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arxiv: 2601.19520 · v2 · pith:T77QZPNAnew · submitted 2026-01-27 · ✦ hep-lat · hep-th

Intrinsic Width of the Flux Tube as a tool to explore confining mechanisms in Lattice Gauge Theories

Michele Caselle , Elia Cellini , Alessandro Nada , Dario Panfalone , Lorenzo Verzichelli This is my paper

Pith reviewed 2026-05-25 07:08 UTC · model grok-4.3

classification ✦ hep-lat hep-th
keywords flux tubeintrinsic widthconfinementlattice gauge theorySU(2)dual superconductorSvetitsky-Yaffe mappingGinzburg-Landau parameter
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The pith

The intrinsic width of the flux tube distinguishes between different models of confinement in SU(2) lattice gauge theory.

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

The paper examines the profile of flux tubes in the SU(2) gauge model in 2+1 dimensions, focusing on the intrinsic width that controls the exponential decay of flux density far from the tube. Data collected over a range of lattice spacings, temperatures, and tube lengths show that near the deconfinement transition an effective Ising model based on the Svetitsky-Yaffe mapping fits perfectly. At lower, more physically relevant temperatures the Ising description fails and the dual superconductor model provides the closest match among tested proposals, yet the fitted Ginzburg-Landau parameter grows with tube length. This length dependence indicates that a more complete model is required and positions the intrinsic width as a diagnostic tool for future confinement theories.

Core claim

Precise measurements of the intrinsic width demonstrate that high-temperature data just below deconfinement are described by an Ising-like effective model from the Svetitsky-Yaffe mapping, while lower-temperature data are better described by the dual superconductor model than by alternatives; however the Ginzburg-Landau parameter extracted from the latter fits increases with flux tube length, revealing that the model is incomplete for non-abelian gauge theories.

What carries the argument

The intrinsic width of the flux tube, which drives the exponential decay of the flux density at large transverse distances and is directly linked to the underlying confining mechanism.

If this is right

  • The Svetitsky-Yaffe mapping accurately describes flux tube profiles near the deconfinement transition.
  • The dual superconductor model outperforms other proposals at lower temperatures.
  • The Ginzburg-Landau parameter in the dual superconductor fits depends on the length of the flux tube.
  • The intrinsic width serves as a benchmark for testing candidate models of confinement.
  • More sophisticated models are needed to fully explain confinement in non-abelian theories.

Where Pith is reading between the lines

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

  • The observed length dependence may signal additional dynamical effects such as string fluctuations or higher-order corrections not captured by the basic dual superconductor picture.
  • Similar width measurements in 3+1 dimensions or with dynamical quarks could test whether the same model hierarchy persists in more realistic settings.
  • Developers of new confinement mechanisms could use these width profiles as quantitative targets for validation.

Load-bearing premise

The intrinsic width drives the exponential decay of the flux density at large transverse distances and this quantity is directly related to the confining mechanism which generates the flux tube.

What would settle it

Observing that the Ginzburg-Landau parameter extracted from dual superconductor fits remains independent of flux tube length would challenge the conclusion that the model requires refinement.

Figures

Figures reproduced from arXiv: 2601.19520 by Alessandro Nada, Dario Panfalone, Elia Cellini, Lorenzo Verzichelli, Michele Caselle.

Figure 1
Figure 1. Figure 1: Schematic representation of the three-point function Fµν in Eq. (6), with ˆµ = ˆ0 and νˆ = ˆ1. Thick lines indicate the traced Wilson lines: the two Polyakov loops (red, blue) and the plaquette operator (black). at a given space point ⃗x: P = 1 2 TrY Nt t=1 U0 (⃗x, t). (4) The primary observable related to the properties of the flux tube between two static sources is the two-point correlator of Polyakov lo… view at source ↗
Figure 2
Figure 2. Figure 2: The profile of the flux tube for fixed lattice spacing and different temperature T (left panel) and vice versa (right panel). Note that, in order to compare between different values of the lattice spacing, the profile has been rescaled with the square of the string tension. The profiles obtained with β = 10.865 (left panel) correspond to R = 9 a = 1.18/ √ σ, while for the one at β = 12.963 the distance was… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the profile from a simulation of the (2 + 1)-dimensional SU(2) gauge theory (at β = 10.865) with one from direct simulation of the Nambu-Got=o EST. In both cases, the length of the string equals 9a and the temporal extent is 60a. goal in this subsection is to compute the moments with sufficient precision and with a limited set of assumptions. The simplest model-independent approach would be t… view at source ↗
Figure 4
Figure 4. Figure 4: we also show an example of this combination of fit and interpolation. −10 −5 0 5 10 y/a 10−5 10−4 ρ Spline and exponential tails fitted exponential spline [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of a fit with the Clem model, see Eq. (13), for a profile obtained at β = 10.865 and T = 0.23 Tc, with R = 11a. The two parameters ξ and λ are particularly important since they allow to connect the dual superconductor picture with the flux tube one. As we mentioned in Section 3.3, in the dual superconductor model they correspond to the two typical lengths of a vortex: ξ is the radius of the core, w… view at source ↗
Figure 6
Figure 6. Figure 6: Different values of λ in units of the string tension, showing no dependence on the distance between the sources R. In the left panel the temperature is fixed to T = 0.23 Tc and we compare different values of the lattice spacing. In the right panel we show different temperatures keeping the lattice spacing fixed by setting β = 10.865. The solid line and the shaded region correspond to our final estimation a… view at source ↗
Figure 7
Figure 7. Figure 7: The logarithmic broadening of the flux tube increasing its length R. The data-point are the values we found at T = 0.23 Tc. The dashed line is a fit to all the plotted data according to Eq. (8) with R0 as the only free parameter. The coefficient multiplying the logarithm was fixed interpolating the values of the string tension in Ref. [52]. 4.4.1 On the logarithmic growth of the total flux tube width Assum… view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Square of the total width and Binder cumulant from the fit assuming the SY mapping of our data at T = 0.68 Tc. The solid line with the confident band is the expansion from Eq. (27) in the left panel and Eq. (29) in the right one. The input value of λ is the one extracted from the Polyakov loop correlator λ = 1/(2E0). which perfectly matches the leading term in Eq. (9). Notice, as a side remark, that with t… view at source ↗
Figure 10
Figure 10. Figure 10: Values of λ obtained fitting with an exponential the tails of the profile using different values of yintr. We compare R/a = 9 on the β = 10.865, Nt = 30 lattice with R/a = 11 on the β = 12.963, Nt = 36. For small values of yintr, λ is considerably overestimated. The shaded regions correspond to the values of the identified plateau. In order to estimate the statistical error, we performed a bootstrap analy… view at source ↗
Figure 11
Figure 11. Figure 11: Optimal values of yintr for the profile we measure, in units of the string tension, plotted against the distance between the Polyakov loops in the same units. Although a general increasing trend is evident, its interpretation as a physical length should be carefully considered. It is not obvious from our data if the best value of yintr can be predicted from the simulation parameters or not. It appears to … view at source ↗
Figure 12
Figure 12. Figure 12: Example of a fit with the convolution model, see Eq. (26), for a profile obtained at β = 10.865 and T = 0.23 Tc, with R = 11a. C Moments of the profile in the Svetitsky–Yaffe mapping Let us consider the distribution p(y) = ρ(y)/ R ρ(y) dy, obtained normalizing the expres￾sion in Eq. (17), for a given value of R. We managed to reduce the moments of this distribution in terms of special functions present in… view at source ↗
read the original abstract

We study the profile of the flux tube in the SU(2) gauge model in 2+1 dimensions, with a particular attention to the so called "intrinsic width" which drives the exponential decay of the flux density at large transverse distances. This quantity is directly related to the confining mechanism which generates the flux tube: to test the properties of the latter we study a wide range of different values of lattice spacing, temperature and flux tube lengths and show that our data are precise enough to distinguish between different confining models. In particular we show that at high temperatures (just below the deconfinement transition) the data are perfectly described by an Ising-like effective model based on the Svetitsky-Yaffe mapping. At lower temperatures this approximation does not hold anymore. In this regime (which is the most interesting one from a physical point of view) we test several alternative proposals and show that the dual superconductor model is the one which better fits the data. However, this proposal is not fully satisfactory, because the values of the Ginzburg-Landau parameter extracted from the fits increase with the length of the flux tube, which is not a feature predicted by the model. This suggests that a more sophisticated model is needed to explain confinement in non-abelian gauge theories and, at the same time, that our data on the intrinsic width may be a powerful tool to benchmark these candidates.

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 / 0 minor

Summary. The manuscript studies the profile of the flux tube in SU(2) lattice gauge theory in 2+1 dimensions, focusing on the intrinsic width that governs the exponential decay of the flux density at large transverse distances. It claims that data across a range of lattice spacings, temperatures, and flux-tube lengths are precise enough to distinguish confining models: an Ising-like effective description based on the Svetitsky-Yaffe mapping fits well just below the deconfinement transition, while at lower temperatures the dual-superconductor model provides the best fit among alternatives, although the extracted Ginzburg-Landau parameter increases with flux-tube length, indicating that a more sophisticated effective theory is required.

Significance. If the results hold, the work supplies a quantitative lattice observable—the intrinsic width—for benchmarking candidate models of confinement in non-Abelian gauge theories. The systematic variation over temperature, spacing, and length, together with the explicit identification of the dual-superconductor model's length-dependent inconsistency, strengthens the utility of the observable as a diagnostic tool rather than as support for any single model.

major comments (1)
  1. [Abstract] Abstract: the central claim that the data 'are precise enough to distinguish between different confining models' rests on the reported superiority of the dual-superconductor fit at low T; however, the same paragraph states that the Ginzburg-Landau parameter grows with flux-tube length, contrary to the model's expectation of length-independent parameters. This tension is load-bearing for the distinction claim and requires explicit quantitative support (e.g., tabulated χ²/dof or AIC values for each model across lengths) to establish that the dual-superconductor description is meaningfully better despite the inconsistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comment. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the data 'are precise enough to distinguish between different confining models' rests on the reported superiority of the dual-superconductor fit at low T; however, the same paragraph states that the Ginzburg-Landau parameter grows with flux-tube length, contrary to the model's expectation of length-independent parameters. This tension is load-bearing for the distinction claim and requires explicit quantitative support (e.g., tabulated χ²/dof or AIC values for each model across lengths) to establish that the dual-superconductor description is meaningfully better despite the inconsistency.

    Authors: We agree that explicit quantitative measures of fit quality are required to substantiate the distinction claim. The manuscript already notes in the abstract and main text that the dual-superconductor model yields the best description among the alternatives tested at low T, while simultaneously flagging its inconsistency with the observed length dependence of the Ginzburg-Landau parameter. To make this comparison transparent, we will add a table in the revised version reporting χ²/dof and AIC values for the Ising-like, dual-superconductor, and other models at each flux-tube length in the low-T regime. This will allow direct assessment of whether the dual-superconductor description remains meaningfully superior despite the parameter variation. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper measures the intrinsic width from lattice simulations in SU(2) 2+1D and fits the resulting profiles to external literature models (Ising effective theory via Svetitsky-Yaffe at high T; dual superconductor at low T). Extracted parameters such as the Ginzburg-Landau parameter are reported with their observed length dependence, which contradicts model expectations, and the authors conclude a more sophisticated theory is needed. No derivation reduces a claimed prediction to a fitted input by construction, no load-bearing self-citation chain is invoked, and the observable is defined directly from the flux-density decay without circular redefinition. The analysis is therefore data-driven model testing against independent benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis rests on the domain assumption linking intrinsic width directly to the confining mechanism and on fitting a model parameter to the lattice data.

free parameters (1)
  • Ginzburg-Landau parameter
    Extracted from fits of the dual superconductor model to the intrinsic width data; observed to increase with flux tube length.
axioms (1)
  • domain assumption The intrinsic width drives the exponential decay of the flux density at large transverse distances and is directly related to the confining mechanism.
    This premise, stated in the abstract, underpins the use of the quantity to test and distinguish confining models.

pith-pipeline@v0.9.0 · 5794 in / 1544 out tokens · 48891 ms · 2026-05-25T07:08:19.295621+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Effective strings and particles interacting in 3D: the Ising model

    hep-th 2026-04 unverdicted novelty 6.0

    In the 3D Ising model, a fluctuating domain wall interacts with bulk particles such that large-distance corrections to free energy and correlation tails are controlled by a single renormalized coupling λ in the nearly...

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