Photon-Photon Diffractive Interaction at High Energies

Transcription

1 Photon-Photon Diffractive Interaction at High Energies Cong-Feng Qiao Graduate University Chinese Academy of Sciences December 17,2007 1

2 Contents Brief About Diffractive Interaction Leading Order Photon Wave function NLO Photon Impact Factor Photon-photon Diffractive Scattering Summary and Outlook 2

3 Diffractive Interaction Non-Diffractive Rapidity y 1 E+ P ln Z ln tan θ = η 2 E P 2 Z 3

4 pp(p) e.g. In Diffractive Scattering Elastic single diffra. double diffra. double P [J.D. Bjorken,1993] 4

5 Regge trajectories Exchanged Objects: 2 mi = t [Donnachie and Landshoff 1992] 5

6 The Single Pole Contribution in Regge Theory σ tot s 1 Im Ast (, = 0) ~ s s s α (0) 1 d σ el 2 α( t) 2 2 α(0) 2 dt = Fts () ~ t 0 s 6

7 Regge trajectories have intercepts which do not exceed 0.5. Their exchange leads to total cross sections decreasing, which contradicts to experiment. In order to account for the experimental results, the reggeon with intercept larger than one was introduced, named Pomeron,, IP [Pomeranchuk,, 1956; Foldy & Peierls,, 1963] 7

8 Pomeron carries the quantum number of the vacuum Is Pomeron IP the glueball? 8

9 QCD Description of Pomeron (BFKL Equation) [Balitsky, Fadin, Lipatov Kuraev,, 1976,1977,1978] In perturbative QCD, in the limit s >> t and in leading logarithm ln(s/t) approximation (LLA). e.g. quark-quark scattering 9

10 The gluon is reggized to all orders α g g 2 μν s Dμν (, s q ) = i q 2 s 0 2 ( q ) 1 α q q g 2 2 ( ) = 1 + ε ( ) is the Regge trajectory of gluon. Pomeron emerges as a reggized gluon ladder in color-singlet. 10

11 The leading logarithm result of Pomeron trajectory α () t 1 4α Ρ = + s ln2 This is much larger than the phenomenologically obtained intercept of the pomeron The leading order (Log) result violate the Froissart 2 bound, that is Ln s, at high energy [Froissart(1961), Martin(1963)] NLO corrections to BFKL Kernel greatly reduce the pomeron trajectory [Fadin and Lipatov,, 1998; Ciafaloni and Camici,, 1998] _ 11

12 Leading order photon wave function [Gieseke & Qiao 2000;Muller 1990; Nikolaev ] 12

13 Diffractive photon-proton proton scattering is of particular interest in testing the perturbative QCD, because of not only the HERA prolific experimental data, but also the theoretical development 13

14 In Sudakov decomposition k = αq' + β p' + k t r = q' + x p' + r s P q x x Bj = 2 pq 2 q ( p p') x P pq B 2 x Q = x = x 2 2 Bt P Q + Mx t β = x P S Q = q r = t M << s ~ ~ x 14

15 In the above limit, the metric tensor decomposes like 2 ( ' ' ' ' ) gμν = s pμqν + pνqμ + gμν for the gluon propagator, only retain the first term, while other terms are suppressed by power of t/s The photon polarization vectors can be chosen as ε 0 = 1 ( q Q ') + x p B ' ε ( ± ) = (0,1, γ i,0) for longitudinal and transverse ( γ = ± ) cases 1 2 The γ P scattering amplitude can be expressed as 2 2 s A= eg δ C α(1 α) ( Ψ( k, α) Ψ ( k+ r, α)) t λ λ B B' 15

16 Here 0 2 e+ α(1 α) Q Ψ ± m ( k, α) = α(1 α) Q + k + m ± Ψ ( k, α ) = ± m 2e α k f α(1 α ) Q + k + m ± Ψ ( k, α) = m± ± Ψ ( K, α) = m± i 2e m 2 e (1 α) k f f α(1 α) Q + k + m α(1 α) Q + K + m ± Ψ ( k, α) =Ψ ( k, α) = 0 ±± k = k + iγ k x mm y 16

17 The Application of Photon wave function To the photon-proton proton diffractive interaction in two gluon model + Using Cutkosky rules the full amplitude can be written in double difference form s ' d l r A= iegs δλλ C α(1 α) B B ' (2 ) 2 2 ( ) 2 t π l r l { Ψ ( k, α) +Ψ ( k+ r, α) Ψ ( k+ l, α) Ψ ( k+ r l, α)} 17

18 To include the non-perturbative coupling of the two gluons for the proton, one need to introduce the unintegrated off- diagonal gluon distribution F( xxl, ', 2, r 2 ) and after integrating over, we would get an off-diagonal gluon ldistribution 0 2 Q dlfxxl 2 (, ', 2, r 2 ) = Gxxl (, ', 2, r 2 ) 18

19 In the limit x x, ' Gxxr (,, = 0, Q) = xgxq (, ) Therefore, the general amplitude for diffractive scattering off the proton is 2 2 π 4 dl r ' 2 2 A= i eg s α(1 α) Fxxl (,,, r) DΨ( krl,, ) π l l ( r l) Here, DΨ (,,) k r l =Ψ (, kα) +Ψ ( k+ r, α) Ψ ( k+ l, α) Ψ ( k+ r l, α) 19

20 The variable conjugate to k is the transverse separation of the qq pair, the ρ, which is called the dipole size The variable conjugated to the momentum transfer between the diffractive system and the proton is the impact parameter b 20

21 The conjugated photon wave functions are 0 1 Ψ ± m( ρα, ) = efα(1 α) QK0 ( δρ) π ± i ρε Ψ ± m( ρα, ) = e fαδ K 1( δρ) π ρ ± i ρε Ψ m± ( ρ, α) = e f (1 α) K 1( δρ) π ρ ± im Ψ ±± ( ρα, ) = ek f 0( δρ) 2π δ = α(1 α) Q + m ; K ( z) Bessel function ν is the modified 21

22 The diffractive amplitude configuration space is 2 2 D dk dr iks irb D A% (,) ρ b = e e A (,) k r 2 2 (2 π) (2 π) 2 2 dr ir b dl 2 ' 2 2 = BΨ (,) ρα e α( )(,,, ) 2 2 s μ F x x l r (2 π) πl e il s i( r l) s [1 ][1 ] =Ψ(, sσσ ) ( ρ, b) qq e The dipole picture of diffractive interaction 22

23 NLO Photon Impact Factor People believe that at LCs things would be much clear To make full NLO prediction, we have the NLO BFKL Kernel, but no NLO Photon impact factor yet 23

24 To calculate the NLO Impact factor, we can use γ q ( qq) q scattering as the starting point, in the high energy limit tq,, t, t, M << S 2 2 a b 24

25 Taking the Regge ansatz for the scattering amplitude a s s ω s ω A =Γ [( ) + ( ) ] Γ γ qq t t t 1+ ω is the gluon trajectory. Expanding all terms in powers on the strong coupling ω = g ω + g ω 2 (1) 4 (2) s s g s Γ = g Γ + g Γ a qq a (0), a 3 (1), a qq s γ γ qq s γ qq Γ = g Γ + g Γ a (0), a 3 (1), a qq s qq s qq 25

26 Then And 2 (0 ) 4 (1) Then T = g T + g T [Bartels,Gieseke, T T s 2 s =Γ ( ) Γ t (0) (0), a (0), a γ qq qq (1) (1), a (0), a (0), a (1), a γ qq qq γ qq qq s =Γ 2s 2s () Γ +Γ () Γ t t +Γ s s s () Γ ω [ln + ln ] Γ t t t Qiao,2001] (0), a (0), a (1) (0), a γ qq qq qq The calculation of virtual correction to involving 14 diagrams (1),a Γ γ qq 26

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28 In practical calculation, we have used the computer algebra system Mathematica with package Feyncalc Using the loop integral technique given by [Bern, Dixon and Kosower,1994],all the results are presented analytically. Finally, we need to take the high energy limit It has been showed that the ultra-violet divergencies are removed in M S scheme; the infrared divergencies are cancelled out by adding real corrections [Bartels, Gieseke and Kyrieleis,, 2002] 28

29 Photon-Photon Diffractive Scattering γ*γ* * diffractive scattering cross section can be expressed as 29

30 γ*γ* * diffractive scattering cross section up to 3 order α s 30

31 γ*γ* * diffractive scattering cross section at order α 3 s 31

32 Photon Impact Factor at NLO 32

33 Because of the complexity, the NLO corrections to photon impact factor are only presented numerically 33

34 The full next-to to-leading order photon 2 impact factor at S0 = r r 34

35 To check the numerical result, one should get the S 0 independence of NLO γ*γ* * total cross section 35

36 Summary & Outlook 3 The full fixed order ( α s ) prediction for γ*γ* * total cross-section section well come soon [Bartels, Chachamis and and Qiao,, in preparation] In the future, our NLO photon impact factor should convoluted with NLO BFKL Green s s function in making predictions 36

37 The ILC provide us a very good play ground for testing and understanding the QCD. The diffractive interaction would be one of the emphases So far, the meaning of the NLO calculation for hard diffractive interaction is more on the theoretical side, rather than for the phenomenological sense 37

38 Thank you for your attention 38