Integral and derivative dispersion relations, analysis of the forward scattering data J.R. Cudell *, E. Martynov *+, O.V.Selyugin *# * Institut de Physique, Universite de Liege, Belgique, + Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine, # Joint Institute of Nuclear Researchers, Dubna, Russia

E.Martynov Aachen, 30 April, 2015 OutlineOutline ● Introduction – COMPETE results – Main assumptions, main goals ● Integral Dispersion Relations (IDR) ● Derivative Dispersion Relations (DDR) ● Phenomenology – Proton-proton and antiproton-proton experimental data – Regge models: Simple pomeron pole, dipole, tripole – Results, comparison of methods and models ● Conclusion

E.Martynov Aachen, 30 April, 2015 COMPETE results CO mputerised M odels, P arameter E valuation for T heory and E xperiment. Protvino, Russia Dubna, Russia, Paris, France, Liege, Belgium, Providence, USA, Kiev, Ukraine, Uzhgorod, Ukraine, Durham, UK Data & Model Base in high energy physics Analytic (Regge type) parametrizations of the forward scattering amplitudes for interactions. Phys. Rev. D61(2001)034019, D63 (2001) ; D65(2002),074024; Phys. Rev. Lett. 89(2002) Number of models with at were analyzed and compared. A selection of acceptable models and of acceptable data is made. The models are then ranked according not only to their, but also to their number of parameters, stability, extendability to other data, etc. The best model accordingly to the COMPETE criteria is the model with universal (for all hadronic processes) behavior of the. The simple pole pomeron model ( ) is excluded from the list of the best models.

E.Martynov Aachen, 30 April, 2015 Main assumptions and goals Assumptions Analyticity, structure of singularities, crossing- symmetry, optical theorem Spin is ignored Pomeron, reggeons (degene- rated crossing-even,, and crossing-odd, ) No (asymptotic) Odderon Unphysical cuts and resonances are unimportant at [ V.Lengyel, A.Lengyel, Yad. Fiz. (1970)] Goals Corrections to the asymptotic form of the derivative dispersion relations. Fit and comparison of the various models for and amplitudes (IDR method vs. DDR method). Extension to other processes,.

Integral Dispersion Relations (IDR) Aachen, 30 April, 2015 E.Martynov If odderon does not contribute asymptotically, then where m and E are the mass and lab. energy of proton, B is a subtract constant (to be determined from the fit to the data), +(-) stands for. Problem High-energy parametrizations cannot be used between m and (usualy ) and the possible solutions: Numerical integration of the data (some defects) Any parametrization (with any number of parameters)

Derivative Dispersion Relations (DDR) Aachen, 30 April, 2015 E.Martynov Input: IDR for cross-even (-odd) parts of the amplitudes Output: derivative (or local) dispersion relations. High-energy form [ V.N.Gribov, A.B.Migdal (1968); J.B. Bronzan et al. (1974); K.Kang, B. Nicolescu (1975)] Corrections: Similar result for crossing-odd part. /Constants C (+,-) can be calculated/

Phenomenology. General remarks Aachen, 30 April, 2015 E.Martynov Standard formulationHigh-energy approximation or IDR at low energies DDR with a subtraction constant Regge poles DDR at h.e.

Phenomenology. Models Aachen, 30 April, 2015 E.Martynov Secondary reggeons Pomeron Simple pole with Double pole with Triple pole with

Phenomenology. Fitting procedure, IDR Aachen, 30 April, 2015 E.Martynov Any good (with minimal ) parametrisation of at The only requirement: Step 1: The given model is fitted to the at Step 2: All high-energy parameters are fixed. Fit to the data on at Step 3: The last free parameter, subtraction constant, is fitted to the data on at Then fit of each pomeron model is performed in three steps:

Phenomenology. Results Aachen, 30 April, 2015 E.Martynov Total cross sections

Phenomenology. Results Aachen, 30 April, 2015 E.Martynov Ratios of the real to imaginary part

Phenomenology. Results Aachen, 30 April, 2015 E.Martynov Standard normalizationAsymptotic normalization IDRDDRIDR‘‘-is’’ rule Simple pole Double pole Triple pole All considered models give the comparable values of. The integration in IDR over physical region only gives very good agreement of the calculated with the data at (all parameters are fixed from the fit at. Conclusions ( preliminary )

Aachen, 30 April, 2015 E.Martynov The standard normalization and parameterizations with the "Regge variable“ lead to a better description. A deviation the DDR as curve for of the IDR one becomes visible even at (and about 10% at 6 GeV ).  The corrections to DDR as must be taken into account. Inclusion of non asymptotic terms improves the description of and may lead to different conclusion regarding simple pole model. However, in order to check that a magnitudes of the resonance contributions as well as of the unphysical cut at relatively low energies must be estimated, the whole set of the amplitudes and data, including and processes, must be considered.