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We present a study of lattice-QCD methods to determine the relevant hadronic form factors for radiative-leptonic decays of pseudoscalar mesons. We provide numerical results for

In this paper, we develop and test lattice-QCD methods for computing the hadronic matrix elements describing

Knowledge of the radiative-leptonic decay rate in the region of small (soft) photon energies is required to include

In the region of hard (experimentally detectable) photon energies, radiative-leptonic decays represent important probes of the internal structure of the mesons and also provide sensitive probes of physics beyond the Standard Model inducing nonstandard currents and/or nonuniversal corrections to the lepton couplings. For example, the rare decays

The experimental status for radiative-leptonic decays (with detected photons of energy above some specified lower limit) can be summarized as follows. For the kaon and pion decays

In the Standard Model, the hadronic contributions to the

For low photon energies, the form factors can be studied using chiral perturbation theory (ChPT), which has been done for light-meson radiative-leptonic decays in Refs.

All of these limitations can be overcome, at least in principle, using lattice gauge theory, a nonperturbative formulation of QCD that does not introduce new parameters beyond those of QCD itself and whose precision is limited only by the available computing resources. Numerical lattice-QCD calculations based on the path-integral formulation are performed in Euclidean spacetime, which may pose challenges for time-dependent matrix elements. As we showed in Ref.

While the present work was in progress, an independent lattice study of radiative-leptonic decays was published in Ref.

Here we extend our preliminary work

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

In this work, we focus on charged-current decays

Unlike the vector form factor, the axial form factor is composed of two pieces, namely a structure-dependent contribution and a pointlike contribution. The pointlike contribution describes the part of the decay amplitude when the photon does not probe the internal structure of

In Sec.

In this section, we show how to extract

Starting with the

Before proceeding, it is worth noting that, on a periodic lattice, one must be careful however when taking the

Schematic visualization of the different time orderings for the Euclidean-time three-point function in Eq.

In this section, we describe two different methods of calculating the time-integrated correlation function

The left (right) figure is a schematic drawing of the 3D (4D) methods. For both methods, the initial noise source is located at the weak-current time. The sequential propagator for the 3D (4D) method is shown in green (orange) and the sequential source is circled in green (orange).

One limitation of the 4D method is that, because the integral over

In this work, in order to control systematic errors from the unwanted exponentials, we perform the calculation for multiple values of

Number of propagator solves required for a single configuration for a single source in terms of the desired number of meson momenta

In this section, we describe the properties of the lattice we perform calculations on as well as the details of our numerical setup. As previously described in the introduction, we have performed two sets of calculations. We start with the common parameters between them and then discuss the differences.

Both calculations were performed on a single RBC/UKQCD ensemble

We use local currents in our calculation. The matching factors of the individual quark components of the electromagnetic current were computed nonperturbatively using charge conservation. We employ “mostly nonperturbative” renormalization of the weak axial-vector and vector currents

The results in Secs.

The methods, sources, and momenta for which we performed calculations in Secs.

The calculations in Secs.

Another set of questions are the particular details of how the time integrals

In this section, we compare the statistical precision of the vector form factor calculated using both noise and point sources.

Note that we did not perform the necessary calculations to extract

Figure

Ratio of statistical uncertainties of point to noise sources as a function of summation range. The left and right plots show

The differences in behavior of the two time orderings can be understood by considering the maximum Euclidean-time separation between any of the three operators in the correlation function. The maximum time separation between any two operators in the

To determine which source offers the best precision to computational cost ratio, we need to refer back to Table

One additional factor to consider is that noise sources benefit from volume averaging, while point sources do not. Because our numerical test was performed on a relatively small lattice with

In this section we describe the fit methods used to remove unwanted exponentials from the form-factor results presented here and in Sec.

We start by studying, in continuum QCD, the quantum numbers of the states that have a nonzero contribution to the sum over states in the spectral decompositions of

Note that because we neglect disconnected diagrams in this present work, certain states will not contribute to the spectral decomposition of

From this discussion we also learn that in general, for a particular momentum and a given time ordering, the same states contribute to all

To help constrain the energy gap

We fit our data as a function of both source-sink separation

The fit form used for the 3D method data includes one exponential to account for the unwanted exponential from the lowest-energy excited state created by the interpolating field and one exponential for the unwanted exponential that comes with the lowest-energy intermediate state. The fit forms used for the

Because the 4D data are a sum of both time orderings, one possible fit form would be a sum of those in Eqs.

Figure

In this section, we compare the 3D, 4D, and

We start by comparing the 4D and

Comparison of the 3D, 4D, and

The right plot in Fig.

If one uses both methods, however, our results suggest that a factor of

In the following, we describe our improved methods of calculating lattice correlators that will be used to extract the form factors using the 3D method. We begin by discussing the infinite-volume approximation, which allows us to calculate the three-point functions at arbitrary photon momentum (i.e., not subject to the usual restriction from the periodic boundary conditions) with errors exponentially small in the lattice volume. Then, in Sec.

In this section we describe our approach to estimate momentum-projected correlation functions at arbitrary momenta (i.e., not restricted to integer multiples of

In practice,

Finally, we would like to point out that the method is similar in spirit to the

The three-point function in Eq.

We define the time integrals of this correlation function for the different time orderings as

Notice that the spectral decompositions of the

This example leads to a clear scenario where having both sets of data would be crucial for the analysis. In particular, consider the possibility where the results of the fits to the

Considering instead the opposite scenario where one observes stability for both time orderings of both

In Sec.

In this section we describe improvements to the original 3D method calculation presented in Secs.

Section

Section

Section

Section

Another modification to the analysis is that we now fit the contributions to the form factors from the separate quark components of the electromagnetic current separately. This is done for two reasons. First, these separate contributions are well-defined QCD form factors and are therefore of phenomenological interest. The second is that, in general, the intermediate states that contribute to the different quark contributions are different. Fitting them separately therefore reduces the possible number of exponential that contribute at finite integration range

To be able to use the infinite-volume approximation in Sec.

Specifically, suppose we calculate a three-point function [either in Eq.

When deciding what ratio to take for the weak axial-vector component of the three-point function, for each value of

no ratio,

ratio using

ratio using

ratio using the two values of

To test the improvements gained using this ratio method, we compare the form factors as a function of

Left (right) compares

Looking first at

One advantage of our improved method is the ability to average over the positive and negative photon momenta for free. In this section, we compare the precision of the form factors calculated by performing this average to form factors calculated using only positive photon momentum. As in Sec.

Looking at Fig.

Left (right) column compares

Another observation from Fig.

In this section, we compare three different methods for calculating

Using method I, the structure-dependent part of the axial form factor is first calculated as a function of

Method II improves upon the first by exactly subtracting the unwanted exponential contribution from the

The third method, originally put forth in Ref.

Another advantage of method III has to do with discretization effects. It was shown in Ref.

In this section, we compare form-factor results calculated using the three-point functions in Eqs.

Starting with the

Figure

Left (right) column compares

To summarize, data calculated using either of the three-point functions in Eq.

In this section we summarize the improved methods used to extract the form factors.

Using the 3D method, we calculate the three-point functions with the weak and electromagnetic currents at the origin using the infinite-volume approximation method described in Sec.

Contrary to the analysis methods outlined in Sec.

For fits to

The vector-form-factor fits are performed using the methods outlined in Sec.

Fit results for

Fit results for

The results of

Results of

In this work, we presented a study of lattice-QCD data-generation and analysis methods to determine the form factors describing radiative-leptonic decays of pseudoscalar mesons. We calculated the relevant nonlocal matrix elements using the 3D, 4D, and

From there, we further improved upon the 3D method by calculating the three-point function using the infinite-volume approximation method, which allows us to access the full range of kinetically allowed photon momenta without having to perform calculations in the moving frame of the meson. We then showed that the hadronic tensor could be extracted using an alternate three-point function with the electromagnetic current at the origin rather than the weak current at origin. The alternate three-point function can be calculated by reusing propagators required for the original three-point function. Performing simultaneous fits to both datasets resulted in reductions in statistical noise for both

Further improvements in the statistical precision were achieved by multiplying the desired three-point function by ratios of three-point functions calculated using noise and point sources. This procedure resulted in significant improvements in precision for

Using the improved lattice methods developed in this work, we plan to perform calculations on more ensembles and perform extrapolations to the physical pion mass and the continuum for the pion, kaon,

Precise determinations of the QCD form factors for radiative-leptonic decays are relevant for a number of phenomenological reasons. At small photon energies, a calculation of the radiative-leptonic decay rate is needed in order to include

S. M. thanks Diego Guadagnoli for asking the question whether the form factors describing radiative-leptonic decays are calculable on the lattice. We thank the RBC and UKQCD Collaborations for providing the gauge-field configurations. C. F. K. is supported by the Department of Energy (DOE) Computational Science Graduate Fellowship under Grant No. DE-SC0020347. S. M. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Grant No. DE-SC0009913. A. S. is supported in part by the U.S. DOE Contract No. DE-SC0012704. This research used resources provided by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and the Extreme Science and Engineering Discovery Environment (XSEDE)

In the following, we discuss the nontrivial limit of the three-point axial-vector correlation function

In general, the lattice vector WI at finite lattice spacing reads

In the case of the correlation function

To implement method III described in Sec.

By differentiating Eq.

As can be seen, the structure of

We observe that the use of a nonconserved electromagnetic current