Azimuthal angle correlations at large rapidities: revisiting density variation mechanism

The European Physical Journal C, Nov 2017

We discuss the angular correlation present in hadron–hadron collisions at large rapidity difference (\(\bar{\alpha }_S\,y_{12}\gg \,1\)). We find that in the CGC/saturation approach the largest contribution stems from the density variation mechanism. Our principal results are that the odd Fourier harmonics (\(v_{2n+1}\)) decrease substantially as a function of \(y_{12}\), while the even harmonics (\(v_{2n}\)) increase considerably with the growth of \(y_{12}\).

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Azimuthal angle correlations at large rapidities: revisiting density variation mechanism

Eur. Phys. J. C Azimuthal angle correlations at large rapidities: revisiting density variation mechanism E. Gotsman 2 E. Levin 0 1 2 0 Centro Científico-Tecnológico de Valparaíso , Avda. Espana 1680, Casilla 110-V, Valparaiso , Chile 1 Departemento de Física, Universidad Técnica Federico Santa María , Valparaiso , Chile 2 Department of Particle Physics, Raymond and Beverly Sackler Faculty of Exact Science, School of Physics and Astronomy, Tel Aviv University , 69978 Tel Aviv , Israel We discuss the angular correlation present in hadron-hadron collisions at large rapidity difference (α¯ S y12 1). We find that in the CGC/saturation approach the largest contribution stems from the density variation mechanism. Our principal results are that the odd Fourier harmonics (v2n+1) decrease substantially as a function of y12, while the even harmonics (v2n ) increase considerably with the growth of y12. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Correlations in the momentum representation . . . . . 3 Single inclusive production in a one parton shower . . 3.1 BFKL Pomeron: the simplest approach for a one parton shower . . . . . . . . . . . . . . . . . . . 3.2 General estimates . . . . . . . . . . . . . . . . . 4 Double inclusive cross section for two parton shower production . . . . . . . . . . . . . . . . . . . . . . . 4.1 The simplest diagram . . . . . . . . . . . . . . . 4.2 The CGC/saturation approach . . . . . . . . . . 5 Azimuthal angle correlations . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . Appendix A: BFKL Pomeron in the mixed representation Appendix B: Calculation of the integrals for the contribution of the simplest diagram . . . . . . . . . . . . . Appendix C: Integration over dipole sizes in the CGC/saturation approach . . . . . . . . . . . . . . . Appendix D: Integration over dipole sizes in the angle correlation function . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . - Contents 1 Introduction In this paper we address the problem of the azimuthal angle correlations of two hadrons with transverse momenta pT 1 and pT 2 and rapidities y1 and y2, at large values of y12 ≡ |y1 − y1| 1/α¯ S . Our main theoretical assumption is that these correlations stem from interactions in the initial state. We are aware that, unlike rapidity correlations which at large rapidities originate from the initial state interactions due to causality reasons [ 1 ], a substantial part of these correlations could be due to the interactions in the final states [ 2–4 ]. On the other hand, it has been demonstrated that at small rapidity difference α¯ S y12 < 1 the interactions in the initial state [ 5–14 ] yield the value of the correlations, which describe the major part of the experimentally observed correlations [ 15– 37 ]. In this paper we concentrate our efforts on calculating the long range rapidity part of angular correlations with large value of the rapidity difference y12. All previous calculations assumed that α¯ S y12 < 1[ 5–14 ]. It turns out that in this kinematic region, the main source of the azimuthal angle correlations is the Bose–Einstein correlations of identical gluons, corresponding to the interference diagram in the production of two partonic showers. Intuitively, we expect that the correlations in the process, where two different gluons are produced from two different partonic showers, should not depend on the difference of rapidities (y12), nor on the values of y1 and y2. Using the AGK cutting rules [38]1 one 1 In the framework of perturbative QCD for the inclusive cross sections, the AGK cutting rules were discussed and proven in Refs. [ 44–52 ]. However, in Ref. [ 47 ] it was shown that the AGK cutting rules are violated for double inclusive production. This violation is intimately related to the enhanced diagrams [ 47–49,52 ] and to the production of gluon from the triple Pomeron vertex. It reflects the fact that different (y1, pT1) lines denote gluons. Figure 1c shows the example of a more complicated structure of the partonic cascades, than the exchange of the BFKL Pomeron. The color of the lines indicates the parton shower can prove that the two gluon correlations can be calculated using the Mueller diagrams [39] of Fig. 1. The diagrams of Fig. 1 lead to correlations which do not depend on y1 and y2, but only for α¯ S y12 1. For large y12 the contributions of Fig. 1 decrease. The main goal of this paper is to find the contributions which survive at large y12 (α¯ S y12 1). At large y12, we have to take into account the emission gluons, with rapidities y2 < yi < y1, which transform the Mueller diagram of Fig. 1b to the more general diagrams of Fig. 2. The general feature of Fig. 1b is that the lower Pomerons carry momenta QT + p12 and −QT − p12 with p12 = pT 1 − pT 2. QT denotes the momentum along the BFKL Pomeron. After integration over QT , we obtain p12 ∼ 1/ Rh , where Rh is the size of the target (projectile), which has a non-perturbative origin. Roughly speaking, the correlation function turns out to be proportional to G ( p12), where G denotes the non-perturbative form factor of the target or projectile [ 12 ]. This conclusion stems from the value of the typical QT for the BFKL Pomeron, which is determined by the size of the largest dipoles in the Pomeron. Figure 2 does not have these features. We will show that the azimuthal angle correlations originate from the integration over QT (see Fig. 2), due to the structure of the vertices of emission of the Footnote 1 continued cuts of the triple BFKL Pomeron vertex with the produced gluon lead to different contributions. We do not consider such diagrams. gluons with pT 1 and pT 2, which have contributions proportional to ( pT 1 · QT )n ( pT 2 · QT )n . Recall that these kinds of vertices are the only possibilities to obtain angular correlations in the classical Regge analysis [ 53 ]. This mechanism for azimuthal angular correlations was suggested in Ref. [ 54 ] (see also Refs. [ 10, 55–57 ]), and in the review of Ref. [57] it was called the density variation mechanism. The paper is organized as follows. In the next section we discuss the contribution of the diagram of Fig. 2 in the momentum representation. In the remainder of the paper, we will use the mixed representation: the dipole sizes and momentum transferred (QT ), which will be introduced in Sect. 3 and appendix A. Section 4 is devoted to the discussion of the single inclusive production in the Color Glass Condensate (CGC)/saturation approach. The double inclusive production is considered in Sect. 4, in which the rapidity dependence of the master diagram of Fig. 2 will be calculated. In Sect. 5, we estimate the angular correlation function and Fourier harmonics vn , and we present our prediction for the dependence of vn on the difference of rapidities (y12). In Sect. 6 we draw our conclusions and outline problems for future investigation. 2 Correlations in the momentum representation The double inclusive cross section of Fig. 2 takes the following form: d2σ (Fig. 2) dy1d2 pT 1 dy2d2 pT 2 2CF αS 2 = (2π )2 −QT )2 × N QT φHG −kT + QT , kT ; Y − y1 where φHG kT , −k + QT , as well as all other functions φ of this type, are the correlation functions, which at QT = 0 give the probability to find a gluon with transverse momentum kT in the hadron (nucleus) of the projectile (target). φ kT , −k + QT ; kT , −kT + QT describes the interaction of two gluons with momenta kT and kT , which scatter at momentum transferred QT . N QT is a pure phenomenological form factor that describes the probability to find two Pomerons in the projectile or target, with transferred moment QT and −QT . CF = Nc2 − 1 /2Nc where Nc is the number of colors. The Lipatov vertex μ (kT , pT 1) has the following form: 1 μ (kT , pT 1) = pT 1 2 Using Eq. (2.2) we obtain 2 ν −k + QT , pT 1 ν k − QT − QT , pT 1 kT2 pT 1,μ − kT,μ pT 1 . 2 (2.2) −k + QT kT − pT 1 − QT − QT 1 = pT 1 2 1 = pT 2 2 2 2 2 2 + −k + pT 1 − QT kT − QT − QT 2 μ −kT − pT 2 + QT , pT 2 μ kT − pT 2 − QT , pT 2 −kT − pT 2 + QT which in Eq. (2.1) is absorbed in the phenomenological form factors N (QT ) and N (QT ). Therefore, the typical QT and QT turn out to be of the order of the soft scale μsoft, which is much smaller that the other typical momenta in Eq. (2.1), assuming that PT 1 and PT 2 are larger than μsoft. Introducing (2.4) 2 μsoft = d2 Q T N QT (2 π )2 we can neglect QT and QT in the BFKL Pomeron Green functions and rewrite Eq. (2.1) in the form d2kT d2kT (2π )2 (2π )2 d(22πQ)T2 kT2 (kT −QT )2 × φHG −kT , kT ; Y − y1 × φHG kT −QT , −kT + QT ; Y − y1 × ν − k +QT , pT 1 ν k − QT , pT 1 × φ − kT , −kT + QT ; kt + pT 2, −kT − p2T − QT ; y12 ×φ − kT + pT 1, kT − pT 1 − QT ; kT + QT , kT − pT 2; y12 × φHG kT + QT , −kT − QT ; y2 G ×φH −kT + QT , kT ; y2 × μ(−kT − pT 2, pT 2) μ kT − pT 2, pT 2 , (2.5) 2 2 2 1 = pT 1 2 1 = pT 2 2 with Eq. (2.3), which takes the following form: 2 ν −k + QT , pT 1 ν k − QT , pT 1 −k + QT kT − pT 1 − QT + −k + pT 1 − QT y1 and y2, and its expression has the following form: The most economical way of calculating the diagram of Fig. 2, is to use the mixed representation of the BFKL Pomeron Green function, G r , R, QT , Y , where r and R are the sizes of two interacting dipoles, QT denotes the momentum transferred by the Pomeron, and Y the rapidity between the two dipoles. This Green function is well known [ 42,43 ], and for the completeness of presentation we discuss it in Appendix A, referring to Refs. [ 42,43,58–60 ] for all details. 3 Single inclusive production in a one parton shower 3.1 BFKL Pomeron: the simplest approach for a one parton shower The single inclusive cross section resulting from the one BFKL Pomeron is well known, and it is equal to d2σ 2CF αS dyd2 pT = (2π)2 (d22πk)T2 φHG kT , QT = 0; Y − y ×φHG kT − pT , QT = 0; y ν kT , pT ν −kT , pT . The relation between the parton densities φ and the Green function of the BFKL Pomeron has been given in Ref. [ 46 ]: N BFKL (r, r1; y, QT = 0) αS = 2 d2kT 1 − eikT ·r φHG kT , QT = 0; y kT2 (3.2) where N BFKL (r, r1; Y ) is given by Eq. (A.1) or by Eq. (A.9), in the high energy limit. Equation (3.2) can be rewritten as follows: We have d2r eikT ·r ∇r2 N BFKL (r, r1; y, QT = 0) . (3.3) kT2 kT − pT pT2 2 . ν kT , pT ν −kT , pT = Substituting Eq. (3.2) and also Eq. (3.4) into Eq. (3.1) we obtain [ 46 ] d2σ 8CF 1 dy d2 pT = αS (2π )2 pT2 × (r, r1; Y − y, QT = 0) ∇r2 NtBrFKL d2r ei pT ·r ∇r2 NpBrFKL × (r, r2; y, QT = 0) where Npr and Ntr denote the probability to find a dipole in the projectile and target, respectively. r1 and r2 are the typical dipoles sizes in the projectile and target. As can be seen from Eq. (2.1) we need to generalize Eq. (3.5) for the case QT = 0. Equation (3.1) has to be replaced by d2σ dyd2 pT (QT = 0) 2CF αS = (2π )2 (d22πk)T2 φHG kT , QT , Y − y × φHG kT − pT , QT ; y ν kT , pT ν −kT + QT , pT . 1 = 2 1 pT2 2 Taking into account Eq. (3.2) for QT = 0 and ν kT , pT ν −kT + QT , pT 2 −y, QT ) NtBrFKL (r, r2; y, QT ) . d2r ei pT ·r NpBrFKL (r, r1; Y (3.8) 3.2 General estimates It should be stressed that the single inclusive production has the form of Eqs. (3.5) and (3.8) for the general structure of the single parton shower, as was shown in Ref. [ 46 ]. For example, for the process shown in Fig. 1c. We need only to substitute NtGr (r, r2; y, QT ) for 2NtBrFKL (r, r2; y, QT ) where 2NtBrFKL (r, r2; y, QT ) → NtGr (r, r2; y, QT ) = 2Ntr (r, r2; y, QT ) − d2 QT Ntr r, r2; y, QT −QT Ntr r, r2; y, QT ; Ntr (r, r2; y, QT ) is a solution to the non-linear evolution equation. For the case of inclusive production, we can considerably simplify the estimates noting that r2 Qs2(y) 1 ∇r2 Ntr (r, r2; y, QT ) −−−−−−−→ NtBrFKL (r, r2; y, QT ) r2 Qs2(y) 1 ∇r2 Ntr (r, r2; y, QT ) −−−−−−−→ 0, where Qs (y) denotes the saturation momentum. In other words, the main contribution to inclusive produc tion comes from the vicinity of the saturation scale, where r 2 Q2 s ≈ 1. Fortunately, the behavior of N in this kinematic region is determined by the linear BFKL evolution equation [ 62–67 ] and has the following form [68]: Ntr (r, r2; y, QT = 0) ∝ r 2 Qs2(y) 1−γcr with Q2 s = (1/r22) exp = (1/r22)eκ y 1 ω γ = 2 + i ν = γcr y where γcr = 0.37. From Eq. (A.8) we see that, for QT = 0, the scattering amplitude decreases at Q4T r 2 r22 1. Therefore, we need to consider rather small values of QT : Q4T r 2 r 2 2 ≤ 1. The product of vertices that determines the amplitude has two terms (see Eq. (A.5)) which are proportional to r 2/r22 iν and to Q4T r 2 r22 I ν . Therefore, the maximum of ∇r2 N can be reached if r 2/r22eκ y ∼ 1 and Q4T r 2 r22eκ y ∼ 1 and the amplitude then has the following form: Ntr (r, r2; y, QT ) ∝ c1 + c2 Q4T r 2 r22 eκ y 1−γcr . r 2 eκ y r22 The first term does not depend on QT and, therefore, the upper limit of the integral over QT , goes up to QmTax 2 ≈ 1/(r r2). The second term, both for Q2T r r2 < e− 21 κ y and for Q2T r r2 > e− 21 κ y , turns out to be small. Indeed, in the first region the amplitude is small, while in the second region we are deep in the saturation domain where ∇r2 N → 0. Hence, we expect that in the integral over QT , the first term gives a larger contribution than the second term, and we will only keep this contribution in our estimates. 4 Double inclusive cross section for two parton shower production 4.1 The simplest diagram In this section we calculate the simplest diagram of Fig. 2. We need to integrate the product of two BFKL Pomerons over QT (see Eq. (2.5)): From Eq. (2.5) in the momentum representation, we see that r1 = r1 (r2 = r2), however, they are close to each other, being determined by the same momentum kT . We assume that pT 1 < kT , since kT ∼ Qs (Y − y1) μsoft. Considering r1 ≈ r1 r2 ≈ r2 we will show that in the integral over QT , the typical QT ∼ 1/r2. In other words, the dependence of QT is determined by the largest of interacting dipoles. From Eq. (A.8) we see that, for large QT , when r12 Q2 T 1 and r22 Q2T 1, the integrand is proportional to 1/Q4T and converges. The main region of interest is r22 Q2T 1 and r12 Q2T 1. In this kinematic region for the vertices Vν1 r1, QT and Vν2 r1, QT , we can use Eq. (A.6), while the conjugated vertices are still in the regime of Eq. (A.8). Eq. (4.1) then takes the form I = 26i(ν1+ν2) −26ν1 ν2 π The appearance of the pole ν1 = −ν2 indicates that the contribution from this kinematic region is large. Closing the contour of integration on ν2 over the pole, we obtain Actually, the double inclusive cross section depends on ∇2 N as we argued in the previous section. Repeating the procedure for Using the method of steepest descent, to integrate over ν1, we obtain the following contribution: d2σ d y1d2 pT 1 dy2d2 pT 2 2 In Eq. (4.8) we have neglected the terms which are pro portional to Q2T (see Eq. (2.5)), since, as we have argued, the typical QT are small, and because these terms do not lead to additional correlations in the azimuthal angles. In Appendix (4.4) 4.2 The CGC/saturation approach B we calculate this integral and obtain the final expression for the double inclusive cross section: The integral over kT in Eq. (B.6) has an infrared singularity with a cutoff at pT 2, since we assume that pT 2 is the smallest momentum. This reflects the principal feature of the BFKL Pomeron parton cascade, which has diffusion, both in the region of small and large transverse momenta. On the other hand, we know that the CGC/saturation approach suppressed the diffusion in the small momenta [ 44 ], providing the natural cutoff for the infrared divergency. We expect that such a cutoff will be the value of the smallest saturation momenta: Qs (Y − y1) or Qs (y2), which will replace one of the pT2 2 in the dominator of Eq. (4.9). Therefore, we anticipate that for a realistic structure of the one parton shower cascade, (see Fig. 1c for example), the contribution for the double inclusive cross section will be different. We need to specify the behavior of the scattering ampli tude in the vicinity of the saturation scale. We have discussed the basic formulas [ 68 ] of Eq. (3.11), but for integration over the dipole sizes we need to know the size of this region. The scattering amplitude can be written in the form where ω(γ , 0) is given by Eq. (A.2), replacing 21 + i ν ≡ γ and ξ = ln r12/r22 . The saturation scale is determined by the line on which the amplitude is a constant (C), of the order one. This leads to the following equation for the saturation scale [ 62,68 ]: ω (γcr , 0) Y − (1 − γcr )ξs = 0; ωγ (γ , 0) Y − ξs = 0, (4.11) which results in the value of γcr given by the equation ω (γcr , 0) = ωγ (γ , 0) and gives γcr = 0.37, with the equation for the saturation momentum: ξs ≡ ln Qs2 r22 = κY = 1ω −(γγcrc)r Y. (4.12) (4.13) Fig. 3 a − N (τ ) = − ∇2 N (τ ) = − 4 τ 1 d τ ddτ N (τ ) τ dτ versus τ for the behavior of the scattering amplitude deep in the saturation domain[ 72 ]. b The example of a more complicated structure of the partonic cascades than the exchange of the BFKL Pomeron, which are shown in Fig. 2. The color of the lines indicates the parton shower 0.8 Q’ (y1, pT1 ) (y2, pT2 ) Expanding the phase ω (γ , 0) Y − (1 − γ )ξ in the vicinity ξ = ξ − ξs and γ = γ − γcr we obtain N (r1, r2; Y ) = C −i∞ +i∞ dγ 2π = r12 Qs2 1−γcr C r12 Qs2 1−γcr π e− (4DξY)2 . DY d γ e 21 ωγγ (γ,0)Y ( γ )2+ γ ξ 2πi (4.14) At first sight, Eq. (4.14) shows that the amplitude has a maximum at τ = r12 Qs2 = 1. However, this is not correct. Equation (4.14) gives the correct behavior for τ < 1, while for τ > 1 we need to take into account the interaction of the BFKL Pomerons and the non-linear evolution, generated by these interactions. The general result of this evolution is the fact that the amplitude depends on one variable [ 69–72 ] τ , i.e. N (τ ) (as it shows geometric scaling behavior). The peak at τ = 1 appears in ∇r21 N (r1, r2; Y ) = 4 Qs2(Y ) 1 d d τ τ dτ dτ N (τ ) . (4.15) From Eq. (4.15) we can conclude that the width of the distribution in r12 is of the order of Qs2, but it depends crucially on the model for the Pomeron interaction. In Fig. 3a we plot this value for the behavior of the scattering amplitude deep in the saturation domain (see Ref. [ 72 ]). This approach is not correct for τ → 1 and −∇2 N = 1.58 at τ = 1, but it starts to be small at τ > 2, which could be large enough to trust the formulas of Ref. [ 72 ]. At least such a conclusion can be justified considering the fit of the DIS data in the saturation model of Refs. [ 73,74 ], which is based on the idea of Ref. [75], and which has the correct behavior of the scattering amplitude, both deep in the saturation domain, and near τ = 1. Hence, we expect that ∇2 N decreases faster than we can see from Eq. (4.14). Bearing these conclusions in mind, we will calculate the contribution of Fig. 2, keeping all N in Eq. (4.8) in the vicinities of the saturation scales, by replacing 0∞ dτ (−∇2 N ) = − 01 dτ (−∇2 N ). We will show in the following that we cannot integrate over the dipole sizes, so that all six Pomerons will be in the vicinity of the saturation scale. At least two of the Pomerons occur either deep in the saturation domain, or in the perturbative QCD region. We believe that the largest contribution stems from the exchange of two Pomerons between rapidities y1 and y2 (see Fig. 3b), which are in the perturbative QCD region. Unfortunately, we cannot use the AGK cutting rules [ 38 ], which state that these Pomerons will not be affected by the Pomeron interaction, and the contributions of these interactions (see the red Pomeron in Fig. 3b) are canceled. Indeed, it has been proven that for the double inclusive production [ 47 ] they are not applicable in perturbative QCD. On the other hand, these Pomerons carry transverse momentum QT , unlike the others in the diagram, which is larger than the saturation scale Qs (y2); hence, their contributions are suppressed in comparison with the other Pomerons in Fig. 2. In addition our choice leads to the natural matching with the region α¯ S y12 < 1. The integration over QT will produce the same result as Eq. (4.7), as in the previous section. In Appendix C we discuss making estimates for the integrals over the dipole sizes which lead for pT 1 Qs2 (y1) to the following cross section: d2σ dy1d2 pT 1 (QT = 0; Eq. (3.5)) 8CF 1 = αS (2π )2 p2 T ×∇r2 NtBrFKL (r, r2; y, QT = 0) = αS (2π )2 p12 C2 (4γ¯ 2)2 exp − 8CF T d2r ei pT ·r ∇r2 NpBrFKL (r, r1; Y − y, QT = 0) ln2 Qs2 (Y − y) /Q2 (y) 4D y pp W 13 TeV Fig. 4 The ratio of Eq. (4.17) at W=13 TeV versus y12, assuming that the experiment has a symmetric pattern with Y − y1 = y2 = 21 (Y − y12). The dotted line in a is for the estimates for the y12 dependence of the Bose–Einstein contribution at small y12 [ 11,76 ]. a and b The estimates in the leading order of perturbative QCD with α¯ S = 0.25. In c we take BFKL = 0.25 and Qs2(Y ) ∝ exp (λY ) with λ = 0.25. These numbers correspond to the BFKL phenomenology × × exp × exp One can see that Eq. (4.17) demonstrates the additional suppression due to the infrared cutoff at Qs (y2) instead of at pT 2, as taken in the calculation of the simplest diagram. The factor exp (2 BFKL y12) reflects the fact that two BFKL Pomerons between rapidities y1 and y2 are taken in the perturbative QCD region. It should be stressed that we can only trust our estimates for values of y12 at which the exchange of the BFKL Pomeron with rapidity y12 give a contribution smaller than C. This condition means that 1 (2 D y12) e2 BFKL y12 < C. (4.18) Taking BFKL = 0.25 and Qs2(Y ) ∝ exp (λY ) with λ = 0.25 (these values correspond to the BFKL phenomenology) we see that the l.h.s. of Eq. (4.18) is smaller than 0.15 for y12 ≤ 7. Therefore, we can trust our estimates shown in Fig. 4 for C > 0.15. We take C = 0.3, which leads to the contribution of the shadowing corrections of the order of 30%. The two last factors in Eq. (4.17) stem from the perturbative QCD nature of two Pomerons in Eq. (C.8) (see Eq. (4.16)). In Fig. 4 we plot the ratio R as a function of y12 for y12 ≤ 7 (see Eq. (4.18). One can see that the ratio increases for large y12. 5 Azimuthal angle correlations The azimuthal angle correlations stem from terms (QT · ri )n in the vertices (see Eqs. (A.5), (A.6)). Indeed, after integrating over ri these terms transform to expressions of the folm1 m2 , which lead lowing type [ 54 ]: QT · pT 1 QT · pT 2 to the term ( pT 1 · pT 2)m . We have illustrated in Eqs. (A.5) and (A.6) how these originate from the general form of the BFKL Pomeron vertices in the coordinate representation. From Eqs. (A.5) and (A.6) only terms proportional to n QT · ri with even n appear in the expansion. Therefore, the azimuthal angle (φ) correlation function contains only terms cos2n (φ), and it is invariant with respect to φ → π −φ. In other words, vn with odd n turn out to be zero. Hence, we have the first prediction: the value vn with odd n should decrease with y12, and their dependence should follow the dotted lines in Fig. 4a. We return to Eq. (4.1) and integrate over QT , collecting terms that depend on the angles between QT and ri , which we have neglected in the previous section. As we have learned, the typical values of QT ∝ 1/r2 ∼ 1/r2 where r2 and r2 are larger than r1 and r1. In other words, we showed that the main contributions stem from the kinematic regions r12 Qs2 (Y − y1) ∼ 1 ( r12 Qs2 (Y − y1) ∼ 1) and r22 Qs2 (y2) ∼ 1 ( r22 Qs2 (y2) ∼ 1). Assuming that Qs (Y − y1) Qs (y2) we conclude that r1(r1) r2(r2). The typical QT is determined by the largest dipoles and, therefore, we expect QT ≈ 1/r2(1/r2), as has been demonstrated above. Bearing these estimates in mind, we can replace vertices Vν1 r1, QT and Vν2 r1, QT in Eq. (4.1) by Eq. (A.6) in which we put QT = 1/r2 and QT = 1/r2, respectively. Taking into account that r1/r2 1(r1/r2 1), we obtain Vν1 r1, QT Vν2 r1, QT 1 + 28 1 + 28 QT · r1 r 2 −iν2 26 QT · r1 4 − 4 At first sight Eq. (5.1) should enter two angles between QT and r1 and r1, respectively. However, in the integrand for integration over ri (see Eq. (B.1)) it depends only on one vector pT 1. Therefore, after integration over all angles, we find that the angle φ in Eq. (5.1) is the angle between QT and pT 1. For vertices Vν∗1 r2, QT and Vν∗2 r2, QT in Eq. (4.1) we use Eq. (A.8). Finally, we need to evaluate the integral IQ = −16 ν1 ν2 QT d QT Vν1 r1, QT Vν2 r1, QT × Q2T −i(ν1+ν2) cos2 Q212TQrT22 · r2 cos QT · r2 r1 =r1=1/Qs(Y−y1) Eq. (5.1) (5.2) with better accuracy that we did in Sect. 5.1, keeping the dependence on the angle between QT and r2. Note that the factor cos QT · r2 comes from exp i QT · r2 in Eq. (4.8). Taking this integral we substitute for the terms in parentheses in Eq. (5.1), |QT | = 1/r22(1/r22). The integral is equal to IQ = 26i(ν1+ν2) −27ν1 ν2 where n = QT /QT , and φ2 is the angle between n and n2 = r2/r2. In Eq. (5.3) the terms in (. . . ) (. . . ) stem from the expansion with respect to r12/r22 1. However, for the terms in {. . . } there are no such small parameters, and we expand the function of φ2 in a Fourier series. Integrating over n one obtains 3 r14 + 212 r 4 (n1 · n2)4 + 2 2 n1 · n2 4 (5.4) where ϕ is the angle between pT 1 and pT 2. vn is calculated from vn,n ( pT 1, pT 2), 1. vn ( pT ) = 2. vn ( pT ) = vn,n ( pT , pT ); vn,n pT , pTRef vn,n pTRef , pTRef . Equations (5.8)-1 and (5.8)-2 depict two methods of how the values of vn have been extracted from the experimentally measured vn,n ( pT 1, pT 2), where pRef denotes T the momentum of the reference trigger. These two definitions are equivalent if vn,n ( pT 1, pT 2) can be factorized as vn,n ( pT 1, pT 2) = vn ( pT 1) vn ( pT 2). In this paper we use the definition in Eq. (5.8)-1. Introducing the angular correlation function as C ( pT , φ) ≡ we obtain vn,n = In Eq. (4.17) we have calculated the part of C ( pT , φ) which does not depend on φ, which coincides with C ( pT , φ = 0) = R of Eq. (4.17) for Qs (Y − y1) Qs (y2). To calculate the contribution to C , which depends on φ, we need to take the separate integrals over ν1 and ν2, since the terms, which are proportional to cos2 (φ) and cos4 (φ) do not have a pole at ν1 = −ν2 (see Eq. (5.5)). These integrations lead to the following extra factor in C ( pT , φ) − C ( pT , φ = 0): C ( pT , φ) − C ( pT , φ = 0) pT2 pT2 ∝ R Qs2 (Y − y1) Qs2 (Y − y1) R = 2 ξ 2 1 (2 D y12)3 exp −2ξ 2/ (4 D y12) C ( pT , φ = 0) ; (5.11) where ξ = ln Qs2 (Y − y1) /Qs2 (y2) . We took factors proportional to pT from the expression for A (kT , pT 1) and A(4) (kT , pT 1) putting pT 1 = pT 2 = pT . To find the final correlation function and v2,2 and v4,4, we need to collect all numerical factors that come from A (kT , pT 1), A(4) (kT , pT 1) and Eq. (5.6), and to integrate over φ, as given in Eq. (5.9). Note that in the case of symmetric kinematics, where Y − y1 = y2 = 21 (Y − y12), ξ = 0 and Eq. (D.7 vanishes. In this case, we have to use Eq. (A.5) instead of Eq. (A.6), keeping track of the corrections, which are (5.8) (5.9) (5.10) proportional to νi . As a result, we can consider ξ = 0 in Eq. (D.7), but we need to replace the factor ξ 2 by 1. at α¯ S y12 Equations (4.17) and (D.7) contain numerical uncertainties, which stem both from the values of the soft parameters μ˜ soft and μsoft, as well as the values of the saturation scale at low energies, and from the integration in Eqs. (C.3) and (C.5), which were taken neglecting contribution from the region τ < 1. On the other hand, the contribution to the double inclusive cross sections of the diagram of Fig. 2 1 coincides with the contribution of Fig. 1b, d2σ (Fig. 2) dy1 dy2d2 pT 1d2 pT 2 α¯ S y12→ 1 −−−−−−→ d2σ (Fig. 1b) dy1 dy2d2 pT 1d2 pT 2 (5.12) Therefore, to obtain the realistic estimate we use the following procedure of matching: where v2 ( pT = 5 GeV) |Fig. 1b and v4 ( pT = 5 GeV) |Fig. 1b are taken from Ref. [ 14 ] where the estimates were performed based on the model for soft interaction which describes all features of the soft interaction at high energy and provides an interface with the hard processes. Figure 5 shows the pT and y dependence of the v2 and v4 using Eq. (D.8) for normalization. In addition we take BFKL = 0.25 and Qs2(y) ∝ exp (λ y) with λ = 0.25. These values correspond to the BFKL Pomeron phenomenology. We believe that this figure illustrates the scale of rapidity dependence and will be instructive for future experimental observations. 6 Conclusions In this paper we generalize the interference diagram that described the Bose–Einstein correlation for small rapidity difference α¯ S y12 1, to include the emission of the gluons with rapidities (yi ) between y1 and y2 (y1 , yi < y2). We calculate the resulting diagram in the CGC/saturation approach and make two observations, which we consider as the main result of this paper. The first one is a substantial decrease of the odd Fourier harmonics v2n+1 as a function of the rapidity difference y12 ( see Fig. 4c). The second result is that even Fourier harmonics v2n have a rather strong dependence on y12, showing a considerable increase in the region of large y12 (see Fig. 5). We believe that our calculations, which have been performed both for the simplest diagrams Fig. 5 vn versus pT (a, c) and versus y (b, d) at W = 13 TeV assuming that the experiment has a symmetric pattern with Y − y1 = y2 = 21 (Y − y12). In all these figures we use Eq. (D.8) for normalization and and for the CGC/saturation approach, will be instructive for further development of the approach especially in the part that is related to the integration of the momenta transferred by the BFKL Pomerons. We demonstrated in this paper the general origin of the density variation mechanism, whose nature does not depend on the technique that has been used. This mechanism has to be taken into account, since it leads to the values of the Fourier harmonics that are large and of the order of vn, which have been observed experimentally. We hope that the paper will be useful in the clarification of the origin of the angular correlation, especially for hadron–hadron scattering at high energy. We firmly believe that the experimental observation of both phenomena, the sharp decrease of vn with odd n and the substantial increase of vn with even n as a function of y12, will be a strong argument for the CGC/saturation nature of the angular correlations. Acknowledgements We thank our colleagues at Tel Aviv university and UTFSM for encouraging discussions. Our special thanks go to Carlos Contreras, Alex Kovner and Misha Lublinsky for elucidating discussions on the subject of this paper. This research was supported by the BSF Grant 2012124, by Proyecto Basal FB 0821(Chile), Fondecyt (Chile) Grant 1140842, and by CONICYT grant PIA ACT1406. pp W 13 TeV 0 1 2 3 4 5 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: BFKL Pomeron in the mixed representation In this appendix we discuss the BFKL Pomeron Green function G r , R, QT , Y in the mixed representation, where r and R are the sizes of two interacting dipoles, QT denotes the momentum transferred by the Pomeron, and Y the rapidity between the two dipoles. This Green function has the following form [ 42,43,58–60 ]: G r , R, QT ; Y r R ∞ = 16 n=−∞ −∞ ×Vν,n r , QT ∞ dν 1 ν2 + 41 (n − 1)2 ν2 + 41 (n + 1) Vν∗,n R, QT eω(ν,n) Y (A.1) where ω (ν, n) = 2 α¯ SRe ψ ω (ν, 0) = 2 α¯ SRe ψ where ψ (z) is the Euler ψ -function (see Ref. [ 61 ] formulas 8.36) and BFKL = α¯ S4 ln 2, D = α¯ S14ζ (3), ξ = ln r12/r22 . Each term in Eq. (A.1) has a very simple structure, being the typical contribution of the Regge-pole exchange: the product of two vertices and Regge-pole propagator. From Eq. (A.2) one can see that at large Y the main contribution comes from the term with n = 0, and in what follows we will concentrate on this particular term. The vertices with n = 0 have been determined in Refs. [ 42,43,58–60 ], and they have an elegant form in the complex number representation for the point on the two dimensional plane, viz., For r (x , y) : ρ = x + i y; ρ∗ = x − i y; For QT (Qx , Q y ) : q = Qx + i Q y ; q∗ = Qx − i Q y . (A.3) Using this notation the vertices have the following structure: Vν r , QT = Q2T iν × J−iν − Jiν At QT → 0 this vertex takes the form QT r 1 r2 −iν 26iν Vν r, QT −−−−−→ 26 ⎛ (ν + i) 8(QT · r)4 − 8(QT · r)2 Q2T r2 + 5 21 Q4T r4 + (2i + ν)Q4T r4) × ⎝ 642(ν + 2i)(1 − iν)2 + i(ν + i) (2(QT · r)2 − Q2T r2 32(1 − iν)2 ⎞ + 1⎠ + Q2 iν Q2r2 iν 26 ⎛ (ν − 2i) 8(QT · r)4 − 8(QT · r)2 Q2T r2 + 5 21 Q4T r4 ) × ⎝ 212((2 + iν)(1 + iν))2 + 2(1 + iν) (2(QT · r)2 − Q2T r2 26(1 + iν)2 For small values of ν (which are related to the region of large α¯ SY 1), Eq. (A.5) can be simplified and reduced to the form 26iν Vν r , QT QT r 1 −−−−−→ − − − Using 2(QT · r )2 − Q2T r 2 25 dν 1 ν2 + 1/4 2 Y 1;ν 1 −−−−−−→ 2 r1 r2 −α¯ S14ζ (3)ν2 Y eω(ν,0)Y 2 dν exp ((α¯ S 4 ln 2 r22 iν r12 = r1 r2 D2πY exp ξ 2 BFKLY − 4DY r22 iν r12 (A.6) (A.7) 4 i ν . QT r (A.8) (A.9) (A.5) where BFKL and D are defined in Eq. (A.2). In the derivation of Eq. (A.9) we neglected the contri−iν butions that are proportional to Q4T2r1222 r12 , since this contribution will be the same as in Eq. (A.9), but with ξ = ln Q4T2r1222 r12 1. To integrate over ν, we use the method of steepest descent, and the expansion of ω (ν, 0) at small ν (diffusion approximation; see the second equation in Eq. (A.2)). N (r1, r2; Y ) denotes the imaginary part of the dipoledipole sacttering amplitude at QT = 0, which is related to the cross section. One can check that Eq. (A.9) has the correct dimension. Appendix B: Calculation of the integrals for the contribution of the simplest diagram In this appendix we discuss the integrations in Eq. (2.5). The integral over QT has been considered in Sect. 4.1 and it has the form of Eq. (4.7). The extra eir1·QT give an additional numerical factor, replacing 25 by 27 in Eq. (4.7). To integrate over kT and kT we replace ! dφi e−i pT 1·r˜1 δ(2) r1 + r1 − r˜1 − r˜1 → (2π )4 kT dkT J0 (kT r1) J0 |kT + pT 1| r˜1 J0 (kT r˜1) J0 (kT r˜1) . 0 r˜2 0 ∞ Now we can take the integrals over ri bearing in mind Eq. (4.7) and N pr (ri , Y − y1) = dν 2π 2 2 21 +iνi eω(νi ,0)(Y −y1). μsoft ri The integrals over r˜1and r˜1 have the following form (see Ref. [ 61 ], Eq. 6.511(6)): 1 J0 (kT r˜1) dr˜1 = r˜2 J0 (kr˜2) + 2 πr˜2 J1 (kr˜2) H0 (kr˜2) − J0 (kr˜2) H1 (kr˜2) = Using Eq. (B.2) we obtain r˜2 1 k if kr˜2 if kr˜2 Collecting Eqs. (B.2), (B.3) and (B.4) we see that the main contribution stems from the region kr˜2 1 and the integral over kT has the form r˜2r˜ 2 2 pT1 We see that the integrals over r1 and r1 lead to r1 ∼ 1/Qs (Y − y1) and r1 ∼ 1/Qs (Y − y1). The same holds for the integrals over r2 and r2, leading to r2 ∼ 1/Qs (y2) and r2 ∼ 1/Qs (y2). Assuming that Qs (Y − y1) > Qs (y2) we conclude that ri and ri are much smaller than r2 and r2. Replacing ∇r21 Npr (r1; Y − y1) ei QT ·r˜ 1 ∇r˜21 Npr r˜1; Y − y1 r12 Qs2 (Y − y1) γ¯ r12 Qs2 (Y − y1) γ¯ (C.2) (C.1) 1 1 = Qs 1 + 2γ¯ where γ¯ = 1 − γcr , we see from Eq. (C.1) that integration over r takes the form We need to estimate the diagram of Fig. 1a (see Eq. (2.7)). This diagram can be rewritten as (C.8) Examining Eq. (3.5), one can see that in the general case when Y − y1 = y1 and Y − y2 = y2 all four Pomerons cannot be in the vicinity of the saturation scale. Actually we have two kinematic regions which give the maximal contributions (assuming Qs (Y − y1) > Qs2 (y1)): 1. r 2 Qs2 (Y − y1) ≈ 1 but r 2 Qs2 (y1) → Qs2 (Y − y1) / Qs2 (y1) 1; 2. r 2 Qs2 (y1) ≈ 1 but r 2 Qs2 (y1) → Qs2 (y1) /Qs2 (Y − y1) 1. In region 1 the upper Pomeron is in the vicinity of the saturation scale, while the lower Pomeron is in the perturbative QCD region. In region 2 the lower Pomeron is in the vicinity of the saturation scale, and the upper Pomeron is deep inside the saturation domain. As we have discussed (see Fig. 3a) ∇2 N decreases in the saturation region much faster than in the perturbation QCD region and, therefore, we assume that the kinematic region 1 gives the largest contribution. Hence, for pT 1 Qs2 (y1) we obtain Eq. (4.16). Appendix D: Integration over dipole sizes in the angle correlation function In this appendix we collect tedious integration over the dipole sizes in Eq. (5.6). Each term in this equation can be factorized as a product of two functions which depend on r 1i and on r 2i. Bearing this feature in mind we calculate each term going to the momentum representation using Eq. (C.1). We obtain a product of functions of kT . Each of these functions has the following general form: j d2r eikT ·r ! rμi F (r ) i=1 = (−i ∇kT ) j = 2π (−i ∇kT ) j d2r eikT ·r F (r ) d2r J0 (kT r ) F (r ) . (D.1) Using Eq. (C.5) for pT 2 be reduced to 1 (1 + 2γ¯ )2 × ln τ τ 0 1 dτ 2 1 F2 = 3.50. Qs (y2) the integral over kT can Finally, collecting all numerical coefficients, we obtain where the constant C is the value of the amplitude at τ = 1. This contribution is proportional to ∝ e2 BFKL y12 /Qs2 (Y − y1) for pT 1 Qs (Y − y1) and pT 2 Qs2 (Y − y1) > Qs2 (y2). Qs (y2). Note that As we have seen the dependence on ri stems from the integration over QT or, in other words, from (IQ ). In IQ the dependence on r1 and r1 can be extracted explicitly, leading to F (r ) ∝ 1/r . Hence the momentum image for Eq. (D.1) has a simple form: The expressions for A and B can be written in a general form. Assuming that both pT 1 and pT 2 are smaller than Qs (y2), we can expand the answer, taking into account only terms that are proportional to pT2 1/ kT2 and pT2 2/ kT2. We obtain j i=1 = d2r eikT ·r ! rμi F (r ) = 2π (−i ∇kT ) (D.2) . For j = 2 and j = 4, which we need to calculate considering Eq. (5.6), we have (−i ∇kT ) (−i ∇kT ) . Note that, for integration over r1, Eq. 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E. Gotsman, E. Levin. Azimuthal angle correlations at large rapidities: revisiting density variation mechanism, The European Physical Journal C, 2017, 773, DOI: 10.1140/epjc/s10052-017-5350-3