Unified study of two-meson and axion−meson production from semileptonic tau decays within resonance chiral framework

Abstract

We carry out the joint study of the semileptonic tau decays into the two-meson and axion-meson channels, viz. {\tau }^{-}\to (P_1{P}_2{)}^{-}{\nu }_{\tau } and {\tau }^{-}\to{\pi }^{-}(K^{-})a{\nu }_{\tau } within the framework of resonance chiral theory by including the model-independent axion−gluon−gluon interaction. By utilizing the {\pi }^0-\eta-{\eta }^{\prime }-axion mixing matrix elements from recent studies, we calculate the pertinent two-pseudoscalar boson form factors. To simultaneously fit the experimental spectra measured in the Cabibbo allowed {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau } process and also the Cabibbo suppressed {\tau }^{-}\to (K_S{\pi }^{-},K^{-}\eta ){\nu }_{\tau } ones, we determine all the relevant hadron resonance parameters. Then we give predictions to the spectra and branching ratios for various channels, such as {\tau }^{-}\to ({\pi }^{-}\eta ,{\pi }^{-}{\eta }^{\prime },K^{-}{\eta }^{\prime },{\pi }^{-}a,K^{-}a){\nu }_{\tau }. We also calculate the forward-backward asymmetries for all the aforementioned channels. The interplay between the scalar and vector form factors for different observables is analyzed in detail. Our theoretical predictions supply useful guidance to the future tau experiments, such as those at Belle-II, Super Tau-Charm Facility and Tera-Z factory of Circular Electron−Positron Collider.

Graphical abstract

Keywords

tau lepton / chiral perturbation theory / axion-like particle

1 Introduction

The longstanding strong CP problem in QCD can be elegantly explained by the Peccei-Quinn (PQ) mechanism [1, 2], which predicts an extremely luring particle, viz. the axion [3, 4]. The quest for the latter hypothesized particle has been intensively carried out in a wide scope of physics, including quantum precision measurement, optical cavity, astrophysical observations, rare meson decays, and so on [5-14].

The featured axion interaction is given by the anomalous operator aG\tilde{G}/f_a, where G and \tilde{G} represent the gluon field strength tensor and its dual form in order, and f_a is the axion decay constant. Since the aG\tilde{G}/f_a term is responsible for resolving the strong CP problem and is a universal prediction of various ultraviolet axion models, it is usually deemed as the model-independent axion interaction. The anomalous axion−gluon−gluon operator inevitably induces the axion-hadron interactions, which can be probed in axion−baryon scattering [15-21], axion-meson reactions [22-25], light-flavor meson decays [26-32], etc.

It is reminded that tau, as the only lepton that has large enough mass for the hadronic decays, offers a valuable opportunity to study the hadron interactions. Large amounts of tau events are being collected at the ongoing Belle-II experiment [33] and could be also abundantly produced in the future experiments, such as Super Tau-Charm Facility (STCF) [34-37] and Tera-Z factory at Circular Electron Positron Collider (CEPC) [38, 39]. High sensitivity of the rare tau decay channels can be reached at such future facilities. Therefore it is interesting to explore the axion-meson interaction in the semileptonic tau decays. Such kind of theoretical study is still rare, and we carry out a pioneer investigation in this work.

To be specific, we focus on the {\tau }^{-}\to {\pi }^{-}a{\nu }_{\tau } and {\tau }^{-}\to K^{-}a{\nu }_{\tau } processes. It is quite plausible that hadronic resonances will play crucial roles in such decays. This requires us to estimate not only axion-light pseudoscalar meson couplings but also the axion-resonance ones. To account for the latter types of couplings, the framework of resonance chiral theory (\mathrm{R}\chi \mathrm{T}) [40], together with the a−{\pi }^0−\eta−{\eta }^{\prime } mixing formulas in Refs. [31, 32], will be employed in our study. Comparing with the general setups of the axion weak chiral Lagrangians in Ref. [26], we will stick to the model-independent axion interaction, i.e., the aG\tilde{G} term. The primary aim of the present work is to systematically include the hadronic resonance contributions in the {\tau }^{-}\to {\pi }^{-}(K^{-})a{\nu }_{\tau } decays and to investigate to what extent the axion production rates from the tau decays can be enhanced after including the hadronic resonances. To make a self-consistent study, we also update the calculation of the {\tau }^{-}\to (P_1{P}_2{)}^{-}{\nu }_{\tau } processes in \mathrm{R}\chi \mathrm{T} both for the Cabibbo allowed and suppressed situations, instead of just focusing upon one specific channel. A joint fit of different kinds of experimental data, including the invariant mass distributions of the {\pi }^{-}{\pi }^0, K_S{\pi }^{-} and K^{-}\eta systems from the {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau }, {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau } and {\tau }^{-}\to K^{-}\eta {\nu }_{\tau } decays, respectively, will be performed to determine the unknown resonance couplings. The updated resonance parameters will be further exploited to predict the spectra and branching ratios of the channels that are yet to be measured, such as {\tau }^{-}\to K^{-}{\eta }^{\prime }{\nu }_{\tau }, {\tau }^{-}\to {\pi }^{-}\eta ({\eta }^{\prime }){\nu }_{\tau } and {\tau }^{-}\to {\pi }^{-}(K^{-})a{\nu }_{\tau }. Meanwhile, we also give predictions to the forward-backward asymmetries arising from all the relevant channels, including {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau }, {\tau }^{-}\to K^{-}\eta ({\eta }^{\prime }){\nu }_{\tau }, {\tau }^{-}\to {\pi }^{-}\eta ({\eta }^{\prime }){\nu }_{\tau } and {\tau }^{-}\to {\pi }^{-}(K^{-})a{\nu }_{\tau }.

This article is structured as follows. The general formulas for the theoretical description of the two-boson semileptonic tau decays are given in Section 2, in order to set up the notations. We elaborate calculations of the vector and scalar form factors within the framework of resonance chiral theory in Section 3. Next, we discuss the combined fit to the experimental data from the {\tau }^{-}\to ({\pi }^{-}{\pi }^0,K_S{\pi }^{-},K^{-}\eta ){\nu }_{\tau } decays in Section 4. The predictions to the spectra and branching ratios of the {\tau }^{-}\to ({\pi }^{-}\eta ,{\pi }^{-}{\eta }^{\prime },K^{-}{\eta }^{\prime },{\pi }^{-}a,K^{-}a){\nu }_{\tau } processes are presented in Section 5, where the forward-backward asymmetric distributions are also calculated for all the pertinent channels. We give the summary and conclusions in Section 6.

2 Description of the tau decay amplitudes

The Standard Model (SM) structure for the charged currents (CC) will be employed to calculate the semileptonic tau decays, which can be cast in the conventional form of four-fermion operator as

(1){\mathcal{L}}_{\mathrm{C}\mathrm{C}}^{\mathrm{S}\mathrm{M}}=-{\displaystyle \frac{G_F}{\sqrt{2}}}{V}_{uD}{\overline{\nu }}_{\tau }{\gamma }_{\mu }(1-{\gamma }_5)\tau \overline{D}{\gamma }^{\mu }(1-{\gamma }_5)u,

where G_F denotes the Fermi constant, V_{uD} corresponds to the CKM matrix elements, with the down-type quarks D=d and s. The amplitude of the two-pseudoscalar boson tau decay process, i.e., \tau (p_{\tau })\to P_1(p_1)P_2(p_2){\nu }_{\tau }(p_{\nu }), can be then written as

(2)\mathcal{M}={\displaystyle \frac{G_F{V}_{uD}}{\sqrt{2}}}\overline{u}(p_{\nu }){\gamma }_{\mu }(1-{\gamma }_5)u(p_{\tau })H^{\mu },

where the hadronic matrix element of the current is usually parameterized by

(3)\begin{array}{rl}{H}^{\mu }\equiv & ⟨P_1{P}_2|\overline{D}{\gamma }^{\mu }u|0⟩=\left[(p_2-p_1{)}^{\mu }-\frac{{\mathrm{\Delta }}_{P_2{P}_1}}{s}{q}^{\mu }\right]F_{+}^{P_1{P}_2}(s)\\ & +\frac{{\mathrm{\Delta }}_{Du}}{s}{q}^{\mu }{\hat{F}}_0^{P_1{P}_2}(s),\end{array}

with

(4)\begin{array}{rl}& {\mathrm{\Delta }}_{P_2{P}_1}=m_{P_2}^2-m_{P_1}^2,{\mathrm{\Delta }}_{Du}=B_0(m_D-m_u),\\ & q_{\mu }=(p_1+p_2{)}_{\mu },s=q^2.\end{array}

Here m_{P_i} denotes the mass of the pseudoscalar boson P_i, and m_{D=d,s} and m_u are the light-flavor quark masses. The quantity B_0 relates with the nonperturbative quark condensate via ⟨0|{\overline{q}}_i{q}_j|0⟩=-F^2{B}_0{\delta }_{ij}, with F the pion decay constant in chiral limit. In this parameterization, F_{+}^{P_1{P}_2}(s) and {\hat{F}}_0^{P_1{P}_2}(s) represent the vector and scalar form factors, respectively. The finitude feature of the matrix elements in Eq. (3) at s=0 requires the following normalization condition

(5)F_{+}^{P_1{P}_2}(0)={\displaystyle \frac{{\mathrm{\Delta }}_{Du}}{{\mathrm{\Delta }}_{P_2{P}_1}}}{\hat{F}}_0^{P_1{P}_2}(0).

After the evaluation of |\mathcal{M}{|}^2 in Eq. (2) by taking the spin average/sum of the \tau/{\nu }_{\tau }, one can acquire the conventional form of the differential decay width of the {\tau }^{-}\to (P_1{P}_2{)}^{-}{\nu }_{\tau } process

(6)\begin{array}{rl}& \frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}d\cos \alpha }\\ & =\frac{G_F^2{\left|V_{uD}\right|}^2{S}_{\mathrm{E}\mathrm{W}}{\left(m_{\tau }^2-s\right)}^2}{128{\pi }^3}\{\frac{{\left|{\hat{F}}_0^{P_1{P}_2}(s)\right|}^2{\mathrm{\Delta }}_{Du}^2{q}_{P_1{P}_2}(s)}{m_{\tau }{s}^2}\\ & +\frac{4{\mathrm{\Delta }}_{Du}{q}_{P_1{P}_2}^2(s)\cos \alpha \mathrm{\Re }\left[F_{+}^{P_1{P}_2}(s){\hat{F}}_0^{P_1{P}_2\ast }(s)\right]}{m_{\tau }{s}^{3/2}}\\ & +\frac{4{\left|F_{+}^{P_1{P}_2}(s)\right|}^2{q}_{P_1{P}_2}^3(s)\left(m_{\tau }^2\cos {\alpha }^2+s\sin {\alpha }^2\right)}{m_{\tau }^{3}s}\}.\end{array}

Here we adopt the short-distance electroweak radiative correction S_{\mathrm{E}\mathrm{W}}=1.0201 from Ref. [41]. It consists of (i) the combined logarithmically enhanced electroweak and short-distance QCD corrections, by performing the next-to-leading order resummation of the logarithms through renormalization group equation; (ii) the remaining non-logarithmically enhanced electroweak corrections. The quantity \alpha in Eq. (6) denotes the angle between the momenta of one of the two pseudoscalar bosons and the \tau lepton in the P_1{P}_2 rest frame, and the three-momentum of the pseudoscalar bosons in the latter frame is given by

(7)\begin{array}{rl}& q_{P_1{P}_2}(s)=\frac{\sqrt{s^2-2s{\mathrm{\Sigma }}_{P_1{P}_2}+{\mathrm{\Delta }}_{P_1{P}_2}^2}}{2\sqrt{s}},\\ & {\mathrm{\Sigma }}_{P_1{P}_2}=m_{P_1}^2+m_{P_2}^2.\end{array}

To further integrate out the angle \alpha in Eq. (6), we obtain the differential decay width with respect to \sqrt{s}, i.e., the energy of the P_1{P}_2 system in the center of mass frame,

(8)\begin{array}{rl}\frac{\mathrm{d}{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}}=& \frac{G_F^2{M}_{\tau }^3}{48{\pi }^{3}s}{S}_{\mathrm{E}\mathrm{W}}{\left|V_{uD}\right|}^2{\left(1-\frac{s}{M_{\tau }^2}\right)}^2\\ & \{\left(1+\frac{2s}{M_{\tau }^2}\right)q_{P_1{P}_2}^3(s){\left|F_{+}^{P_1{P}_2}(s)\right|}^2\\ & +\frac{3{\mathrm{\Delta }}_{Du}^2}{4s}{q}_{P_1{P}_2}(s){\left|{\hat{F}}_0^{P_1{P}_2}(s)\right|}^2\}.\end{array}

Alternatively, one can also construct another important type of differential distribution, viz. the so-called forward-backward (FB) asymmetry,

(9)\begin{array}{rl}{A}_{FB}(s)=& \frac{{\displaystyle {\int }_0^1\mathrm{d}\cos \alpha {\displaystyle \frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}\mathrm{d}\cos \alpha }}-{\displaystyle {\int }_{-1}^0\mathrm{d}\cos \alpha {\displaystyle \frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}\mathrm{d}\cos \alpha }}}}}{{\displaystyle {\int }_0^1\mathrm{d}\cos \alpha {\displaystyle \frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}\mathrm{d}\cos \alpha }}+{\displaystyle {\int }_{-1}^0\mathrm{d}\cos \alpha {\displaystyle \frac{{\mathrm{d}}^2{\mathrm{\Gamma }}_{\tau \to P_1{P}_2{\nu }_{\tau }}}{\mathrm{d}\sqrt{s}\mathrm{d}\cos \alpha }}}}}\\ =& \frac{{\mathrm{\Delta }}_{Du}{q}_{P_1{P}_2}(s)\mathrm{\Re }\left[F_{+}^{P_1{P}_2}(s){\hat{F}}_0^{P_1{P}_2\ast }(s)\right]}{{\displaystyle \frac{2\sqrt{s}}{3}}\left(1+{\displaystyle \frac{2s}{M_{\tau }^2}}\right)q_{P_1{P}_2}^2(s){\left|F_{+}^{P_1{P}_2}(s)\right|}^2+{\displaystyle \frac{{\mathrm{\Delta }}_{Du}^2}{2\sqrt{s}}}{\left|{\hat{F}}_0^{P_1{P}_2}(s)\right|}^2},\end{array}

which probes the interference term of the vector and scalar form factors that is however absent in the differential decay width expression in Eq. (8). It is noted that the FB asymmetry can play important roles in establishing the CP violation in hadronic \tau decays [42-44].

From the above discussions, it is clear that under the assumption of SM structure of charged currents the \tau \to P_1{P}_2{\nu }_{\tau } processes are totally determined by the two form factors F_{+}^{P_1{P}_2}(s) and {\hat{F}}_0^{P_1{P}_2}(s), both of which will be calculated within \mathrm{R}\chi \mathrm{T} in the following section.

3 Calculation of the hadronic form factors

3.1 Pertinent resonance chiral Lagrangians

Since the relevant energy region of the form factors F_{+}^{P_1{P}_2}(s) and {\hat{F}}_0^{P_1{P}_2}(s) spans from the two-boson threshold up to m_{\tau }, apart from the contact meson interactions that are dictated by the chiral symmetry in the low energy region, the hadronic meson resonances play decisive roles in the intermediate energy region. In practice, it is convenient to derive the vector form factor F_{+}^{P_1{P}_2}(s) via the matrix elements of the \overline{D}{\gamma }^{\mu }u current and the scalar form factor {\hat{F}}_0^{P_1{P}_2}(s) via the matrix elements of \overline{D}u. \mathrm{R}\chi \mathrm{T} has proven to be able to provide a reliable and sophisticated theoretical framework to calculate pertinent form factors appearing in the \tau decays [45-54]. To make a self-consistent evaluation, we recapitulate the calculation of the relevant two-meson form factors to the \tau decays in \mathrm{R}\chi \mathrm{T}, though they have been individually addressed in previous works, such as those in Refs. [47, 48, 50, 51]. As a novelty, we calculate the axion related form factors appearing in the {\tau }^{-}\to {\pi }^{-}(K^{-})a{\nu }_{\tau } processes, and also update the expressions involving the \eta and {\eta }^{\prime } based on the newly determined {\pi }^0-\eta-{\eta }^{\prime }-a mixing formulas in Refs. [31, 32], where the model-independent axion interaction operator, viz., {\alpha }_{s}aG\tilde{G}/(8\pi f_a), is introduced to describe the axion-meson system.

In this work, it is reiterated that we will also stick to the model-independent axion interaction aG\tilde{G}/f_a. In particular, we do not consider the axion-lepton interaction vertex throughout, while this latter type of interaction has been recently explored in the tau decays in Ref. [55]. It is demonstrated in Ref. [31] that the U(3) axion chiral theory that explicitly includes the QCD U_A(1) anomaly effect provides a viable framework to simultaneously incorporate the axion, \pi, K, \eta and {\eta }^{\prime } states. The leading order (LO) U(3) Lagrangian reads

(10){\mathcal{L}}^{\mathrm{L}\mathrm{O}}={\displaystyle \frac{F^2}{4}}⟨u_{\mu }{u}^{\mu }⟩+{\displaystyle \frac{F^2}{4}}⟨{\chi }_{+}⟩+{\displaystyle \frac{F^2}{12}}{M}_0^2{X}^2,

where ⟨...⟩ stands for the trace over the light-quark flavor space, and the last term corresponds to the explicit U_A(1) anomaly term, responsible for the large mass M_0 of the singlet {\eta }_0. The associated chiral building operators are defined as

(11)\begin{array}{rl}& u_{\mu }=\mathrm{i}{u}^{†}{D}_{\mu }U{u}^{†},{\chi }_{\pm }=u^{†}\chi u^{†}\pm u{\chi }^{†}u,\\ & U=u^2={\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\mathrm{\Phi }}{F}},\end{array}

(12)\begin{array}{rl}& D_{\mu }U={\mathrm{\partial }}_{\mu }U-\mathrm{i}(v_{\mu }+a_{\mu })U+\mathrm{i}U(v_{\mu }-a_{\mu }),\\ & \chi =2B_0(s+\mathrm{i}p),\end{array}

where v_{\mu }, a_{\mu }, s, p are the chiral external sources with types of vector, axial-vector, scalar and pseudoscalar, respectively. The matrix representation of the pseudo-Nambu Goldstone bosons (pNGBs) is given by

(13)\mathrm{\Phi }=\left(\begin{array}{ccc}{\displaystyle \frac{1}{\sqrt{2}}}{\pi }^0+{\displaystyle \frac{1}{\sqrt{6}}}{\eta }_8+{\displaystyle \frac{1}{\sqrt{3}}}{\eta }_0& {\pi }^{+}& K^{+}\\ {\pi }^{-}& {\displaystyle \frac{-1}{\sqrt{2}}}{\pi }^0+{\displaystyle \frac{1}{\sqrt{6}}}{\eta }_8+{\displaystyle \frac{1}{\sqrt{3}}}{\eta }_0& K^0\\ K^{-}& {\overline{K}}^0& {\displaystyle \frac{-2}{\sqrt{6}}}{\eta }_8+{\displaystyle \frac{1}{\sqrt{3}}}{\eta }_0\end{array}\right).

In U(3) chiral theory, the axion field can be introduced into the chiral Lagrangian via [31, 32]

(14)X=\log (detU)-\mathrm{i}{\displaystyle \frac{a}{f_a}},

without performing the axial transformation of the quark fields as commonly used in chiral studies [7, 11].

Next, we provide the relevant resonance chiral Lagrangians to our study. According to the seminal work of \mathrm{R}\chi \mathrm{T} [40], the minimal interacting Lagrangians involving vector and scalar resonances are

(15){\mathcal{L}}_V={\displaystyle \frac{F_V}{2\sqrt{2}}}⟨V_{\mu \nu }{f}_{+}^{\mu \nu }⟩+{\displaystyle \frac{\mathrm{i}{G}_V}{\sqrt{2}}}⟨V_{\mu \nu }{u}^{\mu }{u}^{\nu }⟩,

(16){\mathcal{L}}_S=c_d⟨S{u}_{\mu }{u}^{\mu }⟩+c_m⟨S{\chi }_{+}⟩,

where V_{\mu \nu } is the vector resonance nonet in the anti-symmetric tensor form, S stands for the scalar resonance nonet and the large N_C relations are imposed to both the vector (F_V,G_V) and scalar (c_d,c_m) resonance couplings. Since we investigate the two-boson interactions from their thresholds up to m_{\tau } in the current study, several excited multiplets of hadron resonances, instead of just the low-lying ones whose couplings are denoted by the unprimed symbols, like F_V,G_V,c_{m,d}, can play relevant roles. We will use the same forms of interacting operators, but with the obvious replacement of resonance couplings by introducing primes, such as F_V^{\prime }, G_V^{\prime } and c_{m,d}^{\prime }, to include the contributions from the first excited hadron resonances. We also introduce double-prime couplings, such as F_V^{″} and G_V^{″}, to account for the contributions from the second excited vector resonance states. The scalar resonance masses in the chiral limit will be denoted by M_S and M_{S^{\prime }}. Under the resonance saturation assumption at large N_C, the heavy hadron resonances give the dominant contributions to the next-to-leading order (NLO) chiral low energy constants (LECs), which should not be included any more in the \mathrm{R}\chi \mathrm{T} Lagrangians [40]. Otherwise, there will be double counting issues.

However, in the U(3) case, two additional local NLO operators, namely

(17){\mathcal{L}}_{\mathrm{\Lambda }}^{\mathrm{N}\mathrm{L}\mathrm{O},U(3)}=-{\displaystyle \frac{F^2{\mathrm{\Lambda }}_1}{12}}{D}^{\mu }X{D}_{\mu }X-{\displaystyle \frac{F^2{\mathrm{\Lambda }}_2}{12}}X⟨{\chi }_{-}⟩,

can not be accounted for by the resonance exchanges at large N_C. Therefore we explicitly take them into account in our calculation, together with the resonance Lagrangians in Eqs. (15) and (16), and also those from highly excited multiplets. The Feynman diagrams that contribute to the corresponding form factors are illustrated in Fig. 1. At the same time, we also need to compute the wave renormalization constants and masses of the pNGBs via the self-energy diagrams, as shown in Fig. 2.

The neutral pNGBs, i.e., {\pi }^0,{\eta }_8,{\eta }_0, and the axion a will get mixed, according to the chiral Lagrangians in Eqs. (10), (16) and (17). Before calculating any physical quantities, such as the form factors and scattering amplitudes, one should first perform the field redefinitions to eliminate their mixing in order to get the physical states. In Refs. [31, 32], the mixing pattern among {\pi }^0,{\eta }_8,{\eta }_0,a has been worked out within the U(3) chiral perturbation theory by explicitly incorporating the local NLO LECs in the \delta counting scheme [56]. We can directly take the results of Refs. [31, 32] to handle the mixing in the present work. The mixing formula between the bases of physical states (denoted by \hat{\varphi }) and the flavor states can be written as

(18)\left(\begin{array}{c}{\hat{\pi }}^0\\ \hat{\eta }\\ {\hat{\eta }}^{\prime }\\ \hat{a}\end{array}\right)=\left(\begin{array}{cccc}1+z_{11}& c_{\theta }(-v_{12}+z_{12})+s_{\theta }(-v_{13}+z_{13})& -s_{\theta }(-v_{12}+z_{12})+c_{\theta }(-v_{13}+z_{13})& -v_{14}+z_{14}\\ v_{12}+z_{21}& c_{\theta }(1+z_{22})+s_{\theta }(z_{23}-v_{23})& -s_{\theta }(1+z_{22})+c_{\theta }(z_{23}-v_{23})& -v_{24}+z_{24}\\ v_{13}+z_{31}& c_{\theta }(z_{32}+v_{23})+s_{\theta }(1+z_{33})& -s_{\theta }(z_{32}+v_{23})+c_{\theta }(1+z_{33})& -v_{34}+z_{34}\\ v_{41}+z_{41}& c_{\theta }(v_{42}+z_{42})+s_{\theta }(v_{43}+z_{43})& -s_{\theta }(v_{42}+z_{42})+c_{\theta }(v_{43}+z_{43})& 1+v_{44}+z_{44}\end{array}\right)\left(\begin{array}{c}{\pi }^0\\ {\eta }_8\\ {\eta }_0\\ a\end{array}\right),

where v_{ij} and z_{ij} correspond to the LO and NLO contributions, respectively. In later discussions, we designate {\pi }^0,\eta ,\eta ,a as the physical states for simplicity. c_{\theta } and s_{\theta } are short for \cos \theta and \sin \theta, with \theta the LO \eta-{\eta }^{\prime } mixing angle. The NLO coefficients z_{ij} encode two parts denoted as x_{ij} and y_{ij} in Ref. [31, 32], where x_{ij} are responsible for the kinematic mixing and y_{ij} are introduced to address the mass mixing at NLO. Though the expressions of x_{ij} and y_{ij} are rather lengthy, they only depend on four NLO LECs, namely L_5,L_8,{\mathrm{\Lambda }}_1 and {\mathrm{\Lambda }}_2. To make a consistent calculation, we will use the resonance saturation to estimate L_5 and L_8 in this work, which are given by L_5=c_d{c}_m/M_S^2+c_d^{\prime }{c}_m^{\prime }/M_{S^{\prime }}^2 and L_8=c_m^2/(2M_S^2)+c_m^{\prime 2}/(2M_{S^{\prime }}^2) [40]. The effects of the pseudoscalar resonances are neglected for the latter quantity. The values of L_5 and L_8 by taking the resonance parameters discussed later in our study are compatible with those obtained from the comprehensive Bayesian method in Ref. [57]. The explicit expressions of v_{ij}, x_{ij}, y_{ij} and z_{ij} are given in Ref. [32], and we do not repeat them here.

3.2 Strangeness conserving form factors

The decays of {\tau }^{-}\to {\pi }^{-}({\pi }^0,\eta ,{\eta }^{\prime },a){\nu }_{\tau } are driven by the strangeness conserving currents, belonging to the Cabibbo allowed processes. In this case, we need to evaluate the matrix elements of the strangeness conserving vector current ⟨P_1{P}_2|\overline{d}{\gamma }_{\mu }u|0⟩ to acquire the corresponding vector form factors. Regarding the scalar form factors, the proper object to extract them is ⟨P_1{P}_2|(m_D-m_u)\overline{D}u|0⟩, since the latter quantity corresponds to the divergence of the vector current, i.e., ⟨P_1{P}_2|{\mathrm{\partial }}^{\mu }(\overline{D}{\gamma }_{\mu }u)|0⟩=i⟨P_1{P}_2|(m_D-m_u)\overline{D}u|0⟩, and is renormalization group invariant [58, 59].

In practical calculation, we will substitute the light quark masses m_{D=d,s} and m_u with the physical masses of \pi and K evaluated in \mathrm{R}\chi \mathrm{T}. For the strangeness conserving situation, the isospin breaking quantity m_d-m_u enters in the definition of the scalar form factors. According to the Dashen’s theorem [60], the electromagnetic (EM) corrections to the pion and kaon masses are equal, i.e., {\delta }_{m_{K^0}^2-m_{K^{+}}^2}^{\mathrm{E}\mathrm{M}}={\delta }_{m_{{\pi }^0}^2-m_{{\pi }^{+}}^2}^{\mathrm{E}\mathrm{M}}. By further taking into account the resonance contribution to the kaon masses, up to NLO accuracy one can rewrite the light-quark mass m_d-m_u in terms of the physical meson masses as

(19)\begin{array}{rl}{\mathrm{\Delta }}_{du}=& B_0(m_d-m_u)\\ =& {\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}\left\{1+\frac{m_K^2}{F^2}\left[\frac{16c_m(c_d-c_m)}{M_S^2}+\frac{16c_m^{\prime }(c_d^{\prime }-c_m^{\prime })}{M_{S^{\prime }}^2}\right]\right\},\end{array}

where {\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}=m_{K^0}^2-m_{K^{+}}^2-(m_{{\pi }^0}^2-m_{{\pi }^{+}}^2), m_K^2 inside the curly brackets denotes the isospin-averaged kaon mass squared. In this work we include the effects of isospin breaking (IB) up to the linear order for the scalar component of the hadronic matrix element of \overline{d}{\gamma }^{\mu }u. In this case, it is justified to use the physical isospin averaged mass squared inside the curly brackets of Eq. (19). The strangeness conserving scalar form factor is then defined as

(20)(m_d-m_u)⟨{\pi }^{-}P|\overline{d}u|0⟩={\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}{F}_0^{{\pi }^{-}P}(s),

with P={\pi }^0,\eta ,{\eta }^{\prime },a. Comparing with the definition in Eq. (3), the product of {\mathrm{\Delta }}_{du}{\hat{F}}_0^{{\pi }^{-}P}(s) in fact equals to {\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}{F}_0^{{\pi }^{-}P}(s), which leads to

(21){\hat{F}}_0^{{\pi }^{-}P}(s)={\displaystyle \frac{{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{{\mathrm{\Delta }}_{du}}}{F}_0^{{\pi }^{-}P}(s).

Correspondingly, the matrix elements of the strangeness conserving vector current in Eq. (3) can be recast as

(22)\begin{array}{rl}⟨{\pi }^{-}P|\overline{d}{\gamma }^{\mu }u|0⟩=& \left[(p_P-p_{\pi }{)}^{\mu }-\frac{{\mathrm{\Delta }}_{P\pi }}{s}{q}^{\mu }\right]F_{+}^{{\pi }^{-}P}(s)\\ & +\frac{{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{s}{q}^{\mu }{F}_0^{{\pi }^{-}P}(s).\end{array}

It is pointed out that many other works, e.g., Refs. [61, 62], parameterize the scalar component of the matrix element ⟨P_1{P}_2|\overline{D}{\gamma }^{\mu }u|0⟩ as {\mathrm{\Delta }}_{P_1{P}_2}{\overline{F}}_0(s)q^{\mu }/s, with {\mathrm{\Delta }}_{P_1{P}_2}=m_{P_1}^2-m_{P_2}^2. Since {\mathrm{\Delta }}_{P_1{P}_2} is not always proportional to {\mathrm{\Delta }}_{Du}, this indicates that the scalar form factor {\overline{F}}_0(s) would compromise the flavor symmetry breaking factor in such cases. While, in our convention the factor proportional to m_D-m_u is factored out and the scalar form factors {\hat{F}}_0(s) and F_0(s) do not contain such flavor symmetry breaking terms.

For the decay channel of {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau }, compared to the vector form factor, the contribution from the scalar form factor is much suppressed by the isospin breaking factor. Therefore we will neglect the tiny effect from the scalar form factor. The vector form factor appearing in this decay channel takes the form

(23)F_{+}^{{\pi }^{-}{\pi }^0}(s)=-{\displaystyle \frac{\sqrt{2}}{F^2}}{G}_{\mathrm{L}\mathrm{O}+\rho \mathrm{E}\mathrm{x}}(s),

with

(24)\begin{array}{rl}{G}_{\mathrm{L}\mathrm{O}+\rho \mathrm{E}\mathrm{x}}(s)=& \frac{G_V{F}_{V}s+F^2(M_{\rho }^2-s)}{M_{\rho }^2-s-\mathrm{i}{M}_{\rho }{\mathrm{\Gamma }}_{\rho }(s)}\\ & -\frac{G_V^{\prime }{F}_V^{\prime }s}{M_{{\rho }^{\prime }}^2-s-\mathrm{i}{M}_{{\rho }^{\prime }}{\mathrm{\Gamma }}_{{\rho }^{\prime }}(s)}\\ & -\frac{G_V^{″}{F}_V^{″}s}{M_{{\rho }^{″}}^2-s-\mathrm{i}{M}_{{\rho }^{″}}{\mathrm{\Gamma }}_{{\rho }^{″}}(s)},\end{array}

where we follow the recipe of \mathrm{R}\chi \mathrm{T} Lagrangian in Eq. (15) to also introduce another two excited resonances {\rho }^{\prime }/\rho (1450) and {\rho }^{″}/\rho (1700), in addition to the ground state \rho/\rho (770). We point out the subtle issue about the minus signs in front of the {\rho }^{\prime } and {\rho }^{″} terms in Eq. (24). In fact, it is common to introduce some phase parameters for different resonance states to describe the \tau \to \pi \pi {\nu }_{\tau } process in many existing works [63-67]. We take advantage of these previous studies to fix minus signs for the {\rho }^{\prime } and {\rho }^{″} a posteriori, allowing us to make a good description of the precise \pi \pi spectra from Belle [68]. In this way, the number of free parameters is reduced and the fit tends to be stable.

We further extend the discussion to the decay channel of {\tau }^{-}\to {\pi }^{-}(\eta ,{\eta }^{\prime },a){\nu }_{\tau }. Unlike the {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau } channel, there is no relative suppression between the vector and scalar form factors, although an overall isospin breaking factor appears in each of the three processes of {\tau }^{-}\to {\pi }^{-}(\eta ,{\eta }^{\prime },a){\nu }_{\tau }. As a result, one should consider both contributions from the vector and scalar form factors to these processes. To make a more realistic description of the differential decay widths, we also introduce two scalar nonet resonances. We use c_m and c_d to denote the resonance parameters for the low-lying scalar multiplet containing the relevant a_0(980) (simply denoted as a_0) and K_0^{\ast }(800) (denoted as K_0^{\ast }) in this work, and designate the primed resonance parameters c_m^{\mathrm{\prime }} and c_d^{\mathrm{\prime }} for the second multiplet of scalar resonances containing a_0(1450) (denoted as a_0^{\prime }) and K_0^{\ast }(1430) (denoted as K_0^{{\ast }^{\prime }}) that are pertinent to the present study. The corresponding vector form factors take the form

(25)F_{+}^{{\pi }^{-}\eta }(s)=-{\displaystyle \frac{\sqrt{2}{v}_{12}}{F^2}}{G}_{\mathrm{L}\mathrm{O}+\rho \mathrm{E}\mathrm{x}}(s)-\sqrt{2}\left(y_{12}-v_{13}{y}_{23}^{(0)}\right),

(26)F_{+}^{{\pi }^{-}{\eta }^{\prime }}(s)=-{\displaystyle \frac{\sqrt{2}{v}_{13}}{F^2}}{G}_{\mathrm{L}\mathrm{O}+\rho \mathrm{E}\mathrm{x}}(s)-\sqrt{2}\left(y_{13}+v_{12}{y}_{23}^{(0)}\right),

(27)\begin{array}{rl}{F}_{+}^{{\pi }^{-}a}(s)=& -{\displaystyle \frac{\sqrt{2}{v}_{41}}{F^2}}{G}_{\mathrm{L}\mathrm{O}+\rho \mathrm{E}\mathrm{x}}(s)\\ & -\sqrt{2}\left(y_{14}+v_{12}{y}_{24}^{(0)}+v_{13}{y}_{34}^{(0)}\right),\end{array}

and the scalar form factors are given by

(28)\begin{array}{rl}{F}_0^{{\pi }^{-}\eta }(s)=& \sqrt{\frac{2}{3}}(c_{\theta }-\sqrt{2}{s}_{\theta })+\frac{1}{\sqrt{3}}({\mathrm{\Lambda }}_1-2{\mathrm{\Lambda }}_2)s_{\theta }-\frac{1}{\sqrt{3}}{y}_{23}(2c_{\theta }+\sqrt{2}{s}_{\theta })\\ & +4\sqrt{\frac{2}{3}}\frac{c_{\theta }-\sqrt{2}{s}_{\theta }}{F^2}\{[\frac{c_m(c_m-c_d)2m_{\pi }^2+c_m{c}_d(s+m_{\pi }^2-m_{\eta }^2)}{M_{a_0}^2-s-\mathrm{i}{M}_{a_0}{\mathrm{\Gamma }}_{a_0}(s)}-\frac{2c_m(c_m-c_d)(2m_K^2-m_{\pi }^2)}{M_S^2}]\\ & +[c_{m,d},M_{a_0},{\mathrm{\Gamma }}_{a_0},M_S\to c_{m,d}^{\prime },M_{a_0^{\prime }},{\mathrm{\Gamma }}_{a_0^{\prime }},M_{S^{\prime }}]\},\end{array}

(29)\begin{array}{rl}{F}_0^{{\pi }^{-}{\eta }^{\prime }}(s)=& \sqrt{\frac{2}{3}}(\sqrt{2}{c}_{\theta }+s_{\theta })-\frac{1}{\sqrt{3}}({\mathrm{\Lambda }}_1-2{\mathrm{\Lambda }}_2)c_{\theta }+\frac{1}{\sqrt{3}}{y}_{23}(\sqrt{2}{c}_{\theta }-2s_{\theta })\\ & +4\sqrt{\frac{2}{3}}\frac{\sqrt{2}{c}_{\theta }+s_{\theta }}{F^2}\{[\frac{c_m(c_m-c_d)2m_{\pi }^2+c_m{c}_d(s+m_{\pi }^2-m_{{\eta }^{\prime }}^2)}{M_{a_0}^2-s-\mathrm{i}{M}_{a_0}{\mathrm{\Gamma }}_{a_0}(s)}-\frac{2c_m(c_m-c_d)(2m_K^2-m_{\pi }^2)}{M_S^2}]\\ & +[c_{m,d},M_{a_0},{\mathrm{\Gamma }}_{a_0},M_S\to c_{m,d}^{\prime },M_{a_0^{\prime }},{\mathrm{\Gamma }}_{a_0^{\prime }},M_{S^{\prime }}]\},\end{array}

(30)\begin{array}{rl}{F}_0^{{\pi }^{-}a}(s)=& \frac{(\sqrt{2}{c}_{\theta }-2s_{\theta })}{\sqrt{3}}{v}_{24}^{(0)}+\frac{(2c_{\theta }+\sqrt{2}{s}_{\theta })}{\sqrt{3}}{v}_{34}^{(0)}-\frac{2}{\sqrt{3}}\left(s_{\theta }{v}_{24}^{(0)}-c_{\theta }{v}_{34}^{(0)}+\frac{F}{\sqrt{6}{f}_a}\right)({\mathrm{\Lambda }}_2-\frac{1}{2}{\mathrm{\Lambda }}_1)\\ & +\frac{(\sqrt{2}{c}_{\theta }-2s_{\theta })}{\sqrt{3}}{y}_{24}^{(0)}+\frac{(2c_{\theta }+\sqrt{2}{s}_{\theta })}{\sqrt{3}}{y}_{34}^{(0)}+\frac{4(\sqrt{2}{c}_{\theta }-2s_{\theta })v_{24}^{(0)}+4(2c_{\theta }+\sqrt{2}{s}_{\theta })v_{34}^{(0)}}{\sqrt{3}{F}^2}\\ & \cdot \{[\frac{2c_m^2{m}_{\pi }^2+c_m{c}_d\left(s-m_a^2-m_{\pi }^2\right)}{M_{a_0}^2-s-\mathrm{i}{M}_{a_0}{\mathrm{\Gamma }}_{a_0}(s)}+\frac{2c_m(c_d-c_m)(2m_K^2-m_{\pi }^2)}{M_S^2}]\\ & +[c_{m,d},M_{a_0},{\mathrm{\Gamma }}_{a_0},M_S\to c_{m,d}^{\prime },M_{a_0^{\prime }},{\mathrm{\Gamma }}_{a_0^{\prime }},M_{S^{\prime }}]\}.\end{array}

It is reiterated that the mixing items v_{ij} and y_{ij} appearing in above equations are explicitly given in Ref. [32]. The values of L_5 and L_8 entering in y_{ij} will be estimated by the resonance saturation assumption in this work. The hadronic matrix element H_{\mu } of Eq. (22) should remain finite as s\to 0, which in turn requires the following normalization condition

(31)F_{+}^{{\pi }^{-}(\eta ,{\eta }^{\prime },a)}(0)={\displaystyle \frac{{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{{\mathrm{\Delta }}_{(\eta ,{\eta }^{\prime },a)\pi }}}{F}_0^{{\pi }^{-}(\eta ,{\eta }^{\prime },a)}(0).

By carefully expanding the right hand side of the above equation up to the NLO in the U(3) \delta counting scheme, i.e., the simultaneous expansions of momentum, quark mass and 1/N_C, and combining the results of Ref. [32], we have explicitly verified that the vector and scalar form factors in Eqs. (25)−(30) indeed satisfy the relations in Eq. (31). We point out several subtleties about the calculation of scalar form factors. For the scalar hadron exchanges in the diagram (b) of Fig. 1, i.e., with the vertices between the unflavored scalar hadrons and the vacuum, we will take the scalar masses M_S and M_{S^{\prime }} in the chiral and large N_C limits. While for the scalar resonance exchanges in the diagram (c) of Fig. 1, we use physical parameters for different intermediate scalar resonances in the propagators, including the masses and finite widths, which are expected to offer a more reliable description of the two-pseudoscalar boson spectra. It is noted that there are other more rigorous ways to include the scalar resonances in the form factors [69, 70], while we stick to the Lagrangian method mentioned above, since this latter approach is more flexible to fit the experimental data and is also easier to relate the scalar dynamics in different channels, allowing us to make predictions to the channels yet to be measured.

Regarding the energy-dependent finite decay widths of the vector resonances appearing in Eq. (24), we follow Ref. [67] to construct their forms. The expressions for the three vector resonances \rho, {\rho }^{\prime } and {\rho }^{″} are

(32){\mathrm{\Gamma }}_{\rho }(s)={\displaystyle \frac{M_{\rho }s}{96\pi F_{\pi }^2}}\left[{\sigma }_{\pi \pi }^3(s)+{\displaystyle \frac{1}{2}}{\sigma }_{KK}^3(s)\right],

(33){\mathrm{\Gamma }}_{{\rho }^{\prime },{\rho }^{″}}(s)={\mathrm{\Gamma }}_{{\rho }^{\prime },{\rho }^{″}}{\displaystyle \frac{s}{M_{{\rho }^{\prime },{\rho }^{″}}^2}}{\displaystyle \frac{{\sigma }_{\pi \pi }^3(s)}{{\sigma }_{\pi \pi }^3(M_{{\rho }^{\prime },{\rho }^{″}}^2)}},

with

(34){\sigma }_{P_1{P}_2}(s)={\displaystyle \frac{2q_{P_1{P}_2}(s)}{\sqrt{s}}}\theta [s-(m_{P_1}+m_{P_2}{)}^2],

where the values of the masses and widths related to the \rho-type resonances will be fitted to the \pi \pi vector form factor measured in the \tau \to \pi \pi {\nu }_{\tau } process [68].

For the energy-dependent widths of the scalar resonances of a_0(980) and a_0(1430), we use the forms suggested in Ref. [47]

(35){\mathrm{\Gamma }}_{a_0}(s)={\mathrm{\Gamma }}_{a_0}(M_{a_0}^2){\left({\displaystyle \frac{s}{M_{a_0}^2}}\right)}^{3/2}{\displaystyle \frac{g_{a_0}(s)}{g_{a_0}(M_{a_0}^2)}},

with

(36)\begin{array}{rl}{g}_{a_0}(s)=& {\sigma }_{K^{-}{K}^0}(s)+2{\left(\frac{c_{\theta }-\sqrt{2}{s}_{\theta }}{\sqrt{3}}\right)}^2{\left(1+\frac{{\mathrm{\Delta }}_{{\pi }^{-}\eta }}{s}\right)}^2{\sigma }_{{\pi }^{-}\eta }(s)\\ & +2{\left(\frac{s_{\theta }+\sqrt{2}{c}_{\theta }}{\sqrt{3}}\right)}^2{\left(1+\frac{{\mathrm{\Delta }}_{{\pi }^{-}{\eta }^{\prime }}}{s}\right)}^2{\sigma }_{{\pi }^{-}{\eta }^{\prime }}(s).\end{array}

Since the current available data in the \tau decays are not able to effectively constrain the masses and widths of the scalar resonances a_0(980) and a_0(1430), we will fix their parameters through reproducing the pole positions given by the Particle Data Group (PDG) [71]. The pole positions in the complex energy plane correspond to the zeros of M_R^2-s-\mathrm{i}{M}_R{\mathrm{\Gamma }}_R(s), appearing in the denominators of the resonance exchange terms in the aforementioned form factors, on a given Riemann sheet (RS). Taking the a_0 resonance as an example for illustration, the expression in Eq. (36) corresponds to the first/physical RS. The second RS is obtained by reversing the sign of the {\sigma }_{{\pi }^{-}\eta }(s) term from lightest threshold {\pi }^{-}\eta, while keeping the other terms unchanged. The third RS is obtained by reversing the signs of both {\sigma }_{{\pi }^{-}\eta }(s) and {\sigma }_{K^{-}{K}^0}(s), leaving {\sigma }_{{\pi }^{-}{\eta }^{\prime }}(s) untouched. The first, second and third RSs can be denoted by the notations of (+,+,+),(-,+,+) and (-,-,+), respectively, where +/- in each entry denotes the sign of {\sigma }_{P_1{P}_2} for the thresholds in an ascending order. For the a_0(980), its mass and width parameters are determined to be M_{a_0}=1.022 GeV and {\mathrm{\Gamma }}_{a_0}(M_{a_0})=0.118 GeV, in order to reproduce the pole position at \sqrt{s_{a_0}}=(0.995-\mathrm{i}0.050) GeV [71] on the third RS. The pole position at \sqrt{s_{a_0^{\prime }}}=(1.395-\mathrm{i}0.085) GeV on the third RS requires M_{a_0^{\prime }}=1.412 GeV and {\mathrm{\Gamma }}_{a_0^{\prime }}(M_{a_0^{\prime }})=0.204 GeV. Such values of the masses’ and widths’ parameters will be exploited in later phenomenological studies.

3.3 Strangeness changing form factors

The decays of {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau } and {\tau }^{-}\to K^{-}(\eta ,{\eta }^{\prime },a){\nu }_{\tau } are governed by the strangeness changing currents, belonging to the Cabibbo suppressed reactions. Following the similar procedure elaborated in the previous subsection, we now need to calculate ⟨P_1{P}_2|\overline{s}{\gamma }_{\mu }u|0⟩ and ⟨P_1{P}_2|(m_s-m_u)\overline{s}u|0⟩ to determine the strangeness changing vector and scalar form factors, respectively.

Following the discussion of Eq. (19), we have

(37)\begin{array}{rl}{\mathrm{\Delta }}_{su}=& B_0(m_s-m_u)\\ =& {\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}\left\{1+{\displaystyle \frac{8(m_K^2+m_{\pi }^2)}{F^2}}\left[{\displaystyle \frac{c_m(c_d-c_m)}{M_S^2}}+\frac{c_m^{\prime }(c_d^{\prime }-c_m^{\prime })}{M_{S^{\prime }}^2}\right]\right\},\end{array}

where {\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}=m_K^2-m_{\pi }^2, being m_K and m_{\pi } the physical isospin-averaged masses. The strangeness changing scalar form factor is then defined as

(38)(m_s-m_u)⟨(KP{)}^{-}|\overline{s}u|0⟩={\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}{F}_0^{(KP{)}^{-}}(s),

with P=\pi ,\eta ,{\eta }^{\prime },a. The product of {\mathrm{\Delta }}_{su}{\hat{F}}_0^{(KP{)}^{-}}(s) in fact equals to {\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}{F}_0^{(KP{)}^{-}}(s), which leads to

(39){\hat{F}}_0^{(KP{)}^{-}}(s)={\displaystyle \frac{{\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{{\mathrm{\Delta }}_{su}}}{F}_0^{(KP{)}^{-}}(s).

Correspondingly, the matrix elements of the strangeness changing vector current in Eq. (3) can be recast as

(40)\begin{array}{rl}⟨(KP{)}^{-}|\overline{s}{\gamma }^{\mu }u|0⟩=& \left[(p_P-p_K{)}^{\mu }-\frac{{\mathrm{\Delta }}_{PK}}{s}{q}^{\mu }\right]F_{+}^{(KP{)}^{-}}(s)\\ & +\frac{{\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{s}{q}^{\mu }{F}_0^{(KP{)}^{-}}(s).\end{array}

The normalization conditions for the strangeness changing vector and scalar form factors read

(41)F_{+}^{(KP{)}^{-}}(0)={\displaystyle \frac{{\mathrm{\Delta }}_{su}^{\mathrm{P}\mathrm{h}\mathrm{y}}}{{\mathrm{\Delta }}_{PK}}}{F}_0^{(KP{)}^{-}}(0),

which guarantees the finiteness of the matrix elements in Eq. (40) at s=0.

Before presenting the vector and scalar form factors, it should be noted that for the {\tau }^{-}\to [K_S{\pi }^{-},K^{-}\eta /{\eta }^{\prime }]{\nu }_{\tau } channels, we disregard the isospin breaking contributions, as their effects are tiny. However, for the {\tau }^{-}\to K^{-}a{\nu }_{\tau } process, the isospin breaking is found to give noticeable effect, due to the relatively large contribution via the {\pi }^0-a mixing, i.e., the isospin-breaking term v_{41} in Ref. [32], which is basically proportional to the factor (m_d-m_u)/(m_d+m_u) when taking the vanishing axion mass. Thus, we retain the linear isospin-breaking terms as provided in Ref. [32] for the vector and scalar form factors in the {\tau }^{-}\to K^{-}a{\nu }_{\tau } decay. Similar to the case of strangeness conserving channels in previous subsection, we include three sets of vector resonances, i.e., K^{\ast }(892),K^{\ast }(1410) and K^{\ast }(1680), which are simply denoted by K^{\ast },K^{{\ast }^{\prime }} and K^{{\ast }^{″}} in order, and the corresponding vector form factors are

(42)F_{+}^{K_S{\pi }^{-}}(s)=-{\displaystyle \frac{1}{\sqrt{2}F{F}_K}}{G}_{\mathrm{L}\mathrm{O}+{\mathrm{K}}^{\ast }\mathrm{E}\mathrm{x}}(s),

(43)F_{+}^{K^{-}\eta }(s)=-\sqrt{{\displaystyle \frac{3}{2}}}{\displaystyle \frac{c_{\theta }}{F{F}_K}}{G}_{\mathrm{L}\mathrm{O}+{\mathrm{K}}^{\ast }\mathrm{E}\mathrm{x}}(s)+\sqrt{{\displaystyle \frac{3}{2}}}{s}_{\theta }{y}_{23}^{(0)},

(44)F_{+}^{K^{-}{\eta }^{\prime }}(s)=-\sqrt{{\displaystyle \frac{3}{2}}}{\displaystyle \frac{s_{\theta }}{F{F}_K}}{G}_{\mathrm{L}\mathrm{O}+{\mathrm{K}}^{\ast }\mathrm{E}\mathrm{x}}(s)-\sqrt{{\displaystyle \frac{3}{2}}}{c}_{\theta }{y}_{23}^{(0)},

(45)\begin{array}{rl}{F}_{+}^{K^{-}a}(s)=& -\frac{1}{\sqrt{2}F{F}_K}{G}_{\mathrm{L}\mathrm{O}+{\mathrm{K}}^{\ast }\mathrm{E}\mathrm{x}}(s)(v_{41}+\sqrt{3}{c}_{\theta }{v}_{42}+\sqrt{3}{s}_{\theta }{v}_{43})\\ & -\frac{1}{\sqrt{6}}[\sqrt{3}{y}_{14}+\sqrt{3}{v}_{12}{y}_{24}^{(0)}+\sqrt{3}{v}_{13}{y}_{34}^{(0)}+3c_{\theta }(y_{24}+v_{23}{y}_{34}^{(0)})+3s_{\theta }(y_{34}-v_{23}{y}_{24}^{(0)})],\end{array}

with

(46)G_{\mathrm{L}\mathrm{O}+{\mathrm{K}}^{\ast }\mathrm{E}\mathrm{x}}(s)={\displaystyle \frac{G_V{F}_{V}s+F{F}_K(M_{K^{\ast }}^2-s)}{M_{K^{\ast }}^2-s-\mathrm{i}{M}_{K^{\ast }}{\mathrm{\Gamma }}_{K^{\ast }}(s)}}+{\displaystyle \frac{G_V^{\prime }{F}_V^{\prime }s}{M_{K^{{\ast }^{\prime }}}^2-s-\mathrm{i}{M}_{K^{{\ast }^{\prime }}}{\mathrm{\Gamma }}_{K^{{\ast }^{\prime }}}(s)}}+{\displaystyle \frac{G_V^{″}{F}_V^{″}s}{M_{K^{{\ast }^{″}}}^2-s-\mathrm{i}{M}_{K^{{\ast }^{″}}}{\mathrm{\Gamma }}_{K^{{\ast }^{″}}}(s)}}.

It is pointed out that we have replaced one of the F with F_K for the resonance exchange contributions to the form factors entering the {\tau }^{-}\to [K_S{\pi }^{-},K^{-}\eta /{\eta }^{\prime }/a]{\nu }_{\tau } decays. Although formally such a replacement only has influences beyond NLO accuracy, practically it can lead to noticeable effects. This is because the resonance exchanges play significant roles in the \tau decays and the replacement of F (estimated by F_{\pi }) by F_K can cause around 20% changes of the resonance contributions.

Two sets of scalar resonances, i.e., K_0^{\ast }(700) (labeled as K_0^{\ast }) and K_0^{\ast }(1430) (labeled as K_0^{\ast \mathrm{\prime }}), will be included in the scalar form factors, whose explicit expressions are given by

(47)\begin{array}{rl}& \\ & F_0^{K_S{\pi }^{-}}(s)=\frac{1}{\sqrt{2}}+\frac{2\sqrt{2}}{F{F}_K}\{[\frac{c_m^2(m_K^2+m_{\pi }^2)+c_m{c}_d(s-m_{\pi }^2-m_K^2)}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}-\frac{c_m(c_m-c_d)(m_K^2+m_{\pi }^2)}{M_S^2}]\\ & +[c_{m,d},M_{K_0^{\ast }},{\mathrm{\Gamma }}_{K_0^{\ast }},M_S\to c_{m,d}^{\prime },M_{K_0^{{\ast }^{\prime }}},{\mathrm{\Gamma }}_{K_0^{{\ast }^{\prime }}},M_{S^{\mathrm{\prime }}}]\},\end{array}

(48)\begin{array}{rl}{F}_0^{K^{-}\eta }(s)=& -\frac{1}{\sqrt{6}}(c_{\theta }+2\sqrt{2}{s}_{\theta })+\frac{1}{\sqrt{3}}({\mathrm{\Lambda }}_1-2{\mathrm{\Lambda }}_2)s_{\theta }-\frac{1}{\sqrt{6}}(2\sqrt{2}{c}_{\theta }-s_{\theta })y_{23}^{(0)}\\ & -\frac{1}{F{F}_K}\{[\frac{4c_{\theta }}{\sqrt{6}}\frac{c_m{c}_d(s-m_K^2-m_{\eta }^2)+c_m^2(5m_K^2-3m_{\pi }^2)}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}\\ & +\frac{8s_{\theta }}{\sqrt{3}}\frac{c_m{c}_d(s-m_K^2-m_{\eta }^2)+2c_m^2{m}_K^2}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}+\frac{4c_{\theta }}{\sqrt{6}}\frac{c_m(c_m-c_d)}{M_S^2}(3m_K^2-5m_{\pi }^2)\\ & -\frac{8s_{\theta }}{\sqrt{3}}\frac{2c_m(c_m-c_d)m_{\pi }^2}{M_S^2}]+[c_{m,d},M_{K_0^{\ast }},{\mathrm{\Gamma }}_{K_0^{\ast }},M_S\to c_{m,d}^{\prime },M_{K_0^{{\ast }^{\prime }}},{\mathrm{\Gamma }}_{K_0^{{\ast }^{\prime }}},M_{S^{\prime }}]\},\end{array}

(49)\begin{array}{rl}{F}_0^{K^{-}{\eta }^{\prime }}(s)=& \frac{1}{\sqrt{6}}(2\sqrt{2}{c}_{\theta }-s_{\theta })-\frac{1}{\sqrt{3}}({\mathrm{\Lambda }}_1-2{\mathrm{\Lambda }}_2)c_{\theta }-\frac{1}{\sqrt{6}}(2\sqrt{2}{s}_{\theta }+c_{\theta })y_{23}^{(0)}\\ & +\frac{1}{F{F}_K}\{[\frac{8c_{\theta }}{\sqrt{3}}\frac{c_m{c}_d(s-m_K^2-m_{{\eta }^{\prime }}^2)+2c_m^2{m}_K^2}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}\\ & -\frac{4s_{\theta }}{\sqrt{6}}\frac{c_m{c}_d(s-m_K^2-m_{{\eta }^{\prime }}^2)+c_m^2(5m_K^2-3m_{\pi }^2)}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}-\frac{8c_{\theta }}{\sqrt{3}}\frac{2c_m(c_m-c_d)m_{\pi }^2}{M_S^2}\\ & -\frac{4s_{\theta }}{\sqrt{6}}\frac{c_m(c_m-c_d)}{M_S^2}(3m_K^2-5m_{\pi }^2)]+[c_{m,d},M_{K_0^{\ast }},{\mathrm{\Gamma }}_{K_0^{\ast }},M_S\to c_{m,d}^{\prime },M_{K_0^{{\ast }^{\prime }}},{\mathrm{\Gamma }}_{K_0^{{\ast }^{\prime }}},M_{S^{\prime }}]\},\end{array}

(50)\begin{array}{rl}{F}_0^{K^{-}a}(s)=& \frac{v_{41}}{\sqrt{2}}-v_{42}(\frac{c_{\theta }}{\sqrt{6}}+\frac{2s_{\theta }}{\sqrt{3}})-v_{43}(-\frac{2c_{\theta }}{\sqrt{3}}+\frac{s_{\theta }}{\sqrt{6}})-(\frac{\sqrt{2}F}{3f_a}+v_{42}\frac{2s_{\theta }}{\sqrt{3}}-v_{43}\frac{2c_{\theta }}{\sqrt{3}})({\mathrm{\Lambda }}_2-\frac{1}{2}{\mathrm{\Lambda }}_1)\\ & -\frac{1}{6}[-3\sqrt{2}{y}_{14}+4\sqrt{3}{s}_{\theta }{y}_{24}-3\sqrt{2}{v}_{12}{y}_{24}^{(0)}-\sqrt{6}{s}_{\theta }{v}_{23}{y}_{24}^{(0)}+\sqrt{6}{s}_{\theta }{y}_{34}\\ & -3\sqrt{2}{v}_{13}{y}_{34}^{(0)}+4\sqrt{3}{s}_{\theta }{v}_{23}{y}_{34}^{(0)}+\sqrt{3}{c}_{\theta }(\sqrt{2}{y}_{24}+4v_{23}{y}_{24}^{(0)}-4y_{34}+\sqrt{2}{v}_{23}{y}_{34}^{(0)})]\\ & +\frac{1}{F{F}_K}\{[\frac{4v_{41}}{\sqrt{2}}\frac{c_m^2(m_K^2+m_{\pi }^2)+c_m{c}_d(s-m_a^2-m_K^2)}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}-\frac{4v_{41}}{\sqrt{2}}\frac{c_m(c_m-c_d)(m_K^2+m_{\pi }^2)}{M_S^2}\\ & -\frac{2(\sqrt{2}{c}_{\theta }+4s_{\theta })v_{42}+2(-4c_{\theta }+\sqrt{2}{s}_{\theta })v_{43}}{\sqrt{3}}\frac{c_m{c}_d(s-m_a^2-m_K^2+{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}})}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}\\ & +\frac{16}{\sqrt{3}}\frac{(-s_{\theta }{v}_{42}+c_{\theta }{v}_{43})c_m^2(m_K^2-{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}})}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}-2\sqrt{\frac{2}{3}}\frac{(s_{\theta }{v}_{43}+c_{\theta }{v}_{42})c_m^2(5m_K^2-3m_{\pi }^2-2{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}})}{M_{K_0^{\ast }}^2-s-\mathrm{i}{M}_{K_0^{\ast }}{\mathrm{\Gamma }}_{K_0^{\ast }}(s)}\\ & -\frac{4(c_{\theta }{v}_{42}+s_{\theta }{v}_{43})c_m(c_m-c_d)(3m_K^2-5m_{\pi }^2-2{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}})}{\sqrt{6}{M}_S^2}\\ & +\frac{8(s_{\theta }{v}_{42}-c_{\theta }{v}_{43})c_m(c_m-c_d)(2m_{\pi }^2+2{\mathrm{\Delta }}_{du}^{\mathrm{P}\mathrm{h}\mathrm{y}})}{\sqrt{3}{M}_S^2}]\\ & +[c_{m,d},M_{K_0^{\ast }},{\mathrm{\Gamma }}_{K_0^{\ast }},M_S\to c_{m,d}^{\prime },M_{K_0^{{\ast }^{\prime }}},{\mathrm{\Gamma }}_{K_0^{{\ast }^{\prime }}},M_{S^{\mathrm{\prime }}}]\}.\end{array}

According to Ref. [48], the expressions of the energy-dependent widths for three vector resonances K^{\ast },K^{{\ast }^{\prime }} and K^{{\ast }^{″}} in the strangeness changing form factors are taken as

(51){\mathrm{\Gamma }}_{K^{\ast }}(s)={\mathrm{\Gamma }}_{K^{\ast }}{\displaystyle \frac{s}{M_{K^{\ast }}^2}}{\displaystyle \frac{{\sigma }_{K\pi }^3(s)+c_{\theta }^2{\sigma }_{K\eta }^3(s)+s_{\theta }^2{\sigma }_{K{\eta }^{\prime }}^3(s)}{{\sigma }_{K\pi }^3(M_{K^{\ast }}^2)}},

(52){\mathrm{\Gamma }}_{K^{{\ast }^{\prime }{(}^{\prime })}}(s)={\mathrm{\Gamma }}_{K^{{\ast }^{\prime }{(}^{\prime })}}{\displaystyle \frac{s}{M_{K^{{\ast }^{\prime }{(}^{\prime })}}^2}}{\displaystyle \frac{{\sigma }_{K\pi }^3(s)}{{\sigma }_{K\pi }^3(M_{K^{{\ast }^{\prime }{(}^{\prime })}}^2)}}.

The masses’ and widths’ parameters for the vector strange resonances in the above equations will be fitted to the experimental data from the \tau \to K_S\pi {\nu }_{\tau } and \tau \to K\eta {\nu }_{\tau } processes in Refs. [72, 73]. And for the K_0^{\ast }(700) and K_0^{\ast }(1430) scalar resonances, their energy-dependent widths read [48]

(53)\begin{array}{rl}& {\mathrm{\Gamma }}_S(s)={\mathrm{\Gamma }}_S(M_{K_0^{\ast }}^2){\left(\frac{s}{M_S^2}\right)}^n\frac{g_{K_0^{\ast }}(s)}{g_{K_0^{\ast }}(M_{K_0^{\ast }}^2)},\\ & [n=0:K_0^{\ast }(700)\mathrm{a}\mathrm{n}\mathrm{d}n=\frac{3}{2}:K_0^{\ast }(1430)],\end{array}

with

(54)\begin{array}{rl}{g}_{K_0^{\ast }}(s)=& \frac{3}{2}{\sigma }_{K\pi }(s)+\frac{1}{6}{\sigma }_{K\eta }(s)[c_{\theta }(1+\frac{3{\mathrm{\Delta }}_{K\pi }+{\mathrm{\Delta }}_{K\eta }}{s})\\ & +2\sqrt{2}{s}_{\theta }\left(1+\frac{{\mathrm{\Delta }}_{K\eta }}{s}\right){]}^2\\ & +\frac{4}{3}{\sigma }_{K{\eta }^{\prime }}(s)[c_{\theta }\left(1+\frac{{\mathrm{\Delta }}_{K{\eta }^{\prime }}}{s}\right)\\ & -\frac{s_{\theta }}{2\sqrt{2}}\left(1+\frac{3{\mathrm{\Delta }}_{K\pi }+{\mathrm{\Delta }}_{K{\eta }^{\prime }}}{s}\right){]}^2.\end{array}

Similar to the discussions at the end of the previous subsection, we fix the parameters of the masses and widths for K_0^{\ast } and K_0^{{\ast }^{\prime }} appearing in the form factors by requiring their pole positions to be consistent the values given by PDG [71]. The results turn out to be M_{K_0^{\ast }}=0.771 GeV and {\mathrm{\Gamma }}_{K_0^{\ast }}(M_{K_0^{\ast }})=0.387 GeV when the K_0^{\ast } pole is located at \sqrt{s_{K_0^{\ast }}}=(0.680-\mathrm{i}0.300) GeV on the second RS. Similarly, we obtain M_{K_0^{{\ast }^{\prime }}}=1.454 GeV and {\mathrm{\Gamma }}_{K_0^{{\ast }^{\prime }}}(M_{K_0^{{\ast }^{\prime }}})=0.233 GeV to reproduce the K_0^{{\ast }^{\prime }} pole at \sqrt{s_{K_0^{{\ast }^{\prime }}}}=(1.431-\mathrm{i}0.110) GeV on the third RS.

4 Combined fit to the {\mathit{\tau }}^{\mathbf{-}}\mathbf{\to }\mathbf{(}{\mathit{\pi }}^{\mathbf{-}}{\mathit{\pi }}^{\mathbf{0}}\mathbf{,}{\mathit{K}}_{\mathit{S}}{\mathit{\pi }}^{\mathbf{-}}\mathbf{,}{\mathit{K}}^{\mathbf{-}}\mathit{\eta }\mathbf{)}{\mathit{\nu }}_{\mathit{\tau }} decays

In this section, we perform the combined fit to three kinds of experimental data, including the pion vector form factors F_{+}^{{\pi }^{-}{\pi }^0} measured in the {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau } decay in Ref. [68], and the invariant-mass distributions of K_S{\pi }^{-} [72] and K^{-}\eta [73] in the {\tau }^{-}\to {\nu }_{\tau }(K_S{\pi }^{-},K^{-}\eta ) processes.

We use the standard {\chi }^2 function in the fit to incorporate the modulus squared of the normalized pion vector form factor

(55){\chi }_1^2={\displaystyle \sum _i{\left({\displaystyle \frac{|{\tilde{F}}_{+}^{{\pi }^{-}{\pi }^0}(s_i){|}_{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}}^2-|{\tilde{F}}_{+}^{{\pi }^{-}{\pi }^0}(s_i){|}_{\mathrm{E}\mathrm{x}\mathrm{p}}^2}{{\sigma }_{|{\tilde{F}}_{+}^{{\pi }^{-}{\pi }^0}(s_i){|}_{\mathrm{E}\mathrm{x}\mathrm{p}}^2}}}\right)}^2,}

where the normalized form factor is given by {\tilde{F}}_{+}^{{\pi }^{-}{\pi }^0}(s)=F_{+}^{{\pi }^{-}{\pi }^0}(s)/\sqrt{2} and 62 data points are taken from Ref. [68]. For the two-meson invariant-mass spectra, we use the following formula to perform the fit of the normalized experimental event distributions

(56){\mathcal{N}}_{P_1{P}_2}(E)\equiv {\displaystyle \frac{\mathrm{d}{N}_{\mathrm{E}\mathrm{v}\mathrm{e}}^{\mathrm{b}\mathrm{i}\mathrm{n}}}{N_{\mathrm{E}\mathrm{v}\mathrm{e}}^{\mathrm{T}\mathrm{o}\mathrm{t}}\mathrm{d}E}}={\displaystyle \frac{\mathrm{d}\mathrm{\Gamma }(\tau \to P_1{P}_2{\nu }_{\tau })}{\mathrm{d}E}}{\displaystyle \frac{1}{{\mathrm{\Gamma }}_{\tau }{\overline{B}}_{\tau \to P_1{P}_2{\nu }_{\tau }}}},

where E=\sqrt{s}, N_{\mathrm{E}\mathrm{v}\mathrm{e}}^{\mathrm{b}\mathrm{i}\mathrm{n}} stands for the observed number of events in each bin, N_{\mathrm{E}\mathrm{v}\mathrm{e}}^{\mathrm{T}\mathrm{o}\mathrm{t}} is the total number of events, and {\mathrm{\Gamma }}_{\tau } corresponds to total decay width or the inverse of lifetime for \tau. Energy bin sizes of 11.5 MeV and 25 MeV are used for {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau } [72] and {\tau }^{-}\to K^{-}\eta {\nu }_{\tau }[73], respectively, in the experimental study. For {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau }, the analysis incorporates experimental data up to bin 90 (\sqrt{s} = 1.65925 \mathrm{G}\mathrm{e}\mathrm{V}), consistent with prior analyses [74-76]. Similar as the latter references, data points corresponding to bins 5, 6, and 7 are also excluded from the {\chi }^2 minimization due to their obvious incompatibility with theoretical expectations. It is verified that the inclusion of these three data points yields a substantially increased {\chi }^2 while inducing negligible changes in the fitted parameters. For {\tau }^{-}\to K^{-}\eta {\nu }_{\tau }, the first two data points from Belle [73] are excluded as they lie significantly below the K^{-}\eta production threshold, and data points significantly exceeding the \tau mass are also excluded. The quantity of {\overline{B}}_{\tau \to P_1{P}_2{\nu }_{\tau }} in Eq. (56) is introduced as a normalization factor in our fit, which would be identical to the branching ratio of the \tau \to P_1{P}_2{\nu }_{\tau } channel for the ideal description of the experimental event distributions. The Belle analysis in Ref. [72] for the {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau } decay channel yields a branching ratio of B_{K_S{\pi }^{-}}^{\mathrm{E}\mathrm{x}\mathrm{p}}=(4.04\pm 0.13)\times {10}^{-3}. Regarding the {\tau }^{-}\to K^{-}\eta {\nu }_{\tau } process, Belle Collaboration reports a branching fraction of B_{K^{-}\eta }^{\mathrm{E}\mathrm{x}\mathrm{p}}=(1.58\pm 0.10)\times {10}^{-4} [73]. To circumvent the imperfect mismatch of the normalization between theoretical and experimental event distributions, we also fit the {\overline{B}}_{\tau \to P_1{P}_2{\nu }_{\tau }} quantities in Eq. (56) to the aforementioned branching ratios from Belle. Thus, the {\chi }^2 function minimized in our fit for the event distributions takes the form

(57)\begin{array}{rl}{\chi }_2^2=& \sum _{P_1{P}_2=\begin{array}{c}{K}_S{\pi }^{-},\\ K^{-}\eta \end{array}}\underset{i}{{\sum }^{\prime }}{\left[\frac{{\mathcal{N}}_{P_1{P}_2}^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}}(E_i)-{\mathcal{N}}_{P_1{P}_2}^{\mathrm{E}\mathrm{x}\mathrm{p}}(E_i)}{{\sigma }_{\mathcal{N},P_1{P}_2}^{\mathrm{E}\mathrm{x}\mathrm{p}}(E_i)}\right]}^2\\ & +\sum _{P_1{P}_2=\begin{array}{c}{K}_S{\pi }^{-},\\ K^{-}\eta \end{array}}{\left(\frac{{\overline{B}}_{P_1{P}_2}^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}}-B_{P_1{P}_2}^{\mathrm{E}\mathrm{x}\mathrm{p}}}{{\sigma }_{B_{P_1{P}_2}}^{\mathrm{E}\mathrm{x}\mathrm{p}}}\right)}^2,\end{array}

where {\sigma }^{\mathrm{E}\mathrm{x}\mathrm{p}} stand for the corresponding experimental uncertainties. The prime symbol in the summation notation denotes the exclusion of specific points from the minimization process as mentioned previously. The number of fitted data points amounts to 87 for the K_S{\pi }^{-} spectrum and 31 for the K^{-}\eta spectrum, each accompanied by their corresponding branching fractions. The total {\chi }^2 function minimized in the fit is given by the sum of those from the three channels, i.e., {\chi }^2={\chi }_1^2+{\chi }_2^2, with {\chi }_1^2 and {\chi }_2^2 in Eqs. (55) and (57), respectively.

Next we specify the values of the parameters used in the fit. The isospin-averaged pion and kaon masses are taken as m_{\pi }=0.137\mathrm{G}\mathrm{e}\mathrm{V} and m_K=0.495\mathrm{G}\mathrm{e}\mathrm{V}. Other relevant parameter values are taken from PDG [71]: m_{\eta }=0.547\mathrm{G}\mathrm{e}\mathrm{V},m_{{\eta }^{\prime }}=0.957\mathrm{G}\mathrm{e}\mathrm{V}, G_F=1.16637\times {10}^{-5}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-2}, {\mathrm{\Gamma }}_{\tau }=2.265\times {10}^{-12}\mathrm{G}\mathrm{e}\mathrm{V}, m_{\tau }=1.7769\mathrm{G}\mathrm{e}\mathrm{V}, V_{us}=0.2243,V_{ud}=0.9737,F_K=0.110\mathrm{G}\mathrm{e}\mathrm{V}. The value of the chiral-limit quantity F will be estimated by the pion decay constant F_{\pi }=0.0923\mathrm{G}\mathrm{e}\mathrm{V}. For the LO mass M_0 of the singlet {\eta }_0 in Eq. (10), we take M_0=0.820\mathrm{G}\mathrm{e}\mathrm{V} from Ref. [31], which leads to the LO \eta-{\eta }^{\prime } mixing angle \theta =-{19.6}^{\circ }. The values of the low-energy coupling constants {\mathrm{\Lambda }}_1 and {\mathrm{\Lambda }}_2 in Eq. (17) are fixed as {\mathrm{\Lambda }}_1=-0.17, and {\mathrm{\Lambda }}_2=0.06, according to Ref. [31]. The masses for the scalar resonances in the chiral limit are estimated by M_S=1.0 GeV and M_{S^{\prime }}=1.4 GeV. Concerning the scalar resonance couplings, we use the values of {\tilde{c}}_{d,m} in Ref. [77] to fix c_m=\sqrt{3}{\tilde{c}}_m=0.027 GeV and c_d=\sqrt{3}{\tilde{c}}_d=0.015 GeV, and further exploit the relationship of c_m{c}_d+c_m^{\prime }{c}_d^{\prime }=\frac{F^2}{4}, dictated by the high energy behavior of the scalar form factors [59]. Thus, among the four scalar resonance couplings c_m,c_d,c_m^{\prime } and c_d^{\prime }, only one of them is a free parameter. For definiteness, we choose to fit c_m^{\prime } in the following discussion.

Furthermore, we find that the K^{{\ast }^{″}} resonance has little impact on our fit. Thus, we fix its mass and width to be M_{K^{{\ast }^{″}}}=1.718\mathrm{G}\mathrm{e}\mathrm{V} and {\mathrm{\Gamma }}_{K^{{\ast }^{″}}}=0.320\mathrm{G}\mathrm{e}\mathrm{V}, according to PDG [71]. When allowing the width parameter of the K^{{\ast }^{\prime }} resonance to vary freely in the fit, we observe that it easily floats to a rather large value. However, fixing this parameter to its PDG value at {\mathrm{\Gamma }}_{K^{{\ast }^{\prime }}}=0.232\mathrm{G}\mathrm{e}\mathrm{V} in the fit turns out to barely change the {\chi }^2. Therefore we opt to fix this width parameter in our analysis.

In total, we have 14 free parameters and their values resulting from the joint fit, together with error bars, are presented in Table 1. The fitted values of M_{\rho }, M_{K^{\ast }} and {\mathrm{\Gamma }}_{K^{\ast }} are consistent with the PDG averages [71], while the fitted M_{K^{{\ast }^{\prime }}} with (1.339\pm 0.009) GeV is smaller than the PDG average with (1.414\pm 0.015) GeV. It is pointed out that similarly smaller values for M_{K^{{\ast }^{\prime }}} are also obtained in the \tau decays in Refs. [48, 76]. We have further tried other types of fits by fixing the parameters of M_{\rho }, M_{K^{\ast }} and {\mathrm{\Gamma }}_{K^{\ast }} at their PDG averages, which turn out to marginally affect the results presented in Table 1. In contrast, the tentative fit by fixing M_{K^{{\ast }^{\prime }}} at its PDG average leads to obviously increased {\chi }^2. The parameters of {\rho }^{\prime } are compatible with the PDG averages within uncertainties. While, the resulting mass of {\rho }^{″} seems larger than the PDG average and its width is consistent with PDG. Since the {\rho }^{″} resides in the boundary of the kinematically allowed region of \tau decays, it plays a marginal role. In the following, we will take the values of parameters in Table 1 to proceed the phenomenological discussions. We use bootstrap method to estimate the uncertainties of the fitted parameters. Namely, by taking Gaussian sampling of the experimental data, large random pseudo-data sets are generated, which are then used to redo the fits. The large samples of the refitted parameters are exploited to perform all the remaining uncertainty analyses, including the error bars of the parameters, uncertainty bands of the various curves and the predictions of branching ratios that will be addressed later.

The curves and the error bands derived from the parameters in Table 1, together with the experimental data, are shown in Figs. 3 and 4, where black solid circles denote experimental data points used in our fit. In Fig. 3, the blue curve represents the modulus squared of normalized vector form factor {\tilde{F}}_{+}^{{\pi }^{-}{\pi }^0}(s)=F_{+}^{{\pi }^{-}{\pi }^0}(s)/\sqrt{2}, while in Fig. 4, the blue lines depict the total event yield distributions and black dotted lines illustrate the contributions from scalar form factors.

At the end of this section, we present the predictions to the branching ratios by taking our theoretical form factors as inputs for the {\tau }^{-}\to ({\pi }^{-}{\pi }^0,K_S{\pi }^{-},K^{-}\eta ){\nu }_{\tau } processes in Table 2, along with the separate contributions from the vector and scalar form factors. Such theoretical predictions to the branching ratios based on form factors are found to be compatible with the fitted normalization factors in Table 1 for the K_S{\pi }^{-} and K^{-}\eta channels, which reflects the self consistency of our theoretical framework. The theoretical branching ratio for the {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau } channel is also nicely consistent with the result of (25.24\pm 0.39)\mathrm{\%} from Belle [68].

5 Predictions to the spectra in other channels and the forward-backward asymmetries

After fixing all the relevant resonance parameters through the fit elaborated in the previous section, we are ready to present predictions to the differential decay widths for other decay channels, such as {\tau }^{-}\to ({\pi }^{-}\eta ,{\pi }^{-}{\eta }^{\prime },K^{-}{\eta }^{\prime },{\pi }^{-}a,K^{-}a){\nu }_{\tau }, that are not measured yet by experiments. The separate contributions from the vector and scalar form factors are also analyzed for different observables. The differential decay widths with respect to the two-boson energies are shown in Fig. 5 for {\pi }^{-}\eta and {\pi }^{-}{\eta }^{\prime }, Fig. 6 for K^{-}{\eta }^{\prime } and Fig. 7 for {\pi }^{-}a and K^{-}a. In the latter figure, we have taken the QCD axion scenario for illustration by setting the bare axion mass m_{a,0}=0, and in this case m_a can be set as zero as a perfect approximation in the phenomenological studies [31, 32].

According to Figs. 5-7, it is obvious that the spectra for the {\pi }^{-}\eta and {\pi }^{-}/K^{-}a channels are primarily governed by the vector contributions associated with the \rho (770) and \rho (770)/K^{\ast }(892) resonances, respectively. In contrast, the spectra of the {\pi }^{-}{\eta }^{\prime } and K^{-}{\eta }^{\prime } decay channels are clearly dominated by the scalar form factors. In particular, for the {\pi }^{-}{\eta }^{\prime } and K^{-}{\eta }^{\prime } spectra, prominent peaks can be easily identified around the energies of 1.4-1.5 GeV, which correspond to the scalar resonances a_0(1450) and K_0^{\ast }(1430), respectively. The a_0(980) manifests as a noticeable kink structure in the {\pi }^{-}\eta invariant-mass distribution from the {\tau }^{-}\to {\pi }^{-}\eta {\nu }_{\tau } decay. It is interesting to explain the somewhat large uncertainties in the \tau \to \pi {\eta }^{\prime }{\nu }_{\tau } and \tau \to K{\eta }^{\prime }{\nu }_{\tau } processes and small error bands for other channels. The reason behind is that the free parameters are fitted to the three types of data sets from the \tau \to \pi \pi {\nu }_{\tau }, \tau \to K_S\pi {\nu }_{\tau } and \tau \to K\eta {\nu }_{\tau } processes, which are all dominated by the vector resonances. The scalar resonances do not enter \tau \to \pi \pi {\nu }_{\tau } and play marginal roles in the \tau \to K_S\pi {\nu }_{\tau } and \tau \to K\eta {\nu }_{\tau } decays. As a result, the vector resonance parameters are strongly constrained by the data, but the scalar resonance coupling is somewhat loosely constrained in the fit. Consequently, the \tau \to \pi {\eta }^{\prime }{\nu }_{\tau } and \tau \to K{\eta }^{\prime }{\nu }_{\tau } processes that are dominated by the contributions from the scalar resonances bear large uncertainties. While, for other channels that are dominated by the vector resonances, the corresponding uncertainties are small.

Additionally, the predicted branching ratios for the five decay channels are tabulated in Table 3, where the separate contributions of the vector and scalar form factors to these ratios are given as well. Several recent experimental studies have reported upper limits for some of the decay channels, which are also summarized in Table 3 for easy comparisons. For the decay mode {\tau }^{-}\to {\pi }^{-}\eta {\nu }_{\tau }, the BaBar and Belle collaborations presented upper limits of 9.9\times {10}^{-5} [78] at 95\mathrm{\%} confident level (C.L.) and 7.3\times {10}^{-5} [79] at 90\mathrm{\%} C.L., respectively. Our theoretical model predicts the branching ratio of this channel to be {1.63}_{-0.14}^{+0.14}\times {10}^{-5}, which is in good agreement with the existing experimental upper limits. For the {\tau }^{-}\to {\pi }^{-}{\eta }^{\prime }{\nu }_{\tau } decay mode, the BaBar collaboration has also made some progress, updating the upper limit to be 4.0\times {10}^{-6} [80] at 90\mathrm{\%} C.L., which outperforms its previous result of 7.2\times {10}^{-6} [81] at 90\mathrm{\%} C.L. Our predicted branching ratio for this channel is {1.17}_{-0.07}^{+0.36}\times {10}^{-7}, which is much lower than the current experimental limit. As for the {\tau }^{-}\to K^{-}{\eta }^{\prime }{\nu }_{\tau } decay mode, the BaBar collaboration [80] has provided a measurement, determining an upper branching ratio limit as 2.4\times {10}^{-6} at 90\mathrm{\%} C.L. Our theoretical prediction for this channel is about {2.00}_{-0.70}^{+1.01}\times {10}^{-6}, which is around the edge of the experimental limit. Regarding the axion production rates, we verify that the hadronic resonance contributions give significant enhancements, compared to the case by only including the LO chiral results. Taking the situation of m_a=0 for illustration, the enhanced ratio of the \tau \to \pi a{\nu }_{\tau } decay widths by using the complete hadronic amplitude and the LO chiral amplitude is found to be 7.5, while such ratio in the \tau \to Ka{\nu }_{\tau } channel is enlarged to be 19.2.

Apart from the QCD axion case, we further explore the phenomenological consequences by introducing the bare mass term m_{a,0} for the axion, which is usually deemed as the axion-like particle (ALP) scenario. In this situation, the ALP mass m_a is predominated by the bare mass m_{a_{,}0}. We present the differential decay widths of {\tau }^{-}\to {\pi }^{-}/K^{-}a{\nu }_{\tau } channels by taking m_a=0.1 and 0.3 GeV in Fig. 8, where the curves corresponding to m_a=0 are also included for comparison. By continuously varying the ALP masses from 0 to 1 GeV, we give predictions to the branching ratios of the {\tau }^{-}\to {\pi }^{-}/K^{-}a{\nu }_{\tau } processes as a function of m_a in Fig. 9. The singularities in the latter figure emerge at specific mass thresholds, i.e., when m_a\sim m_{\overline{\pi }},m_{\overline{\eta }},m_{{\overline{\eta }}^{\prime }}, due to the fact that the axion-meson mixing matrix elements v_{4i/i4} and y_{i4} (with i=1,2,3) are proportional to \frac{1}{m_a^2-m_i^2}, with m_i=m_{\overline{\pi }},m_{\overline{\eta }},m_{{\overline{\eta }}^{\prime }}, according to Refs. [31, 32]. It is pointed out that we have taken the same LO isospin-limit masses in the denominators of both v_{4i/i4} and y_{i4} here, while the isospin breaking corrected masses are used for y_{i4} in Ref. [32]. The seemingly magnificent enhancements around m_a=m_{\overline{\pi }},m_{\overline{\eta }},m_{{\overline{\eta }}^{\prime }} in Fig. 9 are expected to be artificial, merely reflecting the deficiency of the perturbative treatment of the axion-meson mixing around these regions, and therefore are shaded in the plots. As a result, only the predictions distant from the regions of m_a=m_{\overline{\pi }},m_{\overline{\eta }},m_{{\overline{\eta }}^{\prime }} are considered to be meaningful.

It should be noted that our study presents a complementary calculation of the ALP production rates in the tau decays to the work of Ref. [55], since the latter reference only incorporates the ALP-lepton coupling, while ours assumes that the predominant mechanism in the \tau \to \pi (K)a{\nu }_{\tau } processes is the model-independent aG\tilde{G} interaction. According to the branching ratios of the ALP production from the \tau decays illustrated in Fig. 9, although it could be difficult to observe such \tau decay processes with tiny absolute branching ratios in the current experiments, our study reveals an alternative interesting feature that the ratios of \frac{{\mathrm{\Gamma }}_{\tau \to Ka{\nu }_{\tau }}/V_{us}^2}{{\mathrm{\Gamma }}_{\tau \to \pi a{\nu }_{\tau }}/V_{ud}^2}\simeq({6.3}_{(m_a=0)},{41}_{(m_a=0.3\mathrm{G}\mathrm{e}\mathrm{V})},{2513}_{(m_a=0.8\mathrm{G}\mathrm{e}\mathrm{V})}) are much larger than those of \frac{{\mathrm{\Gamma }}_{\tau \to K{\nu }_{\tau }}/V_{us}^2}{{\mathrm{\Gamma }}_{\tau \to \pi {\nu }_{\tau }}/V_{ud}^2}\simeq 1.2 and \frac{{\mathrm{\Gamma }}_{\tau \to K\pi {\nu }_{\tau }}/V_{us}^2}{{\mathrm{\Gamma }}_{\tau \to \pi \pi {\nu }_{\tau }}/V_{ud}^2}\simeq 0.3. This significant enhancement in the aK system, compared to the \pi a one, is in accord with the recent finding in Ref. [82] that explores the aK\to \pi K and a\pi \to \pi \pi interactions. We expect that such enhanced ratio of \frac{{\mathrm{\Gamma }}_{\tau \to Ka{\nu }_{\tau }}/V_{us}^2}{{\mathrm{\Gamma }}_{\tau \to \pi a{\nu }_{\tau }}/V_{ud}^2} will not persist in the axion-lepton dominant scenario in Ref. [55]. Therefore this ratio could provide a useful quantity to distinguish different theoretical axion models. Interestingly, the authors of Ref. [55] show that their branching ratio of \tau \to \pi a{\nu }_{\tau } is around {10}^{-8}|g_{a\tau \tau }{|}^2 by setting f_a=100 GeV, which is roughly similar to ours when taking the axion-tau coupling g_{a\tau \tau } as O(1). As illustrated in Ref. [55], in order to set realistic constraints on the ALP parameters, one would also need the ALP decay information as inputs to match the experimental conditions, which is beyond the scope of the present study. We believe that such an analysis definitely deserves another independent work in future. It is reiterated that one of the key novelties of this work is to systematically include the hadronic resonance contributions to the ALP production in tau decays with the model-independent interaction of the aG\tilde{G} type.

Last but not least, we present predictions to the forward-backward asymmetries arising from the two-pseudoscalar boson decays of the \tau lepton, viz., A_{FB} defined in Eq. (9). The FB asymmetry A_{FB} provides a valuable quantity to probe the interference term between the vector and scalar form factors, which is absent in the differential decay width with respect to the two-boson energy given in Eq. (8). For the decay channel {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau }, as discussed earlier, the contribution of the scalar form factor to this channel is much suppressed, compared to the vector form factor, and therefore it is neglected in our study, indicating that A_{FB} in this channel is zero. Regarding the FB asymmetries for the other channels, the resulting curves are given in Fig. 10 for the Cabibbo allowed channels, including {\tau }^{-}\to (\eta ,{\eta }^{\prime },a){\pi }^{-}{\nu }_{\tau }, and Fig. 11 for the Cabibbo suppressed channels, such as {\tau }^{-}\to (K_S{\pi }^{-},K^{-}\eta ,K^{-}{\eta }^{\prime },K^{-}a){\nu }_{\tau }.

6 Summary and conclusions

In this work, we exploit resonance chiral theory augmented with model-independent axion interaction operator aG\tilde{G}/f_a to calculate the form factors for the two-meson and axion-meson production from the semileptonic \tau decays, by utilizing the \pi−\eta−{\eta }^{\prime }−a four-particle mixing matrix elements in Ref. [32]. Our calculations simultaneously include the following channels for both Cabibbo-allowed and Cabibbo-suppressed cases: {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau },{\tau }^{-}\to {\pi }^{-}\eta /{\eta }^{\prime }{\nu }_{\tau },{\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau },{\tau }^{-}\to K^{-}\eta /{\eta }^{\prime }{\nu }_{\tau } and {\tau }^{-}\to {\pi }^{-}/K^{-}a{\nu }_{\tau }.

We have successfully performed the joint fit by reasonably reproducing the experimental data from the Belle collaboration on the two-meson invariant-mass spectra in the {\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau } [68], {\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau } [72] and {\tau }^{-}\to K^{-}\eta {\nu }_{\tau } [73], where the relevant unknown resonance parameters are determined. We then utilize those fitted resonance parameters to make predictions to the two-boson spectra and branching ratios for other five decay channels, including {\tau }^{-}\to {\pi }^{-}\eta /{\eta }^{\prime }{\nu }_{\tau },{\tau }^{-}\to K^{-}{\eta }^{\prime }{\nu }_{\tau } and {\tau }^{-}\to {\pi }^{-}/K^{-}a{\nu }_{\tau }. Apart from the QCD axion scenario, we extend our analysis to the axion-like particle case by introducing a nonzero axion mass m_a. We compute differential decay widths for {\tau }^{-}\to {\pi }^{-}/K^{-}a{\nu }_{\tau } channels at m_a=0, 0.1 GeV and 0.3 GeV for illustrations, and calculate these branching ratios as a function of m_a across a continuous range from 0 to 1 GeV. Our study reveals that to properly include the hadronic resonance contributions in the axion/axion-like particle productions from the tau decays are critical, since the production rates can be enlarged by around one order of magnitude, compared to the case by only including the leading-order chiral amplitudes.

Our predicted branching ratios to {\tau }^{-}\to {\pi }^{-}\eta /{\eta }^{\prime }{\nu }_{\tau } and {\tau }^{-}\to K^{-}{\eta }^{\prime }{\nu }_{\tau } are in agreement with the current experimental upper limits. Furthermore, we present predictions to forward−backward asymmetries in the two-pseudoscalar boson decays of the \tau lepton, which probe vector-scalar form factor interference that is absent in the differential decay widths with respect to the two-boson energy. Our predictions to the branching ratios, invariant-mass distributions and the forward−backward asymmetries can provide useful guidelines to the future experimental measurements, such as the ongoing Belle II and future facilities like the Super Tau-Charm Facility and the Circular Electron Positron Collider.

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