Leptogenesis from magnetic helicity of gauged $\text{U}(1)_{B-L}$

Let us now review the generation of the helicity of the gauge field. The simplest way to generate the helicity is to consider the axion inflation Turner and Widrow (1988); Garretson et al. (1992); Anber and Sorbo (2006); Barnaby and Peloso (2011); Barnaby et al. (2011). Here, we review it following the ideas given in Refs. Jiménez et al. (2017); Domcke and Mukaida (2018). See also Ref. Kamada et al. (2023) for a review.

The Lagrangian of the inflaton $\phi$ and the $B-L$ gauge field is given by

$\displaystyle\frac{\mathcal{L}}{\sqrt{-g}}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}+\frac{g_{\phi XX}}{4}\phi X_{\mu\nu}\tilde{X}^{\mu\nu}$

(II.4)

where $g_{\phi XX}$ is coupling constant, and we adopt the conformal metric, $ds^{2}=a^{2}(t)(d\eta^{2}-d\vec{x}^{2})$ where $\eta$ is conformal time, $a(t)$ is scale factor, and the index, e.g., $\mu$ runs for $\eta,x,y,z$. Here the field strength tensor and its dual are defined covariantly, namely, $X^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}X_{\rho\sigma}$ and $\tilde{X}^{\mu\nu}=\frac{1}{2\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma}X_{\rho\sigma}$ with $\epsilon^{\mu\nu\rho\sigma}$ being the totally asymmetric Levi-Civita symbol, $\epsilon^{0123}=+1$. We assume the inflation potential $V(\phi)$ such that $\dot{\phi}$ is almost constant and its change is much slower than the change of the gauge field; this assumption can be satisfied in slow-roll inflation. We discuss the gauge field $X_{\mu}=(X_{0},\vec{X})$ with the radiation gauge $X_{0}=0,\ \nabla\cdot\vec{X}=0$ and perform mode expansion of in the circular polarization basis as

$$\vec{X}(\eta,\vec{x})=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\sigma=\pm}\left[\vec{\epsilon}^{\,(\sigma)}(\vec{k})a_{\vec{k}}^{(\sigma)}X_{\sigma}(\eta,\vec{k})e^{i\vec{k}\cdot\vec{x}}+\mathrm{h.c.}\right],$$

(II.5)

where polarization vector $\vec{\epsilon}^{\,(\pm)}$ satisfying $\vec{\epsilon}^{\,(\pm)}(\vec{k})\cdot\vec{k}=0,\ i\vec{k}\times\vec{\epsilon}^{\,(\pm)}(\vec{k})=\pm k\vec{\epsilon}^{\,(\pm)}(\vec{k})$ and $\vec{\epsilon}^{\,(\sigma)*}(\vec{k})\cdot\vec{\epsilon}^{\,(\sigma^{\prime})}(\vec{k})=\delta^{\sigma\sigma^{\prime}}$ with $k=|\vec{k}|$. Regarding quantization, we require the creation and annihilation operators, $a_{\vec{k}}^{(\sigma)}$ and $a_{\vec{k}}^{(\sigma)\dagger}$, to satisfy the usual canonical commutation relations, $[a^{(\sigma)}_{\vec{k}},a_{\vec{q}}^{(\sigma^{\prime})\dagger}]=\delta^{\sigma\sigma^{\prime}}\delta(\vec{k}-\vec{q})$.

While in the realistic case we need to take into account the induced current from non-perturbative fermion production, let us first neglect it to investigate the dynamics of the gauge field amplification. The equation of motion for the mode function in the inflationary background then reads

$$0=[\partial_{\eta}^{2}+k(k\mp 2\xi aH_{\rm inf})]X_{\pm}(\eta,k),$$

(II.6)

where

$$\xi\equiv-\frac{g_{\phi XX}\dot{\phi}}{2H_{\rm inf}}$$

(II.7)

is the instability parameter and $H_{\rm inf}$ is the Hubble constant during inflation. We can easily see that $\pm$ polarization has instability for $\xi\gtrless 0$ and grows exponentially at $k/a<2|\xi|H_{\mathrm{inf}}$. Since $H_{\rm inf}$ and $\xi$ vary only slowly with time during the slow-roll inflation, let us take them constants, such that $a(t)=e^{H_{\rm inf}t}$ and $\eta=-1/aH_{\rm inf}$. We find the analytical solution of Eq. (II.6) in this approximation as Jiménez et al. (2017); Domcke and Mukaida (2018)

$$X_{\sigma}(\eta,\vec{k})=\frac{e^{\sigma\pi\xi/2}}{\sqrt{2k}}W_{-i\sigma\xi,1/2}(2ik\eta)$$

(II.8)

where $W_{\kappa,\mu}$ is the Whittaker function with $\sigma=\pm$, and we have taken the Bunch-Davis vacuum, $\lim_{-k\eta\rightarrow\infty}X_{\sigma}(\eta,\vec{k})=e^{-ik\eta}/\sqrt{2k}$. The exponentially amplified mode is $\pm$ polarization for $\xi\gtrless 0$. The physical electric and magnetic fields are given by $\vec{E}_{X}=-\partial_{\eta}\vec{X}/a^{2},\ \vec{B}_{X}=\vec{\nabla}\times\vec{X}/a^{2}.$ From now on, we assume that $\xi>0$ which is going to ensure the sign of the resultant baryon asymmetry to be positive and correct, as we will see. Substituting the solution into the mode expansion, for $\xi>0$, we obtain the physical electromagnetic field as

	$$\displaystyle\langle\vec{E}_{X}^{2}\rangle=\frac{1}{2a^{4}}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\ |\partial_{\eta}X_{+}|^{2}\simeq 2.6\times 10^{-4}\frac{e^{2\pi\xi}}{\xi^{3}}H_{\rm inf}^{4},$$		(II.9)
	$$\displaystyle\langle\vec{B}_{X}^{2}\rangle=\frac{1}{2a^{4}}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}k^{2}\ |X_{+}|^{2}\simeq 3.0\times 10^{-4}\frac{e^{2\pi\xi}}{\xi^{5}}H_{\rm inf}^{4}$$		(II.10)
	$$\displaystyle\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle=-\frac{1}{2a^{4}}\int\frac{d^{3}k}{(2\pi)^{3}}\ k\partial_{\eta}|X_{+}|^{2}\simeq-2.6\times 10^{-4}\frac{e^{2\pi\xi}}{\xi^{4}}H_{\rm inf}^{4},$$		(II.11)

for sufficiently large $\xi$, where $\langle\cdot\rangle$ denotes the quantum vacuum expectation value and the cutoff of the momentum is taken to be $2\xi aH_{\mathrm{inf}}$. See also Ref. Ballardini et al. (2019) for the renormalization of the gauge fields in axion inflation. We emphasize that the dominant contribution comes from superhorizon mode $k\lesssim aH_{\rm inf}$, so that the produced gauge field can be regarded to be constant over the Hubble scale, while the electric field and magnetic field point in an anti-parallel direction. The production rate of the physical helicity is given by

$$\partial_{\eta}(a^{3}\langle\mathcal{H}_{X}\rangle)=-2\int d^{3}xa^{4}\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle,$$

(II.12)

with which we can identify that a positive magnetic helicity is induced at the end of the inflation. The spatial average of the magnetic helicity or the magnetic helicity density, $h_{X}\equiv\langle{\cal H}_{X}\rangle/\mathbb{V}$ with $\mathbb{V}=a^{3}\int d^{3}x$ being the volume of spatial hypersurface in Friedmann– Lemaître-Robertson-Walker coordinates, at the end of inflation is roughly estimated as

$$h_{X}\sim-\left.\frac{2\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle}{3H}\right|_{\mathrm{end}}\simeq 1.7\times 10^{-4}\frac{e^{2\pi\xi}}{\xi^{4}}H_{\mathrm{inf}}^{3}.$$

(II.13)

Next, we discuss the effect when there is, e.g, a Dirac fermion $\psi$ interacting with gauge field during the inflation through $-g_{B-L}q_{\psi}\bar{\psi}\gamma^{\mu}X_{\mu}\psi$ where $q_{\psi}$ is the $B-L$ charge of this fermion. The amplified gauge fields induce the current of fermions, which backreacts on the dynamics of gauge fields. The extension to include many fermion species can be done by summing up their contributions. With one Dirac fermion species, the equation governing the energy transfer of the gauge field is given by

$$\dot{\rho}_{X}=-4H_{\mathrm{inf}}\rho_{X}-2\xi H_{\rm inf}\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle-g_{B-L}q_{\psi}\langle\vec{E}_{X}\cdot\vec{J}_{\psi}\rangle,$$

(II.14)

which is derived from Eq. (II.6). Here, $\rho_{X}$ is the expectation value of the energy density of the gauge field, $\rho_{X}=(1/2\mathbb{V})\int d^{3}x\langle\vec{E}_{X}^{2}+\vec{B}_{X}^{2}\rangle$, and $\vec{J}_{\psi}$ is the induced current of fermion $\psi$. While the gauge fields evolve as stochastic variables Fujita et al. (2022a), they can be approximated as constant and anti-parallel electric and magnetic fields at the horizon scale, as previously discussed. In the presence of a constant magnetic field background, the spectrum of fermions forms Landau levels, with the lowest level exhibiting chiral asymmetry and the higher level exhibiting chiral symmetry. When electric fields are applied in parallel to magnetic fields, states are excited along the Landau levels, inducing a fermion current Nielsen and Ninomiya (1983); Fukushima et al. (2008). The induced current arises from contributions of the lowest Landau level, which describes the chiral anomaly Nielsen and Ninomiya (1983), and from the higher Landau levels, which account for the Schwinger effect Heisenberg and Euler (1936); Schwinger (1951), representing tunneling processes between the discrete energy levels. Collectively, by taking $\vec{E}_{X}=(0,0,E_{X})$ and $\vec{B}_{X}=(0,0,-B_{X})$ this current is evaluated as Domcke and Mukaida (2018)

$$g_{B-L}q_{\psi}\langle J^{z}_{\psi}\rangle=\frac{(g_{B-L}q_{\psi})^{3}}{6\pi^{2}}\coth\left(\frac{\pi B_{X}}{E_{X}}\right)E_{X}B_{X}\frac{1}{H_{\rm inf}},$$

(II.15)

where scattering among particles is neglected. In the case of the standard model contents with, e.g., the addition of 3 generation right-handed neutrinos, we can modify Eq. (II.15) as

$$q^{3}_{\psi}\rightarrow 3\left(3\times\left(\frac{1}{3}\right)^{3}+3\times\left(\frac{1}{3}\right)^{3}+1+1\right),$$

(II.16)

where the first factor of 3 outside big parenthesis accounts for 3 generations. The first and the second contributions come from up and down quarks of each color, while the remaining terms are from electrons and neutrinos. Here, for simplicity, we assume the $B-L$ charge of the right-handed neutrinos is $1$.

Let us now examine how to include the effect of the induced current. While there are several approaches on this problem Domcke and Mukaida (2018); Gorbar et al. (2021, 2022); Fujita et al. (2022b), no quantitatively precise consensus has been obtained. Here we adopt the “equilibrium estimate” Domcke and Mukaida (2018) as an example to give a concrete estimate. Its idea is described as follows. In case there is no fermion, the energy feeding from the axion is balanced by the cosmic expansion giving a constant electromagnetic field configuration. On the other hand, in the presence of charged fermions, the induced current leads to additional energy transfer from the gauge field to the fermion sector, as reflected in the last term of the right-hand side of Eq. (II.14). Then, we expect the dynamical equilibrium $\dot{\rho}_{X}=0$, meaning that the energy in gauge field sector is balanced between the energy feeding from axion dynamics and the energy drain to both the fermion sector and cosmic expansion,

$$0=-2H_{\rm inf}\langle\vec{E}_{X}^{2}+\vec{B}_{X}^{2}\rangle-2\xi H_{\rm inf}\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle-g_{B-L}q_{\psi}\langle\vec{E}_{X}\cdot\vec{J}_{\psi}\rangle,$$

(II.17)

Approximating $\langle\vec{E}_{X}^{2}\rangle\simeq E_{X}^{2},\langle\vec{B}_{X}^{2}\rangle\simeq B_{X}^{2},\langle\vec{E}_{X}\cdot\vec{B}_{X}\rangle\simeq-E_{X}B_{X},$ and $\langle\vec{E}_{X}\cdot\vec{J}_{\psi}\rangle\simeq E_{X}\langle J^{z}_{\psi}\rangle$, and substituting induced current, Eq. (II.15), into Eq. (II.17), we obtain

$$0=-2H_{\rm inf}(E_{X}^{2}+B_{X}^{2})+2\xi_{\rm eff}H_{\rm inf}E_{X}B_{X},$$

(II.18)

with

$$\xi_{\rm eff}\equiv\xi-\frac{(g_{B-L}q_{\psi})^{3}}{12\pi^{2}}\coth\left(\frac{\pi B_{X}}{E_{X}}\right)\frac{E_{X}}{H_{\rm inf}^{2}}$$

(II.19)

denoting the effective instability parameter. The effect of fermion production is now implemented as a suppressed production of the helicity of the gauge field. We then estimate the resultant gauge field strength as follows. By assuming constant electromagnetic configuration, $\xi_{\rm eff}$ is also taken to be constant. The dynamical equilibrium equation is identical to the case where there are no fermions with $\xi\rightarrow\xi_{\rm eff}$ substituted in analytical solution, Eqs. (II.9) and (II.10). Given a specific value of $\xi$, equivalently $\dot{\phi}$, as it appears in Eq. (II.18), and assuming that the solutions for $E_{X}$ and $B_{X}$ take the same form as the analytical solutions in Eqs. (II.9) and (II.10), with $\xi$ replaced by $\xi_{\rm eff}$, one can determine the value of $\xi_{\rm eff}$ that satisfies the dynamical equilibrium condition, thereby obtaining the solution for the electromagnetic field configuration that includes the backreaction of fermions. The magnetic helicity at the end of inflation is also evaluated by Eq. (II.13) with $\xi$ replaced by $\xi_{\mathrm{eff}}$. As has been discussed, the produced fermions carry $B-L$ asymmetries. From Eq. (II.3), we evaluate the $B-L$ number density in the standard model sector at the end of inflation as

$$n_{B-L,SM}\equiv\frac{N_{B-L,SM}}{\mathbb{V}}=-\frac{3g_{B-L}^{2}}{8\pi^{2}}h_{X}\simeq-0.6\times 10^{-4}\frac{g_{B-L}^{2}e^{2\pi\xi_{\mathrm{eff}}}}{\pi^{2}\xi^{4}_{\mathrm{eff}}}H_{\mathrm{inf}}^{3}.$$

(II.20)

Next, we discuss the dynamics of the plasma with $B-L$ interaction and the $B-L$ gauge field after inflation, with instant reheating in mind. In this subsection, we focus on the case where the $B-L$ gauge field is massless. Throughout this paper, we require that the magnetohydrodynamics (MHD) description is valid for the $B-L$ gauge field and the magnetic helicity is conserved if the $B-L$ symmetry is not broken. We clarify the necessary conditions for these requirements in the following discussion. As is the case for the standard model hypercharge gauge field Giovannini and Shaposhnikov (1998b); Son (1999); Jedamzik et al. (1998); Banerjee and Jedamzik (2004); Domcke et al. (2019), the following MHD equation is to be satisfied;

	$\displaystyle\vec{J}_{B-L}$	$\displaystyle=\sigma(\vec{E}_{X}+\vec{v}\times\vec{B}_{X}),$		(II.21)
	$\displaystyle\nabla\times\vec{B}_{X}$	$\displaystyle=\vec{J}_{B-L},$		(II.22)
	$\displaystyle\frac{\partial}{\partial t}\vec{v}+(\vec{v}\cdot\nabla)\vec{v}$	$\displaystyle=\frac{\eta_{\mathrm{vis}}}{\rho+p}\nabla^{2}\vec{v}+\frac{1}{\rho+p}\quantity(\vec{J}_{B-L}\times\vec{B}_{X}),$		(II.23)

where $\vec{J}_{B-L}$ is the total $B-L$ current, $\sigma$ is the $B-L$ conductivity, $\vec{v}$ is the velocity of the fluid, $\eta_{m}athrm{vis}$ is the shear viscosity, $\rho$ is the energy density of the fluid, and $p$ is the pressure of the fluid. We have ignored the chiral magnetic effect Fukushima et al. (2008); Son and Surowka (2009); Neiman and Oz (2011) which does not change the evolution of the system unless the chiral plasma instability Joyce and Shaposhnikov (1997); Akamatsu and Yamamoto (2013); Rogachevskii et al. (2017); Schober et al. (2018); Kamada (2018) or the anomalous chirality cancellation Brandenburg et al. (2023a, b) becomes effective. In order to explain the present baryon asymmetry of the universe, the mechanism discussed in the subsequent subsections should take place. The $B-L$ magnetic field must be subdominant to the radiation energy density of the plasma, $B_{X}^{2}\ll\rho$. For simplicity, we assume all particles in the plasma are massless and in thermal equilibrium. Additionally, we assume that the typical scale of the dynamics, $L$, is much larger than the mean-free path of the $B-L$ charge carriers. The cosmic expansion does not appear in the MHD equations, because for massless gauge fields it can be removed by moving to the conformal frame Brandenburg et al. (1996). Here we use a conventional treatment following Ref. Domcke et al. (2019), which assumes that the typical spatial scales for the magnetic and velocity fields are the same. Although recent studies show that it is not always the case Uchida et al. (2023, 2024), this approach is enough to give a rough estimate.

Let us consider the validity of these equations. The first equation is the Ohm’s law for the $B-L$ current and can be qualitatively understood from the Drude model Arnold et al. (2000). In the Drude model, the conductivity of the $B-L$ current in the conformal frame is

$\displaystyle\sigma\simeq a(T)\sum_{i}q_{i}^{2}g_{B-L}^{2}g_{i}T^{2}\tau_{i},$

(II.24)

where $a(T)$ is the scale factor of the universe at $T$, $q_{i}$ is the $B-L$ charge of the charge carrier $i$, $T$ is the temperature of the universe, $g_{i}$ is the number of the internal degrees of freedom of $i$ and $\tau_{i}$ is the mean-free time for $i$. For the standard model particles, the dominant contribution comes from the right-handed charged leptons, where $\tau\sim 1/(g^{\prime 4}T)$, where $g^{\prime}$ is the $\text{U}(1)_{Y}$ gauge coupling constant. With new particles, such as a sterile neutrino, which interacts with the standard model particles only via the $B-L$ gauge field, $\tau$ can be longer and $\sigma$ can be bigger. However, as we have assumed $L\gg\tau$, there is an upper bound for $\sigma$, $\sigma\lesssim a(T)g_{B-L}^{2}T^{2}L.$ Thus, as a typical value, we take $\sigma\sim a(T)g_{B-L}^{2}T/g^{\prime 4}$ in the subsequent discussion.

The second equation is the Ampère’s law for the $B-L$ current; it is derived from one of the Maxwell’s equations,

$\displaystyle\nabla\times\vec{B}_{X}=\vec{J}_{B-L}+\frac{\partial}{\partial t}\vec{E}_{X}.$

(II.25)

Using the Ohm’s law, Eq. (II.21), we can eliminate the displacement current $\frac{\partial}{\partial t}\vec{E}_{X}$ if

	$\displaystyle\frac{1}{\sigma T^{\prime}}$	$\displaystyle\ll 1,$		(II.26)
	$\displaystyle|\vec{v}|\cdot\frac{L}{T^{\prime}}$	$\displaystyle\ll 1,$		(II.27)

where $T^{\prime}$ is the typical timescale of the dynamics. In the expanding universe, if the magnetic field is created during the inflation, $T^{\prime}$ is identified as the conformal time, $T^{\prime}\simeq\frac{a(T)}{a^{2}(T_{\text{re}})}H(T_{\text{re}})^{-1}$, where $a(T)$ is the scale factor of the universe, $H(T)$ is the Hubble parameter, and $T_{\text{re}}$ is the reheating temperature. In such a case, the first condition is satisfied if

$\displaystyle\frac{g^{\prime 4}}{g_{B-L}^{2}}\ll\frac{M_{p}}{T_{\text{re}}},$

(II.28)

where $M_{p}$ is the reduced Planck mass. The second condition is satisfied if $|\vec{v}|\ll 1$, which we will see is satisfied in the later discussion.

The third equation is the Navier-Stokes equation for the fluid and can be derived from the conservation of the energy-momentum tensor of the imperfect fluid Weinberg (1972). We have assumed that the fluid is uniform, $\rho,p=\text{constant}$. The viscosity is written as Weinberg (1971); Arnold et al. (2000)

$\displaystyle\eta_{\mathrm{vis}}=\frac{4}{15}\rho\tau.$

(II.29)

Therefore, we take the kinetic viscosity, $\nu$, defined as $\nu\equiv\eta_{\mathrm{vis}}/(\rho+p)$, to be around

$\displaystyle\nu\sim g^{\prime-4}(a(T)T)^{-1}$

(II.30)

with the contributions from the standard model particles being dominant.

Next, we discuss the dynamics of the magnetic field and the fluid. With the Ampère’s law, Eq. (II.22), the Ohm’s law, Eq. (II.21), and the other Maxwell equations, the equation of motion of the magnetic field is

$\displaystyle\frac{\partial}{\partial t}\vec{B}_{X}=\nabla\times\quantity(\vec{v}\times\vec{B}_{X})+\frac{1}{\sigma}\nabla^{2}\vec{B}_{X}.$

(II.31)

The first term on the right-hand side is the advection term and the second term is the diffusion term. If the diffusion term is much larger than the advection term, the equation is the diffusion equation and the magnetic field is quickly damped. To avoid this, we require the advection term to be much larger than the diffusion term:

$\displaystyle R_{m}\equiv\sigma L|\vec{v}|\gg 1,$

(II.32)

where we have defined the magnetic Reynolds number $R_{m}$. With this requirement, the magnetic helicity is indeed conserved:

$\displaystyle\frac{d}{dt}\mathcal{H}_{X}$

$\displaystyle\simeq-\frac{2}{T^{\prime}}R_{m}^{-1}\quantity(\frac{|\vec{v}|}{L/T^{\prime}})\mathcal{H}_{X}\ll-\frac{1}{T^{\prime}}\mathcal{H}_{X},$

(II.33)

just as the case for the standard model hypercharge gauge field Giovannini and Shaposhnikov (1998b); Son (1999); Jedamzik et al. (1998); Banerjee and Jedamzik (2004); Domcke et al. (2019).

To estimate the magnetic Reynolds number, we need to know the typical velocity of the fluid. Although we need numerical simulations for more precise evaluations, we can roughly estimate the velocity of the fluid from the Navier-Stokes equation, Eq. (II.23) Banerjee and Jedamzik (2004); Domcke et al. (2019). Using the Ampère’s law, Eq. (II.22), the Navier-Stokes equation is written as

$\displaystyle\frac{\partial}{\partial t}\vec{v}=-\quantity(\vec{v}\cdot\vec{\nabla})\vec{v}+\nu\nabla^{2}\vec{v}+\frac{1}{\rho+p}\quantity(-\frac{1}{2}\vec{\nabla}B_{X}^{2}+\quantity(\vec{B}_{X}\cdot\vec{\nabla})\vec{B}_{X}).$

(II.34)

If the first term on the right-hand side is much larger than the second term, namely the kinetic Reynolds number is much larger than unity,

$\displaystyle R_{e}\equiv\frac{|\vec{v}|L}{\nu}\gg 1,$

(II.35)

the velocity of the fluid is determined by the balance between the first and the third terms on the right-hand side as

$\displaystyle v\sim\frac{B_{X}}{T^{2}},$

(II.36)

and the equipartition between the magnetic energy and the kinetic energy is satisfied. On the other hand, if the kinetic Reynolds number is much smaller than unity, the velocity of the fluid is determined by the balance between the second and the third terms on the right-hand side as

$\displaystyle v\sim\sqrt{R_{e}}\frac{B_{X}}{T^{2}}\sim\frac{L}{\nu}\frac{B_{X}^{2}}{T^{4}}.$

(II.37)

In any case, the velocity of the fluid is much smaller than unity as the magnetic field is subdominant to the radiation energy density of the plasma, $B_{X}^{2}\ll\rho$, and the requirement, Eq. (II.27), is satisfied.

To summarize this subsection, we would like to clarify the requirements we have used. In the above discussion, we have required the large conductivity, Eq. (II.26), the small velocity of the fluid, Eq. (II.27) and the large magnetic Reynolds number, Eq. (II.32). We have already discussed that the first and the second requirements are easily met; the first requirement is satisfied if the reheating temperature is much larger than the Planck mass, Eq. (II.28), and the second requirement is automatically satisfied as the magnetic field is subdominant to the radiation energy density of the plasma. Assuming that the magnetic field is created during the inflation, the third assumption is reduced to

$\displaystyle\frac{g_{B-L}^{2}B_{X0}H(T_{\text{re}})^{-1}}{g^{\prime 4}T_{\text{re}}}\times\min\quantity(1,\frac{g^{\prime 4}B_{X0}H(T_{\text{re}})^{-1}}{T_{\text{re}}})\gg 1,$

(II.38)

where $B_{X0}$ is the amplitude of the magnetic field at the reheating and we have used $L\sim(a(T_{\mathrm{re}})H(T_{\text{re}}))^{-1}$.

In Fig. 1, we illustrate the parameter space where the $B-L$ helicity is well-conserved, under the assumption that the magnetic field is generated during inflation and the reheating process is instantaneous. The light-shaded regions indicate where the energy density of the $B-L$ gauge field is sufficiently small (lower boundary) and where the helicity-to-entropy ratio is sufficiently large (upper boundary). Specifically, for the lower boundary, we require that the energy density of the $B-L$ gauge field is less than $1\%$ of the total energy density at the reheating temperature. For the upper boundary, we define the helicity-to-entropy ratio as

$\displaystyle\eta_{\mathcal{H}}\equiv\quantity|\frac{3g_{B-L}^{2}}{8\pi^{2}}\frac{h_{X}}{s(T_{\text{re}})}|,$

(II.39)

where $s$ is the entropy density of the universe. We require that the helicity-to-entropy ratio is more than $\frac{79}{28}\eta_{0}$, where $\eta_{0}\simeq 8.7\times 10^{-11}$ is the observed baryon-to-entropy ratio in the current universe Aghanim et al. (2020), to ensure that the $B-L$ helicity decay would explain the present baryon asymmetry of the universe as we will discuss in the next subsection. The dark-shaded regions indicate where the magnetic Reynolds number is larger than the unity (Eq. (II.38)) such that the $B-L$ helicity is well-conserved among the corresponding light-shaded regions. In these regions, we show the region where the kinetic Reynolds number is larger than unity, $R_{e}>1$, as the meshed regions. We also show the contours of the effective inflaton-$B-L$ instability parameter, $\xi_{\text{eff}}=3,4,\cdots,10$, according to the equilibrium estimation reviewed in the previous subsection.

The contours of $\xi_{\text{eff}}$ show that the effective instability parameter can be large if the gauge coupling $g_{B-L}$ is small. This is because the backreaction to the effective instability parameter is smaller for smaller $g_{B-L}$. For the larger $\xi_{\text{eff}}$, the helicity-to-entropy ratio can be larger despite the smaller gauge coupling. For the $B-L$ helicity to be well-conserved, the magnetic Reynolds number should be much larger than unity, $R_{m}\gg 1$. This requires that the magnetic field is sufficiently large, and therefore the viable parameter space lies in the smaller $g_{B-L}$ and larger $\xi_{\text{eff}}$ regions.

Before moving to the next subsection, let us comment on the uncertainty of our discussion about the MHD evolution. Since the MHD equations describe a fully non-linear system, the estimates are not very precise to determine its evolution. While a sufficiently large magnetic Reynolds number is a necessary condition for the survival of the magnetic helicity, our estimate is based on rough order estimates and hence is not quantitatively precise. For example, the estimate of the velocity field at a small kinetic Reynolds number might not be correct since it is not clear if a turbulent dynamics takes place, which is essential for our order estimate. This is the reason why Ref. Domcke et al. (2023) adopted several ways to estimate the magnetic Reynolds number. Moreover, recent studies suggest that the typical length scale for the magnetic field and velocity field might be different Uchida et al. (2023, 2024), which violates the first assumption of our estimate. Therefore, one should regard the parameter space for the survival of the $B-L$ helicity in Fig. 1 as just a demonstration for a rough estimate. The origin of this uncertainty comes from the hierarchy between the magnetic and kinetic Reynolds number, $R_{m}/R_{e}\sim g^{2}_{B-L}/g^{\prime 8}$, for not so small $g_{B-L}$, which is difficult to be implemented in the numerical simulation. In order to determine precisely the condition for the helicity to be well-conserved, more precise studies in the case where the magnetic and kinetic Reynolds numbers are hierarchical are needed, but it is beyond the scope of the present study.

Let us discuss the fate of the $B-L$ helicity in the presence of a Higgs field with a negative mass squared. It is important to note that even if such a Higgs field exists, the phase of the system may differ from the one without helicity due to the anomaly equation, Eq. (II.3), which links the helicity to the fermion number Rubakov (1986). However, in our case of interest, where the helicity-to-entropy ratio is much smaller than unity, the phase of the system is not significantly affected by the helicity, and the $B-L$ symmetry is broken.

In the Higgs phase, the magnetic helicity can decay through two mechanisms: the imaginary part of its self-energy and Ohmic dissipation. The former would be analogous to the decay of a $B-L$ gauge boson particle in a vacuum. Consequently, we expect the decay rate to be similar to that of the $B-L$ gauge boson in a vacuum. The lifetime $\tau_{X}$ is thus given by

$\displaystyle\tau_{X}^{-1}\simeq\Gamma_{X}\equiv\frac{g_{B-L}^{2}\sum q_{i}^{2}}{24\pi}m_{X}.$

(II.40)

Here, we assume the fermions are massless.

To estimate the decay rate due to Ohmic dissipation, we first write down the equation of motion for the $B-L$ gauge field and the matter fields, which may replace Eq. (II.21) and Eq. (II.22) in the previous subsection. For the time being, let us consider the Minkowski spacetime, $a=1$ for simplicity. We will back to this point later. With the existence of a would-be Nambu-Goldstone boson $\phi_{M}$ for the $B-L$ gauge symmetry, the effective Lagrangian of the $B-L$ gauge field is

$\displaystyle\mathcal{L}=-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}+\frac{1}{2}m_{X}^{2}\quantity(X_{\mu}-g_{B-L}^{-1}v_{B-L}^{-1}\partial_{\mu}\phi_{M})^{2}-X^{\mu}J_{B-L,\mu},$

(II.41)

where $m_{X}$ is the mass of the $B-L$ gauge field and $v_{B-L}$ is the VEV of the Higgs field breaking the $B-L$ gauge symmetry ^‡‡‡Here, for simplicity, we ignore the Chern-Simons term between $X_{\mu}$ and $\phi_{M}$.. The mass of the $B-L$ gauge field is $m_{X}=qg_{B-L}v_{B-L}$, where $q$ is the $B-L$ charge of the Higgs field. We may redefine the $B-L$ gauge field as $X_{\mu}\to X_{\mu}+g_{B-L}^{-1}v_{B-L}^{-1}\partial_{\mu}\phi_{M}$ to get rid of the would-be Nambu-Goldstone boson. The equation of motion of the $B-L$ gauge field is then

	$\displaystyle\nabla\cdot\vec{E}_{X}$	$\displaystyle=\rho_{B-L}-m_{X}^{2}X_{0},$		(II.42)
	$\displaystyle\nabla\times\vec{B}_{X}-\frac{\partial}{\partial t}\vec{E}_{X}$	$\displaystyle=\vec{J}_{B-L}-m_{X}^{2}\vec{X}.$		(II.43)

or, equivalently,

$\displaystyle\quantity(\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}+m_{X}^{2})X^{\mu}=J_{B-L}^{\mu}.$

(II.44)

Here, again we have ignored the chiral magnetic effect. In this paper, we assume that the $B-L$ symmetry is Higgsed before the anomalous chirality cancellation Brandenburg et al. (2023a, b) would occur. The gauge transformation of the matter fields can absorb the redefinition, and the equations of motion of the matter fields are not changed. Thus, we assume that the Ohm’s law, Eq. (II.21), is still valid after the $B-L$ gauge symmetry is broken. The time dependence of the electric field is not negligible, and the electric conductivity may be different from the one before the $B-L$ gauge symmetry is broken. However, for the time being, we assume $\sigma\sim g_{B-L}^{2}T^{2}\tau\sim g_{B-L}^{2}T/g^{\prime 4}$ if the temperature $T$ is not much smaller than $v_{B-L}$. We will return to this point later.

From Eq. (II.21) and Eq. (II.44), we can eliminate the current and obtain the time evolution of the gauge field as

$\displaystyle\quantity(\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}+m_{X}^{2}+\sigma\frac{\partial}{\partial t})\vec{X}=-\sigma\vec{\nabla}X^{0}+\sigma\vec{v}\times\quantity(\vec{\nabla}\times\vec{X}).$

(II.45)

To further simplify the equation, we assume the typical spatial scale of the $B-L$ gauge field is much larger than the typical time scale of the $B-L$ gauge field, $L\gg T^{\prime}$. We will discuss the validity of this assumption later. Also, in the situation of our interest, the net $B-L$ charge is zero or, at least, much smaller than the current and we may neglect the term $X^{0}$. The equation then becomes

$\displaystyle\quantity(\frac{\partial^{2}}{\partial t^{2}}+m_{X}^{2}+\sigma\frac{\partial}{\partial t})\vec{X}\simeq 0.$

(II.46)

This equation is the damped harmonic oscillator equation and the solution is

$\displaystyle\quantity|\vec{X}|\simeq\quantity|\vec{X}(0)|e^{-\frac{t}{\tau_{\text{MHD}}}},$

(II.47)

where

$\displaystyle\tau_{\text{MHD}}=\begin{cases}\frac{2}{\sigma}\quad&\text{for }2m_{X}>\sigma,\\ \frac{\sigma+\sqrt{\sigma^{2}-4m_{X}^{2}}}{2m_{X}^{2}}\quad&\text{for }2m_{X}<\sigma.\end{cases}.$

(II.48)

Here, for $2m_{X}<\sigma$, we have taken a solution with a decreasing lifetime as the mass increases.

Let us now make a few comments. First, we have assumed that the electric conductivity is not much different from the one before the $B-L$ gauge symmetry is broken. However, in the broken phase, the time dependence of the electric field can be much larger than the spatial dependence of the electric field; if the fluid is non-conducting, the time frequency of the electric field is $m_{X}$. When the gauge field is oscillating, the electric conductivity is different from the one for non-oscillating gauge field. According to the Drude model Ashcroft and Mermin (2011), the electric conductivity, $\sigma(\omega)$, for oscillating gauge field with the frequency $\omega$ is now complex and

$\displaystyle\sigma(\omega)=\frac{\sigma}{1-i\omega\tau}.$

(II.49)

For $T\ll v_{B-L}$, $m_{X}\tau\gg 1$ and the fluid becomes non-conducting. Second, we have assumed that the typical spatial scale of the $B-L$ gauge field is much larger than the typical time scale of the $B-L$ gauge field. This is justified for

$\displaystyle m_{X}\gtrsim\sqrt{\frac{|\vec{v}|\sigma}{L}}=\frac{\sqrt{R_{m}}}{L}.$

(II.50)

Otherwise, the decay rate might be less effective. In the following, however, we assume that the discussion in the previous paragraph is unchanged.

Taking both decay effects of the gauge field itself into account, we, very roughly, conclude the time evolution of the magnetic helicity is given as

$\displaystyle\mathcal{H}_{X}\simeq\mathcal{H}_{X}(0)e^{-\frac{t}{\tau_{\mathcal{H}}}},$

(II.51)

where

$\displaystyle\tau_{\mathcal{H}}\equiv\frac{1}{2}\min\quantity(\tau_{\text{MHD}},\tau_{X}).$

(II.52)

However, we emphasize that for more precise estimation, we need numerical simulations of MHD and more careful analyses of the decay rate of the helicity.

We have so far ignored the cosmic expansion and the $B-L$ phase transition. Let us estimate when the helicity decays in the cosmic history and examine its validity. Here we assume that the gauge field obtains the mass via the Higgs mechanism,

$\displaystyle m_{X}=qg_{B-L}v_{B-L}(T),$

(II.53)

where $v_{B-L}(T)$ is the VEV of the Higgs field at the temperature $T$. We also assume that the $B-L$ phase transition is of the second order. $v_{B-L}(T)$ depends on the critical exponent of the $B-L$ phase transition, $\beta$, $v_{B-L}(T)\propto(T_{c}-T)^{\beta}$, where $T_{c}$ is the critical temperature of the $B-L$ phase transition. Let $T_{HD}$ be the temperature when the helicity decays. As we have discussed in this section, it is difficult to estimate the decay rate of the helicity precisely, but for simplicity, we assume that $T_{HD}$ is the temperature when the helicity decay time is equal to the Hubble time,

$\displaystyle H(T_{HD})^{-1}=\tau_{\mathcal{H}}|_{m_{X}=qg_{B-L}v_{B-L}}.$

(II.54)

Note that near the critical temperature, the universe undergoes a quenching process where the system cannot keep up with the rapid changes in temperature Kibble (1976); Zurek (1985). This quenching and the formation of the topological defects could potentially impact the MHD description and the helicity decay. However, for simplicity, we ignore these effects and assume our estimation remains valid.

To evaluate the decay time, let us focus on the decay time of the helicity from the MHD, $\tau_{\text{MHD}}$; we use $H(T_{HD})^{-1}\lesssim\tau_{\text{MHD}}\sim\sigma/m_{X}^{2}$. Then, we obtain the following inequality:

$\displaystyle q\frac{v_{B-L}(T_{HD})}{T_{HD}}\lesssim\frac{1}{g^{\prime 2}}\sqrt{\frac{H(T_{HD})}{T_{HD}}}\ll 1.$

(II.55)

Here, because $L\gtrsim H(T_{HD})^{-1}|\vec{v}|$ due to the MHD dynamics Banerjee and Jedamzik (2004)^§§§Here we suppose that the coherence length of magnetic field is larger than the Alfvén scale. If we adopt the reconnection-driven turbulence Uchida et al. (2023, 2024), the coherence length can be shorter., the requirement, Eq. (II.50), is satisfied at least for $q\frac{v_{B-L}(T_{HD})}{T_{HD}}\sim\frac{1}{g^{\prime 2}}\sqrt{\frac{H(T_{HD})}{T_{HD}}}$. If we assume the Landau-Ginzburg theory,

$\displaystyle v_{B-L}(T)=v_{B-L}^{0}\quantity|1-\frac{T}{T_{c}}|^{\frac{1}{2}}$

(II.56)

with $v_{B-L}^{0}\sim T_{c}$, we can conclude $T_{c}-T_{HD}\ll T_{c}$; the helicity quickly decays after the phase transition, in a shorter time scale than the Hubble time. Therefore, we expect that the effect of the cosmic expansion is negligible. If this condition is not satisfied, for example, if the mass of the $B-L$ gauge field is Stueckelberg-like ^¶¶¶The ’t Hooft anomaly may be canceled by the four-dimensional analogue of the Green-Schwarz mechanism Green and Schwarz (1984); Faddeev and Shatashvili (1986); Krasnikov (1985); Babelon et al. (1986); Harada and Tsutsui (1987); Preskill (1991), but the cutoff scale cannot be arbitrarily large in that case. and no phase transition occurs, the helicity may also decay due to cosmic expansion.