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(\omega, \phi) P^- decays of tau leptons

The \tau ->(\omega, \phi)P^-\nu decays, where P^-=\pi^-, K^-, are considered within a phenomenological model with dominance of meson intermediate states. We assume SU(3) flavor symmetry to fix some of the unknown strong interaction couplings. Our predictions for the \tau ->\phi (\pi^-, K^-)\nu branching fractions are in good agreement with recent measurements of the BABAR and BELLE Collaborations.

Earlier theoretical estimates for some of these decays were considered in references [4,5,6].
Based on the Conserved Vector Current (CVC) hypothesis and using bounds for the cross section of e + e − → φπ 0 , the loosely limit B(τ − → φπ − ν) ≤ 9.0 × 10 −4 at 90% c.l. was derived in [5]. On another hand, by assuming that the form factor of this decay is dominated by the contribution of two vector resonances (ρ and ρ ′ ) and that flavor SU(3) is a good symmetry, the value B(τ − → φπ − ν) = (1.20 ± 0.48) × 10 −5 was obtained in Ref. [6]. This prediction clearly underestimates the measured fraction of φπ − (see Table I). Concerning the φK − modes only rough estimates are available based on phase-space and quark mixing angles considerations [10].
In this paper we revisit this subject and provide a unified description of the four V P − decays shown in Eqs.
In this paper we consider the possibility that a meson dominance model with a few intermediate states can account, in a unified way, for the observed branching fractions of reactions (1)-(4). As a simplifying assumption we will rely on SU(3) flavor symmetry for the strong couplings and we will assume an ideal value tan θ V = 1/ √ 2 of the ω − φ mixing angle (i.e., φ is assumed to be an almost puress state). On the basis of these assumptions we conclude that present data on τ − → V P − ν decays can be easily accommodated within the meson dominance model.

II. MESON DOMINANCE MODEL FOR TAU DECAYS
Thus, let us first consider the decay τ − (p τ ) → V (p V )P − (p P )ν(p ν ), where p i denote the fourmomenta of particle i. The hadronic matrix element can be decomposed in terms of four form factors (Q = s or d) [13]: where ǫ * β is the polarization four-vector of the outgoing vector meson (p V · ǫ * = 0), and q ± = p V ± p P . The vector (g) and axial (f, a ± ) form factors are functions of s = q 2 + only. If we define Σ 2 = m 2 V + m 2 P , ∆ 2 = m 2 V − m 2 P , and β V P = (1 − 2Σ 2 /s + ∆ 4 /s 2 ) 1/2 , the differential decay rate can be written in the simple form: where G F is the Fermi constant, V uQ is the uQ entry of the Cabibbo-Kobayashi-Maskawa matrix The Feynman diagrams with the intermediate mesons that connect the weak current and the strong vertex in τ → V P ν decays are shown in Figure 1. Using the Feynman rules for the elements of these diagrams, we get the following expressions for the form factors (V j , A j and P j denote vector, axial and pseudoscalar intermediate meson states, respectively): where f M j denotes the weak coupling of the M j intermediate meson, is the mass (width) parameter of the corresponding intermediate state.

III. STRANGENESS-CONSERVING DECAYS
The G-parity properties of the weak currents and V π − system in this case, impose f = a − = a + = 0. As in previous papers [4,6], we will assume that the non-vanishing vector form factor is saturated by the exchange of two vector resonances (the ρ(770) and the ρ ′ (1523)). Thus we where α V π = f ρ ′ g ρ ′ V π /f ρ g ρV π is the only free parameter of the model at this stage. We chose the ρ ′ (1523) state (m ρ ′ = 1523 MeV and Γ ρ ′ = 400 MeV [8,14,15]) as the second vector resonance, instead of the ρ(1450) [1], because its larger width allows for a better fit to data on the spectral function [8,14,15].
As usual [15], we can define a vector spectral function whose expression becomes very simple in this case: This spectral function has been measured by the ALEPH [7] and CLEO [8] Collaborations for the dominant ωπ − final state. In order to fit the data on the spectral function, we use: and ω → π 0 γ decays, but they agree within their error bars). A fit to the spectral function reported in Ref. [7] gives us α ωπ = −0.57 ± 0.11. The data of Ref. [8] and the fitted curves of the spectral function are shown in Figure 2. Using the above value of α ωπ we can derive the following branching fraction by integration of Eq. (6): which is in very good agreement with the experimental value shown in Table I. Now we focus on the φπ decay channel. We will assume that flavor SU (3) is a good symmetry for the V V P couplings of the octet of vector mesons and of their radial excitations. Under this assumption, we can get α φπ = α ωπ for the relative weights of ρ and ρ ′ contributions in Eq. (9). In addition we use g ρφπ = −(1.57 ± 0.03) × 10 −3 MeV −1 , which is obtained from the φ → ρπ → π + π − π 0 decay rate [1]. Now, if we insert these parameters into Eq. (6), we get: which clearly favors the result reported by BABAR [11] (see Table I).

IV. COUPLING CONSTANTS FOR ∆S = −1 DECAYS
The case of ∆S = −1 tau decays is more difficult to deal with because the vector and axial weak currents contribute to the decay amplitude. Consequently, more independent information is needed to specify the input coupling constants. The vector form factor g will be assumed to be dominated by the K * = K * (892) and the K ′ * = K * (1410) intermediate vector mesons (the coupling of the K ′′ * = K * (1680) to the V K − system is more suppressed). The axial form factors f, a ± will originate from the exchange of the K − pseudoscalar and the K 1 = K 1 (1270), The expressions for the form factors were given in Eq. (9), and the numerical values of coupling constants will be discussed in the following subsections.

A. Weak couplings
The value of the K − weak coupling is known with good precision, f K = (159.8 ± 1.5) MeV [1], and is extracted from K → µν decays by including the effects of radiative corrections. The weak couplings of the other mesons can be extracted from the measurements of τ → Kν decays (where K denotes either state among K * , K ′ * , K 1 , K ′ 1 mesons). We get: The weak coupling with the largest uncertainty corresponds to the K ′ 1 meson. Since the contribution of the K ′ 1 intermediate state will provide an important contribution, a more precise measurement of the τ → K 1 (1400)ν would be suitable to improve the accuracy of our prediction.

B. KV K couplings
The determination of the strong couplings of the intermediate resonances to the V K − system in a reliable way is also a difficult task, either because such decays are not allowed by kinematics or because there are not independent processes where their contribution can be studied. Thus we will strongly rely on SU(3) flavor symmetry to fix their values when necessary.
From the experimental value of the φ → K + K − branching fraction [1], we get g K + φK − = (4.48 ±0.04). Now, using the SU(3) symmetry and assuming an ideal value for the ω −φ mixing angle (tan θ V = 1/ √ 2), we get: An alternative calculation of the KV K couplings can be obtained by assuming the vector-meson dominance model of the kaon electromagnetic form factors at zero momentum transfer: where em V /γ V defines the coupling of the neutral vector meson V to the photon. Now, we can use the SU(3) relations between the P V P ′ couplings (we assume again tan θ V = 1/ √ 2): Solving the set of eqs. (14)-(16), we finally get: which are quite similar values to the ones computed from the φ → KK decays (see above).

C. V ′ V P strong couplings
Flavor SU(3) symmetry predicts the following relations among V ′ V P − couplings (we assume later below tan θ V = 1/ √ 2): where r ≡ G 0

D. AV P couplings
The couplings of axial-vector mesons (A) to the V K − system are the most difficult to determine. One may attempt to compute them from the measured branching fractions of K 1 , K ′ 1 into the ωK − channel (which is the only allowed by kinematics). We get from this 1 : However, the K 1 φK couplings can not be obtained in this way given that K 1 , K ′ 1 → φK decays are not allowed by kinematics.
Given the poor quality of the measurements used to extract the above couplings, we can resort again to the SU(3) flavor symmetry. Since the (K 1 , K ′ 1 ) physical states are a mixture of the (K 1A , K 1B ) states that belong to different 1 3 P 1 (A) and 1 1 P 1 (B) multiplets of axial 1 In order to extract g K1ωK we have taken the maximum values for the mass and width of K 1 that are allowed by their error bars [1]. mesons, we propose as the starting point the following AV P interaction Lagrangian: The physical strange axial mesons are defined in terms of flavor SU(3) states as follows: The determination of the K 1A − K 1B mixing angle is still controversial [16]. According to different authors its value can be in the range 30 0 ≤ α ≤ 60 0 [16]. For illustrative purposes, in the present paper we will use α = 45 0 [1] . If in addition we assume a nonet symmetry for the 1 1 P 1 couplings, namely g 8 BV P = g 0 BV P and the ideal value for the ω − φ mixing angle (tan θ V = 1/ √ 2), we get the following simplified expressions for the couplings that involve the φ and ω mesons (the expressions of the couplings constants for arbitrary values of the α and θ V mixing angles are given in the Appendix): where we have defined Σ ± ≡ g 8 AV P ± g 8 BV P . We can fix the values of the effective couplings Σ ± by using the decay rates of K 1 , K ′ 1 axial mesons in the same limit where Eqs. (30)-(32) were obtained. Using the expressions given in the Appendix and comparing with the measured rates of K ′ 1 → K * π decays [1], we obtain: Similarly, from the measured rate of K ′ 1 → ωK we get: Σ − = (1.36 ± 0.67) × 10 −3 MeV −1 . Finally, if we insert these values into Eqs. (30)-(33), we get: Observe that the K 1 ωK coupling in Eqs.

V. STRANGENESS-CHANGING DECAYS
With the information on the coupling constants given in the previous section, the only free parameters to our disposal are the relative contributions of the vector meson contributions in the form factor g(s): We can further attempt the use of SU(3) symmetry to derive such couplings. Instead, we will fix the values of α ωK by requiring that it reproduces the experimental branching fraction for the well measured τ − → ωK − ν decay (see Table I which are consistent with the experimental values measured by BABAR [11] and BELLE [10] Collaborations (see Table I).
Note that the large error bars quoted in Eq. (39) are dominated by the uncertainty in the K ′ 1 = K 1 (1400) weak decay constant given in Eq. (13), which was extracted from the poorly measured τ → K 1 (1400)ν decay. This error bar can be reduced if we use a weak coupling constant obtained from a phenomenological quark model. Thus, for example, if we assume α = 45 0 , from the covariant quark model of ref. [16] we obtain f cqm This value of the weak coupling does not affect in a sensitive way the ratio of K ′ * /K * couplings extracted from τ − → ωK − ν branching fraction, which now become: α ωK − = 0.55 ± 0.38 or α ωK − = −0.78 ± 0.39. Using the value of f K ′ 1 obtained above, we get:  Table I).
In Figure 3 we compare the invariant mass distribution of the φK − system, with the measurements reported by the BABAR Collaboration [11]. As we can observe, our model (with our numbers multiplied by an arbitrary scale) nicely reproduces the data on the invariant mass distribution. Although it is difficult to discriminate between values of the two-fold ambiguity in α φK , data seems to favor the solution with α φK = 0.55.

VI. CONCLUSIONS
The decays τ − → (ω, φ)P − ν τ , where P is a charged pseudoscalar meson, are studied in a phenomenological model where the form factors are dominated by the exchange of meson states with the appropriate quantum numbers. We rely on SU(3) flavor symmetry to fix the strong interaction coupling constants, and assume an ideal value for the ω − φ mixing angle.