Type II Seesaw Leptogenesis

Firstly, consider the following field redefinitions of $h$ and $\Delta^{0}$,

using the polar coordinate parametrization given in Eq. (15). Here we will temporarily ignore the motion of the angular directions, $\eta$ and $\theta$, since the potential is dominated by the dynamics of the radial directions, $\rho_{H}$ and $\rho_{\Delta}$. Under this redefinition, the kinetic terms of the Lagrangian become,

In our analysis, we require that the non-minimal couplings are large, $\xi_{H}$ and $\xi_{\Delta}\gg 1$, such that at leading order in $1/\xi$ the kinetic terms are given by,

For large $\xi$ parameters, the mixing term $(\partial_{\mu}\chi)(\partial^{\mu}\kappa)$ is suppressed, resulting in a canonically normalised $\chi$ and a suppression of the kinetic term of $\kappa$. Below we show the three main regimes for $\kappa$, that each have a corresponding canonically normalized variable $\kappa^{\prime}$,

Now consider the scalar potential under this field redefinition, in terms of $\kappa$,

where we have taken the large field limit for $\chi$ and assume that the $U(1)_{L}$ breaking terms can be ignored. This potential has the following minima, dependent upon the relations between the different non-minimal couplings and quartic self-couplings,

In Cases 2 and 3, we arrive at the usual single field non-minimally coupled inflationary potential, $\lambda_{\Delta}/(4\xi_{\Delta}^{2})M_{p}^{4}$ and $\lambda_{H}/(4\xi_{H}^{2})M_{p}^{4}$, respectively. The inflaton in Case 1 is characterised by a mixture of the two scalars which is the typical scenario we require to ensure that the lepton number violating interactions are relevant during inflation. Importantly in Case 1, the potential has the following minimum,

The numerator of Eq. (23) must be positive to ensure that we do not obtain a negative vacuum energy at large field values. In each of the above cases, the canonical field $\kappa^{\prime}$ has a large mass which is of order $M_{p}/\sqrt{\xi}$. This mass will always be greater than the Hubble parameter during inflation, so we can safely integrate out the $\kappa$ field, establishing the inflationary trajectory.

Having derived this inflationary trajectory, we can now proceed to analyzing the inflaton dynamics in our Lagrangian given in Eq. (13). Since we only wish to consider Case 1, we can simplify our analysis by defining the inflaton as $\varphi$, through the following relations to the polar coordinate fields,

The Lagrangian in Eq. (13) is now given by,

where

and

The Affleck-Dine mechanism will be realised in our scenario through the motion of the dynamical field $\theta$. The size of the generated lepton asymmetry will be determined by the size of the non-trivial motion induced in $\theta$ sourced by inflation. During inflation, $m\ll\varphi$, which means that the quartic term in the Jordan frame potential dominates the inflationary dynamics.

We will choose input parameters that preserve the inflationary trajectory and ensure that it is negligibly affected by the dynamics of $\theta$ and the lepton asymmetry production. In this case, the inflationary observables in our framework are consistent with those of the Starobinsky inflationary model, and as such, are in excellent agreement with current observational constraints Planck:2018jri .

Since we have derived the approximate single field inflationary behaviour above, we can analyse the evolution of the inflationary phase and its predictions for the inflationary parameters. To do this, we will first translate the Lagrangian in Eq. (25) from the Jordan frame to the Einstein frame, using the following transformations Wald:1984rg ; Faraoni:1998qx ,

and subsequently reparametrizing $\varphi$ in terms of the canonically normalized scalar $\chi$.

In transforming from the Jordan to Einstein frame, the Jordan frame field $\varphi$ no longer has a canonically normalised kinetic term. We must make the following field redefinition to obtain the canonically normalized scalar,

which gives the definition of $\chi$ in terms of $\varphi$ as,

Three distinct regimes can be observed in the evolution of the relation between the $\chi$ and $\varphi$ fields, namely,

Thus, we now obtain the final Einstein frame Lagrangian,

where

As with the relation between the $\chi$ and $\varphi$ fields, the scalar potential exhibits three distinct regimes in each field definition. Importantly, it is evident that the non-minimal coupling leads to a flattening of the potential in the large field limit. The scalar potential in each regime is,

Therefore, the $\chi$ potential replicates the Starobinsky form in the large field limit,

where the Starobinsky mass scale is required by observational constraints to be $m_{S}=\sqrt{\frac{\lambda M_{p}^{2}}{3\xi^{2}}}\simeq 3\cdot 10^{13}$ GeV Faulkner:2006ub ; Planck:2018jri . This will be the inflationary framework we will use in our scenario, and from which we will now determine the expected observational signatures.

Before calculating the inflationary observables, we provide a plot in Figure 1 of the evolution of $\chi$ during and after inflation, where we have numerically solved the equations of motion derived from the Lagrangian in Eq. (32). The input parameters are fixed to $\xi=300$ and $\lambda=4.5\cdot 10^{-5}$, we take $\tilde{\mu}$ to be sufficiently small to not affect the dynamics, and the following initial conditions are chosen, $\dot{\theta}_{0}=0$, $\chi_{0}=6.0M_{p}$, and $\dot{\chi}_{0}=0$ with the definition $H_{0}\equiv m_{S}/2$. The figure shows that the end of inflation occurs near $tH_{0}=100$, and thus, horizon crossing occurs at approximately $tH_{0}\approx 40-50$. After inflation, the universe quickly enters into a matter-like evolution until $tH_{0}\approx 400$. The line $\chi=M_{p}/\xi$ in Figure 1 is included to indicate the end of the matter-like epoch.

It is important to note that, the reason we choose $\xi=300$ and $\lambda=4.5\cdot 10^{-5}$ is that for larger $\xi$ values, or equivalently small $\lambda$, issues can arise regarding unitarity during the oscillatory stage that follows inflation in Higgs inflation models Ema:2016dny ; DeCross:2015uza ; DeCross:2016cbs ; DeCross:2016fdz . It has been found that this problem can be avoided if these models are UV completed by the addition of an $R^{2}$ term Giudice:2010ka ; Gorbunov:2018llf ; Ema:2017rqn ; Ema:2019fdd ; Ema:2020zvg .

Another subtlety is that the $\lambda$ coupling will obtain radiative corrections from the gauge couplings and Yukawa couplings of the SM and triplet Higgs’. A naive estimation of the size of these radiative corrections is given approximately by - gauge couplings: $\frac{g^{4}}{16\pi^{2}}$ ($\mathcal{O}(10^{-(3-4)})$); neutrino Yukawa couplings: $-\frac{y^{4}}{16\pi^{2}}$; or the top Yukawa coupling: $-\frac{y_{t}^{4}}{16\pi^{2}}$, for a general mixing angle $\alpha$. To obtain $\lambda=4.5\cdot 10^{-5}$, a small tuning of $y$ or $y_{t}$ might be necessary to cancel the contributions from the Higgs’ gauge couplings. However, we stress that such a small $\lambda$ is not a necessary condition for our mechanism to work, and we will give a comment on our final result for larger couplings. In any case, for couplings of $\lambda\sim 10^{-2}$, the dynamics will be stable against these radiative corrections. Nevertheless, we will use this benchmark coupling for our study to simplify the numerical analysis during the oscillation stage.

Considering the Einstein frame inflationary potential in Eq. (35), we derive the following characteristic slow-roll parameters,

where we have given them in terms of the number of e-foldings of expansion from horizon crossing to the end of the inflation, $N_{*}$. Subsequently, we obtain the predicted spectral index and tensor-to-scalar ratio in our inflationary scenario that should be observed in CMB measurements,

respectively. The required number of inflationary e-folds to solve the flatness and horizon problems is between $50<N_{*}<60$. Thus, our model gives the following predicted ranges for the spectral index and tensor-to-scalar ratio,

We can now compare these to the current best constraints on the inflationary predictions from the recent Planck results Planck:2018jri ,

for the $\Lambda-$CDM$+r$ model. Comparing these with those in Eq. (38) and (39), we see that our scenario is in extremely good agreement with current observational results for the required number of inflationary e-folds. The prediction for the tensor-to-scalar ratio is well below the current constraints, but it will be observable by the upcoming LiteBIRD telescope Hazumi:2019lys .

In addition to these results, the scalar perturbations observed in the CMB provide a constraint on the Starobinsky mass scale $m_{S}$. It is necessary to satisfy the following relation for consistency with this observation,

with the corresponding Starobinsky mass scale being $m_{S}\simeq 3\cdot 10^{13}$ GeV. This relation shall inform our parameter choices in the upcoming sections, and was used to make the parameter choices in Figure 1.

To begin our analysis, consider the Lagrangian of our model in the Einstein frame, as given in Eq. (32),

where

and

The equations of motion for $\chi$ and $\theta$ can be simply derived as follows,

Evaluating these equations of motion during the inflationary epoch, it is found that both $\chi$ and $\theta$ enter approximate slow-roll regimes described by,

To simplify our analysis, we make use of the following parameterization $\tau=tH_{0}$, where $H_{0}=m_{S}/2$ and define prime as the derivative with respect to $\tau$. We also define the reduced Hubble parameter $\tilde{H}$ by,

Under these redefinitions, the equations of motion from Eq. (48) become,

and the associated lepton number density is,

In Figure 2, we depict the evolution of $\theta$ and $\dot{\theta}$ during inflation, providing a comparison with the estimation of $\dot{\theta}$ presented in Eq. (49). We find that $\dot{\theta}$ is well described by the derived slow-roll approximation. Due to the non-zero $\dot{\theta}$ induced, there is an increase in $\theta$ during inflation. However, as shown in the right panel of Figure 2, for fixed $\theta_{0}$, the rate of increase is proportional to $\tilde{\lambda}_{5}$, which is indicated by the parameter dependencies found in Eq. (49).

During inflation, the $\chi$ field approaches $M_{p}$ and so we can ignore the subdominant $m$ and $\tilde{\mu}$ terms that are much less than $M_{p}$. Using these approximations, we can determine the lepton number asymmetry generated by the motion of the triplet Higgs phase $\theta$ at the end of inflation,

where the $\mathcal{O}(1)$ factor that appears in the second step is typically $\sim 3$ from numerical calculations, and accounts for the breakdown of the slow-roll approximation at the end of inflation, which we have defined as when the slow-roll parameter satisfies $\epsilon=1$. In the last step, we assume the quartic term dominates the inflationary potential and that the $\tilde{\lambda}_{5}$ coupling dominates the lepton number violating interactions. After inflation, the lepton number density is solely red-shifted by $a^{3}$ alongside another $\mathcal{O}(1)$ factor given by $\Omega$.

When inflation ends, at $\chi_{\textrm{end}}\simeq 0.67M_{p}$, the oscillatory epoch begins for which the universe is characterized by an approximately matter-like evolution. The inflationary potential at this time is given in Eq. (34). At the end of the inflationary stage, we define the lepton asymmetry number density as,

Now we wish to analyse the evolution of the lepton number density after inflation. To determine this, first consider the following relation derived from the equation of motion in Eq. (48),

During the matter-like phase, if the term $U_{,\theta}$ red-shifts faster than matter-like, $a^{-3}$, then it can be ignored and the quantity $a^{3}n_{L}/(Q_{L}\Omega^{2})$ will be conserved. This is guaranteed when the dominant $U(1)_{L}$ breaking term in $V(\varphi,\theta)$ is a polynomial function of $\varphi$ larger than degree four because the quartic term of $\varphi$ in $V(\varphi,\theta)$ is equivalent to the mass term of the $\chi$ field, which red-shifts as $a^{3}$ after inflation. Thus, we consider the quintic $\tilde{\lambda}_{5}$ term to be the dominant $U(1)_{L}$ breaking term at large field values.

If instead, the cubic term was to dominate, the $U(1)_{L}$ breaking term would become increasingly relevant as the expansion of the universe proceeds, and subsequently destroy the lepton number generated during inflation. In the following subsections, we will show that the cubic term is already required to be small to avoid the washout of the lepton asymmetry after reheating.

As shown in Eq. (55), after inflation the lepton asymmetry number density is red-shifted by the usual matter-like scale factor dependence, with the addition of a factor of $1/\Omega^{2}$ which is of order of $\mathcal{O}(1)$. It is then possible to estimate the lepton asymmetry number density at any time after inflation using,

In Figure 3, we depict a comparison of the numerical simulations to the analytical result given in Eq. (56) to demonstrate the accuracy of this relation. The input parameters have been chosen to fit the current CMB constraints.

In the preceding analysis, it has been assumed that the mixing angle $\alpha$ is fixed throughout the inflation and oscillation epochs. It is observed that the $\Omega^{2}$ factor quickly approaches 1 after inflation, and thus, the scalar potential can be approximated by $V(h,\Delta^{0})$ in Eq. (8). This potential also exhibits a characteristic minimum direction which is defined by,

where we require $2\lambda_{\Delta}-\lambda_{H\Delta}>0$ and $2\lambda_{H}-\lambda_{H\Delta}>0$ for stability. By comparing this relation to Eq. (16), we see that if $\xi_{H}=\xi_{\Delta}$, then the two mixing angles $\alpha$ and $\beta$ converge, and we can use the same mixing angle to describe the inflationary and oscillation epochs. If instead, we have $\xi_{H}\neq\xi_{\Delta}$, the $\alpha$ and $\beta$ are generally different and a detailed understanding of the evolution during the oscillation stage after inflation requires a dedicated analysis. However, we believe that the lepton asymmetry would only be negligibly affected if this is the case because almost all of the lepton asymmetry is generated during inflation, with its evolution just determined by red-shifting due to the expansion of the universe in the oscillatory phase. It should be noted that the detail of the oscillation epoch, and the transition between unequal $\alpha$ and $\beta$, might affect the expected preheating dynamics and the reheating temperature. The analysis of these dependencies is beyond the scope of this work, and so, we adopt $\xi_{H}=\xi_{\Delta}$ for simplicity.

We now provide a discussion of the dynamics of the cubic coupling $\tilde{\mu}$. Unlike the quartic and quintic terms, the cubic term becomes increasingly relevant as $\varphi$ decreases. If the cubic term is dominant during the oscillatory phase, the $\theta$ field will not exhibit directed motion in phase space, and will oscillate instead. This means that the lepton asymmetry starts to oscillate and the predictability of our model is lost. This is depicted in the top-left panel of Figure 4, where the $U(1)_{L}$ breaking term is taken to consist only of the $\tilde{\mu}$ term.

We must determine the condition that preserves the lepton asymmetry and does not allow this oscillatory behaviour to become significant. During the matter-like epoch, the lepton number density generated from the cubic term during one oscillation time $1/m_{S}$ can be given approximately by,

To preserve the lepton number density generated during inflation, it is necessary to ensure that $\Delta n_{L}\lesssim n_{L}$ throughout the matter-like epoch, prior to the completion of reheating. That is,

Numerically, we find the following constraint $\tilde{\mu}\lesssim 10^{-13}M_{p}$ when requiring $\Delta n_{L}\lesssim n_{L}$, where we have made the parameter choices $\theta_{0}=0.1$, $\tilde{\lambda}_{5}=10^{-9},$ and subsequently ${n_{L}}_{\textrm{end}}=1.1\cdot 10^{-11}M_{p}^{3}$. In Figure 4, we depict the evolution of the lepton asymmetry number density for different values of $\tilde{\mu}$. It is found that when $\tilde{\mu}<10^{-13}M_{p}$ the lepton asymmetry does indeed become stable, showing consistency with the analytical constraint. For parameter values that lead to the observed baryon asymmetry today, we obtain the following upper bound $\tilde{\mu}\lesssim 10^{-18}M_{p}$.

Once reheating is completed, the generated non-zero lepton number density $n_{L}$ will be present in the form of neutrinos. The lepton number carried by these neutrinos will be redistributed into baryons through the equilibrium electroweak sphalerons, with the ratio defined by $n_{B}\simeq-\frac{28}{79}n_{L}$ Klinkhamer:1984di ; Kuzmin:1985mm ; Trodden:1998ym ; Sugamoto:1982cn . To calculate the baryon asymmetry parameter generated in our scenario, we must determine the reheating temperature.

The process of reheating in standard Higgs inflation was first analysed in Ref. Garcia-Bellido:2008ycs ; Bezrukov:2008ut , where it was found that the parametric resonance production of $W/Z$ bosons plays a significant role in the preheating process. It has since been found that the preheating process is more violent than was previously expected Ema:2016dny ; DeCross:2015uza ; DeCross:2016cbs ; DeCross:2016fdz , with unitarity being violated for SM Higgs non-minimal couplings of $\xi>350$. However, recently it has been found that for such large $\xi$ values, it is possible for the model to be UV complete in Higgs-$R^{2}$ scenarios of inflation Giudice:2010ka ; Gorbunov:2018llf ; Ema:2017rqn ; Ema:2019fdd ; Ema:2020zvg for which the preheating process must be recalculated He:2018mgb ; He:2020ivk ; He:2020qcb . In our scenario, we choose $\xi=300$ to avoid any possible issues with unitarity. A recent analysis of the preheating process in SM Higgs inflation Sfakianakis:2018lzf has shown that for models with $\xi>100$, the reheating happens at approximately $\sim 3$ e-folds after the end of inflation, so we adopt $\Delta N=N_{\textrm{reh}}-N_{\textrm{end}}=3$ for simplicity. The details of reheating may be different in our scenario due to the mixing of the SM and triplet Higgs’, and thus, it is necessary to undertake a comprehensive analysis, which will be left to future work.

For the typical parameters chosen, the reheating process occurs at $t_{\textrm{reh}}=223/H_{0}$, which has the corresponding Hubble parameter $H_{\textrm{reh}}=0.0047H_{0}$. From the relation $H_{\textrm{reh}}^{2}\simeq\frac{\pi^{2}}{90}g_{*}\frac{T_{\textrm{reh}}^{4}}{M_{p}^{2}}$, we then obtain the reheating temperature $T_{\textrm{reh}}\approx 2.2\cdot 10^{14}$ GeV. Considering the entropy density just after reheating defined by $s=\frac{2\pi^{2}}{45}g_{*}T_{\textrm{reh}}^{3}$ Husdal:2016haj , we then determine that the baryon asymmetry parameter generated in our model is given by,

where $\eta_{B}^{\textrm{obs}}\simeq 8.5\cdot 10^{-11}$ is the observed baryon asymmetry parameter Aghanim:2018eyx . This relation shows that at the end of inflation, it is necessary to have generated a lepton number asymmetry density of $1.3\cdot 10^{-16}M_{p}^{3}$ to produce the observed baryon asymmetry, which corresponds to the example parameter sets: $\tilde{\lambda}_{5}=7\cdot 10^{-15}$ for $\theta_{0}=0.1$ and $\tilde{\lambda}_{5}=10^{-10}$ for $\theta_{0}=6.5\cdot 10^{-6}$ from numerical calculations. These example parameters illustrate the wide range of parameters over which successful Leptogenesis can occur. Note that in both of these example cases, the typical parameters escape the isocurvature constraints Byrnes:2006fr ; Gordon:2000hv ; Kaiser:2012ak placed by CMB observations Planck:2018jri ; see the next section for the detailed calculations of the expected isocurvature perturbations in our framework.

As detailed in the previous subsection, if the $\tilde{\mu}$ coupling is too large, the lepton asymmetry will begin to rapidly oscillate during the matter-like epoch and the predictability of our model breaks down. On the other hand, a smaller $\tilde{\mu}$ term helps to avoid the washout of the lepton asymmetry after reheating. In the following, we require $|\tilde{\mu}|\lesssim 10^{-18}M_{p}$ for the initial $\theta_{0}=0.1$ to ensure the generated baryon asymmetry is preserved and consistent with observation.

In our analysis, we choose the parameter inputs $\lambda_{H}=0.1$, and $\xi_{H}=\xi_{\Delta}=300$ based on the arguments laid out above. For the other parameters, we set $\lambda_{\Delta}=4.5\cdot 10^{-5}$ to accommodate the current CMB measurements, while there exists two options for $\lambda_{H\Delta}$ that ensure the required inflationary trajectory is preserved. In the case of $\lambda_{H\Delta}>0$, we require $2\lambda_{\Delta}\xi_{H}-\lambda_{H\Delta}\xi_{\Delta}>0$ and $2\lambda_{H}\xi_{\Delta}-\lambda_{H\Delta}\xi_{H}>0$ so that we have a mixed state of $h$ and $\Delta^{0}$ as the inflaton. The typical value we can consider is $\lambda_{H\Delta}=10^{-5}$, which gives the mixing angle as $\alpha\simeq 0.02$. In the case of $\lambda_{H\Delta}<0$, we require that $|\lambda_{H\Delta}|<2\sqrt{\lambda_{H}\lambda_{\Delta}}$ to avoid the potential becoming unbounded from below. We can then choose $\lambda_{\Delta}=4.5\cdot 10^{-5}$, $\lambda_{H\Delta}=-0.001$, which gives $\alpha\simeq 0.07$.

We stress here that the above parameter choices are just a simplified version of our benchmark model considered in previous subsections. For the parameters in a real model, we need to also consider the contributions of radiative corrections, as well as the satisfaction of the vacuum stability conditions. As we have mentioned before, the selection of $\lambda_{H\Delta}\sim\lambda_{\Delta}\sim\mathcal{O}(0.01-0.1)$ may be a more realistic coupling choice to avoid any issues that could arise from radiative corrections.

Numerically, an example set of parameters that successfully generates the observed baryon asymmetry is $\lambda_{5}\sim\lambda_{5}^{\prime}=8.8\cdot 10^{-11}~{}(7.9\cdot 10^{-12})$ with $\theta_{0}\sim 0.1$ in combination with the typical parameter choices given above, for the $\lambda_{H\Delta}>0$ ($\lambda_{H\Delta}<0$) cases respectively. In addition, we obtain an upper limit on the cubic term coupling of $|\mu|\lesssim 15~{}(1.4)$ TeV. Note that, all these parameters are defined at a renormalization scale near $M_{p}$. Given that the $U(1)_{L}$ breaking terms exhibit small couplings, it is natural to consider that they originate from a spurion field that carries a $U(1)_{L}$ charge of $+2$. The $U(1)_{L}$ breaking terms are subsequently generated by requiring that the spurion field obtains a VEV of the order of $\mathcal{O}(10^{4})$ TeV.