Bound state poles of the form factors in eikonal approximation and infinite component wave equations

We investigate the form factor in eikonal approximation for the whole region of momentum transfer t in electrodynamics and in φ³ theory, and determine the poles in the time-like region below the threshold. A covariant mass formula is obtained similar to the one derived from four point functions in eikonal approximation. The mass formula does not become imaginary for large values of the coupling constant. The relation to positronium Regge trajectory and infinite component wave equations is discussed.     

On the two-reggeon-two-particle amplitude at large momentum transfer

The role of sister trajectories in the fixedangle limit of reggeon amplitudes is analyzed. The conventional α-sector of the two-reggeon-two-particle scattering amplitude has been proved to be negligeable, in the non-Regge asymptotic limit, as compared to the unconventional β-sector, which behaves as exp{?β(s)f(θ)}. It is conjectured that the fixed-angle limit of a general reggeon amplitude is dominated by the less sloped sister trajectory exchanged in the Regge region.     

Phase transitions and the high temperature expansion for the reggeon calculus

We study a recently proposed lattice version of the reggeon field theory. First a simple one-dimensional system possessing many of the features of the reggeon calculus (Ising model in an imaginary magnetic field) is solved. Surprisingly, the system is found to undergo a phase transition at a non-zero critical temperature, which, although of first order obeys the universality and scaling laws previously postulated for the reggeon calculus. Returning to the full lattice theory, the machinery for performing a high temperature expansion is set up, and initial calculations carried out to order T^?3. In this order, best estimates for the critical indices η and v in the asymptotic elastic amplitude A(s,t) ～ is (ln s)^ηf(t(lns)^v) yield $\text{η = 0 (}\text{1}\text{2}\text{～ 1), v = 0(1}\text{1}\text{2}\text{～ 2)}$. Check on the method, including comparisons with known Ising model results, are also discussed.     

Is reggen field theory rendered obsolete byqcd?

We affirm the power of the eikonal approximation in reggeon field theory (rft) and point out its merit as contrasted with renormalization group method for calculating reggeon self-energy correction due to pomeron exchanges. Relative merits ofrft andqcd in appropriate momentum regions of hadronic interactions are also examined to suggest a negative answer to the title.     

s-channel discontinuities of the torus insertion in the dual unitarisation scheme

The refined analysis of the s-channel discontinuities of the torus insertion in the topological expansion is performed. Assuming the dependence on reggeon masses of the fixed-pole residue similar to that given by the Veneziano model we obtain almost exact cancellations of the discontinuities which in the one-dimensional approximation were argued to be responsible for the exchange degeneracy breaking between ? and A₂ trajectories. Implications on different f-IP schemes are discussed.     

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.     

Mass generation in the O(3) non-linear σ model

We study the two-point function of the azimuthal angle, G^(φ)(x) = 〈e^iφ(x)e^?iφ(0)〉_inst [φ = arg (q¹ + iq²), where q^a is a three-component unit vector field], in the dense instanton gas approximation for the two-dimensional O(3) non-linear σ model. We find that G^(φ) (x) decreases exponentially as |x| → ∞. This suggests that the dense instanton gas may generate a mass gap in the O(3) non-linear σ model. The physical mechanism of this mass generation is also discussed.     

Processes at high energy and large momentum transfer: the momentum-space approach to the asymptotics ofφ 3 ladder diagrams

The Sudakov method is extended to study the scattering amplitudes in the high-energy and large-momentum-transfer limit. The behaviour of arbitraryφ ³ ladder diagrams in the leading logarithm approximation is reconsidered. This leads to a simple physical picture of the asymptotic behaviour: the leading-term contributions are related to the different paths along which the large momentum is transferred through the ladder diagrams.     

A non-perturbative study of regularized bosonic string amplitudes

The Wick-rotated, light-cone gauge U(n) Veneziano model of open and closed strings is examined in the limit as n → ∞, with g₀²n fixed. The amplitudes in both the “pomeron sector” (whose graphs are cylinders with holes punched out) and the “reggeon sector” (whose graphs are rectangles with holes punched out) are regularized using the lattice method of Giles and Thorn, as well as an alternative method. It is found that this string model is trivial. When the cut-off is removed, the pomeron and reggeon sector spectra are those of the free closed and open strings, respectively. This result is independent of the choice of g₀²n. A possible extension to fermionic strings, and the implications for large-n gauge theories are briefly discussed.     

The :φ2: Field in theP(φ)2 model

Euclidean Field Theory techniques are used to study the Schwinger functions and characteristic function of the :φ²: field in evenP(φ)₂ models. The infinite volume limit is obtained for Half-Dirichlet boundary conditions by means of correlation inequalities. Analytic continuation yields Lorentz invariant Wightman functions. It is shown that, in the infinite volume limit, <:φ(x)²:>≧0 for both the Half and the Full-Dirichlet (λφ⁴)₂ model. This result also holds for a finite volume with periodic boundary conditions.     

Hysteresis in the nonlinear driven drift and KdV equations

The nonlinear driven/damped differential equation ∂φ/∂t+a∂³φ/∂t∂²y+b∂³φ/∂³y+c∂φ/∂y+fφ∂φ/∂y= −ϵ sin (Ky-ωt)−γφ with a=0 (KdV equation) or b=0 (drift equation) is numerically studied in a parameter region where the energy tends to a constant, E_s, for t→∞. It is found that E_s(ϵ) traces a hysteresis curve when the driving amplitude ϵ is cyclically varied.     

A generalized leading-term approximation of high-energy scattering amplitudes in Yang-Mills theory

In this paper, the high-energy vector meson-vector meson elastic scattering amplitudes of a Yang-Mills theory of an SU(2) model are calculated in a generalized leading-term approximation (s→∞ with t fixed). The resulting amplitude, through the 10th order, coincides with the expansion of an eikonal formula. The present method is generalizable to other non-Abelian gauge theories.     

s-channel discontinuities of reggeon diagrams with planar vertices

The mutual relations between s-channel discontinuities of reggeon diagrams with planar triple-Regge vertices are discussed. It is shown that, in general, contributions coming from different cuttings are not simply related to each other as certain vertices acquire non-trivial phase factors depending on the nature of the cutting. For diagrams with only triple-Regge vertices this phase may be related to the triple-Regge phase. The complete four-reggeon vertices are not known in general. An approximation is considered which amounts to preserving only the absorptive part of triple- or four-reggeon amplitudes in certain parts of the diagrams. This approximation turns out to be equivalent to a simple absorbed multiperipheral model. The problem of exchange degeneracy breaking within the dual unitarisation Scheme is briefly discussed as an example.     

Higher order reggeon field theory results for dσ/dt

The critical reggeon field theory (RFT) asymptotic scaling law for dσ/dt can be evaluated using the ε expansion for renormalization group (RG) quaatities at t ≠ 0 along with the two variable RG equations found by Abarbanel, Bartels, Bronzan and Sidhu (ABBS). The amplitude thus obtained has an ε-exponentiated form. Interestingly, it reduces in a certain limit to the Bessel function pomeron model advocated by Kane. We perform a systematic numerical study here.Our results are as follows. Although the asymptotic RFT scaling is violated by the data, introduction of a mild external particle vertex with one parameter improvesthe situation considerably at current energies. Moreover, the shape of the ABBS scaling curve predicted by the O(ε²) results of Dash and Grandou is in at least qualitative agreement with the data, including the absence of a second dip. Two additional parameters (the t = 0 normalization and the scale in t) have to be fixed by hand.The overall conclusion is that in spite of many a priori objections the asymptotic RFT does a surprisingly good job of describing both the s and t dependence of dσ/dt at current energies.     

Second-order eikonal corrections for

The first-order eikonal approximation is frequently adopted in interpreting the results of A(e,e^′p) measurements. Glauber calculations, for example, typically adopt the first-order eikonal approximation. We present an extension of the relativistic eikonal approach to A(e,e^′p) which accounts for second-order eikonal corrections. The numerical calculations are performed within the relativistic optical model eikonal approximation. The nuclear transparency results indicate that the effect of the second-order eikonal corrections is rather modest, even at Q²≈0.2 (GeV/c)². The same applies to polarization observables, left–right asymmetries, and differential cross sections at low missing momenta. At high missing momenta, however, the second-order eikonal corrections are significant and bring the calculations in closer agreement with the data and/or the exact results from models adopting partial-wave expansions.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

Unitarizing high-energy scattering amplitudes in field theories I: QED

Using a procedure which incorporates the s-channel unitarity and t-channel unitarity at every step, we calculate the scattering amplitudes of QED in the high-energy limit. We find that all scattering amplitudes, elastic and inelastic, are summarized by a single eikonal formula which is explicitly unitary.     

Anharmonic oscillator model for first order structural phase transition

Exact solutions for the motion of a classical anharmonic oscillator in the potentialV(φ)=Bφ ² ? |A|φ ⁴ +Cφ ⁶ are obtained in (1 + 1) dimensions. Instanton-like solutions in (imaginary time) which takes the particle from one maximum of the potential to the other are obtained in addition to the usual oscillatory solutions. The energy dependence of the frequencies of oscillation is discussed in detail. This can be used as a model for the first order structural phase transition in the mean field approximation. The high and low temperature behaviour of the static susceptibility is obtained. Finally, a qualitative explanation is offered for the observed central peak in ferroelectrics like SrTiO₂.     

An eikonal perturbation theory having applications in hadron dynamics. II. Massive quantum electrodynamics

An eikonal perturbation theory (EPT), derived in previous work for a superrenormalizable coupling, is here developed for massive quantum electrodynamics (MQED) involving scalar or spinor matter fields minimally coupled to neutral massive vector gluons. After summarizing the functional method, we present the EPT for the external field problem. In agreement with results known within ordinary perturbation theory (OPT) in the eikonal approximation (EA), from an exact eikonal equation derived here we show that the EPT for the external field problem provides an excellent approximation method for Green's functions at large momenta. We then discuss some general features of the EPT for MQED, and show that it leads to a renormalizable approximation method. Our approach is then illustrated by deriving explicit expressions for various renormalized Green's functions in lowest order EPT. We also discuss some asymptotic properties of such Green's functions and indicate how to proceed with calculations in higher orders. As in our previous work, we again find that the renormalization procedure in EPT bears close resemblance to the one for OPT. Contrary to what happens with the EA, the inclusion of self-interactions and of other field-theoretic effects does not spoil the virtues of the EPT as a far better high-momenta approximation than the OPT. As a typical example, if s is an energy parameter and g the coupling constant with $\text{g}{}^2\text{4π}\text{< 1}$, OPT to order g²ⁿ often fails to be a good approximation as soon as $\text{(}\text{g}{}^2\text{4π}\text{)}\text{ln}\text{s ～ 1}$, while in such cases EPT to order g²ⁿ is still a good approximation as long as $\text{(}\text{g}{}^2\text{4π}\text{)}{}^{\mathrm{n+1}}\text{ln}\text{s < 1}$. We also find that the EPT is superior to the EA in that, contrary to the EA, it provides a step-by-step rigorous and renormalizable iterative approximation method which can account for self-interactions and other field-theoretic effects. We emphasize that the EPT is much simpler and more general than other explicit approximate summation methods of classes of OPT Feynman graphs.In field theory, we consider the use of the EPT as a generalization of the EA for discussing, e.g. high-energy behaviors in MQED as well as infrared divergence and bound-state problems in the limit of massless gluons. It is also suggested that, in view of its nice field-theoretic and high-energy properties, the EPT for MQED might provide a useful laboratory where ideas and problems in hadron dynamics could be meaningfully investigated within a Lagrangian field theory.     

Cut contributions in the triple-pomeron limit

We investigate pomeron cut contributions in the triple-pomeron limit of one-particle inclusive cross sections, near t = 0 where the triple-pole coupling vanishes. We find that at t = 0 the cuts themselves are suppressed, contributing factors (lnM²)^?2, rather than the single logarithms characteristic of cut contributions in two-body processes. We construct a simple reggeon calculus model in which all important cuts near t = 0 can be calculated, and suggest a simple way of parametrizing the data in that region.