Hydrodynamics of the Polyakov Line in SU$(N_c)$ Yang-Mills

We discuss a hydrodynamical description of the eigenvalues of the Polyakov line at large but finite $N_c$ for Yang-Mills theory in even and odd space-time dimensions. The hydro-static solutions for the eigenvalue densities are shown to interpolate between a uniform distribution in the confined phase and a localized distribution in the de-confined phase. The resulting critical temperatures are in overall agreement with those measured on the lattice over a broad range of $N_c$, and are consistent with the string model results at $N_c=\infty$. The stochastic relaxation of the eigenvalues of the Polyakov line out of equilibrium is captured by a hydrodynamical instanton. An estimate of the probability of formation of a Z(N$_c)$ bubble using a piece-wise sound wave is suggested.

1. Introduction. Lattice simulations of Yang-Mills theory in even and odd dimensions show that the confined phase is center symmetric [1,2]. At high temperature Yang-Mills theory is in a deconfined phase with broken center symmetry. The transition from a center symmetric to a center broken phase is non-perturbative and is the topic of intense numerical and effective model calculations [3] (and references therein). Of particular interest are the semi-classical descriptions and matrix models.
In the semi-classical approximations, the confinementdeconfinement transition is understood as the breaking of instantons into a dense plasma of dyons in the confined phase and their re-assembly into instanton molecules in the deconfined phase [4,5]. A mechanism similar to the Berezinsky-Kosterlitz-Thouless transition in lower dimensions [6], and to the transition from insulators to superconductors in topological materials [7]. In matrix models, the Yang-Mills theory is simplified to the eigenvalues of the Polyakov line and an effective potential is used with parameters fitted to the bulk pressure to study such a transition [8,9], in the spirit of the strong coupling transition in the Gross-Witten model [10].
Matrix models for the Polyakov line share much in common with unitary matrix models in the general context of random matrix theory [11]. The canonical example is Dyson [12] with its hydrodynamical interpretation [13,14]. In this letter we develop a hydrodynamical description of the gauge invariant eigenvalues of the Polyakov line for an SU(N c ) Yang-Mills theory at large but finite N c . We will use it to derive the following new results: 1/ a hydrostatic solution for the eigenvalue density that interpolates between a confining (uniform) and de-confining (localized) phase; 2/ explicit critical temperatures for the Yang-Mills transitions in 1 + 2 and 1 + 3 dimensions; 3/ a hydrodynamical instanton for the density distribution that captures the stochastic relaxation of the eigenvalues of the Polyakov line; 4/ an estimate of the fugacity or probability to form a Z(N c ) bubble using a piece-wise sound-wave.
2. Polyakov line in 1 + 2 dimensions. The matrix model partition function for the eigenvalues of the Polyakov line for SU(N c ) in 1 + 2 dimensions was discussed in [8]. If we denote by diag(e iθ1 , ..., e iθ Nc ) with i θ i = 0 the gauge invariant eigenvalues of the Polyakov line, then [8] with z ij = z i − z j and z i = e iθi . The perturbative potential V (z ij ) is center symmetric and quadratic in leading order or V (|z ij |) ≈ |z ij | 2 , with α(T ) = T 2 V 2 /2π and V 2 the spatial 2-volume [8]. N c g 2 T (ln(T /m D ) + C)/2π [15] to tame all infra-red divergences, with C ≈ 1.3 from lattice simulations [16,17].
(1) can be regarded as the normalization of the squared and real many-body wave-function Ψ 0 [z i ] which is the zero-mode solution to the Shrodinger equation H 0 Ψ 0 = 0 with the self-adjoint squared Hamiltonian with ∂ i ≡ ∂/∂θ i and the pure gauge potential a i ≡ ∂ i S.
is half the energy in the defining partition function in (1). In (2) the mass parameter is 1/2.

Hydrodynamics.
We can use the collective coordinate method in [12] to re-write (2) in terms of the density of eigenvalues as a collective variable ρ(θ) = Nc i=1 δ(θ − θ i ). For that, we re-define H 0 → H through a similarity transformation to re-absorb the diverging 2body part induced by the Vandermond contribution ∆ = [12] and is amenable after some algebra to with the potential-like contribution Here and ρ H is the periodic Hilbert transform of ρ As conjugate pairs, π(θ) and ρ(θ) satisfy the equal-time We identify the collective fluid velocity with v = ∂ θ π and re-write (3) in the more familiar hydrodynamical form modulo ultra-local terms. The Heisenberg equation for ρ yields the current conservation law ∂ t ρ = −2∂ θ (ρv), and the Heisenberg equation for v gives the Euler equation Note that all the relations hold for large but finite N c .
4. Hydro-static solution. The static hydrodynamical density follows from the minimum of (6) with v(θ) = 0, To solve (9), we insert the leading quadratic contribution with a ≡ 1/πβ(T ), b ≡ 2α(T )/β(T ) and c 1 the first moment of the density or πc 1 ≡ 2π 0 ρ(θ)cosθdθ. Let ρ 0 = N c /2π be the uniform eigenvalue density and ρ 1 = ρ − ρ 0 its deviation. Consider the Cauchy transform with η = e iθ . The contour C is counter-clockwise along the unit circle. G(z) is a holomorphic function in the complex z-plane. Let G + and G − be its realization inside and outside C respectively, so that We now carry the Hilbert transform on both sides of (10). Setting G(z) = G + (z) and using 2[ρ 1 ρ H ] H = ρ 2 H − ρ 2 1 , we have for (10) on the boundary C, thus within the circle. Here, we should require G(z = 0) = 0 to ensure that ρ 1 integrates to zero. a ≈ 1/V 2 is subleading and will be dropped. Thus (13) is algebraic in G(z). Since ρ(θ) = ρ 0 + Re G + (z = e iθ ), careful considerations of the singularity structures of the quadratic solutions to (13) yield (Θ is a step function) The analytic properties of G(z) fix c 1 /ρ 0 = 1+(1−1/b) 1 2 and θ 0 at cos θ 0 = 1−2ρ 0 /bc 1 . For b < 1 the non-uniform solution with ρ 1 = 0 is absent. For b 1, c 1 → 2ρ 0 and Therefore (14) interpolates between a uniform density distribution ρ 0 (confined phase) and a Wigner semicircle (deconfined phase) with a transition at b = 1 or T c = m D . In 1 + 2 dimensions the fundamental string tension is given to a good accuracy by √ σ 1 /g 2 N c = [20]. Thus the ratio in 1+2 dimensions with C ≈ 1.3 [16,17]. In Fig. 1 we show the behavior of (16) (upper curve) versus N c , in comparison to the numerical fit T c / √ σ 1 = 0.9026 + 0.880/N 2 c to the lattice results (lower curve) in [21]. Amusingly, (16) at large N c is consistent with 3/π in the string model [18].

Dyson Coulomb gas.
We note that (9) coincides with the saddle point equation to (1) by re-writing it using Dyson charged particle analogy on S 1 with the energy 2S[z] = i<j G(z ij ) and the pair interaction (17) At large N c the ensemble described by (1) is sufficiently dense to allow the change in the measure. Following Dyson [11] we obtain with the effective action The β contribution is the self Coulomb subtraction and is consistent with the subtraction in the Hilbert transform. The saddle point equation δΓ/δρ = 0 following from (18)(19) is in agreement with the hydro-static equation (9), 6. Hydrodynamical instanton. The fixed time zero energy solution to (7) is an instanton with imaginary velocity v = −iA. We have checked that this is a solution to (8) for all times. The current j ≡ ρv = −iρA is conserved.Thus ∂ τ ρ − 2∂ θ (ρA) = 0 or for Euclidean times τ = it. For A = 0 and β(T ) = 2, (21) agrees with the viscid Burger s equation describing large Wilson loops in 1 + 1 dimensions [19]. Following [11] we identify τ with the stochastic (Langevin) time. (21) describes the stochastic relaxation of the eigenvalue density of the Polyakov line (out of equilibrium) to its asymptotic (in equilibrium) hydro-static solution.
7. Sound waves. The hydrodynamical action follows from standard procedure.The momentum π(θ) = (1/∂ θ )v is canonically conjugate to the density ρ, and the Lagrange density is L = π∂ t ρ − H. Thus the action S = dtdθ ρ(θ) v 2 − u[ρ] , which is linearized by Inserting (22) into S yields with the potential For constant ρ 0 and large N c , (23) simplifies to after the rescaling v s t → t with v s = πρ 0 β(T ). (25) describes sound waves in the large N c space of holonomies.
8. Z(N c ) bubble. In a de-confined phase of infinite volume, the Yang-Mills ground state settles in one of the degenerate Z(N c ) vacua. In a finite volume, bubbles of different vacua may form [23]. Consider a de-confined bubble of volume V 2 immersed in a confined volume V 2 . In V 2 all the eigenvalues are localized initially within a small ∆θ around the origin with ρ(τ = 0, θ) = N c /∆θ ≡ ρ B , and zero otherwise.
Using this piece-wise wave as an initial condition we solve (21) with A = 0 for simplicity. For large times τ , the result is which shows the relaxation of the piece-wise wave over a time τ ≈ 1/v s set by the speed of sound. Using (26) in S yields the Euclidean action estimate for small ∆θ The bubble formation probability or fugacity is e −S E (V2) .
9. Polyakov line in 1 + 3 dimensions. To extend our analysis to 1 + 3 dimensions, we approximate the Yang-Mills thermal state by a dense plasma of dyons and anti-dyons [4,5]. This semi-classical description reproduces a number of key features of the Yang-Mills phase both in the confined (center-symmetric) and de-confined (center-broken) phase. There are two key differences with the 1 + 2 dimensional partition function in (1). First the many-body energy 2S with γ(T ) = 4πN c f V 3 and f = 4πΛ 4 /T g 4 the dyon fugacity [4]. Second and more importantly β(T ) = 2 and is not extensive with the spatial 3-volume V 3 . Finally, α(T ) = T 3 V 3 /3. Since (θ i+1 − θ i ) ≈ 1/2πρ(θ i ), then in the continuum the additional string of factors in (28) is With this in mind, a re-run of the preceding arguments yields the Hamiltonian in (3)(4) with the shifted potential and N c lnγ 0 [ρ] = dθρ(θ)ln(ρ(θ)/N c ). The hydro-static equation (9)  The β = 2 contribution is now sub-leading and can be dropped. The corresponding solution to (31) is a localized density for πc 1 = 2π 0 dθρ(θ)cosθ = 0, and a uniform density ρ 0 = N c /2π for c 1 = 0. Specifically with c = c 1 /N c and γ = γ/N 3 c . The two parameters η = 8πα(T )/γ and x = c ηγ 0 are fixed by the transcendental equations A solution exists only for γ < 2α(T )/π. Else the density is uniform. Thus the transition temperature from center symmetric (confining) to center-broken (deconfining) occurs for α(T c )/γ(T c ) = π/2N 3 c or T 4 c = 3 8π Λ 4 λ 2 with λ = g 2 N c /8π 2 . For the dyon model, the fundamental string tension is given by σ 1 = (N c /π) sin(π/N c ) Λ 2 /λ [4]. Thus the model independent ratio in 1 + 3 dimensions (34) compares favorably to the lattice results [22] even for small N c as shown in Fig. 2. At large N c , (34) is consistent with the value of 3/2π in the string model [18]. 10. Conclusions. The hydrodynamical description of the Polyakov line captures aspects of the center dynamics in Yang-Mills theory in terms of the gauge invariant density of eigenvalues. The hydro-static equations yield solutions that interpolate between a center symmetric (confining) and a center-broken (de-confining) phase. The transition temperatures normalized to the string tension compare well to the lattice results over a broad range of N c , and asymptote the string model results at N c = ∞. The hydrodynamical set-up supports a hydrodynamical instanton that describes the stochastic relaxation of the eigenvalues of the Polyakov line viewed as a fluid. The fluid supports sound waves that can be used to estimate the probability of formation of Z(N c ) bubbles. The relaxation of a fluid of holonomies across the critical temperature may prove useful for understanding the onset of equilibration in a Yang-Mills plasma.