# Vacuum bosonic currents induced by a compactified cosmic string in dS
background

E. A. F. Bragança1 , E. R. Bezerra de Mello2 and A. Mohammadi 3

1,3Departamento de Física, Universidade Federal de Pernambuco,
52171-900, Recife-PE, Brazil

2Departamento de Física, Universidade Federal da Paraíba
58.059-970, Caixa Postal 5.008, João Pessoa, PB, Brazil E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the present paper, we study the vacuum bosonic currents in the geometry of
a compactified cosmic string in the background of the de Sitter spacetime. The
currents are induced by magnetic fluxes, one running along the cosmic string
and another one enclosed by the compact dimension. To develop the analysis, we
obtain the complete set of normalized bosonic wave-functions obeying a
quasiperiodicity condition. In this context, we calculate the azimuthal and
axial current densities and we show that these quantities are explicitly
decomposed into two contributions: one originating from the geometry of a
straight uncompactified cosmic string and the other induced by the
compactification. We also compare the results with the literature in the case
of a massive fermionic field in the same geometry.

PACS numbers: 98.80.Cq, 11.10.Gh, 11.27.+d, 04.62.+v, 03.70.+k

##  1 Introduction

De Sitter (dS) is a curved spacetime that has been most analyzed in the
context of quantum field theory. The main reason is due to its degree of
symmetry which allows that many physical problems to be exactly solvable.111De
Sitter space enjoys the same number of degrees of symmetries as the Minkowski
one [1]. Besides that, by the appearance of the inflationary cosmological
scenario for the early Universe expansion which is best described by the dS
Universe, the importance of these theoretical analyses increased [2]. In this
scenario, the cosmological constant plays the role of the inflaton field, and
quantum fluctuations in the field give rise to inhomogeneities which are
important in the generation of large scale structures from cosmic inflation.

The topological defects formation is predicted in many field theoretical and
cosmological models due to symmetry breaking phase transitions. The early
Universe was hotter and consequently in a more symmetric state. Therefore, in
the expansion process, its temperature dropped and faced a series of phase
transitions accompanied by a spontaneous breakdown of symmetries, which could
leave topological defects as remnants. According to the Higgs-Kibble
mechanism, different types of solitons may have formed in this process. Cosmic
strings are amongst the most probable ones. Cosmic strings are linear
gravitational topological defects which are predicted in the context of the
standard gauge field theory of elementary particle physics [3, 4, 5]. They are
lines of trapped energy density analogous to topological defects such as
vortex lines in condensed matter systems. Even though the observations of
anisotropies in the CMB radiation by COBE, WMAP, and the Planck satellite have
ruled them out as the primary source for primordial density perturbations [6],
they can result in a number of fascinating physical effects including
gravitational lensing, the generation of highly energetic cosmic rays and
gravitational waves emission (see, for instance, [7, 8, 9]). The spacetime
geometry associated with an infinitely long and straight cosmic string is
locally flat, although not globally. The presence of a cosmic string leads to
an azimuthal deficit angle and, as a result, this spacetime has a conical
singularity [10, 11, 12]. One of the most novel features of this spacetime is
the fact that quantum fields are sensitive to its global structure and this
may give rise to compelling phenomena. For instance, in the context of quantum
field theory in curved spacetimes, the presence of a cosmic string polarizes
the vacuum state of a quantum field.

The presence of a nontrivial topology or geometry of the spacetime or even a
background gauge field may affect the vacuum in quantum field theory
considerably. This means that many consequent physical observables can provide
important information about the structure of the physical system under
investigation. Among the important ones is the vacuum expectation value (VEV)
of the current density for the charged fields. In the present paper, we study
the vacuum bosonic current induced by a magnetic flux running along an
idealized cosmic string in an expanding Universe, represented by dS spacetime.
As we shall see, besides the natural dependence on the distance to the string,
there appears an additional explicit time dependence on the current,
reflecting the expansive character of the dS background. In addition, we
assume that the coordinate along the string is compactified to a circle. As
one knows, the compact spatial dimensions are an inherent feature of most
high-energy theories of fundamental physics, including supergravity and
superstring theories. In this way, we are investigating induced bosonic
current in a dS space as general as possible. Obviously, when the size of the
compact dimension tends to infinity we recover the results without
compactification.

Cosmic string spacetime modifies the vacuum fluctuations associated with
quantum fields due to its topologically conical structure. Non-vanishing
results for the renormalized VEV of physical observables, such as the energy-
momentum tensor, considering scalar and fermionic quantum fields were obtained
in [13, 14, 15, 16, 17, 18, 19, 20, 21]. Moreover, taking into account the
presence of a magnetic flux running through the string’s core, there appear
additional contributions to the above VEVs associated with charged fields [22,
23, 24, 25, 26, 27, 28]. The magnetic flux can also induce vacuum current
densities. The phenomenon was addressed in [29] and [30] for massless and
massive scalar fields, respectively. It was shown that the fractional part of
the ratio of the magnetic flux by the quantum one leads to non-vanishing
induced vacuum current densities along the azimuthal direction. In [31, 32]
the authors studied induced bosonic and fermionic currents in higher-
dimensional cosmic string spacetime in the presence of magnetic flux. The
induced fermionic current by a magnetic flux in $(2+1)$-dimensional conical
spacetime was also calculated in [33]. In all these calculations that we
mentioned above the cosmic string was considered to be ideal, without any
inner structure. A nonvanishing core was studied in [34, 35] for the scalar
and fermionic vacuum current densities induced by a magnetic flux in a cosmic
string. Moreover, the calculation of the VEV of fermionic energy-momentum
tensor was addressed in [36].

In general, quantum effects associated with matter fields in a cosmic string
spacetime are mostly analyzed in a flat background geometry. The effects
associated with a scalar field in a curved background in the presence of a
cosmic string were discussed in [37]. In [38, 39], the authors studied the
vacuum polarization in Schwarzschild spacetime pierced by an ideal cosmic
string. In this line of work, the vacuum polarization induced by a cosmic
string in anti-dS spacetime was developed in [42] and [43] for massive scalar
and fermionic fields, respectively. A similar analysis was done in [40, 41],
considering the dS spacetime. We studied the vacuum fermionic current induced
by a magnetic flux and the VEV of the energy-momentum tensor in dS spacetime
in the presence of a compactified cosmic string [44, 45]. In the current
paper, we continue the same line of analysis by calculating the induced scalar
current in the same geometry. We also compare the results with the ones for a
massive fermionic field obtained in [44].

The paper is organized in the following order. In section 2 we describe the
background geometry and construct the positive frequency Wightman function for
a massive charged scalar quantum field in dS spacetime in the presence of a
cosmic string presenting a magnetic flux along its axis. Besides that, we
consider that the $z$-axis along the string is compactified to a circle and
carries an extra magnetic flux. By using the Wightman function we evaluate the
renormalized vacuum current density induced by the magnetic fluxes and the
compactification. The only nonzero components of the current densities
correspond to the azimuthal and axial ones. These quantities are investigated
in sections 2.1 and 2.2. The most relevant conclusions of our study are
summarized in section 3. We also dedicate an Appendix to provide some
important expressions used for the development of our study for the induced
current densities. We consider natural units $G=\hbar=c=1$ throughout the
paper.

##  2 Induced bosonic currents

Let us consider a charged massive scalar field in a $(3+1)$ dS spacetime with
an ideal cosmic string assuming that the direction along the string is
compactified to a circle with the length $L$. The geometry associated with the
corresponding background spacetime is given by the following line element

$ds^{2}=dt^{2}-e^{2t/\alpha}(dr^{2}+r^{2}d\phi+dz^{2})\ .$ (2.1)

The coordinates take values in the intervals $-\infty<t<+\infty$, $r\geq 0$,
$0\leq\phi\leq\phi_{0}=2\pi/q$ and $0\leqslant z\leqslant L$. The parameter
$q\geq 1$ encodes the deficit angle due to the conical structure of the cosmic
string, keeping in mind that the minimum value $q=1$ shows its absence. The
parameter $\alpha$ is related to the cosmological constant and Ricci scalar
through the expression, $R=4\Lambda=12/\alpha^{2}$. For further analysis, in
addition to the synchronous time coordinate $t$, it is more convenient to
introduce the conformal time $\tau$ according to $\tau=-\alpha e^{-t/\alpha}$
($-\infty<\tau<0$) [40]. This leads to the following line element

$ds^{2}=(\alpha/\tau)^{2}(d\tau^{2}-dr^{2}-r^{2}d\phi^{2}-dz^{2})\ .$ (2.2)

which shows a conformally flat spacetime.

In this paper we are interested in calculating the induced vacuum current
density, $\langle j^{\mu}\rangle$, associated with a charged scalar quantum
field, $\varphi(x)$, in the presence of magnetic flux running along the core
of the string. The equation which governs the quantum dynamics of a charged
bosonic field with mass $m$, interacting with an electromagnetic potential
vector, $A_{\mu}$, in a curved background reads

$\left({\cal D}^{2}+m^{2}+\xi R\right)\varphi(x)=0\
,\quad\quad\textrm{with}\quad\quad{\cal
D}^{2}=\frac{1}{\sqrt{|g|}}\,D_{\mu}\left(\sqrt{|g|}\,g^{\mu\nu}D_{\nu}\right)\
,$ (2.3)

being $D_{\mu}=\partial_{\mu}+ieA_{\mu}$ and $g={\rm det}(g_{\mu\nu})$. In
(2.3) we also considered the presence of a non-minimal coupling, $\xi$,
between the scalar field and the geometry represented by the Ricci scalar $R$.
Here, we are mainly interested in two specific values of the curvature
coupling, $\xi=0$ and $\xi=1/6$, which correspond to the minimal and conformal
coupling, respectively. It is known that for a bosonic field in a conformally
flat spacetime only in the case of a massless scalar field with a conformal
coupling $\xi=\frac{(D-1)}{4D}$ where D is the spatial dimension, the physical
quantities are related with the corresponding flat one multiplied by some
power of the conformal factor.

The compactification along the $z$-axis is achieved by imposing the
quasiperiodicity condition on the matter field,

$\varphi(t,r,\phi,z+L)=e^{2\pi i\beta}\varphi(t,r,\phi,z)\ ,$ (2.4)

with a constant phase $\beta$, $0\leqslant\beta\leqslant 1$, along the
$z$-direction. For the rotation around the $z$-axis we adopt the periodic
boundary condition

$\varphi(t,r,\phi+\phi_{0},z)=\varphi(t,r,\phi,z)\ .$ (2.5)

In addition, we consider the existence of a constant vector potential
$A_{\mu}=(0,0,A_{\phi},A_{z})$ with $A_{\phi}=-q\Phi_{\phi}/(2\pi)$ and
$A_{z}=-\Phi_{z}/L$, being $\Phi_{\phi}$ and $\Phi_{z}$ the corresponding
magnetic fluxes. In quantum field theory the condition (2.4) changes the
spectrum of the vacuum fluctuations compared with the case of uncompactified
dimension ($L\to\infty$) and, as a consequence, the induced vacuum current
densities are dependent on the compactification parameter $L$, in general.

The positive energy solution of (2.3) obeying the boundary conditions (2.4)
and (2.5) can be found in a similar way as it was calculated in [40] where the
geometry of a straight cosmic string in dS spacetime, although without the
magnetic flux and compactification, was considered. In the geometry we
consider here, the normalized complete set of solutions generalizes to

$\displaystyle\varphi_{\sigma}(x)=\left(\frac{qp}{8L\alpha^{2}}\right)^{1/2}e^{-i\pi\nu/2}\tau^{3/2}H_{\nu}^{(1)}(\lambda\tau)J_{q|n+a|}(pr)e^{inq\phi+ik_{z}z}\
,$ (2.6)

where $J_{\mu}(x)$ and $H_{\mu}^{(1)}(x)$ are the Bessel and Hankel functions
[46], respectively, and

$\displaystyle\lambda=\sqrt{p^{2}+{\tilde{k}}_{z}^{2}}\ ,\ {\rm with}\
{\tilde{k}}_{z}=k_{z}+eA_{z}\ ,\ p\geq 0\ .$ (2.7)

The parameters in the order of Hankel and Bessel functions are

$\displaystyle\nu=\frac{1}{2}\sqrt{9-48\xi-4m^{2}\alpha^{2}}\,,\quad\quad
a=\frac{eA_{\phi}}{q}=-\frac{\Phi_{\phi}}{\Phi_{0}}$ (2.8)

with $\Phi_{0}=2\pi/e$ being the quantum flux.

The quasiperiodicity condition (2.4) results in a discretization of the
quantum number $k_{z}$ as shown below

$k_{z}=k_{l}=\frac{2\pi}{L}(l+\beta)\quad{\rm with}\quad l=0\ ,\pm 1,\pm
2,...\ ,$ (2.9)

which gives

$\displaystyle{\tilde{k}}_{z}={\tilde{k}}_{l}=\frac{2\pi}{L}(l+\tilde{\beta})\quad{\rm
with}\quad\tilde{\beta}=\beta+\frac{eA_{z}L}{2\pi}=\beta-\frac{\Phi_{z}}{\Phi_{0}}\
.$ (2.10)

As a result, the positive-energy solution (2.6) is characterized by the set of
quantum numbers, $\sigma=\\{p,n,l\\}$.

In order to find the induced bosonic current densities we need first to
calculate the Wightman function which describes the properties of the vacuum
state. The positive frequency Wightman function is given by
$W(x,x^{\prime})=\left\langle
0|{\hat{\varphi}}(x){\hat{\varphi}}^{*}(x^{\prime})|0\right\rangle$, where
$|0\rangle$ stands for the vacuum state and $\hat{\varphi}(x)$ the field
operator. For the evaluation of the Wightman function, we use the mode sum
formula

$W(x,x^{\prime})=\sum_{n=-\infty}^{+\infty}\ \int_{0}^{\infty}\,dp\
\sum_{l=-\infty}^{+\infty}\varphi_{\sigma}(x)\varphi_{\sigma}^{*}(x^{\prime})\
.$ (2.11)

The set $\\{\varphi_{\sigma}(x),\ \varphi_{\sigma}^{*}(x^{\prime})\\}$
represents a complete set of normalized mode functions satisfying the
quasiperiodicity condition (2.4).

Substituting (2.6) into the sum (2.11), and after several intermediate steps
we obtain

$\displaystyle W(x,x^{\prime})$
$\displaystyle=\frac{2q(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}(z-z^{\prime})}}{(2\pi\alpha)^{2}L}\int_{0}^{\infty}dp\,p\sum_{l=-\infty}^{\infty}e^{\frac{2\pi}{L}i(l+\tilde{\beta)}(z-z^{\prime})}$
$\displaystyle\times\sum_{n=-\infty}^{\infty}e^{iqn(\phi-\phi^{\prime})}J_{q|n+a|}(pr)J_{q|n+a|}(pr^{\prime})K_{\nu}(i\lambda\tau^{\prime})K_{\nu}(-i\lambda\tau)\
.$ (2.12)

To obtain the above expression we have used the relation between the Hankel
function, $H_{\nu}^{(1)}(z)$, and the modified Bessel function with imaginary
argument, $K_{\mu}(-iz)$ [46]. In order to simplify the summation over $l$, we
have introduced an exponential function $e^{-ieA_{z}(z-z^{\prime})}$ to
replace the exponent factor $\beta$ by $\tilde{\beta}$, using relation (2.10),
in the summation over the quantum number $l$. With the objective to develop
the summation over the quantum number $l$ we apply the Abel-Plana summation
formula [47] in the form below

$\displaystyle\sum_{l=-\infty}^{\infty}g(l+\tilde{\beta})f(|l+\tilde{\beta}|)$
$\displaystyle=\int_{0}^{\infty}du\,\left[g(u)+g(-u)\right]f(u)$
$\displaystyle+i\int_{0}^{\infty}du\left[f(iu)-f(-iu)\right]\sum_{\chi=\pm
1}\frac{g(i\chi u)}{e^{2\pi(u+i\chi\tilde{\beta})}-1}\ .$ (2.13)

For the case under investigation, we consider $u\equiv\frac{L}{2\pi}k_{z}$ as
well as

$\displaystyle
g(l+\tilde{\beta})=e^{\frac{2\pi}{L}i(l+\tilde{\beta)}(z-z^{\prime})}\quad\quad\textrm{and}\quad\quad
f(|l+\tilde{\beta}|)=K_{\nu}(i\lambda\tau^{\prime})K_{\nu}(-i\lambda\tau)\ ,$
(2.14)

where

$\lambda=\lambda_{l}=\sqrt{\left({2\pi}(l+\tilde{\beta})/L)\right)^{2}+p^{2}}\
.$ (2.15)

At this point, we can express the Wightman function as the sum of two terms as
shown below

$\displaystyle W(x,x^{\prime})=W_{s}(x,x^{\prime})+W_{c}(x,x^{\prime})\ ,$
(2.16)

where the first contribution, $W_{s}(x,x^{\prime})$, represents the Wightman
function in dS spacetime in the presence of an uncompactified cosmic string,
and the second, $W_{c}(x,x^{\prime})$, is induced by the compactification. Let
us call the former and the latter, the string and the compactified Wightman
functions, respectively.

Substituting (2.14) into (2.13), and also substituting the result into (2.12)
we get

$\displaystyle W_{s}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2q(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}(z-z^{\prime})}}{(2\pi)^{3}\alpha^{2}}\int
dk_{z}\ e^{ik_{z}\Delta
z}\sum_{n=-\infty}^{\infty}e^{iqn(\phi-\phi^{\prime})}\int_{0}^{\infty}dp\,p$
(2.17)
$\displaystyle\times\,J_{q|n+a|}(pr)J_{q|n+a|}(pr^{\prime})K_{\nu}\left(i\tau^{\prime}\sqrt{p^{2}+k_{z}^{2}}\right)K_{\nu}\left(-i\tau\sqrt{p^{2}+k_{z}^{2}}\right)\
,$

the string Wightman function, originating from the first term on the right
hand side of (2.13).

The string Wightman function (2.17) can be expressed in terms of a more
workable integral representation in a similar way as developed in Appendix A
of [40]. The final expression is,

$\displaystyle W_{s}(x,x^{\prime})$
$\displaystyle=\frac{q(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}\Delta
z}}{2\pi^{5/2}\alpha^{2}}\sum_{n=-\infty}^{\infty}e^{iqn\Delta\phi}\int_{0}^{\infty}dx\,x^{1/2}e^{-{\cal{V}}x}\,I_{q|n+a|}(2xrr^{\prime})K_{\nu}(2\tau^{\prime}\tau
x)\ ,$ (2.18)

with

$\displaystyle{\cal{V}}=(z-z^{\prime})^{2}+r^{2}+r^{\prime
2}-(\tau^{2}+\tau^{\prime 2})\ .$ (2.19)

As to the compactified Wightman function, it is given by the second term on
the right hand side of (2.13). In Appendix A we show that
$W_{c}(x,x^{\prime})$ can be expressed in the following form

$\displaystyle W_{c}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2q(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}\Delta
z}}{(2\pi\alpha)^{2}}\sum_{n=-\infty}^{\infty}e^{iqn(\phi-\phi^{\prime})}\int_{0}^{\infty}dp\,p\,J_{q|n+a|}(pr)J_{q|n+a|}(pr^{\prime})$
(2.20)
$\displaystyle\times\,\int_{0}^{\infty}\frac{d\gamma\,\gamma}{\sqrt{\gamma^{2}+p^{2}}}\left[K_{\nu}(\tau^{\prime}\gamma)I_{-\nu}(\tau\gamma)+I_{\nu}(\tau^{\prime}\gamma)K_{\nu}(\tau\gamma)\right]$
$\displaystyle\times\,\sum_{l=1}^{\infty}e^{-lL\sqrt{\gamma^{2}+p^{2}}}\cos\left(2\pi
l\tilde{\beta}-i\Delta z\sqrt{\gamma^{2}+p^{2}}\right)\ .$

With the Wightman functions (2.18) and (2.20) in hands, we are in position to
evaluate the induced current densities.

The bosonic current density operator is given by

$\displaystyle\hat{j}_{\mu}(x)$ $\displaystyle=$ $\displaystyle
ie\left[{\hat{\varphi}}^{*}(x)D_{\mu}{\hat{\varphi}}(x)-(D_{\mu}{\hat{\varphi}})^{*}{\hat{\varphi}}(x)\right]$
(2.21) $\displaystyle=$ $\displaystyle
ie\left[{\hat{\varphi}}^{*}(x)\partial_{\mu}{\hat{\varphi}}(x)-\varphi(x)(\partial_{\mu}{\hat{\varphi}}(x))^{*}\right]-2e^{2}A_{\mu}(x)|{\hat{\varphi}}(x)|^{2}\
.$

Its vacuum expectation value (VEV) can be evaluated in terms of the positive
frequency Wightman function as follows

$\left\langle j_{\mu}(x)\right\rangle=ie\lim_{x^{\prime}\rightarrow
x}\left\\{(\partial_{\mu}-\partial_{\mu^{\prime}})W(x,x^{\prime})+2ieA_{\mu}W(x,x^{\prime})\right\\}\
.$ (2.22)

Due to the decomposition (2.16), the current density can be written as,

$\displaystyle\left\langle j_{\mu}(x)\right\rangle=\left\langle
j_{\mu}(x)\right\rangle_{s}+\left\langle j_{\mu}(x)\right\rangle_{c}\ ,$
(2.23)

where the first term corresponds to the contribution of an uncompactified
cosmic string in dS spacetime, while the second one is the contribution
induced by the compactification. As we will see shortly, the non-vanishing
components of the current density are periodic functions of the magnetic
fluxes $\Phi_{\phi}$ and $\Phi_{z}$ with the period equal to the quantum flux.
This can be observed easily writing the parameter $a$ in (2.8) in the form
$a=n_{0}+a_{0}\ {\rm with}\ |a_{0}|<\frac{1}{2}$, where $n_{0}$ is an integer
number. In this case the VEV of the current density depends on $a_{0}$ only.

In the problem under consideration, the only nonzero components of the current
density are the azimuthal and axial ones.

###  2.1 Azimuthal current

The VEV of the azimuthal current density is given by

$\left\langle j_{\phi}(x)\right\rangle=ie\lim_{x^{\prime}\rightarrow
x}\left\\{(\partial_{\phi}-\partial_{\phi^{\prime}})W(x,x^{\prime})+2ieA_{\phi}W(x,x^{\prime})\right\\}\
.$ (2.24)

The above expression can be decomposed as in (2.23). Let us start with the
string contribution. Substituting (2.18) into (2.24) as well as taking the
derivative and coincidence limit, one finds

$\displaystyle\left\langle j_{\phi}(x)\right\rangle_{s}$
$\displaystyle=-\frac{q^{2}e\tau^{3}}{\pi^{5/2}\alpha^{2}}\int_{0}^{\infty}dx\,x^{1/2}e^{-2(r^{2}-\tau^{2})x}K_{\nu}(2\tau^{2}x)\mathcal{J}(q,a_{0},2xr^{2})\
,$ (2.25)

where

$\mathcal{J}(q,a_{0},2xr^{2})=\sum_{n=-\infty}^{\infty}(n+a_{0})I_{q|n+a_{0}|}(2xr^{2})\
.$ (2.26)

The summation above has been developed in [31]. The result is

$\displaystyle\mathcal{J}(q,a_{0},w)$
$\displaystyle=\frac{2w}{q^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})e^{w\cos(2k\pi/q)}+\frac{w}{q\pi}\int_{0}^{\infty}dy\,\sinh
y\frac{e^{-w\cosh y}g(q,a_{0},y)}{\cosh(qy)-\cos(q\pi)}\ ,$

with

$g(q,a_{0},y)=\sin(q\pi
a_{0})\sinh[(1-|a_{0}|)qy]-\sinh(qya_{0})\sin[(1-|a_{0}|)q\pi].$ (2.28)

In the expression (LABEL:Summation2), $[q/2]$ means the integer part of this
ratio and the prime in the summation symbol means that in the case of $q/2$ is
an integer, the term should be taken with the coefficient $1/2$.

Taking into account (LABEL:Summation2), the Eq. (2.25) becomes

$\displaystyle\left\langle j_{\phi}(x)\right\rangle_{s}$
$\displaystyle=-\frac{e}{2^{1/2}\pi^{5/2}}\left(\frac{r}{\alpha\tau}\right)^{2}\int_{0}^{\infty}dz\,z^{3/2}K_{\nu}(z)\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})\,e^{-\left(\frac{2r^{2}s_{k}^{2}}{\tau^{2}}-1\right)z}\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}e^{-\left(\frac{2r^{2}c_{y}^{2}}{\tau^{2}}-1\right)z}\right],$
(2.29)

where we have introduced $z=2\tau^{2}x$ and defined

$s_{k}\equiv\sin(k\pi/q)\ \ \ {\rm and}\ \ \ c_{y}\equiv\cosh(y).$ (2.30)

The integral over $z$ can be evaluated with the help of the formula [40]

$\displaystyle\int_{0}^{\infty}dx\,x^{(\mu-3)/2}e^{-cx}K_{\nu}(\beta x)$
$\displaystyle=$
$\displaystyle\frac{2^{1-\mu}\sqrt{\pi}}{\Gamma(\mu/2)(\beta/2)^{(\mu-1)/2}}\Gamma\left(\frac{\mu-1}{2}-\nu\right)\Gamma\left(\frac{\mu-1}{2}+\nu\right)$
(2.31)
$\displaystyle\times\,F\left(\frac{\mu-1}{2}+\nu,\frac{\mu-1}{2}-\nu,\mu/2;\frac{1}{2}-\frac{c}{2\beta}\right),$

with $F(a,b,c;x)$ being the hypergeometric function. After the evaluation of
the integral over $z$ the final expression for the azimuthal current induced
by the cosmic string in dS background is given by

$\displaystyle\left\langle j^{\phi}(x)\right\rangle_{s}$
$\displaystyle=\frac{e}{(4\pi)^{2}\alpha^{4}}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})\,\mathcal{B}_{0}\left(r/\tau,s_{k}\right)\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{B}_{0}\left(r/\tau,c_{y}\right)\right],$
(2.32)

which is an odd function of the magnetic flux $a_{0}$ and depends on the ratio
$r/\tau$. This ratio is the proper distance from the string in units of the dS
curvature ratio $\alpha$. In addition, we have used the definition

$\mathcal{B}_{l}(u,v)\equiv\Gamma\left(\frac{5}{2}+\nu\right)\Gamma\left(\frac{5}{2}-\nu\right)\,F\left(\frac{5}{2}+\nu,\frac{5}{2}-\nu;3;1-u^{2}v^{2}-\frac{l^{2}L^{2}}{4\tau^{2}}\right),$
(2.33)

with $l=0$. The above definition will also be useful in the rest of the
calculations. Notice that the parameter $\nu$, defined in (2.8), is related to
the mass of the scalar field and can be a real or an imaginary quantity.

At this point we would like to analyze some asymptotic cases of the azimuthal
current density. Let us first consider the situation where, $r/\tau\gg 1$. For
a fixed conformal time this corresponds to the region far from the string. In
this region the the main contribution to the integral over $z$ in (2.29) comes
from small values of the argument of the Macdonald function. By considering
the corresponding expansion of the Macdonald function for small arguments and
$\nu$ as imaginary, we find

$\displaystyle\left\langle j^{\phi}(x)\right\rangle_{s}$
$\displaystyle\approx\frac{e}{8\pi^{5/2}\alpha^{4}}\left(\frac{\tau}{r}\right)^{5}\left\\{\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\sin(2k\pi/q)\sin(2k\pi
a_{0})}{s_{k}^{5}}\,{\rm
Re}\left[\left(\frac{2rs_{k}}{\tau}\right)^{2\nu}\Gamma(\nu)\Gamma(5/2-\nu)\right]\right.$
$\displaystyle\left.+\,\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)c_{y}^{-5}}{\cosh(2qy)-\cos(q\pi)}\,{\rm
Re}\left[\left(\frac{2rc_{y}}{\tau}\right)^{2\nu}\Gamma(\nu)\Gamma(5/2-\nu)\right]\right\\}.$
(2.34)

From the above expression we note an oscillatory behavior at large distances
from the string when $\nu$ is imaginary. As can be seen, for large distances
the string contribution of the azimuthal current density tends to zero with
the fifth power of $\tau/r$. If $\nu$ is real, we have the same equation,
(2.34), with the coefficient $1/2$. However, knowing that for the case of a
minimal coulpling $(\xi=0),$ $0\leq m\alpha\leq 3/2$ for real $\nu$, depending
on the mass of the bosonic field, the decay behavior for large distances can
be really different. As $\nu$ becomes closer to the maximum value (as mass
decreases to zero), the decay in $\left\langle j^{\phi}(x)\right\rangle_{s}$
becomes slower. It tends to zero with the second power of $\tau/r$ for the
massless field. In contrast, $\left\langle j^{\phi}(x)\right\rangle_{s}$ for
the fermionic field in the same geometry decays with the fourth power of
$\tau/r$ independent of the mass, and while decaying oscillates as it was
shown in [44]. However, in the massless limit the oscillating behavior
disappears. Now, in the case of a conformal coulpling, $\xi=1/6$ resulting in
$0\leq m\alpha\leq 1/2$ for real $\nu$, the azimuthal current density goes to
zero with the fourth power of $\tau/r$ in the massless limit, similar to the
fermionic case.

Moreover, by explicit derivation we can show that in the limit $r/\tau\ll 1$,
i.e., in the regions near the string, Eq. (2.32) presents a divergence on the
string with the inverse fourth power of the proper distance matching exactly
with the result for the fermionic field studied in [44].

In Fig. 1 we plot the string part of the azimuthal current density (in the
absence of the compactification) as a function of $r/\tau$ considering
different values of $q$ and a minimal coupling ($\xi=0$). In the left plot,
the parameter $\nu$ is real and in the right one is imaginary. In the latter
case we show the oscillatory behavior of the azimuthal current density at
large distances from the string.

Figure 1: String part of the azimuthal current density as a function of
$r/\tau$ considering several values of the parameter $q$. In the left plot
$m\alpha=1$ while in the right one $m\alpha=2$. In both plots $a_{0}=0.25$ and
$\xi=0$.

Now, for the evaluation of the azimuthal current density induced by the
compactification, we need to start again with the Eq. (2.20). Substituting
this equation into (2.24), after taking the derivatives and the coincidence
limit, one obtains

$\displaystyle\left\langle j_{\phi}(x)\right\rangle_{c}$ $\displaystyle=$
$\displaystyle-\frac{4eq^{2}\tau^{3}}{(2\pi\alpha)^{2}}\sum_{l=1}^{\infty}\cos(2\pi
l\tilde{\beta})\sum_{n=-\infty}^{\infty}(n+a_{0})\int_{0}^{\infty}dp\,p\,J^{2}_{q|n+a|}(pr)$
(2.35)
$\displaystyle\times\int_{0}^{\infty}d\gamma\,\gamma\,\frac{e^{-lL\sqrt{\gamma^{2}+p^{2}}}}{\sqrt{\gamma^{2}+p^{2}}}K_{\nu}(\tau\gamma)[I_{\nu}(\tau\gamma)+I_{-\nu}(\tau\gamma)].$

After evaluating the integrals over $p$ and $\gamma$ [48, 49], the expression
becomes

$\displaystyle\left\langle j_{\phi}(x)\right\rangle_{c}$
$\displaystyle=-\frac{2eq^{2}\tau^{3}}{\alpha^{2}\pi^{5/2}}\sum_{l=1}^{\infty}\cos(2\pi
l\tilde{\beta})\int_{0}^{\infty}dx\,x^{1/2}e^{-(l^{2}L^{2}+2r^{2}-2\tau^{2})x}\,K_{\nu}(2\tau^{2}x)\mathcal{J}(q,a_{0},2r^{2}x),$
(2.36)

where $x=1/(4s^{2})$. In the above equation, it is not difficult to see that
in the limit $L\to\infty$ the contribution of the current originated from the
compactification vanishes which is the expected result. By using the
representation (LABEL:Summation2) and after the integration over $x$, one
finds

$\displaystyle\left\langle j^{\phi}(x)\right\rangle_{c}$ $\displaystyle=$
$\displaystyle\frac{2e}{(4\pi)^{2}\alpha^{4}}\sum_{l=1}^{\infty}\cos(2\pi
l\tilde{\beta})\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})\,\mathcal{B}_{l}\left(r/\tau,s_{k}\right)\right.$ (2.37)
$\displaystyle\left.+\,\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{B}_{l}\left(r/\tau,c_{y}\right)\right],$

where we have used the definition (2.33). The above expression is an even
function of the magnetic flux enclosed by the compactification and an odd
function of the magnetic flux running along the string.

Some asymptotic cases can be studied for the current density induced by the
compactification. Considering small distances from the string, we can use the
following expression for the hypergeometric function [46]

$\displaystyle F(a,b,c;z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\frac{F\left(a,c-b,a-b+1;\frac{1}{1-z}\right)}{(1-z)^{a}}$
(2.38)
$\displaystyle+\,\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}\frac{F\left(b,c-a,b-a+1;\frac{1}{1-z}\right)}{(1-z)^{b}}.$

For the case where the parameter $\nu$ is imaginary, in the leading term we
have

$\displaystyle\left\langle j^{\phi}(x)\right\rangle_{c}$
$\displaystyle\approx$ $\displaystyle\frac{eq\sin(q\pi
a_{0})}{(2\pi)^{3}\alpha^{4}}\left(\frac{\tau}{r}\right)^{2(1-q|a_{0}|)}\Gamma(1-q|a_{0}|)\sum_{l=1}^{\infty}\cos(2\pi
l\tilde{\beta})$ (2.39) $\displaystyle\times\,{\rm
Re}\left[\frac{\Gamma(2\nu)\Gamma\left(\frac{3}{2}+q|a_{0}|-\nu\right)}{\Gamma\left(\frac{1}{2}+\nu\right)}\left(\frac{2\tau}{lL}\right)^{2(q|a_{0}|-1-\nu)+5}\right].$

If $\nu$ is real, the leading contribution comes from the second term on the
right-hand side of (2.38) where the result is $1/2$ of the imaginary $\nu$
case. In this limit, the VEV of the azimuthal current density induced by the
compactification diverges on the string if $|a_{0}|\leq 1/q$ and is finite on
the string if $|a_{0}|>1/q$, unlike the fermionic case where the threshold
value is $|a_{0}|=(1-1/q)/2$ [44]. Now, in the limit where $L/\tau\gg 1$ and
$r/\tau$ fixed, again we can use (2.38) and for $\nu$ being imaginary one
finds

$\displaystyle\left\langle j^{\phi}(x)\right\rangle_{c}$
$\displaystyle\approx$
$\displaystyle\frac{e}{2\pi^{2}\alpha^{4}}\left(\frac{\tau}{L}\right)^{5}\sum_{l=1}^{\infty}\cos(2\pi
l\tilde{\beta})\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})\,\mathcal{G}_{l}(r/L,s_{k})\right.$ (2.40)
$\displaystyle\left.+\,\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{G}_{l}\left(r/L,c_{y}\right)\right],$

where we use the notation

$\mathcal{G}_{l}(u,v)\equiv{\rm
Re}\left[\frac{\Gamma(2\nu)\Gamma(5/2-\nu)}{\Gamma(1/2+\nu)}\left(\frac{L}{\tau}\right)^{2\nu}\left(\frac{l^{2}}{4}+u^{2}v^{2}\right)^{\nu-5/2}\right].$
(2.41)

Clearly, the compactification contribution of the azimuthal current density
tends to zero when $L\to\infty$.

In Fig. 2, taking into consideration a minimal couping, we plot the VEV of the
azimuthal current density induced by the compactification. In the left plot we
have the induced current as a function of $r/\tau$ for three different values
of $q$. Note that the the left plot shows both cases where the azimuthal
current density diverges or not on the string depending on the value of $q$.
Moreover, when $r/\tau\to\infty$ it goes to zero. In the right plot of Fig. 2
the azimuthal current density induced by the compactification is shown as a
function of $\tilde{\beta}$ for the same values of the parameter $q$. It
exhibits the periodic behavior in $\tilde{\beta}$, as expected.

Figure 2: Azimuthal current density induced by the compactification as a
function of $r/\tau$ (left plot) and $\tilde{\beta}$ (right plot) for
different values of the parameter $q$. In the left plot we consider
$\tilde{\beta}=0.25$, $a_{0}=0.45$ and in the right one $r/\tau=a_{0}=0.25$.
In both plots we have $m\alpha=1.5$, $L/\tau=0.75$ and $\xi=0$.

We note that taking into account the Eqs. (2.32) and (2.37) is possible to
write the total azimuthal current density as

$\displaystyle\left\langle j^{\phi}(x)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{2e}{(4\pi)^{2}\alpha^{4}}\sideset{}{{}^{\prime}}{\sum}_{l=0}^{\infty}\cos(2\pi
l\tilde{\beta})\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin(2k\pi/q)\sin(2k\pi
a_{0})\,\mathcal{B}_{l}\left(r/\tau,s_{k}\right)\right.$ (2.42)
$\displaystyle\left.+\,\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{\sinh(2y)g(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{B}_{l}\left(r/\tau,c_{y}\right)\right].$

The term $l=0$ has to be taken with the coefficient $1/2$ and give us the
azimuthal current density induced by the uncompactified string and the
curvature of the dS spacetime. For a massless scalar field and considering the
conformal coupling, $\xi=1/6$, we have $\nu=1/2$. For this specific case, by
using the properties of the hypergeometric function [46], the total azimuthal
current matches exactly to the one for the flat spacetime in the presence of a
compactified cosmic string [31], for the $(1+3)-$dimension, times the
conformal factor $(\tau/\alpha)^{4}$ which is the expected result.

###  2.2 Axial current

Now, we focus on the axial current density, which its VEV is given by

$\left\langle j_{z}(x)\right\rangle=ie\lim_{x^{\prime}\rightarrow
x}\left\\{(\partial_{z}-\partial_{z^{\prime}})W(x,x^{\prime})+2ieA_{z}W(x,x^{\prime})\right\\}\
.$ (2.43)

Replacing the Wightman function (2.18) in the above equation, after taking the
derivatives and the coincidence limit we note that the string contribution
vanishes.

The contribution of the axial current density induced by the compactification
after several intermediate steps, similar to the ones done in the last
sections, is given in the following form

$\displaystyle\left\langle j_{z}(x)\right\rangle_{c}$ $\displaystyle=$
$\displaystyle-\frac{eq\tau^{3}}{(\pi\alpha)^{2}}\sum_{l=1}\sin(2\pi
l\tilde{\beta})\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}dp\,pJ^{2}_{q|n+a|}(pr)\int_{0}^{\infty}d\gamma\,\gamma
K_{\nu}(\nu\gamma)$ (2.44) $\displaystyle\times$
$\displaystyle[I_{\nu}(\tau\gamma)+I_{-\nu}(\tau\gamma)]e^{-lL\sqrt{\gamma^{2}+p^{2}}}.$

The integral representation

$e^{-lL\sqrt{\gamma^{2}+p^{2}}}=\frac{lL}{\sqrt{\pi}}\int_{0}^{\infty}ds\,s^{-2}\,e^{-(\gamma^{2}+p^{2})s^{2}-l^{2}L^{2}/(4s^{2})}.$
(2.45)

allows us to perform the integration over $p$ and $\gamma$ in the expression
(2.44) where the result is given by

$\displaystyle\left\langle j_{z}(x)\right\rangle_{c}$
$\displaystyle=-\frac{4eqL\tau^{3}}{\alpha^{2}\pi^{5/2}}\sum_{l=1}^{\infty}l\sin(2\pi
l\tilde{\beta})\int_{0}^{\infty}dx\,x^{3/2}e^{-(l^{2}L^{2}+2r^{2}-2\tau^{2})x}K_{\nu}(2\tau^{2}x)\mathcal{I}(q,a_{0},2r^{2}x),$
(2.46)

with $x\equiv 1/(4s^{2})$ and

$\displaystyle{\cal{I}}(q,a_{0},w)=\sum_{n=-\infty}^{\infty}I_{q|n+a_{0}|}(w)\
.$ (2.47)

In [31], it has been shown that (2.47) can be written as

$\displaystyle{\cal{I}}(q,a_{0},w)$
$\displaystyle=\frac{e^{w}}{q}-\frac{1}{\pi}\int_{0}^{\infty}dy\frac{e^{-w\cosh
y}f(q,a_{0},y)}{\cosh(qy)-\cos(\pi
q)}+\frac{2}{q}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2k\pi
a_{0})e^{w\cos(2k\pi/q)}\ ,$ (2.48)

where $[q/2]$ represents the integer part of $q/2$, and the prime on the sign
of the summation means that in the case $q=2p$ the term $k=q/2$ should be
taken with the coefficient $1/2$. Obviously, if $q<2$ the summation term must
be omitted. Moreover, the function $f(q,a_{0},y)$ is given by

$\displaystyle f(q,a_{0},y)=\sin[(1-|a_{0}|)\pi
q]\cosh(|a_{0}|qy)+\sin(|a_{0}|\pi q)\cosh[(1-|a_{0}|)qy]\ .$ (2.49)

After the integration over $x$ [48], the final expression for the axial
current density is given by

$\displaystyle\left\langle j^{z}(x)\right\rangle_{c}$ $\displaystyle=$
$\displaystyle\frac{eL}{2(2\pi)^{2}\alpha^{4}}\sum_{l=1}^{\infty}l\sin(2\pi
l\tilde{\beta})\left[\sideset{}{{}^{\prime}}{\sum}_{k=0}^{[q/2]}\cos(2k\pi
a_{0})\,\mathcal{B}_{l}\left(r/\tau,s_{k}\right)\right.$ (2.50)
$\displaystyle\left.\,-\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{f(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{B}_{l}\left(r/\tau,c_{y}\right)\right].$

For the case of a massless scalar field and considering a conformal coupling,
we have $\nu=1/2$. By similar considerations done for the azimuthal current,
we find that the axial current density matches to the result for the Minkowski
spacetime times the conformal factor $(\tau/\alpha)^{4}$ (the sign of the
axial current obtained in [31] should be corrected to the opposite one).

Considering the term $k=0$ of the Eq. (2.50) we have

$\displaystyle\left\langle j^{z}(x)\right\rangle_{c}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{eL}{\pi^{2}(2\alpha)^{4}}\sum_{l=1}^{\infty}l\sin(2\pi
l\tilde{\beta})\,\mathcal{B}_{l}\left(r/\tau,s_{0}\right).$ (2.51)

This expression is the VEV of the axial current density with no dependence on
$q$ and $a_{0}$. It is a purely topological term being dependent only on the
compactification of the $z-$axis and obviously goes to zero in the limit
$L\to\infty$. The remaining in (2.50), magnetic flux and planar angle deficit
contribution, in the axial current density is given by

$\displaystyle\left\langle j^{z}(x)\right\rangle_{c}^{(q,a_{0})}$
$\displaystyle=$
$\displaystyle\frac{eL}{2(2\pi)^{2}\alpha^{4}}\sum_{l=1}^{\infty}l\sin(2\pi
l\tilde{\beta})\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\cos(2k\pi
a_{0})\,\mathcal{B}_{l}\left(r/\tau,s_{k}\right)\right.$ (2.52)
$\displaystyle\left.-\frac{q}{\pi}\int_{0}^{\infty}dy\,\frac{f(q,a_{0},2y)}{\cosh(2qy)-\cos(q\pi)}\,\mathcal{B}_{l}\left(r/\tau,c_{y}\right)\right].$

This expression is finite on the string. Besides that, it is an odd function
of the magnetic flux enclosed by the compactification along the string axis
and an even function of the magnetic flux along the string’s core.

We can note that in the case of $L/\tau\gg 1$ by using (2.38), the axial
current goes to zero, as expected, with the fourth power of $\tau/L$. The
behavior of the axial current density is shown in Fig. 3 as a function of
$r/\tau$ (left plot) and as a function of the compactification length $L/\tau$
(right plot) considering a minimal coulpling. Comparing the left plot with the
result in the fermionic field case studied in [44], it is easy to see that
they have opposite signs. This is not characteristic of the dS spacetime due
to the fact that this also happens in the Minkowski spacetime studied in [31]
and [50]. In both references the sign of the final axial currents should be
corrected to the opposit one. The right plot in Fig. 3 shows the behavior of
$\left\langle j^{z}(x)\right\rangle_{c}$ as a function of the compactification
length $L/\tau$ for several values of $a_{0}$. As one would expect, it
vanishes in the limit $L/\tau\to\infty$, i.e. in the absence of
compactification.

Figure 3: Axial induced current density as a function of $r/\tau$ (left plot)
for three values of $q$ and as a function of $L/\tau$ (right plot) for three
different values of $a_{0}$. In the left graph $L/\tau=0.75$, while in the
right one $r/\tau=0.25$ and $q=1.5$. In both plots $\tilde{\beta}=a_{0}=0.25$,
$m\alpha=1.5$ and $\xi=0$.

##  3 Conclusion

In this paper we have studied the VEVs of the induced bosonic currents in dS
spacetime considering the presence of a compactified cosmic string. These
currents are induced by the presence of magnetic fluxes, one along the
string’s core and another one enclosed by the compactified dimension. In order
to perform our analysis, we constructed the positive frequency Wightman
function associated with a massive scalar field considering that the field
obeys the quasiperiodicity condition (2.4), with a constant arbitrary phase
$\beta$, and that the dimension along the $z$-direction is compactified to a
circle with lenght $L$. The quasiperiodicity condition imposes that the
quantum number related with the compactified string becomes discrete, and in
order to consider the summation over this quantum number, we have employed the
Abel-Plana summation formula (2.13). By using this formula, the Wightman
function and consequently the induced currents, are decomposed into two
contributions: one induced by a cosmic string in dS spacetime with no
compactification, Eq. (2.18), and another one induced by the compactification,
Eq. (2.20). Taking into account these functions, we have calculate the
azimuthal and axial current densities.

The string part of the azimuthal current density was obtained by substituting
the Wightman function (2.18) into (2.22). Its final expression is given by
(2.32) and is an odd function of the magnetic flux along the string’s core. In
addition, the azimuthal current depends on the ratio $r/\tau$, which is the
proper distance from the string measured in units of the dS curvature scale
$\alpha$. Some particular asymptotic cases of (2.32) are considered. For the
region $r/\tau\gg 1$ the leading term of the azimuthal current induced by an
uncompactified string, considering $\nu$ imaginary, is given by (2.34). From
this expression we note that for large distances from the string, the
azimuthal current density besides decaying with the fifth power in $\tau/r$,
has an oscillatory behavior if the parameter $\nu$ is imaginary. The parameter
$\nu$ is related to the mass of the scalar field. If $\nu$ is real, we have
the same equation, but with the coefficient $1/2$. However, depending on the
mass of the bosonic field, the decay behavior for large distances can be very
different. Considering a minimal coulpling, $\xi=0$, as mass tends to zero the
string contribution of the azimuthal current density goes to zero with the
second power of $\tau/r$. This is in contrast with the result for the
fermionic field in the same geometry. In the fermionic field case the current
decays with the fourth power of $\tau/r$ independent of the mass, and while
decaying oscillates. However, in the massless limit the oscillation
disappears. In the case of a conformal coupling and a massless scalar field,
the azimuthal current density goes to zero with the fourth power of $\tau/r$,
the same as for the fermionic field. These results for the bosonic field are
shown in Fig. 1 where we have shown the string contribution of the azimuthal
current density as a function of $r/\tau$ considering just the case of a
minimal coupling.

For the azimuthal current density induced by the compactification, we have
used the Wightman function (2.20) and obtained the Eq. (2.37). This expression
is an odd function of the magnetic flux along the string and an even function
of the magnetic flux enclosed by the string axis. For regions close to the
string, $r/\tau\ll 1$, and considering that the parameter $\nu$ is imaginary,
the leading term of the azimuthal current density induced by the
compactification is given by (2.39). This expression presents a divergence on
the string if $|a_{0}|\leq 1/q$ and is finite on the string if $|a_{0}|>1/q$,
in contrast with the fermionic case where the threshold value is
$|a_{0}|=(1-1/q)/2$. We have also considered the case of $L/\tau\gg 1$, for
$r/\tau$ fixed, where the leading contribution is given by (2.40). In this
regime, the azimuthal current density goes to zero, as one expects, with the
fifth power of $\tau/L$. The behavior of the azimuthal current density induced
by the compactification as functions of $r/\tau$ and the magnetic flux
enclosed by the string axis has been shown in the Fig. 2 for a minimal
coupling, $\xi=0$. The expression for the total azimuthal current density is
presented in (2.42).

Our next step was to consider the axial current density. We have shown that
the only non-zero contribution for the axial current is due to the
compactification. The total axial current density is given by (2.50). We were
able to decompose it into two contributions. The first one is given by (2.51)
which is a purely topological contribution being dependent only on the
compactification along the string axis. The other contribution for the
azimuthal current density, given by (2.52), is induced by the planar angle
deficit, the magnetic flux and also the compactification. This contribution is
an odd function of the magnetic flux enclosed by the compactified string and
an even function of the magnetic flux running along the string’s core. Besides
that, this contribution is finite on the string. We have also considered some
particular cases in the axial current density expression. In Fig. 3 we have
plotted the axial current induced by the compactification as functions of
$r/\tau$ and $L/\tau$ considering a minimal coupling, $\xi=0$, for several
values the parameter $q$ and the magnetic flux along the string’s core. We
have shown that the induced axial current densities considering bosonic and
fermionic fields have opposite signs which is a shared characteristic with the
Minkowski spacetime. Moreover, in the limit where $L/\tau\gg 1$, both the
azimuthal and axial current densities induced by the compactification go to
zero, as one expects. Furthermore, we have shown that for a massless field and
considering conformal coupling, $\xi=1/6$, the expressions for both current
densities match the Minkowski one with the same geometry, found in [31], times
the fourth power of the conformal factor $\tau/\alpha$.

Before finishing this paper we would like to mention that all expressions
obtained for the current densities associated with the uncompactified
contribution present the natural dependence on the distance to the string,
besides an explicit time dependence. In fact these dependencies appear in the
combination $\tau/r$. As to the compactified contribution, an extra dependence
on the compactification length $L$ and conformal time appears in the
combination $\tau/L$. Finally we should say that our results may provide
relevant information about the vacuum bosonic current in the inflationary
phase of the Universe.

## Acknowledgments

E.A.F. B and A.M. would like to thank the Brazilian agency CAPES for the
financial support. A.M. also thanks the Brazilian agency CNPq and Universidade
Federal de Pernambuco Edital Qualis A for financial support.

##  Appendix A Integral representation for the compactified Wightman function

Here, we develop the intermediate steps in the calculation of the compactified
Wightman function (2.20). In order to obtain this function, we substitute
(2.14) into (2.13), and also substitute the result into (2.12). From the
second term on the right hand side of (2.13), the compactified Wightman
function is given by

$\displaystyle W_{c}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2iq(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}\Delta
z}}{(2\pi)^{3}\alpha^{2}}\sum_{n=-\infty}^{\infty}e^{iqn\Delta\phi}\int_{0}^{\infty}dppJ_{q|n+a|}(pr)J_{q|n+a|}(pr^{\prime})$
$\displaystyle\times\int_{0}^{\infty}dk_{z}\left[K_{\nu}(i\tau^{\prime}\sqrt{(ik_{z})^{2}+p^{2}})K_{\nu}(-i\tau\sqrt{(ik_{z})^{2}+p^{2}})\right.$
$\displaystyle\left.-K_{\nu}(i\tau^{\prime}\sqrt{(-ik_{z})^{2}+p^{2}})K_{\nu}(-i\tau\sqrt{(-ik_{z})^{2}+p^{2}})\right]\sum_{\lambda=\pm
1}\frac{e^{-\lambda k_{z}\Delta z}}{e^{Lk_{z}+2\pi
i\lambda{\tilde{\beta}}}-1},$

defining $u\equiv\frac{L}{2\pi}k_{z}$. The integral over $k_{z}$ must be
considered in two different situations as follows

  * •

For $p>k_{z}$:

$\displaystyle\sqrt{(\pm ik_{z})^{2}+p^{2}}=\sqrt{p^{2}-(k_{z})^{2}}\ .$ (A.2)

  * •

For $p<k_{z}$:

$\displaystyle\sqrt{(\pm ik_{z})^{2}+p^{2}}=e^{\pm
i\frac{\pi}{2}}\sqrt{(k_{z})^{2}-p^{2}}\ .$ (A.3)

The integral over the segment $[0,\ p]$ vanishes in contrast with the segment
$[p,\ \infty)$. In fact in the latter interval the integrand reads

$\displaystyle
K_{\nu}(e^{i\pi}\tau^{\prime}\sqrt{(k_{z})^{2}-p^{2}})K_{\nu}(\tau\sqrt{(k_{z})^{2}-p^{2}})-K_{\nu}(\tau^{\prime}\sqrt{(k_{z})^{2}-p^{2}})K_{\nu}(e^{-i\pi}\tau\sqrt{(k_{z})^{2}-p^{2}})\
.$ (A.4)

Now using the the identity [46],

$\displaystyle
K_{\nu}(ze^{im\pi})=e^{-im\nu\pi}K_{\nu}(z)-i\pi\sin(m\nu\pi)\csc(\nu\pi)I_{\nu}(z)\
,$ (A.5)

for $m$ integer, using the well-known relation involving the modified Bessel
functions, $K_{\nu}(z)=\frac{\pi}{2}(I_{-\nu}(z)-I_{\nu}(z))/\sin{\nu\pi}$ as
well as defining a new variable $\gamma^{2}=k_{z}^{2}-p^{2}$, Eq. (A) can be
expressed as

$\displaystyle W_{c}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2q(\tau\tau^{\prime})^{3/2}e^{-ieA_{z}\Delta
z}}{(2\pi\alpha)^{2}}\sum_{n=-\infty}^{\infty}e^{iqn\Delta\phi}\int_{0}^{\infty}dp\,pJ_{q|n+a|}(pr)J_{q|n+a|}(pr^{\prime})$
(A.6)
$\displaystyle\times\int_{0}^{\infty}\frac{d\gamma\,\gamma}{\sqrt{\gamma^{2}+p^{2}}}\left[K_{\nu}(\tau^{\prime}\gamma)I_{-\nu}(\tau\gamma)+I_{\nu}(\tau^{\prime}\gamma)K_{\nu}(\tau\gamma)\right]$
$\displaystyle\times\sum_{l=1}^{\infty}e^{-lL\sqrt{\gamma^{2}+p^{2}}}\cos(2\pi
l{\tilde{\beta}}-i\Delta z\sqrt{\gamma^{2}+p^{2}}),$

where we have also used the expansion
$(e^{u}-1)^{-1}=\sum_{l=1}^{\infty}e^{-lu}$.

## References

  * [1] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge U.K. (1982).
  * [2] A. Linde, Particle physics and inflationary cosmology, Harwood Academic Publishers, Chur Switzerland (1990).
  * [3] A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects.  Cambridge monographs on mathematical physics. Cambridge Univ. Press, Cambridge, 1994.
  * [4] M. Hindmarsh and T. Kibble, Cosmic strings, Rept. Prog. Phys. 58, 477–562 (1995).
  * [5] J. M. Hyde, A. J. Long, and T. Vachaspati, Dark Strings and their Couplings to the Standard Model, Phys. Rev. D 89, 065031 (2014).
  * [6] Planck Collaboration Collaboration, P. Ade et al., Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, (2013).
  * [7] T. Damour and A. Vilenkin, Gravitational Wave Bursts from Cosmic Strings, Phys. Rev. Lett. 85, 3761 (2000).
  * [8] P. Battacharjee and G. Sigl, Origin and propagation of extremely high-energy cosmic rays, Phys. Rep. 327, 109 (2000).
  * [9] V. Berezinski, B. Hnatyk, and A. Vilenkin, Gamma ray bursts from superconducting cosmic strings, Phys. Rev. D 64, 043004 (2001).
  * [10] H. B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61, 45–61 (1973).
  * [11] D. Garfinkle, General Relativistic Strings, Phys. Rev. D 32, 1323–1329 (1985).
  * [12] B. Linet, A Vortex Line Model for Infinite Straight Cosmic Strings, Phys. Lett. A 124, 240–242 (1987).
  * [13] B. Linet, Quantum Field Theory in the Space-time of a Cosmic String, Phys. Rev. D 35, 536–539 (1987).
  * [14] B. Allen and E. P. S. Shellard, On the evolution of cosmic strings, in The formation and evolution of cosmic strings : proceedings of a workshop supported by the SERC and held in Cambridge, 3-7 July, 1989 (G. W. Gibbons, S. W. Hawking, and T. Vachaspati, eds.), (Cambridge), pp. 421–448, Cambridge University Press, 1990.
  * [15] M. Guimarães and B. Linet, Selfinteraction and quantum effects near a point mass in three-dimensional gravitation, Class. Quant. Grav. 10, 1665–1680 (1993).
  * [16] P. Davies and V. Sahni, Quantum gravitational effects near cosmic strings, Class. Quant. Grav. 5, 1 (1988).
  * [17] T. Souradeep and V. Sahni, Quantum effects near a point mass in (2+1)-Dimensional gravity, Phys. Rev. D 46, 1616–1633 (1992).
  * [18] V. P. Frolov and E. Serebryanyi, Vacuum Polarization in the Gravitational Field of a Cosmic String, Phys. Rev. D 35, 3779–3782 (1987).
  * [19] B. Linet, Euclidean spinor Green’s functions in the space-time of a straight cosmic string, J. Math. Phys. 36, 3694–3703 (1995).
  * [20] E. S. Moreira Júnior, Massive quantum fields in a conical background, Nucl. Phys. B 451, 365–378 (1995).
  * [21] V. B. Bezerra and N. R. Khusnutdinov, Vacuum expectation value of the spinor massive field in the cosmic string space-time, Class. Quant. Grav. 23, 3449–3462 (2006).
  * [22] J. Dowker, Vacuum Averages for Arbitrary Spin Around a Cosmic String, Phys. Rev. D 36, 3742 (1987).
  * [23] M. Guimarães and B. Linet, Scalar Green’s functions in an Euclidean space with a conical-type line singularity, Commun. Math. Phys. 165, 297–310 (1994).
  * [24] J. Spinelly and E. R. Bezerra de Mello, Vacuum polarization of a charged massless scalar field on cosmic string space-time in the presence of a magnetic field, Class. Quant. Grav. 20, 873–888 (2003).
  * [25] J. Spinelly and E. R. Bezerra de Mello, Vacuum polarization by a magnetic field in the cosmic string space-time, Int. J. Mod. Phys. A 17, 4375–4384 (2002).
  * [26] J. Spinelly and E. R. Bezerra de Mello, Vacuum polarization of a charged massless fermionic field by a magnetic flux in the cosmic string space-time, Int. J. Mod. Phys. D 13, 607–624 (2004).
  * [27] J. Spinelly and E. R. Bezerra de Mello, Vacuum polarization by a magnetic flux in a cosmic string background, Nucl. Phys. Proc. Suppl. 127, 77–83 (2004).
  * [28] J. Spinelly and E. R. Bezerra de Mello, Spinor Green function in higher-dimensional cosmic string space-time in the presence of magnetic flux, JHEP 0809, 005 (2008).
  * [29] L. Sriramkumar, Fluctuations in the current and energy densities around a magnetic flux carrying cosmic string, Class. Quant. Grav. 18, 1015–1025 (2001).
  * [30] Y. Sitenko and N. Vlasii, Induced vacuum current and magnetic field in the background of a cosmic string, Class. Quant. Grav. 26 (2009).
  * [31] E. A. F. Bragança, H. F. Santana Mota and E. R. Bezerra de Mello, Induced vacuum bosonic current by a magnetic flux in a higher-dimensional compactified cosmic string spacetime, Int. J. Mod. Phys. D 24, 1550055 (2015).
  * [32] E. R. Bezerra de Mello, Induced fermionic current densities by magnetic flux in higher dimensional cosmic string spacetime, Class. Quant. Grav. 27, 095017 (2010).
  * [33] E. R. Bezerra de Mello, V. Bezerra, A. Saharian and V. Bardeghyan, Fermionic current densities induced by magnetic flux in a conical space with a circular boundary, Phys. Rev. D 82, 085033 (2010).
  * [34] E. R. Bezerra de Mello, V. B. Bezerra, A. A. Saharian and H. H. Harutyunyan, Vacuum currents induced by a magnetic flux around a cosmic string with finite core, Phys. Rev. D 91, 064034 (2015).
  * [35] M. S. Maior de Sousa, R. F. Ribeiro and E. R. Bezerra de Mello, Induced fermionic current by a magnetic tube in the cosmic string spacetime, Phys. Rev. D 93, 043545 (2016).
  * [36] M. S. Maior de Sousa, R. F. Ribeiro and E. R. Bezerra de Mello, Fermionic vacuum polarization by an Abelian magnetic tube in the cosmic string spacetime, Phys. Rev. D 95, 045005 (2017).
  * [37] P. C. W. Davies and V. Sahni, Quantum gravitational effects near cosmic strings, Class. Quantum Grav. 5, 1 (1988).
  * [38] A. C. Ottewill and P. Taylor, Vacuum polarization on the Schwarzschild metric threaded by a cosmic string, Phys. Rev. D 82, 104013 (2010).
  * [39] A. C. Ottewill and P. Taylor, Renormalized vacuum polarization and stress tensor on the horizon of a Schwarzschild black hole threaded by a cosmic string, Class. Quantum Grav. 28, 015007 (2011).
  * [40] E. R. Bezerra de Mello and A. A. Saharian, Vacuum polarization by a cosmic string in de Sitter spacetime, JHEP 04, 046 (2009).
  * [41] E. R. Bezerra de Mello and A. A. Saharian, Fermionic vacuum polarization by a cosmic string in de Sitter spacetime, JHEP 08, 038 (2010).
  * [42] E. R. Bezerra de Mello and A. A. Saharian, Vacuum polarization induced by a cosmic string in anti-de Sitter spacetime, J. Phys. A: Math. Theor. 45 115402 (2012).
  * [43] E. R. Bezerra de Mello and A. A. Saharian, Fermionic vacuum polarization by a cosmic string in anti-de Sitter spacetime, Class. Quantum. Grav. 30, 175001 (2013).
  * [44] A. Mohammadi, E. R. Bezerra de Mello and A. A. Saharian, Induced fermionic currents in de Sitter spacetime in the presence of a compactified cosmic string, Class. Quantum Grav. 32, 135002 (2015).
  * [45] E. A. F. Bragança, E. R. Bezerra de Mello and A. Mohammadi, Induced fermionic vacuum polarization in dS spacetime with a compactified cosmic string, Phys. Rev. D 101, 045019 (2020).
  * [46] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  * [47] S. Bellucci, A. Saharian, and V. Bardeghyan, Induced fermionic current in toroidally compactified spacetimes with applications to cylindrical and toroidal nanotubes, Phys. Rev. D 82, 065011 (2010).
  * [48] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products.  Academic Press, 1980.
  * [49] S. Bellucci, E. R. Bezerra de Mello and A. A. Saharian, Finite temperature fermionic condensate and currents in topologically nontrivial spaces, Phys. Rev. D 89, 085002 (2014).
  * [50] E. R. Bezerra de Mello and A. A. Saharian, Fermionic current induced by magnetic flux in compactified cosmic string spacetime, Eur. Phys. J. C 73, 2532 (2013).


MITP/21-058, LTH 1283

#  Explaining excesses in four-leptons at the LHC with a double peak from a CP
violating Two Higgs Doublet Model

Stefan Antusch<EMAIL_ADDRESS>Department of Physics, University of
Basel,
Klingelbergstr. 82, CH-4056 Basel, Switzerland
Oliver Fischer<EMAIL_ADDRESS>Department of Mathematical
Sciences, University of Liverpool
Liverpool, L69 7ZL, UK
A. Hammad<EMAIL_ADDRESS>Centre for Theoretical Physics,
the British University in Egypt, P.O. Box 43, Cairo 11837, Egypt
Christiane Scherb<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence $\&$
Mainz Institute for Theoretical Physics,
Johannes Gutenberg University, 55099 Mainz, Germany

###### Abstract

Extended scalar sectors with additional degrees of freedom appear in many
scenarios beyond the Standard Model. Heavy scalar resonances that interact
with the neutral current could be discovered via broad resonances in the tails
of the four-lepton invariant mass spectrum, where the Standard Model
background is small and well understood. In this article we consider a recent
ATLAS measurement of four-lepton final states, where the data is in excess
over the background for invariant masses above 500 GeV. We discuss the
possibility that this excess could be interpreted as a “double peak” from the
two extra heavy neutral scalars of a CP violating Two Higgs Doublet Model,
both coupling to the $Z$ boson. We apply an iterative fitting procedure to
find viable model parameters that can match the excess, resulting in a
benchmark point where the observed four-lepton invariant mass spectrum can be
explained by two scalar particles $H_{2}$ and $H_{3}$, with masses of 540 GeV
and 631 GeV, respectively, being admixtures of the CP eigenstates. Our
explanation predicts additional production processes for $t\bar{t}$,
$W^{+}W^{-}$, $4b$ and $\gamma\gamma$, some of which have cross sections close
to the current experimental limits. Our results further imply that the
electric dipole moment of the electron should be close to the present bounds.

##  1 Introduction

The discovery of the scalar resonance with a mass of about 125 GeV, compatible
with the predicted Higgs boson [1], completes the Standard Model (SM). While
currently most searches for physics beyond the SM do not point at new physics
beyond the SM, excesses in final states with leptons [2, 3] and di-photons [4,
5, 6] indicate the possible existence of additional scalar degrees of freedom.
The four-lepton final states are often referred to as the ‘golden channel’
when searching for heavy scalar resonances, due to small and controllable SM
backgrounds. The four-lepton analyses from ATLAS [7] and CMS [8, 9] show
enhanced event rates in final states with high invariant mass. These analyses
were combined in ref. [10] to demonstrate the compatibility of the data with a
broad resonance structure around 700 GeV. This is compatible with a very
recent interpretation of ATLAS data as a second Higgs excitation at 680 GeV
[11].

Searches for heavy scalars above 600 GeV in $\gamma\gamma$, $Z\gamma$, $ZZ\to
4\ell$, top quarks, pairs of Higgs and $W$ bosons were reviewed and discussed
in ref. [12], supporting the claim made in ref. [10] of a resonance in $ZZ$
around 700 GeV. This possible resonant enhancement is visible both in gluon
and vector boson fusion channels as reported by ATLAS in ref. [7] The more
recent analysis of four-lepton final states by the ATLAS collaboration also
shows an enhancement of event rates with invariant masses above about 500 GeV
[13]. This observation was considered in ref. [14] to corroborate the possible
resonance around 700 GeV. The theoretical framework was the Georgi-Machacek
model and a cross section for the heavy resonance was found to be $\sim 160$
fb. It is interesting to note that a broader enhancement of the
$b\bar{b}b\bar{b}$ final state with invariant mass above 500 GeV is visible,
which might be compatible with the observed $4\ell$ excess from ref. [15]. On
the other hand, a recent analysis searching for heavy diboson resonances in
semi-leptonic final states is compatible with the SM prediction [16], however
here the backgrounds are at a much higher level and might cover up a possible
enhancement.

In this article we consider an explanation of the four-lepton excess at
invariant masses above 500 GeV by a “double peak” from the two extra heavy
neutral scalars of a CP violating Two Higgs Doublet Model (THDM), where the
latter extends the scalar sector of the SM by an additional scalar $SU(2)_{L}$
doublet field [17]. Due to CP violation, both extra heavy neutral scalars can
couple to the $Z$ boson, thus showing up in the four-lepton invariant mass
spectrum. Recently, we studied a class of THDMs with CP violation in ref.
[18], exploring the testability of CP violation at the LHC and evaluating the
current constraints on the model parameters, pointing out the importance of
the the four-lepton final state as “discovery channel”. We will make use of
the results of [18] to find a viable set of model parameters that can match
the current four-lepton excess. The article is structured as follows: in
section 2 we briefly review the model framework, in section 3 we describe our
analysis and discuss the results, and in section 4 we conclude.

##  2 The model

The THDM was first discussed in ref. [19] to discuss the phenomenon of CP
violation in the scalar sector. For a comprehensive review we refer the reader
e.g. to ref. [20]. In THDMs, the scalar sector contains two
$SU(2)_{L}$-doublet fields, $\phi_{1}$ and $\phi_{2}$, with identical quantum
numbers under the SM gauge symmetry group:

$\phi_{1}=\begin{pmatrix}\eta_{1}^{+}\\\ (v_{1}+h_{1}+ih_{3})/\sqrt{2}\\\
\end{pmatrix}\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\phi_{2}=\begin{pmatrix}\eta_{2}^{+}\\\
(v_{2}+h_{2}+ih_{4})/\sqrt{2}\\\ \end{pmatrix}\,.$ (1)

The components $h_{i},\,i=1,...,4$, are real neutral fields,
$\eta_{i}^{+},\,i=1,2$ are complex charged fields, and $v_{i},\,i=1,2$ are the
vacuum expectation values (vevs). The most general Lagrangian density for the
model can be decomposed as

$\mathcal{L}_{\text{THDM}}=\mathcal{L}_{\text{SM},\text{kin}}+\mathcal{L}_{\phi,\text{kin}}+V_{\phi}+Y_{\phi}\,,$
(2)

where $\mathcal{L}_{\text{SM},\text{kin}}$ denotes the kinetic terms for SM
gauge fields and fermions, $\mathcal{L}_{\phi,\text{kin}}$ denotes the kinetic
terms for the two scalar fields $\phi_{i},\,i=1,2$, $V_{\phi}$ denotes the
scalar potential, and $Y_{\phi}$ contains the Yukawa terms that give rise to
the couplings between the SM fermions and the scalar fields.

Since scalar decays into four-lepton final states (henceforth referred to as
$4\ell$) come about from scalar decays into two $Z$ bosons that in turn decay
into leptons, we are interested in the interaction of the scalar fields with
the neutral current. There are only two interaction eigenstates that can decay
into two $Z$ bosons. However, the most general (CP violating) form of the THDM
leads to scalar mixing, and in the presence of CP violation all the three
neutral mass eigenstates $H_{i}$ of the THDM can mediate the process $pp\to
H_{i}\to ZZ\to 4\ell$, with $i=1,2,3$.

The final state from the process $pp\to H_{i}\to ZZ\to 4\ell$, with
$4\ell=\ell_{\alpha}^{+}\ell_{\alpha}^{-}\ell_{\beta}^{+}\ell_{\beta}^{-}$
(and where the considered lepton flavors are
$\ell_{\alpha,\beta}^{\pm}=e^{\pm},\,\mu^{\pm}$) features an invariant mass
that reflects the mass of the mediating $H_{i}$. The mass eigenstate $H_{1}$
corresponds to the SM-like Higgs boson with $m_{H_{1}}\simeq 125$ GeV, whose
four-lepton signal has been studied (see examples for ATLAS [21] and CMS [22]
analyses) and $H_{2}$ and $H_{3}$ are assumed to be heavier. As discussed
above, in a CP violating THDM the extra neutral scalars $H_{2}$ and $H_{3}$
give rise to two additional broad peaks in the $4\ell$ invariant mass spectrum
- or to a broad “double peak” if the masses are not too separated. This is a
feature that is usually not considered when fitting the THDM to the data.

##  3 Analysis and results

We consider the measurements of differential cross-sections in $4\ell$ events
in the 139 fb-1 data set by the ATLAS collaboration [13]. From the results in
the ATLAS publication we use the $4\ell$ differential cross sections, and in
particular the invariant mass spectrum ($M_{4\ell}$). We digitise the observed
event rates, their errors, and the theory prediction.

We use the nine bins from $M_{4\ell}$ between 500 GeV and 900 GeV, six of
which show event counts in excess of the theory prediction. We create a sample
of excess events by subtracting the theory prediction from the observed event
rates (which means that “excess events” per bin can also be negative).

###  3.1 Numerical setup

As an explicit example we consider the same class of THDMs with CP violation
and type I Yukawa structure as in ref. [18], to which we refer the reader for
model details and notation. We calculate testable properties of the THDM via
numerical tools that include the following recent constraints:111We use the
same limits as detailed in sec. 3 of ref. [18]. $B$ physics data using
FlavorKit [23], Higgs data using HiggsBounds [24] and HiggsSignals [25], and
electric dipole moments with the formulae from refs. [29, 30].

The process $pp\to H_{i}\to 4\ell$ for the heavy scalars with indices $i=2,3$
is calculated in MadGraph [31], including the effective gluon-Higgs vertex via
SPheno [32, 33] and QCD corrections [34]. We note at this point that the
inclusive $4\ell$ invariant mass spectrum is simulated to include the
interference between the scalars. A fast detector simulation is done with 500k
events per sample with Delphes [35] using the standard ATLAS detector card.
From the reconstructed events we read out the invariant mass spectra.

###  3.2 First approximation

We searched for points with masses $m_{H_{2}}$ and $m_{H_{3}}$ around 500 GeV
and 700 GeV, respectively, and with total cross sections for the $4\ell$ final
state that are of similar magnitude. We found the benchmark point $P_{1}$ with
$m_{H_{2}}^{P_{1}}=535$ GeV and $m_{H_{3}}^{P_{1}}=703$ GeV and total $4\ell$
cross sections $\sigma_{H_{2}\to 4\ell}^{P_{1}}=1.3$ fb and $\sigma_{H_{3}\to
4\ell}^{P_{1}}=0.86$ fb. For $H_{2}$ and $H_{3}$ the signal selection
efficiency based on the experimental selection criteria is found to be
$\epsilon_{4\ell}\sim 0.3$ and not too dependent on the scalar mass. We
simulated exclusively $pp\to H_{2}\to 4\ell$ and $pp\to H_{3}\to 4\ell$ from
which we obtained the invariant mass spectra $\rho_{2}$ and $\rho_{3}$,
respectively. Notice that at this step, the interference between the scalars
is not taken into account.

Next we performed a simple $\chi^{2}$ analysis. We varied the signal peaks of
the two spectra with the two parameters $\delta m_{j}$, such that the masses
are given by $m_{H_{j}}=m_{H_{j}}^{P_{1}}+\delta m_{j}$. We also introduce the
signal strength multipliers $s_{j}$ which are multiplied with the invariant
mass spectra $\rho_{H_{j}}$. We construct the $\chi^{2}$:

$\chi^{2}_{\rm sig}(\delta m_{2},\delta
m_{3},s_{2},s_{3})=\sum_{i}\frac{(b_{sig,i}(\delta m_{2},\delta
m_{3},s_{2},s_{3})-b_{i})^{2}}{\delta_{obs,i}^{2}+\delta_{sys,i}^{2}}$ (3)

where $i$ is the bin number, $b_{i}$ is the measured event rate
$\delta_{obs,i}=\sqrt{b_{i}}$, $\delta_{syst,i}=10\%$. The signal rate for bin
$i$ is

$b_{sig,i}(\delta m_{2},\delta
m_{3},s_{2},s_{3})=b_{SM,i}+\int_{i}(s_{2}\cdot\rho_{H_{2}}(E-\delta
m_{2})+s_{3}\cdot\rho_{H_{3}}(E-\delta m_{3}))dE\,,$ (4)

with $b_{SM,i}$ being the SM prediction, $\rho_{H_{j}}$ the signal
distributions, and the parameters $\delta m_{j}$ and $s_{j}$ are varied to
minimise the $\chi^{2}$.

###  3.3 Iterative analysis

Notice that the signal rate as described in (4) distorts the invariant mass
spectrum and thus disconnects it from the underlying benchmark point. However,
the distorted spectrum can be used to locate the masses and event rates that
are preferred by the fit to the data.

Consequently we use the best-fit values for the masses and the total $4\ell$
cross sections (converted from the fiducial cross sections using the signal
selection efficiency and the integrated luminosity) as selection criteria to
find a new benchmark point. From a fine grained scan in the model parameters
we select a point $P_{n}$ that has masses $m_{H_{j}}$ and cross sections
$\sigma_{H_{j}\to 4\ell}$ that are as close to the best-fit results for the
masses and event rates of the previous point $P_{n-1}$ as possible.

For each new point $P_{n}$ we create an inclusive $4\ell$ invariant mass
spectrum $\rho_{incl}^{P_{n}}$ with two peaks around $m_{H_{2}}$ and
$m_{H_{3}}$, including the interference between $H_{2}$ and $H_{3}$. We remark
that the interference with $H_{1}$ is negligible for the here relevant mass
scales of $H_{2}$ and $H_{3}$, which we verified by computation. We separate
the spectrum into $\rho_{H_{2}}^{P_{n}}$ and $\rho_{H_{3}}^{P_{n}}$ at the
minimum between the two peaks and fit the parameters $\delta m_{j}$ and
$s_{j}$ via the two partial spectra as in eq. (3) to the data.

Once we have a benchmark point with a spectrum that provides a good fit, we
carry out a Bayesian fit of the parameters $\delta m_{j},\,j=2,3$ and
$s_{j},\,j=2,3$ to establish the Bayesian confidence limits on the parameters.

###  3.4 Results

Our analysis converged sufficiently after six iterations. We perform an even
more fine-grained parameter space scan and select parameter space points with
$m_{H_{2}},m_{H_{3}}$, $\sigma_{H_{2}\to 4\ell}$ and $\sigma_{H_{3}\to 4\ell}$
within the 90% Bayesian confidence interval around the last iteration’s best
fit parameters, which are:

$\begin{array}[]{cc}521.1\leavevmode\nobreak\ \text{GeV}\leq m_{H_{2}}\leq
562.9\leavevmode\nobreak\ \text{GeV},&\qquad 0.3\leavevmode\nobreak\
\text{fb}\leq\sigma_{H_{2}\to 4\ell}\leq 0.6\leavevmode\nobreak\ \text{fb},\\\
602.2\leavevmode\nobreak\ \text{GeV}\leq m_{H_{3}}\leq
655.9\leavevmode\nobreak\ \text{GeV},&\qquad 0.3\leavevmode\nobreak\
\text{fb}\leq\sigma_{H_{3}\to 4\ell}\leq 0.6\leavevmode\nobreak\
\text{fb}.\end{array}$ (5)

In this scan we find the benchmark point $P_{7}$ that is defined by the
following model parameters: $\tan\beta=20$, $\eta=0.659$, $\lambda_{1}=0.7$,
$\lambda_{2}=0.099$, $\lambda_{3}=4.97$, $\lambda_{4}=6.45$ and
$\Re(\lambda_{5})=1.792$. These parameters give rise to
$m_{H_{2}}^{P_{7}}=540$ GeV, $m_{H_{3}}^{P_{7}}=631$ GeV, and
$\sigma_{H_{2}\to 4\ell}^{P_{7}}=0.45$ fb, $\sigma_{H_{3}\to
4\ell}^{P_{7}}=0.42$ fb. We call this point the “best-fit benchmark point” as
is provides a very good fit to the spectrum with a $\chi^{2}=5.76$, which is
to be compared to the SM value for all bins above 500 GeV. For the SM we get a
$\chi^{2}_{SM}=21.0\leavevmode\nobreak\ (16.9)$ corresponding to an upward
fluctuation with a p-value of 0.007 (0.03) considering statistical errors only
(all errors). The contribution of the two neutral scalar particles to the
four-lepton invariant mass spectrum for the “best fit benchmark point” $P_{7}$
is shown in fig. 1. The striking feature of the spectrum is the wide range of
$M_{4\ell}$ that receives contributions from the two heavy scalars.

Figure 1: Invariant mass spectrum of the best-fit points for the “best-fit
benchmark point” $P_{7}$, for details, see text. The black dots with error
bars denote the difference between observed and predicted data from the four-
lepton invariant mass spectrum in ref. [13].

In general, scalars that mix with the Higgs doublet can also decay into other
SM particles than $Z$ bosons. Thus, after finding a good benchmark point to
match the $4\ell$ invariant mass spectrum as reported by ATLAS, we explore the
possibility of making quantitative predictions for the $H_{i}$ decays into
$t\bar{t}$, $W^{+}W^{-}$ and $\gamma\gamma$. Therefore, we show in the four
panels of fig. 2 the projections of the total cross sections for the $\sim
2000$ parameter space points from the very fine grained scan over THDM model
parameters. The figure shows the $4\ell$ final state of $H_{2}$ and $H_{3}$
into $ZZ$ and the color code of the points denotes the total production cross
sections for $W^{+}W^{-}$ (upper panels), $t\bar{t}$ (middle pannels) and
$\gamma\gamma$ (lower panels).

Top: $WW$

Middle: $t\bar{t}$

Bottom: $\gamma\gamma$

Figure 2: Results from a very fine-grained parameter space scan within the
class of THDMs with CP violation and type I Yukawa structure (cf. [18]). Shown
points feature masses $m_{H_{i}}$ and $\sigma_{pp\to H_{i}\to 4\ell}$, $i=2,3$
within the 90% Bayesian confidence limit of the best-fit point. The left and
right panels are for $H_{2}$ and $H_{3}$, respectively, showing total cross
sections for the $W^{+}W^{-}$ (top), $t\bar{t}$ (middle) and $\gamma\gamma$
(bottom) final states.

###  3.5 Discussion

Our “best-fit benchmark point” $P_{7}$ provides an excellent fit to the
excesses of events in the $4\ell$ spectrum with invariant masses between 500
GeV and 700 GeV observed by the ATLAS collaboration. The fit prefers $H_{2}$
and $H_{3}$ with similar masses and production cross sections for the $4\ell$
final state. Combined with the facts that (i) the THDM adds one CP-even and
one CP-odd scalar to the SM field content and (ii) that the CP-odd field does
not decay into $ZZ$ this implies that the mass eigenstates must be strongly
CP-mixed. The CP properties of the two scalars could be tested at the HL-LHC
via correlations in final states with two tau leptons from the processes
$pp\to H_{i}\to\tau^{+}\tau^{-}$ as discussed in ref. [18].

A comment on our statistical analysis is in order at this point. The central
aim of this article is not to establish statistical evidence for the existence
of a THDM signal in the $M_{4\ell}$ spectrum. We rather want to point out that
if the here discussed signal were to become statistically significant the THDM
with CP violation is a suitable model to explain it. We thus decided against
combining the data sets from ATLAS with a similar analysis by the CMS
collaboration [36], which also observes excesses in $4\ell$ final states at
high invariant mass. Since the CMS analysis uses a smaller data set, including
this result would at best increase the statistical significance of our result
very slightly.

As mentioned above, our analysis allows us to make quantitative predictions
for decays of $H_{j},\,j=2,3$ into $t\bar{t}$. Our “best-fit benchmark point”
has cross sections $\sigma_{H_{2}\to t\bar{t}}=22$ fb and $\sigma_{H_{3}\to
t\bar{t}}=13$ fb. This is smaller than the current uncertainty of the recent
measurements of the total production cross section $\sigma_{t\bar{t}}=830\pm
36{\rm(stat)}\pm 14{\rm(syst)}$ pb by ATLAS [26] and $791\pm 25$ pb by CMS
[27].

Let us now confront our “best-fit benchmark point” with the fact that no
enhancement of semi-leptonic final states from $WW$ or $ZZ$ decays has been
reported in ref. [16]. For our “best-fit benchmark point”, the sum of $WW$ and
$ZZ$ cross sections yields:

$\sum_{i,V}\sigma_{pp\to H_{i}\to VV}=392\leavevmode\nobreak\ \rm{fb}\,.$ (6)

This cross section is comparable with the $2\sigma$ upper limits on the
production cross section from ref. [16], which are 470 fb for a mass of $500$
GeV (for the example of a scalar radion).

The small apparent enhancement of the $b\bar{b}b\bar{b}=4b$ final state for
invariant masses above 500 GeV as observed in ref. [15] could be another
indication for the process $pp\to H_{i}\to ZZ$, $i=2,3$. For the corresponding
cross section one would expect that

$\sigma_{pp\to H_{i}\to 4b}\geq\left(\frac{{\rm Br}(Z\to b\bar{b})}{{\rm
Br}(Z\to\ell^{+}\ell^{-})}\right)^{2}\epsilon_{b}^{4}\frac{\epsilon_{4b}}{\epsilon_{4\ell}}$
(7)

with the $b$-tagging efficiency $\epsilon_{b}\simeq 0.7$, and the selection
efficiencies $\epsilon_{4b}\sim 0.1$, and where additional $4b$ production
could come from $pp\to H_{i}\to 2H_{1}\to 4b$. Eq. (7) results in a lower
limit of 16.5 additional events in the $4b$ final state, which matches quite
well the observed $\sim 20$ events in excess of the background for $M_{4b}\geq
500$ GeV.

Next we comment on the $H_{i}\to\gamma\gamma$ channel, for which our “best-fit
benchmark point” has cross sections $\sigma_{H_{j}\to\gamma\gamma}=1.2$ fb and
$1.05$ fb for $j=2,3$, respectively. The ATLAS search for resonances in
diphoton final states places at $2\sigma$ upper limits on the production cross
section of 1.15 fb and 0.83 fb for resonances with masses corresponding to 540
GeV and 631 GeV, respectively [28]. This implies that our “best-fit benchmark
point” is slightly in tension with these limits, but may still be regarded as
compatible. We also note that in the current data there indeed exist some
upward fluctuations of the observed event counts at 540 GeV and at 680 GeV. In
any case, future analyses of the diphoton spectrum with more data should be
able to test this prediction of our “best-fit benchmark point”.

Last but not least we consider the impact that CP violation in the scalar
sector has for low-energy observables. As mentioned above, large CP-violating
phases are an implication of $H_{2}$ and $H_{3}$ having similar signal
strengths in the $4\ell$ final state. Large CP phases imply that the THDM
fields give rise to the Electric Dipole Moments (EDM) of SM particles, in
particular for the electron, cf. e.g. refs. [29, 30, 37]. In our fine-grained
parameter space scan we reject EDM for the electrons that are above the
current exclusion limit of $|d_{e}|<1.1\times 10^{-29}$ ecm [38], and we find
that the majority ($\gtrsim 90\%$) of all points has
$10^{-30}\leq\frac{|d_{e}|}{\rm ecm}\leq 1.09\cdot 10^{-29}$.

##  4 Conclusions

In this paper we have considered a “double peak” from a CP violating Two Higgs
Doublet Model as explanation for the local excess in four-lepton events with
invariant masses above 500 GeV as observed by ATLAS and CMS. Within a class of
THDMs, we used an iterative fitting procedure to search for model parameters
that give rise to heavy Higgs masses and signal strengths towards explaining
the excess.

The “best-fit benchmark point” (called P7) we found this way provides an
excellent explanation for the data, with a $\chi^{2}=5.76$ for 8 bins,
predicting events in excess of the SM in the range from 500 GeV to around 700
GeV. It prefers a broad “double peak” in the invariant mass spectrum, with two
resonances at 540 GeV and 631 GeV, respectively.

Interpreted in the context of a THDM, this would be an indication for CP
violation in the scalar sector. The CP mixing is required to be close to
maximal due to the comparable production cross section for the two processes
$pp\to H_{2}\to 4\ell$ and $pp\to H_{3}\to 4\ell$. The CP mixing of $H_{2}$
and $H_{3}$ could in principle be tested at the HL-LHC in the future, e.g. via
the di-tau final state (cf. [18]).

Our “best-fit benchmark point” predicts additional $t\bar{t}$, $VV$ and
$\gamma\gamma$ production channels with cross sections (summed over $H_{2}$
and $H_{3}$) of about 35 fb for $t\bar{t}$, 199 fb for $WW$ and 2.2 fb for
$\gamma\gamma$. Our results are compatible with present limits, and may be
responsible for minor excesses in the $4b$ and $\gamma\gamma$ channels.
Moreover, the parameter space that leads to an explanation of the observed
$4\ell$ spectrum gives rise to electron EDMs that are close to the current
experimental bounds, providing an example for a complementary way to test the
scenario by low energy experiments.

### Acknowledgements

This work was supported by the Swiss National Science Foundation. A. Hammad is
supported from the Basic Science Research Program through the National
Research Foundation of Korea Research Grant No. NRF-2021R1A2C4002551. CS was
supported by the Cluster of Excellence Precision Physics, Fundamental
Interactions, and Structure of Matter (PRISMA+ EXC 2118/1) funded by the
German Research Foundation (DFG) within the German Excellence Strategy
(Project ID 39083149), and by grant 05H18UMCA1 of the German Federal Ministry
for Education and Research (BMBF).

## References

  * [1] G. Aad et al. [ATLAS and CMS], JHEP 08 (2016), 045 [arXiv:1606.02266 [hep-ex]].
  * [2] W. Abdallah, S. Khalil and S. Moretti, Phys. Rev. D 91 (2015) no.1, 014001 [arXiv:1409.7837 [hep-ph]].
  * [3] S. Buddenbrock, A. S. Cornell, Y. Fang, A. Fadol Mohammed, M. Kumar, B. Mellado and K. G. Tomiwa, JHEP 10 (2019), 157 [arXiv:1901.05300 [hep-ph]].
  * [4] P. J. Fox and N. Weiner, JHEP 08 (2018), 025 [arXiv:1710.07649 [hep-ph]].
  * [5] U. Haisch and A. Malinauskas, JHEP 03 (2018), 135 [arXiv:1712.06599 [hep-ph]].
  * [6] T. Biekötter, M. Chakraborti and S. Heinemeyer, [arXiv:2003.05422 [hep-ph]].
  * [7] M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78 (2018) no.4, 293 [arXiv:1712.06386 [hep-ex]].
  * [8] [CMS], CMS-PAS-HIG-18-001.
  * [9] [CMS], CMS-PAS-HIG-16-041.
  * [10] P. Cea, Mod. Phys. Lett. A 34 (2019) no.18, 1950137 [arXiv:1806.04529 [hep-ph]].
  * [11] L. Cosmai and M. Consoli, [arXiv:2111.08962 [hep-ph]].
  * [12] F. Richard, [arXiv:2001.04770 [hep-ex]].
  * [13] G. Aad et al. [ATLAS], JHEP 07 (2021), 005 [arXiv:2103.01918 [hep-ex]].
  * [14] F. Richard, [arXiv:2103.12639 [hep-ph]].
  * [15] G. Aad et al. [ATLAS], JHEP 07 (2020), 108 [erratum: JHEP 01 (2021), 145; erratum: JHEP 05 (2021), 207] [arXiv:2001.05178 [hep-ex]].
  * [16] G. Aad et al. [ATLAS], Eur. Phys. J. C 80 (2020) no.12, 1165 [arXiv:2004.14636 [hep-ex]].
  * [17] H. E. Haber, G. L. Kane and T. Sterling, Nucl. Phys. B 161 (1979), 493-532
  * [18] S. Antusch, O. Fischer, A. Hammad and C. Scherb, JHEP 03 (2021), 200 [arXiv:2011.10388 [hep-ph]].
  * [19] T. D. Lee, Phys. Rev. D 8 (1973), 1226-1239
  * [20] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012), 1-102 doi:10.1016/j.physrep.2012.02.002 [arXiv:1106.0034 [hep-ph]].
  * [21] G. Aad et al. [ATLAS], Eur. Phys. J. C 80 (2020) no.10, 957 [erratum: Eur. Phys. J. C 81 (2021) no.1, 29; erratum: Eur. Phys. J. C 81 (2021) no.5, 398] [arXiv:2004.03447 [hep-ex]].
  * [22] A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 81 (2021) no.6, 488 [arXiv:2103.04956 [hep-ex]].
  * [23] W. Porod, F. Staub and A. Vicente, Eur. Phys. J. C 74 (2014) no.8, 2992 [arXiv:1405.1434 [hep-ph]].
  * [24] P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein and K. E. Williams, Eur. Phys. J. C 74 (2014) no.3, 2693 [arXiv:1311.0055 [hep-ph]].
  * [25] P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak and G. Weiglein, Eur. Phys. J. C 74 (2014) no.2, 2711 [arXiv:1305.1933 [hep-ph]].
  * [26] G. Aad et al. [ATLAS], Phys. Lett. B 810 (2020), 135797 [arXiv:2006.13076 [hep-ex]].
  * [27] A. Tumasyan et al. [CMS], [arXiv:2108.02803 [hep-ex]].
  * [28] G. Aad et al. [ATLAS], Phys. Lett. B 822 (2021), 136651 [arXiv:2102.13405 [hep-ex]].
  * [29] T. Abe, J. Hisano, T. Kitahara and K. Tobioka, JHEP 01 (2014), 106 [erratum: JHEP 04 (2016), 161] [arXiv:1311.4704 [hep-ph]].
  * [30] E. J. Chun, J. Kim and T. Mondal, JHEP 12 (2019), 068 [arXiv:1906.00612 [hep-ph]].
  * [31] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli and M. Zaro, JHEP 07 (2014), 079 [arXiv:1405.0301 [hep-ph]].
  * [32] W. Porod, Comput. Phys. Commun. 153 (2003), 275-315 [arXiv:hep-ph/0301101 [hep-ph]].
  * [33] W. Porod and F. Staub, Comput. Phys. Commun. 183 (2012), 2458-2469 [arXiv:1104.1573 [hep-ph]].
  * [34] A. Belyaev, N. D. Christensen and A. Pukhov, Comput. Phys. Commun. 184 (2013), 1729-1769 [arXiv:1207.6082 [hep-ph]].
  * [35] J. de Favereau et al. [DELPHES 3], JHEP 02 (2014), 057 [arXiv:1307.6346 [hep-ex]].
  * [36] A. M. Sirunyan et al. [CMS], JHEP 06 (2018), 127 [erratum: JHEP 03 (2019), 128] [arXiv:1804.01939 [hep-ex]].
  * [37] W. Altmannshofer, S. Gori, N. Hamer and H. H. Patel, Phys. Rev. D 102 (2020), 115042 [arXiv:2009.01258 [hep-ph]].
  * [38] V. Andreev et al. [ACME], Nature 562 (2018) no.7727, 355-360