collinear splitting process $c\rightarrow a_{1}a$ is known since a long time
Bern:1998sc ; Bern:1999ry ; Kosower:1999rx ; Sborlini:2013jba . The original
results in the literature regard collinear factorization at the amplitude
level and they refer to the TL collinear region. The extension to the SL
region was worked out and discussed in ref. Catani:2011st , by noticing two
main features: one-loop factorization for the collinear splitting
${c}\rightarrow a_{1}a$ is process dependent at the level of the scattering
amplitudes, and it becomes process independent at the squared amplitude level.
We have computed the SL kernels ${\cal P}_{{c}\rightarrow a_{1}a}^{(1)}$ by
squaring the amplitude level results of refs. Bern:1998sc ; Bern:1999ry ;
Kosower:1999rx ; Sborlini:2013jba  and by applying the prescription of ref.
Catani:2011st  (see section 7.2 therein) to derive the results in the SL
region.

The expression of ${\cal F}_{ga,\,{\rm corr.}}^{(1R1V)}(p,k;n)$ is directly
proportional to ${\cal P}_{{g}\rightarrow a_{1}a}^{(1)}$ (see eqs. (26) and
(28)) and we obtain the following results:

$\displaystyle\mathcal{F}^{(1R1V)}_{gg,\,{\rm corr.}}(p,k;n)=$
$\displaystyle\left(\frac{\alpha^{u}_{s}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\right)^{2}\frac{e^{2\epsilon\gamma_{E}}{\rm\Gamma}(1+\epsilon){\rm\Gamma^{2}(1-\epsilon)}}{\pi^{1-\epsilon}{\rm\Gamma(1-2\epsilon)}}\frac{C_{A}\delta_{+}(k^{2})}{(2pk)^{1+\epsilon}}\bigg{\\{}\\!\\!-\frac{N_{f}}{6}+\frac{C_{A}}{6}$
(101)
$\displaystyle+C_{A}\frac{1-z}{z^{2}}\left[\frac{1}{\epsilon^{2}}-\frac{1}{\epsilon}\Big{(}\ln(1-z)-2\ln(z)\Big{)}-\frac{\pi^{2}}{3}+\frac{1}{2}\ln^{2}(1-z)\right]\\!\\!\bigg{\\}}+{\cal
O}(\epsilon),$ $\displaystyle\mathcal{F}^{(1R1V)}_{ga,\,{\rm corr.}}(p,k;n)=$
$\displaystyle-\\!\\!\left(\\!\frac{\alpha^{u}_{s}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\\!\right)^{2}\frac{e^{2\epsilon\gamma_{E}}{\rm\Gamma}(1+\epsilon){\rm\Gamma^{2}(1-\epsilon)}}{\pi^{1-\epsilon}{\rm\Gamma(1-2\epsilon)}}\frac{C_{F}\delta_{+}(k^{2})}{(2pk)^{1+\epsilon}}\frac{1-z}{z^{2}}\bigg{\\{}\\!C_{F}\\!\left[\\!-\frac{2}{\epsilon^{2}}-\frac{3}{\epsilon}-8\right]$
$\displaystyle+C_{A}\left[\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\left(\frac{11}{3}-2\ln(z)+\ln(1-z)\right)+\frac{76}{9}+\frac{\pi^{2}}{3}-\frac{1}{2}\ln^{2}(1-z)\right]$
$\displaystyle+N_{f}\left[-\frac{2}{3\epsilon}-\frac{10}{9}\right]\\!\bigg{\\}}+{\cal
O}(\epsilon)\;,\quad\quad\quad\quad\quad\quad\quad\quad(a=q,{\bar{q}}).$ (102)

We note that in eqs. (101) and (5.1) we have neglected contributions of ${\cal
O}(\epsilon)$, since they are not relevant to the four-dimensional limit at
${\cal O}(\alpha_{\rm S}^{2})$. Moreover, we have used the momentum fraction
$z$ that is defined as

$z=\frac{\bar{n}(p-k)}{\bar{n}p},$ (103)

where the light-like auxiliary vector ${\bar{n}}^{\mu}$ is

$\bar{n}^{\mu}\equiv n^{\mu}-\frac{n^{2}p^{\mu}}{2np}.$ (104)

In our customary reference frame we have $p^{\mu}=(p^{+},\mathbf{0_{T}},0)$,
$n^{\mu}=(n^{+},\mathbf{0_{T}},n^{-})$ and, therefore,
${\bar{n}}^{\mu}=(0,\mathbf{0_{T}},n^{-})$ and $z=1-k^{+}/p^{+}$.

The use of the variable $z=\bar{n}(p-k)/\bar{n}p$ in eqs. (101) and (5.1)
deserves some comments, since it is practically equivalent to compute
$\mathcal{F}^{(1R1V)}_{ga,\,{\rm corr.}}$ ($a=g,q,{\bar{q}}$) by setting
$n^{2}=0$. As discussed in sections 4.2 and 4.3, the use of the time-like
auxiliary vector $n^{\mu}$ in the computation of the collinear functions is,
in general, essential to avoid unphysical behaviour outside the collinear
region. Nonetheless, already at ${\cal O}(\alpha_{\rm S})$ we noticed that we
can correctly compute the collinear functions by setting $n^{2}=0$ in most of
the $n$-dependent terms of $\mathcal{F}^{(1R)}_{ca,\,{\rm az.in.}}$ in eq.
(43) and in the entire $n$ dependence of $\mathcal{F}^{(1R)}_{ga,\,{\rm
corr.}}$ in eq. (44). Therefore, in the expressions of eqs. (101) and (5.1) we
have directly implemented the approximation of setting $n^{2}=0$ in terms with
a harmless dependence on $n^{2}$. The exact dependence of
$\mathcal{F}^{(1R1V)}_{ga,\,{\rm corr.}}$ on $n^{2}$ can be recovered from
eqs. (101) and (5.1) by replacing the variable $z$ of eq. (103) with the
variable $z_{n}$ of eq. (45).

The term ${\cal F}_{ga,\,{\rm corr.}}^{(2R)}(p,k;n)$ in eq. (41) is due to the
final-state emission of two collinear partons at the tree level. It is
obtained starting from the azimuthal-correlation component of the tree-level
collinear kernel ${\cal P}_{{g}\rightarrow
a_{1}a_{2}a}^{(0)\mu\nu}(k_{1},k_{2},p;n)$ (see eqs. (26) and (28)). The tree-
level kernel is process independent, and the TL and SL regions are
straightforwardly related through the replacement $p\leftrightarrow-p$. The
explicit expressions of ${\cal P}_{{g}\rightarrow
a_{1}a_{2}a}^{(0)\mu\nu}(k_{1},k_{2},p;n)$ for the various partonic channels
were obtained in refs. Catani:1998nv ; Catani:1999ss . We evaluate ${\cal
F}_{ga,\,{\rm corr.}}^{(2R)}(p,k;n)$ in eq. (28) by performing the
integrations of the azimuthal-correlation component of ${\cal
P}_{{g}\rightarrow a_{1}a_{2}a}^{(0)}(k_{1},k_{2},p;n)$ over $k_{1}$ and
$k_{2}$ at fixed momentum $k=k_{1}+k_{2}$. Such integrations can be carried
out to all orders in $\epsilon$ and for arbitrary values of $n^{2}$ by using
the $d$-dimenional angular integrals of ref. Somogyi:2011ir . In the following
we present the results in a simplified form that amounts to neglect
contributions at high orders in $\epsilon$ and to set $n^{2}=0$ in all the
contributions with a harmless (smooth and non-singular) dependence on $n^{2}$.
In particular, we use the variable $z$ of eq. (103) to express part of the $n$
dependence of $\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$.

We write $\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}$ in terms of contributions
with different colour factors:

$\displaystyle\mathcal{F}^{(2R)}_{gg,\,\text{corr.}}(p,k;n)=\left(\frac{\alpha_{s}^{u}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\right)^{2}\frac{e^{2\epsilon\gamma_{E}}\,{\rm\Gamma}(1-\epsilon)}{2\pi^{1-\epsilon}\,{\rm\Gamma}(1-2\epsilon)}\,(k^{2})^{-\epsilon}$
(105)
$\displaystyle\\!\times\\!\\!\left[C_{F}T_{R}N_{f}\mathcal{F}^{(2R),\,C_{F}T_{R}N_{f}}_{gg,\,\text{corr.}}(p,k;n)+C_{A}T_{R}N_{f}\mathcal{F}^{(2R),\,C_{A}T_{R}N_{f}}_{gg,\,\text{corr.}}(p,k;n)+C_{A}^{2}\mathcal{F}^{(2R),\,C_{A}^{2}}_{gg,\,\text{corr.}}(p,k;n)\right]\\!,$
$\displaystyle\mathcal{F}^{(2R)}_{gq,\,\text{corr.}}(p,k;n)=\left(\frac{\alpha^{u}_{s}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\right)^{2}\frac{e^{2\epsilon\gamma_{E}}\,{\rm\Gamma}(1-\epsilon)}{2\pi^{1-\epsilon}\,{\rm\Gamma}(1-2\epsilon)}\,(k^{2})^{-\epsilon}$
$\displaystyle\times\left[C_{F}^{2}\mathcal{F}^{(2R),\,C_{F}^{2}}_{gq,\,\text{corr.}}(p,k;n)+C_{A}C_{F}\mathcal{F}^{(2R),\,C_{A}C_{F}}_{gq,\,\text{corr.}}(p,k;n)\right],$
(106)

and we recall that
$\mathcal{F}^{(2R)}_{g{\bar{q}},\,\text{corr.}}(p,k;n)=\mathcal{F}^{(2R)}_{gq,\,\text{corr.}}(p,k;n)$.
The individual coefficients
$\mathcal{F}^{(2R),\,C_{F}T_{R}N_{f}}_{gg,\,\text{corr.}}$,
$\mathcal{F}^{(2R),\,C_{A}T_{R}N_{f}}_{gg,\,\text{corr.}}$ and
$\mathcal{F}^{(2R),\,C_{A}^{2}}_{gg,\,\text{corr.}}$ of eq. (105) for the
gluon-gluon channel are given by the following expressions:

$\displaystyle\mathcal{F}^{(2R),\,C_{F}T_{R}N_{f}}_{gg,\,\text{corr.}}(p,k;n)=$
$\displaystyle\frac{(k^{2}-2pk)^{2}}{2z^{2}(pk)^{4}}\left[\frac{1}{\epsilon}+4\right]+\frac{(1+z)(k^{2}-2pk)}{z^{2}(pk)^{3}}\left[\frac{1}{\epsilon}+4\right]$
$\displaystyle+\frac{1}{z^{2}(pk)^{2}}\left[\frac{1+2z}{\epsilon}+1+8z\right]+\frac{2}{zpk(k^{2}-2pk)}\left[\frac{1}{\epsilon}+1\right],$
(107)
$\displaystyle\mathcal{F}^{(2R),\,C_{A}T_{R}N_{f}}_{gg,\,\text{corr.}}(p,k;n)=$
$\displaystyle\frac{4(1-z)(3+5\epsilon)}{9z^{2}k^{2}(k^{2}-2pk)}+\frac{(k^{2}-2pk)^{2}}{6z^{2}(pk)^{4}}-\frac{k^{2}-2pk}{6(1-z)(pk)^{3}}-\frac{1}{3(1-z)(pk)^{2}}$
$\displaystyle+\frac{(1+2z)(k^{2}-2pk)}{6z^{2}(pk)^{3}}+\frac{2+z+z^{2}}{3z^{2}(pk)^{2}}-\frac{2(1-2z)}{3z^{2}pk(k^{2}-2pk)},$
(108)
$\displaystyle\mathcal{F}^{(2R),\,C_{A}^{2}}_{gg,\,\text{corr.}}(p,k;n)=$
$\displaystyle\\!-\\!\frac{1-z}{z^{2}k^{2}(k^{2}-2pk)}\\!\left[\frac{2}{\epsilon}\\!+\\!\frac{11}{3}\\!+\\!\frac{67}{9}\epsilon\right]\\!-\frac{4}{z(k^{2}-2pk)^{2}}\frac{1}{\epsilon}\\!+\\!\frac{k^{2}-2pk}{12(1-z)(pk)^{3}}$
$\displaystyle-\frac{(k^{2}-2pk)^{2}}{4z^{2}(pk)^{4}}\left[\frac{1}{\epsilon}+\frac{10}{3}\right]-\frac{k^{2}-2pk}{2z^{2}(pk)^{3}}\left[\frac{1+z}{\epsilon}+\frac{19+20z}{6}\right]$
$\displaystyle+\frac{1}{6(1-z)(pk)^{2}}-\frac{1}{z^{2}(pk)^{2}}\left[\frac{2+z}{\epsilon}+\frac{14+19z+z^{2}}{6}\right]$
$\displaystyle-\frac{1}{z^{2}pk(k^{2}-2pk)}\left(\frac{1+4z}{\epsilon}-\frac{11-28z}{6}\right)+\mathcal{K}(p,k;n)$
$\displaystyle+\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n),$ (109)

where the functions $\mathcal{K}$ and $\mathcal{F}^{n^{2}-\text{sing.}}$ are
presented in eqs. (5.1)–(114). The individual coefficients
$\mathcal{F}^{(2R),\,C_{F}^{2}}_{gq,\,\text{corr.}}$ and
$\mathcal{F}^{(2R),\,C_{A}C_{F}}_{gq,\,\text{corr.}}$ of eq. (5.1) for the
gluon-quark channel are

$\displaystyle\mathcal{F}^{(2R),\,C_{F}^{2}}_{gq,\,\text{corr.}}(p,k;n)=$
$\displaystyle-\frac{1-z}{z^{2}k^{2}(k^{2}-2pk)}\left[\frac{4}{\epsilon}+3+7\epsilon\right]+\frac{k^{2}-2pk}{4z^{2}(pk)^{3}}\left[\frac{1}{\epsilon}+2\right]$
(110)
$\displaystyle-\frac{1}{2z^{2}(pk)^{2}}\left[\\!\frac{1-z}{\epsilon}\\!+\\!3-2z\\!\right]\\!+\\!\frac{1}{2z^{2}pk(k^{2}-2pk)}\\!\\!\left[\\!\frac{2(2-z)}{\epsilon}\\!+\\!3(1-2z)\right]\\!,$
$\displaystyle\mathcal{F}^{(2R),\,C_{A}C_{F}}_{gq,\,\text{corr.}}(p,k;n)=$
$\displaystyle\frac{2(1-z)}{z^{2}k^{2}(k^{2}-2pk)}\frac{1}{\epsilon}-\frac{4}{z(k^{2}-2pk)^{2}}\frac{1}{\epsilon}-\frac{k^{2}-2pk}{4z^{2}(pk)^{3}}\left[\frac{1}{\epsilon}+2\right]$
$\displaystyle-\frac{1}{2z^{2}(pk)^{2}}\left[\frac{2+z}{\epsilon}+1+2z\right]-\frac{1}{z^{2}pk(k^{2}-2pk)}\left[\frac{3+2z}{\epsilon}+z\right]$
$\displaystyle+\mathcal{K}(p,k;n)+\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n).$
(111)

The function $\mathcal{K}$ in eqs. (5.1) and (5.1) has the following
expression:

$\displaystyle\mathcal{K}(p,k;n)=$
$\displaystyle-\frac{1}{z^{2}(k^{2}-2pk)}\left[\frac{2(1-z)}{k^{2}}+\frac{1}{pk}\right]\frac{1}{\epsilon}-\frac{2}{z^{2}pk(k^{2}-2pk)}\left(\frac{\mathbf{k_{T}}^{2}}{k^{2}}\right)^{1+\epsilon}\left[\frac{1}{\epsilon}+\frac{\pi^{2}}{6}\epsilon\right]$
$\displaystyle-\frac{1}{z^{2}(k^{2}-2pk)}\left[\frac{6(1-z)}{k^{2}}-\frac{1}{pk}\right]\ln\left(1+\frac{k^{2}}{\mathbf{k_{T}}^{2}}\right)-\frac{2}{z}\frac{k^{2}}{pk(k^{2}-2pk)^{2}}$
$\displaystyle+\ln(z)\left[\frac{4(2-z)pk}{(k^{2}-2pk)^{3}}-\frac{2(4-z)}{z(k^{2}-2pk)^{2}}\right]+\frac{1}{z^{2}\mathbf{k_{T}}^{2}pk}\ln\left(1-\frac{\mathbf{k_{T}}^{2}}{2pk}\right)$
$\displaystyle+\left[\frac{1}{z^{2}pk(k^{2}-2pk)}-\frac{2(1+z)(1-z)}{z^{2}k^{2}(k^{2}-2pk)}\right]\frac{k^{2}}{\mathbf{k_{T}}^{2}}\ln\left(1+\frac{\mathbf{k_{T}}^{2}}{k^{2}}\right)-\ln\left(1-\frac{k^{2}}{2pk}\right)$
$\displaystyle\times\left[\frac{8(1-z)(2-z)(pk)^{2}}{k^{2}(k^{2}-2pk)^{3}}-\frac{1}{z^{2}k^{2}pk}-\frac{4(6-3z+z^{2})pk}{zk^{2}(k^{2}-2pk)^{2}}-\frac{2(2+5z-z^{2})}{z^{2}k^{2}(k^{2}-2pk)}\right]$
$\displaystyle+\frac{8z(1-z)(2-z)(pk)^{2}}{\mathbf{k_{T}}^{2}(k^{2}-2pk)^{3}}\left[\frac{\ln(z)}{1-z}-\frac{2pk}{k^{2}}\ln\left(1-\frac{k^{2}}{2pk}\right)\right].$
(112)

The function $\mathcal{F}^{n^{2}-\text{sing.}}$ in eqs. (5.1) and (5.1)
embodies the non-smooth dependence of $\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}$
on $n^{2}$. In the case of a time-like auxiliary vector we find the following
result:

$\displaystyle\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)=\frac{4(1-z)}{z^{2}k^{2}(k^{2}-2pk)}\left(\frac{n^{2}k^{2}}{4(np)^{2}(1-z)^{2}}\right)^{-\epsilon}\left[\frac{1}{\epsilon}+\frac{\pi^{2}}{3}\epsilon\right]-\frac{2}{z(k^{2}-2pk)}$
(113)
$\displaystyle\times\left[\frac{k^{2}}{pk\left(\mathbf{k_{T}}^{2}+\frac{n^{2}(pk)^{2}}{(np)^{2}}\right)}+\frac{2(1-z)}{\mathbf{k_{T}}^{2}}\ln\left(\frac{n^{2}k^{2}(pk)^{2}}{(np)^{2}(k^{2}+\mathbf{k_{T}}^{2})\left(\mathbf{k_{T}}^{2}+\frac{n^{2}(pk)^{2}}{(np)^{2}}\right)}\right)\right],\quad(n^{2}>0).$

Performing the calculation of $\mathcal{F}^{(2R)}_{ga,\text{corr.}}$ with a
light-like auxiliary vector we obtain

$\displaystyle\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$
$\displaystyle=-\frac{2}{z(k^{2}-2pk)}$ (114)
$\displaystyle\times\left\\{-\frac{1}{pk}+\frac{2(1-z)}{\mathbf{k_{T}}^{2}}\left[\ln\left(1+\frac{\mathbf{k_{T}}^{2}}{k^{2}}\right)+\left(\frac{k^{2}}{\mathbf{k_{T}}^{2}}\right)^{\epsilon}\frac{1+\epsilon}{\epsilon}\right]\right\\},\quad(n^{2}=0).$

We comment on the results in eqs. (105)–(114) by discussing the $\epsilon$
dependence and the $n^{2}$ dependence in turn.

The differential collinear function
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ is singular in the limit
$k^{2}\rightarrow 0$. The singularities are due to contributions that are
proportional to the factors $(k^{2})^{-1-\epsilon}$ and
$(k^{2})^{-1-2\epsilon}$. Such factors can be expressed in terms of plus-
distributions of the variable $k^{2}$, which are integrable at $k^{2}=0$, and
contact terms of the type $\delta_{+}(k^{2})$, whose coefficients are single
poles $1/\epsilon$. Owing to these single-pole effective contributions, in
eqs. (5.1)–(113) the coefficients of the factors that are singular at
$k^{2}=0$ are reported up to ${\cal O}(\epsilon)$, by neglecting terms at
${\cal O}(\epsilon^{2})$ and higher orders in $\epsilon$.333Other comments on
the singularities in the limit $k^{2}\rightarrow 0$ are presented in section
5.2. In all the other contributions to eqs. (5.1)–(114) that are not singular
in the limit $k^{2}\rightarrow 0$, we limit ourselves to presenting the
results up to ${\cal O}(\epsilon^{0})$.

The contributions of the function $\mathcal{F}^{n^{2}-\text{sing.}}$ to
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}$ have a non-smooth dependence on
$n^{2}$. All the other contributions of eqs. (5.1)–(5.1) to
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}$ have a smooth and harmless dependence
on $n^{2}$ and, similarly to $\mathcal{F}^{(1R1V)}_{ga,\,\text{corr.}}$ in
eqs. (101) and (5.1), they are presented by setting $n^{2}=0$.

By direct inspection and comparison of eqs. (113) and (114), we can see that
$\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$ has a non-smooth and singular
dependence on $n^{2}$ in the limit $n^{2}\rightarrow 0$. The $n^{2}$
dependence of $\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$ is also related to the
issue of the rapidity divergences, which we have already considered in
sections 4.2 and 4.3. If $n^{2}=0$, $\mathcal{F}^{n^{2}-\text{sing.}}$ is
given by eq. (114) and its contribution to
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ in the limit
$k^{-}\rightarrow+\infty$ is due to the following factor:

$\left(k^{2}\right)^{-\epsilon}\;\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)\simeq\frac{2(1-z)}{z^{2}}\;\frac{\left(\mathbf{k_{T}}^{2}\right)^{-1-\epsilon}}{p^{+}\;k^{-}}\;\frac{1+\epsilon}{\epsilon},\quad\quad
k^{-}\rightarrow+\infty\quad(n^{2}=0),$ (115)

which is proportional to $1/k^{-}$. This contribution is not integrable over
$k^{-}$ in the region where $k^{-}\rightarrow+\infty$. This singular behaviour
is a manifestation at ${\cal O}(\alpha_{\rm S}^{2})$ of the rapidity
divergences that we have discussed in section 4.2. If $n^{2}>0$, the
expression of $\mathcal{F}^{n^{2}-\text{sing.}}$ is given by eq. (113) and in
the limit $k^{-}\rightarrow+\infty$ we have

$\displaystyle\left(k^{2}\right)^{-\epsilon}\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$
$\displaystyle\simeq\frac{2(1-z)\left(2k^{+}k^{-}\right)^{-\epsilon}}{z^{2}\,\mathbf{k_{T}}^{2}\,p^{+}\;k^{-}}$
(116)
$\displaystyle\times\left[\frac{1}{1+\frac{n^{2}(pk)^{2}}{(np)^{2}\mathbf{k_{T}}^{2}}}-\ln\left(1+\frac{\mathbf{k_{T}}^{2}(np)^{2}}{n^{2}(pk)^{2}}\right)\right],\;k^{-}\rightarrow+\infty\quad(n^{2}>0),$

where we have neglected contributions of ${\cal O}((k^{-})^{-2-2\epsilon})$
and ${\cal O}((k^{-})^{-2-\epsilon})$ that are $n^{2}$-independent at large
values of $k^{-}$. The overall factor $(k^{-})^{-1-\epsilon}$ in the right-
hand side of eq. (116) is not integrable444In our computation we have
$\epsilon<0$ to regularize the IR divergences. Therefore, the
$\epsilon$-dependent factor $(k^{-})^{-\epsilon}$ does not remove the rapidity
divergence due to $1/k^{-}$ at $k^{-}\rightarrow+\infty$. as
$k^{-}\rightarrow+\infty$. However, since
$n^{2}(pk)^{2}/(np)^{2}=n^{2}(k^{-})^{2}/(n^{-})^{2}$, both terms in the
square bracket of eq. (116) screen the non-integrable behaviour of the factor
$(k^{-})^{-1-\epsilon}$. Therefore, $\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$
does not produce rapidity divergences in the expression of
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$. The $n^{2}$ dependence of the
square bracket term in eq. (116) leads to a dynamical damping at large values
of $k^{-}$, and the non-integrable behaviour of the type
$(k^{-})^{-1-\epsilon}$ is effective only inside the region where
$(k^{-})^{2}\raisebox{-2.58334pt}{\hbox
to0.0pt{$\,\sim\,$\hss}}\raisebox{1.72218pt}{$\,<\,$}(n^{-})^{2}\,\mathbf{k_{T}}^{2}/n^{2}$.
Since $\mathbf{k_{T}}^{2}\leq 2k^{+}k^{-}$ (i.e., $k^{2}\geq 0$), this region
is contained in the region with $k^{-}\raisebox{-2.58334pt}{\hbox
to0.0pt{$\,\sim\,$\hss}}\raisebox{1.72218pt}{$\,<\,$}2(n^{-})^{2}\,k^{+}/n^{2}$,
where the collinear approximation of the squared amplitudes is certainly
valid. Obviously, the issue of rapidity divergences is not relevant by
considering the differential collinear functions
$\mathcal{F}_{ga,\,\text{corr.}}(p,k;n)$ at finite and not large values of
$k^{-}$.

As discussed at the end of sections 3.2 and 4.1, setting $n^{2}=0$,
$\mathbf{q_{T}}=\mathbf{k_{T}}$ and $t=2zpk$, the collinear function
$\mathcal{F}_{ca}(p,k;n)$ is related to the SCET differential beam function
${\cal B}_{ca}(z,t,\mathbf{q_{T}})$ Jain:2011iu ; Mantry:2009qz ;
Mantry:2010mk  at the partonic level. At ${\cal O}(\alpha_{\rm S}^{2})$ we
have ${\cal
B}_{ca}^{(2)}(z,t,\mathbf{q_{T}})\\!=\\!{F}^{(1R1V)}_{ca}(p,k;n)+{F}^{(2R)}_{ca}(p,k;n)$
($a=q,\bar{q},g$), where ${\cal B}_{ca}^{(2)}$ is the second-order
contribution to ${\cal B}_{ca}$. The azimuthal-independent component ${\cal
B}_{ca,\,\text{az.in.}}^{(2)}$ at ${\cal O}(\alpha_{\rm S}^{2})$ was computed
in refs. Gaunt:2014xxa ; Gaunt:2020xlc . Our expressions in eqs. (101), (5.1),
(105) and (5.1) give the result at ${\cal O}(\alpha_{\rm S}^{2})$ for the
azimuthal-correlation component ${\cal B}_{ga,\,\text{corr.}}^{(2)}$ of the
SCET differential beam function.

###  5.2 TMD functions

We use our results at ${\cal O}(\alpha_{\rm S}^{2})$ for the differential
collinear functions $\mathcal{F}_{ga,\,\text{corr.}}(p,k;n)$ to compute the
TMD functions ${F}_{ga,\,\text{corr.}}$ at the corresponding perturbative
order. Similarly to our computations of the TMD functions at ${\cal
O}(\alpha_{\rm S})$ (see section 4.2), we use a time-like auxiliary vector
$n^{\mu}$ to avoid rapidity divergences. However, we comment on the $n^{2}$
dependence of the various contributions to ${F}_{ga,\,\text{corr.}}$.

The ${\cal O}(\alpha_{\rm S}^{2})$ contributions to ${F}_{ga,\,\text{corr.}}$
are denoted by ${F}^{(1R1V)}_{ga,\,\text{corr.}}$ and
${F}^{(2R)}_{ga,\,\text{corr.}}$, and they are obtained according to the
definition in eq. (3.3) by integrating
$\mathcal{F}^{(1R1V)}_{ga,\,\text{corr.}}(p,k;n)$ and
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ over the momentum $k^{\mu}$.
The integrations over $k^{+}$ and $\mathbf{k_{T}}$ are trivial, and the
integration over $k^{-}$ extends up to $+\infty$.

The $k^{-}$ integration of the expressions of
$\mathcal{F}^{(1R1V)}_{ga,\,\text{corr.}}(p,k;n)$ ($a=g,q,{\bar{q}}$) in eqs.
(101) and (5.1) is elementary since it simply sets
$k^{-}=\mathbf{q_{T}}^{2}/(2(1-z)p^{+})$ by using the delta function
$\delta_{+}(k^{2})$. The ensuing expressions of
${F}^{(1R1V)}_{ga,\,\text{corr.}}$ contain double and single poles,
$1/\epsilon^{2}$ and $1/\epsilon$, and the dominant contributions in the limit
$\mathbf{q_{T}}^{2}\rightarrow 0$ do not depend on $n^{2}$. These
contributions can be computed by setting $n^{2}=0$, since the $n^{2}$
dependence leads to terms of ${\cal O}(n^{2}\mathbf{q_{T}}^{2}/(np)^{2})$.

The $k^{-}$ integration of $\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$
extends over the region $k^{-}_{\rm min}\leq k^{-}<+\infty$. The lower limit
$k^{-}_{\rm min}=\mathbf{q_{T}}^{2}/(2(1-z)p^{+})$ derives from $k^{2}\geq 0$.
As noticed in section 5.1, the expressions of
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ contain contributions of the
type $(k^{2})^{-1-\epsilon}$ and $(k^{2})^{-1-2\epsilon}$. The integration of
these contributions over $k^{-}$ produces single poles $1/\epsilon$ from the
region where $k^{-}\rightarrow k^{-}_{\rm min}$ (i.e., $k^{2}\rightarrow 0$).
The $k^{-}$ integration in the region $k^{-}\rightarrow+\infty$ produces
rapidity divergences if $n^{2}=0$. We use an auxiliary vector with $n^{2}>0$
and the corresponding expression of $\mathcal{F}^{n^{2}-\text{sing.}}(p,k;n)$
in eq. (113), which eventually leads to contributions of the type
$\ln(n^{2}\mathbf{q_{T}}^{2}/(np)^{2})$ to ${F}^{(2R)}_{ga,\,\text{corr.}}$.
The remaining contributions of eqs. (5.1)–(5.1) to
$\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ produce terms with a smooth
$n^{2}$ dependence in ${F}^{(2R)}_{ga,\,\text{corr.}}$, which can be evaluated
by setting $n^{2}=0$ (i.e., neglecting subdominant terms of ${\cal
O}(n^{2}\mathbf{q_{T}}^{2}/(np)^{2})$). Regarding the dependence of
${F}^{(2R)}_{ga,\,\text{corr.}}$ on the momentum fraction $z=1-k^{+}/p^{+}$,
the $k^{-}$ integration of $\mathcal{F}^{(2R)}_{ga,\,\text{corr.}}(p,k;n)$ can
be carried out in terms of rational, logarithmic and di-logarithmic functions
of the variable $z$.

Having computed the ${\cal O}(\alpha_{\rm S}^{2})$ contributions
${F}^{(1R1V)}_{ga,\,\text{corr.}}$ and ${F}^{(2R)}_{ga,\,\text{corr.}}$ to the
TMD function ${F}_{ga,\,\text{corr.}}$ in $\mathbf{q_{T}}$ space, we perform
their Fourier transformations to obtain the corresponding contributions
${\widetilde{F}}^{(1R1V)}_{ga,\,{\rm corr.}}$ and
${\widetilde{F}}^{(2R)}_{ga,\,{\rm corr.}}$ of eq. (68) to the $\mathbf{b}$
space function ${\widetilde{F}}_{ga,\,{\rm corr.}}$ in eqs. (60) and (4.2).
The explicit Fourier integrals that are required at ${\cal O}(\alpha_{\rm
S}^{2})$ are given in eq. (4.2) with $\delta=\epsilon,2\epsilon$ and $m=0,1$.

The expressions of ${\widetilde{F}}^{(1R1V)}_{ga,\,{\rm corr.}}$ and
${\widetilde{F}}^{(2R)}_{ga,\,{\rm corr.}}$ have double and single poles,
$1/\epsilon^{2}$ and $1/\epsilon$, in the limit $\epsilon\rightarrow 0$. Part
of the single-pole terms are cancelled by performing the UV renormalization of
the bare coupling: we use eq. (42) up to ${\cal O}(\alpha_{\rm S})$ and we
choose the renormalization scale $\mu_{R}^{2}=b_{0}^{2}/\mathbf{b}^{2}$. The
remaining $\epsilon$ poles have to be treated by using the IR factorization
formula in eq. (71), which we expand up to ${\cal O}(\alpha_{\rm S}^{2})$:

$\displaystyle{\widetilde{F}}_{ga,\,{\rm
corr.}}(z;\mathbf{b}^{2}/b_{0}^{2},{\widetilde{\lambda}})=$
$\displaystyle\frac{\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2})}{\pi}\;{\widetilde{G}}^{(1)}_{ga}(z;\epsilon,{\widetilde{\lambda}})+\left(\frac{\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2})}{\pi}\right)^{2}\Bigl{[}\;Z_{g}^{(1)}({\widetilde{\lambda}})\;{\widetilde{G}}^{(1)}_{ga}(z;\epsilon,{\widetilde{\lambda}})\Bigr{.}$
$\displaystyle-\Bigl{.}\frac{1}{\epsilon}\sum_{b}\int_{z}^{1}\frac{dx}{x}\;{\widetilde{G}}^{(1)}_{gb}(x;\epsilon,{\widetilde{\lambda}})\;P^{(1)}_{ba}(z/x)+{\widetilde{G}}^{(2)}_{ga}(z;\epsilon,{\widetilde{\lambda}})\;\Bigr{]}+{\cal
O}(\alpha_{\rm S}^{3}),$ (117)

where ${\widetilde{\lambda}}=n^{2}(b_{0}^{2}/\mathbf{b}^{2})/(2zpn)^{2}$. The
first-order functions $P^{(1)}_{ba},Z_{g}^{(1)}$ and
${\widetilde{G}}^{(1)}_{ga}$ are given in eqs. (66), (75) and (85). Our
computation of ${\widetilde{F}}^{(1R1V)}_{ga,\,{\rm corr.}}$ and
${\widetilde{F}}^{(2R)}_{ga,\,{\rm corr.}}$ gives an explicit expression up to
${\cal O}(\epsilon^{0})$ of the entire contribution in the square bracket on
the right-hand side of eq. (5.2) and, knowing $P^{(1)}_{ba},Z_{g}^{(1)}$ and
${\widetilde{G}}^{(1)}_{ga}$, we can explicitly determine
${\widetilde{G}}^{(2)}_{ga}$ up to ${\cal O}(\epsilon^{0})$. Our result for
${\widetilde{G}}^{(2)}_{ga}$ is finite and independent of
${\widetilde{\lambda}}$ in the four-dimensional limit $\epsilon\rightarrow 0$,
therefore confirming the validity of the IR factorization structure of eq.
(71).

Setting $\epsilon=0$ we have

${\widetilde{G}}^{(2)}_{ga}(z;\epsilon=0,{\widetilde{\lambda}})={G}^{(2)}_{ga}(z),$
(118)

where ${G}^{(2)}_{ga}(z)$ ($a=g,q,{\bar{q}}$) are the contributions at ${\cal
O}(\alpha_{\rm S}^{2})$ to the transverse-momentum resummation functions in
eq. (6). We find the following results:

$\displaystyle G^{(2)}_{gg}(z)=$ $\displaystyle
C_{A}^{2}\bigg{\\{}\\!\\!-\frac{37}{36z}+\frac{31}{18}-\frac{13z}{12}+\frac{11z^{2}}{36}-\ln(z)\left[\frac{1}{z}+\frac{19}{12}\right]\\!+\\!\frac{\ln^{2}(z)}{2}\\!+\\!\frac{1-z}{z}\left[\text{Li}_{2}(z)-\frac{\pi^{2}}{6}\right]\\!\\!\bigg{\\}}$
$\displaystyle+C_{F}N_{f}\left\\{\frac{(1-z)^{3}}{2z}-\frac{1}{4}\ln^{2}(z)\right\\}+C_{A}N_{f}\left\\{-\frac{17}{36z}+\frac{4}{9}+\frac{z}{12}+\frac{z^{2}}{36}-\frac{1}{6}\ln(z)\right\\}$
$\displaystyle-h^{(1)}_{g}\,C_{A}\,\frac{1-z}{z},$ (119) $\displaystyle
G^{(2)}_{gq}(z)=$ $\displaystyle
C_{F}^{2}\left\\{-\frac{1-z}{2}+\frac{5}{4}\ln(z)-\frac{1}{4}\ln^{2}(z)-\frac{1-z}{2z}\bigg{[}\ln(1-z)+\ln^{2}(1-z)\bigg{]}\right\\}$
$\displaystyle+C_{F}N_{f}\left\\{-\frac{1-z}{3z}\left[\frac{2}{3}+\ln(1-z)\right]\right\\}+C_{A}C_{F}\bigg{\\{}\\!-\frac{11}{18z}+\frac{10}{9}-\frac{z}{2}-\ln(z)\left[\frac{1}{z}+\frac{5}{2}\right]$
$\displaystyle+\frac{1}{2}\ln^{2}(z)+\frac{1-z}{z}\left[\frac{5}{6}\ln(1-z)+\frac{1}{2}\ln^{2}(1-z)+\text{Li}_{2}(z)-\frac{\pi^{2}}{6}\right]\\!\bigg{\\}}-h^{(1)}_{g}\,C_{F}\,\frac{1-z}{z},$
(120)

and $G^{(2)}_{g{\bar{q}}}(z)=G^{(2)}_{gq}(z)$. These results regard a generic
resummation scheme, which is specified by the coefficient $h^{(1)}_{g}$ of the
IR factorization function $Z_{c}^{(1)}$ with $c=g$ in eq. (75). We recall that
the hard scheme Catani:2013tia  is defined by setting $h^{(1)}_{g}=0$, and the
‘minimal’ scheme has $h^{(1)}_{g}=\pi^{2}C_{A}/24$. We find full agreement
with the results for $G^{(2)}_{ga}(z)$ that were obtained in refs. Luo:2019bmw
; Gutierrez-Reyes:2019rug  (ref. Luo:2019bmw  uses the hard scheme, while ref.
Gutierrez-Reyes:2019rug  uses the minimal scheme) by using TMD functions
defined in a SCET framework.

We note that our results at ${\cal O}(\alpha_{\rm S}^{2})$ for the
azimuthally-correlated TMD function ${\widetilde{F}}_{ga,\,{\rm corr.}}$ are
an important and highly non-trivial check of the IR factorization formulae in
eqs. (70) and (71) and of our framework to define and compute TMD functions.
In particular, we find that the same IR singular and $n^{2}$ dependent
contribution $Z_{g}^{(1)}$ of eq. (75) is involved in the IR factorization of
both the azimuthal-independent function ${\widetilde{F}}_{ga,\,{\rm az.in.}}$
at ${\cal O}(\alpha_{\rm S})$ (see section 4.2) and the azimuthally-correlated
function ${\widetilde{F}}_{ga,\,{\rm corr.}}$ at ${\cal O}(\alpha_{\rm
S}^{2})$. At the purely technical level, we also note that the $n^{2}$
dependence of $Z_{g}^{(1)}$ has an entirely different origin in the
calculations of ${\widetilde{F}}_{ga,\,{\rm az.in.}}$ at ${\cal O}(\alpha_{\rm
S})$ and ${\widetilde{F}}_{ga,\,{\rm corr.}}$ at ${\cal O}(\alpha_{\rm
S}^{2})$ (see eq. (4.2) vs. eq. (113) and related comments).

The agreement between our results in eqs. (5.2) and (5.2) and those in refs.
Luo:2019bmw ; Gutierrez-Reyes:2019rug  is also highly non-trivial. Indeed,
this agreement is obtained by using fully independent methods and, in
particular, very different procedures to deal with the issue of rapidity
divergences.

##  6 TL collinear functions

In this section we consider the perturbative computation of the differential
collinear functions ${\cal F}^{{\rm TL}}_{ca}$ and of the TMD functions
$F^{{\rm TL}}_{ca}$ in the TL collinear region. We closely follow the
presentation of the analogous SL results in sections 4.1, 4.2 and 5, with
brief comments to highlight the main differences between SL and TL functions.

###  6.1 TL differential collinear functions at ${\cal O}(\alpha_{\rm S})$

The perturbative expansion of the TL collinear functions in eqs. (3.2), (22)
and (3.2) can be written in the following form:

${\cal F}^{\rm TL}(p,k;n)={\cal F}^{{\rm TL}\,(1R)}(p,k;n)+\left[\;{\cal
F}^{{\rm TL}\,(2R)}(p,k;n)+{\cal F}^{{\rm TL}\,(1R1V)}(p,k;n)\;\right]+{\cal
O}(\alpha_{\rm S}^{3}),$ (121)

where, analogously to eq. (41), we have not explicitly denoted subscripts and
superscripts in ${\cal F}^{\rm TL}$.

The first-order term ${\cal F}^{{\rm TL}\,(1R)}_{ca}(p,k;n)$ in eq. (121) is
obtained from the contribution of the tree-level collinear kernel ${\cal
P}^{(0)}_{{c}\rightarrow a_{1}a}(k,p;n)$ to eqs. (3.2) and (3.2). Comparing
eqs. (3.2) and (3.2) with the analogous expression in eq. (28) for the SL
function ${\cal F}^{(1R)}_{ca}(p,k;n)$, we straightforwardly obtain the
following relation:

${\cal F}_{ca}^{{\rm
TL}\,(1R)}(p,k;n)=\frac{\mathcal{N}_{a}(\epsilon)}{\mathcal{N}_{c}(\epsilon)}\;\left[{\cal
F}_{ca}^{(1R)}(p,k;n)\right]_{p\rightarrow-p},$ (122)

which is valid for both the azimuthal-independent and azimuthal-correlation
components of the differential collinear functions. We note that the SL-TL
crossing relation in eq. (122) is valid since the tree-level kernel ${\cal
P}^{(0)}_{{c}\rightarrow a_{1}a}(k,p;n)$ is process independent and it has a
rational dependence on the external momenta, which permits a straightforward
replacement $p\leftrightarrow-p$. A similar comment applies to any tree-level
kernel ${\cal P}^{(0)}_{{c}\rightarrow a_{1}a_{2}\dots a}$, and this leads to
ensuing SL-TL crossing relations at higher perturbative orders (see, e.g., eq.
(151)).

Using the SL results in eqs. (43) and (44) and the relation in eq. (122), we
obtain the following explicit expressions for the azimuthal-
independent555Similarly to the case of the SL functions, we introduce the
notation ${\cal F}^{{\rm TL}}_{ca}={\cal F}^{{\rm TL}}_{ca,\,\text{az.in.}}$
and $F^{{\rm TL}}_{ca}=F^{{\rm TL}}_{ca,\,\text{az.in.}}$ ($c=q,{\bar{q}}$)
for the TL collinear functions in the quark and antiquark partonic channels.
and azimuthal-correlation components of the TL collinear functions at ${\cal
O}(\alpha_{\rm S})$:

$\displaystyle{\cal F}^{{\rm TL}\,(1R)}_{ca,\,\text{az.in.}}(p,k;n)$
$\displaystyle=\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}}\;\frac{\delta_{+}(k^{2})}{pk}\;{\widehat{P}}_{ac}(z_{n}^{\rm
TL};\epsilon),\;\;\;\quad c=g,q,{\bar{q}},$ (123)
$\displaystyle\mathcal{F}^{{\rm TL}\,(1R)}_{gg,\,\text{corr.}}(p,k;n)$
$\displaystyle=-\;\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}}\;\frac{\delta_{+}(k^{2})}{pk}\;C_{A}\;z_{n}^{\rm
TL}(1-z_{n}^{\rm TL}),$ (124) $\displaystyle\mathcal{F}^{{\rm
TL}\,(1R)}_{ga,\,\text{corr.}}(p,k;n)$ $\displaystyle=\;\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}}\;\frac{\delta_{+}(k^{2})}{pk}\;T_{R}\;\frac{z_{n}^{\rm
TL}(1-z_{n}^{\rm TL})}{1-\epsilon},\quad\quad a=q,{\bar{q}},$ (125)

where we have introduced the $n$-dependent variable $z_{n}^{\rm TL}$,

$z_{n}^{\rm TL}=\frac{np}{n(p+k)}.$ (126)

The right-hand side of eq. (123) is proportional to the $d$-dimensional
Altarelli–Parisi kernel ${\widehat{P}}_{ac}(x;\epsilon)$ of eqs. (46)–(50),
and we note that we have directly used the following well-known (see, e.g.,
ref. Catani:1996vz ) crossing symmetry relation:

$\frac{\mathcal{N}_{a}(\epsilon)}{\mathcal{N}_{c}(\epsilon)}\;\left[{\widehat{P}}_{ca}(x;\epsilon)\right]_{x=1/z}=-\frac{1}{z}\;{\widehat{P}}_{ac}(z;\epsilon).$
(127)

As discussed at the end of section 3.2, the collinear function ${\cal F}^{{\rm
TL}}(p,k;n)$ is related to the SCET fragmenting jet function ${\cal
G}(z,s,\mathbf{p_{\perp}})$ of ref. Jain:2011iu . More precisely, setting
$n^{2}=0$, $s=(p+k)^{2}$ and $\mathbf{p_{\perp}}=z\mathbf{k_{T}}$, the first-
order collinear function ${\cal F}^{{\rm TL}\,(1R)}_{ca}(p,k;n)$ is equal to
the first-order contribution ${\cal G}_{ca}^{(1)}(z,s,\mathbf{p_{\perp}})$ to
$\cal G$ at the partonic level. The azimuthal-independent component of ${\cal
G}_{ca}^{(1)}$ was computed in ref. Jain:2011iu , and the expression in eq.
(123) agrees with the results presented therein. The azimuthal-correlation
component of ${\cal G}_{ga}^{(1)}$ is given by eqs. (124) and (125).

###  6.2 TMD functions and IR factorization in the TL region

The TL TMD functions $F^{{\rm TL}}_{ca}$ are obtained from the corresponding
differential collinear functions ${\cal F}^{{\rm TL}}_{ca}(p,k;n)$ by using
eqs. (3.3) and (3.3)–(3.3).

The contribution of ${\cal O}(\alpha_{\rm S})$ to $F^{{\rm TL}}_{ca}$ is
denoted by $F^{{\rm TL}\,(1R)}_{ca}$, and it is computed by using the
expressions of ${\cal F}^{{\rm TL}\,(1R)}_{ca}(p,k;n)$ in eqs. (123)–(125). We
define and compute the TL functions $F^{{\rm TL}}_{ca}$ by using a time-like
auxiliary vector $n^{\mu}$, since the use of a light-like vector produces
rapidity divergences. The origin of the rapidity divergences is exactly
similar in the TL and SL regions, and we can briefly follow the discussion in
section 4.2. At ${\cal O}(\alpha_{\rm S})$ the non-smooth dependence of
$F^{{\rm TL}\,(1R)}_{ca}$ on $n^{2}$ is produced by the contribution
${\widehat{P}}^{\rm\,sing}_{ac}(z_{n}^{\rm TL})$ (see eqs. (51)–(53)) to the
kernel ${\widehat{P}}_{ac}(z_{n}^{\rm TL};\epsilon)$ in the right-hand side of
eq. (123). Indeed, such contribution is proportional to the following factor:

$\displaystyle\frac{1}{1-z_{n}^{\rm TL}}$
$\displaystyle=1+\frac{p^{+}}{k^{+}+\frac{n^{2}}{2np}\frac{k^{-}}{n^{-}}p^{+}}=1+\frac{z}{1-z+\frac{z^{2}n^{2}\mathbf{q_{T}}^{2}}{(1-z)(2np)^{2}}}$
(128)
$\displaystyle=\frac{1}{2}\;\ln\left(\frac{1}{\lambda}\right)\;\delta(1-z)+\left(\frac{1}{1-z}\right)_{+}+{\cal
O}({\sqrt{\lambda}}),$ (129)

where $\lambda=n^{2}\mathbf{q_{T}}^{2}/{(2pn)^{2}}$. In the last equality of
eq. (128) we have implemented the kinematics of the TL TMD function at ${\cal
O}(\alpha_{\rm S})$ (i.e.,
$k^{2}=0,k^{+}=p^{+}(1-z)/z,\mathbf{k_{T}}=-\mathbf{q_{T}}$), and in eq. (129)
we have properly performed the limit $\mathbf{q_{T}}\rightarrow 0$ by using
the same procedure as in eq. (4.2). The small-$\mathbf{q_{T}}$ limit of the
remaining dependence of $F^{{\rm TL}\,(1R)}_{ca}$ on $z_{n}^{\rm TL}$ can be
evaluated by simply setting $z_{n}^{\rm TL}=z+{\cal O}(\mathbf{q_{T}}^{2})$.
Eventually, we obtain the following final results for the first-order TMD
functions in the TL region:

$\displaystyle F^{{\rm TL}\,(1R)}_{ca,\,{\rm
az.in.}}\\!\left(z;\mathbf{q_{T}}^{2},\frac{n^{2}\mathbf{q_{T}}^{2}}{(2pn/z)^{2}}\right)=\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}\,\mathbf{q_{T}}^{2}}$
(130)
$\displaystyle\quad\times\left\\{{\widehat{P}}^{\rm\,reg}_{ac}(z;\epsilon)+\delta_{ca}\;A_{c}^{(1)}\left[\left(\frac{1}{1-z}\right)_{+}-\frac{1}{2}\;\ln\left(\frac{n^{2}\mathbf{q_{T}}^{2}}{(2pn/z)^{2}}\right)\,\delta(1-z)\right]\right\\},\;\;\;\;\;\;c=g,q,{\bar{q}},$
$\displaystyle F^{{\rm TL}\,(1R)}_{gg,\,{\rm
corr.}}\\!\left(z;\mathbf{q_{T}}^{2},\frac{n^{2}\mathbf{q_{T}}^{2}}{(2pn/z)^{2}}\right)=-\;\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}\,\mathbf{q_{T}}^{2}}\;C_{A}\;z(1-z),$
(131) $\displaystyle F^{{\rm TL}\,(1R)}_{ga,\,{\rm
corr.}}\\!\left(z;\mathbf{q_{T}}^{2},\frac{n^{2}\mathbf{q_{T}}^{2}}{(2pn/z)^{2}}\right)=\frac{\alpha^{u}_{\rm
S}\,\mu_{0}^{2\epsilon}\,S_{\epsilon}}{\pi}\;\frac{e^{\epsilon\gamma_{E}}}{\pi^{1-\epsilon}\,\mathbf{q_{T}}^{2}}\;T_{R}\;\frac{z(1-z)}{1-\epsilon}\;\;,\quad\quad
a=q,{\bar{q}}.$ (132)

The Fourier transformation ${\widetilde{F}}^{\rm TL}_{ca}$ of the TL TMD
function ${F}^{\rm TL}_{ca}$ is defined analogous to the SL case: we simply
use eqs. (59)–(4.2) with the replacements ${F}_{ca}\rightarrow{F}^{\rm
TL}_{ca}$ and ${\widetilde{F}}_{ca}\rightarrow{\widetilde{F}}^{\rm TL}_{ca}$.
The perturbative expansion of the $\mathbf{b}$ space functions
${\widetilde{F}}^{\rm TL}_{ca}$ can be written as follows:

$\displaystyle{\widetilde{F}}^{\rm TL}_{ca,\,{\rm
az.in.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)\\!=$
$\displaystyle\delta_{ca}\;\delta(1-z)+{\widetilde{F}}^{{\rm
TL}\,(1R)}_{ca,\,{\rm
az.in.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)+{\cal
O}(\alpha_{\rm S}^{2}),$ $\displaystyle\hskip
142.26378pt\,\,\,\,\,c=g,q,{\bar{q}},$ (133)
$\displaystyle{\widetilde{F}}^{\rm TL}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=$
$\displaystyle{\widetilde{F}}^{{\rm TL}\,(1R)}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)$
(134) $\displaystyle+\left[{\widetilde{F}}^{{\rm TL}\,(2R)}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)+{\widetilde{F}}^{{\rm
TL}\,(1R1V)}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)\right]+{\cal
O}(\alpha_{\rm S}^{3}),$ $\displaystyle\hskip
142.26378pt\,\,\,\,\,a=g,q,{\bar{q}}.$

The first-order contributions ${\widetilde{F}}^{{\rm TL}\,(1R)}_{ca,\,{\rm
az.in.}}$ and ${\widetilde{F}}^{{\rm TL}\,(1R)}_{ga,\,{\rm corr.}}$ are
directly computed from eqs. (130)–(132) by using the Fourier transformations
in eqs. (62) and (4.2). We obtain the following results:

$\displaystyle{\widetilde{F}}^{{\rm TL}\,(1R)}_{ca,\,{\rm
az.in.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=\frac{\alpha^{u}_{\rm
S}\,S_{\epsilon}}{\pi}\;\left(\frac{\mu_{0}^{2}\,\mathbf{b}^{2}}{b_{0}^{2}}\right)^{+\epsilon}\;\frac{e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)}{(-\epsilon)}\Bigg{\\{}{P}^{(1)}_{ac}(z)+\epsilon\;{\widehat{P}}^{\prime}_{ac}(z;\epsilon)$
$\displaystyle\hskip
14.22636pt-\delta_{ca}\,\delta(1-z)\left[\frac{A_{c}^{(1)}}{2}\left(\frac{1}{\epsilon}+\psi(1-\epsilon)-\psi(1)+\ln\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)+\frac{\gamma_{c}}{2}\right]\Bigg{\\}},\quad
c=g,q,{\bar{q}},$ (135) $\displaystyle{\widetilde{F}}^{{\rm
TL}\,(1R)}_{gg,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=\frac{\alpha^{u}_{\rm
S}\,S_{\epsilon}}{\pi}\;\left(\frac{\mu_{0}^{2}\,\mathbf{b}^{2}}{b_{0}^{2}}\right)^{\\!+\epsilon}\;e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)\;C_{A}\;z(1-z),$
(136) $\displaystyle{\widetilde{F}}^{{\rm TL}\,(1R)}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=-\frac{\alpha^{u}_{\rm
S}\,S_{\epsilon}}{\pi}\;\left(\frac{\mu_{0}^{2}\,\mathbf{b}^{2}}{b_{0}^{2}}\right)^{\\!+\epsilon}\;e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)\;T_{R}\;\frac{z(1-z)}{1-\epsilon},\,\,a=q,{\bar{q}}.$
(137)

In the right-hand side of eq. (135) we have used the lowest-order
Altarelli–Parisi kernel ${P}^{(1)}_{ac}(x)$ (see eq. (66)) and the
contribution ${\widehat{P}}^{\prime}_{ac}$ to the $d$-dimensional real
emission kernel ${\widehat{P}}_{ac}$ (see eqs. (51)–(53)).

The perturbative expansion of ${\widetilde{F}}^{\rm TL}_{ca}$ in powers of the
bare QCD coupling $\alpha^{u}_{\rm S}$ has $\epsilon$-pole contributions of UV
and IR origins. The UV poles are removed by using eq. (42) and introducing the
renormalized coupling $\alpha_{\rm S}(\mu_{R}^{2})$. The IR poles can be
factorized, similarly to the SL case in eqs. (70) and (71). The IR
factorization formulae for the $\mathbf{b}$ space TMD functions
${\widetilde{F}}^{\rm TL}_{ca}$ in the TL region are

$\displaystyle{\widetilde{F}}^{\rm TL}_{ca,\,{\rm
az.in.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=Z_{c}^{\rm
TL}\\!\left(\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2}),\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)$
$\displaystyle\quad\quad\times\sum_{b}\int_{z}^{1}\frac{dx}{x^{1-2\epsilon}}\;{\widetilde{C}}^{\rm
TL}_{cb}\\!\left(z/x;\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2}),\epsilon,\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)\;\;{\widetilde{\Gamma}}^{\rm
TL}_{ba}(x;b_{0}^{2}/\mathbf{b}^{2}),$ (138)
$\displaystyle{\widetilde{F}}^{\rm TL}_{ga,\,{\rm
corr.}}\\!\left(z;\frac{\mathbf{b}^{2}}{b_{0}^{2}},\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)=Z_{g}^{\rm
TL}\\!\left(\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2}),\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)$
$\displaystyle\quad\quad\times\sum_{b}\int_{z}^{1}\frac{dx}{x^{1-2\epsilon}}\;{\widetilde{G}}^{\rm
TL}_{gb}\\!\left(z/x;\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2}),\epsilon,\frac{n^{2}b_{0}^{2}}{(2pn/z)^{2}\,\mathbf{b}^{2}}\right)\;\;{\widetilde{\Gamma}}^{\rm
TL}_{ba}(x;b_{0}^{2}/\mathbf{b}^{2}).$ (139)

The factor ${\widetilde{\Gamma}}^{\rm TL}_{ba}(x;\mu_{F}^{2})$ in the right-
hand side of eqs. (138) and (139) is the customary collinear-divergent
function that defines the scale-dependent PFF $d_{b}(z;\mu_{F}^{2})$ in the
$\overline{{\rm MS}}$ factorization scheme. The relation between
$d_{b}(z;\mu_{F}^{2})$ and the bare PFF $d^{(0)}_{a}(x)$ is

$d_{b}(z;\mu_{F}^{2})=\sum_{a}\int_{z}^{1}\frac{dx}{x}\;{\widetilde{\Gamma}}^{\rm
TL}_{ba}(z/x;\mu_{F}^{2})\;d^{(0)}_{a}(x).$ (140)

The perturbative expansion of ${\widetilde{\Gamma}}^{\rm TL}_{ba}$ is

${\widetilde{\Gamma}}^{\rm
TL}_{ba}(z;\mu_{F}^{2})=\delta_{ba}\;\delta(1-z)-\frac{\alpha_{\rm
S}(\mu^{2}_{F})}{\pi}\;\frac{{P}^{{\rm TL}\,(1)}_{ba}(z)}{\epsilon}+{\cal
O}(\alpha_{\rm S}^{2}),$ (141)

with

${P}^{{\rm TL}\,(1)}_{ba}(z)={P}^{(1)}_{ab}(z),$ (142)

where ${P}^{(1)}_{ba}(z)$ is the lowest-order Altarelli–Parisi kernel in eq.
(66). We recall that the SL and TL functions ${\widetilde{\Gamma}}_{ba}$ and
${\widetilde{\Gamma}}^{\rm TL}_{ba}$ at ${\cal O}(\alpha_{\rm S})$ are
directly related through the Gribov–Lipatov relation Gribov:1972ri ;
Gribov:1972rt , namely, through the transposition $ba\leftrightarrow ab$ of
the flavour indices (see eqs. (73), (141) and (142)).

The structure of the eqs. (138) and (139) for the TL functions is analogous to
that of the corresponding eqs. (70) and (71) for the SL functions. The IR
$\epsilon$-poles of ${\widetilde{F}}^{\rm TL}_{ca}$ are factorized in the
fragmentation related function ${\widetilde{\Gamma}}^{\rm
TL}_{ba}(x;b_{0}^{2}/\mathbf{b}^{2})$ and in the overall perturbative function
$Z_{c}^{\rm TL}$. The remaining functions ${\widetilde{C}}^{\rm TL}_{cb}$ and
${\widetilde{G}}^{\rm TL}_{gb}$ are finite and independent of $n^{2}$ (i.e.,
$n^{2}(b_{0}^{2}/\mathbf{b}^{2})/(2np/z)^{2}$) in the limit
$\epsilon\rightarrow 0$, order-by-order in the perturbative expansion in
powers of $\alpha_{\rm S}(b_{0}^{2}/\mathbf{b}^{2})$. The main difference
between eqs. (70), (71) and eqs. (138), (139) is due to the phase space
convolution factor $x^{-1+2\epsilon}$ in the IR factorization formulae for the
TL functions. The $\epsilon$ dependence of such convolution factor has a
general origin from the $d$-dimensional kinematics of inclusive single-
particle production and fragmentation (see, e.g., ref. Catani:1996vz ).

The first-order perturbative results for the TL factors $Z_{c}^{\rm
TL},{\widetilde{C}}^{\rm TL}_{cb}$ and ${\widetilde{G}}^{\rm TL}_{gb}$ in eqs.
(138) and (139) are obtained by using the expressions of
${\widetilde{F}}^{{\rm TL}\,(1R)}_{ca}$ in eqs. (135)–(137) and they are
reported below. We note that these TL factors depend on $n^{2}$ through the
variable ${\widetilde{\lambda}}=n^{2}(b_{0}^{2}/\mathbf{b}^{2})/(2np/z)^{2}$.

The IR factor $Z_{c}^{\rm TL}$ in eqs. (138) and (139) has the following
perturbative expansion:

$Z_{c}^{\rm TL}(\alpha_{\rm S},{\widetilde{\lambda}})=1+\frac{\alpha_{\rm
S}}{\pi}\;Z_{c}^{{\rm TL}\,(1)}({\widetilde{\lambda}})+{\cal O}(\alpha_{\rm
S}^{2}),$ (143)

and its first-order contribution is

$Z_{c}^{{\rm
TL}\,(1)}({\widetilde{\lambda}})=\frac{1}{2}\left[A^{(1)}_{c}\left(\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\ln{\widetilde{\lambda}}\right)+\frac{1}{\epsilon}\,\gamma_{c}\right]-\frac{\pi^{2}}{24}A^{(1)}_{c}+h_{c}^{{\rm
TL}\,(1)}+{\widetilde{h}}_{c}^{{\rm
TL}\,(1)}(\epsilon,{\widetilde{\lambda}}).$ (144)

Similarly to eq. (75) for the SL case, the terms $h_{c}^{{\rm TL}\,(1)}$ and
${\widetilde{h}}_{c}^{{\rm TL}\,(1)}$ in eq. (144) specify the resummation-
scheme dependence. In particular, we have $h_{c}^{{\rm TL}\,(1)}=0$ in the
hard scheme Catani:2013tia  and $h_{c}^{{\rm TL}\,(1)}=\pi^{2}A^{(1)}_{c}/24$
in a minimal scheme in which $Z_{c}^{{\rm TL}\,(1)}$ contains only $\epsilon$
pole contributions in the limit $\epsilon\rightarrow 0$. The term
${\widetilde{h}}_{c}^{{\rm TL}\,(1)}(\epsilon,{\widetilde{\lambda}})$ vanishes
in the limit $\epsilon\rightarrow 0$. We note that the SL contribution
$Z_{c}^{(1)}$ in eq. (75) and the TL contribution $Z_{c}^{{\rm TL}\,(1)}$ in
eq. (144) are equal, modulo their scheme-dependence arbitrariness.

The perturbative expansion of the IR finite function
${\widetilde{C}}_{ca}^{\rm TL}$ in eq. (138) is

$\displaystyle{\widetilde{C}}_{ca}^{\rm TL}(z;\alpha_{\rm
S},\epsilon,{\widetilde{\lambda}})=$
$\displaystyle\delta_{ca}\;\delta(1-z)+\frac{\alpha_{\rm
S}}{\pi}\;{\widetilde{C}}_{ca}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})+\left(\frac{\alpha_{\rm
S}}{\pi}\right)^{2}\;{\widetilde{C}}_{ca}^{{\rm
TL}\,(2)}(z;\epsilon,{\widetilde{\lambda}})+{\cal O}(\alpha_{\rm S}^{3}),$
$\displaystyle\hskip 227.62204ptc=q,{\bar{q}},g,$ (145)

and the limit $\epsilon\rightarrow 0$ gives the TL collinear function
${C}_{ca}^{\rm TL}(z;\alpha_{\rm S})={\widetilde{C}}_{ca}^{\rm
TL}(z;\alpha_{\rm S},\epsilon=0,{\widetilde{\lambda}})$ for transverse-
momentum resummation (see section 2). At ${\cal O}(\alpha_{\rm S})$ we
explicitly obtain

${C}_{ca}^{{\rm
TL}\,(1)}(z)=-{\widehat{P}}^{\prime}_{ac}(z;\epsilon=0)+2{P}^{(1)}_{ac}(z)\,\ln
z-\delta_{ca}\;\delta(1-z)\;{h}_{c}^{{\rm TL}\,(1)},\quad\quad
c=q,{\bar{q}},g,$ (146)

which agrees with the known results in the literature Nadolsky:1999kb ;
Echevarria:2016scs ; Luo:2019hmp ; Luo:2019bmw . The complete $\epsilon$
dependence of ${\widetilde{C}}_{ca}^{{\rm TL}\,(1)}$ is

$\displaystyle{\widetilde{C}}_{ca}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})=-\,e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)\;{\widehat{P}}^{\prime}_{ac}(z;\epsilon)+\frac{z^{2\epsilon}-e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)}{\epsilon}{P}^{(1)}_{ac}(z)$
$\displaystyle-\delta_{ca}\;\delta(1-z)\;\Bigg{\\{}\frac{1-e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)}{2\epsilon}\left[A_{c}^{(1)}\left(\frac{1}{\epsilon}+\ln{\widetilde{\lambda}}\right)+\gamma_{c}\right]$
(147)
$\displaystyle-\frac{e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)}{2\epsilon}A_{c}^{(1)}\left(\psi(1-\epsilon)-\psi(1)\right)-\frac{\pi^{2}}{24}A^{(1)}_{c}+h_{c}^{{\rm
TL}\,(1)}+{\widetilde{h}}_{c}^{{\rm
TL}\,(1)}(\epsilon,{\widetilde{\lambda}})\Bigg{\\}},\quad c=q,{\bar{q}},g.$

Comparing the SL function in eq. (77) with the TL function in eq. (146), we
note that they both involve the contribution ${\widehat{P}}^{\prime}_{ca}$ of
${\cal O}(\epsilon)$ to the $d$-dimensional kernel
${\widehat{P}}_{ca}(z;\epsilon)$. In particular, the transposition of flavour
indices (i.e.,
${\widehat{P}}^{\prime}_{ca}\leftrightarrow{\widehat{P}}^{\prime}_{ac}$) in
eqs. (77) and (146) is due to the crossing relation in eq. (127). The TL
function ${\widetilde{C}}_{ca}^{{\rm TL}\,(1)}$ also includes the term
$2{P}^{(1)}_{ac}(z)\,\ln z$, which is due to the collinear factorization of
the fragmentation contributions in the $\overline{{\rm MS}}$ factorization
scheme (see the convolution factor $x^{-1+2\epsilon}$ in eq. (138)).

The azimuthal-correlation function ${\widetilde{G}}_{ga}^{\rm TL}$ in eq.
(139) has the following perturbative expansion:

${\widetilde{G}}_{ga}^{\rm TL}(z;\alpha_{\rm
S},\epsilon,{\widetilde{\lambda}})=\frac{\alpha_{\rm
S}}{\pi}\;{\widetilde{G}}_{ga}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})+\left(\frac{\alpha_{\rm
S}}{\pi}\right)^{2}\;{\widetilde{G}}_{ga}^{{\rm
TL}\,(2)}(z;\epsilon,{\widetilde{\lambda}})+{\cal O}(\alpha_{\rm S}^{3}).$
(148)

At ${\cal O}(\alpha_{\rm S})$ the azimuthal-correlation component
${\widetilde{F}}_{ga,\,{\rm corr.}}^{\rm TL}$ of the $\mathbf{b}$ space TMD
function does not require the factorization of IR divergent contributions, and
we directly obtain the following results:

${\widetilde{G}}_{gg}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})=e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)\;C_{A}\;z(1-z),$
(149) ${\widetilde{G}}_{gq}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})=-e^{-\epsilon\gamma_{E}}\,\Gamma(1-\epsilon)\;T_{R}\;\frac{z(1-z)}{1-\epsilon},$
(150)

and ${\widetilde{G}}_{g{\bar{q}}}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})={\widetilde{G}}_{gq}^{{\rm
TL}\,(1)}(z;\epsilon,{\widetilde{\lambda}})$. The limit $\epsilon\rightarrow
0$ of eq. (148) gives the transverse-momentum resummation function
$G_{ga}^{{\rm TL}}(z;\alpha_{\rm S})$ of section 2, and we have
${\widetilde{G}}_{ga}^{{\rm
TL}\,(m)}(z;\epsilon=0,{\widetilde{\lambda}})=G_{ga}^{{\rm TL}\,(m)}(z)$.
Setting $\epsilon=0$ in eqs. (149) and (150), we find agreement with the
results for $G_{ga}^{{\rm TL}\,(1)}(z)$ in eqs. (10) and (11). The computation
of the TL azimuthal-correlation functions $G_{ga}^{{\rm TL}\,(2)}(z)$ at
${\cal O}(\alpha_{\rm S}^{2})$ is discussed in the following subsection.

###  6.3 TL azimuthal correlations at ${\cal O}(\alpha_{\rm S}^{2})$

We briefly describe the calculation at ${\cal O}(\alpha_{\rm S}^{2})$ of the
azimuthal-correlation components ${\cal F}_{ga,\,{\rm corr.}}^{\rm TL}$ and
$F_{ga,\,{\rm corr.}}^{\rm TL}$ of the differential and TMD collinear
functions in the TL collinear region.

At ${\cal O}(\alpha_{\rm S}^{2})$ the differential collinear function ${\cal
F}_{ga,\,{\rm corr.}}^{\rm TL}(p,k;n)$ receives the two contributions, ${\cal
F}_{ga,\,{\rm corr.}}^{{\rm TL}\,(2R)}$ and ${\cal F}_{ga,\,{\rm corr.}}^{{\rm
TL}\,(1R1V)}$, in eq. (121).

The term ${\cal F}_{ga,\,{\rm corr.}}^{{\rm TL}\,(2R)}$ is obtained by
inserting the tree-level collinear kernel ${\cal P}_{{g}\rightarrow
a_{1}a_{2}a}^{(0)}(k_{1},k_{2},p;n)$ in eq. (3.2) and performing the
integration over $k_{1}$ and $k_{2}$ at fixed momentum $k=k_{1}+k_{2}$. This
integration procedure is completely similar to that involved in the SL
collinear region and, therefore, by direct comparison of eqs. (3.2) and (28)
we obtain

${\cal F}_{ga,\,{\rm corr.}}^{{\rm
TL}\,(2R)}(p,k;n)=\frac{\mathcal{N}_{a}(\epsilon)}{\mathcal{N}_{g}(\epsilon)}\;\left[{\cal
F}_{ga,\,{\rm corr.}}^{(2R)}(p,k;n)\right]_{p\rightarrow-p},$ (151)

where ${\cal F}_{ga,\,{\rm corr.}}^{(2R)}(p,k;n)$ ($a=g,q,{\bar{q}}$) are the
contributions in eqs. (105)–(114) to the SL collinear functions ${\cal
F}_{ga,\,{\rm corr.}}$. The SL-TL crossing relation in eq. (151) is analogous
to the corresponding relation at ${\cal O}(\alpha_{\rm S})$ (see eq. (122) and
accompanying comments).

The term ${\cal F}_{ga,\,{\rm corr.}}^{{\rm TL}\,(1R1V)}(p,k;n)$ is directly
proportional (see eqs. (3.2) and (26)) to the azimuthal-correlation component
of the one-loop collinear kernel ${\cal P}_{{g}\rightarrow
a_{1}a}^{(1)\,\mu\nu}(k,p;n)$. This kernel can be evaluated by squaring the
corresponding collinear-factorization results Bern:1998sc ; Bern:1999ry ;
Kosower:1999rx ; Sborlini:2013jba  at the amplitude level. We obtain the
following explicit expressions:

$\displaystyle\mathcal{F}^{{\rm TL}\,(1R1V)}_{gg,\,{\rm corr.}}(p,k;n)=$
$\displaystyle-\left(\frac{\alpha^{u}_{s}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\right)^{2}\frac{e^{2\epsilon\gamma_{E}}{\rm\Gamma}(1+\epsilon){\rm\Gamma^{2}(1-\epsilon)}}{\pi^{1-\epsilon}{\rm\Gamma(1-2\epsilon)}}\frac{C_{A}\,\delta_{+}(k^{2})}{(2pk)^{1+\epsilon}}\bigg{\\{}\\!\\!-\frac{N_{f}}{6}+\frac{C_{A}}{6}$
(152) $\displaystyle+C_{A}\,z_{\rm TL}(1-z_{\rm
TL})\left[-\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\ln\left(z_{\rm
TL}(1-z_{\rm
TL})\right)-\frac{\pi^{2}}{6}-\frac{1}{2}\ln^{2}\Big{(}\frac{1-z_{\rm
TL}}{z_{\rm TL}}\Big{)}\right]\\!\\!\bigg{\\}}\cos(\pi\epsilon)+{\cal
O}(\epsilon),$ $\displaystyle\mathcal{F}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm
corr.}}(p,k;n)=$
$\displaystyle\left(\frac{\alpha^{u}_{s}\mu_{0}^{2\epsilon}S_{\epsilon}}{\pi}\right)^{2}\frac{e^{2\epsilon\gamma_{E}}{\rm\Gamma}(1+\epsilon){\rm\Gamma^{2}(1-\epsilon)}}{\pi^{1-\epsilon}{\rm\Gamma(1-2\epsilon)}}\frac{T_{R}\,\delta_{+}(k^{2})}{(2pk)^{1+\epsilon}}\,\frac{z_{\rm
TL}(1-z_{\rm TL})}{1-\epsilon}$
$\displaystyle\times\bigg{\\{}C_{A}\left[\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\left(\frac{11}{3}+\ln\left(z_{\rm
TL}(1-z_{\rm
TL})\right)\right)+\frac{76}{9}-\frac{\pi^{2}}{6}-\frac{1}{2}\ln^{2}\Big{(}\frac{1-z_{\rm
TL}}{z_{\rm TL}}\Big{)}\right]$
$\displaystyle+C_{F}\left[-\frac{2}{\epsilon^{2}}-\frac{3}{\epsilon}-8\right]+N_{f}\left[-\frac{2}{3\epsilon}-\frac{10}{9}\right]\bigg{\\}}\cos(\pi\epsilon)+{\cal
O}(\epsilon),\quad\quad(a=q,{\bar{q}}),$ (153)

where the momentum fraction $z_{\rm TL}$ is

$z_{\rm TL}=\frac{\bar{n}p}{\bar{n}(p+k)},$ (154)

and ${\bar{n}}^{\mu}$ is the light-like auxiliary vector in eq. (104).
Similarly to the SL results in eqs. (101) and (5.1), we have neglected
contributions to $\mathcal{F}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm corr.}}(p,k;n)$
with a harmless dependence on $n^{2}$. We note that the SL expressions in eqs.
(101) and (5.1) and the corresponding TL expressions in eqs. (152) and (6.3)
are not related by a crossing relation similar to eqs. (122) and (151). This
is due to the fact that the corresponding SL and TL one-loop kernels ${\cal
P}_{{g}\rightarrow a_{1}a}^{(1)}(k,p;n)$ cannot be directly related by the
replacement $p\leftrightarrow-p$ Catani:2011st , as recalled at the beginning
of section 5.1.

As discussed at the end of sections 3.2 and 6.1, setting $n^{2}=0$,
$s=(p+k)^{2}$ and $\mathbf{p_{\perp}}=z\mathbf{k_{T}}$, the collinear function
$\mathcal{F}^{{\rm TL}}_{ca}(p,k;n)$ is directly related to the SCET
fragmenting jet function ${\cal G}_{ca}(z,s,\mathbf{p_{\perp}})$ Jain:2011iu .
At ${\cal O}(\alpha_{\rm S}^{2})$ we have ${\cal
G}_{ca}^{(2)}(z,s,\mathbf{p_{\perp}})=\mathcal{F}^{{\rm
TL}\,(1R1V)}_{ca}(p,k;n)+\mathcal{F}^{{\rm TL}\,(2R)}_{ca}(p,k;n)$, where
${\cal G}_{ca}^{(2)}$ is the second-order contribution to ${\cal G}_{ca}$. The
expressions in eqs. (151)–(6.3) give the explicit result for the azimuthal-
correlation component of ${\cal G}_{ca}$.

The azimuthal-correlation component of the TL TMD function at ${\cal
O}(\alpha_{\rm S}^{2})$ can be evaluated by integrating over $k$ (see eq.
(3.3)) the results for $\mathcal{F}^{{\rm TL}\,(2R)}_{ga,\,{\rm
corr.}}(p,k;n)$ and $\mathcal{F}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm corr.}}(p,k;n)$
that we have just presented in eqs. (151)–(6.3). The integration procedure is
completely similar to that of section 5.2 for the case of the SL collinear
function. In particular, we recall that we use a time-like auxiliary vector
$n^{\mu}$ to avoid rapidity divergences. Using eqs. (151)–(6.3) we compute the
corresponding contributions ${F}^{{\rm TL}\,(2R)}_{ga,\,{\rm corr.}}$ and
${F}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm corr.}}$ to the TMD function in
$\mathbf{q_{T}}$ space, and then the $\mathbf{b}$-space terms
${\widetilde{F}}^{{\rm TL}\,(2R)}_{ga,\,{\rm corr.}}$ and
${\widetilde{F}}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm corr.}}$ in eq. (134).

The expression of ${\widetilde{F}}^{{\rm TL}}_{ga,\,{\rm corr.}}$ at ${\cal
O}(\alpha_{\rm S}^{2})$ has $\epsilon$-pole divergences of UV and IR origins.
The UV divergences are removed by renormalizing the bare coupling at the scale
$\mu_{R}^{2}=b_{0}^{2}/\mathbf{b}^{2}$. The IR divergences are treated by
expanding the IR factorization formula in eq. (139) up to ${\cal
O}(\alpha_{\rm S}^{2})$, and we have

$\displaystyle{\widetilde{F}}^{\rm TL}_{ga,\,{\rm
corr.}}(z;\mathbf{b}^{2}/b_{0}^{2},{\widetilde{\lambda}})\\!$
$\displaystyle=\\!\frac{\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2})}{\pi}{\widetilde{G}}^{{\rm
TL}\,(1)}_{ga}(z;\epsilon,{\widetilde{\lambda}})\\!+\\!\left(\frac{\alpha_{\rm
S}(b_{0}^{2}/\mathbf{b}^{2})}{\pi}\right)^{2}\\!\Bigl{[}Z_{g}^{{\rm
TL}\,(1)}({\widetilde{\lambda}})\;{\widetilde{G}}^{{\rm
TL}\,(1)}_{ga}(z;\epsilon,{\widetilde{\lambda}})\Bigr{.}$
$\displaystyle-\Bigl{.}\frac{1}{\epsilon}\sum_{b}\int_{z}^{1}\\!\frac{dx}{x^{1-2\epsilon}}{\widetilde{G}}^{{\rm
TL}\,(1)}_{gb}(z/x;\epsilon,{\widetilde{\lambda}})P^{{\rm
TL}\,(1)}_{ba}(x)\\!+\\!{\widetilde{G}}^{{\rm
TL}\,(2)}_{ga}(z;\epsilon,{\widetilde{\lambda}})\Bigr{]}\\!+\\!{\cal
O}(\alpha_{\rm S}^{3}),$ (155)

where ${\widetilde{\lambda}}=n^{2}(b_{0}^{2}/\mathbf{b}^{2})/(2np/z)^{2}$. The
first-order functions $P^{{\rm TL}\,(1)}_{ba},\,Z_{g}^{{\rm TL}\,(1)}$ and
${\widetilde{G}}^{{\rm TL}\,(1)}_{ga}$ are given in eqs. (142), (144), (149)
and (150). Therefore, using eq. (6.3) and our results for
${\widetilde{F}}^{{\rm TL}\,(2R)}_{ga,\,{\rm corr.}}$ and
${\widetilde{F}}^{{\rm TL}\,(1R1V)}_{ga,\,{\rm corr.}}$, we obtain the
explicit expressions of the functions ${\widetilde{G}}^{{\rm
TL}\,(2)}_{ga}(z;\epsilon,{\widetilde{\lambda}})$ ($a=g,q,{\bar{q}}$) up to
${\cal O}(\epsilon^{0})$. We find that ${\widetilde{G}}^{{\rm TL}\,(2)}_{ga}$
are finite and independent of ${\widetilde{\lambda}}$ in the limit
$\epsilon\rightarrow 0$, namely ${\widetilde{G}}^{{\rm
TL}\,(2)}_{ga}(z;\epsilon=0,{\widetilde{\lambda}})={G}^{{\rm
TL}\,(2)}_{ga}(z)$. Our results for the TL azimuthal-correlation functions
${G}^{{\rm TL}\,(2)}_{ga}(z)$ are

$\displaystyle G^{{\rm TL}\,(2)}_{gg}(z)=$ $\displaystyle
C_{F}N_{f}\left\\{\frac{1}{18z}+\frac{1}{2}+z-\frac{14z^{2}}{9}+\ln(z)\left[-\frac{1}{3z}+1+\frac{3z}{2}\right]+\frac{3z}{4}\ln^{2}(z)\right\\}$
$\displaystyle+\\!C_{A}N_{f}\left\\{\\!-\frac{1}{36z}-\frac{1}{12}-\frac{4z}{9}\\!+\\!\frac{17z^{2}}{36}-\frac{z}{6}\ln(z)\\!\right\\}\\!+\\!C_{A}^{2}\bigg{\\{}\\!-\frac{1}{36z}-\frac{5}{12}-\frac{20z}{9}\\!+\\!\frac{11z^{2}}{4}$
$\displaystyle+\ln(z)\left[\frac{1}{3z}-1-\frac{67z}{12}+z^{2}\right]+z(1-z)\bigg{[}\ln(z)\ln(1-z)-\text{Li}_{2}(1-z)\bigg{]}$
$\displaystyle-\ln^{2}(z)\left[3z-\frac{z^{2}}{2}\right]\bigg{\\}}-{h}_{g}^{{\rm
TL}\,(1)}\,C_{A}\,z(1-z),$ (156) $\displaystyle G^{{\rm TL}\,(2)}_{gq}(z)=$
$\displaystyle
C_{F}\bigg{\\{}-\frac{1}{8}+\frac{3z}{4}-\frac{5z^{2}}{8}-\ln(z)\left[\frac{1}{4}+\frac{3z}{8}-\frac{z^{2}}{4}\right]+\ln^{2}(z)\left[\frac{3z}{8}-\frac{3z^{2}}{4}\right]$
$\displaystyle+z(1-z)\bigg{[}\frac{1}{4}\ln(1-z)-\frac{3}{2}\ln(z)\ln(1-z)+\frac{1}{4}\ln^{2}(1-z)-\text{Li}_{2}(z)-\frac{\pi^{2}}{12}\bigg{]}\bigg{\\}}$
$\displaystyle+N_{f}\left\\{z(1-z)\left[\frac{1}{9}-\frac{1}{6}\ln(z)+\frac{1}{6}\ln(1-z)\right]\right\\}+C_{A}\bigg{\\{}\ln(z)\left[\frac{1}{4}+\frac{13z}{6}-\frac{17z^{2}}{12}\right]$
$\displaystyle+\ln^{2}(z)\left[\frac{3z}{4}+\frac{z^{2}}{2}\right]+z(1-z)\bigg{[}-\frac{25}{36}-\frac{5}{12}\ln(1-z)-\frac{1}{2}\text{Li}_{2}(1-z)$
$\displaystyle-\frac{1}{4}\ln^{2}(1-z)+\frac{\pi^{2}}{4}\bigg{]}\bigg{\\}}+{h}_{g}^{{\rm
TL}\,(1)}\;\frac{1}{2}\,z(1-z),$ (157)

and $G^{{\rm TL}\,(2)}_{g{\bar{q}}}(z)=G^{{\rm TL}\,(2)}_{gq}(z)$. The
coefficient ${h}_{g}^{{\rm TL}\,(1)}$ in eqs. (156) and (157) parametrizes the
resummation-scheme dependence, and we recall that ${h}_{g}^{{\rm TL}\,(1)}=0$
in the hard scheme (see eqs. (144) and (146)).

The transverse-momentum resummation functions ${G}^{{\rm TL}\,(2)}_{ga}(z)$
were first computed in ref. Luo:2019bmw  (the hard scheme is used therein), by
using a SCET framework with properly regularized rapidity divergences. We have
presented the first recomputation of ${G}^{{\rm TL}\,(2)}_{ga}(z)$, and we
have used a fully independent method. Our results in eqs. (156) and (157)
agree with those in ref. Luo:2019bmw .

##  7 Summary

QCD squared amplitudes are singular in the multiparton collinear limit, and
the singular behaviour is controlled by the splitting kernels of a
factorization formula that has a process-independent structure.

In this paper we have exploited the collinear factorization of QCD amplitudes
to introduce collinear functions that contribute to QCD resummation formulae
for hard-scattering cross sections. The collinear functions are defined
through the integration of the multiparton splitting kernels over a
constrained phase space that depends on the hard-scattering observable of
interest. Considering different phase-space constraints, one can define
different collinear functions, which embody the logarithmically-enhanced
contributions of collinear origin to the corresponding hard-scattering
observables. In this paper we have explicitly considered differential
collinear functions that can in turn be used to evaluate TMD functions for
transverse-momentum resummation and beam functions for $N$-jettiness
resummation.

A distinctive and relevant feature of our collinear functions is their
dependence on an auxiliary vector $n^{\mu}$, which directly follows from the
corresponding $n$ dependence of the splitting kernels. In applications of
collinear factorization of QCD amplitudes, the auxiliary vector is usually
chosen to be light-like. We use both light-like and time-like auxiliary
vectors. In the paper we have discussed how the $n$ dependence controls the
behaviour of the splitting kernels in kinematical regions that are far from
the collinear region. In particular, in the case of TMD functions we have
shown that the use of a time-like vector $n^{\mu}$ avoids the rapidity
divergences that are instead present if $n^{\mu}$ is light-like.

The collinear functions can be introduced for the cases of both final-state
fragmenting partons and initial-state colliding partons. The final-state and
initial-state collinear functions use the splitting kernels in the
corresponding TL and SL collinear regions, respectively. The TL splitting
kernels and, consequently, the final-state collinear functions are process
independent. In contrast, the initial-state collinear functions are, in
general, process dependent beyond ${\cal O}(\alpha_{\rm S}^{2})$ (although
they have a process-independent structure). Such process dependence is a
consequence of the violation of strict collinear factorization of the QCD
squared amplitudes in the SL collinear regions.

As discussed in the paper, our TMD and beam functions with a light-like
auxiliary vector $n^{\mu}$ can be related to the analogous SCET functions in
the literature. In the TL region our collinear functions are equivalent to the
parton level SCET functions. In the SL region the same equivalence is limited
up to ${\cal O}(\alpha_{\rm S}^{2})$, since at higher perturbative orders our
collinear functions are process dependent.

We have discussed the perturbative computation of the collinear functions.
Such computation leads to UV and IR divergences, which we have regularized by
the customary procedure of analytic continuation in $d=4-2\epsilon$ space-time
dimensions. The $\epsilon$-pole divergences of UV origin are removed by the
renormalization of the QCD coupling $\alpha_{\rm S}$. The IR divergences can
instead be factorized from IR finite collinear terms that directly contribute
to QCD resummation formulae of hard-scattering observables. At the cross
section level the IR divergences of the collinear functions are partly removed
by the ‘renormalization’ of the bare parton densities and fragmentation
functions, and the remaining part is then cancelled by the IR terms due to the
soft and purely-virtual contributions to the hard-scattering observable.

We have illustrated the perturbative features of the collinear functions by
performing their calculation at ${\cal O}(\alpha_{\rm S})$ and discussing the
explicit dependence on the auxiliary vector $n^{\mu}$.

In the case of TMD observables, the collinear functions have both azimuthal-
independent and azimuthal-correlation components. The azimuthal-correlation
component is specific of the gluon partonic channels and is also known as the
contribution of linearly-polarized gluons. We have presented the calculation
at ${\cal O}(\alpha_{\rm S}^{2})$ of the azimuthal-correlation component of
the differential and TMD collinear functions. Performing UV renormalization
and factorization of the IR divergences, we have computed the ${\cal
O}(\alpha_{\rm S}^{2})$ contribution of linearly-polarized gluons to
transverse-momentum resummation. Our result for both the SL and TL regions
agree with the results obtained by other authors using SCET functions and
related theoretical methods.

The computation at ${\cal O}(\alpha_{\rm S}^{2})$ of the azimuthal-independent
component of the collinear functions will be presented in future work, where
we shall also discuss the related computation of the $n$-dependent soft factor
for transverse-momentum resummation. We also plan to study collinear functions
for other hard-scattering observables.

Certainly, an important future step can be the explicit extension of our
theoretical framework to ${\cal O}(\alpha_{\rm S}^{3})$ and higher
perturbative orders. In particular, the explicit computation of the process
dependence of the collinear functions in the SL region is very relevant. This
computation requires the preliminary calculation at ${\cal O}(\alpha_{\rm
S}^{3})$ of the splitting kernels for the SL collinear limit of the QCD
scattering amplitudes. At present, such SL splitting kernels are not known at
the required order in the $\epsilon$ expansion.

## Acknowledgement

We would like to thank Roberto Bonciani, Leandro Cieri, Giancarlo Ferrera,
Massimiliano Grazzini, and Germán Rodrigo for useful discussions.

## References

  * (1) G. Luisoni and S. Marzani, _QCD resummation for hadronic final states_ , _J. Phys. G_ 42 (2015) 103101 [1505.04084].
  * (2) T. Becher, A. Broggio and A. Ferroglia, _Introduction to Soft-Collinear Effective Theory_ , vol. 896, Springer (2015), 10.1007/978-3-319-14848-9, [1410.1892].
  * (3) J.C. Collins, D.E. Soper and G.F. Sterman, _Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production_ , _Nucl. Phys. B_ 250 (1985) 199.
  * (4) I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, _N-Jettiness: An Inclusive Event Shape to Veto Jets_ , _Phys. Rev. Lett._ 105 (2010) 092002 [1004.2489].
  * (5) Z. Bern, G. Chalmers, L.J. Dixon and D.A. Kosower, _One loop N gluon amplitudes with maximal helicity violation via collinear limits_ , _Phys. Rev. Lett._ 72 (1994) 2134 [hep-ph/9312333].
  * (6) Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, _One loop n point gauge theory amplitudes, unitarity and collinear limits_ , _Nucl. Phys. B_ 425 (1994) 217 [hep-ph/9403226].
  * (7) Z. Bern and G. Chalmers, _Factorization in one loop gauge theory_ , _Nucl. Phys. B_ 447 (1995) 465 [hep-ph/9503236].
  * (8) D.A. Kosower, _All order collinear behavior in gauge theories_ , _Nucl. Phys. B_ 552 (1999) 319 [hep-ph/9901201].
  * (9) Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, _The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order_ , _Phys. Rev. D_ 60 (1999) 116001 [hep-ph/9903516].
  * (10) S. Catani and M. Grazzini, _Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond_ , _Nucl. Phys. B_ 570 (2000) 287 [hep-ph/9908523].
  * (11) S. Catani, D. de Florian and G. Rodrigo, _The Triple collinear limit of one loop QCD amplitudes_ , _Phys. Lett. B_ 586 (2004) 323 [hep-ph/0312067].
  * (12) S. Catani, D. de Florian and G. Rodrigo, _Space-like (versus time-like) collinear limits in QCD: Is factorization violated?_ , _JHEP_ 07 (2012) 026 [1112.4405].
  * (13) J. Collins, _Rapidity divergences and valid definitions of parton densities_ , _PoS_ LC2008 (2008) 028 [0808.2665].
  * (14) J. Collins, _Foundations of perturbative QCD_ , vol. 32, Cambridge University Press (11, 2013).
  * (15) T. Becher and M. Neubert, _Drell-Yan Production at Small $q_{T}$, Transverse Parton Distributions and the Collinear Anomaly_, _Eur. Phys. J. C_ 71 (2011) 1665 [1007.4005].
  * (16) M.G. Echevarria, A. Idilbi and I. Scimemi, _Factorization Theorem For Drell-Yan At Low $q_{T}$ And Transverse Momentum Distributions On-The-Light-Cone_, _JHEP_ 07 (2012) 002 [1111.4996].
  * (17) J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, _A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory_ , _JHEP_ 05 (2012) 084 [1202.0814].
  * (18) C.W. Bauer, S. Fleming and M.E. Luke, _Summing Sudakov logarithms in $B\rightarrow X_{s}\gamma$in effective field theory._, _Phys. Rev. D_ 63 (2000) 014006 [hep-ph/0005275].
  * (19) C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, _An Effective field theory for collinear and soft gluons: Heavy to light decays_ , _Phys. Rev. D_ 63 (2001) 114020 [hep-ph/0011336].
  * (20) C.W. Bauer, D. Pirjol and I.W. Stewart, _Soft collinear factorization in effective field theory_ , _Phys. Rev. D_ 65 (2002) 054022 [hep-ph/0109045].
  * (21) C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, _Hard scattering factorization from effective field theory_ , _Phys. Rev. D_ 66 (2002) 014017 [hep-ph/0202088].
  * (22) M. Ritzmann and W.J. Waalewijn, _Fragmentation in Jets at NNLO_ , _Phys. Rev. D_ 90 (2014) 054029 [1407.3272].
  * (23) M.-X. Luo, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, _Transverse Parton Distribution and Fragmentation Functions at NNLO: the Gluon Case_ , _JHEP_ 01 (2020) 040 [1909.13820].
  * (24) D. Gutierrez-Reyes, S. Leal-Gomez, I. Scimemi and A. Vladimirov, _Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution_ , _JHEP_ 11 (2019) 121 [1907.03780].
  * (25) S. Catani, D. de Florian and M. Grazzini, _Universality of nonleading logarithmic contributions in transverse momentum distributions_ , _Nucl. Phys. B_ 596 (2001) 299 [hep-ph/0008184].
  * (26) S. Catani and M. Grazzini, _QCD transverse-momentum resummation in gluon fusion processes_ , _Nucl. Phys. B_ 845 (2011) 297 [1011.3918].
  * (27) S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, _Universality of transverse-momentum resummation and hard factors at the NNLO_ , _Nucl. Phys. B_ 881 (2014) 414 [1311.1654].
  * (28) S. Catani, M. Grazzini and H. Sargsyan, _Azimuthal asymmetries in QCD hard scattering: infrared safe but divergent_ , _JHEP_ 06 (2017) 017 [1703.08468].
  * (29) P.J. Mulders and J. Rodrigues, _Transverse momentum dependence in gluon distribution and fragmentation functions_ , _Phys. Rev. D_ 63 (2001) 094021 [hep-ph/0009343].
  * (30) D. de Florian and M. Grazzini, _The Structure of large logarithmic corrections at small transverse momentum in hadronic collisions_ , _Nucl. Phys. B_ 616 (2001) 247 [hep-ph/0108273].
  * (31) S. Catani and M. Grazzini, _Higgs Boson Production at Hadron Colliders: Hard-Collinear Coefficients at the NNLO_ , _Eur. Phys. J. C_ 72 (2012) 2013 [1106.4652].
  * (32) S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, _Vector boson production at hadron colliders: hard-collinear coefficients at the NNLO_ , _Eur. Phys. J. C_ 72 (2012) 2195 [1209.0158].
  * (33) T. Gehrmann, T. Lubbert and L.L. Yang, _Transverse parton distribution functions at next-to-next-to-leading order: the quark-to-quark case_ , _Phys. Rev. Lett._ 109 (2012) 242003 [1209.0682].
  * (34) T. Gehrmann, T. Luebbert and L.L. Yang, _Calculation of the transverse parton distribution functions at next-to-next-to-leading order_ , _JHEP_ 06 (2014) 155 [1403.6451].
  * (35) M.G. Echevarria, I. Scimemi and A. Vladimirov, _Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order_ , _JHEP_ 09 (2016) 004 [1604.07869].
  * (36) M.-X. Luo, X. Wang, X. Xu, L.L. Yang, T.-Z. Yang and H.X. Zhu, _Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case_ , _JHEP_ 10 (2019) 083 [1908.03831].
  * (37) M.-x. Luo, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, _Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order_ , _Phys. Rev. Lett._ 124 (2020) 092001 [1912.05778].
  * (38) M.A. Ebert, B. Mistlberger and G. Vita, _Transverse momentum dependent PDFs at N 3LO_, _JHEP_ 09 (2020) 146 [2006.05329].
  * (39) M.-x. Luo, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, _Unpolarized quark and gluon TMD PDFs and FFs at N 3LO_, _JHEP_ 06 (2021) 115 [2012.03256].
  * (40) J.C. Collins and D.E. Soper, _Back-To-Back Jets in QCD_ , _Nucl. Phys. B_ 193 (1981) 381.
  * (41) P.M. Nadolsky, D.R. Stump and C.P. Yuan, _Semiinclusive hadron production at HERA: The Effect of QCD gluon resummation_ , _Phys. Rev. D_ 61 (2000) 014003 [hep-ph/9906280].
  * (42) X.-d. Ji, J.-p. Ma and F. Yuan, _QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum_ , _Phys. Rev. D_ 71 (2005) 034005 [hep-ph/0404183].
  * (43) M.A. Ebert, B. Mistlberger and G. Vita, _TMD fragmentation functions at N 3LO_, _JHEP_ 07 (2021) 121 [2012.07853].
  * (44) P. Sun, B. Yan, C.P. Yuan and F. Yuan, _Resummation of High Order Corrections in $Z$ Boson Plus Jet Production at the LHC_, _Phys. Rev. D_ 100 (2019) 054032 [1810.03804].
  * (45) P. Sun, J. Isaacson, C.P. Yuan and F. Yuan, _Resummation of High Order Corrections in Higgs Boson Plus Jet Production at the LHC_ , _Phys. Lett. B_ 769 (2017) 57 [1602.08133].
  * (46) Y.-T. Chien, D.Y. Shao and B. Wu, _Resummation of Boson-Jet Correlation at Hadron Colliders_ , _JHEP_ 11 (2019) 025 [1905.01335].
  * (47) Y. Hatta, B.-W. Xiao, F. Yuan and J. Zhou, _Azimuthal angular asymmetry of soft gluon radiation in jet production_ , _Phys. Rev. D_ 104 (2021) 054037 [2106.05307].
  * (48) L. Buonocore, M. Grazzini, J. Haag and L. Rottoli, _Transverse-momentum resummation for boson plus jet production at hadron colliders_ , _Eur. Phys. J. C_ 82 (2022) 27 [2110.06913].
  * (49) Y.-T. Chien, R. Rahn, D.Y. Shao, W.J. Waalewijn and B. Wu, _Precision boson-jet azimuthal decorrelation at hadron colliders_ , _JHEP_ 02 (2023) 256 [2205.05104].
  * (50) H.T. Li, C.S. Li, D.Y. Shao, L.L. Yang and H.X. Zhu, _Top quark pair production at small transverse momentum in hadronic collisions_ , _Phys. Rev. D_ 88 (2013) 074004 [1307.2464].
  * (51) S. Catani, M. Grazzini and A. Torre, _Transverse-momentum resummation for heavy-quark hadroproduction_ , _Nucl. Phys. B_ 890 (2014) 518 [1408.4564].
  * (52) S. Catani, I. Fabre, M. Grazzini and S. Kallweit, _${t{{\bar{t}}}H}$ production at NNLO: the flavour off-diagonal channels_, _Eur. Phys. J. C_ 81 (2021) 491 [2102.03256].
  * (53) R. Angeles-Martinez, M. Czakon and S. Sapeta, _NNLO soft function for top quark pair production at small transverse momentum_ , _JHEP_ 10 (2018) 201 [1809.01459].
  * (54) S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli and H. Sargsyan, _Top-quark pair hadroproduction at next-to-next-to-leading order in QCD_ , _Phys. Rev. D_ 99 (2019) 051501 [1901.04005].
  * (55) J.R. Forshaw, M.H. Seymour and A. Siodmok, _On the Breaking of Collinear Factorization in QCD_ , _JHEP_ 11 (2012) 066 [1206.6363].
  * (56) L.J. Dixon, E. Herrmann, K. Yan and H.X. Zhu, _Soft gluon emission at two loops in full color_ , _JHEP_ 05 (2020) 135 [1912.09370].
  * (57) J. Collins and J.-W. Qiu, _$k_{T}$ factorization is violated in production of high-transverse-momentum particles in hadron-hadron collisions_, _Phys. Rev. D_ 75 (2007) 114014 [0705.2141].
  * (58) T.C. Rogers and P.J. Mulders, _No Generalized TMD-Factorization in Hadro-Production of High Transverse Momentum Hadrons_ , _Phys. Rev. D_ 81 (2010) 094006 [1001.2977].
  * (59) I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, _Factorization at the LHC: From PDFs to Initial State Jets_ , _Phys. Rev. D_ 81 (2010) 094035 [0910.0467].
  * (60) I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, _The Quark Beam Function at NNLL_ , _JHEP_ 09 (2010) 005 [1002.2213].
  * (61) C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, _Higgs Production with a Central Jet Veto at NNLL+NNLO_ , _JHEP_ 04 (2011) 092 [1012.4480].
  * (62) J.R. Gaunt, M. Stahlhofen and F.J. Tackmann, _The Quark Beam Function at Two Loops_ , _JHEP_ 04 (2014) 113 [1401.5478].
  * (63) J. Gaunt, M. Stahlhofen and F.J. Tackmann, _The Gluon Beam Function at Two Loops_ , _JHEP_ 08 (2014) 020 [1405.1044].
  * (64) K. Melnikov, R. Rietkerk, L. Tancredi and C. Wever, _Double-real contribution to the quark beam function at N 3LO QCD_, _JHEP_ 02 (2019) 159 [1809.06300].
  * (65) K. Melnikov, R. Rietkerk, L. Tancredi and C. Wever, _Triple-real contribution to the quark beam function in QCD at next-to-next-to-next-to-leading order_ , _JHEP_ 06 (2019) 033 [1904.02433].
  * (66) A. Behring, K. Melnikov, R. Rietkerk, L. Tancredi and C. Wever, _Quark beam function at next-to-next-to-next-to-leading order in perturbative QCD in the generalized large- $N_{c}$ approximation_, _Phys. Rev. D_ 100 (2019) 114034 [1910.10059].
  * (67) D. Baranowski, _NNLO zero-jettiness beam and soft functions to higher orders in the dimensional-regularization parameter $\epsilon$_, _Eur. Phys. J. C_ 80 (2020) 523 [2004.03285].
  * (68) M.A. Ebert, B. Mistlberger and G. Vita, _$N$ -jettiness beam functions at N3LO_, _JHEP_ 09 (2020) 143 [2006.03056].
  * (69) M. Procura and I.W. Stewart, _Quark Fragmentation within an Identified Jet_ , _Phys. Rev. D_ 81 (2010) 074009 [0911.4980].
  * (70) C.G. Bollini and J.J. Giambiagi, _Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter_ , _Nuovo Cim. B_ 12 (1972) 20.
  * (71) J.F. Ashmore, _A Method of Gauge Invariant Regularization_ , _Lett. Nuovo Cim._ 4 (1972) 289.
  * (72) G.M. Cicuta and E. Montaldi, _Analytic renormalization via continuous space dimension_ , _Lett. Nuovo Cim._ 4 (1972) 329.
  * (73) R. Gastmans and R. Meuldermans, _Dimensional regularization of the infrared problem_ , _Nucl. Phys. B_ 63 (1973) 277.
  * (74) S. Catani and M.H. Seymour, _A General algorithm for calculating jet cross-sections in NLO QCD_ , _Nucl. Phys. B_ 485 (1997) 291 [hep-ph/9605323].
  * (75) J.M. Campbell and E.W.N. Glover, _Double unresolved approximations to multiparton scattering amplitudes_ , _Nucl. Phys. B_ 527 (1998) 264 [hep-ph/9710255].
  * (76) S. Catani and M. Grazzini, _Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations_ , _Phys. Lett. B_ 446 (1999) 143 [hep-ph/9810389].
  * (77) Z. Bern, V. Del Duca and C.R. Schmidt, _The Infrared behavior of one loop gluon amplitudes at next-to-next-to-leading order_ , _Phys. Lett. B_ 445 (1998) 168 [hep-ph/9810409].
  * (78) D.A. Kosower and P. Uwer, _One loop splitting amplitudes in gauge theory_ , _Nucl. Phys. B_ 563 (1999) 477 [hep-ph/9903515].
  * (79) G.F.R. Sborlini, D. de Florian and G. Rodrigo, _Double collinear splitting amplitudes at next-to-leading order_ , _JHEP_ 01 (2014) 018 [1310.6841].
  * (80) V. Del Duca, A. Frizzo and F. Maltoni, _Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit_ , _Nucl. Phys. B_ 568 (2000) 211 [hep-ph/9909464].
  * (81) T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, _Multi-gluon collinear limits from MHV diagrams_ , _JHEP_ 05 (2005) 013 [hep-ph/0503063].
  * (82) T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, _Collinear limits in QCD from MHV rules_ , _JHEP_ 07 (2005) 068 [hep-ph/0505219].
  * (83) V. Del Duca, C. Duhr, R. Haindl, A. Lazopoulos and M. Michel, _Tree-level splitting amplitudes for a quark into four collinear partons_ , _JHEP_ 02 (2020) 189 [1912.06425].
  * (84) V. Del Duca, C. Duhr, R. Haindl, A. Lazopoulos and M. Michel, _Tree-level splitting amplitudes for a gluon into four collinear partons_ , _JHEP_ 10 (2020) 093 [2007.05345].
  * (85) G.F.R. Sborlini, D. de Florian and G. Rodrigo, _Triple collinear splitting functions at NLO for scattering processes with photons_ , _JHEP_ 10 (2014) 161 [1408.4821].
  * (86) G.F.R. Sborlini, D. de Florian and G. Rodrigo, _Polarized triple-collinear splitting functions at NLO for processes with photons_ , _JHEP_ 03 (2015) 021 [1409.6137].
  * (87) S. Badger, F. Buciuni and T. Peraro, _One-loop triple collinear splitting amplitudes in QCD_ , _JHEP_ 09 (2015) 188 [1507.05070].
  * (88) M. Czakon and S. Sapeta, _Complete collection of one-loop triple-collinear splitting operators for dimensionally-regulated QCD_ , _JHEP_ 07 (2022) 052 [2204.11801].
  * (89) Z. Bern, L.J. Dixon and D.A. Kosower, _Two-loop g — $>$ gg splitting amplitudes in QCD_, _JHEP_ 08 (2004) 012 [hep-ph/0404293].
  * (90) S.D. Badger and E.W.N. Glover, _Two loop splitting functions in QCD_ , _JHEP_ 07 (2004) 040 [hep-ph/0405236].
  * (91) C. Duhr, T. Gehrmann and M. Jaquier, _Two-loop splitting amplitudes and the single-real contribution to inclusive Higgs production at N 3LO_, _JHEP_ 02 (2015) 077 [1411.3587].
  * (92) A. Jain, M. Procura and W.J. Waalewijn, _Fully-Unintegrated Parton Distribution and Fragmentation Functions at Perturbative $k_{T}$_, _JHEP_ 04 (2012) 132 [1110.0839].
  * (93) S. Mantry and F. Petriello, _Factorization and Resummation of Higgs Boson Differential Distributions in Soft-Collinear Effective Theory_ , _Phys. Rev. D_ 81 (2010) 093007 [0911.4135].
  * (94) S. Mantry and F. Petriello, _Transverse Momentum Distributions from Effective Field Theory with Numerical Results_ , _Phys. Rev. D_ 83 (2011) 053007 [1007.3773].
  * (95) J.R. Gaunt and M. Stahlhofen, _The Fully-Differential Quark Beam Function at NNLO_ , _JHEP_ 12 (2014) 146 [1409.8281].
  * (96) J.R. Gaunt and M. Stahlhofen, _The fully-differential gluon beam function at NNLO_ , _JHEP_ 07 (2020) 234 [2004.11915].
  * (97) M.A. Ebert, B. Mistlberger and G. Vita, _Collinear expansion for color singlet cross sections_ , _JHEP_ 09 (2020) 181 [2006.03055].
  * (98) J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, _The Rapidity Renormalization Group_ , _Phys. Rev. Lett._ 108 (2012) 151601 [1104.0881].
  * (99) T. Becher and G. Bell, _Analytic Regularization in Soft-Collinear Effective Theory_ , _Phys. Lett. B_ 713 (2012) 41 [1112.3907].
  * (100) M.G. Echevarria, I. Scimemi and A. Vladimirov, _Transverse momentum dependent fragmentation function at next-to–next-to–leading order_ , _Phys. Rev. D_ 93 (2016) 011502 [1509.06392].
  * (101) Y. Li, D. Neill and H.X. Zhu, _An exponential regulator for rapidity divergences_ , _Nucl. Phys. B_ 960 (2020) 115193 [1604.00392].
  * (102) M.A. Ebert, I. Moult, I.W. Stewart, F.J. Tackmann, G. Vita and H.X. Zhu, _Subleading power rapidity divergences and power corrections for q T_, _JHEP_ 04 (2019) 123 [1812.08189].
  * (103) S. Catani and L. Trentadue, _Resummation of the QCD Perturbative Series for Hard Processes_ , _Nucl. Phys. B_ 327 (1989) 323.
  * (104) S. Catani, L. Trentadue, G. Turnock and B.R. Webber, _Resummation of large logarithms in e+ e- event shape distributions_ , _Nucl. Phys. B_ 407 (1993) 3.
  * (105) G. Somogyi, _Angular integrals in d dimensions_ , _J. Math. Phys._ 52 (2011) 083501 [1101.3557].
  * (106) V.N. Gribov and L.N. Lipatov, _Deep inelastic e p scattering in perturbation theory_ , _Sov. J. Nucl. Phys._ 15 (1972) 438\.
  * (107) V.N. Gribov and L.N. Lipatov, _e+ e- pair annihilation and deep inelastic e p scattering in perturbation theory_ , _Sov. J. Nucl. Phys._ 15 (1972) 675.