Branch cuts in two-point function

Question

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is identified as the root of the argument of the logarithmic piece. Is this not a scheme dependent piece? At least, at the outset it looks so and it is also the same that one gets under dimensional regularization. Is there a general argument to prove that it is scheme independent?

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user MadKal

Comments

It is not clear what you mean by this... the exact propagator function (along with all other Green functions in QFT) is renormalization scheme independent. Therefore, any singularities such as cuts and poles would also be scheme independent.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user QuantumDot

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I agree with your statement. That was why eqn.7.19 in section 7.1 of Peskin & Schroeder confused me. The log[(1-x)m_0^2+...] term that determines the location of branch cut is part of the finite pieces of the subtractions - which are the scheme dependent pieces. My only question was whether there are some general arguments to show that this piece or the location of the branch cuts therein are scheme independent. I have to look at what is suggested below.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user MadKal

Answer 1

The logs that you get at one loop are scheme independent pretty much like the beta function at one loop is. There is a nice and neat discussion about it on the second volume of Weinberg.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user TwoBs