Long range rapidity correlations in soft interaction at high energies
aa r X i v : . [ h e p - ph ] J u l Preprint typeset in JHEP style - HYPER VERSION
TAUP -2973/13September 12, 2018
Long range rapidity correlations in soft interaction at highenergies. E. Gotsman a ∗ , E. Levin a,b † and U. Maor a ‡ a) Department of Particle Physics, School of Physics and Astronomy, Raymond and Beverly SacklerFaculty of Exact Science, Tel Aviv University, Tel Aviv, 69978, Israelb) Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Avda. Espa˜na 1680and Centro Cientifico-Tecnol ´ o gico de Valparaiso,Casilla 110-V, Valparaiso, Chile Abstract:
In this paper we take the next step (following the successful description of inclusive hadronproduction) in describing the structure of the bias events without the aid of Monte Carlo codes. Twonew results are presented :(i) a method for calculating the two particle correlation functions in the BFKLPomeron calculus in zero transverse dimension; and (ii) an estimation of the values of these correlations ina model of soft interactions. Comparison with the multiplicity data at the LHC is given.
Keywords:
Soft Pomeron, BFKL Pomeron, Diffractive Cross Sections, Survival Probability.
PACS: 13.85.-t, 13.85.Hd, 11.55.-m, 11.55.Bq ∗ Email: [email protected]. † Email: [email protected] ‡ Email: [email protected]. ontents
1. Introduction 12. Correlation function in the BFKL Pomeron Calculus in zero transverse dimensions 2
3. Correlations in a model for soft interactions 11
4. Conclusions 19
1. Introduction
The goal of this paper is twofold: to consider the two hadron long range rapidity correlations in theBFKL Pomeron Calculus in zero transverse dimensions; and to calculate these correlations in a model ofsoft interactions at high energy. The BFKL Pomeron Calculus in zero transverse dimension describes theinteraction of the Pomerons through the triple Pomeron vertex ( G IP ) with a Pomeron intercept ∆ IP ≡ – 1 – > α ′ IP = 0. The theory that includes all these ingredients can be formulated ina functional integral form [1]: Z [Φ , Φ + ] = Z D Φ D Φ + e S with S = S + S I + S E , (1.1)where, S describes free Pomerons, S I corresponds to their mutual interaction and S E relates to theinteraction with the external sources (target and projectile). Since α ′ IP = 0, S has the form S = Z dY Φ + ( Y ) (cid:26) − ddY + ∆ (cid:27) Φ( Y ) . (1.2) S I includes only triple Pomeron interactions and has the form S I = G IP Z dY (cid:8) Φ( Y ) Φ + ( Y ) Φ + ( Y ) + h.c. (cid:9) . (1.3)For S E we have local interactions both in rapidity and in impact parameter space, S E = − Z dY X i =1 (cid:8) Φ( Y ) g i ( b ) + Φ + ( Y ) g i ( b ) (cid:9) , (1.4)where, g i ( b ) stands for the interaction vertex with the hadrons at fixed b .At the moment this theory has two facets. First, it is a toy-model describing the interaction of theBFKL Pomerons in QCD. Many problems can be solved analytically in this simple model leading to a setof possible scenarios for the solution in BFKL Pomeron calculus [1–7]. Our first goal is to find an analyticalsolution for the correlation function in rapidity defined as R ( y , y ) = σ in d σdy dy σ in dσdy σ in dσdy , (1.5)where, σ in , d σ/dy dy and dσ/dy are inelastic, double and single inclusive cross sections. We considerthis problem as the most natural starting point to search for a solution for R ( y , y ), in a more generaland more difficult approach based on high density QCD.On the other hand, recent experience in building models for high energy scattering [8–13] shows thata Pomeron with α ′ IP = 0 can describe the experimental data including that at the LHC. It also appearsin N=4 SYM [14–18] with a large coupling, which at the moment, is the only theory that allows one totreat the strong interaction on a theoretical basis. Therefore, our second goal is to evaluate the correlationfunction R ( y , y ) in our model for soft high energy interactions (see [8–10]).
2. Correlation function in the BFKL Pomeron Calculus in zero transverse dimensions
It is well known [19] that the most appropriate framework to discuss the inclusive processes has beendeveloped by A.H. Mueller [20] (Mueller diagrams). In Fig. we show the most general Mueller diagram– 2 – y y 0 N( Y − y , Y − y )
N( y , y ) x xx x a) Y0N( y , y ) xx x y y b) Y0 xx c) y a P a P a P Figure 1:
The Mueller diagram [20] for double (Fig. -a and b) and single (Fig. -c) inclusive cross section. Thewavy lines denote Pomerons. The cross on a wavy line indicates that this line describes a cut Pomeron. = xx y = y a) b) a P x Figure 2:
Shows the main ingredients ofFig. : the cut Pomeron that describes theprocess of multiparticle (multigluon) pro-duction (Fig. -a) and the single inclusiveproduction from the cut Pomeron (Fig. -b). for the double inclusive cross section (see also Fig. ). From Fig. -a one can see that it is necessary tocalculate the amplitudes of the cut Pomeron interaction with the hadrons, denoted by N ( Y − y , Y − y )and N ( y , y ). The fact that we can reduce the calculation of the double inclusive production to anevaluation of N ( Y − y , Y − y ) and N ( y , y ) stems from the AGK cutting rules [21] which state that theexchanges of the Pomerons from the top to the bottom of the Mueller diagram cancel each other leadingto the general structure of Fig. -a. Recall that the AGK cutting rules are violated in QCD due to theemission diagrams from the triple Pomeron vertex (see Fig. -b) (see Ref. [22]). In our treatment weneglect such a violation since Γ ( Y − y ) turns out to be smaller at high energy than N ( Y − y , Y − y ) .Indeed, in the first approximation Γ ( Y − y ) ∝ ge ∆( Y − y ) while N ( Y − y , Y − y ) ∝ (cid:16) ge ∆( Y − y ) (cid:17) andΓ ( Y − y ) . N ( Y − y , Y − y ) → Y − y . Analyzing the diagrams one can see thattheir contributions are proportional to two parameters which are large at high energy: L ( Y ) = g ( b ) G IP ∆ e ∆ Y ; and T ( Y ) = G IP ∆ e ∆ Y . (2.1)Note that ∆ in the dominator stems from the integration over internal rapidities of the triple Pomeronvertices. We consider the first three diagrams (see Fig. ) for Γ ( Y − y ) (see Fig. ) to illustrate how thesetwo parameters appear in the calculations. For the diagrams of Fig. -a, Fig. -b and Fig. -c we have,– 3 – (b) Y G a) g(b) G g(b)G g(b)yy’ y’y’’yy b) c) Figure 3:
Low order diagrams forΓ ( Y − y ) (see Fig. ). Wavy lines denotethe Pomerons. respectively, A (Pomeron) = g ( b ) e ∆( Y − y ) ; (2.2) A (’fan’ diagram) = − g ( b ) G IP Z Y dy ′ e Y − y ′ ) e ∆ y ′ = − g ( b ) e ∆( Y − y ) (cid:16) L ( Y − y ) − g ( b ) G IP ∆ (cid:17) Y − y ≫ −−−−−→ − A (Pomeron) L ( Y − y ) ; (2.3) A (enhanced diagram) = − g ( b ) G IP Z Y dy ′ Z y ′ dy ′′ e ∆( Y − y ′ ) e y ′ − y ′′ ) e ∆ y ′′ = − g ( b ) e ∆( Y − y ) (cid:16) T ( Y − y ) − g ( b ) G IP ∆ (1 + ∆( Y − y ) (cid:17) Y − y ≫ −−−−−→ − A (Pomeron) T ( Y − y ) ; (2.4)At high energy both L ( Y ) ≫ T ( Y ) ≫ g(b)Y xxxx y x x y G G g(b)Y xxxx y x x y G G a) b) G Figure 4:
The main diagrams relatingto L ( Y − y i ) (see text), that contributeto the function N ( Y − y , Y − y ) (Fig. -a); and the first diagram with correctionthat is proportional to T ( Y − y i ) (Fig. -b). Wavy lines denote the Pomerons. Thecross on the wavy line indicates that thisline describes the cut Pomeron. In this kinematic region each Pomeron diagram is proportional to powers of L ( Y ) and T ( Y ). Therefore,the first approximation is to sum the largest contributions at high energies in every Pomeron diagram.Such an approach to high energy scattering was proposed by Mueller, Patel, Salam and Iancu (MPSIapproximation [23]). It turns out that the value of G IP is rather small (see discussion below). Basedon this fact we propose that the leading approximation shall be to sum all contributions proportional to L n ( Y − y ) having in mind the following kinematic region: L ( Y − y ) ≥ T ( Y − y ) ≪ g ( b ) ≪ G IP ≪ . (2.5)– 4 –or the scattering with nuclei g ( b ) ∝ A / , and in this region which covers all reasonable energies, the maincontribution emanates from ’fan’ diagrams (see Fig. and Fig. - b for the first diagram of this kind). Theexpression for Γ ( Y − y ) is known [24, 25]:Γ ( Y − y ) = 2 g e ∆( Y − y ) L ( Y − y ) ; . (2.6)As we shall see below the factor 2 stems from the initial cut Pomeron. Below we shall obtain theseexpressions using a more general technique in which we find the sum of the diagrams in a more generalkinematic region: L ( Y − y ) ≥ T ( Y − y ) ≥ g ( b ) ≪ G IP ≪ , (2.7)selecting contributions of the order of L m ( Y − y ) T n − m ( Y − y ). i.e. we shall find the scattering amplitudein the kinematic region of Eq. (2.7) using MPSI approximation.The most important diagrams for N ( Y − y , Y − y ) are shown in Fig. -a. One can see that thekinematic region of Eq. (2.5) N ( Y − y , Y − y ) is: N ( Y − y , Y − y ) = Γ ( Y − y ) Γ ( Y − y ) . (2.8) We believe that the method of a generating function (functional) is the most appropriate method forsumming Pomeron diagrams. In the MPSI approach, one can explicitly see the conservation of probability(unitarity constraints) in each step of the evolution in rapidity. This method was proposed by Muellerin Ref. [4] and has been developed in a number of publications(see Ref. [26] and references therein). InRef. [27] it was generalized to account for the contribution to the inelastic processes by summing both cutand uncut Pomeron contributions. For completeness of the presentation, in this section we shall discussthe main features of this method, referring to Refs. [9, 10, 27] for essential details. Following Ref. [27], weintroduce the generating function Z ( w, ¯ w, v | Y ) = X k =0 X l =0 X m =0 P ( k, l, m | Y ) w k ¯ w l v m , (2.9)where, P ( k, l, m | Y ) stands for the probability to find k uncut Pomerons in the amplitude, l uncut Pomeronsin the conjugate amplitude and m cut Pomerons at some rapidity Y . w, ¯ w and v are independent variables.Restricting ourselves by taking into account only a Pomeron splitting into two Pomerons, we can write thefollowing simple evolution equation: ∂Z∂Y = − ∆ n w (1 − w ) ∂Z∂w − ¯ w (1 − ¯ w ) ∂Z∂ ¯ w o − ∆ n w ¯ w − wv − wv + v + v ) ∂Z∂v o . (2.10)Fig. illustrates the two steps of evolution in rapidity for Z ( w, ¯ w, v ; Y ). The general solution toEq. (2.10) has the form C Z ( w ) + C Z ( ¯ w ) + C Z ( w + ¯ w − v ) , (2.11)– 5 –here, C and C are constants and Z ( ξ ) is the solution to the equation: ∂Z∂Y = − ∆ ξ (1 − ξ ) ∂Z∂ξ . (2.12)The particular form of Z and the values of C i are determined by the initial condition at Y = 0. XX XX X w vv v v v
XXX w vv v w _ a) b)w _ w _ w _ Figure 5:
Two examples for two stepsof evolution in rapidity for the generatingfunction Z ( w, ¯ w, v ; Y ). Wavy lines denotethe Pomerons. The cross on the wavy lineindicates that this line describes the cutPomeron. The general formula for the amplitude in the MPSI approach has the form (see Ref. [27]) N MP SI ( γ, γ in | Y ) = (cid:16) exp ( − γ ∂∂γ (1) ∂∂γ (2) − γ ∂∂ ¯ γ (1) ∂∂ ¯ γ (2) + γ in ∂∂γ (1) in ∂∂γ (2) in ) − (cid:17) Z (cid:16) γ (1) , ¯ γ (1) , γ (1) in | Y − Y ′ (cid:17) Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) | γ ( i ) =¯ γ ( i ) = γ ( i ) in = 0 , (2.13)where, w = 1 − γ , ¯ w = 1 − ¯ γ and v = 1 − γ in . YY’ y y g(b) x xx x xxxxx x x inin in _ in _ in Figure 6:
An example of dia-grams that contribute to the function N ( Y − y , Y − y ) (see Fig. ). Wavylines denote the Pomerons. The crosson the wavy line indicates that thisline describes a cut Pomeron. γ is theamplitude of the dipole-dipole interactionat low energies . The particular set ofdiagrams shown in this figure, correspondsto the MPSI approach [23]. Eq. (2.13) has a very simple meaning which is clear from Fig. . The derivatives of the generatingfunctional Z (cid:16) γ (1) , ¯ γ (1) , γ (1) in | Y − Y ′ (cid:17) determine the probability to have cut and uncut Pomerons at Y = Y ′ ,while the derivatives of Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) lead to the probabilities of the creation of cut and uncut– 6 –omerons from two initial cut Pomerons at rapidity Y ′ . Two uncut Pomerons interact with the amplitude γ at rapidity Y ′ and with the amplitude γ in in the case of cut Pomerons. The phases of the amplitude aregiven by related signs in Eq. (2.13): minus for γ and plus for γ in . In addition, we assume that the lowenergy at which the wee partons from two Pomerons interact is large enough to assume that γ and γ in are purely imaginary. We denote the imaginary part of the amplitude, by γ ’s. It follows from the AGKcutting rules that γ in = 2 γ. (2.14)According to Eq. (2.13), the contribution to the scattering amplitude of one Pomeron exchange is equalto ∗ ˜ g e ∆( Y − Y ′ ) γ e ∆ Y ′ ˜ g. (2.15)For the first ’fan’ diagram, Eq. (2.13) leads to the following contribution:˜ g Z YY ′ dy ′ e ∆( Y − y ′ ) ∆ e y ′ − Y ′ ) γ , e Y ′ ) ˜ g , (2.16)while the first enhanced diagram can be written as˜ g Z YY ′ dy ′ e ∆( Y − y ′ ) ∆ e y ′ − Y ′ ) γ Z Y ′ dy ′′ e Y ′ − y ′′ ) ∆ e ∆ y ′′ ˜ g. (2.17)Comparing these expressions with the Pomeron diagrams (see Eq. (2.2),Eq. (2.3) and Eq. (2.4)), we havethe correspondence between these two approaches,˜ g = g/ √ γ ; γ = G IP ∆ . (2.18) The pattern of calculation of Glauber-Gribov rescatterings due to Pomeron exchanges is shown in Fig. -a.The forms of the generating functions Z (cid:16) γ (1) , ¯ γ (1) , γ (1) in | Y − Y ′ (cid:17) and Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) are simple, Z (cid:16) γ (1) , ¯ γ (1) , γ (1) in | Y − Y ′ (cid:17) = e ˜ g e ∆( Y − Y ′ ) ( w (1) + ¯ w (1) − v (1) − ) = e ˜ g e ∆( Y − Y ′ ) (cid:16) γ (1) + ¯ γ (1) − γ (1) in (cid:17) ; (2.19) Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) = e ˜ g e ∆( Y ′ ) ( w (2) + ¯ w (2) − v (2) − ) = e ˜ g e ∆( Y − Y ′ ) (cid:16) γ (2) + ¯ γ (2) − γ (2) in (cid:17) . (2.20)These generating functions describe the independent (without correlations) interaction of Pomerons withthe target and the projectile. In the case of nuclei, Pomerons interact with different nucleons in thenucleus, and the correlations between nucleons in the wave function of the nucleus are neglected. Notethat Eq. (2.13) with Z ’s from Eq. (2.19) and Eq. (2.20) do not depend on the sign of v ( γ in ). However, ∗ We suppress the notation of the impact parameter, which if needed can be easily be replaced. – 7 –e shall see below that the choice of the above equation is correct since it reproduces Eq. (2.6), which hasbeen derived by summing the Pomeron diagrams.Using Eq. (2.13), we can calculate the inelastic cross section requiring that at rapidity Y ′ we have atleast one cut Pomeron (one γ in ). The result is: σ in = 1 − e − γ in ˜ g e ∆ Y = 1 − e − g e ∆ Y . (2.21)which reproduces the well known expression for the inelastic cross section in the Glauber-Gribov approach.We can also calculate the contribution which has no cut Pomeron at rapidity Y ′ (elastic cross sections).It has the form σ el = (cid:16) − e − γ ˜ g e ∆ Y (cid:17) (cid:16) − e − ¯ γ ˜ g e ∆ Y (cid:17) = (cid:16) − e − g e ∆ Y (cid:17) . (2.22)The total cross section is given by: σ tot = σ el + σ in = 2 (cid:16) − e − g e ∆ Y (cid:17) . (2.23) YY’0 g X in in gga) b) X XX XXX X X X Y’ in X g Y c)
XX XXXX
Figure 7:
MPSI approximation: Glauber-Gribov rescattering (Fig. -a), summation of ’fan’ diagrams (Fig. -b)and the diagrams for single inclusive cross section (Fig. -c) Wavy lines denote the Pomerons. The cross on the wavyline indicates that this line describes the cut Pomeron. γ is the amplitude of the dipole-dipole interaction at lowenergies. As one can see from Fig. -b, the form of Z (cid:16) γ (1) , ¯ γ (1) , γ (1) in | Y − Y ′ (cid:17) is the same as in the previous problem.It is given by Eq. (2.19). To obtain an expression for Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) , we need to find Z ’s and C i inEq. (2.11) with the initial condition Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ = 0 (cid:17) = v. (2.24)– 8 –he resulting solution is of the form (see more details in Ref. [27]) Z (cid:0) w, ¯ w, v ; Y ′ (cid:1) = Z el (cid:0) w, ¯ w ; Y ′ (cid:1) + Z in (cid:0) w, ¯ w, v ; Y ′ (cid:1) ; (2.25) Z el (cid:0) w, ¯ w ; Y ′ (cid:1) = w e − ∆ Y ′ w ( e − ∆ Y ′ −
1) + ¯ w e − ∆ Y ′ w ( e − ∆ Y ′ − − ( w + ¯ w ) e − ∆ Y ′ w + ¯ w )( e − ∆ Y ′ −
1) ; (2.26) Z in (cid:0) w, ¯ w, v ; Y ′ (cid:1) = ( w + ¯ w ) e − ∆ Y ′ w + ¯ w )( e − ∆ Y ′ − − ( w + ¯ w − v ) e − ∆ Y ′ w + ¯ w − v )( e − ∆ Y ′ − . (2.27)Substituting for Z in in Eq. (2.13) we obtain for the inelastic part of Γ ( Y − y ) (see Fig. -b),Γ in ( Y − y ) = 2 ˜ gγe ∆( Y − y ) gγe ∆( Y − y ) = 2 L ( Y − y )1 + 2 L ( Y − y ) . (2.28)Eq. (2.28) has been derived from the direct summation of the Pomeron diagrams in Ref. [25]. The factthat we reproduce the results of Ref. [25] , vindicates our choice of the generating functions in Eq. (2.19)and Eq. (2.20).Using Z el we obtain the elastic contribution which is intimately related to the processes of diffractionproduction:Γ el ( Y − y ) = 2 ˜ gγe ∆( Y − y ) gγe ∆( Y − y ) − gγe ∆( Y − y ) gγe ∆( Y − y ) = 2 L ( Y − y )1 + L ( Y − y ) − L ( Y − y )1 + 2 L ( Y − y ) . (2.29)The resulting Γ ( Y − y ) is given by: Γ ( Y − y ) = 2 L ( Y − y )1 + L ( Y − y ) . (2.30)Actually Eq. (2.30) gives the same expression as Eq. (2.6). The difference in an extra factor, √ γ , stemsfrom the fact that, we need to take ˜ g rather than g in the vertex for the Pomeron-hadron interaction. As one can see from Fig. -c, to evaluate the single inclusive cross section, we need to calculate Γ ( Y − y ).We have done so in the previous section, however, we now want to take into account both L n ( Y − y ) and T n ( Y − y ) contributions. From Fig. -c we see that Z (cid:0) w (2) , ¯ w (2) , v (2) ; Y ′ (cid:1) has the form given in Eq. (2.25).However, in Z (cid:0) w (1) , ¯ w (1) , v (1) ; Y ′ (cid:1) , we need to take into account that each Pomeron at Y − Y ′ = 0, createsa cascade of Pomerons that is described by Eq. (2.10). In other words, we need to replace w (1) , ¯ w (1) and v (1) in Eq. (2.19) by w (1) → w (1) e − ∆( Y − Y ′ ) w (1) ( e − ∆( Y − Y ′ ) −
1) ; ¯ w (1) → ¯ w (1) e − ∆( Y − Y ′ ) w (1) ( e − ∆( Y − Y ′ ) −
1) ; (2.31) v (1) → w (1) e − ∆( Y − Y ′ ) w (1) ( e − ∆( Y − Y ′ ) −
1) + ¯ w (1) e − ∆( Y − Y ′ ) w (1) ( e − ∆( Y − Y ′ ) − − ( w (1) + ¯ w (1) − v (1) ) e − ∆( Y − Y ′ ) w + ¯ w − v )( e − ∆ ( Y − Y ′ − . (2.32)– 9 –sing these substitutions we obtain Z (cid:16) w (1) , ¯ w (1) , v (1) ; Y − Y ′ (cid:17) = exp ( w (1) + ¯ w (1) − v (1) ) e − ∆( Y − Y ′ ) w + ¯ w − v )( e − ∆ ( Y − Y ′ − ! . (2.33)Using the generating function for Laguerre polynomials (see Ref. [29] formula ),(1 − z ) − α − exp (cid:18) x zz − (cid:19) = ∞ X n =0 L αn ( x ) z n . (2.34)We obtain for Eq. (2.33) Z (cid:16) w (1) , ¯ w (1) , v (1) ; Y ′ (cid:17) = − ∞ X n =0 L − n (˜ g i ) (cid:16) − (cid:16) γ (1) + ¯ γ (1) − γ (1) in (cid:17) e ∆( Y − Y ′ ) (cid:17) n . (2.35)From Eq. (2.13) using ∂ l ∂ l γ (1) ∂ m ∂ m ¯ γ (1) ∂ n − l − m ∂ n − l − m γ (1) in (cid:16) γ (1) + ¯ γ (1) − γ (1) in (cid:17) n = ( − n − l − m n ! . (2.36)We obtain Γ ( Y − y ) = ∞ X n =1 L − n (˜ g i ) n ! (cid:0) − γ e ∆ Y (cid:1) n = ∞ X n =1 L − n (˜ g i ) n ! ( − n T n ( Y − y ) . (2.37)Introducing n ! = R ∞ dξξ n exp ( − ξ ) we reduce Eq. (2.37) to the formΓ ( L ( Y − y ) , T ( Y − y )) = Z ∞ dξ e − ξ e − ξ ˜ g γ e ∆( Y − y )1 + ξ γ e ∆( Y − y ) − ! = Z ∞ dξ e − ξ (cid:18) e − ξ L ( Y − y )1 + ξ T ( Y − y ) − (cid:19) . (2.38)Using Eq. (2.38) we obtain the following result for the single inclusive cross section: dσdy = a IP Γ ( L ( Y − y ) , T ( Y − y )) Γ ( L ( y ) , T ( y )) , (2.39)where, a IP denotes the vertex of emission of the hadron from Pomeron (see Fig. and Fig. -b). Calculating N ( Y − y , Y − y ) (see Fig. -a) we use Z (cid:0) w (1) , ¯ w (1) , v (1) ; Y − Y ′ (cid:1) , given by Eq. (2.33), as onecan see from Fig. . However, Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ (cid:17) is different from the expression which has been usedin the calculation of the single inclusive cross section, and it can be written as: Z (cid:16) γ (2) , ¯ γ (2) , γ (2) in | Y ′ − y , Y ′ − y (cid:17) = (2.40) Z (cid:16) Eq. (2.25) | γ (2) , ¯ γ (2) , γ (2) in | Y − y (cid:17) Z (cid:16) Eq. (2.25) | γ (2) , ¯ γ (2) , γ (2) in | Y − y (cid:17) . – 10 –irst, we calculate N ( Y − y , Y − y ) at y = y . Using Eq. (2.35) and Eq. (2.36) we obtain from Eq. (2.13)that N ( Y − y , Y − y ) = ∞ X n =1 L − n (˜ g i ) n ! ( n −
1) ( − n T n ( Y − y ) (2.41)= T ddT (1 /T ) n ∞ X n =1 L − n (˜ g i ) n ! ( − n T n ( Y − y ) o = Z ∞ dξ e − ξ (cid:26) e − ξL ( Y − y ξ T ( Y − y (cid:18) − − ξT ( Y − y ) − ξL ( Y − y )(1 + ξT ( Y − y )) (cid:19)(cid:27) . (2.42)At L ( Y − y ) ≫ T ( Y − y ). The first four terms are given by: N ( L ( Y − y ) , T ( Y − y ) ; L ( Y − y ) , T ( Y − y )) = L ( Y − y )(1 + L ( Y − y )) (2.43) − L ( Y − y ) T ( Y − y )(1 + L ( Y − y )) − L ( Y − y ) T ( Y − y )(1 + L ( Y − y )) − L ( Y − y ) T ( Y − y )(1 + L ( Y − y )) − . . . = L ( Y − y )(1 + L ( Y − y )) − X n =2 n ! L n ( Y − y ) T n − ( Y − y )(1 + L ( Y − y )) n . (2.44)Note that all corrections have minus signs and the function of Eq. (2.42) gives the analytical summationof the asymptotic series of Eq. (2.43). For y = y we have a more complex answer, namely, N ( L ( Y − y ) , T ( Y − y ) ; L ( Y − y ) , T ( Y − y )) = (2.45) n T ( Y − y ) Γ ( L ( Y − y ) , T ( Y − y )) − T ( Y − y ) Γ ( L ( Y − y ) , T ( Y − y )) o ( T ( Y − y ) − T ( Y − y )) . The double inclusive cross section can be written as (see Fig. -a) d σdy dy = (2.46) a IP N ( L ( Y − y ) , T ( Y − y ) ; L ( Y − y ) , T ( Y − y )) N ( L ( y ) , T ( y ) ; L ( y ) , T ( y )) .
3. Correlations in a model for soft interactions
Recently considerable progress has been achieved in building models for soft scattering at high energies[8–13]. The main ingredient of these models is the soft Pomeron with a relatively large intercept ∆ IP = α IP − . − . α ′ IP ≃ . GeV − . Such a Pomeron appears in N=4SYM [14–18] with a large coupling. This is, at present , is the only theory that allows us to treat thestrong interaction on the theoretical basis. Having α ′ IP →
0, the Pomeron in these models has a naturalmatching with the hard Pomeron that occurs in perturbative QCD. Therefore, these models could be a– 11 – IP β g ( GeV − ) g ( GeV − ) m (GeV) m (GeV) γ G IP / ∆ IP ( GeV − )5 1.71 0.0045 0.03 Table 1:
Fitted parameters for our model. α ′ IP = 0 . GeV − . ) first step in building a selfconsistent theoretical description of the soft interaction at high energy, in spiteof its many phenomenological parameters (of the order of 10-15) in every model.In this section we shall discuss the size of the correlation function in our model [8–10]. This modeldescribes the LHC data (see Refs. [30–33]), including the single inclusive cross section. Thus our next stepis to try, to understand the predicted size of the long range rapidity correlations in this model. In Table 1 we present the main parameters of our model. The parameter T ( Y ) = γe ∆ IP Y is small in ourmodel reaching about 0.3 at the LHC energies. However, L i ( Y ; b ) = g i ( b ) G IP / ∆ IP e ∆ IP Y is large (seeRef. [8, 9]). g i ( b ) = g i S i ( b ) = g i π m i b K ( m i b ) . (3.1)One can see that L ( Y, b = 0) is as large as 25 at Y = 17 .
7. Therefore, we can evaluate the influence ofthe corrections with respect to T ( Y ), by calculating the contributions of two diagrams: Fig. -a (the maincontribution) and Fig. - b (the corrections ∝ T ( Y )). We need to use the first two terms of Eq. (2.43) tocalculate N ( L ( Y − y ) , T ( Y − y ) ; L ( Y − y ) , T ( Y − y )) while being careful to account for the correct b dependence.Introducing two functions,Γ (1) ( L i ( Y − y ; b )) = ∆ IP L i ( Y − y ; b )1 + L i ( Y − y ; b ) ; Γ (2) ( L i ( Y − y ; b )) = ∆ IP L i ( Y − y ; b )(1 + L i ( Y − y ; b )) . (3.2)We can see that Fig. -a has the following contributions: d σ (0) dy dy = Z d b n Z d b ′ Γ (1) (cid:16) L i (cid:16) Y − y ; ~b ′ (cid:17)(cid:17) Γ (1) (cid:16) L i (cid:16) y ; ~b − ~b ′ (cid:17)(cid:17) o × n Z d b ′ Γ (1) (cid:16) L i (cid:16) Y − y ; ~b ′ (cid:17)(cid:17) Γ (1) (cid:16) L i (cid:16) y ; ~b − ~b ′ (cid:17)(cid:17) o , (3.3)while for Fig. -b we have, for y > y : d σ (1) dy dy = − T ( Y − y ) (3.4) × Z d bd b ′ Γ (2) (cid:16) L i (cid:16) Y − y ; ~b − ~b ′ (cid:17)(cid:17) Γ (2) (cid:16) L i (cid:16) Y − y ; ~b − ~b ′ (cid:17)(cid:17) Γ (1) (cid:16) L i (cid:16) y ; ~b ′ (cid:17)(cid:17) Γ (1) (cid:16) L i (cid:16) y ; ~b ′ (cid:17)(cid:17) – 12 – y yy Figure 8:
The general diagram for the production of two hadrons(gluons) with rapidities y and y . Performing the calculations, we found that the correlation function R ( y = Y / , y = Y /
2) (see Eq. (1.5))is equal to R (0) ( y = Y / , y = Y /
2) = 13 at the Tevatron energy and R (0) ( y = Y / , y = Y /
2) = 16 at W = 7 T eV . The corrections turn out to be small ( < Eq. (3.3) is written without taking into account any corrections due to energy conservation. As hasbeen discussed in the 80’th (see Refs. [35, 36]), these corrections are important for the calculation of thecorrelations. Generally speaking, in Pomeron calculus the long range correlations in rapidity stem fromthe production of two hadrons from two different Pomerons (two different parton showers, see Fig. ). Inother words, two hadrons in the central rapidity region can be produced in an event with more than twoparton showers (see Fig. ). This is shown in Fig. -a in an eikonal type model, where the proton-protonscattering amplitude is written as: A ( s, b ) = i (cid:16) − e − Ω( s,b ) (cid:17) . (3.5)The cross section of n parton showers production is equal to (see Refs. [35, 36] and references therein) σ n − showers = Z d b Ω n ( s, b ) n ! e − Ω( s,b ) . (3.6)Eq. (3.6) shows that the parton showers are distributed according to Poisson distribution with an averagenumber of parton showers Ω ( s, b ) which has the following form in the simple model of Eq. (3.5):Ω ( s, b ) = Z d b ′ g (cid:0) b ′ (cid:1) g (cid:16) ~b − ~b ′ (cid:17) (cid:18) ss (cid:19) ∆ IP . (3.7)However, the simple Eq. (3.6) has to be modified to account for the fact that the energy of the partonshower is not equal to W = √ s , but it is smaller or equal to ˜ W = √ x x s (see Fig. ). The easiest way– 13 –o find x and x is to assume that both p = p = − ¯ Q ≫ µ soft, where µ soft is the scale of the softinteractions µ soft ∼ Λ QCD . In Ref. [37] we have argued that for a Pomeron ¯ Q ≈ GeV ≫ µ soft.Bearing this in mind, the energy variable x ( x ) for gluon-hadron scattering is equal to0 = ( x P + p ) = − ¯ Q + x p · P ; p = − ¯ Q ; x = ¯ Q M + ¯ Q . (3.8) p , P and x P are the momenta of the gluon, the hadron and the parton (quark or gluon) with whichthe initial gluon interacts. From Eq. (3.8) one can see that˜ s = x x S = s ¯ Q M (3.9)For the second parton shower ˜ s = ( q + q ) (see Fig. ), where q i = (cid:16) x q i P , x q i P , ~q i, ⊥ (cid:17) . Using P P p p MM s P MM s p p q q a) b) P P p p MM s P x P x P Figure 9:
Production of one(Fig. -a) and two(Fig. -b) parton showers. the conservation of momentum we see that x q = x + x g and x q = x + x g . Note that g and g denote the gluons with momenta p and p respectively (see Fig. ). Vectors p and p take the form: p = (cid:16) x g P , x g P , ~p , ⊥ (cid:17) and p = (cid:16) x g P , x g P , ~p , ⊥ (cid:17) . Bear in mind the following equations:( p + P ) = M ; x g = M s ; p = x g , x g s + p , ⊥ = ¯ Q ; x g < ¯ Q x g s = ¯ Q M ≪ x . ( p + P ) = M ; x g = M s ; p = x g , x g s + p , ⊥ , = ¯ Q ; x g < ¯ Q x g s = ¯ Q M ≪ x . (3.10)Therefore, the value of ˜ s for the second parton shower turns out to be the same as for the first one for M ≫ ¯ Q . The value of M can be estimated using the quark structure function as it has been suggestedin Ref. [37]. Indeed, h| M |i = R dM M M q (cid:16) ¯ Q M + ¯ Q , ¯ Q (cid:17)R dM M q (cid:16) ¯ Q M + ¯ Q , ¯ Q (cid:17) . (3.11)– 14 –sing ¯ Q = 1 GeV and q (cid:0) x, ¯ Q (cid:1) given by a combined fit [38] of H1 and ZEUS data (HERAPDF01) weobtain that h| M |i ≈ GeV which is much larger than ¯ Q .However, the scale of hardness ¯ Q in CGC/saturation approach is proportional to the saturation mo-mentum Q s ( ¯ Q ∝ Q s ) and, therefore , depends on energy. Such energy dependence of ¯ Q induces thedependence of average mass M on energy. Assuming that Q s ∝ s λ with λ = 0 .
24 we found that in theenergy range W = 0 . ÷ T eV the typical M = M ( W = 0 . T eV ) s β with β = 0 . h| M |i = M (cid:16) ss (cid:17) β with β = 0 . √ s = 0 . T eV . M is equal to 10 GeV . σ n − showers ( s ) = Z d b Ω n (˜ s, b ) n ! e − Ω( s,b ) . (3.12)We need to sum over n ≥ d σdy dy = 2 a IP X n =2 σ n − showers ( s ) = a IP Z d b Ω (˜ s, b ) e Ω(˜ s,b ) − Ω( s,b ) , (3.13)where, a IP is a new vertex defined as shown in Fig. -b). The factor 2 stems from the possibility to emita hadron with rapidity y from each of two parton showers. One can see that the double inclusive crosssection does not depend on y and y , leading to the long range rapidity correlation. In the model for soft interactions that has been suggested in Refs. [8–10] (GLM model) we evaluate morecomplicated sum of diagrams than in Eq. (3.5). The different contributions to the two particle correlationin this model are shown in Fig. . Eikonal diagrams:
In order to account for diffraction dissociation in the states with masses that are much smaller than theinitial energy, we use the simple two channel Good-Walker model. In this model we introduce two eigenwave functions, ψ and ψ , which diagonalize the 2x2 interaction matrix T , A i,k = < ψ i ψ k | T | ψ i ′ ψ k ′ > = A i,k δ i,i ′ δ k,k ′ . (3.14)The two observed states are an hadron whose wave function we denote by ψ h , and a diffractive state witha wave function ψ D , which is the sum of all the Fock diffractive states. These two observed states can bewritten in the form ψ h = α ψ + β ψ , ψ D = − β ψ + α ψ , (3.15)where, α + β = 1. For each state we sum the eikonal diagrams of Fig. -a using Eq. (3.5). The firstcontribution to Ω ( s, b ) is the exchange of a single Pomeron. However, the Pomeron interaction leads to amore complicated expression for Ω ( s, b ). – 15 – nhanced diagrams: In our model [10], the Pomeron’s Green function which includes all enhanced diagrams, is approximatedusing the MPSI procedure [23], in which a multi Pomeron interaction (taking into account only triplePomeron vertices) is approximated by large Pomeron loops of rapidity size of ln s . We obtain G IP ( Y ) = 1 − exp (cid:18) T ( Y ) (cid:19) T ( Y ) Γ (cid:18) , T ( Y ) (cid:19) , (3.16)in which: T ( Y ) = γ e ∆ IP Y . (3.17)Γ (0 , /T ) is the incomplete gamma function (see formulae in Ref. [29]). Yy 0 x a) Y0 b) Y0 c) a P y’y a P y y y y’y’’ x xx y x xx x x xxx xxx Figure 10:
Mueller diagrams for double inclusive production in the GLM model [8–10]. Crosses mark the cutPomerons. Γ ( y ) is given by Eq. (2.6). All rapidities are in the laboratory reference frame. Semi-enhanced (net) diagrams:
A brief glance at the values of the parameters of our model (see Ref. [8] and Table 1), shows that we havea new small parameter, T ( Y ) = G IP ( s/s ) ∆ IP ≪
1, while, L i ( Y, b ) = G IP g i ( b ) ( s/s ) ∆ IP ≈
1. Wecall the diagrams which are proportional to L ni ( Y, b ), but do not contain any of the T n ( Y, b ) contributions,net diagrams. Summing the net diagrams [9], we obtain the following expression for Ω i,k ( s, b ):Ω i,kIP ( Y ; b ) = Z d b ′ g i (cid:16) ~b ′ (cid:17) g k (cid:16) ~b − ~b ′ (cid:17) (cid:16) /γ G IP ( T ( Y )) (cid:17) G IP /γ ) G IP (cid:16) T ( Y ) (cid:17) h g i (cid:16) ~b ′ (cid:17) + g k (cid:16) ~b − ~b ′ (cid:17)i . (3.18) G IP is the triple Pomeron vertex, and γ = R d k t π G IP . Mueller diagrams for the different contributions to the double inclusive production are shown in Fig. .The diagram of Fig. -a is the same as we have discussed in section 3.1. The main ingredient for thiscontribution is Γ i ( Y ; b ), which is given by a slight modification of Eq. (3.2):Γ i ( Y − y, b ) = g i ( b ) γ G IP ( T ( Y − y ))1 + ( G IP /γ ) g i ( b ) G IP ( T ( Y − y )) . (3.19)– 16 –ntroducing, H ik ( Y ; Y ; b ) ≡ Z d b ′ Γ i (cid:16) ~b − ~b ′ , Y (cid:17) Γ k (cid:16) ~b ′ , Y (cid:17) , (3.20)we can rewrite the contribution of the diagram of Fig. -a in the form I ( y , y ) = a IP ( Z d b n α exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) H (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17) H (cid:16) ˜ Y / − y , Y / y ; b (cid:17) +2 α β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) H (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17) H (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17) + β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) H (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17) H (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17)o , (3.21)where, a IP is shown in Fig. . In Eq. (3.21) we used the following notations: Y = ln ( s/s ) and ˜ Y =ln (˜ s/s ). y and y are rapidities of the produced hadrons in the c.m.frame. For the contribution of thediagram of Fig. -b, we need to change H ik (cid:16) ˜ Y / − y , ˜ Y / y ; b (cid:17) in Eq. (3.21) to J ik ( y , y ; b ) whichis defined as J ik ( y , y ; b ) = Z ˜ Y ˜ Y / − y dy ′ Z d b ′ Γ i (cid:16) ˜ Y − y ′ ; ~b − ~b ′ (cid:17) G IP (cid:16) T (cid:16) y ′ − ˜ Y / y (cid:17)(cid:17) G IP (cid:16) T (cid:16) y ′ − ˜ Y / y (cid:17)(cid:17) × Γ k (cid:16) ˜ Y / − y , b ′ (cid:17) Γ k (cid:16) ˜ Y / − y , b ′ (cid:17) . (3.22)Therefore, this contribution takes the form I ( y , y ) = a IP G IP Z d b n α exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) J ( y , y ; b ) (3.23)+2 α β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) J ( y , y ; b ) + β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) J ( y , y ; b ) o . Introducing, K ik ( y , y ; b ) = Z d b ′ Z ˜ Y ˜ Y/ − y dy ′ Γ i (cid:16) ˜ Y − y ′ ; ~b − ~b ′ (cid:17) G IP (cid:16) T (cid:16) y ′ − ˜ Y / y (cid:17)(cid:17) G IP (cid:16) T (cid:16) y ′ − ˜ Y / y (cid:17)(cid:17) × Z ˜ Y / − y dy ′′ Γ k (cid:16) y ′′ ; ~b − ~b ′ (cid:17) G IP (cid:16) T (cid:16) ˜ Y / − y − y ′′ (cid:17)(cid:17) G IP (cid:16) T (cid:16) ˜ Y / − y − y ′′ (cid:17)(cid:17) . (3.24)We can reduce the contribution of the diagram of Fig. -c to the form I ( y , y ) = a IP G IP Z d b n α exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) K ( y , y ; b ) (3.25)+2 α β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) K ( y , y ; b ) + β exp (cid:16) Ω (cid:16) ˜ Y ; b (cid:17) − Ω ( Y ; b ) (cid:17) K ( y , y ; b ) o . Collecting all contributions, the long range rapidity correlation function has the following form:– 17 –(TeV) 0.9 1.8 2.36 7 R ( y = 0 , y = 0) 1.0 1.12 1.026 1.034 Table 2: R ( y = 0 , y = 0) versus energy. R ( η , η ) = h ( η,Q ) σ in ( Y ( η )) { I ( y ( η ) , y ( η )) + I ( y ( η ) , y ( η )) + I ( y ( η ) , y ( η )) } σ in ( Y ) dσdy σ in ( Y ) dσdy − . (3.26)Expressions for the single inclusive cross section σ in ( Y ) dσdy i as well as the Jacobian h and the definition ofthe pseudo-rapidity η can be found in Ref. [39]. Using the formulae of the previous section we calculate the correlations in the GLM model. It turns outthat R (0 ,
0) is a constant in the energy range W = 0.9 to 7 TeV, and it is equal to R (0 , ≈ C = h n i / h n i was measured for the rapidity window | η | < . C ≈
2. For this small range of rapidity, we can consider that C = R (0 ,
0) + 1. It is worthwhile mentioning that using our calculation of R (0 , σ n σ in = (cid:18) rr + h n i (cid:19) r Γ ( n + r ) n ! Γ ( r ) (cid:18) h n i r + h n i (cid:19) n . (3.27)In our model, given | η | ≤ . h n i = 5.8 (see Ref. [39]) and r = 1 .
25. Using this distribution we calculate C q = h n q i / h n i q . They equal C = 5.65, C = 21 .
18 and C =98.2. They are in good agreement withthe experimental data of Ref. [40] except C which experimentally is about 70. In Fig. we compareEq. (3.27) with the CMS experimental data at W = 7 T eV . In Fig. we plot the correlation function R ( η , η ) as a function of η . One can see that this function falls steeply at large η . At first sight, suchform of η dependence looks strange since all diagrams of Fig. generate long range rapidity correlations.It turns out that the main contribution comes from the enhanced diagram of Fig. -c. The eikonal-typediagram of Fig. -a leads to long range rapidity correlations which do not depend on the values of η and η . The diagram of Fig. -b gives a negligible contribution. Let us consider Fig. -c in a simple modelreplacing Γ( y ) by the exchange of the Pomeron, and considering all Pomeron exchanges as the exchangeof a ‘bare’ Pomeron. In this model the diagram of Fig. has the form: g p a IP G IP Z Yy dy ′′ Z y dy ” G IP (cid:0) Y − y ′ (cid:1) G IP (cid:0) y ′ − y (cid:1) G IP (cid:0) y ′ − y (cid:1) G IP (cid:0) y ′ − y (cid:1) × G IP ( y − y ”) G IP ( y − y ”) G IP ( y ”) = g p a IP G IP ∆ IP e IP Y (cid:16) − e ∆ IP ( y − Y ) − e ∆ IP ( y − Y ) − e ∆ IP ( − y ) + e ∆ IP ( y − y − Y ) (cid:17) , (3.28)– 18 – n / s (W=7 TeV)n -7 -6 -5 -4 -3 -2 -1 Figure 11:
Multiplicity distribution measured by CMS collaboration [40] and Eq. (3.27) with our parameters. where, we used G IP ( Y ) = exp (∆ IP Y ). Recalling that the single inclusive cross section dσ/dy = g p a p exp (∆ IP Y ),in this simple model, the correlation function of Eq. (1.5) is equal to R ( y , y ) = σ in G IP ∆ IP (cid:16) − e ∆ IP ( y − Y ) − e ∆ IP ( y − Y ) − e ∆ IP ( − y ) + e ∆ IP ( y − y − Y ) (cid:17) − . (3.29)In Fig. -b the correlation function is plotted with σ in G IP / ∆ IP = 2 and ∆ IP = 0 .
08 which correspond tothe effective behaviour of the dressed Pomeron in our model at high energies ( W = 1 . − T eV ). One cansee that simple formula of Eq. (3.29) reproduces the short-range correlation type behaviour of Fig. -a.
4. Conclusions
In this paper we taken the next step, following the single inclusive cross section [39], in the description of themulti particle production processes in the framework of our soft interaction model. The main ingredients– 19 – R( h , h ) (W=7 TeV) h =0 h =1 h =2 h R( h , h ) (W=7 TeV) h =0 h =1 h Fig. -a Fig. -b Figure 12:
Our prediction for R ( η , η ) versus η at different values of η at W = 7 T eV (Fig. -a) and theestimates of the simple model (see Eq. (3.28) with ∆ IP = 0 .
08) for the correlation function (Fig. -b). of our model are the large Pomeron intercept (∆ IP = 0 .
23 ) and α ′ IP = 0. The model gives a practicalrealization of the BFKL Pomeron Calculus in zero transverse dimensions. The model reproduces quite wellall classical soft scattering data: total, elastic and diffractive cross sections and the energy dependence ofthe elastic slope in wide range of energy W = 20 GeV to 7 TeV. The attraction of the Pomeron approachreveals itself in the possibility to discuss not only the forward scattering data but, also, to make predictionsrelating to multiparticle production processes using the AGK cutting rules [21].In this paper we have developed a procedure for calculating the correlation function in the MPSI ap-proximation utilizing the BFKL Pomeron Calculus in zero transverse dimensions. The theoretical formulaeobtained allow us to calculate the rapidity correlation function in our model for soft interactions. We com-pare our prediction with the multiplicity distribution at W = 7 TeV measured by CMS collaboration [40],which we describe quite well. In Fig. we present our prediction for the rapidity dependence of thecorrelation function.We believe that our approach opens the way to discuss the structure of the bias events without buildingMonte Carlo codes. At the moment we demonstrate that our model describes all standard soft data onforward scattering, inclusive cross sections and multiplicity distribution. We also predict the rapiditycorrelation function.We thank all participants of “Low x’2013 WS” for fruifful discussions on the subject. This research ofE.L. was supported by the Fondecyt (Chile) grant 1100648.– 20 – eferences [1] M. A. Braun, Phys. Lett. B632 (2006) 297 [arXiv:hep-ph/0512057]; Eur. Phys. J.
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