Seesaw Model of Neutrino Mass Estimation with a Dirac Mass of 585 GeV Electroweak Fermion and the Unified Stoney Mass

1. Introduction

The origin and absolute scale of neutrino masses remain one of the central mysteries in particle physics. We approach this problem through a Dirac seesaw framework [1,2,3,4,5,6,7,8] inspired by our proposed 4G model of final unification having four different gravitational constants and an electroweak fermion of rest energy 585 GeV [9,10,11]. Within this model, we derive neutrino masses based on a high-scale Dirac mass term of 585 GeV and a unified gravitational mass associated with the Stoney scale [12,13]. To better match the mass observations, a fitting coefficient of 0.88 is applied. It can have a range of (0.85 to 0.9).

2. Modified Seesaw Mass Formula

Based on the Seesaw model, we adopt a modified formula in the following way. It needs a critical review at fundamental level. For the time being it can be considered as a reference mass formula.

$$m_{l\upsilon }\cong k*n*\sqrt{{\displaystyle \frac{m_{lepton}}{m_e}}}\left({\displaystyle \frac{M_D^2}{M_{GUT}}}\right)\cong 0.88*n*\sqrt{{\displaystyle \frac{m_{lepton}}{m_e}}}\left({\displaystyle \frac{M_{wf}^2}{M_{Stoney}}}\right)$$

(1)

$\begin{array}{l}\mathrm{where},\\ m_{l\upsilon }\cong \mathrm{Mass}\mathrm{of}\mathrm{lepton}\mathrm{neutrino}\\ k\cong \mathrm{A}\mathrm{coefficient}\mathrm{needs}\mathrm{attention}\mathrm{and}\mathrm{review}\cong 0.88\\ n=1,2,3\mathrm{for}\mathrm{electron},\mathrm{muon}\mathrm{and}\mathrm{tau}\mathrm{respectively}.\\ {\displaystyle \frac{m_{lepton}}{m_e}}\cong {\displaystyle \frac{\mathrm{Mass}\mathrm{of}\mathrm{electron}\mathrm{or}\mathrm{Muon}\mathrm{or}\mathrm{Tau}}{\mathrm{Mass}\mathrm{of}\mathrm{electron}}}\\ M_D\cong \mathrm{Dirac}\mathrm{mass}\cong M_{wf}\cong 584.725\mathrm{GeV}/c^2\end{array}$

$\cong \mathrm{Proposed}\mathrm{electroweak}\mathrm{fermion}\mathrm{of}\mathrm{our}4\mathrm{G}\mathrm{model}\mathrm{of}\mathrm{final}\mathrm{unification}$ [9,10,11]

$M_{Gut}\cong \mathrm{Grand}\mathrm{Unified}\mathrm{mass}\mathrm{unit}$

$\cong M_{Stoney}\cong \sqrt{{\displaystyle \frac{e^2}{4\pi {\epsilon }_0{G}_N}}}\cong 1.85921\times {10}^{-9}\mathrm{kg}\cong \mathrm{Stoney}\mathrm{mass}$ [12,13]

G_N ≅ Newtonian gravitational constant

The factor 0.88 is included as a phenomenological correction factor, likely tied to radiative or cosmological damping effects. These values suggest a hierarchical pattern among neutrino masses, with the electron neutrino being the lightest and tau neutrino the heaviest. This mass distribution is a natural outcome of the formula used and reflects the scaling with lepton mass and generation index. In terms of gravitational and electromagnetic force ratio associated with $M_{wf}$ having a charge $e$ can be expressed as,

$$m_{l\upsilon }\cong 0.88*n*\sqrt{{\displaystyle \frac{4\pi {\epsilon }_0{G}_N{M}_{wf}^2}{e^2}}}\sqrt{{\displaystyle \frac{m_{lepton}}{m_e}}}\left(M_{wf}\right)$$

(2)

3. Calculated Neutrino Masses and the Cosmological Observations

At n=1, Electron neutrino mass is $m_{e\upsilon }\cong 0.289\mathrm{meV}/c^2$

At n=2, Muon neutrino mass is $m_{\mu \upsilon }\cong 8.297\mathrm{meV}/c^2$

At n=3, Tau neutrino mass is $m_{\tau \upsilon }\cong 51.03\mathrm{meV}/c^2$

Sum of the three neutrino rest masses can be expressed as,

$${\displaystyle \sum m_{\upsilon }{c}^2}\cong \left(0.289\right)+\left(8.297\right)+\left(51.03\right)\cong 59.63\mathrm{meV}$$

(3)

Sum of the three anti neutrino rest masses can also be expressed as,

$${\displaystyle \sum \overline{m_{\upsilon }}{c}^2}\cong \left(0.289\right)+\left(8.3\right)+\left(51.0\right)\cong 59.63\mathrm{meV}$$

(4)

$${\displaystyle \sum m_{\upsilon }{c}^2}+{\displaystyle \sum \overline{m_{\upsilon }}{c}^2}\cong 2*59.63\cong 119.26\mathrm{meV}$$

(5)

This value is consistent with cosmological observations (Planck 2018, DESI 2024) [17,18].

4. Mass Splitting of the Estimated Neutrinos Using the Squared Mass Differences

For the above estimated neutrino masses,

$$\left(m_{\tau \upsilon }^2-m_{\mu \upsilon }^2\right)c^4\cong 2.54\times {10}^{-3}{\mathrm{eV}}^2$$

(6)

$$\left(m_{\mu \upsilon }^2-m_{e\upsilon }^2\right)c^4\cong 6.875\times {10}^{-5}{\mathrm{eV}}^2$$

(7)

These values are well-aligned with experimental data from solar and atmospheric neutrino oscillations [19,20]. The mass-squared differences are key observables in neutrino physics. Their compatibility with experimental results provides support for the proposed mass model and helps validate the scaling relations applied [14,15,16,17,18].

5. To Replace the Stoney Scale and the Planck Scale

The Stoney scale [11,12], introduced by George Stoney before the advent of quantum theory, is based on the elementary constants $e,G_N\mathrm{and}c$ and defines a natural gravitational-electromagnetic mass scale. It is given by,

$$M_{Stoney}\cong \sqrt{{\displaystyle \frac{e^2}{4\pi {\epsilon }_0{G}_N}}}\cong 1.85921\times {10}^{-9}\mathrm{kg}$$

(8)

Using the definition of the fine-structure constant,

$$\alpha \cong {\displaystyle \frac{e^2}{4\pi {\epsilon }_0\hslash c}}$$

(9)

We obtain the relation,

$$M_{Stoney}\cong \sqrt{\alpha }\left(\sqrt{{\displaystyle \frac{\hslash c}{G_N}}}\right)\cong \sqrt{\alpha }\left(M_{Planck}\right)$$

(10)

This shows that the Stoney mass is naturally suppressed compared to the Planck mass by a factor of $\sqrt{\alpha }\cong {\displaystyle \frac{1}{11.7}}$. In this context, the Stoney mass may be viewed as an intermediate scale connecting gravity and electromagnetism, and its appearance in our seesaw formulation highlights a possible coupling between charged lepton mass ratios and unified gravitational effects. Thus, our proposed seesaw relation can be expressed as,

$$\begin{array}{l}{m}_{l\upsilon }\cong {\displaystyle \frac{k*n}{\sqrt{\alpha }}}*\sqrt{{\displaystyle \frac{m_{lepton}}{m_e}}}\left({\displaystyle \frac{M_{wf}^2}{M_{Planck}}}\right)\\ \mathrm{where}k\cong \left(0.85\mathrm{to}0.9\right)\cong 0.88\end{array}$$

(11)

Following this relation, quantitatively, in a trial-error method, we have noticed that,

$$\begin{array}{l}{\displaystyle \frac{k}{\sqrt{\alpha }}}\cong \ln \left({\displaystyle \frac{M_{wf}}{\sqrt{m_p{m}_e}}}\right)\cong 10.193\\ \to k\cong \sqrt{\alpha }\times \ln \left({\displaystyle \frac{M_{wf}}{\sqrt{m_p{m}_e}}}\right)\cong 0.871\end{array}$$

(12)

Now ${\displaystyle \frac{k*n}{\sqrt{\alpha }}}$ can be expressed as,

$${\displaystyle \frac{k\ast n}{\sqrt{\alpha }}}\cong n\ast \ln \left({\displaystyle \frac{M_{wf}}{\sqrt{m_p{m}_e}}}\right)\cong \ln {\left({\displaystyle \frac{M_{wf}}{\sqrt{m_p{m}_e}}}\right)}^n$$

(13)

Thus,

$$m_{l\upsilon }\cong \ln {\left({\displaystyle \frac{M_{wf}}{\sqrt{m_p{m}_e}}}\right)}^n*\sqrt{{\displaystyle \frac{m_{lepton}}{m_e}}}\left({\displaystyle \frac{M_{wf}^2}{M_{Planck}}}\right)$$

(14)

6. General Discussion

Our approach assumes Dirac neutrinos, supported by the absence of neutrino less double beta decay and lepton number violation. This contrasts with conventional Type-I seesaw models which typically favour Majorana neutrinos due to lepton number violation considerations. The model’s consistency with cosmological bounds (total mass < 0.12 eV) strengthens its viability. The introduction of the 0.88 coefficient seems to direct towards a deeper unification of physics or understated suppression effects inherent in cosmic evolution.

The use of a Dirac framework suggests that neutrinos have distinct antiparticles and conserve lepton number. This choice is not only conceptually economical but also observationally motivated. Furthermore, the cosmological data from Planck and DESI provides strong upper limits on the neutrino mass sum, and our results satisfy these bounds comfortably.

Currently, there is limited clarity on the cosmological distinction between neutrinos and antineutrinos-particularly regarding their thermal histories, possible annihilation, and relic asymmetries. Further theoretical and observational studies are required to explore these aspects, especially in the context of Dirac neutrino models where lepton number conservation plays a crucial role.

Although neutrinos and antineutrinos can, in principle, annihilate via weak interactions, such annihilations become highly suppressed after thermal decoupling in the early universe. In the context of Dirac neutrinos, where lepton number is conserved and neutrinos are distinct from antineutrinos, it remains unclear whether any relic asymmetry could have influenced their annihilation history. This issue merits deeper theoretical investigation.

7. Applications and Significance of the 585 GeV Electroweak (Dirac) Fermion

The 585 GeV electroweak Dirac fermion posited in our 4G unification model is supported by multiple independent lines of evidence spanning nuclear physics and astrophysical observations [9,10,11]. Nuclear structural analyses suggest signatures consistent with the physical existence of such a heavy fermion, offering new insight into the interplay between nuclear phenomena and electroweak scale physics.

From an astrophysical perspective [9], the presence of unexplained TeV-scale gamma rays observed in galactic and extragalactic sources can be naturally accounted for by particle processes involving this 585 GeV fermion (in place of electron or proton), including annihilation and inverse Compton scattering, thereby bridging particle physics and high-energy astrophysics in a unified framework.

A key phenomenological scaling factor appearing in our model is the ratio of the geometric mean of the charged and neutral pion masses (~137.26 MeV) to that of the weak boson masses (~85.61 GeV), which numerically evaluates to approximately 0.0016. This dimensionless ratio encapsulates the profound hierarchical gap between the strong interaction scale and the electroweak scale and forms a cornerstone of the mass relations underlying our Dirac seesaw construction involving the 585 GeV electroweak fermion. Importantly, this ratio is not merely a numerical coincidence but has substantive implications for understanding nuclear stability and nuclear binding energy. The interplay of these fundamental mass scales suggests that the dynamics governing nuclear forces and nucleon interactions may be intimately connected to electroweak-scale physics mediated by the 585 GeV fermion. For a deeper exploration of how this mass ratio informs nuclear binding mechanisms and stability criteria, interested readers are encouraged to refer our recent preprints and other peer-reviewed publications [21,22], where these connections are discussed in detail with complementary theoretical and phenomenological analyses.

$$\begin{array}{l}{\displaystyle \frac{m_p}{M_{wf}}}\cong 0.001605\cong \left({\displaystyle \frac{\sqrt{{\left(m_{\pi }{c}^2\right)}^0{\left(m_{\pi }{c}^2\right)}^{\pm }}}{\sqrt{{\left(m_w{c}^2\right)}^{\pm }{\left(m_z{c}^2\right)}^0}}}\right)\\ \cong \left({\displaystyle \frac{\sqrt{134.98\times 139.57}\mathrm{MeV}}{\sqrt{80379.0\times 91187.6}\mathrm{MeV}}}\right)\cong 0.0016032\cong \beta \mathrm{....}\left(\mathrm{say}\right)\end{array}$$

(15)

Based on this electroweak coefficient $\beta \cong 0.001605$, stability corresponding to nuclear beta decay can be understood with the following relation.

$$A_s\cong 2Z+\beta {\left(2Z\right)}^2\cong 2Z+0.00642Z^2$$

(16A)

$${\displaystyle \frac{A_s-2Z}{{\left(2Z\right)}^2}}\cong {\displaystyle \frac{A_s-2Z}{4Z^2}}\cong \beta$$

(16B)

One can find a similar relation in the literature [23]. This relation can be well tested for Z=21 to 92. For example,

$$\begin{array}{l}{\displaystyle \frac{45-\left(2\times 21\right)}{4{\left(21\right)}^2}}\cong 0.00170;{\displaystyle \frac{63-\left(2\times 29\right)}{4{\left(29\right)}^2}}\cong 0.00149;{\displaystyle \frac{89-\left(2\times 39\right)}{4{\left(39\right)}^2}}\cong 0.00181;\\ {\displaystyle \frac{109-\left(2\times 47\right)}{4{\left(47\right)}^2}}\cong 0.0017;{\displaystyle \frac{169-\left(2\times 69\right)}{4{\left(69\right)}^2}}\cong 0.00163;{\displaystyle \frac{238-\left(2\times 92\right)}{4{\left(92\right)}^2}}\cong 0.001595;\end{array}$$

This is one best practical and quantitative application of our proposed electroweak fermion and bosons. Following this relation and based on various semi empirical mass formulae [23,24,25,26,27,28,29,30,31], by knowing any stable mass number, its corresponding proton number can be estimated with,

$$Z\cong {\displaystyle \frac{A_s}{1+\sqrt{1+0.0064A_s}}}\cong {\displaystyle \frac{A_s}{2+0.0153A_s^{2/3}}}$$

(17)

$\mathrm{where}{\displaystyle \frac{a_c}{2a_{asy}}}\cong {\displaystyle \frac{0.71\mathrm{MeV}}{2\times 23.21\mathrm{MeV}}}\cong {\displaystyle \frac{0.6615\mathrm{MeV}}{2\times 21.6091\mathrm{MeV}}}\cong 0.0153$Considering this relation, we are working on understanding the stable super heavy elements.

One important point to be noted here is that, self-attractive (potential) energy of proton having a nuclear charge [9,10] of $e_n\cong 2.9464e$ and root mean square radius of 0.83 fm [11,32], can be expressed as,

$$\begin{array}{l}\left({\displaystyle \frac{3}{5}}\right){\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_p}}\cong \left({\displaystyle \frac{3}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0\left(0.83\mathrm{fm}\right)}}\cong 9.04\mathrm{MeV}\\ \mathrm{where},R_p\cong \mathrm{Root}\mathrm{mean}\mathrm{square}\mathrm{radius}\mathrm{of}\mathrm{proton}\\ {\alpha }_s\cong {\left({\displaystyle \frac{e}{e_n}}\right)}^2\cong \mathrm{Strong}\mathrm{coupling}\mathrm{constant}\cong 0.1152\end{array}$$

(18)

[9,10]

With reference to the nuclear binding energy scheme, maximum binding energy per nucleon observed in the cases of Iron and Nickel atomic nuclei is around 8.8 MeV. Following this observation, independent of total binding energy calculations, maximum binding energy per nucleon for medium and heavy atomic nuclides can be expressed directly as,

$$\begin{array}{l}{〈{\displaystyle \frac{BE}{A_s}}〉}_{\max }\cong \left[1-{\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right]\times \left(9.1\pm 0.05\right)\mathrm{MeV}\\ \mathrm{where}{A}_s>56,\beta \cong 0.001605\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(19)

Maximum binding energy per nucleon for light atomic nuclides can be expressed directly as,

$${〈{\displaystyle \frac{BE}{A_s}}〉}_{\max }\cong {\left({\displaystyle \frac{A_s}{56}}\right)}^{{\alpha }_s}\times \left(1-\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)\right)\times \left(9.1\pm 0.05\right)\mathrm{MeV}$$

(20)

$\begin{array}{l}\mathrm{where},4\le A_s\le 56\\ {\alpha }_s\cong \mathrm{Strong}\mathrm{coupling}\mathrm{constant}\cong 0.1152\\ \beta \cong 0.001605\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$

Accuracy point of view, for light, medium and heavy atomic nuclides, energy coefficient seems to be around 9.15 MeV and for super heavy atomic nuclides, energy coefficient seems to be around 9.05 MeV. See the following Table 1 for the estimated data. It needs further study.

Proceeding further, investigations into the role of fundamental constants, notably the Planck length and the unified Stoney mass, reveal their significance in low-energy nuclear phenomena such as neutron lifetime measurements. Their inclusion in our seesaw neutrino mass formula is thus well motivated, providing an intermediate gravitational-electromagnetic mass scale that mediates neutrino mass suppression consistently with the Dirac framework.

Collectively, these theoretical and phenomenological findings position the 585 GeV electroweak Dirac fermion as a well-founded component connecting neutrino mass generation, nuclear structure, and cosmic high-energy phenomena. This synthesis motivates targeted experimental strategies both at collider experiments and in astrophysical observations to test the existence and properties of this particle, advancing our understanding of fundamental mass hierarchies and unification.

8. Conclusion

Our 4G model of final unification, combined with a Dirac seesaw mechanism and lepton mass ratios, yields physically meaningful neutrino mass estimates consistent with known neutrino oscillations and cosmological data. Further exploration of the proposed 585 GeV Dirac mass and the unified gravitational Stoney mass may uncover deeper structures in neutrino physics.

This approach offers a compelling alternative to conventional Majorana-based models, and its alignment with data encourages further theoretical and phenomenological development. Future studies may focus on refining the damping coefficient, exploring links with quantum gravity, and identifying potential experimental signals.