Triangle Graph Research Articles

Graphs associated with triangulations of lattice polygons

AbstractTwo graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

In a search of symmetry behind topological chiral current

We try to identify symmetry associated with the topological current of gluon field K μ . We argue that in the case of supersymmetric Yang-Mills theory the matrix elements of ∂ μ K μ can be evaluated by varying the effective action with respect to the bare coupling constant. The triangle graph does not obey this relation and represent an anomaly. In the case of pure Yang-Mills fields the notion of conserved chirality is introduced for a special background in one-loop order. In particular the conservation of the chirality implies again the “naive” vanishing of the triangle graph.

Axial and vector anomalies in a broad class of invariant regularization and renormalization schemes

Within the framework of the spinor QED we describe a broad class of gauge invariant ultraviolet cut-off schemes, which provide a unifying view of the regularization of a Feynman graph in general and which may also be generalized to other gauge theories. Extending slightly this class of regularizations, we recover the standard anomalies in the axial-vector and vector Ward identities for the VVA triangle diagram. A cancellation of ultraviolet divergences for this graph is demonstrated, using an explicitα-parametric representation, for a broad sub-class of our regularizations. Renormalization of the triangle graph in terms of an auxiliary “soft” mass is also discussed. Some earlier results concerning properties of the VVA triangle graph and perturbative aspects of the corresponding anomalies are thereby generalized considerably.

Provenance Study of Venezuelan Pre-Columbian Ceramics Employing Radioisotope X-Ray Fluorescence

Radioisotope x-ray fluorescence analysis was performed to determine normalized elemental intensities for pre-Columbian sherds and roller stamps from different archaeological sites in the Middle Orinoco region of Venezuela. These normalized intensities were used to construct three-component (Triangle) graphs separating the sherds into three groups based on their known origin. The three-component graphs were: (Zn, Sr, Zr), (Rb, Sr, Zr), and ( A, Sr, Zr), where A = Cu + Zn + Pb normalized intensities. The total precision was shown to be better than 5% absolute for all cases studied. It was concluded that two of the roller stamps found at one site were probably manufactured in a different source area. This lends support to the notion that the roller stamps were a trade item in late pre-Columbian times (1000–1400 A.D.). The method applied in this study using a 109Cd (2 mCi) source seems to be appropriate for provenance studies of archaeological ceramics where many samples are processed, because it has the inherent advantages of being simple, economical, and rapid.

Dimensional regularization of the VVA triangle graph as a continuous superposition of Pauli-Villars regularizations

It is pointed out that dimensional regularization of the VVA triangle diagram is equivalent (for various approaches to γ5) to a continuous superposition of Pauli-Villars cut-offs, defined by means of an integration over the regulator mass smeared with an appropriate weight function. The idea of a “dimensional regularization in four dimensions” is briefly discussed.

N-quantum calculation of the Nambu-Jona-Lasinio model with isospin.

We use the N-quantum approach to quantum field theory to study the Nambu--Jona-Lasinio model with isospin. Our main purpose is to resolve a paradox found earlier. The paradox is that the form factor of a massless particle diverges as 1/Q/sup 2/ for small momentum transfer, so that the charge diverges and the amplitude cannot be normalized. We study this problem in one-loop approximation. We show that for a charged boson which is massless by virtue of being a Nambu-Goldstone boson the triangle graph which previously diverged now vanishes identically for all momentum transfers. We exhibit two other graphs which also contribute to the form factor in one-loop approximation and give a finite, nonvanishing contribution to the charge. For a charged boson whose mass is arbitrarily set equal to zero and whose bound-state amplitude does not have the structure of the Nambu-Goldstone amplitude the triangle graph still diverges as 1/Q/sup 2/. On the basis of this result we raise the question whether the only massless composite spin-zero particles which have finite charge are those which arise through the Nambu-Goldstone mechanism, at least in nonsupersymmetric theories.

A simple approach to the ABJ axial anomaly

A very simple semi-quantitative derivation of the Adler-Bell-Jackiw (ABJ) axial anomaly is given, based on an investigation of the absorptive part of the VVA triangle graph and dispersion relations. Essential ingredients of the author's discussion are: normal Ward identities for the absorptive part of the relevant diagram, dimensional analysis, unitarity and energy-momentum conservation. An explanation of the physical origin of the axial anomaly, offered in some earlier treatments within such a dispersive framework, is critically examined. In particular, the interpretation of the ABJ anomaly as an analogue of the Lee-Nauenberg effect occurring in the massless limit of spinor electrodynamics is shown to be fallacious.

The tensor analyzing power in backward pd elastic scattering and its relation to pp → πd at intermediate energies

The tensor analyzing power for pd backward elastic scattering is calculated in the energy range 150 ≤ T p ≤ 800 MeV. Two main contributions are considered: the one-nucleon-exchange and the so-called triangle graph, including pp → πd as a subprocess. The pd backward elastic cross section and tensor analyzing power are fairly well reproduced by the model.

Couplings of low-lying glueballs to photons and to heavy quarks.

We extend earlier work on couplings of the idealized pure-glue scalar glueball S and pseudoscalar glueball P to include couplings to heavy quarks and to photons. Using sum rules for the scalar and pseudoscalar densities of QCD we show that the couplings of charmed or heavier quarks to S and P decrease with quark mass M like ${M}^{\mathrm{\ensuremath{-}}3}$. Since for light quarks the coupling is proportional to the (constituent) mass, it turns out that S and P couple preferentially to strange quarks. We give a quantitative application to the decay J/\ensuremath{\psi}(3100)\ensuremath{\rightarrow}\ensuremath{\iota}(1440)\ensuremath{\gamma} (identifying \ensuremath{\iota} with P and ignoring qq\ifmmode\bar\else\textasciimacron\fi{} mixing). Naive use of the light-quark coupling rising like M gives a result thousands of times larger than experiment, while our coupling going like ${M}^{\mathrm{\ensuremath{-}}3}$ is in accord with data. In the course of deriving the heavy-quark coupling we give a new and simple derivation of Schwinger's results for a fermion propagator in a constant external field. S-\ensuremath{\gamma}\ensuremath{\gamma} and P-\ensuremath{\gamma}\ensuremath{\gamma} couplings are estimated with triangle graphs; we find \ensuremath{\Gamma}(P\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})\ensuremath{\sim}4 to 9 keV, and \ensuremath{\Gamma}(P\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\gamma}) ranges from 0.1 to 1 MeV, depending on various uncertainties. We contrast our calculations with canonical zero-momentum theorems, based on the anomaly, for coupling gluons to photons.

Absorptive part of the VVA triangle graph: A collection of formulae

A set of formulae is presented for the absorptive part of the familiar VVA triangle graph. Properties of the pseudotensor basis used for the definition of the corresponding invariant amplitudes and the ambiguity of such a definition are discussed in detail. There are essentially two independent invariant amplitudes and we give the imaginary parts of both of them explicitly. We thereby extend and generalize the results of previous treatments dealing with the subject. The limiting cases of vanishing fermion mass and/or some of the external momenta being lightlike are briefly discussed, as well as the recovering of the axial anomaly via dispersion relations.

Contribution of the triangle graphs to the neutron electric dipole moment

In the Kobayashi-Maskawa model we compute a new contribution to the neutron electric dipole moment induced byCP violating flavour-changing transitions of the typess→d+gluon+γ through heavy quark loops. This contribution, essentially given by the triangle anomaly graph, is by itself gauge invariant. We obtain from this mechanismDn/e=3.10−32 cm, one order of magnitude below our previous estimate from usual penguin diagram contributions.

VVA triangle graph ambiguities in four and N dimensions

We demonstrate that ambiguities arising from (i) arbitrariness in the internal momentum variable in four-dimensional Feynman integrals, (ii) arbitrariness in the location of γ 5 within γ-matrix traces under dimensional regularization, and (iii) arbitrariness associated with which vertex occurs first in γ-matrix traces under dimensional reduction all lead to equivalent relations between vector-current and axial-vector-current divergences in the absence of externally imposed vector-current constraints. We find, however, that further ambiguities, associated with the projection of less-than-four-dimensional momenta on four-dimensional γ-matrices, occur within regularization by dimensional reduction.

Macrocausality, unitary and physical-region structure of the multiparticleS matrix

The physical-region analytic structure of theS-matrix is established on the basis of refined macrocausality, unitary and a weak supplementary analyticity assumption for a 3→3 process below the four-particle threshold, and away from points where two initial, or two final, four-momenta are colinear: the connectedS matrix is decomposed as a sum of contributions that generalize Feynman graphs and are associated with the various +α-Landau surfaces encountered (=surfaces of graphs with one internal line and triangle graphs). The property of “separation of singularities” in unitarity equations, which was used as a crucial ad hoc assumption in traditional derivations of related results inS-matrix theory, is obtained as a byproduct in the region considered.

Deuteron stripping reactions at intermediate energies

The deuteron stripping reaction A(d,p)B is estimated in terms of the cross section for pion production with a proton beam A(p, pi +)B for the same nuclear states A and B. Extending the triangle graph model, which was constructed to explain large-angle proton-deuteron elastic scattering, data on 2H, 3He and 11C are described successfully for deuteron energies of about 800 MeV.

Fictitious-particle vertex in quantum gravity

We derive an exact expression for the fictitious-particle contribution to the pure graviton triangle diagram which is needed to verify, through the appropriate Slavnov-Taylor identities, the gauge invariance of the scattering matrix to third order in the gravitational coupling constant $\ensuremath{\kappa}$. The calculation is performed by working in the framework of covariant quantization and exploiting the technique of dimensional regularization which is known to preserve the local gauge symmetry of the underlying Einstein-Hilbert Lagrangian. The derivation is simplified by setting the energy-momentum of one of the external graviton lines equal to zero. It is shown that the final expression for the fictitious vertex is characterized by 14 invariant amplitudes and that the original divergences of the triangle graph reassert themselves as poles of Weierstrass's gamma function.

Single Pion Production in Proton-Deuteron Collisions

The reaction p+d-π++t is studied in terms of the p+pπ++d amplitude and the triangle graph along the line of Yao's model. Though the predicted cross section has a free parameter, the shapes are in good agreement with the experimental data.

The stress tensor in QED

The gauge-invariant ‘point-splitting’ version of the energy-momentum tensor in QED is discussed with respect to ‘Poincare identities’. In one-loop approximation an anomaly in the conservation law has been found for the triangle graph.

Infrared structure of the leading conformal contribution to the electromagnetic vertex function

We treat the infrared problem directly related to the asymptotic behavior of the form factor of a particle belonging to the fermion-antifermion channel, with anomalous dimension $\overline{d}$. For $\overline{d}<2$ we relate it to the off-shell vertex function; for $\overline{d}>2$ we examine the triangle graph which is considered to be the most infrared divergent. We show that the leading ${\ensuremath{\gamma}}_{5}$-even conformal contribution to the wave function does not give rise to infrared divergences, provided the dimensions of the various fields satisfy the conformal bounds. Thus the ensuing contribution to the asymptotic form factor is ${(\ensuremath{-}{q}^{2})}^{1\ensuremath{-}\overline{d}}$.

Infinities of the massless spin-1/2 triangle diagram in quantum gravity

In a massless fermion-gravity interaction field theory, the triangle graph with three external graviton lines is considered. The infinities arising from this and associated graphs are calculated in the dimensional regularization scheme, putting the energy-momentum of one of the external legs equal to zero. These infinities are exactly cancelled by those of the three-graviton graph originating from the graviton self-energy counter-Lagrangian density.

Energy dependence of the cross sections for the [formula omitted] and the [formula omitted] reactions

The excitation function for the 10B(d, α) 8Be g.s. reaction has been measured with both single counter and coincidence techniques in the bombarding energy range 2.5–4.5 MeV. The excitation function for the 11B(p, α) 8Be ∗(2.9 MeV) reaction has been measured in the bombarding energy range 7.5–10.5 MeV. The observed structures in the excitation functions have been compared with the prediction of a simple theoretical model involving pole and triangle graphs.