Massless $\phi^4$ theory

Asked Jan 15 '19 at 18:18

Active Jan 15 '19 at 18:26

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Most of the standard textbooks on QFT discuss in some detail the massive $\phi^4$ theory in 4d space-time.

I would be interested to see a discussion of massless $\phi^4$ theory (in fact other non-vanishing renormalizable interactions would be of interest too). Particularly I am curious in the following two questions:

1) how to compute the S-matrix in low order of perturbation (taking into account both UV and IR divergences).

2) if the physical mass vanishes, does the bare mass vanish?

UPDATE: Essentially the same question was asked more than 4 years ago here: Massless $\lambda \phi^4$ QFT However it is still unanswered.

edited Jan 15 '19 at 18:26

asked Jan 15 '19 at 18:18

MKO

2,140

If you know a reason why this would not just be the limit of massive $\phi^4$ theory for $m\to 0$, it would be nice to mention that since that would dissolve the question. – ACuriousMind Jan 15 '19 at 18:25
@ACuriousMind: The reason is that the relevant expressions for S-matrix diverge in that limit. They should diverge because the IR divergences should be taken into account like in QED. – MKO Jan 15 '19 at 18:28
It might also be relevant to mention that $\phi^4$ theory is known to be trivial in >4d, making perturbative computations inherently questionable. – ACuriousMind Jan 15 '19 at 18:34

Massless $\phi^4$ theory

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