#  Vacuum Radiation in $z=2$ Lifshitz QED

R. Bufalo111E-mail<EMAIL_ADDRESS>and T. Cardoso e Bufalo222E-mail:
<EMAIL_ADDRESS>
1 Departamento de Física, Universidade Federal de Lavras,
Caixa Postal 3037, 37200-900 Lavras, MG, Brazil

###### Abstract

We discuss in this paper the vacuum Cherenkov radiation in the $z=2$ Lifshitz
electrodynamics. The improved ultraviolet behavior, in terms of higher spatial
derivatives, and the renormalizable couplings, due to the time-space
anisotropic scaling, present in the Lifshitz setting are extremely important
in fulfilling the physical constraints in order to this vacuum process to
happen. We evaluate in details the instantaneous rate of energy loss for a
charge, and also analyze the emission of very soft photons in this framework.

##  1 Introduction

In the past decade we have witnessed an important era of precision experiments
in both particle physics and astrophysics, which has deepened our
understanding of the Standard Model of particle physics (SM) In addition to
the experimental verification of several theoretical mechanisms and
predictions of SM, physical phenomena that are not adequately explained by SM,
the so-called Physics Beyond the Standard Model, have received attention and
been scrutinized to great accuracy [1]. From the many possibilities to make
contact with phenomena related to physics beyond the standard model, the most
compelling ones are those proposals of violation of exact symmetries in field
theories, that aim to make contact with Planck-scale physics. In particular,
models involving Lorentz violation have reached an important milestone in
recent years due to systematic development and subjection to precision tests
[2, 3].

Anomalous decay processes are valuable probes in the study of departures from
Lorentz symmetry. The main interest in decay processes is that they are
affected in unexpected ways by Lorentz violation, meaning that forbidden
processes can occur in certain regions of the parameter space [2, 4]. In the
context of anomalous decays, highly energetic particles are the most
interesting candidates to examine Lorentz violation because they are usually
subject to instabilities; in particular, instabilities involving photons in
vacuum have caught interest in recent years because sufficiently energetic
photon (usually from gamma-ray bursts) may decay as a manifestation of Lorentz
violation [5, 6, 7]. A widely explored anomalous process in the framework of
Lorentz violating photons is the emission of vacuum Cherenkov radiation, an
extremely important energy loss process for high-energy particles [8, 9, 10].

It is well known that ordinary Cherenkov radiation can only occur for
particles propagating in a medium, since a Lorentz-invariant vacuum prevents
it by energy-momentum conservation [11, 12]. However, some Lorentz violating
scenarios provide sufficient instabilities so that particles can radiate
through the Cherenkov process even in vacuum [8, 13]. Many aspects about the
possibility of vacuum Cherenkov radiation by Lorentz violating effects have
been discussed in the framework of the Standard-Model Extension (SME), within
the classical approach to electromagnetic particle radiation [14, 15, 16, 17,
18], as well as in the field theory [9, 10, 19, 20]. In general, the TeV
photons data used to constraint Lorentz violation from the vacuum Cherenkov
radiation come from extremely energetic astronomical sources, and therefore
even tiny changes in electromagnetic wave propagation can be scrutinized.

In this paper we examine the problem of energy loss for a charged particle in
vacuum through the Cherenkov effect in the Lifshitz field theory framework
[21], establishing an alternative point of view for previous Lorentz symmetry
violation studies. This proposal is mainly motivated by the fact that Lifshitz
field theories have a better ultraviolet behavior at the expenses of breaking
Lorentz invariance. This improved behavior is achieved by means of higher
spatial derivative terms that are introduced in such a way to avoid the
appearance of _ghosts_ (negative energy modes), resulting in a theory that
exhibits an anisotropic scaling of space and time, i.e. the scaling
$x^{i}\rightarrow\lambda x^{i}$ whereas $t\rightarrow\lambda^{z}t$. Therefore
, this setting is a highly fascinating scenario to examine Lorentz violating
effects in phenomenological analyses [22, 23, 24, 25, 26, 27].

The interest of this approach in the description of the problem of computing
the rate of radiated energy by a charged particle is threefold: i) the theory
is constructed in such a way to have an improved ultraviolet behavior, then we
can show that the $z=2$ Lifshitz electrodynamics satisfies the vacuum
Cherenkov effect kinematical constraint: the fermion group velocity exceeds
the photon velocity for a given value of the three-momentum; ii) this theory
also establish physical region in the phase space for this anomalous decay to
happen; and iii) this theory admit by construction the definition of only
renormalizable interactions, this point will be very important in the
computation of the rate of energy loss by a charge in the field theory
approach.

Hence, since the time-space asymmetry is a phenomenologically appealing
feature to be studied in high energy physics, we will examine the possibility
of the vacuum Cherenkov radiation within the $z=2$ Lifshitz quantum
electrodynamics (QED) [28, 29, 30, 31]. We start Sec. 2 by reviewing the main
aspects of the dynamics for the fermion and gauge fields within the $z=2$
Lifshitz QED. Moreover, we discuss the general plane wave solutions and
coupling for the fields, establishing the modified dispersion relations,
fermionic energy projection operators, and the photon polarization tensor. We
discuss in Sec. 3 the features of the vacuum process $e^{-}\to\gamma+e^{-}$ in
the Lifshitz framework. First, we examine the vacuum Cherenkov kinematical
constraint, and show that for a certain a given value of the three-momentum
the fermion group velocity exceeds the photon velocity. Furthermore, we
compute the rate of radiated energy for the $z=2$ Lifshitz QED, and discuss
the behavior of the instantaneous energy loss and also the emission of soft
photons of this decay process. In Sec.4 we summarize the results, and present
our final remarks.

##  2 $z=2$ Lifshitz electrodynamics

We start this section by reviewing and discussing the main points regarding
the $z=2$ Lifshitz electrodynamics, more importantly the free field solutions,
fermion and gauge fields dispersion relations and completeness relations [29,
30]. In particular, the analysis of the dispersion relations is very important
since the first constraint upon the vacuum Cherenkov radiation is kinematical,
where the fermion group velocity must exceed the photon phase velocity for a
certain range of the three-momentum for this anomalous decay to happen. We
define a gauge and $z=2$ Lifshitz-invariant QED Lagrangian density as

$\mathcal{L}=\frac{1}{2}F_{0i}F_{0i}-\frac{1}{4}F_{ij}\left(\mu^{2}-\Delta\right)F_{ij}+\overline{\psi}\left(i\gamma_{0}D_{0}-i\mu\gamma_{k}D_{k}-D_{k}D_{k}-m^{2}\right)\psi,$
(2.1)

where the covariant derivative is $D_{\mu}=\partial_{\mu}+igA_{\mu}$, and the
field strenght is defined as usual
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The model (2.1) is
invariant under the $U\left(1\right)$ gauge symmetry

$\psi\rightarrow e^{i\sigma}\psi,\quad A_{\mu}\rightarrow
A_{\mu}+\frac{1}{g}\partial_{\mu}\sigma.$ (2.2)

This theory (2.1) is known to be super-renormalizable [29]. From the
Lagrangian density (2.1) we may observe that the length dimensions are, in
$3+1$ dimensions,

$\displaystyle\left[A_{0}\right]$
$\displaystyle=\left[\psi\right]=L^{-\frac{3}{2}},\quad\left[A_{i}\right]=L^{-\frac{1}{2}},$
(2.3) $\displaystyle\left[g\right]$
$\displaystyle=L^{-\frac{1}{2}},\quad\left[m\right]=\left[\mu\right]=L^{-1}.$
(2.4)

Since we are interested in evaluate the matrix element related to the
$e^{-}\rightarrow\gamma+e^{-}$ process, corresponding to the Cherenkov
radiation, we need to establish the free field solutions for the fermionic and
gauge field equations. Hence, in order to construct the solutions for the
Dirac fields, we shall consider the following modified free field equation,
obtained from (2.1), which reads

$\left(i\gamma_{0}\partial_{0}-i\mu\gamma_{k}\partial_{k}-\partial_{k}\partial_{k}-m^{2}\right)\psi=0,$
(2.5)

the coefficients of the differential equation are constants, thus
$\psi\left(x\right)=e^{-ipx}\chi\left(p\right)$, will be a solution. Hence, we
obtain

$\left(\gamma_{0}p_{0}-\mu\gamma_{k}p_{k}+p^{2}-m^{2}\right)\chi\left(p\right)=0,$
(2.6)

where $p^{2}=p_{i}p_{i}$. For the energy eigenvalues, we have

$p_{0}\equiv\pm\mathcal{E}_{p}=\pm\sqrt{\mu^{2}p^{2}+\left(p^{2}-m^{2}\right)^{2}}.$
(2.7)

For each value of $p_{0}$, the solution (2.6) has a two-dimensional solution
space. Thus, for the explicit calculation, we choose

$\chi\left(p\right)=\left(\begin{array}[]{c}u_{s}\left(p\right)\\\
v_{s}\left(p\right)\end{array}\right),$ (2.8)

and we also consider the following representation for the $\gamma$-matrices,

$\gamma^{0}=\left(\begin{array}[]{cc}\mathbf{1}&0\\\
0&-\mathbf{1}\end{array}\right),\quad\vec{\gamma}=\left(\begin{array}[]{cc}0&\vec{\sigma}\\\
-\vec{\sigma}&0\end{array}\right),$

where $\mathbf{1}$ is a two-dimensional identity matrix, and $\vec{\sigma}$
are the set of Pauli matrices. All these considerations lead to the
expressions

$\displaystyle\left(p_{0}+p^{2}-m^{2}\right)u_{s}\left(p\right)$
$\displaystyle=\mu\left(\vec{\sigma}.\vec{p}\right)v_{s}\left(p\right),$ (2.9)
$\displaystyle\left(p_{0}-p^{2}+m^{2}\right)v_{s}\left(p\right)$
$\displaystyle=\mu\left(\vec{\sigma}.\vec{p}\right)u_{s}\left(p\right).$
(2.10)

Nevertheless, it is not difficult to show that $u\left(p\right)$ corresponds
to those solutions with positive energy, $p_{0}=+\mathcal{E}_{p}$, whereas
$v\left(p\right)$ are solutions with negative energy,
$p_{0}=-\mathcal{E}_{p}$. The next step involves the definition of the energy
projection operators, so that the following relations hold

$\displaystyle\Pi^{+}\left(p\right)u_{s}\left(p\right)$
$\displaystyle=u_{s}\left(p\right),$ (2.11)
$\displaystyle\Pi^{-}\left(p\right)v_{s}\left(p\right)$
$\displaystyle=v_{s}\left(p\right),$ (2.12)

and also

$\Pi^{-}\left(p\right)u_{s}\left(p\right)=\Pi^{+}\left(p\right)v_{s}\left(p\right)=0.$
(2.13)

After some algebraic construction, we can show that the operators

$\Pi^{\pm}\left(p\right)=\frac{\mp\gamma_{0}p_{0}\pm\mu\gamma_{k}p_{k}+p^{2}-m^{2}}{2\left(p^{2}-m^{2}\right)},$
(2.14)

satisfy the above relations, as well as the following identities

$\displaystyle\Pi^{+}\left(p\right)+\Pi^{-}\left(p\right)=\mathbf{1},\quad\Pi^{+}\left(p\right)\Pi^{-}\left(p\right)=0,$
(2.15)
$\displaystyle\left[\Pi^{\pm}\left(p\right)\right]^{2}=\Pi^{\pm}\left(p\right).$
(2.16)

Furthermore, from the definitions (2.14), we can also show that these
solutions satisfy the completeness relations

$\displaystyle\Pi_{\alpha\beta}^{+}\left(p\right)$
$\displaystyle=\sum_{s=1}^{2}u_{\alpha}\left(p,s\right)\overline{u}_{\beta}\left(p,s\right)=\left[\frac{-\gamma_{0}p_{0}+\mu\gamma_{k}p_{k}+p^{2}-m^{2}}{2\left(p^{2}-m^{2}\right)}\right]_{\alpha\beta},$
(2.17) $\displaystyle\Pi_{\alpha\beta}^{-}\left(p\right)$
$\displaystyle=-\sum_{s=1}^{2}v_{\alpha}\left(p,s\right)\overline{v}_{\beta}\left(p,s\right)=\left[\frac{\gamma_{0}p_{0}-\mu\gamma_{k}p_{k}+p^{2}-m^{2}}{2\left(p^{2}-m^{2}\right)}\right]_{\alpha\beta}.$
(2.18)

Finally, taking into account all of these results, we can write the free
solutions as the following

$\displaystyle\psi\left(x\right)$
$\displaystyle=\sum_{r}\int\frac{d^{3}p}{\left(2\pi\right)^{\frac{3}{2}}}\left(\frac{p^{2}-m^{2}}{\mathcal{E}_{p}}\right)^{\frac{1}{2}}\left[b_{r}\left(p\right)u_{r}\left(p\right)e^{-ipx}+d_{r}^{\dagger}\left(p\right)v_{r}\left(p\right)e^{ipx}\right],$
(2.19) $\displaystyle\overline{\psi}\left(x\right)$
$\displaystyle=\sum_{r}\int\frac{d^{3}p}{\left(2\pi\right)^{\frac{3}{2}}}\left(\frac{p^{2}-m^{2}}{\mathcal{E}_{p}}\right)^{\frac{1}{2}}\left[b_{r}^{\dagger}\left(p\right)\overline{u}_{r}\left(p\right)e^{ipx}+d_{r}\left(p\right)\overline{v}_{r}\left(p\right)e^{-ipx}\right],$
(2.20)

where, the operators’ anti-commutation algebra is given as usual,

$\left\\{b_{r}\left(p\right),b_{s}^{\dagger}\left(q\right)\right\\}=\left\\{d_{r}\left(p\right),d_{s}^{\dagger}\left(q\right)\right\\}=\delta_{rs}\delta\left(\vec{p}-\vec{q}\right).$
(2.21)

From such construction one can easily show that the equal-time anti-
commutation relations is satisfied

$\left\\{\psi_{\alpha}(x),\psi_{\beta}^{\dagger}(y)\right\\}_{x_{0}=y_{0}}=\delta_{\alpha\beta}\delta^{(3)}\left(\vec{x}-\vec{y}\right).$
(2.22)

Besides, we can also determine the fermion propagator

$S\left(p_{0},p\right)=i\frac{\gamma_{0}p_{0}-\mu\gamma_{k}p_{k}+p^{2}-m^{2}}{p_{0}^{2}-\mu^{2}p^{2}-\left(p^{2}-m^{2}\right)^{2}},$
(2.23)

which is in accordance with the free field solutions and operator algebra. The
development for the gauge field follows closely of the fermionic part. First
we find the energy eigenvalues,

$k_{0}\equiv\pm\Omega_{k}=\pm\sqrt{\mu^{2}k^{2}+k^{4}}.$ (2.24)

In the case of the gauge field, the free field solution is simply

$A_{\mu}\left(x\right)=\sum_{\lambda}\int\frac{d^{3}k}{\left(2\pi\right)^{\frac{3}{2}}}\left(\frac{1}{2\omega_{k}}\right)^{\frac{1}{2}}\left[a_{i}\left(k\right)\epsilon_{\mu}\left(k,\lambda\right)e^{-ikx}+a_{i}^{\dagger}\left(k\right)\epsilon_{\mu}\left(-k,\lambda\right)e^{ikx}\right].$
(2.25)

where the polarization tensor satisfies the normalization condition

$\eta^{\mu\nu}\epsilon_{\mu}^{*}\left(k,\lambda\right)\epsilon_{\nu}\left(k,\lambda\right)=-1$
(2.26)

and also the completeness relation

$\sum_{\lambda}\epsilon_{\mu}^{*}\left(k,\lambda\right)\epsilon_{\nu}\left(k,\lambda\right)=g_{\mu\nu}$
(2.27)

where $g_{\mu\nu}$ is a ‘metric’ with well defined components (the difference
with the Lorentzian metric $\eta_{\mu\nu}$ is due to the different scaling
between time and spatial components). Moreover, in order to define the photon
propagator and determine the polarization tensor, we must impose a gauge
condition, which is a Lorenz-like condition

$\Omega\left[A\right]=\partial_{0}A_{0}-\left(-\Delta+\mu^{2}\right)\partial_{k}A_{k}=0,$
(2.28)

this leads to a non-local gauge-fixing term in the Lagrangian density which,
in the Feynman gauge $\left(\alpha=1\right)$, reads

$\mathcal{L}_{gf}=-\left(\partial_{0}A_{0}-\left(-\Delta+\mu^{2}\right)\partial_{k}A_{k}\right)\frac{1}{2\left(-\Delta+\mu^{2}\right)}\left(\partial_{0}A_{0}-\left(-\Delta+\mu^{2}\right)\partial_{j}A_{j}\right),$
(2.29)

Finally, we see that the nonvanishing components of the gauge field propagator
have a well-behaved expression:

$\displaystyle i\mathcal{D}_{00}\left(k_{0},k\right)$
$\displaystyle=\frac{\mu^{2}+k^{2}}{k_{0}^{2}-\mu^{2}k^{2}-k^{4}},$ (2.30)
$\displaystyle i\mathcal{D}_{ij}\left(k_{0},k\right)$
$\displaystyle=-\frac{\delta_{ij}}{k_{0}^{2}-\mu^{2}k^{2}-k^{4}},$ (2.31)

with no off-diagonal components. At last, we can determine the metric
components $g_{\alpha\beta}$, present in modified completeness relation
(2.27), so that they are in agreement with the Feynman propagators (2.30) and
(2.31) computed in the gauge (2.28). This identification yields

$\displaystyle\varepsilon_{0}^{*}\left(k,\lambda\right)\varepsilon_{0}\left(k,\lambda\right)$
$\displaystyle=\mu^{2}+k^{2},$ (2.32)
$\displaystyle\varepsilon_{0}^{*}\left(k,\lambda\right)\varepsilon_{k}\left(k,\lambda\right)$
$\displaystyle=0,$ (2.33)
$\displaystyle\varepsilon_{s}^{*}\left(k,\lambda\right)\varepsilon_{l}\left(k,\lambda\right)$
$\displaystyle=g_{sl}=-\delta_{sl}.$ (2.34)

These results for the completeness relations of the polarization tensor
$\varepsilon_{\mu}\left(k,\lambda\right)$, eq. (2.27), as well as for the
spinor components $u$ and $v$, eqs. (2.17) and (2.18), will be very important
in the evaluation of the amplitude related to the decay $e\to e+\gamma$.

The last part we need to discuss is about the couplings present in the
Lagrangian density (2.1) and their contribution to the decay process of
interest. One can observe in (2.1) that the $z=2$ QED presents three vertices:
two three-point vertices,

$\left\langle\overline{\psi}A_{0}\psi\right\rangle\rightarrow
ig\gamma_{0},\quad\left\langle\overline{\psi}A_{k}\psi\right\rangle\rightarrow-
ig\left(\gamma_{k}\mu+2p_{k}^{\left(\psi\right)}+p_{k}^{\left(A\right)}\right),$
(2.35)

which can be conveniently rewritten in a compact form,

$\left(\left\langle\overline{\psi}A_{0}\psi\right\rangle,\left\langle\overline{\psi}A_{k}\psi\right\rangle\right)\rightarrow\Lambda_{a}\left(q\right)=ig\left(\gamma_{0},-\gamma_{k}\mu-
q_{k}\right),$ (2.36)

where we have introduced, by means of notation, the index $a=0,...,3$, which
should not be confused with spacetime index, and $q_{k}\equiv
2p_{k}^{\left(\psi\right)}+p_{k}^{\left(A\right)}$; and one four-point vertex,
$\left\langle\overline{\psi}\psi
A_{i}A_{j}\right\rangle\rightarrow-2ig\delta_{ij}$. But since we are
interested in the decay $e\to e+\gamma$, we shall consider only the three-
point vertices (2.36).

In the next section we shall analyze the Cherenkov radiation in the Lorentz
violating framework of the Lifshitz electrodynamics. The interest in this kind
of radiation is due its unique signature of Lorentz violation. We start by
discussing the kinematical constraint related with the fermionic group
velocity and the photon phase velocity; this analysis establishes the
threshold of the three-momentum where the radiation can occur.

Furthermore, after verifying that the Cherenkov kinematical constraint is
satisfied by the $z=2$ Lifshitz electrodynamics, we proceed to the evaluation
of the rate of energy loss through vacuum Cherenkov radiation. The rate of
radiated energy is strongly dependent on the cutoff due to energy-momentum
conservation [15], meaning that two types of processes can occur based on the
energies of the photons involved: i) the instantaneous rate of emission, in
which the charge emits a single energetic photon, drops below the Cherenkov
threshold, and stops emitting, and ii) the emission of very soft photons,
where the particle’s energy is not lowered below the threshold, and so the
charge will continue to radiate afterwards the emission. We shall mainly
discuss the instantaneous emission of photons, but will present some general
remarks about the emission of very soft photons in the Lifshitz framework.

##  3 Vacuum Cherenkov radiation in $z=2$ Lifshitz electrodynamics

###  3.1 Kinematical constraint

Before we start our analysis on the rate of radiated energy of the Cherenkov
process in the Lifshitz electrodynamics, we must be sure that the kinematical
constraint upon this decay is fulfilled, allowing this to occur in this
Lorentz violating framework. First, we can recast the dispersion relations by
a rescaling $(p_{0},k_{0})\to\mu(p_{0},k_{0})$, yielding

$\displaystyle\Omega_{k}$ $\displaystyle=\sqrt{k^{2}+\frac{k^{4}}{\mu^{2}}},$
(3.1) $\displaystyle\mathcal{E}_{p}$
$\displaystyle=\sqrt{\frac{1}{\mu^{2}}p^{4}+\left(1-\eta\right)p^{2}+m_{r}^{2}},$
(3.2)

with the mass $m_{r}^{2}=\frac{1}{\mu^{2}}m^{4}$ and
$\eta=\frac{2m_{r}}{\mu}$. In this form, we see that the usual relativistic
dispersion relations are recovered as $\eta=0$ and $\mu\to\infty$. The free
parameter $\mu^{2}$ can be chosen as: the electron mass $m_{r}\simeq 0,5\,{\rm
MeV}$ and the GUT scale $\mu\simeq 10^{16}\,{\rm GeV}$, which results into
$\eta\simeq 10^{-19}$; these values are within the Lorentz symmetry violation
bounds [29].

It is well known that Cherenkov radiation is possible only if there is a range
of three-momentum for which the fermion group velocity exceeds the photon
phase velocity in medium materials [11], but also for Lorentz-violating vacua
[14, 15, 16]. In our model we have that

$\displaystyle v_{\rm g}$
$\displaystyle=\frac{d\mathcal{E}_{p}}{dp}=\frac{\frac{2}{\mu^{2}}p^{3}+\left(1-\eta\right)p}{\sqrt{\frac{1}{\mu^{2}}p^{4}+\left(1-\eta\right)p^{2}+m_{r}^{2}}}$
(3.3)

and

$\displaystyle v_{\rm ph}$
$\displaystyle=\frac{\Omega_{k}}{k}=\sqrt{1+\frac{k^{2}}{\mu^{2}}}.$ (3.4)

By a simple numerical analysis, we can check that for values above $p>p_{\rm
min}=3,16228\times 10^{3}~{}{\rm TeV}$ we have $v_{g}>v_{ph}$. Hence, as long
as we are in the region $p>p_{\rm min}$, vacuum Cherenkov radiation can occur
in the $z=2$ Lifshitz QED. Actually, in our discussion of the phase space
related to this decay, we will find a tighter bound upon the charge three-
momentum by demanding the reality of the physical region. Now that we have
established the momentum threshold, and consequently the physical region of
interest, we can proceed to the computation of the instantaneous rate of
radiate energy for the $e\to e+\gamma$ in the $z=2$ Lifshitz QED.

###  3.2 Rate of radiated energy

Since we are interested in computing the energy loss associated with the
vacuum Cherenkov scattering in the $z=2$ Lifshitz electrodynamics scenario,
consisting in a decay process $1\rightarrow 2+3$, let us review some general
results necessary for the development of this analysis [9, 10]. The energy-
momentum loss of a Lorentz invariant charged particle per unit of time is
equal to the photon four-momentum $k$ weighted by the scattering amplitude
squared and integrated over phase space,

$\frac{dp^{\mu}}{dt}=\int Dk\left|\mathcal{M}\right|^{2}k^{\mu}$ (3.5)

where $Dk$ is the phase-space invariant measure. The rate of total radiated
energy is obtained from the time component of the above expression.

Figure 1: Tree-level Feynman graphs for vacuum Cherenkov radiation in the
$z=2$ Lifshitz electrodynamics, the left panel corresponds to the coupling
$\left\langle\overline{\psi}A_{0}\psi\right\rangle$ while the right one to the
coupling $\left\langle\overline{\psi}A_{j}\psi\right\rangle$.

In the presence of Lorentz violating effects, however, the above expression is
generally no longer valid, because such effects also modify the energy-
momentum tensor for the particle, and hence the energy of the particle is not
necessarily equal to the time component of its four-vector momentum. On the
other hand, since the emission rate is identically zero without the Lorentz
violation, only the desired term survives at leading $\mu^{2}$ order. Thus,
the rate of radiated energy, in the first-order in $\mu^{2}$, can be expressed
as

$W\approx-\dot{p}_{0}=\int k_{0}d\Gamma,$ (3.6)

where $d\Gamma$ is the differential decay rate for the given process [32]. The
contributions to the Cherenkov radiation process at tree-level are represented
by the Feynman diagrams in Fig. 1.

The differential decay rate of our interest can readily be obtained by
identifying it with the time derivative of the probability for the decay
$1\rightarrow 2+3$, i.e. $d\Gamma=dP/dt$, where $P=\frac{\left|\left\langle
f|i\right\rangle\right|^{2}}{\left\langle f|f\right\rangle\left\langle
i|i\right\rangle}$. Hence, for the process $e^{-}\rightarrow\gamma+e^{-}$ the
differential decay rate is explicitly written as

$d\Gamma=\frac{m_{r}}{\mathcal{E}_{p_{i}}}\frac{d^{3}k_{f}}{\left(2\pi\right)^{3}2\Omega_{k_{f}}}\frac{m_{r}d^{3}p_{f}}{\left(2\pi\right)^{3}\mathcal{E}_{p_{f}}}\left(2\pi\right)^{4}\delta^{4}\left(p_{i}-p_{f}-k_{f}\right)\left|\mathfrak{M}\right|^{2}$
(3.7)

where the normalization factors are chosen accordingly for bosonic and
fermionic fields, whereas $\Omega_{k_{f}}$ and $\mathcal{E}_{p}$ are the
dispersion relations (3.1) and (3.2), respectively, and
$m_{r}^{2}=m^{4}/\mu^{2}$ is the fermionic mass. Thus, substituting the result
(3.7) into the expression (3.6), we can write the rate of radiated energy as
the following

$W=\frac{\mu^{2}}{8\pi^{2}E_{p_{i}}}\int\frac{d^{3}k_{f}d^{3}p_{f}}{E_{p_{f}}}\delta^{4}\left(p_{i}-p_{f}-k_{f}\right)\left|\mathfrak{M}\right|^{2}$
(3.8)

Furthermore, by calculation purposes and to make the reaction kinematics
visible, it is convenient to express the integration over the variables
$p_{f}$ as

$\int\frac{d^{3}p_{f}}{2\mathcal{E}_{p_{f}}}=\int
d^{4}p_{f}\delta\left(p_{f}^{02}-\mathcal{E}_{p_{f}}^{2}\right)\theta\left(p_{f}^{0}\right)$
(3.9)

which allow us to rewrite Eq. (3.8) in a convenient form for the remaining
analysis

$\displaystyle W$
$\displaystyle=\frac{m_{r}^{2}}{4\pi^{2}\mathcal{E}_{p_{i}}}\int
d^{3}k\delta\left(\left(\bar{p}-\bar{k}\right)^{2}-\mathcal{M}_{p-k}^{2}\right)\theta\left(\mathcal{E}_{p}-\Omega_{k}\right)\left|\mathfrak{M}\right|^{2}$
(3.10)

where we have defined the notation

$p^{02}-\mathcal{E}_{p}^{2}=\bar{p}^{2}-\left(\frac{1}{\mu^{2}}p^{4}-\eta
p^{2}+m_{r}^{2}\right)\equiv\bar{p}^{2}-\mathcal{M}_{p}^{2}$ (3.11)

as matter of simplification, since we are using the bar notation $\bar{p}$ for
a “four-dimensional” momentum, also we have rewritten the momenta as $k_{f}=k$
and $p_{i}=p$.

Once we have concluded the formal development of our analysis, by establishing
an expression for the rate of radiated energy (3.10), we should proceed now to
the computation of the scattering matrix $\mathfrak{M}$ in (3.10) due to the
$z=2$ Lifshitz couplings (2.36). Hence, the corresponding element of the
matrix $\mathfrak{M}$ for the $e^{-}\rightarrow\gamma+e^{-}$ process, depicted
in Fig. 1, reads

$\displaystyle\mathfrak{M}$
$\displaystyle=ig\overline{u}\left(p_{f},s_{f}\right)\Lambda\left(k_{f},\lambda\right)u\left(p_{i},s_{i}\right)$
(3.12)

where we have defined the dependence with the polarization tensor components
as

$\displaystyle\Lambda=ig\gamma_{0}\varepsilon_{0}-ig\varepsilon_{k}\left(\gamma_{k}\mu-2\left(p_{i}\right)_{k}+\left(k_{f}\right)_{k}\right)$
(3.13)

Since the electrons are not polarized, in order to compute the squared
scattering matrix in Eq. (3.10), we average over initial spins and sum over
final spins states, that is
$\left|\mathfrak{M}\right|^{2}\rightarrow\frac{1}{2}\sum_{\rm
spin}\left|\mathfrak{M}\right|^{2}$. Hence, we can make use of the
completeness relations eqs. (2.17) and (2.18), so that it yields

$\displaystyle\frac{1}{2}\sum_{spin}\left|\mathfrak{M}\right|^{2}$
$\displaystyle=\frac{g^{2}}{8\left(\left(p-k\right)^{2}-m^{2}\right)\left(p^{2}-m^{2}\right)}$
(3.14) $\displaystyle\times{\rm
Tr}\Big{[}\left(-\gamma_{0}\left(p_{0}-k_{0}\right)+\mu\gamma_{k}\left(p_{k}-k_{k}\right)+\left(p-k\right)^{2}-m^{2}\right)\Lambda\left(k,\lambda^{*}\right)$
(3.15)
$\displaystyle\times\left(-\gamma_{0}p_{0}+\mu\gamma_{j}p_{j}+p^{2}-m^{2}\right)\Lambda\left(k,\lambda\right)\Big{]}$
(3.16)

Furthermore, we can compute the trace over the Dirac matrices in (3.14), use
the polarization states (2.32)–(2.34), and then work the algebraic part of the
process kinematics, to find the expression

$\displaystyle\frac{1}{2}\sum_{spin}\left|\mathfrak{M}\right|^{2}$
$\displaystyle=2g^{2}\left[-p^{2}+pk\cos\theta\right]$ (3.17)
$\displaystyle+\frac{g^{2}}{\left(p^{2}-2pk\cos\theta+k^{2}-\mu
m_{r}\right)\left(p^{2}-\mu
m_{r}\right)}\biggl{\\{}\left(\mathcal{E}_{p}-\varOmega_{k}\right)\mathcal{E}_{p}\left[\mu^{2}+2p^{2}-2pk\cos\theta\right]$
(3.18)
$\displaystyle-\mu^{2}\bigg{[}8p^{4}-16p^{3}k\cos\theta+4p^{2}k^{2}\cos^{2}\theta+3k^{2}p^{2}-k^{3}p\cos\theta\bigg{]}$
(3.19) $\displaystyle-\mu^{2}\bigg{[}-\mu
m_{r}\left(8p^{2}-8pk\cos\theta+k^{2}\right)+2\mu^{2}m_{r}^{2}\bigg{]}\biggr{\\}}.$
(3.20)

where $\theta$ is chosen as the opening angle between the incoming electron
and the outgoing photon, and it is determined from the energy-momentum
conservation given by the delta function in (3.10). The analysis of the
kinematics of the decay is also crucial to establish the physical phase space
where the decay can happen, which corresponds to the allowed integration
interval in the momentum integral in (3.10). Under these considerations, we
shall work with the variables $d^{3}k=2\pi
d\cos\theta~{}|\vec{k}|^{2}d|\vec{k}|$. Hence, the energy-momentum
conservation for the decay process
$\left(\bar{p}-\bar{k}\right)^{2}=\mathcal{M}_{p-k}^{2}$, implies the
following relation

$\displaystyle
2\mathcal{E}_{p}\Omega_{k}-2\left\\{\left(1-\eta\right)pk+\frac{2}{\mu^{2}}\left(p^{3}k-p^{2}k^{2}+pk^{3}\right)\right\\}\cos\theta+\frac{1}{\mu^{2}}2p^{2}k^{2}-\eta
k^{2}$ $\displaystyle=0,$ (3.21)

where $k\equiv|\vec{k}|$. This energy balance equation can be used to arrive
at the radiation condition for the $e^{-}\rightarrow e^{-}+\gamma$ process
within the Lifshitz framework [11]. After some algebraic manipulations, we
find that the opening angle $\theta$, in the regime $\mu^{2}>(p^{2},k^{2})$,
is cast as

$\cos\theta\approx
1+\frac{\eta(p-k)}{2p}-\frac{3}{2\mu^{2}}(p-k)^{2}+\mathcal{O}\left(\frac{(p-k)^{4}}{\mu^{4}}\right).$
(3.22)

It is straightforward to recognize that the removal of the Lorentz violating
effects through $\mu^{2}\to\infty$ , implies that $\cos\theta=1$, which
reflects the fact that the process $e^{-}\rightarrow e^{-}+\gamma$ has a
vanishing radiation rate in the Lorentz invariant QED. On the other hand, for
finite values of the parameters $\mu^{2}\neq 0$ and $\eta\neq 0$, one can
realize that there is a region in the phase space where $\cos\theta<1$, even
in vacuum. This is a very interesting result, since this radiation condition
corresponds to the availability of a physical phase space for the anomalous
decay, and it also corroborates the kinetic condition discussed above, where
we have established the existence of a physical region for the decay
corresponding to $p>p_{\rm min}$ (for smaller momentum values, the radiation
rate is strictly zero).

Furthermore, since the energy conservation requires that the allowed values
for $k$ are such that the relation (3.21) is satisfied for a given value of
$\theta$, thus the integration over $\theta$ restricts the region of
integration over $k$. To determine this constraint upon $k$, it is worth to
rewrite the integration in $k$ as the following

$\displaystyle\mathcal{T}$ $\displaystyle=\int
k^{2}dk\theta\left(\mathcal{E}_{p}-\Omega_{k}\right)\int_{-1}^{+1}d\cos\theta\delta\left(\left(\bar{p}-\bar{k}\right)^{2}-\mathcal{M}_{p-k}^{2}\right)\frac{1}{2}\sum_{spin}\left|\mathfrak{M}\right|^{2}$
(3.23) $\displaystyle=\int
k^{2}dk\theta\left(\mathcal{E}_{p}-\Omega_{k}\right)\frac{1}{2\left\\{\left(1-\eta\right)pk+\frac{2}{\mu^{2}}\left(p^{3}k-p^{2}k^{2}+pk^{3}\right)\right\\}}$
(3.24)
$\displaystyle\times\int_{-1}^{+1}d\cos\theta\delta\left(\cos\theta-\frac{\frac{1}{\mu^{2}}2p^{2}k^{2}-\eta
k^{2}+2\mathcal{E}_{p}\varOmega_{k}}{2\left\\{\left(1-\eta\right)pk+\frac{2}{\mu^{2}}\left(p^{3}k-p^{2}k^{2}+pk^{3}\right)\right\\}}\right)\frac{1}{2}\sum_{spin}\left|\mathfrak{M}\right|^{2}$
(3.25)

Finally, from the expression (3.25) we can conclude that the condition
$\cos\theta\in[-1,1]$ restricts the magnitude of the photon momentum $k$ to
the values333 In fact, eq.(3.21) gives four nonvanishing roots, two of them
are imaginary for any value of $p$, while the remaining two result into in
eqs.(3.27) and (3.29).

$\displaystyle k_{+}=-\frac{\gamma\mu^{4}}{64\xi^{2}}+\frac{1}{2}\sqrt{\beta}$
(3.26)
$\displaystyle+\frac{1}{2}\sqrt{\frac{3\gamma^{2}\mu^{8}}{1024\xi^{4}}-\frac{\delta\mu^{2}}{8\xi^{2}}+\frac{\gamma\mu^{6}}{256\xi^{4}\sqrt{\beta}}\left(\delta+\frac{\gamma^{2}\mu^{6}}{64\xi^{2}}\right)+\frac{\mu^{3}}{2\xi\sqrt{\beta}}\left(12\xi^{4}-8\eta\xi^{2}+6\xi^{2}+\eta^{2}-\eta\right)-\beta}$
(3.27) $\displaystyle
k_{-}=-\frac{\gamma\mu^{4}}{64\xi^{2}}+\frac{1}{2}\sqrt{\beta}$ (3.28)
$\displaystyle-\frac{1}{2}\sqrt{\frac{3\gamma^{2}\mu^{8}}{1024\xi^{4}}-\frac{\delta\mu^{2}}{8\xi^{2}}+\frac{\gamma\mu^{6}}{256\xi^{4}\sqrt{\beta}}\left(\delta+\frac{\gamma^{2}\mu^{6}}{64\xi^{2}}\right)+\frac{\mu^{3}}{2\xi\sqrt{\beta}}\left(12\xi^{4}-8\eta\xi^{2}+6\xi^{2}+\eta^{2}-\eta\right)-\beta}$
(3.29)

where we have introduced the factors $\alpha$, $\beta$, $\gamma$ and $\delta$
for simplicity of notation, and define them in the Appendix A. Moreover, we
have also introduced the parameter $\xi=\frac{p}{\mu}$, which will work as the
perturbative parameter in the analysis of the approximated expression for the
radiation rate.

Hence, the above discussion about availability of the physical phase space for
the electron decay, displayed by Eqs. (3.27) and (3.29), permit us to rewrite
(3.10) as the instantaneous rate

$\displaystyle W_{\rm inst}$
$\displaystyle=\frac{m_{r}^{2}}{4\pi\mathcal{E}_{p}}\int_{k_{-}}^{k_{+}}\frac{k^{2}dk}{\left(1-\eta\right)pk+\frac{2}{\mu^{2}}\left(p^{3}k-p^{2}k^{2}+pk^{3}\right)}$
(3.30)
$\displaystyle\times\int_{-1}^{+1}d\cos\theta\delta\left(\cos\theta-\frac{\frac{1}{\mu^{2}}2p^{2}k^{2}-\eta
k^{2}+2\mathcal{E}_{p}\varOmega_{k}}{2\left\\{\left(1-\eta\right)pk+\frac{2}{\mu^{2}}\left(p^{3}k-p^{2}k^{2}+pk^{3}\right)\right\\}}\right)\frac{1}{2}\sum_{spin}\left|\mathfrak{M}\right|^{2}$
(3.31)

Finally, substituting the expression for the scattering matrix (3.17) into
(3.31), we can perform the integrations over $\theta$ and over the momentum
$k$ with the use of Mathematica. However, the complete result is not
enlightening, but we can discuss some asymptotic regions of interest, in
particular $\xi<1$ and $\xi>1$. It is straightforward to conclude that the
case of $\xi>1$ corresponds to a region where this effective theory stops
working, hence it is not of interest. Hence, we can conclude that the only
region of physical interest for the Cherenkov decay is for $\xi<1$, more
precisely this anomalous decay can happen in the Lifshitz QED only in the
region $p_{\rm min}<p<\mu$. This discussion implies that the instantaneous
rate read

$\displaystyle W_{\rm inst}$ $\displaystyle=0,\quad p<p_{\rm min}$ (3.32)
$\displaystyle W_{\rm inst}$
$\displaystyle\approx\frac{g^{2}}{2560\pi}\sqrt{\frac{3}{2}}\left[1546+2391\eta\right]\frac{m_{r}^{6}}{\mu^{2}\xi^{9}}+\mathcal{O}\left(\frac{g^{2}m_{r}^{6}}{\mu^{2}\xi^{7}}\right),\quad
p_{\rm min}<p<\mu$ (3.33)

We observe that at large energies the instantaneous rate $W_{\rm inst}$ is a
rapidly decreasing function of $\xi$. Moreover, as we would expect, the limit
$\mu\rightarrow\infty$ of eq. (3.33) gives a vanishing result to the Cherenkov
radiation.

As previously discussed, in addition to the instantaneous energy loss, there
is the possibility of the emission of lower-energy photons with energies
$k<k^{-}$. In this regime, the charged particle continue to radiate after the
emission of one of these photons because its energy is still above the
threshold, which makes it reasonable to approximate the energy losses for
these photons as a continuous process [15]. Actually, the instantaneous rate
$W_{\rm inst}=\int_{k_{-}}^{k_{+}}W(k)dk$ can be understood as the rate of
photon emission per unit energy $W(k)=\mathcal{P}(k)/\Omega_{k}$, i.e. the
charged particle radiates a single high-energy photon. Within this
interpretation, the continuous rate of radiating power related with soft
photons can be written as $\mathcal{P}_{\rm
soft}=\int_{0}^{k_{-}}\mathcal{P}(k)dk$ and we can use the previous results
derived for the evaluation of the instantaneous rate, in particular eq.
(3.31), to compute this expression. Hence, in the region $p_{\rm min}<p<\mu$,
we find that the radiating power related with the emission of soft photons is

$\mathcal{P}_{soft}\approx\frac{3g^{2}}{40960\pi}\left[2\left(796\sqrt{6}-1155\right)+\left(2072\sqrt{6}-3585\right)\eta\right]\frac{m_{r}^{7}}{\mu^{2}\xi^{10}}+\mathcal{O}\left(\frac{g^{2}m_{r}^{7}}{\mu^{2}\xi^{8}}\right).$
(3.34)

Moreover, we observe that the ratio of instantaneous by the soft emission is
greater than unity for a finite value of the parameters, that is, we have
$W_{inst}/\left(\mathcal{P}_{soft}\xi/m_{r}\right)\approx\frac{6184\sqrt{\frac{2}{3}}}{796\sqrt{6}-1155}>1$.
This shows that although the two types of losses are comparable in this energy
regime (since the ratio is only slightly greater than the unit), the
instantaneous emission still is more important over the emission of very soft
photons in the $z=2$ Lifshitz framework.

##  4 Final remarks

In this paper, we have studied vacuum effects of a Lorentz violating quantum
electrodynamics in the context of $z=2$ Lifshitz field theory. Anomalous decay
processes are a suitable scenario to study due its unique signature of Lorentz
violation. In particular, we analyzed the rate of radiated energy from a
charged particle through vacuum Cherenkov radiation.

We started our analysis of vacuum Cherenkov radiation described in the $z=2$
Lifshitz framework by considering the instantaneous rate of radiated energy
for the process $e^{-}\rightarrow\gamma+e^{-}$, where the charge emits a
single energetic photon, drops below the Cherenkov threshold, and stops
emitting. In the instantaneous case we observed that it only occurs in the
regime $p_{\rm min}<p<\mu$, and the emission rate at large energies behaves as
a decreasing function of $\xi$ ($W_{\rm inst}\sim\xi^{-9}$). In addition to
the instantaneous emission, emission of very soft photons may also occur,
where the charge’s energy is not lowered below the threshold, and thus the
charge continue to radiate afterwards. However, after computing these
radiation rates, we found that $W_{\rm inst}/W_{\rm soft}>1$ in the high-
energy regime, revealing the importance of instantaneous emission face to the
soft emission of soft photons in the case of $z=2$ Lifshitz electrodynamics.

It is necessary to remark that the $z=2$ Lifshitz contributions to the vacuum
processes are highly nontrivial. We emphasize that the nonvanishing radiated
energy rate has two underlying causes: it is the severe departure of the
dispersion relations of the electron and photon fields in the $z=2$ Lifshitz
electrodynamics in relation to the usual QED, with higher spatial derivative
contributions, that causes this theory to fulfill the kinematical constraint,
allowing this anomalous decay to happen, but also the unique couplings
contributions in the evaluation of the decay amplitude.

Based on the interesting outcome that the $z=2$ Lifshitz effects have in the
vacuum decay processes, in particular in the emission of soft photons, we
believe that the signatures of Lorentz violation in the Bremmstrahlung
contributions to the tree level Coulomb scattering deserve further analysis
and discussion, analysing whether the Bremmstrahlung cancels the radiate
corrections, as it happens in usual QED.

### Acknowledgements

The authors would like to thank the anonymous referee for his/her comments and
suggestions to improve this paper. R.B. acknowledges partial support from
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Projects
No. 305427/2019-9 and No. 421886/2018-8) and Fundação de Amparo à Pesquisa do
Estado de Minas Gerais (FAPEMIG Project No. APQ-01142-17).

##  Appendix A Complementary expressions

In this appendix we present by complementarity some auxiliary lengthy
coefficient expressions, $\alpha$, $\beta$, $\gamma$ and $\delta$, present in
the photon momentum roots, eqs.(3.27) and (3.29),

$\displaystyle\beta=$ $\displaystyle\frac{m_{r}^{4}}{24\
2^{2/3}\xi^{4}\alpha}-\frac{2\mu^{2}\sqrt[3]{2}m_{r}^{2}}{3\alpha}+\frac{\mu^{2}\eta
m_{r}^{2}}{2\ 2^{2/3}\xi^{2}\alpha}-\frac{9\mu^{2}m_{r}^{2}}{4\
2^{2/3}\xi^{2}\alpha}-\frac{\mu^{2}\eta^{2}m_{r}^{2}}{48\
2^{2/3}\xi^{4}\alpha}+\frac{\mu^{2}\eta}{4}-\frac{\mu^{2}}{2}+\frac{\mu^{2}\eta^{2}}{48\xi^{2}}$
(A.1)
$\displaystyle+\frac{m_{r}^{2}}{6\xi^{2}}-\frac{5\mu^{2}\xi^{2}}{12}+\frac{\alpha}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}\mu^{4}\xi^{4}}{3\alpha}-\frac{\mu^{4}\eta\xi^{2}}{2^{2/3}\alpha}+\frac{\mu^{4}\xi^{2}}{2^{2/3}\alpha}+\frac{\mu^{4}\eta^{2}}{3\
2^{2/3}\alpha}-\frac{\mu^{4}\eta}{2\ 2^{2/3}\alpha}+\frac{3\mu^{4}}{8\
2^{2/3}\alpha}$ (A.2) $\displaystyle+\frac{\mu^{4}\eta^{3}}{8\
2^{2/3}\xi^{2}\alpha}-\frac{3\mu^{4}\eta^{2}}{16\
2^{2/3}\xi^{2}\alpha}+\frac{\mu^{4}\eta^{4}}{384\ 2^{2/3}\xi^{4}\alpha}$ (A.3)
$\displaystyle\alpha_{1}=$
$\displaystyle-\frac{m_{r}^{6}}{32\xi^{6}}+\frac{3\mu^{2}m_{r}^{4}}{2\xi^{2}}-\frac{9\mu^{2}\eta
m_{r}^{4}}{16\xi^{4}}-\frac{135\mu^{2}m_{r}^{4}}{32\xi^{4}}+\frac{3\mu^{2}\eta^{2}m_{r}^{4}}{128\xi^{6}}+\frac{57\mu^{4}\xi^{2}m_{r}^{2}}{4}$
(A.4) $\displaystyle-\frac{135\mu^{4}\eta
m_{r}^{2}}{8}+\frac{207\mu^{4}m_{r}^{2}}{8}+\frac{69\mu^{4}\eta^{2}m_{r}^{2}}{16\xi^{2}}-\frac{45\mu^{4}\eta
m_{r}^{2}}{4\xi^{2}}+\frac{405\mu^{4}m_{r}^{2}}{32\xi^{2}}-\frac{27\mu^{4}\eta^{2}m_{r}^{2}}{64\xi^{4}}-\frac{3\mu^{4}\eta^{4}m_{r}^{2}}{512\xi^{6}}$
(A.5)
$\displaystyle+2\mu^{6}\xi^{6}-\frac{9\mu^{6}\eta\xi^{4}}{2}+\frac{9\mu^{6}\xi^{4}}{2}+\frac{51\mu^{6}\eta^{2}\xi^{2}}{16}-\frac{45\mu^{6}\eta\xi^{2}}{8}+\frac{27\mu^{6}\xi^{2}}{8}+\frac{33\mu^{6}\eta^{4}}{64\xi^{2}}$
(A.6)
$\displaystyle-\frac{45\mu^{6}\eta^{3}}{32\xi^{2}}+\frac{135\mu^{6}\eta^{2}}{128\xi^{2}}-\frac{45\mu^{6}\eta^{3}}{32}+\frac{63\mu^{6}\eta^{2}}{32}-\frac{27\mu^{6}\eta}{16}+\frac{27\mu^{6}}{32}+\frac{9\mu^{6}\eta^{5}}{256\xi^{4}}-\frac{27\mu^{6}\eta^{4}}{512\xi^{4}}+\frac{\mu^{6}\eta^{6}}{2048\xi^{6}}$
(A.7) $\displaystyle\alpha_{2}=$
$\displaystyle\frac{3\mu^{4}}{8\xi^{2}}\left(40\xi^{6}-28\eta\xi^{4}+12\xi^{4}+6\eta^{2}\xi^{2}-4\eta\xi^{2}-\eta^{3}+\eta^{2}+\frac{8m_{r}^{2}}{\mu^{2}}\right)$
(A.8)
$\displaystyle-\frac{\mu^{4}}{256\xi^{4}}\left(-64\xi^{4}+24\eta\xi^{2}-12\xi^{2}-\eta^{2}+\frac{4m_{r}^{2}}{\mu^{2}}\right)^{2}$
(A.9) $\displaystyle\gamma=$
$\displaystyle\frac{8\xi}{\mu^{3}}\left(6\xi^{2}-\eta\right)$ (A.10)
$\displaystyle\delta=$ $\displaystyle
64\xi^{4}-24\eta\xi^{2}+12\xi^{2}-\frac{4m_{r}^{2}}{\mu^{2}}+\eta^{2}$ (A.11)

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# FINITE AND DIVERGENT PARTS OF THE SELF-FORCE OF A POINT CHARGE FROM ITS
SPHERICALLY AVERAGED SELF-FIELD

V. Hnizdo<EMAIL_ADDRESS>2044 Georgian Lane, Morgantown, WV 26508, USA
G. Vaman<EMAIL_ADDRESS>Institute of Atomic Physics, 30 Reactorului St.,
P. O. Box MG-6, Bucharest, Romania

###### Abstract

The electromagnetic self-force of a point charge moving arbitrarily on a
rectilinear trajectory is calculated by averaging its retarded electric self-
field over a sphere of infinitesimal radius centered on the charge’s present
position. The finite part of the self-force obtained is the well-established
relativistic radiation reaction, while its divergent part implies the pre-
relativistic longitudinal electromagnetic mass of Abraham.

self-force, radiation reaction, electromagnetic mass, finite part, divergent
part

1\. Introduction

Recently, we have obtained the relativistic Lorentz-Abraham-Dirac (LAD)
radiation-reaction force dir  of a point charge moving arbitrarily on a
rectilinear trajectory by calculating the negative of the time rate of change
of the momentum of the charge’s retarded self-field hni1 . Remarkably, the
non-manifestly-covariant calculation procedure employed in hni1  involved no
inertial-mass renormalization or any other explicit removal of infinities. The
procedure thus amounted to the extraction of a finite part of the divergent
integral for the point charge’s retarded-self-field momentum.

In this paper, we revisit the same problem, but calculate the radiation-
reaction force using the finite part of the average of the charge’s retarded
self-field over a sphere of infinitesimal radius centered on its present
(current) position. The calculation, unlike that in hni1 , yields also the
divergent part of the self-force. It will turn out that the electromagnetic
mass implied by the divergent part is surprisingly the longitudinal
electromagnetic mass that Abraham obtained in his theory of a non-Lorentz-
contractible electron before the advent of special relativity, but the finite
part is again the relativistic LAD radiation reaction for rectilinear motion.

The procedure of calculating the radiation-reaction force of a point charge
using a suitable average of its self-field has already been employed by
several authors tei –man . Teitelboim’s calculation tei  was manifestly
covariant, yielding the full relativistic LAD equation of motion. Boyer boy
and Haque haq  used the procedure to obtain the radiation-reaction force of a
charge in nonrelativistic uniform circular motion. Mansuripur man  employed
the average of what is effectively half the difference of the retarded and
advanced self-fields (following in this Dirac dir ) over a sphere of
vanishingly small radius, centered on the charge’s retarded position in its
instantaneous rest frame. The self-force thus obtained is the well-known
nonrelativistic Lorentz-Abraham radiation-reaction force, which on the Lorentz
transformation to the laboratory frame becomes the exact LAD radiation
reaction. It should be noted that, like the calculation in hni1 , Mansuripur’s
procedure did not require any renormalization of the charge’s inertial mass.

2\. Averaging the retarded electric self-field (Finite part)

We consider a point charge $e$ moving arbitrarily on a rectilinear trajectory
$w(t)$ along the $x$-axis. It is convenient to assume that $w(t)\hat{\bf x}$
vanishes at the given instant of time $t$, so that the charge is located at
the origin ${\bf r}=0$ of the coordinate system at that time instant. The
radius vector $\bf r$ of observation will thus coincide with the displacement
from the position of the charge at the given present time $t$.

The $x$-component of the charge’s retarded electric field can be written as

$E_{x}({\bf r},t)=E_{1\,x}({\bf r},t)+E_{2\,x}({\bf r},t),$ (1)

where

$\displaystyle E_{1\,x}({\bf r},t)$ $\displaystyle=-\frac{\partial\phi({\bf
r},t)}{\partial
x}=-e\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\,\partial_{x}^{n+1}\frac{w^{n}(t-r/c)}{r},$
(2) $\displaystyle E_{2\,x}({\bf r},t)$ $\displaystyle=-\frac{\partial A({\bf
r},t)}{c\,\partial t}=-\frac{e}{c^{2}}\frac{\partial}{\partial
t}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\,\partial_{x}^{n}\frac{w^{n}(t-r/c)\dot{w}(t-r/c)}{r}.$
(3)

Here, $\phi({\bf r},t)$ and $A({\bf r},t)\hat{\bf x}$ are the pertinent
retarded scalar and vector potentials, respectively; $\partial_{x}^{n}$
denotes the partial differentiation with respect to $x$ of order $n$,
$w^{n}(\cdot)\equiv[w(\cdot)]^{n}$ and the overdot indicates time
differentiation. The retarded potentials used here were obtained in hni1  (see
Eqs. (8) and (9) there) using the Taylor-series expansions of delta-function
expressions for the charge and current densities of a moving point charge.

To be able to calculate the average of the charge’s retarded electric field
over a sphere centered on the present position of the charge, we shall need
the following formulas, holding for the time $t$ at which $w(t)\hat{\bf x}$ is
assumed to vanish:

$\displaystyle\frac{d^{p}}{dt^{p}}w^{k}(t)$
$\displaystyle=\left\\{\begin{array}[]{cc}0,&p<k\\\
k!\dot{w}^{k}(t),&p=k\end{array}\right.,$ (6)
$\displaystyle\frac{d^{k+1}}{dt^{k+1}}w^{k}(t)$
$\displaystyle=\frac{k(k+1)!}{2}\dot{w}^{k-1}(t)\ddot{w}(t),$ (7)
$\displaystyle\frac{d^{k+2}}{dt^{k+2}}w^{k}(t)$
$\displaystyle=\frac{k(k-1)(k+2)!}{8}\dot{w}^{k-2}(t)\ddot{w}^{2}(t)+\frac{k(k+2)!}{6}\dot{w}^{k-1}(t)\dddot{w}(t).$
(8)

They can be inferred from the differentiation analogue of the multinomial
theorem,

$\displaystyle\frac{d^{m}}{dt^{m}}f^{k}(t)$
$\displaystyle=\sum_{j_{1}+{\dots+}j_{k}=m}\frac{m!}{j_{1}!\cdot\cdot\cdot
j_{k}!}f^{(j_{1})}(t)\cdot\cdot\cdot f^{(j_{k})}(t)$
$\displaystyle=\sum_{p=0}^{k}\binom{k}{k-p}f^{p}(t)\sum_{\begin{subarray}{c}j_{1}+\dots+j_{k-p}=m\\\
j_{1},\dots,\,j_{k-p}>0\end{subarray}}\frac{m!}{j_{1}!\cdot\cdot\cdot
j_{k-p}!}f^{(j_{1})}(t)\cdot\cdot\cdot f^{(j_{k-p})}(t),$ (9)

written in the second line so that its application to $f(t)=w(t)$ at the time
when $w(t)\hat{\bf x}=0$ is facilitated.

Using the fact that $w^{n}(t)\dot{w}(t)=(n{+}1)^{-1}dw^{n+1}(t)/dt$ and
expanding $w^{n}({t-}r/c)$ in Taylor series, the fields (2) and (3) can be
written as

$\displaystyle E_{1\,x}({\bf
r},t)=-e\sum_{n=0}^{\infty}\sum_{i=n}^{\infty}\frac{(-1)^{n+i}}{n!\;i!\;c^{i}}(\partial_{x}^{n+1}r^{i-1})\frac{d^{i}}{dt^{i}}w^{n}(t),$
(10) $\displaystyle E_{2\,x}({\bf
r},t)=-\frac{e}{c^{2}}\sum_{n=0}^{\infty}\sum_{i=n-1}^{\infty}\frac{(-1)^{n+i}}{(n+1)!\;i!\;c^{i}}(\partial_{x}^{n}r^{i-1})\frac{d^{i+2}}{dt^{i+2}}w^{n+1}(t),$
(11)

where the lower limits of the sums over $i$ are adjusted in accordance with
formula (6). The derivatives $\partial_{x}^{n+1}r^{i-1}$ and
$\partial_{x}^{n}r^{i-1}$ can be calculated using Faa di Bruno’s formula faa ,

$\displaystyle\frac{d^{m}}{dx^{m}}g(f(x))=\sum_{\begin{subarray}{c}b_{1}+2b_{2}+{\dots+}mb_{m}=m\\\
b_{1}+{\dots+}b_{m}=k\end{subarray}}\frac{m!}{b_{1}!\cdot\cdot\cdot
b_{m}!}g^{(k)}(f(x))\left(\frac{f^{(1)}(x)}{1!}\right)^{b_{1}}\cdot\cdot\cdot\left(\frac{f^{(m)}(x)}{m!}\right)^{b_{m}},$
(12)

by choosing $f(x)=x^{2}{+}y^{2}{+}z^{2}$ ($y$ and $z$ thus being regarded as
parameters) and $g(f)=f^{(i-1)/2}$; note that only the first two derivatives
of thus defined $f(x)$ are different from zero: $f^{\prime}(x)=2x$ and
$f^{\prime\prime}(x)=2$. This way, we obtain

$\displaystyle\frac{\partial^{n+1}}{\partial x^{n+1}}r^{i-1}$
$\displaystyle=\sum_{b=0}^{n/2}\frac{(-1)^{b+1+n/2}2^{2b+1}(n+1)!}{(2b+1)!(n/2-b)!}\left(\frac{1-i}{2}\right)_{b+1+n/2}r^{i-n-2b-3}x^{2b+1},\>n\;{\rm
even},$ (13) $\displaystyle\frac{\partial^{n+1}}{\partial x^{n+1}}r^{i-1}$
$\displaystyle=\sum_{b=0}^{(n+1)/2}\frac{(-1)^{b+(n+1)/2}2^{2b}(n+1)!}{(2b)!((n+1)/2-b)!}\left(\frac{1-i}{2}\right)_{b+(n+1)/2}r^{i-n-2b-2}x^{2b},\>n\;{\rm
odd},$ (14)

where $(\tfrac{1}{2}-\frac{i}{2})_{b+1+n/2}$, etc. are the Pochhammer symbols
(pru , Appendix I). The expressions for $\partial_{x}^{n}r^{i-1}$ are obtained
by replacing $n$ in (13) and (14) with $n-1$, and changing “even” to “odd” and
vice versa. These expressions facilitate angular integration of (10) and (11).
Since

$\displaystyle\int d\Omega\,x^{2b}=4\pi\frac{r^{2b}}{2b+1}$ (15)

and the angular integral of $x^{2b+1}$ vanishes, we obtain for the angular
averages of $E_{1\,x}({\bf r},t)$ and $E_{2\,x}({\bf r},t)$:

$\displaystyle\bar{E}_{1\,x}(r,t)$ $\displaystyle=\frac{1}{4\pi}\int
d\Omega\,E_{1\,x}({\bf r},t)$
$\displaystyle=-8e\sum_{n=0}^{\infty}\sum_{i=2n+1}^{\infty}\frac{(-1)^{i+n}2^{2n}(n+1)\left(\frac{i}{2}\right)!\left(\frac{1-i}{2}\right)_{n+1}}{(2n+3)!\,i!\,\left(\frac{i}{2}-n-1\right)!\,c^{i}}\,r^{i-2n-3}\,\frac{d^{i}}{dt^{i}}w^{2n+1}(t),$
(16) $\displaystyle\bar{E}_{2\,x}(r,t)$ $\displaystyle=\frac{1}{4\pi}\int
d\Omega\,E_{2\,x}({\bf r},t)$
$\displaystyle=-e\sum_{n=0}^{\infty}\sum_{i=2n-1}^{\infty}\frac{(-1)^{i+n}2^{2n}\left(\frac{i}{2}\right)!\left(\frac{1-i}{2}\right)_{n}}{(2n+1)(2n+1)!\,i!\left(\frac{i}{2}-n\right)!\,\,c^{i+2}}\,r^{i-2n-1}\,\frac{d^{i+2}}{dt^{i+2}}w^{2n+1}(t).$
(17)

In the limit $r\rightarrow 0$, only the terms with $1/r^{2}$, $1/r$ and
$r^{0}$ can contribute to angular averages (16) and (17). We consider first
the finite parts of the limits $r\rightarrow 0$ of these averages, obtained
accordingly by retaining only the terms with $i=2n+3$ and $i=2n+1$ in (16) and
(17), respectively, and using formulas (7) and (8):

$\displaystyle\bar{E}_{1\,x}^{\,\rm fin}(t)$ $\displaystyle={\rm
f.p.}\lim_{r\rightarrow 0}\bar{E}_{1\,x}(r,t)$
$\displaystyle=e\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\,c^{2n+3}}(2n+1)(-n-1)_{n+1}\Big{[}\frac{n}{2}\dot{w}^{2n-1}(t)\ddot{w}^{2}(t)+\frac{1}{3}\dot{w}^{2n}(t)\dddot{w}(t)\Big{]},$
(18) $\displaystyle\bar{E}_{2\,x}^{\,\rm fin}(t)$ $\displaystyle={\rm
f.p.}\lim_{r\rightarrow 0}\bar{E}_{2\,x}(r,t)$
$\displaystyle=e\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\,c^{2n+3}}(n+1)(2n+3)(-n)_{n}\Big{[}\frac{n}{2}\dot{w}^{2n-1}(t)\ddot{w}^{2}(t)+\frac{1}{3}\dot{w}^{2n}(t)\dddot{w}(t)\Big{]}.$
(19)

The series over $n$ can be summed using

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}n(2n+1)(-n-1)_{n+1}\beta^{2n-1}$
$\displaystyle=-6\,\frac{\beta(1+\beta^{2})}{(1-\beta^{2})^{4}},$ (20)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(2n+1)(-n-1)_{n+1}\beta^{2n}$
$\displaystyle=-\frac{1+3\beta^{2}}{(1-\beta^{2})^{3}},$ (21)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}n(n+1)(2n+3)(-n)_{n}\beta^{2n-1}$
$\displaystyle=2\,\frac{\beta(5+\beta^{2})}{(1-\beta^{2})^{4}},$ (22)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(n+1)(2n+3)(-n)_{n}\beta^{2n}$
$\displaystyle=\frac{3+\beta^{2}}{(1-\beta^{2})^{3}},$ (23)

where $\beta=\dot{w}(t)/c$, and we obtain

$\displaystyle\bar{E}_{1\,x}^{\,\rm fin}(t)$
$\displaystyle=-3e\,\frac{\beta(1+\beta^{2})}{(1-\beta^{2})^{4}}\frac{\ddot{w}^{2}(t)}{c^{4}}-\frac{e}{3}\,\frac{1+3\beta^{2}}{(1-\beta^{2})^{3}}\frac{\dddot{w}(t)}{c^{3}},$
(24) $\displaystyle\bar{E}_{2\,x}^{\,\rm fin}(t)$
$\displaystyle=e\,\frac{\beta(5+\beta^{2})}{(1-\beta^{2})^{4}}\frac{\ddot{w}^{2}(t)}{c^{4}}+\frac{e}{3}\,\frac{3+\beta^{2}}{(1-\beta^{2})^{3}}\frac{\dddot{w}(t)}{c^{3}}.$
(25)

The angular averages of the electric-field components $E_{y}$ and $E_{z}$,
along with those of the magnetic field, can be seen to vanish already on
account of symmetry. The retarded-self-field force of a point charge $e$
moving arbitrarily along the $x$-axis, calculated using the finite part of the
self-field’s average over a sphere of infinitesimal radius, centered on the
charge’s present position, is thus

$\displaystyle{\bf F}_{\rm fin}$ $\displaystyle=e[\bar{E}_{1\,x}^{\,\rm
fin}(t)+\bar{E}_{2\,x}^{\,\rm fin}(t)]\,\hat{\bf x}$
$\displaystyle=\frac{2e^{2}}{3c^{3}}\gamma^{4}\left[\ddot{v}(t)+\frac{3}{c^{2}}\gamma^{2}v(t)\dot{v}^{2}(t)\right]\hat{\bf
x},$ (26)

where we now write the velocity $\dot{w}(t)$ as $v(t)$ and
$\gamma=(1-v^{2}/c^{2})^{-1/2}$. It is exactly the same relativistic
radiation-reaction force as that obtained in hni1 .

3\. Averaging the retarded electric self-field (Divergent part)

We now calculate the divergent parts of the fields (2) and (3) averaged over a
sphere of infinitesimal radius centered on the charge. To do this, we take
$i=2n+1$, $2n+2$ and $2n-1$, $2n$ in Eqs. (16) and (17), respectively. Since
the Pochhammer symbols $(-n)_{n+1}$ and $(1-n)_{n}$ vanish, the resulting
terms with $1/r^{2}$ vanish, too. Using formulas (7) and (8), we thus have for
the divergent parts of the limits $r\rightarrow 0$ of (16) and (17):

$\displaystyle\bar{E}_{1\,x}^{\,\rm div}(t)$
$\displaystyle=-\lim_{r\rightarrow
0}\left(\frac{4e\ddot{w}(t)}{rc^{2}}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}2^{2n}(n+1)(2n+1)(n+1)!\left(-n-\frac{1}{2}\right)_{n+1}}{(2n+3)!\,c^{2n}}\,\dot{w}^{2n}(t),$
(27) $\displaystyle\bar{E}_{2\,x}^{\,\rm div}(t)$
$\displaystyle=-\lim_{r\rightarrow
0}\left(\frac{e\ddot{w}(t)}{rc^{2}}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}2^{2n}(n+1)!\left(\frac{1}{2}-n\right)_{n}}{(2n)!\,c^{2n}}\,\dot{w}^{2n}(t).$
(28)

Using now

$\displaystyle\left(-n-\frac{1}{2}\right)_{n+1}=(-1)^{n+1}\frac{(2n+1)!}{2^{2n+1}n!},\;\;\;\;\left(\frac{1}{2}-n\right)_{n}=(-1)^{n}\frac{(2n)!}{2^{2n}n!},$
(29)

and summing the series over $n$,

$\displaystyle\sum_{n=0}^{\infty}\frac{(n+1)(2n+1)}{2n+3}\beta^{2n}$
$\displaystyle=\gamma^{4}+\frac{1}{\beta^{3}}\left(\frac{1}{2}\ln\frac{1+\beta}{1-\beta}-\frac{\beta}{1-\beta^{2}}\right),$
(30) $\displaystyle\sum_{n=0}^{\infty}(n+1)\beta^{2n}$
$\displaystyle=\gamma^{4},$ (31)

where $\gamma=(1-\beta^{2})^{-1/2}$ and $\beta=\dot{w}(t)/c$, we obtain for
the divergent part of the force of the electric self-field averaged over a
sphere of infinitesimal radius:

$\displaystyle{\bf F}_{\rm div}$ $\displaystyle=e[\bar{E}_{1\,x}^{\,\rm
div}(t)+\bar{E}_{2\,x}^{\,\rm div}(t)]\,\hat{\bf x}$
$\displaystyle=\lim_{r\rightarrow
0}\left(\frac{e^{2}\dot{\beta}}{rc}\right)\frac{1}{\beta^{3}}\left(\frac{1}{2}\ln\frac{1+\beta}{1-\beta}-\frac{\beta}{1-\beta^{2}}\right)\hat{\bf
x}.$ (32)

Before the advent of special relativity, the so-called longitudinal
electromagnetic mass, $m_{\rm{em}\,||}$, was taken to be the proportionality
coefficient of the negative of the electromagnetic self-force proportional to
the acceleration of a charge in rectilinear motion (lor , sec. 27). At a
distance $r=a$ from the charge’s location, this coefficient is according to
Eq. (32):

$\displaystyle
m_{\rm{em}\,||}=\frac{e^{2}}{ac^{2}}\frac{1}{\beta^{3}}\left(\frac{\beta}{1-\beta^{2}}-\frac{1}{2}\ln\frac{1+\beta}{1-\beta}\right).$
(33)

Perhaps surprisingly, it happens to equal the mass $m_{\rm{em}\,||}$ that
Abraham obtained in his model of the electron as a uniformly charged “rigid”
(meaning today non-Lorentz-contractible) spherical shell of radius $a$ (abr ,
p. 191, Eq. (117); see also lor , p. 39, Eq. (68)).

We note here that Abraham’s electromagnetic mass $m_{{\rm em}\,||}$ and the
longitudinal electromagnetic mass $2e^{2}\gamma^{3}/(3ac^{2})$, obtained using
the time derivative of the momentum $(2e^{2}\gamma/3ac^{2}){\bf v}$ of the
well-known Heaviside fields of a uniformly moving charged shell, in its rest
frame spherical with radius $a$ hni2 , agree only when $\beta<<1$. But it is
interesting to note also that it can be shown (see Appendix) that when the
divergent self-field momentum $\bf G$ of a uniformly moving point charge is
evaluated by approaching the field-momentum density’s singularity on a
spherical surface of radius $a$, one obtains a rather more complicated
expression:

$\displaystyle{\bf G}=\lim_{a\rightarrow
0}\frac{e^{2}}{8ac^{2}}\frac{1}{\beta^{3}}[(1+2\beta^{2})\beta-(1-4\beta^{2})\gamma\,\rm{arctg}(\beta\gamma)]{\bf
v}.$ (34)

The “electromagnetic-mass” factor multiplying here the velocity $\bf v$
reduces to the value $2e^{2}\gamma/(3ac^{2})$ only when $\beta<<1$.

4\. Concluding remarks

In a recent Comment hni3  on the derivation of the LAD equation of motion by
Dondera don , we have advanced a conjecture concerning the Hadamard
decomposition of the retarded-self-field momentum of an arbitrarily moving
point charge. According to the conjecture, the divergent electromagnetic mass
implied by the self-field momentum of a uniformly moving point charge is given
by $\lim_{\varepsilon\rightarrow 0}2e^{2}\gamma/(3\varepsilon c^{2})$. The
requisite integration of the divergent integral for the retarded-self-field
momentum of such a charge is thus understood there to be done so that the
singularity in the self-field-momentum density is approached on an oblate-
spheroidal surface whose shape is congruent with that of a Lorentz-contracted
sphere.

Curiously, Rohrlich has attributed to Abraham an equation of motion (ror ,
Eqs. (2.7)–(2.8)) that implies the same decomposition of the retarded-self-
field momentum as our conjecture would for a charged rest-frame-spherical
shell of radius $a$. Rohrlich’s attribution is correct for the equation’s
radiation-reaction force, which Abraham was indeed the first to obtain in its
full relativistic form (abr , p. 123, Eq. (85)), but not for its
electromagnetic-inertia term since Abraham’s “rigid” shell of charge does not
undergo the Lorentz contraction when it is moving. Yaghjian has also
attributed the same equation of motion (sans the “bare-mass” term) to
Abraham, but he has attributed it to Lorentz as well, saying that it was for a
“relativistically rigid” (i.e. Lorentz-contractible, in Lorentz’s parlance
“deformable”) spherical shell of charge (yag , pp. 11-12).

Using a spherical average of the retarded electric self-field of a point
charge moving arbitrarily on a rectilinear trajectory, we obtained in this
paper both the finite and divergent parts of its electromagnetic self-force.
The finite part obtained is the same as the well-established relativistic LAD
radiation reaction, but the longitudinal electromagnetic mass implied by the
divergent part differs at relativistic velocities markedly from that obtained
from the electromagnetic momentum of a uniformly moving rest-frame-spherical
shell of charge. This fact indicates that, in contrast to the divergent part
of the self-force, the force of radiation reaction is not sensitive to the
details of the models for the charge used and the methods of calculation
employed.

Abraham obtained his longitudinal electromagnetic mass as the time derivative
of the electromagnetic momentum he calculated for a uniformly moving “rigid”
spherical shell of charge, the exterior electric field of which is not the
well-known Heaviside electric field of a uniformly moving spheroidal, Lorentz-
contracted shell of charge. The fact that our divergent part of the self-force
implies Abraham’s electromagnetic mass is indeed surprising.

In closing, we should caution that electromagnetic mass is today a concept of
limited applicability and usefulness. It is rooted in the electromagnetic
program of the beginning of the 20th century, in which Abraham and some other
prominent physicists attempted to supplant classical mechanics by a field
theory based on Maxwell’s electromagnetism and the ether as the medium
sustaining it. But after the advent of special relativity the program became
rather quickly outmoded, and its key concepts, the ether and electromagnetic
mass, superseded (an illuminating account of this interesting episode in the
history of physics is given in jan ).

Appendix

The well-known Heaviside expressions for the fields of a point charge $e$
moving with a constant velocity $\bf v$ are given by

$\displaystyle{\bf E}({\bf
r},t)=\frac{e}{\gamma^{2}(1-\beta^{2}\sin^{2}\theta)^{3/2}}\frac{\bf
r}{r^{3}},\;\;\;\;{\bf B}({\bf r},t)=({\bm{\beta}}\times{\bf E}),$ (35)

where ${\bm{\beta}}={\bf v}/c$, $\gamma=(1-\beta^{2})^{-1/2}$ and $\theta$ is
the angle between $\bf v$ and $\bf r$, the latter being the displacement from
the present position of the charge. The momentum of these fields is

$\displaystyle{\bf G}$ $\displaystyle=\frac{1}{4\pi c}\int d^{3}r\,({\bf
E}\times{\bf B})$ $\displaystyle=\frac{1}{4\pi c}\int
d^{3}r\,(1-\cos^{2}\theta)E^{2}{\bm{\beta}},$ (36)

where account is taken of the fact that only the component of the vector
product $({\bf E}\times{\bf B})$ along the direction of $\bm{\beta}$
contributes to the integral. The square of the Heaviside electric field can be
written as

$\displaystyle
E^{2}=\frac{\gamma^{2}}{(1+\beta^{2}\gamma^{2}\cos^{2}\theta)^{3}}\,\frac{e^{2}}{r^{4}},$
(37)

and thus the divergent momentum $\bf G$ calculated by approaching the
singularity at $r=0$ on a spherical surface of radius $a$ is given by

$\displaystyle{\bf G}$ $\displaystyle=\lim_{a\rightarrow
0}\frac{e^{2}\gamma^{2}}{4\pi
c}\int_{a}^{\infty}\frac{dr}{r^{2}}\,\int\frac{d\Omega\,(1-\cos^{2}\theta)}{(1+\beta^{2}\gamma^{2}\cos^{2}\theta)^{3}}\,{\bm{\beta}}$
$\displaystyle=\lim_{a\rightarrow
0}\frac{e^{2}\gamma^{2}}{2ac}\int_{-1}^{1}d\xi\,\frac{(1-\xi^{2})}{(1+\beta^{2}\gamma^{2}\xi^{2})^{3}}\,{\bm{\beta}}$
$\displaystyle=\lim_{a\rightarrow
0}\frac{e^{2}}{8ac^{2}}\frac{1}{\beta^{3}}[(1+2\beta^{2})\beta-(1-4\beta^{2})\gamma\,\rm{arctg}(\beta\gamma)]\,{\bf
v}.$ (38)

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