#### Abstract

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 charge which is associated with the PNA. It is written in terms of the condensation and Green’s function, which is obtained with two-particle irreducible (2PI) closed time path (CTP) effective action (EA). In the spatially flat universe with a time-dependent scale factor, the time evolution of the PNA is computed. We start with an initial condition where only the condensation of the neutral scalar is nonzero. The initial condition for the fields is specified by a density operator parameterized by the temperature of the universe. With the above initial conditions, the PNA vanishes at the initial time and later it is generated through the interaction between the complex scalar and the condensation of the neutral scalar. We investigate the case that both the interaction and the expansion rate of the universe are small and include their effects up to the first order of the perturbation. The expanding universe causes the effects of the dilution of the PNA, freezing interaction, and the redshift of the particle energy. As for the time dependence of the PNA, we found that PNA oscillates at the early time and it begins to dump at the later time. The period and the amplitude of the oscillation depend on the mass spectrum of the model, the temperature, and the expansion rate of the universe.

#### 1. Introduction

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) [1] and cosmic microwave background [2] measurements give the BAU as , where is the baryon number density and is the entropy density. In order to address this issue, many different models and mechanisms have been proposed [3–7]. The mechanisms discussed in the literature satisfy the three Sakharov conditions [3], namely, (i) baryon number () violation, (ii) charge () and charge-parity () violations, and (iii) a departure from the thermal equilibrium. For reviews of different types of models and mechanisms, see, for example, [8–10]. Recently, the variety of the method for the calculation of BAU has been also developed [11–13].

In the present paper, we further extend the model of scalar fields [14] so that it generates the PNA through interactions. In many of the previous works, the mechanism generating BAU relies on the heavy particle decays. Another mechanism uses phase of the complex scalar field [6]. In this work, we develop a new mechanism to generate PNA. The new feature of our approach is briefly explained below.

The model which we have proposed [15] consists of a complex scalar field and a neutral scalar field. The PNA is related to the current of the complex field. In our model, the neutral scalar field has a time-dependent expectation value which is called condensation. In the new mechanism, the oscillating condensation of the neutral scalar interacts with the complex scalar field. Since the complex scalar field carries charge, the interactions with the condensation of the neutral scalar generate PNA. The interactions break symmetry as well as charge conjugation symmetry. At the initial time, the condensation of the neutral scalar is nonzero. We propose a way which realizes such initial condition.

As for the computation of the PNA, we use 2PI formalism combined with density operator formulation of quantum field theory [16]. The initial conditions of the quantum fields are specified with the density operator. The density operator is parameterized by the temperature of the universe at the initial time. We also include the effect of the expansion of the universe. It is treated perturbatively and the leading order term which is proportional to the Hubble parameter at the initial time is considered. With this method, the time dependence of the PNA is computed and the numerical analysis is carried out. The dependence, especially, on the various parameters of the model such as masses and strength of interactions is investigated. We also study the dependence on the temperature and the Hubble parameter at the initial time. We first carry out the numerical simulation without specifying the unit of parameter sets. Later, in a radiation dominated era, we specify the unit of the parameters and estimate the numerical value of the PNA over entropy density.

This paper is organized as follows. In Section 2, we introduce our model with and particle number violating interactions. We also specify the density operator as the initial state. In Section 3, we derive the equation of motion for Green’s function and field by using 2PI CTP EA formalism. We also provide the initial condition for Green’s function and field. In Section 4, using the solution of Green’s function and field, we compute the expectation value of the PNA. Section 5 provides the numerical study of the time dependence of the PNA. We will also discuss the dependence on the parameters of the model. Section 6 is devoted to conclusion and discussion. In Appendix A, we introduce a differential equation which is a prototype for Green’s function and field equations. Applying the solutions of the prototype, we obtain the solutions for both Green’s function and field equations. In Appendices B–D, the useful formulas to obtain the PNA for nonvanishing Hubble parameter case are derived.

#### 2. A Model with CP and Particle Number Violating Interaction

In this section, we present a model which consists of scalar fields [15]. It has both and particle number violating features. As an initial statistical state for scalar fields, we employ the density operator for thermal equilibrium.

Let us start by introducing a model consisting of a neutral scalar, , and a complex scalar, . The action is given bywhere is the metric and is the Riemann curvature. With this Lagrangian, we aim to produce the PNA through the soft-breaking terms of symmetry whose coefficients are denoted by and . One may add the quartic terms to the Lagrangian which are invariant under the symmetry. Though those terms preserve the stability of the potential for large field configuration and are also important for the renormalizability, we assume they do not lead to the leading contribution for the generation of the PNA. We also set the coefficients of the odd power terms for zero in order to obtain a simple oscillating behavior for the time dependence of the condensation of . We assume that our universe is homogeneous for space and employ the Friedmann-Lemaître-Robertson-Walker metric,where is the scale factor at time . Correspondingly the Riemann curvature is given byIn (1), the terms proportional to , , and are the particle number violating interactions. In general, only one of the phases of those parameters can be rotated away. Throughout this paper, we study the special case that and are real numbers and is a complex number. Since only is a complex number, it is a unique source of the violation.

We rewrite all the fields in terms of real scalar fields, (), defined asWith these definitions, the free part of the Lagrangian is rewritten aswhere the kinetic term is given byand their effective masses are given as follows:Nonzero or leads to the nondegenerate mass spectrum for and . The interaction Lagrangian is rewritten with a totally symmetric coefficient ,with . The nonzero components of are written with the couplings for cubic interaction, and , as shown in Table 1. We also summarize the qubic interactions and their properties according to symmetry and symmetry.

Nöether current related to the transformation isIn terms of real scalar fields, the Nöether current alters intoThe ordering of the operators in (11) is arranged so that it is Hermite and the particle number operator,has a normal ordered expression. Then, in the vanishing limit of interaction terms and particle number violating terms, the vacuum expectation value of the particle number vanishes. With the above definition, is the PNA per unit comoving volume. The expectation value of the PNA is written with a density operator,Note that, the PNA is a Heisenberg operator and is a density operator which specifies the state at the initial time . In this work, we use the density operator with zero chemical potential. It is specifically given bywhere denotes inverse temperature, , and is a Hamiltonian which includes linear term of fields,where is a constant. The linear term of fields in (16) is prepared for the nonzero expectation value of fields. Note that the density operator in (15) is not exactly the same as the thermal equilibrium one since, in the Hamiltonian, the interaction parts are not included. Since we assume three-dimensional space is translational invariant, then the expectation value of the PNA depends on time and the initial time . As we will show later, the nonzero expectation value for the field leads to the time-dependent condensation which is the origin of the nonequilibrium time evolution of the system.

Below we consider the matrix element of the density operator given in (15). We start with the following density operator for one component real scalar field as an example,The above Hamiltonian is obtained from that of (16) by keeping only one of the real scalar fields. The matrix element of the initial density operator in (17) is written in terms of the path integral form of the imaginary time formalism given aswhere is an Euclidean action which corresponds to the Hamiltonian in (18) and it is given byAfter carrying out the path integral, the density matrix is written with which is the action for the classical orbit satisfying the boundary conditions, . It is given as the functional of the boundary fields and vacuum expectation value aswhere is given byIn the above expression, we drop the terms which are proportional to because they do not contribute to the normalized density matrix. is defined as [14]Using the above definitions, one can write the density matrix in (21) as the following form,with and . The upper indices and are or . is the metric of CTP formalism [16] and and . In the above expression, the source terms and do not vanish only at the initial time and they are given bywhere is given in (23). In (24), is a normalization constant which is given as

#### 3. Two-Particle Irreducible Closed Time Path Effective Action

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.

##### 3.1. 2PI Formalism in Curved Space-Time

2PI CTP EA in curved space-time has been investigated in [17] and their formulations can be applied to the present model. In 2PI formalism, one introduces nonlocal source term denoted as and local source term denoted as ,where or and is given bywhere and the other components are zero. The upper indices of the field and the source terms distinguish two different time paths in closed time path formalism [16]. One can define the mean fields and Green’s function by taking the functional derivative with respect to the source terms and , respectively,

If one sets the source terms to be the ones given in (25) and (26), one can show that the expectation value of the product of the field operators with the initial density operator is related to the Green function and mean fields. Definitely, we can prove the following relations,with and is a Heisenberg operator which has form as . With (35), one can write the expectation value of the current as the sum of the contribution from Green’s function and the current of the mean fields. Then (14) alters intowhere we have used (12) and the following relationswhere is the Pauli matrix.

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 by Legendre transformation as [18, 19]

Let us write the 2PI EA in our model, in which we only keep the interaction term up to the first order of cubic interaction, . It is given aswhere is the action written in terms of mean fields asIn (40), the interactions are included in the first term as well as in the second term. In the action above, we have also taken into account the surface term at the boundary which corresponds to the last term of (41). and in (41) are the upper bound and the lower bound of the time integration, respectively.

##### 3.2. Schwinger Dyson Equations

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, , with respect to the scalar field and Green’s function .

In the following, we first derive SDEs for the field. The variation of the 2PI EA in (39) with respect to the scalar field leads toUsing (25) and (26), one computes the right-hand side of the above equation aswhere we have used given in (23). The left-hand side of Eq. (42) is computed using (40) and one obtains the following equation of motion of the scalar field ,where the Laplacian of Friedman-Lemaître-Robertson-Walker metric is given by

Next, the equation of motion for Green’s function is derived in the following way. The variation of the 2PI EA in (39) with respect to Green’s function leads toThe left-hand side of the above equation is obtained by taking variation of Eq. (40) with respect to Green’s function as,where the second term of above expression is computed using action in (41). Taking all together (46) and (47), one obtains the following two differential equations for Green’s function,where and .

Next, we rescale Green’s function, field and coupling constant of interaction as follows:where stands for the initial value for the scale factor and we have defined and we have used Fourier transformation for Green’s function asBy using these new definitions, SDEs in (44) are written as

Next SDEs for the rescaled Green’s function in (48) and (49) are written aswhere we have definedNote that the first derivative with respect to time which is originally presented in the expression of Laplacian (45) is now absent in the expression of SDEs for the rescaled fields and Green’s functions.

##### 3.3. The Initial Condition for Green’s Function and Field

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 (see (50)). Using (32) and setting , we compute it as follows:where we have computed the last term of (22) using (25) and is defined as [14]To proceed the calculation, we denote as . Then the initial condensation of field is given bywhere we have defined and satisfies

Next we will compute the initial condition for Green’s function (see (51)). By using (34) and setting and , one computes it as follows:Using (53) and (60), (63) becomes

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 . First we integrate the field equation in (54) with respect to time. By setting , we obtainSimilarly, we integrate (55) with respect to time . By setting both and equal to , we obtain the following initial conditionFinally, we integrate (56) with respect to time . By setting both and equal to , we obtain another initial condition,

#### 4. The Expectation Value of PNA

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 4.1, we show how one analytically obtains the solutions of SDEs. We write down the solutions up to the first order of the cubic interaction. In the Subsection 4.2, we also write the expectation value of the PNA up to the first order of the cubic interaction and investigate it by taking into account of the time dependence of the scale factor.

##### 4.1. The Solution of Green’s Function and Fields including Corrections

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 aswhere denotes the free part contribution while is the contribution due to the first order of the cubic interaction. In Appendix A.2, (70)-(72) are derived in detail. and is defined bywhere is defined as and are the solutions which satisfy the following homogeneous differential equations:where is given in (57). In Appendix B, and are derived in detail. and is also defined as follows:

Next we write down the solution of Green’s function as follows:where is an antisymmetric tensor and its nonzero components are given as while denotes a unit step function,where , , and are given asand is given in (23). In Appendices A.4 and A.5, we derive (79) and (80) in detail, respectively.

##### 4.2. The Expectation Value of PNA including Corrections and the First Order of the Hubble Parameter

Next we compute the PNA in (36) including the first order correction with respect to () and the effect of expansion up to the first order of the Hubble parameter. By using rescaled fields, Green’s function, and coupling constant in (50)-(52), one can write down total contribution to the expectation value of PNA with order corrections asThe first line of the above equation is the zeroth order of the cubic interaction while the next three terms are the first order.

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 for as follows: We first assume that when is near . Then one can keep only the following terms:and for () are set to be zero. corresponds to the constant scale factor and corresponds to linear Hubble parameter . Thus it can be written aswhere is given byand . Throughout this study, we only keep first order of as the first nontrivial approximation. For the case that Hubble parameter is positive, it corresponds to the case for the expanding universe. Under this situation, and .

Now let us briefly go back to (58). With these approximations, the second term of (58) is apparently vanished. Since is proportional to linear , the third term of (58) involves second order of . Hence, one can neglect it and the Riemann curvature in (3) is also vanished. Therefore, is simply written as . Now are given asNext we define asWe consider defined in (57). One can expand it around time as

Now let us investigate the expectation value of PNA under these approximations. For the case that and , the nonzero contribution to the expectation value of PNA comes only from corrections to Green’s function. From (84), we can obtainwhere we have used (71) and (80). Following the expression of the scale factor in (87), is also divided into the part of the constant scale factor and the part which is proportional to . In Appendix C, and its derivative are derived in detail. In the above expression, is also included in . Since we are interested in the PNA up to the first order of , we expand it as follows:Furthermore, substituting (87), (95) and and its derivative in (C.7), (C.8), (C.11)-(C.16) into Eq. (94), one can divide the PNA into two parts,