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Unitarisation of EFT Amplitudes for Dark Matter Searches at the LHC

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Nicole F. Bell^{1}^{1}1,
Giorgio Busoni^{2}^{2}2,
Archil Kobakhidze^{3}^{3}3,
David M. Long^{4}^{4}4,
Michael A. Schmidt^{5}^{5}5,

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ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Victoria 3010, Australia

ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, NSW 2006, Australia

School of Physics, The University of Sydney, NSW 2006, Australia

Abstract

We propose a new approach to the LHC dark matter search analysis within the effective field theory framework by utilising the -matrix unitarisation formalism. This approach provides a reasonable estimate of the dark matter production cross section at high energies, and hence allows reliable bounds to be placed on the cut-off scale of relevant operators without running into the problem of perturbative unitarity violation. We exemplify this procedure for the effective operator D5 in monojet dark matter searches in the collinear approximation. We compare our bounds to those obtained using the truncation method and identify a parameter region where the unitarisation prescription leads to more stringent bounds.

## 1 Introduction

A dedicated search for Dark Matter (DM) at the Large Hadron Collider is currently one of the foremost objectives in particle physics. The most generic search channel is the mono-jet plus missing transverse energy signal, which searches for a single jet recoiling against the momentum of the DM particles which escape the detector unseen [1, 2, 3, 4, 5, 6]. In order to make such a search possible, it is necessarily to have a framework in which to describe the interactions of dark matter particles with SM fields. Given the plethora of possible dark matter models in the literature, it is impractical to perform a dedicated analysis of each model. It is thus imperative to work with a small number of models that capture the essential aspects of the physics in some approximate way. Effective field theories (EFTs) achieve this aim, by parameterising the DM interactions with SM particles by a small set of non-renormalizable operators. For instance, the lowest order operators that describe the interaction of a pair of fermionic DM particles, , with a pair of SM fermions, , are of the form

(1.1) |

where the Lorentz structure can be . A full set of operators can be found in [7, 8], where a standard naming convention has been defined. Such operators are not intended to be complete description of DM interactions, valid at arbitrarily high energy. They would be obtained as a low energy approximation of some more complete theory by integrating out heavy degrees of freedom. The energy scale is related to the parameters of that high energy theory as , where is a coupling constant and is the mass of a heavy mediator.

The EFT description will clearly break down at energies comparable to , at which scale we expect the mediators to be produced on-shell or give rise to cross section resonances. Moreover, while the EFT will provide physically well-behaved cross sections at low energies, they will give rise to bad high energy behaviour if used outside their region of validity. This manifests as a violation of perturbative unitarity [9, 10, 11, 12]. While these issues may be remedied with a Simplified Model [13] in which a mediator is explicitly introduced, issues of unitarity violation can persist if gauge invariance is not respected. The shortcoming of EFTs and Simplified Models that violate SM gauge invariance [14, 15, 16, 17] or dark-sector gauge invariance [18, 19] have recently been discussed.

Given the usefulness of the EFT and Simplified Model description of DM interactions, they will continue to be used in collider DM search analyses. Therefore, it is important to limit analyses to parameters that respect perturbative unitarity. One such approach is to use a truncation technique [20, 21, 22], which introduces a momentum cutoff equal to the mass of the would-be integrated-out mediator. In this paper we will instead use a procedure known as -matrix unitarisation [23, 24, 25, 26, 27, 28] to enforce unitarisation of all scattering amplitudes. Although this procedure will not capture the resonance structure of the true high energy theory, it will force scattering amplitudes to be well behaved at high energies, allowing us to derive meaningful limits on EFT models from LHC collisions with high centre of mass energies.

We will use the -matrix approach to unitarise the 2 to 2 scattering amplitudes, such as . This will allow us to determine unitarised cross sections for the 2 to 3 mono-jet processes such as , under the assumption that the gluon can be treated with the collinear approximation. We will also compare the results obtained from this unitarisation technique with those obtained with truncation. The rest of the paper is organised as follows: in Section 2 we summarise the theoretical framework for the unitarisation procedure. We illustrate the unitarisation procedure in two toy models in Section 3 and apply it to the standard vector operator D5 in Section 4. Section 5 contains the conclusions, while in Appendix A we derive the relevant cross sections in the collinear limit.

## 2 K-Matrix Unitarisation

The -matrix formalism was first introduced in
Ref. [23, 24]. It is a technique to impose
unitarity on amplitudes which naively violate unitarity. In the
derivation we largely follow the notation and arguments in
Refs. [25, 26, 29] ^{1}^{1}1See
Ref. [27, 28] for further details..
Unitarity of the -matrix,

(2.1) |

implies the well-known relation for the -matrix

(2.2) |

Note the factor of in the definition of the -matrix which has been introduced for convenience.

Following the seminal work by Jacob and Wick [27], for scattering processes we can describe both the initial and the final state in terms of two-particle helicity states which are characterised by the helicities of the two particles and two angles and , collectively denoted . Choosing the initial state to align with the -axis, the individual -matrix element for a process with fixed helicities in the initial and final state is given by

(2.3) |

in terms of the partial waves

(2.4) |

the Wigner -functions with total angular momentum , and the resultant helicity of the two-particle states and , where we used the normalisation of the Wigner -functions in Ref. [26]. Assuming that no three-particle states are kinematically accessible, an analogous unitarity relation holds for each partial wave separately,

(2.5) |

in terms of matrices with components . This condition can be rewritten in terms of

(2.6) |

which motivates the definition of the -matrix for the J^{th} partial wave,
. The -matrix is hermitean, .
If the -matrix is invariant under time reversal, the -matrix is symmetric
and thus and are real. Hence can be
considered as the real part of and the imaginary part of is
determined by the term in Eq. (2.6). We can invert the
relation in Eq. (2.6) to obtain

(2.7) |