Radiation of Closed Strings between the Parallel Dynamical-Dressed Unstable
D$p$-Branes

Hamidreza Daniali and Davoud Kamani

Department of Physics, Amirkabir University of Technology (Tehran Polytechnic)
P.O.Box: 15875-4413, Tehran, Iran
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We introduce a boundary state which is corresponding to a D$p$-brane with
tangential dynamics in the presence of the Kalb-Ramond field, a tachyonic
field and a $U(1)$ gauge potential in a special gauge. From the interaction of
such branes radiation amplitude of a general massless closed string will be
computed. The effects of the large distances of the branes on this radiation
will be studied. In the large distances of the branes, the possibility of
axion radiation will be investigated. Our calculations will be in the context
of the bosonic string theory.

PACS numbers: 11.25.-w; 11.25.Uv

Keywords: Boundary state; Background fields; Dynamics; Radiation amplitude;
Axion production.

##  1 Introduction

By studying the D-branes properties and their interactions, some of the
important physical results in the string theory were obtained [1, 2]. One of
the convenient methods for calculating the interaction of two branes, in the
closed-channel, is the boundary state formalism [3]-[15]. The interaction of
two bare-static D-branes in the superstring theory is zero [16]. By adding
dynamics and various background fields, some nonzero and valuable interaction
amplitudes were obtained [17]-[27].

Closed strings can be radiated from the D$p$-branes with various
configurations. One of the most important setups is the radiation of closed
strings from a single unstable D$p$-brane [28, 29, 30]. Besides the tachyonic
background, which prominently induces instability on the branes, closed string
radiation from a single unstable brane in the presence of the various
background fields has been also investigated [31, 32]. In addition, the
supersymmetric version of the closed string radiation was also constructed
[33]. Another interesting configuration is the closed string radiation from
the interacting D-branes. This kind of radiation was studied only in specific
setups [34, 35]. We shall add dynamics, internal and background fields to the
interacting unstable branes to extract the closed string radiation from a
generalized configuration.

The background fields and dynamics of the branes motivated us to obtain the
effects of these variables on the closed string radiation from the interacting
branes. Therefore, in this paper, in the context of the bosonic string theory,
we investigate the massless closed string radiation from the interaction of
two unstable D$p$-branes with tangential dynamics in the presence of the Kalb-
Ramond field, a quadratic tachyonic field and a U(1) gauge potential in a
special gauge. For computing the radiation amplitude, we shall apply the
boundary state formalism. Thus, by inserting an appropriate vertex operator in
the worldsheet of the exchanged closed string between the branes, we produce a
radiated closed string. We accurately use the eikonal approximation in which
the recoil of the branes is ignored. After calculations for radiating a
general massless state, we concentrate on the axion radiation from the distant
branes. We shall observe that the radiated axion is directly emitted by one of
the two interacting branes. Note that the interaction between two D-branes in
the large distance limit occurs only via the exchange of the massless states.

This paper is organized as follows. In Sec. 2, the boundary state,
corresponding to a dynamical-dressed unstable D$p$-brane, will be introduced.
In Sec. 3.1, the radiation amplitude of a general massless closed string from
the interaction of two parallel D$p$-branes will be constructed. In Sec. 3.3,
the foregoing radiation amplitude will be deformed for the distant branes. In
Sec. 4, the radiation of an axion from the distant branes will be obtained.
Sec. 5 is devoted to the conclusions.

##  2 The boundary state of the dynamical-dressed unstable D$p$-brane

In this section we introduce the boundary state corresponding to a dynamical-
dressed unstable D$p$-brane. Hence, we start with the following sigma-model
action for closed string

$\displaystyle S=$ $\displaystyle-$
$\displaystyle\frac{1}{4\pi\alpha^{\prime}}{\int}_{\Sigma}{\rm
d}^{2}\sigma\left(\sqrt{-g}g^{ab}G_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}+\varepsilon^{ab}B_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\right)$
(2.1) $\displaystyle+$
$\displaystyle\frac{1}{2\pi\alpha^{\prime}}{\int}_{\partial\Sigma}{\rm
d}\sigma\left(A_{\alpha}\partial_{\sigma}X^{\alpha}+\omega_{\alpha\beta}J^{\alpha\beta}_{\tau}+T(X^{\alpha})\right),$

where $\mu,\nu\in\\{0,1,\cdots,25\\}$ represent the spacetime indices, which
are split into the indices for the worldvolume directions, i.e.
$\alpha,\beta\in\\{0,\cdots,p\\}$, and for the perpendicular directions to it,
i.e. $i,j\in\\{p+1,\cdots,25\\}$. Besides, $a,b\in\\{0,1\\}$ specify the
worldsheet directions. $\Sigma$ is the closed string worldsheet, while
$\partial\Sigma$ is its boundary. $G_{\mu\nu}$ and $g_{ab}$ are the metrics of
the target spacetime and string worldsheet, respectively. The background
fields are the Kalb-Ramond field $B_{\mu\nu}$, the $U(1)$ internal gauge field
$A_{\alpha}$ and the open string tachyon field $T(X^{\alpha})$. The constant
antisymmetric tensor $\omega_{\alpha\beta}$ shows the angular velocity of the
brane. Hence, the $\omega$-term obviously expresses the tangential dynamics of
the brane, whose explicit form is given by
${\omega}_{\alpha\beta}J^{\alpha\beta}_{\tau}=2{\omega}_{\alpha\beta}X^{\alpha}{\partial}_{\tau}X^{\beta}$.
Besides the tangential dynamics, one can also impose a transverse dynamics to
the brane. This can be solely exerted by the boost transformations on the
boundary state equations [22], [32], [34], [36], [37], [38]. However, for
simplicity, we shall consider only the tangential motion and rotation.

We apply the flat spacetime with the metric $G_{\mu\nu}=\eta_{\mu\nu}={\rm
diag}(-1,+1,\cdots,+1)$ and a constant Kalb-Ramond field $B_{\mu\nu}$.
Besides, we utilize the trusty gauge
$A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}X^{\beta}$ with the constant field
strength $F_{\alpha\beta}$. In addition, we use the quadratic tachyon profile
$T=\frac{1}{2}U_{\alpha\beta}X^{\alpha}X^{\beta}$ where $U_{\alpha\beta}$ is a
constant symmetric matrix. We should note that the origin and the conformal
invariance of the action (2.1) have been widely studied in various papers,
e.g., see the Refs. [3], [39], [40], [41], [42], [43] and [44] (and also
references therein).

By setting the variation of the action to zero we receive the equation of
motion and the following boundary state equations

$\displaystyle\left[{\mathbf{A}}_{\alpha\beta}{\partial}_{\tau}X^{\beta}+{\mathcal{F}}_{\alpha\beta}{\partial}_{\sigma}X^{\beta}+U_{\alpha\beta}X^{\beta}\right]_{\tau=0}\
\ |B\rangle\ =0,$ (2.2) $\displaystyle{\delta X}^{i}|_{\tau=0}|B\rangle\ =0,$

where the total field strength is
${\cal{F}}_{\alpha\beta}=F_{\alpha\beta}-B_{\alpha\beta}$ and
${\mathbf{A}}_{\alpha\beta}=\eta_{\alpha\beta}+4\omega_{\alpha\beta}$. For the
next purpose we bring in the solution of the equation of motion of
$X^{\mu}(\sigma,\tau)$,

$\displaystyle
X^{\mu}(\sigma,\tau)=x^{\mu}+2\alpha^{\prime}Q^{\mu}\tau+\frac{i}{2}\sqrt{2\alpha^{\prime}}\sum_{m\neq
0}\frac{1}{m}\left({\alpha}^{\mu}_{m}e^{-2im\left(\tau-\sigma\right)}+{\widetilde{\alpha}}^{\mu}_{m}e^{-2im(\tau+\sigma)}\right).$
(2.3)

In fact, because of the presence of the background fields on the brane
worldvolume, the Lorentz symmetry has been clearly broken. Thus, the
tangential dynamics of the brane along its worldvolume directions is sensible.
Now we prove this. The effect of the Lorentz generators on the boundary state
can be obtained from Eq. (2.2),

$\displaystyle J^{\alpha\beta}|B\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}d\sigma\bigg{[}(\mathbf{A}^{-1}\mathcal{F})^{\alpha}_{\
\gamma}X^{\beta}\partial_{\sigma}X^{\gamma}-(\mathbf{A}^{-1}\mathcal{F})^{\beta}_{\
\gamma}X^{\alpha}\partial_{\sigma}X^{\gamma}$ (2.4) $\displaystyle+$
$\displaystyle(\mathbf{A}^{-1}U)^{\alpha}_{\
\gamma}X^{\beta}X^{\gamma}-(\mathbf{A}^{-1}U)^{\beta}_{\
\gamma}X^{\alpha}X^{\gamma}\bigg{]}_{\tau=0}|B\rangle.$

This equation prominently demonstrates that for restoring the Lorentz
symmetry, the tachyon matrix $U_{\alpha\beta}$ and the total field strength
$\mathcal{F}_{\alpha\beta}$ should vanish. We see that even in the absence of
the electric and magnetic fields, the tachyon field independently breaks the
Lorentz symmetry along the worldvolume of the brane. However, since the RHS of
Eq. (2.4) depends on the spacetime coordinates along the brane worldvolume we
conclude that the Lorentz symmetry breaking is local. Therefore, the internal
linear motion and rotation of the brane in any direction clearly are sensible.

For receiving more perception of the tangential dynamics of a D$p$-brane,
beside the Lorentz symmetry breaking, we should note that such setups are
T-dual of some imaginable systems. In other words, a D$p$-brane with the
tangential dynamics can be constructed via T-duality from a D$(p-1)$-brane
which moves and rotates parallel and perpendicular to its volume. For example,
consider a D-string along the direction $x^{1}$ with the velocity components
$V^{1}$ and $V^{2}$ in the $x^{1}$\- and $x^{2}$-directions, respectively. Now
apply the T-duality in the direction $x^{2}$, in which we suppose that it is
compact. The resultant system is a D2-brane, which has been expanded along the
plane $x^{1}x^{2}$ with the velocity $V^{1}$ in the direction $x^{1}$ and an
electric field $E=V^{2}$ in the direction $x^{2}$. In the same way, one can
acquire a rotating D2-brane through the T-duality from another setup of a
D-brane. Note that after exerting the T-duality all compact coordinates can be
decompactified.

Combining Eqs. (2.2) and (2.3), the boundary state equations are conveniently
expressed in terms of the zero modes and oscillators. The coherent state
method enables us to obtain the solution of the oscillatory part of the
boundary state equations

$\displaystyle{|B\rangle}^{({\rm osc})}\ =\prod^{\infty}_{n=1}{[\det
Q_{(n)}]^{-1}}{\exp\left[-\sum^{\infty}_{m=1}{\frac{1}{m}\left({\alpha}^{\mu}_{-m}S_{(m)\mu\nu}{\widetilde{\alpha}}^{\nu}_{-m}\right)}\right]\
}{|0\rangle}_{\alpha}\otimes{|0\rangle}_{\widetilde{\alpha}}\;,$ (2.5)

where the normalization factor $\prod^{\infty}_{n=1}{{[\det Q_{(n)}]}^{-1}}$
comes from the disk partition function. In fact, the advent of this prefactor
is due to the square structure of the action (2.1). That is, in addition to
the gauge choice $A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}X^{\beta}$, the other
terms in the action also possess the squared form of $X^{\alpha}$s. Thus, this
configuration path-integrally is feasible and the normalizing prefactor must
be inserted [11]. The matrices possess the following definitions

$\displaystyle
Q_{(m){\alpha\beta}}={\mathbf{A}}_{\alpha\beta}-{{\mathcal{F}}}_{{\mathbf{\alpha}}{\mathbf{\beta}}}+\frac{i}{2m}U_{\alpha\beta},$
(2.6) $\displaystyle
S_{(m)\mu\nu}=(\Delta_{(m)\alpha\beta}\;,\;-{\delta}_{ij}),$ (2.7)
$\displaystyle\Delta_{(m)\alpha\beta}=(Q_{(m)}^{-1}N_{(m)})_{\alpha\beta},$
(2.8) $\displaystyle
N_{(m){\alpha\beta}}={\mathbf{A}}_{\alpha\beta}+{{\mathcal{F}}}_{{\mathbf{\alpha}}{\mathbf{\beta}}}-\frac{i}{2m}U_{\alpha\beta},$
(2.9)

For the zero-mode part of the boundary state we receive the following
expression

$\displaystyle{{\rm|}B\rangle}^{\left(0\right)}$ $\displaystyle=$
$\displaystyle\prod_{i}{\delta{\rm(}x^{i}}{\rm-}y^{i}{\rm)}{\rm|}p^{i}{\rm=0}\rangle\;\int^{\infty}_{{\rm-}\infty}{\prod^{p}_{\alpha=0}\bigg{\\{}{\rm
d}p^{\alpha}}\exp\Bigg{[}i{\alpha}^{{\rm{}^{\prime}}}\Bigg{(}{\left(U^{{\rm-}{\rm
1}}{\mathbf{A}}\right)}_{\alpha\alpha}{\left(p^{\alpha}\right)}^{{\rm 2}}$
(2.10) $\displaystyle+$
$\displaystyle\sum_{\beta\neq\alpha}{{\left(U^{{\rm-}{\rm
1}}{\mathbf{A}}+{\mathbf{A}}^{T}U^{-1}\right)}_{\alpha\beta}p^{\alpha}p^{\beta}}\Bigg{)}\Bigg{]}{\rm\
\ }{\rm|}p^{\alpha}\rangle\bigg{\\}}.\ \ \ \ $

By considering the contribution of the conformal ghosts to the boundary state,
which is

$\displaystyle{|B\rangle}^{(\rm
gh)}=\exp\left[\sum^{\infty}_{m=1}{\left(c_{-m}{\tilde{b\
}}_{-m}-b_{-m}{\tilde{c}}_{-m}\right)}\right]\frac{c_{0}+{\tilde{c}}_{0}}{2}|q=1\rangle\
\otimes|\tilde{q}=1\rangle,$ (2.11)

one can construct the total boundary state

$\displaystyle{|B\rangle}^{(\rm tot)}=\frac{T_{p}}{2}{|B\rangle}^{(\rm
osc)}\otimes{|B\rangle}^{(0)}\otimes{|B\rangle}^{(\rm gh)},$ (2.12)

where $T_{p}$ is the D$p$-brane tension.

##  3 Radiation of a general massless closed string

In this section we calculate the radiation amplitude of a general massless
closed string from the interaction of two parallel dynamical-dressed unstable
D$p$-branes. To generalize our calculations let us assume that the fields and
dynamics of the two interacting branes to be different. Therefore, the
subscripts (1) and (2) will be used to exhibit these differences.

###  3.1 The radiation amplitude

In the closed string channel, the interaction of two D-branes takes place by
exchanging a closed string between the branes. The geometry of the exchanged
string worldsheet is a cylinder with $\tau$ as the coordinate along the length
of the cylinder, $0\leq\tau\leq t$, and $\sigma$ as the periodic coordinate,
i.e. $0\leq\sigma\leq\pi$. The radiation of a closed string state is
elaborated by inserting the corresponding vertex operator of the string state,
i.e. $V(\tau,\sigma)$, into the amplitude of the interaction. Precisely, one
should compute

$\mathcal{A}=\int_{0}^{\infty}dt\int_{0}^{t}d\tau\ ^{(\rm tot)}\langle
B_{1}|e^{-tH_{\rm(tot)}}V(\tau,\sigma)|B_{2}\rangle^{(\rm tot)},$ (3.1)

where $H_{\rm(tot)}$ comprises the ghost and matter parts of the closed string
Hamiltonian

$\displaystyle H_{\rm(tot)}=H_{\rm ghost}+{\alpha^{\prime}}Q^{{\rm 2}}{\rm+\
2}\sum^{\infty}_{n{\rm=1}}{{\rm(}{\alpha}_{{\rm-}n}\cdot{\alpha}_{n}}{\rm+}{\widetilde{\alpha}}_{{\rm-}n}\cdot{\widetilde{\alpha}}_{n}{\rm)-}4.$
(3.2)

Let us apply $z=\sigma+i\tau$ and $\partial=\partial_{z}$. Subsequently, the
vertex operator for a general massless string possesses the feature

$V(z,\bar{z})=\epsilon_{\mu\nu}\partial X^{\mu}\bar{\partial}X^{\nu}e^{ip\cdot
X},$ (3.3)

where $\epsilon_{\mu\nu}$ is the polarization tensor and $p^{\mu}$ (with
$p^{\mu}p_{\mu}=0$) is the momentum of the radiated closed string.

Since each string coordinate is the summation of the zero-mode part and
oscillating portion, i.e.
$X^{\mu}(\sigma,\tau)=X^{\mu}_{0}(\tau)+X^{\mu}_{{\rm osc}}(\sigma,\tau)$, we
can write

$\displaystyle{\partial}X^{\mu}\bar{\partial}X^{\nu}e^{ip\cdot X}$
$\displaystyle=$
$\displaystyle\big{(}{\partial}X^{\mu}_{0}\bar{\partial}X^{\nu}_{0}+{\partial}X^{\mu}_{0}\bar{\partial}X^{\nu}_{{\rm
osc}}$ (3.4) $\displaystyle+$ $\displaystyle{\partial}X^{\mu}_{{\rm
osc}}\bar{\partial}X^{\nu}_{0}+{\partial}X^{\mu}_{{\rm
osc}}\bar{\partial}X^{\nu}_{{\rm osc}}\big{)}e^{ip\cdot X_{0}}e^{ip\cdot
X_{{\rm osc}}}.$

This implies that the appearing terms in the vertex operator can be written in
the general form

$\epsilon_{\mu\nu}\left[\Gamma(X_{0})\Lambda({X_{{\rm
osc}}})\right]^{\mu\nu}e^{ip\cdot X_{0}}e^{ip\cdot X_{{\rm osc}}},$ (3.5)

where $\Gamma(X_{0})\in\\{1,\;\partial X_{0},\;\bar{\partial}X_{0},\;\partial
X_{0}\bar{\partial}X_{0}\\}$ and $\Lambda({X_{{\rm osc}}})\in\\{1,\;\partial
X_{\rm osc},\;\bar{\partial}X_{\rm osc},\;\partial X_{\rm
osc}\bar{\partial}X_{\rm osc}\\}$. This notation allows us to separate the
computation as follows

${}^{(\rm tot)}\langle B_{1}|e^{-tH_{\rm
tot}}V(\tau,\sigma)|B_{2}\rangle^{(\rm tot)}$ $\displaystyle=$
$\displaystyle\frac{T_{p}^{2}}{4}\epsilon_{\mu\nu}\Big{[}\ ^{(0)}\langle
B_{1}|e^{-tH(X_{0})}\Gamma(X_{0})e^{ip\cdot X_{0}}|B_{2}\rangle^{(0)}$ (3.6)
$\displaystyle\times$ ${}^{\rm{(osc)}}\langle
B_{1}|e^{-tH_{\rm(osc)}}\Lambda({X_{\rm osc}})e^{ip\cdot X_{\rm
osc}}|B_{2}\rangle^{\rm{(osc)}}\Big{]}^{\mu\nu}$ $\displaystyle\times$
${}^{\rm{(g)}}\langle B_{1}|e^{-tH_{\rm(g)}}|B_{2}\rangle^{\rm{(g)}},$

where $H(X_{0})=\alpha^{\prime}Q^{2}-4$.

After very long calculations, for the various forms of $\Gamma(X_{0})$, we
receive

${}^{(0)}\langle B_{1}|e^{-tH(X_{0})}e^{ip\cdot X_{0}}|B_{2}\rangle^{(0)}$
$\displaystyle=$
$\displaystyle(2\pi)^{26}\mathcal{D}(y_{1},y_{2})e^{4t}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\alpha=0}^{p}dk^{\alpha}dk^{\prime\alpha}$
$\displaystyle\times$
$\displaystyle\mathfrak{D}(k;k^{\prime})e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau
k^{\prime^{2}})},$ ${}^{(0)}\langle B_{1}|e^{-tH(X_{0})}\partial
X^{\mu}_{0}e^{ip\cdot X_{0}}|B_{2}\rangle^{(0)}$ $\displaystyle=$
$\displaystyle-\alpha^{\prime}(2\pi)^{26}\mathcal{D}(y_{1},y_{2})e^{4t}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\alpha=0}^{p}dk^{\alpha}dk^{\prime\alpha}$
$\displaystyle\times$
$\displaystyle\mathfrak{D}(k;k^{\prime})k^{\mu}e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau
k^{\prime^{2}})},$ ${}^{(0)}\langle
B_{1}|e^{-tH(X_{0})}\bar{\partial}X^{\mu}_{0}e^{ip\cdot
X_{0}}|B_{2}\rangle^{(0)}$ $\displaystyle=$
$\displaystyle\alpha^{\prime}(2\pi)^{26}\mathcal{D}(y_{1},y_{2})e^{4t}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\alpha=0}^{p}dk^{\alpha}dk^{\prime\alpha}$
$\displaystyle\times$
$\displaystyle\mathfrak{D}(k;k^{\prime})k^{\mu}e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau
k^{\prime^{2}})},$ ${}^{(0)}\langle B_{1}|e^{-tH(X_{0})}\partial
X^{\mu}_{0}\bar{\partial}X^{\nu}_{0}e^{ip\cdot X_{0}}|B_{2}\rangle^{(0)}$
$\displaystyle=$ $\displaystyle-\alpha^{\prime
2}(2\pi)^{26}\mathcal{D}(y_{1},y_{2})e^{4t}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\alpha=0}^{p}dk^{\alpha}dk^{\prime\alpha}$
(3.7) $\displaystyle\times$
$\displaystyle\mathfrak{D}(k;k^{\prime})k^{\mu}k^{\nu}e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau
k^{\prime^{2}})},$

where, by applying the Wick’s rotation $\tau\rightarrow-i\tau$, we introduced
another proper time $l^{\prime}=t-\tau$. Additionally, the functions
$\mathcal{D}(y_{1},y_{2})$ and $\mathfrak{D}(k;k^{\prime})$ have the following
definitions

$\displaystyle\mathcal{D}(y_{1},y_{2})$ $\displaystyle\equiv$
$\displaystyle\prod_{i=p+1}^{25}\delta(x^{i}-y_{1}^{i})\delta(x^{i}-y_{2}^{i})\delta(p^{i}),$
(3.8) $\displaystyle\mathfrak{D}(k;k^{\prime})$ $\displaystyle\equiv$
$\displaystyle\prod_{\alpha=0}^{p}\delta\left(p^{\alpha}+k^{\prime\alpha}-k^{\alpha}\right)$
(3.9) $\displaystyle\times$
$\displaystyle\exp\Bigg{[}-i\alpha^{\prime}\Bigg{(}\sum_{\alpha=0}^{p}\left[(U_{1}^{-1}\mathbf{A}_{1})_{\alpha\alpha}(k^{\alpha})^{2}-(U_{2}^{-1}\mathbf{A}_{2})_{\alpha\alpha}(k^{\prime\alpha})^{2}\right]$
(3.10) $\displaystyle+$ $\displaystyle
2\sum_{\alpha\neq\beta}\left[(U_{1}^{-1}\mathbf{A}_{1})_{\alpha\beta}k^{\alpha}k^{\beta}-(U_{2}^{-1}\mathbf{A}_{2})_{\alpha\beta}k^{\prime\alpha}k^{\prime\beta}\right]\Bigg{)}\Bigg{]}.\quad\
$ (3.11)

In Eq. (3.6), for the factor which includes the oscillators, we write

${}^{\rm(osc)}\langle B_{1}|e^{-tH{\rm(osc)}}\Lambda({X_{\rm osc}})e^{ip\cdot
X_{\rm osc}}|B_{2}\rangle^{\rm(osc)}\equiv\langle\Lambda({X_{\rm
osc}})e^{ip\cdot X_{\rm osc}}\rangle Z^{\rm(osc)},$ (3.12)

where $Z^{\rm(osc)}$ is the oscillation part of the partition function

$Z^{\rm(osc)}=\ ^{\rm(osc)}\langle
B_{1}|e^{-tH_{\rm(osc)}}|B_{2}\rangle^{\rm(osc)}.$ (3.13)

The other factor in Eq. (3.12) is given by

$\langle\Lambda({X_{\rm osc}})e^{ip\cdot X_{\rm
osc}}\rangle\equiv\dfrac{{}^{\rm(osc)}\langle
B_{1}|e^{-tH_{\rm(osc)}}\Lambda({X_{\rm osc}})e^{ip\cdot X_{\rm
osc}}|B_{2}\rangle^{\rm(osc)}}{{}^{\rm(osc)}\langle
B_{1}|e^{-tH_{\rm(osc)}}|B_{2}\rangle^{\rm(osc)}}\;.$ (3.14)

For the various forms of $\Lambda({X_{\rm osc}})$ Eq. (3.14) finds the
following features

$\displaystyle\langle\partial X^{\mu}_{\rm osc}e^{ip\cdot X_{\rm osc}}\rangle$
$\displaystyle=$ $\displaystyle i\langle\partial X^{\mu}p\cdot X\rangle_{\rm
osc}\langle e^{ip\cdot X_{\rm osc}}\rangle,$ (3.15)
$\displaystyle\langle\bar{\partial}X^{\mu}_{\rm osc}e^{ip\cdot X_{\rm
osc}}\rangle$ $\displaystyle=$ $\displaystyle
i\langle\bar{\partial}X^{\mu}p\cdot X\rangle_{\rm osc}\langle e^{ip\cdot
X_{\rm osc}}\rangle,$ (3.16) $\displaystyle\langle\partial X^{\mu}_{\rm
osc}\bar{\partial}X^{\nu}_{\rm osc}e^{ip\cdot X_{\rm osc}}\rangle$
$\displaystyle=$ $\displaystyle\left[\langle\partial
X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm osc}-\langle\partial
X^{\mu}p.X\rangle_{\rm osc}\langle\bar{\partial}X^{\nu}p\cdot X\rangle_{\rm
osc}\right]\langle e^{ip\cdot X_{\rm osc}}\rangle.\ \ \ $ (3.17)

The new proper time $l^{\prime}=t-\tau$ enables us to change the integrations
as

$\int_{0}^{\infty}dt\int_{0}^{t}d\tau=\int_{0}^{\infty}d\tau\int_{0}^{\infty}dl^{\prime}.$
(3.18)

In fact, $l^{\prime}$ and $\tau$ indicate the proper times of the radiated
closed string from the right (first) brane and the left (second) brane,
respectively. Therefore, $\tau=0$ ($l^{\prime}=0$) specifies the radiation
from the first (the second) brane, i.e. that brane with the boundary state
$|B_{1}\rangle$ ($|B_{2}\rangle$). The case $\tau,l^{\prime}>0$ obviously
corresponds to radiation that has been occurred between the branes.

Adding all these together we can write the general radiation amplitude as

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\dfrac{T_{p}^{2}}{4}(2\pi)^{26}\int_{0}^{\infty}d\tau\int_{0}^{\infty}dl^{\prime}\
\mathcal{D}(y_{1},y_{2})e^{4(l^{\prime}+\tau)}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\alpha=0}^{p}dk^{\alpha}dk^{\prime\alpha}\mathfrak{D}(k;k^{\prime})$
(3.19) $\displaystyle\times$ $\displaystyle
e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau k^{\prime^{2}})}Z^{\rm(osc,g)}\langle
e^{ip\cdot X_{osc}}\rangle\mathcal{M},\ \ \ $

where $\mathcal{M}$ and $Z^{\rm(osc,g)}$ are given by

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}\Big{\\{}\langle\partial
X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm osc}-\langle\partial
X^{\mu}p.X\rangle_{\rm osc}\langle\bar{\partial}X^{\nu}p\cdot X\rangle_{\rm
osc}-i\alpha^{\prime}k^{\mu}\langle\bar{\partial}X^{\nu}p\cdot X\rangle_{\rm
osc}$ (3.20) $\displaystyle+$ $\displaystyle
i\alpha^{\prime}k^{\nu}\langle\partial X^{\mu}p\cdot X\rangle_{\rm
osc}-\alpha^{\prime 2}k^{\mu}k^{\nu}\Big{\\}},$ $\displaystyle
Z^{\rm(osc,g)}=\ ^{\rm(osc)}\langle
B_{1}|e^{-(\tau+l^{\prime})H_{\rm(osc)}}|B_{2}\rangle^{\rm(osc)}\;\;{}^{\rm{(g)}}\langle
B_{1}|e^{-(\tau+l^{\prime})H_{\rm(g)}}|B_{2}\rangle^{\rm{(g)}}.$ (3.21)

According to the factor $e^{-\alpha^{\prime}(l^{\prime}k^{2}+\tau
k^{\prime^{2}})}$ the integration over $l^{\prime}$ and $\tau$ leads to the
factors similar to $1/k^{2}$ or $1/k^{\prime 2}$ or both, which correspond to
the propagators of the radiated massless strings by the branes.

###  3.2 The correlators

Now we should compute all correlators in Eq. (3.19). In Eqs. (3.7) we employed
the Wick’s rotation to determine the zero-mode correlators. Now we should
extract the oscillatory correlators in this frame. Thus, after some heavy
calculations we acquire

$\displaystyle Z^{\rm(osc,g)}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\det\left[Q^{\dagger}_{(n)1}Q_{(n)2}\right]^{-1}\;\prod_{n=1}^{\infty}\dfrac{(1-q^{2n})^{2}}{\det\left({\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}\right)}\;,$
(3.22) $\displaystyle\langle\partial X^{\mu}X^{\nu}\rangle_{\rm osc}$
$\displaystyle=$ $\displaystyle
i\alpha^{\prime}\sum_{n=1}^{\infty}\Bigg{\\{}\eta^{\mu\nu}\text{Tr}\left(\dfrac{{\mathbf{1}}+S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}\right)$
(3.23) $\displaystyle-$ $\displaystyle
S^{(1)\dagger\mu\eta}_{(n)}S^{(2)\nu}_{(n)\
\eta}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}\right)$
$\displaystyle-$ $\displaystyle
S^{(2)\mu\nu}_{(n)}\text{Tr}\Bigg{[}\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4l^{\prime}}}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4l^{\prime}}}$
$\displaystyle\times$
$\displaystyle\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4l^{\prime}}}\right)\Bigg{]}$
$\displaystyle+$ $\displaystyle
S^{(1)\dagger\mu\nu}_{(n)}\text{Tr}\Bigg{[}\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4\tau}}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4\tau}}$
$\displaystyle\times$
$\displaystyle\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4\tau}}\right)\Bigg{]}\Bigg{\\}},\
\ \ \ \ \ \ \ $ $\displaystyle\langle\partial
X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm osc}$ $\displaystyle=$ $\displaystyle
2\alpha^{\prime}\sum_{n=1}^{\infty}\Bigg{\\{}2n\eta^{\mu\nu}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}\right)^{2}}\right)$
$\displaystyle-$ $\displaystyle nS^{(1)\dagger\mu\eta}_{(n)}S^{(2)\nu}_{(n)\
\eta}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2n}\right)^{2}}\right)$
$\displaystyle+$ $\displaystyle
2nS^{(2)\mu\nu}_{(n)}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4l^{\prime}}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4l^{\prime}}\right)^{3}}\right)$
$\displaystyle+$ $\displaystyle
2nS^{(1)\dagger\mu\nu}_{(n)}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4\tau}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}q^{2(n-1)}e^{-4\tau}\right)^{3}}\right)\Bigg{\\}},$
(3.24)

where $q=e^{-2(\tau+l^{\prime})}$. One can conveniently show that
$\langle\partial X^{\mu}X^{\nu}\rangle_{\rm
osc}=-\langle\bar{\partial}X^{\mu}X^{\nu}\rangle_{\rm osc}$.

Using the Cumulant expansion, the exponential $\langle e^{ip\cdot X_{\rm
osc}}\rangle$ can be elaborated in terms of the correlators of $X$’s.
According to the boundary states formulation and Eq. (3.14), the correlators
with odd number of $X$’s vanish. Hence, only the correlators with even number
of $X$’s remain. Now let us assume that the momentum of the radiated string is
small. This implies that in the Cumulant expansion one should only compute the
factor $\exp\left(-\frac{1}{2}p_{\mu}p_{\nu}\langle X^{\mu}X^{\nu}\rangle_{\rm
osc}\right)$. Subsequently, for the massless radiated strings, the following
result is received

$\displaystyle\langle e^{ip.X_{\rm osc}}\rangle$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\bigg{\\{}\det\left(\mathbf{1}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}\;q^{2n}\right)^{\frac{\alpha^{\prime}}{2n}p_{\mu}p_{\nu}S^{(1)\dagger\mu\eta}_{(n)}S^{(2)\nu}_{(n)\
\eta}}$ (3.25) $\displaystyle\times$
$\displaystyle\det\left(\mathbf{1}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}\;q^{2(n-1)}e^{-4l^{\prime}}\right)^{-\frac{\alpha^{\prime}}{2n}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(n)}}$
$\displaystyle\times$
$\displaystyle\det\left(\mathbf{1}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}\;q^{2(n-1)}e^{-4\tau}\right)^{-\frac{\alpha^{\prime}}{2n}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(n)}}$
$\displaystyle\times$
$\displaystyle\det\left[\exp\left(\frac{\alpha^{\prime}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(n)}}{2n}\left(\mathbf{1}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}\;q^{2(n-1)}e^{-4l^{\prime}}\right)^{-1}\right)\right]\
\ \ $ $\displaystyle\times$
$\displaystyle\det\left[\exp\left(\frac{\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(n)}}{2n}\left(\mathbf{1}-S^{(1)\dagger}_{(n)}S^{(2)}_{(n)}\;q^{2(n-1)}e^{-4\tau}\right)^{-1}\right)\right]\bigg{\\}}.\
\ \ \ \ \qquad\qquad$

###  3.3 The large distance branes

Here we consider radiation of the closed strings from the branes which are
located far from each other. In this case only the exchange of the massless
closed strings possesses the dominant contribution to the interaction. Since
the long enough time corresponds to the large distance of the branes, the
limit $t\to\infty$ should merely exert on the oscillating part of the
amplitude, i.e. on Eqs. (3.22)-(3.25). Accordingly, $Z^{\rm(osc,g)}$ and the
correlators in the amplitude (3.19) find the following forms

$\displaystyle
Z^{\rm(osc,g)}|_{t\rightarrow\infty}=\prod_{n=1}^{\infty}\det\left[Q^{\dagger}_{(n)1}Q_{(n)2}\right]^{-1},$
(3.26) $\displaystyle\langle\partial X^{\mu}X^{\nu}\rangle_{\rm
osc}|_{t\rightarrow\infty}$ $\displaystyle=$ $\displaystyle
i\alpha^{\prime}\Bigg{\\{}-13\eta^{\mu\nu}$ (3.27) $\displaystyle-$
$\displaystyle
S^{(2)\mu\nu}_{(1)}\text{Tr}\Bigg{[}\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}}{{\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}}\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}}\right)\Bigg{]}$
$\displaystyle+$ $\displaystyle
S^{(1)\dagger\mu\nu}_{(1)}\text{Tr}\Bigg{[}\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}}{{\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}}\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}}\right)\Bigg{]}\Bigg{\\}},\
\ \ \ \ \ \ \ $ $\displaystyle\langle\partial
X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm osc}|_{t\rightarrow\infty}$
$\displaystyle=$ $\displaystyle
4\alpha^{\prime}\Bigg{\\{}S^{(2)\mu\nu}_{(1)}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}\right)^{3}}\right)$
$\displaystyle+$ $\displaystyle
S^{(1)\dagger\mu\nu}_{(1)}\text{Tr}\left(\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}}{\left({\mathbf{1}}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}\right)^{3}}\right)\Bigg{\\}},$
(3.28) $\displaystyle\langle e^{ip.X_{\rm osc}}\rangle|_{t\rightarrow\infty}$
$\displaystyle=$
$\displaystyle\det\left[\left(\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}\right)^{-\frac{\alpha^{\prime}}{2}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}}\right]$
(3.29) $\displaystyle\times$
$\displaystyle\det\left[\left(\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}\right)^{-\frac{\alpha^{\prime}}{2}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}}\right]$
$\displaystyle\times$
$\displaystyle\exp\left[\frac{\alpha^{\prime}}{2}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}{\rm
Tr}\left(\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4l^{\prime}}\right)^{-1}\right]\
\ \ $ $\displaystyle\times$
$\displaystyle\exp\left[\frac{\alpha^{\prime}}{2}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}{\rm
Tr}\left(\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4\tau}\right)^{-1}\right].\
\ \ \ \ \qquad\qquad$

In fact, for obtaining Eq. (3.29) we assumed that the parameters of the setup
and the momentum of the radiated closed string satisfy the condition

$\displaystyle
p_{\mu}p_{\nu}\sum^{\infty}_{n=2}\left(S^{(1)*}_{(n)}+S^{(2)}_{(n)}\right)^{\mu\nu}=0.$
(3.30)

This implies that for the given setup parameters, two components of the
momentum of the radiated closed string are specified in terms of the other
components.

Substitute Eqs. (3.26)-(3.29) into Eq. (3.19), the amplitude for the closed
string radiation between the large distance branes is acquired.

##  4 The axion radiation

In this section we shall accomplish the amplitude of the Kalb-Ramond (axion)
radiation from the interacting distant branes. The axion polarization tensor
satisfies $\epsilon_{\mu\nu}=-\epsilon_{\nu\mu}$ and
$p^{\mu}\epsilon_{\mu\nu}=0$. Thus, Eq. (3.20) becomes

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}\Biggl{\\{}4\alpha^{\prime}\left(S^{(2)\mu\nu}_{(1)}K_{l^{\prime}}+S^{(1)\dagger\mu\nu}_{(1)}K_{\tau}\right)-\alpha^{\prime
2}\bigg{[}p_{\lambda}p_{\delta}\bigg{(}S^{(2)\mu\lambda}_{(1)}S^{(2)\nu\delta}_{(1)}R_{l^{\prime}}^{2}$
$\displaystyle-$
$\displaystyle\left(S^{(2)\mu\lambda}_{(1)}S^{(1)\dagger\nu\delta}_{(1)}+S^{(1)\dagger\mu\lambda}_{(1)}S^{(2)\nu\delta}_{(1)}\right)R_{\tau}R_{l^{\prime}}+S^{(1)\dagger\mu\lambda}_{(1)}S^{(1)\dagger\nu\delta}_{(1)}R_{\tau}^{2}\bigg{)}$
$\displaystyle+$ $\displaystyle
k^{\mu}p_{\lambda}\left(S^{(2)\nu\lambda}_{(1)}R_{l^{\prime}}-S^{(1)\dagger\nu\lambda}_{(1)}R_{\tau}\right)+k^{\nu}p_{\lambda}\left(S^{(2)\mu\lambda}_{(1)}R_{l^{\prime}}-S^{(1)\dagger\mu\lambda}_{(1)}R_{\tau}\right)\bigg{]}\Biggr{\\}},\
\ \ \ \ $ (4.1)

where we have defined

$\displaystyle R_{(l^{\prime},\tau)}$ $\displaystyle=$
$\displaystyle\text{Tr}\left[\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4(l^{\prime},\tau)}}{\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4(l^{\prime},\tau)}}\left(\mathbf{1}+\dfrac{1}{\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4(l^{\prime},\tau)}}\right)\right],$
$\displaystyle K_{(l^{\prime},\tau)}$ $\displaystyle=$
$\displaystyle\text{Tr}\left[\dfrac{S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4(l^{\prime},\tau)}}{\left(\mathbf{1}-S^{(1)\dagger}_{(1)}S^{(2)}_{(1)}e^{-4(l^{\prime},\tau)}\right)^{3}}\right].$
(4.2)

For comparing our results with the Ref. [34] we can express $\mathcal{M}$ only
in terms of the functions $R_{(l^{\prime},\tau)}$. For this purpose we impose
the following condition on the setup parameters
$\Delta^{(1)\dagger}_{(1)}\Delta^{(2)}_{(1)}=\mathbf{1}$. Afterward, for large
distance limit the two-derivative term in Eq. (3.20) can then be written as

$\displaystyle\langle\partial X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm
osc}|_{t\rightarrow\infty}=-\langle\partial X^{\mu}X^{\nu}\rangle_{\rm
osc}|_{t\rightarrow\infty}\left[\langle p\cdot\partial Xp\cdot X\rangle_{\rm
osc}|_{t\rightarrow\infty}+\dfrac{i\alpha^{\prime}}{2}(k^{2}-k^{\prime
2})\right].$ (4.3)

Exerting Eq. (3.27) into the one-derivative terms of Eq. (4.3), we receive Eq.
(3.20) only in terms of the functions
$R_{(l^{\prime},\tau)}|_{\Delta^{(1)\dagger}_{(1)}\Delta^{(2)}_{(1)}=\mathbf{1}}$.
Besides, we have the following identity [34],[35]

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\rm
d}\tau\partial_{\tau}\left[e^{-\tau(\alpha^{\prime}k^{\prime^{2}}-4)}\exp\left(-\dfrac{1}{2}\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}\text{Tr}\left[\ln(\mathbf{1}-e^{-4\tau})-(\mathbf{1}-e^{-4\tau})^{-1}\right]\right)\right]$
(4.4) $\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\rm
d}\tau\Bigg{[}\left(\alpha^{\prime}k^{\prime
2}-4+2\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}{\tilde{R}}_{\tau}\right)e^{-\tau(\alpha^{\prime}k^{\prime^{2}}-4)}$
$\displaystyle\times$
$\displaystyle\exp\left(-\dfrac{\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}}{2}\text{Tr}\left[\ln(\mathbf{1}-e^{-4\tau})-(\mathbf{1}-e^{-4\tau})^{-1}\right]\right)\Bigg{]}.$

Similar identity can be obtained for the integration over $l^{\prime}$ with
${\tilde{R}}_{l^{\prime}}$, where

$\displaystyle{\tilde{R}}_{\tau}$ $\displaystyle=$
$\displaystyle-\dfrac{\alpha^{\prime}k^{\prime
2}-4}{2\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}}\;,$
$\displaystyle{\tilde{R}}_{l^{\prime}}$ $\displaystyle=$
$\displaystyle-\dfrac{\alpha^{\prime}k^{2}-4}{2\alpha^{\prime}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}}\;.$
(4.5)

Now we acquired $\mathcal{M}$ in terms of the functions ${\tilde{R}}_{\tau}$
and ${\tilde{R}}_{l^{\prime}}$.

For receiving the final form of the axion amplitude we should use the
following integral too

$\displaystyle\int_{0}^{\infty}d\tau\int_{0}^{\infty}dl^{\prime}e^{-l^{\prime}(\alpha^{\prime}k^{2}-4)}e^{-\tau(\alpha^{\prime}k^{\prime^{2}}-4)}\langle
e^{ip\cdot
X_{osc}}\rangle|_{\Delta^{(1)\dagger}_{(1)}\Delta^{(2)}_{(1)}=\mathbf{1}}\approx\dfrac{1}{(\alpha^{\prime}k^{2}-4)(\alpha^{\prime}k^{\prime
2}-4)},$ (4.6)

where for the correlator of the exponential we used the following form

$\displaystyle\langle e^{ip\cdot
X_{osc}}\rangle|_{\Delta^{(1)\dagger}_{(1)}\Delta^{(2)}_{(1)}=\mathbf{1}}$
$\displaystyle=$
$\displaystyle\exp\left\\{-\alpha^{\prime}p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}\text{Tr}\left[\ln(\mathbf{1}-e^{-4l^{\prime}})-(\mathbf{1}-e^{-4l^{\prime}})^{-1}\right]\right\\}$
(4.7) $\displaystyle\times$
$\displaystyle\exp\left\\{-\alpha^{\prime}p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}\text{Tr}\left[\ln(\mathbf{1}-e^{-4\tau})-(\mathbf{1}-e^{-4\tau})^{-1}\right]\right\\}.$

Let us apply the constant shifts on the momenta, i.e.
$\mathcal{K}^{\alpha}=k^{\alpha}+l^{\alpha}$ and
$\mathcal{K}^{\prime\alpha}=k^{\prime\alpha}+l^{\alpha}$, in which the
internal vector $l^{\alpha}$ satisfies the following conditions

$\displaystyle k.l=k^{\prime}.l=0\;,\;\;\;\;l^{2}=-\frac{4}{\alpha^{\prime}}.$
(4.8)

Adding all these together we obtain the ultimate feature of the amplitude as
in the following

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}(2\pi)^{26}T_{p}^{2}\mathcal{D}(y_{1},y_{2})\prod_{n=1}^{\infty}\left[\det\left(Q^{\dagger}_{(n)1}Q_{(n)2}\right)\right]^{-1}$
(4.9) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\prod_{\gamma=0}^{p}d\mathcal{K}^{\gamma}d\mathcal{K}^{\prime\gamma}\mathfrak{D}(\mathcal{K};\mathcal{K}^{\prime})\epsilon_{\alpha\beta}\left(\dfrac{\Gamma^{\alpha\beta}}{\mathcal{K}^{2}}-\dfrac{\Upsilon^{\alpha\beta}}{\mathcal{K}^{\prime
2}}\right),$

where $\Gamma^{\alpha\beta}$ and $\Upsilon^{\alpha\beta}$ are given by

$\displaystyle\Gamma^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\dfrac{1}{p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}}\left[p_{\gamma}\left((\mathcal{K}-l)^{\alpha}\Delta^{(1)\dagger\beta\gamma}_{(1)}+(\mathcal{K}-l)^{\beta}\Delta^{(1)\dagger\alpha\gamma}_{(1)}\right)-\frac{1}{2}\left(\mathcal{K}^{2}-\mathcal{K}^{\prime
2}\right)\Delta^{(1)\dagger\alpha\beta}_{(1)}\right]$ $\displaystyle-$
$\displaystyle\dfrac{\mathcal{K}^{\prime
2}}{\left(p_{\mu}p_{\nu}S^{(1)\dagger\mu\nu}_{(1)}\right)^{2}}\left[p_{\gamma}p_{\eta}\left(\Delta^{(1)\dagger\alpha\gamma}_{(1)}\Delta^{(1)\dagger\beta\eta}_{(1)}-\Delta^{(1)\dagger\alpha\beta}_{(1)}\Delta^{(1)\dagger\gamma\eta}_{(1)}\right)+p^{\lambda}p_{\lambda}\Delta^{(1)\dagger\alpha\beta}_{(1)}\right],$
$\displaystyle\Upsilon^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\dfrac{1}{p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}}\left[p_{\gamma}\left((\mathcal{K}-l)^{\alpha}\Delta^{(2)\beta\gamma}_{(1)}+(\mathcal{K}-l)^{\beta}\Delta^{(2)\alpha\gamma}_{(1)}\right)-\frac{1}{2}\left(\mathcal{K}^{2}-\mathcal{K}^{\prime
2}\right)\Delta^{(2)\alpha\beta}_{(1)}\right]$ (4.10) $\displaystyle+$
$\displaystyle\dfrac{\mathcal{K}^{2}}{\left(p_{\mu}p_{\nu}S^{(2)\mu\nu}_{(1)}\right)^{2}}\left[p_{\gamma}p_{\eta}\left(\Delta^{(2)\alpha\gamma}_{(1)}\Delta^{(2)\beta\eta}_{(1)}-\Delta^{(2)\alpha\beta}_{(1)}\Delta^{(2)\gamma\eta}_{(1)}\right)-p^{\lambda}p_{\lambda}\Delta^{(2)\alpha\beta}_{(1)}\right].$

The indices $\alpha,\beta,\gamma,\lambda,\eta$ belong to the set
$\\{0,1,\cdots,p\\}$ while $\mu,\nu$ are the spacetime indices. Since our
calculations were in the lowest order Eq. (4.9) accurately is in the tree-
level diagram.

The amplitude (4.9) manifestly represents the axion radiation from the
interaction of two parallel dynamical-dressed unstable D$p$-branes in the
large distance. As we see, according to the upper index $(1)$ ($(2)$) the
tensor $\Gamma^{\alpha\beta}$ ($\Upsilon^{\alpha\beta}$) depends on the
parameters of the first (second) brane. Therefore, the first and second terms
of (4.9) elaborate the axion radiation by the first and the second brane,
respectively. According to the factors in Eq. (4.9), a massless closed string
state is emitted by one of the branes, then it is absorbed by the other brane,
afterward it travels as an excited state on the brane for a while, it finally
decays by radiating an axion state. This radiation is a bremsstrahlung-like
process. We should mention that our results completely are in accordance with
the Ref. [34].

Note that in Eq. (4.9) the term $1/\mathcal{K}^{2}\mathcal{K}^{\prime 2}$ is
absent, which clarifies that there is no axion radiation near the middle
points between the branes. For the distant branes the middle region is far
from the both branes.

In fact, in some physical quantities the squared form of the amplitude, i.e.
$|\mathcal{A}|^{2}$, is appeared. Let us call the amplitude (3.1) as
$\mathcal{A}_{12}$. By exchanging the branes we receive $\mathcal{A}_{21}$. We
observe that the squared versions of them are not equal, i.e.,
$|\mathcal{A}_{12}|^{2}\neq|\mathcal{A}_{21}|^{2}$. This asymmetry precisely
is due to the presence of the vertex operator of the radiated closed string.
In other words, in the absence of any closed string radiation we acquire
$|\mathcal{A}_{12}|^{2}=|\mathcal{A}_{21}|^{2}$. In this case we have only the
pure interaction between the branes. The amplitude of the pure interaction is
given by Eq. (3.1) without the vertex operator and the $\tau$-integration.
Note that even for the pure interaction there is
$\mathcal{A}_{12}\neq\mathcal{A}_{21}$. Therefore, the amplitude (for the pure
interaction and / or radiation) is exactly elaborated by $\mathcal{A}_{12}$,
but not with the arithmetic mean
$\left(\mathcal{A}_{12}+\mathcal{A}_{21}\right)/2$, e.g. see the Refs.
[3]-[27], [34].

##  5 Conclusions

We introduced a boundary state which is corresponding to a dynamical
D$p$-brane in the presence of the antisymmetric tensor field, an internal
$U(1)$ gauge potential with constant field strength and a specific open string
tachyon field. For computing the amplitude of closed string radiation between
two parallel D$p$-branes, we combined the associated vertex operator of the
radiated string and the boundary state formalism. We acquired the radiation
amplitude for a general massless closed string. Besides the arbitrary
distance, the amplitude for large distances of the branes was also calculated.
Presence of the various parameters drastically generalized the amplitude. By
varying the parameters the value of the radiation amplitude can be accurately
adjusted to any desirable value.

From the radiation amplitude of a general massless state we explicitly
computed the axion production between the large distance D$p$-branes. We
observed that this radiation occurs only from one of the branes. Finally, we
demonstrated that the axion production cannot occur near the middle region
between the branes. This is due to the fact that the middle points are far
from the both branes.

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# The resonances $X(4140)$, $X(4160)$, and $P_{cs}(4459)$ in the decay of
$\Lambda_{b}\to J/\psi\Lambda\phi$

Wen-Ying Liu  School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China  Wei Hao  CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190,China  University of Chinese Academy of Sciences (UCAS),
Beijing 100049, China  Guan-Ying Wang  School of Physics and Electronics,
Henan University, Kaifeng 475004, China  Yan-Yan Wang  School of Materials
Science and Engineering, Zhengzhou University of Aeronautics, Henan 450001,
China  En Wang<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China  De-Min Li
<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China

###### Abstract

We study the decay of $\Lambda_{b}\to J/\psi\Lambda\phi$ by taking into
account the intermediate resonances $X(4140)$, $X(4160)$, and $P_{cs}(4459)$.
In addition to the peak of the $X(4140)$, we also find a bump structure around
4160 MeV followed by a cusp structure around $D^{*}_{s}\bar{D}^{*}_{s}$
threshold in the $J/\psi\phi$ invariant mass distribution of the
$\Lambda_{b}\to J/\psi\Lambda\phi$ process, which can be associated to the
$D^{*}_{s}\bar{D}^{*}_{s}$ molecular state $X(4160)$. We also show that this
reaction can also be used to confirm the existence of the hidden-charm
pentaquark $P_{cs}(4459)$ with strangeness. We predict an enhancement
structure close to the threshold in the $D^{*}_{s}\bar{D}^{*}_{s}$ invariant
mass distribution of the $\Lambda_{b}\to D^{*}_{s}\bar{D}^{*}_{s}\Lambda$
process, which is the reflection of the $X(4160)$ and should not be
misidentified with a new resonance. We strongly suggest experimentalists to
further measure these processes, which can be useful to clarify the nature of
$X(4140)$ and $X(4160)$ resonances, and to confirm the existence of the
$P_{cs}(4459)$.

##  I INTRODUCTION

In the last decades, a lot of exotic multi-quark states were observed
experimentally, which demonstrates that there exist more complex
configurations than the conventional mesons and baryons predicted by the
classical quark model Brambilla:2019esw ; Chen:2016qju ; Liu:2019zoy . Many of
them are reported in the weak decays of the heavy hadrons, and the theoretical
studies on the weak decays of the heavy hadrons have also increased our
understanding of hadron-hadron interactions and the nature of many
intermediate exotic states Guo:2017jvc ; Oset:2016lyh .

It should be stressed that the weak decays of the $\Lambda_{b}$ have been
proved to be a rich source of exotic spectroscopy, such as the new pentaquark
states $P_{c}(4312)^{+}$, $P_{c}(4380)^{+}$, and $P_{c}(4350)^{+}$ observed by
the LHCb Collaboration Aaij:2015tga ; Aaij:2019vzc . Very recently, the CMS
Collaboration has reported the observation of the $\Lambda_{b}\to
J/\psi\Lambda\phi$ decay using proton-proton collision data at the LHC, the
ratio of the branching fraction $\mathcal{B}(\Lambda_{b}\to
J/\psi\Lambda\phi)/{\mathcal{B}(\Lambda_{b}\to\psi(2s)\phi)}=(8.26\pm 0.90\pm
0.68\pm 0.11)\times{10}^{-2}$ Sirunyan:2019dwp . The observation of this
reaction has opened a window on future searches for new resonances in the
$J/\psi\Lambda$ and $J/\psi\phi$ systems, once a sufficient number of signal
events is accumulated, as mentioned by the CMS Collaboration Sirunyan:2019dwp
.

A series of meson-baryon interactions in the hidden charm sector, including
the $J/\psi\Lambda$, were studied within the framework of the coupled channel
unitary approach with the local hidden gauge formalism in Ref. Wu:2010vk , and
two hidden charmonium pentaquark states were predicted in the isospin $I=0$
and strangeness $S=-1$ sector. Since the predicted hidden charm $S=-1$
pentaqurak couples in $S$-wave to $J/\psi\Lambda$ with a reasonable strength,
some reactions involving the $J/\psi\Lambda$ final state system are proposed
to search for the hidden charmonium pentaquark state, such as $\Lambda_{b}\to
J/\psi K^{0}\Lambda$ Lu:2016roh , $\Xi_{b}\to J/\psi K^{-}\Lambda$
Chen:2015sxa , $\Lambda_{b}\to J/\psi\eta\Lambda$ Feijoo:2015kts . There are
also some studies about the hidden-charm pentaquark state with strangeness
Anisovich:2015zqa ; Wang:2015wsa ; Chen:2016ryt ; Xiao:2019gjd ; Zhang:2020cdi
; Shen:2020gpw . Basing the suggestion proposed in Ref. Chen:2015sxa , the
LHCb has started to search for the hidden-charm pentaquark with strangeness in
Ref. Aaij:2017bef . Very recently, the LHCb Collaboration reported a hidden-
charm pentaquark state with strangeness $P_{cs}(4459)$ in the $J/\psi\Lambda$
invariant mass spectrum of the $\Xi^{-}_{b}\to J/\psi K^{-}\Lambda$ process,
and the mass and width of $P_{cs}(4459)$ were measured to be LHCbPcs

$\displaystyle M_{P_{cs}}$ $\displaystyle=$ $\displaystyle 4458.8\pm
2.9^{+4.7}_{-1.1}~{}{\rm MeV},$ $\displaystyle\Gamma_{P_{cs}}$
$\displaystyle=$ $\displaystyle 17.3\pm 6.5^{+8.0}_{-5.7}~{}{\rm MeV}.$ (1)

Although the quantum numbers of the $P_{cs}(4459)$ have not been determined, a
hidden-charm pentaquark with strangeness $S=-1$ and $J^{P}=1/2^{-}$ was
predicted to have a mass much close to the $P_{cs}(4459)$ mass in many works
Chen:2016ryt ; Xiao:2019gjd ; Chen:2020uif . Thus, we will take into account
the contribution from the $P_{cs}(4459)$ in the $J/\psi\Lambda$ system of the
$\Lambda_{b}\to J/\psi\Lambda\phi$ process.

For the $J/\psi\phi$ system, one of the most important resonances is the
charmonium-like state $X(4140)$, which was first reported in the $B^{+}\to
J/\psi\phi K^{+}$ process by the CDF Collaboration Aaltonen:2009tz , and later
confirmed by the CMS Chatrchyan:2013dma , D0 Abazov:2013xda ; Abazov:2015sxa ,
and LHCb Collaborations Aaij:2016iza ; Aaij:2016nsc . However, the width of
the $X(4140)$ is measured to be $83\pm 21^{+4.6}_{-2.8}$ MeV by LHCb
Aaij:2016iza ; Aaij:2016nsc , much larger than the data from the other
experiments Zyla:2020zbs , which confuses the understanding of the $X(4140)$
nature Hao:2019fjg ; Chen:2016iua . We have provided a reasonable explanation
of the LHCb measurements Aaij:2016iza ; Aaij:2016nsc  by considering the
contributions from the $X(4160)$ and the $X(4140)$ with a narrow width, and
have predicted a broad bump around 4160 MeV followed by a cusp at
$D^{*}_{s}\bar{D}^{*}_{s}$ threshold in the $J/\psi\phi$ invariant mass
distribution, which is one feature of the $D^{*}_{s}\bar{D}^{*}_{s}$ molecular
state $X(4160)$ Wang:2017mrt . In addition, the predicted $J/\psi\phi$
invariant mass distribution of the $e^{+}e^{-}\to\gamma J/\psi\phi$ involving
the intermediate $X(4140)$ and $X(4160)$ are also compatible with the BESIII
measurement Wang:2018djr .

The $S$-wave interaction of ten vector-vector channels with charm $C=0$ and
strangeness $S=0$ ($J/\psi\phi$, $D_{s}^{*}\bar{D}_{s}^{*}$,
$D^{*}\bar{D}^{*}$, $K^{*}\bar{K}^{*}$, $\rho\rho$, $\omega\omega$,
$\phi\phi$, $J/\psi J/\psi$, $\omega J/\psi$, $\omega\phi$) were studied
within the hidden gauge formalism, and one $D^{*}\bar{D}^{*}$ molecular state
with $I(J^{PC})=0(2^{++})$ was predicted, which could be identified as
$X(4160)$ Molina:2009ct . Based on the above discussions, we expect that the
$X(4140)$ and $X(4160)$ play an important role in the reaction of
$\Lambda_{b}\to J/\psi\Lambda\phi$.

In this work, we will study the reaction of $\Lambda_{b}\to J/\psi\Lambda\phi$
by taking into account the contributions from the $X(4140)$, $X(4160)$, and
$P_{cs}(4459)$, and also propose to test the molecular nature of the $X(4160)$
by looking for the enhancement structure near the $D^{*}_{s}\bar{D}^{*}_{s}$
threshold in the $\Lambda_{b}\to D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$
reaction, since more data samples of those two processes are crucial to learn
more about the charmonium-like states $X(4140)$ and $X(4160)$ and especially
to confirm the existence of the recently observed $P_{cs}(4459)$.

This paper is organized as follows. In Sec. II, we will present the mechanism
for the reaction of $\Lambda_{b}\to J/\psi\Lambda\phi$ and $\Lambda_{b}\to
D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$, and in Sec. III, we will show our
results and discussions, followed by a short summary in the last section.

##  II FORMALISM

###  II.1 The $\Lambda_{b}\to J/\psi\Lambda\phi$ process

Figure 1: The $\Lambda_{b}$ weak decay at the microscopic quark picture. (a)
The $\Lambda{{}_{b}}\to J/\psi\Lambda\phi$ decay through the internal $W^{-}$
emission and hadronization of $sud$ through $q\bar{q}$ created with vacuum
quantum numbers; (b) the $\Lambda_{b}\to
D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$ decay through the internal $W^{-}$
emission and hadronization of $c\bar{c}$ through $\bar{s}s$; (c) the
$\Lambda_{b}\to D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$ decay through the
external $W^{-}$ emission and hadronization of $cud$ with $\bar{s}s$.

Analogous to the Refs. Lu:2016roh ; Chen:2015sxa ; Feijoo:2015kts ;
Wang:2017mrt ; Sirunyan:2019dwp ; Magas:2020zuo , the $\Lambda_{b}\to
J/\psi\Lambda\phi$ process can perform through the $W^{-}$ internal emission
mechanism. In the first step, the $b$ quark of the initial $\Lambda_{b}$
weakly decays into a $c$ quark and a $W^{-}$ boson, followed by the $W^{-}$
decaying into $\bar{c}s$ pair, which can be expressed as,

$\displaystyle\left|\Lambda_{b}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}b(ud-du)$ (2) $\displaystyle\Rightarrow$
$\displaystyle V_{bc}\frac{1}{\sqrt{2}}cW^{-}(ud-du)$
$\displaystyle\Rightarrow$ $\displaystyle
V_{bc}V_{cs}\frac{1}{\sqrt{2}}c\bar{c}s(ud-du),$

where we take the flavor wave function $\Lambda_{b}\ =b(ud-du)/\sqrt{2}$,
$V_{bc}$ and $V_{cs}$ are the elements of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix. In order to give rise to the final states of this process, the quark
components of Eq. (2), together with an extra $\bar{q}q$
($\equiv\bar{u}u+\bar{d}d+\bar{s}s$) created from the vacuum, should hadronize
into hadrons, as depicted in Fig. 1,

$\displaystyle\left|H^{(a)}\right\rangle$ $\displaystyle=$ $\displaystyle
V_{bc}V_{cs}J/\psi\left|s(\bar{s}s)\frac{(ud-du)}{\sqrt{2}}\right\rangle$ (3)
$\displaystyle=$ $\displaystyle V_{bc}V_{cs}J/\psi\phi\Lambda.$

On the other hand, since the $X(4160)$ was predicted to strongly couple to the
$D^{*}_{s}\bar{D}^{*}_{s}$ channel and weakly couple to the $J/\psi\phi$
channel Molina:2009ct , the $c\bar{c}sud$ component from the $\Lambda_{b}$
weak decay, together with the extra $\bar{s}s$ created from the vacuum, could
hadronize to the $D^{*}_{s}\bar{D}^{*}_{s}\Lambda$ in two ways: the $W^{-}$
internal emission of Fig. 1 and the $W^{-}$ external emission of Fig. 1.
Although this is not the final state of $J/\psi\Lambda\phi$ that one observes,
the idea is that the $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}$ will interact and
create a resonance that couples to $J/\psi\phi$. We have,

$\displaystyle\left|H^{(b)}\right\rangle$ $\displaystyle=$ $\displaystyle
V_{bc}V_{cs}\left|c(\bar{s}s)\bar{c}s\frac{(ud-du)}{\sqrt{2}}\right\rangle$
(4) $\displaystyle=$ $\displaystyle
V_{bc}V_{cs}D^{*}_{s}\bar{D}^{*}_{s}\Lambda,$
$\displaystyle\left|H^{(c)}\right\rangle$ $\displaystyle=$ $\displaystyle
V_{bc}V_{cs}\left|\bar{c}sc(\bar{s}s)\frac{(ud-du)}{\sqrt{2}}\right\rangle$
(5) $\displaystyle=$ $\displaystyle
V_{bc}V_{cs}\bar{D}^{*}_{s}D^{*}_{s}\Lambda,$

which finally produce $J/\psi\phi$ via the $D_{s}^{*}\bar{D}_{s}^{*}$ final
state interaction.

Although we can straightly get the final state of $J/\psi\phi\Lambda$ via the
mechanism of Fig. 1, this mechanism is penalized by a color factor with
respect to the $W^{-}$ external emission of Fig. 1, and this term with
intermediate state $J/\psi\phi$ instead of $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}$
would involve the extra factor
${g_{J/\psi\phi}}/{g_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}}=|-2617-i5151|/|18927-i5524|\approx
0.3$ versus the amplitudes of Figs. 1 and 1, where
${g_{J/\psi\phi}}=(-2617-i515)$ MeV and
${g_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}}=(18927-i5524)$ MeV are the couplings
of the $D^{*}_{s}\bar{D}^{*}_{s}$ molecule to $J/\psi\phi$ and
$D^{*}_{s}\bar{D}^{*}_{s}$ Molina:2009ct . Thus, we will neglect the
contribution from the Fig. 1 in this work.

Now, we need the final state interaction of $D_{s}^{*}\bar{D}_{s}^{*}$
produced in Figs. 1 and 1 to give rise to the $J/\psi\phi$ final state, and
also generate dynamically the $D_{s}^{*}\bar{D}_{s}^{*}$ molecular state
$X(4160)$, as shown in Fig. 2. Since the quantum numbers of $X(4160)$ are
predicted to be $J^{PC}=2^{++}$ in Refs. Molina:2009ct ; Torres:2016oyz , the
$\Lambda$ should be in $P$-wave to match to the spin $J=1/2$ of the initial
$\Lambda_{b}$, and the amplitude can be expressed as,

$\mathcal{M}^{P}=A\left(\vec{\epsilon}_{J/\psi}\times\vec{\epsilon}_{\phi}\right)\cdot\vec{k}\,G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\,t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to
J/\psi\phi},$ (6)

where the unknown constant $A$ represents the total weight of the microscopic
decay in the quark level showed in Figs. 1 and 1; $\vec{\epsilon}_{J/\psi}$
and $\vec{\epsilon}_{\phi}$ are the polarization vectors of the $J/\psi$ and
$\phi$, respectively; $\vec{k}$ is the $\Lambda$ momentum in the rest frame of
the $J/\psi\phi$ system; $G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}$ is the loop
function of the $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}$ channel, and
$t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to J/\psi\phi}$ is the transition
amplitude of $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to J/\psi\phi$, both of which
are the functions of the $J/\psi\phi$ invariant mass $M_{J/\psi\phi}$. To
avoid potential dangers using the dimensional regularizations as pointed out
in Ref. Wu:2010rv , in this work we use the cut-off method and take $q_{\rm
max}=630$ MeV followed Ref. Wang:2018djr ,

$\displaystyle G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}$ $\displaystyle=$
$\displaystyle
i\int{\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}\frac{1}{(p-q)^{2}-m_{2}^{2}+i\epsilon}}$
$\displaystyle=$ $\displaystyle\int_{0}^{q_{\rm
max}}{\frac{q^{2}dq}{(2\pi)^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}[(P^{0})^{2}-(\omega_{1}+\omega_{2})^{2}+i\epsilon]}},$

where $m_{1,2}$ are the masses of $D^{*}_{s}$ and $\bar{D}^{*}_{s}$,
$\omega_{i}=\sqrt{|\vec{q}_{i}|^{2}+m^{2}_{i}}$ are the energies of
$D^{*}_{s}$ and $\bar{D}^{*}_{s}$, and $P^{0}=\sqrt{s}$ is the center-of-mass
energy. And the transition amplitude is given by,

$t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to
J/\psi\phi}=\frac{g_{J/\psi\phi}g_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}}{M_{J/\psi\phi}^{2}-M_{X(4160)}^{2}+iM_{X(4160)}\Gamma_{X(4160)}}$
(8)

with the couplings $g_{D^{*}_{s}\bar{D}^{*}_{s}}$ and $g_{J/\psi\phi}$
obtained in Ref. Molina:2009ct , the $X(4160)$ mass $M_{X(4160)}=4156$ MeV
Zyla:2020zbs . For the $X(4160)$ width $\Gamma_{X(4160)}$, we take into
account the Flatté effect for the opening of the important
$D^{*}_{s}\bar{D}^{*}_{s}$ channel above the $D^{*}_{s}\bar{D}^{*}_{s}$
threshold as follows,

$\displaystyle\Gamma_{X(4160)}$ $\displaystyle=$
$\displaystyle\Gamma_{0}+\Gamma_{J/\psi\phi}+\Gamma_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}},$
(9) $\displaystyle\Gamma_{J/\psi\phi}$ $\displaystyle=$
$\displaystyle\frac{{|g_{J/\psi\phi}|}^{2}\,\widetilde{p}_{\phi}}{8\pi{M_{X(4160)}^{2}}},$
(10) $\displaystyle\Gamma_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}$ $\displaystyle=$
$\displaystyle\frac{{|g_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}|}^{2}\,\widetilde{p}_{D_{s}^{*}}}{8\pi{M_{X(4160)}^{2}}}\Theta\left(M_{J/\psi\phi}-2m_{D^{*}_{s}}\right),$
(11) $\displaystyle\widetilde{p}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M_{J/\psi\phi}^{2},m_{J/\psi}^{2},m_{\phi}^{2})}{2M_{J/\psi\phi}},$
(12) $\displaystyle\widetilde{p}_{D_{s}^{*}}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}(M_{J/\psi\phi}^{2},m_{D_{s}^{*}}^{2},m_{\bar{D}_{s}^{*}}^{2})}{2M_{J/\psi\phi}},$
(13)

where $\Gamma_{0}$ is taken as $65.0$ MeV by fitting to the LHCb data of the
$B^{+}\to J/\psi\phi K^{+}$ process Wang:2017mrt , $\widetilde{p}_{\phi}$ and
$\widetilde{p}_{D_{s}^{*}}$ are respectively the momenta of $\phi$ and
$D_{s}^{*}$ in the rest frame of $J/\psi\phi$.
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ is the Källen function.

Figure 2: Mechanism to produce the $J/\psi\Lambda\phi$ final state through
$X(4160)$, $X(4140)$, and $P_{cs}(4459)$.

On the other hand, the $X(4140)$ state was observed in the $J/\psi\phi$
invariant mass distribution of the process $B^{+}\to J/\psi\phi K^{+}$
Zyla:2020zbs , and could be interpreted as the $\chi_{c1}(3P)$ state
Hao:2019fjg ; Chen:2016iua , thus the decay $\Lambda_{b}\to J/\psi\phi\Lambda$
can also happen through the intermediate resonance $X(4140)$ with $\Lambda$ in
$S$-wave, and the corresponding amplitude can be expressed as,

$\mathcal{M}^{S}=B\times\frac{M_{X(4140)}^{3}\,\vec{\epsilon}_{J/\psi}\cdot\vec{\epsilon}_{\phi}}{M_{J/\psi\phi}^{2}-M_{X(4140)}^{2}+iM_{X(4140)}\Gamma_{X(4140)}},$
(14)

where the parameter $B$, with the same dimension as $A$, accounts for the
relative weight of the contribution from the intermediate resonance $X(4140)$.
We take $M_{X(4140)}=4135$ MeV and $\Gamma_{X(4140)}=19$ MeV Zyla:2020zbs .
Since the $\Lambda_{b}\to J/\psi\phi\Lambda$ process has been observed by the
CMS Collaboration Sirunyan:2019dwp , we will show our results up to an
arbitrary normalization, and our predictions can be tested while enough data
sample are available in future.

As we discussed above, many studies favor the quantum numbers $J^{P}=1/2^{-}$
for the $P_{cs}(4459)$ Chen:2020uif ; Peng:2020hql ; Wang:2020eep , which
implies that the $P_{cs}(4459)$ couples to $J/\psi\Lambda$ in $S$-wave, and
the $\Lambda_{b}$ weakly decays into $P_{cs}(4459)\phi$ in $S$-wave, as
depicted in Fig. 2. The amplitude for the contribution of $P_{cs}(4459)$ can
be expressed as,

$\displaystyle\mathcal{M}^{P_{cs}}=C\times\frac{M_{P_{cs}}^{3}\,\vec{\epsilon}_{J/\psi}\cdot\vec{\epsilon}_{\phi}}{M_{J/\psi\Lambda}^{2}-M_{P_{cs}}^{2}+iM_{P_{cs}}\Gamma_{P_{cs}}},$
(15)

where we take $M_{P_{cs}}=4458.8$ MeV and $\Gamma_{P_{cs}}=17.3$ MeV LHCbPcs ,
and the parameter $C$ stands for the relative weight of the contribution from
the $P_{cs}(4459)$.

With the amplitudes of Eqs. (6), (14) and (15), the mass distribution of the
$\Lambda_{b}\to J/\psi\phi\Lambda$ process is given by,

$\displaystyle\frac{d^{2}\Gamma}{dM^{2}_{J/\psi\phi}dM^{2}_{J/\psi\Lambda}}$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32M^{3}_{\Lambda_{b}}}\sum|\mathcal{M}|^{2},$
(16)

with

$\displaystyle\sum|\mathcal{M}|^{2}$ $\displaystyle=$
$\displaystyle\sum\left(|\mathcal{M}^{S}|^{2}+|\mathcal{M}^{P}|^{2}\right)$
(17) $\displaystyle=$ $\displaystyle
B^{2}\left(3|\tilde{\mathcal{M}}^{S}|^{2}+2|\vec{k}|^{2}|\tilde{\mathcal{M}}^{P}|^{2}\right),$
(18) $\displaystyle\tilde{\mathcal{M}}^{P}$ $\displaystyle=$
$\displaystyle\alpha
G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\,t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to
J/\psi\phi},$ (19) $\displaystyle\tilde{\mathcal{M}}^{S}$ $\displaystyle=$
$\displaystyle\frac{M_{X(4140)}^{3}}{M_{J/\psi\phi}^{2}-M_{X(4140)}^{2}+iM_{X(4140)}\Gamma_{X(4140)}}$
(20) $\displaystyle+$ $\displaystyle\frac{\beta
M_{P_{cs}}^{3}}{M_{J/\psi\Lambda}^{2}-M_{P_{cs}}^{2}+iM_{P_{cs}}\Gamma_{P_{cs}}},$

where $\alpha=A/B$ and $\beta=C/B$ stand for the relative weights of the
$X(4160)$ and $P_{cs}(4459)$ terms with respect to the $X(4140)$ term, and we
take $\alpha=\alpha_{0}=2100$ in order to give roughly equal strengths for
$X(4140)$ and $X(4160)$, as done in Refs. Wang:2017mrt ; Magas:2020zuo . It
should be pointed out that there is no interference between the $S$-wave and
$P$-wave, and the sum over the $J/\psi$ and $\phi$ polarizations is,

$\displaystyle\sum_{\rm
pol}\left[\left(\vec{\epsilon}_{J/\psi}\times\vec{\epsilon}_{\phi}\right)\cdot\vec{k}\right]^{2}=2\delta^{il}k^{i}k^{l}=2|\vec{k}|^{2},$
(21) $\displaystyle\sum_{\rm
pol}\left(\vec{\epsilon}_{J/\psi}\cdot\vec{\epsilon}_{\phi}\right)^{2}=3.$
(22)

###  II.2 The $\Lambda_{b}\to D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$ process

Since the width of the $X(4160)$ is $139^{+110}_{-60}$ MeV Zyla:2020zbs , and
its mass is close to the one of the $X(4140)$, the signal of the bump
structure from the $X(4160)$ is usually difficult to be identified. Because
the $X(4160)$ is predicted to couple strongly to the
$D^{*}_{s}\bar{D}^{*}_{s}$ channel, it is expected to give rise to an
enhancement structure near threshold in the $D^{*}_{s}\bar{D}^{*}_{s}$
invariant mass distribution. Thus, it is also interesting to investigate the
$\Lambda_{b}\to D^{*}_{s}\bar{D}^{*}_{s}\Lambda$ process. The mechanism of
this process can also be depicted in Figs. 1 and 1, and we will take into
account the contributions from the tree level diagram of Fig. 3 and the
$D^{*}_{s}\bar{D}^{*}_{s}$ final state interaction of Fig. 3. The amplitudes
for both diagrams can be expressed as,

$\mathcal{M}^{S}=D\times k_{\rm
ave}\,\vec{\epsilon}_{D^{*}_{s}}\cdot\vec{\epsilon}_{\bar{D}^{*}_{s}}$ (23)

for $\Lambda$ in $S$-wave, and,

$\mathcal{M}^{P}=E\left(\vec{\epsilon}_{D^{*}_{s}}\times\vec{\epsilon}_{\bar{D}^{*}_{s}}\right)\cdot\vec{k}^{\,\prime}\,\left(1+G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\,t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to
D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\right)$ (24)

for $\Lambda$ in $P$-wave, where $\vec{\epsilon}_{D^{*}_{s}}$ and
$\vec{\epsilon}_{\bar{D}^{*}_{s}}$ are the polarization vectors of the
$D^{*}_{s}$ and $\bar{D}^{*}_{s}$, respectively, and $\vec{k}^{\,\prime}$ is
the $\Lambda$ momentum in the $D^{*}_{s}\bar{D}^{*}_{s}$ system. We introduce
$k_{\rm ave}=1000$ MeV to have the parameters $D$ and $E$ with same dimension.
The loop function is given by Eq. (LABEL:eq:loop), and the transition
amplitude can be expressed as,

$t_{D_{s}^{*}\bar{D}_{s}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}=\frac{g_{D_{s}^{*}\bar{D}_{s}^{*}}g_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}}{M_{D_{s}^{*}\bar{D}_{s}^{*}}^{2}-M_{X(4160)}^{2}+iM_{X(4160)}\Gamma_{X(4160)}}.$
(25)

Figure 3: Mechanism to produce the $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}$ final
state

Finally, the $D^{*}_{s}\bar{D}^{*}_{s}$ invariant mass distribution of the
$\Lambda_{b}\to D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$ process is given by,

$\displaystyle\frac{d\Gamma}{dM_{D_{s}^{*}\bar{D}_{s}^{*}}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{4M^{2}_{\Lambda_{b}}}p_{\Lambda}\tilde{p}_{D^{*}_{s}}\sum|\mathcal{M}|^{2},$
(26) $\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=$ $\displaystyle
D^{2}\left(3|k^{2}_{\rm
ave}|+2|\vec{k}^{\,\prime}|^{2}|\tilde{\mathcal{M}}^{P}|^{2}\right),$
$\displaystyle\tilde{\mathcal{M}}^{P}$ $\displaystyle=$
$\displaystyle\gamma\left(1+G_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\,t_{D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\to
D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}}\right),$ (27)

with $\gamma=E/D$. $p_{\Lambda}$ is the $\Lambda$ momentum in the rest frame
of the $\Lambda_{b}$, $\tilde{p}_{D^{*}_{s}}$ is the $D^{*}_{s}$ momentum in
the rest frame of the $D^{*}_{s}\bar{D}^{*}_{s}$ system.

##  III RESULTS AND DISCUSSIONS

Figure 4: The $J/\psi\phi$ invariant mass distribution of the $\Lambda_{b}\to
J/\psi\Lambda\phi$ process. The curves labeled as ‘$X(4160)$’, ‘$X(4140)$’ and
‘$P_{cs}(4459)$’ show the contributions from the the molecular state
$X(4160)$, the intermediate resonances $X(4140)$ and $P_{cs}(4459)$,
respectively. The curve labeled as ‘total’ corresponds to the total
contributions of Eq. (17). Figure 5: The Dalitz plot of the $\Lambda_{b}\to
J/\psi\Lambda\phi$ process. Figure 6: The $J/\psi\phi$ invariant mass
distribution of the $\Lambda_{b}\to J/\psi\phi\Lambda$ process with different
values of $\alpha$ ($=0.8\alpha_{0},~{}1.0\alpha_{0},~{}1.2\alpha_{0}$).

In this section, we will present our results. With the $\beta=1$, we show the
$J/\psi\phi$ mass distribution in Fig. 4. We can find a peak around 4140 MeV,
corresponding to $X(4140)$ resonance, and a bump around 4160 MeV followed by a
cusp around $D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}$ threshold, which can be
associated to the resonance $X(4160)$. It should be stressed that a molecular
state that couples to several hadron-hadron channels may develop a strong and
unexpected cusp in the invariant mass distribution in one of the weakly
coupled channels at the threshold of the channels which are the main
components of the state Wang:2018djr ; Wang:2017mrt ; Dai:2018tgo ;
Dai:2018nmw . The intermediate $P_{cs}(4459)$ only plays an important role
above the 4160 MeV, which can be easily understood according to the Dalitz
plot of Fig. 5, and will shift the bump position of the $X(4160)$ to high
energies. In Fig. 6, we also show the $J/\psi\phi$ invariant mass distribution
for the different values of $\alpha$, and one can easily find the bump of
$X(4160)$ is clearer for the larger $\alpha$.

Figure 7: The $J/\psi\Lambda$ invariant mass distribution of the
$\Lambda_{b}\to J/\psi\phi\Lambda$ process. The explanations of the curves are
the same as those of Fig. 4. Figure 8: The $J/\psi\Lambda$ invariant mass
distribution of the $\Lambda_{b}\to J/\psi\phi\Lambda$ process with different
values of $\beta$ ($=0.8,~{}1.0,~{}1.2$).

In Fig. 7, we show the $J/\psi\Lambda$ invariant mass distribution of the
$\Lambda_{b}\to J/\psi\Lambda\phi$, and predict a clear peak around 4460 MeV,
associated to the hidden-charm pentaquark $P_{cs}(4459)$. The $X(4140)$ and
$X(4160)$ resonances only have contribution in the region of
$M_{J/\psi\Lambda}>4500$ MeV and $M_{J/\psi\Lambda}>4400$ MeV, respectively,
and they do not significantly affect the peak position of the $P_{cs}(4459)$,
which can be understood through the Dalitz plot of Fig. 5. We also show the
$J/\psi\Lambda$ invariant mass distribution with the different values of
$\beta$, the relative production weight of the $P_{cs}(4459)$, in Fig. 8. It
should be pointed out that we do not know the exact value of $\beta$, which
should be determined by comparing with the measurements. Since the
$P_{cs}(4459)$ was observed in the $J/\psi\Lambda$ invariant mass spectrum of
the $\Xi_{b}\to J/\psi\Lambda K$, it is expected that there is a signal of
$P_{cs}(4459)$ in the $J/\psi\Lambda$ invariant mass distribution of
$\Lambda_{b}\to J/\psi\Lambda\phi$. To confirm the existence of the
$P_{cs}(4459)$, it is necessary to observe this state in another different
process, which would strengthen its interpretation as exotic state
Chen:2020uif ; Peng:2020hql ; Wang:2020eep , rather than possible kinematical
effects.

As we mentioned above, the signal of the bump structure from the $X(4160)$ is
usually difficult to be identified in the $J/\psi\phi$ mass spectrum, because
the width of the $X(4160)$ is too large and its mass is close to the one of
the $X(4140)$. Searching for the signal of the $X(4160)$ in other channels is
also important. Although the $X(4160)$ mass is less than the
$D^{*}_{s}\bar{D}^{*}_{s}$ threshold, we expect it causes an enhancement
structure near the $D_{s}^{*}\bar{D}_{s}^{*}$ threshold in the
$D_{s}^{*}\bar{D}_{s}^{*}$ mass spectrum, since the $X(4160)$ is predicted to
couple strongly to the $D_{s}^{*}\bar{D}_{s}^{*}$ channel within the hidden
gauge formalism Molina:2009ct . In Fig. 9, we present the
$D_{s}^{*}\bar{D}_{s}^{*}$ invariant mass distribution of the $\Lambda_{b}\to
D^{*}_{s}\bar{D}^{*}_{s}\Lambda$ process with $\gamma=1$. There is a
significant enhancement close to the threshold with respect to the phase space
distribution, which is the reflection of the $X(4160)$ and should not be
misidentified with a new resonance. We also show the results for the different
values of $\gamma$ in Fig. 10, where one can see that the enhancement
structure close to the threshold is always clear for the different values of
$\gamma$. Our prediction can be tested by future measurements about the
$\Lambda_{b}\to D{{}_{s}^{*}}\bar{D}{{}_{s}^{*}}\Lambda$ reaction.

Figure 9: The $D_{s}^{*}\bar{D}_{s}^{*}$ invariant mass distribution of the
$\Lambda_{b}\to D^{*}_{s}\bar{D}^{*}_{s}\Lambda$ process. The curves labeled
as ‘$S$-wave’ and ‘$P$-wave’ show the contributions from the $S$-wave of Eq.
(23) and the $P$-wave of Eq. (24). The curve labeled as ‘total’ corresponds to
the total contributions of Eq. (26), and the curve labeled as ‘phase space’ is
the phase space distribution, which is normalized to the same area from
threshold to 4350 MeV. Figure 10: The $D_{s}^{*}\bar{D}_{s}^{*}$ invariant
mass distribution of the $\Lambda_{b}\to D^{*}_{s}\bar{D}^{*}_{s}\Lambda$
process with different values of $\gamma$.

##  IV Conclusions

In this work, we have investigated the $\Lambda_{b}\to J/\psi\Lambda\phi$, by
taking into account the $X(4140)$, $X(4160)$, and $P_{cs}(4459)$, and
including the mechanisms of the $W^{-}$ internal and external emissions. In
the $J/\psi\phi$ mass distribution, one can find a clear peak of $X(4140)$,
and a bump structure around 4160 MeV followed by a cusp structure around
$D^{*}_{s}\bar{D}^{*}_{s}$ threshold, which can be associated to the
$X(4160)$. We also expect that the hidden-charm pentaquark with strangeness
$P_{cs}(4459)$, recently observed by the LHCb Collaboration, causes a clear
peak in the $J/\psi\Lambda$ invariant mass distribution. It should be noted
that it is necessary to confirm this state in another process, which would
strengthen its interpretation as exotic state, rather than possible
kinematical effects.

Indeed, the signal of the bump structure from the $X(4160)$ is usually
difficult to be identified in the $J/\psi\phi$ mass spectrum with the low data
sample, because the width of the $X(4160)$ is too large, and its mass is close
to the one of the $X(4140)$. Although the $X(4160)$ mass is less than the
$D^{*}_{s}\bar{D}^{*}_{s}$ threshold, we have predicted that there is an
enhancement structure near the $D_{s}^{*}\bar{D}_{s}^{*}$ threshold in the
$D_{s}^{*}\bar{D}_{s}^{*}$ mass spectrum, since the $X(4160)$ is predicted to
couple strongly to the $D_{s}^{*}\bar{D}_{s}^{*}$ channel with in the hidden
gauge formalism Molina:2009ct .

In summary, we strongly suggest further measurements of these two processes to
clarify the nature of $X(4140)$ and $X(4160)$ resonances, and to confirm the
existence of the $P_{cs}(4459)$.

###### Acknowledgements.

This work is partly supported by the National Natural Science Foundation of
China under Grants No. 11947089. It is also supported by the Key Research
Projects of Henan Higher Education Institutions under No. 20A140027, the
Project of Youth Backbone Teachers of Colleges and Universities of Henan
Province (2020GGJS017), the Natural Science Foundation of Henan
(212300410123), the Fundamental Research Cultivation Fund for Young Teachers
of Zhengzhou University (JC202041042), and the Academic Improvement Project of
Zhengzhou University.

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# Supporting FAIR Principles in the Astrophysics Community:
the European Experience

Marco Molinaro,1 Mark Allen,2 François Bonnarel,2 Françoise Genova,2 Markus
Demleitner,3 Kay Graf,4 Dave Morris,5 Enrique Solano,6 and André Schaaff2

###### Abstract

FAIR principles have the intent to act as a guideline for those wishing to
enhance the reusability of their data holdings and put specific emphasis on
enhancing the ability of machines to automatically find and use the data, in
addition to supporting its reuse by individuals. Interoperability, one core of
these principles, especially when dealing with automated systems’ ability to
interface with each other, requires open standards to avoid restrictions that
negatively impact the user’s experience. Open-ness of standards is best
supported when the governance itself is open and includes a wide range of
community participation. In this contribution we report our experience with
the FAIR principles, interoperable systems and open governance in
astrophysics. We report on activities that have matured within the ESCAPE
project with a focus on interfacing the EOSC architecture and Interoperability
Framework.

1INAF - Osservatorio Astronomico di Trieste, Trieste, Italy;
<EMAIL_ADDRESS>
2Centre de Données astronomiques de Strasbourg, Strasbourg, France

3Universität Heidelberg, Astronomisches Rechen-Institut, Heidelberg, Germany

4Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for
Astroparticle Physics, Erlangen, Germany

5University of Edinburgh, Edinburgh, Scotland, UK

6Centro de Astrobiología (INTA-CSIC), Madrid, Spain

##  1 Introduction

The Virtual Observatory (VO) ecosystem of standards and technologies,
discussed and defined by the International Virtual Observatory Alliance
(IVOA), has, since the setup of the Alliance itself (2002), focused on
interoperability as a way to provide discovery of and access to distributed
resources in the astrophysics research domain and to enable re-use of those
same resources.

The above statement explains why VO standards provide a direct support of FAIR
principles (Fig. 1 provides a visual support on how FAIR principles fit in the
IVOA architecture) and, adding the fact that they’re open in definition and
governance, allow for an Open Science scenario in astrophysics. VO also pre-
dates the FAIR principles formalisation (Wilkinson et al. 2016) by a timespan
long enough that let its architecture grow mature.

Figure 1.: The IVOA architecture (in black) and the top FAIR principles
concepts superimposed (in green). Interoperability fits the core of the Level
1 IVOA Architecture (Dowler et al. 2021) while the pillars of Registry and
Data Access Protocols embody the Findability and Accessibility. Re-usability
pops out of the full architecture, taking advantage of the entire ecosystem
the VO architecture defines.

In Europe, the VO community coordinates itself around EuroVO, roughly spanning
the same timeline of the gloabl IVOA efforts it contributes to. The succession
of EuroVO projects (sketched in Fig. 2, adapted from Genova et al. 2015) is
currently continued within the H2020 ESCAPE111EU Horizon 2020 funded:
https://projectescape.eu/ project (see Sec. 2), in particular within its Work
Package 4 (WP4, CEVO “Connecting ESFRI projects to EOSC through the VO”).

Figure 2.: Timeline of EuroVO projects, adapted from Genova et al. (2015)

In this contribution the focus will be on the activities (Sections 3 to 5)
carried on within ESCAPE that deal with integrating the VO architecture in the
European Open Science Cloud (EOSC) and supporting ESFRI projects in improving
the interoperability of their resources, tackling common issues towards FAIR
maturity.

##  2 ESCAPE

The ESCAPE project (European Science Cluster of Astronomy & Particle Physics
ESFRI Research Infrastructures) puts together astronomy and particle physics,
through their Research Infrastructures considered crucial by the European
Science Forum, towards the common goal of implementing a functional link
between them and European Open Science Cloud (EOSC). Within the project, the
CEVO WP has the goal to connect ESFRI projects to EOSC using the VO framework
as the means to do so. This involves three main acitivities: integration of
astronomy VO data and services into the EOSC, implementation of FAIR
principles for ESFRI data through the Virtual Observatory, and adding value to
trusted content in astronomy archives. The activities reported in the
subsequent sections of this contribution are mainly performed within CEVO but
include cross activities with other ESCAPE WPs.

##  3 IVOA architecture integration

###  3.1 Registry integration

Integration of the IVOA Registry of Resources in EOSC has happened with the
EUDAT B2FIND service harvesting222Currently harvested records are visible at
http://b2find.eudat.eu/dataset?groups=ivoa, through the OAI-PMH protocol,
DataCite metadata from the IVOA Registry. Work is ongoing towards adding more
metadata, using the B2FIND metadata schema at harvesting, instead of DataCite
only. The mapping from VOResource metadata to DataCite is not complete, the
most important differences being about mapping per-protocol access URLs and
tablesets. The Unified Astronomy Thesaurus top level concepts have been mapped
to VOResource subject keywords. A further goal of such mapping and integration
could be trying to bridge different disciplines of research. Initial use cases
have been drawn for this purpose333You can find them here:
https://github.com/msdemlei/cross-discipline-discovery.

###  3.2 EOSC onboarding

The EOSC onboarding procedure shows another possible way of integration, with
resources ending up directly visible in the EOSC portal. To test this solution
a Provider has been set up (for INAF) and a couple of services (in
collaboration with the NEANIAS project) have been onboarded successfully. The
onboarding procedure will be tested also for registered VO resources, to
report guidelines on the procedure in view of avoiding duplication of efforts
for providers. The differences in architecture, metadata goals and granularity
are the challenges offered to this activity.

##  4 IVOA standards updates for ESFRI

Interoperability and FAIR-ness inclusion in standards require continuous
updates. In ESCAPE, CEVO continues gathering requirements from ESFRIs, using
the IVOA interoperability meetings as milestones to report result and help
keeping the IVOA standards up to date. Radio astronomy data access, tools and
standards to support follow-up of gravitational waves events are among the
main results alongside common support to projects and communities.

###  4.1 Science with interoperable data schools

Re-usability, besides its metadata implications, also requires a community of
users that are aware of standards and tools. ESCAPE CEVO continues the
sequence of Virtual Observatory schools in Europe that has been an integral
part of the EuroVO projects. Gathering feedback from the community is a goal
of these events alongside dissemination (see Jiménez-Esteban 2022, for
details).

##  5 Software interoperability

FAIR principles and interoperability are also useful guidelines for open
solutions to discover, re-use and interoperate software and services. Open-
source scientific Software and Service Repository (OSSR, ESCAPE WP3)
activities include easing publication and integration of these non-data
resources. Work moves from code versioning on to subsequent publishing in
dedicated repositories (like Zenodo and Docker Hub) and, finally, integration
in the EOSC marketplace. Software interoperability is also investigated in
terms of standardisation of the current heterogeneous scenario of environments
and tasks (Jupyter notebooks, containers and other) that differ in terms of
interfaces, behaviour, configuration, authentication. Identifying a minimal
set of common functionalities, trying to constrain the interfaces require test
implementations to help in answering the first questions in interoperability
of these environments444 For basic concepts see
https://wiki.ivoa.net/internal/IVOA/InterOpMay2021SPW/20210527-IVOA-
canidothis.pdf.

##  6 Connection to other projects, initiatives

Connections that ESCAPE members are making with other projects and initiatives
help in keeping interoperability alive across domain boundaries (project,
system or geographical). Among these connections are worth mentioning that
with the FAIRsFAIR project, with contributions to activities in turning FAIR
principles into reality, and with FREYA project, including discussion on
Persistent identifiers usage starting from the experience of IVOA IDs. Part of
the activities have been also testing and discussion at RDA level about the
FAIR Data Maturity Model and the proposed metrics connected, and the
participation to LISA conference to report stewardship, curation and
publication related to the ESFRI archives.

#### Acknowledgments

ESCAPE - The European Science Cluster of Astronomy & Particle Physics ESFRI
Research Infrastructures has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Grant Agreement n°
824064. MM acknowledges support form the NEANIAS project in the EOSC
onboarding procedure.

## References

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