Paul J Dervan
Jon described Maximum Likelihood fits first. Pointing out the necessity to normalise the likelihood function. The likelihood tends to be Gaussian. John then went on the discuss log likelihoods. The advantage with log likelihoods is that the likelihoods are summed instead of multiplied. This also helps in the computation as addition is more accurate than multiplication. John then went on to describe the errors on Likelihood fits. If the Likelihood is Gaussian then the errors can be defined three ways (see slides) (a) the root mean square deviation of the liklihood about its mean .(b) SQRT((-d^2L)/(dp^2)) where p is the parameter we wish to determine. (c) The change in p required to reduce L from its maximum value by 0.5. But the error determination is much harder if L is not Gaussian. There are two are two methods. (i) Find a range of p values p_1.le.p.le.p_2 where L(p_i) = L(p_0) - 1/2, i=1,2 probability that parameter lies in this range = 60%. (ii) <SQRT((-d^2L)/(dp^2)) > Where the average extends over a suitable range of p e.g over the range of p for which L is large. Jon suggested trying to choose the varying parameters which are Gaussian. In summary Jon said the maximum likelihood method is very good for low statistics measurements because they use every single event - no need to use histograms. But that background subtraction is a problem and large amounts of computation is often required and that the normalisation factor cannot be determined easily (usually numerically).
CP
Violating TGC's -- Paul Dervan
Paul summarised his initial investigation into
CP-violating TGC's. There were a number of technical herdals to jump before
any analysis could proceed. The first was the standard definition of the
CP-violating parameters (cf. $\alpha_{w\phi}$, $\alpha_{W}$, $\alpha_{B\phi}$
for the CP-conserving analysis). There is no LEP standard for the CP-violating
parameters. C. Papadopalous in "J.
Phys. G: Nucl. Part. 24 (1998) 405-419," suggested the use of
$\\tilde{\alpha_{BW}}$ and $\\tilde{\alpha_{W}}$. These parameters were
used by DELPHI. Hence this parameter set has be adopted. The second problem
was the Monte Carlo generators. At present there are only to options. The
first is EXCALIBUR. There is a theorist working on adding the CP-violating
parameters into the MC so they can be varied. When this work will be complete
is unknown. The second option is ERATO which does already incorporate the
CP-violating parameters, unfortunetly ERATO is not OPALised.
Paul then moved on to the possible options for the analysis base. He listed four. (a) Steve Bett's nearest neighbour analysis. (b) Jon Couchman's spin density method. (c) Alan Skillman's unbinned maximum likelihood method. (d) An asymmetry method. Jim suggested we try them all and compare results. Paul had been investigating the asymmetry method suggested by V. Spanos et al "Phys. Lett. B388 (1996) 371-375." The paper suggests that there will be an asymmetry in the lepton polar angle distribution if there are CP-violating TGC's. Paul described how the asymmetry was formed (see slides). As there is no MC with CP-violating TGC's availible yet, Paul produced measurements of the aysmmetry using Standard Model EXCALIBUR MC. The distributions looked flat as was expected. There was some discussion about the points in the first bin, but it was decided that this was a statistical problem.
Paul plans to continue the investigation into
the asymmetry method, imparticular to see if the CP-conserving parameters
introduce an asymmetry, as we already have the MC for these. If they do
introduce an aysmmetry this particular method will not be used. He also
planned to try and get ERATO working at UCL.
Nearest Neighbour Analysis -- Steve Betts
Steve gave a quick review of the work he is presently working on. In particular the background determination. Steve said most analysis (Delphi) assume the background to be a flat distribution for all distributions. After studying his background distributions this is clearly not the case. This needs to be understood better. Steve mentioned a problem with the 2-photon background which he will discuss with Jan Lauber.
Next Meeting: 2:00pm 12/06/98
Jon Couchman will present his work on spin density analysis and the asymmetry stuff will be discussed some more.