***These authors made main contributions equally to this work.††institutetext:
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute
of Modern Physics, Fudan University,
Shanghai 200433, China

# 1-jettiness with jet axis at $O(\alpha_{s})$ in deep inelastic scattering

Zexuan Chu * Yunlu Wang * June-Haak Ee  Jinhui Chen  Daekyoung Kang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present $O(\alpha_{s})$ analytic predictions for event shape 1-jettiness
$\tau_{1}$ distribution aiming measurements in deep inelastic scattering
process at future Electron Ion Colliders. The result depends on conventional
variables $x$ and $Q$ as well as on $\tau_{1}$ and is relatively compact and
easy to implement for numerical calculation. Three different choices of axis,
with respect to which $\tau_{1}$ is measured are considered in the Breit
frame. The first is the one optimally adjusted to minimize $\tau_{1}$ and the
second and third are taken from anti-$k_{T}$ and Centauro jet algorithms
defined with a jet radius parameter $R$, respectively. We find that the first
and second give the same result at this order and are independent of $R$,
while the third depends on the radius. This fixed-order result provides a
nonsingular contribution to be combined with a singular log-resummed
contribution to give the full spectrum in $\tau_{1}$ space and also shows how
fixed-order and resummation regions change as a function of $x$ and $Q$.

##  1 Introduction

Jets, energetic hadron bunches produced in high-energy collisions, are useful
tool to study the strong dynamics induced by Quantum Chromodynamics (QCD) and
to probe new physics beyond the Standard Model. Its application to future QCD
machines that probe internal structure of ions with electron beam is
extensively investigated in Electron-Ion-Collider in the US (EIC)
AbdulKhalek:2021gbh ; Accardi:2012qut  and in Electron-ion-collider in China
(EicC) Anderle:2021wcy . The jets can be defined and studied exclusively using
jet finding algorithms Catani:1991hj ; Catani:1993hr ; Ellis:1993tq ;
Dokshitzer:1997in ; Salam:2007xv ; Cacciari:2008gp , which conventionally
require parameters like a jet radius and jet veto. On the other hand, they can
also be studied inclusively with classic observables called event shapes
Dasgupta:2003iq , which are theoretically simpler with a small number of
parameters, hence easier to achieve higher accuracy compared to exclusive jet
study. An example of event shapes is a thrust which has been predicted up to
N3LL$+\mathcal{O}(\alpha_{s}^{3})$ accuracy GehrmannDeRidder:2007bj ;
GehrmannDeRidder:2007hr ; Weinzierl:2008iv ; Weinzierl:2009ms ; Becher:2008cf
; Abbate:2010xh  in $e^{+}e^{-}$ annihilation. From the event shape, the
strong coupling constant was determined at 1% precision Becher:2008cf ;
Chien:2010kc ; Abbate:2010xh , which is one among the most precise
determinations listed in Particle Data Book ParticleDataGroup:2020ssz .

The thrust as well as other event shapes in deep-inelastic scattering (DIS)
was studied Dasgupta:2003iq  and measured in HERA experiment Adloff:1997gq ;
Adloff:1999gn ; Aktas:2005tz ; Breitweg:1997ug ; Chekanov:2002xk ;
Chekanov:2006hv . Due to limited detector coverage they were defined from
particles in a current hemisphere, to which products of hard scattering
usually belong while initial-state radiations and beam remnants do not. On the
other hand, the definition in $e^{+}e^{-}$ is done with both hemispheres. More
recent developments including improved accuracy and/or new event shape
predictions Kang:2013nha ; Kang:2013lga ; Aschenauer:2019uex ; Li:2020bub ;
Kang:2020fka ; Li:2021txc ; Zhu:2021xjn  assume that future machines can cover
both regions. With the assumption, DIS thrust, which we call a 1-jettiness
Kang:2013nha ; Kang:2013lga  can be defined as 111It is called 1-jettiness
since it is $N=1$ version of N-jettiness Stewart:2010tn  that is generalized
thrust for N-jet events

$\tau_{1}=\frac{2}{Q^{2}}\sum_{i\in
X}\min\\{q_{B}\\!\cdot\\!p_{i},q_{J}\\!\cdot\\!p_{i}\\}\,,$ (1)

where $p_{i}$ is the momentum of particle $i$ in the final state $X$ and $Q$
is a hard momentum transferred by a virtual photon. $q_{B}$ and $q_{J}$ are
beam and jet axes, onto which momentum $p_{i}$ is projected. The operator
‘min’ takes smaller one among two scalar products and this makes particles
grouped into one of two regions, beam or jet. The value of $\tau_{1}$ is small
when the final state $X$ contains two collimated bunches along each of beam
and jet axes. Otherwise, for multi-jet final state the value is not small. In
Kang:2013nha , a version called $\tau_{1}^{b}$ was defined by $z$ axis in the
Breit frame for $q_{J}$ and known to be same as a version of DIS thrust called
$\tau_{Q}$ measured in HERA Antonelli:1999kx . It was computed analytically at
the first order in $\alpha_{s}$ Kang:2014qba .

In this paper, we study another version called $\tau_{1}^{a}$ Kang:2012zr ;
Kang:2013nha  for which the jet axis $q_{J}$ is determined by axis finding
algorithms, while $q_{B}$ defined next section is held fixed to the beam axis.
Its distribution was numerically computed in the Lab frame Kang:2013lga  by
using the axis obtained from anti-$k_{T}$ algorithms. In this paper we
consider three different algorithms in the Breit frame in Sec. 2 and show
analytic expressions of our fixed-order predictions at the first order in
$\alpha_{s}$ and separately next-to-leading power (NLP) terms in small
$\tau_{1}$ limit in Sec. 3 as well as numerical results in Sec. 4. Finally, we
summarize in Sec. 5.

##  2 1-jettiness with three jet axes

In this section we define the jet axis $q_{J}$ and write down the expression
of 1-jettiness defined with the axis for 2-body final state. We consider three
different jet axes in the Breit frame. The first axis is the axis that
minimizes the value of 1-jettiness as in $e^{+}e^{-}$ thrust and we call it a
1-jettiness axis. Next two axes are obtained from jet momentum defined by
anti-$k_{T}$ algorithm Cacciari:2008gp  and by Centauro algorithm
Arratia:2020ssx .

We first make a quick review of kinematics in the Breit frame. In the frame,
the virtual photon with momentum $q$ and the proton with momentum $P$ are
aligned along the $z$ axis. One of advantages of this frame is that the
initial state radiation moving along the proton direction is well separated
from the other particles produced by a hard scattering. We take the proton to
move against $z$ direction

$P^{\mu}=\frac{Q}{x}\frac{\bar{n}_{z}^{\mu}}{2}\,,$ (2)

where the unit vectors $n_{z}=(1,0,0,1)$ and $\bar{n}_{z}=(1,0,0,-1)$, the
hard scale $Q$ is the photon virtuality $Q^{2}=-q^{2}$ and the Björken
variable $x=Q^{2}/(2P\\!\cdot\\!q)$. The virtual photon $q$ is spacelike

$q^{\mu}=Q\frac{n_{z}^{\mu}-\bar{n}_{z}^{\mu}}{2}\,.$ (3)

A parton from the proton taking a fractional momentum $p=\xi
P=Q/z\tfrac{\bar{n}_{z}}{2}$, where $z=x/\xi$ and $x<z<1$, scatters off the
photon. At order $\mathcal{O}(\alpha_{s})$, the final states contain two
particles with momenta $p_{1}$ and $p_{2}$. Using the momentum conservation
$p+q=p_{1}+p_{2}$ and the onshell condition $p_{1}^{2}=p_{2}^{2}=0$ we can
express the final momenta in terms of $z$ and a dimensionless variable
$v=p^{-}_{2}/Q$ as

$\displaystyle p_{1}^{\mu}$
$\displaystyle=Q(1-v)\frac{n_{z}^{\mu}}{2}+Q\frac{1-z}{z}v\frac{\bar{n}_{z}^{\mu}}{2}-Q\sqrt{\frac{1-z}{z}(1-v)v}~{}n_{\perp}^{\mu},$
(4) $\displaystyle p_{2}^{\mu}$
$\displaystyle=Qv\frac{n_{z}^{\mu}}{2}+Q\frac{1-z}{z}(1-v)\frac{\bar{n}_{z}^{\mu}}{2}+Q\sqrt{\frac{1-z}{z}(1-v)v}~{}n_{\perp}^{\mu},$

where $0<v<1$ and $n_{\perp}$ is an orthogonal unit vector to $n_{z}$ and
$\bar{n}_{z}$, i.e.,
$n_{\perp}\\!\cdot\\!n_{z}=n_{\perp}\\!\cdot\\!\bar{n}_{z}=0$ and
$n_{\perp}\cdot n_{\perp}=-1$.

The beam axis $q_{B}$ is defined to be proportional to the proton momentum

$q_{B}^{\mu}=xP^{\mu}=Q\frac{\bar{n}_{z}^{\mu}}{2}\,.$ (5)

The scalar products of the beam axis $q_{B}$ and $p_{i}$ contributing to
1-jettiness can be expressed as

$\displaystyle 2q_{B}\\!\cdot\\!p_{1}=(1-v)Q^{2}\,,$ $\displaystyle
2q_{B}\\!\cdot\\!p_{2}=vQ^{2}\,,$ $\displaystyle
2q_{B}\\!\cdot\\!(p_{1}+p_{2})=Q^{2}\,.$ (6)

###  2.1 1-jettiness axis $\tau^{\text{jt}}$

In the determination of 1-jettiness axis, we allow the direction of jet axis
to be varied, while the absolute magnitude of three-momentum part to be held
fixed to $Q/2$. Then, the $q_{J}$ can be expressed as

$q_{J}^{\mu}=Q\frac{n_{J}^{\mu}}{2}=\frac{Q}{2}(1,\hat{n}_{J})^{\mu}\,,$ (7)

where $\hat{n}_{J}$ is a unit vector adjusted event by event in such a way
that minimizes the value of 1-jettiness. Rewriting Eq. (1) with $p_{1}$ and
$p_{2}$ explicitly gives222From now on we drop the subscript “1” in
1-jettiness $\tau_{1}$ for a conventional simplicity hence, $\tau^{\text{jt}}$
implies $\tau_{1}^{jt}$ and same for $\tau^{b}$

$\displaystyle\tau^{\text{jt}}$
$\displaystyle=\frac{2}{Q^{2}}\mathop{\textrm{min}}_{\hat{n}_{J}}\left\\{\sum^{2}_{i=1}\textrm{min}\\{q_{B}\\!\cdot\\!p_{i},q_{J}\\!\cdot\\!p_{i}\\}\right\\}$
$\displaystyle=\text{min}\left\\{q_{B}\\!\cdot\\!(p_{1}+p_{2}),\;\mathop{\textrm{min}}_{\hat{n}_{J}}\\{q_{J}\\!\cdot\\!(p_{1}+p_{2})\\},\;q_{B}\\!\cdot\\!p_{1}+\mathop{\textrm{min}}_{\hat{n}_{J}}\\{q_{J}\\!\cdot\\!p_{2}\\},\;\mathop{\textrm{min}}_{\hat{n}_{J}}\\{q_{J}\\!\cdot\\!p_{1}\\}+q_{B}\\!\cdot\\!p_{2}\right\\}\,$
$\displaystyle=\textrm{min}\left\\{\;1\,,\;\theta\left(-z+\frac{1}{2}\right)+\frac{1-z}{z}\theta\left({z-\frac{1}{2}}\right)\,,\;1-v\,,\;v\right\\}$
$\displaystyle=(1-v)~{}\Theta_{\text{I}}(v,z)+v~{}\Theta_{\text{II}}(v,z)+\frac{1-z}{z}~{}\Theta_{\text{III}}(v,z)\,.$
(8)

The outer ‘min’ operator with $\hat{n}_{J}$ beneath it in the first line
implies that it is adjusted to minimize quantities in the braces. In the
second line, we listed all four combinations of scalar products between
$q_{B,J}$ and $p_{1,2}$ then, moved the outer ‘min’ inside so that the axis
$\hat{n}_{J}$ is determined by minimizing each scalar product. For the first
product there is no product with $q_{J}$ and $\hat{n}_{J}$ is not determined.
From the second to the last $\hat{n}_{J}=\hat{z},\hat{p}_{2},$ and
$\hat{p}_{1}$, respectively. We used Eq. (2) to express the scalar products in
terms of $v$ and $z$ and obtained the third line. Then, the outer ‘min’
selects the smallest among those four and it leads to three regions denoted by
$\Theta_{i}$

$\displaystyle\Theta_{\text{I}}$
$\displaystyle=\theta\left(v-\frac{1}{2}\right)\theta\left(v-\frac{2z-1}{z}\right)\,,$
$\displaystyle\Theta_{\text{II}}$
$\displaystyle=\theta\left(-v+\frac{1}{2}\right)\theta\left(-v+\frac{1-z}{z}\right)\,,$
$\displaystyle\Theta_{\text{III}}$
$\displaystyle=\theta\left(-v+\frac{2z-1}{z}\right)\theta\left(v-\frac{1-z}{z}\right)\,,$
(9)

where the step function $\theta(x)$ is 1 for $x>0$ and 0 otherwise. The three
regions are plotted in the left panel of Fig. 1. From the expression we find
the maximum value of $\tau^{\text{jt}}$

$\tau^{\text{jt}}_{\max}=\frac{1}{2}\theta\left(-x+\frac{2}{3}\right)+\frac{1-x}{x}\theta\left(x-\frac{2}{3}\right),$
(10)

which is a constant in $v$. We used the hadronic variable $x$ in advance
instead of the partonic variable $z$, which is replaced by $x$ at the end. The
maximum occurs on the heavy lines in Fig. 1.

Figure 1:  Two-body phase space for 1-jettiness, $\tau^{\text{jt,kt}}$ with
jettiness/anti-$k_{T}$ axis (left) and $\tau^{\text{ct}}$ with Centauro axis
(right). 1-jettiness takes the same expression in first three regions while
the fourth region is only for $\tau^{\text{ct}}$. The bound of fourth region
for jet radius $R=1$ and $2$ is shown in red dashed and red solid,
respectively.

###  2.2 Anti-$k_{T}$ algorithm $\tau^{\text{kt}}$

In the $k_{T}$-type algorithms Cacciari:2008gp  designed to be invariant under
longitudinal boosts, the distance between particles is defined in term of
transverse momentum $\mathbf{p}_{T}$ and invariant angular distances

$\displaystyle d_{ij}=d_{ji}$
$\displaystyle=\min\big{\\{}\mathbf{p}_{Ti}^{2p}\,,\mathbf{p}_{Tj}^{2p}\big{\\}}\frac{\Delta
R^{2}_{ij}}{R^{2}},\quad d_{iB}=\mathbf{p}_{Ti}^{2p}\,,$ (11)
$\displaystyle\Delta R^{2}_{ij}$
$\displaystyle=(y_{i}-y_{j})^{2}+(\varphi_{i}-\varphi_{j})^{2}\,,$ (12)

where $d_{ij}$ is the distance between outgoing particles $i$ and $j$, and
$d_{iB}$ is the distance of a particle $i$ from the beam. $\mathbf{p}_{Ti}$ is
the magnitude of the transverse momentum of particle $i$, $R$ is a jet radius
parameter of $\mathcal{O}(1)$ and $\Delta R_{ij}$ is an angular distance
defined in terms of distances in rapidity $y$ and in azimuth $\varphi$. The
power $p$ on the momentum is a real valued parameter. When $p=1,0,-1$, it is
called $k_{T}$, Cambridge/Aachen and anti-$k_{T}$, respectively. In this paper
we take the anti-$k_{T}$ algorithm $p=-1$, which is a conventional choice by
many experiments.

With above distances, the jet algorithm finds minimum $d_{\text{min}}$ among
all the $d_{ij}$ and $d_{iB}$. If $d_{\text{min}}$ is a $d_{ij}$, particle $i$
and $j$ are merged into a single particle and if $d_{\text{min}}$ is a
$d_{iB}$, the particle is declared to be a final jet and removed from the list
of final particles. We repeat this with new list of particles until all the
particles in the list are gone.

For a two-body final state, we need to compare $d_{12},d_{1B}$ and $d_{2B}$ to
classify jets. Recall that in the Breit frame the transverse momenta of two
particles in Eq. (4) are back-to-back so that

$d_{1B}=d_{2B}=|{\mathbf{p}_{T1}}|^{-2}\,.$ (13)

The distance in azimuthal angle between $p_{1,2}$ is
$(\varphi_{1}-\varphi_{2})^{2}=\pi^{2}$ and the distance in rapidity is
$(y_{1}-y_{2})^{2}=\ln^{2}\tfrac{1-v}{v}$. Then, the distance $d_{ij}$ is

$d_{12}=|{\mathbf{p}_{T1}}|^{-2}\frac{\pi^{2}+\ln^{2}\frac{1-v}{v}}{R^{2}}\,.$
(14)

Typical size of jet radius $R$ is smaller than $\pi$ and
$\ln^{2}\tfrac{1-v}{v}>0$ hence, $d_{ij}$ is always greater than $d_{1B}$.
This means that there is no merging in the two-body final state. The
anti-$k_{T}$ algorithm gives two jets and each of them is $p_{1}$ and $p_{2}$
and the jet momentum of each jet is $\mathbf{p}_{i}$. The jet axis $q_{J}$ is
defined from the jet momentum as

$q_{J}^{kt}=\big{\\{}|\mathbf{p}_{1}|(1,\hat{\mathbf{p}}_{1}),|\mathbf{p}_{2}|(1,\hat{\mathbf{p}}_{2})\big{\\}}=\big{\\{}p_{1},p_{2}\big{\\}}\,,$
(15)

where in the second equality, we used the fact that $p_{i}$ is massless. In
higher multiplicity, the jet algorithm returns more jets. Among those jets,
what we need is collinear jets with a large momentum that are candidates for
the jet axis $q_{J}$ and by implementing a veto condition we can reject
unnecessary soft jets with a small momentum. Among the candidates, the one
that gives the smallest value of 1-jettiness will be selected. Then,
1-jettiness with the candidates $p_{1}$ and $p_{2}$ is given by

$\displaystyle\tau^{\text{kt}}$
$\displaystyle=\frac{2}{Q^{2}}\min_{q_{J}\in\left\\{p_{1},p_{2}\right\\}}\bigg{\\{}\sum_{i=1}^{2}\min\\{q_{B}\\!\cdot\\!p_{i}\,,q_{J}\\!\cdot\\!p_{i}\\}\bigg{\\}}$
(16)
$\displaystyle=\frac{2}{Q^{2}}\min\bigg{\\{}\sum_{i=1}^{2}\min\\{q_{B}\\!\cdot\\!p_{i}\,,q_{1}\\!\cdot\\!p_{i}\\}\,,\,\sum_{i=1}^{2}\min\\{q_{B}\\!\cdot\\!p_{i}\,,q_{2}\\!\cdot\\!p_{i}\\}\bigg{\\}}$
$\displaystyle=\frac{2}{Q^{2}}\textrm{min}\bigg{\\{}\min\\{q_{B}\\!\cdot\\!(p_{1}+p_{2}),\,p_{1}\\!\cdot\\!(p_{1}+p_{2}),\,q_{B}\\!\cdot\\!p_{1}+p_{1}\\!\cdot\\!p_{2}\,,\,q_{B}\\!\cdot\\!p_{2}+p_{1}\\!\cdot\\!p_{1}\\}\,,\,$
$\displaystyle\qquad\qquad\qquad\min\\{q_{B}\\!\cdot\\!(p_{1}+p_{2}),\,p_{2}\\!\cdot\\!(p_{1}+p_{2}),\,q_{B}\\!\cdot\\!p_{2}+p_{1}\\!\cdot\\!p_{2}\,,\,q_{B}\\!\cdot\\!p_{1}+p_{2}\\!\cdot\\!p_{2}\\}\bigg{\\}}$
$\displaystyle=\min\bigg{\\{}\frac{1-z}{z},\,v,\,1-v\bigg{\\}}$
$\displaystyle=\tau^{\text{jt}}\,,$

where in the second line we set $q_{J}=p_{1}$ and $q_{J}=p_{2}$ after first
and second summation symbols, respectively. Then, as was done in Eq. (2.1),
all possible combinations are taken into account in the third line. To get the
fifth line, we used Eq. (2) then, deleted trivial terms that cannot be taken
as a minimum. Interestingly, we come up with a fact that $\tau^{\text{kt}}$ is
equal to $\tau^{\text{jt}}$ in the two-body case. This is because of
simplification with a small number of final particles. In general, the radius
$R$ dependence enters in the case of anti-$k_{T}$ algorithm and we do not
expect that $\tau^{\text{jt}}$ and $\tau^{\text{kt}}$ are the same with higher
multiplicity. We do not consider higher multiplicity in this paper and from
now on, we do not separately treat $\tau^{\text{kt}}$ except for the case when
we need to distinguish $\tau^{\text{kt}}$ from $\tau^{\text{jt}}$.

###  2.3 Centauro algorithm $\tau^{\text{ct}}$

For the third jet axis we would like to adopt the Centauro algorithm
Arratia:2020ssx , which is more recently introduced algorithm for DIS taking
into account the target-current asymmetry in the Breit frame. It is still
longitudinal-boost invariant like the anti-$k_{T}$ algorithm but it allows to
capture jets close to beam axis while the anti-$k_{T}$ algorithm cannot form a
jet in the region since the rapidity distances between particles across the
beam line become very large $(y_{i}-y_{j})\to\pm\infty$.

The Centauro algorithm defines the following distance measure

$\displaystyle d_{ij}$ $\displaystyle=\left[(\Delta
f_{ij})^{2}+2f_{i}f_{j}(1-\cos\Delta\phi_{ij})\right]/R^{2},\quad d_{iB}=1\,,$
(17)

where $\Delta f_{ij}$ and $\Delta\phi_{ij}$ are differences between $f_{i}$
and $f_{j}$ and $\phi_{i}$ and $\phi_{j}$, respectively. The function $f_{i}$
is a function of an angular variable $\bar{\eta}_{i}$ of particle $i$

$\displaystyle f_{i}$
$\displaystyle=f(\bar{\eta}_{i})\,,\qquad\qquad\bar{\eta}_{i}=2\frac{|\mathbf{p}_{iT}|}{\bar{n}_{z}\\!\cdot\\!p_{i}}\,,$
(18)

where $\bar{\eta}$ diverges in forward region as
$\bar{n}_{z}\\!\cdot\\!p_{i}\to 0$ and thus prevents jets from enclosing the
proton beam direction, while it decreases as particles become closer in
backward region ${\bar{n}_{z}\\!\cdot\\!p_{i}}\gg|\mathbf{p}_{iT}|$. For the
function $f$, we take the simplest form in Arratia:2020ssx

$f(\bar{\eta})=\bar{\eta}\,.$ (19)

Then, $f_{1}$ and $f_{2}$ and the distance $d_{12}$ are given by

$\displaystyle f_{1}$
$\displaystyle=2\sqrt{\frac{1-z}{z}}\sqrt{\frac{v}{1-v}}\,,\qquad\qquad
f_{2}=2\sqrt{\frac{1-z}{z}}\sqrt{\frac{1-v}{v}}\,,$ (20) $\displaystyle
d_{12}$
$\displaystyle=\frac{(f_{1}+f_{2})^{2}}{R^{2}}=\frac{4}{R^{2}}\frac{1-z}{z}\frac{1}{v(1-v)}\,,$
(21)

where $R$ is a radius parameter of order $\mathcal{O}(1)$. If $d_{\text{min}}$
is a $d_{iB}$, we have the same jet as the anti-$k_{T}$ jet. If
$d_{\text{min}}$ is a $d_{12}$, we have a jet with momentum
${\mathbf{p}}_{1}+{\mathbf{p}}_{2}$, which is not allowed in the anti-$k_{T}$
algorithm. The condition for two particles merging into a single jet is
$d_{12}<d_{iB}$, which indicates the following region

$\displaystyle{\Theta_{\text{IV}}}(z,v)=\theta\left(z-z_{c}(R)\right)\theta\left(v-v_{-}(z,R)\right)\theta\left(-v+v_{+}(z,R)\right)\,,$
(22)

where $v_{\pm}$ are upper and lower bounds of the variable $v$ and
$v_{\pm}=1/2$ at the value $z=z_{c}(R)$

$\displaystyle
v_{\pm}(z,R)=\frac{1}{2}\pm\frac{1}{2}\sqrt{1-\frac{1-z}{z}\frac{16}{R^{2}}}\,,\quad
z_{c}(R)=\frac{16}{16+R^{2}}\,.$ (23)

In this region, the jet momentum is $\mathbf{p}_{1}+\mathbf{p}_{2}$ and
$q_{J}$ always is in the opposite to the proton direction

$\displaystyle
q_{J}^{\text{ct}}=|\mathbf{p}_{1}+\mathbf{p}_{2}|(1,\hat{n}_{12})=Q\frac{2z-1}{z}\,\frac{n_{z}}{2}\,,$
(24)

where
$\hat{n}_{12}=(\mathbf{p}_{1}+\mathbf{p}_{2})/|\mathbf{p}_{1}+\mathbf{p}_{2}|$.
Otherwise $q_{J}^{\text{ct}}=q_{J}^{\text{kt}}$ is one of $p_{1}$ and $p_{2}$
as in Eq. (15). The right panel in Fig. 1 indicates the phase space for
1-jettiness with the Centauro algorithm. The bounds of region IV for $R=1,2$
are shown in the figure in red curves. The region reduces to zero as $R\to 0$
and invades the regions I and II for $R>2$. In many experiments, $R$ is taken
to be less than or, equal to 2 and the phase space in this range is also
relatively simple. So, we constrain the region of our interest to be

$R\leq 2\,.$ (25)

Then, in the region IV the value of $\tau^{\text{ct}}$ can be computed with
Centauro-jet axis in Eq. (24) and in the regions I, II, and III
$\tau^{\text{ct}}=\tau^{\text{jt}}$. Now $\tau^{\text{ct}}$ in any region can
be expressed as

$\displaystyle\tau^{\text{ct}}={\Theta_{\text{IV}}}\frac{(2z-1)(1-z)}{z^{2}}+(1-{\Theta_{\text{IV}}})\tau^{\text{jt}}\,.$
(26)

Note that $\tau^{\text{ct}}$ is always positive in the region IV as well as
the other regions because the minimum value of $z$ is $z_{c}(R)\geq 4/5$ in
the range given by Eq. (25).

##  3 Analytic cross section at order $\alpha_{s}$

In this section we summarize the analytic result for 1-jettiness cross section
in DIS computed at the first order in $\alpha_{s}$. Details of the
calculations are given in App. A. Here, our results are given in the form of
cumulative cross section obtained by integrating the differential cross
section

$\sigma^{\rm
c}(\tau,x,Q)=\int_{0}^{\tau}d\tau^{\prime}\frac{d\sigma}{dx\,dQ^{2}\,d\tau^{\prime}}\,.$
(27)

The DIS cross section is conventionally decomposed into the structure
functions $F_{i}$

$\displaystyle\sigma^{\rm c}(\tau,x,Q)$
$\displaystyle=\frac{4\pi\alpha^{2}}{Q^{4}}\left[(1+(1-y)^{2})F_{1}+\frac{1-y}{x}F_{L}\right]\,.$
(28)

This can be expressed in terms of $F_{1},F_{2}$ by using the relation between
structure functions $F_{2}=F_{L}+2xF_{1}$. Note that the structure functions
are the functions of $\tau$ as well as $x$ and $Q$. At the point
$\tau=\tau_{\text{max}}$, the cumulative functions $\sigma^{\rm c}$ and
$F_{i}$ reduce to the inclusive cross section and inclusive structure
functions, respectively. The $F_{i}$ are coefficients of basis tensors that
decompose a current-current correlator called the hadronic tensor $W^{\mu\nu}$
defined in Eq. (A) and one can read off the individual coefficients by
projecting with orthogonal tensors like $P_{\mu}P_{\nu}W^{\mu\nu}$ and
$g_{\mu\nu}W^{\mu\nu}$ as shown in Eq. (A).

The structure functions are written in terms of these projected correlators

$\displaystyle F_{1}$ $\displaystyle=\sum_{i=q,\bar{q},g}(A_{i}+B_{i})\,,$
(29) $\displaystyle F_{L}$ $\displaystyle=\sum_{i=q,\bar{q},g}4x\,A_{i}\,,$

where $A_{i}$ and $B_{i}$ are respectively equivalent to
$P_{\mu}P_{\nu}W^{\mu\nu}$ and $g_{\mu\nu}W^{\mu\nu}$ up to multiplicative
factors that can be found in Eqs. (98) and (101). They contain logarithmic
terms (singular) that were obtained by the factorization formula in
Kang:2013nha ; Kang:2012zr . Non-logarithmic terms (nonsingular) are obtained
in this study by fixed-order QCD calculations. Each of $A,B$ can be written
like $A=A^{\text{sing}}+A^{\text{ns}}$ and $B=B^{\text{sing}}+B^{\text{ns}}$.
For the completeness we copy and paste the singular parts

$\displaystyle A_{q}^{\text{sing}}$ $\displaystyle=A^{\text{sing}}_{g}=0\,,$
$\displaystyle B_{q}^{\text{sing}}$
$\displaystyle=\sum_{f}Q_{f}^{2}\biggl{\\{}f_{q}(x)\bigg{[}\frac{1}{2}-\frac{\alpha_{s}C_{F}}{4\pi}\biggl{(}\frac{9}{2}+\frac{\pi^{2}}{3}+3\ln\tau+2\ln^{2}\tau\biggr{)}\bigg{]}$
$\displaystyle\quad+\frac{\alpha_{s}C_{F}}{4\pi}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\biggl{[}\mathcal{L}_{1}(1-z)\,(1+z^{2})+1-z+P_{qq}(z)\ln\frac{Q^{2}\tau}{\mu^{2}z}\biggr{]}\biggr{\\}}\,,$
$\displaystyle B_{g}^{\text{sing}}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\int_{x}^{1}\frac{dz}{z}f_{g}\left(\frac{x}{z}\right)\Biggl{[}1-P_{qg}(z)+P_{qg}(z)\ln\left(\frac{Q^{2}\tau}{\mu^{2}}\frac{1-z}{z}\right)\Biggr{]}\,,$
(30)

where $P_{qq}(z)$ and $P_{qg}(z)$ are the splitting functions and
$\mathcal{L}_{n}(1-z)$ is a plus distribution

$\displaystyle P_{qq}(z)$
$\displaystyle=\bigg{[}\theta(1-z)\frac{1+z^{2}}{1-z}\bigg{]}_{+}\,,$
$\displaystyle P_{qg}(z)$ $\displaystyle=[(1-z)^{2}+z^{2}]\,,$
$\displaystyle\mathcal{L}_{n}(1-z)$
$\displaystyle=\bigg{[}\frac{\theta(1-z)\ln^{n}(1-z)}{1-z}\bigg{]}_{+}\,.$
(31)

Note that the anti-quark contributions $A_{\bar{q}}$ and $B_{\bar{q}}$ are the
same as $A_{q}$ and $B_{q}$ except for the quark PDF replaced by the anti-
quark PDF.

The nonsingular parts are computed in App. A. The final expressions are given
by

$\displaystyle A^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\biggl{\\{}(2\tau-1)\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dz\,f_{q}\left(\frac{x}{z}\right)+\int_{x}^{1}dz\,f_{q}\left(\frac{x}{z}\right)\biggr{\\}}\,,$
$\displaystyle A^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\biggl{\\{}(2\tau-1)\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dzf_{g}\left(\frac{x}{z}\right)(1-z)+\int_{x}^{1}dz\,f_{g}\left(\frac{x}{z}\right)(1-z)\biggr{\\}}\,,$
$\displaystyle B^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\Biggl{(}\Theta_{0}\biggl{\\{}\int_{x}^{\frac{1}{1+\tau}}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\Bigl{[}\frac{1-4z}{1-z}(\tau-1/2)-P_{qq}(z)\ln\frac{1-\tau}{\tau}\Bigr{]}\biggr{\\}}$
$\displaystyle\quad+\int^{1}_{x}\frac{dz}{z}\,f_{q}\left(\frac{x}{z}\right)\bigg{[}\mathcal{L}_{0}(1-z)\frac{1-4z}{2}-P_{qq}(z)\ln\tau\bigg{]}+f_{q}(x)\big{(}2\ln^{2}\tau+3\ln\tau\big{)}\bigg{)}\,,$
$\displaystyle B^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\biggl{\\{}\Theta_{0}\int^{\frac{1}{1+\tau}}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\bigg{[}1-2\tau-
P_{qg}(z)\,\ln\frac{1-\tau}{\tau}\bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\int^{1}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\big{[}1+P_{qg}(z)\ln\tau\big{]}\biggr{\\}}\,,$
(32)

where $\Theta_{0}$ represents physical region of $\tau$ for a given value of
$x$

$\displaystyle\Theta_{0}(\tau,x)$
$\displaystyle\equiv\theta(\tau)\,\theta\bigg{(}-\tau+\frac{1}{2}\bigg{)}\,\theta\bigg{(}-\tau+\frac{1-x}{x}\bigg{)}\,.$
(33)

In comparison to $\tau^{b}$, the singular and nonsingular parts of
$\tau^{\text{jt}}$ have many terms in common with those of $\tau^{b}$ in
Kang:2014qba . Their differences are summarized in App. C.

As shown in Eq. (26), $\tau^{\text{ct}}$ is different from $\tau^{\text{jt}}$
in the region IV. We take their differences in the region and denote them by
$\delta A_{i}$ and $\delta B_{i}$. Then, the structure functions for
$\tau^{\text{ct}}$ are obtained by replacing $A$ by $A+\delta A$ and $B$ by
$B+\delta B$ in Eq. (29). Their final expressions are given by

$\displaystyle\delta A^{\text{ns}}_{q}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\,\theta_{\text{ct}}(\tau)\int_{z_{\text{ct}}}^{z_{\text{jt}}}dz\,f_{q}\left(\frac{x}{z}\right)r(z,R)\,,$
$\displaystyle\delta A^{\text{ns}}_{g}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{\pi}\,\theta_{\text{ct}}(\tau)\int_{z_{\text{ct}}}^{z_{\text{jt}}}dz\,f_{g}\left(\frac{x}{z}\right)(1-z)r(z,R)\,,$
$\displaystyle\delta B^{\text{ns}}_{q}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\,\theta_{\text{ct}}(\tau)\int_{z_{\text{ct}}}^{z_{\text{jt}}}\frac{dz}{z}\,f_{q}\left(\frac{x}{z}\right)\left\\{\frac{1-4z}{2(1-z)}r(z,R)+\frac{1+z^{2}}{1-z}\ln\frac{1+r(z,R)}{1-r(z,R)}\right\\}\,,$
$\displaystyle\delta B^{\text{ns}}_{g}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{2\pi}\theta_{\text{ct}}(\tau)\int_{z_{\text{ct}}}^{z_{\text{jt}}}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\left\\{-r(z,R)+P_{qg}(z)\ln\frac{1+r(z,R)}{1-r(z,R)}\right\\}\,,$
(34)

where $\theta_{\text{ct}}$ is upper limit of $\tau$

$\theta_{\text{ct}}(\tau)=\theta\left(-\tau+\frac{R^{2}}{16}\right)\,,$ (35)

and the parameters $z_{\text{jt},\text{ct}}$ and $r(z,R)$ are given by

$\displaystyle z_{\text{jt}}$
$\displaystyle=\max\left\\{x,\frac{1}{1+\tau}\right\\}\,,$ (36) $\displaystyle
z_{\text{ct}}$
$\displaystyle=\max\left\\{x\,,\frac{16}{16+R^{2}}\,,\frac{3+\sqrt{1-4\tau}}{2(2+\tau)}\right\\}\,,$
(37) $\displaystyle r(z,R)$
$\displaystyle=v_{+}-v_{-}=\sqrt{1-\frac{1-z}{z}\frac{16}{R^{2}}}\,.$ (38)

The differential distributions can be obtained by differentiating Eqs. (32)
and (34). We also give their explicit expressions for $\tau^{\text{jt}}$ in
Eqs. (A.5) and (A.5) and Eqs. (A.5) and (A.5) and for $\tau^{\text{ct}}$ in
Eq. (B).

In addition, by expanding $A^{\text{ns}}_{q}$, $A^{\text{ns}}_{g}$,
$B^{\text{ns}}_{g}$ and $B^{\text{ns}}_{q}$ in the $\tau\rightarrow 0$ limit
we can obtain the NLP term which is the power correction to the singular terms
in Eq. (30). At the leading power the corrections contain terms like $\tau$
and $\tau\ln\tau$, which is suppressed by $\tau$ compared to the singular
terms. The NLP obtained from Eq. (32) can be expressed as

$\displaystyle A^{\text{ns}}_{q}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\tau\biggl{\\{}f_{q}(x)+2\int^{1}_{x}dz\,f_{q}\left(\frac{x}{z}\right)\biggr{\\}}+\mathcal{O}(\tau^{2})\,,$
$\displaystyle A^{\text{ns}}_{g}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\,\tau\biggl{\\{}2\int^{1}_{x}dz\,f_{g}\left(\frac{x}{z}\right)(1-z)\biggr{\\}}+\mathcal{O}(\tau^{2})\,,$
$\displaystyle B^{\text{ns}}_{q}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\tau\biggl{\\{}\bigg{(}\frac{1}{2}+3\ln{\tau}\bigg{)}\,f_{q}(x)-\bigg{(}\frac{3}{2}+2\ln{\tau}\bigg{)}\,x\,f^{{}^{\prime}}_{q}(x)$
$\displaystyle\qquad\qquad\qquad+\,\int^{1}_{x}\frac{dz}{z}\,f_{q}\,\left(\frac{x}{z}\right)\bigg{[}(1-4z)\mathcal{L}_{0}(1-z)+P_{qq}(z)\bigg{]}\biggr{\\}}+\mathcal{O}(\tau^{2})\,,$
$\displaystyle B^{\text{ns}}_{g}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\,\tau\biggl{\\{}-f_{g}\left(x\right)\big{(}1+\ln\tau\big{)}+\int^{1}_{x}\frac{dz}{z}f_{g}\,\left(\frac{x}{z}\right)\big{[}P_{qg}(z)-2\big{]}\biggr{\\}}+\mathcal{O}(\tau^{2})\,,$
(39)

where $f^{\prime}(x)=df(x)/dx$.

In expansions of Eq. (34), we assume that $\tau<R^{2}/16$ then,
$z_{\text{ct}}=1-\tau-\tau^{3}+\mathcal{O}(\tau^{4})$. Otherwise, Eq. (34)
vanishes. At $\mathcal{O}(\tau)$ the power corrections are zero except for
$\delta B^{\text{ns}}_{q}$. So, expanding up to first nonzero correction we
have

$\displaystyle\delta A^{\text{ns}}_{q}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\,\tau^{2}\,f_{q}(x)+\mathcal{O}(\tau^{3})\,,$
$\displaystyle\delta A^{\text{ns}}_{g}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{\pi}\,\tau^{3}\,f_{g}(x)+\mathcal{O}(\tau^{4})\,,$
$\displaystyle\delta B^{\text{ns}}_{q}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\tau\bigg{(}-\frac{3}{2}+2\ln\frac{R^{2}}{4\tau}\bigg{)}\,f_{q}(x)+\mathcal{O}(\tau^{2})\,,$
$\displaystyle\delta B^{\text{ns}}_{g}\big{|}_{\tau\rightarrow 0}$
$\displaystyle=-\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{2\pi}\tau^{2}\bigg{(}\ln\frac{R^{2}}{4\tau}-1\bigg{)}\,f_{g}(x)+\mathcal{O}(\tau^{3})\,.$
(40)

It is worth to note that PDFs derivatives appear in the NLP results which
should be identified in a factorization at the subleading power and
$\mathcal{O}(\sqrt{\tau_{1}})$ corrections are absent for any of the jet
algorithms presented since power corrections of 1-jettiness scales like
$\mathcal{O}(\tau_{1}^{n})$ with integer $n$. Another feature to note is that
the appearance of a jet radius dependence in Eq. (40) is expected from the
effective theory analysis Ellis:2010rw  that implies that power corrections
are sensitive to the jet algorithms and the logarithmic structure due to the
phase-space cutoff typically appear at the leading power in jet substructure
observables Hornig:2016ahz  and in transverse vetos Hornig:2017pud .

In the context of high-order calculations in perturbative QCD, the observable
N-jettiness is used to control and to subtract the infrared singularities in
numerical computation Boughezal:2015dva ; Boughezal:2015aha ; Gaunt:2015pea ,
the results in Eqs. (39) and (40) are essential ingredients required to
improve the subtraction accuracy to subleading order Moult:2016fqy ;
Ebert:2018lzn , while the singular part in Eq. (30) is for subtraction at the
leading order. The results also serve as an important crosscheck to be
reproduced by effective field theory approach using the factorization at
subleading power on the way to the higher-order in $\alpha_{s}$. It is worth
to point out an interesting observation in our results. The algorithms,
1-jettiness and anti-$k_{t}$, are the same at the order $\alpha_{s}$ hence,
their subtractions to arbitrary accuracy are identical at this order. On the
hand, the jet algorithms, anti-$k_{t}$ and Centauro, are different in the
subtraction at the subleading power because their difference in Eq. (40)
contains non-vanishing $\tau\ln\tau$ terms.

##  4 Numerical results

In this section, we show our numerical results obtained by using analytic
1-jettiness cross sections at order $\alpha_{s}$ in Sec. 3. We compare
singular and nonsingular parts at different values of $x$ and $Q$. Numerical
difference between $\tau^{\text{jt}}$ and $\tau^{\text{ct}}$ are also
presented for several values of $R$. We also compare our result to $\tau^{b}$
result given in Kang:2014qba . The crossing point between singular and
nonsingular parts implies a boundary between fixed-order and resummation
regions and the point as a function of $x$ at several values of $Q$ is
presented. In choosing the values for the kinematic parameters $x$ and $Q$ in
the section, we considered the region that can be studied in future
experiments such as EIC or EicC.

Figure 2:  Differential $\tau^{\text{jt}}$ cross section at $Q=30$ GeV and
$x=0.05$ and 0.7. Singular (blue dashed), nonsingular (green dotted), and
their full (red solid) distributions

Main achievement of this paper is the analytic expressions of nonsingular part
of $\tau^{\text{jt}}$ and $\tau^{\text{ct}}$. In Fig. 2 we show the
differential $\tau^{\text{jt}}$ cross sections where singular and nonsingular
parts as well as full fixed-order results are presented at two values of $x$
with a fixed value of $Q$. The result at different values of $Q$ shows the
similar pattern. The cross section shown in the plots is normalized as

$\frac{d\hat{\sigma}}{d\tau}=\frac{1}{\sigma_{0}}\frac{d\sigma}{d\tau}\,,$
(41)

where $\sigma_{0}=2\pi\alpha^{2}\left[1+(1-y)^{2}\right]/Q^{4}$. Note that
$\tau^{\text{jt}}$ for 2-body final state vanishes at $\tau=1/2$ when $x<2/3$
and at $\tau=(1-x)/x$ when $x>2/3$. Beyond the point the singular and
nonsingular cancel in the sum and the full cross section is zero.

Figure 3: Relative difference between $\tau^{\text{ct}}$ and
$\tau^{\text{jt}}$ distributions with three values of $R=1.0\,,1.4\,,1.8$ at
$x=0.01$ and $Q=10$ GeV.

About the other variant $\tau^{\text{ct}}$, its singular part is the same as
that of $\tau^{\text{jt}}$ and the nonsingular is similar to
$\tau^{\text{jt}}$. Instead of showing the similar style of plots for
$\tau^{\text{ct}}$, we compare difference between $\tau^{\text{jt}}$ and
$\tau^{\text{ct}}$ in Fig. 3 by using the expression in Eq. (34). The size of
difference increases with $R$ increasing and it becomes as large as 25% when
$R=1.8$. With smaller values of $R$ the difference quickly reduces to zero and
this is consistent with the $R$ dependence of the region IV in Fig. 1 left
panel, where the region shrinks to zero as $R\to 0$. A feature of sharp
turning upwards in the plot is associated with the contributions from
$\tau^{\text{ct}}$ vanishing in $\tau\geq(16-R^{2})R^{2}/256$. Then, when the
other contribution from $\tau^{\text{jt}}$ vanishes in $\tau\geq R^{2}/16$ the
difference becomes zero as shown in the figure.

Figure 4: Upper two figures are $\tau^{\text{jt}}$ and $\tau^{\text{ct}}$
distribution in comparison to $\tau^{b}$ at $Q=30$ GeV (left) and 4 GeV
(right) with fixed values of $x=0.05$ and $R=1$ and lower ones are
corresponding uncertainties.

In Fig. 4 we compare our result to another version of 1-jettiness called
$\tau^{b}$ in Kang:2014qba . Because of difference in factorized formula
between them, their singular parts are different Kang:2013nha . However the
difference in singular part is proportional to $\delta(\tau)$ and is not
visible in a differential distribution like Fig. 4. Therefore, the difference
from $\tau^{b}$ shown in the plot is from nonsingular parts and the size of
difference is larger at lower value of $Q$. Note that as shown in the plot
$\tau^{b}$ does not vanish beyond $\tau=1/2$ because the maximum of $\tau^{b}$
is 1, while it is 1/2 for $\tau^{\text{jt}}$ and $\tau^{\text{ct}}$. For the
uncertainty figure, the uncertainties are obtained by varying the
renormalization scale from $\mu=Q$ by a factor of two up and down
$(d\hat{\sigma}(Q)-d\hat{\sigma}(2Q))/d\hat{\sigma}(Q)$ and
$(d\hat{\sigma}(Q)-d\hat{\sigma}(Q/2))/d\hat{\sigma}(Q)$ as upper and lower
boundary, respectively. We can find that $\tau^{\text{jt}}$ and
$\tau^{\text{ct}}$ share almost same uncertainty. They only have slightly
difference with each other within $\tau=1/2$. This uncertainty increases with
the decrease of the values of $Q$. We also compare uncertainties for
$\tau^{\text{ct}}$ with different $R$ value and find that the effect is around
$0.01\%$ changes.

The crossing point where singular and nonsingular parts meet each other can be
understood as a boundary between fixed-order and resummation regions. In the
fixed-order region, the singular part is smaller than the nonsingular part
hence, an ordinary fixed-order QCD result is valid, while in the resummation
region, the singular part is larger than nonsingular part due to increasing
logarithms and resummation of those logarithms is necessary. In Fig. 5 the
crossing points in $\tau$ as a function of $x$ are shown and the results for
$\tau^{\text{jt}}$ (solid) and $\tau^{b}$ (dashed) are similar as implied from
Fig. 4.

The color density plot in Fig. 5 represents a relative size of the nonsingluar
part to the full differential cross section at $Q=15$ GeV for
$\tau^{\text{jt}}$. An absolute values are taken for simplicity. In the blue
left and lower corner the nonsingular is small while the singular dominates
the cross section. In the light-colored region the singular and nonsingular
are comparable to each other. Finally, in deep-red right and upper corner
unphysical singular and nonsingular are largely cancelled to give the full
cross section. In the region their absolute magnitudes easily become greater
than full cross section and the deep-red region should be understood as 100%
or, greater relative to the full cross section.

An important feature in the plot is that the resummation region increases with
decreasing value of $x$ and the region gets close to the maximum of
$\tau^{\text{jt}}$ near $x=0.01$ while it does near $x=10^{-5}$ for
$\tau^{b}$. These crossing points imply when the resummation should be turnoff
in $\tau$ spectrum and specifically in the scale profile function Kang:2013nha
; Kang:2014qba  the points can be taken to be the value of a parameter $t_{2}$
which is the point where the resummation begins to be turned off.

Figure 5: Values of $\tau^{\text{jt}}$ (solid lines) and $\tau^{b}$ (dashed
lines) when the singular and nonsingular parts cross each other as a function
of $x$ at $Q=15$ GeV (black), $30$ GeV (orange), and $45$ GeV (green). The
color density plot represents the portion of the contributions from the
nonsingular part to the total differential cross section at $Q=15$ GeV for
$\tau^{\text{jt}}$. An approximate crossing (dotted gray lines) are obtained
by making use of Eq. (44) with the values of $c_{\text{ns}}$ between $0$ and
$-\beta(x)$.

In order to understand the behavior shown in Fig. 5, we take a following form

$\alpha(x)\frac{\ln{\tau}}{\tau}+\beta(x)\frac{1}{\tau}=c_{\text{ns}}\,,$ (42)

where the left side is from the singular part and the coefficients $\alpha(x)$
and $\beta(x)$ are obtained by differentiating Eq. (30).

$\displaystyle\alpha(x)$ $\displaystyle=-8C_{F}\sum_{f}Q_{f}^{2}f_{q}(x)\,,$
$\displaystyle\beta(x)$
$\displaystyle=\sum_{f}Q_{f}^{2}\left[-6C_{F}f_{q}(x)+2C_{F}\int_{x}^{1}\frac{dz}{z}P_{qq}(z)f_{q}\left(\frac{x}{z}\right)+2T_{F}\int_{x}^{1}\frac{dz}{z}P_{qg}(z)f_{g}\left(\frac{x}{z}\right)\right]\,.$
(43)

On the right-hand side of Eq. (42), the parameter $c_{\text{ns}}$ corresponds
to the nonsingular part in Eq. (32) multiplied by a proper normalization
factor. Instead of using the known result we assume $c_{\text{ns}}$ is a
unknown constant of $\tau$ and would like to find an approximate solution to
Eq. (42) by using an iterative approach. First, we set $c_{\text{ns}}$ to
zero, then find the solution to Eq. (42) is
$\tau=\exp[{-\beta(x)/\alpha(x)}]$. Second by introducing a small correction
term $\delta(x)$ to the solution for zero $c_{\text{ns}}$ as in Eq. (44), we
solve Eq. (42) for $\delta(x)$ with nonzero $c_{\text{ns}}$. The approximate
solution can be expressed as

$\tau_{\text{cross}}(x,c_{\text{ns}})=e^{-\frac{\beta(x)}{\alpha(x)}+\delta(x,c_{\text{ns}})}\,,$
(44)

where $\delta(x)=(c_{\text{ns}}/\alpha(x))\exp[{-\beta(x)/\alpha(x)}]$. We can
find the values of $c_{\text{ns}}$ that makes the approximate solution fitted
to the curves in Fig. 5 obtained using the known nonsingular part. We find
that $c_{\text{ns}}$ between $0$ and $-1$ times $\beta(x)$, represented by the
grey dotted lines in Fig. 5, shows a reasonable description to the curves.
Therefore, the singular part is mainly responsible to the behavior of crossing
point as a function of $x$ and in the absence of nonsingular part, the
approximation in Eq. (44) can be used as a first time estimate for the
crossing points and to fix a corresponding parameter of scale profile
function.

Our numerical results given in this section are purely perturbative results.
Including nonperturbative corrections and hadronization effects are important
in precision predictions. The hadronic effects are power suppressed by
$\Lambda_{\text{QCD}}/(\tau Q)$ for $\tau\gg\Lambda_{\text{QCD}}/Q$ in small
$\tau$ region and the correction at the leading power can be parameterized
using a nonperturbative parameter $\Omega_{1}$ Lee:2006nr ; Abbate:2010xh ;
Mateu:2012nk . The parameter is universal for any version of 1-jettiness and
the dependence on the jet algorithm would remain small in this region. One
also can take a shape function method that takes into account nonperturbative
behavior as well as hadronic power correction Hoang:2007vb ; Kang:2013nha .
However more recent studies imply that more careful considerations are
required in a scheme related to renormalon subtraction SCET2021_CLee  and in
hadronization effect away from dijet region Luisoni:2020efy . Furthermore,
recent analysis using Monte Carlo simulations implies its fine-tuning is
required to explain HERA measurements PANIC2021_SHLEE . Therefore, a
quantitative analysis on these effect in 1-jettiness would be beyond the scope
of this work and could be done in future project.

##  5 Conclusions

In this paper we study the event shape 1-jettiness in DIS at the first order
in $\alpha_{s}$ that can be measured in future experiments. We considered
three different jet axes, onto which a particle momentum is projected to
compute the value of 1-jettiness. $\tau^{\text{jt}}$ is a version with the
jettiness axis that is optimally adjusted to minimize the value of
1-jettiness. The other two versions are $\tau^{\text{kt}}$ and
$\tau^{\text{ct}}$ that take their axes from exclusive jet algorithms such as
anti-$k_{T}$ and the Centauro algorithms in the Breit frame, respectively. We
find that $\tau^{\text{jt}}$ and $\tau^{\text{kt}}$ are equivalent for the
two-body final state, $\textit{i}.\textit{e}.,$ at the order $\alpha_{s}$.

Our main result is the predictions for $\tau^{\text{jt}}$ and
$\tau^{\text{ct}}$ distributions at the first order in $\alpha_{s}$
analytically expressed in Eqs. (32) and (34). They are expressed in the form
of the cumulative distribution and in the Appendix the differential
distribution is also given. The results are expressed such that one can easily
write the structure functions such as $F_{1}(x,Q^{2},\tau)$ and
$F_{L}(x,Q^{2},\tau)$ as well as the cross section. The results of
$\tau^{\text{ct}}$ share many terms in common with those of $\tau^{\text{jt}}$
and their difference depending on $R$ is given so that $\tau^{\text{ct}}$ is
obtained by adding the difference on the top of $\tau^{\text{jt}}$ result.
Comparison to the analytic result for $\tau^{b}$ using Breit frame axis are
also given in App. C.

Numerical results of our predictions are presented at different values of $x$
and $Q$. Singular and nonsingular parts are compared in $\tau^{\text{jt}}$
distributions. For $\tau^{\text{ct}}$, we studied the difference from
$\tau^{\text{jt}}$ and found that it is sensitive to the value of radius $R$
and the magnitude is in the range of $5\sim 25\%$ of $\tau^{\text{jt}}$
distribution when the value of $R$ is from $1$ to $1.8$. We also studied the
singular-nonsingular crossing point $\tau_{\text{cross}}$ as a function of $x$
and $Q$. The crossing implies a boundary between resummation and fixed-order
regions and the value can be used as a reference point for turning off the
resummation. The value of $\tau_{\text{cross}}$ hence, the resummation region
increases with the decreasing value of $x$, and the region gets close to
$\tau_{\text{max}}=1/2$ at $x=0.01$, while the value is less sensitive to $Q$.
We found that the $x$ dependence is well explained by the singular part and
obtained an approximate expression for $\tau_{\text{cross}}$, which can be
used in the absence of nonsingular part.

Our results provide an important piece of information toward precision
predictions of event shapes in DIS. Our prediction for nonsingular part
combined with resummed singular part can be measured in the future EIC and
EicC and can be used to determine the strong coupling constant and a universal
hadronic nonpertubative parameter.

###### Acknowledgements.

D.K. likes to thank to Iain Stewart and Christopher Lee for fruitful
discussions on the singular and nonsingular crossing and feedback. The work of
Z.C. and J.C. are supported in part by the Guangdong Major Project of Basic
and Applied Basic Research No. 2020B0301030008. The work of D.K., J.-H.E., and
Y.L. is supported by the National Key Research and Development Program of
China under Contracts No. 2020YFA0406301 and by the National Natural Science
Foundation of China (NSFC) through Grant Nos. 12150610461 and 11875112. The
work of J.-H.E. is supported by the NSFC through Grant No. 12105051. The work
of J.C. is also supported by the NSFC through Grant Nos. 12025501, 11890714,
12147114, the Strategic Priority Research Program of Chinese Academy of
Science under Grant No. XDB34030000

## APPENDIX

##  Appendix A Details of calculations

In this section we calculate 1-jettiness QCD cross section. In derivations of
the cross section, we follow the steps done in Kang:2013nha . The cross
section is expressed in the product of lepton and hadronic tensors.

$\frac{d\sigma}{dx\,dQ^{2}\,d\tau}=L^{II^{\prime}}_{\mu\nu}(x,Q^{2})W^{II^{\prime}\mu\nu}(x,Q^{2},\tau)\,,$
(45)

where $L^{II^{\prime}}_{\mu\nu}$ is the lepton tensor given in Kang:2013nha
and the index $I={V,A}$ implies vector and axial currents. Although, we
consider $II^{\prime}=VV$ in this paper but we keep these index to maintain
generality for a moment. For vector currents, we have

$L^{VV}_{\mu\nu}(x,Q^{2})=-\frac{\alpha^{2}}{2x^{2}s^{2}}g_{\mu\nu}^{T}\,,$
(46)

where
$-g_{\mu\nu}^{T}=-g_{\mu\nu}+2/Q^{2}(k^{\mu}k^{\prime\nu}+k^{\prime\mu}k^{\nu})$,
with $k^{\mu}$ and $k^{\prime\mu}$ being the momenta of incoming and outgoing
leptons and the photon momentum is given by $q^{\mu}=k^{\mu}-k^{\prime\mu}$.
One can show that the lepton tensor is transverse to virtual photon as implied
by the Ward identity $q^{\mu}L^{VV}_{\mu\nu}\propto q^{\mu}g_{\mu\nu}^{T}=0$.
The hadronic tensor $W^{\mu\nu}$ that measures the 1-jettiness from final
hadronic state $X$ can be expressed as

$\displaystyle W_{II^{\prime}}^{\mu\nu}(x,Q^{2},\tau)$
$\displaystyle=\sum_{X}\left\langle P\right\rvert
J_{I}^{\mu{\dagger}}(x)\left\lvert X\right\rangle\left\langle X\right\rvert
J_{I^{\prime}}^{\nu}(x)\left\lvert
P\right\rangle(2\pi)^{4}\delta^{4}(P+q-p_{X})\delta(\tau-\tau(X))\,,$
$\displaystyle=\sum_{n=1}\int\frac{d\Phi_{n}}{d\tau}\left\langle P\right\rvert
J_{I}^{\mu{\dagger}}(x)\left\lvert p_{1},\cdots,p_{n}\right\rangle\left\langle
p_{1},\cdots,p_{n}\right\rvert J_{I^{\prime}}^{\nu}(0)\left\lvert
P\right\rangle\,,$ (47)

where there is an average over incoming spins implicit. In the second line we
specify the number of final particles and the corresponding phase-space
integral.

The hadronic tensor is decomposed into several tensors

$\displaystyle W_{II^{\prime}}^{\mu\nu}(x,Q^{2},\tau)$
$\displaystyle=(4\pi)\bigg{[}T^{\mu\nu}_{1}F_{1}(x,Q^{2},\tau)+T^{\mu\nu}_{2}\frac{F_{2}(x,Q^{2},\tau)}{P\cdot
q}+T^{\mu\nu}_{3}\frac{F_{3}(x,Q^{2},\tau)}{2P\cdot q}\bigg{]}\,.$ (48)

where $F_{i}$ are called structure functions, which is differential in $\tau$
here and integrating $F_{i}$ over $\tau$ gives ordinary structure function.
The tensor $T^{\mu\nu}_{i}$ are given by

$\displaystyle T_{1\,\mu\nu}$
$\displaystyle=-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$ $\displaystyle
T_{2\,\mu\nu}$ $\displaystyle=\bigg{(}P_{\mu}-q_{\mu}\frac{P\cdot
q}{q^{2}}\bigg{)}\bigg{(}P_{\nu}-q_{\nu}\frac{P\cdot q}{q^{2}}\bigg{)}\,,$
$\displaystyle T_{3\,\mu\nu}$
$\displaystyle=-i\epsilon_{\mu\nu\alpha\beta}q^{\alpha}P^{\beta}\,.$ (49)

Multiplying $g_{\mu\nu}$ and $P_{\mu}P_{\nu}$ by Eq. (A), one obtains linear
combinations of $F_{1}$ and $F_{2}$. Solving for $F_{1}$ and $F_{2}$, they are
expressed in terms of the hadronic tensor. Similarly, $F_{3}$ can be obtained
by multiplying $T_{3\,\mu\nu}$ by Eq. (A).

$\displaystyle F_{1}(x,Q^{2},\tau)$
$\displaystyle=\frac{1}{8\pi(1-\epsilon)}\bigg{(}-g_{\mu\nu}+\frac{4x^{2}}{Q^{2}}P_{\mu}P_{\nu}\bigg{)}W^{\mu\nu}_{II^{\prime}}\,,$
$\displaystyle F_{2}(x,Q^{2},\tau)$
$\displaystyle=\frac{x}{4\pi(1-\epsilon)}\bigg{(}-g_{\mu\nu}+(3-2\epsilon)\frac{4x^{2}}{Q^{2}}P_{\mu}P_{\nu}\bigg{)}W^{\mu\nu}_{II^{\prime}}\,,$
$\displaystyle F_{L}(x,Q^{2},\tau)$ $\displaystyle=\frac{2x^{3}}{\pi
Q^{2}}\frac{-q^{2}}{(P\cdot
q)^{2}}P_{\mu}P_{\nu}W^{\mu\nu}_{II^{\prime}}=F_{2}-2x\,F_{1}\,,$
$\displaystyle F_{3}(x,Q^{2},\tau)$
$\displaystyle=\frac{x}{2\pi(1-\epsilon)(1-2\epsilon)}\frac{q^{\alpha}P^{\beta}\epsilon_{\alpha\beta\mu\nu}}{Q^{2}}W^{\mu\nu}_{II^{\prime}}\,.$
(50)

In this paper we consider $II^{\prime}=VV$, there is no $F_{3}$ contribution.
From hereafter we drop the index for the hadronic tensor
$W^{\mu\nu}=W^{\mu\nu}_{VV}$ and one needs to calculate
$g_{\mu\nu}W^{\mu\nu}$, $P_{\mu}P_{\nu}W^{\mu\nu}$. The cross section in Eq.
(45) in terms of the structure functions $F_{i}$ is given by Eq. (28).

###  A.1 Phase space integral

The phase-space integral can be explicitly written as

$\displaystyle\int\frac{d\Phi_{n}}{d\tau}$
$\displaystyle=\mu^{2\epsilon(n-1)}\int\prod_{i=1}^{n}\frac{d^{d}p_{i}}{(2\pi)^{d}}(2\pi)\delta(p_{i}^{2})\,(2\pi)^{d}\delta^{d}(P+q-\sum_{i}p_{i})\delta(\tau-\tau(\\{p_{i}\\}))\,,$
(51)

where $d=4-2\epsilon$. For $n=1$,

$\displaystyle\int\frac{d\Phi_{1}}{d\tau}$
$\displaystyle=\frac{2\pi}{Q^{2}}\delta(1-z)\delta(\tau)\,.$ (52)

For $n=2$,

$\displaystyle\int\frac{d\Phi_{2}}{d\tau}$ $\displaystyle=\frac{1}{8\pi
Q}\frac{(4\pi\mu^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\,\int\frac{dp_{2}^{-}}{(p_{2}^{+}p_{2}^{-})^{\epsilon}}\delta\big{(}\tau-\tau(p_{1},p_{2})\big{)}$
$\displaystyle=M(\epsilon)\,\,\int
dv\frac{\theta(v)\,\theta(1-v)}{v^{\epsilon}(1-v)^{\epsilon}}\delta\big{(}\tau-\tau(v,z)\big{)}\,,$
(53) $\displaystyle M(\epsilon)$
$\displaystyle=\frac{1}{8\pi}\frac{(4\pi\mu^{2}/Q^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\bigg{(}\frac{z}{1-z}\bigg{)}^{\epsilon}\,.$

Fig. 1 shows three and four regions in $v$-$z$ space divided for
$\tau^{\text{jt,kt}}$ and for $\tau^{\text{ct}}$, respectively. Then the
phase-space integral in Eq. (A.1) splits into three pieces as

$\displaystyle\int_{\text{I}}\frac{d\Phi_{2}}{d\tau^{\text{jt}}}$
$\displaystyle=M(\epsilon)\,\Theta_{0}(\tau,z)\frac{1}{(1-\tau)^{\epsilon}\tau^{\epsilon}}\int
dv\,\delta(v-1+\tau)\,,$ (54a)
$\displaystyle\int_{\text{II}}\frac{d\Phi_{2}}{d\tau^{\text{jt}}}$
$\displaystyle=M(\epsilon)\,\Theta_{0}(\tau,z)\frac{1}{(1-\tau)^{\epsilon}\tau^{\epsilon}}\int
dv\,\delta(v-\tau)\,,$ (54b)
$\displaystyle\int_{\text{III}}\frac{d\Phi_{2}}{d\tau^{\text{jt}}}$
$\displaystyle=M(\epsilon)\theta\left(z-\frac{2}{3}\right)\delta\left(\tau-\frac{1-z}{z}\right)\int^{1-\tau}_{\tau}\frac{dv}{v^{\epsilon}(1-v)^{\epsilon}}\,,$
(54c)
$\displaystyle\int_{\text{IV}}\left(\frac{d\Phi_{2}}{d\tau^{\text{ct}}}-\frac{d\Phi_{2}}{d\tau^{\text{jt}}}\right)$
$\displaystyle=M(0)\left[\delta\left(\tau-\frac{(2z-1)(1-z)}{z^{2}}\right)-\delta\left(\tau-\frac{1-z}{z}\right)\right]\int^{v_{+}(z,R)}_{v_{-}(z,R)}dv\,,$
(54d)

where $\Theta_{0}$ and $v_{\pm}$ are given in Eqs. (33) and (23),
respectively. Note that the region III in Eq. (54c) is that of
$\tau^{\text{jt}}$ shown in Fig. 1. In $\tau^{\text{ct}}$ computation we use
the same expression, which incorrectly include the region IV then, in Eq.
(54d) the incorrect contribution from $\tau^{\text{jt}}$ is subtracted. It
makes Eq. (54d) finite so that we can set $\epsilon$ to zero and also makes
the expression of $\tau^{\text{ct}}$ cross section simple with an additional
term added to $\tau^{\text{jt}}$ cross section.

###  A.2 Hadronic tensor for incoming quark

The hadronic tensor defined in Eq. (A), the matrix element with incoming
proton can be factorized into convolutions of proton PDF and Wilson
coefficients. The Wilson coefficients can be determined by matching the
factorization formula with incoming parton to the hadronic tensor for the
parton state. The hadronic tensors $W^{q}_{\mu\nu}$ and $W^{g}_{\mu\nu}$ for
initial quark and gluon, respectively should be calculated. To
$\mathcal{O}(\alpha_{s})$ $W^{q}_{\mu\nu}$ involves one-loop calculation and
$W^{g}_{\mu\nu}$ is simply tree level calculation. In this section, we
calculate $W^{q}_{\mu\nu}$. $W^{q}_{\mu\nu}$ receives contributions from tree,
virtual, and real diagrams so can be written as

$\displaystyle
W^{q}_{\mu\nu}=W_{\mu\nu}^{(0)}+W_{\mu\nu}^{\text{vir}}+W_{\mu\nu}^{\text{real}}\,.$
(55)

Here, we suppressed the superscript $q$ on right side and we will suppress it
in the middle of calculations.

####  A.2.1 Tree-level and virtual contributions

The tree-level amplitude is
$\mathcal{M}_{\mu}^{(0)}=Q_{f}\bar{u}(p_{1})\gamma_{\mu}u(P)$. The tree-level
hadronic tensor is given by

$\displaystyle W_{\mu\nu}^{(0)}$
$\displaystyle=\frac{1}{2}\sum_{\sigma}\int\frac{d\Phi_{1}}{d\tau}\,\mathcal{M}^{(0)}_{\mu}\mathcal{M}^{(0)\,*}_{\nu}=-2\pi
Q_{f}^{2}g^{T}_{\mu\nu}\delta(1-x)\delta(\tau)\,,$

where we performed the spin average explicitly over quark spins $\sigma$ then
used the 1-body phase space in Eq. (52). The transverse metric is
$g^{T}_{\mu\nu}=g_{\mu\nu}-(n_{z\mu}{\bar{n}}_{z\nu}+n_{z\nu}{\bar{n}}_{z\mu})/2$.
This gives the projected hadronic tensors

$\displaystyle-g^{\mu\nu}W_{\mu\nu}^{(0)}$ $\displaystyle=4\pi
Q_{f}^{2}\delta(1-z)\delta(\tau)\,,$ (56) $\displaystyle
P^{\mu}P^{\nu}\,W_{\mu\nu}^{(0)}$ $\displaystyle=0\,,$ (57)

where the second line vanishes by the Dirac equation of massless particle
$P\\!\\!\\!\\!/\,u(P)=0$.

The virtual contribution can be found from many literature for instance
(14.19) Sterman:1994ce  and the contribution is the same as that of $\tau^{b}$
in Kang:2014qba

$\displaystyle-g^{\mu\nu}W_{\mu\nu}^{\text{vir}}$
$\displaystyle=-4\alpha_{s}Q_{f}^{2}C_{F}(1-\epsilon)\bigg{(}\frac{4\pi\mu^{2}}{Q^{2}}\bigg{)}^{\epsilon}\frac{\Gamma(1+\epsilon)\Gamma(1-\epsilon)^{2}}{\Gamma(1-2\epsilon)}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+4\bigg{)}\delta(1-z)\delta(\tau)\,$
$\displaystyle=2\alpha_{s}Q_{f}^{2}C_{F}(1-\epsilon)\bigg{[}-\frac{2}{\epsilon^{2}}-\frac{1}{\epsilon}\bigg{(}2L+3\bigg{)}-L^{2}-3L+\frac{\pi^{2}}{6}-8\bigg{]}\delta(1-z)\delta(\tau)\,,$
$\displaystyle P^{\mu}P^{\nu}\,W_{\mu\nu}^{\text{vir}}$ $\displaystyle=0\,,$
(58)

where $L=\ln\frac{\mu^{2}}{Q^{2}}$. The factor $(1-\epsilon)$ is not expanded
because it is to cancel $1/(1-\epsilon)$ in Eq. (A). In the second line of Eq.
(A.2.1) we have used the $\overline{\text{MS}}$ scheme by re-scaling the scale
$\mu^{2}$ by $e^{\gamma_{E}}/(4\pi)$ such that we use following replacement

$\displaystyle\frac{(4\pi\mu^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\to\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}=\mu^{2\epsilon}\bigg{(}1-\epsilon^{2}\frac{\pi^{2}}{12}+\mathcal{O}(\epsilon^{3})\bigg{)}\,.$
(59)

Note that the finite part in Eq. (A.2.1) is the $\alpha_{s}$ term in the hard
function in Kang:2013nha , derived in SCET in Bauer:2003di ; Manohar:2003vb .
The $P^{\mu}P^{\nu}\,W_{\mu\nu}^{\text{vir}}$ is zero because
$P\\!\\!\\!\\!/\,u(P)=0$.

####  A.2.2 Real contribution

The definition of $\tau$ in Eqs. (2.1) and (26) divide the phase space into
three or four regions as shown in Fig. 1 hence the 2-body phase-space is
accordingly divided as follows

$\displaystyle W_{\mu\nu}^{\text{real}}$
$\displaystyle=\sum_{i=\text{I}}^{\text{IV}}W_{\mu\nu}^{(i)}\,,$ (60)
$\displaystyle W_{\mu\nu}^{(i)}$
$\displaystyle=\frac{1}{2}\sum_{\sigma}\int_{(i)}\frac{d\Phi_{2}}{d\tau}\,\,\mathcal{M}_{\mu}^{\text{real}}\mathcal{M}_{\nu}^{\text{real}\,*}\hskip
28.45274pti\in\\{\text{I},\text{II},\text{III},\text{IV}\\}\,,$ (61)

where in the last line the factor $1/2$ takes account the average over quark
spins $\sigma$. In this subsection, we discuss the results of regions I,II,
and III, which is relevant to $\tau^{\text{jt}}$ and for the region IV, which
is IR finite and simpler, we just give the final result in App. B.

The real emission contribution to the two projections of the color-averaged
squared amplitudes that we need are given by

$\displaystyle-g^{\mu\nu}\mathcal{M}_{\mu}^{\text{real}}\mathcal{M}_{\nu}^{\text{real}\,*}$
$\displaystyle=32\pi\alpha_{s}C_{F}Q_{f}^{2}(1-\epsilon)\bigg{[}(1-\epsilon)\bigg{(}\frac{1-z}{v}+\frac{v}{1-z}\bigg{)}+2\frac{z}{1-z}\frac{1-v}{v}+2\epsilon\bigg{]}\,,$
(62) $\displaystyle
P^{\mu}P^{\nu}\,\mathcal{M}_{\mu}^{\text{real}}\mathcal{M}_{\nu}^{\text{real}\,*}$
$\displaystyle=16\pi\alpha_{s}C_{F}Q_{f}^{2}Q^{2}(1-\epsilon)\frac{1-v}{z}\,,$
(63)

where the first line can be found from (14.21) in Sterman:1994ce .

For $P^{\mu}P^{\nu}\,W_{\mu\nu}^{(i)}$, there is no singular term and we can
safely set $\epsilon=0$. Including the spin average, we obtain

$\displaystyle P^{\mu}P^{\nu}\,W_{\mu\nu}^{\text{I}}$
$\displaystyle=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\,\frac{\tau}{z}\Theta_{0}(\tau,z)\,,$
(64a) $\displaystyle\vspace{5pts}P^{\mu}P^{\nu}\,W_{\mu\nu}^{\text{II}}$
$\displaystyle=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\,\frac{1-\tau}{z}\,\Theta_{0}(\tau,z)\,,$
(64b) $\displaystyle\vspace{5pts}P^{\mu}P^{\nu}\,W_{\mu\nu}^{\text{III}}$
$\displaystyle=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\,\theta(z-2/3)\delta\left(\tau-\frac{1-z}{z}\right)\frac{(1-2\tau)(1+\tau)}{2}\,,$
(64c)

where the $\Theta_{0}(\tau,z)$ is given in Eq. (33). The sum of Eq. (64) is
given by

$\displaystyle
P^{\mu}P^{\nu}\,W^{q}_{\mu\nu}=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\,\Theta_{0}(\tau,z)\left[\frac{1}{z}+\theta\left(z-\frac{2}{3}\right)\delta\left(\tau-\frac{1-z}{z}\right)\frac{\left(1-2\tau\right)\left(1+\tau\right)}{2}\right]\,,$
(65)

where the superscript $q$ representing the incoming quark is made explicit,
again. Note that the tree-level and virtual contributions are zero and the
real contribution is the total.

$\displaystyle-g^{\mu\nu}W^{\text{(I)}}_{\mu\nu}$
$\displaystyle=16\pi\alpha_{s}C_{F}Q^{2}_{f}M(\epsilon)\Theta_{0}(\tau,z)(1-\tau)^{-\epsilon}\tau^{-\epsilon}(1-\epsilon)$
$\displaystyle\quad\times\,\left[(1-\epsilon)\left(\frac{1-z}{1-\tau}+\frac{1-\tau}{1-z}\right)+2\frac{z}{1-z}\frac{\tau}{1-\tau}+2\epsilon\right],$
(66a) $\displaystyle-g^{\mu\nu}W^{{\textrm{(II)}}}$
$\displaystyle=16\pi\alpha_{s}C_{F}Q^{2}_{f}M(\epsilon)\Theta_{0}(\tau,z)(1-\tau)^{-\epsilon}\tau^{-\epsilon}(1-\epsilon)$
$\displaystyle\quad\times\,\left[(1-\epsilon)\left(\frac{1-z}{\tau}+\frac{\tau}{1-z}\right)+2\frac{z}{1-z}\frac{1-\tau}{\tau}+2\epsilon\right],$
(66b) $\displaystyle-g^{\mu\nu}W^{{\textrm{(III)}}}$
$\displaystyle=16\pi\alpha_{s}C_{F}Q^{2}_{f}M(\epsilon)\theta\left(z-\frac{2}{3}\right)\delta\left(\tau-\frac{1-z}{z}\right)(1-\epsilon)$
$\displaystyle\quad\times\int^{1-\tau}_{\tau}\frac{dv}{v^{\epsilon}(1-v)^{\epsilon}}\left[(1-\epsilon)\left(\frac{1-z}{v}+\frac{v}{1-z}\right)+2\frac{z}{1-z}\frac{1-v}{v}+2\epsilon\right].$
(66c)

The regions I and II share the same $\Theta_{0}(\tau,z)$, so we combine them
and expand in $\epsilon$

$\displaystyle-g^{\mu\nu}W^{\textrm{(I)}+\textrm{(II)}}_{\mu\nu}=2\alpha_{s}C_{F}Q^{2}_{f}\left(\frac{\mu^{2}}{Q^{2}}\right)^{\epsilon}\frac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}z^{\epsilon}(1-\epsilon)$
$\displaystyle\quad\times\left[\theta\left(-\tau+\frac{1-z}{z}\right)\theta\left(z-\frac{2}{3}\right)+\theta\left(-\tau+\frac{1}{2}\right)\theta\left(-z+\frac{2}{3}\right)\right]$
$\displaystyle\quad\times\left\\{(1-\epsilon)\left((1-z)^{1-\epsilon}\frac{1}{\tau^{1+\epsilon}}\frac{1}{(1-\tau)^{1+\epsilon}}+\left(1-\tau\right)^{-\epsilon}\tau^{-\epsilon}\frac{1}{(1-z)^{1+\epsilon}}\right)\right.$
$\displaystyle\quad\quad+2z\tau^{1-\epsilon}{\frac{1}{(1-z)^{1+\epsilon}}}\frac{1}{(1-\tau)^{1+\epsilon}}$
$\displaystyle\quad\quad\left.+2z(1-\tau)^{1-\epsilon}{\frac{1}{(1-z)^{1+\epsilon}}}{{\frac{1}{\tau^{1+\epsilon}}}}+4\epsilon(1-\tau)^{-\epsilon}\tau^{-\epsilon}\frac{1}{(1-z)^{\epsilon}}\right\\},$
(67)

where $\Theta_{0}(\tau,z)$ was divided into two parts, and the first part
multiplied by terms $1/\tau^{1+\epsilon}$ and $1/(1-z)^{1+\epsilon}$ gives
$1/\epsilon$ pole, while with the other part the only term like
$1/\tau^{1+\epsilon}$ gives the pole. We now expand above equation in powers
of $\epsilon$. For the term like
$\theta(-\tau+\tfrac{z}{1-z})/\tau^{1+\epsilon}/(1-z)^{1+\epsilon}$ we use a
plus distribution identity in Appendix A of Kang:2014qba .

$\displaystyle-g^{\mu\nu}W^{\textrm{(I+II)}}_{\mu\nu}$
$\displaystyle=2\alpha_{s}C_{F}Q^{2}_{f}\left(\frac{\mu^{2}}{Q^{2}}\right)^{\epsilon}(1-\epsilon)\Theta_{0}(\tau,z)$
$\displaystyle\quad\times\,\left[\left(\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}\right)\delta(\tau)\delta(1-z)-\frac{P_{\text{qq}}(z)}{\epsilon}\delta(\tau)+I^{s}_{\textrm{fin}}+I^{ns}_{\textrm{fin}}\right],$
(68)

where $P_{qq}$ is the splitting function in Eq. (3). The functions
$I^{s}_{\textrm{fin}}$ and $I^{ns}_{\textrm{fin}}$ contribute to singular and
nonsingular parts, respectively.

$\displaystyle I^{s}_{\textrm{fin}}$
$\displaystyle=\delta(\tau)\left[(1+z^{2})\mathcal{L}_{1}(1-z)-\frac{\pi^{2}}{12}\delta(1-z)+1-z-(1+z^{2})\ln
z~{}\mathcal{L}_{0}(1-z)\right]$
$\displaystyle\quad+\mathcal{L}_{0}(\tau)\left[P_{\mathrm{qq}}(z)-\frac{3}{2}\delta(1-z)\right]-2\mathcal{L}_{1}(\tau)\delta(1-z),$
$\displaystyle I^{ns}_{\textrm{fin}}$
$\displaystyle=\frac{(3\tau-1)\mathcal{L}_{0}(1-z)-4\tau-z+3}{1-\tau}+2z~{}I_{ns}(\tau,1-z)\,,$
(69)

where $I_{ns}$ is identical to corresponding part in $\tau^{b}$ and is given
in Eq. (A.8) of Kang:2014qba . We use following relations to compress Eq.
(A.2.2)

$\displaystyle
P_{qq}(z)=\bigg{[}\frac{1+z^{2}}{1-z}\bigg{]}_{+}=2\mathcal{L}_{0}(1-z)+\frac{3}{2}\delta(1-z)-(1+z)=(1+z^{2})\mathcal{L}_{0}(1-z)+\frac{3}{2}\delta(1-z)\,,$
$\displaystyle(1+z^{2})\mathcal{L}_{0}(1-z)=2\mathcal{L}_{0}(1-z)-(1+z)\,,$
$\displaystyle(1+z^{2})\mathcal{L}_{1}(1-z)=2\mathcal{L}_{1}(1-z)-(1+z)\ln(1-z)\,,$
(70)

where standard plus distributions $\mathcal{L}_{n}(z)$ are defined by
Kang:2014qba

$\displaystyle\mathcal{L}_{n}(z)\equiv\lim_{\varepsilon\to
0}\frac{d}{dz}\bigg{[}\frac{\theta(z-\varepsilon)\ln^{n+1}z}{n+1}\bigg{]}=\bigg{[}\frac{\theta(z)\ln^{n}(z)}{z}\bigg{]}_{+}\,.$
(71)

With a test function $g(z)$ well behaving near $z=0$, the integration against
the function gives

$\displaystyle\int^{z}_{0}dz^{{}^{\prime}}\mathcal{L}_{n}(z^{{}^{\prime}})g(z^{{}^{\prime}})=\int^{z}_{0}dz^{{}^{\prime}}\frac{\ln^{n}z^{{}^{\prime}}}{z^{{}^{\prime}}}[g(z^{{}^{\prime}})-g(0)]+g(0)\frac{\ln^{n+1}z}{n+1}\,.$
(72)

A variation of $\mathcal{L}_{n}$ with a variable lower bound $z_{0}$ and its
integral is given by

$\displaystyle\mathcal{L}_{n}(z,z_{0})$ $\displaystyle=\lim_{\varepsilon\to
0}\frac{d}{dz}\bigg{[}\frac{\theta(z-z_{0}-\varepsilon)\ln^{n+1}z}{n+1}\bigg{]}\,,$
$\displaystyle\int^{z}_{0}dz^{{}^{\prime}}\mathcal{L}_{n}(z^{{}^{\prime}},z_{0})g(z^{{}^{\prime}})$
$\displaystyle=\int^{z}_{z_{0}}dz^{{}^{\prime}}\frac{\ln^{n}z^{{}^{\prime}}}{z^{{}^{\prime}}}[g(z^{{}^{\prime}})-g(z_{0})]+g(z_{0})\frac{\ln^{n+1}z}{n+1}\,.$
(73)

For the contribution from the region III, it contains singular terms when
$\tau$ or, $1-z$ approaches zero. We isolate the singular part carefully by
subtracting and adding singular terms

$\displaystyle-g^{\mu\nu}W^{{\textrm{(III)}}}_{\mu\nu}$
$\displaystyle=2\alpha_{s}C_{F}Q^{2}_{f}\left(\frac{\mu^{2}}{Q^{2}}\right)^{\epsilon}\frac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\theta\left(z-\frac{2}{3}\right)\delta\left(\tau-\frac{1-z}{z}\right)(1-\epsilon)$
$\displaystyle\quad\times\left[(1-\epsilon)(I_{1}+I_{2})+2I_{3}\right],$ (74a)
$\displaystyle\vspace{5pts}I_{1}$
$\displaystyle=\frac{\tau^{1-\epsilon}}{1+\tau}\int^{1-\tau}_{\tau}dv\frac{1}{v^{1+\epsilon}(1-v)^{\epsilon}}=\frac{\tau}{1+\tau}\ln\bigg{(}\frac{1-\tau}{\tau}\bigg{)},$
(74b) $\displaystyle\vspace{5pts}I_{2}$
$\displaystyle=\frac{1+\tau}{\tau^{1+\epsilon}}\int^{1-\tau}_{\tau}dv\frac{v^{1-\epsilon}}{(1-v)^{\epsilon}}$
$\displaystyle=\frac{1+\tau}{\tau^{1+\epsilon}}\left[\frac{1}{2}-\tau+\epsilon\big{(}1-2\tau)+\epsilon(\tau-1)\ln(1-\tau)+\epsilon\tau\ln\tau\big{)}\right],\
$ (74c) $\displaystyle\vspace{5pts}I_{3}$
$\displaystyle=\frac{1}{\tau^{1+\epsilon}}\int^{1-\tau}_{\tau}dv\,\frac{(1-v)^{1-\epsilon}}{v^{1+\epsilon}}$
$\displaystyle=\frac{1}{\tau^{1+\epsilon}}\int^{1-\tau}_{\tau}dv\frac{(1-v)^{1-\epsilon}-(1-\tau)^{1-\epsilon}}{v^{1+\epsilon}}+\frac{1}{\epsilon}\left[\frac{(1-\tau)^{1-\epsilon}}{\tau^{1+2\epsilon}}-\frac{(1-\tau)^{1-2\epsilon}}{\tau^{1+\epsilon}}\right]\,,$
(74d)

where we used the delta function in the first line and replace $z$ by $\tau$.
Finally we expand Eq. (74a) in $\epsilon$

$\displaystyle-g^{\mu\nu}W^{(\text{III})}_{\mu\nu}=$ $\displaystyle
2\alpha_{s}C_{F}Q_{f}\left(\frac{\mu^{2}}{Q^{2}}\right)^{\epsilon}(1-\epsilon)\,\theta\left(z-\frac{3}{2}\right)\delta\left(\tau-\frac{1-z}{z}\right)$
$\displaystyle\times\left[\delta(\tau)\big{(}\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{7}{2}-\frac{5\pi^{2}}{12}\big{)}-\frac{3}{2}\mathcal{L}_{0}(\tau)-2\mathcal{L}_{1}(\tau)+\frac{7}{2}\right.$
$\displaystyle\qquad\qquad\left.+\frac{\tau^{2}+2\tau+2}{\tau(1+\tau)}\ln(1-\tau)-\frac{\tau}{1+\tau}\ln\tau-\tau\right]\,.$
(75)

It is a non-trivial cross check to show that sum of
$-g^{\mu\nu}W_{\mu\nu}^{(i)}(\tau)$ and $P^{\mu}P^{\nu}W_{\mu\nu}^{(i)}(\tau)$
over $i$ and its integration over $\tau$ are equivalent to the inclusive
results that simply obtained by integrating the phase space measures in Eqs.
(54a), (54b), and (54c) over $\tau$ then by carrying out integral against
amplitudes Eqs. (62) and (63) over $v$.

Now we collect all the pieces and put them together. All IR divergences with
$1/\epsilon^{2}$ and $1/\epsilon$ are cancelled when virtual part in Eq.
(A.2.1) and real parts in Eqs. (56), (A.2.2), and (75) are combined except for
the IR divergence associated with 1-loop quark PDF. Let us first highlight a
few terms

$\displaystyle-g^{\mu\nu}\,W_{\mu\nu}$
$\displaystyle\propto\bigg{[}\delta(1-z)\big{(}L^{2}+3L\big{)}-\frac{P_{qq}(z)}{\epsilon}-P_{qq}(z)L\bigg{]}$
$\displaystyle\qquad+\text{terms from finite part}\,,$ (76)

where $L=\ln\frac{\mu^{2}}{Q^{2}}$. First, the two logarithmic terms above are
cancelled by the same terms in virtual part Eq. (A.2.1). The $1/\epsilon$ term
above will be replaced by 1-loop correction of the proton PDF during matching
procedure. The logarithmic scale dependence proportional to $P_{qq}$ in the
last term should cancel the same scale dependence from RG evolution of the
proton PDF. Therefore, all $\mu$ dependence at $\mathcal{O}(\alpha_{s})$ are
cancelled.

Finally we have

$\displaystyle-g^{\mu\nu}\,W^{q}_{\mu\nu}(z,\tau,Q^{2})$ $\displaystyle=4\pi
Q_{f}^{2}\delta(1-z)\delta(\tau)$
$\displaystyle\quad+2\alpha_{s}Q_{f}^{2}C_{F}(1-\epsilon)\Theta_{0}(\tau,z)\bigg{\\{}\delta(\tau)\,\left[-\frac{P_{qq}(z)}{\epsilon}+S_{-1}^{q}(z)\right]+\mathcal{L}_{0}(\tau)\,S_{0}^{q}(\tau,z)$
$\displaystyle\quad\quad+\mathcal{L}_{1}(\tau)\,S_{1}^{q}(\tau,z)+R^{q}(\tau,z)+\delta\left(\tau-\frac{1-z}{z}\right)\,\Delta^{q}(\tau)\bigg{\\}}\,,$
(77)

where the superscript $q$ denotes the contribution from the incoming quark.
The functions $S,~{}R,~{}\Delta$ are given by

$\displaystyle S_{-1}^{q}(z)$
$\displaystyle=-P_{qq}(z)\ln\frac{\mu^{2}}{Q^{2}}+(1+z^{2})\mathcal{L}_{1}(1-z)-\bigg{(}\frac{9}{2}+\frac{\pi^{2}}{3}\bigg{)}\delta(1-z)+1-z-\frac{1+z^{2}}{1-z}\ln
z\,,$ $\displaystyle S_{0}^{q}(\tau,z)$
$\displaystyle=2z\,\mathcal{L}_{0}\left(1-z,\tfrac{\tau}{1+\tau}\right)-\frac{3}{2}\delta\left(\tau-\frac{1-z}{z}\right)+(1-z)\,,$
$\displaystyle S_{1}^{q}(\tau,z)$
$\displaystyle=-2(2+\tau)\,\delta\left(\tau-\frac{1-z}{z}\right)\,,$
$\displaystyle R^{q}(\tau,z)$
$\displaystyle=\frac{1-4z}{1-z}+\frac{1+z^{2}}{1-z}\frac{1}{1-\tau}\,,$
$\displaystyle\Delta^{q}(\tau)$
$\displaystyle=\frac{\tau^{2}+2\tau+2}{\tau(1+\tau)}\ln(1-\tau)-\frac{\tau}{1+\tau}\ln\tau+\frac{2(1+\tau)}{\tau}\ln(1+\tau)-\tau+\frac{7}{2}\,.$
(78)

The functions $S^{q}_{n}$ are similar to those in $\tau^{b}$. $S^{q}_{-1}$ is
the same except for the last $\ln z$ term and in $S^{q}_{0,1}$ the delta
function terms contains additional factors $1+\tau$ associated with change of
variable in the delta function
$\delta(\tau-\frac{1-z}{z})=(1+\tau)^{-2}\delta(\tfrac{1}{1+\tau}-z)$.

###  A.3 Hadronic tensor for incoming gluon

A process with the initial gluon $g+\gamma^{*}\to q\bar{q}$ starts at
$\mathcal{O}(\alpha_{s})$ and this process is the tree level. The tree level
amplitude averaged over incoming colors is given by

$\displaystyle-g^{\mu\nu}\mathcal{M}_{\mu}^{g}\mathcal{M}_{\nu}^{g\,*}$
$\displaystyle=32\pi\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}(1-\epsilon)\bigg{[}(1-\epsilon)\bigg{(}\frac{1-v}{v}+\frac{v}{1-v}\bigg{)}-2\frac{z(1-z)}{v(1-v)}-2\epsilon\bigg{]}\,,$
(79) $\displaystyle
P^{\mu}P^{\nu}\,\mathcal{M}_{\mu}^{g}\mathcal{M}_{\nu}^{g\,*}$
$\displaystyle=32\pi\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}\,Q^{2}(1-\epsilon)\frac{1-z}{z}\,,$
(80)

where the sums go over the all flavors $f\in\\{u,d,s,c,b\\}$ allowed by the
energy. Note the $(1-\epsilon)$ factors in Eqs. (79) and (80) are going to be
canceled by the same factor in the structure functions Eq. (A). Integrating
Eq. (80) over the two-body final-state phase space,

$W_{\mu\nu}^{g}=\frac{1}{2-2\epsilon}\sum_{\lambda}\int\frac{d\Phi_{2}}{d\tau}\mathcal{M}_{\mu}^{g}\mathcal{M}_{\nu}^{g*}\,,$
(81)

where the superscript $g$ representing the incoming gluon and the prefactor
$(2-2\epsilon)$ accounts for the average over incoming gluon polarization
$\lambda$ in $D=4-2\epsilon$ dimensions. Then, we have

$\displaystyle
P^{\mu}P^{\nu}\,W^{g}_{\mu\nu}=4\alpha_{s}T_{F}Q_{f}^{2}Q^{2}\Theta_{0}\,(1-z)\left[\frac{1}{z}+\theta\left(z-\frac{2}{3}\right)\delta\left(\tau-\frac{1-z}{z}\right)\,\frac{(1-2\tau)(1+\tau)}{2}\right]\,,$
(82)

The contributions $-g^{\mu\nu}W_{\mu\nu}^{(i)}$ for $i=\text{I}$ and for II
are identical and involve singular terms

$\displaystyle-g^{\mu\nu}\,W_{\mu\nu}^{(\text{I})}$
$\displaystyle=-g^{\mu\nu}\,W_{\mu\nu}^{(\text{II})}$
$\displaystyle=2\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}\frac{(\mu^{2}e^{\gamma_{E}}/Q^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\bigg{(}\frac{z}{1-z}\bigg{)}^{\epsilon}(1-\epsilon)\Theta_{0}(\tau,z)$
$\displaystyle\quad\times\left[\bigg{(}\frac{\tau^{1-\epsilon}}{(1-\tau)^{1+\epsilon}}+\frac{(1-\tau)^{1-\epsilon}}{{\tau^{1+\epsilon}}}\bigg{)}-\frac{2}{1-\epsilon}\frac{z(1-z)}{{\tau^{1+\epsilon}}(1-\tau)^{1+\epsilon}}\right]$
$\displaystyle=2\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}^{\epsilon}(1-\epsilon)\,\Theta_{0}(\tau,z)\bigg{[}-\frac{P_{qg}(z)}{\epsilon}\delta(\tau)$
$\displaystyle\quad+\bigg{(}1-P_{qg}(z)+P_{qg}(z)\ln\frac{1-z}{z}\bigg{)}\,\delta(\tau)+P_{qg}(z)\mathcal{L}_{0}(\tau)+\frac{P_{qg}(z)}{1-\tau}-2\bigg{]}\,,$
(83)

where the splitting function $P_{qg}(z)$ is given in Eq. (3).

The contraction $-g^{\mu\nu}\,W_{\mu\nu}^{(\text{III})}$ also involves
singular terms when $\tau\rightarrow 0$ and $\tau\rightarrow 1$, so doing
similar expansions as in Eq. (74d), we get

$\displaystyle-g^{\mu\nu}\,W_{\mu\nu}^{(\text{III})}$
$\displaystyle=2\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}\frac{(\mu^{2}e^{\gamma_{E}}/Q^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\bigg{(}\frac{z}{1-z}\bigg{)}^{\epsilon}\,\theta\left(z-\tfrac{2}{3}\right)\delta\left(\tau-\tfrac{1-z}{z}\right)$
$\displaystyle\quad\times\int^{1-\tau}_{\tau}dv\left[(1-\epsilon)\bigg{(}{\tfrac{(1-v)^{1-\epsilon}}{v^{1+\epsilon}}}+{\tfrac{v^{1-\epsilon}}{(1-v)^{1+\epsilon}}}\bigg{)}-2\tfrac{z(1-z)}{{v^{1+\epsilon}}{(1-v)^{1+\epsilon}}}-\tfrac{2\epsilon}{v^{\epsilon}(1-v)^{\epsilon}}\right]$
$\displaystyle=2\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}\left(\tfrac{\mu^{2}}{Q^{2}}\right)^{\epsilon}(1-\epsilon)\,\theta\left(z-\tfrac{2}{3}\right)\delta\left(\tau-\tfrac{1-z}{z}\right)\,\left[4\tau-2+2P_{qg}(z)\ln\tfrac{1-\tau}{\tau}\right]\,.$
(84)

By putting all the $-g^{\mu\nu}\,W_{\mu\nu}^{(i)}$ in Eqs. (A.3) and (A.3)
together we have

$\displaystyle-g^{\mu\nu}\,W_{\mu\nu}^{g}$
$\displaystyle=4\alpha_{s}T_{F}\sum_{f}Q_{f}^{2}(1-\epsilon)\Theta_{0}(\tau,z)\bigg{[}-\frac{P_{qg}(z)}{\epsilon}\delta(\tau)$
$\displaystyle\qquad+S^{g}_{-1}\delta(\tau)+S^{g}_{0}\mathcal{L}_{0}(\tau)+R^{g}(\tau,z)+\delta\bigg{(}\tau-\frac{1-z}{z}\bigg{)}\Delta^{g}(\tau,z)\bigg{]}\,,$
(85) $\displaystyle S^{g}_{-1}(z)$
$\displaystyle=-P_{qg}(z)+1+P_{qg}(z)\bigg{(}\ln\frac{1-z}{z}-\ln\frac{\mu^{2}}{Q^{2}}\bigg{)}\,,$
$\displaystyle S^{g}_{0}(\tau,z)$ $\displaystyle=P_{qg}(z)\,,$ $\displaystyle
R^{g}(\tau,z)$ $\displaystyle=\frac{P_{qg}(z)}{1-\tau}-2\,,$
$\displaystyle\Delta^{g}(\tau,z)$
$\displaystyle=2\tau-1+P_{qg}(z)\ln\frac{1-\tau}{\tau}\,.$ (86)

###  A.4 Nonsingular part and cumulative results

In this subsection, we give the expressions for the nonsingular terms of the
hadronic tensors obtained in previous sections. The tensors reduce to singular
terms having $\delta(\tau)$, or $\mathcal{L}_{n}(\tau)$ as $\tau\to 0$. In the
contributions $P_{\mu}P_{\nu}W_{\mu\nu}$ in Eqs. (65) and (82) there is no
such terms and they are purely nonsingular.

In the gluon process $-g^{\mu\nu}W_{\mu\nu}^{g}$ in Eq. (A.3) contains terms
with $\delta(\tau)$ and $\mathcal{L}_{0}(\tau)$. Subtracting those terms from
Eq. (A.3), nonsingular part of the gluon tensor is given by

$\displaystyle-g^{\mu\nu}\,W_{\mu\nu}^{g,\text{ns}}$
$\displaystyle=4\alpha_{s}T_{F}Q_{f}^{2}(1-\epsilon)\bigg{\\{}\Theta_{0}(\tau,z)\big{[}R^{g}(\tau,z)+\delta(\tau-\tfrac{1-z}{z})\Delta^{g}(z)\big{]}$
$\displaystyle\qquad+(\Theta_{0}(\tau,z)-\theta(\tau))P_{qg}(z)\frac{1}{\tau}\bigg{\\}}\,.$
(87)

Note that the last term in Eq. (A.4) gives $1/\tau$ term in unphysical region
where $\Theta_{0}$ is zero and this term cancel against the same term in
singular part in the region.

Similarly, in the quark process $-g^{\mu\nu}W_{\mu\nu}^{q}$ in Eq. (A.3)
contains terms with $\delta(\tau)$, $\mathcal{L}_{0}(\tau)$, and
$\mathcal{L}_{1}(\tau)$ as $\tau\to 0$. Subtracting those terms from Eq. (77),
the nonsingular part is given by

$\displaystyle-g^{\mu\nu}\,W^{q,\text{ns}}_{\mu\nu}$
$\displaystyle=2\alpha_{s}Q_{f}^{2}C_{F}(1-\epsilon)\bigg{\\{}\frac{1}{\tau}\,N_{0}(\tau,z)+\frac{\ln\tau}{\tau}\,N_{1}(\tau,z)$
$\displaystyle\quad\quad+\Theta_{0}(\tau,z)\,\big{[}R^{q}(\tau,z)+\delta\left(\tau-\frac{1-z}{z}\right)\,\Delta^{q}(\tau)\big{]}\bigg{\\}}\,,$
(88)

where the plus distribution $\mathcal{L}_{n}$ is replaced by
$\ln^{n}\tau/\tau$ and the function $N_{0,1}(\tau,z)$ is defined from
differences as

$\displaystyle N_{0}(\tau,z)$
$\displaystyle=\Theta_{0}(\tau,z)\,S_{0}^{q}(\tau,z)-\theta(\tau)S_{0}^{q}(0,z)$
$\displaystyle=2z\,[\Theta_{0}\,\mathcal{L}_{0}(1-z,\tfrac{\tau}{1+\tau})-\theta(\tau)\mathcal{L}_{0}(1-z)]-\frac{3}{2}\bigg{[}\Theta_{0}\,\delta\left(\tau-\tfrac{1-z}{z}\right)$
$\displaystyle\qquad-\theta(\tau)\delta(1-z)\bigg{]}+(1-z)[\Theta_{0}(\tau,z)-\theta(\tau)]\,,$
$\displaystyle N_{1}(\tau,z)$
$\displaystyle=\Theta_{0}(\tau,z)\,S_{1}^{q}(\tau,z)-\theta(\tau)S_{1}^{q}(0,z)$
$\displaystyle=-4\Big{[}\Theta_{0}\,\bigg{(}1+\frac{\tau}{2}\bigg{)}\,\delta\left(\tau-\tfrac{1-z}{z}\right)-\theta(\tau)\delta(1-z)\Big{]}\,.$
(89)

Now we calculate the cumulative of the nonsingular parts by using following
integral

$\displaystyle{\widetilde{W}}_{\mu\nu}(z,\tau,Q^{2})=\int^{\tau}_{0}d\tau^{\prime}\,W_{\mu\nu}(z,\tau^{\prime},Q^{2})\,.$
(90)

Integration of Eqs. (65) and (82) is given by

$\displaystyle P^{\mu}P^{\nu}\,{\widetilde{W}}^{q,\text{ns}}_{\mu\nu}$
$\displaystyle=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\bigg{[}\frac{1}{z}\min[\tau,\tfrac{1}{2},\tfrac{1-z}{z}]+\Theta_{2}\frac{3z-2}{2z^{2}}\bigg{]}\,$
$\displaystyle=\alpha_{s}C_{F}Q_{f}^{2}Q^{2}\left[\frac{\tau}{z}\,\Theta_{0}+\frac{1}{2z}(\Theta_{1}+\Theta_{2})\right]\,,$
$\displaystyle P^{\mu}P^{\nu}\,{\widetilde{W}}^{g,\text{ns}}_{\mu\nu}$
$\displaystyle=4\alpha_{s}T_{F}Q_{f}^{2}Q^{2}(1-z)\left[\frac{\tau}{z}\,\Theta_{0}+\frac{1}{2z}(\Theta_{1}+\Theta_{2})\right]\,,$
(91)

where $\Theta_{0}$ is defined in Eq. (33) and

$\Theta_{1}=\theta(-z+2/3)\,\theta(\tau-1/2)\,,\qquad\qquad\Theta_{2}=\theta(z-2/3)\,\theta(\tau-\tfrac{1-z}{z})\,.$
(92)

They cover all regions in $\tau$ and $z$ space
$\Theta_{0}+\Theta_{1}+\Theta_{2}=1$. One finds that Eq. (91) are
discontinuous at $\tau=(1-z)/z$ when $z>2/3$ because $\delta(\tau-(1-z)/z)$ in
Eqs. (65) and (82) are turned on at the value and the discontinuity implies
the end of physical region.

The cumulant of Eq. (A.4) is given by

$\displaystyle-g^{\mu\nu}\,{\widetilde{W}}_{\mu\nu}^{g,\text{ns}}$
$\displaystyle=4\alpha_{s}T_{F}Q_{f}^{2}(1-\epsilon)\Bigl{\\{}\Theta_{0}\bigg{[}1-2\tau-
P_{qg}(z)\ln\frac{1-\tau}{\tau}\bigg{]}-\big{[}1+P_{qg}(z)\ln\tau\big{]}\Bigr{\\}}\,.$
(93)

The cumulant of Eq. (88) is given by

$-g^{\mu\nu}\,{\widetilde{W}}^{q,\text{ns}}_{\mu\nu}=2\alpha_{s}Q_{f}^{2}C_{F}(1-\epsilon)\bigg{\\{}{\widetilde{N}}_{0}(\tau,z)+{\widetilde{N}}_{1}(\tau,z)+{\widetilde{R}}^{q}(\tau_{\text{m}},z)+\Theta_{2}\Delta^{q}\Bigl{(}\frac{1-z}{z}\Bigr{)}\bigg{\\}}\,,$
(94)

where $\tau_{\text{m}}$ is the cumulative integral over $\Theta_{0}$

$\tau_{\text{m}}\equiv\int
d\tau^{\prime}\Theta_{0}(\tau^{\prime},z)=\min\bigg{\\{}\tau,\frac{1}{2},\frac{1-z}{z}\bigg{\\}}=\Theta_{0}\,\tau+\Theta_{1}\,\frac{1}{2}+\Theta_{2}\,\frac{1-z}{z}\,.$
(95)

The functions ${\widetilde{N}}_{0,1},\,{\widetilde{R}}$ are defined as

$\displaystyle\begin{split}{\widetilde{N}}_{n}(\tau,z)&=\int_{0}^{\tau}d\tau^{\prime}\,\frac{\ln^{n}\tau^{\prime}}{\tau^{\prime}}\,N_{n}(\tau^{\prime},z)\\\
&=\theta(\tau)\int_{0}^{\tau_{\text{m}}}\frac{d\tau^{\prime}}{\tau^{\prime}}\ln^{n}\tau^{\prime}[S_{n}^{q}(\tau^{\prime},z)-S_{n}^{q}(0,z)]+S_{n}^{q}(0,z)\Biggl{[}\frac{\ln^{n+1}\tau-\ln^{n+1}\tau_{\text{m}}}{n+1}\Biggr{]}\,,\end{split}$
(96a) $\displaystyle{\widetilde{R}}^{q}(\tau_{\text{m}},z)$
$\displaystyle=\int^{\tau}_{0}d\tau^{\prime}\Theta_{0}(\tau^{\prime},z)R^{q}(\tau^{\prime},z)=\frac{1-4z}{1-z}\tau_{\text{m}}-\frac{1+z^{2}}{1-z}\ln(1-\tau_{\text{m}})\,.$
(96b)

###  A.5 Convolution with PDF

The hadronic tensor with a hadron state $h$ is expressed in the factorized
form as

$\displaystyle
W^{h}_{\mu\nu}(x,\tau,Q^{2})=\sum_{i\in\\{q,\bar{q},g\\}}\int_{x}^{1}\frac{d\xi}{\xi}f_{i/h}(\xi,\mu)\,w^{i}_{\mu\nu}(x/\xi,\tau,Q^{2},\mu)\,,$
(97)

where $f_{i/h}$ is the PDF for the initial hadron $h$ into a parton $i$ and
the superscript $i$ on the coefficient $w^{i}$ represents the contribution
from the parton $i$. Note that we obtained the tensors $W_{\mu\nu}^{q,g}$ for
incoming quark and gluon and by the perturbative matching performed in
Kang:2014qba  one can translate them into the coefficients $w^{i}$. We do not
describe the matching procedure, and the expressions for $w^{i}$ are
essentially the same as $W_{\mu\nu}^{q,g}$. Instead of writing $w^{i}$, we
give expressions for the hadronic tensor Eq. (97) convolved with the PDF. The
nonsingular part of the hadronic tensor is given by

$\displaystyle P^{\mu}P^{\nu}W^{h,\text{ns}}_{\mu\nu}=\frac{2\pi
Q^{2}}{x^{2}}(\mathcal{A}^{\text{ns}}_{q}+\mathcal{A}^{\text{ns}}_{\bar{q}}+\mathcal{A}^{\text{ns}}_{g})\,,$
(98)
$\displaystyle\mathcal{A}^{\text{ns}}_{q}=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\left[\Theta_{0}(\tau,z)\,2z+\delta(\tau-\tfrac{1-z}{z})\theta(z-2/3)\,(1-2\tau)(1+\tau)z^{2})\right]$
$\displaystyle\quad\quad=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\Theta_{0}(\tau,x)\left\\{\int_{x}^{\tfrac{1}{1+\tau}}dz\,2\,f_{q}\bigg{(}\frac{x}{z}\bigg{)}+\frac{1-2\tau}{(1+\tau)^{2}}f_{q}\big{(}x(1+\tau)\big{)}\right\\}\,,$
(99)
$\displaystyle\mathcal{A}^{\text{ns}}_{g}=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\int_{x}^{1}\frac{dz}{z}f_{g}\left(\frac{x}{z}\right)\Theta_{0}(\tau,z)(1-z)\left[2z+\delta(\tau-\tfrac{1-z}{z})\,(1-2\tau)(1+\tau)z^{2}\right]$
$\displaystyle\quad\quad=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\,\Theta_{0}(\tau,x)\left\\{\int_{x}^{\tfrac{1}{1+\tau}}dz\,2(1-z)f_{g}\bigg{(}\frac{x}{z}\bigg{)}+\frac{\tau(1-2\tau)}{(1+\tau)^{3}}f_{g}\big{(}x(1+\tau)\big{)}\right\\}\,,$
(100)

The sums goes over the flavors $f\in\\{u,d,s,c,b\\}$.
$\mathcal{A}^{\text{ns}}_{\bar{q}}$ is obtained by replacing quark PDF by
anti-quark PDF in $\mathcal{A}^{\text{ns}}_{q}$ and this is true for
$\mathcal{B}^{\text{ns}}_{q}$ and their cumulative. The other tensor
projection is expressed as

$-g^{\mu\nu}W_{\mu\nu}^{h,\text{ns}}=8\pi(1-\epsilon)(\mathcal{B}^{\text{ns}}_{q}+\mathcal{B}^{\text{ns}}_{\bar{q}}+\mathcal{B}^{\text{ns}}_{g})\,,$
(101)

where $\mathcal{B}^{\text{ns}}_{q}$ and $\mathcal{B}^{\text{ns}}_{g}$ are

$\displaystyle\mathcal{B}^{\text{ns}}_{q}=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\bigg{\\{}\int_{x}^{1}\frac{dz}{z}f_{q,\bar{q}}\left(\frac{x}{z}\right)\bigg{[}\frac{1}{\tau}\,N_{0}(\tau,z)+\frac{\ln\tau}{\tau}\,N_{1}(\tau,z)\bigg{]}$
$\displaystyle\qquad\qquad+\Theta_{0}(\tau,x)\int_{x}^{\tfrac{1}{1+\tau}}\frac{dz}{z}f_{q,\bar{q}}\left(\frac{x}{z}\right)R^{q}(\tau,z)+\Theta_{0}(\tau,x)\,f_{q,\bar{q}}(x(1+\tau))\frac{\Delta^{q}(\tau)}{1+\tau}\bigg{\\}}\,,$
(102)
$\displaystyle\mathcal{B}^{\text{ns}}_{g}=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\bigg{\\{}\Theta_{0}\int_{x}^{\tfrac{1}{1+\tau}}\frac{dz}{z}f_{g}\big{(}x/z\big{)}\bigg{(}\frac{P_{qg}(z)}{1-\tau}-2\bigg{)}$
$\displaystyle\hskip
85.35826pt+\Theta_{0}f_{g}(x(1+\tau))\bigg{(}\frac{2\tau-1}{1+\tau}+\frac{1+\tau^{2}}{(1+\tau)^{3}}\ln\frac{1-\tau}{\tau}\bigg{)}$
$\displaystyle\hskip
85.35826pt+\frac{1}{\tau}\Theta_{0}\int^{\frac{1}{1+\tau}}_{x}\frac{dz}{z}f_{g}\left(\frac{x}{z}\right)P_{qg}(z)-\frac{1}{\tau}\int^{1}_{x}\frac{dz}{z}f_{g}\left(\frac{x}{z}\right)P_{qg}(z)\bigg{\\}}\,,$
(103)

where the integrals involving $N_{0,1}$ are given by

$\displaystyle\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\frac{\ln\tau}{\tau}\,N_{1}(\tau,z)=-4\frac{\ln\tau}{\tau}\bigg{[}\Theta_{0}\,\frac{2+\tau}{2(1+\tau)}f_{q}\big{(}x(1+\tau)\big{)}-f_{q}(x)\bigg{]}\,,$
(104)
$\displaystyle\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\frac{1}{\tau}\,N_{0}(\tau,z)=-\frac{3}{2\tau}\bigg{[}\Theta_{0}\,\frac{f_{q}\big{(}x(1+\tau)\big{)}}{1+\tau}-f_{q}(x)\bigg{]}$
$\displaystyle\quad+\frac{2}{\tau}\bigg{\\{}\Theta_{0}\bigg{[}\int^{\frac{1}{1+\tau}}_{x}dz\frac{f_{q}\big{(}x/z\big{)}-f_{q}\big{(}x(1+\tau)\big{)}}{1-z}+f_{q}\big{(}x(1+\tau)\big{)}\ln(1-x)\bigg{]}$
$\displaystyle\hskip
56.9055pt-\int^{1}_{x}dz\frac{f_{q}\big{(}x/z\big{)}-f_{q}(x)}{1-z}-f_{q}(x)\ln(1-x)\bigg{\\}}$
$\displaystyle\quad+\frac{1}{\tau}\bigg{\\{}\Theta_{0}\,\int^{\tfrac{1}{1+\tau}}_{x}dz\,f_{q}\left(\frac{x}{z}\right)\frac{1-z}{z}-\int_{x}^{1}dz\,f_{q}\left(\frac{x}{z}\right)\frac{1-z}{z}\bigg{\\}}\,.$
(105)

Note that terms with $1-\Theta_{0}=\Theta_{1}+\Theta_{2}$ are contributions
from unphysical regions and they cancel the contributions from singular parts
in the same region.

Next let us calculate the convolution with cumulants of Eq. (91)

$\displaystyle P^{\mu}P^{\nu}{\widetilde{W}}^{h,\text{ns}}_{\mu\nu}$
$\displaystyle=\frac{2\pi
Q^{2}}{x^{2}}(A^{\text{ns}}_{q}+A^{\text{ns}}_{\bar{q}}+A^{\text{ns}}_{g})\,,$
(106) $\displaystyle A^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\int_{x}^{1}\frac{dz}{z}f_{q,\bar{q}}\left(\frac{x}{z}\right)\,\bigg{[}2z\tau_{\text{m}}+\Theta_{2}\,(3z-2)\bigg{]}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\bigg{\\{}2\tau\Theta_{0}\,\int^{\tfrac{1}{1+\tau}}_{x}dz\,f_{q}\left(\frac{x}{z}\right)+\Theta_{1}\int^{\tfrac{2}{3}}_{x}dz\,f_{q}\left(\frac{x}{z}\right)$
$\displaystyle\qquad\hskip
170.71652pt+\int^{1}_{\max\big{[}\tfrac{1}{1+\tau},\tfrac{2}{3},x\big{]}}dz\,f_{q}\left(\frac{x}{z}\right)\bigg{\\}}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\biggl{\\{}(2\tau-1)\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dz\,f_{q}\left(\frac{x}{z}\right)+\int_{x}^{1}dz\,f_{q}\left(\frac{x}{z}\right)\biggr{\\}}\,,$
(107) $\displaystyle A^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\biggl{\\{}(2\tau-1)\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dzf_{g}\left(\frac{x}{z}\right)(1-z)+\int_{x}^{1}dz\,f_{g}\left(\frac{x}{z}\right)(1-z)\biggr{\\}}\,.$
(108)

Here the extra factor $(1-z)$ in the gluon integral compared with the quark
part corresponds to the extra factor $(1-z)$ in Eq. (91).

Now we calculate convolution with cumulants in Eqs. (93) and (94):

$-g^{\mu\nu}{\widetilde{W}}^{h,\text{ns}}_{\mu\nu}=8\pi(1-\epsilon)(B^{\text{ns}}_{q}+B^{\text{ns}}_{\bar{q}}+B^{\text{ns}}_{g})\,,$
(109)

where

$\displaystyle B^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\biggl{\\{}\Theta_{0}\int^{\frac{1}{1+\tau}}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\big{[}1-2\tau-
P_{qg}(z)\,\ln\tfrac{1-\tau}{\tau}\big{]}$
$\displaystyle\qquad\qquad\qquad\qquad-\int^{1}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\big{[}1+P_{qg}(z)\ln\tau\big{]}\biggr{\\}}\,,$
(110) $\displaystyle B^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\bigg{\\{}\int_{0}^{\tau}d\tau^{\prime}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\bigg{[}\frac{1}{\tau}N_{0}(\tau^{\prime},z)+\frac{\ln\tau^{\prime}}{\tau^{\prime}}N_{1}(\tau^{\prime},z)\bigg{]}$
$\displaystyle\hskip
85.35826pt+\,\int^{1}_{x}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\widetilde{R}^{q}(\tau^{\prime},z)+\,\int^{1}_{x}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\widetilde{\Delta}^{q}(\tau,z)\bigg{\\}}\,,$
(111)

where ${\widetilde{R}}^{q}$ is cumulant of $R^{q}$ and is defined in Eq. (96b)
and $\widetilde{\Delta}^{q}$ is cumulant of $\delta^{q}$ term, that is

$\widetilde{\Delta}^{q}(\tau,z)=\Theta_{2}(\tau,z)\Delta^{q}\big{(}(1-z)/z\big{)}.$
(112)

Plugging Eqs. (96b) and (112) into last two integrals in Eq. (A.5), we obtain

$\displaystyle\int^{1}_{x}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\bigg{(}\widetilde{R}^{q}(\tau,z)+\widetilde{\Delta}^{q}(\tau,z)\bigg{)}$
$\displaystyle\quad=\Theta_{0}\int^{\frac{1}{1+\tau}}_{x}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\bigg{[}\frac{1-4z}{1-z}\tau-
P_{qq}(z)\,\ln(1-\tau)\bigg{]}$
$\displaystyle\qquad+\Theta_{1}\int^{\frac{2}{3}}_{x}\frac{dz}{z}\,f_{q}\left(\frac{x}{z}\right)\bigg{[}\frac{1-4z}{2(1-z)}+P_{qq}(z)\,\ln
2\bigg{]}$
$\displaystyle\quad\quad+\int^{1}_{\max\big{[}\frac{1}{1+\tau},\frac{2}{3},x\big{]}}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\bigg{[}\frac{1}{2}+\frac{2\ln
z}{z-1}+(z-1)\ln\frac{1-z}{z}\bigg{]}\,.$ (113)

The terms $N_{i}$ are given in Eq. (89). Integral of each term is given by

$\displaystyle\int_{0}^{1}d\tau^{\prime}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\frac{\ln\tau^{\prime}}{\tau^{\prime}}N_{1}(\tau^{\prime},z)$
$\displaystyle\quad=-4\bigg{[}\int_{0}^{\tau_{\text{m}}}d\tau^{\prime}\,\frac{\ln\tau^{\prime}}{\tau^{\prime}}\bigg{(}\frac{2+\tau^{\prime}}{2(1+\tau^{\prime})}f_{q}(x(1+\tau^{\prime}))-f_{q}(x)\bigg{)}-\frac{f_{q}(x)}{2}\left(\ln^{2}\tau-\ln^{2}\tau_{\text{m}}]\right)\bigg{]}$
$\displaystyle\quad=-2\int^{1}_{\max\big{[}\frac{1}{1+\tau},\frac{2}{3},x\big{]}}\frac{dz}{z}\frac{\ln\frac{1-z}{z}}{1-z}\bigg{[}(1+z)f_{q}\left(\frac{x}{z}\right)-2f_{q}(x)\bigg{]}+2f_{q}(x)\,\bigg{[}\ln^{2}\tau-\ln^{2}\tau_{\text{m}}\bigg{]}\,,$
(114)
$\displaystyle\int_{0}^{\tau}d\tau^{\prime}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\frac{1}{\tau^{\prime}}N_{0}(\tau^{\prime},z)$
$\displaystyle\quad=2\\!\\!\int_{0}^{\tau_{\text{m}}}d\tau^{\prime}\,\bigg{[}\frac{\ln\tfrac{\tau^{\prime}}{1+\tau^{\prime}}}{\tau^{\prime}}\big{[}f_{q}(x(1+\tau^{\prime}))-f_{q}(x)\big{]}-\frac{1}{\tau^{\prime}}\\!\\!\int^{1}_{\tfrac{1}{1+\tau^{\prime}}}\,dz\frac{f_{q}\left(\frac{x}{z}\right)-f_{q}(x)}{1-z}\bigg{]}$
$\displaystyle\hskip
28.45274pt-2\ln\bigg{(}\frac{\tau}{\tau_{\text{m}}}\bigg{)}\bigg{(}\int_{x}^{1}dz\,\frac{f_{q}\left(\frac{x}{z}\right)-f_{q}(x)}{1-z}+f_{q}(x)\ln(1-x)\bigg{)}$
$\displaystyle\hskip
28.45274pt-\frac{3}{2}\bigg{[}\int_{0}^{\tau_{\text{m}}}d\tau^{\prime}\,\frac{1}{\tau^{\prime}}\bigg{[}\frac{1}{(1+\tau^{\prime})}f_{q}(x(1+\tau^{\prime}))-f_{q}(x)\bigg{]}-f_{q}(x)\left(\ln\tau-\ln\tau_{\text{m}}\right)\bigg{]}$
$\displaystyle\hskip
28.45274pt+\int_{x}^{1}dz\,\frac{(1-z)\,f_{q}\left(\frac{x}{z}\right)}{z}\ln\frac{\tau_{\text{m}}}{\tau}\,.$
(115)

The double integrals over $\tau^{\prime}$ and $z$ that still exist in Eq.
(115) can be simplified by switching the order of integration. Doing so, and
also changing variables in the remaining integrals over $\tau^{\prime}$ using
$z=1/(1+\tau^{\prime})$ so that all integrals are over $z$, we obtain simpler
expression

$\displaystyle\int_{0}^{\tau}d\tau^{\prime}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\,\frac{1}{\tau^{\prime}}N_{0}(\tau^{\prime},z)$
$\displaystyle=2\int_{\max\big{[}\frac{1}{1+\tau},\frac{2}{3},x\big{]}}^{1}\frac{dz}{z}\Bigl{[}\ln(1-z)+z\ln\frac{1-z}{z\tau}\Bigr{]}\frac{f_{q}\left(\frac{x}{z}\right)-f_{q}(x)}{1-z}$
(116)
$\displaystyle\quad-2\Theta_{1}\ln(2\tau)\int_{x}^{\frac{2}{3}}dz\frac{f_{q}\left(\frac{x}{z}\right)-f_{q}(x)}{1-z}-2\Bigl{[}\Theta_{1}\ln(2\tau)+\Theta_{2}\ln\frac{x\tau}{1-x}\Bigr{]}f_{q}(x)\ln(1-x)$
$\displaystyle\quad-\frac{3}{2}\int_{\max\big{[}\frac{1}{1+\tau},\frac{2}{3},x\big{]}}^{1}\frac{dz}{z}\frac{zf_{q}\left(\frac{x}{z}\right)-f_{q}(x)}{1-z}+\frac{3}{2}f_{q}(x)\Bigl{[}\Theta_{1}\ln(2\tau)+\Theta_{2}\ln\frac{\tau
x}{1-x}\Bigr{]}$
$\displaystyle\quad+\int_{\max\big{[}\frac{1}{1+\tau},\frac{2}{3},x\big{]}}^{1}dz\,f_{q}\left(\frac{x}{z}\right)\frac{1-z}{z}\ln\frac{1-z}{z\tau}-\Theta_{1}\ln(2\tau)\int_{x}^{2/3}dz\,f_{q}\left(\frac{x}{z}\right)\frac{1-z}{z}\,.$
(117)

Finally, we combine all terms contributing to $B^{\text{ns}}_{q}$ in Eq.
(A.5). Some further simplifications occur upon summing Eqs. (A.5), (114), and
(A.5). Doing so, we obtain the final result for $B^{\text{ns}}_{q}$, which can
be written as in Eq. (32).

##  Appendix B Additional contribution for $\tau^{\text{ct}}$

Here, we summarize the results of additional contributions for
$\tau^{\text{ct}}$ which is to be added onto $\tau^{\text{jt}}$ results.

Since we take the difference between two 1-jettiness, all divergences are
cancelled. Hence, the necessary $v$ integrals are finite and all $\epsilon$
can be dropped as done in Eq. (54d) and in the quark amplitudes Eqs. (62) and
(63) and in the gluon amplitudes Eqs. (79) and (80). Integrating over $v$ in
these amplitudes and performing the matching, we obtain the final results. To
avoid duplication we omit showing intermediate steps and actually the
intermediate expressions are similar to final results because there are not
much simplifications.

The final expressions of differential version of $\delta A$ and $\delta B$ are
given by

$\displaystyle\delta\mathcal{A}^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\,\left\\{\frac{2z_{0}^{4}}{3z_{0}-2}\,\delta\hat{A}_{q}(R,z_{0})-2z_{1}^{3}\,\delta\hat{A}_{q}(R,z_{1})\right\\}\,,$
$\displaystyle\delta\mathcal{A}^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{\pi}\,\left\\{\frac{z_{0}^{4}}{3z_{0}-2}\,\delta\hat{A}_{g}(R,z_{0})-z_{1}^{3}\,\delta\hat{A}_{g}(R,z_{1})\right\\}\,,$
$\displaystyle\delta\mathcal{B}^{\text{ns}}_{q}$
$\displaystyle=\sum_{f}Q^{2}_{f}\frac{\alpha_{s}C_{F}}{4\pi}\,\left\\{\frac{z_{0}^{2}}{3z_{0}-2}\,\delta\hat{B}_{q}(R,z_{0})-z_{1}\,\delta\hat{B}_{q}(R,z_{1})\right\\}\,,$
$\displaystyle\delta\mathcal{B}^{\text{ns}}_{g}$
$\displaystyle=\sum_{f}Q^{2}_{f}\frac{\alpha_{s}T_{F}}{2\pi}\,\left\\{\frac{z_{0}^{2}}{2(3z_{0}-2)}\,\delta\hat{B}_{g}(R,z_{0})-\frac{z_{1}}{2}\,\delta\hat{B}_{g}(R,z_{1})\right\\}\,.$
(118)

The parameters $z_{0,1}$ are the relations between $z$ and $\tau$ in region IV

$\displaystyle z_{0}=\frac{3+\sqrt{1-4\tau}}{2(2+\tau)}\,,\quad
z_{1}=\frac{1}{1+\tau}\,.$ (119)

The functions $\delta\hat{A},\delta\hat{B}$ are

$\displaystyle\delta\hat{A}_{q}(R,z)$
$\displaystyle=\theta(z-x)\theta(z-z_{c})\theta(-z+1)\frac{r(z,R)}{2z}f_{q}\left(\frac{x}{z}\right)\,,$
$\displaystyle\delta\hat{A}_{g}(R,z)$
$\displaystyle=\theta(z-x)\theta(z-z_{c})\theta(-z+1)\frac{1-z}{z}r(z,R)f_{g}\left(\frac{x}{z}\right)\,,$
$\displaystyle\delta\hat{B}_{q}(R,z)$
$\displaystyle=\theta(z-x)\theta(z-z_{c})\theta(-z+1)\left[\frac{1-4z}{2(1-z)}r(z,R)+\frac{1+z^{2}}{1-z}\ln\frac{1+r(z,R)}{1-r(z,R)}\right]f_{q}\left(\frac{x}{z}\right)\,,$
$\displaystyle\delta\hat{B}_{g}(R,z)$
$\displaystyle=\theta(z-x)\theta(z-z_{c})\theta(-z+1)\left[2P_{qg}(z)\ln\frac{1+r(z,R)}{1-r(z,R)}-2r(z,R)\right]f_{g}\left(\frac{x}{z}\right)\,,$
(120)

where $z_{c}(R)$ is given in Eq. (23) and $r(z,R)$ is in Eq. (38). If we
insert $z_{0}$ and $z_{1}$ into the constraint $\theta(z-z_{c})$, we obtain
two upper limits of $\tau$ $(16-R^{2})R^{2}/256$ and $R^{2}/16$, beyond which
each of two contributions in Eq. (B) becomes zero.

##  Appendix C Difference between $\tau^{b}$ and $\tau^{\text{jt}}$

Here we give the difference between $\tau^{b}$ and $\tau^{\text{jt}}$ results.
The cumulative results for $\tau^{\text{jt}}$ are given in Eqs. (30) and (32)
and results for $\tau^{b}$ in Kang:2014qba . Their differences in singular and
nonsingular parts are given by

$\displaystyle
B_{q}^{\text{sing}}(\tau^{b})-B_{q}^{\text{sing}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\int_{x}^{1}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)P_{qq}(z)\ln
z\,,$ $\displaystyle
B_{g}^{\text{sing}}(\tau^{b})-B_{g}^{\text{sing}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\int_{x}^{1}\frac{dz}{z}f_{g}\left(\frac{x}{z}\right)P_{qg}(z)\ln
z\,,$ $\displaystyle
A_{q}^{\text{ns}}(\tau^{b})-A_{q}^{\text{ns}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dz\,f_{q}\left(\frac{x}{z}\right)2\tau(z-1)\,,$
$\displaystyle
A_{g}^{\text{ns}}(\tau^{b})-A_{g}^{\text{ns}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\,\Theta_{0}\int_{x}^{\frac{1}{1+\tau}}dzf_{g}\left(\frac{x}{z}\right)2\tau[-(1-z)^{2}]\,,$
$\displaystyle
B_{q}^{\text{ns}}(\tau^{b})-B_{q}^{\text{ns}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\Biggl{(}\Theta_{0}\biggl{\\{}\int_{x}^{\frac{1}{1+\tau}}\frac{dz}{z}f_{q}\left(\frac{x}{z}\right)\Bigl{[}-\tau(1-4z)+P_{qq}(z)\ln\frac{z(1-\tau)}{1-z\tau}\Bigr{]}\biggr{\\}}$
$\displaystyle\quad+\int^{1}_{x}\frac{dz}{z}\,f_{q}\left(\frac{x}{z}\right)\bigg{[}-P_{qq}(z)\ln
z\bigg{]}\Biggr{)}\,,$ $\displaystyle
B_{g}^{\text{ns}}(\tau^{b})-B_{g}^{\text{ns}}(\tau^{\text{jt}})$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\biggl{\\{}\Theta_{0}\int^{\frac{1}{1+\tau}}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)\bigg{[}2\tau(1-z)+P_{qg}(z)\,\ln\frac{z(1-\tau)}{1-z\tau}\bigg{]}$
$\displaystyle\qquad\qquad-\int^{1}_{x}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)P_{qg}(z)\ln
z\biggr{\\}}\,,$ (121)

where those not listed above are zero.

The NLP obtained from Eq. (C) can be expressed as

$\displaystyle
A_{q}^{\text{ns}}(\tau^{b})-A_{q}^{\text{ns}}(\tau^{\text{jt}})\big{|}_{\tau\rightarrow
0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\,\tau\int_{x}^{1}dz\,f_{q}\left(\frac{x}{z}\right)2(z-1)+\mathcal{O}(\tau^{2})\,,$
$\displaystyle
A_{g}^{\text{ns}}(\tau^{b})-A_{g}^{\text{ns}}(\tau^{\text{jt}})\big{|}_{\tau\rightarrow
0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{\pi}\,\tau\int_{x}^{1}dzf_{g}\left(\frac{x}{z}\right)[-2(1-z)^{2}]+\mathcal{O}(\tau^{2})\,,$
$\displaystyle
B_{q}^{\text{ns}}(\tau^{b})-B_{q}^{\text{ns}}(\tau^{\text{jt}})\big{|}_{\tau\rightarrow
0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}C_{F}}{4\pi}\tau\biggr{\\{}\int_{x}^{1}\frac{dz}{z}\,f_{q}\left(\frac{x}{z}\right)\bigg{[}-1+4z+P_{qq}(z)(z-1)\bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad+2f_{q}\left(x\right)\biggr{\\}}+\mathcal{O}(\tau^{2})\,,$
$\displaystyle
B_{g}^{\text{ns}}(\tau^{b})-B_{g}^{\text{ns}}(\tau^{\text{jt}})\big{|}_{\tau\rightarrow
0}$
$\displaystyle=\sum_{f}Q_{f}^{2}\frac{\alpha_{s}T_{F}}{2\pi}\tau\biggr{\\{}\int_{x}^{1}\frac{dz}{z}\,f_{g}\left(\frac{x}{z}\right)(1-z)\big{[}2-P_{qg}(z)\big{]}\biggr{\\}}+\mathcal{O}(\tau^{2})\,.$
(122)

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