Perturbative and Nonperturbative Contributions to a Simple Model for Baryogenesis

Single field baryogenesis, a scenario for Dirac leptogenesis sourced by a time-dependent scalar condensate, is studied. We compare the creation of the charge asymmetry by the perturbative decay of the condensate with the nonperturbative decay, a process of particle production commonly known in the context of inflation as preheating. The nonperturbative channel dominates when the coupling of the scalar field to leptons is sufficiently large.


INTRODUCTION
Models for baryogenesis tie together cosmology and particle physics [1]. The discovery of small neutrino masses [2] and their explanation via lepton-number violating Majorana masses and the see-saw mechanism strongly supports the leptogenesis mechanism [3]. An alternative to this scenario is to assume pure Dirac mass terms for the neutrinos, that is the absence of Majorana masses, and to induce an asymmetry between the left-and right-handed neutrinos, which is subsequently turned into baryon number through sphaleron transitions.
This idea is referred to as Dirac leptogenesis [4].
In many scenarios for baryogenesis the necessary C and CP violation occur simultaneously, induced by a matrix of Yukawa couplings. An exception is electroweak baryogenesis [5,6], where in first place a CP -violating axial asymmetry is produced which is subsequently not erased via equilibration. Parity violation is then contributed in a second step due to sphaleron interactions.
Also Dirac leptogenesis relies on the transition of an axial asymmetry into baryons through the sphaleron process. The initial asymmetry is stored within Dirac neutrinos and does not get erased due to equilibration until electroweak symmetry breaking since the Yukawa coupling to the Standard Model Higss field is tiny. While the CP asymmetry can be provided through a matrix of Yukawa couplings and the out-of equilibrium decay of heavy scalar particles into Dirac neutrinos [4], it has been suggested that also a single Dirac mass term can source CP , provided it is time dependent. This mass term can arise due to Yukawa couplings of neutrinos to a rolling scalar field, and the resulting mechanism has been named single field baryogenesis [7]. Recently, an interesting realisation of this mechanism through a decaying Affleck Dine condensate has been proposed [8].
Interpreting the scalar condensate oscillating around zero as a large amount of scalar quanta at zero momentum, the axial asymmetry can be generated due to the perturbative decay of these particles, as commonly assumed in scenarios for Affleck-Dine baryogenesis [9].
However particles can also be produced nonperturbatively, as first pointed out in Ref. [10], a process which is often referred to as preheating in the context of the decay of the inflaton [11].
Various aspects of preheating from the decay of flat directions are discussed in [12].
In parallel, in the coherent baryogenesis [13] scenario, the oscillating condensate leads directly and at tree-level to the production of a charge asymmetry during preheating when it couples to matter such that a time-dependent C and CP violating mass matrix arises.
Consequently, in the case of a single time-dependent mass term a preheating process can lead to an axial asymmetry [14] and thereby source single field baryogenesis.
In the present analysis, we focus on the importance of non-perturbative contributions to the baryon asymmetry, and we choose the single field model due to its simplicity. We emphasise nonetheless that nonperturbative particle production may be of relevance for other scenarios, e.g Affleck-Dine baryogenesis.

PERTURBATIVELY SOURCED SINGLE FIELD BARYOGENESIS
Let us begin by considering a simple toy-model potential The fields φ u and φ d are scalar and are multiplets of the electroweak group G EW = SU(2) L × and ν R = (1, 0) are Weyl fermions. The scalar mass eigenstates are then 1 The inflationary Hubble rate is given by H I , and we assume that both of these mass eigenvalues are slightly below this value. Therefore, at horizon exit the scalar fields get amplified up to a magnitude ∼ H I and a random direction in SU(2) L -space. Inflationary expansion then leaves behind a homogeneous vacuum expectation value for the scalar fields in our patch of the Universe. This induces large neutrino masses, such that they initially do not thermalise. In turn, the potential for φ u,d does not get altered by thermal corrections.
Since we are interested in neutrino production, in the following, we consider only the neutral components φ 0 u,d . In particular, since the asymmetry is produced from φ 0 u , we take this to be the source field as in the single field baryogenesis scenario. Furthermore, we assume m ≪ µ.
Coherent oscillations begin at the time when the Hubble rate has decreased to the value µ, and the solution for φ 0 u can be approximated for small small µ/H by where the values of A R,I 1,2 are random initial values arising from inflation, as described above, and a(t) denotes the scale factor of the Universe, t denotes comoving time. In order to keep the present discussion simple, we assume A R 1 = A R , A R 2 = 0, A I 2 = A I and A I 1 = 0 in our patch of the Universe. Under these conditions, the charge density carried by the field φ 0 u is given by where we expanded in µ/m and neglected time derivatives acting on the scale factor. According to the interaction term with the leptons in the potential (1), Q φ is transferred to a charge asymmetry within the left handed neutrinos when φ 0 u decays. Due to conservation of total lepton number, a precisely opposite amount of the asymmetry is stored within the right handed neutrinos. However, this asymmetry in the right-handed sector is not transferred into baryons by sphalerons due to the left-handed nature of interactions.
We assume that the Universe is radiation dominated when coherent oscillations commence and that this remains so until the scalar fields decay, such that they contribute only negligibly to the entropy density s. Just like Q φ , s scales down as a −3 , such that we find for the asymmetry within left-handed neutrinos at the time Γ −1 , when the scalar field decays, Here, we have used the relations H = 1.66g 1/2 * T 2 /m P l , s = 2π 2 45 g * T 3 and have taken H ≈ µ. The number of relativistic degrees of freedom is denoted by g * , such that α is a numerical constant of order one for realistic values of g * . Furthermore, we have assumed that φ 0 u decays at a total rate Γ, whereas the decay rate into neutrinos is given by Γ ν = λ 2 ν µ/(8π), such that a branching factor of Γ ν /Γ arises.

NONPERTURBATIVE SOURCE
Following Ref. [14], we calculate the axial asymmetry induced by the nonperturbative decay of φ 0 u . We do so by solving numerically the conformally rescaled Dirac equation We take such that by the potential (1), we have Furthermore, we introduce the conformal time η, which is related to comoving time as dt = adη, and we take ∂ 0 = ∂ η .
Let us introduce the positive and negative frequency mode functions, u h (k, η) and v h (k, η) = −iγ 2 (u h (k, η)) * , respectively. They form a basis for the Dirac field, where ξ h is the helicity two-eigenspinor,ĥξ h = hξ h . The Dirac equation then decomposes From L h and R h , we can define the quantities where f 0h is the charge density, f 3h the axial charge density, f 1h the scalar density and f 2h the pseudoscalar density. Note that one can easily show that f 0h is conserved by Eq. (10), reflecting the charge conservation of the Dirac neutrinos.
The initial conditions corresponding to a particle number n h (k) = |β 0 | 2 are where and ω(k) = k 2 + |m| 2 . Since we assume to have initially zero neutrinos, we take β 0 = 0 in the following.
When φ 0 u ceases to oscillate, the particle number is given by Of course there is no charge asymmetry, since there is an opposite amount of antiparticles.
However, when m I = 0, an asymmetry in the number of particles with positive (h = +) and negative (h = −) helicity may be generated. Note that in the limit m R , m I → 0, n kh = 1 2 hf 3h + 1 2 , since then chirality and helicity coincide. Therefore, when the masses vanish. The factor two on the left hand side occurs because the total axial asymmetry gets contributions from particles and antiparticles, while n h (k) counts just the particles.
With a prime denoting a derivative w.r.t. η, the scalar equation of motion reads During radiation expansion, a = a R η, and when H = a ′ /a 2 ≪ √ µ 2 ± m 2 , the solution to this equation is well approximated by with the same assumptions for the real and imaginary parts as in the previous section. We use this solution to obtain the Dirac neutrino mass term (8) and numerically solve Eq. (10) by integrating up to the time when Γ > H, such that the Dirac mass term ceases to oscillate and the axial charges f 3h (k) get frozen in. A typical plot of the spectrum of the generated charge charge asymmetry is given in FIG. 1. Particle production occurs at a time t Res when the fermionic mode is in resonance with the coherently oscillating field. The production of the soft modes with small momentum k is suppressed because the initial charge asymmetry in the scalar field is small (m 2 /µ)t Res ≪ 1, cf. Eq. (5). Consequently, the production of asymmetry within modes which resonate later becomes stronger first. Eventually, there is a damping effect due to the red-shifting of the oscillating condensate and finally due to its decay at the rate Γ. Note that due to Pauli blocking −2 ≤ f 3+ (k) + f 3− (k) ≤ 2.
Of course, the axial asymmetry vanishes in the case when the scalar charge (4) is zero.
When A I = 0, the term ∝ γ 5 in the Dirac equation (6)  CP -violation. When A I = 0 but m = 0 there is also zero scalar charge. Since then the phase is constant, ∂ η arg(m r + im I ) = 0, the γ 5 -term can in principle be removed at all times by a rephasing of the fermionic field. Consequently, even if we do not perform this rephasing, we expect to find f 3+ (k) + f 3− (k) ≡ 0, which can also be verified numerically.
The axial charge density stored within the neutrinos is is the integral over the asymmetry within the modes where k is to be understood as the physical momentum at the time when coherent oscillations begin. The axial asymmetry to entropy ratio then turns out to be which we want to compare with the perturbative result (5).
We denote the axial densities n(ν L ) − n(ν L ) by ρ res for the nonperturbative or resonant case of Eq. (18) and by ρ pert for the perturbative decay as expressed in Eq. (5). The initial amplitudes of the scalar field are chosen to be A R = A I . We display the produced axial asymmetries over the initial amplitudes in FIGs. 2 and 3, where we have taken different values for the damping rate Γ.
Clearly, the perturbative source ρ pert gets enhanced by the factor Γ ν /Γ in (5) as the damping Γ becomes smaller. Note that we have consistently chosen Γ ν ≤ Γ with the case Γ ν = Γ displayed in FIG. 3. But also the nonperturbative contribution grows for smaller decay rates, because coherent oscillations last longer and a larger phase space volume may be filled as the fermionic modes are red-shifted. However, while initially ρ pert and ρ res grow as the square of the initial scalar amplitude, ρ res gets suppressed for large amplitudes due to Pauli blocking, which we do not take into account in our formula for the perturbative asymmetry (5). In either case as displayed in FIGs. 2 and 3, we note that the total asymmetry is the sum of the individual contributions, ρ pert + ρ res .

CONCLUSIONS
Nonperturbatively sourced single field baryogenesis is a viable scenario. We have shown that it contributes over a wide range of parameter space by the same order of magnitude as the perturbative source to the baryon asymmetry. Note that a rescaling of the Yukawa coupling λ ν can be absorbed into different intitial amplitudes A R,I , such that the effect can be read of from FIGs. 2 and 3. Besides the model presented here, the coherent baryogenesis mechanism is an example for generating the bayon asymmetry directly from preheating. We conclude that processes of nonperturbative particle production may be of importance for explaining the baryon asymmetry of the Universe.