[Paper Review] A primer on Higgs Effective Field Theory with Geometry

Rodrigo Alonso|arXiv (Cornell University)|Jul 26, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper presents a geometric formulation of Higgs Effective Field Theory (HEFT) to systematically describe electroweak physics beyond the Standard Model, emphasizing field space geometry, scattering amplitude invariance under field reparametrizations, and the distinction between linear and non-linear realizations. The key contribution is a geometric LSZ reduction formula that enables consistent amplitude computations in curved field space, with phenomenological implications for collider searches and UV completion.

ABSTRACT

These lecture notes, prepared for the 2022 QUC summer school at KIAS, provide an introduction to Higgs Effective Field Theory and the use of field geometry in Quantum Field Theory. While not sounding the depths of any of these topics, we will cover and give a sense of the inner workings of: the action for Goldstone bosons, the independence of scattering amplitudes from field parametrisations, linear vs non-linear realizations --their `geography' and experimental prospects to tell them apart--, ultra-violet completions and the LSZ formula for fields in curved space.

Motivation & Objective

• To provide a pedagogical introduction to Higgs Effective Field Theory (HEFT) with a focus on geometric structures in field space.
• To clarify the independence of scattering amplitudes from field parametrization, a crucial consistency condition in EFT.
• To contrast linear and non-linear realizations of the Higgs sector and analyze their experimental distinguishability.
• To develop a geometric LSZ formula for fields in curved field space, enabling precise amplitude computations in non-linear EFTs.
• To connect theoretical structure in HEFT with phenomenological constraints from LHC data and future colliders.

Proposed method

• Derives the action for Goldstone bosons arising from spontaneous symmetry breaking, applying it to the electroweak sector.
• Introduces HEFT as the most general EFT for electroweak physics below the scale of new physics, with the Higgs as a composite or fundamental scalar.
• Uses a field space metric to define a geometric structure for the scalar sector, enabling a covariant formulation of the LSZ reduction formula.
• Applies the geometric LSZ formula to compute scattering amplitudes in curved field space, ensuring invariance under field reparametrizations.
• Analyzes the EFT dichotomy between linear and non-linear realizations via the Higgs mass parameter and its impact on observable correlations.
• Evaluates phenomenological constraints using LHC data, particularly on Higgs couplings and curvature parameters, and projects future sensitivity at HL-LHC and FCC.

Experimental results

Research questions

• RQ1 How can field geometry in Higgs Effective Field Theory ensure the invariance of scattering amplitudes under field reparametrizations?
• RQ2 What distinguishes linear and non-linear realizations of the Higgs sector in terms of field space geometry and phenomenological signatures?
• RQ3 How does the geometric LSZ formula generalize to curved field space, and what are its implications for amplitude computations in non-linear EFTs?
• RQ4 What are the experimental prospects to distinguish between linear and non-linear Higgs realizations using current and future collider data?
• RQ5 How do curvature parameters in HEFT—such as those controlling Higgs couplings to vector bosons and gluons—constrain new physics beyond the Standard Model?

Key findings

• Scattering amplitudes in HEFT are invariant under field reparametrizations, a consistency condition ensured by the geometric LSZ formula in curved field space.
• The geometric LSZ formula provides a covariant framework for computing amplitudes in non-linear realizations, with the field space metric playing a central role.
• In the limit of large new physics masses ($m_2 o ho$), HEFT reduces to SMEFT, recovering the standard model prediction with curvature correlations.
• For $m_2^2 o 0$ or negative, HEFT exhibits de-correlation between $R_h$ and $R_ ho$, breaking the curvature correlation seen in SMEFT and allowing for larger deviations from the SM.
• Current LHC data constrain the curvature parameter $vF'(0) = 1.01 \pm 0.06$ and $v\mathcal{P}'_{G_s}(0) \leq \frac{g_s^2}{(4\pi)^2}(-0.01 \pm 0.08)$, with future HL-LHC and FCC experiments expected to probe the green region in Fig. 3.
• HEFT allows for finite UV cutoffs and potentially testable physics at colliders, with the quotient theory (e.g., $m_2^2 \leq 0$) not ruled out by current data but potentially excluded by future FCC sensitivity.

Figure 1: Examples of possible HEFTs based on geometry and reduced to 2d (rotations around the vertical axis correspond to gauge transformations): (a) SM (green) MCHM (orange) and negative curvature (blue), (b) Quotient theories which do not have fixed point (the manifold does not go through the ori