1. Reconnection of magnetic field in neutron stars driven by electron mass term in triangle anomaly
Maxim Dvornikov
(in collaboration with V.B.Semikoz)
IZMIRAN, Moscow, Russia
Tomsk State University, Russia

2. Outline Adler-Bell-Jackiw anomaly
Mean spin of plasma (chiral separation effect)
Pseudoscalar contribution in quasiclassical limit
Chemical potential of plasma in magnetic field
Evolution of magnetic helicity in finite volume
Reconnection of magnetic field in neutron star
Conclusion

3. References
M. Dvornikov & V.B. Semikoz, Magnetic helicity evolution in a neutron star accounting for the Adler-Bell-Jackiw anomaly, JCAP 08 (2018) 021 [arXiv: ]

4. Conservation of axial current for classical massless fermions
For chiral fermions with m = 0, γ5 = iγ0 γ1γ2γ3 is conserved operator: [γ5,H]=0, where H = (αp) is Hamiltonian
Noether theorem implies that the classical chiral current should be conserved

5. Adler anomaly for chiral charged fermions
If chiral particle has nonzero electric charge e, in presence of electromagnetic field Fμν = (E,B) the axial current is no longer conserved due to quantum effects (Adler, 1969; Bell & Jackiw, 1969) γν
γαγ5 γμ

6. Mass correction to Adler anomaly
Non-conservation of axial current also holds true for massive fermions (Ioffe, 2006)
If we represent ψT = (ψR,ψL), then
Chiral imbalance
Mean spin of plasma
New pseudoscalar

7. Mean spin External magnetic field B = Bez
Mean spin reads (Semikoz & Valle, 1997)
If fermions are ultrarelativistic
This result is known as the chiral separation effect (Metlitski & Zhitnitsky, 2005) since S = JA

8. Computation of pseudoscalar in external magnetic field
If magnetic field is weak, eB << Ep2, we can use plane waves approximation (Wentzel-Kramers-Brillouin approximation) for the calculation of the pseudoscalar 2im<ψ+γ0 γ5 ψ > = -div(S5)
Dvornikov & Semikoz (2018) obtained that
Equilibrium spin distribution function (Silin, 1968)
- γ= E/m is the Lorentz factor
- Pseudoscalar computation in E||B was made by Fukushima et al. (2018)

9. Magnetic helicity |L| = 5
Magnetic helicity was first introduced by Gauss (1833)
Magnetic helicity is conserved in the perfectly conducting fluid
Magnetic helicity is gauge invariant
In the system of two linked magnetic fluxes, magnetic helicity takes the form (Berger, 1999)
|L| = 5
In classical MHD, magnetic helicity in finite volume evolves as

10. Quantum contributions to helicity evolution
Dvornikov & Semikoz (2018):
Seff = 0 in nonrelativistic plasma, Ep = m
Seff = S = - eμeB/2π2 in ultrarelativistic plasma, Ep >> m

11. Chemical potential of degenerate plasma in external magnetic field
Nunokawa, et al. (1997):
We assume the axially symmetric magnetic field B = B(r,θ)

12. Helicity evolution in neutron star
Dvornikov (2016) found that chiral imbalance in NS vanishes for s. Thus we take that nR = nL.
We assume the quadrupole configuration of the magnetic field B(r,θ)=Bp(r)[cos(2θ) er +sin(2θ)eθ]+Bϕ(r)cos(θ)eϕ
The explicit form of function A was obtained by Dvornikov & Semikoz (2018)

13. Comparison of classical and quantum contributions
Evolution of total helicity consists of two terms: dH/dt = (dH/dt)class + (dH/dt)quant
Classical contribution arises from the surface term known in MHD (see above): (dH/dt)class ~ Bp R3 <v>
We consider core of NS (μe = 100 MeV and R = 105 cm) and assume rigid rotation
In this case, (dH/dt)quant >> (dH/dt)class

14. Magnetar bursts
Magnetars are supposed to be highly magnetized compact stars with B > 1015 G (Turolla et al., 2016)
Origin of such strong magnetic fields is an open question of modern astrophysics
Magnetars are observed by electromagnetic emission in X-ray and gamma-ray regions ranging from short bursts to giant flares
Mechanism of electromagnetic emission of magnetars is unclear 14

15. Implication for reconnection of magnetic field lines
Reconnection is change of magnetic field topology resulting in dissipation of magnetic energy (Priest & Forbes, 2000)
Reconnection is pulsar magnetosphere can result in magnetar bursts (Thompson, et al, 2002)
- Calculated quantum correction to helicity change can be interpreted as intertwining of two thin magnetic tubes: (dH/dt)quant = dθ/dt FpFt, where dθ/dt is angular velocity with which magnetic loop bases are twisting one around other causing interlacing of flux tubes
- Released energy can cause magnetar bursts

16. Conclusion We studied Adler-Bell-Jackiw anomaly for massive particles
Both mean spin and pseudoscalar contributions were accounted for in external magnetic field
Pseudoscalar was computed in WKB approximation in relatively weak magnetic field
These terms are quantum correction to magnetic helicity evolution
We estimated this quantum correction in core of NS and found that it can be greater than classical contribution known in MHD
Results can be relevant for reconnection of magnetic field lines and magnetar bursts

17. Acknowledgements I am thankful to
Organizers of ICPPA-2018 for invitation
RFBR (Russia)