The well-known discrepancy in the muon $g-2$ between experiment and theory demands further theory investigations in view of the upcoming new experiments. One of the leading uncertainties lies in the hadronic light-by-light scattering contribution (HLbL), that we address with our position-space approach. We focus on exploratory studies of the pion-pole contribution in a simple model and the fermion loop without gluon exchanges in the continuum and in infinite volume. These studies provide us with useful information for our planned computation of HLbL in the muon $g-2$ using full QCD.
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.
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.
I present a preliminary calculation of the hadronic vacuum polarization for 2+1 flavors of improved Kogut-Susskind quarks by utilizing a set of gauge configurations recently generated by the MILC collaboration. The polarization function $Π(q^2)$ is then used to calculate the lowest order (in $α_{QED}$) hadronic contribution to the muon anomalous magnetic moment.
Hadronic light-by-light scattering in the anomalous magnetic moment of the muon $a_μ$ is one of two hadronic effects limiting the precision of the Standard Model prediction for this precision observable, and hence the new-physics discovery potential of direct experimental determinations of $a_μ$. In this contribution, we report on recent progress in the calculation of this effect achieved both via dispersive and lattice QCD methods.
After a brief self-contained introduction to the muon anomalous magnetic moment, (g-2), we review the status of lattice calculations of the hadronic vacuum polarization contribution and present first results from lattice QCD for the hadronic light-by-light scattering contribution. The signal for the latter is consistent with model calculations. While encouraging, the statistical error is large and systematic errors are mostly uncontrolled. The method is applied first to pure QED as a check.
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 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{…
The form factor that yields the light-by-light scattering contribution to the muon anomalous magnetic moment is computed in lattice QCD+QED and QED. A non-perturbative treatment of QED is used and is checked against perturbation theory. The hadronic contribution is calculated for unphysical quark and muon masses, and only the diagram with a single quark loop is computed. Statistically significant signals are obtained. Initial results appear promising, and the prospect for a complete calculation with physical masses and controlled errors is discussed.
We report preliminary results for the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment. Several ensembles using 2+1 flavors of Möbius domain-wall fermions, generated by the RBC/UKQCD collaborations, are employed to take the continuum and infinite volume limits of finite volume lattice QED+QCD. We find $a_μ^{\rm HLbL} = (7.41\pm6.33)\times 10^{-10}$
The lowest order hadronic contribution to the $g-2$ factor of the muon is analyzed in the framework of the operator product expansion at short distances, and a QCD finite energy sum rule designed to quench the role of the $e^+ e^-$ data. This procedure reduces the discrepancy between experiment and theory, $Δa_μ\equiv a^{EXP}_μ- a^{SM}_μ$, from $Δa_μ= 28.7 (8.0) \times 10^{-10}$ to $Δa_μ= 19.2 (8.0) \times 10^{-10}$, i.e. without changing the uncertainty.
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 the first result for the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment with all errors systematically controlled. Several ensembles using 2+1 flavors of physical mass Möbius domain-wall fermions, generated by the RBC/UKQCD collaborations, are employed to take the continuum and infinite volume limits of finite volume lattice QED+QCD. We find $a_μ^{\rm HLbL} = 7.87(3.06)_\text{stat}(1.77)_\text{sys}\times 10^{-10}$. Our value is consistent with previous model results and leaves little room for this notoriously difficult hadronic contribution to e…
The hadronic light-by-light scattering contribution to the muon g-2 is the most troublesome component of its theoretical prediction; (1) it cannot be determined from the other measurable quantities, (2) the dimensional argument and the estimation based on hadronic models imply that the magnitude of this contribution may be comparable to the discrepancy between the standard model prediction and the experimental value. The direct approach to evaluate the hadronic light-by-light scattering contribution requires the evaluation of the correlation function of {\it four} hadronic electromagnetic curr…
The quark-connected part of the hadronic light-by-light scattering contribution to the muon's anomalous magnetic moment is computed using lattice QCD with chiral fermions. We report several significant algorithmic improvements and demonstrate their effectiveness through specific calculations which show a reduction in statistical errors by more than an order of magnitude. The most realistic of these calculations is performed with a near-physical, $171$ MeV pion mass on a $(4.6\;\mathrm{fm})^3$ spatial volume using the $32^3\times 64$ Iwasaki+DSDR gauge ensemble of the RBC/UKQCD Collaboration.
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 anomalous magnetic moment of the muon currently exhibits a discrepancy of about three standard deviations between the experimental value and recent Standard Model predictions. The theoretical uncertainty is dominated by the hadronic vacuum polarization and the hadronic light-by-light (HLbL) scattering contributions, where the latter has so far only been fully evaluated using different models. To pave the way for a lattice calculation of HLbL, we present an expression for the HLbL contribution to $g-2$ that involves a multidimensional integral over a position-space QED kernel function in th…
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…
The anomalous magnetic moment of the muon, $g_μ-2$, is one of the most promising observables to identify hints for physics beyond the Standard Model. QCD contributions are currently responsible for the largest fraction of the overall theoretical uncertainty in $g_μ-2$. The possibility to determine these hadronic contributions from first principles through lattice QCD calculations has triggered a number of recent studies. Recent proposals to improve the accuracy of lattice determinations are reported. We present an update of our studies of the leading-order hadronic contribution to $g_μ-2$ with…
We present a quenched lattice calculation of the lowest order (alpha^2) hadronic contribution to the anomalous magnetic moment of the muon which arises from the hadronic vacuum polarization. A general method is presented for computing entirely in Euclidean space, obviating the need for the usual dispersive treatment which relies on experimental data for e^+e^- annihilation to hadrons. While the result is not yet of comparable accuracy to those state-of-the-art calculations, systematic improvement of the quenched lattice computation to this level of accuracy is straightforward and well within t…
We discuss the hadronic contributions to the muon anomalous magnetic moment. They are dominated by light quark contributions which are constrained by the mechanism of chiral symmetry breaking. Using the leading order result based on $e^+ e^-$ scattering data, we show that the next-to-leading order contributions in the fine structure constant $α$ can be reliably calculated. Extending this idea to the hadronic four-point function we give a prediction for the light-by-light contribution.
We present preliminary results from our calculation of the pseudoscalar transition form factors of the $η$ and $η'$ mesons using staggered quarks on $N_f=2+1+1$ gauge ensembles generated by the Budapest-Marseille-Wuppertal collaboration. These transition form factors are an important input for the hadronic light-by-light contribution to the muon $(g-2)$. We first elaborate on the extraction of the masses of the $η$ and $η'$ mesons, that mix under the dynamics of QCD. Thereafter, we show our preliminary results for the pseudoscalar transition form factors, focusing on the $η$ meson in the absen…
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 review some of the issues that arise in attempts to compute the hadronic corrections to the muon anomalous magnetic moment using Lattice QCD. We concentrate on the dominant contribution, which requires an accurate evaluation of the hadronic vacuum polarization.
A lattice computation of the leading-order hadronic contribution to the muon anomalous magnetic moment can potentially help reduce the error on the Standard Model prediction for this quantity, if sufficient control of all systematic errors affecting such a computation can be achieved. One of these systematic errors is that associated with the extrapolation to the physical pion mass from values on the lattice larger than the physical pion mass. We investigate this extrapolation assuming lattice pion masses in the range of 200 to 400MeV with the help of two-loop chiral perturbation theory, and f…
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 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 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…
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…
We report on our progress toward the computation of the $π^0$, $η$ and $η^{\prime}$ transition form factors using staggered quarks on $N_f=2+1+1$ gauge ensembles generated by the Budapest-Marseille-Wuppertal collaboration. These form factors are essential ingredients to evaluate the pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon $g-2$. Preliminary results for the pseudoscalar-pole contributions are presented, at finite lattice spacing, for all three light mesons.
Higher-order hadronic corrections to the anomalous magnetic moment of the muon have been evaluated including next-to-next-to-leading-order insertions of hadronic vacuum polarization and next-to-leading-order corrections to hadronic light-by-light scattering. This leaves a set of mixed leptonic and hadronic corrections in the form of double-bubble topologies as the only remaining hadronic effect at ${\mathcal O}(α^4)$. Here, we estimate these contributions by analyzing the respective cuts of the diagrams, suggesting that the impact is limited to $\lesssim 1\times 10^{-11}$ and thus negligible a…
I present quenched domain wall fermion and 2+1 flavor improved Kogut-Susskind fermion calculations of the hadronic vacuum polarization which are used to calculate the ${\cal O}(α^2)$ hadronic contribution to the anomalous magnetic moment of the muon. Together with previous quenched calcuations, the new results confirm that in the quenched theory the hadronic contribution is signifcantly smaller ($\sim 30%$) than the value obtained from the total cross section of $e^+e^-$ annhilation to hadrons. The 2+1 flavor results show an increasing contribution to $g-2$ as the quark mass is reduced.
We present a calculation of the hadronic vacuum polarization (HVP) tensor within the framework of Dyson--Schwinger equations. To this end we use a well-established phenomenological model for the quark-gluon interaction with parameters fixed to reproduce hadronic observables. From the HVP tensor we compute both the Adler function and the HVP contribution to the anomalous magnetic moment of the muon, $a_μ$. We find $a_μ^{HVP}= 6760\times 10^{-11}$ which deviates about two percent from the value extracted from experiment. Additionally, we make comparison with a recent lattice determination of $a_…
We give a status report on the hadronic light-by-light scattering contribution to the muon's anomalous magnetic moment from the Dyson-Schwinger approach. We discuss novel, model-independent properties of the light-by-light amplitude: we give its covariant decomposition in view of electromagnetic gauge invariance and Bose symmetry, and we identify the relevant kinematic regions that are probed under the integral. The decomposition of the amplitude at the quark level and the importance of its various diagrams are discussed and related to model approaches.
We present the most recent lattice results for the lowest-order hadronic contribution to the muon anomalous magnetic moment using 2+1 flavor 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 with the inclusion of the vector particles as resonances, to evaluate the vacuum polarization. We discuss the preliminary fit results and attendant systematic uncertainties, paying particular attention to the relative contributions of the pions and vecto…
We extend our analysis of the leading hadronic contribution to the anomalous magnetic moment of the muon using the mixed representation method to study its chiral behavior. We present results 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 with pion masses as low as 190 MeV. The data is analyzed also using the more standard four-momentum method. Both methods are systematically compared as the calculations approach the physical point.
The anomalous magnetic moment of muon, $g-2$, is a very precisely measured quantity. However, the current measurement disagrees with standard model by about 3 standard deviations. Hadronic vacuum polarization and hadronic light by light are the two types of processes that contribute most to the theoretical uncertainty. I will describe how lattice methods are well-suited to provide a first-principle's result for the hadronic light by light contribution, the various numerical strategies that are presently being used to evaluate it, our current results and the important remaining challenges which…
We present a refined calculation of the quark-loop contribution to hadronic light-by-light scatter- ing that focuses upon the impact of the transverse components of the quark-photon vertex. These structures are compared and contrasted with those found within the extended NJL-models. We discuss similarities and differences between the two approaches and further clarify the important role of momentum dependent dressing functions. As expected we find that the transverse structures of the quark-photon vertex lead to a suppression of the quark-loop contribution to the anomalous magnetic moment of t…
The current measurement of muonic $g - 2$ disagrees with the theoretical calculation by about 3 standard deviations. Hadronic vacuum polarization (HVP) and hadronic light by light (HLbL) are the two types of processes that contribute most to the theoretical uncertainty. The current value for HLbL is still given by models. I will describe results from a first-principles lattice calculation with a 139 MeV pion in a box of 5.5 fm extent. Our current numerical strategies, including noise reduction techniques, evaluating the HLbL amplitude at zero external momentum transfer, and important remaining…
We report our recent lattice calculation of hadronic light-by-light contribution to muon $g-2$ using our recently developed moment method. The connected diagrams and the leading disconnected diagrams are included. The calculation is performed on a $48^3 \times 96$ lattice with physical pion mass and 5.5 fm box size. We expect sizable finite volume and finite lattice spacing corrections to the results of these calculations which will be estimated in calculations to be carried out over the next 1-2 years.
The leading order hadronic contribution to the muon magnetic moment anomaly, $a^{HAD}_μ$, is determined entirely in the framework of QCD. The result in the light-quark sector, in units of $10^{-10}$, is $a^{HAD}_μ|_{uds} =686 \pm 26$, and in the heavy-quark sector $a^{HAD}_μ|_{c} =14.4 \pm 0.1$, and $a^{HAD}_μ|_{b} =0.29 \pm 0.01$, resulting in $a^{HAD}_μ= 701 \pm 26$. The main uncertainty is due to the current lattice QCD value of the first and second derivative of the electromagnetic current correlator at the origin. Expected improvement in the precision of these derivatives may render this …
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.
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.
One of the most promising quantities for the search of signatures of physics beyond the Standard Model is the anomalous magnetic moment $g-2$ of the muon, where a comparison of the experimental result with the Standard Model estimate yields a deviation of about $3.5~σ$. On the theory side, the largest uncertainty arises from the hadronic sector, namely the hadronic vacuum polarisation and the hadronic light-by-light scattering. I review recent progress in calculating the hadronic contributions to the muon $g-2$ from the lattice and discuss the prospects and challenges to match the precision of…
In our previous work, the connected and leading disconnected hadronic light-by-light contributions to the muon anomalous magnetic moment (g - 2) have been computed using lattice QCD ensembles corresponding to physical pion mass generated by the RBC/UKQCD collaboration. However, the calculation is expected to suffer from a significant finite volume error that scales like $1/L^2$ where $L$ is the spatial size of the lattice. In this paper, we demonstrate that this problem is cured by treating the muon and photons in infinite volume, continuum QED, resulting in a weighting function that is pre-co…
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…
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 …
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…
We present the mini-proceedings of the workshops Hadronic contributions to the muon anomalous magnetic moment: strategies for improvements of the accuracy of the theoretical prediction and $(g-2)_μ$: Quo vadis?, both held in Mainz from April 1$^{\rm rst}$ to 5$^{\rm th}$ and from April 7$^{\rm th}$ to 10$^{\rm th}$, 2014, respectively.
