Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 7.8

Feynman rules — Schwartz M.D. z-lib.org

Physics — QFT · radioactive-decay-muon

Problem

Radioactive decay. The muon decays to an electron and two neutrinos through an intermediate massive particle called the

W^-

boson. The muon, electron and

W^-

all have charge

-1

. (a) Write down a Lagrangian that would allow for

\mu^- \to e^- \bar{\nu}_e \nu_\mu

. Assume the

W

and other particles are all scalars, and the

e^-

\nu_e

and

\nu_\mu

are massless. Call the coupling

g

. (b) Calculate

|\mathcal{M}|^2

for this decay in the limit that the

W

mass,

m_W

, is large. (c) The decay rate

\Gamma

(

= \frac{1}{\text{lifetime}}

) is proportional to

|\mathcal{M}|^2

. The coupling

g

should be dimensionless (like the coupling

e

for the photon), but appears dimensionful because we ignore spin. If the

W

spin were included, you would get extra factors of

p^\mu

, which would turn into a factor of

\sqrt{s} = m_\mu

|\mathcal{M}|^2

. Use dimensional analysis to figure out what power of

m_\mu

should be there. Also, throw in a

\frac{1}{192\pi^3}

for the three-body phase space, as in Eq. (5.55) from Problem 5.3. (d) Pick some reasonable perturbative value for

g

and use the muon mass (

m_\mu = 105\,\mathrm{MeV}

) and lifetime (

2.2 \times 10^{-6}\,\mathrm{s}

) to estimate the

W

mass. (e) The tauon,

\tau

, also decays to

e^- \nu_e \nu_\mu

. Use the

\tau

lifetime

\Gamma_{\mathrm{tot}}^{-1} = 2.9 \times 10^{-13}\,\mathrm{s}

and previous parts to estimate the

\tau

mass. Which of

m_W

g

m_\mu

, the muon lifetime, or the

192\pi^3

we threw in does your prediction depend on? (f) In reality, the tauon only decays as

\tau \to e^- \nu_e \nu_\mu

17.8\%

of the time. Use this fact to refine your

\tau

mass estimate. (g) How could you measure

g

and

M_W

separately using very precise measurements of the

\mu

and

\tau

decay distributions? What precision would you need (in \%)?