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 Incoming links: Past exam of the quantum field theory course of the University of Oxford / 2024 / 2 / d

Past exam of the quantum field theory course of the University of Oxford / 2024 / 2 / c / Solution 2026-10-25

Interpret the repeated species label in the Yukawa interaction as summed over

i = 1, 2

: each Feynman vertex is diagonal in species and equals

- i g

. Define the invariant scattering amplitude by

S - 1 = i (2 π)^{4} δ^{(4)} (P_{f} - P_{i}) M

. There is one tree-level Feynman diagram, an

s

-channel scalar exchange. In the left panel,

j = 2

for this part; the right panel only occurs in d. Arrows show fermion-number flow, so an antifermion arrow opposes its physical momentum.

Figure 1.
Labelled scalar-exchange annihilation and elastic diagrams, with arrows following fermion-number flow
. Details.

k = p_{1} + p_{2} = p_{3} + p_{4}

and

s = k^{2}

, the momentum-space Feynman rules give

i M = [\overset{v}{ˉ}^{s_{2}} (p_{2}) (- i g) u^{s_{1}} (p_{1})] \frac{i}{s - M ^{2} + i 0} [\overset{u}{ˉ}^{s_{3}} (p_{3}) (- i g) v^{s_{4}} (p_{4})], M = - \frac{g ^{2} [ v ˉ _{2} u _{1} ] [ u ˉ _{3} v _{4} ]}{s - M ^{2} + i 0} .

The scalar carries no spin label. Average over the

2 \times 2

incoming spin states and sum the four outgoing combinations. For the real scalar Yukawa interaction, the conjugate of

\overset{v}{ˉ}_{2} u_{1}

is

\overset{u}{ˉ}_{1} v_{2}

. The fermion spin sums then yield

\overline{∣ M ∣^{2}} = \frac{g ^{4}}{4 ( s - M ^{2} ) ^{2}} tr [(\neq p_{2} - m) (\neq p_{1} + m)] tr [(\neq p_{3} + m) (\neq p_{4} - m)] .

The bar denotes the initial spin average, not a Dirac adjoint. The gamma matrix trace identities give

tr [(\neq p_{2} - m) (\neq p_{1} + m)] = 4 (p_{1} \cdot p_{2} - m^{2}) = 2 (s - 4 m^{2}),

and the second trace has the same value because

p_{3} + p_{4} = p_{1} + p_{2}

and all external masses are

m

. Therefore

\overline{∣ M ∣^{2}} = g^{4} \frac{( s - 4 m ^{2} ) ^{2}}{( s - M ^{2} ) ^{2}} = g^{4} (\frac{4 m ^{2} - s}{M ^{2} - s})^{2} .

This is the fixed-order expression away from

s = M^{2}

. If that pole is accessible, a physical scattering resonance requires a scalar self-energy and decay width; the tree formula alone is not finite there.

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