Baryogenesis in false vacuum

The null result in the LHC may indicate that the standard model is not drastically modified up to very high scales such as the GUT/string scale. Having this in the mind, we suggest a novel leptogenesis scenario realized in the false vacuum of the Higgs field. If the Higgs field develops a large vacuum expectation value in the early universe, a lepton number violating process is enhanced, which we use for baryogenesis. To demonstrate the scenario, several models are discussed. For example, we show that the observed baryon asymmetry is successfully generated in the standard model with higher-dimensional operators.


I. INTRODUCTION
Although the standard model (SM) is complete after the discovery of the Higgs boson at the Large Hadron Collider [1,2], there are still mysteries in elementary particle physics such as the finite neutrino mass and dark matter. Besides them the baryon asymmetry in the universe (BAU) is also one of unsolved problems. That is, how had baryogenesis been realized in evolution of the universe? The latest cosmological result from the Planck observations [3] tells us that the BAU is where n B is the baryon number density and s is the entropy density.
In order to theoretically explain the BAU within elementary particle physics, the Sakharov conditions [4] have to be satisfied: There exists a process violating the baryon number; C and CP invariances are violated; the system leaves from equilibrium state. The SM does not accommodate the departure from equilibrium. Although the baryon number is violated through the sphaleron process and CP symmetry is violated in the weak interaction, it is not enough to reproduce the BAU. Therefore the SM cannot satisfy these conditions and must be extended.
Some baryogenesis mechanisms satisfying the Sakharov conditions have been suggested, e.g. the grand unified theory [5] and the Affleck-Dine mechanism [6]. Leptogenesis is also one of the well-known mechanisms for baryogenesis [7] (see also reviews [8,9]) where we use the fact that through the sphaleron process [10][11][12][13], the difference B − L between the baryon number B and the lepton number L is conserved whereas their sum B + L is not.
The baryon number density in thermal equilibrium is provided by the B − L number density via the sphaleron process: where N F is generation of quarks and leptons and N S is that of scalar doublets. For instance, in case of the SM where N F = 3 and N S = 1, the factor in the right-hand side is 28/79.
Through the decay of the heavy particle, the lepton number is generated, and then its number density changes to the B − L number density n B−L , whose process is described by the coupled Boltzmann equations for these number densities.
In this paper, we study the leptogenesis realized in the false vacuum of the Higgs field.
The mass of the particles coupled to the Higgs field becomes massive. There are the lepton number violating processes where the heavy particles decay into the lighter ones. At the same time, the phase transition of the Higgs takes place and the Higgs moves from the false vacuum to the true electroweak one.
To demonstrate this scenario, two models are investigated. We first consider a minimal model depending on the SM with a high dimensional operator, where L i is the lepton doublet, Λ is a cutoff scale with O(10 14 ) GeV, 1 and the Higgs doublet is defined as Such a operator is generated in typically the type I seesaw model by integrating out the right-hand neutrino. This effective interaction breaks the lepton number and thus is used as the source of the lepton asymmetry. In particular, we consider the decay of the left-hand neutrino, which is given by the mode ν → − W + . Note that in the broken phase H = 0, this operator turns to a neutrino mass term, where we have assumed that the coupling constant λ ij is of order one since neutrino can have a finite mass m ν ∼ 0.1 eV. Therefore, the leptogenesis takes place in the false vacuum where the neutrino mass h 2 /Λ becomes larger value than the charged lepton and W boson ones.
As will be seen in next section, in such a minimal model, the baryon asymmetry produced by this process actually is not adequate for the observed value (1).
Next, we consider an extended system in which the second Higgs doublet is introduced.
In this case, we will see that the lepton asymmetry is caused by the second Higgs boson and it is possible to explain the observation. 1 In Ref. [14,15], the operator (3) is used to realize leptogenesis as well as the CP violating operator L i γ µ L jLi γ µ L j . These operators are naturally generated in a low energy effective theory of various seesaw models.
We have to see whether or not the phase transition of the Higgs field from the false vacuum to the electroweak one occurs after the lepton asymmetry is produced. To this end, we investigate the thermal history of the Higgs potential. Including a new singlet-scalar field coupled to the SM Higgs field, there exists a certain parameter space where the phase transition appropriately takes place.
We organize this paper as follow: In next section, we present the formulation of the Boltzmann equations in order to calculate the baryon asymmetry. Numerically solving them, we investigate the produced baryon asymmetry for two cases explained above. Section III is devoted to investigate the thermal history of the Higgs potential. We summarize and discuss our study and obtained results, and we comment on the possibility of the high scale electroweak baryogenesis in section IV. In appendix A, the thermal effects on the Higgs potential and their formulations are shown.

II. MECHANISM AND BOLTZMANN EQUATIONS
First, we consider the situation where the decay of the left-handed neutrino produces the baryon asymmetry. In this section, we present the Boltzmann equations and quantitatively evaluate the baryon asymmetry by numerically solving them. We evaluate the baryon asymmetry produced by the left-handed neutrino decay; however we see that not enough baryon asymmetry is produced. To ameliorate the situation, next we add the second Higgs doublet.
We demonstrate that the decay of new charged Higgs boson can reproduce the observed amount of asymmetry.

A. The derivation of Boltzmann equations
In this subsection, to calculate the asymmetry of the universe, we follow Ref. [9,16,17] and derive the Boltzmann equations for general case of leptogenesis. The change of the number density of a heavy particle is governed bẏ where X and Y represent the heavy particles without lepton number; the number 1 · · · N denotes lighter particle; the dot on n X in the left-hand side denotes the time derivative; we have neglected the effects of the Pauli blocking and stimulated emission; is the phase space integral; H =Ṙ/R is the Hubble parameter given by the scale factor R which is governed by the Friedmann equation. f is the distribution function, and especially in the case where the system is in thermal equilibrium, f is given as the Maxwell-Boltzmann distribution.
The first and second terms of the right-hand side in Eq. (6) correspond to the decay and annihilation of heavy particle, respectively. Let us rewrite the first term by using the definition of the decay rate, We use the fact that the kinetic equilibrium allows us to make the replacement, 2 Furthermore, at the leading order, |M(X → 12)| 2 = |M(12 → X)| 2 . Hence, we find that the first term in the right-hand side becomes The second term in Eq. (6) can be written in terms of the thermal average cross section of the pair annihilation σ ann v : 2 Here we neglect the chemical potential of particle 1 and 2 as the effect is subleading.
We assume that f (p i ) ∝ f EQ (p i ) thanks to the kinetic equilibrium, so that the second term in Eq. (6) becomes To summarize, the Boltzmann equation of n X is given bẏ In a similar manner, we can write the Boltzmann equation governing the lepton number density: where the first and second terms in the right-hand side describe the decay of the heavy particle and annihilation of the leptons, respectively; W is a particle without the lepton number; l i is a particle having the lepton number. Furthermore, we rewrite this equation as one for B − L asymmetry, which is given bẏ where is the parameter which denotes the CP asymmetry; Br is the branching ratio of X → lW ; n γ is the number density of photon; and σ L v is the thermally-averaged scattering cross section which does not conserve the lepton number.
It is convenient to introduce N i ≡ n i /n γ because this quantity is conserved under the cosmic expansion. We also introduce z ≡ M X /T as a variable. Using these variables, let us now rewrite the Boltzmann equations. For instance, the left-hand side becomeṡ where in the second equality, we have used The right-hand side is In terms of N i and z, we can write the set of the Boltzmann equations as follows: where ζ(3) ≈ 1.20205 is the Riemann zeta function of 3; GeV is the reduced Planck scale; K 2 is the modified Bessel functions of second kind; g * (z) is the total number of effectively massless degree of freedom; and g is the internal degree of freedom of the heavy particle. We neglect the z dependence of g * (z) and use g * = 106.75.
Simultaneously solving the Boltzmann equations, we can evaluate the value of the lepton asymmetry due to the decay of left-handed neutrino which is identified as the heavy particle X. In order to perform numerical calculations, we have to specify Γ X , Br, , σ ann v and σ L v . In next subsection, we give these variables for the minimal model.

B. Minimal model case
We evaluate the baryon asymmetry in the minimal model whose Lagrangian is given as where L SM is the Lagrangian of the SM and ∆L 5 is the higher dimensional operator given in Eq. (3). The lepton number is produced by the decay of the left-handed neutrino. We now show the variables given in the Boltzmann equations in order.
respectively, where λ is the quartic coupling constant of the Higgs field, and g 2 is the SU(2) L gauge coupling constant The decay rate of the left-handed neutrino and the branching ratio to the longitudinal gauge boson are calculated as where 1/γ = K 1 (z)/K 2 (z) in the thermal bath, K 1 is the modified Bessel functions of first kind, and y τ is the tau Yukawa coupling. We note that the branching ratio to the transverse game boson is important. This is because, in order to pick up the imaginary part of the amplitude, it is needed to use lepton Yukawa coupling rather than SU (2) gauge coupling.
The CP asymmetry comes from the interference between the tree and loop diagram shown in Fig. 1, whose order is given by  26) is related to the Jarlskog invariant in the lepton sector [19], and the order is estimated as [19] i ∼ 1 8π where δ is Dirac CP phase of neutrino sector.
We note that, by using the renormalization group equations, we obtain the values of coupling constants at the high scale: 4 Numerical result in minimal model

The Planck observation [3] tells us
where the factor 2387/86 is the photon production factor. If this value comes from the sphaleron effect, we should have Therefore, we numerically solve the Boltzmann equations given in Eq. (18)- (22) and investigate whether or not the appropriate parameter space which satisfies the value (31) exists.
Unfortunately, we can easily see that the baryon asymmetry cannot be reproduced in this framework. We obtain 3 Even if M ν < M W + M τ , the imaginary part appears in higher order. However, it is too small to obtain the enough baryon asymmetry. 4 See e.g. Ref. [18].

C. Extended model: Two Higgs doublets
A way to improve the situation is to add new particle. If the second Higgs doublet H 2 is introduced, we have new interaction terms: Then, the decay of the charged Higgs boson can generate the baryon asymmetry. The diagram contributing to the asymmetry is shown in Fig. 3. In this case, the decay rate is , where M H 2 is the mass of the second Higgs doublet.
We evaluate the resultant asymmetry obtained by the decay of the SM Higgs. The set of the initial conditions of the Boltzmann equations is and we set the cutoff scale as Λ = 6 × 10 14 GeV. In Fig. 2, we show the result assuming that y 2,ij ∼ 1 and the CP phase is of the order of one, i.e. e iδ ∼ 1. We can see that the BAU is reproduced in this extension. In this model, the mass of H 2 should be around 10 13-14 GeV in order to obtain consistent thermal history, as we will see in later. Note that since the new Higgs boson is added, the factor in Eq. (2) slightly changes. The factor becomes 8/23 for N F = 3 and N S = 2.

III. THERMAL HISTORY
In this section, we discuss the thermal history of the universe. We introduce a new scalar S to make Higgs field stay at false vacuum in the early universe, where S is singlet under the SM gauge group.
First, we explain the zero temperature scalar potential of the extended model with S and the thermal correction to it. Then, we discuss how the Higgs field is in false vacuum in the early universe.

A. Zero temperature Higgs potential
The tree level scalar potential is given by where S is the new singlet scalar field. We omit the field H 2 since it is not relevant to discussion here. We consider the region where all couplings take O(0.1-1) value. Although λ becomes small or negative at high scale in the SM (see e.g. Ref. [20]), now the running of λ is modified, λ can take O(0.1-1) value since some scalar fields are added.
We note that the one-loop Coleman-Weinberg potential can be sefely neglected because of O(0.1-1) couplings, and therefore we do not include it for simplicity.
The potential (37) has an absolute minimum at 5 The quadratic term of the SM Higgs is added in order to make the Higgs massless in this vacuum.

B. Thermal potential
We follow the Ref. The NG bosons χ i in the Higgs doublet field (4) are neglected since their effects are small. 5 The potential (37) has a minimum at h = κm 2 S 2λλ S = 2κ λ v S , S = 0. This minimum does not becomes the absolute minimum but the local one for the parameter space we consider here.
As the thermal effects, there are two components, namely V FT (h, T ) and V ring (h, T ). 6 The main contribution of thermal effects comes from V FT (h, T ), which is where the mass for each particle is given by the thermal functions are defined as Remember here that the coupling constants g 2 , g Y and y t are SU(2) L , U(1) Y and top-Yukawa coupling constants, respectively. Since one cannot analytically and exactly evaluate these functions, the approximated expressions are made. 7 There are contributions to the ring diagrams (or the daisy diagrams) from the Higgs boson and the gauge boson: where the first and second terms correspond to the contribution from the Higgs and the 6 The derivation of these functions is shown in appendix A. 7 The high temperature expansion is often used. However, they are not useful for the case where we see the large field value of h. Therefore, the fitting functions (A23) are also employed [22]. See appendix A for detail. scalar S; 8 the thermal masses of the Higgs and scalar S are Π h (T ) = T 2 12 (45) and we have defined To summarize, in order to trace the thermal history of the Higgs potential in the SM, we analyze the effective potential where V tree (h, S) is given in Eq. (37). In next subsection, we investigate the phase transition of Higgs field by using this potential.

C. Thermal history
In the early universe, due to the finite temperature effect, S and H does not have the vacuum expectation value(VEV). 9 They develop the VEV at the temperature when the thermal mass term becomes comparable with the negative mass term. By utilizing the high temperature expansion (A21) and (A22), we estimate the critical temperatures which are given as the vanishing curvature of V eff (h, S, T ) at the origin (h, S) = (0, 0), namely Solving these equations for T , we find .
(49) 8 Combining the ring contribution of the Higgs boson and the first term of Eq. (39), we can write wherem 2 (T ) =m 2 h + Π h (T ) is the Debye mass of the Higgs boson. In the same manner, the thermal effects for the scalar S also can be written as the same form. 9 Our thermal scenario is similar with Ref. [23] where the gravitational wave from electroweak phase transition at the high scale is discussed.
Here T S and T h denote the critical temperatures of the phase transition of S and h, respectively. Our scenario is as follows. The phase transition of Higgs field happens at T = T h . At this time, S and h are in the false vacuum, S = 0, h = 2κ λ v S , and the lepton number is created by the decay of heavy charged Higgs. After that, at T = T S , S develops VEV S = v S , and then h comes back to the true vacuum Eq. (38).
In order to work our scenario, we require Moreover, S must have the negative mass at S = 0, h = 2κ λ v S , namelym S < 0 which yields As an example of successful parameters, we take κ = 0.7, λ S 1.5, λ = 0.4 and h = 2 × 10 13 GeV. T h and T S become and the Eq. (51) is satisfied. Here g Y = g 2 = y t = 0.5 is used.
Therefore, by solving the Boltzmann equations with we can calculate the asymmetry. For example, we obtain 10 with |y 2,ij | = 1, M H 2 = 1.5v S and the CP phase being one. We note that M H 2 should be close to the temperature of phase transition, otherwise the decay of H 2 is not effective. Here we have taken into account the washout factor [15] in the symmetric phase, This implies that we can realize the observed value, N B−L,obs = 4.8 × 10 −8 , by slightly changing the value of CP phase.

IV. SUMMARY AND DISCUSSION
We have considered the possibility of the baryogenesis in the false vacuum where the Higgs field develops the large field value compared with the electroweak scale. Since all the SM particles receives mass from the coupling with the Higgs boson, the large field value of the Higgs field means that they are super heavy. We have estimated the asymmetry produced by the decay of the heavy left-handed neutrino. It have turned out that the decay of neutrino can not realize the observed baryon asymmetry. If the new Higgs doublets H 2 is introduced, the decay of new charged Higgs boson can provide the enough asymmetry.
We have also presented the thermal history where the Higgs develops the large field value in the early universe. It have been found that, by adding the singlet scalar S, our scenario safely works.
Finally, we briefly mention the possibility of the high scale electroweak baryogenesis. So far, we pursue the possibility that the baryon asymmetry is created by the heavy particle while the lepton number violation is given by Majorana mass term of left-handed neutrino.
However, if the coupling λ is small, the electroweak phase transition at high scale becomes first-order. Since our extended model has many CP phases, there is chance to generate the B + L asymmetry. If the L asymmetry is washed out in the false vacuum, the net B asymmetry survives. The condition of L wash out would be roughly given by By putting T H , M W H , we obtain Hence, we have chance to create the baryon asymmetry by the electroweak baryogenesis in addition to the decay of heavy particle. This is one-loop contribution at vanishing temperature, i.e. the Coleman-Weinberg potential. The second term is the thermal potential at one-loop level and becomes where the thermal functions for boson and fermion are defined as with x ≡ | k|/T and r ≡ m(h) /T . Note that in general case, the operator k 2 + m 2 (h) is not diagonal, i.e. k 2 δ ij + m 2 ij (h). Therefore, the mass matrix m 2 ij (h) has to be diagonalized. In the SM case, taking account of the degrees of freedom of particles, the thermal potential is given by where n W = 6, n Z = 3, n t = 12 and n h = 1.
Next, we consider the ring (or daisy) contributions shown in Fig. 4, which are the nexthigher-order corrections and are related to the infrared divergence; see e.g. [24] for a detailed discussion. The ring contribution for the Higgs field is given by where the thermal mass comes from the diagrams in the limit m(h) /T 1 shown in Fig. 5 and becomes Note that these contributions are evaluated by setting the external momentum to zero since we are interested in the infrared limit.
In a similar manner, one can obtain the ring contributions from the gauge bosons, which becomes V gb Here the mass matrices in the original gauge field basis (A i µ , B µ ) are where Π 00 (T ) is the (00) component of the polarization tensor in the infrared limit, namely Π µν (p = 0, T ) and with Π (1) These thermal masses are obtained by calculating the two-point functions of SU(2) and U (1) gauge fields shown in Fig. 6. Evaluating the eigenvalues of M 2 (h) + Π 00 (T ) and M 2 (h) to the three-half power, and then taking trace of them, we have where a g , b g and c g are given in Eq. (46). thus the larger field value than temperature. This is because the ring contribution vanishes for the larger mass.
In case where the scalar S is introduced, the contribution from the diagram shown in Fig. 7 is added, and then the thermal masses of the Higgs field and S are given as Eq. (44) and (45), respectively.

The thermal functions and their approximation
The high temperature expansion is often applied to the thermal functions (A8). However, it is not useful for investigating the large field value m(φ) /T ≡ r ≥ 1. In this subsection we compare the exact forms of the the thermal functions (A8) numerically evaluated with their approximated forms and investigate the effectiveness of them.