# Improved description of di-lepton production in $\tau^{-}\to\nu_{\tau}P^{-}$
decays

Adolfo Guevara1 2, Gabriel López Castro3 and Pablo Roig3
1 Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.
2 Departament de Física Teòrica, IFIC,
Universitat de València - CSIC,
Apt. Correus 22085, E-46071 València, Spain
3 Departamento de Física, Centro de Investigación
y de Estudios Avanzados del Instituto Politécnico Nacional.
AP 14-740, 07000, Ciudad de México, México

Abstract Recently, the Belle collaboration reported the first measurements of
the $\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ branching fraction and the
spectrum of the pion-dielectron system. In an analysis previous to Belle’s
results, we evaluated this branching fraction which turned out to be
compatible with that reported by Belle, although with a large uncertainty.
This is the motivation to seek for improvements on our previous evaluation of
$\tau^{-}\to\nu_{\tau}\pi^{-}\ell^{+}\ell^{-}$ decays ($\ell=e,\,\mu$). In
this paper we improve our calculation of the $WP^{-}\gamma^{*}$ vertex by
including flavor symmetry breaking effects in the framework of the Resonance
Chiral Theory. We impose QCD short-distance behavior to constrain most
parameters and data on the $\pi^{-}e^{+}e^{-}$ spectrum reported by Belle to
fix the remaining free ones. As a result, improved predictions for the
branching ratios and hadronic/leptonic spectra are reported, in good agreement
with observations. Analogous calculations for the strangeness-changing
$\tau^{-}\to\nu_{\tau}K^{-}\ell^{+}\ell^{-}$ transitions are reported for the
first time. Albeit one expects the $m_{\pi\mu^{+}\mu^{-}}$ spectrum to be
measured in Belle-II and the observables with $\ell=e$ can be improved, it is
rather unlikely that the $K$ channels can be measured due to the suppression
factor $|V_{ud}/V_{us}|^{2}=0.05$.

##  1 Introduction

The search for signals of physics beyond the Standard Model (SM) requires a
good understanding of SM processes either to discard possible backgrounds
coming from it, such as large radiative corrections [1, 2, 3] or to have under
good control hadronic contamination in precision tests of the SM [4]. In
addition to offering a clean laboratory to test the hadronization of the weak
currents, some semileptonic $\tau$ lepton decays, such as
$\tau\to\nu_{\tau}P(\gamma)$ for $P=\pi,K$, provide a good example where SM
effects can be reliably calculated to disentangle possible New Physics signals
hidden in precision observables.

In Ref. [5] we reported the first prediction of
$\mathcal{B}(\tau\to\nu_{\tau}\pi\ell\overline{\ell})$ and the corresponding
di-lepton spectrum where $\ell=e,\mu$ (this can be viewed as the crossed
channels of lepton pairs produced in $\pi_{\ell 2}$ decays [6] in a larger
kinematical domain); later on the Belle collaboration [7] announced the first
searches of these decays. Recently, some of us have also reported similar
studies of $\tau^{-}\to\nu_{\tau}\pi^{-}\pi^{0}\ell\bar{\ell}$ decays [8].
Together with the five lepton decays of tau leptons [9], they provide a better
description of possible backgrounds in Lepton Number- or Lepton Flavor
Violation searches in $\tau$ decays. Motivated by the Belle Collaboration
studies [7], in this work we revisit our predictions for
$\tau\to\nu_{\tau}\pi\ell\overline{\ell}$ decays with the aim of improving the
theoretical description of structure-dependent effects and to get reduced
uncertainties. In addition, we make for the first time an analogous analysis
of the strangeness-changing processes $\tau\to\nu_{\tau}K\ell\overline{\ell}$
as well.

In these phenomena, the $W\gamma^{\star}P$ vertex plays a central role and its
description is necessary to understand the radiative corrections to the
$\tau^{-}\to\nu_{\tau}P^{-}$ decays [10]. This vertex also involves parameters
which are needed to describe the pion transition form factor (TFF), which is
required to compute the dominant piece (the pion pole) of the hadronic light-
by-light contribution to the anomalous magnetic moment of the $\mu$ lepton,
$a_{\mu}$. (The TFF can be obtained by our vector form factor (see section
3.2) by considering Bosé symmetry.) Although knowledge on these parameters
could in principle help reduce the uncertainty on the hadronic part of
$a_{\mu}$ [11], the $\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ data does not
(and is not foreseeable to) have the necessary precision to improve actual
predictions on the $\pi$-pole contribution to $a_{\mu}$.

The problem with the description of these effective vertices arises when one
tries to describe them in terms of the fundamental fields of the Standard
Model, since at energies below the $m_{\tau}$ scale, one can not give a proper
perturbative description of color interactions. However, the decay amplitude
involving these vertices can be, for the sake of convenience, split into a
part where the hadronic current $\langle
0|\overline{u}\gamma_{\mu}(1-\gamma_{5})d|\pi^{-}\rangle=-i\sqrt{2}f_{\pi}p_{\mu}$
and the electromagnetic interactions are computed using scalar QED (sQED),
which we call structure-independent, and a part where more involved hadronic
interactions are computed using an Effective Field Theory, called structure-
dependent. Thus, we try to surpass the difficulties of calculating the
structure-dependent part using Resonance Chiral Theory (R$\chi$T) [12, 13],
which is an extension of Chiral Perturbation Theory ($\chi$PT) [14, 15, 16]
that includes resonances as active degrees of freedom. $\chi$PT relies on the
chiral symmetry group $G=U(3)_{L}\otimes U(3)_{R}$ of the massless QCD
Lagrangian. After it gets spontaneously broken, $G\to U(3)_{V}$, the remaining
symmetry gets explicitly broken when the masses of the light quarks are
considered to be non-vanishing. The
$\mathcal{B}(\tau^{-}\to\nu_{\tau}\pi^{-}\ell\overline{\ell})$ and di-lepton
spectrum were computed previously in ref. [5] using such techniques, however,
the novelty in the present treatment is that we include the effects of finite
different light-quark masses as done for the Transition Form Factor of the
pseudo Goldstone bosons for the Hadronic Light-by-Light part of the $a_{\mu}$
in ref. [17] (over [18], where these were neglected). We also give a more
thorough treatment of the uncertainties than those in ref. [5], thus obtaining
consistent results comparing with the corresponding form factors given in ref.
[19]. Furthermore, the recent measurement of the branching fraction with a
lower limit in the invariant mass of the pion and di-lepton pair [7], $m_{\pi
e^{-}e^{+}}$, motivates further this re-analysis, since in the $m_{\pi
e^{-}e^{+}}\geq 1.05$ GeV region the branching fraction gets saturated by the
structure-dependent contribution. While most of the parameters of the model
can be constrained by means of the high-energy behavior of QCD, some of them
remain loose. We fit these to the measured invariant mass
$m_{\pi^{-}e^{-}e^{+}}$ spectra and also to the measurement of the branching
fraction $\mathcal{B}(m_{\pi^{-}e^{-}e^{+}}\geq 1.05\text{ GeV})=(5.90\pm
1.01)\times 10^{-6}$ [7]. Despite the access to the invariant mass spectra
data for the $\tau^{+}\to\nu_{\tau}\pi^{+}\overline{\ell}\ell$ decay, we will
only make use of the data for the $\tau^{-}$ decay. The reason not to use both
sets is that the spectra have incompatibilities in several bins; also, when
fitting individually the $\pi^{+}$ data set leads to unphysical conditions
(see discussion in subsection 4.2). As a result, we improve our predictions,
with correspondingly reduced uncertainties.

The outline of the paper is as follows. In section 2 the different
contributions to the matrix element are collected. In section 3 we introduce
the Lagrangian used for computing the structure-dependent corrections,
calculate the corresponding form factors (including flavor-breaking
corrections to our previous results) and derive the short-distance constraints
among resonance couplings. In section 4 we carry out our phenomenological
analysis, including a fit to Belle $\tau^{-}\to\pi^{-}e^{+}e^{-}\nu_{\tau}$
data and predicting the partner $(\pi\leftrightarrow K,e\leftrightarrow\mu)$
modes, yet to be discovered. We give our conclusions in section 5.

##  2 Amplitudes

For convenience, we take three kinds of contributions to the decay amplitude:
the first called inner bremsstrahlung (IB) or structure independent (SI). The
other two are the structure dependent (SD) ones, namely the polar- (V) and
axial-vector (A) parts of the left-handed weak charged current. The IB
amplitude can be obtained using the sQED Lagrangian, where the photon is
either radiated by the $\tau$ lepton, off the pseudo Goldstone boson ($\pi$ or
$K$) or by the longitudinal propagation mode of the $W^{-}$ boson, a
contribution which is needed to achieve gauge invariance of the total IB
amplitude. The total IB contribution is shown in eq. (1), along with the
parametrization of the SD parts as given in ref. [5]. The momenta definition
is given in Figure 1.

Figure 1: Feynman diagrams of the SI contributions (only scalar QED is used
for the radiation off the $P^{-}$ meson) to the
$\tau^{-}(p_{\tau})\to\nu_{\tau}(q)P^{-}(p)\ell(p_{-})\overline{\ell}(p_{+})$
decay amplitude. The diamond vertex is an effective vertex meaning the $W$
boson has been integrated out.

The different contributions to the matrix element are ($D=d,s$ for
$P=\pi,\,K$)

$\displaystyle\mathcal{M}_{IB}$
$\displaystyle=-iG_{F}V_{uD}f_{\pi}m_{\tau}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}\overline{u}_{\nu_{\tau}}(1+\gamma_{5})\left[\frac{2p_{\nu}}{2p\cdot
k+k^{2}}+\frac{2{p_{\tau}}_{\nu}-\not{k}\gamma_{\nu}}{-2p_{\tau}\cdot
k+k^{2}}\right]u_{\tau},$ (1a) $\displaystyle\mathcal{M}_{V}$
$\displaystyle=-G_{F}V_{uD}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}J_{\tau}^{\mu}F_{V}(W^{2},k^{2})\varepsilon_{\mu\nu\alpha\beta}k^{\alpha}p^{\beta},$
(1b) $\displaystyle\mathcal{M}_{A}$
$\displaystyle=iG_{F}V_{uD}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}J_{\tau}^{\mu}\left\\{F_{A}(W^{2},k^{2})\left[(W^{2}+k^{2}-m_{\pi}^{2})g_{\mu\nu}-2k_{\mu}p_{\nu}\right]\frac{}{}\right.$
$\displaystyle\left.\hskip
77.49976pt-\frac{}{}A_{2}(W^{2},k^{2})k^{2}g_{\mu\nu}+A_{4}(W^{2},k^{2})k^{2}(p+k)_{\mu}p_{\nu}\right\\}.$
(1c)

Here, $J_{\ell}^{\nu}=\overline{u}(p_{-})\gamma^{\nu}v(p_{+})$ and
$J_{\tau}^{\mu}=\overline{u}(q)(1+\gamma_{5})\gamma^{\mu}u(p_{\tau})$ are the
lepton electromagnetic and $\tau$ weak charged currents, respectively. We use
$W^{2}\equiv(p_{\tau}-q)^{2}$ and $k^{2}\equiv(p_{-}+p_{+})^{2}$ as the two
independent Lorentz-invariants upon which the form factors
($F_{V},F_{A},A_{2},A_{4}$) depend. In ref. [5] the axial amplitude was given
only in terms of three form factors ($F_{V},F_{A}$ and a combination of
$A_{2}$ and $A_{4}$ called $B$), since at chiral order $p^{4}$, the $A_{2}$
and $A_{4}$ form factors are linearly dependent and can be written in terms of
the pseudo Goldstone electromagnetic form factor $F_{V}^{P}(k^{2})$ [6]. Here,
$A_{2}$ and $A_{4}$ cannot be recast in terms of $F_{V}^{P}(k^{2})$, since we
are considering contributions of chiral order $p^{6}$. Furthermore, including
the complete set of leading-order chiral symmetry breaking contributions will
change the pion pole for the massive pion propagator. As a result, the $A_{2}$
and $A_{4}$ form factors become linearly independent and the axial-vector part
of the left hadronic current cannot be expressed in terms of the two form
factors $\mathcal{F}(W^{2},k^{2},p^{2})$ and $\mathcal{G}(W^{2},k^{2},p^{2})$
of refs. [19, 20, 21] (see discussion after eq. (24)).

##  3 Structure dependent form factors

###  3.1 The relevant operators

In this section we will present, for the sake of simplicity, only the relevant
operators in the R$\chi$T Lagrangian needed to compute the form factors, which
are given in the next subsection. We will be concise here, for a more extended
discussion see eg. ref. [17]. R$\chi$T extends the domain of applicability of
Chiral Perturbation Theory [16, 14, 15] ($\chi$PT) by adding the light-
flavored resonances as active degrees of freedom.

We start with operators involving no resonances, these being 111Although these
terms also appear in the $\chi$PT Lagrangian, their couplings get shifted in
the presence of resonance contributions (see for instance [22, 23, 24]).

$\mathcal{L}_{0\,Res}=\frac{f^{2}}{4}\langle
u^{\mu}u_{\mu}+\chi_{+}\rangle+\mathcal{L}_{WZW}+C_{7}^{W}\mathcal{O}_{7}^{W}+C_{11}^{W}\mathcal{O}_{11}^{W}+C_{22}^{W}\mathcal{O}_{22}^{W}\,,$
(2)

where the first term is given by the leading $\chi$PT Lagrangian operators of
chiral order $p^{2}$ [16, 14, 15], the second one is the anomalous Wess-
Zumino-Witten Lagrangian of $\mathcal{O}(p^{4})$ [25, 26] and the last three
operators belong to the subleading odd-intrinsic parity sector
$\mathcal{O}(p^{6})$ Lagrangian [27]. We neglect operators not included in
this Lagrangian. Congruently with refs. [19], [21] and [28], we will not
consider any $\mathcal{O}(p^{8})$ contribution whatsoever. In the first term,
$f$ is the decay constant in the chiral limit, which we will set to
$f=f_{\pi}\sim 92$ MeV, $u^{\mu}$ and $\chi_{+}$ are chiral tensors [29], the
former containing derivatives of the $\pi/K$ fields and external spin-one
currents and the latter scalar currents involving the previous fields masses
squared, $m_{\pi/K}^{2}$, times even powers of such fields.

The equations of motion of the resonances give their classical fields in terms
of series of chiral tensors of different order. The resonances are said to be
integrated out (tree-level integration) when the classic fields are
substituted in favor of chiral tensors in the resonant Lagrangian. Integrating
the resonances out using the leading-order terms of the equations of motion
very approximately saturates the $\mathcal{O}(p^{4})$ (and leading
$\mathcal{O}(p^{6})$) contributions in the even-intrinsic parity sector [12,
13, 19]; therefore, we will not use the non-resonant $\mathcal{O}(p^{4})$ set
of operators for the sake of simplicity, since they are considered to yield
negligible contributions. Since we will only consider leading-order terms in
the resonances equations of motion, the $\mathcal{O}(p^{6})$ chiral low-energy
constants in the odd-intrinsic parity sector cannot be saturated upon
resonance exchange [28], therefore we have to include the three contributing
$C_{i}^{W}\mathcal{O}_{i}^{W}$ terms [27]:

$\displaystyle\mathcal{O}_{7}^{W}$ $\displaystyle=$ $\displaystyle
i\epsilon_{\mu\nu\alpha\beta}\langle\,\chi_{-}f_{+}^{\mu\nu}f_{+}^{\alpha\beta}\,\rangle,$
$\displaystyle\mathcal{O}_{11}^{W}$ $\displaystyle=$ $\displaystyle
i\epsilon_{\mu\nu\alpha\beta}\langle\,\chi_{+}[f_{+}^{\mu\nu},f_{-}^{\alpha\beta}]\,\rangle,$
(3) $\displaystyle\mathcal{O}_{22}^{W}$ $\displaystyle=$ $\displaystyle
i\epsilon_{\mu\nu\alpha\beta}\langle\,u^{\mu}\\{\nabla_{\rho}f_{+}^{\rho\nu},f_{+}^{\alpha\beta}\\}\,\rangle\,,$

where the following chiral tensors [29] enter: $\chi_{-}$ gives odd powers of
the $\pi/K$ fields with factors involving $m_{\pi}^{2}$ or $m_{K}^{2}$,
$\nabla_{\mu}$ is the covariant derivative and includes spin-one left and
vector external currents through the connection and $f_{\pm}^{\mu\nu}$ yields
the field-strength tensors of the charged-weak or electromagnetic fields.

We turn next to those operators with one resonance field, in either intrinsic
parity sector,

$\mathcal{L}_{1\,Res}=\mathcal{L}_{1\,Res}^{even}+\mathcal{L}_{1\,Res}^{odd}\,.$
(4)

In turn, the first piece can be further divided according to the quantum
numbers of this resonance

$\mathcal{L}_{1\,Res}^{even}=\sum_{R_{i}=V,A,P}\mathcal{L}_{1\,R_{i}}^{even}\,.$
(5)

The contributions with one vector resonance read 222$V_{\mu\nu}$ (analogously
$A_{\mu\nu}$ for axial resonances below) is a matrix in flavor ($u,d,s$) space
and we use the antisymmetric tensor formalism for spin-one fields for
convenience [12, 13]. [12, 19]

$\mathcal{L}_{1\,V}^{even}=\frac{F_{V}}{2\sqrt{2}}\langle\,V_{\mu\nu}f_{+}^{\mu\nu}\,\rangle+\frac{i}{2\sqrt{2}}G_{V}\langle\,V_{\mu\nu}[u^{\mu},u^{\nu}]\,\rangle+\frac{\lambda_{V}}{\sqrt{2}}\langle\,V_{\mu\nu}\\{f_{+}^{\mu\nu},\chi_{+}\\}\,\rangle\,,$
(6)

where the $V$ field (we assume ideal mixing of neutral mesons) has an
analogous flavor structure as the pseudo Goldstone field $\phi$, namely

$V_{\mu\nu}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\rho^{0}_{\mu\nu}+\omega_{\mu\nu}\right)&\rho^{+}_{\mu\nu}&{K^{\star}}^{+}_{\mu\nu}\\\
\rho^{-}_{\mu\nu}&\frac{1}{\sqrt{2}}\left(-\rho^{0}_{\mu\nu}+\omega_{\mu\nu}\right)&{K^{\star}}^{0}_{\mu\nu}\\\
{K^{\star}}^{-}_{\mu\nu}&{\overline{K}^{\hskip
1.50694pt\star}}^{0}_{\mu\nu}&\phi_{\mu\nu}\end{array}\right).$ (7)

In eq. (6), the first two operators give the contribution from the coupling of
vector resonances to external fields in the chiral limit and the last term
gives the flavor-breaking corrections to such couplings. Our
$\lambda_{V}=\sqrt{2}\lambda_{6}^{V}$, using the notation in ref. [19]. This
last operator is the only one included from the full basis of
$\mathcal{O}(p^{4})$ even-intrinsic parity operators in ref. [19] since it is
the single one that can contribute to the $U(3)_{V}$ breaking in the
$V-\gamma$ coupling. There are, however, two reason to disregard basis of
operators in the even-intrinsic parity sector: The operators that are relevant
to the process can be dismissed on the basis of resonance field
redefinitions333Through the redefinition of the vector resonance field $V\to
V+g\\{V,\chi_{+}\\}$ it is possible to cancel the $\lambda_{V}$ operator [19],
however, we keep it in order to show the full basis of possible $U(3)_{V}$
breaking operators since we do not consider the full even-intrinsic parity
basis of ref. [19]. We will show later that this is consistent, since the
short distance constraints give $\lambda_{V}=0$.; if we, however, keep such
operators, they will only give subleading contributions to those from the
first two operators in eq. (6) with no contribution to $U(3)_{V}$-breaking
vertices.

The axial resonance operators present a similar feature and an analogous
flavor-space structure to that of the vector mesons. This is, the
$\mathcal{O}(p^{4})$ one-resonance even-intrinsic parity operators for axial
resonances in ref. [19] can be absorbed through field redefinitions. We will
therefore disregard any contribution from this part of the Lagrangian,
including the $U(3)_{V}$ breaking terms to the axial-vector resonance coupling
to external currents, namely, the $JA$ vertex. The remaining contributions
with one resonance field are [12]

$\mathcal{L}_{1\,A/P}^{even}=\frac{F_{A}}{2\sqrt{2}}\langle
A_{\mu\nu}f_{-}^{\mu\nu}\rangle+id_{m}\langle P\chi_{-}\rangle,$ (8)

with $P$ a matrix in three-flavors space containing the lightest pseudoscalar
resonances. The inclusion of the pseudoscalar resonance is necessary in order
to obtain consistent short-distance constraints in $\langle VAP\rangle$ and
$\langle VJP\rangle$ Green’s functions [19, 28, 30, 31]. All Feynman diagrams
involving these resonances will give $U(3)$ breaking contributions to the
amplitude due to the last term in eq. (8). We have neglected other spin-zero
resonance contributions (scalar and heavier pseudoscalar resonances [5]),
which are not needed for theoretical consistency and are irrelevant
phenomenologically. The odd-intrinsic parity contributions to
$\mathcal{L}_{1\,Res}$ are [32] 444Since we are only considering operators
with one $\pi/K$ field, these constitute a basis. In the general case, the
basis is given in ref. [28]. The translation between them can be read from
ref. [30].

$\mathcal{L}_{1\,Res}^{odd}=\sum_{j=1}^{7}\frac{c_{j}}{M_{V}}\mathcal{O}^{j}_{V}+\varepsilon_{\mu\nu\alpha\beta}\langle\kappa_{5}^{P}\\{f_{+}^{\mu\nu},f_{+}^{\alpha\beta}\\}P\rangle\,,$
(9)

with the operators

$\displaystyle\mathcal{O}^{1}_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{V^{\mu\nu},f^{\rho\alpha}_{+}\right\\}\nabla_{\alpha}u^{\sigma}\rangle\,,$
$\displaystyle\mathcal{O}^{2}_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{V^{\mu\alpha},f^{\rho\sigma}_{+}\right\\}\nabla_{\alpha}u^{\nu}\rangle\,,$
$\displaystyle\mathcal{O}^{3}_{V}$ $\displaystyle=$ $\displaystyle
i\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{V^{\mu\nu},f^{\rho\sigma}_{+}\right\\}\chi_{-}\rangle\,,$
(10) $\displaystyle\mathcal{O}^{4}_{V}$ $\displaystyle=$ $\displaystyle
i\varepsilon_{\mu\nu\rho\sigma}\langle
V^{\mu\nu}\left[f^{\rho\sigma}_{-},\chi_{+}\right]\rangle\,,$
$\displaystyle\mathcal{O}^{5}_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{\nabla_{\alpha}V^{\mu\nu},f^{\rho\alpha}_{+}\right\\}u^{\sigma}\rangle\,,$
$\displaystyle\mathcal{O}^{6}_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{\nabla_{\alpha}V^{\mu\alpha},f^{\rho\sigma}_{+}\right\\}u^{\nu}\rangle\,,$
$\displaystyle\mathcal{O}^{7}_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{\nabla^{\sigma}V^{\mu\nu},f^{\rho\alpha}_{+}\right\\}u_{\alpha}\rangle.$

In the following, we quote those terms bilinear in resonance fields (we do not
display the kinetic terms for the resonances, which can be found in ref. [12],
as they do not contribute to the effective vertices).

$\mathcal{L}_{2\,Res}=\mathcal{L}_{2\,Res}^{even}+\mathcal{L}_{2\,Res}^{odd}\,,$
(11)

with [19, 33, 34, 35] 555The operator with coefficient $e^{V}_{M}$ allows to
account for $U(3)$ breaking effects in the vector resonance masses, in
agreement with phenomenology.

$\mathcal{L}_{2\,Res}^{even}=-e^{V}_{M}\langle
V_{\mu\nu}V^{\mu\nu}\chi_{+}\rangle+\lambda_{1}^{PV}\mathcal{O}_{1}^{PV}+\lambda_{2}^{PV}\mathcal{O}_{2}^{PV}+\lambda_{1}^{PA}\mathcal{O}_{1}^{PA}+\sum_{i=1}^{5}\lambda^{VA}_{i}\mathcal{O}^{VA}_{i}\,,$
(12)

and [28, 32]

$\mathcal{L}_{2\,Res}^{odd}=\sum_{i=1}^{3}d_{i}\mathcal{O}_{i}^{VV}+\kappa_{3}^{PV}\mathcal{O}_{3}^{PV}\,.$
(13)

The operators appearing in the two previous equations are
($h^{\mu\nu}=\nabla^{\mu}u^{\nu}+\nabla^{\nu}u^{\mu}$)

$\displaystyle\mathcal{O}_{1}^{PV}$ $\displaystyle=$ $\displaystyle
i\langle[\nabla^{\mu}P,V_{\mu\nu}]u^{\nu}\rangle,$
$\displaystyle\mathcal{O}_{2}^{PV}$ $\displaystyle=$ $\displaystyle
i\langle[P,V_{\mu\nu}]f_{-}^{\mu\nu}\rangle;$
$\displaystyle\mathcal{O}_{1}^{PA}$ $\displaystyle=$ $\displaystyle
i\langle[P,A_{\mu\nu}]f_{+}^{\mu\nu}\rangle;$
$\displaystyle\mathcal{O}_{1}^{VA}$ $\displaystyle=$
$\displaystyle\langle[V^{\mu\nu},A_{\mu\nu}]\chi_{-}\rangle,$
$\displaystyle\mathcal{O}_{2}^{VA}$ $\displaystyle=$ $\displaystyle
i\langle[V^{\mu\nu},A_{\nu\alpha}]h_{\mu}^{\alpha}\rangle,$
$\displaystyle\mathcal{O}_{3}^{VA}$ $\displaystyle=$ $\displaystyle
i\langle[\nabla^{\mu}V_{\mu\nu},A^{\nu\alpha}]u_{\alpha}\rangle,$
$\displaystyle\mathcal{O}_{4}^{VA}$ $\displaystyle=$ $\displaystyle
i\langle\nabla^{\alpha}V_{\mu\nu},A_{\alpha}^{\;\nu}]u^{\mu}\rangle,$ (14)
$\displaystyle\mathcal{O}_{5}^{VA}$ $\displaystyle=$ $\displaystyle
i\langle[\nabla^{\alpha}V_{\mu\nu},A^{\mu\nu}]u_{\alpha}\rangle;$
$\displaystyle\mathcal{O}_{1}^{VV}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{V^{\mu\nu},V^{\rho\alpha}\right\\}\nabla_{\alpha}u^{\sigma}\rangle,$
$\displaystyle\mathcal{O}_{2}^{VV}$ $\displaystyle=$ $\displaystyle
i\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{V^{\mu\nu},V^{\rho\sigma}\right\\}\chi_{-}\rangle,$
$\displaystyle\mathcal{O}_{3}^{VV}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\\{\nabla_{\alpha}V^{\mu\nu},V^{\rho\alpha}\right\\}u^{\sigma}\rangle,$
$\displaystyle\mathcal{O}_{3}^{PV}$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu\nu\alpha\beta}\langle\\{V^{\mu\nu},f_{+}^{\alpha\beta}\\}P\rangle.$

There is only one relevant operator with three resonance fields in either
parity sector

$\mathcal{L}_{3\,Res}=i\lambda^{VAP}\langle[V_{\mu\nu},A^{\mu\nu}]P\rangle+\kappa^{PVV}\varepsilon_{\mu\nu\rho\sigma}\langle
V^{\mu\nu}V^{\alpha\beta}P\rangle.$ (15)

Operators with a higer number of resonant fields will not be included, since
otherwise one has to include subleading diagrams with loops where some of the
internal lines are given by resonances. The present analysis is restricted to
tree-level diagrams, which should already capture the leading effects
associated with resonance exchange. One-loop diagrams with resonances are
expected to be a numerically small correction since these would be subleading
in the $1/N_{C}$ expansion [4]. Such corrections will be neglected due to the
already sizeable number of parameters involved in the tree-level analysis and
the current precision of the experimental data.

###  3.2 Form Factors

In this section we quote our results for the different contributions to the
$F_{V}$, $F_{A}$, $A_{2}$ ,$A_{4}$ form factors, for $P=\pi,K$. All resonance
propagators are to be understood as provided with an energy-dependent width
($M_{R}^{2}-x\to M_{R}^{2}-x-iM_{R}\Gamma_{R}(x)$, $x=W^{2},k^{2}$) computed
within $R\chi T$, using those in refs. [36] ($\rho(770)$), [37] ($K^{*}(892)$)
and [38, 39] ($a_{1}(1260)$, including the $KK\pi$ cuts [40]). A constant
width will suffice for the very narrow $\omega(782)$ and $\phi(1020)$ mesons
(their PDG [41] values will be taken). For the $K_{1}(1270/1400)$ states we
will follow [42].

Figure 2: Feynman diagrams contributing to the vector part of the left
hadronic current. The circled cross vertex indicates vector current. The
resonance $P^{\star}$ is the pseudoscalar resonance corresponding to
$\pi(1300)\equiv\pi^{\prime}$ ($K(1460)\equiv K^{\prime}$) for $P=\pi(K)$. The
resonance $V^{0}$ means $\omega$ for $P=\pi$ and $\rho^{0},\omega,\phi$ for
$P=K$.

The vector form factors are ($N_{C}=3$ in QCD)

$F_{V}^{(\pi)}(W^{2},k^{2})=\frac{1}{3f}\left\\{-\frac{N_{C}\frac{}{}}{8\pi^{2}}+64m_{\pi}^{2}C_{7}^{W\star}-8C_{22}^{W}(W^{2}+k^{2})\hskip
43.05542pt\right.$ $\hskip
43.05542pt+\frac{4{F_{V}^{ud}}^{2}}{M_{\rho}^{2}-W^{2}}\frac{d_{3}(W^{2}+k^{2})+d_{123}^{\star}m_{\pi}^{2}}{M_{\omega}^{2}-k^{2}}$
$\hskip
60.27759pt+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}W^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}k^{2}}{M_{\rho}^{2}-W^{2}}$
$\left.\hskip
60.27759pt+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}k^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}W^{2}}{M_{\omega}^{2}-k^{2}}\right\\},$
(16)
$F_{V}^{(K)}(W^{2},k^{2})=\frac{1}{f}\left\\{-\frac{N_{C}}{24\pi^{2}}+\frac{64}{3}m_{K}^{2}C_{7}^{W\star}+32C_{11}^{W}\Delta_{K\pi}^{2}-\frac{8}{3}C_{22}^{W}(W^{2}+k^{2})\right.$
$+\frac{2F_{V}^{us}\left[d_{3}(W^{2}+k^{2})+d_{123}^{\star}m_{K}^{2}\right]}{M_{K^{\star}}^{2}-W^{2}}\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}-\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right)$
$+\frac{2\sqrt{2}F_{V}^{us}}{3M_{V}}\frac{c_{1256}W^{2}-c_{1235}^{\star}m_{K}^{2}-c_{125}k^{2}+24c_{4}\Delta_{K\pi}^{2}}{M_{K^{\star}}^{2}-W^{2}}$
(17)
$\left.+\frac{\sqrt{2}\left(c_{1256}k^{2}-c_{1235}^{\star}m_{K}^{2}-c_{125}W^{2}\right)}{M_{V}}\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}-\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right)\right\\},$

where $\Delta_{K\pi}^{2}=m_{K}^{2}-m_{\pi}^{2}$ and we have used the
combinations of coupling constants [43]

$\displaystyle c_{125}=c_{1}-c_{2}+c_{5},$ $\displaystyle
c_{1256}=c_{1}-c_{2}-c_{5}+2c_{6},$ $\displaystyle
c_{1235}=c_{1}+c_{2}+8c_{3}-c_{5},$ (18) $\displaystyle
d_{123}=d_{1}+8d_{2}-d_{3}.$

$F_{V}^{uD,ss}$ and starred coefficients absorb $U(3)$ breaking contributions
induced by $\lambda_{V}$ in eq. (6) and pseudoscalar resonances, respectively.
Their expressions are given at the end of this subsection, after eq. (24).

Figure 3: Feynman diagrams contributing to the axial part of the left hadronic
current. The circled cross vertex indicates axial current. Conventions for
$P^{\star}$ is the same as in the previous figure, the resonance $V^{0}$ means
$\rho^{0}$ for $P=\pi$ and $\rho^{0},\omega,\phi$ for $P=K$.

The axial form factors are 666We note two mistakes in writing $F_{A}^{(\pi)}$
in ref. [5], see Appendix A. The result written here agrees with the one in
ref. [44] for $k^{2}\to 0$.

$F_{A}^{(\pi)}(W^{2},k^{2})=\frac{F_{V}^{ud}}{2f}\frac{F_{V}^{ud}-2G_{V}-m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}(\lambda_{1}^{PV}+2\lambda_{2}^{PV})}{M_{\rho}^{2}-k^{2}}$
$-\frac{F_{A}}{2f}\frac{F_{A}-2m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PA}}{M_{a_{1}}^{2}-W^{2}}+\frac{\sqrt{2}}{f}\frac{F_{A}F_{V}^{ud}}{M_{a_{1}}^{2}-W^{2}}\frac{\lambda_{0}^{\star}m_{\pi}^{2}-\lambda^{\prime}k^{2}-\lambda^{\prime\prime}W^{2}}{M_{\rho}^{2}-k^{2}},$
(19)

$F_{A}^{(K)}(W^{2},k^{2})=-\frac{F_{A}}{2f}\frac{F_{A}-2m_{K}^{2}\frac{4\sqrt{2}d_{m}}{M_{K^{\prime}}^{2}}\lambda_{1}^{PA}}{M_{K_{1}}^{2}-W^{2}}$
$+\left[\frac{\sqrt{2}F_{A}}{2f}\frac{\lambda^{\star}_{0}m_{K}^{2}-\lambda^{\prime}k^{2}-\lambda^{\prime\prime}W^{2}}{M_{K_{1}}^{2}-W^{2}}+\frac{F_{V}^{us}\left(F_{V}^{us}-2G_{V}+m_{K}^{2}\frac{4\sqrt{2}d_{m}}{M_{K^{\prime}}^{2}}(\lambda_{1}^{PV}+2\lambda_{2}^{PV})\right)}{4f}\right]$
$\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$
(20)

$A_{2}^{(\pi)}(W^{2},k^{2})=\hskip 284.16577pt$
$\frac{2}{f}\left(G_{V}+\frac{2\sqrt{2}m_{\pi}^{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PV}+\frac{\sqrt{2}F_{A}}{M_{a_{1}}^{2}-W^{2}}W^{2}(\lambda^{\prime}+\lambda^{\prime\prime})\right)\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}},$
(21)

$A_{2}^{(K)}(W^{2},k^{2})=\left(\frac{G_{V}}{f}+\frac{2\sqrt{2}m_{K}^{2}d_{m}}{M_{K^{\prime}}^{2}}\frac{\lambda_{1}^{PV}}{f}+\frac{\sqrt{2}F_{A}}{M_{K_{1}}^{2}-W^{2}}\frac{W^{2}(\lambda^{\prime}+\lambda^{\prime\prime})}{f}\right)$
$\hskip
120.55518pt\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$
(22)

$A_{4}^{(\pi)}(W^{2},k^{2})=\frac{2}{f}\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}\left[\frac{G_{V}}{W^{2}-m_{\pi}^{2}}+\frac{2\sqrt{2}d_{m}m_{\pi}^{2}\lambda_{1}^{PV}}{M_{\pi^{\prime}}^{2}\left(W^{2}-m_{\pi}^{2}\right)}+\frac{\sqrt{2}F_{A}(\lambda^{\prime}+\lambda^{\prime\prime})}{M_{a_{1}}^{2}-W^{2}}\right],$
(23)

$A_{4}^{(K)}(W^{2},k^{2})=\frac{1}{f}\left(\frac{G_{V}}{W^{2}-m_{K}^{2}}+\frac{2\sqrt{2}d_{m}m_{K}^{2}\lambda_{1}^{PV}}{M_{K^{\prime}}^{2}\left(W^{2}-m_{\pi}^{2}\right)}+\frac{\sqrt{2}F_{A}(\lambda^{\prime}+\lambda^{\prime\prime})}{M_{K_{1}}^{2}-W^{2}}\right)$
$\hskip
120.55518pt\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$
(24)

It is worth to notice that by replacing the $P$ propagator in $A_{2}^{(P)}$
and $A_{4}^{(P)}$ with the massless pole propagator, one recovers the linear
dependence between both form factors, thus getting a congruent expression with
those in reference [19]. Therefore, the short-distance constraints obtained in
this reference can be used as shown there if the Weinberg’s sum rules are
imposed. We, however, do not make use of these sum rules, as $F_{V/A}$ are
fitted to data (see discussion in sections 3.3 and 4.2).

We introduced the short-hand notation

$\displaystyle F_{V}^{ud}\equiv F_{V}+8m_{\pi}^{2}\lambda_{V},$ $\displaystyle
F_{V}^{us}\equiv F_{V}+8m_{K}^{2}\lambda_{V},$ (25) $\displaystyle
F_{V}^{ss}\equiv F_{V}+8(2m_{K}^{2}-m_{\pi}^{2})\lambda_{V},$

for the shifts appearing also in [17]. 777As mentioned in section 3.1, a
similar shift can be introduced in $F_{A}$, however, the operator responsible
for such shift can be absorbed through axial resonance field redefinitions
[19].

We also used [33]

$\displaystyle-\sqrt{2}\lambda_{0}^{\star}=4\lambda_{1}^{\star}+\lambda_{2}+\frac{\lambda_{4}}{2}+\lambda_{5},$
$\displaystyle\sqrt{2}\lambda^{\prime}=\lambda_{2}-\lambda_{3}+\frac{\lambda_{4}}{2}+\lambda_{5}$
(26)

and

$\sqrt{2}\lambda^{\prime\prime}=\lambda_{2}-\frac{\lambda_{4}}{2}-\lambda_{5}.$

We employed several starred coefficients including $U(3)$ breaking
contributions, as given below:

$\displaystyle\lambda_{1}^{\star}=\lambda_{1}-\frac{\lambda^{VAP}d_{m}}{M_{P}^{2}},$
$\displaystyle
C_{7}^{W\star}=C_{7}^{W}+\frac{\kappa_{5}^{P}d_{m}}{M_{P}^{2}},$
$\displaystyle
c_{3}^{\star}=c_{3}+\frac{\kappa_{3}^{PV}d_{m}M_{V}}{M_{P}^{2}},$ (27)

implying

$\displaystyle
c_{1235}^{\star}=c_{1}+c_{2}+8c_{3}^{\star}-c_{5},\;\mathrm{and}$ (28)
$\displaystyle d_{2}^{\star}=d_{2}+\frac{\kappa^{VVP}d_{m}}{2M_{P}^{2}},$ (29)

yielding

$d_{123}^{\star}=d_{1}+8d_{2}^{\star}-d_{3}.$ (30)

We have first shown here the correction to $\lambda_{1}$ appearing in
$\lambda_{1}^{\star}$, while the remaining starred couplings were already
introduced in ref. [17].

We will follow the scheme explained in ref. [42] to account for the mixing
between the $K_{1}(1270)=K_{1L}$ and the $K_{1}(1400)=K_{1H}$ states. This
amounts to replacing, in eq. (3.2),
$(M_{K_{1}}^{2}-W^{2})^{-1}\to\mathrm{cos}^{2}\theta_{A}(M_{K_{1H}}^{2}-W^{2})^{-1}+\mathrm{sin}^{2}\theta_{A}(M_{K_{1L}}^{2}-W^{2})^{-1}$,
with mixing angle $\theta_{A}\in[37,58]^{\circ}$.

###  3.3 Short-distance constraints

We will demand that the different form factors have an asymptotic behaviour in
agreement with QCD [45, 46]. Specifically, we will require their vanishing for
large $\lambda$ in the $\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},0)$ and
$\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},\lambda k^{2})$ cases. We will do
this first in the chiral limit and then at $\mathcal{O}(m_{P}^{2})$ 888Since
we are considering a complete basis of chiral symmetry breaking operators at
order $m_{P}^{2}$, we neglect higher-order chiral corrections., paralleling
the discussion in ref. [17] for the neutral pseudoscalar transition form
factors. In this way, we find the following relations:

  * •

$F_{V}^{(\pi)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{0})$:

$\displaystyle C_{22}^{W}$ $\displaystyle=$ $\displaystyle 0\,,$ (31)
$\displaystyle c_{125}$ $\displaystyle=$ $\displaystyle 0\,,$ (32)
$\displaystyle c_{1256}$ $\displaystyle=$
$\displaystyle-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}}\,,$ (33)
$\displaystyle d_{3}$ $\displaystyle=$
$\displaystyle-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}\,.$ (34)

  * •

$F_{V}^{(\pi)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{2})$:

$\displaystyle\lambda_{V}$ $\displaystyle=$
$\displaystyle-\frac{64\pi^{2}F_{V}}{N_{C}}C_{7}^{W\star}\,,$ (35)
$\displaystyle c_{1235}^{\star}$ $\displaystyle=$
$\displaystyle\frac{N_{C}M_{V}e^{V}_{m}}{8\sqrt{2}\pi^{2}F_{V}}+\frac{N_{C}M_{V}^{3}\lambda_{V}}{4\sqrt{2}\pi^{2}F_{V}^{2}}.$
(36)

  * •

$F_{V}^{(K)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{0})$: Same constraints as for
the $\pi$ case, since both form factors 999$F_{A}^{P}$ and $A_{2,4}^{P}$ form
factors are also identical in this limit for $P=\pi$ or $K$, obviously. are
identical in the $U(3)$ symmetry limit.

  * •

$F_{V}^{(K)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{2})$:

$\displaystyle C_{11}^{W}$ $\displaystyle=$
$\displaystyle\frac{N_{C}\lambda_{V}}{64\pi^{2}F_{V}}\,.$ (37)

For the sake of predictability and in order to further constrain the
parameters in the form factor, we use the $VVP$ Green function,
$\Pi_{VVP}(r^{2},p^{2},q^{2})$, constraints [28] obtained from the high-energy
behaviour when $r^{2}\to\infty,p^{2}\to\infty$ and $q^{2}\to\infty$ and
matching to the Operator Product Expansion (OPE) leading terms in the chiral
and large-$N_{C}$ limits. These give

$\displaystyle c_{125}=c_{1235}=0,\hskip
107.63855ptc_{1256}=-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}},$
$\displaystyle\kappa_{5}^{P}=0,\hskip
73.19421ptd_{3}=-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}+\frac{F^{2}}{8F_{V}^{2}}+\frac{4\sqrt{2}d_{m}\kappa_{3}^{PV}}{F_{V}},$
$\displaystyle C_{7}^{W}=C_{22}^{W}=0,\hskip
142.08289ptd_{123}=\frac{F^{2}}{8F_{V}^{2}}.$ (38)

Notice that these constraints coincide with our expressions in eqs. (31-34)
and that they imply

$\displaystyle C_{11}^{W}$ $\displaystyle=$
$\displaystyle\lambda_{V}=C_{7}^{W\star}=0,$ $\displaystyle d_{123}$
$\displaystyle=$ $\displaystyle\frac{F^{2}}{8F_{V}^{2}},$ $\displaystyle
d_{m}\kappa_{3}^{PV}$ $\displaystyle=$
$\displaystyle\frac{N_{C}M_{\pi^{\prime}}^{2}e_{m}^{V}}{64\sqrt{2}\pi^{2}F_{V}}.$
(39)

One can see that from the definition of $c_{1235}^{\star}$ (eqs. (27) and
(28)) combined with the last expression and the short-distance constraints
eqs. (34), (36) and (38) would imply a relation of $e_{m}^{V}$ in terms of $F$
and $M_{\pi^{\prime}}$, namely

$e_{m}^{V}=-\frac{2\pi^{2}F^{2}}{N_{C}M_{\pi^{\prime}}^{2}}$ (40)

however, we do not rely on this relation since comparison with previous
phenomenology [34, 35] shows that the absolute value of eq. (40) obtained for
$f\approx 92$ MeV and $M_{\pi^{\prime}}=1.3$ GeV is an order of magnitude
smaller than required by phenomenology.

On the other hand, no relation among parameters of the axial form factors can
be obtained by taking the infinite virtualities limit, since it already has
the right asymptotic behavior. Instead, we will rely on the relations obtained
using the $VAP$ Green Function101010See, however, the discussion in section
6.2 of [47] comparing these short-distance constraints to the results in refs.
[48, 20]. $\Pi_{VAP}(p^{2},q^{2},(p+q)^{2})$ [21] in an analogous manner to
that done for the $\Pi_{VVP}(r^{2},p^{2},q^{2})$ Green Function.

We recall that the simultaneous analysis of the scalar form factor [49, 50]
and the SS-PP sum rules [51] yields $d_{m}=f/(2\sqrt{2})$. Additionally,
notice that $A_{2,4}^{(P)}$ depend on $\lambda_{1}^{PV}$. In turn,
$F_{A}^{(P)}$ depends on $\lambda_{1}^{PA}$ and
$\lambda_{1}^{PV}+2\lambda_{2}^{PV}$. The appropriate short-distance behaviour
of the $VAP$ Green Function [21] fixes all of them but $\lambda_{0}^{\star}$
or, in other words $\lambda^{VAP}$, as noted in ref. [19]

$\displaystyle\lambda_{0}=\frac{f^{2}}{4\sqrt{2}F_{V}F_{A}},$
$\displaystyle\lambda^{\prime}=\frac{f^{2}+F_{A}^{2}}{2\sqrt{2}F_{V}F_{A}},$
$\displaystyle\lambda^{\prime\prime}=-\frac{f^{2}+F_{A}^{2}-2F_{V}G_{V}}{2\sqrt{2}F_{V}F_{A}},$
$\displaystyle d_{m}\lambda_{1}^{PV}=-\frac{f^{2}}{4\sqrt{2}F_{V}},$
$\displaystyle
d_{m}\lambda_{2}^{PV}=\frac{3f^{2}+2F_{A}^{2}-2F_{V}^{2}}{16\sqrt{2}F_{V}},$
$\displaystyle d_{m}\lambda_{1}^{PA}=\frac{f^{2}}{16\sqrt{2}F_{A}}\,.$ (41)

Despite the relation for $d_{m}$ from the scalar form factor and the SS-PP
Green’s function, notice that there is no need for one since $d_{m}$ always
appears multiplied by one of the other parameters to be constrained. We will
also make use of the constraint [12]

$F_{V}G_{V}=f^{2}.$ (42)

In order to gain predictability, we will use the values of $d_{123}^{\star}$,
$M_{V}$ and $e_{m}^{V}$ given for the best fit of reference [17], namely
(their correlations are given in the quoted reference)

$\displaystyle d_{123}^{\star}$ $\displaystyle=$ $\displaystyle-(2.3\pm
1.5)\times 10^{-1},$ $\displaystyle M_{V}$ $\displaystyle=$
$\displaystyle(791\pm 6)\mathrm{\hskip 4.30554ptMeV},$ (43) $\displaystyle
e_{m}^{V}$ $\displaystyle=$ $\displaystyle-(0.36\pm 0.10).$

##  4 Phenomenological analysis

###  4.1 Phase space

In order to compare our results with those of ref. [5] we use the same phase
space configuration. We recall that the variables in ref. [5] are the
invariant mass squared of the pseudo Goldstone and the neutrino,
$s_{12}=m_{P\nu_{\tau}}^{2}$, the invariant mass squared of the charged lepton
pair, $s_{34}=m_{\ell\overline{\ell}}^{2}$, two polar angles
${\theta_{1}},\,{\theta_{3}}$ and one azimuthal angle $\phi_{3}$, with the
integration limits given by

$\displaystyle(m_{3}+m_{4})^{2}\leq$ $\displaystyle
s_{34}\leq(M-m_{1}-m_{2})^{2},$ (44a) $\displaystyle(m_{1}+m_{2})^{2}\leq$
$\displaystyle s_{12}\leq(M-\sqrt{s_{34}})^{2},$ (44b) $\displaystyle
0\leq\theta_{1,3}\leq\pi,$ $\displaystyle\qquad 0\leq\phi_{3}\leq 2\pi.$ (44c)

If we identify the particle with tag 1 with $\nu_{\tau}$, the invariant mass
of the weak gauge boson can be related to the Lorentz invariants of eqs. (44)
via

$W^{2}=M_{234}^{2}=M^{2}+m_{1}^{2}-\frac{(M^{2}+s_{12}-s_{34})(s_{12}+m_{1}^{2}-m_{2}^{2})}{2s_{12}}-X\beta_{12}\cos{\theta_{1}}\,,$
(45)

where $\beta_{ij}=\lambda^{1/2}(s_{ij},m_{i}^{2},m_{j}^{2})/s_{ij}$ and
$X=\lambda^{1/2}(M^{2},s_{12},s_{34})/2$, being
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$, the Källén lambda function.
Eq. (45) allows us to eliminate $\theta_{1}$ in favor of $M_{234}$. The
importance of the phase space configuration with $W^{2}$ instead of
$\theta_{1}$ relies on the need to compute the $m_{\pi e^{+}e^{-}}$ spectrum
in order to fit unconstrained parameters to the Belle invariant mass spectrum
[7]. The kinematic limits on the non-angular variables for this phase space
configuration read

$\displaystyle m_{2}+m_{3}+m_{4}$ $\displaystyle\leq\hskip
4.30554ptM_{234}\hskip 4.30554pt\leq M,$ (46a)
$\displaystyle(m_{3}+m_{4})^{2}$ $\displaystyle\leq\hskip
4.30554pts_{34}\hskip 4.30554pt\leq\left(M_{234}-m_{2}\right)^{2},$ (46b)
$\displaystyle s_{12}^{-}$ $\displaystyle\leq\hskip 4.30554pts_{12}\hskip
4.30554pt\leq s_{12}^{+},$ (46c)

where

$\displaystyle s_{12}^{\pm}$ $\displaystyle=$ $\displaystyle
M^{2}+m_{1}^{2}+m_{2}^{2}+s_{34}-W^{2}+\frac{(M^{2}-m_{1}^{2})(m_{2}^{2}-s_{34})}{W^{2}}$
(47)
$\displaystyle\pm\frac{1}{2W^{2}}\lambda^{1/2}(M^{2},W^{2},m_{1}^{2})\lambda^{1/2}(W^{2},s_{34},m_{2}^{2})\,.$

With this, the differential decay rate is given as

$\displaystyle d\Gamma(\tau^{-}\to\nu_{\tau}P^{-}\ell\overline{\ell})$
$\displaystyle=$
$\displaystyle\frac{X\beta_{12}\beta_{34}}{4(4\pi)^{6}m_{\tau}^{3}}\overline{\left|\mathcal{M}\right|^{2}}ds_{34}ds_{12}d(\cos{\theta_{1}})d(\cos{\theta_{3}})d\phi_{3}$
(48) $\displaystyle=$
$\displaystyle\frac{\beta_{34}}{4(4\pi)^{6}m_{\tau}^{3}}\overline{\left|\mathcal{M}\right|^{2}}dM_{P\ell\overline{\ell}}^{2}ds_{34}ds_{12}d(\cos{\theta_{3}})d\phi_{3}\,.$

###  4.2 Fit to data

Short-distance QCD behaviour [17, 19] does not constrain all parameters. Thus,
we fit some of the remaining unknowns using the invariant mass spectra of the
$W$ boson, $m_{\pi^{-}e^{+}e^{-}}$, measured by the Belle collaboration [7].
We start with a four parameters fit ($F_{V}$, $F_{A}$, $\lambda_{0}$ and
$\mathcal{B}$, the branching fraction, are floated). Despite the fact that the
whole $m_{\pi^{-}e^{+}e^{-}}$ spectrum has been measured, not all the data is
available for this minimization since points below
$m_{\pi^{-}e^{+}e^{-}}<1.05$ GeV were used as a control region to validate the
Monte Carlo simulation, leaving thus the most sensitive part to SD
contributions as the signal region [7]; therefore we use for the minimization
the data above the cut $m_{\pi^{-}e^{+}e^{-}}=1.05$ GeV.

We use only the data set for the $\tau^{-}\to\pi^{-}e^{+}e^{-}\nu_{\tau}$ mode
111111This is the one shown in the plots of ref. [7]. We have checked better
agreement with the Monte Carlo simulation (based on our previous paper [5],
see also [52, 53]) for this mode with respect to its charge-conjugated mode..
Comparison of this with the expected signal events distribution in this
reference allows us to roughly quantify the deconvolution of signal from
detector, which we ignore. We have assumed this to be an energy-independent
effect for simplicity, and taken it into account as a systematic uncertainty
in the data. This error turns out to be comparable to the one reported by
Belle for the branching fraction measured above the cut. In addition to this,
the Belle collaboration used the expressions of our ref. [5]; which however
had typos in some of the $F_{A}(t,k^{2})$ terms (see Appendix A). Besides,
trying to keep our previous analysis as simple as possible, it resulted
incomplete in the sense of $VAP$ Green’s function analysis121212Pseudoscalar
resonance exchange and $\mathcal{O}(p^{6})$ operators in the $A_{2}^{(\pi)}$
and $A_{4}^{(\pi)}$ form factors are lacking in our ref. [5]. This leads to
relating both form factors to the $\pi$ electromagnetic form factor [6].,
which altogether could lead to biased estimations of the decay observables.
Both reasons motivate our choice of fitting the total branching fraction,
$\mathcal{B}$, as an additional parameter instead of simply computing it from
the decay width expression in eq. (48).

We used the relation

$\int
dm_{\pi^{-}e^{+}e^{-}}\frac{1}{\Gamma}\frac{d\Gamma}{dm_{\pi^{-}e^{+}e^{-}}}=1=\int
dm_{\pi^{-}e^{+}e^{-}}\frac{1}{N}\frac{dN}{dm_{\pi^{-}e^{+}e^{-}}}=\sum_{\mathrm{bins}}\frac{1}{N}\frac{N_{\mathrm{bin}}}{\Delta
m_{\pi^{-}e^{+}e^{-}}},$ (49)

where $N$ is the total number of events, $N_{\mathrm{bin}}$ is the number of
events in a given bin and $\Delta m_{\pi^{-}e^{+}e^{-}}$ is the bin width. We
thus minimize the $\chi^{2}$ given by

$\chi^{2}=\left(\frac{N\hskip 4.30554pt\Delta
m_{\pi^{-}e^{+}e^{-}}}{\Gamma\hskip
4.30554pt\varepsilon_{\mathrm{bin}}}\frac{d\Gamma}{dm_{\pi^{-}e^{+}e^{-}}}-\frac{N_{\mathrm{bin}}}{\varepsilon_{\mathrm{bin}}}\right)^{2}+\left(\frac{\mathcal{B}-BR}{\varepsilon_{BR}}\right)^{2},$
(50)

where $\varepsilon_{\mathrm{bin}}$ is the experimental uncertainty in a given
bin, $BR$ is the branching ratio for $m_{\pi^{-}e^{+}e^{-}}\geq 1.05$ GeV
reported by Belle [7], $\varepsilon_{BR}$ its error and $\mathcal{B}$ the one
obtained integrating our differential decay width above this cut. We recall
that $BR(\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-})=(5.90\pm 0.53\pm 0.85\pm
0.11)\times 10^{-6}$ [7], where the first uncertainty is statistical, the
second is systematic, and the third is due to the model dependence. In our
fits, we have first realized that, varying all four parameters, there happen
to be many quasidegenerate best fits and that the correlations among the fit
parameters ($F_{V}$, $F_{A}$, $\lambda_{0}$ and $\mathcal{B}$) are always
large. We interpret this as the data not being precise enough to disentangle
the physical solutions among all close to the global minimum. Then, we
proceeded to further simplify the fits by making constant one of these four
parameters (not $\mathcal{B}$, as our systematic error due to unfolding is
comparable to the overall uncertainty of $BR$).

We present two sets of fits as our reference results. One fixing $F_{A}=130$
MeV ($\sim\sqrt{2}F$, in agreement with [30]), and the other setting
$\lambda_{0}^{\star}=102\times 10^{-3}$ (which is in the ballpark of previous
estimates for $\lambda_{0}$, and neglects the contribution of pseudoscalar
resonances to the starred coupling, see [54] and refs. therein). Considering
individually the $\pi^{+}$ or $\pi^{-}$ sets we find that, apart from the
incompatibility among both sets in several bins, the $\pi^{+}$ data set leads
to the unphysical condition $F_{A}>F_{V}$. Also, fixing $F_{V}$ to its short-
distance prediction $F_{V}\sim\sqrt{3}F\sim 159$ MeV [30], yields fits with
larger $\chi^{2}$ that we disregard. One way of interpreting this feature
would be that excited resonances (at least the $\rho(1450)$ state and its
interference with the $\rho(1700)$ resonance) are needed for an improved
description of the data. However, given the errors of the measurement and the
lack of $m_{e^{+}e^{-}}$ invariant mass distribution data we are not able to
test such more sophisticated theory input, which introduces several additional
parameters that remain unconstrained after applying the short-distance
conditions (see, e. g. ref. [55]). We thus understand that our fitted values
of $F_{V}$ are effectively capturing missing dynamics in our description (such
as the $\rho(770)$ excitations).

Figure 4: Normalized invariant mass spectra obtained with the two sets of
parameters obtained from fitting to the Belle data. The purple line
corresponds to the data with $F_{A}$ fixed, while the green one stands for
that with $\lambda_{0}^{\star}$ fixed. The blue data corresponds to
measurements of $\tau^{-}$ decays, which show best agreement with our model.
[7]

Considering the Weinberg sum rules [56]

$\displaystyle\sum_{i}\left(F_{V_{i}}^{2}-F_{A_{i}}^{2}\right)$
$\displaystyle=f^{2},$ (51a)
$\displaystyle\sum_{i}\left(F_{V_{i}}^{2}M_{V_{i}}^{2}-F_{A_{i}}^{2}M_{A_{i}}^{2}\right)$
$\displaystyle=0,$ (51b)

and taking only the contributions from the lightest nonet of resonances (along
with $M_{A}=\sqrt{2}M_{V}$), one finds that the fitted value for $F_{V}$
approaches the prediction from these relations, namely that131313Also found
from short distance constrictions of vector and axial form factors considering
only one multiplet [12, 13]. $F_{V}=\sqrt{2}f$. However, and as has been
stated above, the consistent short distance limit when operators contributing
to 2 and 3-point Green’s Functions are considered should be [30]
$F_{V}=\sqrt{3}f$. This makes us believe that the dynamics from heavier copies
of the $\rho$ meson must be affecting the constraints on the decay constant
$F_{V}$. Of course, these copies will undoubtedly affect the contribution to
the chiral-order $p^{4}$ LECs of the non-resonant Lagrangian when integrating
the resonances out, however, we still assume they approximately saturate them
and neglect the operators of such Lagrangian.141414Since the copies of the
$\rho$ must have analogous dynamics, the operators contributing to the LECs of
$\mathcal{O}(p^{4})$ must be the same with the sustitution
$\rho\to\rho^{\prime}$ (and so on for heavier copies). Thus, their
contributions must read  $\displaystyle
L_{1}^{V}=\sum_{i}\frac{G_{V_{i}}^{2}}{8M_{V_{i}}^{2}},\hskip
43.05542ptL_{9}^{V}=\sum_{i}\frac{F_{V_{i}}G_{V_{i}}}{2M_{V_{i}}^{2}},\hskip
43.05542ptL_{10}^{V}=-\sum_{i}\frac{F_{V_{i}}^{2}}{4M_{V_{i}}^{2}},$ (52) with
$L_{2}^{2}=2L_{1}^{V}$ and $L_{3}^{V}=-6L_{1}^{V}$.

Our results are shown in table 1, the corresponding $\chi^{2}/dof\sim 1.2$ is
reasonably good and $\mathcal{B}$ is in agreement with the Belle data within
less than 1 standard deviation in both cases. According to these results, we
cannot exclude that pseudoscalar resonances give sizeable contributions to
$\lambda_{0}^{\star}$. We consider both fit results in table 1 as benchmarks
for our predictions in the remainder of this work (we will refer to them as
’the two sets’). The difference among the corresponding two results can be
taken as a first, rough estimate of our model-dependent error.

| set 1 | set 2
---|---|---
|  $F_{A}=130$ MeV | $\lambda_{0}^{\star}=102\times 10^{-3}$
$F_{A}$ | 130 MeV | ($122\pm 0$) MeV
$F_{V}$ | (135.5$\pm 1.1$) MeV | ($137.4\pm 1.6$) MeV
$\lambda_{0}^{\star}$ | $(384\pm 0)\times 10^{-3}$ | 102$\times 10^{-3}$
$\mathcal{B}$ | $(6.01\pm 0)\times 10^{-6}$ | $(6.36\pm 0.12)\times 10^{-6}$
$\chi^{2}/dof$ | 31.1/26 | 31.4/26
Table 1: Our best fit results for $F_{A}$ , $F_{V}$, $\lambda_{0}^{*}$ and
$\mathcal{B}$. For the fit results shown on the left (right) columns we fix
$F_{A}=130$ MeV ($\lambda_{0}^{\star}=102\times 10^{-3}$), respectively. A $0$
error means that the fit uncertainty in the parameter is negligible with
respect to its central value.

###  4.3 Predictions for the $\tau^{-}\to\nu_{\tau}P^{-}\ell\overline{\ell}$
decays

|  | $\mathcal{B}(\tau\to\nu_{\tau}P\ell\overline{\ell})$ |
---|---|---|---
$P,\ell$ | set 1 | set 2 | IB
$\pi,e$ | $(2.38\pm 0.28\pm 0.11)\cdot 10^{-5}$ | $(2.45\pm 0.45\pm 0.04)\cdot 10^{-5}$ | $1.457(5)\cdot 10^{-5}$
$\pi,\mu$ | $(8.45\pm 2.45\pm 1.09)\cdot 10^{-6}$ | $(9.15\pm 3.25\pm 0.25)\cdot 10^{-6}$ | $1.5935(4)\cdot 10^{-7}$
$K,e$ | $(1.17\pm 0.26\pm 0.09)\cdot 10^{-6}$ | $(1.11\pm 0.28\pm 0.04)\cdot 10^{-6}$ | $3.225(5)\cdot 10^{-7}$
$K,\mu$ | $(6.4\pm 1.9\pm 0.8)\cdot 10^{-7}$ | $(5.85\pm 1.75\pm 0.20)\cdot 10^{-7}$ | $3.4191(8)\cdot 10^{-9}$
Table 2: Branching ratios for the different $\tau$ decay channels. In the
middle columns, our prediction for the full branching ratio accounting for
both (dominant) systematic and statistical uncertainties (see main text). In
the right column we show the SI contribution with the error arising from
numerical integration of the differential decay width.

By generating 2400 points in the parameter space making a Gaussian variation
of parameters, taking into account the correlations among them, we computed
the sum of the SD and the SD-SI interference contributions to the branching
fractions for the full phase space. We also computed for the $P=\pi$ and
$\ell=e$ channel the SI contribution to $\mathcal{B}$ with the cut
$m_{\pi^{-}e^{+}e^{-}}\geq 1.05$ GeV,

$\left.\frac{}{}\mathcal{B}(IB)\right|_{m_{\pi^{-}e^{+}e^{-}}\geq
1.05\mathrm{GeV}}=(1.599\pm 0.003)\times 10^{-7}.$ (53)

As expected, it is a factor $\sim 37$ smaller than the Belle measurement,
which confirms $m_{\pi^{-}e^{+}e^{-}}\geq 1.05$ GeV is a good cut to study
structure-dependent effects.

Figure 5: Invariant mass spectra $m_{\pi^{-}e^{+}e^{-}}$ for $P=\pi$, the
thickness represents the error band obtained from the difference between the
two sets. The plot in the left is for $\ell=e$, while the other is for
$\ell=\mu$.

Thus, the $\mathcal{B}$ adding the SI contribution gives the total branching
ratios shown in table 2, where the first (dominant) error includes the
uncertainty from unfolding and from the difference between $\mathcal{B}$ and
BR and the second error was obtained from the Gaussian distribution of the
fitted parameters. Also, in the same table, we show the SI contributions to
the branching fractions for the different decay channels in the complete phase
space.

Figure 6: Same as Figure 5 for $P=K$.

From the discussion at the beginning of section 4.2 and from the results shown
in table 2, we take the $\mathcal{B}$ of each decay channel to be within the
range obtained from the union of the intervals given by each set of fitted
parameters, the latter ranges defined as the intervals given by each central
value of Table 2 and its uncertainties. Also, we computed the $W^{2}$ spectra
for the different decay channels shown in Figures 5 and 6 using both sets of
fitted parameters, where the error band was obtained by taking the difference
between them. The same was done for the dilepton spectra in Figures 7 and 8.
Measurement of these observables at Belle-II [57] will be crucial for further
reducing the uncertainties shown.

Figure 7: Invariant mass spectra $m_{\ell\overline{\ell}}$ for $P=\pi$, the
thickness of the purple line represents the error band obtained from the
difference between the two sets. The green line is the prediction of ref. [5].
The left-hand plot is for $\ell=e$, while the other is for $\ell=\mu$ .

Figure 8: Invariant mass spectra $m_{\ell\overline{\ell}}$ for $P=K$, the
thickness of the purple line represents the error band obtained from the
difference between the two sets. The plot on the left is for $\ell=e$, while
the other is for $\ell=\mu$.

##  5 Conclusions

Motivated by recent measurements of the Belle Collaboration [7], in this paper
we present an improved prediction of the
$\tau^{-}\to\pi^{-}e^{+}e^{-}\nu_{\tau}$ decay. This includes a more accurate
description of the structure-dependent parts of the decay amplitudes, by
taking into account the first order flavor-breaking corrections to the form
factors involved in the $W\pi\gamma^{*}$ vertex. As done for the $VVP$ Green’s
function [30], we found that the inclusion of the pseudoscalar resonance is
needed in order to obtain compatible expressions between the $VAP$ Green’s
function and the form factors. In the high-energy limit we find that these
expressions give the same constraints on the parameters of the resonance
Lagrangians, as happens in the $VVP$ case.

$P,\ell$ | $\mathcal{B}(\tau^{-}\to\nu_{\tau}P^{-}\ell\overline{\ell})$
---|---
$\pi,e$ | $(2.41\pm 0.40\pm 0.12)\cdot 10^{-5}$
$\pi,\mu$ | $(9.15\pm 3.25\pm 1.12)\cdot 10^{-6}$
$K,e$ | $(1.13\pm 0.30\pm 0.09)\cdot 10^{-6}$
$K,\mu$ | $(6.2\pm 2.1\pm 0.8)\cdot 10^{-7}$
Table 3: Branching ratios for the different decay channels. The central value
is the mean of the union of intervals given in both columns of Table 2, the
first error covers the width of such union of ranges (see discussion below eq.
(53)) and the second error is the quadratic mean of statistical uncertainties
in Table 2.

We have obtained a reasonably good fit of the parameters that remain
unconstrained after applying the SD behaviour to the form factors. A better
set of data for the invariant mass spectrum of the hadronic current would
allow to determine physically meaningful parameters in a unique way. It is
worth to recall that despite the fact that we had access to the
$\tau^{+}\to\bar{\nu}_{\tau}\pi^{+}e^{+}e^{-}$ spectra obtained from Belle, we
found some inconsistencies that make unreliable the fits to data from both
positive and negative tau decays (see discussion in section 4.2). We have
therefore only considered the data set of the $\tau^{-}$ decays. From the
results in Table 2, we conclude that our best result for the branching
fractions is the union of ranges given for both fitting sets. This is shown in
Table 3, where the central value is the mean of the union of these intervals.
The results for the $P=\pi$ case agree with those in ref. [5], where
$(1.7^{+1.1}_{-0.3})\cdot 10^{-5}$ for $\ell=e$ and $[3\cdot 10^{-7},1\cdot
10^{-5}]$ for $\ell=\mu$ were predicted. Thus, we have reached a more precise
determination of the branching ratios for the $\pi$ decay channels than the
previous ones in ref. [5]. Also, similar observables for the
$\tau^{\pm}\to\nu_{\tau}K^{\pm}\ell\overline{\ell}$ channels are predicted for
the first time.

Despite the great achievement of the Belle collaboration [7] discovering the
$\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ decays, our study shows the need for
better data (hopefully from Belle-II [57] and forthcoming facilities) in order
to increase our knowledge of these decay modes. The $m_{\pi^{-}e^{+}e^{-}}$
spectrum shown in Figure 4 is consistent with the destructive interference of
the $\rho(1450)$ and $\rho(1700)$ resonances; however, current data
uncertainties prevent investigating the dynamics involved in the interplay of
such resonances. The effect of $\rho$ excitations does not seem, however,
negligible, since by imposing the known behaviour [30] $F_{V}=\sqrt{3}F$ to
the fit gives a far worse $\chi^{2}$ than those in Table 1, which are closer
to $F_{V}=\sqrt{2}F$ (which holds with a minimal resonance Lagrangian beyond
which we go in this and in our previous work on the subject). We assume that
the effect of these heavier copies of the $\rho$ meson are responsible for
this shift in the value of $F_{V}$.

## Acknowledgements

We are indebted to Denis Epifanov and Yifan Jin for leading the Belle analysis
of these decays, and measuring for the first time the $\tau\to\pi
e^{+}e^{-}\nu_{\tau}$ decays. We specially acknowledge Yifan Jin for sharing
with us detailed information on their study and providing us with the
simulated Monte Carlo generation. We also aknowledge Pablo Sánchez Puertas for
usefull comments on short distance constraints. A.G. was supported partly by
the Spanish MINECO and European FEDER funds (grant FIS2017-85053-C2-1-P) and
Junta de Andalucía (grant FQM-225) and partly by the Generalitat Valenciana
(grant Prometeo/2017/053). G.L.C. ackowledges funding from Ciencia de Frontera
Conacyt project No. 428218 and perfil PRODEP IDPTC 162336, and P.R. by the
SEP-Cinvestav Fund (project number 142), grant PID2020-114473GB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by Cátedras Marcos Moshinsky (Fundación
Marcos Moshinsky), that also supported A. G.

##  Appendix A Previous and current expressions of axial form factors

When we compare the expressions for the axial part of the decay amplitude we
see that the following relations should be fulfilled

$\displaystyle
F_{A}^{\text{new}}(W^{2},k^{2})=\frac{1}{2}F_{A}^{\text{old}}(t,k^{2}),$ (54a)
$\displaystyle
A_{2}^{\text{new}}(W^{2},k^{2})\leftrightarrow\frac{1}{2}A_{2}^{\text{old}}(k^{2}),$
(54b) $\displaystyle
A_{4}^{\text{new}}(W^{2},k^{2})\leftrightarrow\frac{1}{2}A_{4}^{\text{old}}(k^{2}),$
(54c)

where $t=W^{2}$. There is, however, some mistakes in the form factors of
reference [5]. There we have

$\displaystyle
F_{A}^{\text{old}}(t,k^{2})=\frac{F_{V}^{2}}{F}\left(1-2\frac{G_{V}}{F_{V}}\right)D_{\rho}(k^{2})-\frac{F_{A}^{2}}{F}D_{a_{1}}(t)$
$\displaystyle\hskip
64.58313pt+\frac{F_{A}F_{V}}{\sqrt{2}F}D_{\rho}(k^{2})D_{a_{1}}(t)(-\lambda^{\prime\prime}t+\lambda_{0}m_{\pi}^{2}),$
(55)

where $D_{\rho}$ and $D_{a_{1}}$ are the propagators of the $\rho$ and $a_{1}$
respectively. However, when we neglect the contributions stemming from the
$U(3)_{V}$-breaking contributions in eq. (19) we get

$\displaystyle
2F_{A}^{\text{new}}(t,k^{2})=\frac{F_{V}^{2}}{F}\left(1-2\frac{G_{V}}{F_{V}}\right)D_{\rho}(k^{2})-\frac{F_{A}^{2}}{F}D_{a_{1}}(t)$
$\displaystyle\hskip
64.58313pt+{\color[rgb]{1,0,0}2\sqrt{2}}\frac{F_{A}F_{V}}{F}D_{\rho}(k^{2})D_{a_{1}}(t)(-\lambda^{\prime\prime}t{\color[rgb]{1,0,0}-\lambda^{\prime}k^{2}}+\lambda_{0}m_{\pi}^{2}),$
(56)

where we show in red the factors and terms missing in the expression for
$F_{A}^{\text{old}}(t,k^{2})$ of ref. [5].

Furthermore, since ref. [5] works in the chiral limit
$A_{2}^{\text{old}}(t,k^{2})$ and $A_{4}^{\text{old}}(t,k^{2})$ are linearly
dependent and are replace with the form factor $B$, such that

$\displaystyle A_{2}^{\text{old}}(t,k^{2})$ $\displaystyle\to-2B(k^{2}),$
(57a) $\displaystyle A_{4}^{\text{old}}(t,k^{2})$
$\displaystyle\to-\frac{2B(k^{2})}{k^{2}+2p\cdot k}.$ (57b)

As said previously, ref. [5] works on the chiral limit, therefore the
$\lambda_{1}^{PV}$ term in eqs. (21) and eqs. (23) do not contribute.
Nevertheless and for the sake of simplicity, ref. [5] gives $B$ in terms
entirely of the $I=1$ part of the $\pi^{+}\pi^{-}$ vector form factor. This
means that the contribution from the $a_{1}$ meson is being neglected.

Figure 9: Invariant mass spectra in a double logarithmic scale for the
complete amplitude (purple band), its contribution of the axial part (green
band) and the same contribution but with the mistakes of ref. [5] (pale-blue
band). The width of the band is an uncertainty computed as for Figures 7
Figure 10: Invariant mass spectra of Figure 9 using the same color code, with
a logarithmic scale for the vertical axis, showing the region for their
maximum contribution to $\mathcal{B}$.

As a consequence of all this, we can see that the total spectrum gets really
affected by such differences: We computed only the contribution of the axial
amplitude to the $m_{e^{+}e^{-}}$ spectrum (i.e., turning off the SI and
vector contributions in eqs. (1) and keeping only that of eq. (1c)), and
computed also such spectrum but with the axial form factors of reference [5].
The complete spectrum is shown in Figure 9 with a double logarithmic scale and
a zoom with a logarithmic scale for the vertical axis in Figure 10.

As shown in ref. [5], the importance of the SD contributions start at
$m_{e^{+}e^{-}}^{2}\sim 0.1\text{ GeV}^{2}$. Thus, as shown in Figures 9 and
10, the total invariant mass spectrum (purple band) gets almost completely
saturated by the axial contribution151515at lower energies, the total
invariant mass spectrum gets saturated by the SI contribution (green band) at
$m_{e^{+}e^{-}}^{2}\gtrsim 0.1\text{ GeV}^{2}$. In these Figures we also show
the spectrum obtained with the mistakes shown in eq. (A) (pale-blue band),
keeping our constrained couplings and values for fitted parameters, we see
that there is an important difference between the spectra with the new and old
form factors. Their contribution to the branching fraction of the
$\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ decay obtained from the spectra are
an order of magnitude away

$\displaystyle\left.\frac{}{}\mathcal{B}\right|_{IB,V\to 0}^{\text{new}}$
$\displaystyle=1.09\times 10^{-5},$ (58a)
$\displaystyle\left.\frac{}{}\mathcal{B}\right|_{IB,V\to 0}^{\text{old}}$
$\displaystyle=1.66\times 10^{-6},$ (58b)

In contrast, we computed also the contribution from the axial amplitude to the
branching ratio of the $\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ decay as done
for the complete $\mathcal{B}$ in Table 2 for the new form factors both,
considering (fb) and neglecting (fc) the $U(3)_{V}$ breaking terms

$\displaystyle\left.\frac{}{}\mathcal{B}\right|_{IB,V\to 0}^{\text{fb}}$
$\displaystyle=(1.03\pm 0.10)\times 10^{-5},$ (59a)
$\displaystyle\left.\frac{}{}\mathcal{B}\right|_{IB,V\to 0}^{\text{fc}}$
$\displaystyle=(1.02\pm 0.10)\times 10^{-5},$ (59b)

showing thus that the difference between our results and those in ref. [5]
stems from the differences between eqs. (A) and (A), and not from the
$U(3)_{V}$ breaking terms.

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