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Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections

Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections. Martin Block Northwestern University. Prior Restraint! the Froissart Bound. OUTLINE. Data selection: The “Sieve” Algorithm- --“Sifting data in the real world”,

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Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections

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

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

  3. OUTLINE • Data selection: The “Sieve” Algorithm---“Sifting data in the real world”, • M. Block, Nucl. Instr. and Meth. A, 556, 308 (2006). • 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! 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(2005). 4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished). ) The Glauber calculation: M. Block, Aspen Workshop Cosmic Ray Physics 2007

  4. Conclusions From hadron-hadron scattering 1) The Froissart bound for gp, pp and pp collisions is saturated at high energies. • 2) At the LHC, • stot = 107.3 ± 1.2 mb, r = 0.132±0.001. 3) At cosmic ray energies,we can make accurate estimates of spp and Bpp from collider data. 4) Using a Glauber calculation of sp-air from spp and Bpp, we now have a reliable benchmark tying together colliders to cosmic rays. M. 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). M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

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

  14. M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

  16. We can also prove that for odd amplitudes: sodd(n0) = sodd (n0). so that:sexp’t (n0) = s(n0), dsexp’t (n0)/dn = ds(n0) /dn, or, its practical equivalent, sexp’t (n0) = s(n0), sexp’t (n1) = s(n1), for n1>n0 for both pp and pbar-p exp’t cross sections M. 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). M. Block, Aspen Workshop Cosmic Ray Physics 2007

  18. m=0.5, Regge-descending trajectory ln2(s/s0) fit 7 parameters needed, including f+(0), a dispersion relation subtraction constant M. Block, Aspen Workshop Cosmic Ray Physics 2007

  19. These anchoring conditions, just above the resonance regions, are analyticity conditions! Only 3 Free Parameters However, only2, c1andc2, are needed in cross section fits ! M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

  21. r-value fits for Ecms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm M. 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: c2/d.f.=5.7 for 209 degrees of freedom; Total c2=1182.3. After imposing the “Sieve” algorithm: Renormalized c2/d.f.=1.09 for 184 degrees of freedom, for Dc2i > 6 cut; Total c2=201.4. Probability of fit ~0.2. The 25 rejected points contributed 981 to the total c2 , an average Dc2i of ~39 per point. M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

  24. M. Block and F. Halzen, Phys Rev D 70, 091901, (2004) gp log2(n/m) fit, compared to the pp even amplitude fit M. Block, Aspen Workshop Cosmic Ray Physics 2007

  25. Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV, p+p and p-p, after applying “Sieve” algorithm M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

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

  28. Monte Carlo Example Fly’s Eye Shower Profile Logarithmic slope, Lm, is measured 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 Xmax =727 ± 33 g cm-2 give the best fit. Fig. 7 Xmax distribution with exponential trailing edge EXPERIMENTAL PROCEDURE: Fly’s Eye and AGASA M. Block, Aspen Workshop Cosmic Ray Physics 2007

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

  30. HiRes Measurement of Xmax Distribution: Xmax = X1 + X’ M. Block, Aspen Workshop Cosmic Ray Physics 2007

  31. B, from Aspen (eikonal) Model s, from ln2s fit Ingredients needed for Glauber Model M. Block, Aspen Workshop Cosmic Ray Physics 2007

  32. spp from ln2(s) fit and B from QCD-fit HiRes Point Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished) B (nuclear slope) vs. spp, as a function of sp-air M. Block, Aspen Workshop Cosmic Ray Physics 2007

  33. sp-airinel = 460±14(stat)+39(sys)-11(sys) mb sp-air as a function ofÖs, with inelastic screening We find: k = 1.28 ± 0.07 Belov, this conference, k = 1.21 + 0.14 - 0.09 sp-airinel = 460±14(stat)+39(sys)-11(sys) mb M. Block, Aspen Workshop Cosmic Ray Physics 2007

  34. Conclusions From hadron-hadron scattering 1) The Froissart bound for gp, pp and pp collisions is saturated at high energies. • 2) At the LHC, • stot = 107.3 ± 1.2 mb, r = 0.132±0.001. 3) At cosmic ray energies,we can make accurate estimates of spp and Bpp from collider data. 4) Using a Glauber calculation of sp-air from spp and Bpp, we now have a reliable benchmark tying together colliders to cosmic rays. M. Block, Aspen Workshop Cosmic Ray Physics 2007

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