Re: muon kinetics equations

[Tom Banks <tbanks@socrates.berkeley.edu>]

> I updated my pure kinetics studies on
>    http://twiki.npl.uiuc.edu/twiki/bin/view/Main/GeneralPK
> In the end I get a 26 Hz correction, a bit higher then the 24 Hz
> I quoted last Friday. The reason is that I used the old lower
> capture rates for last week's estimates. I did not include
> accidentals in my fit.

I have just discovered that the inclusion/exclusion of an accidental
background is the source of the discrepancy between my 1st moment and
best-fit estimates for the ppmu "correction."  Using the following
parameter values,

  lambda_0  = 455160 Hz
  Lambda_S  = 710 Hz
  Lambda_T  = 12 Hz
  phi       = 0.0116;
  gamma_o   = 1.009/2
  gamma_p   = 1.143/2
  lambda_of = 2.3e-6
  lambda_pf = 0.0074e-6
  lambda_op = 8e-4

I performed the following 3 studies where I varied the statistics N and
the background B; each study was comprised of 500 MC best-fit simulations
to fill out the distribution.

     Study 1: N=1e10, B=0
        lambda(1st moment) - lambda_0 - Lambda_S = -25.98
        lambda(best-fit, 2-parameter) - lambda_0 - Lambda_S = -25.5 +- 0.2
  -->   lambda(1st moment) - lambda(fit) ~ -0.5 +- 0.2

     Study 2: N=1.6e9, B=0
        lambda(1st moment) - lambda_0 - Lambda_S = -25.98
        lambda(best-fit, 2-parameter) - lambda_0 - Lambda_S = -25.7 +- 0.6
  -->   lambda(1st moment) - lambda(fit) ~ -0.3 +- 0.6

     Study 3: N=1.6e9, B=2000
        lambda(1st moment) - lambda_0 - Lambda_S = -25.98
        lambda(best-fit, 3-parameter) - lambda_0 - Lambda_S = -19.6 +- 0.7
  -->   lambda(1st moment) - lambda(fit) ~ -6.4 +- 0.7

Notice that changing the statistics N appears to have little effect, but
when I incorporate the observed Run8 Prod-50 background level the two
methods give significantly different results.  This explains why I have
been seeing discrepancies between the two approaches: I have been using
B=2000 for the past week or so, and did not analyze the B=0 case until
Peter described his studies this afternoon.  We need to discuss this.

One other thing: Is any one else disturbed by the fact that our extracted,
ostensibly experimental, capture rate is sensitive to the theoretical
prediction for the capture rate?

Regards,
Tom