## Decay Rates

The definition of the decay rate (also called decay width) of an unstable particle is best defined by considering its self-energy,

( p) = 1 Z

ﬀ

, (11.1)

where the diagram for the 1P I two-point function is now to be evaluated using the Feynman rules in Table 9.1 (pg. 63). The relation with the selfenergy follows from the fact that, apart from the overall factor i/(2π )4, one has for each of the two external lines an extra factor −i(2π )2√Z as compared to the amputated 1P I two-point function; in total one therefore has −i Z times the amputated two-point function. The latter indeed equals i( p); see Equation (9.9). We will now consider a simple example of a scalar field theory with two types of fields, a field ϕ(x) associated with a light particle (mass m) and a field σ (x) associated with a heavy particle (mass M > 2m), which can decay in the lighter particles if we allow for a coupling between one σ and two ϕ fields,

V(σ, ϕ) = 12 gσϕ2 ,

ϕ

ϕσ ≡ g. (11.2)

For the σ two-point function in lowest order we find

ﬀ

p p =

ﬀ

p p

p−k

k

+

ﬀ

p p + · · · . (11.3)

If σ is a stable particle (i.e., M < 2m), the loop in the first diagram corresponds to virtual ϕ particles moving between the vertices, since always k2 = m2 and (k − p)2 = m2. However, as soon as M > 2m, the loop integral will contain contributions where the ϕ particles can be on the mass-shell and behave as real particles, e.g., k2 = m2 and (k − p)2 = m2 in the first diagram. The real ϕ particles can escape to infinity, thereby describing the decay of the σ particle. Its number will reduce as a function of time.