Session Z7: Advances in Field Theory and String Theory

Standard Model and beyond within the framework of Non-Commutative Geometry

We study the minimal Standard Model, Left-right Symmetric Model and their extensions within the framework of a discretized Kaluza-Klein version of a two-sheeted space-time. Left- and right- chiral fields live on two separate sheets. The main virtue of this approach is that it treats gauge fields and Higgs fields on a similar geometrical footing leading to a definite spontaneous symmetry breaking pattern. This results in a number of predictions concerning the Higgs sector of our theory.   

QEDxQCD Exponentiation and Shower/ME Matching at the LHC

We present the theory of QEDxQCD exponentiation for LHC processes with an eye toward shower/ME matching. We show that we can systematically improve the attendant MC or semi-analytical results order by order in perturbation theory without double counting while treating the respective phase space exactly.   

Comparisons of Virtual Corrections to Bremsstrahlung in Radiative Return at High Energy $e^+ e^-$ Colliders

Radiating a photon from the initial state provides a useful tool for studying a range of low energy physics using a high-energy $e^+ e^-$ accelerator. Accurate results require careful calculation of the first order virtual photon corrections. We compare two exact results for initial state radiative corrections, finding agreement to within $10^{-5}$ or better as a fraction of the Born cross section.   

Virasoro generators and the dS$_3$/CFT$_2$ correspondence

We discuss the quantization of a scalar field in three-dimensional de Sitter space. By extending the isometry group to include asymptotic symmetries, we obtain explicit expressions for the corresponding charges and show that they satisfy the Virasoro algebra with c=3l/2G. The charges act on the scalar field near the boundary in a way prescribed by the dS/CFT correspondence.   

Two-dimensional super Yang-Mills theory investigated with improved resolution

In earlier work, N=(1,1) super Yang--Mills theory in two dimensions was found to have several interesting properties, though these properties could not be investigated in any detail. In this paper we analyze two of these properties. First, we investigate the spectrum of the theory. We calculate the masses of the low-lying states using the supersymmetric discrete light-cone (SDLCQ) approximation and obtain their continuum values. The spectrum exhibits an interesting distribution of masses, which we discuss along with a toy model for this pattern. We also discuss how the average number of partons grows in the bound states. Second, we determine the number of fermions and bosons in the N=(1,1) and N=(2,2) theories in each symmetry sector as a function of the resolution. Our finding that the numbers of fermions and bosons in each sector are the same is part of the answer to the question of why the SDLCQ approximation exactly preserves supersymmetry.   

On the nonperturbative solution of quantum field theories

Light-cone quantization and numerical methods can be used to approximate nonperturbative solutions of quantum field theories. For non-supersymmetric theories, massive Pauli--Villars particles are included in the Lagrangian to provide ultraviolet regularization. Applications to quantum electrodynamics and super Yang--Mills theory will be summarized; results of more extensive calculations for Yukawa theory will be presented in some detail.   

Verification of the Ward Identites and Generalization to the Comma Field Theory

The proof for the Ward identities in the gauge invariant open string field theory of Witten requires checking a set of mathematical identities. These identities are cumbersome to establish analytically. In this paper we construct a numerical procedure capable of checking the Ward identities to any desired level. Furthermore, we discuss the generalization of the Ward identities to the Comma field theory of the open bosonic string and discuss how the generalized Ward identities can be established numerically.   

Bound States in the Noncommutative Plane

We demonstrate that the (non-relativistic) electrostatic potential felt by fermions or bosons - minimally coupled to a Maxwell-Chern-Simons theory in the noncommutative plane, can generate bound states between like charges. The result is reminiscent of models in commutative spacetime with non-minimal Pauli magnetic coupling. Results up to one loop and all orders in $\theta$- the noncommutative parameter, are under study.   

Variational Method for Few-Body Bound States in QFT

We discuss a variational method for deriving relativistic few-body equations in QFT. The method is illustrated on two, three and four body systems in a strongly coupled model, namely the Scalar Yukawa (Wick-Cutkosky) model, and on QED (the muonium, positronium and muonic hydrogen systems). A reformulation of the models is employed, in which covariant Green functions are used to solve the field equations partially, so as to express the mediating field in terms of the particle fields. The resulting reformulated Hamiltonian of the quantized system thereby contains an interaction term in which the mediating-field Green function is sandwiched between the particle currents. Numerical solutions of two, three and four body equations are presented for the scalar Yukawa model for various strengths of the coupling. The results are compared, where possible, to Bethe-Salpeter based and other calculations. Perturbative solutions of the two-fermion equations are presented and compared to other calculations as well as to experimental results for some states of muonium.   

Session Z8: Tevatron B Physics Topics

Observation and Measurements of Semileptonic$B$ decays to $D^{**}$ Mesons

Using 460 pb$^{-1}$ of integrated luminosity accumulated with the D\O\ detector, we have observed the semileptonic decays $B \rightarrow \mu \mu D^0_1(2420) X$ and $B \rightarrow \mu \nu D^{*0}_2(2460) X$. The results on their branching rates are presented.   

Semileptonic Lifetimes of B Mesons from CDF RUN II Data

A measurement of the B+/-, B0, and Bs mesons' lifetimes are presented using semileptonic decay from $\sim$ 350 pb-1 of data collected by CDF's lepton plus displaced track trigger. The decays of B$\rightarrow$l nu DX, where D is either D0, D*, Ds, are partially reconstructed from a lepton, a displaced track, and a fully reconstructed charm meson. The B+/-, B0, and Bs lifetimes are obtained from an unbinned maximum-likelihood fit to the proper decay length distribution.   

First Measurements of polarization in Charmless B to Vector Vector decays at CDF

We study samples of $B^0\to\phi K^{\ast 0}$ and $B^0_s\to\phi \phi$ collected in approximately 400 pb-1 of CDF run II data to measure the longitudinal and transverse polarization in these b-s penguin dominated decays. Comparison with analogous B-factories measurement and prospect for future developments are also given.   

Ratios of Branching Ratios for Hadronic $b$ Hadron Decays

The addition of the displaced track trigger at the second trigger level has enabled CDF in Run II to study purely hadronic $b$ hadron decays. We present ratio of branching ratios for $B^+\to\overline{D}^0 \pi^+$, $B^0\to D^-\pi^+(\pi^+\pi^-)$, $B_s\to D_s^-\pi^+(\pi^+\pi^-)$ and $\Lambda_b\to \Lambda_c^-\pi^+$ and first observations of doubly charmed $B$ decays at Hadron Colliders.   

$\Upsilon$ Production and Spin Alignment in $p \overline{p}$ Collisions at $\sqrt{s} = 1.96$ TeV

We report on measurements of the $\Upsilon$(1S), $\Upsilon$(2S) and $\Upsilon$(3S) differential cross sections ($d^2\sigma/dp_T dy)_{|y|<0.6}$, as well as on the $\Upsilon$ spin alignment in $p \overline{p}$ collisions at $\sqrt{s} = 1.96$ TeV using a sample of 360~pb$^{-1}$ collected by the CDF II detector. The three resonances were reconstructed through the decay $\Upsilon \rightarrow \mu^{+} \mu^{-}$.   

Study of orbitally excited B mesons at CDF

Using over $330~pb^{-1}$ of data collected by the Run II Collider Detector at Fermilab (CDF), we study the properties of the lowest orbitally excited (L = 1) B mesons, collectively denoted as the $B^{**}$. We reconstruct $B^{**}$ candidates in the decay channel $B^{**} \rightarrow B \pi$, $B \rightarrow J/\psi K$, $J/\psi \rightarrow \mu^+ \mu^-$.   

Lifetimes in Exclusive $b$ Hadron Decays to $J/\psi$ at CDF

Using exclusive decays such as $B^+ \to J/\psi K^+$ and $B_s \to J/\psi \phi$ collected with the CDF detector,the lifetimes of several different bottom mesons and baryons are measured. Results are comparied with previous measurements and theoretical predictions.   

B cross section measurements in lD

We present a new measurement of the B meson production cross section using $80 pb^-1$ of p-pbar collisions at $\sqrt{s}=1.96$ TeV with the CDF detector. Semileptonic B mesons decay modes such as $mu^+D^{*-}$ and $mu^+D0$ are used to isolate a clean sample of semileptonic B decays. The data sample utilized in this analysis was accumulated using a semileptonic trigger which takes advantage of CDF's ability to trigger on displaced tracks.   

Measurement of $b$~Quark Fragmentation Fractions in $p\bar p$ Collisions at $\sqrt{s} = 1.96$~TeV

Using $p\bar p$ collision data at $\sqrt{s} = 1.96$~TeV collected with the CDF\,II experiment at the Fermilab Tevatron collider, we present measurements of the fragmentation fractions $f_u$, $f_d$, $f_s$ and $f_{baryon}$ of produced $b$ quarks that yield $B^+$, $B0$, $B0_s$ and $\Lambda_b$ hadrons. Using a data sample enriched in semileptonic $B$ hadron decays, we reconstruct five lepton-charm final states, $\ell D0$, $\ell D^+$, $\ell D^*+$, $\ell D^+_s$ and $\ell \Lambda_c$ to obtain precise measurements of $f_u / f_d$, $f_s / (f_u+f_d)$ and $f_{baryon} / (f_u+f_d)$.   

Session Z9: New Ideas in Gravitation and Astrophysics

Energy distribution of Reissner-Nordstrom-anti-de Sitter black holes with negative or vanishing curvature

We use the Einstein energy-momentum complex to investigate the energy distributions of the generalized Reissner-Nordstrom family of spacetimes in 3+1 dimensions with asymptotically anti-de Sitter behavior. We replace the standard spherically-symmetric metric with two alternative geometries, each a solution of the Einstein-Maxwell equations with a negative cosmological constant. The first is a static ``black plane'' solution with vanishing two-dimensional curvature. The second possesses two-dimensional surfaces with constant negative curvature. We calculate the energy distributions of these models as functions of the mass, charge and cosmological constant. Comparing these expressions to those of the standard Reissner-Nordstrom-anti-de Sitter solution, we comment on the qualitative differences.   

Spin motion of a string

The equation of motion of a string is derived, and it is shown that deviations from the geodesic include a spin like character. In fact, it is shown that strings have spin and it is quantized.   

Gravity as Nonmetricity

It is shown that in the presence of an arbitrary affine connection, the gravitational field is described as nonmetricity of the affine connection. An affine connection can be interpreted as induced by a frame of reference (FR), in which the gravitational field is considered. This leads to an alternative geometrical interpretation of GR wherein gravity is a nonmetricity of space-time. Although the gravitational field equations are identical to Einstein's equations of GR, this formulation leads to a covariant tensor (instead of the pseudotensor) of energy-momentum of the gravitational field and covariant conservation laws. We further develop a geometric representation of FR as a metric-affine space ($L_{4}$,$g)$, with transition between FR represented as affine deformation of the connection. We show that the affine connection of a NIFR has curvature and may have torsion. We calculate the curvature for the uniformly accelerated FR. Finally, we show that GR is inadequate to describe gravitational field in a NIFR. We propose a generalization of GR, which describes gravity as nonmetricity of the affine connection induced in a FR. This generalization contains GR as a special case of the inertial FR.   

Extended Cellular Automata Models of Particles and Space-Time

Models of particles and space-time are explored through simulations and theoretical models that use Extended Cellular Automata models. The expanded Cellular Automata Models consist go beyond simple scalar binary cell-fields, into discrete multi-level group representations like S0(2), SU(2), SU(3), SPIN(3,1). The propagation and evolution of these expanded cellular automatas are then compared to quantum field theories based on the "harmonic paradigm" i.e. built by an infinite number of harmonic oscillators, and with gravitational models.   

Non-Local Gauge Field Theory: a Model

Some consequences of Non-Local Gauge (NLG) Invariance are explored with a NLG generalization of QED for which the Unitary group is generated by local functions and differential operators. One of the main features of the model is the existence of spin-two field with gravity-like interaction. Some of the most attractive features of this model are found when we try to quantize it. Most of the obstacles to covariant quantization that are found in General Relativity and other theories of gravity are avoided. This is due to the following features of the model: (1) unique dimensionless perturbation constant, (2) placement of the dimensionful Newton's constant into the gauge-field propagator, (3) polynomial form of the field self-interaction, and (4) the infinite number of local field functions. All these features render this model renormalizable.   

A massive graviton generalization of general relativity

The gauge bosons of Einstein's theory of general relativity are thought to be massless, with gravity waves propagating at the speed of light. It may turn out, however, that gravitons possess a very small mass. Developing a geometric theory of massive gravitons, therefore, may aid current researchers in gravity wave physics, who are currently mounting great efforts to observe these waves directly. One such theory, together with some basic solutions, is presented here. The theory is a generalization of Weyl's geometry, and could conceivably provide another avenue in the development of inflationary cosmologies.   

Precession of Mercury and Bending of Starlight Using Gravitational Emission Theory

The precession of the perihelion of Mercury and the bending of starlight by the Sun are calculated exactly by modifying Newton's Theory of Gravitation. Assuming for gravity a propagation speed equal to that of light, we derive and solve an equation that includes the relative velocity between two objects. Analytical and numerical results are presented and the relevance to experiment is discussed.   

Gravity as a Grand Unification of Forces

Any of all possible types of charges corresponds in nature to a kind of the inertial mass. Such a mass - charge duality of matter explains the coexistence of grand united rest mass and charge for the same neutrino equal respectively to its all the gravitational mass and charge which consist of the gravitoelectric, gravitoweak, gravitostrong and a range of others, innate components. From their point of view, a new grand unification theory has been created at the discussion of a question about unification of forces of a different nature. In this theory, the gravitational field must be naturally united gauge field of the unified system of the most diverse combinations of electromagnetic photons, weak bosons and strong gluons where the four pairs of forces of the micro world fundamental interactions are united. Some consequences and laboratory confirmations of the suggested theory have been listed which allow also to define the structure of the graviton as a grand united boson. Thereby it gives the possibility to directly look at the nature of the gravitational matter elucidating the interratio of intraneutrino forces and the problem of elementary particles chiral and mirror symmetries.   

Session Z12: Mini-symposium B: Pentaquarks and Other Exotics II

Search for the Theta(1540) Pentaquark in Electro-Production with the BaBar Detector

Since early in 2003, several experiments have presented evidence for the existence of a positive strangeness baryon state of mass around 1540 $MeV/c^2$ and width $< 8$ MeV, the Theta(1540), which decays to $K^+ n$ and $K^0 p$. Such a state has minimum quark content u d u d sbar and consequently has been interpreted as the S = +1 member of the anti-decouplet of pentaquark states proposed by Diakonov et al. Six of the claimed observations involve real or virtual photoproduction. We present preliminary results from a search for the production of the Theta(1540) in $e^+ e^-$ interactions (i.e. virtual photoproduction) in the Be beampipe of the BaBar Detector at the PEP-II Collider. Event selection procedures are summarised and $K^0_S$ p invariant mass distributions in the threshold region are presented.   

Searches for exotic particles in d+Au collisions at $\sqrt{s_{_{NN}}}=200$~GeV and Au+Au collisions at $\sqrt{s_{_{NN}}}=62.4$~GeV

Exotic particles such as pentaquark states and di-baryons are allowed to exist within the framework of Quantum Chromo Dynamics (QCD). Several experiments reported the evidence for the existence of a pentaquark state, $\Theta^{+}(uudd\bar{s})$, however, several other experiments reported null results. We present results on the search of pentaquark states in d+Au collisions at $\sqrt{s_{_{NN}}} = 200$~GeV and Au+Au collisions at $\sqrt{s_{_{NN}}}=62.4$~GeV from the STAR experiment at RHIC. Statistical significance of our searches will be discussed.   

Constraints on narrow exotic states from K+p and K0$_{Lp}$ scattering data

We consider the effect of exotic S=+1 resonances Theta+ and Theta++ on K+p elastic scattering data (total cross section) and the process K0$_{Lp}$--$>$K0$_{Sp}$. Data near the observed Theta+ (1540) are examined for evidence of additional states. The width limit for a Theta++ state is reconsidered.   

All-Charm Tetraquarks

We report on the possibility of all-charm tetraquark states bound with respect to breakup threshold into mesons with the same quark and total angular momentum content. We use a set of Hamiltonians tuned to fit the charmonia spectroscopy and employ recently developed many-body methods that are free of spurious center-of-mass motion effects. Global color symmetry is guaranteed for all solutions. We extend results reported recently (1) by using more realistic Hamiltonians (2) and carrying out renormalization procedures to reduce truncation effects. The renormalization procedures are adapted from the successful ab-initio no-core shell model (3). This work was supported in part by a USDOE grant DE-FG02-87ER-40371. \begin{enumerate} \item R. Lloyd and J.P. Vary, Phys. Rev. D 70, 014009 (2004). \item J.R. Spence and J.P. Vary, Phys. Rev. C 59, 1762 (1999); and to be published. \item P. Navratil, J. P. Vary and B.R. Barrett, Phys. Rev. Lett. 84, 5728 (2000); Phys. Rev. C 62, 054311 (2000). \end{enumerate}   

Session Z13: Nuclear Theory II

R-Matrix Method Using the Polar Form of the Schroedinger Operator

Any linear operator ($\hat A$) can be decomposed into a product of an Hermitian times a Unitary operator [P-O L\"owdin, ${\bf Linear\ Algebra\ for\ QuantumTheory}$,Wiley 1998, New York], as per ${\hat A} \equiv {\hat H}_1{\hat U_1} ={\hat U_2}{\hat H_2}$ where ${\hat H}_1 = ( {\hat A}{\hat A}^{\dagger})^{1/2}$, ${\hat U}_1 = {\hat H}_1^{-1} {\hat A}$, ${\hat H}_2 = ( {\hat A}^{\dagger}{\hat A})^{1/2}$, ${\hat U}_2 = {\hat A}{\hat H}_2^{-1} $. Such decompositions constitute what is called the polar form of the operator. A version of the R-matrix scattering theory will be presented employing the polar form of the time-independent Schroedinger operator (SO). The SO is not Hermitian in the continuum within the finite R-matrix sphere. An approximate inverse of the SO is constructed by diagonalizing its positive definite Hermitian component. An approximate expression for the scattered wave is obtained by projecting onto these eigenstates. In the process, a new type of minimum principal obtains such that minimization of the positive definite eigenvalues produces the most rapidly convergent series for the operator inverse and therefore, the solution. This algorithm is described and applied to several model problems which constitute a proof of principle.   

DWBA (d,N) Calculations Including Dirac Phenomenological Potentials and an Exact Treatment of Finite-range Effects

An algorithm for the inclusion of both Dirac phenomenological potentials and an exact treatment of finite-range effects within the DWBA is presented. The numerical implementation of this algorithm is used to calculate low-energy deuteron stripping cross sections, analyzing powers, and polarizations. These calculations are compared with experimental data where available. The impact of using several commonly employed nuclear potentials (Reid soft-core, Bonn, Argonne v18) for the internal deuteron wave function is also examined.   

The number of $J=0$ pairs in $^{44,46,48}$Ti

In the single $j$-shell, the configuration of an even--even Ti isotope consists of 2 protons and $n$ neutrons. The $I=0$ wave function can be written as $$\Psi=\sum_{Jv} D(J,Jv) [(j^2)^J_\pi (j^n)^J_\nu ]^{I=0},$$ where $v$ is the seniority quantum number. There are several states with isospin $T_{\rm min}=|(N-Z)/2|$, but only one with $T_{\rm max}=T_{\rm min}+2$. By demanding that the $T_{\rm max}$ wave function be orthogonal to the $T_{\rm min}$ ones, we obtain the following relation involving a one-particle cfp: $$ D(00)=\frac{n}{2j+1} \sum_J D(J,Jv)(j^{n-1}(jv=1)j|}j^nJ) \sqrt{2J+1} $$ This leads to the following simple expressions for the number of $J=0$ $np$ pairs in these Ti isotopes: \begin{itemize} \item For $T=T_{\rm min}$, \ \ \ \# of pairs $(J_{12}=0)=2|D(00)|^2/n$ \item For $T=T_{\rm max}$, \ \ \# of pairs $(J_{12}=0)=2n|D(00)|^2=\frac{2n(2j+1-n)}{(2j+1)(n+1)}$ \end{itemize} For $^{44}$Ti we have also the result for {\em even} $J_{12}$ $$ \#\ {\rm of}\ nn\ {\rm pairs}\ =\ \#\ {\rm of}\ pp\ {\rm pairs}\ =\ \#\ {\rm of}\ np\ {\rm pairs}\ =\ |D(J_{12},J_{12})|^2 $$   

Alternate derivation of the Ginocchio-Haxton relation $[(2j-3)/6]$

We want the number of states with total angular momentum $J=j$ for 3 identical particles (e.g. neutrons) in a $j$ shell. We form states $M_1>M_2>M_3$ with total $M=M_1+M_2+M_3$. Consider first all states with $M=j+1$. Next form states by lowering $M_3$ by one. All such states exist because the lowest value of $M_3$ is $(j+1)-j-(j-1)=-j+2$. So far we have the total number of states with $J > j$ and $M=j$. The additional states with $M=j$ are the states with $J=j$. These additional states have the structure $M_1,M_2,M_2-1$ because if we try to raise $M_3$ we get a state not allowed by the Pauli principle, namely $M_1,M_2,M_2$. The possible values of $M_1,M_2$ are respectively $j-2n$ and $1/2+n$, where $n=0,1,2\cdots$. The total number of $J=j$ states is $N=\bar{n}+1$ (with $\bar{n}=n_{\rm max}$), while $\bar{n}$ itself is the number of seniority 3 states. The condition $M_1>M_2$ leads to $\bar{n}<(2j-1)/6$ or $N<(2j+5)/6$. This is our main result. It is easy to show that this is the same as the G-H relation\footnote{J.N.~Ginocchio and W.C.~Haxton, {\em Symmetries in Science VI}, ed. by B.~Gruber and M~Ramek, Plenum, New York (1993)} (see also Talmi's 1993 book) $\bar{n}=[(2j-3)/6]$, where $[]$ means the largest integer. Since $2j$ is an odd integer, $(2j-1)/6$ is either $I, I-1/3$ or $I-2/3$, where $I$ is an integer. If the value is $I$, then $\bar{n}=[(2j-3)/6]=[I-1/3]=I-1$. It is easy to show agreement in the other 2 cases as well. The number of $J=j$ states for the 3-particle system is equal to the number of $J=0$ states for a 4-particle system.   

Bound nucleons have unique masses that govern elemental properties

It is know that measured binding energies associated with elements require equivalent energy to break the nuclear bond of a nucleus. Based upon the proposals contained in a recent published work [1] and with support from experimental high-energy data, it can be shown that a portion of listed binding energies are attributed to bound nucleons having a unique mass for every element. The figures show, relative to the hydrogen proton, that of the: a) 1.112 MeV binding energy per nucleon for $^{2}$H, 44{\%} or 0.486 MeV represents a change in mass for the proton and neutron; b) of 5.629 MeV binding energy per nucleon for $^{7}$Li, 87{\%} or 4.890 MeV represents a change of mass for each nucleon; c) likewise, $^{56}$Fe has 8.811 MeV binding energy per nucleon and of this 92{\%} or 8.119 MeV represents a change in mass for each nucleon; and $^{232}$Th has 7.639 MeV binding energy per nucleon and of this, 90{\%} or 6.848 MeV represents a change in mass for each nucleon. This demonstrates that the nucleons of each element have unique masses. It can be shown that if three protons are removed from $_{82}$Pb the result is not $_{79}$Au. We conclude and predict that in addition to the Z number, elemental properties are determined by the unique proton and neutron masses for each element. [1] \href{mailto:megforce@physast.uga.edu}{megforce@physast.uga.edu} ``The Order of the Forces''   

Chaos as a Bridge between Classical Determinism and Quantum Probability

Chaos provides the fundamental determinism so dear to Einstein, yet for all practical purposes it must be treated statistically, as proposed by the Copenhagen school. Thus, both Einstein and Bohr could have been correct in their debates. In a series of papers I have demonstrated that a number of the so-called imponderables or paradoxes generated in the Copenhagen interpretation of quantum mechanics have parallel explanations in the realm of nonlinear dynamics and chaos theory [i.a., J. Opt. B: Quantum and Semiclass. Opt. {\bf 5}, S442 (2003)]. These include exponential decay laws, interpretations of Bell-type inequalities, spontaneous symmetry breaking, and even diffraction. I give a brief overview of these, concentrating on the interpretation of the CHSH inequality (an experimentally friendly Bell-type inequality), demonstrating that here one is comparing correlated versus uncorrelated statistics more than quantum versus classical mechanics--- nonlinear classical dynamical systems have been shown to have sufficient long-range correlations, as codified by the entropy of nonextensive thermodynamics, to raise the upper bound imposed by Bell-type inequalities into the range of quantum mechanics. As a result, many of the experiments ruling out ``local reality'' are perhaps moot.   

Variable Nuclear Barrier Heights as Irregular Potential Waves Due to Various Nuclear Motions

The nuclear potential barrier height is an irregular wave due to random and periodic motion nuclear motions such as vibration, rotation and orbiting. Due to the vibrations and other nuclear motions, the potential well is vibrating irreglarly also. Assume the nuclear motion is a three dimensional oscillator were r = \{(AcosX)$^2 +$ (AcosY$^2 +$ AcosZ)$^2$)\}$^{1/2}$. For cos =0, r = 0 min, cos =RMScos, r=1.22A average, cos=1, r= 1.707A max. Therefore, using V = kq(1)q(2)/r the barrier height ranges from V=infinitely high, = 0.816q(1)q(2)/A on average, to a low of 0.577q(1)q(2)/A where A= average amplitude of nuclear vibration, q(1) is the nuclear charge, q(2) = charge of incoming or outgoing particle. Nuclear motion makes the gravitational and magnetic fields irregular wave also.   

Variable Cross Sections Due to Nuclear Vibrations

Due to random nuclear vibrations the cross sectional area for inoming particles to a nucleus is a variable. If b = AcosY is the impact parameter in one dimension, then the cross section $\sigma$=$\pi$(A cosY)$^{2}$ where A=amplitude of vibration. Therefore, $\sigma$=$\pi$(A)$^{2}$ maximum, $\sigma$=$\pi$(0.707A)$^{2}$ average rms, and $\sigma$=0 minimum values for the variable nuclear cross sections per nucleus.   

Classical Model of Rutherford-Santilli Neutron

Model n(RS), isomer of n, Bohr's real ground state of H$^{+}$: proton p and electron e, separated 0.81 Fermi, in circular orbits about c.m., with parallel magnetic dipole moments (MDMs) normal to orbital plane. Binding force: Coulomb less magnetic. Each momentum: mass x velocity less charge x (vector potential A at particle). Assuming unmutated p, and n data: orbital e velocity v is 0; slight mutation: v/c $<<<$ 1. (Ref. 1.) Mutated e: mass, 2.5 x (mass of free e); spin, 0.038$\hbar$/2; g, 0.52; MDM, 3.6 x E(-26) S.I. Stability requires external A, 0[0.01 S.I.], found in atomic nuclei. The n(RS) \textbf{$\to $} n by spin flip of e; e captured by positive constituent of p; gamma photon emitted. (Ref. 2.) \newline \newline 1. R. M. Santilli, \textit{Hadronic Journal 13}, 513 (1990); \textit{Chinese Journal Sys. Eng. {\&} Elec. 6}, No. 4, 177 (1995) \newline 2. R. B. Driscoll, \textit{Hadronic Journal 27}, No. 6 (2004)