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On four-point interactions in massless higher spin theory in flat space

R. Roiban, A. A. Tseytlin

TL;DR

The paper addresses whether a local, Lorentz-covariant quartic interaction can complete a minimal flat-space higher-spin theory with a single tower of even spins by enforcing on-shell gauge invariance of the tree-level 000j amplitude. Using the S-matrix gauge-invariance approach with known local cubic vertices, it shows that local 000j terms exist only for j=2 and j=4, while for j≥6 the required quartics must be nonlocal. It then demonstrates how nonlocalities can be interpreted as exchanges of a ghost-like tower of higher-spin fields and discusses a conformal off-shell extension that parallels the Einstein case, suggesting a possible extended theory with a trivial S-matrix. The results illuminate the tension between locality and higher-spin gauge invariance in flat space and motivate the exploration of conformal and ghost-extended frameworks as potential resolutions. Overall, the work clarifies locality constraints for massless higher-spin theories and connects them to conformal higher-spin symmetry and AdS/CFT-inspired ideas about trivial S-matrices.

Abstract

We consider the minimal interacting theory of a single tower of spin j=0,2,4,... massless Fronsdal fields in flat space for which consistent covariant cubic interaction vertices are known. We address the question of constraints on possible quartic interaction vertices imposed by the condition of on-shell gauge invariance of the tree-level four-point scattering amplitudes involving three spin 0 and one spin j particle. We find that these constraints admit a local solution for quartic 000j interaction term in the action only for j=2 and j=4. We determine the non-local terms in four-vertices required in the case of spin j greater than 4 and show that these non-localities can be interpreted as a result of integrating out a tower of auxiliary ghost-like massless higher spin fields in an extended theory with a local action. We also consider the conformal off-shell extension of the Einstein theory and show that its perturbative expansion is the same as of the the non-local action resulting from integrating out the trace of the graviton field from the Einstein action. Motivated by this example, we conjecture the existence of a similar conformal off-shell extension of a massless higher spin theory that may be related to the above extended theory and may have the same infinite-dimensional symmetry as the conformal higher spin theory and thus may lead to a trivial S matrix.

On four-point interactions in massless higher spin theory in flat space

TL;DR

The paper addresses whether a local, Lorentz-covariant quartic interaction can complete a minimal flat-space higher-spin theory with a single tower of even spins by enforcing on-shell gauge invariance of the tree-level 000j amplitude. Using the S-matrix gauge-invariance approach with known local cubic vertices, it shows that local 000j terms exist only for j=2 and j=4, while for j≥6 the required quartics must be nonlocal. It then demonstrates how nonlocalities can be interpreted as exchanges of a ghost-like tower of higher-spin fields and discusses a conformal off-shell extension that parallels the Einstein case, suggesting a possible extended theory with a trivial S-matrix. The results illuminate the tension between locality and higher-spin gauge invariance in flat space and motivate the exploration of conformal and ghost-extended frameworks as potential resolutions. Overall, the work clarifies locality constraints for massless higher-spin theories and connects them to conformal higher-spin symmetry and AdS/CFT-inspired ideas about trivial S-matrices.

Abstract

We consider the minimal interacting theory of a single tower of spin j=0,2,4,... massless Fronsdal fields in flat space for which consistent covariant cubic interaction vertices are known. We address the question of constraints on possible quartic interaction vertices imposed by the condition of on-shell gauge invariance of the tree-level four-point scattering amplitudes involving three spin 0 and one spin j particle. We find that these constraints admit a local solution for quartic 000j interaction term in the action only for j=2 and j=4. We determine the non-local terms in four-vertices required in the case of spin j greater than 4 and show that these non-localities can be interpreted as a result of integrating out a tower of auxiliary ghost-like massless higher spin fields in an extended theory with a local action. We also consider the conformal off-shell extension of the Einstein theory and show that its perturbative expansion is the same as of the the non-local action resulting from integrating out the trace of the graviton field from the Einstein action. Motivated by this example, we conjecture the existence of a similar conformal off-shell extension of a massless higher spin theory that may be related to the above extended theory and may have the same infinite-dimensional symmetry as the conformal higher spin theory and thus may lead to a trivial S matrix.

Paper Structure

This paper contains 23 sections, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lagrangian vs. S-matrix gauge symmetries and massless higher spin interaction vertices
  3. Gauge transformations from the perspective of the S matrix
  4. Cubic and quartic terms in a massless higher spin action
  5. The $000j$ scattering amplitude
  6. Exchange contribution and its gauge transformation
  7. Contact term contribution and its gauge transformation
  8. Constraints from gauge invariance of the amplitude
  9. Solution of the gauge invariance constraints on four-vertex coefficient functions
  10. $j=2$
  11. $j=4$
  12. $j\ge 6$
  13. Non-local terms in $000j$ vertex for $j \ge 6$
  14. Non-local actions and conformal off shell extension: spin 2 example
  15. Integrating out the trace from the Einstein action
  16. ...and 8 more sections

Figures (1)

  • Figure 1: The two types of contributions to a soft limit of an amplitude with $n+1$ particles. The first has a pole when $p_{n+1}\rightarrow 0$ while the second does not. The zig-zag line represents a particle of spin $j$ whose momentum is taken to be soft; straight lines represent particles of spin $j_i$ ($i=1, 2, ..., n$).