# Study on higher moments of net-charge multiplicity distributions using a
multiphase transport model

Ling Huang  Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai 201800, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Guo-Liang Ma<EMAIL_ADDRESS>Key Laboratory of Nuclear
Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan
University, Shanghai 200433, China  Shanghai Institute of Applied Physics,
Chinese Academy of Sciences, Shanghai 201800, China

###### Abstract

The moments and moment products of conserved charges are believed to be
sensitive to critical fluctuations, which have been adopted in determining the
QCD critical point. Using a dynamical multiphase transport model, we reproduce
the centrality and energy dependences of moments and moment products of net-
charge multiplicity distributions in Au+Au collisions measured by the Beam
Energy Scan program at the RHIC. No non-monotonic energy dependence is
observed. We infer that the moment products develop during the dynamical
evolution of heavy-ion collisions. The observed difference based on the
expectation of the Poisson baseline indicates a positive two-particle
correlation between positively and negatively charged particles, which can
arise from different dynamical processes at different stages. Therefore, to
adopt moments and moment products of net-charge multiplicity distributions in
determining the QCD critical point of relativistic heavy-ion collisions, it is
essential to take the dynamical evolution.

##  I Introduction

In recent years, the study on the structure of the QCD phase diagram has been
a key research area of relativistic heavy-ion collisions Adams:2005dq ;
Bzdak:2019pkr . Lattice QCD calculations indicate that the phase transition
from quark gluon plasma to hadron gas is smooth when the baryon chemical
potential $\mu_{B}$ disappears Aoki:2006we ; deForcrand:2002hgr ; Ding:2015ona
; however, this transition is a first order phase transition at a large
$\mu_{B}$ Ejiri:2008xt . As an analog of the critical point’s role in the
phase diagram of water, the critical point, which is located at the end point
of the first order phase transition line connected with the crossover region,
is a significantly important feature of the QCD phase diagram Stephanov:1998dy
; Stephanov:1999zu ; Stephanov:2004wx ; Asakawa:1989bq ; Barducci:1989wi ;
Berges:1998rc ; Halasz:1998qr ; Berdnikov:1999ph ; Fodor:2001pe ; Hatta:2002sj
. Several ideas have been proposed to search for the QCD critical point, which
include approaches via ratio of HBT-radii Lacey:2014wqa , yield ratio of light
nuclei Sun:2017xrx ; Yu:2018kvh ; Shao:2019xpj , and directed flow slope of
net proton Stoecker:2004qu . It has been suggested that the moments of
conserved charges are proportional to some powers of the correlation length
closely related to susceptibilities Stephanov:2008qz ; Athanasiou:2010kw ;
Stephanov:2011pb ; Gavai:2010zn ; Cheng:2008zh . Because susceptibilities
diverge at the critical point and the fluctuations of conserved charges are
proportional to their corresponding susceptibilities, moments and moment
products of conserved charges have been proposed as sensitive probes to search
for the QCD critical point Stephanov:2008qz ; Luo:2011ts ; Luo:2011rg ;
Luo:2010by .

In Lattice QCD, the susceptibilities of conserved charges, e.g. baryon number
$B$, strangeness $S$ and electric charge $Q$ , are defined as

$\chi_{lmn}^{BSQ}=\frac{\partial^{\,l+m+n}(P/T^{4})}{\partial(\mu_{B}/T)^{l}\partial(\mu_{S}/T)^{m}\partial(\mu_{Q}/T)^{n}},$
(1)

where P is pressure, T is temperature, $\mu_{BSQ}$ are chemical potentials of
conserved charges (baryon number $B$, strangeness $S$, and electric charge
$Q$). To connect the susceptibilities with the measured products of moments of
the distributions of the corresponding conserved charges, we can conveniently
introduce the following volume-independent ratios:

$\chi_{2}/\chi_{1}=\sigma^{2}/M;\chi_{3}/\chi_{2}=~{}S\sigma;\chi_{4}/\chi_{2}=\kappa\sigma^{2}.$
(2)

Here, the moments of mean($M$), standard deviation($\sigma$), skewness($S$),
and kurtosis($\kappa$) are measurable experimental observerbles, defined as
follows Luo:2011ts ; Adamczyk:2017wsl ; Luo:2013bmi ; Sahoo:2012wn :

$M=K_{1}=\langle X\rangle,$ (3)
$\sigma=\sqrt{K_{2}}=\sqrt{\langle(X-M)^{2}\rangle},$ (4)
$S=\frac{K_{3}}{K_{2}^{3/2}}=\frac{\langle(X-M)^{3}\rangle}{\sigma^{3}},$ (5)
$\kappa=\frac{K_{4}}{K_{2}^{2}}=\frac{\langle(X-M)^{4}\rangle}{\sigma^{4}}-3,$
(6)

where $X$ is the net charge, i.e. the difference between the positive and
negative conserved charges, $K_{n}$ is the $n$th cumulant of the net-charge
distribution, e.g. $K_{n}=\langle(X-M)^{n}\rangle$ for n=1,2 and 3, and
$K_{4}=\langle(X-M)^{4}\rangle-3\langle(X-M)^{2}\rangle^{2}$, and
$\langle...\rangle$ means the average value obtained over all events.

The BNL Relativistic Heavy Ion Collider (RHIC) has been performing a Beam
Energy Scan program Aggarwal:2010wy ; Aggarwal:2010cw ; Adamczyk:2017iwn  with
the task to search for the critical point of the QCD phase diagram
Stephanov:1998dy . Several moment types of net-proton, net-strangeness and
net-charge have been reported Aggarwal:2010wy ; Adamczyk:2014fia ;
Adamczyk:2017wsl ; Adamczyk:2013dal ; Adam:2021fby ; Pandav:2020uzx ;
Abdallah:2021fzj . As the proxy of baryon number, a non-monotonic energy
dependence $\kappa\sigma$ of the net-proton multiplicity distribution has been
observed from the first phase of BES measurements, which is consistent with
the expected QCD critical fluctuations Adam:2021fby ; Abdallah:2021fzj .
However, for the proxy of strangeness, the measured net-kaon cumulant ratios
do not exhibit any significant non-monotonic energy dependence
Adamczyk:2017wsl . In 2014, the STAR collaboration published the first
measurement on the moments of the net-charge ($X\equiv Q_{+}-Q_{-}$)
distribution fluctuation Adamczyk:2014fia , which deviated from the
expectations of Poisson and negative binomial distributions; however no non-
monotonic behavior was observed as a function of the colliding energy.

In this study, we focus on the dynamical evolution of the moments of net-
charge multiplicity distribution using a multi-phase transport model, because
heavy-ion collisions consist of dynamical evolution stages that exist between
the Lattice QCD calculation and final experimental measurement. This paper is
organized as follows. We introduce the AMPT model and our method of
calculating moments in Section II. Our results and discussions are presented
in Section III. Finally, a summary is presented in Section IV.

##  II Model and calculation method

###  II.1 The AMPT model

A multiphase transport model (AMPT), has been widely used to investigate
several aspects of the physics of relativistic heavy-ion collisions Lin:2004en
; Ma:2016fve ; Ma:2013gga ; Ma:2013uqa ; Bzdak:2014dia . The AMPT model with a
string melting mechanism Lin:2004en  primarily comprises four stages: initial
condition, partonic cascade, hadronization and hadronic rescatterings. The
initial condition obtained from the HIJING model Wang:1991hta ;
Gyulassy:1994ew  primarily provides the spatial and momentum distributions of
minijet partons and soft strings. Under the string melting mechanism, both
excited strings and minijet partons are melted into partons, i.e. decomposed
into constituent quarks according to their flavor and spin structures. The
strong interactions among partons are simulated by Zhang’s parton cascade
model Zhang:1997ej , which includes elastic partonic scatterings with a fixed
cross section (3 mb). A simple quark coalescence model describes the
conversion of these partons to hadrons, i.e. hadronization. The ART model
simulates interactions among the hadrons and corresponding inverse reactions,
as well as resonance decays Li:1995pra . In old versions of AMPT model, charge
conservation is violated due to certain problems Lin:2014uwa . In this study,
a new version of the AMPT model which ensures the strict charge conservation
of electric charge for each hadronic reaction channel, is used to investigate
the net-charge fluctuations. In order to study the energy dependence of the
moments of net-charge fluctuations, we simulated Au+Au minus bias collisions
for seven different energies ($\sqrt{s_{NN}}$=200, 62.4, 39, 27, 19.6, 11.5,
7.7 GeV) of the BES program at RHIC.

###  II.2 Calculation method

The mean($M$), standard deviation($\sigma$), skewness($S$), and
kurtosis($\kappa$) can be used to characterize different features of the
multiplicity distribution. The mean represents the average value of the
distribution, standard deviation represents the degree of dispersion of the
distribution, the skewness represents the asymmetry of the distribution, and
the kurtosis describes the degree to which the distribution is peaked relative
to the normal distribution. The formulas of $M$, $\sigma$, $S$ and $\kappa$
are shown in Eqs. (3) -(6). To calculate the aforementioned different moments
of net-charge multiplicity distribution, we select charged particles with
transverse momenta $0.2<p_{T}<2.0$ GeV/c and a pseudorapidity range of
$-0.5<\eta<0.5$. We adopt the charged particle multiplicity distribution to
define centrality bins. To avoid the self-correlation effect(ACE) Luo:2013bmi
; Chatterjee:2019fey , we select the $\eta$ range of charged particles, which
are used to define centrality bins to exclude $|\eta|<0.5$. We apply the Delta
method to estimate the statistical errors of moments or moment products,
according to the procedure of the experimentalists in Luo:2011tp .

In addition, the initial volume fluctuations can contribute to the moments as
a background, which can induce a centrality bin width effect(CBWE) Luo:2011ts
; Luo:2013bmi  and a centrality resolution effect(CRE) Luo:2013bmi . The CBWE
is triggered by the finite width of centrality bin. In other words, if the
width of centrality bin is sufficiently small , the influence of the CBWE will
be weakened. The CRE originates from the uncertainty of the collision
centrality determination, as different initial collision geometries will
trigger the uncertainty of results. To reduce the contribution of these
background effects, the centrality bin width correction (CBWC) and the
centrality resolution correction (CRC) are applied in a similar manner to the
experiments in Luo:2011ts ; Luo:2013bmi ; Sahoo:2012wn . We compare the
results with and without the CBWC and CRC in subsection III.1 and III.2,
respectively.

##  III Results and Discussions

###  III.1 Centrality bin width effect and correction

To eliminate the influence of the CBWE, CBWC has been proposed Luo:2011ts ;
Luo:2013bmi ; Sahoo:2012wn . This method is defined as:

$X=\frac{\sum_{r}n_{r}X_{r}}{\sum_{r}\ n_{r}},$ (7)

where $X$ represents any moments or moment products, such as $M$, $\sigma$,
and $S\sigma$, $n_{r}$ is the number of events in the $r$th multiplicity bin,
and $X_{r}$ represents moments or moment products in the $r$th multiplicity
bin.

In order to clearly see the influence of the CBWE on moments, the results
without and with the CBWC are compared. Figure 1 presents the centrality
dependences of mean($M$) and standard deviation($\sigma$) of the net-charge
multiplicity distributions in Au+Au collisions at seven different colliding
energies($\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$), which are presented as a
function of the average number of participant nucleons $\langle
N_{part}\rangle$. The moments $M$ and $\sigma$ without the CBWC are consistent
with those with the CBWC, which indicates that the CBWE has negligible
influence on the mean and standard deviation of the net-charge distribution.
As the $\langle N_{part}\rangle$ increases, the $M$ and $\sigma$ both exhibit
an increasing dependence of $\langle N_{part}\rangle$ for each energy. When
the colliding energy increases, the mean decreases but the standard deviation
increases for a given $\langle N_{part}\rangle$ value. This indicates that the
QCD matter created at higher energies is in a state with less electric charge
chemical potential but with larger fluctuation.

Figure 1: (Color online) Centrality dependence of mean($M$)(top plots) and
standard deviation($\sigma$)(bottom plots) for net-charge multiplicity
distributions in Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$,
where the STAR data are obtained from Ref. Adamczyk:2014fia .

Figure 2 illustrates the centrality dependences of the skewness($S$) and
kurtosis($\kappa$) for net-charge multiplicity distributions in Au+Au
collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. For both results of
skewness and kurtosis, the values with the CBWC are systematically smaller
than those without the CBWC. However, when the energy increases, the
difference between the results with and without the CBWC becomes increasingly
smaller. These results with the CBWC can describe experimental data better
than those without the CBWC, which indicates that the CBWE makes a significant
contribution to skewness and kurtosis. The CBWE has a larger influence on the
low energies than on the high energies, which indicates that the effect of
volume fluctuations inside one centrality bin increases with the decreasing of
colliding energy. Therefore, the CBWC is very essential, especially for low
energies.

Figure 2: (Color online) Centrality dependence of skewness($S$)(top plots) and
kurtosis($\kappa$)(bottom plots) for net-charge multiplicity distributions in
Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$, where the STAR data
are obtained from Ref. Adamczyk:2014fia

Figure 3 presents the centrality dependence of $S\sigma$ on net-charge
multiplicity distributions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. It can be observed that the moment
product $S\sigma$ of net charge distribution with the CBWC can describe
experimental data better than the $S\sigma$ without the CBWC which
overestimates the experimental data. This can be easily understood because $S$
with the CBWC does a better job for describing the experimental data than
without the CBWC, as shown in Fig. 2.

Figure 3: (Color online) Centrality dependence of moment products $S\sigma$
for net-charge multiplicity distributions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$, where the STAR data are obtained from
Ref. Adamczyk:2014fia .

Figure 4 presents the dependence results of the $\kappa\sigma^{2}$ on net-
charge multiplicity distributions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. The AMPT results with and without the
CBWC can both describe the experimental data within the statistical errors.
However, the $\kappa\sigma^{2}$ with the CBWC is slightly smaller than that
without the CBWC, which is approximately the constant of unity within the
statistical errors.

Figure 4: (Color online) Centrality dependence of moment products
$\kappa\sigma^{2}$ for net-charge multiplicity distributions in Au+Au
collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$, where the STAR data are
obtained from Ref. Adamczyk:2014fia .

###  III.2 The centrality resolution effect and correction

We also study the influence of the centrality resolution effect (CRE) on the
moments and moment products of net-charge multiplicity distributions in Au+Au
collisions. Because the CRE is induced by initial volume fluctuations of
impact parameter or collision centrality, using different $\eta$ ranges to
define collision centrality is considered a possible way to weaken this
effect. It should be stated that to avoid the possible interference of the
CBWE, all the AMPT results presented in this subsection are obtained with the
CBWC.

Figure 5 illustrates that the centrality dependence of the mean($M$) and
standard deviation($\sigma$) for net-charge multiplicity distributions with
different centrality definitions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. To define centrality bins, the charge
multiplicities from three different pseudorapidity ranges (0.5$<|\eta|<$1.0,
0.5$<|\eta|<$1.5 and 0.5$<|\eta|<$2.0) are adopted, all of which are out of
midpseudorapidity such that possible autocorrelations are avoided. We find
that the AMPT results of mean and standard deviation with three different
centrality definitions are overlapped and approximate to the experiment data.
This indicates that the CRE has little influence on the $M$ and $\sigma$ of
net-charge multiplicity distributions from the AMPT model.

Figure 5: (Color online) Centrality dependence of mean($M$)(top plots) and
standard deviation($\sigma$)(bottom plots) for net-charge multiplicity
distributions in Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$
with different centrality definitions, where the STAR data are obtained from
Ref. Adamczyk:2014fia .

Figure 6 presents that the centrality dependence of skewness($S$) and
kurtosis($\kappa$) for the net-charge multiplicity distributions with
different centrality definitions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. Regarding skewness, the AMPT results
can describe the experimental data well when we select the centrality
definition as same as in experiment. However, when the $\eta$ range increases
below 27 GeV, we observe that the value of $S$ decreases and seems to saturate
for $0.5<|\eta|<1.5$, which indicates the centrality resolution effect will
enhance the skewness value of net-charge multiplicity distributions. From 27
GeV to 200 GeV, the differences between the results from three different
centrality definitions appear increasingly smaller. This indicates that the
CRE has a larger influence on the value of skewness for lower energies. For
kurtosis, the AMPT results from different centrality definitions are all
similar to the experimental data. This indicates that the CRE has no
significant contribution to the value of kurtosis.

Figure 6: (Color online) Centrality dependence of skewness($S$)(top plots) and
kurtosis($\kappa$)(bottom plots) for net-charge multiplicity distributions in
Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$ with different
centrality definitions, where the STAR data are obtained from Ref.
Adamczyk:2014fia .

Figure 7 presents that the centrality dependence of moment products($S\sigma$)
for the net-charge multiplicity distributions with different centrality
definitions in Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. At
low energies(below 27 GeV), as the $\eta$ range increases, the value of
$S\sigma$ decreases, which can be understood as a result of the CRE on $S$.
The $S\sigma$ describes the experimental data well when the $\eta$ range of
the centrality definition is $0.5<|\eta|<1.0$. At high energies(above 27 GeV),
the values of $S\sigma$ from three different centrality definitions all appear
similar to experimental data.

Figure 7: (Color online) Centrality dependence of moment products $S\sigma$
for net-charge multiplicity distributions in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$ with different centrality definitions,
where the STAR data are obtained from Ref. Adamczyk:2014fia .

Figure 8 presents that the centrality dependence of $\kappa\sigma^{2}$ for the
net-charge multiplicity distributions with different centrality definitions in
Au+Au collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$. From low to high
energy, the values of $\kappa\sigma^{2}$ from three different centrality
definitions are all approximately the constant of unity, which are all
consistent with the experimental data within errors.

Figure 8: (Color online) Centrality dependence of moment products
$\kappa\sigma^{2}$ for net-charge multiplicity distributions in Au+Au
collisions at $\sqrt{s_{NN}}=7.7-200\,\mathrm{GeV}$ with different centrality
definitions, where the STAR data are obtained from Ref. Adamczyk:2014fia .

###  III.3 Energy dependence

As mentioned above, it is believed that moment products, which can eliminate
the volume fluctuation effect, are sensitive to the critical fluctuations.
Hence, we study the energy dependence of moment products, as presented in
Figs. 9 and 10, where both AMPT results with and without the CBWC, are
presented in comparison with the experimental data.

Figure 9: (Color online) Beam-energy dependence of moment products
$\sigma^{2}/M$, $S\sigma$, and $\kappa\sigma^{2}$ in most central ($0-5\%$)
and peripheral ($70-80\%$) bins of Au+Au collisions, where the STAR data are
obtained from Ref. Adamczyk:2014fia .

Figure 9 illustrates the energy dependence of the moment products, which
includes $\sigma^{2}/M$, $S\sigma$, and $\kappa\sigma^{2}$. Regarding
$\sigma^{2}/M$, this moment product exhibits an exponential response to the
energy for a given collision centrality. The $\sigma^{2}/M$ for peripheral
centrality bin ($70-80\%$) is greater than that for central centrality bin
($0-5\%$). In addition, the values of $\sigma^{2}/M$ are similar to the
experimental data whether the CBWC is considered or not. For $S\sigma$ and
$\kappa\sigma^{2}$, we observe that $S\sigma$ decreases as energy increases;
however, $\kappa\sigma^{2}$ is independent of energy. The values of $S\sigma$
or $\kappa\sigma^{2}$ with the CBWC appear systematically smaller than those
without the CBWC. This is consistent with the above result, which indicates
that although the CBWE has little influence on $\sigma$, it enhances $S$ and
$\kappa$ slightly.

Figure 10: (Color online) Beam-energy dependence of moment products
$\sigma^{2}/M$, $S\sigma$, and in most central ($0-5\%$) and peripheral
($70-80\%$) bins of Au+Au collisions with different centrality definitions,
where the STAR data are obtained from Ref. Adamczyk:2014fia .

To study the influence of the CRE on the energy dependence of the moment
products, different centrality definitions have been applied. Figure 10
presents the beam-energy dependence of moment products $\sigma^{2}/M$,
$S\sigma$, and $\kappa\sigma^{2}$ with the CBWC for most central ($0-5\%$) and
peripheral ($70-80\%$) centrality bins with three different centrality
definitions. For $\sigma^{2}/M$, the results with different centrality
definitions are almost same, which indicates the CRE has little influence on
$\sigma^{2}/M$. However, for $S\sigma$ and $\kappa\sigma^{2}$, the value of
the two moment products systematically become smaller as the $\eta$ range
becomes wider, which is consistent with the above result, which indicates that
although the CRE has little influence on $M$ and $\sigma$, it enhances $S$ and
$\kappa$ slightly.

In general, after considering the CBWC and CRE, the AMPT results of moment
products of net-charge multiplicity distribution are consistent with the
experimental measurements. No non-monotonic energy dependence is observed,
which is not surprising because no QCD critical fluctuations are included in
the AMPT model at all.

###  III.4 Stage evolution

It is well known that heavy-ion collisions are actually a complicated
dynamical evolution comprising several important evolution stages. Therefore,
it is important to study the stage evolution of moment products, which can
help us understand the dynamics of the fluctuation observables. This
complements to the results from the Lattice QCD which basically belongs to a
static thermal approach without evolution dynamics. Using the AMPT model, we
focus on the moment products at four evolution stages (i.e. initial stage,
after parton cascade, after coalescence, and after hadron rescatterings) in
Au+Au collisions for two typical energies, 7.7 GeV and 200 GeV.

To elucidate the possible contribution from any dynamics to the moments or
moment products, a reference of Poisson baseline is crucially required. To
obtain the expected moments of net-charge distribution from two independent
Poisson distributions of positively and negatively charged particles, we first
calculate $K_{n}^{Q_{+}}$ and $K_{n}^{Q_{-}}$ for the multiplicity
distributions of positively and negatively charged particles ($Q_{+}$ and
$Q_{-}$), then obtain the Poisson expectation of net charge
$K_{n}^{Poisson}$=$K_{n}^{Q_{+}}+(-1)^{n}K_{n}^{Q_{-}}$, which reflects the
case without any correlation between positively and negatively charged
particles Luo:2017faz ; Tarnowsky:2012vu . To illustrate the close relation
between fluctuation and correlation, $\Delta
K_{n}=K_{2}^{Poisson}-K_{2}^{netq}$ is defined to disclose the difference
between the net-change cumulant $K_{n}^{netq}$ and their corresponding Poisson
expectation $K_{n}^{Poisson}$, with the following relations with correlation
functions:

$\displaystyle\Delta K_{2}$ $\displaystyle\simeq$ $\displaystyle
2C_{2}^{(1,1)}$ (8) $\displaystyle\simeq$ $\displaystyle 2(\langle
Q_{-}Q_{+}\rangle-\langle Q_{-}\rangle\langle Q_{+}\rangle),$

$\displaystyle\Delta K_{3}$ $\displaystyle\simeq$ $\displaystyle
3C_{3}^{(2,1)}-3C_{3}^{(1,2)}$ (9) $\displaystyle\simeq$ $\displaystyle
3(\langle Q_{-}Q_{+}^{2}\rangle-\langle Q_{-}\rangle\langle Q_{+}^{2}\rangle)$
$\displaystyle-3(\langle Q_{-}^{2}Q_{+}\rangle-\langle Q_{-}^{2}\rangle\langle
Q_{+}\rangle)$ $\displaystyle+6\langle Q_{-}\rangle(\langle
Q_{-}Q_{+}\rangle-\langle Q_{-}\rangle\langle Q_{+}\rangle)$
$\displaystyle-6\langle Q_{+}\rangle(\langle Q_{-}Q_{+}\rangle-\langle
Q_{-}\rangle\langle Q_{+}\rangle),$

$\displaystyle\Delta K_{4}$ $\displaystyle\simeq$ $\displaystyle
2C_{2}^{(1,1)}+6C_{3}^{(2,1)}+6C_{3}^{(1,2)}$ (10)
$\displaystyle+4C_{4}^{(3,1)}+4C_{4}^{(1,3)}-6C_{4}^{(2,2)},$

where where $C_{n+m}^{(n,m)}$ represents $n+m$ correlation function for $n$
positively charged particles and $m$ negatively charged particles, which can
be calculated via the relations with factorial moments (See Ref. Bzdak:2016sxg
for details). Note that we only focus on the correlation between positively
and negatively charged particles and ignore the correlation between same-sign
charged particles. Even with the assumption, we could not explicitly express
the formula for Eq. (10) because it is very complicated. It can be easily
observed that the differences between the net-charge moments and their Poisson
expectations originate from two-particle, three-particle or four-particle
correlations between positively and negatively charged particles.

Figure 11: (Color online) $Q_{-}$ vs $Q_{+}$ at different evolution stages of
Au+Au collisions (minus bias) at 7.7 and 200 GeV.

To directly illustrate the two-particle correlation between the multiplicities
of positively and negatively charged particles, figure 11 presents the AMPT
results on $Q_{-}$ vs $Q_{+}$ at four different stages of the Au+Au collisions
(minus bias) at 7.7 and 200 GeV. It can be easily observed that the initial
correlation is weakened by parton cascade and then reduced by coalescence,
however, it is enhanced by hadronic rescatterings. As indicated in Eq. (8),
the change of two-particle correlation could lead to the dynamical evolutions
of $\sigma^{2}/M$ for the net charge distributions in relativistic heavy-ion
collisions.

Figure 12: (Color online) Centrality dependence of moment product
$\sigma^{2}/M$ at different evolution stages of Au+Au collisions at 7.7 and
200 GeV, where the STAR data are obtained from Ref. Adamczyk:2014fia .

Figure 12 presents the centrality dependence of the moment product
$\sigma^{2}/M$ at four different evolution stages of Au+Au collisions at 7.7
and 200 GeV, where both experimental results and Poisson expectations are also
shown. Note that because the experimental results are obtained from the
measured final hadrons, it is only meaningful to compare them with the AMPT
results after hadronic rescatterings. The AMPT model can properly describe the
data for Au+Au collisions at 7.7 GeV, as well as the mid-central and central
bins of Au+Au collisions at 200 GeV; however, it overestimates the data for
peripheral bins of Au+Au collisions at 200 GeV. We also observe that
$\sigma^{2}/M$ gradually increases from initial stage to final stage. When
comparing with the Poisson expectation, it can be observed that the AMPT
result is always lower than the corresponding Poisson expectation, which
indicates that there is a positive correlation between positively and
negatively charged particles, i.e. the two-particle correlation on the right
hand of Eq. (8) is always positive. This indicates that two oppositely charged
particles tend to appear or disappear together. Since the difference changes
gradually stage by stage, which means that the positive correlation develops
during the dynamical evolution of heavy-ion collisions. The possible sources
of the positive two-particle correlation in the AMPT model are presented as
follows. The positive correlation at the initial stage may originate from the
melting process of strings during which the excited strings are decomposed
into their constituent quarks and antiquarks with opposite electric charges.
After parton cascade, the correlation can be modified because parton
collisions can alter the kinematics of partons. However, because the degree of
freedom is altered from the parton to hadron after the hadronization of
coalescence, the correlation is found to be significantly modified. The
difference has almost disappeared, which implies that there is hardly any
correlation after the hadronization. The final hadronic rescatterings can not
only modify the correlation via hadron rescatterings, similar to parton
cascade, but also provide additional correlations owing to resonance decays
that usually transform a mother hadron into two daughter hadrons with opposite
charges. It appears that the residue correlation in the experimental
observable of $\sigma^{2}/M$ is mainly produced in the process of hadronic
rescatterings.

Figure 13: (Color online) Centrality dependence of moment product $S\sigma$ at
different evolution stages of Au+Au collisions at 7.7 and 200 GeV, where the
STAR data are obtained from Ref. Adamczyk:2014fia .

Figure 13 presents the centrality dependence of moment product $S\sigma$ at
four different evolution stages in Au+Au collisions at 7.7 and 200 GeV. The
AMPT model can well describe the experimental data for both energies. For
Au+Au collisions at 200 GeV, the $S\sigma$ at each stages tends to remain
approximately zero. However, we observe there is a more significant stage
evolution of $S\sigma$ in Au+Au collisions at 7.7 GeV, which indicates that
the $S\sigma$ decreases slightly after parton cascade, but increases owing to
the hadronization of coalescence, and decreases again after hadronic
rescatterings. Because the AMPT results are greater than the Poisson
expectations, the total correlation from the third cumulant $K_{3}$ should be
negative in Au+Au collisions at 7.7 GeV. From Eq. (9), it can be observed that
if positively and negatively charged particles are symmetrically related, the
total correlation from $K_{3}$ must be zero. This feature could explain why
the deviation from the Poisson expectation is very small and the stage
evolution of $S\sigma$ is not quite significant in Au+Au collisions at 200
GeV, because it should be expected that the system of Au+Au collisions at 200
GeV is more electrically symmetric (with a lower electric chemical potential)
than that of Au+Au collisions at 7.7 GeV. In contrast, because there are more
positively charged particles than negatively charged particles in the system
of Au+Au collisions at 7.7 GeV, the forementioned positive two-particle
correlation could have a negative impact on the overall correlation of
$K_{3}$, as indicated by the last two lines of Eq. (9).

Figure 14: (Color online) Centrality dependence of of moment product
$\kappa\sigma^{2}$ at different evolution stages of Au+Au collisions at 7.7
GeV and 200 GeV, where the STAR data are obtained from Ref. Adamczyk:2014fia .

Figure 14 presents the centrality dependence of the moment product
$\kappa\sigma^{2}$ at four different evolution stages in Au+Au collisions at
7.7 GeV and 200 GeV. It can be observed that the AMPT results after hadronic
rescatterings can basically describe the experimental data of two energies,
which is slightly higher than the Poisson expectation of unity. From the AMPT
results at different stages, they appear basically consistent with the Poisson
expectation, except those before and after parton cascade for central
centrality bins which could be caused by the two-particle component of four-
particle correlation. According to Eq. (10), the sources of the correlation in
the fourth cumulant $K_{4}$ are significantly complicated, which may be caused
by two-particle, three-particle or four-particle correlations. It is believed
that the four-particle correlation of net-proton multiplicity distribution is
very sensitive to QCD critical fluctuations Abdallah:2021fzj ; Bzdak:2019pkr ;
Bzdak:2016sxg . However, the presence or absence of four-particle correlation
in net-charge multiplicity distribution remains an open question that requires
more detailed investigations in the future.

It is important to mention that the hadron resonance gas (HRG) model
BraunMunzinger:2003zd  has indicated that resonance decays could play an
important role in influencing these moment products, which enhance
$\sigma^{2}/M$ at high colliding energies, suppress $S\sigma$ at low colliding
energies, and almost have no impact on $\kappa\sigma^{2}$ for the entire range
of colliding energy Garg:2013ata ; Mishra:2016qyj . We infer that our results
are basically consistent with the observed effect from the HRG model, when
compared with the results from the last two stages. However, the extent of
impact could be different, as our model includes not only resonance decays but
also hadronic elastic and inelastic reactions during the hadronic phase
evolution. Therefore, it is interesting to separately study the effects from
resonance decays and hadronic reactions in the future.

##  IV Summary

Using a string melting version of a multi-phase transport model, moments and
moment products of net-charge multiplicity distributions in Au+Au collisions
at seven different energies of the BES program at RHIC have been studied. It
was determined that the AMPT model can basically describe the measured
centrality dependences at different energies. After considering the impacts of
the CBWE and CRE, we determined that they can affect the values of moments
especially for more peripheral collisions at lower energies. The measured
energy dependences of the moment products were also optimally reproduced by
the AMPT model, and no non-monotonic energy dependence was observed. Through
the stage evolution of the moment products, we determined that the moment
products develop during the dynamical evolution of heavy-ion collisions. The
deviation from the Poisson expectation indicates the existence of the positive
two-particle correlation between positively and negatively charged particles,
which may be caused by different dynamical processes at different evolution
stages. Therefore, to adopt the moments and moment products of net-charge
multiplicity distributions in determining the QCD critical point of
relativistic heavy-ion collisions, it is important to consider the dynamical
evolution.

## ACKNOWLEDGMENTS

G.L.M. thanks the Department of Physics at East Carolina University for its
hospitality, where part of the work was performed. This work is supported in
part by the Strategic Priority Research Program of Chinese Academy of Sciences
under Grant No. XDB34030000, the National Natural Science Foundation of China
under Contracts No. 11890710, No. 11890714, No. 11835002, and No. 11961131011,
and the Guangdong Major Project of Basic and Applied Basic Research under
Grant No. 2020B0301030008.

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#  __
Two-body hadronic decay of $K^{+}$ in the presence of a circularly polarized
laser field

M Baouahi1, I Dahiri1, M Ouali1, B Manaut1, R Benbrik2 and S Taj1
1 Recherche Laboratory in Physics and Engineering Sciences, Team of Modern
and Applied Physics, FPBM, USMS, Morocco
2 LPFAS, Polydisciplinary Faculty of Safi, UCAM, Morocco.
Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

In this study, we have investigated the two-body hadronic decay of the charged
kaon, $K^{+}\rightarrow\pi^{+}+\pi^{0}$, in the presence of a laser field with
circular polarization. We have derived, by analytical techniques, the laser-
assisted decay width and the branching ratio of the charged kaon decay via the
two-body hadronic channel. We have also taken into consideration the
impressive results obtained for the laser-assisted charged kaon decay via the
leptonic mode in order to understand more clearly the effect of the laser
field on the quantities related to the charged kaon decay such as the decay
width, the branching ratio and lifetime. A precise comparison of the ratios of
hadronic to muonic decay in the presence of the laser field is made to show
that the hadronic mode becomes slightly more important by increasing the laser
field intensity.

Keywords: QED and Weak processes, Laser-assisted processes.

##  1\. Introduction

The interaction between the electromagnetic field and charged particles is a
very interesting research topic in the study of particle physics processes,
especially after the introduction of the laser which is one of the most
important technological discovery in 1960 by Maiman[1]. Today, the
breakthrough progress in laser technology allow us to observe in real time the
electrons and atoms that compose matter, and to deform, in an ultra-fast way,
the molecular structure around the atoms in order to transform materials into
magnetic materials[2]. Moreover some studies and discoveries, have been made
in 2018, have demonstrated that the energy of sub-relativistic electrons is
strongly modified on the scale of a few femtoseconds by the interaction with a
progressive wave created in the vacuum by the collision of two laser pulses of
different frequencies[3].

Studies in the field of laser-matter interactions usually deal with non-
relativistic [4] and relativistic atomic physics [5, 6]. For example, in the
study of Mott scattering of an electron by a charged nucleus in the presence
of an electromagnetic field with a given polarization (linear, circular or
elliptical), they showed that the intensity of the electric field, the
kinetics of the incident electron and the collision geometry influence the
behavior of the cross section[7]. Today, the technological progress gives us
the opportunity to introduce an external electromagnetic field in high energy
physics interactions. Indeed, several studies have been made to study the
impact of the laser field on the decay width[8, 9, 10, 11, 12] and on the
effective cross section [7, 13, 14, 15] by introducing the phenomenon of
emission and/or absorption of photons[16]. For instance, in the study of
nuclear decay $\beta$, there have been debates on the possibility of having
significant effects by the presence of a strong laser pulse [8, 9, 10, 11]. In
[17, 18], the authors illustrated the effect of an electromagnetic field with
circular polarization on the decay widths and lifetime of the charged pion and
the $Z$ boson. In [19], it is found that the circularly polarized laser field
has a great impact on the cross section of the charged Higgs pair production
via electron-positron annihilation. These processes are called laser-assisted
processes for that they may occur also in the presence of the electromagnetic
field.

There is another category of very well known processes in laser physics, and
they are induced by the laser field. We cite, here, some examples in which the
creation of pairs $\\{e^{+},e^{-}\\}$ is induced by the presence of an
external field. In [21, 22, 23], the authors showed that the creation of
electron-positron pairs is possible by a strong plane electromagnetic wave in
the presence of an additional electromagnetic energy source. In [24], the
pairs of $\\{e^{+},e^{-}\\}$ are produced when a high energy photon of wave
four-vector $k^{\prime\mu}$ is propagating in a strong laser field of wave
four-vector $k^{\mu}$. However, the pair production, in this case, is possible
only under the condition that $n(k^{\prime}k)>2m_{e}$ with $n$ representing
the minimal integer for this condition to be valid. Another example of
electron-positron pair production is illustrated in [25, 26] where two strong
laser beams propagate in opposite directions and generate a stationary light
wave. For all these studies, the production of ultrashort pulses is important,
not only because time compression obviously implies an increase of the
intensity at a given laser energy, but also because high intensities allow, in
general, to control fast physical processes[27, 28].

In our previous work [12], we have found impressive results on the laser-
assisted charged kaon decay in the leptonic channel. In addition, we have
indicated that the $CPT$ symmetry might be preserved or broken in both the
muonic and the electronic channels. This is due to the fact that the parameter
associated with the $CPT$ symmetry is affected by the circularly polarized
laser field. Therefore, we can control the dominance of matter over antimatter
or vice versa. In this respect, we have investigated the two-body hadronic
decay of the charged kaon in the presence of an electromagnetic field of
circular polarization by using the same laser parameters as in [12]. Thus, the
aim of this work is to present the analytical and numerical results about the
laser-assisted decay of the charged kaon, which has free a lifetime
$\tau=(1.2380\pm 0.0020)\times 10^{-8}\,s$[29]. Indeed, we have illustrated
the effect of the circularly polarized laser field on different quantities
associated to the two-body hadronic decay of the positive kaon.

The remainder of this work is organized as follows: In the next section (2.),
we will deal with the theoretical and analytical calculation of the charged
kaon decay width in the absence and presence of a circularly polarized laser
field. Then, we will present in section (3.) the obtained data and results. A
short conclusion is given in section (4.). Some theoretical details of the
positive kaon decay width calculations are given in the appendix. We mention
that, in the laboratory system, this disintegration study is represented in
the system of natural units $\hbar=c=1$, and the signature metric of space-
time is chosen as $(+---)$.

##  2\. Theoretical framework

In this part, we will start with the theoretical calculation of the laser-free
two-body hadronic decay of the charged kaon, $K^{+}$, at the lowest order
without taking into account the quarks composition of mesons. In order to
study and analyze the effect of an electromagnetic field on this decay, the
second section will deal with the decay process by dressing the charged
mesons. The plane wave functions used to describe the neutral meson,
$\pi^{0}$, and charged mesons that are involved in the studied decay, in the
absence of the electromagnetic field, are derived from the free Klein-Gordon
equation. However, in the presence of an electromagnetic field, the charged
particle are described by the Volkov functions[16].

###  2.1. Laser-free kaon decay

In the absence of an external field, the transition matrix element $S_{fi}$
which is associated with the decay of the charged kaon $K^{+}$ to
$\\{\pi^{+},\pi^{0}\\}$ is written as a product of two currents such that:

$\displaystyle S_{fi}=\dfrac{-iG_{F}}{\sqrt{2}}\int
d^{4}x\mathcal{J}_{\mu}^{\langle\,K^{+}|\,\mathcal{O}_{1}|\pi^{0}\rangle\dagger}(x)\times\mathcal{J}^{{\langle\,0|\,\mathcal{O}_{2}|\,\pi^{+}\rangle}\mu}(x),$
(1)

with $G_{F}$ represents the Fermi constant. The two hadronic currents
$\mathcal{J}$ [31, 30] are defined as follows:

$\displaystyle\mathcal{J}_{\mu}^{\langle\,K^{+}|\,\mathcal{O}_{1}|\pi^{0}\rangle}(x)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2V\sqrt{E_{1}E_{3}}}\langle\,K^{+}|\,\mathcal{O}_{1}|\pi^{0}\rangle\,e^{i(P_{1}-P_{3}).x}$
(2) $\displaystyle=$
$\displaystyle\dfrac{f_{+}(\Delta)[P_{1}+P_{3}]_{\mu}+f_{-}(\Delta)[P_{1}-P_{3}]_{\mu}}{2V\sqrt{E_{1}E_{3}}}\,e^{i(P_{1}-P_{3}).x},$
(3)
$\displaystyle\mathcal{J}^{{\langle\,0|\,\mathcal{O}_{2}|\,\pi^{+}\rangle}\mu}(x)$
$\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{2E_{2}V}}\langle\,0|\,\mathcal{O}_{2}|\pi^{+}\rangle\,e^{iP_{2}.x}$
(4) $\displaystyle=$
$\displaystyle\dfrac{iP_{2}^{\mu}F_{\pi^{+}}}{\sqrt{2E_{2}V}}\,e^{iP_{2}.x},$
(5)

where $V$, $P_{i}=(E_{i},\vec{p_{i}})$ ($i=\\{1,2,3\\}$) and $\Delta$ are
respectively the quantum volume, the 4-momentum of the hadrons (successively
for $K^{+},\,\pi^{+}$ and $\pi^{0}$) and the transfer momentum such that
$\Delta=(P_{1}-P_{3})^{2}$. $F_{\pi^{+}}$ is the decay constant associated
with the particle $\pi^{+}$. The operator $\mathcal{O}_{1}$ corresponds to the
transition in the vector current $[K^{+},\pi^{0}]$, and $\mathcal{O}_{2}$
corresponds to that in the weak vector-axial current. The form factors
$f_{\pm}$ are given, as in the case of $K^{+}_{e3}$ and $K^{+}_{\mu 3}$
decays, by the following expression[29]:

$\displaystyle
f_{\pm}(\Delta)=f_{\pm}(0)\left[1+\lambda_{\pm}\dfrac{\Delta}{m_{\pi^{+}}^{2}}\right].$
(6)

We use the parameterization $\\{\lambda_{+},\lambda_{0}\\}$ to define the
factors $f_{\pm}$ by the function $f_{0}$ [29] which is given by:

$\displaystyle
f_{0}(\Delta)=f_{+}(\Delta)+\dfrac{\Delta}{m_{K^{+}}^{2}-m_{\pi^{0}}^{2}}f_{-}(\Delta)\qquad\text{with}\qquad
f_{0}(\Delta)=f_{0}(0)\left[1+\lambda_{0}\dfrac{\Delta}{m_{\pi^{+}}^{2}}\right].$
(7)

In this parameterization, we use the universality assumption $\mu-e$ to define
$\lambda_{+}$ and $\lambda_{0}$[29]. After weighting the square of the matrix
element $S_{fi}$ by the phase space and per unit time $T$, the decay width
[32] becomes as follows:

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\dfrac{1}{T}\int\dfrac{Vd^{3}\vec{P}_{2}}{(2\pi)^{3}}\int\dfrac{Vd^{3}\vec{P}_{3}}{(2\pi)^{3}}|\bar{S}_{fi}|^{2}$
(8) $\displaystyle=$ $\displaystyle\dfrac{G_{F}^{2}F_{\pi^{+}}^{2}}{32\pi
m_{K^{+}}}\sqrt{\dfrac{m_{K^{+}}^{4}+m_{\pi^{+}}^{4}+m_{\pi^{0}}^{4}-2m_{K^{+}}^{2}m_{\pi^{+}}^{2}-2m_{K^{+}}^{2}m_{\pi^{0}}^{2}-2m_{\pi^{+}}^{2}m_{\pi^{0}}^{2}}{(m_{K^{+}}^{2}-m_{\pi^{+}}^{2}+m_{\pi^{0}}^{2})^{2}}}$
$\displaystyle\left[(m_{K^{+}}^{2}-m_{\pi^{0}}^{2})f_{+}(m_{\pi^{+}}^{2})+m_{\pi^{+}}^{2}f_{-}(m_{\pi^{+}}^{2})\right]^{2}.$

###  2.2. Laser-assisted kaon decay

An electromagnetic field with a circular polarization is described by the
classical 4-vector potential
$A^{\mu}(\phi)=a_{1}^{\mu}\cos(\phi)+a_{2}^{\mu}\sin(\phi)$, where $\phi=k.x$
is its phase. The 4-vector $A^{\mu}$ verifies the transversality condition
$k.A=0$ (Lorentz gauge), with $k=(\omega,\vec{k})$ is the 4-vector wave. The
quadri-polarizations $a_{1}^{\mu}=|a|(0,1,0,0)$ and
$a_{2}^{\mu}=|a||(0,0,1,0)$ are equal in magnitude such that
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|a|^{2}=-(\varepsilon_{0}/\omega)^{2}$, and they
verify the orthogonality relation $a_{1}a_{2}=0$. The quantity
$\varepsilon_{0}$ represents the amplitude of the electric field of the laser
while $\omega$ is its frequency. By adding the formalism of the laser field to
describe the decay $K^{+}\rightarrow\pi^{+}+\pi^{0}$, the Klein-Gordon
equation in the presence of an electromagnetic field is given by:

$\displaystyle\left[(i\partial-eA)^{2}-m_{p}^{2}\right]\psi_{p}(x)=0,$ (9)

with $m_{p}$ may be either the mass $m_{K^{+}}$ of the charged kaon or
$m_{\pi^{+}}$ of the charged pion. The equation (9) admits as solution the
following Volkov functions:

$\displaystyle\psi_{K^{+}}(x)=\dfrac{1}{\sqrt{2Q_{1}V}}e^{iS(q_{1},x)},\qquad\text{and}\qquad\psi_{\pi^{+}}(x)=\dfrac{1}{\sqrt{2Q_{2}V}}e^{iS(q_{2},x)},$
(10)

where:

$\displaystyle
S(q_{j},x)=q_{j}.x-e\dfrac{a_{1}.P_{j}}{k.P_{j}}\sin\phi+e\dfrac{a_{2}.P_{j}}{k.P_{j}}\cos\phi,$
(11)

$q_{j}=(Q_{j},\vec{q_{j}})$ ($j=\\{1,2\\}$) is the four-momentum of the
charged particle in the presence of the laser field with $Q_{j}$ denotes its
effective energy. It is related to its corresponding free momentum by:
$q_{j}=P_{j}-[(e^{2}a^{2})/(2k.P_{j})]k$, such that
${q_{j}}^{2}={P_{j}}^{2}-e^{2}a^{2}=m_{p}^{2}-e^{2}a^{2}={m_{p}^{*}}^{2}$,
where $m_{p}^{*}$ is the effective mass of the charged particle. In the
presence of the electromagnetic field, the matrix element $S^{\prime}_{fi}$ in
the first Born approximation is given by the following expression:

$\displaystyle S^{\prime}_{fi}$ $\displaystyle=$
$\displaystyle\dfrac{G_{F}}{\sqrt{2}}\int
d^{4}x\dfrac{F_{\pi^{+}}}{\sqrt{8Q_{1}Q_{2}E_{3}V^{3}}}\left[f_{+}(\Delta^{\prime})[q_{1}+P_{3}]_{\mu}q_{2}^{\mu}+f_{-}(\Delta^{\prime})[q_{1}-P_{3}]_{\mu}q_{2}^{\mu}\right]$
(12) $\displaystyle\times$ $\displaystyle
e^{i(-S(q_{1},x)+S(q_{2},x)+P_{3}.x)},$

where $\Delta^{\prime}$ is the new 4-momentum transfer. To expand the element
$S^{\prime}_{fi}$, we will perform some transformations in order to find an
element expandable into a series of ordinary Bessel functions, with argument
$z$ and phase $\phi_{0}$ such that:

$\displaystyle
z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\qquad\text{and}\qquad\phi_{0}=\arctan(\dfrac{\alpha_{2}}{\alpha_{1}}),$
(13)

where:

$\displaystyle\alpha_{1}=e(\dfrac{a_{1}.P_{1}}{k.P_{1}}-\dfrac{a_{1}.P_{2}}{k.P_{2}})\qquad\text{and}\qquad\alpha_{2}=e(\dfrac{a_{2}.P_{1}}{k.P_{1}}-\dfrac{a_{2}.P_{2}}{k.P_{2}}),$
(14)

The exponential term in equation (12) takes the following form:

$\displaystyle-S(q_{1},x)+S(q_{2},x)+P_{3}.x=-(q_{1}-q_{2}-P_{3}).x+z\sin(\phi-\phi_{0}).$
(15)

The decay matrix element can be developed using the following ordinary Bessel
transformation:

$\displaystyle
e^{iz\sin(\phi-\phi_{0})}=\sum_{n=-\infty}^{n=+\infty}J_{n}(z)e^{-in\phi_{0}}e^{in\phi}=\sum_{n=-\infty}^{n=+\infty}\boldsymbol{B}n(z)e^{in\phi}.$
(16)

After some substitutions using the equations (15) and (16), the matrix element
$S^{\prime}_{fi}$ becomes:

$\displaystyle S^{\prime}_{fi}$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\dfrac{G_{F}F_{\pi^{+}}(2\pi)^{4}\delta^{4}(q_{2}+P_{3}-q_{1}+nk)}{4\sqrt{Q_{1}Q_{2}E_{3}V^{3}}}\left[f_{+}(\Delta^{\prime})[q_{1}+P_{3}]_{\mu}q_{2}^{\mu}+f_{-}(\Delta^{\prime})[q_{1}-P_{3}]_{\mu}q_{2}^{\mu}\right]\boldsymbol{B}n(z)$
(17) $\displaystyle=$ $\displaystyle\sum_{n=-\infty}^{n=+\infty}S^{\prime
n}_{fi}.$

Following the same steps as in the absence of the laser field, the decay width
takes the following form:

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\dfrac{G_{F}^{2}F_{\pi^{+}}^{2}}{64\pi^{2}Q_{1}}\int\dfrac{d^{3}q_{2}}{Q_{2}}\int\dfrac{d^{3}P_{3}}{E_{3}}\delta^{4}(q_{2}+P_{3}-q_{1}+nk)|\mathcal{M}_{fi}^{n}|^{2}=\sum_{n=-\infty}^{n=+\infty}\Gamma^{n}$
(18) $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\dfrac{G_{F}^{2}F_{\pi^{+}}^{2}}{64\pi^{2}Q_{1}}\int
d|\vec{q}_{2}|F(|\vec{q}_{2}|)\delta(G(|\vec{q}_{2}|)),$

where $\mathcal{M}_{fi}^{n}$ represents the laser-assisted decay amplitude,
and it is given by:

$\displaystyle|\mathcal{M}_{fi}^{n}|^{2}=\left[q_{1}q_{2}\left(f_{+}(\Delta^{\prime})+f_{-}(\Delta^{\prime})\right)+q_{2}P_{3}\left(f_{+}(\Delta^{\prime})-f_{-}(\Delta^{\prime})\right)\right]^{2},$
(19)

with $\Delta^{\prime}={m_{K^{+}}^{*}}^{2}+m_{\pi^{0}}^{2}-2q_{1}P_{3}$ is the
4-momentum transfer in the presence of the laser field. The functions
$F(|\vec{q}_{2}|)$ and $G(|\vec{q}_{2}|)$ are defined in the appendix. Since
the charged kaon may decay via fifty decay modes[29], and since our purpose is
to study the effect of the laser field on both the hadronic channel
$\\{\pi^{+},\pi^{0}\\}$ and the leptonic channel, we define the total decay
width of the charged kaon $\Gamma^{T}$ as follows:

$\displaystyle\Gamma^{T}=\Gamma+\Gamma^{Lep.}+\Gamma^{O.C.},$ (20)

where $\Gamma$ is the two-body hadronic decay width given by (18),
$\Gamma^{Lep}$ represents the laser-assisted leptonic decay width of the
positive kaon, and $\Gamma^{O.C.}=8.38365\times 10^{-9}\,eV$ is the sum of
decay widths which correspond to other channels. The branching ratio $BR$ and
the lifetime $\tau$ are defined as:

$\displaystyle
BR=\dfrac{\Gamma}{\Gamma^{T}},\qquad\qquad\tau=\dfrac{1}{\Gamma^{T}}.$ (21)

Experimentally the ratio of the partial free-decay width of
$K^{+}\longrightarrow\pi^{+}+\pi^{0}$ to that of the leptonic free-decay
$K^{+}\longrightarrow\mu^{+}+\nu_{\mu}$ of positive kaon is equal to
$0.3325\pm 0.0032$[29].

##  3\. Results and discussion

After the theoretical treatment of the laser-assisted two-body hadronic decay
of the charged kaon, we will now discuss the different obtained results about
the action of a monochromatic and discrete electromagnetic field with circular
polarization on this decay. To do this, we will represent theses results in a
spherical geometry chosen such that the angle $\varphi$ associated with the
produced particle $\pi^{+}$ be zero, and the wave vector $\vec{k}$ of the
laser field propagates along the direction of the $z$-axis. We have calculated
the parameters $f_{+}(\Delta)$ and $f_{-}(\Delta)$ in the absence of the laser
field by using the value of $f_{+}(0)=0.982\pm 0.008$[29] in the
parameterization $\\{\lambda_{+},\lambda_{0}\\}$. The constant $(G_{F}\times
F_{\pi^{+}})^{2}$ is obtained by normalizing the free decay width, expressed
in equation (8), by its experimental value $1.09897\times 10^{-8}\,eV$[29]. As
a first step, we will present the effect of the electromagnetic field on the
number of emissions and absorptions of photons $n$ in order to introduce the
notion of cut-off.

Figure 1: A and B (top): Differential partial decay width of the charged kaon,
$\Gamma^{n}(K^{+}\rightarrow\pi^{+}+\pi^{0})$ (18), as a function of the
photons number $n$ for $\hbar\omega=1.17\,eV$ and for different values of
$\varepsilon_{0}$. The spherical coordinates are chosen such that
$\theta=90^{\circ}$ and $\varphi=0^{\circ}$. C and D (bottom): Dependence of
the laser-assisted differential decay width on the angle $\theta$ for
different numbers of exchanged photons $n$ and for different values of
$\hbar\omega$ and $\varepsilon_{0}$.

According to figure 1, we observe that the variation of the laser field
parameters induces a variation of the partial decay width,
$d\Gamma^{n}/d\theta$, as a function of the number of photons transferred
between the laser field and the decaying system. We start with the figure
(1-A), where the dressing of the charged particles has been made by a Nd:YAG
laser ($\hbar\omega=1.17\,eV$) with a strength $\varepsilon_{0}=2\times
10^{5}\,V/cm$. In this case, the number of possible photons to be exchanged is
$n=3$. However, when the strength of this laser (Nd:YAG laser) is increased to
reach $10^{7}\,V/cm$ as it is shown in the figure (1-B), the possible number
of photons $n$ that can be exchanged may be more than $100$ photons. Outside
of this range of photons numbers, we can see that the value of
$d\Gamma^{n}/d\theta$ becomes zero. This means that the integral of
$d\Gamma^{n}/d\theta$ over the angle $\theta$, formed by the direction of the
outgoing particle $\pi^{+}$ with the $z$-axis (for a value of $\theta$ equal
to $90^{\circ}$), becomes a constant for any number of photons $n$ greater
than the possible number of photons that can be exchanged. The maximum number
of photons that can be exchanged (absolute value of $n$) represents the notion
of ”cut-off” for the laser field applied with a well defined intensity.
Moreover, for a number of photons which is higher than the cut-off, the
obtained constant value of $\Gamma$ represents also the value of the decay
width in the absence of an external field. To illustrate more clearly these
results, we have shown in figures (1-C) and (1-D) the effect of the number of
transferred photons $n$ on the differential decay width $d\Gamma/d\theta$ for
a well-defined laser field strength and by varying the angle $\theta$ from
$1^{\circ}$ to $179^{\circ}$. The results presented in these figures are in
full agreement with that presented in figures (1-A) and (1-B). Indeed, the
differential decay width in the presence of the laser field coincides with
that in the absence of the laser field (red curve in figure (1-C) for
$\theta=90^{\circ}$) when we sum over $n$ from $-4$ to $4$ or from $-20$ to
$20$. In addition, we observe that in the case of a summation over $n$ from
$-20$ to $20$ in both figure (1-C) and (1-D), the laser-assisted differential
decay width (green curve) is equal to its corresponding laser-free
differential decay width for all values of $\theta$ between $1^{\circ}$ and
$179^{\circ}$. This result indicates that $|n|=20$ exceeds the cut-off which
corresponds to both ($\varepsilon_{0}=2\times 10^{5}\,V/cm$;
$\hbar\omega=1.17\,eV$) and ($\varepsilon_{0}=10^{6}\,V/cm$;
$\hbar\omega=2\,eV$). In the following section, we will focus on the impact of
the laser field on both the branching ratio and the lifetime, which are
expressed in the equation (21).

Figure 2: A and B (top): Branching ratio of the charged kaon decay as a
function of the laser field strength $\varepsilon_{0}$. A: for different
values of $\hbar\omega$ and by summing over $n$ from $-10$ to $10$. B: for
different values of $n$ and for $\hbar\omega=1.17\,eV$. C and D (bottom):
Lifetime of the positive kaon as a fucntion of the laser field strength. C:
for different known laser frequencies and by taking the number of exchanged
photons as ranging from $-10\leq n\leq 10$. D: for different values of $n$ and
for $\hbar\omega=1.17\,eV$.

Figure (2-A) represents the variation of the branching ratio as a function of
the laser field strength for different frequencies
$\hbar\omega=\\{0.117\,eV,\,1.17\,eV\,\text{and}\,2\,eV\\}$ and by summing
over the exchanged photons number from $-10$ to $10$. It is well known that in
the absence of the laser field the branching ratio is experimentally equal to
$(20.67\pm 0.08)\times 10^{-2}$[29]. According to this figure (2-A), we
observe that, for low intensities, the laser field doesn’t affect the
branching ratio. However, the branching ratio begins to decrease progressively
from the laser field strength $\varepsilon_{0}=10^{3}\,V/cm$ for
$\hbar\omega=0.117\,eV$. For the $CO_{2}$ laser ($\hbar\omega=0.117\,eV$), and
for $\varepsilon_{0}=10^{12}\,V/cm$, the branching ratio is equal to
$1.686\times 10^{-8}\,s$. Moreover, with this strength value, we mention that
the decay process can exchange more than $100000$ photons. Therefore, the
interpretation of the branching ratio decrease is more significant for
strengths in the order of $10^{4}\,V/cm$. For the two other lasers, the
branching ratio begins to decrease from $\varepsilon_{0}=10^{5}\,V/cm$ for a
Nd:YAG laser ($\hbar\omega=1.17\,eV$), and also from the
$\varepsilon_{0}=10^{6}\,V/cm$ for the He:Ne laser ($\hbar\omega=2\,eV$). In
the figure (2-B) and for the Nd:YAG laser, the branching ratio always
decreases by increasing the laser field strength, regardless of the number of
exchanged photons $n$. The cases where the branching ratio is constant
corresponds to the strengths (low laser field strengths) for which the process
can only exchange a number of photons $|n|$ lower than that shown in the
figure. This means that the number of photons exchanged exceeds the cut-off.
Now, let’s move to the figures (2-C) and (2-D) that illustrate the variation
of the laser-assisted lifetime of the charged kaon. It is clear that the
lifetime increases as a function of the laser field strength especially when
its strength overcome a threshold value. Then, it stagnates as the laser field
strength reaches a certain value. As we have seen for the branching ratio, the
lifetime of the charged kaon (figure (2-C)) in the presence of the laser field
is more significant for the laser field strengths
$\varepsilon_{0}=10^{4}\,V/cm$, $\varepsilon_{0}=10^{6}\,V/cm$ and
$\varepsilon_{0}=10^{7}\,V/cm$ and for the frequencies
$\hbar\omega=0.117\,eV$, $\hbar\omega=1.17\,eV$ and $\hbar\omega=2\,eV$,
respectively. By comparing these results with those obtained in the case of
laser-assisted leptonic decay, we notice that the branching ratio and lifetime
of the laser-assisted two-body hadronic decay of the charged kaon behaves in
the same manner as in the case of leptonic dacy
($K^{+}\longrightarrow\mu^{+}+\nu_{\mu}$). However, we remark that the
lifetime is longer in the case of the hadronic dressing channel[12]. In figure
(3) the ratio of the partial decay width of
$K^{+}\longrightarrow\pi^{+}+\pi^{0}$ to that of
$K^{+}\longrightarrow\mu^{+}+\nu_{\mu}$ is considered.

Figure 3: A: Ratio of the two-body charged kaon haronic decay width to the
muonic decay width as a function of the laser field strength,
$\varepsilon_{0}$, for different values of $\hbar\omega$ and by summing over
$n$ from $-10$ to $10$. B: The decay width of
$K^{+}\rightarrow\pi^{+}+\pi^{0}$ as a function of the laser field strength
for different values of $\hbar\omega$ and by summing over $n$ from $-10$ to
$10$.

Figure (3-A) illustrates the ratio of the $K^{+}\rightarrow\pi^{+}+\pi^{0}$ to
that of $K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ for different known laser
frequencies and by taking $n$ as ranging from $-10$ to $10$. We recall that
the experimental ratio of the two decay widths is equal to $0.3252\pm
0.0016$[29], and the branching ratio in the muonic channel is equal to
$(63.56\pm 0.11)\%$ in the absence of the laser. We also mention that the
results of the two-body muonic decay of the charged kaon in the presence of a
laser field are taken from our previous work[12]. According to figure (3-A),
it is clear that, for different laser field frequencies, the studied ratio
undergoes an augmentation especially for the laser field strengths that allow
the exchange of a number $|n|=10$ of photons (e.g. the strengths which are
greater than $\varepsilon_{0}=\\{10^{3}\,V/cm$ for the case of
$\hbar\omega=0.117\,eV$ ). As an hypothesis, the increase of the ratio of the
hadronic decay width to the muonic decay width may be interpreted by the fact
that the laser field increases the two-body hadronic decay width of the
charged kaon, and decreases its muonic decay width. Another hypothesis is that
both the hadronic and muonic decay width increase or decrease, but they change
with different rhythms. To check the validity of our hypotheses, we have
presented different values of the ratio
$\Gamma\\{\pi^{+},\pi^{0}\\}/\Gamma\\{\mu^{+},\nu_{\mu}\\}$ in the table
associated to the figure (3-B) to illustrate the variation of the decay width
of $K^{+}\rightarrow\pi^{+}+\pi^{0}$ as a function of the laser field
strength. As presented in figure (3-B), the two-body hadronic decay width of
the charged kaon decreases by increasing the laser field strength. This means
that our second hypothesis is the correct one. Therefore, the laser field has
a strong effect of decreasing the charged kaon muonic decay width,
$K^{+}\longrightarrow\mu^{+}+\nu_{\mu}$, as compared to its effect on the
hadronic decay width presented in figure (3-B). We should mention that in the
histogram which corresponds to $\hbar\omega=2\,eV$ and for the intensities
$\varepsilon_{0}=\\{10^{3}\,V/cm,\,10^{4}\,V/cm,\,10^{5}\,V/cm\,\text{and}\,10^{6}\,V/cm\\}$
the decay width has different values others than the ones represented in
(3-B). The real value of this decay width are successively
$\Gamma=1.09897\times 10^{-8}\,eV$ and $\Gamma=1.09894\times 10^{-8}\,eV$ for
the intensities
$\varepsilon_{0}=\\{10^{3}\,V/cm,\,10^{4}\,V/cm,\,10^{5}\,V/cm\\}$ and
$\varepsilon_{0}=10^{6}\,V/cm$. This change is made to illustrate clearly the
decrease of the decay width. After studying the effect of the circularly
polarized laser field on various measurable quantities, we will now move to
the last obtained result which represents the simultaneous dependence of the
charged kaon decay width on both the laser field strength and the number of
exchanged photons for $\theta=90^{\circ}$.

Figure 4: Variation of the charged kaon decay width for different values of
the laser field strength and for different ranges of transferred photons $n$.
Left pannel (A): $\hbar\omega=1.17\,eV$, right pannel (B):
$\hbar\omega=2\,eV$.

Figure 4 represents some values of the charged kaon decay width for different
combinations of the laser field and the number of exchanged photons. The range
of laser field strengths
$\varepsilon_{0}=10^{5}\,V/cm\rightarrow\varepsilon_{0}=10^{7}\,V/cm$ have
been selected based on the results obtained in figures (2) and (3). As figure
4 shows, the red zone corresponds to the maximum decay width. By comparing the
results obtained in figures (1-A) and (1-B) with those of figures (4-A) and
(4-B), we can deduce that the number of photons in this zone are higher than
its corresponding cutoffs. The violet zone in both figures (4-A) and (4-B)
corresponds to values of the charged kaon decay width which are near to zero.
This zone is less probable since the required photons number representing the
cutoff in this zone is very large than that shown in these figures. Therefore,
we predict that the charged kaon decay width is more significant in the region
between the red and violet zones. Another important point to be mentioned,
here, is that by comparing the results of figure (4-A) with those of figure
(4-B), for the same range of laser strengths, it appears that the use of high
frequencies requires high laser field strengths in order to introduce the
phenomena of emission and/or absorption of photons. In contrast, for the
Nd:YAG laser, low laser field strengths can induce the exchange of photons
with the decaying system.

##  4\. Conclusion

In this paper and based on the interesting results found for laser-assisted
leptonic decay [12], we have investigated the charged kaon decay in the
hadronic channel in the presence of a circularly polarized laser field. By
theoretically analyzing the effect of the laser field on different quantities
related to the decay, we have found that the two-body positive kaon decay in
the hadronic channel is modified by a classical monochromatic laser field with
circular polarization. In addition, the decay width decreases by increasing
the laser field strength, and the emission and/or absorption phenomena appear
and increase by increasing the laser field strength. Moreover, the study of
mesons as particles without structure introduces the decrease of the branching
ratio with an increase in the lifetime. We have also found that the ratio of
the hadronic decay width to the muonic decay width increases by increasing the
strength of the laser field. This result is due to the strong and rapid
decrease of the muonic decay width as compared to the two-body hadronic decay
width. All these results invite us to study other channels of the charged kaon
decay in the presence of the electromagnetic field in order to have a complete
picture about its effect on the decay.

## Appendix

To find the final expression of $\Gamma$ in the equation (18), we use the
following integral property:

$\displaystyle\int
F(x)\delta(G(x))dx={\dfrac{F(x_{0})}{|G^{\prime}(x_{0})|}}\qquad\text{with}\qquad
G(x_{0})=0,$ (22)

where the expression of $F$ and $G$ are determined as:

$\displaystyle F(|\vec{q}_{2}|)$ $\displaystyle=$
$\displaystyle\dfrac{|\mathcal{M}_{fi}^{n}|^{2}|\vec{q}_{2}|^{2}}{\sqrt{|\vec{q}_{2}|^{2}+m_{\pi^{+}}^{2}}\sqrt{m_{\pi^{0}}^{2}+|\vec{q}_{2}|^{2}+\\{(n+\dfrac{e^{2}a^{2}}{2kP_{1}})\omega\\}^{2}-2(n+\dfrac{e^{2}a^{2}}{2kP_{1}})\omega|\vec{q}_{2}|\cos(\theta)}},$
(23) $\displaystyle G(|\vec{q}_{2}|)$ $\displaystyle=$ $\displaystyle n\omega-
Q_{1}+\sqrt{|\vec{q}_{2}|^{2}+m_{\pi^{+}}^{2}}+\sqrt{m_{\pi^{0}}^{2}+|\vec{q}_{2}|^{2}+\\{(n+\dfrac{e^{2}a^{2}}{2kP_{1}})\omega\\}^{2}-2(n+\dfrac{e^{2}a^{2}}{2kP_{1}})\omega|\vec{q}_{2}|\cos(\theta)}.$

In this case, the expression of $|\vec{q}_{2}|$ will be as follows:

$\displaystyle|\vec{q}_{2}|$ $\displaystyle=$
$\displaystyle\\{2n\omega(m_{\pi^{0}}^{2}-{m_{\pi^{+}}^{*}}^{2}-E_{1}^{2})(kP_{1})^{2}\cos(\theta)+e^{4}a^{4}\omega^{2}E_{1}\cos(\theta)$
(25) $\displaystyle+$ $\displaystyle
4n^{2}\omega^{2}E_{1}(kP_{1})^{2}\cos(\theta)+e^{2}a^{2}\omega(kP_{1})\cos(\theta)\times(m_{\pi^{0}}^{2}-{m_{\pi^{+}}^{*}}^{2}-E_{1}^{2}+4n\omega
E_{1})$ $\displaystyle+$
$\displaystyle[(e^{2}a^{2}\omega-2(kP_{1})(E_{1}-n\omega))^{2}\times(e^{4}a^{4}\omega^{2}(E_{1}^{2}-{m_{\pi^{+}}^{*}}^{2}\sin(\theta)^{2})+2e^{2}a^{2}\omega(kP_{1})(E_{1}(m_{\pi^{0}}^{2}$
$\displaystyle-$ $\displaystyle E_{1}^{2}+2n\omega
E_{1})+{m_{\pi^{+}}^{*}}^{2}(E_{1}-2n\omega\sin(\theta)^{2}))+(kP_{1})^{2}(m_{\pi^{+}}^{4}+(m_{\pi^{0}}^{2}-E_{1}^{2}+2n\omega
E_{1})^{2}$ $\displaystyle-$ $\displaystyle
2m_{\pi^{+}}^{2}(m_{\pi^{0}}^{2}+E_{1}^{2}-2n\omega
E_{1}+2n^{2}\omega^{2}\sin(\theta)^{2})))]^{1/2}\\}/\\{e^{4}a^{4}\omega^{2}\sin(\theta)^{2}+4e^{2}a^{2}\omega(kP_{1})(n\omega\sin(\theta)^{2}$
$\displaystyle-$ $\displaystyle E_{1})+4(kP_{1})^{2}(E_{1}^{2}-2n\omega
E_{1}+n^{2}\omega^{2}\sin(\theta)^{2})\\}.$

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