A lattice calculation of the hadronic vacuum polarization contribution to $(g-2)_\mu$

We present results of calculations of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment. Specifically, we focus on controlling the infrared regime of the vacuum polarisation function. Our results are corrected for finite-size effects by combining the Gounaris-Sakurai parameterisation of the timelike pion form factor with the L\"uscher formalism. The impact of quark-disconnected diagrams and the precision of the scale determination is discussed and included in our final result in two-flavour QCD, which carries an overall uncertainty of 6%. We present preliminary results computed on ensembles with $N_f=2+1$ dynamical flavours and discuss how the long-distance contribution can be accurately constrained by a dedicated spectrum calculation in the iso-vector channel.


Introduction
The persistent deviation of 3.5 − 4 standard deviations between the direct measurement of the muon's anomalous magnetic moment a µ = 1 2 (g − 2) µ and the value predicted by theory [1] may signal a limitation in the ability of the Standard Model (SM) to provide an accurate and precise description of particle properties. Two new experiments, E989 at Fermilab and E34 at J-PARC, will increase the precision of the direct experimental determination of a µ by up to a factor of four, which calls for a similar reduction in the uncertainty of the SM prediction. Since the latter is dominated by the contributions from low-energy QCD, it is particularly timely to revisit the determinations of the hadronic vacuum polarisation and hadronic light-by-light scattering contributions, a hvp µ and a hlbl µ , respectively. Since the current estimates for these quantities either rely on experimental data or on model assumptions, it is highly desirable to compute them via a first-principles approach such as lattice QCD. However, in order to be competitive with the dispersive approach, any lattice QCD calculation of a hvp µ must be able to control all sources of error at the level of 0.5% or better, if such studies are to have an impact on testing the limits of the SM. By contrast, a determination of a hlbl µ with an overall error of about 15% will constitute a major step forward.
Speaker, e-mail: hartmut.wittig@uni-mainz.de arXiv:1710.10072v1 [hep-lat] 27 Oct 2017 Here we report on our ongoing effort in determining a hvp µ in lattice QCD with fully controlled errors. In particular, we describe our calculation in QCD with two flavours, which is focussed on methodology, and which has been published in [2]. We also discuss preliminary results obtained on gauge ensembles with 2 + 1 dynamical quark flavours.

Methodology
The leading hadronic vacuum polarisation (HVP) is accessible in lattice QCD via two types of integral representations. The first is formulated in terms of a convolution integral over Euclidean momenta [3,4] and reads where K(Q 2 ) is a known kernel function, andΠ(Q 2 ) = 4π 2 Π(Q 2 ) − Π(0) denotes the subtracted vacuum polarisation which is obtained from the correlator of the electromagnetic current Alternatively, one can express a hvp µ via the "time-momentum representation" (TMR) [5], by integrating the spatially summed correlator G(x 0 ) multiplied by a kernelK(x 0 ) over Euclidean time according to A major difficulty, which is encountered in both approaches, is associated with controlling the infrared regime of the vacuum polarisation. In the first approach, one must determine the additive renormalisation Π(0) and a sufficiently precise representation of Π(Q 2 ) for values of Q 2 that do not exceed the muon mass by much. While the use of partially twisted boundary conditions is helpful [6] in this regard, it does not remove the need for simulating large lattice volumes in order to reliably constrain Π(Q 2 ) in the low-momentum regime [7]. If one employs the TMR, one is confronted with the problem of controlling the long-distance regime of G(x 0 ), which is difficult owing to its rapidly rising noise-to-signal ratio. Therefore one has to resort to some kind of model for the large-x 0 behaviour. Furthermore, in the light quark sector, G(x 0 ) is dominated by a two-pion state in a p-wave as x 0 → ∞. Possible choices for the description of the iso-vector component of G(x 0 ) at large distances include a naive single exponential function that falls off with the mass of the ρ-resonance or an ansatz that supplements the single exponential with the contribution from the two-pion state. A third possibility is based on the Gounaris-Sakurai parameterisation [8] of the timelike pion form factor that enters the iso-vector spectral function [9,10]. In ref. [2] we presented a detailed discussion of the different strategies to control the low-energy regime.
has been published in [2]. This study was performed using two degenerate flavours of O(a) improved Wilson quarks and the Wilson plaquette action, using the parameterisation of the clover coefficient c sw from [11]. Extrapolations to the physical point have been performed using data at three different values of the lattice spacing (i.e. a = 0.049, 0.066 and 0.079 fm) and pion masses ranging from 185 − 495 MeV, always keeping m π L ≥ 4. In addition, we present preliminary results for QCD with N f = 2 + 1 dynamical flavours [12], based on the tree-level Symanzik improved gauge action with c sw tuned according to ref. [13]. The current range of lattice spacings and pion masses comprises a = 0.050, 0.065, 0.085 fm and m π = 200 − 420 MeV, which is similar to the two-flavour case. We postpone the inclusion of isospin-breaking effects to future work (see [14,15] for pilot studies).
In the following we focus on the discussion of systematic effects arising from the long-distance behaviour of the vector correlator, which is also connected with the issue of finite-volume effects. The accuracy of the scale determination is also an important factor for the overall precision of the lattice prediction of a hvp µ . For this purpose we restrict the discussion to the TMR. It is then convenient to define the quark-connected contribution of flavour f = ud, s, c to the vector correlator via where q f is the quark electric charge 1 , Z V denotes the renormalisation factor of the local vector current The corresponding contribution to the hadronic vacuum polarisation of flavour f is then given by The statistical precision of the light quark contribution (a hvp µ ) ud , which dominates the total HVP, is limited by the exponentially growing noise-to-signal ratio of the integrand in eq. (6). This is shown in Figure 1 where we have plotted the productK(x 0 ) G ud (x 0 ) in units of m µ versus the Euclidean time separation x 0 [fm]. Above a certain value, i.e. x cut 0 ≈ 1.3 fm, the signal has deteriorated to such an extent that one has to resort to a model for G ud (x 0 ). The various coloured bands compare the extension based on the single exponential and the Gounaris-Sakurai (GS) parameterisation. Overall, one finds that both types of extension yield consistent results for (a hvp µ ) ud within errors. It is also possible to determine finite-volume corrections based on the GS model. To be more explicit, it is useful to start with the isospin decomposition of the vector correlator, i.e.
where G ρρ (x 0 ) denotes the iso-vector contribution, which is proportional to the connected light-quark contribution G ud (x 0 ). In infinite volume the iso-vector part is given by where ρ(ω 2 ) is the (continuous) spectral function, and F π (ω) denotes the pion form factor in the timelike regime. In a finite volume, characterised by a box size L, one encounters a discrete spectrum, i.e. G ρρ (x 0 , L) x 0 →∞ = n |A n | 2 e −ω n x 0 , ω n = 2 m 2 π + k 2 . The energies ω n are related to the scattering momentum via the Lüscher condition [16]: In the inelastic region the amplitudes A n can be expressed in terms of the timelike pion form factor [9] |A n | 2 = 2k 2 3πω 2 n |F π (ω n )| 2 kφ (k) + kδ 1 (k) .
Given input data for F π (ω) ≡ |F π (ω)|e iδ 1 (k) , one may then determine the finite-volume shift by forming the difference G ρρ (x 0 , ∞) − G ρρ (x 0 , L) and inserting it into eq. (3). In the absence of any direct calculation of ω n and |A n |, one can resort to the GS parameterisation of F π (ω) in terms of the resonance mass m ρ and the width Γ ρ . Both parameters can be determined from lattice data via the following procedure: In a first step, the mass of the ground state is identified with the ρ-meson mass extracted from a smeared vector correlation function computed in an auxiliary lattice calculation via a two-state fit with the lowest level set to 2 m 2 π + (2π/L) 2 . Then, by inserting the GS model for |F π | and δ 1 into eq. (11) one obtains an expression for the correlator G ρρ (x 0 ) in terms of the GS parameters (m ρ , Γ ρ ). Fitting G ρρ (x 0 ) to the form in eq. (9) with m ρ fixed to the estimate extracted from the smeared correlator yields the width parameter Γ ρ . During this fit one inserts the GS parameterisation for δ 1 into the Lüscher condition and determines the scattering momenta k of excited states in the iso-vector channel, which, in turn, yield the corresponding energy levels and matrix elements ω n and |A n | 2 . In this way one can evaluate G ρρ (x 0 , L) in eq. (9) for, say, a handful of states. The procedure is completed by inserting the GS parameterisation into eq. (8), which yields the correlator G ρρ (x 0 , ∞). Since the isoscalar part in eq. (7) is sub-dominant, one may approximate it by a single exponential whose fall-off is given by m ω ≈ m ρ . We have determined the finite-volume shift in (a hvp µ ) ud according to the above procedure, for all of our two-flavour ensembles with pion masses m π ≤ 340 MeV. Unsurprisingly, the biggest correction of +3 % was encountered at our smallest pion mass of 185 MeV.
Another source of uncertainty that has received relatively little attention in most of the earlier calculations of a It is useful to replace M u , M d . . . and their derivatives by suitable meson masses in units of Λ. In the isospin limit the above expression can then be rewritten as When working with the TMR one can determine the derivative term involving the muon mass via [2] M µ ∂a where the kernel J(x 0 ) can be easily computed using the series expansion ofK(x 0 ) from appendix B in [2]. The derivative w.r.t. the pion mass M π can be estimated from the slope of the chiral extrapolation of (a hvp µ ) ud at m π = m phys π (see left panel of Figure 2). With these results at hand one finds The factor multiplying the scale setting uncertainty is thus dominated by the contribution from the muon mass, with only a 10% reduction coming from the light quarks. Heavier quark flavours are likely to have an even smaller effect. The lesson one can draw from this analysis is the fact that the proportionality between the relative uncertainties of a hvp µ and the lattice scale Λ is a number of order 1. Therefore, the lattice scale must be known to within a fraction of a percent, if one is to reach the precision goal in the determination of a hvp µ . We have subjected our data for the various flavour contributions to a hvp µ to combined chiral and continuum extrapolations, using a variety of different ansätze, such as Fit A: α 1 + α 2 m 2 π + α 3 m 2 π ln m 2 π + α 4 a, Fit B: β 1 + β 2 m 2 π + β 3 m 4 π + β 4 a, Fit C: γ 1 + γ 2 m 2 π + γ 3 a, Fit D: Since we did not include the O(a) improvement term in the vector currents, we expect that the leading cutoff effects are linear in the lattice spacing, which accounts for the term of order a in the above expressions. Fits A and B each contain a term allowing for a curvature in the chiral behaviour of a hvp µ . Other models describing the pion mass dependence contain terms that diverge in the chiral limit, such as 1/m 2 π or ln m 2 π . While the latter is only justified in the region where m π < m µ , inverse powers of m 2 π may over-amplify the pion mass dependence around the physical pion mass, as was noted in [17]. We have therefore excluded singular ansätze in our final analysis. After adding the contributions from the light, strange and charm quarks, we arrive at our final estimate for a  where the statistical and systematic errors have been estimated via the "extended frequentist's method" which combines the standard bootstrap technique with a number of procedural variations in the analysis. The remaining systematic errors refer to the scale uncertainty, the error assigned to the estimation of the finite-volume shift and the upper bound on the effect of including quark-disconnected diagrams.
The calculation of the latter on a subset of gauge ensembles is described in appendix D of ref. [2]. A crucial ingredient for obtaining accurate results is the cancellation of stochastic noise between the contributions of the light and strange quark flavours, discussed in [18]. A compilation of results for a hvp µ from different collaborations is shown in Figure 3. Overall there is good agreement among different groups, with the largest observed deviations between individual calculations amounting to about 2.5 standard deviations. It is clear, though, that the overall errors have to be further reduced in order to be competitive with the dispersive analysis.
We have started to compute a hvp µ on a set of CLS ensembles generated with N f = 2 + 1 flavours of dynamical quarks. In order to combat the problem of topology freezing, a large fraction of the ensembles have open boundary conditions [26,27], which precludes the determination of the vacuum polarisation via the four-dimensional Fourier transform of eq. (2). Since the TMR is independent of the type of boundary conditions, we exclusively focus on this approach in all our calculations for N f = 2 + 1. One important additional ingredient relative to the earlier two-flavour calculation is the use of the O(a) improved versions of the local and point-split vector currents [28].
In Figure 4 we show once more the integrand of eq. (3), computed at a pion mass of 200 MeV at a = 0.065 fm. The data confirm that a single exponential provides a good approximation of the long-distance behaviour of the integrand. We also display the results of a state-of-the-art calculation aimed at determining the energies ω n and corresponding matrix elements A n of the vector current for the n th energy eigenstate in the iso-vector channel. This not only allows for a precise direct calculation of the long-distance behaviour of G ρρ (x 0 , L) but also provides the determination of the timelike pion form factor that is necessary to compute the finite-volume shift. An account of our method has been provided in [29]. The results from the spectrum calculation demonstrates that the iso-vector correlator G ρρ (x 0 ) is saturated by the first four states in that channel, for x 0 1.7 fm. It is also interesting to note that the two-pion state starts to dominate the correlator for x 0 3 fm. The explicit spectrum calculation provides a highly accurate and precise determination of the vector correlator at long distances, especially since the error grows only linearly with x 0 , in contrast to the exponential error growth of G ρρ itself. The three panels represent calculations with different numbers of quarks in the sea. Squares denote estimates including the contributions from u, d, s, c quarks in the valence sector, while triangles represent results for u, d, s quarks only. The meaning of the labels is: Mainz/CLS 11 [6], Mainz/CLS 17 [2], Aubin+Blum 07 [19], RBC/UKQCD 11 [20], ETM 13 [21], ETM 15 [22], HPQCD 16 [23] and BMW 17 [24]. See [25] for another recent calculation. The vertical band represents the result from dispersion theory.