Strongly-coupled anisotropic gauge theories and holography in 5D Einstein–Gauss–Bonnet gravity

Eur. Phys. J. C (2023) 83:750 https://doi.org/10.1140/epjc/s10052-023-11913-0 Regular Article - Theoretical Physics Strongly-coupled anisotropic gauge theories and holography in 5D Einstein–Gauss–Bonnet gravity S. N. Sajadia , H. R. Safarib School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran Received: 8 May 2023 / Accepted: 9 August 2023 © The Author(s) 2023 Abstract In this paper we study uncharged, non-conformal and anisotropic systems with strong interactions using the gauge-gravity duality by considering Einstein-QuadraticAxion-Dilaton action in five dimension. In fact we would like to gain insight into the influence of higher derivative gravity on the QCD system. At finite temperature, we obtain an anisotropic black brane solution to a 5D Einstein–Gauss– Bonnet-Axion-Dilaton system. The system has been investigated and the effect of the parameter of theory has been considered. The blackening function supports the thermodynamical phase transition between small/large and AdS/large black brane for suitable parameters. We also study transport and diffusion properties, and observe in particular that the butterfly velocity that characterizes both diffusion and growth of chaos transverse to the anisotropic direction saturates a constant value in the IR which can exceed the bound given by the conformal value. We also determine the imaginary part of the heavy quark potential in a strongly coupled plasma dual to Gauss–Bonnet gravity. 1 Introduction Quantum Chromodynamics (QCD) is a non-abelian gauge theory that describes the behavior of the strong nuclear force. One of the key features of QCD is the idea of color confinement, which states that quarks and gluons cannot exist in isolation, but are always bound together to form particles such as protons and neutrons. Another important aspect of QCD is the concept of asymptotic freedom, which describes how the strong force becomes weaker at higher energies or shorter distances. The mechanism behind these two features is not well understood, and remains an active area of research. A a e-mail: (corresponding author) b e-mail: 0123456789().: V,-vol possible way to study is gauge/gravity duality which is a duality between certain strongly interacting quantum field theories in one dimension and a weakly interacting gravitational theory in one higher dimension. In this framework, the extra dimension is interpreted as a scale parameter that controls the energy scale of the QCD theory [1–5]. Quark-gluon plasma (QGP)-is believed to have existed in the early universe, in the cores of compact stars, and can also be created in high-energy heavy ion collisions in anisotropic way- is a state of matter in which, the quarks and gluons are no longer confined within the hadrons, and exist as a deconfined. The anisotropic black hole solutions in the gauge/gravity correspondence and applications to the QGP have been investigated in [6–26]. Higherorder gravitational models have recently received attention [27–29], because of string theory predicts that at low energies Einstein’s equations are subject to first-order corrections due to the interactions of the strings with the additional dimensions. The natural correction of Lovelock gravity to the Einstein–Hilbert gravity appears in five and higher dimensions and is given by a precise combination of quadratic curvature terms yields the second-order field equations known as the Gauss–Bonnet term [30–32]. The different aspect of this theory from cosmology to black hole solutions have been studied in [33–37]. It was noticed some time ago that the GB coupling is constrained by causality and unitarity reasons in AdS to lie in a dimension-dependent interval [39]. Moreover, and most importantly, this interval is further narrowed by more detailed and subtle causality arguments connected to Eikonal graviton scattering [40]. In this paper, we extend the work of [6] to the Einstein Quadratic Gravity, which is general relativity extended by quadratic curvature invariant in the action to find the effect of higher derivative terms on QGP, with the difference that in [6], they were fixed V (φ) and Z (φ), which were chosen in order to obtain specific properties in the IR and UV. In the present paper, however, we fix both V (φ) and Z (φ) from the equations of motion, because 123 750 Page 2 of 13 the functionality of these potentials in terms of the couplings of QGP is unknown to us. The paper is organized as follows. In Sect. 2 we construct the anisotropic 5-dimensional solution with an arbitrary dynamical exponent, an exponential quadratic warp function, in the framework of EGB gravity. We obtained the approximately solution for blackening function and other unknown quantities up to the first order of the theoretical parameter. We have shown the behavior of the quantities with plots and we discuss the thermodynamics of the constructed background in Sect. 2.1. In Sect. 2.2, we obtain transport and diffusion properties in anisotropic theories and observe in particular that the butterfly velocity that characterizes both diffusion and growth of chaos transverse to the anisotropic direction saturates a constant value in the IR which can exceed the bound given by the conformal value. In Sect. 2.3, the imaginary part of the quark–antiquark potential in different directions. In Sect. 2.4, we obtained the Jet quenching related to the suppression of high-energy jets of particles produced in high energy heavy-ion collisions due to their interaction with QGP. We finish the paper with some concluding remarks in Sect. 3. 2 Basic formalism We consider a 5-dimensional Einstein-Quadratic-AxionDilaton action as follows  √ 1 (1) d 5 x −gL , S= 16π G 5 where the Lagrangian is L = R + γ Rabcd R abcd + β Rab R ab + α R 2 1 − ∂μ φ∂ μ φ + V (φ) 2 1 − Z (φ)∂μ χ ∂ μ χ , (2) 2 and V (φ) is the potential energy for the dilaton field φ, Z (φ) is the coupling φ to the axion field χ . G 5 is the Newton constant in five dimensions and α, β and γ are coupling constants of theory. The Einstein and Quadratic terms describe the gravitational interaction between matter in QCD, here the matter is quarks, gluons and plasma. The dilaton field modifies the running of the QCD coupling constant which reflected the non conformality of QCD in gravity. The axion, which is dual to the gauge theory θ -term (is a term in the QCD Lagrangian that describes the possibility of CP violation in strong interactions), is responsible for inducing the anisotropy [51,52]. The variation of the action (1) over metric gμν , the scalar field φ and χ gives the field equations as follows E μν = G μν 123 Eur. Phys. J. C (2023) 83:750     1 +α 2R Rμν − gμν R + 2(gμν  − ∇μ ∇ν )R 4  +β (gμν  − ∇μ ∇ν )R + G μν + 2R λρ   1 × Rμλνρ − gμν Rλρ 4  1 +γ − gμν Rαβγ η R αβγ η + 2Rμλρσ Rν λρσ 2  + 4Rμλνρ R λρ − 4Rμσ Rνσ + 4Rμν − 2∇μ ∇ν R = 1 1 ∇α φ∇β φ + gαβ V (φ) 2 2 1 1 γ − gαβ ∇γ φ∇ φ + Z (φ)∇α χ ∇ (...truncated)