Mandelstam Representation Research Articles

Proof that Scattering Implies Production in Quantum Field Theory

A proof of the necessity of production processes in quantum field theory is carried out in the axiomatic framework. It is shown that a field theory that is assumed to have a nontrivial scattering amplitude violates crossing symmetry, if production processes are absent. The proof is based on the rigorous analytic properties of the scattering amplitude, particularly the analyticity in the invariant-scattering variables s, t, and u. In the case of a scalar theory with pairing symmetry, the scattering amplitude is known to be analytic within the domain |stu| < 7168m6, except for the usual cuts. Under the working assumption that production processes are null, it is shown that this domain can be enlarged by applying the elastic unitarity conditions beyond the (usual) elastic region. The domain is enlarged sufficiently to include the first Landau singularity of the absorptive part of the scattering amplitude. This singularity is not symmetric in s and t within the extended domain, and this is incompatible with the crossing symmetry of the scattering amplitude. In order to avoid a contradiction, the discontinuity across this Landau singularity must be null. It follows that the scattering amplitude must itself be null. In the course of the proof it is shown that the conclusion is valid for a scattering amplitude satisfying the requirements of an S-matrix theory embodied in the Mandelstam representation.

An example of the mandelstam representation in perturbation theory: Paul Federbush, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

An example of the mandelstam representation in perturbation theory: Paul Federbush, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

Upper Bound for the Double Spectral Function in Potential Scattering

For potentials V(z) holomorphic in Re z > 0 and bounded by |V(z)|<K/|z|ρ ρ<2 for |z|≤1,|V(z)|<K|z|−γe−μ0Re z for |z|≥1,γ>7/4,we show that the double spectral ``function'' ρ(s, t) is a continuous function of s and t in s > 0, t > 0, and we obtain an upper bound for it. This upper bound shows clearly that the double integral of the Mandelstam representation in fact exists and defines an analytic function of s and t in two cut planes. We indicate how to generalize these results to the case when ρ(s, t) is no longer a function but a distribution.

Analytic Continuation ofa(ℓ,s)for Re ℓ<N

The positions of the singularities of the partial wave amplitude in the complex l plane are investigated for that case of a relativistic, two-particle elastic scattering amplitude satisfying the Mandelstam representation. For a finite number of intermediate states it is shown that the convex hull of singularities in the l plane is related to a growth indicator function. For an infinite number of intermediate states it is demonstrated that the imaginary parts of the singularities in the l plane are unbounded from either above or below.

ANALYTICITY AND THRESHOLD BEHAVIOURS OF FERMION REGGE TRAJECTORIES

Starting from π-N scattering amplitude and basing on the Mandelstam's representation and unitary condition, analyticity and threshold properties of fermion Regge trajectories of baryon quantum number 1 and strangeness 0 are investigated. The following preliminary results are obtained: (1) The position parameter of the Regge trajectories a(s) has a right-hand physical cut and no left-hand dynamical cut, but has a kinematical left-hand cut at s=0; (2) a(s) in the region s + (s) and α - (s), being of opposite space parity, are complex conjugate to each other. In the region 0 2 , a ± (s) are real but the dependence on s may be different; (3) The energy dependence of a(s) at the threshold is obtained; (4) Two kinds of Regge poles are distinguished. Those with a(W 0 )≠0 (W 0 -threshold energy) are responsible for dynamical resonances and bounded states. There are other kinds of the Regge trajectories for which a(W 0 )=0 and whose threshold behaviours are also investigated. As energy approaches the threshold from energy below the threshold, there are infinite pairs of poles approaching Re J=0 from the left half J-plane, each pair containing a couple of conjugate complex poles. While as energy approaches from above the threshold, there are also infinite pairs of poles approaching Re J=0, each pair containing poles in the first and third quadrant of the J-plane. For all these poles, Re α(s)→0 faster than Im a(s)→0 by one or two orders. The poles condense at the origin of the J-plane. We thus obtain the Gribov-Pomenranchuk condensation of the Regge poles in the case of Fermion trajectories. This kind of poles has a purely kinematical origin. They reflect the general properties of the S-matrix at the threshold.

The two-pion-exchange contribution toN-Λ and Λ-Λ interactions-Λ and Λ-Λ interactions

The two-pion-exchangeN-Λ and Λ-Λ scattering amplitudes are calculated on the basis of the Mandelstam representation. The results contain parameters referring to theN3,3* and Y*1 baryon excited states and the pion-baryon coupling constants. The effects of π-π interaction are also discussed. TheN-Λ and Λ-Λ potentials are then derived from these field-theoretic scattering amplitudes. The shapes of the various parts (central, spin-spin, tensor, spin-orbit, both symmetric and antisymmetric for theN-Λ case) of these potentials are presented.

Pion Exchange Currents in Deuteron Photodisintegration. Dispersion Theory

A dispersion-theoretic treatment of the amplitudes describing photomagnetic disintegration of the deuteron at energies very near threshold has been developed. The effects of meson exchange currents in the photodisintegration or equivalent $n\ensuremath{-}p$ capture process are calculated. At threshold the photomagnetic transition is dominant. This transition is essentially the isovector $^{3}S_{0}\ensuremath{\rightarrow}^{1}S_{0}$ transition. A fixed-angle dispersion relation for the dipole amplitude describing this transition can be written in terms of the related covariant amplitudes. Solutions to the dispersion relation are found using several different approximations. First, one may neglect all high-order effects which serve to define the unphysical cut, and consider only the contribution from the Born poles. Next, one may condense all the higher-order effects into a single interaction pole. The single experimental cross section value for thermal neutrons may be used to relate the position and residue of this pole in a single functional relationship by recalculating the solution to the dispersion relation with this pole included. Finally one may treat the contribution of the pion exchange currents using the Mandelstam representation, and recalculate the dispersion relations once again. This latter treatment is based upon the approximation that only the anomalous tip of the spectral function is effective in providing modifications in the physical region close to threshold. It is shown that the pion exchange terms are the dominant contributors to the spectral function in the anomalous region. The solution obtained in this approximation yields a cross section which is in agreement with the experimental value.

Positive Photopion Production from Hydrogen Near Threshold

Differential cross sections for photopion production from protons were measured for 32 photon energies from 154 to 185 MeV with the Alvarez 4-in. liquid-hydrogen bubble chamber in the bremsstrahlung beam of the Berkeley electron synchrotron. High photon intensities of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{7}$ MeV/pulse were used by collimating the beam down to a narrow "ribbon" which was viewed edge on by the camera. Although the pion origins were obscured by the heavy electron background, the remainder of the $\ensuremath{\pi}$ track was visible for most events. Of 5000 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\mu}$ decays seen, 3400 were deemed suitable for calculations. The results are in excellent agreement with the theoretical calculations by Ball, who applied the Mandelstam representation to the process. The measured values of the square of the matrix element, averaged over c.m. angles, are: All data are subject to an additional correlated error of 4.1% because of the uncertainty in the beam normalization. The cross section for the process is the product of the square of the matrix element times the kinematic factor $\frac{{p}^{*}}{{k}^{*}}$, where ${p}^{*}$ and ${k}^{*}$ are the c.m. momenta of the pion and the photon. Also a value of $\ensuremath{\Lambda}=(+0.931\ifmmode\pm\else\textpm\fi{}0.59)e$ was obtained, where $\ensuremath{\Lambda}$ is the multiplicative factor associated with the matrix element for photopion production from pions as calculated by Wong.

High-energy behaviour of partial-wave amplitudes

It is shown that the assumptions of Regge behaviour and the Mandelstam representation do not immediately yield useful information on the high-energy properties of partial-wave amplitudes. An additional assumption is suggested and this leads to a decreasing behaviour, dominated on the right, but apparently not on the left, by crossedchannel Regge poles.

The low-energy Λ-N and Λ-Λ interaction

The Λ-N and Λ-Λ potentials are derived from field theory by means of a ≪ perturbation-dispersion ≫ method. The two-pion-exchange potential is calculated from the Mandelstam representation for the ∧-N (and ∧-∧) scattering amplitude in one-channel formalism. In the unitarity condition the π∧ scattering amplitude is introduced as an extension of the C.G.L.N. model for πN scattering, where contributions from Σ and Y1* are represented by appropriate pole terms. Since the T.P.E. potential alone is not sufficient to reproduce the low-energy A-JV scattering data, no attempt is made to fit the π∧Σ coupling constant. Forces due to scalar, pseudoscalar, and vector boson exchange are discussed and corresponding potentials are deduced in one-particle-exchange approximation.

The long-range parts of the Λ-N and Λ-λ potentials

The two-pion exchange contributions to the Λ-N and Λ-Λ potentials areobtained on the basis of the Mandelstam representation of the scattering amplitude. Formulae for V 0, V s and V T in the expression V = V 0+ V s( σ 1· σ 2)+ V T S 12+… are given in terms of the parameters related to the π-N and π-Λ scattering amplitudes. The quantity V 0 involves the S-wave π-λ scattering amplitude which is as yet experimentally unknown; V S and V T depend only on the P-wave gp-N and π-λ interactions, which are assumed to be dominated by the 3-3 resonance N ∗ and Y 1 ∗ resonance (1385 MeV), respectively. Higher partial waves are ignored. Only V S and V T are evaluated numerically. It is found that V S and V T are extremely small. This implies that the strongspin-independence of the phenomenological Λ-N force is characteristic of the short-range part of the potential.

Remarks on the Polynomial Boundedness in the Mandelstam Representation

The Mandelstam representation is a statement about the region of analyticity and asymptotic behavior (polynomial boundedness) of a scattering amplitude. In virtue of the unitarity condition, however, these two are not completely independent. Some physical consequences, e.g., uniqueness, polynomial boundedness of the total cross section, etc., which have been already derived from the Mandelstam representation, are shown to be preserved, even if the polynomial boundedness is replaced by a somewhat weaker assumption. By making use of unitarity, analyticity, and crossing symmetry, the following type of scattering amplitude F = E + M, where E is an entire function in both variables s and t, while M denotes a Mandelstam-type function with finite number of subtraction, is shown to be ruled out. Similarly, F = EM is also ruled out, if one imposes the additional restriction that E should increase less fast than an exponential in one variable while the other is finite.

Analyticity in the Angular Momentum for Singular Potentials

The analytic properties in the angular momentum variable of the Jost functions and of the partial-waveS-matrix elements are investigated for a large class of repulsive potentials behaving near the origin as arbitrary inverse powers of the radial variable dominating over the centrifugal term. The asymptotic behaviour of the Jost function for large angular momenta is obtained by means of a W.K.B. analysis and the implications of the asymptotic distribution of poles in the λ-plane on the possibility of writing a Mandelstam representation are also shortly discussed.

A note on the signatures of the dominant regge trajectories

By the requirements of the unitarity condition and the positive definiteness of the imaginary parts of the forward amplitudes, it has been shown that the dominant Regge trajectories of amplitudes that contribute to forward scattering passing through integer values of angular momenta in the interval 0 ⩽t < 4μ 2, wheret is the square of the momentum transfer and 4μ2 is the square of the mass of the lowest intermediate state in the t-channel, must have definite signatures. This result is derived for amplitudes satisfying the Mandelstam representation.

Dispersion relations for analytically continued partial-wave amplitudes in potential theory

The dispersion relations of Fivel are extended to physically meaningful situations. On the basis of the Mandelstam representation and a modified form of the partial-wave expansion a closed system of equations is obtained for the partial-wave amplitude for values of the angular momentum defined by reall equal to a constant positive number. A solution by iteration is indicated. The assumption is made that the partial-wave amplitude has a rightmost singularity in the complex angular momentum plane. The methods of the paper are extended to provide a new derivation of the integral equation satisfied by the Mandelstam spectral function.

Some consequences of a reflection property in the complex λ-plane

An S -matrix theory satisfying unitarity and the reflection propertyS(- λ) = S(λ) exp [-2iπλ] in the complex λ = l + 1/2 plane is examined and the possibility of establishing analyticity and the asymptotic behaviour of the scattering amplitude in cos θ is discussed. If meromorphy of S(λ) is added, it is concluded that a useful theory is possible only under special conditions on the behaviour of S for |λ| → 8 and on the asymptotic distribution of the Regge poles; but then, the background term in the Sommerfeld-Watson transform disappears. The special conditions are shown to be due to the existence of an essential singularity at λ = -i 8. In the presence of appropriate branch cuts this last singularity appears only on the one side of the relevant cut. As a result, the Sommerfeld-Watson transform can be proved and the double spectral function of the Mandelstam representation can be explicitly determined in terms of the A-plane singularities.

New Form of Strip Approximation

A detailed set of equations is formulated for zero-spin external particles based on a combination of the $\frac{N}{D}$ method with the superposition of top-ranking Regge poles in all three reactions of a four-line connected part. The contribution from each pole arises from a distinct strip in the Mandelstam representation so that double counting is avoided. Only real values of $l$ with $ll~1$ need be considered in the bootstrap calculation. The amplitude emerging from our $\frac{N}{D}$ equations is meromorphic in the right-half $l$ plane, and the Regge poles approach high-energy limits that are dynamically determined and which in some cases may lie to the right of $l=0$. The reduced residues vanish in the high-energy limit.

A new Regge representation for potential scattering

For a superposition of Yukawa potentials such thatN Regge trajectories enter the right half of the angular-momentum plane we derive a decomposition of the scattering amplitude intoN+2 terms. These are, respectively, the Born term, a «background term», andN «Regge terms». A Regge term belongs to a specific trajectory and represents the effect of only that part of the trajectory, which lies in the right half of the angular momentum plane. Each Regge term satisfies a Mandelstam representation with a boundary curve for the spectral function as is usual for potential scattering. Moreover, the spectral function is nonvanishing only in a strip, parallel to thet-axis (t is momentum transfer), the width of which is determined by the energy interval for which the corresponding Regge trajectory is in the right half-plane. This corresponds exactly to the «new strip concept» ofChew, Frautschi andMandelstam. Contrary to other Regge representations a Regge term as defined in this paper has no unphysical left-hand cuts. A study of the contribution of a given Regge term to thel-th partial wave reveals that the Regge term describes all bound states and resonances that are being «caused» by the corresponding trajectory. It is shown that the partial-wave projection of a Regge term can be defined throughout the right half of the angular momentum plane.

Dispersive approach to photon-photon scattering

The scattering of light by light is considered with the aid of double dispersion relations. The spectral functions are easily obtained, in the lowest perturbative order, by means of the Cutkosky rules and a Mandelstam representation is explicitly written. This approach is very suitable for the symmetry properties of this scattering reaction and allows to obtain the differential cross-section in a straightforward way for all the values of the energy and of the angle. In order to give a detailed exposition of the method we consider in this paper only that amplitude which describes the scattering of linearly polarized photons in a direction perpendicular to the scattering plane.

A mandelstam representation in hard-core potential scattering

The difference between the scattering amplitude for scattering from a potential with a hard core, and the scattering amplitude for scattering from the hard core alone, is studied. Provided the potential outside the hard core is a superposition of Yukawa potentials, it is shown that this difference satisfies a Mandelstam representation of conventional form.