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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 1 Imposing the Froissart Bound on Hadronic Interactions: Part II, Deep Inelastic Scattering & new structure functions for ultra-high energy -n cross sections Martin Block Northwestern University

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 2 Prior Restraint! the Froissart Bound

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 3 1) 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). 2) Global Structure Function Fit and Predictions for Ultra-high Energy Neutrino-nucleon Cross Sections, Work in progress for this meeting ! M. M. B, Edmund L. Berger and Chung- I Tan Outline

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 4 Deep inelastic electron scattering ee’ ** p *p scattering

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 5 p log 2 (s/s 0 ) fit, compared to the p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, 091901, (2004) Now employ same formalism for p scattering!

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 6 Proton Structure Function F 2 (x,Q 2 ), from Deep Inelastic Scattering, Block, Berger & Tan, PRL 99, 88 (2006). Bjorken x Virtuality

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 7 Reduced Virtual Photon Total Cross Section

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 9 Only ln 2 (s/s 0 ) term needed!

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 12 Froissart bound fit, ln 2 W, to reduced cross sections

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 13 Individual fits of the proton structure F 2 (x,Q 2 ) vs. x, for 15 different Q 2 values, in GeV 2 Bjorken x ONE functional form, ln 2 s, fits ALL!

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 16 Scaling Point Global (Simultaneous) Fit of F 2 (x,Q 2 ) to x and Q 2

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 17 F 2 (x, Q 2 ): Global Fit Parameters for Q 2 /x >>m 2, x<0.09 0.11 < Q 2 < 1200 GeV 2 Excellent 2 /d.f. for 169 degrees of freedom

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 18 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 =229.4. After imposing the “Sieve” algorithm: Renormalized 2 /d.f.=1.09 for 169 degrees of freedom, for 2 i > 6 cut; Total 2 =184.2. Probability of fit ~0.2. The 8 rejected points contributed 63.45 to the total 2, an average 2 i of ~8 per point.

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 19

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 20 Predictions are made using ZEUS data in global fit Experimental data are from H1 collaboration NO RENORMALIZATION made! 1 Q 2 GeV 2

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 21 Comparing ln 2 x and CTEQ6.5 fits

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 22 Factor of 8 ! Standard CTEQ3 n Froissart Bound n n estimates, courtesy F. Halzen

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 23 SUMMARY

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 24 To be done: 1.Include H1 in global fit, simultaneously fitting F 2 and dF 2 /d(logQ2), d(logF 2 ) /d(log x) 2. Constrained gluon particle distribution functions (p.d.f.’s) for LHC &ILC 3. Constrained quark p.d.f’s for LHC & ILC 4. Recalculate cosmic ray neutrino cross sections using bounded p.d.f.’s; current values are much too big! Needs x~10 -8 and Q 2 ~6400 GeV 2. Enormous extrapolation !

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 25 q Curvature = 2 F 2 (x, Q 2 )/ q 2, where q = log 10 (1+ 2Q 2 )

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 26 Scaling point

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 27 Extraction of Gluon Distributions xg(x,Q 2 ) from DIS

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 30 NLO Gluon Distributions, xg(x,Q 2 ) gluon scaling point ! x=0.09,x g=1.8

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 32 Bootstrap Method of Extracting Gluon Distributions

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April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 36 Bootstrap Equation for Gluon Distribution G(x,Q 2 )=xg(x,Q 2 )

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