Time-evolution of magnetic field in hot nuclear matter with fluctuating topological charge

The time-evolution of the magnetic field in hot homogeneous nuclear matter has two qualitatively different stages separated by the sphaleron transition time $\tau_c$. At early times the axial chemical potential and the corresponding chiral conductivity $\sigma_\chi$ are slow functions of time. The soft chiral modes $k<\sigma_\chi$ of the magnetic field grow exponentially with time, which is known as the chiral instability. At later times $\sigma_\chi$ fluctuates due to the sphaleron transitions and can be regarded as a random process. It is argued that the average magnetic field is exponentially damped at later times. The time-evolution of the average magnetic energy is more complicated and depends on the electrical conductivity of the chiral matter but does not depend on chirality. It exhibits instability only if the matter is a poor electrical conductor, such as the quark-gluon plasma near the critical temperature. The precise conditions for the instability and the growth rate of the unstable modes are derived.


I. INTRODUCTION
The topological configurations of gluon fields form P and T -odd domains in hot nuclear matter.
Their topological charge couples to the electromagnetic fields by way of the chiral anomaly. Electrodynamics, modified in the presence of the topological domains, exhibits a number of novel effects such as the chiral magnetic effect [1,2] and the anomalous Hall effect [3]. One of its features that attracted a considerable interest is the emergence of the soft magnetic field modes exponentially growing in time known as the chiral instability [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. These unstable modes transfer helicity from the chiral medium, such as the quark-gluon plasma, to the magnetic field in a process known as the inverse cascade [4,20].
The chiral instability can be most easily derived using the effective Maxwell-Chern-Simons theory which adds to the Maxwell Lagrangian a term that couples FF directly to the topological domains by means of the chiral anomaly [21][22][23]. The dynamics of the electromagnetic field in the spatially uniform chiral matter with electrical conductivity σ can be described by the vector potential A which satisfies the following equation in the radiation gauge

arXiv:1911.01357v1 [hep-ph] 4 Nov 2019
Here σ χ = c A µ 5 is the chiral conductivity, with c A being the anomaly coefficient and µ 5 the axial chemical potential [2,24]. Seeking a solution in the form A = a kλ (t) λ e ik·x , where λ are the circular polarization vectors of helicity λ = ±1 and using the identity ik × λ = kλ λ one deduces that the amplitude a kλ satisfies the equation Assuming that the chiral conductivity is a slow function of time and neglecting for the sake of brevity the electrical conductivity, the magnetic field for each plane wave mode reads, up to the pre-exponential factors, Clearly, when λσ χ < 0 the magnetic field modes k < |λσ χ | are unstable [5]. This instability is manifestly chiral.
It is remarkable, that since (3) satisfies the modified Maxwell equations, the exponential growth of the magnetic field is not constrained by the energy conservation [17,28]. However, it is constrained by the conservation of the total helicity of the magnetic field and the chiral matter, which follows from the chiral anomaly equation [25,26]. As the magnetic field increases, the helicity flows from the chiral matter to the field until the chiral conductivity vanishes [4,17,27,28]. As argued in [28], this helicity transfer is an adiabatic process, meaning that σ χ is a slowly varying function of time.
Let us now examine the assumptions that led us from a MCS equation (1) to the unstable solution (3). Dropping the first assumption σ = 0 results in a more complicated expression for the magnetic field. However, the finite value of the electrical conductivity does not have any significant effect upon the instability dynamics at constant (or slowly varying) σ χ [28]. The second and crucial assumption is the slow variation of the chiral conductivity. It holds while t < τ c where τ c is the sphaleron transition time, i.e. the transition time between the gauge field configurations of different topological charge. By the time t = τ c , the exponent in (3) reaches its maximal value exp(σ χ τ c ).
The estimates of the sphaleron transition time τ c ∼ 1/(g 4 T ) [29] and the chiral conductivity σ χ ∼ e 2 µ 5 [2,24], at temperature T , indicate that σ χ τ c 1 * . The implication is that during the time interval t < τ c when the chiral conductivity (and the corresponding axial chemical potential) may be regarded as slowly varying functions of time, the magnetic field instability simply does not have enough time to grow. Thus, in order to study the magnetic field instability in the hot nuclear matter, one is required to examine the opposite limit of t τ c , when many sphaleron transition can occur causing fluctuations of the topological charge of the chiral domains.
The main goal of this paper is to examine this limit by regarding the chiral conductivity as a stochastic process with vanishing expectation value σ χ = 0. I assume that its dispersion Σ χ = σ 2 χ equals c A µ 5 and the auto-correlation function σ χ (t)σ χ (t − τ ) has support at t < τ c , meaning that the sphaleron transition time τ c is the correlation time. Eq. (2) then describes the harmonic oscillator with a random frequency. Using the Van Kampen's theory [32] one can deduce from (2) the ordinary differential equations for the average values of the field amplitude a kλ and its second moments (a kλ ) 2 , (ȧ kλ ) 2 . The main result of this theory is that the average value of the magnetic field decays exponentially with time. The average value of the magnetic field energy also decays if the plasma is a good electric conductor, which occurs at high temperatures. However, at temperatures close to the critical temperature the plasma is a poor conductor and the average energy is unstable.
The paper is structured as follows. In Sec. II the Van Kampen theory is reviewed and employed to reduce the stochastic equation (2) to the ordinary differential equations for the averages of the amplitude and its second moment. The solutions to these equations are used in Sec. III and Sec. IV to analyze the time evolution of the average magnetic field, electromagnetic energy and magnetic helicity with time. The summary and discussion is presented in Sec. V.

II. EVOLUTION EQUATIONS FOR THE AMPLITUDE MOMENTS
To study the late time behavior of the magnetic field, one can regard the chiral conductivity σ χ (t) as a random process and hence (2) becomes a stochastic equation describing time-evolution of the field amplitude with momentum k and polarization λ. Eq. (2) does not have an analytical solution. However, one can deduce from it ordinary differential equations for the expectation value of the amplitude moments using the Van Kampen theory [32]. This section represents a brief summary of the relevant results. For the reasons explained below, the underdamped and the overdamped modes with k σ/2 and k σ/2 respectively are considered separately from the critically damped ones with k ≈ σ/2.
To begin with, I define, for a given k and λ, a new variable x = a kλ e σt/2 which satisfies the where The random process ξ(t) is such that ξ = 0. The correlation time of ξ is τ c , meaning that the auto-correlation function ξ(t)ξ(t − τ ) vanishes at τ > τ c . For the modes with ω = 0 it is convenient to introduce the dimensionless time variable t = ωt.
Eq. (4) can be cast in the matrix form The main result of [32] is that given Eq. (6) with arbitrary u, A 0 and B, at late times t τ c the expectation value of u satisfies the equation provided that which allows treating the fluctuating term in (4) as a perturbation. In view of ω-dependence of α in (5), the condition (9) excludes modes with small ω's. Namely, expanding k = σ/2 + δk one obtains that (9) is satisfied if Since Σ χ τ c ∼ e 2 µ 5 /g 4 T 1, (9) holds for all k's except in the narrow interval around k = σ/2.
Eq. (8) can now be applied to the underdamped and overdamped modes. Computing with the help of (7) substituting into (8) and converting the result into the second order differential equation for x yields where The evolution equations for the second moments of x can be obtained by applying (6), (8) to The corresponding matrices are which produces Eq where In general, the Van Kampen theory cannot be applied to all critically damped modes because ω 2 is not negligible as compared to the fluctuating term λkΣ χ ξ in (4). The exception is the mode with momentum k = σ/2 in which case case ω = 0 and (4) takes form where ξ is defined as in (5) and Passing to the dimensionless time variable t = βt, one can write (18) in the form (6) with α = 1 which implies The fluctuating term is a small perturbation of the initial condition if |β|τ c 1. In this case, at and c=coefficients given by (22a),(22b) and It is useful for future reference to estimate the coefficients c 1 , c 2 , c 3 appearing in the evolution equations in the previous subsections. Their ω-dependence is qualitatively different for good and poor electric conductors. The hot nuclear matter is a good electric conductor at high temperatures.
Its electrical conductivity is of the order σ ∼ T /e 2 † . However, at temperatures above but close to the critical temperature the hot nuclear matter is a poor conductor with the electrical conductivity This implies that c 2 ≈ c 3 c 1 .
The coefficient c 2 may become negative if the spectrum of ξ contains a mode oscillating with frequency 2ω. Indeed the contribution of such mode to c 2 can be estimated as This causes the parametric resonance, i.e. the exponential growth of x with time, see (32) below. † More precisely σ = 13 T /e 2 ln e −1 [33].
As a specific example, consider the Ornstein-Uhlenbeck random process with the autocorrelation function The corresponding coefficients read Multiplying (28) by cos(2ωτ ) one can check that c 1 → 1/8, c 2 → −ωτ c /2 and c 3 → ωτ c /2 at ωτ c 1. In the next subsection it is shown that at small |ω|τ c , the coefficient c 2 is always positive and so there is no parametric resonance.
(b) The Van Kampen theory fails to describe the critically damped modes of the well-conducting plasma because |β|τ c ∼ µ 5 /T /g 4 1 (unless µ 5 is very small).

Poorly conducting chiral matter
(a) The parameter ωτ c of the underdamped modes k σ/2 of a poorly conducting plasma can be estimated as ωτ c ∼ k/g 4 T . The coefficients of the modes with k g 4 T are given by (26), while the ones with k g 4 T are estimated from (13) and (17) as Estimates (30) apply also to the overdamped modes k σ/2 because |ω|τ c < στ c /2 ∼ e 2 /g 4 1.
Incidentally, this implies that the coefficients (30) are strongly ordered as c 2 c 1 c 3 1.
(b) The c-coefficients for the critically damped mode k = σ/2 can be estimated using (13), (17) with the result similar to (30) In contrast to the well conducting matter, the Van Kampen theory is valid in this case since |β|τ c ∼ (e 2 /g 4 ) µ 5 /T 1.

III. TIME-EVOLUTION OF AVERAGE MAGNETIC FIELD
In this and the following sections the results of the previous section are employed to analyze the late-time behavior of the average magnetic field and average magnetic energy.
The general solution of (12) is a linear combination of the functions where only terms of the order α 2 are retained. The corresponding magnetic field amplitudes are A. The underdamped modes k > σ/2 The condition for the instability of the underdamped modes can be satisfied only for the parametric resonance modes with c 2 < 0. Using the definition of α from (5) this yields As mentioned below (29), the parametric resonance occurs in good conductors at ωτ c 1. Using the estimate (27) in (35) and noting that the best chance to satisfy it is at smaller ω, expand it near k = σ/2 to obtain δk < Σ 2 χ τ c . On the other hand, ω 1/τ c implies that δk 1/στ 2 c . The two conditions contradict each other since Σ 2 χ στ 3 c < 1 . Thus, there are no unstable underdamped modes of the average magnetic field. Even the parametric resonance mode in a well-conducting chiral matter is rapidly diffused due to the large electric conductivity.

B. The overdamped modes k < σ/2
The amplitude of the overdamped modes can be written down as a linear combination of the The second term in the exponent is a small correction to the first one. Clearly, a kλ (t) + is stable.
In a kλ (t) − the first and the third terms in the exponent cancel out at k σ/2. Expanding to the second order in k/σ yields Thus, the overdamped modes are stable at any conductivity.
C. The critically damped mode k = σ/2 It remains to examine the critically damped mode ω = 0. As noted below (29) and (31) only poor conductors can be consistently considered in the present framework. As shown in Sec. II B, the average amplitude can be obtained from (33) by replacing ω → β and α → 1: The amplitude (38) of the critically damped mode exponentially decays at late time because c 2 > 0 as indicated by (31).

D. Conclusion
It follows from the analysis in this section that all magnetic field modes are stable at long times t τ c . This of course also includes the modes with k < σ χ /2 that exhibit exponential growth at early times.

IV. TIME-EVOLUTION OF MAGNETIC FIELD ENERGY AND HELICITY
Having examined the time-evolution of the magnetic field which is determined by the first moment of the amplitudes, we turn to the evolution of the electromagnetic energy and magnetic helicity which is determined by the second moments. The electromagnetic field energy and magnetic helicity read where each mode with given k and λ is normalized to one particle in a unit volume and the amplitudes a kλ are real. The corresponding expressions for the expectation values of energy and helicity of each mode can be express in terms of the moments of the variable x as The time evolution of the second moments of x is given by (16). The eigenvalues of the matrix in (16) to the order α 2 are Transforming (16) to the diagonal form and integrating one finds −ω 2 x 2 + ẋ 2 + 2iω xẋ = 2u + e ν + ωt , where u 0 , u ± are constants. Since ν + = ν * − the second moments and hence the amplitudes are real if u + = −u * − . The solution thus reads When plugged into (41),(42) it yields the time-evolution of the average energy and magnetic helicity.
I turn now to the analysis of the instabilities. Note, that the instabilities of (45) are different from those of (33).
At the parametric resonance (c 2 < 0), the instability can arise when either ν 0 ω > σ or Re ν ± ω > σ. In the former case, using c 3 − c 2 ∼ ωτ c and repeating the arguments of the previous paragraph yields δk < Σ 2 χ τ c which contradicts ω 1/τ c as was argued beneath (35). It actually also contradicts (10). In the later case, using c 3 + 3c 2 ∼ −ωτ c again implies the contradictory Thus, the underdamped modes of a well conducting chiral matter are stable. The dissipation beats the growth.
The unstable modes of average energy read To make a more quantitative estimate it is useful to employ the model (29). Eq. (46) reads Denote y = 2ωτ c , a = στ c and b = Σ χ /σ. With this notations one can write (47) as The expression in the exponent has a maximum However, since ν 0 is proportional to α 2 and hence helicity independent, the total average magnetic helicity of the magnetic field (40) vanishes identically To be sure, this equation does not preclude the magnetic helicity fluctuations. It only indicates that the contributions of the opposite helicity states to the magnetic helicity are on average of equal magnitude and have opposite sign.
Thus, the overdamped modes are stable.
Thus, the critical mode is stable.
It has been argued in this section that the average magnetic field becomes unstable and grows exponentially if the chiral matter has small enough electrical conductivity σ < g 4 T and large dispersion of the chiral conductivity Σ χ σ. The more precise expression is given by (50). The average magnetic helicity vanishes.

V. SUMMARY AND DISCUSSION
This paper addresses the time-evolution of the magnetic field in hot nuclear matter at times much longer than the sphaleron transition time τ c . In this limit the chiral conductivity σ χ is treated as the random process. By expanding the electromagnetic field in the basis of the circularly polarized waves it is shown that the field amplitudes obey the stochastic harmonic oscillator equation.
The relevant observable quantities are statistical averages such as the average magnetic field and average magnetic field energy.
Using the method of [32] the equations for the time-evolution of the average amplitude and its second moments were derived and solved. The solutions indicate that the average magnetic field is exponentially damped, see (12). The time-evolution of the average magnetic energy is more complicated and depends on the electrical conductivity of the chiral matter. At high temperatures when the quark-gluon plasma is expected to be highly conductive, the dissipation effects prevent the development of instabilities. However, at temperatures higher but comparable to the critical temperature, the quark-gluon plasma is known to be a poor electric conductor. It is argued that if the electrical conductivity satisfies σ < g 4 T , the average magnetic energy is unstable, provided that the dispersion of the chiral conductivity fulfills the condition Σ χ σ. The corresponding unstable modes are k > σ/2. A more accurate condition is derived in Sec. IV A.
The magnetic helicity of the right and left-handed modes also increases exponentially. However, the total magnetic helicity vanishes identically. This implies that the conservation of the total helicity cannot tame the chiral instability at later times as it does at early times [4,17,27,28].
Combined with the fact that the growth rate of the average magnetic energy is independent of λ indicates that this instability is not chiral.
These results are in a striking contrast with the constant σ χ approximation which is valid at early times t τ c when the sphaleron transitions can be neglected. In that case, the magnetic field itself is unstable and the unstable modes are soft k < σ χ . Moreover, the instability is chiral as it occurs for only one of the helicity modes at a time depending on the sign of σ χ .