In some dark matter models, the coupling of the dark matter particle to the standard model Higgs determines the dark matter relic density while it is also consistent with dark matter direct detection experiments. On the other hand, the seesaw for generating the neutrino masses probably arises from a spontaneous symmetry breaking of global lepton number. The dark matter particle thus can significantly annihilate into massless Majorons when the lepton number breaking scale and hence the seesaw scale is near the electroweak scale. This leads to an interesting interplay between neutrino physics and dark matter physics and the annihilation mode has an interesting implication on dark matter searches.
The Dimension-5 Seesaw Portal is a Type-I Seesaw model extended by $d=5$ operators involving the sterile neutrino states, leading to new interactions between all neutrinos and the Standard Model neutral bosons. In this work we focus primarily on the implications of these new operators at the GeV-scale. In particular, we recalculate the heavy neutrino full decay width, up to three-body decays. We also review bounds on the dipole operator, and revisit LEP constraints on its coefficient. Finally, we turn to heavy neutrino pair production from Higgs decays, where the former are long-lived and disi…
The type II seesaw mechanism is an attractive way to generate the observed light neutrino masses. It postulates a SU(2)$_\mathrm{L}$-triplet scalar field, which develops an induced vacuum expectation value after electroweak symmetry breaking, giving masses to the neutrinos via its couplings to the lepton SU(2)$_\mathrm{L}$-doublets. When the components of the triplet field have masses around the electroweak scale, the model features a rich phenomenology. We discuss the current allowed parameter space of the minimal low scale type II seesaw model, taking into account all relevant constraints, i…
We present a reinterpretation of CMS searches for type-III seesaw lepton triplets at the LHC, both for the normal and the inverse seesaw. We find that, in contrast with previous expectations, these searches in the trilepton final state have a good sensitivity to triplets with predominant coupling to the tau lepton. We also show that the limits resulting from direct searches can be neatly presented for arbitrary masses and general flavour mixings. Thus, it turns out that the common (and often unrealistic) simplifying assumptions about the flavour couplings of the triplets used by experimental c…
Neutrino are massless in the Standard Model. The most popular mechanism to generate neutrino masses are the type I and type II seesaw, where right-handed neutrinos and a scalar triplet are augmented to the Standard Model, respectively. In this work, we discuss a model where a type I + II seesaw mechanism naturally arises via spontaneous symmetry breaking of an enlarged gauge group. Lepton flavor violation is a common feature in such setup and for this reason, we compute the model contribution to the $μ\rightarrow eγ$ and $μ\rightarrow 3e$ decays. Moreover, we explore the connection between the…
The comparison of samples with different number of charged leptons shows that trilepton signals are the most significant ones for seesaw mediators. As previously pointed out, this is indeed the case for scalar (type II) and fermion (type III) triplets at LHC, which can be discovered in this channel for masses up to 500-700 GeV and an integrated luminosity of 30 fb^-1; whereas fermion singlets (type I) are marginally observable if there are no further new physics near the TeV scale. However, if there are new gauge interactions at this scale coupling to right-handed neutrinos, as in left-right m…
If it can be ascertained experimentally that the 3X3 Majorana neutrino mass matrix M_nu has vanishing determinants for one or more of its 2X2 submatrices, it may be interpreted as supporting evidence for the theoretically well-known canonical seesaw mechanism. I show how these two things are connected and offer a realistic M_nu with two zero subdeterminants as an example.
We propose a mechanism to suppress proton decay induced by dimension-5 operators in a supersymmetric SO(10) model. Proton lifetime is directly connected with the intermediate vacuum expectation value which is responsible for the seesaw mechanism. The model shows many consistencies with the present theoretical results such as the components of the two Higgs doublets in the minimal supersymmetric standard model.
Upcoming Lepton Flavour Violation experiments searching for $μ\rightarrow 3e$ and $μ$ to $ e $ conversion in nuclei processes will provide new opportunities to test the fundamental properties of the neutrino sector, and possibly the origin of matter. In recent work, it was shown that the Type II Seesaw mechanism alone can simultaneously explain the neutrino masses, Leptogenesis, and inflation. A key prediction of this model was the possibility of signals being produced in Lepton Flavour Violation decays. Searches at future experiments such as Mu3e and COMET will be integral to determining the …
The signatures of heavy fermionic triplets ($Σ$) arising in scenarios like Type III seesaw model are probed through their direct production and subsequent decay at high energy electron-positron collider. Unlike the case of LHC, the production process has strong dependence on the mixing parameter ($V_{e,μ}$), making the leptonic collider unique to probe such mixing. We have established that with suitably chosen kinematic cuts, a 1 TeV $e^+e^-$ collider could probe the presence of $Σ$ of mass in the range of 500 GeV having $V_e=0.05$ with a few inverse femto barn luminosity through single produc…
Renewed interest in GeV-scale sterile neutrinos capable of explaining active neutrino oscillations via see-saw type I mechanism has been expressed in several proposals of direct searches. Given this activity we estimate the minimal values of sterile-active mixing angles provided one, two, or three sterile neutrinos are lighter than D-meson.
Type-III seesaw is a simple extension of the Standard Model~(SM) with the SU$(2)_\text{L}$ triplet fermion with zero hypercharge. It can explain the origin of the tiny neutrino mass and flavor mixing. After the electroweak symmetry breaking the light neutrino mass is generated by the seesaw mechanism which further ensures the mixings between the light neutrino and heavy neutral lepton mass eigenstates. If the triplet fermions are around the electroweak scale having sizable mixings with the SM sector allowed by the correct gauge symmetry, they can be produced at the high energy colliders leavin…
We investigate the pair production of right-handed neutrinos from the decay of a light $B-L$ scalar in the $U(1)_{B-L}$ model. The $B-L$ scalar mixes to the SM Higgs, and the physical scalar is required to be lighter than the observed Higgs. The produced right-handed neutrinos are predicted to be long-lived according to the type-I seesaw mechanism, and yield potentially distinct signatures such as displaced vertex and time-delayed leptons at the CMS/ATLAS/LHCb, as well as signatures at the far detectors including the CODEX-b, FACET, FASER, MoEDAL-MAPP and MATHUSLA. We analyze the sensitivity r…
We analyze the sensitivity of low-energy fundamental symmetry tests to interactions mediated by doubly-charged scalars that arise in type-II seesaw models of neutrino mass and their left-right symmetric extensions. We focus on the next generation measurement of the parity-violating asymmetry in Møller scattering planned by the MOLLER collaboration at Jefferson Laboratory. We compare the MOLLER sensitivity to that of searches for charged lepton flavor violation (CLFV) and neutrinoless double beta-decay ($0νββ$-decay) as well as present and possible future high-energy collider probes. We show th…
In the canonical up-quark seesaw, the ratios of light neutrino masses are more easily predicted than the masses themselves. Under explicitly enumerated neccesary but minimal assumptions, these ratios are obtained, including radiative corrections. The predictions remain uncertain only by the top quark mass and triviality mass limit, and the power law of the seesaw. The tau neutrino is specially affected by the non-linear effect of the heavy top quark mass. The derived ratios have application to solar, atmospheric, and cosmological neutrinos and laboratory neutrino oscillation searches. The sees…
The neutrino masses and flavor mixings, which are missing in the Standard Model (SM), can be naturally incorporated in the type-I seesaw extension of the SM with heavy Majorana neutrinos being singlet under the SM gauge group. If the heavy Majorana neutrinos are around the electroweak scale and their mixings with the SM neutrinos are sizable, they can be produced at high energy colliders, leaving characteristic signatures with lepton-number violations. Employing the general parametrization for the neutrino Dirac mass matrix in the minimal seesaw scenario, we perform a parameter scan and identi…
We review and update the current phenomenological constraints on minimal type I seesaw extensions of the Standard Model in which New Physics content can be probed at the electroweak scale. In this class of models, the flavour structure of the neutrino Yukawa couplings is determined by the requirement of reproducing neutrino oscillation data. The strongest constraints on the seesaw parameter space are imposed by the very recent upper limit on μ --> e γ decay rate. Searches of non-standard Higgs boson decays into a light and a heavy neutrino may also provide and independent test of these seesaw …
A collider probe of the radiative seesaw models are considered. Two key ingredients of these models, the extended Higgs sector and the source of the Majorana mass, although these details differ model by model, would be studied at the TeV-scale electron-positron and electron-electron colliders. The searches and mass determinations in the inert doublet model, which is the extended Higgs sector of the Ma model, are summarized as an example.
I review the present understanding of connection between the non-zero neutrino masses and the baryon asymmetry of the Universe. The state-of-art results are presented for the standard thermal leptogenesis.
We discuss ways to probe the origin of neutrino masses at the Energy and Intensity frontiers, in TeV-scale left-right seesaw models where small neutrino masses arise via type-I seesaw mechanism. We consider generic ('vanilla') version of such models as well as a version which leads to 'large' light-heavy neutrino mixing while keeping the neutrino masses small in a natural manner. We point out specific observable implications of these classes of models at the LHC as well as in searches for lepton flavor violating processes such as $μ\to eγ$ and $μ\to 3e$, and also in searches for lepton number …
Motivated by the possibility that new (gauge) symmetries which are broken at the grand- (string-) unification scale give rise to texture zeros in the fermion mass matrices which are at the origin of the hierarchy of masses and mixings we explore the effect of such zeros on the neutrino spectrum of SUSY-GUT models. We find that the quadratic-seesaw spectrum on which most expectations are focused is neither the only nor the most interesting possibility. Cases of strong $ν_μ - ν_τ$ or $ν_e - ν_τ$ mixing are present for a specific texture structure of the Yukawa matrices and experimental evidence …
We show that a TeV scale inverse seesaw model for neutrino masses can be realized within the framework of a supersymmetric SO(10) model consistent with gauge coupling unification and observed neutrino masses and mixing. We present our expectations for non-unitarity effects in the leptonic mixing matrix some of which are observable at future neutrino factories as well as the next generation searches for lepton flavor violating processes such as μ--> e + γ. The model has TeV scale W_R and Z' bosons which are accessible at the Large Hadron Collider.
In this paper we work in the framework of a radiative seesaw model with triplet fermion $Σ$. Due to the $Z_2$ discrete flavor symmetry, the lightest neutral component of $Σ$ is stable and thus can be a dark matter candidate. Its mass can be solely determined by the dark matter relic abundance, which is bout 2.594 TeV. The model also predict a dark matter-nucleus scattering cross section that would be accessible with future dark matter direct detection searches. We further investigate constraints on the parameter space of the model from the lepton-flavor-violating processes and neutrino transit…
Searches at the LHC of lepton flavour violation (LFV) in slepton decays can indirectly test both type-I and II seesaw mechanisms. Assuming universal flavour-blind boundary conditions, LFV in the neutrino sector is related to LFV in the slepton sector by means of the renormalization group equations. Ratios of LFV slepton decay rates result to be a very effective way to extract the imprint left by the neutrino sector. Some neutrino scenarios within the type-I seesaw mechanism are studied. Moreover, for both type-I and II seesaw mechanisms, a scan over the minimal supergravity parameter space is …
We discuss a radiative type-I seesaw. In these models, the radiative generation of Dirac neutrino masses allows to explain the smallness of the observed neutrino mass scale for rather light right-handed neutrino masses in a type-I seesaw. We first present the general idea in a model independent way. This allows us to estimate the typical scale of right-handed neutrino mass as a function of the number of loops. We then present two example models, one at one-loop and another one at two-loop, in which we discuss neutrino masses and lepton flavour violating constraints in more detail. For the two-…
The type III seesaw mechanism for neutrino mass generation usually makes use of at least two $Y = 0$, $SU(2)_L$ lepton triplets. We augment such a model with a third triplet and a sterile neutrino, both of which are odd under a conserved $\Z_2$ symmetry. With all new physics confined to the $\Z_2$-odd sector, whose low energy manifestation is in some higher-dimensional operators, a fermionic dark matter candidate is found to emerge. We identify the region of the parameter space of the scenario, which is consistent with all constraints from relic density and direct searches, and allows a wide r…
We perform a detailed analysis of the one-loop corrections to the light neutrino mass matrix within low scale type I seesaw extensions of the Standard Model and their implications in experimental searches for neutrinoless double beta decay. We show that a sizable contribution to the effective Majorana neutrino mass from the exchange of heavy Majorana neutrinos is always possible, provided one requires a fine-tuned cancellation between the tree-level and one-loop contribution to the light neutrino masses. We quantify the level of fine-tuning as a function of the seesaw parameters and introduce …
Within low-scale seesaw mechanisms, such as the inverse and linear seesaw, one expects (i) potentially large lepton flavor violation (LFV) and (ii) sizeable non-standard neutrino interactions (NSI). We consider the interplay between the magnitude of non-unitarity effects in the lepton mixing matrix, and the constraints that follow from LFV searches in the laboratory. We find that NSI parameters can be sizeable, up to percent level in some cases, while LFV rates, such as that for μ-> e γ, lie within current limits, including the recent one set by the MEG collaboration. As a result the upcoming …
We propose an economical model addressing the generation of the Inverse Seesaw mechanism from the spontaneous breaking of a local $U(1)_{B-L}$, with the Majorana masses of the sterile neutrinos radiatively generated from the dark sector. The field content of the Standard Model is extended by neutral scalars and fermionic singlets, and the gauge group is extended with a $U(1)_{B-L}$ and a discrete $\mathbb{Z}_4$ symmetries. Besides dynamically generating the Inverse Seesaw and thus small masses to the active neutrinos, our model offers two possible dark matter candidates, one scalar and one fer…
The present work investigates the possibility that both dark matter and the anomalous magnetic moment of the muon may be explained within the context of the inverse seesaw extended Next-to-Minimal Supersymmetric Standard Model (ISS-NMSSM). In ISS-NMSSM, the newly introduced Higgs-neutrino Yukawa coupling $Y_ν$ provides additional Higgsino-sneutrino loop contribution to $(g-2)_μ$. If the deviation between the experimental observations and the Standard Model predictions of the anomalous muon magnetic moment is confirmed by the further experimental and theoretical studies, it can be explained nat…
The seesaw mechanism to derive the light masses of left-handed neutrinos using heavy masses of right-handed neutrinos gives rise to a connection between low-energy measurables and GUT-scale mechanism. We expresses the neutrino mixing angles in terms of a single variable $\sinθ_{13}$, whose size was measured recently. The lepton asymmetry from heavy neutrinos via Yukawa coupling is described by CP phases in both Dirac and Majorana type. It is shown that the seesaw scale relevant to the lepton asymmetry can be constrained by CP phase in this minimal model.
We introduce three right-handed neutrinos and three sterile neutrinos, and consider an inverse seesaw mechanism for neutrino mass generation. From naturalness point of view, their Majorana masses should be small, while it induces a large neutrino Yukawa coupling. Then, a neutrinoless double beta decay rate can be enhanced, and a sizable Higgs mass correction is inevitable. We find that the enhancement rate can be more than ten times compared with a standard prediction from light neutrino contribution alone, and an analytic form of heavy neutrino contributions to the Higgs mass correction. In a…
CMS and ATLAS have searched for a doubly-charged boson $H^{\pm\pm}$ which may arise from type II seesaw in the 7 TeV run at the LHC by considering pair or associated production of doubly-charged bosons under the assumption of degenerate triplet scalars. In this work, we consider non-degenerate triplet components with the mass gap $ΔM \sim 1 - 40$ GeV which leads to enhanced pair-production cross-sections of $H^{\pm\pm}$ added by the gauge decays of the heavier neutral and singly-charged bosons. We reevaluate the constraints in the $ΔM-M_{H^{++}}$ plane depending on the triplet vacuum expectati…
We propose a model for the vector-like lepton to explain the small muon mass by a seesaw mechanism, based on lepton-specific two Higgs doublet models with a local $U(1)'$ symmetry. There is no bare muon mass for a nonzero $U(1)'$ charge of the leptophilic Higgs doublet, so the physical muon mass is generated due to the mixing between the vector-like lepton and the muon after the leptophilic Higgs doublet and the dark Higgs get VEVs. In this scenario, the non-decoupling effects of the vector-like lepton give rise to leading contributions to the muon $g-2$, thanks to the light $Z'$ and the light…
An extended seesaw model proposed to achieve low scale leptogenesis can resolve the excess positron and electron fluxes observed from PAMELA, ATIC and/or Fermi-LAT, and simultaneously accommodate some of recent experimental results for dark matter (DM) signals. In this approach, in addition to SU(2)_L doublet and the (light) singlet Higgs fields, an extra vector-like singlet neutrino and a singlet scalar field, which are coexisting two-particle dark matter candidates, are responsible for the origin of the excess positron and electron fluxes to resolve the PAMELA, ATIC and/or Fermi-LAT anomalie…
We study the phenomenology of the minimal $(2,2)$ inverse-seesaw model supplemented with Abelian flavour symmetries. To ensure maximal predictability, we establish the most restrictive flavour patterns which can be realised by those symmetries. This setup requires adding an extra scalar doublet and two complex scalar singlets to the Standard Model, paving the way to implement spontaneous CP violation. It is shown that such CP-violating effects can be successfully communicated to the lepton sector through couplings of the scalar singlets to the new sterile fermions. The Majorana and Dirac CP ph…
We analyze the lepton flavor violation processes $μ-e$ conversion, $l_i\rightarrow l_jγ$ and $l_i\rightarrow 3l_j$ in framework of the Standard Model (SM) extended with inverse seesaw mechanism, as a function of $\tildeη = 1 -|Det(\tilde {U}_{PMNS})|$ that parameterizes the departure from unitary of the light neutrino mixing sub-matrix $\tilde {U}_{PMNS}$. In a wide range of $\tildeη$, the predictions on the $μ-e$ conversion rates and the branching ratio of $μ\rightarrow eγ$ are sizeable to be compatible with the experimental upper limits or future experimental sensitivities. For large scale o…
We consider neutrino Minimal extension of the Standard Model ($ν$MSM), which by introducing only three sterile neutrinos in sub electroweak region can explain active neutrino oscillations (via seesaw type I mechanism), baryon asymmetry of the Universe (leptogenesis via oscillations) and dark matter phenomena (with keV-scale sterile neutrino forming dark matter). We estimate sterile neutrino virtual contributions to various lepton flavor and lepton number violating processes. The contributions are too small, giving no chance for indirect searches to compete with direct measurements in exploring…
We reconsider the role that the possible detection of lepton flavour violating (LFV) decays of supersymmetric particles at the Large Hadron Collider (LHC) can play in helping reconstruct the underlying neutrino mass generation mechanism within the simplest high-scale minimal supergravity (mSUGRA) seesaw schemes. We study in detail the LFV scalar tau decays at the LHC, assuming that the observed neutrino masses arise either through the pure type-I or the simpler type-II seesaw mechanism. We perform a scan over the mSUGRA parameter space in order to identify regions where lepton flavour violatin…
Supersymmetric SO(10) grand unified models with renormalizable Yukawa couplings involving {\bf 10}, {\bf 126} and {\bf 120} Higgs fields have been shown to give a very economical theory for understanding quark-lepton flavor in a unified framework. In previous papers, we showed how nucleon decay can be suppressed in these models without invoking cancellation, by choice of Yukawa flavor texture within a type II seesaw framework for neutrinos that explains all mixings and masses including the recently observed "large" $θ_{13}$. In this follow-up paper, we extend our earlier work to the case of ty…
Given its briefness and predictability, the minimal seesaw -- a simplified version of the canonical seesaw mechanism with only two right-handed neutrino fields -- has been studied in depth and from many perspectives, and now it is being pushed close to a position of directly facing experimental tests. This article is intended to provide an up-to-date review of various phenomenological aspects of the minimal seesaw and its associated leptogenesis mechanism in neutrino physics and cosmology. Our focus is on possible flavor structures of such benchmark seesaw and leptogenesis scenarios and confro…
The hypothetical dark photon portal connecting the visible and dark sectors of the Universe has received considerable attention in recent years, with a focus on CP-conserving kinetic mixing between the Standard Model (SM) hypercharge gauge boson and a new U(1)$_X$ gauge boson. In the effective field theory context, one may write down non-renormalizable CP-violating kinetic mixing interactions involving the $X$ and SU(2)$_L$ gauge bosons. We construct for the first time a renormalizable model for CP-violating kinetic mixing that induces CP-violating non-Abelian kinetic mixing at mass dimension …
Neutrino mass generation through the Higgs mechanism not only suggests the need to reconsider the physics of electroweak symmetry breaking from a new perspective, but also provides a new theoretically consistent and experimentally viable paradigm. We illustrate this by describing the main features of the electroweak symmetry breaking sector of the simplest type-II seesaw model with spontaneous breaking of lepton number. After reviewing the relevant "theoretical" and astrophysical restrictions on the Higgs sector, we perform an analysis of the sensitivities of Higgs boson searches at the ongoin…
The Inverse Seesaw naturally explains the smallness of neutrino masses via an approximate $B-L$ symmetry broken only by a correspondingly small parameter. In this work the possible dynamical generation of the Inverse Seesaw neutrino mass mechanism from the spontaneous breaking of a gauged $U(1)$ $B-L$ symmetry is investigated. Interestingly, the Inverse Seesaw pattern requires a chiral content such that anomaly cancellation predicts the existence of extra fermions belonging to a dark sector with large, non-trivial, charges under the $U(1)$ $B-L$. We investigate the phenomenology associated to …
We study the type-I seesaw model with three right-handed neutrinos and Majorana masses below the pion mass. In this mass range, the model parameter space is not only strongly constrained by the requirement to explain the light neutrino masses, but also by experimental searches and cosmological considerations. In the existing literature, three disjoint regions of potentially viable parameter space have been identified. In one of them, all heavy neutrinos decay shortly before big bang nucleosynthesis. In the other two regions, one of the heavy neutrinos either decays between BBN and the CMB deco…
The triplet or type-II seesaw mechanism is the simplest way to endow neutrinos with mass in the Standard Model (SM). Here we review its associated theory and phenomenology, including restrictions from $S$, $T$, $U$ parameters, neutrino experiments, charged lepton flavour violations as well as collider searches. We also examine restrictions coming from requiring consistency of electroweak symmetry breaking, i.e. perturbative unitarity and stability of the vacuum. Finally, we discuss novel effects associated to the scalar mediator of neutrino mass generation namely, (i) rare processes, e.g. $l_α…
Insights into 1-loop corrections to neutrino low-scale type-I seesaw mechanism
Università degli studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；INFN - Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）
The Standard Model augmented by the presence of gauge-singlet right-handed neutrinos proves to be an ideal scenario for accommodating nonzero neutrino masses. Among the new parameters of this ``New Standard Model'' are right-handed neutrino Majorana masses M. Theoretical prejudice points to M much larger than the electroweak symmetry breaking scale, but it has recently been emphasized that all M values are technically natural and should be explored. Indeed, M around 1-10 eV can accommodate an elegant oscillation solution to the LSND anomaly, while other M values lead to several observable cons…
Strong correlations between leptogenesis and low energy CP violating leptonic processes have been shown by us to exist in the minimal left-right symmetric model with spontaneous CP violation. In this note, we investigate the implications of this model for the electric dipole moment of the electron. With an additional broken U(1)_H symmetry, the seesaw scale can be lowered to close to the electroweak scale. This additional symmetry also makes the connection between CP violation in quark sector to that in the lepton sector possible.
We analyze the consistency of electroweak breaking, neutrino and dark matter phenomenology within the simplest scoto-seesaw model. By adding the minimal dark sector to the simplest "missing partner" type-I seesaw one has a physical picture for the neutrino oscillation lengths: the "atmospheric" mass scale arises from the tree-level seesaw, while the "solar" scale is induced radiatively, mediated by the dark sector. We identify parameter regions consistent with theoretical constraints, as well as dark matter relic abundance and direct detection searches. Using two-loop renormalization group equ…
We analyze the present bounds of a scotogenic model, the Radiative Type III Seesaw (RSIII), in which an additional scalar doublet and at least two fermion triplets of $SU(2)_L$ are added to the Standard Model (SM). In the RSIII the new physics (NP) sector is odd under an exact global $Z_2$ symmetry. This symmetry guaranties that the lightest NP neutral particle is stable, providing a natural dark matter (DM) candidate, and leads to naturally suppressed neutrino masses generated by a one-loop realization of an effective Weinberg operator. We focus on the region with the highest sensitivity in p…
The cosmological evolution of neutrino energy densities driven by cosmon-type field equations is introduced assuming that active and sterile neutrinos are intrinsically connected by cosmon fields through the {\em seesaw} mechanism. Interpreting sterile neutrinos as dark matter adiabatically coupled with dark energy results in a natural decoupling of (active) mass varying neutrino (MaVaN) equations. Identifying the dimensionless scale of the {\em seesaw} mechanism, $m/M$, with a power of the cosmological scale factor, $a$, allows for embedding the resulting masses into the generalized Chaplygin…
