Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 50 (2017)

Coulomb sum rule in a covariant eﬀective quark theory†∗2∗3and A.W.
A.W. ThomasThomas∗4∗4I.I.C.
C.Cloët,
Cloët,∗1 W. Bentz,∗2,∗3andSL (|q|) =|q|dω
ω+RL (ω, |q|),Z G2Ep (Q2 ) + N G2En (Q2 )compared to the non-relativistic expectation (SL (|q|) =1 for |q| much greater than the Fermi momentum),was observed1); on the proviso that the nucleon form
factors are not modiﬁed by the nuclear medium. (Inthe above expression, RL (ω, |q|) is the longitudinal response function, GEp and GEn are the free nucleon
Sachs form factors, Q2 = ω 2 + q2 is the 4-momentum
transfer, and ω + excludes the elastic peak.)In this work we extend our description of the freenucleon form factors2) , which were obtained by using
the Nambu-Jona-Lasinio (NJL) model as an eﬀective
quark theory of QCD, to the in-medium case, including
the self consistent scalar and vector potentials in thenucleon propagators. As a result, we ﬁnd that at nuclear matter saturation density the proton Dirac andcharge radii each increase by about 8%. Using thesein-medium nucleon form factors and propagators obtained in the eﬀective quark theory, we calculate thequasi-elastic longitudinal response function in nuclear
matter by solving the Dyson equation for the polarization propagator in the relativistic random phase approximation (RPA)3) on the level of nucleons. For
the nucleon-nucleon interaction we take into accountthe exchange of σ, ω and ρ mesons, described in theframework of the NJL model.
Our Hartree and RPA results for the longitudinal response function are shown in Fig. 1 for |q| = 0.5 and
0.8 GeV. We ﬁnd that the longitudinal response function determined with in-medium nucleon form factors
is quenched relative to the result obtained using thefree form factors. In our calculation, this quenchingis directly associated with a softer proton Dirac form
factor (F1p ) in the medium. We observe a qualitative
agreement with the 208 Pb data of Ref.1)1..
Results for the CSR, using the nucleon form factors evaluated at three baryon densities (ρB =0, 0.1, 0.16 fm−3 ) are presented in Fig.2. At |q|  1
GeV we ﬁnd relativistic corrections of about 20% (relative to the nonrelativistic value SL = 1), and an additional 30% reduction by the nuclear medium for the†∗1
∗2
∗3
∗4Condensed from an article by I. C. Cloët et al., Phys. Rev.
Lett. 116, 032701 (2016).
Physics Division, Argonne National Laboratory
Radiation Laboratory, RIKEN
Department of Physics, Tokai University
Department of Physics, University of Adelaide- 125 -case ρB = 0.16 fm−3 . We observe a qualitative agreement with the 208 Pb data, but not with the state-ofthe-art Green function Monte Carlo (GFMC) result for121212The 12
data fromfrom Ref.
Ref.5)5 shown inC from Ref.4)4.). TheCC data
the ﬁgure still cannot distinguish between our and theGFMC results.

Fig. 1. Hartree and RPA results for the longitudinal response function in symmetric nuclear matter. Resultslabeled free current are obtained using the free nucleon
form factors, whereas the NM current results use thein-medium form factors. The 208 Pb data at |q| = 0.5
GeV are also shown for comparison.

1.00.8

SL (|q|)Important information on QCD eﬀects in nuclei
came from quasielastic electron scattering on nuclear
targets, where a signiﬁcant quenching of the Coulomb
sum rule (CSR)

0.6
free current – Hartree
free current – RPA
12
C current – RPA
NM current – RPA
208
Pb – experiment
12C – experiment
12C – GFMC

0.4
0.2
00

0.2

0.4

0.6

|q| [GeV]

0.8

1.0Fig. 2. CSR determined using the nucleon form factors atbaryon density ρB = 0 (free current), ρB = 0.1 fm−3
(typical of 12 C), and ρB = 0.16 fm−3 (NM current).
The 208 Pb and 12 C data as well as the Green Function Monte Carlo
Carlo (GFMC)(GFMC) results
results fromfrom Ref.
Ref.4)4) are also
shown for comparison.
This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (Kakenhi Grant
No. 25400270).

References
1) J. Morgenstern and Z. Meziani, Phys. Lett. B 515,269 (2001).
2) I.C. Cloët, W. Bentz and A.W. Thomas, Phys. Rev.
C90, 045202 (2014).
3) K. Wehrberger, Phys. Rept. 225, 273 (1993).
4) A. Lovato et al., Phys. Rev. Lett. 111, 092501 (2013).
5) P. Barreau et al., Nucl. Phys. A 402, 515 (1983).