# Unitarity tests at high energies

S. M. Roy

^{1}Abstract: We derive unitarity inequalities involving Re F ( s , t ) and Im F ( s , t ) in a region 0 ⩽ | t | ⩽ | T | which exhaust the content of unitarity given only the elastic amplitude F ( s , t ) in this range of t . If the Froissart bound is saturated, these inequalities lead to inequalities on Martin's scaling functions. The inequalities may be used to check compatibility with unitarity of theoretical models or experimental data.

Topics: Unitarity (62%)

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Abstract: The real-part dynamics of the proton-proton/antiproton scattering amplitude within the framework of dispersive diffraction theory is presented, from 0.02 through 0.54 to 40 TeV. Specific results are: (1) the explicit corrections to Martin's still useful real-part formula; (2) the correlation between cross-over and vanishing of the real part; (3) dip dynamics; (4) the objective nature of these results.

16 citations

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Abstract: We review recent progresses in the development and application of unitarity bounds based on experimentally determined quantities, the high-energy theorems for global quantities and the scalings for the differential elastic cross section and the inelastic processes of strongly interac particles.

8 citations

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Abstract: Using a novel representation for the Martin's real-part formula without the full scaling property, an almost model-independent description of the proton–proton differential cross-section data at high energies (19.4 GeV–62.5 GeV) is obtained. In the impact parameter and eikonal frameworks, the extracted inelastic overlap function presents a peripheral effect (tail) above 2 fm and the extracted opacity function is characterized by a zero (change of sign) in the momentum transfer space, confirming results from previous model-independent analyses. Analytical parametrization for these empirical results are introduced and discussed. The importance of investigations on the inverse problems in high-energy elastic hadron scattering is stressed and the relevance of the proposed representation is commented. A short critical review on the use of Martin's formula is also presented.

5 citations

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Abstract: Using a novel representation for the Martin's real-part formula without the full scaling property, an almost model-independent description of the proton-proton differential cross section data at high energies (19.4 GeV - 62.5 GeV) is obtained. In the impact parameter and eikonal frameworks, the extracted inelastic overlap function presents a peripheral effect (tail) above 2 fm and the extracted opacity function is characterized by a zero (change of sign) in the momentum transfer space, confirming results from previous model-independent analyses. Analytical parametrization for these empirical results are introduced and discussed. The importance of investigations on the inverse problems in high-energy elastic hadron scattering is stressed and the relevance of the proposed representation is commented. A short critical review on the use of Martin's formula is also presented.

##### References

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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\(t = \mu ^2 \).

348 citations

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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.

150 citations

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Abstract: From unitarity alone a lower bound for the derivative of the absorptive part of the forward scattering amplitude with respect to the momentum transfer is obtained, in terms of the elastic and total cross sections. Comparison with high-energy scattering experiments shows that the actual value of this derivative is rather close to the lower bound, which provides some information on the partial-wave distribution. Our result can also be used to obtain consistency requirements on theoretical models. If Regge behavior is assumed for high-energy scattering, namely, $F(s, t)\ensuremath{\simeq}f(t){s}^{\ensuremath{\alpha}(t)}$, then one can show that either ${\ensuremath{\alpha}}^{\ensuremath{'}}(0)g~\ensuremath{\epsilon}g0$ or $\ensuremath{\alpha}(t)\ensuremath{\equiv}\mathrm{const}.$

61 citations

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60 citations

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Abstract: An analysis of total cross sections and ratios of real to imaginary parts of the forward hadronic scattering amplitude for pp andu¯p is given. The data above s = 25GeV 2 , including the latestp¯p results from the ISR, are simultaneously fit for both real and imaginary parts using proper analytic forms. The resulting fits provide an excellent interpolation of the data using five parameters. Extrapolations are done to collider energies. Limits are placed on the magnitude of odd amplitudes with unconventional energy dependence.

28 citations