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S8e. Renormalization scale and experimental energy scale
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The picture drawn in the preceding is somewhat incomplete with
regard to the practice of computing, due to the fact that we cannot
compute this renormalized theory at any E, since it is exceedingly
complicated.
Thus we need to consider approximations. These approximations are
no longer independent of E, since the approximation errors depend
on it. It turns out that the approximation errors are small only
when the energy scale of the experiment for which a prediction is
made is close to the renormalization scale E, since the perturbative
expansion contains arbitrary powers of log(E_experiment/E) which
therfore must be kept small. See, e.g., Weinberg's QFT book,
Vol. 2, Chapter 18.1.
Thus one needs to evaluate the theory near the scale of interest.
However, perturbation theory is valid only near a fixed point E^* of
the renormalization group equations. Therefore, one determines
approximate formulas for the quantities q_ren(mu,E) with E close to
the appropriate fixed point E^*, and then uses (also approximate)
renormalization group equations to transform the result to the
scale of interest.
Thus there are two different scales involved, the energy scale
E_exp where the experiments are done, and the renormalization scale
E_ren (previously denoted by E).
On the experimental side, coupling constants (such as the charge)
are determined with reference to some effective, coarse grained theory
(such as the nonrelativistic Schroedinger equation). This effective
theory depends on E_exp (for QED, the charge is traditionally defined
in the low energy limit E_exp --> 0). This effective theory behaves
like any other coarse-grained theory, giving rise to running coupling
constants such as e=e_exp(E_exp). But these depend on the details of
the coarse-graining scheme, and the computed results depend on the
coarse-graining, too, and hence on E_exp.
The experimental running coupling constants are only loosely related
to the running coupling constants such as e=e_ren(E_ren) obtained by
the Callan-Symanzik equation (= the renormalization group equation
in terms of the renormalization scale E_ren). The latter are, in theory,
uniquely defined by the renormalization prescription. There the
coupling constants are defined not by an experimental prescription
but as parameters in the renormalization prescription. For example,
in Phi^4 theory, lambda=lambda(M) is defined by equation (12.30) in
Peskin/Schroeder (and E_ren=Mc^2), and the charge e=e(M) in QED
by (10.39) [but at spacelike momentum p^2=-M^2 as in Chapter 12].
As discussed, the physical predictions at any energy are completely
independent of M if e(M) and the other renormalized parameters slide
with M. At least this would be the case in a fully nonperturbative
calculation (which we cannot do). However, the few-loop approximations
depend heavily on M, and give a reasonable approximation to the
exact theory _only_ at energies close to E_ren=Mc^2. Thus the few-loop
approximation behaves just like an effective theory, provided we choose
E_ren = E_exp (or close). But the analogy is not complete since
in a true effective theory we could choose the coarse-graining scale
anywhere at or above E_exp, while for good few-loop approximations
we need to choose it always close to E_exp.
Thus, if one could solve the equations exactly, the dependence on M
and the Callan-Symanzik equation would be completely irrelevant,
and nothing at all could be extracted from it. But in practice one
can work only at few loops, and then different values of M may give
vastly different results, and the equation is very useful since it
enables one to work with the right M.
The renormalization group equations are used to move from an
M near the fixed point (where one can do perturbation theory and has
reliable few-loop calculations but where the approximation errors
given by the higher order terms in the perturbation series are huge)
to an M near the experimental scale (where the approximation error
is small, and the few-loop calculation therefore reasonably accurate).
This is often expressed by saying, loosely, that the renormalization
group approach partially resum the perturbation series.
One gets what is called ''renormalization group improved perturbation
theory'', which is predictive about a much larger range of coupling
constants than simple renormalized perturbation theory (which only
works for very weak coupling).