Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-304/4/ii/solution

Take Euclidean gamma matrices with , and independent Grassmann fields . The minimally coupled Dirac action is
Its mass term is the Dirac mass term, and the interaction is . With Fourier convention , the free kinetic operator is . The Euclidean adjoint Dirac Feynman rules are therefore
The vertex sign follows by expanding , not . Momentum conservation supplies the usual Dirac delta function at each vertex. There is only a one-gluon fermion vertex, with no two-gluon seagull because the fermionic covariant derivative is first order. A closed fermion loop has the usual fermionic sign . The identity checks the Dirac propagator.

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