Rigorous sum rules and bounds on pion-pion scattering amplitudes with positivity

We derive the complete set of crossing symmetric forward pion-pion amplitudes with non-negative absorptive parts obeying twice-subtracted dispersion relations. We give a method to eliminate subtraction constants for these amplitudes by making a simultaneous use of subtracted dispersion relations for the amplitude and its inverse, and unitarity. We thus deduce from axiomatic field theory (i) lower bounds on forward real parts in terms of physical forward amplitudes, (ii) lower bounds on S- and P-wave scattering lengths in terms of forward amplitudes in any part of the physical region, (iii) sum rules on scattering lengths in terms of physical forward amplitudes which provide a complete test of forward dispersion relations, (iv) upper bounds on scattering lengths in terms of total cross sections and (v) absolute sum rule inequalities on total and forward differential cross sections in terms of the pion mass alone which provide new tests of dispersion relations not involving scattering lengths.     

Rigorous absolute bounds for pion-pion scattering (1). Solving a first extremum problem

This paper is the first of a series of three where we reconsider the derivation of rigorous absolute bounds for strong interactions. It is devoted to the solution of a preliminary extremum problem. From the knowledge, at a given physical energy s and unphysical angle cos θ₀ > 1, of the absorptive part A (s, cos θ₀) of the elastic scattering amplitude for two spinless particles, we find, by exploiting the unitarity condition, the least upper bound of the modulus |F(s, cos θ₁)| of the scattering amplitude for the same energy s and various angles θ₁. Upper (but not least upper) bounds given by previous authors are numerically compared with ours.     

Dispersion relations for forward pion-pion scattering and their consistency with Wolf pion-pion phase shifts

It is shown that the Wolf pion-pion phase shifts are in reasonable agreement with dispersion relations for forward scattering. The extent to which the inaccuracies of the experimental data might modify this agreement is examined.At attempt is made to explain some discrepancies which arise probably from too broad a -meson peak.The author is indebted to Dr. M. Petrá for valuable discussions, to P. Bóna for help with some calculations and to Dr. J. Mikloko from the Computing Centre of the Slovak Academy of Sciences for programming and computation. He is also indebted to Dr. G. Wolf and to Dr. F. Meiere for sending a copy of their papers prior to publication.     

New lower bounds on the self-avoiding-walk connective constant

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice ^d . The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for highd, and in fact agree with the first four terms of the 1/d expansion for the connective constant. The bounds are the best to date for dimensionsd 3, but do not produce good results in two dimensions. Ford=3, 4, 5, and 6, respectively, our lower bound is within 2.4%, 0.43%, 0.12%, and 0.044% of the value estimated by series extrapolation.     

Rigorous bounds for the complex dielectric constant of a two-component composite

Rigorous bounds are systematically derived for the bulk effective dielectric constant of a two-phase composite material ε_e under various assumptions about the available information. The bounds are drawn for some special cases in the complex ε_e-plane. The question of optimality of the bounds and their connection to solvable microgeometries is discussed. It is shown how linear programming theory can be used to aid in the derivation of these bounds.     

Effective method for calculating the analytic QCD coupling constant

The analytic running coupling constant α _an for strong interactions is considered for approximations of standard perturbation theory up to the three-loop level. Nonperturbative contributions are singled out explicitly in α _an. They are represented in the form of an expansion in a series in inverse powers of the Euclidean momentum squared. It is shown that two-and three-loop corrections lead to a partial compensation of the nonperturbative one-loop contribution of order 1/q ², which is leading in the ultraviolet region. An efficient method for calculating the analytic running coupling constant for all q>Λ is developed on the basis of the above expansion. A comparative analysis of perturbative and nonperturbative contributions is performed in the infrared region, where the latter play the most important role. A simultaneous consideration of the momentum dependence of α _an and its perturbative component for one-to three-loop cases leads to the conclusion that the analytic running coupling constant is stable with respect to higher corrections and that it depends only slightly on conditions imposed in matching solutions that involve different numbers n _f of active-quark flavors.     

Comments on upper bounds on the inflationary Hubble constant

This is a comment on a letter by Hawking establishing an upper bound on the value of the Hubble parameter during inflation. Here we show that his derivation of the result is based on an invalid mathematical approximation, and demonstrate that a similar bound can be obtained from the standard “slow-rollover” constraint. We also present a derivation of the same result that is completely independent of the form of the scalar field potential.     

New estimate of the pairing coupling constant

The pairing interaction strength is determined from empirical odd-even mass differences for rare-earth and actinide nuclei. The calculation was performed with and without projection on the given particle number when solving the BCS equations.     

Upper bounds on the wave function renormalization constant of pion

Using unitarity, analyticity and the hypothesis of Bjorken scaling inequalities have been derived for the upper bounds on the wave function renormalization constant of pion.     

The pion nucleon coupling constant

The latest elastic scattering data are re-analysed to determine the coupling constant g_c of the charged pion, using the dispersion relation for the invariant amplitude B^{(+ )}. Depending on the choice of data-base, values to 13.65 are obtained with errors of . The mass difference between charge states of is MeV, close to twice the mass difference between neutron and proton. The difference in widths on resonance is MeV. One may account for a width difference of 4.5 MeV from phase space for decays and the extra channel .Received: 21 January 2004, Published online: 3 March 2004     

Optical method for measuring the coupling constant of surface acoustic waves

The coupling constant of common or thin-layered-piezoelectric interdigital transducers for surface acoustic waves up to the highest frequencies can be determined in an optical way with reproducibility better than 3%. Being a nondestructive and noncontact method one can apply it in the technological process for fabricating uniform piezoelectric layered transducers.     

Strong coupling constant from the photon structure function

We extract the value of the strong coupling constant alpha(s) from a single-parameter pointlike fit to the photon structure function F(gamma)(2) at large x and Q(2) and from a first five-parameter full (pointlike and hadronic) fit to the complete F(gamma)(2)data set taken at PETRA, TRISTAN, and LEP. In next-to-leading order and the MS renormalization and factorization schemes, we obtain alpha(s)(m(Z))=0.1183+/-0.0050(expt)(+0.0029)(-0.0028)(theor) (pointlike) and alpha(s)(m(Z))=0.1198+/-0.0028(expt)(+0.0034)(-0.0046)(theor) (pointlike and hadronic). We demonstrate that the data taken at LEP have reduced the experimental error by about a factor of 2, so that a competitive determination of alpha(s) from F(gamma)(2) is now possible.     

Experimental determination of the effective strong coupling constant

We present a first attempt to experimentally extract an effective strong coupling constant that we define to be a low Q²$Q^2$ extension of a previous definition by S. Brodsky et al. following an initial work of G. Grunberg. Using Jefferson Lab data and sum rules, we establish its Q²$Q^2$-behavior over the complete Q²$Q^2$-range. The result is compared to effective coupling constants inferred from different processes and to calculations based on Schwinger–Dyson equations, hadron spectroscopy or lattice QCD. Although the connection between the experimentally extracted effective coupling constants and the calculations is not established it is interesting to note that their behaviors are similar.