Stability of strings dual to flux tubes between static quarks in N=4 SYM

Abstract

Computing heavy quark-antiquark potentials within the AdS/CFT correspondence often leads to behaviors that differ from what one expects on general physical grounds and field-theory considerations. To isolate the configurations of physical interest, it is of utmost importance to examine the stability of the string solutions dual to the flux tubes between the quark and antiquark. Here, we formulate and prove several general statements concerning the perturbative stability of such string solutions, relevant for static quark-antiquark pairs in a general class of backgrounds, and we apply the results to N = 4 SYM at finite temperature and at generic points of the Coulomb branch. In all cases, the problematic regions are found to be unstable and hence physically irrelevant.

Stability of string configurations dual to quarkonium states in AdS/CFT

“…In this section we generalize the results of [13] to non-diagonal metrics of the form (2.1) and we establish a series of results which will ultimately allow us to identify the stable and unstable regions by a combination of exact and approximate analytic methods.…”

“…Therefore, a necessary condition for the appearance of instabilities is the existence of values u (i) 0c of u 0 for which ω 2 0 (u 0 ) = 0. To verify that these zero modes really correspond to points where ω 2 0 changes sign, we may solve the full Sturm-Liouville system near each critical point u (i) 0c , a task that may be accomplished using perturbative [13] or numerical methods. In fact, as we shall see later on, in all examples considered in this paper, a zero mode always signifies a change of sign of ω 2 0 i.e.…”

“…In particular, in [13] it was proven that the longitudinal zero-mode solution can be expressed in terms of the u 0 -derivative of the length function as follows…”

“…Therefore, the solution exists only if 13) or, equivalently, if [13] 14) which requires that the derivative term should change sign at least once as u ranges in the interval u ∈ [u 0 , ∞). Therefore, given the critical points of the length function, we may determine all values u 0c of u 0 where the system (3.6) has a zero mode.…”

“…This description allows us to determine the zero modes to quite high accuracy by using approximate methods. Following [13], we outline the procedure below.…”