# Chiral dynamos and magnetogenesis induced by torsionful Maxwell–Chern Simons electrodynamics

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## Abstract

Recently chiral anomalous currents have been investigated by Boyarsky et al. and Brandenburg et al. with respect to applications to the early universe. In this paper we show that these magnetic field anomalies, which can give rise to dynamo magnetic field amplification can also be linked to spacetime torsion through the use of a chemical potential and Maxwell electrodynamics with torsion firstly proposed by de Sabbata and Gasperini. When the axial torsion is constant this electrodynamics acquires the form of a Maxwell–Chern–Simmons (MCS) equations where the chiral current appears naturally and the zero component of torsion plays the role of a chemical potential, while the other components play the role of anisotropic conductivity. The chiral dynamo equation in torsionful spacetime is derived here from MSC electrodynamics. Here we have used a recently derived a torsion LV bound of \(T^{0}\sim {10^{-26}}\) GeV and the constraint that this chiral magnetic field is a seed for galactic dynamo. This estimate is weaker than the one obtained from the chiral battery seed of \(\sim {10^{30}}\) G without making use of Cartan torsion. The torsion obtained here was derived at 500 pc coherence scale. When a chiral MF is forced to seed a galactic dynamo one obtains a yet weaker MF, of the order of \(B\sim {10^{12}}\) G, which is the value of a MF at nucleosynthesis. By the use of chiral dynamo equations from parity-violating torsion one obtains a seed field of \(B\sim {10^{27}}\) G, which is a much stronger MF closer to the one obtained by making use of chiral batteries. Chiral vortical currents in non-Riemannian spacetimes derived in Riemannian spaces previously by Flaschi and Fukushima are extended to include minimal coupling with torsion. The present universe yields \(B\sim {10^{-24}}\) G, still sufficient to seed galactic dynamos.

## 1 Introduction

Earlier Vilenkin [1] has shown that chiral massless fermions in parity-violating theories could generate chiral currents where the magnetic field would be proportional to that current under equilibrium conditions. That paper laid down the foundations of chiral dynamo theory as later developed by Froehlich and Pedrini [2] and by Boyarsky et al. [3], culminating in more recent investigations by Brandenburg et al. [4]. However, the main problem with the Vilenkin paper pointed out by himself is that when fermions are massive the flat chiral current vanishes. In this paper we show that really, as pointed out by Vilenkin, when one shifts to Kharzeev parity-violating QCD called local P-violation endowed with Cartan torsion, massive fermions do indeed give rise to a non-vanishing chiral current. Actually as is shown in the next section the chiral current axial torsion vector gives rise to an anisotropic tensor conductivity in chiral axial current when the chiral current can be shown to be expressed as the chiral component plus torsion contributions like the vector product between torsion vector and the magnetic field [5]. Previously Garretson et al. [6] have tried to obtain primordial magnetic fields (PMFs) that seed galactic dynamos without success by making use of an axion version of electrodynamics where the time derivative of the axion scalar fields plays the role of the chemical potential. In order to obtain a PMF of the order suitable to seed microgauss galactic magnetic fields (GMFs) via a dynamo mechanism we make use here of a MCCS version of electrodynamics and of a chiral dynamo equation derived from this electrodynamics to obtain a MF of the order of \(B_\mathrm{PMF}\sim {10^{14}}\) G. This has been obtained by making use of a LV torsion as obtained previously by the author [7] considering GMF at the scale of 500 pc. One of the main advantages of using torsion here is that we may generalize work and results by Boyarsky et al. in SM of particle physics to standard model extensions (SMEs) like LV [8, 9] or supersymmetry, where torsion plays a major role. All these results are contained in Sect. 2. In Sect. 3 we address a brand new kind of chiral currents in nR spaces, which extends the recent result by Flasci and Fukushima [10] of chiral currents in Riemannian spaces, by minimal coupling with torsion, showing that the vortical nR expression appears in the second order—a microscopical result which does not appear in traditional Einstein Cartan gravity (EC) [11]. Other types of gravitational anomalies linked to gravity in torsionful space have been considered recently by Mavromatos and Plafitsis [12]. Section 3 addresses the problem of chirality and the determination of some primordial magnetic fields. Section 4 presents computations of chiral dynamos, taking into account the expansion of the universe in a Friedman universe. Discussions are left to Sect. 5, where we briefly present a very simple example of how to generalize gravitational anomalous currents in Riemannian spaces into non-Riemannian and show that a chiral magnetic effect can be obtained from a simple use of the Larmor frequency.

## 2 Chiral dynamo equation from torsion modes in CS electrodynamics and magnetogenesis

*K*is the Cartan contortion. If one compares these last two expressions one notes that the vector \(k_\mathrm{LV}\) plays a similar role of torsion

*T*. Thus this analogy makes us conclude that Cartan torsion can be a faithful representation of the LV. LV could be interesting also in the early universe as can be seen in several recent papers. Now let us review in more detail the QED virtual pair of matter–antimatter, electron–positron pair decay, when photon is placed on a strong torsion field in the early universe for example. Matter–antimatter asymmetry is an important ingredient for the study of chiral anomalies where one handness type of fermions either massive or massless like neutrinos appears in different amounts in the universe. De Sabbata and Gasperini have long ago shown that if a photon, in second order perturbation of a QFT process, disintegrates into virtual pairs of matter–antimatter electron–positron like systems via the vacuum polarization effect, and since these particles are massive fermions, they couple with the Cartan torsion. They certainly justifies the presence of a torsional background. Thus the e.m. field itself is also affected by the presence of torsion. Actually even if torsion does not interact directly with the photon as advocated by Hehl et al. [18], it does interact quantum mechanically with a photon through a virtual pair of matter–antimatter. This is the reason most physicists believe that torsion in Einstein–Cartan theory is a low energy version of a quantum torsion as investigated recently by Mavromatos and Plafitsis [12]. This theory would be a sort of quantum gravity with torsion, as investigated in detail by Shapiro [19, 20]. This kind of process preserves gauge invariance and the second Maxwell equations modified by a quantum contribution of the order, having the classical field as a zeroth order which does not interact with torsion, and Maxwell equations acquire the classical electrodynamical format. Since this lends torsion a QED quantum effect; it seems suitable to address the problems of quantum anomalies in spacetimes endowed with Cartan torsion modes. Now let us not go into details of de Sabbata Gasperini derivation of Maxwells equations using the QED approach. The invariant in the creation of virtual pairs generates an extra current

*B*is the modulus of the MF. Since the wave vector

*k*is given by the inverse of the coherent length

*L*, one may reexpress this formula as

## 3 Small-scale dynamo equation from axial anomalies with torsion

## 4 Axial anomalies with torsion trace vectors in Friedmann universe

*H*is the Hubble parameter, a is the expansion of the universe. The metric is given by

## 5 Discussions and conclusions

Torsion fields introduced in CP-violating cosmic axion \({\alpha }^{2}\) dynamos by the author [26] in order to obtain Lorentz violating bounds for torsion were revisited. Oscillating axion solutions of the dynamo equation with torsion modes were obtained taking into account dissipative torsion fields. Magnetic helicity torsion oscillatory contributions were also obtained. In this paper we use chiral global anomaly ideas of Zanelli and Chandia expressed as the topological invariant \(<j^{5}>\sim {\int {F\wedge {F}}}\) where \(F=dA\), and we transport them to cosmological axial anomalies in a contorted Friedmann universe, while this universe is isotropic and we deal with the early universe, another work in progress would be to consider the Maxwell chiral equations in a anisotropic spacetime as Bianchi type I solutions for example. Other interesting nice properties of gravitational anomalies are the \(\int {{R^{\mu }}_{\nu }\wedge {R^{\nu }}_{\nu }}\) applied to computed gravitational anomalies associated to these spaces and one might consider the chiral dynamos [4] associated to them. The investigations here may pave the way to building models for quantum gravity based on torsion as well as quantum anomalies. In the remaining of the discussion we shall briefly show how to transform Flaschi–Fujikawa gravitational and chiral anomalies in Riemannian space to non-Riemannian. Take for example the simplest example of the CS current vortical current \(j_{A}\sim {-\omega R}\) where \(\omega \) is the frame rotation and *R* is the Ricci tensor. The new current formula in terms of the torsion trace becomes \(j_{A}\sim {-\omega (R-T^{2}}).\) Note that if one takes into account the Larmor frequency formula and substitutes in this expression the important chiral magnetic effect, that would appear in this anomalous current as \(j_{A}\sim {T^{2}B}\) where *B* is the magnetic field. This is of course very similar to the formulas we discuss above in MCCS electrodynamics. The basic difference is that now that the dependence on the torsion is quadratic and not linear. It is interesting how a result of non-Riemannian gravitational anomaly gave rise to a CME. More investigations are actually the subject of a recent preprint, called non-Riemannian gravitational and mixed anomalies [27, 28, 29, 30, 31, 32, 33], and we study their implications to astrophysics and cosmology. It is still in preparation. Emphasis shall be given to topological induced currents. In most of the computations in magnetogenesis we use data from the estimates of chiral batteries [34]; see also work by Ahonen and Enqvist [35]. Just before completion of this paper we were told about the paper of Cesare et al. [36] where leptogenesis induced by torsion is analyzed and a very weak torsion is called to attention being the cause of non-detectability at the CMB signatures. Torsion bounds are consistent with the bounds used here to investigate magnetogenesis induced by torsionful electrodynamics. Their torsion upper bound is of the order of \(T\sim {{{\mathcal {O}}}(10^{-11})}\) GeV, which is consistent with the bounds used here. Ending our discussions, one could say that our estimates in chiral magnetogenesis are consistent with previous computations, which give support and strength to torsionful theories, not only the ones of the string type by inspiration of the Kalb–Ramond torsion, but also at the lower energy limit of Einstein–Cartan gravity and of torsionful Maxwell equations. In the QCD chiral case it is easy to show that by minimally coupling torsion with the non-Abelian gauge field one obtains a massive gauge particle where the square of the torsion induces mass [37]. The consequences of these torsion massive Yang–Mills bosons to cosmology from chiral anomalies may be further investigated. Recently Schober et al. [38], by using numerical simulations, have obtained \(B_{Ch}\sim {10^{-18}}\) G at 1 Mpc scales. This result is stronger than the result obtained here at the same scale but it has the advantage of being analytical and coming with the strong simplification of a constant chiral chemical potential. Note that previously one [39, 40, 41, 42, 43, 44] used several mechanisms to obtain results in magnetogenesis in torsionful spacetimes. Unfortunately in most of these papers one did not address the chiral magnetogenesis, problems with the breaking of gauge invariance and the introduction of massive photons to say the least. In one of the papers with torsion, no dynamo mechanism is achieved to seed galactic dynamos.

## Notes

### Acknowledgements

We would like to express gratitude to A. Boyarsky and J. Schober for helpful discussions on the subject of this paper. We also thank colleagues of the workshop on Magnetic Fields in the Universe at the international institute of physics. Special thanks go to E. Mielke and F.W. Hehl for long conversations on EC gravity and chiral anomalies over the past three decades. Financial support from the University of State of Rio de Janeiro (UERJ) is gratefully acknowledged.

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