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We present a nonperturbative lattice calculation of the form factors which contribute to the amplitudes for the radiative decays

The unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is one of the most precise tests of the Standard Model. Indeed, CKM unitarity may rule out many theoretically well-motivated models for new physics and put severe constraints on the energy scale where new phenomena might occur, well beyond the range accessible to direct experimental searches. In this respect, leptonic decay rates of light and heavy pseudoscalar mesons are essential ingredients for the extraction of the CKM matrix elements. A first-principles calculation of these quantities requires nonperturbative accuracy and hence numerical lattice simulations. Moreover, in order to fully exploit the presently available experimental information and to perform the next generation of flavor-physics tests,

Knowledge of the radiative leptonic decay rate in the region of small (soft) photon energies is required in order to properly define the infrared-safe measurable decay rate for the process

On the one hand, in the limit of ultrasoft photon energy, the radiative decay rate can be reliably calculated in an effective theory in which the meson is treated as a pointlike particle. This is a manifestation of the well-known mechanism known as the “universality of infrared divergences” (see e.g. Refs.

In the region of hard (experimentally detectable) photon energies, radiative leptonic decays represent important probes of the internal structure of the mesons. Moreover, radiative decays can provide independent determinations of CKM matrix elements with respect to the purely leptonic channels. A nonperturbative calculation of the radiative decay rates can be particularly important for heavy mesons since, unlike the case of pions and kaons where such decays have been studied using chiral perturbation theory (ChPT)

In Ref.

In this paper, we present the first nonperturbative lattice calculation of the rates for the radiative decays

The plan of the remainder of this paper is as follows. In Sec.

The nonperturbative contribution to the radiative leptonic decay rate for the processes

Feynman diagrams representing the amplitudes with the emission of a real photon from the

The decomposition of

By setting

Our definition of the form factor

In order to relate the hadronic matrix element to Euclidean correlation functions, the primary quantities computed in lattice calculations, it is useful to express the

The important observation that makes the lattice calculation possible by using standard effective-mass/residue techniques is that the integrals over

The expressions for the correlation function

Schematic diagrams representing the correlation function

In Fig.

The diagram on the left represents the contributions to the correlation functions arising from the emission of the photon by the sea quarks. In our numerical simulations, we work in the electroquenched approximation and neglect such diagrams. The diagram on the right explains our choice of the spatial boundary conditions, which allows us to set arbitrary values for the meson and photon spatial momenta. The spatial momenta of the valence quarks, modulo

The quark-connected diagram in the right panel of Fig.

The numerical results presented in the following sections have been obtained by setting the nonzero components of the spatial momenta along the third direction, that is,

The ratios

In this section, we want to stress a very important issue associated with infrared divergent cutoff effects which can jeopardize the extraction of

In Fig.

The blue circles represent

At finite lattice spacing, the axial form factor is constrained, as in the continuum [see Eq.

We assume here that we are using a lattice discretization in which the leading artifacts are

Study of

We now present an alternative strategy that avoids this problem. In Appendix

The knowledge of

The results presented in this paper were obtained using the ETMC gauge ensembles with

The scale setting is taken from Ref.

In Figs.

Examples of fits to plateaux for the ratio

Examples of fits to plateaux for the ratio

In order to extract the form factors

For pions and kaons, we have covered the full physical range of

For the pion, guided by ChPT, we fit to the formula

When using the simpler expression in Eq.

Dependence of

In Fig.

Extracted values of the pion (left) and kaon (right) form factors

Extracted values of the pion (left) and kaon (right) form factors

For heavy mesons

In this first study, we only have results for the

The form factors

In Fig.

The form factors

The form factors

We also study our physical results (i.e those obtained after the continuum and chiral extrapolations) as a function of

For the axial form factors, we find

For the pion and kaon, we can compare the constants

The pion vector form factor

In the remainder of this section, we present a brief comparison of our results with experimental data. A more detailed phenomenological analysis will be presented in a separate paper.

For the pion, the Particle Data Group (PDG)

For the kaon, the PDG quotes the two combinations

The results in Eqs.

For completeness, we also present the constants

In conclusion, we have shown that by using lattice QCD, even with moderate statistics, it is possible to predict with good precision the structure-dependent form factors

We found that the extraction of the axial form factor

Although the present study clearly can and will be improved by, for example, increasing the statistics, covering the full range of

In future, we also plan to study the emission of off-shell photons (

We gratefully acknowledge helpful discussions with M. Testa. We acknowledge PRACE for awarding us access to Marconi at CINECA, Italy under the Grant No. Pra17-4394. We also acknowledge use of CPU time provided by CINECA under the specific initiative INFN-LQCD123. V. L., G. M., and S. S. thank MIUR (Italy) for partial support under the contract PRIN 2015. The work of C. T. S. was partially supported by STFC (UK) Grant No. ST/P000711/1 and by an Emeritus Fellowship from the Leverhulme Trust. N. T. and R. F. acknowledge the University of Rome Tor Vergata for the support granted to the project PLNUGAMMA. The work of F. S. and S. S. was supported by the Italian Ministry of Research (MIUR) under Grant No. PRIN 20172LNEEZ. The work of F. S. was supported by INFN under GRANT73/CALAT.

In this appendix, we present the explicit formulas needed to evaluate the total and differential decay rates at order

The exchange of a virtual photon depends on the hadron structure, since all momentum modes are included, and the amplitude must therefore be computed nonperturbatively. On the other hand, the nonperturbative evaluation of the amplitude for the emission of a real photon is not strictly necessary

To calculate the partial rates for the emission of a hard real photon, it is sufficient to know the SD form factors,

The terms in the first parentheses on the right-hand side of Eq.

In the terms in the second parentheses on the right-hand side of Eq.

Finally, the term on second line of the right-hand side of Eq.

We express the differential decay rate in terms of the following quantities:

The two dimensionless kinematical variables

The decay constant of the meson

The two SD axial and vector form factors

The differential decay rate is given by the sum of three contributions,

The quantities in the braces on the right-hand side of Eq.

Equations

In this appendix, we derive some useful formulas for the extraction of the two relevant form factors,

In order to construct the finite

The “topology” of the correlation function in Eq.

The incoming meson is interpolated at fixed spatial momentum

The hadronic weak current

The electromagnetic current

We have used

A technical subtlety needs to be stressed here. As discussed in the main text, in order to choose arbitrary (nondiscretized) values of the spatial momenta for the meson and for the photon, we have introduced a “flavored” extension of the electromagnetic current (see the explanation in the caption of Fig.

We are now ready to define the finite-

In the continuum and large-

For

It is useful to note that, in order to separate the axial and vector form factors, it is enough to compute separately the correlation functions corresponding to the vector,

In all the formulas of this appendix, we have used continuum notation for the four vectors but the momentum and energy carried by the current (including the associated projectors) have to be read by performing the following substitutions:

In the lattice regularization that we are using (i.e. Wilson quarks at maximal twist), Eqs.

In this appendix, we study the WI that relates the axial correlation function

We start with a remark about the matrix element of the axial current, determined at a finite lattice spacing

In this appendix, as in Sec.

Consider the following correlation function which is relevant to our study:

The WI can be rewritten in the form

Consider first the case with

By iterating order by order in

From the above discussion, we conclude that the lattice

In order to implement the strategy described in Eq.

The problem we now address is to study the limit

By using the previous two expressions and by differentiating Eq.

As can be seen, the structure of

In this appendix, we present some numerical information that may be useful to the reader. We start by listing in Tables

There was a mistake in the values of the lattice spacing values of the simulated sea and valence quark bare masses for each ensemble used in this work. The table is the same as in Ref.

Central values of the pion mass

Given the smooth behavior that we find for the form factors as functions of

Pion

Kaon

D-meson (linear)

D-meson (pole)