Physics Letters B 743 (2015) 340–346 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Analog geometry in an expanding fluid from AdS/CFT perspective ∗ Neven Bilic´ , Silvije Domazet, Dijana Tolic´ Division of Theoretical Physics, Rudjer Boškovi´c Institute, P.O. Box 180, 10001 Zagreb, Croatia a r t i c l e i n f o a b s t r a c t Article history: The dynamics of an expanding hadron fluid at temperatures below the chiral transition is studied in Received 16 February 2015 the framework of AdS/CFT correspondence. We establish a correspondence between the asymptotic AdS Accepted 5 March 2015 geometry in the 4 + 1dimensional bulk with the analog spacetime geometry on its 3 + 1dimensional Available online 9 March 2015 boundary with the background fluid undergoing a spherical Bjorken type expansion. The analog metric Editor: L. Alvarez-Gaumé tensor on the boundary depends locally on the soft pion dispersion relation and the four-velocity of Keywords: the fluid. The AdS/CFT correspondence provides a relation between the pion velocity and the critical Analogue gravity temperature of the chiral phase transition. AdS/CFT correspondence © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 Chiral phase transition (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Linear sigma model Bjorken expansion 1. Introduction found in the QCD vacuum and its non-vanishing value is a man- ifestation of chiral symmetry breaking [8]. The chiral symmetry The original formulation of gauge-gravity duality in the form of is restored at finite temperature through a chiral phase transition the so called AdS/CFT correspondence establishes an equivalence of which is believed to be of first or second order depending on the a four dimensional N = 4 supersymmetric Yang–Mills theory and underlying global symmetry [9]. string theory in a ten dimensional AdS5 × S5 bulk [1–3]. However, In the temperature range below the chiral transition point the the AdS/CFT correspondence goes beyond pure string theory as it thermodynamics of quarks and gluons may be investigated using links many other important theoretical and phenomenological is- the linear sigma model [10] which serves as an effective model sues such as fluid dynamics [4], thermal field theories, black hole for the low-temperature phase of QCD [11,12]. The original sigma 4 physics, quark-gluon plasma [5], gravity and cosmology. In partic- model is formulated as spontaneously broken ϕ theory with four ular, the AdS/CFT correspondence proved to be useful in studying 1 1 real scalar fields which constitute the ( 2 , 2 ) representation of the some properties of strongly interacting matter [6] described at the chiral SU(2) × SU(2). Hence, the model falls in the O(4) universal- fundamental level by a theory called quantum chromodynamics ity class owing to the isomorphism between the groups O(4) and N = (QCD), although 4 supersymmetric Yang–Mills theory differs SU(2) × SU(2). We shall consider here a linear sigma model with substantially from QCD. spontaneously broken O(N) symmetry, where N ≥ 2. According to Our purpose is to study in terms of the AdS/CFT correspondence the Goldstone theorem, the spontaneous symmetry breaking yields a class of field theories with spontaneously broken symmetry re- massless particles called Goldstone bosons the number of which de- stored at finite temperature. Spontaneous symmetry breaking is pends on the rank of the remaining unbroken symmetry. In the related to many phenomena in physics, such as, superfluidity, su- case of the O(N) group in the symmetry broken phase, i.e., at perconductivity, ferro-magnetism, Bose–Einstein condensation etc. temperatures below the point of the phase transition, there will Awellknown example which will be studied here in some detail be N − 1Goldstone bosons which we will call the pions. In the is the chiral symmetry breaking in strong interactions. At low en- symmetry broken phase the pions, in spite of being massless, prop- ergies, the QCD vacuum is characterized by a non-vanishing expec- agate slower than light owing to finite temperature effects [13–16]. tation value [7]: ψψ¯ ≈ (235 MeV)3, the so-called chiral conden- Moreover, the pion velocity approaches zero at the critical temper- sate. This quantity describes the density of quark-antiquark pairs ature. In the following we will use the term “chiral fluid” to denote a hadronic fluid in the symmetry broken phase consisting predom- inantly of massless pions. In our previous papers [17,18] we have * Corresponding author. E-mail addresses: [email protected] (N. Bilic),´ [email protected] (S. Domazet), demonstrated that perturbations in the chiral fluid undergoing a [email protected] (D. Tolic).´ radial Bjorken expansion propagate in curved geometry described http://dx.doi.org/10.1016/j.physletb.2015.03.009 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. N. Bili´c et al. / Physics Letters B 743 (2015) 340–346 341 2 by an effective analog metric of the Friedmann Robertson Walker m 2 =− 0 ≡ f 2 . (5) (FRW) type with hyperbolic spatial geometry. i λ π i As an application of the AdS/CFT duality in terms of D7-brane embeddings [19] the chiral phase transition can be regarded as a Redefining the fields i(x) →i + ϕi(x), the fields ϕi repre- transition from the Minkowski to black hole embeddings of the sent quantum fluctuations around the vacuum expectation values D7-brane in a D3-brane background. This has been exploited by i. It is convenient to choose here i = 0for i = 1, 2, .., N − 1, Mateos, Myers, and Thomson [20] who find a strong first order and N = fπ . At nonzero temperature the quantity N , usu- phase transition. Similarly, a first order chiral phase transition was ally referred to as the condensate, is temperature dependent and found by Aharony, Sonnenschein, and Yankielowicz [21] and Par- vanishes at the point of phase transition. In view of the usual phys- nachev and Sahakyan [22] in the Sakai–Sugimoto model [23]. In ical meaning of the ϕ-fields in the chiral SU(2) × SU(2) sigma this paper we consider a model of a brane world universe in which model it is customary to denote the N − 1 dimensional vector + the chiral fluid lives on the 3 1 dimensional boundary of the AdS5 (ϕ1, ..., ϕN−1) by π , the field ϕN by σ , and the condensate N bulk. We will combine the linear sigma model with a boost in- by σ . In this way one obtains the effective Lagrangian in which variant spherically symmetric Bjorken type expansion [24] and use the O(N) symmetry is explicitly broken down to O(N − 1). AdS/CFT techniques to establish a relation between the effective At and above a critical temperature Tc the symmetry will be re- analog geometry on the boundary and the bulk geometry which stored and all the mesons will have the same mass. Below the crit- 2 = 2 = 2 satisfies the field equations with negative cosmological constant. ical temperature the meson masses are mπ 0, and mσ 2λ σ . The formalism presented here could also be applied to the cal- The temperature dependence of σ is obtained by minimizing the culation of two point functions, Willson loops, and entanglement thermodynamic potential with respect to σ . Given N, fπ , and entropy for a spherically expanding Yang–Mills plasma as it was mσ , the extremum condition can be solved numerically at one N = recently done by Pedraza [25] for a linearly expanding 4su- loop order [12]. In this way, the value Tc = 183 MeV of the crit- persymmetric Yang–Mills plasma. ical temperature was found [18] for N = 4, fπ = 92.4MeV, and The remainder of the paper is organized as follows. In Section 2 mσ = 1GeVas a phenomenological input. we describe the dynamics of the chiral fluid and the corresponding Consider first a homogeneous steady flow of the medium con- + 3 1 dimensional analog geometry. In Section 3 we solve the Ein- sisting of pions at finite temperature in the Minkowski background stein equations in the bulk using the metric ansatz that respects with gμν = ημν . In a comoving reference frame (characterized by the spherical boost invariance of the fluid energy–momentum ten- uμ = (1, 0, 0, 0)) the effective metric (2) is diagonal with corre- sor at the boundary. We demonstrate a relationship of our solution sponding line element with the D3-brane solution of 10 dimensional supergravity. In Sec- 3 tion 4 we establish a connection of the bulk geometry with the f 2 = μ ν = 2 − 2 analog geometry on the boundary and derive the temperature de- ds Gμνdx dx fcπ dt dxi . (6) cπ pendence of the pion velocity. In the concluding section, Section 5, i=1 we summarize our results and discuss physical consequences. At nonzero temperature, f and cπ can be derived from the finite temperature self energy (q, T ) in the limit when the external 2. Expanding hadronic fluid momentum q approaches zero and can be expressed in terms of second derivatives of (q, T ) with respect to q and q . The quan- Consider a linear sigma model in a background medium at fi- 0 i nite temperature in a general curved spacetime. The dynamics of tities f and cπ as functions of temperature have been calculated at mesons in such a medium is described by an effective action with one loop level by Pisarski and Tytgat [13] in the low temperature O(N) symmetry [16] approximation √ 1 T 2 π 2 T 4 4π 2 T 4 = 4 − μν f ∼ 1 − − , c ∼ 1 − , (7) Seff d x G G ∂μ∂ν 2 2 2 π 2 2 2 12 fπ 9 fπ mσ 45 fπ mσ 2 cπ m λ and by Son and Stephanov for temperatures close to the chiral − 0 2 + (2)2 , (1) f 2 2 4 transition point [14,15] (see also [17]).