Non-thermal Leptogenesis Research Articles

Leptogenesis in parity solutions to the strong CP problem and Standard Model parameters

We study the simplest theories with exact spacetime parity that solve the strong CP problem and successfully generate the cosmological baryon asymmetry via decays of right-handed neutrinos. Lower bounds are derived for the masses of the right-handed neutrinos and for the scale of spontaneous parity breaking, vR. For generic thermal leptogenesis, vR ≳ 1012 GeV, unless the small observed neutrino masses arise from fine-tuning. We compute vR in terms of the top quark mass, the QCD coupling, and the Higgs boson mass and find this bound is consistent with current data at 1σ. Future precision measurements of these parameters may provide support for the theory or, if vR is determined to be below 1012 GeV, force modifications. However, modified cosmologies do not easily allow reductions in vR — no reduction is possible if leptogenesis occurs in the collisions of domain walls formed at parity breaking, and at most a factor 10 reduction is possible with non-thermal leptogenesis. Standard Model parameters that yield low values for vR can only be accommodated by having a high degree of degeneracy among the right-handed neutrinos involved in leptogenesis. If future precision measurements determine vR to be above 1012 GeV, it is likely that higher-dimensional operators of the theory will yield a neutron electric dipole moment accessible to ongoing experiments. This is especially true in a simple UV completion of the neutrino sector, involving gauge singlet fermions, where the bound from successful leptogenesis is strengthened to vR ≳ 1013 GeV.

Towards a systematic study of non-thermal leptogenesis from inflaton decays

This paper investigates non-thermal leptogenesis from inflaton decays in the minimal extension of the canonical type-I seesaw model, where a complex singlet scalar ϕ is introduced to generate the Majorana masses of right-handed neutrinos (RHNs) and to play the role of inflaton. First, we systematically study non-thermal leptogenesis with the least model dependence. We give a general classification of the parameter space and find four characteristic limits by carefully examining the interplay between inflaton decay into RHNs and the decay of RHNs into the standard-model particles. Three of the four limits are truly non-thermal, with a final efficiency larger than that of thermal leptogenesis. Two analytic estimates for these three limits are provided with working conditions to examine the validity. In particular, we find that the strongly non-thermal RHNs scenario occupies a large parameter space, including the oscillation-preferred K range, and works well for a relatively-low reheating temperature TRH ≥ 103 GeV, extending the lower bound on the RHN mass to 2 × 107 GeV. The lepton flavor effects are discussed. Second, we demonstrate that such a unified picture for inflation, neutrino masses, and baryon number asymmetry can be realized by either a Coleman-Weinberg potential (for the real part of ϕ) or a natural inflation potential (for the imaginary part of ϕ). The allowed parameter ranges for successful inflation and non-thermal leptogenesis are much more constrained than those without inflationary observations. We find that non-thermal leptogenesis from inflaton decay offers a testable framework for the early Universe. It can be further tested with upcoming cosmological and neutrino data. The model-independent investigation of non-thermal leptogenesis should be useful in exploring this direction.

Parameter space of leptogenesis in polynomial inflation

Polynomial inflation is a very simple and well motivated scenario. A potential with a concave “almost” saddle point at field value ϕ = ϕ 0 fits well the cosmic microwave background (CMB) data and makes testable predictions for the running of the spectral index and the tensor to scalar ratio. In this work we analyze leptogenesis in the polynomial inflation framework. We delineate the allowed parameter space giving rise to the correct baryon asymmetry as well as being consistent with data on neutrino oscillations. To that end we consider two different reheating scenarios. (i) If the inflaton decays into two bosons, the reheating temperature can be as high as T rh ∼ 1014 GeV without spoiling the flatness of the potential, allowing vanilla N 1 thermal leptogenesis to work if T rh > M 1 where N 1 is the lightest right-handed neutrino and M 1 its mass. Moreover, if the dominant decay of the inflaton is into Higgs bosons of the Standard Model, we find that rare three-body inflaton decays into a Higgs boson plus one light and one heavy neutrino allow leptogenesis even for T rh < M 1 if the inflaton mass is of order 1012 GeV or higher; in the polynomial inflation scenario this requires ϕ 0 ≳ 2.5 MP . This novel mechanism of non-thermal leptogenesis is quite generic, since the coupling leading to the three-body final state is required in the type I see-saw mechanism. (ii) If the inflaton decays into two fermions, the flatness of the potential implies a lower reheating temperature. In this case inflaton decay to two N 1 still allows successful non-thermal leptogenesis if ϕ 0 ≳ 0.1 MP and T rh ≳ 106 GeV.

Gravitational waves from metastable cosmic strings in the Pati-Salam model in light of new pulsar timing array data

A series of pulsar timing arrays (PTAs) recently observed gravitational waves at the nanohertz frequencies. Motivated by this remarkable result, we present a novel class of Pati-Salam models that give rise to a network of metastable cosmic strings, offering a plausible explanation for the observed PTA data. Additionally, we introduce a hybrid inflationary scenario to eliminate magnetic monopoles that arise during the subsequent phase transitions from the Pati-Salam symmetry to the Standard Model gauge group. The resulting scalar spectral index is compatible with Planck data, and the tensor-to-scalar ratio is anticipated to be extremely small. Moreover, we incorporate a nonthermal leptogenesis to generate the required baryon asymmetry in our framework. Finally, the gravitational wave spectra generated by the metastable cosmic strings not only correspond to signals observed in recent PTAs, including NANOGrav, but are also within the exploration capacity of both present and future ground- and space-based experiments. Published by the American Physical Society 2024

Primordial non-Gaussianity as a probe of seesaw and leptogenesis

We present the possibility that the seesaw mechanism and nonthermal leptogenesis can be investigated via primordial non-Gaussianities in the context of a majoron curvaton model. Originating as a massless Nambu-Goldstone boson from the spontaneous breaking of the global baryon (B) minus lepton (L) number symmetry at a scale vB−L, majoron becomes massive when it couples to a new confining sector through anomaly. Acting as a curvaton, majoron produces the observed red-tilted curvature power spectrum without relying on any inflaton contribution, and its decay in the post-inflationary era gives rise to a nonthermal population of right-handed neutrinos that participate in leptogenesis. A distinctive feature of the mechanism is the generation of observable non-Gaussianity, in the parameter space where the red-tilted power spectrum and sufficient baryon asymmetry are produced. We find that the non-Gaussianity parameter fNL ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(0.1) is produced for high-scale seesaw (vB−L at O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(1014−17) GeV) and leptogenesis (M1 ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(106) GeV) where the latter represents the lightest right-handed neutrino mass. While the current bounds on local non-Gaussianity excludes some part of parameter space, the rest can be fully probed by future experiments like CMB-S4, LSST, and 21 cm tomography.

Hilltop sneutrino hybrid inflation

In this work, we consider a hilltop version of the (supersymmetric) sneutrino hybrid inflation where the right-handed sneutrino field plays the role of the inflaton field. This model is a type III hilltop inflation that can produce a spectral index ns=0.96 which fits perfectly to experimental observations without fine-tuning of parameters. We also briefly consider nonthermal leptogenesis via the decay of the right-handed sneutrino inflaton field after inflation.

Dai-Freed anomaly in the standard model and topological inflation

When we impose discrete symmetries in the standard model we have Dai-Freed global anomalies. However, interestingly if we introduce three right-handed neutrinos we can have an anomaly-free discrete Z4 gauge symmetry. This Z4 symmetry should be spontaneously broken down to the Z2 symmetry to generate the heavy Majorana masses for the right-handed neutrinos. We show that this symmetry breaking naturally generates topological inflation, which is consistent with the CMB observations at present and predicts a significant tensor mode with scalar-tensor ratio r > 0.03. The right-handed neutrinos play an important role in reheating processes. The reheating temperature is as high as ∼ 108 GeV, and non-thermal leptogenesis successfully takes place.

Observable r, gravitino dark matter, and non-thermal leptogenesis in no-scale supergravity

We analyse the shifted hybrid inflation in a no-scale supersymmetric SU(5) GUT model which naturally circumvents the monopole problem. The no-scale framework is derivable as the effective field theory of the supersymmetric (SUSY) compactifications of string theory, and yields a flat potential with no anti-de Sitter vacua, resolving the η problem. The model predicts a scalar spectral tilt ns compatible with the most recent measurements by the Planck satellite, while also accommodating large values of the tensor-to-scalar ratio r (~ 0.0015), potentially measurable by the near-future experiments. Moreover, the proton decay lifetime in the presence of the dimension-5 operators is found to lie above the current limit imposed by the Super-Kamiokande experiment. A realistic scenario of reheating and non-thermal leptogenesis is employed, wherein the reheating temperature Tr lies in the (2 × 106 ≲ Tr ≲ 2 × 109) GeV range, and at the same time realizing gravitino as a viable dark matter (DM) candidate.

Muon g − 2 and non-thermal leptogenesis in textrm{U}{(1)}_{L_{mu }-{L}_{tau }} model

The gauged textrm{U}{(1)}_{L_{mu }-{L}_{tau }} symmetry is the simplest possibility to explain the observed muon g − 2, while being consistent with the neutrino oscillations through the seesaw mechanism. In this paper, we investigate if leptogenesis can work at the same time. At first glance, leptogenesis seems challenging because the right-handed neutrino masses are related to the textrm{U}{(1)}_{L_{mu }-{L}_{tau }} breaking scale of 10 – 100 GeV as required from the muon g − 2. Contrary to this expectation, we find that non-thermal leptogenesis with the right-handed neutrino masses of mathcal{O} (107) GeV is possible. The successful scenario results in strict predictions on the neutrino oscillation parameters, which will be tested in future experiments.

A predictive SO(10) model

We discuss some testable predictions of a non-supersymmetric SO(10) model supplemented by a Peccei-Quinn symmetry. We utilize a symmetry breaking pattern of SO(10) that yields unification of the Standard Model gauge couplings, with the unification scale also linked to inflation driven by an SO(10) singlet scalar field with a Coleman-Weinberg potential. Proton decay mediated by the superheavy gauge bosons may be observable at the proposed Hyper-Kamiokande experiment. Due to an unbroken Z 2 gauge symmetry from SO(10), the model predicts the presence of a stable intermediate mass fermion which, together with the axion, provides the desired relic abundance of dark matter. The model also predicts the presence of intermediate scale topologically stable monopoles and strings that survive inflation. The monopoles may be present in the Universe at an observable level. We estimate the stochastic gravitational wave background emitted by the strings and show that it should be testable in a number of planned and proposed space and land based experiments. Finally, we show how the observed baryon asymmetry in the Universe is realized via non-thermal leptogenesis.

Constraining the cosmic strings gravitational wave spectra in no-scale inflation with viable gravitino dark matter and nonthermal leptogenesis

We revisit the hybrid inflation model gauged by $U(1)_{B-L}$ extension of the Minimal Supersymmetric Standard Model (MSSM) in a no-scale background. Considering a single predictive framework, we study inflation, leptogenesis, gavitino cosmology, and the stochastic background of gravitational waves produced by metastable cosmic strings. The spontaneous breaking of $U(1)_{B-L}$ at the end of inflation produces a network of metastable cosmic strings while, the interaction between $U(1)_{B-L}$ Higgs field and the neutrinos generate heavy Majorana masses for the right-handed neutrinos. The heavy Majorana masses explain the tiny neutrino masses via the seesaw mechanism, a realistic scenario for reheating and non-thermal leptogenesis. We show that a successful non-thermal leptogenesis and a stable gravitino as a dark matter candidate can be achieved for a wide range of reheating temperatures and $U(1)_{B-L}$ symmetry breaking scales. The possibility of realizing metastable cosmic strings in a grand unified theory (GUT) setup is briefly discussed. We find that a successful reheating with non-thermal leptogenesis and gravitino dark matter restrict the allowed values of string tension to a narrow range $10^{-9} \lesssim G\mu_{CS} \lesssim 8 \times 10^{-6}$, predicting a stochastic gravitational-wave background that lies within the 1-$\sigma$ bounds of the recent NANOGrav 12.5-yr data, as well as within the sensitivity bounds of future GW experiments.

μ -hybrid inflation, gravitino dark matter, and stochastic gravitational wave background from cosmic strings

We present a successful realization of supersymmetric $\mu$-hybrid inflation model based on a gauged $U(1)_{B-L}$ extension of the minimal supersymmetric standard model, with the soft supersymmetry breaking terms are playing an important role. Successful non-thermal leptogenesis with gravitino dark matter yields a reheat temperature in the range $2 \times 10^{7} \lesssim T_R \lesssim 5 \times 10^{9}$ GeV. This corresponds to the predictions $2 \times 10^{-18} \lesssim r\lesssim 4 \times 10^{-13}$ for the tensor to scalar ratio, and $-2 \times 10^{-6} \lesssim dn_s/d\ln k \lesssim -5 \times 10^{-11}$ for the running of the scalar spectral index. The $B-L$ breaking scale is estimated as $ 6 \times 10^{14}\lesssim M/ \text{GeV}\lesssim 10^{16}$, calculated at the central value of the scalar spectral index, $n_s =0.9655$, reported by Planck 2018. Finally, in a grand unified theory setup the dimensionless string tension parameter associated with the metastable strings is in the range $ 10^{-9} \lesssim G\mu_\text{cs} \lesssim 10^{-6}$ corresponding to a stochastic gravitational wave background lying within the 2$\sigma$ bound of the recent NANOGrav 12.5-yr data.

Sneutrino tribrid inflation, metastable cosmic strings and gravitational waves

We present a successful realization of sneutrino tribrid inflation model based on a gauged U(1)_B-L extension of Minimal Supersymmetric Standard Model (MSSM). A single interaction term involving the B-L Higgs field and the right-handed neutrinos serves multiple purposes. These include the generation of heavy Majorana masses for the right-handed neutrinos to provide an explanation for the tiny neutrino masses via the seesaw mechanism, a realistic scenario for reheating and non-thermal leptogenesis with a reheat temperature as low as 106 GeV, and a successful realization of inflation with right-handed sneutrino as the inflaton. The matter parity which helps avoid rapid proton decay survives as a Z2 subgroup of a U(1) R-symmetry. Depending on the choice of model parameters yields the following predicted range of the tensor to scalar ratio, 3 × 10-11≲ r≲ 7× 10-4 (6 × 10-7≲ r ≲ 0.01), and the running of the scalar spectral index, -0.00022 ≲ dn_s/dln k ≲ -0.0026 (-0.00014 ≲ dn_s/dln k ≲ 0.005), along with the B-L breaking scale, 3 × 1014≲ M/ GeV≲ 5 × 1015 (6 × 1015≲ M/ GeV≲ 2 × 1016), calculated at the central value of the scalar spectral index, n_s =0.966, reported by Planck 2018. The possibility of realizing metastable cosmic strings in a grand unified theory setup is briefly discussed. The metastable cosmic string network admits string tension values in the range 10-8≲ Gμs ≲ 10-6, and predicts a stochastic gravitational wave background lying within the 2-σ bounds of the recent NANOGrav 12.5-yr data.

Realistic inflation in no-scale U(1) R symmetric flipped SU(5)

We have realized non-minimal Higgs inflation and standard hybrid inflation in the supersymmetric flipped SU(5) model with U(1) R symmetry using the no-scale form of the Kähler potential. In non-minimal Higgs inflation the waterfall Higgs field plays the role of inflaton, and in standard hybrid inflation the gauge singlet field S is employed as an inflaton. The predictions of both models are in good agreement with the Planck 2018 data. For numerical calculations we have fixed the gauge symmetry breaking scale, M, around 2 × 1016 GeV. In both models the inflaton field values are constrained below mP . The tensor to scalar ratio r in non-minimal inflation is of the order of 10-3 and for standard hybrid inflation r is tiny, of order 10-15–10-4. The scalar spectral index in both cases lie within the Planck 1-σ bounds, and the running of the scalar spectral index lies in the range, -dns /d ln k ∼ 6 × 10-4 for non-minimal model and 10-9–10-3 for the standard hybrid model. A realistic scenario of reheating and non-thermal leptogenesis is employed with reheat temperature Tr ∼ 109 GeV for non-minimal model and 106–1010 GeV for standard hybrid model. The R-symmetry plays a vital role in forbidding rapid proton decay, but at the same time it also suppresses terms responsible for generating right handed neutrino masses. A realistic scenario of right handed neutrino masses is obtained by considering effective R symmetry breaking at the nonrenormalizable level with adequate suppression of rapid proton decay.

Gravitino dark matter, nonthermal leptogenesis, and low reheating temperature in no-scale Higgs inflation

We revisit Higgs inflation in the framework of a minimal extension of the Standard Model gauge symmetry by a $U(1)_{B-L}$ factor. Various aspects are taken into account with particular focus on the role of the supersymmetry breaking (SUSY) scale and the cosmological constraints associated with the gravitino. The scalar potential of the model is considered in the context of no-scale supergravity consisting of the F-part constructed from the K\"ahler function, the D-terms and soft SUSY contributions. We investigate several limiting cases and by varying the SUSY scale from a few TeV up to intermediate energies, for a spectral index around $n_s\sim 0.9655$ and reheating temperature $T_r\le 10^{9}$ GeV we find that the value of the tensor-to-scalar ratio ranges from $r\approx 10^{-3}$ to $10^{-2}$. Furthermore, it is shown that for certain regions of the parameter space the gravitino can live sufficiently long and as such is a potential candidate for a dark matter component. In general, the inflationary scenario is naturally implemented and it is consistent with non-thermal leptogenesis whereas the dominant decay channel of the inflaton yields right-handed neutrinos. Other aspects of cosmology and particle physics phenomenology are briefly discussed. Finally, we investigate the case where the inflaton is initially relaxed in a false minimum and estimate its probability to decay to the true vacuum.

Nonthermal leptogenesis and UV freeze-in of dark matter: Impact of inflationary reheating

We study a minimal scenario to realize non-thermal leptogenesis and UV freeze-in of a Standard Model (SM) gauge singlet fermionic dark matter (DM) simultaneously, with inflaton field playing a non-trivial role in their yields. The renormalizable interactions are restricted to the SM fields, two right handed neutrinos (RHN) and inflaton coupling exclusively to the RHNs, while the DM couples to both the SM and the RHNs only via operators of dimension $d>4$. Considering two separate cases of $d=\{5,6\}$, we show that for $d=5$, inflaton decay into RHNs followed by their subsequent decay into SM particles lead to both reheating as well as DM production from the SM bath. This requires a cut-off scale as large as $\Lambda\sim 10^{17}~\rm GeV$ depending on the DM mass. On the other hand, for $d=6$, DM production happens directly from scattering of RHNs (for $\Lambda\gtrsim 10^{14}~\rm GeV$) that results in a very non-trivial evolution of the DM yield. In both these cases, it is possible to explain the observed baryon asymmetry through successful non-thermal leptogenesis via the decay of the RHNs, together with the PLANCK observed relic density of the DM via pure UV freeze-in mechanism. Taking into account both instantaneous as well as non-instantaneous reheating separately, we constrain the parameter space of this minimal scenario from relevant phenomenological requirements including sub-eV scale active neutrino masses and their mixing.

Reheating in small-field inflation on the brane: the swampland criteria and observational constraints in light of the PLANCK 2018 results

We study cosmological inflation and its dynamics in the framework of the Randall–Sundrum II brane model. In particular, we analyze in detail four representative small-field inflationary potentials, namely Natural inflation, Hilltop inflation, Higgs-like inflation, and Exponential SUSY inflation, each characterized by two mass scales. We constrain the parameters for which a viable inflationary Universe emerges using the latest PLANCK results. Furthermore, we investigate whether or not those models in brane cosmology are consistent with the recently proposed Swampland Criteria, and give predictions for the duration of reheating as well as for the reheating temperature after inflation. Our results show that (i) the distance conjecture is satisfied, (ii) the de Sitter conjecture and its refined version may be avoided, and (iii) the allowed range for the five-dimensional Planck mass, M_5, is found to be 10^5,text {TeV}lesssim M_5lesssim 10^{12},text {TeV}. Our main findings indicate that non-thermal leptogenesis cannot work within the framework of RS-II brane cosmology, at least for the inflationary potentials considered here.

Cosmic inflation in minimal U(1)_{B-L} model: implications for (non) thermal dark matter and leptogenesis

We study the possibility of realising cosmic inflation, dark matter (DM), baryon asymmetry of the universe (BAU) and light neutrino masses in non-supersymmetric minimal gauged B-L extension of the standard model with three right handed neutrinos. The singlet scalar field responsible for spontaneous breaking of B-L gauge symmetry also plays the role of inflaton by virtue of its non-minimal coupling to gravity. While the lightest right handed neutrino is the DM candidate, being stabilised by an additional Z_2 symmetry, we show by performing a detailed renormalisation group evolution (RGE) improved study of inflationary dynamics that thermal DM is generally overproduced due to insufficient annihilations through gauge and scalar portals. This happens due to strict upper limits obtained on gauge and other dimensionless couplings responsible for DM annihilation while assuming the non-minimal coupling to gravity to be at most of order unity. The non-thermal DM scenario is viable, with or without Z_2 symmetry, although in such a case the B-L gauge sector remains decoupled from the inflationary dynamics due to tiny couplings. We also show that the reheat temperature predicted by the model prefers non-thermal leptogenesis with hierarchical right handed neutrinos while being consistent with other requirements.

Inflationary Universe with a Coleman-Weinberg potential meets non-thermal leptogenesis

Baryon asymmetry via non-thermal leptogenesis is explained in the framework of the standard Coleman-Weinberg inflationary model. The CP-violation asymmetry factor arises from the out-of-equilibrium decays of heavy right-handed neutrinos in supersymmetric SO(10) GUT models. All quantities of interest are computed in terms of the two parameters of the inflationary potential, namely the vacuum expectation value of the inflaton, M, and the amplitude A. For successful baryogenesis we obtain analytical expressions relating A to M, and also the reheating temperature after inflation as well as the mass of the inflaton to M. Assuming typical values of CP-violation asymmetry obtained in concrete supersymmetric SO(10) unified models, we show graphically all three quantities of interest versus M. Interestingly enough, we obtain for (A,M) numerical values comparable to the ones previously obtained in the slow-roll approximation of inflationary dynamics upon comparison with CMB data, such as amplitude of scalar perturbations and spectral index. Proton decay is also briefly discussed. Our results may be of interest to model builders.

New inflation in supersymmetric SU(5) and flipped SU(5) GUT models

We study the feasibility of realizing supersymmetric new inflation model, introduced by Senoguz and Shafi in [1], for SU(5) and flipped SU(5) models of grand unified theories (GUTs). This realization requires an additional U(1)R × Zn symmetry for its successful implementation. The standard model (SM) gauge singlet scalar components of 24H and 10H GUT Higgs superfields are respectively employed to realize successful inflation in SU(5) and flipped SU(5) models. The predictions of the various inflationary observables lie within the recent Planck bounds on the scalar spectral index, ns, for n ⩾ 5 in SU(5) model and for n ⩾ 6 in flipped SU(5) model. In particular, the tensor to scalar ratio r and the running of spectral index d ns/ d ln k are negligibly small and lie in the range, 10−12 ≲ r ≲ 10−8 and 10−9 ≲ dns/d ln k ≲ 10−3, for realistic values of n. In numerical estimation of the various predictions, we fix the gauge symmetry breaking scale, M, around 2 × 1016 GeV . The issue of gauge coupling unification in R-symmetric SU(5) is evaded by adding vectorlike families with mass splitting within their multiplets. The dilution of monopoles beyond the observable limit is naturally achieved in the breaking of SU(5) gauge symmetry during inflation. A realistic scenario of reheating with non-thermal leptogenesis is employed for both models. The predicted range of reheat temperature within Planck bounds, 3 × 107 GeV ≲ Tr ≲ 2 × 109 GeV, is safe from the gravitino problem for the gravitino mass, m3/2 ≳ 10 TeV . Finally, the U(1)R × Zn symmetry is also observed to play a crucial role in suppressing the various fast proton decay operators.