Test of the universal rise of total cross sections at super high energies
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Test of the Universal Rise of Total Cross Sections at Super-high Energies. Muneyuki ISHIDA Meisei Univ. KEKPH07. Mar. 1-3, 2007 In collabotation with Keiji IGI. Introduction. Increase of tot. cross section σ tot is at most log 2 ν : Froissart-Martin Unitarity bound

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Test of the universal rise of total cross sections at super high energies

Test of the Universal Rise of Total Cross Sections atSuper-high Energies

Muneyuki ISHIDA

Meisei Univ.

KEKPH07. Mar. 1-3, 2007

In collabotation with Keiji IGI


  • Increase of tot. cross section σtot is

    at most log2ν: Froissart-Martin Unitarity bound

  • However, before 2002,

    it was not known whether this increase is

    described by logνor log2ν in πp scattering

  • Therefore we have proposed to use rich inf. of σtot(πp) in low energy reg. through FESR.

     log2ν preferred

    σtot = B log2ν+・・・

    at Super-high energies

Universal rise of tot
Universal rise of σtot?

Statement :

Rise of σtot at super-high energies is universal

by COMPETE collab.,that is,

the coefficient of log2(s/s0) term is universal

for all processes with N and γ targets

Particle Data Group’06

(by COMPETE collab.)

Assuming universal B,σtot is fitted by log2ν for various processes:

pp, Σ-p, πp, Kp, γp

ν: energy in lab.system

Result in pdg 06 by compete
Result in PDG’06 by COMPETE

B is taken to be universal from the beginning.

 σπN~ σNN~・・・assumed at super-high energies!

Analysis guided strongly by theory !

Particle data group 2006
Particle Data Group 2006

  • “Boththese refs., however, questioned the statement (by [COMPETE Collab.]) on the universality of the coeff. of the log2(s/s0).The two refs. give different predictions at superhigh energies:

    σπN > σNN [Igi,Ishida’02,’05]

    σπN~ 2/3 σNN [Block,Halzen’04,’05]”

Purpose of my talk
Purpose of my talk

is to investigate the value of

Bfor pp, pp, π±p, K±p

in order to test the universality of B

(the coeff. of log2(s/s0) terms)

with no theoretical bias.

The σtot and ρ ratio(Re f/Im f) are fitted simultaneously, using FESR as a constraint.


  • Crossing-even/odd forward scatt.amplitude:

Imaginary part  σtot

Real part  ρ ratio


  • We have obtainedFESR in the spirit of P’ sum rule:

This gives directly a constraint for πp scattering:

For pp, Kp scatterings, problem of unphysical region. Considering N=N1 and N=N2, taking the difference.


  • Integral of cross sections are estimated with sufficient accuracy (less than 1%).

  • We regard these rels. as exact constraints between high energy parameters:

    βP’, c0, c1, c2

The general approach
The general approach

  • The σtot (k > 20GeV) and ρ(k > 5GeV) are fitted simultly. for resp. processes:

  • High-energy params. c2,c1,c0,βP’,βV are treated as process-dependent.(F(+)(0) : additional param.)

  • FESR used as a constraint βP’=βP’(c2,c1,c0)

  • # of fitting params. is 5 for resp. processes.

  • COMPETEB = (4π / m2 ) c2 ; m = Mp, μ, mK

  •  Check the universality of B parameter.

Result of pp
Result of pp




Fajardo 80



Result of
Result of π



Burq 78




Result of kp
Result of Kp









The 2 in the best fit
The χ2 in the best fit

  • ρ(pp) Fajardo80, Belletini65 removed.

  • ρ(π-p) Apokin76,75,78 removed.

  • Reduced χ2 less than unity both for total χ2 and respective χ2.

     Fits are successful .

The values of b parameters mb
The values of B parameters(mb)

Bpp is somewhat smaller than Bπp, but consistent within two standard deviation. Cons.with BKp(large error).


  • Present experimental data are consistent with the universality of B, that is, the universal rise of the σtot in super-high energies.

  • Especially, σπN~2/3 σNN[Block,Halzen’05], which seems natural from quark model, is disfavoured.

Comparison with other groups
Comparison with Other Groups

  • Our Bpp=0.289(23)mb (αP’=0.5 case) is consistent with B=0.308(10) by COMPETE, obtained by assuming universality.

  • Our Bpp is also consistent with

    0.2817(64) or 0.2792(59)mb byBlock,Halzen, 0.263(23), 0.249(40)sys(23)stat byIgi,Ishida’06,’05

  • Our Bpp is located between the results by COMPETE’02 and Block,Halzen’05.

Our prediction at lhc 14tev
Our Prediction at LHC(14TeV)

  • consistent with our previous predictions:

    σtot =107.1±2.6mb, ρ=0.127±0.004 in’06


    ρ=0.126±0.007syst±0.004stat , in ‘05

    (including Tevatron discrepancy as syst. error.)

    Obtained by analyzing only crossing-even amplitudes using limited data set.

  • Located between predictions by other two groups: COMPETE’02 and Block,Halzen’05