New Developments 2010

The Subatomic Particle Mass Spectrum [January 2010]

Summary:  We discover that the subatomic particle mass spectrum in the 100 MeV to 7,000 MeV range can be retrodicted to a first approximation using the Kerr solution of General Relativity.  The particle masses appear to form a restricted set of quantized values of the Kerr solution: n1/2 M, where values of n are a set of discrete integers and M is the revised Planck mass.  The accuracy of the 27 retrodicted masses averages 98.4%.  Discrete Scale Relativity is required to evaluate M, which differs from the conventional Planck mass by a factor of roughly 1019.

1. Introduction

Retrodictions and predictions of subatomic particle masses have been highly valued desiderata ever since these unanticipated highly compact systems were discovered empirically.  It is widely acknowledged that the particle masses have to be put into the Standard Model of particle physics “by hand”, and further, that this lack of predictive/retrodictive capability is considered to be a significant problem that eventually must be resolved.

Here we will consider an unorthodox but promising approach to addressing the enigma posed by the particle mass spectrum.  The main underlying idea of the approach is that gravitational interactions are much stronger within subatomic particles than was previously realized, by a factor of ~ 1038.  In Section 3 the justification for this “strong gravity” approach will be discussed.  We consider the hypothesis that these ultra-compact subatomic particles can be approximated as quantized allowed values, i.e., eigenstates or excited states, of the basic angular momentum-mass relationship of the Kerr solution of the Einstein field equations of General Relativity.

2 . The Kerr Solution In The Subatomic Realm

For the Kerr solution of the Einstein field equations of General Relativity one finds the following simplifying relationship (McClintock, Shafee, Narayan, Remillard, Davis, and Li, 2006) between the angular momentum (J) of an ultra-compact object and its mass (M):

   J = kGM2/c  .       (1)

The parameter k is referred to as a dimensionless spin parameter associated with the rotational properties of the ultra-compact object.  G is the gravitational coupling factor and c is the velocity of light. Since we are interested in the masses of the ultra-compact objects, we rewrite Eq. (1) in the form:

   M = (Jc/kG)1/2 .  (2)

Since we are interested in applying Eq. (2) in the subatomic domain, we hypothesize that the unit of J in this domain is given by Planck’s constant ћ, and in the subatomic realm we expect J to be restricted to a discrete set of values, i.e., nћ.  As is well-known (Regge, 1959; Eden, 1971; Irving and Worden, 1977), in the 1960s Tullio Regge demonstrated that the masses and total spins of families of baryon and meson resonances were linearly related by J = kM2 relations.  While Regge’s heuristic phenomenology is well-documented, it has never found an adequate explanation in Quantum Chromodynamics or any other part of the Standard Model of particle physics.

With the above assumptions concerning J, we can rearrange Eq. (2) to yield:

   Mn = n1/2 (ћc/G)1/2  ,      (3)

where n = 1/k, and we notice that (ћc/G)1/2 is just the definition for the Planck mass.  Therefore according to Eq. (3) the allowed values of the Kerr-derived J versus M2 relation in the subatomic domain are the square roots of quantized multiples of the Planck mass.  The applicability of the Kerr solution of General Relativity in the subatomic realm, and our initial assumptions concerning k and J, can be tested by attempting to retrodict the subatomic particle mass spectrum using Eq. (3).

3. Evaluating (ћc/G)1/2 With Discrete Scale Relativity

The first step in testing Eq. (3) in the subatomic domain is to re-evaluate the Planck mass.  Motivation for questioning the conventional Planck scale can be found in: (1) the fact that the conventional Planck mass of 2.176 x 10-5 g  is not associated with any particle or phenomenon observed in nature, (2) the fact that the conventional Planck scale results in many forms of the closely related “hierarchy problem”, and (3) the fact that it leads to a vacuum energy density (VED) crisis in which there is a disparity of 120 orders of magnitude between the  VED estimates of particle physics and cosmology. [See “Technical Notes”: Nov 2006 and Feb 2009 for more on these topics.]

A way to avoid these problems, and many more, can be found in a new cosmological paradigm for understanding nature’s structural organization and dynamics.  This new paradigm is called the Discrete Self-Similar Cosmological Paradigm (DSSCP).  It is the product of a very thorough and careful empirical study of the actual objects that comprise nature, and the paradigm is based on the fundamental principle of discrete scale invariance.  The discrete self-similar systems that comprise nature, and the fact that fractal structures are so common in nature, are the physical manifestations of the discrete scale invariance of nature’s most fundamental laws and geometry.  The DSSCP was reviewed in two papers published in 1989 (Oldershaw, 1989a,b; “Selected Papers” #1 and #2).

Discrete Scale Relativity (DSR) is the variation of the general DSSCP which postulates that the discrete self-similarity is exact, and has been discussed recently in a brief paper (Oldershaw, 2007; “Selected Papers” #12).  According to DSR, gravitation scales in the following manner:

   GΨ = (Λ1-D) Ψ G0  ,         (4)

where G0 is the conventional Newtonian gravitational constant, Λ and D are empirically determined dimensionless self-similarity constants equaling 5.2 x 1017 and 3.174, respectively, and Ψ is a discrete index denoting the specific cosmological Scale under consideration.  For the evaluation of Eq. (3) we have Ψ = -1, which designates the Atomic Scale, and therefore G-1 = Λ2.174 G0 = 2.18 x 1031 cm3/g sec2.  According to DSR, G-1 is the proper gravitational coupling constant between matter and spacetime geometry within Atomic Scale systems. Evaluating the Planck mass relation (ћc/G)1/2 using G-1 and the usual values of ћ and c yields a value of 1.203 x 10-24 g, or 674.8 MeV.  This revised Planck mass is identified below by the symbol M.  To a first approximation, the subatomic particle mass spectrum should have peaks at the mass values:

   Mn = n1/2 M = n1/2 (674.8 MeV)  .          (5)

4. Testing Mn = n1/2 M

Table 1 presents relevant data for testing Eq. (5) in terms of a representative set of subatomic particles from a mass/energy range of 100 MeV to 7,000 MeV.  The particles appearing in Table 1 are among the most abundant, well-known and stable members of the particle/resonance “zoo”.  For each integer of n there appears to be an associated particle, or set of related particles, that agrees with mass values generated by Eq. (5) at about the 93 to 99.99 %  level.  The average relative error for the full set of 27 particles is 1.6 %.

Table 1



n1/2 (674.8 MeV)

Particle / MeV

Relative Error

1/36  =  (1/9)/4



μ / 105.66

6.4 %

1/25   (1/6)/4



π / 134.98

0.01 %

1/2  =  2/4



κ / 497.65

4.1 %




η / 547.75

6.7 %

1  =  4/4



M / 674.8





ρ / ~ 770

1.1 %




ω/ ~ 783

2.8 %




p+ / 938.27

1.7 %




n / 939.57

1.6 %




η' / 957.75

0.4 %




Λ0  / 1115.68

4.7 %




Σi /  <1192>

2.0 %




Ξ0 / 1314.83

2.6 %




N(1440)/ 1430-1470

~ 4.8 %




Ω- / 1672.45

1.2 %




τ- / 1784.1

0.05 %




D0 / 1864.5

2.4 %




D+/- /1869.3

2.1 %




2H / 1889.77

1.0 %




Dsi / 2112.1

1.0 %




Λci / 2284.9

2.3 %




Ξci / <2522.75>

~ 0.1 %




Ωc0 / 2697.5

0.1 %




3H / 2829.87

1.2 %




3He / 2829.84

1.2 %




4He / 3727.38

0.9 %




Bji / <5313.25>

~ 1.6 %




Bci / <6400>

~ 0.1 %

Table 1 lists the n values, the retrodicted masses, the empirical masses, and the relative errors for 27 subatomic particles.  Here we will discuss these 27 test particles in two separate groups: those particles that have masses > mp, where mp is the proton mass, and those particles that have masses < mp.  For the former group we see that integer n-values generate good first approximation retrodictions with <98.4 %> accuracy of the particle masses with mp ≤ m < 7,000MeV, with a preponderance of even values of n.

For the much smaller group of particles with m < mp, the set of n-values is not as simple and regular as it is for the m ≥ mp group.  The unit M obviously has n = 1 but other members of this group have fractional values of n.  The μ, π, κ, η, M, ρ and ω particles can be assigned n = (1/9)/4, (1/6)/4, 2/4, 3/4, 4/4, 5/4 and 5/4, respectively, or n = 1/36, 1/25, 1/2, 3/4, 1, 5/4 and 5/4.  One gets the definite impression that there is an underlying order to this set of n-values, but a unique pattern is not obvious.  The distinct possibility exists that n-values for the m < mp group are compound terms such as n = i / j, or i · j, where i and j could be integers, multiples of π, and/or multiple rational fractions, i.e., nη = [3/2 · 1/2].  Rather than explore these possibilities numerologically, an approach with a long and checkered history, it seems more prudent to go directly to a second approximation analysis of the subatomic particle mass spectrum using the Kerr-Newman solution of the Einstein-Maxwell equations to provide a more sophisticated model of the particles.  This more complete and rigorous analysis would include charge, mass, electrodynamic considerations and spin-related phenomena.  The results of this second approximation analysis should provide considerable guidance in understanding the most appropriate set(s) of n-values for all particles, as well as fostering an understanding the more subtle properties of the underlying order that generates the very regular patterns observed in the particle mass spectrum.

5. Implications

As demonstrated by the results presented in Table 1, the subatomic particle mass spectrum appears to manifest a simple, consistent and orderly pattern extending over a considerable range of particle masses and a diversity of family types, i.e., leptons, mesons, and baryons.

To a first approximation the masses and angular momenta of the particles appear to be the primary or dominant physical determinants of the particle mass spectrum.  Charge and other physical phenomena appear to be second order effects that  determine the fine structure of the mass spectrum.

A critical factor in determining the first approximation mass spectrum is the revised Planck mass (≈ 674.8 MeV) which is uniquely obtained via the scaling relations of the Discrete Self-Similar Cosmological Paradigm.

In atomic and nuclear physics, there are well-known examples (Rohlf, 1994; Garai, 2007) of phenomena wherein “magic numbers” appear in the stable solutions of the fundamental equations.  This is especially evident in the isotopic stability of subatomic nuclei, and in the filling of electron “shells” in atoms.  Perhaps the results shown in Table 1 identify an analogous case of a “magic numbers” phenomenon that applies in the context of the subatomic particle mass spectrum.

A second approximation of the particle mass spectrum will clearly require a Kerr-Newman solution of the Einstein-Maxwell equations in order to fully take charge, spin and related phenomena into account.  It can be predicted on the basis of the results discussed in this paper that the full Kerr-Newman solution will permit a much more accurate retrodiction of the mass spectrum that includes more of the spectrum’s fine structure.  The Geometrodynamics approach to working with the Kerr-Newman solution, as developed by Misner, Thorne and Wheeler (1973), would seem to offer a simple method for conducting initial tests of  this prediction.  Interested readers are strongly encouraged to participate in this effort.



Eden, R.J. (1971). Regge poles and elementary particles. Rep. Prog. Phys. 34, 995-1053.

Garai, J. (2007). Mathematical formulas describing the sequences of the periodic table. Internat. J. Quant. Chem. 108, 667-670.

Irving, A.C. and Worden, R.P. (1977). Regge Phenomenology. Phys. Rept. 34, 117-231.

McClintock, J.E., Shafee, R., Narayan, R., Remillard, R.A., Davis, S.W., Li, L.-X. (2006). The

Spin of the Near-Extreme Kerr Black Hole GRS 1915+105. Astrophysical Journal 652, 518-539.

Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973). Gravitation. San Francisco: W.H. Freeman.

Oldershaw, R.L. (1989a). The Self-Similar Cosmological Model: Introduction And Empirical Tests. International Journal of Theoretical Physics 28, 669- 694.

Oldershaw, R.L. (1989b). The Self-Similar Cosmological Model: Technical Details, Predictions, Unresolved Issues, And Implications. International Journal of Theoretical Physics 28, 1503-1532.

Oldershaw, R.L. (2007). Discrete Scale Relativity. Astrophysics and Space Science 311, 431-

433. [DOI: 10.107/s10509-007-9557-x]; also available at as  arXiv:physics/0701132v3.

Regge, T. (1959). Introduction to complex orbital momenta. Nuovo Cimento 14, 951-976.

Rohlf, J.W. (1994). Modern Physics from α to Z0. New York: Wiley.


Acknowledgement: I would like to thank Dr. Jonathan Thornburg for numerous suggestions regarding the technical presentation of this research.