Pole Diagrams Research Articles

Pole and triangle diagram rearrangement mechanisms in an exact three-body formulation of transfer and exchange nuclear reactions

Pole and triangle diagram rearrangement mechanisms in an exact three-body formulation of transfer and exchange nuclear reactions

Application of Hard-Pion Four-Point Functions to Pion-Pion Scattering

Application of hard-pion four-point functions is made to $\ensuremath{\pi}\ensuremath{\pi}$ scattering on the basis of the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ current algebra, a conserved vector current, a partially conserved axial-vector current, and the hypothesis of single-meson dominance of intermediate states in $T$ products. The calculation uses techniques previously developed for exploiting the content of the current algebra and the subsidiary conditions for an $N$-point process. The $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude is shown to include, besides the well-known pole diagrams, a set of seagull terms. The Weinberg scattering lengths and effective ranges are found to be accurate to within a few percent, since the hard-pion corrections at threshold are only of $O(\frac{m_{\ensuremath{\pi}}^{}{}_{}{}^{2}}{m_{\ensuremath{\rho}}^{}{}_{}{}^{2}})$. In the low-energy region, the scattering-phase shifts are seen to be generally small and essentially model-independent, while at the $K$-meson mass we find $\ensuremath{\delta}_{0}^{}{}_{}{}^{0}\ensuremath{-}\ensuremath{\delta}_{0}^{}{}_{}{}^{2}\ensuremath{\simeq}{35}^{\ensuremath{\circ}}$. All existing data (up to 1 GeV) can be fitted by adjusting one model-dependent parameter.

Phenomenological Chiral Model for Nonleptonic Hyperon Decays

The $S$- and $P$-wave nonleptonic hyperon decay amplitudes are calculated using a chiral-invariant effective Lagrangian. This reproduces the current-algebra results in a very simple way, the baryon pole diagram arising without any ambiguity whatsoever. As is well known, this leaves us with the wrong ratio of $S$- to $P$-wave amplitudes. It is noted that one way out of this dilemma is to add another term to the effective Lagrangian which vanishes in the limit of zero pion four-momentum, and hence contributes to the physical answer without contributing to the current-algebra result. The addition of the ${K}^{*}$ pole diagram leaves us with a rough three-parameter fit to all the amplitudes plus the $K\ensuremath{\rightarrow}2\ensuremath{\pi}$ amplitudes.

O(4)Symmetry in Feynman Amplitudes

It is shown that any Feynman amplitude of pole diagrams at zero invariant mass does indeed possess the O(4) symmetry and that there are poles which do not correspond to real particles.Received 29 April 1968DOI:https://doi.org/10.1103/PhysRev.173.1608©1968 American Physical Society

Graphical Stability Analysis of Slopes in Jointed Rock

Graphic projections on a reference hemisphere, using either stereo nets or equal-area nets, are utilized in geologic analyses. The same graphic concept can be used in practical engineering evaluations of the stability of slopes affected by a three-dimensional but planar geologic structure. The effects of remedial measures can also be evaluated. The elements of hemispheric projections such as equatorial nets, meridional nets, pole diagrams, great circles, small circles, friction cones etc. are illustrated on both equal-area nets and spatial views. An example shows the use of these elements in both geologic and engineering phases of a slope stability study. Both wedge-type slides along two planes and block-type movements on one plane with the opening of the other planes are considered. Blanks forms of the three nets used are included.

Feynman diagrams for unitary representations ofSL 2,C

Weinberg’s derivation of Feynman rules for finite-dimensional representation of the Lorentz group is generalized for massive particles to infinite-dimensional irreducible representations of the Lorentz group. Following Feldman and Matthews, local and causal infinite-dimensional fields, quantized by a commutator, are written down for irreducible unitary representations of the principal series ofSL2,C. Feynman rules for those infinite-dimensional fields are stated and three-point functions and pole diagrams are given explicitly for arbitrary unitary irreducible representations of the principal series ofSL2,C.

Nonleptonic Hyperon Decay in the Quark Model

Nonleptonic hyperon and tr decays are discussed in the nonrelativistic quark model and the SU(6) theory, where we consider the pole diagram for quark. the four-body weak current­ current interaction J1 2 J3 1 and the meson exchange interaction_ We assume two models, and can explain the experiments for parity-violating and parity-conserving amplitudes of nonleptonic hyperon decays. In particular the L1I = 1/2 rule is derived, though the weak current-current interaction J1 2 J3 1 does not satisfy £11 = 1/2. As to the decay of tr two models predict the different ratio of r(SJ~A+K) and r(SJ~S+ll). One is consistent with experiment, the other forbids SJ~A+K.

Dynamics of the Weak InteractionΛN→NN

An attempt has been made to understand the spin and isospin dependence of the weak interaction $\ensuremath{\Lambda}N\ensuremath{\rightarrow}\mathrm{NN}$ as determined by Block and Dalitz from a phenomenological analysis of the nonmesonic decays of the light hypernuclei $_{\ensuremath{\Lambda}}\mathrm{He}^{4}$ and $_{\ensuremath{\Lambda}}\mathrm{H}^{4}$. The calculation is an extension of the pole model used by Lee and Swift for nonleptonic hyperon decays. The $\mathrm{CP}$-invariant weak Hamiltonian has been taken to transform like ${\ensuremath{\lambda}}_{6}$, a member of the octet representation of $\mathrm{SU}(3)$. One-boson-exchange forces have been considered for the strong-interaction part; $\mathrm{SU}(3)$-symmetric meson-baryon couplings have been used. We introduce the weak scalarpseudoscalar-meson pole in addition to the vector-pseudoscalar-meson pole for the parity-violating channel. For the parity-conserving channel, we also introduce scalar-meson exchanges in addition to (a) the pseudoscalar and the vector-meson exchanges for the weak-baryon pole diagram, and (b) the weak pseudoscalar-pseudoscalar-meson pole diagram. The results are in fair agreement with the deductions of Block and Dalitz.

Mass-breaking in $$\tilde U_{12} $$

Because the\(\tilde U_{12} \) degeneracy is badly broken experimentally several authors introduce an ad hoc mass splitting between theSU3 multiplets within the external\(\tilde U_{12} \) multiplets in calculations of scattering amplitudes in an inhomogeneous\(\tilde U_{12} \) theory. It is shown that if this mass splitting is consistently extended to internal particle exchange higher-order spurion amplitudes are introduced which, in regions of the energy variables nearSU3 masses, are likely to give important contributions to the amplitudes. The form of these is exhibited for pole diagrams. Restricting ourselves to the elastic scattering ofSU3 multiplets in the forward direction for pole diagrams it is found that in many cases the\(\tilde U_{12} \) predictions for the amplitudes are unaltered. A prescription is given for the general case.

V-θ sector of the lee model

By a slight extension of Amado's method, the matrix elements of the current operators between physical scattering states in the V-θ sector of the Lee model are determined. Then also the coefficients of the expansion of the scattering states in terms of the noninteracting states and theS-matrix elements are computed. The production amplitude is exhibited as a product of the contribution of pole diagrams times a simple correction factor which contains the effect of initial- and final-state interactions.

Consequences of analyticity and unitarity for partial-wave amplitudes

Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with l ≧ 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission factor and the real part of the phase shift. (d) If the transmission factor goes to zero (maximal inelasticity), the rate at which it approaches this limit is sometimes bounded by a known function. In these cases, the unphysical singularities and the transmission factor together determine the leading asymptotic term of the phase shift. (e) In some situations, a sign inconsistency in the asymptotic constraints implies the existence of poles in any N D solution with residues whose signs prevent their interpretation as bound states. These poles are then “ghosts.” (f) In an exact partial-wave dispersion relation, at least when l ≧ 2, there is no simple way by which to determine, from the sign of the unphysical function at threshold or at infinity, whether the corresponding “force” is attractive or repulsive. This sign is fixed at these energies by general arguments. As a result of these constraints, one can conclude that the usual approximations of the unphysical cut in terms of the singularities of pole diagrams have a chance of satisfying consistency requirements only for the s- and p-waves with spin zero and one particle exchanges. For p-waves, the exchange of a spin one particle is required. Certain rapidly convergent integral identities are developed which can be used to test for the presence of ghosts or of bound states in N D solutions or to test the accuracy of solutions. Finally, the relativistic Chew-Low model is used to demonstrate numerically the importance of some of the restrictions obtained.

The [formula omitted] reaction and low energy P-wave πK scattering

Field models are proposed to explain the ππ → K K ̄ cross-sections in the I = 0 and 1 states, derived from the π − + p → K + K + p (or n) experiments. (i) The I = 0 cross-section is explained through the assumption of an elastic KK̄ rescattering in the S-state. (ii) It is shown that the assumption that the I = 1 reaction proceeds predominantly through the intermediate ϱ-meson state fits the I = 1 cross-sections. (iii) Finally the low energy P-wave πK elastic scattering is studied in the framework of a model involving only the ϱ and the K ∗-pole diagrams.

Analyticity of the Production Amplitude in the Higher Order Diagram

Partial-wave s-singularity of the production amplitude is analyzed in the cases of pi + N yields 2 pi + N and 2 pi + N yields 2 pi + N. It is shown that the complex singularities of the fourthorder diagram coincide completely with those obtained from the corresponding pole diagram. Although the left-hand and right-hand singularities simultaneously appear on some region of the s-t plane, they are shown to be separated unambigrously. Detailed discussions about more general diagrams are also made and similar results are obtained. (auth)

MECHANICAL AID FOR PLOTTING AND COUNTING OUT POLE DIAGRAMS

Research Article| December 01, 1955 MECHANICAL AID FOR PLOTTING AND COUNTING OUT POLE DIAGRAMS ROBERT W DUSCHATKO ROBERT W DUSCHATKO COLUMBIA UNIVERSITY, NEW YORK, N. Y. Search for other works by this author on: GSW Google Scholar GSA Bulletin (1955) 66 (12): 1521–1524. https://doi.org/10.1130/0016-7606(1955)66[1521:MAFPAC]2.0.CO;2 Article history first online: 02 Mar 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation ROBERT W DUSCHATKO; MECHANICAL AID FOR PLOTTING AND COUNTING OUT POLE DIAGRAMS. GSA Bulletin 1955;; 66 (12): 1521–1524. doi: https://doi.org/10.1130/0016-7606(1955)66[1521:MAFPAC]2.0.CO;2 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyGSA Bulletin Search Advanced Search Abstract No abstract available. This content is PDF only. Please click on the PDF icon to access. First Page Preview Close Modal You do not have access to this content, please speak to your institutional administrator if you feel you should have access.