Download - The Elusive p-air Cross Section
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 1
The Elusive p-air Cross Section
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 2
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:
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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3) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished)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
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Part 1: “Sifting Data in the Real World”,
M. Block, arXiv:physics/0506010 (2005).
“Fishing” for Data
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 5
All cross section data for Ecms > 6 GeV,
pp and pbar p, from Particle Data Group
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All data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,
pp and pbar p, from Particle Data Group
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 7
All cross section data for Ecms > 6 GeV,
+p and -p, from Particle Data Group
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 8
All data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,
+p and -p, from Particle Data Group
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 9
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“Sieve’’ Algorithm: SUMMARY
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2renorm = 2
obs/R-1
renorm = r2 obs,
where is the parameter error
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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).
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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 Dual equivalents to
finite energy sum rules (FESR)!
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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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log2(/mp) fit compared to log(/mp) fit: All known n-n data
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 30
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!
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Sept. 7-13, 2005 M. Block, Prague, c2cr 2005 31
Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,
+p and -p, after applying “Sieve” algorithm
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-value fits for Ecms > 6 GeV, anchored at 2.6 GeV,
+p and -p, after applying “Sieve” algorithm
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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 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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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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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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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 = 45617(stat)+39(sys)-11(sys) mb
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Measured k = 1.29
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To obtain pp from p-air
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Generalization of the Maximum Likelihood Function
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Hence,minimize i (z), or equivalently, we minimize 2 i 2i
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Problem with Gaussian Fit when there are Outliers
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Robust Feature:
w(z) 1/i2 for large i
2
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Lorentzian Fit used in “Sieve” Algorithm
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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”.
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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