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

Martin BlockNorthwestern University

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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, 036006 (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!

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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,

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Part 1: “Sifting Data in the Real World”,

M. Block, arXiv:physics/0506010 (2005); Nucl. Instr. and Meth. A, 556, 308 (2006).

“Fishing” for Data

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Lorentzian Fit used in “Sieve” Algorithm

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You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.

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This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

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so that:exp’t ( (0),

dexp’t (dd (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).

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Francis, Francis, personally personally funding ICE funding ICE CUBECUBE

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).

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ln2(s/s0) fit=0.5, Regge-

descending trajectory

7 parameters needed, including f+(0), a dispersion relation subtraction constant

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Only 3 Free Parameters

However, only 2, c1 and c2, are needed in cross section fits !

These anchoring conditions, just above the resonance regions, are analyticity conditions!

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Cross section fits for Ecms > 6 GeV, anchored at 4 GeV,

pp and pbar p, after applying “Sieve” algorithm

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-value fits for Ecms > 6 GeV, anchored at 4 GeV,

pp and pbar p, after applying “Sieve” algorithm

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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=1182.3.

After imposing the “Sieve” algorithm:

Renormalized 2/d.f.=1.09 for 184 degrees of freedom, for 2i > 6 cut;

Total 2=201.4.

Probability of fit ~0.2.

The 25 rejected points contributed 981 to the total 2 , an average 2i

of ~39 per point.

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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

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More LHC predictions, from the Aspen Eikonal Model

Nuclear slope B = 19.39 ± 0.13 (GeV/c)-2

elastic = 30.79 ± 0.34 mb

Differential Elastic Scattering

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Saturating the Froissart Boundpp and pbar-p log2(/m) fits, with world’s supply of data

Cosmic ray points & QCD-fit from Block, Halzen and Stanev: Phys. Rev. D 66, 077501 (2000).

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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 Bpp from collider data.

4) Using a Glauber calculation of p-air from pp and Bpp, we now have a reliable benchmark tying together colliders to cosmic rays.

2) At the LHC,

tot = 107.3 1.2 mb, = 0.1320.001.

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

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Reduced Virtual Photon Total Cross Section

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Froissart bound fit, ln2 W, to reduced cross sections

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Scaling Point

Global (Simultaneous) Fit of F2(x,Q2) to x and Q2

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What the “Sieve” algorithm accomplished for F2(x,Q2)

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 2i > 6 cut;

Total 2=184.2.

Probability of fit ~0.2.

The 8 rejected points contributed 63.45 to the total 2 , an average 2i

of ~8 per point.

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Predictions are made using ZEUS data in

global fit

Experimental data are from H1 collaboration

NO RENORMALIZATION

made!

1 Q2 GeV2

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SUMMARY

.

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To be done:

1. 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.