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Test of the Universal Rise of Total Cross Sections at
Super-high Energies and LHC
Keiji IGI
RIKEN, Japan
August 10, 2007
Summer Institute 2007, Fuji-Yoshida
In collaboration with Muneyuki ISHIDAK.Igi and M.Ishida: hep-ph/0703038(to be published in Euro.Phys. J. C)
Phys.Rev.D66 (2002) 034023; Phys.Lett. B 622 (2005) 286; Prg.Theor.Phys. 116 (2006) 1097
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Introduction• As is well-known as Froissart-Martin unitarity bound, Incre
ase of tot. cross section σtot is
at most log2ν: • However, before 2002, it was not known whether this in
crease is described by logνor log2ν in πp scattering.
• Therefore we have proposed to use rich inf. of σtot(πp) in accel. energy reg. through FESR.
log2ν preferred• This preference has also been confirmed by Bloc
k,Halzen’04,’05.
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• For , we searched for the simultaneous best fit of
and up to some energy(e.g.,ISR) in terms of high-energy parameters constrained by FESR.
• We then predicted and in the LHC and high-energy cosmic-ray regions.
tot
tot
14s TeV17( 5 10 )up to eV
,pp p p
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(a) : All region tot (c) : High energy region
tot
(d)
Fig.1. Predictions for and
The fit is done for data up to ISR
11.5 62.7GeV s GeV
as shown by the arrow.
It is very important to notice that energy range of predcted tot
several orders of mag. larger than energy region of input.
| LHC(ECM=14TeV)
| LHC(ECM=14TeV)
p=70GeV| |ISR (p=2100GeV)
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Universal rise of σtot?
Statement :
Rise of σtot at super-high energies is universal
by COMPETE collab., that is,
the coefficient B in front of log2(s/s0) term is universal
for all processes with N and γ targets
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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
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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 !
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Particle Data Group 2006• stated that models with asymp. terms wo
rks much better than models with or was confirmed by [Igi,Ishida’02,’05], [Block,Halzen’04,’05].
• “Both these 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] ”
0log s s
0s s
20log s s
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Purpose of my talk
is to investigate the value of B for pp, pp, π±p, K±pin 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.
ー
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Formula• Crossing-even/odd forward scatt.amplitude:
Imaginary part σtot
Real part ρ ratio
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FESR
• We have obtained FESR 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,
unphysical regions
between these two relations, we obtain
1962
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FESR
• 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
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The general approach• The σtot (k > 20GeV) and ρ(k > 5GeV) are f
itted 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.
COMPETE B = (4π / m2 ) c2 ; m = Mp, μ, mK
Check the universality of B parameter.
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Result of pp
σtot
σtot
ρ
ρFajardo 80
Bellettini65
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Result of π ｐ
σtot
σtot
Burq 78
Apokin76,75,78
ρ
ρ
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Result of Kp
σtotK-p
σtotK+p
ρ K-p
ρ K+p
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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 .
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The values of B parameters(mb)
process B αP’=0.500 αP’=0.542
pp Bpp 0.289(23) 0.268(24)
πp Bπp 0.351(36) 0.333(39)
Kp BKp 0.37(21) 0.37(22)
Bpp is somewhat smaller than Bπp, but consistent within two standard deviation. Cons.with BKp(large error).
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Conclusions
• Present experimental data are consistent with the universality of B, that is, the universal rise of the σtot in super-high energies.
• Especially, σπ Ｎ～ 2/3 σ ＮＮ [Block,Halzen’0
5], which seems natural from quark model, is disfavoured.
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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 by Block,Halzen, 0.263(23), 0.249(40)sys(23)stat by Igi,Ishida’06,’05
• Our Bpp is located between the results by COMPETE’02 and Block,Halzen’05.
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Our Prediction at LHC(14TeV)
• consistent with our previous predictions:
σtot =107.1±2.6mb, ρ=0.127±0.004 in’06
σtot = 106.3±5.1syst±2.4statmb,
ρ=0.126±0.007syst±0.004stat , in ‘05
• Located between predictions by other two groups: COMPETE’02 and Block,Halzen’05
Our pred.contradicts with Donnachie-L. σ=127mb
