Lattice Inputs to Flavor Physics



CP-Origins and Danish IAS, Campusvej 55, 5230 Odense M, Denmark and

IFIC (CSIC), Calle Catedrático José Beltran, 2, 46980 Paterna, Valencia, Spain

We review recent lattice results for quark masses and low-energy hadronic parameters relevant for flavor physics. We do that by describing the FLAG initiative, with emphasis on its scope and rating criteria. The emerging picture is that while for light quantities a large number of computations using different approaches exist, and this increases the overall confidence on the final averages/estimates, in the heavy-light case the field is less advanced and, with the exception of decay constants, only a few computations are available.

The precision reached for the light quantities is such that electromagnetic (EM) corrections, beyond the point-like approximation, are becoming relevant. We discuss recent computations of the spectrum based on direct simulations of QED+QCD. We also present theoretical developments for including EM effects in leptonic decays.

We conclude describing recent results for the transition amplitudes and prospects for tackling hadronic decays on the lattice.

1 Introduction and FLAG

After its discovery in 2012, the Higgs boson was believed to provide a portal to New Physics. This is even somehow assumed when formulating the hierarchy problem of the Standard Model (SM). However, this far, all measurements of the Higgs boson properties lie within 20% of the SM expectations, as reported by ATLAS  and CMS . Instead, there is a number of 2-3 sigmas tensions in rare processes (see for example ), with the most prominent examples being in the angular analysis of the decay  and in the enhancement of the decays  . Significances depend on treatment of several non-perturbative effects. Extrapolating to the future, (some of) these rare decays won’t be so rare anymore. Belle 2 will report results from about 2018 and coexist with the LHC and High Luminosity (HL)-LHC, after Long Shutdown 3 in 2023-2025. Progress on the theoretical side is needed in many instances, to match the expected experimental accuracy.

There are many different groups, all over the world, using different lattice methods, that calculate hadronic matrix elements relevant for a number of weak decay processes of , , and mesons. With so many groups calculating similar matrix elements using different methods, and all providing phenomenologically relevant results with complete error budgets, it is useful to try to produce global averages/estimates and to review virtues and shortcomings of the different computations in a transparent way, which should be accessible also to the non-experts. This is the goal of the FLAG initiative.

1.1 The FLAG review

The Flavor Lattice Averaging Group started its activity in 2010 focusing on light-quark quantities and providing averages from lattice results with comprehensive error budgets . A second similar initiative was started at around the same time , focusing on both heavy- and light-quark quantities. The two groups joined for the second edition of the FLAG-review  (FLAG-2). One of the main goals of FLAG is to assess the reliability of systematic error estimates, in particular concerning continuum extrapolations, chiral extrapolations, finite volume effects and renormalization. This is done through quality criteria by assigning to each computation a symbol for each one of the systematics above. For example, the symbols and the criteria adopted for the light-quark quantities in FLAG-2 are:

  • Chiral extrapolation:
        200 MeV 400 MeV
        400 MeV
    in addition it is assumed that the chiral extrapolation is done using at least three points.

  • Continuum extrapolation:
      3 or more lattice spacings, at least 2 points below 0.1 fm
        2 or more lattice spacings, at least 1 point below 0.1 fm
    in addition it is assumed that the action is -improved (i.e. the discretization errors vanish quadratically with the lattice spacing).

  • Finite-volume effects:
    or at least 3 volumes
        and at least 2 volumes

  • Renormalization (where applicable):
        1-loop perturbation theory or higher with a reasonable estimate of truncation errors

For heavy-light quantities the criteria are similar, with some additional ones concerning discretization effects and treatment of heavy quarks. In general criteria are expected to change in time and possibly become stricter as lattice computations reach new levels of accuracy. In the end, all the published (in journals) results with no red symbols enter the final estimates/averages. In some cases, the averaging procedure leads to results which in the opinion of the authors do not cover all uncertainties. In these cases, in order to stay on the conservative side, averages are replaced by estimates (or ranges), which are considered fair assessments of the current knowledge acquired on the lattice. These estimates are based on a critical (and to some extent subjective) analysis of the available information.

In detail, the FLAG-2 collaboration counted 28 members representing the major lattice groups in the world. Different Working Groups were in charge of reviewing different sets of quantities: Quark masses (WG1), (WG2), PT Low Energy Constants (WG3), (WG4), , , (WG5, our group), , semileptonic and radiative decays (WG6), and finally (WG6). In the following we will focus on the subset of quantities presented during the talk. A more recent update of lattice results concerning heavy-light quantities can be found in . In Fig. 1 we show the results for the strange quark mass and the average up and down quark mass.

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