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 Incoming links: Total decay width

An extra active generation with the same weak neutral current couplings would increase the Z boson total decay width if any new particle–antiparticle pair were light enough. A simple sufficient kinematic condition for closing each such two-body channel is

m_{T}, m_{B}, m_{σ}, m_{N} ≳ \frac{m _{Z}}{2} ≃ 45.6 GeV / c^{2} .

The corresponding within-generation W boson channels close when

m_{T} + m_{B} ≳ m_{W}, m_{σ} + m_{N} ≳ m_{W}, m_{W} ≃ 80.4 GeV / c^{2} .

If both members of each doublet already exceed

m_{Z} /2

, these two sum constraints follow automatically. Mixing with lighter generations can also open charged-current channels involving one heavy and one light particle, depending on the mixing amplitudes; that information is not specified.

These are threshold estimates, not precise confidence limits. Close to threshold, two-body decay phase space suppresses a new partial decay width, and a quantitative lower mass bound requires the experimental width uncertainties and the fermion's couplings. The given observations yield no upper mass bound. A sterile neutrino with no standard active coupling would not obey the stated

Z

-width inference, whereas the question specifies the same quantum numbers as the existing generations.

Under the stipulated peak-energy approximation, set

E = m_{V}

separately for

V = W, Z

σ_{V} (m_{V}) \propto \frac{3 B _{in}^{V} B _{out}^{V}}{m _{V}^{2}} = \frac{3 Γ _{in}^{V} Γ _{out}^{V}}{m _{V}^{2} Γ _{V}^{2}} .

Define the measured ratio as

R = σ (W \to ℓ ν) / σ (Z \to ℓℓ)

. With the known production prefactor ratio neglected as instructed, the spin factors cancel and

R = \frac{m _{Z}^{2}}{m _{W}^{2}} \frac{Γ _{in}^{W} Γ _{out}^{W}}{Γ _{in}^{Z} Γ _{out}^{Z}} \frac{Γ _{Z}^{2}}{Γ _{W}^{2}} .

Solving for the positive physical width gives

Γ_{W} = Γ_{Z} \frac{m _{Z}}{m _{W}} \frac{Γ _{in}^{W} Γ _{out}^{W}}{R Γ _{in}^{Z} Γ _{out}^{Z}} .

If the production prefactors are retained as

C_{W}, C_{Z}

, multiply the quantity under the square root by

C_{W} / C_{Z}

. The partial decay widths in this extraction are calculated inputs, while the total decay width being inferred need not be fixed to its Standard Model value.

The peak and integrated resonance width dependence explains why the square root is essential to the specified peak approximation. An energy-integrated narrow-width cross section contains one inverse power of the total decay width, because integrating over the resonance supplies a factor proportional to its width. That different observable leads to a different extraction formula. Real hadron-collider analyses also include parton distribution functions and the energy convolution; the present result follows the explicit approximation in the source.