the jet function are Sudakov logarithms and have been studied extensively
before Sterman:1986aj ; Catani:1989ne ; deFlorian:2007fv ; Liu:2012sz ;
Liu:2013hba ; Kumar:2013hia ; Dai:2017dpc ; Liu:2017pbb . Their resummation
can be achieved by re-factorizing the siJF into a exclusive jet function and a
collinear-soft function Liu:2017pbb ; Xie:future1 . The procedure is standard
in the language of SCET and we will leave the derivation to future work
Xie:future1 . At the leading logarithmic accuracy, the resummed jet function
is

$\displaystyle J_{thr.,resum}=\exp\left[-\frac{\alpha_{s}}{\pi}{\bf
T}_{j}^{2}\,\left(-\ln{\bar{N}}\ln\frac{p_{J\perp}^{2}R^{2}}{\mu^{2}}+\ln^{2}{\bar{N}}\right)\right]\,,$
(129)

which is the direct exponentiation of the Sudakov logarithms.

  * •

It is more involved to resum the logarithmic term from the interference of the
ISR and FSR

$\displaystyle\langle{\cal
M}_{0}|2\,\frac{\alpha_{s}}{\pi}\,\ln{\bar{N}}\int\frac{d^{2}r_{\perp}}{\pi}\left(\frac{x_{\perp}\cdot
y_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\right)_{+}{\bf
T}_{i}^{a}W_{aa^{\prime}}(r_{\perp}){\bf T}_{j}^{a^{\prime}}|{\cal
M}_{0}\rangle\,,$ (130)

which calls for the threshold resummation technique different from the
standard ones in the collinear factorization Becher:2006nr ; Becher:2006mr ;
Liu:2017pbb . One way to see this is to notice that this term shares the same
color structure ${\bf T}_{i}^{a}W_{aa^{\prime}}(r_{\perp}){\bf
T}_{j}^{a^{\prime}}$ as the non-linear BK evolution, see Eq. (48). Indeed, as
we have seen in Section 3.4, at higher orders, extra Wilson lines $W$’s arise,
which suggests the need for a non-linear evolution to resum this contribution.
The standard Sudakov resummation techniques only turn the color charge ${\bf
T}$ into a resummed form, see for instance Catani:2013vaa ; Becher:2014oda ;
Hinderer:2018nkb ; Liu:2017pbb , while will leave the CGC Wilson line $W$
untouched.

  * •

A systematic method to resum the logs proportional to ${\bf
T}_{i}^{a}W_{aa^{\prime}}(r_{\perp}){\bf T}_{j}^{a^{\prime}}$ in Eq. (127) has
been developed recently in Liu:2020mpy , which will be detailed in Xie:future1
for the jet production. However, we note that for the leading log resummation
it is sufficient to consider the $n$ independent soft gluon emissions at
N(n)LO, with strong ordering $q^{-}_{1}\gg q^{-}_{2}\gg\dots\gg q^{-}_{m}$,
and $p^{-}_{1}\gg p^{-}_{2}\gg\dots p^{-}_{n-m}$, where $q_{i}$ and $p_{i}$
are the ISR and FSR momenta, respectively. It can be shown that the matrix
could be written as

$\displaystyle\frac{1}{n!}\left|\sum_{m=0}^{n}\vbox{\hbox{\includegraphics[scale={0.23}]{real-
soft-multi-resum.pdf}}}\right|^{2}$ (131) $\displaystyle=$
$\displaystyle\frac{1}{n!}\langle{\cal
M}_{0}|\left\\{-\frac{\alpha_{s}}{\pi}\int\frac{d^{2}r_{\perp}}{\pi}\Bigg{[}-2\ln{\bar{N}}\left(\frac{x_{\perp}\cdot
y_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\right)_{+}\right.$ $\displaystyle\hskip
86.11084pt\left.+\ln\frac{X_{f}}{X_{A}}\left(\frac{z^{2}_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\right)_{+}\Bigg{]}{\bf
T}_{j}^{a^{\prime}}W_{a^{\prime}a}(r_{\perp}){\bf
T}_{i}^{a}\right\\}^{n}|{\cal M}_{0}\rangle\,,$

where integration over the phase space is implicitly understood and we have
used the property of the color charge operator in Eq. (31). When sum up all
order contributions, the above equation leads to the exponentiation of the
logarithms, which reads

$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\left|\sum_{m=0}^{n}\vbox{\hbox{\includegraphics[scale={0.23}]{real-
soft-multi-resum.pdf}}}\right|^{2}$ (132) $\displaystyle=$
$\displaystyle\langle{\cal
M}_{0}|\exp\left\\{-\frac{\alpha_{s}}{\pi}\int\frac{d^{2}r_{\perp}}{\pi}\Bigg{[}-2\ln{\bar{N}}\left(\frac{x_{\perp}\cdot
y_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\right)_{+}\right.$ $\displaystyle\hskip
86.11084pt\left.+\ln\frac{X_{f}}{X_{A}}\left(\frac{z^{2}_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\right)_{+}\Bigg{]}{\bf
T}_{j}^{a^{\prime}}W_{a^{\prime}a}(r_{\perp}){\bf T}_{i}^{a}\right\\}|{\cal
M}_{0}\rangle\,.$

One realizes that the above equation reproduce what was found in Liu:2020mpy
based on factorization and the rapidity renormalization group equations. The
resummed formula suggests that when approaching the threshold, instead of
choosing the CGC scale $X_{f}=X_{A}$, one should use a dynamic scale such that
$X_{f}$ minimizes the exponent Liu:2020mpy .

We briefly highlight how we derive the resummation form by going to the soft
strong ordering limit. We start with NLO. At NLO, we can have for the matrix
element

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.43}]{LL-soft-
fsr.pdf}}}=-g_{s}{\bf T}_{j}^{b}\frac{n_{\beta}}{p_{1}^{-}}|{\cal
M}_{0}\rangle\,\equiv{\bfcal J}_{\beta}^{b,(1)}(p_{1})|{\cal M}_{0}\rangle\,,$
(133)

which is the eikonal approximation, presented by the soft eikonal current
${\bfcal J}$ acting on $|{\cal M}_{0}\rangle$. We also have the ISR
contribution, which is

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.43}]{LL-soft-
isr.pdf}}}\equiv{\bfcal K}_{\beta}^{b,(1)}(q_{1})|{\cal M}_{0}\rangle$ (134)
$\displaystyle=$ $\displaystyle ig_{s}{\bf
T}_{i}^{a}\int\frac{dl^{-}dl_{\perp}}{(2\pi)^{D-1}}\frac{n^{\alpha}}{-l^{-}}e^{il_{\perp}\cdot
b_{\perp}}d_{\alpha\beta}(l)\frac{(-q_{1}^{+})}{-l^{2}}\int
d^{2}r_{\perp}e^{-i(q_{1\perp}+l_{\perp})\cdot
r_{\perp}}W_{ba}(r_{\perp})|{\cal M}_{0}\rangle$ $\displaystyle=$
$\displaystyle ig_{s}{\bf
T}_{i}^{a}\int\frac{dl^{-}dl_{\perp}}{(2\pi)^{D-1}}\frac{2e^{il_{\perp}\cdot
b_{\perp}}}{-l^{-}l^{+}}\left(-\frac{n_{\beta}}{2}l^{+}+l_{\beta}\right)\frac{(-q_{1}^{+})}{l^{2}}\int
d^{2}r_{\perp}e^{-i(q_{1\perp}+l_{\perp})\cdot
r_{\perp}}W_{ba}(r_{\perp})|{\cal M}_{0}\rangle$ $\displaystyle=$
$\displaystyle ig_{s}{\bf
T}_{i}^{a}\int\frac{dl^{-}dl_{\perp}}{(2\pi)^{D-1}}\frac{e^{il_{\perp}\cdot
b_{\perp}}}{-l^{-}l^{+}}\frac{2l_{\perp\beta}}{l^{2}}(-q_{1}^{+})\int
d^{2}r_{\perp}e^{-i(q_{1\perp}+l_{\perp})\cdot
r_{\perp}}W_{ba}(r_{\perp})|{\cal M}_{0}\rangle$ $\displaystyle=$
$\displaystyle-
g_{s}\int\frac{dl_{\perp}}{(2\pi)^{D-2}}\frac{2l_{\perp\beta}}{l_{\perp}^{2}}e^{il_{\perp}\cdot
b_{\perp}}\int d^{2}r_{\perp}e^{-i(q_{1\perp}+l_{\perp})\cdot
r_{\perp}}W_{ba}(r_{\perp}){\bf T}_{i}^{a}|{\cal M}_{0}\rangle\,,$

where we introduced $d_{\alpha\beta}(l)\equiv-
g_{\alpha\beta}+\frac{{\bar{n}}_{\alpha}l_{\beta}+{\bar{n}}_{\beta}l_{\alpha}}{\bar{n}\cdot
l}$ and have used the light-cone gauge condition ${\bar{n}}\cdot\epsilon=0$.
We performed the contour integration on $l^{-}$ similar to Eq. (145) to get
the final result. Here we have suppressed the rapidity regulators.

The LL approximation to the NLO cross section will then be

$\displaystyle d\hat{\sigma}^{(1)}_{LL}$ $\displaystyle=$ $\displaystyle\int
d\Phi_{p_{1}}\langle{\cal M}_{0}|\left({\bfcal
J}^{(1)\dagger}_{\beta}(p_{1})+{\bfcal
K}^{(1)\dagger}_{\beta}(p_{1})\right)\cdot\left({\bfcal
J}^{(1)}_{\beta}(p_{1})+{\bfcal K}^{(1)}_{\beta}(p_{1})\right)|{\cal
M}_{0}\rangle$ (135) $\displaystyle\equiv$ $\displaystyle\langle{\cal
M}_{0}|d\hat{\bm{\sigma}}^{(1)}|{\cal M}_{0}\rangle\,.$

At NNLO, to produce LLs, we only need to consider $2$ independent soft gluon
emissions with momenta $p_{1}$ and $p_{2}$. We require $p_{1}^{-}\gg
p_{2}^{-}$. For the double FSR, the eiknoal current in this strongly order
limit is well known, which gives

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.4}]{LL-
soft-2fsr.pdf}}}={\bfcal J}_{\beta}^{b_{1},(1)}(p_{1}){\bfcal
J}_{\alpha}^{b_{2},(1)}(p_{2})|{\cal M}_{0}\rangle\,,$ (136)

with $p_{1}^{-}\gg p_{2}^{-}$. And for double ISR, the matrix element reads

$\displaystyle=$ $\displaystyle ig_{s}{\bf
T}_{i}^{a_{2}}\int\frac{dl_{2}^{-}dl_{2\perp}}{(2\pi)^{D-1}}\frac{n^{\rho}}{-l_{2}^{-}}e^{il_{2\perp}\cdot
b_{\perp}}d_{\rho\alpha}(l_{2})\frac{(-q_{2}^{+})}{-l_{2}^{2}}\int
d^{2}r_{2\perp}e^{-i(q_{2\perp}+l_{2\perp})\cdot
r_{2\perp}}W_{b_{2}a_{2}}(r_{2\perp})|{\cal M}_{0}\rangle$ (137)
$\displaystyle\times ig_{s}{\bf
T}_{i}^{a_{1}}\int\frac{dl_{1}^{-}dl_{1\perp}}{(2\pi)^{D-1}}\frac{n^{\sigma}}{-(l_{1}^{-}+l_{2}^{-})}e^{il_{1\perp}\cdot
b_{\perp}}d_{\sigma\beta}(l_{1})\frac{(-q_{1}^{+})}{-l_{1}^{2}}\int
d^{2}r_{1\perp}e^{-i(q_{1\perp}+l_{1\perp})\cdot
r_{1\perp}}W_{b_{1}a_{1}}(r_{1\perp})$ $\displaystyle\approx$ $\displaystyle
ig_{s}{\bf
T}_{i}^{a_{1}}\int\frac{dl_{1}^{-}dl_{1\perp}}{(2\pi)^{D-1}}\frac{n^{\sigma}}{-l_{1}^{-}}e^{il_{1\perp}\cdot
b_{\perp}}d_{\sigma\beta}(l_{1})\frac{(-q_{1}^{+})}{-l_{1}^{2}}\int
d^{2}r_{1\perp}e^{-i(q_{1\perp}+l_{1\perp})\cdot
r_{1\perp}}W_{b_{1}a_{1}}(r_{1\perp})$ $\displaystyle\times ig_{s}{\bf
T}_{i}^{a_{2}}\int\frac{dl_{2}^{-}dl_{2\perp}}{(2\pi)^{D-1}}\frac{n^{\rho}}{-l_{2}^{-}}e^{il_{2\perp}\cdot
b_{\perp}}d_{\rho\alpha}(l_{2})\frac{(-q_{2}^{+})}{-l_{2}^{2}}\int
d^{2}r_{2\perp}e^{-i(q_{2\perp}+l_{2\perp})\cdot
r_{2\perp}}W_{b_{2}a_{2}}(r_{2\perp})|{\cal M}_{0}\rangle$ $\displaystyle=$
$\displaystyle-
g_{s}\int\frac{dl_{1\perp}}{(2\pi)^{D-2}}\frac{2l_{1\perp\beta}}{l_{1\perp}^{2}}e^{il_{1\perp}\cdot
b_{\perp}}\int d^{2}r_{1\perp}e^{-i(q_{1\perp}+l_{1\perp})\cdot
r_{1\perp}}W_{ba}(r_{1\perp}){\bf T}_{i}^{a}$ $\displaystyle\times$
$\displaystyle-
g_{s}\int\frac{dl_{2\perp}}{(2\pi)^{D-2}}\frac{2l_{2\perp\alpha}}{l_{2\perp}^{2}}e^{il_{2\perp}\cdot
b_{\perp}}\int d^{2}r_{2\perp}e^{-i(q_{2\perp}+l_{2\perp})\cdot
r_{2\perp}}W_{ba}(r_{2\perp}){\bf T}_{i}^{a}|{\cal M}_{0}\rangle$
$\displaystyle=$ $\displaystyle{{\bfcal K}}_{\beta}^{b_{1},(1)}(q_{1}){{\bfcal
K}}_{\alpha}^{b_{2},(1)}(q_{2})|{\cal M}_{0}\rangle\,,$

in the strongly ordered limit, where to get the approximation we have used the
strongly ordered limit that $l_{2}^{-}\ll l_{1}^{-}$.

It is easy to get the matrx element with one ISR and one FSR which gives

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.4}]{LL-soft-isr-
fsr.pdf}}}=\left({\bfcal J}^{b_{1},(1)}_{\beta}(p_{1}){\bfcal
K}^{b_{2},(1)}_{\alpha}(q_{1})+{\bfcal K}^{b_{1},(1)}_{\beta}(q_{1}){\bfcal
J}^{b_{2},(1)}_{\alpha}(p_{1})\right)|{\cal M}_{0}\rangle\,.\quad\quad$ (138)

Here since there is no strong ordering requirements in $q_{1}$ and $p_{1}$, we
have splitted the contribution to 2 terms with $p_{1}>q_{1}$ and
$q_{1}>p_{1}$, respectively.

Therefore, one could find the LL approximation to the NNLO matrix element be

$\displaystyle|{\cal M}_{2}\rangle=\left({\bfcal
J}^{b_{1},(1)}_{\beta}+{\bfcal K}^{b_{1},(1)}_{\beta}\right)\left({\bfcal
J}^{b_{2},(1)}_{\alpha}+{\bfcal K}^{b_{2},(1)}_{\alpha}\right)|{\cal
M}_{0}\rangle\,,$ (139)

and thus

$\displaystyle d\hat{\sigma}^{(2)}_{LL}=\langle{\cal
M}_{0}|\frac{1}{2!}\left(d{\bm{\sigma}}^{(1)}\right)^{2}|{\cal
M}_{0}\rangle\,.$ (140)

Here the factor $1/2!$ accounts for symmetry factor of $2$ indistinguishable
gluons in the final state. The calculation is extendable to arbitrary $n$-th
order to find

$\displaystyle d\hat{\sigma}^{(n)}_{LL}=\langle{\cal
M}_{0}|\frac{1}{n!}\left(d\hat{\bm{\sigma}}^{(1)}\right)^{n}|{\cal
M}_{0}\rangle\,,$ (141)

which is exactly Eq. (131). Once one sums up all orders, one realizes the LL
resummation

$\displaystyle d\hat{\sigma}_{LL}=\langle{\cal
M}_{0}|\sum_{n=0}^{\infty}\frac{1}{n!}\left(d\hat{\bm{\sigma}}^{(1)}\right)^{n}|{\cal
M}_{0}\rangle=\langle{\cal M}_{0}|e^{d\hat{\bm{\sigma}}^{(1)}}|{\cal
M}_{0}\rangle\,,$ (142)

which is exactly Eq. (132) when going to the Mellin space.

##  8 Numerical Analyses

In this section, we perform numerical analyses. For the time being, we pay
attention to the numerical validation of the theoretical framework developed
previously, while we leave the comparison of the phenomenological predictions
with the LHC inclusive jet data to another work Xie:future1 .

Figure 12: Comparison between the NLO calculations with full jet algorithm
dependence and the small-$R$ limits. The open points represent the NLO
calculations in quark channel with the full jet algorithm dependence, using
different shapes to represent different jet radius $R$. The lower panels are
the ratio of these two results defined in Eq. (143). The solid lines are the
corresponding results in the small-$R$ limit. Left panel shows the inclusive
jet $p_{J\perp}$ spectra. Right panel for the jet energy $E_{J}$ distribution.
These are for jets constructed via the anti-$k_{T}$ algorithm with $R=0.5$ in
p+Pb collisions at center-of-mass energy $\sqrt{s}=5.02$ TeV per nucleon pair
and rapidity $5.2<\eta<6.6$.

For the numerical calculation, we use the NLO MSTW2008 PDF sets Martin:2009iq
for the proton PDFs. We applied the LL BK evolution with $\alpha_{s}$ running
Kovchegov:2006vj ; Balitsky:2006wa ; Fujii:2013gxa  to the nucleus dipole
distributions. As for the initial condition, the MV-like model
“$\gamma_{1119}$” are used and we stick to the parameter settings in
Fujii:2013gxa . As for the $S_{\perp}$, we follow the choice in
Watanabe:2015tja . In the calculation, we implement the kinematic cuts
following strictly the CMS experimental set-ups in CMS:2018yhi , in which the
proton and the Pb nucleus are colliding at the center of mass energy
$\sqrt{s}=5.02\,{\rm~{}TeV}$ per each nucleon pair. The jets are constructed
by the anti-$k_{T}$ algorithm with the radius parameter $R=0.5$ and required
to have a minimum transverse momentum $p_{J\perp}>3~{}{\rm GeV}$. We select
the jets at very forward pseudo-rapidities $5.2<\eta_{J}<6.6$ in the
laboratory frame, the boost between the laboratory frame and center-of-mass
frame is $\delta\eta_{J}=0.465$. Throughout the analyses, we set the
factorization scale $\mu=p_{J\perp}$ and the rapidity scale $X_{f}=X_{A}$
where $X_{A}$ is the momentum fraction carried by the gluon from the nucleus.

In fig. 12, we first display the comparison between the NLO single inclusive
jet cross section with full jet algorithm dependence and its analytical
small-$R$ approximation. We stick to the quark channel for this study. The NLO
distributions are obtained by the histogram procedure described in Section
5.9. The error bars reflect the numerical uncertainties. We have performed the
comparisons for $R=0.2$, $0.4$, $0.6$ and $1.0$.

In the left top panel of fig. 12, we plot the inclusive jet $p_{T\perp}$
distributions from the full NLO predictions (in open dots) and the small-$R$
approximations (in solid lines). From the plot, we can observe that for larger
values of $R$, jet spectrum spans to larger transverse momentum. This is
expected since with a larger jet radius, one clusters more particles into one
jet and thus more effortlessly generates larger jet transverse momentum to
pass the $3\,{\rm GeV}$ threshold than the small jet radius case. We also
observe that just like the single hadron production in $pA$ collisions, the
inclusive jet $p_{J\perp}$ spectrum manifests the negative cross section
problem for large $p_{J\perp}$. The issue can be resolved by the threshold
resummation Liu:2020mpy ; Xie:future1  as sketched in Section 7. As we go to
the smaller values of the jet radius $R$, the better the small-$R$
approximation becomes. This can be demonstrated from the ratio

$\displaystyle\rho_{R,p_{J\perp}}\equiv\frac{d\sigma_{\rm
full}/dp_{J,\perp}}{d\sigma_{\text{small $R$}}/dp_{J,\perp}}\,,$ (143)

where the numerator and denominator represent the jet cross section with the
full jet algorithm and its small-$R$ limit, respectively. This ratio is shown
in the lower part of the left panel, which shows strongly that as $R$ becomes
smaller, the ratio $\rho_{R,p_{J\perp}}$ approaches 1. The comparison serves
as a strong validation of the analytic small-$R$ approximation and the
factorization we derived. From the ratio, we can see that for small
$p_{J\perp}$, the small-$R$ approximation makes up $\geq 90\%$ of the complete
NLO result for narrow jets with $R=0.2$, $0.4$ and $0.6$, while the percentage
drops to $80\%$ for fat jets with $R=1.0$. The approximation is even better
for large $p_{J\perp}$. The small-$R$ approximation could break down when
strong cancellation kicks in between different contributions, and therefore
one should be careful when invoking the approximation for phenomenology
studies. In the present case, it happens to be when the cross section becomes
negative.

We plot the comparison using the inclusive jet $E_{J}$ spectrum in the right
panel of fig. 12. The division of the energy bins follows the CMS CMS:2018yhi
. The plot demonstrates the feasibility of our NLO calculation for various
different observables and experimental set-ups. Similar behaviours are
observed in the $E_{J}$ distribution as the $p_{J\perp}$ spectrum. The NLO
cross section eventually turns negative and the threshold resummation comes to
rescue Xie:future1 . The small-$R$ approximation did a better job in the
$E_{J}$ prediction than the $p_{J\perp}$ case, which can be understood that it
is mostly the low $p_{J\perp}$ jet events that fill up the $E_{J}$ spectrum.
This also explains why the NLO $E_{J}$ spectrum can stay positive for a wider
range.

Figure 13: This figure shows the comparison of the threshold contribution with
the NLO result. The error band represents the numerical uncertainty.

Now we turn to study the threshold limit in fig. 13. Here we include all
partonic channels which contain threshold logs (the $q\to q$ and $g\to g$
channels). The upper panel displays the jet $p_{J\perp}$ spectrum predicted by
the complete NLO calculation (in red dots) and its threshold approximation (in
green triangles). The NLO cross section becomes negative when
$p_{J\perp}>20\,{\rm GeV}$. To investigate the size of the threshold
contributions, we plot the ratio of the threshold limit to the full NLO
prediction $\rho_{thr.}=\frac{d\sigma_{thr.}/dp_{J\perp}}{d\sigma_{{\rm
NLO}}/dp_{J\perp}}$ as a function of the jet transverse momentum $p_{J\perp}$.
To manifest the effect of the threshold logarithms
$\left(\frac{\ln^{i}(1-\xi)}{1-\xi}\right)_{+}$, we have removed from the
ratio the common $\delta(1-\xi)$ terms shared by both $d\sigma_{thr.}$ and
$d\sigma_{\rm NLO}$. We see that when we crank up $p_{J\perp}$, the threshold
logarithms are overwhelmingly dominant and the ratio $\rho_{thr.}$ approaches
$1$.

##  9 Conclusions

In this work, we applied the recently proposed computational techniques
Kang:2019ysm ; Liu:2020mpy  to derive the complete NLO corrections to the
single inclusive jet production in proton-nucleus ($pA$) collisions at forward
rapidities, using the CGC effective theory. Within the framework, the cross
section is factorized into the product of the proton PDFs, the small-$x$
nucleus multi-pole distributions and the perturbatively calculable partonic
cross sections. Our result clearly shows that the CGC factorization is valid
at the NLO level for jet production.

The root of the factorization is the consistent and homogeneous power
expansion in the power counting parameter $\lambda\sim{\cal
O}\left(\sqrt{-t/s}\right)$. The power expansion relaxes the phase space
kinematic constraint which in turn is compensated by the existence of the soft
contribution. Our explicit NLO calculation showed that the soft mode is
required in order to generate the kinematic restrictions due to the jet
clustering in addition to the constraint in the hadron production case
Watanabe:2015tja .

Our ${\cal O}(\alpha_{s})$ calculation of the jet cross sections is fully
differential over the final state physical kinematics. The calculation is thus
not limited to the jet spectra predictions but is able to predict any
distribution of infra-red safe quantities. The full NLO jet cross section is
given in Eq. (105) and Appendix B. To perform this computation, we have
discussed a subtraction method to single out the singular contributions from
the phase space integration. The singular terms have been evaluated
analytically. The remaining non-singular part contains the jet algorithm and
the experimental cuts. Since the remaining integration is finite and
integrable in $4$-dimension, it was obtained numerically.

We further investigated the small jet radius limit of jet production. Given
that in practice the jet radius $R$ is usually chosen as ${\cal O}(0.4)$, the
small-$R$ limit in general is believed to be a very good approximation to the
full $R$ result and has been applied in several small-$x$ jet studies. In this
work, we are able to carry out the fully analytic small-$R$ jet cross section
in Eq. (• ‣ 6) and compare it with the full jet algorithm dependent cross
section to validate the approximation. We showed that in the small jet radius
limit, the single inclusive jet cross section can be further factorized into
the same short-distance cross section as the single inclusive hadron
production, with only the fragmentation functions replaced by the semi-
inclusive jet functions (siJFs). The siJFs share the same form as the ones
that show up in the jet production in the central region within the collinear
factorization with slightly different $\zeta$ dependence in the logarithm (see
Eq. (126)) due to the different jet transverse momentum range. We argued that
this factorization feature of the small-$R$ jet cross section holds for
generic jet processes and jet substructure observables in the CGC framework.
We carefully examined numerically when the small-$R$ limit reliably
approximates the full result. The obtained analytic result serves as a guide
to appropriately using the small-$R$ approximation in the forward scatterings,
meanwhile the observed factorization can facilitate high order and resummation
calculations of the jet cross sections and open up the opportunities to
realize the computations of event shapes in other forward scatterings.
Furthermore, the observed factorized form lays down the foundation for
combining the CGC fixed order calculations with the existing parton shower
techniques.

Like the forward hadron production, the obtained NLO jet spectrum also
exhibits the negative cross section feature when the jet transverse momentum
becomes large. Following the suggestions in Liu:2020mpy , we resolved the
negative cross section problem by the threshold resummation, where additional
Sudakov logarithms arising in the jet production have also been resummed.
Different from Liu:2020mpy , in this manuscript, we accomplish the leading
logarithmic threshold resummation using a different approach by considering
the strongly ordered independent emissions to all orders. The achieved
resummation agrees with the one obtained by the renormalizaton group equation
Liu:2020mpy ; Xie:future1 , which suggests the plausibility of both
approaches.

We look forward to comparing the NLO and resummation predictions against the
LHC $pA$ jet production data Xie:future1  in the future. Meanwhile, we expect
the computational set-ups developed in the present work and in Kang:2019ysm ;
Liu:2020mpy  lay down the theoretical foundation of the perturbative
predictions for the CGC phenomenology involving jet clustering procedures and
any other jet substructure and event shape observables.

###### Acknowledgements.

We thank Farid Salazar for suggestive comments on our manuscript. This work is
supported by the National Natural Science Foundation of China under Grant No.
12175016 (H.L., K.X. and X.L.), China Postdoctoral Science Foudation under
Grant No. 212400211 (H.L.) and the National Science Foundation under grant No.
PHY-1945471 (Z.B.).

##  Appendix A Useful Formulae

###  A.1 Contour Integral

The integral of the form

$\displaystyle
I=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{(k^{2}+i0^{+})((p_{j}-k)^{2}+i0^{+})}\,,$
(144)

is frequently encountered in this work. The $k$-loop integration can be
performed by contour, which gives

$\displaystyle\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{(k^{2}+i0^{+})((p_{j}-k)^{2}+i0^{+})}$
(145) $\displaystyle=$
$\displaystyle-\frac{1}{2}\int\frac{dk^{+}dk^{-}}{(2\pi)^{2}}\frac{d^{D-2}k_{\perp}}{(2\pi)^{D-2}}\frac{(k^{+}(p_{j}^{+}-k^{+}))^{-1}}{(k^{-}-\frac{k_{\perp}^{2}-i0^{+}}{k^{+}})(k^{-}-(p_{j}^{-}-\frac{(p_{j\perp}-k_{\perp})^{2}-i0^{+}}{p_{j}^{+}-k^{+}}))}$
$\displaystyle=$ $\displaystyle-(2\pi
i)\frac{1}{2}p_{j}^{+}\int_{0}^{1}\frac{dz}{(2\pi)^{2}}\int\frac{d^{D-2}k_{\perp}}{(2\pi)^{D-2}}\frac{(k^{+}(p_{j}^{+}-k^{+}))^{-1}}{\frac{p_{j\perp}^{2}}{p_{j}^{+}}-\frac{(p_{j\perp}-k_{\perp})^{2}}{p_{j}^{+}-k^{+}}-\frac{k_{\perp}^{2}}{k^{+}}}$
$\displaystyle=$
$\displaystyle\frac{i}{4\pi}\int_{0}^{1}dz\int\frac{d^{D-2}k_{\perp}}{(2\pi)^{D-2}}\frac{1}{(k_{\perp}-(1-z)p_{j\perp})^{2}}\,,$

We first did the $k^{-}$-integral. The integral is only non-vanishing when
$0<k^{+}<p_{j}^{+}$ otherwise all the poles occur at the same side of the
complex $k^{-}$-plain and one can choose the contour in the other side of the
plain to enclose no poles. To get the 3rd line, we have performed the
$k^{-}$-contour integral around the upper plane which picks the pole
$k^{-}=p_{j}^{-}-\frac{(p_{j\perp}-k_{\perp})^{2}-i0^{+}}{p_{j}^{+}-k^{+}}$.
and we let $p_{j}^{-}=p_{\perp}^{2}/p_{j}^{+}$ to reach the final result.

###  A.2 Fourier Transformation

We list some integral formulae for deriving the NLO results.

$\displaystyle\int d^{D-2}q_{\perp}\frac{e^{-iq_{\perp}\cdot
r_{\perp}}}{q_{\perp}^{2+2\alpha}}$ $\displaystyle=$
$\displaystyle\Omega_{D-3}\int
d^{D-2}q_{\perp}\,q_{\perp}^{D-3}\,\int_{0}^{\pi}d\phi\,\,\sin^{-2\epsilon}\phi\,\frac{e^{-iq_{\perp}r_{\perp}\cos\phi}}{q_{\perp}^{2+2\alpha}}$
(146) $\displaystyle=$ $\displaystyle\int
dq_{\perp}q_{\perp}^{D-5-2\alpha}\,\Omega_{D-3}\,\sqrt{\pi}\frac{\Gamma\left(\frac{1}{2}-\epsilon\right)}{\Gamma(1-\epsilon)}{}_{0}F_{1}\left(1-\epsilon,-\frac{1}{4}q_{\perp}^{2}r_{\perp}^{2}\right)$
$\displaystyle=$
$\displaystyle\pi^{1-\epsilon}\,\frac{\Gamma(-\epsilon-\alpha)}{\Gamma(1+\alpha)}\left(\frac{r_{\perp}}{2}\right)^{2\epsilon+2\alpha}\,.$

Here $\Omega_{D-3}=\frac{2\pi^{(D-3)/2}}{\Gamma((D-3)/2)}$ the
$(D-3)$-dimensional solid angle, and from the result we can easily deduce that

$\displaystyle\int\mathrm{d}^{D-2}q_{\perp}\frac{e^{-iq_{\perp}\cdot
r_{\perp}}}{q_{\perp}^{2}}q_{\perp}^{\alpha}=i\partial_{r_{\perp\alpha}}\int\mathrm{d}^{D-2}q_{\perp}\frac{e^{-iq_{\perp}\cdot
r_{\perp}}}{q_{\perp}^{2}}=-i\,2^{1-2\epsilon}\pi^{1-\epsilon}\Gamma(1-\epsilon)\frac{r_{\perp}^{\alpha}}{r_{\perp}^{2}}r_{\perp}^{2\epsilon}\,.$
(147)

###  A.3 Mellin Transformation

The Mellin transformation or the $N$th-moment of a function is given by

$\displaystyle M_{N}(f(\xi))=\int_{0}^{1}d\xi\xi^{N-1}f(\xi)\,.$ (148)

If we apply the Mellin transform to the convolution of $2$ functions

$\displaystyle f(\xi)\otimes
g(\xi)=\int_{0}^{1}\frac{d\xi^{\prime}}{\xi^{\prime}}f(\xi^{\prime})g\left(\frac{\xi}{\xi^{\prime}}\right)\,,$
(149)

we will have

$\displaystyle M_{N}(f(\xi)\otimes g(\xi))=M_{N}(f(\xi))M_{N}(g(\xi))\,.$
(150)

In the resummation part of this work, the transformation we used are

$\displaystyle M_{N}(\delta(1-\xi))=1\,,$ $\displaystyle
M_{N}(1)=\frac{1}{N}\,,$ $\displaystyle M_{N}(1-\xi)=\frac{1}{N(N-1)}\,,$
$\displaystyle M_{N}\left(\frac{1}{(1-\xi)_{+}}\right)=-H_{N-1}\,,$
$\displaystyle
M_{N}\left(\left[\frac{\ln(1-\xi)}{1-\xi}\right]_{+}\right)=\sum_{i=1}^{N-1}\frac{H_{i}}{i}\,,$
(151)

where $H_{n}$ is the harmonic number, defined as

$\displaystyle H_{n}=\sum_{i=1}^{n}\frac{1}{i}\,.$ (152)

The threshold limit $\xi\to 1$ will map onto the large $N$ limit, which leads
to the limit

$\displaystyle M_{N}(\delta(1-\xi))\to 1\,,$ $\displaystyle M_{N}(1)\to 0\,,$
$\displaystyle M_{N}(1-\xi)\to 0\,,$ $\displaystyle
M_{N}\left(\frac{1}{(1-\xi)_{+}}\right)\to-\ln{\bar{N}}\,,$ $\displaystyle
M_{N}\left(\left[\frac{\ln(1-\xi)}{1-\xi}\right]_{+}\right)\to\frac{1}{2}\ln^{2}{\bar{N}}+\frac{\pi^{2}}{12}\,,$
(153)

where ${\bar{N}}=Ne^{\gamma_{E}}$.

###  A.4 Color identities

We provide several color identities used in the CGC multi-point correlators.

Through the Fiertz identity, one can derive that

$\displaystyle{\rm
Tr}[W^{\dagger}(b^{\prime}_{\perp})t^{a}W^{\dagger}(b_{\perp}^{\prime})t^{b}]W_{ab}(r_{\perp})$
(154) $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}[W(b_{\perp})W^{\dagger}(r_{\perp})]{\rm
Tr}[W(r_{\perp})W^{\dagger}(b^{\prime}_{\perp})]-\frac{1}{2N_{C}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(b_{\perp}^{\prime})]\,,$

which relates the $3$-point functions to dipole distributions.

Also one needs the following relation that

$\displaystyle\frac{2}{N_{C}^{2}}{\rm
Tr}\left[t^{b^{\prime}}W^{\dagger}(b_{\perp})W(b_{\perp})t^{b}\right]W^{\dagger}_{b^{\prime}a}(r_{\perp}^{\prime})W_{ab}(r_{\perp})$
$\displaystyle=$ $\displaystyle\left(\frac{1}{N_{C}^{2}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(b_{\perp}^{\prime})W(r_{\perp}^{\prime})W^{\dagger}(r_{\perp})]{\rm
Tr}[W(r_{\perp})W^{\dagger}(r^{\prime}_{\perp})]-\frac{1}{N^{3}_{C}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(b^{\prime}_{\perp})]\right)\,.$

##  Appendix B Contributions from the other channels

In this section, we list all the remaining channels that contribute to the
inclusive jet production, which are the $g\to g$,$q\to g$ and $g\to q$ sub-
processes.

###  B.1 $g\to g$ channel

The LO $g\to g$ is given by

$\displaystyle d\sigma^{(0)}_{gg}=\tau
G(\tau)\int\frac{d^{D-2}b_{\perp}d^{D-2}b_{\perp}^{\prime}}{(2\pi)^{D-2}}\,e^{-ip_{J\perp}\cdot
z_{\perp}}S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})\,,$
(156)

whose form in the momentum space is

$\displaystyle d\sigma^{(0)}_{gg}=\tau G(\tau)S_{\perp}{\cal
F}_{A}(p_{J\perp};X_{f})\,,$ (157)

where $\tau=p^{+}_{J}/p_{p}^{+}$, $G(x)$ denotes the gluon PDF of the proton
and $\mathcal{F}_{A}(k_{\perp};X_{f})$ represents the momentum space nucleus
dipole distribution in the adjoint representation, whose definition is

$\displaystyle(2\pi)^{D-2}S_{\perp}{\cal F}_{A}(k_{\perp},X_{f})=\int
db_{\perp}db_{\perp}^{\prime}e^{-ik_{\perp}\cdot
z_{\perp}}S^{(2)}_{X_{f}}(b_{\perp},b_{\perp}^{\prime})S^{(2)}_{X_{f}}(b_{\perp},b_{\perp}^{\prime})\,,\quad$
(158)

where $z_{\perp}=b_{\perp}-b^{\prime}_{\perp}$. The relation between the
dipole in the adjoint representation ${\cal F}_{A}$ and the fundamental
representation ${\cal F}_{F}$ is given by

$\displaystyle\mathcal{F}_{A}(k_{\perp};X_{f})=\int
d^{D-2}k_{1\perp}\mathcal{F}_{F}(k_{\perp};X_{f})\mathcal{F}_{F}(k_{\perp}-k_{1\perp};X_{f}).$
(159)

The NLO correction of the $g\to g$ channel to the jet cross section is found
to be

$\displaystyle d\sigma^{(1)}_{gg}$
$\displaystyle=\frac{\alpha_{s}}{2\pi}N_{C}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xG(x)\,e^{-ip_{J\perp}\cdot
z_{\perp}}$ $\displaystyle\times\left({\cal
H}_{g,2}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+{\cal
H}_{q\bar{q},2}\,\,S_{X}^{(3)}(b_{\perp},r_{\perp},b_{\perp}^{\prime})\right.$
$\displaystyle\left.+\int\frac{dr_{\perp}}{\pi}\Big{(}{\cal H}_{g,BK}+{\cal
H}_{g,3}+{\cal
H}_{g,kin.}\Big{)}S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},r_{\perp})S_{X}^{(2)}(r_{\perp},b_{\perp}^{\prime})\right)$
$\displaystyle+(d\sigma_{g,fsr}-d\sigma^{c}_{g,fsr})+(d\sigma_{g,isr}-d\sigma^{c}_{g,isr})+(d\sigma_{g,inter.}-d\sigma^{c}_{g,inter.})+(d\sigma_{gR,soft}-d\sigma_{gR,soft}^{c}),\,$

where $x=\tau/\xi$ and $\xi=\frac{p_{jg}^{+}}{p_{g}^{+}}$. Here $p_{g}$ is the
momentum of the initial state gluon, $p_{jg}$ denotes the momentum of one of
the final state gluons that will become the signal jet in the 2-jets case.

The hard factor ${\cal H}_{g,2}$, ${\cal H}_{q\bar{q},2}$ and ${\cal H}_{g,3}$
are found as the following,

$\displaystyle{\cal H}_{g,2}$ $\displaystyle=$
$\displaystyle\left\\{-P_{gg}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\left[\frac{2N_{f}T_{R}}{3N_{C}}\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}\right.\right.$
(161)
$\displaystyle\left.\left.+\left(\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{C}}\right)\ln\frac{z_{\perp}^{2}p_{J\perp}^{2}}{c_{0}^{2}}-\frac{2N_{f}T_{R}}{3N_{C}}\right]\delta(1-\xi)\right\\}\,,$
$\displaystyle{\cal
H}_{q\bar{q},2}=\frac{2N_{f}T_{R}}{N_{C}}\delta(1-\xi)\int\frac{dr_{\perp}}{\pi}\left[\frac{e^{ip_{J\perp}\cdot
y_{\perp}}}{y_{\perp}^{2}}\right]_{+}\int_{0}^{1}d{\xi^{\prime}}[\xi^{\prime
2}+(1-\xi^{\prime})^{2}]e^{-i\xi^{\prime}p_{J\perp}\cdot y_{\perp}}\,,$ (162)
$\displaystyle{\cal H}_{g,3}=$ $\displaystyle
4\frac{[1-\xi(1-\xi)]^{2}}{(1-\xi)_{+}}\frac{x_{\perp}\cdot
y_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\,\frac{1}{\xi^{2}}e^{-i\frac{1-\xi}{\xi}p_{J\perp}\cdot
y_{\perp}}$ (163) $\displaystyle+4\delta(1-\xi)\left[\frac{e^{ip_{J\perp}\cdot
y_{\perp}}}{y_{\perp}^{2}}\right]_{+}\int_{0}^{1}d{\xi^{\prime}}[\frac{\xi^{\prime}}{(1-\xi^{\prime})_{+}}-\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})]e^{-i\xi^{\prime}p_{J\perp}\cdot
y_{\perp}}\,,\quad\quad$

where $x_{\perp}=b_{\perp}-r_{\perp}$,
$y_{\perp}=r_{\perp}-b_{\perp}^{\prime}$ and

$\displaystyle
P^{(1)}_{gg}(\xi)=2\left[\frac{\xi}{(1-\xi)^{+}}+\frac{1-\xi}{\xi}+\xi(1-\xi)\right]+\left(\frac{11}{6}-\frac{2N_{f}T_{R}}{3N_{C}}\right)\delta(1-\xi),$
(164)

in Eq. 164, $T_{R}=\frac{1}{2}$, $N_{f}$ denotes the number of flavors in the
quark loop, which is taken to be $3$ in our calculation.

The $\nu$-related logarithmic term is

$\displaystyle{\cal H}_{g,BK}=2\left(\ln\frac{\nu
p_{g}^{+}}{p_{J\perp}^{2}}\right)\left[\frac{z_{\perp}^{2}}{x_{\perp}^{2}y_{\perp}^{2}}\right]_{+}\,\delta(1-\xi)\,,$
(165)

which is also proportional to the LO kinematics, similar to the $q\to q$
channel case.

The kinematic constraint term is given by

$\displaystyle{\cal
H}_{g,kin.}=2\left[\frac{\ln(x_{\perp}^{2}p_{J\perp}^{2}/c_{0}^{2})}{x_{\perp}^{2}}+\frac{\ln(y_{\perp}^{2}p_{J\perp}^{2}/c_{0}^{2})}{y_{\perp}^{2}}+\frac{2x_{\perp}\cdot
y_{\perp}}{x_{\perp}^{2}y_{\perp}^{2}}\ln\left(\frac{x_{\perp}y_{\perp}p_{J\perp}^{2}}{c_{0}^{2}}\right)\right]_{+}\delta(1-\xi)\,.\quad$
(166)

Finally, the terms in the last line of Eq. B.1 are

$\displaystyle d\sigma_{g,fsr}-d\sigma^{c}_{g,fsr}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}S_{\perp}}{2\pi^{2}}N_{C}\int_{0}^{1}d\xi\frac{[1-\xi(1-\xi)]^{2})}{\xi(1-\xi)}\int
d^{2}p_{k\perp}\Bigg{\\{}2\times\Theta_{2}\,xG(x)\frac{\mathcal{F}_{A}(p_{k\perp}+p_{J\perp};X_{f})}{[\xi
p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}$ (167) $\displaystyle+\Theta_{1}\,\tau
G(\tau)\frac{\mathcal{F}_{A}(p_{J\perp};X_{f})}{[p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}-\tau
G(\tau)\frac{\mathcal{F}_{A}(p_{k\perp}+\xi
p_{J\perp};X_{f})}{[p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}\Bigg{\\}}\,,\quad\quad$

where the factor $2$ before $\Theta_{2}$ accounts for the contribution from
filling the second jet to the histogram if it passes the jet threshold, where
we have used the fact that the filling is symmetric between the $2$ gluon
jets.

The interference term is given by

$\displaystyle d\sigma_{g,inter.}-d\sigma_{g,inter.}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{s}S_{\perp}}{2\pi^{2}}N_{C}\int_{\tau}^{1}d\xi\frac{[1-\xi(1-\xi)]^{2})}{\xi(1-\xi)}$
$\displaystyle\int
d^{2}p_{k\perp}\frac{d^{D-2}l_{\perp}}{(2\pi)^{D-4}}\frac{d^{D-2}l_{\perp}^{\prime}}{(2\pi)^{D-4}}\,xG(x)\,\mathcal{F}_{F}(l_{\perp}^{\prime};X_{f})\mathcal{F}_{F}(l_{\perp}-l_{\perp}^{\prime};X_{f})$
$\displaystyle\Bigg{\\{}(2\Theta_{2}+\Theta_{1}-2)\,\,\mathcal{F}_{F}(p_{k\perp}+p_{j\perp}-l_{\perp}^{\prime};X_{f})\frac{2(\xi
p_{k\perp}-(1-\xi)p_{j\perp})\cdot(l_{\perp}-p_{j\perp})}{(\xi
p_{k\perp}-(1-\xi)p_{J\perp})^{2}(l_{\perp}-p_{j\perp})^{2}}\Bigg{\\}}\,,$
(169)

where a factor $2$ is again included in the $2$-jet case to account for
filling both of the jets in the final state. In this case, the counter
contribution will also be filled twice. The same situation also happens to the
ISR contribution which gives

$\displaystyle
d\sigma_{g,isr}-d\sigma_{g,isr}^{c}=\frac{\alpha_{s}}{2\pi^{2}}N_{C}\int^{1}d\xi\,\frac{[1-\xi(1-\xi)]^{2})}{\xi(1-\xi)}\int
d^{2}p_{k\perp}xG(x)\Big{(}\Theta_{1}+2\Theta_{2}-2\Big{)}$
$\displaystyle\times\int\frac{db_{\perp}db_{\perp}^{\prime}}{4\pi^{2}}\,\frac{dr_{\perp}dr_{\perp}^{\prime}}{4\pi^{2}}\,e^{-ip_{j\perp}\cdot
z_{\perp}}e^{-ip_{k\perp}\cdot z_{\perp}^{\prime}}\frac{x_{\perp}\cdot
x_{\perp}^{\prime}}{x_{\perp}^{2}{x_{\perp}^{\prime}}^{2}}S_{X_{f}}^{(8)}(b_{\perp},r_{\perp},b_{\perp}^{\prime},r_{\perp}^{\prime})\,.$
(170)

The soft contribution gives

$\displaystyle d\sigma_{gR,soft}-d\sigma_{gR,soft}^{c}$ (171) $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{\pi^{2}}\frac{N_{C}}{2}\tau
G(\tau)\,2\,\int_{-\infty}^{\infty}d\eta_{k}\int
d^{2}p_{k\perp}\frac{d^{D-2}l_{\perp}^{\prime}}{(2\pi)^{D-4}}\,\left(\Theta_{1,soft}+\Theta_{2,soft}-1\right)\,$
$\displaystyle\,\Bigg{\\{}S_{\perp}\mathcal{F}_{F}(l_{\perp}^{\prime};X_{f})\mathcal{F}_{F}(p_{k\perp}+p_{j\perp}-l_{\perp}^{\prime};X_{f})\left[\frac{2}{p_{k\perp}^{2}}-\int
d^{D-2}l_{\perp}\mathcal{F}_{F}(l_{\perp}+p_{j\perp}-l_{\perp}^{\prime};X_{f})\frac{2p_{k\perp}\cdot
l_{\perp}}{p_{k\perp}^{2}l_{\perp}^{2}}\right]\,$ $\displaystyle+$
$\displaystyle\int\frac{db_{\perp}db_{\perp}^{\prime}}{4\pi^{2}}\frac{dr_{\perp}dr_{\perp}^{\prime}}{4\pi^{2}}\frac{x_{\perp}\cdot
x_{\perp}^{\prime}}{x^{2}_{\perp}\,{x^{\prime}_{\perp}}^{2}}e^{-ip_{j\perp}\cdot
z_{\perp}}e^{-ip_{k\perp}\cdot
z^{\prime}_{\perp}}S_{X_{f}}^{(8)}(b_{\perp},r_{\perp},b_{\perp}^{\prime},r_{\perp}^{\prime})\Bigg{\\}}\,,$

where $x_{\perp}^{\prime}=b_{\perp}^{\prime}-r_{\perp}^{\prime}$ and
$z_{\perp}^{\prime}=r_{\perp}-r_{\perp}^{\prime}$ and

$\displaystyle
S_{X_{f}}^{(8)}(b_{\perp},r_{\perp},b_{\perp}^{\prime},r_{\perp}^{\prime})=\frac{1}{N_{C}^{3}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(b_{\perp}^{\prime})W(r_{\perp}^{\prime})W^{\dagger}(r_{\perp})]{\rm
Tr}[W(r_{\perp})W^{\dagger}(r^{\prime}_{\perp})]{\rm
Tr}[W(b_{\perp})W^{\dagger}(b^{\prime}_{\perp})].\,$

Here $p_{k}$ and $\eta_{k}$ denote the momentum and the rapidity of the final
state soft gluon, respectively. We note that in this work, we assume all the
signal jets are energetic with $E_{J}\gg Q_{s}\sim p_{J\perp}$, therefore the
soft gluon itself could not form a jet that passes the jet criteria and
therefore the soft gluon jet will not contribute to the jet spectrum and the
$\Theta_{2}$ term will only be filled once to the histogram. However, if the
condition is relieved, the soft jet could contribute to the $p_{J\perp}$
spectrum.

The $8$-point correlator is defined as

$\displaystyle
S^{(8)}_{X_{f}}(b_{\perp},r_{\perp},b_{\perp}^{\prime},r_{\perp}^{\prime})=\frac{1}{N_{C}^{3}}{\rm
Tr}\left[f_{cde}W_{bd}^{\dagger}(r_{\perp})W_{ae}(r_{\perp}^{\prime})f_{cfg}W^{\dagger}_{bf}(b_{\perp})W_{ag}(b_{\perp}^{\prime})\right].$
(173)

Similar to $S_{X_{f}}^{(6)}$, in the large $N_{C}$ limit, the
$S_{X_{f}}^{(8)}$ can be written as products of the $4$-point function and the
dipoles such that

$\displaystyle S_{X_{f}}^{(8)}\approx
S_{X_{f}}^{(4)}(b_{\perp},b_{\perp}^{\prime},r_{\perp}^{\prime},r_{\perp})S_{X_{f}}^{(2)}(r_{\perp},r_{\perp}^{\prime})S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})\,,$
(174)

which is originated from the color identity that

$\displaystyle\frac{1}{N_{C}^{3}}{\rm
Tr}\left[f_{cde}W_{bd}^{\dagger}(r_{\perp})W_{ae}(r_{\perp}^{\prime})f_{cfg}W^{\dagger}_{bf}(b_{\perp})W_{ag}(b_{\perp}^{\prime})\right].$
(175) $\displaystyle=$ $\displaystyle\left(\frac{1}{N_{C}^{3}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(b_{\perp}^{\prime})W(r_{\perp}^{\prime})W^{\dagger}(r_{\perp})]{\rm
Tr}[W(r_{\perp})W^{\dagger}(r^{\prime}_{\perp})]{\rm
Tr}[W(b_{\perp})W^{\dagger}(b^{\prime}_{\perp})]\right.$
$\displaystyle\left.-\frac{1}{N^{3}_{C}}{\rm
Tr}[W(b_{\perp})W^{\dagger}(r^{\prime}_{\perp})W(r_{\perp})W^{\dagger}(b_{\perp})W(b^{\prime}_{\perp})W^{\dagger}(r_{\perp})W(r^{\prime}_{\perp})W^{\dagger}(b^{\prime}_{\perp})]\right)\,.$

We can see that the $g\to g$ channel is very similar to the result of the
$q\to q$ channel except for the splitting function and the color structure. In
QCD, for the collinear limit, the only difference between cross sections of
different channels is with respect to the splitting functions Catani:1998nv  .
And the shock-wave introduced in CGC, which is the origin of the difference in
dipole structure, will not change this property. For the soft limit, according
to the conclusion of the soft theorem Weinberg:1965nx , the only difference
between these two channels is the color factor.

###  B.2 $g\to q$ channel

For the $g\to q+\bar{q}$ channel case, we will suppose that the final state
quark becomes the jet in 2-jets case, and then we can get the NLO correction
of this channel to the jet cross section as

$\displaystyle d\sigma^{(1)}_{gq}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{f}T_{R}}{\pi}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xG(x)\,e^{-ip_{J\perp}\cdot
z_{\perp}}$ (176) $\displaystyle\times\left({\cal
H}_{gq,12}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+{\cal
H}_{gq,22}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+\int\frac{dr_{\perp}}{\pi}{\cal
H}_{gq,3}S_{X}^{(3)}(b_{\perp},r_{\perp},b_{\perp}^{\prime})\right)$
$\displaystyle+(d\sigma_{gq,fsr}-d\sigma^{c}_{gq,fsr})+(d\sigma_{gq,isr}-d\sigma^{c}_{gq,isr})+(d\sigma_{gq,inter.}-d\sigma^{c}_{gq,inter.})\,,$

where $\xi=\frac{p_{jq}^{+}}{p_{g}^{+}}$, $p_{g}$ is the momentum of the
initial state gluon, $p_{jq}$ denotes the momentum of the final state quark.

The hard factors are

$\displaystyle{\cal
H}_{gq,12}=\left(-P_{qg}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}-2\xi^{2}+2\xi\right)\,,$
(177) $\displaystyle{\cal
H}_{gq,22}=\left(-\frac{1}{3}\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\frac{1}{6}\right)\delta(1-\xi)\,,$
(178) $\displaystyle{\cal H}_{gq,3}=$ $\displaystyle
P_{qg}^{(1)}(\xi)\frac{z_{\perp}\cdot
y_{\perp}}{z_{\perp}^{2}y_{\perp}^{2}}\,\frac{2}{\xi}e^{-i\frac{p_{J\perp}}{\xi}\cdot
y_{\perp}},$ (179)

where $P^{(1)}_{qg}(\xi)=[\xi^{2}+(1-\xi)^{2}]$.

The results of terms in the last line of Eq. 176 are

$\displaystyle d\sigma_{gq,fsr}-d\sigma^{c}_{gq,fsr}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{f}T_{R}}{2\pi^{2}}\int_{0}^{1}d\xi
P_{qg}^{(1)}(\xi)\int
d^{2}p_{k\perp}\Bigg{\\{}2\Theta_{2}\,xG(x)\frac{\mathcal{F}_{A}(p_{k\perp}+p_{J\perp};X_{f})}{[\xi
p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}$ (180) $\displaystyle+\Theta_{1}\,\tau
G(\tau)\frac{\mathcal{F}_{A}(p_{J\perp};X_{f})}{[p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}-\tau
G(\tau)\frac{\mathcal{F}_{A}(p_{k\perp}+\xi
p_{J\perp};X_{f})}{[p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}\Bigg{\\}}\,,\quad\quad$

where the factor of $2$ before $\Theta_{2}$ accounts for the contribution from
the ${\bar{q}}$ jet, which at NLO is identical to the $q$ jet case.

The interference term is found to be

$\displaystyle d\sigma_{gq,inter.}-d\sigma_{gq,inter.}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{s}N_{f}T_{R}}{2\pi^{2}}\int_{\tau}^{1}d\xi
P_{qg}^{(1)}(\xi)\int
d^{2}p_{k\perp}\frac{d^{D-2}l_{\perp}}{(2\pi)^{D-4}}\,xG(x)\,\mathcal{F}_{F}(l_{\perp};X_{f})$
$\displaystyle\Bigg{\\{}(2\Theta_{2}+\Theta_{1}-2)\,\,\mathcal{F}_{F}(p_{k\perp}+p_{J\perp}-l_{\perp};X_{f})\frac{2(\xi
p_{k\perp}-(1-\xi)p_{j\perp})\cdot(l_{\perp}-p_{j\perp})}{(\xi
p_{k\perp}-(1-\xi)p_{J\perp})^{2}(l_{\perp}-p_{j\perp})^{2}}\Bigg{\\}},\,$
(181)

and the contribution from the ISR gives

$\displaystyle
d\sigma_{gq,isr}-d\sigma_{gq,isr}^{c}=\frac{\alpha_{s}N_{f}T_{R}}{2\pi^{2}}\int^{1}d\xi\,P_{qg}^{(1)}(\xi)\int
d^{2}p_{k\perp}xG(x)\Big{(}\Theta_{1}+2\Theta_{2}-2\Big{)}$
$\displaystyle\times\int\frac{db_{\perp}db_{\perp}^{\prime}}{4\pi^{2}}\,\frac{dr_{\perp}dr_{\perp}^{\prime}}{4\pi^{2}}\,e^{-ip_{j\perp}\cdot
z_{\perp}}e^{-ip_{k\perp}\cdot z_{\perp}^{\prime}}\frac{x_{\perp}\cdot
x_{\perp}^{\prime}}{x_{\perp}^{2}{x_{\perp}^{\prime}}^{2}}S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X_{f}}^{(2)}(r_{\perp},r_{\perp}^{\prime})\,.$
(182)

In both cases, the contribution from the ${\bar{q}}$ jet is included through
the factor of $2$ before $\Theta_{2}$ and the counter terms.

###  B.3 $q\to g$ channel

For the $q\to g$ channel which means the gluon becomes the jet in 2-jets case
of $q\to q+g$ process, the NLO correction to the jet cross section is found to
be

$\displaystyle d\sigma^{(1)}_{qg}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{C}}{2\pi}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xf(x)\,e^{-ip_{J\perp}\cdot
z_{\perp}}{\cal
H}_{qg,22}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})$
(183)
$\displaystyle+d\sigma_{qg,fsr}+(d\sigma_{qg,isr}-d\sigma^{c}_{qg,isr})+d\sigma_{qg,inter.}\,,$

where $\xi=\frac{p_{jg}^{+}}{p_{q}^{+}}$ and $p_{q}$ is the momentum of the
initial state quark, $p_{jg}$ denotes the momentum of the final state gluon.

The hard factor is

$\displaystyle{\cal
H}_{qg,22}=\left(-P_{gq}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\xi\right)\,,$
(184)

where $\mathcal{P}_{gq}(\xi)=\frac{1}{\xi}[1+(1-\xi)^{2}]$.

The results of terms in the last line of Eq. 183 are

$\displaystyle d\sigma_{qg,fsr}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{C}}{2\pi^{2}}\int_{\tau}^{1}d\xi
P_{gq}^{(1)}(\xi)\int
d^{2}p_{k\perp}\Theta_{2}\,xf(x)\frac{\mathcal{F}_{F}(p_{k\perp}+p_{J\perp};X_{f})}{[\xi
p_{k\perp}-(1-\xi)p_{J\perp}]^{2}}\,.$ (185) $\displaystyle
d\sigma_{qg,inter.}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{s}N_{C}}{2\pi^{2}}\int_{\tau}^{1}d\xi
P_{gq}^{(1)}(\xi)\int
d^{2}p_{k\perp}\frac{d^{D-2}l_{\perp}}{(2\pi)^{D-4}}\,xf(x)\,\Theta_{2}\mathcal{F}_{F}(p_{k\perp}+p_{J\perp};X_{f})$
$\displaystyle\,\,\mathcal{F}_{F}(p_{k\perp}+p_{J\perp}-l_{\perp};X_{f})\frac{2(\xi
p_{k\perp}-(1-\xi)p_{J\perp})\cdot(l_{\perp}-p_{J\perp})}{(\xi
p_{k\perp}-(1-\xi)p_{J\perp})^{2}(l_{\perp}-p_{J\perp})^{2}}\,,$ (186)

and

$\displaystyle
d\sigma_{qg,isr}-d\sigma_{qg,isr}^{c}=\frac{\alpha_{s}N_{C}}{2\pi^{2}}\int^{1}d\xi\,P_{gq}^{(1)}(\xi)\int
d^{2}p_{k\perp}xf(x)\Big{(}\Theta_{2}-1\Big{)}$ (187)
$\displaystyle\times\int\frac{db_{\perp}db_{\perp}^{\prime}}{4\pi^{2}}\,\frac{dr_{\perp}dr_{\perp}^{\prime}}{4\pi^{2}}\,e^{-ip_{j\perp}\cdot
z_{\perp}}e^{-ip_{k\perp}\cdot z_{\perp}^{\prime}}\frac{x_{\perp}\cdot
x_{\perp}^{\prime}}{x_{\perp}^{2}{x_{\perp}^{\prime}}^{2}}S_{X_{f}}^{(6)}(r_{\perp},b_{\perp},r_{\perp}^{\prime},b_{\perp}^{\prime})\,.$

Here $p_{k}$ denotes the momentum of the final state quark. Under this
definition $\Theta_{2}$ has the same form as the $q\to q$ channel cases in
terms of $p_{k}$ ,$\xi$ and $p_{J}$.

##  Appendix C The small-$R$ limit for other channels channel

In this section, we list all the remaining channels that contribute to the
result of jet production in the small-$R$ limit, which are the $g\to g$,$q\to
g$ and $g\to q$ sub-processes.

###  C.1 $g\to g$ channel

In the small-$R$ limit, by performing subtractions similar to the method in
Section 6 and omitting ${\cal O}(R^{2})$ contributions, we can get the
factorized form for the $g\to g$ channel as

$\displaystyle d\sigma_{g,R}=\int
d\xi\frac{d\zeta}{\zeta^{2}}xG(x)d\hat{\sigma}_{g\to
g}(\xi,p_{J}/\zeta)\,J_{g}(\zeta)\,,$ (188)

where $x=\tau/\xi\zeta$ and $J_{g}(\zeta)$ is the gluon siJF Kang:2016mcy ;
Dai:2016hzf  in the large $N_{C}$ limit, which is

$\displaystyle J_{g}(\zeta)=\delta(1-\zeta)$ $\displaystyle-$
$\displaystyle\frac{\alpha_{s}}{2\pi}N_{C}\,\left\\{\Big{(}P^{(1)}_{gg}(\zeta)+\frac{2N_{f}T_{R}}{N_{c}}P_{qg}(\zeta)\Big{)}\ln\frac{p^{2}_{J\perp}R^{2}}{\zeta^{2}\mu^{2}}+4\frac{(1-\zeta+\zeta^{2})^{2}}{\zeta}\left(\frac{\ln(1-\zeta)}{1-\zeta}\right)_{+}\right.$
$\displaystyle\left.-\left(\frac{67}{9}-\frac{2}{3}\pi^{2}-\frac{N_{f}T_{R}}{N_{c}}\frac{23}{9}\right)\delta(1-\zeta)+\frac{4N_{f}T_{R}}{N_{c}}[P^{(1)}_{qg}(\zeta)\ln(1-\zeta)+\zeta(1-\zeta)]\right\\},$
(189)

and we have included the contribution from the $g\to q\bar{q}$ channel,
$P^{(1)}_{qg}(\zeta)=[\zeta^{2}+(1-\zeta)^{2}]$.

$d\hat{\sigma}_{g\to g}(p_{J\perp}/\zeta)$ is the cross section to produce a
gluon with momentum $p_{j\perp}=p_{J\perp}/\zeta$. The explicit results at LO
and NLO are given by

$\displaystyle d\hat{\sigma}_{g\to
g}^{(0)}(\xi,p_{j})=\int\frac{d^{2}b_{\perp}d^{2}b_{\perp}^{\prime}}{(2\pi)^{2}}\,e^{-ip_{j\perp}\cdot
z_{\perp}}S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X_{f}}^{(2)}(b_{\perp},b_{\perp}^{\prime})\,,$
(190)

and

$\displaystyle d\hat{\sigma}^{(1)}_{g\to g}(p_{j})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}N_{C}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xG(x)\,e^{-ip_{j\perp}\cdot
z_{\perp}}$ $\displaystyle\times\left({\cal H}^{g\to
g}_{2}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+{\cal
H}^{g\to
q\bar{q}}_{2}\,\,S_{X}^{(3)}(b_{\perp},r_{\perp},b_{\perp}^{\prime})\right.$
$\displaystyle\left.+\int\frac{dr_{\perp}}{\pi}\Big{(}{\cal H}^{g\to
g}_{BK}+{\cal H}^{g\to g}_{3}+{\cal H}^{g\to
g}_{kin.}\Big{)}S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},r_{\perp})S_{X}^{(2)}(r_{\perp},b_{\perp}^{\prime})\right),$

where ${\cal H}_{2}^{g\to q\bar{q}}$, ${\cal H}_{3}^{g\to g}$, ${\cal H}^{g\to
g}_{BK}$ and ${\cal H}^{g\to g}_{kin.}$ can be obtained by replacing
$p_{J\perp}$ with $p_{j\perp}$ in Eq. (162), Eq. (163), Eq. (165) and Eq.
(166), respectively. The ${\cal H}^{g\to g}_{2}$ is given by

$\displaystyle{\cal H}^{g\to g}_{2}$ $\displaystyle=$
$\displaystyle\left\\{-P_{gg}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\left[\left(\frac{11}{6}-\frac{2N_{f}T_{R}}{3N_{C}}\right)\ln\frac{z_{\perp}^{2}p_{J\perp}^{2}}{c_{0}^{2}}\right.\right.$
(192)
$\displaystyle\left.\left.-\frac{N_{f}T_{R}}{3N_{C}}\right]\delta(1-\xi)\right\\}\left(1+\frac{1}{\xi^{2}}e^{-i\frac{1-\xi}{\xi}p_{j\perp}\cdot
z_{\perp}}\right)\,.$

We note that $d\hat{\sigma}_{g\to g}$ is the gluonic cross section for the
single hadron production.

###  C.2 $g\to q(\bar{q})$ channel

The contribution of the $g\to q$ channel to the small-$R$ limit result is
found to be

$\displaystyle d\sigma_{gq,R}=2\int
d\xi\frac{d\zeta}{\zeta^{2}}xG(x)d\hat{\sigma}_{g\to
q}(\xi,p_{J}/\zeta)\,J_{q}(\zeta)\,,$ (193)

where $x=\tau/\xi\zeta$ and $J_{q}(\zeta)$ is the quark siJF Kang:2016mcy ;
Dai:2016hzf  in the large $N_{C}$ limit whose form is given by Eq. (126). The
factor 2 comes from summing up the contributions from the $g\to q$ channel and
$g\to\bar{q}$ channel, which are the same at this order. Here,
$d\hat{\sigma}_{g\to q}(p_{J\perp}/\zeta)$ is the cross section to produce a
quark(anti-quark) with momentum $p_{j\perp}=p_{J\perp}/\zeta$ when the initial
state parton is a gluon . The explicit results at NLO is given by

$\displaystyle d\sigma^{(1)}_{g\to q}(p_{j})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{f}T_{R}}{2\pi}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xG(x)\,e^{-ip_{j\perp}\cdot
z_{\perp}}$ $\displaystyle\times\left({\cal H}^{g\to
q}_{12}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+{\cal H}^{g\to
q}_{22}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+\int\frac{dr_{\perp}}{\pi}{\cal
H}^{g\to q}_{3}S_{X}^{(3)}(b_{\perp},r_{\perp},b_{\perp}^{\prime})\right),$

where ${\cal H}_{12}^{g\to q}$ and ${\cal H}^{g\to q}_{3}$ can be obtained by
replacing $p_{J\perp}$ with $p_{j\perp}$ in Eq. (177) and Eq. (179)
respectively. The ${\cal H}^{g\to q}_{22}$ is given by

$\displaystyle{\cal H}^{g\to
q}_{22}=\left(-P_{qg}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}-2\xi^{2}+2\xi\right)\frac{1}{\xi^{2}}e^{-i\frac{1-\xi}{\xi}p_{j\perp}\cdot
z_{\perp}}\,,$ (195)

###  C.3 $q\to g$ channel

The contribution of the $q\to g$ channel to the small-$R$ limit result is
found to be

$\displaystyle d\sigma_{qg,R}=\int
d\xi\frac{d\zeta}{\zeta^{2}}xf(x)d\hat{\sigma}_{q\to
g}(\xi,p_{J}/\zeta)\,J_{g}(\zeta)\,,$ (196)

where $x=\tau/\xi\zeta$ and $J_{g}(\zeta)$ is the gluon siJF Kang:2016mcy ;
Dai:2016hzf  in the large $N_{C}$ limit whose form is given by Eq. (C.1).

$d\hat{\sigma}_{q\to g}(p_{J\perp}/\zeta)$ is the cross section to produce a
gluon with momentum $p_{j\perp}=p_{J\perp}/\zeta$ when the initial state
parton is a qurak. The explicit results at NLO is given by

$\displaystyle d\sigma^{(1)}_{qg}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}N_{C}}{2\pi}\int\frac{db_{\perp}db^{\prime}_{\perp}}{4\pi^{2}}\int_{\tau}^{1}d\xi\,xf(x)\,e^{-ip_{j\perp}\cdot
z_{\perp}}$ $\displaystyle\times\left({\cal H}^{q\to
g}_{12}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+{\cal H}^{q\to
g}_{22}\,\,S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})S_{X}^{(2)}(b_{\perp},b_{\perp}^{\prime})+\int\frac{dr_{\perp}}{\pi}{\cal
H}^{q\to g}_{3}S_{X}^{(3)}(b_{\perp},r_{\perp},b_{\perp}^{\prime})\right),$

where ${\cal H}_{22}^{q\to g}$ can be obtained by replacing $p_{J\perp}$ with
$p_{j\perp}$ in Eq. (184). The ${\cal H}^{q\to g}_{12}$ and ${\cal H}^{q\to
g}_{3}$ are given by

$\displaystyle{\cal H}^{q\to
g}_{12}=\left(-P_{gq}^{(1)}(\xi)\ln\frac{z_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\xi\right)\frac{1}{\xi^{2}}e^{-i\frac{1-\xi}{\xi}p_{j\perp}\cdot
z_{\perp}}\,,$ (198)

and

$\displaystyle{\cal H}^{q\to g}_{3}=$ $\displaystyle
P_{gq}^{(1)}(\xi)\frac{z_{\perp}\cdot
y_{\perp}}{z_{\perp}^{2}y_{\perp}^{2}}\,\frac{2}{\xi}e^{-i\frac{1-\xi}{\xi}p_{J\perp}\cdot
z_{\perp}-ip_{J\perp}\cdot y_{\perp}},$ (199)

respectively.

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