In previous work [D. Giusti, et al., Methods for high-precision determinations of radiative-leptonic decay form factors using lattice QCD, <a href="http://dx.doi.org/10.1103/PhysRevD.107.074507">Phys. Rev. D 107, 074507 (2023)</a>], we showed that form factors for radiative leptonic decays of pseudoscalar mesons can be determined efficiently and with high precision from lattice QCD using the “three-dimensional (3D) method,” in which three-point functions are computed for all values of the current insertion time and the time integral is performed at the data-analysis stage. Here, we demonstrate another benefit of the 3D method: the form factors can be extracted for any number of nonzero photon virtualities from the same three-point functions at no extra cost. We present results for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub><mo stretchy="false">→</mo><mo>ℓ</mo><mi>ν</mi><msup><mi>γ</mi><mo>*</mo></msup></math> vector form factor as a function of photon energy and photon virtuality, for both positive and negative virtuality, for a single ensemble with 340 MeV pion mass and 0.11 fm lattice spacing. In our analysis, we separately consider the two different time orderings and the different quark flavors in the electromagnetic current. We discuss in detail the behavior of the unwanted exponentials contributing to the three-point functions, as well as the choice of fit models and fit ranges used to remove them for various values of the virtuality. While positive photon virtuality is relevant for decays to multiple charged leptons, negative photon virtuality suppresses soft contributions and is of interest in QCD-factorization studies of the form factors.

Germany

Davide Giusti

United States

Christopher F. Kane

Christoph Lehner

Stefan Meinel

Amarjit Soni

Efficient lattice QCD computation of radiative-leptonic-decay form factors at multiple positive and negative photon virtualities

We develop an extended framework for the hybrid Monte Carlo (HMC) algorithm in lattice gauge theory by embedding the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> group into the space of general complex matrices, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">C</mi><mo stretchy="false">)</mo></math>. Auxiliary directions will be completely factorized in the path integral, and the embedding does not alter the expectation values of the original theory. We perform the molecular dynamics updates by using the matrix elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>W</mi><mo>∈</mo><msub><mi>M</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi mathvariant="double-struck">C</mi><mo stretchy="false">)</mo></math> as the dynamical variables without group theoretic constraints. The framework enables us to introduce nonseparable Hamiltonians for the HMC in lattice gauge theory exactly, whose immediate application includes the Riemannian manifold HMC.

Norman H. Christ

Lu-Chang Jin

Erik Lundstrum

Nobuyuki Matsumoto

Extended framework for the hybrid Monte Carlo in lattice gauge theory

The separation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment into Euclidean windows allows for a tailored approach to address the different dominant challenges at short, intermediate, and long distances. We present a novel approach to compute the short-distance window without the need for using perturbative QCD. We combine a quenched continuum extrapolation using 18 lattice spacings (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1.6</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>≲</mo><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≲</mo><mn>6.1</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>) with a separate continuum extrapolation of the sea quark effects. This method allows for the computationally expensive sea quark effects to be estimated using only a smaller number of ensembles at coarser lattice spacings, while largely confining the logarithmic dependency of the continuum extrapolation to the quenched component.

Sebastian Spiegel

High-precision continuum limit study of the HVP short-distance window

The hadronic light-by-light scattering contribution to the muon anomalous magnetic moment, <math><mrow><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><mo>)</mo><mrow><mo>/</mo><mn>2</mn></mrow></mrow></math>, is computed in the infinite volume QED framework with lattice QCD. We report <math><msubsup><mi>a</mi><mi>μ</mi><mtext>HLbL</mtext></msubsup><mo>=</mo><mn>12.47</mn><mo>(</mo><mn>1.15</mn><mo>)</mo><mo>(</mo><mn>0.95</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math> where the first error is statistical and the second systematic. The result is mainly based on the <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor Möbius domain wall fermion ensemble with inverse lattice spacing <math><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1.73</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, lattice size <math><mi>L</mi><mo>=</mo><mn>5.5</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>139</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, generated by the RBC-UKQCD collaborations. The leading systematic error of this result comes from the lattice discretization. This result is consistent with previous determinations.

Thomas Blum

Norman Christ

Japan

Masashi Hayakawa

Taku Izubuchi

Luchang Jin

Chulwoo Jung

Cheng Tu

Hadronic light-by-light contribution to the muon anomaly from lattice QCD with infinite volume QED at physical pion mass

We present a lattice calculation of the <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> matrix elements and amplitudes with both the <math><mi>Δ</mi><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> and <math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math> channels and <math><msup><mi>ϵ</mi><mo>′</mo></msup></math>, the measure of direct <math><mi>C</mi><mi>P</mi></math> violation. We use periodic boundary conditions (PBC), where the correct kinematics of <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> can be achieved via an excited two-pion final state. To overcome the difficulty associated with the extraction of excited states, our previous work [Z. Bai et al., Standard Model prediction for direct <math><mi>C</mi><mi>P</mi></math> violation in <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay, Phys. Rev. Lett. 115, 212001 (2015); R. Abbott et al., Direct <math><mi>C</mi><mi>P</mi></math> violation and the <math><mi>Δ</mi><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> rule in <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay from the Standard Model, Phys. Rev. D 102, 054509 (2020).] successfully employed G-parity boundary conditions, where pions are forced to have nonzero momentum enabling the <math><mi>I</mi><mo>=</mo><mn>0</mn></math> two-pion ground state to express the on shell kinematics of the <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay. Here instead we overcome the problem using the variational method which allows us to resolve the two-pion spectrum and matrix elements up to the relevant energy where the decay amplitude is on shell. In this paper we report an exploratory calculation of <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay amplitudes and <math><msup><mi>ϵ</mi><mo>′</mo></msup></math> using PBC on a coarser lattice size of <math><msup><mn>24</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> with inverse lattice spacing <math><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1.023</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and the physical pion and kaon masses. The results are promising enough to motivate us to continue our measurements on finer lattice ensembles in order to improve the precision in the near future.

Peter A. Boyle

Switzerland

Daniel Hoying

Christopher Kelly

Masaaki Tomii

<math><mi>Δ</mi><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> and <math><mi>Δ</mi><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> channels of <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay at the physical point with periodic boundary conditions

A study of two-pion scattering for the isospin channels, <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math><mi>I</mi><mo>=</mo><mn>2</mn></math>, using lattice QCD is presented. Möbius domain-wall fermions, on top of the Iwasaki-DSDR gauge action for gluons with periodic boundary conditions, are used for the lattice computations, which are carried out on two ensembles of gauge field configurations generated by the RBC and UKQCD Collaborations with physical masses, inverse lattice spacings of 1.023 and 1.378 GeV, and spatial extents of <math><mi>L</mi><mo>=</mo><mn>4.63</mn></math> and 4.58 fm, respectively. The all-to-all propagator method is employed to compute a matrix of correlation functions of two-pion operators. The generalized eigenvalue problem (GEVP) is solved for a matrix of correlation functions to extract phase shifts with multiple states—two pions with a nonzero relative momentum, as well as two pions at rest. Our results for phase shifts for both the <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math><mi>I</mi><mo>=</mo><mn>2</mn></math> channels are consistent with the Roy equation and chiral perturbation theory, though at this preliminary stage our errors for <math><mi>I</mi><mo>=</mo><mn>0</mn></math> are large. An important outcome of this work is that we are successful in extracting two-pion excited states, which are useful for studying <math><mi>K</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> decay, on physical-mass ensembles using the GEVP.

Italy

Mattia Bruno

Aaron S. Meyer

Isospin 0 and 2 two-pion scattering at physical pion mass using all-to-all propagators with periodic boundary conditions in lattice QCD

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 <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi><mi>γ</mi></math>. Our calculation is performed using a domain-wall action for all quark flavors and on a single RBC/UKQCD lattice gauge-field ensemble. The first part of the study is how to best control two sources of systematic error inherent in the calculation, specifically the unwanted excited states created by the meson interpolating field and unwanted exponentials in the sum over intermediate states. Using a 3D sequential propagator allows for better control over unwanted exponentials from intermediate states, while using a 4D sequential propagator allows for better control over excited states. We perform individual analyses of the 3D and 4D methods, as well as a combined analysis using both methods, and find that the 3D sequential propagator offers good control over both sources of systematic uncertainties for the smallest number of propagator solves. From there, we further improve the use of a 3D sequential propagator by employing an infinite-volume approximation method, which allows us to calculate the relevant form factors over the entire allowed range of photon energies. We then study improvements gained by performing the calculation using a different three-point function, using ratios of three-point functions, averaging over positive and negative photon momentum, and using an improved method for extracting the structure-dependent part of the axial form factor. The optimal combination of methods yields results for the <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi><mi>γ</mi></math> structure-dependent vector and axial form factors in the entire kinematic range with statistical plus fitting uncertainties of order 5%, using 25 gauge configurations with 64 samples per configuration.

Methods for high-precision determinations of radiative-leptonic decay form factors using lattice QCD

There are emerging tensions for theory results of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment both within recent lattice QCD calculations and between some lattice QCD calculations and R-ratio results. In this paper, we work toward scrutinizing critical aspects of these calculations. We focus in particular on a precise calculation of Euclidean position-space windows defined by RBC/UKQCD that are ideal quantities for cross-checks within the lattice community and with R-ratio results. We perform a lattice QCD calculation using physical up, down, strange, and charm sea quark gauge ensembles generated in the staggered formalism by the MILC Collaboration. We study the continuum limit using inverse lattice spacings from <math><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≈</mo><mn>1.6</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> to 3.5 GeV, identical to recent studies by FNAL/HPQCD/MILC and Aubin et al. and similar to the recent study of BMW. Our calculation exhibits a tension for the particularly interesting window result of <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>ud</mi><mo>,</mo><mi>conn</mi><mo>.</mo><mo>,</mo><mi>isospin</mi><mo>,</mo><mi>W</mi></mrow></msubsup></math> from 0.4 to 1.0 fm with previous results obtained with a different discretization of the vector current on the same gauge configurations. Our results may indicate a difficulty related to estimating uncertainties of the continuum extrapolation that deserves further attention. In this work, we also provide results for <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>ud</mi><mo>,</mo><mi>conn</mi><mo>.</mo><mo>,</mo><mi>isospin</mi></mrow></msubsup></math>, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>s</mi><mo>,</mo><mi>conn</mi><mo>.</mo><mo>,</mo><mi>isospin</mi></mrow></msubsup></math>, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>SIB</mi><mo>,</mo><mi>conn</mi><mo>.</mo></mrow></msubsup></math> for the total contribution and a large set of windows. For the total contribution, we find <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi><mtext> </mtext><mi>LO</mi></mrow></msubsup><mo>=</mo><mn>714</mn><mo>(</mo><mn>27</mn><mo>)</mo><mo>(</mo><mn>13</mn><mo>)</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>ud</mi><mo>,</mo><mi>conn</mi><mo>.</mo><mo>,</mo><mi>isospin</mi></mrow></msubsup><mo>=</mo><mn>657</mn><mo>(</mo><mn>26</mn><mo>)</mo><mo>(</mo><mn>12</mn><mo>)</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>s</mi><mo>,</mo><mi>conn</mi><mo>.</mo><mo>,</mo><mi>isospin</mi></mrow></msubsup><mo>=</mo><mn>52.83</mn><mo>(</mo><mn>22</mn><mo>)</mo><mo>(</mo><mn>65</mn><mo>)</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, and <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>SIB</mi><mo>,</mo><mi>conn</mi><mo>.</mo></mrow></msubsup><mo>=</mo><mn>9.0</mn><mo>(</mo><mn>0.8</mn><mo>)</mo><mo>(</mo><mn>1.2</mn><mo>)</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, where the first uncertainty is statistical and the second systematic. We also comment on finite-volume corrections for the strong-isospin-breaking corrections.

Consistency of hadronic vacuum polarization between lattice QCD and the R ratio

We report the first result for the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment with all errors systematically controlled. Several ensembles using <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of physical mass Möbius domain-wall fermions, generated by the RBC and UKQCD collaborations, are employed to take the continuum and infinite volume limits of finite volume lattice <math><mrow><mi>QED</mi><mo>+</mo><mi>QCD</mi></mrow></math>. We find <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HLbL</mtext></mrow></msubsup><mo>=</mo><mn>7.87</mn><mo>(</mo><mn>3.06</mn><msub><mrow><mo>)</mo></mrow><mrow><mtext>stat</mtext></mrow></msub><mo>(</mo><mn>1.77</mn><msub><mrow><mo>)</mo></mrow><mrow><mtext>sys</mtext></mrow></msub><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>. Our value is consistent with previous model results and leaves little room for this notoriously difficult hadronic contribution to explain the difference between the standard model and the BNL experiment.

Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD

We propose and apply a new approach to determining <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> using dispersion relations with weight functions having poles at Euclidean (spacelike) momentum which relate strange hadronic <math><mi>τ</mi></math> decay distributions to hadronic vacuum polarization (HVP) functions obtained from lattice quantum chromodynamics. We show examples where spectral integral contributions from the region where experimental data have large errors or do not exist are strongly suppressed but accurate determinations of the relevant lattice HVP combinations remain possible. The resulting <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> agrees well with determinations from <math><mi>K</mi></math> physics and three-family Cabibbo-Kobayashi-Maskawa unitarity. Advantages of this new approach over the conventional hadronic <math><mi>τ</mi></math> decay determination employing flavor-breaking sum rules are also discussed.

United Kingdom

Peter Boyle

Canada

Renwick James Hudspith

Andreas Jüttner

Randy Lewis

Australia

Kim Maltman

Hiroshi Ohki

Antonin Portelli

Matthew Spraggs

Novel <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> Determination Using Inclusive Strange <math><mi>τ</mi></math> Decay and Lattice Hadronic Vacuum Polarization Functions