ONE-LOOP EFFECTIVE MULTI-GLUON LAGRANGIAN IN ARBITRARY DIMENSIONS E. RODULFO and R. DELBOURGO

The work presented in this paper has been motivated by a reconsideration of photon-photon scattering, generalized to arbitrary dimensional spacetime [1] and to any number of scattering photons. Though an extremely feeble process and therefore very difficult to observe, this nonlinear process of scattering photons by photons has attracted considerable attention [2, 3, 4, 5, 6, 7, 8] since Euler first considered its low-energy limit in 1935. In this paper, we generalize to a non-Abelian analogue of this phenomenon: multi-gluon scattering in arbitrary dimensional spacetime which is not as weak a process as its Abelian counterpart, but is significantly richer in that gluons have classical self-interaction; also the gluon number need no longer be even, unlike photons. Multi-gluon scattering is mediated not only by meson and fermion virtual particles, but by vector and ghost loops too, quite aside from the tree-level contributions. A process involving an arbitrary number of gluons may be represented by a sum over permutations of the Feynman diagram with those many legs (including pinched diagrams). Clearly, direct evaluation of the sum over Feynman amplitudes is very involved, so instead we shall attack this problem in the background-field formalism [9, 10, 11, 12, 13, 14] where, if appropriate quasilocal conditions are assumed [15, 16, 17], one can find exact expressions for the effective action without the tedious computation of Feynman diagrams. We shall revisit this method in section 2 where we shall be led to an expression for the one-loop effective Lagrangian for any gauge field theory with a quasilocal background in arbitrary dimensions. We shall then extract from this Lagrangian specific results including counterterms up to 12 spacetime dimensions and arrive at some conclusions. (We have limited ourselves to D ≤ 12 since higher