Finite-volume QED corrections to decay amplitudes in lattice QCD

We demonstrate that the leading and next-to-leading finite-volume effects in the evaluation of leptonic decay widths of pseudoscalar mesons at $O(\alpha )$ are universal; i.e. they are independent of the structure of the meson. This is analogous to a similar result for the spectrum but with some fundamental differences, most notably the presence of infrared divergences in decay amplitudes. The leading nonuniversal, structure-dependent terms are of $O(1/L^2)$ [compared to the $O(1/L^3)$ leading nonuniversal corrections in the spectrum]. We calculate the universal finite-volume effects, which requires an extension of previously developed techniques to include a dependence on an external three-momentum (in our case, the momentum of the final-state lepton). The result can be included in the strategy proposed in Ref. [N. Carrasco et al.,Phys. Rev. D 91, 074506 (2015).] for using lattice simulations to compute the decay widths at $O(\alpha )$, with the remaining finite-volume effects starting at order $O(1/L^2)$. The methods developed in this paper can be generalized to other decay processes, most notably to semileptonic decays, and hence open the possibility of a new era in precision flavor physics.

Figure 1

Feynman diagram for the leptonic decay of a pseudoscalar meson (the ${\pi }^{-}$ in this example) in pure QCD. The two black filled circles represent the local current-current operator $(\overline{u}{\gamma }_L^{\mu }d)(\overline{\ell }{\gamma }_{\mu }^L{\nu }_{\ell })$; the circles are displaced for convenience and the index $L$ represents “left.”

Figure 2

Virtual diagrams contributing at $O(\alpha )$ to the renormalization of the four-fermion operator Eq. (5). The same diagrams enter in the evaluation of ${\mathrm{\Gamma }}^0(L)$ for the decay ${\pi }^{-}\to {\ell }^{-}{\overline{\nu }}_l$.

Figure 3

Skeleton diagrams contributing at $O(\alpha )$ to ${\mathrm{\Gamma }}_0$ for the decay $P^{-}\to {\ell }^{-}{\overline{\nu }}_l$. The thick black line represents the pseudoscalar meson and the broken green line represents the leptons. The photon is represented by the wavy line. The vertices marked $\mathrm{\Gamma }$ and $W$ represent the coupling of the photon(s) to the meson or weak Hamiltonian respectively. Their definitions are given in the Appendix. The diagrams (a)–(g) are discussed in detail in the text and in the Appendix.