#  $Z^{0}$-Boson Decays in a Strong Electromagnetic Field 111Published in:
Yad. Fiz. 72 (2009) 1078 , [Phys. At. Nucl. 72 (2009) 1034].

A.V.Kurilin222E-mail address<EMAIL_ADDRESS>
Moscow Technological Institute 333on leave from Moscow State Open Pedagogical
University,
Leninsky Prospect, 38a, Moscow, 119334, Russia

The probability of $Z^{0}$-boson decay to a pair of charged fermions in a
strong electromagnetic field, $Z^{0}\rightarrow\bar{f}f$, is calculated. On
the basis of a method that employs exact solutions to relativistic wave
equations for charged particles, an analytic expression for the partial decay
width $\Gamma(\varkappa)=\Gamma(Z^{0}\rightarrow\bar{f}f)$ is obtained at an
arbitrary value of the parameter
$\varkappa=eM_{Z}^{-3}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}}$, which characterizes
the external-field strength. The total $Z^{0}$-boson decay width in an intense
electromagnetic field, $\Gamma_{Z}(\varkappa)$, is calculated by summing these
results over all known generations of charged leptons and quarks. It is found
that, in the region of relatively weak fields ($\varkappa<0,06$), the field-
induced corrections to the standard $Z^{0}$-boson decay width in a vacuum do
not exceed 2%. As $\varkappa$ increases, the total decay width
$\Gamma_{Z}(\varkappa)$ develops oscillations against the background of its
gradual decrease to the absolute-minimum point. At $\varkappa_{\rm
min}=0,445$, the total $Z^{0}$-boson decay width reaches the minimum value of
$\Gamma_{Z}(\varkappa_{\rm min})=2,164$ GeV, which is smaller than the
$Z^{0}$-boson decay width in a vacuum by more than 10%. In the region of
superstrong fields ($\varkappa>1$), $\Gamma_{Z}(\varkappa)$ grows
monotonically with increasing external-field strength. In the region
$\varkappa>5$, the $t$-quark-production process $Z^{0}\rightarrow\bar{t}t$,
which is forbidden in the absence of an external field, begins contributing
significantly to the total decay width of the $Z^{0}$\- boson.

##  1\. INTRODUCTION

It is not an exaggeration to say that $Z^{0}$-bosons along with their charged
partners, $W^{\pm}$-bosons, have been at the focus of attention in high-energy
physics over more than the past two decades. The properties of these
particles, which are mediators of weak interaction between leptons and quarks,
were studied in detail in experiments at the LEP and SLC electron-positron
colliders. In a series of experiments performed between 1989 and 1995 at CERN
[1], enormous statistics of observations of $Z^{0}$-boson production and
decays were collected. In all, approximately $2\cdot 10^{6}$ events associated
with leptonic modes of $Z^{0}$-boson decay and more than $1.5\cdot 10^{7}$
events of transformation of these unstable particles into hadrons were
detected. The results of these investigations made it possible to determine,
among other things, the mass of the $Z^{0}$-boson, $M_{Z}=91,1876\pm 0,0021\
\mbox{GeV}$, and its total decay width, $\Gamma_{Z}=2,4952\pm 0,0023\
\mbox{GeV}$, to an extremely high degree of precision [2]. Many other
parameters of the process $e^{+}e^{-}\rightarrow Z^{0}\rightarrow\bar{f}f$
were also analyzed both in the vicinity of the $Z$-resonance and far from it
[3]. The precision of experimental data reached in recent years has given a
motivation to theoretical physicists for performing formidable work on
calculating radiative corrections to the processes involving $Z^{0}$-boson
production and decay (see, for example, [4, 5, 6, 7] and references therein).

Not only does $Z^{0}$-boson physics provide a precision test for Standard
Model predictions concerning particle interactions, but it is also a unique
tool for seeking manifestations of new physics (beyond the Standard Model),
which will replace sooner or later the generally accepted $SU(3)\times
SU(2)\times U(1)$ scheme. Despite the impressive successes of the Standard
Model in adequately describing a formidable set of experimental data, there is
no doubt among the physics community that the modern model of fundamental
interactions cannot be conclusive because of a number of fundamental
theoretical drawbacks. These include, first, an enormous number (more than
19!) of independent input parameters — coupling constants, the masses of quark
and leptons, the parameters of the Cabibbo-Kobayashi-Maskawa mixing matrix,
etc; second, the instability of the mass of the as-yet-undiscovered Higgs
boson with respect to quadratically divergent radiative corrections and the
allied hierarchy problem; and, third, the isolated character of gravitational
interactions, which do not fit in the existing quantum model. This is not the
whole story, however. Paradoxically as it is, the main drawback of the
Standard Model at the present time is that it is in perfect agreement with all
experimental data accumulated thus far, providing no hint as to the nature of
new physics that will have to replace it. This kind of ”recess” in the
experimental development of physics theory caused a flash of creative activity
in theoretical physics in attempts at guessing vague outlines of a new theory
of everything. By no means do Grand Unification theories, supersymmetry,
supergravity, superstrings, supermembranes, and $M$-theory exhaust the list of
ideas that require an experimental verification. The questions that a future
theory should answer include that of the origin of particle masses, that of
why our space-time is four-dimensional, that of why several generations of
leptons exist, that of how many flavors quarks have, and that of how many
neutrino flavors exist. Even today, physicists try to find answers to many of
these questions by analyzing available experimental data.

For example, the total invisible width of $Z^{0}$-bosons, $\Gamma_{\rm
inv}=499,0\pm 1,5\ \mbox{MeV}$, with respect to the decays of these particles
through channels that are rather difficult to detect plays an important role
in $Z^{0}$-boson physics. In the experiments performed at the LEP accelerator,
the invisible width $\Gamma_{\rm inv}$ was determined indirectly as the
difference of the total decay width of the $Z^{0}$-boson and its total width
with respect to all observed decays. In the Standard Model, the invisible
decay width of the $Z^{0}$-boson is interpreted as the partial width with
respect to decays to neutrinos of various flavors:

$\Gamma_{\rm
inv}=\sum_{i}^{N_{\nu}}\Gamma(Z^{0}\rightarrow\nu_{i}\tilde{\nu}_{i})=N_{\nu}\Gamma_{\nu\tilde{\nu}}$

This quantity can be considered as some kind of a counter of neutrino
generations, and its precise measurement is of particular interest for new
physics (see, for example, [8]). It is intriguing that the present-day
experimental value of the number of neutrinos from the invisible decay width
of the $Z^{0}$-boson, $N_{\nu}=2.92\pm 0,06$ [2], differs somewhat from the
result expected in the Standard Model, $N_{\nu}=3$.

Exotic channels of $Z^{0}$-boson decay frequently provide unique tests for
some fundamental principles of modern physics. For example, analysis of the
forbidden (in a vacuum) $Z^{0}$-boson decays to photons
($Z^{0}\rightarrow\gamma\gamma$) and gluons ($Z^{0}\rightarrow gg$) makes it
possible to establish a quantitative measure of the principle of quantum
indistinguishability of integer-spin particles — Bose symmetry for photons and
gluons [9]. According to the Landau-Yang theorem [10], a massive vector
particle cannot decay to massless vector states. In an external
electromagnetic field, however, the reactions $Z^{0}\rightarrow\gamma\gamma$
and $Z^{0}\rightarrow gg$ become possible [11].

The problem of searches for new possible manifestations of supersymmetry is
also tightly related to $Z^{0}$-boson physics. There is still a hope for
finding relatively light superparticles produced in rare channels of
$Z^{0}$-boson decay. As an example, one can consider the reaction
$Z^{0}\rightarrow\tilde{g}\tilde{g}$, in which a pair of light gluinos,
$\tilde{g}$, having a mass of about $m_{\tilde{g}}=12\div 16$ GeV appear. This
process, which is extensively discussed in the literature (see [12] and
references therein), is of special interest in view of prospects for observing
new-physics manifestations at next-generation $p\bar{p}$-colliders. Projects
of experiments with $Z^{0}$-bosons at the Large Hadron Collider(LHC) also
caused interest in studying the properties of these particles in the presence
of a quark-gluon plasma, which is quite an unusual environment of [13].

In view of the aforesaid, it is of particular interest to discuss some other
new phenomena in $Z^{0}$-boson physics that manifest themselves under unusual
conditions. In this study, we investigate changes in partial decay widths of
the $Z^{0}$-boson in the presence of strong electromagnetic fields. Interest
in such investigations is explained both by astrophysical applications and by
applications in the physics of relativistic particles channeling through
single crystals (so-called channeling of particles [14]). It is well known
that the strength of electric fields generated by the axes and planes of
single crystals may reach formidable values of ($E\geq 10^{10}\;$V/m) [15]
over macroscopic distances. At the same time, it is well known that
astrophysics studies exotic objects (neutron stars, white dwarfs) that, at the
latest stages of their evolution, can undergo a strong compression, and this
leads to a significant increase in the magnetic field strength inside them
($H\geq 10^{8}\div 10^{13}\;$ G [16]). At such enormous values of the external
field strength, the physics of quantum processes changes significantly.
External fields frequently remove the forbiddance of specific reactions whose
occurrence is impossible in a vacuum.

The first investigations into the physics of relativistic particles in the
presence of a strong external field were performed within quantum
electrodynamics (see, for example, [17, 18, 19]). Later, the technique of
calculations that was developed there was also used in non-Abelian gauge
theories [20]. Respective calculations rely on the method of exact solutions
to the wave equations for charged particles, which makes it possible to take
into account the interaction with the electromagnetic field beyond standard
perturbation theory. In this case, the wave functions for all charged
particles and their propagators are modified in quite a unwieldy manner,
depending on the configuration of an external field [21]. Although the
resulting expressions are cumbersome and although the calculations are quite
involved, the ultimate results, which are valid at arbitrarily high values of
the external-electromagnetic-field strength, carry information that cannot be
obtained by perturbative methods, and this is an obvious advantage of the
method. From this point of view, investigation of quantum processes in
superstrong fields provides a unique possibility of analyzing the self-
consistency of the physical theory globally in the case where an expansion in
a small parameter is impossible.

The method of calculations that is used below is based on the crossed-field
model, which was successfully applied in the previous studies of the present
author to studying photino-pair production [22] and to analyzing the decays of
$W$-bosons in an external field [23].

##  2\. PROBABILITY FOR $Z^{0}$-BOSON DECAY
IN EXTERNAL FIELDS

In the leading order of perturbation theory in the coupling constants $g_{V}$
and $g_{A}$, the matrix element of $Z^{0}$-boson decay to a pair of charged
leptons, $\ell^{\pm}$, is given by

$S_{fi}=i\int
d^{4}x\overline{\Psi}_{\ell^{-}}(x,p)\gamma^{\mu}(g_{V}+\gamma^{5}g_{A})\Psi_{\ell^{+}}^{c}(x,p^{\prime})Z_{\mu}(x,q).$
(1)

The presence of an external electromagnetic field is taken into account via
choosing specific wave functions $\Psi_{\ell^{-}}(x,p)$ and
$\Psi_{\ell^{+}}(x,p^{\prime})$ for the charged leptons $\ell^{\pm}$. This
approach, which, in particle physics, is referred to as the Furry
representation, yields, for the probability of $Z^{0}$-boson decay, an
expression in which electromagnetic interactions are taken into account
exactly, beyond standard perturbation theory in the electromagnetic coupling
constant $\alpha=e^{2}/4\pi\simeq 1/137$. The explicit form of the wave
functions for charged particles in an external electromagnetic field depends
on the field configuration. In the present study, we restrict ourselves to the
crossed-field configuration, in which case the strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ has the form

$F_{\mu\nu}=C(k_{\mu}a_{\nu}-k_{\nu}a_{\mu}).$ (2)

It is assumed in this case that the external-electromagnetic-field potential
$A_{\mu}(x)$ is chosen in the gauge

$A_{\mu}(x)=a_{\mu}C(k_{\nu}x^{\nu}),$ (3)

Here, the unit constant $4$-vector $a_{\mu}$ determines the spatial field
configuration and satisfies the conditions

$a_{\mu}a^{\mu}=-1,\qquad
F_{\mu\nu}=(a_{\mu}F_{\nu\lambda}-a_{\nu}F_{\mu\lambda})a^{\lambda}.$ (4)

A crossed field is a particular case of the field of a plane electromagnetic
wave having an isotropic wave $4$-vector $k_{\mu}\ (k^{2}=0)$. The invariant
parameter $C$ characterizes the crossed-field strength and has the dimensions
of energy. Recall that, in an arbitrary reference frame, a crossed field is a
superposition of constant electric and magnetic fields whose strength vectors
are orthogonal and are equal in absolute value: $E=H=k_{0}C$. Moreover, the
two relativistic invariants of the electromagnetic field are zero:

$F_{\mu\nu}F^{\mu\nu}=F_{\mu\nu}\tilde{F}^{\mu\nu}=0.$ (5)

In the crossed-field model, the choice of wave $4$-vector $k_{\mu}$ is quite
arbitrarily. It is only necessary that this vector be isotropic, have
dimensions of energy, and satisfy the conditions

$k_{\mu}a^{\mu}=0,\qquad F^{\mu\lambda}F_{\lambda\nu}=C^{2}k^{\mu}k_{\nu}.$
(6)

In particular, the wave $4$-vector $k^{\mu}$ can be expressed in terms of the
electromagnetic-field-strength tensor and the constant electromagnetic-
potential $4$-vector $a_{\lambda}$ as

$k^{\mu}=-\frac{1}{C}\cdot F^{\mu\lambda}a_{\lambda}.$ (7)

In dealing with the problem of $Z^{0}$-boson decay, it is advisable to express
the wave vector $k^{\mu}$ in terms of the $Z^{0}$-boson 4-momentum $q^{\nu}$
and the electromagnetic-field-strength tensor $F^{\mu\lambda}$ (see next
section) as

$k^{\mu}=\frac{e^{2}\Delta}{2M_{Z}^{6}\varkappa^{2}}\cdot
F^{\mu\lambda}F_{\lambda\nu}q^{\nu}.$ (8)

In an arbitrary constant uniform electromagnetic field, the probability of the
decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ is a function of three invariant
dimensionless parameters,
$P(Z^{0}\rightarrow\ell^{+}\ell^{-})=P(\varkappa,a,b)$. These parameters are
determined by the strength tensor of an external macroscopic field as

$\displaystyle\varkappa=\frac{e}{M_{Z}^{3}}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}},$
(9) $\displaystyle a=-\frac{e^{2}}{4M_{Z}^{4}}F_{\mu\nu}\tilde{F}^{\mu\nu},$
(10) $\displaystyle b=\frac{e^{2}}{4M_{Z}^{4}}F_{\mu\nu}F^{\mu\nu}.$ (11)

In the crossed-field model, the parameter $\varkappa$ has the simplest form

$\varkappa=\frac{eC}{M_{Z}^{3}}(q_{\mu}k^{\mu}),$ (12)

the square of the wave vector (8) having the form

$k^{2}=\frac{\Delta^{2}}{4M_{Z}^{2}}\left({a^{2}\over\varkappa^{4}}-{2b\over\varkappa^{2}}\right).$
(13)

In a crossed field, the two parameters (10) and (11) of the external-field
strength vanish by virtue of the equality in (5), $a=b=0$; therefore, the
right-hand side of Eq.(13) also vanishes. For a constant uniform
electromagnetic field of general form, the condition that the wave vector in
(8) is isotropic is not satisfied: $k^{2}\neq 0$. Even in this case, however,
we can use the semiclassical crossed-field model, which ensures a satisfactory
description of the decay probability in an electromagnetic field of general
form, provided that $a\ll\varkappa^{2}$ and $b\ll\varkappa^{2}$. The closer to
zero the right-hand side of Eq.(13) and the higher the precision to which the
condition of isotropicity is satisfied, the more precise the results that are
obtained in this approximation. The above constraints are obviously satisfied
in the region of relatively weak electromagnetic fields for $a\ll 1$ and $b\ll
1$. Thus, the decay probability $P(Z^{0}\rightarrow\ell^{+}\ell^{-})$ in a
crossed field will correspond to the first leading term $P_{0}$ in the
semiclassical expansion of the total decay probability $P(\varkappa,a,b)$ in a
power series in the vectors of the external-field strengths at small values of
the parameters $a$ and $b$; that is,

$P(\varkappa,a,b)=P_{0}(\varkappa,0,0)+a\frac{\partial P}{\partial
a}(\varkappa,0,0)+b\frac{\partial P}{\partial b}(\varkappa,0,0)+\ldots$ (14)

This relation between the probabilities of quantum processes in a crossed
field and a constant electromagnetic field of general form was obtained for
the first time within quantum electrodynamics [18].

The wave functions for charged fermions in the field of a plane
electromagnetic wave were obtained as far back as 1937 by D.M.Volkov (see,for
example, [24]). In the case of a crossed field, these exact solutions to the
Dirac equations for leptons $\ell^{\pm}$ in an external field have the form

$\displaystyle\Psi_{\ell^{-}}(x,p)=\exp\biggl{[}-ipx+\frac{ieC(pa)}{2(pk)}(kx)^{2}-\frac{ie^{2}C^{2}}{6(pk)}(kx)^{3}\biggr{]}\times$
$\displaystyle\times\biggl{\\{}1-\frac{e(kx)}{4(pk)}(F_{\mu\lambda}\gamma^{\mu}\gamma^{\lambda})\biggr{\\}}\frac{u(p)}{\sqrt{2p_{0}V}};$
(15)

$\displaystyle\Psi_{\ell^{+}}^{c}(x,p^{\prime})=\exp\biggl{[}ip^{\prime}x+\frac{ieC(p^{\prime}a)}{2(p^{\prime}k)}(kx)^{2}-\frac{ie^{2}C^{2}}{6(p^{\prime}k)}(kx)^{3}\biggr{]}\times$
$\displaystyle\times\biggl{\\{}1+\frac{e(kx)}{4(p^{\prime}k)}(F_{\mu\lambda}\gamma^{\mu}\gamma^{\lambda})\biggr{\\}}\frac{u^{c}(p^{\prime})}{\sqrt{2p^{\prime}_{0}V}};$
(16)

The spin part of these wave functions normalized to a three-dimensional
spatial volume $V$ is determined by the constant Dirac bispinors $u(p)$ and
$u^{c}(p^{\prime})$. The antilepton wave function
$\Psi_{\ell^{+}}(x,p^{\prime})$ differs from expression (2.) only by the
reversal of the sign of the electric charge $e$: $e\rightarrow-e$. In the
expression for the matrix element of the decay in (1), the antilepton wave
function appears in the charge-conjugate form (2.), which can be derived on
the basis of the Dirac-conjugate bispinor
$\bar{\Psi}_{\ell^{+}}(x,p^{\prime})$ with the aid of the charge-conjugation
matrix $U_{C}=-i\gamma^{0}\gamma^{2}$; that is,

$\Psi^{c}_{\ell^{+}}(x,p^{\prime})=U_{C}\bar{\Psi}^{T}_{\ell^{+}}(x,p^{\prime})$
(17)

As for the $Z^{0}$-boson, its wave function does not change in an
electromagnetic field because it is electrically neutral:

$Z_{\mu}(x,q)=\exp\biggl{(}-iqx\biggr{)}\frac{v_{\mu}(q)}{\sqrt{2q_{0}V}}.$
(18)

The spin states of the $Z^{0}$-boson are characterized by the complex
polarization $4$-vector $v_{\mu}(q)$ satisfying standard conditions for
massive vector fields; that is,

$\displaystyle v_{\mu}(q)q^{\mu}=0,\quad v^{*}_{\mu}(q)v^{\mu}(q)=-1,$ (19)
$\displaystyle\sum_{\sigma=1}^{3}v_{\mu}(q,\sigma)v^{*}_{\nu}(q,\sigma)=-g_{\mu\nu}+q_{\mu}q_{\nu}/M_{Z}^{2}.$

We substitute the particle wave functions (2.), (2.), and (18) into expression
(1) for the $S$-matrix element and perform integration of $\mid
S_{fi}\mid^{2}$ over the phase space of final leptons. Upon summation over the
spin states of the lepton-antilepton pair and averaging over polarizations of
the $Z^{0}$-boson, we arrive at

$\displaystyle
P(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)=\frac{(g_{V}^{2}+g_{A}^{2})M_{Z}^{2}}{12\pi^{2}q_{0}}\int\limits_{0}^{1}du\biggl{\\{}\biggl{[}1-(1+3\lambda)\frac{m_{\ell}^{2}}{M_{Z}^{2}}\biggr{]}\Phi_{1}(z)-$
$\displaystyle-\frac{2\varkappa^{2/3}}{\left[u(1-u)\right]^{1/3}}\biggl{[}1-2u+2u^{2}+(1+\lambda)\frac{m_{\ell}^{2}}{M_{Z}^{2}}\biggr{]}\Phi^{\prime}(z)\biggr{\\}}.$
(20)

The probability of $Z^{0}$-boson decay is expressed in terms of the Airy
functions $\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ (for necessary details
concerning these special mathematical functions, see the Appendix) depending
on the argument

$z=\frac{m_{\ell}^{2}-M_{Z}^{2}u(1-u)}{M_{Z}^{2}[\varkappa u(1-u)]^{2/3}}.$
(21)

The couplings of the $Z^{0}$-boson to the charged leptons of all three
generations, $e^{\pm},\mu^{\pm},$ and $\tau^{\pm}$, can be expressed in terms
of the Weinberg angle $\theta_{\rm W}$ and the electroweak coupling constant
$g=e/\sin\theta_{\rm W}$ as

$g_{V}=-\frac{g(1-4\sin^{2}\theta_{\rm W})}{4\cos\theta_{\rm W}},\qquad
g_{A}=-\frac{g}{4\cos\theta_{\rm W}}.$ (22)

The dimensionless parameter $\lambda$ in expression (2.) is given by

$\lambda=\frac{g_{A}^{2}-g_{V}^{2}}{g_{A}^{2}+g_{V}^{2}}=\frac{1-(1-4\sin^{2}\theta_{\rm
W})^{2}}{1+(1-4\sin^{2}\theta_{\rm W})^{2}},$ (23)

at $\sin^{2}\theta_{\rm W}=0,23$, it is very close to unity ($\lambda\simeq
0,987$).

##  3\. KINEMATICS OF $Z^{0}$-BOSON DECAY
IN A CROSSED FIELD

The fact that the law of energy-momentum conservation for particles involved
in the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ does not have a conventional
form,

$q_{\mu}+k_{\mu}=p_{\mu}+p^{\prime}_{\mu}.$ (24)

is a distinctive feature of processes occurring in an external electromagnetic
field. Along with the $Z^{0}$-boson and final-lepton energy-momentum
$4$-vectors $q_{\mu}$, $p_{\mu}(\ell^{-})$, and $p^{\prime}_{\mu}(\ell^{+})$,
expression (24) also involves the wave vector (8) determining the energy
contribution of the external field. It should be noted that, in this case, we
are dealing with asymptotic momenta that charged particles possess at rather
large distances, where the effect of the external field is negligible, rather
than with their dynamical $4$-momenta in a crossed field. This is the meaning
that the parameters $p_{\mu}$ and $p^{\prime}_{\mu}$ appearing in the Volkov
solutions (2.) and (2.) to the Dirac equation in a crossed field have. It is
indeed extremely difficult to observe the kinematics of the decay
$Z^{0}\rightarrow\ell^{+}\ell^{-}$ in the field itself because the charged-
lepton $4$-momenta are not integrals of the motion. As soon as the leptons
escape from the region of the external field, their energies and momenta do
not change any longer, so that one can measure them experimentally. In this
case, the $4$-momenta of all particles satisfy the ordinary kinematical
conditions

$p^{2}=p^{\prime 2}=m_{\ell}^{2},\quad q^{2}=M_{Z}^{2},\quad k^{2}=0,$ (25)

and this makes it possible to represent the isotropic vector $k^{\mu}$ in the
form

$k^{\mu}=\frac{\Delta}{2(q_{\alpha}F^{\alpha\beta}F_{\beta\sigma}q^{\sigma})}\cdot
F^{\mu\lambda}F_{\lambda\nu}q^{\nu},$ (26)

where the parameter $\Delta$ characterizes the energy imbalance of the
reaction $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in the external field,

$\Delta=2(qk)=(p+p^{\prime})^{2}-q^{2}=2m_{\ell}^{2}+2(pp^{\prime})-M_{Z}^{2}$
(27)

Expression (24) arises in a natural way as the argument of the four-
dimensional Dirac delta function in calculating the $S$-matrix element (1). In
performing integration over the phase space of final leptons, there also
occurs summation of the external-field contributions characterized by various
values of the parameter $\Delta$. This is the circumstance that distinguishes
the kinematics of $Z^{0}$-boson decay in an external field from a similar
process involving a real photon, $\gamma+Z^{0}\rightarrow\ell^{+}\ell^{-}$,
where the $4$-momentum $k_{\mu}$ is fixed and is independent of the final-
lepton momenta. The physical meaning of the above formulas can most easily be
understood in the $Z^{0}$-boson rest frame, where the law of energy-momentum
conservation in (24) can be expressed in terms of the following $4$-vectors:

$q^{\mu}=(M_{Z};{\bf 0}),\quad p^{\mu}=(\varepsilon_{1};{\bf p}_{1}),\quad
p^{\prime\mu}=(\varepsilon_{2};{\bf p}_{2}),\quad k^{\mu}=(k^{0};{\bf k}).$
(28)

We chose the axes of the three-dimensional Cartesian coordinate system in such
a way that the vectors of the electric- and magnetic-field strength, ${\bf E}$
and ${\bf H}$, are directed along the $x$ and $y$ axes, respectively. The
direction of the $3$-momentum ${\bf k}$ formed by the spatial components of
the wave $4$-vector $k^{\mu}$ (26) then coincides with the positive direction
of the $z$ axis, which is parallel to the vector product ${\bf[E\times H]}$.
In the Cartesian coordinate system chosen above, the orientation of the lepton
($\ell^{-}$) and antilepton ($\ell^{+}$) $3$-momenta ${\bf p}_{1}$ and ${\bf
p}_{2}$ with respect to the $z$ axis can be described by two azimuthal angles
$\vartheta_{1}$ and $\vartheta_{2}$ ($0\leq\vartheta_{1,2}\leq\pi$). Figure 1
shows schematically the three-dimensional configuration formed by all of the
aforementioned vectors.

Figure 1: Kinematics of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a
crossed field in the $Z^{0}$-boson rest frame.

If the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ occurs in the vacuum and if
the external electromagnetic field is inoperative, the lepton and antilepton
have momenta that are equal in absolute value but are oppositely directed and
which can be expressed in terms of their masses and the $Z^{0}$-boson mass.
The total energy taken away by the lepton-antilepton pair is shared between
the particles in equal parts and its total amount is equal to the rest energy
of the $Z^{0}$-boson,

$p_{1}=p_{2}=\frac{M_{Z}}{2}\left(1-{4m_{\ell}^{2}\over
M_{Z}^{2}}\right)^{1/2},\quad\varepsilon_{1}=\varepsilon_{2}=M_{Z}/2,\quad\vartheta_{1}+\vartheta_{2}=\pi.$
(29)

In a crossed field, each of the equalities in (29) is in general violated, and
the energy $\varepsilon_{1}$ that is taken away in the decay by the lepton is
not equal to the antilepton energy $\varepsilon_{2}$. Only fulfillment of the
following conservation laws is guaranteed:

$\displaystyle\varepsilon_{1}+\varepsilon_{2}-M_{Z}=p_{1}\cos\vartheta_{1}+p_{2}\cos\vartheta_{2},$
(30) $\displaystyle p_{1}\sin\vartheta_{1}-p_{2}\sin\vartheta_{2}=0,$ (31)

The vectors ${\bf p}_{1}$, ${\bf p}_{2}$, and ${\bf k}$ lie in the same plane,
but, in this case, it is not mandatory that the lepton and the antilepton fly
apart along the same straight line, $\vartheta_{1}+\vartheta_{2}\neq\pi$. The
deficit in the ordinary law of energy-momentum conservation is compensated in
an external field by the isotropic wave vector $k^{\mu}$ (8).

Formula (30) represents the three-dimensional form of the conservation law of
the covariant projections of the $4$-momenta of all particles on to the
direction of the wave vector $(qk)=(pk)+(p^{\prime}k)$, while relation (31) is
the conservation law for the ordinary three-dimensional projections of the
particle momenta on to the axis that is orthogonal to the vector ${\bf k}$ and
which lies in the plane spanned by the $3$-vectors ${\bf p}_{1}$, ${\bf
p}_{2}$, and ${\bf k}$.

We can now interpret physically the variable $u$ appearing in the integrand on
the right-hand side of (2.) and in the argument of the Airy functions in (21).
This variable is associated with the angles $\vartheta_{1}$ and
$\vartheta_{2}$ of divergence of the lepton-antilepton pair with respect to
the $z$ axis in the rest frame of the initial $Z^{0}$-boson:

$u=\frac{(pk)}{(qk)}=\frac{\varepsilon_{1}-p_{1}\cos\vartheta_{1}}{M_{Z}}=1-\frac{\varepsilon_{2}-p_{2}\cos\vartheta_{2}}{M_{Z}}$
(32)

If the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ is unaffected by an external
field, then, by virtue of relations (29), the variable $u$ can be represented
in the form

$u={1\over 2}\left[1-\left(1-{4m_{\ell}^{2}\over
M_{Z}^{2}}\right)^{1/2}\cos\vartheta_{1}\right]={1\over
2}\left[1+\left(1-{4m_{\ell}^{2}\over
M_{Z}^{2}}\right)^{1/2}\cos\vartheta_{2}\right].$ (33)

The interval of its values admissible from the point of view of decay
kinematics in a vacuum is determined by the inequality

$u_{1}\leq u\leq u_{2},\quad\mbox{where}\quad u_{1,2}={1\over
2}\left[1\mp\left(1-{4m_{\ell}^{2}\over M_{Z}^{2}}\right)^{1/2}\right]$ (34)

The limiting transition $\varkappa\rightarrow 0$ in (2.) reproduces exactly
well-known results for the ordinary probability of the decay
$Z^{0}\rightarrow\ell^{+}\ell^{-}$ without any field. The kinematically
allowed region of values of the variable $u$ (34) corresponds to negative
values of the argument of the Airy functions in (21); therefore, the limiting
transition to zero values of the external-electromagnetic-field strength is
implemented by means of the formal substitution of the Heaviside step
$\theta$-function for the Airy function $\Phi_{1}(z)$:

$\Phi_{1}(z)\rightarrow\pi\theta(-z)$ (35)

In the limit $\varkappa\rightarrow 0$, the differential probability of
$Z^{0}$-boson decay with respect to the variable $u$ does not depend on the
direction of divergence of leptons and has the form

$\frac{dP}{du}(Z^{0}\rightarrow\ell^{-}\ell^{+})=\frac{M_{Z}^{2}}{12\pi
q_{0}}\left[g_{V}^{2}+g_{A}^{2}+{2m_{\ell}^{2}\over
M_{Z}^{2}}(g_{V}^{2}-2g_{A}^{2})\right]$ (36)

As a matter of fact, integration with respect to the invariant variable $u$
now reduces to multiplying expression (36) by the phase space of this
variable, $(u_{2}-u_{1})$. As a result, we obtain the well-known formula for
the leptonic-decay width of the $Z^{0}$-boson in a vacuum,

$\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})=\frac{G_{\rm
F}M_{Z}^{3}}{12\pi\sqrt{2}}\sqrt{1-4\delta_{\ell}^{2}}\biggl{(}1-c_{\ell}-\delta_{\ell}^{2}(1+2c_{\ell})\biggr{)},$
(37)

where we have introduced the notation

$\delta_{\ell}={m_{\ell}\over M_{Z}},\qquad c_{\ell}=4\sin^{2}\theta_{\rm
W}-8\sin^{4}\theta_{\rm W}\approx 0,497$ (38)

In a crossed electromagnetic field, the kinematics of $Z^{0}$-boson decay
changes, first of all, owing to the increase in the phase space of possible
states of final leptons $\ell^{\pm}$. In the three-dimensional space of the
momenta $(p_{x},p_{y},p_{z})$ of one of the leptons — for example, $\ell^{-}$
— the kinematically allowed region of the decay
$Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see (29)] can be represented
as a three-dimensional sphere, each point of this sphere corresponding to one
of the final states of this lepton. In an external electromagnetic field, all
possible final states of the lepton $\ell^{-}$ already fill some volume in
momentum space. At a fixed value of the azimuthal angle $\vartheta_{1}$, the
momentum of the lepton $\ell^{-}$ can take any values from the interval

$\displaystyle 0\leq p_{1}<p_{\rm max}(\vartheta_{1}),\quad\mbox{where}$
$\displaystyle\ p_{\rm
max}(\vartheta)=\frac{M_{Z}}{\sin^{2}\vartheta}\left(\cos\vartheta+\sqrt{1-\delta_{\ell}^{2}\sin^{2}\vartheta}\right).$
(39)

The energy $\varepsilon_{1}$ of this particle will be bounded by the
inequality

$\displaystyle m_{\ell}\leq\varepsilon_{1}<\varepsilon_{\rm
max}(\vartheta_{1}),\quad\mbox{where}$ $\displaystyle\varepsilon_{\rm
max}(\vartheta)=\frac{M_{Z}}{\sin^{2}\vartheta}\left(1+\cos\vartheta\sqrt{1-\delta_{\ell}^{2}\sin^{2}\vartheta}\right).$
(40)

The variables $\varepsilon_{1}$ and $\vartheta_{1}$ determine unambiguously
decay kinematics in a crossed field. The energy $\varepsilon_{2}$ and the
azimuthal angle $\vartheta_{2}$ of the antilepton $\ell^{+}$ can be expressed
in terms of these variables with the aid of the kinematical relations (30) and
(31) and the standard relativistic relations

$\varepsilon_{1}=\sqrt{p_{1}^{2}+m_{\ell}^{2}},\quad\varepsilon_{2}=\sqrt{p_{2}^{2}+m_{\ell}^{2}}.$
(41)

After some simple algebra, we obtain

$\displaystyle\varepsilon_{2}=M_{Z}-\varepsilon_{1}+\frac{M_{Z}(\varepsilon_{1}-M_{Z}/2)}{M_{Z}-\varepsilon_{1}+p_{1}\cos\vartheta_{1}}$
(42)
$\displaystyle\vartheta_{2}=\arccos\left(\frac{\varepsilon_{1}+\varepsilon_{2}-M_{Z}-p_{1}\cos\vartheta_{1}}{\sqrt{\varepsilon_{2}^{2}-m_{\ell}^{2}}}\right)$
(43)

An analysis of the resulting formulas shows that, as a rule, the leptons
$\ell^{\pm}$ diverge at arbitrary angles $\vartheta_{1}$ and $\vartheta_{2}$
with respect to the $z$ axis. This is true for almost all of the azimuthal
angles, with the exception of the case where the lepton and antilepton move in
the direction parallel to the $z$ axis. In the case of $\sin\vartheta_{1}\neq
0$ and $\sin\vartheta_{2}\neq 0$, the lepton and antilepton energies
$\varepsilon_{1}$ and $\varepsilon_{2}$, respectively, can be expressed in
terms of the emission angles $\vartheta_{1}$ and $\vartheta_{2}$ as

$\displaystyle\varepsilon_{1}\approx
p_{1}\simeq\frac{M_{Z}\sin\vartheta_{2}}{\sin\vartheta_{1}+\sin\vartheta_{2}-\sin(\vartheta_{1}+\vartheta_{2})}$
(44) $\displaystyle\varepsilon_{2}\approx
p_{2}\simeq\frac{M_{Z}\sin\vartheta_{1}}{\sin\vartheta_{1}+\sin\vartheta_{2}-\sin(\vartheta_{1}+\vartheta_{2})}$
(45)

Figure 2: Lepton energy $\varepsilon_{1}$ as a function of the antilepton
emission angle $\vartheta_{2}$ with respect to the $z$ axis at
$\vartheta_{1}=2\pi/3$.

These relations were obtained in the relativistic approximation, where
$\varepsilon_{1}\gg m_{\ell}$, $\varepsilon_{2}\gg m_{\ell}$ and where the
parameter $\delta_{\ell}$ (38) can be disregarded. We can now draw qualitative
conclusions on a typical behavior of the energies of the lepton-antilepton
pair almost over the entire azimuthal-angle range $0<\vartheta_{1,2}<\pi$. It
was indicated above that, at a fixed value of the emission angle
($\vartheta_{1}={\rm const}$) of the lepton $\ell^{-}$, its energy can vary
from the rest energy to the maximum value $\varepsilon_{\rm
max}(\vartheta_{1})$ (3.). Concurrently, the emission angle $\vartheta_{2}$ of
the antilepton $\ell^{+}$ can take an arbitrary value between zero and $\pi$.
As for its energy $\varepsilon_{2}$, one can see that, as the energy
$\varepsilon_{1}$ increases, $\varepsilon_{2}$ first decreases from the value

$\varepsilon_{0}=\frac{M_{Z}}{2}\frac{(1-2\delta_{\ell}+2\delta_{\ell}^{2})}{(1-\delta_{\ell})}\approx\frac{M_{Z}}{2}\left(1-\delta_{\ell}\right)$
(46)

to its local minimum

$\varepsilon_{2}=\frac{M_{Z}\cos\left(\vartheta_{1}/2\right)}{1+\cos\left(\vartheta_{1}/2\right)},$
(47)

whereupon it increases monotonically to infinity as the angle $\vartheta_{2}$
tends to zero, while the lepton energy $\varepsilon_{1}$ tends to its maximum
value $\varepsilon_{\rm max}(\vartheta_{1})$. The above relationship between
the energies of the lepton-antilepton pair and the directions of lepton and
antilepton emission with respect to the $z$ axis is illustrated by the graphs
in Figs. 2 and 3. It should be noted that the minimum value of the antilepton
energy $\varepsilon_{2}$ (47) is reached at the emission angle of
$\vartheta_{2}=\pi-\vartheta_{1}/2$, in which case the energy of the other
particle, $\ell^{-}$,is

$\varepsilon_{1}=\frac{M_{Z}/2}{1+\cos\left(\vartheta_{1}/2\right)}.$ (48)

Figure 3:  Antilepton energy $\varepsilon_{2}$ versus the antilepton emission
angle $\vartheta_{2}$ with respect to the $z$ axis at a fixed value of the
lepton emission angle ($\vartheta_{1}=2\pi/3$).

If the lepton escapes in the direction parallel to the $z$ axis, expressions
(44) and (45) are inapplicable, so that one must use the exact kinematical
relations (30) and (31) or their consequences (42) and (43). In this case, two
situations are possible — that in which the leptons fly apart in opposite
directions and that in which the leptons move in the same direction.

In the first case (at $\vartheta_{1}=0$ and $\vartheta_{2}=\pi$ or at
$\vartheta_{1}=\pi$ and $\vartheta_{2}=0$), the energy of the lepton moving
along the positive direction of the $z$ axis can take any value from the rest
energy $\varepsilon_{\rm min}=m_{\ell}$ to infinity. The energy of the second
lepton, which flies in the opposite direction, is virtually constant — it
ranges between the initial value $\varepsilon_{0}$ (46) and its maximum value

$\varepsilon_{\pi}=\lim_{\vartheta\rightarrow\pi}\varepsilon_{\rm
max}(\vartheta)=\frac{M_{Z}}{2}\left(1+\delta_{\ell}^{2}\right).$ (49)

which is very close to $\varepsilon_{0}$. The case where both particles
$\ell^{+}$ and $\ell^{-}$ produced in $Z^{0}$-boson decay move in the same
direction opposite to the positive direction of the $z$ axis
($\vartheta_{1}=\vartheta_{2}=\pi$) is also possible. In this case, their
energies $\varepsilon_{1}$ and $\varepsilon_{2}$ can vary from the rest energy
to the limiting value $\varepsilon_{0}$ (46). In addition, the following
relation can be obtained for this case from (30) in the relativistic
approximation (for $\delta_{\ell}\rightarrow 0$):

$\varepsilon_{1}+\varepsilon_{2}=\frac{M_{Z}}{2}.$ (50)

Thus, one can see that, in relation to the kinematics of the process
$Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see(29)], the physics of
$Z^{0}$-boson decay in an external electromagnetic field is richer in the
number of possible final lepton states. From the mathematical point of view,
this manifests itself in the fact that all values of the invariant variable
$u$ (32) from the domain $0<u<1$ are now admissible. Moreover, it is
noteworthy that, in a crossed field, the physical meaning of this variable
changes because it now characterizes two independent kinematical quantities
simultaneously - for example, $\varepsilon_{1}$ and $\vartheta_{1}$.
Disregarding the cases in which leptons fly apart in the direction parallel to
the $z$ axis, we can express the variable $u$ in terms of the azimuthal angles
$\vartheta_{1}$ and $\vartheta_{2}$ alone by using formulas (44) and (45). The
result is

$u\simeq\frac{\tan({\vartheta_{1}/2})}{\tan({\vartheta_{1}/2})+\tan({\vartheta_{2}/2})}.$
(51)

Figure 4: Differential width with respect to the decay
$Z^{0}\rightarrow\ell^{-}\ell^{+}$ as a function of the invariant angular
variable $u$ (32) in a weak crossed field (at $\varkappa=0,01$)

An analysis of the integrand on the right-hand side of (2.) shows that, in an
external field, not all of the directions of divergence of the lepton and
antilepton are equiprobable. The graphs in Fig. 4 that represent the
dependence of the differential decay width of the $Z^{0}$-boson on the
invariant variable $u$ corroborate that, even in a relatively weak external
field, there are noticeable deviations from a uniformly distributed
probability of the leptonic decay mode in a vacuum [see (36)]. In the case
where leptons produced in the process $Z^{0}\rightarrow\ell^{+}\ell^{-}$ fly
apart at specific angles $(\vartheta_{1,2}\neq 0,\pi)$ in diametrically
opposite directions $(\vartheta_{1}+\vartheta_{2}\approx\pi)$, the invariant
variable $u$ takes values of $u\approx 0,5$. In this region, which is
kinematically allowed for $Z^{0}$-boson decay in a vacuum, the deviations of
the differential decay width $\Gamma^{\prime}_{u}(u,\varkappa)$ from the
corresponding static value of $\Gamma^{\prime}_{u}(u,0)\approx 83$ MeV are in
fact unobservable. If one of the leptons flies away at a specific angle
$\vartheta$ with respect to the $z$ axis and at an energy close to the maximum
value $\varepsilon_{\rm max}(\vartheta)$ in (3.) and if the second lepton
moves along the direction nearly parallel to this axis (in the positive
direction), the variable $u$ takes values in the region around $u\approx 0$ or
$u\approx 1$. In this case, one can observe oscillations of the differential
decay width of the $Z^{0}$-boson that are characterized by a growing
amplitude, which increases and decreases the static vacuum value in (36) by
more than 15%. Similar oscillations of probabilities of quantum processes in
an external electromagnetic field are characteristic of all particle
transformations allowed in a vacuum and were described in detail in the
literature [18, 19].

##  4\. LEPTONIC DECAY WIDTH OF THE $Z^{0}$-BOSON

The integral representation of the partial width of the $Z^{0}$-boson with
respect to its decay to a pair of charged leptons in an external
electromagnetic field can be derived from expression (2.) for the probability
by going over to the rest frame ($q_{0}=M_{Z}$) and by making the change of
the integration variable $x=u(1-u)$. The result is

$\displaystyle\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)$
$\displaystyle=$
$\displaystyle\frac{(g_{V}^{2}+g_{A}^{2})M_{Z}}{6\pi^{2}}\int\limits_{0}^{1/4}\frac{dx}{\sqrt{1-4x}}\biggl{\\{}\biggl{[}1-\delta_{\ell}^{2}(1+3\lambda)\biggr{]}\Phi_{1}(z)-$
(52) $\displaystyle-$ $\displaystyle
2\left(\frac{\varkappa^{2}}{x}\right)^{1/3}\biggl{[}1-2x+\delta_{\ell}^{2}(1+\lambda)\biggr{]}\Phi^{\prime}(z)\biggr{\\}},$

where the argument of the Airy functions is now determined by the expression

$z=\frac{\delta_{\ell}^{2}-x}{(\varkappa x)^{2/3}}.$ (53)

Let us now consider the asymptotic estimates obtained from this formula at
various values of the external-field-strength parameter $\varkappa$ (12).

In weak electromagnetic fields such that $\varkappa\ll\delta_{\ell}^{2}\ll 1$,
the partial decay width of the $Z^{0}$-boson can be written as the sum of two
terms,

$\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)=\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})+\Delta\Gamma(\varkappa).$
(54)

The first term in this expression is the $Z^{0}$-boson width with respect to
the leptonic decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see
expression (37) above]. The experimental value of this quantity is known to a
precision of 0.1%: $\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})=83,984\pm 0,086\
\mbox{MeV}$ [2]. The second term characterizes the effect of the external
electromagnetic field. In the leading order of the expansion in the small
parameters $\varkappa$ and $\delta_{\ell}$, this term can be represented in
the form

$\displaystyle\Delta\Gamma(\varkappa)=-\frac{G_{\rm
F}M_{Z}^{3}}{4\pi\sqrt{6}}\Biggl{[}\
\delta_{\ell}^{2}(1-6c_{\ell})\varkappa\cos\left({1\over 3\varkappa}\right)-$
$\displaystyle-(1-c_{\ell})\varkappa^{2}\sin\left({1\over
3\varkappa}\right)+\frac{8\varkappa^{2}}{\sqrt{3}}(1-c_{\ell})\ \Biggr{]}$
(55)

Similar calculations of the probability of $Z^{0}$-boson decay in a magnetic
field were performed in [25], where the contribution of the virtual electron-
positron loop to the amplitude for elastic $Z^{0}$-boson scattering was
considered, its imaginary part being directly related to the
$Z^{0}\rightarrow\ell^{+}\ell^{-}$ decay width being studied. It is worth
noting, however, that the results of the present calculations [expressions
(52) and (4.)] differ somewhat from their counter parts in [25].

For the parameter of the external-electromagnetic-field strength, we will now
consider the region specified by the condition
$\varkappa\gg\delta_{\ell}^{2}$. We use the fact that the masses of the
leptons of all three generations are much smaller than the $Z^{0}$-boson mass
and go over to the limit $\delta_{\ell}\rightarrow 0$ in formulas (52) and
(53). We can then calculate analytically the integral of the Airy function on
the right-hand side of (52) at arbitrary values of the external-field-strength
parameter $\varkappa$. The final expression for the leptonic-decay width of
the $Z^{0}$-boson is written in terms of the Bessel functions $J_{\nu}(w)$
carrying the noninteger indices $\nu=\pm 1/6$ and $\nu=\pm 5/6$ and depending
on the argument $w=(6\varkappa)^{-1}$. We now consider a universal function
$R(\varkappa)$ that describes the degree of influence of the external field on
the leptonic mode of $Z^{0}$-boson decay:

$R(\varkappa)=\frac{\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)}{\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})}.$
(56)

Figure 5: Oscillations of the partial width of the $Z$-boson with respect to
its decay to a pair of charged leptons in a weak electromagnetic field.

In the massless-lepton approximation ($\varkappa\gg\delta_{\ell}^{2}$), this
function, which is a relative width of the $Z^{0}$-boson with respect to the
decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in an electromagnetic field, can be
represented in the form

$R(\varkappa)=\frac{1}{3}\Biggl{[}1+R_{1}(\varkappa)+R_{2}(\varkappa)+R_{3}(\varkappa)\Biggr{]},$
(57)

where the following notation has been introduced:

$R_{1}(\varkappa)=\frac{\pi(1+3\varkappa^{2})}{12\varkappa}\left[J^{2}_{1/6}\left({1\over
6\varkappa}\right)+J^{2}_{-1/6}\left({1\over 6\varkappa}\right)\right],$ (58)
$R_{2}(\varkappa)=\frac{\pi(1-15\varkappa^{2})}{12\varkappa}\left[J^{2}_{5/6}\left({1\over
6\varkappa}\right)+J^{2}_{-5/6}\left({1\over 6\varkappa}\right)\right],$ (59)
$\displaystyle R_{3}(\varkappa)$ $\displaystyle=$
$\displaystyle\frac{\pi(2+15\varkappa^{2})}{6}\Biggl{[}\ J_{1/6}\left({1\over
6\varkappa}\right)J_{-5/6}\left({1\over 6\varkappa}\right)-$ (60)
$\displaystyle-$ $\displaystyle J_{-1/6}\left({1\over
6\varkappa}\right)J_{5/6}\left({1\over 6\varkappa}\right)\ \Biggr{]}.$

Figure 6: Relative $Z^{0}\rightarrow\ell^{+}\ell^{-}$ decay width
$R(\varkappa)$ (56) as a function of the parameter $\varkappa$ at the boundary
of the region of oscillations.

The graphs representing the dependence of the relative leptonic-decay width
(56) normalized to unity at zero external field on the invariant field-
strength parameter $\varkappa$ (9) are on display in Figs. 5, 6, 7. In
relatively weak fields (such that $\varkappa<0,06$), there arise oscillations
of the probability of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$, their
amplitude increasing quadratically with the parameter $\varkappa$. In this
region, one can use the asymptotic expansion for the Bessel functions and
approximate formula (57) by the expression

Figure 7: Leptonic-decay width of the $Z^{0}$-boson as a function of the
external-field-strength parameter $\varkappa$ (12) in the vicinity of the
absolute-minimum point.

$\displaystyle
R(\varkappa)=1-\frac{16}{3}\varkappa^{2}+\frac{3\varkappa^{2}}{\sqrt{3}}\sin\left({1\over
3\varkappa}\right)+\frac{65\varkappa^{3}}{\sqrt{3}}\cos\left({1\over
3\varkappa}\right)+$
$\displaystyle+\frac{880\varkappa^{4}}{3\sqrt{3}}\sin\left({1\over
3\varkappa}\right)-\frac{160}{3}\varkappa^{4}+\ldots$ (61)

It is obvious from Figs. 5 and 6 that the maximum deviation of the decay width
oscillating in an external field from its vacuum value in (37) does not exceed
2%. A further increase in the parameter of the external-electromagnetic-field
strength to values in the region $\varkappa>0,06$ leads to a complete
disappearance of oscillations and a monotonic decrease in the relative width
$R(\varkappa)$ down to the absolute minimum $R_{\min}=0,843$ in the vicinity
of the point $\varkappa_{\rm min}=0,213$ (see Fig. 7). In the region of strong
fields (such that $\varkappa>0,213$), the relative width of the $Z^{0}$-boson
with respect to its leptonic decay mode grows monotonically with the field
strength. In the region $\varkappa\gg 1$, this growth can be approximated by
the asymptotic expression.

$R(\varkappa)=\frac{15\
\Gamma^{4}(2/3)}{14\pi^{2}}(3\varkappa)^{2/3}+\frac{1}{3}+\frac{3\
\Gamma^{4}(1/3)}{110\pi^{2}\ (3\varkappa)^{2/3}}+\ldots$ (62)

##  5\. HADRONIC MODES OF $Z^{0}$-BOSON DECAY

The results obtained in the preceding section admit a straight forward
generalization to the case of the decay $Z^{0}\rightarrow\bar{f}f$, where, by
$f$, we denote an arbitrarily charged fermion, $q_{f}$ standing for the
absolute value of its electric charge. As is well known, the charge is
$q_{f}=2/3$ for the up quarks ($u,c,t$) and $q_{f}=1/3$ for the down quarks
($d,s,b$). It is necessary to consider that, for the $Z\bar{f}f$ vertex, the
coupling constants $g_{V}$ and $g_{A}$ (1) also depend on the absolute value
of the electric charge ($q_{f}$). In the Glashow-Weinberg-Salam model of
electroweak interactions, they are determined by the relations

$\mid g_{V}(f)\mid=\frac{g(1-4q_{f}\sin^{2}\theta_{\rm W})}{4\cos\theta_{\rm
W}},\qquad\mid g_{A}(f)\mid=\frac{g}{4\cos\theta_{\rm W}}.$ (63)

Upon making the formal substitution $e\rightarrow q_{f}e$ or
$\varkappa\rightarrow q_{f}\varkappa$ in (2.), the differential
$Z^{0}\rightarrow\bar{f}f$ decay width as a function of the invariant
variables $\varkappa$ (9) and $u$ (32) assumes the form

$\displaystyle\frac{d\Gamma}{du}(Z^{0}\rightarrow\bar{f}f)=\frac{G_{\rm
F}M_{Z}^{3}}{12\pi^{2}\sqrt{2}}\Biggl{\\{}\left[1-c_{f}-\delta_{f}^{2}(1+2c_{f})\right]\Phi_{1}(z_{f})-$
(64)
$\displaystyle-2\left[\frac{q_{f}^{2}\varkappa^{2}}{u(1-u)}\right]^{1/3}\left[(1-c_{f})(1-2u+2u^{2})+\delta_{f}^{2}\right]\Phi^{\prime}(z_{f})\Biggr{\\}}.$

In this expression, use is made of the same notation as in (21) and (38):

$\delta_{f}={m_{f}\over M_{Z}},\qquad c_{f}=4q_{f}\sin^{2}\theta_{\rm
W}(1-2q_{f}\sin^{2}\theta_{\rm W}),$ (65)
$z_{f}=\frac{\delta_{f}^{2}-u(1-u)}{[q_{f}\varkappa u(1-u)]^{2/3}}.$ (66)

The limiting transition to an infinitely weak external field
($\varkappa\rightarrow 0$) leads to reproducing the well-known results for the
partial widths of the $Z^{0}$-boson with respect to its decay to light quarks;
that is,

$\Gamma(Z^{0}\rightarrow\bar{u}u)=\frac{G_{\rm
F}M_{Z}^{3}}{4\pi\sqrt{2}}\left[1-\frac{8}{3}\sin^{2}\theta_{\rm
W}+\frac{32}{9}\sin^{4}\theta_{\rm W}\right].$ (67)

$\Gamma(Z^{0}\rightarrow\bar{d}d)=\frac{G_{\rm
F}M_{Z}^{3}}{4\pi\sqrt{2}}\left[1-\frac{4}{3}\sin^{2}\theta_{\rm
W}+\frac{8}{9}\sin^{4}\theta_{\rm W}\right].$ (68)

These relations are obtained from (37) upon the formal substitution
$c_{\ell}\rightarrow c_{f}$ and $\delta_{\ell}\rightarrow\delta_{f}$ and the
multiplication by the common factor $N_{c}=3$ (number of color quark states).
The inclusion of radiative corrections leads to various modifications of
expressions (67) and (68), whose explicit form depends on the choice of input
parameters in electroweak theory and renormalization scheme for divergent loop
diagrams [4, 7]. For example, Sirlin, who was the author of one of the
pioneering studies in these realms [26], proposed the parametrization

$G_{\rm F}=\frac{\pi\alpha}{\sqrt{2}M_{W}^{2}\sin^{2}\theta_{\rm
W}}\frac{1}{(1-\Delta r)},\qquad\sin^{2}\theta_{\rm
W}=1-\frac{M_{W}^{2}}{M_{Z}^{2}},$ (69)

which has been widely used since then. Here, $\Delta r$ includes all radiative
corrections. For the sake of simplicity, we restrict ourselves to the case of
massless quarks ($\delta_{f}\ll\varkappa$) and, without going in to details of
the calculation of radiative corrections, try to analyze the effect of
external electromagnetic fields on the hadronic decay modes of the
$Z^{0}$-boson. Without allowance for the external-field effect, the width

$\Gamma_{had}=\sum^{5}_{i=1}\Gamma(Z^{0}\rightarrow\bar{q}_{i}q_{i})\simeq
2\Gamma(Z^{0}\rightarrow\bar{u}u)+3\Gamma(Z^{0}\rightarrow\bar{d}d)$

agrees with the experimentally measured value of $\Gamma(Z^{0}\rightarrow
hadrons)=1744,4\pm 2,0$ MeV [2] to a high degree of precision. We use this
approximation in the region of relatively strong electromagnetic fields
($\varkappa\sim 1$) and estimate the width of the $Z^{0}$-boson with respect
to the hadronic decay mode as

$\Gamma_{had}(\varkappa)=2\Gamma(Z^{0}\rightarrow\bar{u}u)R\left(\frac{2}{3}\varkappa\right)+3\Gamma(Z^{0}\rightarrow\bar{d}d)R\left(\frac{1}{3}\varkappa\right),$
(70)

Figure 8: Hadronic-decay width of the $Z^{0}$-boson in an external
electromagnetic field as a function of the parameter $\varkappa$ (12).

where the function $R(\varkappa)$ is determined by expressions (56) – (60).
Figure 8 shows the results of numerical calculations performed for the total
width of the $Z^{0}$-boson with respect to hadronic decay modes in an external
field on the basis of expression (70) with allowance for the production of
only five quarks ($u,d,s,c,$ and $b$). In this region, the contribution of the
heavy $t$ quark is insignificant. This contribution will be considered
individually in the next section.

One can see from Fig. 8 that, in just the same way as the width with respect
to leptonic decay modes (see Fig. 7), the width with respect to the hadronic
modes of $Z^{0}$-boson decay in an intense external field behaves
nonmonotonically. As the parameter $\varkappa$ increases, the width
$\Gamma_{had}(\varkappa)$ first decreases, reaching the absolute minimum of
$\Gamma_{had}(\varkappa_{\rm min})=1497$ MeV at the point $\varkappa_{\rm
min}=0,501$, where upon it begins growing. In a weak electromagnetic field
($\varkappa\ll 1$), the gradual decrease in the hadronic-decay width
$\Gamma_{had}(\varkappa)$ is accompanied by quite complicated oscillations.

##  6\. TOTAL DECAY WIDTH OF THE $Z^{0}$-BOSON AND $t$-QUARK CONTRIBUTION

In a vacuum, $Z^{0}$-boson decay to $t$ quarks via the process
$Z^{0}\rightarrow\bar{t}t$ is forbidden by the energy-conservation law because
the mass of the heavy $t$ quark is rather high, $m_{t}=174,2\pm 3,3$ GeV [2].
The presence of an external electromagnetic field removes this forbiddance,
opening new channels of $Z^{0}$-boson decay. In principle, the production of
arbitrary charged fermions whose masses satisfy the relation
$\delta_{f}=m_{f}/M_{Z}>1/2$ becomes possible. The missing energy for these
reactions comes from the external field, and this resembles tunneling through
a potential barrier in nonrelativistic quantum mechanics. At small values of
the electromagnetic-field strength (such that $\varkappa\ll 1$), the
$Z^{0}\rightarrow\bar{f}f$ decay width is exponentially small:

Figure 9: Partial width of the $Z^{0}$-boson with respect to its decay to
$t$-quarks in an external electromagnetic field (for the description of the
curves, see the main body of the text).

$\Gamma(Z^{0}\rightarrow\bar{f}f)=\frac{G_{\rm
F}M_{Z}^{3}\left(8\delta_{f}^{2}+1-6c_{f}\right)\delta_{f}^{2}q_{f}}{8\pi\sqrt{6}\left(4\delta_{f}^{2}-1\right)\sqrt{8\delta_{f}^{2}+1}}\varkappa\
\exp\left[-\frac{\left(4\delta_{f}^{2}-1\right)^{3/2}}{3q_{f}\varkappa}\right].$
(71)

This formula was derived from expression (64) upon integration with respect to
the invariant variable $u$ (32) in the approximation $z_{f}\gg 1$ (66).
Substituting the $t$-quark charge $q_{f}=2/3$ and performing summation over
three color states, we obtain an asymptotic estimate for the partial width of
the $Z^{0}$-boson with respect to its decay to $t$-quarks in a relatively weak
electromagnetic field; that is,

$\displaystyle\Gamma(Z^{0}\rightarrow\bar{t}t)=\frac{G_{\rm
F}m_{t}^{2}M_{Z}^{2}\left[8m_{t}^{2}+\left(1-16\sin^{2}\theta_{\rm
W}+\frac{64}{3}\sin^{4}\theta_{\rm
W}\right)M_{Z}^{2}\right]}{4\pi\sqrt{6}\left(4m_{t}^{2}-M_{Z}^{2}\right)\sqrt{8m_{t}^{2}+M_{Z}^{2}}}\times$
$\displaystyle\varkappa\
\exp\left[-\frac{\left(4m_{t}^{2}-M_{Z}^{2}\right)^{3/2}}{2\varkappa
M_{Z}^{3}}\right].$ (72)

The results of numerical calculations show that the above approximate formula
for $\Gamma(Z^{0}\rightarrow\bar{t}t)$ leads to an error not greater than 4%
for all values of the external-field-strength parameter from the region
$\varkappa\leq 5$. As the parameter $\varkappa$ grows, the deviation of the
results obtained according to the asymptotic formula (6.) from the precise
value of the partial width of the $Z^{0}$-boson with respect to the decay
$Z^{0}\rightarrow\bar{t}t$ in an external field increases [see Fig.9, where
the dotted curve corresponds to the calculation according to formula (6.),
while the solid curve represents the results of a numerical integration of
expression (64)].

Figure 10: Total decay width of the $Z^{0}$-boson in an external
electromagnetic field (for the description of the curves, see the main body of
the text).

The $t$-quark contribution to the total decay width of the $Z^{0}$-boson
remains negligible (less than 1%) up to strength-parameter values of about
$\varkappa=6,4$. Figure 10 displays the total decay width of the $Z^{0}$-boson
as a function of the parameter $\varkappa$ in the region of strong fields
($\varkappa\leq 10$). The solid curve corresponds to the total decay width of
the $Z^{0}$-boson with allowance for the $t$-quark contribution, while the
dotted curve represents the same quantity calculated without taking into
account the $t$-quark contribution. One can clearly see a trend toward the
growth of both the absolute value of the total decay width of the
$Z^{0}$-boson and the fraction of the $t$-quark contribution in this decay
width. As the external-field strength increases, the process
$Z^{0}\rightarrow\bar{t}t$, which is forbidden in a vacuum, becomes dominant
in superstrong electromagnetic fields. By way of example, it can be indicated
that, in the $Z^{0}$-boson decay width at $\varkappa=100$, the fraction
associated with $t$-quarks is as large as 50%. This circumstance clearly
demonstrates how an external electromagnetic field can change drastically the
standard physics of quantum processes in a vacuum and serve as some kind of a
catalyst for a number of new and nontrivial phenomena.

##  7\. CONCLUSIONS

The effect of a strong electromagnetic field on the probability of
$Z^{0}$-boson decay has been investigated. The present calculations have been
performed within the crossed-field model, which makes it possible to trace
characteristic variations in the modes of $Z^{0}$-boson decay to known leptons
and quarks. We have found that, in the region of relatively strong fields
($\varkappa\sim 1$), all partial decay widths
$\Gamma(Z^{0}\rightarrow\bar{f}f)$ decrease in the same manner by about $12$
to $15\%$. This decrease can be described by a universal function
$R(\varkappa)$ [see (56)–(60)] depending only on the energy-momentum of the
$Z^{0}$-boson and on the external-electromagnetic-field strength. This
circumstance becomes quite obvious if we consider that all known fermions
observed in $Z^{0}$-boson decays have masses at least an order of magnitude
smaller than the $Z^{0}$-boson mass. Therefore, it comes as no surprise that,
in the region of strong electromagnetic fields, the massless-fermion
approximation works well, also making it possible to calculate the dependence
of the total decay width $\Gamma_{Z}(\varkappa)$ of the $Z^{0}$-boson on the
external-field-strength parameter $\varkappa$ (9).

The results of the present calculations have revealed that, in just the same
way as in the case of the $W$-boson [23], the total decay width of the
$Z^{0}$-boson in a strong electromagnetic field depends nonmonotonically on
the strength parameter $\varkappa$, this dependence featuring the point of a
local minimum, $\Gamma_{Z}(\varkappa_{\rm min})=2,164$ GeV, at $\varkappa_{\rm
min}=0,445$. The graph in Fig. 11, which represents the behavior of
$\Gamma_{Z}(\varkappa)$ in the region $\varkappa\leq 1$, illustrates this
observation.

Figure 11: Minimal value of the total decay width of the $Z^{0}$-boson in an
external field.

For the sake of comparison, it can be indicated that the analogous $W$-boson-
decay process $W^{-}\rightarrow\ell\bar{\nu}_{\ell}$ in an external field is
also characterized by a nonmonotonic dependence of the decay width on the
parameter $\varkappa$ with the point of a local minimum at $\varkappa_{\rm
W}=0,6112$ [23]. Upon rescaling this value to the $Z^{0}$-boson energy scale,
it corresponds to $\varkappa=\varkappa_{\rm
W}\left(M_{W}/M_{Z}\right)^{3}=0.419$. Thus, the probabilities for the decays
of gauge bosons in an external field, $Z^{0}\rightarrow\bar{f}f$ and
$W^{-}\rightarrow\ell\bar{\nu}_{\ell}$, take their minimum values at nearly
the same field strength. This agreement is not accidental since the two
reactions in question proceed under similar kinematical conditions and since
the masses of the $W^{\pm}$ and $Z^{0}$ bosons are almost indistinguishable in
order of magnitude. It is also intriguing that, in the region around
$\varkappa\sim 1$, the presence or the absence of the initial-gauge-boson
electric charge is immaterial because the change in the decay width is due
primarily to the increase in the phase space of final particles.

In the region of superstrong fields ($\varkappa\geq 10$), a sizable
contribution to the total decay width of the $Z^{0}$-boson comes from the
$t$-quark-production process $Z^{0}\rightarrow\bar{t}t$, which is forbidden in
a vacuum by conservation laws. As the external-field strength increases, this
reaction becomes dominant in relation to other modes of $Z^{0}$-boson decay,
this being due to a very large mass of the $t$-quark. The external field
serves as some kind of a catalyst for super heavy-particle-production
processes, which cannot occur under ordinary conditions. In view of the
aforesaid, it would be of interest to discuss searches for new (as-yet-
undiscovered) hypothetical particles (supersymmetry and so on) among new
$Z^{0}$-boson decay modes that arise in an external electromagnetic field.
Although external-field strengths necessary for directly observing such
reactions are not yet available in experiments (see, for example, [21]), these
investigations are of course very interesting from the point of view of
fundamental principles of physical theory.

## APPENDIX

In the present study, we have employed the special mathematical functions
$\Phi(z),\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ generically referred to as Airy
functions. The function $\Phi(z)$ is a particular solution to a second-order
linear differential equation for specific initial conditions:

$\Phi^{\prime\prime}(z)-z\Phi(z)=0,\quad\Phi(0)=\frac{\pi}{3^{2/3}\Gamma(2/3)},\quad\Phi^{\prime}(0)=-\frac{\pi}{3^{1/3}\Gamma(1/3)}.$
(A.1)

It has the well-known integral representation

$\Phi(z)=\int\limits_{0}^{\infty}\cos\left(zt+{t^{3}\over 3}\right)dt.$ (A.2)

Two other functions $\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ can be expressed in
terms of this function as

$\Phi_{1}(z)=\int\limits_{z}^{\infty}\Phi(t)dt,\qquad\Phi^{\prime}(z)=\frac{d\Phi(z)}{dz}.$
(A.3)

At small values of the argument $z$ ($z\ll 1$), the Airy function $\Phi(z)$
can be calculated on the basis of the following expansion in a numerical
series:

$\Phi(z)={1\over 3^{2/3}}\sum_{n=0}^{\infty}\Gamma\biggl{(}{n+1\over
3}\biggr{)}\sin\biggl{(}{2\pi\over 3}+{2\pi n\over
3}\biggr{)}\frac{3^{n/3}}{n!}\cdot z^{n}.$ (A.4)

At large values of the argument ($z\gg 1$), the Airy function can be evaluated
by using the asymptotic expressions

$\Phi(z)={1\over 2z^{1/4}}\exp\left(-{2\over
3}z^{3/2}\right)\sum_{n=0}^{\infty}(-1)^{n}\frac{\Gamma(3n+1/2)}{(2n)!\
9^{n}}\cdot\frac{1}{z^{3n/2}},$ (A.5)

$\Phi(-z)={1\over z^{1/4}}\sum_{n=0}^{\infty}\sin\left({2\over
3}z^{3/2}+{\pi\over 4}-\frac{\pi n}{2}\right)\frac{\Gamma(3n+1/2)}{(2n)!\
9^{n}}\cdot\frac{1}{z^{3n/2}}.$ (A.6)

The properties of the Airy functions are well known and can be found in
mathematical handbooks (see, for example, [27]).

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