# Next-to-leading power corrections to the event shape variables

Neelima Agarwal1<EMAIL_ADDRESS>Melissa van Beekveld2
<EMAIL_ADDRESS>Eric Laenen3,4,5<EMAIL_ADDRESS>Shubham Mishra6<EMAIL_ADDRESS>Ayan Mukhopadhyay6
<EMAIL_ADDRESS>Anurag Tripathi6<EMAIL_ADDRESS>1Department of Physics, Chaitanya Bharathi Institute of Technology, Gandipet,
Hyderabad, Telangana State 500075, India  2Rudolf Peierls Centre for
Theoretical Physics, Clarendon Laboratory, Parks Road, University of Oxford,
Oxford OX1 3PU, UK  3Institute of Physics, University of Amsterdam, Science
Park 904,
1098 XH Amsterdam, The Netherlands  4Nikhef, Theory Group, Science Park 105,
1098 XG,
Amsterdam, The Netherlands  5Institute for Theoretical Physics, Utrecht
University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands  6Department of
Physics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy,
Telangana State 502285, India

###### Abstract

We investigate the origin of next-to-leading power corrections to the event
shapes thrust and $c$-parameter, at next-to-leading order. For both event
shapes we trace the origin of such terms in the exact calculation, and compare
with a recent approach involving the eikonal approximation and momentum shifts
that follow from the Low-Burnett-Kroll-Del Duca theorem. We assess the
differences both analytically and numerically. For the $c$-parameter both
exact and approximate results are expressed in terms of elliptic integrals,
but near the elastic limit it exhibits patterns similar to the thrust results.

###### Contents

  1. I Introduction
  2. II Next-to-leading power terms and kinematical shifts
  3. III Thrust
    1. III.1 Thrust distribution at NLO
    2. III.2 Next-to-leading power corrections for thrust from kinematical shifts
    3. III.3 Numerical assessment of the shifted kinematics approximations for thrust
  4. IV $C$-parameter
    1. IV.1 The $c$-parameter distribution at NLO
    2. IV.2 Next-to-leading power corrections to $c$-parameter from shifted kinematics
    3. IV.3 Numerical assessment of the shifted kinematics approximations for $c$-parameter
  5. V Conclusions
  6. A Transformation of incomplete elliptic integrals
    1. A.1 Non-reducible incomplete elliptic integrals
    2. A.2 Reducible incomplete elliptic integrals
  7. B Eikonal approximation and result summary
    1. B.1 Eikonal approximation to thrust and $c$-parameter distribution
    2. B.2 Table of results

##  I Introduction

Providing precise estimates for cross-sections in perturbative QCD is needed
to match the ever increasing precision of collider physics measurements. Two
complementary directions are pursued in this endeavour. In one, exact higher
order calculations in the coupling $\alpha_{s}$ are performed, and methods
thereto developed. In the other, all-order results are derived in certain
kinematic limits, where associated classes of logarithmic terms are enhanced;
the latter can often be resummed to all orders, using a varied and ever
expanding set of methods. A highly relevant region in this regard is the near-
elastic region (which we loosely refer to here also as the threshold region),
where the phase space for emitted particles is limited. In such a situation,
the cancellation of infrared singularities, guaranteed by the KLN theorem [1,
2], is incomplete in the sense that it leaves large logarithmic remainders at
any order in perturbation theory. Our analysis in this paper is relevant for
the second direction.

To be more specific, if $\xi$ is a dimensionless kinematic variable, such that
$\xi\rightarrow 0$ towards the elastic region, the corresponding differential
cross-section has the generic form

$\frac{d\sigma}{d\xi}\,=\,\sum_{n=0}^{\infty}\left(\frac{\alpha_{s}}{\pi}\right)^{n}\,\left[\sum_{m=0}^{2n-1}c_{nm}^{\text{LP}}\left(\frac{\log^{m}\xi}{\xi}\right)_{+}+\,c_{n}^{(\delta)}\delta(\xi)+\,\sum_{m=0}^{2n-1}c_{nm}^{\text{NLP}}\,\log^{m}\xi\,+\,\ldots\right]\,.$
(1)

The first term on the right in the above equation is well-known to originate
from soft and/or collinear radiation and, together with the second term, makes
up the leading power (LP) terms. Much is known about LP terms to arbitrary
order, and there have been numerous approaches towards their resummation [3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and power correction
studies [19, 20]. Reviews on some of these different approaches can be found
in [21, 22, 23, 24, 25].

The last term on the right represents next-to-leading power (NLP) terms. These
originate from both soft gluon and soft quark emissions. Although suppressed
by a power of $\xi$, they can be relevant, since they grow logarithmically
towards threshold. In contrast to the LP terms, the precise organization of
these NLP terms to all orders and arbitrary logarithmic accuracy are not yet
clear. In this paper we focus on such NLP terms for two event shapes in
$e^{+}e^{-}$ collisions: thrust and the $C$-parameter. These observables are
interesting in this regard because, in contrast to most earlier studies, all
QCD effects reside in the final state, and because their definition involves
special phase space constraints that were not considered so far [26, 27, 28,
29]. For these observables our aim is to trace the origin of the NLP terms
near the elastic limit, to examine to what extent there is a common pattern of
NLP terms, and assess their size.

Patterns among NLP terms have been studied for various processes [30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. For a number of observables their
numerical contribution can be significant [43, 44, 45, 46, 47, 48]. The all-
order resummation of NLP terms has been pursued through different approaches,
such as a diagrammatic and a path integral approach given in [49, 50], while a
physical kernel approach is pursued in [51, 52, 53]. From direct QCD
formalism, the development of factorization theorems for NLP terms extending
from LP terms are studied in [30, 31, 54, 55, 56, 57, 58] (for NLP
factorization see, [59]). The study of NLP effects in the framework of SCET
has many aspects, and the active studies are done in the operators
contributing at NLP level [60, 61, 62, 63, 64, 65, 66, 67], the development of
factorization [34, 33, 42, 68], and the explicit studies of physical
observables [69, 70, 71, 32, 72, 73, 36, 74, 75, 76, 77, 39].

Event shapes, which probe final state dynamics through geometrical
constraints, have long been used to develop QCD ideas [78, 79, 80, 81], e.g.
the development of resummation and fixed-order computations [82, 83, 78, 84,
85, 86, 87]. They have also played a vital role in extracting the strong
coupling constant [88, 89, 90, 91]. Here we examine to what extent NLP terms
for two event shapes can be predicted using the kinematical shift method [92,
46], as well as the soft quark emission approximation.

Our paper is organized as follows. We compute the thrust distribution in
shifted approximation in section III to assess the relevance of NLP terms. We
perform a similar assessment in section IV, which is considerably more
complicated. We conclude in section V, while appendices contain certain
technical aspects of elliptic functions, and a summary table of our results.

##  II Next-to-leading power terms and kinematical shifts

It was recently shown [92] that NLP terms, at next-to-leading order (NLO) and
to the extent that they are due to soft gluon radiation, may be derived
efficiently using a combination of the eikonal approximation and kinematical
shifts, in first instance for colour singlet final states. The method holds
for matrix elements and thus for differential distributions, and was
subsequently extended to final state radiation in [46]. It rests in essence on
the Low-Burnett-Kroll-Del Duca (LBKD) theorem [93, 94, 95], which enables the
expression of the one-gluon emission amplitude in terms of the elastic
amplitude (even if the latter contains loops, such as in the case of Higgs, or
multi-Higgs production). The NLO matrix element up to NLP accuracy can be
written as a combination of scalar, spin, and orbital terms:

$\displaystyle\mathcal{M}_{\text{NLP}}\,=\,\mathcal{M}_{\text{scal}}+\mathcal{M}_{\text{spin}}+\mathcal{M}_{\text{orb}}\,.$
(2)

The scalar term is essentially a multiplicative term containing the LP eikonal
approximation. The spin-dependent term is of NLP accuracy, as the eikonal
approximation cannot resolve emitter spin. The orbital term, also of NLP
accuracy, involves derivative operators. These can be represented as a first-
order Taylor expansion of kinematically shifted momenta. Combining these terms
for the production of colour singlet particles yields the expression

$\displaystyle\overline{\sum}|\mathcal{M}_{\text{shift}}|^{2}\,=\,$
$\displaystyle g_{s}^{2}N_{c}(N_{c}^{2}-1)\frac{2p_{1}\cdot p_{2}}{(p_{1}\cdot
p_{3})(p_{2}\cdot p_{3})}$ $\displaystyle\times|\mathcal{M}_{0}(p_{1}-\delta
p_{1},p_{2}-\delta p_{2})|^{2},$ (3)

where $|\mathcal{M}_{0}(p_{1},p_{2})|^{2}$ is the matrix element squared at
the leading order (LO), and $\overline{\sum}$ denotes the sum (average) over
the final (initial) state spins and colours, $p_{3}$ is the momentum of the
emitted radiation, and $p_{1}$, $p_{2}$ are the momenta of the particles
already present at the born level. The shifts in the momenta are given by

$\displaystyle\delta p_{1}^{\mu}$
$\displaystyle\,\,=\,-\frac{1}{2}\left(\frac{p_{2}\cdot p_{3}}{p_{1}\cdot
p_{2}}p_{1}^{\mu}-\frac{p_{1}\cdot p_{3}}{p_{1}\cdot
p_{2}}p_{2}^{\mu}+p_{3}^{\mu}\right),$ $\displaystyle\delta p_{2}^{\mu}$
$\displaystyle\,\,=\,-\frac{1}{2}\left(\frac{p_{1}\cdot p_{3}}{p_{1}\cdot
p_{2}}p_{2}^{\mu}-\frac{p_{2}\cdot p_{3}}{p_{1}\cdot
p_{2}}p_{1}^{\mu}+p_{3}^{\mu}\right).$ (4)

Expressions (3) and (4) yield the dominant NLP contributions to the NLO matrix
element. We now turn to examine the implications of this for the two event
shapes.

##  III Thrust

Thrust [96] in $e^{+}e^{-}$ collisions is defined as

$\displaystyle T\,=\,\underset{\bf{\hat{n}}}{\rm{max}}\
\frac{\sum_{i}|{\bf{p}_{i}}\cdot{\bf{\hat{n}}}|}{\sum_{i}E_{i}}\,,$ (5)

where $\bf{p}_{i}$ and $E_{i}$ are the three-momentum and energy of the
$i^{\text{th}}$ particle present in the final state. The unit vector
$\bf{\hat{n}}$ that maximizes the sum in the numerator is the thrust axis. The
range of thrust is $[1/2,1]$ where $T=1/2$ for a spherically symmetric event
and $T=1$ for a pencil-like (dijet) event. It is an infrared-safe event shape
variable, i.e. it is insensitive to the soft emissions or collinear
splittings. The LO reaction at the parton level is

$\displaystyle e^{+}(p_{b})+e^{-}(p_{a})\rightarrow\gamma^{*}(q)\rightarrow
q(p_{1})+{\overline{q}}(p_{2})\,,$ (6)

where we assume all particles to be massless. For this case, $T=1$. At NLO the
real emission process is

$\displaystyle e^{+}(p_{b})+e^{-}(p_{a})\rightarrow\gamma^{*}(q)\rightarrow
q(p_{1})+{\overline{q}}(p_{2})+g(p_{3})\,.$ (7)

The diagrams for this process are shown in fig. (1). For a three-body final
state $T$ takes values in the range $[2/3,1]$. The limit $T=1$ is approached
when either $(i)$ the emitted gluon is soft ($p_{3}\to 0$), $(ii)$ the quark
or the anti-quark is soft ($p_{1}\to 0\ \text{or}\ p_{2}\to 0$), or $(iii)$
any two final state partons are collinear.

Figure 1: Feynman diagrams for the real emission of a gluon from the final
state quark or anti-quark.

It is standard practice to define the dimensionless energy fractions for the
final state particles,

$\displaystyle x_{i}\,=\,\frac{2E_{i}}{Q}\quad\quad(i=1,2,3)\,,$ (8)

where $Q$ is the total center of mass energy, with $x_{1}+x_{2}+x_{3}\,=\,2$.
One can readily derive the relations

$\displaystyle(p_{2}+p_{3})^{2}\,=\,$ $\displaystyle 2p_{2}\cdotp
p_{3}\,=\,Q^{2}(1-x_{1}),$ $\displaystyle(p_{1}+p_{3})^{2}\,=\,$
$\displaystyle 2p_{1}\cdotp p_{3}\,=\,Q^{2}(1-x_{2}),$ (9)
$\displaystyle(p_{1}+p_{2})^{2}\,=\,$ $\displaystyle 2p_{1}\cdotp
p_{2}\,=\,Q^{2}(1-x_{3}),$

using momentum conservation. For a final state with three massless particles,
eq. (5) takes the simple form

$\displaystyle T\,=\,\text{max}(x_{1},x_{2},x_{3})\,.$ (10)

###  III.1 Thrust distribution at NLO

The thrust distribution at NLO is given by

$\displaystyle\frac{d\sigma}{dT}\,=\,\frac{1}{2s}\int
d\Phi_{3}\overline{\sum}|\mathcal{M}(x_{1},x_{2})|^{2}\delta(T-\text{max}(x_{1},x_{2},x_{3})),$
(11)

where $s$ is the center of mass energy squared. The matrix element squared for
the process in eq. (7) is

$\displaystyle\overline{\sum}$ $\displaystyle|\mathcal{M}(x_{1},x_{2})|^{2}$
$\displaystyle\,=\,$ $\displaystyle
8(e^{2}e_{q})^{2}g_{s}^{2}C_{F}N_{c}\frac{1}{3Q^{2}}\,\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}\,,$
(12)

where $\alpha=\left(e^{2}/4\pi\right)$, $e_{q}$ is the charge of quarks in the
unit of fundamental electric charge $e$, $\alpha_{s}=g_{s}^{2}/4\pi$,
$C_{F}=(N_{c}^{2}-1)/2N_{c}$, and $N_{c}$ is the number of quark colours. The
three-particle phase space measure, expressed in terms of $x_{i}$ reads

$\displaystyle d\Phi_{3}\,=\,$
$\displaystyle\frac{Q^{2}}{16(2\pi)^{3}}\,dx_{1}\,dx_{2}\,dx_{3}\,\delta(x_{1}+x_{2}+x_{3}-2).$
(13)

The phase space in eq. (13) is depicted in fig. (2),

Figure 2: The Dalitz plot for the production of three massless particles. The
phase space can be divided into three different regions depending upon which
energy fraction is the largest. In region ABG, $x_{1}$ is the largest of the
three $x_{i}$, while in region BGC, (AGC) $x_{2}$ ($x_{3}$) is the largest. In
this figure, we set $BA=BC=1$. Along the lines AB, BC, and AC one has $T=1$.

with every point in the plane fulfilling the constraint $x_{1}+x_{2}+x_{3}=2$.
The region surrounding point $A$ ($C$) corresponds to the soft anti-quark
(quark) region, and the region around $B$ to the soft gluon region. On line
$BA$, where $x_{1}=1$, the anti-quark and gluon move collinearly in opposite
directions, while on line $BC$, where $x_{2}=1$, the quark and the gluon are
collinear. On line $AC$, where $x_{3}=1$, the quark and the anti-quark are
collinear, with the gluon moving in the opposite direction. Along these three
lines, $T=1$.

To obtain the thrust distribution one integrates over the $x_{1},x_{2}$
variables in eq. (11) with the appropriate limits:

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}$
$\displaystyle dx_{2}\,\,\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}$
$\displaystyle\times\delta\left(T-\text{max}(x_{1},x_{2},x_{3})\right)\,,$
(14)

where $x_{3}=2-x_{1}-x_{2}$, and $\sigma_{0}(s)$ is the LO cross-section,
given by

$\displaystyle\sigma_{0}(s)\,=\,\frac{4\pi\alpha^{2}e_{q}^{2}\,N_{c}}{3s}\,,$
(15)

For our purposes, we label the contributions from three distinct contributions
in the phase space integration in eq. (III.1), defined by either $x_{1}$,
$x_{2}$, and $x_{3}$ being the largest, as follows:

  1. I :

Quark has the largest energy ($x_{1}>x_{2},x_{3}$)

  2. II :

Anti-quark has the largest energy ($x_{2}>x_{1},x_{3}$)

  3. III :

Gluon has the largest energy ($x_{3}>x_{1},x_{2}$)

The contribution from region I is then

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{I}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}\,$
$\displaystyle\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}\,\,$
$\displaystyle\times$ $\displaystyle\delta(T-x_{1})\,.$ (16)

With the appropriate limits of integration we have

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{I}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{2(1-T)}^{T}dx_{2}\,\frac{T^{2}+x_{2}^{2}}{(1-T)(1-x_{2})}.$
(17)

Instead of $T$, we shall mostly use the variable $\tau\,=\,1-T$, which
vanishes in the zero-radius dijet limit. The integration in eq. (17) leads to

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I}}}$
$\displaystyle\,=\,$ $\displaystyle\
\frac{2\alpha_{s}}{3\pi}\biggl{[}\frac{3\tau^{2}+8\tau-3}{2\tau}+\left(\frac{\tau^{2}-2\tau+2}{\tau}\right)\log\left(\frac{1-2\tau}{\tau}\right)\biggr{]}.$
(18)

Expanding the above expression around $\tau=0$ gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I}}}$
$\displaystyle=\
\frac{2\alpha_{s}}{3\pi}\biggl{(}\frac{-3-4\log\tau}{2\tau}+2\log\tau+\frac{3\tau}{2}-\tau\log\tau+\mathcal{O}(\tau^{2})\biggr{)}.$
(19)

The first term in the brackets forms the LP term (in $\tau$), with both
leading-logarithmic (LL) and next-to-leading logarithmic (NLL) terms. The
former derives from soft and collinear gluon emission, the latter from hard
collinear gluon emission. The next term constitutes the (NLP) term, which is
our main focus here. Subsequent terms are of NNLP and beyond accuracy. The
result for thrust distribution from region I is provided in eq. (19); however,
this expression does not show the correspondence between the various
kinematical configurations and the contributions they produce at the LP and
NLP. In order to extract such information, we shall split the integration
result of eq. (17) into its upper and lower limit contributions separately, as
these limits reflect the different regions of the phase space. The
contributions from the upper (I,u) and lower (I,l) limits of $x_{2}$ from the
region I are, respectively, after expansion around $\tau\,=\,0$

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,u}}}$
$\displaystyle=$ $\displaystyle\
\frac{2\alpha_{s}}{3\pi}\biggl{(}\frac{-2\log\tau}{\tau}+2+2\log\tau-\frac{\tau}{2}+\tau\log\tau+\mathcal{O}\left(\tau^{2}\right)\biggr{)},$
(20)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,l}}}\,=\,$
$\displaystyle\
\frac{2\alpha_{s}}{3\pi}\left(\frac{3}{2\tau}+2-2\tau+\mathcal{O}(\tau^{2})\right).$
(21)

Note that here and henceforth the lower limit term is to be _subtracted_ from
the upper limit result. In region I, the upper limit of $x_{2}$ for small
$\tau$ corresponds to a back-to-back quark and anti-quark configuration
allowing the gluon to go soft while the lower limit corresponds to the quark
and gluon being back-to-back, allowing the gluon to be hard. The LL terms at
LP and NLP follow only from the upper limit ($x_{2}\,=\,1-\tau$), whereas the
lower limit ($x_{2}\,=\,2\tau$) yields NLL terms both at LP and NLP. In fact,
identical NLL contributions at NLP follow from both limits; they therefore
cancel in the final result for region I.

Thus the upper and lower limit contributions unveil the relation between the
particular kinematical configurations and their respective contributions at LP
and NLP. To better comprehend the origin of LP and NLP terms, we will from
here on first provide our full results and then their breakdown into upper and
lower limit contributions.

In region II, due to the symmetry under the interchange of
$x_{1}\leftrightarrow x_{2}$, the contribution from this region is identical
to eq. (19). Thus

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{II}}}\,=\,\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I}}}\,.$
(22)

In region III, the thrust axis is aligned with the momentum of the gluon. The
contribution is

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{III}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}$
$\displaystyle\int_{0}^{1-x_{2}}dx_{1}\,\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}\,\,$
$\displaystyle\times\delta(T+x_{1}+x_{2}-2)\,,$ (23)

which gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{III}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{2(1-T)}^{T}dx_{2}\,\frac{(2-x_{2}-T)^{2}+x_{2}^{2}}{(x_{2}+T-1)(1-x_{2})}\,.$
(24)

Expanding in $\tau$ gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}-2-2\log{\tau}+(2-2\log\tau)\tau+\mathcal{O}(\tau^{2})\bigg{)}.$
(25)

Thus no LP contributions arise from region III, as expected. The $\tau=0$
configurations in this region correspond to either the quark or anti-quark
being soft, or the quark and anti-quark pair moving collinearly opposite to
the gluon. The latter configuration does not lead to a large contribution
because the propagator is then far off the mass-shell. The contribution from
upper and lower limits from this region respectively are, for small $\tau$,

$\displaystyle\frac{1}{\sigma_{0}(s)}$
$\displaystyle\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,u}}}$
$\displaystyle\,=\,$ $\displaystyle\
\frac{2\alpha_{s}}{3\pi}\biggl{(}-2-\log\tau+(1-\log\tau)\tau+\mathcal{O}\left(\tau^{2}\right)\biggr{)},$
(26)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,l}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}\log\tau+(-1+\log\tau)\tau+\mathcal{O}(\tau^{2})\bigg{)}.$
(27)

The upper limit of $x_{2}$ in region III corresponds to the gluon and anti-
quark being back-to-back, and the lower limit has the gluon and quark back-to-
back. LL terms at NLP arise from both limits.

Combining contributions from all three regions, the thrust distribution at NLO
yields the well-known result

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Bigg{|}_{\text{NLO}}\,=\,\frac{2\alpha_{s}}{3\pi}$
$\displaystyle\biggl{[}\frac{2(3\tau^{2}-3\tau+2)}{\tau(1-\tau)}\,\log\left(\frac{1-2\tau}{\tau}\right)$
$\displaystyle\hskip 42.67912pt-\frac{3(1-3\tau)(1+\tau)}{\tau}\biggr{]}\,,$
(28)

with expansion

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Bigg{|}_{\text{NLO}}\,=\,\
\frac{2\alpha_{s}}{3\pi}$
$\displaystyle\biggl{(}\frac{-3-4\log{\tau}}{\tau}-2+2\log{\tau}+(5\tau$
$\displaystyle\hskip
42.67912pt-4\log\tau)\tau+\mathcal{O}(\tau^{2})\biggr{)}.$ (29)

Note that in eq. (29) at NLP all three regions produce LL terms, but a partial
cancellation takes place when combined.

###  III.2 Next-to-leading power corrections for thrust from kinematical
shifts

Next we use the method of kinematical shifts [92, 46] to compute dominant NLP
terms for the thrust distribution due to gluon radiation. The expression for
the shifted matrix elements is given in eq. (3), with the momenta shifts given
by eq. (4). Note that because the emitted gluon is emitted from final
particles the momenta shifts in eq. (3) are subtracted [46] from the radiating
particle momenta, rather than added as for the initial state case. The
squared, spin-summed, and averaged LO matrix element $|\mathcal{M}_{0}|^{2}$
for the process in eq. (6) is

$\displaystyle\overline{\sum}|\mathcal{M}_{0}(p_{1},p_{2})|^{2}\,=\,\frac{4(e^{2}e_{q})^{2}N_{c}}{3s}(2p_{1}\cdot
p_{2})\,.$ (30)

Inserting shifted momenta according to eq. (4) leads to

$\displaystyle\overline{\sum}$ $\displaystyle|\mathcal{M}_{0}(p_{1}-\delta
p_{1},p_{2}-\delta p_{2})|^{2}$ $\displaystyle\,=\,$
$\displaystyle\frac{8(e^{2}e_{q})^{2}N_{c}}{3Q^{2}}\biggl{[}(p_{1}\cdot
p_{2})+p_{3}\cdot(p_{1}+p_{2})+{\cal O}(p_{3}^{2})\biggr{]}\,.$ (31)

The last term contains two powers of the momentum $p_{3}$, and can thus be
omitted. Combining eqs. (3) and (31) gives the expression for shifted matrix
element squared as

$\displaystyle\overline{\sum}$
$\displaystyle|{\mathcal{M}}_{\text{shift}}|^{2}$ $\displaystyle\,=\,$
$\displaystyle 8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\,\frac{1}{3Q^{2}}\,$
$\displaystyle\times\left(\frac{2(p_{1}\cdot p_{2})^{2}}{(p_{1}\cdot
p_{3})(p_{2}\cdot p_{3})}+\frac{2p_{1}\cdot p_{2}}{p_{2}\cdot
p_{3}}+\frac{2p_{1}\cdot p_{2}}{p_{1}\cdot p_{3}}\right)\,,$ (32)

which, expressed in terms of $x_{i}$’s, reads

$\displaystyle\overline{\sum}|{\mathcal{M}}_{\text{shift}}(p_{1},p_{2})|^{2}\,=\,$
$\displaystyle 8(e^{2}e_{q})^{2}N_{c}g_{s}^{2}C_{F}\,\frac{1}{3Q^{2}}\,$
$\displaystyle\hskip
14.22636pt\times\left(\frac{2x_{1}+2x_{2}-2}{(1-x_{1})(1-x_{2})}\right)\,.$
(33)

Note that when the emitted gluon becomes soft (i.e. $x_{1},x_{2}\to 1$), the
above matrix element diverges, whereas when the gluon is hard ($x_{3}\to 1$),
the numerator $(2x_{1}+2x_{2}-2)$ vanishes. The thrust distribution under this
approximation is given by

$\displaystyle\frac{d\sigma}{dT}\Bigg{|}_{\text{shift}}\,=\,\frac{1}{2s}\int
d\Phi_{3}\,\,$ $\displaystyle\overline{\sum}\
|\mathcal{M}_{\text{shift}}(x_{1},x_{2})|^{2}\,\,$
$\displaystyle\times\delta(T-\text{max}(x_{1},x_{2},x_{3})).$ (34)

We repeat the steps to compute the thrust distribution from the exact
computation for each of the three regions. Region I gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{I}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}\,\,$
$\displaystyle\frac{2x_{1}+2x_{2}-2}{(1-x_{1})(1-x_{2})}\,\,$
$\displaystyle\times\delta(T-x_{1}),$ (35)

which leads to the expansion about $\tau\,=\,0$

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\left(\frac{-2-2\log\tau}{\tau}+2+2\log\tau+\mathcal{O}(\tau^{2})\right).$
(36)

When we compare the above expression to eq. (19), we see that the LLs at both
the LP and NLP are correctly captured, while the NLL terms at LP are only
partially reproduced. The contributions from upper and lower limits are

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,u}}}\,=\,$
$\displaystyle\
\frac{2\alpha_{s}}{3\pi}\left[-\frac{2(1-\tau)(1+\log\tau)}{\tau}\right],$
(37)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,l}}}\,=\,$
$\displaystyle\
\frac{2\alpha_{s}}{3\pi}\left[-4-\frac{2(1-\tau)\log(1-2\tau)}{\tau}\right],$
(38)

which, expanded around $\tau\,=\,0$, gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,u}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\biggl{(}\frac{-2-2\log\tau}{\tau}+2+$ $\displaystyle
2\log\tau$ $\displaystyle+\mathcal{O}(\tau^{2})\biggr{)},$ (39)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,l}}}\,=\,$
$\displaystyle\
\frac{2\alpha_{s}}{3\pi}\bigg{(}\mathcal{O}(\tau^{2})\bigg{)}.$ (40)

The upper limit contribution in region I contains the LL and NLL terms at LP
and NLP, while the lower limit acts beyond NLP. The contribution from region
II is identical to I. The contribution from region III is

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{III}}}\,=\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}$
$\displaystyle\int_{0}^{1-x_{2}}dx_{1}\,\,$ $\displaystyle\times\delta$
$\displaystyle(T+x_{1}+x_{2}-2)$ $\displaystyle\times$
$\displaystyle\frac{2x_{1}+2x_{2}-2}{(1-x_{1})(1-x_{2})}\,,$ (41)

which gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\left[\frac{4\tau}{1-\tau}\log\left(\frac{1-2\tau}{\tau}\right)\right]\,.$
(42)

Upon expansion around $\tau\,=\,0$ this yields only terms beyond NLP accuracy

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}-4\tau\log\tau+\mathcal{O}(\tau^{2})\bigg{)}\,,$
(43)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,u}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}-2\tau\log\tau+\mathcal{O}(\tau^{2})\bigg{)},$
(44)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,l}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}2\tau\log\tau+\mathcal{O}(\tau^{2})\bigg{)}\,.$
(45)

Indeed the hard gluon/soft quark region is not part of the shifted kinematics
method.

Combining contributions from all three regions the thrust distribution from
shifted kinematics formalism at NLO reads

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Bigg{|}_{\text{shift}}\,=\,\frac{2\alpha_{s}}{3\pi}\biggl{[}\frac{8\tau^{2}-8\tau+4}{\tau(1-\tau)}$
$\displaystyle\log\left(\frac{1-2\tau}{\tau}\right)$
$\displaystyle-\frac{3\tau^{2}-4\tau+1}{\tau(1-\tau)}\biggr{]}\,,$ (46)

or

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Bigg{|}_{\text{shift}}\,=\,\
\frac{2\alpha_{s}}{3\pi}$
$\displaystyle\bigg{[}\frac{-4-4\log{\tau}}{\tau}+4+4\log{\tau}$
$\displaystyle\hskip 25.6073pt-(4\log\tau)\tau+\mathcal{O}(\tau^{2})\bigg{]}.$
(47)

When compared to the exact computation in eq. (29), we see that the LL terms
at LP have been reproduced, in addition to some NLL terms. At the NLP level,
we have captured some, but not all, of the LL terms, in addition to some NLL
contributions. The missing LL terms at NLP should arise from soft quark (anti-
quark) contributions.

The exact squared matrix element in eq. (12) has the following $x_{i}$
dependence

$\displaystyle\overline{\sum}|{\mathcal{M}}(x_{1},x_{2})|^{2}\,=\,8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}\,\biggl{(}\frac{2}{(1-x_{1})(1-x_{2})}-\frac{2}{1-x_{1}}-\frac{2}{1-x_{2}}+\frac{1-x_{2}}{1-x_{1}}+\frac{1-x_{1}}{1-x_{2}}\biggr{)}.$
(48)

The first three terms constitute the shifted matrix element squared, as shown
in eq. (33). The eikonal approximation in fact consists of only the first term
of eq. (48). The remaining last two terms in eq. (48) we refer to as remainder
matrix element squared and it has following form

$\displaystyle\overline{\sum}$
$\displaystyle|{\mathcal{M}}_{\text{rem}}(x_{1},x_{2})|^{2}$
$\displaystyle\,=\,8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}\biggl{(}\frac{1-x_{2}}{1-x_{1}}+\frac{1-x_{1}}{1-x_{2}}\biggr{)}\,.$
(49)

Let us take the first term on the right, which diverges for $x_{1}\to 1$. It
is most relevant when $x_{2}\to 0$ and $x_{1}\to 1$, i.e. $x_{3}\to 1$, which
corresponds to the emission of a soft anti-quark. Similarly the second term is
dominant when $x_{1}\to 0$ and $x_{2}\to 1$ i.e. the emission of soft quark.

The contributions from soft quark and anti-quark emissions can alternatively
be computed using the quark emission operator defined in [46]. The matrix
element for the emission of a soft quark ($p_{1}\to 0$) in fig. (1) is

$\displaystyle i\mathcal{M}_{1,q}\,=\,$
$\displaystyle\frac{ig_{s}t_{a}}{2p_{1}\cdot
p_{3}}\left[\mathcal{M}_{H_{1}}(p_{1},p_{2},p_{3})\not{p}_{3}\gamma^{\mu}\bar{\epsilon}_{\mu}\bar{u}(p_{1})\right]\,,$
(50)

and similarly for the emission of soft anti-quark ($p_{2}\to 0$)

$\displaystyle i\mathcal{M}_{2,q}\,=\,$
$\displaystyle\frac{ig_{s}t_{a}}{2p_{2}\cdot
p_{3}}\left[\mathcal{M}_{H_{2}}(p_{1},p_{2},p_{3})\not{p}_{3}\gamma^{\mu}\bar{\epsilon}_{\mu}v(p_{2})\right]\,.$
(51)

Here the hard scattering matrix elements $\mathcal{M}_{H_{1}}$ and
$\mathcal{M}_{H_{2}}$ are defined to contain all the external states except
for the polarization vector $\epsilon_{\mu}$ and spinors for soft fermion
emission. The matrix element squared after the sum (average) over the final
(initial) state spins and colours

$\displaystyle\overline{\sum}$
$\displaystyle|\mathcal{M}_{q}|^{2}\,=\,\overline{\sum}|{\mathcal{M}}_{\text{rem}}(x_{1},x_{2})|^{2}\,$
$\displaystyle=\,8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}\left(\frac{1-x_{2}}{1-x_{1}}+\frac{1-x_{1}}{1-x_{2}}\right)\,,$
(52)

which indeed reproduces eq. (49). Thus eq. (49) features a clean separation of
the soft quark and anti-quark contributions from the next-to-soft gluon
contributions. Soft quark emissions contribute to the LL terms at NLP [46].
Their contribution to the thrust distribution from region I is

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{I}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}\int_{0}^{1-x_{2}}$
$\displaystyle dx_{1}\,\,\delta(T-x_{1})$ $\displaystyle\times$
$\displaystyle\left(\frac{1-x_{1}}{1-x_{2}}+\frac{1-x_{2}}{1-x_{1}}\right)\,\,,$
(53)

which gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\left(\frac{1}{2\tau}-2+\frac{3\tau}{2}-\tau\log\tau+\mathcal{O}(\tau^{2})\right).$
(54)

The upper and lower limit components are

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,u}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}\frac{1}{2\tau}+-\frac{\tau}{2}-\tau\log\tau+\mathcal{O}(\tau^{2})\bigg{)},$
(55)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{I,l}}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}2-2\tau+\mathcal{O}(\tau^{2})\bigg{)}\,.$ (56)

In this region, the hard gluon can become collinear to the anti-quark, and the
contribution from this kinematical configuration is captured by the upper
limit of $x_{2}$, which yields the NLL at LP. At the same time, the lower
limit captures the NLL at NLP. Region II again gives a contribution identical
to eq. (54). Region III gives

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dT}{\Bigg{|}_{\text{III}}}\,=\,\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{2}$
$\displaystyle\int_{0}^{1-x_{2}}dx_{1}$
$\displaystyle\times\left(\frac{1-x_{1}}{1-x_{2}}+\frac{1-x_{2}}{1-x_{1}}\right)\,\,$
$\displaystyle\times\delta(T+x_{1}+x_{2}-2),$ (57)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}-2-2\log\tau+$ $\displaystyle
2(1+\log\tau)\tau$ $\displaystyle+\mathcal{O}(\tau^{2})\bigg{)}.$ (58)

The contributions from upper and lower limits are

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,u}}}$
$\displaystyle\,=\,$
$\displaystyle\frac{2\alpha_{s}}{3\pi}\bigg{(}-2-\log\tau-\tau\log\tau+\mathcal{O}(\tau^{2})\bigg{)},$
(59)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{III,l}}}$
$\displaystyle\,=\,$
$\displaystyle\frac{2\alpha_{s}}{3\pi}\bigg{(}\log\tau+(-2-\log\tau)\tau+\mathcal{O}(\tau^{2})\bigg{)}.$
(60)

As the upper limit corresponds to soft quark emission and the lower limit to
soft anti-quark contributions, one finds LL contributions at NLP from both
limits. The remained matrix element squared in region III reproduces the
missing LL contributions at NLP in eq. (47), which the shifted kinematics
methods do not capture.

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}{\Bigg{|}_{\text{rem}}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}\frac{1}{\tau}-6-2\log\tau+5\tau\bigg{)}.$
(61)

These are, as expected, the terms that were missing from eq. (47). It is now
interesting to assess the quality of the shifted kinematics approximation.

###  III.3 Numerical assessment of the shifted kinematics approximations for
thrust

The shifted kinematics approximation at the cross-section level changes the
Born cross-section $\sigma_{0}(s)$ in eq. (15) by replacing $s\to
s(1-\tau)^{-1}$

$\displaystyle\sigma_{0}\left(\frac{s}{1-\tau}\right)\,=\,\sigma_{0}(s)(1-\tau).$
(62)

The LP term of thrust distribution computed under the formalism of shifted
kinematics in eq. (47) is

$\displaystyle\frac{d\sigma}{d\tau}\Bigg{|}_{\text{LP}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\left(\frac{-4-4\log\tau}{\tau}\right)\sigma_{0}(s),$
(63)

when multiplied by the _shifted_ born cross-section in eq. (62) yields

$\displaystyle\sigma_{0}$
$\displaystyle\left(\frac{s}{1-\tau}\right)\,\times\frac{d\sigma}{d\tau}\Bigg{|}_{\text{LP}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\biggl{(}\frac{-4-4\log{\tau}}{\tau}+4+4\log{\tau}\biggr{)}\sigma_{0}(s)\,.$
(64)

which reproduces eq. (47). In fig. (3a) we compare the thrust distribution as
computed from the shifted kinematics approximation in eq. (47), up to NLP,
with the exact computation in eq. (28).111We have taken $\alpha_{s}=0.1193$ at
$172$ GeV from LEP2 from two loop result in [97], the numerical results are
also given in [98].

(a)  (b)

Figure 3:  In (a), we plot comparison of the thrust distribution. We show the
exact result (solid red curve) in eq. (28), the shifted approximation (dashed
blue curve) up to NLP terms given in eq. (47). We also plot the exact LP term
(dotted purple curve). In (b), we show $X(\tau)$ in eq. (65) (solid blue
curve) and $Y(\tau)$ in eq. (66) (purple dashed curve) vs. $\tau$.

In the small $\tau$ region, the LP terms alone provide an excellent
approximation to the exact result. However, as $\tau$ increases, we notice
that the shifted kinematic curve approximates the exact results better. The
reason LP terms are such an excellent approximation for thrust is the small
coefficient of the NLP LL term; as such the NLL at LP for thrust dominates the
LL term at NLP for small values of $\tau$. This is, as we shall see, in
contrast to the $C$-parameter, the LL at NLP dominates the NLL term at LP [99,
100] in the dijet limit.

In fig. (3b) we provide two different ratio plots that involve expressions of
thrust distribution computed from both of the approaches viz. the exact
approach (28) and the shifted approach (47). We define $X(\tau)$ and $Y(\tau)$
as

$\displaystyle X(\tau)\,=\,$
$\displaystyle\frac{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Big{|}_{\text{NLO}}-\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Big{|}_{\text{shift-
NLP}}}{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Big{|}_{\text{NLO}}},$
(65) $\displaystyle Y(\tau)\,=\,$
$\displaystyle\frac{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Big{|}_{\text{NLO-
LP}}}{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Big{|}_{\text{NLO}}}.$ (66)

As is evident from their expressions, $X(\tau)$ measures the accuracy of the
shifted approximation, whereas $Y(\tau)$ measures the importance of NLP terms
(and beyond) compared to the LP terms. The plot of $X(\tau)$ shows that the
shifted kinematics method approximates the exact result well near the dijet
limit. The plot of $Y(\tau)$ shows that the LP contribution is dominant near
$\tau\to 0$. As $\tau$ increases, the denominator falls faster than the
numerator due to the presence of the NLP terms in addition to the LP
contribution, and thus the ratio rises above one.

##  IV $C$-parameter

In the previous section, we compared the shifted kinematic approximation of
the thrust distribution with the exact NLO result, to understand the origin of
NLP corrections. Here we perform the same comparison for the more complicated
$C$-parameter distribution. The $C$-parameter for massless particles [101,
102, 103, 104] is defined as

$\displaystyle C\,=\,3-\frac{3}{2}\sum_{i,\ j}\frac{\left(p_{i}\cdot
p_{j}\right)^{2}}{\left(p_{i}\cdot q\right)\left(p_{j}\cdot q\right)}\,,$ (67)

where $p_{i}$ is the four-momentum of $i$-th particle, the total four-momentum
is given by $q=\sum_{i}p_{i}$, and the sums over $i$ and $j$ run over all the
particles present in the final state. The minimum value taken by $C$ equals
$0$ (for a zero-radius dijet event), and the maximum value is $1$ (for an
isotropic event). Here we consider a three-body final state, for which the
maximal value of $C$ is $3/4$; the value $C=1$ can only be obtained if more
than three particles are present in the final state. For the three-body final
state in fig. (1), it takes the form

$\displaystyle
C\,=\,\frac{6\left(1-x_{1}\right)\left(1-x_{2}\right)\left(1-x_{3}\right)}{x_{1}\
x_{2}\ x_{3}}\,,$ (68)

where $x_{i}$ are the energy fractions given in eq. (9). The $C$-parameter has
a critical point at $C=3/4$, which gives rise to a Sudakov shoulder [105].
However, these first occur at second order in $\alpha_{s}$, where four-
particle final states are possible, hence our study does not encounter them.
Henceforth we work with a rescaled definition of the $C$-parameter

$\displaystyle
c\,=\,\frac{C}{6}\,=\,\frac{\left(1-x_{1}\right)\left(1-x_{2}\right)\left(1-x_{3}\right)}{x_{1}\
x_{2}\ x_{3}}\,,$ (69)

for which the range is $0<c<1/8$. Note that the definition of the
$c$-parameter does not involve the selection of a special axis, which thereby
distinguishes it from a group of other event shapes variables such as thrust,
jet broadening, jet mass, and angularities.

###  IV.1 The $c$-parameter distribution at NLO

The $c$-parameter distribution is defined as

$\displaystyle\frac{d\sigma}{dc}\,=\,\frac{1}{2s}\int d\Phi_{3}\
\overline{\sum}\ |\mathcal{M}(x_{1},x_{2})|^{2}\
\delta\left(c(x_{1},x_{2})-c\right),$ (70)

where $\overline{\sum}\ |\mathcal{M}(x_{1},x_{2})|^{2}$ and $d\Phi_{3}$ are
given in eq. (12) and eq. (13) respectively. The normalized expression at NLO
takes the form

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\,=\,\
\frac{2\alpha_{s}}{3\pi}\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}$ $\displaystyle
dx_{2}\ \frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}$
$\displaystyle\times\delta\left(c(x_{1},x_{2})-c\right).$ (71)

The integrations over energy fractions are considerably more involved than for
thrust due to the rational polynomial form in eq. (69). It is advantageous to
convert our kinematic variables to $(y,z)$ [100] where

$\displaystyle y\,=\,$ $\displaystyle 2-x_{1}-x_{2},$ $\displaystyle z\,=\,$
$\displaystyle\frac{1-x_{2}}{y}\,,$ (72)

with Jacobian $J(z,y)=y$. Note that $y$ is in fact the gluon energy fraction.
The expressions for the matrix element and $c$-parameter in terms of the new
variables read

$\displaystyle\overline{\sum}\ |{\mathcal{M}(y,z)}|^{2}$
$\displaystyle\,=\,8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}\left(\frac{2+y(y-2yz(1-z)-2)}{y^{2}z(1-z)}\right),$
(73)

and

$\displaystyle c(y,z)\,=\,\frac{(1-y)(1-z)yz}{(1-y(1-z))(1-yz)}\,.$ (74)

The $x_{1}\leftrightarrow x_{2}$ symmetry in the matrix element squared in eq.
(12) now appears as $z\leftrightarrow(1-z)$ symmetry in eq. (73). Substitution
into eq. (70) yields

$\displaystyle\frac{1}{\sigma_{0}(s)}$
$\displaystyle\frac{d\sigma}{dc}\Bigg{|}_{\text{NLO}}$
$\displaystyle\,=\,\frac{2\alpha_{s}}{3\pi}\int^{1}_{0}dy\ \int^{1}_{0}dz\
\frac{2+y(y-2yz(1-z)-2)}{yz(1-z)}\ $ $\displaystyle\hskip
85.35826pt\times\delta\left(c(y,z)-c\right).$ (75)

To determine the limits of the $z$-integration, we use the phase space in fig.
(2) as follows. From eq. (72), it is evident that $z$ will attain its lowest
value when $x_{2}=1$, yielding $z=0$ as the lower limit of $z$. Similarly, $z$
will attain its largest value when $x_{2}$ and $y$ both are smallest. However
$x_{2}$ and $y$ cannot be zero at the same time. Thus with $x_{2}\to 0$ (and
thus $y\to 1$), one finds that $z=1$ is the upper limit. The limits of the
$y$-integration can be found from the $\delta$-function constraint and the
limits of the $z$-integration. We first rewrite eq. (75) in the form

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{NLO}}\,=\,\frac{2\alpha_{s}}{3\pi}\int^{1}_{0}dy\int^{1}_{0}dz\frac{2(y(z-1)+1)^{2}(yz-1)^{2}(y(2y(z-1)z+y-2)+2)}{(y-1)^{2}y^{2}(z-1)z(2z-1)}\biggl{(}\delta(z-z_{1})+\delta(z-z_{2})\biggr{)}\,,$
(76)

where we used the argument of the $\delta$-function in eq. (75) and it has
following two roots

$\displaystyle z_{1,2}$
$\displaystyle\,=\,\frac{1}{2}\left(1\pm\frac{\sqrt{y(y(1+c)-1)(c(y-2)^{2}+y(y-1))}}{y(y(1+c)-1)}\right)\,.$
(77)

Here $z_{1}(z_{2})$ corresponds to the $+(-)$ solution. Note that as $c\to 0$,
we have $z_{1}\to 1$ and $z_{2}\to 0$ as expected. The argument of the
$\delta$-function in eq. (75) is a complicated function of $y$ and $c$. As the
integral has a symmetry $z\leftrightarrow(1-z)$, the integral over $z$ in eq.
(76) equals twice the integral between $z=1/2$ and the upper limit, where only
$\delta(z-z_{1})$ is relevant. After the $z$ integration the limits of $y$
change to $(y_{1},y_{2})$, see below in eq. (81). We have now The above
expressions are smooth functions of $z$, as can be seen from fig. (4).

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{\rm{d}\sigma}{\rm{d}c}\Bigg{|}_{\text{NLO}}\,=\,\frac{2\alpha_{s}}{3\pi}\int^{y_{2}}_{y_{1}}dy\frac{2(1-y)\left(y\left(c(y-2)^{2}+(y-3)y+4\right)-2\right)}{c(cy+y-1)\sqrt{y(cy+y-1)\left(c(y-2)^{2}+(y-1)y\right)}}\,.$
(78)

(a)  (b)

Figure 4: Behavior of $y_{1}(z)$ and $y_{2}(z)$ as functions of $z$ at
$c\,=\,0.01$.

To find the integration limits of $y_{1}$ and $y_{2}$ we proceed as follows.
First, going back to eq. (75), the $\delta$-function can be also used to solve
for $y$, for which there are two solutions,

$\displaystyle y_{1}(z)$
$\displaystyle\,=\,\frac{2c}{\left[c^{2}(1-2z)^{2}+2c(z-1)z+(z-1)^{2}z^{2}\right]^{1/2}+c-z^{2}+z}\,,$
(79) $\displaystyle y_{2}(z)$
$\displaystyle\,=\,-\frac{\left[c^{2}(1-2z)^{2}+2c(z-1)z+(z-1)^{2}z^{2}\right]^{1/2}+c-z^{2}+z}{2(c+1)(z-1)z}\,.$
(80)

Clearly, there are extrema for both expressions at $z=1/2$. The second
derivative of $y_{1}(z)$ ($y_{2}(z)$) at $z=1/2$ is positive (negative). Thus
the expression of $y_{1}(z)$ in eq. (79) is the lower limit of $y$ and
$y_{2}(z)$ in eq. (80) is the upper limit of $y$. Substituting $z=1/2$ in eqs.
(79) and (80) we find

$\displaystyle y_{1}\,=\,\frac{1+4c-\sqrt{1-8c}}{2(1+c)}\,,\qquad
y_{2}\,=\,\frac{1+4c+\sqrt{1-8c}}{2(1+c)}\,.$ (81)

(a)  (b)

Figure 5: Lower and upper limits $y_{1}(c)$ and $y_{2}(c)$ respectively as
functions of $c$.

The behavior of lower and upper limits are shown in figs. (5a) and (5b). For
thrust, the integration limits of the final integration in eq. (17) had simple
expressions ($2\tau$ and $1-\tau$ ), and their relation with the respective
kinematic configurations were readily visible in fig. (2). To establish the
relation between kinematic configurations and the integration limits $y_{1}$
and $y_{2}$ for the case of $c$-parameter, we expand their expressions around
$c\,=\,0$ (as $c\to 0$ is the dijet limit),222Note that while performing the
final integration in eq. (78) we use the exact expressions of limits as given
in eq. (81).

$\displaystyle y_{1}$ $\displaystyle\,=\,4c+\mathcal{O}(c^{3}),$
$\displaystyle y_{2}$ $\displaystyle\,=\,1-c-3c^{2}+\mathcal{O}(c^{3}).$ (82)

In the dijet limit, the upper limit of $y$ corresponds to either $(i)$ a back-
to-back hard gluon and quark–anti-quark collinear pair $(ii)$ soft quark (or
soft anti-quark) emission, or $(iii)$ the hard gluon collinear to the quark or
the anti-quark. The lower limit in dijet limit corresponds to a soft gluon
emission and a back-to-back quark and anti-quark. Thus these upper and lower
limits correspond to different points in the phase space, as expected. The
integration over $y$ in eq. (78) produces incomplete elliptic integrals of
three types, each with somewhat involved arguments and coefficients. After
conversion to their so-called complete counterparts333More information about
these transformations of elliptic integrals and their analytical properties
can be found in [106]., and carefully collecting their coefficients, the final
expression can be organized in a compact form [100] as follows

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{NLO}}\,=\,\frac{2\alpha_{s}}{3\pi}\bigg{(}e(c)\
{\rm{E}}[m_{1}(c)]+p(c)\ {\Pi}[n_{1}(c),m_{1}(c)]+k(c)\
{\rm{K}}[m_{1}(c)]\bigg{)}\,,$ (83)

where ${\rm{E}}$, ${\Pi}$, and ${\rm{K}}$ are the complete elliptic integrals
of the first, second, and third kind. The integral representation of these
elliptic integrals are as following

$\displaystyle{\rm{E}}[m]$ $\displaystyle\,=\,\int^{1}_{0}dt\
\frac{\sqrt{1-m^{2}t^{2}}}{\sqrt{1-t^{2}}}\,,$ $\displaystyle{\Pi}[n,m]$
$\displaystyle\,=\,\int^{1}_{0}\frac{dt}{(1-nt^{2})\sqrt{(1-t^{2})(1-m^{2}t^{2})}}\,,$
$\displaystyle{\rm{K}}[m]$
$\displaystyle\,=\,\int^{1}_{0}\frac{dt}{\sqrt{(1-t^{2})(1-m^{2}t^{2})}}\,.$
(84)

Here $m$, and $n$ are the _parameter_ and _characteristic_ of the elliptic
integrals, respectively. The arguments of these elliptic integrals in eq. (83)
have the following form

$\displaystyle n_{1}(c)$
$\displaystyle\,=\,\frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c}\,,$ $\displaystyle
m_{1}(c)$ $\displaystyle\,=\,\frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^{2}}\,.$
(85)

The behavior of these arguments is shown in fig. (6). We computed eq. (83)
throughout with a massless on-shell gluon and it agrees with the
characteristic function derived in [100] for finite gluon virtuality $\xi$, in
the limit $\xi\to 0$.

(a)  (b)

Figure 6: Behavior of arguments $n_{1}(c)$ and $m_{1}(c)$.

Notice that the arguments have monotonic behavior. The elliptic integrals in
eq. (83) have the following asymptotic behavior as $c\to 0$

$\displaystyle{\rm{E}}[m_{1}(c)]\,=\,$ $\displaystyle
1+\mathcal{O}(c^{3}\log{c})\,,$
$\displaystyle{\rm{\Pi}}[n_{1}(c),m_{1}(c)]\,=\,$
$\displaystyle-\frac{\log{c}}{8c^{2}}-\frac{1+\log{c}}{4c}+\mathcal{O}(c^{0}\log{c})\,,$
(86) $\displaystyle{\rm{K}}[m_{1}(c)]\,=\,$
$\displaystyle-\frac{3\log{c}}{2}+\mathcal{O}(c^{3}\log{c})\,.$

The expressions for the coefficients appearing in eq. (83) are

$\displaystyle e(c)\,=\,$
$\displaystyle\frac{-3(1+2c)\sqrt{1-4c(1+2c)+\sqrt{1-8c}}}{\sqrt{2}c(1+c)^{3}}\,,$
$\displaystyle p(c)\,=\,$
$\displaystyle\frac{\sqrt{2}(2+c+2c^{2})(1-\sqrt{1-8c})^{2}}{c(1+c)^{3}\sqrt{1-4c(1+2c)+\sqrt{1-8c}}}\,,$
(87) $\displaystyle k(c)\,=\,$
$\displaystyle\frac{4\sqrt{2}(1-2c(2+c))}{(1+c)^{3}\sqrt{1-4c(1+2c)+\sqrt{1-8c}}}\,.$

Their expansions around $c=0$ are

$\displaystyle e(c)\,=\,$
$\displaystyle-\frac{3}{c}+9+12c+36c^{2}+\mathcal{O}(c^{3})\,,$ $\displaystyle
p(c)\,=\,$ $\displaystyle 32c+112c^{2}+\mathcal{O}(c^{3})\,,$ (88)
$\displaystyle k(c)\,=\,$ $\displaystyle 4-20c+48c^{2}+\mathcal{O}(c^{3})\,.$

The coefficient $e(c)$, when multiplied by the complete elliptic integral of
the second kind $E[m_{1}(c)]$ yields NLL terms at LP, the coefficient $p(c)$,
multiplied with the complete elliptic integral of the third kind
${\Pi}[n_{1}(c),m_{1}(c)]$ produces LL and NLL contributions at both LP and
NLP, while the coefficient $k(c)$ together with the complete elliptic integral
of the first kind $K[m_{1}(c)]$ also produces LL terms at NLP. From eqs. (85),
(86) and (88) the $c$-parameter distribution at NLO for small-$c$ reads,

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{NLO}}\,$
$\displaystyle=\,\frac{2\alpha_{s}}{3\pi}\biggl{(}\frac{-3-4\log{c}}{c}+1-28\log{c}+\mathcal{O}(c)\biggr{)}.$
(89)

We have computed the $c$-parameter distribution here without approximating the
event shape variable or the matrix element squared, similar to what we did for
thrust in eq. (29). The above result, which agrees with [99, 100], contains
the contributions from all regions of the phase space. By comparing eq. (89)
and eq. (29) we observe that the LP terms for both event shape variables have
an identical structure in the dijet limit. Further similarities between these
event-shape variables are discussed in [107, 108, 109, 110, 111]. The
expression eq. (89) does not fully expose the relation to different
kinematical configurations. To do so, we list the upper and lower limit
contributions of the $y$ integral separately. When computing the contributions
from the upper and lower limits separately we have observed that it is only
the coefficients $e(c),p(c)\,\text{and}\,k(c)$ of the elliptic integrals that
differ, while the general form of the elliptic integrals remains the same as
in eq. (83). From the upper limit of eq. (78) we have, for small-$c$,

$\displaystyle e_{\text{u}}(c)$
$\displaystyle\,=\,-\frac{3}{c}+9+12c+36c^{2}+\mathcal{O}(c^{3})\,,$
$\displaystyle p_{\text{u}}(c)$ $\displaystyle\,=\,0\,,$ (90) $\displaystyle
k_{\text{u}}(c)$ $\displaystyle\,=\,-12-24c-108c^{2}+\mathcal{O}(c^{3})\,,$

For the lower limit, the behavior reads

$\displaystyle e_{\text{l}}(c)$ $\displaystyle\,=\,0\,,$ $\displaystyle
p_{\text{l}}(c)$
$\displaystyle\,=\,-32c-112c^{2}-912c^{3}+\mathcal{O}(c^{7/2})\,,$ (91)
$\displaystyle k_{\text{l}}(c)$
$\displaystyle\,=\,-16-4c-156c^{2}+\mathcal{O}(c^{3})\,.$

Since $p_{u}(c)\,=\,e_{l}(c)\,=\,0$, no LL terms at LP derive from the upper
limit, nor do we find NLL terms at LP from the lower limit. Collecting the
upper and lower limit contributions for the $c$-parameter distribution in the
dijet limit leads to

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{u}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\left(-\frac{3}{c}+9(1+2\log
c)+\mathcal{O}(c)\right),$ (92)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{l}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\left(\frac{4\log
c}{c}+2(4+23\log c)+\mathcal{O}(c)\right).$ (93)

Thus the upper limit of $y$ in the dijet limit corresponds to various
kinematic configurations involving a hard gluon such as hard gluon being back-
to-back with quark (anti-quark) and collinear to anti-quark (quark), which
produces NLL terms at LP. As discussed the upper limit also corresponds to the
soft quark/anti-quark emissions, LL and NLL terms at NLP also arise from this
limit. The lower limit of $y$ corresponds to kinematic configurations
involving a soft gluon and thus yields LL terms at both LP and NLP. No NLL
terms at LP are generated here, although there is a NLL contribution at NLP.

###  IV.2 Next-to-leading power corrections to $c$-parameter from shifted
kinematics

We next compute the $c$-parameter distribution using the shifted kinematics
method, and again assess to what extent LP and NLP terms in the exact NLO
calculation are reproduced. The approximation is

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{shift}}\,=\,\frac{1}{2s}\int
d\Phi_{3}\ \overline{\sum}$
$\displaystyle|\mathcal{M}_{\text{shift}}(x_{1},x_{2})|^{2}\ $
$\displaystyle\times\delta\left(c(y,z)-c\right)\,.$ (94)

The shifted matrix element in eq. (33), when written in terms of the
transformed variables ($y,z$), takes the form

$\displaystyle\overline{\sum}$
$\displaystyle|\mathcal{M}_{\text{shift}}(y,z)|^{2}$
$\displaystyle\,=\,8(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}\left(\frac{2(y-1)}{y^{2}(z-1)z}\right).$
(95)

Again, we make no approximation to the event shape definition itself. We then
proceed in the same manner as with the exact matrix element. We have

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{shift}}$
$\displaystyle\,=\,\
\frac{2\alpha_{s}}{3\pi}\int^{1}_{0}dy\int^{1}_{0}dz\frac{2(1-y(1-z))^{2}(1-yz)^{2}}{(1-y)^{2}y^{2}z(z-1)(2z-1)}\bigg{(}\delta(z-z_{1})+\delta(z-z_{2})\bigg{)}\,,$
(96)

where $z_{1}$ and $z_{2}$ are provided in eq. (77). After the $z$ integration
we have

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{shift}}\,=\,\
\frac{2\alpha_{s}}{3\pi}\int^{y_{2}}_{y_{1}}dy\frac{4(y-1)^{2}}{c\sqrt{y\left(y+cy-1\right)\left(c(y-2)^{2}+(y-1)y\right)}}\,.$
(97)

The result of $y$ integration can again be organized in a manner similar to
eq. (83) as

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{shift}}\,=\,\frac{2\alpha_{s}}{3\pi}\bigg{(}e_{\text{s}}(c)E[m_{1}(c)]+p_{\text{s}}(c)\Pi[n_{1}(c),m_{1}(c)]+k_{\text{s}}(c)K[m_{1}(c)]\bigg{)}\,,$
(98)

A comparison of eq. (97) with eq. (78) shows a significant simplification of
the integrand, where the subscript s on the coefficients on the right
indicates the shifted kinematics method. The elliptic integrals and their
arguments are given in eqs. (86) and (85) respectively.

The coefficients in eq. (98) do change from the exact result, and we find

$\displaystyle k_{\text{s}}(c)\,=\,$
$\displaystyle-\frac{\sqrt{2-2\sqrt{1-8c}-8c-16c^{2}}}{c^{3/2}(1+c)^{5/2}}\,,$
$\displaystyle e_{\text{s}}(c)\,=\,$ $\displaystyle-\frac{(1+\sqrt{1-8c}-4c)\
\sqrt{1-\sqrt{1-8c}-4c-8c^{2}}}{2\sqrt{2}c^{5/2}(1+c)^{5/2}}\,,$ (99)
$\displaystyle p_{\text{s}}(c)\,=\,$ $\displaystyle\frac{(1-\sqrt{1-8c}-4c)\
\sqrt{1-\sqrt{1-8c}-4c-8c^{2}}}{\sqrt{2}c^{5/2}(1+c)^{5/2}}\,.$

Their small-$c$ behavior is

$\displaystyle e_{\text{s}}(c)\,=\,$
$\displaystyle-\frac{4}{c}+16-4c+72c^{2}+\mathcal{O}(c^{3})\,,$ $\displaystyle
p_{\text{s}}(c)\,=\,$ $\displaystyle 32c+128c^{2}+\mathcal{O}(c^{3})\,,$ (100)
$\displaystyle k_{\text{s}}(c)\,=\,$ $\displaystyle-8c+\mathcal{O}(c^{3})\,.$

Expanding around $c=0$, the $c$-parameter distribution from shifted kinematics
approximation reads

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{shift}}\,$
$\displaystyle=\,\
\frac{2\alpha_{s}}{3\pi}\bigg{(}\frac{-4-4\log{c}}{c}+8-24\log{c}+\mathcal{O}(c)\bigg{)}.$
(101)

For better insight, it is again useful to examine separately the contributions
of the upper and lower of the $y$ integral. The small-$c$ behavior of the
coefficients of the elliptic integrals appearing from the upper limit of eq.
(97) is

$\displaystyle e_{\text{su}}(c)$
$\displaystyle\,=\,-\frac{4}{c}+16-4c+72c^{2}+252c^{3}+\mathcal{O}(c^{7/2}),$
$\displaystyle p_{\text{su}}(c)$ $\displaystyle\,=\,0,$ (102) $\displaystyle
k_{\text{su}}(c)$
$\displaystyle\,=\,-16-16c-144c^{2}-688c^{3}+\mathcal{O}(c^{7/2})\,.$

For the lower limit we have

$\displaystyle e_{\text{sl}}(c)$ $\displaystyle\,=\,0,$ $\displaystyle
p_{\text{sl}}(c)$
$\displaystyle\,=\,-32c-128c^{2}-928c^{3}+\mathcal{O}(c^{7/2}),$ (103)
$\displaystyle k_{\text{sl}}(c)$
$\displaystyle\,=\,-16-8c-144c^{2}-616c^{3}+\mathcal{O}(c^{7/2}).$

Note that the $\mathcal{O}(c^{3})$ terms contribute at NNLP accuracy. We now
have

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{su}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\left(-\frac{4}{c}+8(2+3\log
c)+\mathcal{O}(c)\right),$ (104)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{sl}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\left(\frac{4\log c}{c}+8(1+6\log
c)+\mathcal{O}(c)\right).$ (105)

From the above two expressions, we observe that the lower (upper) limit fully
captures the LL (NLL) at LP, while the LL and NLL at NLP receive contributions
from both limits. The results for the $c$-parameter distributions at NLO up to
NLP, obtained from the exact and shifted kinematics computation, are given in
eqs. (89) and (101), respectively. The integrand in the shifted kinematics
method was considerably simpler than for the exact computation. At LP, the
method reproduced the LL term. The NLL term at LP was not fully reproduced
because the contribution from the hard-collinear gluon is absent. Similarly,
not all of the LL at NLP is captured, due to the absence of from soft quark
and soft anti-quark contributions. Recall that the shifted kinematics method
does not account for soft quark contributions, which we computed separately
using soft quark emission vertices in section III.2.

We again examine the remaining contributions to the $c$-parameter
distribution, such as those from the soft quark, soft anti-quark, and hard-
collinear gluon configurations. This allows the mapping of all of the
contributions to the $c$-parameter from various regions of the phase space.
When expressed in terms of the transformed $(y,z)$ variables the remainder
matrix element as described in eq. (49) reads

$\displaystyle\overline{\sum}|{\mathcal{M}}_{\text{rem}}(y,z)|^{2}\,=\,8$
$\displaystyle(e^{2}e_{q})^{2}N_{c}C_{F}g_{s}^{2}\frac{1}{3Q^{2}}$
$\displaystyle\times\biggl{(}\frac{1}{z}+\frac{1}{1-z}-2\biggr{)}.$ (106)

The expression for $c$-parameter distribution using the above matrix element
squared is

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{rem}}=\frac{2\alpha_{s}}{3\pi}\int^{1}_{0}dy\int^{1}_{0}dz\frac{1}{z(1-z)}\Big{(}\delta(z-z_{1})+\delta(z-z_{2})\Big{)}\frac{-y(y-1)^{2}\left(1-2z(1-z)\right)}{(cy+y-1)\sqrt{y(cy+y-1)\left(c(y-2)^{2}+(y-1)y\right)}},$
(107)

with $z_{1}$ and $z_{2}$ are given in eq. (77). After performing the $z$
integration we are left with

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{rem}}\,=\,\frac{2\alpha_{s}}{3\pi}\int^{y_{1}}_{y_{2}}dy\
\frac{-2y(y-1)(c((y-2)y+2)+(y-1)y)}{c(cy+y-1)\sqrt{y(cy+y-1)\left(c(y-2)^{2}+(y-1)y\right)}}\,.$
(108)

with $y_{1}$ and $y_{2}$ given in eq. (81). The outcome takes again the form
as in eq. (83), somewhat more complicated than either of the two
approximations. The result can be written as

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{rem}}\,=\,\frac{2\alpha_{s}}{3\pi}\bigg{(}e_{\text{r}}(c)\
E[m_{1}(c)]+p_{\text{r}}(c)\ \Pi[n_{1}(c),m_{1}(c)]+k_{\text{r}}(c)\
K[m_{1}(c)]\bigg{)},$ (109)

The small-$c$ behavior of the coefficients reads

$\displaystyle e_{r}(c)$
$\displaystyle\,=\,\frac{1}{c}-7+16c-36c^{2}+9c^{3}+\mathcal{O}(c^{7/2})\,,$
$\displaystyle p_{r}(c)$
$\displaystyle\,=\,-16c-16c^{3}+\mathcal{O}(c^{7/2})\,,$ (110) $\displaystyle
k_{r}(c)$ $\displaystyle\,=\,4-12c+48c^{2}+16c^{3}+\mathcal{O}(c^{7/2})\,,$
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{rem}}\,=\,\frac{2\alpha_{s}}{3\pi}\left(\frac{1}{c}-7-4\log
c+\mathcal{O}(c)\right)\,.$ (111)

Here we indeed see contributions from the hard-collinear gluon, the soft
quark, and anti-quark configurations, which yield NLL terms at LP (hard-
collinear gluon) as well as LL terms at NLP (soft quark/anti-quark)
respectively. Combining these contributions with the outcome of shifted
kinematics approximation eq. (101) allows us to fully map the distributions of
the $c$-parameter to the different kinematical configurations.

Let us finally break the contributions from remaining term also down into
upper and lower limit components. The coefficients from the upper limit of eq.
(108) yield

$\displaystyle e_{\text{ru}}(c)$
$\displaystyle\,=\,\frac{1}{c}-7+16c-36c^{2}+9c^{3}+\mathcal{O}(c^{7/2})\,,$
$\displaystyle p_{\text{ru}}(c)$ $\displaystyle\,=\,0\,,$ (112) $\displaystyle
k_{\text{ru}}(c)$
$\displaystyle\,=\,4-8c+36c^{2}+64c^{3}+\mathcal{O}(c^{7/2})\,.$

while for the lower limit, we have

$\displaystyle e_{\text{rl}}(c)$ $\displaystyle\,=\,0,$ $\displaystyle
p_{\text{rl}}(c)$ $\displaystyle\,=\,16c+16c^{3}+\mathcal{O}(c^{7/2}),$ (113)
$\displaystyle k_{\text{rl}}(c)$
$\displaystyle\,=\,4c-12c^{2}+48c^{3}+\mathcal{O}(c^{7/2}).$

Using eqs. (85), (86) and above two expressions, the contributions to the
$c$-parameter distribution from the upper and lower limit respectively are

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{ru}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\left(\frac{1}{c}-7-6\log
c+\mathcal{O}(c)\right),$ (114)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{rl}}$
$\displaystyle\,=\,\ \frac{2\alpha_{s}}{3\pi}\bigg{(}-2\log
c+\mathcal{O}(c)\bigg{)}.$ (115)

The missing LL terms at NLP have been generated here. The upper limit
contributes to the missing NLL term at LP (related to hard-collinear gluon
emission) and NLP. Both limits contain LL terms at NLP (soft (anti-)quark
emission).

###  IV.3 Numerical assessment of the shifted kinematics approximations for
$c$-parameter

(a)  (b)

Figure 7:  In (a), we plot comparison of the $c$-parameter distribution. We
show the exact result (solid red curve) in eq. (83), the shifted approximation
(dashed blue curve) up to NLP terms given in eq. (101). We also plot the exact
LP term given in eq. (89) (dotted purple curve). In (b), we show behavior of
$X(c)$ (solid blue curve) and $Y(c)$ (purple dashed curve) from eqs. (116) and
(117) respectively vs. $c$.

We analyze our $c$-parameter results numerically. In the small-$c$ limit, the
effectiveness of the shifted kinematics method is evident from fig. (7a), the
shifted kinematics result provides a better approximation to the exact result,
than the exact result taken up to LP terms only, even though we did not
generate all the dominant terms at LP and NLP from this approximation. The
$c$-parameter distribution in fig. (7a) has a more pronounced departure of the
LP terms from the exact result than the thrust distribution in fig. (3a). This
is due to the $c$-parameter distribution having a larger LL coefficient at NLP
than thrust. We define $X(c)$ and $Y(c)$ in similar fashion to $X(\tau)$ and
$Y(\tau)$ in eqs. (65) and (66) as

$\displaystyle X(c)$
$\displaystyle\,=\,\frac{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Big{|}_{\text{NLO}}-\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Big{|}_{\text{shift-
NLP}}}{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Big{|}_{\text{NLO}}},$ (116)
$\displaystyle Y(c)$
$\displaystyle\,=\,\frac{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Big{|}_{\text{NLO-
LP}}}{\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Big{|}_{\text{NLO}}}\,,$ (117)

and exhibit them in fig. (7b). $X(c)$ exhibits similar behavior as the
analogous case for thrust in fig. (3b), but the deviation of $X(c)$ from zero
is faster with increasing $c$. $Y(c)$ decreases sharply from unity as $c$
increases, a behavior opposite to thrust in fig. (7b). Indeed, due to the
large negative LL coefficient at NLP for the $c$-parameter, the denominator
here increases sharply. More precisely, for the $c$-parameter, the LL
coefficient at NLP is seven times that at LP, with the same sign, while for
thrust, the LL coefficient at NLP is half that of LP, with the opposite sign.

##  V Conclusions

The fixed order results in perturbation theory for massless fields contain
logarithms of the (small) ratio of energy scales, arising from soft or
collinear radiation. They occur at both leading and next-to-leading power of
this ratio. This work focuses on NLP terms appearing in the thrust and
$C$-parameter event shape distribution. We identify in exact NLO results the
origin of large logarithmic terms at LP and NLP. Subsequently we provide an
approximation of these results using the recent shifted kinematics method,
designed to capture large logarithms at NLP accuracy due to soft gluon
emission. Moreover we compute soft quark emission contributions using
corresponding effective Feynman rules.

This formalism indeed reproduces the dominant contributions at LP and NLP from
soft gluon radiation correctly for both thrust and $c$-parameter. The
remaining LL terms at NLP are reproduced the soft (anti-)quark emission, as
anticipated in [46, 112]. We were able to map the various sources of
contributions at LP and NLP, using an integral form, and its limits, for the
distributions.

Our detailed diagnosis of NLP terms in event shape variables, and
demonstration that they can be fully reproduced using a shifted kinematics
approach together with soft fermion emission vertices, should be a useful
resource towards further understanding of NLP terms, in particular those
arising from final state emissions.

###### Acknowledgements.

EL and AT would like to thank the MHRD Government of India, for the SPARC
grant SPARC/2018-2019/P578/SL, Perturbative QCD for Precision Physics at the
LHC. SM would like to thank CSIR, Govt. of India, for the SRF fellowship
(09/1001(0052)/2019-EMR-I).

##  Appendix A Transformation of incomplete elliptic integrals

In this appendix, we demonstrate how we handle the incomplete elliptic
integrals that appear in the expression of $c$-parameter distribution. The
final expression for the $c$-parameter distribution is written in a compact
manner in eq. (83), where $K$, $E$ and $\Pi$ are the complete elliptic
integrals of the first, second and third kind, respectively. However, when we
perform the integration over final variable $y$ in eqs. (78), (97) and (108),
this integration produces incomplete elliptic integrals $F$, $E$, and $\Pi$.
These incomplete elliptic integrals are later converted into complete
integrals to arrive at eq. (83), as first written in [100]. The three kinds of
incomplete elliptic integrals appearing in $c$-parameter distribution are

$\displaystyle F[\phi,m]\,\equiv\,$
$\displaystyle\int_{0}^{\phi}d\theta\frac{1}{\sqrt{1-m\sin^{2}\theta}}$
$\displaystyle\,=\,$
$\displaystyle\int_{0}^{\sin\phi}\frac{dt}{\sqrt{(1-t^{2})(1-mt^{2})}}\,,$
(118) $\displaystyle E[\phi,m]\,\equiv\,$
$\displaystyle\int_{0}^{\phi}d\theta\sqrt{1-m\sin^{2}\theta}\,$
$\displaystyle\,=\,$
$\displaystyle\,\int_{0}^{\sin\phi}dt\sqrt{\frac{1-mt^{2}}{1-t^{2}}}\,,$ (119)
$\displaystyle\hskip 19.91684pt\Pi[n,\phi,m]\,\equiv\,$
$\displaystyle\int_{0}^{\phi}d\theta\frac{1}{(1-m\sin^{2}\theta)\sqrt{1-m\sin^{2}\theta}}$
$\displaystyle\,=\,$
$\displaystyle\int_{0}^{\sin\phi}\frac{dt}{(1-nt^{2})\sqrt{(1-t^{2})(1-mt^{2})}}\,.$
(120)

Here $\phi,m$, and $n$ are called the _amplitude_ , _parameter_ , and
_characteristic_ of the elliptic integrals, respectively. Eq. (IV.1) gives
their respective complete form. The corresponding transformation into a
complete elliptic integral can be performed using the rule

$\displaystyle F[\phi,m]$ $\displaystyle\,=\,K[m],$ $\displaystyle E[\phi,m]$
$\displaystyle\,=\,E[m],$ (121) $\displaystyle\Pi[n,\phi,m]$
$\displaystyle\,=\,\Pi[n,m].$

The above transformation is only possible when the amplitude $\phi=\pi/2$. The
indefinite integration of eqs. (78), (97) and (108) results in multiple
incomplete elliptic integrals with only two unique amplitudes in their
arguments, namely $\phi_{1}(c,y)$ and $\phi_{2}(c,y)$, given by

$\displaystyle\phi_{1}(c,y)$
$\displaystyle\,=\,\left(\frac{-1+\sqrt{1-8c}-4c+8c/y}{2\sqrt{1-8c}}\right)^{1/2},$
(122)

$\displaystyle\phi_{2}(c,y)$
$\displaystyle\,=\,\left(\frac{1+\sqrt{1-8c}+4c-8c/y}{2\sqrt{1-8c}}\right)^{1/2}.$
(123)

The amplitudes $\phi_{1}(c,y)$ and $\phi_{2}(c,y)$ are present in the
arguments of all three kinds of incomplete elliptic integrals. If we directly
substitute the upper limit ($y_{2}$) after integration, then none of the
incomplete elliptic integrals with the amplitude $\phi_{1}(c,y)$ can be
reduced into complete elliptic integral since

$\displaystyle\phi_{1}(c,y)\Big{|}_{y\,=\,y_{2}}\,=\,0.$ (124)

The transformation into complete elliptic integrals holds only when
$\phi_{1}=\pi/2$ as given in eq. (A). However, the incomplete elliptic
integrals with the amplitude $\phi_{2}(c,y)$ upon substitution of upper limit
can be directly reduced into complete elliptic integral as

$\displaystyle\phi_{2}(c,y)\Big{|}_{y\,=\,y_{2}}\,=\,\frac{\pi}{2}.$ (125)

We observe a similar but opposite behavior when we substitute the lower limit
($y_{1}$) into the integration result, this time, the incomplete elliptic
integrals with the amplitude $\phi_{1}(c,y)$ can be reduced into complete
elliptic integral as

$\displaystyle\phi_{1}(c,y)\Big{|}_{y\,=\,y_{1}}\,=\,\frac{\pi}{2},$ (126)

while the elliptic integrals with amplitude $\phi_{2}(c,y)$ cannot be reduced
into complete elliptic integral as

$\displaystyle\phi_{2}(c,y)\Big{|}_{y\,=\,y_{1}}\,=\,0.$ (127)

In order to resolve the issue of transforming the incomplete elliptic
integrals into complete elliptic integrals, we modify the upper and lower
limits of $y$ integration as

$\displaystyle(y_{1},y_{2})\to(y_{1}+e,y_{2}+e)\,,$ (128)

where $e$ is an infinitesimal real off-set parameter that will be taken to
zero at the end. The relative sign of $e$ does not affect the final expression
for $c$-parameter distribution, which is independent of this parameter $e$.
When we substitute the upper limit of integration $y_{2}$ with the off-set
parameter as defined in the above expression, the amplitudes $\phi_{1}(c,y)$
and $\phi_{2}(c,y)$ modify and have the following form

$\displaystyle\phi_{1}(c,y,e)\Big{|}_{y\,=\,y_{2}+e}$
$\displaystyle\,=\,\left(\frac{\left(-4c+\sqrt{1-8c}-1\right)(c+1)e}{\sqrt{1-8c}\left(2c(e+2)+\sqrt{1-8c}+2e+1\right)}\right)^{1/2}\,,$
(129) $\displaystyle\phi_{2}(c,y,e)\Big{|}_{y\,=\,y_{2}+e}$
$\displaystyle\,=\,\left(\frac{-\frac{16(c+1)c}{2c(e+2)+\sqrt{1-8c}+2e+1}+4c+\sqrt{1-8c}+1}{2\sqrt{1-8c}}\right)^{1/2},$
(130)

similarly, from the lower limit, the amplitudes are

$\displaystyle\phi_{1}(c,y,e)\Big{|}_{y\,=\,y_{1}+e}$
$\displaystyle\,=\,\left(\frac{\frac{16(c+1)c}{2c(e+2)-\sqrt{1-8c}+2e+1}-4c+\sqrt{1-8c}-1}{2\sqrt{1-8c}}\right)^{1/2}\,,$
(131) $\displaystyle\phi_{2}(c,y,e)\Big{|}_{y\,=\,y_{1}+e}$
$\displaystyle\,=\,\left(-\frac{(c+1)\left(4c+\sqrt{1-8c}+1\right)e}{\sqrt{1-8c}\left(-2c(e+2)+\sqrt{1-8c}-2e-1\right)}\right)^{1/2}.$
(132)

Note that, it is necessary to use this parameter because the straightforward
substitution of limits did not allow the transformation of every incomplete
elliptic integral. Let us consider a few examples to demonstrate how this off-
set parameter solves the problem with incomplete elliptic integrals that
cannot be reduced into complete elliptic integrals as their amplitude
$\phi\neq\pi/2$. We can categorize all the elliptic integrals appearing after
$y$ integration into two classes according to their amplitudes $\phi$ as $(i)$
non-reducible incomplete elliptic integrals $(\phi\neq\pi/2)$, and $(ii)$
reducible incomplete elliptic integrals $(\phi=\pi/2)$.

###  A.1 Non-reducible incomplete elliptic integrals

Here we consider an incomplete elliptic integral that appears from the upper
limit contributions, the expression of the elliptic integral is

$\displaystyle
E[\phi_{1}(c,e),m_{1}(c)]\,=\,E\left[\sin^{-1}\left(\frac{\left(-4c+\sqrt{1-8c}-1\right)(c+1)e}{\sqrt{1-8c}\left(2c(e+2)+\sqrt{1-8c}+2e+1\right)}\right)^{1/2},\
\frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^{2}}\right]\,,$ (133)

where $m_{1}$ is given in eq. (85), the amplitude of this elliptic function is

$\displaystyle\phi_{1}(c,e)\,=\,\sin^{-1}\left(\frac{\left(-4c+\sqrt{1-8c}-1\right)(c+1)e}{\sqrt{1-8c}\left(2c(e+2)+\sqrt{1-8c}+2e+1\right)}\right)^{1/2}.$
(134)

The elliptic integral in eq. (133) cannot be reduced into a complete elliptic
integral as in the limit $e\to 0$ the amplitude $\phi_{1}(c,e)\to 0$. However,
we can expand this elliptic integral in powers of $e$ around $e=0$, upon
expanding we get

$\displaystyle E[\phi_{1}(c,e),m_{1}(c)]$
$\displaystyle\,=\,\sqrt{\frac{\left(-4c+\sqrt{1-8c}-1\right)(c+1)}{\sqrt{1-8c}\left(4c+\sqrt{1-8c}+1\right)}}\sqrt{e}+\mathcal{O}(e^{3/2})$
(135)

The expansion’s first term, proportional to $e^{1/2}$, is the only significant
term in the limit when $e\to 0$. The next term in the above expression is
proportional to $e^{3/2}$, and can be dropped. The small-$e$ expression for
these specific incomplete elliptic integrals occurring is then stored in the
form

$\displaystyle
E[\phi_{1}(c,e),m_{1}(c)]\,=\,\sqrt{\frac{\left(-4c+\sqrt{1-8c}-1\right)(c+1)}{\sqrt{1-8c}\left(4c+\sqrt{1-8c}+1\right)}}\sqrt{e}.$
(136)

Now, we shift our attention toward the elliptic integrals that appears from
the lower limit contribution, one such incomplete elliptic integral is

$\displaystyle
F[\phi_{2}(c,e),m_{1}(c)]\,=\,F\left[\sin^{-1}\left(\frac{(c+1)\left(4c+\sqrt{1-8c}+1\right)e}{\sqrt{1-8c}\left(2c(e-2)+\sqrt{1-8c}+2e-1\right)}\right)^{1/2},\
\frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^{2}}\right]\,.$ (137)

The above elliptic integral cannot be reduced into a complete elliptic
integral as the amplitude $\phi_{2}(c,e)$ is

$\displaystyle\phi_{2}(c,e)\,=\,\sin^{-1}\left(\frac{(c+1)\left(4c+\sqrt{1-8c}+1\right)e}{\sqrt{1-8c}\left(2c(e-2)+\sqrt{1-8c}+2e-1\right)}\right)^{1/2}$
(138)

and in the limit $e\to 0$ the amplitude $\phi_{2}(c,e)\to 0$. Following the
similar procedure as in eq. (136), the elliptic integral in eq. (137) is
expanded around $e=0$, we get

$\displaystyle
F[\phi_{2}(c,e),m_{1}(c)]=\sqrt{\frac{(c+1)\left(4c+\sqrt{1-8c}+1\right)}{\sqrt{1-8c}\left(-4c+\sqrt{1-8c}-1\right)}}\sqrt{e}\,.$
(139)

The coefficients of these elliptic functions depend on $e$ when the
integration limits are modified in accordance with eq. (128). When we expand
our final result in powers of $e$, the negative powers of $\sqrt{e}$ from the
coefficients combine with the positive powers of $\sqrt{e}$ from the stored
expressions of non-reducible incomplete elliptic integrals to yield few terms
independent of $e$. Subsequently $e$ can be set to zero.

Observable |  Matrix element |  Distribution up to NLP
---|---|---
Thrust |  Exact |  $\frac{-3-4\log{\tau}}{\tau}-2+2\log{\tau}$
Thrust |  Shift |  $\frac{-4-4\log{\tau}}{\tau}+4+4\log{\tau}$
Thrust |  Remainder |  $\frac{1}{\tau}-6-2\log{\tau}$
Thrust |  Eikonal |  $\frac{-4\log{\tau}}{\tau}-8-4\log{\tau}$
c-parameter |  Exact |  $\frac{-3-4\log{c}}{c}+1-28\log{c}$
c -parameter |  Shift |  $\frac{-4-4\log{c}}{c}+8-24\log{c}$
c -parameter |  Remainder |  $\frac{1}{c}-7-4\log{c}$
c -parameter |  Eikonal |  $\frac{-4\log{c}}{c}-8-40\log c$
Table 1: Table for the result of thrust and $c$-parameter distributions
calculated using the exact definition of event shape variables and four
different definitions of matrix elements.

###  A.2 Reducible incomplete elliptic integrals

The category of incomplete elliptic integrals, which can be reduced into
complete elliptic integrals is easier to handle as compared with non-reducible
ones. An elliptic integral appearing from the upper limit contribution is

$\displaystyle
E[\phi_{2}(c,e),m_{1}(c)]\,=\,E\left[\sin^{-1}\left(\frac{-\frac{16(c+1)c}{2c(e+2)+\sqrt{1-8c}+2e+1}+4c+\sqrt{1-8c}+1}{2\sqrt{1-8c}}\right)^{1/2},\
\frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^{2}}\right]\,,$ (140)

where the amplitude $\phi_{2}(c,e)$ reads

$\displaystyle\phi_{2}(c,e)\,=\,\sin^{-1}\left(\frac{-\frac{16(c+1)c}{2c(e+2)+\sqrt{1-8c}+2e+1}+4c+\sqrt{1-8c}+1}{2\sqrt{1-8c}}\right)^{1/2}\,.$
(141)

In the limit $e\to 0$ the amplitude $\phi_{2}(c,e)\to\pi/2$ and this
incomplete elliptic integral can be reduced directly to a complete elliptic
integral using reduction formula in eq. (A)

$\displaystyle E[\phi_{2}(c,e),m_{1}(c)]\,=\,E[m_{1}(c)]\,.$ (142)

In this way all reducible incomplete elliptic integrals can be substituted
with their corresponding complete elliptic integrals. Upon substituting all
the complete elliptic integrals and the stored expressions of incomplete
elliptic integrals, our final result is expanded in powers of $e$. No negative
powers occur, and $e$ can be taken to zero, yielding eq. (83).

##  Appendix B Eikonal approximation and result summary

(a)  (b)

Figure 8:  In (a) and (b), we plot the thrust and $c$-parameter distributions,
respectively. We show the exact result (solid red curves) given in eqs. (28)
and (83), the shifted approximation result (dashed blue curves) up to NLP
terms given in eqs. (47) and (101) for $\tau$ and $c$ respectively. We also
plot the LP term (dotted purple curves) given in eqs. (144) and (145)

###  B.1 Eikonal approximation to thrust and $c$-parameter distribution

In sections III.3 and IV.3, we compared the shifted approximation with the LP
expression of the exact distribution in figs. (3a) and (7a), for the thrust
and $c$-parameter distribution, respectively. Here we add the results from the
simpler eikonal approximation to the comparison. The eikonal case follows from
the approximated matrix element squared

$\displaystyle\overline{\sum}|\mathcal{M}_{\text{eik}}(x_{1},x_{2})|^{2}\,=\,$
$\displaystyle 8(e^{2}e_{q})^{2}g_{s}^{2}C_{F}N_{c}\frac{1}{3Q^{2}}$
$\displaystyle\times\left(\frac{2}{(1-x_{1})(1-x_{2})}\right)\,.$ (143)

Using the above expression and the exact definitions of thrust and
$c$-parameter in eqs. (10) and (69), one finds for their respective
distributions

$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{d\tau}\Bigg{|}_{\text{NLO}}$
$\displaystyle\,=\,\frac{2\alpha_{s}}{3\pi}\left(\frac{-4\log{\tau}}{\tau}-8-4\log{\tau}+\mathcal{O}(\tau)\right),$
(144)
$\displaystyle\frac{1}{\sigma_{0}(s)}\frac{d\sigma}{dc}\Bigg{|}_{\text{NLO}}$
$\displaystyle\,=\,\frac{2\alpha_{s}}{3\pi}\left(\frac{-4\log{c}}{c}-8-40\log{c}+\mathcal{O}(c)\right).$
(145)

The eikonal matrix element squared generates LL terms at LP correctly,
together with some LL and NLL at NLP. It does not capture any NLL term at LP
since it lacks contributions from hard-collinear gluon emission. In figs. (8a)
and (8b), we plot these eikonal results together with the thrust and
c-parameter distributions computed from the exact approach as given in eq.
(28) and eq. (83) along with the shifted approximation results up to NLP from
eqs. (47) and (101). Clearly for both event shapes the shifted kinematics
methods provides a significantly better approximation than the eikonal
approximation.

###  B.2 Table of results

In table 1, we summarize our results for thrust and the $c$-parameter using
the different approximations of the matrix elements squared: eikonal, exact,
shifted kinematics, as well as the remainder/soft quark part.

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