Spectral-weight sum rules for the hadronic vacuum polarization Diogo BoitoaMaarten GoltermanbcKim Maltmandeand Santiago Perisc

2025-05-03 0 0 645.38KB 33 页 10玖币

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Spectral-weight sum rules for the

hadronic vacuum polarization

Diogo Boito,aMaarten Golterman,b,c Kim Maltmand,e and Santiago Peris,c

aInstituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo

CP 369, 13570-970, S˜ao Carlos, SP, Brazil

bDepartment of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

cDepartment of Physics and IFAE-BIST, Universitat Aut`onoma de Barcelona

E-08193 Bellaterra, Barcelona, Spain

dDepartment of Mathematics and Statistics, York University

Toronto, ON Canada M3J 1P3

eCSSM, University of Adelaide, Adelaide, SA 5005 Australia

We develop a number of sum rules comparing spectral integrals involving

judiciously chosen weights to integrals over the corresponding Euclidean two-

point function. The applications we have in mind are to the hadronic vacuum

polarization that determines the most important hadronic correction aHVP

µto the

muon anomalous magnetic moment. First, we point out how spectral weights

may be chosen that emphasize narrow regions in √s, providing a tool to inves-

tigate emerging discrepancies between data-driven and lattice determinations of

aHVP

µ. Alternatively, for a narrow region around the ρmass, they may allow for a

comparison of the dispersive determination of aHVP

µwith lattice determinations

zooming in on the region of the well-known BaBar–KLOE discrepancy. Second,

we show how such sum rules can in principle be used for carrying out preci-

sion comparisons of hadronic-τ-decay-based data and e+e−→hadrons(γ)-based

data, where lattice computations can provide the necessary isospin-breaking cor-

rections.

arXiv:2210.13677v2 [hep-lat] 20 Feb 2023

I. INTRODUCTION

As is well known, the recent FNAL E989 experimental result for the muon anomalous

magnetic moment aµ= (g−2)/2 [1] conﬁrms the earlier BNL E821 result [2] and produces an

updated experimental world average 4.2σlarger than the g−2 Theory Initiative assessment

[3] of the Standard Model (SM) prediction, based on the work of Ref. [4–27]. However,

a lattice result for the hadronic vacuum polarization (HVP) contribution aHVP

µ[28] comes

out 2.1σhigher than the dispersive R-ratio-based estimate which underlies the SM value of

Ref. [3]. Replacing the dispersive estimate for aHVP

µwith the value found in Ref. [28], the

SM-based estimate for aµis only 1.5σlower than the world-average experimental value. As

there is no evidence for discrepancies in other contributions to aµ, the recent focus has been

on understanding the discrepancy between the dispersive and lattice values for aHVP

µ.1

The dispersive vs. lattice discrepancy becomes even more pronounced if we consider the

“intermediate window” quantity introduced by RBC/UKQCD [29], in which the integral

over Euclidean time, t, of the lattice correlator that yields aHVP

µis restricted to a “window”

between t= 0.4 and t= 1 fm (smeared by a width of 0.15 fm on both boundaries to

avoid lattice artifacts) by multiplying the integrand with a (smoothed-out) double step

function in t. While the original RBC/UKQCD study found agreement between the lattice-

based result and the corresponding electro-production-based dispersive estimate, Ref. [30]

found a signiﬁcantly larger lattice value. This larger value was subsequently conﬁrmed in

Ref. [28], which produced a lattice result 3.7σabove the dispersive estimate. The virtue

of the RBC/UKQD intermediate window quantity is that it can be computed with smaller

errors on the lattice than the quantity aHVP

µitself, thus allowing more stringent tests between

diﬀerent lattice computations, as well as between lattice and dispersive results.

Meanwhile, the larger lattice values for the intermediate window quantity found in

Refs. [28, 30] have been conﬁrmed, with diﬀerent lattice discretizations of the QCD action

and the electromagnetic (EM) current, by a number of other groups as well as in updates

of the results of Refs. [29, 30], in Refs. [31–33, 36–39].

Clearly, then, it is important to develop further tools to study the discrepancies between

dispersive and lattice estimates for aHVP

µand closely related quantities such as the inter-

mediate window. To obtain dispersive estimates for the window quantity, which is deﬁned

as a function of Euclidean time, the window function needs to be converted to a window

in √s, the center-of-mass energy in e+e−→hadrons. As a function of √s, however, the

weight which deﬁnes the intermediate window is very broad, ranging from about 0.7 GeV

to 3 GeV (taking the values of √swhere it exceeds roughly half of its maximum value).

To explore the lattice vs. dispersive discrepancy in more detail, it would be useful to have

access to tools to compare data-driven dispersive results with lattice results in narrower,

more “custom-designed” windows in √s. In this paper, we propose a class of sum rules

designed with precisely this goal in mind. Similar ideas were recently explored in Ref. [40]

by considering linear combinations of a set of Euclidean windows, and in Ref. [41]. Here,

instead, we deﬁne the weights we will employ in our weighted spectral integrals directly as

a function of s, and derive sum rules relating these weighted spectral integrals to integrals

in Euclidean time over correlation functions which can be evaluated on the lattice.

1For instance, there is good agreement between data-driven and lattice values for the hadronic light-by-

light contribution, within ∼

<20% errors. This 20% error corresponds to an uncertainty of about 2 ×10−10

in aµ, which is not suﬃcient to explain the discrepancy.

In Sec. II we develop two sets of sum rules starting from weights deﬁned as a function

of s. In Sec. II A we consider a class of rational weights that allow us to deﬁne windows

localized in s. They are similar to those used recently in Ref. [42] to obtain a lattice-based

determination of |Vus|from strange hadronic τ-decay data. In Sec. II B we use these rational

weights as the starting point for deﬁning a set of sum-of-exponential weights with shapes

very similar to those deﬁned by the underlying rational weights, following ideas proposed

in Ref. [43]. In both cases exact sum rules exist relating spectral integrals employing these

weights to quantities that can be directly computed on the lattice. We explain why the

sum-of-exponential weights may lead to smaller errors for the lattice side of the sum rules

than the corresponding rational weights, and provide examples of this reduction in Sec. III.

Narrower windows in √sare also potentially useful for comparing I= 1 contributions

to aHVP

µinferred from I= 1 hadronic τ-decay data with the corresponding contributions

obtained using R-ratio data. An example of the potential usefulness of narrower windows is

the application to the BaBar–KLOE discrepancy in the two-pion spectral distributions (for

a review, see Ref. [3]), where the discrepancy occurs over a fairly narrow range in energy

around the ρpeak. Attempts to use τ-based data have a long history [44–47], but have been

abandoned more recently because of the increased precision of electroproduction data, and

the lack of a solid theoretical framework for evaluating the isospin-breaking (IB) corrections

that must be applied to the τ-based data. It would be interesting to revisit this possibility

since (i) a more precise τ-based non-strange vector spectral function is now available [48],

and (ii) Belle II may provide improved τ-decay-distribution data for at least some of the most

important vector-channel exclusive modes [49, 50]. Moreover, we will argue in Sec. IV of this

paper that if the combined 2πand 4πchannels are taken from τ, then, to good accuracy,

the lattice can be used to compute the necessary IB corrections from ﬁrst principles. We

illustrate the comparison of electroproduction- and hadronic-τ-decay-based 2π+ 4πdata

using two rational weight choices, W1,5and W2,5, deﬁned in Sec. III.

We end the paper with a brief conclusion in Sec. V, and relegate some technical details

to two appendices.

II. SUM RULES

In this section, we develop two types of sum rules, one, in Sec. II A, based on a set of

weights used before in Ref. [42], and one, in Sec. II B, based on the ideas advocated in

Ref. [43]. They are closely related, and we will explore their diﬀerences in examples in

Sec. III.

We start from a Euclidean current-current correlator

Gµν (x) = hjµ(0)j0

ν(x)i=Zd4q

(2π)4eiqx δµν q2−qµqνΠ(q2),(2.1)

for two potentially diﬀerent vector currents jµand j0

ν, and deﬁne from this the time-

momentum correlator C(t) by

C(t) = −1

k=1 Zd3xhjk(0)j0

k(~x, t)i=−ZdQ

2πeiQt Q2Π(Q2),(2.2)

where Q=q4, and we have assumed that a regulator (such as the lattice) has been intro-

duced, so that C(t) and Π(Q2) are ﬁnite. While in straightforward applications to aHVP

µthe

Re q2

FIG. 1: Contour Cused in Eq. (2.8); z=q2=−Q2. The black dot indicates the point q2=sth.

currents jµand j0

νwill both be the hadronic electromagnetic current, they can also be chosen

diﬀerent, as will be done in the application described in Sec. IV below. The corresponding

subtracted polarization ˆ

Π(Q2) = Π(Q2)−Π(0) (2.3)

can be expressed in terms of C(t) by

Π(Q2) = −1

Q2Z∞

−∞ dt eiQt −1C(t)−Π(0) (2.4)

=−2

Q2Z∞

0dt (cos(Qt)−1) C(t)−Π(0)

=Z∞

0dt 4 sin2(Qt/2)

Q2−t2!C(t),

where we used Rdt C(t) = 0 and, in the second step, that C(t) is an even function of t. We

deﬁne the spectral function ρ(s) as usual by

ρ(s) = 1

πIm Π(s),(2.5)

with ˆ

Π(Q2) and ρ(s) satisfying the subtracted dispersion relation

Π(Q2) = −Q2Z∞

sth

ds ρ(s)

s(s+Q2),(2.6)

with sth the relevant threshold value for ρ(s).

A. Rational-weight sum rules

We begin with a set of spectral weights of the form

Wm,n(s;{Q2

`}) = µ2(n−m−1) (s−sth)m

`=1(s+Q2

`), Q2

n> Q2

n−1>··· > Q2

1>0.(2.7)

Here the Q2

`are a set of ﬁxed Euclidean squared momenta, and we will always take m

suﬃciently smaller than nthat the weighted spectral integral with weight (2.7) is ﬁnite.

We multiply by a generic mass scale µ2(n−m−1) to make the weighted spectral integrals

considered below dimensionless; any precisely known scale is suitable for this purpose and

we will employ the choice µ=mτin what follows.

We then consider the integral

2πi ZCdz Wm,n(z;{Q2

`})Π(−z) = (−1)mm2(n−m−1)

k=1

(Q2

k+sth)m

Q`6=k(Q2

`−Q2

k)Π(Q2

k),(2.8)

with Cthe contour in the complex z=q2=−Q2plane shown in Fig. 1, assumed to have

a radius large enough that all the z=−Q2

kpoints lie in its interior on the negative z-axis.

The result on the right-hand side follows from the fact that Π(−z) is analytic in the complex

plane except for a cut starting at sth on the positive real z-axis, as shown in Fig. 1. If we

now take the radius of the circular part of Cto inﬁnity, we obtain the sum rule

Im,n ≡Z∞

sth

ds Wm,n(s;{Q2

`})ρ(s) = (−1)mm2(n−m−1)

k=1

(Q2

k+sth)m

Q`6=k(Q2

`−Q2

k)Π(Q2

k),(2.9)

where the integral over the spectral function comes from the discontinuity along the positive

real z-axis. We will prove in App. A that

k=1

(Q2

k+sth)m

Q`6=k(Q2

`−Q2

k)= 0 ,(2.10)

from which it follows that we can replace Π(Q2

k) by ˆ

Π(Q2

k) on the right-hand side of Eq. (2.9).

This has to be the case, as the left-hand side is ﬁnite by construction, and thus the term

proportional to Π(0) on the right-hand side has to vanish.

A potentially useful application of this sum rule is to evaluate the spectral integral using

data, for instance obtained from R-ratio measurements, and the sum on the right-hand side

using data from lattice QCD. In fact, what is usually computed in lattice QCD is a position-

space correlator; in the case of aHVP

µ, Eq. (2.2) with jµ=j0

µthe hadronic electromagnetic

current. Replacing Π(Q2

k) with ˆ

Π(Q2

k) and using Eq. (2.4), the sum rule (2.9) can then be

recast as Z∞

sth

ds Wm,n(s;{Q2

`})ρ(s) = Z∞

0dt c(m,n)(t)C(t),(2.11)

with

c(m,n)(t)=(−1)mm2(n−m−1)

k=1

(Q2

k+sth)m

Q`6=k(Q2

`−Q2

k) 4 sin2(Qkt/2)

k−t2!.(2.12)

We will refer to the sum rules of Eq. (2.11) as “rational-weight sum rules” (RWSRs). On

the lattice, the integral over ton the right-hand side of Eq. (2.11) would be replaced by a

sum over the discrete values of tavailable on the lattice, using, for example, the trapezoidal

rule. We will discuss examples in Sec. III A below.

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Spectral-weightsumrulesforthehadronicvacuumpolarizationDiogoBoito,aMaartenGolterman,b;cKimMaltmand;eandSantiagoPeris,caInstitutodeFsicadeS~aoCarlos,UniversidadedeS~aoPauloCP369,13570-970,S~aoCarlos,SP,BrazilbDepartmentofPhysicsandAstronomy,SanFranciscoStateUniversity,SanFrancisco,CA94132,USAcDepar...

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Spectral-weight sum rules for the hadronic vacuum polarization Diogo BoitoaMaarten GoltermanbcKim Maltmandeand Santiago Perisc

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