Neutrino Masses, Gravitational Coupling Constant And Cosmological Constant

Neutrino Masses, Gravitational Coupling Constant And Cosmological Constant Deepak Pashilkar∗ (Dated: December 6, 2025) 1 Abstract Masses of the three neutrino mass eigenstates are predicted to be m0, 4m0and 22m0where m0= 2.281 meV/c2. These predictions are arrived at by applying two ad-hoc postulates to neutrino oscillation data. The first postulate is that the mass m0of the lightest neutrino mass eigenstate is the smallest quantum of mass, and that the masses of all the massive elementary particles are positive integer multiples of m0. The dimensionless gravitational coupling constant αgis then defined as, αg=m0 Mp , where Mpis the Planck Mass. The second postulate is that the dark energy driving the accelerated expansion of the universe is represented by a cosmological constant Λ, or, equivalently, a vacuum energy with constant density ρΛ=αg4ρP, where ρPis the Planck Density. These postulates also lead to the prediction of (dimensionless) vacuum energy density to be ΩΛh2= 0.3344, in agreement with the νΛCDM model. Furthermore, the effective electron neutrino mass is predicted to be mβ= 9.185 meV/c2. We also predict the effective Majorana mass of the electron neutrino to be mββ ≤5.803 meV/c2. This upper bound on mββ is then used to calculate lower bounds on half-lifes of various isotopes expected to undergo a neutrinoless double beta decay (0νββ). A new natural system of units is proposed in which both m0and ρΛare unity. I. INTRODUCTION Experimenters have discovered three flavours of neutrinos: electron neutrino νe, muon neutrino νµand tau neutrino ντ. They also observed that these neutrinos keep changing flavours, i.e. they keep transforming into each other, for example, νe⇌νµ. This phenomenon of neutrino oscillations conclusively proves that the neutrinos have mass. In fact, various experiments indicate that the neutrinos have sub-electronvolt masses, making them the lightest known massive elementary particles. However, their masses are not known yet. The aim of this article is to predict the masses of neutrinos by introducing two postulates — one concerning the neutrinos themselves and the other concerning the dark energy responsible for the accelerated expansion of the universe. In the first postulate, we propose a new analogy between mass and charge. We postulate that the masses of massive elementary particles are ‘quantized’ — just like their charges — ∗deepak.pashilk[email protected] 2 and that they are positive integer multiples of the ‘smallest quantum of mass’. We then use this postulate to define a dimensionless fundamental constant, the gravitational coupling constant αg. The second postulate adds a new dimension to the cosmological constant problem.[1] Quantum Field Theory (QFT) provides a natural explanation for the existence of dark energy in the form of a constant vacuum energy. But the observed value of the vacuum energy differs from the value expected in QFT by a factor of about 10−123 This is the well-known cosmological constant problem. In the second postulate, we elevate this factor of 10−123 to the status of a fundamental constant by assuming that it is equal to the gravitational coupling constant raised to the power 4, i.e. αg4∼10−123. The rest of this article is arranged as follows. We will begin in section [II] with a brief overview of the 3νmixing model of neutrino oscillations. In section [III], we will present the Quantum of Mass (QoM) postulate and define the gravitational coupling constant αg. In section [IV], we will present the Cosmological Constant (CC) postulate. We will also discuss how the ongoing Hubble Tension constrains our ability to make predictions using the cosmological data. In section [V], we will predict the masses of neutrino mass eigenstates and the vacuum energy density using the neutrino oscillation data. We will also predict the effective electron neutrino mass mβin β-decay. In addition, we will make predictions about the effective Majorana mass mββ of electron neutrino and half-lifes of various isotopes expected to undergo a neutrinoless double beta decay (0νββ). In section [VI], we will construct a new natural system of units. In section [VII], we will discuss the implications of the two postulates and their predictions. II. THE 3νMODEL The phenomenological three-neutrino mixing model (3νmodel) provides a good fit to the outcome of the neutrino oscillation experiments. In this model, each of the three neutrino flavour eigenstates νe, νµand ντis considered to be an admixture of the three mass eigenstates ν1, ν2and ν3having respective mass eigenvalues m1, m2and m3. The eigenstates are related by |να⟩= 3 X i=1 U∗ αi |νi⟩(1) 3 where Uis the neutrino mixing matrix.[2] If neutrinos are Majorana particles, then Udepends on six independent parameters which are conveniently chosen to be 3 mixing angles θ21, θ31, θ32 ∈0,π 2and 3 phases δ, η1, η2∈[0,2π] so that, U=     1 0 0 0c32 s32 0−s32 c32           c31 0s31 e−iδ 0 1 0 −s31 eiδ 0c31           c21 s21 0 −s21 c21 0 0 0 1           eiη10 0 0eiη20 0 0 1      (2) where, sij = sin θij and cij = cos θij. If neutrinos are Dirac particles, then the Majorana phases ηican be absorbed into the neutrino state and then Uis a function of the remaining 4 parameters, U=     c21 c31 s21 c31 s31 e−iδ −s21 c32 −c21 s32 s31 eiδ c21 c32 −s21 s32 s31 eiδ s32 c31 s21 s32 −c21 c32 s31 eiδ −c21 s32 −s21 c32 s31 eiδ c32 c31      (3) Application of the 3νmodel to the data obtained from various neutrino oscillation experiments provides best fit values of these 4 parameters — the 3 mixing angles θij, the CP phase δ— as well as the squared mass differences, ∆21 =m22−m12 ∆31 =m32−m12 ∆32 =m32−m22 (4) It is not possible to measure the Majorana phases ηiand the absolute neutrino masses mi from the oscillation data. Even the ordering of these masses is not known yet. There are two possible mass orderings: i Normal Ordering (NO): m1< m2< m3. ii Inverted Ordering (IO): m3< m1< m2. In Table I, we show the best fit values and 3σranges of some of the oscillation parameters needed in this article for both the normal and inverted ordering as provided by NuFIT 6.0 (IC24 with SK atmospheric data).[3] We choose the NuFIT 6.0 data for no particular reason. Choosing other recent dataset by Capozzi et al (2020)[4] will not change the outcome of this article. 4 NuFIT 6.0 (w/ atmospheric data) Best Fit 3σRange Ordering ∆21 (meV/c2)274.9 69.2 →80.5 Both ∆31 (meV/c2)22513 2451 →2578 Normal ∆32 (meV/c2)2-2484 -2547 →-2421 Inverted sin2θ21 0.308 0.275 →0.345 Normal sin2θ31 0.02215 0.02030 →0.02388 Normal δCP 212◦124◦→364◦Normal TABLE I: The best fit values and 3σranges of the neutrino oscillation parameters (needed in this article) from NuFIT 6.0 (with SK atmospheric data). Since ∆21,∆31 and ∆32 are non-zero, the heavier two masses (m2and m3for NO, m1and m2for IO) are non-zero. It is not yet known if the lightest mass is zero or not. For later convenience, let’s denote this lightest mass by m0. Then for NO, m1=m0whereas, for IO, m3=m0. III. QUANTUM OF MASS AND GRAVITATIONAL COUPLING CONSTANT All charged particles have constant charges. In addition, these charges are ‘quantized’, i.e. they are integer multiples of e 3, the charge of the down antiquark, where eis the charge of a proton. We may say that e0=e 3is the ‘smallest quantum of charge’. The charge qof a particle is then given by q=Nqe0for some fixed integer Nq. Just like charges, all particles have constant masses. But it is not known whether these masses are quantized in the same 5 sense. In this article, we assume that this is indeed the case. We assume that there exists a ‘smallest quantum of mass’ and that the masses of massive elementary particles are positive integer multiples of it. Since neutrinos are the lightest known massive elementary particles, they provide a natural candidate for this smallest quantum of mass. A. Quantum of Mass Postulate We postulate that Im0, the mass of the lightest neutrino mass eigenstate, is non-zero. II m0is the smallest quantum of mass. III The mass eigenvalues of all the massive elementary particles — leptons, quarks, W and Z bosons and the Higgs boson — are positive integer multiples of m0. We will call this postulate the Quantum of Mass (QoM) postulate. Note that the QoM postulate does not predict the masses of the elementary particles. These masses need to be measured experimentally. For example, the QoM postulate asserts that the mass of an electron is me=Nem0for some fixed natural number Ne, but it does not provide us with the value of Ne. This value can only be determined experimentally. In section [V], we will determine neutrino masses using the experimentally obtained values of neutrino oscillation parameters. B. Gravitational Coupling Constant The dimensionless gravitational coupling constant is defined as[5], αg=mp MP (5) where mpis the mass of an elementary particle like electron and MP=rℏc Gis Planck mass.1. Different choices of mplead to different definitions of αg. In [5], the mass of a proton was used for mpbut an electron was also acceptable. And in [6], the mass of an 1This definition is ‘square root’ of the definition in [5]. 6 electron was used, although any other elementary particle would have done equally well. Since the QoM postulate accords a special status to the mass m0, we choose mp=m0, therefore, αg=m0 MP (6) This relation between αgand m0is analogous to the relationship between the fine structure constant αand proton charge e, √α=e QP (7) where QP=r4πℏ µ0c=√4π ϵ0ℏcis the Planck charge. We can make this analogy more obvious by defining the electromagnetic coupling constant αeas, αe=e0 QP =√α 3(8) Like αe(or, α), we expect αgto play the role of a fundamental constant of nature. IV. COSMOLOGICAL CONSTANT The origin and composition of dark energy is yet unknown. The ΛCDM model is the simplest phenomenological model of the dark energy in good agreement with cosmological observations. According to this model, the dark energy is represented by a cosmological constant Λ or, equivalently, by a vacuum energy with constant density ρΛ, where, ρΛc2=c4 8π G Λ (9) If the present-day observed value of the Hubble constant is H0and that of the vacuum energy density parameter is ΩΛ, then, ρΛ=3 8π G ΩΛH02≈ΩΛh2×1.8688 ×10−26 kg/m3(10) where h=H0 100 km/s/Mpc is the reduced Hubble constant. A. Cosmological Constant Postulate In this article, we will assume a simple relationship between the vacuum energy density ρΛand the gravitational coupling constant αg. As a fundamental constant of nature, αgsets 7 the value of the smallest quantum of mass m0via definition (6). In addition, we claim that αgalso sets the value of the vacuum energy. We postulate that the vacuum energy density is related to the gravitational coupling constant by, ρΛ=αg4ρP(11) where ρP=c5 ℏG2is the Planck density. We will call this postulate the Cosmological Constant (CC) postulate. And we will refer to both the QoM and the CC postulates together as the postulates regarding Gravitational Coupling Constant or GCC postulates in short. An immediate consequence of the CC postulate is that the vacuum energy density is proportional to the fourth power of the lightest neutrino mass. Substituting from (6) in (11), we get, ρΛ=c3 ℏ3m04(12) This kind of a relationship between the vacuum energy and the neutrino mass has been speculated in the past and has sparked new ideas.[7–9] Further, using (9), (11) and (12)), we obtain the relation between the cosmological constant and αg(and m0) as, Λ = 8π LP2αg4=8π G c ℏ3m04(13) where LP=rℏG c3is the Planck length. Both the GCC postulates are ad hoc. Their merit lies in their ability to make accurate predictions. Using these postulates we can calculate the neutrino masses. In fact, the easiest way of calculating m0is by first combining (10) and (12) to get, m0=4 r3ℏ3 8π G c3 4 pΩΛH02≈34 pΩΛh2meV/c2(14) and then substituting the value of ΩΛh2obtained from cosmological observations. But the ongoing Hubble tension gets in way. B. Hubble Tension Planck 2018 data[10] provides the most precise value of ΩΛh2. For the νΛCDM model2, it reports h= 0.6710+0.0120 −0.0067,Ωbh2= 0.02236(15) and Ωch2= 0.1201(13) so that ΩΛh2= 0.3078+0.0177 −0.0104. A combined analysis of data from Planck, ACT and DESI gives h= 0.6836(29) 8 and Ωm= 0.3009(37) so that ΩΛh2= 0.3267(45) for the νΛCDM model.[11] However, these values of hare in significant disagreement with the model independent, direct measurements of hobtained by late time probes using distance ladders. For example, Breuval et al[12] measured h= 0.7317(86). This disagreement in the measured values of his known as the Hubble Tension. A resolution of this tension in favour of Breuval can lead to a value of ΩΛh2that is significantly different from the corresponding Planck value. It may even lead to a modification of the ΛCDM model itself. Therefore, we will not use the values of ΩΛh2obtained from cosmological observations to measure the neutrino masses. Instead, we will use the experimentally obtained values of neutrino oscillation parameters to predict the value of ΩΛh2. The only restriction we will place on ΩΛh2is that it lies in the interval [0.2,0.5]. This interval includes the Planck and the DESI values of ΩΛh2mentioned above and is large enough to encompass any possible future modifications of it. V. THE PREDICTIONS The QoM postulate asserts that m1, m2and m3are positive integer multiples of m0. Let m1=N1m0 m2=N2m0 m3=N3m0 (15) where N1, N2and N3are natural numbers. Substituting (14) and (15) in (4) we get, ∆21 =N22−N12m2 0≈9N22−N12pΩΛh2meV/c22(16) ∆31 =N32−N12m2 0≈9N32−N12pΩΛh2meV/c22(17) ∆32 =N32−N22m2 0≈9N32−N22pΩΛh2meV/c22(18) Exactly one of N1, N2and N3is equal to 1, depending upon the mass ordering. Their values can be calculated by substituting the experimentally obtained values of ∆21,∆31,∆32 and ΩΛh2in (16), (17) and (18). But, as discussed in the previous section, we will assume 2We consider the νΛCDM (ΛCDM + Pmi) model instead of the ΛCDM model because in this model, the value of the sum of the neutrino masses Pmiis allowed to vary. 9 In the new system of units, l0=t0= 1 (38) The Coulomb force between two charges q1and q2separated by a distance rtakes a simple form in the new units, Fe=αe2q1q2 r2(39) Similarly, the Newtonian gravitational force between two masses M1and M2separated by a distance rtakes the form, Fg=αg2M1M2 r2(40) The Einstein-Dirac-Maxwell action is, S=c3 16 π G Zp|g|(R(g)−2 Λ) d4x +1 cZp|g|¯ ψiℏc γµ∇µ−c q γµAµ−m c2ψ d4x −1 4µ0cZp|g|gαβ gγµ Fαγ Fβµ d4x (41) In the new units, the action takes the form, S=1 16 π αg2Zp|g|R(g)−16 π αg2d4x +Zp|g|¯ ψ(i γµ∇µ−q γµAµ−m)ψ d4x −1 16 π αe2Zp|g|gαβ gγµ Fαγ Fβµ d4x (42) Here, the charge qof the elementary fermion is an integer and it’s mass mis a natural number. VII. REMARKS The CC postulate requires that the dark energy is represented by a cosmological constant. This implies that the 6 parameter ΛCDM model and its 6+ parameter extensions like oΛCDM (ΛCDM + ΩK), νΛCDM etc. are viable models of the universe whereas others like wCDM are not. In addition, this rules out quintessence, MOND etc. as viable models of dark energy. 16 A. Vacuum Energy and Oscillation Parameters Figure 1shows the vacuum energy densities ΩΛh2obtained using neutrino oscillation data as well as cosmological data. The bottom three plots in the figure depict the best fit values and 3σranges of ΩΛh2calculated using (22) and (23). We can see that the 3σrange ΩΛh2∈[0.263,0.356] calculated from (23) using ∆21 ∈[69.2,80.5] (meV/c2)2is large enough to encompass the 3σranges ΩΛh2∈[0.3181,0.3519] and ΩΛh2∈[0.2662,0.2945] calculated from (22) using ∆31 ∈[2451,2578] (meV/c2)2for both N3= 22 and 23. This illustrates why we can not choose between the two possible values of N3from the neutrino oscillation data alone. 0.25 0.3 0.35 h 2 Using 21 Using 31 Planck 2018 ( CDM) DESI 2025 ( CDM) N 3= 22 N 3= 23 FIG. 1: Whisker plots of vacuum energy density ΩΛh2. The bottom three plots are created using best fit values and 3σranges of neutrino oscillation parameter ∆21 in (23) and ∆31 in (22) for both N3= 22 (cyan vertical band) and N3= 23 (light pink vertical band). The top two plots are created using the values and 3 ×1σranges of ΩΛh2provided by DESI [11] and Planck [10] for ΛCDM model. 17 But the cosmological data do show preference for N3= 22 as can be seen from the top two plots in the figure. In particular, the value ΩΛh2= 0.3344 calculated using (22) for N3= 22 is closer to the value ΩΛh2= 0.3267 given by DESI at 1.704σthan the value ΩΛh2= 0.2798 for N3= 23 at 10.47σ. Notably, the GCC postulates enforce a strong correlation between the 3νmodel and the νΛCDM model when we choose N2= 4 and N3= 22. Especially for ∆21 = 74.9 (meV/c2)2 in (23) with N2= 4, we get ΩΛh2= 0.308 consistent with the Planck 2018 and the DESI values for the νΛCDM model. For any other natural number N2, ΩΛh2is either too small (0.120 for N2= 5) or too large (1.08 for N2= 3) as compared to the observed values. A. Remarks On QoM Postulate The masses of composite particles like protons are not equal to the sums of masses of their component elementary particles, the quarks. Therefore the masses of the composite particles are not integer multiples of m0. An immediate consequence of the QoM postulate is that there are no massive particles with mass smaller than m0. Any hypothetical massive particle like Axion, Prion, Inflaton etc. must have mass equal to positive integer multiple of m0. In the QoM postulate, the mass of the lightest neutrino mass eigenstate is taken as the smallest quantum of mass (i.e., m1=m0). If we drop this requirement and instead assume that the mass of the lightest neutrino mass eigenstate is a positive integer multiple of the smallest quantum of mass (i e., m1=N1m0where, N1is a natural number greater than 1), then we get other possibilities for the neutrino masses — for example, 7m0,8m0and 23m0 (or 24m0). In this case, is m0the mass of an axion? Although the QoM postulate applies to all the elementary particles, we have used it only for neutrinos. We need to test its viability for other elementary particles. B. Running Gravitational Coupling Constant In Quantum Electrodynamics (QED), the fine structure constant is a function of the energy scale, α(E). It is possible that in a theory of quantum gravity, even αgis a function of the energy scale, αg(E). Then owing to (6), (11) and (13), m0,ρΛand Λ are also functions 18 of the energy scale. 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