# Cosmological radiation density with non-standard neutrino-electron
interactions

Pablo Martínez-Miravé  Departament de Física Teórica, Universitat de València,
and Instituto de Física Corpuscular, CSIC-Universitat de València, 46980
Paterna, Spain<EMAIL_ADDRESS>
###### Abstract

Non-standard interactions (NSI) between neutrinos and electrons can
significantly modify the decoupling of neutrinos from the plasma. These
interactions have two effects on the overall picture: (i) they alter neutrino
oscillations though matter effects and (ii) they modify the scattering and
annihilation processes involving neutrinos and electrons and positrons. We
study the role of non-universal and flavour-changing NSI in the decoupling and
how they impact the determination of the effective number of neutrinos,
$N_{\rm eff}$. We examine the degeneracies between NSI parameters and we
compare the expected sensitivity from future cosmological surveys with the
current limits from terrestrial experiments. We outline the complementarity
between both approaches.

##  1 Introduction

The phenomenon of neutrino oscillations is a clear indication that there is
physics beyond the Standard Model (SM) in the neutrino sector. The
measurements of flavour oscillations performed using solar, reactor,
atmospheric and accelerator neutrinos are well understood in the three
neutrino framework with non-zero masses and mixing (see for instance [1]). In
many theoretical extensions of the SM, including neutrino mass models, the
existence of additional interactions among neutrinos and with other fermions
are predicted. There are two main consequences of these non-standard
interactions (NSI). In the first place, the propagation of neutrinos in a
medium is modified, as it happens due to electroweak interactions.
Consequently, NSI give rise to an effective potential that can alter neutrino
oscillations. Secondly, these interactions would also give rise to
distinguishable signatures in scattering experiments. This scenario has been
extensively studied and since there is no evidence of non-zero NSI, multiple
constraints have been derived. Non-standard interactions are also expected to
play a role in astrophysical processes as well as in cosmology. In these
proceedings, we focus on the latter and in particular, on the impact of NSI in
the process of neutrino decoupling.

It is well-known that when the temperature fell below the MeV scale, the weak
interactions responsible for keeping neutrinos in thermal equilibrium with the
plasma stopped being effective. As a consequence, neutrinos decoupled and
formed the cosmic neutrino background. This complicated process has been
studied with great detail in the framework of the Standard Model, taking into
account all neutrino interactions and finite-temperature corrections to
Quantum Electrodynamics (QED). The effect of neutrino oscillations has also
been included in such calculations [2, 3].

After the epoch of electron-positron annihilation, the radiation density of
the Universe can be expressed as

$\rho_{\rm
rad}=\rho_{\gamma}\Bigg{\\{}1+\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{\rm
eff}\Bigg{\\}}\,,$ (1)

where we have introduced the effective number of neutrinos, $N_{\rm eff}$, as
a parameter that quantifies the ratio between the energy density in form of
neutrinos and photons. According to the latest calculations, the value of the
effective number of neutrinos is found to be $3.0440$ [2, 3]. Nonetheless,
deviations from this value are expected in the presence of additional
relativistic degrees of freedom or new physics in the neutrino sector.

From a combination of the measurements of the anisotropies in the cosmic
microwave background (CMB) performed by Planck with other cosmological
datasets, one determines $N_{\rm eff}=2.99^{+0.34}_{-0.33}$ at $95\%$ C.L.
[4]. Such value already constrains significantly the possibility of having new
degrees of freedom. In the light of the forecasted sensitivity of forthcoming
CMB observatories such as the CMB-S4 project ($\sigma(N_{\rm eff})=0.02-0.03$)
[5] and the Simons Observatory ($\sigma(N_{\rm eff})=0.05-0.07$) [6], we study
the impact of NSI in the decoupling of neutrinos and in the determination of
$N_{\rm eff}$, extending previous analysis [7, 8].

##  2 Non-standard interactions and current bounds

In this work, we only consider neutral current non-standard interactions (NC-
NSI) between neutrinos and electrons. The reason for this choice is twofold.
Firstly, constraints on NC-NSI are considerably weaker than the charged
current ones. Secondly, we are interested in the the decoupling of relic
neutrinos, fo which only electron-neutrino NSI are estimated to be relevant.
The effective Lagrangian parametrising the interactions between neutrinos and
electrons in this scenario is

$\displaystyle\mathcal{L}_{\nu
e}=-2\sqrt{2}G_{F}\left[\left(\bar{\nu}_{e}\gamma^{\mu}P_{L}e\right)\left(\bar{e}\gamma_{\mu}P_{L}\nu_{e}\right)+\sum_{\alpha,X}g_{X}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\alpha}\right)\left(\bar{e}\gamma_{\mu}P_{X}e\right)\right.$
$\displaystyle+\left.\sum_{\alpha,\beta,X}\varepsilon_{\alpha\beta}^{X}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta}\right)\left(\bar{e}\gamma_{\mu}P_{L}e\right)\right]\,,$
(2)

where the first two terms account for charged current and neutral current
interactions in the SM, respectively, and parametrises NC-NSI. In the
effective Lagrangian, $G_{F}$ is the Fermi constant, greek indices indicate
the three different neutrino flavours and $X=\\{L\,,R\\}$, so that $P_{X}$ are
the left-handed and right-handed chiral projectors. The strength of the
neutral current interactions in the Standard Model depends on the parameters
$g_{L}=\sin^{2}\theta_{W}-1/2$ and $g_{R}=\sin^{2}\theta_{W}$, where
$\theta_{W}$ is the weak angle. In the case of NSI, their strength is
determined by the dimensionless coefficients $\varepsilon_{\alpha\beta}^{X}$.

Non-standard interactions can be classified in two categories. We refer to
$\varepsilon_{\alpha\beta}^{X}$, with $\beta\neq\alpha$, as flavour-changing
NSI, since for non-zero values of these parameters, lepton flavour symmetry is
broken. The coefficients $\varepsilon_{\alpha\alpha}^{X}$ account for non-
universal NSI, since in case the difference between two of them is different
from zero, $\varepsilon_{\alpha\alpha}^{X}-\varepsilon_{\beta\beta}^{X}\neq
0$, lepton flavour universality does not hold.

Current limits on non-standard interactions come from a variety of terrestrial
experiments. They rely on the fact that NSI modify the cross-section measured
at scattering experiments. They also affect neutrino propagation (through
matter effects) so that they can be probed in oscillation experiments. For a
recent review on the existing bounds, see [9].

##  3 Non-standard interactions and Neff - what to expect

Non-standard interactions affect the thermalisation of neutrinos in two
different ways. In the first place, the modify neutrino oscillations through
matter effects. Since the change on $N_{\rm eff}$ arising from neutrino
oscillations is known to be of the order of 0.001 (see [3]), we expect that
the modification introduced in flavour oscillations by NSI will be negligible.
Nonetheless, they are included in our calculations and we will discuss them in
the following sections. The second modification in the picture of neutrino
decoupling is due to the changes in the collision integrals encoding the
interactions involving neutrinos and electrons/positrons. The scattering and
annihilation processes involved are proportional to the coefficients
$g_{L}^{2}$, $g_{R}^{2}$ and the combination $g_{L}g_{R}$. The presence of NSI
introduces the following shifts in the coefficients:

$\displaystyle
g_{L}^{2}\longrightarrow\left(g_{L}+\varepsilon_{\alpha\alpha}^{L}\right)^{2}+\sum_{\beta\neq\alpha}|\varepsilon_{\alpha\beta}^{L}|^{2}$
(3) $\displaystyle
g_{R}^{2}\longrightarrow\left(g_{R}+\varepsilon_{\alpha\alpha}^{R}\right)^{2}+\sum_{\beta\neq\alpha}|\varepsilon_{\alpha\beta}^{R}|^{2}$
(4) $\displaystyle
g_{L}g_{R}\longrightarrow\left(g_{L}+\varepsilon_{\alpha\alpha}^{L}\right)\left(g_{R}+\varepsilon_{\alpha\alpha}^{R}\right)+\sum_{\beta\neq\alpha}|\varepsilon_{\alpha\beta}^{L}||\varepsilon_{\alpha\beta}^{R}|.$
(5)

We use the publicly available code ForEPiaNO
111https://bitbucket.org/ahep_cosmo/fortepiano_public [10] to compute the
evolution of the density matrix $\varrho$ according to

$\frac{d}{dx}\varrho(y)=\sqrt{\frac{3M_{\rm
Pl}^{2}}{8\pi\rho}}\Bigg{\\{}\frac{-ix^{3}}{m_{e}^{3}}\left[\frac{\mathbb{M}_{F}}{2y}-\frac{2\sqrt{2}G_{F}ym_{e}^{2}}{x^{6}}\left(\frac{\mathbb{E}_{l}+\mathbb{P}_{l}}{m_{W}^{2}}+\frac{4\mathbb{E_{\nu}}}{3m_{Z}^{2}}\right),\varrho\right]+\frac{m_{e}^{3}}{x^{4}}\mathcal{I}(\varrho)\Bigg{\\}}$
(6)

where we have introduced the comoving variables $x=m_{e}a$, $y=pa$ and
$z=T_{\gamma}a$, which depend on the electron mass $m_{e}$, the neutrino
momentum $p$, the photon temperature $T_{\gamma}$, and the scale factor $a$.
In the evolution equation above, $M_{\rm Pl}$ denotes the Planck mass, $m_{Z}$
and $m_{W}$ are the masses of the Z and W bosons respectively and $\rho$ is
the total energy density of the universe. In addition, $\mathbb{M}_{F}$ is the
neutrino mass matrix in the flavour basis, $\mathbb{E}_{l}$, $\mathbb{P}_{l}$
and $\mathbb{E}_{\nu}$ are the energy density and pressure of charged leptons
and neutrinos. Finally, $\mathcal{I}(\varrho)$ accounts for the collision
integrals. For a more detailed discussion of the formalism and the procedure
employed to compute the evolution of the density matrix, we refer the reader
to Refs. [3, 10]. In our calculations we take the oscillation parameters from
[1] and adopt the same numerical settings described in [11], where it was
shown that the contributions from muons can be safely ignored.

We first analyse the effect on non-universal NSI. In the left panel of Figure
1, we show how $N_{\rm eff}$ varies in the presence of non-zero
$\varepsilon^{L}_{ee}$ and $\varepsilon^{R}_{ee}$. One can see that for
negative values of $\varepsilon^{L}_{ee}$, $N_{\rm eff}$ is found to be
smaller than in the Standard Model. A minimum is found for
$\varepsilon^{L}_{ee}=-1/2-\sin^{2}\theta_{W}$, when the shifted couplings in
equations (3) and (5) have a minimum. In the case of
$\varepsilon^{L}_{\tau\tau}$, the minimum is found, as expected, for
$\varepsilon^{L}_{\tau\tau}=1/2-\sin^{2}\theta_{W}$, following the same
argument.

Regarding flavour-changing NSI, one can see their impact on $N_{\rm eff}$ for
the particular case of $\varepsilon^{L}_{e\tau}$ in the right panel of Figure
1. Such NSI lead to higher value of $N_{\rm eff}$ due to the fact that they
increase the strength of the interactions between neutrinos and electrons, as
it can be seen in equations (3) to (5). Moreover, the sign of the NSI
parameter is expected to be irrelevant so the values of $N_{\rm eff}$ are
expected to be symmetric with respect to $\varepsilon^{X}_{\alpha\beta}=0$.
This is confirmed by the numerical calculations performed and shown in Figure
1.

Figure 1: Values of $N_{\rm eff}$ in the presence of non-standard
interactions. The left panel corresponds to non-universal NSI, with the effect
of a non-zero $\varepsilon^{L}_{ee}$ in purple and that of a non-zero
$\varepsilon^{L}_{\tau\tau}$ in pink. The right panel corresponds to flavour
changing NSI, in particular, a non-zero $\varepsilon^{L}_{e\tau}$. The value
of $N_{\rm eff}=3.044$ expected in the absence of NSI is indicated by the
black dashed line. The horizontal grey-shaded region corresponds to $\pm 0.02$
which is the expected sensitivity from future cosmological observations. The
vertical coloured bands are the current terrestrial limits at 90$\%$ C.L. Both
panels are adapted from Figures in [11] and reproduced with the authorisation
of the original authors.

The numerical calculations performed and shown in Figure 1 validate the
hypothesis that the main impact of NSI in the decoupling process is due to the
changes in the scattering and in the annihilation between neutrinos and
electrons. Besides that, in Ref. [11], the contribution of NSI trough matter
effects in oscillations to the value of $N_{\rm eff}$ was shown to be three
orders of magnitude smaller than the contribution from the scattering and
annihilation.

It is also interesting to explore the possible interplay between pairs of NSI
parameters. In Figure 2 we show the isocountours of $N_{\rm eff}$ when two NSI
parameters are allowed to vary simultaneously. Notice that the contours of
equal $N_{\rm eff}$ are ellipses in parameters space. Again, this can be well
understood in the light of the shitfs introduced in the coupling by NSI
parameters. Figure 2 also proves that an analysis varying two or more
parameters at the same time is feasible, whereas many of the here presented
terrestrial bounds only allowed for a non-zero NSI parameter at a time.

Figure 2: Isocontours of $N_{\rm eff}$ when two NSI parameters are allowed to
vary simultaneously. The left panel shows the interplay between
$\varepsilon^{L}_{ee}$ and $\varepsilon^{L}_{e\tau}$, and the right panel
shows the combined effect of $\varepsilon^{L}_{\tau\tau}$ and
$\varepsilon^{L}_{e\tau}$. The white point corresponds to the value of $N_{\rm
eff}=3.044$ in the absence of NSI. The white shaded bands are the 95$\%$ C.L.
limits derived in terrestrial experiments varying one parameter at a time.
Both panels are adapted from Figures in [11] and reproduced with the
authorisation of the original authors.

Notice that although the expected constraints from future cosmological
observations are not very competitive, they would provide an independent
confirmation of the terrestrial constraints. In particular, it is in general
complicated to constrain some sets of parameters simultaneously from
terrestrial experiments, for instance $\varepsilon^{X}_{\tau\tau}$ and
$\varepsilon^{X}_{e\tau}$, whereas we have shown that it is feasible using
comsological data.

##  4 Discussion

In these proceedings, we have reported on a precision calculation of the
process of neutrino decoupling in the presence of non-standard interactions
between neutrinos and electrons. We found that the main impact of NSI is due
to the change in the scattering and annihilation processes involving neutrinos
and electrons, while matter effects altering neutrino oscillations play a very
subleading role.

We have shown that, whereas flavour-changing NSI can only enhance the value of
$N_{\rm eff}$, non-universal NSI can both increase or decrease the energy
density transfer between neutrinos and electrons, which translates into a
larger or smaller value of $N_{\rm eff}$ with respect to the value expected
from the SM.

Finally, we commented on how the forecasted sensitivity of $\sigma(N_{\rm
eff})=0.02$ from future cosmological observatories will allow to constraint
these non-standard interactions. The expected limits on NSI parameters would
provide a complementary way to probe these interactions.

I would like to thank my collaborators Stefano Gariazzo, Pablo F. de Salas,
Sergio Pastor and Mariam Tórtola for their contributions to the work here
presented. I am also grateful for the hospitality of the Max-Planck-Institut
für Kernphysik (Heidelberg) during the preparation of these proceedings. This
work is supported by FPU18/04571, FPA2017-85216P (AEI/FEDER, UE) and
PROMETEO/2018/165 (Generalitat Valenciana).

## References

## References

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# Phenomenology of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma$ using LCSR

Anshika Bansal<EMAIL_ADDRESS>Physical Research Laboratory, Ahmedabad,
380009, India.  Indian Institute of Technology, Gandhinagar, 382424, India.
Namit Mahajan<EMAIL_ADDRESS>Physical Research Laboratory, Ahmedabad,
380009, India.

###### Abstract

We present the study of radiative tau decay
($\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma$), computing the structure
dependent contribution using Light Cone Sum Rules. This decay includes the
same form factors as the radiative pion decay with the crucial difference that
the momentum transfer squared, $t$, between the pion-photon system is
positive, which makes these form factors timelike and also as $t$ can now take
values upto $m_{\tau}^{2}$ , it can produce real hadronic resonances. The
analytical form for these form factors has been calculated using light cone
sum rules method and the invariant mass-spectrum in the $\pi-\gamma$ system
and the decay width are presented. The structure dependent parameter,
$\gamma$, the ratio of the axial-vector to vector form factor is found to be
in good agreement with the experimental determination.

##  1 Introduction

$\tau$ is the heaviest lepton with $m_{\tau}=1776.86\pm 0.12MeV$ [1] and has
numerous decay channels because of its heavy mass (see for example [2, 3, 4,
5, 6] for different aspects of $\tau$ lepton physics.). It is the only lepton
which can decay into hadrons. Theoretically, electroweak part is reasonably
well established while one is still lacking in developing a proper methodology
to understand the strong interactions. The study of hadronic $\tau$-decays
helps us to understand the dynamics of strong interaction involved in the
hadronisation of QCD currents in a cleaner environment.
In particular, we are interested in the study of radiative tau decay in the
present work i.e. $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma$. The branching
ratio of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$ is $(10.82\pm 0.05)\%$ [1].
Hence, one expects the branching ratio for radiative tau decay to be
$\mathcal{O}(10^{-3})$. To get a sense for this expectation, one can write the
branching ratio as a product of branching ratios of $\tau\to\rho\nu_{\tau}$
and $\rho\to\pi\gamma$, and using the values from [1], one gets $\sim
10^{-3}$, which is about $10^{-2}$ of the non-radiative branching ratio. Even
though the branching ratio is not very small, these decays are not observed
experimentally yet which makes the study of this mode important.
The decay amplitude of this process includes two contributions [7, 8, 9, 10,
11]:

  * •

Internal Bremsstrahlung (IB): The contribution coming from the emission from
either the incoming or the outgoing particles. This contribution can be
calculated trivially with the use of scalar QED for the point-like charged
pion while the emission from the $\tau$ leg is calculated straightforwardly
using QED. Diagramatically this is shown in (a) and (b) of Fig.(1).

  * •

Structure Dependent (SD): This contribution is governed by the strong
interactions and contains non-trivial parts. Pion can no longer be taken as a
point-like particle. The partonic structure will play a role. This
contribution appears because of the hadronisation of $J^{P}=1^{-}$
($\gamma^{\mu}$) and $1^{+}$($\gamma^{\mu}\gamma_{5}$) intermediate quark-
antiquark currents of the matrix element ((c) of Fig.(1)) and hence depends on
the long distance dynamics. Using the Lorentz symmetry, it can be parametrised
by vector and axial-vector form factors $F_{V}^{(\pi)}$ and $F_{A}^{(\pi)}$,
respectively. These form factors encode the information of strong dynamics
involved in the hadronisation of these currents and their evaluation requires
a non-perturbative treatment such as, Light Cone Sum Rules (LCSR), Chiral
Perturbation Theory $\chi$PT or Lattice QCD. SD contribution also includes a
Contact Term (CT), which emerges as a consequence of gauge invariance and
graphically represented in (d) of Fig.(1).

Figure 1: Feynman diagrams showing different contributions to the radiative
tau decay. (a) and (b) represents the IB contribution, (c) represents the SD
contribution and (d) represents the CT contribution.

The explicit form of these contributions will be calculated in Section-2 where
we will see that the IB part consists of two contributions: one independent of
$m_{\tau}$ and another proportional to $m_{\tau}$. The $m_{\tau}$ independent
contribution turns out to be equal and opposite to the CT contribution and
hence gets cancelled in the total amplitude.
The amplitude for the process of interest is related to that of the radiative
pion decay by crossing symmetry with a major difference that comes at the
level of kinematics as the square of the momentum transferred between the
pion-photon and leptonic system can now take values up to $m_{\tau}^{2}$.
While in the radiative pion decay, it can take values only upto $m_{\pi}^{2}$
which is almost negligible. Also, as both pion and photon are in the final
state, the form factors involved in this process are time-like, and hence
complicated, unlike the form factors involved in the radiative pion decay
which are space-like. As a consequence, the light flavoured mesons will be
created on-shell and give resonant structures in the pion-photon invariant
mass spectrum.
Hence to understand this process, the main task is to calculate the time-like
form factors involved in the process. These form factors probe the structure
of the pion. The information about the pion structure can be obtained by
determining the ratio of $F_{A}^{(\pi)}(0)$ to $F_{V}^{(\pi)}(0)$ which is
defined as the structure dependent parameter, $\gamma$ i.e.
$\gamma=\frac{F_{A}^{(\pi)}(0)}{F_{V}^{(\pi)}(0)}$. We know the values of
$F_{A}^{(\pi)}(0)$ and $F_{V}^{(\pi)}(0)$ from the experimental determination
of radiative pion decay to be equal to ($0.0119\pm 0.0001$) and ($0.0254\pm
0.0017$), respectively[1], which results into the value of $\gamma$ equal to
($0.4685\pm 0.0353$). The value of $\gamma$, which is the ratio of form
factors evaluated at zero momentum transfer, will be same for radiative tau
and pion decays. The calculation of radiative tau decay helps in determining
this structure dependent parameter theoretically in a consistent way. This
decay is also useful to understand the light-by-light hadronic contribution to
the muon anomalous magnetic moment, $(g-2)_{\mu}$ [12]. In [13], the authors
have discussed how this decay can provide the means for the tau neutrino mass
determination. These gauge invariant form factors for the radiative tau decay
have been parametrised using Breit-Wigner type resonances [14], light front
quark model [10] and resonance $\chi$PT [11] in the past.
The differences in the literature stem from the vastly different approaches
adopted to determine or estimate the form factors, which affect the
predictions for the rate and spectrum, as well as extraction of $\gamma$,
including the sign. As an example, whenever the resonances are included via
Breit-Wigner method, a suspecting issue always is the relative phase between
the different contributions. The main aim of this paper is to calculate these
form factors using the method of LCSR in a consistent way.

The rest of the paper is organised as follows; in Section-2, we present the
amplitude calculation for the process and explicitly write the forms of
different contributions mentioned above. In Section-3, we present the
calculation of the form factors using the method of LCSR and in Section-4 we
report our results. Finally, in Section-5 we conclude our results with some
remarks. Various definitions and conventions used are reported in Appendix-A.
The values of various parameters used for numerical calculation are collected
in Appendix-B and the kinematical details are provided in Appendix-C.

##  2 Amplitude Computation

Photon can be emitted by any charged particle. Hence in the present case, the
photon can be emitted from either the pion or tau-lepton, as shown in Fig.(1).
The pion is a composite object with quark-antiquark pair. Therefore, the
internal structure of the pion will also contribute to the process. This gives
rise to two non-perturbative Form Factors. As mentioned above, the amplitude
of radiative tau decay includes various contributions: Internal Bremsstrahlung
(IB), Structure Dependent (SD) and Contact term (CT). IB contribution comes
from the emission of the photon from tau and pion (considering pion to be the
point object). SD contribution comes from the emission of photon from the
internal structure of the pion. The contact term in an interesting effective
contribution and has its origin in the gauge invariance of a QED amplitude
[15]. We follow this approach here.
The amplitude of the process
$\tau^{-}(p_{1})\rightarrow\pi^{-}(p_{2})\nu_{\tau}(p_{3})\gamma(k)$ can be
written as (employing the low energy four-Fermi effective Hamiltonian obtained
by integrating out the heavy W-boson),

$\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)=\frac{G_{F}}{\sqrt{2}}V_{ud}\left<\pi^{-}\nu_{\tau}\gamma|(\bar{\nu}_{\tau}\Gamma^{\mu}\tau)(\bar{d}\Gamma_{\mu}u)|\tau^{-}\right>$
(1)

where, $\Gamma^{\mu}=\gamma^{\mu}(1-\gamma_{5})$.
This amplitude can be factorised in two parts; one where the photon is emitted
from the final state pion and another where the photon gets emitted from the
initial state tau lepton.

$\displaystyle\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{ud}\left[\left<\pi^{-}\gamma|(\bar{d}\Gamma_{\mu}u)|0\right>\left<\nu_{\tau}|(\bar{\nu}_{\tau}\Gamma^{\mu}\tau)|\tau^{-}\right>+\left<\nu_{\tau}\gamma|(\bar{\nu}_{\tau}\Gamma^{\mu}\tau)|\tau^{-}\right>\left<\pi^{-}|(\bar{d}\Gamma_{\mu}u)|0\right>\right]$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{ud}\left[-ie\epsilon^{*}_{\alpha}(\bar{u}_{\nu}\Gamma_{\mu}u_{\tau})\int
d^{4}xe^{ikx}\left<\pi^{-}|T\\{j_{em}^{\alpha}(x)\bar{d}\Gamma^{\mu}u(0)\\}|0\right>\right.$
$\displaystyle\hskip 56.9055pt-\left.ef_{\pi}p_{2\mu}\epsilon^{*}_{\alpha}\int
d^{4}xe^{ikx}\left<\nu_{\tau}|T\\{j_{em}^{\alpha}(x)\bar{\nu}_{\tau}\Gamma^{\mu}\tau(0)\\}|\tau^{-}\right>\right]$
(2)

where,
$j_{em}^{\alpha}(x)=Q_{\psi}\bar{\psi}(x)\gamma^{\alpha}\psi(x)=-\bar{\tau}\gamma^{\alpha}\tau+Q_{u}\bar{u}\gamma^{\alpha}u+Q_{d}\bar{d}\gamma^{\alpha}d$
and $f_{\pi}$ is the pion decay constant. The conventions and definitions are
given in Appendix-A. This factorization of the amplitude holds for energetic
photons and at the leading order in $\frac{1}{m_{\tau}}$ and $\alpha_{em}$.
For the computation of the first term of Eq.(2), define the hadronic matrix
element as,

$T^{\alpha\mu}(p_{2},k)=i\int
d^{4}xe^{ikx}\left<\pi^{-}|T\\{j_{em}^{\alpha}(x)\bar{d}\Gamma^{\mu}u(0)\\}|0\right>.$
(3)

Using the conservation of electromagnetic current, one can apply the Ward
identity which results into

$\displaystyle k_{\alpha}T^{\alpha\mu}(p_{2},k)$
$\displaystyle=\left<\pi^{-}|\bar{d}(0)\Gamma^{\mu}u(0)|0\right>=if_{\pi}p_{2}^{\mu}$
(4)

in the momentum space.
Also, one can write the hadronic matrix element (defined in Eq.(3)) using the
general covariant decomposition in terms of the pion and photon momentum i.e.
$p_{2}$ and $k$ respectively, as

$\displaystyle
T^{\alpha\mu}(p_{2},k)=Ag^{\alpha\mu}+Bp^{2\alpha}p^{2\mu}+Cp^{2\alpha}k^{\mu}+Dk^{\alpha}p^{2\mu}+Ek^{\alpha}k^{\mu}+iF_{V}^{(\pi)}\epsilon^{\alpha\mu\beta\nu}p_{2\beta}k_{\nu}$
(5)

where, $A,B,C,D,E,F_{V}^{(\pi)}$ are gauge invariant scalar functions of
$(p_{2}+k)^{2}$. Contraction of Eq.(5) with $k_{\alpha}$, results in (for on-
shell photon $k^{2}=0$ and Levi-civita tensor is anti-symmetric in $\alpha$
and $\nu$):

$k_{\alpha}T^{\alpha\mu}(p_{2},k)=Ak^{\mu}+B(p_{2}.k)p_{2}^{\mu}+C(p_{2}.k)k^{\mu}.$
(6)

On equating Eq.(4) and Eq.(6), we get

$C=\frac{-A}{(p_{2}.k)},\hskip 14.22636pt\text{ and}\hskip
56.9055ptB=\frac{if_{\pi}}{(p_{2}.k)}$ (7)

which results in the final form of hadronic matrix element to be,

$T^{\alpha\mu}(p_{2},k)=F_{A}^{(\pi)}\left[g^{\alpha\mu}(P.k)-P^{\alpha}k^{\mu}\right]+iF_{V}^{(\pi)}\epsilon^{\alpha\mu\beta\nu}P_{\beta}k_{\nu}-if_{\pi}g^{\alpha\mu}+if_{\pi}\frac{P^{\alpha}P^{\mu}}{P.k}.$
(8)

Here, $F_{A}^{(\pi)}=\frac{A+if_{\pi}}{P.k}$ and $P=p_{1}-p_{3}=p_{2}+k$ and
$p_{2}.k=P.k$. Hence, the first term in Eq.(2) reads,

$\displaystyle\left<\pi^{-}\gamma|\bar{d}\Gamma_{\mu}u|0\right>\left<\nu_{\tau}|\bar{\nu}_{\tau}\Gamma^{\mu}\tau|\tau^{-}\right>$
$\displaystyle=ie\epsilon^{*\alpha}\left[\bar{u}_{\nu}\Gamma^{\mu}u_{\tau}\right]\left[iF_{A}^{(\pi)}\left\\{g_{\alpha\mu}(P.k)-P_{\alpha}k_{\mu}\right\\}-F_{V}^{(\pi)}\epsilon_{\alpha\mu\beta\nu}P^{\beta}k^{\nu}\right]$
$\displaystyle+ie\epsilon^{*\mu}f_{\pi}\bar{u}_{\nu}\Gamma_{\mu}u_{\tau}-ief_{\pi}\frac{\epsilon^{*}.P}{P.k}\bar{u}_{\nu}\not{P}(1-\gamma_{5})u_{\tau}.$
(9)

The second term in Eq.(2), using QED Feynman rules, takes the form

$\displaystyle\left<\nu_{\tau}\gamma|\bar{\nu}_{\tau}\Gamma^{\mu}\tau|\tau^{-}\right>\left<\pi^{-}|\bar{d}\Gamma_{\mu}u|0\right>$
$\displaystyle=-ief_{\pi}\bar{u}_{\nu}(p_{3})\not{\epsilon}^{*}(1-\gamma_{5})u_{\tau}(p_{1})$
$\displaystyle+\frac{ief_{\pi}m_{\tau}}{2p_{1}.k}\left\\{\bar{u}_{\nu}(p_{3})\left[(2\epsilon^{*}.p_{1})-\not{k}\not{\epsilon}^{*}\right](1+\gamma_{5})u_{\tau}(p_{1})\right\\}.$
(10)

Adding the two, the final form of the amplitude is:

$\displaystyle\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{ud}\left[ie\epsilon^{*\alpha}(\bar{u}_{\nu}\Gamma^{\mu}u_{\tau})\left\\{iF_{A}^{(\pi)}\left[g_{\alpha\mu}(P.k)-P_{\mu}k_{\alpha}\right]-F_{V}^{(\pi)}\epsilon_{\alpha\mu\beta\nu}P^{\beta}k^{\nu}\right\\}\right.$
(11)
$\displaystyle+\left.ief_{\pi}m_{\tau}\bar{u}_{\nu}\left\\{\frac{\epsilon^{*}.p_{1}}{p_{1}.k}-\frac{\not{k}\not{\epsilon}^{*}}{2p_{1}.k}-\frac{\epsilon^{*}.p_{2}}{p_{2}.k}\right\\}(1+\gamma_{5})u_{\tau}\right].$

Here, $F_{A}^{(\pi)}$ and $F_{V}^{(\pi)}$ are the gauge invariant axial-vector
and vector form factors, respectively. The contact term appears explicitly by
the use of Ward identity and cancels against the $m_{\tau}$ independent
contribution of photon emission from $\tau$.
For further simplification, we have divided the full amplitude as,

$\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)=\mathcal{A}_{IB}+\mathcal{A}_{V}+\mathcal{A}_{A}\hskip
5.69046pt=\mathcal{A}_{IB}+\mathcal{A}_{SD}.$ (12)

Here,

$\mathcal{A}_{IB}=\frac{G_{F}}{\sqrt{2}}V_{ud}\left[ief_{\pi}m_{\tau}\bar{u}_{\nu}\left\\{\frac{\epsilon^{*}.p_{1}}{p_{1}.k}-\frac{\not{k}\not{\epsilon}^{*}}{2p_{1}.k}-\frac{\epsilon^{*}.p_{2}}{p_{2}.k}\right\\}(1+\gamma_{5})u_{\tau}\right],$
(13)
$\mathcal{A}_{V}=-\frac{G_{F}}{\sqrt{2}}V_{ud}\left[ie\epsilon^{*\alpha}(\bar{u}_{\nu}\Gamma^{\mu}u_{\tau})\left(F_{V}^{(\pi)}\epsilon_{\alpha\mu\beta\nu}P^{\beta}k^{\nu}\right)\right],\text{
and}$ (14)
$\mathcal{A}_{A}=\frac{G_{F}}{\sqrt{2}}V_{ud}\left[ie\epsilon^{*\alpha}(\bar{u}_{\nu}\Gamma^{\mu}u_{\tau})\left(iF_{A}^{(\pi)}\left[g_{\alpha\mu}(P.k)-P_{\mu}k_{\alpha}\right]\right)\right].$
(15)

$\mathcal{A}_{V}$ and $\mathcal{A}_{A}$ combined gives the structure dependent
contribution, while $\mathcal{A}_{IB}$ is the internal bresstrahlung
contribution.

##  3 Form Factors in LCSR framework

In the previous section, we saw that the amplitude of the radiative tau decay
depends on two gauge invariant form factors; $F_{A}^{(\pi)}$ and
$F_{V}^{(\pi)}$. These form factors are the non-perturbative objects and need
a non-perturbative treatment. In this section, we will calculate these form
factors using the method of LCSR.
The method of sum rules was developed in 1979 by Shifman, Vainshtein and
Zakharov (SVZ) [16, 17]. Their basic idea was to use the analytic properties
of a correlation function (treated in the framework of operator product
expansion (OPE)) to derive the hadronic parameter involved in a process. Below
we briefly outline the method (for details, see [18, 19, 20]).
The important tools for deriving the sum rules are: dispersion relation,
operator product expansion(OPE), quark-hadron duality and the Borel
transformation. The dispersion relation relates the real part of the
correlation function to its imaginary part using the Cauchy’s integral
formula. According to OPE, the correlation function can be written as a sum of
products of long distance matrix elements of operators of increasing dimension
and short distance Wilson coefficients which can be calculated using
perturbation theory. The higher dimension operators capture the information of
QCD vacuum fields in the form of vacuum condensates. Both dispersion relation
and OPE gives the same physics and hence can be equated.
Operationally, quark hadron duality means,

$q^{2}\int_{s_{0}^{h}}^{\infty}ds\frac{\rho^{h}(s)}{s(s-q^{2})}\simeq\frac{q^{2}}{\pi}\int_{4m^{2}}^{\infty}ds\frac{Im\Pi^{(pert)}(s)}{s(s-q^{2})}.$
(16)

Here, $\rho^{h}$ is the hadronic spectral density function, while
$\Pi^{pert}(s)$ (or $\Pi^{QCD}(s)$) is the perturbatively calculated
correlation function. We will use this duality approximation below.
As the correlation function has contribution from all the resonance states as
well as the continuum, one performs Borel transformation to suppress the
effect of higher resonances and continuum. Mathematically, the Borel transform
is given by,

$\Pi(M^{2})\equiv\mathcal{B}_{M^{2}}\Pi(k^{2})=\lim_{-k^{2},n\rightarrow\infty,-k^{2}/n=M^{2}}\frac{(-k^{2})^{(n+1)}}{n!}\left(\frac{d}{dk^{2}}\right)^{n}\Pi(k^{2}),$
(17)

where M is known as the Borel parameter.
It was noticed that these SVZ sum rules have some limitations such as: the OPE
upsets the power counting in large $Q^{2}$ and that even after performing the
Borel transformation, practical calculations suffer from unsuppressed
contributions. These limitations can be overcome by using light cone sum rules
(LCSR). In LCSR, one expands the products of the currents near the light cone.
LCSR give vacuum-to-hadron correlation function while by SVZ sum rules one get
vacuum-to-vacuum correlation functions. In LCSR, OPE at short distances is
replaced by systematic expansion in the transverse direction in infinite
momentum frame.
In the lightcone limit, the bi-local operator sandwiched between the pion
state and vacuum is expressed as,

$\left<\pi^{0}(p)|\bar{u}(y)\gamma_{\mu}\gamma_{5}u(x)|0\right>_{x^{2}=0}=-if_{\pi}p_{\mu}\int_{0}^{1}du\hskip
2.84544pte^{i(up_{2}.y+\bar{u}p.x)}\phi(u,\mu)$ (18)

where, $\bar{u}=1-u$ and $\phi(u,\mu)$ is leading twist-2 distribution
amplitude given by

$\phi_{\pi}(u,\mu)=6u\bar{u}\left[1+\sum_{n=2,4\ldots}a_{n}(\mu)C_{n}^{3/2}(u-\bar{u})\right].$
(19)

Here, $C_{n}^{3/2}$ are the Gegenbauer polynomials and $a_{n}$ are the
multiplicatively renormalisable coefficient defined as,

$a_{n}(\mu)=a_{n}(\mu_{0})\left(\frac{\alpha_{s}(\mu)}{\alpha_{s}(\mu_{0})}\right)^{\gamma_{n}/\beta_{0}}$
(20)

with $\alpha_{s}=\frac{g_{s}^{2}}{4\pi}$ ($g_{s}$ is the strong coupling
constant), $\beta_{0}$ is the leading QCD $\beta$-function and

$\gamma_{n}=\frac{4}{3}\left[-3-\frac{2}{(n+1)(n+2)}+4\left(\sum_{k=1}^{(n+1)}\frac{1}{k}\right)\right].$
(21)

The remaining process for computation is same as for SVZ sum rules. We are now
ready to derive the form factors, $F_{V}^{(\pi)}$ and $F_{A}^{(\pi)}$ using
this technique.
As we know, these form factors arises from the computation of the hadronic
matrix element defined in Eq.(3), i.e.,

$T^{\alpha\mu}(p_{2},k)=i\int
d^{4}xe^{ikx}\left<\pi^{-}|T\left\\{Q_{u}\bar{u}\gamma^{\alpha}u(x)\bar{d}\Gamma^{\mu}u(0)+Q_{d}\bar{d}\gamma^{\alpha}d(x)\bar{d}\Gamma^{\mu}u(0)\right\\}|0\right>$
(22)

where, $Q_{u}$ and $Q_{d}$ are the charges of up and down quark respectively
in units of $e$. Using the definitions and identities given in Appendix-A, we
get

$\displaystyle T^{\alpha\mu}(p_{2},k)$ $\displaystyle=if_{\pi}\int
d^{4}x\frac{e^{ikx}}{2\pi^{2}x^{4}}\int_{0}^{1}du\phi(u,\mu)\left[i\epsilon^{\mu\beta\alpha\rho}x_{\beta}p_{2\rho}\left(Q_{u}e^{i\bar{u}p_{2}x}+Q_{d}e^{iup_{2}x}\right)\right.$
$\displaystyle+\left.\left(x^{\mu}p_{2}^{\alpha}-g^{\mu\alpha}(x.p_{2})+x^{\alpha}p_{2}^{\mu}\right)\left(Q_{u}e^{i\bar{u}p_{2}x}-Q_{d}e^{iup_{2}x}\right)\right],$
(23)

where, as mentioned above, $\phi(u,\mu)$ is the pion distribution amplitude
and $\bar{u}=1-u$. The integration over $x$ results into,

$\displaystyle T^{\alpha\mu}(P,k)$
$\displaystyle=if_{\pi}\left[\frac{i\epsilon^{\mu\beta\alpha\rho}}{3}P_{\rho}k_{\beta}\int_{0}^{1}du\frac{\phi(u,\mu)}{P^{2}\bar{u}+k^{2}u}+2\left\\{P^{\alpha}P^{\mu}-(P.k)g^{\mu\alpha}\right\\}\int_{0}^{1}du\frac{\phi(u,\mu)\bar{u}}{P^{2}\bar{u}+k^{2}u}\right.$
$\displaystyle-\left.\left\\{g^{\mu\alpha}(P.k)-P^{\alpha}k^{\mu}\right\\}\left\\{\int_{0}^{1}du\phi(u,\mu)\left(\frac{1-2\bar{u}}{P^{2}\bar{u}+k^{2}u}\right)\right\\}\right].$
(24)

Here, $p_{2}+k=P$ and we have used the fact that the distribution amplitude is
a symmetric function of $u$ and $\bar{u}$.
A comparison with the general decomposition of the hadronic tensor given in
Eq.(8) yields the following forms of vector and axial-vector form factors from
QCD calculation.

$F_{V}^{QCD}(t)=\frac{if_{\pi}}{3}\int_{0}^{1}du\frac{\phi(u,\mu)}{t\bar{u}+k^{2}u}\hskip
28.45274pt$ (25)
$\implies\frac{1}{\pi}Im\\{F_{V}^{QCD}(t)\\}=\frac{if_{\pi}}{3}\int_{0}^{1}du\phi(u,\mu)\delta(t\bar{u}+k^{2}u),\text{
and}$
$F_{A}^{QCD}(t)=-if_{\pi}\int_{0}^{1}du\phi(u,\mu)\left(\frac{1-2\bar{u}}{t\bar{u}+k^{2}u}\right)$
(26)
$\implies\frac{1}{\pi}Im\\{F_{A}^{QCD}(t)\\}=-if_{\pi}\int_{0}^{1}du\phi(u,\mu)(1-2\bar{u})\delta(t\bar{u}+k^{2}u).$

Here, $t\equiv P^{2}=(p_{2}+k)^{2}=(p_{1}-p_{3})^{2}$ is the invariant mass
square of the photon-pion system.
Now, after computing the perturbative QCD contribution, the analytic
properties of this hadronic matrix element are used to derive the contribution
of various hadronic states. It will get contribution from
($\rho,\omega,a_{1}$-mesons)$+$ higher resonances and the continuum. In the
present case, contributions coming from $\rho,\omega,a_{1}$-mesons will
saturate the sum rules and thus will be the focus here.111 The contribution of
the higher resonances, at the present level of accuracy, is roughly 20% of
these resonances because of the Borel suppression.

Considering the matrix element
$\left<\pi^{-}|T\\{j_{em}^{\alpha}(x)j_{ew}^{\mu}(0)\\}|0\right>$ and
inserting a complete set of states, we get,

$\left<\pi^{-}|T\\{j_{em}^{\alpha}(x)j_{ew}^{\mu}(0)\\}|0\right>=\left<\pi^{-}|j_{em}^{\alpha}(x)|n\right>\left<n|j_{ew}^{\mu}(0)|0\right>$
(27)

where, $\lvert n\rangle=\lvert\rho\rangle+\lvert\omega\rangle+\lvert
a_{1}\rangle+$ higher resonances + continuum.

  * •

$\rho$ and $\omega$-meson contribution: The $\rho$-meson contribution will
come from,

$\left<\pi^{-}(p_{2})|j_{em}^{\alpha}(x)|\rho(p_{2}+k)\right>\left<\rho(p_{2}+k)|j_{ew}^{\mu}(0)|0\right>.$
(28)

Using the definitions given in Appendix-A,

$\left<\pi^{-}(p_{2})|j_{em}^{\alpha}(x)|\rho(p_{2}+k)\right>\left<\rho(p_{2}+k)|j_{ew}^{\mu}(0)|0\right>=im_{\rho}f_{\rho}\epsilon^{\alpha\lambda\beta\nu}g^{\mu}_{\lambda}p_{2\beta}k_{\nu}F_{\rho\pi}(k^{2})$
(29)

where, $m_{\rho}$ and $f_{\rho}$ are the mass and decay constant of
$\rho$-meson respectively. Neglecting the very small difference between the
masses of $\rho$ and $\omega$, the contribution of $\omega$ will be equal to
the contribution of $\rho$ and hence multiplying $\rho$ contribution by a
factor of $2$ will incorporate the contribution of $\omega$-meson.

  * •

$a_{1}$-meson contribution: The $a_{1}$-meson contribution will come from,

$\left<\pi^{-}(p_{2})|j_{em}^{\alpha}(x)|a_{1}(p_{2}+k)\right>\left<a_{1}(p_{2}+k)|j_{ew}^{\mu}(0)|0\right>,$
(30)

which results into

$\left<\pi^{-}(p_{2})|j_{em}^{\alpha}(x)|a_{1}(p_{2}+k)\right>\left<a_{1}(p_{2}+k)|j_{ew}^{\mu}(0)|0\right>=im_{a_{1}}f_{a_{1}}\left[2p_{2}.kg^{\alpha\mu}-2p_{2}^{\alpha}k^{\mu}\right]G_{a_{1}\pi}(k^{2})$
(31)

using the definitions given in Appendix-A. Here, $m_{a_{1}}$ and $f_{a_{1}}$
are the mass and decay constant of $a_{1}$-meson respectively.

Here, $F_{\rho\pi}$($G_{a_{1}\pi}$) captures the physics of transition of
$\rho$($a_{1}$)-meson to the $\pi$-meson. Using the optical theorem in Eq.(3),
we get,

$2Im\\{T^{\alpha\mu}(p_{2},k)\\}=\sum_{n}\left<\pi^{-}|j_{em}^{\alpha}(x)|n\right>\left<n|j_{ew}^{\mu}|0\right>d\tau_{n}(2\pi)^{4}\delta^{4}(k-p_{n}),$
(32)

and from Cauchy’s theorem,

$T(k^{2})=\frac{1}{\pi}\int_{t_{min}}^{\infty}ds\frac{Im\\{T(s)\\}}{s-k^{2}-i\epsilon}.$
(33)

Substituting the contributions of $\rho$ and $a_{1}$, we get,

$\displaystyle T^{\alpha\mu}(p_{2},k)$
$\displaystyle=\frac{2im_{\rho}f_{\rho}\epsilon^{\alpha\lambda\beta\nu}g^{\mu}_{\lambda}p_{2\beta}k_{\nu}F_{\rho\pi}(k^{2})}{m_{\rho}^{2}-(p_{2}+k)^{2}-im_{\rho}\Gamma_{\rho}}+\frac{im_{a_{1}}f_{a_{1}}\left[2p_{2}.kg^{\alpha\mu}-2p_{2}^{\alpha}k^{\mu}\right]G_{a_{1}\pi}(k^{2})}{m_{a_{1}}^{2}-(p_{2}+k)^{2}-im_{a_{1}}\Gamma_{a_{1}}}$
$\displaystyle+\frac{1}{\pi}\int_{s_{0}^{h}}^{\infty}ds\frac{Im\\{T^{\alpha\mu}(s,k)\\}}{s-k^{2}-i\epsilon}.$
(34)

Here, $s_{0}^{h}$ is the threshold of the lowest continuum state and
$\Gamma_{\rho}$ and $\Gamma_{a_{1}}$ are the decay widths of $\rho$ and
$a_{1}$ mesons, respectively. This is the dispersion relation which relates
the imaginary part to the real part. Now, the light cone sum rules can be
derived by taking the form of $F_{V}^{(\pi)}(t)$ from this dispersion relation
and equating it with the form obtained in Eq.(25), i.e.

$\frac{2m_{\rho}f_{\rho}F_{\rho\pi}(k^{2})}{m_{\rho}^{2}-t-im_{\rho}\Gamma_{\rho}}+\frac{1}{\pi}\int_{s_{0}^{h}}^{\infty}ds\frac{Im\\{F_{V}(s)\\}}{s-t-i\epsilon}=\frac{if_{\pi}}{3}\int_{0}^{1}du\frac{\phi(u,\mu)}{t\bar{u}+k^{2}u}.$
(35)

Using the duality approximation and the Chauchy’s integral,

$\frac{1}{\pi}\int_{s_{0}}^{\infty}ds\frac{Im\\{F_{V}(s,k)\\}}{s-t-i\epsilon}=\frac{1}{\pi}\int_{s_{0}^{\rho}}^{\infty}ds\frac{Im\\{F_{V}^{QCD}(s,k)\\}}{s-t-i\epsilon}=\frac{if_{\pi}}{3}\int_{u_{0}}^{1}du\frac{\phi(u)}{t\bar{u}+k^{2}u},$
(36)

with $u_{0}=\frac{s_{0}}{k^{2}+s_{0}}=1$ (as $k^{2}=0$). As a result, the sum
rule for $F_{V}^{(\pi)}(t)$ turns out to be,

$\frac{2m_{\rho}f_{\rho}F_{\rho\pi}(k^{2})}{m_{\rho}^{2}-t}=\frac{if_{\pi}}{3}\int_{0}^{u_{0}}du\frac{\phi(u)}{t\bar{u}+k^{2}u}.$
(37)

Similarly, by equating the form of $F_{A}^{(\pi)}(t)$ obtained from the
dispersion relation with the form given in Eq.(26) and using the duality
approximation, the sum rule for $F_{A}^{(\pi)}(t)$ reads,

$\frac{2im_{a_{1}}f_{a_{1}}G_{a_{1}\pi}(k^{2})}{m_{a_{1}}^{2}-t}=-if_{\pi}\int_{0}^{u_{0}}\phi(u)\left(\frac{1-2\bar{u}}{t\bar{u}+k^{2}u}\right).$
(38)

After Borelisation and substituting these sum rules back in Eq.(34), we get
the following analytical forms for $F_{V}^{(\pi)}$ and $F_{A}^{(\pi)}$,

$F_{V}^{(\pi)}(t)=-i\frac{f_{\pi}}{3(m_{\rho}^{2}-t-im_{\rho}\Gamma_{\rho})}\int_{0}^{1}du\frac{\phi(u)}{\bar{u}}e^{\frac{m_{\rho}^{2}}{M^{2}}},$
(39)
$F_{A}^{(\pi)}(t)=-i\frac{f_{\pi}}{m_{a_{1}}^{2}-t-im_{a_{1}}\Gamma_{a_{1}}}\int_{0}^{1}\frac{\phi(u)}{\bar{u}}(1-2\bar{u})e^{\frac{m_{a_{1}}^{2}}{M^{2}}}.$
(40)

Here, $M$ is the Borel parameter and we have used the on-shell condition for
photon (i.e. $k^{2}=0$).222 It is to be noted that these form factors have
dimension of inverse mass and there is an extra factor of $-i$ due to the way
initial amplitude is defined:
$\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)$ instead of
$i\mathcal{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)$ as is often done.

For the present calculation, we will use the asymptotic form (where
$\mu\rightarrow\infty$) and Chernyak-Zhitnisky form (where $C_{2}$ term will
be considered) of the pion distribution amplitude given in Eq.(19). Explicitly
these forms are given by,

$\phi_{\pi}^{asym}(u,\mu)=6u\bar{u},\text{ and}$ (41)
$\phi_{\pi}^{CZ}(u,\mu)=6u\bar{u}\left[1+\frac{3a_{2}(\mu)}{2}\\{5(u-\bar{u})^{2}-1\\}\right]$
(42)

where, $a_{2}(\mu)$ is defined in Eq.(20) with $n=2$. All the structure
dependent information of the pion involved in the radiative tau decay is
contained in the ratio of axial vector form factor and the vector form factor
at zero invariant mass square of the photon-pion system, i.e.,

$\gamma=\frac{F_{A}^{(\pi)}(0)}{F_{V}^{(\pi)}(0)}$ (43)

where, $\gamma$ is known as the structure dependent parameter (SDP). The
vector form factor at $t=0$ can be related to the anomaly term (or Wess-
Zumino-Witten term) in $\pi\gamma\gamma$ vertex ($1/(4\pi^{2}f_{\pi})$). Using
what is referred to as KSFR-II relation ([21, 22]),
$m_{\rho}^{2}=2g_{\rho\pi\pi}^{2}f_{\pi}^{2}$, along with the assumptions of
universality of $\rho$-coupling ($g_{\rho\pi\pi}=g_{\rho
NN}=g_{\rho\gamma}=g=2\pi\sqrt{3/N_{c}}$) and $\rho$ meson dominance of the
pion electromagnetic form factor, one finds the right form emerging from
$F_{V}^{(\pi)}(0)$, up to the overall factor $e^{\frac{m_{\rho}^{2}}{M^{2}}}$
which should tend to unity. As we see later, the choice of the Borel parameter
that provides a stable window, trivially yields unity for this factor within a
few percent.
Before discussing the results, it may be worthwhile to ponder over possible
duality violations. Such contributions arise from our use of perturbatively
evaluated spectral functions, imaginary parts of the form factors here, over
the entire kinematical range. It is notoriously difficult to exactly quantify
the magnitude of such duality violating terms. However, it is rather important
to have some estimate or an educated guess since these would otherwise cause
large uncertainties in the final results. For the case at hand, the
perturbative effects occus at $1/Q$, where hard scale $Q\sim m_{\tau}$ while
the time scale over which the partons come together to form final hadrons
$\sim Q/\Lambda^{2}_{QCD}$. One possible way to evaluate the duality
violations could be to use an instanton model, where the light quark
amplitudes will be suppressed. A rough calculation yields a quantity that in
the Euclidean domain has the form $Exp[-Q\rho]/Q^{n}$, where $\rho$ denotes
the mean instanton size. Analytically continued to the Minkowski space, this
would have an oscillating factor multipled by negative powers of the energy
released in the hard process $m_{\tau}$. Alternatively, one could assume a
comb of hadronic resonances that would contribute and carry out the algebra.
Both lead to similar conclusions that the violations are $\sim 10\%$ [23]
(also see [24, 25] for detailed analyses for inclusive tau decays). This is
the typical duality violation contribution that we expect, though a more
detailed calculation can reveal the actual amount of such violations.

##  4 Results

The analytic expressions for the vector and axial-vector form factors
calculated using LCSR are given in Eq.(39) and Eq.(40). Both these form
factors have the asymptotic $\frac{1}{t}$ dependence on the invariant mass
squared, $t$ of the photon pion system, as expected from QCD in the
perturbative (asymptotic) regime. We have used two forms of pion distribution
amplitude; the asymptotic form and the CZ form as given in Eq.(41) and
Eq.(42), respectively. The structure dependent parameter defined in Eq.(43) is
also calculated using both the forms for pion distribution amplitudes. The
values of the various parameters used for the numerical computation are
collected in Appendix-B. The form factors depend on the value of the Borel
parameter, $M$, and hence also the structure dependent parameter, $\gamma$.
Fig.(2) shows the variation of $F_{A}^{(\pi)}(0)$, $F_{V}^{(\pi)}(0)$ and SDP
($\gamma$) with the variation in the value of $M$. The variation of the
observables with $M$ dictates the model dependence here. As can be seen from
the plot, all the observables are quite stable in the chosen Borel window. The
value of $\gamma$ for $M=3.35$ GeV is 0.469 (using CZ distribution amplitude)
which matches well, including the sign, with the experimental value of
$\gamma$ obtained from the radiative pion decay [1].

Figure 2: The dependence of structure dependent parameter (SDP),
$F_{A}^{(\pi)}(0)$ and $F_{V}^{(\pi)}(0)$ on the Borel parameter $M$ (in GeV
units) is shown in Blue, Magenta and Green, respectively. In this plot, form
factors have been multiplied by $im_{\pi}$ to make them dimensionless in and
take care of the extra $-i$ in the form factors as noted in the Footnote1.

Further, we calculate the decay width contribution for the radiative tau decay
using $M=3.35$ GeV and the form factors given in Eq.(39) and Eq.(40). The
differential decay rate for the radiative tau decay is given by,

$d\Gamma(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)=\frac{1}{512\pi^{5}}E_{\tau}\delta^{(4)}(k+p_{2}+p_{3}-p_{1})\overline{|\mathcal{A}|^{2}}\frac{d^{3}kd^{3}p_{2}d^{3}p_{3}}{E_{\gamma}E_{\pi}E_{\nu}}$
(44)

where, $E_{\tau},E_{\pi},E_{\gamma},E_{\nu}$ are the energies of tau-lepton,
pion, photon and neutrino, respectively. $\overline{|\mathcal{A}|^{2}}$ is the
spin averaged square of the amplitude which has been calculated in Section-2.
In terms of the functions used in Eq.(12),

$\overline{|\mathcal{A}|^{2}}=\overline{|\mathcal{A}_{IB}|^{2}}+\overline{|\mathcal{A}_{SD}|^{2}}+2\mathcal{R}e\\{\overline{\mathcal{A}_{IB}^{*}\mathcal{A}_{SD}}\\}$
(45)

where,
$\overline{|\mathcal{A}_{SD}|^{2}}=\overline{|\mathcal{A}_{A}|^{2}}+\overline{|\mathcal{A}_{V}|^{2}}+2\mathcal{R}e\\{\overline{\mathcal{A}_{A}^{*}\mathcal{A}_{V}}\\}$.
The kinematical details to compute the decay rate can be found in Appendix-C.

Figure 3: The total Structure Dependent Contribution (blue) to the photon
spectrum is shown along with the individual contributions from the vector
(magenta), axial vector (green) and the interference (red) of the two are also
shown for the two distribution amplitudes. Solid lines are for asymptotic
distribution amplitude while dashed ones are for Chernyak-Zitnisky
distribution amplitude.

The structure dependent contribution to the photon spectrum is shown in
Fig.(3) using both forms of pion distribution amplitudes.

Figure 4: The dependence of the IB (solid) contribution on the minimum energy
threshold of the photon is shown here. Along with that, the same dependence
for total decay width including form factors using asymptotic (dashed) and CZ
(dotted) pion distribution amplitude is also shown.

The IB contribution suffers from the infrared divergences which can be taken
care of by putting a threshold on the photon energy. Fig.(4) shows the
threshold energy dependence of the IB contribution as well as the full decay
width of the radiative tau decay. The SD contribution is free from any kind of
divergences.
$F_{A}^{(\pi)}(t)$ gets contribution from $a_{1}$ meson while
$F_{V}^{(\pi)}(t)$ from $\rho$ (and $\omega$) meson.

(a)

(b)

Figure 5: (a): The Structure Dependent contribution (blue) to the invariant
mass spectrum of $\pi-\gamma$ system is shown here for asymptotic (solid) and
Chernyak-Zhitnisky (dashed) pion distribution amplitudes. The contribution
from the vector (magenta), axial vector (green) and the interference (red) of
the two is also shown. (b): Zoomed in version of (a).

Fig.(5) shows the SD contribution to the invariant mass spectrum of
$\pi-\gamma$ system. The higher and sharper peak corresponds on the
contribution coming from the vector mesons while the shorter and broader peak
corresponds to the axial vector contribution. The vector contribution to the
total decay width dominates over the axial-vector contribution.
As $\rho$ and $a_{1}$-mesons are not very narrow, the effect of $t$ dependence
of the widths is also studied using the prescription provided in [26]. The $t$
dependence of $\Gamma_{\rho}$ does not have significant effect as it is not
that wide while the effect of $\Gamma_{a_{1}}$ is clearly visible as one can
see from Fig.(6). The explicit forms of $t$ dependence of the decay widths are
collected in Appendix-A. We have also computed the effect of decay width of
$a_{1}$-meson $\Gamma_{a_{1}}$, as it has huge uncertainty, and found that the
decay width of radiative tau decay decreases with an increase in
$\Gamma_{a_{1}}$. The results reported here are calculated using
$\Gamma_{a_{1}}=425$ MeV.
Fig.(7) represents all the contributions to the invariant mass spectrum of the
$\pi-\gamma$ system. The IB contribution dominates at the low photon energy
for which we have used the minimum energy threshold of $50$ MeV.

(a)

(b)

Figure 6: The SD contribution (blue) considering (a) $\Gamma_{\rho}$ and
$\Gamma_{a_{1}}$ to be constant and (b) the $t$ dependence of $\Gamma_{\rho}$
and $\Gamma_{a_{1}}$ is shown here for asymptotic (solid) and Chernyak-
Zhitnisky (dashed) pion distribution amplitudes. The contribution from the
vector (magenta), axial vector (green) and the interference (red) of the two
is also shown.

After integrating over the full phase space and applying a energy threshold of
50 MeV for the IB contribution, we get the following values for the different
contributions to the decay width (normalised to the non-radiative decay width
Eq.(C.8) ie $\bar{\Gamma}=\Gamma(\tau\to\pi\nu\gamma)/\Gamma(\tau\to\pi\nu)$):
(i) Asymptotic pion distribution amplitude:

$\bar{\Gamma}_{IB}=1.36\times 10^{-2},\hskip
14.22636pt\bar{\Gamma}_{VV}=(1.47\pm 0.06)\times 10^{-3},\hskip
14.22636pt\bar{\Gamma}_{AA}=(3.97\pm 2.45)\times 10^{-4},\hskip
14.22636pt\bar{\Gamma}_{AV}\approx 0$ $\bar{\Gamma}_{SD}=(1.87\pm 0.30)\times
10^{-3},\hskip 14.22636pt\bar{\Gamma}_{int}=(3.82\pm 2.14)\times
10^{-4},\hskip 14.22636pt\bar{\Gamma}_{all}=(1.56\pm 0.04)\times 10^{-2}$

(ii) CZ pion distribution amplitude:

$\bar{\Gamma}_{IB}=1.36\times 10^{-2},\hskip
14.22636pt\bar{\Gamma}_{VV}=(1.70\pm 0.07)\times 10^{-3},\hskip
14.22636pt\bar{\Gamma}_{AA}=(5.91\pm 3.62)\times 10^{-4},\hskip
14.22636pt\bar{\Gamma}_{AV}\approx 0$ $\bar{\Gamma}_{SD}=(2.29\pm 0.43)\times
10^{-3},\hskip 14.22636pt\bar{\Gamma}_{int}=(4.90\pm 2.60)\times
10^{-4},\hskip 14.22636pt\bar{\Gamma}_{all}=(1.61\pm 0.06)\times 10^{-2}$

Since we consider radiative rate normalised to the non-radiative one, the
uncertainty in IB contribution is negligible compared to the SD contribution
which dominates the error budget therefore no uncertainty is shown for the IB
part. The final uncertainties are about $10\%$. From the above it is evident
that there is a dependence on the form of the distribution amplitude chosen to
evaluate these form factors. However, the difference is not too large, which
is reassuring.
Having obtained detailed predictions for the pion in the final state, it is
also instructive to have an estimate of the decay width for the kaon in the
final state. Again, normalising to the appropriate non-radiative width, and
employing the asymptotic distribution amplitude (keeping the Borel parameter,
$M=3.35$ GeV), we get

$\bar{\Gamma}^{K}=\Gamma(\tau\to K\nu\gamma)/\Gamma(\tau\to K\nu)\sim 8\times
10^{-3}$ (46)

This (approriately normalised) rate is roughly half of that for the pion.

(a)

(b)

Figure 7: The invariant mass spectrum of $\pi-\gamma$ system for radiative tau
decay is shown here considering (a) asymptotic and (b) CZ pion distribution
amplitude. The contributions from the IB (magenta), SD (green) and the
interference (red) of the two is also shown. The shaded region shows the
uncertainties.

##  5 Discussion and Conclusions

In the present paper, we have provided detailed predictions for the rate and
photon spectrum for the process $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma$.
Employing Ward identity from the beginning, the amplitude was written so as to
include the contact term which is necessitated by gauge invariance. The decay
involves two time like form factors. These have been calculated in the present
work employing the Light Cone Sum Rules, to twist-2 accuracy. The form
factors, which automatically via the dispersion relations, encode the
contributions from the vector and axial-vector mesons, have the right
asymptotic behaviour expected from perturbative QCD. The ratio of the axial-
vector to vector form factor at zero momentum transfer defines the pion
structure dependent parameter, $\gamma$. Our evaluation of this parameter,
along with the sign, matches very well with the experimental value obtained
from $\pi\to\ell\nu\gamma$, where the relevant pion-photon form factors,
unlike the present case, are space like. The obtained values for the
normalised rate and the photon spectrum are similar to those obtained in [11].
This provides a cross-check on the theoretical predictions employing a totally
different method for computing the non-perturbative quantities. We have also
provided an estimate for the appropriately normalised rate with kaon in the
final state instead of a pion. This normalised rate is approximately half of
that for the pion. The present study employed distribution amplitudes to
twist-2 accuracy. The uncertainties reported here are the uncertainties
associated with the uncertainities of the various parameters used. There will
be further uncertainties associated with quark hadron duality approximation,
and higher twist and hight order contributions. The pion is considered to be
massless here. The effect of such an assumption is less than 1$\%$ on the
values of the form factors. The uncertainties associated with quark hadron
duality violation, like in inclusive tau decays are expected to be at $10\%$
level, and can be calculated in a particular model to parametrise the spectral
density. Precise calculations of these duality violations is indeed an
important missing piece but is out of the scope of present work. It would be
interesting to consider both higher twist contributions as well as
contributions higher order in $\alpha_{s}$. These can have a significant
impact on the phenomenology of radiative one meson tau decays.

##  Appendix A Conventions, Definitions and Identities

Here, we are reporting the various conventions and definitions used for the
sake of completeness,

  1. 1.

The matrix element of the pion is defined as;

$\left<\pi^{-}(p_{2})|(\bar{d}\gamma^{\mu}(1-\gamma_{5})u)|0\right>=if_{\pi}p_{2}^{\mu}$
(A.1)

where, $f_{\pi}$ is the pion decay constant.

  2. 2.

The outgoing photon state can be obtained by the use of creation operator on
the vacuum which results into,

$\left<\nu_{\tau}\gamma|\bar{\nu}_{\tau}\gamma_{\mu}(1-\gamma_{5})|\tau^{-}\right>=-ie\epsilon_{\mu}^{*}\int
d^{4}xe^{ikx}\left<\nu_{\tau}|T\\{j_{em}^{\alpha}(x)\bar{\nu}_{\tau}\Gamma_{\mu}\tau(0)\\}|\tau^{-}\right>$
(A.2)

where,
$j_{em}^{\alpha}(x)=Q_{\psi}\bar{\psi}(x)\gamma^{\alpha}\psi(x)=-\bar{\tau}\gamma^{\alpha}\tau+Q_{u}\bar{u}\gamma^{\alpha}u+Q_{d}\bar{d}\gamma^{\alpha}d$
is the electromagnetic current. $Q_{u}$ and $Q_{d}$ are the electromagnetic
charges of $u$ and $d$ quarks, respectively in the units of $e$.

  3. 3.

The commutator of the electromagnetic charge operator and electroweak current
of the pion is given by,

$\left[j_{em}^{0}(x),\bar{d}\Gamma^{\mu}u(0)\right]=-Q_{u}\delta^{3}(x)\bar{d}(0)\Gamma^{\mu}u(x)+Q_{d}\partial^{3}(x)\bar{d}(x)\Gamma^{\mu}u(0)$
(A.3)

  4. 4.

The propagator of the massless fermions in position space is given by,

$iS_{0}(x)=\left<0|T\left\\{u(x)\bar{u}(0)\right\\}|0\right>=\frac{i\not{x}}{2\pi^{2}x^{4}}=-\left<0|T\left\\{u(0)\bar{u}(x)\right\\}|0\right>$
(A.4)

  5. 5.

$\gamma_{\mu}\gamma_{\beta}\gamma_{\alpha}=g_{\mu\beta}\gamma_{\beta}-g_{\mu\alpha}\gamma_{\beta}+g_{\beta\alpha}\gamma_{\mu}-i\epsilon_{\mu\beta\alpha\rho}\gamma^{\rho}\gamma_{5}$

  6. 6.

The leading order expansion (twist-2) of the matrix element
$\left<\pi^{-}(p_{2})|\bar{d}(y)\gamma_{\mu}\gamma_{5}u(x)|0\right>$ in the
light cone limit ($x^{2}=0$) is given by,

$\left<\pi^{-}(p_{2})|\bar{d}(y)\gamma_{\mu}\gamma_{5}u(x)|0\right>=if_{\pi}p_{2\mu}\int_{0}^{1}due^{i(up_{2}y+\bar{u}p_{2}x)}\phi(u,\mu)$
(A.5)

where, $\bar{u}=1-u$ and $\phi(u,\mu)$ is pion distribution amplitude of
twist-2.

  7. 7.

The matrix elements of $\rho$ and $a_{1}$ mesons are defined as,

$\displaystyle\left<V(p_{2}+k)|\bar{d}\gamma_{\mu}u|0\right>=-im_{V}f_{V}\epsilon_{\mu}^{(V)*}$
(A.6)
$\displaystyle\left<\pi^{-}(p_{2})|j_{em}^{\alpha}(x)|\rho(p_{2}+k)\right>=\epsilon^{\alpha\lambda\beta\nu}\epsilon_{\lambda}^{(\rho)}p_{2\beta}k_{\nu}F_{\rho\pi}(k^{2})$
(A.7)
$\displaystyle\left<\pi^{-}(p_{2})|j_{em}^{\mu}(x)|a_{1}(p_{2}+k)\right>=\left[(2p_{2}-k).kg^{\mu\lambda}-(2p_{2}-k)^{\mu}k^{\lambda}\right]\epsilon_{\lambda}^{(a_{1})*}G_{a_{1}\pi}(k^{2})$
(A.8)

where, $V$ can be $\rho$ or $a_{1}$-meson, $m_{V}$ and $f_{V}$ are the mass
and decay constant of $V$-meson, respectively. $\epsilon_{\lambda}^{(\rho)}$
and $\epsilon_{\lambda}^{(a_{1})*}$ are the polarisation vectors of $\rho$ and
$a_{1}$-meson, respectively. $F_{\rho\pi}(k^{2})$ and $G_{a_{1}\pi}(k^{2})$
are the scalar functions of $k^{2}$ which contains the information of
$\rho\rightarrow\pi$ and $a_{1}\rightarrow\pi$ transitions, respectively.

  8. 8.

The sum over polarisation of $\rho$ or $a_{1}$-meson is given by,

$\epsilon_{\lambda}^{(V)}\epsilon_{\nu}^{(V)*}=-g_{\lambda\nu}+\frac{(p_{2}+k)_{\lambda}(p_{2}+k)_{\nu}}{m_{V}^{2}}$
(A.9)

  9. 9.

The $t$-dependence of the deacy widths of $\rho$ and $a_{1}$ mesons are given
by [26],

$\Gamma_{\rho}(t)=\Gamma_{\rho}\frac{m_{\rho}^{2}}{p_{\rho}^{3}}\frac{p^{3}}{t}$
(A.10)

with, $2p=(t-4m_{\pi}^{2})^{1/2}$ and
$2p_{\rho}=(m_{\rho}^{2}-4m_{\pi}^{2})^{1/2}$.

$\Gamma_{a_{1}}(t)=\frac{m_{a_{1}}\Gamma_{a_{1}}}{\sqrt{t}}\frac{g(t)}{g(m_{a_{1}}^{2})}$
(A.11)

with,

$g(t)=\begin{cases}4.1(t-9m_{\pi}^{2})^{3}(1-3.3(t-9m_{\pi}^{2})+5.8(t-9m_{\pi}^{2})^{2})&\text{if
\hskip 8.5359pt $t<(m_{\rho}+m_{\pi})^{2}$}\\\
t(1.623+\frac{10.38}{t}-\frac{9.38}{t^{2}}+\frac{0.65}{t^{3}})&\text{else}\end{cases}$

##  Appendix B Values of parameters used

Here, we tabulate the values of the various parameters used for numerical
calculation.

S.No. | Parameter | Symbol | Value
---|---|---|---
1. | Fine structure constant | $\alpha$ | ${\frac{1}{133.6}}$
2. | Fermi’s Constant | $G_{F}$ |  $1.166\times 10^{-5}\text{ GeV}^{-2}$[1]
3. | Mass of $\tau$-lepton | $m_{\tau}$ |  $(1776.86\pm 0.12)$ MeV [1]
4. | Pion decay constant | $f_{\pi}$ |  $(130.41\pm 0.23)$ MeV
5. | CKM Matrix element | $V_{ud}$ | $(0.9745\pm 0.0001)$ [1]
6. | Mass of $\rho$-meson | $m_{\rho}$ |  $(775.26\pm 0.25)$ MeV [1]
7. | Decay width of $\rho$-meson | $\Gamma_{\rho}$ |  $(149.1\pm 0.8)$ MeV [1]
8. | Mass of $a_{1}$-meson | $m_{a_{1}}$ |  $(1230\pm 40)$ MeV [1]
9. | Decay width of $a_{1}$-meson | $\Gamma_{a_{1}}$ |  $(425\pm 175)$ MeV [1]
10. | Vector form factor | $F_{V}^{(\pi)}(0)$ |  $0.0254\pm 0.0017$ [1]
11. | Axial-vector form factor | $F_{A}^{(\pi)}(0)$ |  $0.0119\pm 0.0001$ [1]
12. | $\alpha_{s}(1GeV)$ | $\alpha_{s}(1GeV)$ | $\sim 0.7$
13. | $\alpha_{s}(m_{\tau})$ | $\alpha_{s}(m_{\tau})$ | $0.325$
14. | $a_{2}(1GeV)$ | $a_{2}(1GeV)$ | $0.12$

The value of the fine structure constant is taken at the scale $m_{\tau}$ and
the decay width of $a_{1}$ meson is taken to the central value of the range
given in [1].

##  Appendix C Kinematics and Decay width

The differential decay width can be written as a sum of different components
[14]: $\Gamma_{IB}$ coming from $\overline{|\mathcal{A}_{IB}|}^{2}$,
$\Gamma_{SD}$ coming from $\overline{|\mathcal{A}_{SD}|}^{2}$ and
$\Gamma_{int}$ coming from
$2\overline{\mathcal{R}e(\mathcal{A}_{IB}^{*}\mathcal{A}_{SD})}$.
$\Gamma_{SD}$ is further divided into three parts: $\Gamma_{VV}$ coming from
$\overline{|\mathcal{A}_{v}|}^{2}$, $\Gamma_{AA}$ coming from
$\overline{|\mathcal{A}_{A}|}^{2}$ and $\Gamma_{AV}$ coming from
$2\overline{\mathcal{R}e(\mathcal{A}_{V}\mathcal{A}_{A}^{*})}$.

$\displaystyle\Gamma_{all}=\Gamma_{IB}+\Gamma_{int}+\Gamma_{SD},$
$\displaystyle\Gamma_{SD}=\Gamma_{VV}+\Gamma_{AV}+\Gamma_{AA},$
$\displaystyle\Gamma_{int}=\Gamma_{IB-A}+\Gamma_{IB-V}.$ (C.1)

For convenience, we use the dimensionless variables $x$ and $y$ defined as,

$x=\frac{2p_{1}.k}{m_{\tau}^{2}},\hskip
14.22636pty=\frac{2p_{1}.p_{2}}{m_{\tau}^{2}}.$ (C.2)

In the rest frame of tau, $x$ and $y$ are simply the energies of photon and
pion respectively in units of $\frac{m_{\tau}}{2}$. The kinematical boundaries
of $x$ and $y$ are given by,

$0\leq x\leq 1-r_{p}^{2},\hskip 14.22636pt1-x+\frac{r_{p}^{2}}{1-x}\leq y\leq
1+r_{p}^{2}$ (C.3)

where, $r_{p}^{2}=\frac{m_{\pi}^{2}}{m_{\tau}^{2}}$. We have considered pion
to be massless for form factor calculations and hence we will use
$r_{p}\rightarrow 0$ in our final answers.
The variable $t$, the invariant mass square of the pion-photon system, can be
written in terms of $x$ and $y$ as

$t=P^{2}=(p_{2}+k)^{2}=m_{\tau}^{2}(x+y-1)\hskip 14.22636pt\implies
P.k=\frac{m_{\tau}^{2}}{2}(x+y-1-r_{p}^{2}).$ (C.4)

In terms of variables $x$ and $y$, the differential decay width in the rest
frame of tau is,

$\frac{d^{2}\Gamma}{dxdy}=\frac{m_{\tau}}{256\pi^{3}}\overline{|\mathcal{A}|^{2}},$
(C.5)

where different contribution to the differential decay width are (calculated
using FeynCalc [27]):

$\displaystyle\frac{d^{2}\Gamma_{IB}}{dxdy}=\frac{\alpha}{2\pi}f_{IB}(x,y,r_{p}^{2})\frac{\Gamma_{\tau^{-}\rightarrow\pi^{-}\nu_{\tau}}}{(1-r_{p}^{2})^{2}},$
$\displaystyle\frac{d^{2}\Gamma_{SD}}{dxdy}=\frac{\alpha}{8\pi}\frac{m_{\tau}^{4}}{f_{\pi}^{2}}\left\\{|F_{V}^{(\pi)}|^{2}f_{VV}(x,y,r_{p}^{2})+2\mathcal{R}e(F_{A}^{(\pi)*}F_{V}^{(\pi)})f_{AV}(x,y,r_{p}^{2})+|F_{A}^{(\pi)}|^{2}f_{AA}(x,y,r_{p}^{2})\right\\}\frac{\Gamma_{\tau^{-}\rightarrow\pi^{-}\nu_{\tau}}}{(1-r_{p}^{2})^{2}},$
$\displaystyle\frac{d^{2}\Gamma_{int}}{dxdy}=\frac{\alpha}{2\pi}\frac{m_{\tau}^{2}}{f_{\pi}}\left[f_{IB-V}(x,y,r_{p}^{2})\mathcal{R}e(F_{V}^{(\pi)})+f_{IB-A}(x,y,r_{p}^{2})\mathcal{R}e(F_{A}^{(\pi)})\right]\frac{\Gamma_{\tau^{-}\rightarrow\pi^{-}\nu_{\tau}}}{(1-r_{p}^{2})^{2}},$
(C.6)

with $\alpha=\frac{e^{2}}{4\pi}$, being the fine structure constant,

$\displaystyle
f_{IB}(x,y,r_{p}^{2})=\frac{\left[r_{p}^{4}(x+2)-2r_{p}^{2}(x+y)+(x+y-1)(2-3x+x^{2}+xy)\right](r_{p}^{2}-y+1)}{(r_{p}^{2}-x-y+1)^{2}x^{2}},$
$\displaystyle
f_{VV}(x,y,r_{p}^{2})=-\left[r_{p}^{4}(x+y)+2r_{p}^{2}(1-y)(x+y)+(x+y-1)(-x+x^{2}-y+y^{2})\right],$
$\displaystyle
f_{AV}(x,y,r_{p}^{2})=-\left[r_{p}^{2}(x+y)+(1-x-y)(y-x)\right](r_{p}^{2}-x-y+1),$
$\displaystyle f_{AA}(x,y,r_{p}^{2})=f_{VV}(x,y,r_{p}^{2}),$ $\displaystyle
f_{IB-V}(x,y,r_{p}^{2})=-\frac{(r_{p}^{2}-x-y+1)(r_{p}^{2}-y+1)}{x},$
$\displaystyle
f_{IB-A}(x,y,r_{p}^{2})=-\frac{\left[r_{p}^{4}-2r_{p}^{2}(x+y)+(1-x+y)(x+y-1)\right](r_{p}^{2}-y+1)}{(r_{p}^{2}-x-y+1)x},$
(C.7)

and $\Gamma_{\tau^{-}\rightarrow\pi^{-}\nu_{\tau}}$ is the non-radiative decay
width given by,

$\Gamma_{\tau^{-}\rightarrow\pi^{-}\nu_{\tau}}=\frac{G_{F}^{2}|V_{ud}|^{2}f_{\pi}^{2}}{8\pi}m_{\tau}^{3}(1-r_{p}^{2})^{2}.$
(C.8)

The photon spectrum is obtained by integrating over $y$. Integration over $x$
will give the total decay width for radiative tau decay. The IB contribution
has the infrared divergences which can be fixed by putting a threshold on the
minimum energy of the emitted photon. SD contribution does not face any such
divergence and hence can be integrated over the full phase space.

$\Gamma(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\gamma)=\int_{x_{0}}^{1-r_{p}^{2}}dx\int_{1-x+\frac{r_{p}^{2}}{1-x}}^{1+r_{p}^{2}}dy\frac{d^{2}\Gamma}{dxdy}$
(C.9)

where, $x_{0}$ is the minimum energy cut for the photon energy in unit of
$\frac{m_{\tau}}{2}$.
To get the invariant mass spectrum of $\pi\gamma$ system, define another
dimensionless variable $z$ (as used in ref-[14]) as,

$z=\frac{t}{m_{\tau}^{2}}=x+y-1.$ (C.10)

The kinematical boundaries for the new variable are:

$z-r_{p}^{2}\leq x\leq 1-\frac{r_{p}^{2}}{z},\hskip 14.22636ptr_{p}^{2}\leq
z\leq 1.$ (C.11)

The $\pi\gamma$ spectrum can be obtained by substituting $y$ in terms of $z$
in $\frac{d^{2}\Gamma}{dxdy}$ and integrating it over $x$, i.e,

$\frac{d\Gamma}{dz}=\int_{z-r_{p}^{2}}^{1-\frac{r_{p}^{2}}{z}}dx\frac{d^{2}\Gamma}{dxdy}(x,y=z-x+1).$
(C.12)

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