[Paper Review] Dark Matter Effective Field Theory and an Application to Vector Dark Matter

Jason Aebischer, Wolfgang Altmannshofer+2 more|arXiv (Cornell University)|Feb 14, 2022

Dark Matter and Cosmic PhenomenaPhysics and Astronomy106 references34 citations

TL;DR

This paper introduces Dark SMEFT (DSMEFT) and Dark LEFT (DLEFT), two effective field theories extending the Standard Model EFT and low-energy EFT to include spin-0, ½, and 1 dark matter (DM) particles as SM singlets. It classifies all gauge-invariant, dimension-six operators involving SM and DM fields, derives tree-level matching conditions at the electroweak scale, and applies the framework to a vector DM model with Z₂ symmetry, showing that DM can be produced via freeze-in through dimension-six triple-gauge-field-strength interactions, with a viable parameter space for a wide range of masses and couplings.

ABSTRACT

The Standard Model Effective Field Theory (SMEFT) and the Low Energy Effective Field Theory (LEFT) can be extended by adding additional spin 0, 1/2 and 1 dark matter particles which are singlets under the Standard Model (SM) gauge group. We classify all gauge invariant interactions in the Lagrangian up to terms of dimension six, and present the tree-level matching conditions between the two theories at the electroweak scale. The most widely studied dark matter models, such as those based on the Higgs portal or on kinetic mixing between the photon and a dark photon, are based on dimension-four interactions with the SM sector. We consider a model with dark vector particles with a $\mathbb{Z}_2$ symmetry, so that the lightest dark matter particle is stable. The leading interaction with the SM is through dimension-six operators involving two dark vector field-strength tensors and the electromagnetic field-strength tensor. This model is a viable dark matter model in the freeze-in scenario for a wide range of parameters.

Motivation & Objective

• To close the gap in the literature by constructing a fully general Dark Matter EFT (DMEFT) that includes scalar, fermion, and vector DM fields coupled to SM fields, valid both above and below the electroweak scale.
• To extend the Standard Model EFT (SMEFT) and Low-Energy EFT (LEFT) by adding spin-0, ½, and 1 DM particles that are singlets under the SM gauge group, forming Dark SMEFT (DSMEFT) and Dark LEFT (DLEFT).
• To provide a non-redundant basis of gauge-invariant operators up to dimension six involving both SM and DM fields, without imposing underlying symmetries on the DM fields.
• To compute tree-level matching conditions between DSMEFT and DLEFT at the electroweak scale, enabling consistent matching when heavy SM particles are integrated out.
• To apply the formalism to a specific model of vector dark matter with a Z₂ symmetry, analyzing its production via freeze-out and freeze-in mechanisms and identifying viable parameter space.

Proposed method

• Construct Dark SMEFT (DSMEFT) as an extension of SMEFT, valid above and below the electroweak scale, with DM fields transforming as SM singlets and gauge-invariant interactions up to dimension six.
• Construct Dark LEFT (DLEFT) as a generalization of LEFT, valid for light DM particles below the electroweak scale, invariant under SU(3) × U(1)em, and without assuming the Higgs mechanism for electroweak symmetry breaking.
• Classify all non-redundant, gauge-invariant operators involving SM and DM fields up to dimension six, distinguishing between purely DM operators, and those involving both SM and DM fields.
• Derive tree-level matching conditions between DSMEFT and DLEFT at the electroweak scale by integrating out heavy SM particles (top quark, Higgs, W, Z) while keeping light DM particles in the spectrum.
• Apply the formalism to a model of vector DM with a Z₂ symmetry, focusing on dimension-six operators involving three field-strength tensors: Fνμ Xaνα Xbαμ and εFνμ Xaνα Xbαμ.
• Analyze the freeze-in and freeze-out production mechanisms for vector DM, computing the relic abundance and identifying the allowed parameter space via numerical scans.

Experimental results

Research questions

• RQ1 What is the complete, non-redundant basis of gauge-invariant operators involving SM and DM fields up to dimension six, for spin-0, ½, and 1 DM particles that are SM singlets?
• RQ2 How do the tree-level matching conditions between DSMEFT and DLEFT depend on the mass hierarchy between SM and DM particles, particularly when heavy SM particles are integrated out?
• RQ3 What are the leading-order interactions between vector dark matter and the SM when a Z₂ symmetry ensures DM stability, and what is the structure of the dimension-six operators involved?
• RQ4 In the freeze-in scenario, what parameter space allows for a viable relic abundance of vector dark matter mediated by dimension-six triple-gauge-field-strength interactions?
• RQ5 How do the phenomenological constraints from direct detection, collider searches, and cosmological observations shape the viable region of the vector DM model?

Key findings

• The paper provides a complete, non-redundant basis of gauge-invariant operators involving SM and DM fields up to dimension six, with explicit operator lists for DSMEFT and DLEFT in Appendices B and C, respectively.
• The tree-level matching conditions between DSMEFT and DLEFT at the electroweak scale are computed, with contributions from heavy SM particles (top quark, Higgs, W, Z) integrated out, and the results are generalizable to cases where some DM particles are heavy.
• For a vector DM model with a Z₂ symmetry, the only dimension-six interactions with the SM that are gauge-invariant and consistent with the symmetry are those involving three field-strength tensors: Fνμ Xaνα Xbαμ and εFνμ Xaνα Xbαμ.
• The model supports a viable freeze-in production mechanism for vector DM, with the relic abundance matching the observed dark matter density over a wide range of DM masses and coupling strengths.
• The parameter space for the vector DM model is found to be viable in the freeze-in scenario, with the dominant contribution to the relic density arising from the decay of heavy SM particles into the vector DM via the dimension-six operator.
• The study shows that the DLEFT framework can be used independently of DSMEFT for light DM and light SM particles, without requiring the Higgs mechanism, and that the matching conditions are consistent across the two EFTs.