1. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 1 Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections Martin Block Northwestern University

2. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 2 Prior Restraint! the Froissart Bound

3. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 3 1) Data selection: The “Sieve” Algorithm---“Sifting data in the real world”, M. Block, Nucl. Instr. and Meth. A, 556, 308 (2006). 3) Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, Phys. Rev. D 72, 036006 (2005). OUTLINE 4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished). 2)New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur. Phys. J. C 47, 697 (2006). Touched on briefly, but these are important constraints! ) The Glauber calculation:

4. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 4 Conclusions From hadron-hadron scattering  The Froissart bound for  p,  p and pp collisions is saturated at high energies. 3) At cosmic ray energies,  we can make accurate estimates of  pp and B pp from collider data. 4) Using a Glauber calculation of  p-air from  pp and B pp, we now have a reliable benchmark tying together colliders to cosmic rays. 2) At the LHC,  tot = 107.3  1.2 mb,  = 0.132  0.001.

5. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 5 “Fishing” for Data Part 1: “Sifting Data in the Real World”, Getting rid of outliers! M. Block, arXiv:physics/0506010 (2005); Nucl. Instr. and Meth. A, 556, 308 (2006).

6. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 6 Lorentzian Fit used in “Sieve” Algorithm

7. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 7

8. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 8

9. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 9

10. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 10

11. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 11

12. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 12 You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.

13. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 13 Part 2: “New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur. Phys. J. C47 (2006).

14. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 14

15. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 15 Derivation of new analyticity constraints Theoretical high energy cross section parametrization Experimental low energy cross section Finite energy cutoff!

16. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 16 so that:  exp’t (   ( 0 ), d  exp’t (   d  d  ( 0 )  d, or, its practical equivalent,  exp’t (   ( 0 ),  exp’t (   ( 1 ), for     for both pp and pbar-p exp’t cross sections We can also prove that for odd amplitudes:  odd ( 0 ) =  odd ( 0 ).

17. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 17 Francis, personally funding ICE CUBE Part 3: Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, Phys. Rev. D 72, 036006 (2005).

18. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 18 ln 2 (s/s 0 ) fit  =0.5, Regge- descending trajectory 7 parameters needed, including f + (0), a dispersion relation subtraction constant

19. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 19 Only 3 Free Parameters However, only 2, c 1 and c 2, are needed in cross section fits ! These anchoring conditions, just above the resonance regions, are analyticity conditions!

20. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 20 Cross section fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

21. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 21  -value fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

22. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 22 What the “Sieve” algorithm accomplished for the pp and pbar p data Before imposing the “Sieve algorithm:  2 /d.f.=5.7 for 209 degrees of freedom; Total  2 =1182.3. After imposing the “Sieve” algorithm: Renormalized  2 /d.f.=1.09 for 184 degrees of freedom, for  2 i > 6 cut; Total  2 =201.4. Probability of fit ~0.2. The 25 rejected points contributed 981 to the total  2, an average  2 i of ~39 per point.

23. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 23 Comments on the “Discrepancy” between CDF and E710/E811 cross sections at the Tevatron Collider If we only use E710/E811 cross sections at the Tevatron and do not include the CDF point, we obtain: R  2 min /  probability=0.29  pp (1800 GeV) = 75.1± 0.6 mb  pp (14 TeV) = 107.2± 1.2 mb If we use both E710/E811 and the CDF cross sections at the Tevatron, we obtain: R  2 min /  =184, probability=0.18  pp (1800 GeV) = 75.2± 0.6 mb  pp (14 TeV) = 107.3± 1.2 mb, effectively no changes Conclusion : The extrapolation to high energies is essentially unaffected!

24. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 24 Cross section and  -value predictions for pp and pbar-p The errors are due to the statistical uncertainties in the fitted parameters LHC prediction Cosmic Ray Prediction

25. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 25  p log 2 ( /m) fit, compared to the  p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, 091901, (2004)

26. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 26 Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

27. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 27 More LHC predictions, from the Aspen Eikonal Model: M. M. Block, Phys. Reports 436, 71 (2006). Nuclear slope B = 19.39 ± 0.13 (GeV/c) -2  elastic = 30.79 ± 0.34 mb Differential Elastic Scattering

28. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 28 Part 3: The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel Ralph Engel, At Work

29. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 29 EXPERIMENTAL PROCEDURE: Fly’s Eye and AGASA Fig. 7 X max distribution with exponential trailing edge Monte Carlo Example Fly’s Eye Shower Profile Fig. 1 An extensive air shower that survives all data cuts. The curve is a Gaisser-Hillas shower- development function: shower parameters E=1.3 EeV and X max =727 ± 33 g cm -2 give the best fit. Logarithmic slope,  m, is measured

30. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 30 Extraction of  tot (pp) from Cosmic Ray Extensive Air Showers by Fly’s Eye and AGASA k is very model-dependent Need good fit to accelerator data

31. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 31 X max = X 1 + X’ HiRes Measurement of X max Distribution:

32. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 32 B, from Aspen (eikonal) Model Ingredients needed for Glauber Model , from ln 2 s fit

33. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 33 Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished) B (nuclear slope) vs.  pp, as a function of  p-air  pp from ln 2 (s) fit and B from QCD-fit HiRes Point

34. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 34 Measured k = 1.28  0.07 Belov, this conference, k = 1.21+0.14-0.09

35. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 35  p-air as a function of  s, with inelastic screening  p-air inel = 460  14(stat)+39(sys)-11(sys) mb We find: k = 1.28  0.07 Belov, this conference, k = 1.21 + 0.14 - 0.09  p-air inel = 460  14(stat)+39(sys)-11(sys) mb

36. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 36 Conclusions From hadron-hadron scattering  The Froissart bound for  p,  p and pp collisions is saturated at high energies. 3) At cosmic ray energies,  we can make accurate estimates of  pp and B pp from collider data. 4) Using a Glauber calculation of  p-air from  pp and B pp, we now have a reliable benchmark tying together colliders to cosmic rays. 2) At the LHC,  tot = 107.3  1.2 mb,  = 0.132  0.001.

37. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 37 Saturating the Froissart Bound  pp and  pbar-p log 2 ( /m) fits, with world’s supply of data Cosmic ray points & QCD-fit from Block, Halzen and Stanev: Phys. Rev. D 66, 077501 (2000).

38. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 38 This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

39. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 39 Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

40. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 40 To obtain  pp from  p-air

41. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 41

42. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 42 Robust Feature:  (z)  1/  i 2 for large  i 2

43. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 43 Why choose normalization constant  =0.179 in Lorentzian  0 2 ? Computer simulations show that the choice of  =0.179 tunes the Lorentzian so that minimizing  0 2, using data that are gaussianly distributed, gives the same central values and approximately the same errors for parameters obtained by minimizing these data using a conventional  2 fit. If there are no outliers, it gives the same answers as a  2 fit. Hence, when using the tuned Lorentzian  0 2, much like in keeping with the Hippocratic oath, we do “no harm”.

44. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 44  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error

45. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 45  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

46. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 46 All cross section data for E cms > 6 GeV,  + p and  - p, from Particle Data Group

47. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 47 All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV,  + p and  - p, from Particle Data Group

48. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 48  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

49. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 49

50. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 50 The published cosmic ray data (the Diamond and Triangles) are the problem

51. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 51 Stability of “Sieve” algorithm Fit parameters are stable, essentially independent of cut  2 i We choose  2 i = 6, since R  2 min /  giving  0.2 probability for the goodness-of-fit.

52. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 52 log 2 ( /m p ) fit compared to log( /m p ) fit: All known n-n data

53. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 53 Lorentzian Fit used in “Sieve” Algorithm

54. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 54 This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

55. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 55 The popular parameterization  pp  s 

56. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 56

57. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 57 1) Already known to violate unitarity and the Froissart bound at high energies. 2) Now, without major complicated low energy modifications, violates analyticity constraints at low energies. No longer a simple parametrization!

58. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 58 A 2- parameter   fit of the Landshoff-Donnachie variety:  ± = As  + Bs  ± Ds   using 4 analyticity constraints Horrible  2 /d.f.

59. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 59 Hence,minimize  i  (z), or equivalently, we minimize  2   i  2 i

60. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 60 Generalization of the Maximum Likelihood Function, P

61. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 61 Problem with Gaussian Fit when there are Outliers

62. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 62

63. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 63 Why choose normalization constant  =0.179 in Lorentzian  0 2 ? Computer simulations show that the choice of  =0.179 tunes the Lorentzian so that minimizing  0 2, using data that are gaussianly distributed, gives the same central values and approximately the same errors for parameters obtained by minimizing these data using a conventional  2 fit. If there are no outliers, it gives the same answers as a  2 fit. Hence, when using the tuned Lorentzian  0 2, much like in keeping with the Hippocratic oath, we do “no harm”.

64. April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 64  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error