arXiv subjects
Publications and source records attributed to R Delbourgo.
A brief review is given of particle physics, gauge fields and gravity, based on a scheme whereby five complex (Lorentz scalar) anticommuting coordinates are appended to four dimensional space-time; the resulting model is effectively zero-dimensional.
A self-consistent treatment of two and three point functions in models with trilinear interactions forces them to have opposite anomalous dimensions. We indicate how the anomalous dimension can be extracted nonperturbatively by solving and suitably truncating the topologies of the full set of Dyson-Schwinger equations. The first step requires a sensible ansatz for the full vertex part which conforms to first order perturbation theory at least. We model this vertex to obtain typical transcendental equations between anomalous dimension and coupling constant $g$ which coincide with know results to order $g^4$.
We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius $R$ and taking the limit as $R \to \infty$. These relations take the form of mass integrals, associated with extraneous momenta (relative to the lower dimension), and produce the result in the higher dimension.
By describing strong interactions between hadrons via a relativistic supermultiplet scheme and regarding weak interactions as a perturbation thereof, we derive expressions for nonleptonic weak decay amplitudes in terms of constituent quark masses and CKM angles, with no other parameters. Application of this method leads to $ΔI=1/2$ dominance in some pseudoscalar meson decays if one scales down the couplings of heavy particles by $\sqrt{M}$ mass factors, in keeping with heavy quark theory expectations. However, certain B and D decay processes to kaons are badly predicted and point to substantial soft gluon renormalization effects in W-quark interactions.
Unitary group branchings appropriate to the calculation of local invariants of density matrices of composite quantum systems are formulated using the method of $S$-function plethysms. From this, the generating function for the number of invariants at each degree in the density matrix can be computed. For the case of two two-level systems the generating function is $F(q) = 1 + q + 4q^{2} + 6 q^{3} + 16 q^{4} + 23 q^{5} + 52 q^{6} + 77 q^{7} + 150 q^{8} + 224 q^{9} + 396 q^{10} + 583 q^{11}+ O(q^{12})$. Factorisation of such series leads in principle to the identification of an integrity basis of algebraically independent invariants. This note replaces Appendix B of our paper\cite{us} J Phys {\bf A33} (2000) 1895-1914 (\texttt{quant-ph/0001076}) which is incorrect.
Some new identities for quantum variance and covariance involving commutators are presented, in which the density matrix and the operators are treated symmetrically. A measure of entanglement is proposed for bipartite systems, based on covariance. This works for two- and three-component systems but produces ambiguities for multicomponent systems of composite dimension. Its relationship to angular momentum dispersion for symmetric symmetric spin states is described.
The fermion and meson sectors of the quark-level SU(2) linear sigma model are dynamically generated from a meson-quark Lagrangian, with the quark (q) and meson ($σ,π$) fields all treated as elementary, having neither bare masses nor expectation values. In the chiral limit, the masses are predicted to be $m_q = f_πg, m_π= 0, m_σ= 2m_q$, and we also find that the quark-meson coupling $g = 2π/sqrt{N_c}$, the three-meson coupling is $g' = m_σ^2/2f_π= 2gm_q$ and the four-meson coupling is $λ= 2g^2 = g'/f_π$, where $f_π\simeq 90$ MeV is the pion decay constant and $N_c = 3$ is the color number. By gauging this model one can generate the couplings to the vector mesons $ρ$ and $A_1$, including the quark-vector coupling constant $g_ρ= 2π, g_{ρππ}, g_{A_1ρπ}$ and the masses $M_ρ\sim 700$ MeV, $m_{A_1}\simeq \sqrt{3}m_ρ$; of course the vector and axial vector currents remain conserved throughout.
We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts.
A summary of the successes of and obstacles to the gauge technique (a non-perturbative method of solving Dyson-Schwinger equations in gauge theories) is given, as well as an outline of how progress may be achieved in this field.
We demonstrate that a $u-d$ mass difference (which we estimate to be about 4 MeV) within quantum loops, can reproduce the effects of the Coleman-Glashow electromagnetic tadpole operator.
We prove that recent data demonstrates unequivocally that the scalar meson $f_0$(980) is mostly composed of $s\bar{s}$ quarks and that the coupling of $f_0$ to photons and mesons is in agreement with the linear sigma model.
We give best covariant amplitude decompositions for two-body decay processes involving ground state hadrons ($0^-, 1^-, {1/2}^+, {3/2}^+$) and show how these are simply related to helicity amplitudes. After discussing how electromagnetic interactions are incorporated via meson dominance in a relativistic supermultiplet scheme, we extend the analysis to weak flavour changing nonleptonic processes. Such weak interactions are described by three generic amplitudes, which we estimate according to the rules of calculation within covariant SU(2N$_f$).
We provide succinct covariant amplitude decompositions of 2-body weak hadronic decays, with which to compare data, including exclusive rates, helicity amplitudes and polarizations. For weak decays, the systematic dependence of these amplitudes on masses and quantum numbers of participating particles are determined within a factor of about two by the CKM angles and the Fermi constant so theoretical models need to be much more accurate if they are to be convincing.
We exhibit the one-loop multi-gluon effective Lagrangian in any dimension for a field theory with a quasilocal background, using the background-field formalism. Specific results, including counter terms (up to 12 spacetime dimensions), have been derived, applied to the Yang-Mills theory and found to be in agreement with other string-inspired approaches.
The sum of all ladder and rainbow diagrams in $ϕ^3$ theory near 6 dimensions leads to self-consistent higher order differential equations in coordinate space which are not particularly simple for arbitrary dimension D. We have now succeeded in solving these equations, expressing the results in terms of generalized hypergeometric functions; the expansion and representation of these functions can then be used to prove the absence of renormalization factors which are transcendental for this theory and this topology to all orders in perturbation theory. The correct anomalous scaling dimensions of the Green functions are also obtained in the six-dimensional limit.
Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4.
It is possible to express all the strong and electromagnetic interactions of ground state hadrons in terms of a single coupling constant and the constituent quark masses, using spin-flavour relativistic supermultiplet theory. Results are generally accurate to within 10%.
Allowing for the inclusion of the parity operator, it is possible to construct an oscillator model whose Hamiltonian admits an EXACT square root, which is different from the conventional approach based on creation and annihilation operators. We outline such a model, the method of solution and some generalizations.