The impact of domain walls on the chiral magnetic effect in hot QCD matter

The Chiral Magnetic Effect (CME) -- the separation of positive and negative electric charges along the direction of the external magnetic field in quark-gluon plasma and other topologically non-trivial media -- is a consequence of the coupling of electrodynamics to the topological gluon field fluctuations that form metastable $CP$-odd domains. In phenomenological models it is usually assumed that the domains are uniform and the influence of the domain walls on the electric current flow is not essential. This paper challenges the latter assumption. A simple model consisting of a uniform spherical domain in a uniform time-dependent magnetic field is introduced and analytically solved. It is shown that (i) no electric current flows into or out of the domain, (ii) the charge separation current, viz. the total electric current flowing inside the domain in the external field direction, is a dissipative Ohm current, (iii) the CME effect can be produced either by the anomalous current or by the boundary conditions on the domain wall and (iv) the charge separation current oscillates in plasma long after the external field decays. These properties are qualitatively different from the CME in an infinite medium.

the boundary conditions on the domain wall and (iv) the charge separation current oscillates in plasma long after the external field decays. These properties are qualitatively different from the CME in an infinite medium.
Quantitative analysis of the charge separation requires knowledge of the medium response to the external electromagnetic field. The simplest model is to add a new anomalous current j A = σ χ B to the Amper law, where the chiral conductivity σ χ is assumed to be weakly dependent on position and time [3,5,20]. The time dependence of the chiral conductivity arises primarily due to the sphaleron transitions, finite quark mass and the helicity exchange between the magnetic field and QGP. All these effects have very long characteristic time scales compared to the QGP lifetime [21-arXiv:1802.09629v1 [hep-ph] 26 Feb 2018 25], which justifies treating σ χ as time-independent. * The assumption of the spatial uniformity is less sound however. The topological CP -odd fluctuations of the hot nuclear matter occupy a region of a typical size ∼ 1/g 2 T which is of the order of a fm. This implies that a typical heavyion collision can produce a large number of topologically different metastable CP -odd domains.
Electric current varies steeply between the domain interior and the surrounding plasma. Thus, the charge separation effect is expected to be strongly dependent on the domain size and topology.
The main goal of this paper is to compute the charge separation current taking into account these finite size effects.
In order to study the charge separation effect in a finite size domain, it is advantageous to consider an exactly solvable model. The model considered in this paper consists of a spatially uniform spherical domain of radius R immersed into a topologically trivial environment. The electrodynamics with the chiral anomaly is described by the Maxwell-Chern-Simons theory (MCS) in which the anomalous terms are associated with the background pseudoscalar field Θ whose dynamical extension is the axion [5,[28][29][30]. The role of the chiral anomaly is twofold: it induces a new anomalous current into the Amper law and causes a discontinuity of the normal electric and tangential magnetic field components at the domain wall even in the absence of the surface currents.

II. FIELD EQUATIONS AND BOUNDARY CONDITIONS
The field equations of electrodynamics coupled to the topological charge carried by the gluon field read [5,[28][29][30] where c A = N c f q 2 f e 2 /2π 2 is the chiral anomaly coefficient. The plasma is assumed to be electrically neutral. The Ohm current is j = σE where σ is the electrical conductivity. The background field Θ is regarded as spatially uniform everywhere except the domain wall where ∇Θ is discontinuous.
As explained in Introduction, the time-variation of Θ is too slow to be important for the heavy-ion phenomenology. Nevertheless, since the chiral conductivity is proportional to the timederivative of Θ one needs to keep track of its small variations. Hence Θ is approximated by [25] Θ ≈ Θ 0 + µ 5 t , where µ 5 is the axial chemical potential related to the chiral conductivity σ χ as µ 5 = σ χ /c A [3,5].
Estimating the chiral conductivity optimistically as σ χ = 10 −2 fm −1 and using c A = 1/129 one obtains µ 5 = 1.3 fm −1 . Thus, the time-dependent term in (2) is smaller than 2π for t < 3 fm. From now on it is assumed that this condition is satisfied.
With the assumptions outlined in the preceding paragraphs one can simplify equations (1a)-(1d), which read at any point in space except the domain wall The assumption of the uniformity of the domain interior means that its wall width is neglected.
The boundary conditions on the domain wall can be obtained directly from equations (1a)-(1d).
Denoting by ∆ the discontinuity of a field component across the domain wall and neglecting the time-dependent term in (2) one obtains [30] ∆B ⊥ = 0 , (4a) where E ⊥ , B ⊥ and E , B are components of the electromagnetic field normal and tangential to the domain wall respectively.
A more stringent boundary condition can be derived using the continuity equation ∇ · j = 0, which implies that ∆j ⊥ = 0 [31]. Projecting (1d) onto the normal direction and using (1c) one The third term on the left-hand side vanishes because ∇Θ points in the normal direction. The terms on the right-hand side are continuous in view of (4b). Now, solutions of (3d) is a complete set of eigenstates of the curl operator satisfying the equation ∇ × B = αB, where α depends on medium properties. Consider such an eigenstate of frequency ω. Then (5) implies that B ⊥ (α + iωc A Θ) is continuous across the wall. However, B ⊥ is also continuous, whereas α and Θ are discontinuous.
These conditions can only be satisfied if B ⊥ vanishes on the wall:

A. General solution inside and outside domain
The external homogeneous magnetic field of frequency ω induces electromagnetic field in the domain which is governed by equations (3) and boundary conditions (4), (6). Since electric and magnetic fields are divergentless, it is convenient to use the radiation gauge ∇ · A = 0, A 0 = 0 which allows one to write (3d) as an equation for the vector potential Separation of the temporal dependence of the vector-potential A(x, t) = A ω (x)e −iωt yields for its monochromatic component The general solution of (8) can be written as a superposition of the eigenfunctions of the curl operator. These functions are denoted by W ± lm (x, α) and satisfy the equation Their explicit form in the spherical coordinates reads [31] where f l is a linear combination of the spherical Bessel functions j l and n l . The z-axis is chosen in the direction of the external magnetic field which is given by The corresponding vector potential is The symmetry considerations imply that in a spherical domain the only nontrivial component of the induced field is proportional to the linear combination of the functions The general solution to (8) inside the domain reads where q ± are the roots of the equations −q 2 The boundary conditions at the origin require that f l (q ± r) = j l (q ± r). In view of (9), the magnetic field inside the domain is The general solution to (7) outside the domain, where Θ = 0, reads where k = ω(ω + iσ) and f l (kr) = cos δ l j 1 (kr) − sin δ l n l (kr). The magnetic field outside the domain is Note that (19) and (20) do not include the external field.

B. Matching the solutions on the domain wall
The boundary conditions (4),(6) on the spherical domain wall of radius R read, after replacing Since the external magnetic field can be written as B ext ω = − √ 6πB 0 P 10 (x, 0), the only non-trivial solution to (21) is for the partial amplitudes with l = 1 and m = 0. It easy to verify, using (16), (18), (19), (20) that the boundary conditions (21a) and (21d) are identical. Also, vanishing of † The other two roots give linearly dependent solutions. They can be obtained by replacing q± → −q∓ which corresponds to T lm → (−1) l T lm , P lm → (−1) l+1 P lm .
B in ωr on the wall, i.e. (21a), implies vanishing of A ωφ on the wall, which in turn indicates that (21b) and (21f) are identical. Thus, there are five equations to determine five unknown amplitudes g 10 , h 10 , c 10 , d 10 and δ 1 . It is understood that Θ = 0 inside the domain for otherwise some of the equations (21) become redundant.
To write the solution of the boundary conditions (21) in a compact form denote ∂ r [j 1 (αr)r]| r=R ≡ [j 1 (αR)R] and define three auxiliary functions After tedious but straightforward algebraic manipulations one obtains Eq. (24) follows directly from the boundary condition (6), or equivalently, (21a). Other amplitudes can be expressed in terms of g 10 . Define two more auxiliary functions The amplitudes of the positive helicity component of the magnetic field outside the domain, see (20), are The ratio of these equations immediately yields tan δ 1 . The remaining amplitudes, corresponding to the negative helicity component of the magnetic field outside the domain, read −d 10 sin δ 1 = iRB 0 2π/3 − a 2[− cot δ 1 j 1 (Rk) + n 1 (Rk)] .
Substitution of equations (22a)-(26d) into (18) and (20) furnishes the analytic expressions for the electromagnetic field of the spherical domain in the monochromatic uniform magnetic field.

C. Electric current and magnetic moment
Using the results of the previous section one can compute the total current flowing in the direction of the external magnetic field through any cross sectional area of the domain: The magnetic field flux can be written as where ρ is the radial coordinate in the cross-sectional plane and in the second line the integration variable has been changed to r = ρ 2 + z 2 . Using (18) and (15) one derives Integrating the second integral by parts and using the boundary condition (24) yields Thus, the anomalous component of the current does not contribute to the charge separation current.
The computation of the electric flux can be done along the same lines by noting that E ωz = iωA ωz and using (16) in place of (18). The result is This constitutes the charge separation effect. The current I ω does not identically vanish as long as Θ = 0, i.e. either Θ 0 or σ χ is finite.
The magnetic moment of the domain is given by and can be computed using the same steps as were employed in the calculation of the current. The result is µ ω =iẑ 2π 3 It vanishes if σ χ → 0, i.e. existence of the domain magnetic moment requires the anomalous current.
Eventually, however, the helicity conservation puts a cap on the inverse cascade [59,60]. As explained in Sec. II, this interesting effect is not really phenomenologically relevant for heavy-ion collisions. In fact, (2) explicitly neglects any significant long-time evolution effects.

V. SUMMARY AND DISCUSSION
Metastable CP -odd topological domains emerge in the hot QCD matter. The external magnetic field applied to these domains generates an anomalous current and charge densities. This paper focused on one such domain. To simplify the calculations, the domain was assumed to be a uniform sphere, while the surrounding medium to be spatially uniform and topologically trivial.
The electromagnetic field in entire space was analytically calculated by employing a standard technique. The electric and magnetic components of the field induce Ohm and anomalous currents respectively. Their main properties are as follows.
1) The normal component of the electric current vanishes on the domain wall regardless of the domain geometry and uniformity. Thus no electric current flows into or out of the domain.
2) The charge separation current, i.e. the total electric current flowing in the direction of the external magnetic field through any cross sectional area of the domain is the Ohm current, as shown in Sec. III C. The contribution of the total anomalous current is zero. In particular, the total current vanishes in an electric insulator σ → 0. This may appear counterintuitive because a CP -odd effect cannot be generated by the CP -even current. There is no contradiction though, as the the total current vanishes when Θ → 0. Even so, it is interesting to note that the current is finite if either Θ 0 or σ χ is finite. This is especially important if σ χ turns out to be much smaller than a few MeV as assumed in most applications; in that case the CME is generated by the domain walls.
3) The total current is finite long after the external field decayed, owing to the low electrical conductivity of QGP, which implies small dissipation. The current oscillates with roughly the characteristic time t 0 of the external field. However, since no charge leaves the domain, the final charge separation within the domain depends on the current magnitude and direction at the time of the freeze-out.
4) The resonance frequencies of a spherical domain are ω n = x n /R, where x n are zeros of the spherical Bessel function j 1 (x). The current frequency modes with ω ω 1 do not contribute to the total current as the corresponding wavelength does not fit in the domain. In the static limit I ω → 0 as ω → 0. ‡ Finally, the author believes that the present model, despite its simplicity, gives a reasonably accurate idea about a possible effect of the domain size on the charge separation effect. It has been seen throughout the paper that the properties enumerated above a fairly geometry independent.
The gradients ∇Θ also seem to be a minor effect [62]. It thus appears that giving up the spherical symmetry and spatial uniformity would not have a large impact on the above conclusions.

ACKNOWLEDGMENTS
This work was was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-87ER40371.
[1] D. Kharzeev, "Parity violation in hot QCD: Why it can happen, and how to look for it," Phys. Lett.
B 633, 260 (2006) ‡ Actually, the MCS equations do have non-trivial solutions even in the absence of the external field. These are given by the CK states (10)- (12) with α = σχ. However, their radii Rn = xn/σχ are way too big to fit into the QGP, as was first pointed out in [61].