^{1}

^{2}

^{3}

^{4}

^{1}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

^{3}.

A new mechanism for generating particle number asymmetry (PNA) has been developed. This mechanism is realized with a Lagrangian including a complex scalar field and a neutral scalar field. The complex scalar carries

The origin of BAU has long been a question of great interest in explaining why there is more baryon than antibaryon in nature. Big bang nucleosynthesis (BBN) [

In the present paper, we further extend the model of scalar fields [

The model which we have proposed [

As for the computation of the PNA, we use 2PI formalism combined with density operator formulation of quantum field theory [

This paper is organized as follows. In Section

In this section, we present a model which consists of scalar fields [

Let us start by introducing a model consisting of a neutral scalar,

We rewrite all the fields in terms of real scalar fields,

The cubic interactions and their properties.

Cubic interaction coupling | Property |
---|---|

| – |

| |

| – |

| |

| |

| |

| |

Nöether current related to the

Below we consider the matrix element of the density operator given in (

In this section, we derive the equations of motion, i.e., the Schwinger-Dyson equations (SDEs) for both Green’s function and field. SDEs are obtained by taking the variation of 2PI EA with respect to fields and Green’s functions, respectively. In addition, we also provide the initial condition for Green’s function and field to solve SDEs.

2PI CTP EA in curved space-time has been investigated in [

If one sets the source terms to be the ones given in (

The Green’s functions and expectation value of fields are derived as solutions of the SDEs which are obtained with 2PI EA. The 2PI EA is related to the generating functional

Let us write the 2PI EA

Now let us derive SDEs for both Green’s function and field. These equations can be obtained by taking the variation of the 2PI EA,

In the following, we first derive SDEs for the field. The variation of the 2PI EA in (

Next, the equation of motion for Green’s function is derived in the following way. The variation of the 2PI EA in (

Next, we rescale Green’s function, field and coupling constant of interaction as follows:

Next SDEs for the rescaled Green’s function in (

In this subsection, the initial conditions for Green’s function and field are determined. For simplicity, let us look back to example model for one real scalar field. We first compute the initial condensation of the field

Next we will compute the initial condition for Green’s function

The above results with the example model can be extended to our model and we summarize them as

Next we derive the time derivative of the field and Green’s function at the initial time

The SDEs obtained in the previous section allow us to write the solutions for both Green’s functions and fields in the form of integral equations. In this section, we present the correction to the expectation value of the PNA up to the first order contribution with respect to the cubic interaction. For this purpose, in Subsection

The SDEs in present work are inhomogeneous differential equations of the second order. To solve the differential equation, the variation of constants method is used. With the method, the solutions of SDEs are written in the form of integral equations. We solve the integral equation perturbatively and the solutions up to the first order of the cubic interaction are obtained. We first write the solutions of fields as

Next we write down the solution of Green’s function as follows:

Next we compute the PNA in (

As was indicated previously, we will further investigate the expectation value of the PNA for the case of time-dependent scale factor. For that purpose, one can expand scale factor around

Now let us briefly go back to (

Now let us investigate the expectation value of PNA under these approximations. For the case that

Before closing this section, we compute the production rate of the PNA per unit time which is a useful expression when we understand the numerical results of the PNA. We compute the time derivative of the PNA for the case of the constant scale factor

In this section, we numerically study the time dependence of the PNA. The PNA depends on the parameters of the model such as masses and coupling constants. It also depends on the initial conditions and the expansion rates of the universe. Since the PNA is linearly proportional to the coupling constant

How the PNA behaves with respect to time is discussed in the following Subsections

Let us now consider the PNA which has the longer period. While we investigate the dependence of several parameters, we fix two parameters as

Dependence on temperature

Dependence on parameter

The

Dependence on the expansion rate

Now we investigate the PNA with the shorter period. In Figure

Dependence on temperature

In Figure

Dependence on frequency

The expansion rate

In this subsection, we present a comparison of two different periods of the time evolution of the PNA. In Figure

Comparison between two different periods of the time evolution of the PNA. We use the dimensionless time

In this subsection, we interpret the numerical simulation in a specific situation. We assume that the time dependence of the scale factor is given by the one in radiation dominated era. We also specify the unit of the parameters, time and temperature. By doing so, we can clarify implication of the numerical simulation in a more concrete situation.

Specifically, the time dependence of the scale factor is given as follows:

From the expression in (

As an example, we study the implication of the numerical simulation shown in Figure

The mass parameters in GeV unit for both longer and shorter period cases in Figure

Mass parameter (GeV) | The shorter period | The longer period |
---|---|---|

| | |

| ||

| | |

| ||

| | |

One can also estimate the size of PNA. Here, we consider the maximum value of the PNA for the longer period case in Figure

In this paper, we developed a new mechanism for generating the PNA. This mechanism is realized with the specific model Lagrangian which we have proposed. The model includes a complex scalar. The PNA is associated with

The results show that the PNA depends on the interaction coupling

The classification of

The effect | The origin |
---|---|

Dilution | The increase of volume of the universe due to expansion, |

| |

Freezing interaction | The decrease of the strength of the cubic interaction |

| |

Redshift | The effective energy of particle as indicated in Eq. ( |

We have numerically calculated time evolution of the PNA and have investigated its dependence on the temperature, parameter

To show how the mechanism can be applied to a realistic situation, we study the simulated results for radiation dominated era when the degree of freedom of light particles is assumed to be

Compared with the previous works [

In our model, even for the longer period case, the oscillation period is shorter than the Hubble time

In this subsection, we provide the general solution of SDEs. Let us introduce the following differential equation for a field

The method of variation of constants has been employed to determine the solution of (

Let us now write

Next we move to consider the SDEs for the field in (

In this subsection, we consider the SDEs for Green’s function in (

In the following, we will obtain SDEs for Green’s functions at

Two paths to obtain

Now let us explain how one can derive (

We first consider the differential equation in (

where

Next we consider the differential equation in (

where

Substituting (

Equation (

In the following subsection, we derive the free parts of Green’s function which are the zeroth order of cubic interaction. From (

Now we move to consider the interaction parts of Green’s function. From (

To summarize this subsection, let us write the SDEs of Green’s functions in the form of integral equations. Omitting the upper and lower indices

One first considers a couple of general homogeneous differential equations given in (

We first compute solution for

Now we move to compute for

In this appendix, we present

In this section, we provide both time and momentum integrations in the expression of the expectation value of the PNA. Let us first consider (

Below we first carry out time integration of

We move now to compute

The next task is to integrate (

Below we consider time and momentum integrations of the second part of PNA which is (