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.

United States

Thomas Blum

Norman Christ

Japan

Masashi Hayakawa

Taku Izubuchi

Luchang Jin

Chulwoo Jung

Germany

Christoph Lehner

Cheng Tu

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

We report a state-of-the-art lattice QCD calculation of the isovector quark transversity distribution of the proton in the continuum and physical mass limit using large-momentum effective theory. The calculation is done at four lattice spacings <math><mrow><mi>a</mi><mo>=</mo><mo>{</mo><mn>0.098</mn><mo>,</mo><mn>0.085</mn><mo>,</mo><mn>0.064</mn><mo>,</mo><mn>0.049</mn><mo>}</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and various pion masses ranging between 220 and 350 MeV, with proton momenta up to 2.8 GeV. The result is nonperturbatively renormalized in the hybrid scheme with self-renormalization, which treats the infrared physics at large correlation distance properly, and extrapolated to the continuum, physical mass, and infinite momentum limit. We also compare with recent global analyses for the nucleon isovector quark transversity distribution.

China

Fei Yao

Lisa Walter

Taiwan

Jiunn-Wei Chen

Jun Hua

Xiangdong Ji

Sebastian Lahrtz

Lingquan Ma

India

Protick Mohanta

Andreas Schäfer

Hai-Tao Shu

Yushan Su

Peng Sun

Xiaonu Xiong

Yi-Bo Yang

Hong Kong

Jian-Hui Zhang

Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD

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

Amarjit Soni

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

It was recently estimated that the strong isospin-symmetry breaking (ISB) corrections to the Fermi matrix element in free neutron decay could be of the order <math><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, one order of magnitude larger than the naïve estimate based on the Behrends-Sirlin-Ademollo-Gatto theorem. To investigate this claim, we derive a general expression of the leading ISB correction to hadronic Fermi matrix elements, which takes the form of a four-point correlation function in lattice gauge theory and is straightforward to compute from first principles. Our formalism paves the way for the first determination of such correction in the neutron sector with fully-controlled theory uncertainties.

Chien-Yeah Seng

Vincenzo Cirigliano

Xu Feng

Mikhail Gorchtein

Gerald A. Miller

Quark mass difference effects in hadronic Fermi matrix elements from first principles

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 extend the application of lattice QCD to the two-photon-mediated, order <math><msup><mi>α</mi><mn>2</mn></msup></math> rare decay <math><msup><mi>π</mi><mn>0</mn></msup><mo>→</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>. By combining Minkowski- and Euclidean-space methods we are able to calculate the complex amplitude describing this decay directly from the underlying theories (QCD and QED) which predict this decay. The leading connected and disconnected diagrams are considered; a continuum limit is evaluated and the systematic errors are estimated. We find <math><mrow><mi>Re</mi><mi>A</mi><mo>=</mo><mn>18.60</mn><mo>(</mo><mn>1.19</mn><mo>)</mo><mo>(</mo><mn>1.05</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>eV</mi></mrow></math>, <math><mi>Im</mi><mi>A</mi><mo>=</mo><mn>32.59</mn><mo>(</mo><mn>1.50</mn><mo>)</mo><mo>(</mo><mn>1.65</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>eV</mi></math>, a more accurate value for the ratio <math><mrow><mo>(</mo><mi>Re</mi><mi>A</mi><mo>/</mo><mi>Im</mi><mi>A</mi><mo>)</mo><mo>=</mo><mn>0.571</mn><mo>(</mo><mn>10</mn><mo>)</mo><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math>, and a result for the partial width <math><mrow><mi>Γ</mi><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>→</mo><mi>γ</mi><mi>γ</mi><mo>)</mo><mo>=</mo><mn>6.60</mn><mo>(</mo><mn>0.61</mn><mo>)</mo><mo>(</mo><mn>0.67</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>eV</mi></mrow></math>. Here the first errors are statistical and the second systematic. This calculation is the first step in determining the more challenging, two-photon-mediated decay amplitude that contributes to the rare decay <math><mi>K</mi><mo>→</mo><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>.

Yidi Zhao

Lattice QCD Calculation of <math><msup><mi>π</mi><mn>0</mn></msup><mo>→</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> Decay

We present the leading-order mixed-action effect <math><msub><mi>Δ</mi><mrow><mi>mix</mi></mrow></msub><mo>≡</mo><msubsup><mi>m</mi><mrow><mi>π</mi><mo>,</mo><mrow><mi>vs</mi></mrow></mrow><mn>2</mn></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>m</mi><mrow><mi>π</mi><mo>,</mo><mrow><mi>vv</mi></mrow></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mrow><mi>π</mi><mo>,</mo><mrow><mi>ss</mi></mrow></mrow><mn>2</mn></msubsup></mrow><mn>2</mn></mfrac></math> using highly improved staggered quarks (HISQ), clover, or overlap valence fermion actions on gauge ensembles using various sea fermion actions across a widely used lattice spacing range <math><mi>a</mi><mo>∈</mo><mo>[</mo><mn>0.04</mn><mo>,</mo><mn>0.19</mn><mo>]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. The results suggest that <math><msub><mi>Δ</mi><mrow><mi>mix</mi></mrow></msub></math> decreases as the fourth order of the lattice spacing on the gauge ensembles with dynamical chiral sea fermions, such as domain wall or HISQ fermions. When a clover sea fermion action that has explicit chiral symmetry breaking is used in the ensemble, <math><msub><mi>Δ</mi><mrow><mi>mix</mi></mrow></msub></math> can be much larger regardless of the valence fermion action used.

Dian-Jun Zhao

France

Gen Wang

Fangcheng He

Kuan Zhang

Distance between various discretized fermion actions

We use the infinite volume reconstruction method to calculate the charged and neutral pion mass difference. The hadronic tensor is calculated using lattice QCD and then combined with an analytic photon propagator, and the mass splitting is calculated with exponentially suppressed finite-volume errors. The calculation is performed using six gauge ensembles generated with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-flavor domain wall fermions, and five ensembles are at the physical pion mass. Both Feynman and Coulomb gauges are adopted in the calculation and agree well when the lattice spacing approaches zero. After performing the continuum extrapolation and examining the residual finite-volume effects, we obtain the pion mass splitting <math><mrow><mi>Δ</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>4.534</mn><mo>(</mo><mn>42</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>43</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, which agrees well with experimental measurements.

Michael Joseph Riberdy

Lattice QCD Calculation of the Pion Mass Splitting

We present a high-statistics lattice QCD determination of the valence parton distribution function (PDF) of the pion, with a mass of 300 MeV, using two very fine lattice spacings of <math><mrow><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and 0.04 fm. We reconstruct the <math><mi>x</mi></math>-dependent PDF, as well as infer the first few even moments of the PDF using leading-twist 1-loop perturbative matching framework. Our analyses use both RI-MOM and ratio-based schemes to renormalize the equal-time bilocal quark-bilinear matrix elements of pions boosted up to 2.4 GeV momenta. We use various model-independent and model-dependent analyses to infer the large-<math><mi>x</mi></math> behavior of the valence PDF. We also present technical studies on lattice spacing and higher-twist corrections present in the boosted pion matrix elements.

Xiang Gao

Christos Kallidonis

Nikhil Karthik

Swagato Mukherjee

Peter Petreczky

Charles Shugert

Sergey Syritsyn

Yong Zhao

Valence parton distribution of the pion from lattice QCD: Approaching the continuum limit

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