Past exam of the quantum field theory course of the University of Oxford / 2021 / 1 / b / i

Solution

Label the external four-momenta by

Ψ (p_{1}) + χ (p_{2}) \to Ψ (p_{3}) + χ (p_{4})

. The only tree-level Feynman diagram exchanges a scalar field particle between the transitions

Ψ (p_{1}) \to χ (p_{4})

and

χ (p_{2}) \to Ψ (p_{3})

. Its internal four-momentum is

q = p_{1} - p_{4} = p_{3} - p_{2}

. Thus its scattering amplitude, with a consistent external-state ordering, is

i M = (- i g)^{2} [\overset{u}{ˉ}_{χ} (p_{4}) u_{Ψ} (p_{1})] \frac{i}{( p _{1} - p _{4} ) ^{2} - m ^{2} + i 0} [\overset{u}{ˉ}_{Ψ} (p_{3}) u_{χ} (p_{2})] .

It is the crossed, or

u

-channel, exchange with these labels. A direct

t

-channel scalar field exchange would need a flavour-preserving Yukawa coupling, which the Lagrangian density does not contain. There is no annihilation Feynman diagram for two incoming fermions.

There are no discussions about this article yet.

There are currently no matching articles.