Time variation of particle and anti-particle asymmetry
in an expanding universe
Particle number violating interactions wash out the primordial asymmetry of particle number density generated by some interaction satisfying Sakharov conditions for baryogenesis. In this paper, we study how the primordial asymmetry evolves in time under the presence of particle number violating interactions and in the environment of expanding universe. We introduce a complex scalar model with particle number violating mass terms and calculate the time evolution of the particle number density with non-equilibrium quantum field theory. We show how the time evolution of the number density depends on parameters, including the chemical potential related with the particle number, temperature, size of the particle number violating mass terms, and the expansion rate of the universe. Depending upon whether the chemical potential is larger or smaller than the rest mass of the scalar particle, behaviors of the number density are very different to each other. When the chemical potential is smaller than the mass, the interference among the contribution of oscillators with various momenta reduces the number density in addition to the dilution due to the expansion of universe. In opposite case, the oscillation of the particle number density lasts for a long time and the cancellation due to the interference does not occur.
Exploring the origin of the matter and anti-matter asymmetry of our universe, its production mechanism and time evolution are very important issues. In many scenarios of baryogenesis Sakharov:1967dj () Yoshimura:1978ex () and leptogenesis Fukugita:1986hr (), baryon number and lepton number interactions are required so that the primordial asymmetry of the particle number can be generated. After it is generated, the particle number violating interactions must be frozen. Otherwise, the primordial asymmetry created will be washed out. In this regards, in the context of leptogenesis Fukugita:1986hr (), there are studies of the effect operator of the mass dimension on the primordial () asymmetries. If the coefficient of the operator is too large, the primordial asymmetries will be completely washed out. Since the same operator generates the Majorana mass matrix for light neutrinos at low energy, constraints on its elements are obtained from the condition that a leptogenesis scenario succeeds. One can also argue whether they are compatible with neutrino masses, lepton mixing matrix, and the experimental limit on the neutrino-less double beta decay rate Hasegawa:2003vh ().
We introduce a scalar model with particle number violating mass terms to investigate the time dependence of the particle number density in expanding universe, where the scale factor has arbitrary time dependence. In numerical study, we focus on the case that the scale factor grows exponentially and study how a given initial particle number asymmetry evolves under the influence of the particle number violating mass terms and the expansion. The scalar field is written in terms of a complex Klein Gordon field and one can identify the time component of U(1) current as a particle number density. Baryogenesis with a complex field has been discussed in several literatures Dimopoulos:1978kv (), Affleck:1984fy (), Takeuchi:2010tm (), where the time derivative of the phase of the scalar is identified with the baryon number density.
We adopt the non-equilibrium field theory which has been developed in the literatures Schwinger:1960qe (),Bakshi:1962dv (), Bakshi:1963bn (),Keldysh:1964ud (),Ramsey:1997qc (), Calzetta:1986cq () so that one can study the time evolution of the expectation value of the particle number density. We employ functional method since one can naturally extend the present study so that interactions and condensates are incorporated. The present work, therefore, serves as a starting point when we include interactions besides the terms quadratic with respect to the field. Despite of no interactions beyond quadratic terms, under the environment of the expanding universe, the initial condition with non-zero asymmetry and the particle number violating mass terms lead to non-trivial time evolution of the particle and anti-particle asymmetry. In the expectation value, the weight of each state is specified by a density matrix. The density matrix is written with the grand canonical form and it is specified with temperature and chemical potential. The functional form for the density matrix with non-zero chemical potential is constructed explicitly. In our study, the primordial asymmetry of the particle number density is given by choosing the value and the sign of the chemical potential.
We derive formulae for the expectation value of the particle number density in an analytic form. For exponentially expanding universe, the formulae are written with Hankel functions. The various limiting cases, e.g., the case of the vanishing and/or small expansion rate and the case for vanishing particle number violating mass, etc., can be easily obtained. In numerical study, one can change the coefficient of the particle number violating mass term and the expansion rate of the universe. One can also change initial conditions by specifying the temperature and the chemical potential in the density matrix. Therefore, in an unified way, one can study its time evolutions for the cases with different sets of parameters.
The paper is organized as follows. In section II, Lagrangian for the scalar model is given. The initial density matrix is also specified. In section III, using a two particle irreducible effective action, we solve the Schwinger Dyson equation for Green functions and obtain the particle number density at arbitrary time. In section IV, we present the numerical results and section V is devoted to summary. In appendix A , the derivation of particle number density for small expansion rate is given and in appendix B, that for the vanishing limit of the particle number violating mass term is obtained.
Ii The complex scalar model with U(1) breaking
We start with a complex scalar model including a soft U(1) symmetry breaking mass term. The time component of the U(1) current is a particle number density,
U(1) breaking terms are denoted with their coefficients ; and . The metric is given by that of Friedmann Robertson Walker,
The Riemann curvature is given as with . When and are real parameters, the mass eigenstates of the scalar are the real part and the imaginary part of the complex scalar . By decomposing it into a real part and an imaginary part as , their masses are given as follows,
The current associated with U(1) transformation is Dimopoulos:1978kv (),
Next we study the density matrix which specifies the initial state. Since we have non-vanishing primordial asymmetry of the particle number density, the statistical density matrix has the following form with non-zero chemical potential,
where corresponds to the Hamiltonian obtained by taking the U(1) breaking terms and curvature dependent terms turned off and is a particle number operator defined as follows,
The expectation value of the U(1) current is written with the density matrix in Eq.(5).
In section III, we compute the expectation value with the Green function of 2 PI (particle irreducible) formalism. From the definition of the U(1) current in Eq.(4), the expectation value defined in Eq.(7) can be written in terms of the Green function,
The resulting formulae for the expectation value of the current is given as follows,
Iii Schwinger Dyson equation from 2 particle irreducible effective action
In this section, we derive 2 PI effective action and obtain the Schwinger Dyson equations for Green functions. By solving the Schwinger Dyson equations, we obtain an analytic form for the expectation value of the current for the case that the scale factor of the universe of arbitrary time dependence.
2 PI effective action in curved space time for O(N) theory is derived in Ramsey:1997qc () and the method employed can be also applied to the present model. In 2 PI formalism, one first introduces non-local source term denoted by in addition to the usual local source term .
where is the metric of in-in formalism Calzetta:1986cq () and and .
The Legendre transformation of leads to the 2 PI effective action, which is a functional of Green function.
where and are given by,
The variation of the 2 PI effective action with respect to the scalar field leads to,
and one obtains the following equation of motion for the scalar field .
When the single source term vanishes, the equation of motion for is homogeneous and linear with respect to . Therefore is a solution in this case. The variation of the 2 PI effective action with respect to Green function is the source term ,
Eq.(16) leads to two differential equations,
where is a normalization factor and is determined so that the density matrix is normalized as . is non-zero only if both and are the initial time. The resulting has the following form,
where specifies the space dependent part. Since it is invariant under translation, one can carry out the Fourier transformation on it.
Let us derive the functional representation for the density matrix of Eq.(5) and determine .
Note that represents two components scalars.
We assume that the particle number violating term turned on when the universe begins to expand at . The initial value for the scale factor is . Since the Hamiltonian and the particle number commute with each other, the exponential factors in the grand canonical distribution function are factorized as shown in Eq.(21). is a functional derivative acting on and corresponds to the number operator in Eq.(6).
We first investigate the functional representation for the density matrix with zero chemical potential.
where is a Euclidean action for the complex scalar field and is the one for the classical trajectory with the boundary conditions at the Euclidean time and . is a constant. Explicitly is given as,
represents defined in Eq.(20) for the zero chemical potential case. One can find,
where . To obtain the functional representation of the density matrix for non-zero chemical potential, one notes the action of generates rotation among with a complex angle ,
where is a rotation matrix,
Therefore the action of replaces with . The resulting functional representation of the density matrix for non-zero chemical potential is,
for non-zero chemical potential is given as,
The normalization factor can be determined by the condition .
with defined as,
The functional representation of the density matrix in Eq.(33) is used for obtaining the initial condition of the Green functions which are needed to solve the differential equations of Eq.(17). The Green function at is defined as,
and it can be computed with the generating functional,
Differentiating with the source term twice, one obtains,
The Fourier transformation of and its inverse are,
Next we define the Fourier transform of the Green functions,
Since we obtain the initial condition of Green function, one can use it to solve the Schwinger Dyson equations.
In Friedman Robertson Walker metric, the Laplacian is given as,
Therefore, the Fourier transformation of Green functions satisfy,
To solve Eq.(44), we introduce through the following equantion.
The differential equations are rewritten as,
where In the following, we denote two independent solutions of the homogeneous differential equation of Eq.(III) as and .
To solve the differential equations for Green functions, we introduce the following four by four matrices.
where each is given by a two by two matrix.
In this notation, and are given as,
where each is a two by two matrix and is given by,
Next we write with as,
where and are defined as,
Using them, one can write the solution for as,
For , one can also write the solution in the matrix form similar to Eq.(60). The result is,