Re: Question; Renormalization Conditions

Flip wrote:

Hi everyone! I have a few questions regarding renormalization in QFT.

1. In Peskin chapter 10, he renormalizes phi^4 theory using the
renormalization conditions in equation (10.19), which basically say
that the propagator has a pole at p^2 = m^2 and that the 4-point
interaction is exact for s=4m^2, t=u=0. These are reasonable
assumptions (I think). However, in equation (12.30) of chapter 12, he
introduces a different set of renormalization conditions defined at a
spacelike momentum. I.e. the propagator is defined at p^2 = -M^2 nd
the four point function is defined at s = t = u = -M^2. These are
unphysical values, why are these renormalization conditions valid (or
reasonable)? Why not use physically accessible conditions?

To make the results finite, in a renormalizable QFT you need to choose a
finite set of renormalization conditions. These are fixed only up to a
finite counter term for a finite number of wave-function normalization,
mass and coupling constant parameters. Using a space like
renormalization point can be advantageous (or even mandatory if you
have massless particles involved) since there you have no trouble with
branch cuts of your vertex functions.

Then you must choose some observable quantities to fix the physical
(finite) parameters of the theory to experiment anyway. The numerical
values of these physical parameters will depend on the chosen
renormalization scale, M, in a way such that the S-matrix elements,
i.e., the physical observables do not change. However, within
perturbation theory the whole thing makes only sense if the coupling
constants with respect to which you are expanding the perturbation
series is small.

Peskin Schroeder is not such a good choice to learn renormalization
theory since there are a lot of points treated not carefully enough,
especially in the chapter about the linear sigma model where he chooses
troublesome renormalization conditions. I recommend to look the issue
up in Weinberg's Quantum Theory of Fields and in the paper

T. Kugo, Symmetric and mass-independent renormalization, Prog. Theor.
Phys. 57 (1977) 593,
URL http://www-lib.kek.jp/cgi-bin/img_index?197606150


2. In chapter 12 section 4 he describes the renormalization of local
operators. Is it correct to define a local operator as one that is
composed of a product of fields at the same spacetime point? In the
diagrams for the Greens function with a local operator on page 431,
the diagrams being summed have different numbers of legs! (Similar to
page 601-603) I don't quite understand what's going on here and why
these diagrams with different in/out states can be summed together.

I do not have the book here at home. So I cannot help with this specific
issue now.

3. In chapter 11, p. 355, why is it acceptable to use the "tadpole
diagram = 0" renormalization condition in place of the usual one for
the propagator? How is this equivalent to the proagator condition?

Without knowing the context, because I don't have the book here, I can
only say that the tadpole selfenergy diagram of phi^4 theory can be set
to 0 (in vacuum QFT, not at finite temperature by the way!) savely
since it is momentum independent and subtracted by the renormalization
condition for the mass counter term anyway.


--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:hees@xxxxxxxxxxxxx

.

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