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A robust prediction of the LHC cross section

A robust prediction of the LHC cross section. Martin Block Northwestern University. OUTLINE. Data selection: The “Sieve” Algorithm- --“Sifting data in the real world”, M. Block, Nucl. Instr. and Meth. A, 556 , 308 (2006).

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A robust prediction of the LHC cross section

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  1. A robust prediction of the LHC cross section Martin BlockNorthwestern University M. Block, Aspen Winter Physics Conference

  2. OUTLINE • Data selection: The “Sieve” Algorithm---“Sifting data in the real world”, • M. Block, Nucl. Instr. and Meth. A, 556, 308 (2006). 2) New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, arXiv:hep-ph/0601210 (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). M. Block, Aspen Winter Physics Conference

  3. “Fishing” for Data Part 1: “Sifting Data in the Real World”, M. Block, arXiv:physics/0506010 (2005); Nucl. Instr. and Meth. A, 556, 308 (2006). M. Block, Aspen Winter Physics Conference

  4. Generalization of the Maximum Likelihood Function, P M. Block, Aspen Winter Physics Conference

  5. Hence,minimize Sir(z), or equivalently, we minimize c2 º Si Dc2i M. Block, Aspen Winter Physics Conference

  6. Problem with Gaussian Fit when there are Outliers M. Block, Aspen Winter Physics Conference

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  8. Robust Feature: y(z) µ1/ÖDci2 for large Dci2 M. Block, Aspen Winter Physics Conference

  9. Lorentzian Fit used in “Sieve” Algorithm M. Block, Aspen Winter Physics Conference

  10. Why choose normalization constant g=0.179 in Lorentzian L02? Computer simulations show that the choice of g=0.179 tunes the Lorentzian so that minimizing L02, 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 c2 fit. If there are nooutliers, it gives the same answers as a c2 fit.Hence, when using the tuned Lorentzian L02, much like in keeping with the Hippocratic oath, we do “no harm”. M. Block, Aspen Winter Physics Conference

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  16. You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix. M. Block, Aspen Winter Physics Conference

  17. Part 2: “New analyticity constraints on hadron-hadron cross sections”, M. Block, arXiv:hep-ph/0601210 (2006) M. Block, Aspen Winter Physics Conference

  18. M. Block and F. Halzen, Phys. Rev. D 72, 036006 (2005); arXiv:hep-ph/0510238 (2005). K. Igi and M. Ishida, Phys. Lett. B 262, 286 (2005). M. Block, Aspen Winter Physics Conference

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  21. This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do. M. Block, Aspen Winter Physics Conference

  22. Experimental low energy cross section Theoretical high energy cross section parametrization Derivation of new analyticity constraints M. Block, Aspen Winter Physics Conference

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  24. 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 Winter Physics Conference

  25. 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 Winter Physics Conference

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

  27. 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 Winter Physics Conference

  28. Cross section fits for Ecms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm M. Block, Aspen Winter Physics Conference

  29. r-value fits for Ecms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm M. Block, Aspen Winter Physics Conference

  30. 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 Winter Physics Conference

  31. log2(n/mp) fit compared to log(n/mp) fit: All known n-n data M. Block, Aspen Winter Physics Conference

  32. 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´ c2min/n=1.055, n=183, probability=0.29 spp(1800 GeV) = 75.1± 0.6 mb spp(14 TeV) = 107.2± 1.2 mb If we use both E710/E811 and the CDF cross sections at the Tevatron, we obtain: R´ c2min/n=1.095, n=184, probability=0.18 spp(1800 GeV) = 75.2± 0.6 mbspp(14 TeV) = 107.3± 1.2 mb, effectively no changes Conclusion: The extrapolation to high energies is essentially unaffected! M. Block, Aspen Winter Physics Conference

  33. 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 Winter Physics Conference

  34. The popular parameterization sppµ s0.08 M. Block, Aspen Winter Physics Conference

  35. Horrible c2/d.f. A 2- parameter c2 fit of the Landshoff-Donnachie variety: s± = Asa-1 + Bsb-1 ± Dsa-1 , using 4 analyticity constraints M. Block, Aspen Winter Physics Conference

  36. 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! M. Block, Aspen Winter Physics Conference

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  38. More LHC predictions Differential Elastic Scattering Nuclear slope B = 19.39 ± 0.13 (GeV/c)-2 selastic = 30.79 ± 0.34 mb M. Block, Aspen Winter Physics Conference

  39. Cosmic ray points & QCD-fit from Block, Halzen and Stanev: Phys. Rev. D 66, 077501 (2000). Saturating the Froissart Bound: spp andspbar-p log2(n/m) fits, with world’s supply of data M. Block, Aspen Winter Physics Conference

  40. CONCLUSIONS 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 Winter Physics Conference

  41. Cosmic ray points & QCD-fit from Block, Halzen and Stanev: Phys. Rev. D 66, 077501 (2000). Saturating the Froissart Bound: spp andspbar-p log2(n/m) fits, with world’s supply of data M. Block, Aspen Winter Physics Conference

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