Confining strings in supersymmetric theories with Higgs branches

We study flux tubes (strings) on the Higgs branches in supersymmetric gauge theories. In generic vacua on the Higgs branches, strings were shown to develop long-range “tails” associated with massless fields, a characteristic feature of the Higgs branch (the only exception is the vacuum at the base of the Higgs branch). A natural infrared regularization for the above tails is provided by a finite string length $L$. We perform a numerical study of these strings in generic vacua. We focus on the simplest example of strings in $\mathcal{N}=1$ supersymmetric QED with the Fayet-Iliopoulos term. In particular, we examine the accuracy of a logarithmic approximation (proposed earlier by Evlampiev and Yung) for the tension of such string solutions. In the Evlampiev-Yung formula, the dependence of tension on the string length is logarithmic, and the dependence on the geodesic length from the base of the Higgs branch is quadratic. We observe a remarkable agreement of our numerical results for the string tension with the Evlampiev-Yung analytic expression.

Figure 1

Numerical solutions for field profile functions varying $m_L$; the curve labels in (a) also apply to plot (b). In (a) we use ${\rho }_0=0.2$ and in (b) we pick ${\rho }_0=0.3$. The values of $m_L$ used are reported in Tables 1 and 2.

Figure 2

Field profiles varying ${\rho }_0$ at $m_L=0.017$. The plots correspond to ${\rho }_0=0.1,0.3,0.4,0.7$.

Figure 3

Difference between numerical and theoretical tensions $\mathrm{\Delta }T/\xi$ for varying $l$. Solid line corresponds to numerical result, dashed line to theoretical. The values of ${\rho }_0$ used are 0.1 to 0.7 in steps of 0.1.

Figure 4

Difference between numerical and theoretical tensions $\mathrm{\Delta }T$ for varying $m_L$. Solid line corresponds to numerical result, dashed line to theoretical. (a) ${\rho }_0=0.2$. (b) ${\rho }_0=0.3$. The values of $m_L$ used are reported in Table. 1 and 2, plot (a) excludes the point in which our numerical procedure could not determine a solution.