[Paper Review] Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace

Carlos R. Mafra, Oliver Schlotterer|arXiv (Cornell University)|May 11, 2015

Black Holes and Theoretical Physics43 references19 citations

TL;DR

This paper constructs the two-loop five-point amplitudes in ten-dimensional super Yang-Mills (SYM) and type II supergravity using pure spinor superspace, proposing manifestly local kinematic numerators that satisfy the BCJ duality between color and kinematics. The results are UV-finite in D < 7 and reproduce the critical D = 7 UV divergence in terms of SYM tree-level amplitudes, matching the low-energy limit of superstring amplitudes and resolving an apparent R-symmetry paradox via dimensional reduction with a D-dimensional dilaton factor.

ABSTRACT

Supersymmetric integrands for the two-loop five-point amplitudes in ten-dimensional super Yang--Mills and type II supergravity are proposed. The kinematic numerators are manifestly local and satisfy the duality between color and kinematics described by Bern, Carrasco and Johansson. Our results are expected to reproduce the integrated two-loop amplitudes in dimensions $D&lt;7$. The UV divergence in the critical dimension $D=7$ matches the low-energy limit of the corresponding superstring amplitudes and is written in terms of SYM tree amplitudes.

Motivation & Objective

• Develop a manifestly local and supersymmetric two-loop five-point amplitude framework in ten-dimensional super Yang-Mills (SYM) and supergravity.
• Construct kinematic numerators that satisfy the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics.
• Ensure the amplitudes are UV-finite in dimensions D < 7 and reproduce the critical D = 7 UV divergence.
• Resolve the apparent R-symmetry violation in supergravity by identifying the role of the D-dimensional dilaton in dimensional reduction.
• Verify consistency with the low-energy limit of type IIB superstring amplitudes via matching UV divergences.

Proposed method

• Utilize pure spinor superspace to construct local kinematic building blocks T12,3|4,5 and T m1,2,3|4,5, ensuring manifest supersymmetry and gauge invariance.
• Apply BRST cohomology techniques to ensure the kinematic numerators are on-shell and consistent with the pure spinor formalism.
• Implement the BCJ duality by constructing numerators that satisfy the Jacobi identity, mirroring color factors in cubic graphs.
• Derive the two-loop five-point SYM amplitude in (4.15) using planar and non-planar cubic graphs with local numerators.
• Extend the SYM result to supergravity via the BCJ double-copy construction, yielding the supergravity amplitude in (4.16).
• Perform dimensional reduction from ten dimensions to D = 7 to analyze UV divergences, identifying the role of the D-dimensional dilaton in resolving R-symmetry anomalies.

Experimental results

Research questions

• RQ1 How can two-loop five-point amplitudes in ten-dimensional SYM and supergravity be constructed with manifestly local kinematic numerators that satisfy BCJ duality?
• RQ2 What is the structure of the two-loop five-point amplitude in pure spinor superspace, and how does it ensure supersymmetry and gauge invariance?
• RQ3 How do the UV divergences in D = 7 dimensions arise, and how do they match the low-energy limit of superstring amplitudes?
• RQ4 Why does the supergravity amplitude appear to violate R-symmetry in D = 7, and how is this resolved via dimensional reduction?
• RQ5 What is the role of the D-dimensional dilaton in reconciling S-duality properties of the superstring with R-symmetry conservation in supergravity?

Key findings

• The two-loop five-point amplitude in ten-dimensional SYM is explicitly constructed in (4.15) using pure spinor superspace, with local kinematic numerators satisfying BCJ duality.
• The supergravity amplitude is derived via the double-copy construction, resulting in (4.16), and is UV-finite in dimensions D < 7.
• The critical UV divergence in D = 7 is reproduced and expressed in terms of SYM tree amplitudes, matching the low-energy limit of type IIB superstring amplitudes.
• The apparent R-symmetry violation in supergravity is resolved by the presence of a (7 − D) prefactor in the R-symmetry-violating components, arising from dimensional reduction with a D-dimensional dilaton.
• The one-loop UV divergence in supergravity features a (8 − D)/6 factor in the φh4 component, which vanishes in D = 8, preserving R-symmetry and matching the superstring amplitude.
• The structure of the UV divergence in both one- and two-loop amplitudes is consistent with the superfield identities and the BCJ duality, confirming the robustness of the pure spinor approach.