Meson turbulence at quark deconfinement from AdS/CFT†

K. Hashimoto,∗1,∗2 S. Kinoshita,∗3 K. Murata,∗4 and T. Oka∗5

How the quarks are confined at the vacuum of quan-tum chromodynamics (QCD) is one of the most fun-damental questions in the standard model of particlephysics. We would like to find a universal feature ofthe deconfinement. To understand the nature of thequark confinement, we need a proper observable whichexhibits a universal behavior irrespective of how webreak the confinement. In this paper, we propose auniversal behavior of resonant mesons and name it me-son turbulence.

Following our previous paper1), we find that a par-ticular behavior of resonant mesons (excited states ofmesons) can be an indicator of the deconfinement. Themeson turbulence is a power-law scaling of the reso-nant meson condensations. For the the resonant me-son level n (n = 0, 1, 2, · · ·), the condensation of themeson ⟨cn(x, t)⟩ with its mass ωn causes the n-th me-son energy εn scaling as (ωn)α with a constant powerα. This coefficient α will be unique for a given theory,and does not depend on how one breaks the confine-ment. In particular, for the theory which we analyze inthis paper, that is N = 2 supersymmetric QCD withN = 4 supersymmetric Yang-Mills as its gluon sectorat large Nc at strong coupling, the universal power-lawscaling parameter α is found to be

⟨εn⟩ ∝ (ωn)α, α = −5 . (1)

where εn is the energy of the n-th meson resonance.Normally, for example at a finite temperature, the en-ergy stored at the n-th level of the resonant mesonshould be a thermal distribution, εn ∝ exp[−ωn/T ].The thermal distribution is Maxwell-Boltzmann statis-tics, in which the higher (more massive) meson modesare exponentially suppressed. However, we conjecturethat this standard exponential suppression will be re-placed by a power-law near any kind of the deconfine-ment transitions. If we think of the meson resonantlevel n as a kind of internal momentum, then the en-ergy flow to higher n can be regarded as a so-calledweak turbulence. This is why we call the phenomenonmeson turbulence, and the level n can be indeed re-garded as a momentum in holographic direction in theAdS/CFT correspondence.

The reason we came to the universal power behavioris quite simple. We combined two well-known things,

• Mesons are excitations of an open QCD string.† Condensed from the article in arXiv:1412.4964∗1 RIKEN Nishina Center∗2 Department of Physics, Osaka University∗3 Osaka City University Advanced Mathematical Institute∗4 Keio University∗5 Department of Applied Physics, University of Tokyo

Fig. 1. A schematic picture of the deconfinement phase as

condensation of QCD strings. Left: we add a meson (a

pair of a quark and an anti-quark connected by a QCD

string) to the system. Right: due to the background

condensed QCD strings, the QCD string can be recon-

nected, and the quark can freely propagate away from

the anti-quark.

• Deconfinement phase is described by a condensa-tion of long strings.

Combining these two leads us to the conjecture that thedeconfinement of quarks is indicated by a condensationof higher meson resonances. More precisely, we claimthat the condensation should be turbulent: the highermode condensation is not suppressed exponentially butbehaves with a power-law.

We shall investigate various deconfining transitionsin this paper, to check the universality of our conjec-ture of the meson turbulence. First, we work with astatic case. A nonzero electric field is a good exam-ple since a strong electric field can make the quark-antiquark pair dissociate. Then we investigate time-dependent setup. The virtue of the AdS/CFT corre-spondence is that time-dependent analysis is possible,as opposed to lattice simulations of QCD. To demon-strate the universality of the meson turbulence, wework in two examples: (1) electric-field quench, and(2) quark-mass quench. In all cases, we find numer-ically the meson turbulence and the universal powerlaw with the power α = −5.

The universality we found in this paper strongly in-dicates that the meson turbulence is a universal phe-nomena which is independent of how one breaks theconfinement.

References1) K. Hashimoto, S. Kinoshita, K. Murata and T. Oka,

arXiv:1408.6293 [hep-th].

Electromagnetic instability in holographic QCD†

K. Hashimoto,∗1,∗3 T. Oka,∗2 and A. Sonoda∗3

Schwinger effect2) is one of the most interesting phe-nomena in particle physics. This is a phenomenon thata pair creation of charged particles occur under an ex-ternal field such as an electromagnetic field. Schwingerobtained the creation rate of an electron positron pairby evaluating the imaginary part of Euler-HeisenbergLagrangian1), which is an effective Lagrangian for aconstant electric field. This rate Γ is derived asΓ ∼ exp

(−πm2

e/eE)

to leading order and has a formwith a negative power in the gauge coupling e. So theScwinger effect is a non-perturbative effect. Here, me

is the electron mass and E is an electric field. A criticalelectric field necessary energy for the electron positronpair creation is Ecr ∼ m2

ec3/eh̄, and the strength is

about 1018 [V/m]. So, it is a phenomenon which showsup only under strong electromagnetic fields.

Recently, we have seen advance in research on astrong electromagnetic field in both theoretical and ex-perimental aspects of hadron physics. At the heavy ioncollision in RHIC and LHC, it is expected that a strongmagnetic field is generated by a collision of chargedparticles accelerated at about the speed of light.

Within the AdS/CFT framework, the quark paircreation rate in the strongly coupled N = 4 super-symmetric Yang-Mills theory was obtained in3). Onthe other hand, two of the present authors obtainedthe vacuum decay rate, which can be identified as thecreation rate of quark-antiquark pairs, in N = 2 super-symmetric QCD(SQCD) by using a different method4)

in AdS/CFT correspondence: the imaginary part ofthe probe D-brane action. D3-D7 brane system corre-sponds to N = 4 supersymmetric SU(Nc) Yang-Millstheory including an N = 2 hypermultiplet in the fun-damental representation of the SU(Nc) gauge group.They obtained the creation rate of the quark antiquarkin the N = 2 SQCD under a constant electric fieldby evaluating the imaginary part of the D7-brane ac-tion. Then, the present authors evaluated the imag-inary part of the D7-brane action including not onlya constant electric field but also a constant magneticfield and obtained the creation rate of the quarks andantiquarks in the N = 2 SQCD5).

We summarize the properties of the creation ratein both electric and magnetic fields obtained in5) forN = 2 SQCD as follows. We derived the Euler-Heisenberg Lagrangian for a constant electromagneticfield in N = 2 SQCD at large Nc and at strong cou-pling. Then, we obtained the creation rate of thequarks and antiquarks by evaluating the imaginary

† Condensed from the article in arXiv:1412.4254∗1 RIKEN Nishina Center∗2 Department of Applied Physics, University of Tokyo∗3 Department of Physics, Osaka University

part of the Lagrangian. We found that the creationrate diverges at a zero temperature in the masslessquark limit while it becomes finite when we introducea nonzero temperature. The divergence of the creationrate is influenced not only by a constant electric fieldbut also by a constant magnetic field. The results inSQCD showed similarities with the creation rate ofthe electron positron pair in N = 2 supersymmetricQED(SQED) in constant electromagnetic field.

In this paper, we study the quark antiquark paircreation in non-supersymmetric QCD at large Nc atstrong coupling, and the imaginary part of D8-braneaction in a constant electromagnetic field. The holo-graphic models are the Sakai-Sugimoto model6) andits deformed version. Our findings in this paper are asfollows:

• We derive the Euler-Heisenberg Lagrangian forconfining gauge theories: the Sakai-Sugimotomodel and the deformed Sakai-Sugimoto model.We obtain the creation rate of the quark antiquarkpair under the electromagnetic field, by evaluatingthe imaginary part of the D-brane actions.

• The imaginary part is found to increase with themagnetic field parallel to the electric field, whileit decreases with the magnetic field perpendicularto the electric field. So the vacuum instabilitystrongly depends on the direction of the appliedmagnetic field relative to the electric field.

• We obtain a critical value of the electric field, i.e.,the Schwinger limit, by using the condition thatthe D-brane action has the imaginary part. Inthe case of the Sakai-Sugimoto model, the criticalelectric field corresponds to a QCD string tensionbetween a quark and an antiquark.

As for the first part among above, a result with onlyan electric field was reported in Ref. 7). We analyzegeneric electric and magnetic fields in this paper.

References1) J. S. Schwinger, Phys. Rev. 82 (1951) 664.2) W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714

[physics/0605038].3) G. W. Semenoff and K. Zarembo, Phys. Rev. Lett. 107

(2011) 171601 [arXiv:1109.2920 [hep-th]].4) K. Hashimoto and T. Oka, JHEP 1310, 116 (2013)

[arXiv:1307.7423].5) K. Hashimoto, T. Oka and A. Sonoda, JHEP 1406

(2014) 085 [arXiv:1403.6336 [hep-th]].6) T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113

(2005) 843 [hep-th/0412141].7) K. Y. Kim, S. J. Sin and I. Zahed, JHEP 0807, 096

(2008) [arXiv:0803.0318 [hep-th]].

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