Search

We present complete results for the hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment $a_\mu$ in the short- and intermediate-distance window regions, which account for roughly 10% and 35% of the total HVP contribution to $a_\mu$, respectively. In particular, we perform lattice-QCD calculations for the isospin-symmetric connected and disconnected contributions, as well as corrections due to strong isospin-breaking. For the short-distance window observables, we investigate the so-called log-enhancement effects as well as the significant oscillations associated with staggered quarks in this region. For the dominant, isospin-symmetric light-quark connected contribution, we obtain $a^{ll,\,{\mathrm{SD}}}_{\mu}(\mathrm{conn.}) = 48.116(16)(94)[96] \times 10^{-10}$ and $a^{ll,\,{\mathrm{W}}}_{\mu}(\mathrm{conn.}) = 207.06(17)(63)[66] \times 10^{-10}$. We use Bayesian model averaging combined with a global bootstrap to fully estimate the covariance matrix between the individual contributions. Our determinations of the complete window contributions are $a^{{\mathrm{SD}}}_{\mu} = 69.01(2)(21)[21] \times 10^{-10}$ and $a^{{\mathrm{W}}}_{\mu} = 236.57(20)(94)[96] \times 10^{-10}$. This work is part of our ongoing effort to compute all contributions to HVP with an overall uncertainty at the few permille level.

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay-constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. We review the determinations of nucleon charges from the matrix elements of both isovector and flavour-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model., Comment: 435 pages, 53 Figures, 190 tables. arXiv admin note: substantial text overlap with arXiv:2111.09849, arXiv:1902.08191

We present results from the first lattice QCD calculation of the two-pion contributions to the light-quark connected vector-current correlation function obtained from staggered-quark operators. We employ the MILC collaboration's gauge-field ensemble with $2+1+1$ flavors of highly improved staggered sea quarks at a lattice spacing of $a\approx 0.15$ fm with a light sea-quark mass at its physical value. The two-pion contributions allow for a refined determination of the noisy long-distance tail of the vector-current correlation function, which we use to compute the light-quark connected contribution to HVP with improved statistical precision. We compare our results with traditional noise-reduction techniques used in lattice QCD calculations of the light-quark connected HVP, namely the so-called fit and bounding methods. We observe a factor of roughly three improvement in the statistical precision in the determination of the HVP contribution to the muon's anomalous magnetic moment over these approaches. We also lay the group theoretical groundwork for extending this calculation to finer lattice spacings with increased numbers of staggered two-pion taste states.

We compute the vector, scalar, and tensor form factors for the $B\to \pi$, $B\to K$, and $B_s\to K$ amplitudes, which are needed to describe semileptonic $B$-meson decay rates for both the charged and neutral current cases. We use the highly improved staggered quark (HISQ) action for the sea and light valence quarks. The bottom quark is described by the clover action in the Fermilab interpretation. Simulations are carried out on $N_f = 2+1+1$ MILC HISQ ensembles at approximate lattice spacings from $0.15$ fm down to $0.057$ fm. We present blinded preliminary results for the form factors., Comment: 10 pages, 5 figures, proceedings of the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31 - August 4, 2023, Fermilab, USA

We present an update for results on $B$-meson semileptonic decays using the highly improved staggered quark (HISQ) action for both valence and 2+1+1 sea quarks. The use of the highly improved action, combined with the MILC collaboration's gauge ensembles with lattice spacings down to $\sim$0.03 fm, allows the $b$ quark to be treated with the same discretization as the lighter quarks. The talk will focus on updated results for $B_{(s)} \to D_{(s)}$, $B_{(s)} \to K$ scalar and vector form factors., Comment: Proceedings of the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermilab, Batavia, Illinois, USA

We report on the ongoing effort of improving the determination of the gradient flow scale on the (2+1+1)-flavor HISQ ensembles generated by the MILC collaboration. We compute the scales $\sqrt{t_0}/a$ and $w_0/a$ with the Wilson and Symanzik flow using three discretizations for the action density: clover, Wilson and tree-level Symanzik improved. For the absolute scale setting, we intend to employ the $\Omega$-baryon mass, but are also using the pion decay constant while the $\Omega$-mass calculations are in progress., Comment: 10 pages, 6 figures, contribution to the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31 - August 4, 2023, Fermilab, USA

We present an update of the Fermilab Lattice and MILC Collaborations project to compute the form factors for semileptonic $B_{(s)}$-meson decays. Our calculation uses the highly improved staggered quark (HISQ) action for sea and valence quarks, and ensembles with up, down, strange, and charm quarks in the sea. Using a highly improved action with the MILC Collaboration's gauge ensembles with lattice spacings down to $a\approx0.03$ fm, allows the heavy valence quarks to be treated with the same discretization as the light and strange quarks. This unified treatment of the valence quarks allows for absolutely normalized vector currents, bypassing the need for perturbative matching, which has been a source of uncertainty in previous calculations of $B$-meson decay form factors by our collaboration. All preliminary form-factor results are blinded., Comment: Contribution to the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8th-13th August 2022, Bonn, Germany. arXiv admin note: substantial text overlap with arXiv:2111.05184

We present a lattice-QCD calculation of the light-quark connected contribution to window observables associated with the leading-order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, $a_\mu^{\mathrm{HVP,LO}}$. We employ the MILC Collaboration's isospin-symmetric QCD gauge-field ensembles, which contain four flavors of dynamical highly-improved-staggered quarks with four lattice spacings between $a\approx 0.06$-$0.15$~fm and close-to-physical quark masses. We consider several effective-field-theory-based schemes for finite-volume and other lattice corrections and combine the results via Bayesian model averaging to obtain robust estimates of the associated systematic uncertainties. After unblinding, our final results for the intermediate and ``W2'' windows are $a^{ll,{\mathrm W}}_{\mu}(\mathrm{conn.})=206.6(1.0) \times 10^{-10}$ and $a^{ll,\mathrm {W2}}_{\mu}(\mathrm{conn.}) = 100.7(3.2)\times 10^{-10}$, respectively., Comment: v3: Updated to reflect published version, which includes updates to the text in Sections II.B, III.C,D,E. Numerical results unchanged

The 2021-22 High-Energy Physics Community Planning Exercise (a.k.a. ``Snowmass 2021'') was organized by the Division of Particles and Fields of the American Physical Society. Snowmass 2021 was a scientific study that provided an opportunity for the entire U.S. particle physics community, along with its international partners, to identify the most important scientific questions in High Energy Physics for the following decade, with an eye to the decade after that, and the experiments, facilities, infrastructure, and R&D needed to pursue them. This Snowmass summary report synthesizes the lessons learned and the main conclusions of the Community Planning Exercise as a whole and presents a community-informed synopsis of U.S. particle physics at the beginning of 2023. This document, along with the Snowmass reports from the various subfields, will provide input to the 2023 Particle Physics Project Prioritization Panel (P5) subpanel of the U.S. High-Energy Physics Advisory Panel (HEPAP), and will help to guide and inform the activity of the U.S. particle physics community during the next decade and beyond., Comment: 75 pages, 3 figures, 2 tables. This is the first chapter and summary of the full report of the Snowmass 2021 Workshop. This version fixes an important omission from Table 2, adds two references that were not available at the time of the original version, fixes a minor few typos, and adds a small amount of material to section 1.1.3

Gauge covariant smearing based on the 3D lattice Laplacian can be used to create extended operators that have better overlap with hadronic ground states. For staggered quarks, we make use of two-link parallel transport to preserve taste properties. We have implemented the procedure in QUDA. We present the performance of this code on the NVIDIA A100 GPUs in Indiana University's Big Red 200 supercomputer and on the AMD MI250X GPUs in Oak Ridge Leadership Computer Facility's (OLCF's) Crusher and discuss its scalability. We also study the performance improvement from using NVSHMEM on OLCF's Summit. Reusing precomputed two-link products for all sources and sinks, it reduces the total smearing time for a baryon correlator measurement by a factor of 100-120 as compared with the original MILC code and reduces the overall time by 60-70%., Comment: 10 pages, 6 figures, proceedings of the 39th International Symposium on Lattice Field Theory (Lattice 2022), 8-13 August, 2022, Bonn, Germany

We present lattice-QCD calculations of the hadronic form factors for the semileptonic decays $D\to\pi\ell\nu$, $D\to K\ell\nu$, and $D_s\to K\ell\nu$. Our calculation uses the highly improved staggered quark (HISQ) action for all valence and sea quarks and includes $N_f=2+1+1$ MILC ensembles with lattice spacings ranging from $a\approx0.12$ fm down to $0.042$ fm. At most lattice spacings, an ensemble with physical-mass light quarks is included. The HISQ action allows all the quarks to be treated with the same relativistic light-quark action, allowing for nonperturbative renormalization using partial conservation of the vector current. We combine our results with experimental measurements of the differential decay rates to determine $|V_{cd}|^{D\to\pi}=0.2238(11)^{\rm Expt}(15)^{\rm QCD}(04)^{\rm EW}(02)^{\rm SIB}[22]^{\rm QED}$ and $|V_{cs}|^{D\to K}=0.9589(23)^{\rm Expt}(40)^{\rm QCD}(15)^{\rm EW}(05)^{\rm SIB}[95]^{\rm QED}$ This result for $|V_{cd}|$ is the most precise to date, with a lattice-QCD error that is, for the first time for the semileptonic extraction, at the same level as the experimental error. Using recent measurements from BES III, we also give the first-ever determination of $|V_{cd}|^{D_s\to K}=0.258(15)^{\rm Expt}(01)^{\rm QCD}[03]^{\rm QED}$ from $D_s\to K \ell\nu$. Our results also furnish new Standard Model calculations of the lepton flavor universality ratios $R^{D\to\pi}=0.98671(17)^{\rm QCD}[500]^{\rm QED}$, $R^{D\to K}=0.97606(16)^{\rm QCD}[500]^{\rm QED}$, and $R^{D_s\to K}=0.98099(10)^{\rm QCD}[500]^{\rm QED}$, which are consistent within $2\sigma$ with experimental measurements. Our extractions of $|V_{cd}|$ and $|V_{cs}|$, when combined with a value for $|V_{cb}|$, provide the most precise test of second-row CKM unitarity, finding agreement with unitarity at the level of one standard deviation., Comment: 92 pages, V2 matches version accepted for publication in PRD. Expanded supplementary material for reconstructing our final results. An implementation of nonlinear shrinkage is also included

Isospin-breaking corrections to the hadron vacuum polarization component of the anomalous magnetic moment of the muon are needed to ensure the theoretical precision of $g_\mu -2$ is below the experimental precision. We describe the status of our work calculating, using lattice QCD, the QED correction to the light and strange connected hadronic vacuum polarization in a Dashen scheme. We report results using physical $N_f=2+1+1$ HISQ ensembles at three lattice spacings and three heavier-than-light valence quark masses., Comment: 9 pages, 4 figures, to be published in Proceedings of Science, contribution to the 39th International Symposium on Lattice Field Theory, LATTICE2022, Bonn, Germany

This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed., Comment: Invited volume for the EJPC; 567 pages if text/figures and 4783 references occupying about 160 additional pages. arXiv abstract abridged, for the complete abstract please see the full text

The generation of ensembles of gauge configurations is a considerable expense. The preservation and curation of these ensembles constitutes a valuable shared resource for the lattice field theory community. The organizers of Lattice 2022 dedicated a parallel session to the presentation of gauge ensembles and their generation, plans for ensemble publication and data management/storage activities of different collaborations. A summary of the twelve contributions is presented here., Comment: Contribution to the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8-13 August, 2022, Bonn, Germany, 23 pages

This report summarizes the recent progress and promising future directions in theoretical high-energy physics (HEP) identified within the Theory Frontier of the 2021 Snowmass Process., Comment: Contribution to the US Community Study on the Future of Particle Physics (Snowmass 2021), v2: fixed typos

Software and Computing (S&C) are essential to all High Energy Physics (HEP) experiments and many theoretical studies. The size and complexity of S&C are now commensurate with that of experimental instruments, playing a critical role in experimental design, data acquisition/instrumental control, reconstruction, and analysis. Furthermore, S&C often plays a leading role in driving the precision of theoretical calculations and simulations. Within this central role in HEP, S&C has been immensely successful over the last decade. This report looks forward to the next decade and beyond, in the context of the 2021 Particle Physics Community Planning Exercise ("Snowmass") organized by the Division of Particles and Fields (DPF) of the American Physical Society., Comment: Computational Frontier Report Contribution to Snowmass 2021; 41 pages, 1 figure. v2: missing ref and added missing topical group conveners. v3: fixed typos

Lattice gauge theory continues to be a powerful theoretical and computational approach to simulating strongly interacting quantum field theories, whose applications permeate almost all disciplines of modern-day research in High-Energy Physics. Whether it is to enable precision quark- and lepton-flavor physics, to uncover signals of new physics in nucleons and nuclei, to elucidate hadron structure and spectrum, to serve as a numerical laboratory to reach beyond the Standard Model, or to invent and improve state-of-the-art computational paradigms, the lattice-gauge-theory program is in a prime position to impact the course of developments and enhance discovery potential of a vibrant experimental program in High-Energy Physics over the coming decade. This projection is based on abundant successful results that have emerged using lattice gauge theory over the years: on continued improvement in theoretical frameworks and algorithmic suits; on the forthcoming transition into the exascale era of high-performance computing; and on a skillful, dedicated, and organized community of lattice gauge theorists in the U.S. and worldwide. The prospects of this effort in pushing the frontiers of research in High-Energy Physics have recently been studied within the U.S. decadal Particle Physics Planning Exercise (Snowmass 2021), and the conclusions are summarized in this Topical Report., Comment: 57 pages, 1 figure. Submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021). Topical Group Report for TF05 - Lattice Gauge Theory

Contribution from the USQCD Collaboration to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)., Comment: 27 pp. main text, 4 pp. appendices, 29 pp. references, 1 p. index

An accurate determination of the leading-order hadronic vacuum polarisation (HVP) contribution to the anomalous magnetic moment of the muon is critical to understanding the size and significance of any discrepancy between the Standard Model prediction and experimental results being obtained by the Muon g-2 experiment at Fermilab. The Standard Model prediction is currently based on a data-driven approach to the HVP using experimental results for $\sigma(e^+e^-\rightarrow\,\mathrm{hadrons})$. Lattice QCD aims to provide a result with similar uncertainty from calculated vector-vector correlation functions, but the growth of statistical and systematic errors in the $u/d$ quark correlation functions at large Euclidean time has made this difficult to achieve. We show that restricting the lattice contributions to a one-sided window $0

Lattice quantum chromodynamics has proven to be an indispensable method to determine nonperturbative strong contributions to weak decay processes. In this white paper for the Snowmass community planning process we highlight achievements and future avenues of research for lattice calculations of weak $b$ and $c$ quark decays, and point out how these calculations will help to address the anomalies currently in the spotlight of the particle physics community. With future increases in computational resources and algorithmic improvements, percent level (and below) lattice determinations will play a central role in constraining the standard model or identifying new physics., Comment: contribution to Snowmass 2021; 19 pages; v2 corrected typo and added references

The search for new physics requires a joint experimental and theoretical effort. Lattice QCD is already an essential tool for obtaining precise model-free theoretical predictions of the hadronic processes underlying many key experimental searches, such as those involving heavy flavor physics, the anomalous magnetic moment of the muon, nucleon-neutrino scattering, and rare, second-order electroweak processes. As experimental measurements become more precise over the next decade, lattice QCD will play an increasing role in providing the needed matching theoretical precision. Achieving the needed precision requires simulations with lattices with substantially increased resolution. As we push to finer lattice spacing we encounter an array of new challenges. They include algorithmic and software-engineering challenges, challenges in computer technology and design, and challenges in maintaining the necessary human resources. In this white paper we describe those challenges and discuss ways they are being dealt with. Overcoming them is key to supporting the community effort required to deliver the needed theoretical support for experiments in the coming decade., Comment: Contribution to Snowmass 2021. 22 pages

We discuss the prospects for improving the precision on the hadronic corrections to the anomalous magnetic moment of the muon, and the plans of the Muon $g-2$ Theory Initiative to update the Standard Model prediction., Comment: Contribution to the US Community Study on the Future of Particle Physics (Snowmass 2021)

We investigate the state-of-the-art Lanczos eigensolvers available in the Grid and QUDA libraries. They include Implicitly Restarted Lanczos, Thick-Restart Lanczos, and Block Lanczos. We measure and analyze their performance for the Highly Improved Staggered Quark (HISQ) Dirac operator. We also discuss optimization of Chebyshev acceleration., Comment: 9 pages, 4 figures, Proceedings of the 38th International Symposium on Lattice Field Theory (LATTICE2021), 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We give an update on the status of the Fermilab Lattice-HPQCD-MILC calculation of the contribution to the muon's anomolous magnetic moment from the light-quark, connected hadronic vacuum polarization. We present preliminary, blinded results in the intermediate window for this contribution, $a_{\mu, \textrm{W}}^{ll}$. The calculation is performed on $N_f =2+1+1$ highly-improved staggered quark (HISQ) ensembles from the MILC collaboration with physical pion mass at four lattice spacings between 0.15 fm and 0.06 fm. We also present preliminary results for a study of the two-pion contributions to the vector-current correlation function performed on the 0.15 fm ensemble where we see a factor of four improvement over traditional noise reduction techniques., Comment: 9 pages, 4 figures, contribution to the 38th International Symposium on Lattice Field Theory (LATTICE2021)

We report progress on calculating the contribution to the anomalous magnetic moment of the muon from the disconnected hadronic diagrams with light and strange quarks and the valence QED contribution to the connected diagrams. The lattice QCD calculations use the highly-improved staggered quark (HISQ) formulation. The gauge configurations were generated by the MILC Collaboration with four flavors of HISQ sea quarks with physical sea-quark masses., Comment: Proceedings for a talk at the 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities., Comment: 418 pages, 53 figures, 200 tables, 1056 references. Minor changes, version as published in EPJC. arXiv admin note: substantial text overlap with arXiv:1902.08191, arXiv:1607.00299, arXiv:1310.8555

We present results for $B_{(s)}$- and $D_{(s)}$-meson semileptonic decays from ongoing calculations by the Fermilab Lattice and MILC Collaborations. Our calculation employs the highly improved staggered quark (HISQ) action for both sea and valence quarks and includes several ensembles with physical-mass up, down, strange, and charm quarks and lattice spacings ranging from $a\approx0.15$ fm down to 0.06 fm. At most lattice spacings, an ensemble with physical-mass light quarks is included. The use of the highly improved action, combined with the MILC Collaboration's gauge ensembles with lattice spacings down to $a\approx0.042$ fm, allows heavy valence quarks to be treated with the same discretization as the light and strange quarks. This unified treatment of the valence quarks allows (in some cases) for absolutely normalized currents, bypassing the need for perturbative matching, which has been a leading source of uncertainty in previous calculations of $B$-meson decay form factors by our collaboration. All preliminary form-factor results are blinded., Comment: Joint proceedings for talks at the 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We present the first unquenched lattice-QCD calculation of the form factors for the decay $B\rightarrow D^\ast\ell\nu$ at nonzero recoil. Our analysis includes 15 MILC ensembles with $N_f=2+1$ flavors of asqtad sea quarks, with a strange quark mass close to its physical mass. The lattice spacings range from $a\approx 0.15$ fm down to $0.045$ fm, while the ratio between the light- and the strange-quark masses ranges from 0.05 to 0.4. The valence $b$ and $c$ quarks are treated using the Wilson-clover action with the Fermilab interpretation, whereas the light sector employs asqtad staggered fermions. We extrapolate our results to the physical point in the continuum limit using rooted staggered heavy-light meson chiral perturbation theory. Then we apply a model-independent parametrization to extend the form factors to the full kinematic range. With this parametrization we perform a joint lattice-QCD/experiment fit using several experimental datasets to determine the CKM matrix element $|V_{cb}|$. We obtain $\left|V_{cb}\right| = (38.40 \pm 0.68_{\textrm{th}} \pm 0.34_{\textrm{exp}} \pm 0.18_{\textrm{EM}})\times 10^{-3}$. The first error is theoretical, the second comes from experiment and the last one includes electromagnetic and electroweak uncertainties, with an overall $\chi^2\text{/dof} = 126/84$, which illustrates the tensions between the experimental data sets, and between theory and experiment. This result is in agreement with previous exclusive determinations, but the tension with the inclusive determination remains. Finally, we integrate the differential decay rate obtained solely from lattice data to predict $R(D^\ast) = 0.265 \pm 0.013$, which confirms the current tension between theory and experiment., Comment: 46 pages, 14 figures. Synthetic data, results and full correlation matrices available in the ancillary files. Version accepted for publication in EPJ C

This work continues our program of lattice-QCD baryon physics using staggered fermions for both the sea and valence quarks. We present a proof-of-concept study that demonstrates, for the first time, how to calculate baryon matrix elements using staggered quarks for the valence sector. We show how to relate the representations of the continuum staggered flavor-taste group $\text{SU}(8)_{FT}$ to those of the discrete lattice symmetry group. The resulting calculations yield the normalization factors relating staggered baryon matrix elements to their physical counterparts. We verify this methodology by calculating the isovector vector and axial-vector charges $g_V$ and $g_A$. We use a single ensemble from the MILC Collaboration with 2+1+1 flavors of sea quark, lattice spacing $a\approx 0.12$ fm, and a pion mass $M_\pi\approx305$ MeV. On this ensemble, we find results consistent with expectations from current conservation and neutron beta decay. Thus, this work demonstrates how highly-improved staggered quarks can be used for precision calculations of baryon properties, and, in particular, the isovector nucleon charges., Comment: 19 pages, 9 figures

We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. This is performed in a perturbative expansion in the fine-structure constant $\alpha$ and is broken down into pure QED, electroweak, and hadronic contributions. The pure QED contribution is by far the largest and has been evaluated up to and including $\mathcal{O}(\alpha^5)$ with negligible numerical uncertainty. The electroweak contribution is suppressed by $(m_\mu/M_W)^2$ and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at $\mathcal{O}(\alpha^2)$ and is due to hadronic vacuum polarization, whereas at $\mathcal{O}(\alpha^3)$ the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads $a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$ and is smaller than the Brookhaven measurement by 3.7$\sigma$. The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics., Comment: 196 pages, 103 figures, version published in Phys. Rept., bib files for the citation references are available from: https://muon-gm2-theory.illinois.edu

Lattice QCD is an important tool for theoretical input for flavor physics. There have been four reviews by the Flavour Lattice Averaging Group (FLAG). This talk will review the current status of the magnitude of eight of the nine CKM matrix elements, borrowing heavily from the most recent FLAG review (co-authored by the speaker). Future prospects for improving the determination of the CKM matrix will be discussed., Comment: 20 pages, 17 figures, Plenary talk at 37th International Symposium on Lattice Field Theory - Lattice2019; 16-22 June 2019; Wuhan, China

We report updates to an ongoing lattice-QCD calculation of the form factors for the semileptonic decays $B \to \pi \ell \nu$, $B_s \to K \ell \nu$, $B \to \pi \ell^+ \ell^-$, and $B \to K \ell^+ \ell^-$. The tree-level decays $B_{(s)} \to \pi (K) \ell \nu$ enable precise determinations of the CKM matrix element $|V_{ub}|$, while the flavor-changing neutral-current interactions $B \to \pi (K) \ell^+ \ell^-$ are sensitive to contributions from new physics. This work uses MILC's (2+1+1)-flavor HISQ ensembles at approximate lattice spacings between $0.057$ and $0.15$ fm, with physical sea-quark masses on four out of the seven ensembles. The valence sector is comprised of a clover $b$ quark (in the Fermilab interpretation) and HISQ light and $s$ quarks. We present preliminary results for the form factors $f_0$, $f_+$, and $f_T$, including studies of systematic errors., Comment: 7 pages; 3 figures; presented at the 37th International Symposium on Lattice Field Theory, 16-22 June 2019, Wuhan, China

We present an update on the ongoing calculations by the Fermilab Lattice, HPQCD, and MILC Collaboration of the leading-order (in electromagnetism) hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. Our project employs ensembles with four flavors of highly improved staggered fermions, physical light-quark masses, and four lattice spacings ranging from $a \approx 0.06$ to 0.15 fm for most of the results thus far., Comment: LATTICE 2019, 7 pages, 7 figures

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for $D$- and $B$-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal., Comment: 458 pages, 46 figures, 209 tables, 1146 references. Minor changes, version as published in EPJC. arXiv admin note: substantial text overlap with arXiv:1607.00299, arXiv:1310.8555

We calculate the contribution to the muon anomalous magnetic moment hadronic vacuum polarization from {the} connected diagrams of up and down quarks, omitting electromagnetism. We employ QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks and the physical pion mass, and analyze five ensembles with lattice spacings ranging from $a \approx 0.06$ to~0.15~fm. The up- and down-quark masses in our simulations have equal masses $m_l$. We obtain, in this world where all pions have the mass of the $\pi^0$, $10^{10} a_\mu^{ll}({\rm conn.}) = 637.8\,(8.8)$, in agreement with independent lattice-QCD calculations. We then combine this value with published lattice-QCD results for the connected contributions from strange, charm, and bottom quarks, and an estimate of the uncertainty due to the fact that our calculation does not include strong-isospin breaking, electromagnetism, or contributions from quark-disconnected diagrams. Our final result for the total $\mathcal{O}(\alpha^2)$ hadronic vacuum polarization to the muon's anomalous magnetic moment is~$10^{10}a_\mu^{\rm HVP,LO} = 699(15)_{u,d}(1)_{s,c,b}$, where the errors are from the light-quark and heavy-quark contributions, respectively. Our result agrees with both {\it ab-initio} lattice-QCD calculations and phenomenological determinations from experimental $e^+e^-$-scattering data. It is $1.3\sigma$ below the "no new physics" value of the hadronic-vacuum-polarization contribution inferred from combining the BNL E821 measurement of $a_\mu$ with theoretical calculations of the other contributions., Comment: 19 pages, 12 figures, 6 tables; updated to correct a small mistake in the finite volume correction resulting in small changes to the results, matches published version

We discuss preliminary results for the vector form factors $f_+^{\{\pi,K\}}$ at zero-momentum transfer for the decays $D\to\pi\ell\nu$ and $D\to K \ell\nu$ using MILC's $N_f = 2+1+1$ HISQ ensembles at four lattice spacings, $a \approx 0.042, 0.06, 0.09$, and 0.12 fm, and various HISQ quark masses down to the (degenerate) physical light quark mass. We use the kinematic constraint $f_+(q^2)= f_0(q^2)$ at $q^2 = 0$ to determine the vector form factor from our study of the scalar current, which yields $f_0(0)$. Results are extrapolated to the continuum physical point in the framework of hard pion/kaon SU(3) heavy-meson-staggered $\chi$PT and Symanzik effective theory. Our calculation improves upon the precision achieved in existing lattice-QCD calculations of the vector form factors at $q^2=0$. We show the values of the CKM matrix elements $|V_{cs}|$ and $|V_{cd}|$ that we would obtain using our preliminary results for the form factors together with recent experimental results, and discuss the implications of these values for the second row CKM unitarity., Comment: 10 pages, 3 figures, proceeding of The 36th Annual International Symposium on Lattice Field Theory

We use lattice QCD to calculate the form factors $f_+(q^2)$ and $f_0(q^2)$ for the semileptonic decay $B_s\to K\ell\nu$. Our calculation uses six MILC asqtad 2+1 flavor gauge-field ensembles with three lattice spacings. At the smallest and largest lattice spacing the light-quark sea mass is set to 1/10 the strange-quark mass. At the intermediate lattice spacing, we use four values for the light-quark sea mass ranging from 1/5 to 1/20 of the strange-quark mass. We use the asqtad improved staggered action for the light valence quarks, and the clover action with the Fermilab interpolation for the heavy valence bottom quark. We use SU(2) hard-kaon heavy-meson rooted staggered chiral perturbation theory to take the chiral-continuum limit. A functional $z$ expansion is used to extend the form factors to the full kinematic range. We present predictions for the differential decay rate for both $B_s\to K\mu\nu$ and $B_s\to K\tau\nu$. We also present results for the forward-backward asymmetry, the lepton polarization asymmetry, ratios of the scalar and vector form factors for the decays $B_s\to K\ell\nu$ and $B_s\to D_s \ell\nu$. Our results, together with future experimental measurements, can be used to determine the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$., Comment: 57 pages, 22 figures, 13 tables

We review many lattice QCD calculations that impact the precise determination of the CKM matrix. We focus on decay constants and semileptonic form factors of both light ($\pi$ and K) and heavy-light ($D_{(s)}$ and $B_{(s)}$) mesons. Implication of $\Lambda_b$ form factors will be shown. When combined with experimental results for branching fractions and differential decay rates, the above calculations strongly constrain the first two rows of the CKM matrix. We discuss a long standing difference between $|V_{ub}|$ and $|V_{cb}|$ as determined from exclusive or inclusive decays., Comment: 10 pages, 6 figures, Proceedings of 16th Conference on Flavor Physics and CP Violation (FPCP 2018), July 14-18, 2018, Hyderabad, India

We describe a recent lattice-QCD calculation of the leptonic decay constants of heavy-light pseudoscalar mesons containing charm and bottom quarks and of the masses of the up, down, strange, charm, and bottom quarks. Results for these quantities are of the highest precision to date. Calculations use 24 isospin-symmetric ensembles of gauge-field configurations with six different lattice spacings as small as approximately 0.03 fm and several values of the light quark masses down to physical values of the average up- and down-sea-quark masses. We use the highly-improved staggered quark (HISQ) formulation for valence and sea quarks, including the bottom quark. The analysis employs heavy-quark effective theory (HQET). A novel HQET method is used in the determination of the quark masses., Comment: Talk presented CIPANP2018. 11 pages, LaTeX, 8 pdf figures

Using HISQ $N_f=2+1+1$ MILC ensembles with five different values of the lattice spacing, including four ensembles with physical quark masses, we have performed the most precise computation to date of the $K\to\pi\ell\nu$ vector form factor at zero momentum transfer, $f_+^{K^0\pi^-}(0)=0.9696(15)_\text{stat}(12)_\text{syst}$. This is the first calculation that includes the dominant finite-volume effects, as calculated in chiral perturbation theory at next-to-leading order. Our result for the form factor provides a direct determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{us}|=0.22333(44)_{f_+(0)}(42)_\text{exp}$, with a theory error that is, for the first time, at the same level as the experimental error. The uncertainty of the semileptonic determination is now similar to that from leptonic decays and the ratio $f_{K^+}/f_{\pi^+}$, which uses $|V_{ud}|$ as input. Our value of $|V_{us}|$ is in tension at the 2--$2.6\sigma$ level both with the determinations from leptonic decays and with the unitarity of the CKM matrix. In the test of CKM unitarity in the first row, the current limiting factor is the error in $|V_{ud}|$, although a recent determination of the nucleus-independent radiative corrections to superallowed nuclear $\beta$ decays could reduce the $|V_{ud}|^2$ uncertainty nearly to that of $|V_{us}|^2$. Alternative unitarity tests using only kaon decays, for which improvements in the theory and experimental inputs are likely in the next few years, reveal similar tensions. As part of our analysis, we calculated the correction to $f_+^{K\pi}(0)$ due to nonequilibrated topological charge at leading order in chiral perturbation theory, for both the full-QCD and the partially-quenched cases. We also obtain the combination of low-energy constants in the chiral effective Lagrangian $[C_{12}^r+C_{34}^r-(L_5^r)^2](M_\rho)=(2.92\pm0.31)\cdot10^{-6}$., Comment: 42 pages and 12 figures. Expanded discussion of fit methodology. Finite volume error increased, conclusions unchanged. Version accepted by Phys. Rev. D

We present a lattice calculation of the electromagnetic (EM) effects on the masses of light pseudoscalar mesons. The simulations employ 2+1 dynamical flavors of asqtad QCD quarks, and quenched photons. Lattice spacings vary from $\approx 0.12$ fm to $\approx 0.045$ fm. We compute the quantity $\epsilon$, which parameterizes the corrections to Dashen's theorem for the $K^+$-$K^0$ EM mass splitting, as well as $\epsilon_{K^0}$, which parameterizes the EM contribution to the mass of the $K^0$ itself. An extension of the nonperturbative EM renormalization scheme introduced by the BMW group is used in separating EM effects from isospin-violating quark mass effects. We correct for leading finite-volume effects in our realization of lattice electrodynamics in chiral perturbation theory, and remaining finite-volume errors are relatively small. While electroquenched effects are under control for $\epsilon$, they are estimated only qualitatively for $\epsilon_{K^0}$, and constitute one of the largest sources of uncertainty for that quantity. We find $\epsilon = 0.78(1)_{\rm stat}({}^{+\phantom{1}8}_{-11})_{\rm syst}$ and $\epsilon_{K^0}=0.035(3)_{\rm stat}(20)_{\rm syst}$. We then use these results on 2+1+1 flavor pure QCD HISQ ensembles and find $m_u/m_d = 0.4529(48)_{\rm stat}( {}_{-\phantom{1}67}^{+150})_{\rm syst}$., Comment: Version published in Phys. Rev. D. Compared to v1, more discussion of nonperturbative EM renormalization scheme, of statistical errors (with 3 added figures), and of choice QED_{TL} in finite volume. 87 pages, 35 figures

We calculate the up-, down-, strange-, charm-, and bottom-quark masses using the MILC highly improved staggered-quark ensembles with four flavors of dynamical quarks. We use ensembles at six lattice spacings ranging from $a\approx0.15$~fm to $0.03$~fm and with both physical and unphysical values of the two light and the strange sea-quark masses. We use a new method based on heavy-quark effective theory (HQET) to extract quark masses from heavy-light pseudoscalar meson masses. Combining our analysis with our separate determination of ratios of light-quark masses we present masses of the up, down, strange, charm, and bottom quarks. Our results for the $\overline{\text{MS}}$-renormalized masses are $m_u(2~\text{GeV}) = 2.130(41)$~MeV, $m_d(2~\text{GeV}) = 4.675(56)$~MeV, $m_s(2~\text{GeV}) = 92.47(69)$~MeV, $m_c(3~\text{GeV}) = 983.7(5.6)$~MeV, and $m_c(m_c) = 1273(10)$~MeV, with four active flavors; and $m_b(m_b) = 4195(14)$~MeV with five active flavors. We also obtain ratios of quark masses $m_c/m_s = 11.783(25)$, $m_b/m_s = 53.94(12)$, and $m_b/m_c = 4.578(8)$. The result for $m_c$ matches the precision of the most precise calculation to date, and the other masses and all quoted ratios are the most precise to date. Moreover, these results are the first with a perturbative accuracy of $\alpha_s^4$. As byproducts of our method, we obtain the matrix elements of HQET operators with dimension 4 and 5: $\overline{\Lambda}_\text{MRS}=555(31)$~MeV in the minimal renormalon-subtracted (MRS) scheme, $\mu_\pi^2 = 0.05(22)~\text{GeV}^2$, and $\mu_G^2(m_b)=0.38(2)~\text{GeV}^2$. The MRS scheme [Phys. Rev. D97, 034503 (2018), arXiv:1712.04983 [hep-ph]] is the key new aspect of our method., Comment: The published version; 32 pages and 7 figures

We calculate the leptonic decay constants of heavy-light pseudoscalar mesons with charm and bottom quarks in lattice quantum chromodynamics on four-flavor QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks. We analyze over twenty isospin-symmetric ensembles with six lattice spacings down to $a\approx 0.03$~fm and several values of the light-quark mass down to the physical value $\frac{1}{2}(m_u+m_d)$. We employ the highly-improved staggered-quark (HISQ) action for the sea and valence quarks; on the finest lattice spacings, discretization errors are sufficiently small that we can calculate the $B$-meson decay constants with the HISQ action for the first time directly at the physical $b$-quark mass. We obtain the most precise determinations to-date of the $D$- and $B$-meson decay constants and their ratios, $f_{D^+} = 212.7(0.6)$~MeV, $f_{D_s} = 249.9(0.4)$~MeV, $f_{D_s}/f_{D^+} = 1.1749(16)$, $f_{B^+} = 189.4 (1.4)$~MeV, $f_{B_s} = 230.7(1.3)$~MeV, $f_{B_s}/f_{B^+} = 1.2180(47)$, where the errors include statistical and all systematic uncertainties. Our results for the $B$-meson decay constants are three times more precise than the previous best lattice-QCD calculations, and bring the QCD errors in the Standard-Model predictions for the rare leptonic decays $\overline{\mathcal{B}}(B_s \to \mu^+\mu^-) = 3.64(11) \times 10^{-9}$, $\overline{\mathcal{B}}(B^0 \to \mu^+\mu^-) = 1.00(3) \times 10^{-10}$, and $\overline{\mathcal{B}}(B^0 \to \mu^+\mu^-)/\overline{\mathcal{B}}(B_s \to \mu^+\mu^-) = 0.0273(9)$ to well below other sources of uncertainty. As a byproduct of our analysis, we also update our previously published results for the light-quark-mass ratios and the scale-setting quantities $f_{p4s}$, $M_{p4s}$, and $R_{p4s}$. We obtain the most precise lattice-QCD determination to date of the ratio $f_{K^+}/f_{\pi^+} = 1.1950(^{+16}_{-23})$~MeV., Comment: Errors related to the standard model prediction for the rare leptonic decays are fixed in the abstract and Eqs. (7.44), (7.45), and (8.3)

With recent developments in parallel supercomputing architecture, many core, multi-core, and GPU processors are now commonplace, resulting in more levels of parallelism, memory hierarchy, and programming complexity. It has been necessary to adapt the MILC code to these new processors starting with NVIDIA GPUs, and more recently, the Intel Xeon Phi processors. We report on our efforts to port and optimize our code for the Intel Knights Landing architecture. We consider performance of the MILC code with MPI and OpenMP, and optimizations with QOPQDP and QPhiX. For the latter approach, we concentrate on the staggered conjugate gradient and gauge force. We also consider performance on recent NVIDIA GPUs using the QUDA library.

Using the MILC 2+1 flavor asqtad quark action ensembles, we are calculating the form factors $f_0$ and $f_+$ for the semileptonic $B_s \rightarrow K \ell\nu$ decay. A total of six ensembles with lattice spacing from $\approx0.12$ to 0.06 fm are being used. At the coarsest and finest lattice spacings, the light quark mass $m'_l$ is one-tenth the strange quark mass $m'_s$. At the intermediate lattice spacing, the ratio $m'_l/m'_s$ ranges from 0.05 to 0.2. The valence $b$ quark is treated using the Sheikholeslami-Wohlert Wilson-clover action with the Fermilab interpretation. The other valence quarks use the asqtad action. When combined with (future) measurements from the LHCb and Belle II experiments, these calculations will provide an alternate determination of the CKM matrix element $|V_{ub}|$., Comment: 8 pages, 6 figures, to appear in the Proceedings of Lattice 2017, June 18-24, Granada, Spain

All lattice-QCD calculations of the hadronic-vacuum-polarization contribution to the muon's anomalous magnetic moment to-date have been performed with degenerate up- and down-quark masses. Here we calculate directly the strong-isospin-breaking correction to $a_\mu^{\rm HVP}$ for the first time with physical values of $m_u$ and $m_d$ and dynamical $u$, $d$, $s$, and $c$ quarks, thereby removing this important source of systematic uncertainty. We obtain a relative shift to be applied to lattice-QCD results obtained with degenerate light-quark masses of $\delta a_\mu^{{\rm HVP,} m_u \neq m_d}$= +1.5(7)%, in agreement with estimates from phenomenology and a recent lattice-QCD calculation with unphysically heavy pions., Comment: v2: 6 pages, 2 tables, 2 figures. Additional references and expanded discussion of systematic errors. Version accepted to Physical Review Letters

We report the status of an ongoing lattice-QCD calculation of form factors for exclusive semileptonic decays of $B$ mesons with both charged currents ($B\to\pi\ell\nu$, $B_s\to K\ell\nu$) and neutral currents ($B\to\pi\ell^+\ell^-$, $B\to K\ell^+\ell^-$). The results are important for constraining or revealing physics beyond the Standard Model. This work uses MILC's (2+1+1)-flavor ensembles with the HISQ action for the sea and light valence quarks and the clover action in the Fermilab interpretation for the $b$ quark. Simulations are carried out at three lattice spacings down to $0.088$ fm, with both physical and unphysical sea-quark masses. We present preliminary results for correlation-function fits., Comment: 8 pages, 3 figures; presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We calculate in three-flavor lattice QCD the short-distance hadronic matrix elements of all five $\Delta C=2$ four-fermion operators that contribute to neutral $D$-meson mixing both in and beyond the Standard Model. We use the MILC Collaboration's $N_f = 2+1$ lattice gauge-field configurations generated with asqtad-improved staggered sea quarks. We also employ the asqtad action for the valence light quarks and use the clover action with the Fermilab interpretation for the charm quark. We analyze a large set of ensembles with pions as light as $M_\pi \approx 180$ MeV and lattice spacings as fine as $a\approx 0.045$ fm, thereby enabling good control over the extrapolation to the physical pion mass and continuum limit. We obtain for the matrix elements in the $\overline{\text{MS}}$-NDR scheme using the choice of evanescent operators proposed by Beneke \emph{et al.}, evaluated at 3 GeV, $\langle D^0|\mathcal{O}_i|\bar{D}^0 \rangle = \{0.0805(55)(16), -0.1561(70)(31), 0.0464(31)(9), 0.2747(129)(55), 0.1035(71)(21)\}~\text{GeV}^4$ ($i=1$--5). The errors shown are from statistics and lattice systematics, and the omission of charmed sea quarks, respectively. To illustrate the utility of our matrix-element results, we place bounds on the scale of CP-violating new physics in $D^0$~mixing, finding lower limits of about 10--50$\times 10^3$ TeV for couplings of $\mathrm{O}(1)$. To enable our results to be employed in more sophisticated or model-specific phenomenological studies, we provide the correlations among our matrix-element results. For convenience, we also present numerical results in the other commonly-used scheme of Buras, Misiak, and Urban., Comment: Published version, 42 pages, 18 figures

We present a progress report on our calculation of the decay constants $f_B$ and $f_{B_s}$ from lattice-QCD simulations with highly-improved staggered quarks. Simulations are carried out with several heavy valence-quark masses on $(2+1+1)$-flavor ensembles that include charm sea quarks. We include data at six lattice spacings and several light sea-quark masses, including an approximately physical-mass ensemble at all but the smallest lattice spacing, 0.03 fm. This range of parameters provides excellent control of the continuum extrapolation to zero lattice spacing and of heavy-quark discretization errors. Finally, using the heavy-quark effective theory expansion we present a method of extracting from the same correlation functions the charm- and bottom-quark masses as well as some low-energy constants appearing in the heavy-quark expansion., Comment: 7 pages, 3 figures, Lattice 2016

We discuss the reduction of errors in the calculation of the form factor $f_+^{K \pi}(0)$ with HISQ fermions on the $N_f=2+1+1$ MILC configurations from increased statistics on some key ensembles, new data on ensembles with lattice spacings down to 0.042 fm and the study of finite-volume effects within staggered ChPT. We also study the implications for the unitarity of the CKM matrix in the first row and for current tensions with leptonic determinations of $\vert V_{us}\vert$., Comment: 7 pages, 2 figures, to appear in the Proceedings of Lattice 2016, The 34th International Symposium on Lattice Field Theory, held 24-30 July 2016, at the University of Southampton, UK. v2: references corrected