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Chapter XI: The UG Theory and Deep Inelastic Scattering (DIS)

Deep inelastic scattering experiments (DIS) provided the initial motivation for the development of a hadron constituent model. Deep inelastic electron-proton scattering experiments of the 1960s1 revealed a surprising weak fall-off of the deep inelastic cross sections with increasing and exhibited a scaling behavior previously predicted by Bjorken (Bjorken, 1969). Both observations strongly suggested the existence of a substructure in protons and neutrons. A preceding constituent model proposed by Richard Feynman offered a simple dynamical interpretation of the DIS results. Feynman postulated partons as the elementary constituents of hadrons, yet specified their identity only as far as basic particles relating under the strong interaction. Further development linked the parton model with the concept of quarks. Quarks, which were proposed independently by Gell-Mann (Gell-Mann, 1964) and Zweig as mathematically convenient building blocks of unitary symmetry, were the natural candidates for these elementary constituents. The early constituent quark model faced significant obstacles. To begin with, quarks had never been observed directly. The fact that quarks were never observed should have been explained by very strong interactions in their final state. The theory, however, required that the constituents of hadrons behave as free particles during virtual photon absorption. In addition, experimental results demanded significantly more than three constituents within the nucleon substructure. Resolving these, as well as additional inconsistencies, led to enhancements in the quark model and to the development of quantum chromodynamics (QCD). This resulted in a powerful, yet significantly more complex theory, postulating the additional concepts of gluons, colors, sea of quarks, quark confinement and the property of asymptotic freedom, with heavy reliance on renormalization techniques.

The quark and QCD models seem to provide adequate explanations for all available experimental data. Therefore, a choice has been made to first establish the credibility of the UG theory in the cosmic realm, where inconsistencies within the current theory are well documented. In this chapter, the UG theory will be employed on nuclear scale. It will be shown that the UG zonal structure generated by the interaction between two massive particles creates a dynamic substructure that naturally leads to the observed large-angle scattering, as well as to the weak fall-off of the DIS cross section, and provides a scaling phenomenon that resembles Bjorken scaling. Since the substructure is due to the interaction between a pair of particles, rather than due to constituent sub-particles, the UG theory does not require the assumption of quarks as real elementary particles.2

Since the acceptance of quantum chromodynamics in the early 1970s, the quark structure of hadrons has become the dominant framework for theory development and experimental design. As a result, a significant portion of data collected from deep inelastic scattering has generally been analyzed within this framework, and reflects its embedded assumptions. Some parameters commonly used in data presentation are

Equation 11-1


and

Equation 11-2


where denotes the four-momentum transfer function, represents the energy of the incident lepton, represents the energy of the scattered lepton, provides the energy loss of the scattered lepton, or , indicates the scattering angle, is the mass of the nucleon, is the Bjorken scale variable, provides the fraction of the leptonís energy loss in the rest frame of the nucleon, and is the speed of light. Reported experimental data commonly provides the values of the nucleon structure functions as functions of some of the above parameters, as demonstrated in figure 11-7. According to the quark-QCD model, in the case of a high-energy DIS mediated by virtual photon exchange, the structure functions relate to the double differential cross section via the equation (see Devenish & Cooper-Sarkar, 2004)

Equation 11-3


where is the fine structure constant. The experimental data is often presented in the form of the nucleon structure functions and after being subjected to a series of data processing routines, such as radiative corrections and Monte Carlo simulations, which are heavily dependent on the assumptions of the QCD model. The incorporation of these underlying assumptions within the data complicates the ability to apply the data to a new theory with an entirely different set of assumptions. Therefore, the following discussion will present evidence that the UG theory predicts DIS behavior using a related, yet not identical set of parameters.

According to the QCD model, the strong force acts between quarks and is mediated by gluons, while lepton-nucleon interactions are viewed as an electromagnetic interaction between charged leptons and charged quarks within the nucleons. Therefore, according to quantum chromodynamics, the fundamental process in DIS experiments is the electromagnetic scattering of two spin- point-like particles. In contrast, the UG model holds that there is no fundamental difference in the interactions between two nucleons, two leptons, or interactions involving a lepton and a nucleon. All of the above particles have a charge,3 a mass and a spin, and are assumed to interact with each other in accordance with the theory of unified gravitation and the electromagnetic theory.4 According to the proposed UG model, the large scattering angles observed in high-energy collisions between a nucleon and a lepton are mainly attributed to the UG zone structure between the two colliding particles, rather than to electromagnetic interactions between the lepton and a hadron with a substructure composed of several elementary constituents. Historically, the quark-parton model (QPM) grew out of an attempt to provide a simple model to explain the results of the early deep inelastic scattering experiments, where the function was discovered to be independent of at . QPM assumes that the nucleon consists of non-interacting point-like particles which serve as scattering centers. In contrast, according to the UG model, the UG gravitational field produced by the interaction between a rapidly approaching particle and a nucleon is viewed by the particle as an Ďonioní with an infinite number of repulsive layers separated by attractive layers, where the layers consist of the potential energy maxima or minima respectively. It will further be demonstrated that due to relativistic spacetime distortion, the shape and size of the Ďonioní and its layers are strongly affected by the momentary speed of the approaching particle, and may change drastically as the speed of the particle is reduced near the point of closest approach during a DIS event. The number of layers penetrated by the electron will be shown to depend mainly on the initial energy of the probing particle and on its impact parameter. Therefore, the idea of scattering by a sea of quarks is replaced by the concept of scattering via some of an infinite number of UG zones, where the zonal indices that contribute to the scattering are mainly determined by the impact parameter and by the overall energy of the probing particle.

The following analysis deals with a DIS collision between a nucleon and an electron. Note that the discussion is not limited to a collision between an electron and a nucleon, and may be extended to cover collisions between any two leptons, two nucleons, a lepton and a nucleon, or between any two massive particles. However, the discussion is limited to cases where the probing particle survives the collision.


Section XI-1-1: The Effect of the Velocity of Colliding Particles on Their UG Interaction


Based on the second UG postulate, equation 2-1-1 is valid when the interaction between a nucleon and an electron is viewed in the rest frame of the nucleon. For simplicity, the origin of the frame is set to coincide with the location of the nucleon. According to special relativity, the rest mass of the electron and the distance should be replaced by and respectively,5 where is the velocity of the electron relative to the nucleon and is the distance component that is parallel to . Figure 11-1 provides the initial geometry of an electron-nucleon scattering, where a highly relativistic electron with the initial velocity of and an impact parameter of approaches the nucleon. The contribution of the nucleon to the UG potential energy of the electron is given by6

Equation 11-4


where is the rest mass of the nucleon and when the axis is selected to be perpendicular to the plane containing the vectors and . Note that at non-relativistic velocities, or when the vectors and are perpendicular, coincides with the distance between the electron and the nucleon. At relativistic velocities, the true distance between these two particles is less than , and is further dependent on the magnitude and the direction of the relative velocity . According to special relativity, only the distance component that is parallel to the relative velocity is contracted. This parallel component is given by

Equation 11-5


Therefore, the variable (or ) should be considered as a parameter of the electronís trajectory, rather than the distance between the electron and the nucleon. In cases where the velocities are relativistic (and therefore, may not be equal to the actual distance as viewed in the inertial frame of the nucleon) will be referred to as the parametric distance, while the actual distance viewed in the nucleonís inertial frame of reference will be referred to as the apparent distance. At distances where , , and equation 11-4 reduces to

Equation 11-6




Figure 11-1a: An electron with an impact parameter of B approaches the nucleon at a velocity that corresponds with in the direction parallel to the axis. At this speed, the potential energy due to the UG force is provided in cyan and the electromagnetic force is provided in red for the case where the nucleon is a proton. The UG pattern (cyan) remains stable as long as the electron is located at sufficient distance from the nucleon and its velocity is about constant. As the electron approaches the nucleon, its UG potential energy increases abruptly to equal the overall energy of the electron (at about ), causing the electron to decelerate. As the electronís velocity is reduced, the UG pattern and the Coulomb potential energy collapse toward the proton, demonstrating the folding of the zonal structure described in section XI-2-2.



Figure 11-1b: Provides the contributions of the UG and the electromagnetic potential energy patterns as perceived by the electron after its speed is reduced to non-relativistic velocity, where .


Of special interest is the point of closest approach of the electron to the nucleon, measured in the particle acceleratorís frame of reference. As is the closest point on the electronís trajectory toward the nucleon, its parallel component must be equal to zero, and the perpendicular component must be equal to . Therefore, in the vicinity of the point of closest approach, equation 11-4 reduces to

Equation 11-7


At non-relativistic electron velocities, equation 11-4 becomes independent of and reduces to7

Equation 11-8


It is important to realize that if the velocity of the electron remains relativistic throughout its entire journey, the scattering angle would be relatively small, and the scattering of the electron by the nucleon would not be highly inelastic (for example, see the trajectory of the electron in figure 11-3b, indicated in black). On the other extreme, in cases of deep inelastic scattering at the limit where the energy lost by the electron and , the velocity of the electron in the vicinity of the point of impact (or the point of closest approach), denoted , must become negligible, thus (see the electron trajectories in figure 11-3b, indicated in blue and green).8 Due to their importance, events that provide extremely high values of and will be classified as Ďhigh-loss deep inelastic events.í

The maxima and minima of the electronís potential energy can be derived via . At non-relativistic velocities (which is the case in the immediate vicinity of the electronís point of closest approach during a high-loss DIS event), and the potential energy minima and maxima occur only when



Equation 11-9a


Therefore, at the maxima or minima, , or

Equation 11-9b


At distances where the velocity of the electron is relativistic, the distortion of spacetime may introduce dependency on through the velocity terms, and may modify the dependency of the potential energy on the parametric distance as well. As the velocity has no component along the inclination angle, does not depend on , and therefore . At distances , the dependency of on the azimuth angle becomes insignificant, leading to approximately and . Therefore, at , . However, at the maxima or minima of the potential energy, , and the force and acceleration must be equal to zero. Zero acceleration leads to , resulting in . Consequently, the maxima or minima of at parametric distances of , where the velocity of the electron is relativistic, comply approximately with

Equation 11-10a


Therefore, at the maxima or minima, , or

Equation 11-10b


where for the case of . Note that at non-relativistic velocities, equations 11-10a and b reduce to equations 10-9a and 10-9b respectively. Similarly, in the case where is perpendicular to the relativistic velocity of the electron,

Equation 11-11


Therefore, in the case where is perpendicular to the relativistic velocity of the electron,

maxima or minima occur when


where and , or

Equation 11-12


where . Note, however, that at distances less than or of the order of the electronís impact parameter , the azimuth angle starts to vary substantially along the trajectory of the electron. Consequently, at these small distances, equations 11-10 to 11-12 are not entirely accurate for relativistic electron velocities, while 11-9a and 11-9b remain accurate at non-relativistic velocities.


Section XI-1-2: The Effect of Quantum Mechanics on DIS


Before diving deeper into a UG analysis of deep inelastic scattering, it is important to determine the highest level of precision that can be achieved in calculating how close the electron came to the nucleon. The natural tendency is to assume that the quantum uncertainty principle restricts the level of precision of this distance. This is the case, according to quantum mechanics, when the forces acting between the particles are Newtonís gravitational force and/or the electromagnetic force. However, this is not the case for the exponential UG force.

As discussed above, the velocity of the probing electron of a high-loss deep inelastic event (an event where and ) should become non-relativistic at the point of closest approach . Given a high-loss DIS event, is located between the minimum and the maximum of the most external zone indexed with potential energy maximum higher than the local overall energy of the electron. The UG potential energy of the electron at the maximum can therefore be calculated by substituting equation 11-9b in equation 11-8, providing

Equation 11-1-1


Given the high level of energy required for deep inelastic scattering, the term is negligible relative to the exponent, thus

Equation 11-1-2

or

Equation 11-1-3a


where . Maxima occur when the integer is odd, thus . Therefore, the potential energy at the nearest maximum is approximately given by

Equation 11-1-3b

In the case of a neutron, the electromagnetic interaction between the probing electron and the neutron is zero. In the case of a proton, the electromagnetic potential energy at the point of closest approach is given approximately by the Coulomb equation at the nearest maximum indexed ,

Equation 11-4

which can be reduced to at non-relativistic electron velocities.

is a monotonic function of the parametric distance and does not have any maxima or minima. Consequently, the value of at the maximum or minimum of the UG potential energy in the case of a non-relativistic electron velocity is given by

Equation 11-1-5


The uncertainty principle restricts the certainty of simultaneously measuring the electronís momentum and position via the inequality , where denotes the coordinates of , or . The uncertainty of the location of the electron can be approximated by , or . A similar calculation will yield . Therefore, the amount of kinetic energy required by the uncertainty principle is given by



At the maxima and minima of ,

Equation 11-1-6


In figure 11-2 the absolute value of the UG gravitational potential energy of the electron of equation 11-1-2 is displayed in blue, the absolute value of the electromagnetic potential of the electron of equation 11-1-5 is displayed in red, and the absolute value of the uncertainty of the amount of energy due to the uncertainty principle of equation 11-1-6 is provided in violet, as functions of the zonal index .9



Figure 11-2


As demonstrated in the figure, below the UG force is virtually zero, and is thus negligible relative to the Coulomb force in the case where the interacting nucleon is a proton.10 Moreover, throughout the range of distances associated with the maxima and minima to , the uncertainty principle forces the energies to become so high that any effort to accurately localize the electron in relation to the nucleon is doomed to fail. However, due to the rapid exponential growth of the UG interaction, the UG potential energy dominates over the Coulomb interaction at , and over the kinetic energy associated with the uncertainty principle at , which is roughly associated with energy levels above . At energies higher than or of the order of (for ), quantum effects become negligible and the scattering can be localized and viewed using classical physics methodology, thus making it possible to locate the zone of closest approach in spite of the uncertainty principle.11 Consequently, classical (non-quantum) methodology should provide a reasonable approximation and will be used here for the analysis of high-energy DIS events.


Section XI-2-1: The Effect of Special Relativity on the Range of the UG Interaction


According to equation 11-8 for the non-relativistic case , the strength of the UG interaction between two nucleons, or between a nucleon and an electron at a distance of is negligible. Therefore, the effective range of the UG force on the nuclear scale, for the case of non-relativistic electrons, is less than . Consequently, a non-relativistic electron of an impact parameter of would demonstrate exclusively an electromagnetic interaction with a proton, and virtually no interaction with a neutron. Under these conditions, the interaction is too weak to create substantial inelastic scattering. According to equation 2-1-1 and 2-1-2, at non-relativistic velocities the UG interaction is comprised of a central force that is independent of the speed of the electron relative to the nucleon. However, as shown in equations 11-4, 11-6 and 11-7, at relativistic velocities the UG potential energy is indirectly dependent on the electronís velocity due to relativistic spacetime distortions and due to relativistic massĖenergy equivalence. Therefore, at parametric distances where a low-velocity electron is anticipated to remain unaffected by its UG interaction with the nucleon, a relativistic electron may be significantly influenced. Hence, the effective range of the UG interaction may depend heavily on the relative velocity of the electron.

Consider an electron moving toward a nucleon with a kinetic energy of at a sufficiently large distance that the overall potential energy of the electron is negligible.12 The Lorentz factor of the electron before the impact with the nucleon is therefore given by

Equation 11-2-1

, or (or approximately


In the case of highly relativistic electrons, the effect of the electromagnetic interaction on the value of the Lorentz factor is relatively small, even for the case of interaction with a proton. According to equation 11-5, at distances of the velocity of the electron is virtually parallel to the vector. Therefore, the distance between the electron and the proton measured in the rest frame of the proton is given by .13 According to equation 11-6, the distance at which the UG interaction is about equal to the electromagnetic interaction is given by

Equation 11-2-2a

or

Equation 11-2-2b


The equation may be solved for by either a titration technique, or via graphical calculator. For the case of a non-relativistic electron (), the solution of equation 11-2-2b provides

Equation 11-2-2c

or


Therefore, at distances of less than , the UG interaction between a proton and a non-relativistic electron (relative to the proton) becomes stronger than the electromagnetic interaction between them, yet becomes weaker than their electromagnetic interaction at distances greater than . The same calculation shows that the range of the UG dominance is extended by a factor of about 2,172 to in the case of a relativistic electron, where .


Section XI-2-2: DIS and the Relativistic Effect of the Folding and Unfolding of the Particlesí Zonal Structure


In high-energy head on collisions, a large amount of energy is converted to potential energy by the electron as it encounters the point of closest approach with the nucleon. During this process, the momentum and the velocity of the electron relative to the particle acceleratorís frame of reference are drastically reduced. To preserve the overall momentum, the nucleonís momentum, and therefore velocity, must increase significantly. However, since the nucleon is strongly bonded to other nucleons within the atomic nucleus, and the atom itself is bonded to other atoms or molecules, a substantial portion of the transferred energy and momentum are dispersed to the surrounding nucleons and molecules. These losses of energy and momentum, as well as additional loss of energy and momentum in the form of radiation, result in an inelastic scattering, where the final energy of the electron (as viewed in either the reference frame of the particle accelerator or of the nucleon) reduces significantly. The amount of change in the energy and momentum of the electron depends strongly on its initial energy and momentum, and on its impact parameter. The explosive growth of the exponential term of equation 11-4 as a function of at distances of , as well as the sharp increase in the frequency of the oscillations of the cosine term as , cause the trajectories of the probing electrons to behave in a chaotic manner. This behavior appears to be random, as a minute difference in an electronís initial impact parameter or velocity may result in drastically different scattering angles, as well as in a different final momentum and energy. Therefore, although the overall magnitude of the electromagnetic interaction in high-energy deep inelastic scattering is very small compared with the UG interaction, the electromagnetic effect may be significant in such a chaotic system, where a tiny perturbation may result in an entirely different trajectory. Consequently, regardless of the fact that the initial energy and momentum of all of the electrons in the beam are virtually identical, the trajectories of different electrons may vary considerably. Thus, the point of closest approach between different electrons, and their amount of energy and momentum at these points, may vary substantially. In experiments using particle accelerators the velocities of the probing electrons within a beam are all virtually equal. Therefore, the most influential factor in determining the characteristics of the scattering event is the impact parameter of the electron, and to a lesser degree, whether the nucleon is a proton or a neutron.

In cases where the impact parameter of the electron is sufficiently small, the electron will approach the first impenetrable maximum barrier, where the electronís potential energy at its peak exceeds the overall energy of the electron. This outer maximum is expected to have an odd integer index, denoted as the maximum. As the overall energy of the electron is insufficient to penetrate the barrier, the electron will either be deflected to the side or scattered backward by the barrier, in either case demonstrating a large scattering angle and a substantial loss of energy. In high-energy DIS collisions, the rate of reduction of the UG amplitude as a function of an increasing parametric distance is high, and the slopes of the barrier at high zonal indices are extremely steep. Consequently, the electronís energy must have been greater than the maxima of most of the preceding zonal maxima encountered prior to the point of closest approach, and the electronís trajectory should have been minimally affected by the majority of these preceding zones. Therefore, the electron will be scattered by the maximum zone in the same manner as it would be scattered by a hard object of the same approximate shape as the contour of the zone barrier, where the electronís kinetic energy is equal to zero.

The UG zonal structure generated by the interaction between the nucleon and the electron consists of an infinite series of concentric barriers around the nucleon, where the height and density of the barriers approach infinity as . At non-relativistic velocities, these concentric barriers are spherical and their distance from the nucleon is independent of the electronís velocity. However, in the case of relativistic electron velocities (relative to the nucleon), the shape of the barriers begins to resemble a series of elongated ellipses, where their major axes are aligned perpendicular to the velocity of the electron. The heights and locations of these barriers, which are fixed in the case of non-relativistic electron velocities, become highly dependent on the speed of the electron at relativistic electron velocities.

According to equation 11-5, at parametric distances , and . Therefore, the potential energy of the electron is given by equation 11-6, and the coordinates of the potential energy maxima (at odd ) and minima (at even ) are given by 11-10b, or , where . For a high-loss deep inelastic scattering event with a large deflection angle to occur, the electronís potential energy at the barrier maximum must match or exceed its overall energy at the point of impact.14 The equations may be simplified by defining the function . Since the lowest possible value for the Lorentz factor is , , and as the electron is expected to lose energy in a DIS event, is typically less than . Denote as the electric charge of the nucleon and define as the amount of energy lost by the electron by the time it arrived at the parametric distance (when viewed in the rest frame of the nucleon).15 Using equation 11-6 for the UG potential energy, the overall energy of the electron at any given parametric distance along its trajectory is given by

Equation 11-2-4


Where, the small contribution of the particlesí spin is neglected.

At DIS events, the UG maxima must be located at sufficiently short distances, where the Coulomb term is negligible relative to the UG potential energy. Furthermore, at such high energies the term within the parentheses is insignificant compared with the large exponent term required for DIS and can be dropped.

In cases where the energy of the electron on impact is just short of being equal to its potential energy at the zonal maximum, as viewed in the rest frame of the nucleon, we can substitute (where is an odd positive integer) in equation 11-2-4 to provide

Equation 11-2-5


Multiplying both sides by and then taking the log of both sides of the equation yields

Equation 11-2-6


The point of closest approach, denoted as , generally occurs in the case of DIS somewhere within the zone, but is assumed for the moment to occur very close to the zonal maximum. Therefore, at this point in space and time, and . Consequently, provides the actual distance between the electron and the nucleon, as viewed in the rest frame of the nucleon. In the general case, the electronís velocity may be substantial, or even relativistic, and the resultant electron scattering would not be highly inelastic. However, as discussed, for a DIS event to occur where and , the electron should become non-relativistic at the point of closest approach , with and . Therefore, , , and .

As almost all of the energy of the electron is delivered to the nucleon, while a significant portion of the nucleonís energy dissipates into the network of surrounding nucleons, atoms and molecules, the electronís loss of energy must be substantial. The energy loss may cause the electron to stop its forward motion toward the nuclei earlier, at a more external repulsive zone, and may consequently lower the value of . Substituting the above equalities in equation 11-2-6 yields


Equation 11-2-7



For example, provided that , the first maximum contour at which the potential energy of the electron exceeds its initial overall energy is indexed . Note that energy losses may reduce to an odd . Intuition would suggest that as the electron was stopped by the maximum, it should have passed through all minima and maxima of , and could not have reached any of the minima and maxima of index . This will not be shown to be the case, however. While the electron indeed had to pass all minima and maxima of , it will be shown that the electron also had to pass maxima and minima between and .

At parametric distances , where the potential energy of the electron is still small relative to its overall energy, ; hence is still approximately . Therefore, no substantial energy loss has yet occurred, and is negligible, leading to . As , , and consequently, . Following equation 11-2-6,

Equation 11-2-8


For the example of , at the point where or, the calculated index of the last encountered maximum is given by and respectively, which are obviously much larger than the index of .

Therefore, before being stopped by the nucleon at the maximum, the electron managed to pass substantially more than zones. The question arises as to how it is possible that an electron in motion towards a nucleon, after successfully passing through zones, ended up being stopped by a maximum barrier of an index as low as . After all, the electron could not have reached the zone without first passing through all of the maxima. Additionally, the electronís potential energy at the maximum should be substantially higher than its potential energy at the maximum. Therefore, it seems odd that the electron could be stopped by a low energy maximum barrier after having passed a substantially higher energy barrier earlier.

The amplitude ratio between any two successive maxima (or minima) of indices and can be calculated by substituting and of equation 11-10bin equation 11-6. As long as the overall energy of the electron is substantially greater than the local potential energy maxima , or as long as the electronís velocity is non-relativistic (therefore, , changes only slightly between two successive maxima, and . Therefore, provided that at high-energy collisions , the ratio of two successive maxima or minima peaks is given by

Equation 11-2-9

or

Equation 11-2-10

or

Equation 11-2-11



With the example used above, where , the ratio in regions where is given by , compared with in regions where the electron moves at non-relativistic velocities. The ratio of the electronís potential energy between the maximum viewed by the electron at close to zero velocity (where ) and the maximum viewed at is given by . Therefore, the potential energy of the non-relativistic electron at the time it was stopped by the maximum substantially exceeds the potential energy of the relativistic electron of at the maximum. This explains how an electron can have a sufficient amount of energy to pass the maximum barrier, yet not enough energy to pass the earlier maximum barrier.

However, the above explanation does not account for how an electron that just passed the zonal maximum indexed 135,115 can encounter the same maximum for a second time while still moving toward the nucleon. The explanation is as follows: While the parametric distance is reduced monotonically as the electron approaches the nucleon, the apparent distance of the electron from the nucleon fluctuates wildly, at times increasing and at times decreasing with the reduction of . As it moves through the zones toward the nucleon, the electron accelerates when approaching a potential energy minimum and decelerates as it approaches a maximum. When approaching a potential energy maximum, and decrease. Consequently, the apparent distance between the electron and the nucleon (as viewed by the nucleon), as well as the potential energy ratio between two successive maxima (or minima), increases. The opposite occurs when the electron approaches a minimum, in which case and increase, and the apparent distance between the electron and the nucleon (as viewed by the nucleon), as well as the potential energy ratio between two successive maxima or minima, decreases. In effect, as demonstrated by figures 11-1a and 11-1b, the entire zonal pattern folds in when the velocity of the electron is decreased, causing the patternís maxima and minima to move inward toward the nucleon. Conversely, the pattern unfolds when the velocity of the electron is increased, as the maxima and the minima move outward, away from the nucleon. The speed of folding or unfolding depends on the rate of change of the electronís Lorentz factor . Early on, when the distance between the electron and the nucleon is relatively large and the UG interaction is quite weak, the relative velocity of the electron is about constant, and the rate of folding and unfolding of the UG pattern is negligible. In a DIS event, as the electron approaches the point of closest approach the maxima and the minima fluctuations become enormous, causing related fluctuations in the value of . The size and rate of these fluctuations determine the speed and the extent of the folding and unfolding of the pattern. The speeds of the few folds preceding become large and may exceed the velocity of the electron. Hence, some of the maxima already passed by the electron may fold or unfold at a significantly faster rate than the speed of the electron, thus overtaking the electron. It is therefore feasible, and even expected, that in a DIS event the electron will encounter the same maximum or minimum multiple times as it moves toward the nucleon.


Section XI-3: The Weak Fall-off of the DIS Cross Section


Prior to calculating the approximate UG equation that describes the weak fall-off of the DIS cross section, the underlying principle responsible for this phenomenon can be demonstrated by the set of figures 11-3a to 11-3d. Again, consider a high-loss DIS event at the limit and . As discussed, in such an event the electron is scattered by the outermost impenetrable zone indexed in the same manner as it would have been scattered by hard object. However, the velocity of the electron becomes non-relativistic at the point of impact. At non-relativistic velocities the zone structure is spherically symmetric around the nucleon, and therefore depends on the absolute value of the distance between the electron and the nucleon, . In addition, at non-relativistic speeds, when , the ratio between two successive potential energy maxima was shown by equation 11-2-11 to be approximately 13.59. The UG potential energy waveform, indicated in violet in figure 11-3a, is displayed as a function of the distance between the electron and the nucleon along an arbitrary axis within the plane defined by the electronís velocity vector and the distance vector between the electron and the nucleon. The green concentric circles provide the UG potential energy maxima contours within this plane. Note that while the potential energy of successive maxima increases by a factor of 13.59, the distance between them decreases proportionally to (see equation 2-1-16), where denotes the zonal index. Therefore, the UG slopes rapidly become extremely steep.16



Figure 11-3a


The overall energy of the electron illustrated in figure 11-3a is lower than its potential energy at the maximum. As the electron passes the maximum, its overall energy is significantly larger than its potential energy. Note that the electron retains most of its kinetic energy; therefore only a small portion of its energy can be transferred to the nucleon, and subsequently dispersed to nearby nucleons, atoms and molecules. When the impact parameter of the electron is larger than , the electron will not reach the maximum , and the amount of energy lost is relatively small. As a result, the scattering event is only slightly inelastic. At an impact parameter of , the electron passes the higher potential energy maximum , and a greater portion of its kinetic energy is transferred directly to the nucleon and indirectly to nearby nucleons, atoms and molecules. Consequently, the scattering becomes increasingly inelastic. At an impact parameter of , the electron proceeds toward the maximum. Since the electronís potential energy at this maximum exceeds its overall energy, the electronís motion toward the nucleon is halted as it reaches the point at which its potential energy is equal its overall energy, at .17 As the electron does not have sufficient energy to continue to move toward the nucleon, it is scattered in the same manner as it would have been scattered by a hard object. Note that as the overall energy of the electron at the impact point is equal to its potential energy, the velocity of the electron relative to the nucleon is reduced to and its Lorentz factor is reduced to . Since the electron becomes non-relativistic on impact, the UG zones viewed in the particle acceleratorís frame of reference appear spherically symmetric around the nucleon, and the hard scattering object becomes a hard scattering sphere of radius . Hence, the high-loss DIS cross section becomes approximately equal to . Note that the higher the energy of the probing electron, the smaller the radius of the hard sphere becomes.

Adhering to equation 11-2-11, in a DIS event, in regions where the electron moves at non-relativistic velocities, the ratio between the potential energy of an electron at two successive maxima is equal to 13.59. Increasing the energy by a factor of will therefore reduce the DIS cross section by a factor of

when . Therefore, as increases, the DIS cross section is reduced. However, as increases, the DIS cross section fall-off becomes exceedingly weak. This phenomenon is demonstrated in figures 11-3b to 11-3d.

Figure 3-11b depicts three electrons with the same energy and different impact parameters. The solid sphere indicates the impenetrable hard sphere at the given level of energy, while the concentric circles indicate the radii of the lower external maxima, which can be penetrated as the potential energy of the electron at the maxima is lower than the overall energy of the probing electrons. In particular, the electronís potential energy at the surface of the hard sphere is roughly given by . As displayed, the two electrons with the smaller impact parameters (indicated by blue and green trajectories) hit the hard sphere and are deflected by it. The electron with the largest impact parameter (indicated by a black trajectory) passes near the nucleon without colliding with the hard sphere, with only slight elastic deflection by the lower indexed maxima and minima. Increasing the energy of the electrons at the impact point () by a factor of (to) results in the reduction of the radius of the hard sphere to , where neither of the outer two electrons (black and green) collide with the hard sphere.18



Figure 11-3b



Figure 11-3c


In figure 11-3d the impact energy of the electrons is further increased to about , resulting in the reduction of the radius of the hard sphere to . As shown in the figure, at high zonal indices the spherical maxima become compacted together to the extent that increasing the electronsí energy by a large factor results in little reduction in the radii of the hard sphere, thus diminishing the rate of reduction of the DIS cross section.



Figure 11-3d


Recall that close to the impact point where the electronís velocity is no longer relativistic, the effect of any zone is smaller than the Ďhard sphereí effect of the zone by a factor of at least . Thus, the effect of any zone can be regarded as negligible.19 The trajectory of the electron can consequently be expected to be deflected slightly by the maxima and , and by the minima and , before colliding with the hard sphere of a radius greater than , where is the radius of the maximum contour. In summary, the highest DIS losses occur when the electronís impact parameter is sufficiently small. As a result, the electron is scattered as it would have been scattered by a hard sphere, where the radius of the sphereis equal to the distance of the point of closest approach . On the other hand, in cases where the impact parameter of the electron is sufficiently large, the energy of the electron at the point of closest approach (located at zone is higher than the maximum potential energy of zone , thus . In this scenario, the electron will not collide with the hard sphere and its velocity at the point of closest approach may still be relativistic. As the potential energy of the electron reduces drastically with its distance from the nucleon, the scattering becomes more elastic with an increased impact parameter.

Armed with the above insights, the task at hand is to apply the same principles and approximations for the development of a more precise mathematical description of the weak fall-off of the DIS cross section. At the point of closest approach (viewed in the frame of reference of the particle accelerator), the electronís velocity is either zero or perpendicular to and ). Therefore, the dynamics of the interaction at that point are determined by equations 11-7 and 11-12. However, the electronís energy at the point of closest approach is not observable. To simplify the analysis, it is assumed that (or alternatively, ) is a monotonic function of the electronís initial energy and its final energy (or ), and that its behavior is consistent with20 and, on average, , , and .

Expanding to the first Taylor term,

Equation 11-3-1


where is a constant.

The index of the maximum of the zone containing will be denoted as . Consider the specific case where the parametric distance between the nucleon and the point of closest approach is infinitesimally larger than the radius of the maximum contour, . In this case, the potential energy of the electron in the nucleonís frame of reference is given by

Equation 11-3-2

and the maxima occur when


Equation 11-3-3


where . Substituting of equation 11-3-3 in equation 11-3-2 yields



Equation 11-3-4

In cases of deep inelastic scattering, , and at a UG maximum, is an odd integer. Therefore,

Equation 11-3-5


To simplify the equation, define

Equation 11-3-6


Therefore, equation 11-3-5 can be stated as

Equation 11-3-7


or, as of equations 11-3-3 and 11-3-7,

Equation 11-3-8



As discussed above, when the kinetic energy of an electron is substantial at , the influence of the nucleon on the electronís trajectory will be relatively small, resulting in either an elastic scattering, or in a weak inelastic scattering. For a substantial DIS event to take place at the limits of and , the kinetic energy of the electron at should be minimal, leading to and . Therefore, in the case of a substantial DIS event, the parametric (and apparent) distance of is given by

Equation 11-3-9


However, in the usual case of a high-loss deep inelastic scattering event, the electron will not collide directly with the peak of the maximum, and will instead reach somewhere between , where .21 As discussed above, for a substantial inelastic event to occur, the electron must become non-relativistic on impact with , somewhere within the zone or at the tip of the maximum. As may be anywhere within the range of , the maximum potential energy at the maximum is of the order of . For the general maxima, the maximum potential energy at (at non-relativistic velocities) for an integer is given by. Therefore, in the case of deep inelastic scattering, only the contributions of the preceding two or three maxima have a significant effect. This is in agreement with experimental results, where 1 to 3 resonances are generally found in graphs of the double differential cross section as function of energy (or vs. energy lost, ).22

To calculate the approximate scattering cross section, we begin once again with a high-loss DIS event where the impact energy of the electron is slightly lower than and its impact parameter is sufficiently small to produce hard sphere scattering. Therefore, is slightly larger than . In this case, the maximum contour cannot be penetrated by the probing electron, which can only reach as deep as . As the electronís velocity is already reduced to a non-relativistic speed near the point of closest approach, the shape of the maximum is perceived by the electron to be spherically symmetric.

As discussed, in the case of a DIS event, the sharp steepness of the UG exponent as a function of distance is so explosive near the point of closest approach that the electronís course and direction of motion remain largely unaffected by most of the UG maxima and minima encountered, with the exception of the last few extremely narrow zones between and . Therefore, the interaction distance and the interaction time associated with DIS events, during which the electronís overall energy and direction of motion have been significantly altered, are extremely short. For this reason, although the electronís direction of motion may have changed substantially, the effect of this change on its impact parameter is minimal. Consequently, all electrons with an impact parameter of and an impact energy of will scatter in the same manner as they would have been scattered by a hard sphere of radius . Electrons of the same range of energy, with impact parameters within the range (as calculated for non-relativistic electrons) may undergo a weak inelastic scattering, while electrons with impact parameters of will undergo an elastic scattering, or will remain unaffected.

As the cross section of a hard sphere of radius is given by , the deep inelastic scattering event for the case of an impact energy of just under is given by

Equation 11-3-10


where is less than yet almost equal to .2324 Similarly, if the energy is just under , where and),

Equation 11-3-11


In a DIS event, where , . Therefore,

Equation 11-3-12


and consequently, . Based on the above assumptions, the case of leads to , and therefore to , and subsequently to while . For the general case of an electron with an impact energy of

Equation 11-3-13


As is a continuous function of , there is always a radius where and . At an impact parameter of less than , an electron with an impact energy of will be scattered by the zone containing the maximum in the same manner as it would have been scattered by a hard sphere of radius . Therefore, at the limits of and , the cross section of the DIS event must comply with

Equation 11-3-14


Therefore, according to equations 11-3-11 and 11-3-12, for (or ),

Equation 11-3-15


In order to comply with equations 11-3-13, 11-3-14 and 11-3-15, must comply with

Equation 11-3-16


Consequently, equation 11-3-16 provides a weak fall-off of relative to the fall-off demonstrated by their elastic scattering, which is proportional to . Note that this general behavior is consistent with experimental observations. However, is not observable. Using the approximation given in equation 11-3-1 of , where both and are observables, provides



which also provides a similar weak fall-off of .


Section XI-4: Unified Gravitation and the Phenomenon of Bjorken Scaling


Bjorken scaling, demonstrated in figure 11-7, is the observed high-loss DIS behavior where and when and . As discussed, high-loss DIS events adhering to and occur when the electron is scattered by a hard sphere of radius , where the velocity of the electron at the impact point of yields .

From the UG perspective, scaling becomes apparent when the UG force dominates over the electromagnetic interaction and over the range of energy affected by the uncertainty principle. Therefore, according to section XI-1-2, .

The physical underlying principle that leads to the observed Bjorken scaling is demonstrated in figures 11-4a to 11-4e. The illustration in figure 11-4a provides a tilted three dimensional image of the UG potential energy. The illustration displays only three consecutive maxima and one visible minimum out of an infinite set of maxima and minima in the vicinity of the maximum.

Figure 11-4b provides an illustration of the three dimensional geometry of the deep inelastic scattering of the electron by the zone at , where the blue area resides on the plane and is the energy of the electron on impact (at ). The electronís energy is thus given by , where is small relative to . Consequently, the electron contains sufficient energy to pass the maximum, but not enough energy to penetrate the zone. The DIS cross section is equal to the circular area contained within the outer circumference defined by the intersection of the three dimensional UG surface with the blue plane. At a higher electron energy, where is just below the maximum , as shown in figure 11-4c, the electron collides with the zone at a point slightly external to and the DIS cross section is slightly reduced. Note that as the energy of the probing electrons changes continuously between (approximately between the energy levels shown in figure 11-4b and 11-4c), the DIS cross section reduces continuously, and becomes almost a linear function of the electronís energy level. However, as the energy becomes infinitesimally larger than , the electron has sufficient energy to pass the maximum. As the electron passes through , its potential energy begins to fall drastically, and its velocity increases as its potential energy is converted to kinetic energy. After crossing the minimum, the electronís potential once again increases until colliding with the zone containing the higher maximum at the point locatedbetween . Consequently, the DIS cross section is reduced by more than . Therefore, as the energy of the electron in the vicinity of changes from being infinitesimally lower than the maximum to infinitesimally higher than , the DIS cross section reduces abruptly, resulting in a discontinuity. Such a transition is demonstrated in figure 11-4d. As the electronís impact energy increases continuously between , the cross section will reduce continuously until the energy of the electron becomes equal to , where another discontinuity occurs. Hence, a discontinuity is expected to occur when the energy of the electron at the point of impact is equal to its potential energy at any given non-relativistic UG maximum.

It will be shown that the reduction of the DIS cross section can be described to depend on two independent parameters. The first parameter is the zone index that determines the electronís impact energy levels at which discontinuities in the DIS cross section occur, as well as the DIS cross section values on either side of each discontinuity. The second parameter controls the approximately linear relationship between the DIS cross section and the electronís energy at the point of closest approach, within the range of energies lying between two successive maxima. Note that at high zonal indices the slopes of the UG potential energy become extremely steep (as demonstrated by figure 11-3a), and the difference between the radii of two successive maxima becomes negligible.

At the energy range between the two maxima there is no notable change in the cross section or in the radius of the hard sphere, consequently creating an effect similar to Bjorken scaling, where the cross section depends on a single parameter (in this case, the potential energy maximum index). However, at low zonal indices the slopes are less steep, and the difference between the radii of two successive maxima is significant. Therefore, in the energy range between the two maxima, the radius and cross section of the hard sphere change notably as functions of the electronís impact energy , thereby creating a visible break in scaling at low DIS energies.

The above principles can be applied for the development of a more detailed mathematical description of these scaling behaviors. As stated, Bjorken scaling is the observed high-loss DIS behavior where and when and . Bear in mind that these high-loss DIS events occur when the electron is scattered by a hard sphere of radius , where the velocity of the electron at the impact point of yields . As is located within the zone, the electronís velocity either reduces to a non-relativistic velocity of , or alternatively, the overall energy of

the electron as it passes through the maximum is slightly higher than its potential energy with .25

Consider the case where the electronís impact energy is just under the potential energy of at the maximum. Therefore, an electron with a sufficiently large impact parameter of avoids collision with an impenetrable hard sphere, as demonstrated in black in figure 11-3b, or in black and green in figure 11-3c.



The absolute value of the potential energy of the electron at any of the remaining maxima and minima of is significantly lower than . Therefore, throughout its trajectory the electron retains a substantial portion of its kinetic energy with significantly larger than , and undergoes an elastic scattering with a relatively small scattering angle. However, electrons with an impact parameter of collide with an impenetrable hard sphere with a radius of , as indicated in blue and green in figure 11-3b, or in blue in figure 11-3c. The potential energy of a non-relativistic electron at the maximum indexed is equal to , where is an integer greater than zero. Hence, at least 92.6% of the electronís energy at the maximum, and over 99% of the electronís energy at indices of or lower must have been in the form of kinetic energy when encountered by the electron. Consequently, the effect of the maxima and minima of on the electronís path is relatively small. In addition, the electron retains the majority of its velocity and kinetic energy when it encounters any maximum of , while the velocity and kinetic energy of the electron may increase as it encounters a minimum of .26 Therefore, the electron is abruptly halted within the zone at about , leading to . Consequently, the cross section of electrons with an impact energy of , where is equal to

Equation 11-4-1

Bjorken scaling should take place at distances where the effects of the electromagnetic force and the uncertainty principle are negligible. Note that the effects of the electromagnetic force and the uncertainty principle were shown to become insignificant relative to the UG force at . Since the following holds true:

Equation 11-4-2


Similarly, for the case where is just under the potential energy of the maximum,

Equation 11-4-3


Defining provides

Equation 11-4-4a


The impact energy of the electron can therefore be fully defined by the variables and . Hence, can be expressed as .27

In the case of an electron with an impact energy level falling anywhere between its potential energy at the maximum and the maximum, is given by

Equation 11-4-4b

where

Equation 11-4-4c


In such cases, an electron with an initial impact parameter of will have sufficient energy to penetrate the zone and to reach the zone. As the maximum and the minimum are not negligible relative to , the electron will be deflected as it passes the maximum, and again as it passes the minimum. These deflections influence the energy and momentum of the electron, and therefore its direction of propagation. However, as the combined width of these zones is extremely narrow, the deflections exert little effect on the actual location where the electron encounters the zone. Therefore, whether or not the given electron collides with a hard sphere depends mainly on the electronís initial impact parameter .28 With , the electron does not have sufficient energy to penetrate the zone, and consequently, in the case of , the electron is deflected by a hard sphere and its forward motion toward the nucleon is halted at the impact point. In the specific case of a high-loss DIS event, where the energy lost by the electron and , the scattering angle approaches and the electronís relative velocity at the point of closest approach (or impact point) becomes about zero, leading to . Note that it was taken into consideration that the rest mass of the electron as well as its nearly diminished kinetic energy at the impact point are negligible relative to the impact energy . Since the electron is scattered by a nearly hard sphere of radius , its cross section is given by . Consequently, the minimum value of is at the limit , and as the distance between the point of closest approach and the nucleon converges toward . On the other hand, the maximal value of , denoted , is associated with the minimum energy with which the electron can penetrate the zone as it approaches and recedes from the impact point on its way to the detector. This energy value is significantly more difficult to calculate for several reasons. First, the velocity of the electron as it encounters the maximum is unknown, and may be relativistic, and consequently may affect the height and the perceived location of the maximum. In addition, the amount of energy lost by the electron during the time it spent between the distances of the maximum and is substantial, thus the energy of the electron as it passes through the maximum is uncertain. However, it is clear that , where is the radius of the maximum, as perceived by a non-relativistic electron.

In the case of an electron with an impact energy of , where , and an impact parameter of , the electron will escape the zone without colliding with its impenetrable hard sphere. The electron will then re-enter zone on its way toward the detector, while retaining a non-zero and possibly relativistic velocity at the point of closest approach within the zone. At an impact parameter just above , the scattering may still be significantly inelastic, however, as the value of approaches , the scattering becomes less inelastic.

Consequently, DIS cross sections demonstrate discontinuity at energy levels that match the potential energy of the zonal maxima. In the case of a close encounter between the electron and the maximum, the cross section reduces abruptly from to . As the electronís impact energy continues to increase between and , the cross section decreases gradually toward . The next discontinuity occurs at the energy , where the cross section reduces abruptly from to . This pattern of discontinuities, or scaling gaps, depicted in Figure 11-5 will be shown to occur, and to provide a scaling phenomenon that is similar to Bjorken scaling.

As discussed, since is a continuous function of the parametric distance , for any compliant with , there is a distance at which , where . Therefore,

Equation 11-4-5


where the value of depends on the values of and . For a high-loss deep inelastic scattering to take place, the potential energy at the point of closest approach must comply with and . The zero crossing of the potential energy of a non-relativistic electron preceding can be denoted as . Thus, , where . At the impact point , where , the slope of the potential energy must be negative. Further note that must be positive for the electron to escape the nucleon and reach the detector. Therefore, the range of distances between can be ruled out as possible distances for scattering by a hard sphere. Consequently, the value of is restricted to the range of . This creates a discontinuity in the cross section function, as the cross section of is given by , while the DIS cross section of an infinitesimally higher impact energy results in , where . These discontinuities (or gaps), denoted as , are given by

Equation 11-4-6

or

Equation 11-4-7


Recall that the velocity of the electron upon being scattered by the higher potential energy barrier is non-relativistic. Hence, the above inequalities can also be estimated for the non-relativistic electron,

Equation 11-4-8


As ,

Equation 11-4-9

and

Equation 11-4-10



Figure 11-5:
Provides an illustration of the DIS cross section of particles such as electrons by a nucleon (note that for the purpose of clarity, the mass of the scattered particle used in this example is slightly larger than mass of an electron, allowing for the clear identification of more than three maxima in the graph). The figure further includes the curve of , which is proportional to the hard sphere cross section of a zone. The multiplication factor is included in order to allow the hard sphere cross section to be displayed in the same figure, along the zone energies. Discontinuity in the cross section can be noted at every maximum, creating cross section Ďscaling gapsí between the zones and generating a phenomenon similar to the observed Bjorken scaling. The degree of curvature of the cross section between any two successive discontinuities becomes increasingly insignificant as and reduce, and as increases (where the zone maximum index, given by an odd integer , is analogous to , where is the Bjorken scale variable). This increase in the cross section curvature at low k indices (low values) explains the break in scaling at low values of .

As ,

Equation 11-4-11

as


Therefore, the width of the discontinuity in the cross section is consistent with , which converges to


as .


Consequently, deep inelastic scattering results in a series of discontinuities that resemble the behavior of Bjorken scaling. The next step is to determine the behavior of the double differential cross section as a function of and the zonal index .

Knowledge of the value of is required for the calculation of the cross section scaling pattern. The approach taken here is to approximate by a linear function lying between the potential energy of a non-relativistic electron at the maximum , located at , and the preceding zero crossing , located at . Note that the selection of the non-relativistic values of and is driven by the realization that for a DIS event to occur, the velocity of the electron at the impact location of should be close to zero. The slope of this function is given by

Equation 11-4-12


Therefore, the distance at which a particle with an impact energy of will encounter a potential energy equal to is given approximately by

Equation 11-4-13

or

Equation 11-4-14


At a minimum, cannot be lower than . Otherwise, the electron would remain trapped between the maximum and the maximum, and would not reach the detector. Solving specifically for the minimum at , where , provides

Equation 11-4-15

According to equation 11-8, non-relativistic zero crossings occur where . In a high-energy DIS experiment the UG exponent must be very large. Subsequently, . Thus, . Defining and using equations 11-4-15, 11-2-11 and 11-9b (with at both and , as well as the above expression for , yields

Equation 11-4-16

or

Equation 11-4-17

and

Equation 11-4-18

when is just above

In the general case, in which the electronís impact energy in the rest frame of the nucleon is within the range , the cross section (using equation 11-4-14) is given approximately by

Equation 11-4-19

or

Equation 11-4-20

or

Equation 11-4-21


Substituting equation 11-1-2, where is an odd integer, into equation 11-4-21 yields

Equation 11-4-22

Under the condition where , equation 11-1-3b at the maxima and provides

Equation 11-4-23


Substituting equations 11-4-23 and 11-4-4a in equation 11-4-22 yields

Equation 11-4-24


Substituting the estimated values , and the given masses of the electron and proton into equation 11-4-24, with the realization that at close to a scattering angle the velocity at the point of closest approach is non-relativistic, results in

Equation 11-4-25


where according to equation 11-4-4c, . Figures 11-6a and 11-6b illustrate the resulting scaling behavior by displaying the Ďhard sphereí cross section as a function of . Each waveform is associated with the inelastic scattering produced by electron collisions with a given zone, where is an odd integer.


Figure 11-6a: Display of the UG DISdouble differential cross section scaling, calculated via equation 11-4-25, due to repulsive zones above . Note that within the provided range of energies, where , the effect of the UG scaling may be masked by the much larger electromagnetic interaction, as well as by the effect of the uncertainty principle. Nevertheless, the figure is displayed in order to effectively demonstrate the scaling break that occurs at relatively low values.


Figure 11-6b: Display of the calculated double differential cross section of a UG deep inelastic scattering in the range between and . Across most of this range of energy, the UG interaction dominates over the effects of the electromagnetic interaction, and the uncertainty principle. The displayed cross section provides clear scaling properties, as well as a break in the scaling at low values. The scaling pattern in figures 11-6a and b bears a resemblance to the observed Bjorken scaling pattern of the form factor provided in Figure 11-7 within the range of, where an odd is analogous to the Bjorken variable and the factor is analogous to .

As demonstrated in equation 11-4-25, and in figures 11-6a and 11-6b, the dependency of the DIS double differential cross section on is relatively weak, and therefore provides a scaling pattern similar to the observed Bjorken scaling. Similar to Bjorken scaling, the calculated scaling of equation 11-4-25 displays a break in the scaling at low values of , in which the cross section is increased with increasing . As anticipated, an increase in (signifying that the scattering becomes more inelastic) corresponds with an increase in , while an increase in corresponds with an increase in .29

Using equations 11-3-16 and 11-4-4a, the scaling and the break in the scaling at low values are given by

Equation 11-4-26


where .


Figure 11-7:
The deuteron structure function measured in electromagnetic scattering of electrons (SLAC) and muons (BCDMS, E665, NMC) on a fixed target, shown as a function of for bins of fixed . Statistical and systematic errors added in quadrature are shown. For the purpose of plotting, has been multiplied by , where is the number of the bin, ranging from () to (). References: H1-C. Adolff et al., Eur. Phys. J. C21, 33 (2001); C. Adolff et al., Eur. Phys. J. hep-ex/0304003; ZEUS-S. Chekanov et al., Eur. Phys. J. C21,443 (2001); BCDMS-A.C. Benvenuti et al., Phys. Lett. B223, 485 (1989); E665-M.R. Adams et al., Phys Rev. D54, 3006 (1996); NMC-M. Arneodo et al., Nucl. Phys. B483, 3 (97); SLAC-L.W. Whitlow et. Al., Phys. Lett. B282, 475 (1992). (Source: ď16. Structure FunctionsĒ by B. Foster, A.D. Martin and M.G. Vincter, citation: W. M. Yao et al., Journal of Physics G 33, 1 (2006) available on the PDG WWW pages (URL: http;//pdg.1bl.gov/) July 2006.)


Equation 11-4-26 demonstrates that when for odd ,
is almost independent of (thus resulting in the scaling phenomenon). At lower odd indices, where is not negligible compared with , declines with decreasing , generating the break in scaling observed for low (or low ) values. In either case, the DIS cross section fall-off with decreasing (or increasing and decreasing) is significantly weaker than the elastic scattering (which falls off proportionally to ). As stated above, this weaker fall-off provided by the UG theory is consistent with reported experimental data in the range of .







1 at the Stanford Linear Accelerator Center (SLAC)

2 Note, however, that the UG theory does not necessarily contradict the existence of quarks as elementary constituents of hadrons.


3 The electrical charge is equal to or , where denotes the charge of a proton.

4 Note that this book does not deal with the weak force, which is viewed by the standard model as a separate force.

5 To calculate , , where contracts to , . Thus, , or .

6 Note that in the general case, the electronís potential energy depends on the vectors and. However, in cases where it is clear that either or are relatively insignificant, the notation or may be used. For example, see equations 11-6 for the case of , or equation 11-7 for the case of ). In cases of non-relativistic velocity, where becomes independent of , the notations or may be used.

7 Note that at , or when is perpendicular to , the potential energy is reduced to depend on and (or . At shorter distances, where approaches the impact parameter, becomes dependent on , and . In the non-relativistic case, depends exclusively on . Therefore, the notations and will be used interchangeably throughout this chapter.

8 The logic behind this statement is as follows: The initial energy of the electrons is fixed by the experimental setup. increases as increases and as . The energy loss of the scattered electron is not directly dependent on the scattering angle , but increases with decreasing . Thus, increasing will increase and decrease , and vice versa. Therefore, other than setting the highest possible value of the electronsí initial energy , high values of both and can be achieved when is within a specific range of energy values, and when the scattering angle is as close as possible to. With this scattering geometry, the electron and the nucleon nearly undergo a head on collision, where the relative velocity of the electron nearly reduces to zero at the point of closest approach . Hence, . As will be discussed later on in the chapter, the electron in such a case is effectively scattered by a hard sphere.

9 Using equation 11-9b (or equation 11-12, where ) leads to .

10 Note that the distance between the electron and the nucleon increases with decreasing .

11 Note that this analysis does not imply that the electromagnetic interaction and the uncertainty principle are completely negligible at , as they may still bear some influence on the zero intersections of the gravitational potential energy . In addition, the electromagnetic force may pull the electron closer to the nucleon, thereby reducing its impact parameter. As previously noted, the impact parameter plays a crucial role in determining the outcome of the DIS event.

12 Note that the rest mass energy of the electron (of about ) is relatively negligible.

13 Note that at DIS events the nucleon demonstrates substantial acceleration. Consequently, its rest frame can be inertial for only a brief period of time.

14 Note that in deep inelastic scattering, the energy associated with the electronís rest mass is negligible relative to its overall energy and therefore can be neglected.

15 As the nucleon is accelerated significantly during the DIS event, its rest frame is not inertial, and consequently, the overall energy and momentum of the system is not preserved. Therefore, is possible. While this frame of reference is not inertial over the entire period of the DIS event, it can be viewed as an inertial frame at any parametric distance for a brief period of time. Therefore, equation 11-2-4 is valid.

16 Note that lower mass values were used for the figures in this chapter in order to view more maximum and minimum peaks.

17 Note that the energy at the impact point within the nucleonís frame of reference is not equal to the initial energy of the electron. This is due to energy losses endured, as well as to the fact that the nucleonís frame of reference cannot remain inertial for a sufficient amount of time, due to the acceleration of the nucleon.

18 The index increase from to (by 10), rather than to (by 5) reflects that there are five minima in addition to the five maxima separating the radii of the two hard spheres.

19 Note however, that as the velocity of the electron was higher and may have been relativistic when passing through the previous maximum, the ratio between two successive maxima may reduce to below 13.59.

20 Note that the nucleon is accelerated by the electron. Therefore, its frame of reference can be regarded as an inertial frame for only a brief period of time. Consequently, the law of conservation of energy does not apply.

21The reason for the assumption that is that the electron had to pass the maximum just before impact with the zone, and then had to pass the maximum again immediately after the impact, on its way toward the detector.

22 Note that there is also a split of the potential energy maxima due to the slight difference in mass between the proton and the neutron, which may impact the number of resonances.

23 In cases where , the notations and may be used.

24 The DIS cross section can be written as . However, in the case of a negative value, the electron would remain trapped, and would never reach the detector. Therefore, cannot be negative. In addition, as the electron is extremely unlikely to gain energy during the scattering event, should be negligible. Therefore, the overall UG cross section can be expressed as.

25 In the non-quantum approach, the overall energy of the electron must be somewhat above as it passes through this maximum, otherwise the electron would not be able to enter and subsequently exit the zone, which contains , on its way toward the detector.

26 At even indices, is a minimum and its potential energy is negative. Therefore, it is possible that the velocity and the kinetic energy of the electron as it encounters these minima may be larger than their initial values.

27 Note that the potential energy maxima are odd integers, therefore is not continuous, and instead changes in steps of two.

28 Note that the UG and the electromagnetic interactions are both long-range forces. In proton-electron scattering, at all distances greater than nuclear scale, the UG contribution is negligible compared with the electromagnetic contribution (by a factor of approximately ) and can be neglected. However, at such distances the electromagnetic force reduces the impact parameter by pulling the electron toward the proton. This may explain why the cross section of a proton-electron DIS is observed to be larger than the cross section of a neutron-electron DIS as the Bjorken scale variable .

29 Providing that experimental data becomes available in this form (where the overall DIS cross section is dependent on and on odd , rather than on and ), the values of the UG constants and could be calculated with greater accuracy.




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