A robust prediction of the

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A robust prediction of the LHC cross
section
Martin BlockNorthwestern University
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OUTLINE

• Data selection The Sieve Algorithm---Sifting
  data in the real world,
• M. Block, Nucl. Instr. and Meth. A, 556, 308
  (2006).

2) New fitting constraints---New analyticity
constraints on hadron-hadron cross sections, M.
Block, arXivhep-ph/0601210 (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).
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Part 1 Sifting Data in the Real World, M.
Block, arXivphysics/0506010 (2005) Nucl. Instr.
and Meth. A, 556, 308 (2006).
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Generalization of the Maximum Likelihood
Function, P
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Hence,minimize Si r(z), or equivalently, we
minimize c2 º Si Dc2i
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Problem with Gaussian Fit when there are Outliers
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Lorentzian Fit used in Sieve Algorithm
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Why choose normalization constant g0.179 in
Lorentzian L02?
Computer simulations show that the choice of
g0.179 tunes the Lorentzian so that minimizing
L02, 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 c2
fit.
If there are no outliers, it gives the same
answers as a c2 fit. Hence, when using the tuned
Lorentzian L02 , much like in keeping with the
Hippocratic oath, we do no harm.
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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.
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Part 2 New analyticity constraints on
hadron-hadron cross sections, M. Block,
arXivhep-ph/0601210 (2006)
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M. Block and F. Halzen, Phys. Rev. D 72, 036006
(2005) arXivhep-ph/0510238 (2005). K. Igi and
M. Ishida, Phys. Lett. B 262, 286 (2005).
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This is FESR(2) derived by Igi and Ishida, which
follows from analyticity, just as dispersion
relations do.
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Derivation of new analyticity constraints
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We can also prove that for odd amplitudes sodd
(n0) sodd (n0).
so that sexpt (n0) s (n0),
dsexpt (n0)/dn ds (n0) /dn, or, its practical
equivalent, sexpt (n0) s
(n0), sexpt (n1) s (n1),
for n1gt n0 for both pp and pbar-p expt cross
sections
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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).
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Only 3 Free Parameters
However, only 2, c1 and c2, are needed in cross
section fits !
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Cross section fits for Ecms gt 6 GeV, anchored at
4 GeV, pp and pbar p, after applying Sieve
algorithm
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r-value fits for Ecms gt 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
c2/d.f.5.7 for 209 degrees of freedom Total
c21182.3.
After imposing the Sieve algorithm
Renormalized c2/d.f.1.09 for 184 degrees of
freedom, for Dc2i gt 6 cut Total
c2201.4. Probability of fit 0.2. The 25
rejected points contributed 981 to the total c2
, an average Dc2i of 39 per point.


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Stability of Sieve algorithm
We choose Dc2i 6, since R c2min/n 1.1,
giving 0.2 probability for the goodness-of-fit.
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log2(n/mp) fit compared to log(n/mp) fit All
known n-n data
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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 c2min/n1.055, n183,
probability0.29 spp(1800 GeV) 75.1 0.6 mb
spp(14 TeV) 107.2 1.2 mb
If we use both E710/E811 and the CDF cross
sections at the Tevatron, we obtain R
c2min/n1.095, n184, probability0.18 spp(18
00 GeV) 75.2 0.6 mb spp(14 TeV) 107.3
1.2 mb, effectively no changes
Conclusion The extrapolation to high energies
is essentially unaffected!
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Cross section and r-value predictions for
pp and pbar-p
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The popular parameterization spp µ s0.08
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A 2- parameter c2 fit of the Landshoff-Donnachie
variety s Asa-1 Bsb-1 Dsa-1 , using
4 analyticity constraints
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1) 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!
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More LHC predictions
Differential Elastic Scattering
Nuclear slope B 19.39 0.13 (GeV/c)-2
selastic 30.79 0.34 mb
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Saturating the Froissart Bound spp and spbar-p
log2(n/m) fits, with worlds supply of data
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CONCLUSIONS
1) The Froissart bound for gp, pp and pp
collisions is saturated at high energies.
2) At the LHC, stot 107.3 1.2 mb, r
0.1320.001.
3) At cosmic ray energies, we can make accurate
estimates of spp and Bpp from collider data.
4) Using a Glauber calculation of sp-air from
spp and Bpp, we now have a reliable benchmark
tying together colliders to cosmic rays.
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Saturating the Froissart Bound spp and spbar-p
log2(n/m) fits, with worlds supply of data
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All cross section data for Ecms gt 6 GeV, pp and
pbar p, from Particle Data Group
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All r data (Real/Imaginary of forward scattering
amplitude), for Ecms gt 6 GeV, pp and pbar p,
from Particle Data Group
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Cross section fits for Ecms gt 6 GeV, anchored at
2.6 GeV, pp and p-p, after applying Sieve
algorithm
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r-value fits for Ecms gt 6 GeV, anchored at 2.6
GeV, pp and p-p, after applying Sieve algorithm
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gp log2(n/m) fit, compared to the pp even
amplitude fit
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All cross section data for Ecms gt 6 GeV, pp and
p-p, from Particle Data Group
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All r data (Real/Imaginary of forward scattering
amplitude), for Ecms gt 6 GeV, pp and p-p, from
Particle Data Group
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Cross section fits for Ecms gt 6 GeV, anchored at
2.6 GeV, pp and p-p, after applying Sieve
algorithm
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r-value fits for Ecms gt 6 GeV, anchored at 2.6
GeV, pp and p-p, after applying Sieve algorithm
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gp log2(n/m) fit, compared to the pp even
amplitude fit
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Part 3 The Glauber calculation Obtaining the
p-air cross section from accelerator data, M.
Block and R. Engel
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Glauber calculation B (nuclear slope) vs. spp,
as a function of sp-air
spp from ln2(s) fit and B from QCD-fit
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Glauber calculation with inelastic screening, M.
Block and R. Engel (unpublished)
B (nuclear slope) vs.
spp, as a function of sp-air
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k 1.287
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sp-air as a function of Ö s, with inelastic
screening
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To obtain spp from sp-air
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CONCLUSIONS
1) The Froissart bound for pp collisions is
saturated at high energies.
2) At cosmic ray energies, we have accurate
estimates of spp and Bpp from collider data.
3) The Glauber calculation of sp-air from spp and
Bpp is reliable.
4) The HiRes value (almost model independent) of
sp-air is in reasonable agreement with the
collider prediction.
5) We now have a good benchmark, tying together
colliders with cosmic rays
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