# Renormalons in the energy-energy correlator

Stella T. Schindler  Iain W. Stewart  and Zhiquan Sun

(May 30, 2023)

###### Abstract

The energy-energy correlator (EEC) is an observable of wide interest for
collider physics and Standard Model measurements, due to both its simple
theoretical description in terms of the energy-momentum tensor and its novel
features for experimental studies. Significant progress has been made in both
applications and higher-order perturbative predictions for the EEC. Here, we
analyze the nature of the asymptotic perturbative series for the EEC by
determining its analytic form in Borel space under the bubble-sum
approximation. This result provides information on the leading and subleading
nonperturbative power corrections through renormalon poles. We improve the
perturbative convergence of the $\overline{\mathrm{MS}}$ series for the EEC by
removing its leading renormalon using an R scheme, which is independent of the
bubble-sum approximation. Using the leading R-scheme power correction
determined by fits to thrust, we find good agreement with EEC OPAL data
already at ${\mathcal{O}}(\alpha_{s}^{2})$.

##  1 Introduction

A central challenge of high-energy physics is to develop observables that are
easily accessible experimentally, have a clean and clear connection to
dynamics of interest in the underlying physical theory, and can be computed
theoretically at high precision. An important observable for quantum
chromodynamics that currently satisfies the first two criteria is the energy-
energy correlator (EEC), an infrared and collinear (IRC) safe event shape that
characterizes the angular distribution of particles produced in $e^{+}e^{-}$
collisions [1, 2],

$\displaystyle\frac{d\Sigma}{d\cos\chi}=\sum_{i,j}\int
d\sigma\frac{E_{i}E_{j}}{Q^{2}}\delta(\cos\chi-\cos\theta_{ij})\,.$ (1.1)

The EEC has many useful generalizations, including the transverse EEC (TEEC)
for hadron-hadron collisions [3], multi-point energy correlators [4], as well
as EEC and TEEC observables for electron-hadron colliders [5, 6]; it is also
related to energy correlation functions for jets [7, 8, 9]. Nonperturbative
contributions to energy correlators are expected to be small, making them
compelling candidates for studying strong interactions [10], whether through
extracting the QCD coupling constant [11, 12], investigating transverse
momentum distributions [6, 5], or probing factorization violation [13]. These
observables have been measured at LEP and the LHC at CERN [14, 15, 16, 17, 18,
19], SLD at SLAC [20], and are a target of the planned Electron-Ion Collider
at Brookhaven [21]. To interpret experimental data, it is important to have
corresponding high-precision theoretical predictions.

The EEC was first studied at leading order in the 1970s [2], instigating a
decade of work producing numerical predictions at next-to-leading order (NLO)
[22, 23, 24, 25, 26, 27, 28, 29, 30]. It took two further decades to achieve
the first numerical results at next-to-next-to-leading order (NNLO) [31, 32]
and analytic results at NLO [33]. Higher-order perturbative results for the
EEC exist only for certain cases: in the so-called collinear limit at next-to-
next-to-leading logarithms (NNLL) [34, 35, 36], in the back-to-back limit at
N3LO, N3LL′ and N4LL [37, 38, 39, 32, 40, 41, 42], as well as for
$\mathcal{N}=4$ supersymmetric Yang-Mills theory both at leading power at NNLO
and at subleading power only in the back-to-back limit [43, 44, 45, 46, 47].

There has been much less recent work on nonperturbative power corrections for
the EEC. Ref. [48] pointed out that the EEC has $\Lambda_{\rm QCD}/Q$ power
corrections for all values of the angular variable $\chi$, in constrast to
event shapes for which such power corrections appear in the dijet limit. Ref.
[48] also carried out a renormalon calculation for a weighted integral over
the EEC, finding a result in agreement with the presence of $1/Q$ power
corrections. Ref. [49] initiated the use of operator methods to study EEC
power corrections, and argued that the leading corrections take the form

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma}{d\cos\chi}=\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{d\cos\chi}+\frac{2}{\sin^{3}\\!\chi}\frac{\bar{\Omega}_{1}}{Q}\,,$
(1.2)

with $\bar{\Omega}_{1}\sim\Lambda_{\rm QCD}$. Here
$\bar{\Omega}_{1}=\lambda_{1}/2$ in the notation of ref. [49, 50]. Note that
relative to [49, 50], we multiply by $2\pi$ for the azimuthal angular integral
and include an additional combinatoric factor of $2$, since either of the
energies $E_{i}$ or $E_{j}$ can become nonperturbative. Ref. [51] reached this
same conclusion in the dispersive model for power corrections with the same
factors of $2$ shown in eq. (1.2), and also considered perturbative
corrections to the power correction coefficient. The universality of the power
correction is not violated by gluon splitting [52], an effect known as the
Milan factor [53]. The earliest prediction for the $1/\sin^{3}\chi$ behavior
of EEC power corrections was in the fragmentation model of ref. [2]. Our
$\bar{\Omega}_{1}$ notation follows ref. [54], where a value of
$\bar{\Omega}_{1}$ for thrust was determined by a fit to data. Using massless
kinematics, the parameter $\bar{\Omega}_{1}$ appearing for the EEC is the same
as the one appearing in the dijet limit of thrust and other $e^{+}e^{-}$ event
shapes [49, 50, 55]; however, this universality can be spoiled by ${\cal
O}(1)$ hadron mass corrections, which depend on the method used to reconstruct
particle energies and momenta [56, 57].

Interestingly, to our knowledge, a direct bubble-sum renormalon calculation of
the $\chi$-dependence of the $1/Q$ power correction to the EEC in Borel space
has not yet been carried out, so we do so here. Our expectation is that the
leading Borel-space renormalon will agree with the $1/\sin^{3}\\!\chi$ in eq.
(1.2); however, the renormalon analysis method relies on completely different
approximations than the operator-based approach, and thus provides an
independent nontrivial confirmation. The Borel-space result also gives access
to higher-order renormalons. We find that in the bubble-sum approximation,
there is no renormalon corresponding to an $\mathcal{O}(1/Q^{2})$ power
correction. Additionally, our calculations provide input to assess the
convergence of the ${\overline{\mathrm{MS}}}$-scheme perturbative expansion
for the EEC, which is known to be asymptotic. At ${\cal O}(\alpha_{s}^{2})$,
we observe that the large-$\beta_{0}$ approximation provides a reasonable
approximation to the full result. From fig. 1 of ref. [32] at ${\cal
O}(\alpha_{s}^{3})$, we note that the EEC seems to exhibit slow perturbative
convergence for some values of $\chi$. This is also true for the $z\to 0$
fixed-order and resummed results in ref. [34].

We present a scheme change from ${\overline{\mathrm{MS}}}$ to an R scheme [58,
54, 59], which removes the leading renormalon from both the perturbative EEC
series and its leading power correction. This scheme change yields improved
convergence for the perturbative EEC series. Using the universality between
the thrust and EEC $\bar{\Omega}_{1}$ parameters, we show that our R scheme
results are already consistent at ${\cal O}(\alpha_{s}^{2})$ with EEC data
from the OPAL experiment at LEP [60].

The outline of this paper is as follows. Section 1.1 provides a brief overview
of asymptotic series and the types of divergences that they may exhibit,
including renormalons. In section 2, we utilize the bubble chain formalism for
probing renormalons from the infrared structure of final-state gluons to
calculate a Borel space result for the EEC, and demonstrate that its leading
renormalon is consistent with the $1/Q$ power correction in eq. (1.2), while a
renormalon corresponding to the $1/Q^{2}$ power correction is absent. In
section 3, we show how to remove the $u=1/2$ renormalon from the
${\overline{\mathrm{MS}}}$ perturbative series using an R scheme, and
demonstrate that this improves both perturbative convergence and agreement
with experimental data. We make concluding remarks in section 4.

###  1.1 Borel summation and renormalons

Perturbative expansions of observables in QFT are generally asymptotic [61]:
even if their first few terms appear to converge, at some order of expansion,
the coefficients begin to rapidly grow. Divergences should not be viewed as a
fundamental sickness of a theory; rather, asymptotic series encode all-orders
nonperturbative information [62, 63, 64, 65, 66]. Nonetheless, extracting
useful information from such a series is nontrivial; there are many different
types of divergences, each of which must be handled in different ways.

Perhaps the most conspicuous source of divergence in QFT is factorial growth
in the number of Feynman diagrams at high orders, which may lead to
corresponding growth of the series [67, 68, 69, 70]. The technique of Borel
summation is often sufficient to overcome such growth. First, we take a Borel
transform of an observable $f(\alpha_{s})\to F(u)$, which for QCD is most
often defined by:

$\displaystyle f(\alpha_{s})$
$\displaystyle=\sum_{n=-1}^{\infty}c_{n}\,\alpha_{s}^{n+1}\quad\to\quad
F(u)=c_{-1}\,\delta(u)+\sum_{n=0}^{\infty}\frac{c_{n}}{n!}\Big{(}\frac{4\pi}{\beta_{0}}\Big{)}^{n+1}\,u^{n}\,.$
(1.3)

The factor of $4\pi/\beta_{0}$ provides a convenient normalization for the
Borel variable $u$, where $\beta_{0}=11C_{A}/3-2n_{f}/3$ is the lowest order
QCD $\beta$-function. Next, we compute the new, more rapidly convergent sum
$F(u)$. Finally, we take an inverse Borel transform to recover the initial
observable $f(\alpha_{s})$:

$\displaystyle f(\alpha_{s})$
$\displaystyle=\int_{0}^{\infty}du\,\exp\Big{[}-u\frac{4\pi}{\beta_{0}\alpha_{s}(\mu)}\Big{]}\,F(u)\,.$
(1.4)

For more details of summation methods, see refs. [71, 64].

Borel summation techniques are powerful but not sufficient for every series.
If an observable exhibits poles on the positive real axis in the complex-$u$
Borel plane [72, 73, 74], the inverse Borel transform runs into a problem. In
order to evaluate the integral in eq. (1.4), one must deform the integration
contour above or below the real line to avoid the pole. Depending on where one
chooses to deform the contour, the inverse Borel transform will yield
different results. This leads to an ambiguity in the value of the perturbative
series, the magnitude of which is given by the residue of the pole. Such poles
are called infrared (IR) renormalons, which we hereafter refer to as simply
renormalons (poles on the negative real axis are UV renormalons, which are not
discussed here). Renormalons nearer to the origin create greater uncertainty
in the value of a perturbative series, since the series coefficients grow as
$\sim n!\,(2/p)^{n}$ for large $n$, given a pole at $u=p/2$.

Generally, one wishes to evaluate QCD observables theoretically at a level of
precision comparable to experiment. Luckily, renormalon ambiguities in
observables are unphysical: they cancel out between the perturbative series
and nonpeturbative matrix elements (or parameters of the field theory). We may
view them as merely an artifact of not cleanly separating perturbative and
nonperturbative contributions to an observable when carrying out a coupling
expansion.111The path integral formalism provides further intuition for
divergent series in QFT. We may conceptualize perturbative effects as
representing fluctuations about the vacuum, whereas nonperturbative effects
arise from fluctuations about nontrivial saddle points. Some nonperturbative
saddle points, such as instanton-anti-instanton pairs, are Borel summable [75,
76]. On the other hand, renormalons are connected to the more nuanced bions,
or fractional instanton-anti-instanton pairs [65, 77, 66, 78, 79, 80, 81, 82,
83]. Developments from the rich and evolving field of hyper-asymptotics,
resurgence theory, and trans-series [84, 85, 86, 87] promise to provide
further insight into nonperturbative effects in QFT as well as mathematical
tools for handling asymptotic expansions unambiguously. To construct a trans-
series, one sums up a traditional perturbative series with a nonperturbative
contribution, expressed as a complicated sum involving powers of exponential
and logarithmic functions of the coupling. If one examines only a subset of
terms in the trans-series, an imaginary ambiguity like a renormalon may arise.
However, one expects [84] that a set of the so-called resurgence relations
connecting the perturbative and nonperturbative sectors of the trans-series
will cancel all ambiguities in the overall observable. This appears to be
related to well-known phenomena in QCD where renormalon ambiguities are
removed by using alternate renormalization schemes to define the perturbative
and nonperturbative terms (see [64, 88] for reviews), which we explore later
in this paper with an R scheme.

Renormalons originate in the IR region of momentum integrals in loop diagrams
[64], and their behavior depends crucially on our choice of renormalization
scheme; they appear in many schemes, such as the popular
${\overline{\mathrm{MS}}}$ scheme. We may improve the convergence of a
perturbative expansion by modifying our renormalization scheme choice for
matrix elements (or parameters) in a manner that systematically removes these
instabilities. This modification improves both perturbative behavior and the
stability of the matrix element (or parameter) extraction. Examples of
renormalons in QCD include the heavy-quark mass [89, 90, 91, 92, 93], the
$B^{*}$-$B$ mass splitting governed by the parameter $\lambda_{2}$ [94, 58],
and hadronization parameters describing jet cross-sections [95, 90, 96, 97,
54, 98, 99]. In this paper, we obtain a Borel-space result for EEC
renormalons, and then apply the method of ref. [58] to improve stability for
the EEC.

##  2 EEC renormalons

The EEC describes how the energies of final-state hadrons in an $e^{+}e^{-}$
collision are correlated, as a function of their angle $\chi$ relative to one
another:

$\displaystyle\frac{d\Sigma}{d\cos\chi}=\sum_{i,j}\int d\sigma_{e^{+}e^{-}\to
ijX}\frac{E_{i}E_{j}}{Q^{2}}\delta(\cos\chi-\cos\theta_{ij}).$ (2.1)

The right-hand side of eq. (2.1) is a weighted sum over differential cross-
sections for all possible inclusive processes $e^{+}e^{-}\to ijX$ in which
particles $i$ and $j$ are detected, with $X$ representing the remaining
(arbitrary) final states. We take the total invariant mass of the incoming
$e^{+}e^{-}$ to be $Q^{2}$ and work in the center-of-mass frame, where the
measured final-state particles carry energy $E_{i}$ and $E_{j}$ and are
separated by angle $\theta_{ij}$.

(a)

(b)

Figure -119: Leading-order diagrams for the energy-energy correlator in eq.
(2.1).

###  2.1 Leading order

We begin by briefly reviewing the calculation of the EEC at leading order, as
our renormalon studies directly build off of this result. At LO, the EEC is
defined in terms of $e^{+}e^{-}\to q\bar{q}g$ Feynman diagrams shown in figure
-119, and eq. (2.1) becomes

$\displaystyle\frac{d\hat{\Sigma}}{dz}=\int\\!\\!\frac{d^{3}p}{2E}\frac{d^{3}\bar{p}}{2\bar{E}}\frac{d^{3}k}{2E_{g}}\frac{\delta^{4}(q-p-\bar{p}-k)}{(2\pi)^{5}}\sum_{i,j=q,\bar{q},g}\frac{E_{i}E_{j}}{Q^{4}}\delta(z-z_{ij})\alpha_{s}(\mu)\,\langle|\mathcal{M}_{0}|^{2}\rangle,$
(2.2)

where we write the angular variable as $\cos\chi=1-2z$ and
$\cos\theta_{ij}=1-2z_{ij}$. Here, $p$, $\bar{p}$, and $k$ are the four-
momenta of a final-state quark, anti-quark, and gluon, respectively, with
corresponding energies $E$, $\bar{E}$, and $E_{g}$. The four-momentum of the
virtual photon is $q$, with $Q^{2}=q^{2}$. The initial-state leptons have
four-momenta $l$ and $\bar{l}$, and treating the leptons as unpolarized, we
can average over both spins and leptonic angles relative to a given coordinate
system. The spin-averaged LO matrix element
$\langle|\mathcal{M}_{0}|^{2}\rangle$ from summing figures -119(a) and -119(b)
then evaluates to

$\displaystyle\langle|\mathcal{M}_{0}|^{2}\rangle$
$\displaystyle=\sigma_{0}\,\pi^{2}2^{9}C_{F}\frac{E^{2}+\bar{E}^{2}}{(Q-2E)(Q-2\bar{E})}\,.$
(2.3)

Here, the tree-level cross section is

$\displaystyle\sigma_{0}=\frac{4\pi\alpha^{2}}{Q^{2}}\sum_{q}e_{q}^{2}\,,$
(2.4)

where $e_{q}$ is the charge of a quark of flavor $q$. Plugging eq. (2.3) into
eq. (2.2) and carrying out the phase space integrals, one finds the result for
the EEC at $\mathcal{O}(\alpha_{s})$:

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}_{\rm
LO}}{dz}=\frac{\alpha_{s}C_{F}}{4\pi}\frac{3-2z}{z^{5}(1-z)}\Big{[}3z(2-3z)+2(3-6z+2z^{2})\log(1-z)\Big{]}\,.$
(2.5)

Note that this cross-section scales as $1/z$ as $z\to 0$ and as $1/(1-z)$ as
$z\to 1$ (modulo logarithms), and that this is the generic scaling expected
for perturbative contributions to the EEC in these limits.

###  2.2 Bubble diagram calculation

Computing all possible radiative corrections to an observable and analyzing
them for singularities is prohibitively involved. In QCD, it is conventional
to instead examine the smaller set of so-called bubble diagrams to probe the
existence of a renormalon. We define an $n$-bubble diagram by replacing the
outgoing gluon in figure -119(a) and figure -119(b) by a chain of $n$ fermion
loops, shown in detail in figure -118. We must evaluate eq. (2.2) with
$\mathcal{M}_{0}$ replaced by the sum $\mathcal{M}_{\text{bub}}$ of all
possible $n$-bubble diagrams, from $n=0$ to $\infty$:

$\displaystyle\frac{d\hat{\Sigma}_{\text{bub}}}{dz}=\int\\!\\!\frac{d^{3}p}{2E}\frac{d^{3}\bar{p}}{2\bar{E}}\frac{d^{3}k}{2E_{g}}\frac{\delta^{4}(q-\sum
p_{i})}{(2\pi)^{5}}\sum_{i,j}\frac{E_{i}E_{j}}{Q^{4}}\delta(z-z_{ij})\alpha_{s}(\mu)\,\langle|\mathcal{M}_{\text{bub}}|^{2}\rangle,$
(2.6)

where

$\displaystyle
i\sqrt{\alpha_{s}}\mathcal{M}_{\text{bub}}=\sum_{n=0}^{\infty}\left[\,\,\raisebox{-0.45pt}{\includegraphics[width=144.54pt]{figs/blank_bubble_1.png}
}+\raisebox{-0.45pt}{\includegraphics[width=144.54pt]{figs/blank_bubble_2.png}
}\,\,\right].$ (2.7)

Bubble diagrams comprise a gauge-invariant subset of all possible Feynman
diagrams, which for $n_{f}$ light quark flavors are proportional to
$n_{f}^{k}$, with the highest possible power of $k$ at each order in
$\alpha_{s}$. Thus, they serve as a convenient probe for the existence and
severity of renormalon ambiguities [64]. Though the bubble diagram procedure
is by nature imprecise, it nevertheless provides us sufficient information to
both analyze and improve perturbative convergence, as well as better
understand subleading power corrections.

Figure -118:  EEC bubble diagram, formed by replacing the external gluon in
figure -119(a) with a chain of $n$ fermion bubbles. Evaluating eq. (2.6) over
the sum of all bubble diagrams allows us to probe for renormalon divergences
in the EEC.

It will be convenient to organize results at higher orders in perturbation
theory in terms of color coefficients that involve
$\\{\beta_{0},\,C_{F},\,C_{A},\,...\\}$ rather than
$\\{n_{f},\,C_{F},\,C_{A},\,...\\}$. Here, $\beta_{0}=11C_{A}/3-2n_{f}/3$, and
the ellipses denote terms beyond quadratic Casimirs. The basis with
$\beta_{0}$ is more convenient here because terms with the maximum number of
$\beta_{0}$ factors are known to numerically dominate perturbative series in
many cases [100], while terms with $C_{F}$ or $C_{A}$ in place of a
$\beta_{0}$ are numerically subleading. The renormalon bubble-chain
calculation predicts precisely these leading-$\beta_{0}$ terms. Using the
analytic results at ${\cal O}(\alpha_{s}^{2})$ from ref. [33], one can confirm
that for the EEC the $\beta_{0}$ terms numerically dominate, as we show in
figure -117(a).

####  2.2.1 Modified gluon propagator probe

Our impetus to evaluate eq. (2.6) is that a bubble-sum-modified gluon
propagator probes the infrared structure of the perturbative series, and thus
can reveal renormalons. To make the gluon propagator explicit, we rewrite the
gluon phase-space integral in eq. (2.6) as

$\displaystyle\int\frac{d^{3}k}{2E_{g}}=\>2\int\frac{d^{4}k}{2\pi}\,\Theta(k^{0})\,\text{Im}\left[\frac{1}{-k^{2}-i0}\right]\,.$
(2.8)

For expediency, we define $\Pi_{g}(k)$ as the gluon propagator times $g^{2}$,
i.e. $-(4\pi\alpha_{s})iS^{\alpha\beta}/(k^{2}+i0)=-4\pi
iS^{\alpha\beta}\Pi_{g}(k)$. It is convenient to work in Lorentz gauge, so
that the same transverse Lorentz index structure is present at tree level and
in the presence of quark bubbles, in which case
$S^{\alpha\beta}=(g^{\alpha\beta}-{k^{\alpha}k^{\beta}}/{k^{2}})\delta^{ab}$.
The effect of inserting bubbles into eq. (2.6) is

$\displaystyle\langle|M_{\text{bub}}|\rangle^{2}\Pi_{g}(k)=\langle|M_{0}|\rangle^{2}\Pi_{\text{bub}}(k)\,,$
(2.9)

where $\Pi_{\text{bub}}$ is the bubble-modified gluon propagator defined as

$\displaystyle-4\pi
i\,S^{\alpha\beta}\,\Pi_{\text{bub}}(k)=\sum_{n=0}^{\infty}\raisebox{-0.4pt}{\includegraphics[width=101.17755pt]{figs/n_bubbles_1.png}
}\,,$ (2.10)

where the dots on the ends give a factor of $g^{2}=4\pi\alpha_{s}$.
Substituting eq. (2.9) into eq. (2.6),

$\displaystyle\frac{d\hat{\Sigma}_{\text{bub}}}{dz}=2\\!\\!\int\\!\\!\frac{d^{3}p}{2E}\frac{d^{3}\bar{p}}{2\bar{E}}\frac{d^{4}k}{2\pi}\Theta(k^{0})\,\sum_{i,j}\frac{E_{i}E_{j}}{Q^{4}}\delta(z-z_{ij})\text{Im}\big{[}\\!\\!-\\!\\!\Pi_{\text{bub}}\big{]}\frac{\delta^{4}(q-\sum
p_{i})}{(2\pi)^{5}}\langle|\mathcal{M}_{0}|^{2}\rangle.$ (2.11)

We will make an approximation when evaluating eq. (2.11). Diagrammatically,
the result for Im$\,\Pi_{\text{bub}}$ includes both terms with a cut gluon
propagator and terms where a pair of quarks in a bubble are cut. While both of
these cuts are included in our analysis, we allow the energy factors $E_{i}$
and $E_{j}$ to select only from the original $q$, $\bar{q}$, or a gluon with
$k^{0}$, and never from the individual quarks in the cut quark bubble. This
has the virtue of significantly simplifying the calculation while still
allowing the resulting modified gluon propagator to act as a probe of infrared
dynamics.

Let us evaluate the series for $\Pi_{\text{bub}}$ in eq. (2.10). The $n=0$
term is $-4\pi iS^{\alpha\beta}\Pi_{g}$. The $n=1$ renormalized result is well
known:

$\displaystyle=-\frac{2\alpha_{s}^{2}n_{f}}{3}\frac{(-iS^{\alpha\beta})}{k^{2}+i0}\ln\Big{(}\frac{\mu^{2}e^{c}}{-k^{2}-i0}\Big{)}\,.$
(2.12)

Again, we have included couplings and gluon propagators at either end. We use
the modified minimal subtraction (${\overline{\mathrm{MS}}}$) renormalization
scheme with parameter $\mu$, and define $c=5/3$. It is now straightforward to
evaluate the bubble propagator sum in (2.10):

$\displaystyle\Pi_{\text{bub}}(k)=\frac{1}{k^{2}+i0}\frac{4\pi}{\beta_{0}}\sum_{n=0}^{\infty}\Big{(}\frac{\alpha_{s}\beta_{0}}{4\pi}\Big{)}^{n+1}\bigg{[}\ln\Big{(}\frac{\mu^{2}e^{c}}{-k^{2}-i0}\Big{)}\bigg{]}^{n}\,.$
(2.13)

Here, we moved into the color coefficient basis
$\\{\beta_{0},C_{A},C_{F},\ldots\\}$ by making the standard replacement
$n_{f}\to-3\beta_{0}/2$. Next, we move eq. (2.13) into Borel space using eq.
(1.3), taking $\alpha_{s}\beta_{0}/(4\pi)$ to the Borel variable $u$:

$\displaystyle\Pi_{\text{bub}}(u)=\frac{-1}{(-k^{2}-i0)^{1+u}}\frac{4\pi}{\beta_{0}}(\mu^{2}e^{c})^{u},$
(2.14)

We can now convert the bubble phase-space integral in eq. (2.11) back from
four dimensions into a three-dimensional phase-space integral. When doing so,
it is important to remember that the squared amplitude $\langle|{\cal
M}_{0}|^{2}\rangle$ and EEC weighting factor $E_{i}E_{j}$ are functions of
energies, which we denote by ${\cal A}(k^{0})$. For any ${\cal A}(k^{0})$ we
can carry out the transverse momentum integral to obtain222If we were to
ignore ${\cal A}(k^{0})$, it would be natural to carry out the $k^{0}$
integral to find the modified phase-space integral in the last line of eq.
(2.2.1). This choice corresponds to using the equation of motion
$k^{0}=|\vec{k}|$ to simplify ${\cal A}(k^{0})$ before (rather than after)
carrying out the bubble sum in Borel space. Because eq. (2.2.1) involves an
integral over offshell momenta $k^{2}\neq 0$, these choices lead to different
results, with or without the $(\sin^{2}\theta_{k})^{-u}$ factor. Since the EEC
has linear terms in $k^{0}$, one must use the equations of motion after the
bubble sum, as in eq. (2.2.1), because higher-order terms in the series
$k^{0}=|\vec{k}|+k^{2}/(2|\vec{k}|)+\ldots$ all contribute with equal weight
to the renormalon. This constrasts with offshell terms in ${\cal A}$ like
$k^{2}/(p\cdot\bar{p})$, which can safely be dropped (and could only modify
the residues of poles $u\geq 3/2$).

$\displaystyle 2\\!\int\\!\\!\frac{d^{4}k}{2\pi}\,\Theta(k^{0})\,\text{Im}$
$\displaystyle\bigg{[}\frac{1}{(-k^{2}-i0)^{1+u}}\bigg{]}{\cal
A}(k^{0})=\\!\int\\!\frac{dk^{+}dk^{-}}{2\pi}\Theta(k^{0}){\cal
A}(k^{0})\,\text{Im}\bigg{[}\frac{d^{2}k_{\perp}}{(\vec{k}_{\perp}^{2}-k^{+}k^{-}-i0)^{1+u}}\bigg{]}$
$\displaystyle=\frac{\sin(\pi u)}{\pi
u}\,2\pi\\!\int\\!\frac{dk^{+}dk^{-}}{4}\,\Theta(k^{+})\Theta(k^{-})\,(k^{+}k^{-})^{-u}{\cal
A}\Big{(}\frac{k^{+}+k^{-}}{2}\Big{)}\,$ $\displaystyle=\frac{\sin(\pi u)}{\pi
u}\int\\!\\!\frac{d^{3}k}{2\big{|}\vec{k}\big{|}^{1+2u}}(\sin^{2}\theta_{k})^{-u}{\cal
A}(|\vec{k}|)\,.$ (2.15)

In the last line, we make the change of variable $k^{\pm}=|\vec{k}|\pm k^{z}$
to transform the result back into the form of a standard phase-space integral,
and use $k^{+}k^{-}=|\vec{k}_{\perp}|^{2}=|\vec{k}|^{2}\sin^{2}\theta_{k}$.
Here, $\theta_{k}$ is a polar angle defined relative to any fixed axis
$\hat{l}$, a freedom we use later on. We now have all the ingredients we need
for the bubble-approximated cross-section eq. (2.11).

####  2.2.2 Borel-space result

We now insert the LO matrix element eq. (2.3), the Borel-transformed bubble
propagator eq. (2.14), and the phase-space integral relation eq. (2.2.1) into
the bubble-modified EEC cross-section eq. (2.11), and find

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}_{\text{bub}}}{dz}$
$\displaystyle=\frac{8}{\pi^{2}}\frac{C_{F}(\mu^{2}e^{c})^{u}}{Q^{4}\beta_{0}}\,\frac{\sin(u\pi)}{u\pi}\,\,\int
d^{3}p\,d^{3}\bar{p}\,d^{3}k\,\,\delta^{4}(q-{\textstyle\sum}p_{i})\,\,(\sin^{2}\theta_{k})^{-u}$
(2.16)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\sum_{i,j}\frac{E_{i}E_{j}}{E\bar{E}E_{g}^{1+2u}}\delta(z-z_{ij})\frac{E^{2}+\bar{E}^{2}}{(Q-2E)(Q-2\bar{E})}\,.$

We can break this integral up into two pieces. The first term comes from the
choice $ij=q\bar{q}$. The second term comes from $ij=qg$ and $\bar{q}g$, which
are equal by symmetry of eq. (2.16). We illustrate how to evaluate these
integrals using the $qg$ case. First, we integrate over $d^{3}\bar{p}$ using
$\delta^{3}(\vec{q}-\sum\vec{p}\,)$. Without loss of generality, we can take
the quark along the $\hat{l}$-axis, so $\theta_{k}=\theta_{qg}$, and we can
write $d^{3}pd^{3}k=4\pi E^{2}E_{g}^{2}dEdE_{g}d\cos\theta_{k}d\Delta\phi$,
where $\Delta\phi$ is the difference in azimuthal angle between the quark and
the gluon. We carry out the angular integrals using $\delta(z-z_{qg})$. Next,
we eliminate the remaining energy-conserving delta function using $d\bar{E}$.
This leaves us with a single remaining integral in $E=Qx/2$.

Combining both the $q\bar{q}$ and $qg/\bar{q}g$ terms, we have

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}_{\text{bub}}}{dz}$
$\displaystyle=\frac{C_{F}(\mu^{2}e^{c})^{u}}{\beta_{0}Q^{2u}}\frac{\sin(u\pi)}{u\pi}\frac{1}{[z(1-z)]^{1+u}}\,\int_{0}^{1}dx\,\frac{x(1-x)^{-2u}}{(1-xz)^{4-2u}}$
(2.17)
$\displaystyle\quad\times\bigg{\\{}z(1-x)\Big{[}1-2x+2x^{2}-2x^{3}z+x^{4}z^{2}\Big{]}$
$\displaystyle\qquad+2(1-z)\Big{[}1-4xz+x^{2}(1+2z+4z^{2})-2x^{3}z(1+2z)+2x^{4}z^{2}\Big{]}\bigg{\\}}\,.$

It is possible to perform these integrals analytically. Writing

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma_{\text{bub}}}{dz}$
$\displaystyle=\frac{C_{F}(\mu^{2}e^{c})^{u}}{\beta_{0}Q^{2u}}\frac{\sin(u\pi)}{u\pi}\frac{1}{[z(1-z)]^{1+u}}\Bigl{(}\mathcal{I}_{q\bar{q}}+2\,\mathcal{I}_{qg}\Bigr{)}\,,$
(2.18)

we find

$\displaystyle\mathcal{I}_{q\bar{q}}\equiv\,\,$ $\displaystyle
z\,\int_{0}^{1}dx\frac{x(1-x)^{-2u}}{(1-xz)^{4-2u}}(1-x)\Big{[}1-2x+2x^{2}-2x^{3}z+x^{4}z^{2}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{4u^{3}(1\\!-\\!z)^{3}-18u^{2}(1\\!-\\!z)^{2}-u(1\\!-\\!z)(z^{2}+7z-32)\\!-\\!6(4\\!-\\!3z)}{6(u-2)}\,_{2}F_{1}(1,4-2u,5-2u,z)$
$\displaystyle+\frac{-8u^{4}(1\\!-\\!z)^{2}+8u^{3}(1\\!-\\!z)(5\\!-\\!z)+u^{2}(2z^{2}+50z-86)-u(2z^{2}+15z-90)-36}{6(u-1)(2u-3)}\,,$
$\displaystyle\mathcal{I}_{qg}\equiv\,\,$
$\displaystyle(1-z)\,\int_{0}^{1}dx\,\,\frac{x(1-x)^{-2u}}{(1-xz)^{4-2u}}\Big{[}1-4xz+x^{2}(1+2z+4z^{2})-2x^{3}z(1+2z)+2x^{4}z^{2}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{4u^{2}(1\\!-\\!z)^{3}-2u(1\\!-\\!z)^{2}(7\\!-\\!3z)+(1\\!-\\!z)\bigl{(}4z^{2}-14z+13\bigr{)}}{2(u-2)}\,_{2}F_{1}(1,4-2u,5-2u,z)$
$\displaystyle+\frac{1}{(u-1)(2u-3)(2u-1)}\Bigl{[}-8u^{4}(1\\!-\\!z)^{2}+12u^{3}(1\\!-\\!z)(3\\!-\\!2z)-u^{2}\left(30z^{2}\\!-\\!88z\\!+\\!60\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+u(18z^{2}-55z+42)+4z(3-z)-12\Bigr{]}\,,$
(2.19)

where ${}_{2}F_{1}(a,b,c,x)$ is the standard hypergeometric function. These
results are only valid in the region $0<z<1$, as we have not included the full
set of contributions needed to carry out the calculation at $z=0$ and $z=1$.

###  2.3 Renormalons in the EEC

We can now identify renormalons in the EEC from eq. (2.18). Recall that
renormalons are poles on the positive real $u$-axis; given our normalization
for the Borel transform, they generally appear at half-integer or integer
values of $u$. We find that the leading renormalon appears at $u=1/2$,
associated with a pole in $\mathcal{I}_{qg}$ but not in $\mathcal{I}_{qq}$.
This is consistent with the picture of leading power corrections arising from
the limit in which the measured gluon becomes nonperturbative. Examining the
$u\to 1/2$ limit, we find

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}\bigg{|}_{u\to 1/2}$
$\displaystyle=\frac{\text{Res}_{1/2}}{u-1/2}+\ldots$
$\displaystyle=-\frac{4C_{F}}{\pi\beta_{0}}\frac{\mu
e^{c/2}}{Q}\frac{1}{[z(1-z)]^{3/2}}\frac{1}{u-1/2}+\ldots\,,$ (2.20)

where the ellipses denote terms of ${\cal O}\big{(}(u-1/2)^{0}\big{)}$. We can
compute the size of the ambiguity $\Delta_{1/2}$ in the
${\overline{\mathrm{MS}}}$ perturbative series $d\hat{\Sigma}/dz$ associated
with this renormalon. Specifically, we evaluate the inverse Borel integral
over a contour encircling the pole in complex $u$-space:

$\displaystyle\Delta_{1/2}\bigg{(}\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}\bigg{)}$
$\displaystyle=\oint_{u=1/2}du\,\exp\Big{[}-u\frac{4\pi}{\beta_{0}\alpha_{s}(\mu)}\Big{]}\,\frac{\text{Res}_{1/2}}{u-1/2}$
$\displaystyle=-\frac{8iC_{F}e^{5/6}}{\beta_{0}}\frac{1}{[z(1-z)]^{3/2}}\frac{\Lambda_{\rm
QCD}}{Q}\,.$ (2.21)

This result is valid for $0<z<1$. Note that renormalon ambiguities are always
imaginary.

This form for the ambiguity in eq. (2.3) makes clear its connection to the
$\Lambda_{\rm QCD}/Q$ EEC power correction. Using
$\sin^{3}\chi=8[z(1-z)]^{3/2}$, we write eq. (1.2) in terms of $z$:

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma}{dz}=\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}+\frac{1}{2[z(1-z)]^{3/2}}\frac{\bar{\Omega}_{1}}{Q}\,.$
(2.22)

Thus, the $z$-dependence of the leading renormalon in eq. (2.3) agrees with
the leading EEC power corrections derived in refs. [49, 50]. Since the left-
hand side of eq. (2.22) is a renormalon-free observable, the renormalon in the
perturbative series $d\hat{\Sigma}/dz$ cancels against the renormalon in the
${\overline{\mathrm{MS}}}$ definition of the hadronic parameter
$\bar{\Omega}_{1}$. We can cross-check our result for $\text{Res}_{1/2}$ with
the analogous renormalon bubble-chain calculations for thrust in the dijet
limit [101, 54], which allows us to infer the ambiguity

$\displaystyle\Delta_{1/2}\big{(}\bar{\Omega}_{1}\big{)}$
$\displaystyle=\Lambda_{\rm QCD}\frac{16iC_{F}e^{5/6}}{\beta_{0}}\,.$ (2.23)

Thus, we confirm the cancellation of ambiguities between the two terms in eq.
(2.23), $\Delta_{1/2}(d\Sigma/dz)=0$. This cross-check also provides a test of
the universality of power corrections between thrust and the EEC, though we
caution that the renormalon based calculation does not account for potential
${\cal O}(1)$ hadron-mass corrections which can potentially spoil the
universality.

The EEC observable obeys certain sum rules [34, 102, 35]. A particularly
simple one, which holds nonperturbatively,333Note that results for higher $z$
and $(1-z)$ moments of $d\Sigma/dz$ rely on manipulations with massless four-
vectors and hence can have nonperturbative corrections in QCD. states that

$\displaystyle\int_{0}^{1}\\!dz\frac{d\Sigma}{dz}=\sigma\,.$ (2.24)

Equation (2.24) follows from energy conservation, $\sum_{i}E_{i}=Q$, and
reflects that $d\Sigma/dz$ integrates to the total hadronic cross-section
$\sigma=\sigma(e^{+}e^{-}\to X)$, just like a regular differential
distribution. This cross-section only has nonperturbative corrections at
${\cal O}(\Lambda_{\rm QCD}^{4}/Q^{4})$ for massless quarks, implying that to
properly treat the $z=0$ and $z=1$ endpoints, we must modify the function
$[z(1-z)]^{-3/2}$ such that it integrates to zero. The function remains the
same for $0<z<1$, suggesting we need a plus distribution. Because we are
dealing with two singularities, at $z=0$ and $z=1$, we cannot determine a
functional form uniquely by imposing that the integral vanishes. The residual
ambiguity can be parameterized by a constant $\eta$, where $0<\eta<1$.
Constructing standard plus distributions over the intervals $[0,\eta]$ and
$[\eta,1]$, and then recombining terms to cancel as much of the $\eta$
dependence as possible, we find that the regulated $z$-dependence for the
power correction can be written as

$\displaystyle\Big{[}[z(1-z)]^{-3/2}\Big{]}_{+}$
$\displaystyle\equiv\lim_{\epsilon\to
0}\Big{\\{}\frac{d}{dz}\theta(z-\epsilon)\theta(1-z-\epsilon)G(z)\Big{\\}}+\big{[}\delta(1-z)-\delta(z)\big{]}G(\eta)\,,$
(2.25)

where $G(z)=-G(1-z)$, and

$\displaystyle
G(z)=\int^{z}\\!\\!dz^{\prime}\>[z^{\prime}(1-z^{\prime})]^{-3/2}=\frac{2z-2(1-z)}{z^{1/2}(1-z)^{1/2}}\,.$
(2.26)

It is easy to confirm that
$\int_{0}^{1}dz\>\big{[}[z(1-z)]^{-3/2}\big{]}_{+}=0$ for any value of $\eta$.
A natural choice is to require that the plus distribution be symmetric under
$z\to 1-z$, implying $\eta=1/2$ because $G(1/2)=0$. In this case, the
$G(\eta)$ term in eq. (2.25) would vanish. With this modification the formula
for the EEC in ${\overline{\mathrm{MS}}}$ with its leading power correction
now reads

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma}{dz}=\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}+\frac{1}{2\big{[}[z(1-z)]^{3/2}\big{]}_{+}}\frac{\bar{\Omega}_{1}}{Q}\,.$
(2.27)

(a)

(b)

Figure -117: The $\mathcal{O}(\alpha_{s}^{2})$ (NLO) term in the perturbative
series for the EEC, as indicated by the subscript $(2)$ on
$d\hat{\Sigma}^{(2)}$. The normalization used to define this term is
$\frac{d\hat{\Sigma}}{dz}=2\big{(}\frac{\alpha_{s}}{2\pi}\big{)}\frac{d\hat{\Sigma}^{(1)}}{dz}+2\big{(}\frac{\alpha_{s}}{2\pi}\big{)}^{2}\frac{d\hat{\Sigma}^{(2)}}{dz}+...$.
We take $\mu=Qe^{-5/6}$. Figure (a) shows the results calculated in ref. [33],
organized by color channels $C_{F}^{2}$, $C_{F}C_{A}$, and $C_{F}\beta_{0}$,
as discussed in section 2.2. The $C_{F}\beta_{0}$ term (red) dominates over
the other channels. Figure (b) compares the perturbative expansion of the
$u=1/2$ renormalon in eq. (2.31) (dashed red) and the $C_{F}\beta_{0}$ term of
the EEC in the ${\overline{\mathrm{MS}}}$ scheme given in ref. [33] (solid
black) at $\mathcal{O}(\alpha_{s}^{2})$.

While both $\mathcal{I}_{q\bar{q}}$ and $\mathcal{I}_{gq}$ exhibit simple
poles at $u=1$, the prefactor $\sin(u\pi)$ vanishes here. Thus, the full
$d\hat{\Sigma}_{\text{bub}}/dz$ has no $u=1$ pole,

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}\bigg{|}_{u=1}=\,\,\frac{0}{u-1}+\ldots\,,$
(2.28)

where the ellipses are ${\cal O}\big{(}(u-1)^{0}\big{)}$. This means our
bubble-sum analysis has no sensitivity to renormalons corresponding to
$\Lambda_{\rm QCD}^{2}/Q^{2}$ EEC power corrections. We do not, however, rule
out a non-Abelian contribution to the $u=1$ residue. The presence of $u=1$
poles for event shape calculations which treat power corrections from the hard
and collinear regions at higher orders in perturbation theory have been
discussed in the literature, see for example [103, 99].

Our Borel-space result in eq. (2.18) indeed has no poles at any integer values
of $u=k$, corresponding to $\Lambda_{\rm QCD}^{2k}/Q^{2k}$ power corrections.
We do observe poles at higher half-integer values, $u=k+1/2$, associated with
renormalons in $\Lambda_{\rm QCD}^{2k+1}/Q^{2k+1}$ power corrections. Both the
$\mathcal{I}_{q\bar{q}}$ and $\mathcal{I}_{qg}$ terms contribute; for example,
near $u=3/2$, we find

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}\bigg{|}_{u=3/2}=\frac{C_{F}}{\pi\beta_{0}}\frac{\mu^{3}e^{3c/2}}{Q^{3}}\frac{(2-5z+4z^{2})}{[z(1-z)]^{3/2}}\frac{1}{u-3/2}+\ldots\,,$
(2.29)

where the ellipses are ${\cal O}\big{(}(u-3/2)^{0}\big{)}$. Note that our
calculational procedure renders the residues of $u\geq 3/2$ poles somewhat
ambiguous because we used $k^{2}=0$ to simplify the LO matrix element in eq.
(2.3). If we had instead expanded around $k^{2}=0$, the higher-order terms
${\cal O}(k^{2j}/Q^{2j})$ would modify residues at $u\geq 3/2$ (but not at
$u=1/2$ or $u=1$).

####  2.3.1 Renormalon series in ${\overline{\mathrm{MS}}}$

Next, we investigate the manifesation of renormalons in
${\overline{\mathrm{MS}}}$ perturbative calculations. Recall from figure
-117(a) that the dominant contribution to the EEC comes from the
$C_{F}\beta_{0}$ color structure at ${\mathcal{O}}(\alpha_{s}^{2})$ [33]. We
determine the extent to which the $u=1/2$ pole approximates the full
$C_{F}\beta_{0}$ contribution by expanding about $u=0$. To retain the leading
logarithmic $\mu$-independence when expanding in $u$, we replace
$\text{Res}_{1/2}\to-\frac{1}{\beta_{0}}{\cal Z}\exp[2u\ln(\mu e^{c/2}/Q)]$
(which simplifies back to $\text{Res}_{1/2}=-\frac{1}{\beta_{0}}{\cal Z}\mu
e^{c/2}/Q$ at $u\\!\\!=\\!\\!1/2$), and we also expand this exponential. Here
${\cal Z}=4C_{F}[z(1-z)]^{-3/2}/\pi$. This gives

$\displaystyle\left.\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}(u)\right|_{\mathrm{ren.}}$
$\displaystyle=\frac{\text{Res}_{1/2}}{u-1/2}\approx\frac{{\cal
Z}}{\beta_{0}}\Big{[}2+4u\big{(}1+L_{Q}\big{)}+8u^{2}\big{(}1+L_{Q}+\frac{1}{2}L_{Q}^{2}\big{)}+...\Big{]}\,,$
(2.30)

where $L_{Q}=\ln(\mu e^{c/2}/Q)$. Using the inverse Borel transform in eq.
(1.4),

$\displaystyle\left.\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}(\alpha_{s})\right|_{\mathrm{ren.}}\\!\\!\\!\\!$
$\displaystyle=\frac{{\cal
Z}}{\beta_{0}}\>\sum_{n=1}^{\infty}2^{n}\,\Gamma(n)\bigg{(}\frac{\alpha_{s}(\mu)\beta_{0}}{4\pi}\bigg{)}^{\\!\\!n}\
\ \sum_{k=0}^{n-1}\frac{L_{Q}^{k}}{k!}$ (2.31) $\displaystyle=\frac{{\cal
Z}}{\beta_{0}}\>\Bigg{[}2\frac{\alpha_{s}(\mu)\beta_{0}}{4\pi}+4\bigg{(}\frac{\alpha_{s}(\mu)\beta_{0}}{4\pi}\bigg{)}^{2}(1+L_{Q})+\dots\Bigg{]}.$

In figure -117(b), we compare the perturbative $C_{F}\beta_{0}\alpha_{s}^{2}$
result from ref. [33] to the $\mathcal{O}(\alpha_{s}^{2})$ renormalon term in
eq. (2.31), taking $\mu=Qe^{-c/2}$ so that $L_{Q}=0$. We see that the
renormalon prediction mirrors the full $C_{F}\beta_{0}$ perturbative result in
a substantial portion of $z$-space. This evidentiates the importance of taming
terms in eq. (2.31) at higher orders, which grow as $2^{n}(n-1)!$ .

##  3 Mitigating renormalon effects with an R scheme

In this section, we reorganize the perturbative and nonperturbative
corrections to EEC, to remove leading renormalon effects from both of these
terms. This leads to improved perturbative convergence and a better agreement
with experimental data.

###  3.1 EEC in the R scheme

It is possible to systematically mitigate renormalon effects by switching from
the ${\overline{\mathrm{MS}}}$ scheme to a so-called R renormalization scheme
[58, 93]. This scheme change removes the dominant Borel pole from the
perturbative series, and also modifies matrix elements like
$\bar{\Omega}_{1}$. The full perturbative series for the EEC in the
${\overline{\mathrm{MS}}}$ scheme is

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}}{dz}=\sum_{n=1}^{\infty}c_{n}(z,\mu/Q)\left[\frac{\alpha_{s}(\mu)}{4\pi}\right]^{n}\,,$
(3.1)

where $c_{n}$ is a function of $z$ and $\mu/Q$, eq. (2.5) gives $c_{1}$, and
ref. [33] gives $c_{2}$ analytically. To remove the $u=1/2$ renormalon, we
define an R scheme using a new subtraction scale $R$. In the R scheme, the
leading power correction is [104, 54]

$\displaystyle\Omega_{1}(R)=\bar{\Omega}_{1}-\bar{\delta}(R)\,.$ (3.2)

Here $\bar{\delta}(R)$ is a perturbative series that removes the leading
renormalon,

$\displaystyle\bar{\delta}(R)$
$\displaystyle=R\sum_{n=1}^{\infty}d_{n0}\left[\frac{\alpha_{s}(R)}{4\pi}\right]^{n}=R\sum_{n=1}^{\infty}d_{n}(\mu/R)\left[\frac{\alpha_{s}(\mu)}{4\pi}\right]^{n}\,,$
(3.3)

where $d_{n0}$ are numbers and $d_{n}(\mu/R)$ are simple functions
$d_{n}(\mu/R)=\sum_{j=0}^{n-1}d_{nj}\ln^{j}(\mu/R)$. For $j\geq 1$,
$d_{nj}=(2/j)\sum_{k=j}^{n-1}k\,d_{k(j-1)}\beta_{n-k-1}$, where the QCD
$\beta$-function coefficients $\beta_{n}$ are defined by
$\mu\frac{d}{d\mu}\alpha_{s}(\mu)=-2\alpha_{s}\sum_{n=0}^{\infty}\beta_{n}\alpha_{s}^{n+1}/(4\pi)^{n+1}$.
To ensure that the scheme change in eq. (3.2) does not modify the parametric
size of the power correction $\bar{\Omega}_{1}$, we choose $R$ such that
$\bar{\delta}(R)\sim\Lambda_{\rm QCD}$. A typical choice is $R\simeq 2\,{\rm
GeV}$. In the bubble-sum approximation, the coefficients at large $n$ grow due
to the $u=1/2$ renormalon as $d_{n+1}(\mu/R)\simeq(\mu/R)n!(2\beta_{0})^{n}Z$,
for constant $Z$, in agreement with eq. (2.31).

There are multiple possible choices for the $d_{n0}$ coefficients that remove
the $u=1/2$ renormalon; refs. [101, 104, 54, 59] introduce suitable schemes
using the hemisphere or thrust soft functions. We follow the definition of
ref. [59], whose coefficients $d_{n0}$ stem from perturbatively expanding the
logarithm of the position-space thrust soft function,
$\ln\tilde{S}_{\tau_{2}}(y,\mu)$, and evaluating it at $iy=1/\mu=1/R$.
Universality implies that this perturbative series has the desired renormalon,
and this scheme choice does not rely on the bubble-sum approximation since it
contains terms with all allowed color structures. Numerically, the first two
coefficients in the scheme of ref. [59] are

$\displaystyle d_{1}(\mu/R)$ $\displaystyle=d_{10}=-8.357\,,$ (3.4)
$\displaystyle d_{2}(\mu/R)$
$\displaystyle=d_{20}+2\beta_{0}\,d_{10}\ln\Big{(}\frac{\mu}{R}\Big{)}=-72.443-16.713\beta_{0}\ln\Big{(}\frac{\mu}{R}\Big{)}\,.$

Calculating $d_{30}$ requires the non-logarithmic term in the ${\cal
O}(\alpha_{s}^{3})$ thrust soft function, for which only estimates exist
[105]. Note that the $d_{i}(\mu/R)$ coefficients are independent of hadron
masses and are therefore suitable for removing the renormalon from
$\bar{\Omega}_{1}$ for all observables related by massless universality.

We must also change the EEC perturbative series to the R scheme:

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}^{\rm R}(R)}{dz}$
$\displaystyle=\sum_{n=1}^{\infty}\bigg{\\{}c_{n}(z,\mu/Q)+\frac{R}{2Q}\frac{d_{n}(\mu/R)}{\big{[}[z(1-z)]^{3/2}\big{]}_{+}}\bigg{\\}}\left[\frac{\alpha_{s}(\mu)}{4\pi}\right]^{n}\,.$
(3.5)

Here, the $u=1/2$ renormalon cancels order-by-order between the $c_{n}$ and
$d_{n}$ series. This perturbative R scheme result depends both on the use of
the standard ${\overline{\mathrm{MS}}}$ scheme for the coupling
$\alpha_{s}(\mu)$, and on the R scheme for the power correction. When carrying
out the R scheme change on the full EEC observable, we have:

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma}{dz}=\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}^{\rm
R}(R)}{dz}+\frac{1}{2\big{[}[z(1-z)]^{3/2}\big{]}_{+}}\frac{\Omega_{1}(R)}{Q}\,.$
(3.6)

This represents an improvement over the ${\overline{\mathrm{MS}}}$ EEC in eq.
(2.22): here, neither the perturbative series in the first term nor the power
correction in the second term has a $u\\!\\!=\\!\\!1/2$ renormalon.444One can
conceptualize changing to an R scheme as locating a resurgence-pair of
ambiguities in the perturbative and nonperturbative sectors of a trans-series,
and shuffling the ambiguities so they cancel out.  Notably, this renormalon
removal relies on the predicted $z$-dependence of the power correction, but
does not rely on the bubble-sum approximation of the residue. Indeed, it
captures and subtracts even non-Abelian contributions to the renormalon.

###  3.2 R scheme with resummation

Equation (3.6) still has one remaining issue, associated with large logarithms
induced by the scale $R$. To avoid large logarithms in $c_{n}(z,\mu/Q)$ and
$d_{n}(\mu/R)$, we see from eq. (3.5) that we would need $R\simeq\mu\simeq Q$.
Retaining the scaling $\Omega_{1}(R)\sim\Lambda_{\rm QCD}$, however, requires
a relatively small value of $R$, like $R\simeq 2\,{\rm GeV}$; and this scale
would cause large $\ln(\mu/R)\simeq\ln(Q/R)$ in $d_{n}(\mu/R)$. We can resolve
these conflicting needs by using the RGE for $R$ in ref. [104], which has been
implemented for $\Omega_{1}(R)$ in refs. [54, 59],

$\displaystyle\Omega_{1}(R_{1})=\Omega_{1}(R_{0})+K(R_{1},R_{0})=\Omega_{1}(R_{0})-\sum_{n=0}^{\infty}\gamma_{n}^{\Omega_{1},R}\int_{R_{0}}^{R_{1}}dR\Big{[}\frac{\alpha_{s}(R)}{4\pi}\Big{]}^{n+1}\,.$
(3.7)

Here $K(R_{1},R_{0})$ is a dimension-1 evolution kernel that sums large
logarithms between $R_{0}$ and $R_{1}$. For leading logarithmic (LL)
resummation, we need the anomalous dimension
$\gamma_{0}^{\Omega_{1},R}=d_{10}=-8.357$, while at next-to-leading
logarithmic (NLL), we also need
$\gamma_{1}^{\Omega_{1},R}=d_{20}-2\beta_{0}d_{10}=55.693$. Here, we again
quote numerical values from ref. [59], with $n_{f}=5$ active light flavors,
suitable for the $Q=m_{Z}$ that we use in our numerical analysis. Calculating
$\gamma_{2}^{\Omega_{1},R}$ requires $d_{30}$ as input, which is not yet
available.

This solution to the $R$-RGE enables us to write

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\Sigma}{dz}=\bigg{[}\frac{1}{\sigma_{0}}\frac{d\hat{\Sigma}^{\rm
R}(R_{1})}{dz}+\frac{K(R_{1},R_{0})}{2Q\big{[}[z(1-z)]^{3/2}\big{]}_{+}}\bigg{]}+\frac{\Omega_{1}(R_{0})}{2Q\big{[}[z(1-z)]^{3/2}\big{]}_{+}}\,,$
(3.8)

where the first term in square brackets corresponds to the resummed
perturbative prediction for the EEC, while the last term contains the R scheme
power correction. Here, $R_{0}\simeq 2\,{\rm GeV}$ and $R_{1}\simeq Q$.

x

Figure -116: Comparison of perturbative results for the EEC cross-section in
the ${\overline{\mathrm{MS}}}$ and R schemes in eqs. (2.23) and (3.6),
respectively. Here, LO is ${\cal O}(\alpha_{s})$ and NLO is ${\cal
O}(\alpha_{s}^{2})$. The R scheme results have overlapping uncertainty bands
and better convergence than ${\overline{\mathrm{MS}}}$.

###  3.3 Perturbative convergence

Next, we compute the R scheme perturbative EEC cross-section
$d\hat{\Sigma}^{\rm R}/dz$ at LO and NLO from eq. (3.5). For simplicity, we
take $\mu=R_{1}$ in $d\hat{\Sigma}^{\rm R}/dz$. At LO, we carry out the
resummation in $K(R_{1},R_{0})$ at NLL using $\gamma_{0,1}^{\Omega_{1},R}$. At
NLO, one should use $K$ at NNLL order; however, $\gamma_{2}^{\Omega_{1},R}$ is
not yet known, so we use $K$ at NLL. Based on previous R-RGE studies [58, 93,
106], we anticipate that the NNLL kernel would lead to only slightly smaller
perturbative uncertainties at NLO.

In figure -116, we set $Q=m_{Z}$ and $\alpha_{s}(m_{Z})=0.118$, and use
$R_{0}=2\,{\rm GeV}$ for the R scheme.555A useful feature of R schemes is that
the cutoff $R_{0}$ dependence cancels between the perturbative result and the
$\Omega_{1}$ power correction in eq. (3.8). If we leave out $\Omega_{1}$, we
can estimate the size of this power correction by varying $R_{0}$ in the
perturbative term. We do not make use of this method here. We assess the
uncertainty from higher-order perturbative corrections in the conventional
manner, by varying the renormalization scale dependences $\mu$ in
${\overline{\mathrm{MS}}}$ and $R_{1}=\mu$ in R scheme, both on the range
$Q/2$ to $2Q$, using the LL and NLL running of $\alpha_{s}(\mu)$ at LO and
NLO, respectively. In the left panel of figure -116, we show central values;
in the right panel, we show deviation from the NLO R scheme curve and also
include perturbative uncertainties. The ${\overline{\mathrm{MS}}}$ EEC value
increases significantly from LO to NLO, and the NLO curve is not within the
estimated LO uncertainty. In contrast, the R scheme exhibits improved
convergence, with NLO results contained within the LO uncertainty band.
Furthermore, the estimated perturbative uncertainty at NLO is about a factor
of two smaller for the R scheme than for ${\overline{\mathrm{MS}}}$. Note that
(numerical) results exist for the EEC at NNLO=${\cal O}(\alpha_{s}^{3})$ in
the ${\overline{\mathrm{MS}}}$ scheme [31, 32], and these results lie just
above the upper edge of the NLO ${\overline{\mathrm{MS}}}$ uncertainty band
for almost all values of $z$. We do not analyze these results here as
corresponding NNLO R scheme results are not yet available.

Note that so far, we have only considered the EEC perturbative series, i.e.
the first ${\overline{\mathrm{MS}}}$ term in eq. (2.27) and the bracketed R
scheme terms in eq. (3.8). We do not expect these series to asymptote to the
same value, as the nonperturbative parameters $\bar{\Omega}_{1}$ and
$\Omega_{1}(R_{0})$ differ.

###  3.4 Nonperturbative corrections and hadron masses

To make a more complete prediction for the EEC that includes nonperturbative
information, we capitalize on the universality between the thrust and EEC
power corrections. Ref. [54] determined the thrust power corrections
$\bar{\Omega}_{1}$ and $\Omega_{1}(R_{0})$ by fits to data. These fits also
yielded $\alpha_{s}(m_{Z})=0.114$, which for consistency we use throughout
this section, along with $R_{0}=2\,{\rm GeV}$. At N3LL${}^{\prime}+{\cal
O}(\alpha_{s}^{3})$ order, they obtained $\bar{\Omega}_{1}=0.252\pm
0.069\,{\rm GeV}$ and $\Omega_{1}^{\mathrm{ref.}[51]}(R_{0})=0.323\pm 0.045$,
which translates to $\Omega_{1}(R_{0})=0.739\pm 0.045{\rm GeV}$ in the R
scheme of ref. [59] used here.666Compared to eq. (3.4), the scheme of ref.
[54] has $d_{10}=0$ and non-zero results only at higher orders, which is the
main cause of this numerical difference.

If we treat all hadrons as massless, then universality implies that
$\bar{\Omega}_{1}$ and $\Omega_{1}(R)$ are the same for both thrust and the
EEC [49, 50, 55]. Since $\Omega_{1}$ is the vacuum matrix element of a
measurement operator sandwiched by back-to-back lightcone Wilson lines, power
corrections for such observables are related by boost symmetry [55]. The
assumption that hadrons are massless enters such calculations when
manipulating kinematic variables to translate between energies, momenta,
rapidity, and angles.

Figure -115: Predictions for the full EEC with the inclusion of the leading
nonperturbative correction in the ${\overline{\mathrm{MS}}}$ and R schemes,
compared to OPAL data [60], as discussed in section 3.4. Central values are
displayed at both LO=${\cal O}(\alpha_{s})$ and NLO=${\cal
O}(\alpha_{s}^{2})$.

Hadron mass corrections are known to have a significant impact on universality
relations for $\Omega_{1}$ [56]. This is influenced by assumptions made about
hadron masses when using experimental measurements to determine a given
observable. Ref. [57] developed a field-theoretic method to compute hadron
mass corrections to universality relations. For an observable $e$, we denote
the leading-power mass correction by $\Omega_{1}^{e}=c_{e}\Omega_{1}^{g_{e}}$,
where $\Omega_{1}^{g_{e}}=\int_{0}^{1}dr\,g_{e}(r)\,\Omega_{1}(r)$ and
$r=p_{T}/\sqrt{p_{T}^{2}+m^{2}}$ for a hadron of transverse momentum $p_{T}$
and mass $m$. Here $c_{e}$ and $g_{e}(r)$ are analytically calculable terms,
while $\Omega_{1}(r)$ is a universal hadronic matrix element. Following the
notation of ref. [57], for the EEC we define

$\displaystyle f_{\rm EEC}(r,y)$ $\displaystyle=2\cosh
y\>\delta\Big{(}\frac{1-\cos\theta}{2}-z\Big{)}=2\cosh
y\>\delta\bigg{(}\frac{1}{2}-\frac{\sinh
y}{2\sqrt{r^{2}+\sinh^{2}y}}-z\bigg{)}\,.$ (3.9)

Here, $y$ is rapidity and the prefactor $2$ is the combinatoric factor for two
energies. Then,

$\displaystyle c_{\rm EEC}$ $\displaystyle=\int_{-\infty}^{+\infty}dy\>f_{\rm
EEC}(1,y)=\frac{1}{2[z(1-z)]^{3/2}}\,,$ $\displaystyle g_{\rm EEC}(r)$
$\displaystyle=\frac{1}{c_{\rm EEC}}\int_{-\infty}^{+\infty}dy\>f_{\rm
EEC}(r,y)=r\,.$ (3.10)

Here, $c_{\rm EEC}$ is the coefficient of $\Omega_{1}$ for the massless
universality relation, in agreement with eqs. (2.27) and (3.8).

The result $g_{\rm EEC}(r)=r$ implies that $\Omega_{1}^{g_{\rm EEC}}$ is in
the so-called E-scheme universality class of power corrections, which differs
from the thrust universality class $\Omega_{1}^{g_{\tau}}$ quoted above. Since
$g_{\rm EEC}(r)>g_{\tau}(r)$, we expect the value of the EEC power correction
to be larger than that for thrust. A two-term basis expansion provides a
fairly accurate parametrization for the impact of hadron masses on $g(r)$
[57], enabling us to write any observable as a linear combination of
nonperturbative parameters
$\Omega_{1}^{g_{e}}=b_{0}^{e}\Omega_{1}^{(0)}+b_{1}^{e}\Omega_{1}^{(1)}$.
Using the R scheme thrust fit, and $\Omega_{1}^{(0)}-\Omega_{1}^{(1)}\simeq
0.7\,{\rm GeV}$ from Monte Carlo fits [57], we find $\Omega_{1}^{g_{\rm
EEC}}/\Omega_{1}^{g_{\tau}}=1.21$. Thus, hadron mass effects in the EEC induce
a 21% increase in the value of $\Omega_{1}(R_{0})$, giving
$\Omega_{1}^{g_{\mathrm{EEC}}}(R_{0})=0.895\pm 0.054\,{\rm GeV}$. It is harder
to obtain an analogous ${\overline{\mathrm{MS}}}$ estimate, as Monte Carlo
simulations with hadronization at the shower cutoff are physically similar to
the R scheme, but not to ${\overline{\mathrm{MS}}}$. Thus, we simply assume
that ${\overline{\mathrm{MS}}}$ hadron mass corrections also cause a 21%
increase, yielding $\bar{\Omega}_{1}^{g_{\mathrm{EEC}}}=0.305\pm 0.084\,{\rm
GeV}$.

###  3.5 Comparison to experimental data

Figure -114: Full nonperturbative predictions for the EEC without (left) and
with (right) corrections from hadron mass effects as discussed in sections 3.4
and 3.5, normalized to NLO R scheme. We show the uncertainty bands for NLO
curves with the perturbative and parametric uncertainties added in quadrature.

We now combine the perturbative EEC results and leading nonperturbative
corrections to predict the total EEC cross-section. In figure -115, we compare
our ${\overline{\mathrm{MS}}}$ and R scheme results with OPAL data [60]. Here,
$\bar{\Omega}_{1}$ and $\Omega_{1}(R_{0})$ include the estimated hadron mass
corrections. We normalize the theory results by dividing by the full hadronic
cross-section $\sigma$, to ${\cal O}(\alpha_{s})$ and ${\cal
O}(\alpha_{s}^{2})$ for LO and NLO [107]. Our NLO R scheme result agrees well
with OPAL data, except in the $z\to 0$ and $z\to 1$ regions, where one should
resum large perturbative logarithms to higher orders in $\alpha_{s}$. Note
that the NLO R scheme result is also closer to the data than the LO R scheme
result. For ${\overline{\mathrm{MS}}}$, the power correction shifts
${\overline{\mathrm{MS}}}$ predictions closer to the data, but it is apparent
that one needs higher-order perturbative corrections to improve agreement.

It is difficult to illustrate hadron mass effects and uncertainties in the
theoretical predictions on the scales shown in figure -115. Therefore, we
display the same data as percent deviations from the NLO R scheme result in
figure -114. In the left panel, we use the massless hadron parameter values,
corresponding to the thrust fit values of $\bar{\Omega}_{1}$ and
$\Omega_{1}(R_{0})$.777Though we have only shown predictions for the EEC and
not considered fits to the EEC data, we do note that fits at a single value of
$Q$ are insufficient to break the degeneracy between different values of
$\alpha_{s}(m_{Z})$ and $\Omega_{1}(R_{0})$. For example, had we fixed
$\alpha_{s}(m_{Z})=0.118$, we could find a value of $\Omega_{1}(R_{0})$ that
gives similarly good agreement with the OPAL data, as in the right panel of
figure -114. In the right panel, we include the 21% increase in these
parameters due to the hadron mass effects discussed in section 3.4; we find
that this strongly improves the agreement between theory and data in both
normalization and shape. We show both the perturbative uncertainties at NLO
from varying $\mu$ and $R_{1}$ and the parametric uncertainties from the fit
used to obtain values for $\Omega_{1}(R_{0})$ and $\bar{\Omega}_{1}$, quoted
in section 3.4. These two types of uncertainties are added in quadrature in
figure -114. We note that the ${\overline{\mathrm{MS}}}$ perturbative
uncertainty is about twice as large as the parametric uncertainty of
$\bar{\Omega}_{1}$, and the R scheme perturbative uncertainty is of similar
size as the parametric uncertainty of $\Omega_{1}(R_{0})$. We also remark that
in all cases the power corrections give improved agreement with data compared
to the purely perturbative EEC. Furthermore, in the R scheme, the fixed-order
result using a single $R=R_{0}=R_{1}$ and resummation kernel $K=0$ exhibits
worse convergence and worse agreement with data than the NLO R scheme result
shown here.

##  4 Concluding remarks

Energy correlators show great promise for extracting a breadth of information
about QCD from colliders. To fully capitalize on the opportunity presented by
this class of observables, one must reach a high level of precision in both
perturbative calculations and nonperturbative corrections. In this paper, we
analyzed renormalons in the ${\overline{\mathrm{MS}}}$ EEC to investigate the
nature of its asymptotic perturbative series, and we converted our results to
an R scheme that removes the leading renormalon ambiguity, significantly
improving convergence and agreement with experimental data. Specifically, we
computed the Borel space ${\overline{\mathrm{MS}}}$ EEC in the bubble-sum
approximation, allowing us to calculate the $z$-dependence of the leading
$u=1/2$ renormalon pole. We confirmed the expected $[z(1-z)]^{-3/2}$
dependence of the corresponding $\mathcal{O}(\Lambda_{\rm QCD}/Q)$ power
correction, and showed how to regulate the $z\to 0$ and $z\to 1$ endpoints. We
found no $u=1$ pole under this approximation. Next, we constructed an R
renormalization scheme that removes the leading $u=1/2$ renormalon from both
the perturbative and the nonperturbative contributions to the EEC, with
explicit results given to $\mathcal{O}(\alpha_{s}^{2})$. This scheme change
significantly improves the convergence of the EEC perturbative series and
brings theoretical calculations of the EEC into reasonable agreement with
experimental data already at ${\cal O}(\alpha_{s}^{2})$. This paper lays the
groundwork for improving theoretical predictions for other EEC observables,
such as higher-point EEC correlators (see e.g. ref. [4]) and EEC observables
relevant for electron-ion and hadron-hadron colliders. It would also be
interesting to extend the analysis here to make use of the ${\cal
O}(\alpha_{s}^{3})$ ${\overline{\mathrm{MS}}}$ perturbative results for the
two-point EEC [31, 32].

## Acknowledgments

We thank Andre Hoang, Kyle Lee, Johannes Michel, Ian Moult, Michael Ogilvie,
and Aditya Pathak for helpful conversations. This work was supported by the
U.S. Department of Energy, Office of Science, Office of Nuclear Physics from
DE-SC0011090. I.S. was also supported in part by the Simons Foundation through
the Investigator grant 327942. S.T.S. was partially supported by the U.S.
National Science Foundation through a Graduate Research Fellowship under Grant
No. 1745302, and fellowships from the MIT Department of Physics and MIT School
of Science. Z.S. was also supported by a fellowship from the MIT Department of
Physics.

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# A Note on Projective Modules

Hossein Faridian  H. Faridian, School of Mathematical and Statistical
Sciences, Clemson University, SC 29634, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This expository note delves into the theory of projective modules parallel to
the one developed for injective modules by Matlis. Given a perfect ring $R$,
we present a characterization of indecomposable projective $R$-modules and
describe a one-to-one correspondence between the projective indecomposable
$R$-modules and the simple $R$-modules.

###### Key words and phrases:

Indecomposable module; perfect ring; projective module; sum-irreducible
module.

###### 2010 Mathematics Subject Classification:

16L30; 16D40; 13C05.

##  1\. Introduction

Throughout this note, $R$ denotes an associative ring with identity. All
modules are assumed to be left and unitary.

There is a well-known theorem in the representation theory of finite-
dimensional algebras over a field which provides a 1-1 correspondence between
the isomorphism classes of indecomposable projective modules over the algebra
on the one hand, and the isomorphism classes of simple modules over the
algebra on the other hand. More specifically, let $k$ be a field and $A$ a
finite-dimensional $k$-algebra. Then every simple $A$-module is a quotient of
some indecomposable projective $A$-module which is unique up to isomorphism.
Conversely, for every indecomposable projective $A$-module $P$, there is a
simple $A$-module, unique up to isomorphism, that is a quotient of $P$ by some
maximal submodule; see [Le]. For a generalization of this result to Artin
algebras, see [ARS, Corollary 4.5], and to perfect rings, see [Sh, Proposition
5]. The purpose of this note is to present a novel descriptive proof of this
theorem in the following more comprehensive form; see Theorems 3.5 and 3.8.

######  Theorem 1.1.

Let $R$ be a perfect ring, and $P$ a nonzero $R$-module. Then the following
assertions are equivalent:

  1. (i)

$P$ is an indecomposable projective $R$-module.

  2. (ii)

$P$ is a sum-irreducible projective $R$-module.

  3. (iii)

$P$ is the projective cover of its every nonzero quotient module.

  4. (iv)

$P$ is the projective cover of $R/\mathfrak{m}$ for some maximal left ideal
$\mathfrak{m}$ of $R$.

Further if $R$ is commutative, then the above assertions are equivalent to the
following one:

  1. (v)

$P\cong R_{\mathfrak{m}}$ for some maximal ideal $\mathfrak{m}$ of $R$.

######  Theorem 1.2.

Let $R$ be a perfect ring. There is a one-to-one correspondence between the
indecomposable projective $R$-modules and the simple $R$-modules.

##  2\. Preliminaries

We begin with reminding the classical notion of a projective cover.

######  Definition 2.1.

Let $R$ be a ring, and $M$ an $R$-module. Then:

  1. (i)

A submodule $N$ of $M$ is said to be superfluous, written as
$N\subseteq_{sup}M$, if whenever $N+L=M$ for some submodule $L$ of $M$, then
we have $L=M$.

  2. (ii)

By a projective cover of $M$, we mean a projective $R$-module $P$ together
with an epimorphism $\pi:P\rightarrow M$ such that $\ker\pi\subseteq_{sup}P$.

  3. (ii)

$R$ is said to be perfect if every $R$-module has a projective cover.

Enochs and Jenda have defined the general notion of a cover as follows; see
[EJ, Definition 5.1.1].

######  Definition 2.2.

Let $R$ be a ring, $M$ an $R$-module, and $\mathcal{A}$ a class of
$R$-modules. By an $\mathcal{A}$-cover of $M$, we mean an $R$-module
$A\in\mathcal{A}$ together with an $R$-homomorphism $\varphi:A\rightarrow M$
that satisfy the following conditions:

  1. (i)

Any $R$-homomorphism $\psi:B\rightarrow M$ with $B\in\mathcal{A}$ factors
through $\varphi$, i.e. there is an $R$-homomorphism $f:B\rightarrow A$ that
makes the following diagram commutative:

$B$$A$$M$$\psi$$\varphi$$f$

  2. (ii)

Any $R$-homomorphism $f:A\rightarrow A$ that makes the diagram

$A$$A$$M$$\varphi$$\varphi$$f$

commutative is an automorphism.

The classical definition of a projective cover amounts to the modern
definition as recorded in the following result for the sake of bookkeeping.

######  Lemma 2.3.

Let $R$ be a ring, $M$ an $R$-module, $P$ a projective $R$-module, and
$\pi:P\rightarrow M$ an $R$-homomorphism. Let $\mathcal{P}(R)$ denote the
class of projective $R$-modules. Then the following assertions are equivalent:

  1. (i)

$\pi:P\rightarrow M$ is a projective cover of $M$.

  2. (ii)

$\pi:P\rightarrow M$ is a $\mathcal{P}(R)$-cover of $M$.

###### Proof.

See [Xu, Theorem 1.2.12]. ∎

In light of Lemma 2.3, the projective cover of a given $R$-module $M$ is
unique up to isomorphism if it exists, in the sense that if $\pi:P\rightarrow
M$ and $\pi^{\prime}:P^{\prime}\rightarrow M$ are two projective covers of
$M$, then there is an isomorphism $P^{\prime}\xrightarrow{\cong}P$ that makes
the following diagram commutative:

$P^{\prime}$$P$$M$$\pi^{\prime}$$\pi$$\cong$

Accordingly, we denote a choice of a projective cover of $M$ by
$\pi_{M}:P_{R}(M)\rightarrow M$ whenever one exists.

######  Definition 2.4.

Let $R$ be a ring. Then:

  1. (i)

A nonzero $R$-module $M$ is said to be indecomposable if whenever
$M=N_{1}\oplus N_{2}$ for some submodules $N_{1}$ and $N_{2}$ of $M$, then we
have either $N_{1}=0$ or $N_{2}=0$.

  2. (ii)

An $R$-module $M$ is said to be sum-irreducible if whenever $M=N_{1}+N_{2}$
for some submodules $N_{1}$ and $N_{2}$ of $M$, then we have either $N_{1}=M$
or $N_{2}=M$.

By [AF, Theorem 28.4], one has the following lemma:

######  Lemma 2.5.

Let $R$ be a ring, and $\mathfrak{J}(R)$ denote its Jacobson radical. Then the
following assertions are equivalent:

  1. (i)

$R$ is perfect.

  2. (ii)

Every descending chain of principal right ideals of $R$ stabilizes.

  3. (iii)

Every flat $R$-module is projective.

  4. (iv)

$R/\mathfrak{J}(R)$ is semisimple and $\mathfrak{J}(R)$ is left T-nilpotent.

Recall that a subset $A$ of a ring $R$ is said to be left T-nilpotent if for
every sequence $a_{1},a_{2},...\in A$, there is a natural number $n$ such that
$a_{1}a_{2}\cdots a_{n}=0$.

##  3\. Main Results

In this section, we delve into the structure of indecomposable projective
$R$-modules.

######  Lemma 3.1.

Let $R$ be a perfect ring, and $P$ a nonzero projective $R$-module. Then the
following assertions are equivalent:

  1. (i)

$P$ is indecomposable.

  2. (ii)

Given any proper submodule $N$ of $P$, there is an isomorphism
$P\xrightarrow{\cong}P_{R}(P/N)$ that makes the following diagram commutative:

$P_{R}(P/N)$$P$$P/N$$\pi_{P/N}$$\pi$$\cong$

where $\pi$ is the natural epimorphism.

  3. (iii)

$P$ is sum-irreducible.

###### Proof.

(i) $\Rightarrow$ (ii): Let $N$ be a proper submodule of $P$. By Lemma 2.3,
the $R$-homomorphism $\pi_{P/N}:P_{R}(P/N)\rightarrow P/N$ is a
$\mathcal{P}(R)$-cover, so we can find an $R$-homomorphism $f:P\rightarrow
P_{R}(P/N)$, and also using the projectivity of $P_{R}(P/N)$, we can find an
$R$-homomorphism $g:P_{R}(P/N)\rightarrow P$ that make the following diagram
commutative:

$P_{R}(P/N)$$P$$P/N$$0$$\pi_{P/N}$$\pi$$g$$f$

We thus have $\pi_{P/N}fg=\pi g=\pi_{P/N}$. Another use of Lemma 2.3 implies
that $fg$ is an automorphism, so $f$ is surjective. The short exact sequence

$0\rightarrow\ker f\rightarrow P\xrightarrow{f}P_{R}(P/N)\rightarrow 0$

splits, so $P\cong\ker f\oplus P_{R}(P/N)$. But $P_{R}(P/N)\neq 0$, so the
hypothesis implies that $\ker f=0$, whence $f:P\rightarrow P_{R}(P/N)$ is an
isomorphism.

(ii) $\Rightarrow$ (iii): Suppose that $N_{1}$ and $N_{2}$ are two submodules
of $P$ such that $P=N_{1}+N_{2}$. If $N_{1}$ is proper, then the hypothesis
implies that there is an isomorphism $f:P\rightarrow P_{R}(P/N_{1})$ that
makes the following diagram commutative:

$P_{R}(P/N_{1})$$P$$P/N_{1}$$\pi_{P/N_{1}}$$\pi$$f$

We thus obtain the following commutative diagram with exact rows:

$0$$N_{1}$$P$$P/N_{1}$$0$$0$$\ker\pi_{P/N_{1}}$$P_{R}(P/N_{1})$$P/N_{1}$$0$$\pi$$\pi_{P/N_{1}}$$\cong$$f$

Since $\ker\pi_{P/N_{1}}\subseteq_{sup}P_{R}(P/N_{1})$, we deduces that
$N_{1}\subseteq_{sup}P$. It follows that $N_{2}=P$, so $P$ is sum-irreducible.

(iii) $\Rightarrow$ (i): Suppose that $P=N_{1}\oplus N_{2}$ for some
submodules $N_{1}$ and $N_{2}$ of $P$. The hypothesis implies that either
$N_{1}=P$ or $N_{2}=P$. It follows that either $N_{2}=0$ or $N_{1}=0$, so $P$
is indecomposable. ∎

######  Corollary 3.2.

Let $R$ be a perfect ring, and $M$ an $R$-module. Then $M$ is sum-irreducible
if and only if $P_{R}(M)$ is indecomposable.

###### Proof.

Let $\pi_{M}:P_{R}(M)\rightarrow M$ be a projective cover of $M$. Suppose that
$M$ is sum-irreducible, and let $P_{R}(M)=P_{1}+P_{2}$ for some submodules
$P_{1}$ and $P_{2}$ of $P_{R}(M)$. Then

$M=\pi_{M}\left(P_{R}(M)\right)=\pi_{M}(P_{1})+\pi_{M}(P_{2}).$

Without loss of generality, we can conclude that $M=\pi_{M}(P_{1})$. It
follows that $P_{R}(M)=P_{1}+\ker\pi_{M}$. But
$\ker\pi_{M}\subseteq_{sup}P_{R}(M)$, so we infer that $P_{R}(M)=P_{1}$. It
follows that $P_{R}(M)$ is sum-irreducible. Therefore, by Lemma 3.1,
$P_{R}(M)$ is indecomposable.

Conversely, suppose that $P_{R}(M)$ is indecomposable. Hence by Lemma 3.1,
$P_{R}(M)$ is sum-irreducible. Assume that $M=N_{1}+N_{2}$. It is easy to see
that

$P_{R}(M)=\pi^{-1}_{M}(N_{1})+\pi^{-1}_{M}(N_{2})+\ker\pi_{M}.$

But $\ker\pi_{M}\subseteq_{sup}P_{R}(M)$, so we infer that
$P_{R}(M)=\pi^{-1}_{M}(N_{1})+\pi^{-1}_{M}(N_{2})$. Without loss of
generality, we can conclude that $P_{R}(M)=\pi^{-1}_{M}(N_{1})$. This shows
that $M=N_{1}$, so $M$ is sum-irreducible. ∎

######  Corollary 3.3.

Let $R$ be a perfect ring, and $P$ a nonzero projective $R$-module. Then $P$
is indecomposable if and only if $P\cong P_{R}(S)$ for some simple $R$-module
$S$.

###### Proof.

Suppose that $P$ is indecomposable. By [AF, Proposition 17.14], $P$ has a
maximal submodule $N$. Then by Lemma 3.1, $P\cong P_{R}(P/N)$. It is clear
that $S:=P/N$ is a simple $R$-module.

Conversely, suppose that $P\cong P_{R}(S)$ for some simple $R$-module $S$.
Obviously, $S$ is sum-irreducible. Now, Corollary 3.2 implies that $P_{R}(S)$
is indecomposable. ∎

######  Lemma 3.4.

Let $R$ be a commutative perfect ring. Then the following assertions hold:

  1. (i)

$\operatorname{Spec}(R)=\operatorname{Max}(R)$.

  2. (ii)

$\operatorname{Supp}_{R}(R_{\mathfrak{m}})=\\{\mathfrak{m}\\}$ for every
$\mathfrak{m}\in\operatorname{Max}(R)$.

  3. (iii)

$R_{\mathfrak{m}}$ is an indecomposable $R$-module for every
$\mathfrak{m}\in\operatorname{Max}(R)$.

###### Proof.

(i): Let $\mathfrak{p}\in\operatorname{Spec}(R)$, and $0\neq a+\mathfrak{p}\in
R/\mathfrak{p}$. By Lemma 2.5, the descending chain
$(a)\supseteq(a^{2})\supseteq\cdots$ of principal ideals of $R$ stabilizes,
i.e. there is an integer $n\geq 1$ such that $(a^{n})=(a^{n+1})=\cdots$. In
particular, $a^{n}=ra^{n+1}$ for some $r\in R$. It follows that
$(1-ra)a^{n}=0$. As $a\notin\mathfrak{p}$, we get $1-ra\in\mathfrak{p}$, which
yields that $(r+\mathfrak{p})(a+\mathfrak{p})=1+\mathfrak{p}$, i.e.
$a+\mathfrak{p}\in(R/\mathfrak{p})^{\times}$. Hence $R/\mathfrak{p}$ is a
field, so $\mathfrak{p}\in\operatorname{Max}(R)$.

(ii): Let $\mathfrak{m},\mathfrak{n}\in\operatorname{Max}(R)$ be such that
$\mathfrak{m}\neq\mathfrak{n}$. Set

$\mathfrak{J}_{\mathfrak{u}}(R):=\bigcap_{\mathfrak{v}\in\operatorname{Max}(R)\backslash\\{\mathfrak{u}\\}}\mathfrak{v},$

for any $\mathfrak{u}\in\operatorname{Max}(R)$. By Lemma 2.5, the ring
$R/\mathfrak{J}(R)$ is semisimple. This implies that $R$ is semilocal, and so
for any $i\geq 1$, there are elements
$a_{i}\in\mathfrak{J}_{\mathfrak{m}}(R)^{i}\backslash\mathfrak{m}$ and
$b_{i}\in\mathfrak{J}_{\mathfrak{n}}(R)^{i}\backslash\mathfrak{n}$. Set
$c_{i}:=a_{i}b_{i}$ for every $i\geq 1$. It is clear that
$c_{i}\in\mathfrak{J}(R)$ for every $i\geq 1$. By Lemma 2.5, $\mathfrak{J}(R)$
is T-nilpotent. It follows that there is an integer $n\geq 1$ such that
$c_{1}c_{2}\cdots c_{n}=0$. Set $a:=a_{1}a_{2}\cdots a_{n}$ and
$b:=b_{1}b_{2}\cdots b_{n}$. Then it is obvious that $a\notin\mathfrak{m}$,
$b\notin\mathfrak{n}$, and $ab=0$. This shows that
$(R_{\mathfrak{m}})_{\mathfrak{n}}=0$, so
$\operatorname{Supp}_{R}(R_{\mathfrak{m}})=\\{\mathfrak{m}\\}$.

(iii): Suppose that $R_{\mathfrak{m}}=N\oplus N^{\prime}$ for some
$R$-submodules $N$ and $N^{\prime}$ of $R_{\mathfrak{m}}$. Localizing at
$\mathfrak{m}$, we get $R_{\mathfrak{m}}=N_{\mathfrak{m}}\oplus
N^{\prime}_{\mathfrak{m}}$. But since $R_{\mathfrak{m}}$ is a local ring, it
follows that $R_{\mathfrak{m}}$ is an indecomposable
$R_{\mathfrak{m}}$-module. Hence, $N_{\mathfrak{m}}=0$ or
$N^{\prime}_{\mathfrak{m}}=0$. Say $N_{\mathfrak{m}}=0$. But, (ii) implies
that $\operatorname{Supp}_{R}(N)\subseteq\\{\mathfrak{m}\\}$. Therefore,
$N=0$. ∎

######  Theorem 3.5.

Let $R$ be a perfect ring, and $P$ a nonzero $R$-module. Then the following
assertions are equivalent:

  1. (i)

$P$ is an indecomposable projective $R$-module.

  2. (ii)

$P$ is a sum-irreducible projective $R$-module.

  3. (iii)

$P$ is the projective cover of its every nonzero quotient module.

  4. (iv)

$P\cong P_{R}(R/\mathfrak{m})$ for some maximal left ideal $\mathfrak{m}$ of
$R$.

Further if $R$ is commutative, then the above assertions are equivalent to the
following one:

  1. (v)

$P\cong R_{\mathfrak{m}}$ for some maximal ideal $\mathfrak{m}$ of $R$.

###### Proof.

Lemma 3.1 yields the equivalence of (i), (ii) and (iii).

(i) $\Rightarrow$ (iv): Corollary 3.3 implies that $P\cong P_{R}(S)$ for some
simple $R$-module $S$. But then $S\cong R/\mathfrak{m}$ for some maximal left
ideal $\mathfrak{m}$ of $R$.

(iv) $\Rightarrow$ (v): Suppose that $P\cong P_{R}(R/\mathfrak{m})$ for some
maximal ideal $\mathfrak{m}$ of $R$. By Lemma 3.4, $R_{\mathfrak{m}}$ is an
indecomposable $R$-module. Moreover, Lemma 2.5 yields that $R_{\mathfrak{m}}$
is a projective $R$-module. By Corollary 3.3, we have $R_{\mathfrak{m}}\cong
P_{R}(R/\mathfrak{n})$ for some maximal ideal $\mathfrak{n}$ of $R$. If
$\mathfrak{n}\neq\mathfrak{m}$, then take any
$a\in\mathfrak{n}\backslash\mathfrak{m}$. Therefore, the $R$-homomorphism
$a1^{R_{m}}:R_{\mathfrak{m}}\rightarrow R_{\mathfrak{m}}$ is an isomorphism.
The commutative diagram

$R_{\mathfrak{m}}$$R_{\mathfrak{m}}$$P_{R}(R/\mathfrak{n})$$P_{R}(R/\mathfrak{n})$$R/\mathfrak{n}$$R/\mathfrak{n}$$a$$\cong$$\cong$$a$$\pi_{R/\mathfrak{n}}$$\pi_{R/\mathfrak{n}}$$a$

shows that the $R$-homomorphism $a1^{R/\mathfrak{n}}:R/\mathfrak{n}\rightarrow
R/\mathfrak{n}$ is surjective, which is a contradiction. Hence,
$\mathfrak{n}=\mathfrak{m}$.

(v) $\Rightarrow$ (i): Follows from Lemmas 3.4 and 2.5. ∎

The following result may be considered as dual to Matlis Theorem which asserts
that over a commutative noetherian ring $R$, every injective $R$-module
decomposes uniquely into a direct sum of indecomposable injective $R$-modules;
see [Ma].

######  Corollary 3.6.

Let $R$ be a commutative artinian ring, and $P$ a projective $R$-module. Then

$P\cong\prod_{\mathfrak{p}\in\operatorname{Spec}(R)}P_{R}(R/\mathfrak{p})^{\pi_{0}(\mathfrak{p},P)},$

where $\pi_{0}(\mathfrak{p},P)$ is the zeroth dual Bass number of $P$ with
respect to $\mathfrak{p}$, i.e.

$\pi_{0}(\mathfrak{p},P)=\operatorname{rank}_{R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}}\left(R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}\otimes_{R_{\mathfrak{p}}}\operatorname{Hom}_{R}(R_{\mathfrak{p}},P)\right).$

###### Proof.

By [Xu, Theorem 1.2.13], $R$ is perfect, and thus [Xu, Proposition 3.3.1]
implies that $P$ is a flat cotorsion $R$-module. It follows from [Xu, Theorem
4.1.15] that

$P\cong\prod_{\mathfrak{p}\in\operatorname{Spec}(R)}T_{\mathfrak{p}},$

where $T_{\mathfrak{p}}$ is the $\mathfrak{p}R_{\mathfrak{p}}$-adic completion
of a free $R_{\mathfrak{p}}$-module of rank $\pi_{0}(\mathfrak{p},P)$. But
$\mathfrak{J}(R_{\mathfrak{p}})=\mathfrak{p}R_{\mathfrak{p}}$ is nilpotent, so
every $R_{\mathfrak{p}}$-module is $\mathfrak{p}R_{\mathfrak{p}}$-adically
complete. It follows that $T_{\mathfrak{p}}$ is a free
$R_{\mathfrak{p}}$-module of rank $\pi_{0}(\mathfrak{p},P)$. Now, the result
follows from Theorem 3.5. ∎

######  Lemma 3.7.

Let $R$ be a perfect ring, and $f:M\rightarrow N$ an $R$-homomorphism. Then
the following assertions hold:

  1. (i)

There is an $R$-homomorphism $\tilde{f}:P_{R}(M)\rightarrow P_{R}(N)$ that
makes the following diagram commutative:

$P_{R}(M)$$P_{R}(N)$$M$$N$$\tilde{f}$$\pi_{M}$$\pi_{N}$$f$

  2. (ii)

If $f$ is an epimorphism, then any $R$-homomorphism $g:P_{R}(M)\rightarrow
P_{R}(N)$ that makes the diagram in (i) commutative is an epimorphism.

  3. (iii)

If $f$ is an isomorphism, then any $R$-homomorphism $g:P_{R}(M)\rightarrow
P_{R}(N)$ that makes the diagram in (i) commutative is an isomorphism.

###### Proof.

(i): The existence of $\tilde{f}$ follows readily from the projectivity of
$P_{R}(M)$ in light of the following diagram:

$P_{R}(M)$$M$$P_{R}(N)$$N$$0$$\pi_{M}$$f$$\pi_{N}$$\tilde{f}$

(ii): Suppose that $f$ is an epimorphism and an $R$-homomorphism
$g:P_{R}(M)\rightarrow P_{R}(N)$ makes the following diagram commutative:

$P_{R}(M)$$P_{R}(N)$$M$$N$$g$$\pi_{M}$$\pi_{N}$$f$

The diagram shows that $\pi_{N}g$ is surjective. Therefore, it can be seen by
inspection that

$\operatorname{im}g+\ker\pi_{N}=P_{R}(N).$

But $\ker\pi_{N}\subseteq_{sup}P_{R}(N)$, so $\operatorname{im}g=P_{R}(N)$,
i.e. $g$ is surjective.

(iii): Suppose that $f$ is an isomorphism and an $R$-homomorphism
$g:P_{R}(M)\rightarrow P_{R}(N)$ makes the following diagram commutative:

$P_{R}(M)$$P_{R}(N)$$M$$N$$g$$\pi_{M}$$\pi_{N}$$f$

By (ii), $g$ is surjective. The short exact sequence

$0\rightarrow\ker g\rightarrow P_{R}(M)\xrightarrow{g}P_{R}(N)\rightarrow 0$

splits, so $P_{R}(M)=\ker g+\operatorname{im}g^{\prime}$, where
$g^{\prime}:P_{R}(N)\rightarrow P_{R}(M)$ is an $R$-homomorphism such that
$gg^{\prime}=1^{P_{R}(N)}$. It is clear that $\ker g\subseteq\ker\pi_{M}$, so
$P_{R}(M)=\ker\pi_{M}+\operatorname{im}g^{\prime}$. But
$\ker\pi_{M}\subseteq_{sup}P_{R}(M)$, so
$P_{R}(M)=\operatorname{im}g^{\prime}$. It follows that $g^{\prime}$ is an
isomorphism, so $g$ is an isomorphism. ∎

The next result extends the main result of [Le] from finite-dimensional
algebras over a field to perfect rings.

######  Theorem 3.8.

Let $R$ be a perfect ring, and $\mathfrak{m},\mathfrak{n}$ two maximal left
ideals of $R$. Then $P_{R}(R/\mathfrak{m})\cong P_{R}(R/\mathfrak{n})$ if and
only if $\mathfrak{m}=\mathfrak{n}$. Hence, there is a one-to-one
correspondence between the indecomposable projective $R$-modules and the
simple $R$-modules.

###### Proof.

Let $\varphi:P_{R}(R/\mathfrak{m})\rightarrow P_{R}(R/\mathfrak{n})$ be any
isomorphism. Suppose to the contrary that $\mathfrak{m}\neq\mathfrak{n}$, and
take any element $a\in\mathfrak{m}\backslash\mathfrak{n}$. It follows that the
$R$-homomorphism $a1^{R/\mathfrak{n}}:R/\mathfrak{n}\rightarrow
R/\mathfrak{n}$ is an isomorphism. In view of Lemma 3.7, the commutative
diagram

$P_{R}(R/\mathfrak{n})$$P_{R}(R/\mathfrak{n})$$R/\mathfrak{n}$$R/\mathfrak{n}$$a$$\pi_{R/\mathfrak{n}}$$\pi_{R/\mathfrak{n}}$$a$

shows that the $R$-homomorphism
$a1^{P_{R}(R/\mathfrak{n})}:P_{R}(R/\mathfrak{n})\rightarrow
P_{R}(R/\mathfrak{n})$ is an isomorphism. Therefore, the commutative diagram

$P_{R}(R/\mathfrak{m})$$P_{R}(R/\mathfrak{m})$$P_{R}(R/\mathfrak{n})$$P_{R}(R/\mathfrak{n})$$a$$\varphi$$\varphi$$a$

yields that the $R$-homomorphism
$a1^{P_{R}(R/\mathfrak{m})}:P_{R}(R/\mathfrak{m})\rightarrow
P_{R}(R/\mathfrak{m})$ is an isomorphism. Now, the commutative diagram

$P_{R}(R/\mathfrak{m})$$P_{R}(R/\mathfrak{m})$$R/\mathfrak{m}$$R/\mathfrak{m}$$a$$\pi_{R/\mathfrak{m}}$$\pi_{R/\mathfrak{m}}$$a$

implies that the $R$-homomorphism
$a1^{R/\mathfrak{m}}:R/\mathfrak{m}\rightarrow R/\mathfrak{m}$ is surjective.
But this map is zero, so we arrive at a contradiction. Hence,
$\mathfrak{m}=\mathfrak{n}$.

The converse is immediate. ∎

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