------------------------------- 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.