# Search for charged lepton flavor violation of vector mesons in the
$\mathrm{U}(1)_{X}\mathrm{SSM}$

Xing-Xing<EMAIL_ADDRESS>Shu-Min
<EMAIL_ADDRESS>Jia-Peng Huo1,2,3, Tong-Tong Wang1,2,3, Yi-Tong
Wang1,2,3, Tai-Fu<EMAIL_ADDRESS>1 College of Physics Science
$\&$ Technology, Hebei University, Baoding, 071002, China
2 Hebei Key Laboratory of High-precision Computation and Application of
Quantum Field Theory, 071002, China
3 Hebei Research Center of the Basic Discipline for Computational Physics,
Baoding, 071002, China
4 Department of Physics, Chongqing University, Chongqing, 401331, China

###### Abstract

Charged lepton flavor violation (CLFV) represents a clear new physics (NP)
signal beyond the standard model (SM). In this work, we investigate the CLFV
decays of vector mesons $V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$ in the
$\mathrm{U}(1)_{X}\mathrm{SSM}$. Considering the SM-like Higgs boson mass
within $3\sigma$ region, we discuss the corresponding constraints on the
relevant parameter space of the model, which indicate this model can produce
significant contributions to such CLFV decays. From the numerical analyses,
the main sensitive parameters and CLFV sources originate from the non-diagonal
elements correspond to the initial and final generations of the leptons. And
the branching ratios to these CLFV processes can easily reach the present
experimental upper bounds. Therefore, searching for CLFV processes of vector
mesons may be an effective channel to study new physics.

Supersymmetric Model, Charged Lepton Flavor Violation, Vector Meson Decay

###### pacs:

12.60.-Jv, 13.35.-r, 13.20.Gd

##  I Introduction

The standard model(SM) is considered as a much mature theory. However, the
neutrino oscillation experiments have convinced that neutrinos possess tiny
masses and mix with each otherneutrino1 ; neutrino2 ; neutrino3 ; neutrino4 ;
neutrino5 , which indicate that the charged lepton flavor violation (CLFV) is
strongly suppressed in the SMSMLFV . Therefore, some new physical (NP) models
beyond the SM, which can easily generate CLFV precesses, have emerged. If the
CLFV signals are observed in the future experiments, it is obvious evidence of
the NP beyond the SM.

On the base of the minimal supersymmetric standard model (MSSM)MSSM1 ; MSSM2 ;
MSSM3 ; MSSM4 , $\mathrm{U}(1)_{X}\mathrm{SSM}$ extends the gauge symmetry
group to $SU(3)_{C}\otimes{SU(2)_{L}}\otimes{U(1)_{Y}}\otimes{U(1)_{X}}$SARAH1
; U(1)X2 SARAH ; SARAH2 . In this model, right-handed neutrinos and three
Higgs singlets are added to MSSM. The right-handed neutrinos not only produce
tiny masses to light neutrinos through seesaw mechanism, but also provide a
new dark matter candidate. Besides, through the additional singlet Higgs
states and right-handed (s)neutrinos, this model alleviates the hierarchy
problem that appears in the MSSM. As the singlet Higgs superfield, $\hat{S}$
obtains a non-zero vacuum expectation value (VEV) $(v_{S}/\sqrt{2})$ after
spontaneous breaking. Then the effective
$\mu_{\mathrm{eff}}=\mu+\lambda_{H}v_{S}/\sqrt{2}$ term generated from
$\mu\hat{H}_{u}\hat{H}_{d}$ and $\lambda_{H}\hat{S}\hat{H}_{u}\hat{H}_{d}$ can
relieve the $\mu$ problem existing in the MSSM.

In this work, we investigate the CLFV processes of vector mesons $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$ with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$ in the
$\mathrm{U}(1)_{X}\mathrm{SSM}$. In TABLE 1, we show the latest experimental
data for these CLFV processes at $90\%$ confidence level (CL)Phiemuexp ;
Jpsaiemuexp ; Jpsaietauexp ; Jpsaimutauexp ; R(1S)ljliexp ;
R(2S3S)etaumutauexp ; R(3S)emuexp . The SND detector at the VEPP-2M
$e^{+}e^{-}$ collider reported the branching fraction of $\phi\rightarrow
e\mu$ in 2010. BESII amd BESIII experiments gave the branching ratios of
$J/\Psi\rightarrow\mu\tau$ and $J/\Psi\rightarrow e\mu(e\tau)$ respectively,
especially the upper limit of $Br(J/\Psi\rightarrow e\tau)$ was renewed and
reached to $7.5\times 10^{-8}$ in 2022. The upper limits of CLFV decays
$\Upsilon(1S)\rightarrow l_{i}l_{j}$ were collected by the Belle detector at
the KEKB collider in 2022. The BABAR detector discussed the experimental upper
bounds of $Br(\Upsilon(2S)(\Upsilon(3S))\rightarrow e\tau(\mu\tau))$, and
further researched the experiment limits of $Br(\Upsilon(3S)\rightarrow e\mu)$
in 2022 at the SLAC PEP-II $e^{+}e^{-}$ collider. These processes have been
discussed in various theoretical frameworks, such as MRSSMMRSSM , MSSM with
type I seesaw mechanismMSSM type I1 ; MSSM type I2 , $Z^{\prime}$ modelsZ'
model , leptoquark, SUSY, TC2 modelsTC2 model  and so on. In our previous
work, we investigate the CLFV processes of vector mesons in the
BLMSSMmesondecayBLMSSM , and some of the theoretical evaluations fit better
with the experimental upper bounds, which provide references and guidance for
this work. Considering the SM-like Higgs boson mass within experimental
$3\sigma$ regionHiggsmassexp1 ; Higgsmassexp2 ; Higgsmassexp3  and the
constraints from some CLFV process, such as $\mu\rightarrow e\gamma$mutoerexp
and $Z\rightarrow e\mu$Ztoemuexp , we discuss the corresponding constraints on
the relevant parameter space of the $\mathrm{U}(1)_{X}\mathrm{SSM}$. Through
detailed analyses of these CLFV processes of vector mesons, we hope to reveal
some properties of high energy physics.

Table 1:  The latest experiment limits for the CLFV decay ratios of vector mesons. CLFV process | Present limit | CLFV process | Present limit | CLFV process | Present limit
---|---|---|---|---|---
$\phi\rightarrow e\mu$ |  $2.0\times 10^{-6}$Phiemuexp  | $J/\Psi\rightarrow e\tau$ |  $7.5\times 10^{-8}$Jpsaietauexp  | $J/\Psi\rightarrow\mu\tau$ |  $2.0\times 10^{-6}$Jpsaimutauexp
$J/\Psi\rightarrow e\mu$ |  $1.6\times 10^{-7}$Jpsaiemuexp  | $\Upsilon(1S)\rightarrow e\tau$ |  $2.7\times 10^{-6}$R(1S)ljliexp  | $\Upsilon(1S)\rightarrow\mu\tau$ |  $2.7\times 10^{-6}$ R(1S)ljliexp
$\Upsilon(1S)\rightarrow e\mu$ |  $3.9\times 10^{-7}$R(1S)ljliexp  | $\Upsilon(2S)\rightarrow e\tau$ |  $3.2\times 10^{-6}$R(2S3S)etaumutauexp  | $\Upsilon(2S)\rightarrow\mu\tau$ |  $3.3\times 10^{-6}$ R(2S3S)etaumutauexp
$\Upsilon(3S)\rightarrow e\mu$ |  $3.6\times 10^{-7}$R(3S)emuexp  | $\Upsilon(3S)\rightarrow e\tau$ |  $4.2\times 10^{-6}$R(2S3S)etaumutauexp  | $\Upsilon(3S)\rightarrow\mu\tau$ |  $3.1\times 10^{-6}$R(2S3S)etaumutauexp

This work is organized as follows. In Sec.II, we introduce the
$\mathrm{U}(1)_{X}\mathrm{SSM}$ briefly. In Sec.III, we derive the analytic
expressions for the CLFV ratios of vector mesons in the
$\mathrm{U}(1)_{X}\mathrm{SSM}$. The numerical analyses are given out in
Sec.IV, and the conclusion is discussed in Sec.V. The tedious couplings,
hadron matrix elements and one-loop functions are collected in Appendix A, B
and C, respectively.

##  II The $\mathrm{U}(1)_{X}\mathrm{SSM}$

Adding the $\mathrm{U}(1)$ extension on the MSSM, the local gauge group of
$\mathrm{U}(1)_{X}\mathrm{SSM}$ is
$\mathrm{SU}(3)_{C}\otimes\mathrm{SU}(2)_{L}\otimes$
$\mathrm{U}(1)_{Y}\otimes\mathrm{U}(1)_{X}$. Comparing with MSSM,
$\mathrm{U}(1)_{X}\mathrm{SSM}$ considers new superfields such as three Higgs
singlets $\hat{\eta},\;\hat{\bar{\eta}},\;\hat{S}$ and right-handed neutrinos
$\hat{\nu}_{i}$. The three light neutrinos can obtain tiny masses at tree
level through the seesaw mechanism. The neutral CP-even Higgs
$\phi_{d}^{0},\;\phi_{u}^{0},\;\phi_{\eta}^{0},\;\phi_{\bar{\eta}}^{0},\;\phi_{S}^{0}$
mix together, and form a $5\times 5$ mass squared matrix. Besides, the
lightest CP-even Higgs mass can be improved at tree level due to the
introduction of $\eta,\;\bar{\eta},\;S$. The sneutrinos are departed into CP-
even sneutrinos and CP-odd sneutrinos, whose mass squared matrices are both
$6\times 6$. In TABLE 2, we show the chiral superfields and quantum numbers in
the $\mathrm{U}(1)_{X}\mathrm{SSM}$. Then the superpotential of
$\mathrm{U}(1)_{X}\mathrm{SSM}$ is deduced as

$\displaystyle
W=l_{W}\hat{S}+\mu\hat{H}_{u}\hat{H}_{d}+M_{S}\hat{S}\hat{S}-Y_{d}\hat{d}\hat{q}\hat{H}_{d}-Y_{e}\hat{e}\hat{l}\hat{H}_{d}+\lambda_{H}\hat{S}\hat{H}_{u}\hat{H}_{d}$
$\displaystyle\hskip
17.07182pt+\lambda_{C}\hat{S}\hat{\eta}\hat{\bar{\eta}}+\frac{\kappa}{3}\hat{S}\hat{S}\hat{S}+Y_{u}\hat{u}\hat{q}\hat{H}_{u}+Y_{X}\hat{\nu}\hat{\bar{\eta}}\hat{\nu}+Y_{\nu}\hat{\nu}\hat{l}\hat{H}_{u}.$
(1)

where $l_{W}$ is the parameter with mass squared dimension, $\mu$ and $M_{S}$
are both the parameters with mass dimension. $\mu$ indicates the
supersymmetric mass between $SU(2)_{L}$ Higgs doublets $\hat{H}_{d}$ and
$\hat{H}_{u}$, as well as $M_{S}$ represents the supersymmetric mass of
$U(1)_{X}$ Higgs singlet $\hat{S}$.

Table 2:  The chiral superfields and corresponding quantum numbers in the $\mathrm{U}(1)_{X}\mathrm{SSM}$. Superfields | $\hat{q}_{i}$ | $\hat{u}_{i}^{c}$ | $\hat{d}_{i}^{c}$ | $\hat{l}_{i}$ | $\hat{e}_{i}^{c}$ | $\hat{\nu}_{i}^{c}$ | $\hat{H}_{u}$ | $\hat{H}_{d}$ | $\hat{\eta}$ | $\hat{\bar{\eta}}$ | $\hat{S}$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | 3 | $\bar{3}$ | $\bar{3}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$SU(2)_{L}$ | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1
$U(1)_{Y}$ | 1/6 | -2/3 | 1/3 | -1/2 | 1 | 0 | 1/2 | -1/2 | 0 | 0 | 0
$U(1)_{X}$ | 0 | -1/2 | 1/2 | 0 | 1/2 | -1/2 | 1/2 | -1/2 | -1 | 1 | 0

We show the concrete forms of the two Higgs doublets and three Higgs singlets

$\displaystyle H_{u}=\left(\begin{array}[]{c}H_{u}^{+}\\\
\frac{1}{\sqrt{2}}\left(v_{u}+\phi_{u}^{0}+iP_{u}^{0}\right)\end{array}\right),\hskip
14.22636ptH_{d}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}\left(v_{d}+\phi_{d}^{0}+iP_{d}^{0}\right)\\\
H_{d}^{-}\end{array}\right),$ (6)
$\displaystyle\eta=\frac{1}{\sqrt{2}}\left(v_{\eta}+\phi_{\eta}^{0}+iP_{\eta}^{0}\right),\hskip
42.67912pt\bar{\eta}=\frac{1}{\sqrt{2}}\left(v_{\bar{\eta}}+\phi_{\bar{\eta}}^{0}+iP_{\bar{\eta}}^{0}\right),$
$\displaystyle
S=\frac{1}{\sqrt{2}}\left(v_{S}+\phi_{S}^{0}+iP_{S}^{0}\right).$ (7)

Here, $v_{u},v_{d},v_{\eta},v_{\bar{\eta}}$ and $v_{S}$ represent the VEVs of
the Higgs superfields $H_{u},H_{d},\eta,\bar{\eta}$ and $S$, respectively. We
define angle $\tan\beta_{\eta}=v_{\bar{\eta}}/v_{\eta}$ in analogy to the
angle definition $\tan\beta=\frac{v_{u}}{v_{d}}$ in the MSSM.

The soft SUSY breaking terms in the $\mathrm{U}(1)_{X}\mathrm{SSM}$ are

$\displaystyle\mathcal{L}_{\mathrm{soft}}=\mathcal{L}_{\mathrm{soft}}^{\mathrm{MSSM}}-m_{\eta}^{2}|\eta|^{2}-m_{\bar{\eta}}^{2}|\bar{\eta}|^{2}-m_{S}^{2}S^{2}-\left(m_{\tilde{\nu}_{R}}^{2}\right)^{IJ}\tilde{\nu}_{R}^{I*}\tilde{\nu}_{R}^{J}$
$\displaystyle\hskip 25.6073pt-
B_{S}S^{2}-L_{S}S-\frac{T_{\kappa}}{3}S^{3}-T_{\lambda_{C}}S\eta\bar{\eta}+\epsilon_{ij}T_{\lambda_{H}}SH_{d}^{i}H_{u}^{j}-T_{X}^{IJ}\bar{\eta}\tilde{\nu}_{R}^{*I}\tilde{\nu}_{R}^{*J}$
$\displaystyle\hskip
25.6073pt-\epsilon_{ij}T_{\nu}^{IJ}H_{u}^{i}\tilde{\nu}_{R}^{I*}\tilde{l}_{j}^{J}-\frac{1}{2}\left(M_{S}\lambda_{\tilde{X}}^{2}+2M_{BB^{\prime}}\lambda_{\tilde{B}}\lambda_{\tilde{X}}\right)+\text{
h.c. }$ (8)

where $\lambda_{\tilde{B}}$ and $\lambda_{\tilde{X}}$ are the gauginos of
$U(1)_{Y}$ and $U(1)_{X}$ respectively. Besides, we add the mass squared terms
of sneutrinos and Higgs bosons, the trilinear scalar coupling terms and the
Majorana mass terms to the soft breaking terms of the
$\mathrm{U}(1)_{X}\mathrm{SSM}$.

Comparing with the MSSM, the two Abelian groups $\mathrm{U}(1)_{Y}$ and
$\mathrm{U}(1)_{X}$ in the $\mathrm{U}(1)_{X}\mathrm{SSM}$ produce a new
effect called as the gauge kinetic mixing, which can be induced through RGEs
even with zero value at Grand Unified energy scale $M_{\mathrm{GUT}}$. In the
general form, the covariant derivative of $\mathrm{U}(1)_{X}\mathrm{SSM}$ read
as Dmu1 ; Dmu2 ; Dmu3

$\displaystyle
D_{\mu}=\partial_{\mu}-i\left(Y^{Y},Y^{X}\right)\left(\begin{array}[]{cc}g_{Y},&g^{\prime}_{YX}\\\
g^{\prime}_{XY},&g^{\prime}_{X}\end{array}\right)\left(\begin{array}[]{c}A_{\mu}^{{}^{\prime}Y}\\\
A_{\mu}^{{}^{\prime}X}\end{array}\right),$ (13)

where $Y^{Y}$ denotes the $\mathrm{U}(1)_{Y}$ charge and $Y^{X}$ represents
the $\mathrm{U}(1)_{X}$ charge, as well as $A_{\mu}^{{}^{\prime}Y}$ and
$A_{\mu}^{{}^{\prime}X}$ are the gauge fields of $\mathrm{U}(1)_{Y}$ and
$\mathrm{U}(1)_{X}$. We have proven that $\mathrm{U}(1)_{X}\mathrm{SSM}$ is
anomaly freeU(1)X anomaly free . Then, we use the matrix $R$ to obtain
coupling matrix with the two Abelian gauge groups unbroken condition
$\left(\begin{array}[]{cc}g_{Y},&g_{YX}^{\prime}\\\
g_{XY}^{\prime},&g_{X}^{\prime}\end{array}\right)R^{T}=\left(\begin{array}[]{cc}g_{1},&g_{YX}\\\
0,&g_{X}\end{array}\right)$. Here, $g_{1}$ corresponds to the measured
hypercharge coupling which is modified in $\mathrm{U}(1)_{X}\mathrm{SSM}$ as
given along with $g_{X}$ and $g_{YX}$. Then, we can redefine the $U(1)$ gauge
fields $\left(\begin{array}[]{c}A_{\mu}^{Y}\\\
A_{\mu}^{X}\end{array}\right)=R\left(\begin{array}[]{c}A_{\mu}^{\prime Y}\\\
A_{\mu}^{\prime X}\end{array}\right)$.

In this model, the gauge bosons $A_{\mu}^{X},A_{\mu}^{Y}$ and $V_{\mu}^{3}$
mix together at the tree level, whose mass matrix is shown as follows

$\displaystyle\left(\begin{array}[]{ccc}\frac{1}{8}g_{1}^{2}v^{2}&-\frac{1}{8}g_{1}g_{2}v^{2}&\frac{1}{8}g_{1}\left(g_{YX}+g_{X}\right)v^{2}\\\
-\frac{1}{8}g_{1}g_{2}v^{2}&\frac{1}{8}g_{2}^{2}v^{2}&-\frac{1}{8}g_{2}\left(g_{YX}+g_{X}\right)v^{2}\\\
\frac{1}{8}g_{1}\left(g_{YX}+g_{X}\right)v^{2}&-\frac{1}{8}g_{2}\left(g_{YX}+g_{X}\right)v^{2}&\frac{1}{8}\left(g_{YX}+g_{X}\right)^{2}v^{2}+\frac{1}{8}g_{X}^{2}\xi^{2}\end{array}\right),$
(17)

with $v^{2}=v_{u}^{2}+v_{d}^{2}$ and
$\xi^{2}=v_{\eta}^{2}+v_{\bar{\eta}}^{2}$. The mass matrix in Eq.(17) can be
diagonalized by Weinberg angle $\theta_{W}$ and the new mixing angle
$\theta_{W}^{\prime}$. Here, $\sin^{2}\theta_{W}^{\prime}$ is defined as

$\displaystyle\sin^{2}\theta_{W}^{\prime}$
$\displaystyle=\frac{1}{2}-\frac{\left(\left(g_{YX}+g_{X}\right)^{2}-g_{1}^{2}-g_{2}^{2}\right)v^{2}+4g_{X}^{2}\xi^{2}}{2\sqrt{\left(\left(g_{YX}+g_{X}\right)^{2}+g_{1}^{2}+g_{2}^{2}\right)^{2}v^{4}+8g_{X}^{2}\left(\left(g_{YX}+g_{X}\right)^{2}-g_{1}^{2}-g_{2}^{2}\right)v^{2}\xi^{2}+16g_{X}^{4}\xi^{4}}}.$
(18)

Then the eigenvalues of Eq.(17) can be deduced as

$\displaystyle m_{\gamma}^{2}=0,$ $\displaystyle
m_{Z,Z^{\prime}}^{2}=\frac{1}{8}\Big{[}\left(g_{1}^{2}+g_{2}^{2}+\left(g_{YX}+g_{X}\right)^{2}\right)v^{2}+4g_{X}^{2}\xi^{2}$
$\displaystyle\mp\sqrt{\left(g_{1}^{2}+g_{2}^{2}+\left(g_{YX}+g_{X}\right)^{2}\right)^{2}v^{4}+8\left(\left(g_{YX}+g_{X}\right)^{2}-g_{1}^{2}-g_{2}^{2}\right)g_{X}^{2}v^{2}\xi^{2}+16g_{X}^{4}\xi^{4}}\Big{]}.$
(19)

We discuss the Higgs boson mass matrix because of the strict constraint from
SM-like Higgs boson on the numerical results. The
$\phi_{d}^{0},\phi_{u}^{0},\phi_{\eta}^{0},\phi_{\bar{\eta}}^{0},\phi_{S}^{0}$
mix together at the tree level and form the mass squared matrix for neutral
CP-even Higgs boson $M_{h}^{2}$. The concrete form of the tree-level mass
squared matrix can be generated by SARAHSARAH1 ; SARAH2 . $M_{h}^{2}$ is
diagonalizd by $Z^{H}$, then we can derive the lightest tree-level Higgs boson
mass $m_{h_{1}}^{0}$. Considering the leading-log radiative corrections from
stop and top particles, the SM-like Higgs boson mass can be obtained
leadinglog1 ; leadinglog2 ; leadinglog3 .

$\displaystyle m_{h}=\sqrt{(m_{h_{1}}^{0})^{2}+\Delta m_{h}^{2}},$ (20)

where the leading-log radiative corrections $\Delta m_{h}^{2}$ can be written
as

$\displaystyle\Delta
m_{h}^{2}=\frac{3m_{t}^{4}}{4\pi^{2}v^{2}}\Big{[}\Big{(}\tilde{t}+\frac{1}{2}\tilde{X}_{t}\Big{)}+\frac{1}{16\pi^{2}}\Big{(}\frac{3m_{t}^{2}}{2v^{2}}-32\pi\alpha_{3}\Big{)}(\tilde{t}^{2}+\tilde{X}_{t}\tilde{t})\Big{]},$
$\displaystyle\tilde{t}=\log\frac{M_{SS}^{2}}{m_{t}^{2}},~{}~{}~{}~{}\tilde{X}_{t}=\frac{2\tilde{A}_{t}^{2}}{M_{SS}^{2}}(1-\frac{\tilde{A}_{t}^{2}}{12M_{SS}^{2}}).$
(21)

Here, $\alpha_{3}$ is the strong coupling constant,
$M_{SS}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$ with stop masses
$m_{\tilde{t}_{1,2}}$, trilinear Higgs-stops coupling
$\tilde{A}_{t}=A_{t}-\mu\cot\beta$ with $A_{t}=T_{u_{33}}$.

We also discuss some couplings used in this work. The corresponding contents
can be given out in our Appendix A detailedly.

##  III The amplitudes for the CLFV decays of vector mesons

In the $\mathrm{U}(1)_{X}\mathrm{SSM}$, we study the CLFV decays of vector
mesons $V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$. The relevant
Feynman diagrams can be depicted in FIG.1, FIG.2 and FIG.3.

Figure 1: The self-energy type diagrams for processes $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$, with $q$ representing $c$, $s$, $b$.
Figure 2: The penguin type diagrams for processes $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$.
Figure 3: The box type diagrams for processes $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$.

In the quark picture, mesons are composed of a quark and an anti-quark, such
as $\phi$ is made up of $s\bar{s}$, $J/\Psi$ is constituted of $c\bar{c}$ and
$\Upsilon$ is composed of $b\bar{b}$. Here, a sum rule for light-cone
wavefunctionlight-cone1 ; light-cone2 ; light-cone3 ; light-cone4 ; light-
cone5 ; light-cone6 ; light-cone7  is adopted, which is widely used in the
theoretical research of particle physics and nuclear physics.

We need to calculate the matrix elements of gauge invariant nonlocal operators
$\langle 0|\bar{q}(y)\Gamma[x,y]q(x)|V\rangle$ to obtain the decay amplitudes
of processes involving the vector mesonslight-cone4 ; light-cone5 ; light-
cone6 ; light-cone7 . $\Gamma[x,y]$ is a generic Dirac matrix structure with
$x$ and $y$ representing the coordinates of quark and anti-quark. Then we can
deduce the leading-twist distribution amplitude of vector meson through the
correlatorlight-cone4 ; light-cone5 ; light-cone6 ; light-cone7 :

$\displaystyle\langle
0|\bar{q}_{\alpha}(y)q_{\beta}(x)|V(p)\rangle=\frac{\delta_{ij}}{4N_{c}}\int_{0}^{1}due^{-i(upy+\bar{u}px)}[f_{V}m_{V}{/\\!\\!\\!\varepsilon}_{V}V_{||}(u)$
$\displaystyle\hskip
119.50148pt+\frac{i}{2}\sigma^{\mu^{\prime}\nu^{\prime}}f_{V}^{\mathrm{\top}}(\varepsilon_{V\mu^{\prime}}p_{\nu^{\prime}}-\varepsilon_{V\nu^{\prime}}p_{\mu^{\prime}})V_{\bot}(u)]_{\beta\alpha},$
(22)

where the number of colors $N_{c}=3$. $m_{V}$ and $\varepsilon_{V}$ are
respectively the mass and polarization vector of the vector meson. The vector
meson momentum is on-shell, which indicates $p^{2}=m_{V}^{2}$. Here, the meson
masses are adopted as $m_{\phi}=1.019$ GeV, $m_{J/\Psi}=3.097$ GeV,
$m_{\Upsilon(1S)}=9.460$ GeV, $m_{\Upsilon(2S)}=10.023$ GeV and
$m_{\Upsilon(3S)}=10.355$ GeVPDG2022 ; R(3S)mass . $f_{V}$ and $f_{V}^{\top}$
are both the meson decay constants, for example, $f_{\phi}=0.24$ GeV,
$f_{J/\Psi}=0.42$ GeV, $f_{\Upsilon(1S)}=0.65$ GeV, $f_{\Upsilon(2S)}=0.48$
GeV and $f_{\Upsilon(3S)}=0.54$ GeV, and we assume $f_{V}=f_{V}^{\top}$ in the
follow calculationmeson decay constant . The integration variable $u$ and
$\bar{u}\equiv 1-u$ stand for the momentum fraction carried by the quark and
anti-quark, respectively. $V_{||}(u)$($V_{\bot}(u)$) represents the leading-
twist distribution function of longitudinal(transverse) polarized meson. The
meson amplitudes are similar to their asymptotic formsasymptotic form , so we
make $V_{||}=V_{\bot}=V(u)=6u(1-u)$ in our calculation. Then the hadron matrix
elements used in this work are discussed in Appendix B. In the frame of center
of mass, the decay amplitudes of $V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ can be
summarized at hadron level.

###  III.1 The self-energy type diagrams

We can generally write the effective amplitudes of $\gamma$ self-energy type
diagrams corresponding to FIG.1(1) $\sim$ FIG.1(2):

$\displaystyle{\cal
A}_{\gamma-s}={\bar{u}}_{j}(p_{4})\gamma_{\mu}(A_{L}P_{L}+A_{R}P_{R})v_{i}(p_{3})\frac{e}{p^{2}}{\bar{v}}_{V}(p_{2})H_{V}^{\gamma
q\bar{q}}\gamma^{\mu}u_{V}(p_{1}),$ (23)

where $p$ represents the meson momentum, $p_{1}$ and $p_{2}$($p_{3}$ and
$p_{4}$) stand for the quark and anti-quark (lepton and anti-lepton) momentums
respectively. ${\bar{u}}_{j}$ and $v_{i}$ are the wave functions of the $j$-th
and $i$-th generation out-going leptons. As well as, ${\bar{v}}_{V}$ and
$u_{V}$ are the wave functions of the incident quarks with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$. The coupling
coefficients $H_{V}^{\gamma q\bar{q}}$ can be obtained from the vertexes given
out in Appendix A, so as to the following
$H_{L,R}^{\bar{l}_{j}FS},H_{L,R}^{Zq\bar{q}}\cdot\cdot\cdot$. The Wilson
coefficients $A_{L,R}$ from amplitudes of the FIG.1(1) and FIG.1(2) are shown
as follows.

$\displaystyle
A_{L}=\sum_{F=\chi^{0},\chi^{\pm}}\sum_{S=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}\Big{\\{}\frac{1}{x_{i}-x_{j}}[x_{i}\sqrt{x_{F}x_{j}}H_{L}^{\bar{l}_{j}FS}H_{L}^{l_{i}\bar{F}S^{*}}$
$\displaystyle\hskip 34.14322pt-
x_{j}\sqrt{x_{F}x_{i}}H_{R}^{\bar{l}_{j}FS}H_{R}^{l_{i}\bar{F}S^{*}}][I_{2}(x_{F},x_{S})-I_{3}(x_{F},x_{S})]-\frac{1}{2}H_{R}^{\bar{l}_{j}FS}H_{L}^{l_{i}\bar{F}S^{*}}I_{5}(x_{F},x_{S})\Big{\\}},$
$\displaystyle A_{R}=A_{L}|_{L{\leftrightarrow}R}.$ (24)

Here, $x_{F,S}=m_{F,S}^{2}/\Lambda^{2}$, $m_{F,S}$ are the fermion and scalar
boson mass, $\Lambda$ is the NP energy scale. The one-loop functions
$I_{i}(x_{1},x_{2}),i=1,2,\cdot\cdot\cdot,5$, as well as the following
$G_{i}(x_{1},x_{2},x_{3}),\;i=1,2$ and
$J_{i}(x_{1},x_{2},x_{3},x_{4}),\;i=1,2$ are shown in Appendix C. Since the
neutrino mass is very small, it makes little contribution to the problem we
study, so the contributions to the self-energy type diagrams and penguin type
diagrams from W boson-neutrino are ignore.

In the same way, the effective amplitudes of $Z$ self-energy type diagrams are
deduced from FIG.1(1) and FIG.1(2).

$\displaystyle{\cal
A}_{Z-s}={\bar{u}}_{j}(p_{4})\gamma_{\mu}(B_{L}P_{L}+B_{R}P_{R})v_{i}(p_{3})\frac{1}{p^{2}-m_{Z}^{2}}{\bar{v}}_{V}(p_{2})\gamma^{\mu}(H_{L}^{Zq\bar{q}}P_{L}+H_{R}^{Zq\bar{q}}P_{R})u_{V}(p_{1}).$
(25)

The concrete contributions from FIG. 1(1) and FIG. 1(2) are denoted by
$B_{L,R}$.

$\displaystyle
B_{L}=\sum_{F=\chi^{0},\chi^{\pm}}\sum_{S=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}\Big{\\{}\frac{1}{x_{i}-x_{j}}[x_{i}\sqrt{x_{F}x_{j}}H_{L}^{Z{l}_{i}\bar{l}_{j}}H_{L}^{\bar{l}_{j}FS}H_{L}^{l_{i}\bar{F}S^{*}}-x_{j}\sqrt{x_{F}x_{i}}H_{L}^{Z{l}_{i}\bar{l}_{j}}H_{R}^{\bar{l}_{j}FS}H_{R}^{l_{i}\bar{F}S^{*}}]$
$\displaystyle\hskip
34.14322pt\times[I_{2}(x_{F},x_{S})-I_{3}(x_{F},x_{S})]-\frac{1}{2}H_{L}^{Z{l}_{i}\bar{l}_{j}}H_{R}^{\bar{l}_{j}FS}H_{L}^{l_{i}\bar{F}S^{*}}I_{5}(x_{F},x_{S})\Big{\\}},$
$\displaystyle B_{R}=B_{L}|_{L{\leftrightarrow}R}.$ (26)

###  III.2 The penguin type diagrams

We give out the penguin type diagrams for decays $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$ in FIG.2. The relevant effective amplitudes of
$\gamma$ penguin type diagrams can be shown as

$\displaystyle{\cal
A}_{\gamma-p}={\bar{u}}_{j}(p_{4})[\gamma_{\mu}(C_{1}^{L}P_{L}+C_{1}^{R}P_{R})+i\sigma_{\mu\nu}p^{\nu}(C_{2}^{L}P_{L}+C_{2}^{R}P_{R})]v_{i}(p_{3})$
$\displaystyle\hskip
42.67912pt\times\frac{e}{p^{2}}{\bar{v}}_{V}(p_{2})H_{V}^{\gamma
q\bar{q}}\gamma^{\mu}u_{V}(p_{1}).$ (27)

Firstly, we discuss the contributions $C_{1}^{L,R}(n),C_{2}^{L,R}(n)$ from the
virtual neutral fermion diagram in the FIG.2(1).

$\displaystyle
C_{1}^{L}(n)=\sum_{F=\chi^{0}}\sum_{S=\tilde{e}}H_{R}^{SF\bar{l}_{j}}H_{L}^{S^{*}l_{i}\bar{F}}[\frac{1}{2}I_{5}(x_{F},x_{S})+\frac{p^{2}}{6\Lambda^{2}}I_{4}(x_{F},x_{S})],$
$\displaystyle
C_{2}^{L}(n)=\sum_{F=\chi^{0}}\sum_{S=\tilde{e}}\frac{m_{F}}{\Lambda^{2}}H_{L}^{SF\bar{l}_{j}}H_{L}^{S^{*}l_{i}\bar{F}}[I_{2}(x_{F},x_{S})-I_{3}(x_{F},x_{S})],$
$\displaystyle
C_{\alpha}^{R}(n)=C_{\alpha}^{L}(n)|_{L{\leftrightarrow}R},\alpha=1,2.$ (28)

Secondly, the couplings coefficients $C_{1}^{L,R}(c),C_{2}^{L,R}(c)$
correspond to the contributions from the virtual charged fermion diagram in
the FIG.2(2), which are written as:

$\displaystyle
C_{1}^{L}(c)=\sum_{F=\chi^{\pm}}\sum_{S=\tilde{\nu}^{I}(\tilde{\nu}^{R})}H_{R}^{SF\bar{l}_{j}}H_{L}^{S^{*}l_{i}\bar{F}}[\frac{1}{2}I_{5}(x_{S},x_{F})+x_{F}I_{3}(x_{S},x_{F})$
$\displaystyle+\frac{p^{2}}{6\Lambda^{2}}(3I_{2}(x_{S},x_{F})-I_{4}(x_{S},x_{F}))],$
$\displaystyle
C_{2}^{L}(c)=\sum_{F=\chi^{\pm}}\sum_{S=\tilde{\nu}^{I}(\tilde{\nu}^{R})}\frac{m_{F}}{\Lambda^{2}}H_{L}^{SF\bar{l}_{j}}H_{L}^{S^{*}l_{i}\bar{F}}I_{2}(x_{S},x_{F}),$
$\displaystyle
C_{\alpha}^{R}(c)=C_{\alpha}^{L}(c)|_{L{\leftrightarrow}R},\alpha=1,2.$ (29)

Then, the effective amplitudes of $Z$ penguin type diagrams are deduced in
analogy to that of $\gamma$ penguin type diagrams. In addition, we just
consider the dominate contributions of effective operators.

$\displaystyle{\cal
A}_{Z-p}={\bar{u}}_{j}(p_{4})\gamma_{\mu}(D_{L}P_{L}+D_{R}P_{R})v_{i}(p_{3})\frac{1}{p^{2}-m_{Z}^{2}}{\bar{v}}_{V}(p_{2})\gamma^{\mu}(H_{L}^{Zq\bar{q}}P_{L}+H_{R}^{Zq\bar{q}}P_{R})u_{V}(p_{1}).$
(30)

The concrete contributions $D_{L,R}$ corresponding to FIG.2(1) $\sim$ FIG.2(4)
are encoded as follows.

$\displaystyle
D_{L}=-\frac{1}{2}\sum_{F=\chi^{0},\chi^{\pm}}\sum_{S=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}\big{[}-2\sqrt{x_{F_{1}}x_{F_{2}}}H_{L}^{SF_{2}\bar{l}_{j}}H_{R}^{ZF_{1}\bar{F}_{2}}H_{L}^{S^{*}l_{i}\bar{F}_{1}}G_{1}(x_{S},x_{F_{1}},x_{F_{2}})$
$\displaystyle+H_{R}^{SF_{2}\bar{l}_{j}}H_{R}^{ZF_{1}\bar{F}_{2}}H_{L}^{S^{*}l_{i}\bar{F}_{1}}G_{2}(x_{S},x_{F_{1}},x_{F_{2}})-H_{R}^{S_{2}F\bar{l}_{j}}H^{ZS_{1}S_{2}^{*}}H_{L}^{S_{1}^{*}l_{i}\bar{F}}G_{2}(x_{F},x_{S_{1}},x_{S_{2}})\big{]},$
$\displaystyle D_{R}=D_{L}|_{L{\leftrightarrow}R}.$ (31)

###  III.3 The box type diagrams

In this subsection, we discuss the contributions of box type diagrams. In the
calculation, we need to swap the position of the wave functions $u_{V}(p_{1})$
and $v_{i}(p_{3})$ to simplify the amplitudes. The method is named as Fierz
Rearrangement, and the concrete transformation rules can be learnt from
Refs.Fierz1 ; Fierz2 ; Fierz3 . Then the amplitudes can be deduced as:

$\displaystyle{\cal
A}_{box}=\\{N_{1}^{L}{\bar{u}}_{j}(p_{4})\gamma_{\mu}P_{L}v_{i}(p_{3}){\bar{v}}_{V}(p_{2})\gamma^{\mu}P_{L}u_{V}(p_{1})+(L\leftrightarrow
R)\\}$
$\displaystyle+\\{N_{2}^{L}{\bar{u}}_{j}(p_{4})\sigma_{\mu\nu}v_{i}(p_{3}){\bar{v}}_{V}(p_{2})\sigma^{\mu\nu}P_{L}u_{V}(p_{1})+(L\leftrightarrow
R)\\}$
$\displaystyle+\\{N_{3}^{L}{\bar{u}}_{j}(p_{4})\gamma_{\mu}P_{R}v_{i}(p_{3}){\bar{v}}_{V}(p_{2})\gamma^{\mu}P_{L}u_{V}(p_{1})+(L\leftrightarrow
R)\\},$ (32)

where, the couplings coefficients $N_{i}^{L,R}$ correspond to the
contributions from FIG.3.

$\displaystyle
N_{1}^{L}=\frac{1}{4}\sum_{F_{1},F_{2}=\chi^{0},\chi^{\pm}}\sum_{S_{1}=\tilde{q}}\sum_{S_{2}=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}H_{L}^{qS_{1}\bar{F}_{2}}H_{R}^{\bar{q}S_{1}^{*}F_{1}}H_{L}^{l_{i}S_{2}\bar{F}_{1}}H_{R}^{\bar{l}_{j}S_{2}^{*}F_{2}}J_{1}(x_{F_{1}},x_{F_{2}},x_{S_{1}},x_{S_{2}}),$
$\displaystyle
N_{2}^{L}=-\frac{\sqrt{x_{F_{1}}x_{F_{2}}}}{8}\sum_{F_{1},F_{2}=\chi^{0},\chi^{\pm}}\sum_{S_{1}=\tilde{q}}\sum_{S_{2}=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}H_{L}^{qS_{1}\bar{F}_{2}}H_{L}^{\bar{q}S_{1}^{*}F_{1}}H_{L}^{l_{i}S_{2}\bar{F}_{1}}H_{L}^{\bar{l}_{j}S_{2}^{*}F_{2}}J_{2}(x_{F_{1}},x_{F_{2}},x_{S_{1}},x_{S_{2}}),$
$\displaystyle
N_{3}^{L}=-\frac{\sqrt{x_{F_{1}}x_{F_{2}}}}{2}\sum_{F_{1},F_{2}=\chi^{0},\chi^{\pm}}\sum_{S_{1}=\tilde{q}}\sum_{S_{2}=\tilde{e},\tilde{\nu}^{I}(\tilde{\nu}^{R})}H_{L}^{qS_{1}\bar{F}_{2}}H_{R}^{\bar{q}S_{1}^{*}F_{1}}H_{R}^{l_{i}S_{2}\bar{F}_{1}}H_{L}^{\bar{l}_{j}S_{2}^{*}F_{2}}J_{2}(x_{F_{1}},x_{F_{2}},x_{S_{1}},x_{S_{2}}).$
(33)

The branching ratios for processes $V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ can
be deduced as:

$\displaystyle Br\left(V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}\right)=\frac{\sqrt{[m_{V}^{2}-(m_{l_{i}}+m_{l_{j}})^{2}][m_{V}^{2}-(m_{l_{i}}-m_{l_{j}})^{2}]}}{16\pi
m_{V}^{3}\Gamma_{V}}\times\sum_{\xi}{\cal A}_{V_{\xi}}{\cal A}_{V_{\xi}}^{*},$
(34)

where $\Gamma_{V}$ represents the total decay width of meson, for example
$\Gamma_{\phi}\simeq 4.2\times 10^{-3}$ GeV, $\Gamma_{J/\Psi}\simeq
0.0926\times 10^{-3}$ GeV, $\Gamma_{\Upsilon(1S)}\simeq 0.054\times 10^{-3}$
GeV, $\Gamma_{\Upsilon(2S)}\simeq 0.032\times 10^{-3}$ GeV and
$\Gamma_{\Upsilon(3S)}\simeq 0.020\times 10^{-3}$ GeVPDG2022 . $A_{V_{\xi}}$
are the amplitudes corresponding to FIG.1, FIG.2 and FIG.3.

##  IV The numerical results

In this section, we research the numerical results of the CLFV processes
$V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$. We consider
some experimental constraints: 1. The updated experimental data indicates that
the $Z^{\prime}$ boson mass satisfies $M_{Z^{\prime}}\geq 5.15~{}{\rm TeV}$
with $95\%$ CLZpupper . The upper bound on the ratio between the $Z^{\prime}$
boson mass and gauge coupling $g_{X}$ is approximatively smaller than
$6~{}{\rm TeV}$ at $99\%$ CLZpupper1 ; Zpupper2 , so $g_{X}$ is restricted in
the interval of $0\sim 0.85$. 2. The $\bar{B}\rightarrow X_{s}\gamma$
experiment has excluded the large $\tan\beta$BSgamma1 ; BSgamma2 . 3.
$\tan\beta_{\eta}$ is smaller than 1.5 after taking into account the
constraint from LHC datatbyita . 4. According to the latest LHC data, the
slepton mass is greater than 700 GeV, the chargino mass is greater than 1100
GeV and the squark mass is greater than 1500 GeV in our numerical discussions
scalar mass exp1 ; scalar mass exp2 ; scalar mass exp3 ; scalar mass exp4 ;
scalar mass exp5 ; scalar mass exp6 . 5. The SM-like Higgs boson mass,
$m_{h}=125.25\pm 0.17$ GeV, constrains the parameter space strictlyPDG2022 ;
Higgsmassexp1 ; Higgsmassexp2 ; Higgsmassexp3 . Therefore, we limit the SM-
like Higgs boson mass of the $\mathrm{U}(1)_{X}\mathrm{SSM}$ within
experimental $3\sigma$ region. 6. We consider the constraints from some CLFV
processes, such as $Br(\mu\rightarrow e\gamma)<5.7\times 10^{-13}$ and
$Br(Z\rightarrow e\mu)<7.5\times 10^{-7}$mutoerexp ; Ztoemuexp . Then the
numerical results of the CLFV processes $V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$
in the $\mathrm{U}(1)_{X}\mathrm{SSM}$ are discussed detailedly.

Considering the above experimental constraints, some parameters we used are
shown as

$\displaystyle\lambda_{H}=\lambda_{C}=\kappa=0.1,\;\tan\beta_{\eta}=0.8,\;Y_{X_{ii}}=0.5,\;g_{X}=0.3,\;M_{2}=1.2\;\mathrm{TeV},$
$\displaystyle
M_{BB^{\prime}}=0.4\;\mathrm{TeV},\;\mu=M_{1}=M_{BL}=T_{\lambda_{H}}=T_{\lambda_{C}}=T_{\kappa}=T_{u_{ii}}=T_{d_{ii}}=1\;\mathrm{TeV},$
$\displaystyle
T_{X_{ii}}=-1\;\mathrm{TeV},\;T_{E_{ii}}=-3\;\mathrm{TeV},\;T_{\nu_{ii}}=0.5\;\mathrm{TeV},\;v_{S}=3\;\mathrm{TeV},\;m_{{\nu}_{R_{ii}}}^{2}=0.5\;\mathrm{TeV}^{2},$
$\displaystyle
m_{{Q}_{ii}}^{2}=m_{{U}_{ii}}^{2}=m_{{D}_{ii}}^{2}=10\;\mathrm{TeV}^{2},m_{S}^{2}=m_{{E}_{ii}}^{2}=1\;\mathrm{TeV}^{2},\;l_{W}=B_{\mu}=B_{S}=0.1\;\mathrm{TeV}^{2}.$
(35)

###  IV.1 $V\rightarrow e^{+}\mu^{-}$

Parameters $\tan\beta$, $g_{YX}$, $M_{1}$, $M_{2}$, $M_{S}$,
$M_{BB^{\prime}}$, $M_{BL}$, $m_{{L}_{ii}}^{2}$, $m_{{E}_{ii}}^{2}$,
$m_{{\nu}_{R_{ii}}}^{2}$, $T_{X_{ii}}$, $T_{E_{ii}}$, $T_{\nu_{ii}}$ and
$m_{{L}_{ij}}^{2}$, $m_{{E}_{ij}}^{2}$, $m_{{\nu}_{R_{ij}}}^{2}(i\neq j)$ are
assumed as random variables in the suitable regions. As the SM-like Higgs
boson mass is in $3\sigma$ region, and the branching ratios of $V\rightarrow
e^{+}\mu^{-}$ satisfy the current experiment constraints, the reasonable
parameter space is selected to scatter points, which are shown in FIG.4.
Firstly, we discuss the distribution of $\tan\beta$ versus $g_{YX}$ in
FIG.4(a). On the whole, the upper and lower limits of $g_{YX}$ almost increase
with the enlarged $\tan\beta$. It is worth noting that when $\tan\beta$
changes from 7 to 10, the upper limit of $g_{YX}$ increases significantly, and
then its upper limit increases slowly as $\tan\beta$ increases. And when
$\tan\beta$ is between 14 and 40, $g_{YX}$ can fetch almost any value in the
range of 0.2 to 0.5. FIG.2(b) indicates that $M_{S}$ is larger than 2.4 TeV,
as well as, the upper limit and the suitable parameter space of $g_{YX}$ both
increase obviously with the increase of $M_{S}$. Therefore, in the numerical
analyses below, we take $g_{YX}=0.3$, $\tan\beta=15$ and $M_{S}=2.7$ TeV.

Figure 4: $\tan\beta$ ($M_{S}$) varying with $g_{YX}$ are plotted, where the
vector mesons of $\phi$, $J/\Psi$, $\Upsilon(1S)$, $\Upsilon(2S)$,
$\Upsilon(3S)$ are represented by $\blacklozenge$,  $\blacktriangle$,
$\bullet$,  $\bigstar$,  $\blacktriangledown$ respectively.

The CLFV process $V\rightarrow e^{+}\mu^{-}$ is flavor dependent, which can be
influenced by the parameters
$M_{LL}=m_{{L}_{ii}}^{2},M_{LL}^{ij}=m_{{L}_{ij}}^{2},M_{EE}=m_{{E}_{ii}}^{2},M_{EE}^{ij}=m_{{E}_{ij}}^{2},M_{NU}^{ij}=m_{{\nu}_{R_{ij}}}^{2}\cdot\cdot\cdot$.
When $M_{LL}^{12}=0.4\;\mathrm{TeV}^{2}$ and
$M_{EE}^{12}=M_{NU}^{12}=10^{-2}\;\mathrm{TeV}^{2}$, $Br(V\rightarrow
e^{+}\mu^{-})$ versus the parameter $M_{LL}$ are plotted in FIG.5, as well as
$Br(V\rightarrow e^{+}\mu^{-})$ versus the parameter $M_{LL}^{12}$ are figured
in FIG.6 as $M_{LL}=1\;\mathrm{TeV}^{2}$ and
$M_{EE}^{12}=M_{NU}^{12}=10^{-2}\;\mathrm{TeV}^{2}$. The red line has been
excluded due to exceed experimental constraints of SM-like Higgs mass
$3\sigma$ region or $Br(V\rightarrow e^{+}\mu^{-})$. It is obvious that the
CLFV rates of $V\rightarrow e^{+}\mu^{-}$ all decrease with the enlarged
$M_{LL}$, and increase with the enlarged $M_{LL}^{12}$. Moreover, the present
experimental upper bound of $Br(\Upsilon(3S)\rightarrow e^{+}\mu^{-})$
possesses the most obvious constraints on parameter $M_{LL}$ and
$M_{LL}^{12}$, which are $M_{LL}\geq 0.7\;\mathrm{TeV}^{2}$ and
$M_{LL}^{12}\leq 0.45\;\mathrm{TeV}^{2}$. In the follow discussion, we take
$M_{LL}=1\;\mathrm{TeV}^{2}$.

Figure 5: The contributions to $V\rightarrow
e^{+}\mu^{-}(V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\})$
varying with $M_{LL}$ are respectively plotted.
Figure 6: The contributions to $V\rightarrow
e^{+}\mu^{-}(V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\})$
varying with $M_{LL}^{12}$ are respectively plotted.

###  IV.2 $V\rightarrow e^{+}\tau^{-}$ and $V\rightarrow\mu^{+}\tau^{-}$

In this subsection, we first study the CLFV rates for $V\rightarrow
e^{+}\tau^{-}$ versus $M_{LL}^{13}$ and $M_{EE}^{13}$ respectively in FIG.7
and FIG.8. In general, the numerical results of $Br(V\rightarrow
e^{+}\tau^{-})$ both enlarge with the increase of $M_{LL}^{13}$ and
$M_{EE}^{13}$. Obviously, FIG.7 indicates that when
$M_{EE}^{13}=M_{NU}^{13}=0$, the CLFV rates of the processes $V\rightarrow
e^{+}\tau^{-}(V\in\\{\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\})$ changing with
$M_{LL}^{13}$ are very close, and nearly three orders of magnitude larger than
the $Br(J/\Psi\rightarrow e^{+}\tau^{-})$. Then, we take the process
$\Upsilon(3S)\rightarrow e^{+}\tau^{-}$ as an example, and discuss the CLFV
ratio of this process with the parameter $M_{EE}^{13}$ in FIG.8 as
$M_{NU}^{13}=5\times 10^{-2}\;\mathrm{TeV}^{2}$. We find that when the
parameter $M_{LL}^{13}$ enlarges from $0.3\;\mathrm{TeV}^{2}$ to
$0.8\;\mathrm{TeV}^{2}$, the numerical result increases slowly. However, the
numerical results can be improved about an order of magnitude with the
increase of $M_{EE}^{13}$, which indicates that the effect of parameter
$M_{LL}^{13}$ on the numerical results is slightly smaller than that of
parameter $M_{EE}^{13}$.

Figure 7: The contributions to $V\rightarrow
e^{+}\tau^{-}(V\in\\{J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S),\\})$
varying with $M_{LL}^{13}$ are respectively plotted.
Figure 8: The contributions to $\Upsilon(3S)\rightarrow e^{+}\tau^{-}$ varying
with $M_{EE}^{13}$ are plotted, where the solid, dashed and dot-dashed lines
correspond to $M_{LL}^{13}=0.3,0.5,0.8\;\mathrm{TeV}^{2}$, respectively.

When $M_{EE}^{23}=M_{NU}^{23}=0$, we analyze the transformations of the CLFV
ratios of processes $V\rightarrow\mu^{+}\tau^{-}$ with the parameter
$M_{LL}^{23}$, as shown in FIG.9. By observing this figure, it is easy to find
that although parameter $M_{LL}^{23}$ has the lifting effect on the numerical
results, the effect is not obvious. Therefore, the non-diagonal element
$M_{LL}^{23}$, from slepton matrix $m_{L}^{2}$, possesses weak influence on
the CLFV decays. Taking the process $\Upsilon(3S)\rightarrow\mu^{+}\tau^{-}$
as an example, we discuss how the CLFV ratio of this process varies with the
diagonal element $M_{EE}$ and the non-diagonal element $M_{EE}^{23}$ of the
slepton matrix $m_{E}^{2}$ in FIG.10, where
$M_{LL}^{23}=0.5\;\mathrm{TeV}^{2}$ and $M_{NU}^{23}=0.05\;\mathrm{TeV}^{2}$.
We find that an increase in diagonal element $M_{EE}$ significantly depresses
the numerical result, while an increase in non-diagonal element $M_{EE}^{23}$
significantly increases the numerical result. Therefore, $M_{EE}$ and
$M_{EE}^{23}$ are both the sensitive parameters.

Figure 9: The contributions to
$V\rightarrow\mu^{+}\tau^{-}(V\in\\{J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S),\\})$
varying with $M_{LL}^{23}$ are respectively plotted.
Figure 10: The contributions to $\Upsilon(3S)\rightarrow\mu^{+}\tau^{-}$
varying with $M_{EE}^{23}$ are respectively plotted, where the solid and dot-
dashed lines correspond to $M_{EE}=1,8\;\mathrm{TeV}^{2}$, respectively.

##  V discussion and conclusion

In this work, we focus on the CLFV decays of vector mesons $V\rightarrow
l_{i}^{\pm}l_{j}^{\mp}$ with
$V\in\\{\phi,J/\Psi,\Upsilon(1S),\Upsilon(2S),\Upsilon(3S)\\}$ in the
$\mathrm{U}(1)_{X}\mathrm{SSM}$. We find that the
$\mathrm{U}(1)_{X}\mathrm{SSM}$ contributions for CLFV processes beyond MSSM
are considerable. In the numerical discussion, we constrain the SM-like Higgs
mass within $3\sigma$ region, which possesses strict limit on the CLFV decays.
Firstly, the distribution of $\tan\beta$ versus $g_{YX}$ indicates $g_{YX}$
can fetch almost any value in the range of 0.2 to 0.5 as $\tan\beta$ is
between 14 and 40. Besides, $M_{S}$ is larger than 2.4 TeV, as well as, the
upper limit and the suitable parameter space of $g_{YX}$ both increase
obviously with the increase of $M_{S}$. Not only that, the branching ratios of
$V\rightarrow l_{i}^{\pm}l_{j}^{\mp}$ depend on the slepton flavor mixing
parameters $M_{LL}^{ij}$ and $M_{EE}^{ij}$ obviously, and increase with the
enlarged $M_{LL}^{ij}$ and $M_{EE}^{ij}$. Considering the latest experimental
limits of $Br(\Upsilon(3S)\rightarrow e^{+}\mu^{-})$, the slepton flavor
mixing parameter $M_{LL}^{12}$ is restricted as $M_{LL}^{12}\leq
0.45\;\mathrm{TeV}^{2}$. In addition, $M_{LL}$, as the diagonal element of the
slepton and sneutrino, is limited as $M_{LL}\geq 0.7\;\mathrm{TeV}^{2}$ by the
latest experimental limits of $Br(\Upsilon(3S)\rightarrow e^{+}\mu^{-})$. And
the increase of $M_{LL}$ has the significant inhibitory effect on the
numerical results.

After considering the SM-like Higgs mass within $3\sigma$ region, the
experimental constraints on $Br(\mu\to e+\gamma)$, $Br(Z\to e\mu)$ and
$Br(V\to e+\mu^{-})$, we find that $Br(\phi\rightarrow e\mu)\sim
10^{-11},\;Br(J/\Psi\rightarrow e\mu)\sim 10^{-9}$ and
$Br(\Upsilon(1S)(\Upsilon(2S),\Upsilon(3S))\rightarrow e\mu)\sim 10^{-7}$. The
decays $\Upsilon(1S)(\Upsilon(2S),\Upsilon(3S))\rightarrow e\mu$ are much
easier than $\phi(J/\Psi)\rightarrow e\mu$ to reach the experimental upper
bounds. Similarly, $Br(\Upsilon(3S)\rightarrow e\tau)$ are at the order of
$10^{-9}$ and $Br(\Upsilon(3S)\rightarrow\mu\tau)$ can reach $10^{-7}$. Above
results indicate that processes $\Upsilon\rightarrow e\mu$ and
$\Upsilon(3S)\rightarrow\mu\tau$ are very promising to be observed in the near
future experiments. A summary table of the theoretical predictions of
$Br(J/\Psi(\Upsilon)\rightarrow e\mu)$ and
$Br(\Upsilon(3S)\rightarrow\mu\tau)$ in literatures are presented in TABLE 3,
which indicates that the corresponding theoretical predictions in our model
may be realized much easilier than that in the SUSY, 331 and 211 models.

Table 3:  The theoretical predictions of $Br(J/\Psi(\Upsilon)\rightarrow e\mu)$ and $Br(\Upsilon(3S)\rightarrow\mu\tau)$ in literatures. $Br(V\to l_{i}l_{j})$ | $\mathrm{U}(1)_{X}\mathrm{SSM}$ | SUSY madelTC2 model  | 331 modelZ' model  | 211 modelZ' model
---|---|---|---|---
$Br(J/\Psi\rightarrow e\mu)$ | $\sim 10^{-9}$ | $\sim 10^{-16}$ | $\sim 10^{-19}$ | $\sim 10^{-20}$
$Br(\Upsilon\rightarrow e\mu)$ | $\sim 10^{-7}$ | $\sim 10^{-15}$ | $\sim 10^{-17}$ | $\sim 10^{-14}$
$Br(\Upsilon(3S)\rightarrow\mu\tau)$ | $\sim 10^{-7}$ | $\sim 10^{-8}$ | $\sim 10^{-11}$ | $\sim 10^{-9}$

Acknowledgments

This work is supported by the Major Project of National Natural Science
Foundation of China (NNSFC) (No. 12235008), the National Natural Science
Foundation of China (NNSFC) (No. 12075074, No. 12075073), the Natural Science
Foundation of Hebei province(No. A202201022, No. A2020201002, No.
A2023201041), the Natural Science Foundation of Hebei Education Department(No.
QN2022173).

##  Appendix A The couplings

We discuss some couplings used in this work. We deduce the vertexes of $Z$
boson and two scalar bosons, such as vertexes of
$\tilde{e}_{{i}}$-$\tilde{e}^{*}_{{j}}$-$Z^{{\mu}}$ and
$\tilde{\nu}^{I}_{{i}}$-$\tilde{\nu}^{R}_{{j}}$-$Z^{{\mu}}$.

$\displaystyle\mathcal{L}_{Z\tilde{e}_{{i}}\tilde{e}^{*}_{{j}}}=\frac{i}{2}\tilde{e}^{*}_{{j}}\sum_{a=1}^{3}\Big{[}(-g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}+g_{YX}\sin{\theta^{\prime}}_{W})Z^{E,*}_{ia}Z_{{ja}}^{E}$
$\displaystyle+\Big{(}-2g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+(2g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}Z^{E,*}_{i3+a}Z_{{j3+a}}^{E}\Big{]}\Big{(}-p^{\tilde{e}^{*}_{{j}}}_{\mu}+p^{\tilde{e}_{{i}}}_{\mu}\Big{)}\tilde{e}_{{i}}Z^{\mu},$
$\displaystyle\mathcal{L}_{Z\tilde{\nu}^{I}_{{i}}\tilde{\nu}^{R}_{{j}}}=\frac{1}{2}\tilde{\nu}^{R}_{{j}}\sum_{a=1}^{3}\Big{(}(g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}-g_{YX}\sin{\theta^{\prime}}_{W})Z^{I,*}_{ia}Z^{R,*}_{ja}$
$\displaystyle+g_{X}\sin{\theta^{\prime}}_{W}Z^{I,*}_{i3+a}Z^{R,*}_{j3+a}\Big{)}\Big{(}-p^{\nu^{R}_{{j}}}_{\mu}+p^{\nu^{I}_{{i}}}_{\mu}\Big{)}\tilde{\nu}^{I}_{{i}}Z^{\mu}.$
(36)

The vertexes of $Z$ boson and two fermions are discussed follows, including
the vertexes of ${\chi}^{+}_{{i}}$-${\chi}^{-}_{{j}}$-$Z^{{\mu}}$,
${\chi}^{0}_{{i}}$-${\chi}^{0}_{{j}}$-$Z^{{\mu}}$,
$\bar{d}_{{i\alpha}}$-$d_{{j\beta}}$-$Z_{{\mu}}$,
$\bar{l}_{{i}}$-$l_{{j}}$-$Z^{{\mu}}$ and
$\bar{u}_{{i\alpha}}$-$u_{{j\beta}}$-$Z^{{\mu}}$.

$\displaystyle\mathcal{L}_{Z{\chi}^{+}_{{i}}{\chi}^{-}_{{j}}}=\frac{i}{2}{\chi}^{-}_{{j}}\Big{\\{}\Big{[}2g_{2}\cos\theta_{W}\cos{\theta^{\prime}}_{W}U_{{i1}}U^{*}_{j1}+\Big{(}-g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos\theta_{W}\cos{\theta^{\prime}}_{W}$
$\displaystyle+(g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}U_{{i2}}U^{*}_{j2}\Big{]}\gamma_{\mu}P_{L}+\Big{[}2g_{2}\cos\theta_{W}\cos{\theta^{\prime}}_{W}V^{*}_{i1}V_{{j1}}+\Big{(}-g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}$
$\displaystyle+g_{2}\cos\theta_{W}\cos{\theta^{\prime}}_{W}+(g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}V^{*}_{i2}V_{{j2}}\Big{]}\gamma_{\mu}P_{R}\Big{\\}}{\chi}^{+}_{{i}}Z^{\mu},$
$\displaystyle\mathcal{L}_{Z{\chi}^{0}_{{i}}{\chi}^{0}_{{j}}}=-\frac{i}{2}{\chi}^{0}_{{j}}\Big{\\{}\Big{[}\Big{(}g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}-(g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}\Big{(}N_{{i3}}N^{*}_{j3}$
$\displaystyle-
N_{{i4}}N^{*}_{j4}\Big{)}-2g_{X}\sin{\theta^{\prime}}_{W}\Big{(}N_{{i6}}N^{*}_{j6}-N_{{i7}}N^{*}_{j7}\Big{)}\Big{]}\gamma_{\mu}P_{L}-\Big{[}\Big{(}g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}$
$\displaystyle-(g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}\Big{(}N^{*}_{i3}N_{{j3}}-N^{*}_{i4}N_{{j4}}\Big{)}-2g_{X}\sin{\theta^{\prime}}_{W}\Big{(}N^{*}_{i6}N_{{j6}}-N^{*}_{i7}N_{{j7}}\Big{)}\Big{]}\gamma_{\mu}P_{R}\Big{)}\Big{\\}}{\chi}^{0}_{{i}}Z^{\mu},$
$\displaystyle\mathcal{L}_{Z\bar{d}_{{i\alpha}}d_{{j\beta}}}=\frac{i}{6}\bar{d}_{{i\alpha}}\Big{\\{}\delta_{\alpha\beta}\delta_{ij}\Big{(}3g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}+g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}-g_{YX}\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{L}$
$\displaystyle-\delta_{\alpha\beta}\delta_{ij}\Big{(}2g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}-(2g_{YX}+3g_{X})\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{R}\Big{\\}}d_{{j\beta}}Z^{\mu},$
$\displaystyle\mathcal{L}_{Z\bar{l}_{{i}}l_{{j}}}=\frac{i}{2}\bar{l}_{{i}}\Big{\\{}\delta_{ij}\Big{(}-g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}+g_{YX}\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{L}$
$\displaystyle-\delta_{ij}\Big{(}2g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}-(2g_{YX}+g_{X})\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{R}\Big{\\}}l_{{j}}Z^{\mu},$
$\displaystyle\mathcal{L}_{Z\bar{u}_{{i\alpha}}u_{{j\beta}}}=-\frac{i}{6}\bar{u}_{{i\alpha}}\Big{\\{}\delta_{\alpha\beta}\delta_{ij}\Big{(}3g_{2}\cos{\theta^{\prime}}_{W}\cos\theta_{W}-g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}+g_{YX}\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{L}$
$\displaystyle-\delta_{\alpha\beta}\delta_{ij}\Big{(}4g_{1}\cos{\theta^{\prime}}_{W}\sin\theta_{W}-(3g_{X}+4g_{YX})\sin{\theta^{\prime}}_{W}\Big{)}\gamma_{\mu}P_{R}\Big{\\}}u_{{j\beta}}Z^{\mu}.$
(37)

We also derive the vertexes of one scalar boson and two different fermions,
including vertexes of
$\bar{\chi}^{-}_{{i}}$-$u_{{j\beta}}$-$\tilde{d}^{*}_{{k\gamma}}$,
$\bar{d}_{{i\alpha}}$-${\chi}^{-}_{{j}}$-$\tilde{u}_{{k\gamma}}$,
$\bar{l}_{{i}}$-${\chi}^{-}_{{j}}$-$\tilde{\nu}^{I}_{{k}}$,
$\bar{l}_{{i}}$-${\chi}^{-}_{{j}}$-$\tilde{\nu}^{R}_{{k}}$,
$\bar{\chi}^{0}_{{i}}$-$d_{{j\beta}}$-$\tilde{d}^{*}_{{k\gamma}}$,
$\bar{\chi}^{0}_{{i}}$-$l_{{j}}$-$\tilde{e}^{*}_{{k}}$ and
$\bar{\chi}^{0}_{{i}}$-$u_{{j\beta}}$-$\tilde{u}^{*}_{{k\gamma}}$.

$\displaystyle\mathcal{L}_{\bar{\chi}^{-}_{{i}}u_{{j\beta}}\tilde{d}^{*}_{{k\gamma}}}=i\bar{\chi}^{-}_{{i}}\tilde{d}^{*}_{{k\gamma}}\delta_{\beta\gamma}\Big{[}\Big{(}-g_{2}U^{*}_{i1}Z_{{kj}}^{D}+Y_{d,{j}}U^{*}_{i2}Z_{{k(3+j)}}^{D}\Big{)}P_{L}+Y^{*}_{u,{j}}V_{{i2}}Z_{{kj}}^{D}P_{R}\Big{]}u_{{j\beta}},$
$\displaystyle\mathcal{L}_{\bar{d}_{{i\alpha}}{\chi}^{-}_{{j}}\tilde{u}_{{k\gamma}}}=i\bar{d}_{{i\alpha}}\delta_{\alpha\gamma}\Big{[}Y_{d,{i}}Z^{U,*}_{ki}U^{*}_{j2}P_{L}+\Big{(}-g_{2}Z^{U,*}_{ki}V_{{j1}}+Y^{*}_{u,{i}}Z^{U,*}_{k(3+i)}V_{{j2}}\Big{)}P_{R}\Big{]}{\chi}^{-}_{{j}}\tilde{u}_{{k\gamma}},$
$\displaystyle\mathcal{L}_{\bar{l}_{{i}}{\chi}^{-}_{{j}}\tilde{\nu}^{I}_{{k}}}=-\frac{1}{\sqrt{2}}\bar{l}_{{i}}\Big{(}Y_{e,{i}}Z^{I,*}_{ki}U^{*}_{j2}P_{L}-g_{2}Z^{I,*}_{ki}V_{{j1}}P_{R}\Big{)}{\chi}^{-}_{{j}}\tilde{\nu}^{I}_{{k}},$
$\displaystyle\mathcal{L}_{\bar{l}_{{i}}{\chi}^{-}_{{j}}\tilde{\nu}^{R}_{{k}}}=\frac{i}{\sqrt{2}}\bar{l}_{{i}}\Big{(}Y_{e,{i}}Z^{R,*}_{ki}U^{*}_{j2}P_{L}-g_{2}Z^{R,*}_{ki}V_{{j1}}P_{R}\Big{)}{\chi}^{-}_{{j}}\tilde{\nu}^{R}_{{k}},$
$\displaystyle\mathcal{L}_{\bar{\chi}^{0}_{{i}}d_{{j\beta}}\tilde{d}^{*}_{{k\gamma}}}=-\frac{i}{6}\bar{\chi}^{0}_{{i}}\tilde{d}^{*}_{{k\gamma}}\delta_{\beta\gamma}\Big{\\{}\sqrt{2}\Big{(}g_{1}N^{*}_{i1}-3g_{2}N^{*}_{i2}+g_{YX}N^{*}_{i5}\Big{)}Z_{{kj}}^{D}+6Y_{d,{j}}N^{*}_{i3}Z_{{k(3+j)}}^{D}P_{L}$
$\displaystyle\hskip
48.36958pt+\Big{[}6Y^{*}_{d,{j}}N_{{i3}}Z_{{kj}}^{D}+\sqrt{2}\Big{(}2g_{1}N_{{i1}}+(2g_{YX}+3g_{X})N_{{i5}}\Big{)}Z_{{k(3+j)}}^{D}\Big{]}P_{R}\Big{\\}}d_{{j\beta}},$
$\displaystyle\mathcal{L}_{\bar{\chi}^{0}_{{i}}l_{{j}}\tilde{e}^{*}_{{k}}}=i\bar{\chi}^{0}_{{i}}\tilde{e}^{*}_{{k}}\Big{\\{}\Big{[}\frac{1}{\sqrt{2}}\Big{(}(g_{1}N^{*}_{i1}+g_{2}N^{*}_{i2}+g_{YX}N^{*}_{i5})Z_{{kj}}^{E}\Big{)}-Y_{e,{j}}N^{*}_{i3}Z_{{k(3+j)}}^{E}\Big{]}P_{L}$
$\displaystyle\hskip
38.41139pt-\Big{[}\frac{1}{\sqrt{2}}\Big{(}2g_{1}N_{{i1}}+(2g_{YX}+g_{X})N_{{i5}}\Big{)}Z_{{k(3+j)}}^{E}+Y^{*}_{e,{j}}N_{{i3}}Z_{{kj}}^{E}\Big{]}P_{R}\Big{\\}}l_{{j}},$
$\displaystyle\mathcal{L}_{\bar{\chi}^{0}_{{i}}u_{{j\beta}}\tilde{u}^{*}_{{k\gamma}}}=-\frac{i}{6}\bar{\chi}^{0}_{{i}}\tilde{u}^{*}_{{k\gamma}}\delta_{\beta\gamma}\Big{\\{}\Big{(}\sqrt{2}(g_{1}N^{*}_{i1}+3g_{2}N^{*}_{i2}+g_{YX}N^{*}_{i5})Z_{{kj}}^{U}+6Y_{u,{j}}N^{*}_{i4}Z_{{k(3+j)}}^{U}\Big{)}P_{L}$
$\displaystyle\hskip
48.36958pt-\Big{[}-6Y^{*}_{u,{j}}N_{{i4}}Z_{{kj}}^{U}+\sqrt{2}\Big{(}(3g_{X}+4g_{YX})N_{{i5}}+4g_{1}N_{{i1}}\Big{)}Z_{{k(3+j)}}^{U}\Big{]}P_{R}\Big{\\}}u_{{j\beta}}.$
(38)

##  Appendix B The hadron matrix elements

The corresponding hadron matrix elements we used in our work are encoded as:

$\displaystyle\langle
0|\bar{v}_{V}(p_{2})\gamma^{\mu}u_{V}(p_{1})|V(p)\rangle=\frac{f_{V}m_{V}}{N_{c}}\varepsilon^{\mu*}(p),$
$\displaystyle\langle
0|\bar{v}_{V}(p_{2})\sigma^{\mu\nu}P_{L}u_{V}(p_{1})|V(p)\rangle=-\frac{f_{V}\epsilon^{\mu\nu
p\varepsilon^{*}(p)}+if_{V}(p^{\mu}\varepsilon^{\nu*}(p)-p^{\nu}\varepsilon^{\mu*}(p))}{2N_{c}},$
$\displaystyle\langle
0|\bar{v}_{V}(p_{2})\sigma^{\mu\nu}P_{R}u_{V}(p_{1})|V(p)\rangle=\frac{f_{V}\epsilon^{\mu\nu
p\varepsilon^{*}(p)}-if_{V}(p^{\mu}\varepsilon^{\nu*}(p)-p^{\nu}\varepsilon^{\mu*}(p))}{2N_{c}},$
$\displaystyle\langle 0|\bar{v}_{V}(p_{2})u_{V}(p_{1})|V(p)\rangle=\langle
0|\bar{v}_{V}(p_{2})\gamma_{5}u_{V}(p_{1})|V(p)\rangle=\langle
0|\bar{v}_{V}(p_{2})\gamma^{\mu}\gamma_{5}u_{V}(p_{1})|V(p)\rangle=0.$ (39)

##  Appendix C The one-loop functions

In this section, we give out the corresponding one-loop integral functions,
which are read as:

$\displaystyle
I_{1}(x_{1},x_{2})=\frac{1}{16{\pi}^{2}}[-(\bigtriangleup+1+\ln{x_{\mu}})+\frac{x_{2}\ln{x_{2}}-x_{1}\ln{x_{1}}}{(x_{2}-x_{1})}],$
$\displaystyle
I_{2}(x_{1},x_{2})=\frac{1}{32{\pi}^{2}}[\frac{3+2\ln{x_{2}}}{(x_{2}-x_{1})}-\frac{2x_{2}+4x_{2}\ln{x_{2}}}{(x_{2}-x_{1})^{2}}+\frac{2x_{2}^{2}\ln{x_{2}}-2x_{1}^{2}\ln{x_{1}}}{(x_{2}-x_{1})^{3}}],$
$\displaystyle
I_{3}(x_{1},x_{2})=\frac{1}{16{\pi}^{2}}[\frac{1+\ln{x_{2}}}{(x_{2}-x_{1})}+\frac{x_{1}\ln{x_{1}}-x_{2}\ln{x_{2}}}{(x_{2}-x_{1})^{2}}],$
$\displaystyle
I_{4}(x_{1},x_{2})=\frac{1}{96{\pi}^{2}}[\frac{11+6\ln{x_{2}}}{(x_{2}-x_{1})}-\frac{15x_{2}+18x_{2}\ln{x_{2}}}{(x_{2}-x_{1})^{2}}+\frac{6x_{2}^{2}+18x_{2}^{2}\ln{x_{2}}}{(x_{2}-x_{1})^{3}}$
$\displaystyle\hskip
62.59596pt+\frac{6x_{1}^{3}\ln{x_{1}}-6x_{2}^{3}\ln{x_{2}}}{(x_{2}-x_{1})^{4}}],$
$\displaystyle
I_{5}(x_{1},x_{2})=\frac{1}{16{\pi}^{2}}[(\bigtriangleup+1+\ln{x_{\mu}})+\frac{x_{2}+2x_{2}\ln{x_{2}}}{(x_{1}-x_{2})}+\frac{x_{2}^{2}\ln{x_{2}}-x_{1}^{2}\ln{x_{1}}}{(x_{2}-x_{1})^{2}}],$
$\displaystyle
G_{1}(x_{1},x_{2},x_{3})=\frac{1}{16{\pi}^{2}}[\frac{x_{1}\ln{x_{1}}}{(x_{1}-x_{2})(x_{1}-x_{3})}+\frac{x_{2}\ln{x_{2}}}{(x_{2}-x_{1})(x_{2}-x_{3})}+\frac{x_{3}\ln{x_{3}}}{(x_{3}-x_{1})(x_{3}-x_{2})}],$
$\displaystyle
G_{2}(x_{1},x_{2},x_{3})=\frac{1}{16{\pi}^{2}}[-(\bigtriangleup+1+\ln{x_{\mu}})+\frac{x_{1}^{2}\ln{x_{1}}}{(x_{1}-x_{2})(x_{1}-x_{3})}$
$\displaystyle\hskip
85.35826pt+\frac{x_{2}^{2}\ln{x_{2}}}{(x_{2}-x_{1})(x_{2}-x_{3})}+\frac{x_{3}^{2}\ln{x_{3}}}{(x_{3}-x_{1})(x_{3}-x_{2})}],$
$\displaystyle
J_{1}(x_{1},x_{2},x_{3},x_{4})=\frac{1}{16{\pi}^{2}}[\frac{x_{1}^{2}\ln{x_{1}}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}+\frac{x_{2}^{2}\ln{x_{2}}}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$
$\displaystyle+\frac{x_{3}^{2}\ln{x_{3}}}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}+\frac{x_{4}^{2}\ln{x_{4}}}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}],$
$\displaystyle
J_{2}(x_{1},x_{2},x_{3},x_{4})=\frac{1}{16{\pi}^{2}}[\frac{x_{1}\ln{x_{1}}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}+\frac{x_{2}\ln{x_{2}}}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$
$\displaystyle+\frac{x_{3}\ln{x_{3}}}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}+\frac{x_{4}\ln{x_{4}}}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}],$
(40)

with $\Delta=\frac{1}{\epsilon}-r_{\epsilon}+\ln{4\pi}$.

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The Quantum Field Theory Boundaries Applicability and Black Holes
Thermodynamics
Alexander Shalyt-Margolin 111E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Research Institute for Nuclear Problems,Belarusian State University, 11
Bobruiskaya str., Minsk 220040, Belarus,

###### Abstract

This paper presents a study of the applicability boundary of the well-known
quantum field theory. Based on the results of black hole thermodynamics, it is
shown that this boundary may be lying at a level of the energy scales much
lower than the Planck. The direct inferences from these results are given,
specifically for estimation of a cosmological term within the scope of the
quantum field theory.

PACS: 11.10.-z,11.15.Ha,12.38.Bx
Keywords: quantum field theory, black holes thermodynamics, equivalence
principle

##  1 Introduction

The local quantum field theory (QFT) is understood as a canonical Quantum
Field Theory in flat space-time [1]–[3]. But in what follows it is
demonstrated that a flat geometry of space-time in the processes of high
energy physics is not ensured from the start, being based on validity of the
Einstein’s Strong Equivalence Principle (EP). However, this principle has its
applicability boundaries. The Planck scales present a natural (upper) bound
for applicability because at these scales a natural geometry of space-time,
determined locally by the particular metric $g_{\mu\nu}(\overline{x})$,
disappears due to high fluctuations of this metric and is replaced by space-
time (or quantum) foam.
In [4],[5] the author suggested a hypothesis that actually the real
applicability boundary of EP lies in a domain of the energies $E$ considerably
lower than the Planck energies. The principal objective of this paper is to
demonstrate that the hypothesis is true for some, quite naturally arising,
assumptions. Proceeding from the afore-said, this condition sets the
applicability boundaries for the canonical QFT. Besides, direct inferences
from the obtained result are considered. Hereinafter, EP is understood as a
Strong Equivalence Principle.

##  2 The QFT Applicability Boundaries and Equivalence Principle

The canonical quantum field theory (QFT) [1]– [3] is a local theory considered
in continuous space-time with a plane geometry, i.e with the Minkowskian
metric $\eta_{\mu\nu}(\overline{x})$. In reality, any interaction introduces
some disturbances, introducing an additional local (little) curvature into the
initially flat Minkowskian space $\mathcal{M}$. Then the metric
$\eta_{\mu\nu}(\overline{x})$ is replaced by the metric
$\eta_{\mu\nu}(\overline{x})+o_{\mu\nu}(\overline{x})$, where the increment
$o_{\mu\nu}(\overline{x})$ is small. But, when it is assumed that EP is valid,
the increment $o_{\mu\nu}(\overline{x})$ in the local theory has no important
role and, in a fairly small neighborhood of the point $\overline{x}$ in virtue
EP.
The Einstein Equivalence Principle (EP) is a basic principle not only in the
General Relativity (GR) [6]–[8], but also in the fundamental physics as a
whole. In the standard formulation it is as follows: ([8],p.68):
<<at every space-time point in an arbitrary gravitational field it is possible
to choose a locally inertial coordinate system such that, within a
sufficiently small region of the point in question, the laws of nature take
the same form as in unaccelerated Cartesian coordinate systems in the absence
of gravitation>>.
Then in ([8],p.68) <<…There is also a question, how small is ”sufficiently
small”. Roughly speaking, we mean that the region must be small enough so that
gravitational field in sensible constant throughout it…>>.
However, the statement ”sufficiently small” is associated with another
problem. Indeed, let $\overline{x}$ be a certain point of the space-time
manifold $\mathcal{M}$ (i.e. $\overline{x}\in\mathcal{M}$) with the geometry
given by the metric $g_{\mu\nu}(\overline{x})$. Next, in accordance with EP,
there is some sufficiently small region $\mathcal{V}$ of the point
$\overline{x}$ so that, within $\mathcal{V}$ it is supposed that space-time
has a flat geometry with the Minkowskian metric $\eta_{\mu\nu}(\overline{x})$.
In essence, sufficiently small $\mathcal{V}$ means that the region
$\mathcal{V}^{{}^{\prime}}$, for which
$\overline{x}\in\mathcal{V}^{{}^{\prime}}\subset\mathcal{V}$, satisfies this
condition as well. In this way we can construct the sequence

$\displaystyle...\subset\mathcal{V}^{{}^{\prime\prime}}\subset\mathcal{V}^{{}^{\prime}}\subset\mathcal{V}.$
(1)

The problem arises, is there any lower limit for the sequence in formula (1)?
The answer is positive. Currently, there is no doubt that at very high
energies (on the order of Planck energies $E\approx E_{p}$), i.e. on Planck
scales, $l\approx l_{p}$ quantum fluctuations of any metric
$g_{\mu\nu}(\overline{x})$ are so high that in this case the geometry
determined by $g_{\mu\nu}(\overline{x})$ is replaced by the ”geometry”
following from space-time foam that is defined by great quantum fluctuations
of $g_{\mu\nu}(\overline{x})$,i.e. by the characteristic dimensions of the
quantum-gravitational region (for example, [9]–[14]). The above-mentioned
geometry is drastically differing from the locally smooth geometry of
continuous space-time and EP in it is no longer valid [15]–[22].
From this it follows that the region $\mathcal{V}_{\overline{r},\overline{t}}$
with the characteristic spatial dimension $\overline{r}\approx l_{p}$ (and
hence with the temporal dimension $\overline{t}\approx t_{p}$) is the lower
(approximate) limit for the sequence in (1).
It is difficult to find the exact lower limit for the sequence in formula
(1)–it seems to be dependent on the processes under study. Specifically, when
the involved particles are considered to be point, their dimensions may be
neglected in a definition of the EP applicability limit. When the
characteristic spatial dimension of a particle is $\mathbf{r}$, the lower
limit of the sequence from formula (1) seems to be given by the region
$\mathcal{V}_{\mathbf{r^{\prime}}}$ containing the above-mentioned particle
with the characteristic dimensions $\mathbf{r^{\prime}}>\mathbf{r}$, i.e. the
space EP applicability limit should always be greater than dimensions of the
particles considered in this region. By the present time, it is known that
spatial dimensions of gauge bosons, quarks, and leptons within the limiting
accuracy of the conducted measurements $<10^{-18}m$. Because of this, the
condition $\mathbf{r^{\prime}}\geq 10^{-18}m$ must be fulfilled. In addition,
the radius of interaction of particles $\mathbf{r}_{int}$ must be taken into
account in quantum theory. And this fact also imposes a restriction on
considering concrete processes in quantum theory. However, the interactions
radii of all known processes lie in the energy scales $E\ll E_{p}.$
At the present time there is a series of the results demonstrating that EP may
be violated at the energies $E$ considerably lower than $E_{p}$(for example,
the quantum phenomenon of neutrino oscillations in a gravitational background
[23] [24] [25] and others [26],[27],[28]).
As QFT is a local theory applicable only to space-time with a flat geometry
determined by the Minkowskian metric $\eta_{\mu\nu}(\overline{x})$,the
applicability boundary EP may be considered the applicability boundary of QFT
as well.
Main Hypothesis
It is assumed that in the general case EP, and consequently, QFT is valid for
the locally smooth space-time only if all the energies $E$ of the particles
are satisfied the necessary condition

$\displaystyle E\ll E_{p},$ (2)

In the following section this hypothesis is proven in the assumption that
space-time foam consists of micro black holes (mbh) with the event horizon
radius $r\approx l_{p}$ and mass $m\approx m_{p}$.

Remark 2.1
Why in canonical QFT it is so important never forget about the fact that
space-time has a flat geometry, or the same possesses the Minkowskian metric
$\eta_{\mu\nu}(\overline{x})$? Simply, in the contrary case we should refuse
from some fruitful methods and from the results obtained by these methods in
canonical QFT, in particular from Wick rotation [3]. In fact, in this case the
time variable is replaced by $t\mapsto it\doteq t_{E}$, and the Minkowskian
metric $\eta_{\mu\nu}(\overline{x})$ is replaced by the four-dimensional
Euclidean metric

$\displaystyle ds^{2}=dt^{2}_{E}+dx^{2}+dy^{2}+dz^{2}.$ (3)

Clearly, such replacement is possible only in the case when from the start
space-time (locally) has a flat geometry, i. e. possesses the Minkowskian
metric $\eta_{\mu\nu}(\overline{x})$. This is another argument supporting the
key role of the EP applicability boundary. Otherwise, when we go beyond this
boundary, Wick rotation becomes invalid. Naturally, some other methods of
canonical QFT will lose their force too.

##  3 The Strong Equivalence Principle, Black Holes, and QFT

It is supposed that a large (i. e, classical) four-dimensional Schwarzschild
black hole is existent with the metric

$ds^{2}=\left(1-\frac{2MG}{r}\right)dt^{2}-\left(1-\frac{2MG}{r}\right)^{-1}dr^{2}-r^{2}d\Omega^{2},$
(4)

where $M$ is the mass of this black hole, and the Schwarzschild horizon radius
$r_{BH}$ is defined by

$r_{BH}=2MG.$ (5)

As shown in [29],[30], EP is violated for an observer distant from the black
hole event horizon. Considering our objective, it seems expedient to give in
brief the main results from [29],[30].
In view of the Unruh effect, an accelerating observer does detect thermal
radiation (so-called Unruh radiation) with the Unruh temperature given by [32]

$\displaystyle T_{Unruh}=\frac{\hbar a}{2\pi},$ (6)

where $a=|{\bf a}|$ is a corresponding acceleration.
When an observer is at the fixed distance, $r>r_{BH}$, from a Schwarzschild
black hole of mass $M$ and event horizon radius $r_{BH}=2GM$, then, due to the
existence of Hawking radiation [33], the observer will measure radiation with
thermal spectrum and a temperature given by formula [29]

$\displaystyle T_{H,r}=\frac{\hbar}{8\pi
GM\sqrt{1-\frac{r_{BH}}{r}}}\leavevmode\nobreak\ ,$ (7)

where $r>r_{BH}$.
In the foregoing formulae and in what follows, we use the normalization
$c=k_{B}=1$.
Besides, in [29] it is shown that an observer, positioned at the fixed
distance $r>r_{BH}$ from the above-mentioned black holes and measuring Hawking
temperature with the value $T_{H,r}$, experiences the local acceleration

$\displaystyle
a_{BH,r}=\frac{1}{\sqrt{1-\frac{r_{BH}}{r}}}\left(\frac{r_{BH}}{2r^{2}}\right)\leavevmode\nobreak\
.$ (8)

Another observer in the Einstein elevator, moving with acceleration through
Minkowskian space-time, will measure the same acceleration toward the floor of
the elevator, thermal radiation with the Unruh temperature given by formula
(6). As shown in [29], $a_{BH}$ is coincident with the quantity $a$ from the
formula in (6). Then substituting the acceleration $a=a_{BH}$ from formula (8)
into formula (6), we can obtain a formula for $T_{Unruh,r}$ in this case [30]:

$\displaystyle
T_{Unruh,r}=\frac{\hbar}{2\pi\sqrt{1-\frac{r_{BH}}{r}}}\left(\frac{r_{BH}}{2r^{2}}\right)\leavevmode\nobreak\
.$ (9)

If EP is valid, the quantities $T_{Unruh,r}$ from formula (9) and $T_{H,r}$ in
(7) should be coincident for $r>r_{BH}$ to a high degree of accuracy. However,
we see that this is not true. In [30], e.g. for $r=4GM=2r_{BH}$, we have
$T_{H,r}=4T_{Unruh,r}$.
So, far from the event horizon, EP is not the case. Moreover, violation of EP
is the greater the farther it is from the black hole event horizon. Indeed,
for an observer at the distance $r>r_{BH}$ we can write
$r=\mathcal{\alpha}(r)GM=\frac{1}{2}\mathcal{\alpha}(r)r_{BH},\mathcal{\alpha}(r)>2$.
Then

$\displaystyle
T_{Unruh,r}=\frac{\hbar}{2\pi\mathcal{\alpha}^{2}(r)GM\sqrt{1-\frac{2}{\alpha(r)}}}.$
(10)

In this way $T_{H}/T_{Unruh,r}=\mathcal{\alpha}^{2}(r)/4$. And the ratio is
the greater, the higher $\mathcal{\alpha}(r)$,i.e. the farther from horizon
the observer is. Next, for compactness, we denote $T_{Unruh,r}$ in terms of
$T_{U,r}$. Of course, in this case we bear in mind only an observer at a
sufficiently great but finite distance from a black hole, i.e. only when a
gravitational field is thought significant and must be taken into
consideration. So, in the general case $r_{BH}<r\ll\infty$, whereas in the
case of a distant observer we have

$\displaystyle r_{BH}\ll r\ll\infty.$ (11)

Obviously, this case of violated EP is not directly associated with the Main
Hypothesis concerning the boundaries of EP validity (formula (2)) from the
previous section, because in [29],[30] consideration is given to a large black
hole with the event horizon radius $r_{BH}$ much greater than Planck length
$r_{BH}\gg l_{p}$ at sufficiently low energies.
Really, the resulting distribution of the particles emitted by a black hole
has the form (last formula on p.122 in [34])

$\displaystyle n_{E}=\Gamma_{gb}[exp(\frac{E}{T_{H}})-1]^{-1},$ (12)

where $n_{E}$ is the number of particles with the energy $E$ and
$\Gamma_{gb}<1$ is the so-called greybody factor. As the black hole mass $M$
is large, the temperature $T_{H}$ is low, and then from the last formula it
follows that arbitrary large values of $n_{E}$ will be given only by particles
with the energies $E$ close to a small value of $T_{H}$.
The principal result from the remarkable papers [29],[30] may be summarized as
follows:
Comment 3.1
In any point of space-time that is in a field of a large classical
Schwarzschild black hole, and in the cases when this field must be taken into
consideration, it is impossible to remove this field in the vicinity of the
point even locally, i.e. to consider space-time as flat.

Comment 3.2
It is important to refine some formulations from [29],[30]. Specifically, if
$r\rightarrow r_{BH}$, then
$T_{H,r}\rightarrow\infty,T_{Unruh,r}\rightarrow\infty$ in formulae (7) and
(9), respectively. Note that for $r\rightarrow r_{BH}$ these temperatures
become infinite $T_{H,r}=\infty,T_{Unruh,r}=\infty$. Based on this fact, in
[29],[30] it is inferred ”that the equivalence principle is restored on the
horizon”. But this statement is not correct. Restoration of EP is not
following from the fact that the above temperatures take infinite values. We
can only state that temperature on the BH horizon and in its vicinity cannot
be the parameter detecting a deviation from EP. In the opposite case one can
arrive at violation: on the black hole event horizon, where a gravitational
field is very large in value, EP holds, whereas far from the event horizon,
where a gravitational field is much weaker, this principle is violated.

Let us return to high energy physics and to the subject of the previous
section. One of the preferable models for space-time foam is the model based
on the assumption that its unit cells are mbh, with radius and mass on the
order of the Planck (for example, [14],[19], [20]. Of great importance for mbh
are the quantum-gravitational effects and the corresponding quantum
corrections of black hole thermodynamics at Planck scale (for example, [35]).
Then, in line with formula (20) in [35], we have minimal values for radius and
mass of a black hole

$r_{min}=\sqrt{\frac{e}{2}}\alpha^{{}^{\prime}}l_{p},\quad m_{min}\doteq
m_{0}=\frac{\alpha^{{}^{\prime}}\sqrt{e}}{2\sqrt{2}}m_{p},$ (13)

where the number $\alpha^{{}^{\prime}}$ is on the order of 1, and in [35] we
take the normalization $\hbar=c=k_{B}=1$ in which
$l_{p}=m_{p}^{-1}=T_{p}^{-1}=\sqrt{G}.$
From (13) it directly follows that the formula for the event horizon radius
$r=2MG$, valid for large classical black holes, will be valid in the case when
we include the quantum-gravitational effects for mbh because
$r_{min}=2m_{0}G$. Such a black hole of a minimal size is associated with a
maximal temperature (formula (24) in [35]):

$T_{H}^{max}=\frac{T_{p}}{2\pi\sqrt{2}\alpha^{{}^{\prime}}}.$ (14)

A black hole satisfying the formulae (13),(14) is termed as minimal (or
Planck).
Without loss of generality, it is assumed that for event horizon radii and
masses of mbh the following is valid:

$r_{mbh}\approx r_{min},m_{mbh}\approx m_{0},$ (15)

i.e. mbh are Planck black holes.
For the energies $E$ somewhat lower than the Planck energies (i.e., $E\ll
E_{p}$) involved in the condition (2), a semiclassical approximation is valid.
This means that, on substitution of mbh with the mass $m_{mbh}$ for a large
(classical) black hole with the mass $M$, in the case under study the results,
substantiated when an observer uses the standard Unruh-Dewitt detector in
radiation measurement for coupled to a massless scalar field [36],[37], are
valid with the corresponding quantum corrections [31].
Let us revert to the formulae from [29],[30]. In particular, to formula (8)
for the real acceleration measured by an observer who is positioned at the
fixed distance $r\gg r_{BH}$ in the Schwarzschild space-time, given by
(formula (14) in [29], formula (9.170) in [49])

$a_{BH,r}=a_{S}=\frac{{\sqrt{\nabla_{\mu}V\nabla^{\mu}V}}}{V}=\frac{MG}{r^{2}\sqrt{1-2MG/r}}=\frac{MG}{r^{2}V},$
(16)

where it is supposed that a static observer at the radius $r$ moves along
orbits of the time-like Killing vector $\textit{K}=\partial_{t}$ and
$V=\sqrt{-\textit{K}_{\mu}\textit{K}^{\mu}}=\sqrt{1-2MG/r}$ is the red-shift
factor for the Schwarzschild space-time (p.413 in [49]).
How changes formula (16) on going to mbh? It is clear that the condition $r\gg
r_{BH}$ is replaced by the condition $r\gg r_{mbh}$ (corresponding to the
condition $E\ll E_{p}$ and semiclassical approximation),$M$ is replaced by
$m_{mbh}$,the red-shift factor $V$ should be replaced by $V_{q}$, where
$V_{q}$ is the quantum deformation of $V$ with regard to quantum corrections
in the field mbh. Then, for mbh, formula (16) is of the form

$a_{S,q}=\frac{m_{mbh}G}{r^{2}V_{q}},$ (17)

where $a_{S,q}$ is the real acceleration with regard to the quantum
corrections measured by a distant observer in the field mbh.
Clearly, formula (7) for $T_{H,r}$, due to formulae for the red-shift factor
$V$, in the general case may be given as

$\displaystyle T_{H,r}=\frac{\hbar}{8\pi GMV}\leavevmode\nobreak\ .$ (18)

Then its quantum analog, i.e. the corresponding formula for temperature in the
field mbh, for $r\gg r_{mbh}$ is as follows:

$\displaystyle T_{H,r,q}=\frac{\hbar}{8\pi Gm_{mbh}V_{q}}.$ (19)

In virtue of formula (16), formula (9) takes the form

$\displaystyle
T_{U,r}=\frac{\hbar}{2\pi\sqrt{1-\frac{r_{BH}}{r}}}\left(\frac{r_{BH}}{2r^{2}}\right)=\frac{\hbar}{2\pi
V}\left(\frac{r_{BH}}{2r^{2}}\right).$ (20)

For $r>r_{BH}$ and due to formula (10), we have

$\displaystyle
T_{U,r}=\frac{\hbar}{2\pi\mathcal{\alpha}^{2}(r)GM\sqrt{1-\frac{2}{\alpha(r)}}}=\frac{\hbar}{2\pi\mathcal{\alpha}^{2}(r)GMV}.$
(21)

As indicated above, for $r\gg r_{BH}$ we have $\alpha(r)\gg 1$.
What are the changes on going to mbh?
Considering the case $r\gg r_{mbh}$ and semiclassical picture, we again come
to $\alpha(r)\gg 1$, whereas formula (21) is replaced by formula

$\displaystyle
T_{U,r,q}=\frac{\hbar}{2\pi\mathcal{\alpha}^{2}(r)Gm_{mbh}V_{q}}.$ (22)

Proceeding from the above, we have

$\displaystyle\frac{T_{H,r,q}}{T_{U,r,q}}=\frac{T_{H,r}}{T_{U,r}}=\frac{\mathcal{\alpha}^{2}(r)}{4}$
(23)

Formula (23) points to the fact that, within the scope of a semiclassical
approximation, relations of a black hole temperature to the Unruh temperature
for a distant observer in the case of a large (classical) black hole and mbh
are coincident because these quantities are dependent on the same factors:
first, on $1/MV$ and, second, on $1/m_{mbh}V_{q}$.
Note that in this consideration there s no need to have an explicit formula
for $V_{q}$ as this quantity is not involved in the key expression (23).
Specifically, to derive an explicit expression for $V_{q}$, we can use the
results from [50] on quantum deformation of the Schwarzschild solution due to
spherically symmetric quantum fluctuations of the metric and the matter
fields. In this case the Schwarzschild singularity at $r=0$ is shifted to the
finite radius $r_{min}\approx r_{mbh}\propto l_{p}$, where the scalar
curvature is finite. In this way the results from [50] correlate well with the
results from [35].
Quantum corrections at Planck scales were obtained in [35] proceeding from
validity of the Generalized Uncertainty Principle (GUP)[38]–[41]. But the
results presented in this work are independent of this aspect. Actually,
during studies of black hole thermodynamics at Planck scales with the use of
other methods [42],[43] (differing from those in [35]),in particular, Loop
Quantum Gravity (LQG)[43], the obtained results were similar to [35]. Because
of this, for mbh with all the thermodynamic characteristics
(mass,radius,temperature,…) on the order of the corresponding Planck
quantities, all the calculations in this section are valid. As noted above,
far from horizon of mbh, i.e. at the energies $E\ll E_{p}$ (2), the results
from [29],[30] remain valid in this case as well.
Next, similar to [14], we assume that in every cell of space-time foam a micro
black hole (mbh) with a typical gravitational radius of $r_{min}\propto l_{p}$
may be present. Then, in according with the results in [29],[30] and in virtue
of the formula (23) we come to violation of the strong EP for distance $r$,
satisfying the condition

$\displaystyle l_{p}\ll r\ll\infty,$ (24)

that is equivalent to $\widetilde{E}_{r}\ll E_{p}$ for the energies
$\widetilde{E}_{r}$ associated with the scale of $r$.
In the last formula it is implicitly (and purely conditionally) assumed that a
minimum length is equal to $l_{p}$ and to $r_{min}$. But, as noted above, in
the general case we have $r_{min}\propto l_{p}$, i.e., the order is similar to
that of $l_{p}$. Specifically, in [44] by natural assumptions it has been
demonstrated that the minimum length may be twice and more as great as the
Planck length. It is obvious that all the above calculations and derivations
of the present work are independent of the specific value of $r_{min}$.
In this way, if the quantum foam structure is determined by mbh, the
applicability of QFT is limited to the energies $E<\widetilde{E}_{r}\ll E_{p}$
and the formula (2) is the case. This supports the Main Hypothesis from
Section 2 within the assumption concerning the quantum foam structure made in
this section.
Comment 3.3
In the case of mbh Comment 3.2 is absolutely clear. In fact, at a horizon of
mbh,i.e. for $r=r_{min}$, $T_{H,r}=T_{Unruh,r}=\infty$ similar to large black
holes but , naturally, without any restoration of EP as the domain
$r=r_{mbh}\approx r_{min}\propto l_{p}$ is the region of Planck energies or of
quantum foam, where EP in its canonical formulation becomes invalid. It is
obvious that at the event horizon $r=r_{mbh}$ of mbh and in its vicinity a
gravitational field becomes very strong due to quantum effects and nothing
could destroy it.

##  4 Some Immediate Consequences

4.1 Based on the above results, all the energies $E$ we can classify into 3
groups:

a)low energies $0<E\leq\widetilde{E}\ll E_{p}$ – energies, for which the
Strong Equivalence Principle is valid in virtue of formula (2), and hence this
energy interval sets the QFT applicability boundaries.
a1) Since $\widetilde{E}\ll E_{p}$, it is natural to assume that
$\widetilde{E}\approx 10^{-N}E_{p}$, where $N\geq 2$. Obtaining of more
accurate estimates for $N$ is a separate problem;

b)intermediate energies $\widetilde{E}<E<E_{p}$ – energies, for which the
Strong Equivalence Principle and, consequently QFT, becomes invalid but the
corresponding scales are greater than the Planck. It can be assumed that QFT
in this energy range will be a theory in a gravitational field that could not
be destroyed even locally. In the case under study it is assumed that this
field is created by mbh. Impossibility of destroying this field even locally
is associated with large quantum corrections for the corresponding quantities
which should be taken into consideration at these energies [35],[42],[43].
Let us call the energy scale $\widetilde{E}<E<E_{p}$ as prequantum gravity
phase;

c)high (essentially maximal) energies $E\approx E_{p}$ or $E>E_{p}$. This
interval is the region of quantum gravity energies.

Next note that, as all the experimentally involved energies $E$ are low, they
satisfy condition a) or b). Specifically, for LHC, maximal energies are
$\approx 10TeV=10^{4}GeV$, that is by 15 orders of magnitude lower than the
Planck energy $\approx 10^{19}GeV$. Moreover, the characteristic energy scales
of all fundamental interactions also satisfy condition a). Indeed, in the case
of strong interactions this scale is $\Lambda_{QCD}\sim 200MeV$; for
electroweak interactions this scale is determined by the vacuum average of a
Higgs boson and equals $\upsilon\approx 246GeV$; finally, the scale of the
(Grand Unification Theory (GUT)) $M_{GUT}$ lies in the range of $\sim
10^{14}GeV--10^{16}GeV$.
It should be noted, however, that on validity of assumption a1) the energy
scale $M_{GUT}$ lies within the applicability region of the energy group a)
and hence of QFT. Provided the EP applicability boundaries are lying at
considerably lower energies, a study of GUT necessitates a theory with (even
locally) unremovable curvature.
At the same time, it is clear that the requirement of the Lorentz-invariant
QFT, due to the action of Lorentz boost (or same hyperbolic rotations) (for
example formula (3) in [7]), results in however high momenta and energies. But
it has been demonstrated that unlimited growth of the momenta and energies is
impossible because in this case we fall within the energy region, where the
conventional quantum field theory [1]–[3] is invalid.
Note that at the present time there are experimental indications that Lorentz-
invariance is violated in QFT on passage to higher energies (for example,
[45]). Besides, one should note important recent works associated with EP
applicability boundaries and violation in nuclei and atoms at low energies
(for example [26]). Proceeding from the above, the requirement for Lorentz-
invariance and EP is possible only within the scope of the condition (2).

4.2 Proceeding from the above results, it is inferred that the well-known QFT
[1]–[3], from the start, is a ultraviolet-finite theory with the natural
cutoff parameter $l_{\widetilde{E}}\propto\hbar/\widetilde{E}.$ Note that the
quantum-gravitational parameter $r_{mbh}\propto l_{p}$ is beyond the
applicability limits of QFT.

4.3 In the present approach it is of interest to study the problem of
asymptotic safety introduced by Steven Weinberg in [46]. We use the definition
of this notion given in ([47],p.67): ”A theory is said to be asymptotically
safe if all essential coupling parameters $g_{j}$ (these are the ones that are
invariant under field redefinitions) approach, for energies
$k\rightarrow\infty$, a fixed point where at least one of them does not
vanish.” If initially it has been assumed that $r_{min}$ is considered within
the scope of (GUP)[38]–[41],this definition necessitates certain refinements.
In particular, if (GUP) supposes the existence of a maximum momentum
$p_{max}$, as in [40] or (Section V in [41]),it is clear that the condition
$k\rightarrow\infty$ can not be fulfilled. So, the condition
$k\rightarrow\infty$ should be replaced by the condition $k\rightarrow
p_{max}.$ Most often, it is assumed that the momentum $p_{max}$ is on the
order of the Planck momentum, i.e., we have $p_{max}\propto p_{pl}$. However,
in the most general case the quantity $p_{max}$ may be even trans-Planck.
When $p_{max}$ is inexistent (i.e. $p_{max}=\infty$), still by this approach
of asymptotic safety the problem should be reformulated in accordance with the
fact (shown above) that, beginning with the energies $E,E>\widetilde{E}$, a
theory must be considered as QFT in curved space-time in a field created by
mbh.
In his further works the author is planning to study this problem within the
scope of this approach in greater detail.

4.4 It is possible to correct the estimates obtained within the scope of the
known QFT by means of the condition (2). Let us consider a typical example.
In his well-known lectures [48] at the Cornell University Steven Weinberg
considered an example of calculating, within the scope of QFT, the expected
value for the vacuum energy density $<\rho>$ that is proportional to the
cosmological term $\lambda$. To this end, zero-point energies of all normal
modes of some field with the mass $m$ are summed up to the wave number cutoff
$\Lambda\gg m$ for the selected normalization $\hbar=c=1$ (formula (3.5) in
[48]):

$\displaystyle<\rho>=\int_{0}^{\Lambda}\frac{4\pi
k^{2}dk}{(2\pi)^{3}}\frac{1}{2}\sqrt{k^{2}+m^{2}}\simeq\frac{\Lambda^{4}}{16\pi^{2}}.$
(25)

Assuming, similar to [48], that GR is valid at all the energy scales up to the
Planck, we have the cutoff $\Lambda\simeq(8\pi G)^{-1/2}$ and hence (formula
(3.6) in [48]) leads to the following result:

$\displaystyle<\rho>\approx 2\cdot 10^{71}GeV^{4},$ (26)

that by $10^{118}$ orders of magnitude differs from the well-known
experimental value for the vacuum energy density

$\displaystyle<\rho_{exp}>\preceq 10^{-29}\mathrm{g}/cm^{3}\approx
10^{-47}GeV^{4}.$ (27)

Here $G$ is a gravitational constant.
It is clear that in this case the condition (2) is not fulfilled and this
leads to such a monstrous discrepancy with $<\rho_{exp}>$. Based on the afore-
said, the following estimate for $<\rho>$ is more correct:

$\displaystyle<\rho_{\widetilde{E}}>\doteq\int_{0}^{\Lambda_{\widetilde{E}}}\frac{4\pi
k^{2}dk}{(2\pi)^{3}}\frac{1}{2}\sqrt{k^{2}+m^{2}}\simeq\frac{\Lambda_{\widetilde{E}}^{4}}{16\pi^{2}},$
(28)

where $\Lambda(\widetilde{E})$–cut-off parameter of the corresponding energy
$\widetilde{E}$ from point a) in 4.1.
Of course, the main contribution into the integral in the right side of (25)
is made by high energies $\widetilde{E}<E<E_{p}$ from point b) in 4.1,which
are not involved in formula (28). Consequently, it seems possible that
$<\rho_{\widetilde{E}}>\ll<\rho>$ and hence $<\rho_{\widetilde{E}}>$ may be
much closer to $<\rho_{exp}>$ than $<\rho>$.
In the whole, the methods of a quantum field theory may be effectively used to
obtain the vacuum energy density. In his very interesting work [51] the
author, based on the quantum hoop conjecture, has obtained a natural cutoff
for the vacuum energy of a scalar field, also giving an estimate for $<\rho>$
much closer to $<\rho_{exp}>$ than in formulae(25),(26). It is interesting to
know how close are the estimates for the vacuum energy density in [51] and in
formula (28). To this end, we first need a sufficiently accurate estimate of
the quantity $\widetilde{E}$ in line with the QFT boundaries applicability.
Besides, it is important to find whether the methods of QFT are enough to
obtain $<\rho_{exp}>$ or some additional assumptions will be required,
specifically, the Holographic Principle applying to the Universe [52]–[59] .

##  5 Conclusion

Thus, within the scope of natural assumptions, in this paper it is
demonstrated that the applicability boundary of the well-known QFT is lying in
the region of energies considerably lower than the Planck energies, i.e. the
canonical QFT [1]–[3] is an ultraviolet-finite theory.
In this paradigm it is important to understand the way to transform the well-
known results for ultraviolet regularization, renormalization, and so on from
QFT within the scope of the applicability boundary $\widetilde{E}$ of QFT
indicated in point a) of the preceding section. Possibly, this boundary will
be dependent on a nature of the processes under study in high energy physics.

Conflict of Interests

The author declares that there is no conflict of interests regarding the
publication of this paper.

Acknowledgments

The author would like to express his gratitude to the Reviewers of this work
for their valuable remarks and recommendations.

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