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First-principle evaluation of inclusive hadronic $τ$ decays in QCD+QED

Matteo Di Carlo, Simone Bacchio, Erik Bäske, Alessandro De Santis, Antonio Evangelista, Roberto Frezzotti, Giuseppe Gagliardi, Lukas Holan, Vittorio Lubicz, Lorenzo Maio, Francesca Margari, Agostino Patella, Francesco Sanfilippo, Silvano Simula, Nazario Tantalo

Abstract

We present a strategy to extend lattice calculations of inclusive hadronic $τ$ decays from isosymmetric QCD to QCD+QED. The inclusive decay rate can be related to suitable Euclidean correlation functions, allowing for a first-principles evaluation of electromagnetic and isospin-breaking effects. Within the RM123 framework, radiative corrections are decomposed into leptonic, factorizable and non-factorizable contributions. We report preliminary results for the leptonic and factorizable terms in the electro-quenched approximation and discuss the remaining steps towards a complete calculation. This programme aims at a first-principles determination of inclusive $τ$ decay rates with direct implications for the extraction of the CKM matrix element $|V_{us}|$.

First-principle evaluation of inclusive hadronic $τ$ decays in QCD+QED

Abstract

We present a strategy to extend lattice calculations of inclusive hadronic decays from isosymmetric QCD to QCD+QED. The inclusive decay rate can be related to suitable Euclidean correlation functions, allowing for a first-principles evaluation of electromagnetic and isospin-breaking effects. Within the RM123 framework, radiative corrections are decomposed into leptonic, factorizable and non-factorizable contributions. We report preliminary results for the leptonic and factorizable terms in the electro-quenched approximation and discuss the remaining steps towards a complete calculation. This programme aims at a first-principles determination of inclusive decay rates with direct implications for the extraction of the CKM matrix element .

Paper Structure

This paper contains 7 sections, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inclusive decay rate from a Euclidean correlator
  3. RM123 expansion of the QCD+QED correlator
  4. Factorizable and leptonic contributions
  5. Spectral reconstruction and present status
  6. Next steps: non-factorizable corrections and non-perturbative renormalization
  7. Conclusions

Figures (2)

  • Figure 1: Preliminary reconstruction of the contribution of factorizable corrections to the decay rate at $\sigma/m_\tau = 0.040$, obtained through a variation of the parameter $\lambda$. Top panels: reconstructed longitudinal spectral density from the temporal components of the vector (left) and axial (right) currents. Bottom panels: reconstructed transverse spectral density from the spatial components of the vector (left) and axial (right) currents.
  • Figure 2: Preliminary reconstruction of the contribution of leptonic corrections to the decay rate at $\sigma/m_\tau = 0.040$, obtained through a variation of the parameter $\lambda$. Top panels: reconstructed longitudinal spectral density from the temporal components of the vector (left) and axial (right) currents. Bottom panels: reconstructed transverse spectral density from the spatial components of the vector (left) and axial (right) currents.