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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Prior Restraint! the Froissart Bound
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished).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!
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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.
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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).
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Lorentzian Fit used in “Sieve” Algorithm
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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.
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Part 2: “New analyticity constraints on hadron-hadron cross sections”,
M. Block, Eur. Phys. J. C47 (2006).
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Derivation of new analyticity constraints
Theoretical high energy cross section parametrization
Experimental low energy cross section
Finite energy cutoff!
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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).
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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).
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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ln2(s/s0) fit=0.5, Regge-
descending trajectory
7 parameters needed, including f+(0), a dispersion relation subtraction constant
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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!
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Cross section fits for Ecms > 6 GeV, anchored at 4 GeV,
pp and pbar p, after applying “Sieve” algorithm
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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-value fits for Ecms > 6 GeV, anchored at 4 GeV,
pp and pbar p, after applying “Sieve” algorithm
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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.
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,
+p and -p, after applying “Sieve” algorithm
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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Xmax = X1 + X’
HiRes Measurement of Xmax Distribution:
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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B, from Aspen (eikonal) Model
Ingredients needed for Glauber Model
, from ln2s fit
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
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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.