RG flow of AC Conductivity in Soft Wall Model of QCD
aa r X i v : . [ h e p - t h ] D ec RG flow of AC Conductivity in Soft Wall Model of QCD
Neha Bhatnagar ∗ and Sanjay Siwach † Department of Physics,Banaras Hindu University,Varanasi-221005, India
We study the Renormalization Group (RG) flow of AC conductivity in soft wall model ofholographic QCD. We consider the charged black hole metric and the explicit form of ACconductivity is obtained at the cut-off surface. We plot the numerical solution of conductivityflow as a function of radial co-ordinate. The equation of gauge field is also considered and thenumerical solution is obtained for AC conductivity as a function of frequency. The results for ACconductivity are also obtained for different values of chemical potential and Gauss-Bonnet couplings.
Keywords:
Holographic QCD; Soft wall model; Transport Properties; Gauss-Bonnet coupling
I. INTRODUCTION
AdS/CFT correspondence[1] is widely used to studythe dynamics of strongly correlated systems. AdS/CFTcorrespondence dictates that the dynamics of boundarytheory can be studied by considering the fields in thedual gravity theory in the bulk AdS spacetime. A seriousapplication of these ideas in QCD started from the workof Sakai and Sugimoto [2].Alternatively, the phenomenological models of QCD[3], makes use of some known features of QCD and triedto incorporate these features in the bulk gravity the-ory. In these model the phenomena like chiral symmetrybreaking are introduced by bifundamental fields dual tochiral condensate and confinement is realized by intro-ducing an IR cut-off. This is further modified to includethe Regge trajectories of mesons in the model and pop-ularly known as soft wall model of QCD [4]. The role ofIR cut-off is played by a dynamical wall in this model.These models are studied further to investigate thephase structure and thermodynamics of QCD [5]. Thetransport properties like AC and DC conductivity anddiffusion constant are also investigated in these models.[6–13]. Renormalization Group (RG) flow of transportco-efficients in theories dual to charged black hole is stud-ied using membrane paradigm [14–22],where the dynam-ics of vector and tensor perturbations is used to study theflow equations for conductivity and diffusion constant.Here we adapt this formalism [14, 21, 22] to investigatethe re-normalization flow of AC conductivity in the softwall model of QCD. This is further generalized to includethe effects of Gauss-Bonnet couplings in the bulk.The paper is organized as follows. First, we considerthe charged black hole solution of Einstein-Maxwell the-ory and perturbation equations for the bulk metric andgauge field in the soft wall model. We calculate the ACconductivity using the membrane paradigm and explicitresults are given in the near horizon limit. We also plot ∗ Electronic address: [email protected] † Electronic address: [email protected] the full solution for AC conductivity as a function of fre-quency and the results in the probe limits are also plot-ted. Secondly, the results are obtained for AC conductiv-ity while considering higher order gravity corrections inthe action known as Gauss-Bonnet corrections. We alsogive explicit expression for DC conductivity at the cutoff surface in the appendix.
II. TRANSPORT COEFFICIENTS INEINSTEIN-MAXWELL THEORY
Let us consider the Einstein-Maxwell action in 5-dimensions, S = Z d x √− ge − φ (cid:26) κ ( R − − g F (cid:27) , (1)where F = F mn F mn is the Lagrangian density of theMaxwell field, and φ is the dilaton field. The constant κ is related to five dimensional Newton’s constant G as κ = 8 πG , and the cosmological constant is related toAdS radius, Λ = − /l . The AdS/QCD correspondencerelates five dimensional gravitational constant 2 κ andfive dimensional gauge coupling to the rank of color group( N c ) and number of flavours ( N f ) in the boundary theory, κ = N c π , g = N c N f π .The charged black hole solution[21, 23, 24] of Einstein-Maxwell gravity in five dimensions with negative cosmo-logical constant is given by, ds = r l ( − f ( r ) dt + X i =1 dx i dx i ) + l r f ( r ) dr , (2) A t = µ (1 − r r )where f ( r ) = 1 + a r r − (1 + a ) r r , and the charge of theblack hole is related with parameter ‘a’ and chemicalpotential µ as: a = l κ Q g , Q = µr + .Defining, u = r r for simplification, the above metriccan be written as, ds = r l u ( − f ( u ) dt + X i =1 dx i dx i ) + l du u f (3)where f ( u ) = (1 − u )(1 + u − au ).We take dilaton field in soft wall model as φ = cu andthe constant c = 0 . [4].The equation of motion for the gauge field in the softwall models can be written as,1 √− g ∂ m ( √− ge − φ F mn ) = 0 . (4)Let us consider the metric and gauge field perturbationsas, g mn = g mn + ˜ h mn (5) A m = A m + A m . (6)We scale the metric perturbation as ˜ h mn = e φ h mn andtake the Fourier decomposition of the fields as follows, h mn ( t, z, u ) = Z d k (2 π ) e − iωt + ikz h mn ( k, u ) (7) A m ( t, z, u ) = Z d k (2 π ) e − iωt + ikz A m ( k, u ) . (8)Now, focusing on the linearised theory for h mn and forvector field A m propagating in charged black hole back-ground with the gauge condition h un = 0 and A u = 0,one gets the equation of motion for vector modes of met-ric perturbations h xz and h xt kf h ′ xz + ωh ′ xt − aωuA x (9)0 = h ′′ xt − u h ′ xt − b uf ( ωkh xz + k h xt ) − auA ′ x (10)0 = h ′′ xz + ( u − f ) ′ u − f h ′ xz + b uf ( ω h xz + ωkh ′ xt ) (11)where, b = l r + . Similarly, the gauge field equation (4),becomes,0 = A ′′ x +( f ′ f − c ) A ′ x + b uf ( ω − k f ) A x − e φ A ′ t f h ′ xt . (12)The gauge field equation is coupled with metric pertur-bations in charged black hole background and one has toresort to numerical methods to solve these equations inorder to calculate the AC conductivity. However, in thenear horizon regime an exact solution can be obtained.We consider AC conductivity flow in the soft wallmodel in membrane paradigm [14] and using the factthat, σ A ( ω, u ) = J x iωA x , (13) where the current density is given as, J x = − g √− ge − φ F ux + g xx κ √− gA ′ t h xt . (14)Now, using equations (4), (13) and (14) the conductivityflow can be written as, ∂ u c σ A iω − g σ A √− ge − φ g uu g xx − κ g xx g ω √− g u ( A ′ t ) r + √− ge − φ g xx g tt g = 0 . (15) u R e H Σ L - - - u I m H Σ L FIG. 1: The radial flow of AC conductivity( σ ) atfixed frequency( ω = 0 . µ =0.01(black), 0.05(magenta), 0.1(dotted- blue),0.25(dashed red)) In the near horizon limit, σ A is a constant and can beevaluated by applying regularity condition at the horizon,(u=1) σ A ( u = 1) = e − c g r + l . (16)RG flow plots for AC conductivity has been shown inFig.1. The frequency dependence of AC conductivity hasbeen evaluated numerically and shown in Fig.3, Fig.4,Fig.5 and Fig.6. In the probe brane limit, taking f ( u ) =1 − u , the plots in Fig. 3, show striking similarity withcondensed matter systems [25]. III. TRANSPORT COEFFICIENTS WITHGAUSS BONNET CORRECTIONS
We study the effect of Gauss-Bonnet(GB) coupling onthe RG flow of conductivity in the soft wall model ap-proach. The modified action with the GB term is givenby, S = Z d x √− ge − φ (cid:26) κ ( R −
2Λ + αR GB ) − g F (cid:27) , (17)where R GB = R − R MN R MN + R MNP Q R MNP Q isGauss-Bonnet term and ‘ α ’ is the Gauss Bonnet Couplingconstant.We consider the solution for Einstein-Maxwell-Gauss-Bonnet(EMGB) system as [26–30]. ds = r l u ( − f ( u ) N dt + X i =1 dx i dx i ) + l du u f , (18)where N = 12 (cid:0) √ − α + 1 (cid:1) f ( u ) = 12 λ (1 − p − λ (1 − u )(1 + u − au ) , and λ is related with the Gauss Bonnet coupling termas λ = αl .Using the membrane paradigm as explained above forthe charged black hole case and perturbation of metricand gauge fields as in previous section, the modified equa-tions of motion for vector perturbations are given by,0 = ωh ′ xt − uM ′ M f N kh ′ xz − aN A x ωM (19)0 = h ′′ xt + M ′ M h ′ xt + l b f M ′ M ( ωkh xz + k h xt ) − aN A ′ x M (20)0 = h ′′ xz − u − f ′ f + 2 λ [ fu + uf ′′ + u f ′ f − f ′ ]1 + 2 λ ( uf ′ − f ) h ′ xz + l b N uf ( ω h xz + ωkh xt ) , (21)0 = A ′′ x +( f ′ f − c ) A ′ x + l N f r u ( ω − k f ) A x − e φ N f h ′ xt (22)where M = − λf ( u ) u .In order to determine the flow equation of AC con-ductivity, we consider the current density with the GBcorrections defined as, J x = − g √− gg uu g xx e − φ ∂ u A x + 1 g √− g u r N A ′ t h xt (23) The corresponding RG flow equation for AC conductivity(using definition equation (13))becomes, ∂ u c σ A iω − g σ A √− ge − φ g uu g xx − κ g xx √− gg ω ( A ′ t ) u M r + √− ge − φ g xx g tt g = 0(24)In the near horizon limit, u c = 1(the cut-off horizon),we can get an exact expression for the AC conductivity, σ A ( u c = 1) = r + g e − c l (25)RG flow plots for AC conductivity with GB correctionshas been shown in Fig.3 and we can notice that thequalitative feature of the flow are similar to the casewithout Gauss Bonnet correction. u R e H Σ L - - - u I m H Σ L FIG. 2: The radial flow of AC conductivity( σ ) with GBcorrections( λ = 0 .
01) at fixed frequency( ω =0.999) withdiffernet µ =0.01(blue), 0.05(Red), 0.1(dashed magenta),0.25(dashed blue) IV. CONCLUSIONS
The soft wall model of Holographic QCD is used hereto get the insights into the transport properties like Ω R e H Σ L - Ω I m H Σ L FIG. 3: Frequency dependence of AC conductivity( σ ) in the probe limit with varying chemical potential ( µ = 0.01(blue),0.1(black), 0.9(red), 1.5(brown)) AC and DC conductivity in strongly coupled regime ofQCD. The flow equations of AC and DC conductivityare considered for different values of chemical potential.The numerical solution of these equations enabled usto calculate the value of real and imaginary part of ACconductivity and the results seem to agree with the mod-els, which consider the dynamics of condensate. Thissuggests that the soft wall model successfully capturesthe same features. This has also been noticed recently by[31] independently. In the probe limit our results (Fig.3)agree with existing results in the literature [21, 25, 32].The results at high frequency (Fig.4, Fig.5) show oscil-latory behavior of AC conductivity, which is reminiscentof Shubnikov de Haas effect. At low frequency, theDrude behavior[13] is observed (Fig.6). The Gauss-Bonnet coupling does not change the results significantly.
V. ACKNOWLEDGEMENT
We acknowledge the financial support from the DST,Govt. of India, Young Scientist project.
Appendix: DC CONDUCTIVITY
In order to calculate DC conductivity, we consider thegauge field equation equation (12) in the limit ω, k = 0the equation of motion for gauge-field becomes, A ′′ x + ( f ′ f − c ) A ′ x − e φ A ′ t f h ′ xt = 0 (A.1) The solution of A x ( u ), can be written as, A x ( u ) = A x (0)(1 + 3 au ( e cu − cf ′ − au ( e cu −
1) ) , (A.2)which can be used to calculate DC conductivity using theexpression given in [14, 21] σ DC = √− gg g xx g uu e − φ r g uu g tt | u =1 A x (1) A x (1) A x ( u c ) A x ( u c ) . (A.3)Thus, we get the following expression for DC conductiv-ity flow, σ DC = e − c r + g l ( a − ( a ( − cu c + 3 e cu c + 2 c −
3) + 2 c ) ( a ( c − e c + 3) − c ) ( a (3 u c − − In the near horizon region, u c = 1, the DC conductivitytakes the form, σ DC = r + e − c g l . (A.4)At the boundary u c = 0, DC flow is given as σ DC = r + g l e − c ( a − c ( a ( c − e c + 3) − c ) . (A.5)Following the above procedure, we can consider theDC conductivity with Gauss-Bonnet corrections. Weobtain the identical expression for DC conductivityindicating that the flow is independent of Gauss-Bonnetterms. [1] J. M. Maldacena, “The Large N limit of superconfor-mal field theories and supergravity,” Ad v. Theor. Math.Phys. , 231 (1998) [hep-th/9711200], S. S. Gubser,I. R. Klebanov and A. M. Polyakov, “Gauge theory cor- relators from noncritical string theory,” Phys. Lett. B , 105 (1998) [hep-th/9802109],“Anti-de Sitter spaceand holography,” Adv. Theor. Math. Phys. , 253 (1998)[hep-th/9802150], [2] T. Sakai and S. Sugimoto, “Low energy hadron physics inholographic QCD,” Prog. Theor. Phys. , 843 (2005)[hep-th/0412141],[3] J. 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