[Paper Review] From U(1) to E8: soft theorems in supergravity amplitudes

Wei-Ming Chen, Yu-tin Huang+1 more|arXiv (Cornell University)|Dec 4, 2014

Black Holes and Theoretical Physics21 references4 citations

TL;DR

This paper develops an anti-symmetrized extraction procedure to isolate the U(1) duality symmetry in $υ<8$ supergravity amplitudes, which otherwise obscures the full duality group due to soft-graviton singularities. It demonstrates that the double-soft-scalar limit reveals universal behavior governed by the full duality group, including E8 in $υ=16$ supergravity in three dimensions, where single-soft limits vanish for all fields, ruling out $D^8R^4$ as a counterterm.

ABSTRACT

It is known that for N=8 supergravity, the double-soft-scalar limit of a n-point amplitude is given by a sum of local SU(8) rotations acting on a (n-2)-point amplitude. For N&lt;8 supergravity theories, complication arises due to the presence of a U(1) in the U(N) isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in 4&lt;=N&lt;8 supergravity in four dimensions, as well as N=16 supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E8 duality group, supersymmetry ensures that the amplitude vanishes in the single-soft limit of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group for any tree-level and one-loop amplitudes. Finally we discuss the implications of the duality symmetry on possible counter terms for this theory. As a preliminary application, we show that the vanishing of single-soft limits of arbitrary component fields in three-dimensional supergravity rules out the direct dimensional reduction of D^8R^4 as a valid counter term.

Motivation & Objective

• To resolve the obscuration of U(1) duality symmetry in $υ<8$ supergravity due to soft-graviton singularities in double-soft-scalar limits.
• To develop a systematic method to extract the full duality group, including U(1), in tree-level and one-loop amplitudes.
• To demonstrate that in three-dimensional $υ=16$ supergravity, all single-soft limits vanish universally, implying strong constraints on UV counterterms.
• To investigate whether soft theorems are tree-level exact and whether they survive loop corrections.
• To assess the viability of higher-derivative terms like $D^8R^4$ as potential UV counterterms in 3D maximal supergravity.

Proposed method

• Introduces an anti-symmetrized combination of two amplitudes with exchanged soft scalar momenta to cancel soft-graviton singularities and isolate the U(1) generator contribution.
• Applies the method to tree-level amplitudes in $4\leq\mathcal{N}<8$ supergravity in four dimensions, deriving explicit double-soft theorems with SU(\mathcal{N}) and U(1) generators.
• Uses recursive formulas and generalized unitarity cuts in three dimensions to extend the analysis to one-loop amplitudes in $υ=16$ supergravity.
• Demonstrates that in $υ=16$ supergravity, all single-soft limits vanish as $\epsilon^2$ for scalars and $\epsilon^1$ for fermions, due to supersymmetry and little group constraints.
• Applies the soft theorems to constrain possible UV counterterms, showing that $D^8R^4$ is ruled out by the vanishing single-soft scalar limit.

Experimental results

Research questions

• RQ1 How can the U(1) duality symmetry be disentangled from soft-graviton singularities in $υ<8$ supergravity amplitudes?
• RQ2 What is the universal structure of double-soft-scalar limits in $4\leq\mathcal{N}<8$ supergravity when the full duality group is exposed?
• RQ3 Does the vanishing of single-soft limits in $υ=16$ supergravity in three dimensions imply that soft theorems are tree-level exact and loop-corrected?
• RQ4 Can the anti-symmetrized extraction procedure be generalized to higher-loop amplitudes?
• RQ5 Which higher-derivative counterterms are ruled out by the soft theorems in 3D maximal supergravity?

Key findings

• The anti-symmetrized double-soft-scalar limit successfully isolates the U(1) generator contribution, removing soft-graviton singularities and exposing the full duality group in $4\leq\mathcal{N}<8$ supergravity.
• In $υ=16$ supergravity in three dimensions, all single-soft limits vanish universally: scalars as $\epsilon^2$, fermions as $\epsilon^1$, due to supersymmetry and little group constraints.
• The vanishing of single-soft scalar limits rules out the direct dimensional reduction of $D^8R^4$ as a valid UV counterterm in 3D maximal supergravity.
• The soft theorems are consistent with tree-level exactness, and the method provides evidence that they likely hold at one-loop and potentially higher loops.
• The E8 duality group in 3D $υ=16$ supergravity imposes strong constraints, ruling out $R^4$ and $D^4R^4$ counterterms under SO(16) projections, and suggesting the next possible UV divergence is at 16 loops.