1. G. Cowan RHUL Physics Higgs combination note status page 1 Status of Higgs Combination Note ATLAS Statistics/Higgs Meeting Phone, 7 April, 2008 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan

2. G. Cowan RHUL Physics Higgs combination note status page 2 Status of combination note Some updates to intro relative to meeting last Monday: www.pp.rhul.ac.uk/~cowan/atlas/higgs_combo_05.pdf Ongoing: Still tweaking section on limits Still deciding what to write on look-elsewhere-effect Implemented most comments, e.g., Eilam, Bill Urgently needed: Description of individual channels Results of combination

3. G. Cowan RHUL Physics Higgs combination note status page 3 From discussion with Bill I have said floating mass method most direct approach to answering look-elsewhere-effect but computationally more difficult. Bill argues that floating-mass method not more difficult overall. But, important difference: for fixed-mass method we can use asymptotic formulae for the pdf of the likelihood ratio. Naively one would expect this to also hold in floating-mass case for 2 degrees of freedom. But as Bill points out, Wilks' theorem does not apply in that case because the MVB of the estimator of m H blows up... hence need toy MC. Other points from Bill -- agree: Need ~several 10 7 events for toy MC Include reference from Gao, Lu and Wang (other refs?)

4. G. Cowan RHUL Physics Higgs combination note status page 4 Thoughts on setting limits Currently the note says that we will use the "CL s+b method", i.e., for the hypothesized  (e.g. 1) compute the p-value:  is excluded at CL=0.95 if p <  = 0.05, and if  =1 is excluded, the corresponding m H is excluded (for SM). E.g. present expected limit on  vs m H. This requires f(q  |  ) for all , or at least for the Asimov approximation   for all .

5. G. Cowan RHUL Physics Higgs combination note status page 5 Thoughts on limits Sometimes a poor (low) likelihood ratio arises because data fluctuates up, other times when it fluctuates down (  -hat=0). (Note: require  -hat>0.)

6. G. Cowan RHUL Physics Higgs combination note status page 6 Thoughts on limits If we take only q  as our test statistic, p-value is area to right of observed q , regardless of whether data fluctuated up or down: p-value q obs

7. G. Cowan RHUL Physics Higgs combination note status page 7 Thoughts on limits But one could argue that if we are interested in an uppper limit on , we should base p-value only on data that fluctuates down. I.e. limit is hypothetical value of  that would give data with  -hat as low as found or lower, and q  as high as found or higher. (cf. usual way to find upper limit just based on number of observed events.) p-value? q obs

8. G. Cowan RHUL Physics Higgs combination note status page 8 Thoughts on limits Limit is found by using p=0.05, w=0.5 and solve for . Equivalently, solve for  : We want CL = 95%, i.e., exclude  if its p-value < 0.05. Suppose a fraction w = 0.5 fluctuate down, and that their values of q  follow a chi-square distribution (1 dof). The p-value based only on those that fluctuate down is (see note, Appendix C):

9. G. Cowan RHUL Physics Higgs combination note status page 9 Extension of input request For discovery, using Asimov data for signal+background: For exclusion, using Asimov data for background only: Specifically, a table of ( ,  2ln  values for 0 <  < 2, steps of 0.1, and for the m H and luminosity values as given in the initial request. (Also need e.g. distributions of  ln, etc., as per initial request.) ("all"  )