A topological expansion for high-energy hadronic collisions: (I). General properties and connection with the reggeon calculus

In this and in a companion paper a “topological expansion” for high-energy hadronic processes is proposed and discussed. In this first paper the general properties of the expansion and its connection with Gribov's reggeon calculus are presented. The topological expansion is first defined mathematically for a large class of theories and is shown to be equivalent to a “large N expansion” in some theories which include planar dual models and non-Abelian gauge theories. Next, the definition of the bare parameters is given in terms of graphs on a sphere. The bare pomeron pole and its couplings are thus introduced. The (inclusive) form of s-channel unitarity and its consequences fo the above couplings are recalled. It is then shown how the expansion in the number of “handles” of the graph can be related to Gribov's reggeon calculus and how, with the aid of discontinuity equations in the J-plane, scaling solutions can be obtained and critical indices can be computed to yield known results. Our main new results, including the study of s-channel discontinuities and of positivity constraints as well as the definition of a new fireball expansion, and the discussion of the relevance of this theory at finite (present) energies are presented in the second paper.     

Rising total cross sections and the flavoring of the pomeron

The relationship of the non-diffractive renormalization of the bare pomeron via$\text{K}\text{K}$ and $\text{B}\text{B}$ production - or its “flavoring” by λ quark loops and di-quark loops - and the shape of the NN total cross section is studied in some detail. The “unflavored” bare pomeron P? generated by non-strange quark loops with intercept $\text{α}\text{=0.85}$ is non-diffractively renormalized into the “flavored” (Gribov) bare pomeron P with intercept α above one (α = 1.06 here). We utilize inclusive data on $\text{K}\text{K}\text{and}\text{B}\text{B}$ production as well as inelastic diffraction to constrain parameters, and we fit the combination $\text{1}\text{2}\text{(σ}{}_{\mathrm{<mtext>pp</mtext>}}\text{+ σ}{}_{\mathrm{<mtext>p</mtext><mtext>p</mtext>}}\text{)}$ from s = 10 GeV² through ISR energies, including the new Fermilab data, to high accuracy. No pronounced long wavelength oscillations are observed. We suggest that these data favor the Chew-Rosenzweig realization of the topological expansion over that of Harari-Freund. We show that our scheme is consistent with the rising behavior of 2σ_KN ? σ_πN.     

A new interpretation of the absorption mechanism for high energy scattering

We propose a new s-channel absorption mechanism for high energy processes. A possible small dimensionless physical quantity naturally emerges. Consequently, the proposed s-channel equation can accommodate, at present energies, an elastic amplitude dominated by a simple factorizable bare pomeron pole, as in the t-channel perturbative expansion of the Gribov's Reggeon calculus in the strong coupling limit.     

Non-diffractive Pomeron renormalization,multiplicities, and the topological expansion

The renormalization of the Pomeron due to nondiffractive K$\text{K}$, B$\text{B}$ production thresholds is considered within a simple generalization of the Chew-Rosenzweig multiperipheral realization of Veneziano's topological expansion. The results are consistent with the existence of both the “low-energy” bare Pomeron with intercept $\text{α=0.85}$ and Gribov bare Pomeron with intercept above one. The vacuum exchange part of 2σ_KN?σ_πN basically rises with energy. Qualitatively correct features of shrinkage and breaks of $\text{d}\text{σ}\text{d}\text{t}$ emerge. The multiplicity of product clusters (n) increases with energy faster than 1ns and agrees with experiment for an average number of particles per cluster of 3–4. Independently of our model the Harari-Freund multiperipheral realization of the topological expansion is shown to be in serious difficulty with multiplicities, requiring around 20 particles per cluster.     

Magnetic moments of composite fermions

There have been a considerable number of papers proposing composite models for leptons and quarks. Recently, Glück and Lipkin have stated that reproducing the observed magnetic moments of these fermions presents a serious difficulty for these composite models. We show for a renormalizable theory that, in contrast to Glück's and Lipkin's nonrelativistic arguments, a deeply bound system (with heavy constituent particle masses m_c) of (total) spin $\text{1}\text{2}$, charge e and mass m has the magnetic moment $\text{(e/2m) [1 +}\text{“usual” (QED + QCD + weak) corrections +O}\text{(m/m}{}_{\mathrm{<mtext>c</mtext>}}\text{)}\text{“new” bindng corrections]}$. Although there remains the considerable dynamical problem of obtaining “light” bound fermions from heavy constituents, there is no separate, additional magnetic moment difficulty.     

Stochastic derivation of the Dirac equation in terms of a fluid of spinning tops endowed with random fluctuations at the velocity of light

If one analyzes the stochastic behaviour of classical “rigid” tops imbedded in Dirac's aether (relativistic thermostat) one obtains (for random jumps at the velocity of light) a probability distribution corresponding to the Feynman-Gell-Mann equation for relativistic spin $\text{1}\text{2}$ particles.     

The content of unitarity constraints on a pomeron pole of intercept one

We present a consistent picture of a pomeron pole with intercept one, together with its cuts, which evades the decoupling arguments. We use the reggeon cut discontinuity formulae to introduce Gribov's reggeon calculus as an exact solution of multiparticle t-channel unitarity. We show how, within the calculus, two-pomeron iterations of a singular kernel can be responsible for the zero in the triple-pomeron vertex. Using the concept of a bare pomeron pole as a multiperipheral production process which is subsequently renormalised by other effects, we apply the reggeon calculus analysis to inclusive cross sections. We find that the inclusive sum rule decoupling arguments are avoided because of the addition of enhanced absorptive corrections to the conventional Regge pole contributions. However, we show that in this picture the combined pole and two-pomeron cut contribution to the total cross section factories to order (ln s)^?2.We also show that, when the correct helicity structure of the pomeron is taken into account, the s-channel unitarity condition for pomeron scattering amplitudes does not lead to any serious decouplings.     

The relation between reggeon and photon couplings

We propose (a) the “vector coupling hypothesis” (VCH) that the leading normal-parity Regge trajectories (P, f, ω, ? and A₂) couple to hadrons like vector particles, and (b) the “γ-analogy hypothesis” (GAH), that the couplings of P, f, and ω to any vertex are proportional to the isoscalar photon coupling (the transition form factor), while ? and A₂ couple like the isovector photon. These hypotheses are motivated by duality and vector dominance. They allow us to interconnect such well-known results as (a) the relation between s-channel helicity conservation for P, f and ω at the $\text{N}\text{N}$ vertex and the equality of the nucleon's electric and magnetic isoscalar form factors, (c) the Stodolsky-Sakurai flip ? and A₂ couplings to $\text{N}\text{N}$ and the isovector form factors, (c) the Stodolsky-Sakurai magnetic dipole ?-N$\text{Δ}$ coupling. They also permit us to predict, with just a single free parameter, the differential cross section for $\text{pp}\text{→}\text{pN}{}^1$, at all energies, in terms of that for pp → pp. A detailed method for relating the transition form factors to the reggeon couplings for any spin is presented.     

Electrical conductivity and defect structure of Ni-doped and “CO-reduced” Ni-doped SrTiO3 single crystals

The electrical conductivities of Ni-doped and “CO-reduced” Ni-doped SrTiO₃ single crystals were measured at temperatures 700–1200°C and P_o2's of 10^?7–10^?1 atm. Plots of log σ vs $\text{1}\text{T}$ at constant P_o2's were found to be linear, and the activation energies appeared to be 0.92 eV for Ni-doped SrTiO₃ and 0.50 eV for “CO-reduced” Ni-doped SrTiO₃ single crystals, respectively. Plots of log σ vs log P_o2 at constant temperature were found to be linear with an average slope of ?$\text{1}\text{4}$ for SrTiO₃:Ni and of ?$\text{1}\text{6}$ for “CO-reduced” SrTiO₃:Ni single crystals, respectively. The electrical conductivity dependencies on P_o2 indicate that a triply ionized titanium interstitial and an oxygen vacancy model are applicable to Nidoped and “CO-reduced” Ni-doped SrTiO₃ single crystals, respectively. The small polaron conduction was suggested on “CO-reduced” Ni-doped SrTiO₃ single crystal from the temperature dependence of conductivity.     

Yang-Mills instantons and the S-matrix

We present a scheme for calculating gauge-invariant S-matrix elements in the presence of instantons. We exploit the conformal invariance of the zero-mass field equations. The asymptotic in and out states are defined by their values on null infinity $\text{J}$. We use this method to calculate to lowest-order S-matrix elements for scalar particles and fermions in a dilute gas of SU(2) instantons and anti-instantons. The scalar particles acquire an effective mass and an effective interaction of the form $\text{exp}\text{(?(?}{}^2\text{/16π) ?}\text{?}\text{)}$, where ? is the scale of the instanton, plus other interactions which cannot be presented by a local effective lagrangian. The fermions acquire the effective lagrangian obtained by 't Hooft. In the case of a single flavour of fermions, this corresponds to a mass term.     

In-out semi-classical gravity and 1N quantum gravity

Classical gravity with higher derivatives, when coupled to the in-out matrix element of T_μν: (a) appears to be free from instabilities of the in-in theory; (b) may be considered as the “underlying classical theory” to Tomboulis' $\text{1}\text{N}\text{-}\text{resummed}$ quantum gravity. I explain how these two facts may be adduced as evidence for the viability both of the former semi-classical approximation and of the latter approach to full quantum gravity.     

Chaos in numerical analysis of ordinary differential equations

The discretisation of the ordinary nonlinear differential equation $\text{d}\text{y}\text{d}\text{t}\text{= y(1?y)}$ by the entral difference scheme is studied for fixed mesh size. In the usual numerical computation, this method produces some “ghost solution” for the long range calculation. Regarding this discretisation as a dynamical system in R², these pathological behaviors are shown to be a kind of “chaos” in the dynamical system for any mesh size. Moreover, some combination of the central difference scheme and the Euler's scheme is studied for the above equation. It gives some motivation for Hénon's model. The usual discretisation of a second order differential equation are studied also. It gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by Hénon-Heiles.     

“Solidification” in a soluble model of bosons on a one-dimensional lattice: The “Boson-Hubbard chain”

Lieb and Liniger's soluble model of the 1-D Bose gas is put on a lattice, becoming a “boson-Hubbard model”. It remains soluble by the Bethe ansatz. When the coupling exceeds a critical value ($\text{U}\text{2πT}\text{)>}\text{2√3}\text{π}$, a gap is present when the density n bosons per site is 1.     

Manifest chiral symmetry preservation in the 1N-expanded SU(N) Thirring model

By utilizing manifestly chiral-invariant auxiliary field operators, it is demonstrated that continuous chiral symmetry is preserved explicitly in the $\text{1}\text{N}$ expansion of 2-dimensional theories whose fermions have a Gross-Neveu type of potential. The effective lagrangian derived in the $\text{1}\text{N}$ limit describes a massless scalar field whose derivative coupling to the fermions vanishes as N → ∞, and a decoupled massive scalar field.     

“Exact” shell model calculations in A = 17 and A = 18 nuclei and effective operators in the (2s, 1d) shell

A shell model calculation in an enlarged space spanned by two particles in the (2s, 1d) shell and three-particle-one-hole $\text{2}\text{h}\text{?}\text{ω}$ excitations is performed using the “bare” G-matrix. An effective interaction for the (2s, 1d) shell is generated and its convergence properties are investigated. It is maintained that the series converges reasonably fast in G and that the singleparticle energies used in the calculation are instrumental in this respect. This conclusion is corroborated by the E2 effective charges and B(E2) strengths that we compute for A = 17 and A = 18 nuclei respectively.     

On the zero mass boson in QCD2

The zero-mass boson in QCD₂ is shown to gain mass in the next to leading order in $\text{1}\text{N}$ expansion, in agreement with Coleman's Theorem.     

Polyakov's spinning string from canonical point of view

We give a canonical formulation of Polyakov's modified spinning string theory. This means that we start with the lagrangian , where $\text{L}$₁ is a counter term derived from the general form of the trace anomaly. In the superconformal gauge $\text{L}$₁ reduces to the supersymmetric Liouville lagrangian. A general solution of the supersymmetric Liouville equation is derived as well as appropriate boundary condutions for the Neveu-Schwarz (NS) and Ramond models. Under the assumption that the exact quantization of the Liouville theory does not yield any additional anomalies. We show that relativistic invariance requires the constant C to be $\text{C=}\text{10?D}\text{8π}$, in agreement with Polyakov's result. For D<10 the string acquires longitudinal modes. A semiclassical quantization of the Liouville theory is then performed with the result that the mass spectrum starts with $\text{m}{}^2\text{?}\text{1}\text{2}\text{α′}\text{and}\text{m}{}^2\text{= 0}$ in the NS and Ramond models in any dimension D?10. The longitudinal excitations are determined by a simple harmonic oscillator expression. It is shown that a consistent exact quantization could remove the tachyon state in the NS model.     

Canonical quantization of polyakov's string in arbitrary dimensions

Last year Polyakov discovered the important role the trace anomaly plays in the relativistic string theory. This result means that one has to add counter terms to the string lagrangian

$\text{L}\text{=}\text{L}{}_{\mathrm{<mtext>string</mtext>}}\text{+C}\text{L}{}_1\text{,}$

, where $\text{L}$₁ contains a cosmological term and a term from the trace anomaly. In the conformal gauge we have

$\text{L}{}_1\text{=}\text{L}{}_{\mathrm{<mtext>Liouville</mtext>}}\text{.}$

. We give a conventional GGRT treatment of this modified lagrangian for the bosonic string. Under the assumption that the exact quantization of Liouville's equation does not yield any additional anomalies, we show that relativistic invariance requires the constant C to be $\text{C =}\text{26 ? D}\text{48π}$, in agreement with Polyakov's result. For D < 26 the string acquires longitudinal modes, and our calculations show explicitly how the longitudinal component of the string receives the degrees of freedom from the Liouville variable. Under the boundary conditions yielding the lowest value of the classical Liouville hamiltonian, the mass spectrum starts with a tachyon $\text{m}{}^2\text{= ?}\text{1}\text{α′}$, independent of D. The lowest-lying longitudinal excitation is $\text{m}{}^2\text{=}\text{1}\text{α′}$. These results are semiclassical. It is shown that an exact quantization of Liouville's equation could remove the tachyon state when D < 26.     

On the bare pomeron graph in two-dimensional QCD

We present a detailed study of the bare pomeron graph in two-dimensional QCD in the 1/N_c approximation. The absence of a new singularity unrelated to quark parameters is derived. The “cylinder” graph is shown to induce renormalization of the vertex and intercept associated with quark andti-quark exchange in the vacuum channel.