QCD String formation and the Casimir Energy

TL;DR

The paper investigates how bosonic string formation emerges in gauge theories by analyzing the QCD string spectrum and Casimir energy across quark-antiquark separations. It develops a three-dimensional $Z(2)$ model dual to an Ising–$\phi^4$ theory to realize confinement and a loop expansion around a static soliton, deriving a Dirichlet string spectrum and relating it to an effective string action. It shows string-like excitations at $R \sim 2$–$3$ fm and an early Casimir-energy signal below 1 fm, but finds that ground-state properties do not reliably indicate string formation at small $R$ and that matching to the effective string action remains challenging. The results highlight a Casimir-energy puzzle and call for refined descriptions of string dynamics beyond the Nambu–Goto action.

Abstract

Three distinct scales are identified in the excitation spectrum of the gluon field around a static quark-antiquark pair as the color source separation R is varied. The spectrum, with string-like excitations on the largest length scales of 2-3 fm, provides clues in its rich fine structure for developing an effective bosonic string description. New results are reported from the three-dimensional Z(2) and SU(2) gauge models, providing further insight into the mechanism of bosonic string formation. The precocious onset of string-like behavior in the Casimir energy of the static quark-antiquark ground state is observed below R=1 fm where most of the string eigenmodes do not exist and the few stable excitations above the ground state are displaced. We find no firm theoretical foundation for the widely held view of discovering string formation from high precision ground state properties below the 1 fm scale.

Figures (6)

• Figure 1: Short distance degeneracies which are not string-like and their crossover towards the QCD string spectrum are shown from Ref. 1 where further details are explained. The color coded solid curves with simulation points, which identify energy levels degenerate in the asymptotic string limit, are only shown for visualization and do not represent fits to the data. The yellow line without data points marks a lower bound for the onset of mixing effects with glueball states which requires careful interpretation. The symbol LW indicates the ${\rm R}$ range of high precision Casimir energy calculations from Ref. 2.
• Figure 2: The static soliton solution $\phi_s$ of field equation (\ref{['eq:field_eq']}) is shown in (a) with the choice R=100 and renormalized parameters given in the text; (b) shows the N=8 massless string excitation of the static soliton from the numerical diagonalization of the eigenvaule equation (\ref{['eq:fluct']}); the second massive string excitation (K=2 breathing mode) is shown in (c) from the numerical solution of the eigenvalue equation, and compared in (d) to exact Monte Carlo simulation of the same state with remarkable agreement.
• Figure 3: The soliton profile for R=10 is shown in (a), with the only bound state wave function below the glueball threshold depicted in (b). At R=100, the stretched soliton configuration shown in (c) exhibits several string-like excitation with the N=1 wave function shown in (d), N=4 in (e), and N=8 in (f). The simulation results match the loop expansion of Fig. \ref{['fig:mf']} with common renormalized parameters.
• Figure 4: The energy gap ${\rm \Delta E}$ above the ground state is plotted as ${\rm \Delta E/(N\pi/R) - 1}$ to show percentage deviations from the asymptotic ${\rm N=1}$ string level. Several Z(2) simulations with cyan, blue, red, and green points are combined with good scaling properties. The open circles represent D=3 SU(2) results after readjusting the ratio of the string tension $\sigma$ to the glueball mass in Z(2). The null line coresponds to the tree level ${\rm \pi /N}$ NG string gaps. The dashed blue and green lines are 1--loop and two--loop NG approximations, repectively, and the black line is the full NG prediction.
• Figure 5: The red points are from high precision Ising-$\phi^4$ simulations. The solid black curve with NG label is the full NG prediction, ${\rm C_{eff}(R) = 1 }$ is the asymptotic string result which corresponds to the tree-level NG prediction. The dashed green line shows the 1-loop approximation which includes the first correction to tree level from the ${\rm R^{-1} }$ expansion of Eq. (\ref{['eq:Arvis']}). The solid red line is calculated from the numerical evaluation of Eq. (\ref{['eq:casi1']}). The dashed blue line is obtained from the full red line by subtracting the ${\rm E^{cl}_s}$ contribution.