sPHENIX discussion of software

[Gabor David <david AT bnl.gov>]

On Tue, 2 Feb 2016, Huang, Jin wrote:

        A few clarifications / remarks to Tuesday's discussion:

(snip)
> Gabor: also important to pin down the impact point of photon, which dominate
> the resolution for higher pT pi0 via opening angle measurement. Linear
> weighting to get cluster center as we do now would not yield best
> performance. We can try logarithmic weighting with a threshold on minimal
> energy, which take into account the tail of energy distribution better.

        While the central tower in a cluster is crucial in measuring
energy, it is of surprisingly little use for position.  The catchphrase
I like to use is that "the (position) information is in the tail".

        A simple justification (with caveats later).  Assume orthogonal
impact, roughly Gaussian energy distribution, symmetric (modulo
fluctuations) around the maximum value, and extending in one dimension
to about 5 towers (typical for a well-designed, projective calorimeter,
e.m. showers).  You have five positions (x1...x5, x2-x1=d, the tower
size) given by geometry, and five measured energies (E1...E5).

	Assume that the location of the maximum of the Gaussian (i.e. 
the energy deposit) corresponds to the impact point, and it is in
tower 3.  At this point (as an experimentalist) all you know is
that the impact point is somewhere between x3 - d/2 and x3 + d/2,
and that the integral of the Gaussian is E3 - your measured energy
in the central tower.  Now a simple toy Monte Carlo, where you
vary the impact position (the maximum of the Gaussian) from
x3 - d/2 to x3 + d/2 can convince you, that no matter where the
peak is, as long as it is in this third tower, the integral of
the Gaussian (i.e. E3) barely changes.  In other word, E3 is
of little help telling where the true impact point was on a
scale finer than the tower size itself (d).  -  Btw, if you are
of analytical mind, you can convince yourself about this even
without a toy MC, just by carefully looking at the shape of
erf(x), which is almost linear around the middle.

        At the same time, as you move the impact point from the
left to the right of the central tower (3), the energies in
the neighboring towers (E2, E4) change dramatically: they tell
you much more about the impact point that E3 ever could.  So
you need some weighting which underemphasizes the large energy
deposit in the center, and enhances the weight of the "tails".
A logarithmic (ln(E)) weighting does exactly that.  (It is
not the only possibility, but for various groups proved pretty
effective.)

        Of course you have to be careful: once the energy in
a particular tower gets so small that fluctuations or detector
noise starts to dominate, the method breaks down.  Therefore,
the way it is usually used is to (empirically) figure out an
energy threshold, drop any tower below that, and do the log
weighting for the rest the usual way.

        Of course the method has problems once the impact is
not orthogonal and the third dimension (depth) comes into play,
making the shower asymmetric.  Blindly applying the log 
weighting might give you even worse results that a linear
("center of gravity") weighting would.  Therefore, you have to
work on a correction with the impact angle - but "that's
another story", and also, not really a serious problem for
photons with the nominally projective sPHENIX EMCal.
(Of course they will be an issue for electrons.)

        Note the dichotomy of energy and position measurements.
For energy the central tower is crucial, and noisy towers at
the edge (as long as not too noisy) are a small perturbation.
For position the central tower is of little use, while noise
at the edges can become deadly.  For energy large granularity
would be ideal (if occupancy is not an issue), in fact, in
low energy nuclear physics, when doing gamma spectroscopy,
where energy resolution is everything, and position is not
an issue, we made a point of having all energy deposited in
a single detector element.  For position the finer the granularity,
the better (up to the limit given by fluctuations).  The
usual mantra of EMCals in particle physics - "make the tower
size ~ one Moliere radius" - came largely as a reasonable
compromise between these competing requirements.

        Gabor


--
Gabor David  (david AT bnl.gov)
Brookhaven National Laboratory
Physics Department, Bldg 510/c
UPTON NY 11973
Tel: (631)344-3016
FAX: (631)344-3253