I. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 55 (2022)

RIKEN Accel. Prog. Rep. 55 (2022)

Toward ab initio charge symmetry breaking in nuclear energy density
functionals†
T. Naito,∗1,∗2 G. Colò,∗3,∗4 H. Z. Liang,∗1,∗2 X. Roca-Maza,∗3,∗4 and H. Sagawa∗5,∗2

†
∗1
∗2
∗3
∗4
∗5

Condensed from the article in Phys. Rev. C 105, L021304(2022)
Department of Physics, University of Tokyo
RIKEN Nishina Center
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano
Center for Mathematics and Physics, University of Aizu

−60.0
−62.5

ΔEtot (MeV)

The parameter sets of energy density functionals(EDFs) of the nuclear density functional theory (DFT)
are commonly determined phenomenologically to reproduce experimental masses and charge radii as well as
nuclear matter properties. To establish a link between
microscopic approaches and EDFs, an ab initio determination of the parameters of EDFs, as well as theirfunctional forms, is highly desired. To date, however,a direct correspondence between ab initio and DFT remains elusive. A sophisticated yet practical approachto pin down the EDF parameters is to combine the abinitio and phenomenological EDF calculations.
Even though the isospin symmetry breaking (ISB)
terms are small parts of the nuclear interaction, eﬀects
of the ISB terms on nuclear properties have received attention, while the ISB terms have often been neglected
in EDFs. The ISB interaction can be divided into twoparts: the charge symmetry breaking (CSB) and thecharge independence breaking (CIB) interactions.
Our aim in this study is to propose a comprehensive
methodology to determine the CSB terms in nuclear
EDFs adopting the ab initio results. We implement
the ISB terms in a Skyrme EDF to demonstrate ourmethodology. We will show that the mass diﬀerenceof mirror nuclei ΔEtot and the neutron-skin thickness
ΔRnp of doubly-magic nuclei calculated by ab initio
methods without and with the CSB terms, once they
are available, enable us to determine the CSB strengthin the EDF with an uncertainty of less than 6%, independently from other ISB forces such as CIB andCoulomb forces.
The Skyrme-like CSB and CIB interactions read1)
τ1z + τ2z
s0 (1 + y0 Pσ ) δ (r1 − r2 ) ,vCSB (r1 , r2 ) =4(1a)τ1z τ2z
u0 (1 + z0 Pσ ) δ (r1 − r2 ) , (1b)vCIB (r1 , r2 ) =2
respectively, where τiz is the z-direction of the isospin
operator of nucleon i (i = 1, 2) and Pσ is the spinexchange operator. For simplicity, in the SAMi-ISB
EDF,2) y0 = z0 = −1 are chosen to select the spinsinglet channel. The CIB strength u0 = 25.8 MeV fm3
has been previously determined using the Brueckner
Hartree-Fock calculation of symmetric nuclear matter.2,3) The CSB strength s0 = −26.3 MeV fm3 has

48Ca-48Ni

−65.0
−67.5

SLy4
SLy5
SkM*
SAMi
SAMi-noISB
SAMi-CIB
SAMi-ISB

−70.0
−72.5
−75.0
−77.5
−80.0

0

10

2030−s0 (MeV fm3)

4050Fig. 1. Dependence of mass diﬀerence of mirror nuclei 48 Ca
Ca48
Ni48
and 48 Ni, ΔEtot = Etot
− Etot, on CSB strength s0 .

been determined to reproduce the isobaric analog energy of 208 Pb.2) In this study, as u0 has been alreadydetermined microscopically, we propose a way to determine s0 microscopically. Because the isobaric analog
energy is not available using ab initio methods at thismoment, we focus on alternative well-established observables: the nuclear radius and mass, which are moreeasily accessible than any ab initio method.
The dependence of the mass diﬀerence of mirror nuCa48
Ni48− Etot, on the CSB
clei 48 Ca and 48 Ni, ΔEtot = Etot
strength s0 is shown in Fig. 1. For comparison, results
calculated with the original SAMi-ISB and experimental data (AME2020) are also shown using crosses andhorizontal lines, respectively. Remarkably, one can notethat ΔEtot has a strong linear correlation to the CSB
strength s0 , and the correlation is universal among thefunctionals. Thus, the calculated results are ﬁtted toΔEtot = a − bs0 . As seen in the ﬁgure, s0 and ΔEtot
are highly correlated (r = 1.000), and the slope b isalmost universal among Skyrme EDFs. Note that theparameters a and b are determined with a 0.5% error.
Accordingly, among these functionals, the slope b deviates within 6% around the average value of b. Thus,
once the ab initio results for ΔEtot calculated withw/o CSB
andout and with the bare CSB interaction, ΔEtot
w/ CSB
ΔEtot, are obtained, using b, wecan determine s0w/ CSB
w/o CSBas −s0 = ΔEtot− ΔEtot/b. As the uncertainty of b is 6%, the expected uncertainty of s0
is also 6%, assuming that the uncertainty associatedwith the ab initio calculations is negligible.
References1) H. Sagawa et al., Phys. Lett. B 353, 7 (1995).
2) X. Roca-Maza et al., Phys. Rev. Lett. 120, 202501(2018).
3) H. Müther et al., Phys. Lett. B 445, 259 (1999).

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