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%% subsection 2.1 Introductory remarks [slac-pub-7073-0-0-2-1 in slac-pub-7073-0-0-2: ^slac-pub-7073-0-0-2 >slac-pub-7073-0-0-2-2]
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\sectionLink{slac-pub-7073-0-0-2}{slac-pub-7073-0-0-2}{Above: 2. The jet cross section}%
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\subsection{\usemenu{slac-pub-7073::context::slac-pub-7073-0-0-2-1}{Introductory remarks}}\label{subsection::slac-pub-7073-0-0-2-1}
Thanks to the factorization theorem~[\docLink{slac-pub-7073-0-0-8.tcx}[CSS]{23}], a generic differential
cross section in hadronic collisions can be written in the
following way
\beq
d\sigma^{(H_1 H_2)}(K_1,K_2)=\sum_{ab}\int dx_1 dx_2 f^{(H_1)}_a(x_1)
f^{(H_2)}_b(x_2)d\hat{\sigma}_{ab}(x_1 K_1,x_2 K_2)\,,
\label{factth}
\eeq
where $H_1$ and $H_2$ are the incoming hadrons, $K_1$ and $K_2$
their momentum, and the sum runs over all the parton flavours
which give a non-trivial contribution. The quantities
$d\hat{\sigma}_{ab}$ are the {\it subtracted} partonic cross sections,
in which the singularities due to collinear emission
of massless partons from the incoming partons have been cancelled
by some suitable counterterms.
We first express the subtracted cross sections in terms of
the unsubtracted ones, which can be directly calculated in perturbative
QCD. To this end, we have to write the collinear counterterms.
Due to universality, eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[factth]{2.1}) applies also when
the incoming hadrons are formally substituted with partons.
In this case, we are also able to evaluate the partonic
densities, which at the next-to-leading order read
\beq
f^{(d)}_a(x)=\delta_{ad}\delta(1-x)-\frac{\as}{2\pi}
\left(\frac{1}{\epb}P_{ad}(x,0)-K_{ad}(x)\right)
+{\cal O}\left(\as^2\right),
\eeq
where $P_{ad}(x,0)$ are the Altarelli-Parisi kernels in four
dimensions (since we will usually work in $4-2\ep$ dimensions, the $0$
in the argument of $P_{ad}$ stands for $\ep=0$) and the functions
$K_{ad}$ depend upon the subtraction scheme in which the calculation
is carried out. For ${\rm \overline{MS}}$, $K_{ad}\equiv 0$. Writing
the perturbative expansion of the unsubtracted and subtracted partonic
cross sections at next-to-leading order as
\beq
d\sigma_{ab}=d\sigma_{ab}^{(0)}+d\sigma_{ab}^{(1)}\,,\;\;\;\;
d\hat{\sigma}_{ab}=d\hat{\sigma}_{ab}^{(0)}+d\hat{\sigma}_{ab}^{(1)}\,,
\label{decomposition}
\eeq
where the superscript 0 (1) denotes the leading (next-to-leading)
order contribution, we have
\beqn
d\hat{\sigma}_{ab}^{(0)}(k_1,k_2)&=&d\sigma_{ab}^{(0)}(k_1,k_2)
\\*
d\hat{\sigma}_{ab}^{(1)}(k_1,k_2)&=&d\sigma_{ab}^{(1)}(k_1,k_2)
+\frac{\as}{2\pi}\sum_d\int dx\left(\frac{1}{\epb}P_{da}(x,0)
-K_{da}(x)\right)d\sigma_{db}^{(0)}(xk_1,k_2)
\nonumber \\*&&
+\frac{\as}{2\pi}\sum_d\int dx\left(\frac{1}{\epb}P_{db}(x,0)
-K_{db}(x)\right)d\sigma_{ad}^{(0)}(k_1,xk_2)\,.
\label{counterterms}
\eeqn
The second and the third term in the RHS of eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[counterterms]{2.5})
are the collinear counterterms we were looking for. Notice that in this
equation the Born terms $d\sigma^{(0)}$ are evaluated in
$4-2\ep$ dimensions.
For three-jet production, the leading-order cross section
can get contributions only from the two-to-three partonic
subprocesses. We write this contribution in the following way
\beqn
d\sigma_{\aoat}^{(0)}(k_1,k_2;\Jetlist)&=&\uotfct
\FLTsum\MTz(\ArgTfull)
\nonumber \\*&\times&
{\cal S}_3(\STj;\Jetlist)d\PHITsj\,.
\label{bornjetdef}
\eeqn
Here we denoted with
\beq
\Jetlist\,=\,\{J_1,\,J_2,\,J_3\}
\eeq
the set of the four-momenta of the jets. In the following, we will
almost always suppress the indication of the $\Jetlist$ dependence.
We have indicated
with $k_i$ and $a_i$ respectively the momentum and the
flavour of the parton number $i$ involved in the process; by
definition, partons $1$ and $2$ are the incoming ones.
In the sum $\FLTsum$ every $a_l$, with $3\leq l\leq 5$, takes the values
$g$, $u$, $\bar{u}$, and so on. To avoid overcounting in physical
predictions, we inserted the factor $1/3!$. To shorten as much as
possible the notation, we have collectively indicated the momenta as
\beq
\{k_l\}_{i,j}\,\equiv\,\{k_l\,|\,i\leq l\leq j\,\}\,.
\eeq
It will also turn useful to define
\beq
\SFjexcl\equiv \{k_l\,|\,i\leq l\leq j,\,\,l\neq n,\,l\neq p,\,..\}\,.
\eeq
The same notation will be used for flavours. The quantity
${\cal S}_3$ is the so-called measurement function, which defines
the infrared-safe jet observables in terms of the momenta of the
(unobservable) partons; we will describe it in more details in the
following. $\MTz$ is the two-to-three {\it leading-order} transition
amplitude squared, summed over final state and averaged over
initial state color and spin degrees of freedom, and multiplied
by the flux factor
\beq
\MTz=\frac{1}{2k_1\cdot k_2}\,\frac{1}{\omega(a_1)\omega(a_2)}
\sum_{\stackrel{color}{spin}}\abs{{\cal A}^{(tree)}(2\to 3)}^2\,,
\label{bornampdef}
\eeq
where $\omega(a)$ is the number of color and spin degrees
of freedom for the flavour $a$. We remember that, in $4-2\ep$
dimensions,
\beq
\omega(q)=2N_c\,,\;\;\;\;
\omega(g)=2(1-\ep)\DA\,,
\eeq
where $\DA=N_c^2-1$ is the dimension of the adjoint representation
of the color group $SU(N_c)$. Since the incoming partons can
play a very special r\^ole, we will also write the functional
dependence of $\MTz$ in the following way
\beq
\MTz(a_1,a_2,\FLTj;\,k_1,k_2,\STj)\,.
\eeq
The transition amplitude for processes in which only gluons and quarks
are involved is usually evaluated in the unphysical configuration
in which all the particles are outgoing. The amplitude
${\cal A}^{(tree)}(2\to 3)$ of eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[bornampdef]{2.10}) can be obtained
from the amplitude
\beq
{\cal A}^{(tree)}(0\to 5)={\cal A}^{(tree)}
(\bar{a}_1,\bar{a}_2,\FLTj;\,-k_1,-k_2,\STj)
\label{crosssymm}
\eeq
simply by crossing (for details on crossing, see appendix B);
notice that eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[crosssymm]{2.13}) is crossing invariant.
Finally, in eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[bornjetdef]{2.6}) we denoted with $d\phi_3$ the full
(i.e., with the $\delta$ that enforces the conservation of four-momentum)
three-body phase space.
Coming to the next-to-leading order contribution, both the
two-to-three and two-to-four partonic subprocesses have to
be considered. As customary in perturbative QCD, we denote
as virtual the contribution of the former, and as real the
contribution of the latter:
\beq
d\sigma_{\aoat}^{(1)}=d\sigma_{\aoat}^{(v)}
+d\sigma_{\aoat}^{(r)}\,,
\label{realplusvirt}
\eeq
with
\beqn
d\sigma_{\aoat}^{(v)}(k_1,k_2;\Jetlist)&=&\uotfct
\FLTsum\MTo(\ArgTfull)
\nonumber \\*&\times&
{\cal S}_3(\STj;\Jetlist)d\PHITsj\,,
\label{virtjetdef}
\\
d\sigma_{\aoat}^{(r)}(k_1,k_2;\Jetlist)&=&\uoffct
\FLFsum\MF(\Argfull)
\nonumber \\*&\times&
{\cal S}_4(\SFj;\Jetlist)d\PHIFsj\,,
\label{realjetdef}
\eeqn
where ${\cal S}_4$ is the measurement function, analogous to
${\cal S}_3$, for four partons in the final state. $\MTo$ is due
to the loop contribution to the two-to-three subprocesses
\beqn
\MTo&=&\frac{1}{2k_1\cdot k_2}\,\frac{1}{\omega(a_1)\omega(a_2)}
\nonumber \\*&\times&
\sum_{\stackrel{color}{spin}}\Bigg[{\cal A}^{(tree)}(2\to 3)
\left({\cal A}^{(loop)}(2\to 3)\right)^*
+\left({\cal A}^{(tree)}(2\to 3)\right)^*
{\cal A}^{(loop)}(2\to 3)\Bigg],
\nonumber \\*&&
\label{virtampdef}
\eeqn
while $\MF$ is defined in terms of the two-to-four transition amplitude
\beq
\MF=\frac{1}{2k_1\cdot k_2}\,\frac{1}{\omega(a_1)\omega(a_2)}
\sum_{\stackrel{color}{spin}}\abs{{\cal A}^{(tree)}(2\to 4)}^2\,.
\label{realampdef}
\eeq
Notice that in eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[realjetdef]{2.16}) the dependence upon the momentum
and flavour of the additional parton was inserted. Also, the measurement
function, as well as the combinatorial factor $1/4!$, had to be modified
with respect to eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[bornjetdef]{2.6}) and eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[virtjetdef]{2.15}).
The full four-body phase space was denoted with $d\phi_4$.
We can now go back to eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[counterterms]{2.5}), to write explicitly
the collinear counterterms for the three-jet production. Using
eq.~(\docLink{slac-pub-7073-0-0-2.tcx}[bornjetdef]{2.6}) we get
\beqn
d\sigma_{\aoat}^{(cnt+)}&=&\uotfct\frac{\as}{2\pi}\sum_d\int dx
\left(\frac{1}{\epb}P_{da_1}(x,0)-K_{da_1}(x)\right)
\nonumber \\*&\times&
\FLTsum\MTz(d,a_2,\FLTj;xk_1,k_2,\STj)
{\cal S}_3 d\phi_3(xk_1,k_2\to\STj)\,,
\nonumber \\*&&
\label{cnt1}
\\*
d\sigma_{\aoat}^{(cnt-)}&=&\uotfct\frac{\as}{2\pi}\sum_d\int dx
\left(\frac{1}{\epb}P_{da_2}(x,0)-K_{da_2}(x)\right)
\nonumber \\*&\times&
\FLTsum\MTz(a_1,d,\FLTj;k_1,xk_2,\STj)
{\cal S}_3 d\phi_3(k_1,xk_2\to\STj)\,.
\nonumber \\*&&
\label{cnt2}
\eeqn
By construction, these quantities, when added to the unsubtracted
three-jet partonic cross section, must cancel the collinear
singularities coming from initial state emission.