FRACEP,
J. A. Giannini (7/25/2012)
THE FRACEP MODEL, Part 1b:
What
Is the Size of the Composite Particles?
May 17, 2007 (revised 9/26/10)
The Standard Model (SM) has the fermions as part of its list of fundamental particles despite considerable evidence to the contrary. The FRACEP Model, Part 1a ^{[Ref. 1]} is a new heuristic model. It presents a composite structure for the fermions that is consistent with the SM observations of those particles in mass, spin, electromagnetic charge, and decay components. This paper takes the next step. It shows that those FRACEP built-up composite structures have a classical radius that is consistent with the best maximum estimate by the SM for the observed fermions. This offers further support for the concept of non-fundamental fermions.
1.0 INTRODUCTION
2.0 THE CONCEPT
OF SIZE AND ITS DIFFERENT DESCRIPTIONS
2.1. The Macro World Size
Description
2.2. The Quantum World Size
Description
3.0 FUNDAMENTAL
SCALES OF NATURE
3.1. Historical Basis of the
Fundamental Scales
3.2. The Planck Mass
3.3. The Planck Length
3.4. The Planck Time
4.0 THE COMPOSITE
PARTICLE BUILDING BLOCK SIZES
4.1. Assumptions Used in the Size
Determination
4.2. Sizes of the IBB's and Their
Components
5.0 THE SIZES OF
THE COMPOSITE VERSION OF THE SM FERMIONS
6.0 CONCLUSIONS
7.0 REFERENCES
Return to Book Table of Contents
1.0 INTRODUCTION
The Standard Model of Particle
Physics (SM) is a model of the world of the small (atoms and molecules, and
sub-atomic particles). Its theoretical
part is based on quantum mechanics; and, its data base includes its fundamental
particles (the fermions and bosons). The
generally accepted definition of "fundamental" includes the
requirement of indivisibility of the particles.
However, the observed data for the majority of the fermions (and many of
the bosons) indicate that these particles spontaneously decay. This appears to violate a necessary
requirement for a fundamental nature in a particle. At the same time, it seems to support the
possibility of the picture of fermions as composite particles. In recognition of this state of affairs, the FRACEP
Model was developed to address the feasibility of describing fermions as
composite particles.
In Part 1^{[Ref. 1]},
the structures (required components) for all the fermions and anti-fermions
were developed, based on a set of Intermediate Building Blocks (IBB's) that
were themselves built up from fractal-like structures of only two fundamental
spin-less, charge-less particles (G0p and G0m).
The set of IBB's contains: a positive (SGp) and a negative (SGm) spin
carrier; a negative (QGp) and a positive (QGm) charge carrier; several mass
carrying particles (GXp); and several momentum
carrying rings (RXp). The X in the particle designation indicates
the fractal level of the particle. The X
in the ring structure designation indicates that the ring is made of 6 GXp's particles.
The built-up composite FRACEP fermions agree with the SM fundamental
fermions in four characteristics: mass, spin, electromagnetic charge, and decay
products. But, these four
characteristics are only some of what are necessary to validate the FRACEP
picture.
In order to further
support the FRACEP, it is necessary to consider the sizes of the composite
particles by showing that they are consistent with the SM estimates of the
fermion sizes. The purpose of this
exercise is to provide a best estimate of the expected sizes, defined as
classical radius (R_{class}), of the particles and components. The size computations are necessarily
approximate given the challenge of estimating the size of something that cannot
be seen.
The size discussion, in
this paper, is presented in several steps.
First, there is a discussion of the Concept of Size and Its Different
Descriptions. The natural tendency is to
perceive size in the quantum sense as the SM sees it because that is where the FRACEP
particles lie based on their mass. But,
the FRACEP recognizes the extension of the classical description as also valid
in the context presented here. So this
section discusses the difference in the descriptions between classical and
quantum measures of size.
Next, there is a
discussion of the Fundamental Scales of Nature because the FRACEP is dealing
with a true fundamental particle (G0p) that is profoundly interrelated with the
fundamental scales in determining its size.
After that, the
Composite Particle Building Block Sizes as defined by the FRACEP are computed.
And finally, the Sizes
of the Composite Version of the SM Fermions are computed. In this section, there is also a discussion
of the FRACEP composite Planck particle because of its fundamental scales
relation.
2.0 THE CONCEPT OF SIZE AND ITS DIFFERENT DESCRIPTIONS
On the face of it, size,
in general, seems to be a simple concept.
However, our experience in describing size is based mostly on our
perception of the world we see (the macro world). But when dealing with objects too small to
see, such as the fermions, the description of size changes to fit the rules of
the quantum world. To put the size
question in perspective, it is valuable to briefly discuss what is meant by
size in each of the two regimes, recalling that, although the fermions reside
in the quantum world (they have wave-like behavior), they are not totally
divorced from the macro world (they also have coherent particle-like
behavior). Because of this, the FRACEP
extends the macro world size description (R_{class} which is the
classical radius concept) to its composite versions of the SM fundamental
particles.
2.1. The Macro World Size Description
From the macro
perspective of every day life, size is an easy concept. Objects have a fixed surface or boundary
confining them. The size is determined
by measuring their dimensions (length, width and height) with a ruler. An example of the every day scales is a human
who weighs ~200 lbs (m_{h} ~ 90.0 kg ~ 5x10^{31}
MeV/c^{2}) and is ~ 6 ft (~1.8 m) tall.
(This approximately cylindrical person is roughly equivalent in volume
to a sphere of radius r_{class-h} ~ 0.85 m. The equivalent sphere is useful for providing
a perspective for every day scales of spherical objects).
As one progresses from the every day scales to the astronomical scales, the concept of size is maintained though the nature of the ruler is adjusted to accommodate the larger scales of interest. The first step of the astronomical scales includes the planets ^{[Ref. 2, chapt. 1-13]} that still have discernable, relatively solid surfaces, many surrounded by an atmosphere. For example, earth is roughly spherical with a radius r_{class-e} ~ 6.4x10^{6} m, and a mass m_{e} ~ 6.0x10^{24} kg (~3.4x10^{54} MeV/c^{2}). Its atmosphere has numerous layers of gases and charged particles, and is very rarefied with a thickness on the order of r_{class-e. }
_{ }
_{ }The next step in the astronomical scales is
to the very large, such as stars and galaxies ^{[Ref. 2, chapt. 2, p.18-26]}. These
objects lose their fixed surface, so the boundary defining them is based on
large density changes. For stars, the
bulk of the mass is found within an approximately spherical volume that
contains the very high density, gaseous, energy producing region. This region is then surrounded by a much
lower density chromosphere. For example,
our sun has a radius r_{class}_{-s} ~
7.0x10^{8} m and a mass m_{s} ~
2.0x10^{30} kg (~1x10^{60} MeV/c^{2}). Its chromosphere is thin compared to r_{class}_{-s }(~ 0.007 x
r_{class}_{-s}).
The galaxies are larger than the stars. They have a general elliptical shape, though
some are freeform. Many, like our own
Milky Way, have a spiral structure within a flattened elliptical disc around a
roughly spherical bulge that forms a core. Many are believed to have a super massive
black hole at their center. Our galaxy,
the Milky Way, is estimated to have a radius r_{class}_{-MW}
~ 5x10^{4} lt-yrs (~4.7x10^{20}
m) depending on where the cut-off density boundary is set. (The lt-yr is
defined as the distance travelled at the speed of light in one year. That is, ~3x10^{8} m/s x 365.25
days/year x 86,400 seconds/day = 9.5x10^{15} m.)
The Milky Way mass is m_{MW}
~ 1.7x10^{71} kg, though higher and lower estimates can be
found. Its central core has a radius r_{
class-c-MW} ~ 7.5x10^{3} lt-yrs (~7.1x10^{19}
m). It was generally believed that this
very dense core region contained most of the galactic mass. However this perception is currently a matter
of debate, so core estimates are widely varied.
The black hole at the galaxy's center is estimated to have a mass m_{c-MW}
~ 3.6x10^{45} kg (~2x10^{66} MeV/c^{2}).
The largest of the large scales is for the
entire Universe. Cosmological models
using observations of galaxy densities, estimate the entire Universe has a
radius on the order of r_{class}_{-U}
~ 10^{25} m and a mass about m_{U} ~ 10^{80} kg (~8x10^{79}
MeV/c^{2 [Ref. 3, p10; Ref. 4, p65]}).
This brings us to our
last and exotic scale - the measure of the black hole. A black hole is formed when a massive star
has expended its fuel supply and the outward pressure of the energy production
can no longer counterbalance the inward pull of gravity. Under this condition, the star collapses to
the point of destroying the integrity of matter causing the smallest of
particles to tunnel into each other.
At the more conventional scales considered to
this point, the measurement was one of determining the extent in 3-D space of
visible matter (and possibly including the infamous "dark matter")
using optical, microwave, x-ray, etc. frequency data as the rulers. However, since black holes have no visible
matter (and no obvious boundary), the question of size is defined, by
definition, as the "event horizon" which is computed as the
Schwarzschild Radius (R_{SW} = 2Gm/c^{2}
as indicated in Figure1).
Only matter around the black hole that is outside
its R_{SW} is visible. Within
the R_{SW}, the gravity well of the black hole is so strong that no
light can escape, so its presence is observable only by gravitational effects
and the tell-tale glow of surrounding matter trying to escape the black hole's
pull. (For most objects, the R_{SW}
is well within the observable surface.)
(In general, the R_{SW} is not so much an independent measure of
size, as it is the lowest possible limit of R_{class} at any given
mass.)
2.2. The Quantum World Size Description
Having covered the macro
world from every day scales to the largest and most exotic, it is now
appropriate to turn to the small - which is where the atoms and SM particles
lie. In the world of the large, though
the ruler and boundary definitions changed, R_{class}
is the only practical measure of size.
However, it is here in the world of the small that the concept of size
has more than one practical measure.
In the early 1900's, as quantum mechanics was
being developed, it was discovered that the photon had a dual nature ^{[Ref.
5, p. 128,189]}. When shining a
light thru a diffraction grating onto a screen, the photon acted like a
particle producing a single spot of light when the grating spacing was
large. But, for spacings smaller than
the deBroglie wavelength (or alternatively the Compton wavelength, l_{C} = hc/mc^{2}), the photon demonstrated wavelike properties that
appeared as interference patterns on the screen. This appears as a series of bright and dark
lines on the screen.
Since those earliest results, the experiments
have been repeated with larger particles showing dual behavior for electrons
and atoms. For example, Brezger, et. al.^{ [Ref. 6]}
showed wave diffraction behavior for the carbon-70 molecule. This wave nature, only now becoming obvious
in larger objects, is the origin of the second measure of size appropriate to
the quantum scales. This measure is the
Compton wavelength (l_{C}) and it is the point where wave behavior
becomes obvious.
In addition to l_{C}, however, there is still the classical radius, R_{class},
which is the macro scale concept extended to smaller particles (i.e., an
indication of the physical extent of a particle in 3D space). Early in the 1900's, Rutherford, Thomson, and
others used scattering experiments to study atoms. They verified the theory of a positively
charged core surrounded by a negatively charged electron cloud. The core was shown to be made up of positively
charged protons and zero charged neutrons.
Scattering experiments employ models of the
interaction forces between probe and target ^{[Ref. 7, chapt
5, Ref. 8]}. Using input data (the
energy and approach scattering angle), they predict the expected output (energy
and retreat scattered angle) of probe and target. By correlating the predictions with the
measured results, the charge distribution of the target particle is mapped out.
This provides a measurement referred to by the SM as the scattering cross
section which represents the approximate inverse of the R_{class} of
the target.
According to the SM and such experiments, the
minimum scattering cross section of the electron is ~1.8/fm. This gives a maximum size, in 3D space, for
the radius of a spherical electron, R_{class-electron} ~ 1/1.8 = 0.56
fm (5.6x10^{-}^{16} m). (Note that l_{C-electron} is ~3.86x10^{-}^{13} m which is larger than its R_{class}). However, given the state of technology, the
maximum predicted size for leptons in general (including quarks and electron
family members) is accepted as <10^{-}^{18} m with no
discernable difference in their R_{class}'s. For comparison, the proton is R_{class-proton}
~ 8.42x10^{-}^{16 }m and a carbon atom nucleus is R_{class-C12} ~2.5x10^{-}^{15} m.
Thus, in the quantum world, objects
have a dual nature with two different ways to describe their size: 1) the l_{C} is the wave-like size description; and 2)
the R_{class} is the particle-like size description. Even though the SM concentrates on the l_{C} description, the FRACEP is focusing on the R_{class}
description (i.e., the physical extent in space).
Figure 1 shows a comparison of the two measures of size for a sample of
objects from the largest of the macro world to the smallest of the quantum
world. For all non-black-hole objects,
theory requires the R_{class} to be greater than the R_{SW}
(the lowest macro size limit) for any given mass. This is demonstrated in the figure which
shows the observed sizes (D's) for all
objects except the Milky Way black hole which, because of its collapsed nature,
has a defined size of its R_{SW }(its event horizon), and the Planck
particle (which is hypothesized but not observed) whose size is predicted to be
l_{p}*.
The s-shaped curve shows a nonlinear relation (on the log-log scale) of
R_{class} to mass. The mid region (from about the sun down to C_{12})
is roughly linear with curved tails in the extremes. The dashed curve was obtained from a fit of
the form R = F_{1}.M^{2}
+ F_{2}.M + F_{3} where F_{i}
= A_{i} + B_{i} tanh[ 0.125 . (M - 34.0)]. The M = log_{10}(mass/2). A and B are
3-element vector, where A = (5.9092337x10^{-2}, -3.9778756, 55.0231368) and B = (0.1103927, -5.0404973, 71.8671272). The predicted R = log_{10}(radius). The mass is in MeV/c2 and the radius is in
meters. The plotted values are log_{10}(mass) and log10(radius).)
Note, that given the quantum mechanical definition of discrete
space-time (as discussed in the Fundamental Scales section below), no particle
can have an R_{class} < l_{p}*
(the point where the Planck mass's l_{C} and R_{SW} cross). As stated previously, the FRACEP builds up
all of its composite fermions from fractal-like groupings of G0p. In the FRACEP, G0p is the smallest possible
mass. It is therefore not unreasonable
to assume that it also has the smallest physical extent in space. So, the FRACEP defines the R_{class-G0p}
= l_{p}*, the
smallest length scale supported by space, as indicated by the "Z" on the plot.
FIGURE 1. This shows a
comparison of the sizes of objects spanning from the quantum to the macro
scales. The horizontal dashed lines
indicate the log_{10}(mass) of the indicated
particles (with log_{10} (Mass), log_{10}(R_{class})
indicated). The diagonal line, lc, is the particle's quantum description (left side of equation 1); the
diagonal line, R_{sw}, is lowest possible limit for the particle's
macro description (right side of equation 1).
The s-shaped curve, R_{class}, is a fit thru the observed
classical radius points (D), with the R_{class-G0p} indicated (Z).
3.0 FUNDAMENTAL SCALES OF NATURE
The FRACEP describes the
composite fermions as composed of what are referred to as intermediate building
blocks (IBB's), that are themselves built-up from fractal-like groupings of the
FRACEP fundamental particles (G0p and G0m - note these two particles are
identical except for the sign of the mass value where the p denotes positive
mass, and the m denotes negative mass).
So, before we can adequately estimate the physical size (classical
radius) of the composite fermions, we must first address the size of G0p.
The characteristics of mass, em charge, and
spin were determined, in Part 1^{[Ref. 1]}, by applying the constraints
imposed by the SM fermion observations.
However, there was no adequate way to determine the size of G0p using
those data, so it is necessary to look else where - in the fundamental scales
of nature. By linking G0p to the
fundamental scales, its size (R_{class-G0p}) is established, by the FRACEP,
as l_{p}* , the smallest length scale.
3.1. Historical Basis of the Fundamental Scales
Efforts to find a fully
unified picture of the universe are not new.
Newton's theory of gravity ^{[Ref. 9]}, in the late 1600's,
showed that the interaction of bodies on earth, and the interaction of solar
system bodies follow the same rules.
This was the first success in unifying observed effects that were
believed to be different and independent.
Two hundred years later, Maxwell ^{[Ref. 10]} successfully unified the electric and the magnetic effects. He showed that both fields could be described in terms of one another. This meant that they were different aspects of one unified electromagnetic (EM) field. ^{}
By the early 1900's, two new forces were
identified: the weak (nuclear) field that controls radioactive decay; and the
strong (nuclear) field that holds the atom together. In 1967, Weinberg ^{[Ref. 11]}
provided the first unification of electromagnetism with the weak field (called
electroweak theory). The work was
successfully modified by 'tHofft ^{[Ref. 12]}
in 1971 by removing the problematic infinities in the theory. In 1973, Quantum Chromo-Dynamics (QCD)
entered the scene with its description of quark interactions and the strong
field ^{[Ref. 13]}.
Since then, extensive efforts have been ongoing to unify electroweak theory with the strong field to produce a Grand Unified Theory (GUT); and, further to unify the GUT with gravity to produce a Super-GUT (or the theory of everything). General understanding assumes that full unification to the Super-GUT level is achieved when all of the characteristic fields of nature (electromagnetism, weak, strong and gravity) have approximately equal strength and that a relationship among all the fields to one another is established ^{[Ref. 14, p134; Ref. 15, p83; Ref. 16, p346]}. ^{}
The non-unified fields
vary widely in strength. Consider the
relative fields for particle masses around m ~ 10^{-}^{27} kg (on the order of a few hundred MeV/c^{2}), and at nuclear
ranges around r ~ 10^{-}^{15
}m. The strong field is important on the quantum
scales of the atom and smaller. It is
attractive from the smallest distances up to about 1fm (10^{-}^{15}m), then becomes repulsive and quickly goes to zero at larger
distances. The strong field interaction
is about 8 times stronger than the EM field interaction between em-charges.
The EM field spans the quantum scale thru
macro scales of gravity. It is attractive
between unlike charges, and falls off as the inverse of the distance
squared.
Gravity is an attractive force and is most
evident at the macro scales of every day life and greater. Though its effect is not limited to this
range, it is very difficult to measure at the very small scales. Like the EM field, it falls off as the
inverse of the distance squared; however, it is 39 orders of magnitude weaker
than the EM interaction making it the weakest field by far.
Unifying to the GUT level theoretically
describes the three SM fields (EM, weak and strong) in terms of a single
unified field. There are several
versions of GUT models, none of which are totally consistent at this time. They require over 20 free parameters (e.g.,
the masses and charges of the fundamental particles, plus more), all of which
must be experimentally measured before any predictions are possible; however,
despite this, their prediction capability is impressive.
The GUTs predict that the energy at which
unification should be observed (M_{GUT}c^{2} ~ 10^{15}
GeV) is the point where the energy-dependent interaction coefficients for the
three fields merge to a single value (merging the three fields to a single
unified field at the same time).
However, at this energy, it is believed that gravity is still
insignificant. (A detailed discussion of
the GUT unification can be found in Halzen and Martin
^{[Ref. 16, chapt. 15.7]}).
The step to Super-GUT
has seen numerous approaches (two examples include: super symmetry where each
SM fermion has a super heavy boson partner; and string theory where the
topology of space, the gravity-based description, supports vibrating quantum
strings that represent the SM particles).
Like the GUTs, there is not, as yet, a single self-consistent
Super-GUT. However, there is general
agreement that full unification of all of the fields (i.e., the unified GUT
plus gravity) should be seen at the fundamental Planck scales (i.e., the Planck
mass, m_{p}; the Planck
length, l_{p};
and the Planck time t_{p}).
3.2. The Planck Mass
Recall that at energies up to M_{GUT}c^{2},
gravity is sill negligible; however, it is predicted that as the energy grows
beyond that level, the strength of gravity increases rapidly and ultimately
becomes comparable to the unified GUT force at the Planck energy (m_{p}c^{2} ~ 1x10^{19}GeV). According to quantum-gravity models ^{[Ref.
17, p257]}, the mass of the Planck particle is determined when the
deBroglie wavelength (the quantum wave description) equals the Schwarzschild
radius (the gravity description), that is,
2p .
h-bar
2G
.
m
(2) m_{p} = K . (h-bar . c / G)^{1/2} ~
1.22x10^{22} MeV/c^{2}
This is the mass of the predicted Super-GUT unification. In (2), h-bar
(1.0545887x10^{-}^{34} Js) is Planck's constant divided by 2p, c (2.99792458 x10^{8}) is the speed
of light in a vacuum, and G (6.6726x10^{-}^{11}) is the
empirically determined gravitational constant.
A factor 5.60958494x10^{29} MeV/c^{2}/kg converts units
from one system to the other giving m_{p}
~ 2.18x10^{-}^{8} kg. The leading coefficient, K = \/p, has
been set to unity for the order of magnitude value that is sufficient for the
purposes of most discussions.
Hoyle et.
al. ^{[Ref. 3, chapt 17] ,} however, provide a
different way of estimating m_{p}
that is derived from the cosmological-gravity point-of-view, rather than the
usual quantum-gravity view. They note
that an ambiguity in m_{p}
arises because the quantum mechanical side of (1) (i.e., the left) comes from a
scale-invariant theory; while, the gravitational side (i.e., the right) comes
from General Relativity (GR) ^{[Ref. 18]} which is not scale
invariant.
According to Hoyle, the implication of this
is that under any conformal transformation of the non-scale-invariant theory,
the mass is position dependent. This
position dependence, by implication, would in turn affect a particle's
associated wavelength when compared to its gravitational radius.
To address this difficulty, they developed a scale-invariant
form of the gravity equations that reduces to GR with an appropriate choice of
scale, leading to a unique particle (the Planck particle - to avoid confusion,
the Planck particle derived from the fully scale-invariant theory is referred
to as m_{p}*).
The m_{p}* has the property that its
(2*) m_{p}*
= K . (h-bar . c / G)^{1/2} =
5.96610354x10^{21} MeV/c^{2}
(~1.06x10^{-}^{5 }g) using the same constant values for h-bar, c, and G as in (2). Note that m_{p}* is ~51% smaller than m_{p}.
The concept that m_{p}* is the largest mass that can be supported by the space-time
fabric is applied, by the FRACEP, to a composite version of m_{p}* (referred to here as m_{p}*
') based on G0p. Defining it as:
(3) m_{p}*^{
}'^{ } =
6R44p = 36x9^{44} G0p particles
=
6.02206465x10^{21} MeV/c^{2}
gives a mass ~1% larger than the m_{p}*
mass computed by the scale-invariant theory in (2*). Although it is not a SM fundamental particle
(i.e., a fermion or a boson), incorporating the composite Planck particle into
the FRACEP is used as a method of connecting G0p to the fundamental scales of
nature. Specifically, it is used to
determine the physical size (classical radius) of G0p in the next section.
3.3. The Planck Length
The second fundamental scale is the Planck
length, which is defined as:
(4) l_{p}*
= h-bar / (m_{p}* . c)
= 3.3075190x10^{-}^{35 }m
(This is the same definition as used for the non-scale-invariant
version, l_{p};
except that for l_{p}, m_{p} is used rather than m_{p}* - note also that l_{p}* is ~2 times larger than l_{p}).
The significance of the Planck length can be
found in the concept of quantized space.
In the early 1940s, as described by Jammer ^{[Ref. 19, p240]},
in order to resolve infinities in the quantum field theory development, a
smallest length to space-time was proposed (as opposed to a cut-off
frequency).
The concept of discreetness to space-time was
further elaborated in the early 1960s by appealing to the Heisenberg
uncertainty principle (Dp.Dx ~ h, where Dp is
the particle momentum uncertainty, Dx is its
wave localization uncertainty, and h
is Planck's constant). Wheeler hypothesized
the principle forced the fabric of space-time to behave like a
"fluctuating foam" where the fluctuations became physically
significant on the scale of l_{p}.
It is interesting to note that in string theory,
the fundamental entity, the string, has a length of ~ l_{p}.
This paper does not address the question of quantum fluctuations, but it
does embrace the concept of a smallest length supported by space-time in
defining the size of G0p as R_{class-G0p} = l_{p}*.
3.4. The Planck Time
The third fundamental scale is the Planck
time which is defined as:
(5) t_{p}* = l_{p}*
/ c =
1.1032696x10^{-}^{43
}s
(This is the same definition as used for the non-scale-invariant
version, t_{p}; except that
for t_{p}, l_{p} is used rather than
l_{p}* - note also that t_{p}* is ~2 times larger than t_{p}).
The Planck time sees its importance to
particle physics in "Big Bang" cosmology. In theory, starting at time t = 0 (the bang),
there was an explosive, exponential expansion of the universe, due to the
extreme compression of space, where the thermal energy of matter was greater
than the Planck energy (m_{p}c^{2}). This thermodynamic state prevented particles
from forming as isolated entities.
The "inflationary" expansion caused
cooling until time t = t_{p}*,
when the thermal energy had reduced to the Planck energy. This time, t_{p}*, was the boundary (switch-over) from the inflation
period to the classical expansion we see today.
(The expansion speed is no longer exponential, but has the form v = H_{0 }. d where
H_{0} is the Hubble constant and d is the separation distance of masses
such as stars and galaxies. It is this
distance that is increasing due to the expansion of space.)
After t_{p}*,
with continuing cooling, the universe saw a series of symmetry-breaking events
(times when one-by-one the fundamental fields decoupled from the totally
unified super-GUT field, and particle formation of the fermions and bosons
proceeded). Details of the process can
be found in Povh ^{[Ref. 7, chapt. 19.4]},
Peacock ^{[Ref. 17, chapt. 9]}, and Guth ^{[Ref. 14]}.
The relation of t_{p}*
to the cosmology-based particle formation will not be considered here; however,
the relation of t_{p}* to
fermion and boson instability and decay time is presented in Part 2 ^{[Ref. 20]},
and, to the dynamic properties of the spin effect (Part 4 ^{[Ref. 21] )} and
the charge effect (Part 3 ^{[Ref. 22]}).
There is one final point of interest
regarding the relation of G0p to the fundamental scales. Comparing the values of m_{G0p} and m_{p}* suggests a possible
reciprocal relation. A comparison of 1/m_{p}* with m_{G0p} shows a -2.8% difference. This implies a close reciprocal relationship
between the two values. Further, if the
value of G is increased from 6.6726x10^{-}^{11} to 7.07x10^{-}^{11} (just 6%), the reciprocal mass difference drops to +0.02%. This gives a range for m_{p}* (5.96610354x10^{21 }to 5.79600331x10^{21}). It also gives a range for l_{p}* (3.3075190x10^{-}^{35 }to 3.4045876x10^{-35}), and for t_{p}* (1.1032696x10^{-}^{43 }to 1.1356482x10^{-}^{43}).
It is
intriguing that the smallest (m_{G0p}) and largest (m_{p}*) masses supported by
space have an apparent reciprocal relationship.
It is reminiscent of the super-symmetry pairing of small fermions with
their super partners - one difference being that m_{G0p} and m_{p}* (or its equivalent
composite form m_{p}* ') have no associated spin or charge as
do the fermion-boson pairs. We now
proceed to estimate R_{class} for the IBB's
using R_{class-G0p }and the fermions.
4.0 THE COMPOSITE PARTICLE BUILDING BLOCK SIZES
The structure of the
intermediate building blocks (IBB's) (required components) was determined in
Part 1^{[Ref. 1]}, along with a configuration (component arrangement)
that was show as a planar projection.
Since the particles, whose sizes are being determined, cannot be seen
directly with today's technology, a series of assumptions are made which are
used in the size determination. This
section states those assumptions and then proceeds to estimate the size of the
IBB's.
4.1. Assumptions Used in the Size Determination
The assumptions used in
the size determination affect, not only, the IBB's, but also the composite
fermions. The following assumptions address
the spacing between adjacent particles in groupings, and the particle
configurations that affect how the groupings are seen from the outside.
4.1.1. The Smallest Length Scale of Space Is l_{p}* - This assumption restates the quantum mechanical
picture of discrete space with l_{p}*
as the smallest length.
4.1.2. The Classical Radius of G0p Is R_{class-G0p}
= l_{p}* - This assumption re-states the FRACEP picture that the smallest mass
particle has the smallest physical size.
4.1.3. The Minimum Center-to-Center Spacing for Any
Adjacent Particles in a Built-Up Structure Is the R_{class} of the
Larger Particle - This
assumption is valid for groupings making up the mass carrying particles (GXp’s), the momentum carrying rings (RXp's),
the ring and particle clumps (MRXp's and MGXp's), the spin carriers (SGp),
and the charge carriers (QGp). In the MRXp’s and MGXp’s, the M is the number of RXp
rings clumped together or the number of GXp particles
clumped together.
The assumption also holds for the IBB's and
the components within the composite fermions.
Figure 2 shows the effects of this assumption. For this example, D_{c-to-c }= 5r,
where r is the radius of one particle and 2r is the radius of the other. This gives a separation of 2r between the two
particles because of the radius of the larger particle.
4.1.4. The Components Within a
Built-Up Structure Arrange Themselves in a Roughly Spherical Configuration - This assumption is valid for all
structures with two qualifiers. First,
single ring elements (RXp's - as opposed to clumps) are roughly planar rather
than spherical with a horizontal radius r_{X} and a height of h_{ X (}determined
below in section 4.2.1). Second, the
term "roughly spherical" does not imply a uniform distribution of
particles within the spherical volume except for the clumps (MRXp's or MGXp's
described in section 4.2.4).
4.2. Sizes of the IBB's and Their Components
The IBB's include the
mass carrying particles, the momentum carrying rings, the charge carriers, and
the spin carriers - all of which are used to build up the composite fermions ^{[Ref.
1]}. The basic (lowest order
fractal level) ring element (R0p) and the basic mass element (G1p) are
presented separately first because the higher order structures are built on
them and their configurations and size clarify the higher order size
determinations.
4.2.1. The Basic Ring Element (R0p) Configuration and Size
The R0p is the basic
(lowest fractal order) ring element. It is
composed of six G0p (fundamental) particles arranged in a planar configuration
of radius r_{0} = 3 (normalized by r_{0}) which can be thought of as being the
equatorial plane of a sphere with radius r = r_{0}. (The
r_{0} = l_{p}* is the radius of the G0p particle, and
for simplicity is set to unity (r_{0} = 1) in the normalized
system). All points (particle centers)
on the ring are separated by an internal angle of 60^{o}; and the cord
cutting the circle between any two points has a normalized length of 3 as well
(Figure 3).
In this configuration, any two adjacent
points form an isosceles triangle (equal sides) with the ring center as the
third point. The surface-to-surface
normalized distance between any two particles is d_{s-to-s} = 1 (a
total of 3 center-to-center). The space
occupied by the ring is cylindrical
in shape (Assumption 4) with a radius r_{0} = 4 and a height h_{0} = 2
(from ring center to the outside surface of the particles).
FIGURE
3. This shows the
planar ring configuration for R0p with point coordinates for particle
centers. All lengths are normalized by r_{0}
= l_{p}*
(the radius of the G0p particles making up the ring).
4.2.2. The Basic Mass Element (G1p) Configuration and Size
The G1p is the basic
(lowest fractal order) mass element. It is
composed of an R0p with three additional G0p particles arranged at 120^{o}
apart in the planar projection. However,
the three additional G0p's are located out of the plane and are obtained by
sliding the particles along the surface of the sphere of radius r = 3 to points
that satisfy the center-to-center separation requirement in Assumption 3 (Figure 4).
In this configuration, any two adjacent particles have a
surface-to-surface normalized distance of 3.
The space occupied by the G1p is roughly spherical in shape (Assumption
4) with a radius r = 4 (from sphere center to the outside surface of the
particles). For example, G1p appears to
the next level (G2p) as a spherical particle that is used in the size determination
of G2p.
FIGURE 4. This shows the
G0p particle configuration for G1p with point coordinates for particle
centers. All lengths are normalized by r_{0}
= l_{p}*
(the radius of the G0p particles making up the ring). Note, the particles 8 and 9 are on the sphere
surface and below its equatorial plane.
Particle 7 is on the sphere but above the equatorial plane.
4.2.3. The Higher Order GXp's and RXp's Configurations and
Sizes
The higher levels of GXp and RXp are fractal-like
structures based on the previous levels.
The number of particles in each structure is: GXp
= 9^{X }. G0p's
= 9G(X-1)p's, and RXp =
6GXp's (i.e., R0p = 6G0p's, R1p = 6G1p's, .. ; and G1p = 3G0p's + R0p's, G2p =
3G1p's + R1p's, ..). At each level, the configurations are the
same as shown in Figures 3 and 4 except that the distances increase with each
level (Figure 5). This says that for RXp, the r_{RXp} = r_{G}_{(X+1)p},
and h_{RXp} = 2 . r_{GXp}; and, for GXp the radius is r_{Gx} = 4^{x} . r_{0 }.
4.2.4 Clumps of Higher Order
Rings (MRXp) Configuration and Size
Clumps of the MGXp's and MRXp's are aggregate
groupings of M mass carrying particles (GXp’s) and M
momentum carrying rings (RXp’s) respectively. The clumps form spherical bodies (Assumption
4). Their size is determined as
follows. Suppose the clump in question
is M.GXp where GXp has a radius r_{GXp}
and M is the number of GXp's in the clump.
1) Enclose the GXp in a
cube of side length S1 = 2(r_{GXp} + ½ r_{GXp}). The
½ r_{GXp }assures that the adjacent GXp's in the clump are
separated by at least r_{GXp} (Assumption 3).
2) The small (S1) cubes
are then arranged to fill a larger cube of side length S2 = M^{1/3} S1
where the M^{1/3} is rounded up to the next larger integer (which
represent the number of smaller cubes that make up each side of the large cube.
3) The radius of the
sphere that encloses the large (S2) cube is now the diagonal of the S2 cube
from cube center to any corner. With a
little geometry, it is easy to show, that the diagonal in the horizontal plane
is
d_{h}_{ }= S2 [ (½)^{2} + (½)^{2}
]^{1/2}.
This then gives the cube diagonal from center to corner as r_{d}
= S2 [ d_{h}^{2} + (½)^{2} ]^{1/2}
= 0.866 . S2.
This gives
(6) r_{d} = 0.866 . M^{1/3 .} 3r_{GXp}
which is the radius of the spherical volume that is used in the other IBB's and
fermion size determinations.
4.2.5 The Spin Carrier (SGp) Configuration and Size
Spin in the FRACEP is a
dynamic effect explained more fully in Part 4 ^{[Ref. 21]}. This paper addresses only the size of the
configuration. The spin carrier (SGp)
contains two parts: the spin effect component (S0p) and the spin momentum
component (MSp). The SGp
includes both of these components. The
S0p component is the 16^{th} level of a 5-particle fractal-like
structure. Each element at any level
will be referred to as EX for the discussion here. At the lowest level, E0 = 2G0p's (= G0p - G0p =).
The E1 = 5E0's, E2 = 5E1’s, ...., E16 = 5E15’s
= 5^{16}(2G0p's). Figure 6a
shows the configuration.
The size of S0p is determined much like the
higher level momentum rings. It is a
cylindrical structure, which when carried to higher levels has a radius roughly
r_{X} = 4^{X} . (^{5}/_{2} r_{0}) and
a height h_{x} = 2 . r_{X}_{-}_{1} . However, because r_{X}
and h_{X} are sufficiently close, the spin effect component will be
treated as spherical with radius r_{S0P} ~ 4^{16} . (^{5}/_{2} r_{0}). The spin momentum component MSp is composed of four G16p particles which surround the
S0p. The radius of the full spin carrier
(SGp) is approximated as half of the longest distance across the group,
assuming the surface-to-surface separation between the S0p and any of the four
G16p particles is r_{S0p}, that is:
(7) r_{SGp} = ½ [2r_{G16p}
+ r_{S0p} + 2r_{S0p} + r_{S0p} + 2r_{G16p}]
= 2r_{s0p} + 2r_{G16p}
With r_{0} = 3.3x10^{-35} m, then r_{S0p} =
3.45x10^{-25} m, and r_{G16p} = 1.42x10^{-25} m, the
radius of the spin carrier is approximated as r_{SGP} = 9.92x10^{-25}
m.
4.2.6 The Charge Carrier (QGp) Configuration and Size
Charge in the FRACEP is
a dynamic effect explained more fully in Part 4 ^{[Ref. 22]}. In this paper, only the size of the
configuration is addressed. The charge
carrier (QGp) contains two parts: the charge effect component (Q0p) and the
charge momentum component (MQp). The QGp includes both of these components. The Q0p component contains two charge effect
chains (each are ½ Q0p) where each element of the chains is a pair of G0p's (- G0p = G0p -).
Figure
6. This
shows the configurations for spin carrier and charge carrier. In (a) the fractal pattern of the spin
carrier shown is for some sub-level E where in the total spin carrier, E = 16. In (b) the total charge carrier has two
chains. Each chain has 2x10^{19}
G0p pairs.
There are 4x10^{19} elements in each chain where each element
has the required surface-to-surface separation giving it a length of 6r_{0}
, which makes the total length of a single chain as r_{Q0p} = 6r_{0}
. 4^{19} = 5.5x10^{-23} m. Figure 6a shows the configuration.
The momentum carrying component is MQp = G19p
+ 2G13p + 48R13p + 121R16p. Assuming a
configuration as presented in Part 1^{[Ref. 1]}, and assuming the
component separations are the size of the larger of any two components, the r_{MQp}
equals half of the maximum extent in any direction:
(8) r_{MQp} = ½ [2r_{G19p}
+ r_{G19p} + 2r_{48R13p} + r_{121R16p} + 2r_{R16p}]
= ½ [3x9.07x10^{-24} + 2x6.64x10^{-26}
+ 5.45x10^{-24} + 2x5.89x10^{-24}]
giving the radius of the momentum part of the charge carrier as
approximately r_{MQP} =
2.25x10^{-23} m, and r_{QGP} = 7.75x10^{-23} m.
5.0 THE SIZES OF THE COMPOSITE VERSION OF THE SM
FERMIONS
With the sizes of all of
the components of the FRACEP composite fermions determined, this section
proceeds with the size determination of the fermions themselves. Also, because of its importance to the
fundamental scales, the size of the FRACEP composite version of the Planck
particle is discussed. Note that in the
SM, the Planck particle is considered fundamental in nature though it is not
one of the fermions (or bosons, which are discussed in Part 5 ^{[Ref. 23] )}.
The procedure for size
determination also uses the same philosophy as was used for the components:
i.e., 1) the surface-to-surface separation is the radius of the larger of any
two adjacent particles; and, 2) the fermion components arrange themselves to
occupy a spherical volume (though not necessarily uniformly).
Because of this second assumption and the
fact that the size is intended as a best estimate, the radius of the fermion is
taken as one-half the maximum extent in any direction for the structure based
on the configurations given in Part 1^{[Ref. 1]}. Note that the configuration used is just one
possible configuration where configuration is taken to mean an arrangement of
components.
Table 1 shows the
full structure for each of the fermions where structure refers to the
collection of required components to satisfy the SM data. The components used in the size determination
are indicated in bold. In the following
paragraphs (5.1 thru 5.13), the r_{class} for
each fermion is computed by showing the sums of the r_{class}'s for
each component (indicated Table 1), with the separation indicated as a sized
unit.
The sizes of the relevant GXp's, RXp's and
MRXp's were determined using the expressions and assumptions presented in
section 4 above. And finally, the
particles with negative mass components and anti-particles (e.g., n_{e}_{+}
or QGm) are assumed to have the same radius as their
counter particles (n_{e}_{-} or QGp
respectively). Note that in the maximum
extent calculation, the particle size is twice the radius; and only the
components in the maximum extent are shown.
TABLE 1. This shows the total structure, mass and estimated size of the FRACEP composite fermions. For the large particles, the core mass is the total mass without those momentum that are not required for the decay components. The components in bold (and larger type) represent the longest extent across the structure that are used in the size determination. The size is given in meters, and the mass is in MeV/c^{2}. The maximum expected fermion size estimated by the SM is <10^{-}^{18}m for all fermions without any distinction.^{}
FRACEP
Fermion [mass(MeV/c^{2})] |
FRACEP
Components (with maximum extent components larger and in
bold) |
Estimated
Rc_{lass}(m) |
n_{e}_{-} [1.28e-6] |
SGp |
9.92x10^{-}^{25} |
n_{m-} [0.17]_{} |
n_{e}_{-} + G22p |
8.72x10^{-}^{22} |
e^{-} [0.51] |
2(QGp + n_{e}_{-}+ G22p) + (QGp + n_{e}_{-}+ G22p) |
2.27x10^{-}^{21} |
up (u^{+}) [3.57] |
2(QGm + n_{e}_{+}_{ }+ G22p) + 3R22p + n_{m-} |
1.36x10^{-}^{20} |
down (d^{-}) [5.10] |
u^{+} + [
R22p + (n_{e}_{+} + e-) ] |
1.71x10^{-}^{20} |
mu (m^{-}) 105.66 |
(n_{m-} + 2R22p) + 3R22p + 93R22p + 3R22p + [2
R22p + (n_{e}_{+} + e-) ] |
4.36x10^{-}^{20} |
n_{t-} [15.12] |
n_{m-} + R22p + G24p + G22p |
1.83x10^{-}^{20} |
charm (c^{+}) core [core
= 123.16 tot. = 1250.39] |
2(2R22p + u^{+} + n_{m+ }) + 3R22p + (13R24p + 53R22p -- not part of core) + R22p (u^{-} + n_{m-}_{ }+ 2R22p) +
3R22p + (R22p + R24p + n_{t-}) |
1.14x10^{-}^{19} |
strange (s^{-}) [124.68] |
c^{+} core + [ R22p + (n_{e}_{+} + e-) ] |
1.17x10^{-}^{19} |
tau (t-) [core = 327.27 tot. = 1777.05] |
(2R22p + u+ + n_{m+}) + 5R22p
+ (2R22p + n_{e}_{+} + e-) + 27R22p + 319R19p + 120R16p + 93R22p (u- + n_{m-}_{ }+ 2R22p) +5R22p + (R22p + R24p + n_{t-}) + (17R24p + 45R22p +343R19p -- not part of core) |
1.20x10^{-}^{19} |
top (t^{+}) core [core = 4397.68 tot. = 1.82e+5] |
2(2R24p + c+ + n_{t+}) + 3R22p
+ (25R26p + 57R24p + 12R22p -- not part of core) +
R22p (c- + n_{t-}+ 2R24p) +
3R22p + (R22p + R24p + n_{t-}) |
4.43x10^{-}^{19} |
bottom (b^{-}) [4399.21] |
t^{+} core + [ R22p + (n_{e}_{+} + e-) ] |
4.46x10^{-}^{19} |
5.1. The Electron Neutrino (n_{e}_{-}) -
is the spin carrier computed
in section 4.2.5.
r_{n}_{e}_{-} = r_{SQp}
= 9.92x10^{-25} m.
5.2. The Muon Neutrino (n_{m-}) - is the n_{e}_{-} plus a mass carrier.
r_{nm-} = ½ [2r_{n}_{e}_{-} + (separation +
2r_{G22p})]
= ½ [2x9.92x10^{-25} + 3x5.81x10^{-22 ]} =
8.72x10^{-22 }m.
5.3. The Electron (e^{-}) -
is triangular with 3
charge/spin chains (QGp + n_{e}_{-}+ G22p), and a maximum extent along any side.
r_{e}_{-} = ½ {2x [(2r_{QGp} + sep.) + 2r_{n}_{e}_{-}
+ (sep. + 2r_{G22p})] + sep}
= 3x7.75x10^{-23} + 2x9.92x10^{-25}
+ 3.5x5.81x10^{-22} = 2.27x10^{-21 }m.
5.4. The Up-Quark (u^{+}) - is a central chain with 2 charge/spin chains
at one end and n_{m-} at the other.
r_{u}_{+}
= ½ [(2r_{QGm}+ sep) + 2r_{n}_{e}_{+} + (sep + 2r_{G22p})
+ 3(sep + 2r_{R22p}) + sep + 2r_{nm-}]
= ½ [3x7.75x10-23 + 2x9.92x10-25 +
3x5.81x10-22 + 10x2.32x10^{-21} +
2x8.72x10^{-22}] = 1.36x10^{-20 }m.
5.5. The Down-Quark (d^{-}) - is an u+ connected
to an (e^{-}, n_{e}_{-}) pair by a momentum ring.
r_{d}_{-} = ½ [2r_{u+} + (sep
+ 2r_{R22p})]
= ½ [2x1.36x10^{-20} + 3x2.32x10^{-21}]
= 1.71x10^{-20 }m.
5.6. The Muon - is a central pair of R22p chains connecting a ring
clump at one end, and, component groups off the other end of each chain. Note, the muon is
more massive and larger that the tau neutrino that follows because this list
retains the order (based on number of decay paths) presented in Part 1^{[Ref.
1]}.
r_{m-} = ½ [2r_{nm-} + 5(sep + 2r_{R22p}) + sep + 2r_{93R22p}]
= ½ [2x8.72x10^{-22} + 16x2.23x10^{-21}
+ 2x2.49x10^{-20}] = 4.36x10^{-20 }m.
5.7. The Tau Neutrino (n_{t-}) - is the n_{m-} connected to a mass carrier by a momentum
ring.
r_{nt-} = ½ [(2r_{G24p} + sep)
+ (2r_{R22p} + sep) + 2r_{nm-}
= ½ [3x9.29x10^{-21} +
3x2.32x10^{-21} + 2x8.72x10^{-22}] = 1.83x10^{-20 }m.
5.8. The Charm-Quark (u^{+})- is a central pair of R22p chains that terminate at both ends of each chain
by neutrinos, and, with up quarks and momentum ring clumps attached to the R22p
chains.
r_{c}_{+} = ½ [2r_{nm-} + 6(sep + 2r_{R22p})
+ (sep. + 2r_{R24p} + sep.) + 2 r_{nt-}
= ½ [2x1.83x10^{-20} + 6x3x2.32x10^{-21}
+ 4x3.70x10^{-20} +2x8.72x10^{-22}]
= 1.14x10^{-19 }m.
5.9. The Strange-Quark (s^{-}) - is an c+ connected
to an (e^{-}, n_{e}_{-}) pair by a momentum ring.
r_{s}_{-} = ½ [2r_{c}_{+} + (sep
+ r_{R22p})]
= ½ [2x1.14x10^{-19} + 3x2.32x10^{-21}
= 1.17x10^{-19 }m.
5.10. The Tau (t^{-})- is a central pair of R22p chains that
terminate at both ends of each chain by neutrinos, and, with up-quarks and
momentum ring clumps attached to the R22p chains.
r_{t-}
= ½ [2r_{nm-} + 8(sep
+ 2r_{R22p}) + (sep. + 2r_{R24p} + sep.) + 2 r_{nt- }
= ½ [2x1.83x10^{-20} +
8x3x2.32x10^{-21} + 4x3.70x10^{-20} + 2x8.72x10^{-20}]
= 1.20x10^{-19
}m.
5.11. The Top-Quark (t^{+})- is a central pair of R22p chains that
terminate at both ends of each chain by tau-neutrinos, and, with charm quarks
and momentum ring clumps attached to the R22p chains.
r_{c}_{+} = ½ [2r_{nt-} + (2(sep + 2r_{R24p})
+ sep.) + (2(2r_{R22p} + sep.) + 2r_{R22p})
+ (2(sep + 2r_{R24p} ) + sep.) + 2 r_{nt- }
= ½ [2x8.72x10^{-20} +
7x3.70x10^{-20} + 8x2.32x10^{-21} + 7x3.70x10^{-20}
+ 2x8.72x10^{-20}] =
4.43x10^{-19 }m.
5.12. The Bottom-Quark
b^{-}
= t^{+} + R22p
r_{b}_{-} = ½ [2r_{t}_{+} + (sep. + r_{R22p})]
= ½ [2x4.43x10^{-19} +
3x2.32x10^{-21}] = 4.46x10^{-19 }m.
5.13. The Planck Particle
Finally, although the
Planck particle is not a fermion, and it is a hypothesized particle in the SM,
because of its relevance to the fundamental scales discussed above, a
discussion of its size is made for completeness. As stated in paragraph "a" of the
Fundamental Scales section above, the Planck particle is hypothesized as the
largest possible mass supported by the fabric of space-time as described in
Hoyle ^{[Ref. 3]}. (Hoyle, of
course does no describe the Planck particle as a composite as does the FRACEP.) As such, it can be assumed to form in the FRACEP
as a composite particle.
In determining its size, the
surface-to-surface separations are specified as in sections 4.1.3 and 4.1.4
above. It is assumed to have the
configuration specified in the Fundamental Scales section (i.e., 6R44p). The size of such a particle would have a
radius r_{P} = 2.13x10^{-}^{7} m.
However, it is hypothesized that once formed, the Planck particle would
collapse (just as a black hole collapses of its own weight once energy
production within ceases). It is assumed
that, in the collapsing process, the final state occurs when the G0p's in the
Planck particle are close packed but not degenerate. That means that the ring structure is
collapsed due to the large mass but the G0p's are not tunneling into each other
as one models fundamental matter in a fully collapsed black hole.
At this point the collapsed radius is r_{P}_{-collapsed} = 1.08x10^{-}^{20 }m. The final tunneling by the
G0p's (down to the r_{P-final} = l_{p}*, the Planck length) is considered here as
destroying the integrity and nature of the fundamental particles (the G0p's) which
would represent disruption of the space-time fabric. This would resulting in a creation event -
the beginning a new cycle in the build-up of previously non-existing matter -
analogous to Hoyle's creation event.
6.0 CONCLUSIONS
One of the important
aspects of a composite structure in fermions is size. In Part 1 ^{[Ref. 1]}, a composite
structure was determined for the fermions that agreed with the SM data in mass,
spin, em charge, and decay components.
This paper shows that the FRACEP composite structure (with the stated
assumptions) is consistent with yet another characteristic of the SM, that is,
the maximum expected size of the fermions.
There is a difference between the SM and the FRACEP
on the question of size. The SM
estimates the R_{class} of the fermions to be <10^{-}^{18} m; but, because the fermions are considered fundamental, the SM is
unable to differentiate the sizes of the many particles except in a quantum
mechanical sense, the l_{C}.
The FRACEP, on the other hand, shows a
variation in R_{class} with mass as would be expected, because it
builds up the particles based on their mass and structure. This size-mass variation is seen in nature at
the macro scales (e.g., larger mass galaxies are larger in size that smaller
mass stars). It is also evident at the
atomic scales (uranium is larger than carbon is larger than hydrogen). So it is to be expected that the R_{class}
of a neutrino should be smaller than that of the larger mass quarks. The FRACEP shows sizes that range from
9.92x10^{-25} m for the smallest, the electron neutrino, to 4.46x10^{-19
}m for the largest, the bottom quark; and, all of the composite FRACEP
fermions are less that the stated SM estimate.
In addition to the fermions, the R_{class}
of G0p (the FRACEP fundamental particle with mass m_{G0p} = +1.72x10^{-}^{22 }+1.04x10^{-}^{31}^{ }MeV/c^{2},
where the limits of variability in m_{G0p} that still allow it to
satisfy the SM observations within their measurement uncertainty) was
established by linking it to the fundamental scales of nature. The natural extension of the mass-size
variation to this smallest possible particle, and, showing the larger
structures built up from it satisfy the SM observation on maximum size, lends
validity to the choice of R_{class-G0p} = l_{p}* = 3.3x10^{-}^{35} m (the smallest possible length scale).
One final observation can be made about the FRACEP
construction and a possible explanation of the dual nature of particles in the
quantum world. The dual nature (both
particle-like and wave-like behavior) of particles on the quantum scales was
discussed in section 2.
The particle nature on these scales has long
been assumed as a natural extension from the world we see and understand. But, very early on in the development of
quantum mechanics, it became obvious that these same particles also acted like
waves. Because the SM treats fermions as
fundamental particles (i.e., homogeneous, uniform and indivisible), there is
not practical way to explain the observed diffraction that is characteristic of
waves. The FRACEP, on the other hand,
may offer an explanation
DeBroglie reflected on the particle-wave duality at
the quantum scales, noting that even a simple water wave is granular at the
atomic level. By that he meant that
water waves are composed of the coordinated motion of a horde of water
molecules. We can consider this
situation further as we try to understand the duality issue.
On the macro scale, there are two behaviors
noticeable in water waves. The usual
wave considered by deBroglie can be pictured as a collection
of water molecules moving together for some distance. But over time the molecules begin to separate
and spread out the wave structure until the height of the wave is finally
leveled. This spreading is due to
dispersion.
However, there is another kind of wave that
is observed. It is referred to as a soliton. In that
kind of wave, the motion of the molecules (in a perfect medium) remains
coordinated. This coordinated motion
continues because the dispersion is counteracted by the nonlinear propertied of
the medium ^{[Ref. 24]}. In this
condition, the soliton wave takes on particle-like
properties. This same behavior is
observed on the macro scale in light ^{[Ref. 25, -27]}.
It is possible that this same dual behavior
is exhibited in the fermions because their composite structure. As modeled in the FRACEP, the fermions are
composed of an extremely large number of G0p particles that make up their
components. In one mode, you would get
dispersive wave behavior indicated by interference patterns. In the second mode, you would get the soliton particle-like behavior. However, the details of this possible
explanation are a line of inquiry left to future work.
The successful
validation, to date, in the comparison of the FRACEP with the SM fermion observations
is encouraging, though additional work to further support the composite picture
is underway, including: the explanation of the charge effect ^{[Ref. 22]},
the explanation of the spin effect ^{[Ref. 21]}, and the description of
the potential and the decay properties of the composite structures ^{[Ref.
20]}.
7.0 REFERENCES
^{1 }J.A. Giannini, "The FRACEP Model, Part 1a: A Look Inside The Elementary Particles of the Standard Model", in preparation
^{2 }P. Moore, The Data Book of Astronomy,
(2000).
^{3 }F. Hoyle, G. Burbidge, J.V. Narlikai,
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Press, Cambridge, UK (2000).
^{4 }S. Majid,
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^{5 }P. Tipler,
Foundations of Modern Physics, Worth
Pub.
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^{6 }Brezger, et. al., Phys. Rev Letters, 88, 100404, 2002
^{7 }B. Povh,
K. Rith, C. Scholz, F. Zetsche, Particles
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^{8} T.K. Lim,
J.A. Giannini, "Separable Expansion Method for Potential Scattering
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^{9 }I. Newton, Philosophiae Naturalis
Principia Mathematica (1686), Principia, University of California
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^{10 }J.C. Maxwell, A
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^{11 }S.Weinberg, "A Model Of Leptons", Phys. Rev Letters, V19, pp. 1261-6,
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^{12 }G.^{ }'tHofft,
Renormalization of Massless Yang-Mills Fields", Nuclear Physics, VB33, pp.
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^{13 }D.J. Gross and F. Wilczek, "Ultraviolet Behavior of
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Ref. 16, chapt 10.
^{14 }A. Guth, Inflationary
Universe, Perseus Books,
^{15} C.A. Bertulani, Nuclear Physics in a Nutshell, Princeton
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^{16} F. Halzen, A.D. Martin, Quarks & Leptons, John Wiley &
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^{17 }J.A.^{ }Peacock,
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^{18 }G. Rindler, Essential Relativity, VanNostrand
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^{19 }M. Jammer, Concepts Of Space,
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^{20 }J.A. Giannini, "The FRACEP
Model, Part 2: Why Is There No Evidence Of Internal Structure In Electrons?", in progress.
^{21 }J.A. Giannini, "The FRACEP Model, Part 4: What Is the Spin Effect?", in progress
^{22 }J.A. Giannini, "The FRACEP Model, Part 3: What Is the Charge Effect?", in progress
^{23 }J.A. Giannini, "The FRACEP
Model, Part 5: What Is The Structure Of The Bosons And Field Mechanisms?", in progress.
^{24} J.A. Giannini, J.S. Hansen,
L.W. Hart, "Experimental Measurements
of Temporal Phase Shifts During Soliton
Wave-Wave Interactions," presentation at 35th Annual Meeting Am. Phys.
Soc. - Div. Fluid Dyn., New Brunswick, NJ, November 1982.
^{25} J.A. Giannini and R.I. Joseph,
"The Propagation of Bright and Dark Solitons in Lossy Optical Fibers," IEEE J. Quantum Elec., 26,
(1990) 2109.
^{26} J.A. Giannini, "The
Propagation of Bright and Dark, Spatial and Temporal Solitons
in Nonlinear Optical Materials," Doctoral Dissertation, Johns Hopkins
University, (1991).
^{27} J.A. Giannini and R.I.
Joseph, "Propagation in Cylindrically Symmetric Two-Dimensional Nonlinear
Media," Phys. Lett. A., 160 (1991) 363.
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