In this contribution to the DA
NE physics handbook, we estimate the
accuracies achievable at DA
NE in measuring the theoretical
parameters describing the decay 
.
We refer the reader to the contribution of Bijnens, Ecker, and
Gasser [1]
for a detailed discussion of the theory of this decay,
and the definitions of the notation we use.
We should note that we refer to this decay as K_e4 
; we
consider only this particular decay and not the channels with
neutral pions or kaons.
We will consider two of the possible ways
of extracting the parameters of a theory from a set of data.
The first one is the classic technique of asymmetries.
K_e4 
decays have a partial
decay rate 
of the form
where s_ 
, s_l 
, 
, 
, and 
are the set of
kinematic variables necessary to describe K_e4 
, defined in
Ref. [1].
The quantity 
can be written as an expansion in simple
functions of 
and 
multiplying nine
intensities 
, which can in turn be written in
terms of the three remaining kinematical variables, and three
form factors, F, G, and H, which are also dependent on
these three kinematical variables.
The explicit dependences may be
found in Ref. [1].
As can be seen in the contribution of Colangelo, Knecht and
Stern, [2] the
tangent of the phase shift, 
,
can be neatly extracted from the ratio of the intensity
functions, 
, or, equivalently,
, where by the tilde we denote
intensities integrated over 
and 
as well
as over 
.
All of the 
, in
turn, can be written as asymmetries; in particular we have
and
The asymmetry presents a transparent, elegant and quick (computationally) way to determine a parameter.
The second, and main, method we consider is that of the maximum
likelihood, which we shall refer to as the MLM. [3]
Let 
be the vector of phase space variables specifying
an event in an experiment, and 
be the
probability distribution function predicted by a theory.
is the set of parameters in the theory, to
be determined experimentally. The probability of observing
an event at 
in the interval 
is
. The function f is normalized
to 1 over the whole 
interval in which 
is physical. In particular, if cuts are imposed on the phase space,
f must be normalized over the reduced phase space; we must
also be certain that the normalization is maintained even when
the parameters are varied from their central value.
The likelihood,
or joint probability distribution, of an experiment yielding
N events, each
specified by a set of phase space variables 
, is
then defined as
The best estimate for 
is then simply the value
which maximizes 
(or equivalently, of
, which is easier to compute).
It can be shown that the error matrix, in general non-diagonal
for correlated parameters, is
Here the 
denote matrix inversion. This integral
is, in general, not possible to compute analytically, but is
easily evaluated numerically.
If we start with a non-normalized
probability function 
,
then the following equation is useful:
where 
.
While this method is more complicated and time-consuming computationally, it has the decided advantage that it yields the absolute best possible determination of any given set of parameters. In today's age of fast computers, it is thus the method of choice. It also allows one to determine any parameters chosen, while the asymmetry is only good for certain parameters. Finally, the experimental uncertainties in the parameters can be determined reasonably easily, without need for an actual simulation of the determination of the parameters themselves using this method.
It is simple to see that the MLM should give a better determination of the parameters than an asymmetry: while the asymmetry only uses the information of whether an event is in one or another half of phase space, the MLM benefits from the information of the precise position of the event. For example, consider a process specified by the probability distribution
The parameter a can be determined by the ratio
where 
is the number of events
with 
between 0 and 
, and
is the number of events
with 
between 
and 
.
For a small compared to 1, the error on a in this
determination is thus 
where N is the total
number of events.
For the MLM, the integral in eq. 5 is
and the error on a in this determination is thus
. While this improvement may not seem
very impressive, larger improvements may be expected
as the parametrization becomes more complicated.
Using the MLM directly on the parameter that interests us,
we are sure to use all possible information,
and to take into account the effect of all possible
cancelling uncertainties.
We observe an improvement as above, of about fifteen
percent, in the relative error 
when we go from the asymmetry method to the MLM.
However, the relative error
on 
when 
is determined
directly via the MLM is two-thirds of its error when
determined indirectly by the ratio
, if these 
are determined using the MLM, and their errors are then combined
as uncorrelated. This appears to be due to the information
lost in integrating the 
over s_ 
, s_l 
, and 
\
before using the MLM.
