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.
2.1. The Macro World Size Description
2.2. The Quantum World Size Description
3.1. Historical Basis of the Fundamental Scales
3.2. The Planck Mass
3.3. The Planck Length
3.4. The Planck Time
4.1. Assumptions Used in the Size Determination
4.2. Sizes of the IBB's and Their Components
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 (Rclass), 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.
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 (Rclass 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 (mh ~ 90.0 kg ~ 5x1031 MeV/c2) and is ~ 6 ft (~1.8 m) tall. (This approximately cylindrical person is roughly equivalent in volume to a sphere of radius rclass-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 rclass-e ~ 6.4x106 m, and a mass me ~ 6.0x1024 kg (~3.4x1054 MeV/c2). Its atmosphere has numerous layers of gases and charged particles, and is very rarefied with a thickness on the order of rclass-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 rclass-s ~ 7.0x108 m and a mass ms ~ 2.0x1030 kg (~1x1060 MeV/c2). Its chromosphere is thin compared to rclass-s (~ 0.007 x rclass-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 rclass-MW ~ 5x104 lt-yrs (~4.7x1020 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, ~3x108 m/s x 365.25 days/year x 86,400 seconds/day = 9.5x1015 m.)
The Milky Way mass is mMW ~ 1.7x1071 kg, though higher and lower estimates can be found. Its central core has a radius r class-c-MW ~ 7.5x103 lt-yrs (~7.1x1019 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 mc-MW ~ 3.6x1045 kg (~2x1066 MeV/c2).
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 rclass-U ~ 1025 m and a mass about mU ~ 1080 kg (~8x1079 MeV/c2 [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 (RSW = 2Gm/c2 as indicated in Figure1).
Only matter around the black hole that is outside its RSW is visible. Within the RSW, 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 RSW is well within the observable surface.) (In general, the RSW is not so much an independent measure of size, as it is the lowest possible limit of Rclass 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, Rclass 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, lC = hc/mc2), 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 (lC) and it is the point where wave behavior becomes obvious.
In addition to lC, however, there is still the classical radius, Rclass, 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 Rclass 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, Rclass-electron ~ 1/1.8 = 0.56 fm (5.6x10-16 m). (Note that lC-electron is ~3.86x10-13 m which is larger than its Rclass). 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 Rclass's. For comparison, the proton is Rclass-proton ~ 8.42x10-16 m and a carbon atom nucleus is Rclass-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 lC is the wave-like size description; and 2) the Rclass is the particle-like size description. Even though the SM concentrates on the lC description, the FRACEP is focusing on the Rclass 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 Rclass to be greater than the RSW (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 RSW (its event horizon), and the Planck particle (which is hypothesized but not observed) whose size is predicted to be lp*.
The s-shaped curve shows a nonlinear relation (on the log-log scale) of Rclass to mass. The mid region (from about the sun down to C12) is roughly linear with curved tails in the extremes. The dashed curve was obtained from a fit of the form R = F1.M2 + F2.M + F3 where Fi = Ai + Bi tanh[ 0.125 . (M - 34.0)]. The M = log10(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 = log10(radius). The mass is in MeV/c2 and the radius is in meters. The plotted values are log10(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 Rclass < lp* (the point where the Planck mass's lC and RSW 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 Rclass-G0p = lp*, 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 log10(mass) of the indicated particles (with log10 (Mass), log10(Rclass) indicated). The diagonal line, lc, is the particle's quantum description (left side of equation 1); the diagonal line, Rsw, is lowest possible limit for the particle's macro description (right side of equation 1). The s-shaped curve, Rclass, is a fit thru the observed classical radius points (D), with the Rclass-G0p indicated (Z).
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 (Rclass-G0p) is established, by the FRACEP, as lp* , 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/c2), 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 (10-15m), 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 (MGUTc2 ~ 1015 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, mp; the Planck length, lp; and the Planck time tp).
3.2. The Planck Mass
Recall that at energies up to MGUTc2, 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 (mpc2 ~ 1x1019GeV). 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