In this talk we present some preliminary results and discuss the prospects of determining the leading structure-dependent finite-volume effects in the hadronic vacuum polarisation associated to order $e^2$ electromagnetic corrections. In the quantum electrodynamics prescription $\textrm{QED}_{\textrm{L}}$ these arise at order $1/L^3$ in the large-volume expansion, which is also the leading order because of the neutrality of the currents defining the underlying correlation function. Knowing the size of the finite-volume effects in question is relevant for determinations of the leading isospin-breaking corrections to the muon anomalous magnetic moment coming from the hadronic vacuum polarisation.
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.
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 …
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 $g_μ-2$ 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.
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.
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 $α_s(M_Z)=0.1181(27)^{+8}_{-22}$, …
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 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 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 study isospin-1 P-wave $ππ$ scattering in lattice QCD with two flavours of O($a$) improved Wilson fermions. For pion masses ranging from $m_π=265$ MeV to $m_π=437$ MeV, we determine the energy spectrum in the centre-of-mass frame and in three moving frames. We obtain the scattering phase shifts using Lüscher's finite-volume quantisation condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the resonance parameters $m_ρ$ and $g_{ρππ}$. By combining the scattering phase shifts with the decay matrix element of the vector …
We report on our ongoing project to calculate the leading hadronic contribution to the anomalous magnetic moment of the muon a_mu^HLO using two dynamical flavours of non-perturbatively O(a) improved Wilson fermions. In this study, we changed the vacuum polarisation tensor to a combination of local and point-split currents which significantly reduces the numerical effort. Partially twisted boundary conditions allow us to improve the momentum resolution of the vacuum polarisation tensor and therefore the determination of the leading hadronic contribution to (g-2)_mu. We also extended the range o…
The persistent discrepancy of about 3.5 standard deviations between the experimental measurement and the Standard Model prediction for the muon anomalous magnetic moment, $a_μ$, is one of the most promising hints for the possible existence of new physics. Here we report on our lattice QCD calculation of the hadronic vacuum polarisation contribution $a_μ^{\rm hvp}$, based on gauge ensembles with $N_f=2+1$ flavours of O($a$) improved Wilson quarks. We address the conceptual and numerical challenges that one encounters along the way to a sub-percent determination of the hadronic vacuum polarisati…
BMW/DMZ calculation of the hadronic vacuum polarisation for the muon magnetic moment
Adelaide University（阿德莱德大学）；University of Wuppertal（伍珀塔尔大学）；Pennsylvania State University（宾夕法尼亚州立大学）；University of California, San Diego（加州大学圣地亚哥分校）；Aix Marseille Univ（艾克斯-马赛大学）；Mines Paris - PSL（巴黎高等矿业学校-PSL大学）
We present our results for the leading-order hadronic quark-connected contributions to the electron, the muon, and the tau anomalous magnetic moments obtained with four dynamical quarks. Performing the continuum limit and an analysis of systematic effects, full agreement with phenomenological results is found. To estimate the impact of omitting the quark-disconnected contributions to the hadronic vacuum polarisation we investigate them on one of the four-flavour ensembles. Additionally, the light quark contributions on the four-flavour sea are compared to the values obtained for $N_f=2$ physic…
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 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…
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 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…
We present results for the leading hadronic contribution to the muon $g-2$ from configurations with 2+1+1 flavors of HISQ quarks. The ensembles have been generated by the MILC collaboration at three lattice spacings. Using the time-momentum representation of the electromagnetic current correlator, we calculate the finite volume effects up to next-to-next-to-leading-order in Chiral Perturbation Theory.
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.
We present an update of our lattice QCD study of the vacuum polarization function using O$(a)$-improved $N_ {\rm f} =2$ Wilson fermions with increased statistics and a large set of momenta. The resulting points are highly correlated and thus require a correlated fitting procedure. We employ an extended frequentist method to estimate the systematic uncertainties due to the momentum dependence and to the continuum and chiral extrapolations. We present preliminary results of the leading order hadronic contribution of the anomalous magnetic moment of the muon $\left(a_μ^{\mathrm{HLO}}\right)$ at t…
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 $σ(e^+e^-\rightarrow\,\mathrm{hadrons})$. Lattice QCD aims to provide a result with similar uncertainty from calculated vector-vector correlation funct…
We present a Lorentz-covariant, Euclidean coordinate-space expression for the hadronic vacuum polarisation, the Adler function and the leading hadronic contribution to the anomalous magnetic moment of the muon. The representation offers a lot of flexibility for an implementation in lattice QCD. We expect it to be particularly helpful for the quark-line disconnected contributions.
The quark-line disconnected diagram is a potentially important ingredient in lattice QCD calculations of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. It is also a notoriously difficult one to evaluate. Here, for the first time, we give an estimate of this contribution based on lattice QCD results that have a statistically significant signal, albeit at one value of the lattice spacing and an unphysically heavy value of the $u/d$ quark mass. We use HPQCD's method of determining the anomalous magnetic moment by reconstructing the Adler function from …
Results from the first lattice QCD analysis of vacuum correlators of local hadronic currents using dispersion relations are presented. We have explored the vector, pseudoscalar, axial, and scalar meson channels, and the proton-like and delta-like baryon channels. The lattice results are shown to agree qualitatively with experimental results in channels where experimental data exist, and shed insight into interacting instanton approximations and sum rule calculations in the other channels. 3 figures included (as PostScript files): pi.ps, nd.ps, va.ps : Search for %%% to retrieve the files. Late…
We give a short description of the present situation of lattice QCD simulations. We then focus on the computation of the anomalous magnetic moment of the muon using lattice techniques. We demonstrate that by employing improved observables for the muon anomalous magnetic moment, a significant reduction of the lattice error can be obtained. This provides a promising scenario that the accuracy of lattice calculations can match the experimental errors.
We present a progress report on the calculation of the connected hadronic contribution to the muon g-2 with C* boundary conditions. For that purpose we use a QCD gauge ensemble with 3+1 flavors and two QCD+QED gauge ensembles with 1+2+1 flavors of dynamical quarks generated by the RC* collaboration. We detail the calculation of the vector mass and elaborate on both statistical and systematic errors.
The vacuum polarization tensor and the corresponding vacuum polarization function are the basis for calculations of numerous observables in lattice QCD. Examples are the hadronic contributions to lepton anomalous magnetic moments, the running of the electroweak and strong couplings and quark masses. Quantities which are derived from the vacuum polarization tensor often involve a summation of current correlators over all distances in position space leading thus to the appearance of short-distance terms. The mechanism of O(a) improvement in the presence of such short-distance terms is not direct…
We calculate the strong isospin breaking and QED corrections to meson masses and the hadronic vacuum polarization in an exploratory study on a $64\times24^3$ lattice with an inverse lattice spacing of $a^{-1}=1.78$ GeV and an isospin symmetric pion mass of $m_π=340$ MeV. We include QED in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. We find that the electromagnetic correction to the leading hadronic contribution to the anomalous magnetic moment of the muon is smaller than $1\%$ for the up quark and $0.1\%$ for the strange quark, although it s…
We introduce a new method for calculating the ${\rm O}(α^3)$ hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment from ${ab-initio}$ lattice QCD. We first derive expressions suitable for computing the higher-order contributions either from the renormalized vacuum polarization function $\hatΠ(q^2)$, or directly from the lattice vector-current correlator in Euclidean space. We then demonstrate the approach using previously-published results for the Taylor coefficients of $\hatΠ(q^2)$ that were obtained on four-flavor QCD gauge-field configurations with physical light-q…
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 …
We describe a computation of the contribution to the anomalous magnetic moment of the muon from the disconnected part of the hadronic vacuum polarization. We use the highly-improved staggered quark (HISQ) formulation for the current density with gauge configurations generated with four flavors of HISQ sea quarks. The computation is performed by stochastic estimation of the current density using the truncated solver method combined with deflation of low-modes. The parameters are tuned to minimize the computational cost for a given target uncertainty in the current-current correlation function. …
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, $a_{μ, \textrm{W}}^{ll}$. 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 th…
The `intermediate window quantity' of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon allows for a high-precision comparison between the data-driven approach and lattice QCD. The existing lattice results, which presently show good consistency among each other, are in strong tension with the data-driven determination. In order to check for a potentially common source of systematic error of the lattice calculations, which are all based on the time-momentum representation (TMR), we perform a calculation using a Lorentz-covariant coordinate-space (CCS) re…
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 compute the leading QED corrections to the hadronic vacuum polarization (HVP) of the photon, relevant for the determination of leptonic anomalous magnetic moments, $a_\ell$. We work in the electroquenched approximation and use dynamical QCD configurations generated by the CLS initiative with two degenerate flavors of non-perturbatively O($a$)-improved Wilson fermions. We consider QED$_{\rm L}$ and QED$_{\rm M}$ to deal with the finite-volume zero modes. We compare results for the Wilson loops with exact analytical determinations. In addition we make sure that the volumes and photon masses u…
We present the first four-flavour lattice calculation of the leading-order hadronic vacuum-polarisation contribution to the anomalous magnetic moment of the muon, a_μ^{hvp}, and the hadronic running of the QED coupling constant, Δα_{QED}^{hvp} (Q^2). In the heavy sector a mixed-action setup is employed. The bare quark masses are determined from matching the K- and D-meson masses to their physical values. Several light quark masses are used in order to yield a controlled extrapolation to the physical pion mass by utilising a recently proposed improved method. We demonstrate that this method als…
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.
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.
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 $α_s^3$.
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…
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…
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 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…
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…
