1. Extending forward scattering Regge Theory to internediate energies José R. Peláez Departamento de Física Teórica II. Universidad Complutense de Madrid J. R. Peláez and F.J. Ynduráin in preparation J. R. Peláez and F.J. Ynduráin. PRD68:074005,(2003) F.J. Ynduráin. ‘QCD@Work’, hep-ph/0310206 J. R. Peláez and F.J. Ynduráin. PRD69,114001 (2004)

2. Motivation In recent years great effort to extend a precise Regge description of forward hadron scattering down in energies: Very Systematic analysis of COMPAS, COMPETE Collaborations, etc...: PDG 2000PDG 2002-2004 COMPETE 2002 “Former” ideas in Regge Theory were “rescued” to extend Regge to lower energies. Logarithmic growths, lifting of degeneracy... Regge over a wide energy range helps determining: -The logarithmic growth law - Subleading trajectories - Degeneracy issues -Saturation of unitarity bounds?...

3. We propose (“rescue”) further simple refinements that allow the application of Regge theory down to ~1 GeV in Kinetic Energy for forward hadronic scattering 1) Use of =(s-u)/2 powers instead of s powers. This is the natural Regge variable needed to express symmetric amplitudes. 2) Use of a logarithmic law from an improved unitarity bound (Yndurain, 1972). Faster than s log s, slower than s log 2 s. 3) Phase space corrections in total cross sections 4) Inclusion of  data Data fits show a sizable  2 /d.o.f. improvement of with each suggestion

4. First suggestion: =s-u powers Regge behavior can be deduced from the Froissart-Gribov representation. Dispersion relation for s  u even (similar for odd) amplitudes Partial wave projection in t-channel using P l (cos  t ) with Continued to L complex plane Asimptotically If a complex Regge pole accurs at L=  (t) customarily ~s Regge variable is Forward scattering t=0

5. Second suggestion: Yndurain’s logarithmic law With a similar arguments for the second derivative of D partial wave Improved logarithmic bound (Yndurain 72) suggests improved Pomeron Well known Froissart bound for amplitudes. At high energies... Usual Froissart bound recovered at very high energies, but softer rise at intermediate energies.

6. Our fit Regge Pole 1) Factorization to relate the same Regge pole in different channels Five trajectories: R=Pomeron, f,a, ,  10 processes: NN, KN,  N Data Compilation from COMPAS Collaboration   total cross sections data “rescued” Only P,f,  needed. Good to fix the  New: 3 more processes

7. Data on  total cross sections at high energy Large systematics (not given), mostly below 2 GeV, and the  -- in  -  - In conflict with CERN-Munich in 1.4<  s<2 GeV   -  - (mb)   +  - (mb) All consistent above 2 GeV 4 EXPERIMENTS, ‘67, ’73, ’76,’80: NOT phase shift analysis Pomeranchuk theorem OK

8. Our fit PP 1.87  0.02 fNPfNP 1.285  0.002 fKPfKP 0.738  0.004 A 0.046  0.001 2 1.14  0.02 ff 0.90  0.02 fNffNf 2.29  0.03 fKffKf 0.l1  0.02 ff 0.68  0.01 fNafNa 0.20  0.05 fKafKa 0.59  0.16  1.27  0.10 f N  0.52  0.04 f K  0.48  0.04  0.447  0.006 f N  2.06  0.02 f K  0.65  0.01 - Factorization - Degeneracy: f/ , a/  as in PDG 3) Phase space 1) instead of s powers 2) Improved logarithmic law for Pomeron: (Yndurain 72) We use some usual features Together with our three improvements

9. E kin min 1-1.3 GeV 1.5 GeV2 GeV3 GeV # data points 1186 1002895768  2 /d.o.f. Our parametrization0.99 0.710.680.63 Our fit Global fit to total cross sections ~ Im T, and ReT/ImT Fit to data of COMPASS group +  scattering data with 0.5% systematic error added to NN data 1% systematic error added to  N data 1.5% systematic error added to KN data

10. Our fit: Total pp and pp cross sections The fit extends from Kinetic energies  ~1 GeV to 30 TeV !! -

11. Our fit : Total cross sections

12. Our fit We have also included ReT/ImT

13.   +  - (mb)  s(GeV)   0  - (mb)   -  - (mb) Our fit : Total  cross sections

14.  2 /d.o.f. improvement: first suggestion: =s-u powers E kin min1-1.3 GeV1.5 GeV2 GeV3 GeV # data points11861002895768  2 /d.o.f. Our parametrization0.990.710.680.63 s instead of powers 1.731.241.090.87 Only affects the intermediate/low energy. The fits are similar to existing ones at high energies Larger  2 /d.o.f. if using s, but fits are good if E kin > 3 GeV, as it is already known (COMPAS, COMPETE, PDG...)

15.  2 /d.o.f. improvement: second suggestion: Yndurain’s logarithmic law Improvement: only sizable improvement if extended to intermediate/low energy. smaller than that due to using powers E kin min1-1.3 GeV1.5 GeV2 GeV3 GeV # data points11861002895768  2 /d.o.f. Our parametrization0.990.710.680.63 Log( ) Pomeron 1.150.810.710.61 Log 2 ( - th ) Pomeron 1.130.760.620.66 Larger  2 /d.o.f. if rise given by s log (slower) and s log 2 (faster) Indication of saturation of improved unitarity bound ? If not extended to low energies, both logarithmic laws similar We have checked that other logarithmic variables give slightly worse  2 /d.o.f

16.  2 /d.o.f. improvement: third suggestion: phase space factor E kin min 1-1.3 GeV 1.5 GeV2 GeV3 GeV #  tot data points 998 819725615  2 /d.o.f. Our parametrization0.95 0.580.540.45 s instead of 1.36 0.690.630.56 This approximation only affects cross sections ~ Im T At s>> M 2, well approximated by ~ s, but needed below For NN scattering: flux overestimated by 30% at E kin ~ 1 GeV 5% at Vs ~ 5 GeV _

17. We propose (“rescue”) simple refinements to apply Regge theory down to ~1 GeV in Kinetic Energy for forward hadronic scattering 1) Use of =(s-u)/2 powers instead of s powers. This is the natural Regge variable needed to express symmetric amplitudes. 2) Use of a logarithmic law from an improved unitarity bound (Yndurain, 1972). Faster than s log s, slower than s log 2 s. 3) Phase space corrections in total cross sections 4) Inclusion of  data We have fitted the data with that parametrization showing a sizable  2 /d.o.f. improvement with each suggestion CONCLUSIONS

18. Yndurain’s improved unitarity bound. PLB41,591(1972), Rev. Mod. Phys.44,645(1972) VERY, VERY, SKETCHY!!  scattering FOR SIMPLICITY t channel unitarity ensures exists when t  4M 2 The Froissart-Gribov projection Taking the second derivative as above, there is a term Using the asymptotic properties of P’’: If thus Im f l (s) grows slower than that straigthforward generalization for other processes

19. Sub-subleading trajectories Reaching so low in energies... What about other sub-subleading trajectories? Still, we are studying the effect of another two f’ and  trajectories... Since the  2 /dof was already pretty good, thr\ey are not needed from the statistics point of view The price to pay are 5-6 more parameters We have checked they could help restoring degeneracy. Imposing exact degeneracy (extreme case) for the 4 subleading trajectories  2 /dof =1.02 down to E kin >1 GeV The sub-subleading trajectories come with a natural intercept ~0.25. Still, 4 more parameters...

20. Yndurain’s improved unitarity bound. First, Froissart bound: We expand the amplitude in partial waves we sum up to a finite L(s) and then to infinity setting, Using the asymptotic behavior of P l (cos  ) the infinite sum can be shown to behave as the L(s) 2 dominates at large s and n and we recover Froissart bound PLB41,591(1972), Rev. Mod. Phys.44,645(1972) VERY, VERY, SKETCHY!!