Varieties of electrically charged physical states in SU(2)$\times$U(1) lattice gauge Higgs theory

This paper investigates a quenched SU(2)$\times$U(1) lattice gauge Higgs theory coupled to a static vector-like fermion to identify new types of gauge-invariant electrically charged and neutral states that differ from previous constructions in their field dressing, revealing that while neutral states are significantly lighter, the charged spectrum contains at least two distinct particle states with different masses.

The Gist

Imagine you are trying to describe a person in a crowded, chaotic room. In the world of particle physics, this "person" is an electrically charged particle (like an electron), and the "crowd" is a sea of invisible force fields (the Higgs field and gauge fields).

For decades, physicists thought they knew exactly how to describe these particles. They believed that to make a charged particle "real" and observable, you had to dress it up in a specific way, like putting a specific coat on a mannequin. This paper by Jeff Greensite suggests that we missed a whole new wardrobe. There isn't just one way to dress these particles; there are at least two distinct styles, and surprisingly, one of these styles might hold the key to why particles have different "families" or generations.

Here is the breakdown of the paper using simple analogies:

1. The Problem: You Can't Have a Naked Charge

In physics, a charged particle cannot exist in isolation. It must be surrounded by a "cloud" of fields to be a physical, observable object. Think of it like a celebrity trying to walk through a paparazzi-filled street. They can't just walk out naked; they need a security detail (the field) to shield them and make them a coherent entity.

In the past, physicists (like 't Hooft and Banks) figured out one way to build this security detail. They created a "Type I" particle. It's like a celebrity wearing a standard, well-known suit. Everyone recognized it as "The Electron."

2. The Discovery: The "Type II" Outfit

Greensite argues that there is another, completely different way to dress the particle. He calls this "Type II."

• The Analogy: Imagine the celebrity again. The "Type I" outfit is a classic tuxedo. The "Type II" outfit is a high-tech, futuristic jumpsuit.
• The Twist: Both outfits result in a celebrity who is still "charged" (they still have the same electric charge). To an outsider looking from far away, they might look similar. But if you look closely at how they interact with the crowd (the global symmetry of the universe), they behave completely differently.
• The Result: These two outfits create two different types of physical states that are "orthogonal." In math terms, this means they are like two different songs that share the same melody but have completely different harmonies. They don't mix; they are distinct entities.

3. The Experiment: The Lattice Simulation

Since we can't build a giant particle accelerator in a garage to test this, Greensite used a computer simulation.

• The Setup: He built a digital "grid" (a lattice) representing space. He placed a heavy, static particle (like a frozen statue) on this grid and let the digital fields dance around it.
• The Test: He tried to see how these "statues" moved and what energy they had.
  • Neutral Particles (The "Neutrinos"): He found that the neutral particles (which don't have electric charge) were very light and easy to move. They were like a feather floating in the wind.
  • Charged Particles (The "Electrons"): The charged particles were much heavier, like a brick. This makes sense; dragging a charge through the field takes a lot of energy.

4. The Big Surprise: The Spectrum of Mass

Here is the most exciting part. When Greensite looked at the charged particles, he didn't just find one "brick." He found a spectrum.

• The Analogy: Imagine you have a guitar string. You can pluck it to get a low note (the ground state). But you can also pluck it harder or in a different way to get a higher note (an excited state).
• The Finding: Greensite found that the charged particles have a "low note" (a lighter mass) and a "high note" (a heavier mass).
  • The "Type I" charged particles had a specific mass.
  • The "Type II" charged particles had a similar mass for their "low note," but they also had a distinct "high note" (an excited state) that was heavier.

This suggests that what we think of as a single particle might actually be a family of particles with different energy levels, much like an atom has different electron shells.

5. Why Does This Matter?

The Standard Model of physics (our current best theory) has a weird mystery: Why do we have three "generations" of particles?

• Generation 1: The electron (light, stable).
• Generation 2: The muon (heavier, unstable).
• Generation 3: The tau (very heavy, unstable).

We don't know why these exist. They seem to be copies of each other with different weights.

Greensite's paper suggests a possible explanation: Maybe these aren't different particles at all, but different "excited states" of the same fundamental object. Just like a guitar string can make a low note or a high note, a fundamental particle might have a "Type I" ground state and a "Type II" excited state.

Summary

• Old View: There is only one way to dress a charged particle.
• New View: There are at least two distinct ways (Type I and Type II) to dress them, creating different physical states that are invisible to old methods.
• The Result: Charged particles seem to have a "spectrum" of masses (light and heavy versions), while neutral particles stay light.
• The Hope: This could explain why nature has multiple copies of particles (generations), suggesting they are just different "notes" played on the same cosmic instrument.

In short, the universe might be more musical than we thought, with particles having different "octaves" based on how they are dressed by the invisible fields around them.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in non-Abelian gauge theories with Higgs fields: What is the complete spectrum of physical, locally gauge-invariant states in the Higgs phase of an $SU(2) \times U(1)$ theory?

While the construction of physical charged and neutral states (analogous to electrons and neutrinos) was established decades ago by 't Hooft, Banks, Rabinovici, Fröhlich, Morchio, and Strocchi, the author argues that these early constructions are incomplete. Specifically:

• Existing literature does not exhaust all possible ways to "dress" a static matter source with gauge and Higgs fields to form a physical state.
• It is unclear if static fermions in the Higgs phase possess an excitation spectrum (similar to gluelumps in QCD or atoms in quantum mechanics) beyond the ground state.
• The orthogonality between different types of charged/neutral states in the Higgs phase is subtle because the global center symmetry (which usually guarantees orthogonality in the confined phase) is spontaneously broken in the Higgs phase.

2. Methodology

The author employs Lattice Gauge Theory to investigate these questions non-perturbatively.

• Model: A quenched $SU(2) \times U(1)$ gauge Higgs theory on a lattice.
  • Fields: $SU(2)$ links ($U$), $U(1)$ links ($V$), a Higgs scalar field ($\phi$) in the fundamental representation, and a static, vector-like fermion ($\psi$) in the same representation as the Higgs.
  • Approximation: The fermion is treated as static (infinite mass limit) using a hopping parameter expansion ($\kappa \to 0$), and fermion loops are dropped (quenched approximation).
• Construction of Physical States:
  • Physical states must be locally gauge invariant.
  • The author defines "pseudomatter" fields: non-local functionals of the gauge fields that transform covariantly like matter fields but are invariant under the global center subgroup.
  • Two distinct classes of physical operators are constructed for both charged and neutral sectors:
    • Type I: Uses a pseudomatter field $\rho_n(x; V)$ dependent only on the $U(1)$ field.
    • Type II: Uses a pseudomatter field $\xi_n(x; eU)$ dependent on the full $SU(2) \times U(1)$ product field ($eU = UV$).
  • Orthogonality: The paper demonstrates that while Type I and Type II states may share the same electric charge, they transform differently under a specific global subgroup of the electromagnetic gauge group ($U_{EM}(1)$), rendering them orthogonal.
• Numerical Simulation:
  • Simulations are performed on lattices of varying sizes (up to $14^3 \times 60$) with couplings chosen to ensure the system is in the Higgs phase (verified via a first-order transition at $\gamma \approx 2.02$).
  • Observables: Time correlation functions of particle-antiparticle pairs are calculated. Since single charged particles cannot propagate in a finite periodic volume, the study focuses on well-separated static fermion-antifermion pairs.
  • Analysis: Masses and energy levels are extracted using single-exponential fits and the Generalized Eigenvalue Method (GEVM) to resolve excited states.

3. Key Contributions

1. Identification of Two Distinct State Families: The paper rigorously defines and distinguishes Type I and Type II physical states.
   • Type I: The matter source is dressed by a $U(1)$-dependent pseudomatter field.
   • Type II: The matter source is dressed by a full $SU(2) \times U(1)$-dependent pseudomatter field.
   • Crucially, these two types are orthogonal to each other despite having the same local electric charge, distinguished by their transformation properties under the global electromagnetic subgroup.
2. Discovery of an Excitation Spectrum: The study provides evidence that static charged fermions in the Higgs phase are not single-particle states but possess a discrete spectrum of excitations (analogous to gluelumps).
3. Clarification of Orthogonality in the Higgs Phase: The paper resolves how charged and neutral states remain orthogonal in the Higgs phase (where global center symmetry is broken) by relying on the transformation properties of the pseudomatter dressing under the unbroken global electromagnetic subgroup.

4. Results

The numerical results, derived from time correlators on the lattice, yield the following:

• Mass Hierarchy:
  • Neutral States: Both Type I and Type II neutral states have approximately the same mass ($E \approx 0.148$ in lattice units).
  • Charged States: Both Type I and Type II charged states are significantly heavier than neutral states ($E \approx 0.28$ in lattice units).
• Excitation Spectrum:
  • Charged Spectrum: There is clear evidence of at least two distinct energy levels for charged particles.
    • Ground state (Type I and Type II $n=2$): $E \approx 0.277 - 0.281$.
    • Excited state (Type II $n=10$ and Type I $n=8$): $E \approx 0.33 - 0.34$.
  • The energy difference between the ground and excited charged states ($\Delta E \approx 0.06$) is too small to be explained by the emission of a single photon (which would have a minimum energy of $\approx 0.52$ on the lattice), confirming these are intrinsic particle excitations.
  • Neutral Spectrum: No convincing evidence of excited neutral states was found; Type I and Type II neutral correlators overlap perfectly, suggesting a degenerate ground state with no low-lying excitations.
• Degeneracy: The ground state masses and the first excited state masses appear to be degenerate between Type I and Type II charged states, despite their different transformation properties.

5. Significance

• Theoretical Completeness: The work expands the theoretical understanding of the particle spectrum in gauge-Higgs theories, showing that the "electron" in such a theory is not a unique state but part of a multiplet of orthogonal states with potentially different excitation spectra.
• Analogy to Gluelumps: It establishes a strong parallel between static quarks in QCD (gluelumps) and static fermions in the electroweak Higgs phase, suggesting that composite nature leads to rich excitation spectra even in the Higgs phase.
• Implications for the Standard Model: While the current study uses vector-like fermions and quenched approximations (not the realistic chiral Standard Model), the results suggest that if chiral fermions could be simulated, there might be a correspondence between these distinct charged/neutral state types and the generations of fermions (e.g., electron vs. muon vs. tau) or excited states of leptons.
• Phase Transition Understanding: The paper reinforces the view that the Higgs phase is distinct from the symmetric phase not just by local order parameters, but by non-local properties (like the existence of specific orthogonal state families and the absence of "separation of charge" confinement).

In conclusion, Greensite demonstrates that the spectrum of matter in the $SU(2) \times U(1)$ Higgs phase is richer than previously thought, featuring distinct orthogonal families of charged and neutral states, with charged states exhibiting a discrete excitation spectrum.