Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections
description
Transcript of Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections
![Page 1: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/1.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
1
Imposing the Froissart Bound on Hadronic Interactions:
Part I, p-air cross sections
Martin Block
Northwestern University
![Page 2: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/2.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
2
Prior Restraint! the Froissart Bound
![Page 3: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/3.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
3
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4) The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel (unpublished).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!
![Page 4: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/4.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
4
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.
![Page 5: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/5.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
5
“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).
![Page 6: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/6.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
6
Lorentzian Fit used in “Sieve” Algorithm
![Page 7: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/7.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
7
![Page 8: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/8.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
8
![Page 9: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/9.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
9
![Page 10: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/10.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
10
![Page 11: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/11.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
11
![Page 12: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/12.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
12
You are now finished! No more outliers. You have: 1) optimized parameters 2) corrected goodness-of-fit 3) squared error matrix.
![Page 13: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/13.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
13
Part 2: “New analyticity constraints on hadron-hadron cross sections”,
M. Block, Eur. Phys. J. C47 (2006).
![Page 14: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/14.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
14
![Page 15: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/15.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
15
Derivation of new analyticity constraints
Theoretical high energy cross section parametrization
Experimental low energy cross section
Finite energy cutoff!
![Page 16: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/16.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
16
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).
![Page 17: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/17.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
17
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).
![Page 18: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/18.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
18
ln2(s/s0) fit=0.5, Regge-
descending trajectory
7 parameters needed, including f+(0), a dispersion relation subtraction constant
![Page 19: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/19.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
19
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!
![Page 20: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/20.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
20
Cross section fits for Ecms > 6 GeV, anchored at 4 GeV,
pp and pbar p, after applying “Sieve” algorithm
![Page 21: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/21.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
21
-value fits for Ecms > 6 GeV, anchored at 4 GeV,
pp and pbar p, after applying “Sieve” algorithm
![Page 22: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/22.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
22
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.
![Page 23: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/23.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
24
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
![Page 24: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/24.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
25
p log2(/m) fit, compared to the p even amplitude fit
M. Block and F. Halzen,
Phys Rev D 70, 091901, (2004)
![Page 25: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/25.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
26
Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,
+p and -p, after applying “Sieve” algorithm
![Page 26: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/26.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
27
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
![Page 27: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/27.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
28
Part 3: The Glauber calculation: Obtaining the p-air cross section from accelerator data, M. Block and R. Engel
Ralph Engel, At Work
![Page 28: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/28.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
29
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
![Page 29: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/29.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
30
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
![Page 30: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/30.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
31
Xmax = X1 + X’
HiRes Measurement of Xmax Distribution:
![Page 31: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/31.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
32
B, from Aspen (eikonal) Model
Ingredients needed for Glauber Model
, from ln2s fit
![Page 32: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/32.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
33
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
![Page 33: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/33.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
35
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
![Page 34: Imposing the Froissart Bound on Hadronic Interactions: Part I, p-air cross sections](https://reader036.fdocuments.in/reader036/viewer/2022062409/56814807550346895db538fa/html5/thumbnails/34.jpg)
April 15-19, 2007 M. Block, Aspen Workshop Cosmic Ray Physics 2007
36
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.