The Leptons — Question 8

Question 8

An invisible decay width counts light neutrino species conditionally

A neutral resonance has total width $2.500\pm.010$ GeV and visible width $2.000\pm.008$ GeV, with independent uncertainties. A model predicts .167 GeV per light neutrino-antineutrino species, with negligible model uncertainty.

1. Infer invisible width and its uncertainty.

2. Infer the effective number of light species and state the assumptions needed to interpret it as a species count.

Question 8 - Solution

An invisible decay width counts light neutrino species conditionally

(a) Method and calculation

$$\boxed{\Gamma_{\mathrm{inv}}=\Gamma_{\mathrm{total}}-\Gamma_{\mathrm{vis}}=.500\,\mathrm{GeV}}.$$$$\boxed{\sigma_{\mathrm{inv}}=\sqrt{.010^2+.008^2}=.0128062\,\mathrm{GeV}}.$$

(b) Interpretation and further calculation

$$\boxed{N_{\mathrm{eff}}=.500/.167=2.99401,\quad\sigma_N=.0128062/.167=0.0766841}.$$$$\Gamma_{\mathrm{inv}}=N_\nu\Gamma_{\nu\bar\nu}\quad\text{only if no other invisible channels contribute}.$$

Physical check

The inference counts species light enough to be produced and with the assumed coupling strength. It does not rule out heavy or differently coupled neutral particles.