Infrared Singularities of Non-Abelian Gauge Theories
For some time, we have been interested in the infrared structure of non-Abelian gauge theories.1,2 There are prominent infrared singularities near the mass shell only when the local gauge invariance is unbroken, as in quantum chromodynamics (QCD). We (and other authors3,4) began with the straightforward calculation of Feynman graphs, in leading-logarithm approximation, regulating the infrared singularities either by staying slightly off the mass shell by an amount characterized by a small mass μ, or by introducing a fictitious (group-symmetric) vector mass μ in the Feynman gauge. The results turned out to be much simpler than the complexity of the calculations suggested, and are characterized by certain properties of exponentiation and factorization which are reminiscent of the corresponding results for the Abelian case (quantum electrodynamics, or QED). These properties are summarized by a differential equation in μ (the infrared cutoff) which looks very much like a renormalization group (RG) equation with a special type of anomalous dimension generated by the near-mass-shell infrared singularities.
KeywordsAnomalous Dimension Mass Shell Soft Gluon Gluon Propagator Hadronic Physic
Unable to display preview. Download preview PDF.
- 2.J. M. Cornwall and G. Tiktopoulos, UCLA preprint TEP/75/21, October 1975 (submitted to Physical Review).Google Scholar
- 3.J. J. Carrazone, E. R. Poggio, and H. R. Quinn, Phys. Rev. Dll, 2286 (1975). This work on off-shell form factors contains some errors which are corrected in a later preprint.Google Scholar
- 4.E. R. Poggio and H. R. Quinn, Harvard preprint (1975). This work is on fixed-angle amplitudes for slightly off-shell particles.Google Scholar
- 5.To construct the nearly-on-shell amplitude, first take all the external momenta on shell: pi2=Mi2 (=0 for a gluon). Then change the momenta to pi + ki (∑ki =0), where the components of the ki are O(μ). The differential equation is derived by applying the operator ∑ki , considering the possible routings of the ki through the lines of the graph. Details will be given in a forthcoming paper.Google Scholar
- 6.A pioneering Work is by H. T. Nieh and Y. -P. Yao, Phys. Rev. Letters 32, 1074 (1974) (plus a later preprint), but other workers in the field do not agree with these authors. There is agreement among Lipatov; McCoy and Wu; Tyburski; and Cheng and Lo that the gluon Reggeizes, according to recent preprints.Google Scholar
- 7.As predicted earlier by M. T. Grisaru, H. J. Schnitzer, and H. S. Tsao, Phys. Rev. D8, 4498 (1973). There is a close connection between Reggeization of an elementary particle and its coupling to gauge fields, based on the gauge freedom in choosing the wave-function renormalization constant Z2; see J. M. Cornwall, Phys. Rev. 182, 1610 (1969).Google Scholar
- 8.In addition to a preprint by Appelquist, Carrazone, Roth, and Kluberg-Stern, we have seen preprints by Yao and by Sterman. Other references are given by Appelquist in these proceedings.Google Scholar
- 10.One of us (JMC) has remarked that there are new singularities in the Feynman integrals of the light-cone gauge (n2=0); see Phys. Rev. D10, 500 (197+). It now appears that these singularities cancel out in Green’s functions.Google Scholar
- 11.Some very interesting results have also been gotten by Migdal, in the context of Wilson’s lattice-gauge theory, and by Brezin and Zinn-Justin for the nonlinear σ model in 2+ε dimensions, which Migdal’s work suggests has a similar infrared structure to a gauge theory in 2(2+ε) dimensions.Google Scholar