Confront a dilaton model with the LHC measurements

This paper investigates a geometry-derived dilaton model within the metric affine theory framework to identify parameter spaces where the 125 GeV Higgs boson is dilaton-dominant, demonstrating that High Luminosity LHC measurements of Higgs pair production can confirm or rule out this scenario.

The Gist

Here is an explanation of the paper "Confront a dilaton model with the LHC measurements," translated into everyday language with creative analogies.

The Big Mystery: What is the Higgs Boson?

Imagine the universe as a giant, complex video game. In 2012, scientists at the Large Hadron Collider (LHC) found the "Master Key" to the game: the Higgs boson. This particle is responsible for giving other particles (like electrons and quarks) their mass, allowing atoms to form and life to exist.

However, while we found the key, we don't fully understand how the lock works. Is the Higgs a fundamental building block of nature, or is it actually a composite object made of something deeper?

The New Theory: The "Dilaton" and the "Stretchy Universe"

The authors of this paper propose a new way to look at the Higgs. They suggest that the Higgs we found might actually be a Dilaton.

To understand a Dilaton, imagine the universe is made of a giant, stretchy rubber sheet (spacetime).

• Standard Physics: Usually, we think the size of the universe is fixed.
• Weyl Symmetry (The Paper's Idea): This theory suggests the universe has a "stretching symmetry." It means the laws of physics shouldn't change even if we zoom in or zoom out on the rubber sheet.
• The Dilaton: If you stretch that rubber sheet, you need a "messenger" to tell the particles how to adjust their size. That messenger is the Dilaton.

The paper asks: Is the Higgs boson we found actually this "stretching messenger" (the Dilaton) in disguise?

The Two Scenarios: The "Dance" of Particles

In this model, there are two main characters dancing on the stage:

1. The Doublet: The standard Higgs field we know from textbooks.
2. The Singlet (Dilaton): The new "stretching" field.

Depending on how they dance together, the paper identifies two main scenarios (like two different dance styles):

1. The Trigonometric Dance (TSS): Imagine the two fields are dancing in a circle. Their movements are periodic, like a sine wave. They mix together in a way that creates a repeating pattern.
2. The Hyperbolic Dance (HSS): Imagine the fields are dancing on a saddle shape (like a Pringles chip). Their movements grow exponentially, not in a circle.

The authors want to know: In our universe, is the Higgs we see mostly the "Doublet" (the standard dancer) or mostly the "Dilaton" (the stretching messenger)?

The Test: Checking the Footprints at the LHC

The authors took their mathematical models (the two dance styles) and compared them against real data collected by the LHC over the last decade. They looked at three specific "footprints" left by the Higgs:

• How it talks to other particles: (Yukawa couplings).
• How it talks to force carriers: (W and Z bosons).
• How it talks to itself: (Self-coupling). This is like asking, "If two Higgs bosons bump into each other, how do they react?"

The Results:

• Good News: The model fits the current data surprisingly well. It's possible that the Higgs we found is indeed mostly a Dilaton.
• The Catch: The current data isn't precise enough to say "Yes, it's definitely a Dilaton" or "No, it's definitely not." It's like looking at a blurry photo of a suspect; the face looks right, but you can't be 100% sure.

The Future: The High-Luminosity LHC (HL-LHC)

This is where the paper gets exciting. The authors predict that the High-Luminosity LHC (a super-charged version of the current collider coming online soon) will be the "High-Definition Camera" needed to solve the mystery.

• The Smoking Gun: They found that the way the Higgs interacts with itself (Higgs pair production) is different in their Dilaton model compared to the Standard Model.
• The Verdict: If the HL-LHC measures these interactions with high precision, it will either confirm that the Higgs is a Dilaton or rule it out completely.

The "Tension" in the Theory

The paper also admits a small headache in their theory.

• The Problem: The model needs to explain why the "stretching scale" of the universe is so huge (related to gravity) while the "Higgs scale" is so tiny (related to atoms). It's like trying to balance a skyscraper on a toothpick.
• The Fix: They suggest that maybe there are hidden mechanisms (like extra dimensions or new symmetries) that keep these scales stable, but they need more research to prove it.

The Bottom Line

Think of this paper as a detective story.

• The Suspect: The Higgs boson.
• The Alibi: It might be a Dilaton (a messenger of the universe's size) rather than a standard particle.
• The Evidence: Current data from the LHC is consistent with this alibi, but not strong enough to convict or acquit.
• The Next Move: The upcoming High-Luminosity LHC will act as the judge. By measuring how Higgs bosons interact with each other, it will finally tell us if the Higgs is a fundamental particle or a "stretchy" messenger from a deeper layer of reality.

If the HL-LHC confirms this, it would be a massive leap, connecting the tiny world of particles with the giant world of gravity and the geometry of the universe itself.

Technical Summary

Here is a detailed technical summary of the paper "Confront a dilaton model with the LHC measurements" by J.E. Wu and Q.S. Yan.

1. Problem Statement

The origin of the Higgs boson ($H_{125}$), discovered in 2012, remains an open question regarding whether it is a fundamental scalar or a composite state (e.g., a dilaton). While the Standard Model (SM) and General Relativity (GR) are successful individually, their unification is hindered by the non-renormalizability of GR.
The authors investigate a specific theoretical framework where the Higgs boson arises from a dilaton model embedded within the Metric Affine Theory (MAT) and Weyl geometry. In this framework, local scale symmetry (Weyl symmetry) is gauged, and the Higgs potential is derived from geometric principles involving quadratic gravity terms. The central problem is to determine if the observed 125 GeV Higgs boson can be dilaton-dominant (primarily a singlet scalar from the symmetry breaking of scale invariance) rather than the standard doublet Higgs, and to identify the constraints imposed by current and future Large Hadron Collider (LHC) data.

2. Methodology

The authors construct a model based on Weyl conformal geometry where the Lagrangian includes:

• Quadratic Gravity Terms: Including the Weyl tensor squared ($\hat{C}^2$) and Ricci scalar squared ($\hat{R}^2$).
• Scalar Sector: A real singlet scalar field ($\Phi$, the dilaton) and a complex doublet scalar field ($\phi$, the SM Higgs doublet).
• Non-minimal Couplings: Couplings between the scalars and gravity ($\beta, \gamma$) and gauge fields.

Key Theoretical Steps:

1. Symmetry Breaking: The model employs the Stueckelberg mechanism to spontaneously break local conformal symmetry. This generates mass terms for the Weyl vector boson and the scalar fields.
2. Parametrization: After symmetry breaking, the physical degrees of freedom are parametrized using a Brans-Dicke field $\Theta$. The authors identify two distinct scenarios based on the relative signs of the coupling parameters ($\chi_D^2$ and $\chi_H^2$):
   • TSS (Trigonometric Scalar Scenario): $\chi_D^2 > 0, \chi_H^2 > 0$. The potential involves trigonometric functions, leading to a periodic structure.
   • HSS (Hyperbolic Scalar Scenario): $\chi_D^2 > 0, \chi_H^2 < 0$. The potential involves hyperbolic functions, lacking periodicity.
3. Phenomenological Analysis:
   • The authors map the model parameters to the $\kappa$-framework, which describes deviations of Higgs couplings from SM predictions.
   • They define two key parameters: $s$ (measuring the scale of symmetry breaking by the doublet) and $\chi$ (measuring the discrepancy between the singlet and doublet couplings).
   • Numerical Methods: Two approaches are used:
     • Global Fitting (GF): Fitting all LHC data (including Yukawa couplings $\kappa_f$) to determine the allowed parameter space ($s, \chi$).
     • Specific Solution (SS): Fixing specific observables (W mass, Higgs mass, $\kappa_V$) to solve for parameter boundaries and project future HL-LHC sensitivities.

3. Key Contributions

• Derivation of Two Distinct Scenarios: The paper rigorously derives the TSS and HSS potentials from the quadratic gravity framework, showing how the sign of non-minimal couplings dictates the functional form of the Higgs potential (periodic vs. exponential).
• Comprehensive $\kappa$-Framework Mapping: The authors provide explicit analytical expressions for all relevant $\kappa$ parameters (couplings to vector bosons, fermions, and self-couplings $\kappa_2, \kappa_3, \kappa_4$) for both TSS and HSS scenarios.
• Constraint Analysis: They systematically confront the model with the latest LHC Run 2 data (ATLAS/CMS), specifically focusing on the tension between Yukawa coupling measurements and the dilaton-dominance hypothesis.
• Vacuum Stability: The authors demonstrate that despite the TSS potential having a negative quartic coupling in its Taylor expansion (which would be unstable in a standard $\phi^4$ theory), the full periodic potential ensures vacuum stability due to higher-order terms ($h^6, h^8$).

4. Results

• Parameter Space Constraints:
  • Global Fit: The model can accommodate current LHC data. However, the SM limit ($s \to 0$) is generally favored.
  • Dilaton Dominance: A region where the Higgs is dilaton-dominant exists but is tightly constrained.
    • In HSS, the entire dilaton-dominant region lies within the 95% CL of current Yukawa coupling constraints.
    • In TSS, the dilaton-dominant region is split: one part is allowed for $s < 0.2$, while a larger region ($s > 0.6$) is currently excluded by Yukawa coupling measurements unless the dilaton couples to fermions (a scenario not assumed in the baseline analysis).
• Self-Couplings ($\kappa_3, \kappa_4$):
  • The model predicts distinct behaviors for Higgs self-couplings. In the dilaton-dominant region of TSS ($1/\sqrt{2} \le s \le 0.87$), the trilinear ($\kappa_3$) and quartic ($\kappa_4$) couplings are predicted to be negative.
  • This is a unique signature compared to the SM, where couplings are positive.
• HL-LHC Prospects:
  • The High-Luminosity LHC (HL-LHC) is projected to have sufficient precision to confirm or rule out the dilaton-dominant hypothesis.
  • Specifically, measuring $\kappa_V$ with precision $\pm 0.017$ and probing the negative self-couplings in the TSS scenario will be decisive.
• Tension with Cosmology: The paper notes a significant tension between parameters required for particle physics (Higgs mass) and those required for cosmological inflation (Starobinsky-like inflation) in the broader SMW framework, differing by 6–7 orders of magnitude.

5. Significance

• Testing the Nature of the Higgs: This work provides a concrete, testable framework to distinguish between a fundamental Higgs and a composite dilaton using precision Higgs measurements, moving beyond theoretical speculation.
• Unification of Gravity and SM: It demonstrates that a Weyl-geometric approach with quadratic gravity can successfully reproduce SM phenomenology while offering a geometric origin for the Higgs mass and symmetry breaking.
• Future Discovery Potential: The identification of negative Higgs self-couplings as a specific signature of the TSS dilaton-dominant scenario offers a clear target for future collider experiments (HL-LHC and 100 TeV colliders).
• Dark Matter Connection: The model naturally predicts a massive Weyl vector boson that could serve as a Higgs-portal dark matter candidate, linking the Higgs sector to dark matter and gravitational wave production from phase transitions.

In conclusion, the paper establishes that while the dilaton-dominant Higgs is a viable interpretation of current data, it is highly constrained. The upcoming precision measurements of Higgs self-couplings and vector boson couplings at the HL-LHC will be the definitive test for this geometric origin of the Higgs boson.