On the Chiral imbalance and Weibel Instabilities

We study the chiral-imbalance and the Weibel instabilities in presence of the quantum anomaly using the Berry-curvature modified kinetic equation. We argue that in many realistic situations, e.g. relativistic heavy-ion collisions, both the instabilities can occur simultaneously. The Weibel instability depends on the momentum anisotropy parameter $\xi$ and the angle ($\theta_n$) between the propagation vector and the anisotropy direction. It has maximum growth rate at $\theta_n=0$ while $\theta_n=\pi/2$ corresponds to a damping. On the other hand the pure chiral-imbalance instability occurs in an isotropic plasma and depends on difference between the chiral chemical potentials of right and left-handed particles. It is shown that when $\theta_n=0$, only for a very small values of the anisotropic parameter $\xi\sim \xi_c$, growth rates of the both instabilities are comparable. For the cases $\xi_c<\xi\ll1$, $\xi\approx 1$ or $\xi \geq 1$ at $\theta_n=0$, the Weibel modes dominate over the chiral-imbalance instability if $\mu_5/T\leq1$. However, when $\mu_5/T\geq1$, it is possible to have dominance of the chiral-imbalance modes at certain values of $\theta_n$ for an arbitrary $\xi$.

cited therein] is well-known in condensed matter literature and it can have applications in Weyl semimetal [23], graphene [24] etc. There exists a deep connection between a CP-violating quantum field theory and the kinetic theory with the Berry curvature corrections. In Ref. [25] it was shown that the parity-odd and parity-even correlations calculated using the modified kinetic theory are identical with the perturbative results obtained in next-to-leading order hard dense loop approximation.
In this work we aim to apply the kinetic theory with the Berry curvature corrections to some non-equilibrium situations. We first note that the results obtained in Refs. [6,25] are limited to low temperature regime T µ 5 , where µ 5 is chiral chemical potential, when the Fermi surface is well-defined. Recently Ref. [26] argues that the domain of validity of the modified kinetic theory can be extended beyond the Fermi surface to include the effect of finite temperature. As expected from the considerations of quantum-field theoretic approach [27,28,29] the parity-odd contribution remains temperature independent.
Recently using the modified-kinetic theory [25] in presence of the chiral imbalance the collective modes in electromagnetic or quark-gluon plasmas were analyzed [30]. In such a system CP-violating effect can split transverse waves into two branches [31]. It was found in Ref. [30] that in the quasi-static limit i.e.
for ω k, where ω and k respectively denote frequency and wave-number of the transverse wave, there exists an unstable mode. The instability can lead to the growth of Chern-Simons number (or magnetic-helicity in plasma physics parlance) at expense of the chiral imbalance. Similar kind of instabilities were found in Refs. [32,33,34,35,36] in different context.
It may be possible to observe the instability reported in Ref. [30] in the relativistic heavy-ion collisions. But in a realistic scenario the initial distribution function n 0 p for the strongly interacting matter formed during the collision can be anisotropic in the momentum space. This kind of initial distribution known to lead to the Weibel instability of the transverse modes. In the context of relativistic heavy-ion collision experiments Weibel instability has been extensively studied [37,38,39,40,41]. The Weibel instability is also well-known in the condensed matter [42,43] and plasma physics literatures [44,45,46] and it can generate magnetic fields in the plasma. Further it should be emphasized that both the chiral-imbalance and the Weibel instability can operate in the quasi-static regime. Therefore in the present work we aim to analyze the collective modes in an anisotropic chiral plasma and study how the chiral-imbalance and Weibel instabilities can influence each other. We believe that the results presented here will be useful in studying Weyl metals and the quark-gluon plasma created in relativistic heavyion collisions. We consider weak gauge Field limit and assume the following power counting scheme: ∂ x = O(δ) and Here, and δ are small independent parameters. In this senario we use modified collisonless kinetic (Vlasov) equation at the leading order in A µ as given in Ref. [25]: and Ω p = ±p/2p 3 . Here ± sign corresponds to right and lefted handed fermions respectively. In absence of the Berry curvature term (i.e. Ω p =0) p is independent of x, Eq.(1) reduces to the standard Vlasov equation.
In this case current density j is defined as: where ∂ P = ∂ ∂p and ∂ x = ∂ ∂x . The last term on the right hand side of the above equation represents the anomalous Hall current with σ given as follows: Using Maxwell's equations and linear response theory it is easy to write down the expression for the inverse of the propagator in temporal gauge A 0 = 0 as follows, Here, Π i j (K) is the retarded self energy which follows from expression of the induced current j µ ind = Π µν (K)A ν (K) and 2 [∆ −1 (K)] i j is the inverse of the propagator. Dispersion relation can be obtained by finding the poles of the propagator [∆(K)] i j .
Let us first concentrate on right handed fermions with chemical potential µ R . We consider the background distribution of the form n 0 In a linear response theory we are interested in the induced current by a linear-order deviation in the gauge field. We follow the power counting scheme for gauge field A µ and derivatives ∂ x as discussed earlier, and consider deviations in the current and the distribution function up to O( δ). In this case we can write the distribution in Eq.(1) as follows, where, n 0 p is the background distribution function in presence of Berry curvature while n ( ) p and n ( δ) p are the pertubations of order O( ) and O( δ) around n 0 p . Since n 0 p contains the Berry curvature contribution (Due to p ) therefore, can also be splitted into is the part of background distribution function without Berry curvature correction while n 0( δ) is the part of background distribution with Berry curvature correction. In order to bring in effect of anisotropy we follow the arguments of Ref. [41]. It is assumed that the anisotropic equilibrium distribution function can be obtained from a spherically symmetric distribution function by rescaling of one direction in the momentum space. We consider that there is a momentum anisotropy in direction of a unit vectorn. Noting that p = |p|, we replace p → p 2 + ξ(p ·n) 2 in the expression of n 0 p to get anisotropic distribution function. Here ξ is an adjustable anisotropy parameter satisfying a condition ξ > −1. It is convenient to define a new variablep such thatp = p 1 + ξ(v ·n) 2 .
Using this new variable one can write n 0(0) Since v is a unit vector one can express v = (sinθcosφ, sinθsinφ, cosθ) in spherical coordinates. By choosingn in z−direction, without any loss of generality, one can have v ·n = cosθ. Thus the angular integral in the above equation becomes d(cosθ)dφ v (1+ξcos 2 θ) 1/2 . Therefore σ x and σ y components of Eq.(6) will vanish as 2π 0 sinφdφ = 0 and 2π 0 cosφdφ = 0. While σ z will vanish because integration with respect to cos θ variable will yield it (σ z ) to be zero. Thus the anomalous Hall current term will not contribute for the problem at the hand.
Equation for the current defined in Eq.(2) can also split into O( ) and O( δ) scales as given below, After adding the contribution from all type of species i.e.
right/left fermions with charge e and chemical potential µ R /µ L as well as right/left handed antifermions with charge −e and chemical potential −µ R /µ L , using the expression j µ ind = Π µν (K)A ν (K) and Eqs. (7,8,9,10) one can obtain the expression for self energy, Π i j = Π i j + + Π i j − . The expressions for Π i j + (parity even part of polarization tensor) and Π i j − (parity-odd part) can be written as, where, We would like to mention that the total induced current is, i.e. right-handed particle/antiparticles and left-handed particles/antiparticles. The current j δ arises due to chiral imbalance its contribution from each plasma specie, depends upon e Ω p . Since e Ω p can change sign depending on the plasma specie therefore definition of C E contains both positive and negative signs. Consequently a relative signs of fermion and antifermion are different in m 2 D and C E . After performing above integrations one can get m 2 D = e 2 µ 2 R +µ 2 L 2π 2 + T 2 3 and C E = e 2 µ 5 4π 2 , where µ 5 = µ R − µ L . It should be emphasized here that C E = 0 when there is no chiral imbalance whereas m 2 D 0. It should be also be noted that the terms with anisotropy parameter ξ are contributing in the parity-odd part of the self-energy given by Eq. (12). Introduction of chemical chemical potential µ 5 for chiral fermions requires some qualification. Physically the chiral chemical potential imply an imbalance between the right handed and left handed fermion. This in turn related to the topological charge [17,32]. It should be noted here that due to the axial anomaly chiral chemical potential is not associated with any conserved charge. It can still be regarded as 'chemical potential' if its variation is sufficiently slow [30].
In order to get the expression for the propagator ∆ i j it is necessary to write Π i j in a tensor decomposition. For the present problem we need six independent projectors. For an isotropic parity-even plasmas one may need the transverse P i j T = δ i j −k i k j /k 2 and the longitudinal P i j L = k i k j /k 2 tensor projectors. Due to the presence anisotropy vectorn one needs two more projectors P i j n =ñ iñ j /ñ 2 and P i j kn = k iñ j + k jñi [47]. To account for parity odd effect we have included two anti-symmetric op- Thus we write Π i j into the basis spanned by the above six operators as: where, α,β, γ, δ λ and χ are some scalar functions of k and ω and are yet to be determined. Similarly we can write [∆ −1 (k)] i j appearing in Eq. (4) as Using Eqs. (4,14,15) one can find relationship between C's and the scalar functions appearing in Eq. (14) as: For ξ → 0, using Eqs. (11)(12) Scalar functions Π T , Π L and Π A respectively describe the transverse, longitudinal and the axial parts of the self-energy decomposition when ξ = 0 [30].
Using the orthogonality condition, Eq. (18) is the general dispersion relation and it is quite complicated to solve analytically or numerically. Here we would like to ascertain that α, β, γ and δ appearing in C's are same as those given in Ref. [41]. Eq. (18). In order to obtain the growth-rates for the instabilities, one needs to solve Eq.(18) for ω. By setting ∂ω ∂k = 0 one can find k max for which the instability can grow maximally. Upon substituting k max in the expression for ω and using ω = iΓ, one can find the growth rate Γ for the instability.    It is important to notice that there also exists a situation ξ >> 1 when the chiral-imbalance instability can play a dominant role in anisotropic plasma. This is because the Weibel instability growth rate is dependent on θ n and it is possible to find a particular value of θ n = θ nc when the growth rate of the pure-Weibel mode is close to zero. By setting ω = 0 in the pure Weibel dispersion relation, one can find for ξ >> 1, In the regime ξ < 1 but closer to unity at θ n = 0, a comparison between the growth rates of the chiralimbalance (Γ ch ) and Weibel (Γ w ) instabilities is given in the following Thus the ratio Γ ch Γ w decreases by increasing values ξ while keeping µ 5 /T fixed. This is because Γ w is increases by increasing ξ. For α e = 1 137 and µ 5 /T ≤ 1 one can clearly see from the table that the ratio Γ ch (1 + 7 cos 2θ n ) − 3z 2 (1 + 3 cos 2θ n ) where z = ω k and f (ω, k) is some function k and ω. But in the present analysis exact form of f (ω, k) may not be required.
Using the above equations and Eqs. (16,17) which in turn can give following two branches of the dispersion relation, First, we would like to note that when C A = 0, Eqs. (21)(22) reduces to exactly the same dispersion relation discussed in Ref. [41] for the Weibel instability in an anisotropic plasma when there is no parity violating effect. Let us consider Eq. (21), it can be written as: This equation is a quadratic equation in (k 2 − ω 2 ) and it's solutions can be written as, Now, it is of particular interest to consider the quasi-static limit |ω| << k, in this limit expressions for α ∼ Π T and β ∼ ω 2 k 2 Π L and λ ∼ − Π A 2 . Now Π L , Π T and Π A can be obtained by expanding Eq.(17) in the quasi static limit as: Thus in the quasi-stationary limit one can write positive branch of the transverse modes given by Eq.(24) as: (1 + 5 cos 2θ n ) + ξ 12 Here we emphasize that when ξ = 0, first two terms in the square bracket survives and Eq.(27) matches with the dispersion relation of the chiral instability given in Ref. [30] and when µ 5 = 0, the second and the last term survives to give the Weibel modes considered in Ref. [41]. Term with α e ξµ 5 factor arises due to the interaction between the Weibel and chiral-imbalance modes.
Before we analyse the interplay between the chiralimbalance and the Weibel instabilities, it is instructive to qualitatively understand their origin. First consider the chiralimbalance instability. For a such a plasma 'chiral-charge' density n is given by ∂ t n + ∇ · j = 2α e π E · B. From this one can estimate the axial charge density n ∼ α e kA 2 where A is the gauge-field. Assuming that there are only right handed par-ticles i.e (µ 5 ∼ µ R ) then the number and energy densities of the plasma respectively given by µ 5 T 2 and µ 5 2 T 2 . The fermionic number density associated with the gauge field can be estimated from the Chern-Simon term to be α e kA 2 . The number densities associated with the fields and particles have same value for k 1 ∼ µ 5 T 2 α e A 2 . The typical energy for the gauge field A ∼ k 2 A 2 . For this particular value of k 1 it can be Thus there exists a state satisfying the condition T 2 α 2 e < A 2 for which energy in the gauge field is lower than particle energy. This leads to the chiralimbalance instability [30,34]. The Weibel instability arises when the equilibrium distribution function of the plasma has anisotropy in the momentum space [44,45]. The anisotropy in the momentum space can be regarded as anisotropy in temperature. Suppose there is plasma which is hotter in y-direction than x or z direction one may write the distribution function . If in this situation a disturbance with a magnetic-field B = B 0 cos(kx) which arises say from noise, one can write the Lorentz force term in the kinetic equation as . This Lorentz-force can produce current-sheets where the magnetic field changes its sign. The current-sheet in turn enhance the original magnetic field [44,45].
The Weibel instability is known to grow maximally for θ n = 0. In the quasi-static limit the instability has maximum growth rate Γ w ∼ 8ξ 3/2 27π m D for k = √ ξ 3 m D . For the chiral imbalance instability the maximum growth rates Γ ch ∼ Whereas when µ 5 = 0 only Weibel instability will contribute.
From the condition ρ(k) > 0, one can obtain the range of the instability which can be stated as: Here θ n is the angle between the wave vector k and the anisotropy vector. Real part of dispersion relation is zero. Fig. (3a) show plots for three cases: (i) Pure chiral (no anisotropy), (ii) Pure Weibel (chiral chemical potential=0) and (iii) When both chiral and Weibel instabilities are present. Fig. (3b-3d) represent the case when both the instabilities are present but the anisotropy parameter varies at different values of θ n for fixed µ 5 /T = 1. Fig. (3e-3f) represents the case when both instabilities are present for a fixed anisotropy parameter at different values of θ n when µ 5 /T = 1 and µ 5 /T = 0.1 respectively. Fig. (3g) represents the case when for a particular value of θ n ∼ θ c both the instabilities have equal growth rates. Here frequency is normalized in unit of ω/ and k. Fig.(3b) shows clearly shows for θ n = 0 when condition ξ << ξ c is satisfied, the chiral instability dominates over the Weibel modes. However, such values of ξ are extremely small.
For the cases when ξ ≥ ξ c the Weibel modes are dominating.
Contribution from the Weibel modes is maximum for θ n = 0 and the modes are strongly damped at θ n = π/2. Angular part in the dispersion relation for the pure Weibel modes becomes zero when θ n ≈ 55 0 . In this case one can see that chiral modes can remain dominant. This case is shown in Fig.(3c). It should be noted that for the case when ξ >> ξ c the contribution from the coupling term between the Weibel and chiral modes become sufficiently strong and it can again suppress the instability. In Fig.(3d) we have shown the case when θ n = π/2. The modes with ξ ≥ ξ c are strongly damped and there is no instability.
Here the coupling term between the two modes also contribute in the damping of the instability. In Fig.(3e-3f) we have plot-  Fig.(3g) represents this case where we have shown that the growth rate of pure Weibel case at ξ = 0.15ξ c becomes comparable to pure chiral mode with ξ = 0. The topmost (red) curve in this figure shows the case when both the modes operate together. This case shows that the combined effect of the instability can significantly alter the range and the growth rate of the instability.
In conclusion, we have studied collective modes in an anisotropic chiral plasma where the both Weibel and chiralimbalance instabilities are present. We have demonstrated that for θ n = 0, only for a very small values of the anisotropic parameter ξ ∼ ξ c 1 growth rates of the both instabilities are comparable. For the cases when ξ ≥ 1, ξ < 1 but closer to unity and ξ c < ξ 1, the Weibel modes dominate over the chiral-imbalance instability. We have also shown for the case when ξ 1, the chiral-imbalance can dominate over the Weibel modes for certain values of θ n .