QED corrections to hadronic processes in lattice QCD

In this paper, for the first time to our knowledge, a method is proposed to compute electromagnetic effects in hadronic processes, such as decays, using lattice simulations. The method can be applied, for example, to the leptonic and semileptonic decays of light or heavy pseudoscalar mesons. For these quantities the presence of infrared divergences in intermediate stages of the calculation makes the procedure much more complicated than is the case for the hadronic spectrum, for which calculations already exist. In order to compute the physical widths, diagrams with virtual photons must be combined with those corresponding to the emission of real photons. Only in this way do the infrared divergences cancel as first understood by Bloch and Nordsieck in 1937. We present a detailed analysis of the method for the leptonic decays of a pseudoscalar meson. The implementation of our method, although challenging, is within reach of the present lattice technology.

Figure 1

Tree-level diagram for the process $u\overline{d}\to {\ell }^{+}{\nu }_{\ell }$ (left-hand diagram). In the effective theory the interaction is replaced by a local four-fermion operator (right-hand diagram).

Figure 2

Diagrams contributing to the $O(\alpha )$ corrections to muon decay; see Eq. (8). The curly line represents the photon.

Figure 3

Photon-$W$ box diagrams contributing to the $O(\alpha )$ corrections to muon decay in the Standard Model. The curly line represents the photon.

Figure 4

Correlation function used to calculate the amplitude for the leptonic decay of the pion in pure QCD. The two black filled circles represent the local current-current operator $(\overline{d}{\gamma }_L^{\mu }u)({\overline{\nu }}_{\ell }{\gamma }_{\mu }\ell )$; the circles are displaced for convenience.

Figure 5

Connected diagrams contributing at $O(\alpha )$ to the amplitude for the decay ${\pi }^{+}\to {\ell }^{+}{\nu }_l$. The labels (a)–(f) are introduced to identify the individual diagrams when describing their evaluation in the text.

Figure 6

Disconnected diagrams contributing at $O(\alpha )$ to the amplitude for the decay ${\pi }^{+}\to {\ell }^{+}{\nu }_l$. The curly line represents the photon, and a sum over quark flavors $q$, $q_1$ and $q_2$ is to be performed. The labels (a)–(e) are introduced to identify the individual diagrams when describing their evaluation in the text.

Figure 7

Zoom of the lepton-photon vertex at $x_2$ from the diagrams in Figs. 5 and 5.

Figure 8

One loop diagrams contributing to the wave-function renormalization of a pointlike pion.

Figure 9

Radiative corrections to the pion-lepton vertex. The diagrams represent $O(\alpha )$ contributions to ${\mathrm{\Gamma }}_0^{\mathrm{pt}}$. The left part of each diagram represents a contribution to the amplitude and the right part the tree-level contribution to the Hermitian conjugate of the amplitude. The corresponding diagrams containing the radiative correction on the right-hand side of each diagram are also included.

Figure 10

Diagrams contributing to ${\mathrm{\Gamma }}_1(\mathrm{\Delta }E)$. For diagrams (c), (d) and (e) the “conjugate” contributions in which the photon vertices on the left and right of each diagram are interchanged are also to be included. The labels (a)–(f) are introduced to identify the individual diagrams when describing their evaluation in the text.

Figure 11

One-loop Feynman diagrams computed for the renormalization of the four-fermion operators $O_{XY}=(\overline{d}{\mathrm{\Gamma }}_{X}u)({\overline{\nu }}_{\ell }{\mathrm{\Gamma }}_Y\ell )\equiv {\mathrm{\Gamma }}_X\otimes {\mathrm{\Gamma }}_Y$. The labels (a)–(c) are introduced to identify the individual diagrams when describing their evaluation in the text.

Figure 12

Pointlike (pt), structure-dependent (SD) and interference (INT) contributions to the decay $\pi \to \ell \nu \gamma$. The first (second) row corresponds to $\ell =e$ ($\ell =\mu$).

Figure 13

Pointlike (pt), structure-dependent (SD) and interference (INT) contributions to the decay $K\to \ell \nu \gamma$. The first (second) row corresponds to $\ell =e$ ($\ell =\mu$).

Figure 14

Structure-dependent (SD) and interference (INT) contributions to $R_1$ for the decays $B\to \ell \nu \gamma$. Going from left to right, the plots correspond to $\ell =e$, $\ell =\mu$ and $\ell =\tau$, respectively.