Subtraction Constants Research Articles

Real parts of the pion-proton forward scattering amplitude

The most recent total-cross-section data are used to calculate real parts of the forward elastic ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ scattering amplitudes from threshold to 240 GeV/c. Using statistical and systematic uncertainties of the total cross sections and their momenta, along with uncertainties of the subtraction and coupling constants, unphysical cuts, and cross-section extrapolations, we calculate the uncertainties of the real amplitudes. Our results are compared to experimental and other theoretical determinations of the ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ forward real amplitudes.

Dynamical equations for a Regge theory with crossing symmetry and unitarity. II. The case of strong coupling, and elimination of ghost poles

Equations for the construction of a crossing-symmetric unitary Regge theory of meson-meson scattering are described. In the case of strong coupling, Regge trajectories are to be generated dynamically as zeros of the $D$ function in a nonlinear $\frac{N}{D}$ system. This paper is concerned mainly with writing the inputs to the $\frac{N}{D}$ system in such a way that a convergent theory with exact crossing symmetry is defined. The scheme demands elimination of ghosts, i.e., bound-state poles at energies below threshold where trajectories pass through zero. A method for ghost elimination is proposed which entails an $s$-wave subtraction constant, and allows the physical $s$ wave to be different from the $l$-analytic amplitude evaluated at $l = 0$. A dynamical model is suggested in which the subtraction constant alone generates the meson-meson interaction. An alternative ghost-elimination scheme proposed by Gell-Mann, in which only $l$-analytic amplitudes are involved, can be discussed in a formalism including channels with spin.

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.

Results for the $$\pi {\mathcal{N}}$$ crossing-odd forward-scattering amplitude from a new dispersion relation analysiscrossing-odd forward-scattering amplitude from a new dispersion relation analysis

A simultaneous analysis of total cross-sections, real parts, charge exchange forward differential cross-sections and finite-energy sum rules is performed for the crossing-odd\(\pi {\mathcal{N}}\) forward-scattering amplitude, in order to compare several different high-energy models. It is found that other singularities in theJ-plane are required besides the ρ Regge pole. Some implications of one of the possible alternatives, namely the presence of a subtraction constant in the real part, are also discussed.

On the practical evaluation of the asymptotic real-part behaviour and the search for a possible subtraction constant in the crossing-odd forward pion-nucleon amplitude

On the practical evaluation of the asymptotic real-part behaviour and the search for a possible subtraction constant in the crossing-odd forward pion-nucleon amplitude

Dispersion relations with damping function

Dispersion relations are proposed in a new form containing a certain damping function. They contain no arbitrary parameters (subtraction constants or even field coupling constants) and are highly convenient both for experimental tests at high energies and for the description of strong interactions of particles at low and medium energies.

Spin dependent high frequency inelastic electron scattering and helicity flip couplings

We examine the Bjorken scaling and Regge behaviours expected in the high enery inelastic scattering of polarized electrons on a polarized nucleon target. Attention is concentrated on the occurence of operators or exchanges that couple predominantly to non-zero helicity flips ( M) in forward single hadron matrix elements. Bilocal operators in the light cone expansion, with M ≠ 0, have previously been argued to be detectable through changes in the zero moment sum rule of the Bjorken scaling function g 2. We show that this is not the case when one consistently imposes current conservation on the light cone, and that the usual expansion already fully includes the M ≠ 0 bilocals. Instead, M ≠ 0 exchanges can show up in the Regge limit, and be easily detectable. If cuts occur, the cross section is a factor ν 2 higher in ν-dependence than with ordinary poles. We apply these results to fixed mass and scaling function sum rules, the hydrogen hyperfine structure, and view them in the light of positivity restrictions. Maximal Regge behaviour for spin dependent structure functions excludes the avoidance of the unknown subtraction constant in dispersive calculations of the proton-neutron electromagnetic mass difference.

Pion-pion scattering from an SU(3) × SU(3) invariant Lagrangian

Abstract We calculate the low energy pion-pion phase shifts from the SU(3) × SU(3)invariant meson-baryon Lagrangian in the one-loop approximation. The meson loop diagrams are evaluated using superpropagator methods and symmetry arguments. The derivative nature of the meson-baryon coupling leads to one undetermined subtraction constant in the baryon loop amplitudes which is fixed by the requirement that the p-wave exhibits a resonance at the ρ meson mass. Taking the D / F ratio for the meson-baryon coupling from experiment we derive I = 0 and I = 2 s-waves in good agreement with the available data.

Renormalization of a spontaneous symmetry-breaking model in the unitary gauge

For the Weinberg-Salam model of leptons a comparison is made between the subtraction constants in the unitary gauge and the subtraction constants in a renormalizable gauge. It is also shown that second-order corrections in muon decay are the same in both gauges.

A causal approximation scheme for the commutator of the axial vector-currents

An approximation scheme which fulfils the causality condition is used to compute the matrix elements of the commutator of two axial-vector currents between one proton state. The commutator sum rule is supposed to be saturated by taking into account the nucleon, theN*(3/2, 3/2) resonance, and the partially disconnected pion-proton intermediate states. For the axial current form factors pion pole formulas are assumed. It is shown that only the subtraction constants are relevant for the equal time limit of the commutator and so the equivalence with similar earlier methods is proven.

The subtraction constant in π± electroproduction

A method is proposed for dealing with the subtraction constant required in fixed- t dispersion relations for π ± electroproduction. The method requires a knowledge of the t-channel cut contribution (evaluated at the subtraction points s 0), which can be determined at k 2 = 0 using a gauge invariance sum rule. Estimates by this and other methods show that it is small (compared with the pion pole contribution) for − t ⪅ 1 GeV 2, and in particular it is irrelevant to determinations of the pion form factor using high-energy forward π + data.

Corrections to Weinberg's pion-pion scattering lengths

We derive twice subtracted dispersion sum rules for theππ scattering lengths. The subtraction constants are the Weinberg values, and the dispersion integrals yield small corrections in good agreement with other authors.

Single-Loop Calculations with the Chiral-Invariant Pion Lagrangian

In the one-loop approximation to the chiral-invariant pion theory, it is shown that the unrenormalized on-shell $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude depends on the choice of pion field, while the renormalized amplitude does not. By elementary calculation it is verified that only one arbitrary subtraction constant, in addition to conventional renormalization, is required.

Higher-order terms in theV-A theory of the weak interactions and leptonic scattering processes at high energies

In this paper an attempt is made to evaluate higher-order corrections for theV-A theory of the weak interactions. Instead of the customary cut-off method, a perturbative expansion at the second order inGμ is employed. Then we compute the second-order Feynman diagrams introducing indetermined subtraction constants to make the matrix elements finite. This analysis is applied to the leptonic scattering process at high energies. It is shown that their cross-sections can be parametrized in terms of one unknown constantb1, under the assumption thatb1≫1. Ifb1 is sufficiently large,i.e. b1≅104, the second-order corrections to the first-order leptonic processes and the cross-sections of the second-order processes are sizable and could be detected at NAL. Finally the consequences of the lack of the diagonal interaction in theV-A leptonic Lagrangian are discussed.

A BPH approach to weak interactions

A Bogoliubov-Parasiuk-Hepp renormalization program for the usual Fermi Lagrangian is proposed. All second order processes are given in terms of only two subtraction constants associated with “counter terms” exhibiting a high degree of lepton-hadron universality.

Is theε(700)Meson the Goldstone Boson of Broken Scale Invariance?

We examine low-energy theorems of broken scale invariance, using for our tests phenomenological ($\ensuremath{\pi}\ensuremath{\pi}$) phase shifts and thus improving on the usual pole approximation for the $\ensuremath{\epsilon}(700)$ meson, commonly assumed to act as a Goldstone boson. Approximate consistency is found for a model based on broken S${\mathrm{U}}_{3}$ \ifmmode\times\else\texttimes\fi{} S${\mathrm{U}}_{3}$ symmetry, provided that a certain subtraction constant, already present on the level of pole models, is used as a free parameter to be fitted by the data. Yet there are definite indications that contrary to the role of the pion in broken chiral symmetry, broken scale invariance cannot be understood in terms of the $\ensuremath{\epsilon}(700)$ meson alone and contributions from higher states are needed in an essential manner.

Finite-energy sum rules for partial waves and their formulation for the N/ D equations

We derive constraints for partial wave amplitudes from Regge asymptotics in the form of finite-energy sum rules. For simplicity we consider only the scattering of scalar particles and assume a pure Regge pole behaviour of the total amplitude together with a strong enhancement of forward and backward scattering. We formulate the constraints also for the N/ D equations in order to determine possible subtraction constants and CDD-pole parameters.

Nonleptonic Decays of Hyperons

Once-subtracted dispersion relations are written for the amplitudes of the nonleptonic decays of hyperons. The subtraction constants are given by equal-time commutators and, after the low-mass baryon intermediate states are separated, the remaining dispersion integrals over higher-mass resonances are evaluated in the framework of the Regge theory. A four-parameter best-fit solution to both the $s$- and $p$-wave amplitudes is found to be in very good agreement with experiment.

Unitarized pion-pion Veneziano partial-wave amplitudes

A unitarization method is investigated which is based on dispersion relations for the inverse amplitude; the zeros of the partial-wave amplitudes are taken from the Veneziano model and the discontinuity across the right-hand cut is given by elastic unitarity, whereas crossing symmetry yields the left-hand cut discontinuity (for large negatives-values the latter is assumed to be constant). Two free subtraction constants are fixed by minimizing the violation of crossing symmetry-expressed in terms of a number of exact conditions on the partial-wave amplitudes. It was not possible to satisfy all of the crossing constraints simultaneously, but we arrived at a unique solution of the dispersion relations (with a very broadI = 0S-wave resonance) which is unitary and approximately crossing symmetric.

Weak Interactions at High Energies

We discuss the weak interactions from an $S$-matrix point of view in order to make some qualitative statements about high-energy behavior. In order to separate the weak interactions from others as much as possible, attention is focused on the weak interactions of leptons neglecting electromagnetism. We examine the consequences of imposing the constraints of unitarity and analyticity on weak amplitudes whose low-energy behavior is assumed to be correctly given by the usual Fermi theory (including the possibility of neutral currents). We first study corrections to the low-energy limit by using the Mandelstam iteration to express all lepton-lepton two-body scattering amplitudes to third order in the Fermi constant in terms of a small number of subtraction constants. We then speculate beyond perturbation theory. We discuss some consequences of forward dispersion relations and propose a mechanism for providing the necessary damping of weak amplitudes at high energies. The mechanism is the existence of the intermediate vector boson coupled weakly to leptons but interacting strongly with itself. We examine some consequences of this hypothesis.