Densities and momentum distributions in A12nuclei from chiral eective eld theory interactions M. Piarulli12S. Pastore12yR.B. Wiringa3zS. Brusilow1 and R. Lim1

2025-05-06 0 0 3.17MB 12 页 10玖币
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Densities and momentum distributions in A≤12 nuclei from chiral effective field
theory interactions
M. Piarulli1,2,∗S. Pastore1,2,†R.B. Wiringa3,‡S. Brusilow1, and R. Lim1
1Department of Physics, Washington University in Saint Louis, Saint Louis, MO 63130, USA
2McDonnell Center for the Space Sciences at Washington University in St. Louis, MO 63130, USA
3Physics Division, Argonne National Laboratory, Argonne, IL 60439
(Dated: October 12, 2022)
Current and future electron and neutrino scattering experiments will be greatly aided by a better
understanding of the role played by short-range correlations in nuclei. Two-body physics, including
nucleon-nucleon correlations and two-body electroweak currents, is required to explain the body
of experimental data for both static and dynamical nuclear properties. In this work, we focus on
examining nucleon-nucleon correlations from a chiral effective field theory perspective and provide
a comprehensive set of new variational Monte Carlo calculations of one- and two-body densities
and momentum distributions based on the Norfolk many-body nuclear Hamiltonians for A≤12
systems. Online access to detailed tables and figures is available.
PACS numbers:
I. INTRODUCTION
The coordinate and momentum distributions of nucle-
ons in nuclei are one of the key indicators of short-range
correlations (SRCs) in multinucleon systems. SRCs rep-
resent a fascinating aspect of nuclear dynamics; under-
standing their formation mechanisms and specific charac-
teristics is required to obtain a comprehensive description
of nuclei and nucleonic matter. SRCs tell us much about
i) nuclear forces at short distances and how they are
generated from quantum chromodynamics; ii) the limita-
tions of mean-field models and how to ameliorate them;
iii) the properties of matter at high densities, such as
those found in compact stellar objects and in relativistic
heavy ion collisions; iv) the response functions in hadron
and lepton scattering from nuclei; v) the origin of the
EMC effect, and vi) the sensitivity of neutrinoless dou-
ble beta decay matrix elements to short-range dynamics.
Since the 1950s, many efforts have been devoted to
the study of SRCs and the short-range properties of the
nuclear force. It was only recently that experimental
and theoretical studies of these phenomena were placed
on solid ground, thanks to sophisticated high-energy and
large momentum transfer electron and proton scattering
experiments [1–9], allowing for precision measurements
of small cross sections, together with the enormous
progress made by many-body theories [10–18]. For in-
stance, experiments involving high-energy, semi-inclusive
triple coincidence measurements that successfully probed
the isospin composition of nucleon-nucleon (NN) SRCs
in the relative momentum range of 300–600 MeV/c
∗Electronic address: m.piarulli@wustl.edu
†Electronic address: saori@wustl.edu
‡Electronic address: wiringa@anl.gov
discovered a strong (by a factor of 20) dominance of
neutron-proton (np) pair SRCs in nuclei when compared
with proton–proton (pp) and neutron–neutron (nn)
correlations in both light and heavy nuclei [6–8]. This
was explained on the basis of the large tensor force in
the NN interaction at the above-mentioned momentum
range. As a result of this finding, it was predicted
that the single momentum distributions of the proton
and neutron, weighted by their respective fractions,
are nearly equal, and that the probability of a proton
or neutron being in high momentum NN correlation
is inversely proportional to their relative fractions in
the nucleus. The validity of these predictions were
confirmed by results of ab-initio variational Monte Carlo
(VMC) calculations of the momentum distributions of
light nuclei [16] and of approximate schemes like cluster
expansions [11, 13, 19] and correlated basis function
theory [20–22] for medium to heavy nuclei. Moreover,
calculations of the momentum distributions of different
light nuclei showed high momentum tails that resembled
those of the deuteron, demonstrating a universal nature
of SRCs [11, 13, 16, 19–22].
An extensive library of VMC one- and two-body densi-
ties and momentum distributions for many different light
nuclei using the phenomenological Argonne v18 (AV18)
two-nucleon (NN) [23], and Urbana X (UX) three-
nucleon (3N) interactions was previously constructed
and posted online for the benefit of the nuclear physics
community at large [16]. Additionally, these calculations
have contributed to a novel study of many-body factor-
ization and the position-momentum equivalence of nu-
clear short-range correlations, using a Generalized Con-
tact Formalism (GCF), which was reported in Nature
Physics [24].
In this paper, we provide a comprehensive set of new
results of one- and two-body densities and momentum
arXiv:2210.02421v1 [nucl-th] 5 Oct 2022
2
distributions over a wide range of nuclei from 2H up to
12C, using the Norfolk NN and 3N(NV2+3) forces [25–
29]. These results feature new calculations of the pair
density as a function of both the pair separation and
pair center-of-mass, and calculations of the two-body
momentum distribution coming from short- and long-
range pairs differentiated by a pair separation bound-
ary. The full set of calculations is accessible in graphical
and tabular forms online at www.phy.anl.gov/theory/
research/QMCresults.html.
The paper is structured as follows: a brief review of
Norfolk interactions is given in Sec. II. In Sec. III we
present results for the one- and two-body densities cal-
culated for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C. The pair
density as a function of both the pair separation and pair
center-of-mass is presented for 4He and 12C. In Sec. IV
the results for the one- and two-body momentum distri-
bution are provided for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and
12C. Results for momentum distributions as functions of
the relative momentum and center-of-mass momentum
without and with pair separation boundary are displayed
for 4He and 12C. Additional results are available online.
II. NORFOLK MANY-BODY INTERACTIONS
The Norfolk interactions are obtained from a chiral
effective field theory (χEFT) that uses pions, nucleons
and ∆’s as fundamental degrees of freedom, and con-
sists of long-range parts mediated by one- and two-
pion exchange, and contact terms specified by unknown
low-energy constants (LECs). The LECs entering the
NN contact interactions are constrained to reproduce
NN scattering data from the most recent and up-to-
date database collected by the Granada group [30–32].
The contact terms are regularized via a Gaussian cut-
off function with RSas the Gaussian parameter [25–
27]. The divergences at high-value of momentum trans-
fer in the pion-range operators are removed via a spe-
cial radial function characterized by the cutoff RL[25–
27]. There are two classes of NV2 potentials. Class I
(II) has been fitted to data up to 125 MeV (200 MeV).
For each class, two combinations of short- and long-range
regulators have been used, namely (RS,RL)=(0.8, 1.2)
fm (models NV2-Ia and NV2-IIa) and (RS,RL)=(0.7,
1.0) fm (models NV2-Ib and NV2-IIb). Class I (II) fits
about 2700 (3700) data points with a χ2/datum <
∼1.1
(<
∼1.4) [25, 26]. The short-range component of the 3N
interactions is parametrized in terms of two LECs, cD
and cE. In the first generation of Norfolk potentials
(NV2+3-Ia/b and NV2+3-IIa/b), these LECs have been
determined by simultaneously reproducing the experi-
mental trinucleon ground-state energies and nd doublet
scattering length [33]. Within the χEFT framework, cD
is related to the LEC entering the axial two-body con-
tact current [34–36]. This allows one to adopt a dif-
ferent strategy to constrain cDand cE. In particular,
in Ref. [27] they have been constrained to reproduce the
trinucleon binding energies and the empirical value of the
Gamow-Teller matrix element in tritium βdecay. Nor-
folk models that use this fitting procedure are designated
with a ‘*’ namely, NV2+3-Ia*/b* and NV2+3-IIa*/b*.
These interactions have been recently employed in the
VMC and Green’s function Monte Carlo (GFMC) ap-
proaches [37, 38] to calculate energies [33], charge radii
and electromagnetic form factors [38], beta-decay transi-
tions [27, 39, 40], neutrinoless double beta-decay [41, 42]
of light nuclei, beta decay spectra [43], muon-capture
rates [44] and with the auxiliary field diffusion Monte
Carlo (AFDMC) [38] to study the equation of state of
pure neutron matter [45, 46].
III. DENSITY DISTRIBUTIONS
The one- and two-body densities are evaluated as sim-
ple δ-function expectation values given by
ρN(r) = 1
4πr2Ψ
X
i
PNiδ(r− |ri−Rcm|)
Ψ,(1)
ρNN (r) = 1
4πr2Ψ
X
i<j
PNiPNjδ(r− |ri−rj|)
Ψ,(2)
where PNirepresents the projector operator onto protons
(+) or neutrons (−) defined as PNi= (1 ±τzi)/2, riis
the position of nucleon iand Rcm is the coordinate of
the center of mass.
A detailed survey of one- and two-body densities have
been calculated for a variety of nuclei in the range
A= 2 −12 using variational Monte Carlo wave func-
tions developed for the AV18+UX and the Norfolk local
chiral interactions. The corresponding tables and fig-
ures are available online at www.phy.anl.gov/theory/
research/density/, for the one-nucleon densities, and
at www.phy.anl.gov/theory/research/density2/, for
the two-nucleon densities.
A. One-body density results
In Fig. 1 we present the neutron and proton densities
calculated for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C us-
ing the AV18+UX and the NV2+3-Ia, NV2+3-Ia*, and
NV2+3-IIb* local chiral interactions. Additional den-
sities for 2H, 6He, 8,9Li, 8,10,12Be, 11B and 10,11C may
be found in the online tables, as well as results for the
NV2+3-Ib* and NV2+3-IIa* interactions. We also give
neutron and proton rms radii there.
The VMC wave functions are treated as states of
unique isospin T. Thus for N=Znuclei, proton and
neutron densities are the same and only proton densi-
ties are given in the online tables. However, the wave
functions for nuclei with T > 0 can be different for dif-
ferent isospin projections Tz, so mirror nuclei are not
isospin symmetric. This allows the proton-rich nuclei to
3
0.04
0.08
0.12
0.16
ρn(r)
3H
3He
4He
6Li
7Li
8He
9Be
10B
12C
1 2 3 4
r (fm)
0.04
0.08
0.12
0.16
ρn(r)
1 2 3 4
r (fm)
AV18+UX
NV2+3-Ia*
NV2+3-Ia
NV2+3-IIb*
0.04
0.08
0.12
0.16
ρp(r)
3H
3He
4He
6Li
7Li
8He
9Be
10B
12C
1 2 3 4
r (fm)
0.04
0.08
0.12
0.16
ρp(r)
1 2 3 4
r (fm)
AV18+UX
NV2+3-Ia*
NV2+3-Ia
NV2+3-IIb*
FIG. 1: One-body neutron (left panel) and proton (right panel) densities are shown for 3H, 3,4,8He, 6,7Li, 9Be, 10B,
and 12C using the phenomenological AV18+UX and the local chiral NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb*
interactions.
be slightly more diffuse than neutron-rich nuclei due to
their greater repulsive Coulomb interaction.
Spin-up and spin-down densities are also provided in
the online tables. In J= 0 nuclei, spin-up and spin-down
densities are identical, but not for J > 0 nuclei. If spin-up
and spin-down projections are the same, as in 0+states,
we give only totals. The total number of spin-up/down
protons and neutrons in J > 0 nuclei with MJ=J
are reported in Table I. Unless otherwise indicated by
an error in parentheses, variation among the different
interaction models is less than 0.01. We note that for
these nuclei, the subset with an odd number of neutrons
has (n↑−n↓)≈0.7-0.9, while those with an even number
of neutrons have (n↑ −n↓)≈-0.02. Similar results hold
for nuclei with odd and even proton numbers. The sole
exception is 9Li which has an exceptionally large error
bar.
We also note that the s-shell nuclei (A≤4) exhibit
large peaks at small separation, while the p-shell nuclei
(A≥6) are much reduced at small rand more spread
out. This can be attributed to the cluster structure of
these light p-shell nuclei, e.g., αd in 6Li, αt in 7Li, ααn in
9Be, and 3αin 12C. This puts the center of mass of these
nuclei in between clusters and thus reduces the central
density.
B. Two-body density results
In Fig. 2, we present the relative-distance pair den-
sities, with neutron-proton (np) in the left panel and
proton-proton (pp) in the right panel, for 3H, 3,4,8He,
6,7Li, 9Be, 10B, and 12C using the phenomenological
AV18+UX and the local chiral NV2+3-Ia, NV2+3-Ia*,
and NV2+3-IIb* interactions. The online tables contain
additional results for the NV2+3-Ib* and NV2+3-IIa*
interactions.
TABLE I: Total number of spin-up/down protons and
neutrons in J > 0 nuclei with MJ=Jfor the local
chiral Norfolk NV2+3 interactions. Variation among
the different interactions NV2+3-Ia, -Ia*, -Ib*, -IIa*,
and -IIb* is less than 0.01 unless otherwise indicated by
an error in parentheses.
Nucleus N↑pN↓pN↑nN↓n
2H(1+) 0.96 0.04 0.96 0.04
3He( 1
2
+) 0.98 1.02 0.94 0.06
6Li(1+) 1.93 1.07 1.93 1.07
7Li( 3
2
−) 1.94 1.06 1.99 2.01
8Li(2+) 1.91 1.09 2.85(1) 2.15(1)
9Li( 3
2
−) 1.91 1.09 3.12(7) 2.88(7)
9Be( 3
2
−) 2.00 2.00 2.85(2) 2.15(2)
10B(3+) 2.90(1) 2.10(1) 2.90(1) 2.10(1)
11B( 3
2
−) 2.87(2) 2.13(2) 2.99(1) 3.01(1)
We can see that within a fixed interaction model, the
two-nucleon densities at r<
∼1.5 fm for various nuclei
exhibit a similar behavior, generated by the cooperation
of the short-range repulsion and the intermediate-range
tensor attraction of the NN interaction, with the tensor
force governing the large overshoot at r∼1.0 fm between
np pairs.
As shown in Fig. 3, where all calculations are scaled to
have the same value at ∼1 fm, the two-nucleon densities
at short separations appears to be the same for all values
of A, which leads to the nontrivial conclusion that at
short ranges the two-nucleon motion is not affected by
the presence of the other particles. This is what has
been called universality of SRCs [12]. Moreover, at large
separation the asymptotic behavior of the two-nucleon
densities for different nuclei differs due to the different
surface effects.
摘要：

DensitiesandmomentumdistributionsinA12nucleifromchirale ective eldtheoryinteractionsM.Piarulli1;2,S.Pastore1;2,yR.B.Wiringa3,zS.Brusilow1,andR.Lim11DepartmentofPhysics,WashingtonUniversityinSaintLouis,SaintLouis,MO63130,USA2McDonnellCenterfortheSpaceSciencesatWashingtonUniversityinSt.Louis,MO63130...

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