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

transcript

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

Imposing the Froissart Bound on Hadronic Interactions:

Part I, p-air cross sections

Martin Block

Northwestern University

Prior Restraint! the Froissart Bound

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4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished).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

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.

“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).

Lorentzian Fit used in “Sieve” Algorithm

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

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

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

Derivation of new analyticity constraints

Theoretical high energy cross section parametrization

Experimental low energy cross section

Finite energy cutoff!

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).

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).

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

descending trajectory

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

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!

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

pp and pbar p, after applying “Sieve” algorithm

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

pp and pbar p, after applying “Sieve” algorithm

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.

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

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

M. Block and F. Halzen,

Phys Rev D 70, 091901, (2004)

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

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

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

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

Ralph Engel, At Work

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

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

Xmax = X1 + X’

HiRes Measurement of Xmax Distribution:

B, from Aspen (eikonal) Model

Ingredients needed for Glauber Model

, from ln2s fit

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

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