Test of the Universal Rise of Total Cross Sections atSuper-high Energies and LHC Keiji IGIRIKEN, JapanAugust 10, 2007Summer Institute 2007, Fuji-YoshidaIn collaboration with Muneyuki ISHIDA
K.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
IntroductionAs is well-known as Froissart-Martin unitarity bound, Increase of tot. cross section tot is at most log2: However, before 2002, it was not known whether this increase is described by logor log2 in p scattering.Therefore we have proposed to use rich inf. of tot(p) in accel. energy reg. through FESR. log2 preferredThis preference has also been confirmed by Block,Halzen04,05.
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.
(a) : All region(c) : High energy region(d) Fig.1. Predictions for and The fit is done for data up to ISRas shown by the arrow.It is very important to notice that energy range of predcted several orders of mag. larger than energy region of input. | LHC(ECM=14TeV) | LHC(ECM=14TeV)p=70GeV||ISR (p=2100GeV)
Universal rise of tot?
Statement : Rise of tot 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 targets
Particle Data Group06(by COMPETE collab.)
Assuming universal B, tot is fitted by log2 for various processes:pp, -p, p, Kp, p: energy in lab.system
Result in PDG06 by COMPETEB is taken to be universal from the beginning. N NN assumed at super-high energies! Analysis guided strongly by theory !
Particle Data Group 2006stated that models with asymp. terms works much better than models with or was confirmed by [Igi,Ishida02,05], [Block,Halzen04,05].
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: N > NN [Igi,Ishida02,05] N 2/3 NN [Block,Halzen04,05]
Purpose of my talkis to investigate the value of B for pp, pp, p, Kpin order to test the universality of B(the coeff. of log2(s/s0) terms) with no theoretical bias.
The tot and ratio(Re f/Im f) are fitted simultaneously, using FESR as a constraint.
FormulaCrossing-even/odd forward scatt.amplitude:Imaginary part totReal part ratio
FESRWe have obtained FESR in the spirit of P sum rule:This gives directly a constraint for p scattering:For pp, Kp scatterings, problem of unphysical region. Considering N=N1 and N=N2, taking the difference, unphysical regionsbetween these two relations, we obtain1962
FESRIntegral of cross sections are estimated with sufficient accuracy (less than 1%).We regard these rels. as exact constraints between high energy parameters: P, c0, c1, c2
The general approachThe tot (k > 20GeV) and (k > 5GeV) are fitted simultly. for resp. processes: High-energy params. c2,c1,c0,P,V are treated as process-dependent. (F(+)(0) : additional param.) FESR used as a constraint P=P(c2,c1,c0) # of fitting params. is 5 for resp. processes. COMPETE B = (4 / m2 ) c2 ; m = Mp, , mK Check the universality of B parameter.
Result of pptottotFajardo 80Bellettini65
Result of tottotBurq 78Apokin76,75,78
Result of KptotK-ptotK+pK-pK+p
The 2 in the best fit(pp) Fajardo80, Belletini65 removed.(-p) Apokin76,75,78 removed.Reduced 2 less than unity both for total 2 and respective 2 . Fits are successful .
The values of B parameters(mb)Bpp is somewhat smaller than Bp, but consistent within two standard deviation. Cons.with BKp(large error).
ConclusionsPresent experimental data are consistent with the universality of B, that is, the universal rise of the tot in super-high energies.Especially, 2/3 [Block,Halzen05], which seems natural from quark model, is disfavoured.
Comparison with Other GroupsOur Bpp=0.289(23)mb (P=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,Ishida06,05Our Bpp is located between the results by COMPETE02 and Block,Halzen05.
Our Prediction at LHC(14TeV)consistent with our previous predictions: tot =107.12.6mb, =0.1270.004 in06 tot = 106.35.1syst2.4statmb, =0.1260.007syst0.004stat , in 05 Located between predictions by other two groups: COMPETE02 and Block,Halzen05Our pred.contradicts with Donnachie-L. =127mb