aainstitutetext: Department of Physics, Chaitanya Bharathi Institute of
Technology,
Gandipet, Hyderabad, Telangana State 500075, Indiabbinstitutetext: Department
of Physics, Indian Institute of Technology Hyderabad,
Kandi, Sangareddy, Telangana State 502285, India

# Power corrections to event shapes using Eikonal Dressed Gluon Exponentiation

Neelima Agarwal b Ayan Mukhopadhyay b Sourav Pal b Anurag Tripathi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Event shapes are classical tools for the determination of the strong coupling
and for the study of hadronization effects in electron-positron annihilation.
In the context of analytical studies, hadronization corrections take the form
of power-suppressed contributions to the cross section, which can be extracted
from the perturbative ambiguity of Borel-resummed distributions. We propose a
simplified version of the well-established method of Dressed Gluon
Exponentiation (DGE), which we call Eikonal DGE (EDGE), which determines all
dominant power corrections to event shapes by means of strikingly elementary
calculations. We believe our method can be generalized to hadronic event
shapes and jet shapes of relevance for LHC physics.

##  1 Introduction

Infrared-safe event shape variables, which we will generically denote by $e$,
play a central role in perturbative QCD: they are essential tools for the
precise determination of the strong coupling constant, and they are classic
testing grounds for both analytical and numerical models of hadronization.
Owing to their infrared and collinear safety, they can be computed in
perturbation theory, and furthermore large logarithmic corrections to the
distributions can be resummed to all orders by a variety of methods. At fixed
orders, the state of the art is next-to-next-to-leading order (NNLO) accuracy
GehrmannDeRidder:2007hr ; Weinzierl:2009ms ; Gehrmann:2019hwf ; Kardos:2020igb
; GehrmannDeRidder:2009dp , while the next to leading log (NLL) resummation
has been known for a while Catani:1992ua ; Banfi:2010xy ; Banfi:2004yd ;
Banfi:2001bz . In recent years, the NNLL resummation framework has also been
developed Banfi:2016zlc ; Abbate:2010xh ; Becher:2012qc ; Hoang:2014wka ;
Budhraja:2019mcz ; Bell:2018gce ; Hoang:2015gta ; Kolodrubetz:2016reu ;
Lepenik:2019jjk ; Lee:2009cw ; Hornig:2009vb ; Banfi:2014sua ). Here we will
be concerned with analytic estimates of non-perturbative corrections, which
are suppressed by powers of $\Lambda/Q$ (where $\Lambda$ is the QCD scale and
$Q$ is the center-of-mass energy) with respect to the perturbative result. The
basic idea of such analytic estimates goes back to the Operator Product
Expansion (OPE), and was first applied to observable that do not admit an OPE
in the early papers Bigi:1994em ; Beneke:1994sw ; Webber:1994cp . Very roughly
speaking, one notes that a generic (dimensionless) observable in perturbative
QCD is a sum of a ‘leading power’ perturbative series plus power corrections,
of the general form

$\sigma\left(\frac{Q}{\mu},\alpha_{s}\right)\,=\,\sigma_{{\rm
pert}}\left(\frac{Q}{\mu_{f}},\frac{\mu_{f}}{\mu},\alpha_{s}\right)+\sum_{n}\sigma_{n}\left(\frac{\mu_{f}}{\mu},\alpha_{s}\right)\left(\frac{\mu_{f}}{Q}\right)^{n}\,,$
(1)

where $\mu_{f}$ is a perturbative factorization scale, ultimately to be traded
for the strong interaction scale $\Lambda$. Generically, with a dimensional
regulator, different terms in the sum in Eq. (1) mix with each other under
renormalization. In dimensional regularization, the same effect arises in a
subtler fashion: each term in Eq. (1) is ambiguous due to the divergence of
the corresponding perturbative expansion, which manifests itself via
singularities in the Borel plane. These ambiguities are power-suppressed and
are compensated by corresponding ambiguities in subsequent terms in the sum in
Eq. (1). This opens the way for a perturbative estimate of hadronization
corrections based on the study of singularities in the Borel plane.
Phenomenological studies of event shapes and other basic QCD observables with
these tools were first systematically pursued in Dokshitzer:1995qm , and
subsequently developed in a vast literature, reviewed in Dasgupta:2003iq . The
phenomenological importance of these power-suppressed corrections cannot be
understated: for example, they are crucial for a precise determination of the
strong coupling Abbate:2010xh ; Hoang:2015hka ; Wang:2019isi ; Marzani:2019evv
; Gehrmann:2012sc .

In the case of event shape distributions, denoted by $d\sigma/de$ below, the
situation is more subtle. Such distributions peak in the two-jet region, which
can be taken to correspond to $e\to 0$, and which is dominated by soft and
collinear emissions; in this region, the distributions are typically affected
by enhanced power corrections of the form $\left(\Lambda/(eQ)\right)^{n}$,
associated with the emission of soft gluons, as well as corrections scaling as
$\left(\Lambda^{2}/(eQ^{2})\right)^{n}$, associated with hard collinear gluon
emission. We will refer to the first of these as ‘soft’ power corrections, and
to the second ones as ‘collinear’ power corrections for the sake of brevity.
When $e\sim\Lambda/Q$, which is typically close to the peak of the
distribution at least at LEP energies, all soft power corrections become
equally important and need to be resummed in order to get a stable prediction.
At even smaller values of $e$, $e\sim\Lambda^{2}/Q^{2}$, collinear power
corrections become relevant as well.

An elegant and efficient method to handle simultaneously large perturbative
logarithms (up to NLL accuracy) and power corrections in the two-jet region is
Dressed Gluon Exponentiation (DGE) Gardi:2001di , which has already been
applied to a variety of event shapes Gardi:2002bg ; Gardi:2003iv ;
Berger:2004xf , as well as to other important QCD observables Cacciari:2002xb
; Gardi:2004ia ; Andersen:2005bj . DGE, aside from consistently including the
NLL resummation of Sudakov logarithms, provides a renormalon-based estimate of
both soft and collinear power corrections. Collinear power corrections have
been shown to enjoy a degree of universality Gardi:2002bg ; Gardi:2003iv ;
Gardi:2004ia  across several inclusive observables. When, however, this
universality breaks down, as for example in Berger:2004xf ; Lee:2007jr ,
collinear power corrections can be very cumbersome to compute; furthermore,
they only become relevant at extremely small values of the event shape,
usually out of experimental range, or in a region where very few data points
are available.

These facts suggest that it would be useful to construct a systematic
approximation to DGE which would suffice to capture all soft power
corrections, while remaining simple to implement in practice. In this paper,
we will introduce such an approximation, which essentially consists in
combining DGE with the eikonal approximation for the relevant matrix elements.
We call the resulting construction Eikonal Dressed gluon exponentiation or
EDGE. The universality and simplicity of soft emission can then be used to
express soft power corrections to a large class of event shapes in terms of a
very simple integral, which reproduces known results for all event shapes for
which soft power corrections are known. The paper is structured as follows:
section 2 briefly summarizes the essential aspects of DGE, section 3 shows how
to implement EDGE using energy fractions by taking examples from three very
well known event shapes: thrust, $C$-parameter and angularities, section 4
describes the implementation of EDGE using the transverse momentum and the
rapidity, in section 5 we present Sudakov exponent and discuss power
corrections.

##  2 Dressed Gluon Exponentiation

The starting point for DGE is the event shape distribution in the single
dressed gluon approximation, which is constructed from the one-loop real
emission contribution to the event shape for a gluon with virtuality
$k^{2}\neq 0$. From this, one easily obtains Beneke:1994sw  the (renormalon)
resummation of quark vacuum polarization corrections which dominates in the
large $N_{f}$ limit. One can write the result as

$\frac{1}{\sigma}\frac{d\sigma}{de}(e,Q^{2})\Big{|}_{SDG}\,=\,-\,\frac{C_{F}}{2\beta_{0}}\int_{0}^{1}d\xi\,\frac{d\mathcal{F}(e,\xi)}{d\xi}\,A(\xi
Q^{2})\,,$ (2)

where $\beta_{0}=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}$, $\xi=k^{2}/Q^{2}$, and
$A(\xi Q^{2})$ is the large-$\beta_{0}$ running coupling
$(A=\beta_{0}\alpha_{s}/\pi)$ on the time-like axis. In the $\overline{{\rm
MS}}$ scheme, it admits the Borel representation

$A(\xi Q^{2})\,=\,\int_{0}^{\infty}du(Q^{2}/\Lambda^{2})^{-u}\frac{\sin\pi
u}{\pi u}{\rm e}^{\frac{5}{3}u}\xi^{-u}\,.$ (3)

The cornerstone of Eq. (2) is the characteristic function
$\mathcal{F}(e,\xi)$, which is the one-loop event shape distribution with a
non-vanishing gluon virtuality $k^{2}$ Gardi:2000yh ; Dokshitzer:1995qm ,

$\mathcal{F}(e,\xi)\,=\,\int dx_{1}dx_{2}\,{\cal
M}(x_{1},x_{2},\xi)\,\delta\left(e-\bar{e}(x_{1},x_{2},\xi)\right)\,,$ (4)

where $x_{i}$ are the customary energy fraction variables, ${\cal M}$ is the
matrix element for the emission of a gluon with $k^{2}\neq 0$, and $\bar{e}$
is the explicit expression of the event shape in terms of the kinematic
variables. Interchanging the order of integrations in Eq. (2) we can construct
a Borel representation as

$\frac{1}{\sigma}\frac{d\sigma}{de}(e,Q^{2})\Big{|}_{SDG}\,=\,\frac{C_{F}}{2\beta_{0}}\int_{0}^{\infty}du(Q^{2}/\Lambda^{2})^{-u}B(e,u)\,,$
(5)

where the Borel function $B(e,u)$ is defined by

$B(e,u)\,=\,-\frac{\sin{\pi u}}{\pi
u}e^{\frac{5}{3}u}\int_{0}^{\infty}d\xi~{}\xi^{-u}\frac{d\mathcal{F}(e,\xi)}{d\xi}\,.$
(6)

The Borel function $B(e,u)$ has a simple structure in the $u$ plane, without
renormalon singularities. Renormalon poles are however generated when the
single dressed gluon distribution is exponentiated via a Laplace transform
Gardi:2001di .

The additive property of the event shapes with respect to the multiple gluon
emissions together with the factorization of soft and collinear emissions from
the hard part of the matrix element leads to the exponentiation of the
logarithmically enhanced terms in the Laplace space and the resummed cross
section is given by Gardi:2002bg ; Gardi:2003iv ,

$\displaystyle\frac{1}{\sigma}\frac{d\sigma(e,Q^{2})}{de}=\int_{C-i\infty}^{C+i\infty}\frac{d\nu}{2\pi
i}\,e^{\nu e}\,\text{exp}[S(\nu,Q^{2})]$ , (7)

where $C$ lies to the right of the singularities of the integrand. The Sudakov
exponent has the form Gardi:2001ny ,

$\displaystyle S(\nu,Q^{2})$
$\displaystyle=\int_{0}^{1}de\>\frac{1}{\sigma}\frac{d\sigma(e,Q^{2})}{de}\Big{|}_{SDG}\>(e^{-\nu
e}-1).$ (8)

The Sudakov region $e\rightarrow 0$ corresponds to $\nu\rightarrow\infty$.
Using Eq. (5), the Sudakov exponent takes the form

$\displaystyle
S(\nu,Q^{2})=\frac{C_{F}}{2\beta_{0}}\int_{0}^{\infty}du\,\bigg{(}\frac{Q^{2}}{\Lambda^{2}}\bigg{)}^{-u}\,B_{\nu}^{e}(u),$
(9)

where the Borel function in the Laplace space, $B_{\nu}^{e}(u)$, is defined as

$\displaystyle B_{\nu}^{e}(u)=\int_{0}^{1}de\,B(e,u)\,(e^{-\nu e}-1).$ (10)

This exponentiation effectively resums both large Sudakov logarithms and power
corrections in the two-jet region.

##  3 Borel function using Eikonal Dressed Gluon Exponentiation

In this article we undertake the calculation of the Borel function that was
defined in Eq. (6) for three very well known event shape variables: (a) the
thrust Brandt:1964sa ; Kane:1978nd ; Farhi:1977sg ; Altarelli:1981ax , (b) the
$C$-parameter Donoghue:1979vi ; Fox:1978vu ; Bjorken:1969wi ; Parisi:1978eg
and, (c) the angularities Berger:2003pk ; Berger:2003iw ; Berger:2002ig , and
we propose a simplified version of the well-established method of Dressed
Gluon Exponentiation (DGE), which we call Eikonal DGE (EDGE), which determines
all dominant power corrections to event shapes by means of strikingly
elementary calculations. We believe our method can be generalized to hadronic
event shapes and jet shapes of relevance for LHC physics. There are two
aspects to this simplification. First, as we will see in the later parts of
this article, we only need to work with the squared matrix element in the
eikonal limit. Second, and more importantly, the event shape definitions can
be simplified (eikonalized) to significantly simplify the computations,
however, still capturing the dominant power corrections. The definition of
thrust is simple enough and does not require any eikonalization, however we
will introduce the eikonalized versions of $C$-parameter and angularities for
the computation of their respective Borel functions.

As discussed above we need to construct the characteristic function
$\mathcal{F}(e,\xi)$ for these three event shape variables for the order
$\alpha_{s}$ process $\gamma^{*}\rightarrow q\bar{q}g$. The color stripped
squared matrix element after removing the coupling is

$\displaystyle\mathcal{M}(x_{1},x_{2},\xi)=\frac{(x_{1}+\xi)^{2}+(x_{2}+\xi)^{2}}{(1-x_{1})(1-x_{2})}-\frac{\xi}{(1-x_{1})^{2}}-\frac{\xi}{(1-x_{2})^{2}},$
(11)

where the energy fractions are defined by

$\displaystyle x_{1}=$ $\displaystyle\frac{2p_{1}\cdot Q}{Q^{2}},\quad
x_{2}=\frac{2p_{2}\cdot Q}{Q^{2}},\quad x_{3}=\frac{2k\cdot Q}{Q^{2}}.$ (12)

Here $k$ denotes the momentum of the off-shell gluon and, $p_{1}$ and $p_{2}$
are the momenta of the quark and anti-quark respectively. Momentum
conservation $Q=p_{1}+p_{2}+k$ gives the constraint $x_{1}+x_{2}+x_{3}=2$.
Figure (1) gives the Dalitz plot for this processes. In the soft gluon limit
we approximate the squared matrix element to,

$\displaystyle\mathcal{M}_{{\rm{soft}}}(x_{1},x_{2},\xi)=\frac{2}{(1-x_{1})(1-x_{2})}.$
(13)

Note that this is the same as what we would write in the soft gluon limit for
the case of massless gluon.

Next we will take the mentioned three shape variables in turn and construct
eikonalized versions of them and then compute the corresponding characteristic
functions, and their Borel functions.

Figure 1: Dalitz plot showing phase space for $\gamma\rightarrow q\bar{q}g$
with off-shell gluon. The energy momentum conservation condition
$x_{1}+x_{2}+x_{3}=2$ is satisfied throughout this $x_{1}-x_{2}$ plane and the
actual length along $x_{3}$ axis is $\sqrt{2}$ times the measured length. The
collinear limit (when the gluon is collinear to the quark) corresponds to
$x_{1}=1-\xi$, $x_{2}=0$, while the soft limit (when the gluon is soft to the
quark) corresponds to $x_{1}=x_{2}=1-\sqrt{\xi}$. The soft boundary of the
phase space $1-x_{2}=\xi/(1-x_{1})$ is denoted by the red curve.

###  3.1 Thrust

Thrust is one of the most studied event shapes and it has a historical
connection with the determination of strong coupling constant $\alpha_{s}$. It
is defined as Farhi:1977sg

$\displaystyle T=\underset{{\bf{n}}}{\text{Max}}\,\,\frac{\sum_{i}|{\bf
p}_{i}\cdot{\bf{n}}|}{\sum_{i}E_{i}},$ (14)

where ${\bf p}_{i}$ denotes the 3-momentum of the $i$-th particle in the final
state and n is a unit vector. In order to determine the range of $T$, we need
to consider two extreme cases: a most spherical configuration and a pencil
like configuration. For a spherical configuration, $T$ attains a minimum value
$1/2$ and for a pencil like configuration, $T$ attains a maximum value $1$.
Thus, thrust varies in the range $1/2\leq T\leq 1$. For a three particles
final state, the numerator in Eq. (14) is maximum when ${\bf{n}}$ is along the
direction of the largest ${\bf{p_{i}}}$. Thus, the thrust for all massless
particles final state is given by

$\displaystyle T=\text{Max}\\{x_{1},x_{2},x_{3}\\}.$ (15)

In presence of a massive off-shell gluon in the final state, the definition of
thrust needs some simple modifications which was first given in Gardi:1999dq
and has the form,

$\displaystyle
T=\text{Max}\left\\{x_{1},x_{2},\sqrt{x_{3}^{2}-4\xi}\right\\}.$ (16)

Substituting in the definition Eq. (4) of the characteristic function, the
squared matrix element Eq. (13) and the definition of thrust Eq. (16), we
obtain

$\displaystyle\mathcal{F}(T,\xi)=\int\int dx_{1}\
dx_{2}\frac{2}{(1-x_{1})(1-x_{2})}\delta\left(T-\text{Max}\left\\{x_{1},x_{2},\sqrt{x_{3}^{2}-4\xi}\right\\}\right).$
(17)

When the radiated (dressed) gluon is soft, this integral receives
contributions from the regions I and II as shown in Figure (1). Region I
contributes when $x_{1}$ is the largest, and region II contributes when
$x_{2}$ is the largest of $x_{1},x_{2},\sqrt{x_{3}^{2}-4\xi}$. Naming these
contributions as $\mathcal{F}_{1}(T,\xi)$ and $\mathcal{F}_{2}(T,\xi)$
respectively, we have

$\displaystyle\mathcal{F}(T,\xi)\simeq\mathcal{F}_{1}(T,\xi)+\mathcal{F}_{2}(T,\xi),$
(18)

where

$\displaystyle\mathcal{F}_{1}(T,\xi)=$
$\displaystyle\int_{2-T-\sqrt{T^{2}+4\xi}}^{\frac{1-\xi-T}{1-T}}dx_{2}\
\mathcal{M}(T,x_{2},\xi)$ $\displaystyle\mathcal{F}_{2}(T,\xi)=$
$\displaystyle\int_{2-T-\sqrt{T^{2}+4\xi}}^{\frac{1-\xi-T}{1-T}}dx_{1}\
\mathcal{M}(x_{1},T,\xi),$ (19)

Note that the integrals $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are same due
to the symmetry of $\mathcal{M}(x_{1},x_{2},\xi)$ under interchange of $x_{1}$
and $x_{2}$. The limits of the integration are determined by the boundary of
the phase space shown in red in the Figure (1). The characteristic function
immediately evaluates to

$\displaystyle\mathcal{F}\left(t,\xi\right)=-\frac{4}{t}\text{log}\bigg{(}\frac{\xi}{t(q-t)}\bigg{)},$
(20)

where $t\equiv 1-T$ and $q=\sqrt{T^{2}+4\xi}$. Now, using Eqs. (6) and (20)
the Borel function for the thrust takes the form,

$\displaystyle B(t,u)=$ $\displaystyle\frac{4\sin\pi u}{\pi
u}\frac{1}{t}e^{\frac{5u}{3}}\int_{t^{2}}^{t}\xi^{-u-1}d\xi,$ (21)

where the lower limit is determined using the collinear gluon boundary
conditions, $x_{1}=1-\xi$, $x_{2}=0$ and the upper limit is determined from
the soft gluon boundary condition $x_{1}=x_{2}=1-\sqrt{\xi}$. Evaluating the
integral we immediately obtain

$\displaystyle B(t,u)=$ $\displaystyle\frac{\sin\pi u}{\pi
u}e^{\frac{5u}{3}}\frac{4}{u}\frac{1}{t}\bigg{(}\frac{1}{t^{2u}}-\frac{1}{t^{u}}\bigg{)},$
(22)

this agrees with the leading singular terms of the same function presented in
Gardi:2001ny . Thus, it is possible to calculate the leading singular terms in
$\mathcal{F}\left(t,\xi\right)$ and $B(t,u)$ using the eikonal matrix element.

###  3.2 $C$-parameter

The $C$-parameter was originally defined in Parisi:1978eg ; Donoghue:1979vi
using the eigenvalues of the matrix

$\displaystyle\theta_{\alpha\beta}=\frac{1}{\sum_{j}|\textbf{p}^{(j)}|}\sum_{i}\frac{\textbf{p}_{\alpha}^{(i)}\textbf{p}_{\beta}^{(i)}}{|\textbf{p}^{(i)}|},$
(23)

where $\textbf{p}_{\alpha}^{(i)}$ are the spatial component of the momentum of
$i$-th particle. If the eigenvalues of the above matrix are denoted by
$\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$, then the $C$-parameter is
given by

$\displaystyle
C=3(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{3}).$
(24)

This can be cast into a Lorentz invariant form

$\displaystyle C=3-\frac{3}{2}\sum_{i,j}\frac{\left(p^{(i)}\cdot
p^{(j)}\right)^{2}}{(p^{(i)}\cdot q)(p^{(j)}\cdot q)},$ (25)

where $p^{(i)}$ denotes the four momentum of the $i$-th particle and $q$
denotes the total four-momentum. $C$ takes a minimum value $0$ for a two-jet
event and $C$ attains a maximum value $1$ for a spherical event. If, however,
the final state has a planar configuration the largest value that the
parameter can attain is $3/4$. This upper limit also applies for the case of
3-body final state that concerns us. The above expression of the $C$-parameter
and its rescaled version can be written down using the energy fractions and
the virtuality of the off-shell gluon.

$\displaystyle
c=\frac{C}{6}=\frac{(1-x_{1})(1-x_{2})(x_{1}+x_{2}-1+2\xi)-\xi^{2}}{x_{1}x_{2}(2-x_{1}-x_{2})}.$
(26)

Now, we define the eikonalized version of the $c$-parameter

$\displaystyle
c_{eik}(x_{1},x_{2})=\frac{(1-x_{1})(1-x_{2})}{(1-x_{1})+(1-x_{2})},$ (27)

which coincides with the above definition in the soft gluon limit. Note that
$c_{eik}$ is not a function of the virtuality $\xi$. We will use $c_{eik}$ to
calculate the characteristic function for $C$-parameter; it is convenient to
change variables from $x_{1}$ and $x_{2}$ into $y=2-x_{1}-x_{2}$ and
$z=(1-x_{2})/y$. In these new variables $c_{e}ik(y,z)=yz(1-z)$. The
characteristic function (Eq. (4)) in this limit takes the form,

$\displaystyle\mathcal{F}=\int dy\ dz\ y\ \mathcal{M}_{\text{soft}}(y,z,\xi)\
\delta(c_{eik}(y,z)-c),$ (28)

where,

$\displaystyle\mathcal{M}_{\text{soft}}(y,z,\xi)=\frac{2}{y^{2}z(1-z)}$ (29)

The symmetry of $\mathcal{M}$ under $x_{1}\leftrightarrow x_{2}$ appears as
symmetry under $z\rightarrow 1-z$. In order to perform the integral in Eq.
(28), it is required to determine the limits of the $z$-integration using the
boundary of the soft region that is given by $x_{2}=(1-\xi-x_{1})/(1-x_{1})$.
The integral in Eq. (28) has an explicit form,

$\displaystyle\begin{split}\mathcal{F}=\int_{2\sqrt{\xi}}^{1+\xi}dy&\int_{\frac{1}{2}-\frac{1}{2}\sqrt{1-4\xi/y^{2}}}^{\frac{1}{2}+\frac{1}{2}\sqrt{1-4\xi/y^{2}}}dz\,\,\frac{2}{yz(1-z)}\frac{1}{y\sqrt{1-4c/y}}\,\,\Big{(}\delta(z-z_{1})+\delta(z-z_{2})\Big{)},\end{split}$
(30)

where

$\displaystyle z_{1}=\frac{1}{2}+\frac{1}{2}\sqrt{1-4c/y}\quad\text{and}\quad
z_{2}=\frac{1}{2}-\frac{1}{2}\sqrt{1-4c/y}.$ (31)

This integral has a symmetry under $z\leftrightarrow(1-z)$ interchange,
therefore the integral over $z$ equals twice the integral between $z=1/2$ and
the upper limit in Eq. (30), where only the $\delta(z-z_{1})$ is relevant. The
condition $z\leq\frac{1}{2}+\frac{1}{2}\sqrt{1-4\xi/y^{2}}$ implies that
$y\geq\xi/c$. With this the integral in Eq. (30) takes the form

$\displaystyle\mathcal{F}=\frac{4}{c}\int_{\xi/c}^{1+\xi}dy\,\,\frac{1}{y\sqrt{1-4c/y}}.$
(32)

Evidently, it is only the lower limit of the integration that gives rise to
singular contribution in the $\xi\rightarrow 0$ limit. As we are only
interested in the derivative of ${\cal F}$, we get, without even evaluating
the integral

$\displaystyle\frac{d\mathcal{F}}{d\xi}=-\frac{4}{c\xi}\frac{\sqrt{\xi}}{\sqrt{\xi-4c^{2}}}.$
(33)

Contrast this to the computation of $d{\cal F}/{d\xi}$ presented in
Gardi:2003iv  where the computation proceeds with the full definition of the
$c$-parameter. In that paper the authors had to deal with the complicated
elliptic integrals and had to carefully consider small $c$ and small $\xi$
limits. These complications are completely absent in our method.

Now we are in the position to compute the Borel function $B(c,u)$ for the
$c$-parameter. We have to substitute $d{\cal F}/{d\xi}$ into Eq. (5),

$\displaystyle B(c,u)=\frac{4\sin{\pi u}}{\pi
u}\frac{1}{c}e^{\frac{5u}{3}}\int_{4c^{2}}^{\frac{c}{1-c}}d\xi~{}\frac{\xi^{-u}}{\sqrt{\xi(\xi-4c^{2})}}$
(34)

where the lower limit in the above integral is determined using
$x_{1}=x_{2}=1-\sqrt{\xi}$ (soft limit), and the upper limit is determined
using $x_{1}=1-\xi$, $x_{2}=0$ (collinear limit). We are interested in the
logarithmically enhanced terms, thus we can replace the upper limit of the
integration by $c/(1-c)\approx c$. Carrying out the integral yields the Borel
function

$\displaystyle B(c,u)=4\frac{\text{sin}\pi u}{\pi
u}e^{\frac{5u}{3}}\frac{1}{c}\bigg{[}\frac{1}{(2c)^{2u}}\frac{\sqrt{\pi}\Gamma(u)}{\Gamma(u+\frac{1}{2})}-\frac{1}{uc^{u}}\bigg{]}.$
(35)

Our result agrees with the soft contribution of the same function presented in
Gardi:2003iv . We conclude thus, that the leading singular terms in
$\mathcal{F}(c,\xi)$ and $B(c,\xi)$ can be captured with significant ease if
we use the eikonal version $c_{eik}$ that we have introduced for the
$c$-parameter.

###  3.3 Angularities

As a demonstration of the wide applicability of our method we consider one
more event shape variable – the angularities. Angularities are novel
observables that allow us to transform between recoil-insensitive to recoil-
sensitive observables in a continuous manner. Angularities were first
introduced almost twenty years ago in Berger:2003pk ; Berger:2003iw ;
Berger:2002ig  and they were defined as

$\displaystyle\tau_{a}=\frac{1}{Q}\sum_{i}E_{i}(\sin\theta_{i})^{a}(1-|\cos\theta_{i}|)^{1-a},$
(36)

where $\theta_{i}$ is the angle made by $i$-th particle with the thrust axis,
$E_{i}$ is the energy of the particle $i$ and $a$ is a continuous parameter.
The thrust axis is defined as the axis with respect to which Eq. (36) is
minimized at $a=0$. One can easily realize that angularities with $a=0$
corresponds to $1-T$, where $T$ is the thrust, while $a=1$ refers to jet
broadening Catani:1992jc . The continuous parameter $a$ has a range
$-\infty<a<2$, where the upper limit on $a$ is fixed by infrared safety. In
terms of $x_{i}$ and $\xi$ angularities were defined in Berger:2004xf  as,

$\displaystyle\tau_{a}(x_{1},x_{2},\xi)=$
$\displaystyle\frac{1}{x_{1}}(1-x_{1})^{1-a/2}\bigg{[}(1-x_{2}-\xi)^{1-a/2}(x_{1}+x_{2}-1+\xi)^{a/2}$
$\displaystyle+(1-x_{2}-\xi)^{a/2}(x_{1}+x_{2}-1+\xi)^{1-a/2}\bigg{]},$ (37)

where, thrust axis is considered along $\textbf{p}_{1}$ (quark momentum). As
we did for the $c$-parameter we introduce an eikonal version of the
angularities:

$\displaystyle\tau_{a}^{eik}(x_{1},x_{2},\xi)=(1-x_{1})^{1-a/2}(1-x_{2})^{a/2}.$
(38)

Now, Using Eq. (4) and (13) the characteristics function takes the form,

$\displaystyle\mathcal{F}=\int
dx_{1}dx_{2}\frac{2}{(1-x_{1})(1-x_{2})}\delta\big{(}\tau_{a}^{eik}(x_{1},x_{2},\xi)-\tau_{a}\big{)}.$
(39)

It is straight-forward to perform the $x_{1}$ integration to obtain

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{4}{\tau_{a}(1-\frac{a}{2})}\int
dx_{2}\,\frac{1}{1-x_{2}}.$ (40)

We determine the upper limit of this integration using the soft boundary
$1-x_{2}=\xi/(1-x_{1})$. As shown in Berger:2004xf , the lower limit of this
integration does not contribute to the logarithmically enhanced terms. The
upper limit of the integration is

$\displaystyle
1-\bigg{(}\frac{\xi^{1-\frac{a}{2}}}{\tau_{a}}\bigg{)}^{\frac{1}{1-a}}.$

We finally have the characteristics function

$\displaystyle\mathcal{F}(\tau_{a},\xi)=-\frac{4}{\tau_{a}}\frac{1}{1-a}\log\xi.$
(41)

Taking the derivative with respect to $\xi$ and substituting in Eq. (6) we get
the Borel function

$\displaystyle B(\tau_{a},u)=\frac{4\sin{\pi u}}{\pi
u}\frac{1}{1-a}\frac{1}{\tau_{a}}e^{\frac{5u}{3}}\int_{\tau_{a}^{2}}^{\tau_{a}^{\frac{2}{2-a}}}d\xi~{}\xi^{-u-1},$
(42)

where the limits are determined using the collinear and soft gluon boundary
conditions mentioned in Figure (1). Upon performing the integration in Eq.
(42) we obtain

$\displaystyle B(\tau_{a},u)=\frac{\sin{\pi u}}{\pi
u}e^{\frac{5}{3}u}\frac{4}{1-a}\frac{1}{u}\frac{1}{\tau_{a}}\left[\frac{1}{\tau_{a}^{2u}}-\frac{1}{\tau_{a}^{\frac{2u}{2-a}}}\right]\,,$
(43)

which agrees with the soft contribution of the same function presented in
Berger:2004xf . We have, thus obtained, the leading singular terms in
$\mathcal{F}(\tau_{a},\xi)$ and $B(\tau_{a},\xi)$, which are responsible for
power corrections by considering the eikonal matrix element and the eikonal
version of the angularities $\tau_{a}^{eik}(x_{1},x_{2},\xi)$ which again
substantially simplifies the computation.

##  4 Borel function using Eikonal Dressed Gluon Exponentiation
in the light-cone variables

In this section, we will follow the same steps of Sec. (3) and calculate Borel
function for thrust, $C$-parameter and angularities using a different set of
kinematic variables. Instead of the energy fractions that we used in the
previous section we would employ the transverse momentum $k_{\perp}$ and
rapidity $y$ of the massive eikonal gluon. In the soft gluon approximation, a
number of event shapes for massless particles were defined in Salam:2001bd ;
Mateu:2012nk . We will consider a class of event shapes which, for massive
soft gluon emission, can be written as

$\displaystyle\bar{e}(k,Q)\,=\,\sqrt{\frac{k_{\perp}^{2}+k^{2}}{Q^{2}}}~{}h_{e}(y)\,,$
(44)

where $k_{\perp}$ and $y$ denote transverse momentum of the gluon and pseudo-
rapidity measured with respect to the thrust axis respectively. The function
$h_{e}(y)$ characterizes the given event shape. Some of the approximations
described below apply to more general event shapes as well, but the results
are especially simple for those which can be cast in the form of Eq. (44).

The contribution from the emission of a soft off-shell gluon can easily be
computed applying the eikonal approximation to the vertex for the emission
from the hard parton. Since off-shell soft-gluon phase space factorizes
Gardi:2001di  from the hard partons, and also the matrix element factorizes,
the soft cross section takes on a simple and universal form,

$\frac{d\sigma}{\sigma}\,=\,\frac{1}{3}\,\frac{4}{k^{2}+k_{\perp}^{2}}\,dk_{\perp}^{2}dy\,.$
(45)

The characteristic function is also then given, in the soft limit, by a simple
and universal expression

$\mathcal{F}(e,\xi)=\int
dk_{\perp}^{2}dy\,\frac{2}{k^{2}+k_{\perp}^{2}}\delta\left(e-\bar{e}\left(k^{2},k_{\perp}^{2},y\right)\right)\,,$
(46)

which integrates to the remarkably simple form,

$\mathcal{F}(e,\xi)\,=\,\frac{8}{e}\int_{y_{{\rm min}}}dy\,$ (47)

where the only information on the chosen observable is the phase space
boundary given by the minimum rapidity $y_{{\rm min}}$. The upper limit of
integration is not relevant, since it does not give any singular contributions
in the $\xi\rightarrow 0$ limit, which is the only significant limit for power
corrections.

Up to now, we have kept the discussion generic, for any shape belonging to the
class given in Eq. (44). Let us now illustrate the results by looking at some
specific examples.

###  4.1 Thrust

The thrust for a generic process is defined in Eq. (15). In the two jet events
all event shape variables that we consider tends to $0$, except thrust which
tends to 1. Thus, it is convenient to define $t=1-T$. In the soft gluon
approximation, thrust in terms of the $k_{\perp}$ and rapidity $y$ is given by
Salam:2001bd

$\displaystyle t=\frac{1}{Q}\sqrt{k_{\perp}^{2}+k^{2}}\ e^{-|y|}.$ (48)

Note that, for our case the gluon is massive and $k^{2}\neq 0$. In order to
perform the integral in Eq. (47), we need to determine the lower limit of the
rapidity. The lower limit of rapidity $y$ is determined by putting
$k_{\perp}=0$ in Eq. (48), thus minimum rapidity is given by,

$y_{{\rm min}}\,=\,\,\ln\Big{(}\frac{1}{t}\sqrt{\xi}\Big{)}.$ (49)

Now, using Eq. (47) and (49) the characteristics function has the form,

$\displaystyle\mathcal{F}=-\frac{8}{t}\text{log}(\frac{\sqrt{\xi}}{t}).$ (50)

The Borel function $B(t,u)$ is then given by

$B(t,u)=\frac{\sin{\pi u}}{\pi
u}e^{\frac{5}{3}u}\frac{4}{u}\frac{1}{t}\left[\frac{1}{t^{2u}}-\frac{1}{t^{u}}\right]\,,$
(51)

which is in well agreement with the soft approximated version of the
characteristics function and Borel function presented in Gardi:2001ny .

###  4.2 $C$-parameter

The $C$-parameter for a generic process is defined in Eq. (3.2). The
$C$-parameter in the soft approximation and expressed using $k_{\perp}$ and
$y$ is given by Salam:2001bd

$\displaystyle c=\frac{C}{6}=\frac{1}{2Q}\sqrt{k^{2}+k_{\perp}^{2}}\
\frac{1}{\text{cosh}\ y}.$ (52)

As for the case of thrust we determine the lower limit of rapidity by putting
$k_{\perp}=0$ and obtain

$\displaystyle y_{{\rm min}}=\cosh^{-1}\left({\sqrt{\xi}}/(2c).\right).$ (53)

Now, substituting $y_{\text{min}}$ in Eq. (47) we obtain the characteristic
function in the soft gluon limit:

$\displaystyle\mathcal{F}=-\frac{8}{c}\text{cosh}^{-1}\bigg{(}\frac{\sqrt{\xi}}{2c}\bigg{)}.$
(54)

This yields the Borel function

$B(c,u)=4\frac{\text{sin}\pi u}{\pi
u}e^{5u/3}\frac{1}{c}\bigg{[}\frac{1}{(2c)^{2u}}\frac{\sqrt{\pi}\Gamma(u)}{\Gamma(u+\frac{1}{2})}-\frac{1}{uc^{u}}\bigg{]},$
(55)

in full agreement with the soft contribution to the same function in
Gardi:2003iv . Notice that, while collinear effects present in Gardi:2003iv
are not properly reproduced, as expected, the cancellation of the pole at
$u=0$, which is a consequence of the IR safety of the event shape, is
preserved.

###  4.3 Angularities

In the soft gluon limit, angularities takes the form Salam:2001bd ;
Mateu:2012nk ,

$\displaystyle\tau_{a}=\frac{1}{Q}\sqrt{k^{2}+k_{\perp}^{2}}\ e^{-|y|(1-a)},$
(56)

and the minimum rapidity is given by

$y_{{\rm
min}}\,=\,\frac{1}{1-a}\,\ln\Big{(}\frac{1}{\tau_{a}}\sqrt{\xi}\Big{)}\,.$
(57)

Now, using Eq. (47), one easily finds

$\frac{d\mathcal{F}(\tau_{a},\xi)}{d\xi}\,=\,-\frac{1}{1-a}\,\frac{4}{\tau_{a}\xi}\,.$
(58)

The Borel function $B(\tau_{a},u)$ is then given by

$B(\tau_{a},u)=\frac{\sin{\pi u}}{\pi
u}e^{\frac{5}{3}u}\frac{4}{1-a}\frac{1}{u}\frac{1}{\tau_{a}}\left[\frac{1}{\tau_{a}^{2u}}-\frac{1}{\tau_{a}^{\frac{2u}{2-a}}}\right]\,,$
(59)

again in agreement with the soft contribution to the results of Berger:2004xf
, and reproducing, in the limit $a\to 0$, the results for thrust of Ref.
Gardi:2002bg .

##  5 The Sudakov exponent

In this section, we will describe the computation of Sudakov exponent for
thrust. Similar conclusions hold true for the other two shape variables as
well that we have considered in this article. We can calculate the Borel
function in the Laplace space $B_{\nu}^{t}(u)$ in the eikonal limit using
$B(t,u)$ that we wrote above in Eq. (22), we obtain,

$\displaystyle B_{\nu}^{t,\text{eik}\text{ }}(u)$
$\displaystyle=\frac{4\,\text{sin}\,\pi u}{\pi
u}\,e^{\frac{5u}{3}}\,\frac{1}{u}\,\bigg{[}\bigg{(}\nu^{2u}\gamma(-2u,\nu)+\frac{1}{2u}\bigg{)}-\bigg{(}\nu^{u}\gamma(-u,\nu)+\frac{1}{u}\bigg{)}\bigg{]},$
(60)

where, we have used

$\displaystyle\int_{0}^{1}\frac{dt}{t}\,e^{u\,\text{log}\frac{1}{t}}\,(e^{-\nu
t}-1)=\nu^{u}\gamma(-u,\nu)+\frac{1}{u},$ (61)

and $\gamma(-u,\nu)=\Gamma(-u)-\Gamma(-u,\nu)$. In the Sudakov region
($\nu\rightarrow\infty$), we can replace $\gamma(-u,\nu)$ by $\Gamma(-u,\nu)$.
Retaining only the logarithmically enhanced terms (powers of log $\nu$), in
the small $u$ limit $B_{\nu}^{t}(u)$ takes the form,

$\displaystyle
B_{\nu}^{t,\text{{eik}}}(u)=2\,e^{\frac{5}{3}u}\frac{\,\text{sin\,}\pi u}{\pi
u}\bigg{[}\Gamma(-2u)\>\bigg{(}\nu^{2u}-1\bigg{)}\frac{2}{u}-\Gamma(-u)\>\bigg{(}\nu^{u}-1\bigg{)}\frac{2}{u}\bigg{]}$
. (62)

The first term inside the square brackets corresponds to large-angle soft
gluon emissions and the second term to collinear gluon emissions. Note that
this expression is free from any $u=0$ singularities. There are two sources of
the poles in this expression: $\Gamma(-2u)$ has poles for all positive
integers and half-integers, and $\Gamma(-u)$ has poles for all positive
integers. However, the pre-factor sin $\pi u$ regulates the poles at the
integer values of $u$. Thus, $B_{t}^{\nu,\text{eik}}$ has renormalon poles
only at half-integer values of $u$.

We will now compare our result with the full result for $B_{\nu}^{t}$
presented in Gardi:2001ny ; Gardi:2003iv  which is given by,

$\displaystyle B_{\nu}^{t}(u)=\,2{\rm e}^{\frac{5}{3}u}\,\frac{\sin\pi u}{\pi
u}\left[\Gamma(-2u)\left(\nu^{2u}-1\right)\frac{2}{u}-\Gamma(-u)\left(\nu^{u}-1\right)\left(\frac{2}{u}+\frac{1}{1-u}+\frac{1}{2-u}\right)\right]\,.$
(63)

Note the poles at $u=1$ and $u=2$ which are absent in the collinear term of
our eikonalized $B_{\nu}^{t,\text{eik}}(u)$. We further notice that no
spurious renormalon poles are present in the eikonalized version. Recall that,
for thrust approximation was done only for the matrix element and not for the
definition of the variable. To show that our eikonal versions of the shape
variables do not spoil the above feature we present the results for the
$c$-parameter. The eikonal version and full result Gardi:2003iv  are as
follows:

$\displaystyle B_{\nu}^{c,\text{eik}}(u)$ $\displaystyle=2\,\frac{\sin\,\pi
u}{\pi
u}e^{\frac{5u}{3}}\,\bigg{[}\Gamma(-2u)(\nu^{2u}-1)2^{1-2u}\frac{\sqrt{\pi}\Gamma(u)}{\Gamma(u+\frac{1}{2})}-\frac{2}{u}\Gamma(-u)(\nu^{u}-1)\bigg{]},$
(64) $\displaystyle B_{\nu}^{c}(u)$ $\displaystyle=2\,\frac{\sin\,\pi u}{\pi
u}e^{\frac{5u}{3}}\,\left[\Gamma(-2u)\left(\nu^{2u}-1\right)2^{1-2u}\frac{\sqrt{\pi}\Gamma(u)}{\Gamma(\frac{1}{2}+u)}\right.$
$\displaystyle\hskip
115.0pt-\left.\Gamma(-u)\left({\nu}^{u}-1\right)\left(\frac{2}{u}+\frac{1}{1-u}+\frac{1}{2-u}\right)\right]\,.$
(65)

Again, no spurious poles are introduced. As expected, the EDGE does not
reproduce the collinear renormalon singularities as it cannot capture the
hard-collinear emissions correctly.

The perturbative coefficients of the Sudakov exponent in the large-$\beta_{0}$
limit can be determined by expanding $B_{\nu}^{t}(u)$ in powers of $u$ and
replacing $u^{n}$ by $n!/(\beta_{0}\alpha_{s}/\pi)^{n+1}$. We notice that the
large-angle soft gluon emission terms – the coefficient of $\Gamma(-2u)$, are
identical in the eikonalized and full versions of the Borel function in the
Laplace space. This implies that the leading logs – the terms of the form
$L^{n+1}\alpha_{s}^{n}$ where $L=\text{log $\nu$ }$, will be the same between
the two. The differences in the sub-leading logarithms appear due to the
absence of the $u=1$ and $u=2$ poles in the collinear terms. We will now
expand the two functions to the first few orders to demonstrate the matching
of the LL terms and the discrepancy in the sub-leading logarithms. The
expansion of the full result gives,

$\displaystyle B_{\nu}^{t}\left(u\right)$ $\displaystyle=-2\,L^{2}+0.691\,L$
$\displaystyle+\,\left(-2\,L^{3}-5.297\,L^{2}-6.485\,L\right)\,u$
$\displaystyle+\,\left(-1.167\,L^{4}-5.527\,L^{3}-14.491\,L^{2}-31.655\,L\right)\,u^{2}$
$\displaystyle+\,\left(-0.5\,L^{5}-3.262\,L^{4}-12.329\,L^{3}-39.003\,L^{2}-80.940\,L\right)\,u^{3}$
$\displaystyle+\,\left(-0.172\,L^{6}-1.405\,L^{5}-6.832\,L^{4}-28.452\,L^{3}-87.21\,L^{2}-175.80\,L\right)\,u^{4}$
(66) $\displaystyle+\mathcal{O}(u^{5})+\ldots$

whereas, the expansion of the eikonal result gives,

$\displaystyle B_{\nu}^{t,\text{eik}}(u)=$ $\displaystyle-2\>L^{2}-2.31\>L$
$\displaystyle+\>$ $\displaystyle(-2\,L^{3}-6.79\,L^{2}-15.71\,L)\>u$
$\displaystyle+\,$
$\displaystyle(-1.167\,L^{4}-6.02\,L^{3}-19.10\,L^{2}-44.59\,L)\>u^{2}$
$\displaystyle+\,$
$\displaystyle(-0.5\,L^{5}-3.38\,L^{4}-13.86\,L^{3}-45.47\,L^{2}-93.66\,L)\>u^{3}$
$\displaystyle+\,$
$\displaystyle(-0.172\,L^{6}-1.429\,L^{5}-7.216\,L^{4}-30.61\,L^{3}-93.57\,L^{2}-187.395\,L)\>u^{4}$
(67) $\displaystyle+\,$ $\displaystyle\mathcal{{O}}(u^{5})+\ldots\>\,.$

As expected, the leading logarithms are appearing correctly in the eikonal
approximated version of the Borel function in the Laplace space. We observe
that the differences in NLL and NNLL terms between $B_{\nu}^{t}$ and
$B_{\nu}^{t,\text{eik}}$ are decreasing as we go higher order in $u$.

Let us now discuss the power corrections. The Sudakov exponent is an integral
over $u$ and the poles of $B_{\nu}^{t}$ that occur on the real $u$-axis make
it an ill defined integral. The integral can be defined by shifting the poles
above or below the axis or equivalently indenting the contour below or above
the poles. This however, introduces an ambiguity that is proportional to the
residue of the poles. The poles of $B_{\nu}^{t}$ that occur at $u=m/2$, where
$m$ is an odd integer give the ambiguity Gardi:2003iv  originating from the
large-angle soft gluon emissions. From Eq. (9) we see that the ambiguity would
be of the form $(\Lambda\nu/Q)^{m}$ which implies the existence of non-
perturbative power corrections of the same form. In Table (1) we present the
residues of poles at $u=m/2$ arising from $B_{\nu}^{t,\text{eik}}(u)$ which
contribute to the soft power corrections. The full result $B_{\nu}^{t}(u)$
also has poles at $u=1$ and $u=2$ which give indications to the size of the
collinear power corrections whereas these are absent in
$B_{\nu}^{t,\text{eik}}(u)$. Thus, the collinear power corrections exist only
for $\nu^{1}$ and $\nu^{2}$, and using the full expression for
$B_{\nu}^{t}(u)$ we find that they are given by
$-2\big{(}\frac{\overline{\Lambda}}{Q}\big{)}^{2}$ and
$-\frac{1}{2}\big{(}\frac{\overline{\Lambda}}{Q}\big{)}^{4}$ respectively.
Note that, in the calculation of the residues for the collinear terms we have
ignored the $\mathcal{O}(1)$ factor $C_{F}/2\beta_{0}$. We see, thus, that the
residue of the collinear power correction is suppressed by $\bar{\Lambda}/Q$
as compared to the soft correction for $\nu^{1}$ term. For example at the LEP
where $Q=209$ Gev and $\bar{\Lambda}=200\>e^{\frac{5}{6}}$ Mev, the ratio of
the size of the collinear correction to the soft correction for $\nu^{1}$ is
approximately $-0.0017$. Thus, at colliders like LEP, the soft power
corrections are more important as compared to the collinear corrections. As
pointed out in Gardi:2001ny , the dominant power correction arising from the
residue at $u=1/2$ is proportional to $\nu$ and thus generates a shift in the
resummed cross-section.

For the other event shapes considered in this article one can calculate the
Borel function in Laplace space using EDGE. It remains true that soft power
corrections are dominant over the collinear corrections for all the event
shapes considered in this paper.

Correction | Residue
---|---
$\nu^{1}$ | 8$\frac{\overline{\Lambda}}{Q}$
$\nu^{3}$ | $-\frac{4}{27}\big{(}\frac{\overline{\Lambda}}{Q}\big{)}^{3}$
$\nu^{5}$ | $\frac{1}{375}\big{(}\frac{\overline{\Lambda}}{Q}\big{)}^{5}$
$\nu^{7}$ | $-\frac{1}{30870}\big{(}\frac{\overline{\Lambda}}{Q}\big{)}^{7}$
Table 1: The size of the residues of renormalon singularities for soft power
corrections. The numbers quoted are $\pi$ times the residue and
$\overline{\Lambda}=\Lambda\>e^{5/6}$. We ignore here the $\mathcal{O}(1)$
factor $C_{F}/2\beta_{0}$ in Eq. (9).

##  6 Conclusions

In this paper, we have introduced Eikonal Dressed Gluon Exponentiation which
is a combination of Dressed Gluon Exponentiation and Eikonal approximation.
Using our method, we have demonstrated for several event shapes at
$e^{+}e^{-}$ colliders that the leading singular contributions for the
respective Borel functions in the single dressed gluon approximation are
produced with remarkably simple calculations. It is straightforward to
construct the Sudakov exponent in the large-$\beta_{0}$ limit for the power
corrections using the procedure presented in Gardi:2001di ; Gardi:1999dq .
This exponentiation effectively resums both the large Sudakov logarithms and
the power corrections. We observe that EDGE does not introduce any spurious
renormalons and correctly produces the dominant power corrections originating
from soft emissions.

We have shown that in order to accurately capture the leading singular terms
of the characteristic function $\mathcal{F}(e,\xi)$ and Borel function
$B(e,\xi)$ for an event shape variable $e$, it is sufficient to use the
eikonal squared matrix element $\mathcal{M}$, together with the eikonal
version of the event shape variable. Typically the shape variables such as
$C$-parameter and angularities have complicated expressions especially so when
the final state gluon is off the mass shell. We have demonstrated that the
simplification of the computations is achieved, both when one uses the energy
fractions as the variables, and also when we uses light-cone variables to
parameterize the phase space of the eikonal dressed gluon. When using the
latter variables, we observe that the minimum value of rapidity
$y_{\text{min}}$ is the source of the leading singular terms in
$\mathcal{F}(e,\xi)$. We believe that this method is sufficiently simple and
flexible to be implemented also in the more intricate environment of hadron
collisions, where hadronic event shapes and jet shapes provide important tools
for QCD analyses.

##  7 Acknowledgments

SP and AM would like to thank MHRD Govt. of India for an SRF fellowship. AT
would like to thank Lorenzo Magnea for suggesting this project and for very
fruitful discussions, Einan Gardi for very useful discussions, and the
University of Turin and INFN Turin for warm hospitality during the course of
this work.

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