Imposing the Froissart bound on DIS ---> New PDF's for the LHC

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

No time to talk about these. But they are important! • New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur. Phys. J. C 47, 697(2006). OUTLINE • 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, 036006 (2005). M. Block, Aspen Winter Physics Conference

4) The Proton Structure Function F2p(x,Q2) : “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, M. Block, Aspen Winter Physics Conference

“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

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

M. Block, Aspen Winter Physics Conference

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

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

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

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

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

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

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

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

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

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

More LHC predictions, from the Aspen Eikonal Model 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

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

Conclusions From hadron-hadron scattering 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

Proton Structure Function F2(x,Q2), from Deep Inelastic Scattering, Block, Berger & Tan, PRL 99, 88 (2006). M. Block, Aspen Winter Physics Conference

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

Froissart bound fit, ln2 W, to reduced cross sections M. Block, Aspen Winter Physics Conference

Scaling Point Global (Simultaneous) Fit of F2(x,Q2) to x and Q2 M. Block, Aspen Winter Physics Conference

What the “Sieve” algorithm accomplished for F2(x,Q2) Before imposing the “Sieve algorithm: c2/d.f.=1.30 for 177 degrees of freedom; Total c2=229.4. After imposing the “Sieve” algorithm: Renormalized c2/d.f.=1.09 for 169 degrees of freedom, for Dc2i > 6 cut; Total c2=184.2. Probability of fit ~0.2. The 8 rejected points contributed 63.45 to the total c2 , an average Dc2i of ~8 per point. M. Block, Aspen Winter Physics Conference

1 £Q2£ 100 GeV2 Predictions are made using ZEUS data in global fit Experimental data are from H1 collaboration NO RENORMALIZATION made! M. Block, Aspen Winter Physics Conference

SUMMARY . M. Block, Aspen Winter Physics Conference

To be done: • Include H1 in global fit, simultaneously fitting F2, dF2 /d(logQ2), d(logF2) /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 Q2~6400 GeV2 ! Enormous extrapolation. M. Block, Aspen Winter Physics Conference

Imposing the Froissart bound on DIS ---> New PDF's for the LHC