Positively Identifying HEFT or SMEFT

This paper establishes unitarity and analyticity bounds on Higgs effective field theory (HEFT) Wilson coefficients, demonstrating how projecting these constraints into the standard model effective field theory (SMEFT) framework reveals a distinct subspace that can distinguish between a misapplied EFT and a pathological ultraviolet completion, particularly through specific dimension-eight custodial symmetric operators currently being probed at colliders.

The Gist

Imagine you are a detective trying to solve a mystery about the universe. You know there are new, heavy particles out there (the "suspects") that your current telescopes and particle colliders can't see directly because they are too heavy or too far away.

To catch them, you use a clever trick called Effective Field Theory (EFT). Think of EFT as a "shadow puppet" show. You can't see the puppeteer (the new physics), but you can see the shadows they cast on the wall (how known particles like the Higgs boson behave). By studying the shape of these shadows, you can guess what the puppeteer looks like.

For a long time, physicists have used one specific type of shadow puppet show called SMEFT. It assumes the new physics plays by very strict, symmetrical rules (like a rigid dance where everyone moves in perfect unison).

However, this paper by Grant Remmen and Nicholas Rodd suggests that maybe the puppeteer is actually using a different, more flexible style of show called HEFT. In HEFT, the rules are looser; the dancers don't have to move in perfect unison.

Here is the core of their discovery, broken down with simple analogies:

1. The "No-Go" Zones (Positivity Bounds)

Imagine you are trying to build a house. There are laws of physics (like gravity and the speed of light) that act like building codes. If you try to build a house that violates these codes, it will collapse.

In the world of particle physics, these "building codes" are called Positivity Bounds. They are mathematical rules derived from the fact that the universe must be logical (causal), stable (unitary), and local.

• The Rule: If you measure a certain property of a particle interaction, it must be positive (greater than zero). If you measure a negative value, it means your theory is broken, or the universe is acting strangely.

2. The Two Maps (SMEFT vs. HEFT)

The authors realized that the "building codes" look different depending on which map you are using.

• The SMEFT Map: This is the old, strict map. It says, "If you see a shadow here, it must be positive." If you see a negative shadow, you assume the universe is broken.
• The HEFT Map: This is the new, flexible map. It says, "If you see a shadow here, it might be negative, but that's okay because the underlying rules are different."

The Analogy:
Imagine you are looking at a shadow on a wall.

• SMEFT assumes the object casting the shadow is a rigid cube. If the shadow looks weird, you think the cube is broken.
• HEFT realizes the object might be a soft, squishy ball. A weird shadow doesn't mean the ball is broken; it just means it's squishy!

3. The "Forbidden" Orange Zone

The most exciting part of the paper is a specific region on their graph (shown in Figure 1 of the paper) colored orange.

• The Blue Zone: This is where both maps agree. If you find physics here, everything is fine.
• The Dark Blue (Forbidden) Zone: This is where the universe is definitely broken. Nothing can exist here.
• The Orange Zone (The Discovery): This is the sweet spot.
  • If you use the SMEFT map, this orange zone looks forbidden. It looks like a violation of the laws of physics.
  • But if you use the HEFT map, this orange zone is perfectly legal.

Why does this matter?
Imagine you are at a particle collider (like the Large Hadron Collider) and you see a signal in this "Orange Zone."

• Old Thinking: "Oh no! We violated the laws of physics! The universe is broken!"
• New Thinking (This Paper): "Wait! We didn't break the laws of physics. We just used the wrong map! The new physics isn't a rigid cube; it's a squishy ball. We need to switch to the HEFT map."

4. The Real-World Test

The authors didn't just do this on paper. They looked at real data from the ATLAS experiment at the Large Hadron Collider. They found that current measurements are already poking around the edges of this "Orange Zone."

They identified a specific pair of numbers (called Wilson coefficients) that act like a "smoking gun." If future experiments find values in this orange area, it won't mean the universe is broken. It will mean we finally found evidence that the Standard Model needs to be described by the more flexible HEFT rules, not the rigid SMEFT rules.

Summary

• The Problem: We might see weird particle behavior and think the laws of physics are broken.
• The Solution: Maybe we are just using the wrong "rulebook" (SMEFT) for the job.
• The Discovery: There is a specific "safe zone" (the orange region) where the universe is perfectly healthy, but only if you use the flexible HEFT rulebook.
• The Takeaway: If we find new physics in this zone, it's not a disaster; it's a breakthrough. It tells us the new particles are "squishy" and flexible, not rigid and symmetrical.

This paper gives physicists a new way to interpret their data: Don't panic if the numbers look weird; maybe you just need to change your map.

Technical Summary

Here is a detailed technical summary of the paper "Positively Identifying HEFT or SMEFT" by Grant N. Remmen and Nicholas L. Rodd.

1. Problem Statement

In the absence of direct discoveries of new particles at the energy frontier, physicists rely on Effective Field Theories (EFTs) to parameterize potential new physics (UV) effects in low-energy observables. Two primary frameworks exist:

• SMEFT (Standard Model EFT): Assumes the Standard Model (SM) gauge symmetry $SU(3)_C \times SU(2)_L \times U(1)_Y$ is linearly realized. The Higgs field is part of an electroweak doublet.
• HEFT (Higgs EFT): Assumes only the unbroken symmetry $SU(3)_C \times U(1)_{EM}$ is realized. The Higgs sector is described non-linearly, where the four scalar degrees of freedom do not necessarily transform as a doublet.

The Core Issue:
Positivity bounds—constraints on Wilson coefficients derived from fundamental axioms of UV theories (unitarity, locality, and causality)—have been extensively studied for SMEFT. However, if the true UV physics generates an HEFT, but experimentalists analyze the data assuming SMEFT, they may observe Wilson coefficients that appear to violate SMEFT positivity bounds.

• The Question: Does a violation of SMEFT positivity imply a breakdown of fundamental physics (non-unitary/non-causal UV), or does it simply indicate that the wrong EFT (SMEFT) was used to describe a valid HEFT UV completion?
• The Goal: To derive positivity bounds specifically for HEFT, project them onto the SMEFT parameter space, and identify regions where HEFT is allowed but SMEFT is forbidden.

2. Methodology

The authors employ a rigorous theoretical framework combining scattering amplitudes, dispersion relations, and group theory.

• Operator Basis Selection:

  • They focus on dimension-eight operators involving four derivatives acting on Higgs fields, schematically $(\partial H)^4$. These operators dominate the $s^2$ term in forward scattering amplitudes, which is the quantity constrained by positivity.
  • SMEFT: Under custodial symmetry ($O(4) \to O(3)$), the SMEFT basis reduces to two independent operators: $O_+$ and $O_\times$.
  • HEFT: Under the same custodial symmetry, the HEFT basis (where the Higgs is decomposed into a singlet $h$ and triplet $\pi^i$) contains five independent operators: $O^h_1, O^{h\pi}_1, O^{h\pi}_2, O^\pi_1, O^\pi_2$.

• Derivation of Positivity Bounds:

  • Scattering Amplitudes: They construct the forward elastic scattering amplitude $A(s)$ for arbitrary superpositions of the scalar states ($h$ and $\pi^i$).
  • Dispersion Relations: Using the optical theorem and analyticity, they relate the second derivative of the amplitude at $s=0$ ($A''(0)$) to the integral of the total cross-section over all energies. Since the cross-section is positive, $A''(0)$ must be positive.
  • Generalized Optical Theorem: They verify that the bounds derived from elastic scattering are sufficient by applying the generalized optical theorem (considering inelastic intermediate states and irreducible representations of the custodial group). They find that the bounds are identical, confirming the derived constraints are both necessary and sufficient.

• Projection Strategy:

  • The SMEFT is a subspace of the HEFT. The authors define a projection matrix to map the 5-dimensional HEFT coefficient space onto the 2-dimensional SMEFT plane.
  • They calculate the projection of the HEFT positivity cone onto the SMEFT plane. This involves marginalizing over the "orthogonal" HEFT directions (parameters $d_1, d_2, d_3$) that are zero in pure SMEFT but non-zero in general HEFT.

3. Key Contributions

1. First HEFT Positivity Bounds: The paper derives the complete set of positivity bounds for custodial-symmetric dimension-eight Higgs operators in HEFT (Eq. 11).
2. Identification of the "HEFT Window": By projecting the HEFT bounds onto the SMEFT plane, the authors identify a specific region (the orange region in Fig. 1) that is:
   • Allowed by HEFT positivity (consistent with unitary, local, causal UV).
   • Forbidden by SMEFT positivity (would be interpreted as a violation of fundamental axioms if SMEFT were assumed).
3. UV Completion Models: The authors construct explicit tree-level UV extensions (involving heavy scalars and vectors) that saturate the HEFT bounds. They demonstrate that simple scalar extensions can generate Wilson coefficients that land in the "HEFT-only" region, proving these regions are physically realizable and not mathematical artifacts.
4. Experimental Relevance: They connect these theoretical bounds to current Large Hadron Collider (LHC) data, specifically ATLAS measurements of anomalous quartic gauge couplings (aQGCs) in same-sign $W$ boson production.

4. Key Results

• SMEFT Bounds: For SMEFT, the positivity constraints are linear:
  $$C_+ > 0 \quad \text{and} \quad C_+ + C_\times > 0$$
  (Where $C_+$ and $C_\times$ are the Wilson coefficients for $O_+$ and $O_\times$).

• HEFT Bounds: For HEFT, the constraints are non-linear and involve all five coefficients ($c^h_1, c^{h\pi}_1, c^{h\pi}_2, c^\pi_1, c^\pi_2$). The most critical constraint involves a square root term, creating a curved boundary in the parameter space.

• Projected HEFT Bounds on SMEFT: When the HEFT positivity cone is projected onto the SMEFT plane, the allowed region expands significantly. The new constraints are:
  $$6C_+ + C_\times > 0 \quad \text{and} \quad 19C_+ + 9C_\times > 0$$
  The region defined by these inequalities (but violating the strict SMEFT inequalities) constitutes the "HEFT-only" region.

• Interpretation of Violations:

  • If experimental data falls in the dark blue region (forbidden by both), it implies a pathological UV (violation of unitarity/causality).
  • If data falls in the orange region (allowed by HEFT, forbidden by SMEFT), it implies the UV is healthy, but the SMEFT is the wrong effective description. The new physics is best described by HEFT.

5. Significance

• Diagnostic Tool: This work provides a crucial diagnostic tool for the LHC and future colliders. It prevents the misinterpretation of valid new physics as a breakdown of fundamental field theory axioms.
• Symmetry Structure: It highlights that positivity bounds are sensitive not just to the existence of new physics, but to the symmetry realization (linear vs. non-linear) of the Higgs sector.
• Experimental Guidance: The paper identifies a specific 2D slice of the parameter space that is currently being probed by ATLAS and CMS. If future measurements of anomalous quartic gauge couplings fall into the orange region, it would be a "smoking gun" for the non-linear realization of electroweak symmetry (HEFT) rather than the linear SMEFT.
• Theoretical Foundation: It establishes that HEFT and SMEFT are not disjoint; rather, SMEFT is a specific slice of the HEFT parameter space. The "forbidden" regions of SMEFT are merely shadows of the larger HEFT cone.

In summary, the paper demonstrates that "violating" SMEFT positivity bounds does not necessarily signal a crisis in physics; it may simply signal that the Higgs sector is non-linearly realized, and the correct framework (HEFT) must be used to interpret the data.