^{3}.

The spontaneous baryogenesis scenario explains how a baryon asymmetry can develop while baryon violating interactions are still in thermal equilibrium. However, generation of the chemical potential from the derivative coupling is dubious since the chemical potential may not appear after the Legendre transformation. The geometric phase (Pancharatnam-Berry phase) results from the geometrical properties of the parameter space of the Hamiltonian, which is calculated from the Berry connection. In this paper, using the formalism of the Berry phase, we show that the chemical potential defined by the Berry connection is consistent with the Legendre transformation. The framework of the Berry phase is useful in explaining the mathematical background of the spontaneous baryogenesis and also in calculating the asymmetry of the nonthermal particle production in time-dependent backgrounds. Using the formalism, we show that the mechanism can be extended to more complex situations.

Quantum mechanics is distinguishable from the classical counterpart by the phase factor, which explains many characteristic phenomena of the quantum theory. Among those, the Aharonov-Bohm(AB) effect

See also Ref.

As we explain later, the first term (the Berry connection) gives the chemical potential when the spontaneous baryogenesis scenario is considered in the formalism of the Berry phase. However, since the Berry connection vanishes in the adiabatic limit (although its integral may not vanish in a topological background), the evolution has to be nonadiabatic in order to generate a sensible chemical potential. When we consider the spontaneous baryogenesis scenario, the second term (or the higher terms) gives the particle production due to the time-dependent background.

To show our idea in a simple model, we start with the Schrödinger equation for the state

Our discussion here is implicitly based on a kaon, where

Here the capital “

If

Note that

See also Appendix

Formally, the equivalence class of state vectors or “projective Hilbert space” is defined using an arbitrary function

Here,

The Berry phase is defined by the integral of the Berry connection along the orbit. If the Berry phase is defined for a cyclic process starting from

One will find that the mechanism is similar to the spontaneous baryogenesis scenario

See Refs.

Normally, when one discusses the nonadiabatic effect for the Berry phase, his (her) motivation would be to calculate the Berry and the nonadiabatic Berry phases. However, our present discussion is not for the calculation of the Berry phase in a cyclic process, but for finding the sources of the asymmetry in time-dependent backgrounds. We hope there is no misdirection in our arguments.

In the next section, using simple setups, we are going to discuss why the formalism of the Berry phase can be used to understand the scenario of the spontaneous baryogenesis. Then, we will consider some extension of the scenario, to solve more complex situations.

In the early Universe, a field can be placed away from its true minimum. Then the field starts to roll down on the potential during the evolution of the Universe, and it starts to oscillate around the minimum. Sometimes, the trajectory of the oscillation is not a straight line passing through the minimum, but an oval form, since a

Below, we are going to explain the basic idea using the kaonlike model. From the Schrödinger equation

Above, we have calculated the Berry connection with respect to

If

In the past, the effective chemical potential has been studied in particle cosmology in various ways. Spontaneous baryogenesis scenario uses higher-dimensional operators such as

It is easy to show that the chemical potential in the Hamiltonian can bias the particle number densities in the thermal equilibrium. In that sense, the appearance of the chemical potential in the Hamiltonian explains the asymmetry in the thermal equilibrium.

Before moving forward, it will be useful to show explicitly the relation between the chemical potential and the Berry phase in the Lagrangian. We start with the Hamiltonian

Equation

Note that the replacement

As a useful toy model, we first consider a time-dependent background for a complex scalar field and calculate the perturbative particle production, then examine the sources of the asymmetry.

We start with a complex scalar field

To introduce the bias, we introduce

Let us see the origin of the asymmetry in the light of the chemical potential and the Berry connection, not in terms of the interference between terms. To find the origin of the asymmetry, consider a constant (or a slowly varying) chemical potential to define

Note that in the above case the nonadiabatic Berry phase may also appear since the particle production in the above argument is due to the nonadiabatic transition between states. The phase may not be important in the perturbative calculation discussed above, but it could be important in the nonperturbative limit

Normally, the Berry phase is not defined specifically for the spontaneous violation of a symmetry. A naive expectation is that the formalism based on the Berry phase may be used in wider circumstances than the Nambu-Goldstone effective action. To show how it works, we consider the simplest extension in the following, to show that neither the spontaneous symmetry breaking nor the derivative coupling is needed for generating the effective chemical potential. The model will be used also for the nonequilibrium particle production in Sec.

The spontaneous baryogenesis can be discussed for (1) the chemical potential in the thermal equilibrium and (2) the nonperturbative particle creation caused by the time-dependent background. The latter can be discussed for the thermal equilibrium and may compensate the simple discussion based on the chemical potential in the thermal background. However, in our paper, we are considering the nonperturbative particle production only when the thermal background is negligible. Therefore, we are calling the latter process “nonequilibrium particle production” and discriminate it from the former.

We consider a simple example given by

A straight motion with

Our next topic is the nonequilibrium particle production in a more realistic scenario. To compare our results with the conventional spontaneous baryogenesis, we first review the calculation given in Ref.

In our formalisms of the Berry transformation, the assignment is

In the above, we have followed Ref.

Besides the symmetry discussed above, the Lagrangian is symmetric under the exchange

Remember that in our previous discussion,

Using the calculation in Sec.

The above arguments seem to be suggesting that the assignment

Now consider the particle production in the time-dependent and nonequilibrium background. The production can be biased by the oscillation given by

To check the validity of the above calculation in wider circumstances, let us remove the condition

Using

To avoid the cancellation, or to introduce interference between multiple contributions, one can introduce higher terms. For instance, one can introduce

In the followings, we will consider the Dirac and the Majorana fermions and examine the origin of the asymmetry in the nonperturbative particle production.

Usually, the Dirac mass is defined to be real since the redefinition of the field can remove the phase. However, if the Dirac mass is time dependent, the Berry connection appears. Let us introduce the complex Dirac mass, which is rotating with

To show that our expectation is correct, we start with a simple example. Since the basic idea of the fermionic preheating has already been discussed in ref.

The sign in front of

To introduce the asymmetry, we consider the higher term, which is given by

Particle creation area (nonadiabatic area) for

For the antimatter state, the decoupled equation

To conclude the particle production due to the Dirac mass, there is no total asymmetry even if the higher terms are introduced. The asymmetry of the helicity appears for each (matter and antimatter) state because the event of the particle production splits. Similarly, the matter-antimatter asymmetry appears for each helicity state. However, these partial asymmetries do not cause generation of the total asymmetry.

To avoid the cancellation of the asymmetries, which has been seen for the Dirac mass, we will consider the Majorana fermion in the followings. Note that unlike the Dirac fermion, decoupling of the equations is not well-defined at the massless point. First, we consider the rotational oscillation and compare the perturbative and the nonperturbative particle production. Then we examine the nonrotational motion. We will show that unlike the previous models the asymmetry can be generated without introducing the higher terms.

In this section, we consider simple oscillation of

To understand more about the sources of the asymmetry, we will consider the nonperturbative effect (tunneling) using another schematic calculation. We use the Lagrangian given by

One can find similar calculation in Ref.

This equation can be written as

The top figure shows the original Landau-Zener tunneling. Equation

Let us temporarily assume that the amplitude

When the amplitude decreases with time, the delay is approximately given by

From Eq.

Instead of considering the rotational motion, we are going to introduce the Majorana mass given by

In this section, using the nonperturbative calculation, we will show that the above extension can generate the asymmetry. We consider significant particle production, which is realized when the oscillation starts with

This manipulation is not possible in the standard calculation of preheating, since

Unlike the perturbative expansion discussed in Sec.

In the above models, the source of the phase is designed to be very simple. The phase in the off-diagonal element determines the Berry phase, and there is the obvious correspondence between them. We have also seen that a simple extension of the scenario (i.e,

In the above models, all phases in the Hamiltonian can be removed by the field rotation, which we called “the Berry transformation.” Now our question is very simple. “What happens if the fields are multiplied and the Berry transformation has to be given by a complex function of the original parameters?”

One can examine the above idea in the three-family fermion model. One can introduce the flavor index

Now the

The minimal multifield extension that realizes the above idea is given by a complex scalar field couples to a real scalar field. Consider the following Lagrangian;

The chemical potential may cause a problem in the Legendre transformation if it is explained by the derivative coupling of a field in motion. The reason is very simple. If the chemical potential is introduced using the derivative of the field

In this section, we are going to show a more transparent consistency relation between the Berry connection and the Legendre transformation. It is easy to see that the chemical potential defined using the Berry connection appears in the Hamiltonian (after the Legendre transformation) in the expected form. Note also that the Berry transformation and the Legendre transformation obviously commute. We start with the Lagrangian

Let us consider the Berry transformation. We define

Using the Legendre transformation, one can calculate the Hamiltonian of the system. Since the Berry transformation is nothing but inserting “

As is already discussed in Sec.

In this paper, we examined the spontaneous baryogenesis scenario using the framework of the Berry phase. In this approach, the chemical potential is not the derivative coupling of the Nambu-Goldstone boson but the Berry connection defined for the “Berry transformation”. In this paper, the “Berry transformation” is defined specifically for the transformation, which removes the phase in the Hamiltonian during the evolution.

The merit of this approach is the obvious consistency between the Hamiltonian and the Lagrangian formalisms. The Berry transformation commutes with the Legendre transformation, and the chemical potential in the thermal equilibrium is obvious in this approach.

Then, using the Berry transformation as a useful tool for the calculation, we examined the asymmetry generation during the particle production in time-dependent backgrounds. In the framework of the Berry phase, the chemical potential is given by the Berry connection associated with the conventional Berry phase. The conventional Berry phase may appear both in the adiabatic and in the nonadiabatic evolutions. On the other hand, the particle production in the time-dependent background is caused by the transition between states. In the framework of the Berry phase, this can introduce the nonadiabatic Berry phase, which appears only in the nonadiabatic evolution and vanishes in the adiabatic limit. In this paper, we compared the perturbative and the nonperturbative calculations. Our speculation is that the asymmetry in the nonperturbative particle production can be explained by the resurgence theory

Besides the discrepancy between the perturbative and the nonperturbative calculations, we also examined the effect of the expansion

For the rotational oscillation of the time-dependent Majorana mass term, we calculated the nonperturbative particle production using the Landau-Zener tunneling. In this case, the nonperturbative calculation is explicitly defined for the tunneling process and the source of the asymmetry is the split of the tunneling.

The model can also be extended to multifield models, in which the Berry phases are complex functions of the original parameters. Although the parameter dependence of the

From the results, we found that the Berry phase and the Berry connection are giving a natural framework of the spontaneous baryogenesis scenario. The asymmetry of perturbative and nonperturbative particle production will be understood in the resurgence theory. Although in this paper we have considered only a time-dependent parameter, one can also consider a “space”-dependent parameter as the source of the Berry connection, which may appear on topological defects such as walls and strings.

S. E. is supported by the Heising-Simons Foundation Grant No. 2015-109 and by JSPS KAKENHI Grants No. JP18H03708 and No. JP17H01131.

In this section, we derive the formula

Using these equations of motion, we will calculate the number density with Yang-Feldman formalism where the operator field is represented by an asymptotic filed and Green function.

The formal solution called as Yang-Feldman equation for Eqs.

In

The number density

The net number density of the Majorana fermion can be defined by the difference of helicity. Using

In the following, we will show that the contributions from the zeroth, the first and the second order vanishes and the leading contribution appears from the third order.

At the zeroth order, we just consider with (A36) in

The first order of

Taking into account

Taking into account

The asymmetry can be seen as the biased mixing in the eigenstates. In this section, we are going to review basic strategies in this direction.

It would be useful to remember how the bias appears from the loop corrections. Although many fields are required for the quantum correction, which is not explicitly discussed in this paper, the correction is symbolically given by a Hermite matrix

Similar bias can be introduced by the chemical potential. Of course, the statistical bias is obvious in the thermal background (as far as the chemical potential appears in the Hamiltonian), but the thermal equilibrium is not considered here. Here, the particle production is assumed to be nonthermal, and the decay of the particle is assumed to be much slower than the mixings. Initially, the chemical potential is supposed to be a constant (i.e, the Berry connection gives a Hermite and time-independent contribution). More simply, one may think that the particle production proceeds with the pure eigenstates. Later in Sec.

At the beginning of this section, we have explained that

To see the matter-antimatter asymmetry (bias) in the eigenstates, it is useful to calculate eigenvectors of the matrix given by

The above arguments for the matter-antimatter bias can be applied to the Majorana fermions. In the past, the one-loop correction for the wave function mixing of singlet (Majorana) neutrinos has been calculated in Ref.

Let us consider the bias caused by the chemical potential. Our calculation for the chemical potential can be applied straightforwardly to the Majorana fermion. We introduce the Majorana mass given by

Since we are choosing

The most useful example in this direction is the kaon.

Moreover, these “eigenstates” are not the true eigenstates whenOne can introduce the Dirac mass

If

Note that the above arguments are not valid when the chemical potential is changing fast.