Quark-Gluon Plasma and Topological Quantum Fields Theory
aa r X i v : . [ h e p - ph ] M a r Quark-Gluon Plasma and Topological Quantum Fields Theory
M.J.Luo ∗ Department of Physics, Jiangsu University, Zhenjiang, Jiangsu 212013, China
Based on an analogy with topologically ordered new state of matter in condensed matter systems,we propose a low energy effective field theory for a parity conserving liquid-like quark-gluon plasma(QGP) around critical temperature in quantum chromodynamics (QCD) system. It shows thatbelow a QCD gap which is expected several times of the critical temperature, the QGP behaveslike topological fluid. Many exotic phenomenon of QGP near the critical temperature discovered atRHIC are more readily understood by the suggestion that QGP is a topologically ordered state.
I. INTRODUCTION
Quantum Chromodynamics (QCD) based on non-Abelian gauge field theory had been well established as thefundamental theory of the strong interaction. However, because of its strong coupling nature at low energy or longdistance, from the day the exact theory was written down, its low energy behavior is still being poorly understood bysolving the first principle theory. Although we already have a low energy effective theory based on the Goldstone’stheorem under the condition of chiral symmetry breaking (for a review, see: [1]), there is no analog foundationto formulate a low energy effective theory with neither symmetry breaking nor order parameter, more specifically,applying to the disordered quark-gluon plasma (QGP) phase.A crucial phenomenological clue to the problem comes from the experimental studies of the Relativistic Heavy IonCollision (RHIC), in which new state of matter of strong interaction is produced. It is discovered, at sufficiently hightemperature and density, that the nucleons in the collision are deconfined into constituent quarks and gluons and form asoup of QGP. Important experimental facts of the QGP are the following: (i) The long distance low frequency behaviorof the QGP around the critical temperature T c is well described by the theory of fluid mechanics (hydrodynamics)[2, 3]. (ii) The QGP around T c looks like a strong correlated and near-perfect liquid [4], the pressure, numbers oftransport properties (e.g. conductivity, diffusion and viscosity) are small, strongly deviating from the behavior of aweakly coupled ideal gas predicted from the perturbative QCD. (iii) The charge of the deconfined quark or partonexcitation in the QGP is fractionalized [5]. In the temperature range T c < T . T c where these experimental factsare established, we propose a topological dominant effective field theory for the parity conserving QGP by bosonizingthe gauge invariant QCD that I feel are consistent with the known experimental facts.The connection between a pure gluon plasma in hard thermal loop limit and the topological Chern-Simons theoryhas been discussed in literature [6, 7]. The remarkable relation between the QGP and topological field theories canbe generalized to a more generic level. The QGP phase is a deconfined phase where the bound state hadrons aredecomposed into quarks and gluons at high temperature and density. The situation is analogous with the case incondensed matter physics such as the gauge symmetric Heisenberg spin system that at high enough temperatureand chemical potential where the system is well into a deconfined phase, the Heisenberg spins are decomposed intoconstituent holons and spinons and form a plasma phase which enjoy a new type of order called topological order/phase(see, for example, [8, 9] at low dimensions, [10–12] in 3+1 dimensions). The ground state of the topologically orderedmatter is not associated with spontaneous symmetry breaking, protected by a gap, and topological fluid (see, forexample, [13–15] in 2+1 dimensions) as the low energy degree of freedom in many aspects is analogous with thefluid-like plasma of quark-gluon around the critical temperature and chemical potential. The topological fluid is ingeneral incompressible, with excitation carrying fractional charges [16–18], characterized by having non-trivial groundstate degeneracies [8, 19] on topological non-trivial compact manifold.Bosonization is a proper description to such bosonic fluids system, especially at finite chemical potential becauseof a more controllable sign structure than fermion [20, 21]. The validity of bosonization approach in the real 3+1dimensional QCD is based on the facts: (a) lattice data show that quark (baryon) masses seem to be large in thestrongly coupled QGP near T c [22], (b) a massive fermionic theory and the bosonized descriptions share the samemacroscopic limit which is dominated by topological degrees of freedom [12].The structure of the paper is as follows. In section II, we construct a basic formulation of a bosonized low energyeffective field action for the QGP. In subsection A of the section III, we derive the classical equations of motion and ∗ Electronic address: [email protected] argue the dyonic nature of the fluids. In subsection B, we derive the phenomenological hydrodynamics equationsfrom the effective field theory. In subsection C, linear responses of the theory are studied, more specifically, the ratioof shear viscosity over entropy density of an homogeneous and isotropic QGP is derived at Bogomol’nyi bound. Insubsection D, the fractionalization of charge for quark/parton in the plasma is explained. In section IV, we summarizethe results.
II. BOSONIC EFFECTIVE ACTION
Both for mathematical and physical reasons, in a strongly coupled many-body system, what we observed are gaugeinvariant responses to external (in practice electromagnetic) probe, and hence the notion of response, especially linearresponse, is more proper than the unobserved microscopic fundamental particles. The bosonization approach startsfrom replacing the fermionic fundamental quarks by the responses to an external weak U(1) gauge field being a sourceminimally coupling to them, h j µ ( x ) j µ ( x ) ...j µ n ( x n ) i = δδA exµ ( x ) δδA exµ ( x ) ... δδA exµ n ( x n ) ln Z [ A ex ] , (1)where Z [ A ex ] = ˆ D ¯ ψ D ψ D G exp (cid:18) − ˆ d x ¯ ψ (cid:2) iγ µ (cid:0) ∂ µ + iG µ + µδ µ + iA exµ (cid:1)(cid:3) ψ − S Y M [ G ] (cid:19) . (2)The ψ are quarks, G µ are gluon gauge fields, µ is the chemical potential, S Y M [ G ] is the Yang-Mills action for gluons,and A exµ are external U(1) source coupled to an Abelian subgroup of the Non-Abelian SU ( N c ) of QCD.The form of Z [ A ex ] can be determined as follows. By using the U(1) gauge invariance Z [ A ex ] = Z [ A ex + a ] . a is a to-be-interpreted (see following) 1-form gauge field in the effective QCD under the Abelian projection of SU ( N c ) ∼ U (1) N c − which fixes the gauge partially. So there are N c − such fields, and here a describes an effectivemixing of those coupled to A ex . And we consider it is a pure gauge, i.e. f αβ ( a ) = ∂ α a β − ∂ β a α = 0 . The pure gaugecondition can be imposed by inserting a delta function into the functional integral, Z [ A ex ] = ˆ D aZ [ A ex + a ] Y x Y µ<ν ǫ µναβ δ [ f αβ ( a ( x ))] ! . (3)The delta function can be exponentiated by introducing an auxiliary anti-symmetric 2-form gauge fields b µν asLagrangian multipliers, Z [ A ex ] = ˆ D a D bZ [ A ex + a ] exp (cid:18) − i π ˆ d xǫ µναβ b µν f αβ ( a ) (cid:19) , (4)in which the term in the exponent is purely topological known as the BF action (for a review see: [23]), and thepre-factor π is for convention. The pre-factor can be in general defined as m π , m ∈ Z plays a similar role of a Chern-number in quantum Hall systems characterizing its topological order, the integer-valued m so defined characterizesthe topological order of the ground state of the system, a value m = 1 represents a fractionalized topological order. Insuch setting, under a gauge transformation the BF term is just shifted by an integer multiple of π , so the partitionfunction is not affected.The auxiliary gauge field b is of ( D − )-form where D is spacetime dimensions. In the paper we will focus on thereal QCD system D = 4 , so b field is of a 2-form. By doing a variable replacement a → a − A ex , we obtain Z [ A ex ] = ˆ D a D bZ [ a ] exp (cid:18) − i π ˆ d xǫ µναβ b µν [ f αβ ( a ) − f αβ ( A ex )] (cid:19) . (5)By using Eq.(1), the above partition function indicates a bosonized flow j µ ( x ) = 12 π ǫ µναβ ∂ ν b αβ ( x ) , (6)which is gauge invariant under the transformations b µν → b µν + ∂ µ η ν − ∂ ν η µ . (7)Note that the flow is automatically conserved ∂ µ j µ ≡ . (8)The non-dynamical component a coupled to the density j plays the role of a Lagrangian multiplier similar with achemical potential.Beside a , there is another non-dynamical Lagrangian multiplier b i = − b i , ( i = 1 , , ), which implies the existenceof a spin/angular momentum or magnetization density tensor s µν ( x ) = 12 π ǫ µναβ ∂ α a β ( x ) . (9)It can be interpreted as a spin/angular momentum or magnetization density because it is a response to an externalmagnetic field which will be shown as following. If we add a kinetic term of b µν with infinite mass into the parenthesisof the Eq.(5) as a regulator, S → S + S reg , S reg = lim λ →∞ λ ˆ d xh µνρ h µνρ , (10)where h µνρ = ∂ µ b νρ + ∂ ν b ρµ + ∂ ρ b µν is the field strength of b µν , and λ is a mass parameter. By integrating out the b µν field, we obtain an effective coupling between s µν and external source field, S int = 14 π ˆ d xs µν ˜ F exµν , (11)in which ˜ F exµν = ǫ µναβ ∂ α A exβ is an external magnetic field. So we prove that the tensor s µν is a response to the externalmagnetic field, h s µ ν ( x ) s µ ν ( x ) ...s µ n ν n ( x n ) i = (4 π ) n δδ ˜ F exµ ν ( x ) δδ ˜ F exµ ν ( x ) ... δδ ˜ F exµ n ν n ( x n ) ln Z [ A ex ] . (12)In the rest of the paper we will not strictly distinguish the terminologies spin or angular momentum or magnetization,but using “spin” for convention, and a corresponding currents of s µν we call them spin currents.The spin currents can be defined by a spin density tensor M [ µν ] α = ∂ α s µν . (13)From its definition Eq.(9) and Eq.(13), we can see that the spin currents satisfies 6 continuity equations ∂ α M [ µν ] α ≡ . (14)These continuity equations ensure the spin currents are conserved or/and divergentless. In addition, there is anothercontinuity equation, note that a Pauli-Lubanski pseudovector constructed by the spin current W µ ( x ) = 12 ǫ µναβ M [ αβ ] ν , (15)is also automatically satisfied ∂ µ W µ ≡ . (16)The independence of the continuity equations for charge and spin currents manifests an important fact that thecarriers of charge and spin in the fluids are no longer identified, in contrast to the conventional quasi-particle picture inwhich it carries both. In other words, there are two types of currents in the fluids [10], which is known as spin-chargeseparation in condensed matter physics.So far the bosonization recipe is quite general and is able to apply to a gapped or gapless system, but only in1+1 dimensions the Z [ a ] can be written down exactly, here for 3+1 dimensions Z [ a ] can only be approximatelyevaluated below a cutoff or gap Λ . The Z [ a ] now encodes almost all physics of QCD, which needs some guess work.The bosonization gap Λ is closely related to the gap of the fermionic spectrum, from the lattice date [22, 24] it isshown that the gap is at least several time larger than T c when the temperature of sQGP is around T c , or even muchlarger [25, 26]). In the realistic heavy-ion collision experiments, the gap could be different in different situations: e.g. Λ ∼ p T + µ /π at high temperature and chemical potential µ , Λ ∼ p e | B | at strong magnetic fields. In short, itis safe to consider that Λ will be larger than the temperature range of the validity for our low energy effective fieldtheory T c < T . T c for a typical hydro-like sQGP at RHIC, so that the phenomenological results predicted wellbelow the gap will not depend on the exact value of Λ . Therefore we could expand Z [ a ] by the inverse of the gap, twomost general leading terms constructed by the gauge fields a can be explicitly written down based on the principleof gauge invariance, i.e. a θ -term and a Maxwell term. The two leading terms have no dimensional parameter in3+1 dimensions, so they are marginal in the RG sense. The higher order terms contain higher derivatives which aresuppressed by the gap Λ . We have Z [ a ] = exp (cid:18) iθ π ˆ d xǫ µναβ f µν ( a ) f αβ ( a ) − g ˆ d xf µν ( a ) f µν ( a ) + O (cid:0) ∂ / Λ (cid:1)(cid:19) . (17)The first term is the θ -term which is topological and always marginal at low energy. Here the U (1) symmetry for the a field is an Abelianized compact subgroup of the full SU ( N c ) ∼ U (1) N c − , but the theory is not simply a QED N c − ,something of the non-Abelian character has to survive in the exact Z [ a ] . As the starting point of the effective theoryof QGP, it does not associate with any symmetry breaking of QCD, we expect that the exact Z [ a ] encoding physics ofQCD keeps invariant under SU ( N c ) , and in 3+1 dimensions the homotopy π [ SU ( N c )] is non-trivial, so the parameter θ being an angle (with periodicity π ) could have non-trivial choice in the theory and be physical observable. The θ -term generally breaks the parity and time-reversal symmetries unless it is quantized as θ = νπ , ν ∈ Z is a topologicalnon-trivial winding number. It is worth stressing that the θ here is not necessarily the θ QCD parameter in QCD,we know θ QCD is (from the dipole moment of the neutron) exceedingly small θ QCD < − . However, our followingdiscussions are based on a non-trivial choice θ = π (18)in the effective theory of QGP, which is necessary for a consistent hydrodynamic interpretation of the topologicaleffective fields theory. And the choice of the value can be considered as a topological order parameter characterizinga phase of QGP, which is an important difference between our model and the chiral superfluid model of QGP [29, 30].The consequence of the non-trivial choice is an important topic in the next section.The second Maxwell term contains metric, so it is dynamical and non-topological. The coupling constant g in thesecond term effectively encodes the information of N c and N f of the QCD, i.e. g ( N c , N f ) . As is well known thatfermions and bosons contribute opposite signs to the β -function in renormalization: Fermions screen the couplingconstant while self-interacting bosons anti-screen it. As a result, in the pure bosonized system there are no extrascreening effects from fermions, if the pre-bosonized theory is strongly coupled at low energy, the bosonized versionremains being strongly coupled. For the real QCD: N c = 3 , N f = 6 , the theory is well-known being asymptoticfreedom or strongly coupled at low energy because of the competition between fermions and bosons. Therefore, weexpect that the coupling constant g becomes large at low energy due to the anti-screening effects of a µ field from itsself-interacting. In this case, the second term is expected small compared with the first θ -term at low energy and beconsidered as a perturbation, except that a specific choice of N c and N f , e.g. a Banks-Zaks weakly coupled fixed pointwhere g is small when N f is close to N c / , in which case only the second term is not enough and the contributionsfrom higher order terms are required to be considered. The rest paper focuses only on the strongly coupled case.There are two remarks to Z [ a ] taking the form as Eq.(17): (1) such expansion works at large coupling g and atlow energy below the gap Λ , and the results in the rest of the paper are achieved at strong coupling and below thegap; (2) it is such form that is remarkably successful in re-interpreting the gauge fields a µ , b µν as certain fluids in theground state, and we will see that their classical equations of motion are just hydrodynamic equations of the fluids.In other words, such form of Z [ a ] gives a simple hydrodynamic re-interpretation of the QCD ground state, which willbe shown in the section-III.So put everything together, it contains only relevant and marginal terms and should be compatible with the gaugeinvariance, the bosonized low energy effective action is given by Z [ A ex ] = ˆ D a D b exp ( − S eff ) , (19)with S eff ( a, b ) = i π ˆ d xǫ µναβ b µν [ f αβ ( a ) − f αβ ( A ex )] − iθ π ˆ d xǫ µναβ f µν ( a ) f αβ ( a )+ 14 g ˆ d xf µν ( a ) f µν ( a )+ O (cid:0) ∂ / Λ (cid:1) , (20)in which the fermionic quarks are bosonized by two types of gauge fields: 1-form a µ and 2-form b µν . The first twoterms are topological, the integrals in these terms are metric independent. The topological terms giving rise fromthe bosonization play the role of a non-trivial phase and sign structure of fermions, which is instead responsible forthe notorious sign problem for the original fermionic system at finite chemical potential. While the third term isnon-topological and metric dependent, it is real in Euclidean metric or imaginary in Minkovski metric, the paper isusing the Euclidean metric for convention.If the system is subject to parity violation, which is still an open question in heavy-ion collision experiments, thereare in general other types of responses in the system besides Eq.(1) and Eq.(12). For example in an effective theory[27, 28] there may be response to a pseudoscalar Φ ex source coupled to quarks as ¯ ψγ Φ ex ψ , leading to chiralityimbalance. In such case, the effective fields theory QGP is considered as a chiral superfluid discussed in Ref.[29, 30],which is formally similar with a theory of topological superconductor in condensed matter physics [31]. However, inthe present paper, we focus on an effective fields theory of QGP being parity conserving, which is analogous to beanother topological state of matter, a topological insulator. The topological superconductor and topological insulatorare both fully gapped in the bulk for the quasi-particle excitations, they are both topologically ordered state ofmatter in the sense that their characters are topologically protected. A topological superconductor and topologicalinsulator actually can be turned into each other by a phase transition discussed in Ref.[32]. Based on the analogy, aparity conserving QGP in our paper is comparable to the superfluidity model of QGP. There are essential differencesbetween them: (1) θ in the topological θ -term is a dynamical field in the effective chiral superfluid model which leadsto the chirality imbalance, but in our effective theory θ = π is a constant topological order parameter which doesnot violate parity. (2) In a topological superconductor the gauge field and corresponding fluid is massive while in atopological insulator the gauge field and corresponding fluid remains gapless in the long wavelength limit. (3) Wewill see in the next section that in our model the nearly perfect transport properties is due to the self-duality of thefluids configurations or BPS solution of the theory, but rather due to the superfluidity of the fluids, since there is nosymmetry breaking in our theory.It can be regarded that the Faddeev-Popov ghosts does not play fundamental role in our effective theory in thesense that the building blocks of the effective theory are the gauge invariant linear responses. The system is writtendown to the lowest order which is linearized and Abelianized, although the system is essentially non-linear. Surely thenon-linear interactions coming from the higher order terms may be important to the properties of the system. Thenon-Abelian gluonic fields in QCD defined at high energy makes no sense at the scale of our interest and are effectivelyreplaced by the linearized responses or fluids, in this sense they can be well gauge fixed without Faddeev-Popov ghosts.One may wonder that since the system is gapped, there seems no Fermi surface. But remind that the system isbosonized, the chemical potential µ can also be well-defined which formally shifts the non-dynamical time componentof gauge field A ex , as if the quarks fill the Fermi sphere in the equivalent bosonized QCD. III. GENERAL FEATURESA. Plasma with both Electric and Magnetic Charges
At criticality, the fixed point coupling constant g is large, then the first two topological terms in Eq.(20) dominatethe low energy theory at the criticality, while the dynamical Maxwell term can be seen as a perturbation. The classicalequations of motion of the topological action are given by, δS eff δa µ = 0 : j µ − θ π j mµ = 0 . (21) δS eff δb µν = 0 : f αβ ( a ) − f αβ ( A ex ) = 0 . (22)The first equation of motion gives the electric current in terms of the magnetic monopole current j mµ = π ∂ ν ˜ f µν inthe fluid [33], the monopole taking magnetic charge receives an electric charge q e = qθ/ π , where q is the electriccharge of the system conventionally defined in the covariant derivative. In the theory with θ = π = 0 , the fluid carriesboth electric and magnetic charges [34, 35]. As is well known that in the 2+1 dimensional topological Chern-Simonstheory gauge field plays the role of attaching fluxes to electrons, in 3+1 dimensions the gauge field a µ is attaching amonopole to the electron (known as dyon). We will see in the latter discussion that this dyonic property is crucialfor a small value of shear viscosity over entropy density of the QGP. The second equation of motion is a constraintbecause the b field plays the role of a Lagrangian multiplier, or equivalently, the mass of the b field can be consideredinfinitely heavy shown as Eq.(10). The second equation of motion relates the configurations a and A ex . B. Interpretation of the theory via Hydrodynamics
In this subsection, we discuss the physical interpretation of the effective gauge fields a µ and b µν . Unlike the 1-formgluonic fields defined at asymptotically free regime, in which a weakly coupled photon analogous interpretation canbe directly borrowed. However, what is the gauge fields a µ , and even the 2-form b µν in the effective theory?As is discussed in Section II, the topological dominant effective field theory describes a hydrodynamic theory for twotypes of topological fluids, the charge current and spin current, without mentioning their microscopic origins. Herewe have 1+6 continuity equations, one continuity equation for the charge current Eq.(8) and 6 continuity equationsfor the spin currents Eq.(14), ∂ µ j µ = 0 , ∂ α M [ µν ] α = 0 . (23)In general, there is a relation connecting an energy-momentum tensor to the spin current M [ µν ] α ( x ) = l µ T να ( x ) − l ν T µα ( x ) , (24)in which l is a characteristic size of the spin vortex. By using the relation, we could construct a energy-momentumtensor via the effective spin current without concerning the gluon degrees of freedom making sense at high energy.The continuity equations for the spin current are given by ∂ α M [ µν ] α = T νµ + l µ ∂ α T να − T µν − l ν ∂ α T µα = 0 . (25)Under the condition that the energy-momentum tensor is symmetric T µν = T νµ , one can recognize that the 6 continuityequations for the spin currents naturally contain 4 continuity equations for energy-momentum, ∂ µ T µν = 0 . (26)In addition, the classical equation of motion Eq.(21) j µ = θ π M [ µν ] ν implies a relation between the current and theenergy-momentum tensor so defined, for θ = π , we have T µν = ∂ µ j ν + ∂ ν j µ + g µν T αα . The relation naturally leads toa traceless condition T αα = 0 , so the stress-energy tensor is related to the current by T µν = ∂ µ j ν + ∂ ν j µ . (27)For the zero and near-zero modes of the fluids well below the gap, we can define a potential (curl-free) flow vector v µ by the charge current as j i = ρv i = − D∂ i ρ, (28)where D is a diffusion constant with dimension of length and ρ = j is the flow density, we obtain ∂ ( ρv j ) = D∂ i T ij , ( i, j = 1 , , . (29)By using Eq.(27 and 28), we have D∂ i T ij = D∂ i [ ∂ j ( ρv i ) + ∂ i ( ρv j )] = − ∂ i ( ρv i v j ) + Dρ ∇ v j , (30)so the Eq.(29) (the classical equation of motion Eq.(21)) are nothing but surprisingly the Navier-Stokes equations forincompressible fluid without pressure, ∂ ( ρv j ) + ∂ i ( ρv i v j ) = η ∇ v j , ( i, j = 1 , , . (31)in which η = Dρ here is a viscosity constant.In summary, we show the close relation between the effective field theory and hydrodynamic equations. The groundstate of the theory can be interpreted as a theory of topological fluids satisfying hydrodynamics equations, includingthe continuity equation of current, the conservation of the energy-momentum tensor Eq.(26), the traceless of theenergy-momentum tensor, and the Navier-Stokes equations Eq.(31). The surprising relations shed some light on thefact (i) mentioned in the introduction, namely, why the long distance behavior of QGP around the critical temperatureis so well described by hydrodynamics. C. Nearly Perfect Liquid
It is worth stressing that since the low energy effective theory is dominated by the topological terms, so in principle,there is no strict concepts of metric and energy, one may wonder why we can talk about the the energy-momentumtensor defined above. The reason is that the energy-momentum tensor defined above just describes classical flowsor distributions of the charge and spin currents in the ground state gapped from the excitations, which are perfectfluids. Like the Hall fluids which have vanishing viscosity, if the low energy behavior of quark-gluon fluids here aresolely governed by the topological terms, they have no viscosity either (completely perfect). It is a general propertythat the dissipationless of the topological term does not depend on that it is in Euclidean or Minkovski formalism.When the excitation states coming from the non-topological terms, i.e. Maxwell term, are taken into account, internalfrictions appear and the fluid becomes dissipation. Strictly speaking, dissipation or broken time reversal appears onlyin Masubara but Minkovski formalism of the non-topological part of the theory. At temperature well below the gapwhich protects the topological order, the deviation from viscousless is small, so the fluids still behave nearly perfect.This property of the theory is associated with the fact (ii).The terminology “incompressible” of the fluids is tantamount to “topological”, more precisely, since the partitionfunction given by the topological fixed point action only depends on the topology of the manifold, so it is exactlyresponseless to the compressing or expanding of the 3-volume of the manifold. As a result, the quantities like pressureand bulk viscosity of QGP are identically zero predicted from the lowest order topological theory.To consider the transport beyond the lowest order result, one can first integrate out the a and b field, the effectivestrongly coupled fixed point action becomes S eff ( A ex ) = iθ π ˆ d xǫ µναβ ∂ µ A exν ( t, x ) ∂ α A exβ ( t, x ) + 14 g ˆ d xF µν ( A ex ) F µν ( A ex ) + O (cid:0) ∂ / Λ (cid:1) , (32)which at lowest order can be viewed as an action for the U(1) electromagnetic fields in a QGP medium, complicatednon-linear self-interactions are in the higher order terms.The θ parameter in the first θ -term is inherited from its ancestor action Eq.(20) which is non-trivial θ = π . It maycome as a surprise that there is an extra topological θ -term at leading order in the effective U(1) electromagneticfields in the QGP medium, the reason we will see is closely related to the dyonic nature of the a fluids of the QGPmedium. Because at the classical level the constraint Eq.(22) suggests a close relationship between the A fields andthe dyonic a fields, non-trivial topology of the A fields is also expected which makes the non-trivial θ -term shouldnot be simply removed as an ordinary QED. Such result can be understood intuitively as that by intermediating thedyonic medium the electric fields now are able to interact with magnetic fields effectively (magnetoelectric effect).The second term is the dynamical Maxwell term which is also expected being suppressed at low energy since theintegration over QCD fields gives a Debye gap. The transport of the effective theory e.g. the conductivity, chargesusceptibility and shear viscosity, therefore take small values that are of order of the dynamical Maxwell term. Thefollowing calculations are doing at zero temperature, and the finite temperature results can be generalized by standardMatsubara summation procedures.Since the fluid is incompressible if we only consider the topological term, the topological term only gives a currenttransport on the boundary of the QGP, there is no net current in its bulk. The current transport or conductivity inthe bulk solely gives rise from the Maxwell term, σ = lim ω → ω δ S eff ( A ex ) δA exi ( ω, k = 0) δA exj (0 , P ⊥ ij = lim ω → g ω ω = ω g , (33)in which P ⊥ ij = δ ij − k i k j / | k | is the transverse projector. Note that the chemical potential µ can be seen as a shiftof the non-dynamical A ex , i.e. A ex ( ω = 0 , k ) , then the charge susceptibility can be calculated by χ = δ S eff δµ = lim | k |→ δ S eff ( A ex ) δA ex ( ω = 0 , k ) δA ex (0 ,
0) = | k | g . (34)One can find that the ratio between the conductivity and susceptibility matches the Einstein relation, σχ = ω | k | = D, (35)in which D is the charge diffusion constant. For a special interest, the value of D can be determined on self-dualconfigurations shown as follows.An important observation to the effective theory is that a lower bound can be realized from the non-trivial self-dualconfiguration F µν = ˜ F µν = ǫ µνρσ F ρσ . It is known that there is a Bogomol’nyi bound at the self-dual configuration, g ˆ d xF µν F µν > S S.D.
Self − Dual ≡ π | Q | g , (36)where Q = π ´ d xF µν ˜ F µν = π ´ d xf µν ( a ) ˜ f µν ( a ) ∈ Z and in which the second equal sign is given by theconstraint Eq.(22). So the result could be understood from the fact that the self-duality of A ex fields is closelyconnected to the self-duality of the dyonic fluids a µ fields via the constraint, the integer is effectively relates to thetopological charge of the a µ fluids configurations.As a first consequence of the self-duality, it leads to a relation ω = | k | in Euclidean metric for the self-dual zeromode. Thus the charge diffusion constant D in Eq.(35) equals to D = 1 /ω on the self-dual configurations, which atthe lowest Matsubara frequency ω = 2 πT takes value D = πT , agreeing with [36]. The smallness of the diffusionconstant is because it stems from the zero mode of the fluids, but rather the excited states.If we set g = 4 π/N c and replace ω by the lowest Matsubara frequency ω = 2 πT , then the conductivity andsusceptibility agree with [36] σ = N c T π , χ = N c | k | T πω . (37)The second consequence of the self-duality is about the shear viscosity, which is related to component of correlationfunction of stress-energy tensor being a response to metric. Differ from the topological terms, the Maxwell term isthe only term in the effective action containing metric g µν . Strictly speaking, all physics of QCD are encoded in Z [ a ] , to derive shear viscosity of quark-gluon liquids, we need to vary with respect to the metric perturbations ofMaxwell term in Z [ a ] , but when a and b fields are integrated out, it is equivalent to vary with respect to the metricperturbation in the effective action S eff ( A ex ) = − ln Z [ A ex ] . We will rewrite the action by explicitly including themetric (in order to distinguish between the gauge coupling and metric, we use g ∗ to represent the gauge coupling), S = 14 g ∗ ˆ d x √ gF αβ F αβ , (38)then doing the functional derivative with respect to the metric and finally taking the limit of flat metric back lim √ g → δ S eff ( A ex ) δg xy δg xy = 14 g ∗ (cid:18) F xy F xy − F αβ F αβ (cid:19) . (39)By considering the QGP is homogeneous and isotropic ( | F xy | = | F xz | = | F yz | ), so that the component satisfies F xy F xy = F αβ F αβ . For the field strengths in the action are homogeneous, the two-point correlation function ofstress-energy tensor G xy,xy , is just given by the average action density, G xy,xy ( ω, k ) = lim √ g → δ ln Z [ A ex ] δg xy ( ω, k ) δg xy (0 , > π | Q | g ∗ V , (40)in which V is a Euclidean 4-volume defined as ln Z ( x ) = V ´ d k (2 π ) ln Z ( k ) . Then, by using the Kubo formula, theminimal value of action gives lower bound to a shear viscosity η = lim ω → ω G xy,xy ( ω, k ) > π | Q | g ∗ V . (41)in which we have used lim ω → ωV = πV .On the other hand, the third law of thermodynamics tells us that there are non-vanishing entropy near zerotemperature. Considering the thermodynamic relation for the entropy density s = ∂p/∂T , moreover, because the thesystem is isotropic, p x = p y = p z = p , then the pressure is related to the energy density by ǫ = 3 p , which can be givenby the minimal value of the action ǫ = − TV ln Z = TV S S.D. near the zero temperature. Finally we have an asymptoticentropy density in the vicinity of the zero temperature s = 8 π | Q | g ∗ V . (42)The entropy density averaged shear viscosity is a dimensionless quantity, which measures a pure quantum (zerotemperature) originated intrinsic viscosity, ηs = 14 π , (43)in which is archived when the system is self-dual, or equivalently in a BPS state.An important observation of the section is that the self-duality here is crucial for not only achieving the small valueof diffusion constant D but also for that of the η/s . It reflects the fact that the small value of these quantities areclosely related to a mixture of equal importance of the electric and magnetic components of the plasma, which hasbeen discussed in [34, 37–39]. This small value of η/s is also obtained from another self-dual theory [36, 40], theAdS/CFT correspondence, in which people conjectures that the constant is a universal lower bound. The assumptionof isotropy is also important for the result, since without isotropic the lower bound of the action can not constrain eachcomponent of the shear viscosity, and hence certain component may be lower than / π and breaks the conjecture. D. Fractionalized Charge
In this section we show that the fact (iii) is also a natural consequence of the effective theory. Charge fractionalizationis an important mathematical feature of the topological action, and hence be a characteristic of the QGP phase beinga topological phase. Following the form of the effective action Eq.(20), we drop the Maxwell term and focus on thetopological action. Considering effective field action for each parton field (labelled by flavor index I = 1 , , ..., m ) arein its fractionalized topological order characterized by coefficient m , S ( I ) = i m π ˆ d xǫ µναβ b ( I ) µν ∂ α a ( I ) β + iθ π ˆ d xǫ µναβ ∂ µ a ( I ) ν ∂ α a ( I ) β + ˆ d xj ( I ) µ A exµ (44)in which j ( I ) µ = m π ǫ µναβ ∂ ν b ( I ) αβ is the parton current. Like the Chern-Simons theory of the fractional quantum Hallfluids in which the Chern-number characterizes its topological order, the fractionalized topological order of each partonin the plasma is characterized by the integer number m in its action, which must be odd for the fermion statistic ofeach parton.Since the effective coefficient m eff in an effective action for total partons follows the composition law m − eff = P I m − I from the coefficient of each parton, hence we can prove that the effective action is given by Z [ A ex ] = ˆ Y I D a ( I ) D b ( I ) exp − X I S ( I ) ! = ˆ D a D b exp ( − S eff ) (45)in which S eff = i π ˆ d xǫ µναβ b µν ∂ α a β + iθ π ˆ d xǫ µναβ ∂ µ a ν ∂ α a β + ˆ d xj µ A exµ , (46)with fractionalized charges and parton fields satisfying e = X I e I = 1 , a µ = X I a ( I ) µ , b µν = X I b ( I ) µν , j µ = X I j ( I ) µ . (47)The observed effective charge e is quantized by fundamental unit, but here the current j µ seems as a compositecurrent and can be splited into constitutes with fractionalized charges e I = e/ m in its fractionalized topological order.In this sense, the plasma phase is a deconfined phase. In the soup of QGP, the baryon with integer electric charge issplited into three individual quarks with fractionalized electric charges. In this framework, the exotic assumption thatquarks have fractional electric charge ( e/ ) is a natural consequence of the deconfined topologically ordered phasecharacterized by the odd integer m = 3 .Reminding that there are two independent fluids in the plasma, the charge current and spin current. It is worthnoting that the fractionalization does not apply to the spin current. Opposite to the direct coupling between thecharge current and the external gauge field A ex , the coupling between the spin density and the external magneticfield is indirect, mediating by the infinitely massive b fields. The b field, as is proved, could be splited into constitutes b ( I ) and hence the corresponding coupling between b ( I ) and external magnetic field ˜ F ex is fractionalized, however,the ˜ F ex can not couple to each individual spin constitute s ( I ) µν = m π ǫ µναβ ∂ α a ( I ) β since all intermediate constitute b ( I ) fields must be integrated out. Thus the coupling constant between spin density and external magnetic field is notfractionalized into smaller units, i.e. spin or angular momentum does not split.0 IV. CONCLUSIONS
In this paper, we show remarkable connections between a low energy effective field theory of QGP and a topologicalquantum field theory, and a hydrodynamic theory. In many aspects the deconfined QGP phase of strong interactionsystem is analogous to the topological phase in condensed matter systems. In fact, the central argument of the paper isthat it is indeed a topological phase which is also argued in [41]. The low energy degrees of freedoms in the topologicalphase are incompressible quantum fluids which are analogous with the nearly perfect quark-gluon liquid around T c discovered at RHIC. Bosonization approach makes an effective description to the topological phase and low energytopological fluid possible, due to the fact that a gauge theory with massive fermions and the bosonized descriptionshare the same topological limit. Below the massive scale, which is expected several times of T c , the bosonizationapproach is valid, and the effective action Eq.(20) is the central result of the paper, which is suggested as an infraredfixed point action for the QCD system coupling to an U(1) external source. A hydrodynamic interpretation of thegauge theory requires a non-trivial value θ = π . In such a theory, the QGP is effectively described by two typesof fluids: charge current and spin currents, governed by hydrodynamics continuity equations. They represent thelow energy collective modes about the topologically ordered state. The exotic phenomenon of QGP discovered atRHIC, for instance the smallness of pressure and viscosity around T c , as well as the charge fractionalization of theexcitations, seem much easier to be understood by its topological nature suggested in the effective theory. Both theelectric and magnetic components are equally important and half-half mixing in the quark-gluon liquids, which isalso topologically originated. Indeed, the electric-magnetic self-duality is closely related to achieving a small value ofdiffusion constant and η/s , making the quark-gluon liquids most perfect liquids. Acknowledgments
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