Baryogenesis via leptonic CP-violating phase transition

We propose a new mechanism to generate a lepton asymmetry based on the vacuum CP-violating phase transition (CPPT). This approach differs from classical thermal leptogenesis as a specific seesaw model, and its UV completion, need not be specified. The lepton asymmetry is generated via the dynamically realised coupling of the Weinberg operator during the phase transition. This mechanism provides a connection with low-energy neutrino observables.


Introduction.
The origin of the matter-antimatter asymmetry is one of the most important mysteries of our Universe. One popular mechanism to explain this asymmetry is baryogenesis via leptogenesis. The classic examples of high-scale leptogenesis [1] introduce righthanded neutrinos, N , which are Majorana in nature and therefore break lepton number. The decay of the righthanded neutrinos provide a departure from equilibrium and their coupling to leptonic doublets, , violate CP. A lepton asymmetry, produced from the preferential decays of N , is subsequently converted to a baryon asymmetry by B − L conserving sphaleron processes [2]. In addition to fulfilling Sakharov's criterion [3], this scenario of leptogenesis provides a natural explanation of small neutrino masses via the seesaw mechanism.
The origin and energy scale of CP violation is still unknown and remains a widely studied theoretical issue. There is a rich programme of neutrino experiments such as LBNF/DUNE [4] and T2HK [5] that aim to measure leptonic CP violation. In conjunction, these experiments will investigate the correlations between leptonic observables. The interrelation between mixing angles and phases will play a crucial role in determining the fine structure of leptonic mixing. The observed pattern may be the result of an underlying flavour symmetry which could be continuous, U (1) [6], SU (3), SO(3) [7], or a non-Abelian, discrete symmetry such as A 4 , S 4 [8], et al. In these models, SM-singlet scalars (flavons) acquire vacuum expectation values that lead to the breaking of the flavour symmetry and results in the observed mixing structure and CP violation. The source of CP violation can arise spontaneously or explicitly. Spontaneous CP violation refers to the scenario in which CP conservation is imposed on the Lagrangian but is spontaneously broken by the vacuum [9,10] whilst explicit CP violation results from complex Yukawa couplings. Unlike conventional high-scale leptogenesis, where CP is explicitly violated above the seesaw scale, in this work we investigate the possibility CP violation occurs below such a scale.
There has been recent work that explores varying quark Yukawa couplings at the TeV scale as an origin of a strong first-order phase transition in electroweak baryogenesis [11,12]. This approach offers a link be-tween flavour in the quark sector and the observed baryon asymmetry. In leptonic flavour models, varying coefficients of the Weinberg operator provide a completely new mechanism to link flavour structure and a lepton asymmetry. Moreover, as experimental constraints on the leptonic flavour scale and thus flavon masses are not stringent, this allows the leptonic flavour scale to range from the electroweak [13] to beyond the seesaw scale.
In this paper, we propose a new mechanism for generating a lepton asymmetry based on a CP-violating phase transition (CPPT). This mechanism allows a connection between the baryon asymmetry, neutrino oscillation experiments and leptonic flavour mixing. CPPT differs from conventional scenarios of high-scale leptogenesis in several key aspects: we apply an effective field theory approach, which does not constrain the study to a particular model of neutrino mass generation and consequently CP violation occurs below this energy scale. We simply assume that neutrino masses are generated by the Weinberg operator, the coefficients of which are dynamically realised during CPPT. The lepton asymmetry is produced via the interference of Weinberg operators at different times. To perform the calculation we utilise the closed-time-path (CTP) formalism [14,15]. We focus on the generation of initial asymmetry at the constant temperature, T , and defer a more complete calculation of the final lepton asymmetry, accounting for evolution, to future work [16].
The Basic Idea of CPPT. As previously mentioned, throughout this work, we assume neutrinos acquire their masses via the dimension-5 Weinberg operator where λ αβ = λ βα and C is the charge conjugation matrix. For the purpose of generating a lepton asymmetry, the full UV completion of this operator need not be specified. The coupling of the Weinberg operator, λ αβ , can be associated to a SM-singlet scalar field (or a linear combination of scalar fields) whose vacuum expectation value (VEV) corresponds to physical leptonic masses and mixing. This dynamically generated coupling may be realised as λ αβ = λ 0 values of λ αβ before the flavon φ undergoes a phase transition.
In the Early Universe, the ensemble expectation value (EEV) of φ is dependent upon the finite temperature scalar potential. A phase transition occurs when the minima of this potential becomes metastable. As a consequence, the minima changes to a non-zero and stable value, φ . During this phase transition, bubbles of the leptonically CP-violating, broken phase begin to nucleate and expand within the symmetric phase. At a fixed space point around the bubble wall, λ αβ is time-dependent, λ αβ (t 1 ) = λ αβ (t 2 ). Subsequently, the lepton asymmetry is generated via the interference of the Weinberg operator at different times.
The Closed-Time-Path Formalism. Typically, observables in the thermal bath are derived from expectation values of operators that are not time-ordered. They can be calculated directly in the real time formalism, also known as the CTP formalism [14,15], which is derived from the first principles of Quantum Field Theory [17][18][19]. This method has successfully been applied to leptogenesis based on the decay of right-handed neutrinos (see, e.g. [20][21][22]). In this letter, we apply the CTP formalism to calculate the CPPT-induced lepton asymmetry. In comparison with classical Boltzmann transport equations, in which the collision terms are calculated in zero temperature, an advantage of this formalism is the proper inclusion of quantum memory effects [20]. As will be demonstrated, this effect plays a crucial role in our mechanism.
For the Higgs (H) in the CTP formalism, one defines the following four Green functions: where T (T ) denotes time (anti-time) ordering. The Feynman, Dyson and Wightman propagators are represented by ∆ T , ∆ T and ∆ <,> , respectively. Analogously, the definition of the Green functions for lepton ( α ) with flavour index α is The additional minus sign in S < comes from the anticommutation property of fermions. In Eqs. (2) and (3), electroweak gauge and fermion spinor indices have been suppressed. The Wightman propagators S <,> can be used to define the lepton asymmetry, e.g., the number density difference between lepton and anti-lepton n L ≡ α (n α − n α ) Moreover, they satisfy the Kadanoff-Baym (KB) equation. We follow the convention of [23][24][25] and express the KB equation as where denotes a convolution, Σ is the self energy of the lepton, and S H and Σ H are Hermitian parts of propagator and self energy given by On the LHS of Eq. (5), Σ H S <,> represents the self-energy contribution and Σ <,> S H describes the broadening of the on-shell dispersion relation. On the RHS, 1 2 (Σ > S < − Σ < S > ) is the collision term which includes the CP-violating source [23]. As we focus on the generation of an initial asymmetry, we consider only the collision term.
Lepton Asymmetry. We follow the techniques developed for thermal leptogenesis as presented in [20] and calculate the lepton asymmetry to leading order in a time-independent flavour basis. In order to derive the lepton asymmetry, the Green functions for the Higgs and leptons are Fourier transformed where r ≡ x 1 − x 2 , t 1 ≡ x 0 1 and t 2 ≡ x 0 2 . Subsequently, the lepton asymmetry at a fixed space point in the bubble wall may be written as n L (x) = where t i (t f ) is the initial (final) time and Σ k (t 1 , t 2 ) is the self-energy contribution. In this mechanism, the leading CP-violating contribution to Σ k (t 1 , t 2 ) is a 2-loop diagram as shown in Fig. 1. The memory effect is reflected in the 'memory integral' over t 1 and t 2 , which involves the time-dependent couplings shown in Fig. 1. Using Eq. (1), the lepton asymmetry may be re-expressed as 3 and M αβγδ (t 1 , t 2 , k, k , q, q ) is the finite temperature matrix element. Ignoring the differing flavours of lepton propagators, the total lepton asymmetry L k ≡ α L kαα is given by where the finite temperature matrix element, decomposed in terms of the lepton and Higgs propagators, is expressed as A homogeneous system is a principle assumption in the derivation of Eq. (7) [20]. However, this is clearly not the case for CPPT as the bubble expansion provides a special direction perpendicular to the bubble wall which results in the transport of the lepton asymmetry along this particular direction. We anticipate the directional dependence of the asymmetry will be small and therefore ignore its impact at this stage. As the temperature at which CPPT occurs is significantly higher than the electroweak scale, both leptons and the Higgs are almost in thermal equilibrium and we apply this approximation throughout this work.
The flavon EEV profile, φ , along the CP-violating bubble wall plays an important role in CPPT. An analogous example is the Higgs EEV profile studied in the electroweak strong first-order phase transition. It has been numerically checked the bubble wall profile fits to the form of a tanh function [26][27][28][29][30]. We assume the flavon EEV along the bubble has a similar structure and thus the resulting time-dependent coupling may be parametrised as where v w is the velocity of the bubble wall, L w the width of the wall and z a certain point along the direction of bubble expansion. During this process, the system begins its evolution from the false vacuum at t = −∞ to the true vacuum at t = +∞. Naively one can check that, in the asymptotic limits, the coupling behaves as expected: λ(−∞) = λ 0 and λ(∞) = λ 0 + λ 1 . Assuming this form of the coupling the coefficient, Im λ * (t 1 )λ(t 2 ) , may be factorised as Im tr [λ * (t 1 )λ(t 2 )] = Im tr λ 0 λ * sinh vwy where the average and relative coordinates aret = (t 1 + t 2 )/2 and y = t 1 − t 2 respectively. A convenient change of integration variables can be made such thatt can be integrated over. The lepton asymmetry becomes where we have reparametrised the effective neutrino mass matrices as m 0 As v w and L w have dropped out of Eq. (13), the bubble wall properties do not affect the lepton asymmetry. This is based on the assumption of fast bubble expansion. In the remainder of this work, we shall continue to assume that the bubble is expanding significantly faster than the Universe. As the bubble wall sweeps over a certain region, the temperature, T = 1/β, changes slightly.
To evaluate Eq. (13), firstly we calculate M using the lepton and Higgs propagators. As the scale of the CPPT is significantly higher than that of the electroweak scale, we will assume that leptons and Higgs are in thermal equilibrium. In the massless limit, the lepton and Higgs propagators are written as where k = | k|, q = | q|, c k = cos(ky − ), s k = sin(ky − ), ch k = cosh(kβ/2), sh k = sinh(kβ/2), y − = y − iβ/2, k = k/k and γ H , γ are the thermal damping rates of the Higgs and the leptonic doublets respectively [20]. Substituting Eq. (14) into Eq. (10), M becomes where γ = γ + γ H . In order to simplify Eq. (13), we will first perform the integration of the y variable and then the momentum integration. The y integration may be performed by exploiting that M is an odd function where K η2η3η4 = k + η 2 k + η 3 q + η 4 q .
The evaluation of the momentum integration will closely follow that of [20]. We abstain from re-deriving the details of this calculation and instead refer the reader to the reference. However we will present the simplified form of the momentum integration where p = k − q = k − q , q 2 = k 2 + p 2 − 2pk cos θ and q 2 = k 2 + p 2 − 2pk cos θ have been applied. Using Eq. (16) together with Eq. (17), the final result is written as F (x 1 , x γ ) is a loop factor given by where The loop factor is dependent upon the lepton energy and the thermal width normalised by the temperature, i.e., x 1 and x γ . Discussion. The lepton asymmetry, as shown in Eq. (7), is dependent upon three components: the loop factor F (x 1 , x γ ) derived from the correction to the lepton propagator; the effective neutrino mass matrices m 0 ν , m ν and the temperature, T , at which CPPT occurs. We shall address each of these contributions in turn.
In Fig. 2, we allow x γ to vary and display the numerical results of the loop factor F (x 1 , x γ ) as a function of x 1 . The main contribution to the Standard Model value, x γ ≈ 0.1, comes from electroweak gauge couplings [31]. However, this value may be enlarged due to new physics contributions at high energy scales. For x 1 ∼ 1, we observe the loop factor provides a factor O(10) enhancement to the lepton asymmetry.
We have introduced the effective neutrino mass matrices m 0 ν and m ν . The structure of m 0 ν is dependent on the coupling of the flavons to the Weinberg operator. The form of this coupling is determined by the details of particular flavour models and will be studied elsewhere. After CPPT, the coefficients of the Weinberg operator are fixed and m ν is identified to the measurable low-energy neutrino mass matrix (ignoring RG running effects). This mass matrix is diagonalised by the PMNS matrix, U PMNS , which allows for a strong connection between lepton asymmetry and low-energy leptonic observables.
x Γ 0.05 Finally, we discuss the associated energy scale of CPPT. The lepton asymmetry generated during CPPT is partially converted, via sphaleron processes, to the baryon asymmetry. The resultant baryon asymmetry is of the same order as the lepton asymmetry [2]. In order to estimate the temperature at which CPPT occurs, we will assume Im{tr[m 0 ν m * ν ]} is the same order as m 2 ν ∼ (0.1 eV) 2 and note that F (x 1 , x γ ) provides an O(10) factor enhancement. The temperature for successful CPPT is Using the observed ratio of baryon to photon, η B = (6.19 ± 0.15) × 10 −10 [32], and requiring L k ∼ η B we conclude T ∼ 10 11 GeV. This is a rough estimate and a more detailed calculation, involving the evolution of the initial asymmetry and inclusion of effects such as differing thermal width of charged leptons, may lower this scale. Conclusion. We have proposed a novel mechanism based on CPPT to generate the matter-antimatter asymmetry. It differs from conventional high-scale leptogenesis scenarios as we assume CP is broken below the scale of neutrino mass generation and apply an effective theory approach which permits model independence. Moreover, this mechanism allows for a connection between leptonic flavour structure and the baryon asymmetry.
The essential requirements of this approach are a CPviolating phase transition and the Weinberg operator. We assume the complex coefficients of this operator are dynamically realised. During the phase transition, the lepton asymmetry is generated from the interference of the Weinberg operator at different times. In order to generate the observed baryon asymmetry, the temperature scale of CPPT is approximately T ∼ 10 11 GeV.