Nonperturbative renormalization and O(a) -improvement of the nonsinglet vector current with Nf=2+1 Wilson fermions and tree-level Symanzik improved gauge action

In calculating hadronic contributions to precision observables for tests of the Standard Model in lattice QCD, the electromagnetic current plays a central role. Using a Wilson action with $\mathrm{O}(a)$ improvement in QCD with ${N}_{\mathrm{f}}$ flavors, a counterterm must be added to the vector current in order for its on-shell matrix elements to be $\mathrm{O}(a)$ improved. In addition, the local vector current, which has support on one lattice site, must be renormalized. At $\mathrm{O}(a)$, the breaking of the $\mathrm{SU}({N}_{\mathrm{f}})$ symmetry by the quark mass matrix leads to a mixing between the local currents of different quark flavors. We present a nonperturbative calculation…

Lattice calculation of the pion transition form factor $\pi^0 \to \gamma^* \gamma^*$

We calculate the $\pi^0\to \gamma^*\gamma^*$ transition form factor ${\cal F}_{\pi^0\gamma^*\gamma^*}(q_1^2,q_2^2)$ in lattice QCD with two flavors of quarks. Our main motivation is to provide the input to calculate the $\pi^0$-pole contribution to hadronic light-by-light scattering in the muon $(g-2)$, $a_\mu^{\rm HLbL;\pi^0}$. We therefore focus on the region where both photons are spacelike up to virtualities of about $1.5~$GeV$^2$, which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral anomaly for real photons with an accuracy of about $8-9\%$. We also com…

The hadronic contribution to the running of the electromagnetic coupling and the electroweak mixing angle

37th International Symposium on Lattice Field Theory, Wuhan, China, 16 Jun 2019 - 22 Jun 2019; PoS(LATTICE 2019)010 (2019).

Nonperturbative renormalization and O(a)-improvement of the nonsinglet vector current with Nf=2+1 Wilson fermions and tree-level Symanzik improved gauge action

In calculating hadronic contributions to precision observables for tests of the Standard Model in lattice QCD, the electromagnetic current plays a central role. Using a Wilson action with O(a) improvement in QCD with Nf flavors, a counterterm must be added to the vector current in order for its on-shell matrix elements to be O(a) improved. In addition, the local vector current, which has support on one lattice site, must be renormalized. At O(a), the breaking of the SU(Nf) symmetry by the quark mass matrix leads to a mixing between the local currents of different quark flavors. We present a nonperturbative calculation of all the required improvement and renormalization constants needed for …

Hadronic light-by-light contribution to $$(g-2)_\mu $$ ( g - 2 ) μ from lattice QCD with SU(3) flavor symmetry

We perform a lattice QCD calculation of the hadronic light-by-light contribution to $$(g-2)_\mu $$ ( g - 2 ) μ at the SU(3) flavor-symmetric point $$m_\pi =m_K\simeq 420\,$$ m π = m K ≃ 420 MeV. The representation used is based on coordinate-space perturbation theory, with all QED elements of the relevant Feynman diagrams implemented in continuum, infinite Euclidean space. As a consequence, the effect of using finite lattices to evaluate the QCD four-point function of the electromagnetic current is exponentially suppressed. Thanks to the SU(3)-flavor symmetry, only two topologies of diagrams contribute, the fully connected and the leading disconnected. We show the equivalence in the continu…

Hadronic light-by-light scattering contribution to the muon $g-2$ on the lattice

International Workshop on “Flavour Changing and Conserving Processes, FCCP2017, Anacapri, Italy, 7 Sep 2017 - 9 Sep 2017; The European physical journal / Web of Conferences 179, 01017 (2018). doi:10.1051/epjconf/201817901017

Decay constants of B-mesons from non-perturbative HQET with two light dynamical quarks

We present a computation of B-meson decay constants from lattice QCD simulations within the framework of Heavy Quark Effective Theory for the b-quark. The next-to-leading order corrections in the HQET expansion are included non-perturbatively. Based on Nf=2 gauge field ensembles, covering three lattice spacings a (0.08-0.05)fm and pion masses down to 190MeV, a variational method for extracting hadronic matrix elements is used to keep systematic errors under control. In addition we perform a careful autocorrelation analysis in the extrapolation to the continuum and to the physical pion mass limits. Our final results read fB=186(13)MeV, fBs=224(14)MeV and fBs/fB=1.203(65). A comparison with o…

Lattice calculation of the pion transition form factor with Nf=2+1 Wilson quarks

We present a lattice QCD calculation of the double-virtual neutral pion transition form factor, with the goal to cover the kinematic range relevant to hadronic light-by-light scattering in the muon g−2. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the Nf=2+1 coordinated lattice simulation gauge ensembles. Second, we have implemented the on-shell O(a) improvement of the vector current to reduce the discretization effects associated with Wilson quarks. Finally, in order to have access to a wider range of photon virtualities, we have computed the transition form factor in a moving frame as well as in th…

Lattice calculation of the pion transition form factor with Nf=2+1 Wilson quarks

We present a lattice QCD calculation of the double-virtual neutral pion transition form factor, with the goal to cover the kinematic range relevant to hadronic light-by-light scattering in the muon $g\ensuremath{-}2$. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the ${N}_{f}=2+1$ coordinated lattice simulation gauge ensembles. Second, we have implemented the on-shell $\mathcal{O}(a)$ improvement of the vector current to reduce the discretization effects associated with Wilson quarks. Finally, in order to have access to a wider range of photon virtualities, we have computed the transition form factor …

The leading hadronic contribution to the running of the Weinberg angle using covariant coordinate-space methods

We present a preliminary study of the leading hadronic contribution to the running of the Weinberg angle $\theta_{\mathrm{W}}$. The running is extracted from the correlation function of the electromagnetic current with the vector part of the weak neutral current using both the standard time-momentum representation method and the Lorentz-covariant coordinate-space method recently introduced by Meyer. Both connected and disconnected contributions have been computed on $N_{\mathrm{f}}=2+1$ non-perturbatively $O(a)$-improved Wilson fermions configurations from the CLS initiative. Similar covariant coordinate-space methods can be used to compute the leading hadronic contribution to the anomalous…

Hadronic light-by-light contribution to $(g-2)_\mu$ from lattice QCD with SU(3) flavor symmetry

We perform a lattice QCD calculation of the hadronic light-by-light contribution to $(g-2)_\mu$ at the SU(3) flavor-symmetric point $m_\pi=m_K\simeq 420\,$MeV. The representation used is based on coordinate-space perturbation theory, with all QED elements of the relevant Feynman diagrams implemented in continuum, infinite Euclidean space. As a consequence, the effect of using finite lattices to evaluate the QCD four-point function of the electromagnetic current is exponentially suppressed. Thanks to the SU(3)-flavor symmetry, only two topologies of diagrams contribute, the fully connected and the leading disconnected. We show the equivalence in the continuum limit of two methods of computin…

Lattice calculation of the hadronic leading order contribution to the muon g − 2

The European physical journal / Web of Conferences Web of Conferences : proceedings proceedings 234, 01016 - (2020). doi:10.1051/epjconf/202023401016

The b-quark mass from non-perturbative Nf=2 Heavy Quark Effective Theory at O(1/mh)

Abstract We report our final estimate of the b-quark mass from N f = 2 lattice QCD simulations using Heavy Quark Effective Theory non-perturbatively matched to QCD at O ( 1 / m h ) . Treating systematic and statistical errors in a conservative manner, we obtain m ¯ b MS ¯ ( 2 GeV ) = 4.88 ( 15 ) GeV after an extrapolation to the physical point.

Developments in the position-space approach to the HLbL contribution to the muon $g-2$ on the lattice

The measurement of the anomalous magnetic moment of the muon and its prediction allow for a high-precision test of the Standard Model (SM). In this proceedings article we present ongoing work combining lattice QCD and continuum QED in order to determine an important SM contribution to the magnetic moment, the hadronic light-by-light contribution. We compute the quark-connected contribution in the Mainz position-space approach and investigate the long-distance part of our data using calculations of the $\pi^0$-pole and charged pion loop contributions.

Lattice calculation of the pion transition form factor π0→γ*γ*

The pion transition form factor for the neutral pion double virtual photon decay is computed in two flavor lattice QCD, extrapolated to the continuum physical point. Implications for the computation of the contribution of hadronic light-by-light scattering to the muon anomalous magnetic moment are discussed.

The charm-quark contribution to light-by-light scattering in the muon (−2) from lattice QCD

We compute the hadronic light-by-light scattering contribution to the muon $g-2$ from the charm quark using lattice QCD. The calculation is performed on ensembles generated with dynamical $(u,d,s)$ quarks at the SU(3)$_{\rm f}$ symmetric point with degenerate pion and kaon masses of around 415 MeV. It includes the connected charm contribution, as well as the leading disconnected Wick contraction, involving the correlation between a charm and a light-quark loop. Cutoff effects turn out to be sizeable, which leads us to use lighter-than-physical charm masses, to employ a broad range of lattice spacings reaching down to 0.039 fm and to perform a combined charm-mass and continuum extrapolation.…

Hadronic light-by-light contribution to $(g-2)_\mu $ from lattice QCD: a complete calculation

The European physical journal / C 81(7), 651 (2021). doi:10.1140/epjc/s10052-021-09455-4

Nonperturbative renormalization and O ( a ) -improvement of the nonsinglet vector current with N

Nonperturbative renormalization and $O(a )$-improvement of the nonsinglet vector current with $N_f = 2 + 1$ Wilson fermions and tree-level Symanzik improved gauge action

Physical review / D 99(1), 014519 (2019). doi:10.1103/PhysRevD.99.014519

Exploratory studies for the position-space approach to hadronic light-by-light scattering in the muon g - 2

The well-known discrepancy in the muon $g-2$ between experiment and theory demands further theory investigations in view of the upcoming new experiments. One of the leading uncertainties lies in the hadronic light-by-light scattering contribution (HLbL), that we address with our position-space approach. We focus on exploratory studies of the pion-pole contribution in a simple model and the fermion loop without gluon exchanges in the continuum and in infinite volume. These studies provide us with useful information for our planned computation of HLbL in the muon $g-2$ using full QCD.

Leading hadronic contribution to (g−2)μ from lattice QCD with Nf=2+1 flavors of O(a) improved Wilson quarks

The comparison of the theoretical and experimental determinations of the anomalous magnetic moment of the muon (g−2)μ constitutes one of the strongest tests of the Standard Model at low energies. We compute the leading hadronic contribution to (g−2)μ using lattice QCD simulations employing Wilson quarks. Gauge field ensembles at four different lattice spacings and several values of the pion mass down to its physical value are used. We apply the O(a) improvement program with two discretizations of the vector current to better constrain the approach to the continuum limit. The electromagnetic current correlators are computed in the time-momentum representation. In addition, we perform auxilia…

Light-by-light forward scattering amplitudes in Lattice QCD

We present our preliminary results on the calculation of hadronic light-by-light forward scattering amplitudes using vector four-point correlation functions computed on the lattice. Using a dispersive approach, forward scattering amplitudes can be described by $\gamma^* \gamma^* \to$ hadrons fusion cross sections and then compared with phenomenology. We show that only a few states are needed to reproduce our data. In particular, the sum rules considered in this study imply relations between meson$-\gamma\gamma$ couplings and provide valuable information about individual form factors which are often used to estimate the meson-pole contributions to the hadronic light-by-light contribution to …

A lattice calculation of the hadronic vacuum polarization contribution to (g - 2)μ

We present results of calculations of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment. Specifically, we focus on controlling the infrared regime of the vacuum polarisation function. Our results are corrected for finite-size effects by combining the Gounaris-Sakurai parameterisation of the timelike pion form factor with the Lüscher formalism. The impact of quark-disconnected diagrams and the precision of the scale determination is discussed and included in our final result in two-flavour QCD, which carries an overall uncertainty of 6%. We present preliminary results computed on ensembles with Nf = 2 + 1 dynamical flavours and discuss how the long-distance …

Hadronic Contributions to the Anomalous Magnetic Moment of the Muon from Lattice QCD

The Standard Model of Particle Physics describes three of the four known fundamental interactions: the strong interaction between quarks and gluons, the electromagnetic interaction, and the weak interaction. While the Standard Model is extremely successful, we know that it is not a complete description of nature. One way to search for physics beyond the Standard Model lies in the measurement of precision observables. The anomalous magnetic moment of the muon \(a_\mu \equiv \frac{1}{2}(g-2)_\mu \), quantifying the deviation of the gyromagnetic ratio from the exact value of 2 predicted by the Dirac equation, is one such precision observable. It exhibits a persistent discrepancy of 3.5 standar…

Density distributions in the $B$ meson

We report on a two-flavor lattice QCD study of the axial, charge and matter distributions of the $B$ meson and its first radial excitation. As our framework is the static limit of Heavy Quark Effective Theory (HQET), taking their Fourier transform gives access to several form factors at the kinematical point $q^2=0$. Moreover they provide some useful information on the nature of an excited state, i.e. a radial excitation of a quark-antiquark bound state or a multihadron state.

B-physics from non-perturbatively renormalized HQET in two-flavour lattice QCD.

We report on the ALPHA Collaboration's lattice B-physics programme based on N_f=2 O(a) improved Wilson fermions and HQET, including all NLO effects in the inverse heavy quark mass, as well as non-perturbative renormalization and matching, to fix the parameters of the effective theory. Our simulations in large physical volume cover 3 lattice spacings a ~ (0.08-0.05) fm and pion masses down to 190 MeV to control continuum and chiral extrapolations. We present the status of results for the b-quark mass and the B_(s)-meson decay constants, f_B and f_{B_s}.

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale $\Lambda\sim400\;$MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale $\Lambda$, and a UV-finite long-di…

Hadronic light-by-light scattering amplitudes from lattice QCD versus dispersive sum rules

The hadronic contribution to the eight forward amplitudes of light-by-light scattering ($\gamma^*\gamma^*\to \gamma^*\gamma^*$) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the $\gamma^*\gamma^*\to {\rm hadrons}$ cross sections in which the fusion process is described by hadronic resonances. Our results thus provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, $a_\mu^{\rm HLbL}$. Using simple parametrizations of the resonance $M\to \gamma^*\gamma^*$ transition form factors, we determine the corresponding monopole and dipole masses by performing a global fit to …

The anomalous magnetic moment of the muon in the Standard Model

We are very grateful to the Fermilab Directorate and the Fermilab Theoretical Physics Department for their financial and logistical support of the first workshop of the Muon g -2 Theory Initiative (held near Fermilab in June 2017) [123], which was crucial for its success, and indeed for the successful start of the Initiative. Financial support for this workshop was also provided by the Fermilab Distinguished Scholars program, the Universities Research Association through a URA Visiting Scholar award, the Riken Brookhaven Research Center, and the Japan Society for the Promotion of Science under Grant No. KAKEHNHI-17H02906. We thank Shoji Hashimoto, Toru Iijima, Takashi Kaneko, and Shohei Nis…

Hadronic light-by-light scattering in the anomalous magnetic moment of the muon

15th International Workshop on Tau Lepton Physics, Amsterdam, The Netherlands, 24 Sep 2018 - 28 Sep 2018; SciPost physics 1, 031 (2019). doi:10.21468/SciPostPhysProc.1.031