Exploring center strings in $SU(2)$ and $SU(3)$ relativistic Yang-Mills-Higgs models

We develop numerical tools and apply them to solve the relativistic Yang-Mills-Higgs equations in a model where the $SU(N)$ symmetry is spontaneously broken to its center. In $SU(2)$ and $SU(3)$, we obtain the different field profiles for infinite and finite center strings, with end points at external monopole sources. Exploration of parameter space permits the detection of a region where the equations get Abelianized. Finally, a general parametrization of the color structure of $SU(2)$ fields leads us to a reference point where an Abelian-like Bogomol’nyi-Prasad-Sommereld (BPS) bound is reconciled with $N$-ality.

Figure 1

[$SU(2)$, $g=0.1$, $\mu =1$, $\kappa =-2$, $\lambda =1$, $v=1$.] The plots show the infinite string profiles. The dashed lines include Fourier coefficients up to the fifth order (the term with $\sin 5\pi t$), while the continuous ones include one more Fourier mode. Each of the plots is accompanied by a zoomed-in plot (which zooms in on the part marked with a box in the main plot) to see more clearly how far the two lines are from each other. One can see that the numerics have converged quite well and the two successive approximations are almost on top of each other.

Figure 2

[$SU(3)$, $g=0.1$, $\mu =1.8$, $\kappa =-4$, $\lambda =1$, $v=2.87$.] The plots show, in that order, the infinite string profiles $a$, $h$, $h_0$, $h_1$, and $h_2$, as functions of $\rho$, with accompanying zoom on the marked region. In this case the dashed lines include three Fourier modes for each of the unknown functions, while the continuous ones include four.

Figure 3

[$SU(2)$, $L=2$, $g=0.1$, $\mu =0.9$, $\kappa =-2$, $\lambda =1$, $v=1.47$.] Here, we show the approximation with more Fourier coefficients described in the text. The axes are $x_1$, the radial distance to the $x_1$ axis, and the different profiles. The mesh corresponds to the prolate spheroidal coordinates used in the computations. (d) The energy density, after the quark self-energy density inside the integral in (68) has been subtracted. (e) Its spatial distribution.

Figure 4

Relative errors, in percent, for the first three profiles in Fig. 3.

Figure 5

[$SU(2)$, $g=0.1$, $\kappa =-2$, $\lambda =1$.] $a(\rho )$ (lower blue line), $h(\rho )$ (central red line), $h_1(\rho )$ (upper yellow-green line).