High-energy amplitudes and impact factors at next-to-leading order

Transcription

1 High-energ amplitudes and impact factors at net-to-leading order CPHT, École Poltechnique, CNRS, 9118 Palaiseau Cede, France & LPT, Université Paris-Sud, CNRS, Orsa, France To stud scattering amplitudes at high-energ, the T-product of two currents can e epanded in terms of coefficient functions (impact factors and matri elements of composite color dipoles made of Wilson line operators with rapidit cutoff preserving conformal invariance. In the leading order, the high-energ evolution of color dipoles is governed the non-linear Balitsk- Kovchegov (BK equation. To descrie the high-energ amplitudes in the net-to-leading order (NLO one needs to know the coefficient function ( impact factor and the evolution of corresponding Wilson-line operators. Using the high-energ OPE, we find the net-to-leading order (NLO correction to the BK equation and calculate the impact factor for virtual photons in deep inelastic scattering. PoS(DIS XVIII International Workshop on Deep-Inelastic Scattering and Related Sujects April 19 -, 010 Convitto della Calza, Firenze, Ital Speaker. c Copright owned the author(s under the terms of the Creative Commons Attriution-NonCommercial-ShareAlike Licence.

2 High-energ amplitudes and impact factors at net-to-leading order 1. Introduction Wilson line operators are the effective degrees of freedom for the description of high-energ scattering in gauge theories (for a review, see Ref. [1,. Indeed, at high-energ (Regge limit particles move along their straight-line classical trajector and the onl quantum effect is the eikonal phase factor acquired along this propagation path. In QCD, for fast quarks or gluons scattering off some target, this eikonal phase factor is a Wilson line - an infinite gauge link ordered along the straight line collinear to particle s velocit n µ : { U η ( = Pep ig du n µ A µ (un+ where A µ is the gluon field of the target, is the transverse position of the particle which remains unchanged throughout the collision, and the inde η is the rapidit of the particle. The highenerg ehavior of QCD amplitudes can then e studied in the framework of the evolution of color dipoles. We consider the small- ehavior of structure functions of deep inelastic scattering (DIS: the virtual photon decomposes into quark and antiquark pair which propagate along the straight lines separated transverse distance and forms a color dipole - two-wilson-line operator: (1.1 ˆ U η (, = 1 1 N c tr{û η ( Û η ( (1. The energ dependence of the structure function is translated into the dependence of the color dipole on the rapidit η. Although it appears to e more natural to restrict the rapidit considering the Wilson line with the supporting line collinear to the velocit of the fast-moving particle, we choose to cut the rapidit integrals hand : the method of rigid cutoff in the longitudinal direction is technicall simpler and more efficient in order to get the conformal results. Thus, the small- ehavior of the structure functions is governed the rapidit evolution of color dipoles [, 4. At relativel high energies and for sufficientl small dipoles we can use the leading logarithmic approimation (LLA where α s 1, α s ln B 1 and get the non-linear BK evolution equation for the color dipoles [6, 7: d dη U ˆ η (z 1,z = α sn c π d z 1 z z [ Uˆ η (z 1,z + Uˆ η (z,z 1 z Uˆ η (z 1,z Uˆ η (z 1,z ˆ U η (z,z (1. PoS(DIS where η = ln 1 B and z 1 z 1 z etc. (we denote operators hat. The first three terms in the BK equation correspond to the linear BFKL evolution [5 and descrie the partons emission while the last term is responsile for the partons annihilation. For sufficientl low B the partons emission alances the partons annihilation so the partons reach the state of saturation [8 with the characteristic transverse momentum Q s growing with energ 1/ B (for a review, see [9.. Net-to-leading order photon impact factor In the Regge limit all transverse momenta are of the same order of magnitude and consequentl it is natural to introduce a factorization scale in rapidit: one introduces a rapidit divide

3 High-energ amplitudes and impact factors at net-to-leading order a * * * a * a a (a ( (c (d Figure 1: Leading-order diagrams for the small- evolution of color dipole. Wilson lines are denoted dotted lines. η which separate fast field from slow fields. Thus, the amplitude of the process is given a convolution of contriutions coming from fields with rapidit η < Y ( fast field and contriutions coming from fields with rapidit η > Y ( slow fields. As in the case of the usual Operator Product Epansion (OPE, the integration over the field with rapidit η < Y gives us the coefficients function while the integrations over the field with rapidit η > Y are the matri elements of the operators. Thus, the OPE at high-energ (Regge limit for the T-product of two electromagnetic currents is otained in terms of Wilson lines where T { ĵ µ ( ĵ ν ( = d z 1 d z I LO µν (z 1,z [Tr{Û z η 1 conf + d z 1 d z d z I NLO µν (z 1,z,z [tr{ûz η 1 tr{ûz η N c tr{ûz η 1 (.1 [Tr{Û z η 1 conf = Tr{Û z η 1 + α s π d z 1 z z [Tr{T n Û 1 z z η 1 T n Ûz η N c Tr{Ûz η 1 ln az 1 z 1 z (. PoS(DIS is the composite dipole with the conformal longitudinal cutoff in the net-to-leading order. The appearance of the composite operators is due to the loss of conformal invariance of the Wilson line operator in the NLO. Indeed, the light-like Wilson lines U( are formall Möius invariant and consequentl the leading-order BK equation is also conformal invariant. At NLO the Wilson line operator are divergent and its regularization introduces a dependence on the rapidit and conformal smmetr is lost. In order to restore the conformal invariance we redefine the operator Tr{Û z η 1 adding suitale conterterms. The procedure of finding the dipole with conformall regularized rapidit divergence is analogous to the construction of the composite renormalized local operator adding the appropriate counterterms order order in perturation theor. In equation (.1 the coefficient I LO is the leading-order (LO impact factor which has een known for long time, while

4 High-energ amplitudes and impact factors at net-to-leading order Y > η Y > η Z1 Z 1 Figure : High energ epansion of the T product of two electromagnetic currents I NLO is the NLO impact factor [15 given {[ I NLO µν (, = α snc 8π 7 d z 1 d z U conf 1 (z 1,z Z1 Z µ Z 1 Z ν [ 1 + ln R + 1 R + ν Z 1 Z1 4Z µ ν ln (. [ 1 ln R 1 R 1 [ µ Z 1 Z1 Z ν ln + νz 1 Z1 Z (1 1 1 [ ( µ ln 1 R Z µ Z1 ν R+ ( 1 ν Z1 ( µ R [ 1 ν z 4 1 R + R ( 1 [ µ Z 1 ( + R Z1 Z ν ln + νz 1 Z1 Z ν Z + 4 [4Li Z1 Z (1 R π + ( 1( 1 R ν Z [ + Z1 Z R(1 R 1 R + 4 ν Z 1 [ + Z1 4Z R(1 R 1 R ( µ Z 1 ν ln + νz µ ln [ R(1 R + (z 1 z Z1 Z z 1 Z1 Z Z Z 1 ( µ ln [4Li (1 R π + (ln 1 R + 1 R + 1 R ln 1 R ( R R 4 µ R 1 R µ ν PoS(DIS where (, = + s/, = + s/, R z 1 Z 1 Z Z 1 = ( z 1 + ( z 1, Z = ( z + ( z (.4 Equation (. is the analtic epression for the full NLO impact factor which was not known efore. (A comination of numerical and analtical results can e found in Ref. [16. We plan to perform the Fourier transform in momentum space which will e useful for phenomenological studies. In order to otain the NLO evolution for the DIS amplitude in QCD one needs the NLO evolution equation of color dipoles with respect to rapidit which was found in [14, then solve the corresponding evolution equation, and finall assemle the result for structure functions: take the 4

5 High-energ amplitudes and impact factors at net-to-leading order initial conditions at low energ (rapidit, evolve color dipoles to higher rapidit and multipl the result the corresponding impact factor. The work is in progress. In Ref. [1 the full program for the calculation of the NLO evolution amplitude is performed for the N = 4 SYM theor for two BPS-protected currents. The author is grateful to the organizers of DIS 010 and in particular to D. Colferai for financial support. This work is supported the grant ANR-06-JCJC References [1 I. Balitsk, High-Energ QCD and Wilson Lines, In *Shifman, M. (ed.: At the frontier of particle phsics, vol. *, p (World Scientific, Singapore, 001 [hep-ph/ [ I. Balitsk, arxiv: [hep-ph. [ A.H. Mueller, Nucl. Phs. B415, 7 (1994; A.H. Mueller and Bimal Patel, Nucl. Phs. B45, 471 (1994. [4 N.N. Nikolaev and B.G. Zakharov, Phs. Lett. B, 184 (1994; Z. Phs. C64, 61 (1994; N.N. Nikolaev B.G. Zakharov, and V.R. Zoller, JETP Letters 59, 6 (1994. [5 V.S. Fadin, E.A. Kuraev, and L.N. Lipatov, Phs. Lett. B 60, 50 (1975; I. Balitsk and L.N. Lipatov, Sov. Journ. Nucl. Phs. 8, 8 (1978. [6 I. Balitsk, Nucl. Phs. B46, 99 (1996; [hep-ph/ ; [7 Yu.V. Kovchegov, Phs. Rev. D60, (1999; Phs. Rev. D61, (000. [8 L.V. Griov, E.M. Levin, and M.G. Rskin, Phs. Rept. 100, 1 (198, A.H. Mueller and J.W. Qiu, Nucl. Phs. B68, 47 (1986; A.H. Mueller, Nucl. Phs. B5, 115 (1990. [9 E. Iancu and R. Venugopalan, In *Hwa, R.C. (ed. et al.: Quark gluon plasma* 49-6, [e-print: hep-ph/0004; H. Weigert, Prog.Part.Nucl.Phs. 55, 461 (005; J. Jalilian-Marian and Yu.V. Kovchegov, Prog.Part.Nucl.Phs. 56, 104 (006. [10 I. I. Balitsk, Phs. Lett. B 14, 0 (198. [11 A. V. Efremov and A. V. Radushkin, Phs. Lett. B 94, 45 (1980. G. P. Lepage and S. J. Brodsk, Phs. Rev. D ( M. K. Chase, Nucl. Phs. B 174, 109 (1980. T. Ohrndorf, Nucl. Phs. B 186, 15 (1981. [1 I. I. Balitsk and V. M. Braun, Nucl. Phs. B 11, 541 (1989. [1 I. Balitsk and G. A. Chirilli, Nucl. Phs. B 8 ( [arxiv: [hep-ph. [14 I. Balitsk and G. A. Chirilli, Phs. Rev. D 77 ( [arxiv: [hep-ph. [15 I. Balitsk and G. A. Chirilli, In preparation. [16 J. Bartels and A. Krieleis, Phs. Rev. D70,11400(004; J. Bartels, D. Colferai, S. Gieseke, and A. Krieleis, Phs. Rev. D66, (00. J. Bartels, S. Gieseke, and A. Krieleis, Phs. Rev. D65, (00. PoS(DIS