Status of Higgs Combination Note

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 Higgs combination note status

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 Higgs combination note status

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 mH blows up... hence need toy MC. Other points from Bill -- agree: Need ~several 107 events for toy MC Include reference from Gao, Lu and Wang (other refs?) Higgs combination note status

4. Thoughts on setting limits Currently the note says that we will use the "CLs+b method", i.e., for the hypothesized m (e.g. 1) compute the p-value: m is excluded at CL=0.95 if p < a = 0.05, and if m =1 is excluded, the corresponding mH is excluded (for SM). E.g. present expected limit on m vs mH . This requires f(qm|m) for all m, or at least for the Asimov approximation lA(m) for all m. Higgs combination note status

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

6. Thoughts on limits qobs If we take only qm as our test statistic, p-value is area to right of observed qm, regardless of whether data fluctuated up or down: p-value Higgs combination note status

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

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

9. Extension of input request For discovery, using Asimov data for signal+background: For exclusion, using Asimov data for background only: ("all" m) Specifically, a table of (m,-2lnl) values for 0 < m < 2, steps of 0.1, and for the mH and luminosity values as given in the initial request. (Also need e.g. distributions of -2 lnl, etc., as per initial request.) Higgs combination note status