Subtracted Dispersion Relations Research Articles

Dispersion relations and subtractions in hard exclusive processes

We study analytical properties of the hard exclusive process amplitudes. We found that QCD factorization for deeply virtual Compton scattering and hard exclusive vector meson production results in the subtracted dispersion relation with the subtraction constant determined by the Polyakov-Weiss $D$-term. The relation of this constant to the fixed pole contribution found by Brodsky, Close, and Gunion and defined by parton distributions is studied and proved for momentum transfers exceeding the typical hadronic scale. The continuation to the real photon limit is considered, and the numerical correspondence between lattice simulations of the $D$-term and low energy Thomson amplitude is found. For sufficiently large $t$ the subtraction may be expressed in a form similar to that suggested earlier for real Compton scattering.

Proton spin polarizabilities from polarized Compton scattering

Polarized Compton scattering off the proton is studied within the framework of subtracted dispersion relations for photon energies up to 300 MeV. As a guideline for forthcoming experiments, we focus the attention on the role of the proton's spin polarizabilities and investigate the most favorable conditions to extract them with a minimum of model dependence. We conclude that a complete separation of the four spin polarizabilities is possible, at photon energies between threshold and the $\ensuremath{\Delta}(1232)$ region, provided one can achieve polarization measurements with an accuracy of a few percent.

ALTERNATIVE CALCULATION OF THE PHYSICAL MASS OF THE ρ-MESON

We have derived an expression for the physical mass and width of the ρ-meson in vacuum from its spectral function, calculated in the vector meson dominance model when a ρ0meson couples to two virtual pions π+–π-. The propagator is computed after evaluating the ρ-meson self-energy. The real part of the ρ-meson self-energy is given by a divergent integral and needs to be regularized; the regularization is done by using a double subtracted dispersion relation. The result leads to a closed analytical expression which allow us to evaluate the spectral function in a closed way. The physical mass, defined as the magnitude of the four-momentum |k| for which the spectral function S(k2) attains its maximum value, is obtained, and it gives a value of 770 MeV, which is in total agreement with the reported experimental value of the ρ-meson mass.

Fixed- t subtracted dispersion relations for Compton scattering off the nucleon

Fixed- t subtracted dispersion relations are presented for Compton scattering off the nucleon at energies E γ ≤ 500 MeV, as a formalism to extract the nucleon polarizabilities with a minimum of model dependence. The scalar polarizability difference α - β and the backward spin polarizability γ π enter directly as fit parameters in this formalism.

Fixed-tsubtracted dispersion relations for Compton scattering off the nucleon

Fixed-$t$ subtracted dispersion relations are presented for Compton scattering off the nucleon at energies ${E}_{\ensuremath{\gamma}}<~500 \mathrm{MeV},$ as a formalism to extract the nucleon polarizabilities with a minimum of model dependence. The subtracted dispersion integrals are mainly saturated by $\ensuremath{\pi}N$ intermediate states in the s channel, $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\pi}\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\gamma}N,$ and $\ensuremath{\pi}\ensuremath{\pi}$ intermediate states in the t channel, $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremath{\pi}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\pi}}N\overline{N}.$ For the subprocess $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}\ensuremath{\pi}\ensuremath{\pi},$ we construct a unitarized amplitude and find a good description of the available data. We show results for Compton scattering using the subtracted dispersion relations and display the sensitivity on the scalar polarizability difference $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ and the backward spin polarizability ${\ensuremath{\gamma}}_{\ensuremath{\pi}},$ which enter directly as fit parameters in the present formalism. Double polarization observables are shown to have a unique potential for measuring the spin polarizabilities of the nucleon.

Subtracted dispersion relations for in-medium meson correlators in QCD sum rules

We analyze subtracted dispersion relations for meson correlators at finite baryon density and temperature. Such relations are needed for QCD sum rules. We point out the importance of scattering terms, as well as finite, well-defined subtraction constants. Both are necessary for consistency, in particular for the equality of the longitudinal and transverse correlators in the limit of vanishing three-momentum of mesons relative to the medium. We present detailed calculations in various mesonic channels for the case of the Fermi gas of nucleons.

Possible violation of sum rules for higher-twist parton distributions

Precise measurements of polarized electro-production and Drell-Yan in the deep inelastic limit will soon provide first information on the higher twist parton distributions $g_T(x)$ and $h_L(x)$. Sum rules for higher-twist structure functions are only valid provided the corresponding Compton amplitudes satisfy un-subtracted dispersion relations. Subtracted dispersion relations have to be used when the (real part of the) forward scattering amplitudes does not fall off rapidly enough for $\nu \rightarrow \infty$ (fixed $Q^2$). Formally, such subtractions lead to $\delta$-functions at the origin in the parton distributions, which are not accessible to experiment, and the integral over the data fails to satisfy the sum rule. The $\delta$-functions in the parton distributions can be identified with the zero-modes that appear in light-front quantization. An explicit {\it infinite momentum boost} identifies these soft quark modes with low momentum contributions to the self-energy.

Geometry of spacetime propagation of spinning particles

We develop a unified geometrical description of spacetime propagation of particles of arbitrary spin. We systematize, from the point of view of the theory of induced representations, the various forms of relativistic wave equations. Utilizing Fronsdal's construction of a class of unconstrained wave equations, we obtain closed and explicit forms of the particles' propagators, discuss their analytic properties in the momentum and mass, and express them through subtracted dispersion relations. By requiring unitarity, we then establish the spin-statistics relation. We discuss a class of associative algebras arising in the construction of the propagators and wave functions. These (momentum-space) wave functions are sections of bundles over (complexified) spheres, constructed in analogy to the Yang-Mills instanton bundles. We then derive proper-time and ramdom-walk representations of the propagators. The latter are conveniently expressed in terms of matrix-valued complex “measures” on the spaces of particle's paths. We develop a general method for analyzing fractal properties of sample paths, based on the calculation of ball hitting probabilities (or amplitudes), and find Hausdorff dimensions of paths for half-integer and integer spins.

QCD corrections to vector boson self-energies in the standard model

The O (α s) corrections to the vacuum polarization tensor of the gauge bosons are presented for both light and heavy quark flavours in the standard model of electroweak interactions on the basis of a running coupling constant α s. The influence of ϒ, T b and θ resonance as well as the proper threshold behaviour of the imaginary part of the vacuum polarization function are incorporated in the analysis. The real part is calculated numerically through the properly subtracted dispersion relation. A set of FORTRAN routines that implements the scheme is described.

Concerning dispersion relations for the magnetotelluric impedance tensor

SUMMARY Adopting a purely systems-theoretic viewpoint, it is shown how a small set of well-established general physical principles such as causality, stability, and passivity, applied to the earth system, determines the analytical properties of the associated impedance tensor in the complex frequency plane. These physical principles and their consequences are utilized to obtain an integral representation for the impedance tensor based upon the Cauer representation for positive functions. This representation is developed primarily by the imposition of the passivity and symmetry requirement on the impedance tensor. As an application, the representation is employed to develop a set of inequality constraints (necessary conditions) that must be verified by the impedance tensor. Since linearity and passivity imply causality, the Cauer integral representation for the impedance tensor is then utilized to derive a subtracted dispersion relation that connects the real and imaginary Hermitian parts of the tensor on the real frequency axis. Furthermore, it is shown how subtracted dispersion relations for the impedance tensor may be developed directly from the causality principle without recourse to the Cauer representation. In this fashion, the dispersion relations for the scalar response function, developed by Weidelt and Fischer & Schnegg in the context of 1-D and 2-D earth systems, respectively, have been extended to the tensor response function corresponding to general 3-D earth systems. Although the focus in this paper is limited to the explicit development of dispersion relations for the impedance tensor, the applicability and scope of the methods utilized are of a more general nature and can be employed to develop integral relationships for other Earth response functions (e.g. tipper responses). For the general 3-D earth, the dispersion relations have direct applications in the formulation of consistency tests, the construction of consistent estimates of the impedance tensor, and the identification of non-linearities.

Self-energy, momentum distribution, and effective masses of a dilute Fermi gas

The dependence upon momentum $k$ and upon energy $E$ of the self-energy of a dilute Fermi gas is studied up to terms of order ${({k}_{F}c)}^{2}$, where ${k}_{F}$ denotes the Fermi momentum and $c$ is the positive scattering length. Algebraic expressions are derived for the imaginary part $W(k;E)$ of the self-energy in the whole ($k$,$E$) plane. They are compared with a conjecture recently made by Orland and Schaeffer in their analysis of single-particle states in nuclei. The contributions of core polarization and of ground state correlations to the real part $V(k;E)$ of the self-energy are calculated with the help of subtracted dispersion relations which connect them with $W(k;{E}^{\ensuremath{'}})$. Algebraic expressions are derived for the momentum distribution in the correlated ground state. It is shown that the effective mass of a quasiparticle with momentum $k$ is equal to the bare particle mass at $k=0$ and reaches a local maximum for $k$ close to ${k}_{F}$. This maximum is ascribed to the dependence of $V(k;E)$ upon $E$, which is described in terms of an $E$ mass. We compute the contributions of core polarization and of ground state correlations to this $E$ mass. The dependence of $V(k;E)$ upon $k$ reflects the nonlocality of the self-energy. It is characterized by a $k$ mass that we also calculate. These results shed light on some nuclear matter properties and on the meaningfulness and limitation of nuclear matter calculations that have recently been performed in the framework of the Brueckner-Hartree-Fock approximation.NUCLEAR STRUCTURE Effective mass near the Fermi surface in dilute Fermi gas model.

Dispersion relations for the stark effect in the hydrogen atom

On the basis of the analyticity properties of an approximate expression for the energy level, we obtain a twice subtracted dispersion relation for the Stark shift in the ground state of the hydrogen atom. This relation allows us to obtain (i) the large- n behaviour of expansion coefficients, (ii) the energy shift and level width for a wide range of values of the field.

The slope of the form factor in pseudoscalar Dalitz decays

It is pointed out that a negative slope for the form factor in pseudoscalar Dalitz decays can be explained by vector-meson dominance with a subtracted dispersion relation. Calculations are made for the pi 0 and eta using the experimental omega to pi 0 gamma , phi to pi 0 gamma and phi to eta gamma decay rates and SU(3) symmetry. An incidental consequence of these calculations is that excitable colour gluons do not significantly affect eta Dalitz decay.

Calculation of the nucleon optical potential via dispersion relations

The energy dependence and radial behaviour of the real and imaginary part of the nucleon optical potential is obtained by generalizing a model so far applied to infinite nuclear matter. The method is based on the determination of the imaginary part by its relationship to the total reaction cross section. For the calculation of this quantity the irreducible vertex and the target autocorrelation function of the density fluctuations are needed. The real part emerges from a dispersion relation which contains an energy-dependent contribution and the Hartree-Fock potential. The dispersion relation has been applied in two versions. In the so-called subtracted form-where the extreme energies play a minor role-a phenomenological homogeneous term is used. The calculations were performed using a non-singular force. The agreement with the phenomenological potentials is satisfactory using the subtracted dispersion relation; in the non-subtracted version too strong real potentials ( approximately 10-15 MeV) are obtained.

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.

Low-energy theorems for nuclei

A generalization of low-energy theorems is given to obtain the Compton-scattering amplitude on nuclei up to energies smaller than the excitation of nucleons. By means of this technique the presence of seagull terms coming from exchange forces is predicted and the consistency of their contribution is demonstrated in the framework of subtracted dispersion relations.

A dispersion series for nonlocal potentials

For a class of short−range nonlocal potentials, and for the energy variable E in a certain part of the complex plane, we obtain a generalized subtracted dispersion relation which relates the forward scattering amplitude to contributions from negative energy pole terms, a usual dispersion integral along the positive real axis of the complex energy plane, and a uniformly convergent infinite series, apart from subtraction terms, subject to the condition that no bound state exists with energy less than −γ2, where γ is some parameter of

Constraints imposed by the data above 1 GeV on the low-energy ππ amplitudes

We study the influence of the region above 1 GeV on low-energy ππ amplitudes by using manimum subtracted dispersion relations, in addition to the twice subtracted ones. When our results are compared with the recent phase-shift analysis below 1 GeV, we find that the S-wave isospin-zero scattering length has to lie between 0.5 and 0.8.

The pion form factor at a timelike momentum transfer of approximately 1 fm −2 from π+ electroproduction at threshold

From an extrapolation of threshold π + electroproduction data combined with a kinematical constraint at an unphysical point, we derive the value of the pion form factor at a time-like value of the momentum transfer k 2 π ≅ 1 fm −2. The result is F π ( k 2 π ) = 1.19 ± 0.1. From this value, using a twice subtracted dispersion relation, we obtain r π ⪆ 0.98 ± 0.24 fm .

The derivative of the π0π0 → π0π0 G-wave

Constraints on the derivative of the G-wave a 4( s) for π 0 π 0 → π 0 π 0 are investigated using crossing sum rules which follow from quadruply subtracted dispersion relations. In particular it is shown that da 4(s) ds ⩽0 for 4μ 2 ⩾ s⩾ 1.488 μ 2 .