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Dynamical Higgs potentials with a landscape

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J.L.F. Barbon, C. Hoyos

High Energy Physics - Theory · Submitted 28 Feb 2006 · 20 references

Pith's verdict: Solid derivation of the N-ality band rule; the central result holds, but the residual-degeneracy claim in the Conclusions overcounts.

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arXiv IDhep-th/0602285 (version 1)
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CitationC. Hoyos, J.L.F. Barbon. "Dynamical Higgs potentials with a landscape." arXiv:hep-th/0602285 (2006). https://pith.science/paper/D7DDAVXS
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The paper

The paper claims

A one-loop Higgs potential on a torus can generate a landscape of vacuum energies without fluxes or branes.

The reading

This paper aims to show that a simple higher-dimensional gauge theory, with no fluxes and no branes, can generate a landscape-like effective potential. For an SU(N) gauge theory on $\mathbb{R}^d \times T^n$, the flat Wilson lines along the torus give adjoint Higgs fields whose one-loop potential has $N^n$ degenerate vacua with unbroken gauge symmetry. Adding matter in a representation $R$ with nonzero $N$-ality (its charge under the center $\mathbb{Z}_N$) lifts these $N^n$ vacua to different energies, spreading them into a band of width $\mathrm{dim}(R)\,\Delta$ with level spacings as small as $\mathrm{dim}(R)(\eta_R/N)^2\,\Delta$ at the bottom. Because $N$ can be large and the matter mass $M$ can be tuned so that $ML\gg 1$, the band can become an almost continuous scan of vacuum energies. The point is that a discretuum of cosmological constants can arise from group theory alone, not from a complicated flux landscape.

What carries the argument

The load-bearing object is the one-loop effective potential on the toron-valley moduli space $\mathcal{M}=(T_C^r)^n/W$, expressed through the scalar function $V_0(\vec{\xi}) = \frac{\Gamma((d+n)/2)}{\pi^{(d+n)/2} L^d} \sum_{\vec{\ell}\neq 0} \frac{\sin^2(\tfrac12 \vec{\ell}\cdot\vec{\xi})}{(\vec{\ell}^2)^{(d+n)/2}}$, with a massive generalization involving modified Bessel functions. This function turns the holonomy $\vec{\varphi}$ into frequency shifts of the Kaluza–Klein oscillators; evaluating it on the matter weights at the unbroken vacuum points $\vec{\varphi}_0$ converts the slight misalignment of the weight basis into the band formula for vacuum energies. The same function, with fermion sign factors, controls the orientifold example in which the averaged planar potential cancels to $O(N)$.

Extended reading notes

Core claim

The central discovery is a quantitative band formula for the lifted vacua. In the pure Yang–Mills theory on $\mathbb{R}^d \times T^n$, the one-loop potential vanishes at the lattice of points $\varphi_0 = 4\pi \sum_i \vec{n}_i\, \nu_i$, where $\nu_i$ are the weights of the defining fundamental representation of $SU(N)$; these $N^n$ points are the images of the origin under the $(\mathbb{Z}_N)^n$ group of central conjugations and are locally identical copies of the unbroken vacuum. When matter in a representation $R$ of dimension $\mathrm{dim}(R)$ and $N$-ality $\eta_R$ is integrated out, each of these points acquires a vacuum energy $\Lambda(\vec{n}_i) = \sum_R N_R (-1)^{F_R}\, \mathrm{dim}(R)\, V_{M_R}(2\pi \vec{K}_R/N)$ with $\vec{K}_R = \eta_R \sum_k \vec{n}_k$. Provided $ML$ is sufficiently large, the $N^n$ vacua are spread into a band of width $\Delta\Lambda = \mathrm{dim}(R)\,\Delta$ and bottom-of-band spacings $\delta\Lambda = \mathrm{dim}(R)(\eta_R/N)^2 \Delta$, where $\Delta \sim (ML)^{(d+n-1)/2} L^{-d} e^{-ML}$. The root cause is the slight non-orthogonality of the fundamental weights, $4\pi\,\nu_i\cdot\nu_j = 2\pi\delta_{ij} - 2\pi/N$: the Kronecker-delta term drops out by periodicity of the scalar potential, and the residual $1/N$ term makes the lifted energies depend on the vacuum through $\vec{K}$ modulo $N$.

Load-bearing premise

The load-bearing assumption is that the one-loop effective potential is reliable: the effective dimensionless gauge coupling $\lambda = g^2(L) N L^{4-d}$ must be small enough that the ultraviolet completion at the scale $\Lambda_{\rm UV} = \lambda^{-1/(d+n-4)}L$ is far away and its corrections are suppressed.

Editorial extensions

If this is right

  • With matter in the fundamental representation ($\mathrm{dim}(R)=N$, $\eta_R=1$), the lifted vacuum energies form a band whose low end has spacings of order $\Delta/N^2$ and whose top reaches $O(N)\Delta$, so large $N$ gives an increasingly dense discretuum.
  • For rank-two antisymmetric or symmetric representations, a quasi-continuous band requires the mass hierarchy $ML\gg 1$ to make $\Delta$ exponentially small; without that hierarchy the lifting is too coarse.
  • Representations with zero $N$-ality do not split the energies of the $N^n$ vacua, although they still change the local physics near those points, so the center charge of the representation controls the scanning.
  • Varying the matter mass $M$ tunes the overall quenching factor $e^{-ML}$: heavier matter makes the band finer and can preserve the local minima against being washed out by the matter potential.
  • In the orientifold planar-equivalence example, the generic potential scale is $O(N^2)$ but the average over the toron valley is only $O(N)$, so the $O(N^2)$ peaks and valleys remain narrow rather than forming large plateaus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the band formula survives in a UV-complete setting, the density of vacuum energies is fixed by the representation content; counting how many of the $N^n$ minima fall in a given energy interval would give a concrete 'density of states' for this discretuum as a function of $N$ and $ML$.
  • The same misalignment mechanism should operate for Wilson-line moduli on other compact manifolds, wherever the basic weights are slightly non-orthogonal; this geometric generalization is not explored in the paper.
  • Because the band width scales as $\mathrm{dim}(R)\Delta$ while the bottom spacing is suppressed by $(\eta_R/N)^2$, a combination of a large-$N$ fundamental plus a heavily suppressed higher representation could engineer an almost continuous low-energy spectrum; the paper does not perform this optimization, but its formula makes it explicit.
Original abstract

We consider one-loop effective potentials for adjoint Higgs fields that originate from flat holonomies in toroidal compactification of gauge theories. We show that such potentials are "landscape-like" for large gauge groups and generic non-supersymmetric matter representations. In particular, there is a large number of vacua with similar local properties, scanning a broad band of vacuum energies.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,901 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider one-loop effective potentials for adjoint Higgs fields that originate from flat holonomies in toroidal compactification of gauge theories. We show that such potentials are "landscape-like" for large gauge groups and generic non-supersymmetric matter representations. In particular, there is a large number of vacua with similar local properties, scanning a broad band of vacuum energies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider one-loop effective potentials for adjoint Higgs fields that originate from flat holonomies in toroidal compactification of gauge theories. contribution 0.71 0.43 0.04 asserted 0.83
s1 We show that such potentials are "landscape-like" for large gauge groups and generic non-supersymmetric matter representations. result 0.95 0.93 0.70 asserted 1.00
s2 In particular, there is a large number of vacua with similar local properties, scanning a broad band of vacuum energies. result 0.94 0.83 0.31 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How the literature uses it

6 arXiv papers cite this in 11 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Builds on it 1 paper
Compares with it 1 paper
Cites it as background 1 paper
  • Weak coupling calculations on $S^3 ×S^1$ have been used for determining the phase structure of various supersymmetric and non-supersymmetric gauge theories [ref] , treating each case independently.

    Confinement, Screening and the Center on S^3 x S^1 · Introduction · 0.87

Discussion

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