MITP-22-083 DESY-22-163 Hadronic light-by-light scattering contribution to the muon g2 from lattice QCD semi-analytical calculation of the QED kernel

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MITP-22-083 DESY-22-163

Hadronic light-by-light scattering contribution to the muon g−2

from lattice QCD: semi-analytical calculation of the QED kernel

Nils Asmussen,1En-Hung Chao,2Antoine G´erardin,3Jeremy R. Green,4

Renwick J. Hudspith,5Harvey B. Meyer,6, 2 and Andreas Nyﬀeler∗2

1Department of Physics and Astronomy,

University of Southampton, Southampton SO17 1BJ, UK

2PRISMA+Cluster of Excellence & Institut f¨ur Kernphysik,

Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany

3Aix-Marseille-Universit´e, Universit´e de Toulon, CNRS, CPT, Marseille, France

4Deutsches Elektronen-Synchrotron DESY,

Platanenallee 6, 15738 Zeuthen, Germany

5GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany

6Helmholtz-Institut Mainz, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany

Hadronic light-by-light scattering is one of the virtual processes that causes the

gyromagnetic factor gof the muon to deviate from the value of two predicted by

Dirac’s theory. This process makes one of the largest contributions to the uncertainty

of the Standard Model prediction for the muon (g−2). Lattice QCD allows for a

ﬁrst-principles approach to computing this non-perturbative eﬀect. In order to avoid

power-law ﬁnite-size artifacts generated by virtual photons in lattice simulations, we

follow a coordinate-space approach involving a weighted integral over the vertices of

the QCD four-point function of the electromagnetic current carried by the quarks.

Here we present in detail the semi-analytical calculation of the QED part of the

amplitude, employing position-space perturbation theory in continuous, inﬁnite four-

dimensional Euclidean space. We also provide some useful information about a

computer code for the numerical implementation of our approach that has been

made public at https://github.com/RJHudspith/KQED.

May 2, 2023

∗Currently no aﬃliation.

arXiv:2210.12263v2 [hep-lat] 29 Apr 2023

Contents

I. Introduction 3

II. Master formula for aHLbL

µin position space 5

III. Preparatory steps for the calculation of the QED weight functions in

position space 12

A. Starting point for the calculation of I(p, x, y)IR reg.12

B. Gegenbauer method for angular integrals in position space in four dimensions 14

C. Expansion in Gegenbauer polynomials of propagators in position space, of the

exponential function and of the function J(ˆ, y) 15

D. Average over the direction of the muon momentum 19

E. Calculation of s(x, u) 19

F. Calculation of vδ(x, u) 20

G. Calculation of tαβ (x, u) 21

IV. Direct evaluation of the ﬁnal convolution integral 22

A. Calculation of the weight function ¯

g(0) 24

B. Calculation of the weight functions ¯

g(1,2) 24

C. Calculation of the weight functions ¯

l(1,2,3) 26

V. Final convolution integral via the multipole expansion of the massless

propagator 29

A. Derivation of Eqs. (166)-(168) 30

B. Gegenbauer expansion of the QED weight functions 32

VI. Numerical evaluation of the QED kernel 35

VII. Example calculations of the four-point amplitude ib

Π41

A. General properties of ib

Π 41

B. Pion-pole contribution to hadronic light-by-light scattering in the VMD model 42

1. Vector-meson dominance parametrization of the form factor 42

2. Tests performed 45

C. Lepton-loop contribution to light-by-light scattering in QED 45

1. The coordinate-space four-point function of the electromagnetic current 45

2. Vanishing background gauge ﬁeld: the QED case 46

3. Calculation of ib

Πρ;µνλσ(x, y) 47

D. Pion-loop contribution to light-by-light scattering in scalar QED 50

1. Four-point function of the current 51

2. One-tadpole contributions 52

3. Two-tadpole contributions 53

4. Test of the Ward identity 53

5. The expression for ib

Πρ;µ1µ2µ3σ(X1, X2) 54

VIII. Applications and tests of the QED kernel 56

A. Improved kernels 56

B. Tests in the continuum 59

1. The pion-pole contribution to aHLbL

µ59

2. The lepton-loop contribution to aLbL

µ60

3. The charged-pion loop contribution to aHLbL

µ61

C. The lepton-loop on the lattice 61

D. Overview of lattice QCD results for the quark-connected contribution 65

IX. Conclusions 66

Acknowledgements 67

A. The tensors TA

αβδ(x, y) in terms of the weight functions 67

B. Derivatives of the integrands for the six weight functions with respect to

|x|71

C. Expansion of the kernel for small arguments 73

1. The regime of small |x|73

a. The scalar weight function 75

b. The vector weight functions 76

c. The tensor weight functions 77

d. The limit |x| → 0 for the tensors TA

αβδ(x, y) 79

2. The regime of small |y|79

a. The scalar weight function 80

b. The vector weight functions 80

c. The tensor weight functions 80

d. The limit |y| → 0 for the tensors TA

αβδ(x, y) 81

D. Contribution of the scalar function S(x, y) to the QED kernel: large-|y|

asymptotics 81

E. Our version of the kernel code 82

References 83

I. INTRODUCTION

The anomalous magnetic moment of the muon, aµ≡(g−2)µ/2, characterizes its response

to a magnetic ﬁeld, and is one of the most precisely known quantities in fundamental physics.

Currently, the experimental world average [1, 2] is in tension with the theoretical evaluation

based on the Standard Model (SM) of particle physics. On the basis of the Muon g−2 Theory

Initiative’s 2020 White Paper (WP) [3] with input from Refs. [4–23], the tension is at the

4.2σlevel. Theoretical and experimental uncertainties are practically equal and just under

the level of 0.4 ppm. While a tension between theory and experiment has persisted for about

twenty years, the 2021 result of the Fermilab Muon (g−2) experiment [1] has increased this

tension and thereby revived the general interest in possible explanations involving beyond-

the-Standard-Model physics, see e.g. [24].

The leading prediction for aµin QED is α

2π[25], where αis the ﬁne-structure constant.

Eﬀects of the strong interaction enter at O(α2). Due to the low mass scale of the muon,

strong-interaction eﬀects in the muon (g−2) must be treated in their full, non-perturbative

complexity. As a result, the theory uncertainty of this precision observable is entirely dom-

inated by the hadronic contributions.

The leading hadronic contribution goes under the name of hadronic vacuum polarization

(HVP). The situation around the muon (g−2) has become more intricate with the publica-

tion of a lattice-QCD based calculation [26] of the HVP contribution, which ﬁnds a larger

value than the dispersion-theory based estimate of the WP and would bring the overall the-

ory prediction into far better agreement with the experimental value of aµ. Thus it will be

crucial to resolve the tension between the diﬀerent determinations of the HVP contribution

in order to capitalize on the expected improvements in the experimental determinations of

aµ: a reduction by more than a factor of two is expected from the Fermilab Muon g−2

experiment [27], and further measurements are planned at J-PARC [28] and considered at

PSI [29].

An O(α3) hadronic contribution to aµ, known as the hadronic light-by-light (HLbL)

contribution, also adds signiﬁcantly to the error budget of the SM prediction. It can be

represented as the Feynman diagram depicted in Fig. 1. In the WP error budget for aµ, its

assigned uncertainty is 0.15 ppm. Therefore, anticipating error reductions in the HVP contri-

bution and in the experimental measurements, it is crucial to further reduce the uncertainty

on the HLbL contribution by at least a factor of two.

The HLbL contribution is conceptually more complex than the HVP contribution. On

the other hand, being suppressed by an additional power of the ﬁne-structure constant α,

the requirements on its relative precision are far less stringent: the uncertainty quoted in

the WP corresponds to 20%. In recent years, the HLbL contribution has been evaluated

using either dispersive methods, for which a full result can be found in the WP [3] based

on Refs. [15–21, 23, 30–35], or lattice QCD ([22] and [36]–[37]). Good agreement is found

among the three evaluations within the quoted uncertainties.

The purpose of the present paper is to provide a detailed account of the computational

strategy underlying our recent calculation [36, 37]. Its full development spanned several

years, with progress reported in a number of conferences since 2015 [38–43]. The basic idea

is to treat the muon and photon propagators of Fig. 1 in position-space perturbation theory,

in the continuum and in inﬁnite-volume, while the ‘hadronic blob’ is to be treated in lattice

QCD on a spatial torus. Thus much of this paper is concerned with the semi-analytical

calculation of the QED part of the amplitude.

The idea to compute the HLbL contribution to aµwas ﬁrst proposed in 2005 [44], with a

follow-up three years later [45]. These initial methods ﬁnally led to the 2014 publication [46].

In parallel to the development of our strategy, the RBC/UKQCD collaboration then also

worked on improving its computational methods [47], with a ﬁrst exploratory calculation

at physical quark masses published in [48]. These methods are based on treating the QED

parts of Fig. 1 within the lattice ﬁeld theory set up on a ﬁnite torus. Starting with Ref. [49],

the RBC/UKQCD collaboration also developed its own tools to treat the muon and photon

propagators in inﬁnite volume. We will return in section VIII A to some aspects of the

cross-fertilization that occurred between the two groups.

It is also worth pointing out other, less direct approaches that have been pursued towards

better determining the HLbL contribution to the muon (g−2) using lattice QCD. Of all me-

son exchanges, the neutral-pion pole contribution is by far the largest, and we have published

two lattice calculations of its transition form factor describing its coupling to two (in general)

virtual photons [19, 50]. Since the π0contribution is the numerically dominant one at long

FIG. 1: Hadronic light-by-light scattering diagram in the muon (g−2).

distances, having a dedicated determination thereof also helps control systematic errors at

long distances in the direct lattice calculation [51] based on the formalism presented in this

paper. As a separate line of study, we have investigated the HLbL scattering amplitude at

Euclidean kinematics [52], particularly its eight independent forward-scattering components,

which depend on three invariant kinematic variables. Knowing these amplitudes allows one

to constrain the contributions of various meson exchanges [52, 53] by parametrizing their

transition form factors, information which may subsequently be used to estimate the HLbL

contribution to the muon (g−2).

This manuscript is organized as follows. Section II presents the general features of our

position-space approach and the master-formula for aHLbL

µ. The ingredients necessary for the

evaluation of the ‘QED kernel’ describing all purely QED elements of the amplitude depicted

in Fig. 1 are collected in section III, at the end of which the averaging over the direction

of the muon momentum is performed. A relatively straightforward method of evaluating

the ﬁnal convolution integral yielding the weight functions parametrizing the QED kernel is

described in section IV. An alternative, ultimately favored method based on the multipole

expansion of the photon propagator in Gegenbauer polynomials is presented in section V.

Some technical aspects of the numerical implementation are given in section VI. Then several

models are used in section VII to compute various contributions to the four-point function

of the electromagnetic current in QED and QCD. Since these contributions to the muon

(g−2) have been computed previously (using analytical methods in momentum-space), we

use them to perform tests of our position-space QED kernel in section VIII. Published results

obtained in lattice QCD by the present methods for the quark-connected contribution are

also reviewed in that section. Section IX collects our concluding remarks. The appendices

contain additional material useful for numerical implementations, providing in particular the

kernel asymptotics for various special kinematic regimes. The ﬁnal appendix (E) provides

some information about a computer code available for the numerical implementation of our

approach based on the results of section V.

II. MASTER FORMULA FOR aHLbL

µIN POSITION SPACE

We are interested in the hadronic light-by-light (HLbL) scattering contribution to the

anomalous magnetic moment of the muon, see Fig. 1. The basic idea of our approach is to

treat the four-point function of hadronic electromagnetic currents, represented by the blob

in Fig. 1, in lattice QCD regularization, while for the remaining QED part with photons

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MITP-22-083DESY-22-163Hadroniclight-by-lightscatteringcontributiontothemuong2fromlatticeQCD:semi-analyticalcalculationoftheQEDkernelNilsAsmussen,1En-HungChao,2AntoineGerardin,3JeremyR.Green,4RenwickJ.Hudspith,5HarveyB.Meyer,6,2andAndreasNyeler21DepartmentofPhysicsandAstronomy,UniversityofSouthamp...

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MITP-22-083 DESY-22-163 Hadronic light-by-light scattering contribution to the muon g2 from lattice QCD semi-analytical calculation of the QED kernel

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