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%% subsection 2.4 Infrared singular regions [slac-pub-7073-0-0-2-4 in slac-pub-7073-0-0-2: slac-pub-7073-0-0-2-5]
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\subsection{\usemenu{slac-pub-7073::context::slac-pub-7073-0-0-2-4}{Infrared singular regions}}\label{subsection::slac-pub-7073-0-0-2-4}
It is well known that the perturbatively calculated QCD cross
sections, even after the ultraviolet renormalization, have a divergent
behaviour, arising from the regions in which a parton (either virtual,
that is, exchanged in a loop, or real, that is, emitted and contributing
to the final state kinematics) is soft or collinear to another parton.
In this section, we will investigate the behaviour of the
measurement function in this regions (which we will denote as
infrared singular regions) in the case when there are four partons
in the final state. The singular regions eventually occurring
are as follows
$\bullet$ parton $i$ ($i=3,\,4,\,5,\,6$) is soft ($k_i^0\,\to\,0$);
$\bullet$ parton $i$ is collinear to the incoming parton $1$ or $2$
($i\parallel 1$ or $i\parallel 2$);
$\bullet$ parton $i$ is collinear to parton $j$ ($i\parallel j$).
\noindent
{}From eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[F1]{2.35})-(\docLink{slac-pub-7073-0-0-2.tcx}[F2a]{2.38}), it is quite easy to prove
that the ${\cal F}$ functions are non vanishing only in
the following singular regions
\beqn
{\cal F}_{jkl}^{(i,0)}\;\;\;\;&\Rightarrow&\;\;\;\;
k_i^0\,\to\,0,\;\;i\parallel 1,\;\;i\parallel 2\,,
\label{list1}
\\
{\cal F}_{kl}^{(ij,1)}\;\;\;\;&\Rightarrow&\;\;\;\;
k_i^0\,\to\,0,\;\;k_j^0\,\to\,0,\;\;i\parallel j\,,
\label{list4}
\\
{\cal F}_{jkl}^{(i,2)}\;\;\;\;&\Rightarrow&\;\;\;\;
none\,,
\label{list2}
\\
{\cal F}_{jkl}^{(i,3)}\;\;\;\;&\Rightarrow&\;\;\;\;
none\,.
\label{list3}
\eeqn
Eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[list4]{2.40}) suggests the following decomposition
\beq
{\cal S}_{ij}^{(1)}={\cal S}_{ij}^{(1)}\Th(d_j-d_i)
+{\cal S}_{ij}^{(1)}\Th(d_i-d_j)\,;
\label{Sij3dec}
\eeq
in this way, the first term in the RHS of eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[Sij3dec]{2.43}) does not
get contributions when $j$ is soft, and the second one when $i$ is soft.
This in turn implies that, after some algebra, we can cast
eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S40]{2.26}) in the following form
\beq
{\cal S}_4=\sum_i \left({\cal S}_i^{(sing)} + {\cal S}_i^{(fin)}\right),
\label{S4fin}
\eeq
where
\beqn
{\cal S}_i^{(sing)}&=&{\cal S}_i^{(0)}+
\sum_j^{[i]}{\cal S}_{ij}^{(1)}\Th(d_j-d_i)\,,
\label{S4sing}
\\*
{\cal S}_i^{(fin)}&=&{\cal S}_i^{(2)}+{\cal S}_i^{(3)}\,,
\label{S4ns}
\eeqn
and the $[i]$ in the sum means that $j$ can take the values
$3,\,4,\,5,\,6$ with the exclusion of $i$.
The quantity ${\cal S}_i^{(fin)}$ does not get any contribution
from the singular regions; on the other hand, ${\cal S}_i^{(sing)}$
is different from 0 when $i$ is soft (or $i\parallel 1$, $i\parallel 2$),
but it is equal to zero when any other parton is soft (or
collinear to the incoming partons), thanks to the factor $\Th(d_j-d_i)$.
The region in which $i\parallel j$ contributes to ${\cal S}_i^{(sing)}$
and to ${\cal S}_j^{(sing)}$, but the $\Th(d_j-d_i)$ inserted in
eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4sing]{2.45}) prevents any double counting.
We can finally investigate the form of the limiting behaviour
of the measurement function ${\cal S}_4$ in the infrared singular regions.
$\bullet$~$i$ is soft. From eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[list1]{2.39})-(\docLink{slac-pub-7073-0-0-2.tcx}[list3]{2.42})
and eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4fin]{2.44}), we have
\beq
\lim_{k_i^0\to 0}{\cal S}_4=\lim_{k_i^0\to 0}\Bigg[
{\cal S}_i^{(0)}+\sum_j^{[i]}{\cal S}_{ij}^{(1)}\Bigg].
\label{splitlim}
\eeq
The limit of the ${\cal F}$ functions can be very easily evaluated
from their definition; it simply amounts to make the formal substitution
$i+j\to j$ wherever $i+j$ appears. We still need to do some
combinatorial algebra to get the limit of the full ${\cal S}_4$;
after writing explicitly the terms contributing to the
sum and using
\beq
\lim_{k_i^0\to 0}\Bigg[\Th(m([i])-i)+
\sum_j^{[i]}\Th(m([i j])-i j)\Bigg]=1
\eeq
we get
\beq
\lim_{k_i^0\to 0}{\cal S}_4={\cal S}_3([i])\,,
\label{S4sftlim}
\eeq
where
\beqn
{\cal S}_3([i])&=&\sum_{\sigma(J)}\delta(j,k,l)\Th(m(j,k,l)-f\,M(j,k,l))
\nonumber \\*&&\,\,\times\,
\Th(R_{jk}-D^2)\Th(R_{jl}-D^2)\Th(R_{kl}-D^2).
\eeqn
Notice that this quantity is identical to the ${\cal S}_3$ function
of eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S3def]{2.24}), but for the fact that the indices $j$, $k$
and $l$ can also take the value $6$ (and the value $i$ is excluded).
Eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4sftlim]{2.49}) has an obvious physical meaning: when a parton
gets soft, the remaining partons act as physical jets. The measurement
function has then to coincide with the one defined for three partons
in the final state. We also point out that each term in the sum in
the RHS of eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[splitlim]{2.47}) has a well defined soft limit:
\beqn
\lim_{k_i^0\to 0}{\cal S}_i^{(0)}&=&{\cal S}_3([i])
\Th(R_{ij}-D^2)\Th(R_{ik}-D^2)\Th(R_{il}-D^2),
\\
\lim_{k_i^0\to 0}{\cal S}_{ij}^{(1)}&=&{\cal S}_3([i])
\Th(D^2-R_{ij})\Th(R_{ik}-R_{ij})\Th(R_{il}-R_{ij}).
\eeqn
$\bullet$~$i\parallel 1$. From eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[list1]{2.39})-(\docLink{slac-pub-7073-0-0-2.tcx}[list3]{2.42})
and eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4fin]{2.44}), we have
\beq
\lim_{\vec{k}_i\parallel\vec{k}_1}{\cal S}_4=
\lim_{\vec{k}_i\parallel\vec{k}_1}{\cal S}_i^{(0)}\,.
\eeq
It is quite easy to prove that, in this limit,
the following equation holds
\beq
\lim_{\vec{k}_i\parallel\vec{k}_1}\Th(m([i])-i)=1
\eeq
and therefore
\beq
\lim_{\vec{k}_i\parallel\vec{k}_1}{\cal S}_4={\cal S}_3([i])\,.
\label{S4clllim}
\eeq
The case in which $i\parallel 2$ is completely analogous and
gives an identical result.
$\bullet$~$i\parallel j$. From eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[list1]{2.39})-(\docLink{slac-pub-7073-0-0-2.tcx}[list3]{2.42})
and eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4fin]{2.44}), we have
\beq
\lim_{\vec{k}_i\parallel\vec{k}_j}{\cal S}_4=
\lim_{\vec{k}_i\parallel\vec{k}_j}{\cal S}_{ij}^{(1)}\,,
\label{S4ijlimtmp}
\eeq
where we have used the fact that ${\cal S}_{ij}^{(1)}$ is symmetric
in the exchange of $i$ and $j$ and
\beq
\Th(d_i-d_j)+\Th(d_j-d_i)\equiv 1.
\eeq
In the limit at hand, $d_{ij}\to 0$ and therefore
\beq
\lim_{\vec{k}_i\parallel\vec{k}_j}\Th(m([ij])-ij)=1\,,
\eeq
while all the other $\Th$ functions containing $d_{ij}$
in the list over which the minimum is evaluated are zero.
Therefore we have
\beq
\lim_{\vec{k}_i\parallel\vec{k}_j}{\cal S}_4=
{\cal S}_3([ij])\,,
\label{S4albelim}
\eeq
where
\beqn
{\cal S}_3([ij])&=&\sum_{\sigma(J)}\delta(p,k,l)\Th(m(p,k,l)-f\,M(p,k,l))
\nonumber \\*&&\,\,\times\,
\Th(R_{pk}-D^2)\Th(R_{pl}-D^2)\Th(R_{kl}-D^2).
\eeqn
and the indices $p$, $k$ and $l$ in the sum can now take also
the value $7$, having defined $k_7=k_i+k_j$
(notice that this formal manipulation allows to maintain the
notation used in the previous case; the meaning of eq.(\docLink{slac-pub-7073-0-0-2.tcx}[S4albelim]{2.59})
is that the final state collinear limit of the ${\cal S}_4$
function is again a jet-defining function of three partons into
three jets, where one of the partons has four-momentum equal to
the sum of the four-momenta of the partons becoming collinear).
We finally point out that in the numerical implementation of
the algorithm we will also need to consider the case where one
parton is soft {\it and} collinear to an incoming parton or to a final
state parton. In this case, we find
\beqn
\lim_{\vec{k}_i\parallel\vec{k}_1}
\lim_{k_i^0\to 0}{\cal S}_i^{(0)}&=&{\cal S}_3([i]),
\label{softcolllim1}
\\
\lim_{\vec{k}_i\parallel\vec{k}_j}
\lim_{k_i^0\to 0}{\cal S}_{ij}^{(1)}&=&{\cal S}_3([ij]).
\label{softcolllim2}
\eeqn
Notice that in eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[softcolllim1]{2.61}) and~(\docLink{slac-pub-7073-0-0-2.tcx}[softcolllim2]{2.62})
the order in which the limits are taken is irrelevant. Also,
eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[softcolllim1]{2.61}) holds true when $i\parallel 2$.
In the following, we will use the properties of the measurement
function ${\cal S}_4$ to disentangle the structure of the
singularities in the real contribution. In spite of this fact,
our method is completely general. In fact, we will basically
rely only upon eqs.~(\docLink{slac-pub-7073-0-0-2.tcx}[S4fin]{2.44})-(\docLink{slac-pub-7073-0-0-2.tcx}[S4ns]{2.46}), that is, on the
decomposition of the measurement function into terms which get
contribution from two singular regions at the worst. It should
be clear that such a decomposition can always be performed, being
essentially due to the infrared safeness requirement on the
measurement function.