
o he binding model app oach, whe e in gene al he on-shell
ene gy o he ini ial nucleon is modi ied by sub ac ing a
cons an e m which e ec i ely accoun s o he nucleon
sepa a ion ene gy and o he possibili y ha he esidual
nuclea sys em is le in an (highly)exci ed s a e. Howe e ,
since he exis ing models ei he ocus only on EMC a ios
and/o use mo e ealis ic, al hough gene ally non ela i is ic
wa e unc ions, a p ecise quan i a i e compa ison wi h hose
models is no possible. As we will discuss in he Resul s
INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004)
035502-5

sec ion, ou calcula ions s ill miss some ing edien s, coming
om meson-exchange cu en s, and his makes a de ailed
quan i a i e compa ison wi h expe imen al da a p ema u e; i
is clea ha , when his compa ison will be made in he u-
u e, a mo e in-dep h s udy o binding e ec s will also be
needed.
In he RFG he ene gy shi is usually in oduced by
modi ying he a gumen o he
␦
unc ion appea ing in he
gene al exp ession o he inelas ic had onic enso in Eq.
(12), acco ding o
␻
+E
¯
h−EX→
␻
⬘+E
¯
h−EX, whe e
␻
⬘=
␻
−
␻
shi . Then, in oducing he in a ian mass WX
⬘2⬅EX
2−pX
2
=共
␻
⬘+E
¯
h兲2−pX
2, we can w i e he inelas ic had onic enso in
he o m
Winel
␮
␯
共
␬
,␭兲=3N
4
␲
␩
F
3
冕
d
␮
X
⬘
冕
d
␩
␮
X
⬘
⑀
¯
⑀
Xwinel
␮
␯
共
␩
,
␮
X
⬘;
␬
,␭兲
⫻
␦
共2␭⬘+
⑀
¯
−
⑀
X兲
␪
共
␩
F−
␩
兲,共47兲
whe e
␮
X
⬘=WX
⬘/mNand ␭⬘=
␻
⬘/共2mN兲. As in he unshi ed
analysis, he
␦
unc ion can be used o pe o m he pola
angula in eg a ion, leading o he esul
Winel
␮
␯
共
␬
,␭兲=3N
␶
⬘
2
␩
F
3
␬
冕
␳
1
⬘共
␬
,␭⬘兲
␳
2
⬘共
␬
,␭⬘兲d
␳
⬘
冕
0
2
␲
⫻d⌽
2
␲
冕
⑀
0
⬘共
␳
⬘兲
⑀
Fd
⑀
¯
winel
␮
␯
共
⑀
¯
,
␳
⬘;
␬
,␭兲,共48兲
whe e he a iable
␳
⬘is de ined as
␳
⬘⬅2H·Q⬘
兩Q⬘2兩=
冋
1+ 1
4
␶
⬘共
␮
X
⬘2−1兲
册
共49兲
and
␶
⬘⬅
␬
2−␭⬘2.
The inclusion o he ene gy shi modi ies he in eg a ion
limi s o e
⑀
¯
in he ollowing way:
⑀
¯
艌
⑀
0
⬘共
␳
⬘兲⬅
␬
冑1
␶
⬘+
␳
⬘2−␭⬘
␳
⬘.共50兲
Co espondingly, he egion o he in eg a ion o e
␳
⬘is
gi en by
关
␳
1
⬘共
␬
,␭⬘兲,
␳
2
⬘共
␬
,␭⬘兲兴 =
冋
max
再
␭⬘
⑀
F−
␬
␩
F
␶
⬘,
1+
␮
␲
4
␶
⬘共2+
␮
␲
兲
冎
,␭⬘
⑀
F+
␬
␩
F
␶
⬘
册
.
共51兲
The de ini ion o he inelas ic scaling a iable becomes
now
␺
X
⬘2⬅
⑀
0
⬘共
␳
⬘兲−1
⑀
F−1 共52兲
and he inelas ic longi udinal and ans e se esponse unc-
ions, calcula ed as Rinel
L=Winel
00 and Rinel
T=Winel
11 +Winel
22 , ha e
he ollowing gene al o ms:
Rinel
L,T共
␬
,
␶
兲=3N
␶
⬘
2
␩
F
3
␬
␰
F
冕
␳
1
⬘共
␬
,␭⬘兲
␳
2
⬘共
␬
,␭⬘兲d
␳
⬘共1−
␺
X
⬘2兲UL,T共
␬
,
␶
,
␳
⬘兲.
共53兲
In o de o e alua e he longi udinal and ans e se nuclea
unc ions UL,T共
␬
,
␶
,
␳
⬘兲one needs o assume a speci ic o m
o he inelas ic single-nucleon enso winel
␮
␯
共
⑀
¯
,
␳
⬘;
␬
,␭兲.I is
impo an o ema k ha he e exis s some ambigui y in he
choice made he e: o ins ance, se e al al e na i es in ol ing
di e en exp essions con aining he ou -momen a Q
␮
and/o
Q⬘
␮
a e possible, and hese can lead o di e en esul s. Fo
example, in Re . 关30兴, he modi ied ou momen um ans e
Q⬘
␮
is used, al hough hen a p esc ip ion mus be used in
o de o eco e he gauge in a iance o he nuclea had onic
enso , which is los by making his choice. In Appendix B
we p esen he speci ic exp essions o he inelas ic and QE
esponses ob ained o a gi en selec ion o he single-
nucleon enso s accoun ing o he ene gy shi . Apa om
he speci ic o m o he enso w
␮
␯
, he choice o he a gu-
men s o he single-nucleon inelas ic s uc u e unc ions,
w1,w2, also p esen s some ambigui ies. In ac , he a ailable
pa ame iza ions o w1,w2 ha we employ in ou calcula-
ions a e gi en o ee, on-shell, nucleons, while he inclu-
sion o he ene gy shi e ec i ely in oduces some “o -
shellness” o he ini ial nucleon, by al e ing he ene gy
balance a he e ex whe e i couples o he exchanged i -
ual pho on. In his case he bound-nucleon Bjo ken a iable
is no uniquely de e mined by he inal-s a e in a ian mass
and, since no heo e ically de i ed p esc ip ions exis , one
has o make some assump ions. As shown in Appendix B,
he inelas ici y pa ame e selec ed in his wo k,
␳
˜
, co e-
sponds o he one gi en as 共2mN
␻
˜
兲/兩Q2兩=
␳
⬘共
␶
⬘/
␶
兲, whe e
␻
˜
is he ene gy ans e ed o he nucleon in he sys em in
which he nucleon is a es . This means ha in ou nume i-
cal calcula ions, o a gi en se o alues o
␻
,Q2, and
␳
⬘we
employ ee-nucleon s uc u e unc ions aken a ou -
momen um Q2and Bjo ken a iable 1/
␳
˜
.
C. Ex ended ela i is ic e mi gas
As discussed in p e ious wo ks [8,16,18,20], o a ixed
alue o he in a ian mass
␮
X, he RFG yields a scaling
unc ion
共
␺
X
⬘兲= L共
␺
X
⬘兲= T共
␺
X
⬘兲=3
4共1−
␺
X
⬘2兲
␪
共1−
␺
X
⬘2兲共54兲
which, as a unc ion o he app op ia e scaling a iable
␺
X
⬘,is
he same o all alues o
␮
X.1
In Re . [18] he beha io o he longi udinal scaling unc-
ion was s udied o he exis ing wo ld da a in he QE egion.
This s udy showed ha o a good app oxima ion L共
␺
⬘兲su-
pe scales, ha is, i does no show any signi ican depen-
dence on he momen um ans e
␬
(scaling o he i s kind)
and is app oxima ely he same o all nuclea species (scal-
1The unc ion in Eq. (54)di e s om he one used in p e ious
wo k [20]by a mul iplica i e unc ion 2
␰
F/
␩
F
2关1+1
2
␰
F共1+
␺
X
⬘2兲兴.We
ha e checked ha his is nume ically unimpo an o all o he
kinema ical condi ions conside ed he e.
BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004)
035502-6
ing o he second kind). An exp ession o a phenomenologi-
cal longi udinal scaling unc ion, uni 共
␺
⬘兲, was ob ained by
i ing he da a [21]. Based on hese esul s, we now make he
ollowing hypo hesis: we assume ha his uni 共
␺
⬘兲, de i ed
om he da a, p o ides a good desc ip ion o 共
␺
X
⬘兲= L共
␺
X
⬘兲
= T共
␺
X
⬘兲,(“scaling o he ze o h kind”)as i implici ly con-
ains he ini ial-s a e physics, and hus we make, o any
␮
X,
he ollowing subs i u ion:
3
4共1−
␺
X
⬘2兲
␪
共1−
␺
X
⬘2兲→ ERFG共
␺
X
⬘兲= uni 共
␺
X
⬘兲.共55兲
To be mo e speci ic, we calcula e he esponse unc ions as
RQE
L,T=N
␩
F
3
␬
mN
␰
F model共
␺
⬘兲UQE
L,T,共56兲
Rinel
L,T共
␬
,
␶
兲=N
␩
F
3
␬
␰
F
冕
␮
h esh
1+2␭−
⑀
Sd
␮
X
␮
X model共
␺
X
⬘兲UL,T,共57兲
whe e
⑀
S=ES/mNis he dimensionless sepa a ion ene gy and
model共
␺
X
⬘兲=
再
3
4共1−
␺
X
⬘2兲
␪
共1−
␺
X
⬘2兲model = RFG
uni 共
␺
X
⬘兲model = ERFG.
共58兲
The unc ions RFG and ERFG a e shown in Fig. 1 as unc-
ions o
␺
X
⬘, while he unc ions UQE
L,Tand UL,Tin Eqs. 共56兲
and 共57兲a e gi en in Appendix B.
IV. RESULTS
In his sec ion we p esen ou esul s o c oss sec ions
and esponse and s uc u e unc ions. In compu ing he in-
elas ic had onic enso o Eq. (47), we employ phenomeno-
logical i s o he single-nucleon inelas ic s uc u e unc ions.
The la e a e measu ed in DIS expe imen s and a a ie y o
pa ame iza ions o w1and w2can be ound in he li e a u e
[15,32–36], including some a ia ions a ising om he di -
e en assump ions made o how o ex ac he neu on
s uc u e unc ions om deu e on da a. Unless s a ed o he -
wise, in he ollowing we adop he Bodek e al. i o Re s.
[15,32,33], which desc ibes bo h he deep inelas ic and eso-
nance egions. Fo he QE con ibu ions, we employ he o m
ac o pa ame iza ion o Re . [37]. The sensi i i y o he
esul s o he di e en pa ame iza ion choices will be dis-
cussed la e .
Addi ionally, o he Fe mi momen um and he ene gy
shi we will employ he alues ob ained in Re . [20],
namely, kF=220 MeV/c,
␻
shi =20 MeV o ca bon, kF
=236 MeV/c,
␻
shi =18 MeV o aluminum, kF
=241 MeV/c,
␻
shi =23 MeV o i on, and kF=245 MeV/c,
␻
shi =25 MeV o gold.
A. C oss sec ions
In his sec ion we p esen ou esul s o he c oss sec ions
in he RFG and ERFG models and compa e hem wi h he
a ailable expe imen al da a [38–41].
FIG. 2. Inclusi e c oss sec ion o elec on sca e ing om ca -
bon a Einc=500 MeV and
␪
e=60° s he ene gy ans e . The cal-
cula ion includes an ene gy shi
␻
shi =20 MeV and he sepa a e
QE and inelas ic con ibu ions o he c oss sec ion a e shown. Da a
a e om Re . [41].
FIG. 3. As o Fig. 2, bu a Einc=2.020 GeV and sca e ing
angle
␪
e=15° (a)and
␪
e=20° (b). Da a a e om Re . [39].
FIG. 1. Scaling unc ion model共
␺
X
⬘兲o Eq. (58) o he RFG and
ERFG models.
INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004)
035502-7
In Fig. 2 we show he inclusi e c oss sec ion o a 12C
a ge a Ee=500 MeV and
␪
e=60°. We sepa a e he QE
om he inelas ic con ibu ion. We no ice ha he shi ed
RFG model (solid line)yields oughly he igh posi ion and
heigh o he QE peak, bu ails o ep oduce he ails o he
peak, gi ing in pa icula an unobse ed dip a
␻
⯝200 MeV. On he o he hand he ERFG model (do ed
line), while ep oducing he da a in he ails be e , signi i-
can ly unde es ima es he c oss sec ion a he peak. This is
ela ed o he ac ha , as shown in Fig. 1, he peak o he
ERFG uni e sal unc ion ERFG is lowe han he co espond-
ing RFG alue. Due o he la ge ex ension o ERFG o e
␺
X
⬘
he no maliza ion o he wo unc ions is he same, namely,
兰 RFGd
␺
X
⬘=兰 ERFGd
␺
X
⬘=1. One migh hen nai ely expec he
in eg al in Eq. (57)which yields he inelas ic esponse unc-
ions o be he same in he wo models. Howe e , a close
inspec ion shows ha his is no he case because he in e-
g a ion limi s and/o he weigh ing p o ided by UL,Ta e such
ha he ERFG in eg al does no “sa u a e” as does he RFG
one.
Figu es 3–5 co espond o di e en kinema ical condi-
ions, namely, Ee=2.020 and 3.595 GeV (SLAC)and Ee
=4.045 GeV (JLab)and a ious sca e ing angles. Conce n-
ing Figs. 3(a)and 3(b), a simila end pe sis s, wi h he
FIG. 4. As o Fig. 2, bu a
Ee=3.595 GeV and sca e ing
angle
␪
e=16° (a), 20° (b), 25° (c),
and 30° (d). Da a a e om Re .
[39].
FIG. 5. As o Fig. 2, bu a
Einc=4.045 GeV and sca e ing
angle
␪
e=15° (a), 30° (b), 45° (c),
and 74° (d). Da a a e om Re .
[38].
BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004)
035502-8
ERFG model signi ican ly unde es ima ing he da a in he
egion o he QE peak, whe eas he RFG is close o he da a
(pa icula ly o
␪
e=20°), al hough i lea es no oom o
o he con ibu ions o be added. No e also ha o his sca -
e ing angle he inelas ic channel s a s o be sizable.
Examining Figs. 4 we ema k ha he QE peak, which is
mo e clea ly sepa a ed om he inelas ic egion in Fig. 4(a),
is again well ep oduced in he low-
␻
ail by he ERFG,
while i s maximum ag ees be e wi h he RFG. On he o he
hand he inelas ic c oss sec ion is in all cases unde es ima ed
by he ERFG, while he RFG alone would oughly accoun
o wha is obse ed.
Simila commen s apply o Fig. 5(a), co esponding o
highe ene gy and low sca e ing angle. Fo highe angles
[Figs. 5(b)–5(d)] he da a lie oughly in be ween he p edic-
ions o he ERFG (smalle )and RFG (la ge )models, he
o me again ep oducing he low-
␻
beha io be e . As a
gene al esul we obse e ha as he sca e ing angle in-
c eases he ange o alidi y o he ERFG also inc eases.
Finally, in Figs. 6 and 7 we conside sligh ly di e en
kinema ical condi ions, co esponding o ixed alues o 兩Q2兩
in he ange 2–10 共GeV/c兲2, and a ious elec on ene gies
(in he ange 8–25 GeV)and angles 共12° –22°兲. Theo e ical
esul s o 56Fe a e shown as unc ions o he “labo a o y”
Bjo ken a iable xL. The da a co esponding o a ixed Q2a e
aken a di e en alues o Eeand
␪
e. Fo 兩Q2兩=2 and
10 共GeV/c兲2(Fig. 6) he a ious da a i easonably well on
one plo , whe eas o 兩Q2兩=5 共GeV/c兲2(Fig. 7), o cla i y
we ha e sepa a ed he da a in o h ee se s as indica ed in he
igu e cap ion. We no ice ha a la ge xL共艌0.6兲 he da a a e
close o he ERFG p edic ions, a low xL共0.1–0.3兲 hey a e
close o he RFG calcula ion and o 0.3艋xL艋0.6 hey lie
in be ween he wo models. This gene al end seems o be
espec ed o all alues o Q2(a leas whe e da a a e a ail-
able).
We ha e also analyzed he e ec in oduced by di e en
elec omagne ic o m ac o pa ame iza ions ([42–44]) and
e i ied ha i can p oduce a ±3 % unce ain y a he QE
peak, bu does no change he gene al ag eemen /
disag eemen o he models wi h he da a. Mo eo e , i
should be ema ked ha , a he ene gies conside ed in his
sec ion, he con ibu ion om he esonance egion o he
inelas ic pa o he c oss sec ion is qui e impo an and hus
a compa ison wi h esul s ob ained by using pu ely DIS pa-
ame iza ions [34,36]o he single-nucleon s uc u e unc-
ions is no app op ia e. A he highes 兩Q2兩 alues conside ed
he e [Figs. 6(b)and 7], he use o di e en pa ame iza ions
[34,36]does no p oduce signi ican a ia ions in he esul s.
An impo an commen , al eady an icipa ed in he in o-
duc ion, is in o de . The RFG and ERFG models conside ed
FIG. 6. Di e en ial c oss sec ion d
␴
/d
␻
d⍀e o elec on sca -
e ing on i on, shown as a unc ion o xLa ixed 兩Q2兩. Panel (a):
兩Q2兩=2共GeV/c兲2, he expe imen al poin s, om igh o le , a e
aken a 关Ee共GeV兲,
␪
e共deg兲兴=共8,11.8兲,共8,12.4兲,共8,13.6兲,共9.7,
11.8兲,共12,11.8兲,共15,11.8兲. Panel (b):兩Q2兩=10 共GeV/c兲2, he ex-
pe imen al poin s, om igh o le , a e aken
a 关Ee共GeV兲,
␪
e共deg兲兴⫽(15,16.0),(15,17.1),(17,15.0),(17,16.9),(21,
14.1),(24,14.1). Da a a e om Re . [40].
FIG. 7. As o Fig. 6, bu a 兩Q2兩=5共GeV/c兲2. The wo da a in
he uppe panel, om igh o le , co espond o Ee=8 GeV and
␪
e=22° and o Ee=9.7 GeV and
␪
e=19.7°; he da a in he middle
panel ha e ixed Ee=12 GeV and, om igh o le ,
␪
e
=12.8,13.3,14.2,15.8,20.6°; he da a in he lowe panel, om igh
o le , a e aken a 共Ee=15 GeV,
␪
e=13.2°兲,共Ee=17 GeV,
␪
e
=13.5°兲,共Ee=24.5 GeV
␪
e=11.1°兲. Da a a e om Re . [40].
INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004)
035502-9
␺
⬘2=1
␰
F
冉
␬
冑1
␶
⬘+1−␭⬘−1
冊
共B13兲
and he elec omagne ic s uc u e unc ions by
w1,QE共
␶
兲=
␶
GM
2共
␶
兲,
w2,QE共
␶
兲=GE
2共
␶
兲+
␶
GM
2共
␶
兲
1+
␶
wi h GE,M he p o on o neu on Sachs elec omagne ic o m
ac o s.
Finally, he “ o al” esponse unc ions a e e alua ed by
adding he abo e QE esponses o he inelas ic ones, i.e.,
R o
L,T=RQE
L,T+Rinel
L,T.
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