QCD Effective Lagrangian and Condensation of Chromomagnetic Flux Tubes

Abstract

We compute the effective action for covariantly constant gauge fields that are solutions of the sourceless Yang-Mills equation and have the form of magnetic flux tubes. They represent a superposition of infinite many alternating monopole/ani-monopole pairs situated at infinity, with each pair having a structure similar to the Nielsen-Olesen magnetic flux tube. The chromomagnetic flux tubes condensation is stable and indicates that the Yang-Mills vacuum state is highly degenerate.

Figures (3)

• Figure 1: The figure demonstrates a finite part of an infinite wall of finite thickness ${2\over a}$ in the direction of the $x$ axis of the solution (\ref{['polsol']}), (\ref{['magsheet']}). It is filled by parallel chromomagnetic fluxes of opposite orientation (see Appendix B for details). Each chromomagnetic flux tube cell of the square area ${2\over a} { \pi \over b}$ carries the flux ${4 \pi \over g}$. The circuits in the left figure show the flow of the conserved current $J^a_{k}=g \epsilon^{abc} A^b_{j} G^c_{ik}$ and the vertical arrows show the vorticity directions $\omega^a_i = \epsilon_{ijk} \partial_j J^a_k$ (\ref{['vorticity']}). In the right figure the unit vector $n^a =(\sqrt{1-x^2} \cos y, \sqrt{1-x^2} \sin y,x )$ defines the map of a unit cell ${\bf C}^2: x\in (-1,1); y \in (0,2\pi)$ to a sphere ${\bf S}^2$.
• Figure 2: The figure demonstrates the geometry of a chromomagnetic flux tube of finite thickness ${2\over a}$ in the direction of the $x$ axis and infinite in $y$ and $z$ axis. It is a section of the solution (\ref{['polsol']}), (\ref{['magsheet']}), (\ref{['magsheet1']} ) by the plane $(x,y,0)$ when $g H =0$. A space is filled by parallel chromomagnetic fluxes of opposite orientation. Each chromomagnetic flux tube cell of the square area ${2\over a} { \pi \over b}$ carries the flux ${2 \pi \over g}$. $L_1, L_2$ are the integration contours in the operator $A(L)$ (\ref{['magflux']}).
• Figure 3: The flow of the chromomagnetic currents $(J^a_1(x,y),J^a_2(x,y))$ (\ref{['eleccurr']}) in the plane normal to the $z$ axis in two neighbouring cells $L1$ and $L_2$ defined in Fig.(\ref{['fig11']}).