Application of Current Algebra in Three Pseudoscalar Meson Decays of Τ Lepton

The decays of τ → 3πν and τ → πK∗ν,Kρν are calculated using the hard pion and kaon current algebra and assuming the Axial-Vector meson dominance of the hadronic axial currents. Using the experimental data on their masses and widths, the τ decay branching ratios into these channels are calculated and found to be in a reasonable agreement with the experimental data. In particular, using the available Aleph data on the 3π spectrum, we determine the A1 parameters, mA = 1.24± 0.02GeV , ΓA = 0.43± 0.02 GeV; the hard current algebra calculation yields a 3π branching ratio of 19± 3%. ∗ Laboratoire Propre du CNRS UPR A.0014 Application of hard current algebra in τ decay was initiated soon after its discovery [1, 2, 3]. It was also pointed out that in a related calculation, using the hard current algebra technique, the cross sections for e+e− → 4π± can be calculated and agree with the data in the 1 GeV region within a factor of 4 instead of a factor of 10 using the usual soft current algebra. This technique was extended to the Ke4 decays with an implementation of the unitarity in the 2 pion channels [4], and was later used with success in the resolution of the η → 3π problem [5]. The technique of the hard pion current algebra consists in using the PCAC and the Lehmann Symanzik Zimmerman reduction formula [6] by contracting out the pseudoscalar fields. One would get the expressions for single, double or triple equal time commutator relations (ETCR) whose Fourier transforms are the physical matrix elements involving soft pion emissions. The remaining terms, which cannot be calculated by current algebra technique, involve higher order in the pseudoscalar meson momenta and therefore do not contribute in the hypothetical world where the pseudoscalars have zero four momenta. In the physical world where their four momenta do not vanish, corrections must be taken into account in an approximate manner by using substracted dispersion relations; the substraction constants are given by the current algebra low energy theorems and the substraction points are made at the scales where these current algebra low energy theorems are valid [7]. To illustrate our point let us consider the e+e− → 4π. By contracting out the 2 S-wave pions we obtain the single and the double ETCR. The single ETCR term is related to the physical matrix element τ → 3πν where the axial vector meson A1 dominates; the double ETCR is related to the physical pion form factor which is dominated by the vector meson ρ. This technique enables us to incorporate the effect of the heavy fields ρ and A1 which are the main features of the low energy QCD and in a way which is consistent with chiral symmetry and experimental data. This method is however not unique. Instead of this approach, it is now a fashion to use the effective Chiral Lagrangian which treats elegantly the low energy theorems involving the NambuGoldstone bosons; quantum corrections are treated perturbatively; this method is effectively a power series expansion in momenta and hence cannot take into account of the resonance effect. One is forced to accept the fact that Chiral Perturbation Theory, as is usually practiced, described only the low energy tail of the resonance which is certainly not the gross feature of the physics involved. The remedy for this approach is to build in the theory the heavy fields ρ and A1 and possibly also a heavy σ fields as was done by a number of authors [8, 9]. It is suggested here 1 that the loop corrections should be treated unsystematically in the bubble chain approximation in order to make these heavy fields unstable in a way which is consistent with the unitarity requirement. This procedure is the same as the usual way of handling the W and Z propagators in the standard model calculation. One then can incorporate many nice features of the old Vector Meson Dominance models but taking into account also of the low energy chiral properties of the pseudoscalars. This approach will be explored in the future. The purpose of this letter is to study the simpler processes τ → 3πν and τ → πK∗ν,Kρν and leave the more complicated process e+e− → 4π± or τ → 4πν for a future publication. We do not expect to achieve here the precision of the order of 10% or better which is usually obtained for soft pion emission processes like Kl2, Kl3 and Kl4 or the S wave πN scattering lengths etc. This is so because the matrix elements depend on the scalar product of pion momenta to that of the current which is large in the physical region and considerable correction has to be made in order to reach the chiral limit. Using the hadronic properties (widths and masses) of the Axial-mesons A1, Q1 and Q2 and treat them as unstable particles in the usual way, their branching ratios and spectra in τ decays are calculated and found to be in good agreement with the experimental data. We begin first by recalling the well-known formula [10] for the ratio RH = Γ(τ → H−ν) Γ(τ → eνν) = 6π mτ ( cos θc sin θc ) m2τ ∫ m H dQ(1− Q 2 mτ ) ( a0(Q ) + (1 + 2 Q mτ )a1(Q ) ) (1) where a0(Q ) and a1(Q ) are respectively the spin 0 and spin 1 part of the hadronic spectral functions and θc is the Cabbibo angle. The expression multiplying with cos θc is for the ∆S = 0 hadronic tau decay and that multiplying with sin 2 θc is for ∆S = 1 hadronic tau decay. We want to study the matrix element for the axial vector hadronic current involving three pseudoscalar mesons. For clarity, we first make the simplified approximation that the system of three pseudoscalar mesons can be represented by a pseudoscalar and a vector meson. This assumption is reasonable because in the usual angular momentum decomposition one pair of the pseudoscalars have to be in the relative P state and will be shown to be dominated by the vector meson. We 2 assume furthermore that the axial hadronic currents are dominated by the Axial vector mesons A1, Q1 and Q2, just the same as the vector hadronic currents are dominated by the vector mesons ρ and K∗. A more precise study of τ → 3πν is given, treating the vector meson ρ as a resonant 2π state. I) τ → 3πν Decay Let us first consider the ∆S = 0 decay. As mentionned above we approximate the 3π state by a πρ state. The most general matrix element can be written as: 〈π−(k)ρ0(p)|A1−i2 μ (0)|0〉 = f1(Q)ǫμ + ǫ.k ( (k + p)μf2(Q ) + (k − p)μf3(Q) ) (2) where Q = (k+p) and ǫ is the polarisation vector of ρ. f1, f2, and f3 are complex form factors and are only functions of Q. Current algebra soft pion theorem, which is obtained by taking the limit kμ → 0, gives only information on f1 but not on the other 2 form factors. In an explicit model, it was shown that they contribute little to the τ → πρν. Interested readers are refered to the original article [5]. (We assume here that the decay constant of π′ is sufficiently small and hence can be neglected). Using the standard low energy current algebra theorem and taking the limit kμ → 0 we have: lim kμ→0 〈π−(k)ρ0(p)|A1−i2 μ (0)|0〉 = − √ 2 fρ fπ ǫμ(p) (3) where fπ = 93MeV , and fρ is defined by the rate of ρ → e+e−. Using the experimental data [11] we obtain, fρ = 0.118GeV . This value of fρ is equivalent to writing approximatively the pion form factor as Fπ(s) = m 2 ρ(1+δs/m 2 ρ)/ ( mρ − s− imρΓρ(s) ) . A good fit to the experimental data is obtained with δ = 0.2. In fact the more general form of Eq(3) reads lim kμ→0 〈π(k)π(q1)π(q2)|A μ (0)|0〉 = − √ 2 fπ Fπ(s)(q1 − q2)μ (4) For convenience we shall first use Eq(3). The 3π matrix element below the ρπ threshold can be straightforwardly obtained from Eq(4). Using Eq(2) in (3) we have: f1(m 2 ρ) = − √ 2 fρ fπ (5)