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.




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...