1. 1 Methods of Experimental Particle Physics Alexei Safonov Lecture #25

2. Today Brief reminder on upper limits discussion from last time More on parameter estimation Combining measurements Alternatives to ML Walk through and interpret the CMS results 2

3. One Sided Limits It is typical in HEP to look for things that are at the edge of your sensitivity You frequently can’t “see” the signal, but you still want to say that the cross-section for that new process can’t be larger than X Also very useful information for theorists and model builders as your results can rule out a whole bunch of models or parameter space of the models Can do it for either Bayesian or Frequentist methods Most of the time fairly straightforward – either construct a one-sided intervals with known coverage or calculate the integral from 0 to x in Bayesian case 3

4. Some good to remember numbers Upper 90% and 95% limit on the rate of a Poisson process in the absence of background when you observe n events If you observe no events, the limit is 3 events In Bayesian case, this would also be true for any expected B rate 4

5. Practical Parameter Estimation You usually calculate –log(L) Assuming you are doing a measurement, the minimum of –log(L) is the maximum of L, so that gives the most likely parameter value Changing – log(L) by +/-1/2 (think of taking a log of a gaussian distribution – it will give you (x-x 0 ) 2 /2   so x shifted by sigma gives ½) gives you 1 sigma deviation in the parameter (68% C.L.) With the MLS method, you vary it by 1 instead: MINUIT is the most used minimization package in HEP (it is part of ROOT), easy to use in simple cases, some experience required for more complex ones 5

6. Upper Limits Calculations In case of upper limits, you will need to integrate the distribution to find the point where 95% of the integral is accumulated Any numeric integration method will do as the integral is usually not too complicated I usually just keep halving the bin size until I get accuracy well below a fraction of a percent Surely can be done more efficiently, but if it takes no time who cares 6

7. Combining Measurements The product/limit will be a factor sqrt(2) more narrow/lower Kind of like doubling the sample 7

8. Correlated Systematics 8

9. Un-Binned Likelihood Binned likelihood is easy to interpret and gives consistent and predictable outcomes But there is a choice of bin size and strictly speaking you are loosing information by clamping things together Can you avoid that? 9

10. Un-Binned Likelihood 10

11. WALK THROUGH AND COMMENT ON THE CMS OBSERVATION PAPER http://arxiv.org/pdf/1303.4571.pdf 11

12. Expected Upper Limits Mean expected 95% C.L. upper limit on  /  (SM) in the absence of Higgs The p-value if Higgs exists versus its mass 12

13. Signal in Two Photon and ZZ Channels Pretty compelling, right? 13

14. Observed Upper Limits Three main channels What can you tell about upward downward fluctuations that happened in data based on these plots? 14

15. Two Less Important Channels How do you interpret these? 15

16. Upper Limit and Hypothesis Testing This includes all channels Left is upper limit Right is the “p-value” for the Higgs hypothesis (not null hypothesis) 16

17. Local p-value Two channels 17

18. Combined Channels 18

19. Signal Strength and Mass q on the right is the test statistic (it’s the likelihood ratio based analysis) 19

20. Signal Strength: Channels 20

21. Next Time Review ongoing and future analyses at the LHC 21