Exploratory studies for the position-space approach to hadronic light-by-light scattering in the muon $g-2$

The well-known discrepancy in the muon $g-2$ between experiment and theory demands further theory investigations in view of the upcoming new experiments. One of the leading uncertainties lies in the hadronic light-by-light scattering contribution (HLbL), that we address with our position-space approach. We focus on exploratory studies of the pion-pole contribution in a simple model and the fermion loop without gluon exchanges in the continuum and in infinite volume. These studies provide us with useful information for our planned computation of HLbL in the muon $g-2$ using full QCD.


Introduction
The anomalous magnetic moment of the muon a µ = gµ−2 2 provides a high-precision test of the Standard Model. Current experiments and Standard-Model computations show a discrepancy of about three standard deviations; see Ref. [1] and references therein. This leads to the question whether this is a hint of new physics or just a statistical or systematic fluctuation from the exact value. To address this question, the uncertainty on this value has to be reduced. Experiments at Fermilab and at J-PARC plan to improve on the uncertainty by a factor of four [2]. The theoretical prediction ought to be improved in equal measure. The theoretical uncertainty is dominated by the hadronic vacuum polarization contribution (HVP) and the hadronic light-by-light contribution (HLbL).
Lattice QCD can provide a first-principle estimate of a HLbL µ [3][4][5][6][7][8][9]. Other methods rely on models, because the HLbL is not fully related to any cross section, leading to large uncertainties. Dispersion relations allow one to use experimental data to reduce the uncertainties for the dominant contributions (π 0 , η , η ; ππ), see Colangelo et al. [10][11][12][13][14] and Pauk and Vanderhaeghen [15]. Results from lattice QCD can be used as inputs to [16] or tests of [17] these dispersive approaches. More challenging is a full calculation of a HLbL µ ; in these proceedings, we report our progress towards such a calculation.
For information on the derivation of the master formula, see our Lattice 2016 proceedings contribution [9]. Note, that by treating the QED kernel function in infinite volume, we avoid introducing 1 L 2 finite-volume effects due to the photons. In the derivation we made the Lorentz covariance manifest, which allows us to reduce the eight-dimensional integral to an integral in three dimensions, as annotated in formula (1). In a Lattice QCD computation of the fully connected diagrams, one would evaluate the four-dimensional x integral with the help of sequential propagators and only reduce the y integral to one dimension. The square brackets in formula (1) can be evaluated in one step for one value of y. This makes it affordable to sample the integrand for the remaining one-dimensional integral over y.
The kernel functionL is decomposed into tensors: with e. g.
The trace of the gamma matrices evaluates to sums of products of Kronecker deltas. The tensors T (A) in turn are decomposed into a scalar S, vector V and tensor T contribution, The latter are parametrized by the six weight functionsḡ (0,1,2) ,l (1,2,3) : wherex = x |x| andŷ = y |y| . To evaluate the QED kernelL [ρ,σ];µνλ (x, y) we compute and store all six weight functions; the remaining operations to get the QED kernel are computationally inexpensive and it is convenient to perform them during the lattice computation. Due to the Lorentz covariance, the six weight functionsḡ (0,1,2) andl (1,2,3) are functions of the three parameters x 2 , y 2 and x · y only. Therefore, it is feasible to precompute and store them. Plots of all six weight functions are shown in Fig. 2.

Numerical tests
To verify that the method and its implementation are correct, we computed the π 0 -pole contribution in the vector-meson-dominance model as well as the lepton-loop contribution to a LbL µ in QED. These results can be compared with the known values of these contributions.

The π 0 -pole contribution
The first check is the π 0 -pole contribution assuming a vector-meson-dominance transition form factor (parameters: m V , m π and overall normalization), We construct the correlation function for the π 0 -pole contribution. The result reads where, with the massive propagator in position space G m (x), With the correlation function at hand, we can apply the techniques described in Sec. 2. The result is shown in Fig. 3. In view of the exponential decay ∼ e −cmπ|y| of the correlation function, the observed contribution to a HLbL µ is remarkably long-range. This demands for large lattices at the order of 5 − 10 fm for the physical pion mass.

The lepton loop contribution in QED
The analytic result for the correlation function for a lepton loop with mass m l is: It consists of the two functions Π (1) and Π (r,1) . These functions are sums of products of Bessel functions and traces of gamma matrices. The gamma matrices evaluate to sums of products of Kronecker deltas. The integral in z has already been performed analytically and the evaluation boils down to computing the Bessel functions and evaluating the traces. The and where f ρδ (y) = π 2 m 3 l ŷ ρŷδ m l |y|K 1 (m l |y|) + δ ρδ K 0 (m l |y|) , The integrand f (|y|) of the final |y| integration is shown in Fig. 4. The behaviour for small |y| is numerically compatible with f (|y|) ∝ m µ |y| log 2 (m µ |y|). This is quite steep and means that we probe the kernel precisely also for small distances. With this correlation function, the resulting value for a LbL µ for different loop masses can be reproduced at the percent level; see Table 1.  Table 1. Results for the lepton-loop contribution in QED. For the exact numbers cf. [18,19]. The first uncertainty stems from the three-dimensional integration, the second from the extrapolation to small |y|.

Conclusions
The covariant position-space method remains a promising approach to calculate the HLbL contribution to (g − 2) µ . We did two tests of our QED kernel with the help of semi-analytic computations of the correlation function iΠ. The first test is the π 0 -pole contribution in a vector-meson dominance model for the transition form factor, and the second is a lepton loop. We reproduce the known analytic result for the lepton loop at the percent level. One important observation is that the π 0 -pole contribution is very long-range, but we hope to be able to correct for the finite-size effects on this contribution, by computing the transition form factor [16] on the same ensemble and using Eqs. (12,13). We plan to make the QED kernel publicly available.