Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity-I
CERN1
21 Aug 1966-Nuovo Cimento Della Societa Italiana Di Fisica A-nuclei Particles and Fields (Springer Berlin Heidelberg)-Vol. 42, Iss: 4, pp 930-953
TL;DR: In this article, it was shown that the pion-pion scattering amplitude can be continued until the border-line of the double-spectral function, except for the case that the nearest singularities are induced by two-particle unitarity.
Abstract: This paper, the second of a series on the subject, is entirely devoted to the pion-pion scattering amplitude. The main results are: 1) The scattering amplitude can be continued till the border-line of the double-spectral function, except for\(8\mu ^2< s< 32\mu ^2 \). Hence the nearest singularities are really induced by two-particle unitarity. Another consequence is that the only possible static potential describing low-energy pion-pion scattering is a Yukawa superposition. The domain of validity of fixed-transfer dispersion relations is slightly enlarged, as compared to I. 2) A partial analytic completion is carried out by various methods. As a result one finds a very large domain of analyticity for the fixed-angle amplitude and the partial-wave amplitudes; this domain extends from\(s = - 28\mu ^2 \) tos=+∞. However, only in the interval\( - 28\mu ^2< \operatorname{Re} s< 78\mu ^2 \) the extension in Ims is appreciable (\(\left( {\left[ {\operatorname{Im} s} \right.|\max = 70\mu ^2 } \right)\)). 3) The result of Bros, Epstein and Glaser on the validity of fixed, negative-t quasi-dispersion relations is extended to anyt inside the parabola with focust=0 and summit\(t = \mu ^2 \).
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TL;DR: In this article, an exact relation for ππ scattering which yields the real parts of the π π partial wave amplitudes al(I)(s) in the region 4m2π ⩽ s ⩻ 60m 2π, is given, where s denotes the centre-of-mass energy, l the angular momentum and I the isotopic spin.
333 citations
TL;DR: The sigma meson has been controversial for almost six decades, despite playing a central role in the spontaneous chiral symmetry of QCD or in the nucleon-nucleon attraction as discussed by the authors.
Abstract: The existence and properties of the sigma meson have been controversial for almost six decades, despite playing a central role in the spontaneous chiral symmetry of QCD or in the nucleon-nucleon attraction. This controversy has also been fed by the strong indications that it is not an ordinary quark-antiquark meson. Here we review both the recent and old experimental data and the model independent dispersive formalisms which have provided precise determinations of its mass and width, finally settling the controversy and leading to its new name: $f_0(500)$. We then provide a rather conservative average of the most recent and advanced dispersive determinations of its pole position $\sqrt{s_\sigma}=449^{+22}_{-16}-i(275\pm12)$.
In addition, after comprehensive introductions, we will review within the modern perspective of effective theories and dispersion theory, its relation to chiral symmetry, unitarization techniques, its quark mass dependence, popular models, as well as the recent strong evidence, obtained from the QCD $1/N_c$ expansion or Regge theory, for its non ordinary nature in terms of quarks and gluons.
311 citations
TL;DR: The existence and properties of the sigma meson have been controversial for almost six decades, despite playing a central role in the spontaneous chiral symmetry of QCD or in the nucleon-nucleon attraction.
260 citations
TL;DR: In this article, an effective field theory that allows for stable NEC-violating solutions with exactly luminal excitations only is presented, and it is shown that the theory obeys standard positivity as implied by dispersion relations.
Abstract: In QFT, the null energy condition (NEC) for a classical field configuration is usually associated with that configuration’s stability against small perturbations, and with the sub-luminality of these. Here, we exhibit an effective field theory that allows for stable NEC-violating solutions with exactly luminal excitations only. The model is the recently introduced ‘galileon’, or more precisely its conformally invariant version. We show that the theory’s low-energy S-matrix obeys standard positivity as implied by dispersion relations. However we also show that if the relevant NEC-violating solution is inside the effective theory, then other (generic) solutions allow for superluminal signal propagation. While the usual association between sub-luminality and positivity is not obeyed by our example, that between NEC and sub-luminality is, albeit in a less direct way than usual.
239 citations
Cites background from "Extension of the axiomatic analytic..."
...[10], analyticity in (s, t) holds on the product space (|t| < M)× (cut s plane)....
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...Martin(5) [10], unitarity together with standard properties of Legendre polynomials imply that for real s ∂ ∂tn ImM(s + iǫ, 0) ≥ 0 ∀n , (84) which is a generalization of the optical theorem, ImM(s + iǫ, 0) ∝ σtot > 0....
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TL;DR: In this article, a set of six equations of the Roy and Steiner type for the S- and P-waves of the K-wave scattering amplitudes is derived and the range of validity and the multiplicity of the solutions are discussed.
Abstract: With the aim of generating new constraints on the OZI suppressed couplings of chiral perturbation theory a set of six equations of the Roy and Steiner type for the S- and P-waves of the \(\pi K\) scattering amplitudes is derived. The range of validity and the multiplicity of the solutions are discussed. Precise numerical solutions are obtained in the range \(E\lesssim 1\) GeV which make use as input, for the first time, of the most accurate experimental data available at \(E\gtrsim 1\) GeV for both \(\pi K\to\pi K\) and \(\pi\pi\to K\overline{K}\) amplitudes. Our main result is the determination of a narrow allowed region for the two S-wave scattering lengths. Present experimental data below 1 GeV are found to be in generally poor agreement with our results. A set of threshold expansion parameters, as well as sub-threshold parameters are computed. For the latter, a matching with the SU(3) chiral expansion at NLO is performed.
211 citations
References
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TL;DR: In this article, it was shown that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form Ω( √ log n 2 s ) at forward and backward angles.
Abstract: It is proved that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form $\mathrm{Cs}{\mathrm{ln}}^{2}s$ at the forward and backward angles, and $C{s}^{\frac{3}{4}}{\mathrm{ln}}^{\frac{3}{2}}s$ at any other fixed angle in the physical region, $C$ being a constant, $s$ being the total squared c.m. energy. This corresponds to cross sections increasing at most like ${\mathrm{ln}}^{2}s$. These restrictions limit the freedom of choice of the subtraction terms to six arbitrary single spectral functions and one subtraction constant.
1,071 citations
TL;DR: In this article, a method for using relativistic dispersion relations, together with unitarity, to determine the pion-nucleon scattering amplitude was proposed, and the method makes use of an iteration procedure analogous to that used by Chew and Low for the corresponding problem in static theory.
Abstract: A method is proposed for using relativistic dispersion relations, together with unitarity, to determine the pion-nucleon scattering amplitude. The usual dispersion relations by themselves are not sufficient, and we have to assume a representation which exhibits the analytic properties of the scattering amplitude as a function of the energy and the momentum transfer. Unitarity conditions for the two reactions $\ensuremath{\pi}+N\ensuremath{\rightarrow}\ensuremath{\pi}+N$ and $N+\overline{N}\ensuremath{\rightarrow}2\ensuremath{\pi}$ will be required, and they will be approximated by neglecting states with more than two particles. The method makes use of an iteration procedure analogous to that used by Chew and Low for the corresponding problem in the static theory. One has to introduce two coupling constants; the pion-pion coupling constant can be found by fitting the sum of the threshold scattering lengths with experiment. It is hoped that this method avoids some of the formal difficulties of the Tamm-Dancoff and Bethe-Salpeter methods and, in particular, the existence of ghost states. The assumptions introduced are justified in perturbation theory.As an incidental result, we find the precise limits of the region for which the absorptive part of the scattering amplitude is an analytic function of the momentum transfer, and hence the boundaries of the region in which the partial-wave expansion is valid.
471 citations
TL;DR: In this paper, upper and lower bounds of the imaginary part of a scattering amplitude are obtained for physical and unphysical values of the scattering angle, respectively, from unitarity alone, and it is shown that the total cross section cannot increase faster than the logarithm squared of the energy under assumptions appreciably more general than Mandelstam representation.
Abstract: Upper and lower bounds of the imaginary part of a scattering amplitude are obtained for physical and unphysical values of the scattering angle, respectively, from unitarity alone. This imposes rather stringent conditions on the high-energy behavior of the scattering amplitude. In particular unitarity alone rules out Regge poles with value larger than unity for zero momentum transfer. On the other hand, it is shown that the total cross section cannot increase faster than the logarithm squared of the energy under assumptions appreciably more general than Mandelstam representation. Finally, in the light of the preceding results, we give a few comments on the problem of the shrinking of the diffraction peak and its connection with the decrease of the elastic cross section.
306 citations
TL;DR: The partial wave expansions which define physical scattering amplitudes continue to converge for complex values of the scattering angle, and define uniquely the amplitudes appearing in the unphysical region of non-forward dispersion relations as discussed by the authors.
Abstract: Scattering amplitudes are shown to have analytic properties as functions of momentum transfer. The partial wave expansions which define physical scattering amplitudes continue to converge for complex values of the scattering angle, and define uniquely the amplitudes appearing in the unphysical region of non-forward dispersion relations. The expansions converge for all values of momentum transfer for which dispersion relations have been proved.
165 citations
TL;DR: In this article, the crossing property on the mass shell for amplitudes involving two incoming and two outgoing stable particles with arbitrary masses was proved on the (s, t, u)-plane corresponding to crossed processes.
Abstract: In the framework of the ℒ.l.Z. formalism, the crossing property is proved on the mass shell for amplitudes involving two incoming and two outgoing stable particles with arbitrary masses. Any couple of physical regions in the (s, t, u)-plane corresponding to crossed processes are shown to be connected by a certain domain of analyticity. For every negative value oft, the amplitude is analytic in the cuts-plane outside of a large circle.
121 citations