aainstitutetext: University of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Wien, Austriabbinstitutetext: Instituto de Física de
São Carlos, Universidade de São Paulo,
CP 369, 13560-970, São Carlos, SP, Brazilccinstitutetext: Erwin Schrödinger
International Institute for Mathematics and Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, Austriaddinstitutetext:
Department of Addictive Behaviour, Central Institute of Mental Health, Medical
Faculty Mannheim, Heidelberg University, Mannheim, Germany

# Reconciling the Contour-Improved and Fixed-Order Approaches for $\tau$
Hadronic Spectral Moments II:
Renormalon Norm and Application in $\alpha_{s}$ Determinations

Miguel A. Benitez-Rathgeb a,b Diogo Boito a,c André H. Hoang a,d Matthias
Jamin<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In a previous article, we have shown that the discrepancy between the fixed-
order (FOPT) and contour-improved (CIPT) perturbative expansions for $\tau$
hadronic spectral function moments, which had affected the precision of
$\alpha_{s}$ determinations for many years, may be reconciled by employing a
renormalon-free (RF) scheme for the gluon condensate (GC) matrix element. In
addition, the perturbative convergence of spectral function moments with a
sizeable GC correction can be improved. The RF GC scheme depends on an IR
factorization scale $R$ and the normalization $N_{g}$ of the GC renormalon. In
the present work, we use three different methods to determine $N_{g}$,
yielding a result with an uncertainty of $40\%$. Following two recent state-
of-the-art strong coupling determination analyses at ${\cal
O}(\alpha_{s}^{5})$, we show that using the renormalon-free GC scheme
successfully reconciles the results for $\alpha_{s}({m_{\tau}^{2}})$ based on
CIPT and FOPT. The uncertainties due to variations of $R$ and the uncertainty
of $N_{g}$ only lead to a small or moderate increase of the final uncertainty
of $\alpha_{s}(m_{\tau}^{2})$, and affect mainly the CIPT expansion method.
The FOPT and CIPT results obtained in the RF GC scheme may be consistently
averaged. The RF GC scheme thus constitutes a powerful new ingredient for
future analyses of $\tau$ hadronic spectral function moments.

††preprint:  UWThPh-2022-7

##  1 Introduction

The determination of the QCD strong coupling $\alpha_{s}$ from the inclusive
hadronic $\tau$ decay spectral functions, obtained through the analysis of
finite energy sum rules (FESRs) for spectral function moments, represents one
of the most precise methods to extract this fundamental quantity of quantum
chromodynamics (QCD) from experimental data ParticleDataGroup:2020ssz . For a
long time, however, a systematic theoretical uncertainty associated with the
renormalization scale setting has persisted. The values of $\alpha_{s}$
obtained with a strict fixed-order expansion, which goes under the name fixed-
order perturbation theory (FOPT) (see Ref. Beneke:2008ad ), have been
systematically lower than the values obtained using the so-called contour-
improved perturbation theory (CIPT) Pivovarov:1991rh ; LeDiberder:1992jjr ,
which resums a certain set of logarithmic phase corrections with the use of
the QCD $\beta$-function.

In a recent paper Benitez-Rathgeb:2022yqb , to which we will refer often as
‘Part I’, we laid the foundations of a method to eliminate the CIPT-FOPT
discrepancy. The method was motiviated from the work of Hoang and Regner
Hoang:2020mkw ; Hoang:2021nlz  where it was shown that the discrepancy can be
of infra-red (IR) origin and caused by contributions in the CIPT expansion
terms that are incompatible with the standard form commonly adopted for the
higher dimensional corrections in the operator product expansion (OPE). Any IR
renormalon contributes a certain amount to the discrepancy, which was termed
‘asymptotic separation’. But numerically the asymptotic separation is strongly
dominated by the renormalon associated with the gluon condensate (GC)
contribution in the operator product expansion (OPE). The basic idea of the
method proposed in Ref. Benitez-Rathgeb:2022yqb , consists in a redefinition
of the GC matrix element in order to remove (or subtract) the associated
renormalon from the perturbative expansion of the Adler function. This
redefinition was called the renormalon-free gluon condensate (RF GC) scheme
and involves perturbative subtractions that allow for a straightforward and
transparent implementation in the context of the computations involved for the
FOPT and CIPT expansions of the spectral function moments. This perturbative
and observable-independent subtraction approach is very common in many areas
of particle physics phenomenology where the removal of renormalon divergences
are important, most prominently in the context of using short-distance mass
schemes in heavy-quark physics Hoang:2020iah ; Beneke:2021lkq . In Ref.
Benitez-Rathgeb:2022yqb  such a subtraction scheme was for the first time
applied to the $\tau$ hadronic spectral function moment. The essential
conceptual aspect of this seemingly trivial scheme change, which reaches
beyond the well established use of short-distance quark masses, is that it
substantially modifies the CIPT series coefficients even for spectral function
moments where the GC OPE correction is strongly suppressed or even
vanishes.111We call such spectral function moments gluon condensate suppressed
(GCS). For GCS spectral function moments, when switching to the RF GC scheme,
the changes of the FOPT series terms are tiny and comparable to the GC OPE
corrections. This fact signifies the inconsistency of the CIPT expansion when
the $\overline{\rm MS}$ GC scheme is employed Hoang:2020mkw ; Hoang:2021nlz .
The work of Hoang and Regner laid out the conceptual basis to understand this
paradoxical behavior by showing that the Borel representation of the CIPT
expansion has a structure that is incompatible with the standard Borel
calculus.222Based on plausibility arguments in the context of Borel model
studies, it was stated already in Refs. Beneke:2008ad ; Beneke:2012vb  that
the CIPT expansion is disfavored. However the inconsistency of the CIPT
expansion with the standard form of the OPE in the presence of IR renormalons
was not known prior to the work by Hoang and Regner.

A particularly useful property of the RF GC scheme we have proposed in Part I
is that it is scale-invariant, but at the same time has a freely adaptable IR
factorization scale $R$, which should be chosen to be close to the dynamical
scale of the observable. This freedom of scale choice is controlled by a
renormalization group evolution (called ‘R-evolution’) that is known exactly
to all orders and makes the RF GC scheme also suitable for other processes
where the gluon condensate contributes. The implementation of the scheme
depends on knowledge about the GC renormalon norm, $N_{g}$, which must be
supplemented independently.

In Part I, we have demonstrated that in the RF GC scheme with the correct
value of $N_{g}$, the CIPT-FOPT discrepancy can indeed be resolved and,
moreover, is already significantly reduced at ${\cal O}(\alpha_{s}^{4})$ and
${\cal O}(\alpha_{s}^{5})$, which are the perturbative orders for which QCD
corrections are available. While the CIPT-FOPT discrepancy and its reduction
in our RF GC scheme take place for GCS spectral function moments, the new
scheme is also capable of significantly improving the perturbative behaviour
of moments for which the GC OPE correction is sizeable, i.e. for so-called GC
enhanced (GCE) moments. In Part I, this was shown in specific realizations of
the perturbative series at higher orders, namely the large-$\beta_{0}$ limit
and using all-order renormalon models for the Adler function, where the gluon
condensate renormalon norm $N_{g}$ was exactly known. An application of the RF
GC scheme in the context of a fully consistent phenomenological determination
of the strong coupling and a study on the impact of the uncertainty of $N_{g}$
was missing.

In the present work, which is Part II in this series of papers, we address
these two issues: We carry out a thorough analysis to determine the GC
renormalon norm $N_{g}$ using three different methods, and we conduct
consistent phenomenological determinations of $\alpha_{s}$ following two
state-of-the-art approaches from the recent literature Pich:2016bdg ;
Boito:2020xli , accounting for the uncertainty of the norm $N_{g}$ and
variations of the IR factorization scale $R$. We emphasize that the main aim
of our $\alpha_{s}$ determinations is to demonstrate the effectiveness of the
RF GC scheme to resolve the CIPT-FOPT discrepancy in the context of accounting
for all perturbative uncertainties in a realistic manner. In particular, our
analyses are not intended to provide a discussion on the merits of the
different ways the nonperturbative corrections in Refs. Pich:2016bdg ;
Boito:2020xli  have been accounted for. We emphasize that our two analyses are
not identical copies of the complete set of analyses carried out in Refs.
Pich:2016bdg ; Boito:2020xli  as this would go way beyond the sope of this
article. The main aim of them is to demonstrate the practical use of the RF GC
scheme in achieving precise strong coupling determinations that are not
affected by the CIPT-FOPT discrepancy problem. The essential message is that
in the RF GC scheme the CIPT results closely approach the FOPT results and
that the additional sources of theoretical uncertainties in the RF GC scheme
(related to $R$ and $N_{g}$) are relatively small.

The basis of our analysis is the proposition that the GC renormalon gives a
sizeable contribution to the known QCD corrections of the Adler function at
${\cal O}(\alpha_{s}^{4})$ (including the state-of-the-art estimates for the
${\cal O}(\alpha_{s}^{5})$ coefficient Jamin:2021qxb ; Boito:2018rwt ;
Caprini:2019kwp ; Baikov:2008jh ), such that $N_{g}$ is sizeable as well and
can be determined with reliable uncertainties from the ${\cal
O}(\alpha_{s}^{4})$ (or ${\cal O}(\alpha_{s}^{5})$) perturbative coefficients.
This proposition has been advocated as being natural in Ref. Beneke:2008ad .
We have furthermore shown in Part I through a dedicated analysis, that the
perturbative behavior of the spectral function moments is fully consistent
with this proposition. It should be mentioned, however, that it is in
principle not excluded that this behavior may be mimicked at ${\cal
O}(\alpha_{s}^{4})$ through an interplay of other renormalons and sizeable
lower-order convergent QCD corrections in the presence of an extremely small
(and practically negligible) value of $N_{g}$ DescotesGenon:2010cr . However,
as shown in Ref. Beneke:2012vb  such a scenario would correspond to an
unnatural and implausible fine-tuned renormalon structure of the Adler
function. More light could potentially be shed on this matter through the
actual calculation of the QCD corrections at ${\cal O}(\alpha_{s}^{5})$ and
beyond, which, however, may appear unlikely in the near future. Our analysis
therefore relies on the proposition that the observed consistency of the known
QCD corrections with a sizeable GC norm $N_{g}$ is not caused by some fine-
tuned renormalon scenario with a negligible value for $N_{g}$ or some purely
accidental behavior of the ${\cal O}(\alpha_{s}^{4})$ QCD corrections. The
same proposition is in principle also applied when short-distance quark mass
renormalization schemes employed in order to achieve an improved perturbative
behavior for heavy quark mass sensitive observables Hoang:2020iah ;
Beneke:2021lkq .

We note that, the RF GC scheme is formulated using the $C$-scheme for the
strong coupling Boito:2016pwf  for the concrete value $C=0$, see Sec. 2.2 of
Part I Benitez-Rathgeb:2022yqb . This scheme, which we just refer to as the
$C$-scheme, has the special feature that the QCD $\beta$-function depends only
on the universal one- and two-loop coefficients $\beta_{0}$ and $\beta_{1}$
and can be written down in closed form. Furthermore, the QCD hadronization
scale $\Lambda_{\rm QCD}$ is identical to the one in the usual $\overline{\rm
MS}$ scheme. The simple form of the $\beta$-function in the $C$-scheme has the
important property that the GC renormalon and the associated asymptotic series
behavior used in the subtration definition can be written down in a closed and
compact form without any truncation concerning subleading asymptotic
contributions. This makes the $C$ scheme special and particularly useful for
conceptual renormalon studies in full QCD. Our actual numerical studies are,
however, still carried out in the usual $\overline{\rm MS}$ strong coupling
scheme and are obtained from the $C$-scheme expressions by a finite-order re-
expansion. The relevant formulae can be found in the appendix of Ref. Benitez-
Rathgeb:2022yqb . For the numerical evaluations of the strong coupling in the
complex plane we used the quasi-exact routine from the REvolver library
package Hoang:2021fhn . Throughout this work we employ the strong coupling for
$n_{f}=3$ active flavors unless stated otherwise.

The content of this paper is as follows: In Sec. 2 we set up our notations and
briefly review the theoretical ingredients relevant to the phenomenological
analysis of $\tau$ hadronic spectral function moments as well as the basics of
implementing the RF GC scheme. Section 3 is dedicated to the determination of
the GC renormalon norm $N_{g}$ using three different methods. Two of the
methods are already known from the literature, and the third one is new. The
constructive elements of the new method are the improvements in the
perturbative behavior for the CIPT and FOPT expansions for GCS and GCE
spectral function moments, and we demonstrate that it constitutes a very
powerful and reliable method. In Sec. 4 we finally put the RF GC to a
practical test by applying it in the context of two realistic phenomenological
strong coupling determinations following the analyses of Refs. Pich:2016bdg ;
Boito:2020xli . Here we only rely on the perturbative input that is commonly
used in such phenomenological analyses, and we in particular account for the
uncertainty in the norm $N_{g}$ and variations of the IR factorization scale
$R$. In Sec. 5 we conclude. There are a number of appendices. In App. A we
review the different main conventions concerning the definitions used for the
renormalon calculus in the literature. The information provided there allows
for an easy conversion of our convention for the GC renormalon norm $N_{g}$ to
other conventions used in the literature. While the main focus of our work
lies on the phenomenologically relevant case of $n_{f}=3$ active quark
flavors, our analyses on the determination of the GC renormalon norm $N_{g}$
can also be applied for the quenched approximation $n_{f}=0$, for which a
number of dedicated analyses are available in the literature. In App. B we
compare our results to these previous determinations. We find a particularly
large discrepancy to the results from Ref. Bali:2014fea  based on lattice
perturbation theory, which we comment on in App. C.

##  2 Notation, Strong Coupling and Renormalon Calculus

###  2.1 Theoretical Setup

The total hadronic $\tau$ decay rate can be separated into three different
components. The contributions from the light-quark ($\bar{u}d$) current, which
can be split into a vector and an axial-vector part, $R_{\tau,V}$ and
$R_{\tau,A}$, respectively, and a contribution with net strangeness,
$R_{\tau,S}$. In studies aimed at the extraction of the strong coupling, one
commonly focuses on the light-quark contributions, which receive only tiny
quark-mass corrections in the theoretical descriptions that can be neglected
for all practical purposes. Here we follow this strategy and focus only on the
light-quark components.

On the theory side, the description of the $\tau$ hadronic decay spectral
function moments receives contributions from the four two-point $q\bar{q}$
correlators: vector, axial-vector, scalar, and pseudo-scalar. The
contributions of the latter two are suppressed by at least two powers of the
light-quark masses and are very small. Because of chiral symmetry, the
perturbative contributions for the vector ($V$) and axial-vector ($A$)
currents, which are obtained in the chiral limit, are the same.
nonperturbative contributions, on the other hand, are different for the two
currents.

The $V$ and $A$ correlation functions $\Pi^{\mu\nu}_{V/A}(p)$ are defined as

$\Pi^{\mu\nu}_{V/A}(p)\,\equiv\,i\\!\int\\!{\rm
d}x\,e^{ipx}\,\langle\Omega|\,T\\{j^{\mu}_{V/A}(x)\,j^{\nu}_{V/A}(0)^{\dagger}\\}|\Omega\rangle\,,$
(1)

where $|\Omega\rangle$ is the physical QCD vacuum and the currents are
$j^{\mu}_{V/(A)}(x)=:\\!\bar{u}(x)\gamma^{\mu}(\gamma_{5})d(x)\\!:$. The
correlators $\Pi^{\mu\nu}_{V/A}(p)$ admit the usual decomposition into
transversal, $\Pi_{V/A}^{(1)}$, and longitudinal, $\Pi_{V/A}^{(0)}$,
components. The longitudinal components in $V$ and $A$ are related to the
scalar and pseudoscalar correlators, respectively. We will often work with the
combination
$\Pi_{V/A}(p^{2})\equiv\Pi_{V/A}^{(1+0)}(p^{2})=\Pi_{V/A}^{(1)}(p^{2})+\Pi_{V/A}^{(0)}(p^{2})$
and we frequently drop the subscripts $V$/$A$ when suitable. These correlators
are not physical quantities because they are ultra-violet (UV) divergent and
require a scale and scheme dependent subtraction. Therefore, one often works
with the spectral function, $\rho(s)$, or the (reduced) Adler function,
$D(s)$, which are UV finite and physical. We use the definitions

$\rho(s)\equiv\frac{1}{\pi}{\rm
Im}\,\Pi(s+i0),\qquad\frac{1}{4\pi^{2}}\big{[}1+D(s)\big{]}\,\equiv\,-\,s\,\frac{\rm
d}{{\rm d}s}\,\Pi(s)\,.$ (2)

In the analysis of hadronic $\tau$ spectral function moments the experimental
spectral functions are integrated over weight functions. They can be written,
in general, as

$R_{V/A}^{(w)}(s_{0})\,=\,12\pi^{2}\,S_{\rm
ew}\,|V_{ud}|^{2}\,\int\limits_{0}^{s_{0}}\frac{{\rm
d}s}{s_{0}}w({\textstyle\frac{s}{s_{0}}})\left[\,\rho_{V/A}(s)-\frac{2s/s_{0}}{1+2s/s_{0}}\rho_{V/A}^{(0)}(s)\right],$
(3)

where the weight function $w(s)$ can be any analytic function; in practice one
most often chooses polynomials. To obtain the physical decay rate, one must
set $s_{0}=m_{\tau}^{2}$ and use the weight function determined by the
kinematics of the hadronic $\tau$ decay Benitez-Rathgeb:2022yqb ;
Beneke:2008ad . Smaller values of the upper limit in the integration,
$s_{0}<m_{\tau}^{2}$, are also frequently used in the literature as a means to
further constrain the theoretical description Boito:2020xli ; Boito:2014sta ;
Boito:2012cr ; Maltman:2008nf , but it is important to keep $s_{0}$ within the
perturbative regime.

The different contributions to the theoretical counterpart of Eq. (3) can be
decomposed in the following way

$R_{V/A}^{(w)}(s_{0})\,=\,\frac{N_{c}}{2}\,S_{\rm
ew}\,|V_{ud}|^{2}\Big{[}\,\delta^{\rm
tree}_{w}+\delta^{(0)}_{w}(s_{0})+\sum_{d\geq
4}\delta^{(d)}_{w,V/A}(s_{0})+\delta_{w,V/A}^{(\rm DV)}(s_{0})\Big{]}\,,$ (4)

where $N_{c}=3$, $S_{\rm ew}=1.0201(3)$ are factorizable electroweak
corrections Marciano:1988vm ; Braaten:1990ef ; Erler:2002mv  (which is the
value used in Refs. Pich:2016bdg ; Boito:2020xli ), and $V_{ud}$ is the CKM
matrix element. The tree-level contribution is represented by $\delta_{w}^{\rm
tree}$. The perturbative QCD corrections, exactly known up to 5 loops,
$\mathcal{O}(\alpha_{s}^{4})$, are encoded in $\delta_{w}^{(0)}$. The terms
$\delta_{w,V/A}^{(d)}$ contain the nonperturbative corrections arising from
the power suppressed contributions of OPE condensates of dimension $d$.
Finally, the duality violation (DV) contributions, that account for
nonperturbative corrections that are not captured through the OPE Cata:2005zj
; Cata:2008ye ; Boito:2017cnp  in the region close to the Minkowskian real
$s$-axis, are encoded in $\delta_{w,V/A}^{(\rm DV)}$.

The perturbative contribution, $\delta_{w}^{(0)}$, was extensively discussed
in Part I of this work Benitez-Rathgeb:2022yqb . A review of these results is
given in Sec. 2.2. Here we focus on the nonperturbative terms. The
nonperturbative corrections arising from the OPE condensates, using
analyticity properties of the correlators and of the Adler function, can be
written in terms of contour integrals Braaten:1991qm ; Beneke:2008ad
($x=s/s_{0}$)

$\sum_{d\geq 4}\delta_{w,V/A}^{(d)}(s_{0})=4\pi
i\ointctrclockwise\limits_{|s|=s_{0}}\\!\\!\frac{{\rm
d}s}{s_{0}}w({\textstyle\frac{s}{s_{0}}})\Pi^{\rm OPE}_{V/A}(s)=\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm d}x}{x}\,W(x)\,D^{\rm
OPE}_{V/A}(xs_{0})\,,$ (5)

where the weight function for the Adler function is $W(x)=2\int_{x}^{1}{\rm
d}z\,w(z)$. The general form of OPE power corrections for the Adler function
can be written as Shifman:1978bx

$D^{\rm
OPE}_{V/A}(s)\,=\,\frac{C_{4,0}(\alpha_{s}(-s))}{s^{2}}\langle\bar{\cal
O}_{4,0}\rangle+\sum\limits_{d=6}^{\infty}\frac{1}{(-s)^{d/2}}\sum_{i}C_{d,i}^{V/A}(\alpha_{s}(-s))\langle\bar{\cal
O}_{d,\gamma_{i}}\rangle_{V/A},\,$ (6)

where the condensates $\langle\bar{\cal O}_{d,\gamma_{i}}\rangle$ are
nonperturbative vacuum matrix elements of gauge invariant dimension $d$
operators built from the light-quark and gluon fields, with anomalous
dimension $\gamma_{i}$. The short-distance information is encoded in the
Wilson coefficients $C_{d,i}$ which can be computed in perturbation theory as
a series expansion in powers of $\alpha_{s}(-s)$. In the massless limit, for
$d=4$ the only contribution stems from the gluon condensate but starting from
$d=6$ contributions from several different condensates need to be considered
for each $d$ — this is accounted for by the sum in $i$.

In practical applications of the OPE for the analysis of hadronic $\tau$
spectral function moments, due to the lack of knowledge about terms with
$d\geq 6$ and due to the proliferation of operator matrix elements, an
important approximation is made Davier:2008sk ; Davier:2013sfa ;
Maltman:2008nf ; Pich:2016bdg ; Boito:2012cr ; Boito:2014sta ; Boito:2020xli :
no $\alpha_{s}$ corrections in the Wilson coefficients are considered and the
anomalous dimensions of all operators are neglected, thereby removing the
$\alpha_{s}$-suppressed logarithmic $s$ dependence from the Wilson
coefficients of Eq. (6) which reduce to mere tree-level constants. This
approximation also implies that effectively only a single condensate term
contributes at each $d$. For the gluon condensate correction at $d=4$, we use
the common renormalization scheme invariant definition

$\langle\bar{\cal
O}_{4,0}\rangle\,=\,\frac{2\pi^{2}}{3}\,\langle\Omega|\tilde{\beta}(\alpha_{s})\,G^{\mu\nu}G_{\mu\nu}|\Omega\rangle\,\equiv\,\frac{2\pi^{2}}{3}\,\langle\bar{G}^{2}\rangle\,,$
(7)

where
$\tilde{\beta}(\alpha_{s})\equiv-2\beta(\alpha_{s})/\beta_{0}\alpha_{s}=\alpha_{s}/\pi+\ldots$.
The ${\cal O}(\alpha_{s})$ correction in the Wilson coefficient
$C_{4,0}(\alpha_{s}(-s))$ is known Chetyrkin:1985kn . The latter has sometimes
been included in phenomenological analyses, see e.g. Refs. Davier:2013sfa ;
Boito:2011qt . However, the logarithmic $s$ dependence introduced by this
correction leads to rather small numerical effects for gluon condensate
suppressed (GCS) spectral function moments,333For the GCS spectral function
moments the polynomial weight functions $w(x)$ ($W(x)$) do not contain a
linear term $x$ (quadratic term $x^{2}$). In all recent high-precision
$\alpha_{s}$ determinations from spectral function moments GCS spectral
function moments were employed due to their better perturbative behavior and
to suppress the rather large impact of the gluon condensate correction which
is currently not known with good precision Gubler:2018ctz . so that the
approximation of a constant Wilson coefficient is quite good for the gluon
condensate and has therefore been adopted in most recent spectral function
moment analyses. This motivates the common assumption that such an
approximation (including the neglect of anomalous dimensions) is also reliable
for $d\geq 6$ and that it does not affect the $\alpha_{s}$ determinations, see
e.g. Refs. Boito:2011qt ; Maltman:2008nf , but no strict proof of this
assumption is available. Finally, it is also customary to write the OPE in
terms of the correlators $\Pi_{V/A}^{(1+0)}$ (instead of the Adler function).
In this simplified form, the Wilson coefficients and the respective effective
condensates with dimension $d$ are collectively represented by a constant
$\mathcal{C}_{d}$ and one has

$\Pi_{V/A}^{\rm
OPE}\,=\,\frac{1}{12}\frac{\langle\bar{G}^{2}\rangle}{s^{2}}\,+\,\sum_{d=6}^{\infty}\,\frac{\mathcal{C}_{d}^{V/A}}{(-s)^{d/2}}\,,$
(8)

which gives

$D_{V/A}^{\rm
OPE}\,=\,\frac{2\pi^{2}}{3}\frac{\langle\bar{G}^{2}\rangle}{s^{2}}\,+2\pi^{2}\,\sum_{d=6}^{\infty}\,d\frac{\mathcal{C}_{d}^{V/A}}{(-s)^{d/2}}\,.$
(9)

In this form, Cauchy’s theorem for Eq. (5) implies that a monomial term
$x^{m}$ in the weight function $w(x)$ ($W(x)$) picks up solely the OPE
correction with dimension $d=2(m+1)$ ($d=2m$). For GCS moments the gluon
condensate correction term vanishes.

The DV component of the spectral functions, $\rho_{V/A}^{\rm(DV)}(s)$, cannot
be obtained from first principles, but, for sufficiently large $s$, it can be
parametrized under generally accepted assumptions about the QCD spectrum
(Regge behaviour and large-$N_{c}$ limit considerations) Cata:2005zj ;
Cata:2008ye ; Boito:2017cnp . Exploiting the analyticity properties of the
correlators, it is possible to write the contributions from DV effects as an
integral over the respective spectral function contribution. Since the
approximate form for the DVs is only valid for sufficiently large $s$, away
from the resonance peaks, this contribution is then written as an integral for
$s>s_{0}$ as Cata:2008ye

$\delta_{w,V/A}^{({\rm DV})}=-8\pi^{2}\int_{s_{0}}^{\infty}\frac{{\rm
d}s}{s_{0}}w({\textstyle\frac{s}{s_{0}}})\rho_{V/A}^{\rm(DV)}(s).$ (10)

In this work, when implementing the analysis setup of Ref. Boito:2020xli , the
DV contribution to the spectral functions will be parametrized as

$\rho_{V/A}^{\rm(DV)}(s)=e^{-\delta_{V/A}-\gamma_{V/A}s}\sin\left(a_{V/A}+b_{V/A}s\right),$
(11)

which introduces four additional parameters per channel: $\delta_{V/A}$,
$\gamma_{V/A}$, $a_{V/A}$, and $b_{V/A}$. Following the work of Ref.
Boito:2017cnp , corrections to this ansatz arise in the form of logarithms and
$1/s$ suppressed terms (see also Eq. (3.9) of Ref. dEnterria:2022hzv ). These
corrections have so far not been included in phenomenological analyses.

###  2.2 Renormalon-Free GC Scheme for the Adler Function

Employing the $C$-scheme for the strong coupling Boito:2016pwf  and in the
common approach of using a vanishing IR cutoff, the perturbative series for
the Adler function can be written in the form

$\displaystyle\hat{D}(s)$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\infty}\,\bar{c}_{\ell}\,\bar{a}^{\ell}(-s)\,,$
(12)

where ($\beta_{0}=11-2n_{f}/3$, $n_{f}=3$)

$\displaystyle\bar{c}_{\ell}\,\equiv\,\frac{4^{\ell}\,\bar{c}_{\ell,1}}{\beta_{0}^{\ell}}\,,\qquad\bar{a}(\mu^{2})\,\equiv\,\frac{\beta_{0}\,\bar{\alpha}_{s}(\mu^{2})}{4\pi}\,.$
(13)

The perturbative coefficients are known to ${\cal O}(\alpha_{s}^{4})$ (5-loop)
Baikov:2008jh  and read

$\displaystyle\bar{c}_{1,1}$ $\displaystyle=$ $\displaystyle
1\,,\quad\bar{c}_{2,1}\,=\,\mbox{$\frac{299}{24}$}-9\zeta_{3}\,=\,1.640\,,$
$\displaystyle\vbox{\vskip 17.07164pt}\bar{c}_{3,1}$ $\displaystyle=$
$\displaystyle\mbox{$\frac{262955}{1296}$}-\mbox{$\frac{779}{4}$}\zeta_{3}+\mbox{$\frac{75}{2}$}\zeta_{5}\,=\,7.682\,,$
$\displaystyle\vbox{\vskip 17.07164pt}\bar{c}_{4,1}$ $\displaystyle=$
$\displaystyle\mbox{$\frac{357259199}{93312}$}-\mbox{$\frac{1713103}{432}$}\zeta_{3}+\mbox{$\frac{4185}{8}$}\zeta_{3}^{2}+\mbox{$\frac{34165}{96}$}\zeta_{5}-\mbox{$\frac{1995}{16}$}\zeta_{7}\,=\,61.060\,.$
(14)

As is customary in state-of-the-art analyses of hadronic $\tau$ decay spectral
function moments, we in addition employ an estimate for the 6-loop coefficient
$\bar{c}_{5,1}$ Benitez-Rathgeb:2022yqb ,

$\bar{c}_{5,1}\,=\,345.4774\pm 140\,,$ (15)

which covers all the estimates obtained in the recent literature by dedicated
analyses Boito:2018rwt ; Baikov:2008jh ; Beneke:2008ad ; Caprini:2019kwp ;
Jamin:2021qxb . For the coefficients $\bar{c}_{\ell}$ this gives
$\bar{c}_{1}=0.444$, $\bar{c}_{2}=0.324$, $\bar{c}_{3}=0.674$,
$\bar{c}_{4}=2.382$, $\bar{c}_{5}=5.991\pm 2.428$. We note that the inclusion
of the coefficient $\bar{c}_{5,1}$ (and its uncertainty estimate) is not at
all a minor aspect of the phenomenological analyses, as it leads to a
noticeable reduction of the renormalization scale uncertainties. In this work
we adopt the estimate of Eq. (15).

The perturbative contributions to the spectral function moments are then
determined in analogy to Eq. (5) via the contour integral

$\displaystyle\delta^{(0)}_{w}(s_{0})\,=\,\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm
d}x}{x}\,W(x)\,\hat{D}(xs_{0})\,.$ (16)

For the CIPT method Pivovarov:1991rh  to determine the perturbation series
$\delta^{(0)}_{w}(s_{0})$ the expansion of the Adler function shown in Eq.
(12) is used, and the contour integration is carried out over powers of the
complex-valued strong coupling $\alpha_{s}(-s)$. The CIPT expansion series
arises from truncating the partial sum in Eq. (12). For the FOPT method the
Adler function shown in Eq. (12) is first reexpanded in powers of
$\alpha_{s}(s_{0})$. The contour integration is carried out over the resulting
polynomials of $\ln(-s/s_{0})$ which multiply the powers of
$\alpha_{s}(s_{0})$. The FOPT expansion series arises from truncating the
partial sum in powers of $\alpha_{s}(s_{0})$. From the perspective of the FOPT
expansion, the CIPT method resums powers of $\ln(e^{\pm i\pi})=\pm i\pi$
terms. For the implementation of renormalization scale variations the CIPT and
FOPT methods are actually expansions in powers of $\alpha_{s}(-\xi s)$ and
$\alpha_{s}(\xi s_{0})$, respectively. In our analysis we in general use the
canonical variation interval $1/2\leq\xi\leq 2$.

In the common approach of using a vanishing IR cutoff the coefficients
$\bar{c}_{\ell}$ contain IR renormalon contributions which cause the Adler
function perturbation series to be asymptotic. The different kinds of
divergent contributions are compensated by associated divergent order-by-order
contributions in the condensates of the OPE corrections shown in Eq. (6). We
call this IR factorization scheme for the condensates $\overline{\rm MS}$,
which is indicated by the bar over the condensate operators. The explicit form
of the GC OPE correction is

$\displaystyle\delta D^{\rm OPE}_{4,0}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\frac{2\pi^{2}}{3}\,\Big{[}1-\frac{22}{81}\,\bar{a}(-s)+\ldots\,\Big{]}\,\langle\bar{G}^{2}\rangle\,,$
(17)

where we have shown the ${\cal O}(\alpha_{s})$ corrections to the Wilson
coefficient in the renormalization-scale invariant notation of the GC matrix
element $\langle\bar{G}^{2}\rangle$. The yet unknown higher order corrections
are indicated by the ellipses. The analytic order-dependent form of the IR
renormalon contributions contained in the coefficients $\bar{c}_{\ell}$
associated to the GC in the $\overline{\rm MS}$ scheme can be unambiguously
quantified using the renormalon calculus and can be obtained from the
coefficients of the asymptotic series expression

$\delta\hat{D}_{4,0}(s)\,=\,\frac{2\pi^{2}}{3}\,N_{g}\,[1-\frac{22}{81}\,\bar{a}(-s)+\ldots]\,\sum_{n=1}^{\infty}\,r_{n}^{(4,0)}\,\bar{a}(-s)^{n}\,,$
(18)

where

$r_{\ell}^{(4,0)}=\Big{(}\frac{1}{2}\Big{)}^{\ell+4\hat{b}_{1}}\,\frac{\Gamma(\ell+4\hat{b}_{1})}{\Gamma(1+4\hat{b}_{1})}\,,$
(19)

and $\hat{b}_{1}=\beta_{1}/2\beta_{0}^{2}$, with $\beta_{1}=102-38\,n_{f}/3$
being the two-loop QCD $\beta$-function coefficient. In the $C$-scheme for the
strong coupling the form of $r_{\ell}^{(4,0)}$ is exact. The order-dependence
proportional to $2^{-\ell}\Gamma(\ell+4\hat{b}_{1})$ precisely characterizes
the divergent asymptotic contributions for the ${\cal O}(\Lambda_{\rm
QCD}^{4})$ renormalon associated to the GC OPE correction. The only unknown is
the exact value of the normalization factor $N_{g}$, since it cannot be
inferred from the renormalon calculus.

The purpose of the RF GC scheme we devised in Part I Benitez-Rathgeb:2022yqb
is to eliminate the contribution of the GC renormalon shown in Eq. (18) order-
by-order from the perturbative coefficients $\bar{c}_{\ell}$ of the Adler
function through a scheme change of the GC matrix element while maintaining
its scale-invariance and its natural ${\cal O}(\Lambda_{\rm QCD}^{4})$
scaling. This is achieved in a first step by the rewriting the $\overline{\rm
MS}$ GC $\langle\bar{G}^{2}\rangle$ in terms of the renormalon-free GC matrix
element $\langle G^{2}\rangle(R^{2})$ which depends on the IR factorization
scale $R$:

$\langle\bar{G}^{2}\rangle\,\equiv\,\langle
G^{2}\rangle(R^{2})\,-\,R^{4}\,\sum\limits_{\ell=1}^{\infty}\,N_{g}\,r_{\ell}^{(4,0)}\,\bar{a}^{\ell}(R^{2})\,.$
(20)

Here the $\overline{\rm MS}$ GC $\langle\bar{G}^{2}\rangle$ has a role
analogous to the pole mass in heavy quark physics, while the renormalon-free
GC matrix element $\langle G^{2}\rangle(R^{2})$ has the role analogous to a
short-distance mass Hoang:2020iah ; Beneke:2021lkq . In fact, the definition
of $\langle G^{2}\rangle(R^{2})$ is in close analogy to the RS heavy quark
mass scheme proposed in Ref. Pineda:2001zq . Upon insertion in Eq. (17) the
series on the RHS of Eq. (20) is combined with the original Adler function
series in Eq. (12). For the IR factorization scale $R$ the proper scaling is
$R^{2}\lesssim m_{\tau}^{2}$ for the application to $\tau$ hadronic spectral
function moments to ensure that it is still in the perturbative region and to
avoid the appearance of large logarithms $\ln(R^{2}/s_{0})$. For the
systematic order-by-order cancellation of the divergent renormalon
contributions in the Adler function it is essential that the combined series
is consistently expanded using the strong coupling at the same renormalization
scale either using the CIPT or FOPT renormalization scale prescription
described after Eq. (16).

The GC matrix element $\langle G^{2}\rangle(R^{2})$ is scale-dependent and has
${\cal O}(R^{4})$ scaling, which renders it not very convenient, since the
truncated series values can strongly depend on the values of $R$ and $N_{g}$.
This is remedied in a second step by rewriting $\langle G^{2}\rangle(R^{2})$
in terms of the scale-invariant GC matrix element $\langle G^{2}\rangle^{\rm
RF}$,

$\displaystyle\langle G^{2}\rangle(R^{2})$ $\displaystyle\equiv$
$\displaystyle\langle G^{2}\rangle^{\rm RF}+N_{g}\,\bar{c}_{0}(R^{2})\,,$ (21)

where the function $\bar{c}_{0}(R^{2})$ satisfies the same $R$-evolution
equation as the subtraction series on the RHS in Eq. (20) and thus of $\langle
G^{2}\rangle(R^{2})$. Interestingly, since the divergent subtraction series is
associated to a pure ${\cal O}(\Lambda_{\rm QCD}^{4})$ renormalon ambiguity,
its $R$-derivative is a convergent series (within its radius of convergence)
which can be summed exactly in closed form Benitez-Rathgeb:2022yqb ,

$\displaystyle\frac{{\rm d}}{{\rm d}\ln
R^{2}}\,\bigg{[}\,R^{4}\,\sum\limits_{\ell=1}^{\infty}\,N_{g}\,r_{\ell}^{(4,0)}\,\bar{a}^{\ell}(R^{2})\,\bigg{]}$
$\displaystyle=$ $\displaystyle\frac{{\rm d}}{{\rm d}\ln R^{2}}\,\langle
G^{2}\rangle(R^{2})\,=\,\frac{N_{g}}{2^{4\hat{b}_{1}}}\,\frac{R^{4}\,\bar{a}(R^{2})}{1-2\hat{b}_{1}\bar{a}(R^{2})}\,.$
(22)

A very suitable choice for $\bar{c}_{0}(R^{2})$ is the Borel sum of the
subtraction series itself using the common principle value prescription for
the inverse Borel transform integration,

$\bar{c}_{0}(R^{2})\equiv R^{4}\,\,\,{\rm PV}\,\int_{0}^{\infty}\\!\frac{{\rm
d}u\,\,e^{-\frac{u}{\bar{a}_{R}}}}{(2-u)^{1+4\hat{b}_{1}}}=-\frac{R^{4}\,e^{-\frac{2}{\bar{a}(R^{2})}}}{(\bar{a}(R^{2}))^{4\hat{b}_{1}}}\,{\rm
Re}\left[\,e^{4\pi\hat{b}_{1}i}\,\Gamma\Big{(}\\!-4\hat{b}_{1},-\frac{2}{\bar{a}(R^{2})}\,\Big{)}\,\right]\,,$
(23)

where the analytic result of Eq. (23) is valid for real-valued
$\bar{a}(R^{2})$. Here, $R^{4}/(2-u)^{1+4\hat{b}_{1}}$ is the Borel
(transform) function of the subtraction series on the RHS of Eq. (20) for
$N_{g}=1$ and the integral over the GC renormalon branch cut for $u\geq 2$ is
defined using the average of the contour deformation above and below the real
axis, referred to as the principal value (PV) prescription. In total, the
expression for the Adler function in the RF GC scheme has the form

$\displaystyle\hat{D}^{\rm RF}(s,R^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\,\Big{[}1+\bar{c}_{4,0}^{(1)}\bar{a}(-s)\Big{]}\,\frac{2\pi^{2}}{3}\,N_{g}\,\bar{c}_{0}(R^{2})\,+\,\sum_{\ell=1}^{\infty}\,\bar{c}_{\ell}\,\bar{a}^{\ell}(-s)\,\qquad$
(24)
$\displaystyle\,-\,\Big{[}1+\bar{c}_{4,0}^{(1)}\,\bar{a}(-s)\Big{]}\,\frac{2\pi^{2}}{3}\,N_{g}\,\frac{R^{4}}{s^{2}}\sum\limits_{\ell=1}^{\infty}\Big{(}\frac{1}{2}\Big{)}^{\ell+4\hat{b}_{1}}\,\frac{\Gamma(\ell+4\hat{b}_{1})}{\Gamma(1+4\hat{b}_{1})}\,\bar{a}^{\ell}(R^{2})\,,$

where the first term involving $\bar{c}_{0}(R^{2})$ is treated strictly as a
tree-level term (i.e. not being reexpanded again and numerically evaluated in
the $C$-scheme for the strong coupling). To obtain the perturbation series,
either using the CIPT or the FOPT renormalization scale setting prescription,
it is mandatory to consistently expand and truncate the sum of the two series
in $\bar{a}^{\ell}(-\xi s)$ and $\bar{a}^{\ell}(\xi s_{0})$, i.e. using the
strong coupling at a common renormalization scale. This ensures the systematic
removal of the GC renormalon from the Adler function. The GC OPE correction
adopts the standard form

$\displaystyle\delta D^{\rm OPE,RF}_{4,0}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\frac{2\pi^{2}}{3}\,\Big{[}1-\frac{22}{81}\,\bar{a}(-s)\Big{]}\,\langle
G^{2}\rangle^{\rm RF}\,.$ (25)

The RF scheme entails that the difference between the original $\overline{\rm
MS}$ GC and our new scale-independent RF GC, $\langle G^{2}\rangle^{\rm RF}$,
is formally $\mathcal{O}(\bar{a}_{R}^{n+1})$. In this sense, the modifications
to the GC in the RF scheme are minimal. Furthermore, the $R$-dependence of
$\hat{D}^{\rm RF}(s,R^{2})$ vanishes in the limit of large truncation order
and the Borel sum of $\hat{D}^{\rm RF}(s,R^{2})$ (defined with the PV
prescription above) agrees with the one of the original $\overline{\rm MS}$ GC
scheme Adler function $\hat{D}(s)$. This means that the RF GC matrix element
$\langle G^{2}\rangle^{\rm RF}$ is defined with respect to the Borel sum in
the PV prescription, a scheme that has been considered in the literature
before Caprini:2009vf ; Lee:2010hd ; Caprini:2011ya ; Bali:2014sja ;
Ayala:2020pxq ; Hayashi:2021vdq ; Abbas:2013usa , but has not been realized in
the context of the $\tau$ hadronic spectral function moments in the FOPT and
CIPT expansions prior to Ref. Benitez-Rathgeb:2022yqb .

We note that while the dependence on the renormalon norm in Eq. (20) is an
aspect the RF GC scheme shares with the renormalon-subtraction (RS) approach
proposed in Ref. Pineda:2001zq , the additional constructive element of Eq.
(21), which renders the RF GC to be scale-invariant, provides the practical
advantage that $\langle G^{2}\rangle^{\rm RF}$ is asymptotically independent
of the value of $N_{g}$. This means that the value of $\langle
G^{2}\rangle^{\rm RF}$ is more stable with respect to uncertainties in the
value of $N_{g}$.

##  3 Determination of the GC Renormalon Norm

In Part I Benitez-Rathgeb:2022yqb  we have demonstrated the effectiveness of
the RF GC scheme in the context of concrete Borel function models for the all-
order perturbation series of the Adler function $\hat{D}(s)$. For these models
the value of the GC renormalon norm $N_{g}$ was known exactly. We also
considered a multi-renormalon model for the Adler function in full QCD,
following the construction of Beneke and Jamin Beneke:2008ad , which uses as
input the known 5-loop coefficients of Eq.(2.2) and the central value for
$\bar{c}_{5,1}$ in Eq. (15). The Borel function model was updated to account
for the now known 5-loop coefficient of the QCD $\beta$-function and the use
of the $C$-scheme, see App. A of Part I. The construction principle of the
Beneke-Jamin model is based on the natural assumption that the IR and UV
renormalon terms closest to the origin are sufficient to achieve a realistic
description of the true Borel function. Because the known perturbative
corrections to Adler function are fully consistent with a sizeable GC
renormalon norm, the model predicts a sizeable value for $N_{g}$. We obtained
the concrete value $N_{g}=0.64$ in Part I. (See App. A for how our renormalon
norm convention is related to others employed in the recent literature.)

This result and the constructive elements of the Beneke-Jamin model are
compatible with the proposition that the GC renormalon gives a sizeable
contribution to the known QCD corrections of the Adler function at the 5-loop
level. In this section we apply this proposition to also quantify the norm
$N_{g}$ with a realistic uncertainty. The uncertainty in $N_{g}$ will then
enter our phenomeological analyses in Sec. 4 as an additional parametric
error. We carry out our analysis for $n_{f}=3$ active flavors, which is the
flavor number relevant for the analysis of $\tau$ hadronic spectral functions.

In the following three subsections we use three different methods to determine
$N_{g}$. In Sec. 3.1 we discuss different modifications of the multi-
renormalon model compatible with the naturalness assumption mentioned above to
assess the stability of $N_{g}$ under these variations. In Sec. 3.2 we examine
the ‘conformal mapping approach’ introduced by Lee to determine $N_{g}$ for
$n_{f}=3$ active flavors in Ref. Lee:2011te . He used the known Taylor
expansion terms of the Euclidean Adler function’s Borel function
$B[\hat{D}](u)$ at 5 loops and a conformal mapping to relocate the gluon
condensate renormalon branch point originally located at $u=2$ into the unit
circle such that it becomes the branch point closest to the origin. This
allows to determine the norm through a Taylor expansion in the mapped variable
around the origin. We show that the concrete form of Lee’s conformal
transformation exhibits a rather slow convergence and substantially
undershoots the norm at the 5 loop level. We devise improved conformal
transformations following the works of Cvetic:2001sn ; Contreras:2002kf ;
Campanario:2005np ; Caprini:2009vf , where the convergence is accelerated and
the results at 5 loops are more reliable. Finally, in Sec. 3.3 we discuss a
novel method, called ‘optimal subtraction method’, where $N_{g}$ is determined
from a minimization procedure that encodes the two improvements the RF GC
scheme achieves for FOPT and CIPT expansion series for GCS and GCE spectral
function moments. These improvements are that the CIPT-FOPT discrepancy for
GCS moments is removed and that the perturbative convergence for GCE moments
is substantially improved Benitez-Rathgeb:2022yqb . We find that all three
methods provide consistent estimates for $N_{g}$ compatible with $0.64$ and
with uncertainties of around $30$-$40\%$. We adopt the outcome of the optimal
subtraction method, which is given in Eq. (37), as our final result for
$N_{g}$ because it does not rely on the estimate of the 6-loop coefficient
$\bar{c}_{5,1}$.

Interestingly, in the quenched approximation the GC renormalon norm
$N_{g}^{(n_{f}=0)}$ was determined previously in two dedicated analyses by Lee
Lee:2011te  and Bali et al. Bali:2014fea . Since our methods can also be
applied in the quenched approximation and because these two analyses quoted
incompatible results, we take the opportunity to apply our three methods to
also determine $N_{g}^{(n_{f}=0)}$. The results are given and discussed in App
B. In App. C we also show that the incomplete approximate knowledge on the
description of the large-order asymptotic behavior of the gluon condensate
renormalon series in the lattice strong coupling scheme may be the reason, why
the result for $N_{g}^{(n_{f}=0)}$ obtained by Bali et al. is so much larger
than the one obtained by Lee.

###  3.1 Multi-Renormalon Borel Function Model Approach

Even though the concrete form of the Borel function of the perturbative Adler
function $B(u)$ is unknown, the structure of the non-analytic contributions
related to IR and UV renormalons is known due to a one-to-one association to
the terms in the OPE (see Eq. (6)) for IR renormalons and to higher-
dimensional operator insertions related to integrated out short-distance
quantum corrections for UV renormalons. To the extent that the QCD
$\beta$-function, the anomalous dimensions of the operators and the Wilson
coefficients are known, the associated non-analytic contributions can be
quantified unambiguously fixing the order-by-order relative behavior of the
renormalon contributions in the perturbative coefficients Gross:1974jv ;
tHooft:1977xjm ; David:1983gz ; Mueller:1984vh ; Beneke:1998ui , see also Sec.
2.3 in Part I. The Borel function contains a linear combination of such IR and
UV renormalon terms plus terms that are fully analytic in the
$u$-plane.444There are other non-analytic terms such as instanton
contributions, which have negligible numerical effects and are therefore not
relevant for the discussions here. The non-analytic branch points for IR (UV)
renormalons are located on the positive (negative) real $u$-axis and related
to equal-sign (sign-alternating) asymptotic series contribution. The analytic
term is related to a series contribution with a finite radius of convergence.
Non-analytic renormalon terms with branch points further away from the origin
are associated to high-dimensional and more power-suppressed contributions.
The information that is a priori unknown is the normalization of the different
non-analytic IR or UV contributions.

The previously mentioned proposition and naturalness assumption implies that
the renormalon terms closest to the origin in the Borel plane are sufficient
to achieve a good approximate description of the Adler function’s Borel
function and that the known perturbative coefficients of the Adler function
are sufficient to approximately determine their normalization. The ‘multi-
renormalon’ model construction devised by Beneke and Jamin Beneke:2008ad
follows this proposition. The model includes a term related to the gluon
condensate OPE correction displayed in Eq. (17), which has a branch point at
$u=2$ and is the IR renormalon located closest to the origin. Furthermore it
contains a term associated to the dimension-6 OPE correction according to the
simplified expression in Eq. (9) with a branch point at $u=3$ as well as a
term accounting for the leading UV renormalon with a branch point at $u=-1$
and an anomalous dimension consistent with the leading double-pole structure
known from the large-$\beta_{0}$ approximation Broadhurst:1992si  (see Sec. 4
of part I). Since the QCD series terms for the Euclidean Adler function up to
${\cal O}(\alpha_{s}^{4})$ do not show any significant sign alternation in
either the $\overline{\rm MS}$ or $C$ schemes for the strong coupling, the
normalization of the UV renormalons comes out very small, so that the
inclusion of the leading UV renormalon term is sufficient Beneke:2008ad . The
model is supplemented by a linear polynomial in $u$ to account for the
analytic contribution.

In the analyses of Borel function models of the Adler function $\hat{D}(s)$
Beneke:2008ad ; Beneke:2012vb ; Jamin:2021qxb  it is common practice to
consider the Borel (transform) function, $B[\hat{D}(s)](u)$, to be defined
with respect to the coupling for the renormalization scale $\mu^{2}=-s$. We
adopt this convention here as well and furthermore use the $C$-scheme for the
strong coupling. We also refer the reader to App. A for the precise definition
of the Borel function we employ in this work (and its relation to other
conventions used in the literature) and to Ref. Boito:2016pwf  for the
$C$-scheme.

In this context the Beneke-Jamin model (which was employed in this form in
Part I) adopts the concrete form555 Through integration-by-parts the ${\cal
O}(\bar{a}^{n})$ term of the form $\frac{\bar{a}^{n}}{(p-u)^{\gamma}}$ in the
Borel function is equivalent to a term of the form
$\frac{\Gamma(\gamma-n)}{\Gamma(\gamma)(p-u)^{\gamma-n}}-\sum_{i=0}^{n-1}p^{n-\gamma-i}\frac{\Gamma(\gamma-n+i)}{\Gamma(\gamma)\Gamma(1+i)}u^{i}$,
which is an alternative notation frequently used in the literature.

$\displaystyle\begin{split}B[\hat{D}(s)]_{\rm
mr}(u)&=b^{(0)}+b^{(1)}u+\frac{2\pi^{2}}{3}\frac{N_{g}\left[1-\frac{22}{81}\bar{a}(-s)\right]}{(2-u)^{1+4\hat{b}_{1}}}+\frac{N_{6}}{(3-u)^{1+6\hat{b}_{1}}}+\frac{N_{-2}}{(1+u)^{2-2\hat{b}_{1}}}\,.\end{split}$
(26)

The three renormalon norms $N_{g}$, $N_{6}$ and $N_{-2}$ and the two
polynomial coefficients $b^{(0,1)}$ are then determined from the Adler
function coefficients up to $\bar{c}_{5,1}$, which imply that the first 5
terms of the Taylor series for $B[\hat{D}(s)]_{\rm mr}(u)$ read
$\sum_{n=1}^{5}\bar{c}_{n}/\Gamma(n)u^{n-1}$, see Eq. (13). Using
$\bar{c}_{5,1}\,=\,345.477$ and $\hat{b}_{1}=32/81=0.395$ we obtain
$N_{g}=0.64$, $N_{6,0}=-15.65$, $N_{-2}=-0.027$, $b^{(0)}=0.154$ and
$b^{(1)}=0.008$, which is the set of values related to the GC norm value
mentioned above in the introduction. It is the idea of the ‘renormalon model
approach’ to determine $N_{g}$ with an uncertainty by considering several
modifications in the construction of $B[\hat{D}(s)]_{\rm mr}(u)$ that are
consistent with the central proposition and the naturalness assumption.

Let us first consider the impact of the uncertainty on the 6-loop coefficient
$\bar{c}_{5,1}$, see Eq. (15). Changing the coefficient by $\pm 140$ we obtain
$N_{g}=0.64\pm 0.27$ which corresponds to a relative variation of $43\%$. This
uncertainty is irreducible in the context of only having estimates for the
6-loop coefficient $\bar{c}_{5,1}$ using the Borel model approach. It is now
interesting examining how the uncertainties related to the structure of the
model itself can affect the value of $N_{g}$. Let us study the possible impact
of the ${\cal O}(\bar{a}^{2})$ correction in the Wilson coefficient of the GC
term. Given that the known ${\cal O}(\bar{a})$ correction has the coefficient
$-22/81\approx 0.27$ and that the perturbative coefficients $\bar{c}_{\ell}$
of the Adler function are very nicely behaved with coefficients well below
$0.5$ up to ${\cal O}(\bar{a}^{2})$, see the text below Eq. (15), it is
reasonable to assume that the size of the yet unknown ${\cal O}(\bar{a}^{2})$
coefficient does not exceed $0.5$ as well. We thus consider a modification of
the GC term shown in Eq. (26) by considering a Wilson coefficient of the form
$[1-\frac{22}{81}\bar{a}(-s)+\delta\,\bar{a}^{2}(-s)]$ with
$-0.5\leq\delta\leq 0.5$. This leads to the result
$N_{g}=0.64^{+0.11}_{-0.08}$ which corresponds to a relative variation of
$+18\%$ and $-13\%$. These variations are much smaller than that coming from
$\bar{c}_{5,1}$. Using the modifications $[1+\delta\,\bar{a}(-s)]$ for the
Wilson coefficient of the $d=6$ ($N_{6}$) IR renormalon and the UV renormalon
($N_{-2}$) terms with the same $\delta$-variation, we obtain
$N_{g}=0.64^{+0.09}_{-0.03}$ and $N_{g}=0.64^{+0.01}_{-0.01}$, which are even
smaller variations. Next, let us consider two other structural modifications
of the Borel model of Eq. (26). The fact that the linear coefficient $b^{(1)}$
for the default model of Eq. (26) is rather small indicates that that
structure of the renormalon terms already accounts nicely for all corrections
at ${\cal O}(\alpha_{s}^{2})$ and beyond. We therefore drop the linear term
$b^{(1)}u$ and include a $d=8$ renormalon term of the form
$N_{8}/(4-u)^{1+8\hat{b}_{1}}$. This yields $N_{g}=0.68$ which differs only by
$6\%$ from the norm obtained from the original model. If we include instead of
the $d=8$ renormalon term an additional $u=-2$ UV renormalon term
$N_{-4}/(2+u)^{2-4\hat{b}_{1}}$ we obtain a result for $N_{g}$ that only
differs by $0.5\%$.

We see that the unknown 6-loop coefficient $\bar{c}_{5,1}$ represents the
largest source of uncertainty of $\pm 40\%$ when determining the GC renormalon
norm $N_{g}$ using the Borel model method. Increasing the error in
$\bar{c}_{5,1}$ beyond the one adopted in Eq. (15) would further increase this
uncertainty.666 In Ref. Pich:2016bdg , see footnote 10, a much larger
uncertainty is used. This larger uncertainty is much more conservative but is
disfavoured by recent dedicated analyses Baikov:2008jh ; Boito:2018rwt ;
Caprini:2019kwp ; Jamin:2021qxb , and is also incompatible with our basic
proposition that the GC renormalon has a sizeable contribution to the known
5-loop correction, see Ref. Beneke:2008ad ; Jamin:2021qxb . Structural
uncertainties related to the form of the Borel, model related to the unkown
higher order corrections to the Wilson coefficients or to the form of the
renormalons that are accounted for, lead to much smaller effects. Among these
subleading sources of uncertainty, the still unknown two-loop correction to
the GC Wilson coefficient is the largest with around $15$ to $20\%$.
Modifications related to IR renormalon terms associated to condensates beyond
dimension-4 or to the UV renormalons (which are known to be small due to the
absense of sign alternating contributions) have significantly smaller effects.
These observations are consistent with the assumptions entailed in our
proposition and the naturalness of the Adler function’s renormalon structure.
It should be noted, however, that the possibilities to change the form of the
Borel model are certainly not exhausted with the modifications we discussed
above. Overall, we therefore consider the outcome of this analysis primarily
as a useful starting point and it is appropriate to consider also the two
alternative methods discussed in the following subsections.

###  3.2 Conformal Mapping Approach

The conformal mapping approach Lee:2011te  to determine the GC renormalon norm
$N_{g}$ uses the fact that the function

$\displaystyle\tilde{B}(u)\equiv\frac{3\,(2-u)^{1+4\hat{b}_{1}}}{2\pi^{2}}\,B[\hat{D}(s)](u)$
(27)

is analytic in the vicinity around the point $u=2$. It has, however, still
branch points at $u=3,4,\ldots$ from IR renormalons and at $u=-1,-2,\ldots$
from UV renormalons. The idea of the conformal mapping approach is that one
can apply conformal transformations $z=f(u)$ such that the origin is
unchanged, i.e. $f(0)=0$ and the point $z_{2}=f(2)$ is located within the
radius of convergence of the Taylor expansion of $\tilde{B}(f^{-1}(z))$ around
$z=0$ that is given by the point among the set of mapped branch points
$\\{f(3),f(4),\ldots,f(-1),f(-2),\dots\\}$ that is closest to the origin. It
is then possible to determine $N_{g}$ from that Taylor series evaluated at
$z=z_{2}$. The condition $f(0)=0$ ensures that the $n$-th term in this Taylor
expansion is determined from terms in the Adler function’s perturbation series
up to ${\cal O}(\alpha_{s}^{n})$, i.e. from
$\bar{c}_{1,1},\ldots,\bar{c}_{n,1}$. This means that the Taylor series can be
calculated based purely on perturbation theory and that there is no need to
consider the reconstruction of the Borel function. The only information needed
is the form of the most singular non-analytic structure associated to the GC
renormalon. The latter is known exactly from the renormalon calculus and reads
$1/(2-u)^{1+4\hat{b}_{1}}$. The conformal mapping approach is model-
independent in the sense that is does not rely on assumptions about the form
of the Borel function. However, it depends on the choice of the conformal
transformation and that the resulting series for $N_{g}$ is already close to
saturation for the available orders.

Figure 1:  Series for the $n_{f}=3$ GC renormalon norm $N_{g}$ up to order
$40$ obtained from the conformal mapping approach using the Borel model of Eq.
(26). The first five orders (indicated by the blue starts) come from the known
5-loop corrections quoted in Eqs. (2.2) and the central estimate for the
6-loop coefficient $\bar{c}_{5,1}$ of Eq. (15). Upper left panel: results
based on Lee’s mapping of Eq. (28). The other panels show results for the
mapping function $w(u,p)$ of Eq. (29) for $p=5,10,15$.

In his analysis Lee used the conformal mapping Lee:1996yk ; Lee:1999ws ;
Lee:2011te

$\displaystyle z=\frac{u}{(1+u)}\,,$ (28)

which leads to $z_{2}=2/3\approx 0.667$. Here the convergence radius is
$r_{c}=0.75$ and given by the $d=6$ renormalon branch point at $u=3$. This
results in the Taylor series $N_{g}=0.40-0.15-0.042+0.040+0.071=0.32$ at
${\cal O}(\alpha_{s}^{5})$ using the central value $\bar{c}_{5,1}=345$ of Eq.
(15) for the last term.777Lee carried out his analysis in the
${\overline{\mathrm{MS}}}$ scheme up to ${\cal O}(\alpha_{s}^{4})$ and
determined the GC renormalon norm of the Adler function. Accounting for the
conventional factor $2\pi^{2}/3$ in the GC OPE correction shown in Eq. (17) he
obtained ${\cal N}_{g}=0.048-0.018-0.008+0.001$ in his convention for the GC
renormalon norm. Using the estimate of Eq. (15), which corresponds to the
${\overline{\mathrm{MS}}}$ coefficient $c_{5,1}=280$ Benitez-Rathgeb:2021gvw ,
the ${\cal O}(\alpha_{s}^{5})$ term reads $+0.005$. In our convention for the
GC norm this corresponds to $N_{g}=0.40-0.15-0.065+0.008+0.043=0.24$. The
value of $z_{2}$ is quite close to the convergence radius, and the series does
not yet look to be close to a saturation. This suspicion can be substantiated
by testing the method using the multi-renormalon model of Eq. (26) with the
set of parameter values displayed in the text below that equation. Since the
model can be considered as a reasonable approximation to the full QCD Borel
function, we can check how fast the method converges using the conformal
mapping of Eq. (28). The result as a function of order $n$ is shown in the
left upper panel of Fig. 1. We see that the series is converging to the
correct value of $N_{g}$. However, the series saturates towards $N_{g}=0.64$
only for orders $n\gtrsim 40$. For orders $n\lesssim 5$ (shown with blue stars
in Fig. 1) the series significantly undershoots the correct model value. This
is far from satisfying and motivates considering an alternative conformal
transformation that leads to an improved saturation behavior.

A class of conformal transformations that turns out to be much more suitable
is given by

$\displaystyle
w(u,p)=\frac{\sqrt{1+u}-\sqrt{1-\frac{u}{p}}}{\sqrt{1+u}+\sqrt{1-\frac{u}{p}}}\,,$
(29)

where $p$ is a free parameter. This mapping has been employed for specific
values of $p$ in this and other contexts in Refs. Caprini:1998wg ;
Cvetic:2001sn ; Cvetic:2001ws  and many other works thereafter. We apply it
here for the conformal mapping approach to determine $N_{g}$. For the values
$p=(5,10,15)$ we obtain $z_{2}=(0.38,0.32,0.30)$ and a convergence radius
$r_{c}=(0.52,0.41,0.38)$, which provides an evaluation of the Taylor series
closer to the origin and also has a smaller ratio of $z_{2}/r_{c}$. In the
upper right and the lower panels of Fig. 1 we display the convergence of the
Taylor series for $N_{g}$ for $p=5,10,15$ for the Borel model already
considered for Lee’s mapping. We find a substantially improved convergence
behavior. The series saturate towards $N_{g}=0.64$ already for orders
$n\gtrsim 20$. At ${\cal O}(\alpha_{s}^{4})$ and ${\cal O}(\alpha_{s}^{5})$,
the series values are already within $30\%$ and $10\%$, respectively, of the
correct model value, were the ${\cal O}(\alpha_{s}^{4})$ result is always
below the true value. Furthermore, the model values coming from orders beyond
${\cal O}(\alpha_{s}^{5})$ do not deviate from $0.64$ by more than $20\%$. In
view of the uncertainty for $N_{g}$ from the multi-renormalon model approach
this is satisfactory.

|  $w(u,5)$ |  $w(u,10)$ |  $w(u,15)$
---|---|---|---
${\cal O}(\alpha_{s}^{4})$ |  $0.57$ |  $0.47$ |  $0.45$
${\cal O}(\alpha_{s}^{5})$ |  $0.72\pm 0.24$ |  $0.63\pm 0.17$ |  $0.60\pm 0.15$
Table 1:  Results for the $n_{f}=3$ GC renormalon norm $N_{g}$ at ${\cal
O}(\alpha_{s}^{4})$ and ${\cal O}(\alpha_{s}^{5})$ . The uncertainties at
order $5$ come from varying $\bar{c}_{5,1}$ within the uncertainty estimate
given in Eq. (15).

For our analysis we extract the value for $N_{g}$ at ${\cal
O}(\alpha_{s}^{4})$ and ${\cal O}(\alpha_{s}^{5})$, where for the latter we
include the uncertainty due to the error in $\bar{c}_{5,1}$ given in Eq. (15).
The results are shown in Tab. 1. Within the uncertainties the ${\cal
O}(\alpha_{s}^{5})$ results are nicely compatible with the ${\cal
O}(\alpha_{s}^{4})$ results for all values of $p$. Furthermore at ${\cal
O}(\alpha_{s}^{5})$ we obtain the smallest uncertainties for $p=15$. We have
checked that the results for the three $p$ values we have shown here are
representative for all values of $p>3$ and that, in particular, there is no
significant change with respect to the $p=15$ result for even larger values of
$p$. Overall, we find that the conformal mapping method confirms the result
for $N_{g}$ we obtained from the Borel model approach with respect to the
central value as well as the uncertainty.

###  3.3 Optimal Subtraction Approach

The third method to determine the GC renormalon norm $N_{g}$, which we call
‘optimal subtraction approach’, encodes the two major improvements the RF GC
scheme achieves over the previously used $\overline{\rm MS}$ GC scheme for the
$\tau$ hadronic spectral function moments: (1) the reduction of the CIPT-FOPT
discrepancy for GCS spectral function moments and (2) the improvement of the
badly behaved perturbation series for GCE spectral function moments. As we
have shown in Part I, for the proper choice of $N_{g}$ these two types of
improvements are realized simultaneously for any possible choice of GCS or GCE
spectral function moments.

It is the idea of the optimal subtraction approach to employ an optimization
procedure, based on a $\chi^{2}$-type minimization, which quantifies the
improvements (1) and (2) as a function of $N_{g}$. To explain the construction
of the $\chi^{2}$ function let us write FOPT (FO) and CIPT (CI) spectral
function moment expansion series for a given weight function $w(x)$ and
truncated at ${\cal O}(\alpha_{s}^{m})$ in the form

$\displaystyle\delta^{(0),{\rm
FO/CI}}_{w,m}(N_{g},s_{0};\alpha_{s}(s_{0}))\,=\,\sum_{n=0}^{m}r_{w,n}^{\rm
FO/CI}(N_{g},R^{2},\xi;s_{0},\alpha_{s}(s_{0}))\,,$ (30)

where the index $n$ counts the order in the $\overline{\rm MS}$ strong
coupling expansion (either in terms of $\alpha_{s}(\xi s_{0})$ or
$\alpha_{s}(-\xi s)$ prior to the contour integration),888We remind the reader
that in Sec. 2.2 we have formulated the RF GC scheme in the context of the
$C$-scheme for the strong coupling, but that all concrete phenomenological
analyses are carried out in the common $\overline{\rm MS}$ scheme. The values
for $\alpha_{s}(\xi s_{0})$, $\alpha_{s}(-\xi s)$ and $\bar{a}(R^{2})$ (the
latter to be used in the function $\bar{c}_{0}$) are obtained from the input
$\overline{\rm MS}$ value for $\alpha_{s}(m_{\tau}^{2})$ using the known
$\beta$-function coefficients up to 5 loops. which we collectively refer to as
${\cal O}(\alpha_{s}^{n})$. The known perturbative coefficients in Eq. (2.2)
uniquely quantify $r_{w,n}^{\rm FO/CI}$ for $n=1,2,3,4$, and $r_{w,5}^{\rm
FO/CI}$ is determined from the estimate (15) with an uncertainty. The
coefficients $r_{w,n}^{\rm FO/CI}$ depend on the subtraction scale $R$ and the
renormalization scaling parameter $\xi$. In the formal large order $n$ limit,
the dependence of the truncated sum $\delta^{(0),{\rm FO/CI}}_{w,m}$ on $R$
and $\xi$ formally vanishes due to renormalization group invariance, which is
the reason why we have suppressed the dependence on $R$ and $\xi$ as arguments
in $\delta^{(0),{\rm FO/CI}}_{w,m}$. For any finite order $n$, the truncated
sum has, however, a residual dependence on $R$ and $\xi$, as it is common in
perturbation theory. Note that the ‘tree-level’ term $r_{w,0}^{\rm FO/CI}$,
which arises in the RF GC scheme and is proportional to
$N_{g}\,\bar{c}_{0}(R^{2})$, is independent of the renormalization scale
parameter $\xi$.

Our $\chi^{2}$ function, which depends on the truncation order $m$ and is
constructed from a set of GCS and GCE spectral function moments, consists of
two additive parts

$\displaystyle\chi^{2}_{m}(N_{g})$ $\displaystyle=$
$\displaystyle\chi^{2}_{m,{\rm GCS}}(N_{g})+\chi^{2}_{m,{\rm GCE}}(N_{g})\,.$
(31)

To keep the expressions compact we have suppressed all arguments except for
$N_{g}$, but the $\chi^{2}$ functions also depends on $R$, $\xi$, $s_{0}$ as
well as $\alpha_{s}(s_{0})$. The first term provides a measure for the CIPT-
FOPT discrepancy at order $m$ for a set of GCS spectral function moments:

$\displaystyle\chi^{2}_{m,{\rm GCS}}(N_{g})=\sum_{i}\Big{(}\,\delta^{(0),{\rm
CI}}_{w_{i},m}(N_{g})-\delta^{(0),{\rm FO}}_{w_{i},m}(N_{g})\Big{)}^{2}\,.$
(32)

We remind the reader that for the GCS moments, the polynomial weight functions
$w_{i}$ do not contain a linear $x$ term and that the GC OPE correction is
strongly suppressed. For these spectral function moments, the CIPT as well as
FOPT expansions already provide well-behaved perturbation series even in the
$\overline{\rm MS}$ GC scheme. A contribution from a GCS moment to
$\chi^{2}_{m,{\rm GCS}}$ is small if the discrepancy between the truncated
CIPT and FOPT expansion series is small as well.

The second term provides a measure for the quality of convergence for a set of
GCE moments:

$\displaystyle\chi^{2}_{m,{\rm GCE}}(N_{g})=\sum_{i}\Big{(}\,r_{w_{i},m}^{\rm
FO}(N_{g})-r_{w_{i},m-1}^{\rm FO}(N_{g})\Big{)}^{2}\,.$ (33)

We remind the reader that for the GCE moments, the polynomial weight functions
$w_{i}$ contain a linear $x$ term (with a sizeable coefficient) and that GC
OPE correction is sizeable. For these spectral function moments, the CIPT as
well as FOPT expansion series in the $\overline{\rm MS}$ GC scheme are quite
badly behaved Beneke:2012vb , so the difference between order $m$ and order
$m-1$ series terms is sizeable. A contribution from a GCE moment to
$\chi^{2}_{m,{\rm GCE}}$ is small if the series is well behaved. Since the
CIPT expansion leads to a significant enhancement of the sign-alternation
behavior,999This property of the CIPT expansion is well-known and has been
pointed out in Refs. Beneke:2008ad ; Beneke:2012vb . We have also described
this behavior in Part I, see Fig. 3. which can upset artificially the
difference between the size of order $m$ and order $m-1$ series terms, we only
account for the FOPT series terms for the construction of $\chi^{2}_{m,{\rm
GCE}}$. The properties of the GCS and GCE moment series in the FOPT and CIPT
expansions either in the $\overline{\rm MS}$ or the RF GC scheme mentioned
above, which motivate the forms of $\chi^{2}_{m,{\rm GCS}}$ and
$\chi^{2}_{m,{\rm GCE}}$, have been discussed in detail in Sec. 5 of Part I
(see in particular Fig. 3).

For the construction of $\chi^{2}_{m,{\rm GCS}}$ we consider five
representative GCS weight functions defined from

$w_{n}(x)\equiv
w^{(2,n)}(x)=(1-x)^{2}\sum_{k=0}^{n}(k+1)x^{k}=1-(n+2)x^{n+1}+(n+1)x^{n+2}\,,$
(34)

for $n=1,2,3,4,5$, which were used in the phenomenological analysis in Sec.
5.3 of Ref. Pich:2016bdg . The first weight function in this sequence,
$w_{1}$, is the well-known kinematic weight function relevant for the
inclusive hadronic $\tau$ decay rate. The explicit expressions for the five
weight functions are

$\displaystyle w_{1}(x)$ $\displaystyle=$ $\displaystyle 1-3x^{2}+2x^{3}\,,$
$\displaystyle w_{2}(x)$ $\displaystyle=$ $\displaystyle 1-4x^{3}+3x^{4}\,,$
$\displaystyle w_{3}(x)$ $\displaystyle=$ $\displaystyle 1-5x^{4}+4x^{5}\,,$
$\displaystyle w_{4}(x)$ $\displaystyle=$ $\displaystyle 1-6x^{5}+5x^{6}\,,$
$\displaystyle w_{5}(x)$ $\displaystyle=$ $\displaystyle 1-7x^{6}+6x^{7}\,.$
(35)

They are linearly independent and all doubly pinched. For the construction of
$\chi^{2}_{m,{\rm GCE}}$ we consider moments obtained from the following
representative five GCE weight functions:

$\displaystyle w_{6}(x)$ $\displaystyle=$
$\displaystyle\frac{3}{2}(1-x)^{2}=\frac{3}{2}-3x+\frac{3x^{2}}{2}\,,$
$\displaystyle w_{7}(x)$ $\displaystyle=$
$\displaystyle(1-x)^{2}\left(\frac{13}{12}+\frac{5x}{3}\right)=\frac{13}{12}-\frac{x}{2}-\frac{9x^{2}}{4}+\frac{5x^{3}}{3}\,,$
$\displaystyle w_{8}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-x)^{2}\left(5-8x\right)=\frac{5}{2}-9x+\frac{21x^{2}}{2}-4x^{3}\,,$
$\displaystyle w_{9}(x)$ $\displaystyle=$
$\displaystyle(1-x)^{2}\left(\frac{3}{2}+x-3x^{2}+x^{3}\right)=\frac{3}{2}-2x-\frac{7x^{2}}{2}+8x^{3}-5x^{4}+x^{5}\,,$
$\displaystyle w_{10}(x)$ $\displaystyle=$
$\displaystyle(1-x)\left(1-\frac{x^{3}}{2}+\frac{3x^{4}}{4}\right)=1-x-\frac{x^{3}}{2}+\frac{5x^{4}}{4}-\frac{3x^{5}}{4}\,.$
(36)

They are linearly independent and doubly pinched as well (except for $w_{10}$
which is singly pinched), and we have adopted their form such that the
coefficients of the linear $x$ terms have different values that is still of
order one, i.e. neither too small nor too large. In any case, all 10 weight
functions are in principle also useful for phenomenological analyses with
suppressed DV corrections and we find compatible results for other choices of
moments. Note that the perturbative spectral function moment series associated
to the 10 weight functions (either in the FOPT or the CIPT expansion) all sum
to values at ${\cal O}(\alpha_{s}^{4})$ and ${\cal O}(\alpha_{s}^{5})$ very
close to the kinematic moment series. This ensures that using the same weight
for all moments contributing to the $\chi^{2}_{m}(N_{g})$ does not cause any
particular bias.

Figure 2:  Results for the $n_{f}=3$ GC renormalon norm $N_{g}$ up to order
$m=10$ obtained from the optimal subtraction approach using the Adler function
obtained by the Borel model of Eq. (26). The first five orders, up to ${\cal
O}(\alpha_{s}^{5})$, (indicated by the blue starts) come from the known 5-loop
corrections quoted in Eqs. (2.2) and the central estimate for the 6-loop
coefficient $\bar{c}_{5,1}$ of Eq. (15). The red error bars arise from varying
the IR factorization scale in the range $0.7\sqrt{s_{0}}\leq
R\leq\sqrt{s_{0}}$ and the stars and dots represent the average of the maximal
and minimal results at each order in perturbation theory. The blue error bar
at order $m=5$ is obtained by conservatively including, in addition to the $R$
variations, independent variations of $\bar{c}_{5,1}$ within the uncertainty
estimate of Eq. (15). Left panel: Results for $s_{0}=m_{\tau}^{2}$ and
$\xi=1$. Right panel: Results for $s_{0}=m_{\tau}^{2}$ and $\xi=2$.

To find the best value with an uncertainty for $N_{g}$ at the truncation order
$m$ we determine the minimum of $\chi^{2}_{m}(N_{g})$ for values of the
subtraction scale $R$ in the range $0.7\sqrt{s_{0}}\leq R\leq\sqrt{s_{0}}$.
For the uncertainty for $N_{g}$ we adopt half of the range of $N_{g}$ values
that is covered and we take the average of the maximum and minimum values as
the central value. As a test of the method, we show the outcome for this
analysis for $s_{0}=m_{\tau}^{2}$ up to truncation order $m=10$ in Fig. 2
using the series for the Adler function generated by the Borel model of Eq.
(26) for $\xi=1$ (left panel) and $\xi=2$ (right panel). We see that for
increasing $m$ the results from this method nicely converge to to correct
value for $N_{g}$ (for the Borel model this value is $0.64$ and is indicated
by the horizontal red line). We also see that for orders $m>6$ the method is
slightly more stable for $\xi=2$. This is related to the fact that for strong
coupling renormalization scales larger than $\sqrt{s_{0}}$ the sign-
alternating effects the UV renormalons, which have branch points at $u=-1$,
are suppressed with respect to the GC renormalon with the branch point at
$u=2$. This sign-alternation disturbs the constructive structure of the
$\chi^{2}$ function which exclusively focuses on the impact of the GC
renormalon subtraction. Since, eventually, the UV renormalon will dominate the
Adler function’s perturbation series at very high orders (see Sec. 5 and Fig.
3 of Part I where the moments for $w_{1}$ and $w_{6}$ have been analyzed in
great detail), the implementation of the optimal subtraction method we have
adopted here cannot be applied to all orders $m$. However, for truncation
orders accessible by available or foreseeable calculations the method is
perfectly adequate.

For the Borel model in Eq. (26) the central value for the ${\cal
O}(\alpha_{s}^{5})$ coefficient Eq. (15), $\bar{c}_{5,1}=345$ has been
adopted. Interestingly, the additional error on $N_{g}$ related to the
uncertainty on $\bar{c}_{5,1}$ turns out to be quite small. To demonstrate
that we have at order $m=5$ also determined the range of $N_{g}$ values by
carrying out the $R$ variation described above and in addition also varied
independently $\bar{c}_{5,1}$ by $\pm 140$. The outcome is shown as the blue
error bar in both panels of Fig. 2. We have checked that the outcome of the
analysis remains essentially unchanged, if other moments are used that satisfy
the same criteria as $w_{1-10}$ or if higher values for $s_{0}$ are adopted.
It should be noted, however, that it is not easy to construct completely
different inequivalent sets of analogous moments which are linearly
independent when imposing a limit on the maximal power of $x$.

|  $\sqrt{s_{0}}=m_{\tau}$ |  $\sqrt{s_{0}}=3~{}{\rm GeV}$
---|---|---
$m=4~{}(\xi=1)$ |  $0.60\pm 0.20$ |  $0.51\pm 0.17$
$m=4~{}(\xi=2)$ |  $0.54\pm 0.21$ |  $0.55\pm 0.11$
$m=5~{}(\xi=1)$ |  $0.64\pm 0.16$ |  $0.50\pm 0.19$
$m=5~{}(\xi=2)$ |  $0.59\pm 0.18$ |  $0.52\pm 0.15$
Table 2:  Results for the $n_{f}=3$ GC renormalon norm $N_{g}$ at orders $m=4$
and $5$ obtained from the optimal subtraction approach for
$s_{0}=m_{\tau}^{2},(3\,\mbox{GeV})$, and $\xi=1,2$. The uncertainties at
order $m=4$ arise from varying the IR factorization scale in the range
$0.7\sqrt{s_{0}}\leq R\leq\sqrt{s_{0}}$ and at order $m=5$ from the additional
independent variation of $\bar{c}_{5,1}$ within the uncertainty estimate of
Eq. (15). The central values are the average of the maximal and minimal
results at each order in perturbation theory.

In Tab. 2 we display the results of the optimal subtraction method at
truncation order $m=4$ and $m=5$ for $\sqrt{s_{0}}=m_{\tau}$ and
$\sqrt{s_{0}}=3$ GeV and using $\xi=1$ and $\xi=2$. For $m=5$ the error on
$\bar{c}_{5,1}$ is included as described in the previous paragraph. All
results are equivalent and consistent and we adopt the envelope of the $m=4$
results for $\xi=1$ and $\xi=2$ as our final result:

$N_{g}=0.57\pm 0.23\,.$ (37)

The result, which has a relative uncertainty of $40\%$, is fully consistent
with the estimates we have obtained from the Borel model and the conformal
mapping approaches discussed in Sec. 3.1 and 3.2 and is furthermore
independent of the estimate for the 6-loop coefficient $\bar{c}_{5,1}$.

The equivalence of the results obtained from the three approaches we have
discussed and which all rely on different criteria underlines that the known
perturbative coefficients of the Adler function are fully compatible with a
sizeable GC renormalon norm in the ranges obtained by our estimates. Within
the natural proposition that the GC renormalon has a sizeable contribution to
the coefficient at ${\cal O}(\alpha_{s}^{4})$, i.e. that the size of
$\bar{c}_{4,1}$ is not just accidentally mimicking this property, our results
provide a realistic estimate on the GC renormalon norm $N_{g}$. We therefore
adopt the result given in Eq. (37) as our final result for the GC renormalon
norm.

##  4 Impact on Strong Coupling Determinations

In order to demonstrate the improvements that can be achieved concerning the
CIPT-FOPT discrepancy when using the RF scheme for the GC in realistic
$\alpha_{s}$ analyses of $\tau$ hadronic spectral function moments, we
exemplarily carry out in this section two determinations of
$\alpha_{s}(m_{\tau}^{2})$ following two analysis setups employed in the
recent references by Pich and Rodríguez-Sanchez Pich:2016bdg  and by Boito,
Golterman, Maltman, Peris, Rodrigues, and Schaaf Boito:2020xli . These two
references are representatives of the two major approaches concerning the
treatment of nonperturbative corrections currently used in the literature, one
employing a truncated version of the traditional parametrization in terms of
the OPE corrections, see Refs. LeDiberder:1992zhd ; Davier:2005xq ;
Davier:2008sk ; Davier:2013sfa , and one that includes, besides the OPE
corrections, DV contributions, see Refs. Boito:2011qt ; Boito:2012cr ;
Boito:2014sta . The two approaches are discussed controversially, see e.g.
Refs. Boito:2016oam ; Boito:2018yml ; Boito:2019iwh  and Pich:2016bdg ;
Pich:2022tca . For each approach we first employ the CIPT and FOPT expansions
in the common $\overline{\text{MS}}$ GC scheme, reproducing the respective
published results, and then carry out the analyses in the RF GC scheme, using
our result for the GC renormalon norm $N_{g}$ quoted in Eq. (37). For the
analysis in the RF GC scheme, additional uncertainties due to the approximate
knowledge of $N_{g}$ and due to variations of the IR subtraction scale $R$ are
accounted for. We find that their numerical impact is relatively small. This
is particularly striking given the quite sizeable uncertainty of $N_{g}$. This
outcome is highly encouraging and underlines the practical value of the RF GC
scheme.

We emphasize that the main purpose of the analyses in this section is to
demonstrate that the CIPT-FOPT discrepancy, that is present in the
$\overline{\text{MS}}$ GC scheme, is significantly reduced in the RF GC
scheme, regardless of which approach is employed concerning the treatment of
the nonperturbative corrections. In our analyses we use the central value for
the estimate of the 6-loop coefficient $\bar{c}_{5,1}$ shown in Eq. (15), for
which there is a common agreement in the literature Pich:2016bdg ;
Boito:2020xli ; Ayala:2021yct ; dEnterria:2022hzv .101010In Ref. Pich:2016bdg
$c_{5,1}=275\pm 400$ is used for the 6-loop coefficient using the
$\overline{\rm MS}$ scheme for the strong coupling, which corresponds to
$\bar{c}_{5,1}=340.477\pm 400$ in the $C$-scheme. In Ref. Boito:2020xli
$c_{5,1}=283\pm 142$ is used, which corresponds to $\bar{c}_{5,1}=348.477\pm
142$. In Ref. Ayala:2021yct  $c_{5,1}=275\pm 63$ is used, which corresponds to
$\bar{c}_{5,1}=340.477\pm 63$. When assessing theoretical uncertainties
arising from the estimate of $c_{5,1}$, we use the uncertainty quoted in Eq.
(15). There is no common agreement for this uncertainty in the literature in
the context of $\alpha_{s}$ determinations, but for the purpose of our
analysis this aspect is not essential since using the central value is
sufficient to demonstrate the improvement. We adopt the estimate given in Eq.
(15) as it is consistent with the basic proposition our work is based on and
furthermore not overly agressive, see Refs. Beneke:2008ad ; Jamin:2021qxb .
Our results show that a higher precision can in general be reached in
$\alpha_{s}$ determinations from $\tau$ hadronic spectral function moments
based on the CIPT and FOPT expansions when the RF GC scheme is employed. The
results can be consistently combined in the RF GC scheme, while a combination
in the $\overline{\rm MS}$ GC scheme may lead to inconsistent results due to
the asymptotic separation. We stress that the analyses carried out in this
section should not be interpreted as a supersedure of the results for the
strong coupling given in Refs. Pich:2016bdg  and Boito:2020xli  since they
carried out many more additional examinations we do not repeat here due to
lack of space. Furthermore, our results should not be interpreted as an
approval (or disapproval) of the methods employed in these references. A
dedicated new strong coupling analysis using $\tau$ hadronic spectral function
moments will be carried out by us in a subsequent work.

As far as experimental data is concerned, we note that the LEP experiments,
ALEPH ALEPH:1998rgl ; ALEPH:2005qgp ; Davier:2008sk ; Davier:2013sfa  and OPAL
OPAL:1998rrm ; Boito:2012cr , have produced spectral function data from the
analyses of hadronic tau decays for the $V$, $A$, $V+A$, and $V-A$ channels.
More recently, with the wealth of data produced by experiments measuring
$e^{+}e^{-}\to{\rm hadrons}$ in the past decade (mainly aiming at the
dispersive calculation of the muon $g-2$), it has become possible to produce a
new isovector $V$ spectral function exclusively based on experimental data
Boito:2020xli . This has been achieved by extracting subleading channels from
recently measured cross-sections for $e^{+}e^{-}\to{\rm hadrons}$ using the
conserved vector current (CVC) hypothesis. (We remind the reader that the
ALEPH and OPAL analyses had to rely on Monte Carlo simulated data for these
subleading channels.) Realistic phenomenological analyses are either based on
the 2013 update of the ALEPH spectral functions Davier:2013sfa ; Pich:2016bdg
; Boito:2014sta ; Ayala:2021mwc ; Ayala:2021yct , which allow for the analysis
of the “more inclusive” $V+A$ channel, or on the new purely data based $V$
spectral function, which has significantly smaller uncertainties near the end-
point of the spectrum Boito:2020xli . In order to follow the works of Refs.
Pich:2016bdg  and Boito:2020xli  we employ, in Sec. 4.1, the ALEPH $V+A$ data,
while in Sec. 4.2 we use the new $V$ spectral function of Ref. Boito:2020xli .
This introduces an additional difference, besides the treatment of
nonperturbative corrections, between the results obtained in these two
sections, which makes a consistent combination of them non-trivial.

###  4.1 Truncated OPE Analysis: Multiple Moments at fixed $s_{0}$

|  central value |  $\sigma_{\text{exp}}$ |  $\sigma_{\mu}$ |  $\sigma_{R}$ |  $\sigma_{N_{g}}$ |  $\sigma_{5}$ |  $\sigma_{\text{tot}}$
---|---|---|---|---|---|---|---
CIPT, $\chi^{2}$/dof = 0.82, $p$-value = 0.37
$\alpha_{s}(m_{\tau})$ |  0.3366 |  0.0042 |  0.0051 |  $-$ |  $-$ |  0.0024 |  0.0070
$\mathcal{C}_{6}\times 10^{3}$ |  0.95 |  0.38 |  0.44 |  $-$ |  $-$ |  0.012 |  0.58
$\mathcal{C}_{8}\times 10^{3}$ |  $-$1.07 |  0.46 |  0.22 |  $-$ |  $-$ |  0.055 |  0.51
$\mathcal{C}_{10}\times 10^{3}$ |  0.22 |  0.28 |  0.49 |  $-$ |  $-$ |  0.0052 |  0.56
FOPT, $\chi^{2}$/dof = 1.23, $p$-value = 0.27
$\alpha_{s}(m_{\tau})$ |  0.3169 |  0.0031 |  0.0056 |  $-$ |  $-$ |  0.0014 |  0.0065
$\mathcal{C}_{6}\times 10^{3}$ |  1.37 |  0.43 |  1.22 |  $-$ |  $-$ |  0.076 |  1.29
$\mathcal{C}_{8}\times 10^{3}$ |  $-$0.95 |  0.45 |  1.02 |  $-$ |  $-$ |  0.069 |  1.12
$\mathcal{C}_{10}\times 10^{3}$ |  0.43 |  0.30 |  0.58 |  $-$ |  $-$ |  0.041 |  0.66
RF GC scheme, CIPT, $\chi^{2}$/dof = 2.14, $p$-value = 0.14
$\alpha_{s}(m_{\tau})$ |  0.3197 |  0.0035 |  0.0047 |  0.0054 |  0.0058 |  0.0017 |  0.010
$\mathcal{C}_{6}\times 10^{3}$ |  4.96 |  0.63 |  0.40 |  0.70 |  1.13 |  0.14 |  1.53
$\mathcal{C}_{8}\times 10^{3}$ |  $-$3.40 |  0.54 |  0.42 |  0.37 |  0.76 |  0.017 |  1.09
$\mathcal{C}_{10}\times 10^{3}$ |  1.50 |  0.33 |  0.22 |  0.33 |  0.41 |  0.020 |  0.65
RF GC scheme, FOPT, $\chi^{2}$/dof = 1.21, $p$-value = 0.27
$\alpha_{s}(m_{\tau})$ |  0.3169 |  0.0031 |  0.0059 |  0 |  0 |  0.0014 |  0.0068
$\mathcal{C}_{6}\times 10^{3}$ |  1.39 |  0.43 |  0.96 |  0.11 |  0.0048 |  0.075 |  1.06
$\mathcal{C}_{8}\times 10^{3}$ |  $-$0.80 |  0.45 |  0.99 |  0.14 |  0.059 |  0.065 |  1.10
$\mathcal{C}_{10}\times 10^{3}$ |  0.40 |  0.30 |  0.57 |  0.039 |  0.014 |  0.040 |  0.65
Table 3: Fitted parameters from the ALEPH $V+A$ spectral function used in Ref.
Pich:2016bdg  for CIPT and FOPT in the $\overline{\text{MS}}$ GC scheme and in
the RF GC scheme. For the latter we use $N_{g}=0.57\pm 0.23$. The uncertainty
$\sigma_{5}$ reflects the variation of the 6-loop coefficient as in Eq. (15),
while $\sigma_{\mu}$ and $\sigma_{R}$ refer to variations of the strong-
coupling renormalization scale $\mu$ and of the IR subtraction scale $R$ as
described in the text. The values for $\sigma_{\mu}$, $\sigma_{5}$,
$\sigma_{R}$, and $\sigma_{N_{g}}$ are half the difference between the maximum
and minimum values obtained from the $\xi$, $c_{5,1}$, $R$, and $N_{g}$
variations, respectively, using the central values for the remaining
parameters. The two zeros in the lowest section refer to numbers smaller than
$10^{-5}$. The $\mathcal{C}_{k}$ are given in units of GeVk and are denoted
$\mathcal{O}_{k}$ in Ref. Pich:2016bdg .

We start by revisiting the analysis conducted by Pich et al. in Sec. 5.3 of
Ref. Pich:2016bdg . This analysis follows the so-called "truncated OPE
strategy" (tOPE) (which is also the basis of the earlier analysis in Ref.
Davier:2013sfa ) and consists of the following. One chooses a number of
linearly independent integrated spectral function moments calculated at
$s_{0}=m_{\tau}^{2}$, all based on (at least) doubly-pinched weight functions.
The pinching is argued to be sufficient to allow for a neglect of any
contribution from DV corrections. Furthermore, to allow for a fit procedure,
the number of moments must, of course, be larger than the number of parameters
in the theory description. Since the pinching and the requirement that the
moments are linearly independent bring in additional powers of $x$ in the
weight functions $w(x)$ and the associated OPE corrections within the
approximation of Eq. (9), it becomes necessary to truncate the tower of OPE
condensates in order to have a manageable number of parameters in the fit
procedure. This is based on the argument that the contributions from the
neglected higher power OPE corrections are hierarchical and small.

In the analysis of Sec. 5.3 of Ref. Pich:2016bdg , the five doubly-pinched
weight functions $w_{1,\ldots,5}(x)$ of Eqs. (35) are used. (We remind the
reader that $w_{1}(x)$ corresponds to the kinematic moment relevant for the
inclusive hadronic $\tau$ decay rate.) Due to the absence of any linear term
$x$, all moments are GCS, i.e. the GC OPE correction is highly suppressed (and
absent in the approximation of Eq. (9)), but involve the condensates
$\mathcal{C}_{2k}$ for $k=3,\dots,8$.

The four parameters of the fit are $\alpha_{s}$ as well as the first three OPE
condensates that contribute at leading- order in the strong coupling in the
approximation of Eq. (9): $\mathcal{C}_{6}$, $\mathcal{C}_{8}$, and
$\mathcal{C}_{10}$. The contributions from condensates of dimension 12, 14,
and 16 are neglected, so that for the moments with $w_{4}(x)$ and $w_{5}(x)$
only the perturbative corrections enter the fit. In the analysis of Ref.
Pich:2016bdg , the ALEPH $V$, $A$ and $V+A$ data from the update of Ref.
Davier:2013sfa  were used for three separate fits, which lead to perfectly
equivalent determinations of $\alpha_{s}(m_{\tau}^{2})$. In the analysis below
we therefore only use the $V+A$ data from that reference. As far as the
improvement of the CIPT-FOPT discrepancy in the RF GC scheme is concerned, the
results based on the $V$ or the $A$ data are equivalent.

The results of the fits based on the CIPT and FOPT expansions in the usual
$\overline{\text{MS}}$ scheme for the GC are shown in the upper half of Tab.
3. For the strong coupling the results are111111In Ref. Pich:2016bdg
additional OPE truncation uncertainties of $0.013$ and $0.012$ for
$\alpha_{s}(m_{\tau}^{2})^{\rm CIPT}$ and $\alpha_{s}(m_{\tau}^{2})^{\rm
FOPT}$, respectively, were added (see their table 8). We do not include these
uncertainties in our analysis as our main focus lies on the reconciliation of
the CIPT-FOPT discrepancy.

$\begin{array}[]{l}\alpha_{s}(m_{\tau}^{2})^{\rm
CIPT}\,=\,0.3366\,\pm\,0.0070\\\ \alpha_{s}(m_{\tau}^{2})^{\rm
FOPT}\,=\,0.3169\,\pm\,0.0065\end{array}\,\qquad\mbox{(tOPE, $\overline{\rm
MS}$ GC, ALEPH $V+A$).}$ (38)

The central values are obtained using $\bar{c}_{5,1}=345$ for the 6-loop
coefficient. Apart from the experimental error from the fits, $\sigma_{\rm
exp}$ (which we treat statistically), we also quote the theory errors related
to the truncation of perturbation theory: $\sigma_{\mu}$, arising from
renormalization scale variations, and $\sigma_{5}$, from the uncertainty in
the estimate of the 6-loop coefficient $\bar{c}_{5,1}=345\pm 140$ in Eq. (15).
For the variation of the renormalization scale in CIPT, we expand in powers of
$\alpha_{s}(-\xi s)\equiv\alpha_{s}(-\xi m_{\tau}^{2}\,x)$ prior to carrying
out the contour integration and, in FOPT, in powers of $\alpha_{s}(\xi
m_{\tau}^{2})$, with $1/2\leq\xi\leq 2$ and $\xi=1$ being the choice for the
central value. The total uncertainty is obtained by quadratially adding
$\sigma_{\rm exp}$, $\sigma_{\mu}$, and $\sigma_{5}$. Our results reproduce
very well those quoted in the last two lines of table 7 in Ref. Pich:2016bdg .

We see that the results for $\alpha_{s}(m_{\tau}^{2})$ obtained from the CIPT
and FOPT expansion show a difference of about $0.0197$, which is the typical
size of the CIPT-FOPT discrepancy that has been found in Ref. Pich:2016bdg
and in many other previous phenomenological analyses based on variants of the
tOPE strategy. This is 5 to 6 times larger than the individual experimental
uncertainties, and about 3 times larger than the total uncertainties we obtain
for the CIPT and FOPT results for $\alpha_{s}(m_{\tau}^{2})$ in our analysis.
Interestingly, the fit results for the condensates obtained in the CIPT and
FOPT analyses are perfectly compatible within uncertainties indicating that
the asymptotic separation mostly affects the value of the strong coupling.

In the lower half of Tab. 3 we show the corresponding CIPT and FOPT results
obtained in the RF GC scheme using Eq. (37) for the gluon condensate
renormalon norm $N_{g}$ and $0.7m_{\tau}\leq R\leq m_{\tau}$ for the IR
subtraction scale variation. For the strong coupling the results are

$\begin{array}[]{l}\alpha_{s}(m_{\tau}^{2})^{\rm
CIPT}\,=\,0.3197\,\pm\,0.0100\\\ \alpha_{s}(m_{\tau}^{2})^{\rm
FOPT}\,=\,0.3169\,\pm\,0.0068\end{array}\qquad\mbox{(tOPE, RF GC, ALEPH
$V+A$)}.$ (39)

The central values are obtained from using $N_{g}=0.57$ and $R=0.8\,m_{\tau}$
as well as $\bar{c}_{5,1}=345$. Apart from the uncertainties related to the
truncation of perturbation theory, $\sigma_{\mu}$ and $\sigma_{5}$, which we
have estimated as in the $\overline{\text{MS}}$ GC scheme, we also quote the
uncertainties $\sigma_{R}$ and $\sigma_{N_{g}}$ coming from the $R$-variation
and the error in $N_{g}$, respectively. We see that the FOPT results are
almost unaltered by the subtraction of the GC renormalon and the switch to the
RF GC scheme. At the same time, the uncertainties $\sigma_{R}$ and
$\sigma_{N_{g}}$ are extremely small for $\alpha_{s}$, and lead to only minor
changes of the fit results for the OPE condensates. Overall, switching from
the $\overline{\text{MS}}$ to the RF GC scheme for GCS spectral function
moments has very little impact for the FOPT expansion. So for the FOPT
expansion the suppression of the GC OPE corrections is associated to a
supression of the effects induced by switching to the GC scheme. This
corroborates the conclusions of Refs. Hoang:2020mkw ; Hoang:2021nlz  and our
observations in Part I, that the FOPT expansion is consistent with the
standard form of the OPE.

The CIPT results, on the other hand, change significantly, and now the
difference of the CIPT and FOPT central values for $\alpha_{s}(m_{\tau}^{2})$
is reduced to merely $0.0028$, which is smaller than the individual
experimental uncertainties. In CIPT, the RF GC scheme introduces a new source
of theory uncertainty for the $\alpha_{s}$ value, which now receives non-
negligible contributions from $\sigma_{N_{g}}$ and $\sigma_{R}$ even though
the numerical size of the GC OPE correction itself is still strongly
suppressed (and even vanishing in the approximation of Eq. (9)). This
(apparently) paradoxical behavior corroborates the conclusions of Refs.
Hoang:2020mkw ; Hoang:2021nlz  and our observations in Part I, and illustrates
the incompatibility of the CIPT expansion with the standard form of the OPE as
given in Eq. (6) and (9), and the need to switch to the RF GC scheme to
eliminate the dominating inconsistency that is related to the GC renormalon.
It is remarkable, however, that the final uncertainty for
$\alpha_{s}(m_{\tau}^{2})$, which has to account for $\sigma_{N_{g}}$ and
$\sigma_{R}$, still remains quite small and only grows from $0.0070$ to about
$0.010$. This growth in total uncertainty for the CIPT analysis is more than
compensated by the gain in consistency between the FOPT and CIPT analyses.
Finally, we observe a change in the central values of the OPE condensates in
CIPT when the RF GC is used. The new results are still marginally compatible
with the ones obtained in the $\overline{\text{MS}}$ GC scheme. We also
observe that in the RF GC scheme, the fit quality of the CIPT analysis
slightly worsens. Still, the fit is perfectly acceptable with a $p$-value of
$14\%$.

Given that the asymptotic separation inherent to the CIPT expansion method in
the original $\overline{\rm MS}$ GC scheme is removed in the RF GC scheme, it
now makes good sense to determine a combined $\alpha_{s}$ result, in contrast
to the results obtained in the $\overline{\rm MS}$ GC scheme. We average the
FOPT and CIPT results in the RF GC scheme following a prescription given in
Ref. Pich:2016bdg : we take the average of the two individual central values
as the final central value and the quadratic sum of the smaller individual
uncertainty and half of the central difference as the final combined
uncertainty. (We do not reduce the individual errors in the averaging due to
the potential correlations in central values.) This results in121212In Ref.
Pich:2016bdg , uncertainties of $\pm 400$ were employed for $\bar{c}_{5,1}$.
If we had adopted these variations, $\sigma_{5}$ would increase to $0.0069$
for CIPT and $0.0039$ for FOPT in the $\overline{\rm MS}$ GC scheme, which
would yield $\alpha_{s}(m_{\tau}^{2})^{\rm CIPT}=0.3366\,\pm\,0.0096$ and
$\alpha_{s}(m_{\tau}^{2})^{\rm FOPT}=0.3169\,\pm\,0.0075$. In the RF GC
scheme, a similar increase in uncertainty is observed and the final values
would be $\alpha_{s}(m_{\tau}^{2})^{\rm CIPT}=0.320\,\pm\,0.011$ and
$\alpha_{s}(m_{\tau}^{2})^{\rm FOPT}=0.3169\,\pm\,0.0077$. The average of Eq.
(40) would then read $\alpha_{s}(m_{\tau}^{2})=0.3183\pm 0.0079$.

$\alpha_{s}(m_{\tau}^{2})\,=\,0.3183\,\pm\,0.0069\qquad\mbox{(tOPE, RF GC,
ALEPH $V+A$)}.$ (40)

We adopt this averaging prescription in the RF GC analysis in the next section
as well. The uncertainty of the average reflects the potential prospect of
strong coupling determinations from hadronic $\tau$ spectral function moments
in a situation where an uncertainty in the parametrization of the
nonperturbative corrections in the tOPE method would be absent. In Fig. 3 a
visual comparison of the results for $\alpha_{s}(m_{\tau}^{2})$ in the
$\overline{\text{MS}}$ (left panel, see Eq. (38)) and in the RF GC scheme
(right panel, see Eq. (39)) is displayed. The average of Eq. (40) is also
shown in the right panel.

We note that in Ref. Pich:2016bdg  a number of other moment analyses using the
CIPT and FOPT expansions have been carried out. We have checked that the
observations described above for the improvement concerning the consistency of
the extracted values for $\alpha_{s}(m_{\tau}^{2})$ from the CIPT and FOPT
expansions in the RF GC scheme are exemplary for all analyses carried out by
them and not specific for the moment analysis in their Sec. 5.3.

###  4.2 Single Weigth Function Analysis with Multiple $s_{0}$ values

We turn now to the analysis by Boito et al. in Ref. Boito:2020xli . This
analysis follows the so-called "DV-model strategy" advocated in Refs.
Boito:2011qt  and used in previous analyses of the ALEPH and OPAL data in
Refs. Boito:2014sta ; Boito:2012cr . In order to avoid the truncation of the
OPE, one chooses weight functions with less (or even no) pinching, since this
allows to employ low-degree polynomial weight functions $w(x)$ which strongly
suppress higher-dimension OPE corrections. In the approximation of Eq. (9)
this leads to the absence of OPE corrections coming from higher-dimensional
OPE condensates. However, this choice in general enhances the contributions
from DV effects, which therefore must be included in the fit. This is done
with the parametrization of Eq. (11). Since DVs are related to residual
resonance effects, the parameters $\delta_{V/A}$, $\gamma_{V/A}$, $a_{V/A}$,
and $b_{V/A}$ are channel dependent and have to be extracted from data. The
fits can be carried out using moments for the same weight function using
different values for $s_{0}\leq m_{\tau}^{2}$ under the assumption that the
ansatz for the DV contribution in Eq. (11) is adequate.

In the analysis of Ref. Boito:2020xli , the $\alpha_{s}$ results are based on
the $V$ spectral function, for which a more precise and updated data set was
provided in the same reference, which includes information from recent
$e^{+}e^{-}\to{\rm hadrons}$ cross-section data related by isospin symmetry.
For the main results, moments for the weight function $w(x)=1$ are considered
for different values of $s_{0}$. For this weight function all OPE corrections
are strongly suppressed (and vanish in the approximation of Eq. (9)). As for
the analysis of Ref. Pich:2016bdg  discussed in the previous section, the
approximation of Eq. (9) is used so that no OPE corrections are included.
Using moments with several $s_{0}$ values simultaneously, one can extract
$\alpha_{s}$ together with the DV parameters from the fit. For the main
results of Ref. Boito:2020xli , between fourteen and twenty $s_{0}$ values are
included in each fit, ranging from $1.55$ GeV2 to $m_{\tau}^{2}$, (see the
entries in the last four lines in their table 1). In our analysis below we
consider the fit with twenty $s_{0}$ values, starting at $s_{0}=1.55$ GeV2.
There are five fit parameters: $\alpha_{s}$ and the four parameters of the DV
contribution. Other moments, including pinched moments requiring OPE
condensate corrections, were also considered in Ref. Boito:2020xli  as a
consistency check of the analysis.

|  central value |  $\sigma_{\text{exp}}$ |  $\sigma_{\mu}$ |  $\sigma_{R}$ |  $\sigma_{N_{g}}$ |  $\sigma_{5}$ |  $\sigma_{\text{total}}$
---|---|---|---|---|---|---|---
CIPT, $\chi^{2}$/dof = 12.97/15, $p$-value = 0.60
$\alpha_{s}(m_{\tau})$ |  0.3256 |  0.0089 |  0.0033 |  $-$ |  $-$ |  0.0021 |  0.0097
$\delta_{V}$ |  3.33 |  0.28 |  0.021 |  $-$ |  $-$ |  0.0055 |  0.28
$\gamma_{V}$ |  0.66 |  0.18 |  0.010 |  $-$ |  $-$ |  0.0029 |  0.18
$a_{V}$ |  $-$1.27 |  0.49 |  0.0026 |  $-$ |  $-$ |  0.0011 |  0.49
$b_{V}$ |  3.76 |  0.26 |  0.0023 |  $-$ |  $-$ |  0.0001 |  0.26
FOPT, $\chi^{2}$/dof = 12.55/15, $p$-value = 0.64
$\alpha_{s}(m_{\tau})$ |  0.3083 |  0.0066 |  0.0014 |  $-$ |  $-$ |  0.0025 |  0.0072
$\delta_{V}$ |  3.51 |  0.28 |  0.054 |  $-$ |  $-$ |  0.027 |  0.29
$\gamma_{V}$ |  0.57 |  0.17 |  0.029 |  $-$ |  $-$ |  0.013 |  0.18
$a_{V}$ |  $-1.26$ |  0.48 |  0.022 |  $-$ |  $-$ |  0.0002 |  0.48
$b_{V}$ |  3.77 |  0.26 |  0.0063 |  $-$ |  $-$ |  0.0020 |  0.26
RF GC scheme, CIPT, $\chi^{2}$/dof = 12.70/15, $p$-value = 0.63
$\alpha_{s}(m_{\tau})$ |  0.3159 |  0.0080 |  0.0033 |  0.0015 |  0.0035 |  0.0017 |  0.0096
$\delta_{V}$ |  3.44 |  0.28 |  0.028 |  0.055 |  0.041 |  0.0024 |  0.29
$\gamma_{V}$ |  0.61 |  0.17 |  0.013 |  0.027 |  0.020 |  0.0012 |  0.18
$a_{V}$ |  $-1.26$ |  0.49 |  0.0037 |  0.0023 |  0.0023 |  0.0007 |  0.49
$b_{V}$ |  3.77 |  0.26 |  0.0031 |  0.0054 |  0.0023 |  0.0001 |  0.26
RF GC scheme, FOPT, $\chi^{2}$/dof = 12.53/15, $p$-value = 0.64
$\alpha_{s}(m_{\tau})$ |  0.3081 |  0.0065 |  0.0015 |  0 |  0.0001 |  0.0025 |  0.0072
$\delta_{V}$ |  3.52 |  0.28 |  0.053 |  0.0037 |  0.0038 |  0.026 |  0.29
$\gamma_{V}$ |  0.57 |  0.17 |  0.029 |  0.0016 |  0.0018 |  0.013 |  0.18
$a_{V}$ |  $-1.26$ |  0.48 |  0.024 |  0.0018 |  0.0008 |  0.0003 |  0.48
$b_{V}$ |  3.77 |  0.26 |  0.0076 |  0.0010 |  0.0006 |  0.0019 |  0.26
Table 4: Fitted parameters from the improved $\tau$ vector-isovector spectral
function of Ref. Boito:2020xli  for CIPT and FOPT in the
$\overline{\text{MS}}$ GC scheme and in the RF GC scheme. For the latter case
we use $N_{g}=0.57\pm 0.23$. The uncertainty $\sigma_{c_{5,1}}$ reflects the
variation of the 6-loop coefficient as in Eq. (15), while $\sigma_{\mu}$ and
$\sigma_{R}$ refer to variations of the strong-coupling renormalization scale
$\mu$ and of the IR subtraction scale $R$ as described in the text. The values
for $\sigma_{\mu}$, $\sigma_{5}$, $\sigma_{R}$, and $\sigma_{N_{g}}$ are half
the difference between the maximum and minimum values obtained from the $\xi$,
$c_{5,1}$, $R$, and $N_{g}$ variations, respectively, using the central values
for the remaining parameters. The parameters $b_{V}$ and $\gamma_{V}$ are
given in units of GeV-2; $a_{V}$ and $b_{V}$ are denoted $\alpha_{V}$ and
$\beta_{V}$ in Ref. Boito:2020xli .

The results of the fits based on the CIPT and FOPT expansions in the usual
$\overline{\text{MS}}$ scheme for the GC and using 20 values for $s_{0}$ are
shown in the upper half of Tab. 4. The central values are again obtained using
$\bar{c}_{5,1}=345$ for the 6-loop coefficient. For the strong coupling the
results are

$\begin{array}[]{l}\alpha_{s}(m_{\tau}^{2})^{\rm
CIPT}\,=\,0.3256\,\pm\,0.0097\\\ \alpha_{s}(m_{\tau}^{2})^{\rm
FOPT}\,=\,0.3083\,\pm\,0.0072\end{array}\,\qquad\mbox{(DV mod., $\overline{\rm
MS}$ GC, new $V$ spec. func.)}.$ (41)

The experimental uncertainties $\sigma_{\rm exp}$ and the uncertainty
$\sigma_{5}$, from the estimate of the 6-loop coefficient $\bar{c}_{5,1}$, are
obtained in the same way as in our tOPE analysis. The uncertainty
$\sigma_{\mu}$, arising from renormalization scale variations are obtained in
an analogous way as well, but a lower bound is imposed on the $\xi$ variations
for small $s_{0}$ values to avoid the appearance of nonperturbative scales. As
the strict lower bound for strong coupling renormalization scale variations we
use $\mu_{\rm min}=0.7m_{\tau}$ to safely stay outside the region of
nonperturbative scales. A similar prescription was applied in Ref.
Boito:2020xli .

Our results reproduce very well those quoted in Eq. (4.1) and Eq. (4.2) of
Ref. Boito:2020xli , for FOPT and CIPT, respectively. The discrepancy between
the results for $\alpha_{s}(m_{\tau}^{2})$ from FOPT and CIPT is of $0.017$,
which is 2 to 3 times larger than the experimental uncertainties $\sigma_{\rm
exp}$ which are about twice the size as in the tOPE analysis. The uncertainty
from the renormalization scale variation $\sigma_{\mu}$ is significantly
smaller than the experimental uncertainty and also smaller than for the tOPE
analysis. As for the tOPE analysis, the CIPT-FOPT discrepancy in
$\alpha_{s}(m_{\tau}^{2})$ is much larger than the total uncertainty for both
expansions which is obtained by adding all individual uncertainties
quadratically. Interestingly, the fit results for the DV parameters for the
CIPT and FOPT analysis are perfectly compatible within uncertainties
indicating again that the asymptotic separation mostly affects the value of
the strong coupling.

In the lower half of Tab. 4, we show the results of the fits using the CIPT
and FOPT expansions in the RF GC scheme. The central values are obtained from
using $N_{g}=0.57$ and $R=0.8\,\sqrt{s_{0}}$ as well as $\bar{c}_{5,1}=345$.
For the strong coupling the results are

$\begin{array}[]{l}\alpha_{s}(m_{\tau}^{2})^{\rm
CIPT}\,=\,0.3159\,\pm\,0.0096\\\ \alpha_{s}(m_{\tau}^{2})^{\rm
FOPT}\,=\,0.3081\,\pm\,0.0072\end{array}\,\qquad\mbox{(DV mod., RF GC, new $V$
spec. func.)}.$ (42)

The uncertainties from the truncation of the perturbation series,
$\sigma_{\mu}$ and $\sigma_{c_{5,1}}$ are estimated as in the $\overline{\rm
MS}$ GC analysis. The uncertainty from the IR factorization scale $R$ is based
on the variation range $0.7\sqrt{s_{0}}\leq R\leq\sqrt{s_{0}}$, where for
smaller values of $s_{0}$ the absolute lower bound $R_{\rm min}=0.7m_{\tau}$
is imposed on $R$, in analogy to the renormalization scale. As for the tOPE
analysis, the FOPT results are virtually unchanged and the uncertainties
$\sigma_{R}$ and $\sigma_{N_{g}}$ are negligibly small. This again reflects
the compatibility of the FOPT expansion with the standard form of the OPE.

The CIPT results, on the other hand, change again significantly. The
difference of the CIPT and FOPT central values for $\alpha_{s}(m_{\tau}^{2})$
is reduced to 0.008. This is more than twice than for the tOPE analysis in the
RF GC scheme, but still half of the difference that is obtained when the
$\overline{\rm MS}$ GC scheme is employed. Overall, there is a much better
agreement for the $\alpha_{s}(m_{\tau}^{2})$ determinations in the RF GC
scheme. There are, however, a few notable differences. While in the tOPE
analysis the uncertainty in $\alpha_{s}(m_{\tau}^{2})$ increased moderately
due to $\sigma_{R}$ and $\sigma_{N_{g}}$, here these two uncertainties are
still relatively small and do not lead to any noticeable increase in the total
uncertainty. The total uncertainty for the CIPT results even slightly
decreases as a result of a smaller experimental uncertainty
$\sigma_{\text{exp}}$. Furthermore, while in the tOPE analysis $\sigma_{R}$
and $\sigma_{N_{g}}$ are significant for the CIPT expansion and there are
notable changes in the central values of the OPE condensates, here the impact
of $\sigma_{R}$ and $\sigma_{N_{g}}$ is smaller in comparison and the fit
results for the DV parameters change very little. Finally, while for the tOPE
analysis the $p$-value decreased in the RF GC scheme, no modification of the
$p$-value is observed here.

With the strong suppression of the asymptotic separation in the RF GC scheme,
it becomes possible to average the FOPT and CIPT results of Eq. (42).
Following the prescription described in the previous section we find131313If
we used $\pm 400$ for the uncertainty in $\bar{c}_{5,1}$, as in Ref.
Pich:2016bdg , $\sigma_{5}$ would increase to $0.0059$ for CIPT and $0.0074$
for FOPT in the $\overline{\rm MS}$ GC scheme, which would yield
$\alpha_{s}(m_{\tau}^{2})^{\rm CIPT}=0.326\,\pm\,0.011$ and
$\alpha_{s}(m_{\tau}^{2})^{\rm FOPT}=0.308\,\pm\,0.010$. In the RF GC scheme,
a similar increase in uncertainty is observed and the final values would be
$\alpha_{s}(m_{\tau}^{2})^{\rm CIPT}=0.316\,\pm\,0.011$ and
$\alpha_{s}(m_{\tau}^{2})^{\rm FOPT}=0.308\,\pm\,0.010$. The average of Eq.
(43) would read $\alpha_{s}(m_{\tau}^{2})=0.312\pm 0.011$.

$\alpha_{s}(m_{\tau}^{2})\,=\,0.3120\,\pm\,0.0082\qquad\mbox{(DV mod., RF GC,
new $V$ spec. func.)}.$ (43)

This result reflects the prospect for strong coupling determinations under the
assumptions made in the DV-model strategy. In Fig. 3 a visual comparison of
the results for $\alpha_{s}(m_{\tau}^{2})$ in the $\overline{\text{MS}}$ (left
panel, see Eq. (41)) and in the RF GC scheme (right panel, see Eq. (42)) is
displayed. The average of Eq. (43) is also shown in the right panel.

As already mentioned above, in Ref. Boito:2020xli  the same type of fits have
also been carried out for smaller number of $s_{0}$ values, and also some
analyses based on a combination of different types moments were carried out.
We emphasize that the observations concerning the good agreement of the CIPT
and FOPT determinations of $\alpha_{s}(m_{\tau}^{2})$ is general and not
dependent on the particular analysis set up described above. We also note that
it may appear tempting to combine the tOPE average of Eq. (40) with the DV
model average in Eq. (43) taking their difference, which only amounts to
$0.0063$, as an estimate for the treatment of nonperturbative effects. We
refrain from such a treatment since the $V$ spectral function determined in
Ref. Boito:2020xli  has not yet been analysed in the tOPE approach.
Furthermore, it is well known from the experience with the ALEPH data that the
tOPE and the DV-model strategies can lead to rather discrepant $\alpha_{s}$
results when the same data set is analysed Boito:2014sta ; Boito:2016oam ;
Pich:2016bdg . This discrepancy should be attributed to the treatment of
nonperturbative effects, and a detailed investigation is beyond the scope of
this work.

Figure 3:  Left panel: Results for $\alpha_{s}(m_{\tau}^{2})$ in FOPT (red)
and CIPT (blue) in the $\overline{\rm MS}$ GC scheme, following the strategies
of Ref. Pich:2016bdg  (Pich et al.), based on the tOPE strategy applied to the
$V+A$ ALEPH data Davier:2013sfa , and Ref. Boito:2020xli  (Boito et al.),
based on the DV-model strategy applied to the new vector spectral function of
Ref. Boito:2020xli . Right panel: Results for $\alpha_{s}(m_{\tau}^{2})$ in
FOPT (red) and CIPT (blue) and their average (black) in the RF GC scheme for
the same analysis set-ups. The substantial reduction in the discrepancy
between FOPT and CIPT result is clearly evident.

##  5 Conclusions

In this work, which is Part II of a series of articles, we have applied the
renormalon-free scheme for the gluon condensate (GC) defined in Part I
Benitez-Rathgeb:2022yqb  to the FOPT and CIPT perturbative expansions for
$\tau$ hadronic spectral function moments and examined it from the
phenomenological perspective. The scheme is based on a perturbative
redefinition of the GC matrix clement in close analogy to the well-known
implementation of short-distance heavy quark mass schemes, where the scheme
changes induce perturbative order-by-order subtractions to the original
perturbative coefficients. The renormalon-free scheme, which was discussed in
detail in Part I and which we call the RF gluon condensate scheme, depends on
an IR factorization scale $R$ and the normalization $N_{g}$ of the GC
renormalon. The RF GC matrix element is scale-invariant and can be easily
related to other schemes through its perturbative definition (including the
non-renormalon-free original $\overline{\rm MS}$ scheme). In addition to the
removal of renormalon related divergent perturbative contributions in the
series, which is well-known from the use of short-distance quark mass schemes,
the RF GC scheme also reconciles the long-standing discrepancy between the
CIPT and FOPT expansions for the perturbative series of $\tau$ hadronic
spectral function moments, which appeared for spectral function moments where
the GC OPE correction is strongly suppressed (and which we call GC
suppressed). As was shown by Hoang and Regner Hoang:2020mkw ; Hoang:2021nlz ,
the CIPT expansion for such moments still has a strong quartic (and higher
power) sensitivity to IR moments and is therefore not compatible with the
standard analytic form for OPE corrections. This effect, which is,
numerically, strongly dominated by the GC renormalon, is much larger than the
size of the GC OPE correction itself and thus renders the CIPT expansion
inconsistent if the usual $\overline{\rm MS}$ scheme is employed for the OPE
condensate matrix elements. By switching to the RF GC scheme, this
inconsistency is reduced to a negligible level, such that the CIPT expansion
is practically cured. While switching to the RF GC scheme for such moments has
very little effects for the FOPT expansion, the CIPT expansion is modified
substantially such that the discrepancy between them is resolved.

In this article we explored the impact of the uncertainties of the GC
renormalon norm $N_{g}$ and of variations of the IR subtraction scale $R$ on
the CIPT expansion in the RF gluon condensate scheme for GC suppressed
moments. Analysing three different methods, which all yield consistent
results, we determined $N_{g}^{(n_{f}=3)}=0.57\pm 0.23$, having a relative
uncertainty of $40\%$. We showed that this uncertainty and variations of the
IR subtraction scale $R$ increase the perturbative uncertainties of the CIPT
expansion, but we found that this increase is by far outweighed by the
improved consistency between the CIPT and FOPT expansion.

We have demonstrated the improved consistency in the RF GC scheme by applying
the CIPT and FOPT expansions in the context of two full-fledged state-of-the
art $\alpha_{s}(m_{\tau}^{2})$ determination approaches from the recent
literature by Pich and Rodríguez-Sanchez Pich:2016bdg  and by Boito,
Golterman, Maltman, Peris, Rodrigues and Schaaf Boito:2020xli  and accounting
for the additional uncertainties related to the RF GC scheme. These two
references are representatives of the two major approaches concerning the
treatment of nonperturbative corrections currently used in the literature,
called the truncated OPE approach and the duality-violation-model strategy.
For both approaches we find a substantially improved consistency between the
FOPT and CIPT expansions and that the uncertainties due to $N_{g}$ and $R$,
that mostly affect the CIPT expansion, are quite small. Since the CIPT
expansion has quite different perturbative properties than the FOPT expansion
related to the all-order resummations of phase corrections Pivovarov:1991rh ,
it thus remains a very valuable method that should still be employed in future
phenomenological analysis – but only if it is employed within a renormalon-
free GC scheme.

We believe that – for all practical matters – the long-standing CIPT-FOPT
discrepancy problem can now be considered as resolved. This resolution depends
on the acceptance of the proposition that the values of the known Adler
function perturbative coefficients up to ${\cal O}(\alpha_{s}^{4})$, which are
perfectly consistent with the sizeable GC norm value
$N_{g}^{(n_{f}=3)}=0.57\pm 0.23$, are indeed affected by the GC renormalon and
not created by some accidental unrelated finite-order behavior. This
proposition can, as a matter of principle, not be proven. But the same type of
proposition is used in all phenomenological applications where perturbative
subtractions of renormalons are important, most notably in heavy quark
physics. We therefore believe that it is reasonable to accept this proposition
for future analyses where the GC or the GC renormalon play important roles. As
far as the analyses of $\tau$ hadronic spectral function moments are
concerned, aspects which so far may have been considered subleading, such as
uncertainties related to the treatment of nonperturbative corrections or the
treatment of different data sets ParticleDataGroup:2020ssz ; dEnterria:2022hzv
, can now receive undivided attention.

## Acknowledgments

DB and MJ would like to thank the Particle Physics Group of the University of
Vienna for hospitality. We acknowledge partial support by the FWF Austrian
Science Fund under the Doctoral Program “Particles and Interactions” No.
W1252-N27 and under the Project No. P32383-N27. We also thank the Erwin-
Schrödinger International Institute for Mathematics and Physics for partial
support. DB’s work was supported by by the São Paulo Research Foundation
(FAPESP) Grant No. 2021/06756-6, by CNPq Grant No. 308979/2021-4, and by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) –
Finance Code 001.

##  Appendix A GC Renormalon Norm Conventions

In this appendix we compare our norm convention for the renormalon calculus
with the conventions used in Beneke:2008ad ; Lee:2011te ; Bali:2014fea  and
provide conversion formulae. All expressions below are written down using

$\frac{d\alpha_{s}(Q^{2})}{d\ln
Q}\,=\,\beta(\alpha_{s}(Q^{2}))\,\equiv\,-2\,\alpha_{s}(Q^{2})\,\sum\limits_{n=0}^{\infty}\beta_{n}\Big{(}\frac{\alpha_{s}(Q^{2})}{4\pi}\Big{)}^{n+1}$
(44)

as the definition of the coefficients of the QCD $\beta$-function such that we
have $\beta_{0}=11-2\,n_{f}/3$ and $\beta_{1}=102-38\,n_{f}/3$ for the one-
and two-loop coefficients for $n_{f}$ dynamical flavors. We also adopt the
abbreviation $a\equiv\alpha_{s}\beta_{0}/(4\pi)$.

In our convention we write the perturbation series for a generic quantity as

$\displaystyle\sigma\,=\,\sum_{n=1}^{\infty}\,c_{n}\,a^{n}\,.$ (45)

The corresponding Borel function is defined as

$\displaystyle
B[\sigma](u)\,\equiv\,\sum_{n=1}^{\infty}\,\frac{c_{n}}{\Gamma(n)}\,u^{n-1}\,,$
(46)

which gives

$\sigma=\int_{0}^{\infty}\\!\\!{\rm d}u\,B[\sigma](u)\,e^{-\frac{u}{a}}$ (47)

for the inverse Borel integration that gives back the original series
$\sigma$. The generic form that is adopted for a non-analytic IR renormalon
term in the Borel function related to a dimension $d=2p$ OPE correction reads

$\displaystyle B[\sigma](u)$ $\displaystyle\sim$
$\displaystyle\frac{N_{2p}}{(p-u)^{\gamma}}\,,$ (48)

which has a branch point at $u=p$.

In Ref. Beneke:2008ad  (Beneke and Jamin) the perturbation series for the
generic quantity $\sigma$ is written as

$\displaystyle\sigma\,=\,\sum_{n=0}^{\infty}\,p_{n}\,\alpha_{s}^{n+1}\,,$ (49)

and the Borel function is defined as

$\displaystyle\hat{B}[\sigma](t)\,\equiv\,\sum_{n=0}^{\infty}\,\frac{p_{n}}{\Gamma(n+1)}\,t^{n}\,.$
(50)

The inverse Borel integration has the form

$\sigma=\int_{0}^{\infty}\\!\\!{\rm
d}t\,\hat{B}[\sigma](t)\,e^{-\frac{t}{\alpha_{s}}}\,=\
{\textstyle\frac{4\pi}{\beta_{0}}}\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\hat{B}[\sigma]\Big{(}{\textstyle\frac{4\pi}{\beta_{0}}u}\Big{)}\,e^{-\frac{u}{a}}\,,$
(51)

and the generic form adopted for a non-analytic IR renormalon term in the
Borel function is

$\displaystyle\hat{B}[\sigma]\Big{(}{\textstyle\frac{4\pi}{\beta_{0}}u}\Big{)}$
$\displaystyle\sim$ $\displaystyle\frac{d_{p}^{\rm IR}}{(p-u)^{\gamma}}\,.$
(52)

In Refs. Lee:2011te  (Lee) and Bali:2014fea  (Bali et al.) the perturbation
series for the generic quantity $\sigma$ is written as in Eq. (49) and the
Borel function is defined as

$\displaystyle\tilde{B}[\sigma](u)\,=\,\tilde{\sigma}(u)\,=\,\sum_{n=0}^{\infty}\,\frac{p_{n}}{\Gamma(n+1)}\,\frac{(4\pi)^{n}u^{n}}{\beta_{0}^{n}}\,.$
(53)

The inverse Borel integration reads

$\sigma={\textstyle\frac{4\pi}{\beta_{0}}}\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\tilde{B}[\sigma](u)\,e^{-\frac{u}{a}}\,,$ (54)

and a generic non-analytic IR renormalon term in the Borel function is written
as

$\displaystyle\tilde{B}[\sigma](u)=\tilde{\sigma}(u)$ $\displaystyle\sim$
$\displaystyle\frac{{\cal N}_{p}}{(1-u/p)^{\gamma}}\,.$ (55)

The various Borel function definitions are related to ours through

$\displaystyle
B[\sigma](u)\,=\,{\textstyle\frac{4\pi}{\beta_{0}}}\,\hat{B}[\sigma]\Big{(}{\textstyle\frac{4\pi}{\beta_{0}}u}\Big{)}\,=\,{\textstyle\frac{4\pi}{\beta_{0}}}\,\tilde{B}[\sigma](u)\,=\,{\textstyle\frac{4\pi}{\beta_{0}}}\,\tilde{\sigma}(u)\,,$
(56)

which leads to the following relation of the norms

$\displaystyle N_{2p}\,=\,{\textstyle\frac{4\pi}{\beta_{0}}}\,d_{p}^{\rm
IR}\,=\,{\textstyle\frac{4\pi}{\beta_{0}}}\,p^{\gamma}\,{\cal N}_{p}\,.$ (57)

The norm for the renormalon terms also depends on the scheme used for the
strong coupling. For two strong coupling schemes, $a$ and $a^{*}$, which are
related by $a^{*}/a=1-\lambda\,a+\ldots\sim(1+\lambda\,a+\ldots)^{-1}$, their
QCD scales are related by $\Lambda_{\rm QCD}=\Lambda_{\rm
QCD}^{*}\,e^{\lambda/2}$. This implies that the Borel function in the two
schemes satisfy $B[\sigma](u)=B^{*}[\sigma](u)\,e^{\lambda u}$, so that their
respective IR renormalon norms are related by
$N_{2p}=N_{2p}^{*}\,e^{p\lambda}$. For a UV renormalon term with a branch
point at $u=k<0$ the analogous relation holds for $p$ replaced by $k$. For the
relation between the $C$-scheme (for $C=0$) Boito:2016pwf  and the
$\overline{\rm MS}$ scheme we have $\lambda=0$ (see the appendix of Part I
Benitez-Rathgeb:2022yqb ), so that their QCD scales and renormalon norms are
identical.

##  Appendix B GC Renormalon Norm in the Quenched Approximation

In the quenched approximation the GC renormalon norm $N_{g}^{(n_{f}=0)}$ was
determined previously in two dedicated analyses by Lee Lee:2011te  and Bali et
al. Bali:2014fea . Lee used the conformal mapping approach described in Sec.
3.2 up to ${\cal O}(\alpha_{s}^{4})$ employing the mapping of Eq. (28). He
obtained ${\cal N}_{2}^{\hat{D},(n_{f}=0)}=0.32-0.13+0.026+0.075=0.29$ for the
GC norm of the Adler function. Accounting for the conventional factor
$2\pi^{2}/3$ in the Adler function’s GC OPE correction shown in Eq. (17) and
switching to our normalization convention this corresponds to
$N_{g}^{(n_{f}=0)}=0.36-0.15+0.029+0.084=0.32$ for the GC renormalon norm.
Bali et al. used numerical stochastic perturbation theory to determine the
perturbative series for the average plaquette $P_{\rm pert}(a)\equiv\langle
P\rangle=\sum_{n=0}p^{\rm latt}_{n}\alpha^{n+1}(a)$ for the lattice spacing
$a$ up to ${\cal O}(\alpha^{35})$ in the infinite volume limit based on the
standard Wilson gauge action. Here $\alpha(a)$ stands for the strong coupling
at lattice spacing $a$ which should not be confused with our abbreviation for
the strong coupling in Eq. (13). Using a formula for the asymptotic large-$n$
behavior of the GC renormalon series $p_{n}^{\rm latt(asy)}$, the GC
renormalon norm was determined from the ratio $p^{\rm latt}_{n}/p_{n}^{\rm
latt(asy)}$. They obtained the result ${\cal N}_{2,{\rm
latt}}^{P,(n_{f}=0)}=(42\pm 17)\times 10^{4}$ for the GC renormalon norm of
the plaquette in the strong coupling lattice scheme, which results in ${\cal
N}_{2,{\rm latt}}^{P,(n_{f}=0)}=(0.61\pm 0.25)$ in $\overline{\rm MS}$ strong
coupling scheme. Accounting for the conventional factor $\pi^{2}/36$ in the
OPE corrections of the plaquette and switching to our normalization convention
this corresponds to $N_{g}^{(n_{f}=0)}=(16.4\pm 6.7)$ for the GC renormalon
norm. The results obtained by Lee and Bali et al. are clearly incompatible. It
is therefore worth to apply the methods we have discussed in Sec. 3 in the
quenched approximation. Since theoretical studies of the Adler function in the
quenched approximation have so far not relied on estimates of the ${\cal
O}(\alpha_{s}^{5})$ coefficient $\bar{c}_{5,1}$, we will in the following only
use the known cofficients up to ${\cal O}(\alpha_{s}^{4})$.

|  $m=4$
---|---
$w(u,5)$ |  $0.88$
$w(u,10)$ |  $0.76$
$w(u,15)$ |  $0.73$
$\chi^{2}_{\xi=1}~{}(\sqrt{s_{0}}=m_{\tau})$ |  $0.86\pm 0.28$
$\chi^{2}_{\xi=2}~{}(\sqrt{s_{0}}=m_{\tau})$ |  $0.65\pm 0.25$
$\chi^{2}_{\xi=1}~{}(\sqrt{s_{0}}=3~{}{\rm GeV})$ |  $0.64\pm 0.22$
$\chi^{2}_{\xi=2}~{}(\sqrt{s_{0}}=3~{}{\rm GeV})$ |  $0.55\pm 0.21$
Table 5:  Results for $N_{g}$ for the $n_{f}=0$ flavor scheme at ${\cal
O}(\alpha_{s}^{4})$ using the conformal mapping approach (upper part) as well
as the optimal subtraction approach (lower part).

Let us start with the Borel function model approach. As was already described
in Ref. Beneke:2008ad , one can use the Borel model of Eq. (26) dropping the
linear term $b^{(1)}u$ given that its coefficient is quite small. The
smallness of $b^{(1)}$ obtained for our $n_{f}=3$ analysis in Sec. 3.1
indicates that the other terms contained in the Borel model should be
sufficient to describe the known coefficients, so that dropping this term does
not deteriorate the quality of the model. Using only the Adler function
cofficients up to ${\cal O}(\alpha_{s}^{4})$ this yields $N_{g}=0.63$ for
$n_{f}=3$, which is almost identical to the case when term $b^{(1)}u$ and the
estimate of $\bar{c}_{5,1}$ are included, see the numbers quoted below Eq.
(26). Applying the same method for $n_{f}=0$ ($\hat{b}_{1}=51/121=0.421$) in
the $C$-scheme we obtain

$\displaystyle\begin{split}B[\hat{D}(s)]_{\text{mr}}^{(0)}(u)&=b^{(0)}+\frac{2\pi^{2}}{3}\frac{N_{g}^{(n_{f}=0)}\left[1-\frac{152}{363}\bar{a}(-s)\right]}{(2-u)^{1+4\hat{b}_{1}^{(0)}}}+\frac{N_{6}^{(0)}}{(3-u)^{1+2\hat{b}_{1}^{(0)}}}+\frac{N_{-2}^{(0)}}{(1+u)^{\gamma_{2-2\hat{b}_{1}^{(0)}}}}\,,\end{split}$
(58)

with $N_{g}^{(n_{f}=0)}=0.98$, $N_{6}^{(0)}=-26.12$, $N_{-2}^{(0)}=0.022$ and
$b^{(0)}=-0.12$, where the coefficients of the Adler function are given by
$\bar{c}_{1,1}^{(0)}=1$, $\bar{c}_{2,1}^{(0)}=1.986$,
$\bar{c}_{3,1}^{(0)}=20.985$, $\bar{c}_{4,1}^{(0)}=161.224$. Applying the
conformal mapping approach for that same Borel function model, we again find
that Lee’s mapping function yields a very slowly converging series that
undershoots the actual result significantly at ${\cal O}(\alpha_{s}^{4})$.
This indicates that his estimate $N_{g}^{(n_{f}=0)}=0.32$ is somewhat low. The
mapping functions of Eq. (29) again provide much better results and yield the
results shown in the upper part of Tab. 5 for $p=5,10,15$. Finally, we apply
the optimal subtraction approach, where we can also obtain a reliable result
using only information up to ${\cal O}(\alpha_{s}^{4})$. Following the method
as described in Sec. 3.3, we find the results for $N_{g}$ shown in the lower
part of Tab. 5 at order $m=4$ for $\sqrt{s_{0}}=m_{\tau}$ and $3$ GeV and
$\xi=1,2$. For illustration, in Fig. 4 we also display $N_{g}$ obtained for
$\sqrt{s_{0}}=m_{\tau}$ and $\xi=1,2$ at order $m=2,3,4$. Adopting the
envelope of all order $m=4$ results, our final result for $N_{g}^{(n_{f}=0)}$
reads:

$N_{g}^{(n_{f}=0)}=0.74\pm 0.40\,.$ (59)

The results we obtain from the Borel function model as well as from the
conformal mapping approaches are fully compatible with Eq. (59), and also
Lee’s result is compatible within uncertainties. The result, however,
disagrees with Bali et al. We note that the relative uncertainty in
$N_{g}^{(n_{f}=0)}$ obtained by the optimal subtraction approach is $54\%$ and
somewhat larger than that for the case $n_{f}=3$ in Eq. (37). This is
potentially related to the higher IR sensitivity of QCD perturbation theory in
the quenched approximation.

Figure 4:  Results for $N_{g}$ at ${\cal O}(\alpha_{s}^{2,3,4})$ using the
optimal subtraction approach for $\xi=1$ (left panel) and $\xi=2$ (right
panel) for the $n_{f}=0$ flavor scheme.

In App. C we provide arguments that suggest that Bali et al. may have
underestimated the uncertainty in their result. We therefore consider the
result quoted in Eq. (59) as a reliable determination of the GC renormalon
norm in the quenched approximation. We note that a scenario where the GC
renormalon norm $N_{g}^{(n_{f}=0)}$ would be around $16$ corresponds to a
situation that strongly contradicts the naturalness assumption we discussed in
Sec. 3 and the proposition that the Adler function series at ${\cal
O}(\alpha_{s}^{4})$ is already largely determined by the terms accounted for
in the Borel function model of Eq. (26). In fact, the parameters of the Borel
function in such a scenario would have to be highly fine-tuned, so that the
large size of the $N_{g}^{(n_{f}=0)}$ is hidden in the perturbative
coefficients up to ${\cal O}(\alpha_{s}^{4})$. By constructing and analyzing a
number of Borel function models with such a large value for
$N_{g}^{(n_{f}=0)}$, we have found that in such a scenario the perturbation
series for GCS spectral function moments entirely change their nicely
converging character beyond ${\cal O}(\alpha_{s}^{4})$, so that the values the
moments series approach for orders up to ${\cal O}(\alpha_{s}^{4})$ are far
away from the values the truncated moment series approach at higher orders. In
other words, the truncated series values at ${\cal O}(\alpha_{s}^{4})$
including the estimate of the truncation error would be far away from the true
value of the series.

##  Appendix C Comment on the Gluon Condensate Norm from Lattice QCD

In Ref. Bali:2014fea  Bali et al. determined the normalization of the GC