[Halld-cal] cal energy resolution observed from meson 2g linewidths in radphi

[Kei Moriya]

Hi Richard,

Thanks for the information. In our beam test we only have
3 different energies that cover a small range (~100 - 260 MeV),
and therefore we will not be fitting our data to any functional
form, but it will be nice to be able to compare the resolution
correctly with the final result of RadPhi.

Thanks,
	Kei

On 1/29/13 6:48 PM, Richard Jones wrote:
> Hi Kei,
>
> The fit formula that we used is just that given in Eq(10): a simple sum
> of the "floor" term and the statistical term.  It is tempting to suppose
> that the correct result should be a quadratic sum of a term proportional
> to 1/sqrt(E) and a constant term representing electronic/digitization
> noise.  However we found empirically that the quadrature sum gave a very
> poor fit (large chi-square) compared with the simple sum of sigma(E) =
> A*sqrt(E) + B*E.  For the fit results, please see the attached figure
> taken from Mihajlo's thesis.  You can see that the solid curve which
> represents the above 2-parameter form, is a pretty good fit.  For
> comparison, you can try yourself what you get when you replace this with
> a quadrature sum.  The quadrature sum curve is much closer to a straight
> line in the region of our data points, and you cannot describe the data
> points as well.
>
> We could have gotten an equally good fit with the PDG formula
> sigma(E) = A*sqrt(E)  (+)  B*E  (+)  C
> but it requires more parameters and hence larger uncertainty on the
> parameters, and the data did not justify the use of a more complicated
> empirical form when our data span so limited a range in energy.
>
> Of course one might prefer the obvious statistical interpretation of the
> quadrature sum (independent noise and Poisson photostatistics) but this
> simple interpretation is certainly wrong [crude] in describing
> resolutions in a real experiment.  At low photon statistics of a few
> dozen photons the Poisson model is fine, but for high energy showers (>1
> GeV in Radphi) there are non-Poisson statistics in the energy-dependent
> term coming from shower fluctuations, one needs a log term to describe
> shower penetration depth and leakage, and so on.  Also we had a better
> E-scale calibration for the inner blocks where the rate per block is
> highest and the average energy deposition is larger than in the outer
> blocks, so there is a correlation between the floor term and the
> statistical term that gets ignored when you do a quadratic sum.  I am
> not trying to justify the empirical formula we used, but just to point
> out that one should not expect naive photon statistics plus independent
> electronic noise to describe the actual resolution obtained in the
> reconstruction of particles decaying into photons.  If it empirically
> fits the data then fine, but that doesn't mean that one is seeing just
> pure Poisson photostatistics plus independent noise.  It just means that
> this particular empirical formula works for this case.
>
> -Richard Jones
>
>
> On 1/29/2013 12:32 PM, Kei Moriya wrote:
>>
>> Hi Richard,
>> By the way, I gave my semi-final results on the FCAL beam test
>> today at the calorimetry meeting,
>> https://halldweb1.jlab.org/wiki/index.php/Jan_29,_2013_Calorimetry
>> and I had one question regarding the RadPhi results in
>> NIMA 566 (2006) 366–374.
>>
>> Within the paper you parametrize the energy resolution sigma/E as
>> Eq.(10), B/sqrt(E) + A. Is this a quadrature sum, or is it a simple
>> linear sum? I don't have the reference that you quote for this equation
>> at hand, and there was some discussion on whether the correct form
>> is a simple sum or quadrature sum. One goal of our current beam test
>> is to show that the resolution is indeed somewhat better than RadPhi
>> due to the addition of the optical cookie guides, and I don't want to
>> mess up when quoting the previous results.
>>
>> Thanks,
>>     Kei
>>
>>
>