Test of the Universal Rise of Total Cross Sections at Super-high Energies

1. 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

2. Introduction • 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

3. 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

4. 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

5. 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 !

6. 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]”

7. 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. ー

8. Formula • Crossing-even/odd forward scatt.amplitude: Imaginary part  σtot Real part  ρ ratio

9. FESR • 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.

10. 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

11. 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.

12. Result of pp ρ σtot ρ Fajardo 80 Bellettini65 σtot

13. Result of πｐ σtot ρ Burq 78 Apokin76,75,78 ρ σtot

14. Result of Kp ρ K-p σtot K-p ρ K+p σtot K+p

15. 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 .

16. The values of B parameters(mb) Bpp is somewhat smaller than Bπp, but consistent within two standard deviation. Cons.with BKp(large error).

17. Conclusions • 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.

18. 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.

19. 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 (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