Sensitivity optimization of multichannel searches for new signals

💡 Research Summary

The paper extends the “sensitivity” concept introduced by G. Punzi for searches of new phenomena to the case of multiple independent channels (or bins) and provides a closed‑form figure‑of‑merit (FOM) that can be used to optimise experimental configurations. The original definition of sensitivity is the requirement that the probability of correctly rejecting the null hypothesis (1 – β(μ)) exceeds a chosen confidence level CL for a given signal strength μ. In the single‑channel Poisson counting experiment this leads to a simple expression involving the signal efficiency ε, the expected background B and the Gaussian‑sigma equivalents a (for the chosen type‑I error α) and b (for the confidence level).

In the multi‑channel case the observable is a set of N integer counts {k_i}, each drawn from a Poisson distribution with mean b_i(t) + ε_i(t) μ, where b_i(t) is the known background in bin i and ε_i(t) is the channel‑specific efficiency. Assuming independence, the joint probability is a multinomial‑Poisson product. The authors adopt the Fisher score S_F = ∂ log L/∂μ|_{μ=0} as the test statistic. Because the score is a linear combination of many Poisson variables, the central‑limit theorem justifies approximating its distribution by a normal law with mean Σ ε_i b_i and variance Σ ε_i² b_i under H₀. The critical region for a one‑sided test at significance α is therefore T > a √(Σ ε_i² b_i).

Under the alternative hypothesis Hₘ (μ ≠ 0) the mean and variance acquire additional μ‑dependent terms: the mean becomes Σ ε_i b_i + Σ ε_i² b_i μ and the variance becomes Σ ε_i² b_i + Σ ε_i³ b_i² μ. Inserting these expressions into the master inequality 1 – β(μ) > CL yields the compact relation

A μ ≥ a √A + b p A + B μ,

with A = Σ ε_i² b_i and B = Σ ε_i³ b_i². Solving for μ gives a quadratic equation; the exact solution provides the minimal detectable signal μ_min. For practical purposes the authors note that b≈a (the usual approximation in the single‑channel case) leads to a much simpler expression

μ_min ≈ (a √A + a² B/A)/A.

Consequently the figure of merit to be maximised in an optimisation procedure is

FOM = A^{3/2}/A + (a/2) B/A = √A + (a/2) (Σ ε_i³ b_i²)/(Σ ε_i² b_i).

Crucially, this FOM depends only on the efficiencies ε_i and the background levels b_i, not on the unknown signal cross‑section σ(m). It therefore provides a clean metric for designing cuts, choosing detector regions, or allocating luminosity among channels. The metric scales linearly with an overall rescaling of all efficiencies, as expected.

The authors then illustrate the formalism with a common experimental scenario: a Gaussian‑shaped signal superimposed on a slowly varying (approximately flat) background. They model the per‑bin efficiency as ε_i ∝ exp