April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 2007 1 Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections.

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Presentation transcript:

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections Martin Block Northwestern University

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Prior Restraint! the Froissart Bound

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics ) 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, (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 OUTLINE 4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished). 2)New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur.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! ) The Glauber calculation:

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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 B pp from collider data. 4) Using a Glauber calculation of  p-air from  pp and B pp, we now have a reliable benchmark tying together colliders to cosmic rays. 2) At the LHC,  tot =  1.2 mb,  = 

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics “Fishing” for Data Part 1: “Sifting Data in the Real World”, Getting rid of outliers! M. Block, arXiv:physics/ (2005); Nucl. Instr. and Meth. A, 556, 308 (2006).

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Lorentzian Fit used in “Sieve” Algorithm

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

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

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

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

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Part 2: “New analyticity constraints on hadron-hadron cross sections”, M. Block, Eur. Phys. J. C47 (2006).

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Derivation of new analyticity constraints Theoretical high energy cross section parametrization Experimental low energy cross section Finite energy cutoff!

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics so that:  exp’t (   ( 0 ), d  exp’t (   d  d  ( 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, 2007M. Block, Aspen Workshop Cosmic Ray Physics Francis, personally funding ICE CUBE Part 3: Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, M. Block and F. Halzen, Phys. Rev. D 72, (2005).

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics ln 2 (s/s 0 ) fit  =0.5, Regge- descending trajectory 7 parameters needed, including f + (0), a dispersion relation subtraction constant

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Only 3 Free Parameters However, only 2, c 1 and c 2, are needed in cross section fits ! These anchoring conditions, just above the resonance regions, are analyticity conditions!

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Cross section fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  -value fits for E cms > 6 GeV, anchored at 4 GeV, pp and pbar p, after applying “Sieve” algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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 = After imposing the “Sieve” algorithm: Renormalized  2 /d.f.=1.09 for 184 degrees of freedom, for  2 i > 6 cut; Total  2 = Probability of fit ~0.2. The 25 rejected points contributed 981 to the total  2, an average  2 i of ~39 per point.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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  2 min /  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  2 min /  =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!

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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, 2007M. Block, Aspen Workshop Cosmic Ray Physics  p log 2 ( /m) fit, compared to the  p even amplitude fit M. Block and F. Halzen, Phys Rev D 70, , (2004)

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics More LHC predictions, from the Aspen Eikonal Model: M. M. Block, Phys. Reports 436, 71 (2006). Nuclear slope B = ± 0.13 (GeV/c) -2  elastic = ± 0.34 mb Differential Elastic Scattering

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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, 2007M. Block, Aspen Workshop Cosmic Ray Physics EXPERIMENTAL PROCEDURE: Fly’s Eye and AGASA Fig. 7 X max 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 X max =727 ± 33 g cm -2 give the best fit. Logarithmic slope,  m, is measured

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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, 2007M. Block, Aspen Workshop Cosmic Ray Physics X max = X 1 + X’ HiRes Measurement of X max Distribution:

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics B, from Aspen (eikonal) Model Ingredients needed for Glauber Model , from ln 2 s fit

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished) B (nuclear slope) vs.  pp, as a function of  p-air  pp from ln 2 (s) fit and B from QCD-fit HiRes Point

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Measured k = 1.28  0.07 Belov, this conference, k =

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  p-air as a function of  s, with inelastic screening  p-air inel = 460  14(stat)+39(sys)-11(sys) mb We find: k = 1.28  0.07 Belov, this conference, k =  p-air inel = 460  14(stat)+39(sys)-11(sys) mb

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics 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 B pp from collider data. 4) Using a Glauber calculation of  p-air from  pp and B pp, we now have a reliable benchmark tying together colliders to cosmic rays. 2) At the LHC,  tot =  1.2 mb,  = 

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Saturating the Froissart Bound  pp and  pbar-p log 2 ( /m) fits, with world’s supply of data Cosmic ray points & QCD-fit from Block, Halzen and Stanev: Phys. Rev. D 66, (2000).

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Cross section fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics To obtain  pp from  p-air

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Robust Feature:  (z)  1/  i 2 for large  i 2

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Why choose normalization constant  =0.179 in Lorentzian  0 2 ? 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, when using the tuned Lorentzian  0 2, much like in keeping with the Hippocratic oath, we do “no harm”.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics All cross section data for E cms > 6 GeV,  + p and  - p, from Particle Data Group

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics All  data (Real/Imaginary of forward scattering amplitude), for E cms > 6 GeV,  + p and  - p, from Particle Data Group

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  -value fits for E cms > 6 GeV, anchored at 2.6 GeV,  + p and  - p, after applying “Sieve” algorithm

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics The published cosmic ray data (the Diamond and Triangles) are the problem

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Stability of “Sieve” algorithm Fit parameters are stable, essentially independent of cut  2 i We choose  2 i = 6, since R  2 min /  giving  0.2 probability for the goodness-of-fit.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics log 2 ( /m p ) fit compared to log( /m p ) fit: All known n-n data

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Lorentzian Fit used in “Sieve” Algorithm

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics The popular parameterization  pp  s 

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics ) Already known to violate unitarity and the Froissart bound at high energies. 2) Now, without major complicated low energy modifications, violates analyticity constraints at low energies. No longer a simple parametrization!

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics A 2- parameter   fit of the Landshoff-Donnachie variety:  ± = As  + Bs  ± Ds   using 4 analyticity constraints Horrible  2 /d.f.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Hence,minimize  i  (z), or equivalently, we minimize  2   i  2 i

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Generalization of the Maximum Likelihood Function, P

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Problem with Gaussian Fit when there are Outliers

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

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics Why choose normalization constant  =0.179 in Lorentzian  0 2 ? 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, when using the tuned Lorentzian  0 2, much like in keeping with the Hippocratic oath, we do “no harm”.

April 15-19, 2007M. Block, Aspen Workshop Cosmic Ray Physics  2 renorm =  2 obs / R -1  renorm = r  2  obs, where  is the parameter error