Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003

1. ｐｐ ー Universal Rise of Hadronic Total Cross Sections based on Forward πｐ, Ｋｐ and , ｐｐ Scatterings Keiji IGI (RIKEN) May 28, 2010 at NTU Seminar Phys.Lett.B670(2009)395 Phys.Rev.D79(2009)096003 In collaboration with Muneyuki Ishida 1

2. Introd. of new sum rule(~1962) σ(k) σ(∞) k α(s) f2(1275) P P’ s (GeV2) Tot. cross sec. σ(k) ≈ σ(∞) + βkα-1 Private letter from Gell-Mann to Chew (If α< 0 is proved, theorysimple) • Many young active physicists attempted to prove this but failed. • I derived new sum rule to prove the above conjecture empirically.

3. Such a kind of attemptK.IGI.,PRL,9(1962)76 • The following sum rule has to hold under the assumption that there is no sing.with vac.q.n. except for Pomeron(P). Evid.that this sum rule not hold  pred. of the traj. with and the f meson was discovered on the 0.0015 -0.012 2.22 VIP: The first paper which predicts high-energy from low-energy (FESR1）

4. SCR(Super Conv.Rel.) Fubini et al This work was one of the highlight at Ｂｅrkeley Conf, 1966. Gell-Mann was the first speaker but he gave a summary talk in the beginning. This was an interesting work but the application was limitted to only one. 4

5. FESR andDualityIgi-Matsuda (1967)Dolen-Horn-Schmid (1967) Since satisfies SCR. 5

6. Using the original physical amp., we obtain Igi, Matsuda, PRL(1967) Average of ImF = low-energy extension of Im 6

7. Veneziano amplitude -- String Theory -- Super String Theory 7

8. Brief Survey on the Main Topic • Increase ofhas been shown to be at most by Froissart (1961) using Analyticity and Unitarity. • Squared log behaviour was confirmedby using the duality constraint of FESR . Keiji Igi & M.I. ’02. Block & Halzen, ’05. • COMPETE collab.(PDG) further assumed for all hadronic scattering to reduce the number of adjustable parameters. 8

9. Particle Data Group (by COMPETE collab.) The upper side：σ The lower side：ρ－ratio B (Coeff. of Assumed to be universal Theory:Colour Glass Condensate of QCD sug. Not rigidly proved from QCD Test of univ. of B:necessary even empirically. This is the main point of today’s talk. 9

10. Ｉncreasing tot.c.s. • Consider the crossing-even f.s.a. with • We assume at high energies. M : proton mass ν, k : incident proton energy, momentum in the laboratory system

11. The ratio • The ratio = the ratio of the real to imaginary part of

12. How to predict σ and ρ for ppat LHC based on duality? (as an example) • We searched for simultaneous best fit of σ and ρ up to some energy(e.g.,ISR) in terms of high-energy parameters constrained by FESR. • We then predicted and in the LHC regions. 12

13. Both and data are fitted through two formulas simultaneously with FESR as a constraint. • FESR is used as constraint of and the fitting is done by three parameters: giving the least . • Therefore, we can determine all the parameters These predict at higher energies including LHC energies 13

14. LHC LHC ISR(=2100GeV) (a) All region • : High energy region Predictions for and The fit is done for data up to ISR LHC As shown by arrow. 14 (b)

15. Pred. for and at LHC • Predicted values of agree with pp exptl. data at cosmic-ray regions within errors. • It is very important to notice that energy range of predicted is several orders of magnitude larger than energy region of the input. • Now let us test the universality of B. It turns out that FESR duality plays very important roles. 15

16. ー ｐｐ Main Topic:Universal Rise of σtot ? • B (coeff. of (log s/s0)2) : Universal for all hadronic scatterings ? • Phenomenologically B is taken to be universal in the fit to πｐ,Ｋｐ,,ｐｐ,∑ｐ,γｐ, γγforward scatt. COMPETE collab. (adopted in Particle Data Group) • Theoretically Colour Glass Condensate of QCD suggests the B universality. Ferreiro,Iancu,Itakura,McLerran’02 Not rigourously proved yet only from QCD.  Test of Universality of B is Necessary even empirically. 16

17. Particle Data Group (by COMPETE collab.) The upper side：σ The lower side：ρ－ratio B (Coeff. of Assumed to be universal Theory:Colour Glass Condensate of QCD sug. Not rigidlyproved from QCD Test of univ. of B:necessary even empirically. 17

18. We attempt to obtain for through search for simultaneous best fit toexperimental. 18

19. New Attempt for • In near future, will be measured at LHC energy. So, will be determined with good accuracy. • On the other hand, have been measured only up to k=610 GeV. So, one might doubt to obtain for (as well as ), with reasonable accuracy. • We attack this problem in a new light.

20. σtot(mb) - 100 50 40 - Fig.1 pp, pp Fig.1 pp, pp s = 60 GeV ISR √ √ Tevatron s= 1.8 TeV ● ● ● ● ● SPS s = 0.9 TeV √ ● Non-Regge  comp.(parabola) 1 10 102 103 104 105 106 log ν(GeV) Practical Approachfor search ofB Tot. cross sec.= Non-Reggeon comp. + Reggeon(P’) comp. • Non-Reg. comp. shows shape of parabola as a fn. of logν with a min. • Inf. of low-energy res. gives inf. on P’ term. Subtracting this P’ term from σtot(+), we can obtain the dash-dotted line(parabola). • We have good data for large values of log ν compared with LHS, so (ISR, SPS, Tevatron)-data is most important for det. of c2(pp)(or Bpp) with good accuracy.

21. σtot (mb) Fig.2 πp 50 40 30 Resonances ↓ √ s = 26.4 GeV kL = 610 GeV - √ s = 26.4 GeV (cf. with pp, pp). highest energy Non-Regge  comp(parabola) ● 1 10 102 103logν(GeV) Fig.2 πp • σtot measured only up to • So, estimated Bπp • may have large uncertainty. • The πp has many res. at low energies, however. • So, inf. on LHS of parabola obtained by subtracting P’ term from σtot(+) is very helpful to obtain accurate value of B(πp). • (res. with k < 10GeV). • ( Kp : similar to πp ) .

22. Fitting high-energy data with no use of FESR - , pp Scattering pp σtot ー For pp scatterings We have data in TeV. σtot = Bｐｐ(log s/s0)2 + Z (+ ρ trajectory) in high-energies. parabola of log s Tevatron Ecm=1.8TeV SPS Ecm<0.9TeV ー ｐｐ ISR Ecm<63GeV CDF D0 Bｐｐ= 0.273(19) mb estimated accurately. Depends the data with the highest energy. (CDF D0) 　ｐｐ Fitted energy region 22

23. πｐ , Ｋｐ Scatterings • No Data in TeV  Estimated Bπｐ , BＫｐ have large uncertainties. No Data No Data Ｋ-ｐ Ｋ+ｐ πｐ π+ｐ 23

24. ー ｐｐ Test of Universality of B • Highest energy of Experimetnal data: : Ecm = 0.9TeV SPS; 1.8TeV Tevatron π-ｐ ： Ecm < 26.4GeV Ｋｐ ： Ecm < 24.1GeV No data in TeV B : large errors. Bｐｐ = 0.273(19) mb Bπｐ= 0.411(73) mb  Bｐｐ =? Bπｐ=? BＫｐ　? BＫｐ = 0.535(190) mb No definite conclusion • It is impossible to test of Universality of B only by using data in high-energy regions. • We attack this problem using duality constraint from finite-energy sum rule (FESR). 24

25. Kinematics ν :Laboratory energy of the incident particle s =Ecm2= 2Mν+M2+m2~ 2Mν M : proton mass of the target. Crossing transf. νー ν m : mass of the incident particle m=mπ , mＫ , M for πｐ； Ｋｐ；ｐｐ; k = (ν2 –m2)1/2 : Laboratory momentum~ν Forward scattering amplitudes fap(ν): a = ｐ,π+,Ｋ+ Im fap (ν) = (k / 4 π)σtotap : optical theorem Crossing relation for forward amplitudes: f π-ｐ(-ν)= fπ+ｐ(ν)* , fＫ-ｐ(-ν) = fＫ+ｐ(ν)* 25

26. Kinematics • Crossing-even amplitudes : F(+)(ーν)=F(+)(ν)* average of π-ｐ, π+ｐ； Ｋ-ｐ, Ｋ+ｐ；ｐｐ, Im F(+)asymp(ν) = βＰ’/m (ν/m)α’(0) +(ν/m2)[c0+c1log ν/m +c2(log ν/m)2] βＰ’ term : P’trajecctory (f2(1275)): αＰ’(0) ~ 0.5 : Regge Theory c0,c1,c2terms: corresponds to Z + B (log s/s0)2 c2 is directly related with B . (s～2Mν) • Crossing-odd amplitudes : F(-)(ーν)= ーF(-)(ν)* Im F(-)asymp(ν) = βV/m (ν/m)αV(0) ρ-trajecctory:αV(0) ~0.5 βＰ’ , βV is Negligible to σtot(= 4π/k Im F(ν) ) in high energies. 26

27. ＦＥＳＲ　Ｄｕａｌｉｔｙ • Remind that the P’ sum rule in the introd.. • Take two N’s(FESR1) • Taking their difference, we obtain has open(meson) ch. below ,and div. above th. • If we choose to be fairly larger than we have no difficulty. ( : similar) No such effects in . RHS is calculable from The low-energy ext. of Im Fasymp. LHS is estimated from Low-energy exp.data.

28. Average of in low-energy regions should coincide with the low-energy extension of the asymptotic formula • This relation is used as a constraint between high-energy parameters:

29. Choice of N1 for πｐ Scattering • Many resonances Various values of N1 in π-ｐ & π+ｐ • The smaller N1 is taken, the more accurate c2 (and Bπｐ)obtained. • We take various N1 corresponding to peak and dip positions of resonances. (except for k=N1=0.475GeV) For each N1, FESR is derived. Fitting is performed. The results checked. Δ(1232) N(1520) N(1650,75,80) - Δ(1905,10,20) Δ(1700) 29

30. Analysis of πｐ Scattering • Simulateous best fit to exp. data of σtot for k = 20~370GeV(Ecm=6.2~26.4GeV) and ρ（= Re f(k) / Im f(k)） fork > 5GeV in π-ｐ,π+ｐforward scatterings. • Parameters : c2,c1,c0,βP’,βV, F(+)(0) (describing ρ) • FESR N2=20GeV fixed. Various N1 tried. Integral of σtot estimated very accurately. Example : N1=0.818GeV 0.872βP’+6.27c0+25.7c1+109c2=0.670 (+ー0.0004 negligible)  βP’ = βP’ (c2,c1,c0):constraint.fitting with 5 params 30

31. N1 dependence of the result • # of Data points : 162. • best-fitted c2 : very stable. • We choose N1=0.818GeV • as a representative. • Compared with the fit by • 6 param fit with • No use of FESR(No SR) 31

32. Result of the fit to σtotπｐ No FESR FESR used c2=(164±29)・10-5 c2=(121±13)・10-5 Bπｐ＝0.411±0.073mb Bπｐ＝0.304±0.034mb FESR integral Fitted region Fitted region π-ｐ π+ｐ much improved 32

33. ー ー ｐｐ ｐｐ Choice of N1 in Ｋｐ, ,ｐｐ ー ｐｐ • Exothermic Ｋ-ｐ　　　 open at thres. meson-channels • Exotic Ｋ+ｐ, ｐｐ steep decrease at Ecm~2GeV ?  We take N1 larger than πｐ. N1=5GeV. Ｋ-ｐ ｐｐ Ｋ+ｐ 33

34. Result of the fit to σtotＫｐ No FESR FESR used Fitted region FESR integral Fitted region c2=(266±95)・10-4 c2=(176±49)・10-4 BＫｐ＝0.535±0.190mb BＫｐ＝0.354±0.099mb large uncertainty much improved 34

35. ー Result of the fit to σtotｐｐ,ｐｐ No FESR FESR used FESR integral Fitted region large Fitted region large c2=(491±34)・10-4 c2=(504±26)・10-4 Bｐｐ＝0.273±0.019mb Bｐｐ＝0.280±0.015mb Improvement is not remarkable in this case. 35

36. Test of the Universal Rise • σtot= B (log s/s0)2 + Z FESR used No FESR • Bπｐ= Bｐｐ= BＫｐ within 1σ • Universality suggested. Bπｐ≠? Bｐｐ =? BＫｐ No definite conclusion in this case. 36

37. Concluding Remarks • In order to test the universal rise of σtot , we have analyzed π±ｐ;Ｋ±ｐ; ,ｐｐ independently. • Rich information of low-energy scattering data constrain,through FESR, the high-energy parameters B to fit experimental σtot and ρratios. • The values of B are estimated individually for three processes. 37

38. Ｋｐ πｐ ｐｐ • We obtain Bπｐ= Bｐｐ= BＫｐ. Universality of B suggested. Use of FESR is essential to lead tothis conclusion. • Universality of B suggests gluon scatt. gives dominant cont. at very high energies( flav. ind. ). • It is also interesting to note that Z for approx. satisfy ratio predicted by quark model. 2:2:3 38

39. Our results predicts at 14TeV 102.0(1.7)mb at 10 TeV 96.0(1.4)mb at 7 TeV • Our Conclusions will be tested by LHC TOTEM.

40. Finally, let us compare our pred. at LHC with other pred. ref. Ishida-Igi (this work) Igi-Ishida (2005) Block-Halzen (2005) COMPETE (2002) Landshoff (2007) • Pred. in various models have a wide range. The LHC(TOTEM) will select correct one among these approaches. Ishida-Igi gives pred., not only for pp but also for πｐ and Kp,although experiments are not so easy in the near future. This is very import. point. 40