extracted from one of these two equations:

$\displaystyle m_{T\,ij}\,\cosh(y_{ij})$
$\displaystyle=u\,p_{T\,i}\,\cosh(y_{i})+(1-u)\,p_{T\,j}\,\cosh(y_{j}),$ (B.4)
$\displaystyle m_{T\,ij}\,\sinh(y_{ij})$
$\displaystyle=u\,p_{T\,i}\,\sinh(y_{i})+(1-u)\,p_{T\,j}\,\sinh(y_{j}).$ (B.5)

Furthermore, the scalar product between the four momenta $\Pi_{ij}$ and
$\hat{p}_{N}$ is

$\displaystyle\Pi_{ij}\cdot\hat{p}_{N}$
$\displaystyle=m_{T\,ij}\,\cosh(y_{ij}-y_{N})-\Pi_{T\,ij}\,\cos(\phi_{N}),$
(B.6)

where $\phi_{N}$ is the direct azimuthal angle between the vectors
$\vec{\Pi}_{T\,ij}$ and $\vec{\hat{p}}_{T\,N}$.

The soft integral, defined in eq.(3.3), becomes then

$\displaystyle J^{\text{soft}}_{ij}$ $\displaystyle=p_{i}\cdot
p_{j}\,\int_{0}^{1}du\,\int_{0}^{\pi}d\phi_{N}\,\sin(\phi_{N})^{-2\,\varepsilon}\,$
$\displaystyle\quad{}\times\int^{+\infty}_{-\infty}dy_{N}\,\frac{1}{(m_{T\,ij}\,\cosh(y_{ij}-y_{N})-\Pi_{T\,ij}\,\cos(\phi_{N}))^{2}}.$
(B.7)

A priori, this integral seems more complicated to compute than the equivalent
one using the polar angle as variable because in this case, an extra
integration has to be performed121212There is one integral over the azimuthal
angle and one over the rapidity instead of only one over the polar angle
between $\vec{p}_{N}$ and $\vec{\Pi}_{ij}$.. As we will see, this is not a
problem and the calculation proceeds in the same way as in the standard case.

To start with, let us split the range of the $y_{N}$ integration in two parts

$\displaystyle\int_{-\infty}^{+\infty}dy_{N}\,F(y_{N})$
$\displaystyle=\int_{y_{ij}}^{+\infty}dy_{N}\,F(y_{N})+\int_{-\infty}^{y_{ij}}dy_{N}\,F(y_{N}).$
(B.8)

We make the changes of variable $\Delta y=y_{N}-y_{ij}$ in the first integral
of the right hand side of eq. (B.8) and $\Delta y=y_{ij}-y_{N}$ in the second
one

$\displaystyle\int_{-\infty}^{+\infty}dy_{N}\,F(y_{N})$
$\displaystyle=\int_{0}^{+\infty}d\Delta y\,F(y_{ij}+\Delta
y)+\int^{+\infty}_{0}d\Delta y\,F(y_{ij}-\Delta y).$ (B.9)

The integrands $F(y_{ij}\pm\Delta y)$ depend on $\Delta y$ only through
$\cosh(\pm\Delta y)$ such that the two integrals are equal and the soft
integral becomes simply

$\displaystyle J^{\text{soft}}_{ij}$ $\displaystyle=2\,p_{i}\cdot
p_{j}\,\int_{0}^{1}du\,\int_{0}^{\pi}d\phi_{N}\,\sin(\phi_{N})^{-2\,\varepsilon}\,\int^{+\infty}_{0}d\Delta
y\,\frac{1}{(m_{T\,ij}\,\cosh(\Delta y)-\Pi_{T\,ij}\,\cos(\phi_{N}))^{2}}.$
(B.10)

Then, the change of variable $\cos(\phi_{N})=2\,x-1$ is performed leading to

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\int_{0}^{1}du\,\int^{+\infty}_{0}d\Delta
y\,\left(m_{T\,ij}\,\cosh(\Delta y)+\Pi_{T\,ij}\right)^{-2}\,$
$\displaystyle\quad{}\times\int_{0}^{1}dx\,x^{-1/2-\varepsilon}\,(1-x)^{-1/2-\varepsilon}\,\left(1-\frac{2\,\Pi_{T\,ij}}{m_{T\,ij}\,\cosh(\Delta
y)+\Pi_{T\,ij}}\,x\right)^{-2}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\int_{0}^{1}du\,\int^{+\infty}_{0}d\Delta
y\,\left(m_{T\,ij}\,\cosh(\Delta y)+\Pi_{T\,ij}\right)^{-2}\,$
$\displaystyle\quad{}\times\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\;{}_{2}\text{F}_{1}\\!\left(2,\frac{1}{2}-\varepsilon;1-2\,\varepsilon;z\right),$
(B.11)

where ${}_{2}\text{F}_{1}$ is the Gauss hypergeometric function (cf. ref.
[53]) and

$z=\frac{2\,\Pi_{T\,ij}}{m_{T\,ij}\,\cosh(\Delta y)+\Pi_{T\,ij}}.$ (B.12)

Using the quadratic transformation given by eq. (15.3.16) of ref. [53], yields

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\,\int^{+\infty}_{0}d\Delta
y\,\left(m_{T\,ij}\,\cosh(\Delta y)\right)^{-2}\,$
$\displaystyle\quad{}\times\;{}_{2}\text{F}_{1}\\!\left(1,\frac{3}{2};1-\varepsilon;z^{\prime\,2}\right),$
(B.13)

with

$z^{\prime}=\frac{\Pi_{T\,ij}}{m_{T\,ij}\,\cosh(\Delta y)}.$ (B.14)

The four-momentum $\Pi_{ij}$ is time like because $p_{i}\cdot p_{j}\geq 0$ and
$z^{\prime}\leq 1$, such that the hypergeometric function can be expanded in a
series131313We disregard the case where $p_{i}\cdot p_{j}=0$ which is not
relevant for NLO computations. Furthermore, we assume that $u\neq 0,1$ in such
way that $z^{\prime}<1$ to justify the expansion. Once the integration over
$\Delta y$ has been performed, the result will be a regular function of $u$
for these values.

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\left(m_{T\,ij}\right)^{-2}\,$
$\displaystyle\quad{}\times\frac{\Gamma(1-\varepsilon)}{\Gamma\left(\frac{3}{2}\right)}\,\sum_{n=0}^{\infty}\frac{\Gamma(1+n)\,\Gamma\left(\frac{3}{2}+n\right)}{\Gamma(1-\varepsilon+n)}\,\frac{1}{n!}\,\left(\frac{\Pi_{T\,ij}}{m_{T\,ij}}\right)^{2\,n}\,\int^{+\infty}_{0}d\Delta
y\,\cosh^{-2\,(n+1)}(\Delta y).$ (B.15)

In order to perform the $\Delta y$ integration term by term, let us introduce

$\displaystyle U(n)$ $\displaystyle\equiv\int^{+\infty}_{0}d\Delta
y\,\frac{1}{\cosh^{2\,(n+1)}(\Delta y)}.$ (B.16)

It is easy to get that

$\displaystyle U(0)$ $\displaystyle=\int^{+\infty}_{0}d\Delta
y\,\frac{1}{\cosh^{2}(\Delta y)}$ $\displaystyle=\left[\tanh(\Delta
y)\right]_{0}^{+\infty}$ $\displaystyle=1.$ (B.17)

Let us establish a recurrence relation for $U(n)$. In order to do that, we use
the following relation, valid for $m>1$ and easily established by an
integration by parts:

$\displaystyle\int\frac{dz}{\cosh^{m}(z)}$
$\displaystyle=\frac{1}{m-1}\,\frac{\sinh(z)}{\cosh^{m-1}(z)}+\frac{m-2}{m-1}\,\int\frac{dz}{\cosh^{m-2}(z)}\quad\text{for}\;m\neq
1.$ (B.18)

The first term on the right hand side of eq. (B.18) will always vanish when
the integration bounds are $0$ and $+\infty$ for $m>2$. But this condition is
satisfied if $n\geq 1$, thus we get that

$\displaystyle U(n)$ $\displaystyle=\frac{2\,n}{2\,n+1}\,U(n-1)$
$\displaystyle=\frac{2\,n\;2\,(n-1)}{2\,n+1\;2\,n-1}\,U(n-2)$
$\displaystyle=\frac{2\,n\;2\,(n-1)\cdots 2}{(2\,n+1)\;(2\,n-1)\cdots
3}\,U(0).$ (B.19)

Using the result of eq. (B.17) leads to

$\displaystyle U(n)$
$\displaystyle=\frac{2^{2\,n}\,(n!)^{2}}{(2\,n+1)!}=2^{2\,n}\,\frac{\Gamma^{2}(n+1)}{\Gamma(2\,(n+1))}.$
(B.20)

Applying the duplication formula for the gamma function, we then obtain

$\displaystyle U(n)$
$\displaystyle=\frac{\sqrt{\pi}}{2}\,\frac{\Gamma(n+1)}{\Gamma\left(n+\frac{3}{2}\right)}.$
(B.21)

Putting the result of eq. (B.21) into eq. (B.15) yields

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\left(m_{T\,ij}\right)^{-2}\,$
$\displaystyle\quad{}\times\frac{\Gamma(1-\varepsilon)}{\Gamma\left(\frac{3}{2}\right)}\,\frac{\sqrt{\pi}}{2}\,\sum_{n=0}^{\infty}\frac{\Gamma^{2}\left(1+n\right)}{\Gamma(1-\varepsilon+n)}\,\frac{1}{n!}\,\left(\frac{\Pi_{T\,ij}}{m_{T\,ij}}\right)^{2\,n}.$
(B.22)

The series in the right hand side of eq. (B.22) can be summed up into a new
hypergeometric function

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\left(m_{T\,ij}\right)^{-2}\,{}_{2}\text{F}_{1}\\!\left(1,1;1-\varepsilon;\frac{\Pi_{T\,ij}^{2}}{m_{T\,ij}^{2}}\right).$
(B.23)

Using the linear transformation eq. (15.3.6) of ref. [53], eq. (B.23) becomes

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\left(m_{T\,ij}\right)^{-2}$
$\displaystyle\quad{}\times\left\\{\frac{\Gamma(1-\varepsilon)\,\Gamma(-1-\varepsilon)}{\Gamma(-\varepsilon)\,\Gamma(-\varepsilon)}\,{}_{2}\text{F}_{1}\\!\left(1,1;2+\varepsilon;\frac{\Pi_{ij}^{2}}{m_{T\,ij}^{2}}\right)\right.$
$\displaystyle\qquad\quad{}+\left.\left(\frac{\Pi_{ij}^{2}}{m_{T\,ij}^{2}}\right)^{-1-\varepsilon}\,\Gamma(1-\varepsilon)\,\Gamma(1+\varepsilon)\;{}_{2}\text{F}_{1}\\!\left(-\varepsilon,-\varepsilon;-\varepsilon;\frac{\Pi_{ij}^{2}}{m_{T\,ij}^{2}}\right)\right\\}.$
(B.24)

Remembering that ${}_{2}\text{F}_{1}(a,b;b;z)=(1-z)^{-a}$, eq. (B.24) can be
rearranged in the following way

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du\left(m_{T\,ij}\right)^{-2}$
$\displaystyle\quad{}\times\left\\{\frac{\varepsilon}{1+\varepsilon}\;{}_{2}\text{F}_{1}\\!\left(1,1;2+\varepsilon;\frac{\Pi_{ij}^{2}}{m_{T\,ij}^{2}}\right)\right.$
$\displaystyle\qquad\quad{}+\left.\Gamma(1-\varepsilon)\,\Gamma(1+\varepsilon)\,\left(\Pi_{ij}^{2}\right)^{-1-\varepsilon}\,m_{T\,ij}^{2}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}\vphantom{{}_{2}\text{F}_{1}\\!\left(1,1;2+\varepsilon;\frac{\Pi_{ij}^{2}}{m_{T\,ij}^{2}}\right)}\right\\}.$
(B.25)

Note that since the remaining hypergeometric function is finite when
$\varepsilon=0$ and is multiplied by $\varepsilon$, the arguments of the
latter can be taken at $\varepsilon=0$. Thus, using that
${}_{2}\text{F}_{1}(1,1;2;z)=-\ln(1-z)/z$ and dropping some terms which vanish
as $\varepsilon=0$, eq. (B.25) can be re-written as

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{1-2\,\varepsilon}\,p_{i}\cdot
p_{j}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{1}du$
$\displaystyle\quad{}\times\left\\{-\frac{\varepsilon}{\Pi_{ij}^{2}}\,\ln\left(\frac{\Pi_{T\,ij}^{2}}{m_{T\,ij}^{2}}\right)+\Gamma(1-\varepsilon)\,\Gamma(1+\varepsilon)\,\left(\Pi_{ij}^{2}\right)^{-1-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}\right\\}.$
(B.26)

Let us focus on the first term inside the curly brackets of eq. (B.26). The
quantity $B_{1}$ which is defined to be

$\displaystyle B_{1}$
$\displaystyle\equiv\int_{0}^{1}du\,\frac{1}{\Pi_{ij}^{2}}\,\ln\left(\frac{\Pi_{T\,ij}^{2}}{m_{T\,ij}^{2}}\right),$
(B.27)

can be written as

$\displaystyle B_{1}$ $\displaystyle=\frac{1}{2\,p_{i}\cdot
p_{j}}\,\left\\{\int_{0}^{1}\frac{du}{u}\,\left[\ln(\Pi_{T\,ij}^{2})-\ln(m_{T\,ij}^{2})\right]+\int_{0}^{1}\frac{du}{1-u}\,\left[\ln(\Pi_{T\,ij}^{2})-\ln(m_{T\,ij}^{2})\right]\right\\}.$
(B.28)

The quantities $\Pi_{T\,ij}^{2}$ and $m_{T\,ij}^{2}$ are second order
polynomials in the variable $u$ and can be written as

$\displaystyle\Pi_{T\,ij}^{2}$
$\displaystyle=p_{T\,j}^{2}\,\left(1-\frac{u}{\tilde{u}_{+}}\right)\,\left(1-\frac{u}{\tilde{u}_{-}}\right),$
(B.29) $\displaystyle m_{T\,ij}^{2}$
$\displaystyle=p_{T\,j}^{2}\,\left(1-\frac{u}{\bar{u}_{+}}\right)\,\left(1-\frac{u}{\bar{u}_{-}}\right),$
(B.30)

where

$\displaystyle\tilde{u}_{\pm}$
$\displaystyle=\frac{p_{T\,j}}{p_{T\,j}-p_{T\,i}\,e^{\mp i\,\phi_{ij}}},$
(B.31) $\displaystyle\bar{u}_{\pm}$
$\displaystyle=\frac{p_{T\,j}}{p_{T\,j}-p_{T\,i}\,e^{\mp(y_{i}-y_{j})}}.$
(B.32)

In eqs. (B.29) and (B.30), the way of writing a second order polynomial in
terms of its roots is not standard but it will simplify the computation of the
two integrals of eq. (B.28). Indeed, we can notice that $\tilde{u}_{+}$ and
$\tilde{u}_{-}$ are complex conjugate while $\bar{u}_{+}$ and $\bar{u}_{-}$
are real but do not belong to the range $[0,1]$ as it can be inferred from eq.
(B.32) and that $1-u/\bar{u}_{+}$ and $1-u/\bar{u}_{-}$ are positive when $u$
is in $[0,1]$. Thus the logarithms of $\Pi_{T\,ij}^{2}$ and $m_{T\,ij}^{2}$
can be safely split, namely

$\displaystyle\ln(\Pi_{T\,ij}^{2})$
$\displaystyle=\ln(p_{T\,j}^{2})+\ln\left(1-\frac{u}{\tilde{u}_{+}}\right)+\ln\left(1-\frac{u}{\tilde{u}_{-}}\right),$
(B.33) $\displaystyle\ln(m_{T\,ij}^{2})$
$\displaystyle=\ln(p_{T\,j}^{2})+\ln\left(1-\frac{u}{\bar{u}_{+}}\right)+\ln\left(1-\frac{u}{\bar{u}_{-}}\right).$
(B.34)

The first integral in the curly brackets of eq. (B.28) can be performed easily
leading to

$\displaystyle\int_{0}^{1}\frac{du}{u}\,\left[\ln(\Pi_{T\,ij}^{2})-\ln(m_{T\,ij}^{2})\right]$
$\displaystyle=-\mbox{Li}_{2}\left(\frac{1}{\tilde{u}_{+}}\right)-\mbox{Li}_{2}\left(\frac{1}{\tilde{u}_{-}}\right)+\mbox{Li}_{2}\left(\frac{1}{\bar{u}_{+}}\right)+\mbox{Li}_{2}\left(\frac{1}{\bar{u}_{-}}\right).$
(B.35)

For the second integral, after a change of variable $t=1-u$, the quantities
$\Pi_{T\,ij}^{2}$ and $m_{T\,ij}^{2}$ can be written as

$\displaystyle\Pi_{T\,ij}^{2}$
$\displaystyle=p_{T\,i}^{2}\,\left(1-\frac{t}{1-\tilde{u}_{+}}\right)\,\left(1-\frac{t}{1-\tilde{u}_{-}}\right),$
(B.36) $\displaystyle m_{T\,ij}^{2}$
$\displaystyle=p_{T\,i}^{2}\,\left(1-\frac{t}{1-\bar{u}_{+}}\right)\,\left(1-\frac{t}{1-\bar{u}_{-}}\right).$
(B.37)

For the same reasons, the logarithm of these quantities can be safely split
leading for this second integral to

$\displaystyle\int_{0}^{1}\frac{du}{1-u}\,\left[\ln(\Pi_{T\,ij}^{2})-\ln(m_{T\,ij}^{2})\right]$
$\displaystyle=-\mbox{Li}_{2}\left(\frac{1}{1-\tilde{u}_{+}}\right)-\mbox{Li}_{2}\left(\frac{1}{1-\tilde{u}_{-}}\right)$
$\displaystyle\quad{}+\mbox{Li}_{2}\left(\frac{1}{1-\bar{u}_{+}}\right)+\mbox{Li}_{2}\left(\frac{1}{1-\bar{u}_{-}}\right).$
(B.38)

Using the following property of the dilogarithms, namely

$\displaystyle\mbox{Li}_{2}\left(\frac{1}{z}\right)+\mbox{Li}_{2}\left(\frac{1}{1-z}\right)$
$\displaystyle=-\frac{1}{2}\,\ln^{2}\left(1-\frac{1}{z}\right)\quad\text{for
$z\notin[0,1]$ },$ (B.39)

the quantity $B_{1}$ becomes

$\displaystyle B_{1}$ $\displaystyle=\frac{1}{4\,p_{i}\cdot
p_{j}}\,\left\\{\ln^{2}\left(1-\frac{1}{\tilde{u}_{+}}\right)+\ln^{2}\left(1-\frac{1}{\tilde{u}_{-}}\right)-\ln^{2}\left(1-\frac{1}{\bar{u}_{+}}\right)-\ln^{2}\left(1-\frac{1}{\bar{u}_{-}}\right)\right\\}.$
(B.40)

For the second term in the curly brackets of eq. (B.26), let us introduce

$\displaystyle B_{2}$
$\displaystyle=\int_{0}^{1}du\,\left(\Pi_{ij}^{2}\right)^{-1-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon},$
(B.41)

which can be written as

$\displaystyle B_{2}$ $\displaystyle=(2\,p_{i}\cdot
p_{j})^{-1-\varepsilon}\,\left[\int_{0}^{1}du\,u^{-1-\varepsilon}\,(1-u)^{-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}+\int_{0}^{1}du\,u^{-\varepsilon}\,(1-u)^{-1-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}\right].$
(B.42)

Let us focus on the first integral of eq. (B.42). The distribution
$u^{-1-\varepsilon}$ can be replaced by

$u^{-1-\varepsilon}\simeq-\frac{1}{\varepsilon}\,\delta(u)+\frac{1}{(u)_{+}}-\varepsilon\,\left(\frac{\ln(u)}{u}\right)_{+}+O(\varepsilon^{2}),$
(B.43)

and the rest of the integrand is expanded around $\varepsilon=0$ yielding

$\displaystyle\int_{0}^{1}du\,u^{-1-\varepsilon}\,(1-u)^{-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}$
$\displaystyle=-\frac{1}{\varepsilon}\,\left[1+\varepsilon\,\ln(p_{T\,j}^{2})+\frac{\varepsilon^{2}}{2}\,\ln^{2}(p_{T\,j}^{2})\right]+\int_{0}^{1}\frac{du}{(u)_{+}}$
$\displaystyle\quad{}+\varepsilon\,\left[\int_{0}^{1}\frac{du}{u}\left(\ln\left(1-\frac{u}{\tilde{u}_{+}}\right)+\ln\left(1-\frac{u}{\tilde{u}_{-}}\right)\right)\right.$
$\displaystyle\qquad{}-\left.\int_{0}^{1}\frac{du}{u}\,\ln(1-u)\right]-\varepsilon\,\int_{0}^{1}du\,\left(\frac{\ln(u)}{u}\right)_{+}$
$\displaystyle=-\frac{1}{\varepsilon}\,\left[1+\varepsilon\,\ln(p_{T\,j}^{2})+\frac{\varepsilon^{2}}{2}\,\ln^{2}(p_{T\,j}^{2})\right]$
$\displaystyle\quad{}-\varepsilon\left[\mbox{Li}_{2}\left(\frac{1}{\tilde{u}_{+}}\right)+\mbox{Li}_{2}\left(\frac{1}{\tilde{u}_{-}}\right)-\frac{\pi^{2}}{6}\right].$
(B.44)

The second integral of eq. (B.42), with a change of variable $t=1-u$, gives

$\displaystyle\int_{0}^{1}du\,u^{-\varepsilon}\,(1-u)^{-1-\varepsilon}\,\left(\Pi_{T\,ij}^{2}\right)^{\varepsilon}$
$\displaystyle=-\frac{1}{\varepsilon}\,\left[1+\varepsilon\,\ln(p_{T\,i}^{2})+\frac{\varepsilon^{2}}{2}\,\ln^{2}(p_{T\,i}^{2})\right]$
$\displaystyle\quad{}-\varepsilon\left[\mbox{Li}_{2}\left(\frac{1}{1-\tilde{u}_{+}}\right)+\mbox{Li}_{2}\left(\frac{1}{1-\tilde{u}_{-}}\right)-\frac{\pi^{2}}{6}\right].$
(B.45)

Plugging the results of eqs. (B.44) and (B.45) into eq. (B.42) leads to

$\displaystyle B_{2}$ $\displaystyle=\frac{1}{2\,p_{i}\cdot
p_{j}}\,\left\\{-\frac{1}{\varepsilon}\,\left[2-2\,\varepsilon\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{p_{T\,i}\,p_{T\,j}}\right)+\frac{\varepsilon^{2}}{2}\,\left[\ln^{2}\left(\frac{p_{T\,i}^{2}}{2\,p_{i}\cdot
p_{j}}\right)+\ln^{2}\left(\frac{p_{T\,j}^{2}}{2\,p_{i}\cdot
p_{j}}\right)\right]\right]\right.$
$\displaystyle\quad{}+\left.\varepsilon\,\left[\frac{1}{2}\,\ln^{2}\left(1-\frac{1}{\tilde{u}_{+}}\right)+\frac{1}{2}\,\ln^{2}\left(1-\frac{1}{\tilde{u}_{-}}\right)+\frac{\pi^{2}}{3}\right]\right\\}.$
(B.46)

Using the results of eqs. (B.40) and (B.46) and the fact that
$\Gamma(1-\varepsilon)\,\Gamma(1+\varepsilon)\simeq
1+\varepsilon^{2}\,\pi^{2}/6$, eq. (B.26) becomes

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\left\\{-\frac{2}{\varepsilon}+2\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{p_{T\,i}\,p_{T\,j}}\right)\right.$
$\displaystyle\quad{}-\frac{\varepsilon}{2}\,\left[\ln^{2}\left(\frac{p_{T\,i}^{2}}{2\,p_{i}\cdot
p_{j}}\right)+\ln^{2}\left(\frac{p_{T\,j}^{2}}{2\,p_{i}\cdot
p_{j}}\right)\right]$
$\displaystyle\quad{}+\left.\frac{\varepsilon}{2}\,\left[\ln^{2}\left(1-\frac{1}{\bar{u}_{+}}\right)+\ln^{2}\left(1-\frac{1}{\bar{u}_{-}}\right)\right]\right\\}.$
(B.47)

Finally, using the definition of the two roots $\bar{u}_{+}$ and $\bar{u}_{-}$
as well as introducing $\bar{d}_{ij}=\cosh(2\,y^{\star}_{ij})-\cos(\phi_{ij})$
with $y^{\star}_{ij}=(y_{i}-y_{j})/2$, we end up with

$\displaystyle J^{\text{soft}}_{ij}$
$\displaystyle=2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\left\\{-\frac{2}{\varepsilon}+2\,\ln\left(2\,\bar{d}_{ij}\right)-\varepsilon\,\ln^{2}\left(2\,\bar{d}_{ij}\right)+4\,\varepsilon\,(y^{\star}_{ij})^{2}\right\\}.$
(B.48)

Note that the definition of $\bar{d}_{ij}$ is equivalent to the one given in
subsection 3.1.1. Let us notice that the result of eq. (B.48) is particularly
simple, the dilogarithms combine to logarithms squared and the dependence on
the azimuthal angle is through $\bar{d}_{ij}$. In addition, this result is
explicitly invariant under boosts along the beam direction as it depends only
on difference of rapidities and azimuthal angles.

##  Appendix C The collinear integral inside the cylinder

Let us recall the integral to compute (see eq. (3.11)):

$\displaystyle J^{\text{coll}}$
$\displaystyle=\int_{0}^{\pi}d\phi_{N}\,(\sin\phi_{N})^{-2\,\varepsilon}\,\int_{-\infty}^{+\infty}dy_{N}\,\frac{\cos\phi_{N}}{\cosh(y_{N}-y_{i})-\cos\phi_{N}}.$
(C.1)

As mentioned earlier, the result of this integral will not depend on $y_{i}$
because the range of integration in $y_{N}$ extends to infinity. As in the
appendix B, the $y_{N}$ integration range is divided into two parts

$\displaystyle\int_{-\infty}^{+\infty}dy_{N}\,F(y_{N})$
$\displaystyle=\int_{y_{i}}^{+\infty}dy_{N}\,F(y_{N})+\int_{-\infty}^{y_{i}}dy_{N}\,F(y_{N}).$
(C.2)

We again make the changes of variable $\Delta y=y_{N}-y_{i}$ in the first
integral on the right hand side of eq. (C.2) and $\Delta y=y_{i}-y_{N}$ in the
second one

$\displaystyle\int_{-\infty}^{+\infty}dy_{N}\,F(y_{N})$
$\displaystyle=\int_{0}^{+\infty}d\Delta y\,F(y_{i}+\Delta
y)+\int^{+\infty}_{0}d\Delta y\,F(y_{i}-\Delta y).$ (C.3)

The integrand depends on $\Delta y$ only through $\cosh(\pm\Delta y)$ such
that the two integrals are equal and thus

$\displaystyle J^{\text{coll}}$
$\displaystyle=2\,\int_{0}^{\pi}d\phi_{N}\,\left(\sin\phi_{N}\right)^{-2\,\varepsilon}\,\int_{0}^{\infty}d\Delta
y\,\frac{\cos\phi_{N}}{\cosh(\Delta y)-\cos\phi_{N}},$ (C.4)

which can be written as

$\displaystyle J^{\text{coll}}$ $\displaystyle=2\,\int_{0}^{\infty}d\Delta
y\,\int_{0}^{\pi}d\phi_{N}\,\left(\sin\phi_{N}\right)^{-2\,\varepsilon}\,\left[-1+\frac{\cosh(\Delta
y)}{\cosh(\Delta y)-\cos\phi_{N}}\right].$ (C.5)

Setting $\cos\phi_{N}=2\,x-1$ leads to

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{1-2\,\varepsilon}\,\int_{0}^{\infty}d\Delta
y\,\int_{0}^{1}dx\,x^{-1/2-\varepsilon}\,(1-x)^{-1/2-\varepsilon}\,\left[-1+\frac{\cosh(\Delta
y)}{\cosh(\Delta y)+1}\,(1-z\,x)^{-1}\right],$ (C.6)

where $z=2/(1+\cosh(\Delta y))$. The $x$ integration gives a Gauss
hypergeometric function

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{1-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{\infty}d\Delta
y\,\left[-1+\frac{\cosh(\Delta y)}{\cosh(\Delta
y)+1}\,{}_{2}\text{F}_{1}\left(1,\frac{1}{2}-\varepsilon;1-2\,\varepsilon;z\right)\right].$
(C.7)

Using the quadratic transformation, eq. (15.3.16) of ref. [53], yields

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{1-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\int_{0}^{\infty}d\Delta
y\,\left[-1+{}_{2}\text{F}_{1}\left(\frac{1}{2},1;1-\varepsilon;z^{\prime\,2}\right)\right],$
(C.8)

with $z^{\prime}=1/\cosh(\Delta y)$. As in the preceding case, the
hypergeometric function can be expanded in a series as $z^{\prime}\leq
1$.141414Once more, a rigorous approach would involve introducing a cutoff
$\lambda$ to prevent reaching the value $\Delta y=0$, ensuring $z^{\prime}<1$,
and then taking the limit $\lambda\rightarrow 0$ at the end. The key point
here is that the first term in the square brackets leads to a divergence when
integrated on $\Delta y$. However, this term cancels against the first term of
the hypergeometric series. Doing so and shifting the variable of the series
such that it starts at 0 yields

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{1-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{\Gamma(1-\varepsilon)}{\Gamma\left(\frac{1}{2}\right)}\,\sum_{m=0}^{+\infty}\,\frac{\Gamma\left(\frac{3}{2}+m\right)\,\Gamma(2+m)}{\Gamma(2-\varepsilon+m)}\,\frac{1}{(m+1)!}\,$
$\displaystyle\quad{}\times\int_{0}^{\infty}d\Delta y\,(\cosh(\Delta
y))^{-2\,(m+1)}.$ (C.9)

Using the eq. (B.21) for the integration on $\Delta y$ leads to a series of
the type

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{-2\,\varepsilon}\,\sqrt{\pi}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{\Gamma(1-\varepsilon)}{\Gamma\left(\frac{1}{2}\right)}\,\sum_{m=0}^{+\infty}\,\frac{\Gamma^{2}(1+m)}{\Gamma(2-\varepsilon+m)}\,\frac{1}{m!},$
(C.10)

which can be expressed as a hypergeometric function whose argument is $1$ and
can be rewritten in terms of gamma functions as long as $\varepsilon<0$:

$\displaystyle J^{\text{coll}}$
$\displaystyle=2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{\Gamma(1-\varepsilon)}{\Gamma\left(2-\varepsilon\right)}\,{}_{2}\text{F}_{1}(1,1;2-\varepsilon;1)=\frac{2^{-2\,\varepsilon}}{-\varepsilon}\,\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}.$
(C.11)

##  Appendix D The collinear integral inside the cone

Let us evaluate the collinear integral that arises when $i_{N}$ lies within
the cone $\Gamma_{i}$:

$\displaystyle J^{\text{coll}}_{\text{out}}$
$\displaystyle=\iint_{\Gamma_{i}}\,d\phi_{N}\,dy_{N}\,\,\frac{\phi_{N}^{-2\,\varepsilon}}{(y_{i}-y_{N})^{2}+\phi_{N}^{2}}.$
(D.1)

Similar to Appendix C, the integration range over $y_{N}$ is split into two
parts. We perform the change of variables $\Delta y=y_{N}-y_{i}$ and $\Delta
y=y_{i}-y_{N}$ in the parts where $y_{N}\geq y_{i}$ and $y_{N}\leq y_{i}$,
respectively, resulting in

$\displaystyle J^{\text{coll}}_{\text{out}}$
$\displaystyle=2\,\iint_{\Gamma_{i}}\,d\phi_{N}\,d\Delta
y\,\,\frac{\phi_{N}^{-2\,\varepsilon}}{(\Delta y)^{2}+\phi_{N}^{2}}.$ (D.2)

The condition that $i_{N}$ lies within the cone $\Gamma_{i}$ is expressed as
$(\Delta y)^{2}+\phi_{N}^{2}\leq R_{\text{th}}^{2}$. However, by construction,
$\Delta y$ must be less than
$\min(y_{N\,\text{max}}-y_{i},y_{i}-y_{N\,\text{min}})$. In cases where
$p_{T\,N}$ is large, it’s possible that
$\min(y_{N\,\text{max}}-y_{i},y_{i}-y_{N\,\text{min}})<R_{\text{th}}$, which
implies that the integration domain is no longer a disk in the $\Delta
y$-$\phi_{N}$ plane. As a result, the analytical calculation of the integral
$J^{\text{coll}}_{\text{out}}$ becomes complicated. To address this challenge,
we adopt the following strategy. Firstly, in the subtraction term, we use
$\tilde{y}_{N\,\text{max}}={y_{N\,\text{max}}}_{|{\phi_{N}=0}}$ and
$\tilde{y}_{N\,\text{min}}={y_{N\,\text{min}}}_{|{\phi_{N}=0}}$ as bounds for
the $y_{N}$ integration. These bounds are independent of $\phi_{N}$. Secondly,
we dynamically adjust $R_{\text{th}}$ to
$\min(R_{\text{th}},\Delta\tilde{y}_{\text{max}})$, where
$\Delta\tilde{y}_{\text{max}}=\min(\tilde{y}_{N\,\text{max}}-y_{N},y_{N}-\tilde{y}_{N\,\text{min}})$
represents the largest value taken by $\Delta y$. In other words, the value of
$R_{\text{th}}$ is not fixed but may vary depending on the kinematics. In
summary, by implementing these adjustments, we ensure that our calculation
accounts for the varying nature of the integration domain, overcoming the
complication introduced by large transverse momenta. In this way, the
analytical integration remains easy and the dependence on $R_{\text{th}}$ is
logarithmic.

To perform the integral analytically, we introduce polar coordinates in the
plane $\Delta y$-$\phi$, such that

$\displaystyle\Delta y$ $\displaystyle=r\,\cos(\theta),$ $\displaystyle\phi$
$\displaystyle=r\,\sin(\theta).$

After this change of variable, equation (D.1) becomes

$\displaystyle J^{\text{coll}}_{\text{out}}$
$\displaystyle=2\,\int_{0}^{R_{\text{th}}}dr\,r^{-1-2\,\varepsilon}\,\int_{0}^{\pi/2}d\theta\,(\sin\theta)^{-2\,\varepsilon}.$
(D.3)

Note that, since $\Delta y$ and $\phi$ are positive, the angle $\theta$ runs
between $0$ and $\pi/2$. The $r$ integration is trivial and we introduce a new
variable $x$ such that $\cos\,\theta=2\,x-1$ leading to

$\displaystyle J^{\text{coll}}_{\text{out}}$
$\displaystyle=\frac{R_{\text{th}}^{-2\,\varepsilon}}{-2\,\varepsilon}\,2^{1-2\,\varepsilon}\,\int^{1}_{1/2}dx\,x^{-1/2-\varepsilon}\,(1-x)^{-1/2-\varepsilon}=2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\;\frac{R_{\text{th}}^{-2\,\varepsilon}}{-2\,\varepsilon}.$
(D.4)

##  Appendix E Details of the divergent terms construction

Let us collect the different divergent terms resulting from the analytical
integration of the subtraction terms

$\displaystyle\mathcal{T}_{\text{tot}}^{\text{div}}$
$\displaystyle=\mathcal{T}_{\text{in}}^{(4)\,\text{div}}+\mathcal{T}_{\text{in}}^{(3)\,\text{div}}+\mathcal{T}_{\text{in}}^{(2)\,\text{div}}+\mathcal{T}_{\text{in}}^{(1)\,\text{div}}+\mathcal{T}_{\text{out}}^{(4)\,\text{div}}+\mathcal{T}_{\text{out}}^{(3)\,\text{div}}+\mathcal{T}_{\text{out}}^{(2)\,\text{div}}.$
(E.1)

Note that the quantities on the right-hand side of eq. (E.1) are given by eq.
(3.24), eq. (3.46), eq. (3.63), eq. (3.68) and eq. (3.71), except
$\mathcal{T}_{\text{in}}^{(3)\,\text{div}}$ and
$\mathcal{T}_{\text{out}}^{(3)\,\text{div}}$ which have not been computed
explicitly but can be easily obtained from
$\mathcal{T}_{\text{in}}^{(2)\,\text{div}}$ and
$\mathcal{T}_{\text{out}}^{(2)\,\text{div}}$ by changing the signs of all the
rapidities and the labels $1\leftrightarrow 2$. We will distinguish the soft
part from the collinear ones. The soft part can be obtained from eq. (E.1) by
collecting all the terms containing the functions $f_{ij}(0,0,0)$,
$f_{1j}(0,0,0)$ and $f_{2j}(0,0,0)$:

$\displaystyle\mathcal{T}_{\text{tot}}^{\text{ soft}}$
$\displaystyle=2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}$
$\displaystyle\quad{}\times\left\\{\sum_{i=3}^{N-2}\sum_{j=i+1}^{N-1}f_{ij}(0,0,0)\,\left[\frac{1}{\varepsilon^{2}}-\frac{1}{\varepsilon}\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{Q^{2}}\right)+\ln^{2}(x_{T\,i})+\ln^{2}(x_{T\,j})\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{}+\left.2\,\ln(x_{T\,\text{m}})\,\ln(2\,\bar{d}_{ij})+\frac{1}{2}\,\ln^{2}(2\,\bar{d}_{ij})-2\,\left(y^{\star}_{ij}\right)^{2}\vphantom{\frac{\Gamma^{2}(1-\varepsilon)}{\Gamma(1-2\,\varepsilon)}}\right]$
$\displaystyle\qquad\quad{}+\sum_{j=3}^{N-1}f_{1j}(0,0,0)\,\left[\frac{1}{\varepsilon^{2}}-\frac{1}{\varepsilon}\,\ln\left(\frac{2\,p_{1}\cdot
p_{j}}{Q^{2}}\right)+\Upsilon(\bar{x}_{1},y_{j})+\ln^{2}(x_{T\,j})\right]$
$\displaystyle\qquad\quad{}+\sum_{j=3}^{N-1}f_{2j}(0,0,0)\,\left[\frac{1}{\varepsilon^{2}}-\frac{1}{\varepsilon}\,\ln\left(\frac{2\,p_{2}\cdot
p_{j}}{Q^{2}}\right)+\Upsilon(\bar{x}_{2},-y_{j})+\ln^{2}(x_{T\,j})\right]$
$\displaystyle\qquad\quad{}+\left.f_{12}\left(0,0,0\right)\,\left[\frac{1}{\varepsilon^{2}}-\frac{1}{\varepsilon}\,\ln\left(\frac{2\,p_{1}\cdot
p_{2}}{Q^{2}}\right)+\Upsilon(\bar{x}_{1},y_{0})+\Upsilon(\bar{x}_{2},-y_{0})\right]\vphantom{\frac{f_{34}\left(y_{3},\frac{1-z_{3}}{z_{3}}\,p_{T\,3},0\right)}{(1-z_{3})_{+}}}\right\\}.$
(E.2)

Let us introduce $\sigma_{H}^{\text{soft}}$ defined as

$\displaystyle\sigma_{H}^{\text{soft}}$
$\displaystyle\equiv\sum_{\\{i\\}_{N-1}\in S_{p}}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(x)\,\mathcal{T}_{\text{tot}}^{\text{ soft}}.$
(E.3)

In order to facilitate the reading, only the case where $x_{T\,\text{m}}$
fulfils the condition (3.55) will be presented151515It is easy to have the
formulae with the condition (3.57).. Expanding around $\varepsilon=0$ the
right hand side of eq. (E.2) and keeping only the relevant terms in
$\varepsilon$ leads to161616Note that in the soft limit, the variables $x_{k}$
and $\bar{x}_{k}$ are equivalent.

$\displaystyle\sigma_{H}^{\text{soft}}$
$\displaystyle\equiv\sum_{\\{i\\}_{N-1}\in
S_{p}}\,K^{(n)\,B}_{i_{1}i_{2}}\,\left(\frac{4\,\pi\,\mu^{2}}{Q^{2}}\right)^{\varepsilon}\,\frac{\alpha_{s}}{2\,\pi}\,\frac{1}{\Gamma(1-\varepsilon)}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(\bar{x})\,\hphantom{\delta^{n-2}\left(\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{x_{l}}\right)}\,$
$\displaystyle\quad{}\times
A_{(i)_{N-1}}(\\{\bar{x}\\}_{N-1})\,\delta^{n-2}\left(\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{\bar{x}_{l}}\right)$
$\displaystyle\quad{}\times\left\\{\frac{1}{\varepsilon^{2}}\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)-\frac{1}{\varepsilon}\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{Q^{2}}\right)\right.$
$\displaystyle\qquad\quad{}+2\,\ln(x_{T\,\text{m}})\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{Q^{2}}\right)-2\,\ln^{2}(x_{T\,\text{m}})\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)$
$\displaystyle\qquad\quad{}+\sum_{i=3}^{N-1}\,\ln^{2}\left(\frac{x_{T\,i}}{x_{T\,\text{m}}}\right)\,\left[\sum_{j=1}^{i-1}H^{(n)}_{ji}(0)+\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\right]$
$\displaystyle\qquad\quad{}+\ln^{2}\left(\frac{\bar{x}_{1}}{1-\bar{x}_{1}}\,\right)\,\sum_{j=2}^{N-1}H^{(n)}_{1j}(0)+\ln^{2}\left(\frac{\bar{x}_{2}}{1-\bar{x}_{2}}\,\right)\,\left[H^{(n)}_{12}(0)+\sum_{j=3}^{N-1}H^{(n)}_{2j}(0)\right]$
$\displaystyle\qquad\quad{}+\left.\sum_{i=3}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\,\left[\frac{1}{2}\,\ln^{2}(2\,\bar{d}_{ij})-2\,\left(y^{\star}_{ij}\right)^{2}\right]\right\\}.$
(E.4)

The final state collinear singularity related to hard a parton $i_{k}$ coming
from inside the cylinder is given by

$\displaystyle\mathcal{T}^{\text{ fin coll}}_{k\,\text{in}}$
$\displaystyle=-2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{x_{T\,k}^{-2\,\varepsilon}}{\varepsilon}\,\left\\{\int_{z_{k\,\text{m}}}^{1}\frac{dv_{k}}{v_{k}}\,v_{k}^{2\,\varepsilon}\,\left[\frac{1}{(1-v_{k})_{+}}-2\,\varepsilon\,\left(\frac{\ln(1-v_{k})}{1-v_{k}}\right)_{+}\right]\,\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{}\times\left[\mathcal{F}_{k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)+f_{1k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}+\left.\left.f_{2k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)\right]\right\\},$
(E.5)

and the one coming from outside the cylinder is

$\displaystyle\mathcal{T}^{\text{ fin coll}}_{k\,\text{out}}$
$\displaystyle=-2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{x_{T\,k}^{-2\,\varepsilon}\,R_{\text{th}}^{-2\,\varepsilon}}{\varepsilon}\,\left\\{\int_{z_{k\,\text{min}}}^{z_{k\,\text{m}}}\frac{dv_{k}}{v_{k}}\,v_{k}^{2\,\varepsilon}\,\left[\frac{1}{(1-v_{k})_{+}}-2\,\varepsilon\,\left(\frac{\ln(1-v_{k})}{1-v_{k}}\right)_{+}\right]\,\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{}\times\left[\mathcal{F}_{k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)+f_{1k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}+\left.\left.f_{2k}\left(y_{k},\frac{1-v_{k}}{v_{k}}\,x_{T\,k},0\right)\right]\right\\}.$
(E.6)

In eqs. (E.5) and (E.6), we changed the name of the integration variable
$z_{k}$ into $v_{k}$ for reasons which will become clear later on. The
corresponding cross section $\sigma^{\text{fin coll}}_{H\,k}$ is

$\displaystyle\sigma^{\text{fin coll}}_{H\,k}$
$\displaystyle\equiv\sum_{\\{i\\}_{N-1}\in S_{p}}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(x)\,\left[\mathcal{T}^{\text{ fin
coll}}_{k\,\text{in}}+\mathcal{T}^{\text{ fin coll}}_{k\,\text{out}}\right].$
(E.7)

Remembering the definition of the functions $f_{ij}$, it is easy to realise
that the Dirac distributions coming from the conservation of the transverse
momentum will depend on $v_{k}$ through the product $v_{k}\,x_{k}$ suggesting
that by a suitable change of variables these constraints could be independent
of the variable of the ”+” distributions. Let us sketch the structure of eq.
(E.7) in terms of the variables $x_{k}$ and $v_{k}$. It is of the type

$\displaystyle\int\frac{dx_{k}}{x_{k}^{2-2\,\varepsilon}}\,\int_{z_{k\,\text{min}}}^{1}\frac{dv_{k}}{v_{k}}\,v_{k}^{2\,\varepsilon}\,G(x_{k},v_{k}),$
(E.8)

and the change of variables $v_{k}=z_{k}$ and $x_{k}=\bar{x}_{k}/z_{k}$ whose
Jacobian is $1/z_{k}$ leads to the following structure

$\displaystyle\int\frac{d\bar{x}_{k}}{\bar{x}_{k}^{2-2\,\varepsilon}}\,\int_{\bar{x}_{k}}^{1}\,dz_{k}\;G\left(\frac{\bar{x}_{k}}{z_{k}},z_{k}\right).$
(E.9)

Up to vanishing terms when $\varepsilon\rightarrow 0$, the cross section
$\sigma^{\text{fin coll}}_{H\,k}$ can be written as

$\displaystyle\sigma^{\text{fin coll}}_{H\,k}=\sum_{\\{i\\}_{N-1}\in
S_{p}}K^{(n)\,B}_{i_{1}i_{2}}\,\left(\frac{4\,\pi\,\mu^{2}}{Q^{2}}\right)^{\varepsilon}\,\frac{\alpha_{s}}{2\,\pi}\,\frac{1}{\Gamma(1-\varepsilon)}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(\bar{x})\,\left(\frac{X_{T\,k}}{\bar{x}_{k}}\right)^{-2\,\varepsilon}$
$\displaystyle\quad{}\times\delta^{n-2}\left(\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{\bar{x}_{l}}\right)\left\\{-\frac{1}{\varepsilon}\,\int_{\bar{x}_{k}}^{1}\frac{dz_{k}}{z_{k}^{1+2\,\varepsilon}}\,A_{(i)_{N-1}}\left(\left\\{\bar{x}|\bar{x}_{k}:\frac{\bar{x}_{k}}{z_{k}}\right\\}_{N-1}\right)\,\right.$
$\displaystyle\quad{}\times\left[\frac{1}{(1-z_{k})_{+}}-2\,\varepsilon\,\left(\frac{\ln(1-z_{k})}{1-z_{k}}\right)_{+}\right]\,z_{k}\,\Xi_{k}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)+\ln\left(R_{\text{th}}^{2}\right)$
$\displaystyle\quad{}\times\int_{\bar{x}_{k}}^{\zeta_{k\,\text{m}}}\frac{dz_{k}}{z_{k}^{1+2\,\varepsilon}}\,A_{(i)_{N-1}}\left(\left\\{\bar{x}|\bar{x}_{k}:\frac{\bar{x}_{k}}{z_{k}}\right\\}_{N-1}\right)\,\,\frac{1}{(1-z_{k})_{+}}\left.z_{k}\,\Xi_{k}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)\right\\},$
(E.10)

with

$\displaystyle\Xi_{k}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)$
$\displaystyle=\sum_{j=k+1}^{N-1}H^{(n)}_{kj}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)+\sum_{j=3}^{k-1}H^{(n)}_{jk}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)$
$\displaystyle\quad{}+H^{(n)}_{1k}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right)+H^{(n)}_{2k}\left((1-z_{k})\,\frac{K_{T\,k}}{\bar{x}_{k}}\right).\,$
(E.11)

and $\zeta_{k\,\text{m}}$ given by eq. (4.7).

The initial state collinear contribution can be split into two parts, a
divergent one and a finite one in which the regulator $\varepsilon$ is set to
zero:

$\displaystyle\mathcal{T}^{\text{ ini coll}}_{\text{div}}$
$\displaystyle=-2^{-2\,\varepsilon}\,\frac{\Gamma^{2}\left(\frac{1}{2}-\varepsilon\right)}{\Gamma(1-2\,\varepsilon)}\,\frac{1}{\varepsilon}\left\\{\sum_{j=3}^{N-1}\chi(\bar{x}_{1},y_{j})^{-2\,\varepsilon}\,\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}}\,z_{1}^{2\,\varepsilon}\,\frac{f_{1j}^{c}\left(\frac{\bar{x}_{1}\,e^{-y_{j}}}{\omega}\,\frac{1-z_{1}}{z_{1}}\right)}{(1-z_{1})_{+}}\right.$
$\displaystyle\;\;\;\left.{}+\chi(\bar{x}_{1},y_{0})^{-2\,\varepsilon}\,\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}}\,z_{1}^{2\,\varepsilon}\,\frac{f_{12}^{(1)\,c}\left(\frac{\bar{x}_{1}\,e^{-y_{0}}}{\omega}\,\frac{1-z_{1}}{z_{1}}\right)}{(1-z_{1})_{+}}+1\leftrightarrow
2\right\\},$ (E.12) $\displaystyle\mathcal{T}^{\text{ ini coll}}_{\text{fin}}$
$\displaystyle=2\,\pi\,\left\\{\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}}\,\left(\frac{\ln(1-z_{1})}{1-z_{1}}\right)_{+}\,f_{12}^{(1)\,c}\left(\frac{\bar{x}_{1}\,e^{-y_{0}}}{\omega}\,\frac{1-z_{1}}{z_{1}}\right)\right.$
$\displaystyle\qquad{}\left.+\sum_{j=3}^{N-1}\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}}\,\left(\frac{\ln(1-z_{1})}{1-z_{1}}\right)_{+}\,f_{1j}^{c}\left(\frac{\bar{x}_{1}\,e^{-y_{j}}}{\omega}\,\frac{1-z_{1}}{z_{1}}\right)+1\leftrightarrow
2\right\\}.$ (E.13)

The associated cross section is then

$\displaystyle\sigma^{\text{ini coll}}_{H}=\sum_{\\{i\\}_{N-1}\in
S_{p}}K^{(n)\,B}_{i_{1}i_{2}}\,\left(\frac{4\,\pi\,\mu^{2}}{Q^{2}}\right)^{\varepsilon}\,\frac{\alpha_{s}}{2\,\pi}\,\frac{1}{\Gamma(1-\varepsilon)}\,\frac{1}{-\varepsilon}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(\bar{x})\,$
$\displaystyle\quad{}\times\,\delta^{n-2}\left(\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{\bar{x}_{l}}\right)\,\left\\{\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}^{2}\,(1-z_{1})_{+}}\,A_{(i)_{N-1}}\left(\left\\{\bar{x}|\bar{x}_{1}:\frac{\bar{x}_{1}}{z_{1}}\right\\}_{N-1}\right)\right.$
$\displaystyle\qquad{}\times
z_{1}^{1+2\,\varepsilon}\,\left(\frac{\bar{x}_{1}\,x_{T\,\text{m}}}{1-\bar{x}_{1}}\right)^{-2\,\varepsilon}\,\left[H^{(n)}_{12}((1-z_{1})\,p_{1})+\sum_{l=3}^{N-1}H^{(n)}_{1l}((1-z_{1})\,p_{1})\right]$
$\displaystyle\quad{}-2\varepsilon\,\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}^{2}}\,\left(\frac{\ln(1-z_{1})}{1-z_{1}}\right)_{+}\,\,A_{(i)_{N-1}}\left(\left\\{\bar{x}|\bar{x}_{1}:\frac{\bar{x}_{1}}{z_{1}}\right\\}_{N-1}\right)$
$\displaystyle\qquad{}\times
z_{1}\,\left[H^{(n)}_{12}((1-z_{1})\,p_{1})+\sum_{l=3}^{N-1}H^{(n)}_{1l}((1-z_{1})\,p_{1})\right]$
$\displaystyle\qquad{}+\left.1\leftrightarrow
2\vphantom{\int_{\bar{x}_{1}}^{1}\frac{dz_{1}}{z_{1}^{2}\,(1-z_{1})_{+}}}\right\\}.$
(E.14)

The relations given by eqs. (4.1), (4.2) and (4.4) must be fulfilled in order
to absorb the collinear divergences into a redefinition of the partonic
density functions as well as the fragmentation functions. They are obtained by
comparing the structure of the collinear divergences in the initial and final
states, as derived in sec. 2.1 (see eq. (2.26)), with the results delineated
form the subtraction terms after the integration over the phase space of the
soft/collinear parton, cf. eqs. (E.10) and (E.14). When the variables $z_{l}$
($l\in{1,2,\ldots,N-1}$) approach one, according to the definition of the
functions $a^{(n)}_{ij}(z)$, only the diagonal cases survive, namely when
$i=j$ and the relations (4.1), (4.2) and (4.4) become

$\displaystyle H^{(n)}_{12}(0)+\sum_{l=3}^{N-1}H^{(n)}_{1l}(0)$
$\displaystyle=a^{(n)}_{i_{1}i_{1}}(1)\,|M^{n}_{[i]_{N-1}}|^{2},$ (E.15)
$\displaystyle H^{(n)}_{12}(0)+\sum_{l=3}^{N-1}H^{(n)}_{2l}(0)$
$\displaystyle=a^{(n)}_{i_{2}i_{2}}(1)\,|M^{n}_{[i]_{N-1}}|^{2},$ (E.16)
$\displaystyle\Xi_{k}\left(0\right)$
$\displaystyle=a^{(n)}_{i_{k}i_{k}}(1)\,|M^{n}_{[i]_{N-1}}|^{2}\qquad\text{for
each $k$ in $S_{f}$}.$ (E.17)

With these relations in hand, eqs. (E.14) and (E.10) can be expressed in terms
of DGLAP kernels. To achieve this, we enforce the appearance of the quantity
$\mathcal{H}$ for initial state and final state radiations, utilizing eqs.
(2.21) and (A.1), and expanding the rest around $\varepsilon=0$. This leads to
the eq. (4.3) presented in sec. 4. The terms containing the final state
collinear divergences can be treated in the same manner, yielding eq. (4.6).

Note that in order to enforce the appearance of a DGLAP kernel, we must
include additional soft terms that are not explicitly expressed in eqs. (4.3)
and (4.6). These terms will be incorporated into the soft part which now reads

$\displaystyle\sigma_{H}^{\text{soft}}$
$\displaystyle\equiv\sum_{\\{i\\}_{N-1}\in
S_{p}}\,K^{(n)\,B}_{i_{1}i_{2}}\,\left(\frac{4\,\pi\,\mu^{2}}{Q^{2}}\right)^{\varepsilon}\,\frac{\alpha_{s}}{2\,\pi}\,\frac{1}{\Gamma(1-\varepsilon)}\,\int
d\text{PS}_{N-1\,\text{h}}^{(n)}(\bar{x})\,$ $\displaystyle\quad{}\times
A_{(i)_{N-1}}(\\{\bar{x}\\}_{N-1})\,\delta^{n-2}\left(\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{\bar{x}_{l}}\right)$
$\displaystyle\quad{}\times\left\\{\frac{1}{\varepsilon^{2}}\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)-\frac{1}{\varepsilon}\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\,\ln\left(\frac{2\,p_{i}\cdot
p_{j}}{Q^{2}}\right)\right.$
$\displaystyle\qquad\quad{}+\frac{1}{\varepsilon}\,\left[\,\sum_{k=1}^{2}b_{i_{k}i_{k}}\,\left(\frac{M^{2}}{Q^{2}}\right)^{-\varepsilon}+\,\sum_{k=3}^{N-1}b_{i_{k}i_{k}}\,\left(\frac{M_{f}^{2}}{Q^{2}}\right)^{-\varepsilon}\;\right]\,|M^{(n)}_{[i]_{N-1}}|^{2}$
$\displaystyle\qquad\quad{}+\left.\text{finite terms of
eq.~{}(\ref{eqdefhadcrxsoft1})}\vphantom{\frac{1}{\varepsilon^{2}}\,\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}H_{ij}(0)}\right\\}.$
(E.18)

The soft divergences cancel out with the real emission due to the relations
(4.9), which are valid in $n$ dimensions. Following this cancellation, some
finite terms remain, as shown in equation (4.10). Note that by using eqs.
(E.15), (E.16) and (E.17), we get that

$\displaystyle\left[a^{(n)}_{i_{1}i_{1}}(1)+a^{(n)}_{i_{2}i_{2}}(1)\right]\,|M^{(n)}_{[i]_{N-1}}|^{2}$
$\displaystyle=\left[2\,H^{(n)}_{12}(0)+\sum_{k=3}^{N-1}\left(H^{(n)}_{1k}(0)+H^{(n)}_{2k}(0)\right)\right],$
(E.19)
$\displaystyle\left[\sum_{k=3}^{N-1}a^{(n)}_{i_{k}i_{k}}(1)\right]\,|M^{(n)}_{[i]_{N-1}}|^{2}$
$\displaystyle=\sum_{i=3}^{N-2}\left[\sum_{j=1}^{i-1}H^{(n)}_{ji}(0)+\sum_{j=i+1}^{N-1}H^{(n)}_{ij}(0)\right],$
(E.20) $\displaystyle\mathcal{A}^{(n)}$
$\displaystyle=\frac{1}{2}\,\sum_{k=1}^{N-1}a^{(n)}_{i_{k}i_{k}}(1).$ (E.21)

##  Appendix F A non trivial example to illustrate the Born colour connected
amplitudes

In this appendix, we compute explicitly the soft approximation of the reaction
$q(p_{1})+\bar{q}(p_{2})\rightarrow g(p_{3})+g(p_{4})(+g(p_{5}))$, that is to
say the coefficient of the eikonal factors taken at $p_{5}=0$. We compare them
to the result of the one loop correction of
$q(p_{1})+\bar{q}(p_{2})\rightarrow g(p_{3})+g(p_{4})$. This gives an example
of the relations among the coefficients of the eikonal factors in order to
have a cancellation of the IR divergences. This reaction is taken because it
contains both external fermions and gluons.

The Born amplitude is depicted in fig. 3. At the Born level, the sum of the
three diagrams is represented by the product of a string of $\gamma$ matrices
carrying two Lorentz indices as well as string of colour matrices carrying
colour indices $\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}a_{4}}$ multiplied by
the Dirac spinors and the polarisation vectors describing the external
particles.

$\leavevmode\resizebox{61.17325pt}{}{ \parbox{85.35826pt}{\fmfgraph*(31,24)
\fmfleftn{i}{2} \fmfrightn{o}{2} \fmflabel{$q(p_{1})$}{i2}
\fmflabel{$\bar{q}(p_{2})$}{i1} \fmflabel{$g(p_{4})$}{o2}
\fmflabel{$g(p_{3})$}{o1} \fmf{fermion}{v1,i1} \fmf{fermion}{i2,v1}
\fmf{gluon}{v1,o1} \fmf{gluon}{v1,o2} \fmfblob{0.20w}{v1}
}}\qquad\equiv\qquad\leavevmode\resizebox{61.17325pt}{}{
\parbox{85.35826pt}{\fmfgraph*(31,24) \fmfleftn{i}{2} \fmfrightn{o}{2}
\fmflabel{$q(p_{1})$}{i2} \fmflabel{$\bar{q}(p_{2})$}{i1}
\fmflabel{$g(p_{4})$}{o2} \fmflabel{$g(p_{3})$}{o1} \fmf{fermion}{i2,v2}
\fmf{fermion}{v2,v1} \fmf{fermion}{v1,i1} \fmf{gluon}{v1,o1}
\fmf{gluon}{v2,o2} }}\qquad+\qquad\leavevmode\resizebox{61.17325pt}{}{
\parbox{85.35826pt}{\fmfgraph*(31,24) \fmfleftn{i}{2} \fmfrightn{o}{2}
\fmflabel{$q(p_{1})$}{i2} \fmflabel{$\bar{q}(p_{2})$}{i1}
\fmflabel{$g(p_{4})$}{o2} \fmflabel{$g(p_{3})$}{o1} \fmf{fermion}{i2,v2}
\fmf{fermion}{v2,v1} \fmf{fermion}{v1,i1} \fmf{phantom}{v1,o1}
\fmf{phantom}{v2,o2} \fmffreeze\fmf{gluon}{v2,o1} \fmf{gluon,rubout}{v1,o2}
}}\qquad{}+\qquad\leavevmode\resizebox{61.17325pt}{}{
\parbox{85.35826pt}{\fmfgraph*(31,24) \fmfleftn{i}{2} \fmfrightn{o}{2}
\fmflabel{$q(p_{1})$}{i2} \fmflabel{$\bar{q}(p_{2})$}{i1}
\fmflabel{$g(p_{4})$}{o2} \fmflabel{$g(p_{3})$}{o1} \fmf{fermion}{i2,v2}
\fmf{gluon}{v2,v1} \fmf{fermion}{v2,i1} \fmf{gluon}{v1,o1} \fmf{gluon}{v1,o2}
}}$ Figure 3: Born Feynman diagrams of the reaction $q+\bar{q}\rightarrow
g+g$.

The Born amplitude can be written as

$\displaystyle\mathcal{M}^{B}_{q\bar{q}\rightarrow 2g}$
$\displaystyle=\bar{v}_{i_{2}}(p_{2})\,\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}a_{4}}\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4}).$
(F.1)

In the soft approximation, it is necessary to consider only the emission of an
extra gluon on the external legs as depicted in fig. 4.

(31,24) i2 o3 $q(p_{1})$i2 $\bar{q}(p_{2})$i1 $g(p_{5})$o3 $g(p_{4})$o2
$g(p_{3})$o1 fermionv1,i1 fermioni2,v3,v1 gluonv1,o1 gluonv1,o2 gluonv3,o3
0.20wv1

(a) (M1)

(31,24) i2 o3 $q(p_{1})$i2 $\bar{q}(p_{2})$i1 $g(p_{4})$o3 $g(p_{3})$o2
$g(p_{5})$o1 fermionv1,v3,i1 fermioni2,v1 gluonv1,o2 gluonv1,o3 gluonv3,o1
0.20wv1

(b) (M2)

(31,24) i2 o3 $q(p_{1})$i2 $\bar{q}(p_{2})$i1 $g(p_{4})$o3 $g(p_{5})$o2
$g(p_{3})$o1 fermionv1,i1 fermioni2,v1 gluonv1,v3,o1 gluonv1,o3 gluonv3,o2
0.20wv1

(c) (M3)

(31,24) i2 o3 $q(p_{1})$i2 $\bar{q}(p_{2})$i1 $g(p_{4})$o3 $g(p_{5})$o2
$g(p_{3})$o1 fermionv1,i1 fermioni2,v1 gluonv1,o1 gluonv1,v3,o3 gluonv3,o2
0.20wv1

(d) (M4)

Figure 4: Feynman diagrams depicting the soft gluon emission associated to
$q+\bar{q}\rightarrow g+g$.

The different amplitudes are

$\displaystyle\mathcal{M}_{1}$
$\displaystyle=-g_{s}\,\frac{p_{1}^{\mu_{5}}}{p_{1}\cdot
p_{5}}\,\bar{v}_{i_{2}}(p_{2})\,\Gamma^{\mu_{3}\mu_{4}}_{i_{2}j_{1};a_{3}a_{4}}\,\left(T^{a_{5}}\right)_{j_{1}i_{1}}\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4})\,\varepsilon_{\mu_{5}}^{a_{5}}(p_{5}),$
(F.2) $\displaystyle\mathcal{M}_{2}$
$\displaystyle=g_{s}\,\frac{p_{2}^{\mu_{5}}}{p_{2}\cdot
p_{5}}\,\bar{v}_{i_{2}}(p_{2})\,\Gamma^{\mu_{3}\mu_{4}}_{j_{2}i_{1};a_{3}a_{4}}\,\left(T^{a_{5}}\right)_{i_{2}j_{2}}\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4})\,\varepsilon_{\mu_{5}}^{a_{5}}(p_{5}),$
(F.3) $\displaystyle\mathcal{M}_{3}$
$\displaystyle=-g_{s}\,\frac{p_{3}^{\mu_{5}}}{p_{3}\cdot
p_{5}}\,\bar{v}_{i_{2}}(p_{2})\,\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};b_{3}a_{4}}\,i\,f^{b_{3}a_{5}a_{3}}\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4})\,\varepsilon_{\mu_{5}}^{a_{5}}(p_{5}),$
(F.4) $\displaystyle\mathcal{M}_{4}$
$\displaystyle=-g_{s}\,\frac{p_{4}^{\mu_{5}}}{p_{4}\cdot
p_{5}}\,\bar{v}_{i_{2}}(p_{2})\,\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}b_{4}}\,i\,f^{b_{4}a_{5}a_{4}}\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4})\,\varepsilon_{\mu_{5}}^{a_{5}}(p_{5}).$
(F.5)

In the Born amplitude, using colour decomposition, the string
$\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}a_{4}}$ can be written as

$\displaystyle\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}a_{4}}$
$\displaystyle=\left(T^{a_{4}}\,T^{a_{3}}\right)_{i_{2}i_{1}}\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})+\left(T^{a_{3}}\,T^{a_{4}}\right)_{i_{2}i_{1}}\,A^{\mu_{4}\mu_{3}}(p_{4},p_{3}),$
(F.6)

with

$\displaystyle A^{\mu_{3}\mu_{4}}(p_{3},p_{4})$
$\displaystyle=\left(-i\,g_{s}^{2}\right)\,\left[\gamma^{\mu_{4}}\,\frac{\mbox{$\not{\\!p}$}_{1}-\mbox{$\not{\\!p}$}_{3}}{(p_{1}-p_{3})^{2}}\,\gamma^{\mu_{3}}-\gamma^{\rho}\,\frac{V^{\mu_{3}\mu_{4}\rho}(-p_{3},-p_{4},p_{3}+p_{4})}{(p_{1}+p_{2})^{2}}\right].$
(F.7)

The tensor $V^{\mu\nu\rho}(p,q,r)$ is the usual tensor appearing in the QCD
Feynman rules for the triple gluon vertex

$V^{\mu\nu\rho}(p,q,r)=g^{\mu\,\nu}\,(p-q)^{\rho}+g^{\nu\,\rho}\,(q-r)^{\mu}+g^{\rho\,\mu}\,(r-p)^{\nu}.$
(F.8)

Let us define the soft amplitude $\mathcal{M}$ as
$\mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}+\mathcal{M}_{3}+\mathcal{M}_{4}$.
Because of the gauge invariance, replacing $\varepsilon^{\mu_{5}}(p_{5})$ by
$p_{5}^{\mu_{5}}$ in the soft amplitude leads to the following relation

$\displaystyle\Gamma^{\mu_{3}\mu_{4}}_{i_{2}j_{1};a_{3}a_{4}}\,\left(T^{a_{5}}\right)_{j_{1}i_{1}}-\Gamma^{\mu_{3}\mu_{4}}_{j_{2}i_{1};a_{3}a_{4}}\,\left(T^{a_{5}}\right)_{i_{2}j_{2}}+\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};b_{3}a_{4}}\,i\,f^{b_{3}a_{5}a_{3}}+\Gamma^{\mu_{3}\mu_{4}}_{i_{2}i_{1};a_{3}b_{4}}\,i\,f^{b_{4}a_{5}a_{4}}$
$\displaystyle=0.$ (F.9)

This relation can be checked explicitly by injecting eq. (F.6) into eq. (F.9)
and by using $i\,f^{abc}\,T^{a}=[T^{b},T^{c}]$. The decomposition given by eq.
(F.6) yields for the soft amplitude

$\displaystyle\mathcal{M}$
$\displaystyle=-g_{s}\,\bar{v}_{i_{2}}(p_{2})\,\left\\{\left(T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\right)_{i_{2}i_{1}}\,\left[\frac{p_{1}^{\mu_{5}}}{p_{1}\cdot
p_{5}}-\frac{p_{3}^{\mu_{5}}}{p_{3}\cdot
p_{5}}\right]\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})\right.$
$\displaystyle\qquad\qquad\qquad\quad{}+\left(T^{a_{5}}\,T^{a_{4}}\,T^{a_{3}}\right)_{i_{2}i_{1}}\,\left[\frac{p_{4}^{\mu_{5}}}{p_{4}\cdot
p_{5}}-\frac{p_{2}^{\mu_{5}}}{p_{2}\cdot
p_{5}}\right]\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})$
$\displaystyle\qquad\qquad\qquad\quad{}+\left(T^{a_{4}}\,T^{a_{5}}\,T^{a_{3}}\right)_{i_{2}i_{1}}\,\left[\frac{p_{3}^{\mu_{5}}}{p_{3}\cdot
p_{5}}-\frac{p_{4}^{\mu_{5}}}{p_{4}\cdot
p_{5}}\right]\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})$
$\displaystyle\qquad\qquad\qquad\quad{}+\left.3\leftrightarrow
4\vphantom{\left(T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\right)_{i_{2}i_{1}}\,\left[\frac{p_{1}^{\mu_{5}}}{p_{1}\cdot
p_{5}}-\frac{p_{3}^{\mu_{5}}}{p_{3}\cdot
p_{5}}\right]}\right\\}\,\,u_{i_{1}}(p_{1})\,\varepsilon_{\mu_{3}}^{a_{3}}(p_{3})\,\varepsilon_{\mu_{4}}^{a_{4}}(p_{4})\,\varepsilon_{\mu_{5}}^{a_{5}}(p_{5}).$
(F.10)

Under this form, the invariance
$\varepsilon^{\mu_{5}}(p_{5})\rightarrow\varepsilon^{\mu_{5}}(p_{5})+\lambda\,p_{5}^{\mu_{5}}$
is explicit.

Squaring the soft amplitude generates terms which are the product of a trace
on the colour matrices times a trace on the $\gamma$ matrices times an eikonal
factor $E_{ij}\equiv p_{i}\cdot p_{j}/(p_{i}\cdot p_{5}\,p_{j}\cdot p_{5})$.
There are four independent types of colour matrices which can be easily
evaluated. We want to enforce the appearance of the Casimirs of the
fundamental and the adjoint representations of the $SU(N_{c})$ Lie group
(resp. $C_{F}$ and $C_{A}$) in the colour factors for reasons that will be
clear later on. To do so, we make use of

$\displaystyle(T^{a}\,T^{a})_{ij}$ $\displaystyle=C_{F}\,\delta_{ij},$
$\displaystyle f_{abc}\,f_{dbc}$ $\displaystyle=C_{A}\,\delta_{ad},$ (F.11)

where $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and $C_{A}=N_{c}$. We get

$\displaystyle\text{Tr}\left[T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\,T^{a_{5}}\,T^{a_{3}}\,T^{a_{4}}\right]$
$\displaystyle=C_{F}^{3}\,N_{c},$ (F.12)
$\displaystyle\text{Tr}\left[T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\,T^{a_{5}}\,T^{a_{4}}\,T^{a_{3}}\right]$
$\displaystyle=C_{F}^{2}\,N_{c}\,\left(C_{F}-\frac{C_{A}}{2}\right),$ (F.13)
$\displaystyle\text{Tr}\left[T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\,T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\right]$
$\displaystyle=\frac{1}{2}\,C_{F}\,N_{c}\,(C_{A}-C_{F})\,(C_{A}-2\,C_{F}),$
(F.14)
$\displaystyle\text{Tr}\left[T^{a_{4}}\,T^{a_{3}}\,T^{a_{5}}\,T^{a_{3}}\,T^{a_{4}}\,T^{a_{5}}\right]$
$\displaystyle=\frac{1}{4}\,C_{F}\,N_{c}\,(C_{A}-2\,C_{F})^{2}.$ (F.15)

Concerning the traces of the $\gamma$ matrices, there is no new trace to
compute. It is rather clear from eqs. (F.1), (F.2), (F.3), (F.4), and (F.5)
that they will be the same as those appearing in the computation of the Born
amplitude squared, namely

$\displaystyle\text{Tr}\left[\mbox{$\not{\\!p}$}_{2}\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})\,\mbox{$\not{\\!p}$}_{1}\,A^{\dagger\,\nu_{3}\nu_{4}}(p_{3},p_{4})\right]\,\Pi_{\mu_{3}\nu_{3}}(p_{3},q_{3})\,\Pi_{\mu_{4}\nu_{4}}(p_{4},q_{4})$
$\displaystyle=8\,g_{s}^{4}\,\left(\frac{u}{t}-2\,\frac{u^{2}}{s^{2}}\right),$
(F.16)
$\displaystyle\text{Tr}\left[\mbox{$\not{\\!p}$}_{2}\,A^{\mu_{4}\mu_{3}}(p_{4},p_{3})\,\mbox{$\not{\\!p}$}_{1}\,A^{\dagger\,\nu_{4}\nu_{3}}(p_{4},p_{3})\right]\,\Pi_{\mu_{3}\nu_{3}}(p_{3},q_{3})\,\Pi_{\mu_{4}\nu_{4}}(p_{4},q_{4})$
$\displaystyle=8\,g_{s}^{4}\,\left(\frac{t}{u}-2\,\frac{t^{2}}{s^{2}}\right),$
(F.17)
$\displaystyle\text{Tr}\left[\mbox{$\not{\\!p}$}_{2}\,A^{\mu_{3}\mu_{4}}(p_{3},p_{4})\,\mbox{$\not{\\!p}$}_{1}\,A^{\dagger\,\nu_{4}\nu_{3}}(p_{4},p_{3})\right]\,\Pi_{\mu_{3}\nu_{3}}(p_{3},q_{3})\,\Pi_{\mu_{4}\nu_{4}}(p_{4},q_{4})$
$\displaystyle=8\,g_{s}^{4}\,\frac{t^{2}+u^{2}}{s^{2}},$ (F.18)

where the symbols $s$, $t$ and $u$ are the usual Mandelstam variables:
$s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{3})^{2}$ and $u=(p_{2}-p_{3})^{2}$ and

$\displaystyle\Pi_{\nu\tau}(p_{i},q_{i})$
$\displaystyle=\left(-g_{\nu\tau}+\frac{p_{i\,\nu}\,q_{i\,\tau}+p_{i\,\tau}\,q_{i\,\nu}}{p_{i}\cdot
q_{i}}\right)\quad\text{for $i=3,4$}.$ (F.19)

The arbitrary four-momentum $q_{i}$ in eq. (F.19) is light-like and not
collinear to $p_{i}$. Also, the square matrix element does not depend on the
choice of $q_{3}$ and $q_{4}$ but the different traces do. To get the right
hand side of eqs. (F.16), (F.17) and (F.18), we have used that $q_{3}=p_{4}$
and $q_{4}=p_{3}$. Putting everything together and after some algebra, we end
up with the following result for the squared amplitude not averaged over the
initial polarisations and colours

$\displaystyle|\mathcal{M}_{q\bar{q}\rightarrow 3g}|^{2}$
$\displaystyle=4\,g_{s}^{6}\,C_{F}\,N_{c}\,\left\\{E_{12}\,(u^{2}+t^{2})\,(C_{A}-2\,C_{F})\left[\frac{C_{A}-2\,C_{F}}{t\,u}+\frac{2\,C_{A}}{s^{2}}\right]\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left(E_{13}+E_{24}\right)\,C_{A}\,\left[2\,C_{F}\left(\frac{t}{u}+\frac{u}{t}\right)-C_{A}\left(\frac{t}{u}+2\,\frac{u^{2}}{s^{2}}\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\left(E_{14}+E_{23}\right)\,C_{A}\,\left[2\,C_{F}\left(\frac{t}{u}+\frac{u}{t}\right)-C_{A}\left(\frac{u}{t}+2\,\frac{t^{2}}{s^{2}}\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\left.E_{34}\,(t^{2}+u^{2})\,C_{A}^{2}\,\left[\frac{1}{t\,u}-\frac{2}{s^{2}}\right]\right\\}.$
(F.20)

This result is in agreement with the one given in ref. [54]. The coefficients
$H_{ij}(0)$ can be read directly from eq. (F.20) 171717The factor $g_{s}^{6}$
is not included in the coefficients $H_{ij}(0)$, cf. eq. (2.29).. Note that
the QED case ($q+\bar{q}\rightarrow\gamma+\gamma+\gamma$) is obtained by
setting $C_{A}\equiv 0$, $C_{F}\equiv 1$ and replacing $g_{s}$ by $e\,Q_{f}$,
where $Q_{f}$ is the electric charge of the initial fermions181818Note that
there remains a global factor $N_{c}$, which corresponds to the number of
colours of the initial state $q\,\bar{q}$.. We get:

$\displaystyle|\mathcal{M}_{{\tiny q\bar{q}\rightarrow 3\gamma}}|^{2}$
$\displaystyle=16\,e^{6}\,Q_{f}^{6}\,N_{c}\,E_{12}\,\left(\frac{t}{u}+\frac{u}{t}\right).$

After that, the only surviving factor is $H_{12}(0)$ which is proportional to
the Born squared amplitude. In the QCD case, as discussed previously, due to
the fact that the soft gluon carries colour charge, these coefficients
$H_{ij}(0)$ are not always proportional to the Born squared amplitude as seen
in eq. (F.20). Having the different $H_{ij}(0)$ at our disposal, we can verify
the cancellation of soft divergences between the real emission and the virtual
ones, as described in equation (4.9). We can extract the values for the
factors $\mathcal{A}^{(n)}$ and $\mathcal{C}^{(n)}_{ij}$ in front of the
divergences from reference [55], considering the structure of the virtual
corrections outlined in equation (4.8). We indeed verify that

$\displaystyle H^{(4)}_{12}(0)+H^{(4)}_{34}(0)$
$\displaystyle=8\,C_{F}\,N_{c}\,\left[\frac{(C_{A}-C_{F})^{2}+C_{F}^{2}}{t\,u}-\frac{2\,C_{A}\,C_{F}}{s^{2}}\right]\,(t^{2}+u^{2}),$
(F.21) $\displaystyle H^{(4)}_{13}(0)+H^{(4)}_{24}(0)$
$\displaystyle=8\,C_{F}\,C_{A}\,N_{c}\,\left[2\,C_{F}\,\left(\frac{t}{u}+\frac{u}{t}\right)-C_{A}\,\left(\frac{t}{u}+2\frac{u^{2}}{s^{2}}\right)\right],$
(F.22) $\displaystyle H^{(4)}_{14}(0)+H^{(4)}_{23}(0)$
$\displaystyle=8\,C_{F}\,C_{A}\,N_{c}\,\left[2\,C_{F}\,\left(\frac{t}{u}+\frac{u}{t}\right)-C_{A}\,\left(\frac{u}{t}+2\frac{t^{2}}{s^{2}}\right)\right],$
(F.23) $\displaystyle\sum_{i=1}^{3}\sum_{j=i+1}^{4}H^{(4)}_{ij}(0)$
$\displaystyle=2\,\,(C_{A}+C_{F})\,|M_{q\bar{q}\rightarrow gg}^{B}|^{2},$
(F.24)

where $|M_{q\bar{q}\rightarrow gg}^{B}|^{2}$ is the squared Born amplitude not
averaged over spins and colours and stripped from the coupling constant of the
reaction $q\bar{q}\rightarrow gg$. It is given by

$\displaystyle|M_{q\bar{q}\rightarrow gg}^{B}|^{2}$
$\displaystyle=8\,C_{F}\,N_{c}\,\left[\frac{C_{F}}{t\,u}-\frac{C_{A}}{s^{2}}\right]\,(t^{2}+u^{2}),$
(F.25)

which is in agreement with ref. [55].

##  Appendix G Notations

In this article, we introduced new notations to maintain the formulas as
compact as possible. This appendix serves as a glossary where most of the
notations used are summarised. We have denoted by $\\{x\\}_{N}$ a sequence of
labelled symbols $x_{i}$ where the index $i$ runs from 1 to $N$

$\displaystyle\\{x\\}_{N}\equiv x_{1},\,x_{2},\cdots,\,x_{N}.$ (G.1)

We also denoted by $(x)_{N}$ a collection (a sequence without comma) of
symbols indexed by an integer

$\displaystyle(x)_{N}\equiv x_{1}\,x_{2}\cdots\,x_{N}.$ (G.2)

And a symbol to describe a generic reaction where two bodies produce $N-2$
bodies, with all the bodies labelled by $i$ indexed by an integer running from
1 to $N$, is introduced as follows:

$\displaystyle[i]_{N}\equiv i_{1}+i_{2}\rightarrow i_{3}+\cdots+i_{N}.$ (G.3)

Furthermore, we also introduced a notation to describe a sequence, collection,
or reaction where one symbol is replaced by another, for example:

$\displaystyle\left\\{x|x_{i}:z_{i}\right\\}_{N}$ $\displaystyle\equiv
x_{1},\,x_{2},\cdots,\,x_{i-1},z_{i},\,x_{i+1},\cdots,\,x_{N},$ (G.4)
$\displaystyle(i|i_{k}:j_{k})_{N}$ $\displaystyle\equiv
i_{1}\,i_{2}\cdots\,i_{k-1}\,j_{k}\,i_{k+1}\,\cdots\,i_{N},$ (G.5)
$\displaystyle[i|i_{k}:j_{k}]_{N}$ $\displaystyle\equiv i_{1}+i_{2}\rightarrow
i_{3}+\cdots+i_{k-1}+j_{k}+i_{k+1}+\cdots+i_{N}.$ (G.6)

We also needed a special notation for the convolution. Specifically, when
considering the convolution of two multivariate functions
$f(a_{1},\ldots,a_{N})$ and $g(b_{1},\ldots,b_{K})$ where the convolution
involves only the variables $a_{k}$ and $b_{l}$, we denoted it as:

$\displaystyle\left[f\left(a_{1},\cdots,a_{k-1},*,a_{k+1},\cdots
a_{N}\right)\otimes
g(b_{1},\cdots,b_{l-1},*,b_{l+1},\cdots,b_{K})\right]_{\eta}(x)$
$\displaystyle\equiv\int_{x}^{1}\frac{dz}{z^{\eta}}\,f\left(a_{1},\cdots,a_{k-1},z,a_{k+1},\cdots
a_{N}\right)\,g\left(b_{1},\cdots,b_{l-1},\frac{x}{z},b_{l+1},\cdots,b_{K}\right).$
(G.7)

Note that we also adhere to the convention that if a function $h$ involved in
the convolution has only one argument, we write it in our special notation as
simply $h$ instead of $h(*)$. A word of caution regarding eq. (G.7): when
$\eta\neq 1$, the convolution product defined in this equation is non-
commutative. Therefore, in such cases, the function on the right-hand side of
the product corresponds to the PDFs, while the one on the left-hand side
corresponds to the DGLAP kernels.

To describe the measures for the different phase spaces, we introduced the
notations

$\displaystyle d\text{PS}_{N-1\,\text{h}}^{(n)}(\bar{x})$
$\displaystyle\equiv\prod_{l=3}^{N-1}\frac{d\bar{x}_{l}}{\bar{x}_{l}^{n-2}}\,dy_{l}\,d^{n-2}K_{T\,l},$
(G.8)

which is the measure of the phase space of the hadrons and

$\displaystyle d\text{PS}_{N}^{(n)}$ $\displaystyle\equiv
dy_{N}\,dx_{T\,N}\,x_{T\,N}^{n-5}\,d\phi_{N}\,(\sin\phi_{N})^{n-4},$ (G.9)

which is the measure of the phase space of the parton which can be soft and/or
collinear to the other ones divided by $x_{T\,N}^{2}$.

In addition, we provide a summary of the various variables used in this
article. Let us recall that $K_{i}$ represents the four-momentum of the hadron
labelled by $i$, which is parameterised by the rapidity $y_{i}$, the magnitude
of the transverse momentum $K_{T\,i}$, and the azimuthal angle between the
transverse momentum $\vec{K}_{T\,i}$ and a reference direction in the
transverse plane. We also introduced an arbitrary energy scale $Q$ in order to
build dimensionless variables. Furthermore, the parton $i$ which fragments
into the hadron $i$ carrying a fraction of four-momentum $x_{i}$ has a four-
momentum $p_{i}=K_{i}/x_{i}$. Additional variables introduced include:

$\displaystyle\omega$ $\displaystyle=Q/\sqrt{s},$ $\displaystyle X_{T\,i}$
$\displaystyle=\frac{K_{T\,i}}{Q},$ $\displaystyle
x_{T\,i}=\frac{p_{T\,i}}{Q}=\frac{K_{T\,i}}{x_{i}\,Q}.$

Due to the conservation of the energy and longitudinal momentum, the fractions
of four-momentum $x_{1}$ and $x_{2}$ carried away by the initial partons from
their parent hadrons are fixed

$\displaystyle x_{1}$ $\displaystyle=\hat{x}_{1}+\omega\,x_{T\,N}\,e^{y_{N}},$
$\displaystyle x_{2}$
$\displaystyle=\hat{x}_{2}+\omega\,x_{T\,N}\,e^{-y_{N}},$

with

$\displaystyle\hat{x}_{1}$
$\displaystyle=\sum_{l=3}^{N-1}\frac{K_{T\,l}}{\sqrt{s}\,x_{l}}\,e^{y_{l}},$
$\displaystyle\hat{x}_{2}$
$\displaystyle=\sum_{l=3}^{N-1}\frac{K_{T\,l}}{\sqrt{s}\,x_{l}}\,e^{-y_{l}}.$

Note that the constraint coming from the conservation of the transverse
momentum has not been taken into account. This constraint can be expressed as:

$\displaystyle\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{x_{l}}+\rm{N}$
$\displaystyle=0.$

At leading order, these fractions of four-momentum are denoted $\bar{x}_{1}$
and $\bar{x}_{2}$, and they are given by:

$\displaystyle\bar{x}_{1}$
$\displaystyle=\sum_{l=3}^{N-1}\frac{K_{T\,l}}{\sqrt{s}\,\bar{x}_{l}}\,e^{y_{l}},$
$\displaystyle\bar{x}_{2}$
$\displaystyle=\sum_{l=3}^{N-1}\frac{K_{T\,l}}{\sqrt{s}\,\bar{x}_{l}}\,e^{-y_{l}}.$

It’s worth noting that while the definition of $\bar{x}_{1}$ (and
$\bar{x}_{2}$) may resemble that of $\hat{x}_{1}$ (and $\hat{x}_{2}$), the
constraint on the transverse momenta is different at this order. Specifically,
it reads:

$\displaystyle\sum_{l=3}^{N-1}\frac{\vec{K}_{T\,l}}{\bar{x}_{l}}$
$\displaystyle=0,$

where we have named $\bar{x}_{i}$ the fraction of momentum carried by the
hadron from its parent parton by similarity.

The fact that $x_{1}$ and $x_{2}$ must be less than one determines the bounds
on the rapidity of the parton $i_{N}$ which can be soft and/or collinear.
These bounds are given by

$\displaystyle y_{N\,\text{max}}$
$\displaystyle=\ln\left(\frac{1-\hat{x}_{1}}{\omega\,x_{T\,N}}\right),$
$\displaystyle y_{N\,\text{min}}$
$\displaystyle=\ln\left(\frac{\omega\,x_{T\,N}}{1-\hat{x}_{2}}\right).$

For convenience while integrating over $y_{N}$, we have defined

$\displaystyle\Delta\,Y_{M}$ $\displaystyle=y_{N\,\text{max}}-y_{j},$
$\displaystyle\Delta\,Y_{m}$ $\displaystyle=y_{j}-y_{N\,\text{min}}.$

When the parton $i_{N}$ is emitted collinearly to another parton $i_{k}$, a
new variable $z_{k}$ is introduced which is bounded from above by
$z_{k\,\text{m}}$ outside the cylinder of size $x_{T\,\text{m}}$:

$\displaystyle z_{k}$ $\displaystyle=\frac{x_{T\,k}}{x_{T\,k}+x_{T\,N}},$
$\displaystyle z_{k\,\text{m}}$
$\displaystyle=\frac{x_{T\,k}}{x_{T\,k}+x_{T\,\text{m}}}.$

To enforce that the coefficient of the collinear singularity is a convolution
of a partonic cross section times a DGLAP kernel, we trade $x_{k}$ for
$\bar{x}_{k}=z_{k}\,x_{k}$. This bound becomes, for a hadron or a jet,

$\displaystyle\zeta_{k\,\text{m}}$
$\displaystyle=\frac{X_{T\,k}-\bar{x}_{k}\,x_{T\,\text{m}}}{X_{T\,k}},$
$\displaystyle\zeta_{\text{jet}\,\text{m}}$
$\displaystyle=\frac{X_{T\,\text{jet}}-x_{T\,\text{m}}}{X_{T\,\text{jet}}}.$

To conclude this appendix, we also needed to define quantities that enable us
to distinguish between two regimes in the case of initial state singularities.
Specifically,

$\displaystyle\beta_{1j}$ $\displaystyle=\omega\,\exp(y_{j})/(1-\bar{x}_{1}),$
$\displaystyle\beta_{2j}$
$\displaystyle=\omega\,\exp(-y_{j})/(1-\bar{x}_{2}).$

## References

  * [1] T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen.  A Full next-to-leading order study of direct photon pair production in hadronic collisions.  Eur. Phys. J., C16:311–330, 2000.
  * [2] T. Binoth, J. Ph. Guillet, E. Pilon, and M. Werlen.  Beyond leading order effects in photon pair production at the Tevatron.  Phys. Rev., D63:114016, 2001.
  * [3] S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini.  Diphoton production at the LHC: a QCD study up to NNLO.  JHEP, 04:142, 2018.
  * [4] T. Gehrmann, N. Glover, A. Huss, and J. Whitehead.  Scale and isolation sensitivity of diphoton distributions at the LHC.  JHEP, 01:108, 2021.
  * [5] T. Gehrmann and R. Schürmann.  Photon fragmentation in the antenna subtraction formalism.  JHEP, 04:031, 2022.
  * [6] X. Chen, T. Gehrmann, E. W. N. Glover, M. Höfer, A. Huss, and R. Schürmann.  Single photon production at hadron colliders at NNLO QCD with realistic photon isolation.  JHEP, 08:094, 2022.
  * [7] J. M. Campbell, R. K. Ellis, and C. Williams.  Direct Photon Production at Next-to–Next-to-Leading Order.  Phys. Rev. Lett., 118(22):222001, 2017.  [Erratum: Phys.Rev.Lett. 124, 259901 (2020)].
  * [8] X. Chen, T. Gehrmann, N. Glover, M. Höfer, and A. Huss.  Isolated photon and photon+jet production at NNLO QCD accuracy.  JHEP, 04:166, 2020.
  * [9] B. Mele and P. Nason.  Next-to-leading QCD calculation of the heavy quark fragmentation function.  Phys. Lett. B, 245:635–639, 1990.
  * [10] B. Mele and P. Nason.  The Fragmentation function for heavy quarks in QCD.  Nucl. Phys. B, 361:626–644, 1991.  [Erratum: Nucl.Phys.B 921, 841–842 (2017)].
  * [11] P. Nason and C. Oleari.  A Phenomenological study of heavy quark fragmentation functions in e+ e- annihilation.  Nucl. Phys. B, 565:245–266, 2000.
  * [12] M. Cacciari, P. Nason, and C. Oleari.  A Study of heavy flavored meson fragmentation functions in e+ e- annihilation.  JHEP, 04:006, 2006.
  * [13] D. P. Anderle, T. Kaufmann, M. Stratmann, F. Ringer, and I. Vitev.  Using hadron-in-jet data in a global analysis of $D^{*}$ fragmentation functions.  Phys. Rev. D, 96(3):034028, 2017.
  * [14] M. Czakon, T. Generet, A. Mitov, and R. Poncelet.  NNLO B-fragmentation fits and their application to $t\overline{t}$ production and decay at the LHC.  JHEP, 03:251, 2023.
  * [15] J. Binnewies, Bernd A. Kniehl, and G. Kramer.  Predictions for D*+- - photoproduction at HERA with new fragmentation functions from LEP-1.  Phys. Rev. D, 58:014014, 1998.
  * [16] B. A. Kniehl and G. Kramer.  D0, D+, D+(s), and Lambda+(c) fragmentation functions from CERN LEP1.  Phys. Rev. D, 71:094013, 2005.
  * [17] B. A. Kniehl and G. Kramer.  Charmed-hadron fragmentation functions from CERN LEP1 revisited.  Phys. Rev. D, 74:037502, 2006.
  * [18] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  Finite-mass effects on inclusive $B$ meson hadroproduction.  Phys. Rev. D, 77:014011, 2008.
  * [19] T. Kneesch, B. A. Kniehl, G. Kramer, and I. Schienbein.  Charmed-meson fragmentation functions with finite-mass corrections.  Nucl. Phys. B, 799:34–59, 2008.
  * [20] M. Soleymaninia, H. Khanpour, and S. M. Moosavi Nejad.  First determination of $D^{*+}$-meson fragmentation functions and their uncertainties at next-to-next-to-leading order.  Phys. Rev. D, 97(7):074014, 2018.
  * [21] M. Salajegheh, S. M. Moosavi Nejad, H. Khanpour, B. A. Kniehl, and M. Soleymaninia.  $B$-hadron fragmentation functions at next-to-next-to-leading order from a global analysis of $e^{+}e^{-}$ annihilation data.  Phys. Rev. D, 99(11):114001, 2019.
  * [22] M. Salajegheh, S. . Moosavi Nejad, M. Soleymaninia, H. Khanpour, and S. Atashbar Tehrani.  NNLO charmed-meson fragmentation functions and their uncertainties in the presence of meson mass corrections.  Eur. Phys. J. C, 79(12):999, 2019.
  * [23] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  $\Lambda_{c}^{\pm}$ production in $pp$ collisions with a new fragmentation function.  Phys. Rev. D, 101(11):114021, 2020.
  * [24] M. Delpasand, S. M. Moosavi Nejad, and M. Soleymaninia.  $\Lambda_{c}^{+}$ fragmentation functions from pQCD approach and the Suzuki model.  Phys. Rev. D, 101(11):114022, 2020.
  * [25] M. Cacciari, M. Greco, and P. Nason.  The P(T) spectrum in heavy flavor hadroproduction.  JHEP, 05:007, 1998.
  * [26] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  Inclusive D*+- production in p anti-p collisions with massive charm quarks.  Phys. Rev. D, 71:014018, 2005.
  * [27] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  Collinear subtractions in hadroproduction of heavy quarks.  Eur. Phys. J. C, 41:199–212, 2005.
  * [28] I. Helenius and H. Paukkunen.  Revisiting the D-meson hadroproduction in general-mass variable flavour number scheme.  JHEP, 05:196, 2018.
  * [29] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  Inclusive Charmed-Meson Production at the CERN LHC.  Eur. Phys. J. C, 72:2082, 2012.
  * [30] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  Inclusive $B$-meson production at small $p_{T}$ in the general-mass variable-flavor-number scheme.  Eur. Phys. J. C, 75(3):140, 2015.
  * [31] M. Benzke, B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger.  B-meson production in the general-mass variable-flavour-number scheme and LHC data.  Eur. Phys. J. C, 79(10):814, 2019.
  * [32] G. Altarelli, R. Keith Ellis, G. Martinelli, and So-Young Pi.  Processes Involving Fragmentation Functions Beyond the Leading Order in QCD.  Nucl. Phys. B, 160:301–329, 1979.
  * [33] F. Aversa, P. Chiappetta, Mario Greco, and J. P. Guillet.  QCD Corrections to Parton-Parton Scattering Processes.  Nucl. Phys., B327:105, 1989.
  * [34] S. Kretzer and I. Schienbein.  Heavy quark fragmentation in deep inelastic scattering.  Phys. Rev. D, 59:054004, 1999.
  * [35] M. Cacciari and S. Catani.  Soft gluon resummation for the fragmentation of light and heavy quarks at large x.  Nucl. Phys. B, 617:253–290, 2001.
  * [36] T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen.  A next-to-leading order study of pion pair production and comparison with E706 data.  Eur. Phys. J., C24:245–260, 2002.
  * [37] J. F. Owens.  A Next-to-leading order study of dihadron production.  Phys. Rev. D, 65:034011, 2002.
  * [38] W. T. Giele and E. W. Nigel Glover.  Higher order corrections to jet cross-sections in e+ e- annihilation.  Phys. Rev. D, 46:1980–2010, 1992.
  * [39] W. T. Giele, E. W. Nigel Glover, and David A. Kosower.  Higher order corrections to jet cross-sections in hadron colliders.  Nucl. Phys. B, 403:633–670, 1993.
  * [40] S. Keller and E. Laenen.  Next-to-leading order cross-sections for tagged reactions.  Phys. Rev. D, 59:114004, 1999.
  * [41] B. W. Harris and J. F. Owens.  The Two cutoff phase space slicing method.  Phys. Rev. D, 65:094032, 2002.
  * [42] S. Frixione, Z. Kunszt, and A. Signer.  Three jet cross-sections to next-to-leading order.  Nucl. Phys. B, 467:399–442, 1996.
  * [43] S. Catani and M. H. Seymour.  A General algorithm for calculating jet cross-sections in NLO QCD.  Nucl. Phys. B, 485:291–419, 1997.  [Erratum: Nucl.Phys.B 510, 503–504 (1998)].
  * [44] S. Frixione.  A General approach to jet cross-sections in QCD.  Nucl. Phys. B, 507:295–314, 1997.
  * [45] D. A. Kosower.  Antenna factorization of gauge theory amplitudes.  Phys. Rev. D, 57:5410–5416, 1998.
  * [46] Z. Kunszt and D. E. Soper.  Calculation of jet cross-sections in hadron collisions at order alpha-s**3.  Phys. Rev. D, 46:192–221, 1992.
  * [47] P. Chiappetta, R. Fergani, and J. P. Guillet.  Production of two large p(T) hadrons from hadronic collisions.  Z. Phys., C69:443–457, 1996.
  * [48] C.Y. Liu, X.M. Shen, B. Zhou, and J. Gao.  Automated calculation of jet fragmentation at NLO in QCD.  JHEP, 09:108, 2023.
  * [49] L. Bourhis, M. Fontannaz, and J. P. Guillet.  Quarks and gluon fragmentation functions into photons.  Eur. Phys. J., C2:529–537, 1998.
  * [50] A. Mukherjee and W. Vogelsang.  Jet production in (un)polarized pp collisions: dependence on jet algorithm.  Phys. Rev. D, 86:094009, 2012.  [Erratum: Phys.Rev.D 107, 119901 (2023)].
  * [51] S. Catani, Y. L. Dokshitzer, M. H. Seymour, and B. R. Webber.  Longitudinally invariant $K_{t}$ clustering algorithms for hadron hadron collisions.  Nucl. Phys. B, 406:187–224, 1993.
  * [52] S. D. Ellis and D. E. Soper.  Successive combination jet algorithm for hadron collisions.  Phys. Rev. D, 48:3160–3166, 1993.
  * [53] M. Abramowitz and I. A. Stegun.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.  Dover, New York, ninth dover printing, tenth gpo printing edition, 1964\.
  * [54] R. Keith Ellis, G. Marchesini, and B. R. Webber.  Soft Radiation in Parton Parton Scattering.  Nucl. Phys. B, 286:643, 1987.  [Erratum: Nucl.Phys.B 294, 1180 (1987)].
  * [55] R. Keith Ellis and J. C. Sexton.  QCD Radiative Corrections to Parton Parton Scattering.  Nucl. Phys. B, 269:445–484, 1986.