Hydrodynamics as v s → c

: I present the simplest 3+1 dimensional quantum field theory for which the speed of sound can be arbitrarily close to the speed of light. Examining the hydrodynamics, I find cases where the shear viscosity is finite, but the “shear relaxation coefficient” appears always to be divergently large.


Introduction
It appears that the high-density equation of state of QCD at large baryon chemical potential, such as exists in neutron star interiors, requires a startlingly high speed of sound to support the largest-mass neutron stars so far observed [1].This has raised the general theoretical question -how high can the speed of sound be in physically reasonable quantum field theories?It has been proposed [2] that the speed of sound v s in theories without chemical potentials should not exceed the conformal value 1 v 2 s = 1/3.(We will mention a counterexample in the discussion, for which v 2 s mildly exceeds 1/3.)Recently, Hippert Noronha and Romatschke have argued that there should be an upper bound on the speed of sound, v 2 s < 0.781, based on hydrodynamical arguments [3].Their argument assumes that the theory in question has "ordinary" hydrodynamical behavior -in particular, the shear viscosity relaxation rate τ π should exist.On the other hand, in the presence of chemical potentials, there is a known example where the speed of sound can be arbitrarily close to the speed of light.Son and Stephanov [4] have shown that, within chiral perturbation theory, QCD at nonzero isospin chemical potential obeys P/ε = (µ 2 I − m 2 π )/(µ 2 I + 3m 2 π ), which in the chiral limit m 2 π → 0 returns P/ε → 1 up to corrections suppressed by m 2 π /µ 2 I and µ 2 I /(4πf π ) 2 -the latter arising from the limitations of chiral perturbation theory.Therefore the speed of sound approaches 1 in this theory in the regime m 2 π ≪ µ 2 ≪ (4πf π ) 2 .The physical origin and meaning of this counterexample is not immediately obvious.Nor is it clear how it evades the bound proposed in Ref. [3].
The purpose of this brief note is to present the simplest example of a quantum field theoretical system where v2 s can approach 1, to clearly explain the physical origin of this effect, and to study the hydrodynamics of such theories.The simplest theory we can find where v 2 s → 1 is N -component scalar λφ 4 theory with N ≥ 2, with O(N ) symmetry and a potential which leads to the spontaneous breaking of the O(N ) symmetry.This theory has N (N − 1)/2 Noether currents with associated charges.In the case with a chemical potential µ for one of these charges satisfying µ ∼ T ≪ m r with m r the heavy mass of radial scalar excitations, we show that the speed of sound approaches 1.For N = 2 this theory has an exponentially large shear viscosity, but for N > 2 the shear viscosity proves to be finite.However, τ π , defined in terms of a Kubo relation, proves to diverge in either case, which explains why the arguments of Hippert et al do not apply.
The next section demonstrates why O(N ) scalar field theory can support a speed of sound arbitrarily close to the speed of light.Section 3 investigates the hydrodynamics of this model, and shows that the shear viscosity is finite for N > 2 but the shear relaxation rate diverges.The paper ends with a brief discussion.

Examining the O(N) model with a large chemical potential
Consider N -component scalar field theory with scalar fields φ a , a = 1, . . ., N with the most general O(N ) respecting, renormalizable Lagrangian where repeated indices are summed and where we have written the −m 2 r /4 term with a minus sign so that the potential is symmetry breaking and m r will represent the mass of the "radial" excitation about the symmetry-broken minimum.Defining the vacuum expectation value v 2 = m 2 r /λ, we can rewrite the potential terms (after shifting by a constant) as The associated energy density ε = T 00 is Here as usual i runs over the spatial directions and repeated indices are summed.We will assume that λ ≪ 1 and that the renormalization scale and scheme have been chosen such that the renormalizations of v and λ are taken into account.The classical vacuum state is that one scalar has a vacuum value equal to v and all others are zero.Without loss of generality we can pick the φ 1 direction for the vacuum value, in which case φ 1 excitations have a mass m r , which we refer to as the radial mass 2 , and the other φ 2...N directions are massless Goldstone modes.We will primarily be interested in the case where both the temperature and any chemical potentials are small compared to m r .We can similarly define the pressure, which is The same three terms appear, but with different coefficients: 1. Potential energy ε pot = (λ/8)(φ a φ a −v 2 ) 2 contributes oppositely to the energy density ε and to the pressure P : P pot = −ε pot .This is familiar from cosmology as the behavior of a cosmological constant, which can also arise from a scalar field potential and plays a central role in inflation [5].
A large speed of sound v 2 s ≡ dP/dε ≃ 1 requires P ≃ ε and therefore a domination of the energy density by φ terms, ε kin ≫ ε pot , ε grad .This can occur if the energy is dominated by the spatially homogeneous motion of the scalar field through field-space, provided that this scalar condensate does not explore large values of the potential energy.This will occur if the scalar field revolves around the vacuum manifold, which will occur in the case of a chemical potential µ for a conserved Noether current in the case µ 2 ≪ m 2 r , as we see next.There are N (N − 1)/2 conserved Noether currents The associated charges are Q ab = d 3 xj 0 ab .Suppose we apply a chemical potential µ for . This is like an "angular momentum" in the (φ 1 , φ 2 ) plane which will cause the fields to rotate around in a circle within this plane: (2.6) The mean value of the field need not equal its vacuum value so we have written it as v µ .The frequency of rotation ω and condensate value v µ are solved by minimizing d 3 x ε − µj 0 : (2.9) The frequency of rotation is precisely the chemical potential.The increase in the vacuum expectation value can be understood as follows.The scalar field undergoes a field-space acceleration of ω 2 v µ , which must be compensated by a radial restoring force: (2.10) The resulting kinetic and potential energies are respectively r , so this limit is the same as µ 2 ≪ m 2 r .In this case the departure from the minimum of the potential is small, v 2 µ ≃ v 2 .In this limit, the scalar field rolls around the circular minimum of the potential, as illustrated in Figure 1.The field possesses kinetic but almost no potential energy, and the speed of sound approaches 1.The addition of a thermal bath with T ≳ µ will add a radiation component with energy density ε ≃ (N −1)π 2 T 4 /30, where (N − 1) counts the Goldstone bosons which will participate in the radiation.Provided that T 4 ≪ µ 2 v 2 , this is also a subdominant contribution.Why does a kinetic energy term of this form support P ≃ ε? Imagine changing the system's volume: V → V + δV with |δV| ≪ V.The initial charge density was j 0 12 = µv 2 µ ≃ µv 2 .Charge conservation means that j 0 must increase to keep Vj 0 fixed: Doubling the charge density required doubling the field velocity, which quadruples the energy density.Therefore δε/ε = −2δV/V.In generality, δε = −(ε + P )δV/V, so this represents P = ε.In other words, because ε ∝ (j 0 ) 2 , the energy density increases very rapidly under compression, representing a very large pressure and supporting a speed of sound approaching the speed of light.
To complete the calculation, expanding to include the first µ 4 and T 4 corrections, the number density, energy density, and pressure are given by Under a volume change, δT /T = −δV/3V and δj 0 /j 0 = −δV/V.Therefore δµ/µ = −(1 − 4µ 2 /λv 2 )δV/V.Inserting these shifts to find dP/dV and dε/dV, we arrive at (2.12) We see that scalar O(N ) λφ 4 theory has v 2 s → 1 when T 2 ∼ µ 2 ≪ m 2 r .This occurs because the energy density is dominated by a homogeneous condensate which is maximally incompressible.

Hydrodynamics of the model
Hydrodynamics is by definition the large distance and time behavior of the conserved quantities in the system and their Noether currents.In theories where only stress-energy is conserved, it is a theory of the stress tensor T µν , which typically deviates from its equilibrium form at linear order as ∆ µν ≡ g µν + u µ u ν .
In our case there are N (N − 1)/2 additional conserved Noether currents, one of which carries a large value.This complicates the general hydrodynamical equations, which must be extended to include Goldstone excitations, as discussed in Ref. [8,9].
Here we will concentrate on understanding the viscosity, without worrying about the other transport coefficients which arise.The viscosity and its relaxation time are defined in terms of a Kubo formula, which reads [10] where G xy,xy R is the retarded correlator of two T xy stress tensors in frequency ω and momentum k space.For our case the coupling is small and the interactions between light modes are all derivative interactions, so a quasiparticle or kinetic picture should be valid and we should be able to evaluate η, τ π using the kinetic theory approaches pioneered by Baym et al [11] and by Jeon and Yaffe [12,13] and extended to second-order coefficients by Moore and York [14].
First we need to expand ϵ−µj 0 about the classical solution.There are two approaches.A common approach is to use angular and radial variables.In the absence of a chemical potential, the potential then depends only on the radial variable, making the masslessness and derivative coupling of the Goldstones manifest.But the kinetic term is nonlinear, giving rise to interactions.This approach is somewhat less practical in our case because the chemical potential further breaks the symmetry and not all angular modes will turn out to be massless.So instead we will work in terms of orthogonal field directions, but we choose the fluctuations in the (φ 1 , φ 2 ) plane to co-rotate with the condensate: where h, π represent the radial excitations and the angular Goldstone mode respectively.Inserting this into ϵ − µj 0 12 , and allowing the index a ′ to run over 3, . . .N , we find Several points should be made.First, the φ a ′ field directions become massive with m 2 = µ 2 , and are no longer Goldstone modes.A fluctuation in one of these directions represents a local distortion in the field direction in which the condensate varies, which will naturally rotate at the same angular frequency µ as the condensate itself.However, π, representing a fluctuation in the phase of the condensate, remains massless.Second, the radial excitation h has also increased in mass, m 2 r = λv 2 → λv 2 µ .Finally, in this basis the Goldstone modes appear to possess non-derivative cubic and quartic interactions.We will now see how matrix elements nevertheless prove to represent derivative couplings.Consider first the case N = 2. Assuming T, µ ≪ m r the only constituents in the thermal bath are the π particles.The lowest-order matrix element for ππ → ππ scattering arises from the diagrams shown in Figure 2 and the Feynman rules yield Here (s, t, u) are the usual Mandelstam variables.Recall that m 2 r = λv 2 µ which is much larger than s, |t|, |u| ∼ T 2 .Performing a geometric series expansion, we find: The leading terms cancel, which is how the derivative-coupled nature of Goldstone boson interactions reasserts itself in our field basis.But s + t + u = 0 in massless kinematics, so the second term also cancels.The resulting matrix element is ∝ s 2 and the squared matrix element scales as ∝ s 4 .When one of the particles in a scattering process has a small momentum p ≪ T , we have s 4 ∝ p 4 .Soft modes therefore have a lifetime in the plasma proportional to p −4 .The standard calculations of transport coefficients [15] then predict that the departure from equilibrium in the presence of shear flow, represented by χ(p) in the notation of that reference, will scale as χ(p) ∝ p −3 and the shear viscosity will behave as which is small-p divergent.Therefore the theory for N = 2 does not have a well behaved shear viscosity if we only consider Goldstone bosons.Damping of the most IR Goldstone bosons will be dominated by scattering off the exponentially rare radial modes, leading to an exponentially large shear viscosity.The case N > 2 is better behaved.Consider N = 3 for concreteness.There is now an additional light mode φ 3 which I will simply write φ.The scattering matrix element for ππ → φφ, arising from the diagrams of Figure 3, is For πφ → πφ scattering we have M = t/v 2 µ by crossing.If we had used angular and radial variables for our fields, the relevant interaction would arise from the curvature of the vacuum-manifold sphere, which explains why the interaction is suppressed by two powers of the radius of the vacuum manifold v µ .The form of the matrix element is the same as for scattering between different pion species in the small m π limit [16] and was originally derived by Weinberg [17].Assuming T ∼ µ so that the φ particles are present in the medium, the scattering leads to a damping of long-wavelength π modes with decay rate Γ ∝ p 2 , a departure from equilibrium χ(p) ∝ p −1 , and a convergent value for the viscosity; Eq. (3.7) has two fewer inverse powers of p and therefore converges at small p.
We have carried out a concrete calculation for the case N = 3 and treating T ≫ µ so that both fields obey massless kinematics.Using the methodology of [13,15], we find explicitly that η = 3.38083v 4 µ /T .η/s ∼ (v/T ) 4 is parametrically large, but this is expected given that Goldstone modes are weakly coupled.
Unfortunately the shear-viscous relaxation rate is not well behaved.According to Ref. [14], this is which given our previous result, χ(p) ∝ p −1 , and n b (p) ≃ T /p for the massless Goldstone modes, leads to a linearly IR divergent contribution.Let us explore the physical reason that τ π diverges.Our theory contains Goldstone modes which propagate as hydrodynamical modes with decay rates scaling as Γ(p) ∝ p 2 for small p.Such hydrodynamical modes give rise to so-called "long-time tails" in correlation functions of stress tensors [18,19], such as the one which defines η and ητ π .It was shown by Kovtun, Moore, and Romatschke [20] that sound waves, which also have ∝ p 2 decay rates, also give rise to a contribution to Eq. (3.2) which is nonanalytic, G xy,xy R (ω) = P − iηω + O(ω 3/2 ).Strictly speaking, such a term renders ητ π ill defined, or alternatively, infinite.There are cases where the coefficient in front of ω 3/2 is small enough that it is dominated by both the ω and an ω 2 term over some range of frequencies.In such a case, there is an intermediate range of scales where second-order hydrodynamics can be applied.
In the absence of Goldstone modes, where sound modes are the principal culprit in such long-time tails, such a regime can exist in either of two cases.If the shear viscosity is large η/s ≫ 1, then the sound waves are rather efficiently damped except for very small p. Hence the long-time tails only emerge at very small ω.Also, if the number of degrees of freedom is large, then the contribution to G xy,xy R from sound modes is overwhelmed by the contributions from the large number of degrees of freedom, except again at very small ω.Both of these possibilities are discussed in Ref. [20].
But in our case, the main problem arises from the long-time tails arising from the Goldstone modes.Such modes are unavoidable in this model because we need a scalar condensate to provide the large pressure.Weaker coupling leads to a more weakly-damped Goldstone mode and therefore a larger ω 3/2 coefficient.And the limit of N → ∞, to increase the degree-of-freedom count, risks providing large additional contributions to the pressure and energy density which do not obey P ≃ ε.
In conclusion, we find that the shear viscosity is finite and calculable for N > 2, but that the long-time tails arising from Goldstone modes' contribution to the viscosity make it impossible to define τ π , or rather, the contributions of such modes to τ π diverges.This explains why this model can violate the proposed bound of Hippert, Noronha, and Romatschke.

Discussion
We see that finding theories with v 2 s → 1 is rather easy if one invokes a chemical potential.However, such chemical potentials together with scalar condensates lead to complicated hydrodynamics, and the presence of Goldstone modes generally leads to a divergently large shear relaxation time, due to the "long-time tails" which arise from the contribution of slowly-equilibrating infrared Goldstone modes to the shear viscosity.As emphasized in the last section, such long-time tails actually always occur due to sound modes and τ π is never strictly well defined, as discussed by Kovtun Moore and Romatschke [20].However, in many cases there is a broad range of scales where τ π is approximately scale invariant; but for the theories considered here this will not be the case, since the relevant Goldstone modes contribute an O(1) fraction of the shear viscosity.
One might object that the theories considered are not asymptotically free and suffer from a Landau pole.But there is at least one case where we know a UV completion.SU (3) gauge theory with two massless fundamental vectorlike fermion fields (quarks) exhibits the spontaneous breaking of an SU L (2) × SU R (2) flavor symmetry to SU V (2), with an SU (2) ∼ = S 3 vacuum manifold.The low-energy effective theory is like our N = 4 scalar theory after integrating out the radial mode, and Son and Stephanov have already shown [4] that it features a speed of sound approaching the speed of light.Hopefully this paper will then help to explain the simplicity of the physics which gives rise to v 2 s → 1 in this case.
We emphasize that the role of the chemical potential is central to our example, and indeed, all examples we know where v 2 s really significantly exceeds 1/3.But there are nevertheless examples of theories with v 2 s > 1/3 without the need for chemical potentials, provided that one does not mind a Landau pole at scales orders of magnitude larger than the thermal scale.Massless QED and massless scalar λφ 4 theory both fall in this category.
To see this, note that the pressure can generically be expanded in the temperature and the coupling g as P (g, T ) = (A − Bg 2 + . ..)T 4 (4.1) with A, B some theory dependent constants with A > 0 and B > 0 in most cases, see for instance [21][22][23][24].Calling the beta function which is positive for β 0 > 0, as occurs for scalar field theory and QED.Therefore, in these non-asymptotically-free theories, the speed of sound of an ordinary thermal medium without net conserved charge densities exceeds 1/ √ 3, albeit by a perturbatively small amount.Note that this simple example features theories which are not UV complete.It would be interesting if one could find a UV completion of such a theory, which would represent a rigorous example of a theory with v 2 s > 1/3 without chemical potentials playing any role.

Figure 1 .
Figure1.Cartoon of the effective potential and the path the scalar field takes around the valley in the potential in the presence of a chemical potential.

Figure 2 .
Figure 2. Feynman diagrams for the scattering ππ → ππ.The intermediate states are the radial mode h.