Small x QCD and Multigluon States: a Color Toy Model
aa r X i v : . [ h e p - t h ] O c t Small x QCD and Multigluon States: a Color Toy Model
G.P. Vacca † and P. L. Iafelice INFN - Sez. di Bologna and Univ. of Bologna, Physics Depart., Via Irnerio 46, 40126 Bologna
Abstract
We introduce and study a toy model with a finite number of degrees offreedom whose Hamiltonian presents the same color structure of theBKP system appearing in the studies of QCD in the Regge limit. Weaddress within this toy model the question of the importance of finite N c corrections with respect to the planar limit case. The large N c expansion [1] is a widely popular framework of approximations which has beensuccesfully applied to gauge theories and has given at leading order some analytical results oth-erwise impossible to obtain. Within the Regge limit of QCD scattering amplitudes, L.N. Lipatovfound [2] that systems of reggeized gluons evolving in rapidity in the leading logarithmic approx-imation (LLA) were showing the emergence of an integrable structure in the planar limit. Similarfeature were found later in other kinematical regimes for other QCD observables. Moreover the N = 4 SYM theory has been investigated at different order in perturbation theory and is nowbelieved to be integrable at all orders.But if one considers some QCD observables at the physical point N c = 3 the situation ismuch more complicated and even the order of the corrections with respect to the planar limit arenot really known. This is the situation, for example, for the spectrum of the BKP kernel [3, 4] atone loop, which describes the high energy behavior in the Regge limit of a system of reggeizedgluons.It is the purpose of this talk to discuss a toy model [5] which has a color structure similarto the BKP system but a different “configuration” dynamics with a finite number of d.o.f., con-strained only by the fact that the two Hamiltonians must have the same leading eigenvalues inthe large N c limit for both one and two cylinder topologies. The main motivation to study thismodel is to understand in a simpler case how much the large N c approximation fails to reproducethe dynamics at finite N c . In order to understand this we shall study the spectrum of such modelas a function of N c . x QCD: the LLA BKP kernel
Let us start by giving a brief overview of the LLA kernels encoding the evolution in rapidity ofsystems of interacting reggeized gluons, which provide a convenient perturbative description ofsome relevant QCD degrees of freedom in the Regge limit (small x ). Their dynamics determinethe high energy behavior of the cross sections, typically associated to the so called BFKL (per-turbative) pomeron [6, 7]. In the simplest form, the BFKL pomeron turns out to be a composite † Talk presented at EDS07 tate of two interacting reggeized gluons “living” in the transverse configuration plane in thecolorless configuration. Its kernel or Hamiltonian is infrared finite and in LLA is constructedsumming the perturbative contributions of different Feynman diagrams: in particular the virtualones (reggeized one loop gluon trajectories) ω and the real ones (associated to an effective realgluon emission vertex) V . One writes formally H = ω + ω + ~T ~T V where ~T i are the gen-erators of the color group in adjoint representation . In the colorless case one has ~T ~T = − N c and finally one obtains: H = ln | p | + ln | p | + 1 p p ∗ ln | ρ | p p ∗ + 1 p ∗ p ln | ρ | p ∗ p − , (1)where Ψ( x ) = d ln Γ( x ) /dx , a factor ¯ α s = α s N c /π has been omitted and the gluon holo-morphic momenta and coordinates have been introduced. One has the freedom, because ofgauge invariance to choose a description within the M¨obius space [8–10]. Then the BFKLhamiltonian has the property of the holomorphic separability ( H = h + ¯ h ) and is in-variant under the M¨obius group SL (2 , C ) transformations, whose generators for the holomor-phic sector in the M¨obius space for the principal series of unitary representations are given by M r = ρ r ∂ r , M + r = ∂ r , M − r = − ρ r ∂ r . The associated Casimir operator for two gluons is M = | ~M | = − ρ ∂ ∂ where ~M = P r =1 ~M r and ~M r ≡ ( M + r , M − r , M r ) . Note that, afterdefining formally J ( J −
1) = M , one may write h = ψ ( J ) + ψ (1 − J ) − ψ (1) .The eigenstates and eigenvalues of the full hamiltonian in eq. (1), H E h, ¯ h = 2 χ h E h, ¯ h are labelled by the conformal weights h = n + iν , ¯ h = − n + iν . The leading eigenvalue, atthe point n = ν = 0 , has a value χ max = 4 ln 2 ≈ . , responsible for the rise of the totalcross section as s ¯ α s χ max , which corresponds to a strong violation of unitarity.Let us now consider the evolution in rapidity of composite states of more than reggeizedgluons [3, 4]. The BKP Hamiltonian in LLA, acting on a colorless state, can be written in termsof the BFKL pomeron Hamiltonian and has the form (see [2]) H n = − N c X ≤ k 10+ ¯10 + P + P = P i P [ R i ] , where T rP [ R i ] = d i is the dimension of the corresponding representation and we consider a unique subspace for the and ¯10 representations. This is convenient for our purposes and we shall therefore work with different projectors to span the color space of two gluons.On considering gluons (1 , to be the reference channel we introduce as the base for thecolor vector space the set { P [ R i ] a a a a } of projectors and write v a a a a = X i v i (cid:0) P [ R i ] a a a a (cid:1) or v = X i v i P [ R i ] . (5)Having chosen a color basis, the next step is to write the BKP kernel with respect to it. Since P i ~T i v = 0 one may finally obtain: H = − N c h ~T ~T ( H + H ) + ~T ~T ( H + H ) + ~T ~T ( H + H ) i . (6)Let us now write explicitely the action of the color operators ~T i ~T j = P a T ai T aj which are as-sociated to the interaction between the gluons labelled i and j . We start from the simple “di-agonal channel” for which we have relation ~T i ~T j = − P k a k P ij [ R k ] with coefficients a k =( N c , N c , N c , , − , . Consequently we can write in the (1 , reference base (cid:16) ~T ~T v (cid:17) j = − a j v j = − ( A v ) j , (7)where A = diag ( a k ) . The action on v of the ~T ~T and ~T ~T operators is less trivial and isconstructed in terms of the j symbols of the adjoint representation of SU ( N c ) group: (cid:16) ~T ~T v (cid:17) j = − X i X k C jk a k C ki ! v i = − ( CA C v ) j (8)and (cid:16) ~T ~T v (cid:17) j = − X i X k s j C jk a k C ki s i ! v i = − ( SCA CS v ) j . (9)The matrix C is the symmetric crossing matrix build on the j symbols and S = diag ( s j ) isconstructed on the parities s j = ± of the different representations R j .e can therefore write the general BKP kernel for a four gluon state, given in eq. (6), as H = 1 N c [ A ( H + H ) + CAC ( H + H ) + SCACS ( H + H )] (10)One can check that if we make trivial the transverse space dynamics, replacing the H ij operatorsby a unit operators, the general BKP kernel in eq. (2) becomes H n = n ˆ1 and indeed one canverify that A + CAC + SCACS = N c ˆ1 .Let us make few considerations on the large N c limit approximation. As we have alreadydiscussed, in the Regge limit one faces the factorization of an amplitude in impact factors anda Green’s function which exponentiates the kernel. The topologies resulting from the large N c limit depend on the impact factor structure. In particular one expects the realization of twocases: the one and two cylinder topologies. The former corresponds to the case, well stud-ied, of the integrable kernel, Heisenberg XXX spin chain-like. It is encoded in the relation: ~T i ~T j → − N c δ i +1 ,j which leads to H = ( H + H + H + H ) . It is characterized byeigenvalues corresponding to an intercept less then a pomeron. The latter case instead is expectedto have a leading intercept, corresponding to an energy dependence given by two pomeron ex-change. Consequently one expects at finite N c a contribution with an energy dependence evenstronger. In the two cylinder topology the color structure is associated to two singlets ( δ a a δ a a ,together with the other two possible permutations). Such a structure is indeed present in the anal-ysis, within the framework of extended generalized LLA, of unitarity corrections to the BFKLpomeron exchange [21] and diffractive dissociation in DIS [22], where the perturbative triplepomeron vertex (see also [23, 24]) was discovered and shown to couple exactly to the four gluonBKP kernel.It is therefore of great importance to understand how much the picture derived in the planar N c = ∞ case is far from the real situation with N c = 3 . One clearly expects for example thatthe first corrections to the eigenvalues of the BKP kernel are proportional to /N c , but what isunknown is the multiplicative coefficient as well as the higher order terms. In this section we shall consider a toy model [5], different from the BKP system, but sharingseveral features with it and analysis within it if the large N c approximation might be more or lesssatisfactory.Besides the color space, a state of n reggeized gluons undergoing the BKP evolution be-longs to the configuration space R n , associated to the position or momenta in the transverseplane of the n gluons. The operators H kl act (see eq. (10)) on such a state and, on the M¨obiusspace, can be written in terms of the Casimir of the M¨obius group, i.e. in terms of the scalarproduct of the generators of the non compact spin group SL (2 , C ) : H kl = H kl ( ~M k · ~M l ) .We are therefore led to consider a class of toy models where the BKP configuration space R n is substituted by the space V ns where V s is the finite space spanned by spin states belongingto the irreducible representation of SU (2) with spin s . In particular we shall consider quantumystems with an Hamiltonian: H n = − N c X ≤ k 12 + q − α (4 + 2 x ) (cid:19)(cid:21) − ψ (1) . (12)This form is suggested by the conformal spin n = 0 BFKL Hamiltonian with the substitution + L ij → − α S ij which assures to have the same leading eigenvalue for any α , since bothexpressions have the value zero as upper bound. The parameter α will be chosen in order toconstrain the full 4-particle Hamiltonian (11) to have the same leading eigenvalue as the QCDBKP system in the large N c limit an one cylinder topology (at zero conformal spin), given in eq.(4). This “BKP toy model” will be used to investigate finite N c effects.Since we have chosen to work with states singlet under SU (2) spin conf also for the spinpart we employ the 2 particle subchannel decomposition in irreducible representations, in away similarly adopted for the color part. After that one is left with the problem of diagonal-izing an Hamiltonian which is a matrix × . Therefore we proceed by introducing for2 particle spin states the resolution of unity Q + Q + Q = P i Q [ R i ] which letus write f ( ~S i ~S j ) = P k f ( − b k ) Q ij [ R k ] with b k = (2 , , − . using a power series repre-sentation ( Q ij [ R k ] are projectors). Introducing the crossing matrices D and the parity ma-trix S ′ we obtain the relations (cid:16) f (cid:16) ~S ~S (cid:17) v (cid:17) j = ( B v ) j , (cid:16) f (cid:16) ~T ~T (cid:17) v (cid:17) j = ( DB D v ) j and (cid:16) f (cid:16) ~T ~T (cid:17) v (cid:17) j = ( S ′ DB DS ′ v ) j . It is then straightforward to derive a matrix form for theHamiltonian of this toy model H a = 2 N c (cid:0) A ⊗ B + CA C ⊗ DB D + SCA CS ⊗ S ′ DB DS ′ (cid:1) (13)which depends on N c and on the parameter α through the function f α given in eq. (12).In the large N c limit one faces for the Hamiltonian two possible cases (see [5] for more de-tails): the one cylinder topology (1CT) which corresponds to the simpler Hamiltonian H CT a = B + S ′ DBDS ′ and the two cylinder topology (2CT) corresponding to the even simpler Hamil-tonian H CT a = 2 B . Let us remark that while in the case of N c > we consider a basis for thevector states made of P [ R i ] Q [ R j ] with elements since in the color sector there is also the P projector, the case N c = 3 is characterized by a basis of elements.The last step to obtain the BKP toy model is to fix the parameter α by requiring H CT a tohave the value of eq. (4) so we obtain α = 2 . . We are therefore left with an Hamiltonianwhich is just a function of the number of colors N c .Let us now consider its spectrum for the cases N c = 3 and N c = ∞ . Here we report justthe leading eigenvalues of H a with their multiplicities: ( . , × . , × . , · · · ). Chang-ing N c from to ∞ we observe that the first three move to the 2CT leading eigenvalue . hile the next two move to the 1CT leading eigenvalue . . With very good approximationone finds that the N c dependence of the leading eigenvalue E is given by E ( N c ) = E ( ∞ ) (cid:18) . N c (cid:19) . (14)One can see that for this toy model the large N c approximation corresponds to an error of about , an error which is not negligible because the coefficient of the leading correction to theasymptotic value, proportional to /N c , is a large number. The color- “spin” configuration mix-ing which is encoded in the eigenvectors has been also studied. We have introduced a family of dynamical models describing interacting particles with colorand spin degrees of freedom in order to see how much the large N c approximation is significantwhen one is trying to extract the spectrum of these quantum systems. 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