aainstitutetext: University of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Wien, Austriabbinstitutetext: Erwin Schrödinger
International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, Austria

# Borel Representation of $\tau$ Hadronic Spectral Function Moments in
Contour-Improved Perturbation Theory

André H. Hoang a Christoph Regner<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that the Borel representations of $\tau$ hadronic spectral function
moments based on contour-improved perturbation theory (CIPT) in general differ
from those obtained within fixed-order perturbation theory (FOPT). We
demonstrate that the Borel sums obtained from both types of Borel
representations in general differ as well and that the apparently conflicting
asymptotic behavior of the FOPT and CIPT series, which has been subject to
many studies in the past literature, can be understood quantitatively from
these results. The discrepancy between the CIPT and FOPT Borel sums, which we
call the ’asymptotic separation’, can be computed analytically and is related
to inverse exponential terms in the strong coupling. The asymptotic separation
arises from the singular and non-analytic infrared renormalon structures in
the Borel function of the underlying Adler function where the leading
dimension four gluon condensate renormalon has the highest weight. The size of
the asymptotic difference is in general larger than that of the FOPT Borel sum
ambiguity, but it can be modulated in a predictable way by choosing specific
spectral function moments. Even though moments can be designed where the
asymptotic difference is smaller than the FOPT Borel sum ambiguity, the
asymptotic separation can as a matter of principle not be avoided entirely.
The asymptotic separation has important implications for the standard operator
product expansion approach used for spectral function moment predictions.

††preprint:  UWThPh-2020-19

##  1 Introduction

Moments of the $\tau$ hadronic spectral functions obtained from the ALEPH
Barate:1998uf ; Schael:2005am ; Davier:2013sfa  and the OPAL Ackerstaff:1998yj
collaborations have served as an important tool for precise determinations of
the strong coupling $\alpha_{s}$. Theoretical predictions for the spectral
function moments can be related to the QCD vacuum vector and axial-vector
current correlator111For simplicity we assume massless quarks and neglect
electroweak corrections throughout this letter.

$\left(p^{\mu}p^{\nu}-g^{\mu\nu}p^{2}\right)\Pi(p^{2})\,\equiv\,i\\!\int\\!dx\,e^{ipx}\,\langle\Omega|\,T\\{j_{v/av,jk}^{\mu}(x)\,j_{v/av,jk}^{\nu}(0)^{\dagger}\\}\Omega\rangle\,,$
(1)

where the vector and axial-vector currents are given by
$j_{v/av,jk}^{\nu}=\bar{q}_{j}\gamma^{\mu}(\gamma_{5})q_{k}$. Accounting only
for first generation quarks, the theoretical predictions for the moments are
conventionally parametrized as Braaten:1991qm ; LeDiberder:1992jjr ;
Boito:2014sta ; Pich:2016bdg

$A_{W_{i}}(s_{0})\,=\,\frac{N_{c}}{2}|V_{ud}|^{2}\biggl{[}\,\delta^{\rm
tree}_{W_{i}}+\delta^{(0)}_{W_{i}}(s_{0})+\sum_{d\geq
2}\delta^{(d)}_{W_{i}}(s_{0})+\delta^{\rm DV}_{W_{i}}(s_{0})\,\biggr{]}\,,$
(2)

where $s_{0}$ is the upper bound of the spectral function integration and the
index $W_{i}$ indicates the type of moment considered. Furthermore, $N_{c}=3$
and $V_{ud}$ is a CKM matrix element. The term $\delta^{\rm tree}_{W_{i}}$ is
the tree-level contribution and $\delta^{(0)}_{W_{i}}(s_{0})$ are higher order
perturbative QCD corrections. The terms $\delta^{(d)}_{W_{i}}(s_{0})$
represent (vacuum matrix element) condensate corrections in the framework of
the operator product expansion (OPE) Shifman:1978bx , where the dimension
$d=4$ term is related to the gluon condensate. The standard approach to define
and compute the OPE corrections is to employ the $\overline{\mbox{MS}}$ scheme
to factorize the low-scale non-perturbative vacuum matrix elements and the
high-scale perturbative Wilson coefficients. The term $\delta^{\rm
DV}_{W_{i}}(s_{0})$ stands for duality violation corrections related to non-
perturbative contributions missed by the expansion in local condensate matrix
elements. The validity of the ansatz of Eq. (2) is based on an expansion in
inverse powers of $s_{0}$, where both types of non-perturbative corrections
are defined from the vacuum polarization function $\Pi(p^{2})$. For massless
quarks the perturbative QCD corrections can be written as a counterclockwise
contour integral in the complex $p^{2}$-plane along a circle with radius
$s_{0}$ around the origin Braaten:1991qm ; LeDiberder:1992jjr :

$\delta^{(0)}_{W_{i}}(s_{0})\,=\,\frac{1}{2\pi
i}\,\,\ointctrclockwise\limits_{|s|=s_{0}}\\!\\!\frac{{\rm
d}s}{s}\,W_{i}({\textstyle\frac{s}{s_{0}}})\,\hat{D}(s)\,.$ (3)

Here $\hat{D}(s)$ is the reduced Adler function defined as

$\frac{1}{4\pi^{2}}\Big{(}1+\hat{D}(s)\Big{)}\,\equiv\,-\,s\,\frac{{\rm
d}\Pi(s)}{{\rm d}s}\,.$ (4)

The OPE and duality violation corrections to the spectral function moments are
computed in analogy to Eq. (3) by integration over the corresponding OPE and
duality violation corrections of the Adler function. The integration path with
distance $s_{0}$ from the origin in the complex $s$-plane may be deformed as
long as the beginning and end points are at $s_{0}\pm i0$, respectively, the
path encloses the Landau pole of the strong coupling and does not cross cuts.
The weight function $W_{i}(x)$ is a polynomial that vanishes at $x=1$ and
determines the moment considered.222The weight function $W_{i}(x)$ is
connected to the weight function $w_{i}(x)$ used for integration over the
measured $\tau$ spectral function by the relation $W_{i}(x)=2\int_{x}^{1}{\rm
d}\bar{x}\,w_{i}(\bar{x})$ which arises when using integration by parts to
relate the theoretical spectral function moments to the Adler function. For
$s_{0}=m_{\tau}^{2}$ and $W_{\tau}(x)=(1-x)^{3}(1+x)$ the moment directly
applies to the normalized total hadronic $\tau$ decay width
$R_{\tau}=\Gamma(\tau^{-}\to\mbox{hadrons}\,\nu_{\tau}(\gamma))/\Gamma(\tau^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\tau}(\gamma))$. The choice of the weight function
$W_{i}(x)$ substantially affects the importance and size of the OPE and DV
corrections. For example, if $W_{i}(x)$ does not contain a quadratic term, the
effects of the dimension-4 gluon condensate is suppressed. Furthermore, DV
effects are smaller, if $W_{i}(x)$ is ’pinched’, i.e. if it vanishes
quadratically or with an even higher power at $x=1$ Boito:2014sta . It should
also be noted that the size of the standard OPE and DV terms are in one-to-one
correspondence to contributions in $\delta^{(0)}_{W_{i}}(s_{0})$, computed
within dimensional regularization and the ${\overline{\rm MS}}$
renormalization scheme in the fixed-order expansion, which diverge at large
orders of perturbation theory David:1983gz ; Mueller:1984vh ; Beneke:1998ui .
These make the series for $\delta^{(0)}_{W_{i}}(s_{0})$ asymptotic, such that
a concrete value of the entire series can only be assigned using the
renormalon calculus and based on models for the Borel representation of the
series $\delta^{(0)}_{W_{i}}(s_{0})$ Ball:1995ni ; Neubert:1995gd ;
Beneke:2008ad . This concrete value for $\delta^{(0)}_{W_{i}}(s_{0})$ is
usually called the ’Borel sum’. The size and importance of the different
renormalon contributions in the spectral function moments depend on the choice
of the weight function $W_{i}(x)$.

Using input from QCD multiloop calculations of the vacuum correlator at four
Gorishnii:1990vf ; Surguladze:1990tg  and five loops Baikov:2008jh  an
impressive precision of around $1.5\%$ has been achieved for
$\alpha_{s}(M_{Z})$, see Ref. Pdg2020  for a comprehensive recent review. This
current uncertainty of $\alpha_{s}$ determinations from $\tau$ hadronic
spectral function moments is dominated by perturbative uncertainties
associated to calculations of $\delta^{(0)}_{W_{i}}(s_{0})$. At the present
time one of the major limitations arises from the fact that two different
perturbative prescriptions to evaluate $\delta^{(0)}_{W_{i}}(s_{0})$ lead to
systematic differences that do not seem to be covered by the conventional
perturbative uncertainty estimates related to renormalization scale
variations.

Starting from the QCD perturbation series of the reduced Adler function333We
adopt the notation from Ref. Beneke:2008ad , where the coefficients $c_{n,k}$
are defined from the perturbation series of the vacuum polarization function
$\Pi(s)$.

$\displaystyle\hat{D}(s)$ $\displaystyle\,=\,$
$\displaystyle\,\sum\limits_{n=1}^{\infty}c_{n,1}\,\big{(}{\textstyle\frac{\alpha_{s}(-s)}{\pi}}\big{)}^{n}\,,$
(5) $\displaystyle\,=\,$
$\displaystyle\,\sum\limits_{n=1}^{\infty}\,\big{(}{\textstyle\frac{\alpha_{s}(s_{0})}{\pi}}\big{)}^{n}\,\sum\limits_{k=1}^{n+1}k\,c_{n,k}\,\ln^{k-1}({\textstyle\frac{-s}{s_{0}}})\,,$
(6)

the first prescription, called contour-improved perturbation theory (CIPT), is
directly based on the evaluation of Eq. (5) such that the perturbative
corrections to the moments adopt the form Pivovarov:1991rh

$\delta^{(0),{\rm CIPT}}_{W_{i}}(s_{0})\,=\,\,\frac{1}{2\pi
i}\,\sum\limits_{n=1}^{\infty}c_{n,1}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm
d}x}{x}\,W_{i}(x)\,\big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\pi}}\big{)}^{n}\,,$
(7)

where the contour integral is written in terms of the dimensionless variable
$x=s/s_{0}$ and the CIPT series is obtained by truncating the sum over $n$. It
has been argued in Ref. LeDiberder:1992jjr  that the CIPT approach sums large
perturbative corrections (involving factors of the phase of $s$) along the
path in the complex plane and that this leads to a fast decrease of the size
of the series coefficients with $n$. The second prescription, called fixed-
order perturbation theory (FOPT), is based on Eq. (6) leading to

$\delta^{(0),{\rm FOPT}}_{W_{i}}(s_{0})\,=\,\,\frac{1}{2\pi
i}\,\sum\limits_{n=1}^{\infty}\,\big{(}{\textstyle\frac{\alpha_{s}(s_{0})}{\pi}}\big{)}^{n}\,\sum\limits_{k=1}^{n+1}k\,c_{n,k}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm
d}x}{x}\,W_{i}(x)\,\ln^{k-1}(-x)\,,$ (8)

where the FOPT series is obtained by summing over $k$ as shown, but truncating
the sum over $n$. It has been argued in Ref. Ball:1995ni  that the FOPT
prescription leads to a more efficient realization of cancellations of
asymptotic renormalon contributions at large orders in association to the
suppression of OPE corrections (with respect to the vacuum polarization
function) depending on the choice of the weight function. In general, for
physically well-motivated weight functions (including $W_{\tau}$) the CIPT and
FOPT series each converge quite well. It has also been observed that for many
weight functions the CIPT series seems to converge somewhat more quickly and
leads to smaller renormalization scale variation. However, with the advent of
the five loop coefficient Baikov:2008jh , it became apparent for the total
hadronic $\tau$ width that both convergent series seem to approach values that
appear incompatible within their respective renormalization scale variation,
where CIPT in general leads to the smaller result. As a result, and in absence
of an argument that would imply differences for the OPE and DV corrections
within both approaches, strong coupling determinations based on FOPT lead to
systematically smaller fitted values for $\alpha_{s}$.

This discrepancy has motivated a number of studies Jamin:2005ip ;
Beneke:2008ad ; Caprini:2009vf ; DescotesGenon:2010cr ; Caprini:2011ya ;
Beneke:2012vb  based on concrete models for the Borel (transformation)
function of the Adler function to explore the resulting higher order behavior
of the CIPT and FOPT series beyond the concretely known five-loop level and to
learn more about the underlying systematics and to possibly identify which one
is ’better’, if at all. It was shown in Ref. DescotesGenon:2010cr  that, if
one makes no assumption on whether the known (and apparently convergent) five-
loop series for the Adler function is already close to its asymptotic
limit,444This assumption implies the possibility that the gluon condensate OPE
correction of the Adler function may not be the dominant source of the Adler
function’s ambiguity in perturbation theory. the construction of Borel models
allows for so much freedom that no insight can be gained at all. On the other
hand, if one argues that the known five-loop series is already close to its
asymptotic limit Beneke:2012vb , the terms in the Adler function Borel
function related to the gluon condensate dominate, and definite studies of the
higher-order behavior can be carried out. In this context it was found that
FOPT in general approaches to the Borel sum, while CIPT seems to approach a
value as well, which can, however, be significantly different from the Borel
sum.

Interestingly, in all these studies it has been commonly assumed that the
correct Borel representations (and thus the corresponding Borel sums) for the
perturbative moments in the FOPT as well as the CIPT approach agree and have
the concrete form

$\delta_{W_{i},{\rm Borel}}^{(0),{\rm
FOPT}}(s_{0})=\int_{0}^{\infty}\\!\\!{\rm d}u\,\,\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{|x|=1}\frac{{\rm
d}x}{x}\,W_{i}(x)\,B[\hat{D}](u)\,e^{-\frac{4\pi
u}{\beta_{0}\alpha_{s}(-xs_{0})}}\,,$ (9)

where $B[\hat{D}](u)$ is the Borel function of the reduced Adler function,
defined with respect to $\alpha_{s}(-s_{0})$. Indeed, expanding in powers of
$u$ and carrying out the Borel integral for each term gives the CIPT series in
Eq. (7). Furthermore, expanding the RHS of Eq. (9) in powers of
$\alpha_{s}(s_{0})$ one can, due to the convergence properties of the
exponential function, recover the Borel representation of the FOPT series in
Eq.(8) without changing the value of the integral. Given the contradictory
factual behavior of the CIPT series, the observed discrepancy directed
theoretical studies on the questions why the CIPT series behaves in this way
and how this behavior may be avoided. This is the point from which we start
the discussion of this article. We show that Eq. (9) is not the correct Borel
representation for the CIPT series of the $\tau$ spectral function moments.

A close inspection reveals that, even though an expansion of Eq. (9) in powers
of $u$ recovers the terms in Eq. (7), the expansion in powers of the path
dependent coupling $\alpha_{s}(-xs_{0})$ shown in Eq. (7) does not reflect the
true character of the CIPT series since the contour integration still has to
be carried out. This means that the effects of the contour integration should
be considered as part of the coefficients of the resulting series and not as
part of the expansion parameter. So when defining the Borel function for the
CIPT series for the spectral function moments one should instead of Eq. (7)
consider the series

$\delta^{(0),{\rm CIPT}}_{W_{i}}(s_{0})\,=\,\,\frac{1}{2\pi
i}\,\sum\limits_{n=1}^{\infty}c_{n,1}\,\Big{[}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm
d}x}{x}\,W_{i}(x)\,\big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\big{)}^{n}\,\Big{]}\,\big{(}{\textstyle\frac{\alpha_{s}(s_{0})}{\pi}}\big{)}^{n}\,.$
(10)

and carry out the Borel transformation with respect to an expansion in the
fixed coupling $\alpha_{s}(s_{0})$. The resulting Borel representation differs
from that in Eq. (9) and has the form

$\delta_{W_{i},{\rm Borel}}^{(0),{\rm
CIPT}}(s_{0})=\int_{0}^{\infty}\\!\\!{\rm d}\bar{u}\,\,\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W_{i}(x)\,\big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\big{)}\,B[\hat{D}]\Big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\bar{u}\Big{)}\,e^{-\frac{4\pi\bar{u}}{\beta_{0}\alpha_{s}(s_{0})}}\,,$
(11)

where the path ${\cal C}_{x}$ in the complex $x$-plane has its beginning and
end points at $x_{\mp}=1\pm i0$, respectively, but must be deformed away from
$|x|=1$ to account for the modified singularity structure depending on the
value of the Borel variable $\bar{u}$. It is straightforward to show that Eqs.
(9) and (11) are equivalent when one expands (either one of Eqs. (9) or (11))
in the difference $\alpha_{s}(-xs_{0})-\alpha_{s}(s_{0})$ prior to the contour
integration and employs partial integrations in $u$ or $\bar{u}$. Furthermore,
it seems tempting to identify Eqs. (9) and (11) as they seem to be related by
the simple change of variables
$u=\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}\bar{u}$. However, both
definitions do in general not agree.

We claim that Eq. (11) is the correct Borel representation of the CIPT series
for the $\tau$ spectral function moments. In this letter we explore its
properties and we show that the CIPT series indeed converge to the Borel sum
of Eq. (11) in the same way as the FOPT series converge to the Borel sum of
Eq. (9). The difference, which we call the ’asymototic separation’, can be
computed analytically and is related to terms that are suppressed by powers of
$\Lambda_{\rm QCD}/s_{0}$ and thus involve exponentials of the inverse strong
coupling that vanish to all orders in the fixed-order expansion. The
asymptotic separation arises from the singular and non-analytic infrared
renormalon structures located along the positive real axis in complex Borel
plane of the Adler function, and its size is in general larger than the
ambiguity that is commonly assigned to the Borel sum of the FOPT series. In
this article we examine the anatomy of the CIPT Borel representation, compute
the asymptotic separation in detail and demonstrate that the difference in the
observable asymptotic behavior of the CIPT and the FOPT series is described
correctly by the asymptotic separation. The existence of the asymptotic
separation has important implications for the standard OPE approach for the
spectral function moments because both cannot be reconciled. We believe that
our results may contribute towards a more refined understanding of strong
coupling determinations from hadronic $\tau$ decays and of the OPE concept.

This article is organized as follows: In Sec. 2 we examine the structure and
the analytic properties of the Borel representation and Borel function of the
CIPT series in comparison to the FOPT series. We discuss the perturbative
construction of the CIPT Borel function in powers of the Borel space variable
$\bar{u}$, the deformation of the contour ${\cal C}_{x}$ needed to define the
Borel integral and the analytic form of the CIPT Borel function. In Sec. 3 we
discuss the properties of the asymptotic separation and present its analytic
form in comparison with the expression for the ambiguity of the FOPT Borel
sum. That the asymptotic separation provides the correct description of the
observable difference in the asymptotic behavior of the FOPT and CIPT series
in the large-$\beta_{0}$ approximation is demonstrated in Sec. 4 for
$R_{\tau}$ and several other moments and Borel functions. In Sec. 5 a similar
analysis is carried out accounting for all known corrections of the QCD
$\beta$-function up to $5$ loops. We show that, again, the asymptotic
separation provides the correct description of the observable difference in
the asymptotic behavior of the FOPT and CIPT series. Finally, in Sec. 6 we
briefly discuss some of the implications of the findings of this article for
the standard OPE approach and from a mathematical perspective, and we
conclude.

##  2 Anatomy of the CIPT Borel Function

In this section we discuss the anatomy of the CIPT Borel representation given
in Eq. (11) in comparison to the Borel representation of the FOPT series in
Eq. (9). To be definite, we consider generic terms in the Borel function of
the reduced Adler function of the form

$B^{\rm IR}_{\hat{D},p,\gamma}(u)\,=\,\frac{1}{(p-u)^{\gamma}}$ (12)

for an infrared (IR) renormalon (with $p$ being a positive integer and
$\gamma$ being real) and

$B^{\rm
UV}_{\hat{D},-\tilde{p},\gamma}(u)\,=\,\frac{1}{(\tilde{p}+u)^{\gamma}}$ (13)

for an ultra-violet (UV) renormalon (with $\tilde{p}=-p$ being a positive
integer and $\gamma$ being real). The Adler function’s Borel function is known
to be an infinite linear combination of such generic terms plus possible
functions that are analytic everywhere along the real $u$-axis (see e.g. Ref.
Beneke:1998ui ). The non-analytic (or singular) structure of an IR renormalon
contribution (cut for $u>p$) is located on the positive real axis, while the
non-analytic (or singular) structure of an UV renormalon contribution (cut for
$u<-\tilde{p}=p$) is located on the negative real axis. Typically the
renormalon terms with the smallest values of $|p|$ dominate the behavior of
the perturbative series. The notation allows to formulate analytic expressions
that apply both to IR and UV renormalons since we can write the generic Borel
functions of Eqs. (12) and (13) collectively as $B^{\rm
IR/UV}_{\hat{D},p,\gamma}(u)=1/[{\rm sign}(p)(p-u)]^{\gamma}$. The non-
analytic structure of the generic infrared Borel transform term in Eq. (12) is
in one-to-one correspondence to an equal-sign factorially divergent behavior
of the perturbation series and entails an arbitrariness in the definition of
the inverse Borel integral for $u>p$. The associated renormalon ambiguity is
related to a non-perturbative term in the OPE involving a condensate vacuum
matrix element of dimension $d=2p$. The non-analytic structure of the generic
UV Borel transform term in Eq. (13) is in one-to-one correspondence to a sign-
alternating factorially divergent behavior of the perturbation series and does
not affect the definition of the inverse Borel integral. Thus the
corresponding sign-alternating factorially divergent perturbation series can
be formally summed without an ambiguity by the Borel representation.

###  2.1 Convergence Radius in Borel Space

Let us first discuss the perturbative properties of the Borel functions for
the CIPT and FOPT spectral function moment series when considered as a series
in powers of the Borel space variables obtained from the perturbation series
in $\alpha_{s}(s_{0})$. The aim of this examination is to compare the radius
of convergence for the two types of Borel functions and to introduce the
$t$-variable notation that allows to account for the higher order corrections
of the QCD $\beta$-function in a transparent analytic way.

To determine the convergence radius of the $u$-series for the FOPT Borel
function let us consider the generic FOPT Borel representation for the
polynomial weight function $W(x)=(-x)^{m}$,

$\delta_{\\{(-x)^{m},p,\gamma\\},{\rm Borel}}^{(0),{\rm
FOPT}}(s_{0})=\int_{0}^{\infty}\\!\\!{\rm d}u\,B^{\rm
IR/UV}_{\hat{D},p,\gamma}(u)\,\,\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{|x|=1}\frac{{\rm
d}x}{x}\,(-x)^{m}\,e^{-\frac{u}{a(-x)}}\,,$ (14)

where we define

$\displaystyle a(x)\,$
$\displaystyle\equiv\,\frac{\beta_{0}\,\alpha_{s}(s)}{4\pi}\,=\,\frac{\beta_{0}\,\alpha_{s}(xs_{0})}{4\pi}\,,$
$\displaystyle a_{0}\,$
$\displaystyle\equiv\,\frac{\beta_{0}\,\alpha_{s}(s_{0})}{4\pi}\,=\,a(1)\,.$
(15)

We see that the contour integral modifies the singular term contained in the
generic Borel function $B^{\rm IR/UV}_{\hat{D},p,\gamma}(u)$, but it does in
general not eliminate it555In the large-$\beta_{0}$ approximation $\gamma$ can
only adopt integer values so that the singular structures of the Adler funtion
Borel transform only involve poles. This allows for the possibility of an
elimination of a simple pole, which, however, cannot happen for the cuts that
appear beyond the large-$\beta_{0}$ approximation. since it does not depend on
$p$ and $\gamma$. Thus the radius of convergence of the Borel function’s
$u$-series for the FOPT moment series and the Adler function is $|p|$ and
agrees.

To determine the convergence radius of the $\bar{u}$-series for the CIPT Borel
function let us consider the CIPT perturbation series for the spectral
function moment with the weight function $(-x)^{m}$:

$\displaystyle\delta^{(0),{\rm CIPT}}_{\\{(-x)^{m}\\}}(s_{0})$
$\displaystyle\,=\,\sum\limits_{n=1}^{\infty}c_{n,1}\,J_{n,m}(s_{0})\,\Big{(}{\textstyle\frac{\alpha_{s}(s_{0})}{\pi}}\Big{)}^{n}\,,$
(16)

where

$\displaystyle J_{n,m}(s_{0})$ $\displaystyle\,=\,\frac{1}{2\pi
i}\,\ointctrclockwise\limits_{|x|=1}\\!\\!\frac{{\rm
d}x}{x}\,(-x)^{m}\,\Big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\Big{)}^{n}\,.$
(17)

We now change the integration variable $x$ to an integration over

$t\,\equiv\,-\frac{2\pi}{\beta_{0}\alpha_{s}(-xs_{0})}\,=\,-\frac{1}{2\,a(-x)}\,.$
(18)

We can then write ($t_{0}=t(\alpha_{s}(\mu_{0}^{2}))$,
$t_{1}=t(\alpha_{s}(\mu_{1}^{2}))$)

$\ln\Big{(}\frac{\mu_{1}^{2}}{\mu_{0}^{2}}\Big{)}\,=\,2\int_{\alpha_{s}(\mu_{0}^{2})}^{\alpha_{s}(\mu_{1}^{2})}\,\frac{{\rm
d}\alpha}{\beta(\alpha)}\,=\,-2\int_{t_{0}}^{t_{1}}\,{\rm
d}t\,\hat{b}(t)\,=\,2\,[G(t_{0})-G(t_{1})]\,,$ (19)

where $\beta(\alpha_{s})$ is the QCD $\beta$-function, with
$\beta_{0}=11-2/3n_{\ell}$ being the one-loop coefficient, and

$\displaystyle\hat{b}(t)$ $\displaystyle\equiv
1+\sum_{k=1}^{\infty}\frac{\hat{b}_{k}}{t^{k}}$ (20)

is a series in inverse powers of $t$ arising from the inverse of the QCD
$\beta$-function. The coefficients $\hat{b}_{k}$ are functions of the
$\beta$-function coefficients and the function $G(t)$ is defined as the
indefinite integral of $\hat{b}(t)$. We refer to the appendix of Ref.
Hoang:2017suc  for the explicit analytic expressions. Using the $t$-variable
notation it is straightforward to derive explicit analytic expressions
concerning the contour integral accounting for the evolution of the strong
coupling according to the exact QCD $\beta$-function. Furthermore the change
of variable provides a mapping of the complex $x$-plane onto a band around the
real $t$-axis, where $t\to-\infty$ corresponds to $x\to-\infty$ and
$t\to+\infty$ corresponds to $x\to 0$. The Landau pole at $s=\Lambda_{\rm
QCD}^{2}$ corresponds to $t=0$. The cut along the positive real $x$ axis is
split-mapped onto lines roughly parallel to the real $t$-axis, where the
distance is related to the imaginary part of the strong coupling and $x\pm i0$
correspond to the line above/below the real axis. The mapping of these
characteristics and also of the contour paths relevant for our discussions
below are illustrated in Fig. 1.

(a) complex $x$-plane

(b) complex $t$-plane

Figure 1:  Panel (a): Path of the $|x|=1$ contour integration and
characteristic points and lines in the complex $x$-plane. The upper and lower
parts of the branch cut of the coupling $a(-x)$ are indicated by orange and
red lines, respectively. Panel (b): Corresponding path and lines in the
complex $t$-plane, where the arrows correspond to the directions shown in
panel (a).

Using the definition of the scale $\Lambda_{\rm QCD}$ given by666Note that Eq.
(21) is related to the conventional $\overline{\rm MS}$ definition Pdg2020  by
$\Lambda_{\rm QCD}^{\overline{\rm MS}}=2^{\hat{b}_{1}}\Lambda_{\rm QCD}$.

$\displaystyle\Lambda_{\rm
QCD}\,\equiv\,\mu\,e^{G(t(\alpha_{s}(\mu^{2})))}\,,$ (21)

we can rewrite Eq. (17) as

$\displaystyle J_{n,m}(s_{0})$
$\displaystyle\,=\,\frac{i}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}(-t_{0})^{n}\,\int_{t_{-}}^{t_{+}}{\rm
d}t\,(-t)^{-2m\hat{b}_{1}-\ell-n}\,e^{-2mt}\,$
$\displaystyle\,=\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\tilde{H}(n,m,-2m\hat{b}_{1}-\ell-n,s_{0})\,,$
(22)

with

$\displaystyle t_{\pm}\equiv$
$\displaystyle-\frac{2\pi}{\beta_{0}\alpha_{s}(-s_{0}\pm
i0)}\,=\,-\frac{1}{2\,a(-1\pm i0)}\,,$ $\displaystyle t_{0}\equiv$
$\displaystyle-\frac{2\pi}{\beta_{0}\alpha_{s}(s_{0})}\,=\,-\frac{1}{2\,a(1)}\,.$
(23)

The coefficients $\tilde{g}_{\ell}^{(2m)}$ are defined by the relation

$\displaystyle\hat{b}(t)\,e^{-2mG(t)}e^{2mt}(-t)^{2m\hat{b}_{1}}\,=\,\sum_{\ell=0}^{\infty}\tilde{g}_{\ell}^{(2m)}(-t)^{-\ell}\,.$
(24)

We have $\tilde{g}_{0}^{(2m)}=1$, and the expressions for $\ell=1,2,\ldots$
can be obtained in a straightforward way from the functions $\hat{b}(t)$ and
$G(t)$. In the $\overline{\rm MS}$ scheme for the strong coupling the series
in $\ell$ is infinite. However, the series terminates if a scheme is adopted,
where the series for the function $\hat{b}(t)$ terminates, see e.g. Refs.
Brown:1992pk ; Boito:2016pwf . In the large-$\beta_{0}$ approximation, where
$\hat{b}(t)=1$ and $\hat{b}_{k}=0$ for all $k$, we also have
$\tilde{g}_{\ell}^{(2m)}=\delta_{\ell 0}$. The function $\tilde{H}$ can be
readily evaluated and reads

$\displaystyle\tilde{H}(1,0,\eta,s_{0})$
$\displaystyle\,=\,-\,\frac{i}{\pi}\,(-t_{0})^{n}\,\ln\Big{(}\frac{t_{+}}{t_{-}}\Big{)}\,,$
(25) $\displaystyle\tilde{H}(n\geq 2,0,\eta,s_{0})$
$\displaystyle\,=\,-\,\frac{i}{\pi}\,\frac{(-t_{0})^{1+\eta+n}}{1+\eta}\bigg{[}\,\Big{(}\frac{t_{+}}{t_{0}}\Big{)}^{1+\eta}-\Big{(}\frac{t_{-}}{t_{0}}\Big{)}^{1+\eta}\,\bigg{]}\,,$
(26) $\displaystyle\tilde{H}(n,m\geq 1,\eta,s_{0})$
$\displaystyle\,=\,\,\frac{i}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,(2m)^{-1-\eta}\,(-t_{0})^{n}\,\bigg{[}\,e^{i\pi(1+\eta)}\Gamma(1+\eta,2mt_{+})$
$\displaystyle\hskip
85.35826pt-e^{-i\pi(1+\eta)}\Gamma(1+\eta,2mt_{-})-\frac{2\pi
i}{\Gamma(-\eta)}\,\bigg{]}\,.$ (27)

In the large-$\beta_{0}$ approximation the function $\tilde{H}$ can be
concisely written in the form

$\displaystyle\tilde{H}(n,0,-n,s_{0})$ $\displaystyle\,=\,{}_{2}{\rm
F}_{1}\Big{(}\frac{n}{2},\frac{n+1}{2},\frac{3}{2},-a_{0}^{2}\pi^{2}\Big{)}\,,$
(28) $\displaystyle\tilde{H}(n,m\geq 1,-n,s_{0})$
$\displaystyle\,=\,\,\frac{i}{2\pi}\,(-m)^{n-1}\,(-2t_{0})^{n}\,e^{2mt_{0}}\,\bigg{[}\,\Gamma(1-n,m(2t_{0}-i\pi))$
$\displaystyle\hskip 85.35826pt-\Gamma(1-n,m(2t_{0}+i\pi))+(-1)^{n}\frac{2\pi
i}{\Gamma(n)}\,\bigg{]}\,.$ (29)

These functions together with Eq. (2.1) provide an analytic method to
determine the contour integrals of Eq. (17) and can be used to determine the
radius of convergence. Using the leading asymptotics for the incomplete
$\Gamma$-function when its first argument adopts large negative values
$\Gamma(\alpha\to-\infty,z)\simeq|z|^{\alpha}e^{-x}e^{i\alpha\arg(z)}/\alpha$
and the fact that the large-$n$ behavior of $J_{n,m}(s_{0})$ is dominated by
the $\ell=0$ term, one can see that
$|J_{n,m}(s_{0})|^{1/n}\simeq|\alpha_{s}(-s_{0})/\alpha_{s}(s_{0})|$ for
$n\to\infty$. Applying the root criterion for the $\bar{u}$ series of the
generic CIPT Borel function

$\displaystyle B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
CIPT}}(s_{0})\Big{]}(\bar{u})$
$\displaystyle=\sum\limits_{n=0}^{\infty}\frac{\Gamma(1-\gamma)}{\Gamma(1-\gamma-n)\Gamma(n+1)}\,J_{n+1,m}(s_{0})\,(\text{sign}(p)p)^{-\gamma}\,\Big{(}\frac{-\bar{u}}{p}\Big{)}^{n}$
(30)

we can thus see that the convergence radius is
$p|\alpha_{s}(s_{0})/\alpha_{s}(-s_{0})|=p|a_{0}/a(-1)|>p$. In the
large-$\beta_{0}$ approximation (or at the leading logarithmic approximation
for $\alpha_{s}$) we have $\frac{1}{a(x)}=\frac{1}{a_{0}}+\ln(x)$, and the
convergence radius reads $p|1+ia_{0}\pi|$. The results show that the Borel
function of the FOPT series can be constructed perturbatively for $u<p$, and
relies on an analytic continuation for $u>p$ in the same way as the Borel
transform of the Adler function. In contrast, the Borel function of the CIPT
series can be constructed perturbatively further out into the complex Borel
plane and does not inherit the singularity from the Adler function Borel
transform related to cuts for $|u|>p$. This underlines the different character
of the CIPT Borel function. Furthermore, the explicit evaluation of the series
in Eq. (30) within its radius of convergence provides an essential consistency
check for the analytic form of the CIPT Borel function discussed below.

###  2.2 Path of the Contour Integral in the Invariant Mass Plane

The invariant mass contour integration involved in the computation of the
perturbative QCD corrections to the spectral function moments
$A_{W_{i}}(s_{0})$ conventionally involves a circular path in the complex
$x$-plane with radius $|x|=1$ which begins/ends at the points located at
$x_{\mp}=1\pm i0$, see Fig. 1(a) for a graphical illustration. The path
applies both to the coefficients of the perturbation series for the CIPT and
the FOPT approach, see Eqs. (7) and (8), respectively, as well as for the
Borel representation of the FOPT series given in Eq. (9). There is the
possibility to deform this path without changing the result as long as the
path encloses the Landau pole of the strong coupling (illustrated by the green
dot in Figs. 1), does not cross the analyticity cuts of the Adler function and
the strong coupling along the positive real $x$-axis and stays within the
perturbative regime. Which path among those one actually picks is therefore a
matter of practical choice.

However, for the Borel representation of the CIPT series additional
restrictions arise on the contour of the $x$-integration for IR renormalons
since the non-analytic structures in the Adler function’s Borel function in
the $u$-plane affect the analytic properties of the integrand in the complex
$x$-plane. Let us consider the CIPT Borel representation for a generic non-
analytic term in the Borel transform of the reduced Adler function related to
an IR or a UV renormalon:

$\displaystyle\delta_{\\{W_{i},p,\gamma\\},{\rm Borel}}^{(0),{\rm
CIPT}}(s_{0})=$ $\displaystyle\int_{0}^{\infty}\\!\\!{\rm
d}\bar{u}\,\,\frac{1}{2\pi i}\,\ointctrclockwise\limits_{{\cal
C}_{x}}\frac{{\rm
d}x}{x}\,W_{i}(x)\,\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\Big{)}\,\frac{1}{\big{[}{\rm
sign}(p)\big{(}p-\frac{a(-x)}{a_{0}}\bar{u}\big{)}\big{]}^{\gamma}}\,e^{-\frac{\bar{u}}{a_{0}}}$
(31)

For the case of a UV renormalon ($p<0$) the pattern, where the non-analytic
structures appear in the complex $x$-plane, are the same as for the FOPT Borel
representation. This is because the real part of the strong coupling is always
positive as long as its scale remains in the perturbative regime, and thus the
circular path with $|x|=1$ can also be adopted for UV renormalon. However, for
an IR renormalon ($p>0$) we can see that, apart from the Landau pole and the
cut along the positive real $x$ axis, which arise from the strong coupling
function, there is an additional cut along the negative real $x$-axis for
${\rm Re}[\alpha_{s}(-xs_{0})]>p\alpha_{s}(s_{0})/\bar{u}$. As long as
$\bar{u}<p$ this cut is still within the conventional circular path with
radius $1$, but for $\bar{u}>p$ the path ${\cal C}_{x}$ must be deformed
further into the negative real complex plane to not cross the cut. In the
large-$\beta_{0}$ approximation the corresponding cuts reduce to poles located
at $\tilde{x}(\bar{u})=-e^{\frac{2(p-\bar{u})t_{0}}{p}}=-(\Lambda_{\rm
QCD}^{2}/s_{0})e^{\frac{\bar{u}}{pa_{0}}}$ and the path must cross the
negative real axis for $x<\tilde{x}(\bar{u})$. Interestingly, for
$\bar{u}\to\infty$ the allowed region where the path can cross the negative
real $x$-axis it shifted towards negative infinity. For the computation of the
CIPT Borel sum of Eq. (31) this means that for IR renormalons the path of the
contour integration must be deformed to minus negative real infinity if the
Borel integration in $\bar{u}$ is carried out first. This will be an essential
element for the explicit evaluation of the CIPT Borel sum and the asymptotic
separation for IR renormalons that is discussed in Sec. 3.

###  2.3 Form of the Borel Function

Let us now examine the explicit analytic expressions for the FOPT and CIPT
spectral moment Borel representations arising from the generic IR and UV Borel
function terms in Eqs. (12) and (13). We again consider the polynomial weight
function $W(x)=(-x)^{m}$.

For the FOPT approach the generic expression is given in Eq. (14). Changing to
the $t$ variable for the contour integration the result can be rewritten in
the form

$\displaystyle\delta_{\\{(-x)^{m},p,\gamma\\},{\rm Borel}}^{(0),{\rm
FOPT}}(s_{0})$ $\displaystyle\qquad=\frac{i}{\pi}\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,B^{\rm IR/UV}_{\hat{D},p,\gamma}(u)\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\int_{t_{-}}^{t_{+}}{\rm
d}t\,(-t)^{-2m\hat{b}_{1}-\ell}\,e^{2(u-m)t}\,$
$\displaystyle\qquad=\int_{0}^{\infty}\\!\\!{\rm
d}u\,e^{-\frac{u}{a_{0}}}\,B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
FOPT}}(s_{0})\Big{]}(u)\,$ (32)

where

$\displaystyle B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
FOPT}}(s_{0})\Big{]}(u)=B^{\rm
IR/UV}_{\hat{D},p,\gamma}(u)\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\tilde{F}(m,-2m\hat{b}_{1}-\ell,s_{0};u)\,.$
(33)

In the complex $t$-plane, see Fig. 1(b), the circular path of the $x$-contour
integration corresponds to an essentially straight line connecting the points
$t_{-}$ and $t_{+}$ which are located in the negative real complex half plane
on opposite sides of the real axis. The function $\tilde{F}$ can then be
readily evaluated giving

$\displaystyle\tilde{F}(m,\eta,$ $\displaystyle
s_{0};u)\,=\,\,\frac{i}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,2^{-1-\eta}\,e^{-2ut_{0}}\,\Big{[}(u-m-i0)^{-1-\eta}\,\Gamma(1+\eta,2(m-u)t_{+})$
$\displaystyle-(u-m+i0)^{-1-\eta}\,\Gamma(1+\eta,2(m-u)t_{-})-\Theta(m-u)(m-u)^{-1-\eta}\,\frac{2\pi
i}{\Gamma(-\eta)}\Big{]}\,,$ (34)

for real-valued $u$, where the term involving the Heaviside step function
$\Theta$ arises due to the cut of the incomplete $\Gamma$-function along the
negative real axis in its second argument. As can be seen from the leading
asymptotics for the incomplete $\Gamma$-function when its second argument
becomes large $\Gamma(\alpha,z\to\infty)\simeq z^{\alpha-1}e^{-z}$, we have
$\tilde{F}\sim e^{2u(t_{\pm}-t_{0})}/u$ for large values of $u$. Because ${\rm
Re}[t_{\pm}-t_{0}]<0$, this provides an exponential suppression for large
positive $u$. Furthermore the complex second arguments of the incomplete
$\Gamma$-functions cause an additional oscillatory dependence with zeros at
non-integer values for $u$. Thus the function $\tilde{F}$ modulates (and
partially suppresses) the singular and non-analytic structures of the Borel
transform of the reduced Adler function on the real $u$ axis for $u>p$, but it
does not eliminate them as we have already mentioned in Sec. 2.1. This
visualizes the statement we have made for the convergence radius of the $u$
series for Eqs. (14) and (2.3). The associated arbitrariness in the definition
of the Borel integral of Eq. (2.3) in the case of IR renormalons for $u>p$
constitutes the well-known renormalon ambiguity of the FOPT spectral function
moments. In the large-$\beta_{0}$ approximation the exponential suppression
does not arise because ${\rm Re}[t_{\pm}-t_{0}]=0$ and the $\Gamma$-functions
acquire zeros at integer values for $u$. Here, only the $\ell=0$ term
contributes and we have Ball:1995ni

$\displaystyle\tilde{F}(m,0,$ $\displaystyle
s_{0};u)\,=\,(-1)^{m}\,\frac{\sin(u\pi)}{\pi(u-m)}\,,$ (35)

so $\tilde{F}$ has zeros at integer values for $u$, except for $u=m$. Since in
the large-$\beta_{0}$ approximation only single or double poles arise in the
Borel transform of the reduced Adler function, the function $\tilde{F}$
eliminates the single poles (and their associated renormalon ambiguity) and
reduces the double poles to single poles if $p\neq m$. In the
large-$\beta_{0}$ approximation the Adler function’s Borel function contains
only a single pole at $u=p=2$ which corresponds to the $d=4$ gluon condensate
OPE correction. For the Adler function the $p=2$ IR renormalon dominates the
behavior of the series at low orders and the corresponding gluon condensate
OPE term represents the largest OPE correction. Because the polynomial weight
function for the tau hadronic decay rate $R_{\tau}$ does not contain a term
with $m=2$, the effects of the $p=2$ IR renormalon in
$\delta_{w_{\tau}}^{(0),{\rm FOPT}}(s_{0})$ are eliminated completely. The
complete removal of the $p=2$ renormalon ambiguity is, however, only possible
within the large-$\beta_{0}$ approximation. So, when the higher loop
corrections in the QCD $\beta$ function are accounted for, the effects of the
$p=2$ IR renormalon are strongly suppressed but not eliminated. Interestingly,
the modulation of the IR renormalon structure in the FOPT series caused by the
contour integration (and visualized in the form of the function $\tilde{F}$)
is in one-to-one correspondence to analogous modulations of the OPE
corrections. This is because IR renormalon contributions in the FOPT series at
high orders develop the same dependence on inverse powers of $s_{0}$ (and even
logarithms of $s_{0}$) as the corresponding terms of the OPE corrections. Thus
a suppression (or elimination) of an IR renormalon term in $\delta^{(0),{\rm
FOPT}}(s_{0})$ implies a suppression (or elimination) of the associated OPE
correction term. So the elimination of the $p=2$ IR renormalon in the
large-$\beta_{0}$ approximation also implies the absence of the gluon
condensate correction. We emphasize that this association of IR renormalon
contributions in the FOPT series and OPE corrections is valid when the
standard approach for the OPE corrections is used. We see below that this
association between the Borel representation and the standard OPE corrections
is not preserved in the CIPT approach.

For comparison, let us now have a close look at the spectral moment Borel
representation in the CIPT approach for the generic IR renormalon contribution
given in Eq. (31) and the weight function $W(x)=(-x)^{m}$. Switching again to
the contour integration variable $t$, the result can be rewritten in the form

$\displaystyle\delta_{\\{(-x)^{m},p,\gamma\\},{\rm Borel}}^{(0),{\rm
CIPT}}(s_{0})\,=\,\int_{0}^{\infty}\\!\\!{\rm
d}\bar{u}\,e^{-\frac{\bar{u}}{a_{0}}}\,B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
CIPT}}(s_{0})\Big{]}(\bar{u})\,,$ (36)

where

$\displaystyle B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
CIPT}}(s_{0})\Big{]}(\bar{u})\,=\,$
$\displaystyle-\frac{i\,|p|^{-\gamma}\,t_{0}}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}$ (37)
$\displaystyle\times\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\int\limits_{{\cal
C}_{t}}{\rm
d}t\,(-t)^{-2m\hat{b}_{1}+\gamma-\ell-1}\,\frac{e^{-2mt}}{(\frac{t_{0}\bar{u}}{p}-t)^{\gamma}}\,.$

The integration path ${\cal C}_{t}$ starts and ends at $t_{-}$ and $t_{+}$,
respectively, and furthermore crosses the real axis at $t<t_{0}\bar{u}/p$.
Because $\mbox{Re}(t_{0})<0$, this means that for $\bar{u}>p$ the path can in
general not connect the points $t_{\mp}$ in a straight line and must be
deformed further into the negative real complex half plane – as we have
already mentioned in Sec. 2.2 considering the path ${\cal C}_{x}$ in the
complex $x$-plane. The expression shown in Eq. (37) also applies for a generic
UV renormalon (where $p<0$). For a UV renormalon the path ${\cal C}_{t}$
crosses the real axis at $t<t_{0}\bar{u}/p$ too, but one can adopt a straight
line between $t_{-}$ and $t_{+}$ because $\mbox{Re}(t_{\pm})<0$ and
$t_{0}\bar{u}/p>0$. It is straightforward to evaluate the ${\cal C}_{t}$
contour integrations in Eqs. (31) and (36) numerically, but we are not aware
of a general closed analytic solution. However, one can see that, due to the
cut (or the pole for integer $\gamma$), the integral picks up a contribution
$\simeq(-\frac{t_{0}\bar{u}}{p})^{-2m\hat{b}_{1}+\gamma-\ell-1}\times
e^{-2(\frac{m}{p})t_{0}\bar{u}}$. Interestingly, because $t_{0}$ is negative,
we see that for IR renormalons and $m\geq p>0$ the Borel integral of Eq. (36)
does not have anymore the exponential suppression that is be commonly expected
within the canonical renormalon calculus.777We note that the fact that the
Borel integral is not finite for $u\to+\infty$ does not invalidate the
underlying perturbative series because the renormalon calculus for asymptotic
series always involves some form of analytic continuation far away from the
origin in the Borel plane. The origin of the modified behavior is the
polynomial weight factor $W(x)=(-x)^{m}$ which causes an enhancement when the
contour of the invariant mass integration is deformed further into the
negative real complex half plane. Written in terms of the $t$ variable, this
corresponds to the enhancement factor $e^{-2mt}$. For the large-$\beta_{0}$
approximation ($\hat{b}_{1}=0$, $\tilde{g}_{\ell}^{(2m)}=\delta_{\ell 0}$,
$\gamma=1,2$), where we have

$\displaystyle B\Big{[}\delta_{\\{(-x)^{m},p,\gamma\\}}^{(0),{\rm
CIPT},\beta_{0}}(s_{0})\Big{]}(\bar{u})\,$
$\displaystyle=\,-\frac{i\,|p|^{-\gamma}\,t_{0}}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\int\limits_{{\cal C}_{t}}{\rm
d}t\,(-t)^{\gamma-1}\,\frac{e^{-2mt}}{(\frac{t_{0}\bar{u}}{p}-t)^{\gamma}}$
$\displaystyle=\,\tilde{C}(p,\gamma,m,s_{0};\bar{u})\,,$ (38)

these properties can be seen explicitly, since the contour integration can be
carried out analytically. The expressions for the $\tilde{C}$-functions for
single and double IR renormalon poles read

$\displaystyle\tilde{C}(p,1,m,s_{0};\bar{u})$
$\displaystyle=\frac{2\,t_{0}}{|p|}\,Q\Big{(}1,m,-2t_{0}(1-{\textstyle\frac{\bar{u}}{p}})\Big{)}$
(39) $\displaystyle\tilde{C}(p,2,m,s_{0};\bar{u})$
$\displaystyle=\frac{2\,t_{0}}{p^{2}}\,\Big{[}Q\Big{(}1,m,-2t_{0}(1-{\textstyle\frac{\bar{u}}{p}})\Big{)}-\frac{2\bar{u}t_{0}}{p}\,Q\Big{(}2,m,-2t_{0}(1-{\textstyle\frac{\bar{u}}{p}})\Big{)}\Big{]}\,,$
(40)

where ($n=1,2,3,\ldots$)

$\displaystyle Q(1,0,\rho)$
$\displaystyle=\frac{i}{2\pi}\,\Big{[}\ln(\rho+i\pi)-\ln(\rho-i\pi)\Big{]}\,,$
(41) $\displaystyle Q(n\geq 2,0,\rho)$
$\displaystyle=-\frac{i}{2\pi(n-1)}\,\Big{[}(\rho+i\pi)^{1-n}-(\rho-i\pi)^{1-n}\Big{]}\,,$
(42) $\displaystyle Q(n,m,\rho)$
$\displaystyle=m^{n-1}\,e^{-m\rho}\,\Big{[}\frac{(-1)^{n}\,i}{2\pi}\Big{(}\Gamma(1-n,-m(\rho+i\pi))$
(43)
$\displaystyle\qquad\qquad\qquad-\Gamma(1-n,-m(\rho-i\pi))\Big{)}-\frac{1}{\Gamma(n)}\Big{]}\,.$

The functions $Q$ exhibit non-analytic structures at distance $\pi$ from the
origin in the complex $\rho$-plane. In the complex $\bar{u}$ plane this
corresponds to a distance $p|1+ia_{0}\pi|$ to the origin which reaffirms our
arguments concerning the radius of convergence for the perturbative
construction of the CIPT Borel transform made in Sec. 2.1. Furthermore, it is
a straightforward exercise to check that for $u<p|1+ia_{0}\pi|$ the series of
Eq. (30) converges to the functions given in Eqs. (39) and (40).

The most peculiar aspects of the CIPT Borel functions for the spectral moments
are that (a) they do not show any non-analytic structures along the positive
real $\bar{u}$-axis and (b) the contour integration in the complex $x$-plane
leads to a substantial modification of the exponential suppression of the
Borel integral for large $\bar{u}$ values depending on the value of $m$. Let
us first comment on point (b): Because typically only positive integer values
are adopted for $m$ and we have $t_{0}<0$, for UV renormalons the Borel
integrand approaches zero more strongly than $e^{-\bar{u}/a_{0}}$ for large
$\bar{u}$. For IR renormalons, on the other hand, the exponential suppression
is attenuated as long as $m<p$, but it is still effective. For $m=p$ the
exponential behavior is gone and convergence for $\bar{u}$ approaching
infinity depends on the value of the exponent $-2m\hat{b}_{1}+\gamma-\ell-1$.
For $m>p$ we have an exponential enhancement and the Borel integral diverges.
This entails that for $m\geq p$ the CIPT Borel sum relies on a definition that
must involve an analytic continuation. We come back to this issue in Sec. 3.
Let us now comment on point (a): The absence of non-analytic structures along
the positive real $\bar{u}$-axis implies that the Borel sum, if the
corresponding Borel integral is convergent, is unique and ambiguity-free
irrespective of whether the underlying Borel function of the Adler function
has a non-analytic structure associated to an IR renormalon or not. This is an
important difference to the FOPT approach where the non-analytic structures
contained in the Adler function’s Borel function are inherited directly to the
FOPT Borel function, albeit the modulation through the function $\tilde{F}$.
So in the FOPT approach the existence of OPE corrections (computed in the
standard approach) is directly reflected in an unambiguous way in non-analytic
structures in the FOPT Borel function along the positive real Borel space
axis. We see for the CIPT approach that this association between the Borel
representation and the standard OPE corrections is not preserved. This is an
issue to be kept in mind, because OPE corrections are certainly needed in the
CIPT approach as well in order to account for non-perturbative corrections.
The analytic properties of the CIPT Borel function thus indicate that the
standard method for the computation of OPE corrections may not be appropriate
for the computation of the spectral function moments in the CIPT approach.

##  3 The Borel Sum and the Asymptotic Separation

In this section we examine the properties of the Borel sums for the
perturbative spectral function moments in the FOPT and CIPT approach, and we
derive an analytic expression for the asymptotic separation. The standard
approach to determine the Borel sum is to calculate the Borel function by
computing the contour integration and then perform the Borel integral in the
Borel space variables $u$ (or $\bar{u}$). For the following examinations,
however, we first carry out the integration over the Borel space variables,
because it can be computed analytically for both the FOPT and the CIPT
approach, and consider the contour integration afterwards. The expressions
reveal the major analytic differences with respect to the remaining contour
integration and can be easily evaluated numerically as well as analytically.
We again consider the generic terms in the Borel transform of the reduced
Adler function shown in Eqs. (12) and (13) for an IR and a UV renormalon,
respectively.

###  3.1 Borel Space Integrals

We start considering the Borel space integral for the case of a UV renormalon.
It is straightforward to show that the results for the CIPT and FOPT approach
give the same result, yielding the expression

$\displaystyle\int_{0}^{\infty}\\!\\!{\rm
d}\bar{u}\,\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\Big{)}\,\frac{e^{-\frac{\bar{u}}{a_{0}}}}{\big{(}\tilde{p}+\frac{a(-x)}{a_{0}}\bar{u}\big{)}^{\gamma}}\,=\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{e^{-\frac{u}{a(-x)}}}{(\tilde{p}+u)^{\gamma}}\,=\,(a(-x))^{1-\gamma}\,e^{\frac{\tilde{p}}{a(-x)}}\,\Gamma\Big{(}1-\gamma,{\textstyle\frac{\tilde{p}}{a(-x)}}\Big{)}\,.$
(44)

The result has a cut along the negative real $a(-x)$-axis which is outside the
perturbative regime and a cut along the positive real $x$-axis from the strong
coupling. The remaining contour integration can therefore be carried out along
the path with $|x|=1$.

(a)

(b)

Figure 2:  Graphical illustration of the Borel integration paths involved for
the FOPT and CIPT Borel representations for the cases of a UV renormalon with
$p<0$ (left panel) and an IR renormalon with $p>0$ (right panel).

It is tempting to assign the equality of the CIPT and FOPT Borel integrals to
the fact that both integrals can be formally related through the change of
variable $u=(\frac{a(-x)}{a_{0}})\bar{u}$ as we have already mentioned in Sec.
1. However, one must keep in mind that $a(-x)$ is in general complex such that
the integrals in $\bar{u}$ and $u$ are, in relation to each other, associated
to two completely different paths in the complex Borel space plane: both run
linearly from the origin to complex infinity, but with a relative angle that
depends on the argument of the complex coupling $a(-x)$. This is visualized in
Fig. 2(a) in the complex $u$-plane. Path 1 corresponds to the FOPT Borel
integral and runs along the positive real axis. Path 2 is associated to
$u=(\frac{a(-x)}{a_{0}})\bar{u}$ with real positive $\bar{u}$ and corresponds
to the CIPT Borel interal. It is a straight line passing through the complex
number $(\frac{a(-x)}{a_{0}})$. Both paths in the complex Borel space plane
lead to the same result if the associated closed contour, which results from
closing the two paths at infinity, does not contain any poles or cuts. For a
UV renormalon this is the case because its generic Borel function of Eq. (13)
has a cut along the negative real Borel space axis, while both Borel integrals
approach positive real infinity (because ${\rm Re}[a(-x)]>0$ and ${\rm
Re}[a_{0}]>0$). This results in the equality shown in Eq. (44). For an IR
renormalon, however, the generic Borel function of Eq. (12) has cuts (or
poles) along the positve real Borel space axis as illustrated in Fig. 2(b).
These lead to a difference between the CIPT and FOPT Borel integrals, if cuts
or poles are contained within the closed contour defined from the $u$ and
$\bar{u}$ paths related through the complex relation
$u=(\frac{a(-x)}{a_{0}})\bar{u}$ and closed at real positive infinity.

The CIPT Borel integral for an IR renormalon gives the expression

$\displaystyle\int_{0}^{\infty}\\!\\!{\rm
d}\bar{u}\,({\textstyle\frac{a(-x)}{a_{0}}})\,\frac{e^{-\frac{\bar{u}}{a_{0}}}}{\big{(}p-\frac{a(-x)}{a_{0}}\bar{u}\big{)}^{\gamma}}\,=\,$
$\displaystyle-(-a(-x))^{1-\gamma}\,e^{-\frac{p}{a(-x)}}\,\Gamma\Big{(}1-\gamma,{\textstyle-\frac{p}{a(-x)}}\Big{)}$
(45)

when $a(-x)$ has a finite imaginary part. There is no ambiguity involved in
the Borel $\bar{u}$-integration because, as explained above, for the CIPT
approach the remaining contour integration is deformed such that it never
crosses the cut in the complex $x$-plane. So the imaginary part of $a(-x)$
always remains finite and the cut contained in the Adler function’s Borel
function is not located on the positive real $\bar{u}$-axis. This results in
the compact analytic expression in terms of simple incomplete $\Gamma$ and
exponential functions and means that the RHS of Eq. (45) is ambiguity-free.

In contrast, for the FOPT Borel integration the cut coming from the Borel
function of the reduced Adler function is located on the real $u$-axis and
thus lies on the path of integration. This enforces a prescription to define
the Borel integral and an associated renormalon ambiguity that needs to be
defined as well. The most popular definition is to deform the Borel
integration such that the path is effectively shifted either above or below
the cut. These two choices are equivalent to the infinitesimal shifts $p\to
p\pm i0$, where the IR renormalon cut shown in Eq. (12) is moved slightly
above or below the positive real axis and the Borel integration remains along
the positive real $u$-axis. In this context the canonical approach is that the
value of the Borel sum is defined as the average of both shift prescriptions
and that the ambiguity is defined as half of the difference multiplied by a
factor of $i$ and the conventional factor $1/\pi$. Typically, the resulting
numerical value for the ambiguity roughly agrees in size with the
corresponding non-perturbative correction term $\delta^{(d)}_{W_{i}}$ obtained
in the standard OPE approach, see Eq. (2). We adopt this definition for the
FOPT Borel integral ambiguity in the following. The corresponding analytic
expressions for the Borel integral and its renormalon ambiguity in this
canonical approach for the FOPT Borel integral, also accounting for the
additional contribution of the cut (or pole) in Eq. (12) if located within the
closed Borel path, are given by

$\displaystyle\frac{1}{2}\left[\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{e^{-\frac{u}{a(-x)}}}{(p+i0-u)^{\gamma}}\,+\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{e^{-\frac{u}{a(-x)}}}{(p-i0-u)^{\gamma}}\,\right]$ (46)
$\displaystyle=$
$\displaystyle\,-(-a(-x))^{1-\gamma}\,e^{-\frac{p}{a(-x)}}\,\Gamma\Big{(}1-\gamma,{\textstyle-\frac{p}{a(-x)}}\Big{)}-{\rm
sig}[{\rm
Im}[a(-x)]]\,(i\pi)\,\frac{(a(-x))^{1-\gamma}}{\Gamma(\gamma)}\,e^{-\frac{p}{a(-x)}}$

and

$\displaystyle\frac{i}{2\pi}\left[\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{e^{-\frac{u}{a(-x)}}}{(p+i0-u)^{\gamma}}\,-\,\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{e^{-\frac{u}{a(-x)}}}{(p-i0-u)^{\gamma}}\,\right]\,=\,\frac{(a(-x))^{1-\gamma}}{\Gamma(\gamma)}\,e^{-\frac{p}{a(-x)}}\,,$
(47)

where the function ${\rm sig[z]}$ gives the sign of $z$. Note that the RHS of
Eq. (47) is valid for any complex $a(-x)$ with a positive real part, while the
RHS Eq. (46) applies only if ${\rm Re}(a(-x))>0$ and ${\rm Im}(a(-x))\neq 0$.
For positive real values of $a(-x)$ the result for the Borel integral is
obtained via replacing $-(-a(-x))^{1-\gamma}$ by $(-a(-x)+i0)^{1-\gamma}$ in
the first term and ${\rm sig[{\rm Im}(a)]}$ by unity in the second term on the
RHS of Eq. (46). The expression on the RHS of Eq. (46) provides the most
convenient way to calculate the FOPT Borel sum for the spectral function
moments.

Inspecting the analytic structure of the result for the CIPT Borel integral in
Eq. (45) we see that it exhibits a cut along the entire positive real axis in
the complex $a(-x)$-plane. Together with the cut contained in the strong
coupling this means that the expression has cuts along the entire real
$x$-axis. The cut along the negative real $x$-axis originates from the cut
already discussed for Eq. (36) and therefore stretches into the entire region
accessible by the perturbative evolution of the strong coupling (when the
$\bar{u}$ Borel integral is carried out first). In the complex $t$-plane these
cuts cover the entire real axes as well. Since the contour of the $x$
integration is not allowed to cross the real negative axis at any finite
distance from the origin it must be deformed to infinity. Comparing to the
result for the FOPT Borel integral in Eq. (46) we see that the first term
agrees with the CIPT Borel integral result, and that the second term exhibits
a cut along the entire real axis in the complex $a(-x)$-plane. Interestingly,
the cut along the positive real $a(-x)$-axis precisely cancels in the sum of
both terms in Eq. (46), allowing to do the contour along the circular path
$|x|=1$ when computing the FOPT Borel sum. The same is true for the expression
for the FOPT Borel sum ambiguity given in Eq. (47).

###  3.2 Asymptotic Separation and FOPT Borel Sum Ambiguity

Let us now focus on the determination of the asymptotic separation arising
from the generic IR renormalon Borel function term shown in Eq. (12).
Considering again the polynomial weight function $W(x)=(-x)^{m}$ the
expression for the asymptotic separation reads

$\displaystyle\Delta(m,p,\gamma,s_{0})\,\equiv\,\delta_{\\{(-x)^{m},p,\gamma\\},{\rm
Borel}}^{(0),{\rm CIPT}}(s_{0})\,-\,\delta_{\\{(-x)^{m},p,\gamma\\},{\rm
Borel}}^{(0),{\rm FOPT}}(s_{0})$
$\displaystyle=\,\frac{1}{2\Gamma(\gamma)}\,\ointctrclockwise\limits_{{\cal
C}_{x}}\frac{{\rm d}x}{x}\,(-x)^{m}\,{\rm sig}[{\rm
Im}[a(-x)]]\,(a(-x))^{1-\gamma}\,e^{-\frac{p}{a(-x)}}\,.$ (48)

The expression on the RHS can also be obtained directly from the
considerations illustrated in Fig. 2(b): it arises from the contribution of
the IR renormalon cut within the contour defined by closing either one of the
paths 1a or 1b and that of path 2 at infinity. Interestingly, the expression
for the asymptotic separation has a structure that is quite similar to one of
the ambiguity of the FOPT Borel sum. Using the definition we explained above
it has the form888The numerical values obtained from the integral in Eq. (49)
evaluate to real numbers with either sign. We define the FOPT Borel sum
ambiguity for a given Adler function Borel function model as the size of the
coherent sum of all individual terms $\delta^{\rm FOPT}(m,p,\gamma,s_{0})$
that arise. In Tabs. 1 and 2 we have, however, kept the signs of the resulting
values of the FOPT Borel sum ambiguity.

$\displaystyle\delta^{\rm FOPT}(m,p,\gamma,s_{0})\,\equiv\,\frac{1}{2\pi
i}\frac{1}{\Gamma(\gamma)}\,\ointctrclockwise\limits_{|x|=1}\frac{{\rm
d}x}{x}\,(-x)^{m}\,(a(-x))^{1-\gamma}\,e^{-\frac{p}{a(-x)}}\,.$ (49)

The similarity arises since both originate from the generic IR renormalon
terms in the reduced Adler function Borel transform of Eq. (12). However, the
asymptotic separation differs from the FOPT Borel sum ambiguity due to the
additional factor ${\rm sig}[{\rm Im}[a(-x)]]$. As a consequence, the
asymptotic separation can remain sizeable even when the renormalon ambiguity
of the FOPT approach is strongly suppressed or vanishes. As we show in our
numerical analysis below, this is the origin of the discrepancy between FOPT
and CIPT series for the $\tau$ hadronic spectral function moments concerning
their asymptotic behavior.

What remains to be discussed for the asymptotic separation is how to carry out
the integration over the contour ${\cal C}_{x}$. The discussion is subtle
because the convergence issues that we already discussed at the end of Sec. 2
for $m\geq p$ reemerge. The question of convergence can also be seen in the
form of Eq. (3.2) since the power-suppression coming from the exponential term
$e^{-p/a(-x)}$ competes with the power-enhancement from the polynomial term
$(-x)^{m}$ when the contour is deformed to infinity.

Let us first consider the asymptotic separation for the case $m<p$, where the
exponential suppression wins and the contour integral in Eq. (3.2) is
convergent. Here the path ${\cal C}_{x}$ is split into two contributions. The
first starts at $x_{-}=1+i0$ and ends at negative real infinity in the
positive imaginary half plane, i.e. at $x^{\infty}_{-}=-\infty+i\eta$ with
$\eta$ being some positive real number. The second starts at
$x^{\infty}_{+}=-\infty-i\eta$, runs in the negative imaginary half plane and
ends at $x_{+}=1-i0$. Both paths are also visualized in Fig. 1(a). Changing
again to the $t$ variable defined in Eq. (18) we can rewrite the expression
for the asymptotic separation as

$\displaystyle\Delta(m,p,\gamma,s_{0})$
$\displaystyle\,=\,-\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\frac{2^{\gamma-1}}{\Gamma(\gamma)}\,\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\left[\\!\int_{t_{-}}^{t_{-}^{\infty}}+\int_{t_{+}}^{t_{+}^{\infty}}\,\right]{\rm
d}t\,(-t)^{-2m\hat{b}_{1}+\gamma-\ell-1}\,e^{2(p-m)t}\,,$ (50)

where the upper limits of the two integrals are $t_{\mp}^{\infty}=-\infty\pm
i\eta$. The corresponding paths are visualized in Fig. 1(b). The $t$-integrals
can be readily evaluated giving

$\displaystyle\Delta(m\neq p,p,\gamma,s_{0})$
$\displaystyle\,=\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\frac{2^{2m\hat{b}_{1}+\ell}}{\Gamma(\gamma)}\,$
(51) $\displaystyle\,\times\,{\rm
Re}\Big{[}(p-m+i0)^{2m\hat{b}_{1}-\gamma+\ell}\,\Gamma(-2m\hat{b}_{1}+\gamma-\ell,-2(p-m)t_{-})\Big{]}$

For the case $m>p$ we cannot employ the integration path described above,
because Eq. (3.2) diverges. To define the result for this case we can add an
infinitesimal imaginary contribution to $p$ of the form $p\to p\pm i0$ for the
integral in the upper/lower complex $t$ plane. This allows us to change the
integration limits to $t_{\mp}^{\infty}=\pm i\infty$ without modifying the
integrals for $p>m$. The result can then be analytically continued to $m>p$
and the result is already shown in in Eq. (51), where the corresponding
prescription is accounted for in the term
$(p-m+i0)^{2m\hat{b}_{1}-\gamma+\ell}$. Note that this prescription is
equivalent to setting insertions of the boundaries $t_{\mp}^{\infty}$ into the
indefinite integral functions in Eq. (51) to zero.

We will show in our numerical analyses of Secs. 4 and 5 that this definition
provides a result for the Borel sum of the CIPT $\tau$ spectral function
moment series for $m>p$ that is perfectly compatible with their asymptotic
high-order behavior. For the case $m=p$ we see that Eq. (3.2) is convergent if
$-2p\hat{b}_{1}+\gamma<0$. For a generic non-analytic term $\sim
1/(p-u)^{\gamma}$ in the Borel function associated to pure $\Lambda_{\rm
QCD}^{2p}$ ambiguity the term $\gamma=1+2p\hat{b}_{1}$ always arises, so the
condition cannot be satisfied in a consistent way. We therefore define the
asymptotic separation to be zero for the case $m=p$:

$\displaystyle\Delta(p,p,\gamma,s_{0})\,=\,0\,.$ (52)

Again, we will show in our numerical analyses of Secs. 4 and 5 that this
definition provides a result for the Borel sum of the CIPT $\tau$ spectral
function moment series for $m=p$ that is perfectly compatible with their
asymptotic high-order behavior. In the large-$\beta_{0}$ approximation the
corresponding results are quite compact. For $\gamma=1,2$ they adopt the form:

$\displaystyle\Delta_{\beta_{0}}(m\neq p,p,1,s_{0})\,=\,$
$\displaystyle\frac{(-1)^{p-m}}{p-m}\,e^{-\frac{p}{a_{0}}}\,,$ (53)
$\displaystyle\Delta_{\beta_{0}}(m\neq p,p,2,s_{0})\,=\,$
$\displaystyle(-1)^{p-m}\,\bigg{[}\frac{1}{(p-m)^{2}}+\frac{1}{(p-m)a_{0}}\bigg{]}\,e^{-\frac{p}{a_{0}}}\,,$

It is instructive to compare the results for the analytic separation between
the CIPT and the FOPT Borel sums to the corresponding ones for the ambiguity
of the FOPT Borel sum defined in Eq. (49). After changing variabels from $x$
to $t$, the ambiguity of the FOPT Borel sum can be written as

$\displaystyle\delta^{\rm
FOPT}(m,p,\gamma,s_{0})\,=\,\frac{1}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\frac{2^{\gamma-1}}{\Gamma(\gamma)}\,\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\int_{t_{-}}^{t_{+}}{\rm
d}t\,(-t)^{-2m\hat{b}_{1}+\gamma-\ell-1}\,e^{2(p-m)t}\,.$ (54)

Since ${\rm Re}[t_{\pm}]<0$ and the integrand is analytic in the negative real
complex half plane, one can adopt a straight line for the integration path
between $t_{\pm}$ for any values of $p$ and $m$. It is straightforward to do
the integrals analytically giving

$\displaystyle\delta^{\rm FOPT}(m\neq p,p,\gamma,s_{0})$
$\displaystyle\,=\,\frac{1}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\frac{2^{2m\hat{b}_{1}+\ell}}{\Gamma(\gamma)}$
(55) $\displaystyle\,\times\,\bigg{\\{}\,{\rm
Im}\Big{[}(p-m+i0)^{2m\hat{b}_{1}-\gamma+\ell}\,\Gamma(-2m\hat{b}_{1}+\gamma-\ell,-2(p-m)t_{-})\Big{]}$
$\displaystyle\hskip
28.45274pt+\,\Theta(m-p)\,(m-p)^{2m\hat{b}_{1}-\gamma+\ell}\frac{\pi}{\Gamma(2m\hat{b}_{1}-\gamma+\ell+1)}\,\bigg{\\}}\,.$

and

$\displaystyle\delta^{\rm
FOPT}(p,p,\gamma,s_{0})\,=\,\frac{1}{\pi}\,\Big{(}\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}}\Big{)}^{m}\,\frac{2^{\gamma}}{\Gamma(\gamma)}\,\,\sum_{\ell=0}^{\infty}\,\tilde{g}_{\ell}^{(2m)}\,\frac{1}{2m\hat{b}_{1}-\gamma+\ell}\,{\rm
Im}\Big{[}(-t_{-})^{-2m\hat{b}_{1}+\gamma-\ell}\Big{]}\,.$ (56)

In the large-$\beta_{0}$ approximation the results are compact as well and
read

$\displaystyle\delta^{\rm FOPT}_{\beta_{0}}(m\neq p,p,1,s_{0})$
$\displaystyle=0\,,$ $\displaystyle\delta^{\rm FOPT}_{\beta_{0}}(p,p,1,s_{0})$
$\displaystyle=e^{-\frac{p}{a_{0}}}\,,$ (57) $\displaystyle\delta^{\rm
FOPT}_{\beta_{0}}(m\neq p,p,2,s_{0})$
$\displaystyle=\frac{(-1)^{p-m}}{m-p}\,e^{-\frac{p}{a_{0}}}\,,$
$\displaystyle\delta^{\rm FOPT}_{\beta_{0}}(p,p,2,s_{0})$
$\displaystyle=\frac{1}{a_{0}}\,e^{-\frac{p}{a_{0}}}\,.$

An essential feature of the results for the FOPT Borel sum ambiguity
$\delta^{\rm FOPT}$ for a certain IR renormalon term in the Adler function’s
Borel function is that they can be used as a proxy for the impact and the size
of the corresponding OPE correction (determined within the standard OPE
method) in the spectral function moment.

It is instructive to compare the expressions for the FOPT Borel sum ambiguity
$\delta^{\rm FOPT}$ with those for the asymptotic separation $\Delta$. It is
straightforward to see that for integer values $m\neq p$ we always have
$\Delta>\delta^{\rm FOPT}$. More importantly, it is even possible that
parametrically $\Delta\gg\delta^{\rm FOPT}$, as can be easily seen in the
large-$\beta_{0}$ approximation where a single pole renormalon ambiguity in
the Adler function is eliminated for $m\neq p$ (i.e. $\delta^{\rm
FOPT}_{\beta_{0}}(m\neq p,p,1,s_{0})=0$), while the corresponding asymptotic
separation $\Delta_{\beta_{0}}$ is finite. If the full QCD $\beta$-function is
accounted for, the FOPT Borel sum ambiguity is not zero, but still strongly
suppressed. We thus see that the asympotic separation due to some IR
renormalon in the Adler function can be parametrically larger than the size of
the OPE corrections computed in the standard OPE approach. This observation is
yet one more hint that the close association between non-analytic structures
in the FOPT Borel representation and the standard OPE corrections does not
apply for the CIPT approach. We touch upon this issue again in Sec. 6.

##  4 Application in the Large-$\beta_{0}$ Approximation

In this section we demonstrate that the result for the asymptotic separation
derived in Sec. 3 correctly quantifies the discrepancy in the observable
asymptotic behavior of the FOPT and CIPT series in the large-$\beta_{0}$
approximation. In the large-$\beta_{0}$ approximation the all-order
perturbative series and the corresponding exact Borel functions are known for
the vacuum polarization, the reduced Adler function and many other
observables, and furthermore many results can be given in terms of brief and
simple analytic expressions. The large-$\beta_{0}$ approximation is useful as
it frequently shares many qualitative properties with the corresponding exact
QCD results. We stress that in the following we do not intend to provide a
general discussion of the high-order behavior of the FOPT and CIPT series. The
main aim is to demonstrate the quality of the description provided by the
asymptotic separation. Since the asymptotic separation only arises from IR
renormalon contributions in the Borel function of the reduced Adler function
we in particular do not consider UV renormalons (for which no discrepancy in
the FOPT and CIPT series asymptotic behavior arises). Note that all
calculations (here and in Sec. 5) are carried out in the $n_{f}=3$ flavor
scheme for the strong coupling with $\alpha_{s}(m_{\tau}^{2})=0.34$ and for
$s_{0}=m_{\tau}^{2}$.

###  4.1 Generic $p=2$ Renormalon

(a) Simple pole, p=2, $W(x)=1$, large-$\beta_{0}$

(b) Simple pole, p=2, $W(x)=(-x)$, large-$\beta_{0}$

(c) Simple pole, p=2, $W(x)=(-x)^{2}$, large-$\beta_{0}$

(d) Simple pole, p=2, $W(x)=(-x)^{4}$, large-$\beta_{0}$

Figure 3:  Moments $\delta_{\\{(-x)^{m},2,1\\}}^{(0),{\rm FOPT}}(m_{t}^{2})$
(red) and $\delta_{\\{(-x)^{m},2,1\\}}^{(0),{\rm CIPT}}(m_{t}^{2})$ (blue) in
the large-$\beta_{0}$ approximation for a pure $p=2$ single renormalon pole
and weight functions $W(x)=(-x)^{m}$ with $m=0,1,2,4$ as a function of the
order up to which the series are summed. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red band indicates the conventional Borel ambiguity of the FOPT series for
$m=2$.

Let us start with the spectral function moment series arising from a generic
single pole IR renormalon Borel term related to a $d=4$ gluon condensate term
in the spectral function moment’s OPE, $B^{\rm IR}_{\hat{D},2,1}(u)=1/(2-u)$,
see Eq. (12). It constitutes the dominant contribution in the Borel function
of the reduced Adler function for positive values of $u$. The FOPT series can
be obtained from expanding the $\ell=0$ term in Eq. (2.3) with the expression
of Eq. (35) in powers of $u$ and carrying out the Borel integral. The CIPT
series can be obtained by determining the $\alpha_{s}$-series of Eq. (5)
associated to $B^{\rm IR}_{\hat{D},2,1}(u)$ and then using Eq. (16). The
resulting FOPT and CIPT series are shown in Figs. 3 as the red and blue dots,
respectively, as a function of the truncation order $n$ for some
representative choices of the weight function $W(x)=(-x)^{m}$ for $m=0,1,2,4$.
We note that physical weight functions vanish at $x=1$ and are always linear
combinations of several of such simple power weight functions. The study of
moments based on simple power weight function thus allows to examine the
interplay of their respective contributions. The horizontal red lines
represent the respective FOPT Borel sum from Eqs. (14) and (46), and the
horizontal blue lines indicate the CIPT Borel sum obtained from adding the
appropriate asymptotic separation terms $\Delta$ from Eqs. (53) to the FOPT
Borel sum. The red band represents the FOPT Borel sum ambiguity related to
adding $\pm\delta^{\rm FOPT}$ given in Eqs. (57) to the FOPT Borel sum. We use
the same colored notations in all subsequent figures in this article. The
respective numerical values for this analysis and all the others of Sec. 4 are
collected for convenience in Tab. 1.

large-$\beta_{0}$
---
$B(u)$ | $W(x)$ | Figure | Scheme | $\delta^{(0),{\rm FOPT}}_{\rm Borel}$ | $\delta^{(0),{\rm CIPT}}_{\rm Borel}$ | $\delta^{\rm FOPT}$ | $\Delta$
$\frac{1}{(2-u)}$ | $1$ | 3(a) | $\overline{\mbox{MS}}$ | $0.11487$ | $0.11500$ | $0$ | $0.00014$
$\frac{1}{(2-u)}$ | $(-x)$ | 3(b) | $\overline{\mbox{MS}}$ | $0.04668$ | $0.04641$ | $0$ | $-0.00027$
$\frac{1}{(2-u)}$ | $(-x)^{2}$ | 3(c) | $\overline{\mbox{MS}}$ | $-0.01193$ | $-0.01193$ | $0.00027$ | $0$
$\frac{1}{(2-u)}$ | $(-x)^{4}$ | 3(d) | $\overline{\mbox{MS}}$ | $-0.00587$ | $-0.00600$ | $0$ | $-0.00014$
$\frac{1}{(2-u)^{2}}$ | $1$ | 4(a) | $\overline{\mbox{MS}}$ | $0.06340$ | $0.06402$ | $-0.00014$ | $0.00062$
$\frac{1}{(2-u)^{2}}$ | $(-x)$ | 4(b) | $\overline{\mbox{MS}}$ | $0.03476$ | $0.03337$ | $0.00027$ | $-0.00138$
$\frac{1}{(2-u)^{2}}$ | $(-x)^{2}$ | 4(c) | $\overline{\mbox{MS}}$ | $-0.00663$ | $-0.00663$ | $0.00111$ | $0$
$\frac{1}{(2-u)^{2}}$ | $(-x)^{4}$ | 4(d) | $\overline{\mbox{MS}}$ | $-0.00303$ | $-0.00352$ | $0.00014$ | $-0.00049$
$B_{\hat{D},\lambda=5/3}$ | $W_{\tau}$ | 5(a) | $\overline{\mbox{MS}}$ | $0.26290$ | $0.24269$ | $0.00291$ | $-0.02021$
$B_{\hat{D},\lambda=10/3}$ | $W_{\tau}$ | 5(b) | $\lambda=10/3$ | $0.26290$ | $0.24269$ | $0.00291$ | $-0.02021$
Table 1:  Numerical values for the FOPT Borel sum $\delta^{(0),{\rm
FOPT}}_{\rm Borel}$, the CIPT Borel sum $\delta^{(0),{\rm CIPT}}_{\rm Borel}$,
the FOPT Borel sum ambiguity $\delta^{\rm FOPT}$ and the asymptotic separation
$\Delta=\delta^{(0),{\rm CIPT}}_{\rm Borel}-\delta^{(0),{\rm FOPT}}_{\rm
Borel}$ in the large-$\beta_{0}$ approximation for the analyses in Sec. 4.

We see from Fig. 1 that for single pole IR renormalons in the
large-$\beta_{0}$ approximation $\delta^{\rm FOPT}$ is only non-zero if $m=p$.
For $m=0,1,4$ the FOPT series are convergent (and thus ambiguity-free) and
closely approach their Borel sum value for orders $n\gtrsim 13$. The
corresponding CIPT series are asymptotic. They closely approach the CIPT Borel
sum but finally diverge. For $m=0$, $m=1$ and $m=4$ the order ranges of
closest approach to the CIPT Borel sum are $5\lesssim n\lesssim 17$,
$10\lesssim n\lesssim 17$ and $5\lesssim n\lesssim 8$, respectively. Both, the
FOPT and CIPT series clearly exhibit a stable convergence/asymptotic regime.
The corresponding Borel sum values are distinctly different, and the
difference is correctly quantified by the asymptotic difference derived in
Sec. 3. For $m=p=2$ the FOPT and CIPT series have a quite different character.
Here, the FOPT and CIPT series are both asymptotic. We see that the FOPT
series oscillates around its Borel sum ambiguity band up to order $n\sim 10$
and then diverges. The CIPT series shows an extended linear behavior for
orders $3\lesssim n\lesssim 17$ and the Borel sum, which is equal to that of
the FOPT series, represents the series value somewhere on the lower side of
the linear regime. While the computed Borel sum is certainly compatible with
the behavior of the FOPT and CIPT series, its value and the FOPT ambiguity
could not be easily determined by eye. We conclude: given that the FOPT Borel
sum provides the commonly accepted value of the FOPT series within the
renormalon calculus, the asymptotic separation (which we defined to vanish for
$m=p$) provides an adequate description for the Borel sum of the CIPT series.

(a) Double pole, p=2, $W(x)=1$, large-$\beta_{0}$

(b) Double pole, p=2, $W(x)=(-x)$, large-$\beta_{0}$

(c) Double pole, p=2, $W(x)=(-x)^{2}$, large-$\beta_{0}$

(d) Double pole, p=2, $W(x)=(-x)^{4}$, large-$\beta_{0}$

Figure 4:  Moments $\delta_{\\{(-x)^{m},2,1\\}}^{(0),{\rm FOPT}}(m_{t}^{2})$
(red) and $\delta_{\\{(-x)^{m},2,1\\}}^{(0),{\rm CIPT}}(m_{t}^{2})$ (blue) in
the large-$\beta_{0}$ approximation for a pure $p=2$ double renormalon pole
and weight functions $W(x)=(-x)^{m}$ with $m=0,1,2,4$ as a function of the
order up to which the series are summed. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red bands indicate the conventional Borel ambiguity of the FOPT series.

Next, let us consider a generic double pole IR renormalon Borel term for
$p=2$, $B^{\rm IR}_{\hat{D},2,2}(u)=1/(2-u)^{2}$. Such a term is not contained
in the Borel function of the reduced Adler function, but double poles arise
for all other IR renormalon singularities. For double poles the
large-$\beta_{0}$ contour integration does not eliminate the renormalon and
the associated FOPT as well as CIPT series are both asymptotic. The results
are shown in Figs. 4 , again considering the weight function $W(x)=(-x)^{m}$
for the cases $m=0,1,2,4$ and using the same conventions as in Fig. 3. Here,
the FOPT Borel sums all exhibit an ambiguity band because $\delta^{\rm FOPT}$
is always non-zero, see Eqs. (57). For $m=0,1,4$ the FOPT series show some
oscillatory convergent behavior for orders $n\lesssim 10$ around their
respective Borel sum ambiguity band. On the other hand, the CIPT series
clearly approach their CIPT Borel sums for orders $7\lesssim n\lesssim 15$,
$10\lesssim n\lesssim 15$ and $5\lesssim n\lesssim 8$, respectively – a
behavior very similar to the single pole case. We again see that the
asymptotic separation (added to the FOPT Borel sum) correctly quantifies the
visible asymptotic behavior of the CIPT series. It is also clearly visible
that the asymptotic separation is much larger than the FOPT ambiguity based on
the standard definition used for $\delta^{\rm FOPT}$. For $m=p=2$ the FOPT and
CIPT series again exhibit a quite different character in analogy to the single
pole case. Both series show no clear sign of an asymptotic regime. Neither for
the FOPT nor for the CIPT series the Borel sum (nor the size of the FOPT
ambiguity band) could be easily determined by eye, but the values are
certainly compatible with the behavior of both series. The CIPT series closely
approaches its Borel sum (which is equal to that of the FOPT series) in the
region around order $n=3$, where the series shows a minimum. We again conclude
that the asymptotic separation provides an adequate description for the Borel
sum of the CIPT series.

We have checked that the behavior we just discussed for $p=2$ IR renormalon
poles and weight functions $W(x)=(-x)^{m}$ with $m=0,1,2,4$ is generic for any
(positive integer) values of $p$ and $m$, and that the asymptotic separation
always provides an adequate and predictable description of the asymptotic
behavior of the CIPT spectral function moment series. We in particular find
that for $m\neq p$ the asymptotic separation is always larger than the FOPT
Borel sum ambiguity defined by $\delta^{\rm FOPT}$, and we stress that this is
true in general even if $\delta^{\rm FOPT}$ is not defined with the
conventional factor $1/\pi$.

###  4.2 Hadronic Tau Decay Width

We now discuss the perturbative series for the hadronic tau decay width
$R_{\tau}$, which is the spectral function moment for the weight function
$W_{\tau}(x)=(1-x)^{3}(1+x)=1-2x+2x^{3}-x^{4}$. The Borel function of the
reduced Adler function is known in closed form and reads Broadhurst:1992si

$\displaystyle B_{\hat{D},\lambda}(u)$
$\displaystyle\,=\,\frac{128}{3\beta_{0}}\,\frac{e^{\lambda
u}}{2-u}\,\sum_{k=2}^{\infty}\,\frac{(-1)^{k}\,k}{[k^{2}-(1-u)^{2}]^{2}}\,,$
(58)

where $\lambda=5/3$ in the $\overline{\mbox{MS}}$ scheme for the strong
coupling. The Borel function has a single pole at $u=2$ and double poles at
integer values for $u$ larger than 2 and all negative integers. In the
$\overline{\rm MS}$ scheme the IR renormalon at $p=2$ dominates the behavior
of the series for orders up to $n\approx 7$, while for orders beyond the
oscillating behavior arising from the UV renormalon poles determines the
behavior of the series. The terms of the FOPT and CIPT series as well as the
FOPT Borel sum, its ambiguity and the asymptotic separation can be obtained in
a straightforward way using the method described in Sec. 4.1, identifying the
corresponding residues and summing all terms. The respective numerical values
obtained in the following analysis are shown in Tab. 1.

(a) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $\alpha_{s}^{\overline{\rm MS}}$,
large-$\beta_{0}$

(b) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $\alpha_{s}^{\lambda=10/3}$,
large-$\beta_{0}$

Figure 5:  Moments $\delta^{(0),{\rm FOPT}}_{W_{\tau}}(m_{\tau}^{2})$ (red)
and $\delta^{(0),{\rm CIPT}}_{W_{\tau}}(m_{\tau}^{2})$ (blue) in the
large-$\beta_{0}$ approximation as a function of the order up to which the
series are summed. Panel (a) shows the result in the $\overline{\mbox{MS}}$
scheme and panel (b) in a scheme where the coupling is defined by
$a_{\lambda=10/3}(x)=a(x)/(1+5/3\,a(x))$. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red bands indicate the conventional Borel ambiguity of the FOPT series.

The outcome for $\delta^{(0)}_{W_{\tau}}$ for the FOPT and CIPT series is
shown in Fig. 5(a) using the same conventions as for the generic examinations
in Sec. 4.1. The FOPT series reaches the FOPT Borel sum (red horizontal line)
at orders $n=9$ and $10$ where it also approaches a stable asymptotic
behavior. At orders beyond it starts to oscillate due to the impact of the UV
renormalon contributions. The difference of the values of the FOPT series in
the asymptotic region at orders $n=9$ and $10$ to its Borel sum is consistent
with the FOPT ambiguity indicated by the red band. The CIPT series shows a
convergent and mildly oscillating behavior for orders $n\lesssim 8$. For
orders beyond that the oscillations quickly become large due to the dominance
of UV renormalon contributions. The values of the CIPT series at order $n=7$
is compatible with the CIPT Borel sum determined from the FOPT Borel sum plus
the asymptotic separation. If we adopt the very practical view that the
ambiguity of the CIPT Borel sum is the same as that of the FOPT sum, we find
that the difference between the values of the CIPT series at orders $n=7$ and
$8$ to its Borel sum is also consistent with this ambiguity.

From the behavior of the CIPT series originating from a single pole at $p=2$
shown in Fig. 3 it is actually easy to understand why the CIPT series for
$R_{\tau}$ happens to be systematically below its CIPT Borel sum for orders
where the oscillatory behavior from the UV renormalon poles is not yet
sizeable. This behavior originates from the different properties of the series
associated to the polynomial terms with $m=1$ and $m=3,4$: the former closely
approaches its Borel sum at orders $n\gtrsim 7$ (see Fig. 3(b)) while the
latter approach their corresponding Borel sum already at lower orders and
already deflect from them when $n\gtrsim 7$ (see Fig. 3(d)). The net effect,
accounting for the signs of the polynomial terms in the weight function
$W_{\tau}(x)$, is that the CIPT series bounces back from its Borel sum at a
finite distance before the start of the oscillatory behavior.

It is also straightforward to check that adopting a different scheme for the
strong coupling leaves the results for the Borel sums, the FOPT Borel sum
ambiguity as well as the asymptotic separation unchanged. All qualitative
statements just made remain intact as well, apart from the fact that the onset
of the oscillatory behavior of the series is shifted towards higher orders if
IR renormalon contributions are enhanced by the change of scheme. For example,
if we adopt a scheme where the Adler function’s Borel function has the form of
Eq. (58) with $\lambda=10/3$, which corresponds to using the coupling
$a_{\lambda=10/3}(x)=a(x)/(1+5/3\,a(x))$, the outcome shown in Fig. 5(b) is
obtained. The invariance of the asymptotic separation under changes of
$\lambda$ for this geometric kind of scheme modification follows from the
identify
$f_{p}(\gamma,a)=e^{p\eta}\sum_{i=0}^{\infty}f_{p}(\gamma-i,a/(1+a\eta))(-\eta)^{i}/(i!)$
for $f_{p}(\gamma,a)\equiv e^{-p/a}a^{1-\gamma}/\Gamma(\gamma)$, and also
applies beyond the large-$\beta_{0}$ approximation. Overall we find that the
FOPT as well as the CIPT series each are compatible with their respective
Borel sums and that none of the series behave in any strange way. However, the
essential point is that even though the $p=2$ renormalon does not contribute
to the ambiguity of the FOPT Borel sum, it provides the dominant contribution
to the asymptotic separation. In fact for the case of $R_{\tau}$ the exact
value for the asymptotic separation is $-0.0202094$ from which $-0.0202591$
(which is $99.8\%$) comes from the $p=2$ renormalon.

##  5 Application Accounting for the Full QCD $\beta$-Function

In this section we demonstrate that the expression for the asymptotic
separation derived in Sec. 3 correctly quantifies the discrepancy in the
observable asymptotic behavior of the FOPT and CIPT spectral function moment
series accounting for all known terms in the QCD $\beta$-function up to five
loops. Since at this level of approximation the exact expression for the Borel
function of the Adler function (and of any other quantity computable in
perturbation theory) is unkown, one has to rely on models. Such Borel function
models can be constructed from generic Borel functions that quantify precisely
the higher order behavior associated with the individual terms in the OPE. For
a dimension-$d$ term in the OPE these generic Borel functions are linear
combinations of the IR renormalon terms $B^{\rm
IR}_{\hat{D},p,\gamma}(u)\,=\,\frac{1}{(p-u)^{\gamma}}$ shown in Eq. (12) for
$p=d/2$ and specific values of $\gamma$. The generic Borel functions for UV
renormalons are related to process-dependent matrix elements with insertions
of higher-dimensional local operators and constructed in an analogous way.
Given the dimension of the local operators entering the condensate (or
process-dependent) matrix elements, their anomalous dimension and their Wilson
coefficients, the form of these generic Borel functions can be perturbatively
determined in an unambiguous way. However, their precise weight within the
exact Borel function is unknown. Furthermore, the information on the Wilson
coefficients (or process-dependent matrix elements) and the anomalous
dimension of the local operators is in general very limited (i.e. known with a
precision that is typically much lower than the perturbative orders known for
$\delta^{(0)}_{W_{i}}(s_{0})$). Borel models for the Adler function thus
comprise of simplifying assumptions on the properties of the OPE terms and
particular (preferential) choices of weight factors. The latter are
furthermore fixed such that the model reproduces the coefficients of the
perturbative series that have been computed exactly. In the following we
consider several Borel models for the reduced Adler function. Since the
construction of these models is not the subject of the article, we only
describe their content briefly and refer to Ref. Beneke:2008ad ;
MasterThesisRegner  for details on their construction. The explicit formulae
for the Borel models used in the following examinations can be found in App.
A. We emphasize that we do not intend to enter any considerations on the
phenomenological soundness of the Borel models that are being discussed. The
main focus is on the quality how well the results for asymptotic separation
describe the difference in the asymptotic large-order behavior of the
resulting FOPT and CIPT spectral function moment series.

###  5.1 $R_{\tau}$ for a Multi-Renormalon Borel Model

We first consider a multi-renormalon Borel model for the Adler function
containing generic Borel functions for two IR renormalons related to $d=4$ and
$d=6$ OPE terms, associated to $p=2$ and $p=3$, respectively, and one UV
renormalon related to dimension-$6$ local operators, associated to $p=-1$. The
$p=2$ IR renormalon is related to the gluon condensate OPE term and we account
for the known ${\cal O}(\alpha_{s})$ Wilson coefficient correction. For the UV
renormalon we assume an anomalous dimension consistent with the emergence of a
double pole in the limit of the large-$\beta_{0}$ approximation, which is
known to exist from Eq. (58). For the $p=3$ IR renormalon, higher order
corrections to the Wilson coefficient are neglected and the anomalous
dimension is assumed to vanish. For the construction of the model we follow
Ref. Beneke:2008ad , where the Adler function coefficients $c_{3,1}$,
$c_{4,1}$ and $c_{5,1}$ in Eq. (5)999For $c_{n\leq 4,1}$ we use the known
exact results Gorishnii:1990vf ; Surguladze:1990tg ; Baikov:2008jh  and for
$5$-loop coefficient we adopt the estimate $c_{5,1}=283$ from Ref.
Beneke:2008ad . are used to fix the coefficients of the three generic
renormalon Borel functions and a function linear in $u$ is added to achieve
consistency with the known expressions for the coefficients $c_{1,1}$ and
$c_{2,1}$. The expression for this Borel model, referred to as
$B_{\hat{D},{\rm mr}}(u)$, is shown in Eq. (A), where we included the effects
of the $5$-loop correction to the QCD $\beta$-function that were neglected in
Ref. Beneke:2008ad . In this Borel model the $p=2$ renormalon is dominating
the series behavior at low orders in close analogy to the exact Borel function
in the large-$\beta_{0}$ approximation.

(a) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(b) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\lambda=10/3}$, full $\beta$-function

Figure 6:  Moments $\delta^{(0),{\rm FOPT}}_{W_{\tau}}(m_{\tau}^{2})$ (red)
and $\delta^{(0),{\rm CIPT}}_{W_{\tau}}(m_{\tau}^{2})$ (blue) based on the
multi-renormalon Borel model $B_{\hat{D},{\rm mr}}(u)$ accounting for the
5-loop QCD $\beta$-function as a function of the order up to which the series
are summed. Panel (a) shows the result in the $\overline{\mbox{MS}}$ scheme
and panel (b) in a scheme where the coupling is defined by
$a_{\lambda=10/3}(x)=a(x)/(1+5/3\,a(x))$. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red bands indicate the conventional Borel ambiguity of the FOPT series.

(a) $\delta^{(0)}_{1}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(b) $\delta^{(0)}_{-2x}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(c) $\delta^{(0)}_{2x^{3}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(d) $\delta^{(0)}_{-x^{4}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

Figure 7:  Moments $\delta_{W(x)}^{(0),{\rm FOPT}}(m_{t}^{2})$ (red) and
$\delta_{W(x)}^{(0),{\rm CIPT}}(m_{t}^{2})$ (blue) based on the multi-
renormalon Borel model $B_{\hat{D},{\rm mr}}(u)$ for $W(x)=1$, $-2x$, $2x^{3}$
and $-x^{4}$ accounting for the 5-loop QCD $\beta$-function as a function of
the order up to which the series are summed. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red bands indicate the conventional Borel ambiguity of the FOPT series.

The outcome for the hadronic tau decay width series $\delta^{(0)}_{W_{\tau}}$
in FOPT (red) and CIPT (blue) for $\alpha_{s}$ in the $\overline{\mbox{MS}}$
scheme is shown in Fig. 6(a) as a function of the truncation order using the
same label conventions as in Sec. 4. The FOPT and CIPT Borel sums are
represented by the red and blue horizontal lines, respectively, and the FOPT
Borel sum ambiguity is shown as the red band. The corresponding numerical
values (for Fig. 6(a) and all other figures in this section) are collected in
Tab. 2. We see that the FOPT series evolves within its narrow ambiguity band
for orders $6\lesssim n\lesssim 11$. The CIPT series approaches its Borel sum,
obtained from the FOPT Borel sum plus the asymptotic separation, reaching a
minimal distance for orders $n=6$ and $7$, but then deviates beyond before the
oscillating behavior caused by the UV renormalon sets in for $n\gtrsim 10$.
The minimal distance is almost 4 times larger than the width of the ambiguity
band. The reason for this behavior can be traced back to the interplay of
contributions in the CIPT series coming from the different polynomial terms in
the weight function $W_{\tau}(x)=1-2x+2x^{3}-x^{4}$. The individual
contributions of the four polynomial terms is shown in Figs. 7. We see that
for $W(x)=1$ the CIPT series essentially equals its Borel sum for the large
range of orders $4\lesssim m\lesssim 13$. For $W(x)=-2x$ it approaches the
Borel sum from below and reaches it for orders $9\lesssim m\lesssim 13$. For
$W(x)=2x^{3}$ on the other hand, the series is only close for $n=1$ and then
continually drops down until the UV renormalon behavior sets in for $n\gtrsim
10$. Finally, for $W(x)=-x^{4}$ the series is close to the Borel sum for
orders $n\lesssim 7$ and only drops slightly before the UV renormalon behavior
sets in for $n\gtrsim 10$. We see that the asymptotic separation quantifies
the different asymptotic behavior of the FOPT and CIPT series very well, but
the region of order where the CIPT series approach the respective Borel sums
closely differs. Since it just so happens that the CIPT series for each
polynmial weight function are always below their Borel values for orders
$n\lesssim 10$, the net effect is that the complete CIPT series cannot come
very close to its Borel sum before the UV renormalon behavior sets in. This
behavior is very similar to the corresponding observation we already discussed
for the large-$\beta_{0}$ approximation in Sec. 4.2, but it is much more
pronounced when the complete QCD $\beta$-function is accounted for. So despite
the fact that the complete CIPT series shows a systematic discrepancy to its
Borel sum, which can be traced back to the particular choice of weight
function for $R_{\tau}$, the asymptotic separation quantifies the different
asymptotic behavior of the FOPT and CIPT series very well. Furthermore, the
results shown in Figs. 6 and 7 illustrate very clearly that the asymptotic
separation is much larger than the FOPT Borel sum ambiguity.

It is also instructive to view the FOPT and CIPT series in different schemes
for the strong coupling. The outcome for the FOPT and CIPT series using again
the scheme defined by $a_{\lambda=10/3}(x)=a(x)/(1+5/3\,a(x))$ with
$\lambda=10/3$ is shown in Fig. 6(b). In analogy to the examination in the
large-$\beta_{0}$ approximation in Sec. 4.2 the regions of closest approach to
the respective Borel sums and the onset of the UV renormalon behavior is
pushed towards higher order, but the Borel sums, the FOPT Borel ambiguity as
well as the overall agreement with the visible asymptotic behavior of the
series are unchanged.

$5$-loop $\beta_{0}$-function
---
$B(u)$ | $W(x)$ | Figure | Scheme | $\delta^{(0),{\rm FOPT}}_{\rm Borel}$ | $\delta^{(0),{\rm CIPT}}_{\rm Borel}$ | $\delta^{\rm FOPT}$ | $\Delta$
$B_{\hat{D},{\rm mr}}$ | $W_{\tau}$ | 6(a) | $\overline{\mbox{MS}}$ | 0.237055 | 0.217754 | 0.001781 | -0.019301
$e^{\frac{5}{3}u}B_{\hat{D},{\rm mr}}$ | $W_{\tau}$ | 6(b) | $\lambda=10/3$ | 0.237055 | 0.217754 | 0.001781 | -0.019301
$B_{\hat{D},{\rm mr}}$ | $1$ | 7(a) | $\overline{\mbox{MS}}$ | 0.095924 | 0.099407 | -0.000029 | 0.003484
$B_{\hat{D},{\rm mr}}$ | $-2x$ | 7(b) | $\overline{\mbox{MS}}$ | 0.136775 | 0.123065 | 0.000103 | -0.013710
$B_{\hat{D},{\rm mr}}$ | $2x^{3}$ | 7(c) | $\overline{\mbox{MS}}$ | 0.003205 | -0.009829 | 0.001738 | -0.013034
$B_{\hat{D},{\rm mr}}$ | $-x^{4}$ | 7(d) | $\overline{\mbox{MS}}$ | 0.001152 | 0.005111 | -0.000031 | 0.003959
$B_{\hat{D},{p=2}}$ | $W_{\tau}$ | 8(a) | $\overline{\mbox{MS}}$ | 0.224200 | 0.212073 | 0.000034 | -0.012127
$B_{\hat{D},{p=3}}$ | $W_{\tau}$ | 8(b) | $\overline{\mbox{MS}}$ | 0.202604 | 0.201964 | -0.003809 | -0.000640
$B_{\hat{D},{p=2}}$ | $(-x)^{2}$ | 8(c) | $\overline{\mbox{MS}}$ | -0.002891 | -0.002891 | 0.004035 | 0
$B_{\hat{D},{p=3}}$ | $(-x)^{3}$ | 8(d) | $\overline{\mbox{MS}}$ | 0.010218 | 0.010218 | 0.001909 | 0
$B_{\hat{D},{\rm mr}}$ | $W_{c=-1}$ | 9(a) | $\overline{\mbox{MS}}$ | 0.277555 | 0.278899 | 0.023019 | 0.001343
$B_{\hat{D},{\rm mr}}$ | $W_{c=0}$ | 9(b) | $\overline{\mbox{MS}}$ | 0.219556 | 0.219821 | 0.011102 | 0.000265
$B_{\hat{D},{\rm mr}}$ | $W_{c=0.75}$ | 9(c) | $\overline{\mbox{MS}}$ | 0.176056 | 0.175513 | 0.002164 | -0.000544
$B_{\hat{D},{\rm mr}}$ | $W_{c=1}$ | 9(d) | $\overline{\mbox{MS}}$ | 0.161557 | 0.160743 | -0.000816 | -0.000814
Table 2:  Numerical values for the FOPT Borel sum $\delta^{(0),{\rm
FOPT}}_{\rm Borel}$, the CIPT Borel sum $\delta^{(0),{\rm CIPT}}_{\rm Borel}$,
the FOPT Borel sum ambiguity $\delta^{\rm FOPT}$ and the asymptotic separation
$\Delta=\delta^{(0),{\rm CIPT}}_{\rm Borel}-\delta^{(0),{\rm FOPT}}_{\rm
Borel}$ accounting for the full 5-loop $\beta$-function for the analyses in
Sec. 5.

###  5.2 Single-Renormalon Borel Model

(a) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $B_{\hat{D},p=2}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(b) $\delta^{(0)}_{W_{\tau}}(m_{\tau}^{2})$, $B_{\hat{D},p=3}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(c) $\delta^{(0)}_{(-x)^{2}}(m_{\tau}^{2})$, $B_{\hat{D},p=2}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(d) $\delta^{(0)}_{(-x)^{3}}(m_{\tau}^{2})$, $B_{\hat{D},p=3}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

Figure 8:  Moments $\delta_{W_{\tau}}(m_{\tau}^{2})^{(0),{\rm
FOPT}}(m_{t}^{2})$ (red) and $\delta_{W_{\tau}}^{(0),{\rm CIPT}}(m_{t}^{2})$
(blue) based on the single renormalon Borel models $B_{\hat{D},p=2}(u)$ (panel
(a)) and $B_{\hat{D},p=3}(u)$ (panel (b)) accounting for the 5-loop QCD
$\beta$-function as a function of the order up to which the series are summed.
Panel (c) shows the two series based on $B_{\hat{D},p=2}(u)$ with
$W(x)=(-x)^{2}$ and panel (d) shows the two series based on
$B_{\hat{D},p=3}(u)$ with $W(x)=(-x)^{3}$. The red and blue horizonal lines
represent the Borel sums of the FOPT and CIPT series, respectively, and the
red bands indicate the conventional Borel ambiguity of the FOPT series.

Let us now discuss two Borel models involving only single IR renormalon
contributions, one based on a generic Borel function for a $p=2$ renormalon
and one for a $p=3$ renormalon. In both cases we assume that the Wilson
coefficients of the associated OPE terms are unity and have vanishing
anomalous dimensions. The models are constructed by first fixing the
coefficient of the generic Borel functions such that the $5$-loop reduced
Adler function coefficient is reproduced exactly and then by adding a fourth
order polynmial in $u$ such that also the lower order coefficients are
reproduced as well. The expressions for these Borel models are referred to as
$B_{\hat{D},p=2}(u)$ and $B_{\hat{D},p=3}(u)$ and their explicit expressions
are shown in Eqs. (60) and (61), respectively. Both account for the QCD
$\beta$-function at the $5$-loop level. These models are interesting because
the numerical impact of the asymptotic separation for a $p=3$ renormalon
substantially differs from that of a $p=2$ renormalon and because we can study
the case $m=p$ where the asymptotic separation vanishes. In the following we
refer to these models also as the $p=2$ and $p=3$ models, respectively.

In Figs. 8(a) and 8(b) the outcome for the FOPT and CIPT series for the
hadronic decay rate weight function $W_{\tau}(x)$ are shown for the $p=2$ and
the $p=3$ models, respectively. The results for the $p=2$ model are quite
similar to the multi-renormalon model discussed in Sec. 5.1. This illustrates
the dominance of the $p=2$ renormalon in the multi-renormalon model for low
orders. In particular, the asymptotic separation is again much larger than the
FOPT ambiguity band, and the CIPT series always remains below its Borel sum.
The result shows that these features are inherent properties of the FOPT and
CIPT series if the renormalon associated to the gluon condensate term in the
Adler function Borel function is sizeable. Interestingly, for the $p=3$ model
the CIPT series closely approaches the FOPT Borel sum for orders $5\lesssim
n\lesssim 9$. This property is nicely explained by the asymptotic separation
as well, which just happens to be an order of magnitude smaller than the width
of the FOPT ambiguity band for a $p=3$ renormalon when the hadronic decay
weight function $W_{\tau}(x)$ is used. This arises due to cancellations among
the contributions from the different $m$ values. Interestingly, in this case
the FOPT series does not behave very well, indicating that neglecting a $p=2$
renormalon contribution in a realistic Borel model for the Adler function may
not be adequate.

In Figs. 8(c) and 8(d) the outcome for the FOPT and CIPT series for the $p=2$
and $p=3$ renormalon models are shown for the weight functions
$W(x)=(-x)^{p}$, where the asymptotic separation vanishes and the FOPT and
CIPT Borel sums agree identically. The FOPT series are quite unstable and do
not show any clear asymptotic regime. Neither their Borel sum nor the size of
their ambiguity bands could be determined by eye. In contrast, the CIPT series
in both cases shows a slow and steady linear rise and the Borel sum is well
within this linear regime. We have confirmed that this behavior is generic for
any series generated by a generic $p$ renormalon for the weight function
$W(x)=(-x)^{p}$. This confirms once more that our definition that the
asymptotic separation vanishes for $m=p$ is also adequate if all known
corrections to the QCD $\beta$-function are accounted for.

###  5.3 Moments with Small Asymptotic Separation

Having predictive control over the difference of the Borel sums of the FOPT
and CIPT series it is more than obvious that this provides a very powerful
theoretical tool to design moments where the asymptotic separation is going to
be small. Such a tool was absent prior to this article, and using it in
connection with the already known constraints related to suppressing the
effects of the OPE and duality violation corrections in Eq. (2) may uncover
new classes of spectral function moments useful for phenomenological analyses
and high-precision strong coupling determinations. As we have said earlier, it
is a priori impossible to construct moments where the asymptotic separation is
guaranteed to vanishes exactly since the Borel function of the Adler function
is unkown (beyond the large-$\beta_{0}$ approximation). However, we have also
seen from the examinations above that the $p=2$ renormalon (which is related
to the gluon condensate OPE correction) provides the largest numerical
contribution to the asymptotic separation. So an adequate strategy is to
construct moments based on weight functions which ensure that the contribution
of the $p=2$ renormalon to the asymptotic separation is suppressed. In the
following examination we use the large-$\beta_{0}$ expressions for the
asymptotic separation given in Eq. (53) as a handy analytic identifier for
suitable moments. We stress that the main purpose of the following analysis is
to demonstrate the practicability of the expressions for the analytic
separation and not to provide a thorough phenomenological analysis.

(a) $\delta^{(0)}_{W_{c=-1}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(b) $\delta^{(0)}_{W_{c=0}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(c) $\delta^{(0)}_{W_{c=0.75}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

(d) $\delta^{(0)}_{W_{c=1}}(m_{\tau}^{2})$, $B_{\hat{D},{\rm mr}}$,
$\alpha_{s}^{\overline{\rm MS}}$, full $\beta$-function

Figure 9:  Moments $\delta_{W_{c}}^{(0),{\rm FOPT}}(m_{t}^{2})$ (red) and
$\delta_{W_{c}}^{(0),{\rm CIPT}}(m_{t}^{2})$ (blue) ) based on the multi-
renormalon Borel model $B_{\hat{D},{\rm mr}}(u)$ for
$W_{c}=(1-x)^{2}(1+cx+x^{2})$ for $c=-1,0,0.75,1$ accounting for the 5-loop
QCD $\beta$-function as a function of the order up to which the series are
summed. The red and blue horizonal lines represent the Borel sums of the FOPT
and CIPT series, respectively, and the red bands indicate the conventional
Borel ambiguity of the FOPT series.

Let us consider exemplarily the polynomial weight function
$W_{c}(x)=(1-x)^{2}(1+cx+x^{2})$, which is pinched (i.e. has a double zero at
$x=1$) and leads to spectral function moments with vanishing asymptotic
separation in the large-$\beta_{0}$ approximation. The value of $c$ modulates
the contribution of the $x^{2}$ term in $W_{c}(x)$ and thus the size of the
FOPT Borel sum ambiguity, which is minimal for $c$ values close to unity. In
Fig. 9 the results for the FOPT (red) and CIPT (blue) spectral function moment
series arising from the multi-renormalon Borel model of Sec. 5.1 are shown for
$c=-1$, $0$, $0.75$ and $1$. The colored horizonal lines represent the
respective Borel sums and the red band indicates the FOPT Borel sum ambiguity.
Again the corresponding numerical values for the Borel sums, the FOPT Borel
ambiguity and the asymptotic separation are collected in Tab. 2. Comparing to
Fig. 6 where the series for the hadronic tau decay rate with the weight
function $W_{\tau}(x)$ are displayed101010For easier visual comparison we have
displayed the same range of values on the $y$-axes in all panels of Figs. 6
and 9., the FOPT and CIPT series for the weight function $W_{c}(x)$ are
substantially closer as long as $c<1$. Furthermore, for $c\lesssim 0$ and
orders $4\lesssim n\lesssim 7$ both series provide essentially equivalent
predictions. For $c=1$ the FOPT and CIPT series show some sizeable discrepancy
for orders $2<n<5$ and are compatible with their Borel sum only at order
$n=7$. For all values of $c$ the asymptotic separation is strongly suppressed
and ranges in size between $0.0002$ and $0.001$. On the other hand, the FOPT
Borel sum ambiguity amounts to sizaeable $0.023$ for $c=-1$ and continually
decreases down to $-0.0008$ for $c=1$ where it agrees in size with the
asymptotic separation. From the numbers shown in Tab. 2 we see that for
$c=0.75$ the FOPT Borel sum ambiguity and the asymptotic separation amounts to
$1.2\%$ and $0.3\%$, respectively, of the Borel sum value. Overall, the
results confirm the effectiveness of the expressions for the asymptotic
separation for the targeted design of moments where FOPT and CIPT series do
not lead to discrepancies.

##  6 Implications and Conclusions

In this article we have shown that the Borel functions of the FOPT and CIPT
series for $\tau$ hadronic spectral function moments differ and in particular
have different analytic properties. Both lead to different Borel sums and the
difference can be quantified reliably by the asymptotic separation for which
we have provided explicit analytical expressions. We have also demonstrated by
examining the FOPT and CIPT series for different Borel function models of the
Adler function, that the differing asymptotic behavior of the associated FOPT
and CIPT series is correctly described by the asymptotic separation. The
asymptotic separation is directly tied to the existence of IR renormalons in
the Borel function of the Adler function, and the size of the asymptotic
separation is in general larger than the size of the FOPT Borel sum ambiguity.
The analytic expressions for the asymptotic separation allow for the design of
spectral function moments for which the asymptotic separation is small and for
which phenomenological strong coupling determinations may be obtained that are
not affected by discrepancies between the FOPT and CIPT perturbation series.
However, our findings also reveal clearly that within the CIPT approach the
established renormalon calculus mechanism, where non-analytic IR renormalon
structures in the Borel function are in a one-to-one manner associated with
OPE correction terms, does not seem to be valid. We conclude with a brief
discussion on some of the implications that arise from our findings.

Let us first address the implications for the OPE. Within the standard OPE
approach, where high-scale Wilson coefficient and low-scale condensate
contributions are defined and separated using dimensional regularization and
the $\overline{\mbox{MS}}$ scheme, the character of the FOPT series and its
Borel representation allows for a one-to-one correspondence of IR renormalon
structures and OPE corrections. This means that the contour integration
involved in the definition of the hadronic spectral function moments modulates
both in the same way. So the suppression of an IR renormalon in the FOPT
series is associated with a suppression of the corresponding OPE correction
term. This is according to the established renormalon calculus that is known
for a long time Gross:1974jv ; tHooft:1977xjm ; David:1983gz ; Mueller:1984vh
. The Borel representation for the series in the CIPT approach we have found
in this article, however, shows that this association to the standard OPE
corrections does not seem to be valid, even though it is still implemented in
the Borel representation of the underlying Adler function. This explains why
the asymptotic separation between the FOPT and the CIPT series can be much and
even parametrically larger than the FOPT Borel sum ambiguity. The important
implication we see at this point is that within the CIPT approach the OPE
corrections cannot be computed with the standard OPE approach and that using
the OPE corrections obtained within the FOPT approach for making physical
predictions may not be appropriate.111111There is the possibility to compute
the standard OPE corrections to the spectral function moments carrying out the
contour integral over the standard vacuum polarization OPE terms within the
FOPT and CIPT approach with respect to the QCD corrections in the Wilson
coefficients. The resulting numerical differences are minor and cannot
reconcile the CIPT approach and the standard OPE method. This property of the
CIPT approach is caused by the contour integration with path-dependent
renormalization scales in the strong coupling. At this point we would like to
mention that this issue could well be an argument against the validity of the
CIPT approach, but only if one insists on using the standard approach to
compute OPE corrections. We plan to come back to this question elsewhere.

Another interesting implication is the obvious question about the mathematical
meaning of the asymptotic separation. In this article we discussed a class of
observables where the rearrangement of an asymptotic series leads to different
values of their Borel sum. At this point an analogy to the mathematical issue
of conditionally convergent series arises immediately. Conditionally
convergent series are series that converge, but they do not converge
absolutely. In practice, conditionally convergent series converge owing to the
sign alterations of the series terms, i.e. they would diverge if the absolute
values of the series terms would be summed. A well known example is the
alternating harmonic series $\sum_{n=1}^{\infty}(-1)^{n}/n$. Riemann’s
rearrangement theorem states that a conditionally convergent series of real
numbers can be rearranged in permutations so that the new series converges to
any arbitrary real number, or diverges. The findings of this article imply
that a similar mechanism may be behind the different asymptotic behavior of
the FOPT and CIPT series. At this point it should be mentioned that the FOPT
and the CIPT series are in fact not permutations of each other, but rather two
different arrangements of the double sum of Eq. (6) (upon insertion into the
contour integration of Eq. (3)) into a single sum. So it is the double sum
related to Eq. (6) which is the underlying series. Here the interesting
question arises whether another fundamentally different and physically
meaningful rearrangement of this underlying series may be constructed that
would lead to a series that differs intrinsically from the FOPT and CIPT
series and leads to yet another Borel function and another Borel sum. In any
case, already the existence of the FOPT and CIPT rearrangements implies that
any arbitrary value for the Borel sum can be obtained by combining FOPT and
CIPT series with arbitrary weights in the form
$\delta^{(0),\epsilon}_{W_{i}}(s_{0})\equiv\epsilon\,\delta^{(0),{\rm
FOPT}}_{W_{i}}(s_{0})+(1-\epsilon)\delta^{(0),{\rm CIPT}}_{W_{i}}(s_{0})$ for
an arbitrary real $\epsilon$. The findings of this article also indicate that
using contour improved computations of perturbative quantities can in general
lead to a different value for the Borel sum in comparison to the usual fixed-
order expansion (see e.g. Refs. Hoang:2004xm ; Dehnadi:2011gc ). In this
context, the considerations to determine the contour improved Borel
representation can proceed along the lines used in this article on the $\tau$
hadronic spectral function moments.

## Acknowledgments

We thank Matthias Jamin for helpful discussions. We acknowledge partial
support by the FWF Austrian Science Fund under the Doctoral Program “Particles
and Interactions” No. W1252-N27. We also thank the Erwin-Schrödinger
International Institute for Mathematics and Physics for partial support.

##  Appendix A Borel Models

$\displaystyle B_{\hat{D},{\rm mr}}(u)\,=\,$ $\displaystyle\,b^{(0)}_{\rm
mr}+b^{(1)}_{\rm mr}\,u\,+\,N^{(2)}_{\rm mr}\sum_{i=0}^{3}\,a^{(2,i)}_{\rm
mr}\,B^{\rm IR}_{\hat{D},2,\gamma^{(2,i)}}(u)\,+\,N^{(3)}_{\rm
mr}\sum_{i=0}^{3}\,a^{(3,i)}_{3}\,B^{\rm IR}_{\hat{D},3,\gamma^{(3,i)}}(u)$
$\displaystyle\,+\,N^{(-1)}_{\rm mr}\sum_{i=0}^{3}\,a^{(-1,i)}_{\rm
mr}\,B^{\rm UV}_{\hat{D},-1,\tilde{\gamma}^{(-1,i)}}(u)$ (59) $\displaystyle
B_{\hat{D},p=2}(u)\,=\,$
$\displaystyle\,\sum_{j=0}^{3}b^{(j)}_{2}\,u^{j}\,+\,N^{(2)}_{2}\sum_{i=0}^{3}\,a^{(2,i)}_{2}\,B^{\rm
IR}_{\hat{D},2,\gamma^{(2,i)}}(u)$ (60) $\displaystyle
B_{\hat{D},p=3}(u)\,=\,$
$\displaystyle\,\sum_{j=0}^{3}b^{(j)}_{3}\,u^{j}\,+\,N^{(2)}_{3}\sum_{i=0}^{3}\,a^{(3,i)}_{3}\,B^{\rm
IR}_{\hat{D},3,\gamma^{(3,i)}}(u)$ (61)

$\displaystyle N^{(2)}_{\rm mr}$ $\displaystyle=4.4669405$ $\displaystyle
N^{(3)}_{\rm mr}$ $\displaystyle=-20.9245713\quad$ $\displaystyle
N^{(-1)}_{\rm mr}$ $\displaystyle=-0.023602$ (62) $\displaystyle N^{(2)}_{2}$
$\displaystyle=2.6819313$ $\displaystyle N^{(2)}_{3}$
$\displaystyle=56.5347352$ $\displaystyle\gamma^{(2,0)}$
$\displaystyle=\frac{209}{81}$ $\displaystyle\gamma^{(2,1)}$
$\displaystyle=\frac{128}{81}\quad$ $\displaystyle\gamma^{(2,2)}$
$\displaystyle=\frac{47}{81}\quad$ $\displaystyle\gamma^{(2,3)}$
$\displaystyle=-\frac{34}{81}$ (63) $\displaystyle\gamma^{(3,0)}$
$\displaystyle=\frac{91}{27}$ $\displaystyle\gamma^{(3,1)}$
$\displaystyle=\frac{64}{27}\quad$ $\displaystyle\gamma^{(3,2)}$
$\displaystyle=\frac{37}{27}\quad$ $\displaystyle\gamma^{(3,3)}$
$\displaystyle=\frac{10}{27}$ $\displaystyle\tilde{\gamma}^{(-1,0)}$
$\displaystyle=\frac{98}{81}$ $\displaystyle\tilde{\gamma}^{(-1,1)}$
$\displaystyle=\frac{17}{81}\quad$ $\displaystyle\tilde{\gamma}^{(-1,2)}$
$\displaystyle=-\frac{64}{81}\quad$ $\displaystyle\tilde{\gamma}^{(-1,3)}$
$\displaystyle=-\frac{145}{81}$ $\displaystyle a^{(2,0)}_{\rm mr}$
$\displaystyle=1$ $\displaystyle a^{(2,1)}_{\rm mr}$ $\displaystyle=-0.499518$
$\displaystyle a^{(2,2)}_{\rm mr}$ $\displaystyle=-0.726053$ $\displaystyle
a^{(2,3)}_{\rm mr}$ $\displaystyle=-2.154561$ (64) $\displaystyle
a^{(3,0)}_{3}$ $\displaystyle=1$ $\displaystyle a^{(3,1)}_{3}$
$\displaystyle=-0.327643$ $\displaystyle a^{(3,2)}_{3}$
$\displaystyle=-0.341418$ $\displaystyle a^{(3,3)}_{3}$
$\displaystyle=1.028346$ $\displaystyle a^{(-1,0)}_{\rm mr}$ $\displaystyle=1$
$\displaystyle a^{(-1,1)}_{\rm mr}$ $\displaystyle=-1.233479$ $\displaystyle
a^{(-1,2)}_{\rm mr}$ $\displaystyle=-3.037540$ $\displaystyle a^{(-1,3)}_{\rm
mr}$ $\displaystyle=-0.173072$ $\displaystyle a^{(2,0)}_{2}$ $\displaystyle=1$
$\displaystyle a^{(2,1)}_{2}$ $\displaystyle=-0.327643$ $\displaystyle
a^{(2,2)}_{2}$ $\displaystyle=-0.879417$ $\displaystyle a^{(2,3)}_{2}$
$\displaystyle=-1.585526$ $\displaystyle b^{(0)}_{\rm mr}$
$\displaystyle=28.154241$ $\displaystyle b^{(1)}_{\rm mr}$
$\displaystyle=-0.985951$ (65) $\displaystyle b^{(0)}_{2}$
$\displaystyle=7.555621$ $\displaystyle b^{(1)}_{2}$ $\displaystyle=-0.758615$
$\displaystyle b^{(2)}_{2}=$ $\displaystyle-0.0807798$ $\displaystyle
b^{(3)}_{2}$ $\displaystyle=0.0453495$ $\displaystyle b^{(0)}_{3}$
$\displaystyle=-33.99899$ $\displaystyle b^{(1)}_{3}$
$\displaystyle=-2.981131$ $\displaystyle b^{(2)}_{3}=$
$\displaystyle-0.571259$ $\displaystyle b^{(3)}_{3}$
$\displaystyle=-0.0641748$

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