Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections

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

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Imposing the Froissart Bound on Hadronic Interactions:

Part I, p-air cross sections

Martin Block

Northwestern University

http://user.web.cern.ch/User/Welcome.html


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Prior Restraint! the Froissart Bound


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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

4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished).

2) New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur. Phys. J. C 47, 697 (2006). Touched on briefly , but these are important

constraints!


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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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“Fishing” for Data

Part 1: “Sifting Data in the Real World”, Getting rid of outliers!

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


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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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Part 2: “New analyticity constraints on hadron-hadron cross sections”,

M. Block, Eur. Phys. J. C47 (2006).


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Derivation of new analyticity constraints

Theoretical high energy cross section parametrization

Experimental low energy cross section

Finite energy cutoff!


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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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p log2(/m) fit, compared to the p even amplitude fit

M. Block and F. Halzen,

Phys Rev D 70, 091901, (2004)


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Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,

+p and -p, after applying “Sieve” algorithm


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More LHC predictions, from the Aspen Eikonal Model: M. M. Block, Phys. Reports 436, 71 (2006).

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

elastic = 30.79 ± 0.34 mb

Differential Elastic Scattering


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Part 3: The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel

Ralph Engel, At Work


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EXPERIMENTAL PROCEDURE: Fly’s Eye and AGASA

Fig. 7 Xmax distribution with exponential trailing edge

Monte Carlo Example

Fly’s Eye Shower Profile

Fig. 1 An extensive air shower that survives all data cuts. The curve is a Gaisser-Hillas shower-development function: shower parameters E=1.3 EeV and Xmax =727 ± 33 g cm-2 give the best fit.

Logarithmic

slope, m,

is measured


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Extraction of tot(pp) from Cosmic Ray Extensive Air Showers by Fly’s Eye and AGASA

k is very model-dependent

Need good fit to accelerator data


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Xmax = X1 + X’

HiRes Measurement of Xmax Distribution:


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B, from Aspen (eikonal) Model

Ingredients needed for Glauber Model

, from ln2s fit


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Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished) B (nuclear slope) vs. pp, as a function of p-air

pp from ln2(s) fit and B from

QCD-fit

HiRes Point


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p-air as a function of s, with inelastic screening

p-airinel = 46014(stat)+39(sys)-11(sys) mb

We find: k = 1.28 0.07Belov, this conference, k = 1.21 + 0.14 - 0.09p-air

inel = 46014(stat)+39(sys)-11(sys) mb


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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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Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections

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