Curvature-squared multiplets, evanescent effects, and the U(1) anomaly in N=4 supergravity

Abstract

We evaluate one-loop amplitudes of N = 4 supergravity in D dimensions using the double-copy procedure that expresses gravity integrands in terms of corresponding ones in Yang-Mills theory. We organize the calculation in terms of a set of gauge-invariant tensors, allowing us to identify evanescent contributions. Among the latter, we find the matrix elements of supersymmetric completions of curvature-squared operators. In addition, we find that such evanescent terms and the U(1)anomalous contributions to one-loop N = 4 amplitudes are tightly intertwined. The appearance of evanescent operators in N = 4 supergravity and their relation to anomalies raises the question of their effect on the known four-loop divergence in this theory. We provide bases of gauge-invariant tensors and corresponding projectors useful for Yang-Mills theories as a by-product of our analysis.Recent explicit calculations have shown that gravity theories still have perturbative secrets waiting to be revealed. We have learned a number of surprising lessons from these calculations: results in gravity theories can be obtained directly from their Yang-Mills counterparts via a double-copy procedure [1][2][3][4]; of a curious disconnect between the leading two-loop divergence of graviton amplitudes [5,6] and the corresponding renormalizationscale dependence [7,8]; and about the surprisingly tame ultraviolet behavior of certain supergravity theories [9][10][11][12]. These lessons augur more surprises to come. In this paper we investigate the role of evanescent effects in the one-loop four-point amplitude of N = 4 supergravity, along with its relation to the U(1) anomaly in the duality symmetry of this theory [13][14][15].Evanescent effects arise from operators whose matrix elements vanish when working strictly in four dimensions, but give rise to nonvanishing contributions in dimensional regularization. Such contributions originate from the cancellation of poles against small deviations in the four-dimensional limit; that is, they are due to ǫ/ǫ effects, where ǫ = (4−D)/2 is the dimensional regulator. Although such effects might at first appear to be a mere technicality, they turn out to play an important role [7] in understanding ultraviolet divergences of Einstein gravity in the context of dimensional regularization [5,6]. In particular, the Gauss-Bonnet operator is evanescent and appears as a one-loop counterterm whose insertion at two loops contaminates the ultraviolet divergence, but results in no physical consequences in the renormalized amplitude. An important question therefore is whether a supersymmetric version of the Gauss-Bonnet operator appears in the matrix elements of N = 4 supergravity. If such an operator exists it would be important to determine its effects on the known four-loop divergence [16] of the theory.On the other hand, the N = 4 supergravity theory has an anomaly in its U(1) duality symmetry [13]. The anomaly manifests itself in the failure of certain helicity amplitudes which vanish at tree level to persist in vanishing at l...

Renormalised 3-point functions of stress tensors and conserved currents in CFT

“…This relation resembles the double-copy relation between Yang-Mills and gravity scattering amplitudes, see [57] and references therein. It is likely that analogous relations for type A anomalies exist in all higher even dimensions.…”

Expansion of Einstein-Yang-Mills theory by differential operators

“…Since gauge invariance is a so important and strong constraint, it is better to construct functions which are manifestly gauge invariant by sacrificing other properties [55]. By Feynman rules amplitudes are multilinear functions of polarization vectors.…”

“…In[55], four point amplitude of gauge theory has been rewritten to a form using the Type-II contractions.…”

Unraveling conformal gravity amplitudes

“…Since the work of Marcus [155], it is known that the U(1) symmetry of N = 4 Einstein supergravity is anomalous at the quantum level (see refs. [154,156,157] for studies of the anomaly from the double-copy perspective). U(1) symmetry breaking is also an important feature of non-minimal conformal supergravities, where it happens already at tree level [125].…”