One-loop gauge theory amplitudes in N=4 super Yang–Mills from MHV vertices

Abstract

We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 super Yang-Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N = 4 super Yang-Mills are used as vertices, using an offshell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of supersymmetric MHV one-loop scattering amplitudes with an arbitrary number of external legs in N = 4 super Yang-Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach. 1

One-Loop Maximal Helicity Violating Amplitudes in N=4 Super Yang-Mills Theories

“…We reproduce the same relation between leading N c amplitudes A n;1 and sub-leading amplitudes A n;c as those obtained from conventional field theory calculations [2]. Combining with existing results [19], this establishes the validity of the CSW approach to one-loop in the calculation of MHV amplitudes, without taking the large N c limit.…”

“…A n;1 is identical to the ones obtained by conventional quantum field theory calculation [19]. For a specific sub-leading term with the color factor…”

“…The issue is how to define the corresponding spinor for particles with off-shell momentum k. The CSW prescription is to take an arbitrary spinor of positive helicityξ and define λ a = k aȧξȧ . Extending these rules to one loop and to the leading order of N c , the correct expression of A n;1 is reproduced [19]. For example, the MHV diagram in Figure 1 gives where P L is the sum of momenta on the left of the diagram.…”

MHV-vertices for gravity amplitudes

“…In particular the trees depend upon the shifted, on-shell, momentaP α, (1) . We consider a single term in the summation corresponding to a specific pole, 12) and evaluate this by determining the poles in D α (z 2 ) under the shift,…”

“…Although originally given for gluon scattering only, these rules have been shown to extend to other types of massless particle [10] and indeed to massive particles [11]. It has been shown [12][13][14] that, with the correct off-shell prescription, these vertices can be used to reproduce known one-loop results [15,16] in supersymmetric theories.…”