Sept. 7-13, 2005M. Block, Prague, c2cr The Elusive p-air Cross Section

Sept. 7-13, 2005M. Block, Prague, c2cr The Elusive p-air Cross Section Martin Block Northwestern University The E(xc)lusive p-air (Pierre) Cross Section for cosmic ray conoscenti, the real title is:

Sept. 7-13, 2005M. Block, Prague, c2cr ) Data selection---“Sifting data in the real world”, M. Block, arXiv:physics/ (2005). 2) Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, arXiv:hep-ph/ (2005); Phys. Rev. D 72, (2005). 3) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished) OUTLINE 4) Details of Robust Fitting: Time permitting

Sept. 7-13, 2005M. Block, Prague, c2cr Part 1: “Sifting Data in the Real World”, M. Block, arXiv:physics/ (2005). “Fishing” for Data

Sept. 7-13, 2005M. Block, Prague, c2cr All cross section data for E cms > 6 GeV, pp and pbar p, from Particle Data Group

Sept. 7-13, 2005M. Block, Prague, c2cr All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV, pp and pbar p, from Particle Data Group

Sept. 7-13, 2005M. Block, Prague, c2cr All cross section data for E cms > 6 GeV,  + p and  - p, from Particle Data Group

Sept. 7-13, 2005M. Block, Prague, c2cr All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV,  + p and  - p, from Particle Data Group

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Sept. 7-13, 2005M. Block, Prague, c2cr “Sieve’’ Algorithm: SUMMARY

Sept. 7-13, 2005M. Block, Prague, c2cr  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error

Sept. 7-13, 2005M. Block, Prague, c2cr Francis, personally funding ICE CUBE Part 2: Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, arXiv:hep-ph/ (2005).

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Sept. 7-13, 2005M. Block, Prague, c2cr 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 Dual equivalents to finite energy sum rules (FESR)!

Sept. 7-13, 2005M. Block, Prague, c2cr Cross section fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Sept. 7-13, 2005M. Block, Prague, c2cr  -value fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Sept. 7-13, 2005M. Block, Prague, c2cr 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 = After imposing the “Sieve” algorithm: Renormalized  2 /d.f.=1.09 for 184 degrees of freedom, for  2 i > 6 cut; Total  2 = Probability of fit ~0.2. The 25 rejected points contributed 981 to the total  2, an average  2 i of ~39 per point.

Sept. 7-13, 2005M. Block, Prague, c2cr 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.

Sept. 7-13, 2005M. Block, Prague, c2cr log 2 ( /m p ) fit compared to log( /m p ) fit: All known n-n data

Sept. 7-13, 2005M. Block, Prague, c2cr 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!

Sept. 7-13, 2005M. Block, Prague, c2cr Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Sept. 7-13, 2005M. Block, Prague, c2cr  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Sept. 7-13, 2005M. Block, Prague, c2cr  p log 2 ( /m) fit, compared to the  p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, , (2004)

Sept. 7-13, 2005M. Block, Prague, c2cr 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

Sept. 7-13, 2005M. Block, Prague, c2cr 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, (2000).

Sept. 7-13, 2005M. Block, Prague, c2cr Part 3: The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel Ralph Engel, At Work

Sept. 7-13, 2005M. Block, Prague, c2cr Glauber calculation: B (nuclear slope) vs.  pp, as a function of  p-air  pp from ln 2 (s) fit and B from QCD-fit HiRes Point

Sept. 7-13, 2005M. Block, Prague, c2cr 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

Sept. 7-13, 2005M. Block, Prague, c2cr k = 1.287

Sept. 7-13, 2005M. Block, Prague, c2cr  p-air as a function of  s, with inelastic screening  p-air inel = 456  17(stat)+39(sys)-11(sys) mb

Sept. 7-13, 2005M. Block, Prague, c2cr Measured k = 1.29

Sept. 7-13, 2005M. Block, Prague, c2cr To obtain  pp from  p-air

Sept. 7-13, 2005M. Block, Prague, c2cr Generalization of the Maximum Likelihood Function

Sept. 7-13, 2005M. Block, Prague, c2cr Hence,minimize  i  (z), or equivalently, we minimize  2   i  2 i

Sept. 7-13, 2005M. Block, Prague, c2cr Problem with Gaussian Fit when there are Outliers

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Sept. 7-13, 2005M. Block, Prague, c2cr Robust Feature: w(z)  1/  i 2 for large  i 2

Sept. 7-13, 2005M. Block, Prague, c2cr Lorentzian Fit used in “Sieve” Algorithm

Sept. 7-13, 2005M. Block, Prague, c2cr 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, using the tuned Lorentzian  0 2, much like using the Hippocratic oath, does “no harm”.

Sept. 7-13, 2005M. Block, Prague, c2cr CONCLUSIONS  The Froissart bound for pp collisions is saturated at high energies. 2) At cosmic ray energies,  we have accurate estimates of  pp and B pp from collider data. 3) The Glauber calculation of  p-air from  pp and B pp is reliable. 4) The HiRes value (almost model independent) of  p-air is in reasonable agreement with the collider prediction. 5) We now have a good benchmark, tying together

Sept. 7-13, 2005M. Block, Prague, c2cr

Sept. 7-13, 2005M. Block, Prague, c2cr The published cosmic ray data (the Diamond and Triangles) are the problem