# A perturbative production of massive Z bosons and fermion-antifermion pairs
from the vacuum in the de Sitter Universe

Diana Dumitrele, Mihaela-Andreea Băloi and Cosmin Crucean

Faculty of Physics, West University of Timişoara,
V. Parvan Avenue 4 RO-300223 Timişoara, Romania<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we study the problem of neutral electro-weak interactions in a
de Sitter geometry. We develop the formalism of reduction for the Proca field
with the help of the solutions for the interacting fields and by using
perturbative methods we obtain the definition of the transition amplitudes in
the first order of perturbation theory. As an application to our formalism we
study the generation of massive fermions and Z bosons from vacuum in the
expanding de Sitter universe. Our results are the generalization to the curved
geometry of the Weinberg-Salam electro-weak theory for the case of Z boson
interaction with leptons. The probability is found to be a quantity that
depends on the Hubble parameter and we prove that such perturbative processes
are possible only for large expansion regime of the early Universe. The total
probability and rate of transition are obtained for the case of large
expansion and we use the dimensional regularization for extract finite results
from the momenta integrals. In the Minkowski limit we obtain that the
probability of particle generation from vacuum is vanishing recovering the
well known result that forbids particle production in flat space-time due to
the momentum-energy conservation.

##  1 Introduction

In Minkowski field theory the massive vector fields are described by the Proca
equation [2], which represents one of the fundamental field equations
alongside the Dirac equation and Maxwell equation. The Proca equation in
curved geometries was studied in [3, 25], where the problem of fundamental
solutions and propagators was approached. It is a well established fact that
the massive neutral boson $Z$ and massive charged $W^{\pm}$ bosons were
generated in the early universe [4], but the mechanisms that were involved are
a matter of further investigations. Among these mechanisms one is the
perturbative approach which studies the particle generation for fields
interactions [41, 42, 43, 44, 45, 46, 47, 48, 52, 49, 50, 53, 54, 55]. This is
because the computations related to this phenomenon need to consider quantum
fields in the presence of large expansion conditions of the early universe
[37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 52, 49, 50, 53, 54, 55]. This can
be properly done by using perturbative methods suitable for studying processes
that generate particle production in the presence of strong gravitational
fields [18, 19, 41, 51] because the translational invariance with respect to
time is lost in a non-stationary background. The first steps for considering
the above methods were done in [3, 41, 51], by constructing the theory of free
fields and fields interactions in a de Sitter geometry, and adopting the
scenario of the fields minimally coupled with gravity [41, 51]. This can be
done by considering the Weinberg-Salam theory [2, 4, 16, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15] of electro-weak interactions in de Sitter geometry and
extending the definitions of the transition amplitudes from Minkowski theory
to the curved space-time. In this paper we will study for the first time the
problem of generating from vacuum the triplet Z boson and electron-positron
pair in the inflation of the early universe. In the presenent study we
restrict ourselves to consider only the interactions between neutral massive Z
bosons and massive fermions. The theory of electro-weak interactions that
implies the Z bosons and fermions can be developed by taking into
consideration in the lagrangean density the interaction term with the usual
electrodynamic coupling $-J^{\mu}A_{\mu}$ where the current represents the
neutral current. It is known that the phenomenon of particle production must
be considered along with the inflations but until now there have been no
definitive conclusions related to the mechanisms of particle generation. We
mention that the nonperturbative approach of the problem of massive bosons
generation is less studied in literature, but among a few results we mention
here the ones obtained in [57]. The general framework related to the idea that
the space expansion produces particles was discussed in the fundamental papers
[52, 53, 54, 55], where the authors prove that the phenomenon of particle
generation is possible only in early universe. In this paper we propose a
perturbative mechanism that could explain the generation of massive Z bosons
by computing the perturbative first order amplitude corresponding to the de
Sitter electro-weak theory [51]. The perturbative approach to the problem of
particle generation was proposed in [18, 19], where it was proven that this
mechanism could play an important role in the matter-antimatter generation in
early universe.

The paper is organized as follows: in the second section we present a short
review about the theory of the free Proca Field and the free Dirac field in de
Sitter geometry, the third section is dedicated to the equations of
interaction between fields and in the forth section we present the reduction
formalism for the Proca field as the basis for defining the amplitudes of
electro-weak theory in de Sitter geometry. The fifth section is dedicated to
the study of the amplitudes and probabilities corresponding to the process of
spontaneous generation from vacuum of the triplet Z boson and electron-
positron pair. In this section we obtain the total probability and we use a
combined dimensional regularization and Pauli-Villars regularization for
computing the integrals over the final momenta. In the six section we present
the computation for the rate of transition and we discuss the limit of large
expansion factor and we give an estimation of the density number of Z bosons,
while in the seven section we present our conclusions.

##  2 Free fields

For our study we consider the de Sitter metric element written in conformal
form [1]:

$ds^{2}=dt^{2}-e^{2\omega t}d\vec{x}^{2}=\frac{1}{(\omega
t_{c})^{2}}(dt_{c}^{2}-d\vec{x}^{2}),$ (1)

where the conformal time is related to the proper time by
$t_{c}=\frac{-e^{-\omega t}}{\omega}$ and $\omega$ is the Hubble parameter
($\omega>0$).

In this paper we propose for the first time a perturbative approach for
explaining the generation of massive Z bosons and massive fermions in early
universe. We work in a de Sitter geometry and use the chart with the conformal
time $t_{c}\in(-\infty,0)$, which covers the expanding portion of de Sitter
space-time. For the line element (1), in the Cartesian gauge, we have the
nonvanishing tetrad components,

$e^{0}_{\widehat{0}}=-\omega
t_{c};\,\,\,e^{i}_{\widehat{j}}=-\delta^{i}_{\widehat{j}}\,\omega t_{c}.$ (2)

The theory of the free Dirac field in de Sitter geometry was studied in [36]
where the fundamental spinor solutions in the momentum-helicity basis were
determined. The Dirac equation in de Sitter space-time in Cartesian gauge was
studied in [36], and reads as:

$\Big{(}-i\omega
t_{c}(\gamma^{0}\partial_{t_{c}}+\gamma^{i}\partial_{i})+\frac{3i\omega}{2}\gamma^{0}-m\Big{)}\psi(x)=0$
(3)

The fundamental solutions of the Dirac equation in de Sitter geometry are the
$U,V$ spinors with a defined momentum and helicity [36]:

$\displaystyle U_{\vec{p},\sigma}(t,\vec{x}\,)=\frac{\sqrt{\pi
p/\omega}}{(2\pi)^{3/2}}\left(\begin{array}[]{c}\frac{1}{2}e^{\pi
k/2}H^{(1)}_{\nu_{-}}(\frac{p}{\omega}e^{-\omega t})\xi_{\sigma}(\vec{p}\,)\\\
\sigma e^{-\pi k/2}H^{(1)}_{\nu_{+}}(\frac{p}{\omega}e^{-\omega
t})\xi_{\sigma}(\vec{p}\,)\end{array}\right)e^{i\vec{p}\cdot\vec{x}-2\omega
t};$ (6) $\displaystyle V_{\vec{p},\sigma}(t,\vec{x}\,)=\frac{\sqrt{\pi
p/\omega}}{(2\pi)^{3/2}}\left(\begin{array}[]{c}-\sigma\,e^{-\pi
k/2}H^{(2)}_{\nu_{-}}(\frac{p}{\omega}e^{-\omega
t})\,\eta_{\sigma}(\vec{p}\,)\\\ \frac{1}{2}\,e^{\pi
k/2}H^{(2)}_{\nu_{+}}(\frac{p}{\omega}e^{-\omega
t})\,\eta_{\sigma}(\vec{p}\,)\end{array}\right)e^{-i\vec{p}\cdot\vec{x}-2\omega
t},$ (9)

where $H^{(1)}_{\nu}(z),H^{(2)}_{\nu}(z)$ are Hankel functions of the first
and second kind and $K=\frac{m}{\omega},\nu_{\pm}=\frac{1}{2}\pm iK$. The
helicity spinors satisfy the relation:

$\vec{\sigma}\vec{p}\,\xi_{\sigma}(\vec{p}\,)=2p\sigma\xi_{\sigma}(\vec{p}\,)$
(10)

with $\sigma=\pm 1/2$, where $\vec{\sigma}$ are the Pauli matrices, and
$p=\mid\vec{p}\mid$ is the modulus of the momentum vector, while
$\eta_{\sigma}(\vec{p}\,)=i\sigma_{2}[\xi_{\sigma}(\vec{p}\,)]^{*}$. The
meaning of positive/negative frequencies can be established if we use the
asymptotic expansion of the Hankel functions at large $z$ [23]:

$H^{(1,\,2)}_{\nu}(z)\simeq\left(\frac{2}{\pi z}\right)^{1/2}e^{\pm
i(z-\nu\pi/2-\pi/4)},$ (11)

and we obtain that for $t\rightarrow-\infty$, the modes defined above behave
as a positive/negative frequency modes with respect to conformal time
$t_{c}=-e^{-\omega t}/\omega$:

$U_{\vec{p},\lambda}(t,\vec{x}\,)\sim
e^{-ipt_{c}};\,\,V_{\vec{p},\lambda}(t,\vec{x}\,)\sim e^{ipt_{c}}.$ (12)

This is the behavior in the infinite past for positive/negative frequency
modes which defines the Bunch-Davies vacuum [18].

In the case of the Proca free field the equations for the spatial and temporal
components of the vector potential were obtained in [3]

$\displaystyle\partial_{t_{c}}(\partial_{i}A_{i})-\Delta
A_{0}+\frac{M_{Z}^{2}}{\omega^{2}t_{c}^{2}}A_{0}=0,$ (13)
$\displaystyle\partial_{t_{c}}^{2}A_{k}-\Delta
A_{k}-\partial_{k}(\partial_{t_{c}}A_{0})+\partial_{k}(\partial_{i}A_{i})+\frac{M_{Z}^{2}}{\omega^{2}t_{c}^{2}}A_{k}=0.$
(14)

The above equations and the Lorentz condition [3]

$\partial_{t_{c}}A_{0}-\frac{2}{t_{c}}A_{0}-\partial_{i}A_{i}=0$ (15)

determine the normalised solutions for the free Proca field in de Sitter
geometry. The spatial part of the solution is [3]:

$\displaystyle\vec{f}_{\vec{P},\lambda}(x)=\left\\{\begin{array}[]{cll}\frac{i\sqrt{\pi}\omega
Pe^{-\pi
k/2}}{2M_{Z}(2\pi)^{3/2}}\left[(\frac{1}{2}+ik)\frac{\sqrt{-t_{c}}}{P}H^{(1)}_{ik}\left(-Pt_{c}\right)-(-t_{c})^{3/2}H^{(1)}_{1+ik}\left(-Pt_{c}\right)\right]e^{i\vec{P}\vec{x}}\vec{\epsilon}\,(\vec{n}_{P},\lambda)&{\rm;}&\lambda=0\\\
\frac{\sqrt{\pi}e^{-\pi
k/2}}{2(2\pi)^{3/2}}\sqrt{-t_{c}}H^{(1)}_{ik}\left(-Pt_{c}\right)e^{i\vec{P}\vec{x}}\vec{\epsilon}\,(\vec{n}_{P},\lambda)&{\rm;}&\lambda=\pm
1.\end{array}\right.$ (18)

The temporal component of the solution for the Proca equation is [3]:

$\displaystyle
f_{0\vec{P},\lambda}(x)=\left\\{\begin{array}[]{cll}\frac{\sqrt{\pi}\omega
Pe^{-\pi
k/2}}{2M_{Z}(2\pi)^{3/2}}(-t_{c})^{3/2}H^{(1)}_{ik}\left(-Pt_{c}\right)e^{i\vec{P}\vec{x}}&{\rm
for}&\lambda=0\\\ 0&{\rm for}&\lambda=\pm 1.\end{array}\right.$ (22)

In the above equations for the solutions of the Proca equation
$\vec{n}_{P}=\vec{P}/\mathcal{P}$ and $\vec{\epsilon}\,(\vec{n}_{P},\lambda)$
are the polarization vectors. For $\lambda=\pm 1$ these vectors are
transversal on the momentum
$\vec{P}\cdot\vec{\epsilon}\,(\vec{n}_{P},\lambda=\pm 1)=0$ and for
$\lambda=0$ the polarization vectors are longitudinal on the momentum
$\vec{P}\cdot\vec{\epsilon}\,(\vec{n}_{P},\lambda=0)=P$, since
$\vec{\epsilon}\,(\vec{n}_{P},\lambda=0)=\vec{n}_{P}$. The mass of the Z boson
is denoted by $M_{Z}$, while the parameter
$k=\sqrt{\left(\frac{M_{Z}}{\omega}\right)^{2}-\frac{1}{4}}$ depends on the
ratio $\frac{M_{Z}}{\omega}$, provided that
$\frac{M_{Z}}{\omega}>\frac{1}{2}$.

The above free field solutions will be the basis of our study for writing down
the amplitudes and probabilities corresponding to the processes of particle
generation from vacuum in an expanding background.

##  3 Neutral current interaction in de Sitter space-time

The perturbative QED was developed in [41] by taking into account that the
theory of quantum fields with spin can be correctly constructed only in
orthogonal (non-holonomic) local frames where the half-integer spins have
sense. In [41] it was proposed that the generators of the covariant
representation are the differential operators produced by the Killing vectors
which are associated to isometries according to the generalized Carter and
McLenagan formula [58]. They form an algebra of conserved observables that
commute with the operators of the field equations. With this method the
quantum states were globally defined on the entire manifold, and these states
are independent on the local coordinates, so the vacuum state is unique and
stable [41]. In this approach the quantum states are prepared and measured by
the same global apparatus which consists of the largest freely generated
algebra that includes the conserved operators [41]. This method does not
exclude the cosmological particle creation which may be observed by using
local detectors [59, 60].

For developing the theory of electro-weak interactions we will adopt the same
methods as those used in [41], and we assume that the electro-weak transitions
are measured by the same global apparatus which prepares all the quantum
states, and this includes the $in$ and $out$ asymptotic free fields which
remain minimally coupled to gravity. This apparatus complies with an
asymptotic prescription of frequencies separation that assures the uniqueness
and stability of both the vacuum states of the free Dirac and Proca fields [3,
36]. This means that it cannot record particle creation in the absence of the
electromagnetic interaction, as the local detectors [59, 60].

Let us comment on the problem related to the definition of the S-matrix in de
Sitter geometry. It is well known that in de Sitter geometry the only time-
like Killing vector is not time-like everywhere and this problem generated
concerns related to the possibility of defining the energy operator correctly,
the in and out fields and the scattering operator [61, 62]. Here we consider
that the S matrix can be defined since the time-like Killing vector remains
time-like everywhere inside the light cone where an observer can perform
physical measurements [41, 64]. In this approach the energy operator can be
defined, but it does not commute with the momentum operator [36, 41]. Another
result worth mentioning here was obtained in [63], where an S-matrix formalism
was analysed for weakly coupled field theories in the global de Sitter
manifold. A remarkable result is that from the S-matrix obtained in [63] one
can recover the usual S-matrix from Minkowski space in the flat-space limit, a
result that was also reported in [41].

Let us assume that the theory of interactions between the vector neutral
fields and fermion fields in de Sitter space-time can be developed by using
the prescription of minimal coupling by introducing the interaction term with
the prescription $\mathcal{L}_{int}=-(j_{neutral})^{\mu}A_{\mu}$, because the
Z bosons have no electric charge and the particle coincides with the
antiparticle, and mediates the neutral current interactions
$(j_{neutral})\,^{\mu}$, which is given by:

$\displaystyle(j_{neutral})\,^{\mu}$ $\displaystyle=$
$\displaystyle\overline{\psi}_{\nu_{e}}\,\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\left(\frac{1-\gamma^{5}}{2}\right)\psi_{\nu_{e}}-\cos(2\theta_{W})\overline{\psi}_{e}\,\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\left(\frac{1-\gamma^{5}}{2}\right)\psi_{e}$
(23)
$\displaystyle+2\sin^{2}(\theta_{W})\overline{\psi}_{e}\,\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\left(\frac{1+\gamma^{5}}{2}\right)\psi_{e}.$

where $e_{0}$ is the electric charge, $\theta_{W}$ is the Weinberg angle and,
$\psi_{\nu_{e}}$ designates the neutrino-antineutrino field, $A_{\alpha}(Z)$
designates the Z boson field and $\psi_{e}$ designates the electron-positron
field. Since it is well known that the neutrinos are only left-handed we use
the left projector $\frac{1-\gamma^{5}}{2}$, with the specification that for
the electrons and positrons we also have the right-handed part, and the
corresponding right projector is $\frac{1+\gamma^{5}}{2}$. In our present
paper we study only the interaction of Z bosons with the massive fermions and
the first term from the neutral current will be not taken into account in our
study. The theory of fields interaction is developed in the conformal chart
$\\{t_{c},\vec{x}\\}$ with the Cartesian coordinates. We mention that a
detailed analysis of the interaction between Z boson and neutrinos that
include the definition of the transition amplitudes can be found in [51].

The action can be expressed with the tetrad gauge invariant Lagrangian density
that gives the coupling between $Z$ bosons and massive fermions, written with
point independent Dirac matrices $\gamma^{\hat{\alpha}}$ and the tetrad fields
$e_{\hat{\alpha}}^{\mu}$ and the Lagrangian densities for Proca field and
Dirac fields as:

$S=\int
d^{4}x\sqrt{-g}\bigg{(}\mathcal{L}_{Dirac}+\mathcal{L}_{Proca}+\mathcal{L}_{int}\bigg{)},$
(24)

which can be expanded as:

$S=\int
d^{4}x\sqrt{-g}\Bigg{\\{}\frac{i}{2}\big{(}\overline{\psi}\gamma^{\hat{\alpha}}D_{\hat{\alpha}}\psi-(\overline{D_{\hat{\alpha}}\psi})\gamma^{\hat{\alpha}}\psi\big{)}-m\overline{\psi}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{M_{Z}^{2}}{2}A_{\mu}A^{\mu}\\\
+\bigg{(}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi
A_{\mu}-e_{0}\tan(\theta_{W})\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi
A_{\mu}\Bigg{\\}}.$ (25)

where $m$ is the mass of the Dirac field and $M_{Z}$ is the mass of the Z
boson field, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
field strength. The notation
$D_{\hat{\alpha}}=\partial_{\hat{\alpha}}+\Gamma_{\hat{\alpha}}$ represents
the covariant derivatives in local frames, that depend on spin connections
$\Gamma_{\hat{\alpha}}=\Gamma_{\hat{\alpha}\hat{\mu}\hat{\nu}}S^{\hat{\mu}\hat{\nu}}$,
which are given in terms of basis generators
$S^{\hat{\mu}\hat{\nu}}=i/4\\{\gamma^{\hat{\mu}},\gamma^{\hat{\nu}}\\}$ of the
spinor representation of $SL(2,\textbf{C})$ group. The Euler-Lagrange
equations for fields will give the equations for interaction between the
massive Dirac field and Z bosons

$\frac{\partial\mathcal{L}}{\partial\overline{\psi}}-\partial_{\rho}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\overline{\psi})}\bigg{)}=0$
(26)

$\frac{\partial\mathcal{L}}{\partial
A_{\mu}}-\partial_{\rho}\bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}A_{\mu})}\bigg{)}=0.$
(27)

The equations of interactions are obtained by using equations (26), (27) and
(25). The Dirac equations will couple the potential $A_{\alpha}(x)$

$\big{(}i\gamma^{\hat{\alpha}}D_{\hat{\alpha}}-m\big{)}\psi(x)=\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi(x)A_{\alpha}(x)-\tan(\theta_{W})\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi(x)A_{\alpha}(x).$
(28)

In the de Sitter metric the above equation become:

$\displaystyle\Big{(}-i\omega
t_{c}(\gamma^{0}\partial_{t_{c}}+\gamma^{i}\partial_{i})+\frac{3i\omega}{2}\gamma^{0}-m\Big{)}\psi(x)=\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi(x)A_{\alpha}(x)$
$\displaystyle-\tan(\theta_{W})\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi(x)A_{\alpha}(x).$
(29)

The Proca equation will have the neutral current as the source and this can be
read as:

$\partial_{\rho}\big{(}g^{\rho\alpha}g^{\mu\beta}\sqrt{-g(x)}F_{\alpha\beta}(x)\big{)}+\sqrt{-g}\,M_{Z}^{2}A^{\mu}(x)=-\sqrt{-g(x)}\,e^{\mu}_{\hat{\alpha}}J^{\hat{\alpha}}(x)$
(30)

and we specify that the Lorentz condition is deduced in the case of the free
Proca theory

$\partial_{\mu}(\sqrt{-g}A^{\mu})=0$ (31)

which is maintained for the interacting fields. Equation (30) can be expanded
in terms of the temporal and spatial components of the potential vector
$A^{\mu}$ and current $J^{\hat{\alpha}}$:

$\partial_{t_{c}}(\partial_{i}A_{i})-\Delta
A_{0}+\frac{M_{Z}^{2}}{\omega^{2}t_{c}^{2}}A_{0}=-\sqrt{-g}J_{0}$ (32)

$\partial_{t_{c}}^{2}A_{k}-\Delta
A_{k}-\partial_{k}(\partial_{t_{c}}A_{0})+\partial_{k}(\partial_{i}A_{i})+\frac{M_{Z}^{2}}{\omega^{2}t_{c}^{2}}A_{k}=\sqrt{-g}J_{k}$
(33)

Equations for the fields in interaction (32), (3), are coupled nonlinear
equations and the solutions can be written down by following the methods from
the flat space case. Then the system of coupled equations is replaced with one
system of integral equations that contain information about the initial
conditions. That means that one selects a Green function $G(x,y)$
corresponding to one initial condition, which helps us write the solutions for
the interacting equations as follows:

$A^{\mu}(x)=\hat{A}^{\mu}(x)-\int
d^{4}y\sqrt{-g(y)}G^{\mu\alpha}(x,y)J_{\alpha}(y)$ (34)

where $\hat{A}^{\mu}(x)$ is a free field and $G^{\mu\alpha}$ is a Green
function of the Proca equation. In the case of coupled Dirac equations the
solution is:

$\psi(x)=\hat{\psi}(x)-e\int
d^{4}y\sqrt{-g(y)}S(x-y)\bigg{[}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi(y)A_{\alpha}(y)\\\
-\tan(\theta_{W})\gamma^{\hat{\mu}}e^{\alpha}_{\hat{\mu}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi(y)A_{\alpha}(y)\bigg{]}$
(35)

where $\hat{\psi}(x)$ is a free field and $S(x-y)$ is a Green function of the
Dirac equation. The Green functions for the Dirac field satisfy the equation
[36, 41]:

$E_{D}\,S(x-y)=-\frac{1}{\sqrt{-g(x)}}\,\delta^{4}(x-y).$ (36)

where the Dirac operator reads:

$E_{D}=-i\omega
t_{c}(\gamma^{0}\partial_{t_{c}}+\gamma^{i}\partial_{i})+\frac{3i\omega}{2}\gamma^{0}-m.$
(37)

At the end of this section we mention the important results in regards to the
propagator of the Dirac field in de Sitter space-time. First the propagator of
the Dirac field on the de Sitter space-time in configuration representation
was constructed by Candelas and Reine [27]. The same propagator was obtained
as a mode sum by Koskma and Prokopec in the general
Friedmann–Lemaître–Robertson–Walker spacetimes of arbitrary dimensions [26]. A
new integral representation in momentum space of the Feynman propagator for
massive Dirac field was obtained by Cotăescu [28]. All these results complete
the theory of Dirac field in de Sitter space-time and propose new methods
related to the regularization and will allow one to study the second order
scattering processes in this geometry.

##  4 Reduction formalism and transition amplitudes

In this section we establish the definition of the transition amplitudes of
the theory of interaction between Z bosons and massive fermions. The solutions
for the equation of interacting fields will be used for constructing the
reduction of the fields from the $in/out$ sectors and then by using the
perturbation theory we will define the first order transition amplitudes of
the electro-weak theory.

###  4.1 Transversal modes

Taking into account that for the transversal modes $\lambda=\pm 1$ we have
only the spatial part of the solutions ($A_{0}=0$) the equation of interaction
can be rewritten in the form:

$\bigg{[}\bigg{(}\partial_{t_{c}}^{2}-\Delta+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}\delta_{ik}+\partial_{i}\partial_{k}\bigg{]}A_{k}=-\sqrt{-g}J_{i}$
(38)

where we introduce the Proca operator for the transversal modes
$E_{ik}(x)=\bigg{(}\partial_{t_{c}}^{2}-\Delta+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}\delta_{ik}+\partial_{i}\partial_{k}$.

Since we want to obtain the solutions for the coupled field equation we must
work with the Green function for the Proca equation [3, 25]. The propagator
for massive vector fields on de Sitter background of arbitrary dimension was
constructed in [25]. The propagator was proven to be de Sitter invariant and
possesses the correct flat spacetime and massless limits [25]. For obtaining
useful equations with the Green functions of the Proca equation that help us
verify that the solution proposed in equation (34) is an exact solution of the
interacting equation (38), we use the relation [3, 25]:

$\eta^{\alpha\beta}\partial_{\alpha}\big{[}\partial_{\beta}G_{\mu\nu}(x,y)-\partial_{\mu}G_{\beta\nu}(x,y)\big{]}+\frac{\mu^{2}}{t_{c}^{2}}G_{\mu\nu}(x,y)=\eta_{\mu\nu}\delta^{4}(x-y)$
(39)

Taking the spatial values for the indices $\mu=i,\,\nu=j$ in the above
equation and knowing that the temporal parts of the Green functions are
vanishing, we deduce the following equation:

$\bigg{[}\delta_{ik}\bigg{(}\partial_{t_{c}}^{2}-\Delta+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}+\partial_{k}\partial_{i}\bigg{]}G_{kj}(x,y)=E_{ik}(x)G_{kj}(x,y)=-\delta_{ij}\delta^{4}(x-y)$
(40)

The solution for the equation (40) is written as such that by using equation
(38) one can verify that is an exact solution:

$A_{k}(x)=\hat{A}_{k}(x)-\int d^{4}y\sqrt{-g(y)}\,G_{kj}(x,y)J_{j}(y)$ (41)

where $\hat{A}_{k}(x)$ is a free field such that $E_{ik}(x)\hat{A}_{k}(x)=0$.
The Proca operator applied on the above equation then gives:

${E}_{ik}(x)A_{k}(x)={E}_{ik}(x)\hat{A}_{k}(x)-\int
d^{4}y\sqrt{-g(y)}\,{E}_{ik}(x)G_{kj}(x,y)J_{j}(y)=-\sqrt{-g}J_{i}(x)$ (42)

which complete our proof.

For constructing a theory of interactions, we seek operators that create
independent particle states with each particle propagating with its physical
mass. Let us start constructing these operators, using (41) which offers us
the possibility of constructing free fields, which are asymptotic equal (at
$t\rightarrow\pm\infty$) with solutions of equation (38). The retarded Green
functions $G_{ij\,\,R}(x,y)$ vanish at $t_{c}\rightarrow-\infty$
($t\rightarrow-\infty$) while the advanced ones $G_{ij\,\,A}(x,y)$ vanish for
$t_{c}\rightarrow 0$ ($t\rightarrow\infty$). Further the equation (41) can be
expressed with the retarded and advanced functions as follows:

$\displaystyle A_{k}(x)=\hat{A}_{k\,\,R/A}(x)-\int
d^{4}y\sqrt{-g(y)}\,G_{kj\,\,R/A}(x,y)J_{j}(y)$ (43)

Now we can define the free fields, $\hat{A}_{k\,\,R/A}(x)$ which satisfy:

$\displaystyle\lim_{t\rightarrow\mp\infty}(A_{k}(x)-\hat{A}_{k\,\,R/A}(x))=0.$
(44)

The free fields $\hat{A_{k\,\,R}}(x)$ and $\hat{A_{k\,\,A}}(x)$ have the mass
$M_{Z}$ and are equal at $t\rightarrow\pm\infty$ with exact solutions of the
coupled equations and represent the fields before and after the interaction.
As in Minkowski the free fields $\hat{A_{k\,\,R}}(x)$ and
$\hat{A_{k\,\,A}}(x)$ are defined up to a normalization constant noted with
$\sqrt{z_{3}}$. This allows us to define the $in/out$ fields up to a
normalization constant $\sqrt{z_{3}}$ with the help of retarded and advanced
Green functions:

$\sqrt{z_{3}}A_{k}^{in/out}(x)=A_{k}(x)+\int
d^{4}y\sqrt{-g(y)}\,G_{kj\,\,R/A}(x,y){E}_{ij}(y)A_{i}(y)$ (45)

The reduction formalism can be developed now by using the expansion of the
field operator

$A^{k}(x)=\sum_{\lambda}\int
d^{3}p\,\Big{(}a(\vec{p},\lambda)f_{\vec{p},\lambda}^{k}(x)+b^{+}(\vec{p},\lambda){f_{\vec{p},\lambda}^{k}}^{*}(x)\Big{)}$
(46)

with the observation that from the above normalization we can define the
creation and annihilation operators

$\displaystyle a(\vec{p},\lambda)$ $\displaystyle=$ $\displaystyle\langle
f_{\vec{p},\lambda}^{k}(x),A_{k}(x)\rangle$ (47) $\displaystyle=$
$\displaystyle i\int
d^{3}x{f_{\vec{p},\lambda}^{k}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}A_{k}(x).$

Considering two states $|\alpha\rangle_{in}$ and $|\beta\rangle_{out}$, then
the probability of transition from state $\alpha$ to state $\beta$ is defined
as the scalar product of the two states:
${}_{out}\langle\beta|\alpha\rangle_{in}$. These are the elements of the
matrix for the scattering operator $S_{\beta\alpha}$, which can be used in
applications. The scattering operator assures the stability of the vacuum
state and one particle state, and in addition transforms any $out$ field in
the equivalent $in$ field.

Then the reduction of the Proca particle from the $out$ state gives:

${}_{out}\langle\beta 1(\vec{p}\,^{\prime},\lambda^{\prime})|\alpha
1(\vec{p},\lambda)\rangle_{in}={}_{out}\langle\beta|\big{(}a_{out}(\vec{p}\,^{\prime},\lambda^{\prime})-a_{in}(\vec{p}\,^{\prime},\lambda^{\prime})\big{)}|\alpha
1(\vec{p},\lambda)\rangle_{in}\\\
+{}_{out}\langle\beta|a_{in}(\vec{p}\,^{\prime},\lambda^{\prime})a^{+}_{in}(\vec{p},\lambda)|\alpha\rangle_{in}$
(48)

where the difference is

$\displaystyle
a_{out}(\vec{p}\,^{\prime},\lambda^{\prime})-a_{in}(\vec{p}\,^{\prime},\lambda^{\prime})$
$\displaystyle=$ $\displaystyle i\int
d^{3}x{f_{\vec{p},\lambda}^{k}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}\Big{(}{A_{k}}^{out}(x)-{A_{k}}^{in}(x)\Big{)}$
$\displaystyle=$ $\displaystyle i\int d^{3}x\int
d^{4}y\sqrt{-g(y)}{f_{\vec{p},\lambda}^{k}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}G_{kj}(x,y)E_{ij}(y)A_{i}(y)$

where the functions $G_{kj}(x,y)$ are the total commutator functions defined
as:

$G_{kj}(x,y)=i\sum_{\lambda}\int
d^{3}p\Big{(}f_{{\vec{p},\lambda}_{k}}(x){f_{{\vec{p},\lambda}_{j}}}^{*}(y)-{f_{{\vec{p},\lambda}_{k}}}^{*}(x)f_{{\vec{p},\lambda}_{j}}(y)\Big{)}.$
(50)

The calculation is completed by using the integral

$i\int
d^{3}x\,{f_{\vec{p},\lambda}^{k}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}G_{kj}(x,y)=i{f_{{\vec{p},\lambda}_{j}}}^{*}(y),$
(51)

which helps us write down the final result for the reduction of the Proca
particle from the $out$ state

${}_{out}\langle\beta 1(\vec{p}\,^{\prime},\lambda^{\prime})|\alpha
1(\vec{p},\lambda)\rangle_{in}=\delta_{\lambda\lambda^{\prime}}\delta^{3}(\vec{p}-\vec{p}\,^{\prime}){}_{out}\langle\beta|\alpha\rangle_{in}\\\
+i\int
d^{4}y\sqrt{-g(y)}\cdot{f_{{\vec{p}^{\prime},\lambda^{\prime}}_{i}}}^{*}(y)E_{ij}(x){}_{out}\langle\beta|A_{j}(x)|\alpha,1(\vec{p},\lambda)\rangle_{in},$
(52)

where the first term is dropped since it represents the case where the
particle makes the transition $in-out$ without interacting with other
particles. The reduction of the Proca particle from the $in$ state gives the
final result

${}_{out}\langle\beta 1(\vec{p}\,^{\prime},\lambda^{\prime})|\alpha
1(\vec{p},\lambda)\rangle_{in}=\delta_{\lambda\lambda^{\prime}}\delta^{3}(\vec{p}-\vec{p}\,^{\prime}){}_{out}\langle\beta|\alpha\rangle_{in}\\\
+i\int d^{4}y\sqrt{-g(y)}\cdot{}_{out}\langle\beta
1(\vec{p}\,^{\prime},\lambda^{\prime})|A_{j}(x)|\alpha\rangle_{in}\overleftarrow{E_{ij}}(x)f_{{\vec{p}\,^{\prime},\lambda^{\prime}}_{i}}(y)$
(53)

Then the generalized Green functions can be expressed in terms of free fields
as:

$\langle
0|\Psi(x_{1})\overline{\Psi}(x_{3})\mathcal{A}_{k}(x_{3})...|0\rangle=\frac{\langle
0|\psi(x_{1})\overline{\psi}(x_{3})A_{k}(x_{3})...\mathbf{S}|0\rangle}{\langle
0|\mathbf{S}|0\rangle},$ (54)

where the scattering operator is

$\displaystyle\mathbf{S}=Texp\left[-i\int
d^{4}x\sqrt{-g(x)}\,\mathcal{L}_{int}\right]$ $\displaystyle=Texp\left[-i\int
d^{4}x\sqrt{-g(x)}\left(\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi
A_{\mu}-e_{0}\tan(\theta_{W})\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi
A_{\mu}\right)\right].$

For example, if we consider a process of spontaneous emission from de Sitter
vacuum of the triplet Z boson and the electron-positron pair the reduction
procedure gives:

${}_{out}\langle
1(\vec{P},\lambda),1(\vec{p},\sigma),1(\vec{p}\,^{\prime},\sigma^{\prime})|0\rangle_{in}=\bigg{(}\frac{i}{\sqrt{z_{3}}}\bigg{)}\bigg{(}\frac{i}{\sqrt{z_{2}}}\bigg{)}^{2}\int
d^{4}y_{1}\sqrt{-g(y_{1})}\int d^{4}y_{2}\sqrt{-g(y_{2})}\\\ \times\int
d^{4}y_{3}\sqrt{-g(y_{3})}\overline{U}_{\vec{p},\sigma}(y_{1})E_{D}(y_{1}){f_{{\vec{p},\lambda}_{i}}}^{*}(y_{2})E_{il}(y_{2})\langle
0|T[\psi(y_{1})A_{l}(y_{2})\overline{\psi}(y_{3})]|0\rangle\overleftarrow{E_{D}}(y_{3})V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})$
(56)

The Green functions from the equation can be expressed in terms of Feynmann
propagators by considering all the T contractions from the Wick theorem. Then
by using the first order in perturbation theory with $\mathbf{S}^{(1)}$ and
spatial index $\mu=i$, and by making the T contraction between the cinematical
part and the dynamical part we obtain the definition of the transition
amplitude in the first order of the perturbation theory for the transversal
modes of the Proca field:

$\displaystyle A_{i\rightarrow f}$ $\displaystyle=$ $\displaystyle-i\int
d^{4}x\sqrt{-g(x)}\int d^{4}y_{1}\sqrt{-g(y_{1})}\int
d^{4}y_{2}\sqrt{-g(y_{2})}\int
d^{4}y_{3}\sqrt{-g(y_{3})}\bigg{[}\overline{U}_{\vec{p},\sigma}(y_{1})$ (57)
$\displaystyle\times\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}E_{D}(y_{1})S(y_{1}-x)\gamma^{\hat{j}}e^{k}_{\hat{j}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}{f_{{\vec{p},\lambda}_{i}}}^{*}(y_{2})E_{il}(y_{2})D_{lk}(y_{2},x)E_{D}(y_{3})$
$\displaystyle\times
S(y_{3}-x)V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})-e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(y_{1})E_{D}(y_{1})S(y_{1}-x)\gamma^{\hat{j}}e^{k}_{\hat{j}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}$
$\displaystyle\times{f_{{\vec{p},\lambda}_{i}}}^{*}(y_{2})E_{il}(y_{2})\frac{1}{i}D_{lk}(y_{2},x)E_{D}(y_{3})S(y_{3}-x)V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})\bigg{]}.$

We use the action of the field operators on propagators and solve the
integrals using the properties of the delta Dirac functions, and obtain the
final result

$\displaystyle A_{i\rightarrow f}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g(x)}\bigg{[}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{j}}e^{k}_{\hat{j}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x){f_{{\vec{p},\lambda}\,_{k}}}^{*}(x)$
(58) $\displaystyle-
e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{j}}e^{k}_{\hat{j}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x){f_{{\vec{p},\lambda}\,_{k}}}^{*}(x)\bigg{]}.$

All of the first order processes can be computed by using the above formalism.
The above definition of the transition amplitude holds in both charts
$\\{t,\vec{x}\\}$ and $\\{t_{c},\vec{x}\\}$. In addition in the limit
$\omega\rightarrow 0$ the amplitude is reduced to the well known formula from
the Minkowski theory.

The reduction formalism for the Dirac field in de Sitter space-time was
developed in [40, 41], and here we used the final results for replacing the
$in\ out$ Dirac fields in our amplitudes.

###  4.2 Longitudinal modes

In the case of longitudinal modes the temporal and spatial components of the
four vector potential are combined in coupled equations according to equations
(32), (33) and are useful for writing the equation with all components by
rewriting the Proca operator with temporal and spatial components. In this
setup the Proca equation can be written as:

$\bigg{[}\delta^{\beta}_{\mu}\bigg{(}\partial_{\rho}\partial^{\rho}+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}-\partial^{\beta}\partial_{\mu}\bigg{]}A_{\beta}={E}^{\beta}_{\mu}(x)A_{\beta}=0.$
(59)

The Green function of the Proca field satisfies the equation:

$\bigg{[}\delta^{\beta}_{\mu}\bigg{(}\partial_{\rho}\partial^{\rho}+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}-\partial^{\beta}\partial_{\mu}\bigg{]}G_{\beta\nu}(x,y)=\eta_{\mu\nu}\delta^{4}(x-y)$
(60)

from which we deduce the following relations:

$\bigg{(}-\Delta+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}G_{00}(x,y)+\partial_{i}\partial_{t_{c}}G_{i0}(x,y)=\delta^{4}(x-y),$
(61)
$\bigg{[}\delta_{ik}\bigg{(}\partial^{2}_{t_{c}}-\Delta+\frac{\mu^{2}}{t_{c}^{2}}\bigg{)}+\partial_{k}\partial_{i}\bigg{]}G_{kj}(x,y)-\partial_{t_{c}}\partial_{i}G_{0j}(x,y)=-\delta_{ij}\delta^{4}(x-y)$
(62)

which combine the spatial and temporal components of the field. Then one can
observe that the equations of interaction (32), (33) can be combined in a
single equation:

${E}^{\mu}_{\nu}(x)A^{\nu}(x)=\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}\psi\sqrt{-g(x)}\\\
-e_{0}\tan(\theta_{W})\overline{\psi}\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}\psi\sqrt{-g(x)}=\sqrt{-g(x)}J^{\mu}(x).$
(63)

with the solution written in terms of Green functions:

$A_{\mu}(x)=\hat{A_{\mu}}(x)+\int
d^{4}y\sqrt{-g(y)}G_{\mu\beta}(x,y)J^{\beta}(y),$ (64)

where $\hat{A_{\mu}}(x)$ is a free field such that
${E}^{\mu}_{\nu}(x)\hat{A_{\mu}}(x)=0$. One can verify that this is an exact
solution by applying the Proca operator ${E}^{\mu}_{\nu}(x)$

$\displaystyle{E}^{\mu}_{\nu}(x)A_{\mu}(x)$ $\displaystyle=$
$\displaystyle{E}^{\mu}_{\nu}\hat{A_{\mu}}(x)+\int
d^{4}y\sqrt{-g(y)}{E}^{\mu}_{\nu}G_{\mu\beta}(x,y)J^{\beta}(y)$ (65)
$\displaystyle=$ $\displaystyle\int
d^{4}y\sqrt{-g(y)}\cdot\eta_{\nu\beta}\delta^{4}(x-y)J^{\beta}(y)$
$\displaystyle=$ $\displaystyle\sqrt{-g(x)}J_{\nu}(x)$

Then we can define the $in\ out$ fields in terms of the retarded and advanced
Green functions up to a normalization constant denoted with $\sqrt{z_{3}}$

$\sqrt{z_{3}}{A_{\mu}}^{in/out}(x)=A_{\mu}-\int
d^{4}y\sqrt{-g(y)}{G_{\mu\beta}}_{R/A}(x,y){E}^{\beta}_{\alpha}(y)A^{\alpha}(y).$
(66)

Since we take into account only the longitudinal components for $\lambda=0$
the field operator can be expanded as:

$A^{\mu}(x)=\int
d^{3}p\,\Big{(}a(\vec{p},\lambda)f_{\vec{p},\lambda=0}^{\mu}(x)+b^{+}(\vec{p},\lambda){f_{\vec{p},\lambda=0}^{\mu}}^{*}(x)\Big{)}.$
(67)

The reduction formalism can be constructed by considering the transition
between two states as in the previous section

${}_{out}\langle\beta 1(\vec{p}\,^{\prime},0)|\alpha
1(\vec{p},0)\rangle_{in}={}_{out}\langle\beta|\big{(}a_{out}(\vec{p}\,^{\prime},0)-a_{in}(\vec{p}\,^{\prime},0)\big{)}|\alpha
1(\vec{p},0)\rangle_{in}\\\
+{}_{out}\langle\beta|a_{in}(\vec{p}\,^{\prime},0)a^{+}_{in}(\vec{p},0)|\alpha\rangle_{in}$
(68)

and the definitions of the creation and annihilation operators in terms of
quantities with four components. The difference between the operators gives:

$\displaystyle
a_{out}(\vec{p}\,^{\prime},0)-a_{in}(\vec{p}\,^{\prime},0)=i\int
d^{3}x{f_{\vec{p},0}^{\mu}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}\Big{(}{A_{\mu}}_{out}(x)-{A_{\mu}}_{in}(x)\Big{)}$
$\displaystyle=-i\int
d^{3}x{f_{\vec{p},0}^{\mu}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}\int
d^{4}y\sqrt{-g(y)}\cdot\Big{(}{G_{\mu\beta}}_{A}(x,y)$ (69)
$\displaystyle-{G_{\mu\beta}}_{R}(x,y)\big{)}{E}_{\alpha}^{\beta}(y)A^{\alpha}(y)$
(70)

The commutator function is defined using the mode functions as

$G_{\mu\beta}(x,y)={G_{\mu\beta}}_{A}(x,y)-{G_{\mu\beta}}_{R}(x,y)=i\int
d^{3}p\Big{(}{f_{\vec{p},0}}_{\mu}(x){f_{\vec{p},0}}_{\beta}(y)-{f_{\vec{p},0}}_{\mu}^{*}(x){f_{\vec{p},0}}_{\beta}(y)\Big{)}$
(71)

Then we compute the integral with the commutator function

$i\int
d^{3}x{f_{\vec{p},0}^{\mu}}^{*}(x)\overleftrightarrow{\partial_{t_{c}}}G_{\mu\beta}(x,y)=i\int
d^{3}p\delta^{3}(\vec{p}-\vec{p}^{\prime}){f_{\vec{p},0}}_{\beta}^{*}(y)=i{f_{\vec{p},0}}_{\beta}^{*}(y)$
(72)

and we obtain the final result:

$a_{out}(\vec{p}\,^{\prime},0)-a_{in}(\vec{p}\,^{\prime},0)=i\int
d^{3}y\sqrt{-g(y)}{f_{\vec{p},0}}_{\beta}(y){E}^{\beta}_{\alpha}(y)A^{\alpha}(y)$
(73)

The final result for the reduction of the Proca particle from the $out$ state
gives:

${}_{out}\langle\beta 1(\vec{p}\,^{\prime},0)|\alpha
1(\vec{p},0)\rangle_{in}=\delta_{\lambda\lambda^{\prime}}\delta^{3}(\vec{p}-\vec{p}\,^{\prime})\,{}_{out}\langle\beta|\alpha\rangle_{in}\\\
+i\int
d^{4}y\sqrt{-g(y)}{f_{{\vec{p}^{\prime},0}_{\beta}}}^{*}(y){E}^{\beta}_{\alpha}(y){}_{out}\langle\beta|A^{\alpha}(x)|\alpha,1(\vec{p},\lambda)\rangle_{in}$
(74)

with the specification that the reduction from the $in$ state can be obtained
in a similar manner. Let us consider the amplitude for the same process of Z
boson and electron-positron emission from vacuum, and the associated amplitude
written with the reduction procedure for the modes with $\lambda=0$:

${}_{out}\langle
1(\vec{P},\lambda),1(\vec{p},\sigma),1(\vec{p}\,^{\prime},\sigma^{\prime})|0\rangle_{in}=\bigg{(}\frac{i}{\sqrt{z_{3}}}\bigg{)}\bigg{(}\frac{i}{\sqrt{z_{2}}}\bigg{)}^{2}\int
d^{4}y_{1}\sqrt{-g(y_{1})}\int d^{4}y_{2}\sqrt{-g(y_{2})}\\\ \times\int
d^{4}y_{3}\sqrt{-g(y_{3})}\overline{U}_{\vec{p},\sigma}(y_{1})E_{D}(y_{1}){f_{\vec{p},0}^{\beta}}^{*}(y_{2}){E}^{\mu}_{\beta}(y_{2})\langle
0|T[\psi(y_{1})A_{\mu}(y_{2})\overline{\psi}(y_{3})|0\rangle\overleftarrow{E_{D}}(y_{3})V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})$
(75)

By using the term corresponding to the first order in perturbation theory from
the expansion of the scattering operator (4.1) and performing all possible T
contractions we obtain:

$\displaystyle A_{i\rightarrow f}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g(x)}\int d^{4}y_{1}\sqrt{-g(y_{1})}\int
d^{4}y_{2}\sqrt{-g(y_{2})}\int
d^{4}y_{3}\sqrt{-g(y_{3})}\bigg{[}\overline{U}_{\vec{p},\sigma}(y_{1})$ (76)
$\displaystyle\times\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}E_{D}(y_{1})S(y_{1}-x)\gamma^{\alpha}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}{f_{\vec{p},0}^{\beta}}^{*}(y_{2}){E}^{\mu}_{\beta}(y_{2})D_{\mu\alpha}(x,y_{2})$
$\displaystyle\times
S(y_{3}-x)\overleftarrow{E_{D}}(y_{3})V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})-e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(y_{1})E_{D}(y_{1})S(y_{1}-x)\gamma^{\alpha}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}$
$\displaystyle\times{f_{\vec{p},0}^{\beta}}^{*}(y_{2}){E}^{\mu}_{\beta}(y_{2})D_{\mu\alpha}(x,y_{2})S(y_{3}-x)\overleftarrow{E_{D}}(y_{3})V_{\vec{p}\,^{\prime},\sigma^{\prime}}(y_{3})\bigg{]}$

The final form of the transition amplitude contains both the spatial and
temporal parts of the Proca field and is obtained by using the relations with
Green functions (36), (60). We also note that on a curved background the
definition of the transition amplitude is similar to the flat space case up to
the measure of integration $\sqrt{-g(x)}$:

$A_{i\rightarrow f}=\int
d^{4}x\sqrt{-g(x)}\bigg{[}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{\nu}}e^{\alpha}_{\hat{\nu}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x){f_{{\vec{P},0}_{\alpha}}}^{*}(x)\\\
-e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{\nu}}e^{\alpha}_{\hat{\nu}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x){f_{{\vec{P},0}_{\alpha}}}^{*}(x)\bigg{]}.$
(77)

The amplitudes defined above are the starting point for developing the
perturbative electro-weak interactions in de Sitter metric. This include the
interesting phenomena related to particle production in early universe.
Finally we must point out that the amplitude obtained in the cases
$\lambda=\pm 1$ and $\lambda=0$ could be emerged in a single formula that
contains the contributions of the temporal part and spatial part of the Proca
field.

##  5 Probability of transition

In this section we compute the amplitude of the transition for the process of
electron-positron and Z boson generation from de Sitter vacuum, using the
solutions of the Proca equation corresponding to $\lambda=\pm 1$ or
$\lambda=0$. The first order transition amplitude corresponding to the process
$vac\rightarrow Z+e^{-}+e^{+}$, which is the spontaneous generation from the
de Sitter vacuum of a Z boson and an electron-positron pair is given by :

$\displaystyle A_{Ze\bar{e}}=\int
d^{4}x\sqrt{-g}\Bigg{\\{}\bigg{(}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f^{*}_{\mu\,\vec{P},\lambda}(x)$
(78) $\displaystyle-
e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f^{*}_{\mu\,\vec{P},\lambda}(x)\Bigg{\\}}$
(79)

where $e_{0}$ is the electric charge, $\theta_{W}$ is the Weinberg angle. We
specify that for the electrons and positrons we have the left-handed part with
the projector $\frac{1-\gamma^{5}}{2}$ and also the right-handed part and the
corresponding right projector $\frac{1+\gamma^{5}}{2}$. The Z boson has no
electric charge and the particle coincides with the antiparticle, and mediates
the neutral current interactions.

###  5.1 Amplitude of transition for $\lambda=\pm 1$

The modes with $\lambda=\pm 1$ only have the spatial components non-vanishing,
so the amplitude in this case is:

$\displaystyle\mathcal{A}_{Ze\overline{e}}(\lambda=\pm 1)$ $\displaystyle=$
$\displaystyle\int
d^{4}x\sqrt{-g}\,\biggl{\\{}\left(\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\right)\overline{U}_{\vec{p},\sigma}(x)\,\gamma^{\hat{i}}e_{\hat{i}}^{j}\left(\frac{1-\gamma^{5}}{2}\right)V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f_{j\vec{P},\lambda=\pm
1}^{*}(x)$
$\displaystyle-(e_{0}\tan(\theta_{W}))\overline{U}_{\vec{p},\sigma}(x)\,\gamma^{\hat{i}}e_{\hat{i}}^{j}\left(\frac{1+\gamma^{5}}{2}\right)V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f_{j\vec{P},\lambda=\pm
1}^{*}(x)\biggl{\\}}$

The amplitude of the process is computed using the equations (6), (18) and the
variable change $z=-t_{c}$ in the temporal integral, while the spatial
integral gives the momentum conservation in the process and we introduce the
sign function for the helicities:

$\displaystyle\mathcal{A}_{Ze\overline{e}}(\lambda=\pm
1)=\frac{\pi^{3/2}\sqrt{pp^{\prime}}\,e^{-\pi
k/2}}{8(2\pi)^{3/2}}\,\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})$ (81)
$\displaystyle\times\biggl{\\{}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\,sgn(\sigma^{\prime})\int_{0}^{\infty}dz\left[z^{3/2}H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)H^{(2)}_{-ik}\left(Pz\right)\right]$
$\displaystyle-
e_{0}\tan(\theta_{W})\,sgn(\sigma)\int_{0}^{\infty}dz\left[z^{3/2}H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)H^{(2)}_{-ik}\left(Pz\right)\right]\biggl{\\}}$
$\displaystyle\times\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\vec{\epsilon}\,^{*}\,(\vec{n}_{P},\lambda=\pm
1)\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})$

The delta Dirac function of momenta assures the momentum conservation in this
process. By using the equations (8) from Apendix we arrive at the final result

$\displaystyle\mathcal{A}_{Ze\overline{e}}(\lambda=\pm
1)=\frac{i^{-1/2}e_{0}}{(8\pi)}\,\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{\mathcal{P}})\,\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\vec{\epsilon}\,^{*}\,(\vec{n}_{\mathcal{P}},\lambda=\pm
1)\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})$
$\displaystyle\times\biggl{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\,sgn(\sigma^{\prime})A_{1}(\lambda=\pm
1)-\tan(\theta_{W})\,sgn(\sigma)A_{2}(\lambda=\pm 1)\biggl{\\}},$ (82)

where the functions $A_{1}(\lambda=\pm 1)\,,A_{2}(\lambda=\pm 1)$ are defined
as :

$\displaystyle A_{1}(\lambda=\pm 1)$ $\displaystyle=$ $\displaystyle
p^{\prime}B_{Kk}(pp^{\prime}P)-B_{1Kk}(pp^{\prime}P)$ (83) $\displaystyle-
B_{2Kk}(pp^{\prime}P)+pB_{-Kk}(pp^{\prime}P),$ $\displaystyle
A_{2}(\lambda=\pm 1)$ $\displaystyle=$ $\displaystyle
pB_{Kk}(pp^{\prime}P)-B_{1Kk}(pp^{\prime}P)$ (84) $\displaystyle-
B_{2Kk}(pp^{\prime}P)+p^{\prime}B_{-Kk}(pp^{\prime}P)$

and the $B$ functions are defined in terms of Appel hypergeometric functions
and gamma Euler functions:

$\displaystyle
B_{Kk}(pp^{\prime}P)=\frac{(pp^{\prime})^{-iK}(iP)^{-\frac{5}{2}+2iK}e^{\pi
K}}{\cosh^{2}(\pi
K)\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{5-4iK+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{5-4iK-2ik}{4}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{5-4iK+2ik}{4},\frac{5-4iK-2ik}{4},\frac{3}{2}-iK,\frac{1}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)};$
(85)

$\displaystyle
B_{1Kk}(pp^{\prime}P)=\frac{(p)^{-iK}(p^{\prime})^{iK}(iP)^{-3/2}}{\cosh^{2}(\pi
K)\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{1}{2}+iK\big{)}}\Gamma\bigg{(}\frac{3+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{3-2ik}{4}\bigg{)}$
$\displaystyle\ \times
F_{4}\bigg{(}\frac{3+2ik}{4},\frac{3-2ik}{4},\frac{1}{2}-iK,\frac{1}{2}+iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)};$
(86)

$\displaystyle
B_{2Kk}(pp^{\prime}P)=\frac{(p)^{1+iK}(p^{\prime})^{1-iK}(iP)^{-\frac{7}{2}}}{\cosh^{2}(\pi
K)\Gamma\big{(}\frac{3}{2}+iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{7+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{7-2ik}{4}\bigg{)}$
$\displaystyle\ \times
F_{4}\bigg{(}\frac{7+2ik}{4},\frac{7-2ik}{4},\frac{3}{2}+iK,\frac{3}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}.$
(87)

The Appel hypergeometric functions of double argument are less studied in
literature but the double infinite sums that define these functions are always
convergent when the two algebraic arguments are subunitary. In our case this
is translated in the ratio of momenta less than one and we will use this
observation in our further analysis. Our amplitude depends on both the ratios
between the fermion mass per expansion factor $\frac{m}{\omega}$ and the Z
boson mass per expansion factor $\frac{M_{Z}}{\omega}$ and these are the key
parameters that give the amplitude dependence of the space expansion. From our
mathematical results given in terms of Appel hypergeometric function we must
obtain the physical significance of these amplitudes and how the probabilities
change with the expansion parameter.

The probability of spontaneous generation of the triplet Z boson and electron-
positron pair from vacuum is defined by taking the square modulus of the
amplitude and sum after the helicities. Since the amplitude is proportional
with the delta Dirac function depending on momentum
$|\delta^{3}(\vec{p}\,)|^{2}=V\delta^{3}(\vec{p}\,)$ we will define the
probability in volume unit [51]:

$\displaystyle P(\lambda=\pm 1)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{64\pi^{2}}\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})\,\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}\Bigg{\\{}\bigg{(}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}^{2}|A_{1}|^{2}+\tan^{2}(\theta_{W})|A_{2}|^{2}$
(90)
$\displaystyle-\bigg{(}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}\tan(\theta_{W})sgn(\sigma^{\prime})sgn(\sigma)\big{[}A_{1}\cdot
A_{2}^{*}+A_{1}^{*}\cdot A_{2}\big{]}\Bigg{\\}}$
$\displaystyle\times|\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\vec{\epsilon}\,^{*}\,(\vec{n}_{\mathcal{P}},\lambda=\pm
1)\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})|^{2}.$

For a correct analysis we write each Appel hypergeometric function of double
arguments with its definition given in Appendix by equation (181), and expand
it as an infinite sum [23]. For example the functions from
$B_{Kk}(p,p^{\prime}P)$ could be written as

$\displaystyle
F_{4}\bigg{(}\frac{5-4iK+2ik}{4},\frac{5-4iK-2ik}{4},\frac{3}{2}-iK,\frac{1}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}=$
$\displaystyle\sum_{m,n=0}^{\infty}\frac{\Gamma\left(\frac{5-4iK+2ik}{4}+m+n\right)\Gamma\left(\frac{5-4iK-2ik}{4}+m+n\right)\Gamma\left(\frac{3}{2}-iK\right)\Gamma\left(\frac{1}{2}-iK\right)}{\Gamma\left(\frac{5-4iK+2ik}{4}\right)\Gamma\left(\frac{5-4iK-2ik}{4}\right)\Gamma\left(\frac{3}{2}-iK+m\right)\Gamma\left(\frac{1}{2}-iK+n\right)m!n!}\Big{(}\frac{p}{P}\Big{)}^{2m}\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2n}$

The above series is convergent for fixed values of the ratio
$\frac{m}{\omega}\,,\frac{M_{Z}}{\omega}$ when the momenta ratio are less than
one i.e. $\frac{p}{P}<1,\,\frac{p^{\prime}}{P}<1$. Our numerical calculations
prove that the series converges very rapidly and keeps constant values up to
infinity. For a complete analysis we will plot the probability in terms of the
parameters $\frac{m}{\omega}\,,\frac{M_{Z}}{\omega}$ for fixed ratios between
the momenta $\frac{p}{P}\,,\frac{p^{\prime}}{P}$ and we mention that in our
plots we use the above expansions for the Appel functions taking the sums from
zero to infinity. Thus we recover the exact variation of the probability with
the expansion parameter as follows in Figs.(1),(2).

Figure 1: Probability as a function of $m/\omega$ for $\lambda=\pm 1$ with
$M_{Z}/\omega=0.9$, $\sigma$ and $\sigma^{\prime}$ having the same sign and
opposite signs respectively. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Figure 2: Probability as a function of $M_{Z}/\omega$ for $\lambda=\pm 1$ with
$m/\omega=0.5$, $\sigma$ and $\sigma^{\prime}$ having the same sign and
opposite signs respectively. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Our graphs Figs.(1),(2) prove that the probability of spontaneous generation
from vacuum of the triplet Z boson and electron-positron pair is nonvanishing
only when the ratios between the masses and the expansion parameter have small
values and is vanishing when these ratios become bigger. For
$\frac{m}{\omega}\rightarrow\infty\,,\frac{M_{Z}}{\omega}\rightarrow\infty$
the probabilities vanish and we recover the Minkowski limit where this process
is forbidden as a perturbative process by the energy conservation law [16, 20,
21].

Another issue regarding our study is related to the computation of the total
probability obtained after solving the integrals after the final momenta in
equation (90). The functions that define the probabilities are written in
terms of complicated Appel hypergeometric functions that have as algebraic
argument ratios of momenta and for this reason it will be of interest to find
an approximation which can replace our functions $B$ with a more suitable
expression that allows us to obtain an analytical result. Let us analyse the
result for the probability. First a numerical and graphical analysis proves
that the graphs of the probability in terms of parameters $K,k$ are preserved
if we neglect the contributions of the Appel hypergeometric functions. This
can be verified by an analytical and numerical study and for large values of
the parameters $K,k$ the graphs for the probability remains the same, while
for small values there are small variations of the probabilities with the
specification that the profile of the curves are preserved as in Figs.(3),(4).
In this situation it is useful to work with the simplified $B$ functions that
can describe the behaviour of our amplitudes and probabilities in terms of any
values of parameters $m/\omega,\,M_{Z}/\omega$ as the functions defined in
equation (83):

$B_{Kk}(pp^{\prime}P)=\frac{(pp^{\prime})^{-iK}P^{-\frac{5}{2}+2iK}e^{\pi
K}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{5-4iK+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{5-4iK-2ik}{4}\bigg{)}$
(92)

$B_{1Kk}(pp^{\prime}P)=\frac{(p)^{-iK}(p^{\prime})^{iK}P^{-3/2}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{1}{2}+iK\big{)}}\Gamma\bigg{(}\frac{3+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{3-2ik}{4}\bigg{)}$
(93)

$B_{2Kk}(pp^{\prime}P)=\frac{(p)^{1+iK}(p^{\prime})^{1-iK}P^{-7/2}}{\Gamma\big{(}\frac{3}{2}+iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{7+2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{7-2ik}{4}\bigg{)}$
(94)

$B_{-Kk}(pp^{\prime}P)=\frac{(pp^{\prime})^{iK}P^{-\frac{5}{2}-2iK}e^{-\pi
K}}{\Gamma\big{(}\frac{1}{2}+iK\big{)}\Gamma\big{(}\frac{3}{2}+iK\big{)}}\Gamma\bigg{(}\frac{5+4iK-2ik}{4}\bigg{)}\Gamma\bigg{(}\frac{5+4iK+2ik}{4}\bigg{)}$
(95)

We specify that we present the above simplified functions for a future study
related to the computation of total probability in the general case, which is
a quantity dependent on the parameters $m/\omega,\,M_{Z}/\omega$. Even with
the simplified functions defined above the calculation of the total
probability is a complicated task since the resulted integrals will contain
momenta at imaginary powers and this leads to integrals that are not well
defined. For this reason it will be of interest to discuss the situation when
the expansion parameter is much larger than the particle masses. The above $B$
functions become very simple in the limit where the expansion parameter is
much larger than the particle mass $\omega>>m,M_{Z}$ and this represents the
limit of large expansion that is interesting for computing the probabilities
and density number of particles.

Figure 3: Probability as a function of $m/\omega$ for $\lambda=\pm 1$,
disregarding the $F_{4}$ functions, with $M_{Z}/\omega=0.9$, $\sigma$ and
$\sigma^{\prime}$ having the same sign and opposite signs respectively. The
dotted line is for $p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Figure 4: Probability as a function of $M_{Z}/\omega$ for $\lambda=\pm 1$,
disregarding the $F_{4}$ functions, with $m/\omega=0.5$, $\sigma$ and
$\sigma^{\prime}$ having the same sign and opposite signs respectively. The
dotted line is for $p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Let us comment on the results related to helicity. In the case when the
helicity of Z boson is $\lambda=1$ then for $\sigma=\sigma^{\prime}=-1/2$ the
helicity is conserved in the process and for the rest of the combinations of
$\sigma,\sigma^{\prime}$ the helicity conservation law is broken. The helicity
not being conserved is due to the fact that the particles have mass like in
Minkowski theory. Our graphical results prove that the processes which do not
conserve helicity have higher probability to happen and these results are
presented in the above right figures for $\sigma$ and $\sigma^{\prime}$ having
opposite signs.

###  5.2 Minkowski limit

In this section we will approach the Minkowski limit of our result for the
amplitude and probability. This is the limit when the expansion parameter
vanishes $\omega=0$, or in our notations the ratios between the masses of
particles and the expansion parameter become infinite i.e
$K=\frac{m}{\omega}\rightarrow\infty$,
$\frac{M_{Z}}{\omega}\rightarrow\infty$. First the functions
$B_{Kk}(pp^{\prime}P),B_{1Kk}(pp^{\prime}P),B_{2Kk}(pp^{\prime}P)$ that define
the amplitude could be approximated for large values of the parameters
$K=\frac{m}{\omega}$, $\frac{M_{Z}}{\omega}$. In all hypergeometric Appel
functions and gamma Euler functions we replace $ik=i\frac{M_{Z}}{\omega}$ for
$\frac{M_{Z}}{\omega}>>1$, and in addition we use the Stirling formula for
approximating the gamma Euler functions for large arguments

$\Gamma(z)\simeq e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}.$ (96)

The behaviour of these functions for large $\frac{m}{\omega}>>1$,
$\frac{M_{Z}}{\omega}>>1$ will be mainly determined by the factors in front of
the Appel functions and we obtain:

$\displaystyle
B_{K>>1k>>1}(pp^{\prime}P)=\frac{(pp^{\prime})^{-iK}P^{-\frac{5}{2}+2iK}e^{-(\frac{5-4iK}{2})}(1-i)\frac{\sqrt{2}}{2}}{e^{2\pi
K}e^{-2+2iK}\big{(}\frac{1}{2}-iK\big{)}^{-iK}\big{(}\frac{3}{2}-iK\big{)}^{1-iK}\frac{1}{4}\sqrt{25-16K^{2}-40iK+4\frac{M^{2}}{\omega^{2}}}}$
$\displaystyle\times\bigg{(}\frac{5-4iK+2i\frac{M}{\omega}}{4}\bigg{)}^{\frac{5-4iK+2i\frac{M}{\omega}}{4}}\bigg{(}\frac{5-4iK-2i\frac{M}{\omega}}{4}\bigg{)}^{\frac{5-4iK-2i\frac{M}{\omega}}{4}}$
$\displaystyle\times
F_{4}\bigg{(}\frac{5-4iK+2i\frac{M_{Z}}{\omega}}{4},\frac{5-4iK-2i\frac{M_{Z}}{\omega}}{4},\frac{3}{2}-iK,\frac{1}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(97)

$\displaystyle B_{1K>>1k>>1}(pp^{\prime}P)$ $\displaystyle=$
$\displaystyle\frac{-\big{(}\frac{p^{\prime}}{p}\big{)}^{iK}P^{-3/2}e^{-(\frac{3}{2})}(1+i)\sqrt{2}}{e^{\pi
K}\frac{1}{2}\sqrt{\left(\frac{M_{Z}}{\omega}\right)^{2}+\frac{9}{4}}}\bigg{(}\frac{3+2i\frac{M_{Z}}{\omega}}{4}\bigg{)}^{\frac{3+2i\frac{M_{Z}}{\omega}}{4}}\bigg{(}\frac{3-2i\frac{M_{Z}}{\omega}}{4}\bigg{)}^{\frac{3-2i\frac{M_{Z}}{\omega}}{4}}$
(98) $\displaystyle\times
F_{4}\bigg{(}\frac{3+2i\frac{M_{Z}}{\omega}}{4},\frac{3-2i\frac{M_{Z}}{\omega}}{4},\frac{1}{2}-iK,\frac{1}{2}+iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$

$\displaystyle B_{2K>>1k>>1}(pp^{\prime}P)$ $\displaystyle=$
$\displaystyle\frac{pp^{\prime}\big{(}\frac{p}{p^{\prime}}\big{)}^{iK}P^{-7/2}e^{-(\frac{7}{2})}(1+i)\sqrt{2}}{e^{\pi
K}\frac{1}{4}\sqrt{\left(\frac{M_{Z}}{\omega}\right)^{2}+\frac{49}{4}}}\bigg{(}\frac{7+2i\frac{M_{Z}}{\omega}}{4}\bigg{)}^{\frac{7+2i\frac{M_{Z}}{\omega}}{4}}\bigg{(}\frac{7-2i\frac{M_{Z}}{\omega}}{4}\bigg{)}^{\frac{7-2i\frac{M_{Z}}{\omega}}{4}}$
(99) $\displaystyle\times
F_{4}\bigg{(}\frac{7+2i\frac{M_{Z}}{\omega}}{4},\frac{7-2i\frac{M_{Z}}{\omega}}{4},\frac{3}{2}+iK,\frac{3}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$

From the above equations one can observe that in this limit the $B$ functions
vanish as $e^{-\pi m/\omega}$ and $,\frac{1}{M_{Z}/\omega}$ multiplied by
factors at imaginary powers, while the probabilities vanish as $e^{-2\pi
m/\omega}$ and $\frac{1}{(M_{Z}/\omega)^{2}}$.

A graphical analysis with the above approximated functions proves that both
the real and imaginary parts are very convergent in terms of parameters
$m/\omega,\,M_{Z}/\omega$. We specify that in our graphs we take one of the
ratios between the mass of the particle and the expansion factor to be greater
than one, with the observation that the behaviour remains the same for
subunitary values of $m/\omega,\,M_{Z}/\omega$. In the Minkowski limit the
amplitude and probability are vanishing, and we recover the well known fact
that in flat space-time the spontaneous particle generation from vacuum is
forbidden by the energy-momentum conservation.

Figure 5: The real and imaginary parts of the $B_{Kk},B_{1Kk}$ as functions of
$m/\omega$ with $M_{Z}/\omega=10$, $p=0.1,p^{\prime}=0.2,P=1$. The dotted
lines represent the imaginary parts, and the solid lines the real parts.

Figure 6: The real and imaginary parts of the $B_{2Kk}$ functions, and the
square modulus of the sum $B_{Kk}+B_{1Kk}+B_{2Kk}$ as functions of $m/\omega$
with $M_{Z}/\omega=10$, $p=0.1,p^{\prime}=0.2,P=1$. The dotted lines represent
the imaginary parts, and the solid lines the real parts.

Figure 7: The real and imaginary parts of the $B_{Kk},B_{1Kk}$ as functions of
$M_{Z}/\omega$ with $m/\omega=10$, $p=0.1,p^{\prime}=0.2,P=1$. The dotted
lines represent the imaginary parts, and the solid lines the real parts.

Figure 8: The real and imaginary parts of the $B_{2Kk}$ functions, and the
square modulus of the sum $B_{Kk}+B_{1Kk}+B_{2Kk}$ as functions of
$M_{Z}/\omega$ with $m/\omega=10$, $p=0.1,p^{\prime}=0.2,P=1$. The dotted
lines represent the imaginary parts, and the solid lines the real parts.

###  5.3 Total probability for $\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0$

The limit where $m/\omega\sim M_{Z}/\omega\rightarrow 0$ is important since in
this situation the gravitational field is large, and this limit is interesting
for the phenomena related to the early universe. In this limit the probability
of transition in volume unit for the case $\lambda=\pm 1$ is reduced to:

$\displaystyle P(\lambda=\pm
1)=\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}|A|^{2}=\frac{e^{2}}{64\pi^{3}}\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}|\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\vec{\epsilon}\,^{*}\,(\vec{n}_{\mathcal{P}},\lambda=\pm
1)\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})|^{2}$
$\displaystyle\times\frac{1}{P(p+p^{\prime}+P)^{2}}\left[\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})-2\tan(\theta_{W})\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma)sgn(\sigma^{\prime})\right]$
(100)

since $A_{1}(\lambda=\pm 1)=A_{2}(\lambda=\pm 1)$ in the limit
$m/\omega=M_{Z}/\omega=0$. The total probability is obtained by performing the
integrals over the final momenta. We will consider the case when the electron
and positron are emitted on the same direction that is the $z$ axis, but move
in opposite senses in such a way that the angle between the momenta vectors is
$\pi$ i.e
$\vec{p}=p\,\vec{e}_{3}\,,\,\vec{p}\,^{\prime}=-p^{\prime}\,\vec{e}_{3}$. In
this particular case the bispinor summation is reduced to a number since the
helicity bispinors are reduced to a simple column matrix with elements
$0,\,\pm 1$. The momentum of the Z boson is also considered on the third axis
such that $\vec{P}=P\,\vec{e}_{3}$.

In a curved space the fact that the four momentum has constant length
$g_{\mu\nu}p^{\mu}p^{\nu}=m^{2}$, we have as consequence that
$g_{00}p^{0}=p_{0}-g_{0i}p^{i}$, with
$p_{0}=\sqrt{g_{00}m^{2}+(g_{0i}g_{0j}-g_{00}g_{ij})p^{i}p^{j}}$. Then at a
coordinate transformation the contravariant components of the momentum
$p^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}p^{\nu}$, the
invariant element of volume in the space phase determined by space and
momentum coordinates is [65]:

$\sqrt{-g}\frac{d^{3}p}{p_{0}}$ (101)

This invariant element of volume was used to define the particle distribution
function [65] and we mention that it is the volume element defined with the
contravariant components of momentum $d^{3}p=dp^{1}dp^{2}dp^{3}$. It is also
important to mention that this invariant volume is always multiplied by a
function dependent on coordinate such as the distribution functions in the
theory of relativistic Boltzmann equation [65] and all the equations can be
expressed in terms of the momenta defined in a local frame. We adopt the
definition for the volume elements as in equation (101), and express the
momenta integrals that results from integrating equation (5.3) in terms of
physical momentum $\widehat{p}=e^{-\omega t}p$, up to some factors dependent
on time that came out from the delta Dirac function and the factor
$\frac{1}{P(p+p^{\prime}+P)^{2}}$ which depends on conserved momentum, and we
find that the integral is of the form:

$\widehat{I}\sim\int\frac{d^{3}\widehat{p}}{\sqrt{m^{2}+\widehat{p}^{2}}}\int\frac{d^{3}\widehat{P}}{\sqrt{M_{Z}^{2}+\widehat{P}^{2}}}\int\frac{d^{3}\widehat{p^{\prime}}}{\sqrt{m^{2}+\widehat{p^{\prime}}^{2}}}\frac{\delta^{3}(\widehat{\vec{p}}+\widehat{\vec{p}\,^{\prime}}+\widehat{\vec{P}})}{(2\pi)^{3}\widehat{P}(\widehat{p}+\widehat{p^{\prime}}+\widehat{P})^{2}}.$
(102)

To obtain the correct dimensionality in our total probability we will
implement a similar definition for the volume element in momentum space, but
defined with the conserved momentum so that we will add a factor of
$\frac{1}{p_{0}}=\frac{1}{\sqrt{m^{2}+p^{2}}}$, with $p^{2}=g^{ij}p_{i}p_{j}$,
to each volume element in momentum space. One can also think that
$\frac{1}{p_{0}}$ factors are in fact similar to the density of the states
function which in Minkowski theory has a dimension of $(energy\cdot
volume)^{-1}$.

Then the total probability is defined as the following integral after the
final momenta of the probability obtained in (5.3):

$P_{tot}=\int\frac{d^{3}p}{p_{0}}\int\frac{d^{3}p^{\prime}}{p^{\prime}_{0}}\int\frac{d^{3}P}{P_{0}}\,P(\lambda=\pm
1)$ (103)

This will led us to the correct dimension for the total probability and
transition rate. Since the probability contains ultra-violet divergences, we
will consider the case of ultra-relativistic momenta i.e. $p>>m$ for
$p^{\prime},P$ integrals, such that $p^{\prime}_{0}=p^{\prime},\,P_{0}=P$,
while for the electron momenta we preserve the mass dependence i.e.
$p_{0}=\sqrt{m^{2}+p^{2}}$. The momenta integrals that need to be computed in
order to obtain the total probability are:

$I=\int\frac{d^{3}p}{\sqrt{m^{2}+p^{2}}}\int\frac{d^{3}P}{P}\int\frac{d^{3}p^{\prime}}{p^{\prime}}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{(2\pi)^{3}P(p+p^{\prime}+P)^{2}}=\int\frac{d^{3}p}{\sqrt{m^{2}+p^{2}}}\int\frac{d^{3}P}{P^{2}}\frac{1}{4(2\pi)^{3}(p+P)^{3}}$
(104)

where we use the setup with the particles moving on the same direction
$|\vec{p}+\vec{P}|=p+P$.

Using equation (183) from Appendix we arrive at the final form for the $p$
integral:

$\displaystyle
I=2\pi\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{p^{2}\sqrt{p^{2}+m^{2}}}.$ (105)

The above integrals contain ultraviolet logarithmic divergences. The structure
of the above integrals is not the same as in Minkowski field theory since the
four momentum integral is missing. To obtain the result of the integral we
apply the dimensional regularization proposed in [29, 30, 31, 32], which
allows us to change the measure of momentum integration so that the dimension
D in the integrals can be an arbitrary complex number. We also specify that
this method was used to study the propagators in de Sitter space-time [26, 25,
24]. Let us recall the well known result of the integral after $p$ in $D$
dimensions adapted to our situation:

$\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{p^{2}\sqrt{p^{2}+m^{2}}}\rightarrow
I(D)=\int\frac{d^{D}p}{(2\pi)^{D}}\frac{1}{p^{2}\sqrt{p^{2}+m^{2}}}=\frac{S_{D}}{(2\pi)^{D}}\int_{0}^{\infty}dp\frac{p^{D-1}}{p^{2}\sqrt{p^{2}+m^{2}}},$
(106)

where $S_{D}$ is the result of the angular integrals in $D$ dimensions

$S_{D}=\frac{2\pi^{D/2}}{\Gamma(D/2)}$ (107)

Then the variable is changed such that $p^{2}=ym^{2}$ and we arrive at the new
integrals

$I(D)=\frac{S_{D}\,m^{D-3}}{2(2\pi)^{D}}\int_{0}^{\infty}dy\frac{y^{D/2-2}}{(1+y)^{1/2}}.$
(108)

One can further use the integral of the Beta Euler function [22, 23]

$\displaystyle
B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}=\int_{0}^{\infty}dy\frac{y^{a-1}}{(1+y)^{a+b}}$
(109)

which in our case takes the values for $a=D/2-1\,,b=\frac{3-D}{2}$ and the
final result reads [29, 30, 35]:

$\displaystyle
I(D)=\frac{2\pi^{D/2}m^{D-3}\Gamma(D/2-1)\Gamma(\frac{3-D}{2})}{2(2\pi)^{D}\Gamma(D/2)\Gamma(1/2)}$
(110)

First is important to mention that the same result is obtained if we consider
that the momenta of electron-positron pair is large and in equation (103) we
take $p_{0}=p,\,p^{\prime}_{0}=p^{\prime},\,P_{0}=\sqrt{M_{Z}^{2}+P^{2}}$, and
in all the above equations we make the substitution $m\rightarrow M_{Z}$. The
above integral contains ultraviolet divergences for $D\geq 3$ and infrared
divergences for $D\leq 0$, and these divergences are contained in the poles of
the gamma Euler functions. In general these poles could be extracted by
introducing an arbitrary mass parameter denoted by $\mu$ and an arbitrary
coupling dimensionless constant denoted by $g$ so that our initial integral is
replaced by:

$I(D)_{r}=-\lambda\int\frac{d^{D}p}{(2\pi)^{D}}\frac{1}{p^{2}\sqrt{p^{2}+m^{2}}}=-\lambda\frac{m^{D-3}\Gamma(D/2-1)\Gamma(\frac{3-D}{2})}{\sqrt{\pi}(4\pi)^{D/2}\Gamma(D/2)}$
(111)

where,

$g=\lambda\mu^{D-3}=\lambda\mu^{-\varepsilon}$ (112)

and we write the result of the integral (110) as a function of $g$ and
$\varepsilon$ for $D=3-\varepsilon$ to adapt to our integral given in equation
(105). Then the regularized integral gives:

$I(D)_{r}=-\frac{g}{(4\pi)^{3/2}}\left(\frac{4\pi\mu^{2}}{m^{2}}\right)^{\varepsilon/2}\frac{\Gamma(\frac{1-\varepsilon}{2})\Gamma(\frac{\varepsilon}{2})}{\sqrt{\pi}\Gamma(\frac{3-\varepsilon}{2})}.$
(113)

The expansion in powers of $\varepsilon$

$\left(\frac{4\pi\mu^{2}}{m^{2}}\right)^{\varepsilon/2}=1+\frac{\varepsilon}{2}\ln\left(\frac{4\pi\mu^{2}}{M_{Z}^{2}}\right)+o(\varepsilon^{2})$
(114)

while the gamma Euler functions can be written as:

$\frac{\Gamma(\frac{1-\varepsilon}{2})\Gamma(\frac{\varepsilon}{2})}{\Gamma(\frac{3-\varepsilon}{2})}=\frac{4}{\varepsilon}-2[\gamma+\psi(-1/2)+\psi(3/2)]+o(\varepsilon),$
(115)

The final result for $I(D)_{r}$ is written in terms of the Euler digamma
function $\psi$ and we restrict it to the terms proportional with
$\varepsilon$

$\displaystyle
I(D)_{r}=\frac{g}{(4\pi^{2})}\left[\frac{2}{\varepsilon}+2-\gamma+\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)+o(\varepsilon)\right]$
(116)

The result obtained above is divergent for $\varepsilon=0$ and finite for
arbitrary small values of $\varepsilon$, and we observe that the dimensional
regularization does not removes all divergences from our integral when $D=3$.

In order to resolve this divergence in the total probability we will use the
method of minimal substraction proposed in [32, 33]. Let us recall the result
obtained in equation (110), where we used the relation with gamma Euler
functions $z\Gamma(z)=\Gamma(1+z)$ for the divergent gamma function
$\Gamma\big{(}\frac{3-D}{2}\big{)}$ in $D=3$:

$\displaystyle\bigg{(}\frac{3-D}{2}\bigg{)}\Gamma\bigg{(}\frac{3-D}{2}\bigg{)}=\Gamma\bigg{(}\frac{5-D}{2}\bigg{)};$
$\displaystyle\Gamma\bigg{(}\frac{3-D}{2}\bigg{)}=\frac{2\Gamma\big{(}\frac{5-D}{2}\big{)}}{(3-D)}.$
(117)

We rewrite the result for $I(D)$ in the form:

$I(D)=\frac{2m^{D-3}\Gamma\big{(}\frac{D}{2}-1\big{)}\Gamma\big{(}\frac{5-D}{2}\big{)}}{\sqrt{\pi}\Gamma\big{(}\frac{D}{2}\big{)}(4\pi)^{D/2}(D-3)}.$
(118)

The above function has a pole in $D=3$ with residue:

$R=\lim_{D\rightarrow 3}(D-3)I(D)=\frac{1}{2\pi^{2}}$ (119)

Then we choose the a counter-term where we introduce the mass parameter $\mu$
of the form $\frac{\mu^{s}R}{D-3}$, where $s$ is taken such that this term has
the same dimension as $I(D)$. Then we define the renormalized integral as:

$\displaystyle I(D)_{ren}$ $\displaystyle=$
$\displaystyle-I(D)+\frac{\mu^{D-3}}{2\pi^{2}(D-3)}$ (120) $\displaystyle=-$
$\displaystyle\frac{1}{(D-3)}\Bigg{[}\frac{2m^{D-3}\Gamma\big{(}\frac{5-D}{2}\big{)}\Gamma\big{(}\frac{D}{2}-1\big{)}}{\sqrt{\pi}(4\pi)^{\frac{D}{2}}\Gamma\big{(}\frac{D}{2}-1\big{)}}-\frac{\mu^{D-3}}{2\pi^{2}}\Bigg{]}.$

The parenthesis from the above equation can be expanded around $D=3$

$\displaystyle\frac{2m^{D-3}\Gamma\big{(}\frac{5-D}{2}\big{)}\Gamma\big{(}\frac{D}{2}-1\big{)}}{\sqrt{\pi}(4\pi)^{\frac{D}{2}}\Gamma\big{(}\frac{D}{2}-1\big{)}}-\frac{\mu^{D-3}}{2\pi^{2}}$
$\displaystyle=$
$\displaystyle\frac{D-3}{4\pi^{2}}\left[-\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)-\psi(3/2)+\psi(1/2)\right]+o((D-3)^{2})$
(121) $\displaystyle=$
$\displaystyle\frac{D-3}{4\pi^{2}}\left[-2+\gamma-\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)\right]+o((D-3)^{2})$

The divergent term $D-3$ is canceled by the expansion around $D=3$, and the
final result for the renormalised integral will be finite:

$I(D)_{ren}=\frac{1}{4\pi^{2}}\left[2-\gamma+\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)\right],$
(122)

where we point out that parenthesis contain the exact finite terms obtained in
equation (116).

The total probability is obtained by collecting the results from equation
(122), with the observation that the summation after
$\lambda,\sigma,\sigma^{\prime}$ is included in our result:

$\displaystyle P_{tot}$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{64\cdot
2\pi}\left[2-\gamma+\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)\right]\left[\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})\right]$
$\displaystyle=$
$\displaystyle\frac{M_{W}^{2}G_{F}\sin^{2}\theta_{W}}{16\sqrt{2}\pi}\left[2-\gamma+\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)\right]\left[\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})\right]$

where in the second equality we introduce the Fermi constant $G_{F}$ i.e.
$e^{2}=4\sqrt{2}M_{W}^{2}G_{F}\sin^{2}\theta_{W}$, with $M_{W}$ the mass of
the W boson. We prove that the total probability is finite by using the
dimensional regularization and the substraction method. Our integrals contain
divergences of the form $\frac{1}{(D-3)^{n}}$ that can be removed by using the
minimal substraction method [32, 33].

The counter-term can also be cast in the lagrangean density, but we restrict
to define the regularized total probability defined in equation (5.3). However
in a future work we want to study all the first order processes to obtain all
counter-terms in order to propose a renormalization scheme based on
dimensional regularization, in the limit of large expansion when $\omega>>m$.

###  5.4 Amplitude of transition for $\lambda=0$

In the case of production of Z bosons with longitudinal polarization the
amplitude of transition contains both the spatial part of the solution and
temporal parts of the amplitude reads:

$\displaystyle A_{Ze\bar{e}}(\lambda=0)$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\Bigg{\\{}\bigg{(}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{0}}e^{0}_{\hat{0}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f^{*}_{0\vec{P},\lambda=0}(x)$
$\displaystyle-
e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{0}}e^{0}_{\hat{0}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)f^{*}_{0\vec{P},\lambda=0}(x)\Bigg{\\}}$
$\displaystyle+\int
d^{4}x\sqrt{-g}\Bigg{\\{}\bigg{(}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{i}}e^{j}_{\hat{i}}\bigg{(}\frac{1-\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)\vec{f}^{*}_{\vec{P},\lambda=0}(x)$
$\displaystyle-
e_{0}\tan(\theta_{W})\overline{U}_{\vec{p},\sigma}(x)\gamma^{\hat{i}}e^{j}_{\hat{i}}\bigg{(}\frac{1+\gamma^{5}}{2}\bigg{)}V_{\vec{p}\,^{\prime},\sigma^{\prime}}(x)\vec{f}^{*}_{\vec{P},\lambda=0}(x)\Bigg{\\}}=A_{1}(\lambda=0)+A_{2}(\lambda=0).$

The integrals that define the $A_{1}(\lambda=0)\,,A_{2}(\lambda=0)$ are:

$\displaystyle A_{1}(\lambda=0)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{pp^{\prime}}}{8(2\pi)^{3/2}}\frac{\sqrt{\pi}\omega
Pe^{-\pi
k/2}}{M_{Z}}\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})\int_{0}^{\infty}dz\cdot
z^{5/2}$
$\displaystyle\times\Bigg{\\{}-\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)$
$\displaystyle+e_{0}\tan(\theta_{W})sgn(\sigma)H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)\Bigg{\\}}\xi^{+}_{\sigma}(\vec{p})\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime});$
$\displaystyle A_{2}(\lambda=0)$ $\displaystyle=$
$\displaystyle\frac{-i\sqrt{pp^{\prime}}\omega Pe^{-\pi
k/2}}{8M_{Z}(2\pi)^{3/2}}\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})\Bigg{\\{}\int_{0}^{\infty}dz\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})$
(126)
$\displaystyle\times\bigg{[}\bigg{(}\frac{1}{2}-ik\bigg{)}\frac{z^{3/2}}{P}H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)-z^{/5/2}H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)H^{(2)}_{1-ik}(Pz)\bigg{]}$
$\displaystyle-\int_{0}^{\infty}dz\cdot
e_{0}\tan(\theta_{W})sgn(\sigma)\bigg{[}\bigg{(}\frac{1}{2}-ik\bigg{)}\frac{z^{3/2}}{P}H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)$
$\displaystyle-z^{5/2}H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)H^{(2)}_{1-ik}(Pz)\bigg{]}\Bigg{\\}}\xi^{+}_{\sigma}(\vec{p})\vec{\sigma}\cdot\vec{\epsilon}^{*}(\vec{n}_{P},\lambda)\eta_{\sigma}(\vec{p}\,^{\prime})$

The integrals can be solved by using the same method as in the previous case,
and the final result for obtaining the production of longitudinal modes is
given by:

$A_{1}(\lambda=0)=\frac{i}{4\pi^{2}}\frac{\omega
P}{M_{Z}}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{\cosh^{2}(\pi
K)}\Bigg{\\{}-\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})B_{1}-e_{0}\tan(\theta_{W})sgn(\sigma)B_{2}\Bigg{\\}}\xi^{+}_{\sigma}(\vec{p})\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})$
(127)

$\displaystyle A_{2}(\lambda=0)$ $\displaystyle=$
$\displaystyle\frac{P}{4\pi^{5/2}}\frac{\omega}{M_{Z}}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{\cosh^{2}(\pi
K)}\Bigg{\\{}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})\bigg{[}\bigg{(}\frac{1}{2}-ik\bigg{)}B_{1}+C_{1}\bigg{]}$
$\displaystyle-
e_{0}\tan(\theta_{W})sgn(\sigma)\bigg{[}\bigg{(}\frac{1}{2}-ik\bigg{)}B_{2}+C_{2}\bigg{]}\Bigg{\\}}\xi^{+}_{\sigma}(\vec{p})\vec{\sigma}\cdot\vec{\epsilon}^{*}(\vec{n}_{P},\lambda)\eta_{\sigma}(\vec{p}\,^{\prime})$

where the functions $B\,,C$ are defined as:

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle
p^{\prime}B_{Kk}(pp^{\prime}P)+pB_{-Kk}(pp^{\prime}P)-A_{Kk}(pp^{\prime}P)+C_{Kk}(pp^{\prime}P)$
(129) $\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle
pB_{Kk}(pp^{\prime}P)+p^{\prime}B_{-Kk}(pp^{\prime}P)-A_{Kk}(pp^{\prime}P)+C_{Kk}(pp^{\prime}P)$
(130) $\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
p^{\prime}D_{Kk}(pp^{\prime}P)+pD_{-Kk}(pp^{\prime}P)-F_{Kk}(pp^{\prime}P)+G_{Kk}(pp^{\prime}P)$
(131) $\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
pD_{Kk}(pp^{\prime}P)+p^{\prime}D_{-Kk}(pp^{\prime}P)-F_{Kk}(pp^{\prime}P)+G_{Kk}(pp^{\prime}P).$
(132)

Each function from the above relations is a combination of Appel
hypergeometric function and gamma Euler functions as follows:

$\displaystyle B_{Kk}(pp^{\prime}P)=\frac{i(pp^{\prime})^{-iK}e^{\pi
K}(iP)^{-\frac{7}{2}+2iK}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{7}{4}-iK-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{7}{4}-iK+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{7}{4}-iK-\frac{ik}{2},\frac{7}{4}-iK+\frac{ik}{2},\frac{1}{2}-iK,\frac{3}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(133)

$\displaystyle
A_{Kk}(pp^{\prime}P)=\frac{(p)^{-iK}(p^{\prime})^{iK}(iP)^{-5/2}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{1}{2}+iK\big{)}}\Gamma\bigg{(}\frac{5}{4}-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{5}{4}+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{5}{4}+\frac{ik}{2},\frac{5}{4}-\frac{ik}{2},\frac{1}{2}-iK,\frac{1}{2}+iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(134)

$\displaystyle
C_{Kk}(pp^{\prime}P)=\frac{(p)^{1+iK}(p^{\prime})^{1-iK}(iP)^{-9/2}}{\Gamma\big{(}\frac{3}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}+iK\big{)}}\Gamma\bigg{(}\frac{9}{4}-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{9}{4}+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{9}{4}+\frac{ik}{2},\frac{9}{4}-\frac{ik}{2},\frac{3}{2}-iK,\frac{3}{2}+iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(135)

$\displaystyle D_{Kk}(pp^{\prime}P)=\frac{i(pp^{\prime})^{-iK}e^{\pi
K}(iP)^{-\frac{7}{2}+2iK}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}-iK\big{)}}\Gamma\bigg{(}\frac{9}{4}-iK-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{5}{4}-iK+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{5}{4}-iK+\frac{ik}{2},\frac{9}{4}-iK-\frac{ik}{2},\frac{1}{2}-iK,\frac{3}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(136)

$\displaystyle
F_{Kk}(pp^{\prime}P)=\frac{(p)^{-iK}(p^{\prime})^{iK}(iP)^{-5/2}}{\Gamma\big{(}\frac{1}{2}-iK\big{)}\Gamma\big{(}\frac{1}{2}+iK\big{)}}\Gamma\bigg{(}\frac{7}{4}-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{3}{4}+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{3}{4}+\frac{ik}{2},\frac{7}{4}-\frac{ik}{2},\frac{1}{2}-iK,\frac{1}{2}+iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(137)

$\displaystyle
G_{Kk}(pp^{\prime}P)=\frac{(p)^{1+iK}(p^{\prime})^{1-iK}(iP)^{-9/2}}{\Gamma\big{(}\frac{3}{2}-iK\big{)}\Gamma\big{(}\frac{3}{2}+iK\big{)}}\Gamma\bigg{(}\frac{11}{4}-\frac{ik}{2}\bigg{)}\Gamma\bigg{(}\frac{7}{4}+\frac{ik}{2}\bigg{)}$
$\displaystyle\times
F_{4}\bigg{(}\frac{7}{4}+\frac{ik}{2},\frac{11}{4}-\frac{ik}{2},\frac{3}{2}+iK,\frac{3}{2}-iK;\Big{(}\frac{p}{P}\Big{)}^{2},\Big{(}\frac{p^{\prime}}{P}\Big{)}^{2}\bigg{)}$
(138)

The probability of transition in volume unit is computed by summing after the
helicities $\sigma\,,\sigma^{\prime}$ and we obtain:

$\displaystyle
P(\lambda=0)=\frac{1}{4}\sum_{\sigma\sigma^{\prime}}\left[|A_{1}(\lambda=0)|^{2}+|A_{2}(\lambda=0)|^{2}+A_{1}^{*}(\lambda=0)A_{2}(\lambda=0)+A_{1}(\lambda=0)A_{2}^{*}(\lambda=0)\right]$

The probability from equation (5.4) depends on the masses of the particles,
expansion parameter and particles momenta, and the variation with the
expansion parameter is given in Figs.(9),(10)

Figure 9: Probability as a function of $m/\omega$ for $\lambda=0$ with
$M_{Z}/\omega=0.9$, $\sigma$ and $\sigma^{\prime}$ having the same sign in the
left figure and opposite signs in the right figure. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Figure 10: Probability as a function of $M_{Z}/\omega$ for $\lambda=0$ with
$m/\omega=0.5$, $\sigma$ and $\sigma^{\prime}$ having the same sign in the
left figure and opposite signs in the right figure. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

The above results prove that the emission of longitudinal modes is possible
for small values of the ratio between the masses and the expansion parameter.
The probabilities are also vanishing in the Minkowski limit. For a clear
picture of the ratios between masses and expansion parameter let us give the
ratio $mc^{2}/\hbar\omega\sim 5\cdot 10^{37}$, (where $m$ is the electron
mass). From this we observe that for large masses comparatively to the
expansion factor, the probability of pair production at present is zero and
our results refer to the early universe when the expansion parameter was
bigger or around the same value of the particle mass.

##  6 Transition rates

In this paper we analyse the problem of generating Z bosons and massive
fermions by using the perturbative transition amplitude generalized to a
curved space-time. We use the electro-weak coupling with the neutral currents
for analysing the spontaneous generation from vacuum of the massive Z bosons
and massive fermions. The analytical and graphical results prove that the
generation of massive Z bosons is a process that takes place along the entire
inflation. This result is important in establishing the mechanisms that were
involved in the generation of massive bosons in the early universe. A
remarkable result that we have obtained is that in the Minkowski limit the
probabilities are vanishing for both the transversal modes and longitudinal
modes, a result that confirms the well established results from electro-weak
theory where the process analysed here is forbidden by the energy
conservation. In the non-stationary de Sitter metric this process is possible
only at large expansion when the mass of the particle is close in value to the
expansion parameter. We will compute the transition rate in de Sitter geometry
for any values of the ratios between the particle masses and expansion
parameter. For defining the transition rate we work in the chart with
conformal time where the definitions could be adapted from Minkowski space-
time [56]

$R_{if}=\lim_{t_{c}\rightarrow
0}\frac{1}{2V}\frac{d}{dt_{c}}|A_{if}|^{2}=\lim_{t\rightarrow\infty}\frac{e^{\omega
t}}{2V}\frac{d}{dt}|A_{if}|^{2}.$ (140)

Equation for transition amplitude in the case $\lambda=\pm 1$ can be written
in the form

$A_{i\rightarrow f}(\lambda=\pm
1)=\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})M_{i\rightarrow
f}I_{i\rightarrow f}$ (141)

where $I_{i\rightarrow f}$ are the temporal integrals and $M_{i\rightarrow f}$
contain the constants and the bispinors. These functions are given bellow:

$M_{i\rightarrow
f}=\frac{e_{0}\pi^{3/2}\sqrt{pp^{\prime}}}{8(2\pi)^{3/2}}\,\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\cdot\vec{\epsilon}\,^{*}\,(\vec{n}_{P},\lambda=\pm
1)\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})$ (142)

$\displaystyle I_{i\rightarrow f}$ $\displaystyle=$
$\displaystyle\biggl{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\,sgn(\sigma^{\prime})\int_{0}^{\infty}dz\left[z^{3/2}H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)K_{-ik}\left(iPz\right)\right]$
(143)
$\displaystyle-\tan(\theta_{W})\,sgn(\sigma)\int_{0}^{\infty}dz\left[z^{3/2}H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)K_{-ik}\left(iPz\right)\right]\biggl{\\}}.$

The temporal integrals can be written in the form $I_{i\rightarrow
f}=\int_{0}^{\infty}\mathcal{K}_{if}dz$, where we denote the integrand by
$\mathcal{K}_{if}$

$\displaystyle\mathcal{K}_{if}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{2i}{\pi}\Big{)}\biggl{[}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\,sgn(\sigma^{\prime})z^{3/2}H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)K_{-ik}\left(iPz\right)$
(145)
$\displaystyle-\tan(\theta_{W})\,sgn(\sigma)z^{3/2}H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)K_{-ik}\left(iPz\right)\biggl{]}$

Then the rate of transition for triplet generation from vacuum can be written
in terms of the above quantities as:

$R_{if}=\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}\frac{1}{(2\pi)^{3}}\,\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})|M_{i\rightarrow
f}|^{2}|I_{i\rightarrow f}|\lim_{t\rightarrow\infty}|e^{\omega
t}\mathcal{K}_{if}|,$ (146)

We consider that the transition from $in$ to $out$ state takes place after a
sufficiently large time and we denote this time by $t_{\infty}$ and the limits
from the rate equation (146) will be evaluated for this time. For computing
the limit we use the expansion of Hankel functions for small arguments since
for $t\rightarrow\infty$ the argument $z=e^{-\omega t}$ become very small

$H^{(1,2)}_{\nu}(z)=\mp
i\left(\frac{2}{z}\right)^{\nu}\frac{\Gamma(\nu)}{\pi}.$ (147)

In the case of Z boson we use the approximation when $M_{Z}/\omega<<1/2$ and
the index of Hankel functions becomes $-ik\rightarrow\frac{1}{2}$. The limit
that defines our transition rate in the general case becomes:

$\displaystyle\lim_{t\rightarrow t_{\infty}}|e^{\omega
t}\mathcal{K}_{if}|=\Big{(}\frac{2}{\pi}\Big{)}\lim_{t\rightarrow+t_{\infty}}\bigg{|}\bigg{(}-\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\frac{sgn(\sigma^{\prime})}{\pi^{2}}\Big{(}\frac{e^{-\omega
t}}{\omega}\Big{)}^{3/2}\Big{(}\frac{2}{p\,\frac{e^{-\omega
t}}{\omega}}\Big{)}^{\frac{1}{2}+iK}\Big{(}\frac{2}{p^{\prime}\,\frac{e^{-\omega
t}}{\omega}}\Big{)}^{\frac{1}{2}-iK}$
$\displaystyle\times\frac{\Gamma\big{(}\frac{1}{2}\big{)}\big{|}\Gamma\big{(}\frac{1}{2}-iK\big{)}\big{|}^{2}}{\sqrt{2}\big{(}\frac{iP}{\omega}e^{-\omega
t}\big{)}^{1/2}}+\tan(\theta_{W})\frac{sgn(\sigma^{\prime})}{\pi^{2}}\Big{(}\frac{e^{-\omega
t}}{\omega}\Big{)}^{3/2}\Big{(}\frac{2}{p\,\frac{e^{-\omega
t}}{\omega}}\Big{)}^{\frac{1}{2}-iK}\Big{(}\frac{2}{p^{\prime}\,\frac{e^{-\omega
t}}{\omega}}\Big{)}^{\frac{1}{2}+iK}$
$\displaystyle\times\frac{\Gamma\big{(}\frac{1}{2}\big{)}\big{|}\Gamma\big{(}\frac{1}{2}-iK\big{)}\big{|}^{2}}{\sqrt{2}\big{(}\frac{iP}{\omega}e^{-\omega
t}\big{)}^{1/2}}\bigg{)}\bigg{|}.$ (148)

The final result for the limit is obtained after extracting the modulus from
the imaginary quantity and this gives:

$\displaystyle\lim_{t\rightarrow t_{\infty}}|e^{\omega t}\mathcal{K}_{if}|$
$\displaystyle=$
$\displaystyle\Big{(}\frac{2}{\pi}\Big{)}^{3/2}\frac{1}{\cosh(\pi
K)\sqrt{pp^{\prime}P}}\bigg{[}\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})$
$\displaystyle-
sgn(\sigma)sgn(\sigma^{\prime})\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\tan(\theta_{W})2\cos\Big{(}2K\ln\Big{(}\frac{p^{\prime}}{p}\Big{)}\Big{)}\bigg{]}^{1/2}$

Collecting all the results we obtain for the transition rate

$\displaystyle
R=\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{(2\pi)^{3}}\sqrt{\mathcal{I}_{i\rightarrow
f}\cdot\mathcal{I}^{*}_{i\rightarrow
f}}\sqrt{\frac{2}{\pi}}\,\frac{1}{\cosh(\pi
K)\sqrt{pp^{\prime}P}}\,\frac{e^{2}\,pp^{\prime}}{8\pi^{2}}\sum_{\sigma\sigma^{\prime}\lambda}\big{|}\xi_{\sigma}^{+}(\vec{p}\,)\,\vec{\sigma}\cdot\vec{\epsilon}\,^{*}\,\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime})\big{|}^{2}$
(150)
$\displaystyle\bigg{[}\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})-sgn(\sigma)sgn(\sigma^{\prime})\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\tan(\theta_{W})\,2\cos\Big{(}2K\ln\Big{(}\frac{p^{\prime}}{p}\Big{)}\Big{)}\bigg{]}^{1/2}$

where we introduce the notation $\mathcal{I}_{i\rightarrow
f}=\frac{I_{i\rightarrow f}}{\sqrt{pp^{\prime}}}$ which are the results of the
temporal integrals as obtained in the amplitude equation (5.1):

$\displaystyle\mathcal{I}_{i\rightarrow
f}=\frac{1}{\sqrt{pp^{\prime}}}\biggl{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}\,sgn(\sigma^{\prime})A_{1}(\lambda=\pm
1)-\tan(\theta_{W})\,sgn(\sigma)A_{2}(\lambda=\pm 1)\biggl{\\}},$ (151)

with the specification that the functions $A_{1}(\lambda=\pm
1),\,A_{2}(\lambda=\pm 1)$ were defined in equation (83).

Figure 11: Transition rate as a function of $m/\omega$ for $\lambda=\pm 1$,
with $M_{Z}/\omega=0.9$, $\sigma$ and $\sigma^{\prime}$ having the same sign
and opposite signs respectively. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

Figure 12: Transition rate as a function of $M_{Z}/\omega$ for $\lambda=\pm
1$, with $m/\omega=0.5$, $\sigma$ and $\sigma^{\prime}$ having the same sign
and opposite signs respectively. The dotted line is for
$p/P=0.1,p^{\prime}/P=0.2$, and the solid line is for
$p/P=0.3,p^{\prime}/P=0.4$

A graphical analysis of equation (150) in terms of parameters
$\frac{m}{\omega},\,\frac{M_{Z}}{\omega}$ proves that the rates for the
process of Z boson and fermion-antifermion generation from vacuum are
important only in inflationary regime and are vanishing in the Minkowski limit
when the expansion parameter becomes zero. The rates behaviour in terms of
parameters $K,M_{Z}/\omega$ reproduce the probabilities behaviour proving that
the production of gauge Z bosons was possible only in the large expansion
conditions of early universe.

###  6.1 Rates in the limit $\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0$

Since it is well known that the massive Z and W bosons could exist as stable
particles only in the early universe there are some interesting observations
that can be made to extend our result. The first is related to the fact that
the transition rate in de Sitter case [56] could in principle be computed for
the case $\omega>>M_{Z}$ in which case the functions that define our amplitude
could become simpler. For example in the case $\lambda=\pm 1$ the amplitude is
reduced in the limit $\frac{m}{\omega}\rightarrow
0,\,\frac{M_{Z}}{\omega}\rightarrow 0$ to a simpler form. The functions that
define the amplitude can be written in the limit when the expansion parameter
is much larger than the particle mass by using equation (182) from Appendix:

$\displaystyle
B_{K=0k=0}(pp^{\prime}P)=\frac{P^{-5/2}}{\pi}\left[\frac{1}{\left(1+\frac{p^{\prime}}{P}\right)^{2}-\frac{p^{2}}{P^{2}}}+\frac{1}{\left(1-\frac{p^{\prime}}{P}\right)^{2}-\frac{p^{2}}{P^{2}}}\right];$
$\displaystyle
B_{1K=0k=0}(pp^{\prime}P)=\frac{(P)^{-3/2}}{\pi}\left[1+\frac{p^{2}+p^{\prime
2}}{P^{2}}\right];$ $\displaystyle
B_{2K=0k=0}(pp^{\prime}P)=\frac{2pp^{\prime}(P)^{-7/2}}{\sqrt{\pi}}\left[1+2\frac{p^{2}+p^{\prime
2}}{P^{2}}\right].$ (152)

We observe that these much simpler functions are combinations that depend on
the ratios of momenta and mention that for approximating the Appel
hypergeometric functions in this limit we consider the ratio of momenta to be
less than one. In this limit one obtains a polynomial expression in momenta so
that these functions are integrable when computing the transition rate.
However since we use a perturbative approach the ultraviolet divergences will
appear in our computations. The dependence of the transition rates on the
ratios between the particle masses and the expansion parameter will imply
nontrivial computations since we will try to evaluate the limits from the
definition of the total transition rates given bellow:

$R_{if}=\int d^{3}p_{1f}...\int d^{3}p_{nf}\left[\lim_{t_{c}\rightarrow
0}\frac{1}{2V}\frac{d}{dt_{c}}|A_{Ze\bar{e}}|^{2}\right]=\int
d^{3}p_{1f}...\int d^{3}p_{nf}\left[\lim_{t\rightarrow\infty}\frac{e^{\omega
t}}{2V}\frac{d}{dt}|A_{Ze\bar{e}}|^{2}\right].$ (153)

where we integrate after the momenta of the final particles. In the general
case the computations cannot be done but in the limit discussed above the
transition rate could in principle be obtained thus determining the density
number of particles. In what follows we will present the computations for the
transition rate in the case of emission of massive fermions and Z bosons for
$\omega>>m\,\,,\omega>>M_{Z}$. For our computations it will be convenient to
use the amplitude written in terms of Hankel functions as:

$\displaystyle A_{i\rightarrow f}(\lambda=\pm 1)$ $\displaystyle=$
$\displaystyle\frac{(\pi)^{3/2}\sqrt{pp^{\prime}}e^{-\pi
k/2}}{8(2\pi)^{3/2}}\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})\Bigg{\\{}\int_{0}^{\infty}dz\cdot
z^{3/2}$ (154)
$\displaystyle\times\bigg{(}\frac{e_{0}\cos(2\theta_{W})}{\sin(2\theta_{W})}\bigg{)}sgn(\sigma^{\prime})H^{(2)}_{\nu_{+}}(pz)H^{(2)}_{\nu_{-}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)$
$\displaystyle-\int_{0}^{\infty}dz\cdot
z^{3/2}e_{0}\tan(\theta_{W})sgn(\sigma)H^{(2)}_{\nu_{-}}(pz)H^{(2)}_{\nu_{+}}(p^{\prime}z)H^{(2)}_{-ik}(Pz)\Bigg{\\}}$
$\displaystyle\times\xi^{+}_{\sigma}(\vec{p})\vec{\sigma}\cdot\vec{\epsilon}^{*}(\vec{n}_{P},\lambda)\eta_{\sigma}(\vec{p}\,^{\prime})$

Then in the limit for $m/\omega=0$, $M_{Z}/\omega=0$ the temporal integrals
are of the type,

$I_{i\rightarrow f}(0)=\int_{0}^{\infty}dz\cdot
z^{3/2}H^{(2)}_{\frac{1}{2}}(pz)H^{(2)}_{\frac{1}{2}}(p^{\prime}z)H^{(2)}_{\frac{1}{2}}(Pz)\Bigg{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})-\tan(\theta_{W})sgn(\sigma)\Bigg{\\}}$
(155)

where we can use the relation

$H^{(2)}_{\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\cdot ie^{-iz}.$ (156)

The final result for the temporal integrals now becomes:

$I_{i\rightarrow
f}(0)=\bigg{(}\frac{2}{\pi}\bigg{)}^{3/2}\frac{1}{\sqrt{pp^{\prime}P}(p+p^{\prime}+P)}\Bigg{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})-\tan(\theta_{W})sgn(\sigma)\Bigg{\\}}$
(157)

This way we arrive at the final result for the transition amplitude in the
case $\lambda=\pm 1$ for the case of large expansion where we can consider
that $m/\omega=0$, $M_{Z}/\omega=0$

$\displaystyle A_{i\rightarrow f}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}(1-i)\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{8\pi^{3/2}}\frac{e_{0}}{\sqrt{P}(p+p^{\prime}+P)}\Bigg{\\{}\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})$
(158)
$\displaystyle-\tan(\theta_{W})sgn(\sigma)\Bigg{\\}}\xi^{+}_{\sigma}(\vec{p})\vec{\sigma}\cdot\vec{\epsilon}\,^{*}\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime}).$

We specify that we evaluate the rate at $t_{\infty}$ as in the general case
and the rate of transition for the emission from de Sitter vacuum of the
triplet electron-positron and a Z boson is then [56]:

$R=\frac{1}{8}\sum_{\sigma\sigma^{\prime}\lambda}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{(2\pi)^{3}}\big{|}M_{if}\big{|}^{2}\big{|}I_{if}(0)\big{|}\lim_{t\to
t_{\infty}}\Big{|}e^{\omega
t}\mathcal{K}_{if}\Big{(}\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0\Big{)}\Big{|}$
(159)

where we denote

$M_{i\rightarrow
f}=\frac{e_{0}\pi^{3/2}\sqrt{pp^{\prime}}}{8(2\pi)^{3/2}}\xi^{+}_{\sigma}(\vec{p}\,)\vec{\sigma}\cdot\vec{\epsilon}\,^{*}\eta_{\sigma^{\prime}}(\vec{p}\,^{\prime}),$
(160)

and we introduce the new notation

$S_{\sigma\sigma^{\prime}}=\frac{\cos(2\theta_{W})}{\sin(2\theta_{W})}sgn(\sigma^{\prime})-\tan(\theta_{W})sgn(\sigma).$
(161)

The notation $I_{i\rightarrow f}(0)$ stands for the result of the temporal
integral given in equation (157) and $\mathcal{K}_{if}$ is the integrand from
the temporal integral

$\mathcal{K}_{if}\Big{(}\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0\Big{)}=z^{3/2}H^{(2)}_{\frac{1}{2}}(pz)H^{(2)}_{\frac{1}{2}}(p^{\prime}z)H^{(2)}_{\frac{1}{2}}(Pz)e^{-\omega
t}S_{\sigma\sigma^{\prime}}.$ (162)

The first step is to compute the $\lambda$ sum by using the relation :

$\sum_{\lambda}\epsilon_{i}^{*}(\vec{n}_{\mathcal{P}},\,\lambda)\epsilon_{j}(\vec{n}_{\mathcal{P}},\,\lambda)=\delta_{ij},$
(163)

for $\lambda=0,\pm 1$. Computing the limit from the rate given in (159) we
obtain:

$\lim_{t\to\infty}\Big{|}e^{\omega
t}\mathcal{K}_{if}\Big{(}\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0\Big{)}\Big{|}=\bigg{(}\frac{2}{\pi}\bigg{)}^{3/2}\frac{1}{\sqrt{pp^{\prime}P}}S_{\sigma\sigma^{\prime}},$
(164)

remarkable is that this is the result from equation (6) when
$\frac{m}{\omega}=\frac{M_{Z}}{\omega}=0$. Collecting all the above results we
obtain the transition rate

$R=\frac{1}{8}\sum_{\sigma\sigma^{\prime}}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{64(2\pi)^{6}}\frac{e_{0}^{2}}{P(p+p^{\prime}+P)}\big{|}S_{\sigma\sigma^{\prime}}\big{|}^{2}\cdot\big{|}\xi^{+}_{\sigma}(\vec{p}\,)\sigma_{i}\eta_{\sigma^{\prime}}(\vec{p}\,\,^{\prime})\big{|}^{2}$
(165)

Because in this limit the dependence on momenta is far simpler we will compute
the total transition rate by integrating the rate equation (165) after the
final momenta and we adopt the same definition of the integrals as used for
the total probability:

$R_{tot}=\int\frac{d^{3}p}{p_{0}}\int\frac{d^{3}p^{\prime}}{p^{\prime}_{0}}\int\frac{d^{3}P}{P_{0}}\cdot
R$ (166)

where the factor we take the same setup as in the case of total probability
$p_{0}=\sqrt{m^{2}+p^{2}},\,p_{0}^{\prime}=p^{\prime},\,P_{0}=P$.

For facilitating our computations we choose the electron momentum and positron
momentum on third axis such that
$\vec{p}=p\cdot\vec{e_{3}},\,\vec{p}\,^{\prime}=-p^{\prime}\vec{e}_{3}$, and
for the Z boson moneta $\vec{P}=P\vec{e}_{3}$, such that
$|\vec{p}+\vec{P}|=p+P$.

The sum with the bispinors is simple, since the electron-positron pair is
emitted on the third axis and the sum is reduced to a numerical factor. The
helicity bispinors in this particular case are:

$\xi_{-\frac{1}{2}}(\vec{p}\,)=\begin{pmatrix}0\\\
1\end{pmatrix};\xi_{\frac{1}{2}}(\vec{p}\,)=\begin{pmatrix}1\\\
0\end{pmatrix}$ (167)

Solving the $p^{\prime}$ integrals we obtain:

$\displaystyle
I=\int\frac{d^{3}p}{\sqrt{m^{2}+p^{2}}}\int\frac{d^{3}P}{P}\frac{\int
d^{3}p^{\prime}}{p^{\prime}}\frac{\delta^{3}(\vec{p}+\vec{p}\,^{\prime}+\vec{P})}{(2\pi)^{3}P(p+p^{\prime}+P)}$
$\displaystyle=\int\frac{d^{3}p}{\sqrt{m^{2}+p^{2}}}\int\frac{d^{3}P}{2(2\pi)^{3}P^{2}(p+P)^{2}}=2\pi\int\frac{d^{3}p}{(2\pi)^{3}p\sqrt{m^{2}+p^{2}}}$
(168)

For solving the $p$ integral we will adopt the dimensional regularization [29,
30, 31, 32, 35]. The $D$ dimensional integral in our case is:

$\int\frac{d^{3}p}{(2\pi)^{3}p\sqrt{m^{2}+p^{2}}}\rightarrow\int\frac{d^{D}p}{(2\pi)^{D}p\sqrt{m^{2}+p^{2}}}$
(169)

The $D$ dimensional integral in this case can be written as

$I(D)=\int\frac{d^{D}p}{(2\pi)^{D}p\sqrt{m^{2}+p^{2}}}=\frac{S_{D}}{(2\pi)^{D}}\int_{0}^{\infty}dp\frac{p^{D-1}}{p\sqrt{m^{2}+p^{2}}},$
(170)

and performing the substitution $p^{2}=ym^{2}$, we obtain the integral of the
Beta Euler function

$I(D)=\frac{S_{D}\,m^{D-2}}{2(2\pi)^{D}}\int_{0}^{\infty}dy\frac{y^{(D-3)/2}}{(1+y)^{1/2}}=\frac{2\pi^{D/2}m^{D-2}}{(2\pi)^{D}\Gamma\bigg{(}\frac{D}{2}\bigg{)}\Gamma\bigg{(}\frac{1}{2}\bigg{)}}\Gamma\bigg{(}\frac{D-1}{2}\bigg{)}\Gamma\bigg{(}1-\frac{D}{2}\bigg{)}$
(171)

The regularized integral is obtained if we introduce the dimensionless
constant $g=\lambda\mu^{D-3}$ and write the result in terms of $g$ and
$\varepsilon$ for $D=3-\varepsilon$

$I(D)_{r}=-\lambda
I(D)=-\frac{gm}{\sqrt{\pi}(4\pi)^{3/2}}\left(\frac{4\pi\mu^{2}}{m^{2}}\right)^{\varepsilon/2}\frac{\Gamma\bigg{(}1-\frac{\varepsilon}{2}\bigg{)}\Gamma\bigg{(}\frac{\varepsilon-1}{2}\bigg{)}}{\Gamma\bigg{(}\frac{3}{2}-\frac{\varepsilon}{2}\bigg{)}}$
(172)

Further we expand the term in powers of $\varepsilon$ as in equation (114) and
use the equation (184) from Appendix to expand the gamma function [35]:

$\displaystyle\frac{\Gamma\bigg{(}1-\frac{\varepsilon}{2}\bigg{)}\Gamma\bigg{(}\frac{\varepsilon-1}{2}\bigg{)}}{\Gamma\bigg{(}\frac{3}{2}-\frac{\varepsilon}{2}\bigg{)}}=-4-2\varepsilon[\gamma+\psi(-1/2)+\psi(3/2)]+o(\varepsilon^{2})$
(173)

The final result for the regularized integral is obtained by taking the limit
$\varepsilon=0$ from the following relation:

$\displaystyle
I(D)_{r}=-\frac{gm}{8\pi^{2}}\biggl{\\{}-4-2\varepsilon\left[\gamma+\psi(-1/2)+\psi(3/2)+\ln\left(\frac{4\pi\mu^{2}}{m^{2}}\right)\right]+o(\varepsilon^{2})\biggl{\\}}$

and one can observe that the integral is finite for $\varepsilon=0$. One may
also try to use the Pauli-Villars regularization [34] to solve the divergent
integrals from probability and transition rate, in which case counter-terms
need to be added in order to obtain finite results. The momenta integrals for
$\varepsilon=0$ become:

$I=\frac{gm}{\pi}$ (175)

The final result for the total rate reads as:

$\displaystyle R_{tot}$ $\displaystyle=$
$\displaystyle\frac{e_{0}^{2}gm}{8\pi\cdot
64(2\pi)^{3}}\Bigg{(}\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})\Bigg{)}$
(176) $\displaystyle=$
$\displaystyle\frac{gmM_{W}^{2}G_{F}\sin^{2}(\theta_{W})}{32\sqrt{2}(2\pi)^{4}}\Bigg{(}\frac{\cos^{2}(2\theta_{W})}{\sin^{2}(2\theta_{W})}+\tan^{2}(\theta_{W})\Bigg{)}$

The rate is finite and can be used for defining the density number of
particles in the large expansion conditions. Then the density number of Z
bosons obtained in the process of spontaneous generation from vacuum of a Z
boson and an electron-positron pair is proportional with the ratio between the
rate computed perturbatively $R_{vac\rightarrow Ze^{-}e+}$ and the rate of
decay $R_{des}$ for the Z boson, that also must be computed in the de Sitter
case. Thus one needs to study the decay rates of the Z boson in fermion-anti-
fermion pair in the de Sitter geometry in order to take into account the
influence of space expansion upon the decay processes. All perturbative
processes give contribution to the density number of Z bosons including the
processes in which the bosons can be emitted by fermions. In the case of Z
emission by fermions the density number of Z bosons depends on the density
number of fermions, the rate of the emission and the decay rate of Z boson.

$n_{Z,emission}=\frac{R_{emission}}{R_{des}}n_{f},$ (177)

where $R_{emission}$ is the rate of Z emission by fermions and $n_{f}$ is the
density number of fermions.

We do not approach the problem related to the origin of the massive bosons
masses in the present paper. However in the early universe the mass of the Z
boson is related to the expectation values of the Higgs field. In electroweak
theory during inflation, the massive bosons masses can be generated by a large
condensate of the Higgs field. In [24] it was proven that the generated vector
mass scales as $m_{W}^{2}\sim\omega^{2}$, and that of the Higgs scalar remains
perturbatively small, $m_{H}^{2}\sim g^{2}\omega^{2}$, where $g$ is the gauge
coupling.

This result should be considered alongside the results obtained by using other
methods [57], where the authors consider the nonperturbative methods. Thus for
a clear picture of the mechanisms that were involved in the problem of
particle production in early universe we must take into account both methods
as was discussed in [18, 19]. This is because the space expansion could play a
similar role in the perturbative approach as the thermal bath and the two
effects would probably compete in the early universe.

Another observation related to the rate refers to the fact that the graphical
results for the probabilities prove that the phenomenon of massive bosons
generation is present during the entire inflation. This means that in
principle the rate could be computed for all values of the ratios between the
particles masses and Hubble parameter. We restricted in the present
investigation only to the limit when $m/\omega=M_{Z}/\omega=0$, and this is
due to the mathematical problems that we encounter. We hope to approach this
interesting topic in a future study.

##  7 Conclusions

In this paper we developed the theory of interaction between Z bosons and
massive fermions by using perturbative methods as in the theory of electro-
weak interactions in flat space-time. The equations of interactions between
massive neutral vector field and massive fermions were established and their
solutions were written down with the help of the Green functions. This helps
us introduce the $in\ out$ fields and construct the formalism of reduction for
the Proca field in de Sitter geometry. The definition of the transition
amplitudes in the first order of perturbation theory was established and we
mention that we work with Feynman rules in coordinates since the Proca
propagators in de Sitter geometry need further study.

As an application of our formalism the problem of generating from de Sitter
vacuum of the triplet Z boson and electron-positron was studied. We
established the transition amplitudes for both the transversal and
longitudinal modes and we defined the transition probabilities in volume unit.
Our analytical and graphical results prove that the phenomenon of particle
production is possible only for a large expansion factor that corresponds to
the early universe. In the Minkowski limit we recovered the expected results
that the process of particle production is no longer possible due to energy-
momentum conservation. We computed the total probability and rate in the case
of large expansion factor $\omega>>m,M_{z}$ and we adopted the dimensional
regularization for computing the momenta integrals. By using the minimal
substraction method we succed to remove the divergences from our total
probability, proving that the theory we constructed here is renormable. The
rate of transition was computed in the conditions of large expansion where the
ratios $M_{Z}/\omega,\,m/\omega$ tend to zero and we determine the density
number of Z bosons as the ratio between the production rate and decay rate.

For further study it will be of interest to take into account all possible
processes that generate Z boson as a result of fields interactions including
the processes of Z boson emission by electrons and positrons. The electro-weak
interactions are less studied in curved backgrounds and we hope that our
results will encourage others to study the Proca field and propagators in
curved space-times. This could help us to understand the problems related to
the mechanisms that were involved in generating massive bosons in early
universe.

##  8 Appendix

For solving our integrals we use the relations that relate Hankel functions to
Bessel functions

$\displaystyle
H^{(1)}_{\mu}(z)=\frac{J_{-\mu}(z)-e^{-i\pi\mu}J_{\mu}(z)}{i\sin(\pi\mu)}$
$\displaystyle
H^{(2)}_{\mu}(z)=\frac{e^{i\pi\mu}J_{\mu}(z)-J_{-\mu}(z)}{i\sin(\pi\mu)}.$
(178)

and the relations that transform Hankel function into modified Bessel K
functions [23]:

$H^{(1,2)}_{\nu}(z)=\mp\left(\frac{2i}{\pi}\right)e^{\mp i\pi\nu/2}K_{\nu}(\mp
iz),$ (179)

The integrals that help compute our amplitudes are of the type [23]:

$\displaystyle\int_{0}^{\infty}dzz^{\lambda-1}J_{\mu}(az)J_{\nu}(bz)K_{\rho}(cz)=\frac{2^{\lambda-2}a^{\mu}b\,^{\nu}c^{-\lambda-\mu-\nu}}{\Gamma(1+\mu)\Gamma(1+\nu)}\Gamma\left(\frac{\lambda+\mu+\nu-\rho}{2}\right)\Gamma\left(\frac{\lambda+\mu+\nu+\rho}{2}\right)$
$\displaystyle\times\,F_{4}\left(\frac{\lambda+\mu+\nu-\rho}{2},\frac{\lambda+\mu+\nu+\rho}{2};1+\mu,1+\nu;-\frac{a^{2}}{c^{2}},-\frac{b^{2}}{c^{2}}\right),$
$\displaystyle Re(\lambda+\mu+\nu)>Re(\rho)\,,Re(c)>|Im(a)|+|Im(b)|.$ (180)

The definition of Appel hypergeometric function is given bellow [23]:

$F_{4}(a,b,c,d;x,y)=\sum_{m,n=0}^{\infty}\frac{\Gamma(a+m+n)\Gamma(b+m+n)\Gamma(c)\Gamma(d)x^{m}\cdot
y^{n}}{\Gamma(a)\Gamma(b)\Gamma(c+m)\Gamma(d+n)m!\cdot n!}.$ (181)

In our computations we use the relation between the Appel hypergeometric
functions and Gauss hypergeometric functions [23]

$\displaystyle
F_{4}\left(\alpha,\alpha+\frac{1}{2},\gamma,\frac{1}{2};x,y\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}(1+\sqrt{y})^{-2\alpha}\,_{2}F_{1}\left(\alpha,\alpha+\frac{1}{2};\gamma;\frac{x}{(1+\sqrt{y})^{2}}\right)$
(182)
$\displaystyle+\frac{1}{2}(1-\sqrt{y})^{-2\alpha}\,_{2}F_{1}\left(\alpha,\alpha+\frac{1}{2};\gamma;\frac{x}{(1-\sqrt{y})^{2}}\right).$

The $P$ integral from total probability has the result:

$\int\frac{d^{3}P}{P^{2}}\frac{1}{(p+P)^{3}}=\frac{2\pi}{p^{2}}.$ (183)

The expansion for gamma Euler function for an integer $n$ is [35]:

$\displaystyle\Gamma(-n+\varepsilon)=\frac{(-1)^{n}}{n!}\biggl{\\{}\frac{1}{\varepsilon}+\psi(n+1)+\frac{\varepsilon}{2}\biggl{[}\frac{\pi^{2}}{3}+\psi^{2}(n+1)-\psi^{\prime}(n+1)\biggl{]}+o(\varepsilon^{2})\biggl{\\}}$
(184)

where $\psi$ is the digamma Euler function and $\psi^{\prime}$ its derivative.

Acknowledgements

Diana Dumitrele was supported by a grant of the Romanian Ministry of Research
and Innovation and West University of Timişoara, CCCDI-UEFISCDI, under project
”VESS, 18PCCDI/2018”, within PNCDI III.

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