Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 3 / a

Solution

Repeated insertions of the fermion self-energy form a geometric Dyson resummation:

G G (\neq p) = S_{F} + S_{F} Σ S_{F} + S_{F} Σ S_{F} Σ S_{F} + \dots, = [S_{F} (\neq p)^{- 1} - Σ (\neq p)]^{- 1} = [i \neq p + m - Σ (\neq p)]^{- 1} .

m_{phys}

is determined by the zero of the exact inverse propagator at

p^{2} = - m_{phys}^{2}

. If

Σ (\neq p) = i \neq p Σ_{V} (p^{2}) + m Σ_{S} (p^{2}),

then to all orders the pole obeys

m_{phys} = m \frac{1 - Σ _{S} ( - m _{phys}^{2} )}{1 - Σ _{V} ( - m _{phys}^{2} )}

with the signs fixed by the displayed Dyson convention.

There are no discussions about this article yet.

There are currently no matching articles.