Constrained Hybrid Monte Carlo algorithms for gauge-Higgs models

Francesco Knechtli; Michael G\"unther; Roman H\"ollwieser

REVIEW 3 major objections 6 minor 36 references

Constrained Hybrid Monte Carlo algorithms for gauge-Higgs models

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Measuring the constraint force itself, rather than a histogram of field values, resolves the full effective Higgs potential on the lattice at volume-independent precision.

desk verdict Solid methods paper: the Lagrange-multiplier observable is clean and the 4D implementation is verified, but the 5D gauge-link extension lacks the explicit reversibility check that would make the central claim airtight. read the letter →

arxiv 1908.10950 v2 pith:X4XHXTO7 submitted 2019-08-07 hep-lat cs.DS

Michael Günther , Roman Höllwieser , Francesco Knechtli This is my paper · ORCID

classification hep-latcs.DS

keywords constrainedhybridMonteCarloRattlealgorithmconstrainteffectivepotentialgauge-HiggsmodelsAbelian-Higgsmodelfive-dimensionalSU(2)gaugetheoryPolyakovlooplattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops constrained Hybrid Monte Carlo algorithms that keep a chosen Higgs observable fixed during the simulation, and shows that the derivative of the resulting constraint effective potential is directly given by the expectation value of the Lagrange multiplier enforcing the constraint. This turns potential measurement from a histogram of field fluctuations into a local observable, so the potential can be computed over the entire allowed range of the Higgs variable instead of only a narrow region around its unconstrained expectation value. The precision of the new observable does not deteriorate when the lattice volume is increased, which is what makes infinite-volume and mass determinations feasible. The method is implemented for the 4D Abelian-Higgs model and for 5D SU(2) gauge theory on a torus and on an orbifold, and in the Abelian-Higgs case the measured potential agrees with the one-loop continuum Higgs potential in unitary gauge. A sympathetic reader should take the paper's central claim to be that this observable is a practical, quantitative route to effective potentials in gauge-Higgs theories.

What carries the argument

The load-bearing object is the Rattle algorithm, an extension of the leap-frog/St\"ormer-Verlet method that adds a half step for the momenta so that fields and momenta are available at the same integration time; this makes it possible to impose the hidden constraint and solve for the two Lagrange multipliers $\lambda^{(1)}$ and $\lambda^{(2)}$. For the 4D Abelian-Higgs model the constraint is the spatial average of $\phi^\dagger \phi$, and $\lambda^{(1)}$ can be written in closed form (taking the minus branch); for the 5D SU(2) torus and orbifold cases the constraint is built from Polyakov loops, and $\lambda^{(1)}$ is found by a Secant iteration seeded by a Taylor expansion of the exponential link update. The observable $U'_\Omega(\Phi) = -\langle\lambda^{(1)}\rangle_\Phi/\Omega$ carries the actual physics: $\lambda^{(1)}$ is the force that holds the Higgs observable fixed, and its average is the derivative of the potential. Auxiliary machinery includes orthogonal projection of the initial random momenta onto the hidden-constraint surface and a continuum-form rewriting of $\lambda^{(1)}$ that removes the dependence on the unphysical integration step size.

What would settle it

Run a single Rattle trajectory for the 5D torus model at $\beta_4 = \beta_5 = 1.66$ with the Polyakov-loop constraint, then reverse it step by step with stepsize $-h$; if the SU(2) link variables do not return to their initial values up to round-off, the constrained update is not reversible and the measured derivative $-\langle\lambda^{(1)}\rangle/\Omega$ would not be the derivative of a well-defined potential.

Watch

Extended reading notes

Core claim

Starting from the constrained path integral, the paper derives $U'_\Omega(\Phi) = -\frac{1}{\Omega} \langle \lambda^{(1)} \rangle_\Phi$, where $\lambda^{(1)}$ is the Lagrange multiplier that holds the averaged Higgs field at value $\Phi$. The central discovery is that this identity can be used as an observable: during a constrained HMC simulation one measures the multiplier and obtains the derivative of the constraint effective potential directly, with no histogram binning and no need to fit a distribution. To make the identity usable for gauge theories, the authors extend the Rattle algorithm to constrained Hamiltonian systems whose equations of motion are non-canonical for SU(2) gauge links, solving for the multipliers at each molecular-dynamics step (first $\lambda^{(1)}$ from the constraint, then $\lambda^{(2)}$ from the hidden constraint). They demonstrate that the continuum limit in integration step size exists and that the resulting potentials reproduce the histogram result where the latter is available, while extending it over the full domain of the Higgs field and keeping errors roughly constant as the volume grows. They also verify the 4D result against the one-loop Higgs potential in unitary gauge, extracting Higgs masses consistent with two-point function determinations.

Load-bearing premise

Everything rests on the constrained molecular-dynamics update being exactly time-reversible in the five-dimensional gauge cases, but the paper demonstrates reversibility numerically only for the four-dimensional Abelian-Higgs model and relies on the general Rattle theory for the SU(2) updates.

Editorial extensions

If this is right

In the 4D Abelian-Higgs model, the derivative of the constraint effective potential can be measured over the full range of the Higgs variable, including regions where the potential diverges ($\Phi \to 0$), whereas histograms only cover a neighbourhood of the expectation value.
The statistical error of the derivative observable stays approximately constant as the lattice volume grows, so volume extrapolation to the infinite-volume effective potential becomes practical; the Higgs mass can then be read off from the curvature at the minimum.
For the 5D SU(2) torus model the method reproduces the Mexican-hat form of the potential for the Polyakov-loop Higgs field and confirms the two degenerate minima of $\langle \operatorname{Tr} P \rangle$, which the histogram method cannot resolve.
For the 5D orbifold model, the constrained potential can be measured across the whole interval $-1 < \operatorname{Tr} P / 2 < 1$, giving access to the confining, Higgs, and hybrid phases from one observable.
The same constrained algorithms can be applied to effective Polyakov loop actions, where individual Polyakov lines are constrained locally rather than globally, providing a route to finite-temperature and finite-density QCD effective potentials.

Reading between the lines

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

Because the derivative is a local expectation value rather than a histogram slope, one could in principle join measurements from simulations at different $\Phi$ values into a continuous potential with controlled interpolation error; the paper does not itself perform such a reconstruction.
The reversibility argument for the 5D non-canonical gauge-link updates is inherited from the general Rattle theory; an explicit numerical reversibility test on the torus or orbifold would settle whether the hidden constraint is preserved exactly there, and would be a natural companion check.
A direct extension suggested by the method is to constrain other order parameters, such as topological charge or the density of a conserved current, and read off the corresponding effective potential from the same Lagrange-multiplier identity.
If the volume-independence of the precision persists on larger lattices, the main cost of the method shifts to the Secant solve for the multiplier, making algorithmic tuning of that solve (or a closed-form approximation) the practical bottleneck; the paper does not analyze this scaling.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth reading. This is the first constrained-HMC implementation for gauge fields, and the core observable, the derivative of the constrained effective potential as minus the expectation value of the first Lagrange multiplier, is derived cleanly in Eqs. (2.3)-(2.5). The 4D Abelian-Higgs implementation is the strongest part: the authors check time-reversibility, verify det J = 1, show the h -> 0 limit of the multiplier observable, and demonstrate volume-independent precision against the histogram method in Table 2. That alone is a solid methods result.

The genuinely new content is the adaptation of Rattle to non-canonical gauge-link dynamics with composite Polyakov-loop constraints, for both the 5D torus and the orbifold. That is a real extension, and the 5D potentials shown are plausible. The one-loop comparison in unitary gauge is useful, though the fit of m_H to the measured derivative is partly self-fulfilling; the extracted mass does agree with the correlator mass m_H,R in Table 1, so there is independent support.

The soft spot is the 5D section. Section 6 says time-reversibility was numerically tested, but the explicit checks shown are for the 4D case only. For the 5D updates, lambda^(1) is determined by a Secant root-find on a transcendental constraint. If the selected root depends on integration direction or multiple roots exist, the map may not invert properly, and the Rattle literature guarantees for standard Hamiltonian systems do not automatically transfer to the non-canonical gauge-link form. This is an evidentiary gap, not a demonstrated failure; the 4D results and volume-scaling table stand independently. A minor related point: the orbifold constraint is a proxy for the true Higgs field, and the authors acknowledge this.

Citation pattern is fair. Kuti-Shen and Hairer-Lubich-Wanner are properly credited, and the novelty claim is accurate: the combination of the multiplier observable with Rattle on gauge links is not in the cited literature.

This paper deserves a serious referee. I would recommend conditional acceptance with a request for an explicit numerical reversibility and volume-preservation check on the 5D updates, or at least a precise statement of which tests were run. The central claim is likely correct; the support just needs to reach the 5D cases.

Referee Report

3 major / 6 minor

Summary. The paper develops constrained Hybrid Monte Carlo algorithms for gauge-Higgs models using an extension of the Rattle integrator to non-linear and gauge-field constraints. The central theoretical result is Eq. (2.5), which expresses the derivative of the constraint effective potential as the expectation value of the first Lagrange multiplier: U'_Ω(Φ) = -⟨λ^(1)⟩_Φ/Ω. The method is implemented for the 4D Abelian-Higgs model, where reversibility and det J = 1 are tested numerically, and effective potentials are compared against histogram results and a one-loop perturbative potential. The same framework is then adapted to 5D SU(2) gauge theory on the torus (with constraints on ⟨TrP⟩ and ⟨(TrP)^2⟩) and on the orbifold (constraint on ⟨TrP⟩), with first results for the constraint effective potentials. The paper claims, for the first time, a solution to constrained HMC for theories with gauge fields.

Significance. If the claims hold, the method is a notable technical advance: it allows precise determination of the constraint effective potential over the full range of the Higgs/Polyakov-loop variable, with statistical errors that do not grow with volume, unlike histogram methods. The derivation of Eq. (2.5) is clean and self-contained, and the 4D numerical tests (reversibility, det J = 1, h→0 limit, agreement with correlator masses) are convincing. The 5D extension to non-canonical gauge-link updates is novel and potentially important for gauge-Higgs unification and effective Polyakov-loop actions. However, the 5D results currently lack explicit reversibility and volume-preservation tests, and the one-loop comparison is partly self-fulfilling because m_H is fit. These issues are local and fixable rather than fatal.

major comments (3)

[Secs. 4, 5 and Appendices A.2-A.4] The 5D Rattle updates for SU(2) gauge links are asserted to preserve time-reversibility and volume preservation by citing the general Rattle theory of Refs. [24,25]. The numerical tests reported in Sec. 3.1, however, cover only the 4D Abelian-Higgs model, whose updates are canonical. The 5D updates are non-canonical because the link update is multiplicative (U_{n+1} = exp(h π_{n+1/2}) U_n) and λ^(1) is solved by a Secant root-find on a transcendental equation (Sec. 4.2c and A.2.2c). The unqualified statement in Sec. 6 that the algorithms were 'numerically tested for time-reversibility and volume preservation' is therefore not supported for the 5D cases. Please add explicit reversibility and det J tests for the 5D torus (both constraints) and orbifold updates, or supply a proof that the Rattle geometric properties extend to the Lie-group update used here.
[Sec. 3.3, Eq. (3.8), Table 1] The statement that the constraint effective potential agrees well with the one-loop Higgs potential is weakened by the fact that m_H is a fit parameter: the derivative U'_1loop(Φ) is fitted to the measured U'_Ω(Φ), so the agreement is partly self-fulfilling. The independent correlator masses m_H,R in Table 1 do provide support (e.g., m_H = 1.093(1) vs m_H,R = 1.099(21) for β=8, κ=0.166), but the paper does not quantify the fit quality or state the number of fit parameters and the Φ-range used. Please report χ²/dof, the fit range, and explicitly compare the fitted m_H with m_H,R for each parameter set.
[Eq. (3.6c) and Sec. 4.2c] The choice of the minus sign in front of the square root in Eq. (3.6c) is justified only by the statement that 'during numerical simulations it turns out that only the − sign fulfills the constraint condition.' Because the Rattle map must be single-valued for reversibility, the branch ambiguity should be resolved analytically; the plus sign can in fact be excluded by the requirement of a finite h→0 limit, as follows from the expansion leading to Eq. (3.7). For the 5D Secant root-finding (Eq. 4.2c and A.2.2c), please comment on the uniqueness of the root and demonstrate that the selected root is the physical one, e.g., by a reversibility check on representative configurations.

minor comments (6)

[Abstract and Sec. 5] The orbifold results use the constraint Φ = ⟨TrP/2⟩, which the authors themselves state is not the exact Higgs field H = (1/4Ω)∑Tr[P5−P5†,σ3]^2 (Sec. 5). Please say this explicitly in the abstract or conclusions, and avoid the phrase 'full domain of the Higgs variable' for the orbifold case.
[Sec. 3.3] The text after Eq. (3.11) contains a typo: 'wich' should be 'which'.
[Title page] The affiliation contains a typo: 'Comuter Science' should be 'Computer Science'.
[Fig. 7 caption] The caption contains a typo: 'compcat phase' should be 'compact phase'.
[Eq. (4.3) and A.2.4] The derivation of the initial guess for the Secant method is shown to O(h^2); please state the order of the truncation and report how the Secant iteration converges (typical number of iterations, achieved machine precision).
[Table 2] The 4^4 row reports Φ0 = 1.122(8), which differs from the larger-volume values 1.132-1.133(1); the text says 'except for the smallest volume we don't see an effect.' Please clarify whether this is a finite-volume effect or a statistical fluctuation, and whether the quoted mH fit for 4^4 uses the same Φ-range as the other volumes.

Circularity Check

1 steps flagged · score 3.0 of 10

Central derivative identity Eq. (2.5) is self-contained, but the one-loop verification is weakened because mH is fit to the measured derivative.

fitted input called prediction [Section 3.3, 'Comparison to the one-loop Higgs potential in unitary gauge', around Eq. (3.8), Fig. 4 and Tables 1-2.]
"via fitting the (bare) Higgs mass mH. We actually fit the derivative U′1loop(Φ)=V′1(Φ−Φ0) to our measured U′Ω(Φ), using the bare quartic coupling λ and Z-boson mass given by the quasi-classical perturbative relation mZ=√2κg2⟨ρ2⟩ [29] ... We find that the one-loop formula fits the constraint potential much better than the classical ansatz U0(Φ) =−m2HΦ2/2+λΦ4, while the histogram data cannot differentiate the one-loop corrections within their limited range of Φ."

The claimed one-loop verification is not an independent prediction: mH is fitted to the very derivative data being compared, so the one-loop curve is matched to U′Ω in the vicinity of Φ0 and the stated agreement is partially enforced by construction. The independent content is the functional shape outside the fitted region and the comparison of the extracted mH with the correlator mass mH,R in Table 1, which reduces but does not remove the fitted-input component. The central identity Eq. (2.5) itself is not circular.

full rationale

The core derivation is self-contained: Eq. (2.5) follows by differentiating the constrained path integral with respect to Φ, so U′Ω(Φ)=−⟨λ(1)⟩Φ/Ω is not an input to the method. The Rattle-based algorithms are supported by external references [24,25] and by internal consistency checks against histogram data and the volume-scaling Table 2. The only genuine reduction found is in Sec. 3.3, where the one-loop potential is compared after fitting mH to the measured derivative; this makes that comparison partially self-fulfilling, though the extracted mass is independently cross-checked against two-point-function fits. The 5D gauge-link Rattle updates lack an explicit reversibility test (the numerical reversibility check reported in Sec. 3.1 is for the 4D Abelian-Higgs case; Sec. 6 states the tests were done without specifying coverage), but that is an evidentiary gap, not circularity. Overall the central claim stands independently, with one localized fitted-input comparison.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation is self-contained, but the physical interpretation relies on the standard constraint-effective-potential equivalence to the effective potential, and the 5D gauge extensions assume the Rattle geometric properties carry over without a direct numerical check. The one-loop comparison introduces one fitted parameter (m_H).

free parameters (1)

m_H (fitted bare Higgs mass) = 1.093(1) for beta=8, kappa=0.166, lambda=0.15, 8^3x16
Used to fit the one-loop potential derivative V'_1(Phi-Phi0) to the measured U'_Omega(Phi) in Sec. 3.3; the fit is the verification, not a prediction.

assumptions (4)

domain assumption The constraint effective potential U_Omega(Phi) converges to the true effective potential in the infinite-volume limit.
Invoked in Sec. 2 via Refs. [19,20]; central to interpreting measured U'_Omega as the physical effective potential.
ad hoc to paper The Rattle algorithm for constrained Hamiltonian systems preserves symplecticity, time-reversibility and volume preservation when extended to the non-canonical gauge-field updates used here.
Stated in Appendix A and Sec. 3.1; verified numerically only for the 4D Abelian-Higgs model. The 5D gauge extensions rely on this without an explicit reversibility test.
standard math The composite constraint on the Higgs field (e.g., Eq. 3.3) and the hidden constraint (Eq. 3.4) are sufficient to fix the Lagrange multipliers at each step.
Standard constrained dynamics; the paper solves for lambda^(1) and lambda^(2) in Appendix A.1-A.4.
ad hoc to paper Only the minus sign in Eq. (3.6c) fulfills the constraint, as observed numerically.
Stated after Eq. (3.6c); the plus branch is discarded based on simulation observation, not proof.

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Pith. "Pith review of Constrained Hybrid Monte Carlo algorithms for gauge-Higgs models." pith.science (2026). https://pith.science/paper/X4XHXTO7

@misc{pith2026190810950,
  author       = {Pith},
  title        = {Pith review of: Constrained Hybrid Monte Carlo algorithms for gauge-Higgs models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4XHXTO7}},
  note         = {Machine review of arXiv:1908.10950}
}

read the original abstract

We develop Hybrid Monte Carlo (HMC) algorithms for constrained Hamiltonian systems of gauge- Higgs models and introduce a new observable for the constraint effective Higgs potential. We use an extension of the so-called Rattle algorithm to general Hamiltonians for constrained systems, which we adapt to the 4D Abelian-Higgs model and the 5D SU(2) gauge theory on the torus and on the orbifold. The derivative of the potential is measured via the expectation value of the Lagrange multiplier for the constraint condition and allows a much more precise determination of the effective potential than conventional histogram methods. With the new method, we can access the potential over the full domain of the Higgs variable, while the histogram method is restricted to a short region around the expectation value of the Higgs field in unconstrained simulations, and the statistical precision does not deteriorate when the volume is increased. We further verify our results by comparing to the one-loop Higgs potential of the 4D Abelian-Higgs model in unitary gauge and find good agreement. To our knowledge, this is the first time this problem has been addressed for theories with gauge fields. The algorithm can also be used in four dimensions to study finite temperature and density transitions via effective Polyakov loop actions.

Figures

Figures reproduced from arXiv: 1908.10950 by the authors.

**Figure 1.** (left) From [17]. The phase diagram for N5 = 4 in the region of the Higgs-hybrid phase transition of the 5D orbifold gauge theory (see Sect. 5). The points show the location of a first-order phase transition. The red and blue lines represent the width of the corresponding hystereses, while the dashed orange line represents γ = 1 (β4 = β5). (right) The phase diagram of the Abelian-Higgs model (see Sect. 3) with λ = 1… view at source ↗

**Figure 2.** (left) Well defined continuum limit of U 0 Ω,cnst. (2.5) with respect to step size h → 0, it agrees with the continuum form U 0 Ω,cont. (3.7) at h = 0 for h ≤ 0.01. (right) Effective potential and derivatives from histogram method and constrained simulations for the Abelian-Higgs model at β = 1.4, κ = 0.17, λ = 0.15 on Ω = 84 lattices. The potential and its derivative diverge for Φ → 0. In [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 3.** Effective potential and derivatives from histogram method and constrained simulations for the Abelian [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Derivatives of the effective potential from histogram method and constrained simulations for the Abelian [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: 5D Torus constrained HMC (Rattle) algorithm on a 2 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: 5D Torus Effective Higgs Potential UΩ,hist. from the histogram method, its numerical derivative U 0 Ω,hist. , the derivative from constrained HMC U 0 Ω,cnst. and its numerical integral UΩ,int. for β4 = β5 = 1.66, Ω = 84 , N5 = 4. The unconstrained expectation value of …

**Figure 7.** Figure 7: 5D Torus Effective Higgs Potential UΩ,hist. from the histogram method, its numerical derivative U 0 Ω,hist. , the derivative from constrained HMC U 0 Ω,cnst. and its numerical integral UΩ,int. for β4 = 1.0, β5 = 2.8, Ω = 24×123 , N5 = 4 (upper plots). This point lies i…

**Figure 8.** Figure 8: 5D Orbifold Effective Higgs Potential UΩ,hist. from the histogram method, its numerical derivative U 0 Ω,hist. and the derivative from constrained HMC U 0 Ω,cnst. = −hλ (1) n iΦ/Ω for the symmetric point β4 = β5 = 1.66, Ω = 8 4 , N5 = 5. This point lies in the Higgs ph…

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Works this paper leans on

36 extracted references · 31 canonical work pages

[1]

Broken Symmetry and the Mass of Gauge Vector Mesons,

F. Englert and R. Brout, “Broken Symmetry and the Mass of Gauge Vector Mesons,” Phys.Rev.Lett. 13 (1964) 321–323

work page 1964
[2]

Broken symmetries, massless particles and gauge ﬁelds,

P. W. Higgs, “Broken symmetries, massless particles and gauge ﬁelds,” Phys.Lett. 12 (1964) 132–133

work page 1964
[3]

Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,

A TLAS Collaboration Collaboration, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys.Lett. B716 (2012) 1–29, arXiv:1207.7214 [hep-ex]

arXiv 2012
[4]

Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,

CMS Collaboration Collaboration, S. Chatrchyan et al., “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys.Lett. B716 (2012) 30–61, arXiv:1207.7235 [hep-ex]

arXiv 2012
[5]

A New Six-Dimensional Approach to the Weinberg-Salam Model,

N. Manton, “A New Six-Dimensional Approach to the Weinberg-Salam Model,” Nucl.Phys. B158 (1979) 141

work page 1979
[6]

Higgs’ Fields and the Determination of the Weinberg Angle,

D. Fairlie, “Higgs’ Fields and the Determination of the Weinberg Angle,” Phys.Lett. B82 (1979) 97

work page 1979
[7]

Dynamical Gauge Symmetry Breaking as the Casimir Eﬀect,

Y. Hosotani, “Dynamical Gauge Symmetry Breaking as the Casimir Eﬀect,” Phys.Lett. B129 (1983) 193

work page 1983
[8]

Non-perturbative definition of five-dimensional gauge theories on the R^4 x S^1/Z_2 orbifold

N. Irges and F. Knechtli, “Non-perturbative deﬁnition of ﬁve-dimensional gauge theories on the R4×S1/Z2 orbifold,” Nucl.Phys. B719 (2005) 121–139, arXiv:hep-lat/0411018 [hep-lat]

work page Pith review arXiv 2005

Show all 36 references

[9]

Gauge theories on a ﬁve-dimensional orbifold,

F. Knechtli, B. Bunk, and N. Irges, “Gauge theories on a ﬁve-dimensional orbifold,” PoS LA T2005(2006) 280, arXiv:hep-lat/0509071 [hep-lat]

2006 arXiv
[10]

Non-perturbative mass spectrum of an extra-dimensional orbifold,

N. Irges and F. Knechtli, “Non-perturbative mass spectrum of an extra-dimensional orbifold,” arXiv:hep-lat/0604006 [hep-lat]

arXiv
[11]

Lattice gauge theory approach to spontaneous symmetry breaking from an extra dimension,

N. Irges and F. Knechtli, “Lattice gauge theory approach to spontaneous symmetry breaking from an extra dimension,” Nucl.Phys. B775 (2007) 283–311, arXiv:hep-lat/0609045 [hep-lat]

2007 arXiv
[12]

The Phase Structure of Discrete Abelian Spin and Gauge Systems,

S. Elitzur, R. B. Pearson, and J. Shigemitsu, “The Phase Structure of Discrete Abelian Spin and Gauge Systems,” Phys. Rev. D19 (1979) 3698

1979
[13]

Symmetry and Z (2) Orbifolding Approach in Five-dimensional Lattice Gauge Theory,

K. Ishiyama, M. Murata, H. So, and K. Takenaga, “Symmetry and Z (2) Orbifolding Approach in Five-dimensional Lattice Gauge Theory,” Prog. Theor. Phys. 123 (2010) 257–269, arXiv:0911.4555 [hep-lat]

2010 arXiv
[14]

Non-perturbative Gauge-Higgs Uniﬁcation: Symmetries and Order Parameters,

N. Irges and F. Knechtli, “Non-perturbative Gauge-Higgs Uniﬁcation: Symmetries and Order Parameters,” JHEP 1406 (2014) 070, arXiv:1312.3142 [hep-lat]

2014 arXiv
[15]

Mean-Field Gauge Interactions in Five Dimensions II. The Orbifold,

N. Irges, F. Knechtli, and K. Yoneyama, “Mean-Field Gauge Interactions in Five Dimensions II. The Orbifold,” Nucl.Phys. B865 (2012) 541–567, arXiv:1206.4907 [hep-lat]

2012 arXiv
[16]

Higgs mechanism near the 5d bulk phase transition,

N. Irges, F. Knechtli, and K. Yoneyama, “Higgs mechanism near the 5d bulk phase transition,” Phys.Lett. B722 (2013) 378–383, arXiv:1212.5514

2013 arXiv
[17]

Five-Dimensional Gauge-Higgs Uniﬁcation: A Standard Model-Like Spectrum,

M. Alberti, N. Irges, F. Knechtli, and G. Moir, “Five-Dimensional Gauge-Higgs Uniﬁcation: A Standard Model-Like Spectrum,” JHEP 09 (2015) 159, arXiv:1506.06035 [hep-lat]

2015 arXiv
[18]

Lines of Constant Physics in a Five-Dimensional Gauge-Higgs Uniﬁcation Scenario,

M. Alberti, N. Irges, F. Knechtli, and G. Moir, “Lines of Constant Physics in a Five-Dimensional Gauge-Higgs Uniﬁcation Scenario,” PoS LA TTICE2016(2016) 215, arXiv:1609.07004 [hep-lat]

2016 arXiv
[19]

Derivation of the Eﬀective Potential,

R. Fukuda and E. Kyriakopoulos, “Derivation of the Eﬀective Potential,” Nucl. Phys. B85 (1975) 354–364

1975
[20]

The Constraint Eﬀective Potential,

L. O’Raifeartaigh, A. Wipf, and H. Yoneyama, “The Constraint Eﬀective Potential,” Nucl. Phys. B271 (1986) 653–680

1986
[21]

Supercomputing the Eﬀective Action,

J. Kuti and Y. Shen, “Supercomputing the Eﬀective Action,” Phys. Rev. Lett. 60 (1988) 85

1988
[22]

Hybrid Monte Carlo,

S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B195 (1987) 216–222

1987
[23]

Renormalization of the Abelian-Higgs model in the R ξ and Unitary gauges and the physicality of its scalar potential,

N. Irges and F. Koutroulis, “Renormalization of the Abelian-Higgs model in the R ξ and Unitary gauges and the physicality of its scalar potential,” Nucl. Phys. B924 (2017) 178–278, arXiv:1703.10369 [hep-ph]

2017 arXiv
[24]

Hairer, C

E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Diﬀerential Equations. Springer, Berlin, 2nd ed. ed., 2006. https://www.springer.com/de/book/9783540306634

2006
[25]

Geometric numerical integration illustrated by the St¨ ormer-Verlet method,

E. Hairer, C. Lubich, and G. Wanner, “Geometric numerical integration illustrated by the St¨ ormer-Verlet method,” Acta Numerica 12 (2003) 399–450. http://www.math.kit.edu/ianm3/lehre/geonumint2009s/media/gni_by_stoermer-verlet.pdf

2003
[26]

Montvay and G

I. Montvay and G. Munster, Quantum ﬁelds on a lattice . Cambridge University Press, 1997. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521404320

1997
[27]

New Higgs physics from the lattice,

Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. Schroeder, “New Higgs physics from the lattice,” PoS LA TTICE2007(2007) 056, arXiv:0710.3151 [hep-lat] . 26

2007 arXiv
[28]

Constraint HMC algorithms for gauge-Higgs models,

R. H¨ ollwieser and F. Knechtli, “Constraint HMC algorithms for gauge-Higgs models,” PoS LA TTICE2018 (2018) 052, arXiv:1812.02045 [hep-lat]

2018 arXiv
[29]

Photon and Bosonium Masses in Scalar Lattice QED,

H. G. Evertz, K. Jansen, J. Jersak, C. B. Lang, and T. Neuhaus, “Photon and Bosonium Masses in Scalar Lattice QED,” Nucl. Phys. B285 (1987) 590–605

1987
[30]

A Study on the nonperturbative existence of Yang-Mills theories with large extra dimensions,

S. Ejiri, J. Kubo, and M. Murata, “A Study on the nonperturbative existence of Yang-Mills theories with large extra dimensions,” Phys.Rev. D62 (2000) 105025, arXiv:hep-ph/0006217 [hep-ph]

2000 arXiv
[31]

On the phase structure of ﬁve-dimensional SU(2) gauge theories with anisotropic couplings,

F. Knechtli, M. Luz, and A. Rago, “On the phase structure of ﬁve-dimensional SU(2) gauge theories with anisotropic couplings,” Nucl.Phys. B856 (2012) 74–94, arXiv:1110.4210 [hep-lat]

2012 arXiv
[32]

Extra-dimensional models on the lattice,

F. Knechtli and E. Rinaldi, “Extra-dimensional models on the lattice,” Int. J. Mod. Phys. A31 no. 22, (2016) 1643002, arXiv:1605.04341 [hep-lat]

2016 arXiv
[33]

The phase diagram of Yang-Mills theory with a compact extra dimension,

P. de Forcrand, A. Kurkela, and M. Panero, “The phase diagram of Yang-Mills theory with a compact extra dimension,” JHEP 1006 (2010) 050, arXiv:1003.4643 [hep-lat]

2010 arXiv
[34]

Numerical corrections to the strong coupling eﬀective Polyakov-line action for ﬁnite T Yang-Mills theory,

G. Bergner, J. Langelage, and O. Philipsen, “Numerical corrections to the strong coupling eﬀective Polyakov-line action for ﬁnite T Yang-Mills theory,” JHEP 11 (2015) 010, arXiv:1505.01021 [hep-lat]

2015 arXiv
[35]

Finite-density transition line for QCD with 695 MeV dynamical fermions,

J. Greensite, and R. H¨ ollwieser, “Finite-density transition line for QCD with 695 MeV dynamical fermions,” Phys. Rev. D97 no. 11, (2018) 114504, arXiv:1708.08031 [hep-lat]

2018 arXiv
[36]

Relative weights approach to SU(3) gauge theories with dynamical fermions at ﬁnite density,

J. Greensite, and R. H¨ ollwieser, “Relative weights approach to SU(3) gauge theories with dynamical fermions at ﬁnite density,” Phys. Rev. D94 no. 1, (2016) 014504, arXiv:1603.09654 [hep-lat] . 27

2016 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Broken Symmetry and the Mass of Gauge Vector Mesons,

F. Englert and R. Brout, “Broken Symmetry and the Mass of Gauge Vector Mesons,” Phys.Rev.Lett. 13 (1964) 321–323

work page 1964

[2] [2]

Broken symmetries, massless particles and gauge ﬁelds,

P. W. Higgs, “Broken symmetries, massless particles and gauge ﬁelds,” Phys.Lett. 12 (1964) 132–133

work page 1964

[3] [3]

Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,

A TLAS Collaboration Collaboration, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys.Lett. B716 (2012) 1–29, arXiv:1207.7214 [hep-ex]

arXiv 2012

[4] [4]

Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,

CMS Collaboration Collaboration, S. Chatrchyan et al., “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys.Lett. B716 (2012) 30–61, arXiv:1207.7235 [hep-ex]

arXiv 2012

[5] [5]

A New Six-Dimensional Approach to the Weinberg-Salam Model,

N. Manton, “A New Six-Dimensional Approach to the Weinberg-Salam Model,” Nucl.Phys. B158 (1979) 141

work page 1979

[6] [6]

Higgs’ Fields and the Determination of the Weinberg Angle,

D. Fairlie, “Higgs’ Fields and the Determination of the Weinberg Angle,” Phys.Lett. B82 (1979) 97

work page 1979

[7] [7]

Dynamical Gauge Symmetry Breaking as the Casimir Eﬀect,

Y. Hosotani, “Dynamical Gauge Symmetry Breaking as the Casimir Eﬀect,” Phys.Lett. B129 (1983) 193

work page 1983

[8] [8]

Non-perturbative definition of five-dimensional gauge theories on the R^4 x S^1/Z_2 orbifold

N. Irges and F. Knechtli, “Non-perturbative deﬁnition of ﬁve-dimensional gauge theories on the R4×S1/Z2 orbifold,” Nucl.Phys. B719 (2005) 121–139, arXiv:hep-lat/0411018 [hep-lat]

work page Pith review arXiv 2005