sfrom an improved vector isovector spectral function Diogo Boito1Maarten Golterman2Kim Maltman3Santiago Peris4Marcus V . Rodrigues5 and Wilder Schaaf6

2025-05-03 0 0 819.71KB 8 页 10玖币

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αsfrom an improved τvector isovector spectral function

Diogo Boito1,Maarten Golterman2,Kim Maltman3,Santiago Peris4,∗,∗∗,Marcus V.

Rodrigues5, and Wilder Schaaf6

1Instituto de Física de São Carlos, Universidade de São Paulo,

CP 369, 13570-970, São Carlos, SP, Brazil

2Department of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

3Department of Mathematics and Statistics, York University

Toronto, ON Canada M3J 1P3

4Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona

E-08193 Bellaterra, Barcelona, Spain

5Deutsches Elektronen-Synchrotron (DESY), Notkestraße 85, 22607 Hamburg, Germany

6Department of Physics, University of Washington, Seattle, WA 98195-1560, USA

Abstract. After discussing diﬃculties in determining αsfrom tau decay due

to the existence of Duality Violations and the associated asymptotic nature of

the OPE, we describe a new determination based on an improved vector isovec-

tor spectral function, now based solely on experimental input, obtained by (i)

combining ALEPH and OPAL results for 2π+4πand (ii) replacing K−K0and

higher-multiplicity exclusive-mode contributions, both previously estimated us-

ing Monte Carlo, with new experimental BaBar results for K−K0and results

implied by e+e−cross sections and CVC for the higher-multiplicity modes. We

ﬁnd αs(mτ)=0.3077±0.0075, which corresponds to αs(mZ)=0.1171±0.0010.

Finally, we comment on some of the shortcomings in the criticism of our ap-

proach by Pich and Rodriguez-Sanchez.

It has been clear since the pioneering work of Ref. [1] (see also Ref. [2] for other pre-

1992 references and a then-up-to-date implementation of the approach) that Finite Energy

Sum Rules (FESRs) provide a potentially useful tool for extracting αsfrom hadronic τdecay

data. Subsequent increases in the precision of both the experimental data [3, 4] and the order

to which the dominant perturbative contribution is known [5] have signiﬁcantly reduced the

resulting uncertainty on αs, to the extent that assumptions/approximations which might have

been reasonable in 1992 now need to be revisited.

In our context, the FESR consists of the identity

Zs0

sth

w s

s0!1

πImΠ(s)

| {z }

Iex p

w(s0)

=−1

2iπI|z|=s0

ˆw z

s0!D(z)

| {z }

Ith

w(s0)

,(1)

with Π(s) the correlator associated with the I=1 vector current ¯uγµd,wand ˆwpolynomials

related by an integration by parts, and D(z) the Adler function, D(z)=−zdΠ(z)

dz .1

∗peris@ifae.es

∗∗Speaker

1In general, the axial current can also be considered.

arXiv:2210.13518v1 [hep-ph] 24 Oct 2022

At large enough s0, it is reasonable to approximate the RHS using the Operator Prod-

uct Expansion (OPE)2and extract αsusing experimental data for Im Π(z) as input on the

LHS. The presence of a cut in Π(z), however, ensures that the OPE is at best an asymptotic

expansion in 1/z.3The RHS must, then, in general, contain an additional, non-OPE ("dual-

ity violating", or DV) contribution to compensate for the lack of convergence on the circle

|z|=s0[6]. Explicitly,

−1

2iπI|z|=s0

ˆw z

s0!D(z)=−1

2iπI|z|=s0

ˆw z

s0!DOPE(z)−Z∞

w s

s0!ImΠDV (s),

(2)

where the DV contribution, Im ΠDV (s), accounts for the oscillatory behavior seen at lower

sin the spectrum, before perturbative dominance sets in. This oscillatory behavior, which

cannot be accounted for by any power behavior in the OPE, led Ref. [7] to conclude that

the OPE should not be used "too close" to the cut. For a number of years this stricture

was implemented by "pinching", i.e., by choosing for ˆw(s/s0) polynomials having a higher

order zero at s=s0, thus suppressing contributions from near the cut. One, however, faces

two potential problems. First, it is not known a priori how much pinching is needed for a

determination of αsfree from DV contamination. Second, a polynomial with a higher-degree

zero is necessarily higher degree in s, and generates higher-dimension, D, OPE contributions

on the RHS of Eq. (2). This is potentially problematic, not only because the relevant higher-D

condensates are not known, but also because an asymptotic expansion like the OPE ceases to

be valid at high orders. This leads us to our ﬁrst message:

It is not possible to simultaneously suppress the contribution from DVs and high-order

condensates. One should restrict oneself to low orders of the OPE, but in a consistent manner.

Using a high-degree polynomial with strong pinching, but truncating the OPE at low D,

when unsuppressed higher-Dcontributions are, in principle, present, is thus a dangerous prac-

tice and leads, not surprisingly to inconsistencies [8]. We comment further on this practice,

which we refer to as the "truncated OPE" (tOPE) approach [4, 9, 10], in Sec. 3.1.

The above discussion makes it clear it is not safe to ignore DV contributions without

further investigation. While no ﬁrst-principles derivation of the form of Im ΠDV (s) exists,

some of its general properties are known. As for the asymptotic expansion in powers of the

coupling g2, where terms missed in the expansion are known to behave as e−const/g2,4so terms

missed in the OPE expansion of Im Π(s) are expected to behave as e−const·s×(oscillation).

This expectation was conﬁrmed in Refs. [11, 12], where the combination of a Regge-like

spectrum (M2

n∼n) asymptotically and a stringy relation (Γn∼Mn/Nc) between resonance

masses and widths at large (but ﬁnite) number of colors, Nc, and large resonance excitation

number, n,5was shown to lead to the large-sexpectation

πImΠDV (s)=e−δ−γssin α+βs+Olog s! 1+O 1

log s,1

s!! .(3)

We will use Eq. (3) in conjunction with Eq. (2), and comment on the impact of possible

subleading corrections in Sec. 3.2.

2In what follows, we consider the perturbative series as the contribution from the identity operator.

3A convergent expansion in inverse powers of zmust have a disc of convergence.

4Recall the case of renormalons and the perturbative series.

5These are properties of QCD in 2 dimensions in the large-Nclimit and also born out phenomenologically in the

real world [13–15]

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sfromanimprovedvectorisovectorspectralfunctionDiogoBoito1,MaartenGolterman2,KimMaltman3,SantiagoPeris4;;,MarcusV.Rodrigues5,andWilderSchaaf61InstitutodeFísicadeSãoCarlos,UniversidadedeSãoPaulo,CP369,13570-970,SãoCarlos,SP,Brazil2DepartmentofPhysicsandAstronomy,SanFranciscoStateUniversity,SanFran...

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