###### Abstract

We elaborate upon the model of baryogenesis from decaying magnetic helicity by focusing on the evolution of the baryon number and magnetic field through the Standard Model electroweak crossover. The baryon asymmetry is determined by a competition between the helical hypermagnetic field, which sources baryon number, and the electroweak sphaleron, which tends to wash out baryon number. At the electroweak crossover both of these processes become inactive; the hypermagnetic field is converted into an electromagnetic field, which does not source baryon number, and the weak gauge boson masses grow, suppressing the electroweak sphaleron reaction. An accurate prediction of the relic baryon asymmetry requires a careful treatment of the crossover. We extend our previous study [K. Kamada and A. J. Long, Phys. Rev. D 94, 065301 (2016)], taking into account the gradual conversion of the hypermagnetic into the electromagnetic field. If the conversion is not completed by the time of sphaleron freeze-out, as both analytic and numerical studies suggest, the relic baryon asymmetry is enhanced compared to previous calculations. The observed baryon asymmetry of the Universe can be obtained for a primordial magnetic field that has a present-day field strength and coherence length of and and a positive helicity. For larger the baryon asymmetry is overproduced, which may be in conflict with blazar observations that provide evidence for an intergalactic magnetic field of strength .

Evolution of the Baryon Asymmetry

through the Electroweak Crossover

in the Presence of a Helical Magnetic Field

Kohei Kamada^{*}^{*}*kohei.kamada@asu.edu and Andrew J. Long^{†}^{†}†andrewjlong@kicp.uchicago.edu

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

###### Contents

## 1 Introduction

The origin of the matter/antimatter asymmetry of the Universe [or baryon asymmetry of the Universe (BAU)] remains a long-standing problem at the interface of cosmology and high energy physics. In order to generate a baryon asymmetry from an initially matter/antimatter symmetric universe, it is necessary for the system to contain processes that violate baryon number [1]. Such processes are already present in the Standard Model (SM) due to field theoretic quantum anomalies [2, 3, 4]. These anomalous processes involve either the weak isospin gauge fields or the hypercharge gauge field. Whereas SM baryon-number violation via the gauge field features prominently in many models of baryogenesis, such as electroweak baryogenesis and leptogenesis, we are interested in SM baryon-number violation via the gauge field.

In the symmetric phase of the electroweak (EW) plasma ( in the SM [5]), the anomaly expresses the fact that changes in baryon number () and lepton number () can be induced by changes in Chern-Simons number () or hypermagnetic helicity () as

(1.1) |

The factor of is the number of fermion generations and is the gauge coupling. Thermal fluctuations of the gauge fields (EW sphalerons [6]) allow to diffuse, which pushes and to zero (assuming a vanishing asymmetry). The system may also contain a helical hypermagnetic field, i.e. a primordial magnetic field (PMF) in the symmetric phase of the EW plasma associated with hypercharge that has excess power in either the left- or right-circular polarization mode. A helical PMF can arise, for example, from axion dynamics during inflation [7, 8, 9, 10, 11, 12, 13, 14] (see also Refs. [15, 16]). Due to interactions of the hypermagnetic field with the charged plasma, the hypermagnetic helicity slowly decays. If initially, then implies the generation of a baryon asymmetry . In this way, the BAU may have arisen from a helical hypermagnetic field in the early Universe.

Various studies have explored the relationship between baryon-number violation and magnetic fields in the early Universe. Among the earliest works, Joyce and Shaposhnikov [17] showed that a helical hypermagnetic field can arise in the symmetric phase of the EW plasma from a preexisting lepton asymmetry carried by the right-chiral electron [18] (see also Refs. [19, 20]). This work was soon extended by Giovannini and Shaposhnikov [21, 22, 23, 24] to consider the generation of baryon-number isocurvature fluctuations from a preexisting stochastic hypermagnetic field. These ideas were formulated into a model of baryogenesis by Bamba [25] where the dynamics of an axion field during inflation leads to the growth of a helical hypermagnetic field with a large correlation length, which is partially converted into baryon number by the SM anomalies at the electroweak phase transition (see also Refs. [26, 27]). Other related work has explored the connection between helical magnetic fields in the early Universe and the anomalous violation of chiral charge [28, 29, 30, 31, 32] (see also Refs. [33, 34, 35]) and lepton number [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

Models of baryogenesis from magnetogenesis are interesting in part because the primordial magnetic field is expected to persist today as an intergalactic magnetic field (IGMF). Although the existence and origin of the IGMF have not yet been established, the body of evidence is growing. (See Refs. [47, 48] for recent reviews on cosmological magnetic fields.) Recent measurements of TeV blazar spectra have identified a deficit of secondary cascade photons. These observations can be explained to result from a magnetic broadening of the cascade by the IGMF [49, 50, 51, 52, 53, 54, 55], which thereby provides indirect evidence for the existence of a PMF with a field strength and coherence length today of and . Similarly, searches for GeV pair halos around TeV blazars have also reported evidence for an IGMF [56, 57] (see also Refs. [58, 59]). Additionally, measurements of the diffuse gamma ray flux at Earth suggest a parity-violating character in gamma ray arrival directions, which can be interpreted as evidence for the presence of a helical IGMF [60, 61, 62, 63].

Motivated in part by these new probes of the IGMF, Fujita and Kamada [64] recently revisited baryogenesis from hypermagnetic helicity. By drawing on the results of recent magnetohydrodynamic simulations, they used an improved model for the evolution of the magnetic field (inverse cascade behavior) to calculate the slowly decaying magnetic helicity and corresponding production of baryon number. Their calculation indicates that a maximally helical magnetic field stronger than today would generate a much larger baryon number than what is observed. Since this baryogenesis is an inevitable consequence of SM physics once the helical hypermagnetic field is provided, there is a mild conflict between the observed BAU and blazar observations, which favor .

However, none of these studies directly addresses the conversion of the hypermagnetic field into an electromagnetic field at the EW crossover and the corresponding effect on the relic baryon asymmetry. Since the electromagnetic field has vectorlike interactions, it does not contribute to the baryon-number anomaly. Therefore, if the conversion completes before the EW sphalerons freeze-out, the sphalerons threaten to erase the baryon asymmetry. In the early works of Giovannini and Shaposhnikov and Bamba, et al. [25, 26, 21, 22, 23, 24] it was argued that the EW phase transition must be first order so that the EW sphaleron process is out of equilibrium in the broken phase and washout of baryon number is avoided. The assumption is implicit in later work [27, 64] where baryon-number violation due to both the EW sphaleron and the hypermagnetic field are assumed to shut off simultaneously at EW temperatures.

Kamada and Long [65] recently demonstrated that a complete washout of baryon number is avoided even if there is no asymmetry and the EW phase transition is a continuous crossover as we expect in the SM. Although the EW sphaleron remains in thermal equilibrium until [66] after the hypermagnetic field has been converted to an electromagnetic field, and therefore no longer sources baryon number, washout is avoided because the EM field sources chirality and inhibits the communication of baryon-number violation from the left-chiral to right-chiral fermions. To model the conversion of the hypermagnetic field into electromagnetic field at the EW crossover, Ref. [65] assumed that the transformation occurs abruptly at a fiducial temperature of where the Higgs condensate first starts to deviate from zero (see also Ref. [67]). As discussed in Ref. [65], this is a conservative approach; since the electromagnetic field does not violate baryon number, this approximation can underestimate the relic baryon asymmetry if the conversion of the hypermagnetic field into the electromagnetic field is gradual.

In this work, we develop a more sophisticated treatment for the conversion of hypermagnetic field into electromagnetic field at the EW crossover. By drawing on analytic and lattice results we see that the hypermagnetic field is not fully converted into an electromagnetic field even at temperatures as low as . Therefore, the source term from decaying magnetic helicity remains active, while the washout by EW sphalerons goes out of equilibrium. Consequently, we show that the relic baryon asymmetry can be greatly enhanced as compared to Ref. [65]. It is possible to generate the observed BAU from a maximally helical magnetic field that was generated prior to the EW crossover and has a strength and coherence length today of about and . If the magnetic field strength is larger, such as suggested by blazar observations, the relic baryon asymmetry is generally overproduced. This presents a new constraint for models of magnetogenesis that rely on inflation or cosmological phase transitions prior to the EW epoch.

The rest of the paper is organized as follows. In Sec. 2, we generalize the calculation of Ref. [65] to allow for a gradual conversion of the hypermagnetic field into an electromagnetic field at the EW crossover. In Sec. 3, we present an analytic solution of the kinetic equations, which gives the equilibrium baryon-number abundance. In Sec. 4, we solve the kinetic equations numerically and compare with our analytic formula. We show how the relic baryon asymmetry depends on the field strength and coherence length today. We see that baryon number is overproduced for relatively large magnetic field strength, . In Sec. 5, we discuss ways to avoid the baryon overproduction while also accommodating the IGMF interpretation of blazar observations. Finally we conclude in Sec. 6 and point to directions for future work.

## 2 Derivation of source terms

In this section, we generalize our previous calculation in order to model the gradual conversion of the hypermagnetic field into an electromagnetic field. For definitions and notation, the reader is referred to Ref. [65].

First, let us recall what is the quantity of interest. In the presence of a helical magnetic field, SM quantum anomalies lead to the appearance of source terms in the kinetic equations for the various SM particle asymmetries. These source terms appear in the kinetic equation for fermion species in the following way [65]:

(2.1) |

Here, is the particle number asymmetry in species divided by the entropy density of the cosmological plasma. We use the dimensionless temporal coordinate where is the temperature of the cosmological plasma and with is the Hubble parameter at temperatures where the entire SM particle content is relativistic. The coefficients of the source terms depend on the quantum numbers of ; see Ref. [65]. The dots ( represent other interactions in which a fermion of species participates. These include Yukawa interactions, EW and strong sphalerons, and weak interactions. The source terms take the form (see Eq. (2.44) of Ref. [65])

(2.2a) | ||||

(2.2b) | ||||

(2.2c) |

where and are the field strength tensor of hypercharge and isospin, respectively, and and are their respective coupling parameters. The dual tensor is defined by with normalization . The angled brackets indicate thermal ensemble averaging, and the bar denotes volume averaging. In this section, we seek to evaluate these three sources.

Now, let us recall how we modeled the gauge fields during the EW crossover in Ref. [65]. We assumed that the system passes abruptly from the symmetric phase to a broken phase as the temperature is lowered through in a similar way to Ref. [67]. This numerical value is taken from lattice studies of the EW crossover [68]. In the symmetric phase (), the non-Abelian gauge field is screened due to its self-interactions [69], and it is well known that the corresponding isomagnetic field vanishes (up to thermal fluctuations). Meanwhile, the sector is assumed to carry a hypermagnetic field generated by a magnetogenesis mechanism that occurred before the EW crossover. In the broken phase (), the Higgs condensate induces a mass for charged and neutral gauge fields. We argued that the massive fields decay quickly, leaving only the massless electromagnetic field . We defined the electromagnetic field through the standard electroweak rotation, , where the vacuum weak mixing angle is expressed in terms of the and and gauge couplings, and , respectively, as . This relation furnishes the matching condition , which we used to relate the electromagnetic field just after the crossover to the hypermagnetic field just before the crossover.

The approach described above is not correct in the following sense.
During the EW crossover, the gauge fields acquire mass from both the Higgs condensate and thermal effects in the plasma.
If the thermal effects could be neglected, then we would have four massless fields in the symmetric phase where the Higgs condensate is zero, and we would have one massless field in the broken phase.
If we define the weak mixing angle as the parameter of the matrix that diagonalizes the quadratic gauge field terms in the Lagrangian, then this approximation corresponds to an abrupt change from in the symmetric phase to in the broken phase.
However, this is not the case.^{1}^{1}1We are grateful to Mikhail Shaposhnikov for bringing this point to our attention.
As we have already mentioned above, the non-Abelian gauge fields also acquire mass from their self-interactions in the plasma, which leads to the screening of isomagnetic fields.
Consequently, the mixing angle will change slowly with time while interpolating smoothly between its symmetric and broken phase limiting values.
It continues to deviate appreciably from its zero-temperature value even at relatively low temperatures of .
This behavior is confirmed by analytic calculations [70] and recent numerical lattice simulations [68].
We will study it quantitatively in Sec. 4.

In light of the preceding discussion, we generalize our treatment of the gauge fields at the EW crossover as follows.
At any time, the spectrum consists of three massive and one massless gauge field degrees of freedom.
In general, the massless degree of freedom at time can be written as an rotation of and with parameter .
In other words, is defined as the rotation angle that projects the massless field degree of freedom onto the field axis.
As before, we assume that the massive fields are screened or decay away quickly compared to the time scale on which the baryon asymmetry evolves.^{2}^{2}2This assumption is confirmed with the following rough estimates. Parametrically, the perturbative Z-boson decay width at temperature is given by where is the vacuum expectation value of the Higgs field at temperature . Comparing this decay rate with the Hubble expansion rate during the EW epoch, we have , which supports our assumption that the Z-field decays quickly. We expect this general conclusion to be unchanged when thermal and nonperturbative effects are considered more carefully.
Therefore, the field evolution can be modeled by the ansatz

(2.3a) | ||||

(2.3b) | ||||

(2.3c) |

By requiring the three massive field degrees of freedom to vanish and their decay not to affect the evolution of the massless field degree of freedom, we have reduced the problem to a single degree of freedom as represented by the classical vector field .

The ansatz (2.3) is represented graphically in Fig. 1, which illustrates the conversion from hypermagnetic field to electromagnetic field. Here, we denote the magnetic field of a gauge field as . We have drawn the figure so as to suggest that does not decrease appreciably during the EW crossover. As we will explain later, this is the case because evolves slowly according to the cosmic expansion and the inverse cascade.

Having generalized the gauge field ansatz from our earlier work, we are now prepared to revisit the calculation of source terms (2.2). Using the ansatz in Eq. (2.3), the source terms can be written as

(2.4a) | ||||

(2.4b) | ||||

(2.4c) |

where is the field strength tensor associated with and is the dual tensor. In terms of the 3-vector notation, the two terms in parentheses are

(2.5) |

where is the electric field with , is the magnetic field with , and is the vector potential with . With this replacement, the sources become

(2.6a) | ||||

(2.6b) | ||||

(2.6c) |

The second term in parentheses is new, since we are now allowing . Recall that is the helicity of the gauge field . Under a gauge transformation, we send ,and since , the helicity density transforms into itself up to a total 3-divergence. The volume averaged helicity is gauge invariant provided that the surface term vanishes; for example, see Ref. [71].

To evaluate the electric field , we recognize that the electric current is given by

(2.7) |

The first term is simply Ohm’s law with the conductivity.
The second term is the chiral magnetic effect (CME) current, which we evaluate below.
The current also appears in the equation of motion^{3}^{3}3
Here, we gloss over some subtleties related to gauge invariance. In general, the transformation (2.3) should be generalized to include the orthogonal field direction . Due to the time-dependent linear transformation, the field equations for and acquire “mass terms” of the form and . Nevertheless, one can verify explicitly that the field equations are gauge invariant. This is because the field strength tensors are no longer invariant under the gauge transformation when . Despite these subtleties, we have checked that the source terms appearing in Eq. (2.6) are gauge invariant. In writing Eq. (2.8), we have dropped the mass term from the right-hand side. It is numerically negligible since and the coherence length of the field is much smaller than the Hubble scale .
for the field ,

(2.8) |

Combining these two formulas, we can show that

(2.9) |

where we have neglected the displacement current . This is justified in the magnetohydrodynamic (MHD) approximation [71], where . The term involving fluid velocity does not contribute to the source term (2.6) since .

The chiral magnetic effect is the phenomenon whereby a magnetic field induces an electric current in a medium with a charge-weighted chiral asymmetry [72]. By adapting the standard result for quantum electrodynamics [73] to our problem, the induced electric current can be written as

(2.10) |

where is the effective gauge coupling for and is the charge-weighted chiral chemical potential. The corresponding charge-weighted chiral charge abundance is given by . The chiral charge abundance is constructed from the abundances for the various SM particle species as

(2.11) |

where the sum runs over the three fermion families. The effective charges can be read off of the Lagrangian upon using the ansatz in Eq. (2.3). These charges are found to be

(2.12a) | ||||

(2.12b) | ||||

(2.12c) | ||||

(2.12d) | ||||

(2.12e) | ||||

(2.12f) | ||||

(2.12g) |

where ’s are the corresponding hypercharges.

Finally, we put these pieces together. By combining Eqs. (2.9) and (2.10), we evaluate the electric field. This lets us express the source terms (2.6) as

(2.13a) | ||||

(2.13b) | ||||

(2.13c) |

where we have used and defined

(2.14a) | ||||

(2.14b) | ||||

(2.14c) |

Due to the volume averaging, the source terms are independent of the spatial coordinate. They depend upon the temporal coordinate through the entropy density , the temperature , the conductivity , and the volume-averaged field products.

Equation (2.13) is one of the main results of this paper. It should be compared with Eqs. (2.53) and (2.60) of our earlier work [65]. To regain Eqs. (2.53) and (2.60), we can take to be a step function and set . In the present calculation, we have generalized to an (as yet) arbitrary . As such, it is not necessary to treat the symmetric and broken phase cases separately, as we did in Ref. [65]. Rather, Eq. (2.13) interpolates smoothly between the two solutions that we found previously. The term proportional to was overlooked in previous studies, and we will see that it can provide an efficient source of baryon number.

## 3 Analytic equilibrium solution

Previous studies [21, 22, 27, 64, 65] have shown that a helical hypermagnetic field in the symmetric phase of the EW plasma sources baryon number, which thereby competes against the washout of baryon number by EW sphalerons [74]. Unlike the earlier work, in Sec. 2, we have taken a more careful treatment for the evolution of the magnetic field through the EW crossover, specifically allowing for a time-dependent weak mixing angle . By doing so, we have identified an additional source term in the kinetic equation for baryon number, namely the term in Eq. (2.13). Here, we examine the evolution of the baryon asymmetry analytically with an emphasis on the effect of varying .

We derive the kinetic equation for baryon number by combining the the kinetic equations in Ref. [65] with the sources in Eq. (2.13). Denoting the baryon number-to-entropy ratio as , its kinetic equation takes the form

(3.1) |

where is the time-dependent weak mixing angle. In the presence of a helical magnetic field, the terms containing and (2.14) become nonzero and source baryon number. In the symmetric phase, the weak mixing angle vanishes , and drives the growth of baryon number. During the EW crossover, begins to increase, and contributes to the baryon-number growth. After the crossover, approaches its vacuum value, , and both source terms become inactive; i.e., their coefficients vanish. As we will see, the coefficient of the new source term can vanish more slowly than the coefficient of , and therefore the baryon asymmetry can be enhanced compared to previous calculations.

The growth of baryon number is inhibited by several washout processes. These include the chiral magnetic effect, the EW sphaleron, and the electron spin-flip interaction, which comes into equilibrium below and communicates baryon-number violation to the right-chiral electron [18]. The equilibrium baryon asymmetry is controlled by the slowest (least efficient) washout processes. For , the CME and spin-flip processes are slowest, and for , the EW sphaleron is slowest. Thus, we calculate the equilibrium baryon number separately for these two periods below.

At sufficiently high temperatures, , the EW sphaleron efficiently violates baryon number, and the equilibrium baryon asymmetry is controlled by a combination of the slower chiral magnetic effect and electron spin-flip interactions.
The CME tends to deplete the charge-weighted chiral charge abundance (2.11), and the electron spin-flip interactions tend to equilibrate left- and right-chiral electron abundances.
In this way, EW sphalerons violate baryon number among the left-chiral fermions, and the other washout processes communicate baryon-number violation to the right-chiral fermions.
As in Ref. [65], we calculate the equilibrium baryon asymmetry in the regime where all of the SM processes are in chemical equilibrium except for the CME and electron spin-flip interactions.^{4}^{4}4
This approach assumes that spin-flip interactions with the background Higgs condensate are in equilibrium. At higher temperatures when the Higgs condensate has not yet developed, these interactions do not occur.
In this regime, the baryon asymmetry can be calculated with Eqs. (3.6) and (3.7) in Ref. [65], but those formulas also agree with Eq. (3.3) below up to an prefactor.
It is known that this treatment during EW crossover gives error in the estimate [75], but here we neglect it.
We also require the four conserved charges to vanish; these are number and electromagnetic charge: .
As discussed in Ref. [65], the baryon asymmetry in equilibrium in this regime can be read off from the kinetic equation for the first-generation right-chiral electron abundance.
Under these assumptions, it is reduced to

(3.2) |

The transport coefficients , , and were defined in Ref. [65]. The equilibrium condition gives the behavior of the baryon asymmetry in equilibrium,

(3.3) |

By taking and we regain Eq. (3.10) of Ref. [65]. Notice how the equilibrium solution takes the form of , which expresses the balance between these two competing effects.

At lower temperatures, , the EW sphaleron rate becomes exponentially suppressed as the weak gauge boson masses grow, but nevertheless the sphaleron remains in equilibrium until [66]. In this window, the EW sphaleron is the slowest washout process, and therefore it controls the equilibrium baryon asymmetry. Assuming that all of the SM processes are in equilibrium except for the EW sphaleron, the kinetic equation for baryon number (3.1) reduces to

(3.4) |

where is the transport coefficient associated with the EW sphaleron process [65]. Here, we omit the term that includes since generally it is much smaller than the term with at this period. The baryon asymmetry is well approximated by the equilibrium solution of Eq. (3.4). Solving gives

(3.5) |

This contribution to the baryon asymmetry is only present when , and consequently it was overlooked in previous studies that did not treat the evolution of the magnetic field through the EW crossover so carefully.

Let us summarize the results of the preceding calculation. During the temperature window , all of the SM processes are in thermal equilibrium, including the electron spin-flip interaction and the EW sphaleron. In this regime, the baryon asymmetry is well approximated by

(3.6) |

At lower temperatures, the EW sphaleron has frozen out, and this calculation overestimates the baryon asymmetry. If the source terms are still active when , because the conversion from hypermagnetic field into electromagnetic field is very slow, then there can be a further enhancement of the baryon asymmetry. This is obtained by neglecting the washout term and directly integrating Eq. (3.1) to find

(3.7) |

where is the time of the EW sphaleron freeze-out. If the magnetic field conversion is sufficiently gradual, then remains nonzero for a long time, and the baryon asymmetry can be enhanced by as much as , as we will see in the next section.

## 4 Resultant baryon asymmetry evolution

In this section, we present the quantitative results. We solve the kinetic equations now using the source terms that were derived in Sec. 2. However, we must first clarify a few additional assumptions.

Following Ref. [65], we assume that the magnetic field is maximally helical and that its spectrum is peaked at the length scale where the field strength is . This allows us to estimate the volume-averaged magnetic field products, which appear in Eq. (2.14), as follows:

(4.1a) | ||||

(4.1b) | ||||

(4.1c) |

The sign indicates the helicity of the magnetic field. Hereafter, we assume that the maximally helical magnetic field has a positive helicity [i.e., the + signs in Eq. (4.1) are used]. Flipping the sign of the helicity simply flips the sign of the resultant baryon asymmetry.

It is well known that a freely decaying, maximally helical magnetic field in a turbulent plasma experiences the inverse cascade evolution where power is transported from small scales to large ones [76, 77, 78]. As in Ref. [65], we assume that the primordial magnetic field experiences the inverse cascade from a time well before the EW crossover until recombination, and afterward it evolves adiabatically (simply diluting with the cosmological expansion). Thus, we can relate the field strength and coherence length in the early Universe, and , to their values today, and , via the scaling laws

(4.2) |

where is the scale factor and is conformal time. These formulas apply when with the conformal time at recombination, and for later times, the factors of must be removed to describe the adiabatic evolution of the magnetic field. Implicitly, the scaling law assumes that backreaction from the presence of particle/antiparticle asymmetries in the plasma is negligible, and we justify this assumption in Appendix A. We also impose the constraint , which is expected to hold for causally generated magnetic fields that are processed on small scales by MHD turbulence [79] (see also the discussion in Ref. [65]).

The time-dependent weak mixing angle has been calculated both analytically [70] and numerically [68]. We give these results in Fig. 2. Evidently, the one-loop perturbative analytic calculation and the numerical lattice calculation agree only marginally. However, we can infer from both approaches that the weak mixing angle varies on a scale of during the EW crossover, which takes place at roughly . Since the analytic calculation of Ref. [70] is only a one-loop result, the true behavior of may differ when higher-order corrections are taken into account. Although the numerical lattice calculation is an all-orders calculation that includes nonperturbative effects, the error bars are still quite large. Since neither the analytic nor the numerical results for time dependence of the weak mixing angle appear more reliable, we will instead introduce a phenomenological parametrization for . Specifically, we write as a smoothed step function,

(4.3) |

which interpolates between at low temperature and at high temperature. A few trial functions are also shown in Fig. 2. It is straightforward to obtain in terms of the dimensionless temporal coordinate .

The conductivity of the SM plasma has been calculated in Ref. [80]. In the symmetric phase at temperature , they find the hypermagnetic conductivity to be , and in the broken phase at temperature , the electromagnetic conductivity is given by (see also Ref. [65]). The conductivity that appears in Eq. (2.7) interpolates between these two limiting behaviors. However, for simplicity, we estimate the conductivity instead as in both the symmetric and broken phases.

Adopting Eq. (4.3) to model the time dependence of the weak mixing angle, we solve the kinetic equations [65] using the source terms in Eq. (2.13). The evolution of the baryon asymmetry during the EW crossover is shown in Fig. 3, where we compare the numerical solution with the analytic formula that appears in Eq. (3.6). Evidently, the evolution of depends strongly on how the weak mixing angle evolves through the EW crossover; this behavior can be understood as follows.

Let us first consider the pair of (purple) curves which correspond to Parametrization A () in Fig. 2. In this case, the weak mixing angle quickly transitions between its asymptotic values at . The sudden change in implies an abrupt decrease in the helicity of the hypermagnetic field and a correspondingly large source of baryon number via the term in Eq. (3.1). As predicted in Ref. [65], the baryon number grows suddenly, but soon the hypermagnetic field is fully converted into an electromagnetic field, and the EW sphaleron, which remains in thermal equilibrium until , is able to wash out the injection of baryon number. At temperatures , the analytic formula from Eq. (3.6) (dashed curve) matches the numerical result (solid curve) very well. After EW sphaleron freeze-out, the baryon number is fixed.

The (gray) dotted curve in Fig. 3 corresponds to the calculation of Ref. [65], which assumed that the weak mixing angle changes abruptly and discontinuously at while at all times. The resultant relic baryon asymmetry agrees well with Model Parametrization A, which approximates the change in as a sudden but smooth step. The slight discrepancy between them can be traced to the factor of that arose in the calculation of Ref. [65] where was used to artificially match the hypermagnetic field into the electromagnetic field at the EW crossover.

For the models with a more gradual change in , we see four distinct stages of evolution.
First, begins to grow because (and hence ) start to deviate from zero.
This growth occurs earlier for the models of that have a broader step (larger ).
The increase of continues until where peaks.
The baryon asymmetry then decreases until since the decrease of the source term with is faster than that of the washout rate by the chiral magnetic effect and the electron spin-flip interaction.
At , the EW sphaleron becomes the least efficient washout process.
Afterward, grows as the EW sphaleron becomes less efficient at washout [ term in Eq. (3.5) decreases exponentially, much faster than the decay of the source term with ].
This growth continues until where the EW sphaleron freezes out.
The evolution of down to is well described by the analytic solution in Eq. (3.6), which appears as the dashed lines in Fig. 3.
If the hypermagnetic field is not fully converted into an electromagnetic field by the time
the EW sphaleron freezes out, there can be a continued growth of , which is described by Eq. (3.7).
Eventually, the hypermagnetic field is fully converted into an electromagnetic field, and the relic baryon asymmetry is fixed.
Practically, it is almost saturated^{5}^{5}5Note that the kinetic equations solved here neglect the effect of masses of the Higgs boson, weak bosons and top quarks and hence are not so reliable at low temperatures.
However, since they do not contribute to the source term of the baryon number or the washout effects, we expect that there will not be a significant change of the baryon asymmetry and the numerical result at gives an appropriate estimate for the relic baryon asymmetry. at

The relic baryon asymmetry [analytic formula Eq. (3.6) and numerical results] is shown in Fig. 4 as a function of the relic magnetic field strength today. It depends sensitively the evolution of the weak mixing angle . In Parametrization A where rapidly interpolates between its asymptotic values, the relic baryon asymmetry always falls below the observed baryon asymmetry of the Universe . In the other cases, we allow for a more gradual variation in , and the relic baryon asymmetry is much larger. The observed BAU is obtained for and , depending on the evolution of . For a weaker magnetic field, the baryon asymmetry is underpredicted, and an additional baryogenesis mechanism is required to explain cosmological observations. For a stronger magnetic field, the baryon asymmetry is over-predicted, and the model comes into tension with the observed baryon asymmetry. The relic BAU is particularly sensitive to the value of , and by changing from just to , the relic BAU varies by up to 3 order of magnitude. Therefore, the accurate determination of is necessary to reliably calculate the relic baryon asymmetry. Nevertheless, the qualitative behavior will be unchanged, and the problem of baryon overproduction will persist for large field strengths.

Before we close this section, let us draw attention to the regime .
If the predicted baryon asymmetry is too large, then our calculation is unreliable.
Specifically, in deriving the kinetic equations [65] we have assumed that for the chemical potentials associated with each of the SM particle species.
The corresponding abundance is calculated as with the entropy density and .
Then, the condition implies .
Consequently, the formula in Eq. (3.6) for the equilibrium baryon asymmetry cannot be trusted^{6}^{6}6One might wonder whether the conclusion of baryon-number overproduction can be avoided in the strong field regime where a more sophisticated calculation is required to accurately infer the late-time behavior of . While we cannot exclude this possibility outright, we cannot envisage any mechanism that would suppress back down to order . if , but the calculation is certainly reliable for as large as .
We discuss further in Appendix A the reliability of our calculation in the large regime.

## 5 Avoiding baryon-number overproduction

As we discussed in the Introduction, various blazar observations provide evidence for the existence of an intergalactic magnetic field with strength and coherence length . However, our calculations of the relic baryon asymmetry, which are summarized in Fig. 4, imply that for such a strong PMF the BAU may be dramatically overproduced,