-------------------------------
S8f. Dimensional regularization
-------------------------------
The neatest way to perform regularization, and the only one which
works well in complicated cases such as nonabelian gauge theories
is dimensional regularization. Unfortunately, it is presented
in most textbooks in a way that looks quite mysterious, involving
unphysical fractional dimensions. This is however just sloppiness
on the side of physics tradition, and a more rigorous approach
removes everything strange.
The rules for dimensional regularization are derived in Euclidean
space rather than Minkowski space. To get the latter, one needs an
additional analytic continuation.
For p in Euclidean d-space (d>0 integral), we put p^2=p^Tp.
If d is a positive integer and f(p^2) is integrable (i.e. decays fast
enough), then standard Lebesgue integration gives the formula
integral dp^d/(2 pi)^d f(p^2)
= C_d integral_0^inf dr r^{d-1}f(r^2), (1)
where C_d is given in terms of the Gamma function as
C_d = 2 pi^{d/2}/Gamma(d/2). (2)
We observe that the formula (2) makes sense for arbitrary complex d
with nonnegative real part, and that therefore for
f(s)=r^2j/(r^2+m^2)^n, n>j+d/2,
the well-defined right hand side of (1) is an expression I(d,j,n)
which depends analytically on d,j,n.
In particular, the cases j=0 and j=1 lead to the expressions
given in (7.85/86) of the quantum field theory book by Peskin/Schroeder
(P/S). A similar reasoning produces (7.87) and more complicated rules
analogous to those given in P/S on p.807
(where, however, analytic continuation to Minkowski space has
already been performed). These rules, together with the
Feynman trick stated as (A.39)on p.806 of P/S, can be used to evaluate
integrals of arbitrary rational Lorentz-invariant expressions
provided that they decay fast enough.
Note that the resulting formula
integral dp^d/(2 pi)^d f(p^2) = I(d,j,n) (3)
is valid only if n>j+d/2. This condition is needed to ensure
sufficiently fast decay at infinity to make the Lebesgue integral
well-defined and integral d.
For other values the above computations are meaningless, and any
contradiction derivable from it is therefore irrelevant.
As irrelevant as the well-known fact that a divergent alternating
infinite sum can be given any value whatsoever by formal rearrangements.
Remarkably, however, I(d,j,n) (and the analogous formulas on
p.807) can be analytically continued to smaller values of d.
Unfortunately this analytic continuation has poles at the most
interesting value d=4. Physicists therefore consider d=4-eps, and
take the limit eps --> 0 at a suitable later stage.
The analytic continuation is, of course, unique, and allows us to
define a _generalized_ Lebesgue integral for d=4-eps by the formula
integral (dp/2 pi)^d p^2j/(p^2+m^2)^n:= I(d,j,n) (4)
and similar expressions for arbitrary rational Lorentz-invariant
expressions. If these expressions happen to have good limits for
eps --> 0, which cannot happen for (4) but happens for suitable linear
combinations, this defines the value also for d=4.
The derivation ensures that it gives the correct results in all
cases where the integral makes sense in the traditional (Lebesgue)
way.
Thus we have defined a consistent generalization of the Lebesgue
integral of rational Lorentz-invariant expressions to the singular case.
This is similar in spirit to Lebesgue's extension of the Riemann
integral to the Lebesgue integral.
A good, mathematically rigorous exposition of d-dimensional integration
theory for general complex dimension d is given in
P. Etingof,
Note on dimensional regularization,
Ppp. 597-607 in: Pierre Deligne et al.,
Quantum Fields and Strings, A Course for Mathematicians, Vol. 1,
Amer. Math. Soc., Providence, Rhode Island, 1999
See also
http://wwwthep.physik.uni-mainz.de/~scheck/Meyer.ps
The theory of renormalization now shows that all integrals
occuring in the expressions for S-matrix elements in renormalizable
theories have a well-defined generalized Lebesgue integral for d=4.
This is all that is required for consistency.
For those who dislike unphysical complex dimensions,
the uniqueness of analytic continuation implies that one can
get completely equivalent results by keeping the physical
dimension d=4. In this case, one must replace the propagator
(p^2+m^2)^{-1} by (p^2+m^2)^{-n} with sufficiently large n,
and continue the result analytically to the physical value n=1.
Then all integrals are (in Euclidean space) ordinary Lebesgue
integrals. The formulas used for the extended Lebesgue integral
defined as above still apply; however, computations are now slightly
more involved.
Those who worry about the appropriateness of analytic continuation
might wish to consider the functions f, g defined by
f(d):=(sqrt(2-d))^{-2},
g(d):=1/(2-d)
in the real domain. They are equal for d<2 but f does not make
sense for d>=2. Nevertheless, it makes exceedingly much sense
to extend the definition of f to arguments d>2 by making
f(d):=g(d)
a definition. Indeed, g(d) is the unique meromorphic extension
of f to arbitrary complex arguments.
This uniqueness is in the nature of analytic continuation,
and makes the latter an extremely useful device in many applications.
It is the reason why we have such useful equations as
exp(ix)=sin(x)+i*cos(x),
which one would have no right to use if one would not silently
identify analytic functions defined on part of their domain with the
full analytic function on the associated Riemann surface.