state: 3S signal is ~ 10 counts. – Aim: EsZmate upper limit on signals (and raZos)
for peripheral bin. 1/31/12. Manuel Calderón de la Barca Sánchez. 2. Ra o.
Mo#va#on • We want to es#mate the yield of ϒ states.
– Cross sec#ons, Ra#os ϒ(3S)/ϒ(1S), ϒ(2S+3S)/ϒ(1S), etc.
• ϒ(3S) signal is small!
– ϒ(1S) cross sec#on BRxdσ/dy ~ 680 pb at √s=1.8 TeV – Ra#os (BR x σ(nS))/(BR x σ(1S)): Ra#o
√s=38 GeV
√s=1.8 TeV
ϒ(2S)/ϒ(1S)
0.28
0.26
ϒ(3S)/ϒ(1S)
0.18
0.14
– 2011 CMS DATA:ϒ(1S) signal in peripheral bins ~ 100 counts – Expect excited states to be suppressed more than ground state: 3S signal is ~ 10 counts. – Aim: Es#mate upper limit on signals (and ra#os) for peripheral bin. 1/31/12
Manuel Calderón de la Barca Sánchez
2
Procedure • Create a toy Monte Carlo with Upsilon signal plus background. – Model Upsilon 1S signal with Gaussian
• mean mass = 9.46 GeV/c2, width = 0.092 GeV/c2
– Model Background using erf x exponen#al
• Calculate upper limit bin-‐by-‐bin
– Use Bayesian confidence interval
• Guillermo is studying CLs. Expect that both approaches will give similar results.
• Check behavior:
– Does upper limit increase in 1S signal region? – Does it handle properly cases when background “fluctuated down” (less background than expected)? – Do “1-‐sigma” intervals roughly correspond to sta#s#cal error bars on background in signal-‐free region? – Do 90% and 95% intervals give progressively larger upper limits?
1/31/12
Manuel Calderón de la Barca Sánchez
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ϒ mass distribu#on in peripheral bin • From Guillermo’s analysis: – Peripheral bin: • 60-‐100% centrality.
– Sta#s#cs: ~654 counts. – erf parameters: • erf mean: 8.32 GeV/c2 • erf width: 1.14 GeV/c2
– Bin width: 0.07 GeV/c2 – Range: 7 – 14 GeV/c2 – Counts in highest bin ~ 60 1/31/12
Manuel Calderón de la Barca Sánchez
4
Signal and Background Func#ons
• Signal: Gaussian • Background: erf x exponen#al 1/31/12
Manuel Calderón de la Barca Sánchez
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Signal and Background Histograms
• Signal: Throw ~190 counts randomly taken from signal Gaussian • Background:Throw ~570 counts randomly taken from “erf x exp” 1/31/12
Manuel Calderón de la Barca Sánchez
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Toy MC signal+background
• Obtain histogram of signal + background • Roughly matches Guillermo’s peripheral bin stats. 1/31/12
Manuel Calderón de la Barca Sánchez
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Calcula#on of Upper-‐Limit • Two approaches: “Frequen#st” vs. “Bayesian” – PDG J. Phys. G. 37 (2010) 075021, Ch. 33
• Bayesian intervals for Poisson variables.
– Coun#ng experiment. – Observe n total counts in a given bin. – Expect a total of b counts due to known background sources (e.g. combinatorics). – Likelihood that we observe n total counts is Poisson distributed: • Upper limit sup at confidence level 1-‐α is obtained by: n
∑
– Equa#on for α:
– Numerical solu#on.
α =e
( sup + b)
m
−sup m=0
n
∑b m=0
( s + b) L(n | s) =
1− α =
m!
n!
∫ ∫
sup
−∞ ∞ −∞
n
e
−( s+b )
L(n | s)π (s)ds L(n | s)π (s)ds
m
m!
– Note: the “prior”, π(s), in this approach simply restricts s to posi#ve values, and is regarded as providing an interval whose frequen#st proper#es can be studied. # 0 π (s) = $ % 1 1/31/12
Manuel Calderón de la Barca Sánchez
s10 or so.
• Difference at n=20 is 0.4%
• But also runs into machine precision (nn)... – Can’t do 143!, this number is > 10245 2/1/12
Manuel Calderón de la Barca Sánchez
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α(s; n, b) • Probability: – for a given n, b, alpha should decrease monotonically with increasing s. – for a given n, s, alpha should decrease monotonically with increasing b.
• 1-‐σ : α = 0.3173 • 90% CL: α = 0.1 • 95% CL: α = 0.05 2/1/12
Manuel Calderón de la Barca Sánchez
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Comparison, n=5 vs n=10
• For n=b (i.e. observe exactly the expected background), at a given α, sup is higher for higher n. – Larger fluctua#ons allow for a larger signal to be buried in them. 2/1/12
Manuel Calderón de la Barca Sánchez
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Upper limits
• Le{: Invariant mass distrub#on • Right: Upper limit on a possible signal above background expecta#on. 2/1/12
Manuel Calderón de la Barca Sánchez
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Upper limits: α=0.3713, 0.1, 0.05
• Upper limits should get larger with decreasing α. 2/1/12
Manuel Calderón de la Barca Sánchez
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Upper limits: α=0.3713, 0.1, 0.05
• • • •
Adding upper limit on top of background expecta#on (red line) Upper limit for α=0.3173 should be similar to 1-‐σ error bars in region where there is no signal. Upper limits should get larger with decreasing α. Does not give bad results when observe less counts than expected. – Upper limit decreases, but is always posi#ve.
2/1/12
Manuel Calderón de la Barca Sánchez
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Conclusions • Simple approach to upper limits – Based on Poisson sta#s#cs.
• Compare to approach following Frequen#st CLs. – Pursued by Guillermo. – Includes way to treat fluctua#ng background.
• Can give a cross check between methods. • Note: – D0 CL writeup: Expect both methods to give very similar results. 2/1/12
Manuel Calderón de la Barca Sánchez
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