#  Microscopic collective inertial masses for nuclear reaction in the presence
of nucleonic effective mass

Kai Wen<EMAIL_ADDRESS>Center for Computational Sciences,
University of Tsukuba, Tsukuba 305-8577, Japan  Takashi Nakatsukasa  Center
for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571,
Japan  RIKEN Nishina Center, Wako 351-0198, Japan

###### Abstract

Collective inertial mass coefficients with respect to translational, relative,
and rotational motions are microscopically calculated, along the collective
reaction path self-consistently determined, based on the adiabatic self-
consistent collective coordinate (ASCC) method. The impact of the time-odd
component of the mean-field potential on the inertial masses are investigated.
The results are compared with those calculated with the cranking formulae. The
inertial masses based on the ASCC method reproduce the exact total nuclear
mass for the translational motion as well as the exact reduced masses as the
asymptotic values for the relative and rotational motions. In contrast, the
cranking formulae fail to do so. This is due to the fact that the (local)
Galilean invariance is properly restored in the ASCC method, but violated in
the cranking formulae. A model Hamiltonian for low-energy nuclear reaction is
constructed with the microscopically derived potentials and inertial masses.
The astrophysical $S$-factors are calculated, which indicates the importance
of microscopic calculation of proper inertial masses.

###### pacs:

21.60.Ev, 21.10.Re, 21.60.Jz, 27.50.+e

##  I Introduction

Theoretical description of low-energy nuclear reaction with a solid
microscopic foundation is still a challenging subject in nuclear physics. It
may provide us with a deep insight into the reaction mechanism and the quantum
dynamics of many-nucleon systems.

Starting from a microscopic many-body Hamiltonian (an energy density
functional), the time-dependent Hartree-Fock (TDHF) method Negele (1982);
Simenel (2012); Maruhn  _et al._ (2014) has been established as a useful tool
to study low-energy nuclear reactions. The TDHF real-time simulations have
been intensively carried out with significant developments Nakatsukasa and
Yabana (2005); Simenel (2012); Nakatsukasa  _et al._ (2016); Sekizawa and
Yabana (2016); Umar  _et al._ (2016); Sekizawa (2017); Umar  _et al._ (2017),
including the time-dependent Hartree-Fock-Bogoliubov (TDHFB) calculations
Ebata  _et al._ (2010); Stetcu  _et al._ (2011); Hashimoto (2012, 2013);
Hashimoto and Scamps (2016); Nakatsukasa  _et al._ (2016); Bulgac  _et al._
(2016); Magierski  _et al._ (2017); Scamps and Hashimoto (2019). In the
TDHF(B) simulations, the reaction path is given by a time-dependent
generalized Slater determinant which is determined by the TDHF(B) equations as
an initial-value problem. It is successful in describing the microscopic
dynamics in major reaction channels on average. However, it is known that
important quantum effects are missing in the TDHF(B). For instance the sub-
barrier fusion reaction and spontaneous fission can not be properly described
within the real-time TDHF(B) simulations (Ring and Schuck, 1980; Negele, 1982;
Nakatsukasa  _et al._ , 2016).

To recover the quantum fluctuations of nuclear collective dynamics that is
missing in TDHF(B), we aim at the requantization of TDHF(B) on the collective
subspace Nakatsukasa  _et al._ (2016). First, we find a collective subspace
spanned by a few selected collective canonical variables, which is well
decoupled from the intrinsic excitations. The collective subspace and the
collective variables can be extracted using the adiabatic self-consistent
collective coordinate (ASCC) method Matsuo  _et al._ (2000); Hinohara  _et
al._ (2007, 2009); Nakatsukasa (2012); Nakatsukasa  _et al._ (2016). The ASCC
is derived from the invariance principle of the TDHF(B) equation Marumori  _et
al._ (1980) under an assumption of adiabatic (slow) collective motion, namely,
second-order expansion with respect to the collective momenta Matsuo  _et al._
(2000). It also has a close connection with a theory of large amplitude
collective motion developed in Refs. Klein  _et al._ (1991); Dang  _et al._
(2000). The collective subspace in the adiabatic regime is given by a series
of time-even Slater determinants and generators of collective coordinates and
momenta locally defined at each state. In the present paper, we apply the
method to the low-energy nuclear reaction, to identify the optimal reaction
path and the canonical variables. The requantization is performed on this
self-consistently determined collective reaction path.

One of the key ingredients of the requantization procedure is the inertial
mass parameters with respect to the collective coordinates. This is essential
for the construction of the collective Hamiltonian. For the nuclear reaction,
the relative distance $R$ between two colliding nuclei should be a proper
choice of the collective coordinate in the asymptotic region
($R\rightarrow\infty$). The corresponding inertial mass should be the reduced
mass $\mu_{\rm red}=A_{P}A_{T}m/(A_{P}+A_{T})$, where $A_{P}$ ($A_{T}$) is the
mass number of the projectile (target) nucleus and $m$ is the bare nucleon
mass. Therefore, a theory can be tested by examining its asymptotic limit. On
the other hand, the inertial mass in the interior region where two nuclei
touch each other is highly nontrivial. Thus, a microscopic theory to calculate
the mass on the entire reaction path is necessary.

A commonly used method to calculate the collective mass coefficient is the
cranking formula (Baranger and Kumar, 1968a, b; Yuldashbaeva  _et al._ , 1999)
which is derived based on the adiabatic perturbation theory. However, it is
known that the cranking formula fails to reproduce the total mass of a whole
nucleus for the translational motion, when the mean-field potential has
velocity dependence Ring and Schuck (1980). This is due to violation of the
Galilean invariance in the mean-field level which must be corrected by
residual fields caused by the translational motion Bohr and Mottelson (1975).
From this failure, we may expect that the cranking formula cannot reproduce
the reduced mass for the relative motion of two nuclei at
$R\rightarrow\infty$. Since most of the realistic nuclear energy density
functionals give effective mass, $m^{*}/m<1$, the mean-field potential indeed
has the velocity dependence. Therefore, it is highly desirable to find a
proper method for the calculation of inertial masses. In this paper, we show
that the ASCC method, which is capable of taking into account the residual
effects missing in the cranking formula, provides a promising tool to
microscopically calculate the proper inertial masses.

In our previous works Wen and Nakatsukasa (2016, 2017), we calculated the ASCC
inertial masses for the relative motion between two nuclei, for the velocity-
independent mean-field potential. We also examined those of the cranking
formula, which turned out to be almost identical to the ASCC mass at
$R\rightarrow\infty$. However, this is not true in general. In this paper, we
show that the asymptotic value in the cranking formula for a velocity-
dependent potential does not agree with the reduced mass, while that in the
ASCC method exactly reproduces it.

In general, the non-local mean-field potential produces the effective mass
$m^{*}/m\neq 1$. For the energy density functionals of the Skyrme type, the
effective mass comes from the $t_{1}$ and $t_{2}$ terms (momentum-dependent
terms of the Skyrme interaction). In order to guarantee the Galilean
invariance of the energy density functional, we need densities that are time-
odd with respect to the time-reversal transformation. These time-odd densities
vanish in the ground state of even-even nuclei. Therefore they do not
contribute to the static mean-field potential. However, the time-odd mean
fields may appear and play important roles in time-dependent dynamics, because
the time-odd densities are non-zero in general for the time-dependent states.
In this paper, we investigate the impact of the time-odd mean fields on the
collective inertial masses in the context of low-energy reaction dynamics of
light nuclei.

The paper is organized as follows: In Sec. II, we recapitulate the theoretical
methods, mainly the ASCC method and cranking formulae, for the calculation of
inertial masses. In Sec. III, in the presence of time-odd mean-field
potential, we apply different methods to calculate the mass coefficients for
the translational, relative and rotational motions, the applications are
carried out for the reaction systems of $\alpha+\alpha\rightarrow$ 8Be,
$\alpha+^{16}$O $\rightarrow$ 20Ne and 16O$+^{16}$O $\rightarrow$ 32S. The
impacts of inertial masses on the $S$-factors of sub-barrier fusion are
discussed. A summary and concluding remarks are given in Sec. IV.

##  II Theoretical methods

It is customary to use a model Hamiltonian for studies of low-energy nuclear
reactions. The relative distance $R$ between the projectile and the target
nuclei is assumed to be a dynamical coordinate, then, the Schrödinger equation
is given in a form,

$\left\\{-\frac{1}{2\mu_{\rm
red}}\frac{d^{2}}{dR^{2}}+\frac{\ell(\ell+1)}{2\mu_{\rm
red}R^{2}}+V(R)-E\right\\}u_{\ell}(R)=0,$ (1)

where $\mu_{\rm red}$ is the reduced mass, $\mu_{\rm
red}=A_{P}A_{T}m/(A_{P}+A_{T})$. We use the unit $\hbar=1$ throughout this
paper. Here, in addition to the assumption of the coordinate $R$, the two
nuclei are approximated as structureless “point” particles, which leads to the
trivial inertial masses, $\mu_{\rm red}$ and $\mu_{\rm red}R^{2}$. This
approximation becomes exact in the asymptotic limit ($R\rightarrow\infty$).
However, at finite distance $R$, even if the two nuclei are well apart, it is
not trivial to justify the point-particle approximation. In fact, as we show
later, the rigid-body values of the moments of inertia are significantly
different from those of the point-particle approximation, ${\cal J}_{\rm
rigid}\neq\mu_{\rm red}R^{2}$, even at $R>R_{P}+R_{T}$ where $R_{P}$ ($R_{T}$)
indicates the radius of the projectile (target) nucleus. It should be noted
that the moments of inertia for deformed nuclei are given by the rigid-body
values in the harmonic-oscillator-potential model Bohr and Mottelson (1975).
Therefore, it is of great interest to examine how far the point-particle
approximation can be justified.

To reveal the coordinate ($R$) dependence and dynamical properties of the
inertial masses, we apply the ASCC method to an energy density functional. In
this section we recapitulate the ASCC method for the calculations of inertial
masses. For comparison, we also give a brief description of the cranking
formulae.

###  II.1 ASCC inertial masses

The ASCC method provides a collective subspace which describes nuclear
reaction. The generators of the collective variables $(q,p)$ are locally given
by $(\hat{Q}(q),\hat{P}(q))$ and determined by the ASCC equations. In the
asymptotic region on the reaction path (collective subspace) at $p=0$, these
should correspond to the relative distance and its conjugate momentum. Thus,
we expect that

$\hat{Q}(q)|{\Psi(q)}\rangle\propto\hat{R}|{\Psi(q)}\rangle,\quad\hat{P}(q)|{\Psi(q)}\rangle\propto\frac{\partial}{\partial
R}|{\Psi(q)}\rangle,$ (2)

where the proportional constants are arbitrary as far as they satisfy the weak
canonicity condition

$\displaystyle\langle\Psi(q)|[i\hat{P}(q),\hat{Q}(q)]|\Psi(q)\rangle=1.$ (3)

In this paper, we study systems that preserve the axial symmetry on the
reaction path. The one-body operator $\hat{R}$ for the relative distance
between projectile and target nuclei can be defined in the following way. The
symmetry axis is chosen as the $z$ axis, assuming the projectile on the right
($z>0$) and target on the left ($z<0$). We introduce a separation plane at
$z=z_{s}$, so that

$\displaystyle\int d\mathbf{r}\ \theta(z-z_{s})\rho(\mathbf{r})=A_{P},$ (4)

where $\rho(\mathbf{r})$ is the nucleon density and $\theta(x)$ is the step
function. The operator $\hat{R}$ is given by
$\hat{R}\equiv\hat{Z}_{P}-\hat{Z}_{T}$, with

$\hat{Z}_{P(T)}=\sum_{s=\uparrow,\downarrow}\sum_{q=n,p}\int
d\mathbf{r}\hat{\psi}_{sq}^{\dagger}(\mathbf{r})\hat{\psi}_{sq}(\mathbf{r})z\frac{\theta(\pm(z-z_{\rm
s}))}{A_{P(T)}},$ (5)

where the upper sign is adopted for $\hat{Z}_{P}$ and the lower for
$\hat{Z}_{T}$.

The operator $\hat{R}$ is artificially defined as a convenient measure for the
relative distance. Its meaning is clear and well defined in the asymptotic
region, though it is no longer meaningful when the projectile and the target
are merged into a single nucleus. Nevertheless, this choice of $\hat{R}$ does
not affect the final result as far as there is a one-to-one correspondence
between $q$ and $R$, because the reaction path $|{\Psi(q)}\rangle$ and its
canonical variables $(q,p)$ are self-consistently determined without any
assumption. After determining all the quantities in the collective
Hamiltonian, we perform the coordinate transformation from $q$ to $R$, by
using the relation $R(q)=\langle{\Psi(q)}|\hat{R}|{\Psi(q)}\rangle$. This is
merely a change in the scale (and the dimension) of the coordinate.

In the ASCC method, the reaction path $|{\Psi(q)}\rangle$ at $p=0$ and the
local generators $(\hat{Q}(q),\hat{P}(q))$ are determined by solving the ASCC
equations

$\displaystyle\delta\langle\Psi(q)|\hat{H}_{\rm mv}|\Psi(q)\rangle=0,$ (6)
$\displaystyle\delta\langle\Psi(q)|[\hat{H}_{\rm
mv},\frac{1}{i}\hat{P}(q)]-\frac{\partial^{2}V(q)}{\partial
q^{2}}\hat{Q}(q)|\Psi(q)\rangle=0,$ (7)
$\displaystyle\delta\langle\Psi(q)|[\hat{H}_{\rm
mv},i\hat{Q}(q)]-\frac{1}{M(q)}\hat{P}(q)|\Psi(q)\rangle=0,$ (8)

where curvature terms are neglected for simplicity Matsuo  _et al._ (2000).
The Hamiltonian in the moving frame $\hat{H}_{\rm mv}$ and the potential
$V(q)$ are respectively defined as

$\displaystyle\hat{H}_{\rm mv}\equiv\hat{H}-\frac{\partial V(q)}{\partial
q}\hat{Q}(q),\quad V(q)\equiv\langle\Psi(q)|\hat{H}|\Psi(q)\rangle.$ (9)

$V(q)$ in Eq. (9) is used as $V(R)$ in Eq. (1) at
$R=\langle{\Psi(q)}|\hat{R}|{\Psi(q)}\rangle$.

Equation (6) which depends on $\hat{Q}(q)$ determines the reaction path
$|{\Psi(q)}\rangle$. The local generators, $\hat{P}(q)$ and $\hat{Q}(q)$, are
given by a solution of Eqs. (7), and (8). However, since solutions of Eqs. (7)
and (8) are not unique, we need to select the one corresponding to the
reaction path we study. Normally, we choose a solution with the lowest
frequency, $\omega^{2}=(\partial^{2}V/\partial q^{2})M^{-1}(q)$, except for
the Nambu-Goldstone (NG) modes. The generators should satisfy the asymptotic
condition of Eq. (2).

In Eq. (8) there appears the inertial mass parameter $M(q)$ with respect to
the coordinate $q$. The magnitude of $M(q)$ depends on the scale of the
coordinate $q$ which is arbitrary. We can choose to set the mass $M(q)$ in Eq.
(8) to be a constant, for instance $M_{q}=1$ MeV-1$[q]^{-2}$ without losing
generality Wen and Nakatsukasa (2016), where $[q]$ represents the unit of $q$.

In order to obtain an intuitive picture of the collective dynamics, we map the
collective coordinate $q$ to the relative distance $R$ between projectile and
target nuclei. The inertial mass should be transformed as

$\displaystyle
M(R)=M(q)\left(\frac{dq}{dR}\right)^{2}=M(q)\left(\frac{dR}{dq}\right)^{-2},$
(10)

where the derivative $dR/dq$ can be obtained as

$\displaystyle\frac{dR}{dq}$ $\displaystyle=$
$\displaystyle\frac{d}{dq}\langle\Psi(q)|\hat{R}|\Psi(q)\rangle=\langle\Psi(q)|[\hat{R},\frac{1}{i}\hat{P}(q)]|\Psi(q)\rangle$
(11) $\displaystyle=$ $\displaystyle 2\sum_{n\in p,j\in h}R_{nj}(q)P_{nj}(q),$

with the local generator $\hat{P}(q)$. $R_{nj}(q)$ and $P_{nj}(q)$ are the
particle-hole (ph) matrix elements of $\hat{R}(q)$ and $\hat{P}(q)$, obtained
by solving ASCC equations (7) and (8) Wen and Nakatsukasa (2016). The
calculated mass $M(R)$ will replace the constant mass $\mu_{\rm red}$ in the
first term of Eq. (1).

Apart from the the relative motion of our current interest, Eqs. (7) and (8)
provide solutions for the NG modes, such as the translation and the rotation.
These modes have natural and global generators, namely, the total linear
momentum $\hat{\mathbf{P}}$ for the translation and the total angular momentum
$\hat{\mathbf{J}}$ for the rotation. For example, the rotational motion around
the $x$ axis can be generated by $\hat{J}_{x}$ without any cost of energy
($\omega^{2}=0$), leading to the Thouless-Valatin equations Thouless and
Valatin (1962); Ring and Schuck (1980),

$\displaystyle\delta\langle{\Psi(q)}|\left[\hat{H}_{\rm
mv},\hat{J}_{x}\right]|{\Psi(q)}\rangle=0,$ (12)
$\displaystyle\delta\langle{\Psi(q)}|\left[\hat{H}_{\rm
mv},i\hat{\Theta}(q)\right]-\frac{1}{{\cal
J}(q)}\hat{J}_{x}|\Psi(q)\rangle=0,$ (13)
$\displaystyle\langle{\Psi(q)}|[i\hat{J}_{x},\hat{\Theta}(q)]|{\Psi(q)}\rangle=1,$
(14)

where $\hat{\Theta}$ is the angle variable conjugate to $\hat{J}_{x}$, and
${\cal J}(q)$ is the moment of inertia around $x$ axis. Equations (12) and
(13) indicate that $(\hat{Q}(q),\hat{P}(q))=(\hat{\Theta}(q),\hat{J}_{x})$
correspond to a solution of the ASCC equations (7) and (8) with
$\partial^{2}V/\partial q^{2}=0$ and $M(q)={\cal J}(q)$. The calculated
rotational moments of inertia can be used to replace $\mu_{\rm red}R^{2}$ in
the second term of Eq. (1).

To calculate the inertial masses for the NG modes, an efficient method was
proposed in Ref. Hinohara (2015). The inertial mass of the NG modes $M(q)$ are
given by the zero-frequency limit of the strength function in the linear
response for the momentum operator $\hat{P}$. We apply this method to the
calculations of the translational and rotational inertial masses.

###  II.2 CHF+cranking method

A simple method based on the cranking formula to calculate the inertial mass
is widely used for nuclear collective motion. The collective path (reaction
path) is usually produced by the constrained-Hartree-Fock (CHF) calculation
with a constraining operator $\hat{C}$ given by hand. The CHF states are given
by the variation

$\delta\langle{\Psi_{0}(\lambda)}|\hat{H}_{\lambda}|{\Psi_{0}(\lambda)}\rangle=0,\quad\hat{H}_{\lambda}\equiv\hat{H}-\lambda\hat{C},$
(15)

where $\lambda$ is the Lagrange multiplier. Upon the CHF states
$|{\Psi_{0}(R)}\rangle=|{\Psi_{0}(\lambda)}\rangle$ with
$R=\langle{\Psi_{0}(\lambda)}|\hat{R}|{\Psi_{0}(\lambda)}\rangle$, the
inertial mass with respect to the coordinate $R$ is calculated using the
cranking formula Baran  _et al._ (2011):

$\displaystyle M_{\rm cr}^{\rm np}(R)$ $\displaystyle=$ $\displaystyle
2\sum_{m}\frac{|\langle{\Psi_{m}(R)}|\partial/\partial
R|{\Psi_{0}(R)}\rangle|^{2}}{E_{m}(R)-E_{0}(R)}$ (16) $\displaystyle=$
$\displaystyle 2\sum_{n\in p,j\in
h}\frac{|\langle{\varphi_{n}(R)}|\partial/\partial
R|{\varphi_{j}(R)}\rangle|^{2}}{e_{n}(R)-e_{j}(R)},$ (17)

where the single-particle states $|{\varphi_{\mu}(R)}\rangle$ and energies
$e_{\mu}(R)$ are defined with respect to the CHF single-particle Hamiltonian,
$\hat{h}_{\lambda}=\hat{h}_{\rm HF}[\rho]-\lambda\hat{C}$, as

$\hat{h}_{\lambda}|{\varphi_{\mu}(R)}\rangle=e_{\mu}(R))|{\varphi_{\mu}(R)}\rangle,\quad\mu\in
p,h.$ (18)

Note that the Lagrange multiplier $\lambda$ is a function of $R$ given by the
condition, $R=\langle{\Psi_{0}(\lambda)}|\hat{R}|{\Psi_{0}(\lambda)}\rangle$.
The constraining operator $\hat{C}$ can be in general different from the
relative-distance operator $\hat{R}$. In the following calculations in Sec.
III, we use the mass quadrupole and octupole operators as $\hat{C}$.

Another cranking formula, which is even more frequently used in many
applications, can be derived by neglecting the rearranged fields induced by
the change of $\lambda$, namely, $\delta h_{\rm HF}/\delta\rho\cdot
d\rho/d\lambda=0$. Normally, one chooses the constraining operator as the
collective coordinate, $\hat{C}=\hat{R}$, for the CHF calculation. However, in
the present calculation, since we need to find the separation plane $z=z_{s}$
to determine the operator $\hat{R}$, it is convenient to adopt a different
one-body operator $\hat{C}$. Let us derive the formula for this case. From Eq.
(18) and the orthonormality condition
$\langle\varphi_{\mu}(R)|\varphi_{\nu}(R)\rangle=\delta_{\mu\nu}$, it is easy
to find

$\langle{\varphi_{n}}|\frac{\partial}{\partial
R}|{\varphi_{j}}\rangle=-\frac{\langle{\varphi_{n}}|\partial
h_{\lambda}/\partial
R|{\varphi_{j}}\rangle}{e_{n}-e_{j}}\approx\frac{d\lambda}{dR}\frac{\langle{\varphi_{n}}|\hat{C}|{\varphi_{j}}\rangle}{e_{n}-e_{j}}.$
(19)

Neglecting the rearrangement,
$\hat{h}_{\lambda+\Delta\lambda}\approx\hat{h}_{\lambda}-\Delta\lambda\hat{C}$,
the derivative $dR/d\lambda$ is estimated with the first-order perturbation as
${dR}/{d\lambda}=2S_{1}(\hat{R},\hat{C})$, where

$\displaystyle S_{k}(\hat{Q}_{1},\hat{Q}_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\sum_{m>0}\frac{\langle{\Psi_{0}}|\hat{Q}_{1}|{\Psi_{m}}\rangle\langle{\Psi_{m}}|\hat{Q}_{2}|{\Psi_{0}}\rangle+\mbox{c.c.}}{\left(E_{m}-E_{0}\right)^{k}}$
(20) $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{n\in p,j\in
h}\frac{\langle{\varphi_{j}}|\hat{Q}_{1}|{\varphi_{n}}\rangle\langle{\varphi_{n}}|\hat{Q}_{2}|{\varphi_{j}}\rangle+\mbox{c.c.}}{\left(e_{n}-e_{j}\right)^{k}}$
$\displaystyle=$ $\displaystyle S_{k}(\hat{Q}_{2},\hat{Q}_{1}).$

Thus, the formula reads

$M_{\rm cr}^{\rm
p}(R)=\frac{1}{2}S_{3}(\hat{C},\hat{C})\left\\{S_{1}(\hat{R},\hat{C})\right\\}^{-2}.$
(21)

According to Ref. Baran  _et al._ (2011), we call the former one in Eq. (17)
“non-perturbative” cranking inertia and the latter in Eq. (21) “perturbative”
one. In contrast to the ASCC mass, the cranking masses of Eqs. (17) and (21)
have a drawback that they both neglect residual effect of the time-odd density
fluctuation. As we shall see in Sec. III, when the nucleonic effective masses
are present, the cranking formulae produce wrong masses for the translation
and for the relative motion at $R\rightarrow\infty$.

For the NG modes such as the translation and the rotation, we know the
generator of the collective coordinate. For instance, in the translational
case, the center-of-mass (COM) coordinates $X_{k}$ ($k=x,y,z$) are the trivial
collective coordinates. Then, the non-perturbative cranking formula (17) can
be transformed into

$\displaystyle M_{\rm cr}^{\rm np}(X_{k})$ $\displaystyle=$ $\displaystyle
2\sum_{n\in p,j\in h}\frac{|\langle{\varphi_{n}(X_{k})}|\partial/\partial
X_{k}|{\varphi_{j}(X_{k})}\rangle|^{2}}{e_{n}(X_{k})-e_{j}(X_{k})}$ (22)
$\displaystyle=$ $\displaystyle 2\sum_{n\in p,j\in
h}\frac{|\langle{\varphi_{n}}|\hat{p}_{k}|{\varphi_{j}}\rangle|^{2}}{e_{n}-e_{j}},$
(23)

where $\hat{p}_{k}$ is the single-particle linear momentum. In the second
equation, we take advantage of the fact that the single-particle energies and
the wave functions relative to the COM are invariant with respect to its
position $X_{k}$. For the rotation, we may replace $X_{k}$ by an angle
$\phi_{k}$ and $\hat{p}_{k}$ by the angular momentum $\hat{j}_{k}$. This leads
to nothing but the cranking formula for the moment of inertia, originally
proposed by Inglis Inglis (1954),

$\left({\cal J}_{\rm cr}^{\rm np}\right)_{k}=2\sum_{n\in p,j\in
h}\frac{|\langle{\varphi_{n}}|\hat{j}_{k}|{\varphi_{j}}\rangle|^{2}}{e_{n}-e_{j}}.$
(24)

It is instructive to investigate properties of the cranking mass formula for
the NG modes Ring and Schuck (1980); Bohr and Mottelson (1975); Thouless and
Valatin (1962). Here, let us examine the formula of Eq. (23) for the
translation, following the argument given in Ref. Bohr and Mottelson (1975).
The summation with respect to the index $n$ in Eq. (23) is restricted to the
particle states. This restriction can be removed, because the summation with
respect to the hole-hole components, $\sum_{n\in h}\sum_{j\in h}$, identically
vanishes. When the mean-field (Hartree-Fock) Hamiltonian $\hat{h}_{\rm
HF}[\rho]$ conserves the Galilean invariance,

$\left[\hat{h}_{\rm HF},\hat{x}_{k}\right]=\frac{-i}{m}\hat{p}_{k},\quad
k=x,y,z,$ (25)

the translational mass becomes

$\displaystyle M_{\rm cr}^{\rm np}$ $\displaystyle=$ $\displaystyle
2\sum_{\mu}\sum_{j\in
h}\frac{\langle{\varphi_{j}}|\hat{p}_{k}|{\varphi_{\mu}}\rangle\langle{\varphi_{\mu}}|\hat{p}_{k}|{\varphi_{j}}\rangle}{e_{\mu}-e_{j}}$
(26) $\displaystyle=$ $\displaystyle\sum_{\mu}\sum_{j\in
h}\left(\langle{\varphi_{j}}|im\left[\hat{h}_{\rm
HF},\hat{x}_{k}\right]|{\varphi_{\mu}}\rangle\langle{\varphi_{\mu}}|\hat{p}_{k}|{\varphi_{j}}\rangle\right.$
$\displaystyle\left.+\langle{\varphi_{j}}|\hat{p}_{k}|{\varphi_{\mu}}\rangle\langle{\varphi_{\mu}}|im\left[\hat{h}_{\rm
HF},\hat{x}_{k}\right]|{\varphi_{j}}\rangle\right)\frac{1}{e_{\mu}-e_{j}}$
$\displaystyle=$ $\displaystyle im\sum_{j\in
h}\langle{\varphi_{j}}|\left[\hat{p}_{k},\hat{x}_{k}\right]|{\varphi_{j}}\rangle=Am.$

Therefore, the Galilean invariance of the mean-field Hamiltonian guarantees
that the cranking formula reproduces the total mass $Am$ for the translation.
However, the Galilean invariance is violated by velocity dependence of the
one-body mean-field potential in most of nuclear energy density functionals.
This results in a wrong mass $M_{\rm cr}\neq Am$ in the cranking formula. This
violation should be corrected by the residual fields that depend on the
velocity of the translational motion Bohr and Mottelson (1975). Indeed, the
ASCC mass reproduces the exact total mass even when the mean-field potential
violates the Galilean invariance, which will be shown in Sec. III.

##  III Application

In the following numerical calculations, in order to express the orbital wave
functions, the grid representation is employed, discretizing the rectangular
box into three-dimensional (3D) Cartesian mesh Nakatsukasa and Yabana (2005).
The model space is set to be $12\times 12\times 18$ fm3 for the systems
$\alpha$+$\alpha$ $\rightarrow$ 8Be and 16O+$\alpha$ $\rightarrow$ 20Ne. It is
$12\times 12\times 24$ fm3 for 16O+16O $\rightarrow$ 32S. The mesh size is set
to be 1.0 fm for the system $\alpha$+$\alpha$ $\rightarrow$ 8Be and 1.1 fm for
the other two systems.

For numerical calculations of the ASCC method, we use the the finite amplitude
method (FAM) (Nakatsukasa  _et al._ , 2007; Inakura  _et al._ , 2009; Avogadro
and Nakatsukasa, 2011, 2013). The FAM provides us with high numerical
efficiency with simple computer programs, because only the calculations of the
mean-field (single-particle) Hamiltonian constructed with independent bra and
ket states are required Nakatsukasa  _et al._ (2007). The matrix FAM (m-FAM)
prescription Avogadro and Nakatsukasa (2013) is adopted to solve the moving
RPA equations (7) and (8). On the other hand, the iterative FAM (i-FAM)
(Nakatsukasa  _et al._ , 2007; Inakura  _et al._ , 2009; Avogadro and
Nakatsukasa, 2011) is adopted to calculate the response functions for the NG
modes. The moving mean-field equation (6) is solved by using the imaginary-
time method (Davies  _et al._ , 1980).

###  III.1 Modified BKN energy density functional

In order to investigate the effect of this time-odd mean-field potential on
the collective inertial masses, we adopt an energy density functional of the
simplest choice, namely, the BKN energy density functional Bonche  _et al._
(1976) with the minimum extension.

The original BKN functional assumes the spin-isospin symmetry without the
spin-orbit interaction, which is a functional of the isoscalar kinetic and
local densities, $\tau(\mathbf{r})$ and $\rho(\mathbf{r})$, only. The mean-
field potential is local and has no velocity dependence. Thus, the nucleon’s
effective mass is identical to the bare nucleon mass. However, in most of
realistic energy density functionals, the effective mass is smaller than the
bare mass, typically $m^{*}/m\sim 0.7$. In order to introduce the effective
mass, we extend the energy density by adding terms $\rho\tau-\mathbf{j}^{2}$
where $\mathbf{j}(\mathbf{r})$ is the isoscalar current density. The
appearance of the current density is necessary to preserve the Galilean
invariance.

The modified BKN energy density functional reads,

$\displaystyle E[\rho]$ $\displaystyle=$
$\displaystyle\int\frac{1}{2m}\tau(\mathbf{r})d\mathbf{r}+\int
d\mathbf{r}\left\\{\frac{3}{8}t_{0}\rho^{2}(\mathbf{r})+\frac{1}{16}t_{3}\rho^{3}(\mathbf{r})\right\\}$
(27) $\displaystyle+\int\int
d\mathbf{r}d\mathbf{r}^{\prime}\rho(\mathbf{r})v(\mathbf{r}-\mathbf{r}^{\prime})\rho(\mathbf{r}^{\prime})$
$\displaystyle+B_{3}\int
d\mathbf{r}\left\\{\rho(\mathbf{r})\tau(\mathbf{r})-\mathbf{j}^{2}(\mathbf{r})\right\\},$

where $\rho(\mathbf{r})$, $\tau(\mathbf{r})$, and $\mathbf{j}(\mathbf{r})$ are
calculated as

$\displaystyle\rho(\mathbf{r})$ $\displaystyle=$ $\displaystyle
4\sum_{j=1}^{A/4}|\psi_{j}(\mathbf{r})|^{2},\quad\tau(\mathbf{r})=4\sum_{j=1}^{A/4}|\nabla\psi_{j}(\mathbf{r})|^{2},$
$\displaystyle\textbf{j}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{4}{2i}\sum_{j=1}^{A/4}\left\\{\psi_{j}^{*}(\mathbf{r})\nabla\psi_{j}(\mathbf{r})-\psi_{j}(\mathbf{r})\nabla\psi_{j}^{*}(\mathbf{r})\right\\}.$

In Eq. (27), $v(\mathbf{r})$ is the sum of the Yukawa and the Coulomb
potentials (Bonche  _et al._ , 1976),

$\displaystyle v(\mathbf{r})\equiv
V_{0}a\frac{e^{-r/a}}{r}+\frac{(e/2)^{2}}{r}.$

The new parameter $B_{3}$ controls the effective mass and the velocity
dependence of the mean-field potential.

The variation of the total energy with respect to the density (or equivalently
single-particle wave functions) defines the mean-field (Hartree-Fock)
Hamiltonian,

$\displaystyle\hat{h}_{\rm HF}(\mathbf{r})$ $\displaystyle=$
$\displaystyle-\nabla\frac{1}{2m^{*}(\mathbf{r})}\nabla+\frac{3}{4}t_{0}\rho(\mathbf{r})+\frac{3}{16}t_{3}\rho^{2}(\mathbf{r})$
(28) $\displaystyle+\int
d\mathbf{r^{\prime}}v(\mathbf{r}-\mathbf{r^{\prime}})\rho(\mathbf{r^{\prime}})+B_{3}(\tau(\mathbf{r})+i\nabla\cdot\mathbf{j}(\mathbf{r}))$
$\displaystyle+2iB_{3}\mathbf{j}(\mathbf{r})\cdot\nabla,$

where the effective mass is now deviated from bare nucleon mass,

$\displaystyle\frac{1}{2m^{*}(\mathbf{r})}=\frac{1}{2m}+B_{3}\rho(\mathbf{r}).$
(29)

For $B_{3}\neq 0$, Eq. (28) indicates the velocity (momentum) dependence of
the mean-field potential and the presence of the time-odd mean fields,
$iB_{3}(\nabla\cdot\mathbf{j}+2\mathbf{j}\cdot\nabla)$. For the time-even
states, such as the ground state of even-even nuclei, the current density
disappears, $\mathbf{j}(\mathbf{r})=0$. Nevertheless, as will be shown later,
the terms associated with $\mathbf{j}(\mathbf{r})$ play an important role in
the collective inertial mass.

The parameters, $t_{0}$, $t_{3}$, $V_{0}$, and $a$ are adopted from Ref.
(Bonche  _et al._ , 1976), and we vary $B_{3}$ to change the effective mass
and the time-odd mean fields.

###  III.2 Inertial massses for translational motion: Alpha particle

First, we demonstrate the importance of the time-odd mean fields, taking the
translational total mass as a trivial example. We adopt the simplest case,
namely, the single alpha particle.

Figure 1: (Color online) Translational mass of the $\alpha$ particle in unit
of the nucleon mass as a function of the parameter $B_{3}$. The solid (red),
dashed (green), and dash-dotted (blue) curves show the results of the ASCC,
the non-perturbative cranking formula (23) and $A$ times the average effective
mass, respectively. See the text for details.

In Fig. 1, we show the translational mass of a single alpha particle as a
function of $B_{3}$. Increasing $B_{3}$, the effective mass $m^{*}$ decreases,
as we see in Eq. (29). In the present case, we use the center-of-mass
coordinate $\mathbf{R}_{\rm cm}$ as the collective coordinate $R$ in Sec. II.1
and II.2. Since the system is isotropic, we use its $z$ component $Z_{\rm cm}$
in the numerical calculation. In the present model, neutrons and protons have
the identical mass $m$. Therefore, the total inertial mass of an alpha
particle should be equal to $4m$.

It is clearly demonstrated that the ASCC always reproduces the correct total
mass $Am$, irrespective of values of the parameter $B_{3}$. The non-
perturbative cranking mass reproduces the total mass, $M^{\rm np}_{\rm
cr}=4m$, at $B_{3}=0$, which agree with Eq. (26). This is due to the velocity-
independent mean-field potential, for which the mean-field Hamiltonian
$\hat{h}_{\rm HF}$ preserves the Galilean invariance. However, at $B_{3}>0$,
the effective mass becomes smaller than the bare nucleon mass, $m^{*}<m$,
inside the nucleus with $\rho(\mathbf{r})>0$. This leads to the velocity-
dependent potential, and the violation of the Galilean invariance in the mean-
field level. Figure 1 indicates $M_{\rm cr}^{\rm np}<4m$ at $B_{3}>0$, and
shows a monotonic decrease with respect to $B_{3}$. We calculate the average
effective mass defined by

$\left(\frac{m^{*}}{m}\right)_{\rm
av}^{-1}\equiv\frac{2m}{A}\int\frac{\rho(\mathbf{r})}{2m^{*}(\mathbf{r})}d\mathbf{r}.$
(30)

It turns out that the cranking mass for the translation approximately
coincides with $Am\times(m^{*}/m)_{\rm av}$ as shown in Fig. 1.

For the translational motion, the ASCC inertial mass becomes identical to the
Thouless-Valatin moments of inertia Thouless and Valatin (1962) in Eqs. (12)
and (13), with $(\hat{J}_{x},\hat{\Theta})$ being replaced by
$(\hat{P}_{z},\hat{Z}_{\rm cm})$. The fact that the ASCC inertial mass agrees
with the total mass $Am$ means that the Galilean invariance, violated in the
mean-field level, is properly restored by the residual induced fields in the
leading order. This shows a clear advantageous feature of the ASCC mass over
the cranking formulae.

###  III.3 Inertial massses for relative motions

####  III.3.1 $\alpha$+$\alpha$ $\rightarrow^{8}$Be

In this section we study how the time-odd mean-field changes the inertial mass
for the relative motion between two alpha particles. The reaction path was
obtained and shown in our former paper Wen and Nakatsukasa (2016) for
$B_{3}=0$. With $B_{3}\neq 0$, basic features of the reaction path is the
same, although the ground state of 8Be becomes more elongated, developing a
prominent $2\alpha$ cluster structure. The curves in Fig. 2 show the inertial
mass and total energy for the relative motion between two alpha particles as a
function of the relative distance $R$. The ASCC mass is calculated based on
the self-consistent ASCC collective path. For the cranking masses, the mass
quadrupole operator $\hat{Q}_{20}$ is used as a constraint to construct the
reaction path with the CHF calculation.

Figure 2: (Color online) Inertial mass $M(R)$ for the reaction $\alpha+\alpha$
as a function of the relative distance $R$. The top panel shows the results of
the ASCC method, while the middle panel shows those of the perturbative and
non-perturbative cranking formulae $M_{\rm cr}^{\rm p}$ (thin lines) and
$M_{\rm cr}^{\rm np}$ (thick lines). The bottom panel shows the potential
energies along the ASCC collective paths. Solid (red) and dashed (blue) lines
indicate those of $B_{3}=0$ and 75 MeV fm5, respectively. In the upper panel,
the dots correspond to the positions of the minimum energy in the bottom
panel.

At large $R$ where the two alpha particles can be approximated as point
particles, we expect that the inertial mass with respect to $R$ is identical
to the reduced mass, $2m$. For the non-perturbative cranking mass, this is
true at $B_{3}=0$, while it monotonically decreases as $B_{3}$ increases (Fig.
3). On the other hand, the ASCC mass reproduces the correct reduced mass at
large $R$, irrespective of the value of $B_{3}$. The main difference between
the ASCC and the cranking masses is the inclusion of the effect of time-odd
residual fields. Therefore, we may conclude that the time-odd residual effect
is essential to reproduce the reduced mass in the asymptotic region of
$R\rightarrow\infty$.

The perturbative cranking mass seems to be unable to reproduce the correct
reduced mass. It is larger than $\mu_{\rm red}=2m$ even for $B_{3}=0$. As is
seen in the first equation of (19), the non-perturbative cranking mass comes
from the $R$-dependence of the Lagrange multiplier $\lambda$ and from that of
the HF Hamiltonian (rearrangement) $\hat{h}_{\rm HF}$. When we take the
asymptotic limit of $R\rightarrow\infty$, since the interaction between two
$\alpha$ particles vanishes, we naturally expect $\lambda\rightarrow 0$. Then,
we have $d\lambda/dR\rightarrow 0$, and $M^{\rm np}_{\rm cr}$ solely comes
from the rearrangement effect. In the perturbative treatment, we neglect the
rearrangement in the last equality of Eq. (19). Instead, we replace
$d\lambda/dR$ by the perturbative value $(2S_{1}(\hat{R},\hat{C}))^{-1}$. It
is thus difficult to justify the perturbative approximation in the asymptotic
limit.

Figure 3: (Color online) Non-perturbative cranking inertial masses calculated
at $R=8$ fm for the relative motion between two $\alpha$ particles divided by
the reduced mass $\mu_{\rm red}=2m$, as a function of $B_{3}$. The blue dash-
dotted curve indicates $(m^{*}/m)_{\rm av}$ of Eq. (30).

Figure 3 clearly demonstrates that the cranking inertial mass at $B_{3}\neq 0$
underestimates the exact reduced mass for the relative motion ($\mu_{\rm
red}=2m$). Again, it well agrees with the average effective mass,
$2(m^{*})_{\rm av}$. Thus, the failure of the cranking mass is due to the
violation of the Galilean invariance.

####  III.3.2  $\alpha$+16O $\rightarrow^{20}$Ne and 16O+16O
$\rightarrow^{32}$S

Next we show the inertial masses $M(R)$ of the relative motion for two
reaction systems $\alpha$+16O $\rightarrow^{20}$Ne and 16O+16O
$\rightarrow^{32}$S. The self-consistent reaction paths for these were
presented in our former paper Wen and Nakatsukasa (2017), but only for
$B_{3}=0$. Similar to the $\alpha+\alpha\rightarrow^{8}$Be case, the increase
of $B_{3}$ favors larger deformation. The upper panels in Figs. 4 and 5 show
$M(R)$ for $\alpha$+16O $\rightarrow$ 20Ne and 16O+16O $\rightarrow$ 32S,
respectively. The smaller effective mass enlarges the size of the nucleus,
thus, the touching points between the two nuclei take place at larger $R$. The
minimum energy position is also shifted to larger $R$ for larger $B_{3}$. The
drastic increase in $M(R)$ in the interior region is due to the increase of
$dq/dR$ in Eq. (10). In contrast to variations of $M(R)$ in the interior
region, $M(R)$ is very stable and independent from the value of $B_{3}$ when
two nuclei are separated. In the asymptotic region ($R\rightarrow\infty$), we
always observe the correct limit, $M(R)\rightarrow\mu_{\rm red}$. The reduced
mass for $\alpha$+16O $\rightarrow^{20}$Ne is $\mu_{\rm red}=3.2m$, and
$\mu_{\rm red}=8m$ for 16O+16O $\rightarrow^{32}$S.

The cranking masses are shown in the middle panels in Figs. 4 and 5. The
perturbative and non-perturbative cranking masses are calculated based on the
CHF reaction paths that are constructed with the constraining operator
$\hat{C}=\hat{Q}_{30}$ (mass octupole) for $\alpha$+16O $\rightarrow^{20}$Ne,
and with $\hat{C}=\hat{Q}_{20}$ (mass quadrupole) for 16O$+^{16}$O
$\rightarrow^{32}$S. If we adopt $\hat{C}=\hat{Q}_{20}$ for the former
reaction system, we cannot obtain a continuous reaction path Wen and
Nakatsukasa (2017). Again, for $B_{3}\neq 0$, the cranking mass does not
reproduce the correct reduced mass at $R\rightarrow\infty$, neither with the
perturbative cranking formula nor with the non-perturbative one. Furthermore,
the perturbative cranking mass is larger than the reduced mass, even for
$B_{3}=0$.

Figure 4: (Color online) Inertial mass $M(R)$ for the reaction path
$\alpha$+16O as a function of $R$, in units of the nucleon mass. The top panel
shows the results of the ASCC method, the middle panel shows those of the
perturbative and non-perturbative cranking formulae $M_{\rm cr}^{\rm p}$ (thin
lines) and $M_{\rm cr}^{\rm np}$ (thick lines), and the bottom panel shows the
potential energies along the three ASCC collective paths. Solid (red) and
dashed (blue) lines indicate those of $B_{3}=0$ and 75 MeV fm5. In the bottom
panel, in order to show it in the given scale, the potential for $B_{3}=75$
MeV fm5 (dashed line) is shifted downwards by 60 MeV.  Figure 5: (Color
online) Same as Fig. 4, but for 16O+16O. Solid (red) and dashed (blue) lines
indicate results of $B_{3}=0$ and 30 MeV fm5. In the bottom panel, the
potential for $B_{3}=30$ MeV fm5 (dashed line) is shifted downwards by 45 MeV.

###  III.4 Rotational moments of inertia

Next let us present results for the moment of inertia which is another
important inertial mass for the collective Hamiltonian in Eq. (1). The
rotational motion is a NG mode whose ASCC inertial mass can be calculated with
the strength function for the angular momentum operator at zero frequency
Hinohara (2015), instead of solving the Thouless-Valatin equation (13). For
this purpose, we perform the i-FAM calculation Nakatsukasa  _et al._ (2007);
Inakura  _et al._ (2009); Avogadro and Nakatsukasa (2011) based on the
reaction path self-consistently constructed by the ASCC method. For
comparison, we also apply the cranking formula to the moments of inertia, Eq.
(24), based on the reaction path calculated with the CHF method.

In Eq. (1), the point-particle approximation was adopted, leading to the
moment of inertia, $\mu_{\rm red}R^{2}$. There is another classical limit,
namely, the rigid-body moment of inertia. In the present case, the system
preserves the axial symmetry on the reaction path. Choosing the axis of the
symmetry as the $z$ axis, we have ${\cal J}_{x}={\cal J}_{y}$ whose rigid-body
value is given by

$\displaystyle{\cal J}_{\rm
rig}=m\int(y^{2}+z^{2})\rho(\mathbf{r})d\mathbf{r}=m\int(z^{2}+x^{2})\rho(\mathbf{r})d\mathbf{r}.$
(31)

The rigid-body moment of inertia about the $z$ axis is non-zero, ${\cal
J}_{z}\neq 0$, which contradicts with the trivial quantum mechanical
requirement that there exists no rotation around the symmetry axis. On the
other hand, it is known that the cranking moments of inertia, ${\cal J}_{x}$
and ${\cal J}_{y}$, for axially deformed nuclei in the harmonic-oscillator-
potential model are given by the rigid-body value at the equilibrium Bohr and
Mottelson (1975). The cranking formula also satisfies the quantum mechanical
condition, producing ${\cal J}_{z}=0$ in the axially symmetric case.

When two nuclei are far away, we expect the point-particle approximation is
good. Therefore, we expect that the moments of inertia, ${\cal J}_{x}$ and
${\cal J}_{y}$, on the reaction path changes from the point-particle value
${\cal J}(R)=\mu_{\rm red}R^{2}$ at large $R$ to the rigid-body value ${\cal
J}(R)={\cal J}_{\rm rig}$ near the equilibrium (ground) state. It is of
significant interest and of importance to examine where and how this
transition takes place during the reaction.

In Ref. Wen and Nakatsukasa (2020), we have published the result for
$\alpha+\alpha\rightarrow^{8}$Be. Thus, in this paper, we present the results
for $\alpha$+16O $\rightarrow^{20}$Ne and 16O+16O $\rightarrow^{32}$S. First,
let us show results for the velocity-independent potential with $B_{3}=0$.
Figure 6 shows the moments of inertia calculated for the two reaction systems
with $B_{3}=0$. The ground state of 20Ne corresponds to $R=3.8$ fm in the
upper panel of Fig. 6, and the superdeformed minimum of 32S is located at
$R=5$ fm in the lower panel. We can see that the ASCC and the (non-
perturbative) cranking formulae give results very similar to each other. This
is the consequence of the local Galilean invariance of the mean-field
potential under the momentum transformation
$\mathbf{p}\rightarrow\mathbf{p}-m(\bm{\omega}\times\mathbf{r})$ Bohr and
Mottelson (1975). The calculated moments of inertia are close to their rigid-
body values near these equilibrium states, ${\cal J}(R_{e})\approx{\cal
J}_{\rm rig}(R_{e})$, where $R_{e}$ represents the value of $R$ at the
potential equilibrium ($dV/dR=0$). However, when the nucleus is more elongated
along the reaction path, ${\cal J}(R)$ decreases as $R$ increases. Since the
rigid-body value is a monotonically increasing function of $R$, they become
smaller than the rigid-body values, ${\cal J}(R)<{\cal J}_{\rm rig}(R)$, then,
quickly approach the point-particle values, $\mu_{\rm red}R^{2}$. Beyond the
scission point $R=R_{s}$ where the system splits into two separated nuclei, we
have ${\cal J}(R)\approx\mu_{\rm red}R^{2}$ ($R\geq R_{s}$). The calculated
moments of inertia give parabolic convex lines as functions of $R$, showing
their minima located around the midpoint between $R_{e}$ and $R_{s}$.

It is surprising that the moments of inertia decrease as the deformation
develops near the equilibrium. This contradicts with our naive expectation
based on the classical model. For instance, the rigid-body moments of inertia
linearly increases with the deformation $\delta$, ${\cal J}_{\rm
rig}\approx{\cal J}_{0}(1+\delta/3)$. Another classical model, irrotational
fluid model suggests that it increases quadratically with the deformation as
${\cal J}_{\rm irrot}\approx{\cal J}_{\rm rig}\delta^{2}$ Bohr and Mottelson
(1975). In any case, a single classical model cannot explain the reduction in
${\cal J}(R)$ as a function of $R$. A hint to understand this behavior may
come from the harmonic oscillator model in the quantum mechanics which
reproduces both the rigid-body value and the irrotational-fluid value
depending on the configuration of nucleus Bohr and Mottelson (1975). During
the evolution of deformation as a function of $R$, different configurations
appear, which may lead to moments of inertia corresponding to different
classical models. This suggests the importance of the quantum mechanical
calculation.

Another striking feature in Fig. 6 is ${\cal J}(R)\approx\mu_{\rm red}R^{2}$
at $R\geq R_{s}$, which is significantly smaller than ${\cal J}_{\rm rig}(R)$.
When the two nuclei are separated, the structure of the projectile and the
target nuclei are almost invariant with respect to $R$. Let us denote the
rigid-body moments of inertia of the projectile and the target with respect to
their own center-of-mass coordinates as ${\cal J}_{P}$ and ${\cal J}_{T}$,
respectively. The total rigid-body moments of inertia can be written as

${\cal J}(R)={\cal J}_{P}+{\cal J}_{T}+\mu_{\rm red}R^{2}.$ (32)

This equation is easy to prove in the rigid-body case. In the quantum
mechanical treatment of the present cases, there is another trivial result,
namely, ${\cal J}_{P}={\cal J}_{T}=0$. This is because both the projectile and
the target nuclei ($\alpha$ and 16O) are spherical in the ground state, thus,
the quantum mechanics requires the vanishing moments of inertia (See also Eq.
(24)). Assuming that Eq. (32) is valid for the quantum mechanical systems, we
find ${\cal J}(R)\approx\mu_{\rm red}R^{2}$ in the region where the projectile
and the target become spherical. This is also a consequence of the quantum
mechanics.

Next, let us show how the $B_{3}$ term influences the moments of inertia.
Figure 7 shows the results. For $B_{3}\neq 0$, the velocity dependence in the
mean-field potential violates the local Galilean invariance. The calculated
cranking moments of inertia are significantly smaller than the rigid-body
value ${\cal J}_{\rm rig}$ at $R\approx R_{e}$. They are also smaller than the
point-particle value $\mu_{\rm red}R^{2}$ at $R>R_{s}$. On the other hand, the
ASCC calculation includes the residual effects of the time-odd mean fields
which restore the local Galilean invariance, then, nicely reproduces both
rigid-body and point-particle values at $R\approx R_{e}$ and at $R>R_{s}$,
respectively. In fact, for these reaction systems, the ASCC moments of inertia
are insensitive to the $B_{3}$ value over the entire region.

Figure 6: (Color online) Calculated rotational moments of inertias for
$B_{3}=0$ as a function of relative distance $R$. The upper panel shows the
results for the system $\alpha$+16O, the lower panel shows the results for the
system 16O+16O. The solid (red), dashed (pink), dotted (blue) and dash-dotted
(green) curves indicate the results of ASCC, non-perturbative cranking
formula, rigid body approximation, and point-mass approximation, respectively.
Figure 7: (Color online) Same as Fig. 6, but for $B_{3}=25$ MeV fm5.

###  III.5 Impact on astrophysical $S$ factors

Figure 8: (Color online) Astrophysical $S$ factor calculated with $B_{3}=0$
for $\alpha+^{16}$O (upper panel) and for 16O+16O (lower panel), as a function
of incident energy $E_{\rm c.m.}$. (1) The point-particle approximation both
for the relative and the rotational degrees of freedom, $\mu^{\rm red}$ and
$\mu_{\rm red}R^{2}$, respectively. (2) The ASCC mass $M(R)$ for the relative
motion together with $\mu_{\rm red}R^{2}$ for the rotation. (3) ASCC $M(R)$
and rigid-body ${\cal J}_{\rm rig}(R)$. (4) ASCC $M(R)$ and ASCC ${\cal
J}(R)$. See text for details.  Figure 9: (Color online) Same as Fig. 8, but
calculated with the inertial masses for $B_{3}=25$ MeV fm5. The same
potentials as those of $B_{3}=0$ are used. See text for details.

In this section, we present the calculation of the astrophysical $S$ factor
for fusion reactions, using the potentials and the inertial masses obtained in
Sec. III.3 and III.4. The ASCC calculation provides us with the collective
Hamiltonian with the optimal reaction paths for 16O+$\alpha$ $\rightarrow$
20Ne and 16O+16O$\rightarrow^{32}$S. The Hamiltonian for the reaction is given
as that of Eq. (1) in which the moment of inertia $\mu_{\rm red}R^{2}$ is
replaced by ${\cal J}(R)$, and the reduced mass $\mu_{\rm red}$ by $M(R)$. We
investigate the dependence of the sub-barrier fusion cross sections on these
inertial masses.

The sub-barrier fusion cross section is calculated with the WKB approximation,
following the procedure in Ref. Reinhard  _et al._ (1984). The total sub-
barrier fusion cross section is given by the sum of all the partial waves with
angular moment $\ell$. Under the WKB approximation, the transmission
coefficient for the partial wave $\ell$ at incident energy $E_{\rm c.m.}$ is
given by

$\displaystyle T_{\ell}(E_{\rm c.m.})=[1+\exp(2I_{\ell})]^{-1},$ (33)

with

$\displaystyle I_{\ell}(E_{\rm c.m.})$ $\displaystyle=$
$\displaystyle\int_{a}^{b}dR\Big{\\{}2M(R)$ (34) $\displaystyle\times$
$\displaystyle\left(V(R)+\frac{\ell(\ell+1)}{2{\cal J}(R)}-E_{\rm
c.m.}\right)\Big{\\}}^{1/2},$

where $a$ and $b$ are the classical turning points on the inner and outer
sides of the barrier respectively. In addition to the potential $V(R)$, the
coordinate-dependent inertial mass for the relative motion $M(R)$ and for the
rotational moment of inertia ${\cal J}(R)$ are necessary for the calculation.
The term $\ell(\ell+1)/2{\cal J}(R)$ represents the centrifugal potential.

The fusion cross section is given by

$\displaystyle\sigma(E_{\rm c.m.})=\frac{\pi}{2\mu_{\rm red}E_{\rm
c.m.}}\sum_{\ell}(2\ell+1)T_{\ell}(E_{\rm c.m.}).$ (35)

For identical incident nuclei, 16O+16O, Eq. (35) must be modified according to
the proper symmetrization. Only the partial wave with even $L$ contribute to
the cross section as

$\displaystyle\sigma(E_{\rm c.m.})=\frac{\pi}{2\mu_{\rm red}E_{\rm
c.m.}}\sum_{\ell}[1+(-)^{\ell}](2\ell+1)T_{\ell}(E_{\rm c.m.}).$

Instead of $\sigma(E_{\rm c.m.})$, we show the astrophysical $S$ factor
defined by

$\displaystyle S(E_{\rm c.m.})=E_{\rm c.m.}\sigma(E_{\rm c.m.})\exp[2\pi
Z_{1}Z_{2}e^{2}/\hbar v],$ (37)

where $v$ is the relative velocity at $R\rightarrow\infty$. The astrophysical
$S$ factor is preferred for sub-barrier fusion because it removes the change
by tens of orders of magnitude present in the cross section due to the
penetration through the Coulomb barrier. The $S$ factor can reveal in a more
transparent way the influence of the nuclear structure and dynamics.

Figure 8 shows the calculated $S$ factor with $B_{3}=0$ for the scattering of
$\alpha+^{16}$O (upper panel) and 16O+16O (lower panel), respectively. In
order to clarify the effect of the inertial masses, we use the same potential
$V(R)$ for all the curves in each panel of Fig. 8, which is the ASCC potential
obtained for $B_{3}=0$ (solid lines in the bottom panels of Fig. 4 and 5).
Different curves show the calculations with different inertia masses. For the
case of $B_{3}=0$, the ASCC and the cranking inertial masses are similar to
each other. Thus, the calculation with the cranking inertial masses produces
the astrophysical $S$ factor similar to the ASCC result. Generally speaking,
the larger the inertial mass $M(R)$ for the relative motion is, the smaller
the astrophysical $S$ factor is. The opposite effect can be seen for the
rotational moments of inertia ${\cal J}(R)$. The larger moments of inertia
give the larger $S$ factor. The replacement of the constant mass $\mu_{\rm
red}$ by ASCC $M(R)$ gives a significant effect for the fusion cross section.
For the reaction system 16O+16O, we find a strong suppression of the fusion
cross section at $E_{\rm c.m.}<4$ MeV compared to the calculation with
$\mu_{\rm red}$, which is an order of magnitude or even larger.

Figure 9 shows the same $S$ factors with inertial masses calculated with
$B_{3}=25$ MeV fm5 for the scattering of $\alpha+^{16}$O (upper panel) and
16O+16O (lower panel), respectively. In order to see the effect of inertial
masses, we use the potential $V(R)$ obtained for $B_{3}=0$. Thus, the
differences between Figs. 8 and 9 come from the change in inertial masses.
Again, the change from the reduced mass $\mu_{\rm red}$ into $M(R)$ gives the
largest impact on the $S$ factor. The suppression effect of $M(R)$ is even
more significant than the $B_{3}=0$ case, because the $M(R)$ is larger for
finite values of $B_{3}$, as we can find in Fig. 4 and 5.

Although the BKN density functional provides us only with qualitative results,
these calculations reveal the important roles of the inertial masses, and
suggest significant influence of the inertial masses on the sub-barrier fusion
cross sections.

##  IV Summary

We calculated the ASCC inertial mass coefficients with respect to the
translational, relative and rotational motions. The numerical calculations are
performed using the FAM in the 3D coordinate space representation for the
reaction systems $\alpha$+$\alpha$ $\rightarrow$ 8Be, $\alpha+^{16}$O
$\rightarrow$ 20Ne and 16O$+^{16}$O $\rightarrow$ 32S. We investigated the
impact of time-odd mean-field potentials on the collective inertial masses. In
the presence of nucleonic effective mass, the cranking mass can neither
reproduce the total mass $Am$ of translational motion nor the correct reduced
mass $\mu_{\rm red}$ for the relative motion. In addition, the cranking
formula for the rotation produces neither the point-particle value $\mu_{\rm
red}R^{2}$ in the asymptotic region ($R\rightarrow\infty$), nor the rigid-body
value ${\cal J}_{\rm rid}$ near the equilibrium states. In contrast, the ASCC
masses is able to properly take into account the residual time-odd effects,
producing the total mass $Am$ for translation, the reduced mass $\mu_{\rm
red}$ for relative motion, and the point-particle value $\mu_{\rm red}R^{2}$
(the rigid-body value ${\cal J}_{\rm rid}$) after the scission (near the
equilibrium). We also calculated the astrophysical $S$ factors with these
microscopic inputs as the inertial masses of a reaction model. It turns out
that replacement of the reduced mass $\mu_{\rm red}$ by $M(R)$ obtained with
ASCC gives the largest impact on the $S$ factors. It could lead to a strong
suppression of the fusion cross section at low energy.

In the present study, a schematic BKN interaction plus time-odd terms are
adopted for a qualitative investigation. It is highly desired to use realistic
modern nuclear energy functionals. The paring correlation is also expected to
play a critical role in low-energy collective dynamics. These issues are
currently under investigation.

###### Acknowledgements.

This work is supported in part by JSPS KAKENHI Grant No. 18H01209, No.
19H05142, and No. 20K14458. This research in part used computational resources
provided through the HPCI System Research Project (Project ID: hp200069), and
by Multidisciplinary Cooperative Research Program in Center for Computational
Sciences, University of Tsukuba.

## References

  * Negele (1982) J. W. Negele, Rev. Mod. Phys. 54, 913 (1982).
  * Simenel (2012) C. Simenel, The European Physical Journal A 48, 152 (2012).
  * Maruhn  _et al._ (2014) J. A. Maruhn, P.-G. Reinhard, P. D. Stevenson, and A. S. Umar, Computer Physics Communications 185, 2195 (2014).
  * Nakatsukasa and Yabana (2005) T. Nakatsukasa and K. Yabana, Phys. Rev. C 71, 024301 (2005).
  * Nakatsukasa  _et al._ (2016) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Yabana, Rev. Mod. Phys. 88, 045004 (2016).
  * Sekizawa and Yabana (2016) K. Sekizawa and K. Yabana, Phys. Rev. C 93, 054616 (2016).
  * Umar  _et al._ (2016) A. S. Umar, V. E. Oberacker, and C. Simenel, Phys. Rev. C 94, 024605 (2016).
  * Sekizawa (2017) K. Sekizawa, Phys. Rev. C 96, 014615 (2017).
  * Umar  _et al._ (2017) A. S. Umar, C. Simenel, and W. Ye, Phys. Rev. C 96, 024625 (2017).
  * Ebata  _et al._ (2010) S. Ebata, T. Nakatsukasa, T. Inakura, K. Yoshida, Y. Hashimoto, and K. Yabana, Phys. Rev. C 82, 034306 (2010).
  * Stetcu  _et al._ (2011) I. Stetcu, A. Bulgac, P. Magierski, and K. J. Roche, Phys. Rev. C 84, 051309 (2011).
  * Hashimoto (2012) Y. Hashimoto, The European Physical Journal A 48, 55 (2012).
  * Hashimoto (2013) Y. Hashimoto, Phys. Rev. C 88, 034307 (2013).
  * Hashimoto and Scamps (2016) Y. Hashimoto and G. Scamps, Phys. Rev. C 94, 014610 (2016).
  * Bulgac  _et al._ (2016) A. Bulgac, P. Magierski, K. J. Roche, and I. Stetcu, Phys. Rev. Lett. 116, 122504 (2016).
  * Magierski  _et al._ (2017) P. Magierski, K. Sekizawa, and G. Wlazłowski, Phys. Rev. Lett. 119, 042501 (2017).
  * Scamps and Hashimoto (2019) G. Scamps and Y. Hashimoto, Phys. Rev. C 100, 024623 (2019).
  * Ring and Schuck (1980) P. Ring and P. Schuck,  _The nuclear many-body problems_ , Texts and monographs in physics (Springer-Verlag, New York, 1980).
  * Matsuo  _et al._ (2000) M. Matsuo, T. Nakatsukasa, and K. Matsuyanagi, Prog. Theor. Phys. 103, 959 (2000).
  * Hinohara  _et al._ (2007) N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Matsuyanagi, Prog. Theor. Phys. 117, 451 (2007).
  * Hinohara  _et al._ (2009) N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Matsuyanagi, Phys. Rev. C 80, 014305 (2009).
  * Nakatsukasa (2012) T. Nakatsukasa, Progress of Theoretical and Experimental Physics 2012, 01A207 (2012).
  * Marumori  _et al._ (1980) T. Marumori, T. Maskawa, F. Sakata, and A. Kuriyama, Progress of Theoretical Physics 64, 1294 (1980).
  * Klein  _et al._ (1991) A. Klein, N. R. Walet, and G. Do Dang, Annals of Physics 208, 90 (1991).
  * Dang  _et al._ (2000) G. D. Dang, A. Klein, and N. R. Walet, Physics Reports 335, 93 (2000).
  * Baranger and Kumar (1968a) M. Baranger and K. Kumar, Nuclear Physics A 110, 490 (1968a).
  * Baranger and Kumar (1968b) M. Baranger and K. Kumar, Nuclear Physics A 122, 241 (1968b).
  * Yuldashbaeva  _et al._ (1999) E. K. Yuldashbaeva, J. Libert, P. Quentin, and M. Girod, Physics Letters B 461, 1 (1999).
  * Bohr and Mottelson (1975) A. Bohr and B. R. Mottelson,  _Nuclear Structure, Vol. II_ (W. A. Benjamin, New York, 1975).
  * Wen and Nakatsukasa (2016) K. Wen and T. Nakatsukasa, Phys. Rev. C 94, 054618 (2016).
  * Wen and Nakatsukasa (2017) K. Wen and T. Nakatsukasa, Phys. Rev. C 96, 014610 (2017).
  * Thouless and Valatin (1962) D. J. Thouless and J. G. Valatin, Nucl. Phys. 31, 211 (1962).
  * Hinohara (2015) N. Hinohara, Phys. Rev. C 92, 034321 (2015).
  * Baran  _et al._ (2011) A. Baran, J. A. Sheikh, J. Dobaczewski, W. Nazarewicz, and A. Staszczak, Phys. Rev. C 84, 054321 (2011).
  * Inglis (1954) D. R. Inglis, Phys. Rev. 96, 1059 (1954).
  * Nakatsukasa  _et al._ (2007) T. Nakatsukasa, T. Inakura, and K. Yabana, Phys. Rev. C 76, 024318 (2007).
  * Inakura  _et al._ (2009) T. Inakura, T. Nakatsukasa, and K. Yabana, Phys. Rev. C 80, 044301 (2009).
  * Avogadro and Nakatsukasa (2011) P. Avogadro and T. Nakatsukasa, Phys. Rev. C 84, 014314 (2011).
  * Avogadro and Nakatsukasa (2013) P. Avogadro and T. Nakatsukasa, Phys. Rev. C 87, 014331 (2013).
  * Davies  _et al._ (1980) K. T. R. Davies, H. Flocard, S. Krieger, and M. S. Weiss, Nucl. Phys. A 342, 111 (1980).
  * Bonche  _et al._ (1976) P. Bonche, S. Koonin, and J. W. Negele, Phys. Rev. C 13, 1226 (1976).
  * Wen and Nakatsukasa (2020) K. Wen and T. Nakatsukasa, Frontiers in Physics 8, 16 (2020).
  * Reinhard  _et al._ (1984) P. G. Reinhard, J. Friedrich, K. Goeke, F. Grümmer, and D. H. E. Gross, Phys. Rev. C 30, 878 (1984).