Advances in High Energy Physics

Volume 2019, Article ID 6825104, 28 pages

https://doi.org/10.1155/2019/6825104

## A New Mechanism for Generating Particle Number Asymmetry through Interactions

^{1}Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan^{2}Core of Research for Energetic Universe, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan^{3}National Institute of Technology, Niihama College, Ehime, 792-8580, Japan^{4}Okayama University of Science, Ridaicho, Kita-ku, Okayama, 700-0005, Japan^{5}Tomsk State Pedagogical University, Tomsk, 634061, Russia

Correspondence should be addressed to Apriadi Salim Adam; pj.ca.u-amihsorih@madaidairpa

Received 27 December 2018; Accepted 19 February 2019; Published 7 April 2019

Guest Editor: Subhajit Saha

Copyright © 2019 Takuya Morozumi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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.