Chiral extrapolation of hadronic vacuum polarization

We study the pion-mass dependence of the two-pion channel in the hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon $a_\mu^\text{HVP}$, by using an Omn\`es representation for the pion vector form factor with the phase shift derived from the inverse-amplitude method (IAM). Our results constrain the dominant isospin-$1$ part of the isospin-symmetric light-quark contribution, and should thus allow one to better control the chiral extrapolation of $a_\mu^\text{HVP}$, required for lattice-QCD calculations performed at larger-than-physical pion masses. In particular, the comparison of the one- and two-loop IAM allows us to estimate the associated systematic uncertainties and show that these are under good control.

In this Letter, we do not address the 2.1σ tension with the datadriven approach, 1 see Refs. [56][57][58][59][60], but instead focus on the potential source of systematic uncertainty in lattice calculations that may arise if the simulation is performed at unphysical values of the quark masses.
This effect is most relevant for the isospin-symmetric ud correlator, both because its contribution is by far the largest, and because it is the lightest quarks that make simulations at the physical point expensive. Often, the required quark-mass extrapolation can be controlled using chiral perturbation theory (ChPT), at least for sufficiently small masses, but the analysis of Ref. [61] showed that for the HVP contribution this does not seem to be the case. On the one hand, the presence of a mass scale lighter than M π , namely the muon mass, makes the pure chiral expansion of practical use only for M π m µ [61]. Physically, it is well known that the 2π contribution to HVP is dominated by the ρ(770) meson, see, e.g., Ref. [62] for the implication for lattice calculations, and that controlling the quarkmass dependence of its parameters requires information beyond ChPT. On the other hand, one would not expect the quark-mass dependence of the ρ(770) mass, for example, to be so complicated that it could not be described by a simple parameterization. If this is the case, it is not clear why a simple parameterization of the quark-mass dependence of the 2π contribution to HVP should not be possible, and even allow for a controlled chiral extrapolation of good precision (in fact, finite-volume corrections have been addressed using ChPT methods [63]). Given the high computational cost of simulations at the physical quark masses this is a question of current high interest, which can be addressed from a ChPT/phenomenological point of view and deserves the detailed investigation we aim to provide in this Letter.
Our approach here is to follow Ref. [64] and combine an Omnès description [65] of the pion vector form factor (VFF) with the inverse-amplitude method (IAM) [66][67][68][69][70][71][72][73], to capture the quark-mass dependence of the dominant two-pion intermediate states. To this end, we employ the one-and two-loop IAM to describe the pion-mass dependence of the ππ P-wave phase shift [74], leading to a representation that guides the chiral extrapolation of the I = 1 component of the isospin-symmetric ud contribution to a HVP µ . We stress that our goal is not to show that the IAM is able to predict to high precision the quark-mass dependence of the ππ P-wave phase shift, but rather whether it is able to describe it, and we trust that the analysis in Ref. [74] provides a positive answer to this question. I = 0 and isospin-breaking terms are much smaller in size, in such a way that the systematic uncertainty in their extrapolation becomes less critical. Further, effects from inelastic states (mainly 4π) are sufficiently small that standard polynomial extrapolations should be sufficient. For the dominant 2π contribution, which we can capture with the IAM, the comparison of one-and two-loop extrapolations provides a measure of the systematic uncertainty, and thus allows the complete quantification of uncertainties that arise when ensembles at heavier-thanphysical pion masses are included in the analysis.
We establish the formalism in Sec. 2, reviewing the relevant aspects of HVP, Omnès methods for the pion VFF, and the IAM. For the applications described in Sec. 3, we will first impose the physical point from data and show the resulting quarkmass dependence of the HVP integral, before turning to strategies how the corresponding constraints could be implemented in lattice analyses in Secs. 4 and 5. We conclude in Sec. 6.

Formalism
In the data-driven approach the HVP contribution is calculated as [75,76] a HVP whereK(s) is a known kernel function and the hadronic cross section is photon inclusive. In contrast, lattice QCD does not proceed via the R-ratio, but instead employs a representation [77][78][79] a HVP whereK(t) is another analytically known kernel function and G(t) is determined by the correlator of two electromagnetic currents j em where a is the lattice spacing and the limit a → 0 implied in the end. This shows that in this approach the contributions of particular channels in R had (s) cannot be resolved, while instead the calculation is organized in a flavor decomposition, separated into an isospin-symmetric ud contribution, other quarks flavorby-flavor, and isospin-breaking corrections (both electromagnetic and from the quark-mass difference m u − m d ). For that reason, information on the quark-mass dependence of a particular hadronic channel, in general, does not translate to an extrapolation prescription for lattice calculations. However, the I = 1 component of the isospin-symmetric ud correlator does correspond predominantly to two-pion intermediate states, with effects from other possible states, such as 4π, appreciably suppressed. Since, in addition, the lightest states are expected to be most affected by non-trivial features of the chiral extrapolation, we will assume that such subleading effects can be adequately described by a polynomial, with the chiral behavior of a HVP µ [ππ] thus a proxy for that of a HVP µ [ud, I = 1]. The two-pion contribution to R had (s) can be expressed in terms of the pion VFF F V π (s) with σ(e + e − → π + π − ) = πα 2 3s and σ π (s) = 1 − 4M 2 π /s. F V π (s) is then strongly constrained by ππ scattering, as reflected by the fact that up to a polynomial the combination of analyticity and unitarity equates the elastic contributions to the VFF with the Omnès factor [65] where δ 1 1 (s) is the P-wave ππ scattering phase shift. This connection between the VFF and ππ scattering has been employed in numerous works in the literature, see, e.g., Refs. [80][81][82][83][84][85][86][87][88], and also forms the basis for the present analysis. In general, inelastic and isospin-breaking corrections need to be considered for a phenomenologically viable description [8,81,82] including factors G ω (s) and G in (s) that account for 3π and 4π intermediate states, respectively, with the former dominated by ρ-ω mixing and the latter by the ωπ channel, which justifies the expansion in a conformal polynomial. For the quark-mass extrapolation of a HVP µ [ud] this representation can be simplified in several ways. First, since isospinbreaking effects are booked elsewhere in lattice calculations, we can set G ω (s) = 1 and ignore final-state-radiation corrections to the cross section. The effects of inelastic states on the 2π-channel below 1 GeV are small and well described by a conformal polynomial of low degree [8]: we will truncate the integral in Eq. (6) at Λ = 1 GeV (with the threshold at s thr = 4M 2 π ). In order to simplify the analysis of its quark-mass dependence we will first replace G in (s) by a polynomial and consider its coefficients as parameters in the lattice analysis, meant to subsume inelastic effects. For a linear polynomial, G in (s) = 1 + βs, the free parameter is related to the pion charge radius via whereΩ 1 1 (0) denotes the derivative of the Omnès factor at s = 0. At the physical point all parameters are then determined via Eqs. (11) and (13), using input for δ 1 1 (s) and r 2 π from Ref. [8] (derived from a fit to the data sets of Refs. [89][90][91][92][93][94][95][96][97][98][99][100][101][102], including constraints from ππ Roy equations [103][104][105][106] and the Eidelman-Łukaszuk bound [107,108]). The final representation for the VFF in the isospin limit then reads where, by including information on the charge radius, we have incorporated the dominant inelastic effects. We will show below that the switch from a polynomial in s to one in a conformal variable does not change the results of our analysis. The quark-mass dependence of the resulting a HVP µ [ππ, ≤ 1 GeV] is taken from the IAM, using the analytic expressions from Ref. [74]. The phase shift δ 1 1 (s) is expressed in terms of the pion decay constant F in the chiral limit, the pion mass M π (including quark-mass renormalization), and a set of low-energy constants (LECs): at next-to-leading order (NLO) l r 2 − 2l r 1 , at next-to-next-to-leading order (NNLO) l r 1,2,3 , r a,b,c , and, in both cases, potentially l r 4 (plus r r F at NNLO) to convert F to the physical-point F π . Here, we will illustrate the resulting quarkmass dependence using the lattice results for ππ scattering from Ref. [109], but these LECs could also become free parameters of the lattice analysis. As final ingredient we need the quarkmass dependence of r 2 π , which is also known at two-loop order [110] At NLO the only new LEC, l r 6 , is determined from the physicalpoint r 2 π = 0.429(4) fm 2 [8], 2 while the quark-mass dependence of the Omnès function and its derivative is given by the IAM. At NNLO a new LEC, r r V1 , enters, as discussed in more detail below.

Chiral extrapolation of I = 1 contribution
As phenomenological reference point we start from [8] which gives the two-pion contribution to Eq. (6) up to a cutoff Λ = 1 GeV and includes final-state radiation in the point-like approximation. Numerically, this dominant, infrared-enhanced contribution increases the HVP integral by 4.2 × 10 −10 [113]. In addition, we need to remove the impact of ρ-ω mixing as the second important isospin-breaking effect, which can be done by setting the corresponding mixing parameter ω in G ω (s) to zero, amounting to a shift of 4.3 × 10 −10 . In total, we then arrive at for the two-pion contribution to a HVP µ [ud, I = 1]. As a first step, we may compare to the result if δ 1 1 is solely determined via the IAM fits to the lattice data of Ref. [109] (and the physical pion decay constant F π = 92.28(10) MeV [112]), which gives where the first error derives from the fit parameters, the second one gives the truncation error in the chiral expansion, estimated for an observable X as [74,114] and the third one propagates the uncertainty in r 2 π . The level of agreement between Eqs. (17) and (18) reflects the extent to which the extrapolations of the lattice fits to the physical point via the one-and two-loop IAM reproduce the physical phase shift, see Fig. 1 in Ref. [74].
Next, we consider a variant of the IAM fits that includes the physical δ 1 1 from Ref.
especially in the two-loop fit. In this case, the central value moves close to Eq. (17), mainly, because the functional form has the necessary freedom to reconcile the expected asymptotic behavior of the phase shift δ 1 1 s→∞ → π with the resonant line shape of the ρ(770). The remaining uncertainty originates from the input for the pion charge radius, which in turn is dominated by inelastic effects. 3 Accordingly, the IAM representation reproduces the full result up to the level at which uncertainties from inelastic channels begin to matter, but provides a reliable implementation of the elastic ππ effects. This points the way towards the application in the chiral extrapolation of lattice HVP results: the pion-mass dependence of the ππ physics can be controlled with the IAM, and only the estimate of the pion-mass dependence of inelastic effects needs to rely on a parameterization that is not controlled by effective field theory. For larger-than-physical pion masses the ρ(770) resonance also moves higher in energy, so that a fixed cutoff at Λ = 1 GeV may not be equally well motivated for all pion masses. With the IAM representation successfully benchmarked against phenomenology, we will thus take Λ → ∞ in the following, which changes Eq.
as starting point for a study of the pion-mass dependence.  [1,10,11], but we stress that this extrapolation is not constrained by e + e − → 2π data, and, in the case of the IAM representation, simply serves as a convenient reference point compared to which we will study the quark-mass dependence. In practice, this subsumes some inelastic effects, but their contribution will need to be included as an additional term in any case, see Sec. 5.
To obtain the pion-mass dependence ofā HVP µ [ππ], we need to determine the free parameters in Eq. (15). At NLO, only l r 6 enters, which is then eliminated by imposing r 2 π at the physical point. At NNLO, however, a new LEC (r r V1 ) arises, which describes the quark-mass dependence of r 2 π and is therefore essentially inaccessible to phenomenology. 4 Instead, we turn to the estimate of r r V1 via resonance saturation [110,116,117] 4 Separating r r V1 and l r 6 from the momentum dependence of F V π (s) is possible, in principle, indirectly via loop effects, but comes at the cost of introducing difficult-to-control systematic errors especially in the determination of l r 6 . with parameters that can be determined from ρ → e + e − , ρ → ππ, and K * → Kπ where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc). Numerically, we find and thus r r V1 = 2.0 × 10 −5 , where the difference to Ref. [110] originates from using the kaon decay constant F K instead of F π in Γ[K * → Kπ] to minimize SU(3) breaking effects. Alternatively, and in the future hopefully more reliably, one could constrain the pion-mass dependence of r 2 π directly from lattice QCD [118,119], e.g., at M π = 340.9 MeV ChPT predicts where the first error is propagated from the physical-point r 2 π , the second one estimates the truncation error by the prescription in Eq. (19), and the third one arises when assigning a 100% uncertainty to Eq. (25). In comparison, Ref. [118] quotes in good agreement with Eq. (26). We will thus continue to use Eq. (25) in the following (including a 100% uncertainty). The resulting pion-mass dependence ofā HVP µ [ππ] is shown in Fig. 1. The NLO band is dominated by the truncation of the chiral expansion, while at NNLO this error stays below 5 × 10 −10 up to M π = 0.25 GeV, with the main effect thus from the uncertainty in the physical r 2 π (and, for higher pion masses, increasingly r r V1 ). In particular, we observe good consistency between the NLO and NNLO trajectories.
We also considered a variant in which the polynomial in Eq. (14) is replaced in favor of a conformal polynomial, with conformal variable furthermore s c = −1 GeV 2 , c 1 = −2c 2 to remove the Swave singularities, and the remaining parameter again determined via r 2 π . The pion-mass dependence of M ω is taken from Ref. [120]. The resulting bands, shown in Fig. 2 on the same scale as in Fig. 1, are well consistent with the parameterization in terms of a linear polynomial, especially the NNLO result is very stable under the change of parameterization. In both cases, one could also include a second free parameter in the (conformal) polynomial, to be identified with the curvature s=0 , as the only new LEC, r V2 , would be determined by c π at the physical point.

Space-like formulation
The results shown so far do not indicate any conceptual issues with the pion-mass dependence at least of the two-pion contribution, in contrast to the negative conclusions reached in Ref. [61]. In order to better understand the relation between the two approaches, we will now consider the space-like HVP master formula, as it was used in Ref. [61] (after applying a cutoff Q max ). Starting from [78,121] a HVP µ = −4α 2  with the subtracted vacuum-polarization functionΠ(s) = Π(s)− Π(0) and weight function w(Q 2 ), the ππ contribution can be evaluated by inserting a dispersion relation forΠ(s) and retaining the imaginary part produced by ππ intermediate states.
In the end, this leads to a modification of the time-like master formula (6) by a weight function reminiscent of the definition of Euclidean time windows [40]. The result for Q 2 max = 0.1 GeV 2 shown in Fig. 3 looks very similar to Fig. 1, only with a lower overall scale given that part of the integral has been removed by the space-like cutoff. In Ref. [61] this cutoff is required because ChPT is used directly for the vector correlator [122,123], which does include effects beyond the ππ channel, but restricts the domain of validity in the momentum integral. The approach we are suggesting here does not rely on any cutoffs and amounts to instead concentrating on the ππ channel (including some inelastic effects via the pion charge radius), since the pion-mass dependence can then be controlled via the IAM, with only the remainder to be described by simpler parameterizations.
In the space-like region, the integrandΠ(−Q 2 ) becomes sufficiently smooth that simple descriptions in terms of a few parameters become possible, e.g., we have checked that the ππ contribution toΠ(−Q 2 ) can be represented by the ansatz with an accuracy below 10 −3 for M π ∈ [0.14, 0.25] GeV and Q 2 ∈ [0, 10] GeV 2 . We can thus express the IAM prediction for the pion-mass dependence ofΠ(−Q 2 ) in terms of the fit coeffi-cients in Eq. (32), see Fig. 4. We tried several ansätze cf. also Ref. [61], for the pion-mass dependence of {a, b, c, d} at NLO and NNLO, with the result that f 1 and in most cases f 2 are (strongly) disfavored, indicating that an M −2 π term is necessary. The NLO result for a displays some preference for an additional log M 2 π term, but for all other coefficients, and also for a at NNLO, f 3 and f 4 describe the pion-mass dependence from the IAM equally well. Extending these empirical fits to include pion masses below the physical point increases the sensitivity to the infrared singularities, with results that suggest the presence of a log M 2 π in addition to the M −2 π term. In principle, a similar strategy could also be pursued for the integrand G(t) in the time-momentum representation (7), but due to its more complicated behavior an accurate description in terms of few parameters in analogy to Eq. (32) is difficult to find. In contrast, the pion-mass dependence ofā HVP µ [ππ] is sufficiently smooth to be described by f 2-4 with an error below 1 × 10 −10 for M π ∈ [0.14, 0.25] GeV when fitting the central curves. Again, these empirical fits favor the presence of an M −2 π term in the extrapolation, and when including pion masses below the physical point we also see indications for the presence of an additional log M 2 π singularity. We emphasize that these findings are purely empirical, to describe the IAM results in a finite range of M π , and we do not claim that either fit function represent an analytic approximation to the full IAM.

Possible implementation strategies
The above results suggest two main strategic approaches to the chiral extrapolation of lattice results for HVP, each of which could then be implemented following different variants. The first would explicitly rely on the description of the dominant ππ contribution in terms of the P-wave phase shift, whose quarkmass dependence has been shown to be well described by the IAM [74], and use the LECs appearing in that description directly as fit parameters. Possible variants for the implementation of this strategy include: 1. In the most constrained scenario, the free parameters of the IAM could be taken from an independent lattice calculation of the ππ P-wave and the pion decay constant, leaving only the parameter β as a free parameter. In fact, to use the maximum amount of chiral input, the charge radius r 2 π at the physical point could be identified as fit parameter (instead of β), given that this input yields the dominant uncertainty in predicting the pion-mass dependence ofā HVP towards the chiral limit at higher chiral orders. We thus consider it highly unlikely that these higher intermediate states would affect the parameterization we adopted to describe the chiral extrapolation in any perceivable way, so that a simple phenomenological description should be sufficient for any practical purposes.
We stress that the bands in Fig. 1 appear broad compared to the sub-percent target precision, but also that, crucially, at NNLO the uncertainty is dominated by external input quantities (most notably r 2 π ), with the chiral convergence well under control. This implies that in the opposite direction, extrapolating HVP values at larger-than-physical pion masses towards the physical point, the intrinsic uncertainty to be assigned to the ππ component should be small, with the dominant sources of uncertainty likely from the required LECs and the chiral extrapolation of the non-ππ contribution. In particular, the LECs that enter the extrapolation can be constrained from independent lattice calculations for δ 1 1 , F π , and r 2 π . The second strategic approach to the chiral extrapolation of the HVP contribution only indirectly relies on the description of the two-pion contribution in terms of the P-wave phase shift. The latter is used only to show that in the space-like region the polarization function due to the ππ contribution is sufficiently smooth to allow for an accurate representation in terms of just a few fit parameters, whose quark-mass dependence is again well described by simple parameterizations. One would then adopt one (or a few) of the parameterizations proposed and tested here and fit its parameters to the lattice data to perform the chiral extrapolation. Here the possible variants would depend on the amount and precision of lattice data and would boil down to choosing among the different possible parameterizations discussed here or on further refinements thereof.

Conclusions
In this Letter we studied potential strategies to control the quark-mass dependence of the HVP contribution to the anomalous magnetic moment of the muon based on effective field theory. In particular, we focused on the I = 1 component of the isospin-symmetric ud correlator, which receives its by far dominant contribution from ππ intermediate states, with 4π and other non-ππ contributions appreciably suppressed. A direct application of ChPT is not possible due to the limited range of convergence [61], ultimately, because information on the ρ meson needs to be provided. Here, we argued that this can be achieved based on the IAM, with one-and two-loop implementations allowing one to verify the chiral convergence and thus assess the corresponding systematic uncertainties.
To illustrate the formalism, we addressed the opposite problem, to predict the quark-mass dependence starting from the physical point, with the main result shown in Fig. 1, based on input from the ππ P-wave phase shift δ 1 1 and the pion charge radius r 2 π at the physical point [8], combined with a lattice calculation of δ 1 1 and the pion decay constant F π [109] to determine the ChPT parameters. We found that the IAM representation indeed allows one to reproduce the expected HVP value, and suggested strategies how it could be used to constrain the chiral extrapolation of HVP calculations in lattice QCD performed at larger-than-physical quark masses.
Moreover, we found that in the space-like region the IAM results are sufficiently smooth that the correlator can be described by simple parameterizations. We could successfully reproduce their quark-mass dependence by simple fit functions, with the result that the presence of an infrared singularity as strong as M −2 π seems to be preferred empirically. Altogether, we see no conceptual issues in controlling the quark-mass dependence of HVP in lattice calculations along these lines using effective field theory, with several opportunities to constrain the extrapolation with independent lattice input. Of course, the precision that can be reached in extrapolations to the physical point, or even interpolations around it, will strongly depend on the details of each lattice calculation, but the possibility to achieve high precision even if working away from the physical point and reaching it by an extrapolation stays open.