TTaming instability of magnetic field in chiral medium
Kirill Tuchin Department of Physics and Astronomy, Iowa State University, Ames, Iowa, 50011, USA (Dated: October 8, 2018)Magnetic field is unstable in a medium with time-independent chiral conductivity. Owingto the chiral anomaly, the electromagnetic field and the medium exchange helicity whichresults in time-evolution of the chiral conductivity. Using the fastest growing momentum andhelicity state of the vector potential as an ansatz, the time-evolution of the chiral conductivityand magnetic field is solved analytically. The solution for the hot and cold equations of stateshows that the magnetic field does not develop an instability due to helicity conservation.Moreover, as a function of time, it develops a peak only if a significant part of the initial helicity is stored in the medium. The initial helicity determines the height and position ofthe peak.
I. INTRODUCTION: INSTABILITY OF MAGNETIC FIELD
A medium with chiral anomaly responds to magnetic field by generation of anomalous electriccurrent flowing in the magnetic field direction. The strength of this response is determined bythe chiral conductivity σ χ . The induced current in turn produces the magnetic field in medium.This process obeys the Maxwell equations that yield an equation for the magnetic field in a chiralmedium −∇ B = − ∂ t B + σ χ ∇ × B . (1)Assuming that σ χ is time-independent constant, a solution to (1) can be written as a superpositionof the Chandrasekhar-Kendall states [1, 2]. In particular, for a circularly polarized plane wave withmomentum k B k λ ∼ e i k · r exp (cid:26) − itλ (cid:113) k ( k + λσ χ ) (cid:27) , (2)where λ , = ± λ = 1 and λ = − k < σ χ indicating that the magnetic field is unstable in the chiral medium. This effect and itsconsequences have been discussed in various contexts [2–17].The chiral conductivity is actually a function of time since energy and helicity are exchangedbetween the chiral medium and the magnetic field due to the chiral anomaly. It has been recently a r X i v : . [ nu c l - t h ] O c t argued in [16] that the instability of the magnetic field is curtailed by the total energy and helicityconservation. The goal of this paper is to investigate this mechanism in more detail. To thisend, the vector potential is expanded into a complete set of the helicity states. The circularlypolarized plane waves are used in this paper, but the same result can be obtained with any otherChandrasekhar-Kendall states as well. The vector potential is then approximated by a single fastestgrowing (as a function of time) state. Momentum k of the fastest growing state is proportional tothe chiral conductivity and hence is a function of time. This is different from the monochromaticapproximation discussed in [4, 6], where the momentum of a monochromatic state is constant. Inthe present model, the rate of the state growth is determined by the anomaly equation.In order to solve the Maxwell and the chiral anomaly equations one needs to know the equationof state of the medium that relates the chiral charge density to the axial chemical potential. Weconsider two equations of state (38) and (55) corresponding to hot and cold media. In each casean analytical solution is derived for the chiral conductivity and magnetic field. The details of thisderivation are discussed in the subsequent sections. The results are shown in Fig. 1–Fig. 4. Onecan see that the vector potential increases rapidly with time. However the magnetic field developsa maximum only if at the initial time a large enough fraction of the total helicity is stored in themedium rather than in the field. Eventually, the magnetic field vanishes at later times. II. MAXWELL EQUATIONS
Electrodynamics coupled to the topological charge carried by the gluon field is governed by theMaxwell-Chern-Simons equations [18–21] ∇ · B = 0 , (3) ∇ · E = ρ − c A ∇ θ · B , (4) ∇ × E = − ∂ t B , (5) ∇ × B = ∂ t E + j + c A ( ∂ t θ B + ∇ θ × E ) , (6)where θ is a pseudo-scalar field and c A = N c (cid:80) f q f e / π , N c is the number of colors, f is flavorindex and q f is a quark electric charge in units of e . In a chiral medium, the time-derivative of θ can be identified with the axial chemical potential ˙ θ = µ . We are going to consider an idealizedcase of a homogeneous medium ∇ θ = 0 with vanishing charge density ρ = 0. In this case theanomalous current is given by j A = σ χ B , (7)where the chiral conductivity defined as [21, 22] σ χ = c A µ (8)is a function of only time. In the radiation gauge ∇ · A = 0, A = 0 Eq. (6) can be written as anequation for the vector potential −∇ A = − ∂ t A + j + σ χ ( t ) ∇ × A . (9) III. THE FASTEST GROWING STATE
We proceed by expanding the vector potential into eigenstates of the curl operator W k λ ( x ),known also as the Chandrasekhar-Kendall (CK) states [1]. A particular form of these functions isnot important, but its easier to deal with them in Cartesian coordinates where they are representedby the circularly polarized plane waves W k λ ( x ) = (cid:15) λ √ kV e i k · x , (10)where V is volume and λ = ± ∇ × W k λ ( x ) = λk W k λ ( x ) (11)and the normalization condition (cid:90) W ∗ k λ ( x ) · W k (cid:48) λ (cid:48) ( x ) d x = 12 k δ λλ (cid:48) δ k , k (cid:48) . (12)Expansion of the vector potential into a complete set of the CK states reads A = (cid:88) k ,λ [ a k λ ( t ) W k (cid:48) λ (cid:48) ( x ) + a ∗ k (cid:48) λ (cid:48) ( t ) W ∗ k (cid:48) λ (cid:48) ( x )] . (13)The corresponding electric and magnetic fields are given by E = − ∂ t A = (cid:88) k ,λ [ − ˙ a k λ ( t ) W k (cid:48) λ (cid:48) ( x ) − ˙ a ∗ k λ ( t ) W ∗ k λ ( x )] , (14) B = ∇ × A = (cid:88) k ,λ [ λk a k λ ( t ) W k λ ( x ) + λka ∗ k λ ( t ) W ∗ k λ ( x )] . (15)Substituting (13) into (9) and using the Ohm’s law j = σ E one gets an equation k a k λ = − ¨ a k λ − σ ˙ a k λ + σ χ ( t ) λk a k λ . (16)It can be solved in the adiabatic approximation, which is adequate for analysis of the unstablestates ∗ . Namely, we are seeking a solution in the form a k λ = e − i (cid:82) t ω kλ ( t (cid:48) ) dt (cid:48) (17)and assume that ω kλ ( t ) is a slow varying function, which allows one to neglect terms proportionalto ˙ ω kλ . This yields ω kλ ( t ) = (cid:20) − iσ λ (cid:113) k − σ χ λk − σ / (cid:21) , (18)with λ = ±
1. States with λ = 1 and k such that the expression under the square root isnegative are unstable. A more detailed analyzes can be found in [3]. We are going to concentrateon the fastest growing state, whose momentum k corresponds to the maximum of the function − ( k − σ χ λk − σ / k = σ χ λ . (19)Clearly, σ χ λ is positive in an unstable state. We assume that σ χ > λ = 1. † At k = k Eq. (18) becomes ω ( t ) = − iσ i (cid:113) σ + σ χ ( t ) . (20)Thus, the fastest growing state is a ( t ) = e γ ( t ) / , (21)with γ ( t ) = (cid:90) t (cid:104)(cid:113) σ + σ χ ( t (cid:48) ) − σ (cid:105) dt (cid:48) . (22)The model employed in the ensuing sections of this paper, consists in approximating the vectorpotential by the fastest growing mode given by (21),(22). The corresponding vector potential is A ( r , t ) ≈ a ( t ) W k + ( r ) + c.c. (23)To verify that the ansatz Eq. (23) is indeed a solution to Eq. (9) one has to keep in mind thatwhen taking the time-derivative of A , function W k + is treated as time-independent (even though k depends on time), because its time-derivative is proportional to ˙ ω , which is neglected in theadiabatic approximation. ∗ The self-consistency of this approximation is confirmed in Appendix, † If during the evolution σ χ changes sign, then λ = 1 state stops growing while λ = − IV. ENERGY AND HELICITY
Now let us verify whether the energy conservation restricts the functional form of σ χ ( t ). Energystored in the electromagnetic field is E em = 12 (cid:90) ( E + B ) d x = (cid:88) k ,λ k (cid:0) | ˙ a k λ | + | a k λ | k (cid:1) , (24)where (14),(15),(12) were used. Similarly, magnetic helicity is given by ‡ H em = (cid:90) A · B d x = (cid:88) k ,λ λ | a k λ | , (25)while the energy loss due to Ohm’s currents reads Q = (cid:90) j · E d x = σ (cid:90) E d x = σ (cid:88) k ,λ k | ˙ a k λ | . (26)The energy balance equation follows from Maxwell equations. Subtracting the scalar productof (5) with B from the scalar product of (6) with E and integrating over volume yields12 ∂ t (cid:90) ( E + B ) d x + (cid:90) j · E d x + σ χ ( t ) (cid:90) B · E d x = 0 , (27)where we neglected the surface contribution. Noting that up to a surface term ∂ t H em = − (cid:90) E · B d x , (28)we can write (27) as ∂ t E em + Q − σ χ ( t ) ∂ t H em = 0 . (29)In the framework of our model, i.e. using the fastest growing state (23) as an ansatz for thevector potential, one obtains E em = 12 σ χ (cid:16) σ χ + σ − σ (cid:113) σ + σ χ (cid:17) e γ , (30) H em = e γ , (31) Q = σ σ χ (cid:16)(cid:113) σ + σ χ − σ (cid:17) e γ . (32)Noting that in the adiabatic approximation ˙ E em ≈ ˙ γ E em it is straightforward to verify thatEqs. (30)–(32) satisfy the energy balance equation (29). ‡ The gauge-dependence of magnetic helicity does not affect its time-evolution, see e.g. [23].
We emphasize that Eq. (29) is satisfied for any function σ χ ( t ). Thus, energy conservation doesnot tame the instability of the magnetic field. In particular, for a constant chiral conductivity,Eqs. (30),(31),(32) diverge exponentially with time. This does not contradict the conclusions ofRef. [16], because it explicitly used the chiral anomaly equation. V. CHIRAL ANOMALY
Time-dependence of σ χ is determined by the chiral anomaly equation and the equation of statethat connects the average chiral charge density (cid:104) n A (cid:105) to the axial chemical potential µ . The chiralanomaly equation reads [24, 25] ∂ µ j µA = c A E · B . (33)In a homogeneous medium ∇ · j A = σ χ ∇ · B = 0, so that (33) reduces to an equation for the timecomponent of the anomalous current ˙ n A = c A E · B . (34)Averaging over volume and using (28) gives § ∂ t (cid:104) n A (cid:105) = c A V (cid:90) E · B d x = − c A V ∂ t H em . (35)Integrating, one obtains the total helicity conservation condition2 Vc A (cid:104) n A (cid:105) + H em = H tot , (36)where H tot is a constant.Magnetic helicity H em explicitly depends on σ χ ( t ), rather than on (cid:104) n A (cid:105) . Therefore, in orderto solve (36) one needs an equation of state of the chiral medium that connects the chiral chargedensity and the axial potential. It can be computed from the grand canonical potential Ω as [22] (cid:104) n A (cid:105) = − V ∂ Ω ∂µ . (37)Clearly, the equation of state depends on medium properties and, in general, is complicated evenin the non-interacting approximation. It simplifies in two important limits: when temperatures T and quark chemical potentials µ are much higher or much lower than the axial chemical potential µ . We refer to these limits as the hot and cold medium respectively and consider separately inthe following two sections. § Our notations generally follow [2].
VI. MAGNETIC FIELD EVOLUTION IN HOT MEDIUM
In a hot medium with µ, T (cid:29) µ , the equation of state is linear [22] (cid:104) n A (cid:105) = χ µ , (38)where the susceptibility χ depends on µ and T , but not on time. It follows then from (8) that σ χ ( t ) = c A χ (cid:104) n A ( t ) (cid:105) , (39)The helicity conservation (36) now reads σ χ ( t ) α = 1 − H em ( t ) H tot , (40)where α = H tot c A / (2 V χ ) is a characteristic energy scale.The vector potential in (23) is normalized such that at the initial time the magnetic helicityequals unity H em (0) = 1. Denoting the initial value of the chiral conductivity by σ χ (0) = σ onecan infer from (40) that σ = α (1 − H − ) < α . (41)The ratio σ (cid:48) = σ /α determines the fraction of the total initial helicity stored in the medium. Ifat t = 0 H tot (cid:29)
1, then most of the initial helicity is stored in the medium, implying σ (cid:48) (cid:46) σ (cid:48) never equals 1 for a finite positive total helicity. ¶ In the opposite case, all helicity isinitially magnetic H tot = H em (0) = 1 implying σ (cid:48) = 0.Substituting (31) into (40), using the definition of γ from (22) and taking the time-derivative,one derives an equation for σ χ ˙ σ χ = − (cid:16)(cid:113) σ + σ χ − σ (cid:17) ( α − σ χ ) . (42)It is convenient to use a set of dimensionless quantities σ (cid:48) χ = σ χ /α , σ (cid:48) = σ/α , τ = αt , (43)in terms of which Eq. (42) is cast into the form˙ σ (cid:48) χ = − (cid:16)(cid:113) σ (cid:48) + σ (cid:48) χ − σ (cid:48) (cid:17) (1 − σ (cid:48) χ ) . (44) ¶ In the case of negative helicity, all terms in (36) would change sign, see footnote † . In view of (41), the right-hand-side of (44) is always negative. Perforce, σ χ is a monotonicallydecreasing function of time implying that helicity always flows from the medium to the field untilall of it is stored in the field. This is in contrast to [2, 4] where the helicity can flow in bothdirections. How long it takes to transfer the helicity to the field depends on an equation of stateas discussed in this and the following sections.Eq. (44) can be analytically integrated and yields a transcendental equation for σ (cid:48) χ ( τ ). It is notvery illuminating though, so instead we directly plot its solution in left panel of Fig. 1. α t0.10.20.30.40.5 σ χ / α α t510501005001000A / A FIG. 1: Time dependence of the the chiral conductivity (left panel) and the vector potential (right panel)in hot medium with the initial condition σ (cid:48) χ (0) ≡ σ (cid:48) = 0 . σ (cid:48) = 0 (solid line), 1 (dotted line), 10 (dashedline). Lines on the right panel represent (52) for σ (cid:48) = 0 (solid line), 0 .
01, 0 .
1, 1 (dotted lines left-to-right),10 (dashed line).
Once the chiral conductivity is determined from (44), the vector potential can be computedusing (23) and (10). Its time-dependence is given by A ∝ (cid:112) σ χ ( t ) a ( t ) . (45)Since γ is an increasing function of time, a and A are also increasing functions of time, which isimportant for the model self-consistency. Time-dependence of A is exhibited in the right panel ofFig. 1. The magnetic helicity H em = a grows from its initial value H em (0) = 1 to the final valueof H em ( ∞ ) = H tot .Unlike the vector potential, the magnetic field is not necessarily a monotonic function. Its time-dependence follows from (15) and is proportional to a product of an increasing and decreasingfunctions B = k A ∝ (cid:113) σ χ ( t ) a ( t ) . (46)The time t ∗ at which B ( t ) attains its maximum satisfies ˙ B ( t ∗ ) = 0 implying (cid:20) ˙ γ + ˙ σ (cid:48) χ σ (cid:48) χ (cid:21) τ = τ ∗ = 0 . (47)Using (22) and (44) one derives σ (cid:48) χ ( τ ∗ ) = 12 . (48)Since σ (cid:48) χ ( τ ) is monotonically decreasing from its initial value σ (cid:48) , the magnetic field has maximumonly if σ (cid:48) > /
2, i.e. if most of the initial helicity is in the medium. Otherwise, B ( t ) is a mono-tonically decreasing function of time (despite the fact that A always grows). This is shown inFig. 2. α t0.20.40.60.81.0B / B α t0.51.01.52.0B / B FIG. 2: Evolution of the magnetic field in hot medium for σ (cid:48) = 0 (solid line), 1 (dotted line), 10 (dashedline). The initial condition is σ (cid:48) χ (0) ≡ σ (cid:48) = 0 .
5, i.e. 50% of the initial helicity is in the field (left panel) and σ (cid:48) = 0 .
95, i.e. 95% of the initial helicity is in medium, (right panel).
The self-consistency of the adiabatic approximation employed in this section is verified in Ap-pendix.
A. Insulating medium σ = 0 The chiral conductivity can be explicitly expressed as a function of τ in the case of vanishingelectrical conductivity σ = 0. In this case, solution to Eq. (44) reads σ (cid:48) χ ( τ ) = 11 + (cid:16) σ (cid:48) − (cid:17) e τ . (49)Clearly, at τ (cid:28)
1, the chiral conductivity is constant, while at t > γ ( t ) = (cid:90) t σ χ ( t (cid:48) ) dt (cid:48) = τ − ln (cid:2) e τ (cid:0) − σ (cid:48) (cid:1) + σ (cid:48) (cid:3) . (50)0Plugging this into (21) one derives for an insulating medium a ( t ) = e τ/ (cid:112) e τ (1 − σ (cid:48) ) + σ (cid:48) . (51)Since ˙ a > σ (cid:48) >
0, magnetic helicity H em = a increases from its initial value H em (0) = 1to the final value of H em ( ∞ ) = (1 − σ (cid:48) ) − = H tot , where (41) has been used.In view of (45), time-dependence of the vector potential is given by A ∝ e τ/ . (52)It is an exponentially growing function of time as one would expect for an unstable state. See thesolid line in right panel of Fig. 1. Time-dependence of the corresponding magnetic field followsfrom see (15) B ∝ e τ/ e τ (1 − σ (cid:48) ) + σ (cid:48) . (53)At later times magnetic field exponentially decays even though vector potential grows exponentially.At σ (cid:48) ≤ / σ (cid:48) > /
2, i.e. when most of the initial helicityis in the medium, in agreement with the general conclusion derived after (48). In this case themaximum of magnetic field B ( t ∗ ) ∝ (cid:112) σ (cid:48) (1 − σ (cid:48) ) (54)occurs at τ ∗ = ln[ σ (cid:48) / (1 − σ (cid:48) )]. This is seen in the right panel of Fig. 2.As have been already mentioned, for a finite total helicity σ (cid:48) <
1, therefore the right-hand-sideof (44) does not vanish and σ χ is never a constant. Thus the divergence of the type shown in (2)never arises. It is always tamed by the helicity conservation. VII. MAGNETIC FIELD EVOLUTION IN COLD MEDIUM
In the cold medium, namely
T, µ (cid:28) µ , the equation of state is [22] µ = 3 π (cid:104) n A (cid:105) . (55)Together with the definition (8), this implies that (cid:104) n A (cid:105) = σ χ π c A . (56)1Using this in (36) yields σ χ ( t ) β = 1 − H em ( t ) H tot , (57)where β = 3 π c A H tot / (2 V ) is a characteristic momentum scale. This time, in place of (41) onegets σ = β (1 − H − ) / < β . (58)Once again using (31) in (57) one derives an equation obeyed by the chiral conductivity3 σ χ ˙ σ χ = − (cid:16)(cid:113) σ + σ χ − σ (cid:17) ( β − σ χ ) . (59)It is convenient to define dimensionless quantities σ (cid:48) = σ χ /β , σ (cid:48) = σ/β and τ = βt and write (59)as 3 σ (cid:48) χ ˙ σ (cid:48) χ = − (cid:16)(cid:113) σ (cid:48) + σ (cid:48) χ − σ (cid:48) (cid:17) (1 − σ (cid:48) χ ) , (60)where the dot means the τ -derivative. As in the previous section, solution to this equation is bulkyand is not recorded here. Rather it is shown in Fig. 3. As in the hot medium, σ χ is a decreasingfunction of time, which is also clear from (60) since its right-hand-side is negative. However, incontrast to Fig. 1, it vanishes not asymptotically at τ → ∞ , but at a finial time τ m . β t0.10.20.30.40.5 σ χ / β FIG. 3: Chiral conductivity of cold medium as a function of time for σ (cid:48) = 0 .
5. Lines represent solution of(60) at σ (cid:48) = 0 (solid line), 1 (dotted line), 10 (dashed line). Time dependence of the vector potential and magnetic field can be computed employing (45),(46). They imply that the vector potential diverges whereas the magnetic field vanishes at τ = τ m .The results for the magnetic field are shown in Fig. 4. In order that B ( t ) has a maximum, condition(47) must be satisfied. In the cold medium it yields σ (cid:48) χ ( t ∗ ) = 12 / . (61)2Thus, if σ (cid:48) > / / , magnetic field has a maximum, whereas if σ (cid:48) < / / , it monotonicallydecreases. Thus, in order for the magnetic field to grow in the cold medium, more than 62% of theinitial helicity must be stored in medium. β t0.20.40.60.81.0B / B β t0.51.01.5B / B FIG. 4: Evolution of the magnetic field in cold medium for σ (cid:48) = 0 (solid line), 1 (dotted line), 10 (dashedline). The initial condition is σ (cid:48) χ (0) ≡ σ (cid:48) = 0 .
5, i.e. 50% of the initial helicity is in the field (left panel) and σ (cid:48) = 0 .
95, i.e. 95% of the initial helicity is in medium, (right panel).
A. Insulating medium σ = 0 In the insulating medium, evolution of the chiral conductivity is determined by the equation3 σ (cid:48) χ ˙ σ (cid:48) χ = − (1 − σ (cid:48) χ ) . (62)Its solution in a from of a transcendental equation for σ (cid:48) χ reads τ = 12 ln (1 − σ (cid:48) χ ) σ (cid:48) χ + σ (cid:48) χ + √ σ (cid:48) χ + 1 √ −
12 ln (1 − σ (cid:48) ) σ (cid:48) + σ (cid:48) − √ σ (cid:48) + 1 √ . (63)Explicit dependence of the chiral conductivity σ (cid:48) χ on time is shown in Fig. 3 by the solid line. Itvanishes at τ m that follows directly from (63) τ m = π √ −
12 ln (1 − σ (cid:48) ) σ (cid:48) + σ (cid:48) − √ σ (cid:48) + 1 √ . (64)At later times τ > τ m , the chiral conductivity is identically zero implying that the chiral anomalyceases to play any role in the magnetic field evolution. This is in contrast to the hot medium,where σ (cid:48) χ vanishes only asymptotically at τ → ∞ . B. Conducting medium σ (cid:29) σ χ In this limit (60) reduces to ˙ σ (cid:48) χ = − σ (cid:48) (1 − σ (cid:48) χ ) , (65)3and is solved by τσ (cid:48) = ln (1 − σ (cid:48) χ ) σ (cid:48) χ + σ (cid:48) χ − √ √ σ (cid:48) χ σ (cid:48) χ − ln (1 − σ (cid:48) ) σ (cid:48) + σ (cid:48) + 6 √ √ σ (cid:48) σ (cid:48) . (66)The evolution of the chiral conductivity proceeds from τ = 0 up to τ = τ m where σ (cid:48) χ vanishes.Thus τ m /σ (cid:48) is given by the last two terms in (66). Notice that τ m is proportional to the mediumconductivity. This is seen in Fig. 3. VIII. DISCUSSION AND SUMMARY
Development of magnetic field instability in a chiral medium has been the main subject of thispaper. For a qualitative analytical understanding of the instability dynamics, we developed a modelthat approximates the vector potential by the fastest growing helicity state. Maxwell equations withanomalous currents determine dependence of the magnetic field on the chiral conductivity. Theyhowever, do not determine the functional form of σ χ ( t ) and neither does the energy conservationrequirement, as was verified in Sec. IV. It is the chiral anomaly equation (33) and the resultinghelicity conservation (36), in conjunction with the equation of state that fix the time-dependenceof the chiral conductivity.Two equations of state were considered (38) and (55), which are referred to as the hot andcold matter respectively. Whether the magnetic field increases at the early stages of its evolutiondepends on a single parameter, the fraction of the total helicity initially in the medium σ (cid:48) , whichalso happens to be the initial value of the chiral conductivity (in dimensionless units). In hotmedium it must be more than 1/2 while in the cold medium more than 1 / / . This parameteralso determines the instant t ∗ when the maximum occurs and the peak strength of the magnetic field B ( t ∗ ). If σ (cid:48) is below the values indicated above, the magnetic field is a monotonically decreasingfunction of time. This is seen in Fig. 2 and Fig. 4. Magnetic helicity always monotonically increasesfrom unity at the initial time to H tot at the later times. This is because in the present model helicityflows only from the medium to the magnetic field. The eventual vanishing of the chiral conductivityat late times and the transfer of all helicity to magnetic field is a general property of the MCS theoryin homogeneous medium, even though the details of the time-evolution are model-dependent. Thisis because in a stationary state of the chiral medium with a finite value of σ χ = σ ∞ , the magneticfield contains unstable modes with k < σ ∞ , which increase the magnetic helicity, thereby driving σ χ to smaller values due to the helicity conservation (35). The stability is achieved only when4 σ ∞ = 0. ∗∗ Duration of the chiral evolution strongly depends on the equation of state. In hot matter,which is appropriate for the description of the quark-gluon plasma produced in relativistic heavy-ion collisions, the magnetic helicity approaches the total helicity only asymptotically at t → ∞ .However, in cold matter, the chiral conductivity vanishes at finite time τ m that depends on σ (cid:48) andthe electrical conductivity. At this time, the magnetic field vanishes, while the vector potentialdiverges (this is why the magnetic helicity is finite).Our results for the hot matter are qualitatively similar to the numerical results of [2] (stagesII and III). One however, has to bear in mind that [2] neglected the second time derivative of A in (9), which is important at not very large electrical conductivity. Another numerical calculationof the time-evolution of magnetic helicity in chiral medium was reported in [4]. There a differentapproximation was adopted which precludes the direct comparison with the results of the presentwork. Acknowledgments
The author wishes to thank Dmitri Kharzeev for helpful communications. This work wassupported in part by the U.S. Department of Energy under Grant No. DE-FG02-87ER40371.
Appendix A: Verification of the adiabatic approximation
The adiabatic approximation adopted for the analysis of the time evolution in this paper assumesthat ¨ γ (cid:28) ˙ γ , see the text following Eqs. (17) and (31). Plotted in Fig. 5 is the ratio | ¨ γ/ ˙ γ | for certainvalues of conductivity. Similar results hold for other values of the parameters. Clearly, the adiabaticapproximation is self-consistent. [1] S. Chandrasekhar and P.C. Kendall, “On Force-Free Magnetic Fields”, Astrophysical Journal , 457(1957).[2] Y. Hirono, D. Kharzeev and Y. Yin, “Self-similar inverse cascade of magnetic helicity driven by thechiral anomaly,” Phys. Rev. D , no. 12, 125031 (2015) ∗∗ This conclusion may change in the presence of the helicity non-conserving processes such as discussed in [26]. α t0.0050.0100.0500.100 | γ .. / γ | FIG. 5: Accuracy of the adiabatic approximation for σ (cid:48) = 1 and σ (cid:48) = 0 . .
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