# Relative mass distributions of neutron-rich thermally fissile nuclei within
statistical model

Bharat Kumar1,4<EMAIL_ADDRESS>M.T. Senthil Kannan2 M. Balasubramaniam2 B.
K. Agrawal3,4 S. K. Patra1,4 1Institute of Physics, Sachivalaya Marg,
Bhubaneswar - 751005, India.  2Department of Physics, Bharathiar University,
Coimbatore - 641046, India.  3Saha Institute of Nuclear Physics, 1/AF,
Bidhannagar, Kolkata - 700064, India.  4Homi Bhabha National Institute,
Anushakti Nagar, Mumbai - 400094, India.

###### Abstract

We study the binary mass distribution for the recently predicted thermally
fissile neutron-rich uranium and thorium nuclei using statistical model. The
level density parameters needed for the study are evaluated from the
excitation energies of temperature dependent relativistic mean field
formalism. The excitation energy and the level density parameter for a given
temperature are employed in the convolution integral method to obtain the
probability of the particular fragmentation. As representative cases, we
present the results for the binary yield of 250U and 254Th. The relative
yields are presented for three different temperatures $T=$ 1, 2 and 3 MeV.

###### pacs:

25.85.-w, 21.10.Ma, 21.10.Pc, 24.75.+i

##  I Introduction

Fission phenomenon is one of the most interesting subject in the field of
nuclear physics. To study the fission properties, a large number of models
have been proposed. The fissioning of a nucleus is successfully explained by
the liquid drop model and the semi-empirical mass formula is the best and
simple oldest tool to get a rough estimation of the energy released in a
fission process. The pioneering work of Vautherin and Brink vat70 , who has
applied the Skyrme interaction in a self-consistent method for the calculation
of ground state properties of finite nuclei opened a new dimension in the
quantitative estimation of nuclear properties. Subsequently, the Hartree-Fock
and time dependent Hartree-Fock formalisms pal  are also implemented to study
the properties of fission. Most recently, the microscopic relativistic mean
field approximation, which is another successful theory in nuclear physics is
also used for the study of nuclear fission skp10 .

From last few decades, the availability of neutron rich nuclei in various
laboratories across the globe opened up new research in the field of nuclear
physics, because of their exotic decay properties. The effort for the
synthesis of superheavy nuclei in the laboratories like, Dubna (Russia), GSI
(Germany), RIKEN (Japan) and BNL (USA) is also quite remarkable. Due to all
these, the periodic table is extended till date upto atomic number $Z=118$
ogna . The decay modes of these superheavy nuclei are very different than the
usual modes. Mostly, we understand that, a neutron rich nucleus has a large
number of neutron than the light or medium mass region of the periodic table.
The study of these neutron-rich superheavy nuclei is very interesting, because
of their ground state structures and various mode of decays, including multi-
fragment fission (more than two) skp10 . Another interesting feature of some
neutron rich uranium and thorium nuclei is that similar to 233U, 235U and
239Pu, the nuclei 246-264U and 244-262Th are also thermally fissile, which are
extremely important for the energy production in fission process. If the
neutron rich uranium and thorium nuclei are the viable sources, then these
nuclei will be more effective to achieve the critical condition in a
controlled fission reaction.

Now the question arises, how we can get a reasonable estimation of the mass
yield in the spallation reaction of these neutron rich thermally fissile
nuclei. As mentioned earlier in this section, there are many formalisms
available in the literature to study these cases. Here, we adopt the
statistical model developed by Fong fon56 . The calculation is further
extended by Rajasekaran and Devanathan mrd81  to study the binary mass
distributions using the single particle energies of the Nilsson model. The
obtained results are well in agreement with the experimental data. In the
present study, we would like to replace the single particle energies with the
excitation energies of a successful microscopic approach, the relativistic
mean field (RMF) formalism.

For last few decades, the relativistic mean field (RMF) formalism wal74 ;
seort86 ; horo 81 ; gam90 ; patra91  with various parameter sets have
successfully reproduced the bulk properties, such as binding energies, root
mean square radii, quadrupole deformation etc. not only for nuclei near the
$\beta-$stability line but also for nuclei away from it. Further, the RMF
formalism is successfully applied to the study of clusterization of known
cluster emitting heavy nucleus aru05 ; bks06 ; skp07  and the fission of
hyper-hyper deformed 56Ni rkg08 . Rutz et. al. rutz95  reproduced the double,
triple humped fission barrier of 240Pu, 232Th and the asymmetric ground states
of 226Ra using RMF formalism. Moreover, the symmetric and asymmetric fission
modes are also successfully reproduced. Patra et. al. skp10  studied the neck
configuration in the fission decay of neutron rich U and Th isotopes. The main
goal of this present paper is to understand the binary fragmentation yield of
such neutron rich thermally fissile superheavy nuclei. 250U and 254Th are
taken for further calculations as the representative cases.

The paper is organized as follows: In Section II, the statistical model and
relativistic mean field theory are presented briefly. In subsection A of this
section, the level density parameter and it’s relation with the relative mass
yield are outlined. In subsection B of II, the equation of motion of the
nucleon and meson fields obtained from the relativistic mean field Lagrangian
and the temperature dependent of the equations are adopted through the
occupation number of protons and neutrons. The results are discussed in
Section III and compared with the finite range droplet model (FRDM)
predictions. The summary and concluding remarks are given in Section IV.

##  II Formalism

The possible binary fragments of the considered nucleus is obtained by
equating the charge to mass ratio of the parent nucleus to the fission
fragments as mbs2014 :

$\frac{Z_{P}}{A_{P}}\approx\frac{Z_{i}}{A_{i}},$ (1)

with $A_{P}$, $Z_{P}$ and $A_{i}$, $Z_{i}$ ($i$ = 1 and 2) correspond to mass
and charge numbers of the parent nucleus and the fission fragments mrd81 . The
constraints, $A_{1}+A_{2}=A$, $Z_{1}+Z_{2}=Z$ and $A_{1}\geq A_{2}$ are
imposed to satisfy the conservation of charge and mass number in a nuclear
fission process and to avoid the repetition of fission fragments. Another
constraint i.e., the binary charge numbers from $Z_{2}\geq$ 26 to $Z_{1}\leq$
66 is also taken into consideration from the experimental yield berg71  to
generate the combinations, assuming that the fission fragments lie within
these charge range.

###  II.1 Statistical theory

The statistical theory fon56 ; cole  assumes that the probability of the
particular fragmentation is directly proportional to the folded level density
$\rho_{12}$ of that fragments with the total excitation energy $E^{*}$, i.e.,
$P(A_{j},Z_{j})\propto\rho_{12}(E^{*})$. Where,

$\rho_{12}(E^{*})=\int_{0}^{E^{*}}\rho_{1}(E^{*}_{1})\,\rho_{2}(E^{*}-E_{1}^{*})\,dE^{*}_{1},$
(2)

and $\rho_{i}$ is the level density of two fragments ($i$ = 1, 2). The nuclear
level density bet37 ; mor72  is expressed as a function of fragment excitation
energy $E^{\ast}_{i}$ and the single particle level density parameter $a_{i}$
which is given as:

$\rho_{i}\left(E^{\ast}_{i}\right)=\dfrac{1}{12}\left(\dfrac{\pi^{2}}{a_{i}}\right)^{1/4}E^{\ast(-5/4)}_{i}\exp\left(2\sqrt{a_{i}E^{\ast}_{i}}\right).$
(3)

In Refs. mbs2014 ; sk16 , we calculate the excitation energies of the
fragments using the ground state single particle energies of finite range
droplet model (FRDM) moller97  at a given temperature $T$ keeping the total
number of proton and neutron fixed. In the present study, we apply the self
consistent temperature dependent relativistic mean field theory to calculate
the $E^{*}$ of the fragments. The excitation energy is calculated as,

$E^{*}_{i}(T)=E_{i}(T)-E_{i}(T=0).$ (4)

The level density parameter $a_{i}$ is given as,

$a_{i}=\frac{E^{*}_{i}}{T^{2}}.$ (5)

The relative yield is calculated as the ratio of the probability of a given
binary fragmentation to the sum of the probabilities of all the possible
binary fragmentations and it is given by,

$Y(A_{j},Z_{j})=\frac{P(A_{j},Z_{j})}{\sum_{j}P(A_{j},Z_{j})},$ (6)

where $A_{j}$ and $Z_{j}$ are referred to the binary fragmentations involving
two fragments with mass and charge numbers $A_{1}$, $A_{2}$ and $Z_{1}$,
$Z_{2}$ obtained from Eq. (1). The competing basic decay modes such as
neutron/proton emission, $\alpha$ decay and ternary fragmentation are not
considered. In addition to these approximations, we have also not included the
dynamics of the fission reaction, which are really important to get a
quantitative comparison with the experimental measurements. The presented
results are the prompt disintegration of a parent nucleus into two fragments
(democratic breakup). The resulting excitation energy would be liberated as
prompt particle emission or delayed emission, but such secondary emissions are
also ignored.

###  II.2 RMF Formalism

The RMF theory assume that the nucleons interact with each other via meson
fields. The nucleon - meson interaction is given by the Lagrangian density
patra91 ; wal74 ; seort86 ; horo 81 ; bogu77 ; pric87 ,

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi_{i}}\\{i\gamma^{\mu}\partial_{\mu}-M\\}\psi_{i}+{\frac{1}{2}}\partial^{\mu}\sigma\partial_{\mu}\sigma-{\frac{1}{2}}m_{\sigma}^{2}\sigma^{2}$
(7)
$\displaystyle-{\frac{1}{3}}g_{2}\sigma^{3}-{\frac{1}{4}}g_{3}\sigma^{4}-g_{\sigma}\overline{\psi_{i}}\psi_{i}\sigma$
$\displaystyle-{\frac{1}{4}}\Omega^{\mu\nu}\Omega_{\mu\nu}+{\frac{1}{2}}m_{w}^{2}V^{\mu}V_{\mu}-g_{w}\overline{\psi}_{i}\gamma^{\mu}\psi_{i}V_{\mu}$
$\displaystyle-{\frac{1}{4}}\vec{B}^{\mu\nu}.\vec{B}_{\mu\nu}+{\frac{1}{2}}m_{\rho}^{2}{\vec{R}^{\mu}}.{\vec{R}_{\mu}}-g_{\rho}\overline{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}.\vec{R^{\mu}}$
$\displaystyle-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}-e\overline{\psi}_{i}\gamma^{\mu}\frac{\left(1-\tau_{3i}\right)}{2}\psi_{i}A_{\mu}.$

Where, $\psi_{i}$ is the single particle Dirac spinor. The arrows over the
letters in the above equation represent the isovector quantities. The nucleon,
the $\sigma$, $\omega$, and $\rho$ meson masses are denoted by M,
$m_{\sigma}$, $m_{\omega}$ and $m_{\rho}$ respectively. The meson and the
photon fields are termed as $\sigma$, $V_{\mu}$, $R^{\mu}$ and $A_{\mu}$ for
$\sigma$, $\omega$, $\rho-$ mesons and photon respectively. The $g_{\sigma}$,
$g_{\omega}$, $g_{\rho}$ and $\frac{e^{2}}{4\pi}$ are the coupling constants
for the $\sigma$, $\omega$, $\rho-$mesons and photon fields with nucleons
respectively. The strength of the constants $g_{2}$ and $g_{3}$ are
responsible for the nonlinear couplings of $\sigma$ meson ($\sigma^{3}$ and
$\sigma^{4}$). The field tensors of the isovector mesons and the photon are
given by,

$\displaystyle\Omega^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu},\,$ (8)
$\displaystyle\vec{B}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\vec{R}^{\nu}-\partial^{\nu}\vec{R}^{\mu}-g_{\rho}(\vec{R}^{\mu}\times\vec{R}^{\nu}),\,$
(9) $\displaystyle F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.\,$ (10)

The classical variational principle gives the Euler-Lagrange equation and we
get the Dirac-equation with potential terms for the nucleons and Klein-Gordan
equations with source terms for the mesons. We assume the no-sea
approximation, so we neglect the antiparticle states. We are dealing with the
static nucleus, so the time reversal symmetry and the conservation of parity
simplifies the calculations. After simplifications, the Dirac equation for the
nucleon is given by,

$\\{-i\alpha.\bigtriangledown+V(r)+\beta\left[M+S(r)\right]\\}\,\,\psi_{i}=\epsilon_{i}\,\psi_{i},$
(11)

where V(r) represents the vector potential and S(r) is the scalar potential,

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle
g_{\omega}\omega_{0}+g_{\rho}\tau_{3}\rho_{0}(r)+e\frac{(1-\tau_{3})}{2}A_{0}(r),$
$\displaystyle S(r)$ $\displaystyle=$ $\displaystyle g_{\sigma}\sigma(r),$
(12)

which contributes to the effective mass,

$M^{*}(r)=M+S(r).$ (13)

The Klein-Gordon equations for the mesons and the electromagnetic fields with
the nucleon densities as sources are,

$\displaystyle\\{-\triangle+m_{\sigma}^{2}\\}\sigma(r)=-g_{\sigma}\rho_{s}(r)-g_{2}\sigma^{2}(r)$
$\displaystyle-\,g_{3}\sigma^{3}(r),$ (14)
$\displaystyle\\{-\triangle+m_{\omega}^{2}\\}\omega_{0}(r)=g_{\omega}\rho_{v}(r),$
(15)
$\displaystyle\\{-\triangle+m_{\rho}^{2}\\}\rho_{0}(r)=g_{\rho}\rho_{3}(r),$
(16) $\displaystyle-\triangle A_{0}(r)=e\rho_{c}(r).$ (17)

The corresponding densities such as scalar, baryon (vector), isovector and
proton (charge) are given as

$\displaystyle\rho_{s}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}n_{i}\,\psi_{i}^{\dagger}(r)\,\psi_{i}(r)\,,$ (18)
$\displaystyle\rho_{v}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}n_{i}\,\psi_{i}^{\dagger}(r)\,\gamma_{0}\,\psi_{i}(r)\,,$
(19) $\displaystyle\rho_{3}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}n_{i}\,\psi_{i}^{\dagger}(r)\,\tau_{3}\,\psi_{i}(r)\,,$
(20) $\displaystyle\rho_{\rm c}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}n_{i}\,\psi_{i}^{\dagger}(r)\,\left(\frac{1-\tau_{3}}{2}\right)\,\psi_{i}(r)\,.$
(21)

To solve the Dirac and Klein-Gordan equations, we expand the Boson fields and
the Dirac spinor in an axially deformed harmonic oscillator basis with
$\beta_{0}$ as the initial deformation parameter. The nucleon equation along
with different meson equations form a set of coupled equations, which can be
solved by iterative method. The center of mass correction is calculated with
the non-relativistic approximation. The quadrupole deformation parameter
$\beta_{2}$ is calculated from the resulting quadrupole moments of the proton
and neutron. The total energy is given by blunden87 ; reinhard89 ; gam90 ,

$\displaystyle E(T)=$
$\displaystyle\sum_{i}\epsilon_{i}n_{i}+E_{\sigma}+E_{\sigma
NL}+E_{\omega}+E_{\rho}$ (22) $\displaystyle+E_{C}+E_{pair}+E_{c.m.}-AM,$

with

$E_{\sigma}=-\frac{1}{2}g_{\sigma}\int d^{3}r\rho_{s}(r)\sigma(r),$ (23)
$E_{\sigma NL}=-\frac{1}{2}\int
d^{3}r\left\\{\,\frac{2}{3}g_{2}\,\sigma^{3}(r)+\frac{1}{2}g_{3}\,\sigma^{4}(r)\,\right\\},$
(24) $\displaystyle E_{\omega}=$ $\displaystyle-\frac{1}{2}g_{\omega}\int
d^{3}r\rho_{v}(r)\omega^{0}(r),$ (25) $\displaystyle E_{\rho}=$
$\displaystyle-\frac{1}{2}g_{\rho}\int d^{3}r\rho_{3}(r)\rho^{0}(r),$ (26)
$\displaystyle E_{C}=$ $\displaystyle-\dfrac{e^{2}}{8\pi}\int
d^{3}r\rho_{c}(r)A^{0}(r),$ (27) $\displaystyle
E_{pair}=-\triangle\sum_{i>0}u_{i}v_{i}=-\frac{\triangle^{2}}{G},$ (28)
$E_{c.m.}=-\frac{3}{4}\times 41A^{-1/3}.$ (29)

Here, $\epsilon_{i}$ is the single particle energy, $n_{i}$ is the occupation
probability and $E_{pair}$ is the pairing energy obtained from the simple BCS
formalism.

###  II.3 Pairing and temperature dependent RMF formalism

The pairing correlation plays a distinct role in open-shell nuclei. The effect
of pairing correlation is markedly seen with increase in mass number A.
Moreover it helps in understanding the deformation of medium and heavy nuclei.
It has a lean effect on both bulk and single particles properties of lighter
mass nuclei because of the availability of limited pairs near the Fermi
surface. We take the case of T=1 channel of pairing correlation i.e, pairing
between proton- proton and neutron-neutron. In this case, a nucleon of quantum
states $|jm_{z}\rangle$ pairs with another nucleons having same $I_{z}$ value
with quantum states $|j-m_{z}\rangle$, since it is the time reversal partner
of the other. In both nuclear and atomic domain the ideology of BCS pairing is
the same. The even-odd mass staggering of isotopes was the first evidence of
its kind for the pairing energy. Considering the mean-field formalism, the
violation of the particle number is seen only due to the pairing correlation.
We find terms like $\psi^{\dagger}\psi$ (density) in the RMF Lagrangian
density but we put an embargo on terms of the form
$\psi^{\dagger}\psi^{\dagger}$ or $\psi\psi$ since it violates the particle
number conservation. We apply externally the BCS constant pairing gap
approximation for our calculation to take the pairing correlation into
account. The pairing interaction energy in terms of occupation probabilities
$v_{i}^{2}$ and $u_{i}^{2}=1-v_{i}^{2}$ is written as pres82 ; patra93 :

$E_{pair}=-G\left[\sum_{i>0}u_{i}v_{i}\right]^{2},$ (30)

with $G$ is the pairing force constant. The variational approach with respect
to the occupation number $v_{i}^{2}$ gives the BCS equation pres82 :

$2\epsilon_{i}u_{i}v_{i}-\triangle(u_{i}^{2}-v_{i}^{2})=0,$ (31)

with the pairing gap $\triangle=G\sum_{i>0}u_{i}v_{i}$. The pairing gap
($\triangle$) of proton and neutron is taken from the empirical formula va73 ;
gam90 :

$\triangle=12\times A^{-1/2}.$ (32)

The temperature introduced in the partial occupancies in the BCS approximation
is given by,

$n_{i}=v_{i}^{2}=\frac{1}{2}\left[1-\frac{\epsilon_{i}-\lambda}{\tilde{\epsilon_{i}}}[1-2f(\tilde{\epsilon_{i}},T)]\right],$
(33)

with

$\displaystyle
f(\tilde{\epsilon_{i}},T)=\frac{1}{(1+exp[{\tilde{\epsilon_{i}}/T}])}$
$\displaystyle and$
$\displaystyle\tilde{\epsilon_{i}}=\sqrt{(\epsilon_{i}-\lambda)^{2}+\triangle^{2}}.$
(34)

The function $f(\tilde{\epsilon_{i}},T)$ represents the Fermi Dirac
distribution for quasi particle energy $\tilde{\epsilon_{i}}$. The chemical
potential $\lambda_{p}(\lambda_{n})$ for protons (neutrons) is obtained from
the constraints of particle number equations

$\displaystyle\sum_{i}n_{i}^{Z}=Z,$ $\displaystyle\sum_{i}n_{i}^{N}=N.$ (35)

The sum is taken over all proton and neutron states. The entropy is obtained
by,

$S=-\sum_{i}\left[n_{i}\,ln(n_{i})+(1-n_{i})\,ln(1-n_{i})\right].$ (36)

The total energy and the gap parameter are obtained by minimizing the free
energy,

$F=E-TS.$ (37)

In constant pairing gap calculations, for a particular value of pairing gap
$\triangle$ and force constant $G$, the pairing energy $E_{pair}$ diverges, if
it is extended to an infinite configuration space. In fact, in all realistic
calculations with finite range forces, $\triangle$ is not constant, but
decreases with large angular momenta states above the Fermi surface.
Therefore, a pairing window in all the equations are extended up-to the level
$|\epsilon_{i}-\lambda|\leq 2(41A^{-1/3})$ as a function of the single
particle energy. The factor 2 has been determined so as to reproduce the
pairing correlation energy for neutrons in 118Sn using Gogny force gam90 ;
patra93 ; dech80 .

##  III Results and discussions

In our very recent work mts16 , we have calculated the ternary mass
distributions for 252Cf, 242Pu and 236U with the fixed third fragments
A${}_{3}=~{}^{48}$Ca, 20O and 16O respectively for the three different
temperatures T = 1, 2 and 3 MeV within the TRMF formalism. The structure
effects of binary fragments are also reported in Ref. cinthol . In this
article, we study the mass distribution of 250U and 254Th as a representative
cases from the range of neutron-rich thermally fissile nuclei 246-264U and
244-262Th. Because of the neutron-rich nature of these nuclei, a large number
of neutrons emit during the fission process. These nucleons help to achieve
the critical condition much sooner than the normal fissile nuclei.

To assure the predictability of the statistical model, we also study the
binary fragmentation of naturally occurring 236U and 232Th nuclei. The
possible binary fragments are obtained using the Eq. (1). To calculate the
total binding energy at a given temperature, we use the axially symmetric
harmonic oscillator basis expansion NF and NB for the Fermion and Boson wave-
functions to solve the Dirac Eq. (11) and the Klein Gordon Eqs. (14 \- 17)
iteratively. It is reported bhar15  that the effect of basis space on the
calculated binding energy, quadrupole deformation parameter ($\beta_{2}$) and
the rms radii of nucleus are almost equal for the basis set N${}_{F}=$
N${}_{B}=$ 12 to 20 in the mass region A $\sim$ 200 . Thus, we use the basis
space N${}_{F}=$ 12 and N${}_{B}=$ 20 to study the binary fragments up to mass
number A $\sim$ 182\. The binding energy is obtained by minimizing the free
energy, which gives the most probable quadrupole deformation parameter
$\beta_{2}$ and the proton (neutron) pairing gaps $\triangle_{p}$
($\triangle_{n}$) for the given temperature. At finite temperature, the
continuum corrections due to the excitation of nucleons to be considered. The
level density in the continuum depends on the basis space NF and NB niu09 . It
is shown that the continuum corrections need not be included in the
calculations of level densities up-to the temperature T $\sim$ 3 MeV bka00 ;
bka98 .

Figure 1: (Color online) The level density parameter $a$ for the binary
fragmentation of 236U, 250U, 232Th and 254Th at temperature $T=$ 1, 2 and 3
MeV within the TRMF (solid lines) and FRDM (dash lines) formalisms.

###  III.1 Level density parameter and level density within TRMF and FRDM
formalisms

In TRMF, the excitation energies $E^{*}$ and the level density parameters
$a_{i}$ of the fragments are obtained self consistently from Eqns. (4) to (5).
The FRDM calculations are also done for comparison. In this case, level
density of the fragments are evaluated from the ground state single particle
energies of the finite range droplet model (FRDM) of Möller et. al. moller95
which are retrieved from the Reference Input Parameter Library (RIPL-3) ripl3
. The total energy at a given temperature is calculated as $E(T)=\sum
n_{i}\epsilon_{i}$; $\epsilon_{i}$ are the ground state single particle
energies and $n_{i}$ are the Fermi-Dirac distribution function. The $T$
dependent energies are obtained by varying the occupation numbers at a fixed
particle number for a given temperature and given fragment. The level density
parameter $a$ is a crucial quantity in the statistical theory for the
estimation of yields. These values of $a$ for the binary fragments of 236U,
250U, 232Th and 254Th obtained from TRMF and FRDM are depicted in Fig. 1. The
empirical estimation $a=A/K$ are also given for comparison, with $K$, the
inverse level density parameter. In general, the $K$ value varies from 8 to 13
with the increasing temperature. However, the level density parameter is
considered to be constant up-to $T\approx$ 4 MeV. Hence, we take the practical
value of $K=10$ as mentioned in Ref. ner2002 . The $a$ values of TRMF are
close to the empirical level density parameter. The FRDM level density
parameters are appreciably lower than the referenced $a$. Further, in both
models at $T=$ 1 MeV, there are more fluctuations in the level density
parameter due to the shell effects of the fragments. At $T=$ 2 and 3 MeV, the
variations are small. This may be due to the fact that the shell become
degenerate at the higher temperatures. All fragments becomes spherical at
temperature $T\approx$ 3 MeV as shown in Ref. cinthol .

Figure 2: (Color online) The inverse level density parameters $K_{E}$ (solid
lines) and $K_{S}$ (dash lines) are obtained for 236U, 250U, 232Th and 254Th
at temperatures $T=$ 1, 2 and 3 MeV. Figure 3: (Color online) The level
density of the binary fragmentations of 236U, 250U, 232Th and 254Th at
temperature $T=$ 1, 2 and 3 MeV within the TRMF (solid lines) and FRDM (dash
lines) formalisms.

The level density parameter $a$ is evaluated in two different ways using
excitation energy and the entropy of the system as:

$\displaystyle a_{E}=\dfrac{E^{*}}{T^{2}},$ (38) $\displaystyle
a_{S}=\dfrac{S}{2T}.$

For instance, the inverse level density parameters $K_{E}$ and $K_{S}$ of
236U, 250U, 232Th and 254Th within TRMF formalism are depicted in Fig. 2. Both
$K_{S}$ and $K_{E}$ have maximum fluctuation upto 30 MeV at $T=$1 MeV. These
values reduce to $10-13$ MeV at temperature $T=2$ MeV or above. It is to be
noted that at $T=$ 3 MeV, the inverse level density parameter substantially
lower around the mass number $A\sim$ 130 in all cases. This may be due to the
neutron closed shell ($N=$ 82) in the fission fragments of 236U and 232Th and
the neutron-rich nuclei 250U and 254Th. The level density for the fission
fragments of 236U, 250U, 232Th and 254Th are plotted as a function of mass
number in Fig. 3 within the TRMF and FRDM formalisms at three different
temperatures $T=$ 1, 2 and 3 MeV.

The level density $\rho$ has maximum fluctuations at $T=$ 1 MeV for all
considered nuclei in TRMF model similar to the level density parameter $a$.
The $\rho$ values are substantially lower at mass number $A\sim 130$ for all
nuclei. In Fig. 3, one can notice that the level density has small kinks in
the mass region $A\sim 71-81$ of 236U and $A\sim 77-91$ of 250U, comparing
with the neighboring nuclei at temperature $T=$ 2 MeV. Consequently, the
corresponding partner fragments have also higher $\rho$ values. A further
inspection reveals that the level density of the closed shell nucleus around
$A\sim$ 130 has higher value than the neighboring nuclei for both 236,250U,
but it has lower yield due to the smaller level density of the corresponding
partners. At $T=$ 3 MeV, the level density of the fragments around mass number
$A\sim$72 and 130 have larger values compared to other fragments of 236U. On
the other hand, the level density in the vicinity of neutron number $N=82$ and
proton number $Z=50$ for the fragments of the neutron-rich 250U nucleus is
quite high, because of the close shell of the fragments. This is evident from
the small kink in the level density of 130Cd ($N=$ 82), 132In ($N\sim$ 82) and
135Sn ($Z=$ 50). Again, for 232Th, the level densities are found to be maximum
at around mass number $A\sim$81 and 100 for $T=$ 2 MeV. In case of 254Th, the
$\rho$ values are found to be large for the fragments around $A\sim$ 78 and 97
at $T=$ 2 MeV. Their corresponding partners have also similar behavior. For
higher temperature $T=$ 3 MeV, the higher $\rho$ values of 232Th fragments are
notable around mass number $A\sim 130$. Similarly, for 254Th, the fission
fragments around $A\sim$ 78 has higher level density at $T=$ 3 MeV. In
general, the level density increases towards the neutron closed shell ($N=$
82) nucleus.

###  III.2 Relative fragmentation distribution in binary systems

Figure 4: (Color online) Mass distribution of 236U and 250U at temperatures T
= 1, 2 and 3 MeV. The total yield values are normalized to the scale 2. Figure
5: (Color online) Mass distribution of 232Th and 254Th at temperatures T = 1,
2 and 3 MeV. The total yield values are normalized to the scale 2.

In this section, the mass distributions of 236U, 232Th and the neutron rich
nuclei 250U and 254Th are calculated at temperatures $T=$ 1, 2 and 3 MeV using
TRMF and FRDM excitation energies and the level density parameters $a$ as
explained in Sec. II. The binary mass distributions of 236,250U and 232,254Th
are plotted in Figs. 4 and 5. The total energy at finite temperature and
ground state energy are calculated using the TRMF formalism as discussed in
the section III.1. From the excitation energy E∗ and the temperature $T$, the
level density parameter $a$ and the level density $\rho$ of the fragments are
calculated using Eq. 3. From the fragment level densities $\rho_{i}$, the
folding density $\rho_{12}$ is calculated using the convolution integral as in
Eq. 2 and the relative yields are calculated using Eq. 6. The total yields are
normalized to the scale 2.

The mass yield of normal nuclei 236U and 232Th are briefly explains first,
followed by the detailed description of the neutron rich nuclei. The results
of most favorable fragments yield of 236,250U and 232,254Th are listed in
Table LABEL:tab1 at three different temperatures $T=$ 1, 2 and 3 MeV for both
TRMF and FRDM formalisms. From Figs. 4 and 5, it is shown that the mass
distributions for 236U and 232Th are quite different than the neutron-rich
250U and 254Th isotopes.

The symmetric binary fragmentation 118Pd $+^{118}$Pd for 236U is the most
favorable combination. In TRMF, the fragments with close shell ($N=$ 100 and
$Z=$ 28) combinations are more probable at the temperature $T=$ 2 MeV. The
blend region of neutron and proton close shell ($N\approx$ 82 and $Z\approx$
50) has the considerable yield values at $T=$ 3 MeV. The fragmentations 151Pr
$+^{85}$As, 142Cs $+^{94}$Rb and 144Ba $+^{92}$Kr are the favorable
combinations at temperature $T=$ 1 MeV in FRDM formalism. For higher
temperatures $T=$ 2 and 3 MeV, the closed shell or near closed shell fragments
($N=$ 82, 50 and $Z=$ 28) have larger yields. From Fig. 5 in TRMF formalism,
the combinations 118Pd $+^{114}$Ru and 140Xe $+^{92}$Kr are the possible
fragments at $T=$ 1 MeV for the nucleus 232Th. At $T=$ 2 MeV, we find maximum
yields for the fragments with the close shell or near close shell combinations
($N=$ 82, 50). For higher temperature $T=$ 3 MeV, near the neutron close shell
($N\sim$ 82), 132Sb $+^{100}$Y is the most favorable fragmentation pair
compared with all other yields. Similar fragmentations are found in the FRDM
formalism at $T=$ 2 and 3 MeV. In addition, the probability of the evaluation
of 129Sn $+^{103}$Zr is also quite substantial in the fission process. For
$T=$ 1 MeV, the yield is more or less similar with the TRMF model.

From Fig. 4, for 250U the fragment combinations 140,141Te $+^{110,109}$Zr have
the maximum yields at $T=$ 1 MeV in TRMF. This is also consistent with the
evolution of the sub-close shell proton $Z=40$ in $Zr$ isotopes blatt .
Contrary to this almost symmetric binary yield, the mass distribution of this
nucleus in FRDM formalism have the asymmetric evolution of the fragment
combinations like 160,159Pr $+^{90,91}$As, 163,162Nd $+^{87,88}$Ge and 150Cs
$+^{100}$Rb. Interestingly, at $T=$ 2 and 3 MeV, the more favorable fragment
combinations have one of the closed shell nuclei. At $T=$ 2 MeV, 159Pr
$+^{91}$As, 162Nd $+^{88}$Ge and 173Gd $+^{77}$Ni are the more probable
fragmentations (see Fig. 4(c)). It is reported by Satpathy et al satpathy  and
experimentally verified by Patel et al patel  that $N=$ 100 is a neutron close
shell for the deformed region, where $Z=62$ acts like a magic number. In FRDM,
128Ag $+^{122}$Rh, 132In $+^{118}$Tc, 140Te $+^{110}$Zr and 173Gd $+^{77}$Ni
have larger yield at temperature $T=$ 2 MeV. In TRMF method, the most
favorable fragments are confined in the single region ($A\approx 114-136$)
which is the blend of vicinity of neutron ($N=$ 82) and proton ($Z=$ 50)
closed shell nuclei at $T=$ 3 MeV. The fragment combinations 130Cd
$+^{120}$Ru, 132In $+^{118}$Tc and 135Sn $+^{115}$Mo are the major yields for
250U at T=3 MeV in TRMF calculations. In FRDM method, at $T=$ 3 MeV, more
probable fragments are similar that at $T=$ 2 MeV. A comparison between Fig.
4(c) and 4(d) clears that, although the prediction of FRDM and TRMF at $T=$ 3
MeV are qualitatively similar, but it is quantitatively very different at $T=$
2 MeV in both the predictions. Also, from Fig. 4, it is inferred that the
yields of the fragment combinations in blend region increases and in other
regions decreases at $T=$ 2 MeV.

In the present study, the total energy of the parent nucleus A is more than
the sum of the energies of the daughters $A_{1}$ and $A_{2}$. Here, the
dynamics of entire process starting from the initial stage upto the scission
are ignored. As a result, the energy conservation in the spallation reaction
does not taken into account. The fragment yield can be regarded as the
relative fragmentation probability, which is obtained from Eq. 6. Now we
analyze the fragmentation yields for Th isotopes and the results are depicted
in Fig. 5 and Table LABEL:tab1. In this case, one can see that the mass
distribution broadly spreads through out the region $A_{i}=66-166$. Again, the
most concentrated yields can be divided into two regions I($A_{1}=$ 141-148
and $A_{2}=$ 106-113) and II ($A_{1}=$ 152-158 and $A_{2}=$ 102-96) for 254Th
in TRMF formalism at the temperature $T=$ 1 MeV. The most favorable
fragmentation 142Sn $+^{112}$Zr is obtained from region I. The other
combinations in that region have also considerable yields. In region II, the
isotopes of Ba and Cs appears curiously along with their corresponding
partners. Categorically, in FRDM predictions, region I has larger yields at
$T=$ 1 MeV. The other possible fragmentations are 163Ce $+^{91}$Ge, 168Nd
$+^{86}$Zn and 181Gd $+^{73}$Fe (See Fig. 5 (b,d)). The mass distribution is
different with different temperature and the maximum yields at $T=$ 2 MeV in
TRMF formalism are 174,175,176Sm $+^{80,79,78}$Ni. Apart from these
combinations, there are other considerable yields can be seen in Fig. 5 for
region II. The prediction of maximum probability of the fragments production
in FRDM method are 144Sb $+^{110}$Y, 178Eu $+^{76}$Co and 127Rh $+^{127}$Rh at
$T=$ 2 MeV. Besides these yields, one can find other notable evolution of
masses in region I due to the vicinity of the proton close shell.
Interestingly, at $T=$ 3 MeV, symmetric binary combination 127Rh $+^{127}$Rh
has the largest yield due to the neutron close shell ($N=$ 82) of the fragment
127Rh. The other yield fragments have exactly/nearly a magic nucleon
combination, mostly neutron ($N=$ 82) as one of the fragment. A considerable
yield is also seen for the proton close shell ($Z=$ 28) Ni or/and ($Z=$62) Sm
isotopes supporting our earlier prediction cinthol . This confirms the
prediction of Sm as a deformed magic nucleus satpathy ; patel . Another
observation of the present calculations show that the yields of the neutron-
rich nuclei agree with the symmetric mass distribution of Chaudhuri et. al.
cha2015  at large excitation energy, which contradict the recent prediction of
large asymmetric mass distribution of neutron-deficient Th isotopes pasca .
These two results cha2015 ; pasca  along with our present calculations confirm
that the symmetric or asymmetric mass distribution at different temperature
depends on the proton and neutron combination of the parent nucleus. In
general, both TRMF and FRDM predict maximum yields for both
symmetric/asymmetric binary fragmentations followed by other secondary
fragmentations emission depending on the temperature as well as the mass
number of the parent nucleus. Thus, the binary fragments have larger level
density $\rho$ comparing with other nuclei because of neutron/proton close
shell fragment combinations at $T=$ 2 and 3 MeV. This results ascertain the
fact that most favorable fragments have larger phase space than the
neighboring nuclei as reported earlier mts16 ; cinthol .

Table 1: The relative fragmentation yield (R.Y.) = $Y(A_{j},Z_{j})=\dfrac{P(A_{j},Z_{j})}{\sum P(A_{j},Z_{j})}$ for 236U, 250U, 232Th and 254Th, obtained with TRMF at the temperatures $T=$ 1, 2 and 3 MeV are compared with the FRDM prediction (The yield values are normalized to 2). Parent | T (MeV) | TRMF | FRDM | Parent | T (MeV) | TRMF | FRDM
---|---|---|---|---|---|---|---
Fragment | R.Y. | Fragment | R.Y. | Fragment | R.Y. | Fragment | R.Y.
236U | 1 |  118Pd + 118Pd | 0.949 |  151Pr + 85As | 0.210 | 250U | 1 |  141Te + 109Zr | 1.454 |  160Pr + 90As | 0.248
119Pd + 117Pd | 0.910 |  142Cs + 94Rb | 0.178 |  140Te + 110Zr | 0.491 |  161Pr + 89As | 0.247
143Ba + 93Kr | 0.032 |  144Ba + 92Kr | 0.134 |  148Xe + 102Sr | 0.014 |  159Pr + 91As | 0.166
2 |  165Gd + 71Ni | 0.323 |  132Sb + 104Nb | 0.216 | 2 |  159Pr + 91As | 0.348 |  128Ag + 122Rh | 0.193
164Gd + 72Ni | 0.264 |  133Te + 103Zr | 0.213 |  162Nd + 88Ge | 0.197 |  132In + 118Tc | 0.168
163Gd + 73Ni | 0.0.221 |  151Pr + 85As | 0.210 |  160Pr + 90As | 0.176 |  140Te + 110Zn | 0.140
|  154Nd + 82Ge | 0.240 |  159Sb + 77Zn | 0.087 |  173Gd + 77Ni | 0.175 |  141Te + 109Zn | 0.100
3 |  163Gd + 73Ni | 0.249 |  132Sb + 104Nb | 0.283 | 3 |  130Cd + 120Ru | 0.565 |  128Ag + 122Rh | 0.414
164Gd + 72Ni | 0.214 |  133Te + 103Zr | 0.242 |  132In + 118Tc | 0.255 |  132In + 118Tc | 0.278
136I + 100Y | 0.143 |  134Te + 102Zr | 0.102 |  127Ag + 123Rh | 0.236 |  129Ag + 121Rh | 0.149
131Sb + 105Nb | 0.114 |  129Sn + 107Mo | 0.092 |  135Sn + 115Mo | 0.161 |  130Cd + 120Ru | 0.083
232Th | 1 |  118Pd + 114Ru | 0.773 |  142Cs + 90Br | 0.190 | 254Th | 1 |  142Sn + 112Zr | 0.439 |  145Sb + 109Y | 0.183
140Xe + 92Kr | 0.515 |  144Ba + 88Se | 0.124 |  145Sb + 109Y | 0.291 |  163Ce + 91Ge | 0.118
141Cs + 91Br | 0.174 |  120Ag + 112Tc | 0.123 |  155Cs + 99Br | 0.176 |  144Sb + 110Y | 0.115
120Ag + 112Tc | 0.129 |  158Pm + 74Cu | 0.092 |  157Ba + 97Se | 0.139 |  168Nd + 86Zn | 0.077
2 |  151Pr + 81Ga | 0.505 |  132Sb + 100Y | 0.213 | 2 |  176Sm + 78Ni | 0.370 |  144Sb + 110Y | 0.161
132Sb + 100Y | 0.334 |  134Te + 98Sr | 0.202 |  175Sm + 79Ni | 0.290 |  178Eu + 76Co | 0.141
166Gd + 66Fe | 0.134 |  129Sn + 103Zr | 0.146 |  157Ba + 97Se | 0.172 |  144Sb + 110Y | 0.132
3 |  132Sb + 100Y | 0.886 |  132Sb + 100Y | 0.252 | 3 |  127Rh + 127Rh | 0.803 |  127Rh + 127Rh | 0.325
134Te + 98Sr | 0.148 |  129Sn + 103Zr | 0.207 |  129Pd + 125Ru | 0.350 |  127Rh + 127Rh | 0.210
155Nd + 77Zn | 0.063 |  134Te + 98Sr | 0.153 |  128Rh + 126Rh | 0.307 |  132Ag + 122Tc | 0.120

To this end, it may be mentioned that the differences in the mass
distributions or the relative yields calculated using TRMF and FRDM approaches
mainly arise due to the differences in the level densities associated with
these approaches. The mean values and the fluctuations in the level density
parameter and the corresponding level density are even qualitatively different
in both the approaches considered. This is possibly stemming from the fact
that the single -particle energies in the FRDM based model are temperature
independent. The temperature dependence of the excitation energy as required
to calculate the level density parameter comes only from the modification of
the single-particle occupancy due to the Fermi distribution. In the TRMF
approach, the excitation energy for each fragment at a given temperature is
calculated self-consistently. Therefore, the deformation and the single-
particle energies changes with temperature.

For the neutron-rich nuclei, the fragments having neutron/proton close shell
$N=$ 50, 82 and 100 have maximum possibility of emission at $T=$ 2 and 3 MeV
(for both nuclei 250U and 254Th). This is a general trend, we could expect for
all neutron-rich nuclei. It is worthy to mention some of the recent reports
and predictions of multi-fragment fission for neutron-rich uranium and thorium
nuclei. When such a neutron-rich nucleus breaks into nearly two fragments, the
products exceed the drip-line leaving few nucleons (or light nuclei) free. As
a result, these free particles along with the scission neutrons enhance the
chain reaction in a thermonuclear device. These additional particles (nucleons
or light nuclei) responsible to reach the critical condition much faster than
the usual fission for normal thermally fissile nucleus. Thus, the neutron-rich
thermally fissile nuclei, which are in the case of 246-264U and 244-262Th will
be very useful for energy production.

##  IV Summary and conclusions

The fission mass distributions of $\beta-$stable nuclei 236U and 232Th and the
neutron-rich thermally fissile nuclei 250U and 254Th are studied within the
statistical theory. The possible combinations are obtained by equating the
charge to mass ratio of the parents to that of the fragments. The excitation
energies of fragments are evaluated from the temperature dependent self-
consistent binding energies at the given $T$ and the ground state binding
energies which are calculated from the relativistic mean field model. The
level densities and the yields combinations are manipulated from the
convolution integral approach. The fission mass distributions of the
aforementioned nuclei are also evaluated from the FRDM formalism for
comparison. The level density parameter $a$ and inverse level density
parameter $K$ are also studied to see the difference in results with these two
methods. Besides fission fragments, the level densities are also discussed in
the present paper. For 236U and 232Th, the symmetric and nearly symmetric
fragmentations are more favorable at temperature $T=$ 1 MeV. Interestingly, in
most of the cases we find one of the favorable fragment is a close shell or
near close shell configuration ($N=$ 82,50 and $Z=$ 28) at temperature $T=$ 2
and 3 MeV. This result ascertains with our earlier predictions. Further, Zr
isotopes has larger yield values for 250U and 254Th with their accompanied
possible fragments at $T=$ 1 MeV. The Ba and Cs isotopes with their partners
are also more possible for 254Th. This could be due to the deformed close
shell in the region $Z=52-66$ of the periodic table jonh . The Ni isotopes and
the neutron close shell ($N\sim$ 100) nuclei are some of the prominent yields
for both 250U and 254Th at temperature $T=$ 2 MeV. At $T=$ 3 MeV, the neutron
close shell ($N=$ 82) is one of the largest yield fragments. The symmetric
fragmentation 127Rh $+^{127}$Rh is possible for 254Th due to the $N=$82 close
shell occurs in binary fragmentation. For 250U, the larger yield values are
confined to the junction of neutron and proton closed shell nuclei.

##  V Acknowledgment

The author MTS acknowledge that the financial support from UGC-BSR research
grant award letter no. F.25-1/2014-15(BSR)7-307/2010/(BSR) dated 05.11.2015
and IOP, Bhubhaneswar for the warm hospitality and for providing the necessary
computer facilities.

## References

  * (1) D. Vautherin and D. M. Brink, Phys. Lett. B 32, 149 (1970); Phys. Rev. C 5, 626 (1972).
  * (2) M. K. Pal and A. P. Stamp, Nucl. Phys. A 99, 228 (1967).
  * (3) S. K. Patra, R. K. Choudhury and L. Satpathy, J. Phys. G. 37, 085103 (2010).
  * (4) M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCromick, X. Xu and B. Pfeiffer, Chin. Phys. C 36, 1603 (2012).
  * (5) P. Fong, Phys. Rev. 102, 434 (1956).
  * (6) M. Rajasekaran and V. Devanathan, Phys. Rev. C 24, 2606 (1981).
  * (7) J. D. Walecka, Ann. Phys. 83, 491 (1974).
  * (8) C. J. Horowitz and B. D. Serot, Nucl. Phys. A 368, 503 (1981).
  * (9) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).
  * (10) Y. K. Gambhir, P. Ring and A. Thimet, Ann. of Phys. 198, 132 (1990).
  * (11) S. K. Patra and C. R. Praharaj, Phys. Rev. C 44, 2552 (1991).
  * (12) P. Arumugam, B. K. Sharma, S. K. Patra and R. K. Gupta, Phys. Rev. C 71, 064308 (2005).
  * (13) B. K. Sharma, P. Arumugam, S. K. Patra, P. D. Stevenson, R. K. Gupta and W. Greiner, J. Phys. G. 32, L1 (2006).
  * (14) S. K. Patra, Raj. K. Gupta, B. K. Sharma, P. D. Stevenson and W. Greiner, J. Phys. G. 34, 2073 (2007).
  * (15) Raj. K. Gupta, S. K. Patra, P. D. Stevenson, C. Beck, and W. Greiner, J. Phys. G. 35, 075106 (2008).
  * (16) K. Rutz, J. A. Maruhn, P.-G. Reinhard and W. Greiner, Nucl. Phys. A 590, 680 (1995).
  * (17) M. Balasubramaniam, C. Karthikraj, N. Arunachalam and S. Selvaraj, Phys. Rev. C 90, 054611 (2014); M. Rajasekaran and V. Devanathan, Phys. Rev. C 24, 2606 (1981).
  * (18) D. W. Bergen and R. R. Fullwood, Nucl. Phys. A 163, 577 (1971).
  * (19) A. J. Cole, in Fundamental and Applied Nuclear Physics Series - Statistical models for nuclear decay from evaporation to vaporization, edited by R. R. Betts and W. Greiner, Institute of Physics Publsihing, Bristol and Philadelphia, 2000.
  * (20) J. R. Huizenga and L. G. Moretto, Annu. Rev. Nucl. Sci. 22, 427 (1972).
  * (21) H. Bethe, Rev. Mod. Phys. 9, 69 (1937).
  * (22) M.T. Senthil Kannan and M. Balasubramaniam, Eur. Phys. J. A 53, 164 (2017).
  * (23) P. Möller, J. R. Nix, W. D. Myers and W. J. Swiatecki, At. Data and Nucl. Data Tables 66, 131 (1997).
  * (24) J. Boguta and A. R. Bodmer, Nucl. Phys. A 292, 413 (1977).
  * (25) C. E. Price and G. E. Walker, Phys. Rev. C 36, 354 (1987).
  * (26) P. G. Blunden and M. J. Iqbal, Phys. Lett. B 196, 295 (1987).
  * (27) P. G. Reinhard, Rep. Prog. Phys. 52, 439 (1989).
  * (28) S. K. Patra, Phys. Rev. C 48, 1449 (1993).
  * (29) M. A. Preston and R. K. Bhaduri, Structure of Nucleus, Addison-Wesley Publishing Company, Ch. 8, page 309 (1982).
  * (30) D. Vautherin, Phys. Rev. C 7, 296 (1973).
  * (31) J. Dechargé and D. Gogny, Phys. Rev. C 21, 1568 (1980).
  * (32) M.T. Senthil kannan, Bharat Kumar, M. Balasubramaniam, B. K. Agrawal, S. K. Patra, Phys. Rev. C 95, 064613 (2017).
  * (33) Bharat Kumar, M.T. Senthil kannan, M. Balasubramaniam, B. K. Agrawal and S. K. Patra, arXiv:1701.00731.
  * (34) Bharat Kumar, S. K. Biswal, S. K.Singh and S. K. Patra, Phys. Rev. C 92, 054314 (2015).
  * (35) NIU Yi-Fei, LINAG Hao-Zhao and MENG Jie, Chin. Phys. Lett. 26, 032103 (2009).
  * (36) B. K. Agrawal, Tapas Sil, J. N. De and S. K. Samaddar, Phys. Rev. C 62, 044307 (2000); ibid 63, 024002 (2001).
  * (37) B. K. Agrawal, S. K. Samaddar, J. N. De, and S. Shlomo Phys. Rev. C 580, 3004 (1998).
  * (38) P. Möller, J. R. Nix and K. L. Kratz, At. Data and Nucl. Data Tables 59, 185 (1995).
  * (39) https://www-nds.iaea.org/RIPL-3/.
  * (40) B. Nerlo-Pomorska, K. Pomorski, J. Bartel and K. Dietrich, Phys. Rev. C 66, 051302(R) (2002).
  * (41) J. M. Blatt and V. F. Weisskopf Theoretical Nuclear Physics, Courier Corporation, 1991.
  * (42) L. Satpathy and S. K. Patra, J. Phys. G. 30, 771 (2004).
  * (43) Z. Patel et. al., Phys. Rev. Lett. 113, 262502 (2014).
  * (44) A. Chaudhuri et. al., Phys. Rev. C 91, 044620 (2015).
  * (45) H. Paşca, A. V. Andreev, G. G. Adamian, and N. V. Antonenko, Phys. Rev. C 94, 064614 (2016).
  * (46) E. F. Jones et. al., J. Phys. G 30, L43 (2004).


# Flavor asymmetry from the non-perturbative nucleon sea

Yihan Duan  Department of Modern Physics, University of Sciences and
Technology of China, Hefei 230026, China  Siqi Xu  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China  Shan Cheng  School of Physics and Electronics, Hunan
University, 410082 Changsha, China  Xingbo Zhao  Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science
and Technology, University of Chinese Academy of Sciences, Beijing 100049,
China  CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  Yang Li
Department of Modern Physics, University of Sciences and Technology of China,
Hefei 230026, China  Department of Physics and Astronomy, Iowa State
University, Ames, IA 50011, USA  James P. Vary  Department of Physics and
Astronomy, Iowa State University, Ames, IA 50011, USA

###### Abstract

We demonstrate, in the context of a scalar version of the chiral effective
field theory, that the multi-sea quark contribution to the nucleon is
significant and highly non-trivial in sharp contrast to the prediction of
perturbation theory. The non-perturbative calculation is performed in the Fock
sector dependent renormalization scheme on the light front, in which the non-
perturbative renormalization is incorporated. The calculation suggests that a
fully non-perturbative calculation of the chiral EFT is needed to obtain a
robust result to be compared with the recent experimental measurement of
flavor asymmetry in the proton.

## Introduction

Understanding the sea-quark contribution of hadrons is of fundamental
importance Kumano:1997cy ; Garvey:2001yq ; Chang:2014jba ; Geesaman:2018ixo .
It not only provides a sharper view of the fundamental structure of matter
Gottfried:1967kk , but also serves as a tool for detecting physics beyond
Standard Model NNPDF:2021njg . Recently, the SeaQuest/E906 experiment
published their first measurement of the flavor asymmetry $\bar{d}/\bar{u}$ of
the proton sea from the muon pair production within the Drell-Yan process of
high-energy proton-ion collisions SeaQuest:2021zxb ; Kotlorz:2021goc ;
FNALE906:2022xdu , challenging the previous measurement by the NuSea/E866
experiment at large momentum fraction $x\gtrsim 0.3$ NA51:1994xrz ;
HERMES:1998uvc ; NuSea:1998tww ; NuSea:1998kqi ; NuSea:2001idv ;
Salajegheh:2017iqp . In particular, the NuSea/E866 experiment predicted that
$\bar{d}/\bar{u}$ dips below unity at $x\sim 0.3$, which is contrary to the
SeaQuest/E906 measurement. The tension between these experimental measurements
calls for a better understanding from theories including global analysis
Harland-Lang:2014zoa ; Alekhin:2017kpj ; Accardi:2019ofk ; NNPDF:2021njg ;
Cocuzza:2021cbi ; Hou:2022onq ; Cocuzza:2022jye . It was generally believed
that the flavor asymmetry of the proton encodes the non-perturbative physics
of QCD, making it a particular challenge for calculations from first
principles Scapellato:2019csz ; Ji:2020ect .

Among various theoretical interpretations Henley:1990kw ; Dorokhov:1993fc ;
Steffens:1996bc ; Londergan:1998ai ; Miller:2002ig ; Trevisan:2008zz ;
Song:2011fc ; Kamano:2012yx ; Burkardt:2012hk ; Ji:2013bca ; Ehlers:2014jpa ;
Salamu:2014pka ; Lin:2014zya ; Kofler:2017uzq ; Tadepalli:2019ixa ;
Bourrely:2019ysc ; Soffer:2019gbb ; Leon:2020bvt ; Alexandrou:2021oih ;
Wang:2021ild ; Alberg:2021nmu ; Signal:2021aum ; Cui:2021gzg ; Kotlorz:2021goc
; Chang:2022jri ; Leinweber:2022guz ; He:2022leb ; Bourrely:2022mjf ;
Wang:2022bxo ; Bonvini:2023tnk ; Choudhary:2023unw ; Choudhary:2023bap , the
pion cloud model Theberge:1980ye ; Thomas:1981vc ; Thomas:1983fh  has shown a
trend consistent with the SeaQuest/E906 results as well as other theories at
small and modest $x$. In this model, the sea quark contribution is viewed as
the hadron cloud around the nucleon. The interaction between the pion cloud
and the host nucleon is described by chiral effective field theory
($\chi$EFT). Recently, Alberg et. al. showed that light-cone methods can be
used to compute the pionic longitudinal momentum distribution function (LMDF)
$f_{\pi N}(x)$ Alberg:2012wr ; Alberg:2017ijg ; Alberg:2021nmu . Then, the
quark distribution function can be represented as,

$q_{N}(x)=Zq_{N}^{(0)}(x)+\sum_{B}\int_{x}^{1}\frac{{\mathrm{d}}y}{y}\big{[}f_{\pi
B}(y)q_{\pi}(x/y)\\\ +f_{B\pi}(y)q_{B}(x/y)\big{]}\,.$ (1)

Alberg et. al.’s model is successful in describing the nucleon sea at small
and modest $x$. At large $x\gtrsim 0.3$, this model predicts $\bar{d}/\bar{u}$
is always above unity. However, the results start to show deviation from the
experimental data of SeaQuest/E906 Alberg:2017ijg . The predictions are
improved after taking into the evolution of the sea Alberg:2021nmu . Alberg
et. al.’s calculation is based on leading-order (LO) light cone perturbation
theory which is equivalent to Fock sector expansion up to the one pion sector:
$|N\rangle_{\mathrm{phy}}=|N\rangle+|N\pi\rangle+|\Delta\pi\rangle$. A non-
perturbative extension in a limited Fock space involving a single pion has
also appeared Du:2019ips . One of the immediate questions is how large are the
non-perturbative effects such as multi-pion contributions. Indeed, the
effective coupling, $g_{\pi N}\simeq 13$, is large. Furthermore, the pion
carrying large momentum fraction $y$ is ultra-relativistic and its splitting
function is more likely modified in the non-perturbative regime, which
directly affects the parton distribution functions (PDFs) at large $x$.

In this work, we investigate the non-perturbative effects in the pion cloud
model by exploiting the recent success in solving strongly coupled scalar
field theories in the light-front Hamiltonian formalism Li:2015iaw ;
Karmanov:2016yzu . We perform a systematic Fock expansion and incorporate the
multi-pion component of the nucleon sea,
$|N\rangle_{\text{ph}}=|N\rangle+|N\pi\rangle+|\Delta\pi\rangle+|N\pi\pi\rangle+|\Delta\pi\pi\rangle$,
where the Fock sector expansion is observed to be converged Li:2015iaw . We
show that the multi-pion sea modifies the prediction for the flavor asymmetry
in the region from moderate to large coupling that is relevant for the pion
cloud model. The adoption of the scalar theory certainly simplifies the non-
perturbative calculation at the cost of major differences from $\chi$EFT.
These differences are reflected in the pion longitudinal momentum distribution
at small $x$, which are shown in this work. We instead focus on the non-
perturbative effects within the present model. Our results suggest that a full
non-perturbative calculation starting from the $\chi$EFT is needed to obtain a
robust result to be compared with the recent experimental measurement of the
flavor asymmetry in the nucleon.

## Formalism

We follow Alberg et. al.’s treatment of the pion cloud model Alberg:2021nmu ,
except starting from scalar type pion-nucleon and pion-nucleon-$\Delta$
interactions, viz.,

$\mathscr{L}_{\text{int}}=g_{N\pi}\overline{N}N\pi+g_{\Delta}\overline{N}\Delta\pi+g_{\Delta}\overline{\Delta}N\pi,$
(2)

where, $N,\Delta$ are complex scalar fields and $\pi$ is a real scalar field.
Note that the couplings $g_{N\pi}$ and $g_{\Delta}$ have the dimension of
mass. A dimensionless coupling can be introduced, $\alpha=g^{2}_{N\pi}/(16\pi
M^{2}_{N})$. The theory is a low-energy approximation of $\chi$EFT applicable
to moderate and large $x$. It still encapsulates some general features of the
nucleon sea. Comparing the LO perturbative results for LMDF from $\chi$EFT
Alberg:2021nmu  and from the scalar Yukawa theory, we find that they are
almost identical except for some function that can be absorbed into
phenomenological form factors. In this work, we focus on the non-perturbative
effects within the same model and adopt this simplified model.

To obtain the state vector $|N\rangle_{\text{phy}}$ of the physical nucleon,
we diagonalize the light-front quantized Hamiltonian in Fock space. Fig. 1
shows the first few diagrams of the system of equations derived from the
light-front Schrödinger equation, which are an infinite set of coupled
integral equations in Fock space. A review of the relevant Feynman rules is
collected in Refs. Brodsky:1997de ; Carbonell:1998rj . Note that the non-
perturbative vertex functions (shaded blocks) are employed to extend the
Feynman rules to the non-perturbative regime. The vertex functions are the
$T$-matrix elements and are related to the light-front wave functions by
resolvents Hiller:1998cv . The explicit expressions translated from these
diagrams are shown in Refs. Li:2015iaw ; Karmanov:2016yzu .

To keep the system of equations numerically tractable, we impose the Fock
sector truncation, and investigate the convergence of observables with respect
to the Fock sector expansion Hiller:2016itl . The truncation up to two-body
($|N\rangle+|N\pi\rangle$) is identical to the LO perturbation theory. In Ref.
Li:2015iaw , the state vector of the physical nucleon was solved up to four-
body truncation ($|N\rangle+|N\pi\rangle+|N\pi\pi\rangle+|N\pi\pi\pi\rangle$)
and it was shown that the Fock sector expansion is well converged up to three-
body ($|N\rangle+|N\pi\rangle+|N\pi\pi\rangle$). We thus adopt the four-body
truncation result as the non-perturbative results and estimate the uncertainty
associated with the Fock sector truncation as the difference between the
three-body truncation and the four-body truncation. The Fock states that
contains $\Delta$, i.e.
$|\Delta\pi\rangle+|\Delta\pi\pi\rangle+|\Delta\pi\pi\pi\rangle$, are obtained
from the nucleon wave functions as shown in Fig. 1. Effectively, a small back
reaction of $\Delta$ on the nucleon and pion Fock states is neglected for
numerical simplicity. Note that we excluded sub-diagrams that convert a
nucleon to a $\Delta$ or vice versa since such a process violates the angular
momentum conservation.

Figure 1: Diagrammatic representation of the system of equations with a
truncation up to two pions which is a subset of the three pion equations that
we employ here. The solid lines represent the (mock) bare nucleons. The dashed
lines represent the (mock) $\Delta$’s. And the wavy lines represent the (mock)
pions. The double solid lines represent the physical nucleons.

While the scalar theory in (3+1)D is super-renormalizable, it still contains
divergences. The divergences can be absorbed into the bare parameters by
imposing the mass and coupling constant renormalization conditions Li:2015iaw
. For the mass renormalization, we require the mass eigenvalue equals the
physical mass. For the coupling constant renormalization, we require the
$T$-matrix $\Gamma_{N\pi/N}$ equals the physical coupling multiplied by the
necessary $Z$ factors at the on-shell point. The whole procedure is non-
perturbative. N.B. the values of the bare parameters, i.e., the bare mass
$m_{0}^{2}$ and the bare couplings $g_{0}$, depend on the Fock sector
truncation. Fortunately, boost invariance in light-front dynamics guarantees
that these sector dependent counter-terms can be obtained sector by sector
recursively. This renormalization scheme is known as the Fock sector dependent
renormalization Karmanov:2008br , which has been successful in renormalizing
the light-front Hamiltonian operators of the scalar theory Li:2015iaw ;
Karmanov:2016yzu ; Cao:2023ohj , Yukawa theory Karmanov:2010ih ;
Karmanov:2012aj  and QED Chabysheva:2009vm ; Chabysheva:2009ez  as well as
$\chi$EFT type interactions Tsirova:2010zza .

Figure 2: The light-front wave function representation of the longitudinal
momentum distribution’s contribution to deep inelastic scattering as depicted
by the handbag diagram corresponding to the contributions from the four-body
sector to the physical nucleon.

## Longitudinal momentum distribution functions (LMDF)

The PDF describes the probability of finding a collinear parton with
fractional momentum $x$ inside the proton. In analogy to the QCD definition,
we introduce a LMDF as Diehl:2003ny ,

$f_{B\pi}(x)=\frac{k^{+}}{2}\int{\mathrm{d}}z^{-}e^{{\frac{i}{2}}xp^{+}z^{-}}\\\
\times\langle
p|\bar{B}(-{\frac{z}{2}})B({\frac{z}{2}})|p\rangle\big{|}_{z^{+}=z_{\perp}=0}.$
(3)

where $x={k^{+}}/{p^{+}}$ is the longitudinal momentum fraction of the baryon
$B$ inside the physical nucleon, $B=N,\Delta$. This expression is
diagrammatically shown in Fig. 2. In the light-front wave function
representation Diehl:2000xz ,

$f(x)=f_{1}(x)+f_{2}(x)+\cdots$ (4)

where, the $n$-body contribution is (the $n$-th particle is taken as the
baryon)

$f_{n}(x)=\frac{1}{(n-1)!}\prod_{i=1}^{n}\int\frac{{\mathrm{d}}x_{i}{\mathrm{d}}^{2}k_{i\perp}}{(2\pi)^{3}2x_{i}}2(2\pi)^{3}\delta(x-x_{n})\\\
\times\delta(x_{1}+\cdots+x_{n}-1)\delta^{2}(\vec{k}_{1\perp}+\cdots+\vec{k}_{n\perp})\\\
\times\big{|}\psi_{n}(x_{1},\vec{k}_{1\perp},x_{2},\vec{k}_{2\perp},\cdots,x_{n},\vec{k}_{n\perp})\big{|}^{2}\,.$
(5)

Since the scalar theory is super-renormalizable, the “factorization scale” of
the LMDFs can be taken to infinity Aslan:2022zkz . The PDFs of the physical
nucleon are obtained from convoluting the LMDF with the PDFs of its
constituent hadrons as shown in Eq. (1). We adopt the same pion and bare
proton PDFs as Alberg et. al. at a scale $\mu^{2}=25.5\,\mathrm{GeV}^{2}$
Alberg:2021nmu ; Aicher:2010cb ; Gluck:1999xe ; Szczurek:1997fr .

To demonstrate the effects of the multi-pion sea, we show in Fig. 3 the LMDFs
of the pion inside the physical nucleon for two non-perturbative couplings,
$\alpha=1.0$ and $\alpha=2.0$, up to one nucleon and three pions. We also
turned off the coupling to $\Delta$, i.e. setting $g_{\Delta}=0$, to focus on
the multi-pion sea. Contributions from different Fock sectors are also shown.
Comparing these LMDFs, as may be anticipated, the contribution from the multi-
pion Fock sector increases with the increase in the coupling.

Figure 3: (Colors online) Longitudinal momentum distribution function of the
pion inside the physical scalar nucleon for coupling (Top) $\alpha=1.0$ and
(Bottom) $\alpha=2.0$. The coupling to $\Delta$ is turned off in both cases,
i.e. $g_{\Delta}=0$. The contributions from the multi-pion Fock sectors are
also shown in each panel. For $\alpha=2.0$ there are significant three- and
four-body contributions at the large longitudinal momentum fraction $x$.

To compare with the Drell-Yan data, we adopt $g_{N\pi}=6.0\,\mathrm{GeV}$
($\alpha=0.8$) and $g_{\Delta}=6.0\,\mathrm{GeV}$. These parameters are chosen
to reproduce the $\bar{d}-\bar{u}$ data. Fig. 4 shows the LMDFs of the nucleon
and $\Delta$ obtained from the two-body, three-body and four-body truncations.
As we mentioned, the two-body truncation is equivalent to the light-cone
perturbation theory. The results obtained from three- and four-body
truncations are close to each other, indicating a trend of numerical
convergence of the Fock sector expansion. Thereafter, we will adopt the four-
body truncation as the non-perturbative result, and estimate the uncertainty
using the difference between the three-body and four-body truncations. Alberg
et. al.’s perturbative results based on a chiral Lagrangian are also shown in
Fig. 4 for comparison Alberg:2021nmu . In general, our perturbative results
differ from their perturbative results since we adopt the scalar version of
the theory. Nevertheless, both the perturbative results of Ref. Alberg:2021nmu
(not shown in Fig. 4) and our non-perturbative results reproduce the NuSea
data $\bar{d}-\bar{u}$ as shown in Fig. 5. The integrated flavor asymmetry
$\int{\mathrm{d}}x[\bar{d}(x)-\bar{u}(x)]=0.070(2)$ is consistent with the
NuSea/E866 measurement 0.080(11) within the range $0.015<x<0.35$ NuSea:2001idv
.

Figure 4: (Colors online) Longitudinal momentum distribution function of the
nucleon (Top) and $\Delta$ (Bottom) inside the physical nucleon. Results from
Fock sector truncation up to one baryon plus one pion ($n=2$), one baryon plus
two pions ($n=3$) and one baryon plus three pions ($n=4$) are compared. The
two-body truncation is equivalent to LO light-cone perturbation theory. Alberg
et. al.’s results using LO perturbation theory with a chiral Lagrangian are
included for comparison Alberg:2021nmu .

Figure 5 shows the flavor asymmetry of the “scalar nucleon” sea from this
theory along with available experimental data. While perturbative and non-
perturbative results for $\bar{d}-\bar{u}$ are close to each other, their
difference in $\bar{d}/\bar{u}$ is moderately large, which signifies the high
Fock sector contributions in the non-perturbative regime especially at large
$x$. Within the same theory, our three- (one baryon plus up to two pions) and
four-body (one baryon plus up to three pions) truncation results show a good
convergence pattern as seen in Fig. 4. However, these results are different
from the two-body truncation results. The flavor asymmetry results exhibit
significant sensitivity to the 2-body versus 4-body truncation as shown in
Fig. 5, which signals important non-perturbative effects originating from the
multi-pion sea contributions.

Figure 5: Flavor asymmetry of the nucleon sea: (Top) $\bar{d}-\bar{u}$;
(Bottom) $\bar{d}/\bar{u}$ from the perturbative and non-perturbative
solutions of a pion cloud model with scalar type interactions in comparison
with available experimental data from NuSea/E866 NuSea:1998kqi ; NuSea:2001idv
; Salajegheh:2017iqp  and SeaQuest/E906 SeaQuest:2021zxb . The non-
perturbative result is obtained with a Fock space truncation up to four-body
(one baryon plus up to three pions). The difference between the results with
three-body and four-body truncations are shown as bands to indicate the
convergence of the Fock space expansion.

## Summary and outlook

We investigated the non-perturbative effects of the nucleon sea within a
scalar version of the pion cloud model. The non-perturbative dynamics is
generated by the light-front Schrödinger equation with a systematic Fock
sector expansion. Within this simple model, the convergence is observed
already with three-body contributions (one baryon plus two pions) though we
proceed to include four-body contributions as well. We show that the obtained
longitudinal momentum distributions from the non-perturbative solutions are
considerably different from those obtained from a leading order light-front
perturbation theory. The difference can be attributed to the multi-pion
dynamics, which are the sea contributions within the pion cloud model. We also
applied the solutions to describe the nucleon flavor asymmetry using the pion
cloud model. While both the perturbative and non-perturbative sea can describe
the $\bar{d}-\bar{u}$ data well, the non-perturbative nucleon sea can describe
the $\bar{d}/\bar{u}$ data better for $x\gtrsim 0.1$. These comparisons
between the perturbative and non-perturbative pion clouds indicates that the
non-perturbative effects are important and may help improve the model. Indeed,
part of the multi-pion sea can be taken into account by evolving the sea,
which may help to explain the improved agreement in their updated prediction
Alberg:2021nmu . Nevertheless, a full investigation of the non-perturbative
sea requires a non-perturbative calculation of chiral effective field theory
Tsirova:2010zza , which is a natural next step.

## Acknowledgements

The authors acknowledge valuable discussions with W. Du, V.A. Karmanov and P.
Maris. This work was supported in part by the National Natural Science
Foundation of China (NSFC) under Grant No. 12375081, by the Chinese Academy of
Sciences under Grant No. YSBR-101, and by the US Department of Energy (DOE)
under Grant No. DE-SC0023692.

## References

  * (1) S. Kumano, Phys. Rept. 303, 183-257 (1998) doi:10.1016/S0370-1573(98)00016-7 [arXiv:hep-ph/9702367 [hep-ph]].
  * (2) G. T. Garvey and J. C. Peng, Prog. Part. Nucl. Phys. 47, 203-243 (2001) doi:10.1016/S0146-6410(01)00155-7 [arXiv:nucl-ex/0109010 [nucl-ex]].
  * (3) W. C. Chang and J. C. Peng, Prog. Part. Nucl. Phys. 79, 95-135 (2014) doi:10.1016/j.ppnp.2014.08.002 [arXiv:1406.1260 [hep-ph]].
  * (4) D. F. Geesaman and P. E. Reimer, Rept. Prog. Phys. 82, no.4, 046301 (2019) doi:10.1088/1361-6633/ab05a7 [arXiv:1812.10372 [nucl-ex]].
  * (5) K. Gottfried, Phys. Rev. Lett. 18, 1174 (1967) doi:10.1103/PhysRevLett.18.1174
  * (6) R. D. Ball et al. [NNPDF], Eur. Phys. J. C 82, no.5, 428 (2022) doi:10.1140/epjc/s10052-022-10328-7 [arXiv:2109.02653 [hep-ph]].
  * (7) J. Dove et al. [SeaQuest], Nature 590, no.7847, 561-565 (2021) [erratum: Nature 604, no.7907, E26 (2022)] doi:10.1038/s41586-022-04707-z [arXiv:2103.04024 [hep-ph]].
  * (8) A. Kotlorz, D. Kotlorz and O. V. Teryaev, Int. J. Mod. Phys. A 36, no.36, 2150254 (2021) doi:10.1142/S0217751X21502547
  * (9) J. Dove et al. [FNAL E906 and SeaQuest], Phys. Rev. C 108, no.3, 035202 (2023) doi:10.1103/PhysRevC.108.035202 [arXiv:2212.12160 [hep-ph]].
  * (10) A. Baldit et al. [NA51], Phys. Lett. B 332, 244-250 (1994) doi:10.1016/0370-2693(94)90884-2
  * (11) K. Ackerstaff et al. [HERMES], Phys. Rev. Lett. 81, 5519-5523 (1998) doi:10.1103/PhysRevLett.81.5519 [arXiv:hep-ex/9807013 [hep-ex]].
  * (12) J. C. Peng et al. [NuSea], Phys. Rev. D 58, 092004 (1998) doi:10.1103/PhysRevD.58.092004 [arXiv:hep-ph/9804288 [hep-ph]].
  * (13) E. A. Hawker et al. [NuSea], Phys. Rev. Lett. 80, 3715-3718 (1998) doi:10.1103/PhysRevLett.80.3715 [arXiv:hep-ex/9803011 [hep-ex]].
  * (14) R. S. Towell et al. [NuSea], Phys. Rev. D 64, 052002 (2001) doi:10.1103/PhysRevD.64.052002 [arXiv:hep-ex/0103030 [hep-ex]].
  * (15) M. Salajegheh, H. Khanpour and S. M. Moosavi Nejad, Phys. Rev. C 96, no.6, 065205 (2017) doi:10.1103/PhysRevC.96.065205 [arXiv:1712.07624 [hep-ph]].
  * (16) L. A. Harland-Lang, A. D. Martin, P. Motylinski and R. S. Thorne, Eur. Phys. J. C 75, no.5, 204 (2015) doi:10.1140/epjc/s10052-015-3397-6 [arXiv:1412.3989 [hep-ph]].
  * (17) S. Alekhin, J. Blümlein, S. Moch and R. Placakyte, Phys. Rev. D 96, no.1, 014011 (2017) doi:10.1103/PhysRevD.96.014011 [arXiv:1701.05838 [hep-ph]].
  * (18) A. Accardi et al. [CTEQ-Jefferson Lab (CJ)], Phys. Lett. B 801, 135143 (2020) doi:10.1016/j.physletb.2019.135143 [arXiv:1910.02931 [hep-ph]].
  * (19) C. Cocuzza et al. [Jefferson Lab Angular Momentum (JAM)], Phys. Rev. D 104, no.7, 074031 (2021) doi:10.1103/PhysRevD.104.074031 [arXiv:2109.00677 [hep-ph]].
  * (20) T. J. Hou, H. W. Lin, M. Yan and C. P. Yuan, Phys. Rev. D 107, no.7, 076018 (2023) doi:10.1103/PhysRevD.107.076018 [arXiv:2211.11064 [hep-ph]].
  * (21) C. Cocuzza et al. [Jefferson Lab Angular Momentum (JAM)], Phys. Rev. D 106, no.3, L031502 (2022) doi:10.1103/PhysRevD.106.L031502 [arXiv:2202.03372 [hep-ph]].
  * (22) A. E. Scapellato, doi:10.25926/cwbq-mc10
  * (23) X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang and Y. Zhao, Rev. Mod. Phys. 93, no.3, 035005 (2021) doi:10.1103/RevModPhys.93.035005 [arXiv:2004.03543 [hep-ph]].
  * (24) E. M. Henley and G. A. Miller, Phys. Lett. B 251, 453-454 (1990) doi:10.1016/0370-2693(90)90735-O
  * (25) A. E. Dorokhov and N. I. Kochelev, Phys. Lett. B 304, 167-175 (1993) doi:10.1016/0370-2693(93)91417-L
  * (26) F. M. Steffens and A. W. Thomas, Phys. Rev. C 55, 900-908 (1997) doi:10.1103/PhysRevC.55.900 [arXiv:nucl-th/9612056 [nucl-th]].
  * (27) J. T. Londergan and A. W. Thomas, Prog. Part. Nucl. Phys. 41, 49-124 (1998) doi:10.1016/S0146-6410(98)00055-6 [arXiv:hep-ph/9806510 [hep-ph]].
  * (28) G. A. Miller, Phys. Rev. C 66, 032201 (2002) doi:10.1103/PhysRevC.66.032201 [arXiv:nucl-th/0207007 [nucl-th]].
  * (29) L. A. Trevisan, C. Mirez, T. Frederico and L. Tomio, Eur. Phys. J. C 56, 221-229 (2008) doi:10.1140/epjc/s10052-008-0651-1
  * (30) H. Song, X. Zhang and B. Q. Ma, Eur. Phys. J. C 71, 1542 (2011) doi:10.1140/epjc/s10052-011-1542-4 [arXiv:1101.3378 [hep-ph]].
  * (31) H. Kamano and T. S. H. Lee, Phys. Rev. D 86, 094037 (2012) doi:10.1103/PhysRevD.86.094037 [arXiv:1207.0181 [hep-ph]].
  * (32) M. Burkardt, K. S. Hendricks, C. R. Ji, W. Melnitchouk and A. W. Thomas, Phys. Rev. D 87, no.5, 056009 (2013) doi:10.1103/PhysRevD.87.056009 [arXiv:1211.5853 [hep-ph]].
  * (33) C. R. Ji, W. Melnitchouk and A. W. Thomas, Phys. Rev. D 88, no.7, 076005 (2013) doi:10.1103/PhysRevD.88.076005 [arXiv:1306.6073 [hep-ph]].
  * (34) P. J. Ehlers, A. Accardi, L. T. Brady and W. Melnitchouk, Phys. Rev. D 90, no.1, 014010 (2014) doi:10.1103/PhysRevD.90.014010 [arXiv:1405.2039 [hep-ph]].
  * (35) Y. Salamu, C. R. Ji, W. Melnitchouk and P. Wang, Phys. Rev. Lett. 114, 122001 (2015) doi:10.1103/PhysRevLett.114.122001 [arXiv:1409.5885 [hep-ph]].
  * (36) H. W. Lin, J. W. Chen, S. D. Cohen and X. Ji, Phys. Rev. D 91, 054510 (2015) doi:10.1103/PhysRevD.91.054510 [arXiv:1402.1462 [hep-ph]].
  * (37) S. Kofler and B. Pasquini, Phys. Rev. D 95, no.9, 094015 (2017) doi:10.1103/PhysRevD.95.094015 [arXiv:1701.07839 [hep-ph]].
  * (38) A. S. Tadepalli, doi:10.7282/t3-2ccn-7w70
  * (39) C. Bourrely and J. Soffer, [arXiv:1901.03071 [hep-ph]].
  * (40) J. Soffer and C. Bourrely, Nucl. Phys. A 991, 121607 (2019) doi:10.1016/j.nuclphysa.2019.08.001
  * (41) C. Leon and M. Sargsian, Eur. Phys. J. C 82, no.4, 309 (2022) doi:10.1140/epjc/s10052-022-10243-x [arXiv:2012.14030 [hep-ph]].
  * (42) C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen and F. Manigrasso, Phys. Rev. D 104, no.5, 054503 (2021) doi:10.1103/PhysRevD.104.054503 [arXiv:2106.16065 [hep-lat]].
  * (43) J. B. Wang, G. Li, C. S. An and J. J. Xie, Phys. Rev. D 103, no.11, 114018 (2021) doi:10.1103/PhysRevD.103.114018 [arXiv:2106.00866 [hep-ph]].
  * (44) M. Alberg, L. Ehinger and G. A. Miller, Phys. Rev. D 105, no.11, 114054 (2022) doi:10.1103/PhysRevD.105.114054 [arXiv:2108.12439 [nucl-th]].
  * (45) A. I. Signal and F. G. Cao, Phys. Lett. B 826, 136898 (2022) doi:10.1016/j.physletb.2022.136898 [arXiv:2108.12116 [hep-ph]].
  * (46) Z. F. Cui, F. Gao, D. Binosi, L. Chang, C. D. Roberts and S. M. Schmidt, Chin. Phys. Lett. 39, no.4, 041401 (2022) doi:10.1088/0256-307X/39/4/041401 [arXiv:2108.11493 [hep-ph]].
  * (47) L. Chang, F. Gao and C. D. Roberts, Phys. Lett. B 829, 137078 (2022) doi:10.1016/j.physletb.2022.137078 [arXiv:2201.07870 [hep-ph]].
  * (48) D. B. Leinweber and A. W. Thomas, Phys. Rev. D 109, no.3, L031503 (2024) doi:10.1103/PhysRevD.109.L031503 [arXiv:2211.13423 [hep-ph]].
  * (49) F. He, C. R. Ji, W. Melnitchouk, A. W. Thomas and P. Wang, Phys. Rev. D 106, no.5, 054006 (2022) doi:10.1103/PhysRevD.106.054006 [arXiv:2202.00266 [hep-ph]].
  * (50) C. Bourrely, W. C. Chang and J. C. Peng, Phys. Rev. D 105, no.7, 076018 (2022) doi:10.1103/PhysRevD.105.076018 [arXiv:2202.12547 [hep-ph]].
  * (51) P. Wang, F. He, C. R. Ji and W. Melnitchouk, Prog. Part. Nucl. Phys. 129, 104017 (2023) doi:10.1016/j.ppnp.2022.104017 [arXiv:2206.09361 [hep-ph]].
  * (52) M. Bonvini, F. Buccella, F. Giuli and F. Silvetti, [arXiv:2311.08785 [hep-ph]].
  * (53) P. Choudhary, D. Chakrabarti and C. Mondal, [arXiv:2312.01484 [hep-ph]].
  * (54) S. Choudhary, P. Srivastava and H. Dahiya, [arXiv:2303.11744 [hep-ph]].
  * (55) S. Theberge, A. W. Thomas and G. A. Miller, Phys. Rev. D 22, 2838 (1980) [erratum: Phys. Rev. D 23, 2106 (1981)] doi:10.1103/PhysRevD.22.2838
  * (56) A. W. Thomas, S. Theberge and G. A. Miller, Phys. Rev. D 24, 216 (1981) doi:10.1103/PhysRevD.24.216
  * (57) A. W. Thomas, Phys. Lett. B 126, 97-100 (1983) doi:10.1016/0370-2693(83)90026-6
  * (58) M. Alberg and G. A. Miller, Phys. Rev. Lett. 108, 172001 (2012) doi:10.1103/PhysRevLett.108.172001 [arXiv:1201.4184 [nucl-th]].
  * (59) M. Alberg and G. A. Miller, Phys. Rev. C 100, no.3, 035205 (2019) doi:10.1103/PhysRevC.100.035205 [arXiv:1712.05814 [nucl-th]].
  * (60) W. Du, Y. Li, X. Zhao, G. A. Miller and J. P. Vary, Phys. Rev. C 101, no.3, 035202 (2020) doi:10.1103/PhysRevC.101.035202 [arXiv:1911.10762 [nucl-th]].
  * (61) Y. Li, V. A. Karmanov, P. Maris and J. P. Vary, Phys. Lett. B 748, 278-283 (2015) doi:10.1016/j.physletb.2015.07.014 [arXiv:1504.05233 [nucl-th]].
  * (62) V. A. Karmanov, Y. Li, A. V. Smirnov and J. P. Vary, Phys. Rev. D 94, no.9, 096008 (2016) doi:10.1103/PhysRevD.94.096008 [arXiv:1610.03559 [hep-th]].
  * (63) S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299-486 (1998) doi:10.1016/S0370-1573(97)00089-6 [arXiv:hep-ph/9705477 [hep-ph]].
  * (64) J. Carbonell, B. Desplanques, V. A. Karmanov and J. F. Mathiot, Phys. Rept. 300, 215-347 (1998) doi:10.1016/S0370-1573(97)00090-2 [arXiv:nucl-th/9804029 [nucl-th]].
  * (65) J. R. Hiller and S. J. Brodsky, Phys. Rev. D 59, 016006 (1999) doi:10.1103/PhysRevD.59.016006 [arXiv:hep-ph/9806541 [hep-ph]].
  * (66) J. R. Hiller, Prog. Part. Nucl. Phys. 90, 75-124 (2016) doi:10.1016/j.ppnp.2016.06.002 [arXiv:1606.08348 [hep-ph]].
  * (67) V. A. Karmanov, J. F. Mathiot and A. V. Smirnov, Phys. Rev. D 77, 085028 (2008) doi:10.1103/PhysRevD.77.085028 [arXiv:0801.4507 [hep-th]].
  * (68) X. Cao, Y. Li and J. P. Vary, Phys. Rev. D 108, no.5, 056026 (2023) doi:10.1103/PhysRevD.108.056026 [arXiv:2308.06812 [hep-ph]].
  * (69) V. A. Karmanov, J. F. Mathiot and A. V. Smirnov, Phys. Rev. D 82, 056010 (2010) doi:10.1103/PhysRevD.82.056010 [arXiv:1006.5640 [hep-th]].
  * (70) V. A. Karmanov, J. F. Mathiot and A. V. Smirnov, Phys. Rev. D 86, 085006 (2012) doi:10.1103/PhysRevD.86.085006 [arXiv:1204.3257 [hep-th]].
  * (71) S. S. Chabysheva and J. R. Hiller, Phys. Rev. D 81, 074030 (2010) doi:10.1103/PhysRevD.81.074030 [arXiv:0911.4455 [hep-ph]].
  * (72) S. S. Chabysheva and J. R. Hiller, Annals Phys. 325, 2435-2450 (2010) doi:10.1016/j.aop.2010.05.006 [arXiv:0911.3686 [hep-ph]].
  * (73) N. A. Tsirova, V. A. Karmanov and J. F. Mathiot, Phys. Atom. Nucl. 73, 1952-1955 (2010) doi:10.1134/S1063778810110190
  * (74) M. Diehl, Phys. Rept. 388, 41-277 (2003) doi:10.1016/j.physrep.2003.08.002 [arXiv:hep-ph/0307382 [hep-ph]].
  * (75) M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Nucl. Phys. B 596, 33-65 (2001) [erratum: Nucl. Phys. B 605, 647-647 (2001)] doi:10.1016/S0550-3213(00)00684-2 [arXiv:hep-ph/0009255 [hep-ph]].
  * (76) F. Aslan, L. Gamberg, J. O. Gonzalez-Hernandez, T. Rainaldi and T. C. Rogers, Phys. Rev. D 107, no.7, 074031 (2023) doi:10.1103/PhysRevD.107.074031 [arXiv:2212.00757 [hep-ph]].
  * (77) M. Aicher, A. Schafer and W. Vogelsang, Phys. Rev. Lett. 105, 252003 (2010) doi:10.1103/PhysRevLett.105.252003 [arXiv:1009.2481 [hep-ph]].
  * (78) M. Gluck, E. Reya and I. Schienbein, Eur. Phys. J. C 10, 313-317 (1999) doi:10.1007/s100529900124 [arXiv:hep-ph/9903288 [hep-ph]].
  * (79) A. Szczurek, V. Uleshchenko, H. Holtmann and J. Speth, Nucl. Phys. A 624, 495-516 (1997) doi:10.1016/S0375-9474(97)88959-2