BFKL Pomeron calculus: nucleus-nucleus scattering
aa r X i v : . [ h e p - ph ] D ec Preprint typeset in JHEP style - HYPER VERSION
TAUP 2939/11November 15, 2018
BFKL Pomeron calculus: nucleus-nucleus scattering
Carlos Contreras a ∗ , Eugene Levin a,b † and Jeremy S. Miller b,c ‡ a Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Avda. Espa˜na 1680 and CentroCient´ıfico-Tecnol ´ o gico de Valpara´ıso, Casilla 110-V, Valparaiso, Chile b Department of Particle Physics, School of Physics and Astronomy, Tel Aviv University, Tel Aviv,69978, Israel c CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico (IST), Av. Rovisco Pais, 1049-001Lisboa, Portugal
Abstract:
In this paper the action of the BFKL Pomeron calculus is re-written in momentum represen-tation, and the equations of motion for nucleus-nucleus collisions are derived, in this representation. Wefound the semi-classical solutions to these equations, outside of the saturation domain. Inside this domainthese equations reduce to the set of delay differential equations, and their asymptotic solutions are derived.
Keywords:
BFKL Pomeron calculus, semi-classical approach, Pomeron action, equations of motion.
PACS: 13.85.-t, 13.85.Hd, 11.55.-m, 11.55.Bq ∗ Email: [email protected] † Email: [email protected]., [email protected] ‡ Email: [email protected] ontents
1. Introduction 12. The BFKL Pomeron Calculus 2 S S I and the Field equation 52.3 Classical equations 7
3. Semiclassical solution 9
4. Solution inside the saturation domain 16
5. Conclusions 196. Acknowledgements 19
1. Introduction
High energy QCD has reached a mature stage of development, in its description of dilute-dense scattering.Deep inelastic scattering with nuclei provides a good example of this. The main physical phenomenathat emerges for this type of scattering has been discussed, and the non-linear equations that govern suchprocesses have been derived and discussed in detail [1–8]. On the other hand, the scattering of the densesystem of partons with the dense system of partons has been actively studied [9–15], but with limited– 1 –uccess. This is in spite of the fact that this scattering is closely related to nucleus-nucleus scattering,which is the source of most of the experimental information, for the dense parton system.At the moment there exist two general approaches to high energy QCD: the BFKL Pomeron calculus[1, 2, 9, 16, 17], and the Colour Glass Condensate approach (CGC) [3, 7], which lead to the same non-linearequations [5, 6] for dilute-dense scattering. The interrelation between these two approaches is not clearat the moment. However, the equations that describe nucleus-nucleus collisions have not been derived, inspite of considerable progress made in this direction [10, 12, 13], while in the BFKL Pomeron calculus, suchequations have been proposed in Ref. [9]. These equations have been on the market for some time, butunfortunately, only three attempts to solve them are available in Refs. [18–20].The main goal of this paper is to find the solution to these equations. In the next section we re-derivethe equation of Ref. [9] in momentum representation, which turn out to be the most economical way offinding the solution. In section 3 we will find the semi-classical solutions to the equation, which describedense-dense scattering outside of the saturation region. Section 4 is devoted to finding the solution insideof the saturation region. In the conclusion section we summarize our results.
2. The BFKL Pomeron Calculus
The goal of this section is to find the equation for nucleus-nucleus scattering in the momentum represen-tation based on the BFKL Pomeron calculus, based on the main idea of Ref. [9] that the equation fornucleus-nucleus collisions can be found from the equation of motion for Pomerons. S The BFKL Pomeron calculus can be written through the functional integral [9] Z [Φ , Φ + ] = Z D Φ D Φ + e S with S = S + S I + S E (2.1)where S describes free Pomerons, S I corresponds to their mutual interaction while S E relates to theinteraction with the external sources (target and projectile). Here the free action is given by: S = Z dY ′ Z d x d x Φ † (cid:0) x , x , Y ′ (cid:1) ∇ ∇ [ ∂∂Y + H ]Φ (cid:0) x , x , Y ′ (cid:1) (2.2)Define the following Fourier transform,Φ † (cid:0) x , x , Y ′ (cid:1) = Φ † (cid:0) x , b, Y ′ (cid:1) = x Z d k e − ik · x Φ † (cid:0) k , b, Y ′ (cid:1) (2.3)Φ (cid:0) x , x , Y ′ (cid:1) = Φ (cid:0) x , b, Y ′ (cid:1) = x Z d k e ik · x Φ (cid:0) k , b, Y ′ (cid:1) (2.4)Consider the free action as the sum of two terms:– 2 – = S ′ + S ′′ (2.5)where: S ′ = Z dY ′ Z d x d x Φ † (cid:0) x , x , Y ′ (cid:1) ∇ ∇ ∂∂Y Φ (cid:0) x , x , Y ′ (cid:1) (2.6)This can be re-written as: S ′ = 4 Z dY ′ Z d b d x Z d k d k e ik · x Φ † (cid:0) k , b, Y ′ (cid:1) x ∇ ∇ (cid:18) x e − ik · x ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) (cid:19) (2.7)where in the last step the replacement d x d x = 4 d b d x , where, b = x + x x = x − x (2.8)where b is the the impact parameter. In two-dimensional polar coordinates: ∇ k = ∂ ∂ k + 1 k ∂∂ k + 1 k ∂∂ θ (2.9)Let us introduce the next change of variable l = ln k , such that the two-dimensional Laplacian ∇ k with respect to k , simplifies to the following differential operator: ∇ k = 4 e − l k ∂ ∂l + e − l k ∂∂θ (2.10)According to the definition of Eq. (2.10), the next term x ∇ x ∇ x ( x e − ik · x ) can be recast as, x ∇ x ∇ x (cid:16) x e − ik · x (cid:17) = − x ∇ k ∇ x e − ik · x = − x ∇ k (cid:16) k e − ik · x (cid:17) = ∇ k (cid:16) k ∇ k e − ik · x (cid:17) = 16 (cid:18) ∂∂l + 1 (cid:19) + ∂∂θ ! (cid:18) ∂ ∂l + ∂∂θ (cid:19) e − ik · x (2.11)– 3 –here ∇ k is the two-dimensional Laplacian derivative with respect to k . Inserting Eq. (2.11) backinto Eq. (2.7), leads to the expression: S ′ = 4 Z dY ′ Z d b d x Z d k d k (2.12) × e ik · x Φ † (cid:0) k , b, Y ′ (cid:1) n ∇ k (cid:16) k ∇ k e − ik · x (cid:17)o ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) = 32 Z dY ′ Z d b d x Z d k dθdl e l (2.13) × e ik · x Φ † (cid:0) k , b, Y ′ (cid:1) ( (cid:18) ∂∂l + 1 (cid:19) + ∂∂θ ! (cid:18) ∂ ∂l + ∂∂θ (cid:19) e − ik · x ) ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) where in the last step, the new variable l was introduced. We use the convention that derivatives onlyact on terms inside of the curly brackets {} . Hence in Eq. (2.13) derivatives only act on e − ik · x . Assumingthat the θ -dependent part of the integrand is a purely periodic function of θ , then the derivatives can bere-ordered through integration by parts, to yield: S ′ = 32 Z dY ′ Z d b d x Z d k dθdl e l e ik · x e − ik · x (2.14) × Φ † (cid:0) k , b, Y ′ (cid:1) ( (cid:18) ∂∂l + 1 (cid:19) + ∂∂θ ! (cid:18) ∂ ∂l + ∂∂θ (cid:19) ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) ) = 32 Z dY ′ Z d b d x Z d k dθdl e l e ik · x e − ik · x (2.15) × Φ † (cid:0) k , b, Y ′ (cid:1) ( (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) ) where in the last step it was assumed that Φ ( k , b, Y ′ ) is purely a function of k = e l and not a functionof the angular coordinate θ , such that θ -derivatives vanish. Now returning to the integration variables R dθdl e l = 2 R d k where l = ln k , and integrating over x leads to the delta function (2 π ) δ ( k − k ).The delta function is absorbed by the integral over k -space resulting in the expression: S ′ = 64 (2 π ) Z dY ′ Z d b Z d k Φ † (cid:0) k , b, Y ′ (cid:1) ( (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l ∂∂Y Φ (cid:0) k , b, Y ′ (cid:1) ) (2.16)where l = ln k . This part of the action gives the following contribution to the equation of motion,which stems from the condition δS /δ Φ † ( k, b, Y ) = 0:– 4 – S ′ /δ Φ † ( k, b, Y ) = 64(2 π ) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l ∂∂Y Φ (cid:0) k, b, Y ′ (cid:1) (2.17)Recall that in the above calculation, for the sake of simplicity we considered S = S ′ + S ′′ , and we calcu-lated the Fourier transform of the S ′ part. The full variation δS /δ Φ † ( k, b, Y ) = δ ( S ′ + S ′′ ) /δ Φ † ( k, b, Y )is given by the expression: δS /δ Φ † ( k, b, Y ) = 64(2 π ) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l (cid:18) ∂∂Y − H (cid:19) Φ (cid:0) k, b, Y ′ (cid:1) (2.18)The general properties of the Hamiltonian H have been discussed in Refs. [9, 17] in coordinate repre-sentation, which read as follows in momentum space representation: Hf ( k, Y ) = ¯ α S π Z d l K ( k, l ) { f ( k, l, Y ) − f ( k, Y ) } (2.19) K ( k, l ) = k ⊥ l ⊥ ( ~k − ~l ) ⊥ (2.20) S I and the Field equation The mutual interaction of the Pomerons is described by S I . The interaction which is related to the externalsources (target and projectile) are not consider in this paper. In this approach S I is given by: S I = 2 π ¯ α S N c Z dY ′ Z d x d x d x x x x n(cid:0) L Φ (cid:0) x , x , Y ′ (cid:1)(cid:1) Φ † (cid:0) x , x , Y ′ (cid:1) Φ † (cid:0) x , x , Y ′ (cid:1) (2.21)+ (cid:16) L Φ † (cid:0) x , x , Y ′ (cid:1)(cid:17) Φ (cid:0) x , x , Y ′ (cid:1) Φ (cid:0) x , x , Y ′ (cid:1)o = 2 π ¯ α S N c Z dY ′ Z d b d x d x x x x n(cid:0) L Φ (cid:0) x , x , Y ′ (cid:1)(cid:1) Φ † (cid:0) x , x , Y ′ (cid:1) Φ † (cid:0) x , x , Y ′ (cid:1) (2.22)+ (cid:16) L Φ † (cid:0) x , x , Y ′ (cid:1)(cid:17) Φ (cid:0) x , x , Y ′ (cid:1) Φ (cid:0) x , x , Y ′ (cid:1)o where in the last step the replacement d x d x = 4 d bd x , (see Eq. (2.8)), and where the followingdifferential operator was introduced: L = x ∇ x ∇ x (2.23)– 5 –ow the above defined Fourier transform of Eqs. (2.3) and (2.4) are inserted into Eq. (2.22), whereits assumed that b ≫ x , where b = ( x + x ) /
2, i.e. the impact parameter is much bigger than the sizeof the dipole. Inserting the Fourier transforms of Eqs. (2.3) and (2.4) into Eq. (2.22) gives, S I = 2 π ¯ α S N c Z dY ′ Z d b d x d x Z d k d k d k (2.24) ( exp ( ik · x + ik · x ) L (cid:0) x e − ik · x (cid:1) x ! Φ (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) + exp ( − ik · x − ik · x ) L (cid:0) x e ik · x (cid:1) x ! Φ † (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1)) The integration over x leads to the Dirac delta function (2 π ) δ ( k − k ), where δ ( k − k ) labelsthe delta function in two dimensions. The delta function is absorbed by the integration over k which leadsto: S I = 2 π ¯ α S N c (2 π ) Z dY ′ Z d b d x Z d k d k (2.25) ( exp ( − ik · x ) L (cid:0) x e − ik · x (cid:1) x ! Φ (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) + exp ( ik · x ) L (cid:0) x e ik · x (cid:1) x ! Φ † (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1)) According to the definition of Eq. (2.23), the term L (cid:0) x exp ( − ik · x ) (cid:1) / x can be recast as, L (cid:0) x e − ik · x (cid:1) x = x ∇ x ∇ x (cid:16) x e − ik · x (cid:17) (2.26)= ∇ k (cid:16) k ∇ k e − ik · x (cid:17) = 16 (cid:18) ∂∂l + 1 (cid:19) + ∂∂θ ! (cid:18) ∂ ∂l + ∂∂θ (cid:19) e − ik · x (2.27)where l = ln k and where the last expression was discussed and introduced in the first section (seeEq. (2.11)). Inserting this result into the interaction action of Eq. (2.25), after some algebra one arrives atthe expression: – 6 – I = 16 2 π ¯ α S N c (2 π ) Z dY ′ Z d b d x Z d k d k (2.28) ( exp ( − i ( k + k ) · x ) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) + exp ( i ( k + k ) · x ) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ † (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) k , b, Y ′ (cid:1)) where it was assumed that the functions Φ ( k , b, Y ′ ) and Φ † ( k , b, Y ′ ) are purely functions of the radial k coordinate and do not depend on the angular coordinate θ , such that θ derivatives that appeared inEq. (2.26) have been dropped. The integration over x leads to the Dirac delta function (2 π ) δ ( k l + k ).The delta function is absorbed by the integration over k which leads to: S I = 16 2 π ¯ α S N c (2 π ) Z dY ′ Z d b Z d k (2.29) ( Φ † (cid:0) − k , b, Y ′ (cid:1) Φ † (cid:0) − k , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) k , b, Y ′ (cid:1) Φ (cid:0) − k , b, Y ′ (cid:1) Φ (cid:0) − k , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ † (cid:0) k , b, Y ′ (cid:1)) where l = ln k . Integrating the last term in Eq. (2.29) by parts, and taking into account that d k = dθ dl e l where θ is the azimuthal angle, Eq. (2.29) simplifies to the following expression: S I = 16 2 π ¯ α S N c (2 π ) Z dY ′ Z d b Z d k (2.30) ( Φ † (cid:0) − k , b, Y ′ (cid:1) Φ † (cid:0) − k , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) k , b, Y ′ (cid:1) Φ † (cid:0) k , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) − k , b, Y ′ (cid:1) Φ (cid:0) − k , b, Y ′ (cid:1) ) The last equation Eq. (2.30) allows us to study the classical equation of the action, which can beobtained from the conditions [9, 20]: δSδ Φ( k, b, Y ) = 0 and δSδ Φ † ( k, b, Y ) = 0 The functional derivative of the action for the effective pomeron field theory, with respect to Φ † ( k, b, Y )can be found from the results of Eqs. (2.18) and (2.30), which give:– 7 – ( S + S I ) /δ Φ † ( k, b, Y ) = 64(2 π ) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l (cid:16) ∂∂Y − H (cid:17) Φ ( k, b, Y )+ 16 (cid:18) π ¯ α S N c (cid:19) (2 π ) ( † (cid:0) − k, b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) k, b, Y ′ (cid:1) + (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l Φ (cid:0) − k, b, Y ′ (cid:1)) (2.31)where l = ln k . The equation for nucleus-nucleus scattering can be derived from the following equationof motion [9]: δ ( S + S I ) /δ Φ † ( k, b, Y ) = 0 and δ ( S + S I ) /δ Φ ( k, b, Y ) = 0 (2.32)Now the following approach is used to average these equations: D O ( k, Y ; b ) E = R D Φ D Φ † O ( k, Y ; b ) e S ( Φ , Φ † ) R D Φ D Φ † e S ( Φ , Φ † ) (2.33)Introducing the following new functions: N ( k, Y ; b ) = 2 π α S D Φ ( k, Y ; b ) E N † ( k, Y ; b ) = 2 π α S D Φ † ( k, Y ; b ) E (2.34)then the equation of motion of Eq. (2.31) reduces to:0 = (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l (cid:18) ∂∂Y ′ − H (cid:19) N (cid:0) k l , b, Y ′ (cid:1) (2.35)+ ¯ α S ( N † (cid:0) − k l , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l N (cid:0) k l , b, Y ′ (cid:1) + (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l N (cid:0) − k l , b, Y ′ (cid:1) ) Using the following property of the BFKL Pomeron, namely, p p (cid:18) ∂∂Y − H (cid:19) = (cid:18) ∂∂Y − H † (cid:19) p p , (2.36)we can derive the second equation of motion, by taking the functional derivative with respect to Φ.This generates the following equation of motion: – 8 – = (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l (cid:18) − ∂∂Y − H (cid:19) N † ( k l , b, Y ) (2.37)+ ¯ α S ( N (cid:0) − k l , b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l N † (cid:0) k l , b, Y ′ (cid:1) + (cid:18) ∂∂l + 1 (cid:19) ∂ ∂l (cid:16) N † (cid:17) (cid:0) − k l , b, Y ′ (cid:1) ) In Eq. (2.35) and Eq. (2.37) the following identities were implemented: D Φ ( k, Y ; b ) E = (cid:16)D Φ ( k, Y ; b ) E(cid:17) (2.38) D (cid:16) Φ † (cid:17) ( k, Y ; b ) E = (cid:16)D Φ † ( k, Y ; b ) E(cid:17) D Φ ( k, Y ; b ) Φ † ( k, Y ; b ) E = D Φ ( k, Y ; b ) E × D Φ † ( k, Y ; b ) E For a discussion on why these equations are correct, within an accuracy of about 1 /A / in the case ofnucleus-nucleus scattering, see Refs. [9, 20].
3. Semiclassical solution
In this section we find the semiclassical solutions to Eq. (2.35) and Eq. (2.37). In the semi-classicalapproximation, we are searching for solutions of the following form [21]: N (cid:0) l ≡ ln( k ) , b, Y ′ (cid:1) = exp (cid:0) S (cid:0) l, b, Y ′ (cid:1)(cid:1) where S = ω Y ′ − (1 − γ ) l + β ( b ) N † (cid:0) l ≡ ln( k ) , b, Y ′ (cid:1) = exp (cid:16) S † (cid:0) l, b, Y ′ (cid:1)(cid:17) where S † = ω † Y ′ − (1 − γ † ) l + β † ( b ) (3.1)where ω ( l, b, Y ′ ) and γ ( l, b, Y ′ ) are smooth functions of Y ′ and l , and the following conditions are assumed: dω ( l, b, Y ′ ) dY ′ ≪ ω ( l, b, Y ′ ); dω ( l, b, Y ′ ) dl ≪ ω ( l, b, Y ′ ) (cid:0) − γ ( l, b, Y ′ ) (cid:1) ; (3.2) dγ ( l, b, Y ′ ) dl ≪ (cid:0) − γ ( l, b, Y ′ ) (cid:1) ; dγ ( l, b, Y ′ ) dY ′ ≪ ω ( l, b, Y ′ ) (cid:0) − γ ( l, b, Y ′ ) (cid:1) ; (3.3)with analogous conditions for the functions ω † ( l, b, Y ′ ) and γ † ( l, b, Y ′ ). Assuming that for Eq. (3.1), themethod of characteristics can be applied (see, for example, Ref. [22]) to solve the non-linear equation.Notice that He S = ¯ α S Z K (cid:0) k, k ′ (cid:1) e S ( k ′ ,b,Y ) = ¯ α S χ ( γ ) e S ( k,b,Y ) (3.4)where χ ( γ ) = 2 ψ (1) − ψ ( γ ) − ψ (1 − γ ) (3.5)– 9 –here ψ ( z ) = d ln Γ( z ) /dz is the Di-Gamma function. In the semiclassical approach (for γ a smoothfunction), inserting the definition of Eq. (3.1) into the equations of motion of Eq. (2.35) and Eq. (2.37),and using the conditions of Eqs.(3.2) - (3.3), leads to the following formulae in the semi-classical approach: ω − ¯ α S χ ( γ ) + κ (cid:16) γ † (cid:17) ¯ α S e ˜ S † + ¯ α S e ˜ S = 0 − ω † − ¯ α S χ (cid:16) γ † (cid:17) + κ ( γ ) ¯ α S e ˜ S † + ¯ α S e ˜ S = 0 (3.6)where the following functions were introduced with definitions: κ ( γ ) = 2 γ γ − ˜ S = S − ln κ ( γ ) ˜ S † = S † − ln κ ( γ † ) (3.7)For the equation in the form F ( Y ′ , l, ˜ S, γ, ω ) = 0 (3.8)where S is given by Eq. (3.1), we can introduce the set of characteristic lines on which l ( t ) , Y ′ ( t ) , S ( t ) , ω ( t ) , and γ ( t ) are functions of the variable t (which we call artificial time), that satisfy the following equations:(1 . ) dld t = F γ = − ¯ α S dχ ( γ ) dγ (2 . ) d Y ′ d t = F ω = 1(3 . ) d ˜ Sd t = γ F γ + ω F ω = ¯ α S (1 − γ ) dχ ( γ ) dγ + ω (4 . ) d γd t = − ( F l + γ F S ) = ¯ α S κ (cid:0) γ + (cid:1) (1 − γ † ) e ˜ S † + ¯ α S (1 − γ ) e ˜ S (5 . ) d ωd t = − ( F Y ′ + ω F S ) = − ¯ α S ω † κ (cid:0) γ + (cid:1) e ˜ S † − ¯ α S ω e ˜ S (3.9)where F l = ∂F ( Y ′ , l, ˜ S, γ, ω ) ∂l , and the terms in the first of Eqns. (3.6) that depend on ˜ S † , ω † γ † , are treatedas if they depend explicitly on Y ′ and l . The same five equations we can write for the second of Eqns.(3.6) which have the following form:(1 . ) dl † d t = F γ † = − ¯ α S dχ ( γ † ) dγ † (2 . ) d Y ′ , † d t = F ω † = − , (3 . ) d ˜ S † d t = γ † F γ † + ω † F ω † = ¯ α S (cid:16) − γ † (cid:17) dχ ( γ † ) dγ † − ω † (4 . ) d γ † d t = − ( F l + γ † F ˜ S ) = ¯ α S κ ( γ ) (1 − γ ) e ˜ S + ¯ α S (cid:16) − γ † (cid:17) e ˜ S † (5 . ) d ω † d t = − ( F Y ′ + ω † F S ) = − ¯ α S ωκ ( γ ) e ˜ S − ¯ α S ω † e ˜ S † (3.10)– 10 – .2 The system of equations: linear approximation We start with the solution of Eq. (2.35) and Eq. (2.37) in the kinematic region where the non-linearcorrections are small. In this region, we can reduce this system of equations to the solution of the BFKLequation, both for N ( l, b, Y ′ ) and N † ( l, b, Y ′ ). Actually in order to find the solution, it isn’t necessary tosolve the equation, since the t -channel unitarity constraints for the BFKL Pomeron can be used instead(see Refs. [1, 16, 23]). This is given by: N BF KL ( L, b, Y ) = Z d b ′ Z dl N † BF KL (cid:16) L − l,~b − ~b ′ , Y − Y ′ (cid:17) N BF KL (cid:0) l, b ′ , Y ′ (cid:1) (3.11)Using the explicit form for the BFKL solution [16, 17], then Eq. (3.11) reduces to the following expres-sion: N BF KL ( L, b, Y ) = Z dγ πi n BF KL ( γ, b ) e ¯ α S χ ( γ ) Y − (1 − γ ) L (3.12)= Z d b ′ Z d l Z dγ πi Z dγ ′ πi n † BF KL ( γ,~b − ~b ′ ) e ¯ α S χ ( γ ) ( Y − Y ′ ) − (1 − γ ) ( L − l ) n BF KL (cid:0) γ ′ , b ′ (cid:1) e ¯ α S χ ( γ ′ ) Y ′ − (1 − γ ′ ) l where L = ln (cid:0) k p /k t (cid:1) and l = ln (cid:0) k /k t (cid:1) . k p and k t are the momenta of the dipoles in the projectileand the target, respectively. The function n BF KL ( γ, b ) is determined by the initial condition at Y = 0.The integration over l leads to the delta function (2 π ) δ ( γ − γ ′ ), and hence the γ ′ integral is imme-diately solvable by setting γ = γ ′ everywhere in the integrand. Integrating over the impact parameter b ′ yields: Z d b ′ n † BF KL ( γ,~b − ~b ′ ) n BF KL ( γ, b ′ ) = n BF KL ( γ, b ) (3.13)In light of this Eq. (3.12) simplifies to N BF KL ( L, b, Y ) = Z dγ πi e ¯ α S χ ( γ ) Y − (1 − γ ) L n BF KL ( γ, b ) (3.14)Incidentally Eq. (3.14) satisfies Eq. (3.11). Therefore, we have reproduced Eq. (3.11), where N † and N are given explicitly by: N † BF KL (cid:16) L − l,~b − ~b ′ , Y − Y ′ (cid:17) = Z dγ πi n † BF KL ( γ,~b − ~b ′ ) e ¯ α S χ ( γ ) ( Y − Y ′ ) − (1 − γ ) ( L − l ) (3.15) N BF KL (cid:0) l, b ′ , Y ′ (cid:1) = Z dγ ′ πi n BF KL (cid:0) γ ′ , b ′ (cid:1) e ¯ α S χ ( γ ′ ) Y ′ + (1 − γ ′ ) l (3.16)where the functions n † BF KL ( γ,~b − ~b ′ ) and n BF KL ( γ ′ , b ′ ) can be found from the initial conditions for N † BF KL at Y ′ = Y and N BF KL at Y ′ = 0. – 11 –he integral expressions of Eqs. (3.15) and (3.16) can N(L,Y) N ( L−l,Y− Y’)N(l,Y’) + YY’0
Figure 1:
The BFKL Pomeron: t -channel uni-tarity. be used to derive the explicit expressions for N † BF KL (cid:16) L − l,~b − ~b ′ , Y − Y ′ (cid:17) and N BF KL ( l, b ′ , Y ′ ) in thesemi-classical approach, by integrating using the method ofsteepest decent. This is assuming that n † BF KL ( γ,~b − ~b ′ ) and n BF KL ( γ ′ , b ′ ) are continuous, and differentiable functionsof γ and γ ′ .The saddle point equations, for the functions in the exponent in Eqs. (3.15) and (3.16) are:¯ α S dχ ( γ SP ) dγ SP (cid:0) Y − Y ′ (cid:1) + L − l = 0 and ¯ α S dχ ( γ ′ SP ) dγ ′ SP Y ′ + l = 0 (3.17)The integrals of Eqns. (3.15) and (3.16) can be solved using the method of steepest descents to yield: N † BF KL (cid:16) L − l,~b − ~b ′ , Y − Y ′ (cid:17) = (3.18) s π ¯ α S χ ′′ ( γ SP ) ( Y − Y ′ ) n † BF KL ( γ SP ,~b − ~b ′ ) exp (cid:0) ¯ α S χ ( γ SP ) (cid:0) Y − Y ′ (cid:1) − ( L − l ) (1 − γ SP ) (cid:1) = ˜ n BF KL ( γ SP ,~b − ~b ′ ) e S † N BF KL (cid:16) l,~b − ~b ′ , Y ′ (cid:17) = (3.19) s π ¯ α S χ ′′ (cid:0) γ ′ SP (cid:1) Y ′ n BF KL ( γ ′ SP ,~b − ~b ′ ) exp (cid:0) ¯ α S χ (cid:0) γ ′ SP (cid:1) Y ′ p − l (1 − γ ′ SP ) (cid:1) = ˜ n BF KL (cid:0) γ ′ SP , b ′ (cid:1) e S S = ¯ α S (cid:18) χ ( γ ′ ) − (cid:0) − γ ′ SP (cid:1) dχ ( γ ′ SP ) dγ ′ SP (cid:19) Y ′ ; S † = ¯ α S (cid:18) χ ( γ SP ) − (1 − γ SP ) dχ ( γ SP ) dγ SP (cid:19) (cid:0) Y − Y ′ (cid:1) where all slowly changing terms, have been absorbed by the functions ˜ n † and ˜ n . Eq. (3.9) together withEq. (2.35) have the following form in the linear approximation:(1 . ) dld Y ′ = − ¯ α S dχ ( γ ) dγ ; (2 . ) ω = ¯ α S χ ( γ ); (3 . ) d ˜ Sd Y ′ = ¯ α S (1 − γ ) dχ ( γ ) dγ + ω ; (4 . ) d γd Y ′ = 0; (3.20)It is easy to see that Eq. (3.20) leads to the same S as Eq. (3.18) and Eq. (3.19). One can see that for γ = γ cr for which χ ( γ ′ cr ) + (1 − γ ′ cr ) dχ ( γ ′ cr ) dγ ′ cr = 0 (3.21)– 12 – ’Y l0 L0 N + N Figure 2:
The trajectories of the linear equations for N † and N : L = ln (cid:16) k f /k i (cid:17) , l = ln (cid:0) k /k i (cid:1) and α =¯ α S χ ( γ cr ) /γ cr . Red lines denote the critical trajectories. Only the trajectories to the left (for N † ) and to the right(for N ) are shown. Then S = 0. The equation for this line takes the form: l = ¯ α S χ ( γ cr )1 − γ cr Y ′ (3.22)Repeating all the steps of the calculations above for S † , we find that S † = 0 on the line L − l = ¯ α S χ ( γ cr )1 − γ cr (cid:0) Y − Y ′ (cid:1) (3.23)using the same γ cr from Eq. (3.21). The general pattern of trajectories for the linear equation is shown inFig. 2. It should be stressed that for Eq. (3.9) and Eq. (3.10), we have different trajectories but they areparallel and shifted by ∆ l = ¯ α S ( χ ( γ cr ) /γ cr ) Y .We need to know the initial conditions for S ( S † ) and γ ( γ † ) at Y ′ = 0 ( Y ′ = Y ). We choose theMcLerran-Venugopalan formula [3] which is written for the dipole-target amplitude and is given in termsof N as: N (cid:0) r, b, Y ′ = 0 (cid:1) = 1 − exp (cid:0) − r k i ( b ) / (cid:1) (3.24)where the initial characteristic momentum k i ( b ) ∝ T A ( b ), and T A ( b ) is used to denote the number ofnucleons inside the nucleus with fixed impact parameter b . In momentum representation N is equal to (seeFig. 3): N (cid:0) k, b, Y ′ = 0 (cid:1) = Z rdr J ( kr ) N (cid:0) r, b, Y ′ = 0 (cid:1) /r = 12 Γ (cid:16) , τ = k k i (cid:17) or S = ln (cid:16)
12 Γ (cid:16) τ (cid:17)(cid:17) γ − ∂ ln (cid:16) N ( k, b, Y ′ = 0) (cid:17) ∂ ln (cid:0) k /k i (cid:1) = − e − τ . Γ ( τ ) (3.25)Γ ( τ ) that appears in Eq. (3.25) is the Euler incomplete gamma function ( see formulae inRef. [24]). – 13 –aution should be taken here, since we cannot trust the McLerran-Venugopalan formula at smalldipole sizes. Indeed, we know that in the limit of perturbative QCD, γ is γ → r →
0. The relation in Eq. (3.24) leads to N ∝ r at r →
0, while the correct behaviour should be N ∝ r ln r . Nevertheless we will use Eq. (3.24), because our main interest lyes in the region in the vicinityof the saturation scale, where Eq. (3.24) reproduces the amplitude for N quite well.For N † the initial conditions look the same, but τ † = Τ- H Τ L , Γ H Τ L Figure 3:
Initial conditions: N ( τ )(solid line)and γ (dotted line). k /k f . In other words, N † ( k, b, Y ′ = Y ) = e S † = Γ (cid:0) τ † (cid:1) .One can see that using Eq. (3.9)(4) and Eq. (3.10)(4), weobtain that S † ( l, Y ′ ) = S ( L, Y ) − S ( l, Y ′ ) + S † . Itmeans that ω † = − ω and 1 − γ + = − (1 − γ ). It is easy tosee that the system of equations in Eqns. (3.10) degenerateto the system of equations in (3.9). Therefore, instead of Eq. (3.9) and Eq. (3.10) , we can solvethe following system of equations.(1 . ) dld t = − ¯ α S dχ ( γ ) dγ , (2 . ) d Y ′ d t = 1 , (3 . ) d Sd t = ¯ α S (1 − γ ) dχ ( γ ) dγ + ω, (4 . ) d γd t = − ¯ α S κ ( γ ) (1 − γ ) e S ( L,Y ) − S ( l,Y ′ ) + S + ¯ α S (1 − γ ) e S ( l,Y ′ ) , (5 . ) d ωd t = ¯ α S (cid:16) ωκ ( γ ) e S ( L,Y ) − S ( l,Y ′ ) + S − ω e S ( l,Y ′ ) (cid:17) , (3.26)For the sake of simplicity, the label e has been omitted everywhere in Eq. (3.26). Eq. (3.26) can bere-written in a different form, namely(1 . ) dld Y ′ = − ¯ α S dχ ( γ ) dγ , (2 . ) d Sd Y ′ = ¯ α S (1 − γ ) dχ ( γ ) dγ + ω, (4 . ) d γd ω = − − γω , (5 . ) dγdS = − κ ( γ ) e S ( L,Y ) − S ( l,Y ′ ) + S + e S ( l,Y ) dχ ( γ ) /dγ + ω/ (1 − γ ) (3.27)Eq. (3.27)-4 has the solution ω = ¯ α S Const (1 − γ ) where Const can be determined from the initialconditions. Const = χ ( γ ) − κ ( γ ) e S ( L,Y ) − e S − γ (3.28)Notice that the value of this constant depends on S ( L, Y ), but for large and negative S ( L, Y ) whichwe are dealing with, for γ < γ cr , we can safely neglect this dependence. Introducing ˆ S = S ( l, Y ′ ) − ( S ( L, Y ) + S ) . rajectories a S Y g =0.1 g =0.15 g =0.2 g =0.33 g a S Y g =0.2 g =0.15 g =0.1 g =0.33 -S g = 0.1 g = 0.15 g = 0.2 a S Y g = 0.33 -1 Fig. 4-a Fig. 4-b Fig. 4-c
Figure 4:
Semiclassical solution: trajectories l ( Y ) (Fig. 4-a), γ versus ¯ α S Y (Fig. 4-b ) and S versus ¯ α S Y ′ (Fig. 4-c).The solution with γ close to γ cr is shown in red by dotted line. ¯ α S Y is chosen to be equal 6. dγdS = e ( S ( L,Y )+ S ) / − κ ( γ ) e − ˆ S ( l,Y ′ ) + e ˆ S ( l,Y ′ ) dχ ( γ ) /dγ + Const ! (3.29)We follow the following strategy for solving this equation. Γ- - S H L Figure 5:
Dependence of S ( Y ′ = 0) = S on γ cr from Eq. (3.31) (thick line) and γ cr = 0 .
37 (solution of Eq. (3.21), thinline).
First we assume that we are looking for the solution in the interval { ˆ S , ˆ S max } , where we have selected ˆ S max . Using the initial con-ditions, we can calculate ˆ S ( l, Y = 0) through S and S max , andspecify the coefficient in front of Eq. (3.29). Solving Eq. (3.27)-2we will find Y ′ as a function of ˆ S . In particular Y ′ ( S max ) = Y ,however trying several times we find a value for S max that willgive Y ′ ( S max ). Fig. 4 shows the solutions for four values of γ that are smaller than γ cr . One can see that the trajectories andthe values of S and γ are close to the solution of the linear equa-tion. Indeed, dS/dY ′ < γ < γ cr and therefore, the value of S decreases due to evolution. Starting with S < Y ′ than at Y ′ = 0. Therefore, only at small values of Y ′ we can see a deviationfrom the solution to the linear equation, as shown in Fig. 4 (see Fig. 4-b for example). Eq. (3.27)-4 hasthe form dγdY ′ = ¯ α S (1 − γ ) e ( S ( L,Y )+ S ) / n − κ ( γ ) e − ˆ S ( l,Y ′ ) + e ˆ S ( l,Y ′ ) o (3.30)and due to the smallness of the factor exp (( S ( L, Y ) + S ) / dγ/dY ′ turns out to be small at large Y ′ , leading to a constant γ , as shown in Fig. 4-b. The value of γ cr is slightly different from the one fromEq. (3.21), due to a contribution from the non-linear terms to Eq. (3.27)-3, and it depends on the initialcondition for S ( Y ′ = 0) = S . Indeed, for the trajectory on which S ( l, Y ′ ) = S is constant we have the– 15 –ollowing equation: (1 − γ cr ) dχ ( γ cr ) dγ cr + χ ( γ cr ) − ( κ ( γ cr ) + 1) e S = 0 (3.31)The dependence of S on γ cr from Eq. (3.31) is shown in Fig. 5. One can see that for S < N < γ cr is close to the solution to Eq. (3.21). From Fig. 4 one can see that we start the evolutionfrom the value of γ that is close to γ cr , γ steeply increases to γ = γ cr and freezes at this value leading toconstant S almost in the entire kinematic region of ¯ α S Y ′ . For γ > γ cr , the solution will lead to S ( l, Y ′ )that increases with Y ′ , and we need to search for a different method of finding the solution, other than thesemi-classical approach.
4. Solution inside the saturation domain
As discussed above, the semi-classical approach cannot be used inside of the saturation region. It shouldbe mentioned that at large l , the amplitude behaves as l , which is certainly not the function for whichwe can use the semi-classical approach. However, it has been noticed in Ref. [21] that introducing newfunctions : φ ( l ′ , Y ′ , b ) and φ † ( l ′ , Y ′ , b ) instead of N and N † , defined as N (cid:0) l, Y ′ , b (cid:1) = 12 Z l dl ′ (cid:16) − e − φ ( l ′ ,Y ′ ,b ) (cid:17) , N † (cid:0) l, Y ′ , b (cid:1) = 12 Z l dl ′ (cid:16) − e − φ † ( l ′ ,Y ′ ,b ) (cid:17) (4.1)Then we can indeed use the semi-classical approach for these functions. The first observation is that,from the property of Eq. (3.4) and the definition of Eq. (3.16): H ∂N ( l, Y ′ , b ) ∂l = Z dγ πi ¯ α S χ ( γ ) γ n (cid:0) ( γ − , Y ′ , b (cid:1) exp (cid:0) ¯ α S χ ( γ ) Y ′ − (1 − γ ) l (cid:1) ≡ ¯ α S Z dγ πi (cid:18) χ ( γ ) − − γ (cid:19) ( γ − n (cid:0) γ, Y ′ , b (cid:1) exp (cid:0) ¯ α S χ ( γ ) Y ′ − (1 − γ ) l (cid:1) − ¯ α S N (cid:0) l, Y ′ , b (cid:1) ≡ ¯ α S L (cid:18) − ∂∂l (cid:19) ∂N ( l, Y ′ , b ) ∂l − ¯ α S N (cid:0) l, Y ′ , b (cid:1) The second observation is related to the function L ( γ ), namely that its expansion with respect to(1 − γ ) starts with (1 − γ ) as the first non-zero term. That is: L (cid:18) − ∂∂l (cid:19) = − d ψ ( z ) dz | z =1 ∂ ∂l − d ψ ( z ) dz | z =1 ∂ ∂l − . . . (4.2)– 16 –here ψ ( z ) = d ln Γ( z ) /dz is the Euler ψ -function (see Ref. [24]). Following the definition of Eq. (4.1) andassuming that φ and φ † are smooth functions, we can replace ∗ (cid:18) ∂∂l (cid:19) n N (cid:0) l, Y ′ , b (cid:1) = − (cid:18) ∂∂l (cid:19) n − (cid:16) − e − φ ( l,Y ′ ,b ) (cid:17) = 12 (cid:18) − ∂φ∂l (cid:19) n − e − φ ( l,Y ′ ,b ) (4.3) (cid:18) ∂∂l (cid:19) n N † (cid:0) l, Y ′ , b (cid:1) = − (cid:18) ∂∂l (cid:19) n − (cid:16) − e − φ † ( l,Y ′ ,b ) (cid:17) = 12 (cid:18) − ∂φ † ∂l (cid:19) n − e − φ † ( l,Y ′ ,b ) (4.4)Inserting Eqs. (4.2), (4.3) and (4.4) and into Eq. (2.35) and Eq. (2.37), and introducing the notation ∂φ/∂Y ′ = ω φ and ∂φ † /∂Y ′ = ω † φ we obtain:0 = (cid:16) ω φ + ¯ α S L ( − γ φ ) (cid:17) (1 − γ φ ) γ φ e − φ ( l,b,Y ′ ) + 2 ¯ α S (cid:18) ∂∂l + 1 (cid:19) ∂∂l (cid:16) e − φ ( l,b,Y ′ ) N (cid:0) l, b, Y ′ (cid:1) (cid:17) (4.5)+ ¯ α S N † (cid:0) l, b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂∂l e − φ ( l,b,Y ′ ) (cid:16) − ω † φ + ¯ α S L (cid:16) − γ † φ (cid:17) (cid:17) (cid:16) − γ † φ (cid:17) γ † φ e − φ † ( l,b,Y ′ ) + 2 ¯ α S (cid:18) ∂∂l + 1 (cid:19) ∂∂l n e − φ † ( l,b,Y ′ ) N † (cid:0) l, b, Y ′ (cid:1) o (4.6)+ ¯ α S N (cid:0) l, b, Y ′ (cid:1) (cid:18) ∂∂l + 1 (cid:19) ∂∂l e − φ † ( l,b,Y ′ ) One can see that (cid:18) ∂∂l + 1 (cid:19) ∂∂l n e − φ ( l,b,Y ′ ) N (cid:0) l, b, Y ′ (cid:1) o = e − φ n (cid:0) − γ φ + 7 γ φ (cid:1) e − φ − (cid:0) − γ φ + 3 γ φ (cid:1) (4.7) − γ φ (1 − γ φ ) N (cid:0) l, b, Y ′ (cid:1) o Dividing both sides of Eq. (4.5) by ¯ α S e − φ and introducing the new variable ¯ ω = ω/ ¯ α S , whichcorresponds to the change Y ′ to ¯ Y ′ = ¯ α S Y ′ we obtain: − ¯ ω φ − L ( − γ φ ) + 2 N † (cid:0) l, b, Y ′ (cid:1) − (cid:16) − γ φ + 7 γ φ (cid:17) e − φ − (cid:16) − γ φ + 3 γ φ (cid:17) (1 − γ φ ) γ φ + 2 N (cid:0) l, b, Y ′ (cid:1) = 0 (4.8)A similar equation can be written for φ † . However, we assume that φ † ( l, b, Y ′ ) = φ ( L − l, b, Y − Y ′ )based on our experience with the semi-classical solution. Looking for the solution that has a geometricscaling behaviour [25], we expect that φ ( l, b, Y ′ ) is a function of one variable: z = ln (cid:0) Q s (cid:0) Y ′ , b (cid:1) /k (cid:1) = χ ( γ cr )1 − γ cr Y ′ − l (4.9) ∗ For ∂φ/∂l ( ∂φ † /∂l ) we use the notations γ φ and γ † φ . We hope that it will not lead to any confusions with the notation,due to the similarity with γ and γ † , that was heavily used in the previous section. – 17 – .2 Asymptotic solution Plugging in φ † ( l, b, Y ′ ) = φ ( L − l, b, Y − Y ′ ) and the geometric scaling behavior of Eq. (4.8) that weanticipate, and assuming that φ ≫ G (˜ γ ) = z Y (4.10)where G (˜ γ ) = χ ( γ cr )1 − γ cr ˜ γ + L (˜ γ ) + 1 + 4˜ γ + 3˜ γ (1 + ˜ γ ) ˜ γz Y ≡ ln (cid:0) Q s ( Y ) /k i (cid:1) = χ ( γ cr )1 − γ cr Y − L It should be noticed that ˜ γ is defined as ˜ γ = dφ/dz . The function G ( γ φ ) is shown in Fig. 6. One cansee that we have three types of solutions to Eq. (4.10): - - Γ- - H Γ L - - Γ- - H Γ L Fig. 6-a Fig. 6-b
Figure 6:
Function G ( γ ) (solid line) versus γ . The red line shows the r.h.s. of Eq. (4.10) at z Y = 20. Fig. 6-ashows G ( γ ) as it is written in Eq. (4.10), while G ( γ ) with L = 0 is plotted in Fig. 6-b.
1. At large ˜ γ then G (˜ γ ) → χ ( γ c r ) / (1 − γ cr ) ˜ γ and we have the solution: ˜ γ = (1 − γ cr ) /χ ( γ cr ) z Y which translates into φ = ((1 − γ cr ) /χ ( γ cr )) z Y z ;2. We have solutions at ˜ γ → n where n = 1 , , . . . which lead to φ = nz . We only need to takeinto account n = 1, since other values of n give smaller contributions at large z . In vicinity γ → G ( γ ) = − / (1 − γ ) and the solution to Eq. (4.10) gives φ = (1 + 1 /z Y ) z ;3. Solutions where ˜ γ → − n we do not consider, since they lead to decreasing φ at large Y .It is interesting to notice, that at in the case where we restrict ourselves to the leading twist contribu-tions to the BFKL kernel [26], only the first solution survives (see Fig. 6-b).Using Eq. (4.1) we can obtain the asymptotic solution for the scattering amplitude, namely N ( z Y ) = 12 Z z Y dz (cid:16) − exp ( − (1 − γ cr ) /χ ( γ cr )) z Y z ) (cid:17) ←− leading twist; (4.11) N ( z Y ) = 12 Z z Y dz (cid:16) − exp ( − z ) (cid:17) ←− general asymptotic behaviour; (4.12)– 18 –t should be noticed that the expression in parentheses (cid:16)(cid:17) acxtually gives the scattering amp-litudein the coordinate representation. In general, Eq. (4.8) can be reduced to the differential equation by taking derivatives with respect to z onboth sides of the equation. The resulting expression is: dG (˜ γ ) d ˜ γ d ˜ γdz = e − φ ( z ) − e − φ ( z Y − z ) , dφ ( z ) dz = ˜ γ, (4.13)This equation belongs to the class of delay differential equations, and the solution to this equation wehope to publish in an upcoming paper, since it is a separate and rather difficult problem beyond the scopeof this paper (see for example Ref. [27]).
5. Conclusions
In conclusion, we summarize our results as follows. First we re-wrote the action of the BFKL Pomeroncalculus, and we derived the equations in momentum representation. It turns out that the equations thatwe obtain have a simpler form, than in coordinate representation in the format that they were originallyderived in Ref. [9]. Second, we found the semi-classical solution to these equations, to the right of the criticalline . In the saturation domain, we reduced these equations to the class of delay differential equations, andwe found their asymptotic solution. This solution shows, that the nucleus-nucleus amplitude in coordinaterepresentation approaches unity as 1 − ∆ N , where ∆ N ∝ exp ( − z Y ), where z Y ≡ ln (cid:0) Q s ( Y ) r i (cid:1) , where r i is the size of the colourless dipole in the nucleus. If we take into account only the leading twist part ofthe BFKL kernel, the behaviour of ∆ N is similar to the behaviour of the amplitude of the dilute-denseparton system interaction, given in Ref. [26].We hope that this paper will show, that the problem of the nucleus-nucleus interaction in the frameworkof the BFKL Pomeron calculus, can be solved. We hope that this development will motivate further effortstowards understanding the dense-dense scattering system.
6. Acknowledgements
This research was supported by the Funda¸c˜ a o para ci´ e ncia e a tecnologia (FCT), and CENTRA - InstitutoSuperior T´ e cnico (IST), Lisbon and by the Fondecyt (Chile) grants 1100648, 1095196 and DGIP 11.11.05.One of us (JM) would like to thank Tel Aviv University for their hospitality on this visit, during the timeof the writing of this paper. – 19 – eferences [1] L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. (1983) 1.[2] A. H. Mueller and J. Qiu, Nucl. Phys. B268 (1986) 427.[3] L. McLerran and R. Venugopalan, Phys. Rev.
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