Turbulent meson condensation in quark deconfinement
Koji Hashimoto, Shunichiro Kinoshita, Keiju Murata, Takashi Oka
aa r X i v : . [ h e p - t h ] A ug AP-GR-114, OCU-PHYS-409, OU-HET-821, RIKEN-MP-93
Turbulent meson condensation in quark deconfinement
Koji Hashimoto , , Shunichiro Kinoshita , Keiju Murata , and Takashi Oka Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Mathematical Physics Lab., RIKEN Nishina Center, Saitama 351-0198, Japan Osaka City University Advanced Mathematical Institute, Osaka 558-8585, Japan Keio University, 4-1-1 Hiyoshi, Yokohama 223-8521, Japan and Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
In a QCD-like strongly coupled gauge theory at large N c , using the AdS/CFT correspondence,we find that heavy quark deconfinement is accompanied by a coherent condensation of higher mesonresonances. This is revealed in non-equilibrium deconfinement transitions triggered by static, as wellas, quenched electric fields even below the Schwinger limit. There, we observe a “turbulent” energyflow to higher meson modes, which finally results in the quark deconfinement. Our observation isconsistent with seeing deconfinement as a condensation of long QCD strings. Quark confinement is one of the most fundamental andchallenging problems in elementary particle physics, leftunsolved. Although quantum chromodynamics (QCD) isthe fundamental field theory describing quarks and glu-ons, their clear understanding is limited to the deconfinedphase at high energy or high temperature limits due tothe asymptotic freedom. We may benefit from employ-ing a more natural description of the zero temperaturehadron vacuum. A dual viewpoint of quark confinementin terms of the “fundamental” degrees of freedom at zerotemperature - mesons, is a plausible option.The mesons appear in families: they are categorizedby their spin/flavor quantum numbers, as well as a reso-nant excitation level n giving a resonance tower such as ρ (770) , ρ (1450) , ρ (1700) , ρ (1900) , · · · . In this Letter wefind a novel behavior of the higher meson resonances, i.e. ,mesons with large n . In the confined phase, when the de-confined phase is approached, we observe condensation ofhigher mesons . In this state, macroscopic number of thehigher meson resonances, with a characteristic distribu-tion, are excited. The condensed mesons have the samequantum number as the vacuum. The analysis is donevia the anti-de Sitter space (AdS)/conformal field the-ory (CFT) correspondence [1–3], one of the most reliabletools to study strongly-coupled gauge theories. By shift-ing our viewpoint from quark-gluon to meson degrees offreedom, we gain a simple and universal understanding ofthe confinement/deconfinement transition, with a bonusof solving mysteries in black holes physics through theAdS/CFT.The system we study is the N = 2 supersymmetric SU ( N c ) QCD which allows the simplest AdS/CFT treat-ment [4]. The deconfinement transition is induced by ex-ternal electric fields [26]. In static fields, the confinedphase becomes unstable in electric fields stronger thanthe Schwinger limit E = E Sch beyond which quarks areliberated from the confining force. We find that this in-stability is accompanied by the condensation of highermesons. A striking feature is revealed for the case of anelectric field quench: The kick from the quench triggersa domino-like energy transfer from low to high resonant meson modes. This leads to a dynamical deconfinementtransition [5] even below the Schwinger limit. The trans-fer we find resembles that of turbulence in classical hy-drodynamics as higher modes participate; thus we call ita “turbulent meson condensation” and suggest it beingresponsible for deconfinement.We remind that the N = 2 theory is a toy model:The meson sector is confined and has a discrete spectrumwhile the gluon sector is conformal and is always decon-fined. It resembles heavy quarkonia in a gluon plasma.Generically, quark deconfinement and gluon deconfine-ment can happen separately, as is known through char-monium experiments in heavy ion collisions. Here, weconcentrate on the deconfinement of heavy quarks andnot the gluons.The higher meson resonances are naturally interpretedas long QCD strings, therefore our finding is consistentwith interpreting deconfinement as condensation of QCDstrings [6] (see [7–10]). Under the condensation, a quarkcan propagate away from its partner antiquark by re-connecting the bond QCD string with the backgroundcondensed strings. The gravity dual of the deconfinedphase is with a black hole, so given the relation withlong fundamental strings [8], our result may shed lighton the issue of quantum black holes; In particular, ourtime-dependent analysis gives a singularity formation onthe flavor D-brane in AdS, a probe-brane version of theBizon-Rostworowski turbulent instability in AdS geome-tries [11]. Review: Meson effective action from AdS/CFT. — Theeffective field theory of mesons can be obtained for the N = 2 supersymmetric QCD in the large N c , λ ≡ N c g limits by the AdS/CFT correspondence[12, 13]. Themeson action is nothing but a D7-brane action in theAdS × S geometry; S = − π ) g l s Z d ξ p − det( g ab [ w ] + 2 πl s F ab ) , (1) ds = r R η µν dx µ dx ν + R r (cid:2) dρ + ρ d Ω + dw + d ¯ w (cid:3) , where r ≡ ρ + w + ¯ w , F ab = ∂ a A b − ∂ b A a , and theAdS curvature radius is R ≡ (2 λ ) / l s . For the follow-ing calculations, it is convenient to define a rescaled gaugepotential a a ≡ πl s R − A a . The D7-brane worldvolumefields are w ( x µ , ρ ) and a a ( x µ , ρ ). (We set ¯ w ( x µ , ρ ) = 0consistently because of U (1)-symmetry in ( w, ¯ w )-plane.)We denote the location of the D7-brane at the asymp-totic AdS boundary as w ( x µ , ρ = ∞ ) = R m . Here,the constant m is related to the quark mass m q as m q = ( λ/ π ) / m . A static solution of the D7-brane inthe AdS × S geometry is given by w ( x µ , ρ ) = R m and a a ( x µ , ρ ) = 0. Using the AdS/CFT dictionary, normal-izable fluctuations around the static solutions of w and a a are interpreted as the infinite towers of scalar mesons¯ ψψ and vector mesons ¯ ψγ µ ψ , respectively. (We omit thepseudo scalar mesons which are irrelevant to our discus-sion). In this paper, we focus on the meson condensationinduced by an electric field along the x -direction. Thus,we only consider fluctuations of w and a x .To derive the meson effective action, we use a coor-dinate z defined by ρ = R m √ − z /z (where z = 0 isthe AdS boundary, and z = 1 is the D7-brane center thatis closest to the Poincar´e horizon in the bulk AdS). Theworldvolume is effectively in a finite box along the AdSradial direction, to give a confined discrete spectrum aswe will see below. We expand the D7-brane action up tosecond order in the fluctuations χ ≡ ( R − w − m, a x ) as[12] S = Z dtd x Z dz − z z [ ˙ χ − m (1 − z ) χ ′ ] + O ( χ ) , where · ≡ ∂ t and ′ ≡ ∂ z . An irrelevant overall factor isneglected. The equation of motion for χ is (cid:0) ∂ t + H (cid:1) χ = 0 , H ≡ − m z − z ∂∂z (1 − z ) z ∂∂z . (2)The eigenfunction of H is given by e n ( z ) ≡ p n + 3)( n + 1)( n + 2) z F ( n + 3 , − n, z ), with theeigenvalue ω n = 4( n + 1)( n + 2) m , for the meson levelnumber n = 0 , , , · · · . Here F is the Gaussian hyperge-ometric function. The inner product is defined as ( f, g ) ≡ R dz z − (1 − z ) f ( z ) g ( z ), where ( e n , e m ) = δ mn is satis-fied. Note that an external electric field term, a x = − Et ,satisfies Eq. (2) although it is non-normalizable. Expand-ing the scalar/vector fields as χ = (0 , − Et ) + ∞ X n =0 c n ( t ) e n ( z ) , (3)we find an infinite tower of meson fields c n ( t ) sharing thesame quantum charge - higher meson resonances. Sub-stituting Eq. (3) back to Eq. (1), we obtain the mesoneffective action S = 12 Z d x ∞ X n =0 (cid:2) ˙ c n − ω n c n (cid:3) + interaction , (4) FIG. 1: (Color online) The shape of the probe D7-brane instatic electric fields in the unit of R = m = 1. The linescorrespond respectively to E/E
Sch = 0 . , . , . , e-0101e-0081e-0060.00010.011 0 5 10 15 20 25 30 FIG. 2: (Color online) Decomposed meson conden-sate log[ | c n | / | c | ] in static electric fields. ColorsRed/Green/Blue/Magenta/Cyan correspond respectively to E/E
Sch = 0 . , . , . , where we have omitted a constant term and total deriva-tive terms, while higher order nonlinear terms give riseto meson-meson interactions. From the effective action,we obtain the energy ε n ≡ ( ˙ c n + ω n c n ) stored in the n -th meson resonance, and the linearized total energy ε = P ∞ n =0 ε n . Higher meson condensation and deconfinement. — Theconfined phase becomes unstable in strong static electricfields. Here, we examine this from the viewpoint of mesoncondensation. In Eq. (4), the meson couplings depend onthe external electric field E nonlinearly. Mesons in a sin-gle flavor theory is neutral under E , but it can polarize,and non-linear E may cause a meson condensation. Wefirst solve the equations of motion obtained from the fullnonlinear D-brane action (1) with an external electricfield, and then decompose the solution χ ( t, z ) as Eq. (3).In this way, we can study how the infinite tower of mesons c n ( t ) behave towards the deconfinement transition.The static D7-brane solution in the presence of a con-stant electric field introduced by a x = − Et was obtainedin Refs. [14–16]. Also, the Schwinger limit E = E Sch =0 . m beyond which the first order phase transition todeconfinement occurs was found [15, 16]. Fig. 1 shows theshape of the D7-brane, which is the scalar field configu-ration w ( ρ ), for E/E
Sch = 0 . , . , . , | c n | / | c | as a function of n in Fig. 2, where we define | c n | ≡ p ( c scalar n ) + ( c vector n ) for illustration. As the electricfield increases, the higher meson condensate | c n | ( n ≫ | c | grows rapidly. Note that vec-tor mesons are not excited in static electric fields since thegauge potential is always given by a x = − Et (no highermodes). This implies that the condensed mesons havethe same quantum number as the vacuum. The geomet-rical reason of the condensation of the higher mesons issimple; The D7-brane bends singularly near the Poincar´ehorizon of AdS due to the nature of the metric and highereigenmodes are necessary to reproduce it. It is similar toFourier-decompose a delta-function (narrow Gaussian).“Flow” of energy from low to high resonant mesonmodes takes place at the same time. For static solutions,the energy stored by the n -the meson mode is given by ε n = ω n c n /
2. In Fig. 3, the meson energy distribution instatic E shows enhancement at higher modes as increas-ing E , that is, energy “flow” to higher meson resonances.The exponential behavior of the enegy distribution inFig. 3 for E Sch ≥ E provides a well-defined effective tem-perature [25]. At the critical embedding it turns to apower-law, which exhibits a Hagedorn behavior, and re-minds us of the Kolmogorov scaling.We conclude for the static case that just before thequark deconfinement induced by an applied electric field,meson resonances condense coherently. Meson turbulence in quenched electric fields. — Thehigher meson condensation seems to be a sufficient causeof quark deconfinement. This is clearly seen in a time-dependent, electric field quench that we study below.Starting from the E = 0 vacuum in the confined phase,we turn on the electric field smoothly to reach a finalvalue E f in the duration ∆ V [27]. In our previous work[5], we found that the system deconfines with an emer-gence of a strongly redshifted region on the D7-brane to-ward a naked-singularity formation. This is interpretedas an instability toward deconfinement, which happens, e-0151e-0101e-0051 0 5 10 15 20 25 30 FIG. 3: (Color online) The energy distribution for the n -thmeson resonance. The color of the dots follows that of theprevious figure. The inset is the log-log plot of the energydistribution for the critical embedding, in which we take themeson mass spectrum ω n as the horizontal axis. to our surprise, even when the final field strength is be-low the Schwinger limit. In the following, we choose aweak electric field ( E f /E Sch = 0 . m ∆ V = 2, in which sub-Schwinger-limit decon-finement is realized.The electric field induces nontrivial dynamics in themeson sector. We decompose the time-dependent solu-tions of the meson fields [5] and calculate the energy ε n of the meson resonances. In Fig. 4, the time evolutionof the condensate | c n | is shown for several time slices mt = 15, 40, and 49 .
3, while the time mt = 49 . ε n /ε . We find thatthe condensate and the energy are transferred to highermeson modes during the time evolution. This tendencyis similar to the static case in which they are “trans-ferred” more in stronger electric fields. This indicatesthat higher meson condensation is universally related toquark confinement.The observed time evolution of the distribution sug-gests that turbulence is taking place in the meson sec-tor. This is because higher modes have smaller wavelengths, and the transfer of energy and momentum indi-cates that smaller structure are being organized duringthe time evolution toward deconfinement. Our findingcan be considered as a probe-brane version of the tur-bulent instability of the AdS spacetime [11] in which anon-linear evolution of a perturbed AdS spacetime causesa high-momentum instability resulting in a black hole for-mation. Conclusion and discussions. — In this work we found, viathe AdS/CFT correspondence, that higher meson reso-nances become condensed near the deconfinement transi-tion caused by electric fields. This was confirmed for boththe static electric fields and the time-dependent quenchedelectric field, in N = 2 supersymmetric QCD which mod- e-00 e-0061e-0050.00010.0010.010.11 0 5 10 15 20 25 30 FIG. 4: (Color online) Meson condensate induced by anelectric field quench. Data for times mt = 15, 40, and 49 . E f /E Sch = 0 . m ∆ V = 2. Aclear (non-thermal) growth at large n is found along the timeevolution. e-0101e-0081e-0060.00010.011 0 5 10 15 20 25 30 FIG. 5: (Color online) Turbulent behavior of mesons towarddeconfinement. The energy is transferred to higher modesaccordingly. els heavy quarkonia in a gluon plasma. No internal sym-metry is broken during this process since the mesons thatparticipates in the deconfinement with their condensa-tion have the same quantum numbers as the vacuum andthe external force.The physics in the gravity dual side is simple. Thegravity-dual of the scalar meson condensation is the de-formation of the probe D7-brane. The electric fieldbreaks the supersymmetries of the brane configurationmaking the Poincar´e horizon to attract the D7-brane andto bend it. The tip of the D7-brane becomes sharper asthe electric field increases, and finally when it exceeds thecritical value, deconfinement transition occurs. At theverge of deconfinement (at which we can use the mesonterminology), higher meson resonances condense coher-ently.A single meson can be regarded as quarks connectedby a QCD string and higher meson resonances as its co-herent fluctuation. At the deconfinement transition, oneneeds to dissociate the QCD string to liberate the quarks.The dissociation of the QCD string is described by the coherent dynamics of the excited mesons. Our findingis consistent with this view, and in particular with de-confinement as QCD string condensation [6–10]. Our re-sult seems further consistent with Hagedorn transitionin string theory [17, 18] [28] and the black hole / stringcorrespondence [19, 20].In quark models, the higher resonant meson naivelycorresponds to a state ¯ ψ ( x ) U ( x, y ) ψ ( y ), where U ( x, y ) ≡ P exp[ i R yx G ] is an open Wilson line with the gluon field G µ , which can be expanded as ¯ ψ ( x ) (cid:3) n ψ ( x ) where (cid:3) isthe covariant Laplacian. Our conjecture waits for a con-firmation by lattice QCD on condensation of this opera-tor.The meson turbulence obviously share some featureswith the AdS turbulence [11, 21]. It may have relationwith thermalization and numerical simulations of glasma[22–24]. Our observation is also valid in a temperature-driven deconfinement [25]. Thus, we conjecture thatquark deconfinement is universally associated with a con-densation of higher meson resonances. Acknowledgment. — K. H. would like to thank A. Buchel,H. Fukaya, D.-K. Hong, N. Iqbal, K.-Y. Kim, J. Malda-cena, S. Sugimoto, S. Yamaguchi and P. Yi for valuablediscussions, and APCTP focus week program for its hos-pitality. This research was partially supported by theRIKEN iTHES project. [1] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998)[hep-th/9711200].[2] S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys.Lett. B ,105 (1998).[3] E. Witten, Adv.Theor.Math.Phys. , 253 (1998).[4] A. Karch and E. Katz, JHEP