Contribution of the chiral vortical effect to the evolution of the hypermagnetic field and the matter-antimatter asymmetry in the early Universe
aa r X i v : . [ h e p - ph ] J a n Contribution of the chiral vortical effect to the evolution of thehypermagnetic field and the matter-antimatter asymmetry inthe early Universe
S. Abbaslu ∗ , S. Rostam Zadeh † and S. S. Gousheh ‡ Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839,Iran School of Particles and Accelerators, Institute for Research in FundamentalSciences (IPM), P.O.Box 19395-5531, Tehran, Iran
January 13, 2020
Abstract
In this paper, we study the contribution of the chiral vortical effect, in additionto that of the chiral magnetic effect, to the evolution of the hypermagnetic fieldand the matter-antimatter asymmetry in the symmetric phase of the early Universein the temperature range
GeV ≤ T ≤ TeV. We choose a fully helicalChern-Simons wave configuration for the velocity and the hypermagnetic vectorpotential fields. The latter makes the plasma force-free in the absence of viscosity.We show that the most pronounced effect of the chiral vorticity is the productionand initial growth of the hypermagnetic field. In particular, we show that in thepresence of a non-zero matter asymmetry, the hypermagnetic field can grow fromzero initial value only in the presence of a non-zero vorticity field. Moreover, weshow that larger initial growths not only result in larger maximum values of thehypermagnetic field, but also cause the saturation of the hypermagnetic field andthe conversion of the lepton-baryon asymmetry to occur more quickly, i.e., at ahigher temperature. We show that the damping of the vorticity due to the presenceof viscosity, which typically occurs extremely rapidly, does not significantly affectthe evolution. ∗ s − [email protected] † sh − [email protected] ‡ [email protected] INTRODUCTION
The origin of the matter-antimatter asymmetry in the Universe is an open problemin the particle physics and cosmology. The amplitude of the baryon asymmetryof the Universe (BAU) has been obtained from the observations of the cosmicmicrowave background (CMB) and the big bang nucleosynthesis (BBN) and thecurrent accepted estimate is η B ∼ − [1–3]. Many different mechanisms havebeen suggested for producing this asymmetry from an initial matter-antimattersymmetric state [4–6]. Assuming the CPT invariance, Sakharov stated three nec-essary conditions for generating the BAU (baryogenesis): i) baryon number viola-tion, ii) C and CP violation, iii) departure from thermal equilibrium [7]. However,it has been shown that the third condition is not necessary in the absence of theCPT invariance [8,9]. These conditions can be satisfied within the standard modelof particle physics. Charge conjugation symmetry is violated in the weak inter-actions and CP is slightly violated through the CKM mechanism. The departurefrom thermal equilibrium can occur due to phase transitions and the expansionof the Universe. Furthermore, the baryon and lepton numbers are violated dueto the weak sphaleron processes and the hypercharge Abelian anomaly at finitetemperature [10, 11]. In fact, the matter-antimatter asymmetry generation and themagnetogenesis, which is the generation of long range magnetic fields in the Uni-verse, are strongly intertwined via these hypercharge Abelian anomalous effects.Long range magnetic fields have been widely observed in galaxies, super-clusters, and recently in the intergalactic medium (IGM). The strength of thesemagnetic fields have been measured or estimated by applying different meth-ods [12–14]. The induced Faraday rotation effect is used to measure the strengthof the galactic magnetic fields, which is of the order of the microgauss in the MilkyWay and several spiral galaxies [14]. The temperature anisotropy of CMB puts anupper bound on the strength of the magnetic fields, B ≤ − G on the CMB scales λ ≃ Mpc [15]. The observations of the gamma rays from blazars put the strengthof the intergalactic magnetic fields (IGMFs) in the range B ≃ − − × − Gon the scales as large as λ ≃ Mpc [16–18]. Furthermore, a non-vanishing he-licity of these magnetic fields, with the strength B ≃ . × − G, has beeninferred [19].The origin of these galactic and intergalactic magnetic fields is also an openproblem [20–22]. Different mechanisms have been suggested which generallypursue one of the following two approaches to explain the origin and evolutionof these long range magnetic fields in the Universe. One approach investigatesthe generation of the magnetic fields through different astrophysical mechanismsby assuming that the initial weak magnetic field is produced via a battery mecha-nism [23–25]. The other one assumes that the magnetic fields have cosmologicalorigin, that is, the present magnetic fields are produced from seed fields in the2arly Universe [26–30]. Indeed, the presence of the magnetic fields at high red-shifts everywhere in the Universe reinforces the idea that they have cosmologicalorigin [30]. The calculations of magnetic fields produced after the inflation usu-ally suffer from the small-scale problem, that is, their comoving correlation lengthis much smaller than the observed scales of the magnetic fields in the Universe.In this paper, we pursue the latter approach and present a model that can producehypermagnetic fields before the electroweak phase transition (EWPT).There are different processes that influence the evolution of the magneticfields, such as the adiabatic expansion, the Abelian anomalous effects, the mag-netohydrodynamics turbulent dynamo effect, the viscosity diffusion, the inversecascade, and the direct cascade. Among these, the Abelian anomalous effects areprominent since, as mentioned before, they interconnect the evolution of the mag-netic fields to that of the matter-antimatter asymmetries [31–39]. These anoma-lous effects show up through the Abelian anomaly and the Abelian Chern-Simonsterm. In addition to the Abelian anomalous effects, the magnetohydrodynamicsturbulent dynamo effect also influences the evolution of the cosmological mag-netic fields and their correlation length [40–47]. Indeed, turbulence is a complexphenomenon, and one of the characteristic parameters in the turbulence is theReynolds number Re = Lv/ν , where L is the characteristic length scale, v is thevelocity, and ν is the kinematic viscosity. In this paper, we do not take turbulenceinto account; however, we study the Abelian anomalous effects, while taking intoaccount the effects of velocity, as manifested in the form of vorticity, and theviscosity in the plasma.The Ablelian gauge fields, unlike the non-Abelian ones which acquire massgap ∼ g T , remain massless. This, together with the fact that the plasma hashigh conductivity, make it possible only for the Abelian magnetic fields to survivein the plasma as long range gauge fields. The Abelian U Y (1) anomaly emergesas a result of the chiral coupling of the Abelian hypercharge gauge fields to thefermions in the symmetric phase. The Abelian U Y (1) anomalous processes violatethe baryon number B and the lepton number L , but preserve N i = B/n G − L i andconsequently B − L . Here, L i is the lepton number of the i th generation and n G is the number of generations. Indeed, these charges are the well known conservedcharges of the Standard Model which, along with the ones discussed below, canbe used to describe the plasma in thermal equilibrium.In thermal equilibrium, the electroweak plasma can be described by n G chem-ical potentials µ i , corresponding to the aforementioned conserved charges, where i = 1 , .., n G . Furthermore, due to the hypercharge neutrality of the electroweakplasma, there is also another chemical potential µ Y which corresponds to the hy-percharge of the plasma. Moreover, at temperatures higher than T RL ∼ TeV,the right-handed electron chirality flip rate is much lower than the Hubble expan-sion rate. Therefore, the right-handed electron chirality flip processes are out of3hermal equilibrium [48, 49], and in the absence of the Abelian anomaly, the num-ber of right-handed electrons is perturbatively conserved as well. Therefore, thereis another chemical potential which corresponds to the right-handed electrons (seealso [39]).The Abelian Chern-Simons term emerges in the effective action of the U Y (1) gauge fields, due to the chiral coupling of the Abelian hypercharge gauge fieldsto the fermions in the symmetric phase. The inclusion of the anomalous term inthe magnetohydrodynamics (MHD) equations, results in the anomalous magne-tohydrodynamics (AMHD) equations. This is an important term which leads tothe well known chiral magnetic effect (CME). The AMHD equations describe thecoupling between the hypercharge gauge fields, and the velocity and the numberdensities of the particles. Some authors have extensively studied the leptogene-sis, the baryogenesis and the evolution of hypermagnetic fields in the context ofthe AMHD equations, but without considering the potential role of the velocityfield [31–39].The imbalanced chiral plasma which is affected by the CME in the presence ofa magnetic field, is also influenced by the chiral vortical effect (CVE) when thereis vorticity in the plasma [50–53]. The chiral magnetic and vortical currents corre-sponding to these effects emerge in the AMHD equations and play important rolesin the evolution of the cosmological magnetic fields and the matter-antimatterasymmetries. In the broken phase, the chiral magnetic and vortical currents havethe form ~J cm ∝ ( µ R − µ L ) ~B , and ~J cv ∝ ( µ R − µ L ) ~ω , respectively, where ~B isthe Maxwellian magnetic field, ~ω = ~ ∇ × ~v is the vorticity, and µ R and µ L are theright-handed and the left-handed chemical potentials of the particles [50]. Thesecurrents are the macroscopic manifestation of the triangle anomaly in the chiraltheory [54–59]. The notable fact about these currents is that they have a topologi-cal origin and are non-dissipative; therefore, they do not contribute to the entropyproduction [60].The effects of the chiral magnetic and vortical currents on the evolution ofthe large scale magnetic fields and the matter-antimatter asymmetries have beeninvestigated by the authors of Ref. [51]. They have considered an incompress-ible fluid that has a fully non-helical vorticity field in the same direction as themagnetic field, but have ignored the viscosity damping effect. Recently, the effectof chiral anomaly on the evolution of the magnetohydrodynamic turbulence hasbeen studied, as well [61]. However, the effects of the chiral vorticity and the vis-cosity have not been taken into account. In another work related to the evolutionof the magnetic fields in the neutron stars, the authors have taken the chiral mag-netic effect into account, while considering the axial number density as a timeand space dependent variable [62]. Then, they have added a new pseudoscalar The forms of these currents are different in the symmetric phase, and will be presented later. ~ ∇ .S ( t, x ) to the evolution equation of n ( t, x ) , where S ( t, x ) is the meanspin in the magnetized plasma. They have shown that the new term ~ ∇ .S ( t, x ) produces chirality µ ( t, x ) , and as a result, the chiral magnetic effect leads to theamplification of the seed magnetic field.The main purpose of this paper is to present a simple model which starts withan initial chiral vorticity and describes not only the generation of the hypermag-netic field due to the CVE, but also its subsequent evolution which is mainly dueto the CME, before the electroweak phase transition. Indeed, we present the cor-rect form of the chiral vortical coefficient in the symmetric phase and show that,unlike the previous studies, the hypermagnetic field can be produced from zeroinitial value in the presence of the chiral vorticity. Furthermore, we show thatthe hypermagnetic field can be strengthened due to the CME. We also investigatethe effects of these chiral vortical and magnetic currents on the evolution of thematter-antimatter asymmetries. In our model we use fully helical monochromatichypermagnetic and vorticity fields. Since the hypermagnetic field is fully helical, ~ ∇ × ~B ∝ ~B , it has no influence on the evolution of the velocity or the vorticityfields [63]. We also investigate the effects of the viscosity on the evolution ofthe vorticity and the hypermagnetic fields, and therefore, on the matter-antimatterasymmetries [64]. We also present the correct form of the fluid helicity in thesymmetric phase which has been written incorrectly in some of the previous stud-ies. This paper is organized as follows. In Sec. 2, we present the anomalous mag-netohydrodynamics equations, and obtain the vorticity and the helicity coefficientsin terms of the fermionic chemical potentials in the symmetric phase. In Sec. 3,we consider the Abelian anomalous effects and derive the dynamical evolutionequations of the fermionic asymmetries. In Sec. 4, we solve the set of coupleddifferential equations numerically for the hypermagnetic field, the vorticity field,and the baryon and the first-generation lepton asymmetries. Finally, in Sec. 5, wepresent our results and conclude. In this section, we briefly review the AMHD equations in the expanding Uni-verse. Magnetohydrodynamics is the study of the electrically conducting fluids,combining both the principles of the fluid dynamics and the electromagnetism.In the imbalanced chiral plasma, the magnetic field and the vorticity induce thechiral magnetic effect (CME) and the chiral vortical effect (CVE), respectively.The CME is the generation of the electric current parallel to an external magneticfield, whereas the CVE is the generation of the electric current along the vorticityfield. In the presence of the Abelian anomaly, the MHD equations are generalized5o the AMHD equations. The evolution equations of the neutral plasma in theexpanding Universe are given as (see Refs. [22, 65, 66] and also Appendix A and B for details) R ~ ∇ . ~E Y = 0 , R ~ ∇ . ~B Y = 0 , (2.1) ∂ ~B Y ∂t + 2 H ~B Y = − R ~ ∇ × ~E Y , (2.2) ~J Ohm = σ (cid:16) ~E Y + ~v × ~B Y (cid:17) , (2.3) ~J = ~J Ohm + ~J cv + ~J cm = 1 R ~ ∇ × ~B Y − ∂ ~E Y ∂t + 2 H ~E Y ! , (2.4) ~J cv = c v ~ω, (2.5) ~J cm = c B ~B Y , (2.6) (cid:20) ∂∂t + 1 R ( ~v. ~ ∇ ) + H (cid:21) ~v + ~vρ + p ∂p∂t = − R ~ ∇ pρ + p + ~J × ~B Y ρ + p + νR (cid:20) ∇ ~v + 13 ~ ∇ ( ~ ∇ .~v ) (cid:21) , (2.7) ~ω = 1 R ~ ∇ × ~v, (2.8) ∂ρ∂t + 1
R ~ ∇ . [( ρ + p ) ~v ] + 3 H ( ρ + p ) = 0 , (2.9)where, ρ and p are the energy density and the pressure of the fluid, σ is the electri-cal conductivity, R is the scale factor, H = ˙ R/R is the Hubble parameter, and ν isthe kinematic viscosity. Furthermore, ~v and ~ω are the bulk velocity and the vortic-ity of the plasma, and the currents ~J Ohm , ~J cv , and ~J cm are the Ohmic current, thechiral vortical current, and the chiral magnetic current, respectively. The latter isthe one that promotes the ordinary MHD equations to the AMHD equations. Notealso that the terms like H ~B Y and H ~E Y in Eqs. (2.2) and (2.4) which containthe Hubble parameter H , are due to the expansion of the Universe. The vortic-ity and the helicity coefficients c v and c B appearing in Eqs. (2.5) and (2.6) are asfollows [35, 67, 68] In some work in the literature, only the right-handed currents are considered, resulting in thefollowing simplification: ~J cv = ~J cv , R = ~J and ~J cm = ~J cm , R = ~J , see for example [53]. v ( t ) = g ′ π n G X i =1 (cid:16) − Y R µ R i + Y L µ L i N w − Y d R µ d Ri N c − Y u R µ u Ri N c + Y Q µ Q i N c N w (cid:17) , (2.10) c B ( t ) = − g ′ π n G X i =1 h − (cid:18) (cid:19) Y R µ R i − (cid:18) − (cid:19) Y L µ L i N w − (cid:18) (cid:19) Y d R µ d Ri N c − (cid:18) (cid:19) Y u R µ u Ri N c − (cid:18) − (cid:19) Y Q µ Q i N c N w i , (2.11)where, n G is the number of generations, and N c = 3 and N w = 2 are the ranksof the non-Abelian SU (3) and SU (2) gauge groups, respectively. Furthermore, µ L i ( µ R i ), µ Q i , and µ u Ri ( µ d Ri ) are the common chemical potentials of left-handed(right-handed) leptons, the left-handed quarks with different colors, and up (down)right-handed quarks with different colors, respectively. Moreover, ‘ i ’ is the gen-eration index, and the corresponding hypercharges are Y L = − , Y R = − ,Y u R = 43 , Y d R = − , Y Q = 13 . (2.12)Substituting the above constants in Eqs. (2.10) and (2.12) results in c v ( t ) = g ′ π n G X i =1 (cid:16) µ R i − µ L i + 2 µ d Ri − µ u Ri + 2 µ Q i (cid:17) , (2.13) c B ( t ) = − g ′ π n G X i =1 (cid:18) − µ R i + µ L i − µ d Ri − µ u Ri + 13 µ Q i (cid:19) . (2.14)In a previous study [38], we took the CME into account but neglected theCVE. There, we made some assumptions and simplified the helicity coefficient c B accordingly. In this paper, we take both the CME and the CVE into account. In thefollowing, we make the same assumptions and simplify the vorticity coefficient c v ,as well.We assume that all quark Yukawa interactions are in equilibrium. Moreover,because of the flavor mixing in the quark sector, we assume that all up or down They are: up-type Yukawa in processes u iR ¯ d iL ↔ φ (+) and u iR ¯ u iL ↔ φ (0) , down-type Yukawain processes d jR ¯ u iL ↔ φ ( − ) and d jR ¯ d iL ↔ ˜ φ (0) , and their conjugate reactions [67]. µ u R − µ Q = µ , (2.15) µ d R − µ Q = − µ . (2.16)Where, µ , µ Q , and µ u R ( µ d R ) are the chemical potentials of the Higgs field, theleft-handed up or down quarks, and the right-handed up (down) quarks, respec-tively. Then, for simplicity, we assume that the Higgs asymmetry is zero andobtain [32, 38] µ u R = µ d R = µ Q . (2.17)Using Eq. (2.17), we simplify Eqs. (2.13) and (2.14), and obtain c v ( t ) = g ′ π n G X i =1 (cid:0) µ R i − µ L i (cid:1) (2.18)and c B ( t ) = − g ′ π n G X i =1 ( − µ R i + µ L i − µ Q ) . (2.19)We assume that only the contributions of the baryonic and the first-generationleptonic chemical potentials to the helicity and the vorticity coefficients are non-negligible. Then, Eqs. (2.18) and (2.19) reduce to the forms c v ( t ) = g ′ π (cid:0) µ e R − µ e L (cid:1) , (2.20) c B ( t ) = − g ′ π (cid:18) − µ e R + µ e L − µ B (cid:19) , (2.21)where we have also used the equation µ Q = µ B [38]. The time-dependent coef-ficients c v ( t ) and c B ( t ) evolve in accordance to the evolution of their constituents,the evolution equations of which will be obtained in the next section. Like whatis usually done in the ordinary MHD equations, the displacement current will beneglected in the following. It should be noted that, neglecting the displacementcurrent in the comoving frame is equivalent to neglecting the term ∂ t ~E Y + 2 H ~E Y in the Lab frame. Using Eqs. (2.3) and (2.4) with the mentioned assumption, thehyperelectric field can be obtained as ~E Y = 1 σR ~ ∇ × ~B Y − c v σ ~ω − c B σ ~B Y − ~v × ~B Y . (2.22)8utting the above expression for the hyperelectric field into Eq. (2.2), the evolutionequation of the hypermagnetic field can be obtained as ∂ ~B Y ∂t + ~B Y t = 1 σR ∇ ~B Y + c v σR ~ ∇ × ~ω + c B σR ~ ∇ × ~B Y + 1 R ~ ∇ × ( ~v × ~B Y ) , (2.23)where we have used the equation H = 1 / t for the radiation dominated era.Since ~ ∇ . ~B Y = 0 , the hypermagnetic field can be written as ~B Y = (1 /R ) ~ ∇ × ~A Y , where ~A Y is the vector potential of the hypermagnetic field. Let us consideran incompressible fluid in the comoving frame [51, 61], which leads to the condi-tion of ∂ t ρ + 3 H ( ρ + p ) = 0 in the lab frame. Then, combining this condition withthe continuity equation (2.9) results in ~ ∇ .~v = 0 . Therefore, in analogy with thehypermagnetic field, the velocity field can be written as ~v = (1 /R ) ~ ∇ × ~S , where ~S is the vector potential of the velocity field. In this work, we concentrate on thefully helical hypermagnetic and vorticity fields; To have such fields, we choosethe same non-trivial Chern-Simons wave configuration [69, 70] for both of theirvector potentials, in order to have maximum efficacy. That is, ~A Y = γ ( t )(sin kz, cos kz, , (2.24)and ~S = r ( t )(sin kz, cos kz, , (2.25)where γ ( t ) and r ( t ) are the time-dependent amplitudes of the vector potentials ~A Y and ~S , respectively. Using these configurations, we get ~B Y = (1 /R ) k ~A Y , ~v =(1 /R ) k ~S , and ~ω = (1 /R ) k~v for the hypermagnetic, the velocity and the vorticityfields. In the following, ~ω will be replaced by (1 /R ) k~v , wherever appropriate.Let us compute the ensemble average of the hypermagnetic field energy den-sity by using the aforementioned simple configuration as E B ( t ) = 12 h ~B Y ( x, t ) . ~B Y ( x, t ) i = 12 B Y ( t ) = 12 R k γ ( t ) , (2.26) As mentioned in the Introduction, the coherent magnetic fields in the intergalactic mediumhave been inferred to be helical [19]. Therefore, we have chosen a helical configuration for thehypermagnetic field [69, 70]. This topologically non-trivial configuration, with Chern-Simonsnumber density n CS ∝ k ′ γ ( t ) , has been used extensively to solve the magnetohydrodynamic(MHD) equations [32, 33, 35, 38, 39, 71]. Furthermore, it has been introduced as an exact single-mode solution to the chiral MHD equations [72,73]. Moreover, the four fully helical configurationsare ~A Y = γ ( t )(sin kz, cos kz, , ~A Y = γ ( t )(cos kz, − sin kz, , ~A Y = γ ( t )(cos kz, sin kz, ,and ~A Y = γ ( t )( − sin kz, cos kz, . The first two (last two) have positive (negative) helicity, andtheir unit vectors, along with ˆ z , form orthonormal bases [74]. H B ( t ) = h ~A Y ( x, t ) . ~B Y ( x, t ) i = kR γ ( t ) . (2.27)It can be seen that E B ( t ) = ( k/ R ) H B ( t ) , which indicates that the hypermagneticfield is fully helical.In analogy with the hypermagnetic field, the fluid kinetic energy and the fluidhelicity can be defined as E v ( t ) = ρ h ~v.~v i = ρ v ( t ) , (2.28)and H v ( t ) = n G X i =1 h(cid:16) (cid:17)(cid:16) T R i + T L i N w + T d Ri N c + T u Ri N c + T Q i N c N w (cid:17) + (cid:16) π (cid:17)(cid:16) µ R i + µ L i N w + µ d Ri N c + µ u Ri N c + µ Q i N c N w (cid:17)i h ~v. ~w i = n G X i =1 (cid:20) T + (cid:16) π (cid:17)(cid:16) µ R i + 2 µ L i + 12 µ Q (cid:17)(cid:21) kR v ( t ) , (2.29)respectively [54, 75–77]. In Eq. (2.29), we have assumed that all particles are inthermal equilibrium, and as mentioned earlier, µ d R = µ u R = µ Q . It can be seenthat the time-dependent temperatures and chemical potentials play important rolesin the fluid helicity, and even with constant velocity, the fluid helicity decreases as R − due to the expansion of the Universe.Using the simple configurations for the vector potentials of the hypermagneticand the velocity fields as given by Eqs. (2.24) and (2.25), and their consequentrelations, ~B Y = (1 /R ) ~ ∇ × ~A Y = (1 /R ) k ~A Y and ~ω = (1 /R ) ~ ∇ × ~v = (1 /R ) k~v ,Eqs. (2.22) and (2.23) reduce to the forms ~E Y = k ′ σ ~B Y − c v σ k ′ ~v − c B σ ~B Y , (2.30)and ∂ ~B Y ( t ) ∂t + ~B Y ( t ) t = − k ′ σ ~B Y ( t ) + c v σ k ′ ~v ( t ) + c B σ k ′ ~B Y ( t ) , (2.31)respectively, where k ′ = k/R = kT . It can be seen that the length scale of thehypermagnetic field increases due to the expansion of the Universe. Note that10oth the hypermagnetic and the velocity fields are in the same direction; thus, theadvection term ~v × ~B Y in Eqs. (2.22) and (2.23) has been set to zero.Let us now consider the evolution equation of the velocity field. Neglectingthe displacement current in Eq. (2.4), the total current becomes ~J = (1 /R ) ~ ∇ × ~B Y ; therefore, ~J × ~B Y vanishes in Eq. (2.7). Furthermore, the incompressibilitycondition of the fluid, ∂ t ρ + 3 H ( ρ + p ) = 0 , not only leads to ~ ∇ .~v = 0 , asstated earlier, but also ensures that H~v + ~v∂ t p/ ( ρ + p ) = 0 in Eq. (2.7). Afterneglecting the gradient terms in Eq. (2.7), the evolution equation of the velocityfield simplifies to ∂~v∂t = − νk ′ ~v. (2.32)Note that in the radiation dominated era, only the shear viscosity contributes tothe non-ideal stress energy tensor and the bulk viscosity becomes zero. In thenext section, the evolution equations of the fermion numbers will be obtained. In this section, we briefly review the U Y (1) Abelian anomaly equations, and obtainthe evolution equations of the leptonic and the baryonic asymmetries in the sym-metric phase. Before the electroweak phase transition, in contrast to the brokenphase, the fermion numbers are violated, due to the fact that the coupling of thehypercharge fields to the fermions is chiral. This shows up in the U Y (1) Abeliananomaly equations [31]. These anomaly equations for the first-generation leptonsare ∂ µ j µe R = − (cid:0) Y R (cid:1) g ′ π Y µν ˜ Y µν = g ′ π ~E Y . ~B Y ,∂ µ j µe L = ∂ µ j µν Le = 14 (cid:0) Y L (cid:1) g ′ π Y µν ˜ Y µν = − g ′ π ~E Y . ~B Y . (3.1)In addition to the Abelian anomaly that violates the lepton numbers, the perturba-tive chirality flip reactions should also be considered in the evolution equations ofthe leptonic asymmetries as The term ( ~v.~ ∇ ) ~v is neglected since it is next to leading order. Furthermore, ~ ∇ p = 0 becausethe fluid pressure is only time-dependent. η e R dt = g ′ π s h ~E Y . ~B Y i + 2Γ RL ( η e L − η e R ) ,dη e L dt = dη ν Le dt = − g ′ π s h ~E Y . ~B Y i + Γ RL ( η e R − η e L ) , (3.2)where, η f = ( n f − n ¯ f ) /s with f = e R , e L , ν Le is the fermion asymmetry, s =2 π g ∗ T / is the entropy density, and g ∗ = 106 . is the effective number ofrelativistic degrees of freedom. It should be noted that we are assuming η e L ≈ η ν Le ,based on the fast SU(2) interactions in the SU(2) doublet. The chirality flip rate Γ RL that appears in the above equations is [33] Γ RL = 5 . × − h e ( m T ) T = (cid:18) Γ t EW (cid:19) (cid:18) − x √ x (cid:19) , (3.3)where the variable x = tt EW = ( T EW T ) , in accordance with the Friedmann law, t EW = M T EW , M = M P l / . √ g ∗ , and M P l is the Plank mass. In addition, h e = 2 . × − is the Yukawa coupling of the right-handed electrons, Γ =121 , and m ( T ) = 2 DT (1 − T EW /T ) is the temperature-dependent effectiveHiggs mass at zero momentum and zero Higgs vacuum expectation value. Thecoefficient D ∼ . in the expression for m ( T ) has contributions comingfrom the known masses of gauge bosons m Z and m W , the top quark mass m t ,and the zero-temperature Higgs mass [33]. Using the expression for the fermionicchemical potential, µ f = 6( n f − n ¯ f ) /T , and the changes of variables, ξ f = µ f /T and η f = ξ f T / s , we obtain dξ e R dt = 3 g ′ π T h ~E Y . ~B Y i + 2Γ RL ( ξ e L − ξ e R ) ,dξ e L dt = dξ ν Le dt = − g ′ π T h ~E Y . ~B Y i + Γ RL ( ξ e R − ξ e L ) . (3.4)By considering the conservation law η B / − η L i = const. and the evolution equa-tions of the lepton asymmetries, the evolution equation of the baryon asymmetrycan be obtained as dξ B dt = dξ e R dt + 2 dξ e L dt = 3 g ′ π T h ~E Y . ~B Y i . (3.5)In the above equations, we need to know the exact form of h ~E Y . ~B Y i . We use Eq.(2.30) and obtain h ~E Y . ~B Y i = k ′ σ B Y ( t ) − c B σ B Y ( t ) − c v k ′ σ h ~v ( t ) . ~B Y ( t ) i , (3.6)12here the vorticity coefficient, c v , and the helicity coefficient, c B , are given byEqs. (2.20,2.21), respectively. Then, using σ = 100 T , R = 1 /T , ν ≃ / (5 α Y T ) [64, 78], where α Y = g ′ / π is the fine-structure constant for the U Y (1) gaugefields, and the aforementioned expressions for c v and c B , Eqs. (3.6), (2.31), and(2.32) become h ~E Y . ~B Y i = B Y ( t )100 (cid:20) k ′ T − g ′ π (cid:18) ξ e R − ξ e L ξ B (cid:19)(cid:21) − g ′ π (cid:0) ξ e R − ξ e L (cid:1) k ′ T h ~v ( t ) . ~B Y ( t ) i , (3.7) dB Y ( t ) dt = B Y ( t )100 (cid:20) − k ′ T + k ′ g ′ π (cid:18) ξ e R − ξ e L ξ B (cid:19)(cid:21) − B Y ( t ) t + g ′ π (cid:0) ξ e R − ξ e L (cid:1) k ′ T h ~v ( t ) . ˆ B Y ( t ) i , (3.8) dv ( t ) dt = − k ′ α Y T v ( t ) . (3.9)With the choice of the vector potentials in Eqs. (2.24,2.25), h ~v ( t ) . ~B Y ( t ) i → v ( t ) B Y ( t ) and h ~v ( t ) . ˆ B Y ( t ) i → v ( t ) . Using Eq. (3.7), and the relations y R =10 ξ e R , y L = 10 ξ e L , x = t/t EW = ( T EW /T ) , and Gauss ≃ × − GeV ,we can rewrite Eqs. (3.4), (3.8), and (3.9) in the forms dy R dx = (cid:20) C − C (cid:18) y R − y L y B (cid:19)(cid:21) (cid:18) B Y ( x G (cid:19) x / − C (cid:0) y R − y L (cid:1) v ( x ) (cid:18) B Y ( x )10 G (cid:19) √ x − Γ − x √ x ( y R − y L ) , (3.10) dy L dx = − (cid:20) C − C (cid:18) y R − y L y B (cid:19)(cid:21) (cid:18) B Y ( x )10 G (cid:19) x / + C (cid:0) y R − y L (cid:1) v ( x ) (cid:18) B Y ( x )10 G (cid:19) √ x + Γ − x √ x ( y R − y L ) , (3.11) dB Y dx = C √ x (cid:20) − (cid:18) k − (cid:19) + 10 α Y π (cid:18) y R − y L y B (cid:19)(cid:21) B Y ( x ) − B Y ( x ) x + C (cid:0) y R − y L (cid:1) v ( x ) x / , (3.12)13 v ( x ) dx = − C √ x v ( x ) , (3.13)where C = 25 . (cid:18) k − (cid:19) , C = 77 . , C = 0 . (cid:18) k − (cid:19) g ′ π ,C = 0 . (cid:18) k − (cid:19) , C = 89 × g ′ π (cid:18) k − (cid:19) , C = 7 . α Y (cid:18) k − (cid:19) . (3.14)Following steps analogous to those for the derivation of Eq. (3.5), we obtainthe evolution equation of the baryon asymmetry in the form dy B dx = 32 (cid:20) C − C (cid:18) y R − y L y B (cid:19)(cid:21) (cid:18) B Y ( x )10 G (cid:19) x / − √ x C (cid:0) y R − y L (cid:1) v ( x ) (cid:18) B Y ( x )10 G (cid:19) , (3.15)where y B = 4 × π g ∗ η B / . The terms containing v ( x ) in Eqs. (3.10), (3.11),(3.12), and (3.15) are due to the presence of the chiral vorticity in the plasma. Inthe next section we will solve this set of coupled differential equations numericallyand discuss the results. In this section, we solve the set of coupled differential equations obtained in Sec.3 numerically, and compare the results with the ones obtained in the non-vorticalplasma. The equations are solved with the initial conditions k = 10 − , B (0) Y = 0 , y (0) R = 10 , y (0) L = y (0) B = 0 , and four different values for the initial velocity, v (0) = 0 , − , − , and − . The initial velocities are all within the domainof validity of the non-relativistic approximation. The results are shown in Fig. 1.Figure 1 shows that lepton asymmetries are equalized rather quickly by thechirality flip processes. As shown in the Figs. 1(a-d), if the initial velocity, andhence the vorticity, is zero, nothing else happens. That is, the lepton asymmetriesremain constant, and the baryon asymmetry and the hypermagnetic field ampli-tude remain zero. However, if the initial vorticity is non-zero, the CVE causes B Y to grow extremely rapidly at the start of its evolution, essentially creating a seedfield for it. By increasing the initial velocity, the seed field becomes stronger, and14ts ensuing growth due to the CME leads to yet larger values (see Fig. 1e). Themaximum scale of this initial growth can be seen in Fig. 1f, which shows howquickly the initial velocity is damped by the viscosity.When B Y is produced, it grows until it reaches a maximum or saturation valueat a critical time, and a concurrent transition occurs: the lepton and baryon asym-metries decrease rapidly (see Figs. 1(a-e)). After this transition, the matter asym-metries stay constant, while B Y decreases precisely exponentially and relativelyslowly due to the expansion. The reason for the inclusion of Fig. 1e is to displayclearly the changes of B Y for values below Gauss, and in particular showthat at the end of the time interval, which is the onset of the electroweak phasetransition, B Y ≈ . This final value is almost independent of its initial seed, aslong as it is nonzero, and depends only on the initial matter asymmetries [35, 38].Figure 1 shows that by increasing the initial velocity, and hence the vorticity, thecritical time decreases, or, equivalently, the critical temperature increases.Next, we examine the behavior of the velocity field and the CVE more closely.First we should mention that since the hypermagnetic field is fully helical, i.e. ~ ∇× ~B Y = α ~B Y , it cannot affect the evolution of the velocity or the vorticity fields[63]. This would, in the absence of viscosity, make the plasma force free. Indeed,these fields decrease exponentially due to the kinematic viscosity and rapidly tendto zero, as can be seen in Fig. 1. However, as stated earlier, their very briefpresence can significantly affect the evolution of the hypermagnetic field and thusthe matter asymmetries.The question that we address next is what would happen if the viscosity is zero.For this purpose the set of coupled differential equations are solved with the initialconditions y (0) R = 10 , B (0) Y = 0 , y (0) L = y (0) B = 0 , and v = 10 − , in the presenceand absence of viscosity. Figure 2 shows that in a non-viscose plasma, althoughthe velocity and the vorticity fields remain constant, the aforementioned effectsdue to the chiral vorticity on the matter asymmetries and the hypermagnetic fieldare not significantly altered. The most important effect of the absence of viscosityis that the seed produced for B Y by the vorticity is stronger. Hence, the CME canincrease the amplitude of B Y to its saturation curve sooner, i.e., at higher value ofcritical temperature, as compared to the plasma with the non-zero viscosity. Thedrops in the values of matter asymmetries at the transition are unchanged.This ineffective role of the vorticity after producing the initial seed for B Y is mainly due to the fact that after the electron chirality flip reactions come intoequilibrium, the vorticity coefficient c v vanishes. This in turn is due to the factthat the contributions of the chemical potentials of the right-handed and the left-handed electrons to c v cancel each other. Therefore, after the electron chiralityflip reactions come into equilibrium, the CVE is turned off, even if the vorticity islarge. In fact, the chirality flip reactions in the temperature range under consider-15 - x η e L (a) - x η e R (b) - - - - - - - - x η B (c) - × × × × × x B Y ( x )[ G ] (d) - x B Y ( x )[ G ] (e) × - × - × - × - × - × - x v ( x ) (f) Figure 1:
Time plots of the lepton and the baryon asymmetries and the hypermagnetic fieldamplitude in the presence of the viscosity with the initial conditions k = 10 − , B (0) Y = 0 , y (0) R =10 , and y (0) L = y (0) B = 0 . The solid line is for v = 10 − , large dashed line for v = 10 − ,dashed line for v = 10 − , and dotted line for v = 0 .a: Left-handed lepton asymmetry, η e L .b: Right-handed lepton asymmetry, η e R .c: Baryon asymmetry, η B .d: The hypermagnetic field amplitude, B Y .e: The log plot of B Y (The case for v = 0 yielding B Y = 0 cannot be displayed).f: The velocity field amplitude for v = 10 − and − ≤ x ≤ . × − . ation are important and cannot be neglected. Finally, it should be emphasized thatthe evolution of matter asymmetries and the hypermagnetic field amplitude are16 - x η e L (a) - x η e R (b) - - - - - - - - x η B (c) - x B Y ( x )[ G ] (d) Figure 2:
Time plots of the lepton and the baryon asymmetries and the hypermagnetic fieldamplitude with the initial conditions y (0) R = 10 , B (0) Y = 0 , and y (0) L = y (0) B = 0 , and v = 10 − .Dashed line is obtained for non-zero viscosity and dotted line for zero viscosity.a: Left-handed lepton asymmetry, η e L .b: Right-handed lepton asymmetry, η e R .c: Baryon asymmetry, η B .d: The amplitude of the hypermagnetic field, B Y . almost independent of the initial value of the vorticity and the viscosity, as longas the former is non-zero. In this paper, we have studied the effects of the chiral vorticity on the evolution ofthe hypermagnetic field and the matter asymmetries in the early Universe and inthe temperature range
GeV ≤ T ≤ TeV. Starting with an initial vorticityand large matter asymmetries at TeV, we have investigated the production andgrowth of the hypermagnetic field, and the evolution of the matter asymmetriesand the vorticity till the onset of the electroweak phase transition, i.e.
GeV.We have chosen the non-trivial Chern-Simons configuration with a monochro-matic spectrum for the vector potentials of both the hypermagnetic and the veloc-17ty fields, with the same characteristic wave number k = 10 − in the comovingframe. Since the hypermagnetic field is fully helical, i.e., ~ ∇ × ~B = α ~B , it hasno effect on the evolution of the velocity or the vorticity fields. This is due tothe fact that the term ~J × ~B Y vanishes in the Navier-Stokes equation. By con-sidering an incompressible homogeneous plasma, the only remaining term in theNavier-Stokes equation is the kinematic viscosity, which leads to the exponentialdecrease of the vorticity.Our most important result is that, an initial vorticity and matter asymmetriescan produce a seed for the hypermagnetic field in the plasma via the CVE. Thiscannot occur if only the CME is taken into account. Subsequently, the CME leadsto the growth of the hypermagnetic field amplitude until it reaches its maximumvalue. At this time a transition occurs where the matter asymmetries suddenlychange, while preserving B − L , to attain their constant final values. We haveshown that increasing the vorticity in the plasma leads to a stronger seed of thehypermagnetic field which then grows to yet a larger maximum value. More-over, the critical time decreases, or equivalently the critical temperature increases.Later, the amplitude of the hypermagnetic field decreases gradually due to theexpansion of the Universe, while its length scale increases as λ = π √ xkT EW , where − ≤ x ≤ .As mentioned before, if we choose the vector potentials of the hypermagneticand velocity fields to be two different basis configurations, then their dot prod-uct in Eqs. (3.6,3.8) would vanish. Therefore, The seed hypermagnetic field inEq. (3.8) will not be produced, and the subsequent evolution will be due only tothe chirality flip processes, equalizing the chemical potentials of the right and lefthanded electrons. We have also investigated the case with the same initial mat-ter asymmetry but with both vector potentials having the same negative helicityconfiguration. The result is that the generated B Y and η B are about 23 orders ofmagnitude smaller. For the cases in which the sign of initial matter asymmetriesand helicities are simultaneously reversed, we obtain analogous evolutions butwith the sign of asymmetries reversed. There are two generalization that can beconsidered for the vector potentials. First, one can consider fully helical configu-rations for the vector potentials, but with a superposition of wave numbers k . Inthis case we expect, due to the last term in Eq. (3.12) which acts as a source term,the seed hypermagnetic field to still be generated. Second, one can include a non-helical component in the hypermagnetic field. In that case, ~ ∇ × ~B Y = α ~B Y andconsequently the term ~J × ~B Y will not vanishes in the Navier-Stokes equation.This term acts as a source for vorticity and velocity, i.e. the plasma is no longerforce free. 18 APPENDIX A
It is known that the Universe in large scale is homogeneous and isotropic, soits geometry can be described by a conformally flat metric of the Friedmann-Robertson-Walker (FRW) type with the form ds = dt − R ( t ) δ ij dx i dx j , (6.1)where t is the physical time, x i s are the comoving coordinates, and R ( t ) is thescale factor. Then, the effective Lagrangian density for the hypercharge gaugefields at finite fermion density and in the curved space-time can be written as[79, 80] £ = √− g ˆ £ = √− g (cid:20) − F µν F µν − J µ Ohm A µ + c B ǫ ijk F ij A k R + c v ǫ ijk ω ij A k R (cid:21) , (6.2)where F µν = ∇ µ A ν − ∇ ν A µ is the field strength tensor, A µ is the hyperchargevector potential, g is the determinant of the FRW metric defined in Eq. (6.1), and ∇ µ is the covariant derivative with respect to this metric. Moreover, J µ Ohm =( J , ~J /R ) is the Ohmic four-vector current, ˜ ǫ ijk = − ˜ ǫ ijk is the Levi-Civita sym-bol, ω ij = ∇ i u j − ∇ j u i is the antisymmetric vorticity tensor, and ~u = ~v/R isthe bulk velocity of the plasma in the curved space-time. The vorticity and thehelicity coefficients c v and c B appearing in Eq. (6.2) are given in Eqs. (2.10,2.11),respectively. Using the effective Lagrangian density, as given by Eq. (6.2), in thefollowing equation ∂ ˆ £ ∂A ν − ∇ µ " ∂ ˆ £ ∇ µ A ν = 0 , (6.3)the Euler-Lagrange equations for the hypercharge gauge fields in the curved space-time can be obtained as ∇ µ F µν = J ν Ohm − c B ǫ ijk F ij g kk δ νk R ( t ) − R ( t ) c v ˜ ǫ ijk (cid:0) ∇ i v j (cid:1) δ νk . (6.4)Note also that the only non-vanishing Christoffel symbols of the metric (6.1) are Γ ij = R ˙ Rδ ij and Γ i j = Γ ij = ˙ R/Rδ ij . It can be seen that three different types ofelectric current appear on the rhs of Eq. (6.4). These are the Ohmic current J ν Ohm ,the zeroth component of which is zero due to the hypercharge neutrality in theplasma, the chiral magnetic current J ν cm = (0 , c B ˜ ǫ ijk F ij R/ , and the chiral vor-tical current J ν cv = (0 , − c v ˜ ǫ ijk ( ∇ i v j ) /R ) . By using F ij = − ˜ ǫ ijk (cid:0) B k /R (cid:1) and19 ǫ ijk ( ∇ i v j ) = w k = − w k , these chiral currents simplify to J ν cm = (0 , c B ~B Y /R ) and J ν cv = (0 , c v ~w/R ) , respectively.Considering ν = 0 in Eq. (6.4), the Gauss’s Law is obtained as a ~ ∇ . ~E Y = ρ total = 0 , (6.5)the rhs of which vanishes due to the hypercharge neutrality of the plasma. Then,considering ν = i in Eq. (6.4), the time evolution of the hyperelectric field inthe presence of the CME and the CVE, and in the expanding Universe (Ampere’sLaw) will be obtained as ∂ t ~E Y + 2 H ~E Y = 1 R (cid:16) ~ ∇ × ~B Y (cid:17) − ~J Ohm − c B ~B Y − c v ~ω. (6.6)In the above equation, the term H ~E Y is due to the scaling of the hyperelectricfield in the expanding Universe. In order to obtain the two other Maxwell’s equa-tions, the following Bianchi identity is used ∇ µ F νρ + ∇ ρ F µν + ∇ ν F ρµ = ∂ µ F νρ + ∂ ρ F µν + ∂ ν F ρµ = 0 , (6.7)which results in ~ ∇ . ~B Y = 0 , (6.8)and ∂ t ~B Y + 2 H ~B Y = − R (cid:16) ~ ∇ × ~E Y (cid:17) . (6.9)It can be seen that, similar to the hyperelectric field, the hypermagnetic field isalso scaled as R − . The plasma of the early Universe contains different types of constituents whichare sufficiently strongly coupled to be considered as a fluid [22]. Moreover, Itcan be considered as an ideal fluid with the equation of state p = ρ/ in theradiation dominated era, where p and ρ are the pressure and the energy density ofthe plasma, respectively. The energy momentum tensor of this ideal fluid in thepresence of the hypercharge electromagnetic fields can be written as T µν = T µνf + T µν em , (7.1)where T µνf = ( ρ + p ) U µ U ν − pg µν , (7.2)20nd T µν em = 14 g µν F αβ F αβ − F νσ F µσ . (7.3)In the above equations, F αβ = ∇ α A β − ∇ β A α , U µ = γ (1 , ~v/R ) is the four-velocity of the plasma normalized such that U µ U µ = 1 , and γ is the Lorentzfactor. Due to the ideal fluid assumption, the non-ideal effects are ignored in Eq.(7.2) [81]. Since the Einstein tensor obtained from the metric (6.1) is diagonal,not only the hypercharge electromagnetic field density must be small compared tothe energy density of the Universe [81], but also the bulk velocity should respectthe condition | ~v | ≪ , or equivalently γ ≃ and U µ ≃ (1 , ~v/R ) . Using theconservation equation of the energy momentum tensor ∇ µ T µν = 0 , the conserva-tion equation of the energy density and the continuity equation can be obtained.Considering ν = 0 , we obtain ∂ t ρ + ~ ∇ . (cid:20) ( ρ + p ) ~vR (cid:21) + 3 H ( ρ + p ) (cid:0) v (cid:1) = ~E Y . ~J , (7.4)where ~J = ~J Ohm + c B ~B Y + c v ~ω . The second order term in the velocity field andthe term ~E Y . ~J appearing in the above equation are usually neglected. Considering ν = j , the continuity equation can be obtained as (cid:20) ∂ t ρ + 1 R ~ ∇ . [( ρ + p ) ~v ] + 3 H ( ρ + p ) (cid:21) ~v + [ ∂ t p + H ( ρ + p )] ~v + ( ρ + p ) ∂ t ~v + ( ρ + p ) ~v. ~ ∇ R ~v + ~ ∇ pR = ρ total ~E Y − (cid:16) ~B Y × ~J Ohm (cid:17) − c v ~B Y × ~ω. (7.5)On the rhs of Eq. (7.5), the second and the third terms are obtained from ~B Y × ~J .Furthermore, the term ρ total ~E Y vanishes since J = J = 0 and the plasma iselectrically neutral.Let us now obtain the equations for the anomalous divergence of the mattercurrents in the symmetric phase and in the curved space-time. Due to the chi-ral coupling of the hypercharge fields to the fermions, the fermion numbers areviolated as ∇ µ J µi = C i ~E Y . ~B Y , (7.6)where J µi is the fermionic current and C i is its corresponding Anomaly coefficient.The above equation can also be written in the form ∂ t J i + 1 R ~ ∇ . ~J i + 3 HJ i = C i ~E Y . ~B Y . (7.7)21hen, by integrating over all space and dividing by volume, the second term van-ishes and we obtain ∂ t ( n i − ¯ n i ) + 3 H ( n i − ¯ n i ) = C i h ~E Y . ~B Y i , (7.8)where n i and ¯ n i are the number densities of the ith species of the fermion and theanti-fermion, respectively. Using the relation ˙ s/s = − H , we obtain s∂ t (cid:18) n i − ¯ n i s (cid:19) = C i h ~E Y . ~B Y i , (7.9)where s is the entropy density. References [1] Kazuharu Bamba, C. Q. Geng, S. H. Ho, Hypermagnetic baryogenesis,Physics Letters B (2008), [arXiv:0712.1523 [hep-ph]].[2] B. Fields and S. Sarkar, Big-Bang nucleosynthesis (2006 Particle DataGroup mini-review), J. Phys. G 33 (2006) 1[arXiv:astro-ph/0601514];G. Steigman, Primordial Nucleosynthesis: The Predicted and Ob-served Abundances and Their Consequences, PoS NICXI (2010) 001,[arXiv:1008.4765 [astro-ph.CO]].[3] V. Simha and G. Steigman, Constraining The Early-UniverseBaryon Density And Expansion Rate, JCAP 0806 (2008) 016,[arXiv:0803.3465 [astro-ph]].[4] M. Fukugita and T. Yanagida, Baryogenesis Without Grand Unification,Phys. Lett. B (1986) 45, [DOI: 10.1016/0370-2693(86)91126-3].[5] G. Panotopoulos and N. Videla, Baryogenesis via leptogenesis in multi-fieldinflation, Eur. Phys. J. C78 (2018) 774, [arXiv:1809.07633 [gr-qc]].[6] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, On the AnomalousElectroweak Baryon Number Nonconservation in the Early Universe, Phys.Lett. B (1985) 36,][DOI: 10.1016/0370-2693(85)91028-7].[7] A. D. Sakharov, Violation of CP Invariance, C Asymmetry, and BaryonAsymmetry of the Universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETPLett. 5 (1967) 24] [Sov. Phys. Usp. 34 (1991) 392] [Usp. Fiz. Nauk 161(1991) 61], [DOI: 10.1070/PU1991v034n05ABEH002497].228] A. D. Dolgov, NonGUT baryogenesis, Phys. Rept. , 309 (1992), [DOI:10.1016/0370-1573(92)90107-B].[9] O. Bertolami, D. Colladay, V. A. Kostelecky and R. Potting, CPT violationand baryogenesis, Phys. Lett. B , (1997) 178, [arXiv:hep-ph/9612437].[10] Bjrn Garbrecht , Why is there more matter than antimatter? Cal-culational methods for leptogenesis and electroweak baryogenesis,[arXiv:1812.02651 [hep-ph]].[11] G. t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev.Lett. 37 (1976), [DOI: 10.1103/PhysRevLett.37.8].[12] L. M. Widrow, Origin of galactic and extragalactic magnetic fields, Rev.Mod. Phys. , 775 (2002), [arXiv:astro-ph/0207240].[13] P. P. Kronberg, Extragalactic magnetic fields, Rep. Prog. Phys. , 325(1994), [DOI: 10.1088/0034-4885/57/4/001] .[14] J. P. Vallee, Cosmic magnetic fields as observed in the Universe, in galac-tic dynamos, and in the Milky Way, New Aston. Rev.48,763 (2004), [DOI:10.1016/j.newar.2004.03.017].[15] P. A. R. Ade et al. (Planck Collaboration), Planck 2015 results. XIX. Con-straints on primordial magnetic fields, Astron. Astrophys. , A19 (2016).[16] S. Ando and A. Kusenko, Evidence for Gamma-Ray Halos Around ActiveGalactic Nuclei and the First Measurement of Intergalactic Magnetic Fields ,Astrophys.J. 722 (2010) L39 [arXiv:1005.1924 [astro-ph.HE]].[17] W. Essey, S. Ando and A. Kusenko,
Determination of intergalactic mag-netic fields from gamma ray data , Astropart.Phys. 35 (2011) 135-139[arXiv:1012.5313 [astro-ph.HE]].[18] W. Chen, J. H. Buckley, and F. Ferrer,
Search for GeV Gamma-Ray PairHalos Around Low Redshift Blazars , Phys. Rev. Lett. (2015) 211103[arXiv:1410.7717 [astro-ph.HE]].[19] W. Chen, B. D. Chowdhury, F. Ferrer, H. Tashiro, and T. Vachaspati,
In-tergalactic magnetic field spectra from diffuse gamma-rays , Mon. Not. Roy.Astron. Soc. (2015) 3371 [arXiv:1412.3171 [astro-ph.CO]].[20] A. Brandenburg, D. Sokoloff, and K. Subramanian, Current status of turbu-lent dy- namo theory: From large-scale to small-scale dynamos, Space Sci.Rev. , 123(2012), [arXiv:1203.6195 [astro-ph.SR]].2321] D. Grasso and H. R. Rubinstein, Magnetic fields in the early Universe, Phys.Rep. , 163 (2001), [arXiv:astro-ph/0009061].[22] R. Durrer and A. Neronov, Cosmological magnetic fields: Their genera-tion, evolution and observation, Astron. Astrophys. Rev. , 62 (2013),[arXiv:1303.7121 [astro-ph.CO]].[23] M. J. Rees, The origin and cosmogonic implications of seed magnetic fields,Quart. J. Roy. Astr. Soc., 28, 197-206 (1987).[24] K. Subramanian, D. Narasimha, and S. M. Chitre, Mon. Not. Roy. Astron.Soc. , 15 (1994).[25] R. M. Kulsrud and E. G. Zweibel, The Origin of Astrophysical MagneticFields, Rept. Prog. Phys. , 0046091 (2008), [arXiv:0707.2783 [astro-ph]].[26] M. S. Turner and L. M. Widrow, Inflation Produced, Large Scale MagneticFields, Phys. Rev. D , 2743 (1988), [DOI: 10.1103/PhysRevD.37.2743].[27] K. Enqvist and P. Olesen, On primordial magnetic fields of electroweak ori-gin, Phys. Lett. B , 178 (1993), [arXiv:hep-ph/9308270].[28] K. Bamba, Property of the spectrum of large-scale magnetic fields from in-flation, Phys. Rev. D , 083516 (2007), [arXiv:astro-ph/0703647].[29] M. Joyce and M. Shaposhnikov, Primordial magnetic fields, right-handedelectrons, and the Abelian anomaly, Phys. Rev. Lett. , 1193 (1997),[arXiv:astro-ph/9703005].[30] A. Kandus, K. E. Kunze and C. G. Tsagas, Primordial magnetogenesis, Phys.Reports. , 1 (2011), [arXiv:1007.3891 [astro-ph.CO]].[31] M. E. Shaposhnikov, Structure of the High Temperature Gauge GroundState and Electroweak Production of the Baryon Asymmetry, Nucl. Phys.B , 797 (1988); M. Giovannini and M. E. Shaposhnikov, Primordialhypermagnetic fields and triangle anomaly, Phys. Rev. D , 2186 (1998),[arXiv:hep-ph/9710234].[32] M. Dvornikov and V. B. Semikoz, Leptogenesis via hypermagnetic fields andbaryon asymmetry, J. Cosmol. Astropart. Phys. 1202 (2012) 040; Erratum:JCAP (2012) E01, [arXiv:1111.6876 [hep-ph]].[33] M. Dvornikov and V. B. Semikoz, Lepton asymmetry growth in the sym-metric phase of an electroweak plasma with hypermagnetic fields ver-sus its washing out by sphalerons, Phys. Rev. D , 025023 (2013),[arXiv:1212.1416 [astro-ph.CO]].2434] V. B. Semikoz, A. Yu. Smirnov, and D. D. Sokoloff, Generation of hyper-magnetic helicity and leptogenesis in the early Universe, Phys. Rev. D ,103003 (2016), [arXiv:1604.02273 [hep-ph]].[35] S. Rostam Zadeh and S. S. Gousheh, Contributions to the U Y (1) Chern-Simons term and the evolution of fermionic asymmetries and hypermagneticfields, Phys. Rev. D , 056013 (2016), [arXiv:1512.01942 [hep-ph]].[36] K. Kamada and A. J. Long, Large-scale magnetic fields can explain thebaryon asymmetry of the Universe, Phys. Rev. D , 083520 (2016),[ arXiv:1602.02109 [hep-ph]].[37] K. Kamada and A. J. Long, Baryogenesis from decaying magnetic helicity,Phys. Rev. D , 123509 (2016), [arXiv:1606.08891 [astro-ph.CO]].[38] S. Rostam Zadeh and S. S. Gousheh, Effects of the U Y (1) Chern-Simonsterm and its baryonic contribution on matter asymmetries and hypermagneticfields, Phys. Rev. D , 056001 (2017), [arXiv:1607.00650 [hep-ph]].[39] S. Rostam Zadeh and S. S. Gousheh, A Minimal System IncludingWeak Sphalerons for Investigating the Evolution of Matter Asymme-tries and Hypermagnetic Fields, Phys. Rev. D , 096009, (2019),[arXiv:1812.10092 [hep-ph]].[40] B. E. Goldstein, E. J. Smith, A. Balogh, T. S. Horbury, M. L. Goldstein, andD. A. Roberts, Geophys. Res. Lett. , 3393 (1995).[41] J.W. Armstrong, B. J. Rickett, and S. R. Spangler, Electron density powerspectrum in the local interstellar medium, Astrophys. J. , 209. (1995),[DOI: 10.1086/175515].[42] A. Chepurnov and A. Lazarian, Extending Big Power Law in the Skywith Turbulence Spectra from WHAM data, Astrophys. J. , 853 (2010),[arXiv:0905.4413 [astro-ph.GA]].[43] J. M. Scalo, in Interstellar Processes, edited by D. J. Hollenbach and H. A.Thronson Jr. (Reidel, Dordrecht, 1987), p. 349.[44] A. Brandenburg, K. Enqvist, and P. Olesen, Large scale magnetic fields fromhydromagnetic turbulence in the very early universe, Phys. Rev. D 54, 1291(1996), [arXiv:astro-ph/9602031].[45] A. Brandenburg, K. Enqvist, and P. Olesen, The Effect of Silk damp-ing on primordial magnetic fields, Phys. Lett. B , 395 (1997),[arXiv:hep-ph/9608422]. 2546] P. Olesen, On inverse cascades in astrophysics, Phys. Lett. B , 321(1997), [arXiv:astro-ph/9610154].[47] D. T. Son, Magnetohydrodynamics of the early universe and the evo-lution of primordial magnetic fields, Phys. Rev. D , 063008 (1999),[arXiv:hep-ph/9803412].[48] B. A. Campbell, S. Davidson, J. R. Ellis, and K. A. Olive, On the baryon,lepton flavor and right-handed electron asymmetries of the universe, Phys.Lett. B , 118 (1992).[49] J. M. Cline, K. Kainulainen, and K. A. Olive, Erasure and Regenerationof the Primordial Baryon Asymmetry by Sphalerons, Phys. Rev. Lett. ,2372 (1993), [arXiv:hep-ph/9304321]; J.M. Cline, K. Kainulainen, andK. A. Olive, Protecting the primordial baryon asymmetry from erasure bysphalerons, Phys. Rev. D , 6394 (1994), [arXiv:hep-ph/9401208].[50] A. Vilenkin, Macroscopic Parity Violating Effects: Neutrino Fluxes FromRotating Black Holes And In Rotating Thermal Radiation, Phys. Rev. D ,1807 (1979), [DOI: 10.1103/PhysRevD.20.1807]; A. Vilenkin, EquilibriumParity Violating Current In A Magnetic Field, Phys. Rev. D , 3080 (1980),[DOI: 10.1103/PhysRevD.22.3080].[51] H. Tashiro, T. Vachaspati, and A. Vilenkin, Chiral effectsand cosmic magnetic fields, Phys. Rev. D , 105033 (2012),[arXiv:1206.5549 [astro-ph.CO]].[52] Tamal K. Mukherjee, Soma Sanyal, Particle temperature and the Chiral Vor-tical Effect in the early Universe, Modern Physics Letters A Vol. 32, No. 32(2017), [arXiv:1709.00211 [hep-ph]].[53] S. Anand, J. R. Bhatt, and A. K. Pandey, Chiral Battery, scaling laws andmagnetic fields, JCAP , 051 (2017), [arXiv:1705.03683 [astro-ph.CO]].[54] V.P. Kirilin, A.V. Sadofyev, V.I. Zakharov. Chiral Vortical Effect in Super-fluid , Phys.Rev. D , 025021, (2012),[arXiv:1203.6312 [hep-th]].[55] D.T. Son, P. Surowka, Hydrodynamics with Triangle Anomalies,Phys.Rev.Lett. , 191601, (2009), [arXiv:0906.5044 [hep-th]].[56] D.T. Son, A.R. Zhitnitsky, Quantum anomalies in dense matter, Phys.Rev.D , 074018, (2004), [arXiv:hep-ph/0405216].2657] A. V. Sadofyev, V. I .Shevchenko, V. I. Zakharov, Notes on chiral hydro-dynamics within effective theory approach, Phys.Rev.D , 105025, (2011),[arXiv:1012.1958 [hep-th]].[58] Shi Pu, Jian-hua Gao and Qun Wang, A consistent description of ki-netic equation with triangle anomaly, Phys.Rev. D , 094017 (2011),[arXiv:1008.2418 [nucl-th]].[59] Omer F. Dayi, Eda Kilinarslan, Quantum Kinetic Equation in theRotating Frame and Chiral Kinetic Theory, Phys.Rev. D (2018),[arXiv:1807.05912 [hep-th]].[60] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Progress in Particleand Nuclear Physics, 88, (2016).[61] P. Pavlotic, N. Leite, G. Sigl, Chiral Magnetohydrodynamic Turbulence,Phys Rev. D , 023504 (2017), [arXiv:1612.07382 [astro-ph.CO]].[62] M. Dvornikov, V. B. Semikoz, Generation of strong magneticfields in old neutron stars driven by the chiral magnetic ef-fect,[arXiv:1904.05768 [astro-ph.HE]]. arXiv:[1904.05768].[63] S. Chandrasekhar, P. C. Kendall, ON FORCE-FREE MAGNETIC FIELDS,1957; S. Chandrasekhar and Magnetohydrodynamics, E. N. Parker J. Astro-phys. Astr. (1996) , 147166, [].[64] R. Banerjee, K. jedamzik, The Evolution of cosmic magnetic fields: Fromthe very early universe, to recombination, to the present, Phys Rev. D ,123003 (2004), [ arXiv:astro-ph/0410032].[65] C. P. Dettmann, N. E. Frankel, and V. Kowalenko, Plasma electrodynamicsin the expanding Universe, Phys Rev. D , 12 (1993), [DOI: 10.1103/Phys-RevD.48.5655].[66] K. Subramanian, J. D. Barrow, Magnetohydrodynamics in the Early Uni-verse and the Damping of Non-linear Alfven Waves, Phys. Rev. D ,083502 ,(1998), [arXiv:astro-ph/9712083].[67] A. J. Long, E. Sabancilar, and T. Vachaspati, Leptogenesis and pri-mordial magnetic fields, J. Cosmol. Astropart. Phys. 02 (2014) 036,[arXiv:1309.2315 [astro-ph.CO]].[68] M. Laine, Real-time Chern-Simons term for hypermagnetic fields, J. HighEnergy Phys. 10 (2005) 056, [arXiv:hep-ph/0508195].2769] V. Rubakov and A. Tavkhelidze, Stable anomalous states of superdense mat-ter in gauge theories, Phys. Lett. B , 109 (1985), [DOI: 10.1016/0370-2693(85)90701-4].[70] V. Rubakov, On the electroweak theory at high fermion density, Prog. Theor.Phys. , 366 (1986), [DOI: 10.1143/PTP.75.366].[71] M. Giovannini, Hypermagnetic knots, Chern-Simons waves and the baryonasymmetry, Phys.Rev. D (2000) 063502,[hep-ph/9906241].[72] A. Boyarsky, J. Froehlich, O. Ruchayskiy, Self-consistent evolution of mag-netic fields and chiral asymmetry in the early Universe , Phys. Rev. Lett. ,031301 (2012), [arXiv:1109.3350 [astro-ph.CO]].[73] A. Boyarsky, J. Frohlich, and O. Ruchayskiy,
Magnetohydrodynam-ics of Chiral Relativistic Fluids , Phys. Rev. D (2015) 043004[arXiv:1504.04854 [hep-ph]].[74] M. Giovannini, Spectrum of anomalous magnetohydrodynamics, Phys. Rev.D , 103518 (2016), [arXiv:1509.02126 [hep-th]].[75] A. Avkhadiev V.P.Kirilin, A. V. Sadofyev and V. I. Zakharov, On consis-tency of hydrodynamic approximation for chiral media, Phys. Lett. B (2016),[arXiv:1402.3587 [hep-th]].[76] A. Avkhadiev and A. V. Sadofyev, Chiral Vortical Effect for Bosons, Phys.Rev. D 96, no.4, 045015 (2017),[arXiv:1702.07340 [hep-th]].[77] V. P. Kirilin, A. V. Sadofyev, Anomalous Transport and Generalized AxialCharge, Phys.Rev. D (2017) no.1, 016019, [arXiv:1703.02483 [hep-th]].[78] S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, New york,1972).[79] G. E. Volovik and A. Vilenkin, Macroscopic parity violating effects and He − A , Phys.Rev. D , 025014 (2000) [arXiv:hep-ph/9905460].[80] G. E. Volovik, Superfluid analogies of cosmological phenomena, Phys. Rept.351