Proton elastic impact factors for two, three, and four gluons
aa r X i v : . [ h e p - ph ] A ug Proton elastic impact factors for two, three, andfour gluons
Leszek Motyka , ∗
1- II Institute for Theoretical Physics, University of Hamburg,Luruper Chaussee 149, D-22761, Germany2- Institute of Physics, Jagellonian UniversityReymonta 4, 30-059 Krak´ow, PolandIn this talk [1] we report on recent calculation of high energy baryon scattering am-plitudes in QCD. Elastic baryon impact factors for two, three and four gluons arepresented and their energy evolution is described that incorporates unitarity correc-tions. We find that the baryon couples directly to the BFKL Pomeron, to the BKPodderon and to a new state, a three-gluon BKP Pomeron. The new state may decayinto four gluons with a new 3 → The complete picture of high energy scattering of hadrons in QCD has to incorporate effectsof multiple scattering. In particular, when those effects are properly taken into account,they solve the problem of the rapid increase of Balitsky-Fadin-Kuraev-Lipatov (BFKL) [2]amplitudes with energy, that would eventually lead to a violation of the S -matrix unitar-ity. So far, the most successful realization of the unitarization of BFKL amplitudes wasperformed within the Balitsky-Kovchegov (BK) framework [3], for the scattering of deeplyvirtual photon, γ ∗ , on a large nucleus. This scheme relies on the dipole-like nature of thehard probe and on the large N c limit. Unfortunately, the BK formalism is not sufficient tosolve an important problem of the high energy baryon scattering. Baryons contain at least N c constituents with a non-trivial color connection, and consequently the large N c -limit forthe baryon wave function is much more complex than it was for the γ ∗ (or a color dipole)projectile. This obstacle prohibited the direct solution of the baryon scattering problemwithin the s -channel approach [4].Recently we attempted to analyze the baryon scattering amplitude within the t -channelframework [5], for N c = 3. The formalism applied is derived in perturbative QCD and itresums to all orders the leading logarithmic contributions ( α s ln( s )) n of the collision energysquared, s . It generalizes the Bartels-Kwieci´nski-Prasza lowicz (BKP) evolution scheme [6, 7]valid for a fixed number of reggeized gluons in the t -channel, by inclusion of integral kernelsthat change the number of gluons in the t -channel [6]. In particular, in this approach thetriple-Pomeron vertex was obtained [8] that was shown to match in the large N c limit thevertex defining the BK equation. In this talk [1] results of the t -channel analysis of baryonscattering amplitudes are presented: the baryon impact factors and their small- x evolutionare described. ∗ The support of the DFG grant no. SFB676 is gratefully acknowledged.
DIS 2008 p [k ] ... P j p p +l
32 2 p +lp +l P’ Figure 1: Baryon impact factor.
The baryon impact factors for n gluons are defined by multiple discontinuities of the elasticscattering amplitudes of baryons mediated by n -gluons, see Fig. 1. They are evaluatedin the high energy limit, when only the leading power of the large light-cone momentumcomponent is retained. The impact factors are given by the following formula, B n ;0 ( lll , lll , lll ) = I ( n ) qq X diagrams F ( lll , lll , lll ) C n (diagram) , (1)where the factor I ( n ) qq = ( − ig ) n represents the quark scattering amplitude (without thecolor factor). The overlap of the outgoing and incoming baryon wave functions is decomposedfor each diagram into the color dependent factor C n (diagram), and the momentum dependentfactor, F ( lll , lll , lll ). Note, that the impact factor depends only on the total momentumtransfers, lll , lll and lll to quark lines 1, 2, and 3 respectively. The color part of the baryonwave function is given by the fully antisymmetric tensor, ǫ αβγ . Using the master formula (1)we performed the sum over all relevant diagrams and evaluated the baryon impact factorsup to four external gluons, both in the C -even ( B n ;0 ) and the C -odd ( ˜ B n ;0 ) channel.For two gluons, with transverse momenta kkk and kkk , and color indices a and a , onlythe Pomeron contributes. B can be then represented as a sum of three pieces, B ( kkk , kkk ) = δ a a h D { , } (1 ,
2) + D { , } (1 ,
2) + D { , } (1 , i , (2)where we used a short-hand notation D { , } (1 , ≡ D { i,j } ( kkk , kkk ), D { , } (12 , ≡ D { i,j } ( kkk + kkk , D { i,j } only quarks i and j scatter and the third quark is aspectator. All D { i,j } have the momentum structure of the color dipole impact factor, e.g. D { , } (1 ,
2) = − g
12 [ F (12 , ,
0) + F (0 , , − F (1 , , − F (2 , , , (3)and similarly for D { , } and D { , } .For three gluons in the C -even channel, the impact factor can be decomposed into dipole-like components in an analogous way, B = D { , } + D { , } + D { , } , and the dipole-likecomponents have the reggeizing form known from the color dipole case, D { i,j } (1 , ,
3) = 12 g f a a a h D { i,j } (12 , − D { i,j } (13 ,
2) + D { i,j } (23 , i , (4)In the odderon channel, we see a distinct color-momentum structure,˜ B ( kkk , kkk , kkk ) = d a a a E (1 , , , (5) DIS 2008 here for a symmetric function F (1 , ,
3) and for kkk + kkk + kkk = 0, E takes the form, E (1 , ,
3) = ig F (1 , , − X j =1 F ( j, − j,
0) + F (0 , , . (6)For the case of four gluons ( C = +) we see the further emergence of the gluon reggeizationpattern for each of the dipole-like components, but in addition, a new structure, Q ,appears: B = D { , } + D { , } + D { , } + Q , (7)where D { i,j } have the functional form known from the color dipole case, and Q = − ig (cid:20) d a a b d ba a − δ a a δ a a (cid:21) [ E (12 , ,
4) + E (34 , ,
2) ] + − ig (cid:20) d a a b d ba a − δ a a δ a a (cid:21) [ E (13 , ,
4) + E (24 , ,
3) ] + − ig (cid:20) d a a b d ba a − δ a a δ a a (cid:21) [ E (14 , ,
3) + E (23 , ,
4) ] . (8)In the case of the odderon, for four gluons, one finds that the impact factor is fullyexhausted by the reggeizing contribution, that is ˜ B may be obtained from ˜ B by allpossible splittings of a single gluon into two elementary gluons with the color tensor f abc ,cf. Eq. (4). We demonstrated that the basic objects (modulo reggeization) defining the baryon impactfactor are the dipole-like components, D { i,j } , and the functions E and Q . All thesefunctions vanish if one of the gluon transverse momenta vanishes. They are also fullysymmetric under permutations of the gluon momenta (Bose invariance). Thus, they areproper initial conditions for the BKP evolution: D { i,j } for the BFKL Pomeron, and E forthe BKP odderon spanned by three reggeized gluons. Q may be interpreted as an initialcondition for a C -even three-Reggeon state, where one of the reggeized gluons has the evensignature.In order to analyze the structure of unitarity corrections in the t -channel approach, onehas to go beyond the Reggeon number conserving BKP equation, and include integral kernelsdescribing splittings of 2 → n Reggeons. Then, the small- x evolution of impact factors isgiven by a set of coupled integral equations with the initial conditions given by B n ;0 and˜ B n ;0 . We solved these integral equations for the baryon up to four gluons. In the odderonchannel we found only the BKP evolution preserving the color-momentum structure of E .In the Pomeron channel the situation is more complex. The dipole-like pieces obey theBFKL evolution, preserving their color-momentum structure, but in addition a transitionmay occur to a four-Reggeon state, that may be projected on two BFKL Pomerons (Fig. 2).The amplitude of this transition is given by the V → vertex (related to the triple-Pomeronvertex), well known from the analysis of γ ∗ scattering. In addition, we found the BKPevolution of the three-Reggeon state, Q . The state Q , however, may also decay into four DIS 2008 df f ff f DD Q f f fVf f DDf fD
Figure 2: Transition vertices W → (left) and V → (right); reggeized gluons with the odd( f ) and the even ( d ) signature are indicated.Reggeons with the amplitude given by a new vertex, W → , that may be also be interpretedas a triple Pomeron vertex, but with the three-Reggeon BKP Pomeron, Q , that splitsinto two BFKL Pomerons, see Fig. 2. We point out that the possible direct two-Pomeroncoupling to the baryon was not found. The lack of the direct two-Pomeron coupling, however,essentially relies on taking in account only the lowest Fock component of the baryon, and itholds only in the leading logarithmic ln( s ) approximation. We have analyzed the high energy scattering of a baryon projectile. The baryon, representedby three constituent quarks, was found to couple to the BFKL Pomeron, the BKP (three-Reggeon) odderon and a new state, a BKP Pomeron spanned on three Reggeons, out ofwhich one has an even signature. The BFKL Pomeron may couple to one of dipole-likepieces of the baryon. Each dipole-like component of the baryon has the color-momentumstructure of the genuine color dipole. The evolution of those states was analyzed up tofour reggeized gluons in the t -channel. The dipole-like components were found to evolvein the same way as the color dipoles. Specifically, their evolution is driven by the BFKLequation, followed by a possible splitting of the BFKL Pomeron into four reggeized gluons(two Pomerons). The three-Reggeon Pomeron obeys the BKP equation and it may splitinto four reggeized gluons. This transition is driven by a new 3 → References [1] Slides: http://indico.cern.ch/materialDisplay.py?contribId=32&sessionId=16&materialId=slides&confId=24657 [2] L. N. Lipatov, Phys. Rept. (1997) 131.[3] I. Balitsky, Nucl. Phys. B (1996) 99; Y. V. Kovchegov, Phys. Rev. D (1999) 034008.[4] M. Prasza lowicz and A. Rostworowski, Acta Phys. Polon. B (1998) 745.[5] J. Bartels and L. Motyka, Eur. Phys. J. C (2008) 65.[6] J. Bartels, Nucl. Phys. B (1979) 293; Nucl. Phys. B (1980) 365.[7] J. Kwieci´nski and M. Prasza lowicz, Phys. Lett. B (1980) 413.[8] J. Bartels, Z. Phys. C (1993) 471.(1993) 471.