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Extensions of Theories from Soft Limits

Freddy Cachazo, Peter Cha, Sebastian Mizera

TL;DR

This work reveals that vanishing single soft limits in several effective field theories are governed by larger extension theories that include additional fields and flavor structures. By exploiting the CHY representation and KLT relations, the authors construct complete tree-level S-matrices for extensions of the U(N) NLSM, special Galileon, Born-Infeld, and DBI-Volkov-Akulov theories, and derive new linear-in-$z$ (BCFW-like) recursion relations for the extended NLSM. They provide explicit soft-limit formulas and concrete low-point examples that illustrate how the original theories sit inside richer interacting frameworks. The results suggest a unifying perspective where soft limits reveal the algebraic structure of a broader theory space and hint at master theories that combine multiple sectors, with potential connections to asymptotic symmetries and higher-dimensional realizations.

Abstract

We study a variety of field theories with vanishing single soft limits. In all cases, the structure of the soft limit is controlled by a larger theory, which provides an extension of the original one by adding more fields and interactions. Our main example is the $U(N)$ non-linear sigma model in its CHY representation. Its extension is a theory in which the NLSM Goldstone bosons interact with a cubic biadjoint scalar. Other theories we study and extend are the special Galileon and Born-Infeld theory, including its maximally supersymmetric version in four dimensions, the DBI-Volkov-Akulov theory. In all the cases, we propose the CHY representation of the complete tree-level S-matrix of the extended theories. In fact, CHY formulas are the key technique for studying the single soft limit behavior of the original theories. As a byproduct, we show that the tree-level S-matrix of the extended NLSM theory can be constructed using a very compact BCFW-like recursion relation, where physical poles are at most linear in the deformation parameter.

Extensions of Theories from Soft Limits

TL;DR

This work reveals that vanishing single soft limits in several effective field theories are governed by larger extension theories that include additional fields and flavor structures. By exploiting the CHY representation and KLT relations, the authors construct complete tree-level S-matrices for extensions of the U(N) NLSM, special Galileon, Born-Infeld, and DBI-Volkov-Akulov theories, and derive new linear-in- (BCFW-like) recursion relations for the extended NLSM. They provide explicit soft-limit formulas and concrete low-point examples that illustrate how the original theories sit inside richer interacting frameworks. The results suggest a unifying perspective where soft limits reveal the algebraic structure of a broader theory space and hint at master theories that combine multiple sectors, with potential connections to asymptotic symmetries and higher-dimensional realizations.

Abstract

We study a variety of field theories with vanishing single soft limits. In all cases, the structure of the soft limit is controlled by a larger theory, which provides an extension of the original one by adding more fields and interactions. Our main example is the non-linear sigma model in its CHY representation. Its extension is a theory in which the NLSM Goldstone bosons interact with a cubic biadjoint scalar. Other theories we study and extend are the special Galileon and Born-Infeld theory, including its maximally supersymmetric version in four dimensions, the DBI-Volkov-Akulov theory. In all the cases, we propose the CHY representation of the complete tree-level S-matrix of the extended theories. In fact, CHY formulas are the key technique for studying the single soft limit behavior of the original theories. As a byproduct, we show that the tree-level S-matrix of the extended NLSM theory can be constructed using a very compact BCFW-like recursion relation, where physical poles are at most linear in the deformation parameter.

Paper Structure

This paper contains 13 sections, 77 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Extension of the $U(N)$ Non-linear Sigma Model
  3. Single Soft Limit
  4. Linear BCFW-like Recursion Relation
  5. Examples
  6. Extension of the Special Galileon Theory
  7. Single Soft Limits using KLT Relations
  8. Examples
  9. Extension of Born-Infeld Theory
  10. Extension of DBI-Volkov-Akulov Theory
  11. Discussion and Outlook
  12. Review of CHY Formulas
  13. Large-$z$ Behavior of NLSM

Figures (2)

  • Figure B.1: Example diagram for a NLSM amplitude. Propagators in bold red define a subtree with deformed momenta. All vertices scale as a single power of $z$.
  • Figure B.2: Example diagram for a NLSM $\oplus\, \phi^3$ amplitude. Solid and dashed lines define $\Sigma$ and $\phi$ propagators respectively. Propagators in bold red define a subtree with deformed momenta. Black vertices scale as $z$, while the white one remains constant.