Now turn to the problem of evaluating the kinematic constraints and their gradients. In order to do this, the variables used to describe an event must be specified, as well as the forms of the constraints themselves.
An event consists of a collection of N final state particles with
four-momenta 
. Each particle is described by three variables:
the absolute value of
its three-momentum 
and its direction in 
- 
space 
and
. The masses of the particles are taken to be constants,
and denoted 
.
There can be at most one neutrino in the final state, with
four-momentum 
. The neutrino is also described by three
variables: the z-component of its momentum 
, and
and 
, which are defined as
The neutrino, of course, is considered massless.
Now for the constraints. Two general forms of mass constraint are implemented. The first asserts that some collection of particles has a fixed invariant mass (such as in the W mass constraints). The second asserts that two collections of particles have the same invariant mass (as in the top mass equality constraint). The first of these can be written
and the second
The sums in parentheses can be expanded out as
Note that since the masses are treated as constant, their
gradients are zero. Thus, the problem of calculating the gradients of
the constraints reduces to calculating the gradients of the dot
product of two four-vectors.
If there is no final-state neutrino in the event, then there are the two additional constraints
These constraints will be dealt with separately below.
Unfortunately, the variables used to describe an event are not very
convenient for calculating the gradients. So the strategy used
requires several steps: first, the gradients are calculated with
respect to a more convenient set of variables. Then, the gradients are
converted to the
variables actually desired by a series of Jacobian transformations.
For this initial set of variables, the polar angle 
will be
used instead of the pseudorapidity . In addition, the neutrino
will be treated just like any other final-state particle, and
parameterized using P, 
, and .
So, the problem is to evaluate the gradients of
with respect to the variables 
, 
,
, 
, 
, and 
. Since, by definition,
the dot product
can be written as
The partial derivative with respect to is thus
Similarly for ,
For the angular gradients, the result is
Noting that
(where 
), (A.79) can be rewritten
as
Similarly,
Finally,
and
If there is no neutrino in the final state, then
the gradients of the 
constraints (A.73)
also need to be calculated. This reduces to evaluating the
derivatives of the x and y components of a four-momentum 
,
with respect to P, 
, and .
From (A.74), these gradients are
Also,
And finally,
Now the variables need to be transformed from the set in which
the gradients were evaluated to the final set used for describing
the event. The first step will be to change the variables for
the neutrino from spherical (
, 
,
) to rectangular (
= x, 
= y, = z)
coordinates.
This transformation is independent of all other event variables.
The equations of transformation can be written
Thus, by the chain rule,
The next step is to change from the x and y components of the
neutrino momentum and to and , as
defined by (A.69). That is, the transformation is
and the equations of transformation are
Thus, a Jacobian transform gives
Finally, the polar angles need to be converted from 
to .
This is independent for each object. The transformation is
Thus,
and the associated Jacobian transform is
There is one more transformation which might need to be made. If the lepton in the event is a muon, then the variable used to describe its momentum is not actually P but instead its inverse
The Jacobian transformation for this case is
