The Elusive p-air Cross Section

Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 1

The Elusive p-air Cross Section


The Elusive p-air Cross Section

Martin BlockNorthwestern University

The E(xc)lusive p-air (Pierre) Cross Section

for cosmic ray conoscenti, the real title is:


1) Data selection---“Sifting data in the real world”, M. Block, arXiv:physics/0506010 (2005).

2) Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, arXiv:hep-ph/0506031 (2005); Phys. Rev. D 72, 036006 (2005).

3) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished)

OUTLINE

4) Details of Robust Fitting: Time permitting


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

M. Block, arXiv:physics/0506010 (2005).

“Fishing” for Data


All cross section data for Ecms > 6 GeV,

pp and pbar p, from Particle Data Group


All data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,

pp and pbar p, from Particle Data Group


All cross section data for Ecms > 6 GeV,

+p and -p, from Particle Data Group


All data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,

+p and -p, from Particle Data Group









“Sieve’’ Algorithm: SUMMARY


2renorm = 2

obs/R-1

renorm = r2 obs,

where is the parameter error


Francis, Francis, personally personally funding ICE funding ICE CUBECUBE

Part 2: Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, arXiv:hep-ph/0506031 (2005).







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 Dual equivalents to

finite energy sum rules (FESR)!


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.


log2(/mp) fit compared to log(/mp) fit: All known n-n data


Comments on the “Discrepancy” between CDF and E710/E811 cross sections at the Tevatron Collider

If we only use E710/E811 cross sections at the Tevatron and do not include the CDF point, we obtain:

R 2min/probability=0.29

pp(1800 GeV) = 75.1± 0.6 mb pp(14 TeV) = 107.2± 1.2 mb

If we use both E710/E811 and the CDF cross sections at the Tevatron, we obtain:

R 2min/ =184, probability=0.18

pp(1800 GeV) = 75.2± 0.6 mb pp(14 TeV) = 107.3± 1.2 mb, effectively no changes

Conclusion:

The extrapolation to high energies is essentially unaffected!


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

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


-value fits for Ecms > 6 GeV, anchored at 2.6 GeV,

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


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

M. Block and F. Halzen,

Phys Rev D 70, 091901, (2004)


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


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).


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

Ralph Engel, At Work


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 = 45617(stat)+39(sys)-11(sys) mb


Measured k = 1.29


To obtain pp from p-air


Generalization of the Maximum Likelihood Function


Hence,minimize i (z), or equivalently, we minimize 2 i 2i


Problem with Gaussian Fit when there are Outliers



Robust Feature:

w(z) 1/i2 for large i

2


Lorentzian Fit used in “Sieve” Algorithm


Why choose normalization constant =0.179 in Lorentzian 02?

Computer simulations show that the choice of =0.179 tunes the Lorentzian so that minimizing 0

2, using data that are gaussianly distributed, gives the same central values and approximately the same errors for parameters obtained by minimizing these data using a conventional 2 fit.

If there are no outliers, it gives the same answers as a 2 fit.

Hence, using the tuned Lorentzian 02 , much like using the

Hippocratic oath, does “no harm”.


CONCLUSIONS

The Froissart bound for pp collisions is saturated at high energies.

2) At cosmic ray energies,we have accurate estimates of pp and Bpp from collider data.

3) The Glauber calculation of p-air from pp and Bpp is reliable.

4) The HiRes value (almost model independent) of p-air is in reasonable agreement with the collider prediction.

5) We now have a good benchmark, tying together

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The Elusive p-air Cross Section