Dispersive representation of the pion vector form factor in τ→ππν _τ decays

Abstract

We propose a dispersive representation of the charged pion vector form factor that is consistent with chiral symmetry and fulfills the constraints imposed by analyticity and unitarity. Unknown parameters are fitted to the very precise data on τ ⁻→π ⁻ π ⁰ ν _τ decays obtained by Belle, leading to a good description of the corresponding spectral function up to a ππ squared invariant mass s≃1.5 GeV². We determine the ρ(770) mass and width pole parameters and obtain the values of low-energy observables. The significance of isospin-breaking corrections is also discussed. For larger values of s, this representation is complemented with a phenomenological description to allow its implementation in the new TAUOLA hadronic currents.

Appendix

The explicit form of the loop functions A _PQ (s) can be obtained from Ref. [62]. One has

$$ A_{PQ}(s)=-\frac{192\pi^2 [sM_{PQ}(s)-L_{PQ}(s)]}{s}, $$

(A.1)

where M _PQ (s) and L _PQ (s) can be written in terms of new functions Σ _PQ , Δ _PQ , k _PQ , $\bar{J}_{PQ}$ and $\tilde{J}_{PQ}$ as

$$\begin{aligned} \begin{aligned}[c] &M_{PQ}(s) = \frac{1}{12s}(s-2\varSigma_{PQ}) \bar{J}_{PQ}(s)+ \frac{\Delta_{PQ}^2}{3s^2}\tilde{J}_{PQ}(s) \\ &\phantom{M_{PQ}(s) =}{}-\frac{1}{6}k_{PQ}+ \frac{1}{288\pi^2}, \\ &L_{PQ}(s) = \frac{\Delta_{PQ}^2}{4s}\bar{J}_{PQ}(s) . \end{aligned} \end{aligned}$$

(A.2)

The new functions Σ _PQ and Δ _PQ are defined by $\varSigma_{PQ}=m_{P}^{2}+m_{Q}^{2}$, $\Delta_{PQ}=m_{P}^{2}-m_{Q}^{2}$, while k _PQ includes the renormalization scale μ:

$$ k_{PQ} = \frac{F_\pi^2}{\Delta_{PQ}}(\mu_P-\mu_Q) , $$

(A.3)

where

$$ \mu_P = \frac{m_P^2}{32\pi^2F_\pi^2} \log\biggl(\frac{m_P^2}{\mu ^2} \biggr) $$

(A.4)

(we have taken μ=M _ρ , as in the isospin symmetric case). Finally, the functions $\bar{J}_{PQ}$ and $\tilde{J}_{PQ}$ are given by

$$\begin{aligned} \begin{aligned}[c] & \tilde{J}_{PQ}(s) = \bar{J}_{PQ}(s)-s \bar{J}'_{PQ}(0) \\ & \bar{J}_{PQ}(s) = \frac{1}{32\pi^2} \biggl[ 2 + \biggl( \frac{\Delta_{PQ}}{s} - \frac{\varSigma_{PQ}}{\Delta_{PQ}} \biggr)\log \biggl(\frac{m_Q^2}{m_P^2} \biggr) \\ & \phantom{\bar{J}_{PQ}(s) =} {}- \frac{\nu}{s}\log \biggl(\frac{(s+\nu)^2-\Delta_{PQ}^2}{(s-\nu)^2- \Delta_{PQ}^2} \biggr) \biggr] , \end{aligned} \end{aligned}$$

(A.5)

where $\nu=\lambda^{1/2}(s,m_{P}^{2},m_{Q}^{2})$. We note finally that

$$ s\bar{J}'_{PQ}(0)=\frac{s}{32\pi^2} \biggl( \frac{\varSigma_{PQ}}{\Delta_{PQ}^2} + 2\frac{M_P^2 M_Q^2}{\Delta_{PQ}^3} \log\frac{M_Q^2}{M_P^2} \biggr) . $$

(A.6)