Shear viscosity and instability from third order Lovelock gravity
aa r X i v : . [ h e p - t h ] A ug SHU-Pre2009-08arXiv:0905.2675[hep-th]
August, 2009
Shear viscosity and instability from third order Lovelock gravity
Xian-Hui Ge ∗ , Sang-Jin Sin † , Shao-Feng Wu ∗ and Guo-Hong Yang ∗ ∗ Department of Physics, Shanghai University,Shanghai 200444, China [email protected], [email protected],[email protected] † Department of Physics, Hanyang University, Seoul 133-791, Korea [email protected]
Abstract
We calculate the ratio of shear viscosity to entropy density for charged blackbranes in third order Lovelock theory. For chargeless black branes, the result turnsout to be consistent with the prediction made in arXiv:0808.3498[hep − th]. Wefind that, the third order Lovelock gravity term does not contribute to causalityviolation unlike the Gauss-Bonnet term. The stability of the black brane againrequires the value of the Lovelock coupling constant to be bounded by 1 / The AdS/CFT correspondence [1–3] provides an interesting theoretical framework forstudying relativistic hydrodynamics of strongly coupled gauge theories. The result ofRHIC experiment on the viscosity/entropy ratio turns out to be in favor of the predic-tion of AdS/CFT [4–6]. Some attempt has been made to map the entire process of RHICexperiment in terms of gravity dual [7]. The way to include chemical potential in thetheory was figured out in [8, 9]. 1t had been conjectured that the viscosity value of theories with gravity dual may givea lower bound for the η/s = π for all possible liquid [14]. However, in the presence ofhigher-derivative gravity corrections, the viscosity bound and causality are also violatedas a consequence [15–18]. The ratio of shear viscosity to entropy density are of particularinterest in higher derivative gravity duals because those higher derivative terms can beregarded as generated from stringy corrections given the vastness of the string landscape.In [19–21], the authors computed η/s for general gravity duals by determining the ratio oftwo effective gravitational couplings. The η/s in presence of arbitrary R and R terms inthe bulk action were calculated in [22].The higher derivative terms may be a source of inconsistencies because higher powersof curvature could give rise to fourth or even sixth order differential equation for themetric, and in general would introduce ghosts and violate unitarity. Zwiebach and Zumino[23, 24] found that ghosts can be avoided if the higher derivative terms only consist of thedimensional continuations of the Euler densities, leading to second order field equationsfor the metric. These theories are the so called Lovelock gravity [25]. The zeroth orderof Lovelock gravity correspondences to the cosmological constant. The first order is theEinstein equation and the second order correspondences to Gauss-Bonnet theory. Higherderivative effects on η/s in the presence of a chemical potential have been discussed in[26–33]. In this paper, we discuss shear viscosity in third order Lovelock gravity.Our motivation for this paper is based on the following facts:1). Although people expect that η/s might receive corrections from third and higherorder Lovelock terms, it was conjectured in [19] that η/s gets no corrections at all for higherorder Lovelock terms except the Gauss-Bonnet terms. In this paper, we compute η/s forthird order Lovelock gravity directly by using the standard method developed in [4, 5] andcompare our result with that of [19].2). In [15] and [16], the authors showed that if we consider the Gauss-Bonnet correctionto Einstein equation, the viscosity bound is violated in the hydrodynamics regime. More-over, causality violation happens in the high frequency regime ( k µ → ∞ ), which impliesthat theories in that regime are pathological [17] ∗ . In [26] and [27], some of us consideredmedium effect and the higher derivative correction simultaneously by adding charge andGauss-Bonnet terms and found that the viscosity bound as well as causality violation is ∗ The causality issue in Gauss-Bonnet gravity was further studied in [34]. D -dimensional asymptotically flat Einstein-Gauss-Bonnet and Lovelockblack holes has been discussed by several authors [35–38]. Their results show that forgravitational perturbations of Schwarzschild black holes in D ≥ D = 5 and D = 6 cases at large value of α ′ [36]. In [37], theauthors showed that small black holes in Lovelock gravity are unstable. In this paper,we extend our previous computation to third order Lovelock gravity in D -dimensionalspacetime and show how stability constrains the Lovelock coupling constant.The plan of this paper is as follows. In section 2, we briefly review the thermodynamicproperties of Reissner-Nordstr¨om-AdS black brane solution in third order Lovelock gravity.In section 3, we compute the viscosity to entropy density ratio via Kubo formula and itscharge dependence. In section 4, the causality problem is discussed. We study the stabilityissue of Reissner-Nordstr¨om-AdS black branes in third order Lovelock gravity in section 5.Conclusions and discussions are presented in the last section. We start by introducing the following action in D dimensions which includes Lovelockterms and U (1) gauge field: I = 116 πG D Z d D x √− g (cid:16) −
2Λ + L + α ′ L + α ′ L − πG D F µν F µν (cid:17) , (2.1)where L = R, L = R µνγδ R µνγδ − R µν R µν + R , L = 2 R µνσκ R σκρτ R ρτµν + 8 R µνσρ R σκντ R ρτµκ + 24 R µνσκ R σκνρ R ρµ + 3 RR µνσκ R σκµν +24 R µνσκ R σµ R κν + 16 R µν R νσ R σµ − RR µν R µν + R , (2.2)Λ is the cosmological constant, α ′ and α ′ are Gauss-Bonnet and third order Lovelockcoefficients, respectively. The field strength is defined as F µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ).3he thermodynamics and geometric properties of black objects in Lovelock gravity werestudied in several papers [39–42]. From the action (2.1), we can write down the equationof motion [43], X k =0 k +1 c k δ µc ...c k d ...d k νe ...e k f ...f k R e f c d · · · R e k f k c k d k = 8 πG D T µν , (2.3)where T µν = F µρ F νσ g ρσ − g µν F ρσ F ρσ . Note that for third order Lovelock gravity, we mustdeal with D -dimensional spacetimes with D ≥ α ′ = α ( D − D − , α ′ = α D − · · · ( D − , (2.4)the charged black hole solution in D dimensions for this action is described by [38]d s = − H ( r ) N d t + H − ( r )d r + r l h ij d x i d x j , (2.5a) A t = − Q π ( D − r D − , (2.5b)with H ( r ) = k + r α ( − (cid:20) − αl (cid:18) − ml r D − + q l r D − (cid:19)(cid:21) / ) , Λ = − ( D − D − l , where the parameter l corresponds to AdS radius. The constant N will be fixed later.Note that the constant value of k can be ± h ij d x i d x j represents the line elementof a ( D − D − D − k and volume V D − . The gravitational mass M and the charge Q are expressed as M = ( D − V D − πG D m,Q = 2 π ( D − D − G D q . Taken the limit α ′ , α ′ → k = 0, the solution corresponds to one for Reissner-Nordstr¨om-AdS (RN-AdS). The hydrodynamic analysis in this background has been donein [44, 45].One may notice that here we use a black hole solution by choosing particular valuesof α ′ and α ′ so that our computation can be simplified greatly. Eq.(2.3)with the choice42.4) yields one real and two complex solutions. We use the real solution in (2.5a). Thegeneral solution of third order Lovelock gravity in D dimensions for any arbitrary valuesof α ′ and α ′ was obtained in [38], but the line element of the metric turns out to be verycomplicated. Furthermore, the general solution may present naked singularities, which isnot what we are interested [38]. In this paper, we only focus on the special case given in(2.5a).In the following, we mainly focus on D -dimensional case with k = 0. Defining λ = α/l ,the function H ( r ) becomes H ( r ) = r λl ( − (cid:20) − λ (cid:18) − r D − r D − − a r D − r D − + a r D − r D − (cid:19)(cid:21) / ) , (2.6)where a = q l r D − . The event horizon is located at r = r + . The constant N in the metric(2.5a) can be fixed at the boundary whose geometry would reduce to flat Minkowski metricconformaly, i.e. d s ∝ − c d t + d ~x . On the boundary r → ∞ , we have H ( r ) N → r l , so that N is found to be N = λ − (1 − λ ) / . (2.7)Note that the boundary speed of light is specified to be unity c = 1. From (2.6), onecan assume λ ≤ / λ > / H ( r ) becomes minus in the asymptotic infinity, and wecannot recover the AdS geometry ∗ . In section 4, we will carry out the causality analysisand find the causality constraints imposed on the value of λ .We shall give thermodynamic quantities of this background. The temperature at theevent horizon is defined as T = 12 π √ g rr d √ g tt d r = N r + πl [( D − − ( D − a ] . (2.8) ∗ In the Gauss-Bonnet case, the function H ( r ) has a different from: H ( r ) = r λl (cid:26) − q − λ GB (1 − r r )(1 − r − r )(1 − r r ) (cid:27) , which implies that the significant value of λ liesin the range λ GB ≤ /
4. Beyond this point, the Einstein-Maxwell-Gauss-Bonnet action does not admita vacuum AdS solution, and then the AdS/CFT correspondence is undefined. In [27], it was found thatcausality requires exactly λ GB ≤ / D → ∞ limit. This result matches precisely the assumption(i.e. λ GB ≤ /
4) used in [16, 32]. a → D − D − (i.e. T → k = 0 obeys the area law [38] and thus the entropy density has the form, s = r D − G D l D − . (2.9) We explored the charge dependence of η/s in the presence of Gauss-Bonnet terms for D-dimensional AdS black branes in [27]. In this section, we generalize the previous result on η/s [26, 27] to third order Lovelock gravity. It is convenient to introduce coordinate in thefollowing computation z = rr + , ω = l r + ¯ ω, k = l r ¯ k , f ( z ) = l r H ( r ) ,f ( z ) = z λ h − (cid:18) − λ (cid:16) − a + 1 z D − + az D − (cid:17)(cid:19) / i (3.1)We now study the tensor type perturbation h x x ( t, x , z ) = φ ( t, x , z ) on the black branebackground of the form ds = − f ( z ) N d t + d z f ( z ) + z l φ ( t, x , z )d x d x + D − X i =1 d x i ! , Using Fourier decomposition φ ( t, x , z ) = Z d D − k (2 π ) D − e − i ¯ ωt + i ¯ k x φ ( k, z ) , and expanding the action for tenor type gravitational perturbations φ ( t, x , z ) to the secondorder, we obtain the effective action in the momentum space S = 116 πG D Z dωdk (2 π ) D − dz √− g (cid:0) M ( z ) φ ′ φ ′ + M ( z ) φ (cid:1) , (3.2)where the prime denotes the derivative with respect to z .An easy way to obtain the equation of motion of the tensor type perturbation is tosubstitute the fluctuated metric into Eq. (2.3). One then find the linearized equation ofmotion for φ ( z ) from the third order Lovelock field equation: M ( z ) φ ′′ ( z ) + M ′ ( z ) φ ′ ( z ) + M φ ( z ) = 0 (3.3)6here M ( z ) = z D − f (cid:26) − λD − (cid:2) z − f ′ + z − ( D − f (cid:3) + λ z − D − (cid:2) f ′ + ( D − z − f ′ (cid:3) f (cid:27) M = M ( z ) ω N f − k z D − × (cid:26) − λ ( D − D − (cid:0) f ′′ + ( D − D − z − f + 2( D − z − f ′ (cid:1) + 2 λ ( D − D − (cid:20) z − ( f ′ + f f ′′ ) + 2( D − z − f f ′ + 12 ( D − D − z − f (cid:21)(cid:27) , (3.4)We would like to emphasize that when D = 5 and the λ terms vanished, (3.3) reducesto the main equation obtained in [16, 26]. The shear viscosity involves physics in thelower frequency and lower momentum limit and one can neglect the M ( z ) term in solvingEq.(3.3). For the convenient calculation of the shear viscosity, we would like to introducea new variable u = z and rewrite equation (3.3) in the new coordinate J ( u ) φ ′′ ( u ) + J ′ ( u ) φ ′ ( u ) + J ( u ) φ ( u ) = 0 , (3.5)where J ( u ) = M (1 /z ) z and J ( u ) = M ( z ).In order to solve the equation of motion (3.5) in hydrodynamic regime, let us assumethat the solution yields φ ( u ) = (1 − u ) ν F ( u ) , (3.6)where F ( u ) is regular at the horizon. ν = ± i ω πT can be fixed by substituting (3.6) intothe equation of motion, which we choose ν = − i ω πT , Since we only need to know the behavior at ω → νF ( u ) = F ( u ) + νF ( u ) + O ( ν , k ) . (3.7)The equation governing F ( u ) goes as[ J ( u ) F ′ ( u )] ′ = 0 , (3.8)and can be solved as F ′ ( u ) = C J ( u ) , (3.9)7here C is an integration constant and must be zero as J ( u ) goes zero at the horizon sothat F ( u ) is regular at the horizon. Therefore, F ( u ) is a constant, i.e. F ( u ) = C . Fromthe equation at O ( ν ), [ J ( u ) F ′ ( u )] ′ − (cid:18) C − u J ( u ) (cid:19) ′ = 0 , (3.10)we find that the solution can be written as F ′ ( u ) = C − u + C J ( u ) . (3.11)Regularity of F ( u ) at the horizon requires that C = − h (( D − − ( D − a ) (1 − λD − D − − ( D − a )) i C. (3.12)The value of C can fixed by the boundary condition C = lim u → φ ( u ) = 1. It is worth tonoting that the above calculation is same as the Gauss-Bonnet cases given in [26, 27].Using the equation of motion, we write down the on-shell action I on − shell = − r D − N πG D l D Z d D − k (2 π ) D − (cid:16) J ( u ) φ ( u ) φ ′ ( u ) + · · · (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) u =1 u =0 .. (3.13)The shear viscosity can be read off using the Kubo formula η = lim ω → ImG( ω, ω = r D − N πG D l D lim ω → J ( u ) φ ( u ) φ ′ ( u ) | u =0 iω = 116 πG D (cid:18) r D − l D − (cid:19) (cid:16) − λD − D − − ( D − a ] (cid:17) . (3.14)The ratio of the shear viscosity to the entropy density turns out to be ηs = 14 π (cid:18) − λD − D − − ( D − a ] (cid:19) . (3.15)We obtain the same result as that of [27], which is also consistent with the prediction madein [19] when a = 0. In other words, the third order Lovelock coupling constant α ′ (or λ in our case) does not contribute to the shear viscosity. The shear viscosity above is calculated in the hydrodynamical regime ( i.e. k µ → k µ → ∞ ) anddiscuss the causality issue. 8ue to higher derivative terms in the gravity action, the equation (3.3) for the prop-agation of a transverse graviton differs from that of a minimally coupled massless scalarfield propagating in the same background geometry. Writing the wave function as φ ( x, u ) = e − iωt + ikz + ik x , (4.1)and taking large momenta limit k µ → ∞ , one can find that the equation of motion (3.3)reduces to k µ k ν g eff µν ≃ , (4.2)where the effective metric is given byd s = g eff µν d x µ d x ν = N f ( z ) (cid:18) − d t + 1 c g d x (cid:19) + 1 f ( z ) d z . (4.3)Note that c g can be interpreted as the local speed of graviton: c g ( z ) = N fz h h , (4.4)where h = (cid:26) − λ ( D − D − (cid:0) f ′′ + ( D − D − z − f + 2( D − z − f ′ (cid:1) + 2 λ ( D − D − (cid:20) z − ( f ′ + f f ′′ ) + 2( D − z − f f ′ + 12 ( D − D − z − f (cid:21)(cid:27) ,h = (cid:26) − λD − (cid:2) z − f ′ + z − ( D − f (cid:3) + λ z − D − (cid:2) f ′ + ( D − z − f ′ (cid:3) f (cid:27) . The above equations can exactly reduce to Gauss-Bonnet cases found in [16, 17], if the λ terms vanished. For D = 10, we can expand c g near the boundary z = 0, c g − − (cid:2) − (1 − λ ) / (cid:3) − λ ) / + 2(1 − λ ) / − O ( 1 z ) . (4.5)We can see that c g − D = 7 , , ... , the first term in c g − λ . Figure 1 demonstrates that the value of c g lies in the region 0 . ≤ c g ≤ − .
89s a function of u and λ . Following the procedure of Ref. [17], one can find that the groupvelocity of the graviton is given by v g = d ω d k ∼ c g . (4.6)So different from the Gauss-Bonnet cases [17, 26], there is no causality violation in thirdorder Lovelock theory with the particular choice of α ′ and α ′ . The difference comes fromthe fact that α ′ terms change the causal structure of the boundary CF T . For more thanthird order Lovelock theory, the causal structure might be further modified by α ′ i ( i > Λ u Figure 1: c g as a function of u and λ when we choose D = 7 and a = 1 .
4. The lines correspondto 0 . , . , ..., − .
8, respectively, from left to right.
One may notice that our discussions on the causal structure of third order Lovelockgravity are based on the metric (2.5a) and the equation motion in high frequency limit(4.2). Hence, if we consider the general solution of third order Lovelock gravity witharbitrary values of α ′ and α ′ , we may find totally different causal structure. It remains tobe carried out in the future. In section 4, we have demonstrated that for the RN-AdS black brane in third order Lovelocktheory, causality violation does not happen, which implies that the results obtained in1017, 26, 27] might not be so universal as we expected. In this section, we extend ourprevious work on black brane stability to third order Lovelock gravity.Now, we rewrite the main equation in a Schr¨odinger form, − d ψdr ∗ + V ( z ( r ∗ )) ψ = ω ψ, dr ∗ dz = 1 N f ( z ) , (5.1)where ψ ( z ( r ∗ )) and the potential is defined by ψ = K ( z ) φ, K ( z ) ≡ s M ( z ) N f ( z ) , V = k c g + V ( z ) ,V ( z ) ≡ N "(cid:18) f ( z ) ∂ ln K ( z ) ∂z (cid:19) + f ( z ) ∂∂z (cid:18) f ( z ) ∂ ln K ( z ) ∂z (cid:19) (5.2)In the large momentum limit, the Schr¨odinger potential develops a negative gap near thehorizon and the negative-valued potential in turn leads to instability of the black brane.In the large momenta limit k µ → ∞ , the potential is mainly contributed by k c g . Forcharged black branes, c g can be negative near the horizon and the potential is deep enoughto have bound states living there. The negative-valued potential yields negative energyeigenvalue (i.e. ω < ω can then be positive. Substitutingthe eigenvalue of ω to the wave function for tensor type perturbations, one immediatelyfind that perturbations grow as time goes on and the black branes thus are unstable. Thenegative-valued energy bound state corresponds to modes of tachyonic mass Minkowskislices and signals an instability of the black brane [46]. Let us expand c g in series of(1 − u ), c g = N [( D − − ( D − a ] (cid:8) D (cid:2) λ ( a − + 2( a + 1) λ − (cid:3) − D (cid:2) λ (3 a − a + 1) + 2 λ ( a + 7) − (cid:3) + (cid:2) λ (3 a − − λ ( a − − (cid:3)(cid:9) { ( D −
4) [( D − − λ ( D − − ( D − a )] } − (1 − u ) + O ((1 − u ) ) . (5.3)Since 0 ≤ a ≤ D − D − , and 0 ≤ u ≤ c g will be negative, if (cid:8) D (cid:2) λ ( a − + 2( a + 1) λ − (cid:3) − D (cid:2) λ (3 a − a + 1) + 2 λ ( a + 7) − (cid:3) + (cid:2) λ (3 a − − λ ( a − − (cid:3)(cid:9) { ( D −
4) [( D − − λ ( D − − ( D − a )] } − < . (5.4)11rom the above formula, we find the critical value of λ , λ c = 12 n − ( D − D − − ( D − D + 2) a + n ( D − (3 D − D + 60) + ( D − (3 D − D + 28) a − a ( D − D + 79 D − D + 108) o on D − − ( D − a o − . (5.5)Above the line of λ c , c g can be negative. Eq.(5.5) tells us that the stability of the black branedepends on the charge. The minimal value of λ c can be obtained in the limit a → ( D − D − ), λ c , min = 14 ( D − D − D − D − . (5.6)When D = 5, we recover the result found in [26]. Usually, for the application of AdS/CFTcorrespondence, we do not need to take infinite dimensionality limit. But the stabilityof higher dimensional black holes itself is an important topic in the study of black holephysics.The Einstein-Hilbert action is just the first term in the derivative expansion in a lowenergy effective theory. The Gauss-Bonnet and the third order Lovelock terms can beregarded as higher order corrections to the Einstein gravity. In this sense, the higherderivative gravity coupling constants should be small. In our discussions, we have foundthat the coupling constant λ (= ( D − D − α/l ) depends on the dimensionality D .But it seems that for fixed α and AdS radius l as D approaches infinity, λ would be verylarge. That is not what we want. By doing stability analysis, we will find a way to restrictthe value of λ . As the value of D increases, one finds that λ c , min is bounded by 1 / ( D,a ) → ( ∞ , D − D − ) λ c = 14 (5.7)Thus we reproduce the result of [27]. Third order Lovelock gravity in our case does notadd new constraints on the stability of the black brane. To show explicitly the behaviorof gravitational perturbation in higher dimensions ( D ≥ D increases. Table 2 and 3 tell us the same story as we found in [27], that is to say, lowervalue of charge-( a ) and λ stabilize the perturbation, while the lower value of D strengthens12 λ = 0 . λ = 0 . λ = 0 . λ = 0 . λ = 0 .
187 35 . i . i . i . i . i . i . i . . i . i . i . i . i . i . i
10 22 . i . i . i − − Table 1: Unstable QNMs for third order charged Lovelock black brane perturbation of tensor typefor fixed charge ( a = 1 .
20) and k = 500. As D increases, the unstable modes are suppressed.And also, small λ helps to smooth the perturbation. λ a = 1 . a = 1 . a = 1 . a = 0 . a = 0 . .
33 31 . i . i . i − − .
30 31 . i . i . i − − .
27 28 . i . i . i − − .
24 23 . i . i − − − Table 2: Unstable QNMs for third order charged Lovelock black brane perturbation of tensor typefor fixed dimensionality ( D = 8) and k = 500. This table indicates that instability is increasedby a chemical potential. D a = 1 . a = 1 . a = 1 . a = 0 .
87 42 . i . i . i − . i . i . i − − . i . i − − − . i − Table 3: Unstable QNMs for third order charged Lovelock black brane perturbation of tensor typefor fixed λ ( λ = 0 . a = 1 . D suppresses the gravitational fluctuation isbecause that no matter how big D is, λ is bounded by 1 /
4, which means that for fixedAdS radius l , α ′ → D goes up. The upper bound of λ constrains thegravitational perturbation in the larger D limit. For QNMs of RN-AdS black holes inEinstein and Gauss-Bonnet gravity, one may refer to [36, 47].It would be very interesting to check for fixed value of charge, for which value of λ the black brane becomes stable. In order to do this, one should first fix D in (5.5), thenobtain a formula between λ and a . Actually, (5.5) indicates that for λ < λ c ( D, a ), the blackbrane becomes stable. For 5-dimensional black brane with charge in Gauss-Bonnet gravity,constraints from causality as well as stability separate the physics into four regions in ( a, λ )space: consistent region; only causality violation region; only unstable modes region; bothcausality violation and unstable modes region (see figure 4 in [26] for details). But forthe particular case we are considering here, since causality violation does not occur, wehave only two phases in the ( a, λ ) space: stable and unstable modes regions marked by(5.5). One thing one need to be aware of is that instability of the black brane does notcorrespond to any fundamental pathology with the theory. This is quiet different from thecausality violation which means that a theory is pathological. In the dual gravitationaldescription, the unstable QNMs is identified with unstable uniform plasma with respect tocertain non-uniform perturbation [34].
In conclusion, we derive the main equation for tensor type perturbation in third orderLovelock theory and compute the shear viscosity. The result turns out to be in agreementwith the prediction made in [19] when a = 0, that is to say, the third order Lovelock termdoes not add new ingredients into the shear viscosity of Gauss-Bonnet theory.We notice that an interesting point comes from the causality analysis. While in theGauss-Bonnet theory, causality could be violated in the boundary CFT, we do not findcausality violation in third order Lovelock theory. From (4.4), we can see that the localspeed of graviton depends on both α ′ ( ∼ λ ) and α ′ ( ∼ λ ). Although we are working onlywith a special choice of α ′ and α ′ , Eq. (4.4) implies that causality receives correctionsfrom the α ′ term. Thus, the causal structure in general third order Lovelock gravity must14e different from the Gauss-Bonnet gravity. We also expect that higher than fourth orderLovelock theory may impose more constraints on the causal structure of the boundaryCFT.The instability of charged black brane with third order Lovelock theory shows thesame properties as that of Gauss-Bonnet corrections. We find that higher D suppressesthe unstable modes, but larger value of charge and λ strengthen the perturbation. As D approaches infinity, the stability requires λ to be bounded by 1 /
4. This is an importantobservation in that Eq.(2.6) indicates that λ could be as big as 1 / λ ∼ / λ GB ≤ / λ ≤ / Acknowledgments
This work is supported partly by the Shanghai Leading Academic Discipline Project(project number S30105). The work of SFW is partly supported by NSFC under GrantNos. 10847102, and the Innovation Foundation of Shanghai University. The work of SJSwas supported by KOSEF Grant R01-2007-000-10214-0. This work is also supported byKorea Research Foundation Grant KRF-2007-314-C00052 and SRC Program of the KOSEFthrough the CQUeST with grant number R11-2005-021.
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