1. Recent Developments in the CP Violating TGC Analysis
Tim Barklow
June 4, 2003

2. Modified Log Likelihood Estimator
The standard log likelihood function that was used
in 1999 to fit CP violating TGCs in the
channels was augmented with a correction function
in to account for a variety of effects in
the more complicated
channels.
The correction function is properly reweighted so
that no TGC value is favored over another.
The correction function also makes it straightforward
to add new effects such as EW corrections.

3. Modified Log Likelihood Estimator
However, the correction function has a drawback
in that it disturbs the most important statistical
interpretation of the LL function:
in general
In fact, as far as I can tell it is not possible to define
a LL probability density function in the usual sense
once the correction function has been added.
Because of these considerations, the LL estimator
with the correction function is not a true LL estimator,
and so I call it a Modified Log Likelihood Estimator,
or MLLE.

4. Modified Log Likelihood Estimator
The MLLE is an unbiased estimator (which
I will formally demonstrate in a moment)
The MLLE estimate is equal to the OO
estimate in the limit of small TGC value
(shown at Leukerbad Oct. 2001)
Thus the MLLE estimator and the OO method are
in some sense mathematically equivalent. However
I have not investigated the full set of circumstances
under which the two methods are equivalent

5. The MLLE function f(g) is given by
differential cross section using reconstructed variables
(sum over background MC events)

6. For the moment drop the background term:
Take the limit or
So that the standard normalized LL function is
recovered in this limit.
The correction function provides pdf
normalization in the limit

7. Fit for TGC g by solving where
Note that the function that forms the basis for the OO
method,
appears prominently here.

8. Drop the background term and take the limit
for
where
Note that we would still get if we replaced
by an arbitrary function
However gives smallest error (OO again)

9. Statistical variance of TGC fit variable g
Define so that
for near where
where

10. Statistical variance of TGC fit variable g
Including background the full expression for the statistical error is
(sum over bkgnd MC events)

11. Statistical variance of TGC fit variable g
can be calculated simultaneously using signal MC
while are calculated using background MC
Unlike the situation in , the statistical
variance of the TGC fit variable g can now be
calculated without trial MC runs.

12. Including EW corrections
Till now I have used the following weight for a TGC value of g
With EW corrections included additively this becomes
With EW corrections included multiplicatively this becomes
In all 3 cases the TGC correction is applied multiplicatively

13. Including TGC corrections additively
The recommendation that EW corrections be included
additively raises the question whether TGC corrections
should also be included additively.
With EW additive & TGC additive:
With EW multiplicative & TGC additive:
All 4 combinations of EW & TGC add/mult have been coded

14. STATUS
Software has been converted from version 1.2 ntuples to version 1.62 ntuples
All TGC ntuples version have been downloaded to SLAC and my ntuple version 1.62 software seems to run fine on these ntuples.
Software which incorporates all the new features described in this talk has been written. However this software is still being debugged and so I have no new results to present.

15. From Leukerbad 2001:

16. From Leukerbad 2001: 95% C.L. contour for for
Form Factor for Vector Resonance Enhancement of
95% C.L. contour for for

17. Summary
A formula for the variance of the Modified Log Likelihood Estimator (MLLE) method has been developed. This formula provides a means to calculate the statistical error without using trial MC runs.
O(alpha) EW correction has been added to CP violating TGC fits
Corrections due to TGCs can now be applied additively as well as multiplicatively.
The software for the new features has been written. Results for the CP violating TGCs should be ready in a few weeks.
Results will also be forthcoming for the real and imaginary parts of the CP conserving TGCs without SU(2)XU(1) constraints, and for the “technipion form factor” which describes enhancements in longitudinal W boson production.