Kinematic Factor Research Articles

The relativistic SL (6, C ) strong vertex and kinematic form factors

The SL (6, C ) invariant baryon-baryon-meson vertex, involving three infinite multiplets, is expanded in terms of representations of the maximal compact subgroup SU (6) p for particles of arbitrary momentum. The terms in the expansion which involve the lowest-dimensional SU (6) multiplets are evaluated in closed form. The assumption of SL (6, C ) invariance leads to a breaking of SU (6) by spurions in a well-defined way. Each type of spurion coupling is accompanied by a function of momentum which vanishes rapidly for large momentum transfer and can be interpreted as a kinematic form factor. All calculations are performed with the ‘generalized tensor’ formalism, and a number of mathematical results are obtained which greatly simplify the technical problems of dealing with infinite-dimensional group representations.

Regge-Pole Model withL−SCoupling

A Regge-pole model for $\mathrm{PN}\ensuremath{\rightarrow}{P}^{\ensuremath{'}}N$ and $\mathrm{PN}\ensuremath{\rightarrow}\mathrm{VN}$ is proposed. The model is based on the $L\ensuremath{-}S$ coupling scheme and describes the coupling of the particles involved in the reaction to the trajectory having definite quantum numbers ${J}^{\mathrm{PG}}$. The scheme determines the structure of the amplitude, which in turn characterizes the decay density matrix of the produced resonance. Kinematical factors and the threshold behavior are discussed on the basis of the analyticity assumption.

Nuclear photon scattering

The theory of nuclear photon scattering, which was first presented by Placzek and Fano for electric dipole transitions and extended by Arenhövel to electric quadrupoles, is here derived in a general fashion for all electric and magnetic multipoles. The results are written in terms of dynamical quantities (“nuclear polarizabilities”, expressed by the same nuclear current and magnetization reduced matrix elements that also enter into the theory of photo- and electro-excitation), and of kinematical factors expressed by Racah coefficients. All photon and nuclear polarization effects are included in our results. Electric and magnetic dipoles are considered in detail as special cases. The long wavelength limit is contained in our formulae only implicitly and can be obtained in the form of a Thomson (scattering by the nuclear charge as a whole) and a Rayleigh limit (internal excitations) after separating out the relative-coordinate terms from the A 2 part of the amplitude and combining them with the ( j · A) 2 part, as shown by us in a previous publication for electric dipoles.

Application of the Theory of Padé Approximants to the Solution of the N/D Equations

Approximate methods for the solution of the N/D equations are discussed. Two methods in particular are considered, one being the replacement of the unphysical singularities of the scattering amplitude by a series of poles and the other being a pole approximation to the spectral integral of the kinematical factor over the physical region. It is shown that Padé approximants may be used to define a sequence of the above approximate solutions, which converges to the exact solution of the N/D equations in those cases when the Fredholm method also gives a solution.

Meson-Baryon Couplings in a Quark Model

The widths for decay of low-lying baryons of negative parity into baryon plus pseudoscalar meson are determined in a quark model and extensively compared with experiment. One conclusion is that Ξ * (1816) is not in an octet with N * (1518), Y 1 * (1660). A second prediction is a kinematical factor in s-wave decay enhancing the decay into high-mass mesons (K and η over η), which provides a qualitative reason for η peaks at threshold. Properties of the many missing baryon resonances are discussed. Channels appropriate for the search for some of these are indicated.

Baryon-Meson Couplings in BrokenSU(6)W. II

The baryon-meson vertex is studied within the framework of broken $\mathrm{SU}{(6)}_{W}$ symmetry. The breaking is provided by a $W=1$ spurion transforming like the $I=0=Y$ member of the adjoint 35 representation of $\mathrm{SU}{(6)}_{W}$ and which belongs to an $\mathrm{SU}(3)$ octet and/or $\mathrm{SU}(3)$ singlet. The spin kinematic factor in this case is different from the $W=0$ spurion breaking. The present type of breaking forbids all decuplet decays of the type $D\ensuremath{\rightarrow}B+P$, and in this respect the present situation resembles that of the static $\mathrm{SU}{(6)}_{S}$ in exact symmetry. The $\mathrm{BBP}$ couplings are related, in general, by 5 parameters which, under certain simplifying assumptions, reduce to a smaller number. Sum rules are listed for all the possibilities that might arise, and it is found that the predictions for the symmetry broken by a $W=1$ spurion belonging to an $\mathrm{SU}(3)$ octet plus an $\mathrm{SU}(3)$ singlet---both in the same 35 representation---are consistent with the present knowledge of the couplings.

Diffraction Dissociation and the 1.40-GeVπNPeak inp−pCollisions

The quasidiffraction model of Drell and Hiida is applied in an attempt to understand the 1.40-GeV maximum seen in high-energy, small-angle, inelastic $p\ensuremath{-}p$ scattering. Reasonable agreement is found in absolute magnitude, and in dependence on the energy and scattering angle, but the correct position and width cannot be fitted quite accurately; several different phenomenological form factors are attempted. It is shown that, at very forward angles, there is a cancellation in dependence on the off-shell pion mass between the pion propagator and a kinematic factor arising from the diffraction scattering. This leads to a suppression of waves with angular momentum $j>\frac{1}{2}$ for the recoiling system, and might be an explanation for the prominence at very small angles of the 1.40-GeV bump relative to the ${d}_{13}$ and ${f}_{15}$ isobars.

Approximate Solution to the MultichannelND−1Equations

By making a pole approximation to the spectral integral over the kinematical factor ${\ensuremath{\rho}}_{l}(z)$ it is shown that the partial-wave matrix $N{D}^{\ensuremath{-}1}$ integral equations are reduced to algebra. The approximation depends only on the particular partial wave and not the dynamics of the reaction, and it admits of systematic improvement. The resulting scattering amplitude ${T}_{l}(z)$ is symmetric, is independent of the subtraction point for the $D$ function, has the correct discontinuities on the right- and left-hand cuts, and can moreover be explicitly expressed as an algebraic function of the driving term ${B}_{l}(z)$. This last feature enables us to directly inspect the relation between the driving force and the scattering amplitude, and establishes the general usefulness of the method. We find, for example, that the solution imposes general conditions on ${B}_{l}(z)$ for the existence of bound states, resonances, or possible ghosts. The self-consistency property of bootstrap calculations imposes additional explicit restrictions on acceptable ${B}_{l}(z)$ for the existence of the bootstrap.

Kinematical form factors in the peripheral model

In potential theory the «vertex functions» for resonance formation contain kinematical form factors, which depend on the spin of the resonance and are simply related to penetration factors. Analogous form factors are introduced for the relativistic vertex functions of the one-pion exchange model. The resulting vertex functions behave like the Born term functions near the pion pole, but go to constants as the momentum transfer goes to infinity, for arbitrary resonance spin. For elastic nucleon-nucleon scattering, the total form factor reduces to a simple pole.

The unitary sum rule and the threshold behaviour of the dynamical function

Consistency relations of the partial-wave dispersion relations are derived which are valid under rather weak hypotheses on the forces and in particular without any assumption on theirthreshold behaviour. We apply them to a discussion of the consistency of: 1) Thel=2 wave of a scalar version to the nucleon exchange model in π-N scattering. 2) thel=1 wave of the σ exchange model in π-π scattering. Some interesting results such as a clarification to the role played by the numerator of the kinematical factor which multiplies the π-N partial-wave amplitude and an evaluation of the contribution of the short-range forces in π-π scattering are obtained.

Relativistic particles in a constant magnetic field

The relativistic limit of particles with arbitrary spin in a constant magnetic field is obtained by performing a unitary transformation on the Bargmann-Wigner equations. The new representation proves particularly convenient in studying the behaviour of an electron with anomalous magnetic moment in a constant magnetic field. The energy eigen values may be easily found. Moreover in the high-energy limit (m« ω) the depolarization is shown not to depend on kinematical factors in contrast with the medium-energy case studied by Mendlowitz and Case.

The Spins of Odd-Odd Nuclei

The spins of low-lying states of odd-odd nuclei are investigated in detail under the assumption of the j-j coupling odd-group model. For the sake of simplicity, only such nuclei are treated as have one proton (or hole) and one neutron (or hole) outside (inside) each core. The rule [R-3] reported by Brennan and Bernstein is derived from the rules [R-1] and [R-2]. To investigate the rules [R-1] and [R-2], we decompose two-body matrix elements of the effective interaction into the kinematical and dynamical factors using Talmi's method. The detailed analysis of the former factors imposes a few conditions on the latter factors to reproduce the rules, and it is found that Brueckner's K matrix satisfies these conditions reasonably, while usual phenomenological residual forces do not satisfy some of these conditions. Finally the effects of noncentral forces are investigated.

Positive Photopion Production from Hydrogen Near Threshold

Differential cross sections for photopion production from protons were measured for 32 photon energies from 154 to 185 MeV with the Alvarez 4-in. liquid-hydrogen bubble chamber in the bremsstrahlung beam of the Berkeley electron synchrotron. High photon intensities of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{7}$ MeV/pulse were used by collimating the beam down to a narrow "ribbon" which was viewed edge on by the camera. Although the pion origins were obscured by the heavy electron background, the remainder of the $\ensuremath{\pi}$ track was visible for most events. Of 5000 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\mu}$ decays seen, 3400 were deemed suitable for calculations. The results are in excellent agreement with the theoretical calculations by Ball, who applied the Mandelstam representation to the process. The measured values of the square of the matrix element, averaged over c.m. angles, are: All data are subject to an additional correlated error of 4.1% because of the uncertainty in the beam normalization. The cross section for the process is the product of the square of the matrix element times the kinematic factor $\frac{{p}^{*}}{{k}^{*}}$, where ${p}^{*}$ and ${k}^{*}$ are the c.m. momenta of the pion and the photon. Also a value of $\ensuremath{\Lambda}=(+0.931\ifmmode\pm\else\textpm\fi{}0.59)e$ was obtained, where $\ensuremath{\Lambda}$ is the multiplicative factor associated with the matrix element for photopion production from pions as calculated by Wong.

Scattering length and effective range theory for multichannel processes

The threshold properties of a two-body system which is coupled to other two-body channels are investigated. In particular, we examine the new channel when it is coupled to only one other, open, channel. At the threshold of the new channel we consider three real parameters (which describe the scattering amplitudes): the complex scattering length, a = A − iB, in the new channel and the diagonal K matrix element in the old channel, c. We first establish the quantitative behavior of these parameters with respect to certain averaged strengths of the interactions. An interesting application is, for example, a relation between the three actual parameters and the c that would apply if the coupling between the channels were turned off. Effective range expansions exist for the three parameters. (These can be easily generalized to the problem of many old channels.) These have the same form as the familiar one-channel expansion, except that the coefficient, r 0, of the quadratic term in the expansion is now an effective range type intergral multiplied by a factor depending on a and c. For example, in the effective range expansion of a, we find that if Im( 1 a ) is large (compared to a kinematical factor, roughly the momentum, κ 1, in the old channel) r 0 will be large compared to the range of forces. A similar result applies for the effective range expansion of the length c κ 1 . We then illustrate the usefulness of some of the relations by solving, exactly, the problem in which the 2-channel system is described by 2 coupled Schrödinger equations with square well potentials. The scattering length is also examined by means of a complex potential for the situation where there are many coupled channels.

Participation of Vibration in Exchange Reactions

Exothermic exchange reactions of the type A+BC→AB+C sometimes leave the product AB in a highly excited vibrational state. Simple theoretical considerations related to the description of the molecular collision show that the fraction of the reaction energy available for this vibrational mode is limited by the kinematic factor sin2β, where β is the angle of rotation required to take a coordinate system describing the reactants into one suited to the products, and tan2β=(mB/mA)+(mB/mC)+(mB2/mAmC). In the reaction A+BCD→AB+CD, similar results apply to the newly formed bond AB, and no excitation of vibration in CD is predicted. Although effects to be expected from details of the potential energy surface are ignored, comparison of the kinematic factor alone with experimental results shows excellent agreement. Applied to the reverse reaction, the theory gives a criterion for predicting the effect on the reaction rate of the distribution of energy between translational and vibrational modes in the collision. In the reaction O3+O2*→O+2O2, it is predicted that the oxygen atom comes preferentially from the vibrationally excited oxygen molecule. Incidentally, it is shown that the transformation from a properly normalized center-of-mass coordinate system describing the reactants to one describing the products can always be resolved into a sequence of simple rotations.

Sum Rules for Inelastic Electron Scattering

Sum rules are constructed for the analysis of inelastic electron scattering at high energy (\ensuremath{\sim}150 Mev) from light nuclei. Effects taken into account are: nucleon charge, recoil, and magnetic-moment currents; exchange currents; finite nucleon size; nuclear center-of-mass motion; and the kinematical factors describing the correct relation between initial and final electron energies, the scattering angle, and the nuclear excitation energy. It appears that a sensitive test of the role of exchange currents in the nuclear ground state is provided by a sum rule for the energy-weighted cross section for fixed-momentum transfer: ${\ensuremath{\sigma}}_{E}=\ensuremath{\int}\stackrel{}{|\mathrm{q}|=\mathrm{const}}\ensuremath{\epsilon}\ensuremath{\sigma}(\ensuremath{\epsilon}, q)d\ensuremath{\epsilon}.$