QED Corrections to Hadronic Processes in Lattice QCD

QED Corrections to Hadronic Processes in Lattice QCD

N.Carrasco Dipartimento di Fisica, Università Roma Tre and INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy    V.Lubicz Dipartimento di Fisica, Università Roma Tre and INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy    G.Martinelli SISSA, Via Bonomea 265, I-34136, Trieste, and INFN Sezione di Roma La Sapienza Piazzale Aldo Moro 5, 00185 Roma, Italy    C.T.Sachrajda School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK    N.Tantalo PH-TH, CERN, CH-1211, Geneva 23, Switzerland Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, I-00133 Roma, Italy    C.Tarantino Dipartimento di Fisica, Università Roma Tre and INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy    M.Testa Physics Department and INFN Sezione di Roma La Sapienza Piazzale Aldo Moro 5, 00185 Roma, Italy
Abstract

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.

pacs:
11.15.Ha, 12.15.Lk, 12.38.Gc, 13.20.-v

I Introduction

Precision flavour physics is a particularly powerful tool for exploring the limits of the Standard Model (SM) of particle physics and in searching for inconsistencies which would signal the existence of new physics. An important component of this endeavour is the over-determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix from a wide range of weak processes. The precision in extracting CKM matrix elements is generally limited by our ability to quantify hadronic effects and the main goal of large-scale simulations using the lattice formulation of QCD is the ab-initio evaluation of the non-perturbative QCD effects in physical processes. The recent, very impressive, improvement in lattice computations has led to a precision approaching for a number of quantities (see e.g. Ref. Aoki:2013ldr () and references therein) and therefore in order to make further progress electromagnetic effects (and other isospin-breaking contributions) have to be considered. The question of how to include electromagnetic effects in the hadron spectrum and in the determination of quark masses in ab-initio lattice calculations was addressed for the first time in Duncan:1996xy (). Much theoretical and algorithmic progress has been made following this pioneering work, particularly in recent years, leading to remarkably accurate determinations of the charged-neutral mass splittings of light pseudoscalar mesons and light baryons (see Refs. Borsanyi:2014jba (); Basak:2014vca (); deDivitiis:2013xla (); Ishikawa:2012ix (); Aoki:2012st (); Blum:2010ym () for recent papers on the subject and Refs. Tantalo:2013maa (); Portellilattice () for reviews of these results and a discussion of the different approaches used to perform QED+QCD lattice calculations of the spectrum).

In the computation of the hadron spectrum there is a very significant simplification in that there are no infrared divergences. In this paper we propose a strategy to include electromagnetic effects in processes for which infrared divergences are present but which cancel in the standard way between diagrams containing different numbers of real and virtual photons Bloch:1937pw (). The presence of infrared divergences in intermediate steps of the calculation requires the development of new methods. Indeed, in order to cancel the infrared divergences and obtain results for physical quantities, radiative corrections from virtual and real photons must be combined. We stress that it is not sufficient simply to add the electromagnetic interaction to the quark action because amplitudes with different numbers of real photons must be evaluated separately, before being combined in the inclusive rate for a given process. In this paper for the first time we introduce and discuss a strategy to compute electromagnetic radiative corrections to leptonic decays of pseudoscalar mesons which can then be used to determine the corresponding CKM matrix elements. Although we present the explicit discussion for this specific set of processes, the method is more general and can readily be extended to generic processes including, for example, to semileptonic decays.

We now focus on the leptonic decay of the charged pseudoscalar meson . Let be the partial width for the decay where the charged lepton is an electron or a muon (or possibly a ) and is the corresponding neutrino. The subscript indicates that there are no photons in the final state. In the absence of electromagnetism, the non-perturbative QCD effects are contained in a single number, the decay constant , defined by

(1)

where is composed of the valence quarks and , and the axial current in (1) is composed of the corresponding quark fields. There have been very many lattice calculations of the decay constants , ,  and  Aoki:2013ldr (), some of which are approaching precision. As noted above, in order to determine the corresponding CKM matrix elements at this level of precision isospin breaking effects, including electromagnetic corrections, must be considered. It will become clear in the following, and has been stressed in Bijnens:1993ae (); Gasser:2010wz (), that it is not possible to give a physical definition of the decay constant in the presence of electromagnetism, because of the contributions from diagrams in which the photon is emitted by the hadron and absorbed by the charged lepton. Thus the physical width is not just given in terms of the matrix element of the axial current and can only be obtained by a full calculation of the electromagnetic corrections at a given order.

The calculation of electromagnetic effects leads to an immediate difficulty: contains infrared divergences and by itself is therefore unphysical. The well-known solution to this problem is to include the contributions from real photons. We therefore define to be the partial width for the decay where the energy of the photon in the rest frame of is integrated from to . The sum is free from infrared divergences (although, of course, it does depend on the energy cut-off ). We restrict the discussion to corrections, where is the electromagnetic fine-structure constant, and hence only consider a single photon.

The previous paragraph reminds us that the determination of the CKM matrix elements at (i.e. at or better) from leptonic decays requires the evaluation of amplitudes with a real photon. The main goal of this paper is to suggest how such a calculation might be performed with non-perturbative accuracy. There are a number of technicalities which will be explained in the following sections, but here we present a general outline of the proposed method. We start with the experimental observable , the partial width for . The final state consists either of or of where the energy of the photon in the centre-of-mass frame is smaller than :

(2)

In principle at least, can be evaluated in lattice simulations by computing the amplitudes for a range of photon momenta and using the results to perform the integral over phase space. Such calculations would be very challenging. Since the computations are necessarily performed in finite volumes the available momenta are discrete, so that it would be necessary to choose the volumes appropriately and compute several correlation functions. We choose instead to make use of the fact that a very soft photon couples to a charged hadron as if to an elementary particle; it does not resolve the structure of the hadron. We therefore propose to choose to be sufficiently small that the pointlike approximation can be used to calculate in perturbation theory, treating as an elementary particle. On the other hand, must be sufficiently large that can be measured experimentally. We imagine setting which satisfies both requirements. From Refs. Ambrosino:2005fw (); Ambrosino:2009aa () we learn that resolutions on the energy of the photon in the rest frame of the decaying particle of this order are experimentally accessible. In Appendix B we present a discussion, based on phenomenological analyses, of the uncertainties induced by treating the meson as elementary as a function of .

It is necessary to ensure that the cancellation of infrared divergences occurs with good numerical precision leading to an accurate result for . Since is to be calculated in a Monte-Carlo simulation and in perturbation theory using the pointlike approximation, this requires an intermediate step. We propose to rewrite Eq. (2) in the form

(3)

where is the volume of the lattice. is an unphysical quantity; it is the perturbatively calculated amplitude at for the decay with the treated as an elementary particle. In the finite-volume sum over the momenta of the photon is performed over the full range. The contributions from small momenta to and are the same and thus the infrared divergences cancel in the first term on the right-hand side of Eq. (3). Moreover, the infrared divergences in and are both equal and opposite to that in . The infrared divergences therefore cancel separately in each of the two terms on the right-hand side of Eq. (3) and indeed we treat each of these terms separately. is calculated in perturbation theory directly in infinite volume. The QCD effects in are calculated stochastically in a lattice simulation and the virtual photon is included explicitly in the Feynman gauge. For each photon momentum this is combined with and the difference is summed over the momenta and then the infinite-volume limit is taken. This completes the sketch of the proposed method, and in the remainder of this paper we explain the many technical issues which must be addressed.

It will be helpful in the following to define in terms of the first term on the right-hand side of Eq. (3):

(4)

where we have made the dependence on the volume explicit, and is the length of the lattice in any spacial direction (for simplicity we assume that this length is the same in all three directions). In analogy to Eq. (2) we also define the perturbative quantity

(5)

We note that, since the sum of all the terms in Eq. (3) is gauge invariant as is the perturbative rate , the combination is also gauge invariant, although each of the two terms is not.

The plan of this paper is as follows. In the next section we discuss the effective weak Hamiltonian and its renormalisation in the presence of electromagnetism. The structure of the calculation and the correlation functions which need to be calculated are presented in Sec. III. The evaluation of the second term on the right-hand side of Eq. (3), , directly in infinite volume, is theoretically straightforward and we perform this calculation in Sec. V. Sec. VI contains a detailed discussion of the regularisation and cancellation of infrared divergences in a finite volume. We put all the elements of the calculation together in Sec. VII, where we present a summary and the prospects for the implementation of the method in numerical simulations. There are two appendices. In Appendix A we discuss the matching of the bare lattice operators used in the calculation of correlation functions and those defined in the -regularisation which is a natural scheme used in the definition of the Fermi constant in the presence of electromagnetism. Finally in Appendix B we present some phenomenological estimates of the uncertainties due to the use of the point-like approximation for in the decay .

In the remainder of the paper, to be specific we choose but the discussion generalizes trivially to other pseudoscalar mesons with the obvious changes of flavour labels. The method does not require to be a light psuedo-Goldstone Boson nor on the use of chiral perturbation theory.

Ii Matching the effective local four-quark operator(s) onto the standard model

Figure 1: Tree-level diagram for the process (left-hand diagram). In the effective theory the interaction is replaced by a local four-fermion operator (right-hand diagram).

At lowest order in electromagnetic (and strong) perturbation theory the process proceeds by an -channel exchange, see the left-hand diagram in Fig. 1. Since the energy-momentum exchanges in this process are much smaller than , it is standard practice to rewrite the amplitude in terms of a four-fermion local interaction:

(6)

where the subscript represents left, , and is the Fermi constant. In performing lattice computations this replacement is necessary, since the lattice spacing is much greater than , where is the mass of the -Boson. When including the corrections, the ultra-violet contributions to the matrix element of the local operator are different to those in the Standard Model and in this section we discuss the matching factors which must be computed to determine the corrections to the decay from lattice computations of correlation functions containing the local operator in (6). Since the pion decay width is written in terms of , it is necessary to start by revisiting the determination of the Fermi constant at .

ii.1 Determination of the Fermi constant,

is conventionally taken from the measured value of the muon lifetime using the expression Berman:1958ti (); Kinoshita:1958ru ()

(7)

leading to the value . (For an extension of Eq. (7) to and the inclusion of higher powers of see Sec. 10.2 of Beringer:1900zz (). The Particle Data Group Beringer:1900zz () quote the corresponding value of the Fermi constant to be .)

Eq. (7) can be viewed as the definition of . When calculating the Standard Model corrections to the muon lifetime many of the contributions are absorbed into and the remaining terms on the right-hand side of (7) come from the diagrams in Fig. 2. Specifically in these diagrams the factor in the Feynman-gauge photon propagator is replaced by , where is the momentum in the propagator; this is called the -regularisation of ultra-violet divergences. These diagrams are evaluated in the effective theory with the local four-fermion operator ; the two currents are represented by the filled black circles in Fig. 2.

Figure 2: Diagrams contributing to the corrections to muon decay; see Eq. (7). The curly line represents the photon.

An explanation of the reasoning behind the introduction of the W-regularisation is given in Sirlin:1980nh (). The Feynman-gauge photon propagator is rewritten as two terms:

(8)

and the ultra-violet divergent contributions come from the first term and are absorbed in the definition of . In addition, the Standard-Model - box diagram in Fig. 3 is ultra-violet convergent and is equal to the corresponding diagram in the effective theory (i.e. the third diagram in Fig. 2) with the W-regularisation, up to negligible corrections of , where is the four-momentum of the electron and its neutrino. Other electroweak corrections not explicitly mentioned above are all absorbed into .

ii.2 -regularisation and Weak Decays of Hadrons

Figure 3: Photon- box diagrams contributing to the corrections to muon decay in the Standard Model. The curly line represents the photon.

It is a particularly helpful feature that most of the terms which are absorbed into the definition of are common to other processes, including the leptonic decays of pseudoscalar mesons Sirlin:1981ie (); Braaten:1990ef (). There are however, some short-distance contributions which do depend on the electric charges of the individual fields in the four-fermion operators and these lead to a correction factor of to  Sirlin:1981ie (). This is a tiny correction (), but one which nevertheless can readily be included explicitly.

The conclusion of the above discussion is that the evaluation of the amplitude for the process up to can be performed in the effective theory with the effective Hamiltonian

(9)

and with the Feynman-gauge photon propagator in the W-regularisation. The value of is obtained from the muon lifetime as discussed around Eq. (7).

Of course we are not able to implement the W-regularisation directly in present day lattice simulations in which the inverse lattice spacing is much smaller than . The relation between the operator in eq. (9) in the lattice and W regularisations can be computed in perturbation theory. Thus for example, with the Wilson action for both the gluons and fermions:

(10)

where

(11)

The superscript “bare” indicates that these are bare operators in the lattice theory and the presence of 5 operators on the right-hand side of Eq. (10) is a consequence of the breaking of chiral symmetry in the Wilson theory. Using lattice actions with good chiral symmetry, such as domain wall fermions with a sufficiently large fifth dimension, only would appear on the right-hand side of Eq.(10). The coefficients multiplying the operators depend of course on the lattice action being used. More details of the derivation of Eq. (10) are presented in Appendix A. Eq. (10) is valid up to corrections of  .

Having formulated the problem of calculating in terms of the evaluation of correlation functions involving the effective Hamiltonian in Eq. (9) we are now in a position to discuss the calculation of , the first term on the right-hand side of the master formula Eq. (3).

Iii Structure of the calculation

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 ; the circles are displaced for convenience.

In this section we begin our explanation of how the calculations of the amplitudes for the processes and are to be performed. Before entering into the details however, we discuss more extensively the structure of the different terms appearing in Eq. (3).

Since we add and subtract the same perturbative quantity , we find it convenient to choose this to be the virtual decay rate for a point-like pion computed in the W-regularisation. In this way we obtain the important advantage that the difference of the first two terms () and the sum of the last two terms () on the r.h.s. of Eq. (3) are separately ultraviolet and infrared finite.

Let be the contribution to the decay amplitude from the electromagnetic wave-function renormalisation of the final state lepton (see the diagram in Fig. 5(d)). An important simplifying feature of this calculation is that cancels in the difference . This is because in any scheme and using the same value of the decay constant , the contribution from the diagram in Fig. 5(d) computed non-perturbatively or perturbatively with the point-like approximation are the same. Thus we only need to calculate directly in infinite volume and include it in the second term on the right-hand side of Eq. (3). As a result of this cancellation it is convenient to rewrite and in the form:

(12)

where the superscript tree indicates the width in the absence of electromagnetic effects, denotes the contribution from the leptonic wave function renormalisation and the index represents the remaining contributions of other than those proportional to . In this notation the above discussion can be summarised by saying that and that the calculation of at reduces to that of computing .

Having eliminated the need to include the effects of the lepton’s wave-function renormalisation from the evaluation of , we need to make the corresponding modification in the factor(s) relating the lattice and regularisations. This simply amounts to subtracting the term corresponding to the matching between the lattice to regularisations of the lepton wave function renormalisation diagram. With the Wilson action (for both gluons and fermions) for example, the contribution to this matching factor is

(13)

Thus, with the Wilson action, we can avoid calculating the effects of the lepton’s wave-function renormalisation in by neglecting the diagram in Fig. 5(d) and the corresponding diagram with the point-like pion, and simply replacing in Eq.(10) by

(14)

Such matching factors depend, of course, on the lattice discretisation of QCD and we simply present the results for the Wilson action for illustration.

Of course needs to be computed for the second term on the right-hand side of Eq. (3). This is a straightforward perturbative calculation in infinite-volume and gives

(15)

where we use the -regularisation for the ultra-violet divergences and have introduced a mass for the photon in order to regulate the infrared divergences. The explicit expression for is given in Eq. (20) below. Using the -regularisation we naturally work in the Feynman gauge, but note that with as the infrared regulator the result for is generally gauge-dependent. For example, using dimensional regularisation for the ultraviolet divergences and as the infrared regulator leads to a gauge dependent result for this single diagram (gauge invariance is restored of course for ).

In summary therefore, we need to compute the two quantities

(16)

where corresponds to using instead of . Note that and are separately infrared finite and the result of the calculation of these two quantities does not depend on the infrared cutoff. In particular, this means that the infrared cutoff can be chosen in two different ways for the two quantities. We have decided to give a mass to the photon in the perturbative calculation of , whereas for a possible convenient choice is to use the finite volume as the infrared regulator. This will be explained in more detail in Sec. VI.

In the following two sections we discuss the calculation of and respectively.

Iv Calculation of

In this section we describe the calculation of the first term on the right-hand side of Eq. (3), , at . We start however, by briefly recalling the calculation of at , i.e. without electromagnetism.

iv.1 Calculation of at

Without electromagnetic corrections we need to compute the correlation function sketched in Fig. 4, which is a completely standard calculation. Since the leptonic terms are factorized from the hadronic ones, the amplitude is simply given by

(17)

Here in the matrix element represent the quark fields with the corresponding flavour quantum numbers and and the spinors of the leptons defined by the subscript. The hadronic matrix element, and hence the decay constant , are obtained in the standard way by computing the correlation function

(18)

where is an interpolating operator which can create the pion out of the vacuum, and . We have chosen to place the weak current at the origin and to create the pion at negative time , where and are sufficiently large to suppress the contributions from heavier states and from the backward propagating pions (this latter condition may be convenient but is not necessary). The subscript or superscript here denotes the fact that the calculation is performed at , i.e. in the absence of electromagnetism. is obtained from the two-point correlation function of two operators:

(19)

For convenience we take to be a local operator (e.g. at in Eq. (18)), but this is not necessary for our discussion. Any interpolating operator for the pion on the chosen time slice would do equally well.

Having determined and hence the amplitude , the contribution to the decay width is readily obtained

(20)

In this equation we use the label tree to denote the absence of electromagnetic effects since the subscript here indicates that there are no photons in the final state.

iv.2 Calculation at

Figure 5: Connected diagrams contributing at contribution to the amplitude for the decay .
Figure 6: Disconnected diagrams contributing at contribution to the amplitude for the decay . The curly line represents the photon and a sum over quark flavours , and is to be performed.

We now consider the one-photon exchange contributions to the decay and show the corresponding six connected diagrams in Fig. 5 and the disconnected diagrams in Fig. 6. By “disconnected” here we mean that there is a sea-quark loop connected, as usual, to the remainder of the diagram by a photon and/or gluons (the presence of the gluons is implicit in the diagrams). The photon propagator in these diagrams in the Feynman gauge and in infinite (Euclidean) volume is given by

(21)

In a finite volume the momentum integration is replaced by a summation over the momenta which are allowed by the boundary conditions. For periodic boundary conditions, we can neglect the contributions from the zero-mode since a very soft photon does not resolve the structure of the pion and its effects cancel in in Eq. (3). Although we evaluate (see Eq. (2)) in perturbation theory directly in infinite volume, we note that the same cancellation would happen if one were to compute also in a finite volume. Moreover from a spectral analysis we conclude that such a cancellation also occurs in the Euclidean correlators from which the different contributions to the decay rates are extracted. For this reason in the following and are evaluated separately but using the following expression for the photon propagator in finite volume:

(22)

where all quantities are in lattice units and the expression corresponds to the simplest lattice discretisation. , , and are four component vectors and for illustration we have taken the temporal and spatial extents of the lattice to be the same ().

For other quantities, the presence of zero momentum excitations of the photon field is a subtle issue that has to be handled with some care. In the case of the hadron spectrum the problem has been studied in Hayakawa:2008an () and, more recently in Borsanyi:2014jba (); Basak:2014vca (), where it has been shown, at , that the quenching of zero momentum modes corresponds in the infinite-volume limit to the removal of sets of measure zero from the functional integral and that finite volume effects are different for the different prescriptions.

We now divide the discussion of the diagrams in Fig. 5 and Fig. 6 into three classes: those in which the photon is attached at both ends to the quarks (diagrams 5(a)-5(c) and 6(a), (b) and (d)), those in which the photon propagates between one of the quarks and the outgoing lepton (diagrams 5(e), 5(f) and 6(c)) and finally diagram 5(d) which corresponds to the mass and wave-function normalisation of the charged lepton. We have already discussed the treatment of the wave function renormalisation of the lepton in detail in Sec. III so we now turn to the remaining diagrams.

iv.2.1 The evaluation of diagrams Fig. 5(a)-(c) and Fig. 6(a),(b) and (d)

We start by considering the connected diagrams 5(a)-(c). For these diagrams, the leptonic contribution to the amplitude is contained in the factor and we need to compute the Euclidean hadronic correlation function

(23)

where represents time-ordering, is the current and we take . is the hadronic component of the electromagnetic current and we find it convenient to include the charges of the quarks in the definition of :

(24)

where the sum is over all quark flavours . The factor of is the standard combinatorial one.

The computations are performed in Euclidean space and in a finite-volume with the photon propagator given in Eq. (22) (or the corresponding expression for other lattice discretisations). The absence of the zero mode in the photon propagator implies a gap between and the energies of the other eigenstates. Provided one can separate the contributions of these heavier states from that of the pion, one can perform the continuation of the correlation function in Eq. (23) from Minkowski to Euclidean space without encountering any singularities. From the correlation function we obtain the electromagnetic shift in the mass of the pion and also a contribution to the physical decay amplitude, as we now explain. For sufficiently large the correlation function is dominated by the ground state, i.e. the pion, and we have

(25)

where the electromagnetic terms are included in all factors (up to ). Writing , where is the mass shift,

(26)

so that is of the schematic form

(27)

By determining we obtain the electromagnetic mass shift, , and from we obtain the electromagnetic correction to  . Note that is gauge invariant and infrared finite, whereas the coefficient obtained from these diagrams is neither.

In order to obtain the contribution to the decay amplitude we need to remove the factor on the right-hand side of Eq. (25), including the corrections to this factor. Having determined , we are in a position to subtract the corrections present in . The corrections to are determined in the standard way, by performing the corresponding calculation to but with the axial current replaced by :

(28)
(29)

We finally obtain

(30)

and the contribution to the amplitude from these three diagrams is

(31)

For these three diagrams the term can be simply considered as a correction to . Note however, that such an “” would not be a physical quantity as it contains infrared divergences.

The treatment of the disconnected diagrams in Figs. 6(a), (b) and (d) follows in exactly the same way. These diagrams contribute to the electromagnetic corrections to both the pion mass and the decay amplitude in an analogous way to the discussion of the connected diagrams above . It is standard and straightforward to write down the corresponding correlation functions in terms of quark propagators. We do not discuss here the different possibilities for generating the necessary quark propagators to evaluate the diagrams; for example we can imagine using sequential propagators or some techniques to generate all-to-all quark propagators.

iv.2.2 The evaluation of diagrams Fig. 5(e)-(f)

For these diagrams the leptonic and hadronic contributions do not factorise and indeed the contribution cannot be written simply in terms of the parameter . We start by considering the Minkowski space quantity

where and are the lepton and (Feynman gauge) photon propagators respectively in Minkowski space (more precisely the photon propagator with Lorentz indices is , but the Lorentz indices have been contracted with the electromagnetic currents in (IV.2.2)). In order to demonstrate that we can obtain the corrections to the decay amplitude from a Euclidean space correlation function, we use the reduction formula to rewrite the expression in Eq. (IV.2.2) as

(33)

where is the field which creates a pion with amplitude 1. On the other hand the Euclidean space correlation function which we propose to compute is

(34)

Here and are Euclidean propagators, and are spinor indices. Similarly to the discussion in Sec. IV.2.1, provided that the pion is the lightest hadronic state then for large , is dominated by the matrix element with a single pion in the initial state.

Figure 7: Zoom of the lepton-photon vertex at from the diagrams in Fig. 5(e) and (f).

In view of the factor on the right-hand side of Eq. (34), the new feature in the evaluation of the diagrams in Fig. 5 (e) and (f) is that we need to ensure that the integration converges as . For the convergence of the integral is improved by the presence of the exponential factor and so we limit the discussion to the case . is the energy of the outgoing charged lepton with three-momentum . To determine the behaviour, consider the lepton-photon vertex at from the diagrams in Fig. 5(e) and (f), redrawn in Fig. 7. and are the four-momentum variables in the Fourier transform of the propagators and respectively in Eqs. (IV.2.2) - (34). The integration is indeed convergent as we now show explicitly.
1. The integration over implies three-momentum conservation at this vertex so that in the sum over the momenta , where is the momentum of the outgoing charged lepton.
2. The integrations over the energies and lead to the exponential factor , where , , and is the mass of the photon introduced as an infra-red cut-off. The large behaviour is therefore given by the factor .
3. A simple kinematical exercise shows that in the sum over (with ), the minimum value of is given by

(35)

4. Thus for non-zero , the exponent in for large is negative for every term in the summation over and the integral over is convergent so that the continuation from Minkowski to Euclidean space can be performed.
5. We note that the integration over is also convergent if we set but remove the mode in finite volume. In this case .

In summary the integration is convergent because for every term in the sum over momenta and so for sufficiently large we can write

(36)

and the contribution from the diagrams of Fig. 5(e) and 5(f) is . This completes the demonstration that the Minkowski-space amplitude (33) is equal to the pion contribution to the Euclidean correlation function (34), up to a factor which accounts for the normalisation of the pion field.

Again the evaluation of the correction to the amplitude from the disconnected diagram in Fig. 6(c) follows in an analogous way.

V Calculation of

The evaluation in perturbation theory of the total width in infinite volume, was performed by Berman and Kinoshita in 1958/9 Berman:1958ti (); Kinoshita:1959ha (), using the Pauli-Villars regulator for the ultraviolet divergences and a photon mass to regulate the infrared divergences in both and . is the rate for process for a pointlike pion with the energy of the photon integrated over the full kinematic range. We have added the label pt in to remind us that the integration includes contributions from regions of phase space in which the photon is not sufficiently soft for the structure of the pion to be reliably neglected. We do not include this label when writing because we envisage that is sufficiently small so that the pointlike approximation reproduces the full calculation.

In our calculation, is evaluated in the W-regularisation, so that the ultra-violet divergences are replaced by logarithms of . For convenience we rewrite here the expression for from Eq. (16)

(37)

and have already been presented in Eqs. (20) and (15) respectively. In the following we give separately the results of the remaining contributions to also using a photon mass as the infrared regulator. We neglect powers of in all the results.

In the perturbative calculation we use the following Lagrangian for the interaction of a point-like pion with the leptons:

(38)

The corresponding Feynman rules are:

(39)

In addition we have used the standard Feynman rules of scalar electromagnetism for the interactions of charged pions in an electromagnetic field.

Figure 8: One loop diagrams contributing to the wave-function renormalisation of a point-like pion.

We start by giving the contributions to .
Wave function renormalisation of the pion: The contribution of the pion wave function renormalisation to is obtained from the diagrams in Fig. 8 and is given by

(40)

These diagrams correspond to those in Fig. 5(a), Fig. 5(b) and Fig. 5(c) in the composite case.

Figure 9: Radiative corrections to the pion-lepton vertex. The diagrams represent contributions to . 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.

  -  Vertex: The remaining graphs contributing to are the  -  vertex corrections from the diagrams shown in Fig. 9 and their complex conjugates. The contribution from these diagrams is

(42)