# Two loop Feynman diagram with quartic vertex

• A

## Summary:

I want to calculate a feynman diagram with four fields and two loops
I am trying to calculate the effective potential of two D0 branes scattering in Matrix theory and verify the coefficients in this paper: K. Becker and M. Becker, "A two-loop test of M(atrix) theory", Nucl. Phys. B 506 (1997) 48-60, arXiv:hep-th/9705091. The fields are expanded about a constant background classical field which is treated non perturbatively. As a consequence, the diagrams have no legs and we can calculate the effective action just by evaluating the diagrams.

I need to know if I am following the correct approach, especially because the calculations are tedious and I need to prove a cancellation at the end. Consider the quartic vertex, where the wavy lines indicate bosonic and gauge field propagators.

This diagram corresponds to the quartic vertex $$-\frac{g}{2}\epsilon^{abx} \epsilon^{cdx} A_{a} Y^{i}_{b}A_{c} Y^{i}_{d}$$

−g2ϵabxϵcdxAaYbiAcYdi in a Lagrangian expanded around a background field. We know the Feynman rules for the theory, i.e. we know $<AA>, <YY> \text{ and } <YA>$ propagators. I need to know if the contribution to the effective action is correct: $$-\frac{g}{2} \epsilon^{abx} \epsilon^{cdx}\int d\tau \langle A_{a} A_{c} \rangle \langle Y^{i}_{b} Y^{i}_{d} \rangle + \langle A_{a} Y^{i}_{b} \rangle \langle A_{c} Y^{i}_{d} \rangle$$
\langleYY⟩,\langleAA