- #1

RicardoMP

- 49

- 2

where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices.

For this diagram I've written:

$$i\Pi^{\mu\nu}=\int_{}^{}\frac{d^4k}{(2\pi)^4}.[ig(t_a)_{ij}\gamma^\mu].[\frac{i[(\not\! k-\frac{\not{q}}{2})+m_1]}{(k-\frac{q}{2})^2-m_1+i\epsilon}\delta_{il}].[ig(t_b)_{kl}\gamma^\nu].[\frac{i[(\not\! k+\frac{\not{q}}{2})+m_2]}{(k+\frac{q}{2})^2-m_2+i\epsilon}\delta_{kj}]$$

In Peskin & Schroeder's Introduction to QFT it is said: "a closed fermion loop always gives a factor of -1 and the

**trace of a product of Dirac matrices**". In short, I can write the above expression as:

$$i\Pi^{\mu\nu}=g^2T_F\delta{ab}\int_{}^{}\frac{d^4k}{(2\pi)^4}\frac{tr[\gamma^\mu.((\not\! k-\frac{\not{q}}{2})+m_1).\gamma^\nu.((\not\! k+\frac{\not{q}}{2})+m_2)]}{((k-\frac{q}{2})^2-m_1+i\epsilon).((k+\frac{q}{2})^2-m_2+i\epsilon)}$$

where ##T_F## is the Dynkin index in the fundamental representation I get after contracting the generators and deltas.

What I'm having a hard time with is on how to explicitly write the Dirac indices in each gamma matrix and slashed momenta in order to get something of the sort, e.g, ##(matrix)_{ii}=tr(matrix)##.