New Relations for Gauge-Theory Amplitudes (0805.3993v2)

Abstract: We present an identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this we find new relations between color-ordered partial amplitudes. We discuss applications to multi-loop calculations via the unitarity method. In particular, we illustrate the relations between different contributions to a two-loop four-point QCD amplitude. We also use this identity to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. We expect this to be helpful in higher-loop studies of the ultraviolet properties of gravity theories.

• The paper introduces a kinematic identity that mirrors the Jacobi identity of color factors, simplifying gauge-theory amplitude structures.
• It reduces independent tree-level amplitudes from (n-2)! to (n-3)! and establishes non-trivial relations between planar and non-planar loop contributions.
• The study demonstrates that gravity amplitudes can be expressed as the square of gauge-theory numerators, offering a fresh take on KLT relations.

Relating Gauge Theory to Gravity through Kinematic Identities

The paper "New Relations for Gauge-Theory Amplitudes," authored by Z. Bern, J.J.M. Carrasco, and H. Johansson, provides an intriguing examination of the connections between gauge theories and gravity. The paper proposes a novel kinematic identity that has implications for simplifying both tree and loop-level calculations in gauge theories and offers a new interpretation of the Kawai-Lewellen-Tye (KLT) relations in gravity.

The research investigates the structures underlying scattering amplitudes in gauge theories, particularly focusing on the identities satisfied by the kinematic factors of these amplitudes. The authors base their findings on the observation that the kinematic numerators of gauge-theory amplitudes can be rearranged to satisfy an identity analogous to the Jacobi identity of color factors. This analogy provides a new perspective on the structure of scattering amplitudes and aids in deriving new relations between color-ordered partial amplitudes at tree-level.

Key Findings

1. Kinematic Identity:
   • The paper introduces a kinematic identity for the numerators of Feynman diagrams that resembles the Jacobi identity obeyed by color factors. This identity is pivotal as it is not immediately apparent from conventional Feynman rules and requires a careful arrangement of the contributing diagrams to manifest.
2. Reduction of Independent Amplitudes:
   • Using the newly proposed kinematic identity, the paper reduces the number of independent tree-level $n$-point amplitudes from the $(n-2)!$ determined by Kleiss-Kuijf (KK) relations to $(n-3)!$. This indicates a greater inner structure in the scattering process previously overlooked.
3. Loop-Level Implications:
   • The identity induces non-trivial relations between planar and non-planar contributions at loop-level, providing a simpler and more unified approach to constructing loop amplitudes. This is particularly useful for understanding multiloop perturbative calculations in theories like QCD and supersymmetric gauge theories.
4. Relation to Gravity:
   • The paper reveals that gravity amplitudes can be represented using the square of gauge-theory numerators. Specifically, it proposes that gravity amplitudes can be systematically expressed in terms of gauge-theory amplitudes through the square of the numerators once the kinematic identity is satisfied. This construction provides a new insight into the KLT relations that naturally arise from this decomposition.

Detailed Analysis

The kinematic identity has a profound impact on simplifying calculations by restructuring the gauge-theory amplitudes into forms more compatible with gravity. This is especially critical when analyzing the ultraviolet properties of gravity theories at higher-loop orders, where traditional techniques are computationally intensive. By reorganizing the structure of the gauge theory amplitudes, the paper lays a foundation for efficiently studying higher-loop properties and their implications in not only gauge theories but also gravity.

The authors exemplify the utility of the kinematic identity through explicit calculations, including two-loop examples in pure Yang-Mills theory. These calculations demonstrate that the notorious complexity of QCD calculations can be significantly reduced, aiding in more accurate and feasible theoretical predictions.

Implications and Future Directions

The implications of this research are multifaceted. By enhancing the understanding of amplitude structures in gauge theories and their connections to gravity, it opens new pathways for exploration in theoretical physics, particularly in the paper of supersymmetric theories and potential quantum gravity frameworks. The findings suggest that the unification of gauge theories and gravity at a more fundamental level, possibly through string theory or other quantum gravity theories, could be better understood through the language of amplitude relations.

Future research could focus on further elucidating the connections between gauge and gravity theories, potentially uncovering new symmetries or invariants in scattering processes. Additionally, extending these methods to massive or spontaneously broken theories and exploring their phenomenological implications at both low and high-energy scales would be a natural progression of this work.

In summary, the paper by Bern, Carrasco, and Johansson offers groundbreaking insights into the kinematic origins of scattering amplitudes, paving the way for enhanced computational techniques in both gauge theories and gravity, and potentially illuminating a path towards a deeper understanding of fundamental physical processes.