# Renormalization of $\phi^3$-theory in 3+1 dimensions

Consider the Lagrangian $$\mathcal{L} = \frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi) - \frac{1}{2} m^{2} \phi^2 -\frac{1}{3!}\lambda \phi^{3}.$$ Following Michio Kaku's discussion in his QFT book, I proved that the superficial degree of divergence is given by $$D = 4 - E - V,$$ where $E$ is the number of external legs of a diagram, and $V$ the number of vertices (I'm not completely sure this expression is correct, though).

Using this, I found that the diagrams which are superficially divergent and need to be regularized are the tadpole diagram and the self-energy (again, I'm not completely sure of this, so if I'm wrong by any means tell me, please). I then calculated both amplitudes and introduced counterterms in the Lagrangian in order to render the calculations finite.

However, I'm having serious doubts when adding the counterterm corresponding to the tadpole diagram, since apparently it has to be linear (for instance, the author of these notes (pdf) includes such a term). Is it correct to do this? My main concern is that, since there are no linear terms in my original Lagrangian, the tadpole diagrams do not survive, and my reasoning was that maybe I shouldn't bother to renormalize this diagram at all, since there would be no contributions due to it. Is my reasoning right? Or should I keep the linear counterterm? Any input will be much appreciated.

• Expression of superficial divergence is correct. There are three more diagrams but they are cancelled as normalization factor and only self-energy and tadpole contributes. Tadpole causes a non-zero VEV of the field which need to be made zero in order to carry over the definitions of one-particle field operators from free fields to interacting fields (LSZ reduction) which requires a linear counterterm. Some discussion on tadpoles is given in Srednicki. – ved Oct 2 '16 at 13:17
• Maybe problem set 10 from the advanced QFT lecture I took can help you a bit. It was $\phi^3$ theory in six dimensions. The original problem set is no longer online, but the problem was basically about the steps to regularize and renormalize this theory. – Martin Ueding Oct 2 '16 at 19:47
• Also this might be helpful about the linear terms. I have referenced it in my problem set: physics.stackexchange.com/questions/103328/… – Martin Ueding Oct 2 '16 at 19:49