Weak and Higgs physics from the lattice

This paper reports on ongoing lattice investigations using a two-generation lepton setup to explore the spectrum, internal structure, and spectral functions of the weak and Higgs sectors, aiming to reconcile non-perturbative gauge-invariant results with perturbative phenomenology via the Fröhlich-Morchio-Strocchi mechanism and ultimately compare theoretical cross sections with experimental data.

The Gist

The Big Picture: Checking the Blueprint

Imagine the Standard Model of particle physics as the ultimate blueprint for how the universe works. For decades, physicists have used a method called perturbation theory to read this blueprint. Think of this like trying to understand a complex machine by looking at it while it's running smoothly and making small, predictable adjustments. It works incredibly well and has predicted almost everything we see in experiments.

However, there is a philosophical problem. The blueprint relies on the idea that certain symmetries "break" to give particles mass (the Higgs effect). But in the strict laws of quantum mechanics (specifically a rule called Elitzur's theorem), symmetries can't actually break on their own. It's like saying a spinning top stops spinning because it "decided" to fall over, when in reality, it's just wobbling in a way that looks like falling.

The authors of this paper are asking: "What if our blueprint is slightly wrong because we are ignoring the messy, non-linear reality of how these particles actually hold together?"

They are using a supercomputer to build a "digital universe" (a Lattice) to see if the particles behave exactly as the simple blueprint predicts, or if there are hidden, complex structures underneath.

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The Analogy: The "Composite" Particle

To understand their approach, imagine a magnet.

• The Old View (Standard Theory): We treat the magnet as a single, solid, indivisible object. We calculate how it moves and interacts as if it were a smooth marble.
• The New View (This Paper): The scientists suspect the magnet is actually made of billions of tiny, dancing atoms. If you look closely, the "magnet" isn't a smooth marble; it's a swirling cloud of smaller parts.

In this paper, the scientists are treating particles like the W and Z bosons (which carry the weak force) and the Higgs boson not as smooth marbles, but as composite objects—like a cloud of smaller ingredients glued together. They want to see if this "cloud" structure changes how the particles behave.

What Did They Do? (The Simulation)

They built a digital playground using a supercomputer.

1. The Setup: They created a grid (the lattice) representing space and time. On this grid, they placed:
   • Gauge Fields: The "glue" holding things together.
   • Higgs Field: The "mud" that gives particles mass.
   • Two Generations of Leptons: They simulated two types of particles (like electrons and muons) acting as "test subjects."
2. The Goal: They wanted to see if the "composite" nature of these particles creates any surprises that the simple "smooth marble" theory misses.

Key Findings (The "Aha!" Moments)

1. The Mass Hierarchy (Who is heavier?)

In the standard theory, there's a strict rule about which particles are heavier: the W/Z bosons are usually lighter than or equal to the Higgs.

• The Discovery: When they added the "test subject" particles (fermions) to their simulation, they found that this rule can flip. Under certain conditions, the Higgs became lighter than the W/Z boson.
• The Metaphor: Imagine a seesaw where the heavy side is always the same. The scientists found that if you add a specific weight (the fermions) to the other side, the seesaw flips. This suggests the "rules" of the universe might be more flexible than we thought.

2. The Internal Structure (The "Quasi-PDF")

Physicists usually assume particles are point-like dots. But if they are composite clouds, they should have an internal structure, like a planet has a core and an atmosphere.

• The Discovery: They mapped the inside of the W-boson and found it wasn't a simple dot. It had a complex, "cloud-like" internal structure.
• The Metaphor: If you shine a light through a glass marble, it looks uniform. If you shine it through a cloud of dust, the light scatters in complex patterns. The scientists saw the "dust cloud" pattern, proving the particle has an internal life.

3. Scattering (The Collision Course)

They are also learning how to calculate what happens when these particles crash into each other (scattering).

• The Challenge: Current methods are like trying to predict a car crash by only looking at the cars before they hit. The new methods allow them to look at the "debris" and the complex interaction during the crash.
• The Goal: They want to calculate the "cross-section" (the probability of a crash happening) with extreme precision to see if it matches the simple theory or if the "composite" nature changes the odds.

Why Does This Matter?

This research is a "reality check" for the Standard Model.

• If the simple theory is perfect: Then we are safe, and our current understanding is complete.
• If the "composite" effects are real: It means there are hidden layers of physics we haven't seen yet. It could explain why particles have the masses they do, or even suggest that heavier particles (like the muon) are just "excited versions" of lighter ones (like the electron), similar to how a guitar string can vibrate in a low note or a high note.

The Bottom Line

The scientists are building a digital microscope to look at the fundamental building blocks of the universe. They are checking if the "smooth marble" view of physics is actually a "swirling cloud" of complex interactions.

If they find differences, it won't break physics; it will upgrade it. It will help us understand if the universe is simpler than we think, or if it's a much more intricate, composite masterpiece than we ever imagined. This is crucial for future experiments at the Large Hadron Collider (LHC), ensuring that when we see something new, we know if it's a glitch in our math or a genuine new discovery.

Technical Summary

1. Problem Statement and Motivation

The paper addresses a fundamental conceptual tension in the Standard Model (SM) of particle physics:

• The Gauge Invariance Paradox: Perturbative electroweak phenomenology relies on the Brout-Englert-Higgs (BEH) mechanism, which assumes spontaneous symmetry breaking (SSB). However, Elitzur's theorem states that local gauge symmetries cannot break spontaneously. Consequently, in a manifestly gauge-invariant lattice formulation, quantities like the Higgs vacuum expectation value (VEV) vanish without gauge fixing.
• The FMS Mechanism: The Fröhlich-Morchio-Strocchi (FMS) mechanism resolves this by mapping the perturbative spectrum to physical, gauge-invariant composite operators (similar to hadrons in QCD). While the leading term in this expansion recovers standard perturbation theory, subleading terms (augmented perturbation theory or APT) exist.
• The Open Question: These subleading terms are kinematically suppressed but potentially significant. It is unclear if they can be fully accounted for by APT or if they represent genuine non-perturbative effects that deviate from standard predictions. Furthermore, there is a hypothesis that higher-generation fermions (muons, taus) might be excited states of the first generation, a non-perturbative phenomenon requiring lattice investigation.
• Goal: To perform a systematic, non-perturbative lattice study of the weak and Higgs sector with dynamical fermions to test the FMS mechanism, explore the internal structure of composite states, and calculate scattering cross sections.

2. Methodology

The authors employ a lattice field theory approach using the HiRep code with Hybrid Monte Carlo (HMC) algorithms.

• Theoretical Setup:
  • Gauge Group: $SU(2)$ Wilson gauge action.
  • Scalar Sector: A complex doublet scalar field (Higgs).
  • Fermion Sector: Two generations of unimproved, degenerate Wilson fermions coupled vectorially to the gauge-Higgs system. These represent the 1st (electron/neutrino) and 2nd (muon/neutrino) lepton doublets.
  • Symmetry: The setup possesses an exact symmetry between lepton generations due to degenerate masses and a global custodial $SU(2)$ symmetry.
• Observables and Techniques:
  • Spectrum Extraction: Using operators corresponding to physical states:
    • $O_{Higgs} = \text{tr}(X^\dagger X)$ (Scalar singlet).
    • $O_W = \text{tr}(\tau^a X^\dagger D_\mu X)$ (Vector triplet).
    • $\Psi = X^\dagger \psi$ (Fermionic doublet).
  • Spectral Analysis: Utilizing the HLT (High-Light-Tensor) approach to extract smeared spectral density functions, allowing the distinction between stable particles and unstable resonances.
  • Quasi-PDFs: Computing parton distribution functions (PDFs) for composite states using the ratio of three-point and two-point correlation functions to probe internal structure.
  • Scattering Cross Sections: Moving beyond the Lüscher formalism (limited to elastic thresholds) by employing a method based on spectral densities (from Ref. [23]) to compute exclusive scattering amplitudes directly from four-point correlation functions.

3. Key Contributions

1. Inclusion of Dynamical Fermions: This work extends previous lattice studies of the pure bosonic sector and quenched fermions to a setup with two generations of dynamical fermions, providing a more realistic proxy for the SM lepton sector.
2. Validation of FMS/APT: The study provides a non-perturbative check of the FMS mechanism by comparing lattice spectral densities with Next-to-Leading Order (NLO) APT predictions.
3. Exploration of Excited States: The authors investigate the hypothesis that higher-generation leptons are excited states of the first generation, utilizing variational analysis and spectral methods.
4. Internal Structure Analysis: First lattice results on the Quasi-PDFs of the W-boson, treating it as a composite object rather than an elementary particle.
5. Cross Section Calculation: A novel application of spectral density methods to compute vector boson scattering cross sections below the inelastic threshold, offering a path to compare with experimental data without relying solely on perturbation theory.

4. Key Results

• Mass Hierarchies and Phase Structure:
  • The introduction of fermions has only a mild effect on the mass hierarchy between the Higgs and W/Z bosons.
  • Reflection Positivity: The onset of unphysical correlators (violations of reflection positivity) for Wilson fermions was observed at $\kappa_F \approx 0.14$, lower than the standard $0.15$, likely due to the scalar field presence.
  • Hierarchy Flipping: In specific parameter regions (QCD-like), lowering the fermion mass can flip the mass hierarchy (making the Higgs heavier than the W/Z), challenging the universality of the standard BEH hierarchy.
• Spectral Densities:
  • The spectral density of the composite scalar operator is compatible with NLO APT results.
  • Clear transitions were observed between stable Higgs states (thin peaks) and unstable resonances (threshold behavior).
• Quasi-PDFs:
  • The W-boson exhibits a non-trivial internal structure. The quasi-PDF shows weak dependence on momentum ($P_3$) but deviates significantly from the trivial Dirac delta peak expected for an elementary particle, confirming its composite nature.
• Scattering:
  • Preliminary results for vector boson scattering cross sections were obtained using the spectral density method. The signal-to-noise ratio suggests that precise cross-section calculations are viable with current simulation accuracy.

5. Significance and Outlook

• Testing Quantum Field Theory: This research is crucial for testing QFT beyond the perturbative regime. If discrepancies between lattice results and standard perturbation theory are found, they could signal new physics or confirm the necessity of APT.
• Beyond Standard Model (BSM) Implications: The framework is essential for model building. If higher generations are confirmed as excited states, it would revolutionize the understanding of mass hierarchies and mixing angles (CKM/PMNS).
• Experimental Relevance: By calculating cross sections non-perturbatively, the authors aim to prevent the misinterpretation of standard model effects as BSM signals at future colliders (e.g., High-Luminosity LHC).
• Future Work: The authors plan to increase the precision of spectral densities, extract scattering amplitudes using alternative methods (e.g., Lüscher formalism for cross-checks), and eventually include dynamical Yukawa couplings to fully explore the mixing of excited leptonic states.

In summary, this paper establishes a robust, non-perturbative lattice framework for the weak sector, demonstrating that composite operators successfully reproduce the SM spectrum while revealing rich internal structures and potential deviations from standard perturbative expectations.