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2007.05.30 Efficiency of reconstructing photons in EEMC
Updated on Fri, 20100716 12:08. Originally created by seluzhen on 20100716 11:26.
Under:
Pibero Djawotho Last updated Wed May 30 00:32:16 EDT 2007
Efficiency of reconstructing photons in EEMC
Monte Carlo sample
 10k photons
 STAR y2006 geometry
 zvertex=0
 Flat in pt 1030 GeV
 Flat in eta 1.02.1
SMD gamma/pi0 discrimination algorithm
The following
from the IUCF STAR Web site gives a brief overview of the SMD gamma/pi0 discrimination algorithm using the method of maximal sided fit residual (data  fit). This technique comes to STAR EEMC from the Tevatron via Les Bland via Jason Webb. The specific fit function used in this analysis is:
f(x)=[0]*(0.69*exp(0.5*((x[1])/0.87)**2)/(sqrt(2*pi)*0.87)+0.31*exp(0.5*((x[1])/3.3)**2)/(sqrt(2*pi)*3.3))
x is the strip id in the SMDu or SMDv plane. The widths of the narrow and wide Gaussians are determined from empirical fits of shower shape response in the EEMC from simulation.
Optimizing cuts for gamma/pi0 separation
In the rest of this analysis, only those photons which have reconstructed pt > 5 GeV are kept. There is no requirement that the photon doesn't convert. The dividing curve between photons and pions is:
f(x)=4*x+1e7*x**5
The yaxis is integrated yield over the SMDu and SMDv plane, and the xaxis is the sum of the maximal sided residual of the SMDu and SMDv plane.
Following exchanges with Scott Wissink, the idea is to move from a quintic to a quadratic to reduce the number of parameters. In addition, the perpendicular distance between the curve and a point in the plane is used to estimate the likelihood of a particle being a photon or pion. Distances above the curve are positive and those below are negative. The more positive the distance, the more likely the particle is a photon. The more negative the distance, the more likely the particle is a pion.
Hi Pibero, With your new "linear plus quintic" curve (!) ... how did you choose the coefficients for each term? Or even the form of the curve? I'm not being picky, but how to optimize such curves will be an important issue as we (hopefully soon) move on to quantitative comparisons of efficiency vs purity. As a teaser, please see attached  small loss of efficiency, larger gain in purity. Scott
Hi Pibero, I just worked out the distance of closest approach to a curve of the form y(x) = a + bx^2 and it involves solving a cubic equation  so maybe not so trivial after all. But if you want to pursue this (not sure it is your highest priority right now!), the cubic could be solved numerically and "alpha" could be easily calculated. More fun and games. Scott
Hi Pibero, I played around with the equations a bit more, and I worked out an analytic solution. But a numerical solution may still be better, since it allows more flexibility in the algebraic form of the 'boundary' line between photons and pions. Here's the basic idea: suppose the curved line that cuts between photons and pions can be expressed as y = f(x). If we are now given a point (x0,y0) in the plane, our goal is to find the shortest distance to this line. We can call this distance d (I think on your blackboard we called it alpha). To find the shortest distance, we need a straight line that passes through (x0,y0) and is also perpendicular to the curve f(x). Let's define the point where this straight line intersects the curve as (x1,y1). This means (comparing slopes) (y1  y0) / (x1  x0) = 1 / f'(x1) where f'(x1) is the derivative of f(x) evaluated at the point (x1,y1). Rearranging this, and using y1 = f(x1), yields the general result f(x1) f'(x1)  y0 f'(x1) + x1  x0 = 0 So, given f(x) and the point (x0,y0), the above is an equation in only x1. Solve for x1, use y1 = f(x1), and then the distance d of interest is given by d = sqrt[ (x1  x0)^2 + (y1  y0)^2 ] Example: suppose we got a reasonable separation of photons and pions using a curve of the form y = f(x) = a + bx^2 Using this in the above general equation yields the cubic equation (2b^2) x1^3 + (2ab + 1  2by0) x1  x0 = 0 Dividing through by 2b^2, we have an equation of the form x^3 + px + q = 0 This can actually be solved analytically  but as I mentioned, a numerical approach gives us more flexibility to try other forms for the curve, so this may be the way to go. I think (haven't proved rigorously) that for positive values of the constants a, b, x0, and y0, the cubic will yield three real solutions for x1, but only one will have x1 > 0, which is the solution of interest. Anyway, it has been an interesting intellectual exercise! Scott
I made use of the ROOT function TMath::RootsCubic to solve the cubic equation numerically for computing distances of each point to the curve. With the new quadratic curve f(x)=100+0.1*x^2 the efficiency is 63% and the rejection is 82%.
Efficiency and Rejection
The plot on the left below shows the efficiency of identifying photons over the pt range of 1030 GeV and the one on the right shows the rejection rate of single neutral pions. Both average about 75% over the pt range of interest.
Rejection vs. efficiency at different energies
The plot below shows background rejection vs. signal efficiency for different energy ranges of the thrown gamma/pi0.
Rejection vs. efficiency with preshower cut
Below on the left is a plot of the ratio of the sum of preshower 1 and 2 to tower energy for both photons (red) and pions (blue). On the right is the rejection of pions vs. efficiency of photons as I cut on the ratio of preshower to tower. It is clear from these plots that the preshower layer is not a good gamma/pi0 discriminator, although can be used to add marginal improvement to the separation preovided by the shower max.
ALL ENERGIES 

E=2040 GeV 

E=4060 GeV 

E=6080 GeV 

E=8090 GeV 

Pibero Djawotho Last updated Wed May 30 00:32:16 EDT 2007
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