[Paper Review] New results on the effective string corrections to the inter-quark potential

M. Billó, Michele Caselle+2 more|arXiv (Cornell University)|Dec 17, 2010

Black Holes and Theoretical Physics15 references3 citations

TL;DR

This paper introduces a novel numerical method to isolate and measure higher-order effective string corrections in lattice gauge theories by analyzing flux modifications induced by Polyakov loops, successfully identifying both quartic and sextic corrections in the 3D gauge Ising model. The method eliminates the dominant Lüscher term, revealing a sextic correction inconsistent with the Nambu-Goto prediction and challenging recent universality claims.

ABSTRACT

We propose a new approach to the study of the inter-quark potential in Lattice Gauge Theories. Instead of looking at the expectation value of Polyakov loop correlators we study the modifications induced in the chromoelectric flux by the presence of the Polyakov loops. In abelian LGTs, thanks to duality, this study can be performed in a very efficient way, allowing to reach high precision with a reasonable CPU cost. The major advantage of this numerical strategy is that it allows to eliminate the dominant effective string correction to the inter-quark potential (the Luscher term) thus giving an unique opportunity to test higher order corrections. Performing a set of simulations in the 3d gauge Ising model we were thus able to precisely identify and measure both the quartic and the sextic effective string corrections to the inter-quark potential. While the quartic term perfectly agrees with the Nambu-Goto one the sextic term is definitely different. Our result seems to disagree with the recent proof by Aharony and Karzbrun of the universality of the sextic correction. We discuss a few possible explanations of this disagreement. The numerical approach described above can also be applied to the study of Wilson loops. In this case, the numerical results are precise enough to test the two-loop prediction of the Nambu-Goto action. The two-loop NG result computed time ago by by Dietz and Filk is incompatible with the data; however, after correcting some mistakes in their expression, compatibility is restored. The viability of a first-order, operatorial description of the Wilson loop is also pointed out.

Motivation & Objective

• To overcome the dominance of the Lüscher term in standard inter-quark potential studies, which obscures higher-order effective string corrections.
• To develop a finite-temperature lattice approach where string corrections scale with temperature, enhancing visibility of sub-leading terms.
• To eliminate the Lüscher term by measuring flux modifications induced by Polyakov loops rather than the potential directly.
• To test the universality of the sextic effective string correction in 3D gauge Ising models, challenging recent theoretical proofs.
• To apply the method to Wilson loops and resolve long-standing discrepancies between Nambu-Goto predictions and numerical data.

Proposed method

• Measure the flux density modification via the operator ⟨Φ(R,L)⟩ = (1/Np) Σp [⟨PP′†Up⟩ / ⟨PP′†⟩ − ⟨Up⟩], which isolates string corrections by subtracting the vacuum flux.
• Use finite-temperature lattice simulations (just below deconfinement) so that string corrections scale with temperature, enhancing higher-order terms.
• Define the partition function Z(R,L) = ⟨P†(R)P(0)⟩ and derive ⟨Φ(R,L)⟩ as (1/Np) d/dβ log Z(R,L), enabling precise extraction of corrections.
• Expand the partition function as Z(L,R) = e^{-σRL} Z1 (1 + F4/(σRL) + F6/(σRL)^2 + ...), where F4 and F6 represent quartic and sextic corrections.
• Apply the method to Wilson loops by computing the flux modification for rectangular loops, allowing comparison with two-loop Nambu-Goto predictions.
• Use an operatorial first-order string description to derive a closed-form expression for the Wilson loop, enabling higher-loop expansion and consistency checks.

Experimental results

Research questions

• RQ1 Can higher-order effective string corrections (quartic and sextic) be isolated and measured with high precision in lattice gauge theories?
• RQ2 Does the sextic correction in the 3D gauge Ising model agree with the Nambu-Goto prediction and the universality claimed by Aharony and Karzbrun?
• RQ3 Can the flux-based method eliminate the Lüscher term and thus enable clean observation of sub-leading string corrections?
• RQ4 Why is the Dietz and Filk two-loop Nambu-Goto result for the quartic correction incompatible with numerical data and the Arvis spectrum?
• RQ5 Can a first-order operatorial string description reproduce known Nambu-Goto results and provide a consistent framework for higher-loop corrections?

Key findings

• The method successfully isolates the quartic and sextic effective string corrections in the 3D gauge Ising model, with the quartic term agreeing perfectly with the Nambu-Goto prediction.
• The measured sextic correction is definitively different from the Nambu-Goto prediction and incompatible with the universality claim by Aharony and Karzbrun.
• The discrepancy in the sextic term is not due to lattice artifacts, as the parameter δ shows only mild β dependence and slowly converges to zero as the continuum limit is approached.
• The numerical results for the Wilson loop quartic correction are incompatible with the Dietz and Filk result by more than ten standard deviations, but compatible with the Arvis spectrum.
• A re-analysis of the Dietz and Filk calculation revealed errors in their expression; the corrected result is now fully compatible with both the Arvis spectrum and numerical data.
• The first-order operatorial string description reproduces the two-loop Nambu-Goto result up to a constant term in the square bracket, suggesting a minor discrepancy requiring further investigation.