Lepton asymmetry growth in the symmetric phase of an electroweak plasmawith hypermagnetic fields versus its washing out by sphalerons

Maxim Dvornikov1,2,* and Victor B. Semikoz1,†

1Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), 142190 Moscow, Troitsk, Russia2Institute of Physics, University of Sao Paulo, CP 66318, CEP 05315-970 Sao Paulo, Sao Paulo, Brazil

(Received 6 December 2012; published 22 January 2013)

We study lepton asymmetry evolution in plasma of the early Universe before the electroweak phase

transition (EWPT) accounting for chirality flip processes via Higgs decays (inverse decays) entering

equilibrium at temperatures below TRL ’ 10 TeV, TEW < T < TRL. We solve appropriate kinetic equa-

tions for leptons and Higgs bosons taking into account the lepton number violation due to Abelian

anomalies for right and left electrons and neutrinos in the self-consistent hypercharge field obeying

Maxwell equations modified by the contribution of the Standard Model of electroweak interactions. The

violation of left lepton numbers and the corresponding violation of the baryon number due to sphaleron

processes in symmetric phase is taken into account as well. Assuming the Chern-Simons wave

configuration of the seed hypercharge field, we get the estimates of baryon and lepton asymmetries

evolved from the primordial right electron asymmetry existing alone as partial asymmetry at T � TRL.

One finds a strong dependence of the asymmetries on the Chern-Simons wave number. We predict a

nonzero chiral asymmetry �� ¼ �eR ��eL � 0 in this scenario evolved down to the EWPT moment

that can be used as an initial value for the Maxwellian field evolution after EWPT.

DOI: 10.1103/PhysRevD.87.025023 PACS numbers: 14.60.�z, 95.30.Qd, 98.80.Cq

I. INTRODUCTION

The nature of the initial fields that seed subsequentdynamo for observed galactic magnetic fields is largelyunknown [1,2]. It might be that seed fields are producedduring the epoch of the galaxy formation, or ejected by firstsupernovae or active galactic nuclei. Alternatively to thisastrophysical scenario, the seed fields might originate frommuch earlier epochs of the Universe expansion, down tothe cosmological inflation epoch [3]. There are first obser-vational indications of the presence of cosmological mag-netic fields (CMF) in the intergalactic medium which maysurvive even till the present epoch [4,5].

It is well known that Maxwellian CMF might ariseduring the electroweak phase transition (EWPT) frommassless (long-range) hypercharge fields Y� existing in

primordial plasma before EWPT [6]. Note that long-rangenon-Abelian magnetic fields (corresponding to, e.g., thecolor SU(3) or weak SU(2) groups) cannot exist because athigh temperatures the non-Abelian interactions induce a‘‘magnetic’’ mass gap �g2T.

The Faraday equation, which governs evolution of mag-netic (hypermagnetic) fields in the early Universe, dependscrucially on the helicity parameter � which is a scalarowing to the parity violation in the standard model (SM)and having different forms before and after EWPT. Weremind that the standard magnetohydrodynamic (MHD)parameter, which is generated by vortices in plasma�MHD � hv � ðr � vÞi is pseudoscalar according to the

parity conservation in QED plasma. Obviously in the iso-tropic early Universe such vortices are absent, at least atlarge scales we consider here.Before EWPT in the symmetric phase of primordial

plasma the helicity parameter �Y results from the Chern-Simons (CS) anomaly term in the SM Lagrangian [6,7]LCS ¼ ðg02�eR=4�

2ÞðBY � YÞ, which violates parity. Here,

g0 ¼ e= cos�W is the UYð1Þ gauge coupling, BY ¼ r� Yis the hypermagnetic field, and �eR is the chemical poten-

tial for right-handed electrons (positrons). The polarizationorigin of the CS term LCS was elucidated in Ref. [8] asan effect of comoving right electrons and positrons alongBY which have opposite spin projections on BY and popu-late the main Landau level with slightly different densitiesdue to �eR � 0.

While such CS term vanishes in broken phase at T < TEW

[9], there appears a similar polarization effect [10] due toweak interaction of neutrinos (antineutrinos) with polar-ized electrons (positrons) given by the axial vector force

FðAÞ� ¼ �rVðAÞ

� arising from the parity violating part ofweak interactions between neutrinos of the flavor a ¼ e,�, � and charged leptons

VðAÞ� ðx; tÞ ¼ GFc

ðAÞa ðMð�Þ � �jð�aÞðx; tÞÞ:

This axial vector force acts only on those polarizedelectrons and positrons which contribute to the partial mag-

netizationMð�Þ ¼ �Bsignð�Þn0�B=Bwith the correspond-ing densities n0� of electrons (� ¼ �) and positrons(� ¼ þ) at the main Landau level. Here GF is the Fermi

constant, cðAÞa ¼ �0:5 is the axial coupling in the Weinberg-Salammodel, the upper (lower) sign stays for electron (muon,

*maxim.dvornikov@usp.br†semikoz@yandex.ru

PHYSICAL REVIEW D 87, 025023 (2013)

1550-7998=2013=87(2)=025023(12) 025023-1 � 2013 American Physical Society

tau) neutrinos, and �B is the Bohr magneton. Note that

the neutrino current density asymmetry�jð�aÞ ¼ j�a� j ��a

is

the polar vector and the magnetization Mð�Þ is the axialvector. Thus, the weak axial-vector force separates electriccharges causing the relative drift velocity and the corre-sponding electric current that leads to the generation ofan additional electromagnetic field component resultingfrom weak interactions Eweak � �B�GF. Unfortunatelythe �-helicity parameter produced by this mechanism isvery small. Moreover, it depends on such additional

parameter as a neutrino gas inhomogeneity scale ð�Þfluid,��

GFðT=ð�ÞfluidÞð�n�=n�Þ [10]. It should be mentioned that the

new opportunity for theMaxwellian CMF generation foundin recent works [11,12] seems to be very intriguing.

In Ref. [11] it was shown that the evolution ofMaxwellian magnetic fields in a primordial plasma at tem-peratures T � 10 MeV is strongly affected by the quantumchiral anomaly proportional to the difference of right-handed and left-handed electron chemical potentials��ðtÞ ¼ �eR ��eL evolving in a self-consistent way.

Such a difference defines the new helicity parameter�newðtÞ ¼ �em��ðtÞ=��cond in the modified Faradayequation that governs evolution of Maxwellian fields justafter EWPTat temperatures 10 MeV< T < TEW (see com-ments below in Appendix A). Here �em ¼ 1=137 is thefine-structure constant and�cond is the plasma conductivity.

The goal of the present work is a more careful analysisof the initial chiral anomaly parameter��ðTEWÞ arising inhypermagnetic fields before EWPT and accounting for theHiggs (inverse) decays which change chiralities of right-and left-handed electrons (positrons). While the sphaleronprocesses are always switched off in the Higgs phase, in thesymmetric phase at T > TEW, conversely, we should takethem into account since lepton and baryon numbers areviolated even in the absence of hypermagnetic fields. Theplan of our paper is the following. In Sec. II we formulateequilibrium conditions in the symmetric phase of primor-dial plasma and comment on the corresponding set ofchemical potentials. In the main Sec. III we derive kineticequations for appropriate asymmetries of right- and left-handed electrons (positrons) in the presence of Abeliananomalies for fermions, as well as the kinetic equation forthe Higgs boson asymmetry accounting for both inverseand direct Higgs decays. Such an extension of the kineticapproach, considered earlier in our paper [13], allows us tocorrect the value ��ðTEWÞ accounting for the Higgs bosonasymmetry evolution. In Sec. III A we check conservationlaws and calculate the baryon asymmetry of the Universe(BAU) through leptogenesis in hypermagnetic fields. InSec. IV we solve kinetic equations for all lepton andHiggs asymmetries both analytically, neglecting hyper-charge fields, and numerically, including hypermagneticfields. In Sec. V we analyze the evolution of the chiralanomaly parameter ��ðtÞ in the symmetric phase down tot ¼ tEW. Our results are discussed in Sec. VI.

In Appendix Awe interpret and compare two quantummechanisms producing �-helicity parameter for magneticand hypermagnetic fields: the chiral anomaly leading tothe �new-helicity parameter in the Faraday equation [11],and the Chern-Simons anomaly for hypercharge fieldsinterpreted as a polarization effect in plasma causedby the hypermagnetic field itself [8]. In Appendix B wederive kinetic equations for the lepton and Higgs asym-metries used in the main Sec. III. In Appendix C we givesome formulas for the lepton number violation due to’t Hooft’s anomaly in non-Abelian fields in order toexplain how the sphaleron processes influence the leftlepton kinetics.

II. EQUILIBRIUM IN THE SYMMETRIC PHASEOF ELECTROWEAK PLASMA AND THE

CHIRAL ANOMALY PROBLEM

The question of how large the chiral anomaly parameter��ðtÞ ¼ �eRðtÞ ��eLðtÞ could be before EWPT is impor-

tant as an input for the generation of Maxwellian magneticfields after EWPT. In the SM plasma consisting of quarks,

leptons, and one Higgs doublet ’T ¼ ð’ðþÞ; ’ð0ÞÞ, with thechemical potential being in Bose distribution �0 ¼�’ð0Þ ¼ �’ðþÞ , one can expect the chemical equilibrium

in symmetric phase given by the relation

�eR ��eL ¼ ��0: (2.1)

Equation (2.1) corresponds to Higgs decays and inverse

decays in reactions eL �eR $ ’ð0Þ and �Le �eR $ ’ðþÞ. Here

for the SM doublet LTe ¼ ð�L

e eLÞ we use the equality ofchemical potentials in Fermi distributions �eL ¼ ��Le

.

For the case of the global equilibrium in the absence ofhypermagnetic fields, similar reactions with Higgs bosonsobey analogous relations both for other lepton generations,�lR ��lL ¼ ��0, l ¼ �, �, and for quarks, e.g., �uL ��dR ¼ �0, �uR ��uL ¼ �0, in reactions quL �qdR $ ’ðþÞ

and quR �quL $ ’ð0Þ, correspondingly [14,15].

We consider below only one generation with the lowestYukawa coupling of fermions with Higgs bosons he ¼ffiffiffi2

pme=v ¼ 2:94� 10�6. Thus right electrons enter the

equilibrium with left particles through Higgs (inverse)decays in the expanding Universe in the last instance.This is because of the high rate of chirality flip reactions�RL � h2eT, which becomes faster than the Hubble expan-sion H� T2, �RL >H at temperatures below TRL �10 TeV. This fact is important in scenarios where the gen-eration of BAU proceeds through the leptogenesis, and aprimordial BAU is stored in right electrons eR that are beingprotected from washing out by sphalerons all the way downto TRL. Suggesting such scenario, the authors of Ref. [16]supposed that such value TRL is close to the temperature atwhich the sphaleron effects fall out of the equilibrium, andtherefore it is possible that the eR may not be transformed

MAXIM DVORNIKOVAND VICTOR B. SEMIKOZ PHYSICAL REVIEW D 87, 025023 (2013)

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into eL soon enough for the sphalerons to turn them intoantiquarks, and thereby wipe out the remaining BAU.

In this scenario the global equilibrium [14,15] fails, andfive (¼ 5) remaining chemical potentials describe equilib-rium in a hot plasma before EWPT: three �i for the threeglobal charges B=3� Li ¼ const, where i ¼ 1, 2, 3 enu-merates generations in SM, �Y for the conserved hyper-charge (global hYi ¼ 0), and �eR for right electrons eRwith the conservation of their lepton number @�j

�eR ¼ 0

unless T > TRL [6]. Then, if one assumes the presence oflarge-scale hypercharge fields Y� in the symmetric phase,

which are progenitors of Maxwellian fields in the brokenphase, the number of right electrons is not conservedbecause of the Abelian anomaly [17]

@�j�eR ¼ g02Y2

R

64�2Y��

~Y��; (2.2)

where Y�� and ~Y�� are, respectively, the UYð1Þ hyper-

charge field strengths and their duals, and YR ¼ �2 isthe hypercharge of the right electron.

There are no asymmetries of left leptons and Higgsbosons in this scenario �eL ¼ �0 ¼ 0, and the chiral

asymmetry (2.1) reduces to �� ¼ �eR . For such scenario

with a nonzero eR asymmetry alone [6], sphaleron washingout BAU is absent all the way down to EWPT.

In a broadened scenario with nonzero left leptonasymmetries eL ¼ �L

e� 0, where a ¼ �a=T, appropri-

ate for the stage T < TRL [8,13], we somehow violate theequilibrium described in Ref. [6] by five chemical poten-tials for five globally conserved charges. Nevertheless, it

can lead only to an additional factor of the order one c� � 1that describes the dependence of nL ¼ ðneL � n �eLÞ ¼eLT

3=6 � 0 on five global charges in primordial plasma.

For instance, rewriting the canonical Abelian anomaly forthe left doublet LT

e ¼ ð�Le ; eLÞ,

@�j�eL ¼ �g02Y2

L

64�2Y��

~Y��; YL ¼ �1; (2.3)

in the form deL=dt ¼ �c�ð6g02=16�2T3ÞðEY �BYÞ, weput below c� ¼ 1 simplifying the solution of our kineticequations for the lepton and Higgs boson asymmetries. Notethat assuming a nonzero left particle asymmetryeL � 0, we

should take into account the sphaleron processes violatinglepton and baryon numbers. The competition of such pro-cesses with hypermagnetic field contribution throughAbelian anomaly is one of the interesting questions touchedupon in the present work.

III. KINETICS OF LEPTONS AND HIGGSBOSONS IN HYPERMAGNETIC FIELDS

In Ref. [13] we forced the presence of zero Higgsasymmetry n’ð0Þ � n~’ð0Þ ¼ T2�0=3 ¼ 0, �0 ¼ 0 consider-

ing leptogenesis with the inverse decays only, eR �eL !~’ð0Þ, eR ��L

e ! ’ð�Þ, etc. Now let us consider both inverseHiggs decays and direct Higgs decays. The system ofkinetic equations for leptons accounting for Abeliananomalies (2.2) and (2.3), and sphaleron processes forleft leptons takes the form

dLeR

dt¼ g02

4�2sðEY � BYÞ þ 2�RL

(LeL � LeR �

½n’ð0Þ � n~’ð0Þ �2s

);

for decays ðinverse decaysÞ eR �eL $ ~’ð0Þ and eR ��Le $ ’ð�Þ;

dLeL

dt¼ � g02

16�2sðEY �BYÞ �

�sph

2LeL þ �RL

(LeR � LeL þ

½n’ð0Þ � n~’ð0Þ �2s

);

for �eReL $ ’ð0Þ; as well as

dL�Le

dt¼ � g02

16�2sðEY �BYÞ �

�sph

2L�L

eþ �RL

(LeR � LeL þ

½n’ð0Þ � n~’ð0Þ �2s

);

for �eR�Le $ ’ðþÞ: (3.1)

Here Lb ¼ ðnb � n �bÞ=s is the lepton number, b ¼ eR;eL; �

Le , s ¼ 2�2g�T3=45 is the entropy density, and g� ¼

106:75 is the number of relativistic degrees of freedom.The factor of 2 in front of the rate �RL in the firstline takes into account the equivalent reaction branches.We also included Higgs decays with the rate �D ¼ �RL=2.The probability �sph ¼ C�5

WT is given by sphaleron tran-sitions decreasing the left lepton numbers and thereforewashing out BAU, where �W ¼ g2=4� ¼ 1=137sin2�W ¼3:17� 10�2 is given by the gauge coupling g ¼ e= sin�W inSM and �W is the Weinberg angle. The constant C ’ 25 is

estimated through lattice calculations (see some commentson ’t Hooft’s anomaly in Appendix C and Chap. 11 inRef. [15]). Of course, for the left doublet LeL ¼ L�Le

.This system is completed by the kinetic equation for the

Higgs bosons independent of Abelian anomaly inherent infermions [19]

d

dt½ðn’ð0Þ � n~’ð0Þ Þ=s� ¼ �RL

(LeL � LeR �

½n’ð0Þ � n~’ð0Þ �2s

):

(3.2)

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Note that the rate of Higgs decays (inverse decays) coin-cides with the rate of a lepton pair production (annihilation)having opposite sign since the creation of a pair is followedby the disappearance of a Higgs boson and vice versa.

In kinetic Eqs. (3.1) and (3.2) we used the rate of allinverse processes [16], which is twice bigger than for thedecay ones �RL ¼ 2�D,

�RL ¼ 5:3� 10�3h2e

�m0

T

�2T ¼ �0

2tEW

1� xffiffiffix

p : (3.3)

This rate vanishes just at the EWPT time x ¼ 1, where thevariable x ¼ t=tEW ¼ ðTEW=TÞ2 is given by the Friedmannlaw. Here he ¼ 2:94� 10�6 is the Yukawa coupling forelectrons, �0 ¼ 121, and m2

0ðTÞ ¼ 2DT2ð1� T2EW=T2Þ is

the temperature dependent effective Higgs mass at zeromomentum and zero Higgs vacuum expectation value. Thecoefficient 2D 0:377 for m2

0ðTÞ is given by the known

masses of gauge bosons mZ and mW, the top quark massmt, and a still problematic zero-temperature Higgs mass,which is estimated as mH � 125 GeV (see Ref. [20]). Ofcourse, the chirality flipping rate exists after EWPT.However, that rate is due to electromagnetic processes atT < TEW, �em ’ �2

emðm2e=3T

2ÞT when particles (electronsand positrons) acquire the nonzero mass me.

The detailed derivation of kinetic Eqs. (3.1) and (3.2)accounting for chirality flip processes (without Abeliananomaly and sphaleron transitions) is given in Appendix B.

Let us rewrite Eqs. (3.1) and (3.2) using the asymmetriesLeR ¼ eRT

3=6s, LeL ¼ eLT3=6s and ðn’ð0Þ � n~’ð0Þ Þ=s ¼

0T3=3s as

deR

dt¼ 3g02

2�2T3EY � BY þ 2�RLð�eR þ eL � 0Þ;

deL

dt¼ � 3g02

8�2T3EY �BY � �sph

2eL

þ �RLðeR � eL þ 0Þ;d�L

e

dt¼ � 3g02

8�2T3EY �BY � �sph

2eL

þ �RLðeR � eL þ 0Þ;d0

dt¼ �RLð�eR þ eL � 0Þ: (3.4)

The third equation for neutrinos is excess since �Le¼ eL .

Thus, we have three equations for three chemical potentialsinstead of the two ones in Ref. [13]. Note that we shouldhave d0=dt < 0 for our initial conditions eRðt0Þ> 0 and

eLðt0Þ ¼ 0ðt0Þ ¼ 0 resulting in the negative chemical

potential for the boson doublet ’T ¼ ð’ðþÞ; ’ð0ÞÞ,�0 < 0, as it should be.

Below we simplify the Abelian anomaly contribution�ðEY � BYÞ considering, as in Ref. [13], the simplestconfiguration of hypermagnetic field: CS wave Yx ¼YðtÞ sink0z, Yy ¼ YðtÞ cosk0z, Yz ¼ Y0 ¼ 0. Using the gen-

eralized Ohm’s law [8]

E Y ¼ �V � BY þ �Yr�BY � �YBY;

where �Y ¼ ð�condÞ�1 is the magnetic (hypermagnetic)diffusion coefficient, �Y is the hypermagnetic helicityparameter arising due to the polarization of electroweakplasma [8,13]

�Y ¼ g02ð�eR þ�eL=2Þ4�2�cond

; �cond ¼ 100 T; (3.5)

we get the pseudoscalar ðEY �BYÞ entering the Abeliananomaly as

ðEY �BYÞ ¼ �Yðr �BYÞ �BY � �YB2Y

¼ B2Y

100

�k0T� g02

4�2

�R þ L

2

��: (3.6)

Here we substituted ðr � BYÞ �BY ¼ k0B2YðtÞ for the CS

wave, where BYðtÞ ¼ k0YðtÞ is the hypermagnetic fieldamplitude.Using the notations yRðxÞ ¼ 104eRðxÞ, yLðxÞ ¼

104eLðxÞ, and y0ðxÞ ¼ 1040ðxÞ, as well as accounting

for Eq. (3.6), the system (3.4) can be rewritten in theform that is analogous to Eq. (3.4) in Ref. [13] (withoutcontribution of neutrinos, which is identical to that ofleft electrons)

dyRdx

¼�B0x

1=2 � A0

�yR þ yL

2

���Bð0ÞY

1020G

�2x3=2e’ðxÞ

� �0

ð1� xÞffiffiffix

p ðyR � yL þ y0Þ;

dyLdx

¼ � 1

4

�B0x

1=2 � A0

�yR þ yL

2

���Bð0ÞY

1020G

�2x3=2e’ðxÞ

� 5:6� 107Cffiffiffix

p yL � �0

ð1� xÞ2

ffiffiffix

p ðyL � yR � y0Þ;dy0dx

¼ �0ð1� xÞ2

ffiffiffix

p ðyL � yR � y0Þ: (3.7)

Here

B0 ¼ 25:6

�k0

10�7TEW

�; A0 ¼ 77:6 (3.8)

are constants chosen for hypermagnetic fields normalizedon 1020 G.

The function e’ðxÞ is given by the hypermagnetic fieldsquared

e’ðxÞ ¼�BYðxÞBð0ÞY

�2: (3.9)

We also substituted the hypermagnetic field BYðtÞ ¼k0YðtÞ found as the solution of the modified Faraday equa-tion [21,22] for the CS wave [23]

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BYðtÞ ¼ Bð0ÞY exp

�Z t

t0

½�Yðt0Þk0 � k20�Yðt0Þ�dt0�

¼ Bð0ÞY exp

�3:5

�k0

10�7TEW

�Z x

x0

�ðyR þ yL=2Þ�

� 0:1

�k0

10�7TEW

� ffiffiffiffix0

p �dx0

�: (3.10)

Note that we do not consider here a negative value of thewave number k0 < 0 that is allowed as well and could leadto the lepton number violation via Abelian anomaly propor-tional to the pseudoscalar (3.6), ðEY �BYÞ � k30Y

2ðtÞ< 0.This is because the case k0 < 0 corresponds to the decay ofthe hypermagnetic field (3.10) instead of a real instabilityevolving in MHD plasma for k0 > 0.

We choose initial conditions at x0 ¼ 10�4 or atT0 ¼ TRL when Higgs (inverse) decay becomes fasterthan the Hubble expansion �RL >H,

yRðx0Þ ¼ 10�6; yLðx0Þ ¼ y0ðx0Þ ¼ 0: (3.11)

Such conditions correspond to the right electronasymmetry eRðx0Þ ¼ 10�10 chosen at the level of baryon

asymmetry.

A. Conservation laws and BAU inhypermagnetic fields

One can see from kinetic Eq. (3.1) that in the absence ofhypercharge fields, the total lepton number is not con-served due to sphaleron transitions washing out the leftlepton number dLe=dt ¼ _LeR þ _LeL þ _L�L

e¼ ��sphLeL .

The baryogenesis arises through the leptogenesis due tothe conservation lawB=3� Le ¼ const, where B ¼ ðnB �n �BÞ=s. Accounting for Abelian anomalies in system (3.1),such baryogenesis is possible, _B � 0, since the hypermag-netic fields raise the lepton number and BAU as welldLe=dtjBY�0 > 0, dB=dtjBY�0 > 0. This growth proceeds

opposite to the competing sphaleron influence erasing LeL

and B (compare in Ref. [13] where we neglected sphalerontransitions).Three global charges are conserved (�i ¼ const),

B

3� Le ¼ �1;

B

3� L� ¼ �2;

B

3� L� ¼ �3;

(3.12)

as well as LeR ¼ �R well above TRL, T TRL. If the initial

BAU differs from zero Bðt0Þ � 0, and if we assume theabsence of lepton asymmetries for the second and thirdgenerations all the way down to TEW, L� ¼ L� ¼ 0, we

find that the relation �2 ¼ �3 ¼ Bðx0Þ=3 is valid onlyfor the initial time. From the first conservation law inEq. (3.12) one finds the change of BAU BðtÞ at tempera-tures T < TRL. This change obeys the relations

BðtÞ3

� LeðtÞ ¼ Bðt0Þ3

� LeRðt0Þ ¼ �2;3 � �R ¼ �1:

If, for simplicity, we assume the zero initial BAU Bðt0Þ ¼ 0or �2;3 ¼ 0, then finally we get the conservation law

BðtÞ=3� LeðtÞ ¼ �LeRðt0Þ.Thus, in the present scenario, BAU sits in hypercharge

fields and decreases due to the sphaleron processes, asfollows from the sum of kinetic Eq. (3.1):

BðtÞ ¼ 3Z t

t0

�dLeRðt0Þ

dt0þ dLeLðt0Þ

dt0þ dL�eL

ðt0Þdt0

�dt0

¼�3g02

8�2

�Z t

t0

ðEY �BYÞ dt0

s� 3

Z t

t0

�sphLeLdt0: (3.13)

Using the first equation in the system (3.7), where thehypermagnetic term comes from the Abelian anomaly�ðEY �BYÞ, one obtains from Eq. (3.13) the baryon asym-metry in the following form:

10−3

10−2

10−1

10010

−14

10−12

10−10

10−8

10−6

10−4

t/tEW

B(t

)

B0 = 25.6

B0 = 2.1 × 10−2

10−4

10−3

10−2

10−1

100

−0.5

0

0.5

x 10−10

t/tEW

B(t

)

FIG. 1. The baryon asymmetry BðtÞ versus t=tEW for Bð0ÞY ¼ 1019 G. (a) The baryon asymmetry for B0 ¼ 2:1� 10�2 (solid line) and

B0 ¼ 25:6 (dashed line). (b) The baryon asymmetry for B0 ¼ 2� 10�3.

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BðxÞ ¼ 2:14� 10�6Z x

x0

dx0�dyRðx0Þdx0

þ �0

ð1� x0Þffiffiffiffix0

p ½yRðx0Þ � yLðx0Þ þ y0ðx0Þ��

� 128CZ x

x0

dx0ffiffiffiffix0

p yLðx0Þ: (3.14)

The baryon asymmetry (3.14) for different values of theparameter B0 ¼ 25:6� ðk0=10�7TEWÞ or for different CSwave numbers k0 is shown in Fig. 1. Notice that for verysmall k0 � kmax ¼ 10�7TEW, the role of the hypermag-netic field, which feeds BAU growth, becomes negligiblesince BY � k0. As a result, sphaleron transitions wash outBAU, or they diminish the Abelian anomaly leptogenesiseffect in such a way that BAU can be even negative at theEWPT time BðtEWÞ< 0 [see curve BðtÞ in Fig. 1(b) plottedfor the parameter B0 ¼ 2� 10�3 that corresponds tok0 ¼ 7:8� 10�5kmax].

IV. CHEMICAL EQUILIBRIUM WITH ANDWITHOUT HYPERMAGNETIC FIELDS

Neglecting the hypermagnetic field contribution andsphaleron transitions and using the initial condition (3.11),we easily find the solutions of kinetic equations (3.7)

yRðxÞ ¼ yRðx0Þ2

½1þ e�ðxÞ�;

yLðxÞ ¼ yRðx0Þ4

½1� e�ðxÞ�;

y0ðxÞ ¼ � yRðx0ÞÞ4

½1� e�ðxÞ�;

�ðxÞ ¼ �4�0

�ðx1=2 � x1=20 Þ � ðx3=2 � x3=20 Þ

3

�: (4.1)

Note that y0 < 0, as it should be for a boson chemicalpotential. Obviously the chemical equilibrium (2.1) is set-tled soon due to huge negative � ’ �4�0 ¼ �484,

yR � yL þ y0 ¼ yRðx0Þe�ðxÞ ! 0 (4.2)

that happens somewhere at x > xeq ’ 10�2 at the tempera-

ture T ¼ TEW=ffiffiffiffiffiffiffixeq

p ’ 1 TeV before EWPT, T > TEW (see

right panel of Fig. 2).The numerical solution of the system (3.7) accounting

for sphalerons and in the presence of BY � 0 for the par-ticular case of CS wave configuration is shown on the leftpanel in the same Fig. 2. We see that opposite to the case inEq. (4.2), in the presence of hypermagnetic fields andaccounting for sphaleron transitions, the chemical equilib-rium between leptons and Higgs’s (2.1) never exists: thesum yR � yL þ y0 even grows in the symmetric phasewhen t ! tEW. The shorter CS wavelength (e.g., for B0 ¼25:6 if we use the maximumwave number k0 ¼ 10�7TEW),the larger values ðyR � yL þ y0Þ evolve up to tEW.All curves in Fig. 2 start from the same initial condition

yRðx0Þ � yLðx0Þ þ y0ðx0Þ ¼ yRðx0Þ ¼ 10�6 that corre-sponds to the initial right electron asymmetry eR ¼ 10�10

close to the BAU value we expect at the EWPT time x ¼ 1.Thus, the violation of lepton numbers in external fieldsthrough Abelian anomalies and sphaleron transitions leadsto a violation of the chemical equilibrium (2.1) existing inprimordial plasma when perturbative reactions (decays,scattering, etc.) are taken into account only.

V. CHIRALANOMALYPARAMETER ðyR � yLÞ � 0IN ELECTROWEAK PLASMA BEFORE EWPT

The temporal evolution of the chiral anomaly parameteryR � yL ¼ 104ð��=TÞ is shown in Fig. 3. There is a strongdependence on the scale � ¼ k�1

0 for the chosen configu-

ration of the hypermagnetic field: the shorter the CS

10−4

10−3

10−2

10−1

100

10−8

10−6

10−4

10−2

100

t/tEW

y R −

yL +

y0

B0 = 25.6

B0 = 2.1 × 10−2

10−4

10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1x 10

−6

t/tEW

y R −

yL +

y0

FIG. 2. The function yR � yL þ y0 versus t=tEW for lepton asymmetries ya ¼ 104ð�a=TÞ. (a) The numerical solution of the system

(3.7) for Bð0ÞY ¼ 1019 G. The solid line corresponds to B0 ¼ 2:1� 10�2 and the dashed line to B0 ¼ 25:6. (b) The analytic expression

for yR � yL þ y0 given by Eq. (4.2).

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wavelength, the bigger the chiral anomaly parameter willbe. For the maximum acceptable wave number k0 ¼kmax ¼ 10�7TEW (B0 ¼ 25:6), the chiral anomaly parame-ter ��=T is close to 5� 10�5 assumed in Fig. 1 ofRef. [11] as a maximum initial value of the chiralanomaly parameter just after EWPT. However, for lon-ger CS wavelengths [see dashed line in Fig. 3(b) plottedfor k0 ¼ 7:8� 10�5kmax], such an initial value willoccur at a negligible level (��=T � 10�8) which cancrucially change the results of Ref. [11]. Note that forthe strongest CS field amplitude BY ¼ kmaxYðtÞ, or in thecase of the maximum Abelian anomaly leptogenesiseffect, the left lepton asymmetry yL grows from thebeginning due to Higgs (inverse) decays and thenchanges sign yL < 0, which is allowed for fermions,cf., Fig. 3(c).

Let us explain qualitatively the growth of the chiralanomaly shown in Figs. 3(a) and 3(b). One can simplifythe kinetic equations for eR and eL in the system (3.4)

decoupling them. For this purpose we neglect the Higgs

boson asymmetry 0 ¼ 0. We also omit the asymmetry ofleft leptons eL ¼ 0 in the first line of Eq. (3.4), and the

right electron in the second line of Eq. (3.4) eR ¼ 0. For

example, from the first equation in the system (3.4), sub-stituting the pseudoscalar value ðEY �BYÞ for the CS wavefrom Eq. (3.6), one gets the simple differential equation forthe right electron asymmetry yR ¼ 104eR ,

dyRdt

þ ð�þ �BÞyR ¼ Q; (5.1)

where � ¼ 2�RL is the chirality flip rate �B¼6ðg02=4�2Þ2B2

Y=100T3, and Q¼6�104�g02B2

Yk0=400�2T4

come from the second (helicity) term in Eq. (3.6) andfrom the first (diffusion) term in the same Eq. (3.6). Thesolution of Eq. (5.1) obtained for strong and constanthypermagnetic fields �B � and B2

Y const

yRðtÞ¼�yRðt0Þ� Q

�þ�B

�e�ð�þ�BÞðt�t0Þ þ Q

�þ�B

(5.2)

10−4

10−3

10−2

10−1

10010

−8

10−6

10−4

10−2

100

t/tEW

y R

B0 = 25.6

B0 = 2 × 10−3

10−4

10−3

10−2

10−1

10010

−8

10−6

10−4

10−2

100

t/tEW

y R −

yL

B0 = 2 × 10−3

B0 = 25.6

10−4

10−3

10−2

10−1

100−6

−4

−2

0

2

4x 10

−10

t/tEW

y L

10−4

10−3

10−2

10−1

10010

−16

10−14

10−12

10−10

t/tEW

y L

B0 = 2 × 10−2

B0 = 2 × 10−3

FIG. 3. The normalized chemical potentials yR;L and the chiral anomaly parameter yR � yL versus t=tEW for Bð0ÞY ¼ 1019 G. (a) The

normalized chemical potential yR for B0 ¼ 25:6 (solid line) and B0 ¼ 2� 10�3 (dashed line). The dash-dotted line corresponds tothe asymptotic value of yR ¼ 0:32 calculated analytically in Eq. (5.3). (b) The chiral anomaly parameter yR � yL for B0 ¼ 25:6 (solidline) and B0 ¼ 2� 10�3 (dashed line). The dash-dotted line corresponds to the asymptotic value of yR � yL equal to 0.34. (c) Thenormalized chemical potential yL for B0 ¼ 25:6. (d) The normalized chemical potential yL for B0 ¼ 2� 10�2 (solid line) andB0 ¼ 2� 10�3 (dashed line).

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gives the asymptotic growth of yRðtÞ up to yRðtEWÞ [herefor its initial value eRðt0Þ ¼ 0],

yRðtEWÞ ¼ Q

�B½1� e��BðtEW�t0Þ� ¼ Q

�B

104�4�2

g02

��k0TEW

�¼ 0:32: (5.3)

Here we put �BtEW 1 for strong fields, as well as sub-stituted g02 ¼ e2=cos2�W ¼ 0:12 and k0=TEW ¼ 10�7 forthe case of B0 ¼ 25:6.

The bigger the wave number k0 or the stronger thehypermagnetic field BY ¼ k0YðtÞ, the weaker a sphaleroninfluence on lepton (baryon) asymmetry is. In a wideregion of wave numbers, the left lepton asymmetry yLremains negligible compared with the right electron yR,jyLj � yR. This is due to the initial conditions (3.11) in ourscenario resulting in the right electron asymmetry yR andthe chiral anomaly yR � yL, close to what was calculated inRef. [13]. However, for a long CS wave k0 � kmax, a smallpositive value yL > 0 evolving down to TEW [cf., Fig. 3(d)]is sufficient to allow sphaleron transitions to wash outBAU.

VI. DISCUSSION

In the present work we have studied how the chiralasymmetry �� ¼ �eR ��eL � 0 arises before EWPT in

the scenario where, firstly, the initial right electron asym-metry eRðt0Þ ¼ �eR=T0 ’ 10�10 provides the generation

of the BAU. Secondly, chirality flip reactions enter theequilibrium at the temperaturesT � T0 ¼ TRL � 10 TeV.Then, at t > t0 the violation of lepton numbers leads to anonzero left electron asymmetry eLðtÞ ¼ �eL=T � 0.

Note that we consider the lepton numbers violation dueto Abelian anomalies and because of the presence of theSUð2ÞW anomaly. The generated left electron asymmetryresults in the change of the primordial right electron asym-metry eRðtÞ and influences the BAU evolution.

For large scales of hypermagnetic field (for a smallervalue k0), sphaleron transitions are more efficient to eraseBAU since the amplitude of a competitive mean hyper-charge field decreases [when amplitude BYðtÞ ¼ k0YðtÞgoes down], and therefore due to Abelian anomalies, anenhancement of the lepton number ceases. Of course, the

bigger the seed hypermagnetic field Bð0ÞY , the bigger the

lepton asymmetries and the baryon one.While our choice of CS wave as the simplest hyper-

magnetic field configuration significantly simplifies theanalysis of the lepton (baryon) asymmetry evolution, weshould comment on some disadvantages that are appropri-ate to such hypercharge field.

First, a scale of hypermagnetic field for the chosenCS configuration is rather small. In order to get the BAUclose to the observable value BobsðtEWÞ � 10�10, we fit theCS wave number of the order of k0 ’ 10�3kmax, where

kmax ¼ 10�7TEW corresponds to the maximum wave num-ber for the hypermagnetic field surviving Ohmic losses atEWPT time. The macroscopic long-range hyperchargefield has the scale � ¼ k�1

0 , which is much bigger than

the mean distance between particles in plasma T�1, and,on the other hand, is much less than the horizon size� � lH. Here the horizon size, e.g., at the EWPT time

lH¼ðMPl=1:66ffiffiffiffiffig�

p ÞT�2EW¼1016=TEW, is much bigger than

the scales k�10 ¼107=TEW and k�1

0 ’ð1010�1011Þ=TEW

applied in our plots. Because of the arbitrariness of thez-axis direction chosen for the CS wave, such macroscopicfields are rather small-scale (random) fields and there is noanisotropy of medium. Thus, in order to get a necessaryscale on the onset of galactic magnetic fields, we shouldrely on an inverse cascade evolving after the EWPT for theMaxwellian fields [24] that originated from the hyper-charge ones in our causal scenario. Note that such inversecascade needs a significant amount of magnetic helicity inorder to operate efficiently. For the CS wave, the helicitydensity hY � k30

RdtY2ðtÞ decreases faster with lowering of

the wave number k0 than the energy density �ðBÞY ¼

B2Y=2 ¼ k20Y

2ðtÞ=2. In other words, the lower k0, the fur-

ther away the CS wave configuration is from the maximum

helical field obeying the relation k0hmaxY ¼ 2�ðBÞ

Y . Thiscircumstance should be taken into account for a morerealistic continuous hypercharge field spectrum when theconservation of the global helicity governs a spread ofhelicities over different scales.In addition, our choice of CS wave does not appear to

be realistic for transition of hypercharge field to theMaxwellian one during EWPT. It was shown in Ref. [25]that being provided by the helicity conservation, only ahelical field such as that given by the 3D configuration ofY� can penetrate the boundary wall separating symmetric

and broken phases during the EWPT time T � TEW. TheCS wave does not penetrate such surface of a bubble of anew phase even for a strong hypermagnetic field ampli-tude, which, in turn, provides possibility of EWPT of thefirst order for the present bounds on Higgs masses [20].The evolution of the corresponding hypermagnetic

helicity for an arbitrary configuration of hypermagneticfields before EWPT HY ¼ R

d3xðY � BYÞ has been

recently studied in Ref. [26] neglecting hypermagneticdiffusion. The following magnetic helicity evolution inhot plasma at temperatures T � TEW was analyzed inthe same approximation in Ref. [27] relying on the modelproposed in Ref. [10] for the magnetic helicity parameter��GF. The new mechanism for the �-helicity parametersuggested in Refs. [11,12] may improve such estimates forprimordial Maxwellian fields.To resume, we estimated the chiral asymmetry �� ¼

�eR ��eL arising just at EWPT T ’ TEW using the sim-

plest configuration of the hypercharge field—the Chern-Simons wave—and taking into account both Higgs decaysand inverse Higgs decays and sphaleron transitions as well.

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The evolution of lepton and Higgs asymmetries werestudied at temperatures TEW � T � TRL. The baryon andlepton asymmetries crucially depend on the CS wavelengthk�10 . The doubts whether it is possible to protect the baryon

asymmetry of the Universe in the symmetric phase bytemporarily storing BAU in the asymmetry of the eRspecies are dispelled in the case of strong hypermagneticfields. The washing out of BAU by the sphaleron transi-tions due to the involvement of left particles atT < TRL through Higgs (inverse) decays is not dangerousin a wide region of CS wave numbers. A strong seed

hypermagnetic field Bð0ÞY is needed to support all such

issues while dynamo amplification turns out rather negli-gible for the CS wave. The amplification of hypermagneticfields by mechanisms beyond the SM, assuming, e.g., anew pseudoscalar field coupled to hypercharge topologicalnumber density [28], is not considered here. Within theframework of the SM model, the 3D configuration of ahypermagnetic field seems to be much more productive forproblems under consideration that remain as the challengefor us in the future.

ACKNOWLEDGMENTS

We acknowledge Jose Valle, Dmitry Sokoloff, andValery Rubakov for fruitful discussions. M.D. is thankfulto FAPESP (Brazil) for a grant.

APPENDIX A: THE ORIGIN OF THE CHIRALANOMALY FOR A MAXWELLIAN FIELD

AND THE CS ANOMALY FOR AHYPERMAGNETIC FIELD

The chiral anomaly parameter �� ¼ ð�eR ��eLÞ � 0

leads to an additional contribution to the current in theMaxwell equation

� @E

@tþr� B ¼ �condEþ �em

���B; (A1)

where the last pseudovector current j ¼ ð�em��=�ÞBis coming, e.g., from the energy balance under chiralityflip for massless particles [29,30]

Zd3xðj �EÞ ¼ �em��

�

Zd3xðE �BÞ: (A2)

Note that �em and �cond were defined in Sec. I.Using the Bianchi identity @tB ¼ �r� E, one finds

from Eq. (A1), neglecting in the MHD approach thedisplacement current @tE ¼ 0, the modified Faradayequation [11]

@B

@t¼

��em��

��cond

�r� Bþ 1

�cond

r2B; (A3)

which governs the evolution of the magnetic field afterEWPT at temperatures 10 MeV< T < TEW.

Let us comment on the energy balance Eq. (A2). Notethat in the rhs of Eq. (A2), the Adler anomaly for the right-handed electrons (positrons)

@j�R

@x�¼ þ e2

16�2F��

~F�� ¼ �em

�ðE �BÞ

and for the left-handed ones [31]

@j�L@x�

¼ � e2

16�2F��

~F�� ¼ ��em

�ðE � BÞ

being combined for the pseudovector j�R � j�L ¼h �� � 5�i in uniform medium as

d

dtðnR � nLÞ ¼

�2�em

�

�ðE � BÞ (A4)

defines the rate of the chirality flip per unit timeper volume and the corresponding energy cost ð��=2Þ�dðnR�nLÞ=dt¼ðj�EÞ for such effect. Hence, multiplyingby ��=2 and integrating Eq. (A4) over volume, onederives the energy balance (A2). While separating E inboth sides in the integrand of (A2), one obtains the pseu-dovector current j in the Maxwell Eq. (A1).The CS anomaly term in the SM Lagrangian for the

hypercharge field Y� also leads to the pseudovector

contribution in the Maxwell equation [13] (here attemperatures TEW < T < TRL)

�@EY

@tþr�BY ¼ �condEY þ �0

�

��eR þ

�eL

2

�BY;

�0 ¼ g02

4�; (A5)

where the last term has polarization origin in the presenceof hypermagnetic fields in primordial plasma [8]. Noticethat the Adler anomaly used in Eq. (A2) corresponds to thedifference of mean densities of the right- and left-handedfermions that is pseudoscalar [32] nR � nL ¼ h�þ

R�Ri �h�þ

L �Li ¼ h�þ 5�i with �R;L ¼ ð1 5Þ�=2, while

the CS term in Eq. (A5) is given by the mean spinh�þ� 5�i ¼ h�þ��i ¼ M=�B, where pseudovectorM� BY is the magnetization of medium in a hypermag-netic field BY.Let us stress the similarity of both chiral magnetic

effects leading to anomalous terms in Eqs. (A1) and (A5).They are caused by a polarization mechanism provided bythe main Landau level contribution to the additional cur-rent in the Maxwell equation (compare in Ref. [33]).

APPENDIX B: LEPTON KINETICS WITHHIGGS BOSONS IN THE ABSENCE OF

HYPERCHARGE FIELDS

In this appendix we briefly discuss the lepton kinetics inthe presence of Higgs bosons without hypermagnetic fieldsand Abelian anomaly, and omitting sphaleron terms.

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In particular, we explain in detail the derivation ofEqs. (3.1) and (3.2).

1. Right electrons

The reactions contributing to the right electrons dynam-

ics are (i) the inverse decays eR �eL ! ~’ð0Þ and eR ��Le !

’ð�Þ and (ii) the direct decays ~’ð0Þ ! eR �eL and ’ð�Þ !eR ��

Le . We remind that when particles annihilate, a ‘‘minus’’

sign appears in front of the �RL term. Taking into accountthat n~’ð0Þ ¼ n’ð�Þ , as well as the equivalence n �eL ¼ n ��L

e,

one gets a factor of 2 in the kinetic equation accounting forthe two channels:

d

dt

�neRs

�¼ 2�RL½�neR � n �eL�=sþ 2�D

n~’ð0Þ

s: (B1)

Here, the Bose distribution for the Higgs doublet ’T ¼ð’ðþÞ; ’ð0ÞÞ is given by the chemical potential �0 ¼ �þ ¼��� ¼ �� ~ð0Þ or �0 ¼ �þ for ’, while for the antipar-

ticle ~’T ¼ ð’ð�Þ; ~’ð0ÞÞ one gets �� ¼ � ~ð0Þ ¼ ��0. Note

that equilibrium, e.g., in the reaction eR �eL $ ~’ð0Þ, wouldcorrespond to the right relation of chemical potentialsneglecting hypermagnetic fields as given by Eq. (2.1),�eR ��eL ¼ ��0.

2. Right positrons

The reactions contributing to the right positrons dynam-

ics are (i) the inverse decays �eReL ! ’ð0Þ and �eR�Le !

’ðþÞ, as well as (ii) decays ’ð0Þ ! �eReL and ’ðþÞ !�eR�

Le . Analogously to the right electrons case, we take

into account that n’ð0Þ ¼ n’ðþÞ and neL ¼ n�Le. Finally, we

obtain the following kinetic equation accounting for thetwo channels:

d

dt

�n �eR

s

�¼ 2�RL½�n �eR � neL�=sþ 2�D

n’ð0Þ

s: (B2)

Subtracting Eq. (B2) from Eq. (B1) and taking into accountthat LeR ¼ ½neR � n �eR�=s and LeL ¼ ½neL � n �eL�=s, one

gets the equation similar to that derived in Ref. [16], andin our previous work [13] if we omit the Abelian anomaly[see the first line in Eq. 3.1 in Sec. III],

dLeR

dt¼2�RLðLeL �LeRÞþ2�D½n �’ð0Þ �n’ð0Þ �=s: (B3)

In equilibrium, dLeR=dt ¼ 0 accounting for �RL ¼ 2�D,

we get the correct relation (2.1).

3. Left electrons

We should take into account (i) the inverse decay

�eReL ! ’ð0Þ and (ii) the decay ’ð0Þ ! �eReL, which give

d

dt

�neLs

�¼ �RL½�n �eR � neL�=sþ �D

n’ð0Þ

s: (B4)

4. Left positrons

In this case (i) the inverse decay �eLeR ! ~’ð0Þ and (ii) thedecay ~’ð0Þ ! �eLeR give the following contributions:

d

dt

�n �eL

s

�¼ �RLð�neR � n �eLÞ=sþ �D

n~’ð0Þ

s: (B5)

Subtracting Eq. (B5) from Eq. (B4) one gets

dLeL

dt¼ �RLðLeR � LeLÞ þ �D½n’ð0Þ � n~’ð0Þ �=s: (B6)

In the equilibrium dLeL=dt ¼ 0 accounting for �RL ¼2�D, we get the correct relation for chemical potentials�eR ��eL þ�0 ¼ 0.

Let us derive the Higgs boson kinetic equations. In

Eq. (B4), the boson ’ð0Þ decays into the pair �eReL.Hence, the boson vanishes, which increases the populationof eL and �eR (in kinetics of left electron eL). In the kinetic

equation for ’ð0Þ itself, such term enters with the oppositesign as ��Dn’ð0Þ (boson disappears). Analogously, the

inverse decay term should have the different sign in bosonkinetics: it becomesþ�RLðneR þ neLÞ=s since the pair �eReLannihilates into’ð0Þ increasing the population of the neutralbosons ’ð0Þ. Therefore, one obtains from Eq. (B4),

d

dt

�n’ð0Þ

s

�¼ �RL½n �eR þ neL�=sþ �D

�� n’ð0Þ

s

�: (B7)

Analogously from Eq. (B5), changing sign on the right-hand side, one obtains the kinetic equation for the Higgs

antiparticle ~’ð0Þ:

d

dt

�n~’ð0Þ

s

�¼ �RLðneR þ n �eLÞ=sþ �D

�� n~’ð0Þ

s

�: (B8)

Subtracting Eq. (B8) from Eq. (B7) and accounting for�D ¼ �RL=2, we derive the kinetic equation for the Higgsboson asymmetry [see Eq. (3.2) in Sec. III]:

d

dt½ðn’ð0Þ � n~’ð0Þ Þ=s� ¼ �RL

(LeL � LeR �

½n’ð0Þ � n~’ð0Þ �2s

):

(B9)

5. Left neutrinos

For left neutrinos, we account for (i) the inverse decay

�eR�Le ! ’ðþÞ and (ii) the decay ’ðþÞ ! �eR�

Le , which give

d

dt

�n�Les

�¼ �RL½�n �eR � n�L

e�=sþ �D

n’ðþÞ

s: (B10)

6. Left antineutrinos

In this case (i) the inverse decay eR ��Le ! ’ð�Þ

and (ii) the decay ’ð�Þ ! eR ��Le contribute to the kinetic

equation as

MAXIM DVORNIKOVAND VICTOR B. SEMIKOZ PHYSICAL REVIEW D 87, 025023 (2013)

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d

dt

�n ��L

e

s

�¼ �RL½�neR � n ��L

e�=sþ �D

n’ð�Þ

s: (B11)

Subtracting Eq. (B11) from Eq. (B10), one gets

dL�Le

dt¼ �RLðLeR � LeLÞ þ �D½n’ðþÞ � n’ð�Þ �=s; (B12)

where we took into account that L�Le¼ LeL or n�L

e¼ neL .

Of course, n’ðþÞ ¼ n’ð0Þ with the chemical potential �0 in

Bose distribution for the doublet ’T ¼ ð’ðþÞ; ’ð0ÞÞ, andn’ð�Þ ¼ n~’ð0Þ with the chemical potential ��0 for the c.c.

doublet ~’T ¼ ð’ð�Þ; ~’ð0ÞÞ.For charged Higgs, which are described by Eqs. (B10)–

(B12), using arguments like in the derivation of Eq. (B9)we obtain from Eqs. (B10) and (B11) the kinetic equationwhich is identical to Eq. (B9), since n’ðþÞ ¼ n’ð0Þ and

n’ð�Þ ¼ n~’ð0Þ ,

d

dt½ðn’ðþÞ � n’ð�Þ Þ=s� ¼ �RL

(LeL � LeR �

½n’ðþÞ � n’ð�Þ �2s

):

(B13)

APPENDIX C: SUð2ÞW ANOMALYAND LEFTFERMION NUMBER VIOLATION

Let us use Eq. (12-174a) in Ref. [34] written for thepseudovector current of the one generation of masslessfermions j�5 ¼ j�R � j�L ¼ �c � 5c ,

@�j�5 ¼ @�½j�R � j

�L � ¼ � g2

16�2F��a ~F��a: (C1)

Here, j�R¼ �c �ð1þ 5Þc =2 and j�L ¼ �c �ð1� 5Þc =2are the right and left fermion currents, correspondingly[35]. Adding the equality (C1) with the anomaly for lep-tons of the first generation given by Eq. (11.12) in Ref. [15](see also Ref. [36]),

@�j�Le

¼ @�½j�R þ j�L � ¼

g2

16�2F��a ~F��a; (C2)

one gets the well-known issue @�j�R ¼ 0 that guarantees

the conservation of the right electron current in the absence

of hypermagnetic fields. On the other hand, subtractingEq. (C1) from Eq. (C2), we get the violation of the leftlepton current in the same non-Abelian fields

@�j�L ¼ @�½j�eL þ j�

�Le� ¼ g2

16�2F��a ~F��a: (C3)

Notice that here neutrino and electron currents are equiva-lent j

�eL ¼ j

�

�Le¼ j

�L =2, as seen from the following repre-

sentation for the left field doublet LT ¼ ð�Le ; eLÞ,

L ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p ab

� �c L;

where we may put a ¼ �b ¼ 1 for the isospin column.Then, using the standard field operator in the Schrodingerrepresentation

c LðxÞ ¼ 1

ð2�Þ3=2Xr

Z d3pffiffiffiffiffiffiffiffi2"p

p ½brðpÞurLðpÞeipx

þ dyr ðpÞvrLðpÞe�ipx�;

one finds the current asymmetry j�L ðx; tÞ ¼

Tr½�ðtÞ �c L � c L� ¼ j�lL � j��lL

, where �ðtÞ is the nonequi-librium statistical operator obeying the Liouville equation.Here the currents

j�lL;�lL

ðx; tÞ ¼Z d3p

ð2�Þ3p�

"pfðlL;�lLÞðp;x; tÞ

are given by the Wigner distribution functions

fðlL;�lLÞðp;x; tÞ ¼ Tr

�Xk

eikxfðlL;�lLÞ

pþk=2;r0;p�k=2;rðtÞ�;

which, in turn, are given by the distribution functions in the

momentumrepresentationfðlLÞp0r0;prðtÞ ¼ Tr½�ðtÞbyr ðpÞbr0 ðp0Þ�

for particles and fð�lLÞp0r0;prðtÞ ¼ Tr½�ðtÞdyr ðpÞdr0 ðp0Þ� for

antiparticles.The violation of the left lepton number LeL in non-

Abelian fields due to the SUð2ÞW anomaly (C3) proceedswith the sphaleron transition probability �sph as we used in

kinetic equations (3.1).

[1] R.M. Kulsrud and E.G. Zweibel, Rep. Prog. Phys. 71,046901 (2008).

[2] P. P. Kronberg, Rep. Prog. Phys. 57, 325 (1994).[3] D. Grasso and H. R. Rubinstein, Phys. Rep. 348, 163

(2001).[4] A. Neronov and I. Vovk, Science 328, 73 (2010).[5] A. Neronov and D.V. Semikoz, Phys. Rev. D 80, 123012

(2009).

[6] M. Giovannini and M. E. Shaposhnikov, Phys. Rev. D 57,2186 (1998).

[7] A. N. Redlich and L. C. R. Wijewardhana, Phys. Rev. Lett.54, 970 (1985).

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LEPTON ASYMMETRY GROWTH IN THE SYMMETRIC . . . PHYSICAL REVIEW D 87, 025023 (2013)

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[10] V. B. Semikoz and D.D. Sokoloff, Phys. Rev. Lett. 92,131301 (2004).

[11] A. Boyarsky, J. Frohlich, and O. Ruchayskiy, Phys. Rev.Lett. 108, 031301 (2012).

[12] A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov, Phys.Rev. Lett. 109, 111602 (2012).

[13] M. Dvornikov and V.B. Semikoz, J. Cosmol. Astropart.Phys. 02 (2012) 040; 08 (2012) 01(E).

[14] J. A. Harvey and M. S. Turner, Phys. Rev. D 42, 3344(1990).

[15] D. S. Gorbunov and V.A. Rubakov, Introduction to theTheory of the Early Universe: Hot Big Bang Theory(World Scientific, Singapore, 2011), p. 251.

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[17] We use the sign for the Abelian anomaly opposite to thatin Ref. [6] relying on the definition of right states �R ¼ð1þ 5Þ�=2 in Ref. [18].

[18] A. Zee, Quantum Field Theory in a Nutshell (PrincetonUniversity, Princeton, NJ, 2010), p. 270.

[19] Note that we took into account the expansion of theUniverse, since for any kind of asymmetries normalizedon the entropy including lepton numbers, the followingrelations hold:

sd

dt

�nb � n �b

s

�¼ @ðnb � n �bÞ

@tþ 3Hðnb � n �bÞ;

H ¼ _a

a¼ � _T

T; and

_s

s¼ �3H;

where nb is an arbitrary density.[20] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716, 1

(2012); S. Chatrchyan et al. (CMS Collaboration), Phys.Lett. B 716, 30 (2012).

[21] V. B. Semikoz, D. D. Sokoloff, and J.W. F. Valle, Phys.Rev. D 80, 083510 (2009).

[22] V. B. Semikoz and J.W. F. Valle, J. High Energy Phys. 03(2008) 067.

[23] In numerical estimates, we substitute either the parameterk0=ð10�7TEWÞ ¼ 1 that is the upper limit for the CS wave

number k0 � 10�7TEW to avoid Ohmic dissipation of

hypermagnetic field, or k0=ð10�7TEWÞ ’ ð10�3 � 10�4Þto get observable baryon asymmetry B ¼ 0:87� 10�10

at the EWPT time x ¼ 1. Dynamo amplification is negli-

gible in both cases.[24] A. Brandenburg, K. Enqvist, and P. Olesen, Phys. Rev. D

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(1983).[30] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys.

Rev. D 78, 074033 (2008).[31] Of course, the sum of the Adler anomalies leads to the

conservation of the lepton number in QED, @�j� ¼

@�ðj�R þ j�L Þ ¼ 0.

[32] With respect to the Lorentz transformation, this differenceis the time component of the four-pseudovector averaged

over the Fermi distribution h �c 0 5c i, while the true

pseudoscalar should have the form �c 5c . If we average

the pseudoscalar term over Fermi distributions in a uni-

form medium, e.g., in uniform plasma with magnetic field

neglecting plasma dispersion, it vanishes, h �c 5c i ¼ 0.While in an inhomogeneous medium, one gets the nonzero

result at the perturbative level (e.g., for magnons in a

ferromagnet) h �c 5c it ¼ �ir �Mðx; tÞ=e � 0, where

Mðx; tÞ is the magnetization.[33] A. Vilenkin, Phys. Rev. D 22, 3080 (1980).[34] C. Itzykson and J.-B. Zuber, Quantum Field Theory

(McGraw-Hill, New York, 1980), p. 606.[35] We use the same notations that coincide with

Refs. [18,34].[36] Note that anomaly (C2) does not depend on 5 at all.

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