Unitarity Relation Research Articles

Proton-proton elastic large-angle scattering in the high-energy limit from linearized unitarity relation and Wu and Yang’s conjecture

In 1965, W u and YA~G (1) predicted tha t the differential cross-section of the protonproton elastic large-angle scattering at high energies should be proport ional to the fourth power of the proton magnetic form factor. Since then, there have been published several theoretical papers concerning this prediction (2). Most papers, however, have only taken W u and Yang's conjecture for granted in order to discuss the proton-proton scattering in terms of the proton magnetic form factor instead of testing the conjecture itself. On the other hand, the recent experimental da ta on both pp (a) and ep (4) elastic scattering at high momentum transfers seem to support the original conjecture

Analysis of a Scattering Process with Spin in the Infinite-Momentum Frame

A kinematical analysis is made of a two-particle scattering process with arbitrary spin in an infinite-momentum frame. It is shown how the description of the spin, in particular, can be simplified by going to this frame. The scattering amplitude is expressed as a function over an $E(2)$ algebra in the infinite-momentum frame; this immediately leads to an impact-parameter representation of the amplitude for arbitrary spin. This representation is different from the fixed-energy impact-parameter representation; in particular, the unitarity relation, in the two-body approximation, is seen to have a simple form.

S-Matrix Description ofKLandKSDecays

We show how the usual phenomenological description of the decays of the ${K}_{S}$ and ${K}_{L}$ mesons can be derived in a unified manner beginning from a description of the ${K}_{S}$ and ${K}_{L}$ states as overlapping resonances in a scattering matrix. The unitarity relations for overlapping resonances in a $\mathrm{CPT}$-invariant (but not $\mathrm{CP}$- or $T$-invariant) theory play a crucial role in the discussion, and are treated in detail.

The tensor force and the Kowalski-Noyes representation for the t-matrix

A formulation of the Kowalski-Noyes representation of the t-matrix is developed using Weinberg's quasiparticle formalism. Such a representation leads to non-singular integral equations for the t-matrix. It also suggests a separable approximation to the t-matrix, which reduces the Faddeev equations to coupled equations in one continuous variable. This separable approximation satisfies the off-shell unitarity relation, and has the corre behavior in the neighborhood of bound-state and resonance energies. The formulation given allows for a treatment of tensor forces.

Approximation de la partie hors de la couche d'énergie de la matrice t

The off-shell amplitudes for identical particle scattering are exactly calculated by numerical integration of the Schrödinger equation under the assumption that the two-particle interaction is given by a central parity-dependent Yukawa potential with a hard core. By comparison with these exact values it is shown that the Stern and Chamberlain approximation method previously used by Benoist-Gueutal and the writer enables one to obtain the real part of the off-shell amplitudes with a good accuracy. Yet, the unitarity defect of the method does not permit to insure the correctness of the imaginary-part values. Then we complete the t-matrix by calculating the imaginary part which is the solution of unitarity relation. One can see easily that the accuracy is the same in either the real or the imaginary part. This process gives a good idea of our goal, which is the systematic approach to the off-shell part by the on-shell part of the t-matrix.

Höhere Bornsche Näherungen für die elastische Streuung von Elektronen an Atomen Higher Born Approximations for Elastic Electron Scattering from Atoms

Higher Born Approximations for Elastic Electron Scattering from Atoms. A theorem analogous to the unitarity relation by Glauber and Schomaker is derived for Born’s approximations. For a given potential Φ (r) =μ (ħ2/2 m) V (r) of the scattering atom the n-th approximation fn for the scattering amplitude is defined as the function resulting from the n-th iterative step of Born’s approximation.| If | fn |2 is expanded into a series of powers of the parameter μ, this series up to terms in μn+1 is defined as the n-th approximation for the differential cross section dσn/dΩ, and not |fn|2 itself. In this case an extension of the optical theorem Im fn+1 (0) = (k/4π) σn can be derived. For a potential Φ(r)=-(Z e2/4π ε0 r)·exp(-r/R) ∑ airi exact analytic expressions for the scattering amplitude fn and the differential cross section dσn/dΩ can be given in first and second Born approximations. An important example is the hydrogen-like atomic potential in which, in contradistinction to Wentzel’s model, the electron density σ(r) is limited even for r → 0. For this model, the differential scattering cross section in second Born approximation exceeds the values of the first approximation for all scattering angles, while experimental data indicate a correction of the opposite sign. An estimate of the cross sections in third approximation shows that they are again below’ the first approximation. These results indicate that the Born’s first approximation is not only less complicated but even more accurate than the second one. The absolute values of the scattering amplitudes in first and second approximations diverge considerably even at high energies, mainly due to the imaginary part Im f2(ϑ) of /2(ϑ). Therefore, conclusions on the phase η(ϑ) from Im f2(ϑ) do not seem to be reliable.

Interference effects in the radiative decay of closely spaced levels

In this paper we present a study of the radiative decay of coherently excited, closely spaced, molecular (or atomic) levels. Interference effects in the decay process cannot be treated in general by displaying the damping matrix in a diagonal form, so that each decay channel is assumed to be characterized by its own lifetime. The problem of the radiative decay of coherently excited indistinguishable levels (characterized by the same symmetry) is treated by a self-consistent extension of the Wigner-Weisskopf method, or alternatively, by the ‘unitarity relations’ method, developed in elementary particles physics. Finally, the Fano configuration interaction method is applied for the decay of a manifold of states characterized by the same polarization. General expressions for photon counting rates are derived and the nature of a general quantum beat experiment is discussed.

Relation between the Damping of Large Transverse Momenta in Elastic and Inelastic Collisions from Unitarity

It is shown from the unitarity relation that the damping of large transverse momenta in inelastic collisions necessarily implies their damping in elastic collisions. In this derivation it is assumed that an inelastic amplitude can be written as a product of a slowly varying factor which depends only on incident energy and the longitudinal components ${{q}_{i}}^{L}$ of the momenta ${q}_{i}$ and another factor which incorporates the exponential damping of transverse momenta.

Inelastic Effects inP11-StateπNScattering

It is found that the optimal determinantal approximation, as proposed in an earlier paper, leads to accurate solutions for the $N{D}^{\ensuremath{-}1}$ equations of $\ensuremath{\pi}N$ scattering in the ${P}_{11}$ state. Both left-hand and right-hand cutoff procedures are used in the formulation of the problem, with similar results. Inelasticity is treated by a method proposed by Froissart. It is shown that a qualitatively correct $\ensuremath{\pi}N$ ${P}_{11}$ phase shift in the low-energy region may be reproduced in a one-channel calculation without Castillejo-Dalitz-Dyson poles, provided that the inelasticity factor is chosen properly. A two-channel model with the ($\ensuremath{\pi}{N}^{*}$) as the second channel is discussed, and it is shown that the inelasticity obtained in a rough approximation to the coupled unitarity relations is adequate in generating the "zero" of the ${P}_{11}$ phase shift. Some implications concerning the nonequivalence of one-channel and multichannel $N{D}^{\ensuremath{-}1}$ formalisms are discussed.

Scattering Model with Crossing Symmetry and Minimal Inelastic Effects

The iterative method of solving the dispersion relations and unitarity relations for the two-body scattering amplitude of a neutral scalar field is investigated. A consistent and nontrivial approximating scheme is obtained by incorporating enough inelastic effects that crossing symmetry is maintained at every stage of calculations. If one has initially a twice-subtracted dispersion relation, it can be shown that because of crossing symmetry, the subtraction term linear in $s$, $t$, and $u$ cannot be present, so that it is sufficient to start the iteration process by just one subtraction. The unitarity relations are simplified by taking the scattering particle to have zero mass. Questions concerning infrared divergencies and convergency properties in iterating the single and double density functions are fully discussed. At low energies, the amplitude is given by the first few iterations, which will be carried out explicitly to fourth order. Higher orders of iteration can be carried out in a straightforward but increasingly tedious manner. We study the behavior of the scattering amplitude in the high-energy region, where $s$, $t$, and $u$ are all large in magnitude, while the scattering angles are held at fixed values. The leading terms of the scattering amplitude and its spectral functions are obtained for each order of iteration, and the resulting leading series are subsequently summed into closed forms for weak coupling. The ratio of the elastic cross section to the total cross section is also obtained in the same region. The result indicates that the amount of inelastic effects being included is considerable.

Absorptive corrections and Regge singularities

Two possible alternative methods of computing unitarity corrections to Regge-pole amplitudes are discussed. One method is to correct for rescattering, as motivated directly from the unitarity relation; the other is to correct for absorption, in order to account for the presence of competing channels. It is shown that these two corrections have the same singularity structure, but need not be the same; in the simplest cases the corrections have the same magnitude, but differ in sign.

The DoubleN/DMethod

A method of obtaining the partial-wave amplitude by two steps is proposed, if the amplitude is written as T = t+r where t satisfies the unitarity relation by itself: first t is solved by the N/D method, and then r is solved with the solution of t and the left-hand discontinuity of r as inputs by using the N/D method again. It is proved that the particle pole of T appears in r, not in t. As an interesting application, three examples are discussed: these are the addition of the modified Born term, the channel coupling effects, the effects of the small change of the left-hand discontinuity. For the last case the shifts of the mass and the coupling constant are practically the same as those derived by Dashen and Frautschi, although their derivation is very questionable.

Necessity of Production Amplitudes in Quantum Field Theory

A nonvanishing production amplitude in the pion-nucleon scattering is proved to be a necessary consequence of relativistic invariance, causality, crossing, and unitarity relations in quantum field theory.

General structure of a unitary S-matrix for finite many-channel scattering

The most general solution of the unitarity relations for the S-matrix for elastic, inelastic and rearrangement processes below break-up is given in terms of one many-valued real-analytic function D of the total energy.

Some remarks on the location of singularities inS-matrix theory

It is shown that the standard procedure for locating singularities inS-matrix theory through the analytic continuation of the unitarity relation (making use of the crossing relations and Hermitian analyticity), is not sufficient to determine upon which Riemann sheet certain “noninterseeting” singularities lie. The ambiguity is removed by assuming all such noninterseeting singularities absent from the physical sheet. Arguments are given which suggest that this assumption may be sufficient to “complete” the proofs of Hermitian analyticity and the crossing relations. Finally, it is shown that the assumption follows from a physical-sheet postulate which represents a strengthening of the usuali∈-prescription.

High-Spin Baryons. II. Dynamical Mechanisms for Regge Recurrences of Inelastic Resonances; General Theory

Virtual transitions of the type $\ensuremath{\pi}{N}_{i}\ensuremath{\rightarrow}\ensuremath{\rho}{N}_{j}$, where ${N}_{i}$ represents an arbitrary nucleon isobar, are considered as the driving force for Regge recurrences of inelastic resonances. We consider only the $s$-wave $\ensuremath{\rho}{N}_{j}$ configuration. Since most of the low-lying isobars have even parity, we are therefore concerned mainly with odd-parity resonances. Regge trajectories of the latter are a consequence of Regge recurrences of the even-parity states. An important example is given by taking ${N}_{i}$ to be the sequence: nucleon ${\mathrm{\textonehalf{}}}^{+}$ and its recurrence ${\frac{5}{2}}^{+}$ (quantum numbers ${P}_{11}$ and ${F}_{15}$ in conventional notation). Isospin and centrifugal-barrier considerations lead directly to the existence of the ${D}_{13}\ensuremath{-}{G}_{17}$ sequence. In this paper the general formalism is set up. Numerical results are given separately. First, however, a simpler derivation is given of the elastic forces due to isobar exchange in the quasistatic approximation. Previous work has shown the relation of these forces to the existence of even-parity Regge trajectories. Then the unitarity relations are set up for helicity amplitudes describing the reactions ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\pi}$, ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$, and ${N}_{a}\ensuremath{\rho}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. Next a general explicit expression is given for the onepion-exchange (OPE) approximation for ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. Expressions are given for the above amplitudes in several models, in which only the OPE coupling is considered. The pole approximation is used to solve the many-channel $\frac{N}{D}$ equations. Techniques for handling the general isospin problem are developed and their relevance to the present problem discussed.

On the theory of direct reactions (II)

Distorted wave theories for a direct reaction amplitude M 12 connecting channels 1 and 2 are usually obtained from the formal theory of scattering by means of a long chain of approximations. Here we consider a version of distorted wave theory which involves just one basic assumption and which can be further approximated to yield all the currently used expressions. We show that this basic assumption, which underlies all the usual theories, leads plausibly to the relation Im M 12=M 12f 11 †+M 12 †f 22, where and f 11 are the elastic scattering amplitudes in the initial and final channels. The result is discussed by means of a simple model of nuclear reactions and also on the basis of a more general theory. The implications of this “truncated unitarity relation” were brought out in an earlier paper, where it was shown to give very stringent conditions for the validity of the distorted wave treatment.

Unitarized impulse approximation: KD elastic scattering

An effort is made to incorporate unitarity into the dispersive impulse approximation for KD elastic scattering. It is argued that elastic unitarity suffices since there exist no inelastic two-body states of any kind (involving stable or unstable particles) coupled to the KD system at low energy. The impact parameter representation is used and an approximate unitarity relation is obtained. Correct threshold behavior for the partial waves is built in and a unitarized version of the impulse approximation is constructed. The cross sections are calculated at 230 MeV/ c. Little data is avaiable for comparison with the computed angular distribution; the total cross section is in excellent agreement with experiment.

Three-Body Unitarity and Khuri-Treiman Amplitudes

The $s$-wave Khuri-Treiman (K-T) amplitude for three-body production or decay processes with final-state interactions satisfies a linear single-variable integral equation. The inhomogeneous term is the Watson final-state enhancement factor (or isobar term), while the integral part represents rescattering corrections. It is proved that the presence of this term guarantees that the amplitude satisfies a form of three-body unitarity, in which the three bodies interact in pairs only. This means that two-body unitarity, analyticity, and crossing have generated three-body unitarity (and a three-body amplitude). If the result can be generalized to higher partial waves, it suggests that one can unitarize (in this sense) any inhomogeneous term, provided it has no three-body branch point itself, by simply adding on the integral term, a procedure which could be applied, for instance, to the isobar, peripheral, or stripping models. This is by way of alternative to the usual $\frac{N}{D}$ method in the three-body channel, and seems to imply equations a good deal easier to handle numerically. A comparison between the unitarity relation obtained from the K-T equation and the one normally assumed discloses a difference between them, which would be important if one wished to consider continuations of the relation in the three-body mass.

Completeness, unitarity and dispersion relations in potential scattering

It is shown that the assumption of completeness of the set of scattering states implies the unitarity condition for the total scattering amplitude and dispersion relations for the S-matrix for each value of the angular momentum.