[Paper Review] A Note on Loop Amplitudes in QED

Andreas Brandhuber, Gabriele Travaglini+1 more|ArXiv.org|Aug 10, 2009

Black Holes and Theoretical Physics61 references3 citations

TL;DR

This paper investigates the two-loop four-point amplitude in N=2 super QED and finds an approximate recursive structure analogous to the ABDK/BDS ansatz in N=4 SYM. It shows that the amplitude can be well-approximated by a quadratic polynomial in the one-loop amplitude, with a small remainder function, suggesting hidden simplicity in QED amplitudes despite lacking exact iterative structure.

ABSTRACT

We consider the two-loop four-point amplitude in N=2 super QED, and show that there exists an approximate recursive structure similar to that captured by the ABDK/BDS ansatz for MHV amplitudes in N=4 super Yang-Mills. Furthermore, we present a simple relation between the box coefficients of one-loop photon MHV amplitudes in (super) QED, and sums of box coefficients of one-loop MHV amplitudes in (super) Yang-Mills.

Motivation & Objective

• To investigate whether iterative structures, similar to those in N=4 SYM, appear in QED amplitudes.
• To test if the two-loop four-point MHV amplitude in N=2 super QED can be approximated by a polynomial in the one-loop amplitude.
• To examine the functional form of the remainder function and its behavior across phase space.
• To compare the quality of the ansatz in N=2 and N=1 SQED, and explore the role of maximal transcendentality.
• To explore potential connections to Wilson loop dualities and underlying symmetries in QED.

Proposed method

• Construct an ansatz for the two-loop four-point MHV amplitude in N=2 SQED as a quadratic polynomial in the one-loop amplitude: $ \mathcal{M}^{(2)}_{\text{ansatz}} = b[\mathcal{M}^{(1)}]^2 + c\mathcal{M}^{(1)} + d $, with coefficients $ b, c, d $ to be determined.
• Minimize the squared difference between the exact two-loop amplitude and the ansatz over the kinematic variable $ y = -u/s $, using the functional $ F(b,c,d) = \int_0^1 dy \, \left( \mathcal{M}^{(2)} - \mathcal{M}^{(2)}_{\text{ansatz}} \right)^2 $.
• Perform numerical fitting of $ b, c, d $ at multiple kinematic points (e.g., sets I–VII) to assess the stability and accuracy of the ansatz.
• Define the remainder function as $ \mathcal{R}_4(y) = \text{Re}[\mathcal{M}^{(2)}] - \left( b[\mathcal{M}^{(1)}]^2 + c\mathcal{M}^{(1)} + d \right) $, and analyze its behavior across phase space.
• Compare the quality of the fit in N=2 and N=1 SQED using the $ F(b,c,d) $ metric, and examine the role of maximal transcendentality in the amplitude structure.
• Investigate the relation between box coefficients in QED and Yang-Mills theories, finding a simple sum relation between coefficients.

Experimental results

Research questions

• RQ1 Does the two-loop four-point MHV amplitude in N=2 super QED exhibit an approximate iterative structure similar to the ABDK/BDS ansatz in N=4 SYM?
• RQ2 Can the two-loop amplitude be accurately approximated by a quadratic polynomial in the one-loop amplitude, and how do the coefficients depend on kinematics?
• RQ3 What is the behavior of the remainder function between the exact amplitude and the ansatz, particularly near the phase space boundaries?
• RQ4 How does the quality of the ansatz compare between N=1 and N=2 SQED, and what explains the difference in fit accuracy?
• RQ5 Is there a deeper connection between the structure of box coefficients in QED and those in Yang-Mills theories, and can it be expressed as a sum rule?

Key findings

• The two-loop four-point MHV amplitude in N=2 SQED is well-approximated by a quadratic ansatz in the one-loop amplitude, with a minimal $ F(b,c,d)^{\text{N}=2} = 0.5 $, indicating high accuracy.
• The coefficients $ b, c, d $ in the ansatz are not constant but vary with kinematics, indicating the structure is not exact iterative but approximately recursive.
• The remainder function $ \mathcal{R}_4(y) $ shows small deviations across most of the phase space, but spikes near $ y \to 0 $ and $ y \to 1 $, indicating breakdown at phase space boundaries.
• The fit quality is significantly better in N=2 SQED ($ F = 0.5 $) than in N=1 SQED ($ F = 24.1 $), suggesting enhanced simplicity in the maximally supersymmetric case.
• The N=2 amplitude can be derived from the N=1 amplitude by selecting only terms of maximal transcendentality and removing kinematic ratio-dependent contributions.
• A simple relation is found between box coefficients in one-loop photon MHV amplitudes in QED and sums of box coefficients in one-loop MHV amplitudes in Yang-Mills theories.