Integral and derivative dispersion relations, analysis of the forward scattering data

1. Integraland 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

2. Outline • 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 Aachen, 29 September, 2014 E.Martynov

3. COmputerised Models, Parameter Evaluationfor Theoryand Experiment. Protvino, Russia Dubna, Russia, Paris, France, Liege, Belgium, Providence, USA, Kiev, Ukraine, Uzhgorod, Ukraine, Durham, UK COMPETEresults 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) 059901; D65(2002),074024; Phys. Rev. Lett. 89(2002)201801. 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. Aachen, 29 September, 2014 E.Martynov

4. 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, . 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)] Aachen, 29 September, 2014 E.Martynov

5. Integral Dispersion Relations (IDR) 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) Aachen, 29 September, 2014 E.Martynov

6. Derivative Dispersion Relations(DDR) 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/ Aachen, 29 September, 2014 E.Martynov

7. Phenomenology. General remarks Standard formulation High-energy approximation or or IDR DDR with a subtraction constant Regge poles DDR at h.e. at low energies Aachen, 29 September, 2014 E.Martynov

8. Phenomenology.Models Secondary reggeons Pomeron Simple pole with Double pole with Triple pole with Aachen, 29 September, 2014 E.Martynov

9. Phenomenology.Fitting procedure, IDR Any good (with minimal ) parametrisation of at The only requirement: Then fit of each pomeron model is performed in three steps: • 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 Aachen, 29 September, 2014 E.Martynov

10. Phenomenology.Results Total cross sections Aachen, 29 September, 2014 E.Martynov

11. Phenomenology.Results Ratios of the real to imaginary part Aachen, 29 September, 2014 E.Martynov

12. Phenomenology. Results Conclusions (preliminary) • 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 . Aachen, 29 September, 2014 E.Martynov

13. The standard normalization and parameterizations with the "Regge variable“ • lead to a better description. • A deviation the DDRas curve for of the IDR one becomes visible even at • (and about 10% at 6 GeV ). • N The corrections to DDRas 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. Aachen, 29 September, 2014 E.Martynov