# Leptogenesis driven by majoron

Eung Jin Chun<EMAIL_ADDRESS>Korea Institute for Advanced Study, Seoul
02455, South Korea  Tae Hyun Jung<EMAIL_ADDRESS>Particle Theory and
Cosmology Group, Center for Theoretical Physics of the Universe, Institute for
Basic Science (IBS), Daejeon, 34126, Korea

###### Abstract

We propose a leptogenesis scenario where baryon asymmetry generation is
assisted by the kinetic motion of the majoron, $J$, in the process of lepton-
number violating inverse decays of a right-handed neutrino, $N$. We
investigate two distinct scenarios depending on the sources of majoron kinetic
motion: 1) the misalignment mechanism, and 2) the kinetic misalignment
mechanism. The former case can naturally generate the observed baryon
asymmetry for the majoron mass $m_{J}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\,\mathrm{TeV}$ and
the right-handed neutrino’s mass $M_{N}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{11}\,\mathrm{GeV}$. However, an additional decay channel
of the majoron is required to avoid the overclosure problem of the majoron
oscillation. The later scenario works successfully for $m_{J}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}100\,\mathrm{keV}$, and $M_{N}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{9}\,\mathrm{GeV}$ while $M_{N}$ can be even far below the
temperature of the electroweak phase transition as long as sufficiently large
kinetic misalignment is provided. We also find that a sub-$100\,\mathrm{keV}$
majoron is a viable candidate for dark matter.

††preprint: CTPU-PTC-23-48

##  I Introduction

The seesaw mechanism stands out as one of the most compelling frameworks
explaining the lightness of left-handed neutrinos through the heaviness of
right-handed neutrinos Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._
(1979); Glashow (1980); Mohapatra and Senjanovic (1981); Shrock (1981);
Schechter and Valle (1980). The strength of the seesaw mechanism lies in the
natural realization of the baryon asymmetry of the universe through thermal
leptogenesis Fukugita and Yanagida (1986) (see, e.g. Ref. Davidson  _et al._
(2008) for a review). In this scenario, the CP asymmetric decay of right-
handed neutrinos generates lepton asymmetry which is transferred into the
baryon asymmetry via the weak sphaleron process. However, the amount of the CP
asymmetry is naturally proportional to the mass of the decaying particle
leading to the so-called Davidson-Ibarra bound: $M_{N}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{9}\,\mathrm{GeV}$ Davidson and Ibarra (2002).

As the Majorana mass of neutrinos breaks the $B-L$ number which is an anomaly-
free accidental symmetry in the standard model (SM), an intriguing question is
whether $U(1)_{B-L}$ symmetry breaking is spontaneous or explicit. If it is
broken spontaneously (which is what we assume in this paper), the heavy right-
handed neutrino mass is a consequence of spontaneously broken $U(1)_{B-L}$
symmetry which accompanies a pseudo-Goldstone boson called the majoron
Chikashige  _et al._ (1981); Gelmini and Roncadelli (1981).

In this work, we propose a scenario where a kinetic motion of the majoron,
denoted by $\dot{\theta}$, provides CP asymmetry in the inverse decay of $N$.
This is a realization of spontaneous baryogenesis Cohen and Kaplan (1987,
1988) in the context of the seesaw mechanism endowed with the majoron. Our
scenario can be further characterized by specifying the origin of
$\dot{\theta}$: 1) the (conventional) misalignment mechanism Preskill  _et
al._ (1983); Abbott and Sikivie (1983); Dine and Fischler (1983), and 2) the
kinetic misalignment mechanism Affleck and Dine (1985); Co and Harigaya
(2020); Co  _et al._ (2020).

A similar setup of our first case (conventional misalignment) has been studied
in Ref. Ibe and Kaneta (2015); Domcke  _et al._ (2020) which did not take into
account the dynamics coming from $N$, but considered an effective theory with
the five-dimensional Weinberg operator assuming sufficiently high seesaw
scale. In this case, the $B-L$ number is frozen around its decoupling
temperature $T_{W}\simeq 6\times 10^{12}\,\mathrm{GeV}$, and if the majoron
mass is $O(10^{9})\,\mathrm{GeV}$, the majoron oscillation starts around
$T_{\rm osc}\simeq T_{W}$, which leads to a successful leptogenesis. Unlike
the previous works, we include the effects coming from $N$ which generate the
$B-L$ number more efficiently compared to the processes involving the Weinberg
operator, and consequently, we find how light the majoron can be.

Our second case (kinetic misalignment) has many common features with Refs. Co
and Harigaya (2020); Co  _et al._ (2020); Domcke  _et al._ (2020); Co  _et
al._ (2021a); Harigaya and Wang (2021); Chakraborty  _et al._ (2022); Co  _et
al._ (2022a); Berbig (2023); Chao and Peng (2023) where the final baryon
asymmetry is generically determined at the decoupling temperature of the weak
sphaleron process or a $B-L$ number-changing process (in our case, it is
around $M_{N}$). Variations of axiogenesis augmented by the Weinberg operator
have also been suggested in Refs. Co  _et al._ (2021b); Kawamura and Raby
(2022); Co  _et al._ (2021c). In this work, we not only take into account the
dynamics of $N$, but also include completely different phenomenology that
comes from the majoron property.

##  II Basic features

The seesaw Lagrangian extended with a global $U(1)_{B-L}$ symmetry is written
as

$\displaystyle-\mathcal{L}_{\text{int}}=$
$\displaystyle\frac{1}{2}\sum_{I}y_{N_{I}}\Phi\bar{N}_{I}^{c}N_{I}+\sum_{\alpha,\,I}Y_{N,\alpha
I}\bar{l}_{\alpha}\tilde{H}N_{I}+h.c.,$ (1)

where $\Phi$ is a complex scalar field with the $B-L$ charge $+2$, $N_{I}$ are
the right-handed neutrinos with $I=1,2,3$, $l_{\alpha}$ are the left-handed
lepton doublets with $\alpha=e,\mu,\tau$, and $\tilde{H}\equiv
i\sigma_{2}H^{*}$ is the Higgs doublet coupling to up-type quarks and RHNs.
After the $B-L$ breaking, $\Phi$ is replaced by

$\displaystyle\Phi\to\frac{f_{J}}{\sqrt{2}}e^{iJ/f_{J}},$ (2)

where $J$ is the majoron field. We assume that the reheating temperature after
the inflationary epoch is lower than the $B-L$ phase transition temperature
and the radial mode of $\Phi$ does not affect the physics we discuss in the
following. However, if the reheating temperature is sufficiently high, the
universe undergoes the $B-L$ phase transition which may be first-order and the
radial mode can play a crucial role in the context of leptogenesis Huang and
Xie (2022); Dasgupta  _et al._ (2022); Chun  _et al._ (2023).

Going to the field basis by redefining all the fermionic fields $\psi\to
e^{i(B-L)_{\psi}\,\theta/2}\psi$ where $\theta\equiv J/f_{J}$ and
$(B-L)_{\psi}$ denotes the $B-L$ number of $\psi$ (e.g., $(B-L)_{N_{I}}=-1$),
removed is the $\theta$ dependence in all the Yukawa and scalar potential
terms, and there remains only the derivative coupling of the majoron:
$-\partial_{\mu}\theta\,J^{\mu}_{B-L}/2$ since $B-L$ is anomaly-free. In a
nonzero $\dot{\theta}\equiv d\theta/dt$ background, a perturbation in the
Hamiltonian density, $\dot{\theta}\,n_{B-L}/2$, is generated to act as an
external chemical potential. Thus, the source term of $B-L$ asymmetry in the
Boltzmann equation, proportional to $\dot{\theta}$, is generated in every term
violating the $B-L$ number. This is the origin of the CP violation required
for our leptogenesis.

Unlike the conventional thermal leptogenesis, our scenario generates the
lepton asymmetry via the so-called “wash-out” term which acts to “wash-in” the
CP asymmetry provided by the velocity of the majoron field $\dot{\theta}$.
Assuming a mass hierarchy between right-handed neutrinos: $M_{N_{1}}\ll
M_{N_{2}},\,M_{N_{3}}$, the “wash-in process” is mainly governed by the
lightest one $N_{1}$ (which is denoted by $N$ in the following). Then, the
evolution of the lepton number asymmetry density $n_{\Delta l}\equiv
n_{l}-n_{\bar{l}}$ is determined by (see Appendix. B for the derivation)

$\displaystyle\dot{n}_{\Delta l_{\alpha}}+3Hn_{\Delta
l_{\alpha}}=-\Gamma_{Y_{N,\alpha}}\Bigg{(}\frac{n_{\Delta
l_{\alpha}}}{n_{l_{\alpha}}^{\mathrm{(eq)}}}+\frac{n_{\Delta
H}}{n_{H}^{\mathrm{(eq)}}}-\dot{\theta}T^{2}\Bigg{)}+\cdots,$ (3)

where $n_{\Delta H}=n_{H}-n_{\bar{H}}$, $n_{X}^{\mathrm{(eq)}}$ is the
equilibrium number density of $X$, and the interaction rate
$\Gamma_{Y_{N,\alpha}}$ controlled by the neutrino Yukawa coupling $Y_{N}$ is

$\displaystyle\Gamma_{Y_{N,\alpha}}$
$\displaystyle=n_{N}^{\mathrm{(eq)}}\frac{K_{1}(z)}{K_{2}(z)}\Gamma_{N\to
l_{\alpha}H},$ (4)

with $z=M_{N}/T$, $\Gamma_{N\to l_{\alpha}H}\simeq|Y_{N,\alpha
1}|^{2}M_{N}/16\pi$ (assuming $m_{N}\gg m_{l_{\alpha}},m_{H}$) and $K_{1,2}$
being the modified Bessel functions. We neglect the scattering processes of
$\Delta L=1$ such as $NQ_{3}\leftrightarrow Lt$ since the effect of the
scattering is subdominant to the inverse decay term as in the conventional
thermal leptogenesis.

Note that the interaction involved in Eq. (3) is the inverse decay, and we do
not have a decay term at the tree level. One may wonder about the effect
coming from the helicity asymmetry $n_{\Delta N}=n_{N_{+}}-n_{N_{-}}$ where
$N_{+}$ and $N_{-}$ denote $N$ with positive and negative helicity,
respectively. If the decay term with $n_{\Delta N}$ existed, it would cancel
the $\dot{\theta}$ contribution since $n_{\Delta N}$ is also shifted
proportionally to $\dot{\theta}$, and the helicity of $N$ can be identified by
the chirality (and thus the lepton number) in the $M_{N}\to 0$ limit. However,
although the helicity asymmetry is indeed generated proportionally to
$\dot{\theta}$ at high temperature $T>\gg M_{N}$ (see Appendix. A for detail),
we find that $n_{\Delta N}$ dependence does not appear in Eq. (3) because the
decay rate of $N_{\pm}\to l_{\alpha}H$ is the same as that of
$N_{\pm}\to\bar{l}_{\alpha}\bar{H}$ independently of $N$’s momentum. Remark
that we consider the case where the CP-violating decay of $N$ is absent or
sufficiently suppressed.

##  III Leptogenesis driven by majoron

We focus on the inverse decay which “washes in” the CP asymmetry provided by
$\dot{\theta}$ to the lepton sector. Then, it is transferred to the baryon
asymmetry by the electroweak sphaleron. To maximize the efficiency of the
wash-in process, the inverse decay is required to be in thermal equilibrium
which happens in the so-called strong wash-out regime satisfying
$\Gamma_{N}>H(T_{N})$ with $T_{N}=M_{N}$. Therefore, it is important to
determine the temperature range at which the weak sphaleron rate and wash-in
rate exceed the Hubble expansion rate.

For the weak sphaleron rate, there is a suppression factor of $\exp(-E_{\rm
sph}/T)$ where $E_{\rm sph}$ is the energy of the sphaleron configuration that
rapidly increases in the broken phase proportionally to the Higgs vev $\langle
h\rangle$ at $T$ Kuzmin  _et al._ (1985); D’Onofrio  _et al._ (2014).
Therefore, it gets highly suppressed after the electroweak phase transition,
so we consider it to be turned off at $T<T_{\rm EW}\simeq 130\,\mathrm{GeV}$.
On the other hand, when $\langle h\rangle=0$ at high temperature, the
sphaleron rate is approximately given by $\alpha_{W}^{5}T$, and it gets
decoupled at $T>T_{\rm ws}\simeq 2.5\times 10^{12}\,\mathrm{GeV}$.

Since we use the wash-in term to generate lepton asymmetry, we _have to_ be in
the strong wash-out regime; $\Gamma_{N}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}H(T_{N})$. Taking
the usual parameter of the effective neutrino mass

$\displaystyle\tilde{m}_{\nu}\equiv\sum_{\alpha}|Y_{N,\alpha
1}|^{2}\frac{v_{h}^{2}}{2M_{N}},$ (5)

the strong wash-out condition is $K\equiv\tilde{m}_{\nu}/\mathrm{meV}>1$. For
the atmospheric neutrino mass scale of $\tilde{m}_{\nu}=0.05\,\mathrm{eV}$
($K=50$) and $|Y_{Y,11}|^{2}\simeq|Y_{Y,21}|^{2}\simeq|Y_{Y,31}|^{2}$, the
inverse decay rate is active when

$\displaystyle H<\gamma^{\rm
ID}_{Y_{N},\alpha}=\frac{n_{N}^{\mathrm{(eq)}}}{n_{l_{\alpha}}^{\mathrm{(eq)}}}\frac{K_{1}(z)}{K_{2}(z)}\Gamma_{N\to
l_{\alpha}H}$ $\displaystyle\Leftrightarrow\qquad M_{N}/z_{\rm
fo}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}T\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}M_{N}/z_{\rm in}$ (6)

where $z_{\rm in}\simeq 0.7$ and $z_{\rm fo}\simeq 10$. In order to see the
parametric dependence, we keep $z_{\rm in}$ and $z_{\rm fo}$ unless we
numerically evaluate. Then, the baryon asymmetry generation assisted by the
majoron is determined at $T_{B-L}=M_{N}/z_{\rm fo}$.

When the weak sphaleron and the wash-in processes are strong enough 111 Here,
a “strong enough” reaction means not only to have a reaction rate greater than
the Hubble rate but also to be greater than the inverse time scale of changing
$\dot{\theta}$, $(\Delta
t)^{-1}_{\dot{\theta}}\simeq\ddot{\theta}/\dot{\theta}$. , the baryon number
settles down to the equilibrium value which we parameterize as

$\displaystyle n_{B}^{\rm(eq)}=\frac{c_{B}}{6}\dot{\theta}T^{2},$ (7)
$\displaystyle n_{L}^{\rm(eq)}=\frac{c_{L}}{6}\dot{\theta}T^{2},$ (8)

where $n_{B,\,L}$ are the number density of the baryon and lepton numbers
accounting only for SM fermions. $c_{B}$, $c_{L}$ and $c_{B-L}=c_{B}-c_{L}$
for different temperature range are summarized in appendix. C.

###  III.1 (Conventional) Misalignment mechanism

Let us first consider the initial condition of $\dot{\theta}_{0}=0$ and
$\theta_{0}\neq 0$ at the high temperature $T_{0}$ (which should not be
greater than the critical temperature $\sim f_{J}$ above which the
$U(1)_{B-L}$ symmetry is restored). The classical amplitude $\theta$ starts
coherent oscillation when the Hubble rate becomes comparable to its mass
$m_{J}$. The equation of motion can be written as

$\displaystyle\ddot{\theta}+3H\dot{\theta}=-\frac{1}{f_{J}^{2}}V^{\prime}(\theta)\simeq-
m_{J}^{2}\sin(\theta)$ (9)

where we assumed $V(\theta)=m_{J}^{2}f_{J}^{2}(1-\cos\theta)$ comes from an
explicit breaking term of $B-L$ symmetry in the potential:
$V(\Phi)=\Phi^{n+4}/\Lambda^{n}+h.c.$. When the initial misalignment angle of
$\theta$ is not close to $\pi$, one can approximate
$V(\theta)\simeq\frac{1}{2}m_{J}^{2}f_{J}^{2}\theta^{2}$, and obtain

$\displaystyle\theta(t)\simeq\theta_{0}\Gamma(5/4)\left(\frac{2}{m_{J}t}\right)^{1/4}J_{1/4}(m_{J}t),$
(10)

in the radiation-dominated universe ($H\simeq 1/2t$). Here, $J_{\alpha}$ is
the Bessel function of the first kind.

The behavior of Eq. (10) can be understood separately before and after the
oscillation temperature, $T_{\rm osc}$, which is defined by $3H(T_{\rm
osc})=m_{J}$;

$\displaystyle T_{\rm osc}=5\times
10^{8}\,\mathrm{GeV}\left(\frac{g_{*}}{100}\right)^{-1/4}\left(\frac{m_{J}}{\mathrm{GeV}}\right)^{1/2},$
(11)

where $g_{*}$ is the effective number of relativistic degrees of freedom. By
using $J_{1/4}(x)\simeq\Gamma(5/4)^{-1}(x/2)^{1/4}(1-x^{2}/5)$ for $x\ll 1$,
and $J_{1/4}(x)\sim(2/\pi x)^{1/2}\cos(x-3\pi/8)$ for $x\gg 1$, we obtain an
approximate form of

$\displaystyle\dot{\theta}(T)\sim\begin{cases}\theta_{0}m_{J}\left(\frac{T_{\rm
osc}}{T}\right)^{2}&\text{for $T>T_{\rm osc}$,}\\\
\theta_{0}m_{J}\left(\frac{T}{T_{\rm osc}}\right)^{3/2}\cos(m_{J}t)&\text{for
$T<T_{\rm osc}$,}\end{cases}$ (12)

where we used $H\simeq\frac{m_{J}}{3}(\frac{T}{T_{\rm osc}})^{2}$ and
neglected order one factors (including signs) and phase shift. As $T$
decreases from a high temperature, $\dot{\theta}$ increases and gets maximized
around $T_{\rm osc}$. Then, it starts oscillation with its amplitude being
red-shifted as $(T/T_{\rm osc})^{3/2}$. Therefore, one can expect that the
baryon asymmetry generation will be maximized when $T_{\rm osc}$ coincides
with $T_{\rm B-L}$.

Figure 1: Evolutions of $Y_{B-L}$ for different values of $m_{J}$ and
$M_{N}=10^{11}\,\mathrm{GeV}$ (upper panel) and $10^{12}\,\mathrm{GeV}$ (lower
panel). For a fixed $M_{N}$, $Y_{B-L}$ is maximized when the parameters
satisfy $T_{\rm osc}\simeq T_{B-L}$. The horizontal dashed line corresponds to
the $Y_{B-L}$ value required for the observed baryon asymmetry.

To investigate further detail, let us, first, consider the case when $T_{\rm
osc}<T_{B-L}$. Since the weak sphaleron and the wash-in rates are strong
enough for $T>T_{B-L}$, $n_{B-L}$ follows the equilibrium values (7) and (8)
adiabatically, and gets frozen before the oscillation starts. Therefore, we
can estimate

$\displaystyle Y_{B-L}$
$\displaystyle\simeq\left.\frac{1}{s(T)}\frac{1}{6}c_{B-L}\dot{\theta}(T)T^{2}\right|_{T=T_{B-L}}$
$\displaystyle\sim\frac{c_{B-L}\theta_{0}}{g_{*}(T_{B-L})}\frac{m_{J}}{T_{B-L}}\left(\frac{T_{\rm
osc}}{T_{B-L}}\right)^{2},$ (13)

where $s(T)=\frac{2\pi^{2}}{45}g_{*}T^{3}$ is the total entropy density of the
background plasma with the effective number of relativistic degrees of freedom
$g_{*}$. As $T_{\rm osc}\propto m_{J}^{1/2}$ (see Eq. (11)), we have the
proportionality of $Y_{B-L}\propto m_{J}^{2}$ for $T_{\rm osc}<T_{B-L}$ when
$T_{B-L}$ is fixed.

On the other hand, when $T_{\rm osc}>T_{B-L}$, the oscillation starts first.
Since the oscillation time scale, $\Delta t_{\rm osc}\simeq m_{J}^{-1}$
becomes shorter than the Hubble time scale, it is not guaranteed for the $B-L$
number to settle down at the equilibrium value. Assuming that
$m_{J}>\Gamma_{Y_{N,\alpha}}>H$, $Y_{B-L}$ can be estimated as

$\displaystyle Y_{B-L}(T)$
$\displaystyle\sim\frac{c_{B-L}\Gamma_{Y_{N,\alpha}}}{Ts(T)}\int
dt\,\dot{\theta}$
$\displaystyle\sim\frac{c_{B-L}\theta_{0}}{g_{*}(T)}\frac{\Gamma_{Y_{N,\alpha}}}{T^{4}}\left(\frac{T}{T_{\rm
osc}}\right)^{3/2}\sin(m_{J}t).$ (14)

At the temperature around $T_{B-L}$, $Y_{B-L}$ is frozen during the
oscillation. Taking the approximation of $\Gamma_{Y_{N,\alpha}}\simeq
H(T_{B-L})T_{B-L}^{3}$, we obtain

$\displaystyle|Y_{B-L}|$ $\displaystyle\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{c_{B-L}\theta_{0}}{g_{*}(T_{B-L})}\frac{H(T_{B-L})}{T_{B-L}}\left(\frac{T_{B-L}}{T_{\rm
osc}}\right)^{3/2}$ (15)

which shows the proportionality of $Y_{B-L}\propto m_{J}^{-3/4}$ for a fixed
$T_{B-L}$.

These features can be seen in Fig. 1 where we depict the evolution of
$Y_{B-L}$ as a function of $z=M_{N}/T$ by solving the full set of the
Boltzmann equations summarized in Appendix. B. Considering two different
values of $M_{N}=10^{11}\,\mathrm{GeV}$ (upper panel) and
$10^{12}\,\mathrm{GeV}$ (lower panel), we show the dependence of $Y_{B-L}$ on
the values of $m_{J}$ which is scanned around $T_{\rm osc}\sim T_{\rm B-L}$.
As we discussed previously, the frozen value of $Y_{B-L}$ is maximized when
$T_{\rm osc}\simeq T_{\rm B-L}$.

From the previous estimations, we conclude that $Y_{B-L}$ is bounded from
above for a fixed $T_{B-L}$, and the maximized value at $T_{B-L}\simeq T_{\rm
osc}$ is given by

$\displaystyle Y_{B-L}^{\rm max}(T_{B-L})\sim
10^{-9}\,c_{B-L}\theta_{0}\left(\frac{100}{g_{*}(T_{B-L})}\right)\left(\frac{T_{B-L}}{10^{10}\,\mathrm{GeV}}\right),$
(16)

which we obtain from Eq. (13) or Eq. (15) taking $T_{\rm osc}\simeq T_{B-L}$
(and Eq. (11) to remove $m_{J}$ dependence), and including $O(10)$ factor that
arises from our numerical solution of the Boltzmann equations. This implies
that, for $Y_{B}\simeq\frac{28}{79}Y_{B-L}\simeq 8.7\times 10^{-11}$, we need

$\displaystyle T_{B-L}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{10}\,\mathrm{GeV}\quad\Rightarrow\quad M_{N}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{11}\,\mathrm{GeV}\left(\frac{z_{\rm fo}}{10}\right),$ (17)

considering $\theta_{0}=O(1)$. For the rigorous results, we solve the full
Boltzmann equations, and show the final value of $Y_{B}$ in the plane of
$m_{J}$ and $M_{N}$ in Fig. 2 taking $z_{\rm fo}=10$ and $\theta_{0}=1$. In
the plot, we also show the lifetime of the majoron which is determined by its
dominant decay channel $J\to\nu\nu,\,\bar{\nu}\bar{\nu}$, and thus has the
decay rate proportional to $m_{\nu}^{2}/f_{J}^{2}$. One can see that the
majorons are fairly long-lived in the parameter region of our interest. This
causes a serious problem of overclosing the universe.

The energy density of the majoron oscillation is indeed given by

$\displaystyle\frac{\rho_{\rm osc}(T_{\rm osc})}{s(T_{\rm
osc})}\simeq\frac{\theta_{0}^{2}m_{J}^{2}f_{J}^{2}}{s(T_{\rm osc})}$ (18)
$\displaystyle\sim 0.4\,\mathrm{eV}\,\theta_{0}^{2}\,\left(\frac{10}{z_{\rm
fo}}\right)\left(\frac{0.1}{y_{N}}\right)^{2}\left(\frac{T_{\rm
osc}}{T_{B-L}}\right)\left(\frac{M_{N}}{4\times
10^{8}\,\mathrm{GeV}}\right)^{3},$

which is unacceptably large for $M_{N}>10^{11}{\rm GeV}$.

Figure 2: The expected value of $Y_{B}$ is depicted as a function of $m_{J}$
and $M_{N}$ for $\theta_{0}=1$. Dashed lines show the lifetime of majoron in
the minimal setup.

To circumvent this problem, one may introduce an additional coupling of the
majoron, such as $(J/f_{J})F^{\mu\nu}\tilde{F}_{\mu\nu}$ whose UV completion
can be done by adding vector-like charged leptons (while the $B-L$ charge of
vector-like leptons should be assigned chirally). This can drastically
increase the decay rate to make the majoron decay away to two photons before
Big Bang nucleosynthesis (BBN) but after the leptogenesis era. Introducing
this operator does not change the previous estimation of the leptogenesis
part.

###  III.2 Kinetic misalignment mechanism

Now, let us consider the case when $\dot{\theta}\neq 0$ at $T\gg T_{\rm osc}$.
This can be realized by the so-called kinetic misalignment mechanism Affleck
and Dine (1985); Co and Harigaya (2020); Co  _et al._ (2020). Assuming a
sufficiently flat potential of $B-L$ breaking field $\Phi$, its radial mode
$\phi$ can be stuck at a large field value due to the Hubble friction. When
the Hubble rate becomes comparable to the curvature of the potential, $\phi$
starts rolling down, and an explicit breaking of $B-L$ symmetry, which also
generates $m_{J}$, drives the motion along the majoron direction.

We treat the initial majoron motion as a free parameter since it strongly
depends on the potential shape of $\Phi$. Therefore, our starting point is
taking nonzero $\dot{\theta}(T_{0})=\dot{\theta}_{0}$ at a sufficiently high
temperature $T_{0}$ (but still much lower than $f_{J}$ to avoid thermal
friction). As in Ref. Co  _et al._ (2021a), we take a free parameter
$Y_{\theta}\equiv f_{J}^{2}\dot{\theta}_{0}/s(T_{0})$ which is approximately
conserved throughout the leptogenesis process, i.e.
$f_{J}^{2}\dot{\theta}(T)/s(T)\simeq Y_{\theta}$ or $\dot{\theta}(T)\simeq
Y_{\theta}s(T)/f_{J}^{2}$.

If $T_{\rm EW}<T_{B-L}$, the $B-L$ number is frozen before the electroweak
phase transition. Then, the $B-L$ number is re-distributed, and the baryon
number is finally frozen at $T_{\rm EW}$ as

$\displaystyle
Y_{B}=\frac{28}{79}Y_{B-L}^{\rm(eq)}(T_{B-L})=\frac{14}{237}c_{B-L}Y_{\theta}\left(\frac{y_{N}}{\sqrt{2}z_{\rm
fo}}\right)^{2},$ (19)

where we took the replacement: $\dot{\theta}(T)=Y_{\theta}\,s(T)/f_{J}^{2}$.

On the other hand, if $T_{\rm EW}>T_{B-L}$, the baryon number is frozen at the
electroweak phase transition during the $B-L$ number is changing. Therefore,
the baryon asymmetry is given by the equilibrium value at $T_{\rm EW}$:

$\displaystyle Y_{B}=Y_{B}^{\rm(eq)}(T_{\rm
EW})=\frac{1}{6}c_{B}Y_{\theta}\left(\frac{T_{\rm EW}}{f_{J}}\right)^{2}.$
(20)

which is valid only for $M_{N}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}z_{\rm in}\,T_{\rm EW}$. In this case,
the decay processes such as $N\leftrightarrow lH$ or
$N\leftrightarrow\bar{l}\bar{H}$ may be prohibited if $M_{N}$ is lighter than
the Higgs mass (including thermal corrections). Instead, $H\leftrightarrow lN$
and $\bar{H}\leftrightarrow\bar{l}N$ become responsible for the main $B-L$
number-changing process. Nevertheless, due to the dependence of $z_{\rm
in}\approx(K/2)^{-1/3}$ for large $K$, $z_{\rm in}$ is insensitive to the
detailed dependences, and the validity of Eq. (20) requires $M_{N}$ greater
than $O(10)\,\mathrm{GeV}$ even when we allow a tuning in the $Y_{N,\alpha 1}$
structure.

When $M_{N}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}z_{\rm in}T_{\rm EW}$, there exists an
additional suppression factor of $\gamma^{\rm ID}(T_{\rm EW})/H(T_{\rm EW})$;

$\displaystyle Y_{B}\sim\frac{1}{6}c_{B}Y_{\theta}\left(\frac{T_{\rm
EW}}{f_{J}}\right)^{2}\left(\frac{\gamma^{\rm ID}(T_{\rm EW})}{H(T_{\rm
EW})}\right),$ (21)

where

$\displaystyle\frac{\gamma^{\rm ID}(T_{\rm EW})}{H(T_{\rm EW})}\simeq
4\left(\frac{K}{50}\right)\left(\frac{g_{*}}{100}\right)^{-1/2}\left(\frac{M_{N}}{T_{\rm
EW}}\right)^{3}.$ (22)

Therefore, $Y_{\theta}$ needs to be even greater to compensate for this
suppression factor.

For the validity of our consideration, the kinetic energy density of the
majoron needs to be smaller than the radiation energy density at least when
the $B-L$ number or $B$ number is frozen 222 Our mechanism may work even
during the kination domination with an appropriate change of the Hubble rate,
which needs a further scrutiny. In this article, we limit ourselves to the
radiation domination which does not require too small $y_{N}$. . This implies
$f_{J}^{2}\dot{\theta}(T_{*})^{2}/2<\pi^{2}g_{*}(T_{*})T_{*}^{4}/30$, where
$T_{*}=\max(T_{B-L},T_{\rm EW})$. For $T_{B-L}>T_{\rm EW}$, using the
condition (19), we obtain

$\displaystyle y_{N}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{-7}\left(\frac{g_{*}}{100}\right)^{1/2}\left(\frac{z_{\rm
fo}}{10}\right)\left(\frac{1}{c_{B-L}}\right),$ (23)

where we take $Y_{B}\simeq 8.7\times 10^{-11}$ for the observed baryon
asymmetry Aghanim  _et al._ (2020). Similarly, by using (22) when
$T_{B-L}<T_{\rm EW}$, we obtain the lower bound of $M_{N}$

$\displaystyle
M_{N}>2\,\mathrm{GeV}\left(\frac{f_{J}}{10^{6}\,\mathrm{GeV}}\right)^{1/3}\left(\frac{g_{*}}{100}\right)^{1/3}\left(\frac{1}{c_{B}}\right)^{1/3}\left(\frac{50}{K}\right)^{1/3}.$
(24)

As we will show below, $f_{J}$ needs to be greater than $10^{6}\,\mathrm{GeV}$
to avoid the constraint from CMB and BAO analysis Audren  _et al._ (2014);
Enqvist  _et al._ (2020); Nygaard  _et al._ (2021); Alvi  _et al._ (2022);
Simon  _et al._ (2022), so this puts the lower bound $M_{N}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}2\,\mathrm{GeV}$.

On the other hand, if the temperature $T_{0}$ when $\dot{\theta}$ is initially
generated is large compared to $T_{*}$, there exists a temporary kination
domination (KD) era during which the kinetic energy of $\theta$ dominates the
universe. Although this would not change our baryogenesis analysis, there can
be a significant enhancement of gravitational waves during the transition to
KD from radiation domination (RD) or matter domination (MD) after the
inflation and vice versa Co  _et al._ (2022b); Gouttenoire  _et al._ (2021a,
b); Harigaya  _et al._ (2023).

It is also remarkable that the initial kinetic misalignment required for
successful leptogenesis can generate the right amount of dark matter abundance
from the coherent oscillation of the majoron occurring at a later time. As
$\dot{\theta}$ gets redshifted as $s(T)\sim T^{3}$, the kinetic energy density
of majoron scales as $f_{J}^{2}\dot{\theta}^{2}/2\propto T^{6}$, and becomes
eventually comparable to the potential barrier $m_{J}^{2}f_{J}^{2}$. Once it
happens, the majoron gets trapped in the potential. The trapping temperature
can be estimated by $f_{J}^{2}\dot{\theta}^{2}/2\simeq m_{J}^{2}f_{J}^{2}$
leading to the relation

$\displaystyle s(T_{\rm trap})\simeq\frac{m_{J}f_{J}^{2}}{Y_{\theta}}.$ (25)

Then, the trapped majoron can either 1) start oscillation immediately
($m_{J}>3H(T_{\rm trap})$, i.e. $T_{\rm osc}>T_{\rm trap}$), or 2) start
oscillation after a while ($m_{J}<3H(T_{\rm trap})$, i.e. $T_{\rm osc}<T_{\rm
trap}$). For the first case, the oscillation energy density is frozen as

$\displaystyle\frac{\rho_{\rm osc}}{s}\sim\frac{m_{J}^{2}f_{J}^{2}}{s(T_{\rm
trap})}\sim m_{J}Y_{\theta}.$ (26)

On the other hand, if $T_{\rm osc}<T_{\rm trap}$, the majoron is stuck at an
$O(1)$ intermediate value $\theta=\theta_{0}$, and starts oscillation at
$T_{\rm osc}$. The abundance in this case is given by

$\displaystyle\frac{\rho_{\rm
osc}}{s}\simeq\frac{\theta_{0}^{2}m_{J}^{2}f_{J}^{2}}{s(T_{\rm osc})},$ (27)

from which one finds a fixed relation between $m_{J}$ and $f_{J}$ to explain
the observed dark matter abundance.

In Fig. 3, we show the parameter space (white) that is consistent with the
observed baryon asymmetry and dark matter density at present Aghanim  _et al._
(2020):

$\displaystyle Y_{B}\simeq 8.7\times 10^{-11},\quad\frac{\rho_{J}}{s}\simeq
0.44\,\mathrm{eV}.$ (28)

At each point in the $(m_{J},f_{J})$ plane, these two conditions fix the
parameters $Y_{\theta}$, and $y_{N}$ (gray lines) or $M_{N}$ (colored lines).
Here, we take $z_{\rm fo}\simeq 10$. For the consistency of the scenario, we
requires the following conditions:

  * 1.

The lower limit of the majoron lifetime $\tau_{J}>250\,{\rm Gyr}$ to avoid the
constraint from CMB and BAO analysis Audren  _et al._ (2014); Enqvist  _et
al._ (2020); Nygaard  _et al._ (2021); Alvi  _et al._ (2022); Simon  _et al._
(2022) (see the black line).

  * 2.

When $\Gamma_{HL\leftrightarrow JN}>H$, thermal majorons can be produced
efficiently when they are relativistic and thus their relic energy density may
become too large. The production rate can be approximated as
$\Gamma_{HL\leftrightarrow JN}\approx|Y_{N,\,\alpha 1}|^{2}|y_{N}|^{2}T/8\pi$
for $T>M_{N}$ while $\Gamma_{HL\leftrightarrow JN}$ for $T<M_{N}$ is
negligible due to the Boltzmann suppression and the $T^{2}/f_{J}^{2}$
suppression. Then, the thermalization condition is met for
$1<T/M_{N}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}50\,y_{N}^{2}(K/50)(g_{*}/100)^{-1/2}$. Therefore, we demand
$y_{N}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}0.14\,(K/50)^{-1/2}(g_{*}/100)^{1/4}$ to avoid the
overproduction (see the purple line).

Consequently, we obtain $m_{J}<100\,\mathrm{keV}$ (see the purple line), and
$f_{J}>10^{6}\,\mathrm{GeV}$ (see the black line).

Figure 3: Available parameter space for the kinetic misalignment case. The
gray region is excluded by the overclosure of the universe (above the blue
line), by the condition for the leptogenesis (below the red line), by the
thermal relic of majorons (right to the purple line), by the kination
domination (left to the gray line), or by the constraints from CMB and BAO
(below the black line). Colored lines and vertical gray lines inside the
allowed region (white) show the required values of $M_{N}$ and $y_{N}$,
respectively.

##  IV Discussions on phenomenology

Searching for heavy neutral leptons (HNLs) like $N$ is one of the most active
research fields, and can test the low $M_{N}$ region of our second scenario
(see Chun  _et al._ (2019); Abdullahi  _et al._ (2023) and references
therein). For $M_{N}\approx 0.2-6\,\mathrm{GeV}$, rare meson decays put the
bounds like $K/50\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}30$ for $M_{N}<2\,\mathrm{GeV}$ and
$K/50\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{3}$ for $M_{N}=2-6\,\mathrm{GeV}$ Chun  _et al._ (2019).
The mass range of $M_{N}=O(10)\,\mathrm{GeV}$, can be tested at future
colliders such as FCC-ee and FCC-hh if $K/50\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}O(10)$ Abdullahi
_et al._ (2023).

On the other hand, a direct test of majoron is very challenging. All the
majoron couplings to the SM particles involve $Y_{N}^{2}$ and thus are
generically suppressed by $m_{\nu}/f_{J}$ or $m_{\nu}/v_{h}$. This behavior
can be seen from the Lagrangian (1) where $\Phi$ and $N_{I}$ completely
decouple from the SM sector in the limit of $m_{\nu}\to 0$ corresponding to
$Y_{N,\alpha I}\to 0$. The $m_{\nu}$-suppression at higher loop order can be
explicitly seen in Ref. Heeck and Patel (2019). For instance, majoron to
photon-photon coupling can be generated at two-loop order, but it is very
challenging to leave an observable signature for small $m_{J}$ because of the
$m_{\nu}$ suppression (see Appendix. D for details).

The $m_{\nu}$-suppression makes it (almost) impossible to test the model
except for the high $K$ limit discussed in the beginning of this section.
Although the supernova constraints for $m_{J}<100\,\mathrm{MeV}$ seem strong
in terms of the coupling strength, $g\sim m_{\nu}/f_{J}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{-11}(m_{J}/100\,\mathrm{MeV})^{-1}$ Choi  _et al._ (1988);
Choi and Santamaria (1990); Chang and Choi (1994); Akita  _et al._ (2022);
Fiorillo  _et al._ (2023), the constraint in terms of its decay constant is
only $f_{J}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10\,\mathrm{GeV}\,(m_{\nu}/0.1\,\mathrm{eV})(m_{J}/100\,\mathrm{MeV})$.
Neutrinoless double beta decay experiments also put constraints on the majoron
coupling to $\nu_{e}$ via searching for majoron-emitting channel as
$g_{ee}\sim m_{\nu,ee}/f_{J}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{-5}$ Arnold  _et al._ (2018);
Barabash  _et al._ (2018); Kharusi  _et al._ (2021); Agostini  _et al._
(2022); Azzolini  _et al._ (2023), which becomes stronger than the supernova
bound at $m_{J}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}100\,\mathrm{eV}$.

Despite the intrinsic suppression factor in the coupling strength, the majoron
_dark matter scenarios_ can have some interesting impact in the CMB and BAO
observation Audren  _et al._ (2014); Enqvist  _et al._ (2020); Nygaard  _et
al._ (2021); Alvi  _et al._ (2022); Simon  _et al._ (2022). This puts the
limit $\tau_{J}>250\,{\rm Gyr}$ which was taken into account in the previous
section. Although excluded in our scenario, the majorana mass above MeV is
severely constrained by the measurements of neutrino flux Agostini  _et al._
(2021); Abe  _et al._ (2022); Olivares-Del Campo  _et al._ (2018); Palomares-
Ruiz (2008); Frankiewicz (2016); Bays  _et al._ (2012); Abe  _et al._ (2021);
Abbasi  _et al._ (2021, 2023); Argüelles  _et al._ (2022); Albert  _et al._
(2017) (see Ref. Akita and Niibo (2023) for the analysis in the majoron
parameters).

##  V Summary

In this work, we have investigated a leptogenesis scenario where the lepton
asymmetry is generated via the decay and inverse decay of the lightest right-
handed neutrinos under the CPT violation given by a background majoron motion,
$\dot{\theta}$. To generate nonzero $\dot{\theta}$, we have considered two
scenarios. One is generating it via the conventional misalignment mechanism,
and the other is generating it via the kinetic misalignment mechanism.

For the misalignment scenario, we find that our scenario successfully
generates baryon asymmetry if $M_{N}$ is greater than $10^{11}\,\mathrm{GeV}$.
However, the energy density of the majoron oscillation becomes greater than
the observed dark matter abundance while its lifetime is the order of the age
of the universe. The simplest way to avoid this problem is introducing an
additional interaction such as $JF\tilde{F}$ to make the lifetime much
shorter.

On the other hand, the leptogenesis scenario sourced by the kinetic
misalignment can be realized for $1\,\mathrm{GeV}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}M_{N}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{9}\,\mathrm{GeV}$ and $m_{J}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}100\,\mathrm{keV}$
while the majoron oscillation can be a viable candidate of the dark matter.
Thus, this scenario can be (partially) tested by searching for heavy neutral
leptons. However, the majoron lighter than $100\,\mathrm{keV}$ is hardly
testable as we discussed.

Acknowledgement: This work was supported by IBS under the project code
IBS-R018-D1.

##  Appendix A Majorana fermion and an external chemical potential

In the background of the kinetic motion of majoron field $\dot{\theta}$, the
dispersion relation of a Majorana fermion behaves differently from that of a
Dirac or Weyl fermion due to the Majorana mass term breaking the $U(1)$
symmetry. The Lagrangian of a Majorana fermion $\psi$ whose mass is generated
after the spontaneous breaking of the global $U(1)$ symmetry is

$\displaystyle{\cal L}_{M}$ $\displaystyle={1\over
2}\left(\bar{\psi}_{L}i\gamma^{\mu}\partial_{\mu}\psi_{L}+\bar{\psi}_{R}i\gamma^{\mu}\partial_{\mu}\psi_{R}\right)$
$\displaystyle-{1\over
2}\left(Me^{i\theta}\bar{\psi}_{L}\psi_{R}+Me^{-i\theta}\bar{\psi}_{R}\psi_{L}\right)+\cdots$
(29)

where $\psi_{R}\equiv\psi_{L}^{\;c}=N$ and $\theta=J/f_{J}$ in the case of
majoron under consideration. Removing the $\theta$ dependence in the mass term
by the field redefinition $\psi_{L,R}\to e^{\pm i\theta/2}\psi_{L,R}$, one can
obtain

$\displaystyle{\cal L}_{M}$ $\displaystyle\to{1\over
2}\left(\bar{\psi}_{L}i\gamma^{\mu}\partial_{\mu}\psi_{L}+\bar{\psi}_{R}i\gamma^{\mu}\partial_{\mu}\psi_{R}\right)$
$\displaystyle-{1\over
2}\left(M\bar{\psi}_{L}\psi_{R}+Me^{-i\theta}\bar{\psi}_{R}\psi_{L}\right)$
(30)
$\displaystyle-\frac{1}{2}\partial_{\mu}\theta(\bar{\psi}_{L}\gamma^{\mu}\psi_{L}-\bar{\psi}_{R}\gamma^{\mu}\psi_{R})+\cdots.$

Note that the induced current interaction term is chiral, unlike the case of
other SM fermions where a vector current interaction arises under the
$\exp[i(B-L)\theta/2]$ rotation. This is because of the identity
$\psi_{R}=\psi_{L}^{c}$.

The free equations of motion for the $u$-spinors in $\psi_{L,R}\sim
u_{L,R}e^{-ip\cdot x}$ are given as follows:

$\displaystyle(p_{-}\cdot\bar{\sigma})\,u_{L}=Mu_{R}$ (31)
$\displaystyle(p_{+}\cdot\sigma)\,u_{R}=Mu_{L}$ (32)

where $p_{\mp\mu}=p_{\mu}\mp\partial_{\mu}\theta/2$. Here, we used the chiral
representation of $\gamma^{\mu}$:

$\displaystyle\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{pmatrix}~{}~{}\mbox{with}~{}~{}\begin{cases}\begin{matrix}\sigma^{\mu}=(1,+\vec{\sigma})\\\
\bar{\sigma}^{\mu}=(1,-\vec{\sigma})\end{matrix}\end{cases}$ (33)

where $\vec{\sigma}$ are the Pauli matrices.

From Eq. (31) and (32), we find the dispersion relation

$(p_{+}\cdot\sigma)\,(p_{-}\cdot\bar{\sigma})u_{L}=M^{2}$ (34)

which leads to two distinct solutions for the helicity eigenstates ${\cal
H}=\vec{\sigma}\cdot\vec{p}/{\rm p}=\pm 1$ with ${\rm p}\equiv|\vec{p}|$. We
do not present solutions for $u_{R}$ and $v_{L,R}$ (where $\psi_{L,R}\sim
v_{L,R}\,e^{ip\cdot x}$) because the degrees of freedom is effectively two,
which are identified by the relations with $u_{L}$; for instance, $u_{R}$ is
fixed by Eq. (31), and $v_{L,R}$ are fixed by $\psi_{R}=\psi_{L}^{c}$.

For the homogeneous background, $\dot{\theta}\neq 0$ and
$\partial_{i}\theta=0$, one finds

$E=\sqrt{{\rm p}^{2}+M^{2}+{1\over 4}\dot{\theta}^{2}-{\cal
H}\dot{\theta}\,{\rm p}}.$ (35)

In the limit of $|\dot{\theta}|/E_{0}\ll 1$ with $E_{0}\equiv\sqrt{{\rm
p}^{2}+M^{2}}$, we have $E\approx E_{0}\mp\frac{\dot{\theta}}{2}\frac{\rm
p}{E_{0}}$ where ${\rm p}/{E_{0}}$ approaches 1 in the ultra-relativistic
(Weyl) limit, whereas it gets suppressed as ${\rm p}/M_{N}$ in the non-
relativistic limit.

In the Boltzmann approximation, one finds that the equilibrium number density
of the Majorana fermion $N$ with the external chemical potential is given by

$\displaystyle n_{N_{\pm}}\approx\frac{T^{3}}{2\pi^{2}}\left(z^{2}K_{2}(z)\pm
2{\dot{\theta}\over T}e^{-z}(1+z)\right),$ (36)

where the $+$ sign ($-$ sign) stands for the positive (negative) helicity.

This can be understood as the $B-L$ conservation in the limit of $M_{N}\to 0$
since $N$’s helicity is the lepton number. The scattering processes which does
not vanish at $M_{N}\to 0$, e.g. $NQ_{3}\leftrightarrow lt$, are affected by
the helicity asymmetry of $N$.

On the other hand, the decay and inverse decay processes are always
proportional to $M_{N}$, and therefore it is not affected by the helicity
asymmetry of $N$. This can be explicitly seen from the fact that the decay
rate of $N_{\pm}\to lH$ is the same with $N_{\pm}\to\bar{l}\bar{H}$
independently of the inertial frame.

Since, in this paper, we neglect the scattering terms while we only keep the
decay and inverse decay terms, Eq. (36) will not be used in our Boltzmann
equations. Note, however, that a more precise estimation including scattering
terms should include the helicity asymmetry of $N$, and therefore the
equilibrium values of $Y_{B}$ and $Y_{L}$ are modified accordingly.

Other SM fermions follow different dispersion relation as $\psi_{L}$ and
$\psi_{R}$ are independent degrees (corresponding to particle and anti-
particle states, respectively), and they carry the same $U(1)_{B-L}$ charge.
Thus, the modified four-momenta $p_{\mp}$ that appear in Eqs. (31) and (32)
should be replaced by the same sign ones, which gives the dispersion relation
of

$E_{\psi}=\sqrt{{\rm p}^{2}+m_{\psi}^{2}}\mp{1\over
2}(B-L)_{\psi}\dot{\theta}$ (37)

where $m_{\psi}$ is the Dirac mass.

##  Appendix B Boltzmann equations

###  B.1 Decay and inverse decay of ${N}$

In this section, we approximate that the distribution function of $X$ is given
by $f_{X}(p)\simeq(n_{X}/n_{X}^{\mathrm{(eq)}})f_{X}^{\mathrm{(eq)}}(p)$ with
assuming the kinetic equilibrium. We further approximate
$f_{X}^{\mathrm{(eq)}}(p)$ by the Maxwell-Boltzmann distributions for $X=N$,
$l_{\alpha}$, and $H$ for simplicity. Then, the decay and inverse decay terms
of right-handed neutrinos can be written as

$\displaystyle\dot{n}_{l_{\alpha}}+3Hn_{l_{\alpha}}=$
$\displaystyle+\frac{n_{N}}{n_{N}^{\mathrm{(eq)}}}\Gamma^{\mathrm{(eq)}}(N\to
l_{\alpha}H)$
$\displaystyle-\frac{n_{l}n_{H}}{n_{l}^{\mathrm{(eq)}}n_{H}^{\mathrm{(eq)}}}\Gamma^{\mathrm{(eq)}}(l_{\alpha}H\to
N)+\cdots$ (38)
$\displaystyle\dot{n}_{\bar{l}_{\alpha}}+3Hn_{\bar{l}_{\alpha}}=$
$\displaystyle+\frac{n_{N}}{n_{N}^{\mathrm{(eq)}}}\Gamma^{\mathrm{(eq)}}(N\to\bar{l}_{\alpha}\bar{H})$
$\displaystyle-\frac{n_{\bar{l}_{\alpha}}n_{\bar{H}}}{n_{l_{\alpha}}^{\mathrm{(eq)}}n_{H}^{\mathrm{(eq)}}}\Gamma^{\mathrm{(eq)}}(\bar{l}_{\alpha}\bar{H}\to
N)+\cdots$ (39)

where

$\displaystyle\Gamma_{Y_{N,\alpha}}$
$\displaystyle\equiv\Gamma^{\mathrm{(eq)}}(N\to
l_{\alpha}H)=\Gamma^{\mathrm{(eq)}}(l_{\alpha}H\to N)$
$\displaystyle=\int\frac{d^{3}p_{N}}{(2\pi)^{3}}f_{N}^{\mathrm{(eq)}}(p_{N})\frac{M_{N}}{E_{N}}\Gamma_{N\to
l_{\alpha}H}$ (40)
$\displaystyle=n_{N}^{\mathrm{(eq)}}\frac{K_{1}(z)}{K_{2}(z)}\Gamma_{N\to
l_{\alpha}H}$

and $\Gamma_{N\to l_{\alpha}H}=|Y_{N,\alpha 1}|^{2}M_{N}/16\pi$ (one can use
$n_{N}^{\mathrm{(eq)}}=\frac{2}{(2\pi)^{2}}z^{2}K_{2}(z)$ to further simplify
the equation). Note that since $\Gamma(N_{+}\to l_{\alpha}H)=\Gamma(N_{-}\to
l_{\alpha}H)$ and
$\Gamma(N_{+}\to\bar{l}_{\alpha}\bar{H})=\Gamma(N_{-}\to\bar{l}_{\alpha}\bar{H})$,
the decay terms are combined by $n_{N}=n_{N_{+}}+n_{N_{-}}$.

With nonzero chemical potentials, we can replace $n_{\Delta
X}/n_{X}^{\mathrm{(eq)}}\simeq 2\mu_{X}/T$, where $n_{\Delta X}\equiv
n_{X}-n_{\bar{X}}$. Then, the corresponding term in the Boltzmann equation for
$n_{\Delta l_{\alpha}}$ becomes

$\displaystyle\dot{n}_{\Delta l_{\alpha}}+3Hn_{\Delta
l_{\alpha}}=-\Gamma_{Y_{N,\alpha}}\Bigg{(}\frac{n_{\Delta
l_{\alpha}}}{n_{l_{\alpha}}^{\mathrm{(eq)}}}+\frac{n_{\Delta
H}}{n_{H}^{\mathrm{(eq)}}}\Bigg{)}+\cdots,$ (41)

or equivalently,

$\displaystyle\frac{d}{d\ln
T}\left(\frac{\mu_{l_{\alpha}}}{T}\right)=\gamma^{\rm
ID}_{Y_{N},\alpha}\left(\frac{\mu_{l_{\alpha}}}{T}+\frac{\mu_{H}}{T}\right)+\cdots$
(42)

where

$\displaystyle\gamma^{\rm
ID}_{Y_{N},\alpha}=\frac{n_{N}^{\mathrm{(eq)}}}{n_{l_{\alpha}}^{\mathrm{(eq)}}}\frac{K_{1}(z)}{K_{2}(z)}\Gamma_{N\to
l_{\alpha}H},$ (43)

and $z=M_{N}/T$. Notice that the decay terms do not appear since they were
canceled out when we take $\dot{n}_{l_{\alpha}}-\dot{n}_{\bar{l}_{\alpha}}$.
$\dot{\theta}$ dependence enters with the replacement of
$\mu_{l_{\alpha}}\to\mu_{l_{\alpha}}-\dot{\theta}/2$.

###  B.2 Complete Boltzmann equations

The collision terms for the other SM interactions can be easily derived (see,
e.g., Ref. Domcke  _et al._ (2020)). When there is a nonzero background motion
of the majoron, the Hamiltonian density in the density matrix will be modified
as ${\cal H}\to{\cal H}-\frac{1}{2}\dot{\theta}J_{B-L}^{0}$ (see also section.
A), so we can effectively replace
$\mu_{i}\to\mu_{i}+\frac{1}{2}(B-L)_{i}\dot{\theta}$ for the SM fermions.

Including the Majorana properties discussed above, the complete Boltzmann
equations are

$\displaystyle H\frac{d}{dx}\hat{\mu}_{q_{i}}$
$\displaystyle=\gamma_{Y_{u_{i}}}(\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{H})+\gamma_{Y_{d_{i}}}(\hat{\mu}_{d^{c}_{i}}+\hat{\mu}_{q_{i}}-\hat{\mu}_{H})+3\gamma_{\rm
WS}\sum_{j}(\hat{\mu}_{l_{j}}+3\hat{\mu}_{q_{j}})+2\gamma_{\rm
SS}\sum_{j}(2\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{d^{c}_{i}})$
(44) $\displaystyle H\frac{d}{dx}\hat{\mu}_{u^{c}_{i}}$
$\displaystyle=\gamma_{Y_{u_{i}}}(\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{H})+\gamma_{\rm
SS}\sum_{j}(2\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{d^{c}_{i}})$
(45) $\displaystyle H\frac{d}{dx}\hat{\mu}_{d^{c}_{i}}$
$\displaystyle=\gamma_{Y_{d_{i}}}(\hat{\mu}_{d^{c}_{i}}+\hat{\mu}_{q_{i}}-\hat{\mu}_{H})+\gamma_{\rm
SS}\sum_{j}(2\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{d^{c}_{i}})$
(46) $\displaystyle H\frac{d}{dx}\hat{\mu}_{l_{i}}$
$\displaystyle=\gamma_{Y_{e_{i}}}(\hat{\mu}_{e^{c}_{i}}+\hat{\mu}_{l_{i}}-\hat{\mu}_{H})+\gamma^{\rm
ID}_{Y_{N,i}}\left(\hat{\mu}_{l_{i}}+\hat{\mu}_{H}-\frac{\dot{\theta}}{2T}\right)+\gamma_{\rm
WS}\sum_{j}(\hat{\mu}_{l_{j}}+3\hat{\mu}_{q_{j}})$ (47) $\displaystyle
H\frac{d}{dx}\hat{\mu}_{e^{c}_{i}}$
$\displaystyle=\gamma_{Y_{e_{i}}}(\hat{\mu}_{e^{c}_{i}}+\hat{\mu}_{l_{i}}-\hat{\mu}_{H})$
(48) $\displaystyle H\frac{d}{dx}\hat{\mu}_{H}$
$\displaystyle=\gamma_{Y_{u_{i}}}(\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{H})+\gamma_{Y_{d_{i}}}(-\hat{\mu}_{d^{c}_{i}}-\hat{\mu}_{q_{i}}+\hat{\mu}_{H})+\gamma_{Y_{e_{i}}}(-\hat{\mu}_{e^{c}_{i}}-\hat{\mu}_{l_{i}}+\hat{\mu}_{H})+\gamma^{\rm
ID}_{Y_{N,i}}\left(\hat{\mu}_{l_{i}}+\hat{\mu}_{H}-\frac{\dot{\theta}}{2T}\right)$
(49)

where $\hat{\mu}\equiv\mu_{i}/T$ and $x\equiv\ln T$. The relaxation rates
$\gamma_{\alpha}$ for the SM Yukawa interactions are well-summarized in Ref.
Domcke  _et al._ (2020).

##  Appendix C Equilibrium values

The equilibrium values of $\hat{\mu}_{i}$ can be found by solving
$d\hat{\mu}_{i}/dx=0$. When the relaxation rate $\gamma_{\alpha}>H$, we can
impose equilibration condition $\sum_{j}c_{j}^{\alpha}\hat{\mu}_{j}=0$. These
conditions can be explicitly written as

$\displaystyle\gamma_{Y_{u_{i}}}:$
$\displaystyle~{}~{}\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}-\hat{\mu}_{H}=0$
(50) $\displaystyle\gamma_{Y_{d_{i}}}:$
$\displaystyle~{}~{}\hat{\mu}_{d^{c}_{i}}+\hat{\mu}_{q_{i}}+\hat{\mu}_{H}=0$
(51) $\displaystyle\gamma_{Y_{e_{i}}}:$
$\displaystyle~{}~{}\hat{\mu}_{e^{c}_{i}}+\hat{\mu}_{l_{i}}+\hat{\mu}_{H}=0$
(52) $\displaystyle\gamma_{\rm WS}:$
$\displaystyle~{}~{}\sum_{j}(\hat{\mu}_{l_{j}}+3\hat{\mu}_{q_{j}})=0$ (53)
$\displaystyle\gamma_{\rm SS}:$
$\displaystyle~{}~{}\sum_{j}(2\hat{\mu}_{q_{i}}+\hat{\mu}_{u^{c}_{i}}+\hat{\mu}_{d^{c}_{i}})=0$
(54)

For interactions with $\gamma_{\alpha}<H$, we can neglect the corresponding
term in the Boltzmann equation, and therefore, we do not impose the
equilibration condition for that interaction. We assume
$\gamma^{D}_{Y_{N_{1},i}}$ are always greater than the Hubble rate since we
are investigating the scenario around $T\simeq M_{N}$ in the strong wash-out
regime.

We also impose the (hyper) charge neutrality:

$\displaystyle
0=\sum_{i}(\frac{1}{6}6\mu_{q_{i}}-\frac{2}{3}3\mu_{u^{c}_{i}}+\frac{1}{3}3\mu_{d^{c}_{i}}-\frac{1}{2}2\mu_{l_{i}}+\mu_{e^{c}_{i}})-\frac{1}{2}2\cdot
2\mu_{H}.$ (55)

In addition, there are more conserved numbers depending on the temperature
range. Considering all the effects, one can obtain the baryon and lepton
asymmetries depending on the temperature region as follows.

  * •

$T<10^{5}\,\mathrm{GeV}$: All the interactions are in the thermal bath, and we
obtain the resulting $B$, $L$ and $B-L$ asymmetries as follows.

$\displaystyle c_{B}\simeq-\frac{28}{22},\quad c_{L}\simeq\frac{51}{22},\quad
c_{B-L}\simeq-\frac{79}{22}$ (56)

  * •

$1.1\times 10^{5}<T<4.5\times 10^{6}\,\mathrm{GeV}$: $\gamma_{Y_{e_{1}}}$ (and
$\gamma_{Y_{u_{1}}}$ for $T>10^{6}\,\mathrm{GeV}$) is decoupled. With imposing
$\mu_{e_{1}^{c}}=0$, we obtain

$\displaystyle c_{B}\simeq-\frac{13}{10},\quad c_{L}\simeq\frac{9}{4},\quad
c_{B-L}\simeq-\frac{71}{20}$ (57)

  * •

$4.5\times 10^{6}<T<1.1\times 10^{9}\,\mathrm{GeV}$: $\gamma_{Y_{d_{1}}}$ is
additionally decoupled. With imposing $\mu_{e^{c}_{1}}=0$ and
$\mu_{u^{c}_{1}}=\mu_{d^{c}_{1}}$, we obtain

$\displaystyle c_{B}\simeq-\frac{20}{17},\quad c_{L}\simeq\frac{33}{17},\quad
c_{B-L}\simeq-\frac{53}{17}$ (58)

  * •

$1.1\times 10^{9}<T<4.7\times 10^{9}\,\mathrm{GeV}$: $\gamma_{Y_{d_{2}}}$ is
additionally decoupled. With imposing $\mu_{e^{c}_{1}}=0$,
$\mu_{u^{c}_{1}}=\mu_{d^{c}_{1}}=\mu_{d^{c}_{2}}$, and $B_{1}-B_{2}=0$, we
obtain

$\displaystyle c_{B}\simeq-\frac{34}{31},\quad c_{L}\simeq\frac{54}{31},\quad
c_{B-L}\simeq-\frac{88}{31}$ (59)

  * •

$4.7\times 10^{9}<T<1.2\times 10^{11}\,\mathrm{GeV}$: $\gamma_{Y_{e_{2}}}$ is
additionally decoupled. With imposing $\mu_{e^{c}_{1}}=\mu_{e^{c}_{2}}=0$,
$\mu_{u^{c}_{1}}=\mu_{d^{c}_{1}}=\mu_{d^{c}_{2}}$, and $B_{1}-B_{2}=0$, we
obtain

$\displaystyle c_{B}\simeq-\frac{10}{9},\quad c_{L}\simeq\frac{31}{18},\quad
c_{B-L}\simeq-\frac{17}{6}$ (60)

  * •

$1.2\times 10^{11}<T<1.5\times 10^{12}\,\mathrm{GeV}$: $\gamma_{Y_{u_{2}}}$
(and $\gamma_{Y_{e_{3}}}$ for $T>1.3\times 10^{12}\,\mathrm{GeV}$) is
additionally decoupled. With imposing
$\mu_{e^{c}_{1}}=\mu_{e^{c}_{2}}(=\mu_{e^{c}_{3}})=0$,
$\mu_{u^{c}_{1}}=\mu_{u^{c}_{2}}=\mu_{d^{c}_{1}}=\mu_{d^{c}_{2}}$, and
$B_{1}-B_{2}=0$, we obtain

$\displaystyle c_{B}\simeq-1,\quad c_{L}\simeq\frac{3}{2},\quad
c_{B-L}\simeq-\frac{5}{2}$ (61)

  * •

$1.5\times 10^{12}<T<2.5\times 10^{12}\,\mathrm{GeV}$: $\gamma_{Y_{d_{3}}}$ is
additionally decoupled. With imposing
$\mu_{e^{c}_{1}}=\mu_{e^{c}_{2}}=\mu_{e^{c}_{3}}=0$,
$\mu_{u^{c}_{1}}=\mu_{u^{c}_{2}}=\mu_{d^{c}_{1}}=\mu_{d^{c}_{2}}=\mu_{d^{c}_{3}}$,
and $B_{1}=B_{2}=B_{3}$, we obtain

$\displaystyle c_{B}\simeq-\frac{23}{29},\quad c_{L}\simeq\frac{69}{58},\quad
c_{B-L}\simeq-\frac{115}{58}$ (62)

  * •

$2.5\times 10^{12}<T<6\times 10^{12}\,\mathrm{GeV}$: $\gamma_{\rm WS}$ is
additionally decoupled. With imposing
$\mu_{e^{c}_{1}}=\mu_{e^{c}_{2}}=\mu_{e^{c}_{3}}=0$,
$\mu_{u^{c}_{1}}=\mu_{u^{c}_{2}}=\mu_{d^{c}_{1}}=\mu_{d^{c}_{2}}=\mu_{d^{c}_{3}}$,
and $B_{1}=B_{2}=B_{3}=0$, we obtain

$\displaystyle c_{B}\simeq 0,\quad c_{L}\simeq\frac{69}{44},\quad
c_{B-L}\simeq-\frac{69}{44}$ (63)

Figure 4: The absolute value of $c_{B-L}(T_{B-L})$ as a function of $T_{B-L}$.

##  Appendix D Majoron to photon-photon coupling

For light majoron as in our kinetic misalignment scenario, one may hope that
the photon-photon coupling induced by quantum corrections may have a
phenomenological signature. However, that is not the case as we show in the
following. Since the $B-L$ symmetry is anomaly-free, the majoron couplings to
gauge bosons involve additional derivatives, e.g.
$\partial^{2}aF^{\mu\nu}\tilde{F}_{\mu\nu}$. The photon-photon interaction is
generated at two-loop level Heeck and Patel (2019), and the partial decay rate
is given as

$\displaystyle\Gamma_{J\to\gamma\gamma}=\frac{|g_{J\gamma}^{\rm
eff}|^{2}}{64\pi}m_{J}^{3},$ (64)

where, for $m_{J}\ll\mathrm{MeV}$,

$\displaystyle g_{J\gamma}^{\rm eff}$ $\displaystyle\simeq\frac{\alpha_{\rm
EM}}{16\pi^{3}f}\left(\frac{m_{J}}{\mathrm{MeV}}\right)^{2}\Big{[}-0.15\,{\rm
tr}[Y_{N}Y_{N}^{\dagger}]+0.32\,(Y_{N}Y_{N}^{\dagger})_{ee}$
$\displaystyle+7.5\times 10^{-6}\,(Y_{N}Y_{N}^{\dagger})_{\mu\mu}+2.6\times
10^{-8}\,(Y_{N}Y_{N}^{\dagger})_{\tau\tau}\Big{]}.$ (65)

To derive an aggressive estimation of phenomenological constraints, we choose
the largest $g_{J\gamma}^{\rm eff}$ that is possible along the flavor
structure of $Y_{N}$. First of all, we use $(Y_{N}Y_{N}^{\dagger})_{ll}<{\rm
tr}[Y_{N}Y_{N}^{\dagger}]$ for $l=e$, $\mu,$ and $\tau$ and also ${\rm
tr}[Y_{N}Y_{N}^{\dagger}]\simeq{\rm tr}[Y_{N}Y_{N}^{\rm T}]$ so that we obtain

$\displaystyle{\rm tr}[Y_{N}Y_{N}^{\rm
T}]<\frac{2f_{J}}{v_{h}^{2}}\tilde{m}_{\nu},$ (66)

where we also assumed $M_{1}<M_{2}<M_{3}<f_{J}$ which is true if $N_{I}$
interactions are perturbative. Then, the upper bound of $g_{a\gamma}^{\rm
eff}$ becomes

$\displaystyle g_{a\gamma}^{\rm eff}<g_{a\gamma}^{\rm max}$
$\displaystyle\simeq\frac{\alpha_{\rm
EM}}{16\pi^{3}v_{h}^{2}}\left(\frac{m_{J}}{\mathrm{MeV}}\right)^{2}\Big{[}0.34\,\tilde{m}_{\nu}\Big{]}$
(67) $\displaystyle\simeq 4\times
10^{-21}{\mathrm{GeV}}^{-1}\left(\frac{m_{J}}{\mathrm{MeV}}\right)^{2}\left(\frac{\tilde{m}_{\nu}}{0.05\,\mathrm{eV}}\right).$

Noting that the $X$-ray constraint on an axion-like particle at
$m_{a}\sim\mathrm{keV}$ is roughly $g_{a\gamma}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{-17}\,\mathrm{GeV}^{-1}$ Foster  _et al._ (2021), we
conclude that it is highly challenging to give constraints on majoron by using
the photon-photon interaction.

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# Pion-nucleus elastic scatterings incorporating medium effects
within the Eikonal-Glauber model

Hyeon-dong Han  0000-0001-8645-0982<EMAIL_ADDRESS>Department of
Physics, Pukyong National University (PKNU), Busan 48513, Korea  Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea  Parada
T. P. Hutauruk  0000-0002-4225-7109<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Pukyong National University (PKNU), Busan 48513, Korea
Seung-il Nam  0000-0001-9603-9775<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Pukyong National University (PKNU), Busan 48513, Korea
Center for Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841,
Korea  Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673,
Korea

###### Abstract

In this present investigation, we explore the elastic scattering of pions with
nuclei ($\pi$-$A$), primarily influenced by the $\Delta$(1232) resonance,
within the Eikonal-Glauber model. The medium effects are incorporated by
considering nuclear-density ($\rho_{A}$) dependent masses of baryons and
strong coupling constants. These dependencies are computed and parameterized
up to $\mathcal{O}(\rho_{A}^{2})$ based on the quark-meson coupling (QMC)
model. The Wood-Saxon type density profile is utilized for the bound nucleons
within finite nuclei. The element $\pi^{+}$-$N$ scattering cross section for
the Glauber approach is determined using the conventional effective Lagrangian
method. Subsequently, we analyze the total cross sections for elastic
scattering with 4He and 12C targets. Our numerical results demonstrate a
favorable agreement with JINR data for the 4He target, accurately reproducing
the total cross-section. However, when considering the 12C target, deviations
of approximately $\lesssim 10\%$. We also consider the multiple-scattering
effects inside the nucleus approximately, using the single-channel meson-
baryon Bethe-Salpeter equation, resulting in the effective width broadening of
the $\Delta$ resonance to reproduce the data better.

Elastic pion-nucleus scattering, Eikonal-Glauber model, effective Lagrangian
approach, medium effects, Helium and carbon targets, multiple scattering

††preprint: PKNU-NuHaTh-2024

##  I Introduction

The investigation of low-energy pion-nucleus ($\pi$-$A$) scattering has been
one of the most important subjects within the nuclear physics community since
1950 Serber:1947zza ; Fernbach:1949zz . This field holds significance as it
offers valuable insights into nucleon-nucleon ($N$-$N$) interactions and
nuclear structure. The $\pi$-$A$ interactions manifest through elastic
scatterings, where the target nucleus remains in its ground state
Landau:1977th ; Khankhasayev:1989mp ; Aleksakhin:1998qh , inelastic
scatterings with the nucleus in excited states upon pion emission Lee:2002eq ,
and absorption processes where no pion emerges from the nucleus Weyer:1990ye ;
Golak:2022tjw . Numerous theoretical studies on $\pi$-$N$ and $\pi$-$A$
scatterings have been conducted using diverse approaches, including the
optical model in coordinate space Carr:1982zz ; Seki:1983sh , the optical
model in momentum space Landau:1977th , coupled channel theory Koltun:1980cn ,
extended schematic models Hirata:1977is , cloudy bag models Jennings:1984ej ,
and chiral bag models incorporating quark degrees of freedom Thomas:1982kv .
It is noteworthy that theoretical model calculations often account for pion
absorption by employing an effective potential, typically proportional to the
second order of the nuclear density, denoted as $\mathcal{O}(\rho_{A}^{2})$.

This is akin to the absorption coefficient, signifying that the rate of
particle absorption per unit time at a specific position in the nucleus is
proportionate to the particle density at that location. This coefficient is a
component of the Schrödinger equation within the optical potential,
manifesting as an imaginary potential with absorptive characteristics, while
the real potential is recognized as dispersive. Alternatively, this absorption
coefficient can be derived from the continuity equation, as detailed in Ref.
Bethe:1940zza . Nevertheless, contemporary interest in elastic $\pi$-$A$
scattering has waned due to the intricate nature of nuclear structure. Despite
extensive theoretical endeavors, numerous experiments have been conducted,
amassing a wealth of data on $\pi$-$A$ scattering cross-sections. However,
these experimental findings still bear substantial uncertainties and lack
precision owing to limited information about nuclear structure. Moreover, the
available experimental data pertains only to lower pion energies,
approximately around $70$ MeV. Fortunately, recently, the maximum pion energy
has surged to approximately $350$ MeV for 12C and 208 MeV for 4He,
respectively, for instance Buchle:1989hb .

In this study, we conduct a phenomenological examination of $\pi$-$A$ elastic
scattering, emphasizing the dominance of the $\Delta(1232)$ resonance and
incorporating medium effects within the Eikonal-Glauber model (EGM)l,
utilizing a $\pi^{+}$ beam. The medium effects are introduced through nuclear-
density ($\rho_{A}$) dependent masses of baryons and strong coupling
constants, computed and parameterized based on the quark-meson coupling (QMC)
model Guichon:1987jp ; Saito:2005rv ; Hutauruk:2018qku ; Hutauruk:2019was ;
Hutauruk:2019ipp ; Stone:2016qmi  up to $\mathcal{O}(\rho_{A}^{2})$. QMC
computations reveal that effective nucleon masses for
$A=(^{4}\mathrm{He},^{12}\mathrm{C})$ gradually increase up to $r\simeq 2$ fm
and then become saturated. A similar trend is observed for the $\Delta$ mass,
albeit with differing magnitudes. For the calculation of $\pi^{+}$-$A$ elastic
scattering cross-sections, we utilize the element $\pi^{+}$-$N$ elastic
scattering cross-sections computed via the effective Lagrangian approach at
the tree-level Bonn approximation. It is essential to highlight that
$\pi^{+}$-$N$ scattering cross-sections are predominantly influenced by the
$\Delta(1232)$ resonance, and our numerical results effectively reproduce
existing experimental data, as demonstrated in previous research Han:2021zrh .

Our findings indicate that the total cross-section for the 4He target closely
agrees with the data from the Joint Institute for Nuclear Research (JINR)
Shcherbakov:1975gm . However, for the 12C target, the results tend to
overestimate the data by less than $10\%$, particularly when compared to those
of the Shanghai Institute of Nuclear Research (SINR) Ashery:1981tq ,
attributed to the oversimplification inherent in the current Glauber model for
heavier nuclei. To improve the theoretical results, the multiple-scattering
effects inside the nucleus are taken into account approximately, using the
single-channel meson-baryon Bethe-Salpeter equation. As a result, it is
observed that the effective broadening of the decay width of the $\Delta$
resonance occurs, resulting in the data are described better in comparison to
that without the multiple-scattering effects. We also provide the prediction
for the 16O target case with the same theoretical framework.

This manuscript is structured as follows. Section II provides a brief
introduction to the theoretical framework for the $\pi^{+}$-$A$ elastic
scatterings. In Section III, we present comprehensive numerical results
encompassing various theoretical aspects, such as effective baryon mass,
strong couplings, cross sections, and multiple scatterings for nuclei targets
mainly, accompanied by detailed discussions. The concluding section is
dedicated to summarizing the key findings.

##  II Theoretical framework and numerical results

In this section, we provide a brief overview of the current theoretical
framework governing elastic $\pi^{+}$-$A$ scatterings within the EGM. We
outline the incorporation of medium effects into the model, specifically
addressing the determination of effective baryon masses and relevant strong
couplings. These medium effects are computed using the Quark-Meson Coupling
(QMC) model up to $\mathcal{O}(\rho_{A}^{2})$, with $\rho_{A}$ representing
nuclear density. Additionally, we account for the bound-nucleon density
profile within the nuclei using the Wood-Saxon functional form, given in Ref.
Woods:1954zz . Historically, the Glauber model was initially proposed in Refs.
glauber1970 ; Glauber1959 , proving highly successful as a phenomenological
model for characterizing nuclear properties, particularly in $N$-$A$ and
$A$-$A$ interactions observed in heavy-ion collision (HIC) experiments, as
documented in Ref. Miller:2007ri . This model interprets $N$-$A$ scattering in
terms of multiple elastic $N$-$N$ scatterings. Our focus in this study is on
the total cross-sections of elastic $\pi$-$A$ scatterings involving different
nucleus targets, such as 4He and 12C. To this end, we employ the EGM, an
updated version of the Glauber approach that assumes a projectile follows a
linear path, known as the Eikonal approximation Goldstein:2006sv .

It is crucial to emphasize that, in our computations employing the current
EGM, we rely on the element $\pi$-$N$ scattering cross-sections as input,
where the dominant influence is attributed to the $\Delta(1232)$ resonances.
Within the framework, the expression for the total cross-section is defined as
follows:

$\displaystyle\sigma_{\pi A}(s^{*})=\int
d^{2}\mathbf{b}\Big{[}1-\exp\left[-\tilde{\sigma}_{\pi
N}(E^{*}_{\mathrm{cm}},\rho^{N}_{A})\,T^{N}_{A}(\mathbf{b})\right]\Big{]},\,\,\,\,T^{N}_{A}(\mathbf{b})=\int^{\infty}_{-\infty}dz\,\rho^{N}_{\mathrm{A}}\left(\sqrt{|\mathbf{b}|^{2}+z^{2}}\right),$
(1)

Here, $\tilde{\sigma}_{\pi N}$ represents the sum of the $\pi$-$N$ scattering
cross-section for $N=(n,p)$ within the nucleus, and $T^{N}_{A}$ denotes the
thickness function describing the transverse reaction probability at the two-
dimensional impact-parameter vector $\mathbf{b}$. Meanwhile, $z$ signifies the
longitudinal spatial position. The combined cross-section, accounting for the
numbers of neutrons ($n$) and protons ($p$) within the nuclei, can be
expressed as follows:

$\displaystyle\tilde{\sigma}_{\pi N}=\left(\frac{A-Z}{A}\right)\sigma_{\pi
n}+\left(\frac{Z}{A}\right)\sigma_{\pi p}.$ (2)

It is worth noting that the ratio between the elementary $\pi$-$N$ cross
sections, namely $\sigma_{\pi p}$ and $\sigma_{\pi n}$, is approximately 9:1.
This ratio can be understood by their distinct isospin factors in elastic
scatterings, as elucidated in prior research Han:2021zrh .

To consider the internal density distribution of the nuclei, the Wood-Saxon
density profile, presented as a function in Eq. (1), is expressed as follows:

$\displaystyle\rho^{(n,p)}_{\mathrm{A}}(r)=\left[\frac{\rho^{c}_{A}\left[1+c(r/r_{A})^{2}\right]}{1+\mathrm{exp}[(r-r_{A})/d]}\right]\left[\frac{(A-Z,Z)}{A}\right],\,\,\,\,r=\sqrt{|\mathbf{b}|^{2}+z^{2}},$
(3)

Here, $r$, $\rho^{c}_{A}$, $r_{A}$, $d$, and $c$ represent the radial
distance, center density, average radius, surface thickness, and deformation
parameter for the target nucleus, respectively. The specific values for these
parameters are determined through fitting to experimental data, as detailed in
Ref. DeJager:1974liz ; DeVries:1987atn , and are compiled for 4He, 12C, and
16O in Table 1.

Nuclei | $A$ | $Z$ |  $r_{A}$ (fm) |  $d$ (fm) |  $\rho^{c}_{A}$ (fm-3) | $c$
---|---|---|---|---|---|---
4He | 4 | 2 | 1.01 | 0.327 | 0.2381 | 0.445
12C | 12 | 6 | 2.36 | 0.522 | 0.1823 |  $-$0.149
16O | 16 | 8 | 2.608 | 0.513 | 0.1701 |  $-$0.051
Table 1: $r_{A}$, $d$, $\rho^{c}_{A}$, and $c$ for 4He, 12C, and 16O from the
fitting of experimental data DeJager:1974liz ; DeVries:1987atn .

The density profile in Eq. (3) satisfies the following normalization
condition:

$\sum_{N=(n,p)}\int
dz\,\,d^{2}\mathbf{b}\,\,\rho^{N}_{A}(\sqrt{|\mathbf{b}|^{2}+z^{2}})=A.$ (4)

This implies the conservation of nucleon number density. The density profiles
for 4He (solid) and 12C (dot-dashed) are depicted in panel (a) of Fig. 1,
showcasing their dependence on the radial distance $r$ from the center. Given
that we are specifically examining even-even nuclei, it follows that
$\rho^{n}_{A}=\rho^{p}_{A}=\rho_{A}$.

---
Figure 1: (a Density profile per nucleon $\rho^{N}_{A}$ [fm-3] as a function
of the radial distance $r$ [fm] for 4He (solid) for 12C (dashed). (b)
Effective baryon mass $M^{*}_{B}$ [MeV] for $B=N,\Delta$ in the same manner
with the panel (a).

As previously mentioned, in this study, we determine the density-dependent
effective masses for baryons, along with the corresponding strong coupling
constants, utilizing the QMC model within the mean-field approximation.
Further details regarding these calculations can be found in the prior work
Han:2021zrh  and the references therein. For practical purposes, we
parameterize these quantities as functions of $\rho_{A}$ up to
$\mathcal{O}(\rho^{2}_{A})$, akin to the imaginary potential for $A$-$A$
derived in the Schrödinger equation. Consequently, the parameterization for
the effective baryon mass $M^{*}_{B}$ is expressed as follows:

$\displaystyle M^{*}_{B}$ $\displaystyle=$ $\displaystyle
M_{B}+C_{1}\,\rho_{A}+C_{2}\,\rho^{2}_{A}+\mathcal{O}(\rho^{n>2}_{A}),$ (5)

where $M_{B}$ stands for the vacuum mass, and the relevant coefficients are
given by $C_{1}=-1.543\,\mathrm{GeV\cdot fm}^{3}$ and
$C_{2}=2.036\,\mathrm{MeV\,fm}^{3}$ via the QMC calculation.

The panel (b) of Fig. 1 illustrates the variation of $M^{*}_{B}$ for 4He and
12C with respect to the radial distance $r$. In the QMC model, the effective
nucleon masses $M^{*}_{N}$ exhibit an initial value of approximately $800$ MeV
at the nuclei’s center due to the partial restoration of spontaneous chiral
symmetry breakdown (SCSB) at finite densities. As one moves radially outward,
the medium effects diminish, leading to a convergence toward the nucleon
vacuum mass of approximately $\sim 940$ MeV. It is noteworthy that the
behavior of $M^{*}_{\Delta}$ for the nuclei mirrors that of the nucleon, given
the utilization of the same mass formula in Eq. (5) for both baryons. However,
the $M^{*}_{N,\Delta}$ curves display distinct dependencies on $r$ for
different nuclei. This discrepancy arises from their disparate density
profiles, as depicted in panel (a) of Fig. 1. In addition to examining the
influence of the medium effects on masses, we also investigate alterations in
the strong coupling constants. To address this, we incorporate the Goldberger-
Treiman relation (GTR) concerning the weak axial-vector coupling constant
$f_{A}$, which is expressed as follows:

$\displaystyle f_{A}$ $\displaystyle=$ $\displaystyle\frac{f_{\pi
NN}\,f_{\pi}}{M_{N}}.$ (6)

The numerical outcomes for $M^{*}_{B}$ as functions of $r$ are presented in
panel (b) of Fig. 1 for 4He and 12C. The effective nucleon masses $M_{N}$
commence at approximately $800$ MeV at the nucleus’s center, reflecting the
partial restoration of the spontaneous breakdown of chiral symmetry (SBCS) at
finite densities within the QMC model. With increasing radial distance, medium
effects diminish, converging to the nucleon vacuum mass of $\sim 940$ MeV.
Notably, $M^{*}_{\Delta}$ for the nuclei exhibits similar behaviors to those
of the nucleon, given the employment of the same mass formula in Eq. (5) for
both baryons. However, the $M^{*}_{N,\Delta}$ curves display distinct $r$
dependencies contingent on the nuclei, stemming from their diverse density
profiles as illustrated in panel (a) of Fig. 1.

$f^{*}_{\pi NN}(\rho_{A})=f_{\pi NN}-0.633\,\rho_{A},\,\,\,\,f^{*}_{\pi
N\Delta}(\rho_{A})=f_{\pi N\Delta}-1.360\,\rho_{A}.$ (7)

Here, we adopt the values $f_{\pi NN}=0.989$, $f_{\pi N\Delta}=2.127$, and
$f_{\pi}=93.2$ MeV, sourced from the Nijmegen potential model Gasparyan:2003fp
and experimental data Janssen:1996kx . The determination of $f^{*}_{\pi
N\Delta}$ assumes that the ratio $f_{\pi NN}/f_{\pi N\Delta}\approx 2.15$
remains constant within the medium.

---
Figure 2: (a) Meson-baryon coupling constants $f_{\pi NN}$ (solid) and $f_{\pi
N\Delta}$ (dashed) as functions of $\rho_{A}$ [fm-3]. (b) $\Delta$-baryon
decay width in Eq. (8), $\Gamma(E_{\mathrm{cm}},\rho_{A})$ [MeV] as a function
of $E_{\mathrm{cm}}$ [MeV] and $\rho_{A}$ [fm-3].

Our findings for the in-medium modifications of the pion-nucleon coupling
constants $f^{*}_{\pi NN}$, the pion-delta coupling constant $f^{*}_{\pi
N\Delta}$, and the pion decay constant with medium correction effects, as
functions of $\rho_{A}$, are illustrated in Fig. 2. In panel (a) of Fig. 2, it
is evident that the density dependence of the pion decay constant diminishes
with increasing $\rho_{A}$, aligning with results from other theoretical
models. Similar trends are observed for $f_{\pi NN}^{*}$ and $f_{\pi
N\Delta}^{*}$, both decreasing as $\rho_{A}$ increases, albeit at a slower
rate compared to $f_{\pi}^{*}$. It is important to note that these medium
corrections will be utilized as input when calculating the cross-section of
pion-nucleus scattering for 4He and 12C.

In our calculations, we also account for the momentum-dependent $\Delta$ decay
width $\Gamma$, as employed in Ref. Larionov:2003av . However, in finite
nuclei, the decay width should be determined by the nuclear density
distribution $\rho_{A}$, taking the form:

$\Gamma(E^{*}_{\mathrm{cm}},\rho_{A})=\Gamma_{\mathrm{sp}}\left(\frac{\rho_{A}}{\rho_{0}}\right)+\Gamma_{0}\left[\frac{q\left(M^{*}_{N},M_{\pi},E^{*}_{\mathrm{cm}}\right)}{q\left(M_{N},M_{\pi},M_{\Delta}\right)}\right]^{3}\left[\frac{M^{*}_{\Delta}}{E^{*}_{\mathrm{cm}}}\frac{\beta^{2}_{0}+q^{2}\left(M^{*}_{N},M_{\pi},M^{*}_{\Delta}\right)}{\beta^{2}_{0}+q^{2}\left(M^{*}_{N},M_{\pi},E^{*}_{\mathrm{cm}}\right)}\right],$
(8)

where $\Gamma_{\mathrm{sp}}=80$ MeV represents the spreading density
dependence of the decay width, determined through fitting to the medium
quantity at finite density. Additionally, $\beta_{0}=200$ MeV serves as the
cutoff parameter, consistent with the choice made in Ref. Larionov:2003av .
The results for the 3D plot of the decay width for the $\Delta$ baryon in
finite nuclei, as a function of $\rho_{A}$ and $E^{*}_{\mathrm{cm}}$, are
depicted in Fig. 2. This visualization indicates that the in-medium
modifications of the $\Delta$ baryon decay width differ significantly for 4He
and 12C due to their distinct nuclear charge density distributions of
$\rho_{A}$, as illustrated in Fig. 2. The in-medium modifications of the
$\Delta$ baryon decay width increase with rising nuclear density distribution
$\rho_{A}$. Furthermore, it is observed that as $E_{\mathrm{cm}}$ increases,
the in-medium modifications of the $\Delta$ baryon resonance decay width
$\Gamma^{*}_{\Delta}$ also increase, consistent with the findings in the
calculation of Ref. Cui:2020fhr .

---
Figure 3: (a) Total cross-section [mb] for the 4He target as a function of the
center-of-mass (cm) energy $E_{\mathrm{cm}}$ [MeV]. Here, we consider two
cases for the $\Delta$ decay width in Eq. (8) with (dashed) and without
(dashed) the $(E_{\mathrm{cm}},\rho_{A})$ dependence. The experimental data
are taken from JINR. Shcherbakov:1975gm . (b) The same for the 12C target. The
data are taken from SINR Ashery:1981tq .

Now, we are in a position to present the numerical results for the cross-
section of elastic pion-nucleus scattering at intermediate energies, where
$\Delta$ baryon resonance dominates, for both 4He and 12C in Fig. 3. For a
more in-depth exploration of the elastic pion-proton and pion-neutron
scatterings, interested readers are referred to Ref. Han:2021zrh . The
numerical results for the elastic cross-sections for the 4He target,
accounting for medium corrections up to $\rho_{A}^{2}$ with
$\tilde{\Gamma}^{*}_{\Delta}(E_{\mathrm{cm}},\rho_{A})$ as a function of
center-of-mass (cm) energy, are presented in the panel (a) of Fig. 3,
alongside JINR experimental dataShcherbakov:1975gm . Fig. 3 clearly
illustrates the total cross-section for 4He, being qualitatively consistent
with the data, particularly around the resonance region at
$E_{\mathrm{cm}}\sim$ 1222 MeV. The dashed and solid lines denote the results
with and without the energy- and density-dependent decay width, respectively,
not showing considerable difference. For the 12C, the numerical result,
compared with SINR experimental data Ashery:1981tq , is shown in the panel (b)
of Fig. 3. Here, the the numerical result overestimates the dataAshery:1981tq
by about $10\%$ percent deviations around the resonance peak, in contrast to
the result for 4He. Again, the modified width in Eq. (8) does not make sizable
difference even for the 12C target. In what follows, we would like to discuss
the deviation between the theory and experiment for the 12C target in detail.

In principle, the decay width of the resonance can be modified by the multiple
scattering (MS) inside the nucleus, and this modification can be the cause to
explain the deviation observed above. Hence, we would like to estimate the
effects of the meson-baryon MS inside the nucleus using the single-channel
($\pi$-$N$) Bethe-Salpeter equation, resulting in the broadening decay width
of the $\Delta$ resonance, which dominates the reaction process. The Bethe-
Salpeter equation for the scattering can be defined as follows:

$\tilde{\mathcal{M}}^{\mathrm{MS}}_{ij}\approx\tilde{\mathcal{M}}_{ij}+\sum_{k}\tilde{\mathcal{M}}_{ik}G_{k}\tilde{\mathcal{M}}_{kj}+\cdots=\tilde{\mathcal{M}}_{ij}+\sum_{k}\tilde{\mathcal{M}}_{ik}G_{k}\tilde{\mathcal{M}}^{\mathrm{MS}}_{kj},$
(9)

where $\tilde{\mathcal{M}}$ [mass-1] indicates the reduced amplitude for an
element process with the channel indices $i,j,\cdots$. It relates to the
invariant amplitude by

$\bar{u}_{i}\tilde{\mathcal{M}}_{ij}u_{j}=\mathcal{M}_{ij}\,[\mathrm{mass}^{2}].$
(10)

Here, $G_{k}=G_{k}(s=E^{2}_{\mathrm{cm}})$ stands for the meson-baryon
propagator in the on-mass-shell approximation for the intermediate $k$
channel. In the present work, for simplicity, we only consider the $\pi N$
elastic channel as mentioned previously, i.e., $(i,j)=(\pi N,\pi N)$. Then,
the propagator is given by the dimensional regularization Inoue:2001ip  as
follows:

$\displaystyle G_{\pi N}(s)$ $\displaystyle=$
$\displaystyle\frac{M_{N}}{8\pi^{2}}\Big{[}a+\ln\frac{M^{2}_{N}}{\mu^{2}}+\frac{s-s^{\prime}_{\pi
N}}{2s}\ln\frac{m^{2}_{\pi}}{M^{2}_{N}}$ (11) $\displaystyle+$
$\displaystyle\frac{s_{\pi N}}{2s}\ln\left[(s_{\pi N}+s^{\prime}_{\pi
N}+s)+(s_{\pi N}-s^{\prime}_{\pi N}+s)-(s_{\pi N}+s^{\prime}_{\pi
N}-s)-(s_{\pi N}-s^{\prime}_{\pi N}-s)\right]\Big{]}\,[\mathrm{mass}],$ (12)

where $s_{\pi N}$ and $s^{\prime}_{\pi N}$ are defined by

$s_{\pi N}=s_{\pi
N}(s)=\sqrt{[s-(M_{N}-m_{\pi})^{2}][s-(M_{N}+m_{\pi})^{2}]},\,\,\,\,s^{\prime}_{\pi
N}=M^{2}_{N}-m^{2}_{\pi}.$ (13)

As for the $\pi$-$N$ elastic-scattering channel, the renormalization scale
$\mu$ and subtraction parameter $a$ are determined by $1.2$ GeV and $2$,
respectively, to reproduce available data for the elementary cross-sections
Inoue:2001ip ; Nam:2003ch . In the panel (a) of Fig. 4, we draw the real
(solid) and imaginary (dashed) parts of the $\pi$-$N$ propagator in Eq. (11)
beyond its threshold. For instance, the single-channel elastic scattering for
the $\Delta(1232)$ can be modified by the unitarization of the MS amplitude,
due to the multiple scatterings inside the nuclei as follows:

$\displaystyle\tilde{\mathcal{M}}^{\Delta,\mathrm{MS}}_{\pi N}$
$\displaystyle=$ $\displaystyle\frac{\tilde{\mathcal{M}}^{\Delta}_{\pi
N}}{1-\tilde{\mathcal{M}}^{\Delta}_{\pi N}G_{\pi
N}}\approx\frac{\hat{\mathcal{M}}^{\Delta}_{\pi
N}}{s-M^{2}_{\Delta}-i\Gamma_{\Delta}M_{\Delta}}\left(1-G_{\pi
N}\frac{\hat{\mathcal{M}}^{\Delta}_{\pi
N}}{s-M^{2}_{\Delta}-i\Gamma_{\Delta}M_{\Delta}}\right)^{-1},$ (14)

where, considering the $\Delta$-resonance dominance,

$\hat{\mathcal{M}}^{\Delta}_{\pi N}\propto-i\frac{f_{\pi
N\Delta}^{2}}{M_{\pi}^{2}}(k_{i}\cdot k_{f}).$ (15)

Finally, Eq. (14) turns into

$\displaystyle\tilde{\mathcal{M}}^{\Delta,\mathrm{MS}}_{\pi N}$
$\displaystyle=$ $\displaystyle\frac{\hat{\mathcal{M}}^{\Delta}_{\pi
N}}{s-M^{2}_{\Delta}-i\Gamma_{\Delta}M_{\Delta}+i\xi G_{\pi N}\frac{f_{\pi
N\Delta}^{2}}{M_{\pi}^{2}}(k_{i}\cdot k_{f})}.$ (16)

---
Figure 4: (a) Real (solid) and imaginary (dashed) parts for the propagator
$G_{\pi N}(s)$ in Eq. (11) beyond the $\pi$-$N$ threshold. (b) Numerical
results for the elastic cross sections for the 4He (solid), 12C (dotted), and
16O (dashed) targets as functions of $E_{\mathrm{cm}}$. The data are taken
from Refs. Shcherbakov:1975gm ; Ashery:1981tq .

As understood by Eq. (16), the multiple scatterings change the decay width as
well as the pole position. It is easy to understand from Eq. (9) that other
contributions besides the resonance also get a small but finite effects from
the multiple-scattering mechanism. Note that , in Eq. (16), we introduce a
free parameter $\xi=-15.8\times A$ [MeV] to fit the data. The reasoning for
$\xi\propto A$ is that the elementary MS process will be enhanced by the
number of the nucleons inside the nucleus. In the panel (b) of Fig. 4, we show
the numerical results for the elastic cross-sections for the 4He (solid), 12C
(dotted), and 16O (dashed) targets, being compared with the data. It is
observed that the numerical result is considerably improved for the 12C,
whereas it slightly underestimates the 4He data. We show the theoretical
prediction for the 16O target as well. As a consequence, it turns out that the
width-broadening due to the MS effect is crucial to reproduce the heavier-
nucleus data.

##  III Summary

In summary, our investigation focused on the elastic $\pi^{+}$-$A$ scattering
process at intermediate energies dominated by the $\Delta(1232)$-resonance,
specifically in the $I=$ 3/2 channel, incorporating medium corrections up to
$\rho_{A}^{2}$ parameterized from the quark-meson coupling (QMC) model within
the Eikonal-Glauber model approach. We proceeded to compute the total cross-
sections for 4He and 12C targets, utilizing elementary $\pi$-$N$ cross-
sections as input, which were calculated within the effective Lagrangian
approach at the tree-level Bonn approximation.

For the total cross-section of 4He, our results exhibited relative consistency
with the JINR data Shcherbakov:1975gm , particularly around the resonance
peak. In contrast, that for 12C was found to overestimate the SINR
experimental data Ashery:1981tq . Notably, the resonance peak for 12C shifted
to a lower cm energy, accompanied by a broadened decay width, compared to the
corresponding quantities for 4He. This observation aligns with findings from
other calculations. It was observed that the momentum- and density-dependent
decay width does not considerable modifications to the theoretical results.
Finally, we took into account the multiple-scattering effect inside the
nucleus, using the single-channel Bethe-Salpeter equation, resulting in the
with-broadening of the dominating $\Delta$ resonance. We found that the
heavier-nucleus cross-section is improved much by the MS effect, describing
the 12C data qualitatively well, whereas the light-nucleus data for the 4He is
reproduced as well with some deviations.

To properly assess the results of this study and compare them with existing
theoretical calculations, it is imperative to acquire new data for the elastic
$\pi^{+}$-$A$ scattering reaction process at intermediate energies in future
experiments. Additionally, advancing rigorous theoretical approaches is
essential for a clearer understanding of the intricate interactions within
both light and heavy nuclei. Related works are in progress and will appear
elsewhere.

## Acknowledgements

This work was supported by a research grant from Pukyong National University
(PKNU) (2022).

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