Unsubtracted Dispersion Relations Research Articles

High-energy neutrino-nucleon scattering, current algebra and partons

The structure functions F j which describe high-energy neutrino-nucleon scattering are discussed. The analogue of the Callan-Gross result for electro-production is derived, viz. lim q 2→−∞ F 1 = lim q 2→−∞ F 1 = 0 with the algebra of field commutators and lim q 2→−∞ (F 2−ωF 1) = 0 with the quark model commutators. A sum rule for F 3 is derived assuming that F 3 satisfies an unsubtracted dispersion relation, as suggested by Regge theory. The breakdown of these current algebra results in perturbation theory is dicussed. The F i are also discussed in the ‘parton’ model, in which it is shown that F 3 is a measure of the average baryon number of the partons. In the ‘ quark-parton’ model of Bjorken and Paschos F 3 is small and our sum rule valid. In the ‘field theory parton’ model of Drell, Levy and Yan F 3 is large, in contradiction to Regge theory, and our sum rule invalid. This model predicts that anti-neutrino scattering vanishes in the backward direction.

Pion Lifetime,ρπandσπIntermediate States, and Sum Rules

The pion lifetime is calculated assuming an unsubtracted dispersion relation for the axial-vector matrix element and using $\ensuremath{\rho}\ensuremath{\pi}$ and $\ensuremath{\sigma}\ensuremath{\pi}$ intermediate states as well as the $\overline{N}N$ intermediate state. By imposing convergence conditions, a series of sum rules is obtained, including the generalized Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin relation and modified Goldberger-Treiman relations. By assuming that the $\mathrm{NN}\ensuremath{\pi}$, $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$, and $\ensuremath{\sigma}\ensuremath{\pi}\ensuremath{\pi}$ vertex functions are dominated by a three-pion resonance with $I=1$ and ${J}^{P}={0}^{\ensuremath{-}}$, it is found possible to make the Goldberger-Treiman relations agree with experiment if the resonance mass is ${M}_{R}\ensuremath{\approx}2$ GeV. This resonance is tentatively identified with the $\overline{N}N$ resonance structure observed around 2 GeV.

Unsubtracted Dispersion Relation, Current Algebra, and Momentum-Dependent Vector Form Factor inKl4Decays

Unsubtracted Dispersion Relation, Current Algebra, and Momentum-Dependent Vector Form Factor in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi><mml:mn>4</mml:mn><mml:mn /></mml:mrow></mml:msub></mml:mrow></mml:math>Decays

Pion-Mass Extrapolations and Partial Conservation of Axial-Vector Current

It is proposed, as suggested by a recent study, that the observed 10% corrections to the Goldberger-Treiman relation cannot be accounted for on the basis of unsubtracted dispersion relations. To incorporate the possibility of a small subtraction in the matrix elements of the divergence of the axial-vector current, a supersmooth pion field is defined from the weak interactions by $i{\ensuremath{\partial}}_{\ensuremath{\mu}}{{A}_{\ensuremath{\mu}}}^{(+)}(\mathrm{x})=(\frac{2M{g}_{A}{\ensuremath{\mu}}^{2}}{\sqrt{2}g}){\ensuremath{\pi}}^{(+)}(x)\ensuremath{-}\ensuremath{\Delta}{f}_{\ensuremath{\pi}}\ensuremath{\square}{\ensuremath{\pi}}^{(+)}(x)$, with $\ensuremath{\Delta}=1\ensuremath{-}\frac{2M{g}_{A}}{\sqrt{2}g {f}_{\ensuremath{\pi}}}\ensuremath{\simeq}+0.1$ representing the 10% correction. With this definition and the supersmoothness hypothesis the extrapolation of exact current-algebra threshold theorems for weak amplitudes to relate hadron amplitudes is relatively simple and eliminates a major ambiguity in performing such extrapolations in the virtual pion mass.

Perturbation theory and the goldberger-treiman relation

The Goldberger-Treiman relation was originally derived (~) by postulating that the form factor c~(q), which is the decay constant for a pion of four-momentum qg, satisfies unsubtracted dispersion relations, and that these dispersion relations could be saturated by the contribution of the nueleon-antinueleon intermediate state. Under certain approximations, it was then possible to relate e~ to the pion-nucleon coupling constant g ~ , , the nucleon mass M, and the axial vector coupling constant in neutron beta decay, g~. The resulting relation, cr: =-g .~ M/gr~x.~c, holds to an accuracy of about 10%. The approximations in this derivation were quickly recognized as difficult to justify, but several other derivations were suggested (2), based on assumed properties of matrix elements of the derivative of the axial vector current, taken between single-nucleon states. Here we describe another derivation, related to but distinct from earlier derivations. Following GOLDBERGER and TREIMAN (~), we consider the decay of the pion as occurring through a nueleon-antinueleon loop; we later discuss the extent to which the result depends on this assumption. We assume isospin-invariant derivative coupling at the pion-nucleon vertex; the Lagrangian is (3) *~fi = i O~¢P~u7~5 ~ru. For the weak vertex, we assume that only first-class currents (4) contribute; there is a contribution to the axial vector current of the form g~(q2)~r~,~su, and for the induced pseudosealar term iH(q~)~ybuq~ we assume pion-pole dominance. However, instead of using dispersion relations as in ref. (~), we do simple perturbation theory. After taking a trace, doing some simple algebra and making a translation in momentum space, one obtains

Momentum-transfer dependence of the [formula omitted] vertex in a dispersion theoretical model. I

We study the dependence of the electropion production vertex on λ, the square of the 4-momentum transferred between electrons, by employing unsubtracted dispersion relations written in terms of the λ-variable itself. Adopting a model in which only spinless pions, nucleons, and photons are considered, we concentrate on complications of the λ-plane singularity structure which result from the instability of the final state π- N system. Physical unitarity in the λ-channel is approximated by the three intermediate states ππ, N N , and πN N . A closed system of form factors and scattering amplitudes which are coupled by unitarity is thus obtained. It is assumed that an s-wave gp- N resonance exists, and the π- N center-of-mass s-wave form factor is singled out for special attention. A dispersion relation is written for this form factor and is found to possess several complex anomalous branch cuts when the s-variable is above thresh-old. We calculate the discontinuities across these anomalous cuts and show that the dispersion relations obtained obey both s- and λ-channel unitarity. A version of the coupled channel N D method is used to rewrite the dispersion relations as an equivalent set of partially decoupled, linear integral equations.

Hadronic Corrections to the Goldberger-Treiman Relation

Using unsubtracted dispersion relations in momentum transfer for the matrix element of the divergence of the axial-vector current between nucleon states, we have examined the hadronic continuum corrections to the Goldberger-Treiman relation for ${\ensuremath{\pi}}^{+}$ decay. These are observed to be about + 10%. From a rigorous unitarity bound and the assumption that the pion propagator ${\ensuremath{\Delta}}_{\ensuremath{\pi}}(0)$ is dominated by the pion pole we show that the continuum states of energy greater than two nucleon masses contribute less than \textonehalf{}%. The $\ensuremath{\pi}\ensuremath{\rho}$ and $\ensuremath{\pi}\ensuremath{\sigma}$ states contributed negligibly. Using Weinberg's extrapolation for the $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude and chiral dynamics, we find that the presumably dominant $3\ensuremath{\pi}$ state contributes with opposite sign and is more than an order of magnitude too small. In the absence of any simple explanation for the 10% correction, we conjecture that what is required is a $3\ensuremath{\pi}$ threshold enhancement or possible resonance, the tripion, with the quantum numbers of the pion and mass near threshold at 4.2 BeV/${\mathit{c}}^{2}$ or a possible subtraction in the dispersion relation.

The non-relativistic crossed partial-wave expansion and nucleon-nucleus scattering

The scattering of nucleons and nuclei is expanded in “crossed partial waves”. This expansion is the non-relativistic analogue of the Regge-Joos representation used in relativistic particle physics. It may be interpreted as a decomposition of the scattering amplitude into partial amplitudes, each one of these corresponding to the exchange of a set of pseudostates transforming irreducibly under the Galilei group, in complete analogy with the ordinary partial wave expansion in the direct channel. A small range of the expansion parameter ν gives the small curvature components of the cross section, making this expansion especially appropriate for the description of the energy averaged scattering amplitude. Explicit formulae are given for the elastic scattering of particles with arbitrary spin for the non-forward and forward scattering, where the little groups are E(3) and E(2) respectively. In close analogy with the Regge poles and Regge trajectories which appear in relativistic high- energy scattering, the concept of Euclidian poles and trajectories is introduced. The formalism is applied to the scattering of nucleons by alpha particles. The parametrization of the scattering amplitude is made with two effective multipoles defined in the origin of the ν plane. Assuming that the amplitude obeys unsubtracted dispersion relations in the energy for fixed momentum transfer we obtain a good fit for the differential cross section and polarization for values of the energy in the range from 66 to 312 MeV.

Decay Rate ofΣ+→p+γfrom Unsubtracted Dispersion Relations

Assuming unsubtracted dispersion relations, we set up integral representation for the weak photonic amplitudes ${F}_{1}$ and ${F}_{2}$. The integrals involve the magnetic and electric dipole amplitudes ${M}_{1\ensuremath{-}}$ and ${E}_{0+}$ of the photo-pion production and the pion-nucleon scattering phase shifts of the states ${P}_{11}$, ${P}_{31}$, ${S}_{11}$, and ${S}_{31}$. The integrals were carried out on a computer and the relative decay rate of ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p\ensuremath{\gamma}$ to its total rate was found to be 0.71\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$, which is comparable with the latest experimental value (1.9\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$.

Sum Rules inπΣScattering and theπΛΣandπΣΣCoupling Constants

Following a procedure proposed recently by Atkinson, we derive sum rules for the $A$ and $B$ amplitudes for the reaction ${\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\Sigma}}^{\ensuremath{-}}$ by equating unsubtracted dispersion relations at $t=0$ and at fixed $u$ or fixed backward angle. Combining these with the already known superconvergence relation for the $B$ amplitude, and assuming the sum rules can be saturated with known resonances, we obtain three equations for two unknown coupling constants ${g}_{\ensuremath{\pi}\ensuremath{\Lambda}{\ensuremath{\Sigma}}^{2}}$ and ${g}_{\ensuremath{\pi}\ensuremath{\Sigma}{\ensuremath{\Sigma}}^{2}}$. Choosing to fix $u=0$, one obtains values of the coupling constants an order of magnitude larger than expected on the basis of, say, $\mathrm{SU}(3)$. We argue that this is probably because of the large extrapolation to unphysical values of $cos\ensuremath{\theta}$ required in evaluating the fixed-$u$ dispersion relation for $u=0$. Taking $u$ to be positive in such a way as to minimize the required extrapolations in angle, or choosing fixed $cos\ensuremath{\theta}=\ensuremath{-}1$, one obtains results that are reasonably consistent with one another and with $\mathrm{SU}(3)$, to within estimated uncertainties of 50% or more, resulting from experimental error in the resonance widths, large cancellations between the contributions of different resonances, and unknown nonresonant-background and high-energy contributions.

Threshold Behavior of Partial-Wave Dispersion Relation and Short-Range Force

In a paper on the existence of a ghost-free solution of the unsubtracted partial-wave despersion relation, Frye and Warnock conjectured that the positive-definite value at threshold of the unitarity integral on the physical region must be canceled by the short-range force (i.e., multiparticle-exchange contribution) arising from the first and second double spectral functions in order to guarantee the required $p$ wave and higher threshold behavior. We examined this conjecture explicitly in an unsubtracted dispersion relation for the $I=l=1$ pion-pion scattering amplitude. With certain choices of the $\ensuremath{\rho}$ Regge parameters, we show in our approximation scheme that this conjecture is nearly satisfied, and we can conclude that the so-called subtraction constant or threshold factor which is usually arbitrarily introduced to correct the threshold behavior could be expressed in terms of known physical quantities. However, with some other choice of $\ensuremath{\rho}$ Regge parameters, our numerical result shows that this conjecture is not satisfied by our one-$\ensuremath{\rho}$-Regge-pole approximation and indicates a need for additional Regge poles or Castillejo-Dalitz-Dyson poles.

Subtractions and off-shell continuation of weak form factors

The dependence of the weak form factors on the mass variables is studied in the one-particle approximation, taking the K ℓ3 form factors as an example. The off-shell continuation considered is the same as in soft-pion calculations. The off-shell dependence is obtained in terms of the subtraction constants of the matrix elements of the axial vector current A π o and strong coupling constants. Under the assumption of unsubtracted dispersion relations the correction to the soft-pion result is estimated to be small.

Kℓ3 form factors and the algebra of charges

Starting from the algebra of vector and axial vector charges, we obtain information about the Kl3 form factors by approximating the commutation relations with the one particle intermediate states. We assume a once subtracted dispersion relation for the weak form factors appearing in the calculation. The subtraction constants can then be uniquely determined as a function of the weak decay constants ƒπ, ƒK and ƒK of the mesons contributing in the intermediate states. The momentum transfer dependence of the Kl3 form factors is characterized by the particle masses, the above mentioned decay constants and the two additional constants gπKκ and ƒK∗gπKK∗, where gπKκ∗ and gπKK∗ are strong coupling constants, and ƒK∗ is the weak decay constant of the K∗ meson. Assuming unsubtracted dispersion relations for the various form factors, we are able to make some numerical predictions.

K ℓ3 decay form factors and current algebra

The form factors in K ℓ3 decay are discussed in a current algebra approach by writing unsubtracted dispersion relations for the matrix elements of suitable retarded commutators between single particle states and vacuum, at (arbitrary) fixed linear combination of the two momentum transfers. This allows us to include the contributions of all the relevant poles and a consistent solution of the current algebra equations is then possible.

Analytic Properties of Current-Algebra Vertex Functions

An expansion of the vector-current vertex is obtained in terms of the Joos spinor amplitudes. As these are free of kinematic singularities, one can remove them from the vertex that enters into the algebra-of-currents sum rules. If one assumes unsubtracted dispersion relations for the resulting form factors, then the sum rules can be cast into a form which involves Bessel transforms of these form factors.

Algebra of Currents, Divergence of Strangeness-Changing Vector Currents, Unsubtracted Dispersion Relations, and theKl3form Factors

Algebra of Currents, Divergence of Strangeness-Changing Vector Currents, Unsubtracted Dispersion Relations, and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi><mml:mn>3</mml:mn><mml:mn /></mml:mrow></mml:msub></mml:mrow></mml:math>form Factors

Some remarks on the properties of form factors

We study the properties of a class of form factors, vanishing ast→−∞, and fulfilling in the cut plane |F(t)⩽Me exp[e|t|1/2](1+Δt)−m) for every e>0, where Δt is the distance to the cut. A form factorF in this class is uniquely determined by the discontinuity over the cut, 2i ImF, which exists as a distribution of finite order.F can be expressed in terms of ImF through a generalized unsubtracted dispersion relation. Sum rules and their implications for the asymptotic behaviour of ImF are discussed.

Electromagnetic Mass Differences and Regge Phenomenology

We examine the electromagnetic mass differences of hadrons by a phenomenological analysis of the forward virtual Compton amplitude. In addition to the usual elastic contribution, we consider in detail the contribution from ${{A}_{2}}^{0}$ Regge exchange for $I=1$ mass differences and possible contributions from non-Regge behavior for this weak amplitude in the asymptotic region. In the approximation of neglecting the inelastic resonant spectrum in Compton scattering, the contribution of the pure Regge exchange to the subtraction term for any $I=1$ mass difference can be related to a ratio of the ${{A}_{2}}^{0}$ residue functions (hence independent of the target) and the relevant elastic form factors. This analysis suggests that a large Regge contribution requires comparable longitudinal and transverse electroproduction cross sections at high energy for small spacelike virtual photon mass. The tadpole model is also examined from the observation that certain linear combinations of $I=1$ mass differences with just the ${{A}_{2}}^{0}$ Regge pole removed can be formulated in terms of unsubtracted dispersion relations and are presumably satisfied by retaining just the elastic contribution. The $\frac{F}{D}$ ratio that we calculate in this way for the tensor-meson couplings to baryons agrees with that obtained from high-energy experiments. From a superconvergence relation it is argued that there is a fixed pole in the helicity-flip crossed-channel Compton amplitude at $J=0$, $I=1$, the experimental consequences of which are discussed.

Some Consequences of the Wu-Yang Asymptotic Behavior of Form Factors

We investigate the properties of a class $\mathcal{a}$ of form factors $F(t)$ which include some with the asymptotic behavior suggested by Wu and Yang: $F(t)\ensuremath{\sim}A\mathrm{exp}[\ensuremath{-}a{(\ensuremath{-}t)}^{\frac{1}{2}}]$, $a&gt;0$, for large negative $t$. A simple representation theorem for such form factors is proved. This theorem is then used as the main tool to obtain the results which follow. "Asymptotic lower bounds" are derived for any $F(t)$ of class $\mathcal{a}$, both at large positive and at large negative $t$. It is shown that any form factor belonging to class $\mathcal{a}$ satisfies a generalized unsubtracted dispersion relation. A whole set of sum rules for $\mathrm{Im}F$ and $\mathrm{Re}F$ on the cut are obtained, generalizing those which follow from a superconvergent dispersion relation. An asymptotic lower bound on the number of changes of sign of $\mathrm{Im}F$ and of $\mathrm{Re}F$ along the cut at large positive $t$ is derived. As an illustration, we first construct according to a simple procedure $F(t)'\mathrm{s}$ which have the given asymptotic behavior and whose only singularities are simple poles. We then derive an asymptotic lower bound on the number of these poles at large positive $t$ for any $F(t)$ of this type.

Low-Energy Theorem for Graviton Scattering

A low-energy theorem for the scattering of gravitons from spin-0 particles is derived. We use the dispersion-theoretic method, recently utilized by Abarbanel and Goldberger to derive low-energy theorems for the Compton scattering of photons, to write unsubtracted dispersion relations for physical helicity amplitudes. The scattering amplitude at fixed angle is shown to be given by the Born approximation up to fourth-order terms in the graviton energy.