# Radiative corrections to the
$\boldsymbol{\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}}$
($\boldsymbol{P_{1,2}=\pi,K}$) decays

Rafel Escribano1,2<EMAIL_ADDRESS>Jesús Alejandro Miranda2<EMAIL_ADDRESS>Pablo Roig3<EMAIL_ADDRESS>1Grup de Física Teòrica, Departament de
Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona),
Spain.
2Institut de Física d’Altes Energies (IFAE) and The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain.
3Departamento de Física, Centro de Investigación y de Estudios Avanzados del
Instituto Politécnico Nacional, Apdo. Postal 14-740, 07000 Ciudad de México,
México.

###### Abstract

The radiative corrections to the $\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}$
($P_{1,2}=\pi,K$) decays are calculated for the first time including the
structure-dependent real photon corrections, which are obtained using
Resonance Chiral Theory. Our results, whose uncertainty is dominated by the
model-dependence of the resummation of the radiative corrections and the
missing virtual structure-dependent contributions, allow for precise tests of
CKM unitarity, lepton flavour universality and non-standard interactions.

Effective Field Theories, Semileptonic decays, Tau decays

## Introduction

Semileptonic tau decays are well-known to be a clean laboratory for studying
QCD hadronic matrix elements at energies below $\sim 1.8$ GeV Davier:2005xq ;
Pich:2013lsa , where the light-flavoured resonances play a key role. All non-
perturbative information of the one-meson tau decays is encoded in the
corresponding $P$ decay constants, that are best determined in lattice QCD
FlavourLatticeAveragingGroupFLAG:2021npn . Two-meson tau decays are specified
in terms of two form factors, whose knowledge has improved over the years
thanks to the use of dispersion relations Pich:2001pj ; GomezDumm:2013sib ;
Gonzalez-Solis:2019iod ; Jamin:2001zq ; Moussallam:2007qc ; Boito:2008fq ;
Boito:2010me ; Antonelli:2013usa ; Bernard:2013jxa ; Escribano:2013bca ;
Escribano:2014joa ; Descotes-Genon:2014tla ; Escribano:2016ntp  and nourished
with high quality measurements OPAL:2000fde ; ALEPH:2005qgp ; BaBar:2009lyd ;
OPAL:1998rrm ; CLEO:1999dln ; Belle:2008xpe ; BaBar:2018qry ; Belle:2019bpr .
A similar good understanding has not yet been achieved in three-meson tau
decays GomezDumm:2003ku ; Moussallam:2007qc ; Dumm:2009va ; Dumm:2009kj ;
GomezDumm:2012dpx ; Shekhovtsova:2012ra ; Nugent:2013hxa ; Sanz-
Cillero:2017fvr ; JPAC:2018zwp ; Arteaga:2022xxy  or higher-multiplicity
modes, preventing for the moment their use in searches for new physics.

On the contrary, one- and two-meson tau decays have enabled significant and
promising new physics tests in recent years Garces:2017jpz ;
Cirigliano:2017tqn ; Miranda:2018cpf ; Cirigliano:2018dyk ; Rendon:2019awg ;
Chen:2019vbr ; Gonzalez-Solis:2019lze ; Gonzalez-Solis:2020jlh ; Chen:2020uxi
; Arroyo-Urena:2021dfe ; Arroyo-Urena:2021nil ; Chen:2021udz ;
Cirigliano:2021yto . At the precision attained, radiative corrections for
these decay modes become necessary, which motivated their improved evaluation
for the $\tau^{-}\to P^{-}\nu_{\tau}$ cases Guo:2010dv ; Guevara:2013wwa ;
Guevara:2021tpy ; Arroyo-Urena:2021dfe ; Arroyo-Urena:2021nil . For the di-
pion tau decays, the need for these corrections first stemmed from their use
in the dispersive integral rendering the leading order hadronic vacuum
polarization contribution to the muon $g-2$ Cirigliano:2001er ;
Cirigliano:2002pv ; Flores-Baez:2006yiq , which was again the target of our
recent analysis Miranda:2020wdg  (see also Refs. GutierrezSantiago:2020bhy ;
Chen:2022nxm ; Masjuan:2023qsp ). Ref. Antonelli:2013usa  put forward that,
assuming lepton universality, semileptonic kaon decay measurements could be
used to predict the corresponding (crossing-symmetric) tau decays, yielding a
$V_{us}$ determination closer to unitarity than with the tau decay branching
ratios. In that work, the model-independent radiative corrections were taken
into account and the structure-dependent ones were estimated (see also Ref.
Flores-Baez:2013eba ), resulting in a relative large (conservative)
uncertainty. Including these model-dependent effects is one of our main
motivations: here we focus on those with a real photon and defer the virtual
photon ones to a later dedicated study. Instead of relying on lepton
universality and checking CKM unitarity Antonelli:2013usa , one can in
principle test the latter, comparing the crossed channels, or directly bind
new physics non-standard interactions from $\tau^{-}\to(K\pi)^{-}\nu_{\tau}$
decays Rendon:2019awg . For completeness, we also include the radiative
corrections to the di-kaon tau decays and recall our reference results for the
di-pion mode Miranda:2020wdg . As noted in Ref. Cirigliano:2021yto , see Fig.
1 for instance, bounds on non-standard interactions from hadronic tau decays
are competitive and complementary to those coming from LHC searches and
electroweak precision observables. As a relevant example, the precise
comparison of $\tau\to\pi^{-}\pi^{0}\nu_{\tau}(\gamma)$ with
$e^{+}e^{-}\to\pi^{+}\pi^{-}(\gamma)$ data, which requires the radiative
corrections computed in this work (see also Ref. Miranda:2020wdg ), are able
to reduce the allowed new physics area (in the relevant Wilson coefficients
plane) by a factor $\sim 3$ Cirigliano:2021yto . Real radiation was computed
for the $\tau^{-}\to\eta^{(\prime)}\pi^{-}\nu_{\tau}$ decay channels in Ref.
Guevara:2016trs , showing that it can contend with the non-photon decays, as
$G$-parity and electromagnetic suppressions compete. Finally, we also estimate
the corresponding results for the $K^{-}\eta^{(\prime)}$ channels.

The structure of the paper is as follows. In section I, we recall the model-
independent description of the $\tau^{-}\to
P^{-}_{1}P^{0}_{2}\nu_{\tau}\gamma$ decays and give the leading real-photon
model-dependent corrections for the $K\pi$, $K\bar{K}$ and $\pi\pi$ cases,
where only the latter are known (see e.g. Refs. Cirigliano:2002pv ;
Miranda:2020wdg ). Branching ratios and spectra for the radiative decays are
analysed in section III, and the corresponding radiative correction factors
are computed in section IV. We show the consequences of including them in
section V, where we bind new physics couplings in an effective approach.
Finally, we conclude in section VI. Appendices cover $K_{\ell 3}$ decays (A),
structure-independent virtual corrections to di-meson tau decays (B), the non-
radiative decays (C), and the kinematics of these three- and four-body
processes (D).

##  I The $\boldsymbol{\tau^{-}\to P_{1}^{-}P_{2}^{0}\nu_{\tau}\gamma}$ decays

The most general structure for the decays $\tau(P)\to
P_{1}^{-}(p_{-})P_{2}^{0}(p_{0})\nu_{\tau}(q^{\prime})\gamma(k)$ is given by

$\begin{split}\mathcal{M}&=\frac{eG_{F}V^{*}_{uD}}{\sqrt{2}}\epsilon^{*}_{\mu}\bigg{[}\frac{H_{\nu}(p_{-},p_{0})}{k^{2}-2k\cdot
P}\bar{u}(q^{\prime})\gamma^{\nu}(1-\gamma^{5})(m_{\tau}+\not{P}-\not{k})\gamma^{\mu}u(P)\\\\[4.30554pt]
&\quad+\,(V^{\mu\nu}-A^{\mu\nu})\bar{u}(q^{\prime})\gamma_{\nu}(1-\gamma^{5})u(P)\bigg{]}\
,\end{split}$ (1)

with $V_{uD}$ ($D=d,s$) the corresponding CKM matrix element and where the
hadronic matrix element can be written as

$H^{\nu}(p_{-},p_{0})=C_{V}F_{+}(t)Q^{\nu}+C_{S}\frac{\Delta_{{-}{0}}}{t}q^{\nu}F_{0}(t)\
,$ (2)

with $t=q^{2}$,
$Q^{\nu}=(p_{-}-p_{0})^{\nu}-\frac{\Delta_{{-}{0}}}{t}q^{\nu}$,
$q^{\nu}=(p_{-}+p_{0})^{\nu}$, and $\Delta_{ij}=m_{i}^{2}-m_{j}^{2}$. One
recovers the usual definition of $H_{K\pi}^{\nu}$ Rendon:2019awg  by replacing
$p_{-}\to p_{K}$, $p_{0}\to p_{\pi}$ and $\Delta_{-0}\to\Delta_{K\pi}$ for
$K^{-}\pi^{0}$, and $p_{-}\to p_{\pi}$, $p_{0}\to p_{K}$, $C_{V,S}\to-C_{V,S}$
and $\Delta_{-0}\to-\Delta_{K\pi}$ for $\bar{K}^{0}\pi^{-}$ (we comment on the
identifications for the $P_{1}=P_{2}$ channels below). In all cases, gauge
invariance implies $k_{\mu}V^{\mu\nu}=H^{\nu}(p_{-},p_{0})$ and
$k_{\mu}A^{\mu\nu}=0$.

The structure-independent term is given by

$\begin{split}V^{\mu\nu}_{\rm
SI}&=\frac{H^{\nu}(p_{-}+k,p_{0})(2p_{-}+k)^{\mu}}{2k\cdot
p_{-}+k^{2}}-C_{V}\frac{F_{+}(t^{\prime})-F_{+}(t)}{k\cdot
q}Q^{\nu}q^{\mu}\\\\[4.30554pt]
&+\left\\{-C_{V}F_{+}(t^{\prime})-\frac{\Delta_{-0}}{t^{\prime}}\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]\right\\}g^{\mu\nu}\\\\[4.30554pt]
&+\frac{\Delta_{-0}}{tt^{\prime}}\left\\{2\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]-\frac{C_{S}t^{\prime}}{k\cdot
q}\left[F_{0}(t^{\prime})-F_{0}(t)\right]\right\\}q^{\mu}q^{\nu}\
,\end{split}$ (3)

where $C_{V}^{K^{-}\pi^{0}}=C_{S}^{K^{-}\pi^{0}}=1/\sqrt{2}$ for the
$K^{-}\pi^{0}$ channel, and
$C_{V}^{\bar{K}^{0}\pi^{-}}=C_{S}^{\bar{K}^{0}\pi^{-}}=-1$ for the
$\bar{K}^{0}\pi^{-}$ one, with $t^{\prime}=(P-q^{\prime})^{2}$. The main
difference between these two decay modes comes from the overall sign
difference —except for the first term in the second line of Eq. (3)—, that we
absorbed in our definition of $C_{V,S}^{\bar{K}^{0}\pi^{-}}$, and through the
order of the arguments of $H^{\nu}$ in the above equation. At leading order
(LO) in Chiral Perturbation Theory (ChPT), contributions proportional to
$g^{\mu\nu}$ in $V^{\mu\nu}_{\rm SI}$ stem from the diagrams in Fig. 1.

$\bar{K}^{0}$$\gamma$$\pi^{-}$$\otimes$

$K^{-}$$\gamma$$\pi^{0}$$\otimes$

Figure 1: Feynman diagrams contributing to the term proportional to the metric
tensor $g^{\mu\nu}$ in Eq. (3).

For the other tau decay modes, $V_{\rm SI}^{\mu\nu}$ is also obtained from Eq.
(3). In particular, we are also interested in the $\tau^{-}\to
K^{-}K^{0}\nu_{\tau}\gamma$ decays, where
$C_{V}^{K^{-}K^{0}}=C_{S}^{K^{-}K^{0}}=-1$ 111We discuss briefly the
$\pi^{-}\pi^{0}$ case at the end of Sec. II.0.1; see Ref. Miranda:2020wdg  for
further details..

The structure-dependent part is given by

$\begin{split}V^{\mu\nu}_{\rm SD}&=v_{1}(k\cdot
p_{-}g^{\mu\nu}-k^{\nu}p_{-}^{\mu})+v_{2}(k\cdot
p_{0}g^{\mu\nu}-k^{\nu}p_{0}^{\mu})\\\\[4.30554pt] &+v_{3}(k\cdot
p_{0}p_{-}^{\mu}-k\cdot p_{-}p_{0}^{\mu})p_{-}^{\nu}+v_{4}(k\cdot
p_{0}p_{-}^{\mu}-k\cdot p_{-}p_{0}^{\mu})(p_{-}+p_{0}+k)^{\nu}\ ,\end{split}$
(4)

and

$\begin{split}A_{\mu\nu}&=ia_{1}\epsilon_{\mu\nu\rho\sigma}\left(p_{0}-p_{-}\right)^{\rho}k^{\sigma}+ia_{2}\left(P-q^{\prime}\right)_{\nu}\epsilon_{\mu\rho\sigma\tau}k^{\rho}p_{-}^{\sigma}p_{0}^{\tau}\\\\[4.30554pt]
&+ia_{3}\epsilon_{\mu\nu\rho\sigma}k^{\rho}(P-q^{\prime})^{\sigma}+ia_{4}(p_{0}+k)_{\nu}\epsilon_{\mu\lambda\rho\sigma}k^{\lambda}p_{-}^{\rho}p_{0}^{\sigma}\
,\end{split}$ (5)

where $p_{-}$ and $p_{0}$ refer to the momentum of the charged and neutral
meson, respectively.

From Eq. (3), it is easy to show that the Low’s theorem Low:1958sn  is
manifestly satisfied,

$\begin{split}V^{\mu\nu}&=\frac{p_{-}^{\mu}}{k\cdot
p_{-}}H^{\nu}(p_{-},p_{0})+\left\\{C_{V}F_{+}(t)+\frac{\Delta_{-0}}{t}\left[C_{S}F_{0}(t)-C_{V}F_{+}(t)\right]\right\\}\left(\frac{p_{-}^{\mu}k^{\nu}}{k\cdot
p_{-}}-g^{\mu\nu}\right)\\\\[4.30554pt]
&-\frac{2\Delta_{-0}}{t^{2}}\left[C_{S}F_{0}(t)-C_{V}F_{+}(t)\right]\left(\frac{k\cdot
p_{0}}{k\cdot
p_{-}}p_{-}^{\mu}-p_{0}^{\mu}\right)(p_{-}+p_{0})^{\nu}\\\\[4.30554pt]
&+2\left(\frac{k\cdot p_{0}}{k\cdot
p_{-}}p_{-}^{\mu}-p_{0}^{\mu}\right)\left[C_{V}\frac{dF_{+}(t)}{dt}Q^{\nu}+C_{S}\frac{\Delta_{{-}{0}}}{t}q^{\nu}\frac{dF_{0}(t)}{dt}\right]+\mathcal{O}(k)\
,\end{split}$ (6)

and the amplitude reads

$\begin{split}\mathcal{M}&=\frac{eG_{F}V_{uD}\sqrt{S_{\rm
EW}}}{\sqrt{2}}\epsilon^{*}_{\mu}(k)H_{\nu}(p_{-},p_{0})\bar{u}(q)\gamma^{\nu}(1-\gamma^{5})u(P)\\\\[4.30554pt]
&\quad\times\left(\frac{p_{-}^{\mu}}{k\cdot
p_{-}+\frac{1}{2}M_{\gamma}^{2}}-\frac{P^{\mu}}{k\cdot
P-\frac{1}{2}M_{\gamma}^{2}}\right)+\mathcal{O}(k^{0})\ ,\end{split}$ (7)

where $S_{\rm EW}$ encodes the short-distance electroweak corrections
Sirlin:1974ni ; Sirlin:1977sv ; Sirlin:1981ie ; Marciano:1985pd ;
Marciano:1988vm ; Marciano:1993sh ; Braaten:1990ef ; Erler:2002mv .

In the Low’s limit, one gets

$\begin{split}\overline{\left|\mathcal{M}\right|^{2}}&=2e^{2}G_{F}^{2}\left|V_{uD}\right|^{2}S_{\rm
EW}\left\\{C_{S}^{2}\left|F_{0}(t)\right|^{2}D_{0}^{P^{-}P^{0}}(t,u)\right.+C_{S}C_{V}\mathrm{Re}\left[F_{+}(t)F_{0}^{*}(t)\right]D_{+0}^{P^{-}P^{0}}(t,u)\\\\[4.30554pt]
&\left.+\,C_{V}^{2}\left|F_{+}(t)\right|^{2}D_{+}^{P^{-}P^{0}}(t,u)\right\\}\sum_{\gamma\,\rm
pols.}\left|\frac{p_{-}\cdot\epsilon}{p_{-}\cdot
k}-\frac{P\cdot\epsilon}{P\cdot k}\right|^{2}+\mathcal{O}(k^{0})\
,\end{split}$ (8)

where

$\begin{split}D_{+}^{P^{-}P^{0}}(t,u)&=\frac{m_{\tau}^{2}}{2}(m_{\tau}^{2}-t)+2m_{0}^{2}m_{-}^{2}-2u(m_{\tau}^{2}-t+m_{0}^{2}+m_{-}^{2})+2u^{2}\\\\[4.30554pt]
&+\frac{\Delta_{-0}}{t}m_{\tau}^{2}(2u+t-m_{\tau}^{2}-2m_{0}^{2})+\frac{\Delta_{-0}^{2}}{t^{2}}\frac{m_{\tau}^{2}}{2}(m_{\tau}^{2}-t)\
,\end{split}$ (9)
$D_{0}^{P^{-}P^{0}}(t,u)=\frac{\Delta_{-0}^{2}m_{\tau}^{4}}{2t^{2}}\left(1-\frac{t}{m_{\tau}^{2}}\right)\
,$ (10)
$D_{+0}^{P^{-}P^{0}}(t,u)=\frac{\Delta_{0-}m_{\tau}^{2}}{t}\left[2u+t-m_{\tau}^{2}-2m_{0}^{2}+\frac{\Delta_{-0}}{t}(m_{\tau}^{2}-t)\right]\
,$ (11)

with $u=(P-p_{-})^{2}$. In this way, besides the Low theorem, the Burnet-Kroll
theorem Burnett:1967km  is also explicitly manifest.

Thus, after integration over neutrino and photon 4-momenta, the differential
decay width in this approximation reads

$\begin{split}\left.\frac{d\Gamma^{(0)}}{dt\,du}\right|_{PP\gamma}&=\frac{G_{F}^{2}|V_{uD}|^{2}S_{\rm
EW}}{128\pi^{3}m_{\tau}^{3}}\left\\{C_{S}^{2}\left|F_{0}(t)\right|^{2}D_{0}^{P^{-}P^{0}}(t,u)\right.+C_{V}C_{S}\mathrm{Re}\left[F_{+}^{*}(t)F_{0}(t)\right]D_{+0}^{P^{-}P^{0}}(t,u)\\\\[4.30554pt]
&\quad\left.+\,C_{V}^{2}\left|F_{+}(t)\right|^{2}D_{+}^{P^{-}P^{0}}(t,u)\right\\}g_{\rm
rad}(t,u,M_{\gamma})\ ,\end{split}$ (12)

where (see Refs. Cirigliano:2001er ; Cirigliano:2002pv )

$g_{\rm rad}(t,u,M_{\gamma})=g_{\rm brems}(t,u,M_{\gamma})+g_{\rm rest}(t,u)\
,$ (13)

with

$\displaystyle g_{\rm brems}(t,u,M_{\gamma})$
$\displaystyle=\frac{\alpha}{\pi}\left(J_{11}(t,u,M_{\gamma})+J_{20}(t,u,M_{\gamma})+J_{02}(t,u,M_{\gamma})\right)\
,$ (14a) $\displaystyle g_{\rm rest}(t,u)$
$\displaystyle=\frac{\alpha}{\pi}\left(K_{11}(t,u)+K_{20}(t,u)+K_{02}(t,u)\right)\
.$ (14b)

The relevant expressions for $J_{ij}(t,u,M_{\gamma})$ and $K_{ij}(t,u)$, which
correspond to an integration over $\mathcal{D}^{\rm III}$ and
$\mathcal{D}^{\rm IV/III}$, respectively, can be found in Refs.
Cirigliano:2002pv ; Miranda:2020wdg ; Antonelli:2013usa  and in App. D222The
function $K_{11}(t,u)$ turns out to be numerically negligible and is not
quoted anywhere; see e.g. Refs Cirigliano:2001er ; Miranda:2018cpf ..

Integrating upon the $u$ variable in Eq. (12), one gets

$\begin{split}\left.\frac{d\Gamma}{dt}\right|_{\rm III}&=\frac{G_{F}^{2}S_{\rm
EW}|V_{uD}|^{2}m_{\tau}^{3}}{384\pi^{3}t}\left\\{\frac{1}{2t^{2}}\left(1-\frac{t}{m_{\tau}^{2}}\right)^{2}\lambda^{1/2}(t,m_{-}^{2},m_{0}^{2})\right.\\\\[4.30554pt]
&\times\left[C_{V}^{2}\left|F_{+}(t)\right|^{2}\left(1+\frac{2t}{m_{\tau}^{2}}\right)\lambda(t,m_{-}^{2},m_{0}^{2})\overline{\delta}_{+}(t)+3C_{S}^{2}\Delta_{-0}^{2}\left|F_{0}(t)\right|^{2}\overline{\delta}_{0}(t)\right]\left.+\,C_{S}C_{V}\frac{4}{\sqrt{t}}\overline{\delta}_{+0}(t)\right\\}\
,\end{split}$ (15)

with

$\displaystyle\overline{\delta}_{0}(t)=$ $\displaystyle\
\frac{\int_{u^{-}(t)}^{u^{+}(t)}D_{0}^{P^{-}P^{0}}(t,u)\,g_{\rm
brems}(t,u,M_{\gamma})du}{\int_{u^{-}(t)}^{u^{+}(t)}D_{0}^{P^{-}P^{0}}(t,u)\,du}\
,$ (16a) $\displaystyle\overline{\delta}_{+}(t)=$ $\displaystyle\
\frac{\int_{u^{-}(t)}^{u^{+}(t)}D_{+}^{P^{-}P^{0}}(t,u)\,g_{\rm
brems}(t,u,M_{\gamma})du}{\int_{u^{-}(t)}^{u^{+}(t)}D_{+}^{P^{-}P^{0}}(t,u)\,du}\
,$ (16b) $\displaystyle\overline{\delta}_{+0}(t)=$ $\displaystyle\
\frac{3t\sqrt{t}}{4m_{\tau}^{6}}\int_{u^{-}(t)}^{u^{+}(t)}D_{+0}^{P^{-}P^{0}}(t,u)\,g_{\rm
brems}(t,u,M_{\gamma})\mathrm{Re}\left[F_{+}^{*}(t)F_{0}(t)\right]du\ .$ (16c)

The remaining contribution, $\left.d\Gamma/dt\right|_{\rm IV/III}$, which
corresponds to the integration over $\mathcal{D}_{\rm IV/III}$ with $g_{\rm
rest}(t,u)$ instead of $g_{\rm brems}(t,u,M_{\gamma})$, is almost negligible
and only becomes relevant near threshold. In Ref. Cirigliano:2008wn , the
subleading contributions in the Low’s approximation were studied, showing that
they are not negligible and need to be taken into account to get a reliable
estimation.

##  II Structure-dependent contributions

The evaluation of the structure-dependent tensors, $V^{\mu\nu}_{SD}$ and
$A^{\mu\nu}$ in eqs. (4) and (5), requires non-perturbative methods or lattice
QCD (which has only been explored for the $\pi\pi$ case, see Bruno:2018ono ).
The tau lepton mass value probes the hadronization of QCD currents in its
semileptonic decays beyond the regime of validity of Chiral Perturbation
Theory Weinberg:1978kz ; Gasser:1983yg ; Gasser:1984gg ; Bijnens:1999sh ;
Bijnens:2001bb  ($\chi PT$), which is the low-energy effective field theory of
QCD describing $P$ meson physics. To extend the applicability of the chiral
Lagrangians to the energy region where meson states resonate, a successful
strategy has been to include the corresponding fields as explicit degrees of
freedom into the action, using approximate flavor symmetry, without additional
assumptions affecting the possible resonance dynamics Ecker:1988te ;
Ecker:1989yg . This procedure was later christened Resonance Chiral Theory
($R\chi T$, see e.g. ref. Portoles:2010yt ) and yields as a result the
saturation of the $\chi PT$ low-energy constants upon resonance integration.

Explicitly, the construction of the relevant Lagrangian pieces including
resonances uses the chiral tensors Bijnens:2001bb  (we only quote those
relevant in our computation)

$\displaystyle
u_{\mu}=i\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-i\ell_{\mu})u^{\dagger}\right],$
$\displaystyle f_{\pm}^{\mu\nu}=uF_{L}^{\mu\nu}u^{\dagger}\pm
u^{\dagger}F_{R}^{\mu\nu}u,$ (17) $\displaystyle\chi_{\pm}=u^{\dagger}\chi
u^{\dagger}\pm u\chi u\,,$

where the pseudo-Goldstone bosons are included in $u$ via ($\lambda_{a}$ are
the Gell-Mann matrices, so that $\phi^{3}=\pi^{0}$ in the isospin symmetry
limit)

$u=\mathrm{exp}\left(\frac{i\Phi}{\sqrt{2}f}\right),\qquad\Phi=\sum_{a=0}^{8}\frac{\lambda_{a}}{\sqrt{2}}\phi^{a}\,,$
(18)

and left- and right-handed sources $\ell_{\mu}$ and $r_{\mu}$ enter through

$F_{X}^{\mu\nu}=\partial^{\mu}x^{\nu}-\partial^{\nu}x^{\mu}-i[x^{\mu},x^{\nu}]=eQF^{\mu\nu}+\dots,\quad
x=\ell,\,r;\quad
Q=\mathrm{diag}\left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right),$ (19)

with $x_{\mu}=eQA_{\mu}+\dots$, being $A_{\mu}$ the photon field and
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ the corresponding
field-strength tensor. Spin-zero sources ($s$ and $p$) appear in $\chi_{\pm}$
through $\chi=2B(s+ip)$, where we recall that the two leading chiral low-
energy constants ($f$ and $B$) determine the light-quark condensate in the
chiral limit, $-Bf^{2}=<0|\bar{q}q|0>$, with $f\sim 90$ MeV the pion decay
constant. The numerical value of $B$ is not needed, as it only enters the
pseudoscalar meson squared masses, which are proportional to it. Indeed, in
the isospin symmetry limit,
diag$(2Bs)=(m_{\pi}^{2},\,m_{\pi}^{2},\,2m_{K}^{2}-m_{\pi}^{2})$, where $s$
accounts for the (diagonal) light-quark mass matrix, which ensure chiral
symmetry breaking as in QCD.

The $R\chi T$ Lagrangian includes the lowest order $\chi PT$ Lagrangian in
both parity sectors, which is

$\mathcal{L}_{\mathrm{non-
res}}^{\mathrm{even}}=\frac{f^{2}}{4}<u^{\mu}u_{\mu}+\chi_{+}>\,,\qquad\mathcal{L}_{\mathrm{non-
res}}^{\mathrm{odd}}=\mathcal{L}_{\mathrm{WZW}}\,,\;$ (20)

where WZW stands for the chiral anomaly contribution worked out by Wess-Zumino
and Witten Wess:1971yu ; Witten:1983tw .

In addition, $\mathcal{L}_{R\chi T}$ has pieces including resonance fields and
chiral tensors. These are usually incorporated taking into account the order
(within the chiral counting) of their contributions -upon resonance exchange-
to the $\chi PT$ couplings Ecker:1988te ; Ecker:1989yg ; Cirigliano:2006hb ;
Kampf:2011ty , as well as their behaviour Pich:2002xy  in the limit of a large
number of colors tHooft:1973alw ; tHooft:1974pnl . QCD asymptotic behaviour
forbids (linear combinations of) operators with increasing number of
derivatives. In this way $R\chi T$ bridges between the low-energy behaviour of
$\chi PT$ and the high-energy constraints of perturbative QCD, keeping
predictivity for a set of related observables, to a given precision, without
unnecessary assumptions (like, for instance vector meson dominance
Sakurai:1960ju  or any additional symmetry of gauge type related to them
Bando:1987br ). Within this framework, the need for non-resonant contributions
was explored e. g. in ref. Roig:2014uja  for the $\omega-\pi^{0}$ transition
form factor, where it was found that they could play a rôle above 2 GeV only.
Taking this into account and the fact that we are limited kinematically by the
tau mass, we neglect non-resonant pieces in the following.

Apart from contributions that are suppressed by approximate flavor symmetries
(more on this below), the next-to-leading order $\chi PT$ couplings are
saturated by spin-one resonance exchange Ecker:1988te ; Ecker:1989yg  coming
from the following operators

$\mathcal{L}_{V}=\frac{F_{V}}{2\sqrt{2}}\left\langle
V_{\mu\nu}f_{+}^{\mu\nu}\right\rangle+i\frac{G_{V}}{2\sqrt{2}}\left\langle
V_{\mu\nu}\left[u^{\mu},u^{\nu}\right]\right\rangle\,,\qquad\mathcal{L}_{A}=\frac{F_{A}}{2\sqrt{2}}\left\langle
A_{\mu\nu}f_{-}^{\mu\nu}\right\rangle\,,$ (21)

where $V_{\mu\nu}=\sum_{a=0}^{8}\frac{\lambda^{a}}{\sqrt{2}}V_{\mu\nu}^{a}$,
and analogously for the axial-vector resonances. For convenience, spin-one
resonances are worked out in the antisymmetric tensor field formalism
Ecker:1988te ; Ecker:1989yg . For completeness we note that the kinetic terms
come from the Lagrangian (which also includes interactions, hidden in the
covariant derivative, that are however not needed in what follows, so
$\nabla^{\alpha}\sim\partial^{\alpha}$)

$\mathcal{L}_{\mathrm{Res}}^{\mathrm{Kin}}=-\frac{1}{2}\left\langle\nabla_{\lambda}V^{\lambda\nu}\nabla^{\rho}V_{\rho\nu}\right\rangle+\frac{M_{V}^{2}}{4}\left\langle
V_{\mu\nu}V^{\mu\nu}\right\rangle,$ (22)

with obvious replacements for axial-vector resonances, $A_{\mu\nu}$.

$U(3)$ symmetry in the (axial-)vector resonance masses is broken by operators
of the form $\left\langle V_{\mu\nu}V^{\mu\nu}\chi_{+}\right\rangle$,
$V\leftrightarrow A$ Cirigliano:2003yq ; Guo:2009hi ; Roig:2019reh , which can
accommodate the measured spectra (in this way we will simply replace the
different spin-one resonance masses by the PDG values
ParticleDataGroup:2022pth  in the following). Analogously, breaking of this
flavor symmetry also affects $F_{V,A}$. However, in the vector case, the
coupling giving this shift 333It is $\lambda_{6}^{V}$ in the notation of ref.
Cirigliano:2006hb  and $\lambda_{V}$ in ref. Guevara:2018rhj , with
$\lambda_{6}^{V}=\frac{\lambda_{V}}{\sqrt{2}}$.Cirigliano:2006hb  is
constrained by short-distance QCD to vanish within the scheme considered in
ref. Guevara:2018rhj  (see also ref. Roig:2014uja , where these contributions
were neglected a priori). Based on this, flavor symmetry on $F_{V,A}$ is
assumed in the remainder of the paper.

All resonance contributions to the $v_{i}$ form factors (those to the $a_{i}$
are differed by one chiral order and thus neglected) depend on ratios of
resonance couplings over the meson decay constant
$\left(\frac{F_{V}G_{V}}{f^{2}},\frac{F_{V}^{2}}{f^{2}},\frac{F_{A}^{2}}{f^{2}}\right)$,
which are constrained by asymptotic QCD. Therefore the well-known $SU(3)$
symmetry breaking which causes $f_{K}\sim 1.2f_{\pi}$
ParticleDataGroup:2022pth  cannot be accounted for, within the considered
simplified scheme, without conflicting with short-distance QCD requirements.
Thus, we are using $f\sim 90$ MeV for the $v_{i}$ also in the strangeness-
changing channels 444The associated error is, however, much smaller than the
uncertainty that we will find in our results, see section IV.. The dispersive
constructions giving the $F_{+,0}$ form factors account for flavor symmetry
breaking in the $K\pi$ channels, as required by the precision of the
corresponding measurements, which fed the phaseshifts entering the dispersive
integrals.

####  II.0.1 Vector contributions

Including those Lagrangian terms that, upon resonance integration, contribute
to the ChPT $\mathcal{O}(p^{4})$ low-energy constants (LECs), we have found
the following contributions to the vector form factors $v_{i}$ in Eq. (4),
which are depicted in Fig. 2:

$P^{-}$$P^{(\prime)0}$$\gamma$$\otimes$$V^{-}$

$\gamma$$P^{-}$$P^{(\prime)0}$$\otimes$$V^{-}$

$\otimes$$\gamma$$P^{-}$$P^{(\prime)0}$$V^{-}$$V^{-}$

$\otimes$$\gamma$$P^{-}$$P^{(\prime)0}$$V^{-}$

$\otimes$$\gamma$$P^{(\prime)0}$$P^{-}$$V^{-}$$V^{0}$

$\otimes$$\gamma$$P^{-}$$P^{(\prime)0}$$V^{-}$$V^{0}$

$P^{-}$$P^{(\prime)0}$$\otimes$$\gamma$$V^{0}$

$A^{-}$$P^{(\prime)0}$$\otimes$$P^{-}$$\gamma$

Figure 2: Vector and axial-vector meson exchange diagrams contributing to the
$\tau^{-}\to P_{1}^{-}P_{2}^{0}\nu_{\tau}\gamma$ decays at
$\mathcal{O}\left(p^{4}\right)$. $V^{0}$ stands for the $\rho^{0}$, $\omega$
and $\phi$ resonances, $V^{-}=K^{*-}$ for the $K\pi$ modes and
$V^{-}=\rho^{-}$ for the $K^{-}K^{0}$ one, and $A^{-}=K_{1}^{-}$ in
$K^{-}K^{0}$ and $K^{-}\pi^{0}$, and $A^{-}=a_{1}^{-}$ in
$\pi^{-}\bar{K}^{0}$.

$\displaystyle v_{1}=$ $\displaystyle\
\frac{F_{V}G_{V}}{\sqrt{2}f^{2}M_{\rho}^{2}}\left\\{\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\left[1+\frac{1}{2}\left(t-\Delta_{K\pi}\right)D_{K^{*}}^{-1}(t)\right]\right.$
$\displaystyle+2M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})+M_{\rho}^{2}\left(t-\Delta_{K\pi}\right)D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})\left.\right\\}$
$\displaystyle+\frac{F_{V}^{2}}{2\sqrt{2}f^{2}M_{\rho}^{2}}\left[-\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\left(1-t^{\prime}\,D_{K^{*}}^{-1}(t^{\prime})\right)-M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})\right]$
$\displaystyle+\frac{F_{A}^{2}}{\sqrt{2}f^{2}M_{K_{1}}^{2}}\left(M_{K_{1}}^{2}-\frac{1}{2}\Sigma_{K\pi}+\frac{1}{2}t\right)D_{K_{1}}^{-1}[(p_{K}+k)^{2}]\
,$ (23a) $\displaystyle v_{2}=$ $\displaystyle\
\frac{F_{V}G_{V}}{\sqrt{2}f^{2}M_{\rho}^{2}}\left(t+\Delta_{K\pi}\right)\left[-\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)D_{K^{*}}^{-1}(t)-M_{\rho}^{2}D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})\right]$
$\displaystyle+\frac{F_{V}^{2}}{2\sqrt{2}f^{2}M_{\rho}^{2}}\left[-\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\left(1+t^{\prime}\,D_{K^{*}}^{-1}(t^{\prime})\right)-M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})\right]$
$\displaystyle+\frac{F_{A}^{2}}{\sqrt{2}f^{2}M_{K_{1}}^{2}}\left(M_{K_{1}}^{2}-m_{K}^{2}-k\cdot
p_{K}\right)D_{K_{1}}^{-1}[(p_{K}+k)^{2}]\ ,$ (23b) $\displaystyle v_{3}=$
$\displaystyle\
\frac{F_{A}^{2}}{\sqrt{2}f^{2}M_{K_{1}}^{2}}D_{K_{1}}^{-1}[(p_{K}+k)^{2}]\ ,$
(23c) $\displaystyle v_{4}=$ $\displaystyle\
-\frac{2F_{V}G_{V}}{\sqrt{2}f^{2}}D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})+\frac{F_{V}^{2}}{2\sqrt{2}f^{2}M_{\rho}^{2}}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)D_{K^{*}}^{-1}(t^{\prime})\
,$ (23d)

for $K^{-}\pi^{0}$,

$\displaystyle v_{1}=$
$\displaystyle-\frac{F_{V}G_{V}}{f^{2}M_{\rho}^{2}}\left[2+2M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})+\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\left(t+\Delta_{K\pi}\right)D_{K^{*}}^{-1}(t)\right.$
$\displaystyle+\left(t+\Delta_{K\pi}\right)M_{\rho}^{2}D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})\left.\right]$
$\displaystyle-\frac{F_{V}^{2}}{2f^{2}M_{\rho}^{2}}\left[-M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})+\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)t^{\prime}\,D_{K^{*}}^{-1}(t^{\prime})+\frac{1}{2}\left(-3+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\right]$
$\displaystyle-\frac{F_{A}^{2}}{f^{2}M_{a_{1}}^{2}}\left(M_{a_{1}}^{2}-\frac{1}{2}\Sigma_{K\pi}+\frac{1}{2}t\right)D_{a_{1}}^{-1}[(p_{\pi}+k)^{2}]\
,$ (24a) $\displaystyle v_{2}=$
$\displaystyle-\frac{F_{V}G_{V}}{f^{2}M_{\rho}^{2}}\left\\{\left(t-\Delta_{K\pi}\right)\left[-M_{\rho}^{2}D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})-\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)D_{K^{*}}^{-1}(t)\right]\right.$
$\displaystyle+1-\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}-\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\left.\right\\}$
$\displaystyle-\frac{F_{V}^{2}}{2f^{2}M_{\rho}^{2}}\left[-M_{\rho}^{2}D_{K^{*}}^{-1}(t^{\prime})-\frac{1}{2}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)t^{\prime}\,D_{K^{*}}^{-1}(t^{\prime})+\frac{1}{2}\left(-3+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)\right]$
$\displaystyle-\frac{F_{A}^{2}}{f^{2}M_{a_{1}}^{2}}\left(M_{a_{1}}^{2}-m_{\pi}^{2}k\cdot
p_{\pi}\right)D_{a_{1}}^{-1}[(p_{\pi}+k)^{2}]\ ,$ (24b) $\displaystyle v_{3}$
$\displaystyle=-\frac{F_{A}^{2}}{f^{2}M_{a_{1}}^{2}}D_{a_{1}}^{-1}[(p_{\pi}+k)^{2}]\
,$ (24c) $\displaystyle v_{4}$
$\displaystyle=\frac{2F_{V}G_{V}}{f^{2}}D_{K^{*}}^{-1}(t)D_{K^{*}}^{-1}(t^{\prime})-\frac{F_{V}^{2}}{2f^{2}M_{\rho}^{2}}\left(1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}\right)D_{K^{*}}^{-1}(t^{\prime})\
,$ (24d)

for $\bar{K}^{0}\pi^{-}$, and

$\displaystyle v_{1}=$
$\displaystyle-\frac{F_{V}G_{V}}{f^{2}M_{\rho}^{2}}\left[1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}+\left(t-\Delta_{K^{-}K^{0}}\right)D_{\rho}^{-1}(t)+2M_{\rho}^{2}D_{\rho}^{-1}(t^{\prime})\right.$
$\displaystyle+M_{\rho}^{2}\left(t-\Delta_{K^{-}K^{0}}\right)D_{\rho}^{-1}(t)D_{\rho}^{-1}(t^{\prime})\left.\right]$
$\displaystyle-\frac{F_{V}^{2}}{2f^{2}M_{\rho}^{2}}\left[-\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}-\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}+t^{\prime}\,D_{\rho}^{-1}(t^{\prime})-M_{\rho}^{2}D_{\rho}^{-1}(t^{\prime})\right]$
$\displaystyle-\frac{F_{A}^{2}}{f^{2}M_{K_{1}}^{2}}\left(M_{K_{1}}^{2}-\frac{1}{2}\Sigma_{K^{-}K^{0}}+\frac{1}{2}t\right)D_{K_{1}}^{-1}[(p_{-}+k)^{2}]\
,$ (25a) $\displaystyle v_{2}=$
$\displaystyle-\frac{F_{V}G_{V}}{f^{2}M_{\rho}^{2}}\left[-1+\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}+\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}-\left(t+\Delta_{K^{-}K^{0}}\right)D_{\rho}^{-1}(t)-M_{\rho}^{2}\left(t+\Delta_{K^{-}K^{0}}\right)D_{\rho}^{-1}(t)D_{\rho}^{-1}(t^{\prime})\right]$
$\displaystyle-\frac{F_{V}^{2}}{2f^{2}M_{\rho}^{2}}\left[-\frac{1}{3}\frac{M_{\rho}^{2}}{M_{\omega}^{2}}-\frac{2}{3}\frac{M_{\rho}^{2}}{M_{\phi}^{2}}-t^{\prime}\,D_{\rho}^{-1}(t^{\prime})-M_{\rho}^{2}D_{\rho}^{-1}(t^{\prime})\right]$
$\displaystyle-\frac{F_{A}^{2}}{f^{2}M_{K_{1}}^{2}}\left(M_{K_{1}}^{2}-m_{K^{-}}^{2}-k\cdot
p_{-}\right)D_{K_{1}}^{-1}[(p_{-}+k)^{2}]\ ,$ (25b) $\displaystyle v_{3}=$
$\displaystyle-\frac{F_{A}^{2}}{f^{2}M_{K_{1}}^{2}}D_{K_{1}}^{-1}[(p_{-}+k)^{2}]\
,$ (25c) $\displaystyle v_{4}=$ $\displaystyle\
\frac{2F_{V}G_{V}}{f^{2}}D_{\rho}^{-1}(t)D_{\rho}^{-1}(t^{\prime})-\frac{F_{V}^{2}}{f^{2}M_{\rho}^{2}}D_{\rho}^{-1}(t^{\prime})\
,$ (25d)

for $K^{-}K^{0}$, where $\Sigma_{-0}=m_{-}^{2}+m_{0}^{2}$ and
$D_{R}(x)=M_{R}^{2}-x-iM\Gamma_{R}(x)$. Off-shell resonance widths555The on-
shell width corresponds to the imaginary part of the pole position of the
resonance. The imaginary part of the corresponding loop function provides an
off-shell width function, which extends off the pole GomezDumm:2000fz . are
given in terms of the leading pseudo-Goldstone boson cuts GomezDumm:2000fz ;
Dumm:2009va ; Dumm:2009kj .

It is straightforward to show that, except for a Clebsch-Gordan coefficient
(CGC) factor, one recovers the expressions found in Refs. Cirigliano:2002pv ;
Miranda:2020wdg  for the vector form factors of the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}\gamma$ decays in the isospin-symmetry
limit.

All the former resonance contributions are given in terms of three couplings:
$F_{V}$, responsible for instance of the coupling of the vector resonance to
the vector current; $F_{A}$ for the couplings of the axial resonance; and
$G_{V}$ which yields, among others, vertices between the vector resonance and
a couple of pseudo-Goldstone bosons (see e.g. Ref. Ecker:1988te  for further
details).

####  II.0.2 Axial contributions

The Feynman diagrams that contribute to these decays are depicted in Figs.
3–5. At $\mathcal{O}(p^{4})$, the axial form factors $a_{i}$ in Eq. (5), which
receive contributions from the Wess-Zumino-Witten functional Wess:1971yu ;
Witten:1983tw , are given by

$a_{1}=\frac{N_{c}}{12\sqrt{2}\pi^{2}f^{2}}\ ,\quad
a_{2}=-\frac{N_{c}}{6\sqrt{2}\pi^{2}f^{2}(t^{\prime}-m_{K}^{2})}\ ,\quad
a_{3}=-\frac{N_{c}}{24\sqrt{2}\pi^{2}f^{2}}\ ,$ (26)

for $K^{-}\pi^{0}$,

$a_{3}=-\frac{N_{c}}{24\pi^{2}f^{2}}\ ,$ (27)

for $\bar{K}^{0}\pi^{-}$, and

$a_{3}=\frac{N_{c}}{24\pi^{2}f^{2}}\ ,$ (28)

for $K^{-}K^{0}$, where $N_{c}=3$ is the number of colours and $f$ is the pion
decay constant in the chiral limit, $f\sim 90$ MeV.

$\gamma$$K^{-}$$\pi^{0}$$\otimes$

$\otimes$$\gamma$$K^{-}$$\pi^{0}$$K^{-}$

Figure 3: Axial contributions to the $\tau^{-}\to
K^{-}\pi^{0}\nu_{\tau}\gamma$ decays at $\mathcal{O}\left(p^{4}\right)$.
$\gamma$$\bar{K}^{0}$$\pi^{-}$$\otimes$ Figure 4: Axial contributions to the
$\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma$ decays at
$\mathcal{O}\left(p^{4}\right)$. $\gamma$$K^{0}$$K^{-}$$\otimes$ Figure 5:
Axial contributions to the $\tau^{-}\to K^{-}K^{0}\nu_{\tau}\gamma$ decays at
$\mathcal{O}\left(p^{4}\right)$.

In Fig. 5, only one diagram contributes to the $\tau^{-}\to
K^{-}K^{0}\nu_{\tau}\gamma$ decays similarly to the
$\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma$ decays. This is because the
$K^{-}\to\bar{K}^{0}\pi^{-}\gamma$ (or $\pi^{-}\to K^{-}K^{0}\gamma$) vertex
is absent in the WZW Lagrangian666This feature was already studied for the
$K_{\ell 3}$ decays in Ref. Bijnens:1992en , where the non-local kaon pole
term is only present in $A^{+}_{\mu\nu}$ for
$K^{+}\to\pi^{0}\ell^{+}\nu_{\ell}\gamma$ decays.. We reproduce the known
anomalous contributions Cirigliano:2002pv ; Miranda:2020wdg  for the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}\gamma$ case. We neglect resonance
contributions in the anomalous sector, which start at $\mathcal{O}(p^{6})$ in
the chiral power counting Kampf:2011ty .

##  III Radiative hadronic tau decays

The differential rate for the $\tau^{-}\to P_{1}^{-}P_{2}^{0}\nu_{\tau}\gamma$
decays in the $\tau$ rest frame is given by

$d\Gamma=\frac{(2\pi)^{4}}{4m_{\tau}}\sum_{\text{spin}}\overline{|\mathcal{M}|^{2}}d\Phi_{4}\
,$ (29)

where $d\Phi_{4}$ is the corresponding 4-body phase space, given by

$\begin{split}d\Phi_{4}=&\delta^{(4)}(P-p_{-}-p_{0}-q^{\prime}-k)\frac{d^{3}p_{-}}{(2\pi)^{3}2E_{-}}\frac{d^{3}p_{0}}{(2\pi)^{3}2E_{0}}\frac{d^{3}q^{\prime}}{(2\pi)^{3}2E_{\nu}}\frac{d^{3}k}{(2\pi)^{3}2E_{\gamma}}\
,\end{split}$ (30)

and $\overline{|\mathcal{M}|^{2}}$ is the unpolarized spin-averaged squared
amplitude. Inasmuch as this amplitude is not IR finite, we follow the same
procedure as in Refs. Cirigliano:2002pv ; Miranda:2020wdg  where a photon
energy cut, $E_{\gamma}^{\text{cut}}$, was introduced to study the dynamics of
the $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}\gamma$ decays.

In this analysis, we call “complete bremsstrahlung” or simply “SI” the
amplitude with $v_{1,2,3,4}=a_{1,2,3,4}=0$. For the $\mathcal{O}(p^{4})$
contributions, as in Ref. Miranda:2020wdg , we distinguish between using the
set of short-distance constraints $F_{V}=\sqrt{2}F$, $G_{V}=F/\sqrt{2}$
Ecker:1989yg  and $F_{A}=F$; or $F_{V}=\sqrt{3}F$, $G_{V}=F/\sqrt{3}$ and
$F_{A}=\sqrt{2}F$ Cirigliano:2004ue ; Cirigliano:2006hb ; Kampf:2011ty ;
Roig:2013baa . The former corresponds to the constraints from $2$-point Green
functions and the second to the values consistent up to 3-point Green
functions, which include operators that contribute at $\mathcal{O}(p^{6})$
(that we are not including in this work). The difference between both sets of
constraints has been employed to estimate roughly the model-dependent error of
this approach Roig:2019reh ; Miranda:2020wdg ; Arroyo-Urena:2021nil ; Arroyo-
Urena:2021dfe ; Guevara:2021tpy . In all our subsequent analyses, the
$\mathcal{O}(p^{4})$ results include the SI part and the structure dependent
part (either with the $F_{V}=\sqrt{2}F$ or with the $F_{V}=\sqrt{3}F$ set of
constraints).

Integrating Eq. (29) using the dispersive vector and scalar form factors
Gonzalez-Solis:2019iod ; Boito:2008fq ; Escribano:2014joa ; Escribano:2013bca
; Jamin:2001zq ; Guo:2011pa ; Guo:2012yt ; Guo:2016zep , we get the
$P_{1}^{-}P_{2}^{0}$ invariant mass distribution, the photon energy
distribution and the branching fraction as a function of
$E_{\gamma}^{\text{cut}}$. The outcomes are depicted in Figs. 6, 7, 8 and 9,
and summarized in Tables 1, 2 and 3.

$E_{\gamma}^{\rm cut}$ | Br(Low) | BR(SI) |  BR($F_{V}=\sqrt{2}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$ |  BR($F_{V}=\sqrt{3}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$
---|---|---|---|---
$100\,\mathrm{MeV}$ | $3.4\times 10^{-6}$ | $3.0\times 10^{-6}$ | $3.5\times 10^{-6}$ | $3.8\times 10^{-6}$
$300\,\mathrm{MeV}$ | $6.2\times 10^{-7}$ | $3.4\times 10^{-7}$ | $6.3\times 10^{-7}$ | $9.4\times 10^{-7}$
$500\,\mathrm{MeV}$ | $7.4\times 10^{-8}$ | $3.5\times 10^{-8}$ | $1.5\times 10^{-7}$ | $3.3\times 10^{-7}$
Table 1: Branching ratios Br$(\tau^{-}\to K^{-}\pi^{0}\nu_{\tau}\gamma)$ for different values of $E_{\gamma}^{\rm cut}$. The third column corresponds to the complete bremsstrahlung, and the fourth and fifth to the $\mathcal{O}\left(p^{4}\right)$ contributions. $E_{\gamma}^{\rm cut}$ | Br(Low) | BR(SI) |  BR($F_{V}=\sqrt{2}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$ |  BR($F_{V}=\sqrt{3}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$
---|---|---|---|---
$100\,\mathrm{MeV}$ | $2.6\times 10^{-5}$ | $1.4\times 10^{-5}$ | $1.6\times 10^{-5}$ | $1.6\times 10^{-5}$
$300\,\mathrm{MeV}$ | $6.2\times 10^{-6}$ | $1.1\times 10^{-6}$ | $1.7\times 10^{-6}$ | $1.9\times 10^{-6}$
$500\,\mathrm{MeV}$ | $1.0\times 10^{-6}$ | $7.1\times 10^{-8}$ | $2.0\times 10^{-7}$ | $2.4\times 10^{-7}$
Table 2: Branching ratios Br$(\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma)$ for different values of $E_{\gamma}^{\rm cut}$. The third column corresponds to the complete bremsstrahlung, and the fourth and fifth to the $\mathcal{O}\left(p^{4}\right)$ contributions. $E_{\gamma}^{\rm cut}$ | BR(Low) | BR(SI) |  BR($F_{V}=\sqrt{2}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$ |  BR($F_{V}=\sqrt{3}F$) $\left[\mathcal{O}\left(p^{4}\right)\right]$
---|---|---|---|---
$100\,\mathrm{MeV}$ | $5.3\times 10^{-7}$ | $3.7\times 10^{-7}$ | $6.8\times 10^{-7}$ | $9.4\times 10^{-7}$
$300\,\mathrm{MeV}$ | $4.8\times 10^{-8}$ | $1.9\times 10^{-8}$ | $1.7\times 10^{-7}$ | $3.1\times 10^{-7}$
$500\,\mathrm{MeV}$ | $3.7\times 10^{-10}$ | $3.0\times 10^{-10}$ | $1.1\times 10^{-8}$ | $2.9\times 10^{-8}$
Table 3: Branching ratios Br$(\tau^{-}\to K^{-}K^{0}\nu_{\tau}\gamma)$ for
different values of $E_{\gamma}^{\rm cut}$. The third column corresponds to
the complete bremsstrahlung, and the fourth and fifth to the
$\mathcal{O}\left(p^{4}\right)$ contributions.

The branching fractions of the radiative decays as a function of
$E_{\gamma}^{\text{cut}}$ are shown in Fig. 6.

Figure 6: Branching ratio for the $\tau^{-}\to K^{-}\pi^{0}\nu_{\tau}\gamma$
(top), the $\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma$ (centre) and the
$\tau^{-}\to K^{-}K^{0}\nu_{\tau}\gamma$ (bottom) decays as a function of
$E^{\rm cut}_{\gamma}$. The dotted line represents the bremsstrahlung
contribution, the solid line and dashed line represent the
$\mathcal{O}\left(p^{4}\right)$ corrections using $F_{V}=\sqrt{3}F$ and
$F_{V}=\sqrt{2}F$, respectively. The red one corresponds to the Low
approximation.

In Tables 1 and 2, one can see that for $E_{\gamma}^{\text{cut}}\lesssim
100\,\text{MeV}$ the main contribution at $\mathcal{O}(p^{4})$ comes from the
complete bremsstrahlung (SI) amplitude in agreement with the results in Refs.
Cirigliano:2002pv ; Miranda:2020wdg ; Chen:2022nxm  for the
$\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}\gamma$ decays (see also the recent ref.
Esparza-Arellano:2023dps ). It is also seen that the Low’s approximation is
sufficient to describe the $K^{-}\pi^{0}$ decays up to these energies, while
this is not the case for the $\bar{K}^{0}\pi^{-}$ ones. Contrary to the
$\tau^{-}\to(K\pi)^{-}\nu_{\tau}$ transitions, where the $K^{-}\pi^{0}$ and
$\pi^{-}\bar{K}^{0}$ decay modes differ only by a squared CGC factor in the
isospin symmetry limit, the radiative decays are more subtle. At low photon
energies, these two modes are approximately related by
$\mathrm{Br}(\tau\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma)/\mathrm{Br}(\tau^{-}\to
K^{-}\pi^{0}\nu_{\tau}\gamma)\approx 2(m_{K}/m_{\pi})\sim 7$, which explains
their hierarchy. In both decay channels, the SD contributions seem to be
subdominant, while the $\tau^{-}\to K^{-}K^{0}\nu_{\tau}\gamma$ decays are
more susceptible to these contributions (see Table 3).

Figure 7: The $K^{-}\pi^{0}$ (top), $\bar{K}^{0}\pi^{-}$ (centre) and
$K^{-}K^{0}$ (bottom) SI hadronic invariant mass distributions for several
$E_{\gamma}^{\rm cut}$ values.

In Fig. 7, the decay spectrum is depicted with $v_{i}=a_{i}=0$ for different
$E_{\gamma}^{\text{cut}}$ values. For the
$\tau^{-}\to(K\pi)^{-}\nu_{\tau}\gamma$ decays, the first peak is due to
bremsstrahlung off the charged meson, i.e. $K^{-}$ or $\pi^{-}$, and the
second one receives contributions from bremsstrahlung off the $\tau$ lepton
and resonance exchange.

Figure 8: The $K^{-}\pi^{0}$ (top), $\bar{K}^{0}\pi^{-}$ (centre) and
$K^{-}K^{0}$ (bottom) hadronic invariant mass distributions for
$E_{\gamma}^{\rm cut}\geq 300\,\mathrm{MeV}$. The solid and dashed line
represent the $\mathcal{O}\left(p^{4}\right)$ corrections using
$F_{V}=\sqrt{3}F$ and $F_{V}=\sqrt{2}F$, respectively. The dotted line
represents the bremsstrahlung contribution (SI). The red one corresponds to
the Low approximation.

In Fig. 8, we compare the distributions for
$E_{\gamma}^{\text{cut}}=300\,\text{MeV}$ using the Low’s approximation (red
dashed line), the SI amplitude (dotted line), and the $\mathcal{O}(p^{4})$
amplitude with $F_{V}=\sqrt{2}F$ (dashed line) and $F_{V}=\sqrt{3}F$ (solid
line). The most important contribution for the $(K\pi)^{-}$ decay channels
comes from the $K^{*}(892)$ resonance exchange around $s\sim
0.79\,\text{GeV}^{2}$. It is worth noting that for the
$\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma$ decays, there is a huge
suppression around the $K^{*}(892)$ peak when the full distribution is
compared to the Low one. The reason for that is the following. While the Low
approximation in Eq. (7) includes only the $\mathcal{O}(k^{-1})$ dominant
contributions from the bremsstrahlung off the initial tau lepton and the final
charged meson, the full amplitude in Eq. (1) contains also the
$\mathcal{O}(k^{0})$ subdominant contribution from the first line of this
equation, which is common to both $(K\pi)^{-}$ channels, plus all the
$\mathcal{O}(k^{0})$ contributions from Eq. (6). Among the latter, the leading
numerical contribution comes from the first term in the second line of Eq.
(3), which has different sign depending on the channel, as already mentioned
after that equation. This fact makes that in the case of the $K^{-}\pi^{0}$
mode these two leading subdominant contributions approximately cancel each
other and the Low and full distributions are quite similar at the $K^{*}(892)$
peak. Conversely, for the $\bar{K}^{0}\pi^{-}$ mode, the two contributions
combine with the overall effect of decreasing considerably the peak. Finally,
we just mention that the $K^{-}K^{0}$ invariant mass distribution is more
sensitive to SD contributions, although the $\rho(1450)$ effect is hidden in
the spectrum because of the corresponding kinematical suppression.

Figure 9: Photon energy distribution for the $\tau^{-}\to
K^{-}\pi^{0}\nu_{\tau}\gamma$ (top), the
$\tau^{-}\to\bar{K}^{0}\pi^{-}\nu_{\tau}\gamma$ (centre) and the $\tau^{-}\to
K^{-}K^{0}\nu_{\tau}\gamma$ (bottom) decays normalized with the non-radiative
decay width. The dotted line represents the bremsstrahlung contribution and
the red one the Low approximation. The solid and dashed lines represent the
$\mathcal{O}\left(p^{4}\right)$ corrections using $F_{V}=\sqrt{3}F$ and
$F_{V}=\sqrt{2}F$, respectively.

The photon energy distribution is shown in Fig. 9. The SI amplitude in all
these decays governs the distribution for $E_{\gamma}\lesssim
100\,\text{MeV}$, in agreement with the outcomes for the branching ratio.
However, the SD contributions become relevant for $E_{\gamma}\gtrsim 250\
\text{MeV}$. This feature makes these decays an excellent probe for testing SD
effects. The same analysis for the $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}\gamma$
decays can be found for instance in Ref. Miranda:2020wdg .

##  IV Radiative Corrections

The overall differential decay width is given by

$\left.\frac{d\Gamma}{dt}\right|_{PP(\gamma)}=\left.\frac{d\Gamma}{dt}\right|_{PP}+\left.\frac{d\Gamma}{dt}\right|_{\rm
III}+\left.\frac{d\Gamma}{dt}\right|_{\rm
IV/III}+\left.\frac{d\Gamma}{dt}\right|_{\rm rest}\ ,$ (31)

where the first term is the non-radiative differential width in Eq. (79), the
second and third terms correspond to the Low approximation integrated
according to the kinematics in Refs. Cirigliano:2002pv ; Miranda:2020wdg , Eq.
(15), and the last term includes the remaining contributions.

To evaluate the first term in the right-hand side of Eq. (31) we use two
models for the factorization of the radiative corrections to the form factors.
Both of them were pioneered in $K_{\ell 3}$ decays and have also been employed
in the $\tau^{-}\to(K\pi)^{-}\nu_{\tau}$ processes. As we will see, the
difference between both results will saturate the uncertainty of our real-
photon radiative corrections. To the best of our knowledge, the importance of
the factorization model for the former corrections was not previously
recognized in the literature.

The contributions to the form factors due to virtual corrections can be
written as

$F_{+/0}(t,u)=F_{+/0}(t)+\delta F_{+/0}(t,u)\ ,$ (32)

where $\delta F_{0}(t,u)=\delta F_{+}(t,u)+\frac{\displaystyle
t}{\displaystyle\Delta_{-0}}\delta\bar{f}_{-}(u)$. In model $1$, $\delta
F_{+}(t,u)$ is given by Cirigliano:2002pv

$\frac{\delta F_{+}(t,u)}{F_{+}(t)}=\frac{\displaystyle\alpha}{\displaystyle
4\pi}\left[2(m_{-}^{2}+m_{\tau}^{2}-u)\mathcal{C}(u,M_{\gamma})+2\log\left(\frac{\displaystyle
m_{-}m_{\tau}}{\displaystyle
M_{\gamma}^{2}}\right)\right]+\delta\bar{f}_{+}(u)\ ,$ (33)

while in model 2, it is written as Antonelli:2013usa  (we note that in this
second case, the correction $\delta\bar{f}_{+}(u)$ to $\delta F_{+}(t,u)$ is
not modulated by the vector form factor $F_{+}(t)$)

$\frac{\delta F_{+}(t,u)}{F_{+}(t)}=\frac{\displaystyle\alpha}{\displaystyle
4\pi}\left[2(m_{-}^{2}+m_{\tau}^{2}-u)\mathcal{C}(u,M_{\gamma})+2\log\left(\frac{\displaystyle
m_{-}m_{\tau}}{\displaystyle
M_{\gamma}^{2}}\right)\right]+\frac{\delta\bar{f}_{+}(u)}{F_{+}(t)}\ ,$ (34)

where $\mathcal{C}(u,M_{\gamma})$, $\delta\bar{f}_{+}(u)$ and
$\delta\bar{f}_{-}(u)$ are defined in App. B. A similar factorization
prescription was used in Ref. Cirigliano:2001mk , where model 2 was preferred
over model 1 for the $K_{\mu 3}$ decays since the loop contributions to
$\bar{f}_{+/-}(u)$ are different 777Both prescriptions were studied for the
$K_{e3}$ decays in Ref. Cirigliano:2008wn , their outcomes for
$\delta^{K\ell}_{\text{EM}}(\mathcal{D}_{3})(\%)$ are shifted from 0.41 to
0.56 for $K^{0}_{e3}$ and from -0.564 to -0.410 for $K^{\pm}_{e3}$ modes where
the former numbers correspond to model 2.. We will see here that model 1
factorization warrants smoother corrections than model 2 when resonance
contributions are included, as resonance enhancements will cancel in the long-
distance radiative correction factor $G_{\rm EM}(t)$ in Eq. (35), as opposed
to model 2. This motivates our preference of model 1 over model 2 in our
following phenomenological analysis.

A couple of points are worth to stress in connection to both factorization
models and the preferred one. First, a measurement of di-meson or photon
energy spectra in the considered decays will be really helpful in reducing the
model-dependence of our results (particularly on the factorization
prescription). Second, we will present elsewhere the corresponding
computations of the virtual photon structure-dependent corrections, which will
complete these at the one-loop order. We expect that the model-dependence is
reduced in the sum of all radiative corrections, so having this last missing
piece available will also be valuable for diminishing the model-dependence
(again with particular emphasis on the precise factorization in the considered
decays).

Figure 10: Correction factors $\delta_{+}(t)$ (left) and $\delta_{0}(t)$
(right) to the differential decay rates of the $K^{-}\pi^{0}$,
$\bar{K}^{0}\pi^{-}$ and $K^{-}K^{0}$ modes from top to bottom, according to
models 1 (solid black) and 2 (dashed red).

The correction factors $\overline{\delta}_{A}(t)$ for the four-body decays and
$\widetilde{\delta}_{A}(t)$ for the three-body processes appearing in Eqs.
(16) and (80), respectively, where $A={+,0,+0}$, are both IR divergent when
$M_{\gamma}\to 0$. Nevertheless, the overall contribution,
$\delta_{A}(t)=\overline{\delta}_{A}(t)+\widetilde{\delta}_{A}(t)$, is finite.
In Fig. 10, we show the predictions for $\delta_{A}(t)$ for the
$K^{-}\pi^{0}$, $\bar{K}^{0}\pi^{-}$ and $K^{-}K^{0}$ decay modes using the
form factors in model $1$ and $2$. Whilst our results for $\delta_{+}(t)$ in
model $2$ agree with those in Ref. Antonelli:2013usa  in Figure $2$, the
predictions for $\delta_{0}(t)$ are slightly different as a consequence of the
parameterization of the scalar form factor 888This effect is mainly
responsible for the slight difference between our results for model 2 in Table
10 and those in Ref. Antonelli:2013usa ..

The differential decay width can be written as

$\begin{split}\left.\frac{d\Gamma}{dt}\right|_{PP(\gamma)}&=\frac{G_{F}^{2}\left|V_{uD}F_{+}(0)\right|^{2}S_{\rm
EW}m_{\tau}^{3}}{768\pi^{3}t^{3}}\left(1-\frac{t}{m_{\tau}^{2}}\right)^{2}\lambda^{1/2}(t,m_{-}^{2},m_{0}^{2})\\\\[4.30554pt]
&\times\left[C_{V}^{2}|\tilde{F}_{+}(t)|^{2}\left(1+\frac{2t}{m_{\tau}^{2}}\right)\lambda(t,m_{-}^{2},m_{0}^{2})+3C_{S}^{2}\Delta_{-0}^{2}|\tilde{F}_{0}(t)|^{2}\right]G_{\rm
EM}(t)\ ,\end{split}$ (35)

where $G_{\rm EM}(t)$ encodes the electromagnetic corrections due to real and
virtual photons. For simplicity, we have split $G_{\rm EM}(t)$ in two parts:
the leading Low approximation plus non-radiative contributions, $G_{\rm
EM}^{(0)}(t)$, and the remainder, $\delta G_{\rm EM}(t)$, which includes the
SD contributions to the amplitude. The predictions for both are shown in Fig.
11.

Figure 11: Correction factors $G_{\rm EM}^{(0)}(t)-1$ (left) and $\delta
G_{\rm EM}(t)$ (right) to the differential decay rates of the $K^{-}\pi^{0}$,
$\bar{K}^{0}\pi^{-}$, $K^{-}K^{0}$, and $\pi^{-}\pi^{0}$ modes from top to
bottom.

Integrating upon $t$, we get

$\Gamma_{PP(\gamma)}=\frac{G_{F}^{2}S_{\rm
EW}m_{\tau}^{5}}{96\pi^{3}}\left|V_{uD}F_{+}(0)\right|^{2}I^{\tau}_{PP}\left(1+\delta_{\rm
EM}^{PP}\right)^{2}\ ,$ (36)

where

$\begin{split}I^{\tau}_{PP}=&\frac{1}{8m_{\tau}^{2}}\int^{m_{\tau}^{2}}_{t_{\rm
thr}}\frac{dt}{t^{3}}\left(1-\frac{t}{m_{\tau}^{2}}\right)^{2}\lambda^{1/2}(t,m_{-}^{2},m_{0}^{2})\\\\[4.30554pt]
&\times\left[C_{V}^{2}|\tilde{F}_{+}(t)|^{2}\left(1+\frac{2t}{m_{\tau}^{2}}\right)\lambda(t,m_{-}^{2},m_{0}^{2})+3C_{S}^{2}\Delta_{-0}^{2}|\tilde{F}_{0}(t)|^{2}\right]\
.\end{split}$ (37)

The results for $\delta_{\rm EM}^{PP}$ are shown in Table 4, where the third
and fourth columns correspond to the sum of the first three terms in Eq. (31),
and the last three columns to the fourth term in that equation. The value in
model $1$ for the $\bar{K}^{0}\pi^{-}$ channel agrees with the result in Ref.
Flores-Baez:2013eba , which is related to our definition by $\delta_{\rm
EM}^{\bar{K}^{0}\pi^{-}}=\delta^{\rm m.i.}_{\rm EM}/2\simeq-0.063\%$. Although
our outcomes for the $(K\pi)^{-}$ modes agree within errors with those in
Refs. Flores-Baez:2013eba ; Antonelli:2013usa , the value in model $2$ (and
also model $1$) for the $K^{-}\pi^{0}$ decay channel is larger than the
$K^{0}\pi^{-}$ one, which is at odds with Ref. Antonelli:2013usa
999Incidentally, our results would agree more closely swapping the numbers for
$\delta_{\text{EM}}^{K^{-}\tau}\leftrightarrow\delta_{\text{EM}}^{\bar{K}^{0}\tau}$
in Ref. Antonelli:2013usa ..

$\delta_{\rm EM}^{PP}$ |  Ref. Antonelli:2013usa  | $G_{\rm EM}^{(0)}(t)$ | $\delta G_{\rm EM}(t)$
---|---|---|---
|  | Model 1 | Model 2 | SI | SI + 2F | SI + 3F
$K^{-}\pi^{0}$ | $-0.20(20)$ | $-0.019$ | $-0.137$ | $-0.001$ | $+0.006$ | $+0.010$
$\bar{K}^{0}\pi^{-}$ | $-0.15(20)$ | $-0.086$ | $-0.208$ | $-0.098$ | $-0.085$ | $-0.080$
$K^{-}K^{0}$ | $-$ | $-0.046$ | $-0.223$ | $-0.012$ | $+0.003$ | $+0.016$
$\pi^{-}\pi^{0}$ | $-$ | $-0.196$ | $-0.363$ | $-0.010$ | $-0.002$ | $+0.010$
Table 4: Electromagnetic corrections to hadronic $\tau$ decays in $\%$.

The complete radiative corrections (that we always quote in $\%$) are obtained
adding to the model 1/2 results, which comprise the (negligible)
$\mathcal{D}^{\rm IV/III}$ part, the $2F/3F$ contributions, which include the
SI part. We explained before why we prefer the model 1 over the model 2
results. We will take the difference with respect to model 2 as an asymmetric
error on the model 1 results. For the structure-dependent contributions, we
consider the $3F$ results as our central values and the difference with
respect to $2F$ as a symmetric error for our model-dependence. To be on the
safe side, we will take twice this error as our corresponding uncertainty.
Finally, we have to account for the uncertainty associated to the missing
structure-dependent virtual-photon corrections. According to the results in
Refs. Arroyo-Urena:2021nil ; Arroyo-Urena:2021dfe  for the one-meson tau
decays, this contribution is of the same size as the structure-independent
correction. We will thus estimate its absolute value as the sum of the ‘Model
1’ and ‘SI+3F’ results in Table 4, allowing it to have either sign. We will
assign an additional $60\%$ uncertainty on it according to the results in Ref.
Arroyo-Urena:2021nil ; Arroyo-Urena:2021dfe . Proceeding this way, our main
results are

$\begin{split}&\delta_{\rm
EM}^{K^{-}\pi^{0}}=-\left(0.009^{+0.010}_{-0.118}\right)\ ,\qquad\delta_{\rm
EM}^{\bar{K}^{0}\pi^{-}}=-\left(0.166^{+0.100}_{-0.157}\right)\
,\\\\[4.30554pt] &\delta_{\rm
EM}^{K^{-}K^{0}}=-\left(0.030^{+0.032}_{-0.180}\right)\ ,\qquad\delta_{\rm
EM}^{\pi^{-}\pi^{0}}=-\left(0.186^{+0.114}_{-0.203}\right)\ .\end{split}$ (38)

We see that the model-independent contributions are responsible for the
relatively large radiative corrections obtained for the
$(\bar{K}/\pi)^{0}\pi^{-}$ modes. For the modes with a $K^{-}$, the dominant
(asymmetric) uncertainty comes from the difference between the model 1/2
results, which is much larger than the deviation between the model-dependent
$2F/3F$ values. Instead, for the modes with a $\pi^{-}$, the dominant
uncertainty comes from the missing model-dependent virtual-photon corrections.
Our results for the $\delta_{\rm EM}^{K^{-}\pi^{0}/\bar{K}^{0}\pi^{-}}$ agree
with those in Ref. Antonelli:2013usa , and we reduce the uncertainty band by
$\sim 45\%$ in the $K^{-}$ channel. We note that the estimate of the errors in
this reference yields also an uncertainty band in agreement with ours for
$\delta_{\rm EM}^{K^{-}K^{0}/\pi^{-}\pi^{0}}$ (our errors are smaller by a
factor $\sim 2$ again in the $K^{-}K^{0}$ case) 101010A former estimation of
the $\pi^{-}\pi^{0}$ radiative corrections yielded $\delta_{\rm
EM}^{\pi^{-}\pi^{0}}\sim-0.08\%$ Flores-Baez:2006yiq , where the SD
contributions were evaluated using a vector meson dominance model.. Although
our $\delta_{\rm EM}^{K^{-}\pi^{0}}$ and $\delta_{\rm
EM}^{\bar{K}^{0}\pi^{-}}$ seem to differ (the main reason being the scaling of
the inner bremsstrahlung contribution with the inverse of the charged meson
mass), the corresponding significance of their non-equality is only $\sim
0.7\sigma$, according to our uncertainties. Our radiative corrections in Eq.
(38) improve over previous analysis (where, for instance, the structure-
dependent corrections were not computed) and, as such, should be employed in
precision analysis like, e.g., CKM unitarity or lepton universality tests
HFLAV:2022esi  and searches for non-standard interactions.

For completeness, we have also evaluated these corrections for the
$K^{-}\eta^{(\prime)}$ modes. In the $G_{\rm EM}^{(0)}$ approximation and
using the respective dominance of the vector (scalar) form factor
Escribano:2013bca , we obtain

$\delta_{\rm EM}^{K^{-}\eta}=-\left(0.026^{+0.029}_{-0.163}\right)\
,\qquad\delta_{\rm
EM}^{K^{-}\eta^{\prime}}=-\left(0.304^{+0.422}_{-0.185}\right)\ ,$ (39)

where the uncertainty is saturated by the difference between the model 1/2
results in the $\eta$ channel and by the non yet computed virtual-photon
structure-dependent corrections for the $\eta^{\prime}$ mode. The
$K^{-}\eta^{\prime}$ decay mode is the only one (completely) dominated by the
scalar form factor, which causes the relatively large magnitude of the
corresponding radiative correction.

##  V Impact of radiative corrections on NP bounds

In this section, we update the results of Ref. Gonzalez-Solis:2020jlh
concerning one- and two-meson tau decays including the radiative corrections
computed in this paper 111111One-meson channels were updated in Refs. Arroyo-
Urena:2021dfe ; Arroyo-Urena:2021nil  using the improved radiative corrections
calculated in those papers. It would also be interesting to reanalyse Refs.
Antonelli:2013usa ; Cirigliano:2021yto  with our new radiative corrections..
We recall briefly the main aspects here, but refer the reader to Ref.
Gonzalez-Solis:2020jlh  for details.

The low-energy effective Lagrangian describing the
$\tau^{-}\to\bar{u}D\nu_{\tau}$ decays ($D=d,s$) can be written as
Cirigliano:2009wk ; Bhattacharya:2011qm

$\begin{split}\mathcal{L}_{\rm
eff}&=-\frac{G_{F}V_{uD}}{\sqrt{2}}\Big{[}(1+\epsilon^{\tau}_{L})\bar{\tau}\gamma_{\mu}(1-\gamma_{5})\nu_{\tau}\cdot\bar{u}\gamma^{\mu}(1-\gamma_{5})D+\epsilon^{\tau}_{R}\bar{\tau}\gamma_{\mu}(1-\gamma_{5})\nu_{\tau}\cdot\bar{u}\gamma^{\mu}(1+\gamma_{5})D\\\\[4.30554pt]
&+\bar{\tau}(1-\gamma_{5})\nu_{\tau}\cdot\bar{u}(\epsilon^{\tau}_{S}-\epsilon^{\tau}_{P}\gamma_{5})D+\epsilon^{\tau}_{T}\bar{\tau}\sigma_{\mu\nu}(1-\gamma_{5})\nu_{\tau}\cdot\bar{u}\sigma^{\mu\nu}(1-\gamma_{5})D\Big{]}+\mathrm{h.c.}\
,\end{split}$ (40)

where $G_{F}$ corresponds to the Standard Model (SM) tree-level definition of
the Fermi constant and the non-vanishing $\epsilon_{i}$ $(i=S,P,V,A,T)$ Wilson
coefficients (assumed to be real for simplicity in what follows) determine the
new physics. Beyond the SM, super allowed nuclear Fermi $\beta$ decays do not
depend on $G_{F}V_{ud}$ but rather on
$G_{F}V_{ud}(1+\epsilon^{e}_{L}+\epsilon^{e}_{R})$, as it is accounted for in
our analysis. After using $\mathcal{L}_{\rm eff}$, the relevant (for two-meson
decays) scalar, vector and tensor hadron matrix elements are computed using
dispersion relations, nourished with experimental data, keeping track of the
associated uncertainties.

We will discuss in the following the separate results for the strangeness-
conserving and changing channels and, finally, those of a joint fit.

###  V.1 $\boldsymbol{\Delta S=0}$

From $\tau^{-}\to\pi^{-}\nu_{\tau}(\gamma)$, we restrict Cirigliano:2018dyk ;
Gonzalez-Solis:2020jlh ; Arroyo-Urena:2021dfe ; Arroyo-Urena:2021nil ;
Cirigliano:2021yto

$\epsilon_{L}^{\tau}-\epsilon_{L}^{e}-\epsilon_{R}^{\tau}-\epsilon_{R}^{e}-\frac{m_{\pi}^{2}}{m_{\tau}(m_{u}+m_{d})}\epsilon_{P}^{\tau}=-(0.14\pm
0.72)\cdot 10^{-2}\ ,$ (41)

using $f_{\pi}=130.2(8)\,\text{MeV}$ FlavourLatticeAveragingGroupFLAG:2021npn
, $|V_{ud}|=0.97373(31)$ ParticleDataGroup:2022pth , $S_{\rm EW}=1.0232$
Marciano:1993sh , masses and branching ratios from the PDG
ParticleDataGroup:2022pth , and $\delta_{\text{em}}^{\tau\pi}=-0.24(56)\%$
from Ref. Arroyo-Urena:2021nil .

After performing a fit that includes one ($\pi$) and two meson ($\pi\pi$,
$KK$) strangeness-conserving exclusive tau decays, the constraints for the
non-standard interactions (at $\mu=2\text{ GeV}$ in the $\overline{\text{MS}}$
scheme) are

$\left(\begin{array}[]{c}\epsilon_{L}^{\tau}-\epsilon_{L}^{e}+\epsilon_{R}^{\tau}-\epsilon_{R}^{e}\\\
\epsilon_{R}^{\tau}+\frac{m_{\pi}^{2}}{2m_{\tau}(m_{u}+m_{d})}\epsilon_{P}^{\tau}\\\
\epsilon^{\tau}_{S}\\\
\epsilon^{\tau}_{T}\end{array}\right)=\left(\begin{array}[]{c}\,0.0\pm
0.6\,\,^{+\,\,6.8}_{-\,\,6.4}\pm 0.1\pm 1.7\,\,^{+\,\,0.0}_{-\,\,0.2}\\\
\,0.1\pm 0.5\,\,^{+\,\,3.4}_{-\,\,3.3}\,\,{}^{+\,\,0.0}_{-\,\,0.1}\pm
0.9\,\,\pm 0.1\\\ 10.3\pm
0.5\,\,^{+\,\,\,1.2}_{-\,25.0}\,\,{}^{+\,\,0.0}_{-\,\,0.1}\pm
0.9\,\,^{+\,\,\,6.2}_{-\,22.4}\\\ 0.4\pm
0.2\,\,^{+\,\,4.1}_{-\,\,4.4}\,\,{}^{+\,\,0.0}_{-\,\,0.1}\pm
1.1\,\,^{+\,\,0.3}_{-\,\,0.2}\end{array}\right)\times 10^{-2}\ ,$ (42)

with $\chi^{2}/{\rm dof}\sim 0.8$, and the associated (statistical)
correlation matrix is

$\rho_{ij}=\left(\begin{array}[]{cccc}1&0.662&-0.487&-0.544\\\
&1&-0.323&-0.360\\\ &&1&0.452\\\ &&&1\\\ \end{array}\right)\ .$ (43)

The first error in Eq. (42) is the statistical fit uncertainty, the second
error comes from the theoretical uncertainty associated to the pion form
factor, the third and fourth ones come from the quark masses and from the
uncertainty related to the tensor form factor, respectively, and the last
error is a systematic uncertainty coming from the radiative corrections to
two-meson tau decays.

The results obtained are extremely consistent with those in Ref. Gonzalez-
Solis:2020jlh . In general, the uncertainties induced by the radiative
corrections are negligible with respect to others except for
$\epsilon^{\tau}_{S}$, where they are leading. This is interesting since this
is the largest Wilson coefficient, so its compatibility with the SM gets even
stronger upon including our new radiative corrections computed in this work.

###  V.2 $\boldsymbol{|\Delta S|=1}$

From $\tau^{-}\to K^{-}\nu_{\tau}$, we bind Cirigliano:2018dyk ; Gonzalez-
Solis:2020jlh ; Arroyo-Urena:2021dfe ; Arroyo-Urena:2021nil ;
Cirigliano:2021yto

$\epsilon_{L}^{\tau}-\epsilon_{L}^{e}-\epsilon_{R}^{\tau}-\epsilon_{R}^{e}-\frac{m_{K}^{2}}{m_{\tau}(m_{u}+m_{s})}\epsilon_{P}^{\tau}=-(1.02\pm
0.86)\cdot 10^{-2}\ ,$ (44)

using the lattice result $f_{K}=155.7(3)\,\text{MeV}$
FlavourLatticeAveragingGroupFLAG:2021npn , $|V_{us}|=0.2243(8)$
ParticleDataGroup:2022pth , masses and branching ratios from PDG
ParticleDataGroup:2022pth , and $\delta_{\text{em}}^{\tau K}=-0.15(57)\%$ from
Ref. Arroyo-Urena:2021nil .

The bounds for the non-SM effective couplings resulting from a fit to one
($K$) and two meson ($K\pi,K\eta$) strangeness-changing transitions (at
$\mu=2\ \text{GeV}$ in the $\overline{\text{MS}}$ scheme) are

$\left(\begin{array}[]{c}\epsilon_{L}^{\tau}-\epsilon_{L}^{e}+\epsilon_{R}^{\tau}-\epsilon_{R}^{e}\\\
\epsilon_{R}^{\tau}+\frac{m_{K}^{2}}{2m_{\tau}(m_{u}+m_{s})}\epsilon_{P}^{\tau}\\\
\epsilon^{\tau}_{S}\\\
\epsilon^{\tau}_{T}\end{array}\right)=\left(\begin{array}[]{c}0.4\pm 1.5\pm
0.4\,\,^{+\,\,0.1}_{-\,\,0.0}\\\ 0.7\pm 0.9\pm
0.2\,\,^{+\,\,0.1}_{-\,\,0.0}\\\ 0.8\pm 0.9\pm
0.2\,\,^{+\,\,0.0}_{-\,\,0.1}\\\ 0.5\pm 0.7\pm 0.4\pm
0.0\end{array}\right)\times 10^{-2}\ ,$ (45)

with $\chi^{2}/{\rm dof}\sim 0.9$, and

$\rho_{ij}=\left(\begin{array}[]{cccc}1&0.874&-0.149&0.463\\\
&1&-0.130&0.404\\\ &&1&-0.057\\\ &&&1\\\ \end{array}\right)\ .$ (46)

In this case, the first error in Eq. (45) is the statistical fit uncertainty,
the second one is due to the tensor form factor, and the last uncertainty is
associated to the radiative corrections to two-meson tau decays. The
uncertainties related to the kaon vector form factor and the quark masses are
negligible. These results comply nicely with those in Ref. Gonzalez-
Solis:2020jlh , and the uncertainty induced by the radiative corrections is in
all couplings negligible.

###  V.3 $\boldsymbol{\Delta S=0}$ and $\boldsymbol{|\Delta S|=1}$ joint fit

Our previous fits to the strangeness-conserving and changing channels were not
able to separate $\epsilon^{\tau}_{R}$ and $\epsilon^{\tau}_{P}$ 121212This
was achieved by combining inclusive and exclusive information in Refs.
Cirigliano:2018dyk ; Cirigliano:2021yto .. In Ref. Gonzalez-Solis:2020jlh , we
attained this within Minimal Flavour Violation DAmbrosio:2002vsn  (MFV)
131313MFV assumes that flavour mixing within the Standard Model Effective
Field Theory (SMEFT) is aligned with the SM one (here, in particular, in the
quark sector). This generally allows for orders of magnitude smaller new
physics scales in precision flavour tests., as we will do here. The CKM matrix
elements used in this analysis are obtained from the correlation between
$|V_{ud}|$ and $|V_{us}|$, $|V_{us}/V_{ud}|=0.2313(5)$, and
$|V_{us}|=0.2232(6)$ from Ref. FlavourLatticeAveragingGroupFLAG:2021npn ,
which correspond to the region described by the red ellipse in Fig. 10 of Ref.
FlavourLatticeAveragingGroupFLAG:2021npn .

Performing thus a joint fit that includes both one and two meson strangeness-
conserving and strangeness-changing tau decays (within MFV), the limits for
the NP effective couplings (at $\mu=2\ \text{GeV}$ in the
$\overline{\text{MS}}$ scheme) are

$\left(\begin{array}[]{c}\epsilon_{L}^{\tau}-\epsilon_{L}^{e}+\epsilon_{R}^{\tau}-\epsilon_{R}^{e}\\\
\epsilon_{R}^{\tau}\\\ \epsilon_{P}^{\tau}\\\ \epsilon^{\tau}_{S}\\\
\epsilon^{\tau}_{T}\end{array}\right)=\left(\begin{array}[]{c}\quad 2.7\pm
0.5\,\,^{+\,\,\,2.3}_{-\,\,\,3.1}\,\,{}^{+\,\,\,0.4}_{-\,\,\,0.5}\pm 0.0\pm
0.3\,\,^{+\,\,\,0.0}_{-\,\,\,1.3}\pm 0.0\\\ \quad 7.1\pm
4.7\,\,^{+\,\,\,1.2}_{-\,\,\,1.6}\pm 0.9\pm 1.8\pm
0.2\,\,^{+\,\,12.3}_{-\,\,\,3.6}\pm 0.0\\\ -7.7\pm 6.1\pm
0.0\,\,^{+\,\,\,1.3}_{-\,\,\,1.2}\,\,{}^{+\,\,\,2.4}_{-\,\,\,2.3}\pm
0.0\,\,^{+\,\,\,4.1}_{-\,\,17.0}\pm 0.0\\\ \quad
5.3\,\,^{+\,\,\,0.6}_{-\,\,\,0.7}\,\,{}^{+\,\,\,2.0}_{-\,\,14.9}\,\,{}^{+\,\,\,0.1}_{-\,\,\,0.0}\pm
0.0\pm 0.1\,\,^{+\,\,\,0.1}_{-\,\,15.9}\,\,{}^{+\,\,\,0.1}_{-\,\,\,0.0}\\\
-0.2\pm 0.2\,\,^{+\,\,\,3.6}_{-\,\,\,2.9}\,\,{}^{+\,\,\,0.1}_{-\,\,\,0.0}\pm
0.0\pm
0.4\,\,^{+\,\,\,0.5}_{-\,\,\,0.0}\,\,{}^{+\,\,\,0.2}_{-\,\,\,0.0}\end{array}\right)\times
10^{-2}\ ,$ (47)

with $\chi^{2}/{\rm dof}\sim 1.5$, and

$\rho_{ij}=\left(\begin{array}[]{ccccc}1&0.056&0.000&-0.270&-0.402\\\
&1&-0.997&-0.015&-0.023\\\ &&1&0.000&0.000\\\ &&&1&0.235\\\
&&&&1\end{array}\right)\ .$ (48)

Now, the first error in Eq. (47) corresponds again to the statistical fit
uncertainty, the second one comes from the uncertainty on the pion form
factor, the third error is related to the CKM matrix elements $|V_{ud}|$ and
$|V_{us}|$, the fourth one comes from the radiative corrections
$\delta^{\tau\pi}_{\text{em}}$ and $\delta^{\tau K}_{\text{em}}$, the fifth
error is associated to the tensor form factor, the sixth error comes from the
uncertainty of the quark masses, and the last one is due to the radiative
corrections to two-meson tau decays.

These results accord with Ref. Gonzalez-Solis:2020jlh  closely. As seen, in
this joint fit the uncertainties induced by radiative corrections are always
subdominant.

##  VI Conclusions

Radiative corrections to the one-meson tau decays have been employed in CKM
unitarity, lepton universality and non-standard interactions tests. The
corresponding results for the di-pion tau decays allowed tau-based
computations of the leading-order piece of the hadronic vacuum polarization
part of the muon $g-2$. Even though the model-independent part of these
corrections was available for the $K\pi$ modes, the structure-dependent one
remained to be calculated. We have performed the first step towards bridging
this gap, by computing the real-photon structure-dependent radiative
correction factors with reduced uncertainties, and will continue with the
required virtual-photon model-dependent corrections in a forthcoming work. For
completeness, we also quoted our numbers for the $PP$ ($P=\pi,K$) modes and
estimated them for the $K\eta^{(\prime)}$ cases.

We recapitulate our main results in Eq. (38):

$\begin{split}&\delta_{\rm
EM}^{K^{-}\pi^{0}}=-\left(0.009^{+0.010}_{-0.118}\right)\%\ ,\qquad\delta_{\rm
EM}^{\bar{K}^{0}\pi^{-}}=-\left(0.166^{+0.100}_{-0.157}\right)\%\
,\\\\[4.30554pt] &\delta_{\rm
EM}^{K^{-}K^{0}}=-\left(0.030^{+0.032}_{-0.180}\right)\%\ ,\qquad\delta_{\rm
EM}^{\pi^{-}\pi^{0}}=-\left(0.186^{+0.114}_{-0.203}\right)\%\ ,\end{split}$

which reduce previous uncertainties by a factor of $\sim 2$ for the $K^{-}$
modes and have similar errors to those quoted for the $\bar{K}^{0}\pi^{-}$
channel.

We have put forward the importance of the factorization model for the
radiative corrections, which saturates the uncertainties in Eq. (38) for the
$K^{-}$ channels. Analogous relevance shall be found in the radiative
corrections for processes including diverse final states with hadrons (if the
resonance regime is allowed by phase space), which calls for further devoted
studies. While lattice QCD obtains these complicated radiative corrections, a
deeper understanding of their factorization will probe key in increasing the
reach of new physics searches through processes including hadrons.

We have finally illustrated the application of our results updating our fits
of non-SM interactions in di-meson tau decays (also one-meson channels were
included in the fits), finding results compatible with our earlier work, where
these corrections and their uncertainties were neglected. As interesting
outlooks, the reanalysis of Refs. Miranda:2018cpf ; Rendon:2019awg  (looking
for NP in $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ and
$\tau^{-}\to(K\pi)^{-}\nu_{\tau}$ decays, respectively), of inclusive and
exclusive semileptonic tau decaysCirigliano:2018dyk ; Cirigliano:2021yto  and
of the impact of kaon data on strangeness-violating tau decays
Antonelli:2013usa , are among the important applications of our results that
we can envisage, which can be incorporated into the HFLAV analysis
HeavyFlavorAveragingGroup:2022wzx . All these studies will benefit from our
future results for the virtual-photon structure-dependent radiative
corrections.

###### Acknowledgements.

The work of R. E. has been supported by the European Union’s Horizon 2020
Research and Innovation Programme under grant 824093 (H2020-INFRAIA- 2018-1),
the Ministerio de Ciencia e Innovación under grant PID2020-112965GB-I00, and
by the Departament de Recerca i Universitats from Generalitat de Catalunya to
the Grup de Recerca ‘Grup de Física Teòrica UAB/IFAE’ (Codi: 2021 SGR 00649).
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
J. A. M. is also supported by MICINN with funding from European Union
NextGenerationEU (PRTR-C17.I1) and acknowledges Conacyt for his PhD
scholarship. P. R. was partly funded by Conacyt’s project within ‘Paradigmas y
Controversias de la Ciencia 2022’, number 319395, and by Cátedra Marcos
Moshinsky (Fundación Marcos Moshinsky) 2020.

##  Appendix A $\boldsymbol{K_{\ell 3}}$ decays

The most general amplitude for the
$K(p_{K})\to\pi(p_{\pi})\ell(P)\nu_{\ell}(q^{\prime})\gamma(k)$ decays that
complies with Lorentz invariance and the discrete symmetries of QCD can be
written as

$\begin{split}\mathcal{M}&=\frac{eG_{F}V^{*}_{us}}{\sqrt{2}}\epsilon^{*}_{\mu}\bigg{[}\frac{H_{\nu}(-p_{K},p_{\pi})}{k^{2}+2k\cdot
P}\bar{u}(q^{\prime})\gamma^{\nu}(1-\gamma^{5})(m_{\ell}-\not{P}-\not{k})\gamma^{\mu}v(P)\\\\[4.30554pt]
&+(V^{\mu\nu}-A^{\mu\nu})\bar{u}(q^{\prime})\gamma_{\nu}(1-\gamma^{5})v(P)\bigg{]}\
,\end{split}$ (49)

where

$\begin{split}H^{\nu}(-p_{K},p_{\pi})&\equiv\left\langle\pi(p_{\pi})\left|\bar{s}\gamma^{\nu}u\right|K(p_{K})\right\rangle\\\\[4.30554pt]
&=-C_{V}F_{+}(t)\left[(p_{K}+p_{\pi})^{\nu}-\frac{\Delta_{K\pi}}{t}(p_{K}-p_{\pi})^{\nu}\right]-C_{S}\frac{\Delta_{K\pi}}{t}(p_{K}-p_{\pi})^{\nu}F_{0}(t)\
,\end{split}$ (50)

with $t=(p_{K}-p_{\pi})^{2}$.

The structure-independent term is given by

$\begin{split}V^{\mu\nu}_{\rm
SI}&=\frac{H^{\nu}(-p_{K}+k,p_{\pi})(k-2p_{K})^{\mu}}{k^{2}-2k\cdot
p_{K}}+\left\\{-C_{V}F_{+}(t^{\prime})-\frac{\Delta_{K\pi}}{t^{\prime}}\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]\right\\}g^{\mu\nu}\\\\[4.30554pt]
&+C_{V}\frac{F_{+}(t^{\prime})-F_{+}(t)}{k\cdot(p_{K}-p_{\pi})}\left[(p_{K}+p_{\pi})^{\nu}-\frac{\Delta_{K\pi}}{t}(p_{K}-p_{\pi})^{\nu}\right](p_{K}-p_{\pi})^{\mu}\\\\[4.30554pt]
&+\frac{\Delta_{K\pi}}{tt^{\prime}}\left\\{2\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]+\frac{C_{S}t^{\prime}}{k\cdot(p_{K}-p_{\pi})}\left[F_{0}(t^{\prime})-F_{0}(t)\right]\right\\}\\\\[4.30554pt]
&\times(p_{K}-p_{\pi})^{\mu}(p_{K}-p_{\pi})^{\nu}\ ,\end{split}$ (51)

where $C_{V}^{K^{-}\pi^{0}}=C_{S}^{K^{-}\pi^{0}}=1/\sqrt{2}$ for
$K^{+}\to\pi^{0}\ell^{+}\nu_{\ell}\gamma$, and

$\begin{split}V^{\mu\nu}_{\rm
SI}&=\frac{H^{\nu}(-p_{K},p_{\pi}+k)(k+2p_{\pi})^{\mu}}{k^{2}+2k\cdot
p_{\pi}}+\left\\{C_{V}F_{+}(t^{\prime})-\frac{\Delta_{K\pi}}{t^{\prime}}\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]\right\\}g^{\mu\nu}\\\\[4.30554pt]
&+C_{V}\frac{F_{+}(t^{\prime})-F_{+}(t)}{k\cdot(p_{K}-p_{\pi})}\left[(p_{K}+p_{\pi})^{\nu}-\frac{\Delta_{K\pi}}{t}(p_{K}-p_{\pi})^{\nu}\right](p_{K}-p_{\pi})^{\mu}\\\\[4.30554pt]
&+\frac{\Delta_{K\pi}}{tt^{\prime}}\left\\{2\left[C_{S}F_{0}(t^{\prime})-C_{V}F_{+}(t^{\prime})\right]+\frac{C_{S}t^{\prime}}{k\cdot(p_{K}-p_{\pi})}\left[F_{0}(t^{\prime})-F_{0}(t)\right]\right\\}\\\\[4.30554pt]
&\times(p_{K}-p_{\pi})^{\mu}(p_{K}-p_{\pi})^{\nu}\ ,\end{split}$ (52)

where $C_{V}^{\bar{K}^{0}\pi^{-}}=C_{S}^{\bar{K}^{0}\pi^{-}}=1$ for
$K^{0}\to\pi^{-}\ell^{+}\nu_{\ell}\gamma$, both with
$t^{\prime}\equiv(p_{K}-p_{\pi}-k)^{2}$. All these expressions are obtained
from Eqs. (1)–(3) by replacing
$\\{^{p_{-}}_{p_{0}}\\}\to\\{^{-p_{K}}_{\,\,\,\,\,p_{\pi}}\\}$,
$\Delta_{-0}\to\Delta_{K\pi}$, $P\to-P$ and $m_{\tau}\to m_{\ell}$ for
$K^{+}\to\pi^{0}\ell^{+}\nu_{\ell}\gamma$, and
$\\{^{p_{-}}_{p_{0}}\\}\to\\{_{-p_{K}}^{\,\,\,\,\,p_{\pi}}\\}$,
$\Delta_{-0}\to-\Delta_{K\pi}$, $C_{V,S}\to-C_{V,S}$ $P\to-P$ and $m_{\tau}\to
m_{\ell}$ for $K^{0}\to\pi^{-}\ell^{+}\nu_{\ell}\gamma$. The structure-
dependent terms are analogous to those in Eqs. (4) and (5).

At $\mathcal{O}(p^{0})$, we get

$V_{\text{SI}}^{\mu\nu}=-C_{K^{+}}\frac{p_{K}^{\mu}}{k\cdot
p_{K}}(p_{K}+p_{\pi})^{\nu}-C_{K^{+}}\left(g^{\mu\nu}-\frac{p_{K}^{\mu}k^{\nu}}{k\cdot
p_{K}}\right)\ ,$ (53)

for $K^{+}\to\pi^{0}$, and

$V_{\text{SI}}^{\mu\nu}=-C_{K^{0}}\frac{p_{\pi}^{\mu}}{k\cdot
p_{\pi}}(p_{K}+p_{\pi})^{\nu}+C_{K^{0}}\left(g^{\mu\nu}-\frac{p_{\pi}^{\mu}k^{\nu}}{k\cdot
p_{\pi}}\right)\ ,$ (54)

for $K^{0}\to\pi^{-}$, where $C_{K}=C_{S}=C_{V}$. Thus, the overall amplitude
at $\mathcal{O}(p^{0})$ is given by

$\mathcal{M}_{\gamma}=\frac{eG_{F}}{\sqrt{2}}V_{us}^{*}C_{K^{+}}\bar{u}(q^{\prime})(1+\gamma^{5})(2\not{p}_{\pi}-m_{\ell})\left(\frac{\epsilon\cdot
P}{k\cdot P}-\frac{\epsilon\cdot p_{K}}{k\cdot
p_{K}}+\frac{\not{k}\not{\epsilon}}{2k\cdot P}\right)v(P)\ ,$ (55)

and

$\mathcal{M}_{\gamma}=\frac{eG_{F}}{\sqrt{2}}V_{us}^{*}C_{K^{0}}\bar{u}(q^{\prime})(1+\gamma^{5})(2\not{p}_{K}+m_{\ell})\left(\frac{\epsilon\cdot
P}{k\cdot P}-\frac{\epsilon\cdot p_{\pi}}{k\cdot
p_{\pi}}+\frac{\not{k}\not{\epsilon}}{2k\cdot P}\right)v(P)\ ,$ (56)

respectively. These two expressions agree with the Eqs. (13) and (14) in Ref.
Cirigliano:2008wn . To this order, $V^{\mu\nu}_{\text{SD}}$ and
$A^{\mu\nu}_{\text{SD}}$, which are $\mathcal{O}(p^{2})$, can be neglected.

In the Low limit, we obtain

$\mathcal{M}_{\gamma}=\frac{eG_{F}V_{us}^{*}}{\sqrt{2}}\bar{u}(q^{\prime})\gamma_{\nu}(1-\gamma^{5})v(P)H^{\nu}(-p_{K},p_{\pi})\left(\frac{\epsilon\cdot
p_{+}}{k\cdot p_{+}}-\frac{\epsilon\cdot P}{k\cdot P}\right)\ ,$ (57)

where the subscript ‘$+$’ refers to the charged meson. The spin-averaged
squared matrix element is then given by

$\begin{split}\overline{\left|\mathcal{M}_{\gamma}\right|^{2}}&=4C_{K}^{2}e^{2}G_{F}^{2}\left|V_{us}\right|^{2}S_{\rm
EW}^{K}\left\\{\left[\frac{m_{\ell}^{2}}{2}(t-m_{\ell}^{2})+2m_{K}^{2}m_{\pi}^{2}+2u(m_{\ell}^{2}-t+m_{K}^{2}+m_{\pi}^{2})-2u^{2}\right.\right.\\\\[4.30554pt]
&-\frac{\Delta_{K\pi}}{t}m_{\ell}^{2}(2u+t-m_{\ell}^{2}-2m_{\pi}^{2})+\frac{\Delta_{K\pi}^{2}}{t^{2}}\frac{m_{\ell}^{2}}{2}(t-m_{\ell}^{2})\left.\right]\left|F_{+}(t)\right|^{2}\\\\[4.30554pt]
&+\frac{\Delta_{K\pi}\,m_{\ell}^{2}}{t}\left[2u+t-m_{\ell}^{2}-2m_{\pi}^{2}+\frac{\Delta_{K\pi}}{t}(m_{\ell}^{2}-t)\right]\mathrm{Re}\left[F_{+}(t)F_{0}^{*}(t)\right]\left.\right.\\\\[4.30554pt]
&+\frac{\Delta_{K\pi}^{2}\,m_{\ell}^{2}}{2t^{2}}\left(t-m_{\ell}^{2}\right)\left|F_{0}(t)\right|^{2}\left.\right\\}\sum_{\gamma\
\rm pols.}\left|\frac{p_{-}\cdot\epsilon}{p_{-}\cdot
k}-\frac{P\cdot\epsilon}{P\cdot k}\right|^{2}+\mathcal{O}(k^{0})\
,\end{split}$ (58)

where $u=(p_{K}-P)^{2}$. The last expression can also be written in terms of
$f_{+/-}(t)$,

$\overline{\left|\mathcal{M}_{\gamma}\right|^{2}}=2C_{K}^{2}m_{K}^{4}e^{2}G_{F}^{2}\left|V_{us}\right|^{2}S_{\rm
EW}^{K}\rho^{(0)}(y,z)\sum_{\gamma\ \rm
pols.}\left|\frac{p_{-}\cdot\epsilon}{p_{-}\cdot
k}-\frac{P\cdot\epsilon}{P\cdot k}\right|^{2}+\mathcal{O}(k^{0})\ ,$ (59)

where

$\rho^{(0)}(y,z)=A_{1}^{(0)}(y,z)\left|f_{+}(t)\right|^{2}+A_{2}^{(0)}(y,z)\mathrm{Re}\left[f_{+}(t)f_{-}^{*}(t)\right]+A_{3}^{(0)}(y,z)\left|f_{-}(t)\right|^{2}\
,$ (60)

and the kinematical densities are

$\displaystyle A_{1}^{(0)}$
$\displaystyle=4(y+z-1)(1-y)+r_{\ell}(4y+3z-3)-4r_{\pi}+r_{\ell}(r_{\pi}-r_{\ell})\
,$ (61a) $\displaystyle A_{2}^{(0)}$
$\displaystyle=2r_{\ell}(r_{\ell}-r_{\pi}-2y-z+3)\ ,$ (61b) $\displaystyle
A_{3}^{(0)}$ $\displaystyle=r_{\ell}(r_{\pi}-r_{\ell}+1-z)\ ,$ (61c)

with

$z=\frac{2p_{\pi}\cdot p_{K}}{m_{K}^{2}}=\frac{2E_{\pi}}{m_{k}}\ ,\qquad
y=\frac{2p_{K}\cdot p_{\ell}}{m_{K}^{2}}=\frac{2E_{\ell}}{m_{k}}\ ,$ (62)

$r_{\ell}=(m_{\ell}/m_{K})^{2}$, and $r_{\pi}=(m_{\pi}/m_{K})^{2}$. Here,
$E_{\pi}$ ($E_{\ell}$) is the energy of the pion (charged lepton) in the kaon
rest frame. The expression in Eq. (59) can be compared directly with the
results in Refs. Cirigliano:2001mk ; Cirigliano:2004pv ; Cirigliano:2008wn .

The $K\to\pi\ell\nu_{\ell}$ decay width without radiative corrections
Antonelli:2013usa  is given by

$\Gamma(K\to\pi\ell\nu_{\ell})=\frac{G_{F}^{2}m_{K}^{5}}{192\pi^{3}}S_{\rm
EW}^{K}|V_{us}|^{2}|F_{+}(0)|I^{\ell}_{K}\ ,$ (63)

where

$\begin{split}I^{\ell}_{K}&=\int_{m_{\ell}^{2}}^{t_{\text{max}}}dt\frac{1}{m_{K}^{8}}\lambda^{3/2}(t,m_{K}^{2},m_{\pi}^{2})\left(1-\frac{m_{\ell}^{2}}{t}\right)^{2}\left(1+\frac{m_{\ell}^{2}}{2t}\right)\\\\[4.30554pt]
&\times\left[C_{V}^{2}|\tilde{F}_{+}(t)|^{2}+\frac{3\Delta_{K\pi}^{2}m_{\ell}^{2}}{(2t+m_{\ell}^{2})\lambda(t,m_{K}^{2},m_{\pi}^{2})}C_{S}|\tilde{F}_{0}(t)|^{2}\right]\
,\end{split}$ (64)

and $t_{\text{max}}=(m_{K}-m_{\pi})^{2}$.

##  Appendix B Virtual corrections to the hadronic tau decays

The radiative corrections to the $\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}$
decays at $\mathcal{O}(p^{2})$ in ChPT Weinberg:1978kz ; Gasser:1983yg ;
Gasser:1984gg  are depicted in Fig. 12. The overall contribution is given by
Cirigliano:2001er

$\frac{\delta H^{\mu}(t,u)}{C_{V}}=\delta f_{+}(u)(p_{1}-p_{0})^{\mu}+\delta
f_{-}(u)(p_{1}+p_{0})^{\mu}\ ,$ (65)

where

$\begin{split}\delta
f_{+}(u)&=\frac{\alpha}{4\pi}\left[2+\frac{1}{\epsilon}-\gamma_{E}+\log
4\pi-\log\frac{m_{\tau}^{2}}{\mu^{2}}+(u-m_{-}^{2})\mathcal{A}(u)+(u-m_{-}^{2}-m_{\tau}^{2})\mathcal{B}(u)\right.\\\\[4.30554pt]
&+2(m_{-}^{2}+m_{\tau}^{2}-u)\mathcal{C}(u,M_{\gamma})+2\log\left(\frac{m_{-}m_{\tau}}{M_{\gamma}^{2}}\right)\left.\right],\end{split}$
(66) $\begin{split}\delta
f_{-}(u)&=\frac{\alpha}{4\pi}\left[-5-3\left(\frac{1}{\epsilon}-\gamma_{E}+\log
4\pi\right)+\log\frac{m_{-}^{2}}{\mu^{2}}+2\log\frac{m_{\tau}^{2}}{\mu^{2}}+(3u+m_{-}^{2}-2m_{\tau}^{2})\mathcal{A}(u)\right.\\\\[4.30554pt]
&+(u+m_{-}^{2}-m_{\tau}^{2})\mathcal{B}(u)\left.\right]\ ,\end{split}$ (67)

with

$\mathcal{A}(u)=\frac{1}{u}\left(-\frac{1}{2}\log
r_{\tau}+\frac{2-y}{\sqrt{r_{\tau}}}\frac{x}{1-x^{2}}\log x\right)\ ,$ (68)
$\mathcal{B}(u)=\frac{1}{u}\left(\frac{1}{2}\log
r_{\tau}+\frac{2r_{\tau}-y}{\sqrt{r_{\tau}}}\frac{x}{1-x^{2}}\log x\right)\ ,$
(69)
$\begin{split}\mathcal{C}(u,M_{\gamma})&=\frac{1}{m_{\tau}m_{-}}\frac{x}{1-x^{2}}\bigg{[}-\frac{1}{2}\log^{2}x+2\log
x\log(1-x^{2})-\frac{\pi^{2}}{6}+\frac{1}{8}\log^{2}r_{\tau}\\\\[4.30554pt]
&+\mathrm{Li}_{2}(x^{2})+\mathrm{Li}_{2}\left(1-\frac{x}{\sqrt{r_{\tau}}}\right)+\mathrm{Li}_{2}(1-x\sqrt{r_{\tau}})-\log
x\log\left(\frac{M_{\gamma}^{2}}{m_{\tau}m_{-}}\right)\bigg{]}\ ,\end{split}$
(70)

and
$C_{V}^{\pi\pi,KK,K^{-}\pi^{0},K^{0}\pi^{-}}=\left(\sqrt{2},-1,\frac{1}{\sqrt{2}},-1\right)$.
Here, $\mathcal{A}(u)$, $\mathcal{B}(u)$ and $\mathcal{C}(u,M_{\gamma})$ are
written in terms of the variables

$r_{\tau}=\frac{m_{\tau}^{2}}{m_{-}^{2}}\ ,\qquad
y=1+r_{\tau}-\frac{u}{m_{-}^{2}}\ ,\qquad
x=\frac{1}{2\sqrt{r_{\tau}}}\left(y-\sqrt{y^{2}-4r_{\tau}}\right)\ ,$ (71)

and the dilogarithm

$\mathrm{Li}_{2}(x)=-\int_{0}^{1}\frac{dt}{t}\log(1-xt)\ .$ (72)

$\tau^{-}$$P^{-}$$P^{(\prime)0}$$\nu_{\tau}$$\otimes$

$\tau^{-}$$P^{-}$$P^{(\prime)0}$$\nu_{\tau}$$\otimes$

$\tau^{-}$$P^{-}$$P^{(\prime)0}$$\nu_{\tau}$$\otimes$

Figure 12: Photon loop diagrams that contribute to the
$\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}$ decays.

The radiative corrections to these decays induce a dependence in the $u$
variable. From a comparison with the results in Ref. Antonelli:2013usa , we
get the following relation

$\begin{split}\delta\bar{f}_{+}(u)&=\frac{\alpha}{4\pi}\frac{1}{f_{+}(0)}\left[\Gamma_{1}(u,m_{\tau}^{2},m_{-}^{2})+\Gamma_{2}(u,m_{\tau}^{2},m_{-}^{2})\right]+\cdots\\\\[4.30554pt]
&=\frac{\alpha}{4\pi}\frac{1}{f_{+}(0)}\left[(u-m_{-}^{2})\mathcal{A}(u)+(u-m_{-}^{2}-m_{\tau}^{2})\mathcal{B}(u)\right]+\cdots\
,\end{split}$ (73)

and

$\begin{split}\delta\bar{f}_{-}(u)&=\frac{\alpha}{4\pi}\frac{1}{f_{+}(0)}\left[\Gamma_{1}(u,m_{\tau}^{2},m_{-}^{2})-\Gamma_{2}(u,m_{\tau}^{2},m_{-}^{2})\right]+\cdots\\\\[4.30554pt]
&=\frac{\alpha}{4\pi}\frac{1}{f_{+}(0)}\left[(3u+m_{-}^{2}-2m_{\tau}^{2})\mathcal{A}(u)+(u+m_{-}^{2}-m_{\tau}^{2})\mathcal{B}(u)\right]+\cdots\
.\end{split}$ (74)

##  Appendix C $\boldsymbol{\tau^{-}\to(P_{1}P_{2})^{-}\nu_{\tau}}$ decays

After the inclusion of the virtual-photon radiative corrections to the form
factor in Sect. B, the amplitude for the $\tau^{-}(P)\to
P^{-}_{1}(p_{-})P^{0}_{2}(p_{0})\nu_{\tau}(q^{\prime})$ decays is given by

$\mathcal{M}_{0}=\frac{G_{F}V_{uD}\sqrt{S_{\rm
EW}}}{\sqrt{2}}H_{\nu}(p_{-},p_{0})\bar{u}(q^{\prime})\gamma^{\nu}(1-\gamma^{5})u(P)\
.$ (75)

Thus, the spin-averaged squared amplitude follows

$\begin{split}\overline{\left|\mathcal{M}_{0}\right|^{2}}&=2G_{F}^{2}\left|V_{uD}\right|^{2}S_{\rm
EW}\bigg{\\{}C_{S}^{2}\left|F_{0}(t,u)\right|^{2}D_{0}^{P^{-}P^{0}}(t,u)+C_{S}C_{V}\mathrm{Re}\left[F_{+}(t,u)F_{0}^{*}(t,u)\right]D_{+0}^{P^{-}P^{0}}(t,u)\\\\[4.30554pt]
&+C_{V}^{2}\left|F_{+}(t,u)\right|^{2}D_{+}^{P^{-}P^{0}}(t,u)\bigg{\\}}\
,\end{split}$ (76)

where we have defined $F_{+/0}(t,u)$ in Eq. (32), and the expressions for
$D_{0}^{P^{-}P^{0}}(t,u)$, $D_{+0}^{P^{-}P^{0}}(t,u)$ and
$D_{+}^{P^{-}P^{0}}(t,u)$ are given in Eqs. (9)–(11).

The differential decay width in the tau rest frame is

$\frac{d^{2}\Gamma}{dtdu}=\frac{1}{32(2\pi)^{3}m_{\tau}^{3}}\overline{\left|\mathcal{M}_{0}\right|^{2}}\
,$ (77)

where $t=(p_{-}+p_{0})^{2}$ is the invariant mass and
$u=(P-p_{-})^{2}=(p_{0}+k+q^{\prime})^{2}$. The physical region is limited by
$(m_{-}+m_{0})^{2}\leq t\leq m_{\tau}^{2}$ and $u^{-}(t)\leq u\leq u^{+}(t)$,
with

$u^{\pm}(t)=\frac{1}{2t}\left[2t(m_{\tau}^{2}+m_{0}^{2}-t)-(m_{\tau}^{2}-t)(t+m_{-}^{2}-m_{0}^{2})\pm(m_{\tau}^{2}-t)\sqrt{\lambda(t,m_{-}^{2},m_{0}^{2})}\right]\
,$ (78)

and $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$.

The invariant mass distribution is obtained integrating upon the $u$ variable

$\begin{split}\frac{d\Gamma}{dt}&=\frac{G_{F}^{2}S_{\rm
EW}|V_{uD}|^{2}m_{\tau}^{3}}{384\pi^{3}t}\bigg{\\{}\frac{1}{2t^{2}}\left(1-\frac{t}{m_{\tau}^{2}}\right)^{2}\lambda^{1/2}(t,m_{-}^{2},m_{0}^{2})\\\\[4.30554pt]
&\times\left[C_{V}^{2}\left|F_{+}(t)\right|^{2}\left(1+\frac{2t}{m_{\tau}^{2}}\right)\lambda(t,m_{-}^{2},m_{0}^{2})\left(1+\tilde{\delta}_{+}(t)\right)+3C_{S}^{2}\Delta_{-0}^{2}\left|F_{0}(t)\right|^{2}\left(1+\tilde{\delta}_{0}(t)\right)\right]\\\\[4.30554pt]
&+C_{S}C_{V}\frac{4}{\sqrt{t}}\tilde{\delta}_{+0}(t)\bigg{\\}}\ ,\end{split}$
(79)

where

$\displaystyle\tilde{\delta}_{0}(t)$
$\displaystyle=\frac{\int_{u^{-}(t)}^{u^{+}(t)}D_{0}^{P^{-}P^{0}}(t,u)2\mathrm{Re}\left[F_{0}(t)\delta
F_{0}^{*}(t,u)\right]du}{\int_{u^{-}(t)}^{u^{+}(t)}D_{0}^{P^{-}P^{0}}(t,u)|F_{0}(t)|^{2}du}\
,$ (80a) $\displaystyle\tilde{\delta}_{+}(t)$
$\displaystyle=\frac{\int_{u^{-}(t)}^{u^{+}(t)}D_{+}^{P^{-}P^{0}}(t,u)2\mathrm{Re}\left[F_{+}(t)\delta
F_{+}^{*}(u)\right]du}{\int_{u^{-}(t)}^{u^{+}(t)}D_{+}^{P^{-}P^{0}}(t,u)|F_{+}(t)|^{2}du}\
,$ (80b) $\displaystyle\tilde{\delta}_{+0}(t)$
$\displaystyle=\frac{3t\sqrt{t}}{4m_{\tau}^{6}}\int_{u^{-}(t)}^{u^{+}(t)}D_{+0}^{P^{-}P^{0}}(t,u)\left(\mathrm{Re}\left[F_{+}(t)\delta
F_{0}^{*}(t,u)\right]+\mathrm{Re}\left[F_{0}(t)\delta
F_{+}^{*}(u)\right]\right)du\ .$ (80c)

##  Appendix D Kinematics

As in Refs. Cirigliano:2002pv ; Antonelli:2013usa ; Flores-Tlalpa:2008bws ;
Miranda:2020wdg , after an integration over $\mathcal{D}_{\rm IV/III}$ and
$\mathcal{D}_{\rm III}$, the functions in Eqs. (14) are given by (we note that
$K_{11}$ is numerically negligible)

$\displaystyle J_{11}(t,u)$
$\displaystyle=\log\left(\frac{2x_{+}(t,u)\bar{\gamma}}{M_{\gamma}}\right)\frac{1}{\bar{\beta}}\log\left(\frac{1+\bar{\beta}}{1-\bar{\beta}}\right)$
(81a)
$\displaystyle+\frac{1}{\bar{\beta}}\left[\mathrm{Li}_{2}(1/Y_{2})-\mathrm{Li}_{2}(Y_{1})+\log^{2}(-1/Y_{2})/4-\log^{2}(-1/Y_{1})/4\right]\
,$ (81b) $\displaystyle J_{20}\left(t,u\right)$
$\displaystyle=\log\left(\frac{M_{\gamma}(m_{\tau}^{2}-t)}{m_{\tau}\,x_{+}(t,u)}\right)\
,$ (81c) $\displaystyle J_{02}\left(t,u\right)$
$\displaystyle=\log\left(\frac{M_{\gamma}(m_{\tau}^{2}+m_{0}^{2}-t-u)}{m_{-}\,x_{+}(t,u)}\right)\
,$ (81d) $\displaystyle K_{20}\left(t,u\right)$
$\displaystyle=K_{02}\left(t,u\right)=\log\left(\frac{x_{-}(t,u)}{x_{+}(t,u)}\right)\
,$ (81e)

where

$\begin{split}x_{\pm}\left(t,u\right)=&\frac{-m_{-}^{4}+(m_{{}^{0}}^{2}-t)(m_{\tau}^{2}-u)+m_{-}^{2}(m_{\tau}^{2}+m_{0}^{2}+t+u)}{2m_{-}^{2}}\\\\[4.30554pt]
&\pm\frac{\lambda^{1/2}\left(u,m_{\tau}^{2},m_{-}^{2}\right)\lambda^{1/2}\left(t,m_{-}^{2},m_{0}^{2}\right)}{2m_{-}^{2}}\
.\end{split}$ (82)

These expressions are written in terms of

$Y_{1,2}=\frac{1-2\bar{\alpha}\pm\sqrt{(1-2\bar{\alpha})^{2}-(1-\bar{\beta}^{2})}}{1+\bar{\beta}}\
,$ (83)

with

$\displaystyle\bar{\alpha}=$
$\displaystyle\frac{(m_{\tau}^{2}-t)(m_{\tau}^{2}+m_{0}^{2}-t-u)}{(m_{-}^{2}+m_{\tau}^{2}-u)}\cdot\frac{\lambda(u,m_{-}^{2},m_{\tau}^{2})}{2\bar{\delta}}\
,$ $\displaystyle\bar{\beta}=$
$\displaystyle-\frac{\sqrt{\lambda(u,m_{-}^{2},m_{\tau}^{2})}}{m_{-}^{2}+m_{\tau}^{2}-u}\
,$ $\displaystyle\bar{\gamma}=$
$\displaystyle\frac{\sqrt{\lambda(u,m_{-}^{2},m_{\tau}^{2})}}{2\sqrt{\bar{\delta}}}\
,$ $\displaystyle\bar{\delta}=$ $\displaystyle-
m_{0}^{4}m_{\tau}^{2}+m_{-}^{2}(m_{\tau}^{2}-t)(m_{0}^{2}-u)-tu(-m_{\tau}^{2}+t+u)$
$\displaystyle+m_{0}^{2}(-m_{\tau}^{4}+tu+m_{\tau}^{2}t+m_{\tau}^{2}u)\ .$

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