1. 1 1.Introduction 2.Limitations involved in West and Yennie approach 3.West and Yennie approach and experimental data 4.Approaches based on impact parameter representation 5.General formula and interpretation of experimental data 6.Conclusion Coulomb and hadronic scattering in elastic high-energy nucleon collisions V. Kundrát, M. Lokajíček Institute of Physics, AS CR, Praha, CR

2. 2 1. Introduction Elastic nucleon collisions:  extensive and precise experimental data  accurate analysis  mainly due to hadronic interactions at all t  charged hadrons: due to Coulomb interactions mainly at small |t|  influence of both interactions (spins neglected) … Coulomb (QED),…. hadronic amplitudes mutually bound by relative phase,  West–Yennie (1968) (for large s )

3. 3 some approximations; adding dipole form factors  simplified West-Yennie formula with relative phase, fitting differential cross section data  constant and averaged values of  what approximations used?  their contradictory character  t independence of untenable (V.K., M. Lokajíček, Phys. Lett. B 611 (2005) 102)

4. 4 2. Fundamental limitations involved in West-Yennie  general belief: no limitation of relative phase phase should be real  imaginary part of integral should be zero for all t   quantity should be independent on t in the whole kinematically allowed region of t values!!! for any t, t’ (mentioned among assumptions in: Amaldi et al., Phys. Lett. B 43 (1973) 231)

5. 5 phase real:  introducing phase and modulus ( s depressed)

6. 6 both I 1 (t), I 2 (t) proper integrals if ζ(t) has bounded derivatives for all admissible t modulus F(t) is non-zero in interval (-4p 2,0) similar for higher derivatives

7. 7 key question: what ζ (t) solves representing infinite number of conditions → only one solution ζ(t) = const define functions (heuristic way) → generally near contradiction: asume for ε small, ζ(t)’ ≠0 → f(t,τ)≠0 for any t,τ → ζ(t) = const …any solution !!! 0 ≠ ε [f(t 1,τ 1 ) + f(t 2,τ 2 )] positive number!!! …for finite ε

8. 8 3. West-Yennie approach and data t independence of ρ  first derivative is zero  t independence of ρ  first derivative is zero  valid for all t data: existence of diffractive minimum at  derivative is zero …contradiction  existence of diffr. minimum, i.e., data excludes possibility for to be t independent !!!  purely exponential t dependence of the modulus at all t contradicts experimental data; satisfied only “approximately” in limited region of t  

9. 9  under what assumptions simplified WY derived?  in addition to being constant: purely exponential t dependence of in the whole kinematically allowed region of t integral from WY:

10. 10 high and  as  and small |t|  can be neglected   are assumptions fulfilled by data ?  

11. 11 change of magnitude from optical point to diffractive minimum: ~ 8  9 ~ 7  8 ~ 5  6 ~ 4 ~ 8  9 ~ 7  8 ~ 5  6 ~ 4 change of | | : change of | | : ~ 4.  4.5  3.5  4.  2.5  3  2 ~ 4.  4.5  3.5  4.  2.5  3  2 “approximately” purely exponential for small region becomes narrower when energy increases with |t| ; !!! -t [GeV ] (0., 1.5) (0., 1.35) (0., 0.8) (0., 0.4) 2

12. 12  deviations from purely exponential behavior rise strongly with increasing energy; measure of deviations: diffractive slope constant slope B(s) corresponding to WY formula for simplified WY formula commonly used in inconsistent way if applied to region of small |t| only; discrepancy cannot be removed even if two different formulas are used: simplified WY at small |t| and phenomelogically constructed hadronic amplitude at large |t| simplified WY formula commonly used in inconsistent way if applied to region of small |t| only; discrepancy cannot be removed even if two different formulas are used: simplified WY at small |t| and phenomelogically constructed hadronic amplitude at large |t| integral and simplified WY formulas contradict experimental data integral and simplified WY formulas contradict experimental data however: before ISR experiments nothing was known about diffractive structure  WY amplitude might be used however: before ISR experiments nothing was known about diffractive structure  WY amplitude might be used

13. 13 4. Approaches based on impact param. representation (Franco (1966,1973), Lapidus et al. (1978), Cahn (1982)) (Franco (1966,1973), Lapidus et al. (1978), Cahn (1982)) used eikonal models based on approximate form of Fourier-Bessel transformation valid at asymptotic s and small |t| mathematically rigorous formulation (valid at any s and t ) (Adachi et al., Islam (1965 – 1976)) mathematically rigorous formulation (valid at any s and t ) (Adachi et al., Islam (1965 – 1976)) additivity of potentials  additivity of eikonals (Franco (1973)) additivity of potentials  additivity of eikonals (Franco (1973)) total scattering amplitude 

14. 14 convolution integral convolution integral equation describes simultaneous actions of both Coulomb and hadronic interactions; to the sum of both amplitudes new complex function (convolution integral) is added equation describes simultaneous actions of both Coulomb and hadronic interactions; to the sum of both amplitudes new complex function (convolution integral) is added at difference with WY amplitude (Coulomb amplitude multiplied by phase factor only) at difference with WY amplitude (Coulomb amplitude multiplied by phase factor only) valid at any s and t valid at any s and t

15. 15 general formula valid up to terms linear in α (V. K., M. Lokajíček (1994)) general formula valid up to terms linear in α (V. K., M. Lokajíček (1994)) expression [1 ± iα G(s,t)] … two terms of Talyor series expansion of expression [1 ± iα G(s,t)] … two terms of Talyor series expansion of  complex G(s,t) cannot be simply interpreted as mere change of phase real ↔ constant in t real ↔ constant in t use: (i) determine F N (s,t) (ii) predict F C+N (s,t)

16. 16 5. General formula and interpretation of exp. data  absence of any actual theory  must be derived from data  complex elastic hadronic amplitude characterized by pair of real modulus and phase  only one experimentally determined function  elastic hadron amplitude can be hardly derived from mere exper. data  t dependence of phase needed   distribution of elastic scattering in impact parameter space  if small in Coulomb and interference regions and increases monotony for higher |t| and 

17. 17 has Gaussian shape with maximum at b = 0  central behavior of elastic collisions ( ) has Gaussian shape with maximum at b = 0  central behavior of elastic collisions ( )  current status: elastic diffraction … central inelastic diffraction … peripheral  peripheral distribution of elastic hadron scattering: maximum at b > 0 ; e.g., increases with rising |t| (e.g.,   ); vanishes at |t| 0 ; e.g., increases with rising |t| (e.g.,   ); vanishes at |t| <  similar behavior investigated by: Franco and Yin (1985-6)  analysis of experimental data (pp at 53 GeV, pp at 541 GeV) … t dependent … t dependent influence of Coulomb scattering cannot be neglected at higher |t| influence of Coulomb scattering cannot be neglected at higher |t| peripheral feature of elastic scattering: statistically preferred peripheral feature of elastic scattering: statistically preferred

18. 18 M. Islam (2005) model predictions (J. Kašpar: Diploma thesis (2005)) dσ/dt (difference at t = -.0005 GeV 2 …. 4.5 %) dσ/dt (difference at t = -.0005 GeV 2 …. 4.5 %) ratio of interference to hadronic terms [%]  influence of Coulomb scattering cannot be neglected at higher |t| ratio of interference to hadronic terms [%]  influence of Coulomb scattering cannot be neglected at higher |t|  tot =109.17 mb,  el =21.99 mb, B = 31.5 GeV -2,  =0.123  tot =109.17 mb,  el =21.99 mb, B = 31.5 GeV -2,  =0.123 total WY eikonal

19. 19  Bourrely-Soffer-Wu model (Eur. Phys. J. C 28 (2003) 97)  tot =103.64 mb,  el =28.51 mb, B = 20.19 GeV -2,  =0.121  tot =103.64 mb,  el =28.51 mb, B = 20.19 GeV -2,  =0.121

20. 20  Luminosity different total (eikonal and WY) amplitudes  different luminosity determinations different total (eikonal and WY) amplitudes  different luminosity determinations

21. 21 6. Conclusion  WY integral formula admits only hadronic amplitudes with constant quantity  simplified WY amplitude is in contradiction with data  WY approach leads to false results at high energies  approach based on eikonal model  suitable tool for analyzing high- energy elastic hadron scattering  t dependence of its modulus and phase needed at all allowed t  dynamical characteristics of elastic hadronic amplitude determined by its t dependence, i.e., are model dependent quantities  influence of Coulomb scattering cannot be neglected at higher |t|