# Low-momentum relativistic nucleon-nucleon potentials I: Nuclear matter

Chencan Wang<EMAIL_ADDRESS>School of Physics and Astronomy, Sun
Sat-Sen University, Zhuhai 519082, China  Sibo Wang  Department of Physics,
Chongqing University, Chongqing 401331, China  Hui Tong  Helmholtz-Institut
f$\ddot{u}$r Strahlen- und Kernphysik and Bethe Center for Theoretical
Physics, Universit$\ddot{a}$t Bonn, D-53115 Bonn, Germany  Jinniu Hu  School
of Physics, Nankai University, Tianjin, 300071, China  Jiangming Yao  School
of Physics and Astronomy, Sun Sat-Sen University, Zhuhai 519082, China

###### Abstract

A series of relativistic one-boson-exchange potentials for two-nucleon system,
denoted as OBEP$\Lambda$, is constructed with a momentum cutoff $\Lambda$
ranging from $\infty$ to 2 fm-1. These potentials are developed by
simultaneous fitting to nucleon-nucleon ($NN$) scattering phase shifts, low-
energy scattering length, effective range, and the binding energy of the
deuteron. The momentum-space matrix elements of the low-momentum OBEP$\Lambda$
($\Lambda\leqslant 3$ fm-1) demonstrate consistency with the universal
behaviors observed in other realistic $NN$ potentials evolved by
renormalization group methods. These OBEP$\Lambda$s are applied to calculate
the equation of state of symmetric nuclear matter (SNM) within either the
nonrelativistic (NR) Brueckner-Hartree-Fock (BHF) or relativistic Brueckner-
Hartree-Fock (RBHF) frameworks. The results show that the saturation
properties of SNM are reproduced qualitatively from the RBHF calculation, but
not from the NR-BHF calculation. This study highlights the relativistic
mechanism in explaining the saturation properties of nuclear matter. The
remaining discrepancy in reproducing empirical saturation properties in the
RBHF calculation using the OBEP$\Lambda$s signals the necessity of including
three-nucleon correlations or genuine three-nucleon forces.

##  I INTRODUCTION

The nucleon-nucleon ($NN$) potential serves as a crucial input for nuclear _ab
initio_ calculations. Originating from the 1960s, the meson-exchange model
stands as an effective framework for deriving realistic $NN$ potentials [1, 2,
3]. Within this model, the one-boson exchange potentials (OBEPs), namely the
Bonn potential and the highly accurate charge-dependent Bonn (CD-Bonn)
potential, were proposed and remain frequently utilized in present-day _ab
initio_ calculations [4, 5]. The preservation of Dirac spinors and covariant
operators in Bonn potentials enables the applicability to relativistic _ab
initio_ approaches, such as the relativistic Brueckner-Hartree-Fock (RBHF)
theory. However, the original CD-Bonn potential, utilizing the pseudoscalar
(ps) type of pion-nucleon ($\pi N$) coupling, cannot be employed in RBHF
calculations for nuclear matter due to the unphysically large self-energies
induced by the ps $\pi N$ vertex [6]. Yet, by substituting the ps $\pi N$
coupling with a pseudovector (pv) type, the modified CD-Bonn potentials become
viable for relativistic applications [7, 8].

Over the past two decades, substantial progress has been made in developing
$NN$ potentials rooted in chiral effective field theory (chEFT) and
renormalization group (RG) methods [9, 10, 11, 12, 13]. Consequently, various
nonrelativistic (NR) $NN$ potentials have been formulated at different
resolution scales, characterized by specific momentum cutoffs $\Lambda$ [13,
14, 15]. Additionally, three-nucleon force (3NF) with specified cutoff
$\Lambda_{3N}$ arises naturally, either at the next-next-to-leading (N2LO)
order in chEFT or through RG evolution in the flow equation [16, 17, 18]. In
NR _ab initio_ calculations, the cutoff dependence of few-body observables
directly reflects the residual many-body forces [19, 20]. Furthermore,
variations in many-body observables concerning $\Lambda/\Lambda_{3N}$ provide
insights into estimating theoretical uncertainties [15, 21, 22, 23]. This
encourages the development of OBEPs with different momentum cutoffs $\Lambda$.
Employing these potentials in relativistic _ab initio_ calculations of nuclear
many-body systems, such as nuclear matter, enables the exploration of cutoff-
dependent equation of state (EOS) from a relativistic perspective.

Nuclear matter stands as a research topic of great interest in nuclear physics
since it helps us understand the bulk properties of finite nuclei and the
evolution of astrophysical objects like neutron stars. In particular, the
properties of nuclear matter around the saturation density $n_{0}=0.16$ fm-3
provide benchmarks to test the validity of underlying $NN$ potentials and
many-body methods [24, 25, 26, 27]. In addition, the knowledge of the EOS of
nuclear matter at supra-saturation densities is important to understand the
formation and structure of neutron stars [28, 29, 30, 31], as well as the
particle production in the heavy-ion collision (HIC) [32, 33, 34]. Early
attempts to attack the problem were based on nonperturbative approaches such
as NR variational method or Brueckner theory with traditional $NN$ potentials
[35, 36, 37, 38, 39].

It has been observed that the saturation properties of symmetric nuclear
matter (SNM), derived from various $NN$ potentials, align within the “Coester
band”, systematically differing from the empirical saturation region [40, 41,
42, 43]. This led to the conclusion that relying solely on $NN$ potentials
fails to quantitatively replicate correct saturation properties. This
highlighted the pivotal role of 3NF in understanding the saturation mechanism
of nuclear matter [42, 44]. In contrast, the RBHF framework using only Bonn
potentials, without explicit inclusion of 3NF, nearly reproduced SNM’s
saturation properties [4, 6]. RBHF theory hinges on two primary features: the
effective Dirac spinor of nucleons, where the lower components rely on the
effective Dirac mass, introducing additional density dependence in
relativistic kinetics and the $G$-matrix; and the Lorentz structure of the
self-energy, notably the attractive scalar self-energy and the repulsive
timelike vector self-energy [45, 46, 47]. The emergence of saturation
properties in SNM results from the intricate balance between the linearly
increasing vector self-energy and the gradually diminishing scalar self-energy
[48]. Recent progressions in this field involve the successful application of
RBHF theory, notably in fully self-consistent calculations for finite nuclei
[49, 50], and nuclear matter calculations encompassing the complete Dirac
space [51, 52].

It’s noteworthy that the availability of realistic $NN$ potentials suitable
for RBHF calculations is severely limited, primarily confined to the three
Bonn potentials developed more than thirty years ago [4]. Additionally, these
potentials lack specification regarding resolution scales, and the uncertainty
assessment within the relativistic many-body method remains unexplored. To
revitalize research in relativistic nuclear _ab initio_ calculations, our
initial step involves constructing a series of OBEPs, explicitly incorporating
momentum cutoffs $\Lambda$ (referred to as OBEP$\Lambda$s). These
OBEP$\Lambda$s will be employed within the RBHF framework to compute the EOS
of nuclear matter.

This paper is organized as follows. In Sec. II, the theoretical framework for
OBEP, the scattering equation, and the RBHF theory for nuclear matter will be
briefly reviewed. In Sec. III.1, the fitting protocol and the parameters of
OBEP$\Lambda$s will be given, and the potential matrix elements and the
calculated $NN$ observables with OBEP$\Lambda$s will also be provided. In Sec.
III.2, we will present nuclear matter results from both NR-BHF and RBHF
calculations with these OBEP$\Lambda$s, shedding light on the implications of
relativity in the saturation mechanism of SNM. Finally the summary and
perspectives will be given in Sec. IV.

##  II THEORETICAL FRAMEWORKS

###  II.1 One-boson-exchange potential and NN observables

Analogous to the Bonn potentials [4], present OBEP$\Lambda$s are developed
based on the exchange of $\pi$, $\eta$ in pv coupling, $\sigma$, $\delta$
mesons in scalar (s) coupling, and $\omega$, $\rho$ mesons in vector (v)
coupling. These nucleon-meson interaction Lagrangians are

$\displaystyle\mathcal{L}^{\mathrm{(pv)}}$
$\displaystyle=-\frac{f_{\mathrm{pv}}}{m_{\mathrm{pv}}}\bar{\psi}\gamma^{5}\gamma^{\mu}\psi\cdot\partial_{\mu}\phi^{\mathrm{(pv)}},$
(1a) $\displaystyle\mathcal{L}^{\mathrm{(s)}}$
$\displaystyle=+g_{\mathrm{s}}\bar{\psi}\psi\cdot\phi^{\mathrm{(s)}},$ (1b)
$\displaystyle\mathcal{L}^{\mathrm{(v)}}$
$\displaystyle=-g_{\mathrm{v}}\bar{\psi}\gamma^{\mu}\psi\cdot\phi_{\mu}^{\mathrm{(v)}}-\frac{f_{\mathrm{v}}}{2M}\bar{\psi}\sigma^{\mu\nu}\psi\cdot\partial_{\mu}\phi_{\nu}^{\mathrm{(v)}}.$
(1c)

The $NN$ potential in the center-of-mass (CM) frame is obtained from tree-
level Feynman amplitude:

$\displaystyle V(\mathbf{q}^{\prime},\mathbf{q})=$
$\displaystyle-\sum^{\mathrm{all~{}mes.}}_{a}\bar{u}_{1}(\mathbf{q}^{\prime})\Gamma^{(1)}_{a}u_{1}(\mathbf{q})\frac{\mathcal{F}_{a}(Q^{2})}{Q^{2}+m_{a}^{2}}$
$\displaystyle\times\bar{u}_{2}(-\mathbf{q}^{\prime})\Gamma^{(2)}_{a}u_{2}(-\mathbf{q}).$
(2)

Here, the subscript $a$ represents all the six mesons, $\mathbf{q}$ and
$\mathbf{q^{\prime}}$ representing the incoming and outgoing relative momenta,
$u_{i}~{}(i=1,2)$ for nucleon spinor and $\Gamma^{(i)}_{a}$ for different
meson-nucleon coupling vertices. The
$\mathbf{Q}=\mathbf{q}^{\prime}-\mathbf{q}$ is three-momentum transfer. The
form factor
$\mathcal{F}_{a}(Q^{2})=\exp[-(Q^{2}+m_{a}^{2})^{2}/\Lambda_{a}^{4}]$ is used
on the meson propagator to alter the behavior of local momentum transfer, in
which a meson-dependent parameter $\Lambda_{a}$ is introduced. This choice of
form factor is tested to be more suitable in our fitting procedure, and
different from $(\Lambda_{a}^{2}-m_{a}^{2})^{2}/(\Lambda^{2}+Q^{2})^{2}$ used
in the Bonn potentials.

In addition, we introduce the following non-local regulator

$V_{\Lambda}(\mathbf{q}^{\prime},\mathbf{q})=\mathcal{R}(q^{\prime})V(\mathbf{q}^{\prime},\mathbf{q})\mathcal{R}(q),$
(3)

with

$\mathcal{R}(q)=\exp[-(q^{2n}/\Lambda^{2n})].$ (4)

The regulators above strongly suppress the matrix elements with relative
momenta larger than the cutoff $\Lambda$.

The $T$ matrix for the $NN$ scattering process is obtained by the Thompson
equation [4]:

$\displaystyle T(\mathbf{q}^{\prime},\mathbf{q};W_{q})=$ $\displaystyle
V_{\Lambda}(\mathbf{q}^{\prime},\mathbf{q})+\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{M^{2}}{E^{2}_{p}}V_{\Lambda}(\mathbf{q}^{\prime},\mathbf{p})$
$\displaystyle\times$
$\displaystyle\frac{1}{W_{q}-W_{p}+i\varepsilon}T(\mathbf{p},\mathbf{q};W_{q}),$
(5)

where $E_{p}=\sqrt{\mathbf{p}^{2}+M^{2}}$ is the nucleon on-shell energy.
$W_{q}=2E_{q}$ and $W_{p}=2E_{p}$ are the initial and intermediate two-nucleon
energy in CM frame respectively. The partial-wave scattering matrix is
obtained by

$\mathcal{S}_{\ell^{\prime}\ell}=\delta_{\ell^{\prime}\ell}-i\pi\frac{qM^{2}}{E_{q}}\langle\ell^{\prime}sj|T(W_{q})|\ell
sj\rangle.$ (6)

Corresponding phase shifts $\delta$ in uncoupled channel are given by
$\mathcal{S}_{\ell\ell}=e^{i2\delta_{\ell}}$. For coupled channels, the phase
shifts $\delta_{\ell\pm 1}$ and mixing angle $\varepsilon_{j}$ are obtained by

$\begin{pmatrix}\mathcal{S}_{--}&\mathcal{S}_{-+}\\\
\mathcal{S}_{+-}&\mathcal{S}_{++}\end{pmatrix}=\begin{pmatrix}\cos
2\varepsilon_{j}e^{2i\delta_{-}}&i\sin
2\varepsilon_{j}e^{i(\delta_{-}+\delta_{+})}\\\ i\sin
2\varepsilon_{j}e^{i(\delta_{-}+\delta_{+})}&\cos
2\varepsilon_{j}e^{2i\delta_{+}}\end{pmatrix},$ (7)

where $\pm$ stand for $\ell=j\pm 1$.

The binding energy $E_{d}$ and wave functions $(\psi_{S},\psi_{D})^{T}$ of
deuteron are obtained by solving the following homogeneous Thompson equation

$\displaystyle\begin{pmatrix}\psi_{S}(q)\\\ \psi_{D}(q)\end{pmatrix}=$
$\displaystyle\frac{1}{2M-E_{d}-W_{q}}\int_{0}^{+\infty}p^{2}\mathrm{d}p\frac{M^{2}}{E_{p}^{2}}$
$\displaystyle\times$
$\displaystyle\begin{pmatrix}V_{\Lambda,SS}(q,p)&V_{\Lambda,SD}(q,p)\\\
V_{\Lambda,DS}(q,p)&V_{\Lambda,DD}(q,p)\end{pmatrix}\begin{pmatrix}\psi_{S}(p)\\\
\psi_{D}(p)\end{pmatrix}.$ (8)

###  II.2 The RBHF theory with projection method

The single-nucleon motion in nuclear matter follows the Dirac equation

$[\bm{\alpha}\cdot\mathbf{k}+\beta
M+\beta\Sigma(k)]u(\mathbf{k},\lambda)=E_{k}u(\mathbf{k},\lambda),$ (9)

The self-energy in nuclear matter can be expressed as
$\Sigma=\Sigma_{\mathrm{S}}-\gamma^{0}\Sigma_{0}+\bm{\gamma}\cdot\mathbf{k}\Sigma_{\mathrm{V}}$,
where $\Sigma_{\mathrm{S}}$, $\Sigma_{0}$, and $\Sigma_{\mathrm{V}}$ represent
the scalar self-energy, time-like and space-like vector self-energies,
respectively. Here, $\lambda=\pm 1/2$ denotes helicity.

With the definitions of reduced Dirac mass and effective energy

$M^{*}=\frac{M+\Sigma_{\mathrm{S}}}{1+\Sigma_{\mathrm{V}}},\quad
E^{*}_{k}=\frac{E_{k}-\Sigma_{0}}{1+\Sigma_{\mathrm{V}}},$ (10)

the solutions to the Dirac equation are
$E^{*}_{k}=\sqrt{\mathbf{k}^{2}+M^{*2}}$ and plane-wave spinor

$u(\mathbf{k},\lambda)=\sqrt{\frac{E^{*}_{k}+M^{*}}{2M^{*}}}\begin{pmatrix}1\\\
\frac{\bm{\sigma}\cdot\mathbf{k}}{M^{*}+E^{*}_{k}}\end{pmatrix}|\lambda\rangle.$
(11)

The effective $NN$ potential in nuclear matter is obtained with the Brueckner
$G$-matrix:

$\displaystyle G(\mathbf{q}^{\prime},\mathbf{q}|\mathbf{P},W_{q})$
$\displaystyle=V_{\Lambda}(\mathbf{q}^{\prime},\mathbf{q})+\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{M^{*2}}{E^{*2}_{p}}V_{\Lambda}(\mathbf{q}^{\prime},\mathbf{p})$
$\displaystyle\times\frac{Q(\mathbf{p},\mathbf{P})}{W_{q}-W_{p}+i\varepsilon}G(\mathbf{p},\mathbf{q}|\mathbf{P},W_{q}),$
(12)

where $\mathbf{P}$ is the CM momentum and $Q$ is Pauli blocking operator
prohibiting nucleon scattering into occupied states.

As illustrated in Ref. [6], the scalar and time-component vector self-energies
may exhibit unphysically large values due to the inadequate treatment of the
one-pion-exchange potential $V_{\pi}$. To mitigate this issue, the subtracted
$T$-matrix scheme is proposed, wherein the $G$-matrix is split into two
components, $G=V_{\mathrm{pv}}+\Delta G$, and
$V_{\mathrm{pv}}=V_{\pi}+V_{\eta}$. The transformation of $\Delta G$ from the
$\ell sj$ representation to partial-wave helicity representation enables the
derivation of invariant amplitudes $F$.

$\Delta
G=F_{\mathrm{S}}\Gamma_{\mathrm{S}}+F_{\mathrm{V}}\Gamma_{\mathrm{V}}+F_{\mathrm{T}}\Gamma_{\mathrm{T}}+F_{\mathrm{P}}\Gamma_{\mathrm{P}}+F_{\mathrm{A}}\Gamma_{\mathrm{A}},$
(13)

with the pseudoscalar-type covariant basis

$\displaystyle\Gamma_{\mathrm{S}}$ $\displaystyle=1_{1}\otimes 1_{2},$ (14a)
$\displaystyle\Gamma_{\mathrm{V}}$
$\displaystyle=(\gamma^{\mu})_{1}\otimes(\gamma_{\mu})_{2},$ (14b)
$\displaystyle\Gamma_{\mathrm{T}}$
$\displaystyle=(\sigma^{\mu\nu})_{1}\otimes(\sigma_{\mu\nu})_{2},$ (14c)
$\displaystyle\Gamma_{\mathrm{A}}$
$\displaystyle=(\gamma^{5}\gamma^{\mu})_{1}\otimes(\gamma^{5}\gamma_{\mu})_{2},$
(14d) $\displaystyle\Gamma_{\mathrm{P}}$
$\displaystyle=(\gamma^{5})_{1}\otimes(\gamma^{5})_{2}.$ (14e)

The self-energies generated by $\Delta G$ can be calculated by

$\displaystyle\Sigma_{\mathrm{S}}(k)$
$\displaystyle=\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\frac{M^{*}}{E_{k^{\prime}}^{*}}F_{\mathrm{S}}(q,q),$
(15a) $\displaystyle\Sigma_{0}(k)$
$\displaystyle=-\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}F_{\mathrm{V}}(q,q),$
(15b) $\displaystyle\Sigma_{\mathrm{V}}(k)$
$\displaystyle=-\frac{1}{k^{2}}\int\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}}\frac{\mathbf{k}\cdot\mathbf{k}^{\prime}}{E_{k^{\prime}}^{*}}F_{\mathrm{V}}(q,q),$
(15c)

where $q=\frac{1}{2}\sqrt{s^{*}-4M^{*2}}$ is the relative momentum in two-
nucleon CM frame,
$s^{*}=(E_{k}^{*}+E_{k^{\prime}}^{*})^{2}-(\mathbf{k}+\mathbf{k}^{\prime})^{2}$.

The remaining $V_{\mathrm{pv}}$ is decomposed in complete pseudovector
representation, corresponding formulae for self-energies are complicated, see
Ref. [6] for details. Numerical computations demonstrate that the
contributions from $V_{\mathrm{pv}}$ to self-energies are approximately one
order of magnitude smaller than those originating from $\Delta G$.

The Brueckner G-matrix (12) is solved by iteration. After convergence, the
binding energy per-nucleon in nuclear matter is given by
$E/A=E_{\mathrm{kin}}/A+E_{\mathrm{pot}}/A$, where the kinetic term is
calculated by

$E_{\mathrm{kin}}/A=\frac{1}{n}\sum_{\lambda}\int^{k_{F}}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\langle\bar{u}(\mathbf{k},\lambda)|\bm{\gamma}\cdot\mathbf{k}+M|u(\mathbf{k},\lambda)\rangle-M,$
(16)

with $n$ representing the nucleon number density. The average potential energy
is given by

$\displaystyle E_{\mathrm{pot}}/A$
$\displaystyle=\frac{1}{n}\sum_{\lambda}\int^{k_{F}}\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\langle\bar{u}(\mathbf{k},\lambda)|$
$\displaystyle\times(M^{*}\Sigma_{\mathrm{S}}/E^{*}_{k}-\Sigma_{0}+k^{2}\Sigma_{\mathrm{V}}/E^{*}_{k})|u(\mathbf{k},\lambda)\rangle.$
(17)

##  III RESULTS AND DISCUSSIONS

###  III.1 The fitting procedure and NN observables

Table 1: Mesons parameters in OBEP$\Lambda$s with the variation of cutoffs $\Lambda$ (in unit fm-1). In each row, the coupling parameters are given as $g_{a}~{}(\Lambda_{a})$. The meson masses (in the parenthesis after each meson) and $\Lambda_{a}$ are given in unit GeV. For pion and eta meson, we take $g_{a}=2Mf_{a}/m_{a}$, where the mass of nucleon is the averaged value $M=$938.919 MeV. $\rho$ meson coupling constants are given in the form of $g_{\rho}$–$f_{\rho}/g_{\rho}$. $\Lambda$ |  $\pi$ (0.138) |  $\omega$ (0.783) |  $\rho$ (0.770) |  $\eta$ (0.548) |  $\delta$ (0.983) |  $\sigma$ (0.550)
---|---|---|---|---|---|---
$\infty$ | 12.71 (0.97) | 14.81 (1.05) | 2.38–6.0 (0.90) | 1.29 (0.77) | 9.02 (0.96) | 10.28 (1.83)
5 | 12.69 (0.97) | 14.28 (1.03) | 2.16–6.8 (0.97) | 4.59 (0.77) | 8.07 (0.95) | 10.27 (1.75)
4 | 12.66 (0.93) | 13.55 (1.00) | 2.23–6.9 (1.01) | 5.20 (0.77) | 8.80 (0.87) | 9.95 (1.50)
3 | 12.67 (0.91) | 12.98 (0.97) | 2.42–6.8 (1.03) | 5.06 (0.77) | 10.31 (0.81) | 9.70 (1.32)
2 | 13.20 (0.90) | 12.15 (0.94) | 2.89–5.7 (1.04) | 0.00 (0.77) | 10.38 (0.78) | 9.32 (1.23)

The parameters in the OBEP$\Lambda$ in Eq. (2) are determined by minimizing
the objective function

$\displaystyle f(\mathbf{X})$
$\displaystyle=\sum_{\delta}w_{\delta}^{2}\left(\delta-\delta_{\mathrm{NPWA}}\right)^{2}+w_{d}^{2}\left(E_{d}-E_{d}^{\mathrm{(expt)}}\right)^{2}$
$\displaystyle+\sum_{S}\left[w_{a}^{2}\left(a_{S}-a_{S}^{\mathrm{(expt)}}\right)^{2}+w_{r}^{2}\left(r_{S}-r_{S}^{\mathrm{(expt)}}\right)^{2}\right],$
(18)

where $\mathbf{X}$ contains 12 variables, including $g_{a}~{}(f_{a})$ and
$\Lambda_{a}$ for the six mesons. The neutron-proton (np) phase shifts of
partial wave $j\leqslant 4$ with laboratory energy $E_{\mathrm{lab}}\leqslant
300$ MeV are calculated to compare with phase shifts from Nijmegen partial-
wave analyse (NPWA) [53]. $E_{d}^{\mathrm{(expt)}}=2.2246$ MeV is the binding
energy of deuteron. The low-energy scattering observables [54], namely the
$S$–wave scattering lengths $a_{{}^{1}\\!S_{0}}^{\mathrm{(expt)}}=-23.75$ fm,
$a_{{}^{3}\\!S_{1}}^{\mathrm{(expt)}}=5.42$ fm, and effective ranges
$r_{{}^{1}\\!S_{0}}^{\mathrm{(expt)}}=2.75$ fm,
$r_{{}^{3}\\!S_{1}}^{\mathrm{(expt)}}=1.76$ fm are also included in the
fitting. $w_{\delta}$, $w_{d}$, $w_{a}$, $w_{r}$ are the weighting factors, we
employ $w_{\delta}=1/(\Delta\delta_{\mathrm{NPWA}})$, with
$\Delta\delta_{\mathrm{NPWA}}$ being the phase shift uncertainty in NPWA. We
employ $w_{d}=1000$ and $w_{r},~{}w_{a}=100$ to ensure both partial-wave phase
shifts and low-energy observables can be simultaneously reproduced.

During the fitting, $n=3$ is applied in the regulator (3) for
$\Lambda\geqslant$ 4 fm-1, and $n=4$ is used for $\Lambda\leqslant$ 3 fm-1.
Slightly different parameterizations for the six mesons are allowed to
minimize $f(\mathbf{X})$ at each cutoff, but the nuclear matter results are
insensitive to such small changes in parameters. The mesons’ parameters are
finally determined at each $\Lambda$ by both minimizing $f(\mathbf{X})$ and
considering the continuity between adjacent cutoffs. These parameters are
listed in Tab. 1. In Fig. 1, variations of meson-nucleon coupling constants
with respect to decreasing cutoffs are plotted. As shown in the figure,
$g_{\pi}$, $g_{\rho}$, and $f_{\rho}/g_{\rho}$ only vary a little, while
$g_{\sigma}$ and $g_{\omega}$ are monotonously decreasing with $\Lambda$. The
$g_{\eta}$ and $g_{\delta}$ show nonmonotonic behaviors, and the $\eta$ meson
finally turns out to be redundant at $\Lambda=2$ fm-1 according to the
minimization procedure. For comparison, meson-nucleon coupling constants from
relativistic Hartree-Fock (RHF) density functional parameter sets, such as
PKA1, PKO$i$ ($i=1,2,3$) [55, 56], are also plotted in the form of error bars,
according to their statistical mean values and standard deviations. It can be
found that present coupling constants from OBEP$\Lambda$ have the tendencies
to close those of RHF model at low cutoff.

Figure 1: The meson-nucleon coupling constants $g_{a}$ running with external
relative momentum cutoff $\Lambda$. The error bars are statistic mean values
and standard deviations of meson-nucleon coupling parameters for the RHF
approaches [55, 56]. Figure 2: Neutron-proton partial-wave phase shifts
$j\leqslant 2$ calculated with the OBEP$\Lambda$s.

The phase shifts with $j\leqslant 2$, calculated using OBEP$\Lambda$s and
depicted in Fig. 2, demonstrate good agreement with the NPWA analysis [53] up
to $E_{\mathrm{lab}}=200$ MeV. This congruence extends to the peripheral
partial waves in our calculations. However, for $E_{\mathrm{lab}}>200$ MeV,
discernible deviations arise, particularly noticeable in the ${}^{1}P_{1}$
channel. Although this channel is specially treated in both Bonn potentials
and the CD-Bonn potential, in our work, there are no specific partial-wave
refinements in . Tab. 2 presents the deuteron properties computed using
OBEP$\Lambda$s. Remarkably, properties such as the matter radius $r_{d}$,
quadrupole momentum $Q_{d}$, and asymptotic $D/S$ ratio remain largely
invariant despite variations in cutoffs, while the $D$-state probability
$P_{D}$ gradually diminishes as $\Lambda$ decreases. $P_{D}$ is related to the
tensor components of $NN$ interaction, when the cutoffs decreases, the tensor
components are suppressed. This reduction in $P_{D}$ reflects an understanding
that it is not a directly observable; its dependence on the cutoff is also
observed in the RG evolution as documented in [57].

Table 2: Deuteron properties predicted by OBEP$\Lambda$s. $r_{d}$ is the matter radius, $Q_{d}$ is the quadrupole momentum, $D/S$ is the ratio of asymptotic $D$ and $S$ state amplitudes, and $P_{D}$ is the $D$–state probability. $\Lambda$ |  $r_{d}$ [fm] |  $Q_{d}$ [fm2] | $D/S$ |  $P_{D}$ (%)
---|---|---|---|---
$\infty$ | 1.967 | 0.2634 | 0.0247 | 5.498
5 | 1.967 | 0.2630 | 0.0246 | 5.429
4 | 1.964 | 0.2625 | 0.0246 | 5.205
3 | 1.964 | 0.2634 | 0.0246 | 4.827
2 | 1.967 | 0.2634 | 0.0248 | 4.246
expt. | 1.975 | 0.2859 | 0.0256 |

Figure 3 displays a comparison of momentum-space matrix elements in the
${}^{1}S_{0}$ channels of OBEP$\Lambda$s ($\Lambda\leqslant 4$ fm-1) with
other realistic $NN$ potentials, including OBEP$\infty$, the CD-Bonn potential
[5], Argonne $v18$ (AV18) [58], and the chiral nuclear force at the fifth
order (with the original cutoff $\Lambda_{\chi}=500$ MeV, denoted as N4LO)
evolved by a smooth RG technique [59, 57]. In the left panels (a), (b), and
(c), on-shell ${}^{1}S_{0}$ potential elements are provided. These panels
reveal that as $\Lambda$ decreases to $\Lambda\leqslant 3$ fm-1, the on-shell
potential matrix elements of CD-Bonn, AV18, N4LO, and our OBEP$\Lambda$
converge closely. The corresponding half-on-shell potential matrix elements in
the right panels (a’), (b’), and (c’) generally mirror the trends observed in
the on-shell cases. Similar behavior is noted in potential matrix elements of
other partial-wave channels. Notably, the potential matrix elements of
OBEP$\Lambda$s closely resemble those of other realistic potentials after RG
evolution down to $\Lambda=$ 2 fm-1, clearly demonstrating the universality of
phase-shift equivalent $NN$ potentials [12].

Figure 3: The matrix elements of on-shell (left panels) and half-on-shell
(right panels) OBEP$\Lambda$ potentials in the ${}^{1}S_{0}$ channel, in
comparision with other RG evolved realistic $NN$ potentials, including CD-
Bonn, AV18 and N4LO. Here we use the same convention as Ref. [12],
$\tilde{V}(q^{\prime},q)=\pi MV(q^{\prime},q)/2$.

###  III.2 Nuclear matter results

Figure 4: The energy per nucleon for the symmetric nuclear matter calculated
by BHF and RBHF approaches with different OBEP$\Lambda$s. The shaded area
indicates the empirical saturation region with density $n=0.164\pm 0.007$ fm-3
and $E/A=-15.86\pm 0.57$ MeV [23]. The energy minimum of each curve is marked
by a dot. Figure 5: The kinetic and potential terms obtained by BHF
calculation in panel (a) and RBHF calculation in panel (b), with
OBEP$\Lambda$s as input.

The OBEP$\Lambda$s potentials are utilized in nuclear matter calculations
employing both NR-BHF and RBHF approaches. The resulting EOSs are depicted in
Fig. 4, with the saturation points of each curve indicated. In the NR
calculations, OBEP$\Lambda$s are augmented with minimal relativity [60],

$V_{\Lambda,\mathrm{NR}}(\mathbf{q^{\prime}},\mathbf{q})=\sqrt{\frac{M}{E_{q^{\prime}}}}V_{\Lambda}(\mathbf{q^{\prime}},\mathbf{q})\sqrt{\frac{M}{E_{q}}},$
(19)

while BHF calculations are executed under the continuous choice method [61].

In panel (a) of Fig.4, the NR-BHF results exhibit convergence before reaching
the empirical saturation region, although they notably display an overbound
nature. These findings qualitatively align with the outcomes from Refs. [57,
15], particularly in instances where, at low cutoffs, the bare potentials
demonstrate quantitative proximity, as depicted in Fig.3. However, as
densities surpass the empirical saturation region, the divergence of NR EOSs
becomes evident. The extent of overbinding corresponds to the chosen cutoffs;
notably, for cutoffs lower than 3 fm-1, no saturation points are observed in
the region $n\leqslant 4$ fm-3. This absence underscores the necessity of
incorporating 3NFs to elucidate the saturation mechanism for softer $NN$
potentials in NR frameworks.

In panel (b), the situation regarding relativistic results differs notably.
All RBHF calculations conducted with OBEP$\Lambda$s manifest saturation
phenomena. However, the binding energy and density at each saturation point do
not align adequately with those observed in the empirical saturation region,
even with the softest potential at $\Lambda=2$ fm-1. The convergence patterns
of resulting EOSs in relation to cutoff variation also diverge from their non-
relativistic counterparts; notably, the gaps between adjacent cutoffs decrease
as the cutoff decreases. The disparity between current RBHF calculations and
empirical saturation properties might possibly be attributed to unaccounted
relativistic 3NF, or the exclusion of higher-order contributions in Bethe-
Brueckner-Goldstone (BBG) expansion,which warrants exploration in future
studies. Additionally, it’s important to note that while the original Bonn A
potential can nearly reproduce saturation properties [4], but some of its
phase shift predictions, especially the mixing parameter $\epsilon_{1}$ in the
${}^{3}S_{1}$–${}^{3}D_{1}$ channel, deviate considerably from NPWA even at
very small $E_{\mathrm{lab}}$. In contrast, all the OBEP$\Lambda$s
successfully reproduce $\varepsilon_{1}$ up to $E_{\mathrm{lab}}=200$ MeV.

Figure 6: The self-energy components at Fermi momentum $k_{F}$ obtained in
RBHF calculations with OBEP$\Lambda$s. Figure 7: The pressure of symmetric
nuclear matter at zero temperature obtained by BHF and RBHF calcualtions with
OBEP$\Lambda$s (to distinguish, the NR-BHF results are shaded in grey). The
shaded areas in orange and cyan are experimental constraints from HIC
experiments.

To clarify the differences between BHF and RBHF results with the
OBEP$\Lambda$s, we show in Fig. 5 the kinetic and potential terms obtained
from BHF and RBHF calculations respectively. For BHF calculation in panel (a),
the kinetic terms are all the same as free Fermi gas, while the potential
terms vary with cutoffs, which are in correspondance with the divergence of
EOSs shown in panel (a) of Fig. 4. For RBHF calculations in panel (b), as Eq.
(16) indicates, the kinetic mass equals to the expectation value of
relativistic kinetic operator $\bm{\gamma}\cdot\mathbf{k}+M$ minus rest mass.
Before empirical saturation density, the two kinetic contributions are close.
Since Dirac mass appears in the lower component of spinor
$u(\mathbf{k},\lambda)$, the expectation value of relativistic kinetic
operator can be even smaller than the rest nucleon mass at large densities.

The relativistic potential terms from Eq. (17), plotted in panel (b) of Fig. 5
gain considerable repulsion as compared with BHF results. To understand the
source of repulsion in relativistic potential contributions, we present in
Fig. 6 the self-energy components appearing in Eq. (17). Since
$\Sigma_{\mathrm{V}}$ are one order of magnitude smaller than
$\Sigma_{\mathrm{0}}$ and $\Sigma_{\mathrm{S}}$, we will mainly focus on
$\Sigma_{\mathrm{0}}$ and $\Sigma_{\mathrm{S}}$. By relativistic decomposition
of $\Delta G$ as Eq. (13) and $V_{\mathrm{pv}}$ in complete pv representation,
both attractive $\Sigma_{\mathrm{S}}$ and repulsive $-\Sigma_{0}$ as large to
several hundreds are generated for all cutoffs. The attractive
$\Sigma_{\mathrm{S}}$ gets quenched at large densities, due to a factor
$M^{*}/E_{k^{\prime}}^{*}$ present in the integrand of Eq. (15a), while
$-\Sigma_{0}$ increases almost linearly with increasing density. The
cancellation between $M^{*}\Sigma_{\mathrm{S}}/E_{k}^{*}$ and $-\Sigma_{0}$
finally leads to considerable repulsion in Eq. (17) compared to BHF
calculations.

At zero temperature, the pressures of symmetric nuclear matter, given by
$P=n^{2}\frac{\partial(E/A)}{\partial n}$, are computed via both BHF and RBHF
calculations utilizing OBEP$\Lambda$s as functions of density. Illustrated in
Fig. 7, the shaded areas represent constraints established by HIC experiments,
labeled as ”flow data 2003”[32] and ”FOPI 2016” [33]. BHF calculations yield
EOSs that are too soft to satisfy the constraints imposed by HIC experiments.
Conversely, RBHF calculations, attributed to significant relativistic
repulsion, generate EOSs that better align with the experimental constraints.
Moreover, the causality is automatically encoded in relativistic calculations,
manifested in $c_{s}^{2}=\frac{\partial P}{\partial\epsilon}<1$, with
$\epsilon$ being the energy density.

##  IV SUMMARY AND OUTLOOK

We constructed one-boson-exchange potentials by introducing exponential
regulators with relative momentum cutoffs ranging from $\Lambda=\infty$ to 2
fm-1. The regulator effectively suppress high momenta beyond the given cutoff
$\Lambda$, so we name the potential with each $\Lambda$ “OBEP$\Lambda$”. The
parameters within the OBEP$\Lambda$s were fitted to $NN$ scattering phase
shifts, low-energy scattering data, and deuteron binding energy. Notably, for
$\Lambda\leqslant 3$ fm-1, the potential matrix elements of our OBEP$\Lambda$s
exhibit quantitative agreement with other realistic $NN$ potentials evolved
using the renormalization group (RG) method.

Both NR-BHF and RBHF calculations were conducted using these OBEP$\Lambda$s.
The equations of state obtained from BHF calculations at all cutoffs display
an overbound nature. Notably, for $\Lambda\leqslant 3$ fm-1, the corresponding
OBEP$\Lambda$ demonstrate softness that prevents the production of saturation
phenomena, aligning with prior studies emphasizing the significance of the
three-nucleon force (3NF) in achieving nuclear matter saturation. Conversely,
all EOSs derived from RBHF calculations exhibit saturation behaviors; however,
their saturation densities and binding energies slightly fall short of
accurately reproducing empirical saturation properties. Further investigations
will explore the contributions arising from the relativistic three-hole-line
component in the BBG expansion and the impact of genuine relativistic 3NFs.

We examined the relativistic saturation mechanism by analyzing the kinetic and
potential terms within relativistic definitions. As the Dirac mass impacts the
lower component of the spinor, high-density relativistic kinetic terms exhibit
negativity, while significant repulsion arises in the potential terms due to
the interplay between the attractive scalar self-energy and the repulsive
time-component vector self-energy. Remarkably, even exceedingly soft
potentials can generate substantial scalar and time-component vector self-
energies, reaching several hundreds of MeV. The suppression of the attractive
scalar self-energy and the linear growth of the repulsive time-component
vector self-energy collectively contribute to the stiffening of the
relativistic EOS without the incorporation of 3NFs. This scenario could lead
to divergent interpretations of nuclear matter saturation and related
phenomena.

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