Testing varying coupling constants through multi-Higgs production at the LHC

This paper proposes the One Scalar Theory (1ST), a predictive framework linking Higgs and top couplings to a single scale $\Lambda_0$, and demonstrates that current ATLAS data and future High-Luminosity LHC runs can test this model's dynamical origin of the electroweak sector by probing $\Lambda_0$ up to 4 TeV through multi-Higgs and multi-top resonance searches.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to figure out the "knobs and dials" that control how this machine works. In our current best theory (the Standard Model), some of these dials—like how heavy the top quark is or how the Higgs particle interacts with itself—are just set to specific numbers. We don't know why they are set that way; we just measure them and move on.

This paper proposes a new idea called the One Scalar Theory (1ST). Think of it as a minimalist theory where there isn't a separate dial for every single setting. Instead, there is one master dial (a single invisible field called $S_0$) that controls the most important settings of the machine.

Here is a breakdown of their idea using simple analogies:

1. The "Master Volume Knob"

In this theory, the "knob" isn't just a static number; it's a dynamic field that can change.

• The Analogy: Imagine a radio station where the volume and the bass are usually controlled by two different knobs. In the 1ST model, there is only one knob. If you turn it up, both the volume (the Higgs self-coupling) and the bass (the top quark's interaction) go up together.
• The Result: You can't tweak one without affecting the other. This makes the theory very "predictive" because you can't just fiddle with the settings to hide the answer. If the theory is wrong, the whole machine breaks in a specific, noticeable way.

2. The "Speed Limit" of the Machine

The paper suggests that this master knob is tied to a specific energy scale, which they call $\Lambda_0$ (Lambda-zero).

• The Analogy: Think of $\Lambda_0$ as the "speed limit" of the universe's underlying engine. If you try to drive faster than this limit, the rules of the road change.
• The Constraint: The authors argue that because this single knob controls everything, the production of new particles and how they decay are locked together. You can't tune the "production" dial to be high and the "decay" dial to be low to hide the signal. They are stuck together, like a gear system.

3. The Two "Traffic Zones"

The researchers found that the behavior of this new particle ($S_0$) changes drastically depending on its mass (how heavy it is), creating two distinct zones:

• Zone A (The "Higgs Party"): If the new particle is lighter than a certain threshold (specifically, lighter than two top quarks combined), it mostly breaks apart into pairs of Higgs bosons. It's like a party where everyone is dancing with Higgs partners.
• Zone B (The "Top Quark Rush"): If the new particle is heavier than that threshold, it suddenly switches gears. It stops dancing with Higgs and starts breaking apart into pairs of top quarks.
• The Significance: This "switch" happens at a very specific speed limit ($2m_t$). The paper says we can use this switch to test the theory.

4. Hunting for the Ghost

How do we find this invisible "master knob" particle at the Large Hadron Collider (LHC)?

• The Strategy: The scientists looked at data from the ATLAS experiment (a giant detector at the LHC). They looked for "resonances"—which are like hearing a specific musical note played loudly in a noisy room.
• The Search: They looked for two specific sounds:
  1. Two Higgs bosons (in the lighter zone).
  2. Two top quarks (in the heavier zone).
• The Findings: They didn't find the particle yet, but they used the "silence" (the lack of a signal) to set a speed limit. They calculated that if this "master knob" exists, its energy scale ($\Lambda_0$) must be at least 1 TeV (a very high energy). If it were lower, we would have seen it by now.

5. The Future: The "Super-LHC"

The paper looks ahead to the High-Luminosity LHC (HL-LHC), which is an upgraded version of the current collider that will run in the future.

• The Prediction: With this more powerful machine, they believe they can push the search limit up to 3 or 4 TeV.
• The Analogy: If the current LHC is a flashlight that can see a few meters into the dark, the HL-LHC is a searchlight that can see several kilometers. If the "master knob" exists within that range, the HL-LHC will almost certainly find it or prove it doesn't exist.

6. The "Smoking Gun" Signature

One of the coolest parts of this theory is a unique signature that other theories don't have.

• The Analogy: Most theories allow the new particle to interact with many different things (like W and Z bosons). But because this "master knob" particle is a "singlet" (it's invisible to the standard forces), it can only talk to the top quark.
• The Result: This means if this particle decays into light (photons) or gluons, it does so only through a specific loop involving the top quark. The ratio of these decays is fixed and rigid. If we see a particle that decays in this exact, rigid ratio, it's a "smoking gun" that proves this specific theory is real, ruling out all the other "generic" theories.

Summary

The paper proposes a simple, elegant idea: One field controls the two most important couplings in the universe. Because this field is so tightly constrained, it leaves very specific footprints. By looking at how the LHC produces pairs of Higgs bosons and top quarks, the authors have set new limits on where this field could hide. They predict that the next generation of colliders will be able to definitively say whether the fundamental constants of our universe are "fixed" or if they are "dynamically generated" by this single, hidden field.

Technical Summary

Technical Summary: Testing Varying Coupling Constants through Multi-Higgs Production at the LHC

Problem and Motivation
Fundamental couplings and constants in a theory are expected to emerge from underlying dynamics rather than being arbitrary free parameters. In frameworks such as string theory, these values are determined by the vacuum expectation values (VEVs) of scalar moduli fields. If the Standard Model (SM) parameters, specifically the Higgs self-coupling ($\lambda_h$) and the top Yukawa coupling ($\lambda_t$), are dynamically generated by such fields, their variations could have profound cosmological implications, including triggering strongly first-order electroweak phase transitions or altering QCD confinement. However, the phenomenological consequences of these variations at the TeV scale remain largely unexplored. The interplay between $\lambda_h$ and $\lambda_t$ is central to the hierarchy problem, yet these couplings are among the least constrained in the SM. Their variation significantly impacts di-Higgs production, a process already sensitive to new physics due to delicate cancellations between top-quark box and triangle diagrams in the SM.

Methodology: The One Scalar Theory (1ST)
The authors propose a minimalist framework, the One Scalar Theory (1ST), to test the dynamical origin of these couplings. Unlike generic Higgs-portal models that introduce tunable mixing angles, 1ST posits a single real singlet scalar field, $S_0$, which mediates the variations of both $\lambda_h$ and $\lambda_t$.

The couplings are defined as $\tilde{\lambda}_i = \lambda_i \varepsilon(x)$, where $\varepsilon(x)$ is a dimensionless scalar field with a non-canonical kinetic term typical of moduli fields. Through the field redefinition $\varepsilon = \exp(S_0/\Lambda_0)$, the theory introduces a single new physics scale, $\Lambda_0$, and the scalar mass, $M_0$. The interaction Lagrangian, expanded to leading order in $1/\Lambda_0$, couples $S_0$ to the SM via dimension-five operators:
$$\mathcal{L} \supset -\frac{S_0}{\Lambda_0} \left[ \lambda_h \left( v^2 h^2 + v h^3 + \frac{1}{4}h^4 \right) - \frac{\lambda_t}{\sqrt{2}} (v+h)\bar{t}t \right]$$
Crucially, the 1ST framework locks the production cross-section and decay widths to the same fundamental scale $\Lambda_0$. This removes parametric freedom; a null result cannot be evaded by tuning mixing parameters. The scalar $S_0$ is assumed to be the physical mass eigenstate, with $S_0$-$h$ mixing treated as negligible due to LHC constraints.

Key Contributions and Phenomenology
The paper maps the collider phenomenology of 1ST, which is partitioned by the $2m_t$ kinematic threshold:

1. Production and Decay: $S_0$ is produced predominantly via gluon-gluon fusion (ggF) mediated by a top-quark loop. The primary decay modes are $gg$, $hh$, and $t\bar{t}$.

   • Below $2m_h$: Decay is dominated by $gg$.
   • Between $2m_h$ and $2m_t$: Decay is almost exclusively to $hh$.
   • Above $2m_t$: The $t\bar{t}$ mode dominates, approaching a 100% branching ratio at higher masses.
     Because branching ratios saturate near unity in these regimes, the event rate alone is insufficient to determine $\Lambda_0$. Instead, the scale is constrained through interference effects (e.g., off-shell tails interfering with SM continua) and the total decay width.

2. Unique Signatures:

   • Gauge Singlet Nature: As a singlet, $S_0$ lacks tree-level couplings to $W$ and $Z$ bosons. Consequently, loop-induced decays to $gg$ and $\gamma\gamma$ are mediated exclusively by the top-quark loop. This results in a rigid, parameter-free ratio between partial widths: $\Gamma_{\gamma\gamma}/\Gamma_{gg} = \frac{8}{9}(\alpha/\alpha_s)^2$. This serves as a "smoking-gun" signature distinguishing 1ST from generic portal models.
   • Multi-Higgs and Di-top Channels: The model predicts significant enhancements in resonant di-Higgs ($gg \to S_0 \to hh$) and tri-Higgs production. For $M_0 = 300$ GeV and $\Lambda_0 = 1$ TeV, the tri-Higgs cross-section can be enhanced by a factor of 20 relative to the SM. Above the $t\bar{t}$ threshold, resonant di-top production becomes the dominant probe.

3. Constraints and Bounds:
   The authors recast current ATLAS data for di-Higgs and di-top resonance searches to set lower bounds on $\Lambda_0$.

   • Current Limits: Using Run 2 data, they establish lower bounds on $\Lambda_0$ in the TeV scale. The exclusion regions are shown in the $(M_0, \Lambda_0)$ plane, where the region $M_0 > \Lambda_0$ is excluded as it violates the perturbative validity of the effective field theory.
   • Secondary Channels: Associated production channels like $pp \to tS_0 j$ and $pp \to t\bar{t}S_0$ are suppressed by the $(v/\Lambda_0)^2$ penalty and overwhelmed by backgrounds given the current bounds ($\Lambda_0 \gtrsim 1$ TeV).

Results and Future Projections
The study demonstrates that the High-Luminosity LHC (HL-LHC) will significantly extend the reach of these tests.

• HL-LHC Reach: By scaling current limits with the projected luminosity increase (from 140 fb$^{-1}$ to 3000 fb$^{-1}$), the sensitivity to the cross-section improves by a factor of $\sim 4.6$. Since the theoretical cross-section scales as $1/\Lambda_0^2$, this translates to a factor of $\sim 2$ improvement in the lower bound on $\Lambda_0$.
• Projected Limits: The HL-LHC is projected to probe $\Lambda_0$ up to 3–4 TeV across the mass range. Below $2m_t$, the di-Higgs channel drives the sensitivity; above $2m_t$, the di-top resonance search takes over.
• Tri-Higgs Potential: The substantial enhancement in tri-Higgs production, combined with resonant features, suggests this channel could become observable in future LHC runs, providing a striking signature of the framework.

Significance
The paper claims that the 1ST framework offers a highly predictive avenue to test whether the fundamental constants of the electroweak sector are dynamically generated. By eliminating tunable mixing parameters and locking production and decay to a single scale, the model provides a definitive test case. The rigid structure ensures that the scalar bypasses standard massive vector-boson backgrounds, yielding unique signatures (such as the specific $\gamma\gamma/gg$ ratio) that can veto generic portal theories. Furthermore, the unsuppressed decays of $S_0$ into SM pairs ensure the scalar safely depletes in the early universe, avoiding the cosmological moduli problem. Ultimately, the HL-LHC is positioned to deliver a definitive verdict on the dynamical origin of the Higgs self-coupling and top Yukawa coupling through these multi-Higgs and di-top resonance searches.