Generation of particle number asymmetry in expanding universe
Takuya Morozumi Keiko I. Nagao Apriadi Salim Adam Hiroyuki Takata



Generation of particle number asymmetry in expanding universe[contributed talk at 1st workshop on Phenomenology for Particle and Anti-Particle 2018 (PPAP 2018)]








Apriadi Salim Adam
[apriadiadam@hiroshima-u.ac.jp]
and
Takuya Morozumi
[morozumi@hiroshima-u.ac.jp]





Graduate School of Science Hiroshima University,
1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

Core of Research for Energetic Universe, Hiroshima University, Higashi-Hiroshima,
739-8526, Japan




Keiko I. Nagao
[nagao@dap.ous.ac.jp]





Okayama University of Science, Ridaicho, Kita-ku, Okayama, 700-0005, Japan




Hiroyuki Takata
[takata@tspu.edu.ru]



Tomsk State Pedagogical University, Tomsk, 634061, Russia













We study the creation and time evolution of particle number asymmetry with nonequilibrium quantum field theory. We introduce a model of a neutral scalar and a complex scalar and it has CP violating and particle number violating features. Starting with an initial condition specified by density operator, we show how particle number asymmetry can be generated through interaction. We investigate the time evolution of particle number asymmetry of the universe using the perturbation method. 





§ INTRODUCTION


What still not yet fully understood from the standard model (SM) point of view is why there is more baryon than anti-baryon in the universe <cit.>. In order to address this issue, a number of studies have been proposed <cit.>. In present work, we study a model which generates particle number asymmetry through interactions and develop a formulation which is applicable to various types of expanding universe <cit.>. In addition, we also compute the time evolution of current asymmetry by using quantum field theory with the density operator.
 



§ THE MODEL



The model consists of a neutral scalar, N, and a complex scalar, ϕ. The action S is given by <cit.>,

    S =   ∫ d^4x √(-g)(ℒ_free+ℒ_int + ξ (R - 2 Λ)),  
    ℒ_free =    g^μν∇_μϕ^†∇_νϕ - m_ϕ^2 | ϕ |^2 + 1/2∇_μ N ∇^μ N  - M_N^2/2 N^2+ B^2/2  (ϕ^2 + ϕ^† 2)    
       + (α_2/2ϕ^2 + h.c.)R+ α_3 |ϕ|^2R , 
    ℒ_int =    
    Aϕ^2 N + A^∗ϕ^† 2 N + A_0|ϕ|^2N ,

where A is the interaction coupling of the vertex, B is the parameter which gives the mass difference of the fields and α_2 is the matter-curvature coupling. The metric g_μν is given by the type of Friedmann Robertson Walker with scale factor a(x^0),

    g_μν = (1, -a^2(x^0), -a^2(x^0), -a^2(x^0)),

while R is the Riemann curvature which has form R = 6 [(ä/a)+ (ȧ/a)^2] and H(x^0) = ȧ/a. With the Lagrangian in Eq.(<ref>), we study the particle number asymmetry produced by the soft-breaking terms of U(1) symmetry whose coefficients are denoted by A and B^2. Noticing that, with this Lagrangian, one can also derive the Einstein equations for the scale factor coupled with scalar particles. They are given as,

    T_μν  =    ∂_μϕ_i ∂_νϕ_i - g_μν( 1/2 g^αβ∂_αϕ_i
    ∂_βϕ_i - 1/2 m_i^2 ϕ_i^2 + 1/3 A_ij kϕ_i ϕ_j ϕ_k ) 
    
    	 - 3 (1 - 8 π G β_i ϕ_i^2)  (
    	ȧ/a)^2 + Λ =     - 8 π G T_00 (00 component)
    
    	 (1 - 8 π G β_i ϕ_i^2) (2 a
    	ä + ȧ^2) - a^2 Λ =     - 8 π G T_i i (ii component) 
    
    	 0 =     - 8 π G T_μν (≠μ) (off diagonal component)

However, a full discussion for solving them lies beyond the scope of this study. At present, we work for the case that the time dependence of the scale factor is given.



§ THE CURRENT EXPECTATION VALUE AND ITS TIME EVOLUTION


The particle number is related to U(1) transformation of the complex scalar field, namely, ϕ(x)→ϕ (x) e^iθ. It is U(1) charge represented by particle number operator N <cit.>,


    N(x^0) =   ∫√(-g(x)) j_0(x) d^3 x
    
    j_μ(x) =    i (ϕ^†∂_μϕ -∂_μϕ^†ϕ).

As for the initial condition of the state, it is given by density operator, namely,

    ρ(t_0) = e^- H_0/T/tr e^- H_0/T,

where H_0 is a Hamiltonian includes the linear term of the fields and T denotes the temperature.

It is convenient to write the scalar particles in terms of real fields by using the following relations, ϕ≡ϕ_1 + i ϕ_2/√(2) and ϕ_3 ≡ N. In terms of real fields, the current expectation value written with initial density operator has form,

    ⟨ j_μ(x)⟩   =tr (j_μ (x) ρ(t_0)), 
       = Re.[   ( ∂/∂ x^μ -
    ∂/∂ y^μ) G_12^ (x, y) |_y
    	→ x  + ϕ̅^∗_2 (x)↔∂_μϕ̅_1 (x)  ] ,

where ϕ̅ denotes the mean field with a relation, ϕ=ϕ̅+φ. Both Green's function and field are obtained from 2PI effective action Γ_2 <cit.>,

    Γ_2 [G, ϕ̅, g] = S[ϕ̅, g] + i/2TrLn G^- 1 + Γ_Q - i/2Tr  1 + 1/2∫ d^4 x ∫ d^4 y
    δ^2 S[ϕ̅, g]/δϕ̅_i^a (x) δϕ̅_j^b (y)  G^ab_ij (x, y),

where Γ_Q is the lowest order of 2PI diagram. Their equations of motion are derived by taking derivative Γ_2 with respect to Green's function and field, respectively. 

Before closing this section, let us write down the time evolution of the current asymmetry. First, we consider the solution of Green's function and field up to the first order of interaction coupling A and up to the linear order of the Hubble constant H(t_0). In this case, we focus on the case that the initial expectation value of the field is (ϕ̅_1,ϕ̅_2,ϕ̅_3)=(0,0,v_3). Then, the contribution to the current asymmetry is only given by the first term of Eq.(<ref>),

    ⟨ j_0 (x^0) ⟩_O(A) =    ∫d^3 k/(2 π)^3( ∂/∂ x^0 -
    	∂/∂ y^0) [Re.Ĝ_12^O(A) (x^0, y^0,  k)] |_y^0→ x^0,  
    
    	=   ⟨ j_0 (x^0) ⟩_1st + 	⟨ j_0 (x^0) ⟩_2nd ,

where ⟨ j_0 (x^0) ⟩_1st and ⟨ j_0 (x^0) ⟩_2nd are given by,

    ⟨ j_0 (x^0) ⟩_1st  =   2
    		v_3 A_1 2 3/a_t_0^3∫d^3 
    		 k/(2 π)^3∫_t_0^x^0{ 1 - 3 (x^0 - t_0) H (t_0)  -
    		3/2 (t - t_0) H (t_0) }
       ×[ {(- K̅^(0)'_3, t t_0,
    			 0 )/2 ω_2,  k (t_0) βω_2,  k (t_0)/2[ ( K̅^(0)'_2,^
    		x^0 t_0,  k↔∂ K̅^(0)_1, x^0 t,  k) K̅^(0)'_2, t
    		t_0,  k. . . 
       . . . + ω^2_2,  k (t_0) (
    	K̅^(0)_2, x^0 t_0,  k↔∂K̅^(0)_1, x^0 t,  k) K̅^(0)_2, t
    		t_0,  k] } - { 1 ↔ 2 for lower indices}] d t, 
    
    	
    	⟨ j_0 (x^0) ⟩_2nd =   2
    		v_3 A_1 2 3/a_t_0^3∫d^3 
    		 k/(2 π)^3∫_t_0^x^0[ {(- K̅^(0)'_3, t t_0,  0 )/2 ω_2,
    			 k (t_0) βω_2,  k (t_0)/2. . 
       ×[ ( K̅^(0)'_2,^ x^0 t_0,  k↔∂ K̅^(0)_1, x^0 t,
    		 k) K̅^(1)'_2, t t_0,  k.
    	+ ( K̅^(1)'_2,^ x^0 t_0,  k↔∂ K̅^(0)_1, x^0 t,
    		 k + K̅^(0)'_2,^ x^0 t_0,  k↔∂ K̅^(1)_1, x^0 t,
    		 k) K̅^(0)'_2, t t_0,  k
        + ω^2_2,  k (t_0) [ ( K̅^(0)_2, x^0
    		t_0,  k↔∂ K̅^(0)_1, x^0 t,  k) K̅^(1)_2, t t_0,
    		 k. 
       . . .. + ( K̅^(1)_2, x^0 t_0,  k↔∂ K̅^(0)_1, x^0 t,
    		 k + K̅^(0)_2, x^0 t_0,  k↔∂ K̅^(1)_1, x^0 t,
    		 k) K̅^(0)_2, t t_0,  k] ]
    	}  - { 1 ↔ 2 for lower indices}] d t.


where K̅_i [x^0, y^0] and its derivative are given by,

    K̅^(0)_i [x^0, y^0] =    sin [ω_i,   k (x^0 - y^0)]/ω_i,   k,  ω_i,  k =√( k^2/a(t_0)^2+m̃_i^2),
    m̃_1^2 =    m_ϕ^2 - B^2,  m̃_2^2 = m_ϕ^2 + B^2,  m̃_3^2 = m_N^2, 
    K̅^(1)_i[x^0, y^0] =   H (t_0)  k^2 (x^0 + y^0 - 2 t_0)/2ω_i,   k^2 a (t_0)^2( sin [ω_i,   k (x^0 -
    	y^0)]/ω_i,   k - (x^0 - y^0) cos [ω_i,   k
    (x^0 - y^0)] ), 
    K̅'_i [x^0, y^0]:=   ∂K̅_i[x^0,y^0]/∂ y^0,  K̇̅̇_i [x^0, y^0]:=∂K̅_i[x^0,y^0]/∂ x^0,  K̇̅̇'_i [x^0, y^0]:=∂^2 K̅_i[x^0,y^0]/∂ x^0 ∂ y^0,


From Eqs.(<ref>) and (<ref>), below we remark several findings:

	
  * In the first line of Eq.(<ref>), the first, second and third terms in wave parentheses correspond to the contribution of constant scale factor to the current asymmetry, dilution effect and freezing interaction effect, respectively. 
	
  * Eq.(<ref>) corresponds to the redshift effect for the contribution to the current asymmetry.

 
The above four types of the contribution to the current asymmetry are explained as follows. The constant scale factor which is the zeroth order of H(t_0) is the leading contribution. The rests which are the first order term contribute according to their origins and we summarize them in Table <ref>. In the next section, we will study numerically the time evolution of the current asymmetry.



§ NUMERICAL RESULTS

In the left side of Fig.<ref>, we show the parameter B dependence. Both of the amplitude and the period of the oscillation change when we alter the parameter B. As it increases, the amplitude becomes larger and its period becomes shorter. The right side of Fig.<ref> shows the dependence of the PNA on ω_3,  0. As shown in the black, blue and dot-dashed blue lines, the position of the first node does not change when ω_3,  0 takes its value within the difference of m̃_1 and m̃_2. However, the amplitude of oscillation gradually decreases as ω_3,  0 increases up to the mass difference. The interesting findings were observed when ω_3,  0 becomes larger than the mass difference. As ω_3,  0 becomes larger, the amplitude decreases and the new node is formed at once. The dashed and dotted blue lines show this behavior.    




§ CONCLUSION

We have studied an interacting model in which particle number asymmetry is generated through interactions of scalar fields. The current for the particle and anti-particle asymmetry is given up to the first order of A and linear H(t_0). Time evolution of the particle number asymmetry and its parameter dependence is investigated. 
 






Acknowledgement


This work is supported by JSPS KAKENHI Grant Number JP17K05418 (T.M.) and supported in part by JSPS Grant-in-Aid for Scientific Research for Young Scientists (B) 26800151 (K.I.N.). 

99
	
Kne2004 
J. P. Kneller and G. Steigman, New Journal of Physics 6, 117 (2004); P. A. R. Ade and others, Astron. Astrophys. 594, A13 (2016).	

fuk86
M. Yoshimura, Phys. Rev. Lett. 41, 281 (1978); M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986); D. Zhuridov, Phys. Rev. D 94, 035007 (2016); N. Blinov and A. Hook, Phys. Rev. D 95, 095014 (2017).
	
mor2017
T. Morozumi, K. I. Nagao, A. S. Adam and H. Takata, Russ. Phys. J 59, 1852 (2017), arXiv:1609.02990 [hep-ph].

apri2017
T. Morozumi, K. I. Nagao, A. S. Adam and H. Takata, Advances in High Energy Physics, vol. 2019, Article ID 6825104, 28 pages, 2019. https://doi.org/10.1155/2019/6825104, arXiv:1709.08781 [hep-th].

AffDin85 
I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985).	
	
ramsey97 
S. A. Ramsey and B. L. Hu, Phys. Rev. D 56, 661 (1997); E. A. Calzetta and B. L. Hu, Nonequilibrium Quantum Field Theory (Cambridge Univ. Press, 2008).



	



