Electroweak Restoration: SMEFT and HEFT

Imagine the universe as a giant, high-energy dance floor. For decades, physicists have been watching how particles dance together, trying to figure out the rules of the music. One of the biggest mysteries is how particles get their mass. In our current understanding (the Standard Model), there's a "Higgs field" that acts like a thick, sticky syrup. As particles move through it, they get "stuck" and gain mass.

This paper is about what happens when we crank up the music to an incredibly high volume (high energy). The authors, Ian Lewis, Zhen Liu, and Ishmam Mahbub, are asking: If we turn the energy up high enough, does the "sticky syrup" disappear, and do the particles start dancing as if they were massless again?

They call this the "Electroweak Restoration" regime. It's like turning a crowded, slow dance into a fast, free-form rave where the rules of mass no longer apply.

Here is the breakdown of their investigation using simple analogies:

1. The Two Different Rulebooks (SMEFT vs. HEFT)

The scientists are testing two different theories about how the universe is built, which they call "rulebooks":

The "Linear" Rulebook (SMEFT): Imagine a dance troupe where the Higgs boson (the star dancer) and the Goldstone bosons (the backup dancers) are all part of the same family. They are like siblings in a single, tightly knit group (a "doublet"). In this world, the Higgs and the Goldstones are deeply connected. If you change the Higgs, the Goldstones change with it in a predictable way.
The "Non-Linear" Rulebook (HEFT): Imagine a different troupe where the Higgs is a solo act, and the Goldstone dancers are a separate group entirely. They don't have to follow the same choreography. The Higgs is a "singlet" (a loner), and the Goldstones are a "triplet" (a trio). In this world, there is no strict rule forcing the Higgs and Goldstones to move in sync.

2. The Experiment: The "Goldstone Equivalence"

The authors use a clever trick called the Goldstone Boson Equivalence Theorem. Think of it as a translator.

At low energies, we see heavy, sluggish particles (Longitudinal W and Z bosons).
At high energies, these heavy particles behave exactly like the light, massless "Goldstone" dancers.
So, instead of trying to measure the heavy particles directly, the authors translate the problem: "If we watch the Goldstone dancers, what are they doing?"

3. The Test: The "Ratio" Game

The core of the paper is a simple math game: The Ratio.

The authors look at two specific dance moves:

Move A: Producing a pair of heavy bosons (like $W$ and $Z$ ).
Move B: Producing a heavy boson and a Higgs boson ( $W$ and $h$ ).

They calculate the ratio of how often Move A happens compared to Move B.

In the "Linear" World (SMEFT): Because the Higgs and Goldstones are siblings, the rules force these two moves to happen at almost the exact same rate at high energies. The ratio should be 1. It's like a perfectly choreographed duet where the partners always match steps.
In the "Non-Linear" World (HEFT): Because the Higgs is a solo act and the Goldstones are a separate group, there is no rule forcing them to match. The ratio can be anything. It's like a solo dancer and a group of backup dancers doing their own thing; the ratio of their moves will likely not be 1.

4. The Findings

The authors did the complex math (the "explicit calculations") to prove their point:

Standard Model & Linear Theory: The ratio of these two processes approaches 1 as energy goes up. The "siblings" stay in sync.
Non-Linear Theory: The ratio diverges (drifts away from 1). The "soloist" and the "group" go their separate ways.

They found that if we measure the rate of producing a $W$ boson and a $Z$ boson versus a $W$ boson and a Higgs boson at the Large Hadron Collider (LHC), we can tell which "rulebook" the universe is actually using.

5. The Real-World Check (LHC)

The paper doesn't just stay in theory. They looked at real data from the LHC (the giant particle collider in Europe).

They found that we can already measure the "longitudinal" (heavy) $W$ and $Z$ bosons.
They projected that with future upgrades (the High-Luminosity LHC), we will be able to measure the $W$ and Higgs combination with enough precision to see if the ratio is 1 or not.

The Bottom Line

This paper proposes a new way to check the fundamental structure of the universe. By comparing how often certain particle pairs are created at high speeds, we can determine if the Higgs boson is a "sibling" to the Goldstone bosons (Linear/SMEFT) or a "loner" (Non-Linear/HEFT).

If the ratio of these events is 1, the universe follows the "Linear" rules. If the ratio is not 1, the universe might be following the "Non-Linear" rules, suggesting our understanding of how particles get mass needs a major rewrite. The authors show that the upcoming upgrades to the LHC will have the sensitivity to finally answer this question.

Technical Summary: Electroweak Restoration in SMEFT and HEFT

Problem Statement
As the Large Hadron Collider (LHC) probes electroweak (EW) interactions at increasingly higher energies ( $E \gg m_W, m_Z, m_h$ ), the system enters the "electroweak restoration" regime. In this limit, the broken EW theory is expected to approach the unbroken theory, where longitudinal gauge boson production ( $V_L$ ) corresponds to Goldstone boson production ( $G$ ) via the Goldstone Boson Equivalence Theorem (GBET). While previous studies established that in the Standard Model (SM), amplitudes for processes like $f\bar{f}' \to V_L V'_L$ and $f\bar{f}' \to V_L h$ converge to specific relationships at high energies, the behavior of these correlations in Beyond the Standard Model (BSM) scenarios remains to be fully characterized. Specifically, it is unclear how these high-energy correlations distinguish between linear realizations of EW symmetry (Standard Model Effective Field Theory, SMEFT) and non-linear realizations (Higgs Effective Field Theory, HEFT).

Methodology
The authors investigate electroweak restoration by calculating and comparing scattering amplitudes and cross-sections for longitudinal di-boson production processes: $f\bar{f}' \to W_L^\pm Z_L$ , $f\bar{f}' \to W_L^\pm h$ , $f\bar{f} \to W_L^+ W_L^-$ , and $f\bar{f} \to Z_L h$ .

Theoretical Frameworks: The study employs SMEFT for linear EW symmetry realization (where the Higgs is part of an $SU(2)_L$ doublet) and HEFT for non-linear realization (where the Higgs is a gauge singlet and Goldstones form a triplet).
Calculations: The authors derive amplitudes to linear order in Wilson coefficients for dimension-6 SMEFT operators (Warsaw basis) and a specific subset of HEFT operators that generate quadratic energy growth ( $O(s)$ ). They utilize the GBET to relate longitudinal gauge boson amplitudes to Goldstone boson amplitudes.
Observables: The primary focus is on ratios of amplitudes and partonic cross-sections, specifically $\hat{r}_{Zh} = \sigma(W_L^\pm Z_L) / \sigma(W_L^\pm h)$ . The analysis extends to hadronic cross-sections at the LHC ( $\sqrt{s}=14$ TeV) using parton distribution functions (CTEQ6L1) and projects sensitivities for the High-Luminosity LHC (HL-LHC) with $3000 \text{ fb}^{-1}$ .

Key Contributions and Results

Amplitude Correlations in Linear vs. Non-Linear Realizations:
- SM and SMEFT: In the high-energy limit, the authors demonstrate that the ratio of amplitudes $M(W_L^\pm Z_L) / M(W_L^\pm h)$ approaches unity for specific helicity configurations (e.g., $q^- \bar{q}'_+$ ). This arises because, in linear realizations, the Higgs and Goldstone bosons form a doublet, and the relevant operators (e.g., $Q_{Hq}^{(3)}$ ) induce correlated contact terms for both $W_L Z_L$ and $W_L h$ production.
- HEFT: In non-linear realizations, the Higgs is a singlet, and the Goldstones form a triplet. Consequently, the operators governing $W_L Z_L$ and $W_L h$ production (e.g., $N_7^Q$ and $P_3$ ) are independent. The ratio of amplitudes does not generally approach unity; instead, it depends on arbitrary Wilson coefficients and can diverge at high energies.
Cross-Section Ratios as Discriminators:
- The paper identifies the ratio of longitudinal cross-sections $\hat{r}_{Zh}$ as a robust observable. In SM and SMEFT, this ratio converges to 1 at high energies (after summing initial state quark helicities), whereas in HEFT, it deviates significantly due to uncorrelated energy growth.
- Helicity Summation: The authors note that if gauge boson polarizations are not tagged (i.e., helicities are summed), the distinctive SMEFT/HEFT separation is obscured by dominant transverse contributions. Therefore, tagging longitudinal polarization is crucial for this test.
Phenomenological Projections:
- Using current LHC measurements of $W_L^\pm Z_L$ production and projections for $W_L^\pm h$ measurements, the authors project HL-LHC sensitivities.
- SMEFT: The ratio observable is less sensitive to SMEFT operators (like $C_{Hq}^{(3)}$ ) than the $W_L^\pm Z_L$ channel alone, because the linear energy growth is correlated between the two channels in SMEFT.
- HEFT: The ratio observable significantly improves sensitivity to certain HEFT operators (specifically $n_7^Q$ ) compared to the $W_L^\pm Z_L$ channel alone. This is because HEFT operators can induce constructive interference in one channel and destructive interference in the other, causing the ratio to deviate strongly from unity.

Significance and Claims
The paper claims that comparing the rates of $W_L^\pm Z_L$ and $W_L^\pm h$ production at high energies provides a direct test of the symmetry structure of the scalar sector.

Distinguishing EFTs: The convergence of the $W_L^\pm Z_L / W_L^\pm h$ ratio to unity is a specific prediction of linear EW symmetry (SM and SMEFT) derived from the GBET and the doublet nature of the Higgs. The failure of this ratio to converge is a generic feature of non-linear realizations (HEFT).
Experimental Viability: The authors argue that this test is experimentally feasible at the HL-LHC. With the observation of longitudinal $WZ$ production and projected precision measurements of longitudinal $Wh$, the ratio serves as a natural observable to distinguish between linear and non-linear realizations of EW symmetry without requiring new physics to be directly produced.
Modesty: The authors acknowledge that while the ratio is a powerful discriminator, it relies on the ability to tag longitudinal polarizations. They also note that in HEFT, specific fine-tuning of parameters could mimic SMEFT behavior, but generically, the correlation is broken. The study focuses on dimension-6 operators and specific subsets of HEFT operators sufficient to illustrate the theoretical distinction.