QED Corrections to Hadronic Processes in Lattice QCD

In this paper, for the first time a method is proposed to compute electromagnetic effects in hadronic processes 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.


Introduction
074506) The precision of lattice calculations is such that Isospin-breaking effects, including electromagnetic corrections to hadronic masses, are now being calculated. For a review see A.Portelli at Lattice 2014. arXiv:1505.07057 A highlight has been the Ab initio calculation of the neutron-proton mass difference by the BMW collaboration. S.Borsanyi et al., arXiv:1406.4088 In this talk I review our proposal to calculate electromagnetic corrections to matrix elements.
The new feature is the presence of infrared divergences. This is necessary for further progress in phenomenology, since the results of (some) weak matrix elements obtained from lattice QCD are now being quoted with O(1%) precision or better, e.g. FLAG Collaboration, arXiv:1310.8555 fπ fK fD fD s fB fB s 130.2(1.4) 156.3(0.8) 209.2(3.3) 248.6(2.7) 190.5(4.2) 227.7(4.5) (results given in MeV) For illustration, we consider fπ but the discussion is general; we do not use ChPT. For a ChPT based discussion of fπ, see J.Gasser & G.R.S.Zarnauskas, arXiv:1008.3479

Infrared Divergences
At O(α 0 ) At O(α) infrared divergences are present and we have to consider where the suffix denotes the number of photons in the final state.
Each of the two terms on the rhs is infrared divergent; the divergences cancel in the sum.
The cancelation of infrared divergences between contributions with virtual and real photons is an old and well understood issue. F.Bloch and A.Nordsieck, PR 52 (1937) 54 The question for our community is how best to combine this understanding with lattice calculations of non-perturbative hadronic effects.
This is a generic problem if em corrections are to be included in the evaluation of a decay process.
Lattice computations of Γ(π + → + ν (γ)) at O(α) As techniques and resources improve in the future, it may be better to compute Γ1 nonperturbatively over a larger range of photon energies.
At present we do not propose to compute Γ1 nonperturbatively. Rather we consider only photons which are sufficiently soft for the point-like (pt) approximation to be valid.
A cut-off ∆E of O(10 -20 MeV) appears to be appropriate both experimentally and theoretically.
F.Ambrosino et al., KLOE collaboration,arXiv:0907.3594 We now write The second term on the rhs can be calculated in perturbation theory. It is infrared convergent, but does contain a term proportional to log ∆E. The first term is also free of infrared divergences. Γ0 is calculated nonperturbatively and Γ pt 0 in perturbation theory.

Estimates of structure dependent contributions to Γ1(∆E)
The size of the neglected structure-dependent contributions can be estimated using ChPT.
For the B-meson, for which we cannot use ChPT, we have another small scale < ΛQCD, mB * − mB 45 MeV so that we may expect that we will have to go to smaller ∆E in order to be able to neglect SD effects. The results for the widths are expressed in terms of GF, the Fermi constant (GF = 1.16632(2) × 10 −5 GeV −2 ). This is obtained from the muon lifetime:

Outline of Talk
S.M.Berman, PR 112 (1958)   As an example providing some evidence & intuition that the W-regularization is useful consider the γ − W box diagram.
In the standard model (left-hand diagram) it contains both the γ and W propagators.
In the effective theory this is preserved with the W-regularization where the photon propagator is proportional to We therefore need to calculate the pion-decay diagrams in the effective theory with in the W-regularization.

Proposed calculation of
Consider now the evaluation of Γ0.
The correlation function for this set of diagrams is of the form: where jµ(x) = f Qff (x)γµf (x), JW is the weak current, φ is an interpolating operator for the pion and ∆ is the photon propagator.
Combining C1 with the lowest order correlator: where now O(α) terms are included.
e −mπ t e −m 0 π t (1 − δmπ t) and Z φ is obtained from the two-point function.

Diagrams (e) and (f) are not simply generalisations of the evaluation of fπ.
The lepton's wave function renormalisation cancels in the difference Γ0 − Γ pt 0 . We have to be able to isolate the finite-volume ground state (pion).
The Minkowski ↔ Euclidean continuation can be performed (the time integrations are convergent).

Preliminary Results for "Crossed" Diagrams
Twisted-mass study, 24 3 × 48 lattice with a = 0.086 fm, mπ 500 MeV, 240 configs with 3 stochastic sources per configuration. together with F.Sanfilippo and S.Simula There are also disconnected diagrams to be evaluated.

Calculation of
The total width, Γ pt was calculated in 1958/9 using a Pauli-Villars regulator for the UV divergences and mγ for the infrared divergences. This is a useful check on our perturbative calculation.
In the perturbative calculation we use the following Lagrangian for the interaction of a point-like pion with the leptons: The corresponding Feynman rules are: where rE = 2∆E/mπ and r = m /mπ.
We believe that this is a new result.

Chris Sachrajda
Lattice 2015, 14th July 2015 16 4. Calculation of Γ pt = Γ pt 0 + Γ pt 1 (cont) The total rate is readily computed by setting rE to its maximum value, namely rE = 1 − r 2 , giving This result agrees with the well known results in literature providing an important check of our calculation.

Summary and Conclusions
Lattice calculations of some physical quantities are approaching O(1%) precision ⇒ we need to include isospin-breaking effects, including electromagnetic effects, to make the tests of the SM even more stringent.
For decay widths and scattering cross sections including em effects introduces infrared divergences.
We propose a method for dealing with these divergences, illustrating the procedure by a detailed study of the leptonic (and semileptonic) decays of pseudoscalar mesons.
Although challenging, the method is within reach of present simulations and we are now implementing the procedure in an actual numerical computation.
O(ααs) matching factors to be studied.

Summary and Conclusions (cont.)
One can certainly envisage relaxing the condition ∆E ΛQCD, including the emission of real photons with energies which do resolve the structure of the initial hadron. Such calculations can be performed in Euclidean space under the same conditions as above, i.e. providing that there is a mass gap.
In that case we generalise the master formula to The important point is to organise the calculation into terms, each of which is infrared convergent.
Γ pt 0 + Γ pt 1 (∆E) (in infinite volume) is done. At present we are exploring how best to calculate lim V→∞ (Γ0 − Γ pt 0 ) and exploratory numerical calculations are underway.