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 towards 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 $a^{-1}\approx 1.6$ GeV 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 $a_μ^{\rm ud, conn.,isospin, W}$ from $0.4$ fm 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 $a_μ^{\rm ud,conn.,isospin}$, $a_μ^{\rm s,conn.,isospin}$, $a_μ^{\rm SIB,conn.}$ for the total contribution and a large set of windows. For the total contribution, we find $a_μ^{\rm HVP~LO}=714(27)(13) 10^{-10}$, $a_μ^{\rm ud,conn.,isospin}=657(26)(12) 10^{-10}$, $a_μ^{\rm s,conn.,isospin}=52.83(22)(65) 10^{-10}$, and $a_μ^{\rm SIB,conn.}=9.0(0.8)(1.2) 10^{-10}$, where the first uncertainty is statistical and the second systematic. We also comment on finite-volume corrections for the strong-isospin-breaking corrections.
We propose a method to compute the hadronic vacuum polarization function on the lattice at continuous values of photon momenta bridging between the spacelike and timelike regions. We provide two independent demonstrations to show that this method leads to the desired hadronic vacuum polarization function in Minkowski spacetime. We show with the example of the leading-order QCD correction to the muon anomalous magnetic moment that this approach can provide a valuable alternative method for calculations of physical quantities where the hadronic vacuum polarization function enters.
The dispersive approach to quantum chromodynamics is applied to the study of the hadronic vacuum polarization function and associated quantities. This approach merges the intrinsically nonperturbative constraints, which originate in the kinematic restrictions on the respective physical processes, with corresponding perturbative input. The obtained hadronic vacuum polarization function agrees with pertinent lattice simulation data. The evaluated hadronic contributions to the muon anomalous magnetic moment and to the shift of the electromagnetic fine structure constant conform with recent assess…
We present simple analytic expressions to compute the hadronic vacuum polarization contribution to the muon $g$-2 in the space-like region up to next-to-next-to-leading order. These results can be employed in lattice QCD calculations of this contribution as well as in space-like determinations based on scattering data, like that expected from the proposed MUonE experiment at CERN.
The complete set of relations, which mutually express the spacelike and timelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment $a^{\text{HVP}}_μ$ in terms of each other, is obtained. By making use of the derived relations the explicit expression for the next-to-leading order spacelike kernel function, which enters the representation for $a^{\text{HVP}}_μ$ involving the hadronic vacuum polarization function, is obtained. The corresponding next-to-leading order spacelike kernel function, which appears in the representation for $a^{\text{…
I suggest a new approach to the determination of the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon $a_μ^{\rm HVP}$ in lattice QCD. It is based on properties of the Mellin transform of the hadronic spectral function and their relation to the HVP self energy in the Euclidean. I show how $a_μ^{\rm HVP}$ is very well approximated by a few moments associated to this Mellin transform and how these moments can be evaluated in lattice QCD, providing thus a series of tests when compared with the corresponding determinations using experimental data.
The gradient-flow operator product expansion for QCD current correlators including operators up to mass dimension four is calculated through NNLO. This paves an alternative way for efficient lattice evaluations of hadronic vacuum polarization functions. In addition, flow-time evolution equations for flowed composite operators are derived. Their explicit form for the non-trivial dimension-four operators of QCD is given through order $α_s^3$.
We present steps towards the computation of the leading-order hadronic contribution to the muon anomalous magnetic moment on RBC/UKQCD physical point DWF ensembles. We discuss several methods for controlling and reducing uncertainties associated to the determination of the HVP form factor.
The leading hadronic contribution to the muon anomalous magnetic moment is given by a weighted integral over euclidean momentum of the hadronic vacuum polarization. This integral is dominated by momenta of order the muon mass. Since the finite volume in lattice QCD makes it difficult to compute the vacuum polarization at a large number of low momenta with high statistics (combined with the fact that one cannot compute it at zero momentum), a parametrization of the vacuum polarization is required to extrapolate the data. A much used functional form is based on vector meson dominance, which intr…
We report on the computation of the light quark vacuum polarization with 2+1+1 flavors of H ISQ fermions at the physical point and its contribution to the muon anomalous magnetic moment. Three ensembles, generated by the MILC collaboration, are used to take the continuum limit. We compare our result with recent ones in the literature.
We present and test a new method to compute the hadronic vacuum polarization function in lattice simulations. This can then be used, e.g., to determine the leading hadronic contribution to the anomalous magnetic moment of the muon. The method is based on computing susceptibilities with respect to external electromagnetic plane wave fields and allows for a precision determination of both the connected and the disconnected contributions to the vacuum polarization. We demonstrate that the statistical errors obtained with our method are much smaller than those quoted in previous lattice studies, p…
We present a determination of the QCD coupling constant, $α_s$, obtained by fitting lattice results for the flavour $ud$ hadronic vacuum polarisation function to continuum perturbation theory. We use $n_f=2+1$ flavours of Domain Wall fermions generated by the RBC/UKQCD collaboration and three lattice spacings $a^{-1}=1.79,\, 2.38$ and $3.15\ \text{GeV}$. Several sources of potential systematic error are identified and dealt with. After fitting and removing expected leading cut-off effects, we find for the five-flavour $\overline{\text{MS}}$ coupling the value $α_s(M_Z)=0.1181(27)^{+8}_{-22}$, …
We report the first lattice QCD calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment at physical pion mass. The calculation uses a refined noise-reduction technique which enabled the control of statistical uncertainties at the desired level with modest computational effort. Measurements were performed on the $48^3 \times 96$ physical-pion-mass lattice generated by the RBC and UKQCD collaborations. We find $a_μ^{\rm HVP~(LO)~DISC} = -9.6(3.3)(2.3)\times 10^{-10}$, where the first error is statistical and the second systematic.
The experimental value of g-2 of the muon is larger by $4.2 σ$ than the Standard Model prediction based on the hadronic vacuum polarization contribution (HVP) determined from the measured R-ratio, $σ(e^+e^- \! \rightarrow \! \mathit{hadrons})/σ(e^+e^- \! \rightarrow \!μ^+ μ^-)$; the HVP calculated in lattice QCD also significantly exceeds the measured R-ratio value. We show here that these discrepancies can be explained by an undetected contribution to $e^+e^- \! \rightarrow \! \mathit{hadrons}$ as could arise from production of neutral, long-lived hadrons which have not previously been identi…
In order to reduce the current hadronic uncertainties in the theory prediction for the anomalous magnetic moment of the muon, lattice calculations need to reach sub-percent accuracy on the hadronic-vacuum-polarization contribution. This requires the inclusion of $\mathcal{O}(α)$ electromagnetic corrections. The inclusion of electromagnetic interactions in lattice simulations is known to generate potentially large finite-size effects suppressed only by powers of the inverse spatial extent. In this paper we derive an analytic expression for the $\mathrm{QED}_{\mathrm{L}}$ finite-volume correctio…
It is shown that with a precise determination of a few derivatives of the hadronic vacuum polarization (HVP) self-energy function $Π(Q^2)$ at $Q^2=0$, from lattice QCD (LQCD) or from a dedicated low-energy experiment, one can obtain an evaluation of the lowest order HVP contribution to the anomalous magnetic moment of the muon $a_μ^{\rm HVP}$ with an accuracy comparable to the one reached using the $e^+ e^-$ annihilation cross-section into hadrons. The technique of Mellin-Barnes approximants (MBa) that we propose is illustrated in detail with the example of the two loop vacuum polarization fun…
We discuss a new technique to evaluate integrals of QCD Green's functions in the Euclidean based on their Mellin-Barnes representation. We present as a first application the evaluation of the lowest order Hadronic Vacuum Polarization (HVP) contribution to the anomalous magnetic moment of the muon $\frac{1}{2}(g_μ-2)_{\rm\tiny HVP}\equiv a_μ^{\rm HVP}$. It is shown that with a precise determination of the slope and curvature of the HVP function at the origin from lattice QCD (LQCD), one can already obtain a result for $a_μ^{\rm HVP}$ which may serve as a test of the determinations based on expe…
ALEPH/OPAL data on the V-A spectral functions from hadronic tau decays are used in connection with a set of Laplace transform sum rules (LSR) for fixing the size of the QCD vacuum condensates up to dimension 18. Our results favour the ones from large-N_c QCD within the Minimal Hadronic Approximation (MHA) and show a violation of about a factor 2-5 of the vacuum saturation estimate of the dimension-six to -ten condensates. We scrutinize the different determinations of the QCD vacuum condensates using tau-decays data. After revisiting some of the existing results, we present coherent values of t…
In order to reach (sub-)per cent level precision in lattice calculations of the hadronic vacuum polarisation, isospin breaking corrections must be included. This requires introducing QED on the lattice, and the associated finite-size effects are potentially large due to the absence of a mass gap. This means that the finite-size effects scale as an inverse polynomial in $L$ rather than being exponentially suppressed. Considering the $\mathcal{O}(α)$ corrected hadronic vacuum polarisation in QED$_{\mathrm{L}}$ with scalar QED as an effective theory, we show that the first possible term, which is…
We develop a number of sum rules comparing spectral integrals involving judiciously chosen weights to integrals over the corresponding Euclidean two-point function. The applications we have in mind are to the hadronic vacuum polarization that determines the most important hadronic correction $a_μ^{\rm HVP}$ to the muon anomalous magnetic moment. First, we point out how spectral weights may be chosen that emphasize narrow regions in $\sqrt{s}$, providing a tool to investigate emerging discrepancies between data-driven and lattice determinations of $a_μ^{\rm HVP}$. Alternatively, for a narrow re…
Using experimental spectral data for hadronic tau decays from the OPAL experiment, supplemented by a phenomenologically successful parameterization for the high-s region not covered by the data, we construct a physically constrained model of the isospin-one vector-channel polarization function. Having such a model as a function of Euclidean momentum Q^2 allows us to explore the systematic error associated with fits to the Q^2 dependence of lattice data for the hadronic electromagnetic current polarization function which have been used in attempts to compute the leading order hadronic contribut…
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 …
The renormalization group relations for the higher-order hadronic vacuum polarization function perturbative expansion coefficients are studied. The folded recurrent and unfolded explicit forms of such relations are obtained. The explicit expression for the coefficients, which incorporate the contributions of the $π^{2}$-terms in the perturbative expansion of the R-ratio of electron-positron annihilation into hadrons, is derived at an arbitrary loop level. The obtained results can be employed as an independent crosscheck of the higher-order perturbative calculations of the hadronic vacuum polar…
We compute the vacuum polarisation on the lattice in quenched QCD using non-perturbatively improved Wilson fermions. Above Q^2 of about 2 GeV^2 the results are very close to the predictions of perturbative QCD. Below this scale we see signs of non-perturbative effects which we can describe by the use of dispersion relations. We use our results to estimate the light quark contribution to the muon's anomalous magnetic moment. We find the result 446(23) x 10^{-10}, where the error only includes statistical uncertainties. Finally we make some comments on the applicability of the Operator Product E…
We present a lattice calculation of the hadronic vacuum polarization and the lowest-order hadronic contribution to the muon anomalous magnetic moment, a_μ= (g-2)/2, using 2+1 flavors of improved staggered fermions. A precise fit to the low-q^2 region of the vacuum polarization is necessary to accurately extract the muon g-2. To obtain this fit, we use staggered chiral perturbation theory, including the vector particles as resonances, and compare these to polynomial fits to the lattice data. We discuss the fit results and associated systematic uncertainties, paying particular attention to the r…
We present a first-principles lattice QCD+QED calculation at physical pion mass of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. The total contribution of up, down, strange, and charm quarks including QED and strong isospin breaking effects is found to be $a_μ^{\rm HVP~LO}=715.4(16.3)(9.2) \times 10^{-10}$, where the first error is statistical and the second is systematic. By supplementing lattice data for very short and long distances with experimental R-ratio data using the compilation of Ref. [1], we significantly improve the precision of…
We construct a physically motivated model for the isospin-one non-strange vacuum polarization function Pi(Q^2) based on a spectral function given by vector-channel OPAL data from hadronic tau decays for energies below the tau mass and a successful parametrization, employing perturbation theory and a model for quark-hadron duality violations, for higher energies. Using a covariance matrix and Q^2 values from a recent lattice simulation, we then generate fake data for Pi(Q^2) and use it to test fitting methods currently employed on the lattice for extracting the hadronic vacuum polarization cont…
We present two examples of applications of the analytic continuation method for computing the hadronic vacuum polarization function in space- and time-like momentum regions. These examples are the Adler function and the leading order hadronic contribution to the muon anomalous magnetic moment. We comment on the feasibility of the analytic continuation method and provide an outlook for possible further applications.
We measure the hadronic contribution to the vacuum polarisation tensor, and use it to estimate the hadronic contribution to (g-2)_mu, the muon anomalous magnetic moment.
We investigate finite-volume effects in the hadronic vacuum polarization, with an eye toward the corresponding systematic error in the muon anomalous magnetic moment. While it is well known that leading-order chiral perturbation theory does not provide a good description of the hadronic vacuum polarization, it turns out that it gives a much better representation of finite-volume effects. Indications are that finite-volume effects cannot be ignored when the aim is a few percent level accuracy for the hadronic contribution to the muon anomalous magnetic moment, even when $m_πL \sim 4$ and $m_π\s…
We compute the standard Euclidean window of the hadronic vacuum polarization using multiple independent blinded analyses. We improve the continuum and infinite-volume extrapolations of the dominant quark-connected light-quark isospin-symmetric contribution and address additional sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking from first principles. We find $a_μ^{\rm W} = 235.56(65)(50) \times 10^{-10}$, which is in $3.8σ$ tension with the recently published dispersive result of Colangelo et al., $a_μ^{\rm W} = 229.4(1.4) \times 10^{-10}$, and in agree…
Recently, the Budapest-Marseille-Wuppertal collaboration achieved sub-percent precision in the evaluation of the lowest-order hadronic vacuum polarization contribution to the muon $g_μ-2$ (arXiv:hep-lat/2002.12347v3). At this level of precision, isospin-symmetric QCD is not sufficient. In this contribution we review how QED and strong-isospin-breaking effects have been included in our work. Isospin breaking is implemented by expanding the relevant correlation functions to second order in the electric charge $e$ and to first order in $m_u-m_d$. The correction terms are then computed using isosp…
We present a calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, $a_μ^{\mathrm hvp}$, in lattice QCD employing dynamical up and down quarks. We focus on controlling the infrared regime of the vacuum polarization function. To this end we employ several complementary approaches, including Padé fits, time moments and the time-momentum representation. We correct our results for finite-volume effects by combining the Gounaris-Sakurai parameterization of the timelike pion form factor with the Lüscher formalism. On a subset of our ensembles we have deri…
The Standard Model prediction for $μ$-$e$ scattering at Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD contributions given by diagrams with a hadronic vacuum polarization insertion in the photon propagator. By taking advantage of the hyperspherical integration method, we show that the subset of hadronic NNLO corrections where the vacuum polarization appears inside a loop, the irreducible diagrams, can be calculated employing the hadronic vacuum polarization in the space-like region, without making use of the $R$ ratio and time-like data. We present the analytic expressions …
We determine the contribution to the anomalous magnetic moment of the muon from the $α^2_{\mathrm{QED}}$ hadronic vacuum polarization diagram using full lattice QCD and including $u/d$ quarks with physical masses for the first time. We use gluon field configurations that include $u$, $d$, $s$ and $c$ quarks in the sea at multiple values of the lattice spacing, multiple $u/d$ masses and multiple volumes that allow us to include an analysis of finite-volume effects. We obtain a result for $a_μ^{\mathrm{HVP,LO}}$ of $667(6)(12)$, where the first error is from the lattice calculation and the secon…
We propose a method to evaluate hadronic matrix elements of QCD gluon operators in the instanton vacuum. We construct the ground state of the interacting instanton ensemble for non-zero $\vartheta$--angle using a variational principle. A method to study the $\vartheta$--dependence of observables on the lattice is suggested. We then derive the effective fermion action, which allows to calculate hadronic correlation functions in a $1/N_c$--expansion (Nambu--Jona-Lasinio type effective fermion theory). Gluon operators are systematically represented as effective fermion operators. Physical matrix …
After discussing the relevance of a first principles theory-prediction of the hadronic vacuum polarisation for Standard Model tests, the theoretical challenges for its computation in lattice QCD are reviewed. New ideas that will potentially improve the quality of lattice simulations will be introduced and the status of ongoing simulations will be presented briefly.
Following the publication of the new measurement of the anomalous magnetic moment of the muon, the discrepancy between experiment and the theory prediction from the $g-2$ theory initiative has increased to $4.2\,σ$. Recent lattice QCD calculations predict values for the hadronic vacuum polarization contribution that are larger than the data-driven estimates, bringing the Standard Model prediction closer to the experimental measurement. Euclidean time windows in the time-momentum representation of the hadronic vacuum polarization contribution to the muon $g-2$ can help clarify the discrepancy b…
We study the finite-volume correction on the hadronic vacuum polarization contribution to the muon g-2 ($a_μ^{\rm hvp}$) in lattice QCD at (near) physical pion mass using two different volumes: $(5.4~{\rm fm})^4$ and $(8.1~{\rm fm})^4$. We use an optimized AMA technique for noise reduction on $N_f=2+1$ PACS gauge configurations with stout-smeared clover-Wilson fermion action and Iwasaki gauge action at a single lattice cut-off $a^{-1}=2.33$ GeV. The calculation is performed for the quark-connected light-quark contribution in the isospin symmetric limit. We take into account the effects of back…
Comparison of the hadronic vacuum polarization between hadronic $τ$-decay data and lattice QCD
San Francisco State University（旧金山州立大学）；Instituto de Física de São Carlos, Universidade de São Paulo（圣卡洛斯物理研究所，圣保罗大学）；Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona（巴塞罗那自治大学物理系及IFAE-BIST）；York University（约克大学）；University of Adelaide（阿德莱德大学）；Institut für Kernphysik, Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学核物理研究所）
The leading-order hadronic contribution to the muon anomalous magnetic moment is given by a weighted integral over the subtracted hadronic vacuum polarization. This integral is dominated by euclidean momenta of order the muon mass, i.e., momenta not accessible on current lattice volumes with periodic boundary conditions. Twisted boundary conditions can in principle help in accessing momenta of any size even in a finite volume, but their use leads to a modification of the Ward-Takahashi identity that normally guarantees transversality of the vacuum polarization. As a result, the vacuum polariza…
The MUonE proposal at the CERN SPS consists in extracting the value of the hadronic vacuum polarization self-energy function (HVP) from its contribution to the differential cross-section of elastic muon-electron scattering. The HVP contribution to the muon anomalous magnetic moment can then be obtained from a weighted integral of the measured HVP self-energy function. This, however, requires a knowledge of the HVP function in its full integration domain. This paper discusses a procedure to reconstruct the HVP function in the regions not directly accessible to measurement. The method is based o…
A large body of evidence from lattice calculations indicates that instantons play a major role in the physics of light hadrons. This evidence is summarized, and recent results concerning the instanton content of the SU(3) vacuum, instanton contributions to the static potential, and a new class of instanton solutions at finite temperature are reviewed.
The hadronic contributions to the anomalous magnetic moment of the muon which are relevant for the confrontation between theory and experiment at the present level of accuracy, are evaluated within the same framework: the constituent chiral quark model. This includes the contributions from the dominant hadronic vacuum polarization as well as from the next--to--leading order hadronic vacuum polarization, the contributions from the hadronic light-by-light scattering, and the contributions from the electroweak hadronic $Zγγ$ vertex. They are all evaluated as a function of only one free parameter:…
The leading-order hadronic contribution to the anomalous magnetic moment of the muon is given by a weighted integral over the subtracted hadronic vacuum polarization. This integral is dominated by euclidean momenta of order the muon mass which are not available on current lattice volumes with periodic boundary conditions. Twisted boundary conditions can in principle help access momenta of any size even in a finite volume. We investigate the implementation of twisted boundary conditions both numerically (using all-mode averaging for improved statistics) and analytically, and present our initial…
Lattice determination of the higher-order hadronic vacuum polarization contribution to the muon $g-2$
Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）；GSI Helmholtz Centre for Heavy Ion Research（GSI重离子研究中心）；CERN（欧洲核子研究组织）；Rheinische Friedrich-Wilhelms-Universität Bonn（波恩大学）
We discuss leading isospin breaking effects in the hadronic vacuum polarisation required for the investigation of the hadronic contribution to $(g-2)_μ$. The calculation proceeds by expanding the relevant correlation functions around the isosymmetric limit. Isosymmetric observables are evaluated on gauge ensembles with $N_{f}=2+1$, $O(a)$ improved Wilson fermions and open boundary conditions generated by the CLS effort. Particular emphasis is placed on the relevant quark-disconnected diagrams required for a complete treatment of leading isospin breaking effects in the valence quark sector. We …
We study the pion-mass dependence of the two-pion channel in the hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon $a_μ^\text{HVP}$, by using an Omnès representation for the pion vector form factor with the phase shift derived from the inverse-amplitude method (IAM). Our results constrain the dominant isospin-$1$ part of the isospin-symmetric light-quark contribution, and should thus allow one to better control the chiral extrapolation of $a_μ^\text{HVP}$, required for lattice-QCD calculations performed at larger-than-physical pion masses. In particul…
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 $π^0$, $10^{10} a_μ^{ll}({\rm conn.}) = 637.8\,(8.8)$, in agreement with independe…
We study a representation of the hadronic vacuum polarization based on the time-momentum representation of the vector correlator. This representation suggests a way to compute the hadronic vacuum polarization and the associated Adler function for any value of virtuality, irrespective of the flavor structure of the current. We present results on both of these phenomenologically important functions, derived from local-conserved two-point lattice vector correlation functions, computed on a subset of light two-flavor ensembles made available to us through the CLS effort.
