Preprint number KEK-TH-245 QED on the lattice and numerical perturbative computation of g2

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Preprint number: KEK-TH-245

QED on the lattice and numerical perturbative

computation of g−2

Ryuichiro Kitano1,2Hiromasa Takaura1,∗

1KEK Theory Center, Tsukuba 305-0801, Japan

2Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan

∗E-mail: hiromasa.takaura@yukawa.kyoto-u.ac.jp

17/10/2023

...............................................................................

We compute the electron gfactor to the O(α5) order on the lattice in quenched QED.

We ﬁrst study ﬁnite volume corrections in various IR regularization methods to discuss

which regularization is optimal for our purpose. We ﬁnd that in QEDLthe ﬁnite volume

correction to the eﬀective mass can have diﬀerent parametric dependences depending

on the size of Euclidean time tand match the ‘naive on-shell result’ only at very large

tregion, t≫L. We adopt ﬁnite photon mass regularization to suppress ﬁnite volume

eﬀects exponentially and also discuss our strategy for selecting simulation parameters

and the order of extrapolations to eﬃciently obtain the gfactor. We perform lattice sim-

ulation using small lattices to test feasibility of our calculation strategy. This study can

be regarded as an intermediate step toward giving the ﬁve-loop coeﬃcient independently

of the preceding studies.

..............................................................................................

Subject Index B01, B38, B50, B59

1typeset using PTPT

X.cls

arXiv:2210.05569v2 [hep-lat] 16 Oct 2023

1. Introduction

The anomalous magnetic moment of the electron, the electron g−2, is one of the most

precisely measured observables in particle physics. It thus gives us a good opportunity to

test our understanding of quantum ﬁeld theory. Theoretically the perturbative calculation of

QED contributions has reached the ﬁve-loop [O(α5)] level [1, 2], although a slight discrepancy

in the ﬁve-loop coeﬃcient is reported [3]. The ﬁve-loop contribution is indeed relevant to

the precision of the experimental measurements [4, 5]. In order to examine the consistency

of the SM prediction with the experimental result, the value of the ﬁne structure constant

is crucial, yet the value is not determined consistently among diﬀerent experiments [6].

In ref. [7], we proposed a new method to calculate the perturbative series of the electron

g−2 using numerical stochastic perturbation theory (NSPT) [8–12]. We calculated the

perturbative series to the three-loop level on the small lattices. This was the ﬁrst attempt

to apply NSPT to QED observables. Since it can provide us with a new and alternative

approach to calculating the perturbative series to high orders and can have a wide range of

application, it is worth testing the usefulness of the method further.

In order to perform meaningful numerical simulations, we need to understand ﬁnite volume

(FV) corrections, which are known to be severe in lattice QED because of massless photon.

In the ﬁrst half of this paper, we study FV corrections in various ways of IR regularization

such as subtractions of zero modes (known as QEDLor QEDT L) and ﬁnite photon mass [13–

15] to understand what kind of IR regularization is optimal, although FV corrections are

not understood enough in our previous work [7]. QEDLis a famous and well-adopted regu-

larization, and FV corrections in this regularization have been studied in many papers such

as refs. [13, 14, 16–20]. Mainly based on results in ref. [14], we add a new insight into FV

correction in QEDL. We ﬁnd that the FV correction to the eﬀective mass can have diﬀerent

parametric dependences depending on the size of Euclidean time; at 1/m ≪t≪Lthe FV

correction is given by O(t/(mL2)) while at t≫Lit is given by O(1/(mL)). The latter case

matches the ‘naive on-shell result’ but this is not always true for general t. From the discus-

sions in this part we conclude that the massive photon regularization is the most controlled

method for our computation.

In the second half of this paper, we perform lattice simulation of the electron g−2 in

quenched QED, i.e., QED without the dynamical electron. We adopt ﬁnite photon mass

regularization, which is found to be most suited for our purpose. In quenched QED, i.e.,

in sub-diagrams without lepton loops, there is a discrepancy in the ﬁve-loop perturbative

coeﬃcient between refs. [2, 21] and ref. [3]. Our study can potentially give an independent

result. As an attempt, we perform a ﬁve-loop level calculation on the lattice. Our present

study does not quite give a conclusive result due to small lattice sizes. We regard the study

in this paper as an intermediate step toward obtaining the continuum limit result of the

ﬁve-loop coeﬃcient.

The achievements of the present paper can be stated as follows. First, the higher order

calculation than our previous study [7] is made possible. This is because we use a method

to suppress backward propagations, which are a serious obstacle in the analysis in ref. [7].

The higher order calculation can be done also because numerical costs to generate conﬁgu-

rations are drastically reduced due to quenched QED. In quenched QED, interaction terms

are absent and the Langevin equation becomes trivial. Therefore, conﬁgurations are gen-

erated according to a Gaussian distribution [22–24]. In this sense, the present paper tests

eﬃciency of numerical calculation of perturbative series on the lattice rather than NSPT

itself. Secondly, we discuss in detail the strategy for selecting simulation parameters and

also the order of various extrapolations. This is based on understanding of systematic errors

such as FV corrections, ﬁnite photon mass eﬀects, which are studied in this paper.

The paper is organized as follows. In Sec. 2, we study FV corrections in various IR regular-

ization method to discuss what kind of IR regularization we should adopt. We also add a new

insight into ﬁnite volume corrections in QEDL. In Sec. 3, we perform a lattice simulation.

We ﬁrst explain the outline of our calculation, and then we study systematic uncertainties

of our calculation to discuss the strategy for selecting simulation parameters and the order

of various extrapolations. Then we perform our numerical simulation following the strategy

to examine its feasibility. Sec. 4 is devoted to the conclusions and discussion.

2. Finite volume corrections in lattice QED

Before we attempt to calculate any physical quantities in QED, it is essential to provide a

concrete deﬁnition of QED. Specially, in a ﬁnite volume, the treatment of infrared divergence

requires careful consideration to ensure that predictions agree with QED in continuum and

inﬁnite volume spacetime as a certain limit. We below discuss and compare various deﬁnitions

proposed or used in the literature. We will see that the regularization by ﬁnite photon mass

is the most controlled method for our g−2 computations.

In Sec. 2.1, we study FV corrections in QEDLand QEDT L to a momentum-space correlator

at one loop. This part includes already known facts and can be regarded as a review part.

In Sec. 2.2, we study FV corrections in QEDLto a Euclidean time correlator, i.e., Fourier

transform of the momentum-space correlator, using a result in Sec. 2.1. We point out that a

FV eﬀect has diﬀerent parametric dependences depending on the size of t/L. This is the new

and main result in this section. In Sec. 2.3, we consider massive photon theory and conﬁrm

exponential suppression of FV corrections for clarity.

2.1. Finite volume corrections in QEDLand QEDT L: momentum-space correlator

We consider FV eﬀects in various IR regularization methods: QEDL, QEDT L, and massive

photon regularization [13–15]. The precise meaning of these regularizations is explained

shortly.

We consider the following quantity as an example:

I=X

k2+m2

(k+p)2+m2.(2.1)

Here kdenotes loop momentum and pexternal momentum. The summation/integral symbol

represents the sum/integration of the loop momentum k, and its precise meaning depends

on regularization schemes as discussed below. The above quantity mimics the one-loop cor-

rection to the two-point function in scalar QED. mγdenotes photon mass, which will be set

to zero or non-zero below. In lattice QED, the zero mode of loop momentum k= (0,0,0,0)

makes the result diverge. Therefore some regularization of the zero mode is needed. One way

is to introduce non-zero photon mass. There are alternative methods which do not introduce

photon mass but modify the range of loop momentum sum. QEDLand QEDT L deﬁne P

Ras

follows:

QEDLon T3×R:X

Z=1

L3X

⃗

k∈BZ3

L\{0}Zdk4

2π,

QEDLon T4:X

Z=1

L3TX

k4∈BZTX

⃗

k∈BZ3

L\{0}

QEDT L on T4:X

Z=1

L3TX

k∈BZ4

T L\{0}

,(2.2)

where ⃗

k= (k1, k2, k3), BZL=2π

LZ={2π

Ln|n∈Z},BZT=2π

TZ,BZ4

T L =BZ3

L×BZT, and

\{0}means removal of the element (0,··· ,0). In these regularization methods, the photon

mass is set to zero. We note that, although T3×Rcannot be realized in actual lattice sim-

ulations, it is convenient to consider QEDLon T3×Ras an intermediate step for clarifying

the diﬀerence between QED on R4and an above theory, as explained in ref. [17]. In other

words, we can clarify FV corrections of, for instance, QEDLon T4, considering the chain

(QED on R4)→(QEDLon T3×R)→(QEDLon T4) and studying the diﬀerence of each

deformation.

Equation (2.1) mimics a two point function in scalar QED and is not directly related to

the gfactor. Nevertheless, we understand some features of the regularization methods and

obtain implications by studying this simple integral.

Below we study FV corrections in QEDLand QEDT L setting p= (0,0,0, p4) with p4∈

R. We note that the FV corrections to Ihave been studied in the case of the on-shell

momentum p4=im [14, 17] and in the case of oﬀ-shell momentum p4∈R[14]. We give the

FV corrections for the oﬀ-shell momentum also in this paper in a self-contained manner, as

this result is used in the subsequent subsection. The reason why we are interested in the

oﬀ-shell momentum case rather than the on-shell momentum case is that a Euclidean time

correlator, which is what one actually obtains in lattice simulations, is equivalent to the

Fourier integral of a momentum-space correlator I, where the integral variable p4runs from

−∞ to ∞, namely oﬀ-shell momenta. Although one might expect that this Fourier integral

eﬀectively sets p4to the on-shell value due to the residue theorem, we point out that this

understanding is too naive. This will be discussed in Sec. 2.2 after we review FV corrections

to a momentum-space correlator here.

QED on R4→QEDLon T3×R

First we consider the diﬀerence between QEDLon T3×Rand QED on R4. It is given by

[13]

∆I(L)≡



X

QED on R4

−X

QEDLon T3×R





1

(k+p)2+m2

=−

X

⃗x∈LZ3\{0}−1

L3Zd3x

Zd4k

(2π)4

(k+p)2+m2ei⃗

k·⃗x

=−1

(4π)2Z1

dy Z∞

ds s−1e−s

4π[y(1−y)p2+ym2]L2ϑ3(0, e−π/s)3−1−s3/2.(2.3)

Here we used the Poisson resummation to rewrite the momentum sum P⃗

k∈BZ3

L\0by the

momentum integration Rd3⃗

k/(2π)3while introducing ⃗x ∈LZ, where LZ={Ln|n∈Z}.

Rd3xcorresponds to the subtraction of the spatial zero mode (since it gives (2π)3δ3(⃗

k))

and we rewrote the propagators by the Feynman parameter (y) integral and per-

formed the momentum integration. ϑ3denotes the elliptic theta function, ϑ3(0, e−π/s) =

P∞

n=−∞ e−πn2/s.

To study the asymptotic form of ∆Ifor L→ ∞, we consider

∆I(u)≡Z∞

dL L−u−1∆I(L).(2.4)

We can see the asymptotic behavior of ∆Iby studying singularities of this function. If

∆Ibehaves as ∆I∼L−u0for L≫1, f

∆I(u) develops singularities at u=−u0due to a

divergence of the integral of eq. (2.4) around L∼ ∞.

We obtain

∆I(u)

=−1

32π2

(4π)u/2Γ(−u/2) Z1

dy [y(1 −y)p2+ym2]u/2

×Z∞

ds su/2−1ϑ3(0, e−π/s)3−1−s3/2.(2.5)

The y-integral and the s-integral are factorized. The ﬁrst singularity of f

∆I(u) at negative

uis located at u=−2, where the y-integral diverges. This tells us that the asymptotic

behavior of ∆I(L) is ∆I(L)∼L−2. The s-integral is convergent for u=−2. To examine the

convergence of the s-integral, it is convenient to keep the following relation in mind:

ϑ3(0, e−π/s) =

∞

n=−∞

e−π

sn2=

∞

m=−∞ Zdk

2πe−k2

4πs eikm =s1/2

∞

m=−∞

e−πsm2=s1/2ϑ3(0, e−πs),

(2.6)

where we used the Poisson resummation in the ﬁrst equality and then performed the Gaussian

integral. Thus one can see that

ϑ3(0, e−π/s)3−1∼(1 + 2e−π/s)3−1 for s≪1

ϑ3(0, e−π/s)3−s3/2∼[s1/2(1 + 2e−sπ)]3−s3/2for s≫1 (2.7)

are very suppressed functions. The expansion of f

∆I(u) around u=−2 is given by

∆I(u) = −κ−2

4π

p2+m2

u+ 2 +O((u+ 2)0),(2.8)

where

κ−2≡Z∞

ds s−2ϑ3(0, e−π/s)3−1−s3/2≃ −2.837.(2.9)

The inverted formula of eq. (2.4) is given by

∆I(L) = 1

2πi Zi∞−0

−i∞−0

du Luf

∆I(u).(2.10)

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Preprintnumber:KEK-TH-245QEDonthelatticeandnumericalperturbativecomputationofg−2RyuichiroKitano1,2HiromasaTakaura1,∗1KEKTheoryCenter,Tsukuba305-0801,Japan2GraduateUniversityforAdvancedStudies(Sokendai),Tsukuba305-0801,Japan∗E-mail:hiromasa.takaura@yukawa.kyoto-u.ac.jp17/10/2023.........................

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