[Paper Review] Light-by-Light Scattering Effect in Light-Cone Supergraphs

Рената Каллош, Pierre Ramond|arXiv (Cornell University)|Jun 24, 2010

Black Holes and Theoretical Physics17 references3 citations

TL;DR

This paper explains the ultraviolet (UV) finiteness of N=4 supersymmetric Yang-Mills theory and N=8 supergravity up to 7 loops using a light-cone supergraph approach, where dynamical supersymmetry acts as a field-dependent gauge transformation that enforces a light-by-light scattering-like effect. This effect suppresses UV divergences by introducing non-polynomial transverse momentum dependence in amplitudes, delaying divergences until L=n+3 for n-point functions, with the 4-point amplitude finite up to 7 loops.

ABSTRACT

We give a relatively simple explanation of the light-cone supergraph prediction for the UV properties of the maximally supersymmetric theories. It is based on the existence of a dynamical supersymmetry which is not manifest in the light-cone supergraphs. It suggests that N=4 supersymmetric Yang-Mills theory is UV finite and N=8 supergravity is UV finite at least until 7 loops whereas the $n$-point amplitudes have no UV divergences at least until $L=n+3$. Here we show that this prediction can be deduced from the properties of light-cone supergraphs analogous to the light-by-light scattering effect in QED. A technical aspect of the argument relies on the observation that the dynamical supersymmetry action is, in fact, a compensating field-dependent gauge transformation required for the retaining the light-cone gauge condition $A_+=0$.

Motivation & Objective

• To explain the UV finiteness of maximally supersymmetric theories using light-cone supergraphs.
• To clarify the role of dynamical supersymmetry as a compensating gauge transformation preserving the light-cone gauge condition A+=0.
• To connect the UV behavior of N=4 SYM and N=8 supergravity to the light-by-light scattering effect in QED.
• To show that non-polynomial transverse momentum dependence in amplitudes suppresses UV divergences beyond naive expectations.
• To reconcile the observed finiteness at 3–4 loops with a general prediction of finiteness up to 7 loops in N=8 supergravity.

Proposed method

• Use of light-cone superfield formalism to rewrite the action in terms of scalar superfields, making 8 of 16 supersymmetries manifest.
• Identification of dynamical supersymmetry as a field-dependent gauge transformation required to preserve the light-cone gauge A+=0.
• Application of the helicity formalism to compute amplitudes, where transverse momenta p⊥ arise from linearized dynamical supersymmetry on external chiral superfields.
• Derivation of supergraph amplitudes with δ-functions δ⁴(∑λᵢηᵢ) for N=4 SYM and δ⁸(∑λᵢηᵢ) for N=8 supergravity, encoding transverse momentum dependence.
• Analysis of UV divergence structure by examining the degree of non-polynomial dependence on transverse momenta, which suppresses divergences.
• Comparison of N=4 SYM and N=8 supergravity: in N=8, the dimensionful coupling κ allows removal of non-polynomial dependence at 4 loops higher than naive expectation.

Experimental results

Research questions

• RQ1 How does the light-cone supergraph formalism explain the UV finiteness of N=4 supersymmetric Yang-Mills theory beyond standard perturbative arguments?
• RQ2 What is the role of dynamical supersymmetry in suppressing UV divergences in N=8 supergravity?
• RQ3 Why do 4-point amplitudes in N=8 supergravity remain finite up to 7 loops, and how does this relate to the light-by-light scattering analogy?
• RQ4 How does the presence of transverse momenta from dynamical supersymmetry affect the UV scaling of amplitudes?
• RQ5 Can the E₇(₇) symmetry, which acts non-linearly on chiral superfields, be reconciled with the L=n+3 rule for UV divergence delays?

Key findings

• The N=4 supersymmetric Yang-Mills theory is UV finite due to non-polynomial transverse momentum dependence in amplitudes, which suppresses UV divergences.
• The 4-point amplitude in N=8 supergravity is UV finite up to 7 loops, consistent with the L=n+3 rule for n-point amplitudes.
• The UV finiteness of N=8 supergravity at 3–4 loops is explained by the light-by-light scattering-like effect, where transverse momenta from dynamical supersymmetry delay divergences.
• The dynamical supersymmetry transformations act as compensating gauge transformations to preserve the light-cone gauge A+=0, ensuring gauge invariance is maintained.
• The presence of δ⁸(∑λᵢηᵢ) in N=8 supergravity amplitudes introduces 8 transverse momenta per external leg, which, combined with the κ⁸ coupling, delays divergences by 4 loops beyond naive expectations.
• The L=n+3 rule for UV divergence delays in n-point amplitudes is a direct consequence of the light-by-light scattering analogy in supergraphs, where each leg contributes two transverse momenta.