The light-quark contribution to the leading HVP term of the muon $g - 2$ from twisted-mass fermions

We present a lattice calculation of the leading Hadronic Vacuum Polarization (HVP) contribution of the light u- and d-quarks to the anomalous magnetic moment of the muon, $a_\mu^{\rm HVP}(ud)$, adopting the gauge configurations generated by the European Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing with pion masses in the range 210 - 450 MeV. Thanks to several lattices at fixed values of the light-quark mass and scale but with different sizes we perform a careful investigation of finite-volume effects (FVEs). In order to remove FVEs we develop an analytic representation of the vector correlator, which describes the lattice data for time distances larger than $\simeq 0.2$ fm. The representation is based on quark-hadron duality at small and intermediate time distances and on the two-pion contributions in a finite box at larger time distances. After extrapolation to the physical pion point and to the continuum limit we obtain $a_\mu^{\rm HVP}(ud) = 619.0~(17.8) \cdot 10^{-10}$. Adding the contribution of strange and charm quarks, obtained by ETMC, and an estimate of the isospin-breaking corrections and quark-disconnected diagrams from the literature we get $a_\mu^{\rm HVP}(udsc) = 683~(19) \cdot 10^{-10}$, which is consistent with recent results based on dispersive analyses of the experimental cross section data for $e^+ e^-$ annihilation into hadrons. Using our analytic representation of the vector correlator, taken at the physical pion mass in the continuum and infinite volume limits, we provide the first eleven moments of the polarization function and we compare them with recent results of the dispersive analysis of the $\pi^+ \pi^-$ channels. We estimate also the light-quark contribution to the missing part of $a_\mu^{\rm HVP}$ not covered in the MUonE experiment.


F. Sanfilippo and S. Simula
We present a lattice calculation of the leading hadronic vacuum polarization (HVP) contribution of the light u-and d-quarks to the anomalous magnetic moment of the muon, a HVP μ ðudÞ, adopting the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with N f ¼ 2 þ 1 þ 1 dynamical quarks at three values of the lattice spacing (a ≃ 0.062; 0.082; 0.089 fm) with pion masses in the range M π ≃ 210-450 MeV. Thanks to several lattices at fixed values of the light-quark mass and scale but with different sizes we perform a careful investigation of finite-volume effects (FVEs), which represent one of main source of uncertainty in modern lattice calculations of a HVP μ ðudÞ. In order to remove FVEs we develop an analytic representation of the vector correlator, which describes the lattice data for time distances larger than ≃0.2 fm. The representation is based on quark-hadron duality at small and intermediate time distances and on the two-pion contributions in a finite box at larger time distances. After removing FVEs we extrapolate the corrected lattice data to the physical pion point and to the continuum limit taking into account the chiral logs predicted by Chiral Perturbation Theory (ChPT). We obtain a HVP μ ðudÞ ¼ 619.0ð17.8Þ × 10 −10 . Adding the contribution of strange and charm quarks, obtained by ETMC, and an estimate of the isospin-breaking corrections and quark-disconnected diagrams from the literature we get a HVP μ ðudscÞ ¼ 683ð19Þ × 10 −10 , which is consistent with recent results based on dispersive analyses of the experimental cross section data for e þ e − annihilation into hadrons. Using our analytic representation of the vector correlator, taken at the physical pion mass in the continuum and infinite volume limits, we provide the first eleven moments of the polarization function and we compare them with recent results of the dispersive analysis of the π þ π − channels. We estimate also the light-quark contribution to the missing part of a HVP

I. INTRODUCTION
The anomalous magnetic moment of the muon a μ ≡ ðg − 2Þ=2 is one of the most precisely determined quantities in particle physics. It is known experimentally with an accuracy of 0.54 ppm [1] (BNL E821) and the current precision of the Standard Model (SM) prediction is at the level of 0.4 ppm [2]. The tension between the experimental value a exp μ and the SM prediction a SM μ corresponds to ≃3. 5  where the first error is from the SM theory (mainly the HVP term), the second one from the experiment and the third one corresponds to their sum in quadrature.
Since the tension given in Eq. (1) might be an exciting indication of new physics (NP) beyond the SM, an improvement of the uncertainties is highly desirable. The forthcoming g − 2 experiments at Fermilab (E989) [3] and J-PARC (E34) [4] aim at reducing the experimental uncertainty by a factor of four, down to 0.14 ppm, making the comparison of the experimental value of a μ with the theoretical predictions one of the most important tests of the SM in the quest of NP effects. With such a reduced experimental error, the uncertainty of the hadronic corrections, due to the HVP and hadronic light-by-light (LBL) terms [5], will soon become the main limitation of this SM test.
The theoretical predictions for the hadronic contribution a HVP μ have been traditionally obtained from experimental data using dispersion relations for relating the HVP function to the experimental cross section data for e þ e − annihilation into hadrons [6,7]. An alternative approach was proposed in Refs. [8][9][10], namely to compute a HVP μ in lattice QCD from the Euclidean correlation function of two electromagnetic (em) currents. In this respect an impressive progress in the lattice determinations of a HVP μ has been achieved in the last few years [11][12][13][14][15][16][17][18][19][20][21][22][23] and very interesting attempts to compute also the LBL contribution are under way both on the lattice [24,25] and via dispersion approaches and chiral perturbation theory (ChPT) [26][27][28].
An updated status of lattice (as well as nonlattice) efforts for evaluating the hadronic corrections to a μ can be found in Ref. [29]. The main open issue concerning the most accurate lattice calculations of a HVP μ , performed using gauge configurations at the physical pion point, is a significative tension between the HPQCD [20] result, a HVP μ ¼ 667ð13Þ × 10 −10 , on one hand side and the BMW [22] and RBC/UKQCD [23] findings, a HVP μ ¼ 711.0ð18.9Þ × 10 −10 and a HVP μ ¼ 715.4ð18.7Þ × 10 −10 respectively, on the other hand side. Such a tension originates almost totally from the light u-and d-quark (connected) contribution to the HVP and it turns out to be at the same level of the muon anomaly (1).
Besides the leading HVP correction to the one-loop muon diagram, which is of order Oðα 2 em Þ, the increasing precision of the lattice calculations makes it necessary to include both em and strong isospin-breaking (IB) corrections, which contribute at order Oðα 3 em Þ and Oðα 2 em ðm d − m u ÞÞ to the HVP, respectively. In Ref. [30] a lattice calculation of both the leading and the IB corrections to the HVP contribution due to strange and charm quark intermediate states was carried out using the time-momentum representation for a HVP μ [31] and the expansion method of the path integral in the small parameters α em and ðm d − m u Þ=Λ QCD [32,33]. In the strange and charm sectors the strong IB corrections are absent at leading order in ðm d − m u Þ, while the em corrections have been found to be negligible with respect to present uncertainties. Other recent calculations of the IB corrections to the HVP have been performed in Refs. [23,34,35], while higher-order corrections due to diagrams containing HVP and lepton insertions have been recently estimated on the lattice in Ref. [36].
In this paper we present the results of a new lattice calculation of the leading HVP contribution due to light u-and d-quark (connected) intermediate states, a HVP μ ðudÞ, while the evaluation of the corresponding IB corrections will be addressed in a separate work. We make use of the gauge ensembles generated by the European Twisted Mass Collaboration (ETMC) with N f ¼ 2 þ 1 þ 1 dynamical quarks, which include in the sea, besides two light massdegenerate quarks, also the strange and the charm quarks with masses close to their physical values [37,38].
Thanks to the various lattice volumes of the ETMC gauge ensembles we observe quite relevant finite volume effects (FVEs) for a HVP μ ðudÞ. Thus, we develop an analytic representation of the temporal dependence of the Euclidean vector correlator, based on the quark-hadron duality [39], already observed in Ref. [30], and on the two-pion contributions in a finite box [40][41][42][43][44][45][46]. Using such a representation, which constitutes the original part of this work, we are able to reproduce accurately the temporal dependence of the Euclidean vector correlator for all the ETMC gauge ensembles and, by taking properly the infinite volume limit, we can correct in a systematic way our lattice values of a HVP μ ðudÞ for the FVEs. We point out that our estimate of FVEs takes into account the resonant interaction in the two-pion system at variance with the ChPT prediction at next-to-leading (NLO) order [47].
The main result of the present study is  [50], based on dispersive analyses of the experimental cross section data for e þ e − annihilation into hadrons.
Using our analytic representation of the vector correlator, taken at the physical pion mass in the continuum and infinite volume limits, we provide the slope and curvature of the polarization function, Π ðudÞ 1 ¼ 0.1642 ð33Þ GeV −2 and Π ðudÞ 2 ¼ −0.383 ð16Þ GeV −4 , which are compared with lattice results available in the literature. We also estimate higher-order moments (up to the eleventh moment) and compare them with the values of the dispersive analysis of the π þ π − channels made in Ref. [50]. Finally, we estimate the light-quark contribution to the missing part of a HVP μ not covered in the MUonE experiment [51,52].
The paper is organized as follows. In Sec. II we introduce the basic quantities and notation. After providing the simulation details and addressing the identification of the ground-state, we evaluate a HVP μ ðudÞ for all the ETMC ensembles and show the relevance of FVEs. In Sec. III we develop an analytic representation of the vector correlator, based on the quark-hadron duality and the two-pion contributions, obtaining a quite accurate reproduction of the lattice data of the light-quark vector correlator. In Sec. IV we remove FVEs from the lattice data using the analytic representation, while in Sec. V we perform the extrapolations to the physical pion point and to the continuum limit. Our findings are then compared with lattice results available in the literature. In Sec. VI we discuss some relevant features of the analytic representation extrapolated at the physical pion mass and in the continuum limit. We provide the estimates of the lowest-order moments of the polarization function and compare them with the lattice results available in the literature. We estimate also higher-order moments, which we compare with the values of the dispersive analysis of the π þ π − channels made in Ref. [50], as well as the light-quark contribution to the missing part of a HVP μ not covered in the MUonE experiment [51,52]. Finally, Sec. VII contains our conclusions and outlooks for future developments.

II. TIME-MOMENTUM REPRESENTATION
Following our previous work [30], we adopt the timemomentum representation for the evaluation of a HVP μ , namely where t is the Euclidean time, the kernel function fðtÞ is given by [31] fðtÞ and the (Euclidean) vector correlator VðtÞ is defined as with J μ ðxÞ being the em current operator The vector correlator VðtÞ can be calculated on a lattice with volume L 3 and temporal extension T at discretized values of the time distance t from 0 to T. In this work we will limit ourselves to the contribution of the light u-and dquarks, evaluated in isosymmetric QCD (m u ¼ m d ¼ m ud ) neglecting also off-diagonal flavor terms (i.e., including quark-connected diagrams only). Thus, one gets where the first term in the r.h.s is directly given by the lattice data, while for the second term the identification of the ground-state at large time distances is required (see Refs. [17,18,20,21,30] The value of T data has to be large enough that the ground-state contribution is dominant for t > T data and smaller than T=2 in order to avoid backward signals. An important consistency check is that the sum of the two terms in the r.h.s. of Eq. (8) should be independent of the specific choice of the value of T data , as it will be shown later in Sec. II B.

A. Simulation details
The gauge ensembles used in this work are the same adopted in Ref. [53] to determine the up, down, strange, charm quark masses and the lattice scale. We employ the Iwasaki action [54] for gluons and the Wilson twisted mass action [55][56][57] for sea quarks.
We have considered three values of the inverse bare lattice coupling β and different lattice volumes, as shown in Table I, where the number of configurations analyzed (N cfg ) corresponds to a separation of 20 trajectories. For earlier investigations of FVEs ETMC had produced three dedicated ensembles, A40.20, A40.24 and A40.32, which share the same quark mass and lattice spacing and differ only in the lattice size L. To improve such an investigation, which is crucial in the present work, a further gauge ensemble, A40.40, has been generated at a larger value of the lattice size L.
We work in isosymmetric QCD (m u ¼ m d ¼ m ud ) and at each lattice spacing different values of the light sea quark masses have been considered. The light valence and sea quark masses are always taken to be degenerate (m sea ud ¼ m val ud ¼ m ud ). In this work we made use of the bootstrap samplings elaborated for the input parameters of the quark mass analysis of Ref. [53]. There, eight branches of the analysis were adopted differing in: (i) the continuum extrapolation adopting for the scale parameter either the Sommer parameter r 0 or the mass of a fictitious pseudoscalar meson made up of strange(charm)-like quarks; (ii) the chiral extrapolation performed with fitting functions chosen to be either a polynomial expansion or a ChPT Ansatz in the light-quark mass; (iii) the choice between the methods M1 and M2, which differ by Oða 2 Þ effects, used to determine the mass renormalization constant (RC) Z m ¼ 1=Z P in the RI'-MOM scheme. Throughout this work the renormalized light-quark mass m ud is given in the MS scheme at a renormalization scale equal to 2 GeV. At the physical pion point (M phys π ¼ M π 0 ¼ 135 MeV) the value m phys ud ¼ 3.70ð17Þ MeV was determined in Ref. [53], using the experimental value of the pion decay constant for fixing the lattice scale.

B. Ground-state identification
As in Ref. [30], in the numerical simulations we have adopted the following local version of the vector current where ψ 0 has the same mass and charge of ψ, but it is regularized with an opposite value of the Wilson r-parameter, i.e., r 0 ¼ −r. Being at maximal twist the current (9) renormalizes multiplicatively through the RC Z A determined in Ref. [53]. By construction the local current (9) cannot generate off-diagonal flavor contributions in the vector correlator (6). As discussed in Ref. [30], the properties of the kernel function fðtÞ, given by Eq. (5), guarantee that the contact terms, generated in the HVP tensor by a local vector current, cannot contribute to the evaluation of a HVP μ (see also Ref. [58]).
We have calculated the vector correlator (6) adopting the local current (9) for the light u and d-quarks using 160 stochastic sources (diagonal in the spin variable and dense in the color one) per gauge configuration. For each gauge ensemble the ground-state mass M ðudÞ V and the coupling TABLE I. Values of the simulated quark bare masses (in lattice units), of the pion mass M π , of the lattice size L and of the product M π L for the 16 ETMC gauge ensembles with N f ¼ 2 þ 1 þ 1 dynamical quarks used in this work (see Ref. [53]) and for the gauge ensemble, A40.40 added to improve the investigation of FVEs. The bare twisted masses μ σ and μ δ describe the strange and charm sea doublet according to Ref. [56]. The central values and errors of the pion mass are evaluated using the bootstrap events of the eight branches of the analysis of Ref. [53]. The valence quarks in the pion are regularized with opposite values of the Wilson r-parameter in order to guarantee that discretization effects on the pion mass are of order Oða 2 μ ud Λ QCD Þ.
is illustrated in Fig. 1 by comparing the results obtained using either 40 or 160 stochastic sources per gauge configuration in the case of the ETMC ensembles A80.24, B55.32, and D30. 48. We observe that the increase of the number of stochastic sources is beneficial, but the quality of the plateaux at large time distances is nevertheless still limited.

C. Lattice data and FVEs
We have evaluated Eq. (8) adopting four choices of T data , namely: T data ¼ ðt min þ 2aÞ, ðt min þ t max Þ=2, ðt max − 2aÞ, and ðT=2 − 4aÞ, and using the values of the ground-state mass M ðudÞ V and (squared) matrix elements Z ðudÞ V , determined, as described in the previous subsection, from a single exponential fit of the vector correlator V ðudÞ ðtÞ in the range t min ≤ t ≤ t max , with the values of t min and t max given in Table II.
The results obtained in the case of the ETMC gauge ensembles A40.24, B25.32, and D15.48 are collected in Table III for illustrative purposes. The two terms in the r.h.s. of Eq. (8) depend on the specific value of T data , as expected, but their total sum is almost independent of the specific choice of T data . In order to minimize the impact of the contribution depending on the identification of the groundstate signal and to optimize at the same time the statistical uncertainties the value T data ¼ ðt max − 2aÞ has been adopted in what follows.
The results for a HVP μ ðudÞ for all the ETMC ensembles of Table I versus the simulated pion mass M π are collected in the left panel of Fig. 2, while the right panel contains only our findings in the case of the four ensembles A40.XX with XX ¼ 20, 24, 32, and 40, which share the same quark mass and lattice spacing and differ only in the lattice size L.
The lattice data for a HVP μ ðudÞ exhibit a strong dependence on the pion mass and a remarkable sensitivity to FVEs  at variance with the results obtained in the case of the strange and charm quark contributions to a HVP μ (see Ref. [30]). In particular, the data shown in the right panel of Fig. 2 indicates that at a simulated pion mass M π ≃ 320 MeV the FVEs are at the level of ≃25% for M π L ≃ 3 and they reduce to ≃5% only at M π L ≃ 5. The precision of the lattice data do not allow to distinguish whether the FVEs are exponentially or power-law suppressed [40,41].
The large corrections observed for the ETMC ensembles A40.XX need to be understood and estimated properly. At NLO ChPT is unable to reproduce the value of a HVP μ [59] because of the important role of resonance contributions, which starts only at higher orders. The NLO chiral prediction for the FVEs is believed to be adequate close to the physical pion point [47,60], since it is dominated by pion loops. However, the NLO chiral result for the FVEs coincide with the estimate corresponding to noninteracting two-pion states in a finite box [21,46]. When applied at a pion mass of ≃300 MeV, we find that the NLO chiral prediction for FVEs is off by one order of magnitude with respect to what is observed in the right panel of Fig. 2. The ρ-meson resonant contribution to the interaction between two pions may therefore play an important role not only for a HVP  Table I versus the simulated pion mass M π . Right panel: lattice data in the case of the four ensembles A40.XX with XX ¼ 20, 24, 32 and 40, corresponding to a pion mass M π ≃ 320 MeV and a lattice spacing a ≃ 0.089 fm. The (red) solid and (black) dashed lines correspond, respectively, to an exponential, Að1 − Be −M π L Þ, and a power-law, A 0 ð1 − B 0 =ðM π LÞ 3 Þ, phenomenological fit. states and is given in terms of few quantities exhibiting small FVEs. In this way we may achieve a good, direct control of FVEs in a HVP μ ðudÞ. The analytic representation is described in the next section and the subtraction of FVEs is carried out in Sec. IV.

III. ANALYTIC REPRESENTATION OF THE LIGHT-QUARK VECTOR CORRELATOR
In this section we develop an analytic representation of the temporal dependence of the vector correlator V ðudÞ ðtÞ, based on the quark-hadron duality [39] and on the two-pion contributions in a finite box [40][41][42][43][44][45][46].
Let us start with the two-pion contribution, which in infinite volume is a continuous function above the twoparticle threshold. In a finite box of volume L 3 the two-pion states have been analyzed in detail in Refs. [40][41][42][43]. The energy levels ω n of the two-pion states are given by where the discretized values k n should satisfy the Lüscher condition, which for the case at hand (two pions in a P-wave with total isospin 1) reads as where δ 11 is the (infinite volume) scattering phase shift and ϕðzÞ is a known kinematical function defined as The two-pion contribution to the vector correlator, V ππ ðtÞ, can be written as [44][45][46] V ππ ðtÞ ¼ where ν n is the number of vectors ⃗ z ∈ Z 3 with norm j⃗ zj 2 ¼ n and the squared amplitudes jA n j 2 are related to the square of the timelike pion form factor jF π ðωÞj 2 by For our purposes all we need is a parametrization of the timelike pion form factor F π ðωÞ ¼ jF π ðω n Þje iδ 11 , where its phase coincides with the scattering phase shift according to the Watson theorem. The most popular parametrization is the Gounaris-Sakurai (GS) one [61], which is based on the dominance of the ρ resonance in the amplitude of the pion-pion P-wave elastic scattering (with total isospin 1), namely where the (twice-subtracted [61]) pion-pion amplitude A ππ ðωÞ is given by and k ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . By analytic continuation the GS form factor at ω ¼ 0 is normalized to unity, i.e., F ðGSÞ π ðω ¼ 0Þ ¼ 1. The scattering phase shift δ 11 ðkÞ, i.e., the phase of the form factor, is given by The GS parametrization contains two parameters: the resonance mass, M ρ , and its strong coupling with two pions, g ρππ . At the physical pion point the GS parameterization of the pion form factor provides a reasonable description of the experimental data on the process e þ e − → π þ π − , as shown in Fig. 3.
In what follows we adopt the GS parametrization and treat both M ρ and g ρππ as free parameters to be determined by fitting the vector correlator V ðudÞ ðtÞ. Note that the GS form factor does not contain any effect of the ρ − ω mixing. This is appropriate for our isosymmetric (m u ¼ m d ) QCD lattice setup.
We expect that the low-lying states close to the resonance mass can be properly described by the isovector two-pion contribution (14). This means that we may be able to reproduce the vector correlator V ðudÞ ðtÞ at large time distances. However, we want to achieve an analytic representation of the vector correlator valid also at low and intermediate time distances. To this end we resort to an observation made in Ref. [30], concerning the onset of quark-hadron duality [39]. The matching between perturbative QCD (pQCD) and the vector correlator is expected to occur at enough small values of t, i.e., t ≪ 1=Λ QCD ≈ 1 fm (with Λ QCD ≈ 300 MeV), which correspond to energy scales ≫ Λ QCD . As shown in Ref. [30], the matching with pQCD occurs instead up to time distances of ≈1 fm. Such an agreement holds in the case of the light u-and d-quarks, which can be treated in the massless limit, as well as in the case of the strange and charm quarks, once the corrections due to the nonvanishing quark masses are included. The fact that the matching appears to work up to t ≈ 1 fm is a nice manifestation of the quark-hadron dualityà la SVZ, which states that the sum of the contributions of the excited states is dual to the pQCD behavior [39]. The onset of quark-hadron duality in the vector correlator V ðudÞ ðtÞ, evaluated using our lattice data, is illustrated in Fig. 4.
Thus, inspired by the approach of QCD sum rules we introduce a dual correlator, V dual ðtÞ, defined as where x ≡ ffiffiffiffiffiffiffiffi ffi s dual p t and s dual is an effective threshold above which the hadronic spectral density is considered to be dual to the pQCD prediction R pQCD ðsÞ related to the (one photon) e þ e − annihilation cross section into hadrons.
According to Ref. [39] the value of ffiffiffiffiffiffiffiffi ffi s dual p is expected to be above the ground-state mass by an amount of the order of Λ QCD . Therefore, we assume that  [62] from the process e þ e − → π þ π − (dots). Right panel: the experimental values of the scattering phase shift δ 11 obtained in Ref. [63] (squares) and in Ref. [64] (diamonds). The solid lines represent the results of the GS parametrization (16)-(18) with E dual being treated as a free parameter to be determined by fitting the vector correlator V ðudÞ ðtÞ. Furthermore, we introduce in the r.h.s. of Eq. (23) a multiplicative factor R dual in order to take into account perturbative corrections at order Oðα s Þ (and beyond), discretization effects and an (expected) slight dependence on the light-quark mass m ud . 1 Thus, our final expression for the dual correlator V dual ðtÞ is where both R dual and E dual are free parameters to be determined by fitting the vector correlator V ðudÞ ðtÞ, while M ρ is the same parameter appearing in the two-pion contribution (14)-(15) through the GS parametrization of the timelike pion form factor (16)- (18). To sum up, our analytic representation of the vector correlator V ðudÞ ðtÞ is given by the sum of the dual correlator V dual ðtÞ and the two-pion contribution V ππ ðtÞ, viz.
which contains four free parameters, R dual , E dual , M ρ , and g ρππ . More precisely, we can make use of four dimensionless parameters, namely R dual , E dual =M π , M ρ =M π , and g ρππ , which will be determined by fitting the vector correlator V ðudÞ ðtÞ separately for each of the 17 ETMC gauge ensembles of Table I. In this way the fitting procedure can be carried out entirely in lattice units without requiring the knowledge of the value of the lattice spacing (i.e., the four parameters R dual , E dual =M π , M ρ =M π , and g ρππ are not sensitive to the uncertainty of the scale setting). We find that the inclusion of the (lowest) four two-pion energy levels ω n in Eq. (14) turns out to be sufficient for all of the ETMC ensembles. 2 By means of the analytic representation (26) we reproduce accurately the lattice data for the vector correlator V ðudÞ ðtÞ for t ≳ 0.2 fm for all ETMC ensembles. The fitting region is extended up to larger values of t, where the statistical uncertainties of the lattice correlator V ðudÞ ðtÞ do not exceed ≃10% (i.e., t ≲ 1.7 ÷ 2.0 fm).
The quality of the fits is illustrated in Figs. 5 and 6 in the case of few ETMC gauge ensembles and it is nicely confirmed by the comparison, shown in Fig. 7, between the values of a HVP μ ðudÞ, evaluated using Eq. (8), and those corresponding to the analytic representation (26), namely The high-level accuracy obtained for the reproduction of the vector correlator V ðudÞ ðtÞ using the analytic representation (26) guarantees that the calculated values of 1 A more refined treatment of the perturbative and condensate corrections to V dual ðtÞ is left to future developments. 2 We have explicitly checked that using the (lowest) eight energy levels in Eq. (14) yield results for the four parameters R dual , E dual , M ρ , and g ρππ , which differ well below the uncertainties.
a HVP μ ðudÞ differs form the lattice data less than one standard deviation.
We point out that for all the ETMC ensembles of Table I the first noninteracting two-pion energy level, given by 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi M 2 π þ ð2π=LÞ 2 p , is always well above the position of the resonance mass M ρ . Due to the residual strong interaction between the two pions the first energy level ω n¼1 satisfying the Lüscher condition (12) turns out to be slightly below M ρ . This feature justifies the use of a single exponential fit in Eq. (8), at least for the ETMC ensembles of Table I. Such a situation changes as the simulated pion mass decreases and the single exponential fit is completely ruled out at the physical pion point (see later Sec. VI).
Before closing this section, we address the issue of possible correlations of the vector correlator V ðudÞ ðtÞ at nearby values of t. To this end we have repeated our fitting procedure with reduced numbers of data corresponding to including one out of two (or three) subsequent lattice points. The results obtained for the four parameters R dual , E dual =M π , M ρ =M π , and g ρππ differ within approximately one standard deviation, as shown in Table IV in the case of few ETMC gauge ensembles. The results for the four parameters R dual , E dual =M π , M ρ =M π , and g ρππ obtained in the case of all the ETMC ensembles will be shown later in Figs. 11 and 12.

IV. SUBTRACTION OF FVEs
We start the analysis of FVEs by considering the four ensembles A40.XX, which share the same quark mass (m ud ≃ 17 MeV) and lattice spacing (a ≃ 0.089 fm) and differ only in the lattice size L, namely XX ¼ 20, 24, 32 and 40 (see Table I).

A. Ensembles A40.XX
The values of the four parameters R dual , E dual , M ρ , and g ρππ obtained by fitting the vector correlator V ðudÞ ðtÞ are shown in Fig. 8. It can be seen that the FVEs on all the fitting parameters are definitely more limited with respect to those observed for a HVP μ ðudÞ in the right panel of Fig. 2. This fact allows for a good control of the values of the four parameters in the infinite volume limit, as shown in Fig. 8 by the solid, dashed and dotted lines, whose differences are well within the uncertainties. The solid and dashed lines correspond to the exponentially-suppressed Ansatz with α ¼ 3=2 and α ¼ 0, respectively. In Eq. (28) P stands for fR dual ; E dual ; M ρ ; g ρππ g, while P ∞ and F P are fitting parameters. The dotted lines correspond instead to the power-suppressed Ansatz Besides the four parameters R dual , E dual , M ρ , and g ρππ , also the simulated pion mass M π suffers from FVEs, which have been thoroughly investigated in Ref. [53] using the resummed ChPT approach of Refs. [65,66]. For the purposes of the present work it suffices to consider for M 2 π the exponentially-suppressed Anzatz (28) with α ¼ 3=2, as suggested by the asymptotic behavior of NLO ChPT in the p-regime.
Once the infinite volume limits R ∞ dual , E ∞ dual , M ∞ ρ , g ∞ ρππ , and M ∞ π have been determined, we need to specify the infinite-volume limit of our "dual þ ππ" representation  Values of the four parameters R dual , E dual =M π , M ρ =M π , and g ρππ obtained by fitting the vector correlator V ðudÞ ðtÞ by including all subsequent timeslices (all) or one out of two (or three) subsequent lattice points in the case of the gauge ensembles A40. 24 For the dual contribution one has straightforwardly while the two-pion contribution in the infinite-volume limit becomes [45] V ππ ðtÞ where jF ∞ π ðωÞj can be calculated from the GS parametrization (16)-(18) using M ∞ ρ , g ∞ ρππ , and M ∞ π . We can now correct the lattice data for a HVP μ ðudÞ, obtained at finite volume by means of Eq. (8), for the FVEs evaluated using our representation of the vector correlator V ðudÞ ðtÞ at infinite volume, Eqs. (31)- (32), and the one at finite volume, Eqs. (14) and (25) The results obtained in the case of the ensembles A40.XX are shown in Fig. 9. We observe that most of the FVE correction comes from the ππ contribution. The small residual FVEs can be almost totally taken into account by adding the FVEs related to the dual contribution. We point out that in order to remove properly the FVEs it is important to use in Eqs.
Indeed, if one uses instead the finite volume values (as done, e.g., in Ref. [21]), the correction (34) may be largely underestimated, as shown by the black triangles in Fig. 9.
We have explicitly checked the dependence of our FVE correction (33) on the parametrization adopted for the time-like pion form factor F π ðωÞ. To this end we keep the ρ-meson dominance and consider two simple Breit-Wigner forms in which either Γ ρππ ¼ const. (labeled hereafter as BW) or Γ ρππ ∝ k (labeled as BW 0 ) instead of the GS width (18). Correspondingly, the real part of the twopion amplitude A ππ ðωÞ has been calculated using twicesubtracted dispersion relations, as in the case of the GS parametrization. We have considered also the approximation of neglecting the real part of A ππ ðωÞ. The fitting procedure of the vector correlator V ðudÞ ðtÞ corresponding to the four ensembles A40.XX has been repeated for all the  sensitive to the specific parametrization of the pion form factor than the separate terms. Before closing this subsection, we compare our findings with the results of Ref. [67], where the elastic P-wave ππ phase shifts δ 11 have been extracted from lattice QCD simulations with N f ¼ 2 þ 1 flavors of clover fermions. There the simulated pion mass was M π ≃ 320 MeV, which is quite close to the pion mass corresponding to our A40.XX ensembles (M ∞ π ≃ 315 MeV). The phase shifts δ 11 found in Ref. [67] are compared in Fig. 10 with our A40.XX results corresponding to the infinite volume limit. The comparison is made in terms of the dimensionless variable ω=M ρ , which helps in absorbing the different values of the ρ-meson mass found in Ref. [67], M ρ ≃ 800 MeV, and with our A40.XX ensembles, M ∞ ρ ≃ 850 MeV, as well as in absorbing the statistical fluctuations of the ρ-meson mass. It should be kept in mind that discretization effects are expected to be different between the lattice setup of Ref. [67] and our A40.XX ensembles. Nevertheless, the overall agreement shown in Fig. 10 is quite reassuring.

B. ETMC ensembles
We now address the subtraction of FVEs from the HVP term a HVP μ ðudÞ corresponding to the ETMC ensembles of Table I. The fitting procedure of the vector correlator V ðudÞ ðtÞ provide us with the values of the four dimensionless parameters R dual , ðM π =E dual Þ 2 , ðM π =M ρ Þ 2 and g ρππ , which are collected in Figs. 11 and 12. We stress that dimensionless parameters are not sensitive to the uncertainty FIG. 10. Elastic P-wave ππ scattering phase shift δ 11 obtained in Ref. [67] (blue circles) and with our A40.XX ensembles (red curve) versus the dimensionless variable ω=M ρ . The lattice setup of Ref. [67] corresponds to N f ¼ 2 þ 1 clover fermions with M π ≃ 320 MeV, M ρ ≃ 800 MeV, a ≃ 0.114 fm, and L ≃ 3.65 fm. Our A40.XX setup corresponds to N f ¼ 2 þ 1 þ 1 twisted-mass fermions in the infinite volume limit with of the scale setting. The dependence of the four parameters on the light-quark mass m ud , the lattice spacing a and the lattice size L can be described in terms of combined phenomenological fits, viz. ð35Þ where M 2 ≡ 2B 0 m ud and ξ ≡ M 2 =ð4πf 0 Þ 2 with B 0 and f 0 being the SU(2) low-energy constants (LECs) at LO determined in Ref. [53]. Since the quantities M 2 π =E 2 dual and M 2 π =M 2 ρ have negligible FVEs (see the right panel of Fig. 11 and the left panel of Fig. 12), we have not included in Eqs. (36) and (37) any dependence on the lattice size L. In Eq. (38) the coefficient of the chiral log is the one predicted by ChPT at NLO [68]. Moreover, a nonanalytic term proportional to m 3=2 ud is expected from ChPT [68][69][70] in Eqs. (37)- (38). However, when we tried to include it in the fitting procedure, its coefficient was found to be well compatible with 0.
The quality of the fits based on Eqs. (35)- (38) is quite good with a χ 2 =d:o:f: always less than 1. All the quantities R dual , E dual , M ρ , g ρππ and M π are correlated with each other, since they come from fitting the ETMC vector correlators. Such correlations are properly taken into account in our bootstrap sampling procedure. The results corresponding to the continuum and infinite volume limits are shown in Figs. 11 and 12 as solid lines. In particular, at the physical pion point (M phys π ¼ M π 0 ¼ 135 MeV [53]) the value M phys ρ ≡ M ρ ðm phys ud ; 0; ∞Þ ¼ 760ð19Þ MeV is obtained, in agreement with the experimental ρ-meson mass [2], though within a large uncertainty.
Finally, for the simulated (squared) pion mass M 2 π we adopt an Ansatz consistent with Eqs. (36)-(37), but including a phenomenological term for taking into account FVEs, namely which nicely fits the lattice data and provides results consistent with those of the quark mass analysis of Ref. [53]. Thus, at each value of the light-quark mass m ud and of the lattice spacing a the fitting functions (35)-(39) allow us FIG. 12. The same as in Fig. 11, but for the two-pion parameters ðM π =M ρ Þ 2 and g ρππ . The solid lines represent respectively the fitting functions (37) and (38)  to determine the infinite volume limits R dual ðm ud ; a 2 ; ∞Þ, E dual ðm ud ; a 2 ; ∞Þ, M ρ ðm ud ; a 2 ; ∞Þ, g ρππ ðm ud ; a 2 ; ∞Þ, and M π ðm ud ; a 2 ; ∞Þ, which can be used in Eqs. The ELM procedure was introduced in Ref. [13] in order to weaken the light-quark mass dependence of a HVP μ ðudÞ, improving in this way the reliability of the chiral extrapolation. From Fig. 13 it can be seen that the ELM procedure is able to reduce the light-quark mass dependence, but it does not modify the impact of FVEs. Once the latter are removed, the resulting values of a HVP μ ðudÞj L→∞ (see the full markers in the right panel of Fig. 13) exhibit again a remarkable dependence on the light-quark mass.
The attractive feature of the ELM procedure is based on the fact that a HVP

V. EXTRAPOLATIONS TO THE PHYSICAL PION POINT AND TO THE CONTINUUM LIMIT
In this section we perform the extrapolation to the physical pion point and to the continuum limit of the lattice data corrected by the FVEs as discussed in the previous section (see the full markers in Fig. 13). An important feature of the chiral behavior of a HVP μ ðudÞ is that it diverges in the chiral limit m ud → 0 [71][72][73]. This is connected with the loss of analyticity of the (subtracted) HVP function at vanishing photon virtuality Q 2 ¼ 0 in that limit. As a consequence, ChPT predicts already at NLO the presence of a chiral log proportional to logðm ud Þ [74].
The ChPT expansion can be applied to the HVP form factor Π ðudÞ R ðQ 2 Þ appearing in the covariant decomposition of the HVP tensor related to the u-and d-quark em currents [71][72][73][74]. For the connected part of Π where μ χ is the ChPT renormalization scale and The NLO term (42) is independent of any LECs, while at NNLO two LECs appear in Eq. (43), namely L r 9 ðμ χ Þ and C r 93 ðμ χ Þ. The NLO and NNLO contributions to a HVP μ ðudÞ can be evaluated using the following expression ½a HVP μ ðudÞ NLOðNNLOÞ Thus, we have adopted three different fitting functions, which, besides discretization effects, include in different ways the effects of chiral logs, namely (i) including NLO ChPT: (ii) including NLO and NNLO ChPT: (iii) including free logs: a HVP μ ðudÞ ¼ ðÃ 0 þÃ log 0 logðm ud ÞÞ × ð1 þÃ 1 m ud þÃ log 1 m ud logðm ud ÞÞ · ½1 þD 0 a 2 þD 1 a 2 m ud ; ð49Þ where, for the sake of simplicity, a HVP μ ðudÞ stands from now on for a HVP μ ðudÞj L→∞ [see Eqs. (33)- (34)]. The results of the combined chiral extrapolation and continuum limit obtained using either Eq. (47) or Eq. (48) are shown in Figs. 14 and 15, respectively, with and without the use of the ELM procedure. Similar results are obtained in the case of the "free logs" fitting function (49).
In the case of the NNLO fitting function (48) we get the following values for the LECs L r 9 and C r 93 at the ρ-meson mass scale μ χ ¼ 0.77 GeV: which are consistent (within the uncertainties) with the findings L r 9 ð0.77GeVÞ¼0.00593ð43Þ and C r 93 ð0.77 GeVÞ¼ −0.0154ð4Þ GeV −2 obtained in Ref. [74].
From Figs. 14 and 15 it can be clearly seen that the enhancement due to chiral logs is important close to the physical point. This makes a HVP μ ðudÞ quite sensitive to small changes of the light-quark mass, which may be crucial even for a local interpolation around the physical point. This immediately rises the question of how much we can trust the sharp rise visible in Figs. 14 and 15. In order to address this issue we resort to our "dual þ ππ" analytic representation. At each value of the light-quark mass m ud we can determine the values R dual ðm ud ; 0; ∞Þ, E dual ðm ud ; 0; ∞Þ, M ρ ðm ud ; 0; ∞Þ, g ρππ ðm ud ; 0; ∞Þ and M π ðm ud ; 0; ∞Þ from the fitting functions (35)-(39) (i.e., the solid lines in Figs. 11 and 12). Then, by means of Eqs. (31)-(32) we estimate the light-quark mass dependence of a HVP μ ðudÞ. The corresponding results are shown in Fig. 16 by the blue squares and compared with those obtained using the fitting function (48) in the continuum limit (green dots). A remarkable agreement is observed not only at large values of m ud (where our analytic representation fits very nicely the ETMC vector correlators), but also at values of m ud close and even smaller than the physical point. We point out that our "dual þ ππ" analytic representation does not contain chiral logs explicitly and, therefore, the agreement with the ChPT fit shown in Fig. 16 is reassuring about the reliability of the sharp rise of a HVP where (i) ðÞ statþfitþinput incorporates the uncertainties induced by both the statistical errors and the fitting procedure itself as well as the error coming from the uncertainties of the input parameters of the eight branches of the quark mass analysis of Ref. [53]; (ii) ðÞ chir is the error due to the chiral extrapolation estimated from the spread of the results corresponding to the three fitting functions (47)-(49); (iii) ðÞ disc is the uncertainty due to both discretization effects and scale setting, estimated by comparing the results obtained with and without the ELM procedure (40); (iv) ðÞ FVE is the error due to the subtraction of FVEs, taken conservatively to be twice the uncertainty found in subsection IVA (see Table V).  (47), which includes the NLO ChPT prediction, evaluated at each value of the lattice spacing of the ETMC ensembles. The solid lines represent the same fitting function in the continuum limit. The full (orange) diamonds are the values extrapolated at the physical pion point and in the continuum limit.
FIG. 15. The same as in Fig. 14, but adopting the fitting function (48), which includes also the NNLO ChPT prediction.
Our finding (52) improves the previous ETMC estimate of Ref. [13], a HVP μ ðudÞ ¼ 567 ð11Þ × 10 −10 , thanks to a more accurate treatment of both the FVEs and the extrapolation to the physical pion point. The latter can be clearly avoided using ensembles close to the physical point. Recently ETMC has generated a gauge ensemble close to the physical pion mass with N f ¼ 2 dynamical quarks, obtaining the value a HVP μ ðudÞ ¼ 552 ð39Þ × 10 −10 [75]. The lattice size is L ≃ 4.4 fm, which corresponds to M π L ≃ 3.0. For such a setup we expect large FVEs, which will be discussed in the next Section (see later Fig. 19). For the setup chosen in Ref. [75] we estimate a correction due to FVEs of order of 10%, which would yield a final value a HVP μ ðudÞ ≃ 610 ð40Þ × 10 −10 in agreement with Eq. (52), though within a large uncertainty.
Our result (52) is compared with the most recent ones from other lattice collaborations in the left panel of Fig. 17. Within the errors our value obtained with N f ¼ 2 þ 1 þ 1 dynamical flavors of sea quarks agrees with the corresponding results from HPQCD [20] and RBC/UKQCD [23] Adding the connected contributions from strange and charm quarks, a HVP  [48][49][50] in the right panel of Fig. 17.

VI. LIGHT-QUARK VECTOR CORRELATOR AT THE PHYSICAL PION POINT AND MOMENTS OF THE POLARIZATION FUNCTION
In this section we apply our analytic representation (26) to estimate the connected light-quark vector correlator V ðudÞ ðtÞ at the physical pion point both for finite values of the lattice size L and in the infinite volume limit.
To this end at each value of the lattice size L we determine the values R dual ðm phys ud ; 0; LÞ, E dual ðm phys ud ; 0; LÞ, M ρ ðm phys ud ;0;LÞ, g ρππ ðm phys ud ; 0; LÞ and M π ðm phys ud ; 0; LÞ from the fitting functions (35)- (39), where m phys ud ¼ 3.70ð17Þ MeV as determined in Ref. [53]. We use the above values in  Eqs. (14) and (25) to obtain the connected light-quark vector correlator V ðudÞ ðtÞ at the physical pion point and at finite values of L.
The results obtained for few values of the lattice size L and in the infinite volume limit are shown in the left panel of Fig. 18. The number of elastic energy levels included in Eq. (14) depends on L and, at the physical pion point, it is larger than 4, i.e., of the number of states used in the fitting procedure of the ETMC vector correlators. The right panel of Fig. 18 illustrates this point. There, the full dots represent the position of the energy levels satisfying the Lüscher condition (12) for few values of L and, at the same time, the values of the (squared) GS pion form factor occurring in Eq. (15).
We observe that from the threshold up to ω ∼ 1 GeV the number of energy levels is 5 for L ¼ 5.  (52), and by HPQCD [20], CLS/Mainz [21], BMW [22], and RBC/ UKQCD [23]. Right panel: values of the muon HVP a HVP μ ðudscÞ obtained in the present work (53) and in Refs. [20][21][22][23]. The result obtained in Ref. [23] using the R-ratio method (which turns out to be based on lattice points by ≃30% and on dispersive e þ e − data by ≃70%) is also included as an orange dot. The results of the recent dispersive analysis of Refs. [48][49][50] are shown together with the value of a HV μ corresponding to a vanishing muon anomaly (labeled as "no New Physics").
FIG. 18. Left panel: light-quark vector correlator V ðudÞ ðtÞ, multiplied by the muon kernel fðtÞ, evaluated using our "dual þ ππ" representation (26) extrapolated at the physical pion point and in the continuum limit for three values of the lattice size L (see text). The infinite volume limit, constructed as explained in the text, is also shown by the black solid line. Right panel: the (squared) pion form factor jF π ðωÞj 2 corresponding to the GS parametrization (16)-(21) evaluated in the infinite volume limit (see text) versus the two-pion energy ω. The full dots are located at the position of the energy levels satisfying the Lüscher condition (12) for each value of the lattice size L. L ¼ 8 fm and reaches 14 for L ¼ 10 fm. Therefore, at the physical pion point the spectral decomposition of the vector correlator V ðudÞ ðtÞ is quite involved. Very large time distances should be reached for getting the dominance of the lowest energy level, because the corresponding coupling A n¼1 is quite small. Higher energy levels fall off faster, but they have larger values of the coupling A n up to the location of the ρ-meson resonance. The consequences are that: (i) the FVEs on the tail of V ðudÞ ðtÞ increase significantly as the time distance increases, and (ii) the effective mass of the light-quark vector correlator [see Eq. (10)] does not show any plateau for time distances currently accessible on the lattice.
In Fig. 19 the FVEs estimated at the physical pion mass on a HVP μ ðudÞ by means of Eq. (34) are shown versus M π L and compared with the predictions of ChPT at NLO [47,60]. The latter ones coincide with the FVEs corresponding to non-interacting two-pion states [21,46]. Our determination of FVEs, instead, takes into account the interaction in the two-pion system, and in particular the resonant scattering between two pions in P-wave with total isospin 1. Our estimate of FVEs is significantly larger than the ChPT NLO prediction. Recently, FVEs in the polarization function close to the physical pion point have been analyzed in ChPT at NNLO [60], but the corresponding numerical findings seem to be too small to explain the differences with our determination.
In Fig. 19 we have also shown the results for Δ FVE a HVP μ ðudÞ at a larger pion mass equal to M π ¼ 300 MeV. At fixed values of M π L the FVEs on a HVP μ ðudÞ appear to be only slightly dependent on the pion mass (at variance with what occurs in case of the pion mass and decay constant).
At the physical pion point FVEs of the order of the muon anomaly (i.e., ≃5%) are expected to occur for L ≃ 5.5 fm.
In order to reach a finite volume correction of the order of ≃1% or less a lattice size L larger than ≃8 fm is required.
Recently, in Ref. [76] the slope and the curvature of the leading HVP function at vanishing photon virtuality have been determined on the lattice at the physical pion point and in the continuum and infinite volume limits. These quantities are derivatives of the HVP function evaluated at Q 2 ¼ 0 and they can be easily related to time-moments of the vector correlator. The separate contributions arising from the (connected) light, strange, and charm quarks are also provided in Ref. [76]. Thus, for a comparison with the predictions of our "dual þ ππ" representation of the vector correlator V ðudÞ ðtÞ we consider the following time moments correspond respectively to the slope and the curvature determined in Ref. [76]. There, it has been shown that the time distances that need to be reached to reliably determine the slope and the curvature are above ∼2 and ∼4 fm, respectively. At the physical pion point and in the continuum and infinite volume limits the predictions of our "dual þ ππ" representation are which can be compared with the results Π Ref. [76]. The agreement is quite good in the case of the slope, while our curvature is (in absolute value) larger than the corresponding result of Ref. [76] by ≃20%. We note FIG. 19. Values of Δ FVE a HVP μ ðudÞ [see Eq. (34)], evaluated in the continuum limit according to our "dual þ ππ" representation at the physical pion point (red circles) and at a larger pion mass equal to M π ¼ 300 MeV (blue squares). The dotted line corresponds to the predictions of ChPT at NLO [47,60]. that in Ref. [76] FVEs are estimated using ChPT at NLO and, therefore, the difference with our result is likely to be ascribed to the treatment of FVEs.
In the case of the higher moments Π In the left panel of Fig. 20 we show the FVEs on the ratio of the lowest four moments Π evaluated at finite lattice size L and in the infinite volume limit. Thanks to the correlations between the numerator and the denominator the results for such ratios turn out to be very precise. The impact of FVEs is sizeable and increases significantly as the order of the moment increases. In the case of Π ðudÞ 2 the use of a lattice size L ∼ 10 fm still requires a finite volume correction equal to ≃3-4%.
In the case of higher moments Π ðudÞ n with n > 2 a reliable determination requires to reach very large time distances, i.e., t ≳ 4 fm. This represents a stringent test for the large time-distance tail of the vector correlator V ðudÞ ðtÞ evaluated with our analytic representation. The authors of Ref. [50] have kindly supplied us with the first eleven moments corresponding to the experimental cross section for the e þ e − → π þ π − channels only [77]. The definition of the moments is slightly different from Eq. (54) and follows the notation of Ref. [78], namely M ðudÞ ð−nÞ ≡ 4πα em ð−Þ n ð4M 2 π Þ nþ1 5 9 Π ðudÞ nþ1 : ð57Þ We have evaluated Eq. (57) using the ππ contribution (32) in the infinite volume and continuum limits at the physical pion point. The results are shown in Table VII and in the right panel of Fig. 20 and they are compared with the dispersive values from Refs. [50,77].
Our results agree within the errors with the dispersive ones for n ≤ 4, while they overestimate the dispersive moments at higher values of n. It should be kept in mind that the values of Ref. [77] include the contributions of u and d-quark disconnected diagrams as well as also IB effects. Thus, the differences visible in Table VII (54) evaluated at finite lattice size L and in the infinite volume limit using our "dual þ ππ" analytic representation of the light-quark vector correlator taken in the continuum limit and at the physical pion point. Right panel: values of the first eleven moments (57) evaluated at the physical point using the ππ contribution (32) in the infinite volume limit (red circles), compared with the results of the dispersive analysis of the experimental cross section for the e þ e − → π þ π − channels of Ref. [50]. Courtesy of the authors of Ref. [77].  (57) from the dispersive analysis of the experimental cross section for the e þ e − → π þ π − channels [77] and the corresponding ones evaluated at the physical point using the ππ contribution (32) in the infinite volume limit.
Recently [51,52] it has been proposed to determine a HVP μ by measuring the running of α em ðq 2 Þ for space-like values of the squared four-momentum transfer q 2 using a muon beam on a fixed electron target. The method is based on the following alternative formula for calculating a HVP μ [8]: where Δα HVP em ðq 2 Þ is the hadronic contribution to the running of α em ðq 2 Þ evaluated at The quantity Δα HVP em ðq 2 Þ can be extracted from the q 2dependence of the μe → μe cross section data after the subtraction of the leptonic and weak contributions [51,52]. For the proposed MUonE experiment exploiting the muon beam at the CERN North Area [79] the region x ∈ ½0.93; 1 in Eq. (58) withz ¼x= ffiffiffiffiffiffiffiffiffiffi ffi 1 −x p ≃ 3.5. Using the analytical representation (26) of the vector correlator V ud ðtÞ, evaluated at the physical pion point in the continuum and infinite volume limits, the lightquark (connected) contribution ½a HVP μ > ðudÞ is found to be equal to ½a HVP μ > ðudÞ ¼ ð81.2 AE 1.7Þ × 10 −10 : While the estimate of ½a HVP μ > requires also the addition of the contributions of the connected strange and charm quark terms as well as of disconnected and IB effects, our finding (63) indicates that the uncertainty of ½a HVP μ > should be of the order of ≃2 × 10 −10 . Such a value is close to the statistical uncertainty (≃0.3%) expected in the MUonE experiment for the contribution ½a HVP μ < ≡ ½a HVP μ − ½a HVP μ > after two years of data taking at the CERN North Area [79].

VII. CONCLUSIONS
We have presented a lattice calculation of the leading HVP contribution of the light u-and d-quarks to the anomalous magnetic moment of the muon, a HVP μ ðudÞ. The gauge configurations generated by ETMC with N f ¼ 2 þ 1 þ 1 dynamical quarks at three values of the lattice spacing (a ≃ 0.062; 0.082; 0.089 fm) and with pion masses in the range M π ≃ 210-450 MeV have been adopted.
Thanks to several lattices at fixed values of the lightquark mass and scale but with different sizes, we have performed a careful investigation of FVEs, which represent one of main source of uncertainty in modern lattice calculations of a HVP μ ðudÞ. In order to remove them we have developed an analytic representation of the vector correlator and applied it to describe the lattice data for time distances larger than ≃0.2 fm. The analytic representation is based on quark-hadron duality at small time distances and on the two-pion contributions in a finite box at larger time distances, assuming the GS parameterization [61] for the timelike pion form factor. Our estimate of FVEs takes into account the resonant interaction in the two-pion system at variance with the ChPT prediction at NLO [47].
After removing FVEs we have extrapolated the corrected lattice data to the physical pion point and to the continuum limit taking into account the chiral logs predicted by ChPT, obtaining a HVP μ ðudÞ ¼ 619.0 ð17.8Þ × 10 −10 ; ð64Þ which is consistent with recent lattice results available in the literature [20][21][22][23].
Using our analytic representation of the light-quark vector correlator, taken at the physical pion mass in the continuum and infinite volume limits, we have provided the slope and curvature of the polarization function, Π ðudÞ 1 ¼ 0.1642ð33Þ GeV −2 and Π ðudÞ 2 ¼ −0.383ð16Þ GeV −4 , which have been compared with the corresponding lattice results of Ref. [76]. We have also evaluated the first eleven moments of the polarization function and compared them with the results of the dispersive analysis of the π þ π − channels of Refs. [50,77]. Finally, we have estimated the light-quark contribution to the missing part of a HVP μ not covered in the MUonE experiment [51,52] [see Eq. (63)].
New simulations with N f ¼ 2 þ 1 þ 1 dynamical quarks close to the physical pion point [80], the evaluation of quark disconnected diagrams and of the IB corrections [81] are in progress by ETMC. This will be crucial for improving the determination of the HVP contribution a HVP μ ðudscÞ and for addressing the muon anomaly from first principles.