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High Energy, Nuclear, Particle Physics
Two loop Feynman diagram with quartic vertex
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[QUOTE="saadhusayn, post: 6377213, member: 555489"] [B]TL;DR Summary:[/B] 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", [URL='https://doi.org/10.1016/S0550-3213(97)00518-X']Nucl. Phys. B [B]506[/B] (1997) 48-60[/URL], arXiv:[URL='https://arxiv.org/abs/hep-th/9705091']hep-th/9705091[/URL]. 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 $$ -\frac{g}{2}\epsilon^{abx} \epsilon^{cdx} A_{a} Y^{i}_{b}A_{c} Y^{i}_{d}$$ [ATTACH type="full"]267498[/ATTACH]−g2ϵabxϵcdxAaYbiAcYdi in a Lagrangian expanded around a background field. We know the Feynman rules for the theory, i.e. we know [itex] <AA>, <YY> \text{ and } <YA>[/itex] 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 [/QUOTE]
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Two loop Feynman diagram with quartic vertex
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