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Equivalent photon flux in leptons
With the 'gamma/lepton' option of a PYINIT call,
an
or
event (or corresponding processes with muons)
is factorized into the flux of virtual photons and the subsequent
interactions of such photons. While real photons always are transverse
(T), the virtual photons also allow a longitudinal (L) component.
This corresponds to cross sections

(55) 
and

(56) 
For
events, this factorized ansatz is perfectly general, so long
as azimuthal distributions in the final state are not studied in detail.
In
events, it is not a good approximation when the
virtualities and of both photons become of the order of
the squared invariant mass of the colliding photons
[Sch98]. In this region the cross section have terms that depend
on the relative azimuthal angle of the scattered leptons, and
the transverse and longitudinal polarizations are nontrivially mixed.
However, these terms are of order
and can
be neglected whenever at least one of the photons has low virtuality
compared to .
When is small, one can derive [Bon73,Bud75,Sch98]
where
or .
In
the second term, proportional to
, is not leading log and is therefore often
omitted. Clearly it is irrelevant at large , but around the lower
cutoff
it significantly dampens the small rise
of the first term. (Note that
is dependent,
so properly the dampening is in a region of the plane.)
Overall, under realistic conditions, it reduces event rates by 510%
[Sch98,Fri93].
The variable is defined as the lightcone fraction the photon takes
of the incoming lepton momentum. For instance, for events,

(59) 
where the are the incoming lepton fourmomenta and the
the fourmomenta of the virtual photons.
Alternatively, the energy fraction the photon takes in the rest frame
of the collision can be used,

(60) 
The two are simply related,

(61) 
with
. (Here and in the following formulae we have
omitted the lepton and hadron mass terms when it is not of importance
for the argumentation.)
Since the Jacobian
, either variable
would be an equally valid choice for covering the phase space.
Small values will be of less interest for us, since they lead
to small , so
except in the high tail,
and often the two are used interchangeably. Unless special cuts
are imposed, cross sections obtained with
rather than
differ only at the
per mil level. For comparisons with experimental cuts,
it is sometimes relevant to know which of the two is being used in
an analysis.
In the
kinematics, the and definitions give that

(62) 
The expression for is more complicated, especially
because of the dependence on the relative azimuthal angle of the
scattered leptons,
:
The lepton scattering angle is related to as

(64) 
with
and terms of neglected.
The kinematical limits thus are
unless experimental conditions reduce the ranges.
In summary, we will allow the possibility of experimental cuts in the
, , , and variables. Within the
allowed region, the phase space is Monte Carlo sampled according to
, with the
remaining flux factors combined with the cross section factors to give
the event weight used for eventual acceptance or rejection. This cross
section in its turn can contain the parton densities of a resolved
virtual photon, thus offering an effective convolution that gives
partons inside photons inside electrons.
Next: Kinematics and Cross Section
Up: Parton Distributions
Previous: Leptons
Contents
Stephen Mrenna
20071030