$\tau$ decay into a pseudoscalar and an axial-vector meson

We study theoretically the decay $\tau^- \to \nu_\tau P^- A$, with $P^-$ a $\pi^-$ or $K^-$ and $A$ an axial-vector resonance $b_1(1235)$, $h_1(1170)$, $h_1(1380)$, $a_1(1260)$, $f_1(1285)$ or any of the two poles of the $K_1(1270)$. The process proceeds through a triangle mechanism where a vector meson pair is first produced from the weak current and then one of the vectors produces two pseudoscalars, one of which reinteracts with the other vector to produce the axial resonance. For the initial weak hadronic production we use a recent formalism to account for the hadronization after the initial quark-antiquark pair produced from the weak current, which explicitly filters G-parity states and obtain easy analytic formulas after working out the angular momentum algebra. The model also takes advantage of the chiral unitary theories to evaluate the vector-pseudoscalar amplitudes, where the axial-vector resonances were obtained as dynamically generated from the VP interaction. We make predictions for invariant mass distribution and branching ratios for the channels considered.


I. INTRODUCTION
The fact that the τ is the only lepton heavy enough to decay into hadrons makes the hadronic τ lepton decays a priceless test of the strong interaction at low energy in the light flavor sector [1][2][3][4]. The intermediate and final state decay hadrons are usually produced with lower background than in other low energy processes. Even though there are more than 100 hadronic τ decays experimentally reported by the PDG [5] (which account for about 65% of the τ decay width ), it is also clear that not all the possible ones have been observed or whether there is no room for decays beyond the standard model. While inclusive reactions are well suited for accurate extraction of standard model parameters such as the strong coupling constant [3,6,7], the exclusive ones are much more involved and difficult to predict within QCD, and here is where effective theories for hadronic low energy interactions gain prominence. Special theoretical attention has been devoted to decay channels with two and three pseudoscalar mesons in the final state (see [8] for a brief review). Channels with more pseudoscalars or other mesons like vector or axial-vector ones are less studied [1,9]. Particularly, very poorly understood are the channels with one pseudoscalar plus one axial-vector meson in the final state, which are the aim of the study in the present work. Experimentally only the f 1 (1285)π channel has been measured [5]. It is at this point where effective theories of strong interactions at low energies can stand up. Particularly, the unitary extensions of chiral perturbation theory (UχPT) provide a dynamical and powerful explanation of the generation of the low-lying axial-vector resonances [10][11][12]. With the only input of the lowest order chiral perturbation theory Lagrangians and the implementation of unitarity in coupled channels, most of the lowest mass axial vector resonances (b 1 (1235), h 1 (1170), h 1 (1380), a 1 (1260), f 1 (1285) and two poles for the K 1 (1270)) were dynamically obtained [10][11][12] as poles in the pseudoscalar-vector (PV) scattering amplitudes, without the need to include them as explicit degrees of freedom. With only one free parameter (for regularization of PV loops), this model predicts not only masses and widths of these axial vector resonances but also the full PV scattering amplitudes from where e.g. the coupling of the different resonances to the different PV channels can be obtained. Within this model, the τ decay into one pseudoscalar plus one axial-vector resonance requires the production of one pseudoscalar and one PV pair in the hadronization process, since these axial-vector resonances are dynamically generated from the PV interaction. This can be dominantly accomplished via a triangular mechanism of the kind shown in Fig. 1. Actually, for the f 1 (1285)π case, it was shown in [13] that it was the dominant contribution. Indeed, for some particular kinematic conditions, the triangle diagram benefits from a large enhancement since it develops a singularity [14], which according to the Coleman Norton theorem [15] is related to the classical process in which a particle decays into two particles A and B, then A decays into two other particles, and one of them merges with particle B to produce a third particle.
On the other hand, for the hadronization process from the W − boson to two mesons, we follow the approach of [19], where the 3 P 0 model [20,21] is used to hadronize the primary quarks produced from the weak interaction, working out all the angular momentum and spin algebra. In particular, a Cabibbo favored dū pair is produced from the W − which then hadronizes from an extra qq pair with the quantum numbers of the vacuum. The different possible final meson-meson states are related by SU (3) symmetry. The strength of the formalism in [19] is that it carries on an elaborate calculation of the angular momentum and spin algebra which allows, in the end, to rely upon only one global unknown constant to get all the different channels. This unknown factor is obtained in the present work from the experimental value of the τ − → ν τ π − K * K * decay. Furthermore, the formalism allows for an explicit filter of G-parity states, of special importance in the present work.
Actually, for the K 1 (1270) resonance, it was shown in [11,22] that is has a two pole structure and then we will consider both of them. Since the A resonances are dynamically generated from the PV interaction, for the τ − → ν τ P − A, the way to produce the VP to generate axials and the extra pseudoscalar in the final state is via a triangular mechanism of the kind shown in fig. 1. The well defined G-parity axial-vector states have dominant couplings to either ρπ or K * K [11]. Therefore, for the positive G-parity axial-vector states (f 1 (1285) and b 1 (1235)), the complete Feynman diagrams for the decay with the triangle mechanism are those shown in Fig. 1. Fig. 1(a) shows the process τ − → ν τ K * 0 K * − followed by the K * 0 decay into π − K + and the merging of the K * − K + into A; and Fig.1(b) shows the process τ − → ν τ K * − K 0 followed by the K * − decay into π −K 0 and the merging of the K * 0K 0 into A. The momenta assignment for the decay process is given in Fig. 2. The needed couplings obtained in [11,13] for positive G-parity axial-vector states to the appropriate G-parity V P eigenstates are given in Table I. For the negative G-parity axial-vector states ( h 1 (1170), h 1 (1380) and a 1 (1260)), we need to consider the diagrams in Fig. 3 in addition to those in Fig. 1 because ρρ has G = + and, the fact that G(π) = − demands that the axial has negative G−parity to have G-parity conservation. This mechanism with initial ρρ production is thus not possible for the positive G-parity axials f 1 (1285) and b 1 (1235). Fig. 3 (a) shows the process τ − → ν τ ρ − ρ 0 followed by the ρ − decay into π − π 0 and the merging of the ρ 0 π 0 into A; and Fig. 3 The momenta assignment for the decay process  [11,13] (All the units are MeV).
process τ − → ν τ ρ 0 ρ − followed by the ρ 0 decay into π − π + and the merging of the ρ − π + into A. The needed couplings [11] for negative G-parity states are given in Table II.
In [11,22] two poles for the K 1 (1270) where obtained at complex energy positions 1195 − i123 MeV and 1284 − i73 MeV in unphysical Riemann sheets. The lowest mass pole, which we will call in the following K 1 (1), couples mostly to K * π and the highest one, K 1 (2), to ρK. The dominant couplings are shown in Table III [22].

B. The V V weak production vertex
Next, we address the evaluation of the τ → ν τ K * 0 K * − , ν τ K * − K * 0 , and τ → ν τ ρ − ρ 0 , ν τ ρ 0 ρ − amplitudes. The production is assumed to proceed first from the Cabibbo favoredūd production from the W − which then hadronizes producing an extraqq with the quantum numbers of the vacuum, which are implemented with the 3 P 0 model [20,23,24]. In Ref. [19] the mechanism for hadronization is done in detail. Next, we summarize and adapt the formalism to the present case. The first step corresponds to the flavor combinations in the hadronization. In Ref. [19] it is shown that d(ss)ū = (ds)sū gives rise to [19]). The second step corresponds to the detailed study of the spin-angular momentum algebra to combine the quarks for the 3 P 0q q state (L = 1, S = 1, J = 0) with ad quark in L = 1 to have finally s-wave production of the two mesons. In [18] a similar calculation has been done to discuss the triangle singularity in τ − → ν τ π − f 0 (980) (a 0 (980)) decays, but with pseudoscalar-vector production from the W rather than two vectors, as we have here.
The elementary quark dū production in the tau decay is given by where C contains the couplings of the weak interaction to be determined later on. The leptonic current is given by and the quark current by In the evaluation of the decay widths to three final particles, we find convenient to evaluate the matrix elements in the frame where the two meson system is at rest, and we assume that the quark spinors are at rest for the evaluation of the Q µ matrix element in the same frame [19]. Then we have γ 0 → 1, γ i γ 5 → σ i in terms of bispinors χ, and after the spin angular momentum combination we end up with the following spinor matrix elements: Denoting for simplicity, to obtain the τ width we must evaluate with L µν given by where p, p are the momenta of the τ and ν τ respectively and we use the field normalization for fermions of Ref. [25]. The expression of the amplitudes in terms of the M 0 and N i functions was the main novelty of the work in [19]. This formalism has the advantage of filtering the G-parity contributions since the M 0 and N i operators act with defined G-parity as explained below.
From the work [19] we obtain the results for the V V, J = 1, J = 1 case, where M, M are the third components of J, J respectively, and µ is the index of N i in spherical basis, with C(· · · ) a Clebsch-Gordan coefficient.
In [19], it was shown that the order in which the vector mesons are produced is essential to understand the G-parity symmetry of these reactions, which is given in Table IV. We

C. Evaluation of the constant C
The global unknown constant C in Eq. (1) can be determined from the experimental ratio of τ → ν τ K * 0 K * − , using a similar method as in Ref. [13].
In the present work, the structure of |t| 2 for the τ decay into two vector mesons is taken from the results of [19] for this reaction. If we take the quantization axis along the direction of the neutrino in the τ − rest frame we find where p is the momentum of the τ , or ν τ , in the K * 0 K * − rest frame, given by and E ν = p, E τ = m 2 τ + p 2 . In Eq. (10) the coefficients h i and h i account for the weights of the V V components for M 0 and N µ respectively and their values are listed in Table V.
Note that we are considering only the final s-wave production since, as explained in [19], because of the large vector masses, the expected momenta are very small.
The mass distribution is given by where p K is the momentum of K − in the K * 0 K * − rest frame given by and p ν the neutrino momentum in the τ rest frame The experimental branching ratio of the τ → ν τ K * 0 K * − decay was constructed in Ref. [13] from information in the PDG [5], with the result, from which we can evaluate the value of the constant C 2 , Note that the τ decay into ν τ K * 0 K * − can only proceed because of the finite width of the K * , since otherwise there would be no available phase space for infinitely narrow K * . Hence it is crucial to fold the width with a realistic spectral function of the K * meson, (see Eq. (9) in [13]). Note that, in Ref. [13], the structure of |t| 2 was assumed to proceed with the dominant term E τ E ν − p 2 3 of Eq. (10) alone, and hence a somewhat different C constant was obtained.

D. Evaluation of the triangle diagram
In order to evaluate the triangle loops of Fig. 1, we need first the K * → Kπ vertex obtained from the VPP Lagrangian with the coupling g = 4.31 [13], P and V the SU(3) matrices of the pseudoscalar and vector mesons, by means of which we find We can see that for this vertex we find a relative minus sign from Fig. 1 (a) to Fig. 1 (b). We find convenient to take the z direction along the momentum k of the final pion (see Fig. 2). Indeed, in the πA rest frame, where we evaluate the amplitude, P = 0. The vertex K * → Kπ is of the type · (k + q + k) 1 . On the other hand, the q integration dq i (2k + q) · · · of the triangle loop must necessarily give something proportional to k, which is the only non integrated vector in the loop integral. Then dq i (2k + q) · · · = Ak and contracting with k gives k dq i (2 + q · k/k 2 ) · · · . Hence, we have an effective vertex of the type · k. If the z direction is chosen along k, this selects only the z component ( 0 in spherical basis) and · k = |k| = k. This also means that only M = 0, for Eqs. (8), (9), contributes in the loop and this allows us to calculate trivially the M 0 and N µ amplitudes in that frame.

Evaluation of M 0
For K * 0 (M ) and K * − (M ) of Fig. 1 (a) and M = 0, we get from Eq. (8) On the other hand, in Fig. 1 (b) we will have M of K * − equal to zero and then We can see that the M 0 changes sign from Fig. 1 (a) to Fig. 1 (b). From the sign of Eqs. (18) and (19) and this latter sign, we can see that the global sign is the same for these two diagrams.
Finally, in order to evaluate the final amplitude of the loop diagram we need the vertex A → V P , which is of the type [11] g A,V P V · A with V , A , the polarization vector of the vector and axial-vector resonances. Note that the couplings of the axials to pseudoscalar and vector are given in [11] in terms of well defined 1 Since in the triangle diagram the K * 0 K * − intermediate states have a small momentum compared to the K * mass [13], we neglect the 0 component, which was found in [26] to be an excellent approximation in such a case. Tables I-III). In order to relate those couplings to the charge basis PV states that we are using, we need to write our states for the axial vector mesons in terms of their vector pseudoscalar components forming states of well defined C and G parity:

G-parity PV sates (see
Since G = (−1) I C, we need from the sum of Figs. 1 (a) and 1 (b), the following combinations.
Note that Eq.

Evaluation of N i
In spherical basis, N i → N µ (µ = 0, ±1), is given by Eq.
Contrary to what happens with the M 0 component, we can see that N µ does not change sign when we exchange K * 0 K * − → K * − K * 0 in the loop of Fig. 1 (a) and Fig. 1 (b). Then, considering the different sign in the vertex K * → Kπ of Eqs. (18) and (19), we get the combination g A,K * − K + − g A,K * 0K0 , which in view of Eqs. (23) and (24) provides, This shows explicitly that with G-parity positive axials only the N i term contributes, as we saw at the beginning at the quark level, while for G negative axials only the M 0 term contributes.
From Eq. (30) we can calculate the N i components in cartesian basis and we find

E. Incorporation of intermediate ρρ states
For the production of negative G-parity axial vector mesons, we must also consider the ρ 0 ρ − diagrams of Fig. 3 in addition to Fig. 1. For this we need the h i coefficients of table V. Recall that in this case only the M 0 term contributes. Next we need the ρ − → π − π 0 , ρ 0 → π − π + vertices obtained from the Lagrangian in Eq. (17), Since M 0 changes sign from ρ − ρ 0 to ρ 0 ρ − production (see Table IV), this sign and the relative one of Eq. (35), (36) cancel and we get the factor in the sum of the loops To relate these couplings in charge basis to the coupling of A to ρπ in isospin basis, evaluated in [11], we recall that the isospin multiplets are (−π + , π 0 , π − ), (−ρ + , ρ 0 , ρ − ). Then we have Then The ρπ channel only contributes to these two states that have negative G-parity.
Thus, in order to account for the coherent sum of K * − K * 0 and ρ − ρ 0 we can use t M 0 of Eq. (26) but performing the following substitution, .
Next we must perform the sum of Eq. (6) independently,L 00 M 0 M * 0 for negative G-parity A states andL ij N i N * j for positive G-parity A states. By using Eq. (7) and Eqs. (20), (32), (33), (34) and summing over the M A components we obtain: a) G-parity positive axial states: b) G-parity negative axial states: We should note that the αµβν p α p β term of Eq. (7) does not contribute in M 0 since µ = 0 and p α p β (p τ p ν ) will be spatial and p τ p ν are the same in the frame we work. For N i , α or β must be zero and we have just one vector p ν that cancels in the phase space integration. For the same reason, the term with p νi p νj becomes 1 3 p 2 ν δ ij upon integration over phase space.
In [11] two states corresponding to K 1 were found and the pole positions were refined in [22], one of them at 1195 MeV coupling mostly to K * π, and another one at 1284 MeV coupling mostly to ρK. Proceeding analogously to the previous cases, the terms that go like L 0i , of αµβν cancel again in the integration over phase space, and we obtain: a) K 1 (1) state: b) K 1 (2) state: For τ − → ν τ π − A decay, the differential mass distribution for M inv (π − A) is given by with and p is the momentum of the τ , or ν τ , in the π − A rest frame, given by Similarly, for the τ − → ν τ K − K 1 (1270) decay, we can get the differential mass distribution for M inv (K − K 1 (1270)).
Note that the term m τ m ν in the numerator of Eq. (46) cancels the same factor in the denominator of Eqs. (42), (43), (44), and (45). In Eq. (46) we have the same factor C 2 Γτ from Eq.(16) and thus we can provide absolute values for the mass distributions. Fig. 6 the triangle loop in Eq. (27), for the τ − → ν τ π − f 1 (1285) case, as a function of the π − f 1 invariant mass, M inv (π − f 1 ). Note that there is a large increase of the strength at around the region of interest at the present work, (M inv (π − f 1 ) = m τ = 1777MeV, which will push the invariant mass distributions for the decays considered in the present work to the higher energy region of the spectrum, as we will see below. As already discussed in [13] the origin of this increase is twofold: first because of the presence of a nearby triangular singularity and, second, because of the presence of the K * K * threshold. Both effects are

First we show in
where D(m) is the axial-vector propagator, This folding is particularly relevant for the decays into K 1 because of the little and null available phase space for the K 1 (1) and K 1 (2) respectively. Actually K 1 (2) can only proceed because of its tail. In  which compares well with the value we obtain for that channel within uncertainties. For the channels not yet measured, even though the branching ratios obtained for some of them seem small, they are of the same order as many of the already experimentally measured hadronic decays reported by the PDG [5].
The mass distributions and the branching ratios of table VI are non-trivial and genuine predictions because, first, they crucially depend on the axial-vector resonance couplings to VP which are a non-trivial output of the chiral unitary model [11] and consequence of the dynamical origin of these resonances. And, second, because of the non-trivial shape of the triangular mechanism and the enhancement due to nearby singularities when present.   13. The mass distribution for τ − → ν τ π − K 1 (1) decay 1400 1500 1600 1700 1800 0 14. The mass distribution for τ − → ν τ π − K 1 (2) decay

IV. CONCLUSIONS
We have carried out a theoretical study of the τ decay into a pseudoscalar meson plus an axial-vector resonance. These hadronic decay channels have been very little studied previously, both theoretical and experimentally. Nonetheless, these channels could play an to a lowest order amplitude obtained from chiral Lagrangians, poles of the unitarized PV amplitudes were found which could be associated to the known axial-vector resonances. In particular, of great relevance was the prediction that in the strange sector the K 1 (1270) actually corresponds to two disctinct poles with different coupling intensities to the different VP channels. Within this framework the dominant production mechanism for the τ decays considered in the present work is through a triangle mechanism of the kind shown in Fig. 1, since a vector and a pseudoscalar need to be produced, in addition to the extra final pseudoscalar, to generate the axial-vector resonance. The initial VV production from the weak current has been theoretically determined, up to a global common factor obtained from the experimental τ → ν τ K * 0 K * − branching ratio, from a primary dū formation from the W − boson which then hadronizes producing an extra qq pair within the 3 P 0 model. The spinor algebra is worked out following a recent approach where different G-parity contributions could be easily filtered of special interest in the present work.
The pseudoscalar-axial mass distributions predicted in the present work manifest shifts of the strength to the higher energy region of the spectrum partly due to the special shape of the triangle loop function which is carefully evaluated. We make predictions also for integrated branching ratios and, for the only channel experimentally meassured, τ − → ν τ π − f 1 (1285),