# Calculation of one-point functions in causal perturbation theory

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How are one-point functions evaluated in causal perturbation theory?

I'm not sure where my mistake is in following the standard procedure.

Take the first-order coupling $$T_1=\lambda \phi^3$$. Within the framework of causal perturbation theory, the corresponding tadpole diagram would be $$D^+_m(x_1-x_2):\phi:$$ where $$D^+_m$$ is the positive frequency part of the massive Jordan-Pauli distribution. The singular order is simply that of the given distribution $$\omega=-2$$, so one can proceed with the splitting procedure trivially:

$$t(x)=R_1-R'_1=D^{+(ret)}_m-D^+_m=D^{+(adv)}_m.$$

For two-point functions, the splitting procedure reproduces the usual Feynman rules if $$\omega<0$$, and cases where $$\omega\geq 0$$ are split differently, which avoids UV divergences.

The issue is that the above both does not give a Feynman propagator while having $$\omega<0$$, yet in the "standard" theory with Feynman rules the tadpole loop would be divergent, suggesting $$\omega\geq 0$$. Is that discrepancy a feature of the causal theory or have I misconstrued the procedure? In the latter case, what is the procedure to evaluate such diagrams?

This post imported from StackExchange Physics at 2020-03-18 13:31 (UTC), posted by SE-user Quantumness
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