Feb, 2006M. Block, Aspen Winter Physics Conference 1 A robust prediction of the LHC cross section Martin Block Northwestern University

Feb, 2006M. Block, Aspen Winter Physics Conference 2 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, (2005). OUTLINE 2) New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, arXiv:hep-ph/ (2006).

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

Feb, 2006M. Block, Aspen Winter Physics Conference 4 Generalization of the Maximum Likelihood Function, P

Feb, 2006M. Block, Aspen Winter Physics Conference 5 Hence,minimize  i  (z), or equivalently, we minimize  2   i  2 i

Feb, 2006M. Block, Aspen Winter Physics Conference 6 Problem with Gaussian Fit when there are Outliers

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Feb, 2006M. Block, Aspen Winter Physics Conference 8 Robust Feature:  (z)  1/  i 2 for large  i 2

Feb, 2006M. Block, Aspen Winter Physics Conference 9 Lorentzian Fit used in “Sieve” Algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 10 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”.

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Feb, 2006M. Block, Aspen Winter Physics Conference 16 You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.

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

Feb, 2006M. Block, Aspen Winter Physics Conference 18 M. Block and F. Halzen, Phys. Rev. D 72, (2005); arXiv:hep-ph/ (2005). K. Igi and M. Ishida, Phys. Lett. B 262, 286 (2005).

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Feb, 2006M. Block, Aspen Winter Physics Conference 21 This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

Feb, 2006M. Block, Aspen Winter Physics Conference 22 Derivation of new analyticity constraints Theoretical high energy cross section parametrization Experimental low energy cross section

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Feb, 2006M. Block, Aspen Winter Physics Conference 24 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 ).

Feb, 2006M. 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, (2005).

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Feb, 2006M. Block, Aspen Winter Physics Conference 29 ln 2 (s/s 0 ) fit  =0.5, Regge- descending trajectory 7 parameters needed, including f + (0), a dispersion relation subtraction constant

Feb, 2006M. Block, Aspen Winter Physics Conference 30 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!

Feb, 2006M. Block, Aspen Winter Physics Conference 31 Cross section fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 32  -value fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 33 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.

Feb, 2006M. Block, Aspen Winter Physics Conference 34 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.

Feb, 2006M. Block, Aspen Winter Physics Conference 35 log 2 ( /m p ) fit compared to log( /m p ) fit: All known n-n data

Feb, 2006M. Block, Aspen Winter Physics Conference 36 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!

Feb, 2006M. Block, Aspen Winter Physics Conference 37 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

Feb, 2006M. Block, Aspen Winter Physics Conference 38 The popular parameterization  pp  s 

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Feb, 2006M. Block, Aspen Winter Physics Conference 40 A 2- parameter   fit of the Landshoff-Donnachie variety:  ± = As  + Bs  ± Ds   using 4 analyticity constraints Horrible  2 /d.f.

Feb, 2006M. Block, Aspen Winter Physics Conference 41 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!

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Feb, 2006M. Block, Aspen Winter Physics Conference 43 More LHC predictions Nuclear slope B = ± 0.13 (GeV/c) -2  elastic = ± 0.34 mb Differential Elastic Scattering

Feb, 2006M. Block, Aspen Winter Physics Conference 44 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).

Feb, 2006M. Block, Aspen Winter Physics Conference 45 CONCLUSIONS  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 =  1.2 mb,  = 

Feb, 2006M. Block, Aspen Winter Physics Conference 46 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).

Feb, 2006M. Block, Aspen Winter Physics Conference 47 All cross section data for E cms > 6 GeV, pp and pbar p, from Particle Data Group

Feb, 2006M. Block, Aspen Winter Physics Conference 48 All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV, pp and pbar p, from Particle Data Group

Feb, 2006M. Block, Aspen Winter Physics Conference 49  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error

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Feb, 2006M. Block, Aspen Winter Physics Conference 51 Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 52  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 53  p log 2 ( /m) fit, compared to the  p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, , (2004)

Feb, 2006M. Block, Aspen Winter Physics Conference 54 All cross section data for E cms > 6 GeV,  + p and  - p, from Particle Data Group

Feb, 2006M. Block, Aspen Winter Physics Conference 55 All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV,  + p and  - p, from Particle Data Group

Feb, 2006M. Block, Aspen Winter Physics Conference 56 Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 57  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

Feb, 2006M. Block, Aspen Winter Physics Conference 58  p log 2 ( /m) fit, compared to the  p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, , (2004)

Feb, 2006M. Block, Aspen Winter Physics Conference 59 Part 3: The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel Ralph Engel, At Work

Feb, 2006M. Block, Aspen Winter Physics Conference 60 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

Feb, 2006M. Block, Aspen Winter Physics Conference 61 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

Feb, 2006M. Block, Aspen Winter Physics Conference 62 k = 1.287

Feb, 2006M. Block, Aspen Winter Physics Conference 63  p-air as a function of  s, with inelastic screening  p-air inel = 456  17(stat)+39(sys)-11(sys) mb

Feb, 2006M. Block, Aspen Winter Physics Conference 64 To obtain  pp from  p-air

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Feb, 2006M. Block, Aspen Winter Physics Conference 66 The published cosmic ray data (the Diamond and Triangles) are the problem

Feb, 2006M. Block, Aspen Winter Physics Conference 67 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 colliders with cosmic rays

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