1. The expected confident intervals for triple gauge coupling parameter
Zhijun Liang
Academia Sinica ,Taiwan
5/5/2018
The expected confident intervals for triple gauge coupling parameter

2. The expected confident intervals for triple gauge coupling parameter
Introduction
Section 1 :Comparison of Bayesian approach and Frequentist approach in setting limit on triple gauge coupling parameter
Bayesian approach or Frequentist approach ?
Section 2 : W γ channel radiation zero discovery potential analysis , hypothesis test or consistency test ?
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The expected confident intervals for triple gauge coupling parameter

3. The expected confident intervals for triple gauge coupling parameter
W γ physics
Testing SM model Triple gauge coupling vertex .
Measure magnetic dipole moment and electric quadruple moment of W
Irreducible background
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The expected confident intervals for triple gauge coupling parameter

4. The expected confident intervals for triple gauge coupling parameter
Generator level plots of W γ events with different coupling parameter setup
Lambda and kappa are highly correlated with event kinematic
Try to set limit in triple gauge coupling parameter by doing a counting experiment in event kinematic distribution
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The expected confident intervals for triple gauge coupling parameter

5. W γ events Full simulation result(300 pb-1)
High Pt bin is more sensitive to coupling parameter .
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The expected confident intervals for triple gauge coupling parameter

6. Anomalous Triple gauge coupling study
Try to use number of events in each bin of photon Pt distribution (n_i) to infer the triple gauge coupling parameter (lambda and kappa)
n_i : is observed numbers events in ith pt bin
S_i :is expected signal events in ith pt bin
is Prior of s_i in given λ obtained from BAUR NLO MC
Probability of measured n_i events in ith Pt bin for given expected signal (s_i ) and background (b_i)
Assume π(b_i) follow poison , and π(λ ) is flat .
Final PDF function of λ given obersered pt distribution is :
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The expected confident intervals for triple gauge coupling parameter

7. PDF distribution of λ and Δκ in Bayesian approach
P(Δκ |data)
P(λ |data) Δκ λ
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The expected confident intervals for triple gauge coupling parameter

8. 2D PDF distribution of λ and Δκ (300 pb-1)
Confident interval Δκ Δκ λ λ
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The expected confident intervals for triple gauge coupling parameter

9. Diboson CSC note method (Frequentist approach )
Construct likelihood function by convolved two Gaussian nuisance parameter with Poisson model .
N is number of observed events
v_s ,v_b : expected signal and background events
f_s,f_b : fractional standard deviation of error,
Assume signal systematic uncertainty is 10% , background uncertainty is 20% , that is σ_s=10% , σ_b=20%
The final likelihood function is :
More details could be found in Alan’s talk in SM meeting 13 ,Sep ,2007
where
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The expected confident intervals for triple gauge coupling parameter

10. Log likelihood distribution (300 pb-1)Δκ λ
Table :ΔL corresponseing o a coverage probability
Coverage probability
1 dimension
2 dimension
68%
0.5
1.15
95%
1.92
2.3
Confidence region is
Log(Δκ)>=Log L_max –ΔL.
ΔL=1.92 corresponse to 95% confident level for 1 dimensional case
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The expected confident intervals for triple gauge coupling parameter

11. 2D Log likelihood distribution (300 pb-1)
2D Log likelihood as a function of λ and Δκ
Confident intervals Δκ Δκ λ λ
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The expected confident intervals for triple gauge coupling parameter

12. The expected confident intervals for triple gauge coupling parameter
Confident interval of λ
In 300 pb -1 data
Bayesian
CSC note approach
68% CL
(-0.06,0.07)
(-0.04,0.05)
95% CL
(-0.09,0.1)
(-0.07,0.08)
Confident interval of Δκ
In 300 pb -1 data
Bayesian
CSC note approach
68% CL
(-0.50,0.60)
(-0.31,0.49)
95% CL
(-0.89,1.02)
(-0.58,0.73)
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The expected confident intervals for triple gauge coupling parameter

13. Section 2 : Theory prediction of radiation zero(RAZ)
Destructed interference between S ,t ,u channel will cause a zero in angular distribution of W gamma system (Geobel,Halzen and Leveille,PRD 23:2682,1981)
This zero point is in cos(theta)=(Q_u+Q_d)/ (Q_u-Q_d)=-1/3 for W+
In ATLAS case , theory predict that Eta(e)-Eta(γ) distribution have a dip in 0 .
ATLAS TDR plot
Eta(e)-Eta(γ)
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The expected confident intervals for triple gauge coupling parameter
radiation zero

14. eta(γ)-eta(e) distribution
Theoretical predictions have uncertainty , background will also mass up this radiation zero dip discovery .
Hard to model both signal and background in this delta eta distribution , not easy to do consistency test .
Hypothesis test could be performed to see whether the dip is significant .
Generator level
Central bin
bin2
bin3
Central bin
bin2
bin3
Full simulation
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The expected confident intervals for triple gauge coupling parameter

15. The expected confident intervals for triple gauge coupling parameter
RAZ dip Significant
Define that D1 is number s of events in central bin and D2 is number of events in Minimum neighbor bin .
s1 ,b1 is truth signal and background events numbers in central bin respectively , s2 ,b2 is truth signal and background events numbers in Minimum neighbor bin( bin 2 or bin3)
Our hypothesis test about whether there is a dip in central bin is that H0 :s1<s2 ,H1:s1>=s2
What we would like to know is given D1,D2, what is probability of H0
P(H0|D1,D2)
Central bin
bin2
Bin 3
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The expected confident intervals for triple gauge coupling parameter

16. Hypothesis test for RAZ
P(H0)/ (P(H0)+P(H1)) 1 2
And H0: s1<s2
H1:s1>=s2
In order to get H0, we need to do integration on s1 and s2, b1,b2
but only in region where s1 <s2 3
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The expected confident intervals for triple gauge coupling parameter

17. The expected confident intervals for triple gauge coupling parameter
300pb-1 data: 82% probability for H0:s1<s2
1 fb -1 data : 96% probability for H0:s1<s2
In order to get H0, we need to do integration on s1 and s2, b1,b2
but only in region where s1 <s2
Which is s2/s1>1
P( s2/s1| D1,D2) H1 H0 H0
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The expected confident intervals for triple gauge coupling parameter

18. The expected confident intervals for triple gauge coupling parameter
Summary
The expected 95% confidence intervals for anomalous triple gauge-boson coupling are
Bayesian approach : <λ<0.1 , -0.89<Δκ<1.02
Frequentist approach : -0.07< λ <0.08 , < Δκ <0.73
Frequentist approach could give a tighter limit .
Radiation zero (RAZ)discovery potential was studied using Bayesian Hypothesis test .
Expect to see 2 sigma significant for RAZ discovery in the first 1fb -1 data .
5/5/2018
The expected confident intervals for triple gauge coupling parameter