#  __
New phenomena in laser-assisted leptonic decays of the negatively charged
boson $W^{-}$

S Mouslih2,1, M Jakha1, I Dahiri1, S Taj1, B Manaut1, and E. Siher2
1 Sultan Moulay Slimane University, Polydisciplinary Faculty,
Research Team in Theoretical Physics and Materials (RTTPM), Beni Mellal,
23000, Morocco.
2Faculty of Sciences and Techniques, Laboratory of Materials Physics (LMP),
Beni Mellal, 23000, Morocco. Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

The majority of studies and experiments performed at electron-positron
colliders over the last two decades have focused on studying $W$ and $Z$ weak
force-carrying bosons and accurately measuring all their properties, not only
because they play an important role in establishing Standard Model theory and
providing an accurate test of its predictions of particle interactions, but
also because they are a unique tool for probing manifestations of the new
physics beyond the standard model. Therefore, it would be particularly
important to discuss some of the new phenomena and changes that can arise in
these bosons when their decay occurs under an external electromagnetic field.
In a recent paper, we investigated the laser effect on the final products of
$Z$ boson decay and found that laser had an unprecedented effect on branching
ratios. In this work and within the standard Glashow-Weinberg-Salam model of
electroweak interactions, we study theoretically the leptonic decay of the
$W^{-}$-boson $(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})$ in the presence of
a circularly polarized electromagnetic field and we examine the laser effect,
in terms of its field strength and frequency, on the leptonic decay rate and
the phenomenon of multiphoton processes. The calculations are carried out
using the exact relativistic wave functions of charged particles in an
electromagnetic field. It was found that the laser significantly contributed
to reducing the probability of $W^{-}$-boson decay. We show that the laser-
assisted decay rate is equal to the laser-free one only when the famous Kroll-
Watson sum rule is fulfilled. The notable effect of the laser on the leptonic
decay rate was reasonably interpreted by the well-known quantum Zeno effect or
by the opening of channels other than leptonic ones to decay. This work will
pave the way for an upcoming one to study the hadronic decay of the
$W^{-}$-boson and then explore the laser effect on its lifetime and branching
ratios.

Keywords: electroweak interaction, laser-assisted processes, decay rate.

##  1\. Introduction

The tremendous and rapid progress that laser technology has made since its
invention in the $1960$s has enabled a new field of theoretical and
experimental studies to explore the interactions of electromagnetic (EM)
fields with matter at high intensities [1]. This important development of
laser technology has played an important role in the production of high-power
laser systems with a maximum intensity of $10^{22}\text{W/cm}^{2}$ [2, 3] and
researchers currently expect that higher intensities can be achieved in the
near future [4, 5, 6, 7]. Today, these high-power laser systems have become
indispensable equipment and are used in laboratories all over the world,
thanks to which studies of the basic processes of quantum electrodynamics
(QED) [8, 9, 10, 11, 12, 13] and electroweak theory [14, 15] in the presence
of a strong laser field and realistic in experiments have continued to attract
significantly the attention of physicists. Through all this, scientists are
seeking to know and understand how elementary particles behave and how their
properties change when inserted in an EM field. The theory describing the
fundamental interactions of these particles is called the Standard Model (SM)
and it is a gauge theory, which describes strong, weak and electromagnetic
interactions by exchanging the corresponding spin-$1$ gauge fields: eight
massless gluons and one massless photon, respectively, for the strong and
electromagnetic interactions, and three massive bosons, $W^{\pm}$ and $Z^{0}$,
for the weak interaction. It is one of the most successful achievements of
modern physics in that it provides a very elegant theoretical framework
capable of describing known experimental facts in particle physics with great
precision. The discovery of the $W$ [16, 17] and $Z$ [18, 19] bosons in $1983$
by the UA1 and UA2 collaborations at the CERN’s $p\bar{p}$ collider provided
direct confirmation of the unification of the weak and electromagnetic
interactions in a common framework that describes them together. This common
framework is the so-called electroweak theory predicted by physicists Sheldon
Lee Glashow [20], Steven Weinberg [21] and Abdus Salam [22] in the late
$1960$s, in which they explain that the electromagnetic and weak forces, which
have long been considered as separate entities, are in fact manifestations of
the same fundamental interaction. In recognition of their role in the
discovery of the $W$ and $Z$ particles, the CERN physicist Carlo Rubbia [23]
and engineer Simon van der Meer [24] were awarded the Nobel Prize in physics
in $1984$. Detectable $W$ particles can be produced in proton-antiproton
collision experiments by processes such as $u\bar{d}$ or $\bar{u}d\rightarrow
W$ followed by subsequent leptonic $(W\rightarrow\ell\nu)$ or hadronic decay
$(W\rightarrow q\bar{q}^{\prime})$, where $\ell=e,~{}\mu,~{}\tau$ and $q$ or
$q^{\prime}$ represent one of the quarks $u$, $d$, $c$, $s$, or $b$ except $t$
since top quark is heavier than the $W$ boson. Experiments at the Large
Electron-Positron Collider (LEP) and the Stanford Linear Collider (SLC) refine
the measurements of the $W$-boson properties (mass, total decay rate and cross
sections of their production). At the same time, the properties of $W$ bosons
are studied in the Tevatron proton-antiproton accelerator. All these
experiments have given more precise measurements of the $W$-boson mass and its
total decay rate, which are now known to be $M_{W}=80.379\pm
0.012~{}\text{GeV}$ and $\Gamma_{W}=2.085\pm 0.042~{}\text{GeV}$ [25]. The
accuracy obtained in these experiments makes it possible to test the
predictions of the SM at the level of radiative corrections. This work has, as
an objective, to study the leptonic decay of the negatively charged boson
$W^{-}$ in the presence of a circularly polarized EM field. As mentioned
above, the $W^{-}$-boson can decay into a lepton and antineutrino
$W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell}$ (leptonic decay) or into a pair of
quarks $W^{-}\rightarrow q\bar{q}^{\prime}$ (hadronic decay). Here, we will
only consider the first channel (leptonic channel), where we will try to
realize the effect of the EM field strength and frequency on the decay rate.
In $2004$, the author Kurilin studied the leptonic decays of the $W^{-}$-boson
in a strong crossed EM Field, where he calculated the expression of the decay
rate and studied its changes in terms of the external-field-strength parameter
$\varkappa=eM_{W}^{-3}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}}$, where $F_{\mu\nu}$ is
the EM tensor [26]. Kurilin’s approach is completely different from the
approach followed here. In his study, Kurilin restricts his consideration to
the case of a so-called crossed field, which is a superposition of constant
electric and magnetic fields whose strength vectors are equal and orthogonal.
The wave functions of the charged particles in this case have a simple
expression in terms of the EM tonsor $F_{\mu\nu}$, and this makes the related
mathematical calculations less complicated. Some of the transformations in our
calculations have been simplified by the introduction of the ordinary Bessel
functions, whose order is interpreted as the number of exchanged photons,
while Kurilin calculations have been performed with the help of special
mathematical functions generically termed Airy functions [27]. Despite this
difference in the method and the nature of the EM field, we have compared our
results obtained here with the results obtained in his research and give some
explanations for both results obtained. There is also an earlier paper who did
the same study in brief giving the expression of the decay rate in the
presence of a circularly polarized EM field [28]. An overview of the
interaction of these bosons with the EM field and how to write the expression
for the wave functions describing them can be found in refs. [29, 30].
Although $W$ bosons are best known for their role in radioactive decay, their
decay may provide the opportunity to discover supersymmetric particles and
probe new physics beyond the SM [31, 32]. The present work is part of a series
of studies we have conducted to know and understand the effect of the EM field
on experimentally measurable quantities during electroweak decay processes
(decay rate, lifetime, branching ratio). So far, we have calculated both the
pion [33] and $Z$-boson [34] decays in the presence of an EM field in an
attempt to contribute to enriching the debate on this topic. In this same
context, we try to study the decay of the $W^{-}$-boson in two stages. First,
we will study its leptonic decay and thus the effect of the EM field on the
leptonic decay rate. Secondly, in a forthcoming research, we will study the
hadronic decay and collect the results obtained with the present results in
order to be able to study the effect of the laser on both the lifetime and the
branching ratio for each channel as two important points that must be treated.
It is therefore this work that will pave the way for the next one, and by that
time we will have studied this decay and have sufficiently covered all its
aspects. The rest of the paper is arranged as follows. In section 2., we will
try to formulate, in detail, the theoretical expression of the decay rate,
then in section 3., we will present the results obtained and highlight the
extent of the laser influence on the decay rate. The last section 4. is
devoted to summarizing the conclusions that we have reached. We only mention
here that we used, during our calculations, known natural units $c=\hbar=1$.

##  2\. Theoretical formulation of the decay rate in a laser field

We begin by considering the process of electroweak decay of a boson vector
$W^{-}$, with four-momentum $p$ and mass $M_{W}$, into a lepton $\ell^{-}$ and
corresponding antineutrino $\bar{\nu}_{\ell}$ in the field of a plane EM wave

$W^{-}(p)\longrightarrow\ell^{-}(p_{1})+\bar{\nu}_{\ell}(p_{2}),~{}~{}~{}(\ell=e,\mu,\tau)$
(1)

where the arguments are our labels for the four-momenta. The corresponding
Feynman diagram is shown in figure 1.

Figure 1: Lowest order Feynman diagram of the leptonic $W^{-}$-boson decay.

The laser field, which is assumed to be circularly polarized monochromatic, is
described by the following classical four-potential

$\displaystyle A^{\mu}(\phi)=a^{\mu}_{1}\cos(\phi)+a^{\mu}_{2}\sin(\phi),$ (2)

depending on a single variable, $\phi=(k.x)$, the phase of the laser field.
$k=(\omega,\textbf{k})$ is the wave four-vector $(k^{2}=0)$ and $\omega$ is
the laser frequency. The four-amplitudes $a^{\mu}_{1}=|\textbf{a}|(0,1,0,0)$
and $a^{\mu}_{2}=|\textbf{a}|(0,0,1,0)$ are equal in magnitude and orthogonal,
which implies $(a_{1}.a_{2})=0$ and
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|\textbf{a}|^{2}=-(\mathcal{E}_{0}/\omega)^{2}$
where $\mathcal{E}_{0}$ is the amplitude of the electric field. We suppose
that the four-potential satisfies the Lorentz gauge condition,
$k_{\mu}A^{\mu}=0$, which means $(k.a_{1})=(k.a_{2})=0$, indicating that the
wave vector k is chosen to be along the $z$-axis.
The lowest-order scattering S-matrix element for the laser-assisted leptonic
$W^{-}$ decay reads [35]

$S_{fi}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\dfrac{i\textsl{g}}{2\sqrt{2}}\int
d^{4}x\overline{\psi}_{\ell}(x)\gamma^{\mu}(1-\gamma_{5})\psi_{\bar{\nu}_{\ell}}(x)W^{-}_{\mu}(x),$
(3)

where g is the electroweak coupling constant. $\psi_{\ell}(x)$ is the wave
function of the relativistic lepton $\ell^{-}$ in an EM field given by the
relativistic Dirac-Volkov functions normalized to the volume V [36]

$\psi_{\ell}(x)=\bigg{[}1+\dfrac{e\not{k}\not{A}}{2(k.p_{1})}\bigg{]}\frac{u(p_{1},s_{1})}{\sqrt{2Q_{1}V}}\times
e^{iS(q_{1},x)},$ (4)

where regarding the sign one should note that $e=-|e|$ is the charge of the
electron, and

$S(q_{1},x)=-q_{1}.x-\dfrac{e(a_{1}.p_{1})}{k.p_{1}}\sin(\phi)+\dfrac{e(a_{2}.p_{1})}{k.p_{1}}\cos(\phi).$
(5)

$u(p_{1},s_{1})$ represents the Dirac bispinors for the free lepton with
momentum $p_{1}$ and spin $s_{1}$ satisfying
$\sum_{s_{1}}u(p_{1},s_{1})\overline{u}(p_{1},s_{1})=\not{p}_{1}+m_{\ell}$
where $m_{\ell}$ is the rest mass of the lepton. The 4-vector
$q_{1}=(Q_{1},\textbf{q}_{1})$ is the quasi-momentum that the lepton acquires
in the presence of the EM field

$q_{1}=p_{1}-\frac{e^{2}a^{2}}{2(k.p_{1})}k,~{}~{}~{}q_{1}^{2}=m^{*2}_{\ell}=m_{\ell}^{2}-e^{2}a^{2},$
(6)

where $m^{*}_{\ell}$ is an effective mass of the lepton inside the EM field.
For the laser-dressed $W^{-}$ (1-spin particle), its wave function can be
written in the form [29]

$W^{-}_{\mu}(x)=\bigg{[}\textsl{g}_{\mu\nu}-\dfrac{e}{(k.p)}\big{(}k_{\mu}A_{\nu}-k_{\nu}A_{\mu}\big{)}-\frac{e^{2}}{2(k.p)^{2}}A^{2}k_{\mu}k_{\nu}\bigg{]}\frac{\varepsilon_{\mu}(p,\lambda)}{\sqrt{2p_{0}V}}\times
e^{iS(q,x)},$ (7)

where $\textsl{g}_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ is the metric tensor of
Minkowski space, $\varepsilon_{\mu}(p,\lambda)$ is the $W^{-}$-boson
polarization four-vector such that the summation over all three directions of
polarization $\lambda$ yields
$\sum_{\lambda=1}^{3}\varepsilon_{\mu}(p,\lambda)\varepsilon^{*}_{\nu}(p,\lambda)=-\textsl{g}_{\mu\nu}+p_{\mu}p_{\nu}/M_{W}^{2}$,
and

$S(q,x)=-q.x+\dfrac{e(a_{1}.p)}{k.p}\sin(\phi)-\dfrac{e(a_{2}.p)}{k.p}\cos(\phi),$
(8)

with the 4-vector $q=p-\big{[}e^{2}a^{2}/2(k.p)\big{]}k$ $(q^{2}=M^{*2}_{W})$
and $M^{*}_{W}=\sqrt{M_{W}^{2}-e^{2}a^{2}}$ are, respectively, the quasi-
momentum and the effective mass that the boson $W^{-}$ acquires inside the EM
field. The outgoing antineutrino $\bar{\nu}_{\ell}$ is treated as massless
particle with four-momentum $p_{2}$ and spin $s_{2}$. According to the Feynman
rules, it is represented by an incoming wave function with negative four-
momentum as follows [35]:

$\displaystyle\psi_{\bar{\nu}_{\ell}}(x)=\dfrac{v(p_{2},s_{2})}{\sqrt{2E_{2}V}}e^{ip_{2}.x},$
(9)

where $E_{2}=p_{2}^{0}=|\textbf{p}_{2}|$ and $v(p_{2},s_{2})$ is the Dirac
spinor satisfying
$\sum_{s_{2}}v(p_{2},s_{2})\overline{v}(p_{2},s_{2})=\not{p}_{2}$.
Substituting the lepton, antineutrino and $W^{-}$-boson wave functions (4),
(7) and (9) into expression (3) for the S-matrix element, we obtain

$\begin{split}S_{fi}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=&\dfrac{i\textsl{g}}{2\sqrt{2}\sqrt{8E_{2}Q_{1}p_{0}V^{3}}}\int
d^{4}x\overline{u}(p_{1},s_{1})\Big{\\{}\Big{[}1+C(p_{1})\not{A}\not{k}\Big{]}\gamma^{\mu}(1-\gamma_{5})\\\
&\times\Big{[}\textsl{g}_{\mu\nu}-C(p)\big{(}k_{\mu}A_{\nu}-k_{\nu}A_{\mu}\big{)}-\frac{C(p)^{2}}{2}a^{2}k_{\mu}k_{\nu}\Big{]}\Big{\\}}v(p_{2},s_{2})\\\
&\times\varepsilon_{\mu}(p,\lambda)e^{ip_{2}.x}e^{i(S(q,x)-S(q_{1},x))},\end{split}$
(10)

where $C(p_{1})=e/(2(k.p_{1}))$ and $C(p)=e/(k.p)$. Now, let us transform the
exponential term $e^{i(S(q,x)-S(q_{1},x))}$ by introducing the following
parameter

$z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\quad\text{with}\quad\alpha_{1}=-e\bigg{(}\dfrac{a_{1}.p}{k.p}+\dfrac{a_{1}.p_{1}}{k.p_{1}}\bigg{)}\,;\,\alpha_{2}=-e\bigg{(}\dfrac{a_{2}.p}{k.p}+\dfrac{a_{2}.p_{1}}{k.p_{1}}\bigg{)},$
(11)

this yields

$e^{i(S(q,x)-S(q_{1},x))}=e^{i(q_{1}-q).x}e^{-iz\sin(\phi-\phi_{0})},$ (12)

with $\phi_{0}=\arctan(\alpha_{2}/\alpha_{1})$. Therefore, the S-matrix
element becomes

$\begin{split}S_{fi}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=&\dfrac{i\textsl{g}}{2\sqrt{2}\sqrt{8E_{2}Q_{1}p_{0}V^{3}}}\int
d^{4}x\overline{u}(p_{1},s_{1})\big{[}C_{0}+C_{1}\cos(\phi)+C_{2}\sin(\phi)\big{]}\\\
&\times
v(p_{2},s_{2})\varepsilon_{\mu}(p,\lambda)e^{i(p_{2}+q_{1}-q).x}e^{-iz\sin(\phi-\phi_{0})},\end{split}$
(13)

where the three quantities $C_{0}$, $C_{1}$, and $C_{2}$ are expressed as
follows:

$\begin{split}C_{0}=&\gamma_{\nu}(1-\gamma_{5})-\frac{C(p)^{2}}{2}a^{2}k_{\mu}k_{\nu}\gamma^{\mu}(1-\gamma_{5}),\\\
C_{1}=&C(p_{1})\not{a}_{1}\not{k}\gamma_{\nu}(1-\gamma_{5})-C(p)\gamma^{\mu}(1-\gamma_{5})\big{(}k_{\mu}a_{1\nu}-k_{\nu}a_{1\mu}\big{)}-\frac{C(p)^{2}}{2}C(p_{1})a^{2}k_{\mu}k_{\nu}\\\
&\times\not{a}_{1}\not{k}\gamma^{\mu}(1-\gamma_{5}),\\\
C_{2}=&C(p_{1})\not{a}_{2}\not{k}\gamma_{\nu}(1-\gamma_{5})-C(p)\gamma^{\mu}(1-\gamma_{5})\big{(}k_{\mu}a_{2\nu}-k_{\nu}a_{2\mu}\big{)}-\frac{C(p)^{2}}{2}C(p_{1})a^{2}k_{\mu}k_{\nu}\\\
&\times\not{a}_{2}\not{k}\gamma^{\mu}(1-\gamma_{5}).\end{split}$ (14)

The linear combination of the three different quantities in equation (13) can
be transformed by the well-known identities involving ordinary Bessel
functions $J_{s}(z)$ [37]:

$\displaystyle\begin{bmatrix}1\\\ \cos(\phi)\\\ \sin(\phi)\end{bmatrix}\times
e^{-iz\sin(\phi-\phi_{0})}=\sum_{s=-\infty}^{+\infty}\begin{bmatrix}B_{s}(z)\\\
B_{1s}(z)\\\ B_{2s}(z)\end{bmatrix}e^{-is\phi},$ (15)

where

$\displaystyle\begin{split}\begin{bmatrix}B_{s}(z)\\\ B_{1s}(z)\\\
B_{2s}(z)\end{bmatrix}=\begin{bmatrix}J_{s}(z)e^{is\phi_{0}}\\\
\big{(}J_{s+1}(z)e^{i(s+1)\phi_{0}}+J_{s-1}(z)e^{i(s-1)\phi_{0}}\big{)}/2\\\
\big{(}J_{s+1}(z)e^{i(s+1)\phi_{0}}-J_{s-1}(z)e^{i(s-1)\phi_{0}}\big{)}/2i\end{bmatrix},\end{split}$
(16)

where $z$ is the argument of the Bessel functions defined in equation (11) and
$s$, their order, can be interpreted as the number of exchanged photons. Using
these transformations in equation (13) and integrating over $d^{4}x$, the
matrix element $S_{fi}$ becomes

$S_{fi}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\dfrac{i\textsl{g}}{2\sqrt{2}\sqrt{8E_{2}Q_{1}p_{0}V^{3}}}\sum_{s=-\infty}^{\infty}\mathcal{M}^{s}_{fi}(2\pi)^{4}\delta^{4}(p_{2}+q_{1}-q-sk),$
(17)

where the quantity $\mathcal{M}^{s}_{fi}$ is defined by

$\displaystyle\mathcal{M}^{s}_{fi}=\bar{u}(p_{1},s_{1})\Lambda_{s}v(p_{2},s_{2})\varepsilon_{\mu}(p,\lambda),$
(18)

where

$\displaystyle\Lambda_{s}=C_{0}B_{s}(z)+C_{1}B_{1s}(z)+C_{2}B_{2s}(z).$ (19)

Here, we call the quantity $\mathcal{M}^{s}_{fi}$ ”the spinorial part” since
it is the only part of the matrix element which depends on spin. Multiplying
the squared S-matrix element by the density of final states, summing over
spins of leptons and antineutrinos, averaging over the polarization of the
incoming boson, and finally dividing by the time $T$ yields the decay rate

$\displaystyle\Gamma(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\sum_{s=-\infty}^{+\infty}\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell}),$
(20)

where $\Gamma^{s}$, called the photon-number-resolved decay rate, is defined
by

$\displaystyle\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\dfrac{\textsl{g}^{2}}{64p_{0}}\int\dfrac{d^{3}q_{1}}{(2\pi)^{3}Q_{1}}\int\dfrac{d^{3}p_{2}}{(2\pi)^{3}E_{2}}(2\pi)^{4}\delta^{4}(p_{2}+q_{1}-q-sk)|\overline{\mathcal{M}^{s}_{fi}}|^{2},$
(21)

where we have used
$[(2\pi)^{4}\delta^{4}(p_{2}+q_{1}-q-sk)]^{2}=VT(2\pi)^{4}\delta^{4}(p_{2}+q_{1}-q-sk)$,
and

$\displaystyle|\overline{\mathcal{M}^{s}_{fi}}|^{2}=\frac{1}{3}\sum_{\lambda}\sum_{s_{1},s_{2}}|\mathcal{M}^{s}_{fi}|^{2}=\frac{1}{3}\sum_{\lambda}\sum_{s_{1},s_{2}}|\bar{u}(p_{1},s_{1})\Lambda_{s}v(p_{2},s_{2})\varepsilon_{\mu}(p,\lambda)|^{2}.$
(22)

Performing the integration over $d^{3}p_{2}$ and using
$\delta^{4}(p_{2}+q_{1}-q-sk)=\delta^{3}(\textbf{p}_{2}+\textbf{q}_{1}-\textbf{q}-s\textbf{k})\delta(E_{2}+Q_{1}-Q-s\omega)$,
the photon-number-resolved decay rate $\Gamma^{s}$ becomes

$\displaystyle\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\dfrac{\textsl{g}^{2}}{64(2\pi)^{2}p_{0}}\int\dfrac{d^{3}q_{1}}{Q_{1}E_{2}}\delta(E_{2}+Q_{1}-Q-s\omega)|\overline{\mathcal{M}^{s}_{fi}}|^{2},$
(23)

with $\textbf{p}_{2}+\textbf{q}_{1}-\textbf{q}-s\textbf{k}=0$. We choose the
$W^{-}$-boson rest frame in which $Q=q_{0}=M^{*}_{W}$ and $\textbf{q}=0$, then
$\textbf{p}_{2}=s\textbf{k}-\textbf{q}_{1}$. Hence, using
$d^{3}q_{1}=|\textbf{q}_{1}|^{2}d|\textbf{q}_{1}|d\Omega_{\ell}$, we obtain

$\displaystyle\begin{split}\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=&\dfrac{\textsl{g}^{2}}{64(2\pi)^{2}p_{0}}\int\dfrac{|\textbf{q}_{1}|^{2}d|\textbf{q}_{1}|d\Omega_{\ell}}{Q_{1}E_{2}}\delta\Big{(}\sqrt{(s\omega)^{2}+|\textbf{q}_{1}|^{2}-2s\omega|\textbf{q}_{1}|\cos(\theta)}\\\
&+\sqrt{|\textbf{q}_{1}|^{2}+m_{\ell}^{*2}}-M^{*}_{W}-s\omega\Big{)}|\overline{\mathcal{M}^{s}_{fi}}|^{2}.\end{split}$
(24)

The remaining integral over $d|\textbf{q}_{1}|$ can be solved by using the
familiar formula [35]:

$\displaystyle\int
dxf(x)\delta(g(x))=\dfrac{f(x)}{|g^{\prime}(x)|}\bigg{|}_{g(x)=0}.$ (25)

Thus we get

$\displaystyle\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=$
$\displaystyle\dfrac{\textsl{g}^{2}}{64(2\pi)^{2}p_{0}}\int\dfrac{|\textbf{q}_{1}|^{2}|d\Omega_{\ell}}{Q_{1}E_{2}|g^{\prime}(|\textbf{q}_{1}|)|}|\overline{\mathcal{M}^{s}_{fi}}|^{2}.$
(26)

where $\textsl{g}^{2}=8G_{F}M_{W}^{2}/\sqrt{2}$, with $G_{F}=(1.166~{}37\pm
0.000~{}02)\times 10^{-11}~{}\text{MeV}^{-2}$ is the Fermi coupling constant,
and

$\displaystyle
g^{\prime}(|\textbf{q}_{1}|)=\dfrac{|\textbf{q}_{1}|-s\omega\cos(\theta)}{\sqrt{(s\omega)^{2}+|\textbf{q}_{1}|^{2}-2s\omega|\textbf{q}_{1}|\cos(\theta)}}+\dfrac{|\textbf{q}_{1}|}{\sqrt{|\textbf{q}_{1}|^{2}+m_{\ell}^{*2}}}.$
(27)

Using the fact that $d\Omega_{\ell}=\sin(\theta)d\theta d\varphi$ and $\int
d\varphi=2\pi$, the equation (26) can be written as follows:

$\displaystyle\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=$
$\displaystyle\dfrac{\textsl{g}^{2}}{128\pi
p_{0}}\int\dfrac{|\textbf{q}_{1}|^{2}|\sin(\theta)d\theta}{Q_{1}E_{2}|g^{\prime}(|\textbf{q}_{1}|)|}|\overline{\mathcal{M}^{s}_{fi}}|^{2},$
(28)

where the integral over $d\theta$ is performed with the help of numerical
integration.
The term $|\overline{\mathcal{M}^{s}_{fi}}|^{2}$ can be calculated by
converting the sums over the spins into traces as follows:

$\displaystyle|\overline{\mathcal{M}^{s}_{fi}}|^{2}=\frac{1}{3}\bigg{(}-\textsl{g}_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{M_{W}^{2}}\bigg{)}\text{Tr}\big{[}(\not{p}_{1}+m_{\ell})\Lambda_{s}\not{p}_{2}\overline{\Lambda}_{s}\big{]},$
(29)

where

$\displaystyle\begin{split}\overline{\Lambda}_{s}&=\gamma^{0}\Lambda_{s}^{\dagger}\gamma^{0},\\\
&=\overline{C}_{0}B^{*}_{s}(z)+\overline{C}_{1}B^{*}_{1s}(z)+\overline{C}_{2}B^{*}_{2s}(z),\end{split}$
(30)

and

$\begin{split}\overline{C}_{0}=&\gamma^{0}C_{0}^{\dagger}\gamma^{0}=\gamma_{\mu}(1-\gamma_{5})-\frac{C(p)^{2}}{2}a^{2}k_{\mu}k_{\nu}\gamma^{\nu}(1-\gamma_{5}),\\\
\overline{C}_{1}=&\gamma^{0}C_{1}^{\dagger}\gamma^{0}=C(p_{1})\gamma_{\mu}(1-\gamma_{5})\not{k}\not{a}_{1}-C(p)\gamma^{\nu}(1-\gamma_{5})\big{(}k_{\mu}a_{1\nu}-k_{\nu}a_{1\mu}\big{)}\\\
&-\frac{C(p)^{2}}{2}C(p_{1})a^{2}k_{\mu}k_{\nu}\gamma^{\nu}(1-\gamma_{5})\not{k}\not{a}_{1},\\\
\overline{C}_{2}=&\gamma^{0}C_{2}^{\dagger}\gamma^{0}=C(p_{1})\gamma_{\mu}(1-\gamma_{5})\not{k}\not{a}_{2}-C(p)\gamma^{\nu}(1-\gamma_{5})\big{(}k_{\mu}a_{2\nu}-k_{\nu}a_{2\mu}\big{)}\\\
&-\frac{C(p)^{2}}{2}C(p_{1})a^{2}k_{\mu}k_{\nu}\gamma^{\nu}(1-\gamma_{5})\not{k}\not{a}_{2}.\end{split}$
(31)

The trace calculation is performed with the help of FEYNCALC [38, 39, 40]. The
detailed and explicit expression obtained for the averaged squared ”spinorial
part” $|\overline{\mathcal{M}^{s}_{fi}}|^{2}$ is given by

$\begin{split}|\overline{\mathcal{M}^{s}_{fi}}|^{2}=\dfrac{1}{3}\bigg{[}&AJ_{s}^{2}(z)+BJ_{s+1}^{2}(z)+CJ_{s-1}^{2}(z)+DJ_{s}(z)J_{s+1}(z)+EJ_{s}(z)J_{s-1}(z)\\\
&+FJ_{s+1}(z)J_{s-1}(z)\bigg{]},\end{split}$ (32)

where the six coefficients $A$, $B$, $C$, $D$, $E$ and $F$ are explicitly
expressed by

$\begin{split}A=&\dfrac{4}{(k.p)^{2}M_{W}^{2}}\big{[}a^{4}e^{4}(k.p_{1})(k.p_{2})+2a^{2}e^{2}(2(k.p_{1})(k.p_{2})M_{W}^{2}+(k.p)^{2}(p_{1}.p_{2})\\\
&-(k.p)(k.p_{2})(p.p_{1})-(k.p)(k.p_{1})(p.p_{2}))+2(k.p)^{2}(M_{W}^{2}(p_{1}.p_{2})\\\
&+2(p.p_{1})(p.p_{2}))\big{]},\end{split}$ (33)
$\begin{split}B=&\dfrac{-e^{2}}{2(k.p)(k.p_{1})^{3}M_{W}^{2}}\big{[}4(k.p)(k.p_{1})^{2}(-(a_{1}.p_{1})(a_{1}.p_{2})(k.p)+a^{2}((k.p)(p_{1}.p_{2})\\\
&+(k.p_{2})(M_{W}^{2}-(p.p_{1}))+2(k.p)(p.p_{2})+(k.p_{1})(p.p_{2})))-a^{2}(k.p_{1})\\\
&\times(a^{2}e^{2}((k.p)(k.p_{2})p_{0}-(k.p_{1})(k.p_{2})p_{1}^{0}-2(k.p)^{2}E_{2})+2(k.p)((k.p_{1})p_{0}(p_{1}.p_{2})\\\
&+p_{1}^{0}(2(k.p_{2})M_{W}^{2}+4(k.p)(p.p_{2})+(k.p_{1})(p.p_{2})+2(k.p)(k.p_{1})s)))\omega+a^{4}e^{2}\\\
&\times(2(k.p)(k.p_{2})M_{W}^{2}+(k.p_{1})^{2}(p_{1}.p_{2})+(k.p)(4(k.p)+(k.p_{1}))(p.p_{2}))\omega^{2}\\\
&+|\textbf{a}||\textbf{q}_{1}|(k.p_{1})(2|\textbf{a}|(k.p)(2((k.p_{2})M_{W}^{2}+2(k.p)(p.p_{2}))\omega+(k.p_{1})(-(k.p_{2})p_{0}\\\
&+2(k.p(p_{1}^{0}+E_{2})+(p.p_{2})\omega))\cos(\theta)+(a_{1}.p_{1})(k.p_{1})\omega(-2(k.p)p_{0}+a^{2}e^{2}\omega)\\\
&\times\sin(\theta))\big{]},\end{split}$ (34)
$\begin{split}C=&\dfrac{e^{2}}{2(k.p)(k.p_{1})^{3}M_{W}^{2}}\big{[}-4(k.p)(k.p_{1})^{2}(-(a_{1}.p_{1})(a_{1}.p_{2})(k.p)+a^{2}((k.p)(p_{1}.p_{2})\\\
&+(k.p_{2})(M_{W}^{2}-(p.p_{1}))+2(k.p)(p.p_{2})+(k.p_{1})(p.p_{2})))-a^{2}(k.p_{1})(a^{2}e^{2}((k.p)\\\
&\times(k.p_{2})p_{0}-(k.p_{1})(k.p_{2})p_{1}^{0}-2(k.p)^{2}E_{2})+2(k.p)((k.p_{1})p_{0}(p_{1}.p_{2})+p_{1}^{0}\\\
&\times(2(k.p_{2})M_{W}^{2}+4(k.p)(p.p_{2})+(k.p_{1})(p.p_{2})+2(k.p)(k.p_{1})s)))\omega+a^{4}e^{2}(2(k.p)\\\
&\times(k.p_{2})M_{W}^{2}+(k.p_{1})^{2}(p_{1}.p_{2})+(k.p)(4(k.p)+(k.p_{1}))(p.p_{2}))\omega^{2}+|\textbf{a}||\textbf{q}_{1}|(k.p_{1})\\\
&\times(2|\textbf{a}|(k.p)(2((k.p_{2})M_{W}^{2}+2(k.p)(p.p_{2}))\omega+(k.p_{1})(-(k.p_{2})p_{0}+2(k.p)(p_{1}^{0}\\\
&+E_{2})+(p.p_{2})\omega))\cos(\theta)+(a_{1}.p_{1})(k.p_{1})\omega(-2(k.p)p_{0}+a^{2}e^{2}\omega)\sin(\theta)\big{]},\end{split}$
(35)
$\begin{split}D=&\dfrac{e}{(k.p)^{2}(k.p_{1})^{2}M_{W}^{2}}\big{[}2(k.p)(k.p_{1})(-a^{2}e^{2}(k.p_{1})(2(a_{1}.p_{2})(k.p)+(a_{1}.p_{2})(k.p_{1})\\\
&+(a_{1}.p_{1})(k.p_{2}))+2(k.p)(-(a_{1}.p_{2})(k.p_{1})M_{W}^{2}+(a_{1}.p_{1})(k.p_{2})M_{W}^{2}\\\
&+(a_{1}.p_{2})(k.p_{1})(p.p_{1})+(a_{1}.p_{1})(2(k.p)+(k.p_{1}))(p.p_{2})))+|\textbf{a}||\textbf{q}_{1}|\omega\\\
&\times(a^{4}e^{4}((k.p)(2(k.p)+(k.p_{1}))-2(k.p_{1})(k.p_{2}))\omega-4(k.p)^{2}(k.p_{1})(p_{0}(2(p.p_{2})\\\
&+(k.p_{1})s)+M_{W}^{2}(p_{1}^{0}+-s\omega))+2a^{2}e^{2}(k.p)(-(k.p)(k.p_{1})(p_{0}+2(p_{1}^{0}+E_{2}))\\\
&+(k.p)(M_{W}^{2}+2(k.p_{1})s)\omega+(k.p_{1})(2(k.p_{2})p_{0}+2(p.p_{2})\omega+(k.p_{1})s\omega)))\sin(\theta)\big{]},\end{split}$
(36)
$\begin{split}E=&\dfrac{e}{(k.p)^{2}(k.p_{1})^{2}M_{W}^{2}}\big{[}2(k.p)(k.p_{1})(-a^{2}e^{2}(k.p_{1})(2(a_{1}.p_{2})(k.p)+(a_{1}.p_{2})(k.p_{1})\\\
&+(a_{1}.p_{1})(k.p_{2}))+2(k.p)(-(a_{1}.p_{2})(k.p_{1})M_{W}^{2}+(a_{1}.p_{1})(k.p_{2})M_{W}^{2}\\\
&+(a_{1}.p_{2})(k.p_{1})(p.p_{1})+(a_{1}.p_{1})(2(k.p)+(k.p_{1}))(p.p_{2})))+|\textbf{a}||\textbf{q}_{1}|\omega\\\
&\times(-a^{4}e^{4}((k.p)(2(k.p)+(k.p_{1}))-2(k.p_{1})(k.p_{2}))\omega+2a^{2}e^{2}(k.p)((k.p_{1})\\\
&\times(-2(k.p_{2})p_{0}+(k.p)(p_{0}+2(p_{1}^{0}+E_{2})))-((k.p)M_{W}^{2}+2(k.p_{1})(p.p_{2})\\\
&+(k.p_{1})(2(k.p)+(k.p_{1}))s)\omega)+4(k.p)^{2}(k.p_{1})(p_{0}(2(p.p_{2})+(k.p_{1})s)\\\
&+M_{W}^{2}(p_{1}^{0}+E_{2}-s\omega)))\sin(\theta)\big{]},\end{split}$ (37)
$\begin{split}F&=\dfrac{4e^{2}(a_{1}.p_{1})(a_{1}.p_{2})(k.p)}{(k.p_{1})M_{W}^{2}},\end{split}$
(38)

where the different scalar products are evaluated in the rest frame of
$W^{-}$-boson, and

$\begin{split}p_{0}&=M_{W}^{*}+\frac{e^{2}a^{2}\omega}{2(k.p)},\\\
p_{1}^{0}&=Q_{1}+\frac{e^{2}a^{2}\omega}{2(k.p_{1})}.\end{split}$ (39)

The expression of $|\textbf{q}_{1}|$ is obtained by solving numerically the
equation $g(|\textbf{q}_{1}|)=0$, which is a required condition to apply the
familiar formula given in (25). We mention here that the result obtained in
the trace calculation contains the various antisymmetric tensors,
$\epsilon(a,b,c,d)=\epsilon_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}$
for all four-vectors $a,b,c$ and $d$, which arise when four matrices $\gamma$
meet the fifth matrix $\gamma_{5}$ inside the trace
$(\text{Tr}[\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5}]=-4i\epsilon^{\mu\nu\rho\sigma})$.
Because it was calculated analytically and replaced in our computational
program, it does not appear in the result listed above. We give here an
example showing how we calculate these tensors using Einstein’s summation and
the Grozin convention $\epsilon_{0123}=1$,

$\begin{split}\epsilon(a_{1},a_{2},p_{1},p_{2})&=\epsilon_{\mu\nu\rho\sigma}a_{1}^{\mu}a_{2}^{\nu}p_{1}^{\rho}p_{2}^{\sigma},\\\
&=|\text{{a}}|^{2}\big{[}\epsilon_{1203}p_{1}^{0}p_{2}^{3}+\epsilon_{1230}p_{1}^{3}p_{2}^{0}\big{]},\\\
&=|\text{{a}}|^{2}\Big{[}p_{1}^{0}\big{(}s\omega-|\textbf{q}_{1}|\cos(\theta)\big{)}-\Big{(}|\textbf{q}_{1}|\cos(\theta)+\frac{e^{2}a^{2}\omega}{2(k.p_{1})}\Big{)}p_{2}^{0}\Big{]}.\end{split}$
(40)

So far, we have given a detailed theoretical calculation in order to obtain
the expression of the leptonic decay rate of the $W^{-}$-boson in the presence
of an EM field. The most known and important quantity that comes directly
after calculating the decay rate is the lifetime, and since we only took into
account the leptonic decay without the hadronic, we are not allowed to talk
about the lifetime of the boson $W^{-}$, because this quantity requires the
expression of the total decay rate that combines the leptonic and hadronic
$W^{-}$ decays. This is in our perspectives and will be the subject of an
upcoming research in which we aim to study the hadronic decay of the boson
$W^{-}$ in the presence of an EM field, and then we combine the results
obtained with the present leptonic results so that we can explore the effect
of the laser on the lifetime as a very important point that must be addressed.

##  3\. Results and discussion

This section is devoted to the presentation, analysis, and discussion of the
numerical results obtained in relation to the leptonic decay rate of the
$W^{-}$ boson in the presence of a circularly polarized EM field. Through the
following, we will see the effect that the laser may have in terms of its
parameters (both its field strength and frequency) on the leptonic decay rate.
We chose to have the reference origin on the boson $W$ which is at rest before
decay. Let us recall here that the geometry we have worked on inside is a
general geometry whose spherical coordinates are $\theta$ and $\varphi$. The
angle $\varphi$ was chosen to be zero $(\varphi=0^{\circ})$ in all of the
results obtained. All the results below have been obtained for the following
input parameters:

$\begin{split}M_{W}&=80.379~{}\text{GeV},~{}~{}(\text{from
\cite[cite]{[\@@bibref{}{pdg2020}{}{}]}})\\\ m_{e}&=0.511\times
10^{-3}~{}\text{GeV},~{}~{}m_{\mu}=0.1057~{}\text{GeV},~{}~{}m_{\tau}=1.777~{}\text{GeV},\\\
e&=-8.5424546\times 10^{-2}~{}~{}(\text{charge of electron in natural
units}).\end{split}$ (41)

Remember that we have utilized the following unit transformation: in System
International (SI) units, the electric field strength
$\mathcal{E}_{0}[\text{SI}]=4329.0844~{}[\text{V/cm}]$ corresponds, in natural
units (NU), to $\mathcal{E}_{0}[\text{NU}]=1~{}[\text{eV}^{2}]$. The first
thing we are familiarized to checking in such processes occurring within the
EM field is to make sure that the decay rate in the presence of the laser
field is exactly equal to its corresponding one in the absence of the laser
when we take the zero field strength $(\mathcal{E}_{0}=0~{}\text{eV})$ and
without the exchange of any photons $(s=0)$. In this case, the argument of the
Bessel functions is equal to $z=0$. Thus all the terms that contribute to the
”spinorial part” $|\overline{\mathcal{M}^{s}_{fi}}|^{2}$ given in (32) will be
null except for the term multiplied by the coefficient $A$, since $J_{0}(0)=1$
and $J_{1}(0)=J_{-1}(0)=0$. The expression of
$|\overline{\mathcal{M}^{s}_{fi}}|^{2}$ becomes
$|\overline{\mathcal{M}^{s}_{fi}}|^{2}=AJ_{s}^{2}(z)/3$, and if we take
$a^{2}=0$ we get the trace expression of leptonic $W^{-}$ decay in the absence
of the laser field [35]; that is

$|\overline{\mathcal{M}^{laser-
free}_{fi}}|^{2}=\frac{8}{3}\bigg{(}(p_{1}.p_{2})+\frac{2(p.p_{1})(p.p_{2})}{M_{W}^{2}}\bigg{)}.$
(42)

For the other quantities that compose the decay rate in equation (28), we
finds at $\mathcal{E}_{0}=0$ and $s=0$ their expressions in the absence of the
laser field.

Figure 2: Multiphoton decay rate $d\Gamma^{s}/d\theta~{}(W^{-}\rightarrow
e^{-}\bar{\nu}_{e})$ (28) (in units of $10^{6}$) as a function of the number
of photons exchanged $s$ in the rest frame of the $W$-boson, with the
spherical coordinates $\theta=90^{\circ}$ and $\varphi=0^{\circ}$. The laser
field strength and frequency are $\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ and
$\hbar\omega=0.117~{}\text{eV}$ or $\mathcal{E}_{0}=10^{8}~{}\text{V/cm}$ and
$\hbar\omega=1.17~{}\text{eV}$.

Figure 3: The same as figure 5, but $\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ and
$\hbar\omega=1.17~{}\text{eV}$,

Figure 4: The same as figure 5, but $\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$ and
$\hbar\omega=1.17~{}\text{eV}$,

Figure 5: The same as figure 5, but $\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ and
$\hbar\omega=2~{}\text{eV}$.

Figures 5-5 show the multiphoton decay rate $d\Gamma^{s}/d\theta$ (28) for the
process $(W^{-}\rightarrow e^{-}\bar{\nu}_{e})$ versus the net photon number
$s$ transferred between the decaying system and the laser field. In order to
show numerically the effect of field strength and frequency on the phenomenon
of multiphoton energy transfer, we display the s-resolved decay rate at
different field strengths and frequencies for the angle $\theta=90^{\circ}$.
To examine the effect of the laser field strength in the first step, we set,
in figures 5, 5 and 5 the frequency to the value
$\hbar\omega=1.17~{}\text{eV}$ and changed the field strength
$\mathcal{E}_{0}$ so that it is equal to
$\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ in figure 5,
$\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$ in figure 5 and
$\mathcal{E}_{0}=10^{8}~{}\text{V/cm}$ in figure 5. Comparing these three
figures with each other, we can see that as the laser field strength increases
at a fixed frequency, the number of photons exchanged enhances and increases,
and this is evident through the value of the cutoff number in each case. The
cutoff numbers in figures 5, 5 and 5 are, respectively, $20$, $160$ and $1500$
(for the absorption side) and $-20$, $-160$ and $-1500$ (for the emission
side). It can also be observed from these figures that the magnitude of
$d\Gamma^{s}/d\theta$ decreases as the field strength increases. This is the
effect of laser field strength. For the effect of laser frequency, we have
exactly reversed the previous situation. In figures 5, 5 and 5, we fixed the
field strength to the value $\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ and changed
the laser frequency from $\hbar\omega=0.117~{}\text{eV}$ in figure 5 and
$\hbar\omega=1.17~{}\text{eV}$ in figure 5 to the value
$\hbar\omega=2~{}\text{eV}$ in figure 5. Through these figures, we can see
that, unlike the field strength effect, when the laser frequency increases,
the number of exchanged photons decreases, and this is evident by the decrease
of the cutoff number from the value $1500$ at frequency
$\hbar\omega=0.117~{}\text{eV}$ [figure 5] to $10$ at frequency
$\hbar\omega=2~{}\text{eV}$ [figure 5] with a constant field strength equal to
$\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$. In relation to the magnitude of
$d\Gamma^{s}/d\theta$, it is also noted that it increases with the increase of
the laser frequency until reaching, for example, $d\Gamma^{s}/d\theta=17\times
10^{6}~{}\text{eV}$ at frequency $\hbar\omega=2~{}\text{eV}$ and $s=4$ [see
figure 5]. Based on the foregoing, it can be said that for increasing field
strength and fixed frequency, processes involving large numbers of photons
make significant contributions, in contrast to the case of high frequencies
and fixed field strength where only low-photon exchanging processes are
important. This is demonstrated by the formula of the argument
$z=-e(a_{1}.p_{1})/(k.p_{1})\propto\mathcal{E}_{0}/\omega^{2}$ given in (11),
which is responsible for the effects of the laser, as it decreases with higher
frequencies and vice versa, meaning that at lower frequencies a greater number
of photons can be exchanged; as can be deduced from the behavior of the Bessel
function $J_{s}(z)$ and clearly seen in figures 5-5. Thus, for stronger field
strengths or lower frequencies, an increased effect on the leptonic decay rate
can be expected. The heights of the different photon-number peaks are
critically dependent on the values of the ordinary Bessel functions. In all
the previous results presented in figures 5-5, the contributions of the
different $s$-photon processes are cut off at two edges that are symmetric
with respect to $s=0$. It also appears that all the spectra are symmetric
envelopes with respect to both the positive and negative edges, and this
indicates that the photon absorption processes $(s>0)$ are exactly equal to
the photon emission processes $(s<0)$. The cutoff number can be explained by
the well-known properties of the Bessel function, which decreases considerably
when its argument $z$ (equation 11) is approximately equal to its order $s$.
This has already been referred to in refs. [41, 42]. In fact, our goal in
drawing these envelopes in terms of the number of exchanged photons is to
obtain the cutoff numbers at each specified field strength and frequency. The
cutoff number is defined as a fixed number of photons from which the
s-resolved decay rate $d\Gamma^{s}/d\theta$ suddenly falls off
$(d\Gamma^{s}/d\theta=0)$. This number is very important to us because with it
we make sure that the famous Kroll–Watson sum rule [43] is fulfilled. By the
way, and in this context, we mention here that this sum rule in our case can
be formulated mathematically as follows:

$\sum_{s=-\text{cutoff}}^{+\text{cutoff}}\Gamma^{s}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})=\Gamma^{\text{laser-
free}}(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell}),$ (43)

and is achieved when the decay rate in the presence of the laser field tends
to approach the laser-free results by increasing the number of exchanged
photons until they are exactly equal when we reach the cutoff number. In the
next stage, we will try to make sure that this sum rule is respected. Let us
take for example the case presented in figure 5 where the field strength
$\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$ and frequency
$\hbar\omega=1.17~{}\text{eV}$, the cutoff number in this case is equal, as
shown in figure 5, to about $+160$ (for absorption) and $-160$ (for emission).
In reality, this number of exchanged photons cannot be considered a cutoff
number according to the aforementioned definition, because the decay rate
$d\Gamma^{s}/d\theta$ is not exactly equal to $0$; but it is so small that our
numerical program gives it a zero value when drawing the curves. By checking
the values numerically, we found that the decay rate $d\Gamma^{s}/d\theta$ is
zero exactly at $s\geq 450$ ($s\leq-450$). Therefore, the sum rule must be
satisfied essentially at this number of exchanged photons. To prove this, we
plotted the decay rate $\Gamma(W^{-}\rightarrow e^{-}\bar{\nu}_{e})$ (20), at
$\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$ and $\hbar\omega=1.17~{}\text{eV}$,
summed over an increasingly different numbers of exchanged photons and
compared it with the decay rate in the absence of the laser field. The curves
obtained in connection with this are shown in figure 7, where the numbers of
exchanged photons over which we have summed the leptonic decay rate are as
follows: $-N\leq s\leq+N$, where $N=30,~{}60,~{}120,~{}180,~{}200,~{}450$.
Through this figure, it is evident to us that with the increase in the number
of photons exchanged between the decaying system and the laser field, the
effect of the latter on the leptonic decay rate gradually decreases until it
becomes completely absent when the number of exchanged photons is equal to the
cutoff number $(-450\leq s\leq+450$ in this case$)$, and thus we obtain two
identical curves. In other words, we say that the leptonic decay rate summed
over a number of exchanged photons approaches slowly and tends, as the number
of exchanged photons increases, towards its corresponding one in the absence
of the laser until it is exactly equal to it at a specified number of
exchanged photons, which is the cutoff number. This is exactly what we mean by
the sum rule which is, in our case, respected and successfully achieved.

Figure 6: The variation of the summed leptonic decay rate
$\Gamma(W^{-}\rightarrow e^{-}\bar{\nu}_{e})$ (20) (in units of $10^{6}$) with
and without a laser as a function of the angle $\theta$ for various numbers of
photons exchanged. The spherical coordinate $\varphi=0^{\circ}$. The laser
field strength and frequency are $\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$ and
$\hbar\omega=1.17~{}\text{eV}$.

Figure 7: The same as figure 7 but for $\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$
and $\hbar\omega=2~{}\text{eV}$.

The idea of the sum rule was originally conceived when Kroll and Watson
derived, in $1973$, the differential cross-section formula for an elastic
electron scattering process in the presence of a laser when neglecting the
interaction of the atom (scattering potential) with the laser. Within the
framework of the Kroll-Watson approximation, the information obtained from the
laser-assisted leptonic decay rate regarding the decay system is the same as
the information obtained from the decay rate without the laser field. Figure 7
shows the same thing as figure 7, except that in this case the laser field
strength and frequency are, respectively,
$\mathcal{E}_{0}=10^{6}~{}\text{V/cm}$ and $\hbar\omega=2~{}\text{eV}$, which
are the same values used for each of them to draw the envelope in figure 5. As
shown in figure 5, the cutoff number equals $10$ for absorption ($-10$ for
emission). But its real value, from which the s-resolved decay rate is zero,
is $123$. Therefore, we see that the sum rule in figure 7 is fulfilled exactly
at this number of exchanged photons. These are two clear examples that are
sufficient to make sure that the well-known sum rule is respected and
therefore can be served and considered as a proof of the correctness of our
theoretical calculations. In order to realize the size of the laser effect on
the leptonic decay rate and make it more clear, it is convenient to introduce
here a quantity denoted by $R_{with/without}$ and defined as the ratio between
the summed leptonic decay rate in the EM field given in equation (20) and its
corresponding one in the absence of the laser field; that is,

$R_{with/without}=\frac{\Gamma^{with~{}laser}(W^{-}\rightarrow
e^{-}\bar{\nu}_{e})}{\Gamma^{without~{}laser}(W^{-}\rightarrow
e^{-}\bar{\nu}_{e})}.$ (44)

Figure 8: The variation of the ratio $R_{with/without}$ (44) as a function of
the laser field strength for a Nd:YAG laser ($\hbar\omega=1.17~{}\text{eV}$)
and a $\text{CO}_{2}$ laser ($\hbar\omega=0.117~{}\text{eV}$). The number of
exchanged photons summed over it is $-60\leq s\leq+60$.

Figure 9: The variation of the total leptonic decay rate (sum over leptons) as
a function of the laser field strength for a Nd:YAG laser
($\hbar\omega=1.17~{}\text{eV}$) and a $\text{CO}_{2}$ laser
($\hbar\omega=0.117~{}\text{eV}$). The number of exchanged photons summed over
it is $-60\leq s\leq+60$.

In figures 9 and 9, we aim to illustrate the effect of laser frequency on both
the ratio $R_{with/without}$ (44) and the total leptonic decay rate, which is
equal to the sum of the electron, muon and tau decay rates. To do this, we
plotted their changes in terms of laser field strength for two different
frequencies, $\text{CO}_{2}$ laser frequency ($\hbar\omega=0.117~{}\text{eV}$)
and Nd:YAG laser frequency ($\hbar\omega=1.17~{}\text{eV}$). Based on figure
9, it can be seen that the $\text{CO}_{2}$ laser affects the
$R_{with/without}$ ratio only when the field strength is above the threshold
of $10^{3}~{}\text{V/cm}$, while the effect of the Nd:YAG laser on the ratio
starts only when the field strength is above $10^{5}~{}\text{V/cm}$; so the
$\text{CO}_{2}$ laser, whose frequency is lower than the Nd:YAG one, can
affect the $R_{with/without}$ ratio more than the Nd:YAG laser even at low
field strengths. For example, the $R_{with/without}$ value at
$\mathcal{E}_{0}=10^{5}~{}\text{V/cm}$ in the presence of the $\text{CO}_{2}$
laser $(R_{with/without}=0.4)$ does not attained in the presence of the Nd:YAG
laser except when $\mathcal{E}_{0}=10^{7}~{}\text{V/cm}$. Thus, having a high-
frequency laser effect requires the availability of very high field strengths.
The same can be observed completely with regard to the effect of the laser
frequency on the total leptonic decay rate shown in figure 9. It can be said
that the effect of the laser decreases at high frequency or we say that the
high-frequency laser has little effect compared to the low-frequency laser.

Figure 10: The same as in figure 9, but for different numbers of exchanged
photons and fixed laser frequency $\hbar\omega=1.17~{}\text{eV}$.

Figure 11: The same as in figure 9, but for different numbers of exchanged
photons and fixed laser frequency $\hbar\omega=1.17~{}\text{eV}$.

In figures 11 and 11, we have drawn the same two quantities as in figures 9
and 9 at different numbers of photons exchanged, but now we have set the laser
frequency to $\hbar\omega=1.17~{}\text{eV}$. This is exactly the same as we
did when we wanted to check the sum rule. We will show here via figures 11 and
11 how the sum rule is also respected in both the $R_{with/without}$ ratio and
the total leptonic decay rate. We have summed over $100$ and three cutoff
numbers $170$, $450$ and $1500$. We notice that at small field strengths all
the curves, regardless of the number of exchanged photons, are coincident and
have a constant value of $1$ with respect to $R_{with/without}$ in figure 11
and a value $6.81192\times 10^{8}~{}\text{eV}$ for the total leptonic decay
rate in the absence of the laser in figure 11. Above the strength
$10^{5}~{}\text{V/cm}$, we notice that both quantities, for each specified
number of exchanged photons, decrease nonlinearly with increasing field
strength. It appears clearly to us that with the increase in the number of
exchanged photons, the laser’s influence whether on the ratio or the total
decay rate decreases until it becomes completely absent when we reach the
cutoff $-1500\leq s\leq+1500$ at which the sum rule is fulfilled, and thus we
obtain a constant horizontal curve at the field strengths between $10$ and
$10^{8}~{}\text{V/cm}$. The same thing we have recently examined and verified
concerning the lifetime in the case of the pion and $Z$-boson decays [33, 34],
and we will confirm it in a future study regarding the lifetime of the
$W$-boson. The final result that we can deduce from the above is that the
effect of the laser on the decay rate remains present as long as the number of
exchanged photons that we sum over it does not reach the cutoff number at
which the sum rule is fulfilled.

Figure 12: The 3-dimensional contour-plots of the same quantity as in figure 9
as a function of the laser field strength $\mathcal{E}_{0}$ and frequency
$\omega$ for $-20\leq s\leq+20$ exchanged photons.

Figure 13: The 3-dimensional contour-plots of the same quantity as in figure 9
as a function of the laser field strength $\mathcal{E}_{0}$ and frequency
$\omega$ for $-20\leq s\leq+20$ exchanged photons.

For illustration purposes, similarly to the 2-dimensional plot, the contour-
plots in figures 13 and 13 show more information on the overall variation and
shape of the ratio $R_{with/without}$ and total leptonic decay rate with
respect to both laser field strength and frequency for an exchange of $-20\leq
s\leq+20$ photons. This type of graphics is only a kind of 3-dimensional
representation, in which one studies a certain quantity and its behavior by
changing simultaneously two variables on a 2-dimensional plane and using
colors to represent the variations of the quantity. Apparently, these two
figures 13 and 13 are divided into many colored regions separated by contours,
where each contour separating two different regions corresponds to a value on
the bar legend. Looking at these two figures, we notice that they have the
same feature to some extent, which indicates that the two quantities
represented there change in the same way in terms of field strength and
frequency. For the variation with respect to the laser frequency, we observe
that the effect of the laser field on the two quantities decreases at high
frequencies, while it becomes significant with increasing laser field strength
at each specific frequency. This is completely in agreement with everything we
said above. The important question now is how can we explain the great effect
that the EM field has at strong field strengths. From figure 11, it appears
that the value reached by the ratio $R_{with/without}$, when the number of
exchanged photons is $-100\leq s\leq+100$ and at
$\mathcal{E}_{0}=10^{8}~{}\text{V/cm}$, is $R_{with/without}=0.07$. This means
that, the laser-assisted leptonic decay rate is $93\%$ lower than its
corresponding one in the absence of the laser field. This is a significant
effect of the laser on the decay rate. A computer calculation shows, when
summing over $-10\leq s\leq+10$, that the minimum value reached by the
quantity $R_{with/without}$ is $R_{with/without}^{min}=2.11603\times 10^{-10}$
at field strength $\mathcal{E}_{0}=10^{16}~{}\text{V/cm}$ (Schwinger limit),
meaning that the quantity $R_{with/without}$ is almost zero at this field
strength. This means that the decay rate in the presence of the laser is
negligible compared to its equivalent one in the absence of the laser. Since
the decay rate reflects, in its origin, the probability of occurrence of the
decay, we conclude that the presence of the laser has considerably reduced the
probability of occurrence of the leptonic $W^{-}$-boson decay if we do not
exaggerate and say that it can suppress or prevent its decay. To our
knowledge, there are two explanations for this behavior of the $W$ boson in
the presence of the laser. The first is that the laser has suppressed these
leptonic channels to then completely allow other hadronic ones, whether they
are allowed in the absence of laser, such as
$W^{-}\rightarrow\bar{u}d,~{}\bar{c}s$, or not, such as
$W^{-}\rightarrow\bar{t}b$, especially since the $W$ boson gains in the EM
field an excess of mass (effective mass) which increases with increasing field
strength (e.g. $M_{W}^{*}=186.825~{}\text{GeV}$ at
$\mathcal{E}_{0}=10^{16}~{}\text{V/cm}$ and $\hbar\omega=1.17~{}\text{eV}$).
This first explanation can only be confirmed by studying the $W^{-}$ hadronic
decay in the presence of the EM field. The second explanation is to attribute
this behavior to the so-called quantum Zeno effect. The quantum Zeno effect
can generally be defined by the inhibition of transitions between quantum
states by repeated measurements of the state. This effect has attracted a lot
of attention over the past years [44, 45, 46, 47, 48, 49]. In $1977$, Misra
and Sudarshan showed, based on quantum measurement theory, that an unstable
particle would never decay when observed continuously [44]. They were the
first to give the name to this effect. In our case, the process of the $W$
leptonic decay is observed by inserting it into an EM field. Thus the laser as
an external field here plays the role of the measurement instrument. In
addition, the lifetime of the $W$ boson will certainly, as a result, be
extended and long [45]. In this context, we want to link our results and
compare them with the results obtained by Kurilin in $2004$ when he calculated
the partial width of the $W$-boson leptonic decay in a strong EM crossed field
[26]. He studies the changes of the decay width in terms of a characteristic
parameter $\varkappa=eM_{W}^{-3}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}}$, where
$F_{\mu\nu}$ is the EM tensor. He has revealed that the partial decay width is
a non-monotonic function of the field-strength parameter $\varkappa$. In
superstrong fields $(\varkappa\gg 1)$, the partial width is greater than the
corresponding one in a vacuum by a factor of a few tens, which means that the
laser-assisted leptonic decay occurs faster than that in the absence of a
laser. This result he reached does not constitute any contradiction with what
we obtained. In contrast to the Zeno effect, there is an opposite effect to it
called the quantum anti-Zeno effect, where the decay of some quantum systems
from an initial state might be accelerated by frequent interrogations when the
system remains in that state [50, 51, 52]. This effect was discovered by
Kofman and Kurizki [53, 54]. However, these two results, despite the
difference between them due to the different approach followed in each of
them, can be widely accepted with reference to the quantum Zeno effect or the
anti-Zeno effect, depending on whether the decay is suppressed or enhanced.

##  4\. Conclusion

A detailed theoretical calculation of the leptonic $W^{-}$-boson decay
$(W^{-}\rightarrow\ell^{-}\bar{\nu}_{\ell})$ in the presence of a circularly
polarized electromagnetic field has been performed and an exact expression for
the laser-assisted leptonic decay rate has been analytically derived. Our
obtained numerical results show that the probability of decay to occur in the
presence of a strong electromagnetic field is very small and almost non-
existent, as if the laser at strong strengths and appropriate frequency is
trying to stop the boson and delay its decay. This behavior can be considered
as a result of the so-called quantum Zeno effect. As a consequence, the
lifetime of the $W^{-}$-boson will also be affected by the laser and it will
be long and extended, and it is not excluded that the branching ratios are
also altered by the presence of the laser. In any case, this issue appears to
be of particular interest and deserves a dedicated investigation which we will
undertake in a forthcoming work to study the hadronic decay of the
$W^{-}$-boson.

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# Six years of luminous X-ray emission from the strongly interacting type-Ib
SN 2014C captured by Chandra and NuSTAR

D. Brethauer  Department of Physics and Astronomy and CIERA, Northwestern
University, 2145 Sheridan Road, Evanston, IL 60208, USA  R. Margutti
Department of Physics and Astronomy and CIERA, Northwestern University, 2145
Sheridan Road, Evanston, IL 60208, USA  D. Milisavljevic Department of Physics
and Astronomy, Purdue University, 525 Northwestern Ave., West Lafayette, IN.
47907, USA  M. Bietenholz Hartebeesthoek Radio Astronomy Observatory, PO Box
443, Krugersdorp, 1740, South Africa  Department of Physics and Astronomy,
York University, Toronto, M3J 1P3, Ontario, Canada

###### Abstract

We present the first coordinated soft and hard 0.3-80 keV X-ray campaign of
the extragalactic supernova SN 2014C in the first $\sim$2307 d of its
evolution. SN 2014C initially appeared to be an ordinary type Ib explosion but
evolved into a strongly-interacting hydrogen-rich type IIn SN over $\sim
1\,\rm{yr}$. We observed signatures of interaction with a dense medium across
the X-ray spectrum, which revealed the presence of a
$\sim\,1-2\,\rm{M}_{\odot}$ shell of material at $\sim 6\times
10^{16}\,\rm{cm}$ from the progenitor. This finding challenges current
understanding of hydrogen-poor core-collapse progenitor evolution. Potential
scenarios to interpret these observations include (i) the ejection of the
hydrogen envelope by the progenitor star in the centuries prior to the
explosion; (ii) interaction of the fast Wolf-Rayet (WR) star wind with the
slow, dense wind of the Red Super Giant (RSG) phase, with an anomalously short
WR phase.

supernovae: specific (SN 2014C)

##  1 Introduction

SN 2014C evolved spectroscopically from a type Ib to a type IIn supernova (SN)
as a result of the shock interaction with a H-rich circumstellar material
(CSM) shell (Milisavljevic et al., 2015). The presence of a dense H-rich
medium in close proximity to a H-poor core-collapse SN has been inferred in a
number of cases, including normal type Ib/Ic SNe (e.g., 2001em, 2004dk, Chugai
& Chevalier, 2006; Mauerhan et al., 2018) and superluminous SNe (e.g.,
iPTF13ehe, iPTF15esb, iPTF16bad, Yan et al., 2017). This phenomenology might
require enhanced mass-loss $\lesssim\,2000\,\rm{yrs}$ before explosion due to
physical mechanisms that deviate from the traditional picture of line-driven
winds in single massive stars (e.g. wave-driven mass-loss, envelope ejections
driven by nuclear burning instabilities or binary interaction), or,
alternatively, unusually short WR phases of evolution (e.g., Smith, 2014).

Here we update the analysis of Margutti et al. (2017) and present new broad-
band X-ray observations of SN 2014C up to $t=2307\,\rm{d}$ post explosion,
enabling unprecedented insight into time-dependent mass-loss mechanisms. We
study these mechanisms by using the SN shockwave as a probe of the environment
as described in Chevalier & Fransson (2017).

##  2 Observations and Analysis

We observed SN 2014C with _Chandra_ and _NuSTAR_ at $t\,=\,396-2307\,\rm{d}$
past explosion (December 30th 2013) as part of our joint monitoring program.
Each _Chandra_ observation was paired with a _NuSTAR_ observation taken within
$\Delta\,t/t<10^{-3}$ to provide a complete view of the hard and soft X-ray
emission. We reprocessed the _Chandra_ and _NuSTAR_ data following standard
procedures within CIAO and NuSTARDAS. We find evidence for bright X-ray
emission in each of our exposures both at soft and hard X-rays energies.

For _Chandra_ observations, we extracted a spectrum from a circular region of
1.5″ centered at the location of SN 2014C. We followed a similar procedure for
_NuSTAR_ exposures, but used a 1′ region. We performed a joint spectral fit of
each _CXO+NuSTAR_ observation with an absorbed thermal bremsstrahlung model
(tbabs*ztbabs*bremss) within XSPEC. We accounted for contamination from
unrelated sources inside the _NuSTAR_ PSF following a similar method as
Margutti et al. (2017). _NuSTAR_ observations are heavily dominated by
background emission at energies $>40$ keV and we thus restricted our spectral
analysis to the 3-40 keV range. We adopted a Galactic neutral hydrogen column
density $NH_{\rm MW}\,=\,6.14\,\times 10^{20}\,\rm{cm}^{-2}$ (Kalberla et al.,
2005) in the direction of SN 2014C and a distance of 14.7 Mpc (Freedman et
al., 2001).

Figure 1: _Chandra_ (black circles) and _NuSTAR_ (magenta squares) broadband
(0.5-40 keV) X-ray spectra of SN 2014C acquired at $t=396-2307\,\rm{d}$ post
explosion. Thick blue line: best fitting absorbed bremsstrahlung model. Black
rectangles: excess of emission with respect to this model that we associate
with K$\alpha$ line transitions in H-like or He-like Fe atoms.

##  3 Discussion and Conclusions

A SN shock expanding into the CSM accelerates particles that cool down in the
X-rays by radiating synchrotron and bremsstrahlung emission (Chevalier &
Fransson, 2017). For shocks propagating in high-density media, like in SN
2014C, thermal bremsstrahlung emission dominates over the synchrotron in
X-rays. The observed thermal bremsstrahlung spectrum depends on the emission
temperature $T$, (which directly probes the shock velocity) and on the
electron density of the radiating medium (and hence on the overall density
$\rho$). Additionally, the presence of dense local neutral matter can lead to
photo-electric absorption that leaves a clear imprint at energies $\lesssim
1\,\rm{keV}$ (Fig. 1). We quantify the local absorption with an intrinsic
column density $NH_{\rm int}$. By fitting the thermal bremsstrahlung spectra
of SN 2014C over $\sim 6\,\rm{yrs}$ of evolution, we constrained $T(t)$,
$NH_{\rm{int}}(t)$, and the density profile encountered by the blastwave
$\rho(r)$.

We find $kT$ peaks at $\sim$23 keV at $t\sim 500\,\rm{d}$, and cools as
$T(t)\propto\,t^{-0.5}$. $NH_{\rm{int}}$ also declines with time as the shock
propagates through the dense CSM shell. Specifically, we find
$NH_{\rm{int}}(t)\propto\,t^{-1.4}$ starting from $\approx\,3\times
10^{22}\,\rm{cm^{-2}}$ at $t\sim\,400\,\rm{d}$. The resulting unabsorbed
broad-band 0.3-100 keV X-ray light-curve peaks at $L_{x}\approx 5.5\times
10^{40}\,\rm{erg\,s^{-1}}$ at $t\sim 1000\,\rm{d}$, followed by
$L_{x}(t)\propto\,t^{-1}$, which is significantly less steep than what is
expected from adiabatic expansion. Finally, we note the presence of an excess
of emission at $\sim$6.7 keV (Fig. 1) that we interpret as emission from
H-like or He-like Fe atoms.

Combining the spectral parameters inferred above with direct measurements of
the forward shock radius obtained from VLBI observations of SN 2014C by
Bietenholz et al. (2017) and Bietenholz et al., (_submitted_), we constrain
the properties of its environment. We find that the progenitor star is
surrounded by a low-density cavity (particle density $n<\,100\,\rm{cm}^{-3}$,
Margutti et al. 2017) extending to $\sim\,6\,\times\,10^{16}\,\rm{cm}$. At
$\sim\,6\,\times\,10^{16}\,\rm{cm}$, the blastwave encountered a dense shell
of material with density $n\sim\,2\,\times\,10^{6}\,\rm{cm}^{-3}$ extending to
at least $8\,\times\,10^{16}\,\rm{cm}$ and with a density profile beyond the
shell $\rho(r)\,\propto\,r^{-2.5}$. Our observations sample the medium out to
$r\sim 2.5\times\,10^{17}\,\rm{cm}$. Under the assumption of spherical
symmetry supported by recent VLBI observations of Bietenholz et al,
(_submitted_) (see however Milisavljevic et al. 2015), the total shell mass is
$\sim\,1-2\,\rm{M}_{\odot}$ considering realistic filling factors.

Optical spectroscopy (Milisavljevic et al., 2015) indicates the CSM shell is
H-rich, but it is unclear how this CSM shell was produced at such close
distances from the explosion site. Viable scenarios fall under two broad
categories. A first class of models does _not_ require erratic mass-loss.
Rather, the observed CSM shell is the product of the interaction of a fast
wind with a slow dense wind from the previous phase of stellar evolution, e.g.
the transition from RSG to the WR phase (a known mechanism able to produce
“bubbles” around Galactic WRs at typical distances $\sim$ tens of pc;
significantly larger than our inferred shell radius, e.g., Marston 1997). As a
result, the phenomenology of SN 2014C does require an anomalously short WR
phase of $\lesssim$ 1700 yrs.

A second class of models invoke the ejection of the progenitor’s H-rich
envelope as a result of various mechanisms including binary interaction
(Margutti et al. 2017; Sun et al. 2020 for the specific case of SN 2014C),
wave-driven mass-loss (Quataert & Shiode, 2012; Shiode & Quataert, 2014;
Fuller, 2017; Fuller & Ro, 2018; Wu & Fuller, 2020), or nuclear burning
instabilities similar to those during O and Ne burning phases (Arnett &
Meakin, 2011; Shiode et al., 2013; Smith & Arnett, 2014).

Studies of statistical samples of SN 2014C-like events across the
electromagnetic spectrum hold the key to constraining the physics of the most
spectacular mass-loss histories of evolved massive stars.

We are grateful to the entire _CXO_ and _NuSTAR_ teams for carrying out our
observations, to the Office of Undergraduate Research at Northwestern
University and the Illinois Space Grant for funding this research.

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