Jan, 2007M. Block, Aspen Winter Physics Conference 1 Imposing the Froissart bound on DIS ---> New PDF's for the LHC Martin Block Northwestern University

Jan, 2007M. 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, Eur. Phys. J. C 47, 697 (2006). No time to talk about these. But they are important!

Jan, 2007M. Block, Aspen Winter Physics Conference 3 4) The Proton Structure Function F 2 p (x,Q 2 ) : “Small-x Behavior of Parton Distributions from the Observed Froissart Energy Dependence of the Deep-Inelastic-Scattering Cross Sections”, M. M. Block, Edmund L. Berger and Chung-I tan, Phys.Rev. Lett. 308 (2006). 5) Global Structure Function Fit and New Gluon Distributions using the Froissart Bound : Work in progress for this meeting ! M. M. Block, Edmund L. Berger and Chung-I Tan,

Jan, 2007M. Block, Aspen Winter Physics Conference 4 Part 1: “Sifting Data in the Real World”, M. Block, arXiv:physics/ (2005); Nucl. Instr. and Meth. A, 556, 308 (2006). “Fishing” for Data

Jan, 2007M. Block, Aspen Winter Physics Conference 5 Lorentzian Fit used in “Sieve” Algorithm

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Jan, 2007M. Block, Aspen Winter Physics Conference 11 You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.

Jan, 2007M. Block, Aspen Winter Physics Conference 12 This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

Jan, 2007M. Block, Aspen Winter Physics Conference 13 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 ).

Jan, 2007M. Block, Aspen Winter Physics Conference 14 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).

Jan, 2007M. Block, Aspen Winter Physics Conference 15 ln 2 (s/s 0 ) fit  =0.5, Regge- descending trajectory 7 parameters needed, including f + (0), a dispersion relation subtraction constant

Jan, 2007M. Block, Aspen Winter Physics Conference 16 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!

Jan, 2007M. Block, Aspen Winter Physics Conference 17 Cross section fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Jan, 2007M. Block, Aspen Winter Physics Conference 18  -value fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

Jan, 2007M. Block, Aspen Winter Physics Conference 19 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.

Jan, 2007M. Block, Aspen Winter Physics Conference 20 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

Jan, 2007M. Block, Aspen Winter Physics Conference 21 More LHC predictions, from the Aspen Eikonal Model Nuclear slope B = ± 0.13 (GeV/c) -2  elastic = ± 0.34 mb Differential Elastic Scattering

Jan, 2007M. Block, Aspen Winter Physics Conference 22 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).

Jan, 2007M. Block, Aspen Winter Physics Conference 23 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 =  1.2 mb,  = 

Jan, 2007M. Block, Aspen Winter Physics Conference 24 Proton Structure Function F 2 (x,Q 2 ), from Deep Inelastic Scattering, Block, Berger & Tan, PRL 99, 88 (2006).

Jan, 2007M. Block, Aspen Winter Physics Conference 25 Reduced Virtual Photon Total Cross Section

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Jan, 2007M. Block, Aspen Winter Physics Conference 30 Froissart bound fit, ln 2 W, to reduced cross sections

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Jan, 2007M. Block, Aspen Winter Physics Conference 33 Scaling Point Global (Simultaneous) Fit of F 2 (x,Q 2 ) to x and Q 2

Jan, 2007M. Block, Aspen Winter Physics Conference 34 What the “Sieve” algorithm accomplished for F 2 (x,Q 2 ) Before imposing the “Sieve algorithm:  2 /d.f.=1.30 for 177 degrees of freedom; Total  2 = After imposing the “Sieve” algorithm: Renormalized  2 /d.f.=1.09 for 169 degrees of freedom, for  2 i > 6 cut; Total  2 = Probability of fit ~0.2. The 8 rejected points contributed to the total  2, an average  2 i of ~8 per point.

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Jan, 2007M. Block, Aspen Winter Physics Conference 36 Predictions are made using ZEUS data in global fit Experimental data are from H1 collaboration NO RENORMALIZATION made! 1  Q 2  GeV 2

Jan, 2007M. Block, Aspen Winter Physics Conference 37 SUMMARY.

Jan, 2007M. Block, Aspen Winter Physics Conference 38 To be done: 1.Include H1 in global fit, simultaneously fitting F2, dF 2 /d(logQ2), d(logF 2 ) /d(log x) 2. More gluon distributions 3. Quark distributions 4. Recalculate cosmic ray neutrino cross sections; current values are much too big! Needs x~10 -8 and Q 2 ~6400 GeV 2 ! Enormous extrapolation.