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: n^{1/2 }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 10^{19}.
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 ~ 10^{38}. In Section 3 the justification for this
“strong gravity” approach will be discussed. We consider the hypothesis that these ultracompact subatomic particles
can be approximated as quantized allowed values, i.e., eigenstates or excited
states, of the basic angular momentummass relationship of the Kerr solution of
the Einstein field equations of General Relativity.
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 ultracompact object and its mass (M):
J
= kGM^{2}/c . (1)
The parameter k is referred to as a dimensionless spin
parameter associated with the rotational properties of the ultracompact
object. G is the gravitational coupling
factor and c is the velocity of light. Since we are interested in the masses of
the ultracompact 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 wellknown (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 = kM^{2} relations. While Regge’s heuristic phenomenology is welldocumented, 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:
M_{n} = n^{1/2} (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 Kerrderived J
versus M^{2} 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).
The
first step in testing Eq. (3) in the subatomic domain is to reevaluate 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
SelfSimilar 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 selfsimilar 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 selfsimilarity 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:
G_{Ψ} = (Λ^{1}^{D})^{ Ψ} G_{0} , (4)
where G_{0} is the conventional Newtonian
gravitational constant, Λ and D are empirically determined dimensionless selfsimilarity constants
equaling 5.2 x 10^{17} 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} = Λ^{2.174} G_{0} = 2.18 x 10^{31} cm^{3}/g sec^{2}. 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:
M_{n} = n^{1/2} M = n^{1/2} (674.8 MeV) . (5)
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, wellknown 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
n

n^{1/2}

n^{1/2} (674.8 MeV)

Particle
/ MeV

Relative
Error

1/36 = (1/9)/4 
0.1666 
112.46 
μ / 105.66 
6.4 % 
1/25 ≈ (1/6)/4 
0.2000 
134.96 
π / 134.98 
0.01 % 
1/2 = 2/4 
0.7071 
477.15 
κ / 497.65 
4.1 % 
3/4 
0.8660 
584.39 
η / 547.75 
6.7 % 
1 = 4/4 
1.0000 
674.8 
M / 674.8 
 
5/4 
1.1284 
761.40 
ρ / ~ 770

1.1 % 
5/4 
1.1284 
761.40 
ω/ ~ 783

2.8 % 
2 
1.4142 
954.31 
p^{+} / 938.27 
1.7 % 
2 
1.4142 
954.31 
n / 939.57 
1.6 % 
2 
1.4142 
954.31 
η' / 957.75 
0.4 % 
3 
1.7320 
1167.75 
Λ^{0} ^{ }/ 1115.68 
4.7 % 
3 
1.7320 
1167.75 
Σ^{i} / <1192> 
2.0 % 
4 
2.0000 
1349.60 
Ξ^{0} / 1314.83 
2.6 % 
5 
2.236 
1508.90 
N(1440)/ 14301470 
~ 4.8 % 
6 
2.4495 
1652.91 
Ω^{} / 1672.45 
1.2 % 
7 
2.6458 
1785.35 
τ^{} / 1784.1 
0.05 % 
8 
2.8284 
1908.62 
D^{0} / 1864.5 
2.4 % 
8 
2.8284 
1908.62 
D^{+/} /1869.3 
2.1 % 
8 
2.8284 
1908.62 
^{2}H / 1889.77 
1.0 % 
10 
3.1623 
2133.90 
D_{s}^{i} / 2112.1 
1.0 % 
12 
3.4641 
2337.58 
Λ_{c}^{i} / 2284.9 
2.3 % 
14 
3.7417 
2524.87 
Ξ_{c}^{i} / <2522.75> 
~ 0.1 % 
16 
4.0000 
2699.20 
Ω_{c}^{0} / 2697.5 
0.1 % 
18 
4.2426 
2862.93 
^{3}H / 2829.87 
1.2 % 
18 
4.2426 
2862.93 
^{3}He / 2829.84 
1.2 % 
30 
5.4772 
3696.03 
^{4}He / 3727.38 
0.9 % 
64 
8.000 
5398.40 
B_{j}^{i} / <5313.25> 
~ 1.6 % 
90 
9.4868 
6401.71 
B_{c}^{i} / <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 > m_{p}, where m_{p} is the proton mass, and those particles that have masses < m_{p}. For the former group we see that integer nvalues generate good first approximation retrodictions with <98.4 %> accuracy of the particle masses with m_{p} ≤ m < 7,000MeV, with a preponderance of even values of n.
For
the much smaller group of particles with m < m_{p}, the set of nvalues
is not as simple and regular as it is for the m ≥ m_{p} 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 nvalues, but a unique pattern is not obvious. The distinct possibility exists that nvalues
for the m < m_{p} 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., n_{η} = [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 KerrNewman solution of the EinsteinMaxwell equations to provide a more sophisticated
model of the particles. This more
complete and rigorous analysis would include charge, mass, electrodynamic considerations and spinrelated
phenomena. The results of this second
approximation analysis should provide considerable guidance in understanding
the most appropriate set(s) of nvalues 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.
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 SelfSimilar Cosmological Paradigm.
In
atomic and nuclear physics, there are wellknown 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
KerrNewman solution of the EinsteinMaxwell 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 KerrNewman 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 KerrNewman 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.
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Acknowledgement: I would like to thank Dr. Jonathan Thornburg for numerous suggestions regarding
the technical presentation of this research.