Vacuum Polarization Contribution Research Articles

Model-independent extrapolation of MUonE data with Padé and D-Log approximants

The MUonE experiment is designed to extract the hadronic contribution to the electromagnetic coupling in the spacelike region Δαhad(t) from elastic eμ scattering. The leading-order hadronic vacuum polarization contribution to the muon g−2, aμHVP,LO, can then be obtained from a weighted integral over Δαhad(t). This, however, requires knowledge of Δαhad(t) in the whole domain of integration, which cannot be achieved by experiment. In this work, we propose to use Padé and D-Log Padé approximants as a systematic and model-independent method to fit and reliably extrapolate the future MUonE experimental data, extracting aμHVP,LO with a conservative but competitive uncertainty, using no or very limited external information. The method relies on fundamental analytic properties of the two-point correlator underlying aμHVP,LO and provides lower and upper bounds for the result for aμHVP,LO. We demonstrate the reliability of the method using toy datasets generated from a model for Δαhad(t) reflecting the expected statistics of the MUonE experiment. Published by the American Physical Society 2024

Time-kernel for lattice determinations of NLO hadronic vacuum polarization contributions to the muon g-2

We study the time-momentum representation of the kernel needed to compute the hadronic vacuum polarization contribution to the muon g-2 in the space-like region at next-to-leading order. For small values of the time, we present analytical series expansions; for large values of the time, we present numerical series expansions which overcome the problems showed by naïve asymptotic expansions. These results are to be employed in lattice QCD determination of the hadronic vacuum polarization contribution to the muon g-2 at next-to-leading order.

The detector for the MUonE experiment at CERN

The comparison between the measurement and the Standard Model prediction of the anomalous magnetic moment of the muon (aμ) is a way to search for physics beyond the Standard Model. Recently, the Fermilab Muon g−2 experiment measured aμ with a precision of 200 parts per billion (ppb), while the theoretical prediction of aμ is limited by the uncertainty in the leading-order hadronic vacuum-polarization contribution (aμHLO) as the two calculation methods, namely the dispersive approach and the lattice QCD, yield different results. The MUonE experiment proposes an alternative and independent way to precisely measure aμHLO through a unique direct measurement, which can be used to cross-check the theoretical value.

QCD bounds on leading-order hadronic vacuum polarization contributions to the muon anomalous magnetic moment

QCD bounds on the leading-order (LO) hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon [aμHVP,LO, aμ=(g−2)μ/2] are determined by imposing Hölder inequalities and related inequality constraints on systems of finite-energy QCD sum rules. This novel methodology is complementary to lattice QCD and data-driven approaches to determining aμHVP,LO. For the light-quark (u, d, s) contributions up to five-loop order in perturbation theory in the chiral limit, LO in light-quark mass corrections, next-to-leading order in dimension-four QCD condensates, and to LO in dimension-six QCD condensates, we find that (657.0±34.8)×10−10≤aμHVP,LO≤(788.4±41.8)×10−10, bridging the range between lattice QCD and data-driven values. Published by the American Physical Society 2024

Hadronic vacuum polarization in the muon g − 2: the short-distance contribution from lattice QCD

We present results for the short-distance window observable of the hadronic vacuum polarization contribution to the muon g – 2, computed via the time-momentum representation (TMR) in lattice QCD. A key novelty of our calculation is the reduction of discretization effects by a suitable subtraction applied to the TMR kernel function, which cancels the leading \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${x}_{0}^{4}$$\\end{document}-behaviour at short distances. To compensate for the subtraction, one must substitute a term that can be reliably computed in perturbative QCD. We apply this strategy to our data for the vector current collected on ensembles generated with 2 + 1 flavours of O(a)-improved Wilson quarks at six values of the lattice spacing and pion masses in the range 130 – 420 MeV. Our estimate at the physical point contains a full error budget and reads \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\left({a}_{\\mu }^{{\ ext{hvp}}}\\right)}^{{\ ext{SD}}}$$\\end{document} = 68.85(14)stat (42)syst·10−10, which corresponds to a relative precision of 0.7%. We discuss the implications of our result for the observed tensions between lattice and data-driven evaluations of the hadronic vacuum polarization.

Data-driven estimates for light-quark-connected and strange-plus-disconnected hadronic g−2 window quantities

A number of discrepancies have emerged between lattice computations and data-driven dispersive evaluations of the RBC/UKQCD intermediate-window-hadronic contribution to the muon anomalous magnetic moment. It is therefore interesting to obtain data-driven estimates for the light-quark-connected and strange-plus-disconnected components of this window quantity, allowing for a more detailed comparison between the lattice and data-driven approaches. The aim of this paper is to provide these estimates, extending the analysis to several other window quantities, including two windows designed to focus on the region in which the two-pion contribution is dominant. Clear discrepancies are observed for all light-quark-connected contributions considered, while good agreement with lattice results is found for strange-plus-disconnected contributions to the quantities for which corresponding lattice results exist. The largest of these discrepancies is that for the RBC/UKQCD intermediate window, where, as previously reported, our data-driven result, aμW1,lqc=198.9(1.1)×10−10, is in significant tension with the results of 8 different recent lattice determinations. Our strategy is the same as recently employed in obtaining data-driven estimates for the light-quark-connected and strange-plus-disconnected components of the full leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. Updated versions of those earlier results are also presented, for completeness. Published by the American Physical Society 2024

Tau data-driven evaluation of the Hadronic Vacuum Polarization [formula omitted

Windows in Euclidean time have become a standard tool for comparing lattice QCD and data-driven computations of the hadronic vacuum polarization (HVP) contribution to the muon g−2. Here we review our results, obtained using isospin-rotated τ−→π−π0ντ data instead of e+e−→π+π− measurements, and compare them to other approaches. Consistency of the tau-based and lattice results hints to underestimated uncertainties in the e+e− data. If that is the case, the theory prediction of the muon g−2 would only lie at ≲2.5σ from its measured value.

Effects of the quark flavour thresholds in the hadronic vacuum polarization contributions to the muon anomalous magnetic moment

The equivalent representations for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in the presence of the quark flavour thresholds are studied. Specifically, the explicit relations between the contributions given by the integration over a finite kinematic interval to expressed in terms of the hadronic vacuum polarization function, Adler function, and the R-ratio of electron–positron annihilation into hadrons are derived. It is shown that the quark flavour thresholds of the hadronic vacuum polarization function generate additional contributions to expressed in terms of the Adler function and the R-ratio and the explicit expressions for such contributions are obtained. The commonly employed dispersion relations, which bind together hadronic vacuum polarization function, Adler function, and R-ratio, are extended to account for the effects due to the quark flavour thresholds. The examined additional contributions due to the heavy quark thresholds to expressed in terms of the R-ratio appear to be quite sizable, that can be of a particular relevance for the data-driven method of assessment of the hadronic part of the muon anomalous magnetic moment.

Two-loop radiative corrections to e+e− → γγ∗ cross section

The increasing accuracy of current and planned experiments to measure the anomalous magnetic moment of the muon requires more precision and reliability of its theoretical calculation. For this purpose, we calculate the differential cross section for the process of annihilation of an electron-positron pair into two photons, one of which is virtual, accompanied by the emission of soft photons, taking into account radiative corrections of the order α2. The results obtained can be used to improve the accuracy of calculating the contribution of the hadron vacuum polarization to the muon anomalous moment. It is shown that all logarithmically amplified two-loop corrections can be easily found using modern theorems of soft and collinear factorizations and available one-loop results.

Modeling the R-ratio and hadronic contributions to g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g-2$$\\end{document} with a Treed Gaussian process

The BNL and FNAL measurements of the anomalous magnetic moment of the muon disagree with the Standard Model (SM) prediction by more than 4σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\\sigma $$\\end{document}. The hadronic vacuum polarization (HVP) contributions are the dominant source of uncertainty in the SM prediction. There are, however, tensions between different estimates of the HVP contributions, including data-driven estimates based on measurements of the R-ratio. To investigate that tension, we modeled the unknown R-ratio as a function of CM energy with a treed Gaussian process (TGP). This is a principled and general method grounded in data-science that allows complete uncertainty quantification and automatically balances over- and under-fitting to noisy data. Our tool yields exploratory results are similar to previous ones and we find no indication that the R-ratio was previously mismodeled. Whilst we advance some aspects of modeling the R-ratio and develop new tools for doing so, a competitive estimate of the HVP contributions requires domain-specific expertise and a carefully curated database of measurements (github, https://github.com/qiao688/TGP_for_g-2).

QCD parameters and SM-high precisions from e+e−→ Hadrons

Using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor and the value of gluon condensate 〈αsG2〉 from heavy quarkonia, we extract the value of the four-quark condensate: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and the dimension eight condensate: d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. We show the inconsistency in using at the same time the standard SVZ value of the gluon and the vacuum saturation of the four-quark condensates. Using the previous values of the four-quark and d8 condensates, we re-extract 〈αsG2〉 from R10 to be: (6.12±0.61)×10−2 GeV4 in perfect agreement with the one from heavy quarkonia. We also use the lowest τ-like decay moment Rτee to extract the value of the QCD coupling. We obtain to O(αs4) the mean of FO (fixed order) and CI (contour improved) PT series within the standard OPE: αs(Mτ)=0.3385(50)(136)syst where the last error is an added conservative systematic from the distance of the mean to the FO and CI central values. Assuming that the αs-coefficients of the PT series grow geometrically as observed in the calculated case, we obtain to O(αs5): αs(Mτ)=0.3262(37)(78)syst. These results lead to: αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)] to order αs4 [resp. αs5]. The corresponding value of the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Reciprocally, using αs(Mτ), 〈αsG2〉 and d8 as inputs, we test the stability of the value of the four-quark condensate obtained from the lowest τ-like moment. We complete our analysis by updating our previous determinations of the lowest order hadronic vacuum polarization contributions to the lepton anomalies and to α(MZ2). We obtain in Table 3: aμ|l.ohvp=(7036.5±38.9)×10−11, aτ|l.ohvp=(3494.8±24.7)×10−9 and α(5)(MZ)|had=(2766.3±4.5)×10−5. This new value of aμ leads to: Δaμ≡aμexp−aμth=(142±42th±41exp)×10−11 which reduces the tension between the SM prediction and experiment.

Dark photons and displaced vertices at the MUonE experiment

MUonE is a proposed experiment designed to measure the hadronic vacuum polarization contribution to muon $g-2$ through elastic $\mu-e$ scattering. As such it employs an extremely high-resolution tracking apparatus. We point out that this makes MUonE also a very promising experiment to search for displaced vertices from light, weakly-interacting new particles. We demonstrate its potential by showing how it has excellent sensitivity to dark photons in the mass range $10~\mathrm{MeV} \le m_{A'} \le 100~\mathrm{MeV}$ and kinetic mixing parameter $10^{-5} \le \epsilon e \le 10^{-3}$, through the process $\mu^{\pm}\, e^- \to \mu^{\pm}\, e^-\, A'$ followed by $A'\to e^+e^-$.

Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions

We present a lattice determination of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, ${a}_{\ensuremath{\mu}}^{\mathrm{HVP}}$, in the so-called short and intermediate time-distance windows, ${a}_{\ensuremath{\mu}}^{\mathrm{SD}}$ and ${a}_{\ensuremath{\mu}}^{\mathrm{W}}$, defined by the RBC/UKQCD Collaboration [Phys. Rev. Lett. 121, 022003 (2018)]. We employ gauge ensembles produced by the Extended Twisted Mass Collaboration (ETMC) with ${N}_{f}=2+1+1$ flavors of Wilson-clover twisted-mass quarks with masses of all the dynamical quark flavors tuned close to their physical values. The simulations are carried out at three values of the lattice spacing equal to $\ensuremath{\simeq}0.057$, 0.068 and 0.080 fm with spatial lattice sizes up to $L\ensuremath{\simeq}7.6\text{ }\text{ }\mathrm{fm}$. For the short-distance window we obtain ${a}_{\ensuremath{\mu}}^{\mathrm{SD}}(\mathrm{ETMC})=69.27(34)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, which is consistent with the recent dispersive value of ${a}_{\ensuremath{\mu}}^{\mathrm{SD}}({e}^{+}{e}^{\ensuremath{-}})=68.4(5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ [, Phys. Lett. B 833, 137313 (2022)]. In the case of the intermediate window we get the value ${a}_{\ensuremath{\mu}}^{\mathrm{W}}(\mathrm{ETMC})=236.3(1.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, which is consistent with the result ${a}_{\ensuremath{\mu}}^{\mathrm{W}}(\mathrm{BMW})=236.7(1.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ [, Nature (London) 593, 51 (2021)] by the BMW Collaboration as well as with the recent determination by the CLS/Mainz group of ${a}_{\ensuremath{\mu}}^{\mathrm{W}}(\mathrm{CLS})=237.30(1.46)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ [, Phys. Rev. D 106, 114502 (2022)]. However, it is larger than the dispersive result of ${a}_{\ensuremath{\mu}}^{\mathrm{W}}({e}^{+}{e}^{\ensuremath{-}})=229.4(1.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ by approximately 3.6 standard deviations. The tension increases to approximately 4.5 standard deviations if we average our ETMC result with those by BMW and CLS/Mainz. Our accurate lattice results in the short and intermediate windows point to a possible deviation of the ${e}^{+}{e}^{\ensuremath{-}}$ cross section data with respect to Standard Model predictions in the low- and intermediate-energy regions but not in the high-energy region.

Data-based determination of the isospin-limit light-quark-connected contribution to the anomalous magnetic moment of the muon

We describe how recent determinations of exclusive-mode contributions to $a_\mu^{\rm LO,HVP}$, the leading-order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, can be used to provide, up to small electromagnetic (EM) corrections accessible from the lattice, a data-based dispersive determination of $a_\mu^{\rm lqc;\, IL}$, the isospin-limit, light-quark-connected contribution to $a_\mu^{\rm LO,HVP}$. Such a determination is of interest in view of the existence of a number of lattice results for this quantity, emerging evidence for a tension between lattice and dispersive determinations of $a_\mu^{\rm LO,HVP}$, and the desire to clarify the source of this tension. Taking as input for the small EM correction that must be applied to the purely data-driven dispersive determination the result $-0.93(58)\times 10^{-10}$ obtained in a recent BMW lattice study, we find $a_\mu^{\rm lqc;\, IL}$ to be $635.0(2.7)\times 10^{-10}$ if the results of Keshavarzi, Nomura and Teubner are used for the exclusive-mode contributions and $638.4(4.1)\times 10^{-10}$ if instead those of Davier, H\"ocker, Malaescu and Zhang are used.

Coordinate-space calculation of the window observable for the hadronic vacuum polarization contribution to (g−2)μ

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) representation. We present results for the isovector and the connected strange-quark contributions to the intermediate window quantity at a reference point in the $(m_\pi,m_K)$ plane, in the continuum and infinite-volume limit, based on four different lattice spacings. Our results are in good agreement with those of the recent TMR-based Mainz-CLS publication.

Muon g-2 with overlap valence fermions

We present a lattice calculation of the leading order (LO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment for the connected light and strange quarks, $a^{\rm W}_{{\rm con}, l/s}$ in the widely used window $t_0=0.4~\mathrm{fm}$, $t_1=1.0~\mathrm{fm}$, $\Delta=0.15~\mathrm{fm}$, and also of $a^{\rm S}_{{\rm con}, l/s}$ in the short distance region. We use overlap fermions on 4 physical-point ensembles. Two 2+1 flavor RBC/UKQCD ensembles use domain wall fermions (DWF) and Iwasaki gauge actions at $a = 0.084$ and 0.114 fm, and two 2+1+1 flavor MILC ensembles use the highly improved staggered quark (HISQ) and Symanzik gauge actions at $a = 0.088$ and 0.121 fm. We have incorporated infinite volume corrections from 3 additional DWF ensembles at ${\rm L}$ = 4.8, 6.4 and 9.6 fm and physical pion mass. For $a^{\rm W}_{{\rm con}, l}$, we find that our results on the two smaller lattice spacings are consistent with those using the unitary setup, but those at the two coarser lattice spacings are slightly different. Eventually, we predict $a^{\rm W}_{{\rm con}, l}=206.7(1.5)(1.0)$ and $a^{\rm W}_{{\rm con}, s}=26.8(0.1)(0.3)$, using linear extrapolation in $a^2$, with systematic uncertainties estimated from the difference of the central values from the RBC/UKQCD and MILC ensembles.

Leading hadronic contribution to the muon g − 2 from lattice QCD

We compute the leading order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. The calculations are performed using four flavors of stout smeared staggered quarks, with quark masses at their physical values. The continuum limit is taken using six different lattice spacings ranging from 0.132 fm down to 0.064 fm. All strong isospin breaking and electromagnetic effects are accounted for to leading order. A controlled infinite volume limit is taken thanks to dedicated simulations performed in box sizes up to 11 fm. Putting all these ingredients together, we find [(gµ − 2)/2]LO−HVP = 707.5[5.5] 10−10, which has a total uncertainty of 0.8%. Compared to determinations based on the e+e− → hadrons cross section, our result significantly reduces the tension between the standard model prediction for the muon g − 2 and its experimental value.

MUonE, muon g−2 and electroweak precision constraints within 2HDMs

Two Higgs doublet models are attractive scenarios for physics beyond the Standard Model. In particular, lepton-specific manifestations remain contenders to explain the observed discrepancy between the anomalous magnetic moment of the muon $a_\mu$ predicted within the Standard Model and recent observations at Fermilab and BNL. Dominant uncertainties that affect $a_\mu$ have motivated the MUonE experiment to access the hadronic vacuum polarisation contribution that impacts $a_\mu$ via elastic muon-electron scattering. In this work, we contrast the high precision that is achievable within the MUonE context with constraints from flavour physics, precision electroweak constraints and LHC searches as well as their extrapolations for a range of two Higgs doublet models with a softly broken $\mathbb{Z}_2$ symmetry. We find that the sensitivity of MUonE does not extend beyond the parameter regions that are already excluded by other constraints. MUonE will therefore provide a detailed measurement of the hadronic vacuum polarisation contribution which then transparently informs $a_\mu$ interpretations in 2HDMs without modifications of correlations from beyond the Standard Model interactions. In passing we extend earlier results of LHC and flavour projections to lepton-specific 2HDM (Types X and Y) scenarios, and comment on the possibility of modifying the value of the W-boson mass; we briefly discuss the implications for a strong first-order electroweak phase transition for these models.

Addendum: Timelike and spacelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment (2022 J. Phys. G: Nucl. Part. Phys. 49 055001)

This addendum provides results complementary to those obtained in [J. Phys. G 49, 055001 (2022)]. Specifically, an equivalent form of the relation, which binds together the ‘spacelike’ kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment , is obtained. It is shown that the infrared limiting value of the ‘spacelike’ and ‘timelike’ kernel functions, which enter the representations for involving the Adler function and the R-ratio, is identical to the corresponding QED contribution to the muon anomalous magnetic moment of the preceding order in the electromagnetic coupling. The next-to-leading order contributions (which includes the leptonic and hadronic insertions) and (which includes the double hadronic insertion), are studied. The three kernel functions appearing in the representations for , which involve the hadronic vacuum polarization function, Adler function, and the R-ratio, are presented for the cases of the electron and τ-lepton loop insertions.

Window observable for the hadronic vacuum polarization contribution to the muon g−2 from lattice QCD

Euclidean time windows in the integral representation of the hadronic vacuum polarization contribution to the muon $g-2$ serve to test the consistency of lattice calculations and may help in tracing the origins of a potential tension between lattice and data-driven evaluations. In this paper, we present results for the intermediate time window observable computed using O($a$) improved Wilson fermions at six values of the lattice spacings below 0.1\,fm and pion masses down to the physical value. Using two different sets of improvement coefficients in the definitions of the local and conserved vector currents, we perform a detailed scaling study which results in a fully controlled extrapolation to the continuum limit without any additional treatment of the data, except for the inclusion of finite-volume corrections. To determine the latter, we use a combination of the method of Hansen and Patella and the Meyer-Lellouch-L\"uscher procedure employing the Gounaris-Sakurai parameterization for the pion form factor. We correct our results for isospin-breaking effects via the perturbative expansion of QCD+QED around the isosymmetric theory. Our result at the physical point is $a_\mu^{\mathrm{win}}=(237.30\pm0.79_{\rm stat}\pm1.22_{\rm syst})\times10^{-10}$, where the systematic error includes an estimate of the uncertainty due to the quenched charm quark in our calculation. Our result displays a tension of 3.9$\sigma$ with a recent evaluation of $a_\mu^{\mathrm{win}}$ based on the data-driven method.