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arxiv: 2401.10507 · v4 · pith:LR4H3GVYnew · submitted 2024-01-19 · 🧮 math.PR · hep-th· math-ph· math.MP

A scaling limit of SU(2) lattice Yang-Mills-Higgs theory

Sourav Chatterjee This is my paper

Pith reviewed 2026-05-24 04:04 UTC · model grok-4.3

classification 🧮 math.PR hep-thmath-phmath.MP
keywords scaling limitlattice Yang-MillsHiggs mechanismmass generationGaussian fieldSU(2) gauge theoryunitary gaugecontinuum limit
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The pith

After gauge fixing and tuned scaling, the SU(2) gauge field converges to a massive Gaussian field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a scaling limit of SU(2) lattice Yang-Mills theory coupled to a Higgs field in any dimension d at least 2. After unitary gauge fixing, the lattice spacing is sent to zero while the gauge coupling goes to zero and the Higgs length goes to infinity with their product fixed proportional to the spacing and the coupling kept extremely small. Under these conditions a stereographic projection of the gauge field converges in law to a massive Gaussian field. The same holds for U(1) theory. This supplies the first rigorous scaling limit for a non-Abelian lattice gauge theory above two dimensions and the first proof that the Higgs mechanism produces mass in such a setting.

Core claim

After unitary gauge fixing and taking the lattice spacing ε to zero while sending the gauge coupling g to zero and the Higgs length α to infinity so that αg equals cε for fixed c and g is of order ε to the power 50d, a stereographic projection of the gauge field converges to a massive Gaussian field. The result holds for both SU(2) and U(1) lattice theories.

What carries the argument

Unitary gauge fixing together with the scaling regime αg = cε and g = O(ε^{50d}), under which the stereographic projection of the gauge field is shown to converge to the massive Gaussian field.

If this is right

  • The Higgs mechanism generates a mass for the gauge field in the scaling limit.
  • This yields the first scaling limit of a non-Abelian lattice Yang-Mills theory in dimension greater than two.
  • An analogous Gaussian limit is obtained for U(1) lattice theory under the same scalings.
  • The question of whether a non-Gaussian scaling limit exists remains open.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The extreme smallness required of the gauge coupling suggests that physically interesting continuum limits may demand different tuning or additional structure.
  • The Gaussian character of the limit indicates that interactions are suppressed in this regime, so extensions to other Higgs representations or potentials could test whether non-Gaussianity appears under milder scalings.
  • The construction supplies a concrete benchmark against which future attempts at non-Gaussian limits in four-dimensional Yang-Mills can be compared.

Load-bearing premise

The gauge coupling must be taken extremely small, of order the lattice spacing to a high power, for the convergence argument to work.

What would settle it

Numerical simulation of the lattice model at the stated scalings that shows the stereographic projection failing to converge to a Gaussian field with the predicted mass.

Figures

Figures reproduced from arXiv: 2401.10507 by Sourav Chatterjee.

Figure 1
Figure 1. Figure 1: A plaquette bounded by four directed edges joined end-to-end. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stereographic projection σ1 : S 1 → R. of x on the plane x1 = 1; see [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A plaquette bounded by four positively oriented edges. [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
read the original abstract

The construction of non-Abelian Euclidean Yang-Mills theories in dimension four, as scaling limits of lattice Yang-Mills theories or otherwise, is a central open question of mathematical physics. This paper takes the following small step towards this goal. In any dimension $d\ge 2$, we construct a scaling limit of $\mathrm{SU}(2)$ lattice Yang-Mills theory coupled to a Higgs field (under the degenerate potential) transforming in the fundamental representation of $\mathrm{SU}(2)$. After unitary gauge fixing and taking the lattice spacing $\varepsilon\to 0$, and simultaneously taking the gauge coupling constant $g\to 0$ and the Higgs length $\alpha\to \infty$ in such a manner that $\alpha g$ is always equal to $c\varepsilon$ for some fixed $c$ and $g= O(\varepsilon^{50d})$, a stereographic projection of the gauge field is shown to converge to a massive Gaussian field. This gives the first construction of a scaling limit of a non-Abelian lattice Yang-Mills theory in a dimension higher than two, as well as the first rigorous proof of mass generation by the Higgs mechanism in such a theory. Analogous results are proved for $\mathrm{U}(1)$ theory as well. The question of constructing a non-Gaussian scaling limit remains open.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs scaling limits for SU(2) and U(1) lattice Yang-Mills-Higgs theories. After unitary gauge fixing, as the lattice spacing ε tends to zero, with the gauge coupling g tending to zero at rate O(ε^{50d}) and the Higgs length α tending to infinity such that αg = cε for fixed c, a stereographic projection of the gauge field is shown to converge in distribution to a massive Gaussian free field. This is claimed to be the first scaling limit construction for non-Abelian lattice Yang-Mills in dimensions d > 2 and the first rigorous proof of mass generation via the Higgs mechanism in this context.

Significance. The result, if correct, represents a notable advance in the rigorous construction of continuum limits for lattice gauge theories. It provides an explicit probabilistic construction of a massive Gaussian field as the scaling limit, thereby demonstrating the Higgs mechanism for mass generation in a controlled setting. The strength lies in the direct analysis of the lattice measure under explicit parameter scaling without reliance on previously fitted quantities. However, the ultra-rapid decay of g confines the result to a regime where non-Gaussian interactions are suppressed, so the limit is Gaussian by design. This does not undermine the claim but highlights that extending to order-one or polynomially small g remains open, as the authors note.

major comments (1)
  1. [Scaling regime (abstract and main theorem)] The choice of g = O(ε^{50d}) is load-bearing for suppressing the non-Gaussian terms from the Wilson action and Higgs kinetic term relative to the mass term induced by the Higgs vev. While this enables the convergence proof, it restricts the result to an ultra-weak coupling window. The manuscript should explicitly state in the introduction or theorem statement whether this exponent is an artifact of the proof technique or necessary for the Gaussian limit to hold.
minor comments (1)
  1. [Notation] The definition of the stereographic projection and the precise form of the massive Gaussian field (including the mass parameter in terms of c) should be clarified with an equation reference for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: The choice of g = O(ε^{50d}) is load-bearing for suppressing the non-Gaussian terms from the Wilson action and Higgs kinetic term relative to the mass term induced by the Higgs vev. While this enables the convergence proof, it restricts the result to an ultra-weak coupling window. The manuscript should explicitly state in the introduction or theorem statement whether this exponent is an artifact of the proof technique or necessary for the Gaussian limit to hold.

    Authors: We agree that the exponent 50d is chosen to make the non-Gaussian contributions from the Wilson action and Higgs kinetic term negligible relative to the mass term generated by the Higgs vev; the estimates in the proof are not claimed to be optimal. The manuscript already states in the abstract that the question of a non-Gaussian scaling limit remains open, indicating that the Gaussian character is tied to the present regime. We will revise the introduction to add an explicit sentence clarifying that the specific power is an artifact of the current estimates and that extending the result to polynomially small or order-one g is left open. revision: yes

Circularity Check

0 steps flagged

No circularity: direct rigorous proof under explicit scalings

full rationale

The paper establishes convergence of a stereographic projection to a massive Gaussian field by direct analysis of the lattice measure under the stated parameter regime (ε→0, α→∞, g→0 with αg=cε fixed and g=O(ε^{50d})). These scalings are chosen as hypotheses to control error terms and ensure the quadratic mass term dominates; the Gaussian limit is derived from the resulting measure, not presupposed or fitted. No self-citations, ansatzes, or renamings of known results appear as load-bearing steps in the provided abstract and description. The construction is self-contained against external benchmarks and does not reduce any claimed prediction to its inputs by definition.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard probabilistic definition of lattice Yang-Mills-Higgs measures and on the validity of unitary gauge fixing; the only tunable elements are the fixed constant c and the exponent 50d chosen to close the estimates.

free parameters (2)
  • c
    Fixed positive constant relating αg to ε
  • scaling exponent = 50d
    The power 50d chosen so g decays as O(ε to that power)
axioms (2)
  • domain assumption Standard existence and basic properties of lattice SU(2) Yang-Mills-Higgs measures with degenerate potential
    The paper begins from the usual lattice formulation in probability.
  • domain assumption Unitary gauge fixing can be applied without altering the scaling limit
    Gauge fixing is invoked to reduce the model before taking limits.

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Forward citations

Cited by 1 Pith paper

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    Proves that SO(3) lattice Yang-Mills theory fails Wilson's confinement criterion at strong coupling.

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