Friday, 18 December 2015

quantum field theory - Massless integrals in dim-reg



Consider the massless divergent integral $$ \int dk^4 \frac{1}{k^2}, $$ which occurs in QFT. We can't regularize this integral with dim-reg; the continuation from the massive to the massless case is ill-defined. It can be shown, however, that no "inconsistencies" occur if $$ \int dk^4 \frac{1}{k^2} "=" 0. $$ I think this is the now proven 't Hooft-Veltman conjecture. I don't understand this "equation". The integral is certainly not zero (although inconsistencies might occur in dim-reg if it weren't treated as zero in some contexts.)


Suppose I think about $\lambda\phi^4$ theory, with no bare mass for the field. Is it reasonable to claim that I will chose dim-reg and calculate the one-loop correction to the mass as $$ \delta m^2 \propto \int dk^4 \frac{1}{k^2} = 0 \qquad ? $$ Such that the particle stays massless, with no fine-tuning. I think this is simply incorrect - an unreasonable application of the 't Hooft-Veltman conjecture (I've no reason to worry about the consistency of dim-reg because I'm not regulating any integrals with it). Surely the 't Hooft-Veltman conjecture can only be applied in particular contexts? I can't start making any calculation in QFT, see an integral such as $$ \int dk^n \frac{1}{k^a} = 0 \text{ for $n>a$}, $$ and set it zero, citing dim-reg and 't Hooft-Veltman?


P.S. This is not a straw man. I read people saying such things in the context of classical scale invariance.




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