Volume 246, number 3,4 PHYSICS LETTERS B 30 August 1990 The Chisholm identities in dimensional regularization Jerzy Pawtowski Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, USA Received 14 March 1990; revised manuscript received 9 May 1990 We present a generalization of the Chisholm identities in dimensional regularization, which incorporates the 't Hoofl-Veltman definition of y~ in a natural way. These identities enable the simplification of contracted indices in Dirac matrix products, allowing for great simplification in the calculation of Feynman diagrams. The working knowledge of particle theorists about the algebra of Dirac (gamma) matrices has recently been summarized [ 1 ], and it was pointed out that there is no generalization of Chisholm identities to non-integer dimensions. The purpose of this work is to present a set of such identities for dimensional regularization (DR), which we will call the generalized Chisholm identities (GCI). We derive the GCI using regular Clifford algebra - the algebra of Dirac matrices in d dimensions. We also use the fact that experimentally measurable (external) momenta have only four non-zero components. An imme- diate consequence of this is that antisymmetric products of five or more external momenta vanish identically (and are called evanescent terms). In the final results evanescent terms manifest themselves as Gram determi- nants of rank five or higher. Since evanescent terms vanish identically, the final result of any calculation is invariant under the addition or subtraction of such terms. This invariance, which we will use in the derivation of reduced projection operators, is implicit to DR, and does not require the addition of any postulates to DR. Furthermore, we will assume that loop integrations may be performed before the Dirac algebra, leaving be- hind scalar integrals. Therefore, we assume that Dirac matrices are either contracted with other matrices (such matrices will be called a contracted index, for brevity), or with an external momentum (called an external momentum). Our starting point is the expansion formula for a given string of Dirac matrices (a Clifford product), in terms of a complete set of antisymmetric Grassmann (wedge) products [ 2 ]: S=s~sz...Sm=¼ Tr(S)+~ ~ (--1)esktr(Sk)+¼ ~ (-1)Psi^sjTr(Sij)+...+s~ A...AS m =Po (S) + P, (S) + ...+Pro(S) , ( 1 ) where S,j is the string S with the elements si, sj removed, and P is the number of permutations necessary to commute them to the left of S. Here and in the following the sums extend over combinations of indices or elements of a given string. We will call the above expansion the normal expansion for S. Every term represents a projection operator Pk, which projects out of S all possible wedge products of rank k. By using the normal expansion for the reversed string S n, we express the first four projection operators in terms of the higher rank operators: P1 (S) = If I (S) --Ps (S) -Pg(S) -..., P2(S) = ~C2(S) -P6(S) -P,o(S) --..., P3(S) = -~C2(S) -P7(S) -Pil (S) -..., P4(S) = ~C, (S) -Ps(S) --Pl2(S) --..., (2) where 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 477