PHYSICAL REVIEW D 98, 106007 (2018) Chiral symmetry breaking and restoration in 2 + 1 dimensions from holography: Magnetic and inverse magnetic catalysis † ‡ Diego M. Rodrigues,1,* Danning Li,2, Eduardo Folco Capossoli,1,3, and Henrique Boschi-Filho1,§ 1Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972, Rio de Janeiro RJ, Brazil 2Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China 3Departamento de Física and Mestrado Profissional em Práticas da Educação Básica (MPPEB), Col´egio Pedro II, 20.921-903, Rio de Janeiro RJ, Brazil (Received 3 August 2018; published 9 November 2018) We study the chiral symmetry breaking and restoration in (2 þ 1)-dimensional gauge theories from the holographic hardwall and softwall models. We describe the behavior of the chiral condensate in the presence of an external magnetic field for both models at finite temperature. For the hardwall model we find magnetic catalysis (MC) in different setups. For the softwall model we find inverse magnetic catalysis (IMC) and MC in different situations. We also find for the softwall model a crossover transition from IMC to MC at a pseudocritical magnetic field. This study also shows spontaneous symmetry breaking for both models. Interestingly, for B ¼ 0 in the softwall model, we found a nontrivial expectation value for the chiral condensate. DOI: 10.1103/PhysRevD.98.106007 I. INTRODUCTION in order to reproduce the nonperturbative physics of QCD, one must break the conformal invariance in the Much theoretical progress on the nonperturbative phys- original AdS=CFT correspondence. There are two well- ics of relativistic quantum field theories (RQFT) has been known models which do this symmetry breaking: the achieved recently. In particular, the role played by an hardwall [19–25] and softwall [26–28] models. In par- external magnetic field in these RQFT has been inves- ticular, some researchers have used these models to tigated in many works, especially in connection with QCD discuss the IMC as a perturbation in powers of the [1–12]. In the context of lattice QCD, for the chiral phase magnetic field [9–12], and can only be trusted for weak transition, an inverse magnetic catalysis (IMC) has been 1 2 observed, i.e., the decreasing of the critical temperature fields, that means eB < GeV . Since their approach is (T )with increasing magnetic field (B) for eB ∼ 1 GeV2 perturbative in the magnetic field B, they could not c 1 2 eB ∼ 3 2 predict what would happen for eB > GeV . [13] and more recently for GeV [14]. This is in 2 þ 1 contrast with what would be expected: a magnetic catalysis Concerning free fermions in dimensions in the (MC), meaning the increasing of the critical temperature presence of a magnetic field, we know that, perturbatively, with increasing magnetic field [4]. the phenomenon of magnetic catalysis is due to the effect of A promising approach to investigate the phenomena of the magnetic field itself, which is a strong catalyst of IMC and MC is based on the AdS=CFT correspondence, dynamical chiral symmetry breaking, leading to the gen- or holographic duality [15–18]. Such duality has become eration of a nonzero chiral condensate in the chiral limit → 0 very useful to address strongly coupled gauge theories, (m ), given by [3] including the nonperturbative regime of QCD. However, hψψ¯ i ∝ eB ð Þ 2π : 1 *[email protected] † [email protected] ‡ At finite temperature, also in the perturbative regime, it was [email protected] §[email protected] shown in [29] that the chiral condensate is extremely unstable, meaning that it vanishes as soon as we introduce Published by the American Physical Society under the terms of a heat bath with and without chemical potential. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to In this work we investigate, nonperturbatively, the chiral the author(s) and the published article’s title, journal citation, phase transition and symmetry restoration in (2 þ 1)- and DOI. Funded by SCOAP3. dimensional holographic gauge theories in an external 2470-0010=2018=98(10)=106007(11) 106007-1 Published by the American Physical Society RODRIGUES, LI, CAPOSSOLI, and BOSCHI-FILHO PHYS. REV. D 98, 106007 (2018) magnetic field from the hardwall and softwall models. We with F2 in (2) given by confirm the picture mentioned before, but we also find 2 2 4 IMC, in the softwall model, for small magnetic fields (in 2 ¼ B z ð Þ F 4 : 9 units of the string tension squared), in agreement with the L lattice results in 3 þ 1 dimensions, while the traditional The general solution for Eqs. (7) and (8) is given by MC behavior happens for large magnetic fields in this 2 4 3 nonperturbative approach. In the hardwall model, we only fðzÞ¼1 þ B z þ bz : ð10Þ find MC. At finite temperature our approach predicts a If we impose the horizon condition fðz ¼ z Þ¼0, we find very fast decrease in the chiral condensate as we increase H 3 the temperature, which means that we do not observe the ð Þ¼1 þ 2 3ð − Þ − z ð Þ fBT z B z z zH 3 : 11 instability found in the perturbative thermal field theory zH computation presented in [29]. This solution was found recently in [31,32]. Equation (11) corresponds to the AdS4 in the presence of a background II. THE CHIRAL CONDENSATE AND CHIRAL magnetic field at finite temperature, T, where zH is the ð ¼ Þ¼0 SYMMETRY BREAKING IN THE horizon position, such that fBT z zH . One can note HOLOGRAPHIC FRAMEWORK that this solution indeed satisfies both differential equa- tions (7) and (8). Note that the solution Eq. (11) implies the A. Background geometry existence of an inner and outer horizon. The outer horizon 0 ð ¼ Þ 0 Via Kaluza-Klein dimensional reduction, the supergrav- satisfies fBT z zH < and is the physically relevant 7 ity theory on AdS4 × S may be consistently truncated to one. The temperature is given by the Hawking formula Einstein-Maxwell theory on AdS4 [30]. The action for this j 0 ð ¼ Þj ¼ fBT z zH ð Þ theory, in Euclidean signature, is given by T 4π : 12 Z 1 pffiffiffi Using (11) and the condition f0 ðz ¼ z Þ < 0, we have S ¼ − d4x gðR − 2Λ − L2F FMNÞ; ð Þ BT H 2κ2 MN 2 4 1 3 3 ð Þ¼ − 2 3 4 ð Þ T zH;B B zH ;zH < 2 : 13 2 4π zH B where κ4 is the four-dimensional coupling constant, which is proportional to the four-dimensional Newton’s constant In what follows and in the rest of this work we set the 2 4 (κ4 ≡ 8πG4); d x ≡ dτdx1dx2dz; R is the Ricci scalar; and AdS radius L ¼ 1. Λ is the negative cosmological constant. The field equations coming from the bulk action (2) B. Holographic setup for chiral symmetry breaking together with the Bianchi identity are [30] Here we describe how to realize the chiral symmetry breaking of SUðNfÞ × SUðNfÞ → SUðNfÞ in the 2 1 2 3 L R diag R ¼ 2L FP F − g F − g ; ð3Þ hardwall and softwall models. For both models we consider MN M NP 4 MN L2 MN the action Z MN 1 3 pffiffiffi −Φ † ∇MF ¼ 0: ð4Þ ¼ − ð M þ S 2 d xdz ge Tr DMX D X VX 2κ4 Our ansatz for the metric and the background magnetic − ð 2 þ 2 ÞÞ ð Þ FL FR ; 14 field to solve (3) are given by where X is a complex scalar field; DM is the covariant 2 2 derivative defined as D X ¼ ∂ X þ iAL X − iXAR , 2 ¼ L ð Þ τ2 þ dz þ 2 þ 2 ð Þ M M M M ds 2 f z d dx1 dx2 ; 5 L;R z fðzÞ with AM being the chiral left- and right-handed gauge fields; FMN the field strength defined as FMN ¼ ∂MAN − ∂ − ½ ¼ 2 † þ λð † Þ2 þ … F ¼ Bdx1 ∧ dx2: ð6Þ NAM i AM;AN ; and VX M4X X X X is the potential for the complex scalar field, where M4 is the Using this ansatz, the field equations (3) are simplified mass of the complex scalar field X. From the AdS4=CFT3 and given by correspondence we have 2 00 0 2 4 2 ¼ ΔðΔ − 3Þ ð Þ z f ðzÞ − 4zf ðzÞþ6fðzÞ − 2B z − 6 ¼ 0; ð7Þ M4 : 15 Since the complex scalar field in four spacetime dimen- 0ð Þ − 3 ð Þ − 2 4 þ 3 ¼ 0 ð Þ zf z f z B z ; 8 sions is supposed to be dual to the chiral condensate 106007-2 CHIRAL SYMMETRY BREAKING AND RESTORATION IN … PHYS. REV. D 98, 106007 (2018) σ ≡ hψψ¯ i 2 in three spacetime dimensions, whose dimension (z → 0) behavior of χðzÞ, that is, χðzÞ¼mqz þ σz and Δ ¼ 2 2 ¼ −2 is , we have therefore M4 . So, we have the the regularity of χðzÞ at the horizon, zH, meaning ¼ −2 2 þ λ 4 potential VX X X , where the first term is just the χðzHÞ < ∞. mass term and the second is the term needed to realize the spontaneous symmetry-breaking mechanism [33–35]. C. Zero temperature case The field equations coming from (14) are given by Here the background metric is pure AdS, without pffiffiffi pffiffiffi −Φð Þ −Φð Þ singularity, and there is no natural boundary condition at D ½ ge z gMND X − ge z ∂ V ¼ 0: ð16Þ M N X X the horizon. Numerically, one cannot take the horizon Here, we also assume that the expectation value for X takes exactly at infinity.