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Proposing a new constraint for predictions of tot.c.s. and ratio at LHC Muneyuki ISHIDA Meisei Univ., Japan New Trends in High Energy Physics, Yalta, Sept. 16-23, 2006 In collabotation with Keiji IGI K.Igi and M.Ishida, Phys.Lett. B 622 (2005) 286; Prog.Theor.Phys. 115 (2006) 601; hep-ph/0605337 1 Contents of the talk • We propose to use rich inf. on the tot.cross section in low energy regions through duality to predict the and ratio at very high energies. • Using FESR as constraints for high energy parameters, we search for simultaneous best fit to and ratio up to some energy to determine high-energy parameters. • We then predict and in the LHC energy regions. 2 Last Year : Present. by K. Igi • We analyzed crossing-even forward scatt. ampl. • Using one FESR as a Constraint, n=1 mt. FESR called Constraint(2), and up to Tevatron energy (Ecm=1.8TeV) are fitted simultaneously. • Two conflicting measurements at 1.8TeV. • We analyzed two cases: E710/E811 and CDF, respectively. 3 CDF E710 E811 E710/E811 data input : green lines CDF datum input : blue lines; K. Igi presented in ’05. 4 Table 3 The predictions of tot and at LHC energy , s 14TeV Last Year presented by Prof. Keiji Igi. tot s 14TeV s 14TeV E710 E811 CDF Large systematic difference between E710/E811 input and CDF input. Prediction at LHC : 5 This Year : New topics • We propose a new Constraint(1),corres. to n= -1 mt. FESR, free from unphys. region. • We have used two constraints, (1) and (2), and search for the best fit of , up to SPS energy(0.9TeV). prediction at Tevatron(1.8TeV) to invest. E710/E811 and CDF discrepancy. • We have also analyzed crossing-odd ampl. and obtain pred. at LHC(14TeV). 6 Such a kind of attempt K.Igi, PRL 9(1962), 76 • The following sum rule has to hold under the assumption : there is no sing.with vac.q.n. except for Pomeron(P). f2 N dk tot k tot N 1 a M M 0 0.0015 -0.012 2.22 • Evid.that this sum rule not hold prediction of the P’ traj. with P 0.5 . ' f meson was discovered on the P’ . 7 Brief survey of our prev.work on sat.of Froissart -Martin bound: K.Igi and M.Ishida,PRD66(2002)034023 • As is well known, tot.c.s. increase above 70GeV. • It had not been known, however, if this increase behaved like consistent with Froissart-Martin bound before 2002. 8 • We proposed to use rich inf.of in low energy reg. to investigate high energy behaviours. • FESR(1),(2) as constraints • Searched for best fit to above 70 GeV. • We can conclude that our analysis strongly favours behaviours. 9 Main Topic :Derivation of new constraint(1) • Crossing-even forward scatt.amp. : proton Lab. Energy : Lab. momentum • We assume (log ν)2 behaviour at high energies: • Difference between real ampl. and its asymp. form 10 We can write down FESR in the spirit of P’ sum rule. Unphysical region physical reg. If this eq. reduces to so-called P’ FESR. • This FESR suffers from unphys. region com. from boson poles below thres. 11 • FESR, not at ν＝Ｍ but at some intermediate energy ν＝ν1 . • Difficult to estimate exactly enough. • Taking difference between N=N1 and N=N2. Constraint(1) : relation between . 12 n= -1 mt. FESR, free from unphysical region. Constraint(2) The second FESR corres. to n=1 mt. FESR: Unphysical region Physical region • n=1 mt. FESR. • relation between • Contributions from unphys. region are negligible, due to the suppression factor ν. 13 Contribution to Constraint(2) from unphysical region: negligible • Why? Ave. of im. part from boson poles below p p threshold 2 2 = smooth extrapolation of t-ch q q exch. contribution from high energy to M due to FESR duality. 14 • Since Im Fq q Im F , 2 2 1st term d 0 Im F d M M of LHS of FESR(2) 0 Im Fq 2 q2 M2 d k o M Im f pp Im f pp 3.2GeV 0 2 4 1 2nd term k tot k dk 3403 20GeV N 2 of LHS of 4 0 FESR(2) So, resonance contributions from unphysical regions is less than 0.1% of resonance contributions from phys. reg. Therefore : negligible 15 The ratio • The ratio = the ratio of the real to imaginary part of F : subtraction const. 16 General approach • Constraints(1),(2) , formula of tot and the ratio : our starting point. • In Constraints(1),(2), the integrals of are expressed in terms of high-energy parameters: c0 , c1 , c2 , P' . • The integrals are evaluated with error (less than 1%). Constraints(1),(2) 17 • 3 Parameters : • We search for simultaneous best fit to the data of tot k and k to determine values of params. giving the least . 2 • fitted energy region: and below SPS energy: • We can predict and in Tevatron energy and in LHC energy, using these params. 18 Data: We use rich data (PDG 2004) to evaluate the relevant integrals of cross sections appearing in Constraints(1),(2). 19 Evaluation of integrals: in Constraint(1). in Constraint(2). Phenomenological fits to , in c0 β d f χ2/NF 6.34 11.08 5.28 -0.15 36.6/(70-4) 7pt. below 10GeV are removed by stat. 6.32 4.25 -12.6 24.4 71.8/(103-4) method(Sieve algor) : fixed 20 Based on these phenom. fits, For Constraint(1): (N1,N2,ν1)=(10,70,40) GeV Integrals for Averaging Error is less than 1%, negligible. 21 For Constraint(2): N=10GeV. Integral region is devided into two parts: polygonal line graph↑ ↑phenom. fit Area: Error: Integrals for Consistent with prev. estim.: Error is less than 1%, negligible. 22 Constraints • Constraint(1) : (N1,N2,ν1)=(10,70,40)GeV 3.316βP’+31.98c0+141.1c1+610.9c2=230.81 (normalized : 0.104βP’+c0+4.41c1+19.1c2=7.22) • Constraint(2) : N = 10 GeV 140.7βP’+383.6c0+781.6c1+1635.c2=3396.3 (normalized : 0.367βP’+c0+2.04c1+4.26c2=8.85) 23 data • When and data points are listed at same value of k, we make data points by averaging them. 27(11) points for up to ISR,k=2094GeV( ). • 5(1) points in SPS energy region: • In summary, we have 32(12) points of in . 24 Analysis: • Both tot and (actually Re F ) data in (SPS) are fitted simult. . • Constraints(1),(2) are used. • The fitting is done by three parameters: 25 Result LHC LHC input E. region 26 Table 1 The values of parameters and χ2 in the best fit N F and N N are the degree of freedom and the number of tot data points in the fitted energy region. c2 c1 c0 P ' 0.0473 - 0.173 6.32 7.32 10.01 NF 2 N 2 N 2 15.9/41 10.5/32 5.4/12 The reduced χ2 and respectiveχ2 devided by # of data points for and are less than unity : Fit is successful . 27 Predictions for and at Tevatron and at LHC By using these values of params., we can predict and at CDF . Tevatron and at . E710 E811 LHC. LHC 28 Prediction at Tevatron • We have investigated a possibility to resolve the discrepancy among the E710, E811 and CDF at 1.8TeV. E710 E811 CDF • Our pred. E710 is consistent with the prediction in one standard deviation. Both CDF and E811 are in two standard deviation. • E710 preferable,but CDF,E811 not excluded. 29 Prediction at LHC(14TeV) Based on the result of analysis of crossing-even amplitude, we obtain the prediction at LHC, 30 Analysis of Crossing-odd Amplitude • We can also analyze the crossing- odd amplitude : • It is simply described by exchange of vector-meson trajectories (ρ,ω). • Its Asymptotic form: 31 Its benefit • Analyses of both even and odd amplitudes make it possible to use all the data, not selected for even-amplitude. • Number of fitted data is about 5 times compared with the analysis of even- amplitude. 32 Formula • Using the crossing-odd property, • amplitude 33 General approach • The αV is determined by the fit to the difference between and . • Fixing the value of αV, then, 4 Data sets of are fitted simult.. • Constraints(1),(2) are used for even- amplitude. • Number of parameters is 4: , determined by the fit. Prediction of and at LHC. 34 Value of αV • Using two constrs, both and are fitted simultaneously in 20GeV < k < SPS. : fixed in the following 35 analysis. /GeV TeV Result of the analysis of both even and odd amplitudes using two constraints. Data in 10GeV < k < SPS are fitted simultaneously. LHC Dashed lines : 1-standard deviation 36 Table 2 Result of Analysis of both even and odd amplitudes: ND , Nσ and Nρ are the number of data points in the fitted energy region: 10GeV < k < SPS. c2 c1 c0 P ' 0.0486 -0.201 6.44 7.14 11.43 3.64 266.3/(250-4) 24.4/58 9.8/12 112.3/112 119.8/69 The reduced χ2 for total data is almost unity, although the fit to ρ(pp) is not successful. 37 Table 3 Result of the fit to the data set removing ρ(pp). fitted energy region: 10GeV < k < SPS. c2 c1 c0 P ' 0.0482 -0.203 6.45 7.11 6.36 3.66 142.7/(181-4) 24.7/58 7.3/12 110.7/112 ------ Fit is successful. The result is almost the same as that of the fit including ρ(pp) data. reliable ! 38 Prediction at LHC • By using these values of parameters we can predict the total cross section and ρ ratio at LHC(14TeV). • Consistent with the result of analysis of crossing-even amplitude. # of data becomes 5 times, but error does not reduce so largely, since the added data are below 2100GeV. 39 Comparison with other groups • Predicted central values of pp : tot in good agreement with Block and Halzen tot 107.4 1.2mb, pp pp 0.132 0.001 They adopt a statistical method of data selection, Sieve algorithm, and obtain smaller errors without systematic errors. They also used duality but different method with ours. • In contrary to our results, however, their values are not affected so much about CDF, E710/E811 discrepancy. In our case, the measurement at LHC energy will discriminate which solution is better at Tevatron. 40 • Our prediction has also to be compared with Cudell et al.(PRL 89(2002) 201801) + 4.1 pp tot 111.5 1.2syst 2.1stat mb, + 0.0058 pp 0.1361 0.0015syst 0.0025stat who’s fitting techniques favour the CDF point at s 1.8TeV . • Predictions by Bourrely et al. based on the impact-picture phenomenology. 41 Concluding Remarks • We analyze all the available data in 10GeV < k . • Using FESR as two constraints we obtain the prediction at LHC(14TeV): • We emphasize that precise measurements of both tot and pp in coming LHC exp. will pp resolve Fermilab discrepancy of at 1.8TeV. • It is also important to investigate pp in extremely high-energy regions 5 1019 eV . 42 43 Fit by Cudell et al. • They use the same formula of (log ν)2 . Fitting all the processes, by using common values of parameters. while Z, Y1, Y2 are process-dependent. 44 • Their is smaller than ours αV=0.517. fitted energy region: Ecm > 5GeV (k > 12.3GeV):smaller • Their conjecture : value of B(=0.308+- 0.010 mb) is process-independent. corresponding to c2 = 0.0554+-0.0018 . • Larger value of c2 predicts larger σ at LHC . +4.1 tot 111.5 1.2syst 2.1stat mb, pp • It should be checked if their values of parameters satisfy duality constraints or not. 45 • Our c2 in πp process: c2 = 0.00173 B(πp) = 4π c2 / mπ2 = 0.4345 mb • This B(πp) in our prev. anal. is larger than B(pp)( = 0.270 +- 0.018 mb) in our present analysis. • Conjecture: Universality of B (coeff. of (log ν)2 ) should be checked through the re- analysis of πp in our case. 46 • Since amplitude is crossing-even,we have e i 2c0 c2 2 c1 log log i M M R , 2M 2 e i c2 log 2 log 2 M M e i M P' M P' P' FP' M sin P' and so obtain Re R c1 2c2 log , 2M 2 M P 0.5 Re FP' ' , 47 M M Determination of αV • It is necessary to pay special attention to determine αＶ , since the prediction of tot.c.s. at LHC is sensitive to αv. • We determine αＶ directly by fitting and . • Fitted energy regions are changed: 10,20,30,50,70GeV < k (below SPS). 48 Result Fitted E.range 10GeV<k 102.3/(169-3) 28.6/58 73.7/111 20GeV<k 82.5/(145-3) 21.1/51 61.4/94 30GeV<k 77.2/(126-3) 19.6/45 57.6/81 50GeV<k 66.3/(101-3) 15.3/29 51.0/72 70GeV<k 46.4/(88-3) 13.4/26 33.0/62 Results are mutually consistent for 20,30,50GeV < k . 10GeV seems to be too small as a lower limit. We use αV = 0.517 :fixed to predict the σ and ρ at 49 LHC. Three cosmic-ray data sample 1. N.N.Nikolaev 2. T.K.Gaisser et al.(GSY) 3. M.M.Block et al.(BHS) Nikolaev claims the tot is approximately 30mb larger than BHS’s results. NGSY BHS fit 2 22.9/7 1.3/7 fit 3 18.0/7 0.9/7 Our pred. values of tot are in good agreement with BHS. 50 • There are also very interesting prob. above the GZK cutoff energies 5 1019 eV . • Because of the reaction p CMB n, is supposed to drop off above this pp energy. • According to AGASA, pp is still increasing, but soon Pierre Auger Observatory in Argentina may give new results. • LHC experiments may also be important for searching for this energy regions. 51

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