Adding additional light scalar particle(s) to the Higgs sector at weak scale could lift the Higgs self-coupling to the level of $O(0.1)$ at the inflation scale [14]. For the current study we will just adopt the minimal SM Higgs sector.

An approximate shift symmetry plays a vital role for ensuring flatness of the inflaton potential during inflation, which holds for many inflation models such as the natural inflation [18] and axion monodromy inflation [19, 20], or other models with inflation driven by pseudo-Nambu-Goldstone bosons.

As an illustration, this dimension-5 operator in Eq. (8) may be induced from a UV model having an approximate global $U(1{)}_{B-L}$ symmetry which is spontaneously broken by a new scalar field $\mathrm{\Phi }$ with a $U(1{)}_{B-L}$ charge $-2$ and having a vacuum expectation value (VEV), $⟨\mathrm{\Phi }⟩=f$. After spontaneous symmetry breaking (SSB), the Yukawa interaction $(y_x\overline{N_R^c}{N}_R\mathrm{\Phi }+\mathrm{H}.\mathrm{c}.)$ will generate a Majorana mass $M=y_{x}f$ for the right-handed neutrino $N_R$, where $f=M/y_x=O(10)M$ for a natural Yukawa coupling $y_x=O(0.1)$. The inflaton $\varphi$ emerges as a pseudo-Nambu-Goldstone boson from $\mathrm{\Phi }$ and has its mass generated by soft breaking of the $U(1{)}_{B-L}$ symmetry, and its interactions with fermions are dictated by their U(1) charges. The residual shift symmetry enforces that the inflaton-fermion coupling (such as the $N_R-N_R-\varphi$ coupling) takes the form of a dimension-5 operator. Its cutoff scale $\mathrm{\Lambda }=f$ is given by the $U(1{)}_{B-L}$ breaking scale $f$, which is about a factor of $O(10)$ of the $N_R$ mass scale as shown above since the $N_R$ mass originates from the SSB of $U(1{)}_{B-L}$.

In Ref. [23], the authors introduced a dimension-4 coupling between inflaton ($\varphi$) and right-handed neutrino ($N_R$), which generates the Majorana mass term for $N_R$. They made an assumption that the Dirac mass $y_{h}h\gg M$ with negligible Majorana mass $M$ of the right-handed neutrino. In this way, the inflaton decay is kinematically blocked if the inflaton mass is smaller than twice the Dirac mass. Once the Higgs vacuum expectation value decreases below a certain threshold, the inflaton can decay. This is a different scenario of Higgs modulated reheating and the resultant NG originates from this blocking effect. This scenario assumed negligible Majorana mass $M$ of $N_R$ and did not test high scale seesaw. Besides, the dimension-4 coupling between inflaton and $N_R$ has to be generally suppressed to keep the flatness of inflaton potential (which is realized by imposing an approximate shift symmetry in our framework), so our dimension-5 coupling in Eq. (8) plays the major role for inflaton decay. Hence, our present work has proposed a new scenario of Higgs modulated reheating and can test the high scale seesaw mechanism, which would be complementary to the case of Ref. [23].

Here $t_{\inf }$ is the physical time at the end of inflation, $t_{\mathrm{reh}}$ is the physical time at which reheating occurs, ${\rho }_{\mathrm{f}}$ is a reference energy density and $t_{\mathrm{f}}$ is the reference time where the energy density $\rho ={\rho }_{\mathrm{f}}$ after the completion of reheating.

Here $⟨\delta h_{{\mathbf{k}}_{\mathbf{1}}}\delta h_{{\mathbf{k}}_{\mathbf{2}}}\delta h_{{\mathbf{k}}_{\mathbf{3}}}{⟩}^{\prime }$ is defined as the 3-point correlation function without momentum delta function: $⟨\delta h_{{\mathbf{k}}_{\mathbf{1}}}\delta h_{{\mathbf{k}}_{\mathbf{2}}}\delta h_{{\mathbf{k}}_{\mathbf{3}}}⟩=(2\pi {)}^3{\delta }^3({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3)⟨\delta h_{{\mathbf{k}}_{\mathbf{1}}}\delta h_{{\mathbf{k}}_{\mathbf{2}}}\delta h_{{\mathbf{k}}_{\mathbf{3}}}{⟩}^{\prime }$.

These inputs correspond to the light neutrino mass $m_{\nu }$ at the low-energy electroweak scale. The value of $m_{\nu }$ at the seesaw scale or Hubble scale is connected to its low energy value via renormalization-group (RG) running and is enlarged by about 30% relative to its low energy value [49, 50]. The RG running effects are included in our present analysis.