Modified dispersion relations and sum rules

A systematic treatment of modifications to the usual form of dispersion relations and sum rules is given. The treatment is based on the fact that if a function g(z) has analyticity properties analogous to those of a scattering amplitudef (z), the usual dispersion relation for the productf (z)g(z) holds. After having described the general form of the weight function g(z) we concentrate on cases wheng(z) is large in a pre-chosen region and small elsewhere. Weight functions of this type are useful in practical applications where a weight functiong(z) may enhance a contribution of regions where accurate data are available and suppress those of regions with inaccurate or lacking data.Dedicated to Professor V. Votruba on his sixtieth birthday.     

Sum rule inequalities and subtractions in π π dispersion relations

Sum rule inequalities on the π π scattering amplitudes are derived from analyticity, s ? u crossing properties and positivity of these amplitudes. They connect the real and imaginary parts of the amplitude in the energy region where they may be calculated from phase-shift analysis, and do not require knowledge of these quantities at low energies or in the high energy region. The sum rules are used to confirm earlier results that two subtractions are needed in π π dispersion relations and this conclusion is now much more definite since in contrast to the earlier work, no knowledge of the magnitude of the absorptive part of the amplitude in the low energy region is needed.     

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.     

Partial wave relations from fixed-u dispersion relations

Partial wave relations are derived form fixed-u disperation relations for meson-baryon and meson-meson scattering. General expressions for the integral kernels are given and some problems pertaining to the application of the relations are discussed.     

Sum rules for radiative quarkonium decays from short-range interactions

Sum rules previously derived for proton decay are extended and applied to treat effects of bound states spectroscopy on radiative quarkonium decays. The transition involves boson (photon or pion) emission followed by quark annihilation. Sum rules for the contributions from different intermediate bound states are derived by using closure and the assumptions (1) that the boson emission is described by a plane wave or multipole operator which satisfies a wave equation, and (2) that the annihilation depends on the bound state wave function or its derivative at the origin.     

Sum rules for light-by-light scattering

We derive a set of sum rules for the light-by-light scattering and fusion: γγ→all, and verify them in lowest order QED calculations. A prominent implication of these sum rules is the superconvergence of the helicity-difference total cross section for photon fusion, which in the hadron sector reveals an intricate cancellation between the pseudoscalar and tensor mesons. An experimental verification of superconvergence of the polarized photon fusion into hadrons is called for, but will only be possible at e+ e- and γγ colliders with both beams polarized. We also show how the sum rules can be used to measure various contributions to the low-energy light-by-light scattering.     

Sum rules in perturbation theory

Sum rules derived from isospin current algebra are examined in perturbation theory for nucleons interacting with neutral pseudoscalar mesons and isovector pions. It is shown that the well-known sum rules for electromagnetic isovector form factors of nucleons and pions are correct in zero and second order of the meson-nucleon coupling constant.     

Sum rules for SUSY gluodynamics

The sum rules analysis for SU(N) SUSY gluodynamics is presented, concerning scalar, pseudoscalar and spinor channels. The spectrum obtained is characterized by a relatively dense disposition of excitations. The applicability of the effective lagrangian method to the study of resonance properties is thus made doubtful. Some new non-trivial relations between condensates in the SUSY vacuum are also presented.     

Sum rules in multiphoton coincidence rates

We show that sums of carefully chosen coincidence rates in a multiphoton interferometry experiment can be simplified by replacing the original unitary scattering matrix with a coset matrix containing 0s. The number and placement of these 0s reduces the complexity of each term in the sum without affecting the original sum of rates. In particular, the evaluation of sums of modulus squared of permanents is shown to turn in some cases into a sum of modulus squared of determinants. The sums of rates are shown to be equivalent to the removal of some optical elements in the interferometer.     

Sum rules for the reflectivity coefficients

Sum rules are derived for the logarithm of the absolute value and the phase angle of the amplitude for normally reflected radiation.     

Sum rules for nuclear collective excitations

Characterizations of the response function and of integral properties of the strength function via a moment expansion are discussed. Sum rule expressions for the moments in the RPA are derived. The validity of these sum rules for both density independent and density dependent interactions is proved. For forces of the Skyrme type, analytic expressions for the plus three energy weighted sum rules are given for isoscalar monopole and quadrupole operators. From these, a close relationship between the monopole and quadrupole energies is shown and their dependence on incompressibility and effective mass is studied. The inverse energy weighted sum rule is computed numerically for the monopole operator, and an upper bound for the width of the monopole resonance is given. Finally the reliability of moments given by the RPA with effective interactions is discussed using simple soluble models for the hamiltonian, and also by comparison with experimental data.     

Sum rules in extended RPA theories

Different moments m_k of the excitation strength function are studied in the framework of the second RPA and of the extended RPA in which 2p2h correlations are explicitly introduced into the ground state by using first-order perturbation theory. Formal properties of the equations of motion concerning sum rules are derived and compared with those exhibited by the usual Ip1h RPA. The problem of the separation of the spurious solutions in extended RPA calculations is also discussed.     

Sum rules and analyticity in QCD

QCD sum rules of Shifman, Vainstein and Zakharov are investigated from the point of view of analytical extrapolations: in the weighted sum rule the analyticity connects the low energy hadron parameters with the high energy QCD parameters (coupling constant, quark masses, condensate parameters). It is shown that the optimal weight function in the sum rule is prescribed by the errors of the input information. The optimal tests of analyticity are proposed and applied (under some simplifying assumptions) to the vector and pseudoscalar charmonium channels.     

Sum rules for the vertex function from local commutativity (II) Relations for the causal boundary values in mometum space

The sum rules derived in an earlier paper [1] are applied to the momentum-space boundary values of the vertex function and it is shown how to use the formulas to derive relations for the dynamical on-the-mass-shell quantities, i.e. the matrix elements of locally commuting field operators.     

Sum rules for forward elastic pion-nucleon scattering

A number of energy weighted sum rules relating amplitudes and differential cross sections for forward elastic and charge exchange scattering to the total pion-nucleon cross section are derived from dispersion relations.