Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program (1004.3417v3)

Abstract: Using only the Britto-Cachazo-Feng-Witten(BCFW) on-shell recursion relation we prove color-order reversed relation, $U(1)$-decoupling relation, Kleiss-Kuijf(KK) relation and Bern-Carrasco-Johansson(BCJ) relation for color-ordered gauge amplitude in the framework of S-matrix program without relying on Lagrangian description. Our derivation is the first pure field theory proof of the new discovered BCJ identity, which substantially reduces the color ordered basis from $(n-2)!$ to $(n-3)!$. Our proof gives also its physical interpretation as the mysterious bonus relation with ${1\over z^2}$ behavior under suitable on-shell deformation for no adjacent pair.

• The paper presents a pure field theory derivation of key gauge amplitude identities using the BCFW on-shell recursion technique.
• It rigorously proves identities like color-order reversal, U(1)-decoupling, KK, and the first field-theoretic BCJ relation, reducing computational complexity.
• The findings offer practical insights for simplifying scattering amplitude calculations and advancing unified S-matrix approaches in quantum field theory.

Overview of Gauge Amplitude Identities by On-Shell Recursion Relation in S-matrix Program

The paper "Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program" by Bo Feng, Rijun Huang, and Yin Jia presents a significant advancement in the S-matrix program by utilizing the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion relation. These authors focus on deriving essential identities related to gauge theory, specifically in the context of color-ordered amplitude relations, without relying on traditional Lagrangian formulations. This research contributes to the ongoing development of a unified understanding of quantum field theories using fundamental principles such as Lorentz invariance, locality, causality, and gauge symmetry.

Key Findings

Central to the paper is the pure field theory derivation of several gauge amplitude identities utilizing on-shell recursion techniques:

• Color-Order Reversed Relation: The paper proves the identity $A(1,2,\ldots,n) = (-1)^{n-1} A(n,n-1,\ldots,1)$, showing the symmetry in reversing the order of color indices in amplitude calculations.
• U(1)-Decoupling Relation: The identity $$\sum_{\sigma \in cyclic} A(n1,\sigma(2,3,\ldots,n)) = 0$$ was demonstrated, which illustrates how U(1) gauge bosons decouple from the spectrum of interactions.
• Kleiss-Kuijf (KK) Relation: Using the equation $$A(n1,\{\alpha\},n,\{\beta\}) = (-1)^{|\beta|} \sum_{\sigma \in OP(\{\alpha\}, \{\beta\})} A(n1,\sigma,n)$$, the authors verified the connection between cyclic amplitudes and permutations of particle orderings.
• Bern-Carrasco-Johansson (BCJ) Relation: This paper gives the first purely field-theoretic proof of the BCJ relation, which simplifies the color-ordered basis from $(n-2)!$ to $(n-3)!$. The BCJ relation is shown as a result of the physical interpretation related to the on-shell recursive behavior.

Implications and Future Directions

The implications of this work are multifaceted, impacting both practical and theoretical aspects of particle physics. Practically, the derivation of these identities simplifies the algebraic complexity and computational workload involved in gauge theory amplitude calculations. Theoretically, it enhances the understanding of the interplay between gauge theory and string theory by elucidating the remarkable equivalence observed in on-shell terms between gravity and gauge theory via the KLT relations.

The unification of gauge amplitudes through on-shell recursion aligns with the broader goals of the S-matrix program in establishing a self-consistent framework for scattering amplitudes independent of Lagrangian dynamics. Looking forward, this proof may facilitate further exploration of amplitude relations beyond tree-level and into loop-level calculations, potentially unraveling connections to quantum gravity theories and the finiteness of supergravity models.

Conclusion

This paper represents a pivotal step in refining our comprehension of gauge theory amplitudes using S-matrix methodologies. By eschewing reliance on specific Lagrangian descriptions, the authors have delivered a proof of the BCJ relation with implications that extend beyond traditional field theories, potentially informing approaches to resolving longstanding issues in high-energy physics, such as the construction of a consistent quantum gravity theory. While the exploration of BCJ relations is still in its nascent stages, their application promises substantial advancements in both theoretical investigations and computational efficiency in particle physics and beyond.