Advances in High Energy Physics

Volume 2017 (2017), Article ID 6124619, 11 pages

https://doi.org/10.1155/2017/6124619

## A Study of Confinement for Potentials on D3, M2, and M5 Branes

Instituto de Física, Universidade Federal de Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil

Correspondence should be addressed to Henrique Boschi-Filho; rb.jrfu.fi@ihcsob

Received 16 May 2017; Revised 13 August 2017; Accepted 23 August 2017; Published 3 October 2017

Academic Editor: Piero Nicolini

Copyright © 2017 Edward Quijada and Henrique Boschi-Filho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study analytically and numerically the interaction potentials between a pair of quark and antiquark on D3, M2, and M5 branes. These potentials are obtained using Maldacena’s method involving Wilson loops and present confining and nonconfining behaviours in different situations that we explore in this work. In particular, at the near horizon geometry, the potentials are nonconfining in agreement with conformal field theory expectations. On the other side, far from horizon, the dual field theories are no longer conformal and the potentials present confinement. This is in agreement with the behaviour of strings in flat space where the string mimics the expected flux tube of QCD. A study of the transition between the confining/nonconfining regimes in the three different scenarios (D3, M2, and M5) is also performed.

#### 1. Introduction

Usually, in quantum field theory, the Wilson loop operator is defined aswhere denotes a closed loop in space-time and the trace is over the fundamental representation of the gauge field with symmetry. In the particular case of a rectangular loop (of sides and ), it is possible to calculate (in the limit ) the expectation value for the Wilson loop:where can be identified with the energy of the quark-antiquark pair in the static limit.

Soon after the conjecture about the duality between M/string theory in AdS spaces and conformal gauge field theories [1–5], Maldacena [6] and Rey and Yee [7] (MRY) proposed a method to calculate expectation values of the Wilson loop for the large limit of field theories. This limit is calculated from a string theory in a given background using the gauge/gravity duality.

In this method, the expectation value of the Wilson loop is related to the worldsheet area of a string whose boundary is the loop in question such that

Maldacena used this approach to calculate the quark-antiquark potential for the string in the background [6] obtaining a nonconfining potential for the infinitely massive quark-antiquark pair, consistent with the conformal symmetry of the dual super Yang-Mills theory. In other backgrounds, the quark-antiquark potential can be confining as shown, for instance, in [8], where a confinement criterion was obtained.

This approach can also be extended to the finite-temperature case [9, 10] by considering an AdS Schwarzschild background. In this case, the temperature of the conformal dual theory is identified with the Hawking temperature of the black hole [11]. This situation also leads to a nonconfining potential for the quark-antiquark interaction.

The thermodynamics of D-brane probes in a black hole background were treated in [12]. These systems are holographically dual to a small number of flavours in a finite-temperature gauge theory. First-order phase transitions were found characterised by a confinement/deconfinement transition of quarks.

A phenomenological approach was also considered calculating the Wilson loop for the string in some holographic AdS/QCD models. For instance, the hard-wall model exhibits a confining behaviour [13, 14] reproducing the Cornell potential. At finite temperature, this calculation gives a second-order phase transition describing qualitatively a confinement/deconfinement phase transition [15]. Then, it was shown that a Hawking-Page phase transition [16] should occur for the hard- and soft-wall models at finite temperature [17–21]. In particular, for the soft-wall model, an interesting estimate of the deconfinement temperature was found [17], compatible with QCD expectations.

In a recent paper, some geometric configurations of a static string on a D3 brane background [22] and also a string-like object on M2 and M5 brane backgrounds [23] were studied. These geometric configurations correspond to a gauge theory which describes the quark-antiquark interaction on the branes. For some specific geodesic regimes, we found confining interactions and for others nonconfining potentials were found.

In this paper, we perform a systematic analytical and numerical study of the quark-antiquark potentials in D3, M2, and M5 brane backgrounds analysing their confining/nonconfining behaviours in different situations, always at zero temperature. In particular, at the near horizon geometry, the potentials are nonconfining in agreement with conformal field theory expectations. On the other side, far from horizon, the dual field theories are no longer conformal and the potentials present confinement. This is in agreement with the expected behaviour of strings in flat space where the string mimics the flux tube model of QCD. In the cases of M2 and M5 branes in M-theory, we choose a cigar-shaped membrane object such that stringy picture of the dual flux tube also holds. We also focus in searching for the point in the geodesics at which the zero temperature confinement/deconfinement transition takes place.

#### 2. Wilson Loops in D3, M2, and M5 Brane Spaces

We start this study by considering the Wilson loop on the background generated by a large number of coincident D3 branes in string theory in 10-dimensional space-time. The Nambu-Goto string action [24]is employed on this background, where the scale was set to , are the coordinates of the string worldsheet, and is the background metric. The specific form of the metric is given in the next section. It is considered that the quark-antiquark pair is contained in the D3 brane world which is attached to the ends of the open string that lives in 10 dimensions. For simplicity, we work in a static string configuration that is represented in Figure 1. This is achieved considering the large quark masses limit. As in the original Maldacena’s proposal [6], it is necessary to consider heavy quarks in order to compute the expectation value of the Wilson loop. For this reason, actually the right hand side of (3) contains the contribution of these heavy quark masses so it diverges. As a consequence, in general, it is necessary to subtract the quark masses from the divergent integral of the potential in order to obtain a finite result.