# Presentation: 239311

##      A robust prediction of the 

- LHC cross section

- Martin BlockNorthwestern University

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

##                           OUTLINE

-  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

- 2)  New fitting constraints---“New analyticity constraints on hadron-hadron cross sections”, M. Block, **arXiv:hep-ph/0601210 **(2006).

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Part 1: “Sifting Data in the Real World”, 

- M. Block, arXiv:physics/0506010 (2005); Nucl. Instr. and Meth. A,** 556**, 308 (2006).

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Generalization of the Maximum Likelihood Function, P

- Generalization of the Maximum Likelihood Function, *P*

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Hence,minimize   i (z), or equivalently, we minimize  2 i 2i

- Hence,minimize   i (z), or equivalently, we minimize ** **2 i 2i

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Problem with Gaussian Fit when there are Outliers

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 7

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 8

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Lorentzian Fit used in “Sieve” Algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Why choose normalization constant =0.179 in Lorentzian 02?

- Computer simulations show that the choice of =0.179 tunes the Lorentzian so that minimizing 02, 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 02 , much like in keeping with  the Hippocratic oath, we do “no harm”.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 11

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 12

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 13

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 14

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 15

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## You are now finished!  No more outliers. You have:          1) optimized parameters                                                       2) corrected goodness-of-fit                                                   3) squared error matrix.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Part 2: “New analyticity constraints on hadron-hadron cross sections”,  

- Part 2: “New analyticity constraints on hadron-hadron cross sections”,
- M. Block, arXiv:hep-ph/0601210 (2006)

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## M. Block and F. Halzen, Phys. Rev. D 72, 036006 (2005); 

- M. Block and F. Halzen, Phys. Rev. D **72, **036006 (2005);
- arXiv:hep-ph/0510238 (2005).
- K. Igi and M. Ishida, Phys. Lett. B **262, **286 (2005).

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 19

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 20

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## This is FESR(2) derived by Igi and Ishida, which follows from analyticity, just as dispersion relations do.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Derivation of new analyticity constraints

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 23

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## 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).

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Part 3: Fitting the accelerator data---“New evidence for the Saturation of the Froissart Bound”, 

- 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).

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 26

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 27

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 28

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 29

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Only 3 Free Parameters

***However, only***** *****2*****, *****c******1***** ****and**** *****c******2*****, *****are needed in cross section fits !***

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Cross section fits for Ecms > 6 GeV, anchored at 4 GeV,

- Cross section fits for Ecms > 6 GeV, anchored at 4 GeV,
- pp and pbar p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## -value fits for Ecms > 6 GeV, anchored at 4 GeV,

- -value fits for Ecms > 6 GeV, anchored at 4 GeV,
- pp and pbar p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## 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.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Stability of “Sieve” algorithm

- We choose 2i = 6, since **R** 2min/ giving  0.2 probability for the goodness-of-fit.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## log2(/mp) fit compared to log(/mp) fit: All known n-n data

- log2(/mp) fit compared to log(/mp) fit: All known n-n data

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Conclusion: 

- 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! **

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Cross section and -value predictions for       pp and pbar-p

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## The popular parameterization pp s

- The popular parameterization pp *s*

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 39

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## A  2- parameter  fit of the Landshoff-Donnachie variety:      ± = As+ Bs ± Dsusing 4 analyticity constraints

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## 1) Already known to violate unitarity and the Froissart bound at high energies. 

- 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!

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 42

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## More LHC predictions

- Nuclear slope B = 19.39 ± 0.13 (GeV/c)-2
- elastic  = 30.79 ± 0.34 mb

- Differential Elastic Scattering

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

**Notes:**
Sig(elastic) = SigLHC(tot)*[sig(el)/sig(tot)]QCD=107.3*(30.4/106..19)

## Saturating the Froissart Boundpp and pbar-p log2(/m) fits, with world’s supply of data

- Saturating the Froissart Boundpp and pbar-p log2(/m) fits, with world’s supply of data

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## CONCLUSIONS

- 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,  = 0.1320.001.

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Saturating the Froissart Boundpp and pbar-p log2(/m) fits, with world’s supply of data

- Saturating the Froissart Boundpp and pbar-p log2(/m) fits, with world’s supply of data

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## All cross section data for Ecms > 6 GeV, 

- All cross section data for Ecms > 6 GeV,
- pp and pbar p, from Particle Data Group

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## All  data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV, 

- All  data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,
- pp and pbar p, from Particle Data Group

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 49

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 50

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,

- Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,
- +p and -p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## -value fits for Ecms > 6 GeV, anchored at 2.6 GeV,

- -value fits for Ecms > 6 GeV, anchored at 2.6 GeV,
- +p and -p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## p log2(/m) fit, compared to the p even amplitude fit

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## All cross section data for Ecms > 6 GeV, 

- All cross section data for Ecms > 6 GeV,
- +p and -p, from Particle Data Group

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## All  data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV, 

- All  data (Real/Imaginary of forward scattering amplitude), for Ecms > 6 GeV,
- +p and -p, from Particle Data Group

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,

- Cross section fits for Ecms > 6 GeV, anchored at 2.6 GeV,
- +p and -p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## -value fits for Ecms > 6 GeV, anchored at 2.6 GeV,

- -value fits for Ecms > 6 GeV, anchored at 2.6 GeV,
- +p and -p, after applying “Sieve” algorithm

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## p log2(/m) fit, compared to the p even amplitude fit

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Part 3: The Glauber calculation: 

- Obtaining the p-air cross section from accelerator data,   M. Block and R. Engel

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Glauber calculation:  B (nuclear slope) vs. pp, as a function of p-air

- Glauber calculation:  B (nuclear slope) *vs. *pp, as a function of p-air

- pp from ln2(s) fit and B from QCD-fit

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished)                                            B (nuclear slope) vs. pp, as a function of p-air

- Glauber calculation with inelastic screening, M. Block and R. Engel (unpublished)                                            B (nuclear slope) *vs. *pp, as a function of p-air

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## k = 1.287

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## p-air as a function of  s, with inelastic screening

- p-air as a function of* **** *s, with inelastic screening

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## To obtain pp from p-air

- To obtain pp from p-air

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 65

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 66

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## 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 colliders with cosmic rays

- M. Block, Aspen Winter Physics Conference

- Feb, 2006

## Slide 68

- M. Block, Aspen Winter Physics Conference

- Feb, 2006