Baryon superfluids in AdS/CFT with flavor

Published for SISSA by Springer Received: November 30, 2016 Accepted: January 21, 2017 Published: January 31, 2017 Carlos Hoyos,a Georgios Itsiosa,b and Orestis Vasilakisa a Department of Physics, Universidad de Oviedo, Avda. Calvo Sotelo 18, ES-33007 Oviedo, Spain b Instituto de Fı́sica Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil E-mail: , , Abstract: Baryonic matter is notoriously difficult to deal with in the large-N limit, as baryons become operators of very large dimension with N fields in the fundamental representation. This issue is also present in gauge/gravity duals as baryons are described by very heavy localized objects. There are however alternative large-N extrapolations of QCD where small baryonic operators exist and can be treated on an equal footing to mesons. We explore the possibility of turning on a finite density of “light” baryons in a theory with a hadronic mass gap using a gauge/gravity construction based on the D3/D7 intersection. We find a novel phase with spontaneous breaking of baryon symmetry at zero temperature. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Holography and quark-gluon plasmas ArXiv ePrint: 1611.07029 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP01(2017)139 JHEP01(2017)139 Baryon superfluids in AdS/CFT with flavor Contents 1 2 A holographic model with light baryons 4 3 Effective action of fields dual to light baryons 3.1 Small amplitude expansion 3.2 Integration over S 3 7 8 10 4 Spontaneous breaking of baryon symmetry 4.1 Small amplitude solutions of charged fields 4.2 Backreaction on the gauge field 4.3 Free energy 11 12 15 17 5 Ground state in a simple case 5.1 Effective action and equations 5.2 Solutions 5.3 Free energy and thermodynamics 17 18 20 21 6 Summary and outlook 23 A Explicit form of the orbifold projection 25 B Projected form of covariant derivatives and commutators 26 1 Introduction It is notoriously difficult to describe from first principles dense baryonic matter in QCD at small temperatures and large densities. Perturbation theory can be used only at extremely high densities [1, 2]. Lattice calculations on the other hand are restricted to values of the baryon chemical potential smaller than the temperature [3]. One then has to rely on phenomenological models, but those are usually fitted to describe the physics in very different regimes, so it is far from clear that they can give an accurate description (see e.g. section 7.3 of [4] for a review). A common difficulty is that models are usually adapted to describe either quark or hadronic matter, but there should be a transition (which could be smooth) between the two as the density is changed. –1– JHEP01(2017)139 1 Introduction It would be very interesting to have a theory that can be described in all density regimes from first principles. A natural proposal is to study gauge/gravity models [5–7]. A strongly coupled field theory in the large-Nc limit has a holographic dual description in terms of a classical higher dimensional geometry, which is a black hole if the temperature is nonzero [8]. Flavor degrees of freedom are introduced by adding D-branes in the geometry (we will refer to them as “flavor branes”) [9]. If the number of flavors is much smaller than the number of colors Nf  Nc , the branes can be treated as probes. Flavor currents are dual to gauge fields living in the worldvolume of the branes, in such a way that a state with finite density is realized by having non-zero electric flux on the brane. The second possibility is that the strings extend from the flavor branes to a different kind of brane, dual to a “baryon vertex”, which wraps the internal directions in the geometry and is point-like in the field theory directions [11, 12]. The tension of the strings produces a force such that typically at the equilibrium configuration the brane dual to the baryon vertex lies on the flavor brane and can be described as a solitonic configuration on the flavor brane worldvolume [13–15]. This corresponds to baryonic matter in the dual field theory. Although holographic models can accommodate both quark and baryonic matter in this fashion, there is a clear asymmetry between the two. In order to describe baryonic matter one needs to study multi-soliton solutions [16–19] or use a phenomenological approach if one is interested in homogeneous states [20–29], with the drawback that the physical properties of the state depends on the assumptions one needs to make. Furthermore, stable soliton solutions have sizes that are typically of the order of the string scale [14], thus casting doubts on the validity of the brane action used to find those solutions. Therefore, our understanding of baryonic matter in holographic models is on much more shaky ground than that of quark matter. The difference between quark and baryonic matter in holographic models can be traced back to the large-Nc limit. Baryonic operators are constructed with Nc fundamental fields, thus they are very heavy objects and this is reflected in the holographic dual description, where they are described by branes or solitonic configurations in the flavor branes. Mesonic operators on the other hand can be constructed with a small number of fields and have a holographic dual description in terms of open strings ending on the flavor branes, or small fluctuations of the fields living in their worldvolume. This hierarchy between baryons and mesons is an artifact of the large-Nc limit and is not observed in real QCD (see e.g. [30]). It is thus desirable to study different models where this distinction is erased. –2– JHEP01(2017)139 The worldvolume electric flux has to be sourced by some charges, which we can think of as open strings attached to the brane. In the models that are usually considered, the two string endpoints carry opposite charges, so in order to have a non-zero density one of the endpoints should end on the brane and the other somewhere else. A possibility is that the strings extend from the flavor brane to the horizon. One can think of this situation as having quarks in a plasma. A finite density of them will pull the brane embedding, in such a way that the finite density state can be described by a brane embedding that reaches the horizon from where the electric field is sourced [10]. 1 Other types of large-Nc equivalences have been proposed as well, see e.g. [33]. CFL phases in a model with flavor branes were introduced in [38], however those still preserve a U(1) baryon symmetry. 2 –3– JHEP01(2017)139 The holographic picture gives us a clue about how to do this. Mesons are open strings attached to the flavor branes. They have zero baryon number because the endpoints of each string have opposite charge. This suggests that one could describe states with nonzero baryon number with open strings if both endpoints had the same charge, b (...truncated)