= Solution
The new light active leptons open one additional $W\to\sigma N$ channel and two $Z$ channels. At the stated equal-partial-width approximation,
$$
\boxed{\Gamma_W'=2085+232=2317\ \mathrm{MeV}.}
$$
\b[The quoted $501$ MeV is the total invisible width into the three known <neutrino> flavors], not the width into one flavor. The <invisible Z width per active neutrino> fixes this interpretation. The notation in the numerical input is ambiguous; the <Standard Model> normalization fixes the new active-neutrino contribution to $501/3=167$ MeV. Adding the new charged-lepton width therefore gives
$$
\boxed{\Gamma_Z'=2495+167+84=2746\ \mathrm{MeV}.}
$$
This interpretation is consistent with the usual determination of the effective number of active <neutrinos> from the invisible width divided by the predicted width of a single species.<Number of Neutrino Types by Particle Data Group>[definition of the effective <neutrino> count from the total invisible width and the single-species <neutrino> width]{ref}
Keep the original light-fermion production and observed-decay <partial decay widths>, masses and prefactors fixed. New leptonic channels alter their <branching fractions> through the total widths. From the peak formula in <4/d>,
$$
\frac{R'}{R}=\left(\frac{\Gamma_Z'}{\Gamma_Z}\frac{\Gamma_W}{\Gamma_W'}\right)^2
=\left(\frac{2746}{2495}\frac{2085}{2317}\right)^2
=0.9808893.
$$
Consequently,
$$
\boxed{\frac{R'-R}{R}=-0.0191107\simeq-1.91\%.}
$$
The relatively small change is a cancellation between the increase in the two total widths; each separately changes substantially more. This assumes that the new <quarks> do not contribute and that adding leptons leaves the old <partial decay widths> unchanged.
If the source's $501$ MeV were instead imposed literally as the width of a single <neutrino> flavor, the algebra would give $\Gamma_Z'=3080$ MeV and $R'/R=1.23401$. That would be a different numerical premise, inconsistent with the <Standard Model> interpretation of the supplied invisible-width value. It is recorded to make the normalization issue explicit, rather than silently treating the three-flavor sum as a one-flavor <partial decay width>.
Back to article page
