Contents
###### Abstract

As a result of the Standard Model chiral anomalies, baryon number is violated in the early universe in the presence of a hypermagnetic field with varying helicity. We investigate whether the matter / anti-matter asymmetry of the universe can be created from the decaying helicity of a primordial (hyper)magnetic field before and after the electroweak phase transition. In this model, baryogenesis occurs without -violation, since the asymmetry generated by the hypermagnetic field counteracts the washout by electroweak sphalerons. At the electroweak crossover, the hypermagnetic field becomes an electromagnetic field, which does not source . Although the sphalerons remain in equilibrium for a time, washout is avoided since the decaying magnetic helicity sources chirality. The relic baryon asymmetry is fixed when the electroweak sphaleron freezes out. Under reasonable assumptions, a baryon asymmetry of can be generated from a maximally helical, right-handed (hyper)magnetic field that has a field strength of and coherence length of today. Relaxing an assumption that relates to , the model predicts , which could potentially explain the observed baryon asymmetry of the universe.

Baryogenesis from

Decaying Magnetic Helicity

School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA.

Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA

## 1 Introduction

Among the various problems facing modern cosmology, the origin of the matter / anti-matter asymmetry of the universe is unique in that no direct experimental input is forthcoming. The baryon asymmetry has already been measured – approximately one baryon for every photons – and unlike the problems of dark matter, dark energy, or the primordial density perturbations, there are no dedicated experimental efforts underway to provide additional empirical knowledge about the baryon asymmetry. With this consideration in mind, it is appealing to study models in which the baryon asymmetry is created along with some other cosmological relic, such as the dark matter, a network of topological defects, or the primordial magnetic field. The hope is that future measurements of the secondary relic could provide insight into the origin of the baryon asymmetry.

In fact the prospects are very favorable for future observational probes of primordial magnetic fields (see Ref. [1] for a review). A primordial magnetic field generated in the early universe could persist today in the voids between galaxies and clusters, where it would be largely unprocessed by structure formation. In recent years, TeV blazars have emerged as a potentially powerful tool for measuring the intergalactic magnetic field (IGMF). A deficit of secondary GeV gamma rays observed in blazar spectra points to the presence of an IGMF with field strength [2, 3]. Searches for magnetically broadened cascade halos [4, 5, 6] and parity-odd correlators in diffuse gamma ray data [7, 8] have also suggested the existence of an IGMF, which could be of primordial origin [9]. Expecting that future observations will provide additional evidence for an IGMF, we are motivated to study the implications of a magnetic field in the early universe [10].

From a theory perspective, there is a robust connection between baryon number violation and gauge fields through the Standard Model chiral anomalies [11, 12, 13]. Since the and gauge fields have chiral interactions with the Standard Model fermions, quantum effects lead to the violation of baryon and lepton numbers. This violation is expressed by the current conservation equations,

 (1.1)

which can be integrated over a finite time interval to give

 ΔQB=ΔQL=NgΔN\sc cs−Ngg′216π2ΔHY . (1.2)

Thus changes in baryon and lepton numbers, and , are induced by changes in Chern-Simons number and hypermagnetic helicity . For a coherent magnetic field, helicity , quantifies the excess of power in either the left- or right-circular polarization mode.

Many studies have investigated the connection between a primordial magnetic field (PMF) and the baryon asymmetry of the universe (BAU). Broadly speaking, the literature falls into three categories: PMF-from-BAU [14, 15, 16], BAU-from-PMF [17, 18, 19, 20, 21, 22, 23, 24], and co-evolution [25, 26, 27, 28, 29, 30, 31, 32]. As emphasized in Ref. [16], it is generally difficult to produce a very strong primordial magnetic field starting from a small baryon asymmetry at the level of . On the other hand, Ref. [24] recently pointed out that it is generally easy to over-predict the baryon asymmetry of the Universe from a pre-existing helical magnetic field111Following Ref. [24], we remain agnostic as to the origin of the magnetic field. Many compelling models of magnetogenesis are summarized in the review [1], including inflationary magnetogenesis and magnetogenesis from a first order symmetry breaking phase transition.. This paper builds on the work of Ref. [24] with a more sophisticated model for the evolution of the baryon asymmetry across the electroweak crossover.

When electroweak symmetry breaking occurs at , the primordial hypermagnetic field becomes an electromagnetic field. Since the gauge field has vector-like interactions with the Standard Model fermions, it does not source baryon and lepton number. (There is no term on the right side of Eq. (1.1).) Previously, [24] assumed that the baryon asymmetry freezes out at the electroweak phase transition, since the source for is absent ( term in Eq. (1.2)). However, the electroweak sphaleron ( term in Eq. (1.2)) remains in equilibrium until and threatens to washout the asymmetry [17]. Therefore proper modeling of the epoch is critical to an accurate prediction of the relic baryon asymmetry of the Universe.

The present study builds on earlier work in the following ways:

1. We include kinetic equations for all of the Standard Model fermion species. Many previous studies have focused on simply the electron asymmetries. While the electron asymmetries do play a key role, we find that including the quarks and higher-generation leptons allows us to properly implement the transformation of the hypermagnetic field into an electromagnetic field at the electroweak phase transition.

2. We include the chiral magnetic effect (CME). As we will see, the CME suppresses growth of the baryon asymmetry for models with a strong magnetic field. The CME was not taken into account in some previous studies.

3. We focus on models with vanishing . In this way, we address the question of whether the observed baryon asymmetry of the universe can arise entirely from the decaying magnetic helicity of a primordial magnetic field (i.e., BAU-from-PMF).

Solving the Standard Model kinetic equations in the presence of a primordial (hyper)magnetic field with decaying helicity, we investigate – both analytically and numerically – the evolution of the baryon asymmetry during the critical window . We find that the asymmetry is not washed out by the electroweak sphaleron even though the hypermagnetic field has been transformed into an electromagnetic field, which does not source . Whereas the electroweak sphaleron efficiently erases the asymmetry of the left-chiral fermions, which are charged under the electroweak gauge group , it does not communicate directly with the right-chiral fermions. Thus, the Yukawa interactions or the chiral magnetic effect is necessary to communicate -violation to the right-chiral fermions. However, a total relaxation of both left- and right-chiral fermion asymmetries to zero is prevented by the decaying electromagnetic helicity. Although the electromagnetic field does not source , because of its vector-like interactions, it does source fermion chirality through the standard Adler-Bell-Jackiw anomaly [11, 12], and thereby it avoids a complete washout. Ultimately, the relic baryon asymmetry is determined by a balance between the source term from decaying magnetic helicity and the washout due to electroweak sphaleron in association with either the electron Yukawa interaction or the chiral magnetic effect.

The paper is organized as follows. In the next section, we formulate kinetic equations for the various Standard Model particle asymmetries, paying particular attention to the source terms that arise from the chiral anomaly in the presence of a helical hypermagnetic field. In Sec. 3, we solve the kinetic equations in the equilibrium approximation, which yields an analytic expression for the relic baryon asymmetry in terms of the magnetic field strength and coherence length today. In Sec. 4, we solve the kinetic equations numerically, demonstrate the reliability of the analytic approximation, and determine the magnetic field parameters leading to maximal baryon asymmetry. Section 5 is devoted to the conclusion.

## 2 Kinetic Equations

The baryon asymmetry is distributed among the various Standard Model quarks in the form of particle / anti-particle asymmetries. Interactions between the quarks and other Standard Model particles allow these asymmetries to redistribute and evolve. In this section, we develop a set of kinetic equations to keep track of the evolution of the various asymmetries.

### 2.1 General Discussion

A generic kinetic equation takes the form

In this example, there are three flavors of particles () and anti-particles (). In general, each flavor can carry an asymmetry, i.e. the asymmetry in is quantified by , which equals the number of particles less the number of anti-particles per unit volume. In Sec. 2.2 we present the full system of kinetic equations for the Standard Model particle asymmetries.

We have assumed a homogeneous FRW background spacetime with Hubble parameter . The term accounts for the dilution of density due to cosmological expansion.

Interactions among the three flavors give rise to the source term

 S\sc abc=Γ\sc abcμA−μB−μC6/T2 (2.2)

where is the chemical potential of species , and is the charge transport coefficient. Reactions contributing to include the decay and inverse decay processes,

 classification\omit\span\omit\span\omitreactiontransp. coeff.\multirowsetup decayA→B+C¯B→C+¯A¯C→¯A+B\multirowsetupγ\sc abc=Γ\sc abc% /T¯A→¯B+¯CB→¯C+AC→A+¯B\cline1−4\multirowsetup inverse decayB+C→AC+¯A→¯B¯A+B→¯C¯B+¯C→¯A¯C+A→BA+¯B→C , (2.3)

as well as scattering processes that involve a fourth particle with vanishing chemical potential. In general, some of the reactions will be kinematically forbidden. In Sec. 2.3 we discuss the transport coefficients that appear in the Standard Model kinetic equations due to the Yukawa interactions, Higgs self-interactions, and weak gauge interactions.

The source term represents the rate per unit volume at which the asymmetry in grows due to an external (background) impetus that does not depend on the other particle asymmetries. In Sec. 2.4 we discuss a source term in the Standard Model kinetic equations induced by the decaying helicity of a primordial magnetic field.

It is convenient to express the kinetic equation in an alternate form. We relate the charge density and chemical potential in the relativistic approximation via

 nA≈16kAμAT2 . (2.4)

The statistical factor counts the number of internal degrees of freedom (e.g., spin, color, and isospin). We identify the -abundance

 ηA=nA/s (2.5)

where is the entropy density of the cosmological plasma. While the universe expands adiabatically we have , and Eq. (2.1) becomes

During radiation domination, the Hubble parameter is with , and is the effective number of relativistic species in the early universe. The age of the universe satisfies , and by introducing the dimensionless temporal coordinate we write the kinetic equation as222Here we have assumed that is static, and . In the Standard Model at the temperatures of interest, , it is a good approximation to treat as static. More generally, one could express the kinetic equation in terms of conformal time, .

The dimensionless transport coefficient and source have been written as and .

### 2.2 Standard Model Kinetic Equations

In the Standard Model we are interested in the evolution of various particle asymmetries. In the symmetric phase where the Higgs condensate is vanishing, the relevant fermion degrees of freedom are the chiral fermions. For instance, the charge abundance quantifies the particle / anti-particle asymmetry between the three colors of left-chiral, up-type, first-generation quarks , and their CP-conjugate anti-particles, . That is to say, color is summed: . The abundance is related to the charge density and chemical potential as in Eqs. (2.4) and (2.5).

We enumerate the Standard Model particles as left- and right-chiral up-type quarks , left- and right-chiral down-type quarks , left- and right-chiral electrons , left-chiral neutrinos , charged Higgs boson , neutral Higgs boson , and charged weak boson . The index runs from to and counts the number of fermion generations. For each particle, there is a corresponding anti-particle, which we denote with a bar for the fermions and neutral Higgs (e.g., and ) and we denote as , for the charged Higgs and weak boson. The statistical factors that appear in converting from chemical potential to charge abundance are given by333In general, per degree of freedom for a chiral fermion, for a Dirac fermion, for a complex scalar, and for a complex vector (two spin states).

 kuiL≈kdiL≈kuiR≈kdiR≈Nc ,  kνiL≈keiL≈keiR≈1 ,  kϕ+≈kϕ0≈2 ,  andkW+≈4 (2.8)

where is the number of colors. The neutral gauge bosons ( or in the Higgs phase) are self-conjugate under CP and do not play any role in the charge transport equations. At the temperatures of interest the strong interactions are in thermal equilibrium, and we assume that the universe is color-neutral; we assume that the gluons do not carry a charge asymmetry.

The Standard Model transport equations used in our analysis are summarized below:

 dηuiLdx =−Si\sc udw−Ng∑j=1(Sij\sc uhu+Sij\sc uu+Sij\sc uhd)−Ss,sph−Nc2Sw,sph +(Ncy2QLSbkgy+Nc2Sbkgw+NcyQL2Sbkgyw) (2.9a) dηdiLdx =Si\sc udw−Ng∑j=1(Sij\sc dhd+Sij\sc dd+Sij\sc dhu)−Ss,sph−Nc2Sw,sph +(Ncy2QLSbkgy+Nc2Sbkgw−NcyQL2Sbkgyw) (2.9b) dηνiLdx =−Siν\sc ew−Ng∑j=1Sijνhe−12Sw,sph+(y2LLSbkgy+12Sbkgw+yLL2Sbkgyw) (2.9c) dηeiLdx (2.9d) dηuiRdx =Ng∑j=1(Sji\sc uhu+Sji\sc uu+Sji\sc dhu)+Ss,sph−Ncy2uRSbkgy (2.9e) dηdiRdx =Ng∑j=1(Sji\sc dhd+Sji\sc dd+Sji\sc uhd)+Ss,sph−Ncy2dRSbkgy (2.9f) dηeiRdx =Ng∑j=1(Sji\sc ehe+Sji\sc ee+Sjiνhe)−y2eRSbkgy (2.9g) dηϕ+dx =−(Shh\sc w+Sh\sc w)+Ng∑i,j=1(−Sij\sc d% hu+Sij\sc uhd+Sijνhe) (2.9h) dηϕ0dx =Shh\sc w−Sh+Ng∑i,j=1(−Sij\sc uhu+Sij\sc dhd+Sij\sc ehe) (2.9i) dηW+dx =(Shh\sc w+Sh\sc w)+Ng∑i=1(Si\sc udw% +Siν\sc ew) . (2.9j)

The source terms fall into the following categories. The source terms,

 Si\sc udw ≡γi\sc udw(ηuiLkuiL−ηdiLkdiL−ηW+kW+) , (2.10a) Siν\sc ew ≡γiν\sc ew(ηνiLkνiL−ηeiLkeiL−ηW+kW+) , (2.10b) Shh\sc w ≡γhh\sc w(ηϕ+kϕ+−ηϕ0kϕ0−ηW+kW+) , (2.10c)

arise from the weak gauge interactions. We estimate the corresponding transport coefficients, , , and in Eq. (2.24). The source terms,

 Sij\sc dhu ≡γij\sc dhu2(ηdiLkdiL+ηϕ+kϕ+−ηujRkujR),Sij\sc uhu≡γij\sc uhu2(ηuiLkuiL+ηϕ0kϕ0−ηujRkujR) , (2.11a) Sij\sc uhd ≡γij\sc uhd2(ηuiLkuiL−ηϕ+kϕ+−ηdjRkdjR),Sij\sc dhd≡γij\sc dhd2(ηdiLkdiL−ηϕ0kϕ0−ηdjRkdjR) , (2.11b) Sijνhe ≡γijνhe2(ηνiLkνiL−ηϕ+kϕ+−ηejRkejR),Sij\sc ehe≡γij\sc ehe2(ηeiLkeiL−ηϕ0kϕ0−ηejRkejR) , (2.11c)

arise from the Yukawa interactions. We estimate the transport coefficients in Eq. (2.26). After the electroweak phase transition, the gauge and Yukawa interactions mediate scattering with the Higgs condensate. This gives rise to the additional source terms,

 Sij\sc uu ≡γij\sc uu(ηuiLkuiL−ηujRkujR) , (2.12a) Sij\sc dd ≡γij\sc dd(ηdiLkdiL−ηdjRkdjR) , (2.12b) Sij\sc ee ≡γij\sc ee(ηeiLkeiL−ηejRkejR) , (2.12c) Sh\sc w ≡γh\sc w(ηϕ+kϕ+−ηW+kW+) , (2.12d) Sh ≡γhηϕ0kϕ0 . (2.12e)

We estimate these transport coefficients in Eq. (2.29).

The remaining source terms are associated with the Standard Model chiral anomalies. Thermal fluctuations of the non-Abelian gauge fields lead to the terms

 Ss,sph ≡γs,sphNg∑i=1(ηuiLkuiL+ηdiLkdiL−ηuiRkuiR−ηdiRkdiR) , (2.13a) Sw,sph ≡γw,sphNg∑i=1(Nc2ηuiLkuiL+Nc2ηdiLkdiL+12ηνiLkνiL+12ηeiLkeiL) , (2.13b)

which are known as the strong and electroweak sphalerons. We extract the transport coefficients from the results of lattice simulations in Eqs. (2.43a) and (2.43b). In the presence of a background magnetic field, additional source terms are generated:

 Sbkgw ={0, T>162 GeV12(−Sem+γ\sc cmeemη5,em), T<162 GeV (2.14a) Sbkgy ={−Sy+γ\sc cmeyη5,Y, T>162 GeV−Sem+γ\sc cmeemη5,em, T<162 GeV (2.14b) Sbkgyw ={0, T>162 GeV2(−Sem+γ\sc cmeemη5,em), T<162 GeV . (2.14c)

Above (below) the temperature the system is in the symmetric (broken) phase, see Eq. (B.1). The sources and arise from decaying magnetic helicity, and we estimate them in Eqs. (2.68) and (2.70). The transport coefficients and arise from the chiral magnetic effect, and we estimate them in Eqs. (2.69) and (2.71). The charge-weighted chiral abundances are defined by

 η5,Y =Ng∑i=1[−y2QL(ηuiL+ηdiL)−y2LL(ηνiL+ηeiL)+y2uRηuiR+y2dRηdiR+y2eRηeiR] (2.15) η5,em =Ng∑i=1[−q2uLηuiL−q2dLηdiL−q2νLηνiL−q2eLηeiL+q2uRηuiR+q2dRηdiR+q2eRηeiR] . (2.16)

The hypercharges and electromagnetic charges of the Standard Model particles are

 yQL=16 , yLL=−12 , yuR=23 , ydR=−13 , yeR=−1 , yΦ=12 , yW=0 (2.17) quL=quR=23 , qdL=qdR=−13 , qeL=qeR=−1 , qνL=0 , qϕ+=qW+=1 , qϕ0=0 .

The number of colors is and the number of fermion generations is .

Let us also define the abundances for hypercharge, electromagnetic charge, baryon number, and lepton number:

 ηY =Ng∑i=1[yQL(ηuiL+ηdiL)+yLL(ηνiL+ηeiL)+yuRηuiR+ydRηdiR+yeRηeiR (2.18) ηem =Ng∑i=1[quLηuiL+qdLηdiL+qνLηνiL+qeLηeiL+quRηuiR+qdRηdiR+qeRηeiR +yϕ+ηϕ++qϕ0ηϕ0+qWηW+] (2.19) ηB =1NcNg∑i=1[ηuiL+ηdiL+ηuiR+ηdiR] (2.20) ηL =Ng∑i=1[ηνiL+ηeiL+ηeiR] . (2.21)

From the system of kinetic equations in Eq. (2.9), one can explicitly verify the conservation laws. Both electromagnetic charge and baryon-minus-lepton number are conserved, . Hypercharge is conserved in the symmetric phase, but violated due to the Higgs condensate through the source terms in Eq. (2.12). The sum baryon-plus-lepton number is violated by the weak sphaleron in Eq. (2.13b) and the background field terms in Eq. (2.14):

 d(ηB+ηL)dx=−6Sw,sph+6Sbkgw−3Sbkgy . (2.22)

However, in the broken phase we have as per Eq. (2.14), and the background electromagnetic field does not violate .

### 2.3 Charge Transport

In this section we estimate the charge transport coefficients arising from the charged-current weak interactions, Yukawa interactions, and Higgs condensate.

#### 2.3.1 Charged-Current Weak Interactions

The left-chiral fermions and the Higgs bosons participate in the charged-current weak interactions with the boson. These contributions to the kinetic equations, Eq. (2.9), can be identified by the transport coefficients , , and . The abundances for right-chiral particles (, , and ) do not have source terms associated with the weak interactions. In fact, these source terms are suppressed by an additional factor of Yukawa coupling squared, and we neglect them.

The transport coefficients encode various reactions mediated by the weak interactions. Some examples of decay reactions are shown in the following table:

 classification\omit\span\omit\span\omitreactiontransp. coeff.\multirowsetup decayuiL→W+diLW−→diL¯uiL¯di→¯uiW+\multirowsetupγi\sc udw¯uiL→W−¯diLW+→¯diLuiLdiL→uiLW−\multirowsetup decayνiL→W+eiLW−→eiL¯νiL¯eiL→¯νiLW+% \multirowsetupγiν\sc ew¯νiL→W−¯eiLW+→¯eiLνiLeiL→νiLW−\multirowsetup decayϕ+→W+ϕ0W−→ϕ0ϕ−¯ϕ0→ϕ−W+\multirowsetupγhh\sc wϕ−→W−¯ϕ0W+→¯ϕ0ϕ+ϕ0→ϕ+W− , (2.23)

and the corresponding inverse decay reactions are obtained by reversing the direction of the arrow. Two-to-two scattering reactions are formed by including a photon, gluon, or -boson in the initial state. Depending on the spectrum, some of the decay and inverse decay reactions will be kinematically forbidden. If all decay and inverse decay reactions are forbidden, the transport coefficient arises from scattering reactions, which are suppressed by an additional factor of coupling squared.

In Appendix A we set up the transport coefficient calculation. However, a rigorous evaluation of the transport coefficients is beyond the scope of our work. Moreover, our calculation of the relic baryon asymmetry is insensitive to these parameters, since the weak interactions come into equilibrium early. Thus, we content ourselves with a rough estimate.

If the transport coefficient arises primarily from one of the decay / inverse decay reactions, we estimate from the corresponding decay rate. At zero temperature, the decay rate is where counts the number of decay channels, is the weak fine structure constant, and is the mass of the decaying particle (assumed to be much larger than the mass of the decay products). At finite temperature, we must boost from the rest frame of the particle to the rest frame of the plasma where , and is suppressed by an additional factor of . Thus the dimensionless transport coefficient is parametrically given by .

To obtain a numerical estimate for we must know . At high temperatures, particles in the plasma acquire a mass where the coefficient equals the coupling constant for the interaction between the particle of interest and the plasma. E.g. for quarks is set by the strong coupling, and for the W-boson is set by the weak coupling. Thus the factor depends on the spectrum, since is the mass of the decaying particle. To avoid this detail of the calculation, we write , in which coupling constants are generically taken to be . With this approach, we estimate the dimensionless transport coefficients as

 γi\sc udw∼10−2Ncαw ,γiν\sc ew∼10−2αw ,andγhh\sc w∼10−2αw (2.24)

where is the number of colors and is the weak fine structure constant. While these estimates are rough, we have verified that our numerical results are insensitive to this ambiguity in the calculation of , , and . Even increasing or decreasing by a factor of compared to Eq. (2.24), we find a negligible change in the relic baryon asymmetry.

#### 2.3.2 Yukawa Interactions

The Yukawa interactions allow left-chiral particles to interact with right-chiral particles via a Higgs boson. These contributions to the kinetic equations can be identified by the transport coefficients , , , , , and in Eq. (2.9). There is no source term for the weak boson , because we neglect scattering processes such as that are suppressed by an additional factor of .

Decay reactions contributing to the transport coefficients are shown in the following table,

 classification\omit\span\omit\span\omitreactiontransp. coeff.\multirowsetup decayϕ+→¯diLujRdiL→ϕ−ujR¯ujR→¯diLϕ−% \multirowsetupγij\sc dhuϕ−→diL¯ujR¯diL→ϕ+¯ujRujR→diLϕ+\multirowsetup decayϕ0→¯uiLujRuiL→¯ϕ0ujR¯ujR→¯uiL¯ϕ0\multirowsetupγij\sc uhu¯ϕ0→uiL¯ujR¯uiL→ϕ0¯ujRujR→uiLϕ0\multirowsetup decayϕ+→uiL¯djR¯uiL→ϕ−¯djRdjR→uiLϕ−%\multirowsetup$γij\sc uhd$