[Paper Review] The effect of fluctuations of the vortex core size on the static potentials evaluated by the thick center vortex model

Sedigheh Deldar, Shahnoosh Rafibakhsh|ArXiv.org|Dec 20, 2008

Fluid Dynamics and Turbulent Flows3 citations

TL;DR

This paper proposes that fluctuations in the core size of thick center vortices in the SU(3) gauge group induce a Coulombic potential at short distances, achieving qualitative agreement with Casimir scaling across all regimes. By introducing a Gaussian-distributed fluctuation around a small core size, the model reproduces both the Coulombic and linear potential behaviors within a unified framework.

ABSTRACT

By varying the vortex core size of the thick center vortex model, we have studied the short distance potentials between static sources. It has been found that fluctuations of the vortex core size lead to Coulombic behavior. Furthermore, we discuss the influence of such fluctuations on Casimir scaling for both the Coulombic and the linear part of the potential.

Motivation & Objective

• To address the lack of short-distance Coulombic potential in the thick center vortex model, which otherwise successfully describes intermediate and large-distance linear potentials.
• To investigate whether the trivial center element (I) can contribute to the Wilson loop in a way that generates the Coulombic part of the potential.
• To explore whether fluctuations in vortex core size can enhance the role of the trivial center element, thereby mimicking one-gluon exchange effects.
• To achieve qualitative agreement with Casimir scaling for the Coulombic, linear, and constant terms of the potential across different representations.

Proposed method

• Introduce a Gaussian-distributed fluctuation in the vortex core size around a small central value (a = 0.02), while truncating extreme tails to avoid unphysical values.
• Modify the vortex profile function α_R(x) to depend on the fluctuating core size, altering the distribution of ReG_r[α] near the trivial center element (1) and non-trivial center element (−0.5).
• Use the modified vortex profile in the thick center vortex model's expression for the Wilson loop: ⟨W(C)⟩ = ∏_x {1 − ∑_n f_n (1 − ReG_r[α^n_C(x)])} to compute potentials.
• Fit the resulting potentials to the functional form V(R) = −(A/R) + KR + B to extract Coulombic (A), string tension (K), and constant (B) coefficients for various representations.
• Compare the ratios of these coefficients across representations with the theoretical Casimir ratios C_r/C_f for SU(3).

Experimental results

Research questions

• RQ1 Can fluctuations in the vortex core size generate a Coulombic potential in the thick center vortex model, which otherwise only produces linear potentials?
• RQ2 Does increasing the contribution of the trivial center element (I) through core size fluctuations lead to a physically plausible short-distance behavior?
• RQ3 To what extent does the model's potential for different representations (fundamental, adjoint, etc.) satisfy Casimir scaling for the Coulombic, linear, and constant terms?
• RQ4 How does the choice of core size distribution affect the shape of the potential, particularly in avoiding unphysical concavities in the medium-distance regime?

Key findings

• Fluctuating the vortex core size around a small value (a = 0.02) leads to a Coulombic potential at short distances, as evidenced by the 1/R dependence in the fitted potential V(R) = −(A/R) + KR + B.
• The Coulombic coefficient ratios A_r/A_f for various SU(3) representations (e.g., 8, 6, 15a) show qualitative agreement with Casimir scaling, with values of 2.09(21) for the 8-adjoint, 2.31(23) for the 6, and 4.04(37) for the 10 representation.
• The string tension ratios k_r/k_f also show qualitative agreement with Casimir scaling, ranging from 1.31(8) for the 8-adjoint to 1.63(12) for the 10 representation.
• The constant term ratios B_r/B_f similarly follow Casimir scaling trends, with values from 2.24(12) for the 8-adjoint to 4.93(20) for the 10 representation.
• The model successfully reproduces a smooth potential with a Coulombic regime at short distances and a linear regime at large distances, avoiding unphysical concavities seen with fixed core sizes.
• The results suggest that a single mechanism—fluctuating vortex core size—can unify the description of both Coulombic and linear potentials in the thick center vortex model.