Few-Body Sys (2024) 65:78
h ps://doi.o g/10.1007/s00601-024-01944-y
M. Miho ilo iˇc·L. Do ia ·P. Achenbach ·
A. M. Ankowski ·S. Bacca ·
D. Bosna ·A. Denig ·M.O. Dis le ·
A. Esse ·I. F išˇci´c·C. Gius i ·M. Hoek ·S. Kegel ·
M. Li ich ·G.D. Megias ·H. Me kel ·U. Mülle ·
J. Pochodzalla ·B. S. Schlimme ·M. Scho h ·C. S ien i ·
S. Ši ca ·J. E. Sobczyk ·Y. S ö inge ·M. Thiel
Measu emen o he 12C(e,e)C oss Sec ions a
Q2=0.8GeV2/c2
Recei ed: 20 June 2024 / Accep ed: 1 July 2024 / Published online: 1 Augus 2024
© The Au ho (s) 2024
Abs ac We p esen he indings o a s udy based on a new inelas ic elec on-sca e ing expe imen on he
12C nucleus ocusing on he kinema ic egion o Q2=0.8GeV2/c2. The measu ed c oss sec ion is sensi i e o
he ans e se esponse unc ion and p o ides a s ingen es o heo e ical models, as well as o he heo e ical
assump ions made in Mon e-Ca lo e en -gene a o codes de eloped o he in e p e a ion o neu ino-nucleus
expe imen s,suchasDUNEandHype K.We ind ha mode ngene a o ssuchasGENIEandGiBUU ep oduce
ou new expe imen al da a wi hin 10%.
M. Miho ilo iˇc·S. Ši ca
Jože S e an Ins i u e, 1000 Ljubljana, Slo enia
M. Miho ilo iˇc·S. Ši ca
Facul y o Ma hema ics and Physics, Uni e si y o Ljubljana, 1000 Ljubljana, Slo enia
L. Do ia ·S. Bacca ·A. Denig ·M. O. Dis le ·A. Esse ·M. Hoek ·S. Kegel ·M. Li ich ·H. Me kel·U. Mülle ·
J. Pochodzalla ·B. S. Schlimme ·M. Scho h ·C. S ien i ·J. E. Sobczyk ·Y. S ö inge ·M. Thiel
Ins i u ü Ke nphysik, Johannes Gu enbe g-Uni e si ä Mainz, 55128 Mainz, Ge many
P. Achenbach
Thomas Je e son Na ional Accele a o Facili y, Newpo News, VA 23606, USA
A. M. Ankowski
Ins i u e o Theo e ical Physics, Uni e si y o W ocław, pl. Maxa Bo na 9, 50-204 W ocław, Poland
L. Do ia ·S. Bacca ·A. Denig ·H. Me kel (B
)·B. S. Schlimme ·C. S ien i ·J. E. Sobczyk ·M. Thiel
PRISMA+ Clus e o Excellence, Johannes Gu enbe g-Uni e si ä Mainz, 55128 Mainz, Ge many
E-mail: [email p o ec ed]
D. Bosna ·I. F išˇci´c
Depa men o Physics, Uni e si y o Zag eb, HR-10002 Zag eb, C oa ia
C. Gius i
INFN, Sezione di Pa ia, 27100 Pa ia, I aly
G. D. Megias
Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad de Se illa, 41080 Se ille, Spain
78 Page 2 o 9 M. Miho ilo iˇce al.
1 In oduc ion
Elec ons ep esen a e y p ecise p obe o he in es iga ion o he a omic nucleus [1]. In he pas decades,
expe imen s wi h elec ons ha e p o ided inc easingly accu a e in o ma ion on he s uc u e o nuclei and
hei cons i uen s [2–9]. A he hea o his e o a e he inelas ic sca e ing expe imen s on nuclea a ge s
a ene gies below 1 GeV, which gi e insigh in o he p ope ies and dynamics o nucleons embedded in he
nuclea medium. In such sca e ing p ocesses, an elec on wi h ene gy E0in e ac s wi h he nucleus a es by
exchanging a i ual pho on ans e ing ene gy ωand momen um q, such ha Q2=q2−ω2>0. Using he
nucleon mass mNas a scale, he ene gy and momen um ans e a iables can be ew i en in dimensionless
o m as
λ=ω
2mN
,κ=q
2mN
,τ=Q2
4m2
N=κ2−λ2.
The di e en ial c oss sec ion desc ibing he inclusi ein e ac ion o he elec on wi h he nucleus can be w i en
as
d2σ
ddω=σM[ LRL(ω, q)+ TRT(ω, q)],(1)
whe e σMis he Mo c oss Sec . [1], while Land Ta e kinema ic ac o s gi en by
L=τ
κ22
,
T=τ
2κ2+ an2θe
2,
and θeis he angle o he sca e ed elec on. The c oss sec ion depends on wo nuclea esponses, he longi-
udinal and he ans e se esponse unc ions, which a e bo h unc ions o ωand q=|q|. The longi udinal
esponse unc ion, RL(ω, q), depends on he cha ge ope a o and ca ies in o ma ion on he nucleon-nucleon
co ela ions, while he ans e se esponse unc ion, RT(ω, q), is d i en by he magne ic cu en s [1].
Va ious c oss sec ion measu emen s we e pe o med, mos ly be o e 2000, bu he acqui ed da a we e
domina ed by he longi udinal (cha ge) pa o nuclea esponse. His o ically, he mos ex ensi ely s udied
nucleus has been ca bon. Fo his nucleus, he iches sample o (e,e)da a exis s. I consis s o almos 3500
da a poin s om 12 expe imen s [10,11] o ene gies be ween 0.12 and 17.3 GeV and sca e ing angles up o
145◦. These da a ha e been used o s udy he s uc u e o his nucleus and o de elop models desc ibing i s
elec omagne ic esponse.
In he pas , heo e ical calcula ions o 12C we e o en limi ed o he quasi-elas ic (QE) egion, and he mos
demanding pa was he desc ip ion o he ans e se esponse, which has been o a long ime incomple e [1].
A mo e comp ehensi e desc ip ion o he 12C(e,e)c oss sec ion was de eloped in he mic oscopic calcula ion
byGile al. [12], and mo e ecen ly Megias e al. [13] p oposed a supe scaling model, called SuSA 2-MEC,
which conside s he comple e inelas ic spec um. The model shows qui e good ag eemen wi h da a o e a
b oad ange o ene gy ans e . Howe e , he desc ip ion o he c oss sec ions a la ge sca e ing angles emains
incomple e.
An impo an mo i a ion o new s udies o inclusi e c oss sec ions comes also om he neu ino physics
communi y [14,15]. Sho - and long-baseline neu ino expe imen s de ec neu inos h ough hei in e ac ions
wi hnucleiandaima hep ecisemeasu emen o neu inomasses,mixingangles,andCP- iola ingphasein he
lep on sec o . These measu emen s ep esen one o he highes p io i ies o con empo a y undamen al physics
and hinge on he abili y o he expe imen s o econs uc he neu ino ene gy and on he p ecise knowledge
o he neu ino-nucleus c oss sec ions. Al hough a igo ous expe imen al p og am o he measu emen o
such c oss sec ions is in p og ess [16–20], neu ino expe imen s a e mos ly limi ed by s a is ical unce ain ies
and he lack o knowledge o he neu ino lux. Elec on sca e ing expe imen s, wi h a p ecisely de e mined
beam ene gy and he possibili y o pe o m inclusi e as well as exclusi e measu emen s wi h di e en inal
s a es, ha e he po en ial o p o ide e y p ecise da a o es ing he nuclea models employed in neu ino
expe imen s.
Indeed, i has been demons a ed ha he in e p e a ion o he measu ed neu ino oscilla ions equi es ex en-
si e heo e ical and expe imen al suppo om he nuclea physics communi y. In his con ex he 12C(e,e)
eac ion has played an impo an ole in he de elopmen o eliable models desc ibing c oss Sec . [13,21]in
expe imen s like MiniBooNE [22], MINER A [23], and T2K [24] ha use ca bon-based ma e ials (mine al
Measu emen o he 12C(e,e)C oss Sec ions Page 3 o 9 78
oils, plas ic scin illa o s) as de ec o medium. To ensu e u he in ol emen o mode n neu ino e en gene a-
o s like GENIE [25]andGiBUU[26], he ad ances in he buil -in heo e ical models mus be complemen ed
by he new expe imen al da a on ele an nuclea a ge s and in ele an kinema ics [27,28].
In his pape we ocus on 12C and p esen new da a a Q2=0.8GeV2/c2 o wo easons. On he one hand
ca bon is an in e es ing a ge o neu ino expe imen s as men ioned abo e, and on he o he hand new da a a
la ge Q2in a kinema ic domina ed by he ans e se esponse will boos u he heo e ical p og ess [29–38].
2 Expe imen
The measu emen o he inclusi e c oss sec ion on 12C was pe o med a he Mainz Mic o on (MAMI) acili y
using he spec ome e se up o he A1 Collabo a ion [39]. In he expe imen , an elec on beam wi h ene gy
E0=855MeV was used in combina ion wi h a 43 mg/cm2 hick ca bon oil a ge . Fo measu ing he c oss
sec ion as a unc ion o ene gy o sca e ed elec on Ewe employed a magne ic spec ome e (spec ome e
A) wi h 20% momen um accep ance and 28 ms angula accep ance. The spec ome e was posi ioned a a
ixed angle o 70◦, while i s momen um se ings we e adjus ed o measu e he c oss sec ion as a unc ion o
ω=E0−E. The measu emen s we e made o se en di e en momen um se ings be ween 310 MeV/c
and 650 MeV/cin o de o collec da a in he egion o he QE peak and he - esonance. Fo each se ing
we collec ed 1.8 million e en s. The cen al momen um o each se ing was measu ed o a ela i e accu acy
o 8 ×10−5. The spec ome e was equipped wi h a de ec o package consis ing o wo laye s o e ical
d i chambe s (VDCs) o acking, wo laye s o plas ic scin illa ion de ec o s o igge ing, and a h eshold
Che enko de ec o o elec on iden i ica ion. The beam cu en was be ween 2 and 3 μA and was limi ed by
he maximum da a acquisi ion a e, esul ing in a aw a e o abou 500 Hz. The cu en was de e mined by a
non-in asi e luxga e-magne ome e wi h an accu acy o <0.2%.
The expe imen p o ided new c oss sec ions measu emen s in he egion o beam ene gies and sca e ing
angles, whe e he exis ing measu emen s a e e y spa se, see Fig. 1. The quasi-elas ic peak is cen e ed a
|q|=0.84 GeV/c, hus nicely complemen ing p e ious measu emen s a |q|≈0.8GeV/cpe o med a
560 MeV and 1299 MeV [40,41].
The expe imen al c oss sec ions o he 12C(e,e) eac ion we e ex ac ed om he da a by di iding he
measu ed dis ibu ions o coun s by he in eg a ed luminosi y and he solid angle accep ed by he spec ome e .
The accep ed solid angle was simula ed using a dedica ed simula ion o he h ee spec ome e s acili y o
he A1 Collabo a ion [39], Simul++. To ensu e a eliable compa ison wi h he da a, he simula ion included
ealis ic momen um and spa ial esolu ions o he spec ome e . The ela i e momen um, angula , and e ex
esolu ions (FWHM) we e 2.4×10−4,4.7m ad,and9.4 mm, espec i ely. The simula ion also conside ed he
This wo k
Sealock e al. 1989
Ba eau e al. 1983
Exis ing da a
q
[
GeV
/
c
]
ω[GeV]
10.80.60.40.20
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Fig. 1 Kinema ic con igu a ions o he 12C(e,e)c oss sec ion da a in e ms o ene gy and momen um ans e s ωand |q|. Black
poin s ep esen a ailable c oss sec ion measu emen s wi h ela i e unce ain ies smalle han 10%. The kinema ics co e ed by
his wo k a e p esen ed wi h ed ci cles. The complemen a y measu emen s o Ba eau e al. [40], and Sealock e al. [41], also
a |q|∼0.8GeV/c, a e shown wi h blue diamonds and g een squa es, espec i ely
78 Page 4 o 9 M. Miho ilo iˇce al.
Table 1 De ec ion e iciencies o he se up
Con ibu ion E iciency ac o Unce ain y
Scin illa o e iciency 0.990 0.003
Che enko e iciency 0.999 0.001
VDC e iciency 0.999 0.001
Pa icle iden i ica ion 0.983 0.017
The co ec ion ac o s o he c oss sec ions a e gi en by hei in e ses
Fig. 2 Le : Measu ed c oss sec ion compa ed o he QE calcula ions o Megias e al. [13], Gius i e al. [45,46], Sobczyk e
al. [47] and Benha e al. [10]. The g ay band shows he en elope o he quasi-elas ic c oss-sec ion calcula ions and ep esen s a
measu e o he di e ences be ween di e en models. Cen e : Measu ed c oss sec ion compa ed o he ull heo e ical calcula ions
o Megias e al. [13] based on he SuSA 2-MEC model and p edic ions o Ankowski [48] based on he global Fe mi gas model
(FG) and local Fe mi gas model (LFG). Righ : Compa ison o he new da a wi h he esul s o he Mon e-Ca lo gene a o s
GiBUU [26] and GENIE ( e sion 3.4) [25] employing LFG and SuSA 2-MEC [21] nuclea c oss-sec ion models
elec on ene gy co ec ions due o mul iple sca e ing and adia ion losses. The in e nal and ex e nal adia i e
co ec ions we e included using he o malism o Mo and Tsai [42]. The accompanying mul iple sca e ing
co ec ions in he a ge and su ounding ma e ial we e app oxima ed by a Landau dis ibu ion [43]. Al oge he ,
he ene gy co ec ions ha e less han 4% e ec on he measu ed c oss sec ion.
The in eg a ed luminosi y was de e mined om he p oduc o he accumula ed cha ge and he su ace
densi yo he a ge ma e ial,co ec ed o dead- imeandDAQp escale ac o s.Theluminosi ywasde e mined
sepa a ely o each collec ed da a-sample o ensu e ha dead- ime and p escale co ec ions we e consis en ly
conside ed when weigh ing he measu ed spec a.
The measu ed spec a we e co ec ed o he ine iciencies o he de ec ion sys em, see Table 1.The
e iciencies o he scin illa ion de ec o and he Che enko de ec o we e e alua ed in a pas expe imen [44]
and we e de e mined o be 99.0% and 99.85 %, espec i ely. The e iciency o he ack econs uc ion in
he VDCs was de e mined o be (99.98 ±0.05)%. All h ee co ec ions we e conside ed as mul iplica i e
co ec ion ac o s.
Se e al cu s we e applied o bo h da a and simula ion. Fi s , a cu on he Che enko signal was applied
o iden i y elec ons and minimize he backg ound a ising om cosmic pa icles and by nega i ely cha ged
pions om he 12C(e,π−) eac ion. This cu was ollowed by cu s on he nominal momen um and angula
accep ance o he spec ome e in o de o emo e he a e ac s a i s edges, caused by he ine icien pa s o he
de ec o s, inge ields in he spec ome e s, and seconda y pa icles esca e ed om pa s o he collima o .
The ex ac ed c oss sec ions a e p esen ed in Fig. 2.
The sys ema ic unce ain ies o he ex ac ed c oss sec ions a e a combina ion o a ious con ibu ions.
The unce ain ies ela ed o he de ec o e iciencies a e collec ed in Table 1. The unce ain y o he luminosi y
is gi en by he unce ain y o he absolu e beam cu en calib a ion, which amoun s o 3.3 nA a 855 MeV and
he luc ua ions o he beam cu en we e ela ed o he ins abili ies o accele a o ope a ion. The la e we e
smalle han 5 nA, esul ing in he o al sys ema ic unce ain y smalle han 0.16 %. The dominan con ibu ion
Measu emen o he 12C(e,e)C oss Sec ions Page 5 o 9 78
o he sys ema ic unce ain y is ela ed o he misiden i ica ion o pa icles in he Che enko de ec o and cu s
applied o dis inguish elec ons om pions and muons. This unce ain y was es ima ed o be 1.7%.Thelas
ele an con ibu ion o he sys ema ic unce ain y can be e alua ed by he o malism o Mo and Tsai [42],
employed o desc ibe adia i e and mul iple sca e ing co ec ions o he c oss sec ion. These co ec ions add
0.2 % o he o al unce ain y o he measu ed c oss sec ions. Finally, he unce ain y o he posi ion o he
ex ac ed c oss sec ions on he ene gy scale is ela ed o he ambigui ies in he absolu e ene gy calib a ion o
he accele a o and spec ome e and amoun s o 2.7 MeV, which is less han 1/3 o he employed ene gy bin
size.
3 Compa ison o Models and E en Gene a o s
The ex ac ed c oss sec ion is i s compa ed o QE calcula ions o Gius i e al. [45,46], Sobczyk e al. [47],
Megias e al. [13] and Benha e al. [10]. Figu e 2(Le ) shows ha he calcula ions ag ee wi h each o he a
he le el o 4 % a he op o he QE peak. The compa ison o he expe imen al esul s o he comp ehensi e
calcula ions o Megias e al. [13], which a e based on he SuSA 2-MEC model, a e shown in Fig. 2(Cen e ).
The model exhibi s e y good o e all ag eemen wi h he da a, on a e age a he le el o 7%. Su p isingly
he la ges inconsis ency be ween he da a and he calcula ions appea s a he op o he QE peak, whe e he
disc epancyis 9 %. Since heQEcalcula ionsshow aconsis en pic u e he e, heobse ed disc epancybe ween
he da a and he SuSA 2-MEC model is mos likely ela ed o he incomple e o inconsis en desc ip ion o
he p ocesses in he “dip" egion, which a e he only emaining ele an con ibu ions o he c oss sec ion a
ω≤400 MeV.
An ag eemen a a simila le el has been achie ed by Ankowski [48] who calcula ed c oss sec ions by
using bo h he global Fe mi gas model (FG) and he local Fe mi gas model (LFG) in combina ion wi h he
Bos ed-Ch is y [49,50] app oach o desc ibing pion p oduc ion p ocesses in he “dip” and in he - esonance
egion. The ela i e de ia ion o he FG calcula ion om he da a is on a e age 10%, while he p edic ion o
he LFG model ag ees wi h he da a a he le el o 8%. Bo h calcula ions exhibi a isible inconsis ency a he
op o he QE peak and in he “dip" egion, whe e he calcula ed c oss-sec ion do no ollow he co ec end
o he da a.
Finally, he ex ac ed c oss-sec ions we e compa ed also o he esul s o he Mon e-Ca lo gene a o s
GiBUU [26] and GENIE [11,25]. Figu e2(Righ ) shows ha a he selec ed kinema ic se ing he gene a o s
desc ibe he da a easonably well. GiBUU ag ees wi h he da a a he le el o 9%. The accu acy o he GENIE
gene a o depends on he model used o calcula ing nuclea c oss sec ions. When he local Fe mi gas model is
used, he calcula ed c oss sec ion ag ees on a e age wi h he da a a he le el o 22%. Simila ly o he esul s
o Ankowski’s LFG model, he simula ed c oss sec ion o e es ima es he QE c oss sec ion. Addi ionally, he
GENIE simula ion lacks s eng h in he - egion, whe e he calcula ed c oss sec ion is 17% smalle han he
da a. The simula ed esul s imp o e when GENIE uses he SuSA 2 model o desc ibing QE sca e ing and
p ocesses in he “dip" egion. In his case we obse e much be e ag eemen be ween he da a and simula ion
a he QE peak. No e ha GENIE s ill uses i s de aul model o desc ibing he c oss sec ion in he - esonance
egion, which is he sou ce o di e ence wi h espec o he Megias e al. esul shown in he cen e panel.
Fo a mo e insigh ul analysis o he measu ed c oss sec ion and compa ison wi h he exis ing esul s
ob ained unde di e en kinema ic condi ions, he scaling o malism [51] can be employed. The o malism was
i s de eloped wi hin he amewo k o he ela i is ic Fe mi gas model whe e he cha ac e is ic momen um is
he Fe mi momen um kF, which can be exp essed as a dimensionless scale pa ame e ξF=1+k2
F/m2
N−1.
Building on his o malism, wo dimensionless scaling a iables ψand ψwe e p oposed [52]:
ψ≡1
√ξF
λ−τ
(1+λ)τ +κ√τ(τ +1)
,
ψ≡1
√ξF
λ−τ
(1+λ)τ+κ√τ(τ+1)
.(2)
The a iable ψis co ec ed o an empi ical ene gy shi Eshi co esponding o he a e age o he sepa a ion
ene gies o he a ious shells con ibu ing o he nuclea g ound s a e [51]. Wi h he shi one achie es he cen e
o he QE peak o be a ψ=0. Thenecessa y shi is achie ed by subs i u ing λand τwi h λ=λ−Eshi /2mN
and τ=κ2−λ2.

78 Page 6 o 9 M. Miho ilo iˇce al.
The idea o he scaling o malism is o ac o ize he elas ic c oss sec ion on a single nucleon, ob aining in
his way a uni e sal scaling unc ion which con ains in o ma ion abou he nuclea s uc u e. Fo ha pu pose,
educed longi udinal and ans e se esponse unc ions a e in oduced as [52]:
L=kF
RL
GL(κ, λ), T=kF
RT
GT(κ, λ).
The unc ions GLand GTa e exp essed as:
GL(κ, λ) =(κ2/τ)[˜
G2
E+˜
W2]
2κ[1+ξF(1+ψ2)/2],
GT(κ, λ) =2τ˜
G2
M+˜
W2
2κ[1+ξF(1+ψ2)/2],
whe e
=ξF(1−ψ2)√τ(1+τ
κ+1
3ξF(1−ψ2)τ
κ2,
˜
W1=τ˜
G2
M,
˜
W2=1
1+τ˜
G2
E+τ˜
G2
M,
˜
G2
E=ZG
p
E
2+NG
n
E
2,
˜
G2
M=ZG
p
M
2+NG
n
M
2.
He e Zand N ep esen he numbe o p o ons and neu ons in he nucleus, espec i ely, Gp,n
Eand Gp,n
M
a e nucleon elec ic and magne ic o m ac o s [53]. Using hese unc ions a dimensionless scaling unc ion
o he o al c oss sec ion can be w i en as:
=kF
d2σ/dedω
σM[ LGL(κ, λ) + TGT(κ, λ)].
= Lsin2χTL + Tcos2χTL,(3)
whe e he angle χTL is de ined as
an2χTL = LGL
TGT
.(4)
This angle cha ac e izes he a io be ween he longi udinal and ans e se con ibu ions o he c oss sec ion.
A χTL ≈0 he inclusi e c oss sec ion is domina ed by he ans e se esponse, while a χTL ≈90◦ he c oss
sec ion is go e ned by he longi udinal esponse [51].
Using he scaling a iables in Eqs. (2), (3), and (4), he measu ed c oss sec ions could be compa ed o
he p e ious measu emen s a |q|≈0.8GeV/co Ba eau e al. [40] and Sealock e al. [41]. The ex ac ed
alues o he dimensionless scaling unc ion (ψ)a e shown in Fig. 3. This igu e shows he app oxima e
scaling o he measu ed c oss sec ions which s a s o b eak o ψ>0 when he ans e se con ibu ions
o he - esonance begin o domina e he c oss Sec . [51,52]. A he QE peak he expe imen al alues o
his wo k and Sealock e al. collec ed a sca e ing angles θ<90◦, which co esponds o χTL =36◦and
46◦, espec i ely, ag ee e y well wi h each o he . On he o he hand, he scaling unc ion econs uc ed om
da a o Ba eau e al. a θ=145◦is o e 20 % highe a he op o he QE peak. These da a coincide wi h
a much smalle alue o χTL =12◦, and a e hus domina ed by he ans e se esponse RTwhich is known
o b eak scaling due o a ious nonelas ic con ibu ions anging om inal-s a e-in e ac ion (FSI) e ec s o
con ibu ions om he meson-exchange cu en s (MEC) [51].
Expe imen al alues we e compa ed also o he ull calcula ions o Megias e al.. In e es ingly, a he op
o he QE peak he heo y is consis en wi h he da a o Ba eau e al., bu o e shoo s he expe imen al alues
o his wo k and ha o Sealock e al.. The analysis o he QE c oss sec ions has e ealed ha o hese wo
da a se s he QE pa ma ches he s eng h o he measu ed c oss sec ion, indica ing ha he disc epancy migh
be due o he o e es ima ed MEC con ibu ions.
Measu emen o he 12C(e,e)C oss Sec ions Page 7 o 9 78
SuSA 2 QE (1299 MeV)
SuSA 2 QE (855 MeV)
SuSA 2 QE (560 MeV)
SuSA 2 (1299 MeV)
SuSA 2 (855 MeV)
SuSA 2 (560 MeV)
Sealock e al.
Ba eau e al.
This wo k
ψ
(ψ)
3210−1−2
1.5
1
0.5
0
Fig. 3 The scaling unc ion (ψ)a |q|≈0.8GeV/c. The expe imen al alues o his wo k, Ba eau e al. [40], and Sealock
e al. [41] a e shown oge he wi h he esul s o he SuSA 2-MEC model a co esponding beam ene gies o 560MeV (Ba eau
e al.), 855 MeV ( his wo k) and 1299 MeV (Sealock e al.) and p esen ed wi h he ull lines. The dash-do ed lines a e used o
p esen con ibu ions o he QE p ocesses o he calcula ed scaling unc ions [13]
4 Conclusions
We p esen ed he expe imen al c oss sec ion o he inclusi e eac ion 12C(e,e)a Q2=0.8GeV2/c2.The
measu emen was made a he kinema ics ha is ele an o he accele a o based neu ino expe imen s, bu
whe e he a ailable da a a e sca ce. Since he new da a se has no been conside ed in any o he heo e ical
models, we could use hem o challenge he calcula ions and gene a o s employed in he in e p e a ion o he
expe imen s wi h neu inos. We ha e demons a ed ha he e en gene a o s in combina ion wi h selec ed
nuclea models a e capable o desc ibing da a a he le el o 10 %. Fo e en highe p ecision o he gene a o s
in he u u e, he buil -i nuclea models need o be u he e ined, especially he desc ip ion o he ans e se
pa o he in e ac ion, which go e ns he inclusi e c oss sec ion in he egion o he “dip" and - esonance.
To achie e his goal, u he heo e ical and expe imen al in es iga ions o c oss sec ions a Q2≈1GeV2/c2
a e needed. Kinema ics a lowe Q2would also be use ul o es MEC models. In pa icula , he p ospec o
ha ing ab-ini io calcula ions in he ligh and mid-mass sec o wi h MEC [30,35,54–57] will mo i a e u he
expe imen al ac i i ies in he u u e on a ious a ge s.
Acknowledgemen s The au ho s would like o hank he MAMI accele a o g oup o he excellen beam quali y which made
his expe imen possible. We also hank U. Mosel o use ul discussions. This wo k is suppo ed by he Fede al S a e o Rhineland-
Pala ina e, by he Deu sche Fo schungsgemeinscha (DFG) h ough he Clus e o Excellence “P ecision Physics, Fundamen al
In e ac ions, and S uc u e o Ma e " (PRISMA+EXC 2118/1) unded by he DFG wi hin he Ge man Excellence S a egy
(P ojec ID 390831469), by he DFG g an "Elec on Sca e ing on Nuclei o Neu ino Physics" (P ojec ID 521414474), by he
Slo enian Resea ch Agency unde G an s P1-0102 and J1-4383, by he Eu opean Union’s Ho izon 2020 esea ch and inno a ion
p og amme unde he Ma ie Skłodowska-Cu ie g an ag eemen No. 101026014, by C oa ian Science Founda ion unde he
p ojec IP-2018-01-8570 and by he Uni e si y o Tokyo ICRR’s In e Uni e si y Resea ch P og am FY2024 (Re . 2024i-J-001),
Japan; he Spanish Minis e io de Ciencia, Inno ación y Uni e sidades and ERDF (Eu opean Regional De elopmen Fund) unde
con ac PID2020-114687GB100 and by he Jun a de Andalucía g an No. FQM160.
Au ho Con ibu ions M.M. and L.D. w o e he main manusc ip ex . S.B., A.A., C.G., G.M. and J.S. p o ided he heo e ical
in e p e a ion o he da a, all o he au ho s we e in ol ed in he da a aking. All au ho s e iewed he manusc ip .
Da a A ailabili y Da a can be ob ained om M. Miho ilo ic <miha.miho ilo [email protected] >.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use,
sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal
au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he
hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi
line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed
by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To
iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/.
78 Page 8 o 9 M. Miho ilo iˇce al.
Decla a ions
Con lic o in e es The au ho s decla e no Con lic o in e es .
Re e ences
1. S. Bo i, C. Gius i, F.D. Paca i, Nuclea esponse in elec omagne ic in e ac ionswi h complex nuclei. Phys. Rep . 226, 1–101
(1993). h ps://doi.o g/10.1016/0370-1573(93)90132-W
2. D. And oic e al., S ange qua k con ibu ions o pa i y- iola ing asymme ies in he backwa d angle G0 elec on sca e ing
expe imen . Phys. Re . Le . 104, 012001 (2010). h ps://doi.o g/10.1103/PhysRe Le .104.012001.a Xi :0909.5107 [nucl-
ex]
3. I. Ko o e , P obing he epulsi e co e o he nucleon-nucleon in e ac ion ia he 4He(e,epN) iple-coincidence eac ion.
Phys.Re .Le .113(2), 022501 (2014). h ps://doi.o g/10.1103/PhysRe Le .113.022501.a Xi :1401.6138 [nucl-ex]
4. D. Wang, Measu emen o pa i y iola ion in elec on–qua k sca e ing. Na u e 506(7486), 67–70 (2014). h ps://doi.o g/10.
1038/na u e12964
5. M. De u ne, A glimpse o gluons h ough deeply i ual Comp on sca e ing on he p o on. Na . Commun. 8(1), 1408 (2017).
h ps://doi.o g/10.1038/s41467-017-01819-3.a Xi :1703.09442 [hep-ex]
6. D. Iz aeli e al., Measu emen o pola iza ion- ans e o bound p o ons in ca bon and i s i uali y dependence. Phys. Le .
B781, 95–98 (2018). h ps://doi.o g/10.1016/j.physle b.2018.03.027.a Xi :1711.09680 [nucl-ex]
7. S.Paul e al., Thein luence o Fe mimo ion on he compa isono hepola iza ion ans e o ap o on inelas ic e igh a owp
and quasi-elas ic e igh a owA sca e ing. Phys. Le . B 792, 445–449 (2019). h ps://doi.o g/10.1016/j.physle b.2019.04.
004.a Xi :1901.10958 [nucl-ex]
8. A. Esse , Fi s measu emen o he Q2dependence o he beam-no mal single spin asymme y o elas ic sca e ing o
ca bon. Phys. Re . Le . 121(2), 022503 (2018). h ps://doi.o g/10.1103/PhysRe Le .121.022503
9. R. C uz-To es, P obing ew-body nuclea dynamics ia 3Hand3He (e,ep)pn c oss-sec ion measu emen s. Phys. Re . Le .
124(21), 212501 (2020). h ps://doi.o g/10.1103/PhysRe Le .124.212501.a Xi :2001.07230 [nucl-ex]
10. O. Benha , A. Fab ocini, S. Fan oni, I. Sick, Spec al unc ion o ini e nuclei and sca e ing o GeV elec ons. Nucl. Phys.
A579, 493–517 (1994). h ps://doi.o g/10.1016/0375-9474(94)90920-2
11. A.M. Ankowski, A. F iedland, Assessing he accu acy o he genie e en gene a o wi h elec on-sca e ing da a. Phys. Re .
D102, 053001 (2020). h ps://doi.o g/10.1103/PhysRe D.102.053001
12. A.Gil,J.Nie es,E.Ose ,Manybodyapp oach o he inclusi e(e,e-p ime) eac ion om he quasielas ic o hedel aexci a ion
egion. Nucl. Phys. A 627, 543–598 (1997). h ps://doi.o g/10.1016/S0375-9474(97)00513-7.a Xi :nucl- h/9711009
13. G.D. Megias, J.E. Ama o, M.B. Ba ba o, J.A. Caballe o, T.W. Donnelly, Inclusi e elec on sca e ing wi hin he
SuSA 2 meson-exchange cu en app oach. Phys. Re . D 94, 013012 (2016). h ps://doi.o g/10.1103/PhysRe D.94.013012.
a Xi :1603.08396 [nucl- h]
14. A.M. Ankowski, Elec on sca e ing and neu ino physics. J. Phys. G: Nucl. Pa . Phys. 50, 120501 (2023). h ps://doi.o g/
10.1088/1361-6471/ace 42
15. L.A. Ruso, e al., Theo e ical ools o neu ino sca e ing: in e play be ween la ice QCD, EFTs, nuclea physics, phe-
nomenology, and neu ino e en gene a o s (2022). a Xi :2203.09030 [hep-ph]
16. M. Mu phy, Measu emen o he c oss sec ions o inclusi e elec on sca e ing in he E12–14-012 expe imen a Je e son
Lab. Phys. Re . C 100(5), 054606 (2019). h ps://doi.o g/10.1103/PhysRe C.100.054606.a Xi :1908.01802 [hep-ex]
17. H. Dai, Fi s measu emen o he A (e,e)Xc oss sec ion a Je e son Labo a o y. Phys. Re . C 99(5), 054608 (2019). h ps://
doi.o g/10.1103/PhysRe C.99.054608.a Xi :1810.10575 [nucl-ex]
18. H. Dai, Fi s measu emen o he Ti(e,e)X c oss sec ion a Je e son lab. Phys. Re . C 98(1), 014617 (2018). h ps://doi.
o g/10.1103/PhysRe C.98.014617.a Xi :1803.01910 [nucl-ex]
19. L. Jiang, De e mina ion o he a gon spec al unc ion om (e, e’p) da a. Phys. Re . D 105(11), 112002 (2022). h ps://doi.
o g/10.1103/PhysRe D.105.112002.a Xi :2203.01748 [nucl-ex]
20. L. Jiang, De e mina ion o he i anium spec al unc ion om (e, e’p) da a. Phys. Re . D 107(1), 012005 (2023). h ps://doi.
o g/10.1103/PhysRe D.107.012005.a Xi :2209.14108 [nucl-ex]
21. S. Dolan, G.D. Megias, S. Bolognesi, Implemen a ion o he susa 2-meson exchange cu en 1p1h and 2p2h models in genie
and analysis o nuclea e ec s in 2k measu emen s. Phys. Re . D 101, 033003 (2020). h ps://doi.o g/10.1103/PhysRe D.
101.033003
22. A.A. Aguila -A e alo, A sea ch o elec on neu ino appea ance a he m2∼1eV2scale. Phys. Re . Le . 98, 231801
(2007). h ps://doi.o g/10.1103/PhysRe Le .98.231801.a Xi :0704.1500 [hep-ex]
23. D.D. S ancil, Demons a ion o communica ion using neu inos. Mod. Phys. Le . A 27, 1250077 (2012). h ps://doi.o g/10.
1142/S0217732312500770.a Xi :1203.2847 [hep-ex]
24. K. Abe, The T2K expe imen . Nucl. Ins um. Me h. A659, 106–135 (2011). h ps://doi.o g/10.1016/j.nima.2011.06.067.
a Xi :1106.1238 [physics.ins-de ]
25. C. And eopoulos, The GENIE neu ino Mon e Ca lo gene a o . Nucl. Ins um. Me h. A 614, 87–104 (2010). h ps://doi.o g/
10.1016/j.nima.2009.12.009.a Xi :0905.2517 [hep-ph]
26. O. Buss, T. Gai anos, K. Gallmeis e , H. Hees, M. Kaskulo , O. Lalakulich, A.B. La iono , T. Lei ne , J. Weil, U. Mosel,
T anspo - heo e ical desc ip ion o nuclea eac ions. Phys. Rep . 512, 1–124 (2012). h ps://doi.o g/10.1016/j.phys ep.
2011.12.001.a Xi :1106.1344 [hep-ph]
27. A. Papadopoulou, A. Ashkenazi, S. Ga dine , M. Be ancou , S. Dy man, L.B. Weins ein, E. Piase zky, F. Hauens ein, M.
Khacha yan, S. Dolan, G.D. Megias, O. Hen, Inclusi e elec on sca e ing and he genie neu ino e en gene a o . Phys.
Re . D 103, 113003 (2021). h ps://doi.o g/10.1103/PhysRe D.103.113003
28. M. Khacha yan, Elec on-beam ene gy econs uc ion o neu ino oscilla ion measu emen s. Na u e 599(7886), 565–570
(2021). h ps://doi.o g/10.1038/s41586-021-04046-5
Measu emen o he 12C(e,e)C oss Sec ions Page 9 o 9 78
29. I.C. Cloë , W. Ben z, A.W. Thomas, Rela i is ic and nuclea medium e ec s on he coulomb sum ule. Phys. Re . Le .
116(3), 032701 (2016). h ps://doi.o g/10.1103/PhysRe Le .116.032701.a Xi :1506.05875 [nucl- h]
30. A. Lo a o, S. Gandol i, J. Ca lson, S.C. Piepe , R. Schia illa, Elec omagne ic esponse o 12C: a i s -p inciples calcula ion.
Phys.Re .Le .117, 082501 (2016). h ps://doi.o g/10.1103/PhysRe Le .117.082501
31. A. Lo a o, J. Ca lson, S. Gandol i, N. Rocco, R. Schia illa, Ab ini io s udy o (ν,
−)and (ν,
+)inclusi e sca e ing in
12C: con on ing he miniboone and 2k ccqe da a. Phys. Re . X 10, 031068 (2020). h ps://doi.o g/10.1103/PhysRe X.10.
031068
32. S. Pas o e, J. Ca lson, S. Gandol i, R. Schia illa, R.B. Wi inga, Quasielas ic lep on sca e ing and back- o-back nucleons in
he sho - ime app oxima ion. Phys. Re . C 101, 044612 (2020). h ps://doi.o g/10.1103/PhysRe C.101.044612
33. J.E. Sobczyk, B. Acha ya, S. Bacca, G. Hagen, Coulomb sum ule o 4He and 16O om coupled-clus e heo y. Phys. Re .
C102, 064312 (2020). h ps://doi.o g/10.1103/PhysRe C.102.064312
34. J.E. Sobczyk, B. Acha ya, S. Bacca, G. Hagen, Ab ini io compu a ion o he longi udinal esponse unc ion in 40Ca. Phys.
Re . Le . 127, 072501 (2021). h ps://doi.o g/10.1103/PhysRe Le .127.072501
35. L. And eoli, J. Ca lson, A. Lo a o, S. Pas o e, N. Rocco, R.B. Wi inga, Elec on sca e ing on a=3 nuclei om quan um
mon e ca lo based app oaches. Phys. Re . C 105, 014002 (2022). h ps://doi.o g/10.1103/PhysRe C.105.014002
36. J.E. Sobczyk, S. Bacca, G. Hagen, T. Papenb ock, Spec al unc ion o 4He using he chebyshe expansion in coupled-clus e
heo y. Phys. Re . C 106, 034310 (2022). h ps://doi.o g/10.1103/PhysRe C.106.034310
37. J.E. Sobczyk, B. Acha ya, S. Bacca, G. Hagen, 40Ca ans e se esponse unc ion om coupled-clus e heo y. Phys. Re . C
109, 025502 (2024). h ps://doi.o g/10.1103/PhysRe C.109.025502
38. J.E. Sobczyk, S. Bacca, 16O spec al unc ion om coupled-clus e heo y: applica ions o lep on-nucleus sca e ing. Phys.
Re . C 109, 044314 (2024). h ps://doi.o g/10.1103/PhysRe C.109.044314
39. K.I. Blomq is , The h ee-spec ome e acili y a he mainz mic o on MAMI. Nucl. Ins . Me h. A 403, 263–301 (1998).
h ps://doi.o g/10.1016/S0168-9002(97)01133-9
40. P. Ba eau, M. Be nheim, J. Duclos, J.M. Finn, Z. Meziani, J. Mo gens e n, J. Mougey, D. Roye , B. Saghai, D. Ta nowski,
S. Tu ck-Chieze, M. B ussel, G.P. Capi ani, E. De Sanc is, S. F ullani, F. Ga ibaldi, D.B. Isabelle, E. Jans, I. Sick, P.D.
Zimme man, Deep-inelas ic elec on sca e ing om ca bon. Nucl. Phys. A 402(3), 515–540 (1983). h ps://doi.o g/10.
1016/0375-9474(83)90217-8
41. R.M. Sealock, K.L. Gio ane i, S.T. Tho n on, Z.E. Meziani, O.A. Rondon-A amayo, S. Au e , J.-P. Chen, D.G. Ch is ian,
D.B. Day, J.S. McCa hy, R.C. Mineha , L.C. Dennis, K.W. Kempe , B.A. Mecking, J. Mo gens e n, Elec oexci a ion o
he (1232) in nuclei. Phys. Re . Le . 62, 1350–1353 (1989). h ps://doi.o g/10.1103/PhysRe Le .62.1350
42. L.W. Mo, Y.S. Tsai, Radia i e co ec ions o elas ic and inelas ic ep and up sca e ing. Re . Mod. Phys. 41, 205–235 (1969).
h ps://doi.o g/10.1103/Re ModPhys.41.205
43. L.D. Landau, 56–on he ene gy loss o as pa icles by ionisa ion, in Collec ed Pape s o L.D. Landau. ed. by D. e Haa
(Pe gamon, Ox o d, 1965), pp.417–424. h ps://doi.o g/10.1016/B978-0-08-010586-4.50061-4
44. M. Miho ilo iˇc, The p o on cha ge adius ex ac ed om he ini ial-s a e adia ion expe imen a MAMI. Eu . Phys. J. A
57(3), 107 (2021). h ps://doi.o g/10.1140/epja/s10050-021-00414-x.a Xi :1905.11182 [nucl-ex]
45. A. Meucci, F. Capuzzi, C. Gius i, F.D. Paca i, Inclusi e elec on sca e ing in a ela i is ic g een’s unc ion app oach. Phys.
Re . C 67, 054601 (2003). h ps://doi.o g/10.1103/PhysRe C.67.054601
46. A. Meucci, J.A. Caballe o, C. Gius i, F.D. Paca i, J.M. Udías, Rela i is ic desc ip ions o inclusi e quasielas ic elec on
sca e ing: applica ion o scaling and supe scaling ideas. Phys. Re . C 80, 024605 (2009). h ps://doi.o g/10.1103/PhysRe C.
80.024605
47. J. Nie es, J.E. Sobczyk, In medium dispe sion ela ion e ec s in nuclea inclusi e eac ions a in e media e and low ene gies.
Ann. Phys. 383, 455–496 (2017). h ps://doi.o g/10.1016/j.aop.2017.06.002.a Xi :1701.03628 [nucl- h]
48. A.M. Ankowski, P i a e communica ion
49. P.E. Bos ed, M.E. Ch is y, Empi ical i o inelas ic elec on-deu e on and elec on-neu on esonance egion ans e se c oss
sec ions. Phys. Re . C 77, 065206 (2008). h ps://doi.o g/10.1103/PhysRe C.77.065206
50. M.E. Ch is y, P.E. Bos ed, Empi ical i o p ecision inclusi e elec on-p o on c oss sec ions in he esonance egion. Phys.
Re . C 81, 055213 (2010). h ps://doi.o g/10.1103/PhysRe C.81.055213.a Xi :0712.3731 [hep-ph]
51. T.W. Donnelly, I. Sick, Supe scaling o inclusi e elec on sca e ing om nuclei. Phys. Re . C 60, 065502 (1999). h ps://
doi.o g/10.1103/PhysRe C.60.065502.a Xi :nucl- h/9905060 [nucl- h]
52. C. Maie on, T.W. Donnelly, I. Sick, Ex ended supe scaling o elec on sca e ing om nuclei. Phys. Re . C 65, 025502
(2002). h ps://doi.o g/10.1103/PhysRe C.65.025502.a Xi :nucl- h/0109032
53. J.J. Kelly, Simple pa ame iza ion o nucleon o m ac o s. Phys. Re . C 70, 068202 (2004). h ps://doi.o g/10.1103/
PhysRe C.70.068202
54. R. Seu in, O.J. He nandez, T. Miyagi, S. Bacca, K. Hebele , S. König, A. Schwenk, Magne ic dipole ope a o om chi-
al e ec i e ield heo y o many-body expansion me hods. Phys. Re . C 108, 054005 (2023). h ps://doi.o g/10.1103/
PhysRe C.108.054005
55. B. Acha ya, B.S. Hu, S. Bacca, G. Hagen, P. Na á il, T. Papenb ock, The magne ic dipole ansi ion in 48Ca (2023)
a Xi :2311.11438 [nucl- h]
56. T. Miyagi, X. Cao, R. Seu in, S. Bacca, R.F. Ga cia Ruiz, K. Hebele , J.D. Hol , A. Schwenk, Impac o wo-body cu en s
on magne ic dipole momen s o nuclei (2023) a Xi :2311.14383 [nucl- h]
57. G.B. King, S. Pas o e, Recen p og ess in he elec oweak s uc u e o ligh nuclei using quan um Mon e Ca lo me hods
(2024) a Xi :2402.06602 [nucl- h]
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional
a ilia ions.