Two loop Feynman diagram with quartic vertex

[saadhusayn]

TL;DR 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

 

[LightHero]

⟩,\text{ and }\langleAY⟩ propagators. I need to know if the contribution to the effective action is correct:−g2ϵabxϵcdx∫dτ⟨AaAc⟩⟨YbiYdi⟩+⟨AaYbi⟩⟨AcYdi⟩Yes, your approach is correct. The contribution to the effective action is indeed given by the expression that you have written. The only thing that you need to do is to calculate the propagators for the fields in order to obtain the numerical factors that you need to complete the calculation.