G. CowanRHUL Physics Higgs combination note status page 1

Status of Higgs Combination Note

ATLAS Statistics/Higgs Meeting

Phone, 7 April, 2008

Glen CowanPhysics DepartmentRoyal Holloway, University of Londong.cowan@rhul.ac.ukwww.pp.rhul.ac.uk/~cowan

G. CowanRHUL Physics Higgs combination note status page 2

Status of combination noteSome updates to intro relative to meeting last Monday:

www.pp.rhul.ac.uk/~cowan/atlas/higgs_combo_05.pdf

Ongoing:Still tweaking section on limitsStill deciding what to write on look-elsewhere-effectImplemented most comments, e.g., Eilam, Bill

Urgently needed: Description of individual channelsResults of combination

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From discussion with BillI 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 canuse asymptotic formulae for the pdf of the likelihood ratio.

Naively one would expect this to also hold in floating-masscase for 2 degrees of freedom. But as Bill points out, Wilks'theorem does not apply in that case because the MVB ofthe estimator of mH blows up... hence need toy MC.

Other points from Bill -- agree:Need ~several 107 events for toy MCInclude reference from Gao, Lu and Wang (other refs?)

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Thoughts on setting limits

Currently the note says that we will use the "CLs+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 mH is excluded (for SM).

E.g. present expected limit on vs mH .

This requires f(q|) for all , or at least for the Asimovapproximation for all .

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Thoughts on limits

Sometimes a poor(low) likelihood ratioarises because datafluctuates up, othertimes when it fluctuatesdown (-hat=0).

(Note: require -hat>0.)

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Thoughts on limits

If we take only q as ourtest statistic, p-value is areato right of observed q,regardless of whether datafluctuated up or down: p-value

qobs

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Thoughts on limits

But one could argue that ifwe are interested in an uppper limit on , weshould base p-value only ondata that fluctuates down.

I.e. limit is hypothetical valueof that would give data with-hat as low as found or lower,and q as high as found orhigher. (cf. usual way to find upper limit just based on number of observed events.)

p-value?

qobs

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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 theirvalues of q follow a chi-square distribution (1 dof).

The p-value based only on those that fluctuate down is(see note, Appendix C):

G. CowanRHUL 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 mH and luminosity values as given in the initial request.

(Also need e.g. distributions of ln, etc., as per initial request.)

("all" )