Abstract:We show how to use on-shell unitarity methods to calculate renormalisation group coefficients such as beta functions and anomalous dimensions. The central objects are the form factors of composite operators. Their discontinuities can be calculated via phasespace integrals and are related to corresponding anomalous dimensions. In particular, we find that the dilatation operator, which measures the anomalous dimensions, is given by minus the phase of the S-matrix divided by π. We illustrate our method using several examples from Yang-Mills theory, perturbative QCD and Yukawa theory at one-loop level and beyond.
Abstract. We show that the γ i -deformation, which was proposed as candidate gauge theory for a non-supersymmetric three-parameter deformation of the AdS/CFT correspondence, is not conformally invariant due to a running double-trace couplingnot even in the 't Hooft limit. Moreover, this non-conformality cannot be cured when we extend the theory by adding at tree-level arbitrary multi-trace couplings that obey certain minimal consistency requirements. Our findings suggest a possible connection between this breakdown of conformal invariance and a puzzling divergence recently encountered in the integrability-based descriptions of two-loop finite-size corrections for the single-trace operator of two identical chiral fields. We propose a test to clarify this.
We build the framework for performing loop computations in the defect version of N = 4 super Yang-Mills theory which is dual to the probe D5-D3 brane system with background gauge-field flux. In this dCFT, a codimension-one defect separates two regions of space-time with different ranks of the gauge group and three of the scalar fields acquire non-vanishing and space-time-dependent vacuum expectation values. The latter leads to a highly non-trivial mass mixing problem between different colour and flavour components, which we solve using fuzzy-sphere coordinates. Furthermore, the resulting space-time dependence of the theory's Minkowski space propagators is handled by reformulating these as propagators in an effective AdS 4 . Subsequently, we initiate the computation of quantum corrections. The one-loop correction to the one-point function of any local gauge-invariant scalar operator is shown to receive contributions from only two Feynman diagrams. We regulate these diagrams using dimensional reduction, finding that one of the two diagrams vanishes, and discuss the procedure for calculating the one-point function of a generic operator from the SU(2) subsector. Finally, we explicitly evaluate the one-loop correction to the one-point function of the BPS vacuum state, finding perfect agreement with an earlier string-theory prediction. This constitutes a highly non-trivial test of the gauge-gravity duality in a situation where both supersymmetry and conformal symmetry are partially broken.
We apply on-shell and integrability methods that have been developed in the context of scattering amplitudes in N = 4 SYM theory to tree-level form factors of this theory. Focussing on the colour-ordered super form factors of the chiral part of the stresstensor multiplet as an example, we show how to systematically construct on-shell diagrams for these form factors with the minimal form factor as further building block in addition to the three-point amplitudes. Moreover, we obtain analytic representations in terms of Graßmannian integrals in spinor helicity, twistor and momentum twistor variables. While Yangian invariance is broken by the operator insertion, we find that the form factors are eigenstates of the integrable spin-chain transfer matrix built from the monodromy matrix that yields the Yangian generators. Constructing them via the method of R operators allows to introduce deformations that preserve the integrable structure. We finally show that the integrable properties extend to minimal tree-level form factors of generic composite operators as well as certain leading singularities of their n-point loop-level form factors.
We compute the two-loop minimal form factors of all operators in the SU(2) sector of planar N = 4 SYM theory via on-shell unitarity methods. From the UV divergence of this result, we obtain the two-loop dilatation operator in this sector. Furthermore, we calculate the corresponding finite remainder functions. Since the operators break the supersymmetry, the remainder functions do not have the property of uniform transcendentality. However, the leading transcendentality part turns out to be universal and 1 Moreover, in [22] symmetry was used to show that all tree-level scattering amplitudes are related to certain contributions to the dilatation operator. The picture of [22] is equivalent to taking cuts of form factors.2 In [32], the larger SO(6) sector was actually considered.
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