Short-distance constraints on the hadronic light-by-light

. The muon anomalous magnetic moment continues to attract interest due to the potential tension between experimental measurement [1, 2] and the Standard Model prediction [3]. The hadronic light-by-light contribution to the magnetic moment is one of the two diagrammatic topologies currently sat-urating the theoretical uncertainty. With the aim of improving precision on the hadronic light-by-light in a data-driven approach founded on dispersion theory [4, 5], we derive various short-distance constraints of the underlying correlation function of four electromagnetic currents. Here, we present our previous progress in the purely short-distance regime and current e ﬀ orts in the so-called Melnikov-Vainshtein limit.


Introduction
The muon anomalous magnetic moment, or, a µ = (g − 2) µ /2, is a potential probe for new physics beyond the Standard Model (SM) [3].Comparing the most recent SM prediction from the White Paper [3] with the experimental average from Brookhaven National Laboratory [1] and Fermilab National Laboratory [2]  ( The SM uncertainty is dominated by hadronic contributions, namely the hadronic vacuum polarisation (HVP) and the hadronic light-by-light (HLbL), which can be calculated either in lattice field theory or using a data-driven approach [3].Although the lattice and data-driven predictions for the HLbL agree within uncertainties, there has since the publication of the White Paper arisen a tension for the HVP [6].The updated lattice prediction for the HVP predicts a significantly reduced discrepancy for the muon magnetic moment [6].There is currently much effort in trying to resolve the tension between the two approaches to calculate the HVP.
In the following, we will be concerned with improving the precision on the HLbL contribution, whose White Paper average is [3] a HLbL µ = 92(18) × 10 −11 . ( Diagrammatically, this contribution is given by the topology in Fig. 1 where the photons have momenta q 1 , q 2 , q 3 and q 4 .The external momentum q 4 is for the (g − 2) µ soft, i.e. in the kinematical limit q 4 → 0. The remaining three photons have virtual momenta integrated over, which means that in the evaluation of the HLbL there is an intricate mixing between different energy scales.In the dispersive data-driven approach [4,5], one uses analyticity and unitarity to systematically decompose the HLbL into a sum over hadronic contributions.The appearing form-factors in this decomposition require input from e.g.experiments or the lattice [3].The unknown hadronic contributions can, however, be controlled with the help of so-called short-distance constraints (SDCs) on the underlying correlation function of four electromagnetic currents, see Refs.[7][8][9][10].These kinds of SDCs can be obtained from operator product expansion (OPE) techniques and have been used for model calculations [11][12][13][14][15][16] and a complementary approach building on interpolation techniques [17].
Defining the Euclidean virtualities Q 2 i = −q 2 i for i = 1, 2, 3, we here consider SDCs in the purely short-distance region QCD (based on Refs.[8][9][10]), and preliminary work in the Melnikov-Vainshtein limit These two expansions respectively correspond to three and two of the electromagnetic currents in the underlying correlation function being close.

Some generalities
The HLbL is defined in terms of a correlation function of four electromagnetic currents.The corresponding so-called HLbL tensor is given by Here, the currents are given by J µ (x) = q Q q γ µ q where q = (u, d, s) and Q q = diag(e q ) = diag(2/3, −1/3, −1/3) is the associated charge matrix for light quarks.Here we use the convention q 1 + q 2 + q 3 + q 4 = 0.The above tensor satisfies the Ward identities , Π µ 1 µ 2 µ 3 µ 4 (q 1 , q 2 , q 3 ) = −q 4, ν 4 ∂Π µ 1 µ 2 µ 3 ν 4 ∂q 4, µ 4 (q 1 , q 2 , q 3 ) , which means that one can access a HLbL µ from the derivative tensor on the right-hand side.The derivative can be Lorentz decomposed into a set of 54 scalar functions Π i [4,5].The contribution to the magnetic moment can be written in terms of six linear combinations Π1,4,7,17,39,54 of the Π i , namely where Π i are functions of the six Π j , and the kernels T i are known.Note that the integration in ( 7) is for the full HLbL contribution, whereas the kinematic regions for the two types of SDCs considered here correspond to restricted integration domains.Common for the two limits QCD is the need to define an onset Q min of the asymptotic limits: , respectively.Since there currently is no preferred choice of Q min we keep it variable, and note that this ambiguity induces a source of uncertainty in a HLbL µ .

An OPE for three currents
In the purely short-distance limit QCD , three of the currents in the four-point function defined in (4) are close.Naively performing an OPE of these currents works only to leading order, since the next-to-leading order contribution in the systematic expansion is ill-defined in the static limit q 4 → 0, which is the limit relevant for the magnetic moment [8].The problem arises from propagators of the momentum q 4 which clearly diverge.However, it was shown in Ref. [8] that the problem can be circumvented by treating the static photon as an external electromagnetic field and instead of the four-point function Π µ 1 µ 2 µ 3 µ 4 consider a three-point correlator in the presence of that field.For this reason we study where the static background field is captured in the external state.This is related to the full HLbL tensor through [8] where µ 4 (q 4 ) is the polarisation vector of the external photon and the Ward identity (6) was used.By performing a systematic background field OPE between the three currents in (8) then yields the HLbL derivative through (9).The background field induces non-perturbative condensates in the expansion which are different than those appearing in vacuum OPEs [19].The condensates appear from diagrammatic contributions to the correlator in (8) where some of the fields are left uncontracted in the Wick expansion.Background field OPEs were first used for nucleon magnetic moments in Refs.[20,21], but were in fact later also employed for the electroweak contribution to a µ in Ref. [22]. , Figure 2. Various contributions to the OPE.The leading order contribution is (a) the quark loop.The crossed vertex indicates the interaction of the external field on the perturbative quark line.At next-to leading order the non-perturbative condensates (b 1,2 ) appear, namely q σ µν q , induced by F µν , and qq .At next-to-next-to leading order diagram (c) is a four-quark operator with condensate q Γ 1 q q Γ 2 q , and diagram (d) contains the gluon condensate α s GG .
Here, the gluon field is given by G µν = ig S λ a G a µν as well as its dual Ḡµν = i 2 µνλρ G λρ .The matrix Γ is in flavour, spin and colour space .For more detail, we refer the reader to Ref. [9].The class of four-quark operators S {8}, µν yields flavour mixing, and in the chiral limit there are only 12 independent operators.In Fig. 2 we show the various diagrammatic contributions appearing in the OPE.The leading order term is given by the perturbative quark-loop in Fig. 2(a), which in fact coincides with the leading order term in a naive OPE of the four-point function in (4).This finding in Ref. [8] confirmed that indeed the perturbative quark loop describes the leading behaviour in the short-distance limit [3].
The obtained systematic expansion allows us to quantify for the first time from where the short-distance representation of the HLbL is valid.We study first the impact of the nonperturbative corrections.We note in passing that one also has to renormalise the condensates, which is described in detail in Ref. [9].It suffices here to write the renormalised result on the form where the perturbative short-distance Wilson coefficients are contained in the vector C T, µ 1 µ 2 µ 3 µ 4 ν 4
Numerical estimates for the condensates are obtained from Refs.[9,19,23,24].Analytical formulae for the Πi needed to numerically calculate a HLbL µ in (7) in the shortdistance region Q 2 i > Q 2 min can be found in Refs.[8,9].Performing the numerical integration yields Figs.3-5.The leading order perturbative quark loop corresponds to X 1,0 , which as can be seen is completely dominant by one to two orders of magnitude.This shows that the nonperturbative corrections through next-to-next-to leading order are negligible as compared to the perturbative quark loop.

Gluonic corrections
From the above numerical results it appears that the perturbative quark loop is a good description of the short-distance behaviour of the HLbL.However, perturbative order α s corrections to the quark loop can also be important, which is what we studied in Ref. [10].Including two vertices from the Dyson series with gluon interactions yields two-loop topologies like those in Fig. 6.Details of the calculation are given in Ref. [10] together with the Πi , and we report here only the numerical comparison to the quark loop contribution to a HLbL µ in Fig. 7.As can be seen, the shift in going from the leading order quark loop to including the two-loop corrections is small.Numerically, it corresponds to a shift around −10% compared to the quark loop.This indeed shows that the quark loop describes the short-distance dynamics of the HLbL to within 10% in the region QCD , and can be used by dispersive and model studies as in Ref. [16].
In this OPE we will expand in powers of the large momentum q = (q 1 − q 2 )/2.We further define a Dirac matrix in terms of the massless quark propagator according to Denoting the perturbative quark loop contribution to the HLbL tensor in (4) by Π µ 1 µ 2 µ 3 µ 4 quark−loop , one finds in the chiral limit through next-to-leading order the result q e 2 q(0) Γ µ 1 µ 2 (− q) − Γ µ 2 µ 1 (− q) q(0) γ * (q 3 )γ(q 4 ) We thus see that the two external photons in (16) induce new contributions as compared to (8).There are several technical issues in obtaining the Πi in the Melnikov-Vainshtein limit which will be explained in our upcoming paper.Note that we have made no ordering on the size of Q 2 3 and Λ 2 QCD here, except for an omitted gluon operator contribution to the flavour singlet piece when Λ QCD < Q 3 .However, the perturbative Q corresponds to a special case of the purely short-distance limit Q 2 i Λ 2 QCD discussed above.This means that for Q 2 3 Λ 2 QCD in the Melnikov-Vainshtein limit we should reproduce our previous results for the perturbative quark loop as well as the leading order result of Ref. [7].Evaluating the appearing matrix elements in (19) perturbatively, this is indeed the case.
In the non-perturbative region Q 2 3 / Λ 2 QCD , the matrix elements in ( 19) must be form-factor decomposed.The leading order term (first row in (19)) depends on two form-factors, i.e. the longitudinal and transversal functions discussed in Ref. [7].At next-to-leading order there are six form-factors, and the exact decomposition will be given in our future paper.
We are currently working on gluonic corrections to the OPE in (19).As argued in Ref. [17], the leading perturbative α s correction is expected to be given by the Melnikov-Vainshtein result [7] with an overall factor −α s /π.In our OPE, through leading order in α s the first line in (19) instead yields Although this might seem to be in tension, the Melnikov-Vainshtein result refers to the axial current that preserves chiral symmetry within perturbation theory, related to ours by a finite counterterm [25].We are currently working to complete the calculation, taking these subtleties into account.SDCs in the Melnikov-Vainshtein limit are highly relevant for data-driven approaches to evaluate the HLbL, see e.g.Refs.[7,12,13,16,17].Our anticipated results can be used to further improve the data-driven prediction.

Conclusions
In these proceedings we have reported on recent progress on the derivation of short-distance constraints for the data-driven evaluation of the HLbL.This hadronic contribution to the muon magnetic moment is particularly complicated since it mixes long-and short-distance dynamics.Constraints in the purely short-distance as well as the Melnikov-Vainshtein limit are important to control the systematic uncertainty in the data-driven approach to the HLbL, both of which we have considered herein.Due to the soft external magnetic field, the constraints are derived using background field operator product expansion techniques.Our main result is a precise description for the HLbL in the purely short-distance regime.In the Melnikov-Vainshtein limit, which is still work in progress, we have reproduced important results in certain limits, and are currently evaluating also novel higher-order non-perturbative as well as gluonic corrections.

Figure 7 .
Figure 7. Contributions to a HLbL µ from gluonic corrections to the perturbative quark loop.The uncertainty band is associated to α s (µ = Q min ).