††institutetext: Department of Physics, Harvard University, Cambridge, MA
02138, USA

# Sudakov Shoulder Resummation for
Thrust and Heavy Jet Mass

Arindam Bhattacharya  Matthew D. Schwartz, and  Xiaoyuan Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

When the allowed range of an observable grows order-by-order in perturbation
theory, its perturbative expansion can have discontinuities (as in the C
parameter) or discontinuities in its derivatives (as in thrust or heavy jet
mass) called Sudakov shoulders. We explore the origin of these logarithms
using both perturbation theory and effective field theory. We show that for
thrust and heavy jet mass, the logarithms arise from kinematic configurations
with narrow jets and deduce the next-to-leading logarithmic series. The left-
shoulder logarithms in heavy jet mass ($\rho)$ of the form $r\
\alpha_{s}^{n}\ln^{2n}r$ with $r=\frac{1}{3}-\rho$ are particularly dangerous,
because they invalidate fixed order perturbation theory in regions
traditionally used to extract $\alpha_{s}$. Although the factorization formula
shows there are no non-global logarithms, we find Landau-pole like
singularities in the resummed distribution associated with the cusp anomalous
dimension, and that power corrections are exceptionally important.

##  1 Introduction

It is not uncommon for an observable to have a range that grows order-by-order
in perturbation theory. Traditional $e^{+}e^{-}$ event shapes, such as thrust,
the C parameter, and heavy jet mass Catani:1997xc  have this property as do
some hadron-collider observables like the jet shape Seymour:1997kj ;
Luisoni:2020efy  As observed by Catani and Webber Catani:1997xc , when the
range grows order by order, there can be incomplete cancellations between the
virtual contributions, which are confined to the lower-order range, and the
real-emission contributions, which are not. The results are distributions with
non-analytic behavior at intermediate values of the observable:
discontinuities, cusps or kinks at any given finite order in perturbation
theory, collective called Sudakov shoulders, as shown in Fig. 2. Sudakov
shoulders are caused by large logarithms associated with kinematic regions not
close to the absolute (non-perturbative) phase space boundary. We classify the
Sudakov shoulders as either right shoulders, which have large logarithms
extending into regions accessible only at higher orders in perturbation theory
(i.e. to the right of the shoulder as in thrust or $C$ parameter) or left
shoulders, which have logarithms affecting regions accessible at all orders in
perturbation theory (i.e. to the left of the shoulder, as in heavy jet mass).
Left shoulders are particularly problematic as they can invalidate the use of
fixed order perturbation theory over a wide range of observable values.

To understand Sudakov shoulders, consider first the thrust observable
Farhi:1977sg . Thrust is defined in the center-of-mass frame of an
$e^{+}e^{-}$ collision as

$T\equiv\max_{\vec{n}}\frac{\sum_{j}|\vec{p}_{j}\cdot\vec{n}|}{\sum_{j}|\vec{p}_{j}|}$
(1)

where the sum is over all particles in the event and the maximum is over
3-vectors $\vec{n}$ of unit norm. It is common to use $\tau=1-T$ in place of
$T$. The vector $\vec{n}$ that maximizes thrust is known as the thrust axis.
When there are only 2 particles, they must be back-to-back, and then $\tau=0$
exactly. If there are 3 massless particles, then the phase space is 2
dimensional and can be parameterized with $s_{ij}=(p_{i}+p_{j})^{2}/Q^{2}$
constrained by $s_{12}+s_{23}+s_{13}=1$ with $Q$ the center-of-mass energy.
Then

$\tau=\min(s_{12},s_{13},s_{23})\leq\frac{1}{3}$ (2)

The phase space point that saturates this bound has
$s_{12}=s_{13}=s_{23}=\frac{1}{3}$ and comprises the symmetric trijet
configuration: 3 particles of equal energy and angular separation, as shown in
Fig 1. Near this point the spin-summed 3-body matrix-element-squared is not
exceptional

$|{\mathcal{M}}_{1}|^{2}=|{\mathcal{M}}_{0}|^{2}2C_{F}g_{s}^{2}\frac{s_{12}^{2}+s_{23}^{2}+2Q^{2}s_{13}}{s_{12}s_{13}}\cong|{\mathcal{M}}_{0}|^{2}64\pi
C_{F}\alpha_{s}$ (3)

where $|\mathcal{M}_{0}|^{2}$ is the $\gamma^{*}\to q\bar{q}$ matrix-element-
squared. Because the phase space goes to zero at $\tau=\frac{1}{3}$, the
differential cross section must vanish there. The result is that

$\frac{1}{\sigma_{0}}\frac{d\sigma}{d\tau}\cong 3\times
48C_{F}\frac{\alpha_{s}}{4\pi}\int_{0}^{\frac{1}{3}-\tau}ds_{12}=144C_{F}\frac{\alpha_{s}}{4\pi}\left(\frac{1}{3}-\tau\right)\theta\left(\frac{1}{3}-\tau\right)$
(4)

with the factor of 3 coming from the 3 choices of thrust axis all of which
contribute equally hear $\tau=\frac{1}{3}$. Already here we can see the
Sudakov shoulder: there is a discontinuity in the first derivative of the
distribution from $-144C_{F}\frac{\alpha_{s}}{4\pi}$ for $\tau<\frac{1}{3}$ to
$0$ for $\tau>\frac{1}{3}$.

Given the thrust axis from the maximization in Eq. (1), the event is divided
into two hemispheres. We can compute the invariant masses $m_{1}$ and $m_{2}$
of all the partons in hemisphere 1 and 2 and then heavy jet mass is defined as

$\rho=\frac{1}{Q^{2}}\max(m_{1}^{2},m_{2}^{2})$ (5)

At order $\alpha_{s}$ one hemisphere must be massless and $\tau=\rho$, and
thus $\frac{d\sigma}{d\rho}$ has a discontinuity in its first derivative at
leading order, just like $\tau$.

$p_{1}=\frac{Q}{3}(1,0,0,1)$$~{}~{}p_{2}=\frac{Q}{3}(1,0,\frac{\sqrt{3}}{2},-\frac{1}{2})$$~{}~{}p_{3}=\frac{Q}{3}(1,0,-\frac{\sqrt{3}}{2},-\frac{1}{2})$
Figure 1: The trijet configuration where $\rho=\tau=\frac{1}{3}$ has 3 equally
spaced jets of equal energy

Now, consider what happens at higher order in perturbation theory. The parton
in the light hemisphere will radiate gluons, making the light hemisphere
massive. Since the cross section for the light jet having mass less than $m$
after one emission scales like $\sigma\sim\alpha_{s}\ln^{2}m^{2}$ there is a
Sudakov enhancement to the cross section at small $m^{2}$. As the light
hemisphere jet grows, energy must be drawn away from the heavy hemisphere,
making it lighter. Roughly speaking, setting $Q=1$ for simplicity,
$\rho\lesssim\frac{1}{3}-m^{2}$ (as we will derive). Conversely, the cross
section at $\rho=\frac{1}{3}-m^{2}$ will be enhanced by factors of
$\ln^{2}m^{2}=\ln^{2}(\frac{1}{3}-\rho)$. Thus large Sudakov logs associated
with radiation into the light hemisphere translate into Sudakov shoulder logs.
This is the physical mechanism for the production of large logs in the left
shoulder for heavy jet mass.

To properly and systematically resum the Sudakov shoulder logarithms, we must
understand this mechanism, as well as the consequences of radiation from the
heavy-hemisphere partons. At first glance, the mechanism, which transfers
large logs from the light to the heavy hemisphere using energy conservation
may seem difficult to reconcile with factorization. Indeed, previous work has
noted the recoil sensitivity of Sudakov shoulder logarithms starting at the
next-to-leading logarithmic level Luisoni:2020efy . Nevertheless, as we will
see it is still possible to factorize the matrix elements and phase space near
$\rho=\frac{1}{3}$ to isolate and extract the large logarithms, at least at
the next-to-leading logarithmic level.

One may ask whether Sudakov shoulder resummation is important. For observables
with only a right shoulder, such as thrust, one might argue that it is not so
important, since there is not much data for $\tau>\frac{1}{3}$. However, for
heavy jet mass one should generically expect that logs of the form
$\alpha_{s}\ln^{2}(\frac{1}{3}-\rho)$ are as important away from shoulder
region as logs $\alpha_{s}\ln^{2}\rho$ are away from the threshold $\rho=0$.
This leaves a rather narrow range of intermediate values of $\rho$ where
fixed-order perturbation theory might be trusted. Moreover, looking at Fig. 2
it seems that the Sudakov shoulder effects on the left shoulder of heavy jet
mass curve tend to pull it down (and away from thrust), so that resumming the
left Sudakov shoulder might bring the curves closer together. Thus difference
of the left shoulder in thrust and heavy jet mass could help explain long
standing discrepancies between fits for $\alpha_{s}$ using the two event
shapes Salam:2001bd ; Chien:2010kc .

Figure 2: Thrust (blue) and heavy jet mass (red) at NLO compared to LO
(dashed). The NLO curve does not have LO added in. That is, the LO is the
$\frac{\alpha_{s}}{2\pi}$ times the “A” function and the NLO curves are
$(\frac{\alpha_{s}}{2\pi})^{2}$ times the “B” functions, in the notation of
Gehrmann-DeRidder:2007vsv . Right is a zoom in of Sudakov shoulder region hear
$\frac{1}{3}$. NLO computation is made with the program event2 Catani:1996jh ;
Catani:1996vz . All distributions are normalized to born cross section
$\sigma_{0}$.

In order to resum the Sudakov logs we first explore the regions of phase space
that can contribute logarithms near the shoulder. We do this for the left
shoulder of heavy jet mass in Section 2. We find that the phase space near
$\rho\lesssim\frac{1}{3}$ splits up into regions some of which generate large
logarithms of $\frac{1}{3}-\rho$ and some of which do not. We find that all
the logarithms come from regions with narrow jets in the light and heavy
hemispheres. This is in contrast to threshold region, for which every allowed
point of phase space near $\rho\sim 0$ can contribute logarithms of $\rho$. It
is also in contrast to non-global logarithms, such as for the light jet mass.
There, logarithms of the light jet mass come from regions where the heavy jet
side does not have to contain only narrow jets.

In Section 3 we discuss the factorization of $\rho$ and $\tau$ near
$\frac{1}{3}$. We find that near the shoulder region, the phase space and
matrix elements both neatly factorize. This allows us to define a soft
function, which along with the inclusive jet function, can be used to
reproduce all the logarithms at NLO, and more generally the next-to-leading
logarithmic series. In Section 4 we analyze the resummed expression. We show
that there are no non-global logarithms for the Sudakov shoulder; only regions
related to the trijet configuration by soft or collinear radiation can
generate the shoulder logs. We also find an unusual pole in the the resummed
distribution, qualitatively similar to the Landau pole in the running
coupling. Unlike the QCD Landau pole however, the singularity in the resummed
heavy jet mass shoulder distribution is determined by the cusp anomalous
dimension. Thus it is a kind of Sukdakov Landau pole. Similar poles can be
found in other observables, such as the Drell-Yan spectrum at small $p_{T}$
Frixione:1998dw ; Becher:2010tm ; Monni:2016ktx . We show that for the Sudakov
shoulder case, the large Sudakov anomalous dimension contributing to this pole
also enhances subleading power effects, making them comparable to the leading
power result allowing the pole to be cancelled in the full distribution. We
conclude in Section 6.

##  2 Next-to-leading order analysis

As a first step towards understanding Sudakov shoulder logarithms, we analyze
the matrix elements and phase space near the shoulder region in full QCD. We
concentrate here on the heavy jet mass for concreteness, but the same analysis
works for thrust.

At next-to leading order in QCD, there is the virtual contribution with 3
partons in the final state and a real emission contribution with 4 partons.
The virtual contribution is proportional to the LO cross section and serves to
regularize infrared and collinear divergences. Thus we focus on the real
emission contributions to extract the logarithms.

To have $\rho\lesssim\frac{1}{3}$ we can have configurations which differ from
the trijet configuration by soft and collinear emissions, or configurations
which do not. For example, one could take a non-planar 4-parton configuration
with 4 well-separated partons and $\rho\sim 0.4$, then adjust their momenta to
lower $\rho$. Staring from such a configuration, one would not expect anything
unusual to happen as $\rho$ is lowered through $\frac{1}{3}$. Indeed,
$\rho=\frac{1}{3}$ is only special because it is a kinematic limit for 3-body
phase space. Thus we expect that the only 4-parton configurations which will
contribute Sudakov shoulder logarithms are those close to the trijet
configuration. We will find that this is in fact the case.

###  2.1 Kinematics

Let us define the momenta of the 4 particles in the final state as
$p_{1}^{\mu},p_{2}^{\mu},p_{3}^{\mu}$ and $p_{4}^{\mu}$. After momentum
conservation, on-shell conditions and a frame choice, there are 5 independent
degrees of freedom of these four momenta. Although we will not restrict the
momenta to be soft or collinear, it is helpful to choose variables so that the
soft and collinear limits are transparent. To impose the on-shell constraints,
it is helpful to parameterize the momenta initially in lightcone coordinates:

$p_{1}=z_{1}n^{\mu}+\frac{p^{2}_{\perp}}{4z_{1}}\bar{n}^{\mu}+p^{\mu}_{\perp},\quad
p_{2}=z_{2}n^{\mu}+\frac{p^{2}_{\perp}}{4z_{2}}\bar{n}^{\mu}-p^{\mu}_{\perp}$
(6) $p_{3}=z\omega
n^{\mu}+\frac{q^{2}_{\perp}}{4z\omega}\bar{n}^{\mu}+q^{\mu}_{\perp},\quad
p_{4}=(1-z)\omega
n^{\mu}+\frac{q^{2}_{\perp}}{4(1-z)\omega}\bar{n}^{\mu}-q^{\mu}_{\perp}$ (7)

where $n^{\mu}=(1,0,0,1)$ and $\bar{n}^{\mu}=(1,0,0,-1)$ are back-to-back
lightlike directions. Imposing momentum conservation and defining $\phi$ as
the azimuthal angle between the 1-2 and 3-4 planes we can then express all the
momenta in terms of

$s_{234}=(p_{2}+p_{3}+p_{4})^{2},\quad s_{34}=(p_{3}+p_{4})^{2},\quad
z,\quad\omega\quad\text{and}\quad\phi$ (8)

We conventionally define $\phi$ by
$p_{\perp}^{\mu}=(0,p_{T}\sin\phi,p_{T}\cos\phi,0)$. The variables
$\omega=\frac{1}{2}\bar{n}\cdot(p_{3}+p_{4})$ and $s_{234}$ are hard
variables, approaching $\frac{1}{3}$ at the trijet configuration. $s_{34}$ is
the invariant mass of one of the jets in the collinear limit which approaches
zero in the trijet limit. The collinear momentum fraction $z$ and the
azimuthal angle are order $1$ in the collinear limit, but $z\to 0$ in the
limit that $p_{4}$ is soft. We also find it sometimes convenient to trade
$\cos\phi$ for $s_{23}$ using

$s_{23}=\frac{s_{34}^{2}z+2s_{34}(1+2z\omega-2z-2\omega)\omega+4zs_{234}\omega^{2}+s_{34}s_{234}(z-1+2\omega-4z\omega)}{4\omega^{2}-s_{34}}\\\
+\frac{2\sqrt{s_{34}(1-z)z(2\omega-s_{34})(1-2\omega)(2\omega-s_{234})(2\omega
s_{234}-s_{34})}}{4\omega^{2}-s_{34}}\cos\phi$ (9)

When using $s_{23}$ the physical constraint $-1\leq\cos\phi\leq 1$ must be
imposed on the region of integration. Another useful exact relation is

$s_{12}=1-2\omega+s_{34}\left(1-\frac{1}{2\omega}\right)$ (10)

We can use this relation to trade $\omega$ for $\rho$ when $\rho=s_{12}$.

To compute thrust or heavy jet mass, we need to determine the thrust axis from
the formula in Eq. (1). With 4 partons, the two possibilities are that 3
partons are in one hemisphere and 1 parton in the other, or 2 partons can be
in each hemisphere. If we know that partons $p_{1}\ldots p_{m}$ are to be
clustered in the same hemisphere then

$\max_{\vec{n}}\sum_{j=1}^{m}|\vec{p}_{j}\cdot\vec{n}|=2\max_{\vec{n}}(\sum_{j=1}^{m}\vec{p}_{j})\cdot\vec{n}$
(11)

This dot product will be maximized if
$\vec{n}=|\sum\vec{p}_{j}|^{-1}\sum\vec{p}_{j}$ so that the thrust axis will
always align with the sum of momenta in each hemisphere. So there are 7
possibilities for the thrust axis. For each axis choice

$\sum_{j=1}^{m}|\vec{p}_{j}\cdot\vec{n}|=2\frac{1}{|\sum\vec{p}_{j}|}(\sum\vec{p}_{j})\cdot(\sum\vec{p}_{j})=2\Big{|}\sum_{j=1}^{m}\vec{p}_{j}\Big{|}$
(12)

Thus to determine the thrust axis, we need to find which set of partons has
the largest value of $2|\sum\vec{p}_{j}|$ or equivalently

$T_{j}^{2}\equiv 4|\sum\vec{p}_{j}|^{2}$ (13)

In terms of our variables in Eq. (8), the $T_{j}$ with one parton in one
hemisphere are relatively simple

$\displaystyle T_{1}^{2}$ $\displaystyle=T_{234}^{2}=(1-s_{234})^{2},\quad
T_{2}^{2}=T_{134}^{2}=\left[\frac{2(1+s_{234}-2\omega)\omega-
s_{34}}{2\omega}\right]^{2}$ (14) $\displaystyle T_{3}^{2}$
$\displaystyle=T_{124}^{2}=\left[\frac{4z\omega^{2}+s_{34}(1-z)}{2\omega}\right]^{2},\quad
T_{4}^{2}=T_{123}^{2}=\left[\frac{4(1-z)\omega^{2}+s_{34}z}{2\omega}\right]^{2}$
(15)

We also have

$T_{12}^{2}=\left(\frac{4\omega^{2}-s_{34}}{2\omega}\right)^{2}$ (16)

and

$\displaystyle T_{13}^{2}$
$\displaystyle=\frac{1}{4\omega^{2}}\Big{[}s_{34}^{2}(1-z)^{2}+4\omega^{2}\left(4s_{23}+s_{234}^{2}+(1-2\omega
z)^{2}-2s_{234}(1+2\omega z)\right)$ $\displaystyle\hskip
56.9055pt-4s_{34}\omega\left(1+s_{234}-z-s_{234}z+2\omega\left(z^{2}-z-2\right)\right)\Big{]}$
(17) $\displaystyle T_{14}^{2}$
$\displaystyle=\Big{[}\frac{s_{34}z-2\omega\left(1+s_{234}-2\omega(1-z)\right)}{2\omega}\Big{]}^{2}-4s_{23}$
(18)

All of these $T_{j}$ values are exact.

Figure 3: Example slices of 5D phase space for four massless partons. The
colors indicate which collection of momenta determines the thrust axis. The
left plot has $z=0.06,s_{34}=0.02$ and $s_{234}=\frac{1}{3}-0.01$. The green
region in this plot would contribute to the heavy jet mass distribution at
$r=\frac{1}{3}-\rho=0.01$. The right plot has $\phi=\pi$,
$s_{234}=\frac{1}{3}$ and $s_{12}=\frac{1}{3}-0.01$. The blue region in this
plot contributes also at $r=0.01$.

Now we would like to consider the region $\rho<\frac{1}{3}$. The heavy
hemisphere can have either 2 partons or 3 partons. We can therefore choose it
to be $\rho=s_{234}$ with $T_{1}$ is maximal, or $s_{12}$ with $T_{12}$ is
maximal. The other cases are given by permutation of the indices. Figure 3
shows examples of the phase space regions labeled by which $T_{j}$ is
greatest. All regions in these plots contribute to some value of $\rho$.
However, to avoid overcounting we only need to consider the green region on
the left plot and the blue region in the right plot.

###  2.2 Matrix Elements

Let us define

$r\equiv\frac{1}{3}-\rho$ (19)

As we have discussed, we expect contributions to the NLO heavy jet mass cross
section with factors of $\ln r$ or $\ln^{2}r$ to come from soft or collinear
regions of phase space close to the trijet configuration. We can therefore
power-expand the matrix elements and phase space constraints in soft and
collinear limits. This dramatically simplifies the calculation. There are two
ways to confirm that only soft and collinear limits are relevant. First, we
can extend the integration limits to the full phase space and verify that no
additional logarithms can be generated. Second we can compare the logarithms
we extract with a numerical computation of the heavy jet mass distribution at
NLO.

For power counting we take $r\sim\lambda\ll 1$. In the collinear limit where
$p_{4}\,||\,p_{3}$, the phase space variables scale as

$s_{34}\sim\lambda,\quad x\equiv\omega-\frac{1}{3}\sim\lambda,\quad y\equiv
s_{234}-\frac{1}{3}\sim\lambda,\quad z\sim\lambda^{0},\quad\phi\sim
s_{23}\sim\lambda^{0}$ (20)

In the soft limit, where $p_{3}$ is soft, the scaling is the same except that
$z\sim\lambda$ instead of $z\sim\lambda^{0}$.

First we compute the matrix elements-squared at leading power. We do this by
summing all the relevant Feynman diagrams, squaring the amplitudes and summing
over spins, after which we take the leading power expansion. We cross check
the results against the expectation for soft and collinear limits from
factorization.

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is polarized in the plane of scattering or out of the plane. We find

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}\end{gathered}=|{\mathcal{M}}_{0}|^{2}2g_{s}^{2}C_{F}$ (21)

when the gluon polarization $\epsilon_{\text{in}}=(0,0,1,0)$ in the
conventions of Fig. 1, where $p_{g}=\frac{Q}{3}(1,0,0,1)$. And

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{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
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}\end{gathered}=|{\mathcal{M}}_{0}|^{2}14g_{s}^{2}C_{F}$ (22)

when the gluon polarization is $\epsilon_{\text{out}}=(0,1,0,0)$. The sum of
these agrees with Eq. (3).

The matrix elements depend on which partons are gluons and which are quarks.
If $p_{2}$ is a quark and $p_{4}$ is a gluon, then to leading power in
collinear scaling

$\left|\mathcal{M}^{\text{collinear}}_{\gamma^{*}\rightarrow
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{3.14279pt}{3.14279pt}{5.69046pt}{0.0pt}{5.69046pt}\pgfsys@curveto{-3.14279pt}{5.69046pt}{-5.69046pt}{3.14279pt}{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{-3.14279pt}{-3.14279pt}{-5.69046pt}{0.0pt}{-5.69046pt}\pgfsys@curveto{3.14279pt}{-5.69046pt}{5.69046pt}{-3.14279pt}{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}\end{gathered}\cong\left|\mathcal{M}_{0}\right|^{2}32g_{s}^{4}C_{F}^{2}\frac{1}{s_{34}}\frac{1+z^{2}}{1-z}$
(23)

Here the blob represents all the diagrams that can contribute. We derive this
by squaring the full matrix element for $\gamma^{\star}\to qg\bar{q}g$ using
QgrafNOGUEIRA1993279  and FormVermaseren:2000nd  or FeynCalcHAHN2001418 ;
MERTIG1991345 ; SHTABOVENKO2016432 , summing over spins, and then and power
expanding in the small $s_{34}$ limit. The splitting function naturally
appears.

When $p_{3}$ and $p_{4}$ are gluons, then we find

$\displaystyle\left|\mathcal{M}^{\text{collinear}}_{\gamma^{*}\rightarrow
q\bar{q}gg}\right|^{2}$
$\displaystyle=\begin{gathered}\leavevmode\resizebox{85.35826pt}{}{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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}\end{gathered}\cong\left|\mathcal{M}_{0}\right|^{2}C_{F}C_{A}g_{s}^{4}\frac{64}{s_{34}}\left[\frac{\left(1-z+z^{2}\right)^{2}}{z(1-z)}+\frac{3}{4}z(1-z)\cos
2\phi\right]$ (25)

Note the azimuthal angle dependence is due to the polarization of the gluons.
Indeed, the leading order $\gamma^{\star}\to q\bar{q}g$ matrix element is
polarized, and we must therefore use polarized splitting functions (see
Ellis:1996mzs  for example). We have checked that summing the polarized
leading order matrix elements in Eqs. (21) and (22) with the polarized
splitting functions (see Ellis:1996mzs  for example) reproduces Eq. (25).

And finally when $p_{3}$ and $p_{4}$ are quarks (or antiquarks), the leading
power result is the same whether they are identical or not.

$\displaystyle\left|\mathcal{M}^{\text{collinear}}_{\gamma^{*}\rightarrow
q\bar{q}q^{\prime}\bar{q}^{\prime}}\right|^{2}$
$\displaystyle\cong\left|\mathcal{M}^{\text{collinear}}_{\gamma^{*}\rightarrow
q\bar{q}q\bar{q}}\right|^{2}=\begin{gathered}\leavevmode\resizebox{99.58464pt}{}{
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}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}}
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}\pgfsys@endscope}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{
} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@curveto{5.69046pt}{3.14279pt}{3.14279pt}{5.69046pt}{0.0pt}{5.69046pt}\pgfsys@curveto{-3.14279pt}{5.69046pt}{-5.69046pt}{3.14279pt}{-5.69046pt}{0.0pt}\pgfsys@curveto{-5.69046pt}{-3.14279pt}{-3.14279pt}{-5.69046pt}{0.0pt}{-5.69046pt}\pgfsys@curveto{3.14279pt}{-5.69046pt}{5.69046pt}{-3.14279pt}{5.69046pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\end{gathered}$ (27)
$\displaystyle\cong\left|\mathcal{M}_{0}\right|^{2}C_{F}T_{f}n_{f}g_{s}^{4}\frac{16}{s_{34}}\left(2-z-z^{2}-6z(1-z)\cos^{2}\phi\right)$
(28)

This expression also depends on the azimuthal angle, and like the gluon case,
is consistent with using the polarized 3 parton matrix elements and
polarization-dependent splitting functions.

For the soft limits, we can power expand the full matrix elements in the soft
limit. When $z$ is soft we cannot drop $s_{34}$ with respect to $z$, or vice-
versa. When $p_{3}$ and $p_{4}$ are both gluons, the result can be written as

$\left|\mathcal{M}^{\text{soft}}_{\gamma^{*}\rightarrow
q\bar{q}gg}\right|^{2}\cong|{\mathcal{M}}_{0}|^{2}C_{F}g_{s}^{4}\frac{64}{3}\left[\left(C_{F}-\frac{1}{2}C_{A}\right)\frac{1}{s_{14}s_{24}}+\frac{C_{A}}{2}\frac{1}{s_{14}s_{34}}+\frac{C_{A}}{2}\frac{1}{s_{24}s_{34}}\right]$
(29)

where

$\displaystyle s_{14}s_{24}$
$\displaystyle=9s_{34}^{2}+16z^{2}-24s_{34}z\cos(2\phi)$ (30) $\displaystyle
s_{14}s_{34}$
$\displaystyle=3s_{34}^{2}+4s_{34}z-4s_{34}\sqrt{3s_{34}z}\cos\phi$ (31)
$\displaystyle s_{24}s_{34}$
$\displaystyle=3s_{34}^{2}+4s_{34}z+4s_{34}\sqrt{3s_{34}z}\cos\phi$ (32)

This is consistent with the Eikonal approximation.

To avoid double-counting we also need the soft collinear matrix elements which
come from taking the soft limit (small $z$) of the collinear matrix elements
or equivalently the collinear limit ($s_{34}\ll z$) of the soft matrix
elements. These are therefore the same as the soft matrix elements but keeping
only the final term in Eqs. (30)-(32).

###  2.3 Phase space

For the phase space limits, we will first examine the soft-collinear limit
where $z\sim\lambda$. To leading power in the soft-collinear limit

$\displaystyle T_{1}$ $\displaystyle\cong\frac{1}{9}-\frac{y}{3},\quad
T_{2}\cong\frac{1}{9}-\frac{s_{34}}{2}-\frac{2x}{3}+\frac{y}{3},\quad
T_{3}\cong 0,\quad T_{4}\cong\frac{1}{9}+\frac{2x}{3}-\frac{2z}{9}$ (33)
$\displaystyle T_{12}$
$\displaystyle\cong\frac{1}{9}-\frac{s_{34}}{2}+\frac{2x}{3},\quad
T_{13}\cong\frac{1}{9}+s_{23}-\frac{y}{3}-\frac{4z}{9},\quad
T_{23}\cong\frac{1}{9}-s_{23}-\frac{2x}{3}+\frac{y}{3}+\frac{2z}{9}$ (34)

For the case where $T_{1}$ is maximal, $\rho=s_{234}$ and $y=r$. We can then
impose the constraints $T_{1}>T_{2}$, $T_{1}>T_{3}$, and so on. Since we are
using the variable $s_{23}$ instead of $\cos\phi$ we also have to impose
$-1\leq\cos\phi\leq 1$. Reducing these constraints leads to five integration
regions

$\begin{gathered}\int
d\Pi_{1}=2\int_{0}^{\frac{1}{2}s_{34}^{+}}ds_{34}\int_{z^{+}}^{1-z^{+}}dz\int_{x_{A}}^{x_{B}}dx\int_{s_{23}^{\phi-}}^{s_{23}^{\phi+}}ds_{23}\
J+2\int_{0}^{s_{34}^{+}}ds_{34}\int_{\frac{9}{4}s_{34}}^{z^{+}}dz\int_{x_{C}}^{x_{B}}dx\int_{s_{23}^{\phi-}}^{\frac{4}{9}z}ds_{23}\
J\\\
+2\int_{0}^{s_{34}^{+}}ds_{34}\int_{\frac{9}{4}s_{34}}^{z^{+}}dz\int_{x_{A}}^{x_{C}}dx\int_{s_{23}^{A}}^{\frac{4}{9}z}ds_{23}\
J+2\int_{0}^{s_{34}^{+}}ds_{34}\int_{z^{-}}^{\frac{9}{4}s_{34}}dz\int_{x_{C}}^{x_{D}}dx\int_{s_{23}^{\phi-}}^{\frac{4}{9}z}ds_{23}\
J\\\
+\int_{0}ds_{34}\int_{z^{-}}^{\frac{9}{4}s_{34}}dz\int_{x_{E}}^{x_{C}}dx\int_{s_{23}^{A}}^{\frac{4}{9}z}ds_{23}\
J\end{gathered}$ (35)

where

$\displaystyle s_{34}^{+}$ $\displaystyle=\frac{4}{9}(7-4\sqrt{3}),\quad
z^{\pm}=\frac{9}{4}(7\pm 4\sqrt{3})s_{34}$ (36) $\displaystyle
s_{23}^{\phi\pm}$
$\displaystyle=\left(\sqrt{\frac{s_{34}}{4}}\pm\sqrt{\frac{z}{3}}\right)^{2},\quad
s_{23}^{A}=-\frac{2}{3}x+\frac{2}{3}y+\frac{2}{9}z$ (37) $\displaystyle x_{A}$
$\displaystyle=-\frac{3}{4}s_{34}+y,\quad
x_{B}=\frac{3}{4}s_{34}-\frac{y}{2},\quad
x_{C}=-\frac{3}{8}s_{34}+y+\frac{\sqrt{3s_{34}z}}{2}-\frac{z}{6}$ (38)
$\displaystyle x_{D}$ $\displaystyle=-\frac{y}{2}+\frac{z}{3},\quad
x_{E}=y-\frac{z}{3}$ (39)

The Jacobian

$J=\frac{1}{\sqrt{(s_{23}^{\phi+}-s_{23})(s_{23}-s_{23}^{\phi-})}}$ (40)

scales like $J\sim\lambda^{0}$. For the leading double log we need to compute

$\int d\Pi_{1}|{\mathcal{M}}|^{2}\sim\int d\Pi_{1}\frac{1}{s_{34}z}$ (41)

Analyzing the integrals we find that none of them generate $\ln r$ terms; the
limit $r\to 0$ in each of the integrals is smooth. Thus the region with
$T_{1}$ max does not contribute to the Sudakov shoulder at NLO. The logs must
therefore come from regions with two partons in each hemisphere.

Next, we consider configurations where $T_{12}$ is maximal. As before, we
expand first assuming collinear scaling. In this case, we no longer have
$r=\frac{1}{3}-\rho=y$ but instead

$\rho=s_{12}=1+s_{34}-\frac{s_{34}}{2\omega}-2\omega\approx\frac{1}{3}-\frac{1}{2}s_{34}-2x$
(42)

So that $r=\frac{1}{2}s_{34}+2x$. To hold $r$ fixed we then can use
$r,s_{34},z,y,s_{23}$ as independent variables (instead of
$r,s_{34},z,x,s_{23}$ in the $T_{1}$ max case). Now we find 40 relevant
integration regions. In most of these $r$ can be set to zero without
consequence. Only four can possibly generate logs of $r$:

$\int
d\Pi_{12}=\int_{0}^{r}ds_{34}\int_{z^{+}}^{1-z^{+}}dz\int_{y_{A}}^{y_{B}}dy\int_{0}^{2\pi}d\phi+\int_{0}^{\frac{r}{3}}ds_{34}\int_{\frac{9s_{34}}{4}}^{z^{+}}dz\int_{y_{D}}^{y_{C}}dy\int_{0}^{2\pi}d\phi\\\
+\int_{0}^{\frac{r}{3}}ds_{34}\int_{\frac{9s_{34}}{4}}^{z^{+}}dz\int_{y_{C}}^{y_{B}}dy\int_{s_{23}^{B}}^{s_{\phi}^{+}}ds_{23}\
J+\int_{0}^{\frac{r}{3}}ds_{34}\int_{\frac{9s_{34}}{4}}^{z^{+}}dz\int_{y_{A}}^{y_{D}}dy\int_{s_{23}^{\phi-}}^{s_{23}^{C}}ds_{23}\
J$ (43)

where

$\displaystyle y_{A}$ $\displaystyle=-r+2s_{34},\quad y_{B}=2r-s_{34},$
$\displaystyle y_{C}$
$\displaystyle=2r-\frac{7s_{34}}{4}-\sqrt{3s_{34}z}+\frac{z}{3},\quad
y_{D}=-r+\frac{11s_{34}}{4}+\sqrt{3s_{34}z}-\frac{z}{3}$ $\displaystyle
s_{23}^{B}$
$\displaystyle=-\frac{2r}{3}+\frac{5s_{34}}{6}+\frac{y}{3}+\frac{2z}{9},\quad
s_{23}^{C}=\frac{r}{3}-\frac{2s_{34}}{3}+\frac{y}{3}+\frac{4z}{9}$ (44)

For the $C_{F}^{2}$ color structure, using the power-expanded matrix elements
in the collinear limit, Eq. (23), only the first two integrals in Eq. (43)
contribute. We find

$\mathcal{S}_{\text{c}}^{(C_{F})}=4\int d\Pi_{12}|\mathcal{M}_{\gamma^{*}\to
qgg\bar{q}}^{\text{collinear}}|^{2}=\frac{6\alpha_{s}^{2}}{\pi^{2}}C_{F}^{2}r\left[-2\ln^{2}r+\left(1-8\ln\frac{3}{2}\right)\ln
r+\cdots\right]$ (45)

Similarly, integrating against the soft matrix element and the soft-collinear
overlap region, we find

$\displaystyle\mathcal{S}_{\text{s}}^{(C_{F})}$ $\displaystyle=\int
d\Pi_{12}\left(4|\mathcal{M}_{\gamma^{*}\to
qgg\bar{q}}^{\text{soft}}|^{2}+2|\mathcal{M}_{\gamma^{*}\to
q\bar{q}gg}^{\text{soft}}|^{2}\right)=\frac{12\alpha_{s}^{2}}{\pi^{2}}C_{F}^{2}r\left[-\ln^{2}r+2\left(1+\ln\frac{4}{3}\right)\ln
r+\cdots\right]$ (46) $\displaystyle\mathcal{S}_{\text{sc}}^{(C_{F})}$
$\displaystyle=4\int d\Pi_{12}|\mathcal{M}_{\gamma^{*}\to
qgg\bar{q}}^{\text{soft-
coll}}|^{2}=\frac{12\alpha_{s}^{2}}{\pi^{2}}C_{F}^{2}r\left[-\ln^{2}r+2\left(1-2\ln\frac{3}{2}\right)\ln
r+\cdots\right]$ (47)

The constants in the integrals come from the permutations of final state
particles and we have accounted the symmetry factor for identical gluons. The
total is

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma^{(C_{F})}}{dr}$
$\displaystyle=\mathcal{S}_{\text{c}}^{(C_{F})}+\mathcal{S}_{\text{s}}^{(C_{F})}-\mathcal{S}_{\text{sc}}^{(C_{F})}$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}C_{F}^{2}r\left[-192\ln^{2}r+\left(96+768\ln
2-384\ln 3\right)\ln r+\cdots\right]$ (48)

This is compared to the exact (numerical) NLO calculation in the shoulder
region in Fig. 6.

For the $C_{F}C_{A}$ color structure, there can be single logarithms coming
from both the the $z\sim 0$ and $z\sim 1$ regions. Moreover, the splitting
functions in this case depend on the polarization of the gluon that splits.
However, because the only integration regions that contribute logarithms are
uniform in $\phi$ (the first two in Eq. (43)), one can simply azimuthally-
average the splitting functions, reducing them to the unpolarized case. The
final resumts we find are we find

$\displaystyle\mathcal{S}_{c}^{(C_{A})}$
$\displaystyle=\frac{\alpha_{s}^{2}}{\pi^{2}}C_{F}C_{A}r\left[-6\ln^{2}r+\left(1-24\ln\frac{3}{2}\right)\ln
r+\cdots\right]$ $\displaystyle\mathcal{S}_{s}^{(C_{A})}$
$\displaystyle=6\frac{\alpha_{s}^{2}}{\pi^{2}}C_{F}C_{A}r\left[-\ln^{2}r+2\left(1+\ln\frac{4}{3}\right)\ln
r+\cdots\right]$ $\displaystyle\mathcal{S}_{sc}^{(C_{A})}$
$\displaystyle=6\frac{\alpha_{s}^{2}}{\pi^{2}}C_{F}C_{A}r\left[-\ln^{2}r+2\left(1-2\ln\frac{3}{2}\right)\ln
r+\cdots\right]$ (49)

which gives

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma^{(C_{A})}}{dr}$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}C_{F}C_{A}r\left[-96\ln^{2}r+\left(16+384\ln
2-192\ln 3\right)\ln r+\cdots\right]$ (50)

Again, this is compared to NLO in the shoulder region in Fig. 6.

The $n_{f}T_{F}C_{F}$ color structure only contains a single logarithm since
there is no soft region. Integrating the collinear matrix element Eq. (28)
over power expanded phase space gives

$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma^{(n_{f})}}{dr}$
$\displaystyle=64\left(\frac{\alpha_{s}}{4\pi}\right)^{2}C_{F}T_{f}n_{f}r\ln
r+\cdots$ (51)

No overlap subtraction is needed. This is also shown in Fig. 6.

One can perform a similar leading-power computation for the right shoulder for
thrust and heavy jet mass. For these cases, we find it is only the phase space
regions with 1 parton in one hemisphere and 3 partons in the other hemisphere
that contribute. Since the equivalent calculation is significantly easier
using Soft-Collinear Effective Theory, we skip the details of the right-
shoulder cases using the full theory and turn instead to the effective theory
approach.

##  3 Factorization and Resummation

In Section 2, we computed the Sudakov shoulder logs for heavy jet mass and
thrust at NLO using full QCD expanded to leading power. We now want to
generalize the analysis to all orders leading to a factorization formula. To
do so, we first review the approach of Catani:1997xc  and and discuss recoil
sensitivity. We then different approach inspired by the NLO calculation leads
to a systematically improvable factorization formula.

###  3.1 Recoil sensitivity

One approach to resummation of Sudakov shoulders Catani:1997xc  is that
emissions from one of the hard partons will cause an additive shift in heavy
jet mass (or thrust) from $\rho\to\rho+m^{2}$. Then one could write the
resummed distribution as a convolution. Heuristically,

$\sigma_{\text{resummed}}(\rho)\sim\int
dm^{2}\sigma_{\text{LO}}(\rho-m^{2})J(m^{2})$ (52)

With $J(m^{2})$ representing some sort of jet function and
$\sigma_{\text{LO}}(\rho)$ the leading order cross section.

Unfortunately, when one tries to make this formula more precise it produces
ambiguities beyond the leading logarithmic order. To see this, consider how
$\rho$ changes due to emissions in the light hemisphere making the light
hemisphere have a mass $m^{2}$. With 3 massless partons taking $p_{1}$ and
$p_{2}$ in the heavy hemisphere and $p_{3}$ in the light hemisphere for
concreteness, the heavy jet mass is

$\rho=(p^{\mu}_{1}+p^{\mu}_{2})^{2}=(p^{\mu}_{\text{tot}}-p^{\mu}_{3})^{2}=1-2E_{3}$
(53)

with $E_{3}$ the energy of the light-hemisphere parton. Now say the $p_{3}$
parton becomes massive (i.e. turns into a jet) with $p_{3}^{2}=m^{2}$. Then we
have the exact relation

$\rho=(p^{\mu}_{1}+p^{\mu}_{2})^{2}=(p^{\mu}_{\text{tot}}-p^{\mu}_{3})^{2}=1+m^{2}-2E_{3}$
(54)

So it seems $\rho\to\rho+m^{2}$, as in Eq. (52). However, this was a little
too quick. For suppose instead of expressing $\rho$ in terms of $E_{3}$ we
expressed it in terms of $|\vec{p}_{3}|$. Then when $p_{3}$ is massless,

$\rho=1-2|\vec{p}_{3}|$ (55)

However after the emissions,

$\rho=1+m^{2}-2E_{3}=1+m^{2}-2\sqrt{\vec{p}_{3}^{\,2}+m^{2}}\cong
1+m^{2}-2|\vec{p}_{3}|-\frac{m^{2}}{|\vec{p}_{3}|}+\cdots$ (56)

Now, near threshold $|\vec{p}_{3}|\sim\frac{1}{3}$, so
$\frac{m^{2}}{|\vec{p}_{3}|}\cong 3m^{2}$ and we find $\rho\to\rho-2m^{2}$
instead of $\rho\to\rho+m^{2}$. Thus the way $\rho$ shifts depends on whether
we hold the energy or the momentum of the jet fixed after the emission. This
recoil-sensitivity seems to violate factorization. Moreover, if
$\rho\rightarrow\rho-2m^{2}$ one cannot write down a convolution for the
distribution as in (52), since the shift implies that emissions only decrease
the value of the heavy jet mass. Thus it becomes clear that while one might
use the emission picture for the double-logarithmic analysis of Catani:1997xc
, it is inadequate for NLL resummation.

###  3.2 Factorization

To proceed, recall from Section 2 which configurations contributed to the NLO
logs. With 4 partons, we can have either 2 in each hemisphere or 1 in one
light hemisphere and 3 in the heavy hemisphere. For the left shoulder of heavy
jet mass at NLO we found that only the case with 2 partons in each hemisphere
contributed. Moreover, the two partons in the heavy hemisphere were hard, with
invariant mass $\rho\sim\frac{1}{3}$, while the two partons in the light
hemisphere formed a jet of small invariant mass,
$s_{34}\sim\frac{1}{3}-\rho\ll 1$. In contrast, for the right shoulder of
heavy jet mass or thrust, only the region with 1 parton the light hemisphere
contributed. Moreover, the configuration in the heavy hemipshere had two hard
partons and one parton which was soft or collinear to one of the hard partons.

In the $r=\frac{1}{3}-\rho\ll 1$ region, we found integrals like

$\displaystyle I$
$\displaystyle\sim|{\mathcal{M}}_{0}|^{2}\frac{\alpha_{s}}{4\pi}C_{F}^{2}\int_{0}^{r}\frac{ds_{34}}{s_{34}}\int_{\frac{9}{4}s_{34}}^{1}\frac{dz}{z}\int_{\frac{1}{3}-r}^{\frac{1}{3}+2r}ds_{234}$
(57)
$\displaystyle\cong|{\mathcal{M}}_{0}|^{2}\frac{\alpha_{s}}{4\pi}C_{F}^{2}r\ln^{2}r$
(58)

The integrals over $s_{34}$ (the invariant mass of the $34$ jet) and $z$ (the
collinear splitting fraction in the $34$ jet) are similar to what we would
have in an inclusive jet function. The $s_{234}$ variable is a hard phase
space variable, equal to $s_{23}$ at leading power. The last integral gives
the factor of $r$ which is the same factor in the leading order cross section,
as in Eq. (4). Thus at higher orders it is natural to expect the
generalization of this integral to one with a single integral over hard
kinematic phase space and an integral over the kinematics of the light jet.
Thus, instead of convolution of the hard cross section with the emission cross
section, as in (52), we should expect the phase space to factorize into a part
which depends on the hard kinematics and a part which depends on the
emissions.

The first observation allowing us to factorize the cross section in the region
$r=\frac{1}{3}-\rho\ll 1$ is that only configurations which differ from the
trijet configuration by soft or collinear emissions can generate logarithms of
$r$. The reason for this is that $r=0$ is only special from the point of view
of 3-body massless kinematics. One can have 4-parton configurations with
$\rho$ close to $\frac{1}{3}$ that are not close to the trijet configuration.
However, such configurations contribute to the cross section both for $r<0$
and $r>0$ and will be smooth across $r=0$. Hence they cannot produce large
logarithms (in Section 4.2 we use this same argument to show there are no non-
global logs in the Sudakov shoulders).

So let use consider a generic configuration with 3 jets pointing in the
$n_{1}$, $n_{2}$ and $n_{3}$ directions. Such a configuration can have
particles collinear to the 3 directions as well as soft partons scattered
throughout phase space. At leading power, we can treat the collinear radiation
as generating masses $m_{1}$, $m_{2}$ and $m_{3}$ for the three jets. Thus we
can approximate the state as having three hard, massive particles with momenta
$p_{1},p_{2}$ and $p_{3}$ and soft radiation.

To compute heavy jet mass and thrust, we need to know which direction the
thrust axis points for a given amount of collinear and soft radiation. To
determine this, we first observe that as in Eq. (11), the thrust axis is
determined by the set of momenta in a given hemisphere that maximize

$T_{\\{p_{i}\\}}=2\Big{|}\sum_{p_{i}}\vec{p}\,\,\Big{|}$ (59)

Then $\tau$ and $\rho$ can be computed from the set $\\{p_{i}\\}$.

Let us begin with the case where there is only collinear momenta, so we only
have the 3 massive momenta to consider. In this case, phase space is described
by $s_{12},s_{13}$ and $s_{23}$ subject to
$s_{12}+s_{13}+s_{23}=1+m_{1}^{2}+m_{2}^{2}+m_{3}^{2}$, where
$s_{ij}=(p_{i}+p_{j})^{2}$. Then

$T_{1}^{2}=4\,\vec{p}_{1}^{\,2}=(1-s_{23})^{2}-2m_{1}^{2}(1+s_{23})+m_{1}^{4}$
(60)

and similarly for $T_{2}^{2}$ and $T_{3}^{2}$ by permutation. Let us take the
case where $T_{1}$ sets the thrust axis, so that $r=\frac{1}{3}-s_{23}$. Then
at leading power (assuming $m_{i}^{2}\sim r\sim s_{12}-\frac{1}{3}$)

$T_{1}\cong\frac{2}{3}+r-2m_{1}^{2}\quad
T_{2}\cong\frac{1}{3}-r+s_{12}-m_{1}^{2}-3m_{2}^{2}-m_{3}^{2}\quad T_{3}\cong
1-s_{12}-2m_{3}^{2}$ (61)

So the conditions $T_{1}>T_{2}$ and $T_{1}>T_{3}$ imply

$\frac{1}{3}-r+2m_{1}^{2}-2m_{3}^{2}<s_{12}<\frac{1}{3}+2r-m_{1}^{2}+3m_{2}^{2}+m_{3}^{2}$
(62)

These limits on $s_{12}$ pinch off when $r=m_{2}^{2}+m_{3}^{2}-m_{1}^{2}$. At
$m=0$ the linear scaling with $r$ of the $s_{12}$ integration region is what
generates the linear fall off of the thrust or heavy jet mass cross section as
in Eq. (4). For the integration region to be nonzero we therefore have

$m_{1}^{2}<r+m_{2}^{2}+m_{3}^{2}$ (63)

In other words, at fixed $m_{2},m_{3}$ and $r$, there is an upper limit on the
light-hemisphere jet mass. The probability of finding a light jet of mass at
most $m_{1}$ at leading power is proportional to $\ln^{2}m_{1}$, so for
$m_{2}=m_{3}=0$ the integral over $m_{1}$ up to $r$ will give the $\ln^{2}r$
left Sudakov shoulder logarithms. Combined with the factor of $r$ from the
$s_{12}$ integration gives an overall $r\ln^{2}r$ behavior. If $m_{2}$ and
$m_{3}$ are parametrically larger than $r$ then we can drop $r$ in Eq. (63).
In that case, no logs are generated. Thus the shoulder logs are determined by
the region of small $m_{1}$, $m_{2}$ and $m_{3}$, consistent with a global
observable.

The right shoulder for heavy jet mass is constrained by Eq. (63), but with
$r<0$. For the right shoulder we define $s=\rho-\frac{1}{3}$. Then

$m_{1}^{2}+s<m_{2}^{2}+m_{3}^{2}$ (64)

replaces Eq. (63).

For the thrust case, we define $t=\tau-\frac{1}{3}$. When $T_{1}$ determines
the thrust axis, then at leading power

$t=\frac{2}{3}-T_{1}\cong 2m_{1}^{2}-r$ (65)

and Eq. (63) becomes

$t<m_{1}^{2}+m_{2}^{2}+m_{3}^{2}$ (66)

Thus the right shoulder for thrust is defined by integrals over any of the
masses with a lower limit of $t$. Since the inclusive integral, without this
constraint, has no $t$ dependence, one can equivalently get the right Sudakov
shoulder logarithms by integrating over the masses constrained by
$m_{1}^{2}+m_{2}^{2}+m_{3}^{2}<t$.

For the soft radiation, we first need to determine when it affects the thrust
axis. Let’s start with the configuration with 3 massive partons and suppose
some soft radiation $k$ enters hemisphere $1$. We want to know whether the
thrust axis should shift so that hemisphere $1$ excludes $k$ or if it should
stay fixed, to include $k$. To find out, we need to compare $T_{1k}$, the
thrust value with $p_{1}$ and $k$ included in the hemisphere, to $T_{1}$ where
$k$ is not the 1-hemisphere, but is still included overall. A quick
calculation shows that

$T_{1k}^{2}\cong T_{1}^{2}+\frac{8}{3}(p_{2}\cdot k+p_{3}\cdot k-2p_{1}\cdot
k)$ (67)

Defining $p_{\bar{j}}$ as $p_{j}$ with its 3-momentum reversed, so

$p_{{\bar{1}}}=\frac{Q}{3}(1,0,0,-1)=\frac{2}{3}p_{2}+\frac{2}{3}p_{3}-\frac{1}{3}p_{1}$
(68)

we can write

$T_{1k}^{2}\cong T_{1}^{2}+4(p_{\bar{1}}\cdot k-p_{1}\cdot k)$ (69)

When $k$ is in the $1$ hemisphere it must be closer to $p_{1}$ than
$p_{\bar{1}}$. In that case $p_{\bar{1}}\cdot k>p_{1}\cdot k$. We conclude
that thrust is maximized when all the soft radiation in the hemisphere
centered on $p_{1}$ is included. In other words, if radiation is slightly on
the opposite side of the hemisphere boundary, the thrust axis should not shift
to cluster $k$ with $p_{1}$.

$n_{1}$$n_{2}$$n_{3}$$\bar{n}_{2}\cdot k>n_{2}\cdot k$$\bar{n}_{3}\cdot
k>n_{3}\cdot
k$$k_{1}$$~{}~{}k_{\bar{3}}$$k_{2}$$k_{\bar{1}}$$k_{3}$$k_{\bar{2}}$

Figure 4: Soft radiation from the trijet configuration can be categorized as
entering one of 6 sextant wedges shaped like carpels of an orange. The
boundary of each sextant is determined by two planes orthogonal to the jet
directions $n_{1}$, $n_{2}$ and $n_{3}$. For example, radiation in the sextant
labeled $k_{\bar{1}}$ (backwards to the $1$-jet) is characterized by
$\bar{n}_{2}\cdot k>n_{2}\cdot k$ and $\bar{n}_{3}\cdot k>n_{3}\cdot k$.

Now suppose there is a lot of soft radiation with momenta with
$\\{k_{i}^{\mu}\\}$. Since the thrust value goes up when radiation is included
in a given hemisphere, to find the thrust axis we only have to consider 3 sets
of momenta: for each $j$ the set includes a hard jet’s momentum $p_{j}$ and
all the soft radiation $k_{j}^{\text{hemi}}$ in the jet’s hemisphere. That is,
the maximal value of thrust for a hemisphere containing $p_{j}$ will be given
by

$T_{j}^{\text{max}}=T_{j}+3(p_{\bar{j}}\cdot k_{j}^{\text{hemi}}-p_{j}\cdot
k_{j}^{\text{hemi}})$ (70)

Since the jet hemispheres overlap, there will be some soft radiation included
in both $k_{1}^{\text{hemi}}$ and $k_{2}^{\text{hemi}}$, for example. To avoid
overcounting, let us decompose the soft momenta into 6 regions, as shown in
Fig. 4. So

$k_{1}^{\text{hemi}}=k_{1}+k_{\bar{2}}+k_{\bar{3}}$ (71)

and so on. Here $k_{j}$ is the soft radiation in the sextant centered on
$p_{j}$ and $k_{\bar{j}}$ is the soft radiation in the sextant opposite to
$p_{j}$.

Assuming $p_{1}$ is the thrust axis, then heavy jet mass

$\rho=(p_{2}+p_{3}+k_{\bar{1}}+k_{2}+k_{3})^{2}$ (72)

For $\rho<\frac{1}{3}$ we want to express costraints in terms of
$r=\frac{1}{3}-\rho$. For the hard kinematic variable, we can use anything
equal to $s_{12}$ at leading power. A convenient choice is

$\xi\equiv s_{12}-\frac{1}{3}+r-2m_{1}^{2}+2m_{3}^{2}\\\
+2(p_{2}-p_{1})\cdot(k_{1}+k_{\bar{3}})+2k_{2}\cdot(p_{1}+p_{2})-4k_{\bar{2}}\cdot(p_{1}-p_{3})+4k_{3}\cdot
p_{3}+4k_{\bar{1}}\cdot p_{3}$ (73)

The variable $\xi$ is defined so that
$T_{1k}^{\text{max}}=T_{2k}^{\text{max}}$ at $\xi=0$. Then phase space where
$T_{1k}^{\text{max}}=T_{2k}^{\text{max}}$ and
$T_{1k}^{\text{max}}=T_{3k}^{\text{max}}$ is $0\leq\xi\leq\frac{1}{3}W$ where

$W(r,m_{j},k_{i})=r-m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-2p_{1}k_{1}+2p_{2}k_{2}+2p_{3}k_{3}+2v_{\bar{1}}k_{\bar{1}}-2v_{\bar{2}}k_{\bar{2}}-2v_{\bar{3}}k_{\bar{3}}$
(74)

with

$\displaystyle v_{\bar{1}}$
$\displaystyle=-\frac{1}{3}p_{1}+\frac{2}{3}p_{2}+\frac{2}{3}p_{3}=\frac{Q}{3}(1,0,0,-1)$
(75) $\displaystyle v_{\bar{2}}$
$\displaystyle=\frac{4}{3}p_{1}+\frac{1}{3}p_{2}-\frac{2}{3}p_{3}=\frac{Q}{3}\left(1,0,\frac{\sqrt{3}}{2},\frac{3}{2}\right)$
(76) $\displaystyle v_{\bar{3}}$
$\displaystyle=\frac{4}{3}p_{1}-\frac{2}{3}p_{2}+\frac{1}{3}p_{3}=\frac{Q}{3}\left(1,0,-\frac{\sqrt{3}}{2},\frac{3}{2}\right)$
(77)

We have fixed the signs of the $v_{\bar{j}}$ so that they all have positive
energy. Since $v_{\bar{1}}=p_{\bar{1}}$, and $k_{\bar{1}}$ is close to
$p_{\bar{1}}$, we will have $v_{\bar{1}}\cdot k_{\bar{1}}\geq 0$ for all
$k_{\bar{1}}$. For the other directions, $\vec{v}_{\bar{2}}\cdot\vec{p}_{2}=0$
and $\vec{v}_{\bar{3}}\cdot\vec{p}_{3}=0$, and they will also have
$v_{\bar{2}}\cdot k_{\bar{2}}\geq 0$ and $v_{\bar{3}}\cdot k_{\bar{3}}\geq 0$.

For the integration range over $\xi$ to be nonzero we therefore need

${m_{1}^{2}+2p_{1}k_{1}+2v_{\bar{2}}k_{\bar{2}}+2v_{\bar{3}}k_{\bar{3}}<r+m_{2}^{2}+2p_{2}k_{2}+m_{3}^{2}+2p_{3}k_{3}+2v_{\bar{1}}k_{\bar{1}}}$
(78)

which is the same as $W(r,m_{j},k_{i})>0$, with $W$ in Eq.(74). Every term in
this expression is a positive quantity. This inequality applies to both the
left and right shoulder for heavy jet mass (for the right shoulder we prefer
to use $s=-r=\rho-\frac{1}{3}>0$).

For thrust, defining $t=\tau-\frac{1}{3}=\frac{2}{3}-T_{1k}$ the bound is
$0\leq x\leq T$ where

${T(t,m_{j},k_{i})=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2p_{1}k_{1}+2p_{2}k_{2}+2p_{3}k_{3}+2v_{\bar{1}}k_{\bar{1}}+2v_{\bar{2}}^{\prime}k_{\bar{2}}+2v_{\bar{3}}^{\prime}k_{\bar{3}}-t}$
(79)

where

$\displaystyle
v_{\bar{2}}^{\prime}=2p_{1}-v_{\bar{2}}=\frac{Q}{3}\left(1,0,-\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\frac{Q}{3}\bar{n}_{2}$
(80) $\displaystyle
v_{\bar{3}}^{\prime}=2p_{1}-v_{\bar{3}}=\frac{Q}{3}\left(1,0,\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\frac{Q}{3}\bar{n}_{3}$
(81)

So that

$t<m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+2p_{1}k_{1}+2p_{2}k_{2}+2p_{3}k_{3}+2v_{\bar{1}}k_{\bar{1}}+2v_{\bar{2}}^{\prime}k_{\bar{2}}+2v_{\bar{3}}^{\prime}k_{\bar{3}}$
(82)

For thrust, as for heavy jet mass, every term in this inequality is positive.

As observed in Section 2, we can set $m_{j}=k_{i}=0$ to zero in the hard
matrix elements at leading power. Then the integral over hard phase space
simply gives the maximum value of $\xi$ from Eqs. (74) or (79). That is, each
channel of the LO integral in Eq. (4) gets modified as

$48C_{F}\frac{\alpha_{s}}{4\pi}\int_{0}^{\frac{1}{3}-\tau}ds_{12}\to
48C_{F}\frac{\alpha_{s}}{4\pi}\int_{0}^{R/3}d\xi=48C_{F}\frac{\alpha_{s}}{4\pi}R\,\theta(R)$
(83)

with $\theta(x)$ the Heaviside step function.

The rate for producing collinear radiation is given by splitting functions,
and the cross section for producing collinear radiation of mass $m$ is given
by the inclusive jet function $J(m^{2})$. The rate for soft radiation is given
by a soft function, defined as an integral over emissions from Wilson lines
using a measurement function (see Section 3.3). The key equation, Eq. (78),
lets us then write the factorized expression for the heavy jet mass Sudakov
shoulder as

$\frac{1}{\sigma_{1}}\frac{d\sigma}{dr}=H(Q)\int
d^{3}m^{2}d^{6}qJ(m_{1}^{2})J(m_{2}^{2})J(m_{3}^{2})S_{6}(q_{i})W(m_{j},q_{i},r)\theta[\,W(m_{j},q_{i},r)]$
(84)

where

$\sigma_{1}=48C_{F}\frac{\alpha_{s}}{4\pi}\sigma_{0}$ (85)

The arguments of the 6-parameter soft function $S_{6}(q_{i})$ are the
projections $q_{i}=n_{i}\cdot k_{i}$ and $q_{\bar{i}}=v_{\bar{i}}\cdot
k_{\bar{i}}$. In terms of the $q_{i}$, Eq. (74) becomes

$W(m_{j},q_{i},r)=r-m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+\frac{2Q}{3}(q_{2}+q_{3}+q_{\bar{1}}-q_{1}-q_{\bar{3}}-q_{\bar{2}})$
(86)

We can simplify the factorized expression by defining a 2-parameter trijet
hemisphere soft function

$S(q_{\ell},q_{h})=\int d^{6}q_{i}\
S_{6}(q_{i})\delta(q_{\ell}-q_{1}-q_{\bar{2}}-q_{\bar{3}})\delta(q_{h}-q_{\bar{1}}-q_{2}-q_{3})$
(87)

where $q_{\ell}$ and $q_{h}$ represent the soft radiation in the light and
heavy hemispheres. This soft function contributes to the doubly-differential
distribution of the hemisphere masses as

$\frac{d^{2}\sigma}{dm_{\ell}^{2}dm_{h}^{2}}=H(Q,\mu)\int dm_{1}^{2}\
dm_{2}^{2}\ dm_{3}^{2}\ dq_{\ell}\ dq_{h}\
J(m_{1}^{2},\mu)J(m_{2}^{2},\mu)J(m_{3}^{2},\mu)S(q_{\ell},q_{h},\mu)\\\
\times\delta\left(m_{\ell}^{2}-m_{1}^{2}-\frac{2}{3}q_{\ell}Q\right)\delta\left(m_{h}^{2}-m_{2}^{2}-m_{3}^{2}-\frac{2}{3}q_{h}Q\right)$
(88)

And then

$\frac{d\sigma}{dr}=\int
dm_{h}^{2}dm_{\ell}^{2}\frac{d^{2}\sigma}{dm_{\ell}^{2}dm_{h}^{2}}(r+m_{h}^{2}-m_{\ell}^{2})\Theta(r+m_{h}^{2}-m_{\ell}^{2})$
(89)

One also must sum over channels, corresponding to which jet is the quark jet,
which is antiquark and which is gluon.

The factorization formula for thrust is similar:

$\frac{d\sigma}{dt}=\int
dm_{h}^{2}dm_{\ell}^{2}\frac{d^{2}\sigma}{dm_{\ell}^{2}dm_{h}^{2}}(m_{h}^{2}+m_{\ell}^{2}-t)\Theta(m_{h}^{2}+m_{\ell}^{2}-t)$
(90)

The soft function for thrust is the same as for heavy jet mass after changing
$v\to v^{\prime}$. As we will show in Appendix A changing $v\to v^{\prime}$
has no effect on the parts of the soft function relevant to NLL resummation,
so we will treat the HJM and thrust trijet hemisphere soft functions as being
the same.

###  3.3 Soft function

According to the analysis in the previous section, the factorization formula
requires a soft function giving the rate for producing gluons $k_{i}$ entering
one of 6 sextants, as in Fig. (4). In each sextant we need the projection
$p_{i}\cdot k_{i}$ for $i=1,2,3$ (sextants containing a jet) or $v_{i}\cdots
k_{i}$ for $i={\bar{1}},{\bar{2}},{\bar{3}}$ (sextants between jets). For NLL
resummation, we only need the anomalous dimension of the soft function at
1-loop. This can be determined by RG invariance. However, as a cross check on
the factorization formula, it is important to compute the soft function
explicitly.

It is convenient to introduce the scaleless vectors for the six direction that
appear in the measurment function

$p_{i}=\frac{Q}{3}n_{i},\quad v_{{\bar{i}}}=\frac{Q}{6}N_{i},\quad i=1,2,3$
(91)

where the $n_{i}$ can be read off from Fig. 1 and the $N_{i}$ from Eqs.
(75)-(77) (or see Appendix A). The vectors $p_{i}$, $n_{i}$, $v_{\bar{1}}$ and
$N_{\bar{1}}$ are lightlike while $v_{\bar{2}},v_{\bar{3}},N_{\bar{2}}$ and
$N_{\bar{3}}$ are spacelike. For heavy jet mass, the measurement function
$M(k,q_{i})$ is

$\displaystyle M(k,q_{i})$ $\displaystyle=\theta(n_{\bar{2}}\cdot k-n_{2}\cdot
k)\theta(n_{\bar{3}}\cdot k-n_{3}\cdot
k)\delta\left(q_{1}-\frac{2}{3}n_{1}\cdot k\right)$ (92)
$\displaystyle+\theta(n_{\bar{3}}\cdot k-n_{3}\cdot k)\theta(n_{\bar{1}}\cdot
k-n_{1}\cdot k)\delta\left(q_{2}-\frac{2}{3}n_{2}\cdot k\right)$ (93)
$\displaystyle+\theta(n_{\bar{1}}\cdot k-n_{1}\cdot k)\theta(n_{\bar{2}}\cdot
k-n_{2}\cdot k)\delta\left(q_{3}-\frac{2}{3}n_{3}\cdot k\right)$ (94)
$\displaystyle+\theta(n_{2}\cdot k-n_{\bar{2}}\cdot k)\theta(n_{3}\cdot
k-n_{\bar{3}}\cdot k)\delta\left(q_{\bar{1}}-\frac{2}{3}N_{1}\cdot k\right)$
(95) $\displaystyle+\theta(n_{3}\cdot k-n_{\bar{3}}\cdot k)\theta(n_{1}\cdot
k-n_{\bar{1}}\cdot k)\delta\left(q_{\bar{2}}-\frac{2}{3}N_{2}\cdot k\right)$
(96) $\displaystyle+\theta(n_{1}\cdot k-n_{\bar{1}}\cdot k)\theta(n_{2}\cdot
k-n_{\bar{2}}\cdot k)\delta\left(q_{\bar{3}}-\frac{2}{3}N_{3}\cdot k\right)$
(97)

The matrix element for Eikonal emission of one gluon off of 3 Wilson lines is
the same as for direct photon production Becher:2009th ; Schwartz:2016olw  or
hard $W/Z$ production Becher:2011fc ; Becher:2012xr . There are 3 Wilson lines
in the trijet configuration, pointing in the $n_{1},n_{2}$ and $n_{3}$
directions (see Fig. 1). When the jet in the 1 direction is a gluon, the
1-loop soft function is

$S_{6g}(q_{i})=2g_{s}^{2}\mu^{2\varepsilon}\int\frac{d^{d}k}{(2\pi)^{d-1}}\delta(k^{2})\theta(k_{0})M(k,q_{i})\\\
\times\left[\left(C_{F}-\frac{1}{2}C_{A}\right)\frac{n_{2}\cdot
n_{3}}{(n_{2}\cdot k)(n_{3}\cdot k)}+\frac{1}{2}C_{A}\frac{n_{1}\cdot
n_{2}}{(n_{1}\cdot k)(n_{2}\cdot k)}+\frac{1}{2}C_{A}\frac{n_{1}\cdot
n_{3}}{(n_{1}\cdot k)(n_{3}\cdot k)}\right]$ (98)

The soft function with a quark Wilson line in the 1 direction has the color
structures interchanged:

$S_{6q}(q_{i})=2g_{s}^{2}\mu^{2\varepsilon}\int\frac{d^{d}k}{(2\pi)^{d-1}}\delta(k^{2})\theta(k_{0})M(k,q_{i})\\\
\times\left[\left(C_{F}-\frac{1}{2}C_{A}\right)\frac{n_{1}\cdot
n_{2}}{(n_{1}\cdot k)(n_{2}\cdot k)}+\frac{1}{2}C_{A}\frac{n_{1}\cdot
n_{3}}{(n_{1}\cdot k)(n_{3}\cdot k)}+\frac{1}{2}C_{A}\frac{n_{2}\cdot
n_{3}}{(n_{2}\cdot k)(n_{3}\cdot k)}\right]$ (99)

$n_{1}$$n_{2}$$I_{1}$

$n_{1}$$n_{2}$$I_{2}$

$n_{1}$$n_{2}$$I_{3}$

$n_{1}$$n_{2}$$I_{4}$

Figure 5: There are 4 independent integrals needed for the soft function. The
shaded region indicates the measurement region which can be in various
positions relative to the Wilson lines.

Despite the preponderance of directions, the integrals required are all of the
same general form. By rotational invariance, we can always take the Wilson
lines to be the $n_{1}$ and $n_{2}$ directions. Then all the required
integrals are special cases of the general form

$I_{n_{a},n_{b},N}(q)=\int d^{d}k\frac{{n_{1}}\cdot{n_{2}}}{({n_{1}}\cdot
k)({n_{2}}\cdot k)}\delta(k^{2})\theta(k^{0})\delta\left(q-\frac{2}{3}N\cdot
k\right)\theta({n_{a}}\cdot k-{\bar{n}_{a}}\cdot k)\theta({n_{b}}\cdot
k-{\bar{n}_{b}}\cdot k)$ (100)

This integral is Lorentz invariant, so it can only depend on dot-products of
the 4-vectors involved and is also invariant under separate rescaling of all
the $n_{i}$. Related integrals, with a single $\theta$-function, appear in the
iterative solution of the BMS equation Banfi:2002hw  for non-global logarithms
of the light-jet mass distribution. There, a larger $\text{SL}(2,R)$ symmetry
constrains the functional form even more Schwartz:2014wha . Here, the
$\text{SL}(2,R)$ is broken by the second $\theta$-function, so the integration
region is a cats-eye shaped wedge inside the Poincare disk. However the
conformal coordinates proposed in Schwartz:2014wha  can still provide a useful
change of variables which we used to understand and simplify the integrals.

In the regions without a Wilson line, the anomalous dimension of the soft
function is insensitive to the projection vectors $N_{i}$; it only depends on
the location of the measurement region relative to the Wilson lines. Thus for
NLL resummation there are only 4 independent integrals, as illustrated in Fig.
5. A detailed calculation of the soft integrals can be found in Appendix A.
Here we just summarize the results. We find for the 4 integrals

$\displaystyle I_{1}(q)$
$\displaystyle=I_{n_{2},n_{3},n_{1}}(q)=\frac{1}{q^{1+2\epsilon}}\left(\frac{1}{\epsilon}-\frac{7}{2}\ln
2+\ln 3-\frac{3\kappa}{2\pi}\right)$ (101) $\displaystyle I_{2}(q)$
$\displaystyle=I_{n_{1},n_{2},n_{3}}(q)=\frac{1}{q^{1+2\epsilon}}\left(-\ln
2+\frac{3\kappa}{\pi}\right)$ (102) $\displaystyle I_{3}(q)$
$\displaystyle=I_{\bar{n}_{1},\bar{n}_{2},N_{i}}(q)=\frac{1}{q^{1+2\epsilon}}\left(\ln
2+\frac{3\kappa}{\pi}\right)$ (103) $\displaystyle I_{4}(q)$
$\displaystyle=I_{\bar{n}_{1},\bar{n}_{3},N_{i}}(q)=\frac{1}{q^{1+2\epsilon}}\left(\frac{3}{2}\ln
2-\frac{3\kappa}{2\pi}\right)$ (104)

where

$\kappa=\text{Im}\,\text{Li}_{2}\,e^{\frac{\pi i}{3}}\approx 1.0149$ (105)

is Gieseking’s constant. Gieseking’s constant is a transecendentality-2
number111It has not been proven whether Gieseking’s constant or Catalan’s
constant are transecendental, or even irrational. In this context,
transecendentality-2 refers to the representation of $\kappa$ as a 2-fold
iterated polylogarithic integral.  in the family with Catalan’s constant
$C=\text{Im}\text{Li}_{2}e^{\frac{\pi i}{2}}$ and $\pi^{2}=6\text{Li}_{2}(1)$.

Then, when we add in the colors structures, the soft function is

$I_{6g}(q_{i})\propto\delta(q_{1})\delta(q_{2})\delta(q_{2})\delta(q_{\bar{1}})\delta(q_{\bar{2}})\delta(q_{\bar{3}})\\\
+\delta(q_{2})\delta(q_{2})\delta(q_{\bar{1}})\delta(q_{\bar{2}})\delta(q_{\bar{3}})\left[\left(C_{F}-\frac{1}{2}C_{A}\right)I_{2}(q_{1})+C_{A}I_{1}(q_{1})\right]\\\
+\delta(q_{1})\delta(q_{2})\delta(q_{3})\delta(q_{\bar{2}})\delta(q_{\bar{3}})\left[\left(C_{F}-\frac{1}{2}C_{A}\right)I_{3}(q_{\bar{1}})+C_{A}I_{4}(q_{\bar{1}})\right]+\cdots$
(106)

and so on for the other four $q_{i}$ sectors and for $I_{6q}(q_{i})$. For the
trijet hemisphere soft function in Eq. (87), we can set all the $q_{i}$ in
each hemisphere equal For the channel with a gluon jet in the light hemisphere
we find

$S^{\text{hemi}}_{g}(q_{\ell},q_{h},\mu)\propto\delta(q_{\ell})\delta(q_{h})\\\
+\delta(q_{\ell})\left[\left(C_{F}-\frac{1}{2}C_{A}\right)(I_{2}(q_{h})+2I_{3}(q_{h}))+C_{A}(2I_{1}(q_{h})+I_{4}(q_{h}))\right]\\\
+\delta(q_{h})\left[\left(C_{F}-\frac{1}{2}C_{A}\right)(I_{2}(q_{\ell})+2I_{3}(q_{\ell}))+C_{A}(2I_{1}(q_{\ell})+I_{4}(q_{\ell}))\right]\\\
=\delta(q_{\ell})\delta(q_{h})+\frac{\alpha_{s}(\mu)}{4\pi}\delta(q_{\ell})\left[\frac{-4C_{F}\Gamma_{0}\ln\frac{q_{h}}{\mu}+2\gamma_{sqq}}{q_{h}}\right]_{\star}+\frac{\alpha_{s}(\mu)}{4\pi}\delta(q_{h})\left[\frac{-2C_{A}\Gamma_{0}\ln\frac{q_{\ell}}{\mu}+2\gamma_{sg}}{q_{\ell}}\right]_{\star}$
(107)

with $\Gamma_{0}=4$ and

$\gamma_{sqq}=-4C_{F}\ln 6,\quad\gamma_{sg}=-2C_{A}\ln 3+4C_{F}\ln 2$ (108)

Notation for the $\star$ distributions can be found in Becher:2006mr ;
Schwartz:2007ib ; Becher:2008cf ; Becher:2009th ; Becher:2011fc .

In the channel where the light hemisphere has quark jet, the trijet hemisphere
soft function has terms of the form

$S^{\text{hemi}}_{q}(q_{\ell},q_{h},\mu)\propto\delta(q_{\ell})\delta(q_{h})\\\
+\frac{\alpha_{s}(\mu)}{4\pi}\delta(q_{\ell})\left[\frac{-2(C_{F}+C_{A})\Gamma_{0}\ln\frac{q_{h}}{\mu}+2\gamma_{sqg}}{q_{h}}\right]_{\star}+\frac{\alpha_{s}(\mu)}{4\pi}\delta(q_{h})\left[\frac{-2C_{F}\Gamma_{0}\ln\frac{q_{\ell}}{\mu}+2\gamma_{sq}}{q_{\ell}}\right]_{\star}$
(109)

where

$\gamma_{sqg}=-2(C_{A}+C_{F})\ln
6,\quad\gamma_{sq}=-2C_{F}\ln\frac{3}{2}+2C_{A}\ln 2$ (110)

###  3.4 Resummation

To resum the large Sudakov shoulder logarithms, we convolve the resummed hard,
jet and soft function. The resummation of these individual functions is the
same as for thrust in the threshold limit Fleming:2007xt ; Schwartz:2007ib ;
Becher:2008cf  and other processes Becher:2006mr ; Becher:2007ty ;
Becher:2009th ; Chien:2010kc ; Stewart:2010tn ; Jouttenus:2011wh ;
Feige:2012vc .

The resummed quark and gluon jet functions have the form Becher:2009th :

$J_{i}(m^{2},\mu)=\exp\Big{[}-4C_{i}S(\mu_{j},\mu)+2A_{\gamma_{j}}(\mu_{j},\mu)\Big{]}\widetilde{j}_{q}(\partial_{\eta_{j}})\frac{1}{m^{2}}\left(\frac{m^{2}}{\mu_{j}^{2}}\right)^{\eta_{j}}\frac{e^{-\gamma_{E}\eta_{jq}}}{\Gamma(\eta_{j})}$
(111)

where the Laplace transform of the 1-loop jet functions are

$\tilde{j_{i}}(L)=1+\left(\frac{\alpha_{s}(\mu_{j})}{4\pi}\right)\left[C_{i}\Gamma_{0}\frac{L^{2}}{2}+\gamma_{i}L\right]$
(112)

and the Casimirs and 1-loop anomalous dimensions are

$C_{q}=C_{F},\quad
C_{g}=C_{A},\quad\gamma_{jq}=-3C_{F},\quad\gamma_{jg}=-\beta_{0}$ (113)

The Sudakov RG kernel is

$S(\nu,\mu)=-\int_{\alpha_{s}(\nu)}^{\alpha_{s}(\mu)}d\alpha\frac{\gamma_{\text{cusp}}(\alpha)}{\beta(\alpha)}\int_{\alpha_{s}(\nu)}^{\alpha}\frac{d\alpha^{\prime}}{\beta(\alpha^{\prime})}=-\frac{\alpha_{s}}{8\pi}\Gamma_{0}\ln^{2}\frac{\nu}{\mu}+\cdots$
(114)

with

$\displaystyle\gamma_{\text{cusp}}(\alpha_{s})$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)\Gamma_{0}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Gamma_{1}+\cdots$
(115) $\displaystyle\beta(\alpha_{s})$
$\displaystyle=-2\alpha_{s}\left[\left(\frac{\alpha_{s}}{4\pi}\right)\beta_{0}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\beta_{1}+\cdots\right]$
(116)

where

$\displaystyle\Gamma_{0}$
$\displaystyle=4,\quad\Gamma_{1}=4\left[C_{A}\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)-\frac{20}{9}T_{F}n_{f}\right]$
(117) $\displaystyle\beta_{0}$
$\displaystyle=\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f},\quad\beta_{1}=\frac{34}{3}C_{A}^{2}-\frac{20}{3}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{f}$
(118)

To NLL order

$A_{\gamma_{j}}(\nu,\mu)=-\gamma_{j}\int_{\alpha_{s}(\nu)}^{\alpha_{s}(\mu)}d\alpha\frac{\alpha}{4\pi\beta(\alpha)}=\frac{\gamma_{j}}{2\beta_{0}}\ln\frac{\alpha_{s}(\mu)}{\alpha_{s}(\nu)}$
(119)

Finally,

$\eta_{jq}=2C_{F}A_{\Gamma}(\mu_{j},\mu)\ ,\
\eta_{jg}=2C_{A}A_{\Gamma}(\mu_{j},\mu)\ ,$ (120)

where

$A_{\Gamma}(\nu,\mu)=-\int_{\alpha_{s}(\nu)}^{\alpha_{s}(\mu)}d\alpha\frac{\gamma_{\text{cusp}}(\alpha)}{\beta(\alpha)}=\frac{\alpha_{s}}{4\pi}\Gamma_{0}\ln\frac{\nu}{\mu}+\cdots$
(121)

The hard function can be extracted from Ellis:1980wv , or using the general
forms for hard functions in Becher:2009qa  or from the hard function for
$n$-jettiness Jouttenus:2011wh . It is

$H(Q,\mu)=\exp\Big{[}(4C_{F}+2C_{A})S(\mu_{h},\mu)-2A_{\gamma
h}(\mu_{h},\mu)\Big{]}\left(\frac{Q^{2}}{\mu_{h}^{2}}\right)^{-(2C_{F}+C_{A})A_{\Gamma}(\mu_{h},\mu)}H(Q,\mu_{h})$
(122)

where

$H(Q,\mu_{h})=1+\left(\frac{\alpha_{s}}{4\pi}\right)\left[-(2C_{F}+C_{A})\frac{\Gamma_{0}}{4}\ln^{2}\frac{Q^{2}}{\mu_{h}^{2}}-\gamma_{h}\ln\frac{Q^{2}}{\mu_{h}^{2}}\right],$
(123)

with

$\gamma_{h}=-2(2C_{F}+C_{A})\ln 3-6C_{F}-\beta_{0}$ (124)

The trijet hemisphere soft functions can be resummed in exactly the same
manner as the hemisphere soft function Fleming:2007xt ; Schwartz:2007ib ;
Becher:2008cf ; Hoang:2008fs ; Chien:2010kc . At NLL level they factorize into
the product of soft functions for each hemisphere:

$\displaystyle S_{g}^{\text{hemi}}(k_{\ell},k_{h},\mu)$
$\displaystyle=S_{g}(k_{\ell},\mu)S_{qq}(k_{h},\mu),$ (125) $\displaystyle
S_{q}^{\text{hemi}}(k_{\ell},k_{h},\mu)$
$\displaystyle=S_{q}(k_{\ell},\mu)S_{qg}(k_{h},\mu),$ (126)

The single-variable soft functions all have the same form

$S_{i}(k,\mu)=\exp\Big{[}2C_{i}S(\mu_{s},\mu)+2A_{\gamma_{i}}(\mu_{s},\mu)\Big{]}\widetilde{s}_{i}(\partial_{\eta_{i}})\frac{1}{k}\left(\frac{k}{\mu_{s}}\right)^{\eta_{i}}\frac{e^{-\gamma_{E}\eta_{i}}}{\Gamma(\eta_{i})}\\\
$ (127)

where

$\widetilde{s}_{i}(L)=1+\left(\frac{\alpha_{s}(\mu_{s})}{4\pi}\right)\left[C_{i}\Gamma_{0}\frac{L^{2}}{2}+\gamma_{i}L\right]$
(128)

and

$\eta_{i}=-2C_{i}A_{\Gamma}(\mu_{s},\mu)$ (129)

The only difference is the anomalous dimensions. The coefficient of the
Sudakov logs are determined by Casimir scaling as the sum of the color factors
for each parton in the hemisphere

$C_{g}=C_{A},\quad C_{qq}=2C_{F},\quad C_{q}=C_{F},\quad\text{and}\quad
C_{qg}=C_{F}+C_{A}$ (130)

The anomalous dimensions $\gamma_{sg},\gamma_{sq},\gamma_{sqq}$ and
$\gamma_{sqg}$ are in Eqs. (108) and (110).

Now we just have to put everything together and perform the integrals in Eqs.
(88), (89) and (90). Since the various functions after resummation are simply
powers, e.g. $J(m^{2})\sim(m^{2})^{\eta_{j}-1}$, the integrals are all
products or convolutions of powers, which can be done directly or through
Laplace transforms.

For thrust, with $t=\tau-\frac{1}{3}>0$, the core measurement function
integral following from Eq. (82) is

$\int_{0}^{\infty}dx\int_{0}^{\infty}dy~{}x^{a-1}y^{b-1}(x+y-t)\theta(x+y-t)=t^{1+a+b}\frac{\Gamma(a)\Gamma(b)}{\Gamma(2+a+b)}$
(131)

For the left shoulder of heavy jet mass, the integral is similar, but the sign
flip in Eq. (78) as compared to Eq. (82) gives an important change.

$\int_{0}^{\infty}dx\int_{0}^{\infty}dy\
x^{a-1}y^{b-1}(r+y-x)\theta(r+y-x)=r^{1+a+b}\frac{\Gamma(a)\Gamma(b)}{\Gamma(2+a+b)}\frac{\sin(\pi
a)}{\sin(\pi(a+b))}$ (132)

For the right shoulder of heavy jet mass we define $s=-r=\rho-\frac{1}{3}>0$.
Then the core integral is

$\int_{0}^{\infty}dx\int_{0}^{\infty}dy\
x^{a-1}y^{b-1}(y-x-s)\theta(y-x-s)=s^{1+a+b}\frac{\Gamma(a)\Gamma(b)}{\Gamma(2+a+b)}\frac{\sin(\pi
b)}{\sin(\pi(a+b))}$ (133)

These integrals are all UV and IR divergent, and so analytic continuation has
been used to complete them. We discuss the integrals in more detail in
Sections 4.2.

Putting everything together and applying algebraic simplifications as in
Becher:2008cf ; Becher:2009th , we find that all 3 observables can be written
in terms of the same RG evolution kernel. For the gluon channels

$\displaystyle\frac{1}{\sigma_{1}}\frac{d\sigma_{g}}{dt}$
$\displaystyle=\Pi_{g}(\partial_{{\eta_{\ell}}},\partial_{{\eta_{h}}})t\left(\frac{tQ}{\mu_{s}}\right)^{{{\eta_{\ell}}}}\left(\frac{tQ}{\mu_{s}}\right)^{{\eta_{h}}}\frac{e^{-\gamma_{E}({{\eta_{\ell}}}+{{\eta_{h}}})}}{\Gamma(2+{{\eta_{\ell}}}+\eta_{h})}$
(134) $\displaystyle\frac{1}{\sigma_{1}}\frac{d\sigma_{g}}{dr}$
$\displaystyle=\Pi_{g}(\partial_{{\eta_{\ell}}},\partial_{{\eta_{h}}})r\left(\frac{rQ}{\mu_{s}}\right)^{{{\eta_{\ell}}}}\left(\frac{rQ}{\mu_{s}}\right)^{{\eta_{h}}}\frac{e^{-\gamma_{E}({{\eta_{\ell}}}+{{\eta_{h}}})}}{\Gamma(2+{{\eta_{\ell}}}+\eta_{h})}\frac{\sin(\pi{{\eta_{\ell}}})}{\sin(\pi({{\eta_{\ell}}}+{{\eta_{h}}}))}$