Notes on chiral hydrodynamics within effective theory approach
Abstract
We address the issue of evaluating chiral effects (such as the newly discovered chiral separation) in hydrodynamic approximation. The main tool we use is effective theory which defines interaction in terms of chemical potentials . In the lowest order in we reproduce recent results based on thermodynamic considerations. In higher orders the results depend on details of infrared cutoff. Another point of our interest is an alternative way of the anomaly matching through introduction of effective scalar fields arising in the hydrodynamic approximation.
1 Introduction.
There is a clear renewed interest in the relativistic hydrodynamics, in view of its success in describing properties of the (strongly interacting) quarkgluon plasma (see, e.g., [1] and references therein). Recently some new hydrodynamic phenomena resulted from existence of novel transport coefficients have been found [2, 3]. One considers a medium made of chiral fermions such that there exist two conserved charges and and one can introduce two corresponding chemical potentials: and . Then in the standard hydrodynamic approximation one would use for the phenomenological vector and axialvector currents the following textbook expressions:
(1) 
where and are densities of particles with the corresponding charges, while is the 4velocity of an element of the liquid. As is argued in [2, 3] one can add another term to the axialvector current
(2) 
where and the new term describes chirality separation through rotation. Indeed, if the liquid is rotated with angular velocity then according to (2) there arises chiral current along the vector . A remarkable feature of the new term is that, according to [3] the coefficient (in our approach see (10)) is uniquely fixed by the requirement of positivity of the divergence of the entropy current, .
An intriguing point in the evaluation of the coefficient in Ref. [3] is that it is mostly given in thermodynamic terms. The only fieldtheoretic input is the chiral anomaly in the presence of both electric and magnetic fields. A puzzling observation is that the coefficient is related to the coefficient in front of the chiral anomaly although the chiral separation (2) persists in the absence of any electromagnetic fields. Some progress in understanding relation of Eq. (2) to the field theory was made in Ref. [4] where it was observed that the chirality separation, similar to (2) arises also in the flavorrelated superfluidity (at zero temperature). The derivation [4] is based on the construction [5] of the conserved currents corresponding to WessZumino effective action in the Goldstone phase.
We address the issue of the chiral effects in the hydrodynamic approximation within an effective theory. The interaction in this effective theory is determined by the chemical potentials, and and the perturbation theory looks as an expansion in them. The first term in the expansion reproduces the leading contribution to . Further terms appear to depend on details of the infrared regularization.
2 Effective field theory.
Our set up is as follows. There exists a fundamental Lagrangian constructed on massless lefthanded, and righthanded, fermions. The fundamental interaction is assumed to conserve chirality and be nonanomalous, while in presence of external electromagnetic field there is chiral anomaly. As an example one could think of two flavors of massless quarks interacting with gluons in the standard flavorblind way. This interaction is assumed to be responsible for formation of dense quarkgluon matter state, whose relevant effective longdistance description is given in terms of relativistic hydrodynamics. We choose microscopic currents as:
(3) 
where the quark field , and are the standard Pauli matrices in flavor space. Classically these currents are conserved: .
To imitate the effects of the medium one introduces chemical potentials, conjugated to the corresponding conserved charges:
(4) 
where and . At Lagrangian level (4) corresponds to:
(5) 
where we have introduced the notation , . It is worth mentioning that the condition for the axial current to be nonanomalous with respect to the strong interactions dictates its isovector nature, i.e. one must have . There is no such restriction for the vector current which may contain singlet component. The fact of currents conservation allows to introduce selfconsistently the chemical potentials.
As is well known, the chemical potential conjugated to the charge in quantum field theory brings two aspects absent at zero density: first, it shifts energy levels of the system (since all the levels below are occupied) and second, prescription for propagator poles changes to and dependent one (see, e.g. [6]) and in this sense the theory becomes nonlocal in coordinate space. It is worth stressing that both effects correspond to a change of the vacuum state. However, as we will argue below, from effective theory point of view they play rather different roles.
So far the discussion proceeded in terms of fundamental microscopic fields. We can now address the most important issue of transition to effective theory. First, one is to introduce ”physically microscopic volume” of the medium, , centered around the point , in which infrared, longdistance fields of effective theory may be regarded as being uniform. Performing Lorentz boost to the rest frame one gets:
(6) 
where is the fourvelocity of a given element of the fluid in the centerofenergy rest frame. As an infrared variable, it is supposed to be independent. We have also introduced coupling to the external electromagnetic field with the charge matrix . This field is also treated as longdistance nondynamical one. The term stays for strong interaction part of the total action whose exact form is not relevant here.
To proceed one has to introduce summation over (i.e. integration over effective theory coordinates ) and integrate out microscopic fields: quarks and gluons. However for our purposes we need not to know the final result of this complex procedure. Instead, we notice that the effective currents anomalous (non)conservation cannot be affected by dependence of term in (6). The latter is an infrared effect vanishing at and corresponding to vacuum state change. At large momenta one always comes back to the standard prescription with constant . This is completely analogous to the results on independence of chiral anomaly on temperature/density found in [7]. Speaking more technically, coupling of the effective scalar mass field to fermions is nonanomalous regardless of its dynamics (encoded in ,  and dependence of its complex part in our case), even if the real part of the mass vanishes. Thus the , dependence of the anomaly can come only from the term in the effective action. This term, on the other hand, can be dealt with on exactly the same footing as the standard gauge field term . For finding the anomaly it is convenient to use the FujikawaVergeles method [8]. According to this method, anomalies emerge due to noninvariance of the pathintegral measure under field transformations. For the sake of simplicity we consider below only diagonal components of the currents (i.e. components), however we continue to use invariant isospin notation. Consider the following transformation
(7) 
One readily finds ^{1}^{1}1We could have defined anomaly in such a way that it does not contribute to . In presence of both chemical potentials and there is no physical motivation for such a regularization.:
(8) 
(9) 
In hydrodynamic approximation one has correspondence between microscopic currents , and macroscopic effective ones , carrying the same conserved quantum numbers and given by (1). The collinear nature of the effective currents is an important feature of nonsuperfluid hydrodynamics, where all charges densities propagate with the same velocity . Having in mind that the right hand sides of (8) and (9) contain only effective longdistance fields one can replace divergencies of microscopic currents in the left hand sides by those of the effective currents . In particular, the anomaly equation (8) can be rewritten as:
(10) 
where is the magnetic field in the rest frame. We will come back to discuss the implications of this equation later. Now, let us emphasize that in the effective theory we have an expansion in . The anomalous terms are exhausted by the triangle graph, or by terms quadratic in kept explicit in (9), (10). Another observation is that to keep the coefficient of the term nonvanishing one must have nonzero singlet component in the vector current (otherwise ). The result for the coefficient implied by (10) coincides with that of the paper [3] in the lowest nontrivial order in the chemical potentials. The comparison is not absolutely straightforward, though. The reason is that we introduce two chemical potentials, , and assume interactions to conserve parity (so that is actually a pseudoscalar). On the other hand, the simplest version considered in most detail in [3] deals with a single chiral current, with no parity conservation
It is worth emphasizing that there exist contributions of higher orders in to the hydrodynamic currents. The higherorder terms, however, do not contribute to the anomalous divergences evaluated within the effective theory and for this reason do not enter Eq (8). Moreover, such terms depend, generally speaking, on details of the infrared cutoff and we make no attempt to find them in closed form. For the sake of an estimate, consider the contribution to the term. It is of the order:
(11) 
where we keep the factor just to indicate that it is a oneloop correction and is an infrared cutoff in the energy/momentum integration. In the hydrodynamic limit, the following estimate for seems reasonable:
where are the energy density and pressure respectively and is the particle density. Moreover, the ratio is known to play the role of the mass in the relativistic hydrodynamic. The effectivetheory estimate (11) reproduces then the structure of the explicit contribution to found in [3], .
Our final remark in this section is that the expression (9) for the vector current is closely related to another chiral effect in the hydrodynamic approximation, that is the chiral magnetic effect thoroughly discussed in the literature, for a review and references see [9]. In particular, in case and with one flavor we have
where and are components of the external magnetic field, and reproduce well known results [9] for the chiral magnetic effect. In case of nontrivial Eq (8) demonstrates existence of hydrodynamic corrections. For more details see [10].
3 Comparison with thermodynamic approach
We can compare the results obtained within the effective theory (6) with the results of Ref [3] which are entirely based on thermodynamics (plus equations of state). The results within the thermodynamic approach can be summarized in the following way:
(12) 
where is a chiral current associated with, say, lefthanded fermions. Imposing the positivity of the flow of the entropy one fixes the coefficients uniquely. In particular,
(13) 
where and are the density of the chiral particles and the corresponding chemical potential, respectively.
One can readily see that the terms are the same as within the effectivetheory approach above. However, for higher order terms the predictions vary. According to the effectivetheory approach the terms are dependent on details of the infrared cut off and not fixed in this sense. According to (13), on the other hand, the terms are uniquely determined.
To appreciate the meaning of this discrepancy one should have in mind that some further assumptions were made in [3]. In particular, the idealliquid equations of motion are used. More generally, as is emphasized in [3] the vorticity term appears within the standard hydrodynamic approximation in higher orders in derivatives than the leading term, which is . Keeping systematically all the terms of the next order in derivatives (not only ) would modify equations of motion and destroy (13). Moreover, as is noticed in [11] there is a kind of or gauge invariance in relativistic formulation of hydrodynamics which allows to redefine higherderivatives terms in expansion of the energymomentum tensor and of currents.
Thus, we believe that it is the extra assumptions made which allow to fix the terms within the thermodynamic approach. Within the effective theory approach the terms are infrared dependent. It is worth mentioning that the infrared dependence of higher order terms was found also within a superfluid version of hydrodynamics considered in [4] ^{2}^{2}2Actually the Ref [4] does not state explicitly that the terms are not uniquely fixed. Rather, a particular prescription for an infrared cut off is picked up..
4 Conservation laws in hydrodynamic approximation
To summarize, within the model considered we have two types of anomalies. First, there is the common chiral anomaly in the divergence of the fundamental axial current, see Eq. (8). Second, there are anomalies presented in the effective theory alone.
Let us first consider the case when the product for external fields and the fundamental currents are conserved. The origin of the anomalies in the effective theory is that the symmetry of the effective theory is not the same as of the fundamental interaction. The point is clearly illustrated by the toy model (3), (5). The strong interaction of quarks and gluons is flavor blind. We choose however asymmetric initial conditions by fixing the average number of quarks but not of quarks. Introduction of the effective interaction (5) transfers this flavorasymmetry of the initial condition into the flavordependent effective interaction. Hence, anomalies emerge in the language of the effective theory.
The first example of this type was seemingly given in Ref. [5]. In that case, effective Lagrangian for the interaction of Goldstone bosons is constructed in terms of the matrix
The conservation condition is then a sum of two terms, naive and anomalylike ones:
(14) 
As is noted first in Ref. [4] the analogy between (14) and the hydrodynamic models we are considering might be much more direct than it appears at first sight. We come back to discuss this point later.
Note that the effective theory fixes all the anomalous terms in the lefthand side of the condition (10) in the absence of the genuine chiral anomaly which arises in case of . This is again in analogy with the case considered in Ref. [5] and in contrast with the thermodynamic derivation of Ref. [3] where the genuine chiral anomaly is a necessary input to fix the vorticity coefficient . The simplicity of the Eq. (10) is somewhat deceptive, however, since the current itself, generally speaking, contains divergencefree terms dependent on the infrared cut off.
Turn now to the case of . Then there is an immediate problem that the axial current is no longer conserved and the introduction of the chemical potential, see (6), conjugated to a nonconserved charge is not consistent, for recent discussion see [12] and references therein. As is known since long, see in particular [13] one can introduce a new conserved axial current also in the presence of external fields with :
(15) 
where
(16) 
where . Although the current is not gauge invariant, the corresponding charge, is gauge invariant. However, we are considering the approximation of external electromagnetic fields and the value of is not dynamical. In this sense the chiral charge of the medium is changed under influence of external fields, independent of possible redefinition (16). Therefore Eq (10) is rather to be understood as an approximate, in the sense that we neglect the change of the axial charge due to the (the change of the axial charge of the medium is proportional to time for constant and ).
Turn now to a more delicate point, that is how the ’t Hooft matching condition [14] is realized in the hydrodynamic approximation. Let us begin with commonly accepted points. At vanishing temperatures, the alternatives are that there exist either massless, not confined quarks or massless pseudoscalars. Clearly, in QCD the Goldstone mode is realized at . At small finite temperatures there are well defined corrections to the pseudoscalar decays constants [15]. Finally, at the temperature of deconfining phase transition pions become massive and the matching involves massless, not confined quarks.
Now we are considering hydrodynamic approximation which assumes averaging over distances larger than the free path of the quarks:
As a result of the interaction liquid is formed and one considers hydrodynamics, that is a classical approximation. The central point is then that there are no fermionic classical fields. In other words , in the hydrodynamic approximation there can be no fermionic (massless) excitations which would trivially saturate the ’t Hooft matching condition. Thus, we are led to conclude that in the hydrodynamic approximation the ’t Hooft consistency condition is to be satisfied on the bosonic massless modes ^{3}^{3}3In reality, even the light  and quarks might be too heavy to be considered massless on the distances relevant to the hydrodynamics. .
To reiterate, for fermions interacting only with external magnetic field the chiral anomaly is realized through chiral zero modes, which are nothing else but particular Landau levels, for details see [16, 9]. However, now we consider the case when the fermions interact among themselves and form a liquid as a result of this interaction. This approximation corresponds to
(17) 
where is the freepath length in the liquid and is the radius of the wouldbe zeroenergy Landau level. Note also that the hydrodynamic approximation assumes coarse graining at scale larger than . Then the fermions lose coherence and Landau levels do not exist as solutions any longer. Therefore in the hydrodynamic approximation the matching of the chiral anomaly to physical fermionic excitations is questionable.
Thus, matching of the quark anomaly to massless pseudoscalar degrees of freedom seems to be a viable alternative. In that case one would introduce an analog, , to the matrix above and construct an effective current in terms of a new field which might be called ‘hydrodynamic shadow’ of the pion. As is argued in Ref. [4] the two terms in the current (14) can be interpreted in the hydrodynamic approximation as naive current and the vorticity term .
In our case, introduction of a specifically hydrodynamic massless excitations would result, generally speaking, in a twocomponent liquid:
(18) 
where are two independent 4velocities. Relativistic version of the hydrodynamics of twocomponent liquids has been intensely discussed recently, see in particular [17]. Generically, existence of massless bosonic excitations results in superfluidity.
5 Conclusion.
In this note we have attempted to evaluate the newly discovered hydrodynamic chiral effects within the effectivetheory approach. The perturbation theory within this approach is an expansion in chemical potentials, . The effective theory is anomalous. The chiral anomaly fixes uniquely the terms. Moreover, the anomaly is exhausted, as usual, by its lowest nontrivial order. The terms generated by the anomaly coincide with the terms found earlier within a pure thermodynamic approach [3]. Higherorder terms turn to be infrared dependent within the effectivetheory approach.
Our interest in the problems considered stems not so much from phenomenological applications but rather from the fact that hydrodynamics appears to provide a novel example of realization of chiral symmetries. Namely, the symmetry which is not anomalous in the fundamental theory turns to be anomalous within the hydrodynamic approximation. One of the mechanisms of generating anomalous effective theories is the introduction of chemicalpotential terms. Originally, chemical potentials reflect initial conditions, in terms of conserved charges. The symmetry of the initial conditions does not necessarily coincide with symmetry of the fundamental interactions. Also, we argued that the ’t Hooft matching condition in the hydrodynamics favors massless boson excitations. Thus, even at temperatures above the deconfining phase transition there could exist specific hydrodynamic excitations with quantum numbers of ordinary pions.
We are thankful to A.S. Gorsky, F.V. Gubarev and M.V. Isachenkov for discussions. The work of A.V. Sadofyev and of V.I. Zakharov was partially supported by RFBR grant no. 100201483. The work of Sadofyev was also partially supported by DAAD LeonhardEulerStipendium 20102011.
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