Re: mu+p scattering

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

Dear Peter,

the complexity of this subject probably demands a telephone discussion
rather than an e-mail exchange, but I will try to respond to your major
points below.

> The effiency to "see" a scattering muon is given by
>     eps_seen [theta] = P(Erec>Ethr)[th] P(scattered mu is seen)[th]
> where Erecoil and Ethreshold refer to the recoil proton, P(scattered
> mu is seen) to the muon.
>
> In your estimate, you disucss the P(Erec>Ethr)[th] and try to find
> the correct Ethr to determine this efficiency.
> What about the second term P(scattered mu is seen)[th]. Do you
> assume it to be 1?

I feel that your formula for "eps_seen" is somewhat confusing because it
is not applicable to both the MC and Run8 data.  For MC the scattered
muon is, of course, "seen" because I know its scattering point and
stopping point--which is precisely what makes the MC useful in the first
place.  Thus, I would elect to rewrite your formula for the MC case as

  MC_eps_scatter[th] = P(Erec>Ethr)[th] *
                       P(scattered muon stops outside fidTPC)[th]

The remaining piece of the puzzle is then to make a determination of the
SRIM recoil energy <--> Run8 EH threshold correspondence, so that we can
compare MC numbers with real data.  I used the SRIM and Run8 scatters'
theta distributions to make that connection.

For the Run8 data we are by definition only concerned with scatters that
mimic a muon stop and then stop somewhere outside the fidTPC volume.  Thus
I would write

  Run8_eps_scatter[th] = P(scattered mu is identified)[th]

Due to TPC performance issues, software finder shortcomings, etc., we
expect that

  Run8_eps_scatter < MC_eps_scatter
          1.214e-4 < 1.762e-4 (from my analysis report)

I would not place too much emphasis on the theta distribution.  I used it
primarily to establish the MC <--> Run8 correspondence.  Bear in mind that
I also identify scatters via the MWPC method, which is a good backup means
for catching the most dangerous events: scatters through the bottom of the
TPC, where there is the most stopping material.  MWPC scatters form a
quarter of my total scatters.  I should mention, however, that their
effects on the rate seems to scale in the same way as the TLS scatters.

> What about the second term P(scattered mu is seen)[th]. Do you
> assume it to be 1? It has at least two contributions:
> i) For small theta the scattered muon might not be seen, as it
>    is ignored as a blue point on a valid track. That would
>    effect your observed angular distribution of scattered
>    muons at small theta.

I agree.  This is a somewhat tricky subject, but you may have noticed that
the Run8 theta distribution in my analysis report (p.29) has a small peak
in the valley between the two larger peaks.  It is there because I
performed my scatter search *before* making fiducial cuts, so as not to
miss the sorts of small theta scatters you mention.  However, this means
that I also categorized events which are likely throughgoing muons as
scatters--hence the hump.  However, these are 5% of the total, so I am not
too concerned.  And again, I am only really concerned with the peak to
peak width of the theta distribution.

> ii) At large theta the efficency for seeing the muon will drop,
>     as it goes up or down. By how much is the question. If it
>     is only 50%, your correction would be 50% too small.

See my MWPC argument above.

> Finally, there is the correlation between dacay rate and theta.
> Larger theta will lead to larger decay rate (because up/down
> has the most stopping mass). Thus if your effiency for larger
> theta is small, the decay rate measured by tagging on
> scattered muons leads to an underestimation of the effect
> ("Predominant observation of friendly scattered muons").

In principle you are right.  Disentangling the relationship between
scatter theta, efficiency of detection, efficiency of identification,
effects on the rate, etc., would be nearly impossible.  I have tried to
keep it simple and cover my bases with the TLS and MWPC algorithms.

> P.S. The natural steps to confirm the correction are:
> i) Calculate the scattering fraction carefully for our condition, compare
>    with Tom's Srim calculation.
> ii) Use Srim or Mott cross section as a muon event generator for
>     Bernhard's MC.
> iii) Bernhard should calculate stopping fraction on various
>      materials. This gives an estimate of the absolute scatter fraction
>      and its effect on the decay rate.
> iv) Finally, one could try to track MC generated scatter processes to find
>     "seen" efficiency.
> v) In the future, the "seen" efficieny should be corroborated with run10
>    data, where eps(scattered muons seen) should be high.

You will have to hire another graduate student for this particular
programme.  Although I agree in principle with your characterization of
the problem, I simply cannot afford to devote any more time to this
subject, given my far greater concerns with the other corrections--especially
high-Z.

Regards,
Tom