Novel approach to assess the impact of the Fano factor on the sensitivity of low-mass dark matter experiments

As first suggested by U. Fano in the 1940s, the statistical fluctuation of the number of pairs produced in an ionizing interaction is known to be sub-Poissonian. The dispersion is reduced by the so-called Fano factor, which empirically encapsulates the correlations in the process of ionization. In modelling the energy response of an ionization measurement device, the effect of the Fano factor is commonly folded into the overall energy resolution. While such an approximate treatment is appropriate when a significant number of ionization pairs are expected to be produced, the Fano factor needs to be accounted for directly at the level of pair creation when only a few are expected. To do so, one needs a discrete probability distribution of the number of pairs created $N$ with independent control of both the expectation $mu$ and Fano factor $F$. Although no distribution $P(N|mu,F)$ with this convenient form exists, we propose the use of the COM-Poisson distribution together with strategies for utilizing it to effectively fulfill this need. We then use this distribution to assess the impact that the Fano factor may have on the sensitivity of low-mass WIMP search experiments.