Ratios of the hadronic contributions to the lepton $g-2$ from Lattice QCD+QED simulations

The ratios among the leading-order (LO) hadronic vacuum polarization (HVP) contributions to the anomalous magnetic moments of electron, muon and tau-lepton, $a_{\ell = e, \mu, \tau}^{ HVP, LO}$, are computed using lattice QCD+QED simulations. The results include the effects at order $O(\alpha_{em}^2)$ as well as the electromagnetic and strong isospin-breaking corrections at orders $O(\alpha_{em}^3)$ and $O(\alpha_{em}^2 (m_u - m_d))$, respectively, where $(m_u - m_d)$ is the u- and d-quark mass difference. We employ the gauge configurations generated by the Extended Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089$ fm) with pion masses in the range 210 - 450 MeV. We show that in the case of the electron-muon ratio the hadronic uncertainties in the numerator and in the denominator largely cancel out, while in the cases of the electron-tau and muon-tau ratios such a cancellation does not occur. For the electron-muon ratio we get $R_{e / \mu } \equiv (m_\mu / m_e)^2 (a_e^{HVP, LO} / a_\mu^{HVP, LO}) = 1.1478~(70)$ with an uncertainty of ~ 0.6 %. Our result, which represents an accurate Standard Model (SM) prediction, agrees very well with the estimate obtained using the results of dispersive analyses of the experimental $e^+ e^- \to$ hadrons data. Instead, it differs by ~ 2.7 standard deviations from the value expected from present electron and muon ($g - 2$) experiments after subtraction of the current estimates of the QED, electro-weak, hadronic light-by-light and higher-order HVP contributions, namely $R_{e / \mu} = 0.575~(213)$. An improvement of the precision of both the experiment and the QED contribution to the electron ($g - 2$) by a factor of $\simeq 2$ could be sufficient to reach a tension with the SM value of the ratio $R_{e / \mu }$ at a significance level of ~ 5 standard deviations.

after subtraction of the current estimates of the QED, electro-weak, hadronic light-by-light and higher-order HVP contributions, namely R e/µ = 0.575 (213). An improvement of the precision of both the experiment and the QED contribution to the electron (g − 2) by a factor of 2 could be sufficient to reach a tension with the SM value of the ratio R e/µ at a significance level of 5 standard deviations.

I. INTRODUCTION
Since many years a long standing deviation between experiment and theory persists for the anomalous magnetic moment of the muon, a µ ≡ (g µ − 2)/2. The E821 experiment [1,2] where the first error is statistical, the second one systematic and the third error in brackets is the sum in quadrature corresponding to a final accuracy of 0.54 ppm. An improvement of the uncertainty by a factor of four is in progress thanks to the experiment E989 at FermiLab [3,4] (and later to the experiment E34 at J-PARC [5]). First results from E989 are expected in 2020.
On the theoretical side the present accuracy of the Standard Model (SM) prediction is at a similar level, 0.53 ppm [2]. According to the most recent determinations of the hadronic contributions to a µ , obtained using dispersive analyses of the experimentally measured e + e − → hadrons data [6,7], the muon anomaly, i.e. the difference between a exp µ and a SM µ , is given by where the first error comes from experiment, the second one from theory and the third one is the sum in quadrature corresponding respectively to a final discrepancy of 3.3 [6] and 3.8 [7] standard deviations. Other estimates of the hadronic contributions to a µ , based always on the analysis of e + e − → hadrons data, provide similar discrepancies (see, e.g., Ref. [8]).
A new interesting deviation occurs in the case of the anomalous magnetic moment of the electron a e , which has been measured at the very high level of accuracy of 0.24 ppb [9,10] a exp e = 11 596 521 807 3 [28] · 10 −14 .
Thanks to a precise recent determination of the fine structure constant α −1 em = 137.035 999 046 (27) from Ref. [11], the SM prediction for a e corresponds to an electron anomaly equal to a exp e − a SM e = −89 (28) exp (23) th [36] · 10 −14 [7,12] , where the theory error is dominated by the uncertainty on α em and the final error corresponds to a discrepancy of 2.5 standard deviations. Note that the electron anomaly (4) is opposite in sign with respect to the muon anomaly (2).
On the contrary no direct measurement of the anomalous magnetic moment of the third charged lepton of the SM, the τ lepton, is available due to its short lifetime. Only limits have been set in an indirect way by the DELPHI Collaboration [13] to be −0.052 < a exp τ < 0.013 at the 95% confidence level. The precision is quite poor even with respect to the one-loop QED contribution α em /2π ∼ O(10 −3 ) [14]. Nevertheless, the quantity a τ is considered to be the best candidate for finding physics beyond the SM, since for a large class of theories the contribution of new physics to the lepton anomalous magnetic moments is proportional to the squared lepton mass 1 .
For the three leptons the SM prediction of their anomalous magnetic moments is given by the sum of three contributions a SM = a QED + a EW + a had ( = e, µ, τ ) , where a QED is the QED term known up to five loops [12], a EW represents the electroweak (EW) corrections known up to two loops [16][17][18] and a had is the hadronic term, which includes the hadronic vacuum polarization (HVP) and the light-by-light (LBL) contributions a had = a HV P + a LBL .
Precise determinations of a HV P come from dispersion relations and the experimentally measured e + e − → hadrons data, while a LBL can be estimated through phenomenological models. Both quantities are non-perturbative and, therefore, they should be calculated from first principles, As far as the electron and the τ -lepton are concerned, only two lattice estimates of the HVP contribution from Refs. [21,22] exist to date. 1 In this respect we notice that the absolute value of the electron anomaly (4) is larger by an order of magnitude than the value ≈ 6.5 · 10 −14 expected naively from the muon anomaly (2) and the lepton-mass scaling m 2 e /m 2 µ [15].
The aim of this work is to present a lattice determination of the ratios of the leading-order (LO) HVP contributions to the lepton anomalous magnetic moments a e , a µ and a τ , obtained using the same hadronic input determined by the lattice QCD+QED simulations of Refs. [23][24][25], where the gauge configurations generated by the Extended Twisted Mass Collaboration (ETMC) with N f = 2 + 1 + 1 dynamical quarks at three values of the lattice spacing (a 0.062, 0.082, 0.089 fm) with pion masses in the range 210 − 450 MeV [26,27] were adopted. The lattice framework and details of the simulations are summarized in Appendix A.
Our simulations include the effects at order O(α 2 em ) as well as the electromagnetic (em) and strong isospin-breaking (IB) corrections at orders O(α 3 em ) and O(α 2 em (m u − m d )), respectively, where (m u − m d ) is the u-and d-quark mass difference. The calculations are based on quarkconnected contributions to the HVP in the quenched QED (qQED) approximation, which neglects the charges of the sea quarks. The quark-disconnected terms can be estimated from results available in the literature (see Refs. [20,22,28,29] We stress that the hadronic quantities a HVP,LO for = e, µ, τ share the same hadronic input and differ only in the leptonic kinematical kernel. We show that among the various ratios of a HVP,LO for different leptons the electron-muon ratio play a special role, since in this case the hadronic uncertainties in the numerator and in the denominator are strongly correlated and largely cancel out. The same does not occur in the case of the electron-τ and muon-τ ratios, where the numerator and the denominator turn out to be almost uncorrelated.
For the electron-muon ratio we get 2 where the error includes both statistical and systematic uncertainties and corresponds to a hadronic uncertainty of 0.6%, i.e. a factor ≈ 4 better than the individual precisions of the numerator and the denominator.
Our result (7) (7) we have introduced the factor (mµ/me) 2 so that the ratio R e/µ differs from unity only due to the curvature and higher-order Mellin-Barnes moments (and their derivatives) of the HVP function at vanishing photon virtuality [32]. For the mass ratio mµ/me we adopt the CODATA value mµ/me = 206.7682831 (47) from Ref. [33].
leading to R e + e − e/µ = 1.1483 (41) e (40) µ [57], where the first and second errors are related to the electron and muon contributions separately, while the third error is their sum in quadrature, i.e. without taking into account correlations between the numerator and the denominator.
Let us now introduce the following HVP quantities a HVP,LO defined as where a HV P,HO denotes the higher-order HVP corrections due to multiple insertions of leptonic and hadronic loops. In the case of the electron and the muon, adopting for the quantities in the r.h.s. of Eq. (8) the same inputs from Ref. [7] leading to the anomalies (4) and (2) a HVP,LO µ = 720.8 (6.3) exp (2.9) th [6.9] · 10 −10 , where the theoretical uncertainties come mainly from the QED contribution for the electron and from the hadronic LBL term for the muon.
The results (9-10) imply a value for the electron-muon ratio R e/µ (which for sake of simplicity will be referred to as the "exp -QED" value) equal to which differs from our lattice result (7) by 2.7 standard deviations. Note that an improvement by a factor of 2 in the precision of both the experiment and the QED contribution for the electron might be enough to reach a significance level of 5 standard deviations from our SM value (7).
The plan of the paper is as follows.
In Section II we briefly summarize the way we calculate the LO HVP terms a HVP,LO and present also an explicit comparison among the kinematical kernels for the three leptons = e, µ, τ .
In Section III we describe our results obtained using the same hadronic input shared by all the three leptons, i.e. the vector correlator V (t), adopting the ETMC gauge ensembles described in Appendix A. We define also the electron-muon ratio R e/µ and present our calculations of the light-quark contribution in Section III A. By using the "dual + ππ" representation of the vector correlator V ud (t), developed in Ref. [24] and described in Appendix B, we correct our data for finite-volume effects and extrapolate them to the physical pion mass (see Appendix A) and to the continuum limit. The remaining contributions to R e/µ are evaluated in Section III B.
In Section IV we present our determinations of the three ratios R e/µ , R e/τ and R µ/τ , extrapolated to the physical pion mass and to the continuum and infinite volume limits. We show that our results for the three ratios agree well with those corresponding to the recent analyses of e + e − → hadrons data from Ref. [7] as well as with an estimate of R e/µ , which we derive from the BMW results of Ref. [22].
Section V collects our conclusions and perspectives.

II. THE LO HVP CONTRIBUTION TO THE LEPTON a
As well known, the LO HVP contribution a HVP,LO to the lepton anomalous magnetic moment ( = e, µ, τ ) is related to the Euclidean HVP function Π(Q 2 ) by [34][35][36] a HVP,LO = 4α 2 where Q is the Euclidean four-momentum and the leptonic kernel f (Q 2 ) is given by with m being the lepton mass and ω ≡ Q/m .
The HVP form factor Π(Q 2 ) contains the non-perturbative hadronic effects and it is defined through the HVP tensor as where is the em current operator with q f being the electric charge of the quark with flavor f in units of the electron charge e, while ... means the average of the T -product of the two em currents over gluon and quark fields. In Eq.
appears in order to guarantee that the em coupling α em is the experimental one in the Thomson limit (i.e. Q 2 << m 2 e ). In this work we adopt the time-momentum representation of Ref. [37], in which the HVP where V (t) is the vector current-current Euclidean correlator defined as and t is the Euclidean time distance. Thus, the LO HVP contribution a HV P, LO reads as where with j 0 (y) being the spherical Bessel function j 0 (y) = sin(y)/y.
The LO HVP contributions a HVP,LO , given by Eq. (18), have in common the hadronic input V (t) and differs only in the kernels K (t), which weigh different temporal regions differently according to the lepton masses involved. For purposes of illustration let us use for the hadronic input determined at the physical point in Ref. [24] (see later Section III A and Appendices A and B). In Fig. 1 the t-dependencies of the quantities  [38,39]. Thus, the vector correlator V (t) can be split into two main contributions The three kernels K (t) are given by Eq. (19). The hadronic quantity V ud (t) is the light-quark (connected) contribution to the vector current-current correlator (17), as determined at the physical point in Ref. [24] (see later Section III A and Appendices A and B). The constants N are introduced in order to guarantee the common normalization condition N ∞ 0 dtK (t)V ud (t) = 1 for all leptons, while the uncertainties of V ud (t) are not shown.
where V isoQCD (t) corresponds to the contribution of isosymmetric QCD only (i.e., m u = m d and α em = 0), while δV IB (t) includes the contributions at first order O((m d −m u )/Λ QCD ) and O(α em ).
Terms at higher orders are sub-leading and they can be safely neglected even for a permil-precision calculation of the HVP term a HVP,LO .
It should be stressed that the separation given in Eq. (20) requires a prescription (see Section II of Ref. [40] for an exhaustive discussion), which means that both V isoQCD (t) and δV IB (t) are prescription dependent. Only the complete correlator V (t) (and correspondingly the HVP term a HVP,LO ) is prescription free. In this work we follow Refs. [23][24][25] and adopt the Gasser-Rusetsky-Scimemi prescription [41], in which the renormalized quark masses and strong coupling (evaluated in the MS scheme at a renormalization scale of 2 GeV) are equal in the full QCD+QED and isosymmetric QCD theories.
Since all quark flavors contribute to the em current (15), both V isoQCD (t) and δV IB (t) can be written as δV where the first three terms in the r.h.s. correspond to the contribution of light, strange and charm quark flavor separately (quark-connected contractions), while the fourth term represents the contribution of quark-disconnected diagrams. We have not included any contribution from the bottom quark, since it is sub-leading with respect even to a permil-precision level 3 .
Correspondingly, from Eqs. (20)(21)(22) one has where all the terms in a HVP,LO (isoQCD) are of order O(α 2 em ), while those in a HVP,LO (IB) contain IB contributions at orders O(α 2 em (m d − m u )/Λ QCD ) and O(α 3 em ). We start by considering the electron-muon ratio R e/µ given by Eq. (7). Since the (connected) light-quark contribution a HVP,LO (ud) represents almost 90% of the total LO HVP term a HV P, LO , we rewrite the ratio R e/µ in the following form where In the next two subsections we address separately the determination of R ud e/µ and R e/µ .
The results obtained for the ratio R ud e/µ adopting the N f = 2 + 1 + 1 ETMC gauge ensembles of Appendix A are shown in Fig. 2 as empty markers versus the simulated pion mass M π . Following Ref. [24] in the numerical simulations we have adopted a local version of the em current (15), which in our lattice setup requires a multiplicative renormalization. The latter however cancels out exactly in the ratio R ud e/µ (as well as also in R e/µ ). Results for the (connected) light-quark contribution to the electron-muon ratio, R ud e/µ , versus the simulated pion mass M π for the N f = 2 + 1 + 1 ETMC gauge ensembles of Appendix A. Empty markers correspond to the data computed at finite lattice size L, while full markers represent the ratio R ud e/µ (L → ∞) corrected for FVEs according to Eq. (29) evaluated using the results of Ref. [24]. Errors include (in quadrature) both statistical and systematic uncertainties according to the eight branches of the analyses described in Appendix A.
Few comments are in order.
• the precision of the data ranges from 0.35% to 0.6%, i.e. a reduction by a factor of at least 4 with respect to the precision of the individual HVP terms a HVP,LO µ (ud) and a HVP,LO e (ud) achieved in Refs. [24,31]. This is clearly due to a significative correlation expected between the numerator and the denominator. Using the individual uncertainties we estimate the above correlation to be 0.98, i.e. very close to 100%; • the uncertainties of the data are mainly related to the statistical errors and to a lesser extent to the scale setting; • finite volume effects (FVEs) are clearly visible in the case of the four gauge ensembles A40.XX (see Appendix A), which share the same pion mass and lattice spacing and differ only in the lattice size L; • the pion mass dependence is significative and the extrapolation to the physical pion mass requires a careful treatment, while discretization effects appear to be subleading.
In order to remove FVEs from the data we follow the approach of Ref. [24], where an analytic representation of the temporal dependence of V ud (t) was developed adopting the quark-hadron where the two separate ratios a HVP,LO strongly correlated so that the calculated correction due to FVEs on the ratio R ud e/µ (L) does not exceed 1.3% with an uncertainty not larger than 0.3%. The data for R ud e/µ (L → ∞) are shown in Fig. 2 as full markers.
The final steps are the extrapolations to the physical pion mass and to the continuum limit. In Ref. [24] it was shown that for a proper chiral extrapolation of a HVP,LO µ (ud; L → ∞) the effects of the chiral logs predicted at NLO and NNLO should be taken into account. Therefore we adopt the following Ansatz inspired by ChPT: where A 0 , A 1 and D are free parameters, and the first term in the square brackets corresponds to the ratio of the ChPT predictions at NNLO for the connected part of the light-quark contribution to a HVP,LO in the infinite volume limit [55][56][57][58]. The latter ones contain two low-energy constants (LECs), L r 9 and C r 93 , whose values are taken from the analysis of Ref. [24], namely  At the physical pion mass and in the continuum limit we get R ud e/µ = 1.1543 (54), where the error includes only the uncertainty induced by the statistical Monte Carlo errors of the simulations and its propagation in the fitting procedure. The above result shows that the chiral logs contained in the fitting function (30) yield a significative enhancement of the ratio R ud e/µ toward the chiral limit. However, such an enhancement occurs in a region of pion masses not covered directly by the ETMC data. Therefore, we make use of the recent ETMC ensemble, labelled cB211.072.64, generated with N f = 2 + 1 + 1 dynamical quarks close to the physical pion mass (M π 140 MeV) at a lattice spacing a 0.08 fm and at a lattice size L 5 fm. The lattice setup of the ensemble cB211.072.64 is described in details in Ref. [59] and briefly summarized in Appendix A. The lattice action for cB211.072.64 differs from the one previously adopted by ETMC in Refs. [26,27] and therefore lattice artifacts are expected to be different. The latter ones have however a limited impact on the ratio R ud e/µ . Using 200 gauge configurations and 160 stochastic sources (diagonal in the spin variable and dense in the color one) per each gauge configuration, we get R ud e/µ (cB211.072.64) = 1.1414 (57) 4 . After the subtraction of FVEs estimated through the analytic representation of V ud (t) evaluated at the physical pion mass, we get R ud e/µ (cB211.072.64, L → ∞) = 1.1550 (58), which is shown in Fig. 3 as the blue cross. Our finding nicely confirms the chiral enhancement predicted by the fitting formula (30).
The values of the ratios a HV P,LO (j)/a HVP,LO (ud) for j = s, c, IB, disc are collected in Table I for = e, µ, where the individual uncertainties are calculated by summing in quadrature the errors of the quantities at numerator and denominator. We can now evaluate the ratio R e/µ by considering the four individual contributions corresponding to j = s, c, IB, disc as 98% correlated between the numerator and the denominator. The correlation is taken into account through a bootstrap procedure, which leads to the value where the error includes both statistical and systematic uncertainties. is found. Nevertheless, even if we consider the case in which the IB contribution is completely dropped in the calculation, the value of R e/µ does not change significantly.

IV. RESULTS
Collecting our findings (31) and (32) our estimate of the electron-muon ratio R e/µ is given by where the error includes both statistical and systematic uncertainties and corresponds to a hadronic uncertainty of 0.6%, i.e. a factor ≈ 4 better than the individual precisions of the numerator and the denominator. We can apply the procedure described in Section III B also to the individual results a HV P,LO (j) for j = ud, s, c, IB, disc obtained by the BMW Collaboration in Ref. [22]. Assuming for sake of simplicity a 100% correlation between the individual contributions in the numerator and in the denominator we get which is consistent within the uncertainties with our result (33) as well as with the dispersive one (34). Note that, as far as the individual term a HVP,LO µ is concerned, the result of Ref. [22] exhibits some tension with respect to both the ETMC result [24,25,30,31] and the dispersive ones [6][7][8].
Moreover, the significance of such a tension is remarkably increased by the recent BMW result of Ref. [20]. However, the ratio R e/µ is less sensitive to possible tensions between the results of Before closing this Section we provide our results also for the electron-τ and muon-τ ratios We expect that the above ratios are more sensitive to the hadronic input V (t), since the kinematical kernel K (t) for the τ -lepton differs significantly from the one of the electron(muon), as shown in Fig. 1. Indeed the precision of our lattice data for the (connected) light-quark contributions R µ/τ (ud) and R e/τ (ud) turns out to be at the level of ≈ 3%, while the individual precisions for a HVP,LO e,µ,τ (ud) are at the level of ≈ 2% (see Ref. [31]). This result indicates that the numerator and the denominator in Eq. (36) can be considered almost uncorrelated.
The dependencies of R e(µ)/τ (ud) on the simulated pion mass, on the lattice spacing and on the lattice size is similar to the one shown in Fig. 2 in the case of R e/µ . The analyses of the data for both R e(µ)/τ (ud) and R e(µ)/τ are similar to the ones described in Sections III A and III B in the case of the corresponding electron-muon ratios, respectively. The only difference is that the individual contributions corresponding to j = ud, s, c, IB, disc should be considered to be uncorrelated between the numerator and the denominator. In Table II our final result for the three ratios R e/µ , R e/τ and R µ/τ are collected.
where the third error is the sum in quadrature of the first two, i.e. by considering the numerator and the denominator as uncorrelated.

V. CONCLUSIONS
In this work we have evaluated the ratios among the leading-order (LO) hadronic vacuum polarization (HVP) contributions to the anomalous magnetic moments of electron, muon and τlepton, a HVP,LO =e,µ,τ , using lattice QCD+QED simulations. Our results include the effects at order O(α 2 em ) as well as the electromagnetic and strong-isospin breaking corrections at orders O(α 3 em ) and O(α 2 em (m u − m d )), respectively, where (m u − m d ) is the u-and d-quark mass difference. We have employed the gauge configurations generated by ETMC [26,27] with an uncertainty of 0.6% for the electron-muon ratio and of 3% for the electron-τ and muon-τ ratios. Our results (39)(40)(41) agree very well with the corresponding estimates obtained using the recent results [7] of the dispersive analyses of the experimental e + e − → hadrons data (see Eqs. (34), (37) and (38)).
We stress that the reduced sensitivity of R e/µ to the hadronic uncertainties, present both in the numerator and in the denominator, makes our result (39) an accurate SM prediction.
Using the present determinations of the muon [1] and electron [9,10] (g-2) experiments (see Eqs. (1) and (3)), the updated QED calculations from Ref. [12] and the current estimates of the electro-weak, hadronic LBL and higher-order HVP contributions, the "exp -QED" value of the electron-muon ratio R e/µ (see Eq. (11) of Section I) is equal to which differs from our SM result (39) by 2.7 standard deviations. Thus, an improvement by a factor of 2 in the precision of both the experiment and the QED contribution to the electron (g − 2) could be enough to reach a tension with the SM prediction at a significance level of 5 standard deviations.

ACKNOWLEDGMENTS
We gratefully acknowledge C. Lehner for useful comments. We warmly thank F. Sanfilippo for providing us the code for calculating the light-quark contribution to the vector current-current correlator in the case of the ETMC gauge ensemble cB211.072.64 [59]. We thank also B. Kostrezwa for his help in the production of the gauge ensemble A40.40 with the tmLQCD software package [71][72][73].
The gauge ensembles used in this work are those generated by ETMC with N f = 2 + 1 + 1 dynamical quarks [26,27] and used in Ref. [42] to determine the up, down, strange and charm quark masses. We use the Iwasaki action [60] for the gluons and the Wilson Twisted Mass Action [61][62][63] for the sea quarks. In the valence sector we adopt a non-unitary setup [64] in which the strange quark is regularized as an Osterwalder-Seiler fermion [65], while the up and down quarks have the same action as the sea. Working at maximal twist such a setup guarantees an automatic O(a)-improvement [62,64].
We have performed simulations at three values of the inverse bare lattice coupling β and at several different lattice volumes as shown in Table III and 2.10, respectively. In Ref. [66] it was shown that at the current level of precision the "FLAG" hadronic scheme is equivalent to the Gasser-Rusetsky-Scimemi prescription [41].
In this work we made use of the bootstrap samples generated for the input parameters of the quark mass analysis of Ref. [42]. There, eight branches of the analysis were adopted differing in: • the continuum extrapolation adopting for the matching of the lattice scale either the Sommer parameter r 0 or the mass of a fictitious P-meson made up of two valence strange(charm)-like quarks;    • the chiral extrapolation performed with fitting functions chosen to be either a polynomial expansion or a Chiral Perturbation Theory (ChPT) Ansatz in the light-quark mass; • the choice between the methods M1 and M2, which differ by O(a 2 ) effects, used to determine the mass RC Z m = 1/Z P in the RI -MOM scheme.
The results corresponding to the eight branches are then averaged according to Eq. (28) of Ref. [42].
The statistical accuracy of the meson correlator is based on the use of the so-called "one-end" stochastic method [67], which includes spatial stochastic sources at a single time slice chosen randomly. In the case of the light-quark contribution we have used 160 stochastic sources (diagonal in the spin variable and dense in the color one) per each gauge configuration, while for the strange (charm) quark contribution 4(1) stochastic sources have been employed per each gauge configuration.
In order to check the chiral extrapolation of R ud e/µ in Section III A we have used 200 gauge configurations of the ensemble cB211.072.64 generated by ETMC with N f = 2 + 1 + 1 dynamical quarks close to the physical pion mass [59]. The gauge action is still the Iwasaki action [60], but the fermionic (twisted-mass) actions in both light and heavy sectors contain an additional Clover term with a Sheikoleslami-Wohlert [68] improvement coefficient c SW taken from 1loop tadpole boosted perturbation theory [69]. The presence of the Clover term turns out to be beneficial for reducing cutoff effects, in particular IB effects between the charged and the neutral pions. The masses of the two degenerate light quarks, of the strange and charm quarks are tuned to their physical values. The simulated pion mass turns out to be equal to M π 140 MeV and the lattice spacing is estimated to be a 0.08 fm using as input both mesonic and baryonic quantities. The lattice volume is V = 64 3 × 128 a 4 , so that the product M π L is equal to 3.6.
Appendix B: The dual + ππ representation of the vector correlator V ud (t) Following Ref. [24] the analytic representation, V dual+ππ (t), of the (connected) light-quark contribution V ud (t) is given by the sum of two terms where V ππ (t) represents the two-pion contribution in a finite box, while V dual (t) is the "dual" representation of the tower of the contributions coming from the excited states above the two-pion ones. Therefore, V ππ (t) is expected to dominate at large time distances t, while the contribution of V dual (t) is crucial at low and intermediate time distances, as firstly observed in Ref. [23].
The correlator V dual (t) is defined as where s dual is an effective thresholdà la SVZ, above which the hadronic spectral density is dual to the perturbative QCD (pQCD) prediction R pQCD (s) of the e + e − cross section into hadrons, while R dual is a multiplicative factor introduced mainly to take into account discretization effects.
According to the traditional QCD sum rule framework [45] the value of √ s dual is expected to be above the ground-state mass of the relevant channel by an amount of the order of Λ QCD . Therefore, following Ref. [24] we assume that s dual = (M ρ + E dual ) 2 with M ρ being the mass of the ρ-meson vector resonance and E dual a parameter of order Λ QCD .
Since the effective threshold s dual is well above the light-quark threshold 4m 2 ud , the pQCD density R pQCD (s) is dominated by its leading term of order O(α 0 s ) in the relevant range of the integration over s in the r.h.s. of Eq. (B2). Higher-order corrections (as well as condensates and the slight dependence on the light-quark mass m ud ) should play a sub-leading role and they can be taken into account by the effective parameter R dual in Eq. (B2).
The dual correlator V dual (t) can be explicitly written as [24] V dual (t) = 5 18π 2 R dual t 3 e −(Mρ+E dual )t 1 + (M ρ + E dual )t + where R dual , E dual and M ρ are free parameters to be determined by fitting the lattice data for the light-quark vector correlator V ud (t). Note that the ρ-meson mass M ρ will appear also in the two-pion contribution V ππ (t).
As it is well known after Refs. [46][47][48][49], the energy levels ω n of two pions in a finite box of volume L 3 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 with δ 11 being the (infinite volume) scattering phase shift and φ(z) a known kinematical function given by The two-pion contribution V ππ (t) can be written as [50][51][52] V ππ (t) = n ν n |A n | 2 e −ωnt , where ν n is the number of vectors z ∈ Z 3 with norm | z| 2 = n and the squared amplitudes |A n | 2 are related to the timelike pion form factor F π (ω) = |F π (ω n )|e iδ 11 (kn) by ν n |A n | 2 = 2k 5 n 3πω 2 n |F π (ω n )| 2 k n δ 11 (k n ) + k n L 2π φ k n L 2π −1 . (B8)