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Cutkosky rules and 1-loop $κ$-deformed amplitudes

Andrea Bevilacqua

TL;DR

This paper demonstrates that Cutkosky cutting rules remain valid term-by-term in the $1/\kappa$ expansion of κ-deformed 1-loop amplitudes. It develops both a general mapping to non-deformed amplitudes with heavy propagators and pragmatic reduction identities that render the imaginary part computation tractable, then applies these to compute the κ-deformed imaginary part up to $O(1/\kappa^2)$, showing the deformation induces only modest, frame-dependent corrections (scaling as $p^2/\kappa^2$ in the COM frame) to the standard result. By averaging over incoming momenta to account for experimental indistinguishability of $S(p)$ vs $-p$, the first nontrivial deformation cancels at $O(1/\kappa)$, with the leading correction at $O(1/\kappa^2)$. The results imply that the κ-deformed decay width remains effectively unchanged at leading order, supporting the phenomenological viability of κ-deformed models and providing a general, extendable framework for unitarity analyses in deformed theories.

Abstract

In this paper we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of $κ$ of the $κ$-deformed 1-loop correction to the propagator. We first present a general argument which relates each term in the expansion to a non-deformed amplitude containing additional propagators with mass $M>κ$. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the $κ$-deformed amplitude, to the singularity structure of non-deformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in $1/κ$, but the technique can be generalized to higher orders in $1/κ$. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the $κ$-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in $1/κ$, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behaviour of the decay width of unstable particles in the considered model.

Cutkosky rules and 1-loop $κ$-deformed amplitudes

TL;DR

This paper demonstrates that Cutkosky cutting rules remain valid term-by-term in the expansion of κ-deformed 1-loop amplitudes. It develops both a general mapping to non-deformed amplitudes with heavy propagators and pragmatic reduction identities that render the imaginary part computation tractable, then applies these to compute the κ-deformed imaginary part up to , showing the deformation induces only modest, frame-dependent corrections (scaling as in the COM frame) to the standard result. By averaging over incoming momenta to account for experimental indistinguishability of vs , the first nontrivial deformation cancels at , with the leading correction at . The results imply that the κ-deformed decay width remains effectively unchanged at leading order, supporting the phenomenological viability of κ-deformed models and providing a general, extendable framework for unitarity analyses in deformed theories.

Abstract

In this paper we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of of the -deformed 1-loop correction to the propagator. We first present a general argument which relates each term in the expansion to a non-deformed amplitude containing additional propagators with mass . We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the -deformed amplitude, to the singularity structure of non-deformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in , but the technique can be generalized to higher orders in . Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the -deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in , highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behaviour of the decay width of unstable particles in the considered model.
Paper Structure (16 sections, 96 equations, 4 figures)

This paper contains 16 sections, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Brief review of $\kappa$-deformation and momentum-space picture
  4. $\kappa$-deformed Feynman propagator
  5. Example of loop corrections to the propagator: non-deformed case
  6. Quick review of the direct computation
  7. Quick review of the Cutkosky rules
  8. The $\kappa$-deformed amplitude
  9. Cutkosky rule in the deformed context
  10. Simplified Cutkosky rule in the deformed context
  11. Example of checking the deformed Cutkosky rules
  12. $\kappa$-deformed imaginary part of the 1-loop correction
  13. Centre of mass frame
  14. General frame and decay widths
  15. Conclusions and discussion
  16. ...and 1 more sections

Figures (4)

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