On viscous flow and azimuthal anisotropy of quark-gluon plasma in strong magnetic field
OOn viscous flow and azimuthal anisotropy of quark-gluon plasma in strongmagnetic field
Kirill Tuchin Department of Physics and Astronomy, Iowa State University, Ames, IA 50011 (Dated: November 3, 2018)We calculate the viscous pressure tensor of the quark-gluon plasma in strong magnetic field.It is azimuthally anisotropic and is characterized by five shear viscosity coefficients, four ofwhich vanish when the field strength eB is much larger than the plasma temperature squared.We argue, that the azimuthally anisotropic viscous pressure tensor generates the transverseflow with asymmetry as large as 1/3, even not taking into account the collision geometry.We conclude, that the magnitude of the shear viscosity extracted from the experimental dataignoring the magnetic field must be underestimated. I. INTRODUCTION
Strong magnetic field produced in relativistic heavy-ion collisions [1, 2] has a strong impacton phenomenology of the quark-gluon plasma (QGP). It induces energy loss by fast quarks andcharged leptons via the synchrotron radiation [3] and polarization of the fermion spectra [3]. Itcontributes to the enhancement of the dilepton production [4] and azimuthal anisotropy of thequark-gluon plasma (QGP) [5]. It causes dissociation of the bound states, particularly charmonia,via ionization [6, 7]. Additionally, the magnetic field drives the Chiral Magnetic Effect (CME)[1, 8–11], which is the generation of an electric field parallel to the magnetic one via the axialanomaly in the hot nuclear matter.It has been argued recently in [5] that the magnetic field of strength eB (cid:39) m π [1, 2] is able toinduce the azimuthal anisotropy of the order of 30% on produced particles. This conclusion wasreached by utilizing the solution of the magneto-hydrodynamic equations in weak magnetic field.In this paper we discuss the magneto-hydrodynamics of the QGP in the limit of strong magneticfield. Our goal is to calculate the effect of the magnetic field on viscosity of the plasma. It is well-known that the viscous pressure tensor of magnetoactive plasma is characterized by seven viscositycoefficients, among which five are shear viscosities and two are bulk ones. Generally, calculationof the viscosities requires knowledge of the strong interaction dynamics of the QGP components.However, in strong enough magnetic field these interactions can be considered as a perturbation andviscosities can be analytically calculated using the kinetic equation. Application of this approach a r X i v : . [ nu c l - t h ] A ug to the non-relativistic electro-magnetic plasma is discussed in [12]. A general relativistic approachwas developed in [13]. We apply it in Sec. II to derive the viscosity coefficients of QGP, which aregiven by (22) and (33). As in the non-relativistic case, we found that four viscosities vanish as themagnetic field strength increases.A characteristic feature of the viscous pressure tensor in magnetic field is its azimuthalanisotropy. This anisotropy is the result of suppression of the momentum transfer in QGP inthe direction perpendicular to the magnetic field. Its macroscopic manifestation is decrease ofthe viscous pressure tensor components in the plane perpendicular to the magnetic field, whichcoincides with the “reaction plane” in the heavy-ion phenomenology. Since Lorentz force vanishesin the direction parallel to the field, viscosity along that direction is not affected at all. In fact,the viscous pressure tensor component in the reaction plane is twice as small as the one in thefield direction. As the result, transverse flow of QGP develops azimuthal anisotropy in presence ofthe magnetic field. Clearly, this anisotropy is completely different from the one generated by theanisotropic pressure gradients and exists even if the later are absent.In Sec. III we discuss QGP transverse flow in strong magnetic field using the Navie-Stokes equa-tions. At later times after the heavy-ion collision, flow velocity is proportional to η − / , see (40a)and (40b). If the system is such that in absence of the magnetic field it were azimuthally symmet-ric, then the magnetic field induces azimuthal asymmetry of 1/3, see (44). This is surprisingly closeto the weak field limit recently reported in [5]. The effect of the magnetic field on flow is strongand must be taken into account in phenomenological applications. Neglect of the contribution bythe magnetic field leads to underestimation of the phenomenological value of viscosity extractedfrom the data [14–16]. In other words, more viscous QGP in magnetic field produces the sameazimuthal anisotropy as a less viscous QGP in vacuum. II. VISCOUS PRESSURE IN STRONG MAGNETIC FIELDA. Kinetic equation
Kinetic equation for the distribution function f of a quark flavor of charge ze is p µ ∂ µ f = zeB µν ∂f∂u µ u ν + C [ f, . . . ] (1)where C is the collision integral and B µν is the electro-magnetic tensor, which contains only mag-netic field components in the laboratory frame. Ellipsis in the argument of C indicates the distri-bution functions of other quark flavors and gluons (we will omit them below). The equilibriumdistribution: f = ρ πm T K ( βm ) e − β p · U ( x ) (2)where U ( x ) is the macroscopic velocity of fluid, p µ = mu µ is particle momentum, β = 1 /T and ρ is the mass density. Since ∂f ∂u µ ∝ u µ , the first term on the r.h.s. of (1) vanishes in equilibrium aswell as the collision integral. Therefore, we can write the kinetic equation as an equation for δfp µ ∂ µ f = zeB µν ∂ ( δf ) ∂u µ u ν + C [ δf ] (3)where δf is a deviation from equilibrium. Differentiating (2) we find ∂ µ f = − f T p λ ∂ µ U λ ( x ) (4)Since U λ = ( γ V , γ V V¯ ) and p λ = ( ε, p¯) = ( γ v m, γ v m v¯) it follows p · U = m √ − v √ − V (1 − v¯ · V¯ ) (5)Thus, in the comoving frame ∂ µ f | V¯ =0 = f T p ν ∂ µ V ν (6)Substituting (6) in (3) yields − f T p µ p ν V µν = zeB µν ∂ ( δf ) ∂u µ u ν + C [ δf ] (7)where we defined V µν = 12 ( ∂ µ V ν + ∂ ν V µ ) (8)and used u µ u ν ∂ µ V ν = u µ u ν V µν .Since the time-derivative of f is irrelevant for the calculation of the viscosity we will drop it fromthe kinetic equation. All indexes thus become the usual three-vector ones. To avoid confusion wewill label them by Greek letters from the beginning of the alphabet. Introducing b αβ = B − ε αβγ B γ we cast (7) in the form 1 T p α u β V αβ f = − zeBb αβ v β ∂ ( δf ) ∂v α ε − C [ δf ] . (9)The viscous pressure generated by a deviation from equilibrium is given by the tensor − Π αβ = (cid:90) p α p β δf d pε (10)Effectively it can be parameterized in terms of the viscosity coefficients as follows (we neglect bulkviscosities) Π αβ = (cid:88) n =0 η n V ( n ) αβ (11)where the linearly independent tensors V ( n ) αβ are given by V (0) αβ = (3 b α b β − δ αβ ) (cid:18) b γ b δ V γδ − ∇ · V¯ (cid:19) (12a) V (1) αβ = 2 V αβ + δ αβ V γδ b γ b δ − V αγ b γ b β − V βγ b γ b α + ( b α b β − δ αβ ) ∇ · V¯ + b α b β V γδ b γ b δ (12b) V (2) αβ = 2( V αγ b βγ + V βγ b αγ − V γδ b αγ b β b δ ) (12c) V (3) αβ = V αγ b βγ + V βγ b αγ − V γδ b αδ b αγ b β b δ − V γδ b βγ b α b δ (12d) V (4) αβ = 2( V γδ b αδ b αγ b β b δ + V γδ b βγ b α b δ ) . (12e)For calculation of shear viscosities η n , n = 1 , . . . , ∇ · V¯ = 0 and V αβ b α b β = 0.Let us expand δf to the second order in velocities in terms of the tensors V ( n ) αβ as follows δf = (cid:88) n =0 g n V ( n ) αβ v α v β (13)Then, substituting (13) into (11) and requiring consistency of (10) and (11) yields η n = − (cid:90) εv g n d p (14)This gives the viscosities in the magnetic field in terms of deviation of the distribution functionfrom equilibrium. Transition to the non-relativistic limit in (14) is achieved by the replacement ε → m [12]. B. Viscosity of collisionless plasma
In strong magnetic field we can determine g n by the method of consecutive approximations.Writing δf = δf (1) + δf (2) and substituting into (9) we find1 T p α v β V αβ f = − zeBb αβ v β ∂ ( δf (1) + δf (2) ) ∂v α ε + C [ δf (1) ] . (15)Here we assumed that the deviation from equilibrium due to the strong magnetic field is muchlarger than due to particle collisions. The explicit form of C is determined by the strong interactiondynamics but drops off the equation in the leading oder. The first correction to the equilibriumdistribution obeys the equation1 T p α v β V αβ f = − zeBb αβ v β ∂δf (1) ∂v α ε . (16)Using (13) we get b αβ v β ∂δf (1) ∂v α = 2 b αβ v β (cid:88) n =0 g n V ( n ) αγ v γ (17)Substituting (17) into (16) and using (12) yields: εT zeB p α v β V αβ f = − b βν v α v ν [ g (2 V αβ − V βγ b γ b α ) + 2 g V βγ b γ b α + g ( V αγ b βγ + V βγ b αγ − V γδ b α b δ ) + 2 g V γδ b βγ b α b δ )] (18)where we used the following identities b αβ b α = b αβ b β = b αβ v α v β = 0. Clearly, (18) is satisfied onlyif g = g = 0. Concerning the other two coefficients, we use the identities b αβ b βγ = b γ b α − δ αγ b , (19a) ε αβγ ε δ(cid:15)ζ = δ αδ ( δ β(cid:15) δ γζ − δ βζ δ γ(cid:15) ) − δ α(cid:15) ( δ βδ δ γζ − δ βζ δ γδ ) + δ αζ ( δ βδ δ γ(cid:15) − δ βζ δ γδ ) (19b)that we substitute into (18) to derive − ε T zeB p α v β V αβ f = g [2 V αβ b α b β − V αβ v α b β (b¯ · v¯)] + 2 g V αβ v α b β (b¯ · v¯) . (20)Since p α = εv α we obtain g = g − ε f T zeB (21)Using (2), (21) in (14) in the comoving frame (of course η n ’s do not depend on the frame choice)and integrating using 3.547.9 of [21] we get η = K ( βm ) K ( βm ) ρT zeB (22)The non-relativistic limit corresponds to m (cid:29) T in which case we get η NR3 = ρT zeB . (23)In the opposite ultra-relativistic case m (cid:28) T (high-temperature plasma) η UR3 = 2 nT zeB . (24)where n = ρ/m is the number density. C. Contribution of collisions
In the relaxation-time approximation we can write the collision integral as C [ δf ] = − ν δf (25)where ν is an effective collision rate. Strong field limit means that ω B (cid:29) ν (26)where ω B = zeB/ε is the synchrotron frequency. Whether ν itself is function of the field dependson the relation between the Larmor radius r B = v T /ω B , where v T is the particle velocity in theplane orthogonal to B¯ and the Debye radius r D . If r B (cid:29) r D (27)then the effect of the field on the collision rate ν can be neglected [12]. Assuming that (27) issatisfied the collision rate reads ν = nvσ t (28)where σ t is the transport cross section, which is a function of the saturation momentum Q s [19, 20].We estimate σ t ∼ α s /Q s , with Q s ∼ n = P/T with pressure α s P ∼ we get ν ∼
40 MeV. Inequality (26) is well satisfied since eB (cid:39) m π [1, 2] and m is in the range between thecurrent and the constituent quark masses. On the other hand, applicability of the condition (27) ismarginal and is very sensitive to the interaction details. In this section we assume that (27) holdsin order to obtain the analytic solution. Additionally, the general condition for the applicability ofthe hydrodynamic approach (cid:96) = 1 /ν (cid:28) L , where (cid:96) is the mean free path and L is the plasma sizeis assumed to hold. Altogether we have r D (cid:28) r B (cid:28) (cid:96) (cid:28) L .Equation for the second correction to the equilibrium distribution δf (2) follows from (15) aftersubstitution (25) zeBε b αβ v β ∂δf (2) ∂v α = − νδf (1) (29)Now, plugging δf (1) = [ g V (3) αβ + g V (4) αβ ] v α v β , (30a) δf (2) = [ g V (1) αβ + g V (2) αβ ] v α v β (30b)into (29) yields2 zeBε { g [2 V βα b αγ v β v γ − V βα b αγ v β v γ (v¯ · b¯)] + 2 g V βα b αγ v β v γ (v¯ · b¯) } = − νg {− V βα b αγ v β v γ − V βα b αγ v β v γ (v¯ · b¯) } (31)where we used g = 2 g . It follows that g = g νγ v g ω B (32)With the help of (28),(2),(14) we obtain η = η √ π ρ σ t T / ( zeB ) m / K / ( βm ) K ( βm ) (33) III. TRANSVERSE FLOW
To illustrate the effect of the magnetic field on the viscous flow of the electrically chargedcomponent of the quark-gluon plasma we will assume that the flow is non-relativistic and use theNavie-Stokes equations that read ρ (cid:18) ∂V α ∂t + V β ∂V α ∂x β (cid:19) = − ∂P∂x α + ∂ Π αβ ∂x β (34)where Π αβ is the viscous pressure tensor, ρ = mn is mass-density and P is pressure. We willadditionally assume that the flow is non-turbulent and that the plasma is non-compressible. Theformer assumption amounts to dropping the non-linear in velocity terms, while the later impliesvanishing divergence of velocity ∇ · V¯ = 0 (35)Because of the approximate boost invariance of the heavy-ion collisions, we can restrict our atten-tion to the two dimensional flow in the xz plane corresponding to the central rapidity region.The viscous pressure tensor in vanishing magnetic field is isotropic in the xz -plane and is givenby Π αβ = η (cid:18) ∂V α ∂x β + ∂V β ∂x α (cid:19) = 2 η V xx V xz V zx V zz (36)where the superscript 0 indicates absence of the magnetic field. In the opposite case of very strongmagnetic field the viscous pressure tensor has a different form (11). Neglecting all η n with n ≥ ∞ αβ = η − V zz
00 2 V zz = 2 η V xx V zz (37)where we also used (35). Notice that Π ∞ xx = Π ∞ zz = Π xx indicating that the plasma flows in thedirection perpendicular to the magnetic field with twice as small viscosity as in the direction ofthe field. The later is not affected by the field at all, because the Lorentz force vanishes in thefield direction. Substituting (37) into (34) we derive the following two equations characterizing theplasma velocity in strong magnetic field ρ ∂V x ∂t = − ∂P∂x + η ∂ V x ∂x , ρ ∂V z ∂t = − ∂P∂z + 2 η ∂ V z ∂z (38)Additionally we need to set two initial conditions V x (cid:12)(cid:12) t =0 = ϕ ( x, z ) , V z (cid:12)(cid:12) t =0 = ϕ ( x, z ) (39)The solution to the the problem (38),(39) is V x ( x, z, t ) = (cid:90) ∞−∞ dx (cid:48) ϕ ( x (cid:48) , z ) G ( x − x (cid:48) , t ) − ρ (cid:90) t dt (cid:48) (cid:90) ∞−∞ dx (cid:48) G ( x − x (cid:48) , t − t (cid:48) ) ∂P ( x (cid:48) , z, t (cid:48) ) ∂x (cid:48) (40a) V z ( x, z, t ) = (cid:90) ∞−∞ dz (cid:48) ϕ ( x, z (cid:48) ) G ( z − z (cid:48) , t ) − ρ (cid:90) t dt (cid:48) (cid:90) ∞−∞ dz (cid:48) G ( z − z (cid:48) , t − t (cid:48) ) ∂P ( x, z (cid:48) , t (cid:48) ) ∂z (cid:48) (40b)Here the Green’s function is given by G k ( z, t ) = 1 √ πa kt e − z a kt (41)and the diffusion coefficient by a = 2 η ρ (42)Suppose that the pressure is isotropic, i.e. it depends on the coordinates x , z only via the radialcoordinate r = √ x + z ; accordingly we pass from the integration variables x (cid:48) and z (cid:48) to r in (40a)and (40b) correspondingly. At later times we can expand the Green’s function (41) in inversepowers of t . The first terms in the r.h.s. of (40a) and (40b) are subleasing and we obtain V x ( x, z, t ) ≈ − ρ (cid:90) t ds (cid:90) ∞−∞ dr √ πa s ∂P ( r, t − s ) ∂r = − ρ (cid:90) t ds √ πa s [ P ( R ( s ) , t − s ) − P (0 , t − s )] (43a)and by the same token V z ( x, z, t ) ≈ − ρ (cid:90) t ds √ πa s [ P ( R ( s ) , t − s ) − P (0 , t − s )] (43b)where R ( t ) is the boundary beyond which the density of the plasma is below the critical value. Weobserve that V x /V z = √
2. Consequently, the azimuthal anisotropy of the hydrodynamic flow is V x − V z V x + V z = 1 − = 13 (44)Since we assumed that the initial conditions and the pressure are isotropic, the azimuthal asym-metry (44) is generated exclusively by the magnetic field. IV. SUMMARY
The structure of the viscous stress tensor in a very strong magnetic field (37) is general, modelindependent. However the precise amount of the azimuthal anisotropy that it generates is of coursemodel dependent. We however draw the reader’s attention to the fact that analysis of [5] usingquite different arguments arrived at a very similar estimate. Although a more quantitive numericalcalculation is certainly required before a final conclusion can be made, it looks very plausible thatthe QGP viscosity is significantly higher than the presently accepted value extracted without takinginto account the magnetic field effect [14–16] and is perhaps closer to the value calculated usingthe perturbative theory [17, 18].
Acknowledgments
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