Non-eikonal corrections to multi-particle production in the color glass condensate

Eur. Phys. J. C (2019) 79:600 https://doi.org/10.1140/epjc/s10052-019-7097-5 Regular Article - Theoretical Physics Non-eikonal corrections to multi-particle production in the color glass condensate Pedro Agostini1, Tolga Altinoluk2, Néstor Armesto1,a 1Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain 2National Centre for Nuclear Research, 00-681 Warsaw, Poland Received: 19 February 2019 / Accepted: 1 July 2019 / Published online: 16 July 2019 © The Author(s) 2019 Abstract We consider the non-eikonal corrections to particle production in the color glass condensate stemming from the relaxation of the shockwave approximation for the target that acquires a finite longitudinal dimension. We derive a modified expression of the Lipatov vertex which takes into account this finite target width. This expression is employed to compute single, double and triple gluon production in the Glasma graph limit valid for the scattering of two dilute objects, at all orders in the expansion in the number of colors. We justify and generalize previous results, and discuss the possible implications on two particle correlations of these non-eikonal corrections that induce differences between the away- and near-side peaks. 1 Introduction Particle production at high energies in the soft and semihard regimes is usually computed resourcing to high energy approximations [1], namely the eikonal approximation. This is the case in the color glass condensate (CGC) [2–4]. In this framework, the process of propagation of an energetic parton from the projectile through the target, considered as a background field, is computed in the light cone gauge neglecting its transverse components and considering it as infinitely time dilated and Lorentz contracted (thus treated as a shockwave), see for example the discussion in [5]. Also terms subleading in energy (among them, spin flip ones) are neglected. On the other hand, in the calculation of elastic and radiative energy loss of energetic partons traversing a medium composed of coloured scattering centers – jet quenching – the shockwave approximation is relaxed and the target is considered to have ae-mail: nestor[email protected] a finite length, see e.g. the reviews [6,7].1In this context, a systematic expansion of the gluon propagator in non-eikonal terms was done in [5,11] and applied to particle production in the CGC in [12]. Non-eikonal corrections at high energies have also been treated recently in the context of Transverse Momentum Distributions and spin physics [13–19], and soft gluon exponentiation [20–22]. In the CGC, particle production and correlations have been computed within several approximation schemes, providing an alternative explanation to final state interactions for the ridge phenomenon observed in small systems, proton–proton and proton–nucleus, at the Large Hadron Collider (LHC) at CERN [23–36] and the Relativistic Heavy Ion Collider (RHIC) at BNL [37–41]. The “Glasma graph” approximation [42,43], suitable for collisions between two dilute objects like proton–proton and containing both Bose enhancement and Hanbury–Brown–Twiss effects [44–47], has been used to describe experimental data [48–51], and to compute three and four gluon correlations [52,53]. Quark correlations have also been calculated in this framework [54,55]. It was later extended to dilute–dense (proton–nucleus) collisions both numerically [56] and analytically [57–59], and used to calculate three gluon correlations [58]. A description of data has been obtained [60,61]. Density gradients [62] have also been considered to explain the observed azimuthal structure. Beyond the analytical extension to dense–dense collisions, the remaining key theoretical problem for the description of azimuthal structure in small systems in the CGC lies in odd harmonics that are absent in usual calculations. For this, density corrections in the projectile [63–65], quark correlations [60,66,67] and a more involved description of the target [68,69] than the one provided by the commonly used McLerran–Venugopalan (MV) model [70,71], have been 1The relation between jet quenching and CGC calculations, using the formalism in [8,9], was established in [10] where the validity of the eikonal approximation for this type of computations was also addressed. 123 600 Page 2 of 27 Eur. Phys. J. C (2019) 79 :600 Fig. 1 Diagrams that contribute to the computation of the Lipatov vertex. The black dot represents the Lipatov vertex which is the sum of all real diagrams for gluon production shown on the right hand side of the equation proposed. Using the former, a description of data is possible [72–74]. In this manuscript we deal with non-eikonal corrections to particle production in the CGC that stem from relaxing the shockwave approximation for the target, which becomes of finite length. These are the corrections included in jet quenching calculations and systematically expanded up to next-to-next-to-leading order in [5,11]. In Sect. 2we derive an expression for the Lipatov vertex – one central building block for particle production calculations in the CGC – that takes into account the finite longitudinal extent of the target field. While by itself this result is not new and similar calculations and expressions can be found in the literature, see e.g. Refs. [75,76] or more recently in Ref. [77], its identification for use to include non-eikonal corrections in CGC calculations is done here for the first time. Then, in Sect. 3we apply our corrections to gluon production in the dilute-dilute (Glasma graph) limit, following the notations in [58]. First, in Sect. 3.1 we consider single gluon production, matching the results in [12] and justifying the educated guess done there on the basis of the expansion up to next-to-next-to-leading order. Then we consider double gluon production in Sect. 3.2, where we generalize the results in [12]. Third, in Sect. 3.3 we compute three gluon production. Finally, in Sect. 4we discuss our results. We focus on providing analytical expressions and show a few numerical results; a more complete study of the impact of non-eikonal corrections on particle correlations is left for a forthcoming study [78]. 2 Derivation of the non-eikonal Lipatov vertex As usually done in the CGC, we describe a high energy p-A collision by a right moving dilute projectile which interacts with a left moving dense target described by a random and intense (O(1/g)) classical gluon field Aμ(x). The simplest setup to derive the non-eikonal Lipatov vertex is considering the emission of a gluon from a projectile massless quark in the process of a single scattering with the target (an analogous calculation leading to the same conclusions on the noneikonal corrections holds for a projectile gluon). In light cone coordinates a±=(a0±a3)/√2 and in the light cone gauge (n·A=A+=0, n=(0,1,0⊥)in (+,−,⊥)coordinates), this field can be written as Aμ(x)≈δμ−δ(x+)A−(x⊥), (1) since the transverse component of the gluon field is not altered by the large Lorentz γfactor, the x−dependence disappears due to the time dilatation and the target is shrinked to x+=0 forming a shock-wave. However, in some applications these suppressed terms may be sizeable. For this reason, in this note we will relax the infinite boost approximation, in order to calculate the corresponding non-eikonal corrections to the usual Lipatov vertex computed at O(g2). To proceed, we analyze gluon production in p-A collisions in the quark initiated channel and compute the Lipatov vertex, which is an effective vertex that takes into account all the real contributions to gluon production. For that one needs to sum the amplitudes where the gluon is emitted before, during and after the interaction with the field as shown in Fig. 1. Our setup is such that the right moving quark with momentum p+k−qis generated by some function J(p+k−q)= J(p++k+−q+)at x+ 0=−∞and (x− 0,x0⊥)=0, and then interacts with the classical gluon field Aμ(x)generated by one scattering source located at x1, picking up a momentum q. However, since we are interested in non-eikonal corrections, we consider Aμ(x)with an x+dependence which has a finite support instead of treating it as a shockwave at x+=0, but we still assume that there is no dependence on x−. That is, the new form of Eq. (1)is Aμ(x)≈δμ−Aμ(x+,x⊥), (2) or, in momentum space, Aμ(q)≈δμ−2πδ(q+)A−(q−,q⊥). (3) Furthermore, we assume that the outgoing quark has a large momentum p+compared to all other momenta in the process. The general strategy in this case is to keep the leading terms in +-momenta in the numerator algebra, while taking the full phase corrections coming from the integration of the denominators, see below, as done in the Furry approximation and its non-abelian generalization [79]. We start by computing diagram A where the gluon is emitted with momentum kbefore the quark interaction with the target field as shown in Fig. 2. Using the Feynman rules, we find that the amplitude for fixed gluon and final quark momenta is 123 Eur. Phys. J. C (2019) 79 :600 Page 3 of 27 600 Fig. 2 Diagram A where the gluon is emitted before the interaction of the quark with the target field iMA=¯u(p)(−igγμta) d4q (2π)4Aa μ(q)eiqx1i(/ p−/ q) (p−q)2+i(−igγνtb)b∗ ν(k) ×i(/ p+/ k−/ q) (p+k−q)2+iei(p+k−q)x0J(p+k−q), (4) with tathe SU(Nc)generators in the fundamental representation. Since p+is the largest momentum in our problem, we approximate / p−/ q≈/ pand / p+/ k−/ q≈/ pand write iMA≈¯u(p)ei(p+k)x0g2tatb d4q (2π)4 / Aa(q)/ p/ b∗(k)/ p [(p−q)2+i][(p+k−q)2+i] eiq(x1−x0)J(p++k+−q+). (5) Using again the eikonal approximation (p+much larger than all other momenta), we can approximate (p−q)2≈ −2p+q−and (p+k−q)2≈2p+(k−−q−). Employing / a/ b=2a·b−/ b/ aand the massless Dirac equation ¯u(p)/ p=0, we get ¯u(p)/ Aa(q)/ p/ b∗(k)/ p=¯u(p)4(p·Aa(q))(p·b∗(k)). Therefore, the amplitude for diagram A can be written as iMA≈−¯u(p)ei(p+k)x0g2tatb d4q (2π)4 (p·Aa(q))(p·b∗(k)) [p+q−−i][p+(k−−q−)+i] eiq(x1−x0)J(p++k+−q+) =−¯u(p)ei(p+k)−x+ 0g2tatb d2q⊥ (2π)2e−iq⊥x1⊥(p·b∗(k)) dq+ 2πeiq+x− 1J(p++k+−q+)(2π)δ(q+) ×dq− 2π eiq−(x+ 1−x+ 0)p+A−a(q−,q⊥) (p+)2[q−−i][k−−q−+i],(6) Fig. 3 Diagram B where the gluon is emitted after the interaction of the quark with the target field where in the last line we used Eq. (3) and we have set x0⊥= x− 0=0. Performing the q+and q−integrals we obtain iMA≈−¯u(p)eipx0J(p++k+)g2tatb d2q⊥ (2π)2e−iq⊥x1⊥p·b∗(k) × ieik−x+ 0A−a(0,q⊥)−eik−x+ 1A−a(k−,q⊥) p+k−(x+ 1−x+ 0). (7) Since the outgoing gluon is on-shell, k−=k2 ⊥/2k+and, furthermore, in the light cone gauge we have ∗−(k)= kii/k+. Therefore, making use of pμ∗ μ≈p+∗−,we obtain iMA≈2i¯u(p)eipx0J(p++k+)g2tatb(x+ 1−x+ 0)kibi k2 ⊥ ×d2q⊥ (2π)2e−iq⊥x1⊥ eik−x+ 1A−a(k−,q⊥)−eik−x+ 0A−a(0,q⊥).(8) Now, sending x+ 0→−∞we can finally write iMA≈2i¯u(p)eipx0J(p++k+)g2tatbeik−x+ 1kibi k2 ⊥ d2q⊥ (2π)2e−iq⊥x1⊥A−a(k−,q⊥). (9) Now we proceed to calculate diagram B where the gluon is emitted with momentum kafter the interaction of the quark with the target field, as shown in Fig. 3. Following the previous procedure we find 123 600 Page 4 of 27 Eur. Phys. J. C (2019) 79 :600 Fig. 4 Diagram C where the emitted gluon interacts with the target field iMB≈−2i¯u(p)eipx0J(p++k+)g2tbtaeik−x+ 1kibi k2 ⊥ d2q⊥ (2π)2e−iq⊥x1⊥A−a(k−,q⊥). (10) Diagram C, shown in Fig. 4, where the emitted gluon interacts with the target field, requires dealing with the three-gluon vertex. Applying the Feynman rules we have iMC=¯u(p)(−igγμta)d4q (2π)4 i(/ p+/ k−/ q) (p+k−q)2+i ei(p+k−q)x0J(p+k−q) ×−idμα(k−q) (k−q)2+iVανβ abc Ac β(q)b∗ ν(k)eiqx1,(11) where Vανβ abc =gfabc gαν(q−2k)β+gνβ (k+q)α+gβα (k−2q)ν]is the three-gluon vertex and dμα(k)=gμα − kμnα+kαnμ k·nthe gluon propagator in the light cone gauge. Considering Eq. (3), we only need the Vαν+ abc component of the vertex. Furthermore, using the Dirac equation and the gamma matrices anti-commutation relation we have that ¯u(p)γ μ/ p=¯u(p)2pμ. Thus, iMC=−2i¯u(p)ei(p+k)x0g2ta d4q (2π)4 pμdμα(k−q)Vαν+ abc b∗ ν(k) [(p+k−q)2+i][(k−q)2+i] A−c(q)eiq(x1−x0)J(p++k+−q+). (12) After some algebra we find, in the eikonal approximation, pμdμα(k−q)Vαν+ abc b∗ ν(k)≈−2gfabc p+(k−q)i·bi , (13) and (k−q)2=−2k+q−−k2 ⊥−(k−q)2 ⊥ 2k+ +2q+q−−2k−q+≈ −2k+q−−k2 ⊥−(k−q)2 ⊥ 2k+.(14) Thus, defining ˜ k=k2 ⊥−(k−q)2 ⊥ 2k+, we get iMC≈−i¯u(p)g2tafabcei(p+k)x0 d4q (2π)4(k−q)ibi eiq(x1−x0) k+[k−−q−+i][q−−˜ k−i] A−c(q)J(p++k+−q+). (15) Using Eq. (3) and performing the q+and q−integrals we obtain iMC≈2¯u(p)J(p++k+)eipx0g2tafabc d2q⊥ (2π)2 (k−q)i (k−q)2 ⊥ bi e−iq⊥x1⊥ ×ei˜ kx+ 1 ei(k−−˜ k)x+ 0A−c(˜ k,q⊥) −ei(k−−˜ k)x+ 1A−c(k−,q⊥)(x+ 1−x+ 0). (16) Finally, making use of itafabc =[tb,tc]and sending x+ 0→−∞, we obtain iMC≈−2i¯u(p)J(p++k+)eipx0g2[ta,tb] d2q⊥ (2π)2 (k−q)i (k−q)2 ⊥ bi eik−x+ 1A−a(k−,q⊥)e−iq⊥x1⊥. (17) Summing up the three diagrams we get i(MA+MB+MC) ≈−2i¯u(p)J(p++k+)eipx0g2[ta,tb] d2q⊥ (2π)2Li(k⊥,q⊥)bi eik−x+ 1A−a(k−,q⊥)e−iq⊥x1⊥, (18) where Li(k⊥,q⊥)=(k−q)i (k−q)2 ⊥−ki k2 ⊥ (19) is the eikonal Lipatov vertex. We see that in our calculation, as announced, the non-eikonal corrections result in the sum of the amplitudes simply picking up a phase (important for 123 Eur. Phys. J. C (2019) 79 :600 Page 5 of 27 600 k−x+ 1∼1 with k−∝k2 ⊥e−ηand negligible for k2 ⊥x+ 1/k+ 1 where we recover the eikonal result) that can be absorbed in a redefinition of the Lipatov vertex. Therefore, we define a non-eikonal Lipatov vertex Li NE(k,q⊥;x+ 1)=(k−q)i (k−q)2 ⊥−ki k2 ⊥eik2 ⊥ 2k+x+ 1,(20) with k≡(k−,k⊥). As stated in the Introduction, this result is not new by itself and similar calculations and expressions can be found in the literature, e.g. in Refs. [75,76]orlaterinRef.[77]. But the identification of this building block for its use to include non-eikonal corrections in CGC calculations is done here for the first time. Note that using the non-eikonal expression of the gluon propagator from [5,11], the two first terms of the expansion of the exponential were obtained in [12] and the exponential form guessed. 3 Multi-particle production In the previous section, we have presented the derivation of the non-eikonal Lipatov vertex. Now, we would like to use this expression in order to calculate multi-gluon production cross section at mid rapidity within the Glasma graph approach in order to study the effects of finite target width corrections to those observables. The double and triple inclusive gluon production cross sections in p-A collisions have been recently studied in [52,53] in the Glasma graph approximation, and in [58] going beyond it, i.e. taking into account multiple scattering effects of the dense target. For each observable, the contributions to Bose enhancement of the projectile gluons and HBT contributions of the final state gluons are identified. However, the studies in [52,53,58] are performed within the eikonal approximation without taking into account the corrections due to the finite longitudinal width of the target. In the rest of this section, we take this extra step. Namely, we first expand the single, double and triple inclusive gluon production cross section in powers of the background field of the target which actually corresponds to the original Glasma graph approach. Then, we introduce the non-eikonal Lipatov vertex (20) in the expanded cross sections and get the explicit expressions of the Bose enhancement and HBT contributions beyond the strict eikonal limit for the double and triple inclusive gluon production. Hereafter, in order to alleviate the notation we will drop the ⊥for denoting transverse coordinates and momenta. 3.1 Single inclusive gluon production beyond the eikonal approximation Within the CGC framework, the production cross section of a gluon with transverse momenta kand rapidity ηcan be written dσ d2kdη=4παsz¯z eik(z−¯z)xy Ai(x−z)Ai(¯z−y) ρa(x)ρb(y)PUz−UxacU† ¯z−U† ycbT, (21) where ρa(x)≡ρa xis the colour charge density of the projectile, ···P(T)denote the average over the projectile (target) colour configurations and Aiis the standard Weizäcker- Williams field that is defined as Ai(x−y)=−1 2π (x−y)i (x−y)2=d2p (2π)2e−ip·(x−y)pi p2. (22) Moreover, we have introduced a short hand notation for the transverse coordinate integrals z=d2z. Here, Uab xis the adjoint Wilson line in the colour field of the target representing the scattering matrix of a gluon at transverse position x, whose explicit expression reads Uab x=Peigdx+Tc ab A− c(x+,x),(23) with Tc ab being the SU(Nc)generator in the adjoint representation and A− c(x+,x)the colour field of the target. The Wilson line operator accounts for the multiple scattering effects of the gluon in its interaction with the target. However, as mentioned previously, the Glasma graph approach for double (or multiple) gluon production corresponds to the dilute limit of the target. Therefore, we expand the Wilson lines to first order in the colour field of the target: Uab(x)≈1+igTc ab dx+A− c(x+,x) =1+igTc ab dx+d2q (2π)2eiqx A− c(x+,q). (24) Using Eq. (24) we can write the single inclusive gluon production cross section in the dilute limit as dσ d2kdηdilute =4παsz¯z eik(z−¯z)xy Ai(x−z)Ai(¯z−y) ρa(x)ρb(y)P ×g2dx+ 1dx+ 2d2q1 (2π)2 d2q2 (2π)2 123 600 Page 6 of 27 Eur. Phys. J. C (2019) 79 :600 A− c(x+ 1,q1)A− ¯c(x+ 2,q2)T(TcT¯c)ab e−iq1·z−e−iq1·xeiq2·¯z−eiq2·y.(25) We can now perform the colour averaging over the projectile colour charge densities. For the correlator of two projectile colour charge densities, we use the generalized MV model and write it in the following general form: ρa(x)ρb(y)P=δab μ2(x,y). (26) Inserting Eq. (26) into the expression for the dilute limit of the single inclusive production cross section given in Eq. (25) and integrating over transverse coordinates, we can simply write the dilute limit of the single inclusive production cross section as dσ d2kdηdilute =4πα sCAg2dx+ 1dx+ 2 d2q1 (2π)2 d2q2 (2π)2δc¯cA− c(x+ 1,q1)A− ¯c(x+ 2,q2)T ×μ2k−q1,q2−kLi(k,q1)Li(k,q2), (27) where Li(k,q)is the strict eikonal Lipatov vertex (19). At this point, the effects of finite longitudinal width of the target can be implemented in the single inclusive gluon production cross section. Effectively, the implementation of these effects corresponds to two modifications in the cross section given in Eq. (27). The first modification is to replace the eikonal Lipatov vertices by the non-eikonal ones derived in Sect. 2: Li(k,q)→Li NE(k,q;x+). (28) The non-eikonal Lipatov vertex given in Eq. (20) takes into account the finite longitudinal width of the target to all orders as discussed in Sect. 2. The second modification that is needed to account for the finite longitudinal width of the target is adopting a modified expression for the correlator of two target fields. Since the target has finite longitudinal width, the target fields can be located at two different longitudinal positions. Therefore, for the correlator of two target fields, we consider a generalization of the MV model in which the two colour fields are located at different longitudinal coordinates and are connected via gauge links along the longitudinal axis [12]. In that case, the colour field correlator of two fields can be written as A− c(x+ 1,q1)A− ¯c(x+ 2,q2)T=δc¯cn(x+ 1) 1 2λ+λ+−|x+ 1−x+ 2|(2π)2δ(2)(q1−q2)|a(q1)|2, (29) where λ+is the colour correlation length in the target and much smaller than the total longitudinal width of the target L+. Moreover, function n(x+)defines the one dimensional target density along the longitudinal axis. For simplicity of the calculation, we assume that this function is constant with a finite support, n(x+)=n0for 0 ≤x+≤L+and 0 elsewhere. Finally, function a(q)that appears in the definition of the two field correlator is the functional form of the potential in momentum space which is usually taken to be a Yukawa type potential in jet quenching calculations [6,7,75,76]: |a(q)|2=m2 q2+m22,(30) with msome Debye screening mass or inverse colour correlation length. We would like to emphasise that in the limit of vanishing correlation length λ+together with a constant potential a(q)and a constant longitudinal target density n(x+ 1), the two target field correlator defined in Eq. (29) reduces to the standard MV model correlator. By implementing these two modifications in the single inclusive gluon production cross section and using the expression of the non-eikonal Lipatov vertex given in Eq. (20) together with the two field correlator introduced in Eq. (29), we can write the non-eikonal generalization of the dilute limit of the single inclusive gluon cross section which accounts for the finite longitudinal width of the target as dσ d2kdη NE dilute =4πα sCA(N2 c−1)g2 d2q (2π)2μ2k−q,q−kLi(k,q)Li(k,q)a(q) 2 ×n0 1 2λ+L+ 0 dx+ 1 x+ 1+λ+ x+ 1−λ+ dx+ 2eik2 2k+(x+ 1−x+ 2).(31) In this expression the non-eikonal Lipatov vertex is incorporated via the phase that appears under the longitudinal coordinate integral, the θ-function provides the limits of the integral in x+ 2and the one dimensional target density along the longitudinal axis is taken to be constant, n0for 0 ≤x+ 1≤L+. The integrations over the longitudinal coordinates x+ 1and x+ 2 can be performed in a straight forward manner and the final result for the dilute limit of the non-eikonal single inclusive gluon production cross section reads dσ d2kdη NE dilute =4πα sCA(N2 c−1)g2(noL+) GNE 1(k−;λ+)d2q (2π)2μ2k−q,q−k Li(k,q)Li(k,q)a(q) 2,(32) 123 Eur. Phys. J. C (2019) 79 :600 Page 7 of 27 600 where we have used the fact that λ+L+for the integration over the longitudinal coordinates. Here, GNE 1(k−,λ +)is the function that encodes all the non-eikonal information of the single inclusive gluon production and reads GNE 1(k−;λ+)=1 k−λ+sin(k−λ+), (33) with k−=k2 2k+. We would like to emphasize that the factor (n0L+)in Eq. (32) stands for the the number of scattering centres inside the finite longitudinal extend L+of the target. In the dilute target limit, we only take account one single scattering both in the amplitude and in the complex conjugate amplitude. Therefore, in this limit this factor will be set to one hereafter and we get dσ d2kdη NE dilute =4πα sCA(N2 c−1)g2GNE 1(k−;λ+) d2q (2π)2μ2k−q,q−kLi(k,q)Li(k,q)a(q) 2. (34) Equation (34) is the final result for the dilute target limit of the non-eikonal single inclusive gluon production cross section. Note that in the limit of vanishing correlation length λ+one can expand the non-eikonal single inclusive production cross section to second order in (k−λ+)which corresponds to the single inclusive gluon production cross section at next-to-next-to-eikonal accuracy and the result coincides, as announced, with the expression derived in [12]. Before we conclude this subsection, let us comment on the relative importance of the non-eikonal corrections, that are accounted for in Eq. (34) via the function GNE 1(k−;λ+)that encodes the non-eikonal effects, with respect to the eikonal limit of the single inclusive gluon production cross section in the dilute target limit. First of all, in the limit of vanishing (k−λ+),wehave lim k−λ+→0 GNE 1(k−;λ+)=1 (35) and we recover the well known eikonal limit of the single inclusive gluon production in the limit of the dilute target. In Fig. 5, we have plotted the ratio of the non-eikonal to eikonal single inclusive gluon production cross sections, (33), as a function of the transverse momenta of the produced gluon at fixed pseudorapidity η=2 for different values of the colour correlation length λ+. In the limit of vanishing transverse momenta of the produced gluon, the non-eikonal and eikonal cross sections coincide and the ratio becomes one as expected. The ratio shows up to 20% relative weight of the non-eikonal corrections for λ+=1 fm, for smaller values of λ+the results show a suppression from a few to up to 10%. In Fig. 6, we have plotted the ratio of the non-eikonal to eikonal single inclusive gluon production cross sections, (33), as a function of pseudorapidity for different values of Fig. 5 The ratio of non-eikonal to eikonal single inclusive gluon production cross sections, (33), as a function of the transverse momenta of the produced gluon for different values of the correlation length λ+,at fixed pseudorapidity η=2 Fig. 6 The ratio of non-eikonal to eikonal single inclusive gluon production cross sections, (33), as a function of the pseudorapidity of the produced gluon for different values of its transverse momenta at a fixed correlation length λ+=0.5fm the transverse momenta of the produced gluon at a fixed correlation length λ+=0.5 fm. The ratio of the non-eikonal to eikonal cross sections goes to one with increasing pseudorapidity as expected, since the relative importance of the non-eikonal corrections should vanish for large values of η. The results show that up to pseudorapidity η=2.5, depending on the value of the transverse momenta of the produced gluon, the relative weight of the non-eikonal corrections can vary roughly between 15% and 2%. These results confirm our analytical predictions for the importance of the non-eikonal corrections in certain kinematical regions. 3.2 Double inclusive gluon production beyond the eikonal approximation In this Subsection we consider double inclusive gluon production beyond the eikonal approximation. Our strategy for this subsection is the same as the calculation performed for single inclusive gluon production in the previous Subsection. Namely, we start with the double inclusive gluon produc- 123 600 Page 8 of 27 Eur. Phys. J. C (2019) 79 :600 tion cross section that takes into account multiple scatterings in the target in [58]. Then, we consider the dilute target limit of this expression which effectively corresponds to the Glasma graph approximation by expanding the dipole and quadrupole operators in powers of the background field of the target. Finally, we introduce the finite longitudinal width of the target effects via the non-eikonal Lipatov vertex Eq. (20) and the generalised MV model for the two field correlator Eq. (29) in the expanded expression of the double inclusive gluon production cross section. The general expression for the production of two gluons with pseudorapidities η1and η2, and with transverse momenta k1and k2reads dσ d2k1dη1d2k2dη2=α2 s(4π)2 ×z1¯z1z2¯z2 eik1·(z1−¯z1)+ik2·(¯z2−y2) ×x1x2y1y2 Ai(x1−z1)Ai(¯z1−y1)Aj(x2−z2)Aj(¯z2−y2) ×ρa1 x1ρa2 x2ρb1 y1ρb2 y2PUz1−Ux1a1c ×U† ¯z1−U† y1cb1Uz2−Ux2a2dU† ¯z2−U† y2db2T.(36) In the dilute limit of the target, or equivalently in the Glasma graph approximation, the Wilson lines are expanded in powers of the background field of the target as in Eq. (24). Therefore, in the dilute target limit double inclusive gluon production cross section can be written as dσ d2k1dη1d2k2dη2dilute =α2 s(4π)2 ×z1¯z1z2¯z2 eik1·(z1−¯z1)+ik2·(¯z2−y2) ×x1x2y1y2 Ai(x1−z1)Ai(¯z1−y1)Aj(x2−z2)Aj ×(¯z2−y2)ρa1 x1ρa2 x2ρb1 y1ρb2 y2P ×g4dx+ 1dx+ 2dx+ 3dx+ 4d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 ×A− a(x+ 1,q1)A− b(x+ 2,q2)A− c(x+ 3,q3)A− d(x4,q4)T ×(TaTb)a1b1(TcTd)a2b2e−iq1·z1−e−iq1·x1 ×eiq2·¯z1−eiq2·y1e−iq3·z2−e−iq3·x2 ×eiq4·¯z2−eiq4·y2.(37) Let us now perform the averaging of the double inclusive production cross section with respect to the colour charge densities of the projectile. Since we are using a generalized MV model, the average of any product of the colour charge densities factorize into products of all possible Wick contractions: ρa1 x1ρa2 x2ρb1 y1ρb2 y2P=ρa1 x1ρa2 x2Pρb1 y1ρb2 y2P +ρa1 x1ρb1 y1Pρa2 x2ρb2 y2P +ρa1 x1ρb2 y2Pρa2 x2ρb1 y1P.(38) For the correlator of two colour charge densities, we use the generalized MV model introduced in Eq. (26). After implementing Eq. (38), the dilute limit of the double inclusive gluon production cross section can be written as a sum of three contributions: dσ d2k1dη1d2k2dη2dilute =α2 s(4π)2g4 ×z1¯z1z2¯z2 eik1·(z1−¯z1)+ik2·(¯z2−y2) ×x1x2y1y2 Ai(x1−z1)Ai(¯z1−y1)Aj(x2−z2)Aj(¯z2−y2) ×trTaTbTdTcμ2(x1,x2)μ2(y1,y2) ×+trTaTbtrTcTdμ2(x1,y1)μ2(x2,y2) ×+trTaTbTcTdμ2(x1,y2)μ2(x2,y1) ×dx+ 1dx+ 2dx+ 3dx+ 4d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 ×A− a(x+ 1,q1)A− b(x+ 2,q2)A− c(x+ 3,q3)A− d(x4,q4)T ×e−iq1·z1−e−iq1·x1eiq2·¯z1−eiq2·y1 ×e−iq3·z2−e−iq3·x2 ×eiq4·¯z2−eiq4·y2.(39) In order to preserve the consistency of the notations introduced for different contributions in [58], here after we refer to the first contribution as Type A, the second one as Type B and the last one as Type C, in Eq. (39). Let us focus on Type A contributions to the dilute limit of the double inclusive gluon production cross section and adopt the same procedure applied in single inclusive gluon production in order to incorporate the non-eikonal effects due to the finite longitudinal thickness of the target. The same procedure and arguments hold for the calculation of Type B and Type C contributions. After integrating over the transverse coordinates, the Type A contribution can be written as dσType A d2k1dη1d2k2dη2dilute =α2 s(4π)2g4trTaTbTdTc ×dx+ 1dx+ 2dx+ 3dx+ 4 ×d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 123 Eur. Phys. J. C (2019) 79 :600 Page 9 of 27 600 ×A− a(x+ 1,q1)A− b(x+ 2,q2)A− c(x+ 3,q3)A− d(x4,q4)T ×μ2k1−q1,k2+q4μ2q2−k1,−k2−q3 ×Li(k1,q1)Li(k1,q2)Lj(k2,−q3)Lj(k2,−q4). (40) Moreover, we can factorize the the average of the colour fields of the target into all possible Wick contractions and write it in the following factorized way: A− a(x+ 1,q1)A− b(x+ 2,q2)A− c(x+ 3,q3)A− d(x4,q4)T =A− a(x+ 1,q1)A− b(x+ 2,q2)TA− c(x+ 3,q3)A− d(x4,q4)T +A− a(x+ 1,q1)A− d(x4,q4)TA− c(x+ 3,q3)A− b(x+ 2,q2)T +A− a(x+ 1,q1)A− c(x+ 3,q3)TA− b(x+ 2,q2)A− d(x4,q4)T. (41) We can now incorporate the non-eikonal effects due to the finite width of the target. This is achieved by replacing the Lipatov vertices by the non-eikonal ones and using the generalized MV model for the correlator of two target fields as defined in Eq. (29). After implementing these two modifications, the Type A contribution to the dilute limit of the non-eikonal double inclusive gluon production cross section reads dσType A d2k1dη1d2k2η2 NE dilute =α2 s(4π)2g4C2 A(N2 c−1) ×d2q1 (2π)2 d2q2 (2π)2a(q1) 2a(q2) 2dx+ 1dx+ 2dx+ 3dx+ 4 ×eik− 1(x+ 1−x+ 2)+ik− 2(x+ 4−x+ 3) ×μ2k1−q1,k2−q2μ2q1−k1,q2−k2 ×Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) ×1 2λ+n(x+ 1)λ+−|x+ 1−x+ 2| ×1 2λ+n(x+ 3)λ+−|x+ 3−x+ 4| ×+ μ2k1−q1,k2+q1μ2q2−k1,−k2−q2 ×Li(k1,q1)Li(k1,q2)Lj(k2,−q2)Lj(k2,−q1) ×1 2λ+n(x+ 1)λ+−|x+ 1−x+ 4|1 2λ+n(x+ 2) ×λ+−|x+ 2−x+ 3| ×+ 1 2μ2k1−q1,k2−q2μ2q2−k1,q1−k2 ×Li(k1,q1)Li(k1,q2)Lj(k2,q1)Lj(k2,q2) ×1 2λ+n(x+ 1)λ+−|x+ 1−x+ 3| ×1 2λ+n(x+ 2)λ+−|x+ 2−x+ 4|,(42) where we have used the following colour identities trTaTaTbTb=C2 A(N2 c−1), (43) trTaTbTaTb=1 2C2 A(N2 c−1), (44) with CA=Ncthe quadratic Casimir in the adjoint representation. Now, the integral over the longitudinal coordinates can be performed in the same way as in the single inclusive gluon production. After using the -functions to determine the limits of the integrals, a straightforward integration gives dσType A d2k1dη1d2k2η2 NE dilute =α2 s(4π)2g4C2 A(N2 c−1) ×d2q1 (2π)2 d2q2 (2π)2a(q1) 2a(q2) 2GNE 1(k− 1;λ+)GNE 1(k− 2;λ+) ×μ2k1−q1,k2−q2μ2q1−k1,q2−k2 ×Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) +GNE 2(k− 1,−k− 2;L+)μ2k1−q1,k2+q1μ2 ×q2−k1,−k2−q2Li(k1.q1)Li(k1,q2)Lj(k2,−q2)Lj(k2,−q1) +1 2GNE 2(k− 1,k− 2;L+)μ 2k1−q1,k2−q2μ2 ×q2−k1,q1−k2Li(k1,q1)Li(k2,q2)Lj(k2,q1)Lj(k2,q2), (45) where, on top of the function GNE 1(k−;λ+)that takes into account the non-eikonal effects defined in Eq. (33), we have introduced a new function GNE 2(k− 1,k− 2;L+)that also accounts for the non-eikonal effects in the dilute target limit of the double inclusive gluon production cross section and reads GNE 2(k− 1,k− 2;L+) =2 k− 1−k− 2L+sin k− 1−k− 2 2L+2 . (46) Again, this function goes to 1 when we consider the shockwave (eikonal) limit L+→0. The same procedure can be adopted to calculate Type B and Type C contributions to the dilute target limit of the noneikonal double inclusive gluon production cross section. The results read dσType B d2k1dη1d2k2η2 NE dilute =α2 s(4π)2g4C2 A(N2 c−1) d2q1 (2π)2 d2q2 (2π)2a(q1) 2a(q2) 2GNE 1(k− 1;λ+)GNE 1(k− 2;λ+) ×(N2 c−1)μ 2k1−q1,q1−k1μ2k2−q2,q2−k2 ×Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) +GNE 2(k− 1,k− 2;L+)μ 2k1−q1,q2−k1μ2k2−q2,q1−k2 123 600 Page 16 of 27 Eur. Phys. J. C (2019) 79 :600 The first term in this equation is a contribution to the forward Bose enhancement of the gluons q2and q3in the target wave function with a forward contribution to HBT of gluons k1and k2. The second term in Eq. (72) is a contribution to the forward Bose enhancement of the gluons q2 and q3in the target wave function with a backward contribution to the Bose enhancement of the gluons k1−q1and k2−q2in the projectile wave function. Finally, the third piece of I(2) 2tr,3is proportional to μ2k2−q3,q2−k2⎧ ⎪ ⎩μ2k1−q2,q3−k3 μ2q1−k1,k3−q1 +1 2μ2k1−q2,k3−q1μ2q1−k1,q3−k3⎫ ⎪ ⎭ ∝F|q2−q3|R⎧ ⎪ ⎩F2|k1−k3|R +1 2F2|(k1−q1)+(k3−q3)|R⎫ ⎪ ⎭.(73) The first term here is a contribution to the forward Bose enhancement of the gluons q2and q3 in the target wave function together with a contribution to the forward HBT of the gluons k1and k3. The second term in Eq. (73) is a contribution to the forward Bose enhancement of the gluons q2and q3in the target wave function together with a contribution to backward Bose enhancement to the gluons k1−q1and k3−q3in the projectile wave function. The symmetry partner of the I(2) 2tr,3that is defined in Eq. (A30) can be identified easily in the same manner. (iii) Finally, we can analyze the terms that are originate from the single-trace contribution. They are four of them: (a) The first one, I(2) 1tr,1, is defined in Eq. (A34) with its symmetry partners given in Eq. (A33). The first term is proportional to μ2k1−q1,k2−q2⎧ ⎪ ⎩μ2k3−q3,q1−k1 μ2q2−k2,q3−k3 +μ2k3−q3,q2−k2μ2q1−k1,q3−k3⎫ ⎪ ⎭ ∝F|(k1−q1)+(k2−q2)|R ⎧ ⎪ ⎩F|(k3−q3)−(k1−q1)|R F|(k2−q2)+(k3−q3)|R +F|(k3−q3)−(k2−q2)|R F|(k1−q1)+(k3−q3)|R⎫ ⎪ ⎭.(74) Clearly, the first term in this equation is a contribution to backward Bose enhancement of the gluons k1−q1and k2−q2together with contribution to forward Bose enhancement of the gluons k1−q1and k3−q3as well as a contribution to backward Bose enhancement of the gluons k2−q2and k3−q3, all in the projectile wave function. The second term in Eq. (74)is a contribution to backward Bose enhancement of the gluons k1−q1and k2−q2together with a contribution to forward Bose enhancement of the gluons k3−q3and k2−q2as well as a contribution to backward Bose enhancement of the gluons k1−q1and k3−q3, all in the projectile wave function. The symmetry partners of this term are given in Eq. (A33) and, again, they can be easily identified by using the same procedure. (b) The second term that originates from the singletrace contribution, I(2) 1tr,2, is defined in Eq. (A36) with its symmetric partners given in Eq. (A35). This term has four pieces and the first piece is proportional to μ2k1−q2,k2−q1 ⎧ ⎪ ⎩ 1 2μ2k3−q3,q1−k1μ2q2−k2,q3−k3 +1 2μ2k3−q3,q2−k2μ2q1−k1,q3−k3⎫ ⎪ ⎭ ∝F|(k1−q1)+(k2−q2)|R ⎧ ⎪ ⎩ 1 2F|(k3−q3)−(k1−q1)|R F|(k2−q2)+(k3−q3)|R +1 2F|(k3−q3)−(k2−q2)|R F(k1−q1)+(k3−q3)⎫ ⎪ ⎭.(75) The first term in this equation is a contribution to backward Bose enhancement of the gluons k1−q1and k2−q2as well as k2−q2and k3−q3in the projectile wave function together with a forward contribution to Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function. The second term in Eq. (75) is a contribution to backward Bose enhancement of the gluons k1−q1and k2−q2as well as k1−q1and k3−q3in the projectile wave function together with a forward contribution to Bose enhancement of the gluons k2−q2and k3−q3in the projectile wave function. The second piece of I(2) 1tr,2is proportional to 123 Eur. Phys. J. C (2019) 79 :600 Page 17 of 27 600 μ2k1−q1,k3−q3 ⎧ ⎪ ⎩μ2k2−q2,q2−k1μ2q1−k2,q3−k3 +1 2μ2k2−q2,q3−k3μ2q2−k1,q1−k2⎫ ⎪ ⎭ ∝F|(k1−q1)+(k3−q3)|R ⎧ ⎪ ⎩F|k1−k2|RF|(k1−q1)+(k3−q3)|R +1 2F|(k2−q2)−(k3−q3)|R F|(k1−q1)+(k2−q2)|R⎫ ⎪ ⎭.(76) The first term here is a contribution to backward Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function together with a contribution to forward HBT of the gluons k1and k2. The second term in Eq. (76) is a contribution to backward Bose enhancement of the gluons k1−q1and k3−q3as well as the gluons k1−q1and k2−q2in the projectile wave function together with a contribution to forward Bose enhancement of the gluons k2−q2and k3−q3 in the projectile wave function. The third piece of I(2) 1tr,2is proportional to μ2k1−q1,q1−k2 ⎧ ⎪ ⎩μ2k2−q2,k3−q3μ2q2−k1,q3−k3 +μ2k2−q2,q3−k3μ2k3−q3,q2−k1⎫ ⎪ ⎭(77) ∝F|k1−k2|R⎧ ⎪ ⎩F2|(k2−q2)+(k3−q3)|R +F2|(k2−q2)−(k3−q3)|R⎫ ⎪ ⎭.(78) Clearly, the first term this equation is a contribution to forward HBT of the gluons k1and k2 together with a contribution to backward Bose enhancement of the gluons k2−q2and k3−q3 in the projectile wave function, while the second term is a contribution to forward HBT of the gluons k1and k2together with a contribution to forward Bose enhancement of the gluons k2−q2 and k3−q3in the projectile wave function. The last piece of the I(2) 1tr,2is proportional to μ2k1−q1,q3−k3 ⎧ ⎪ ⎩ 1 2μ2k2−q2,k3−q3μ2q2−k1,q1−k2 +μ2k2−q2,q2−k1μ2k3−q3,q1−k2⎫ ⎪ ⎭(79) ∝F|(k1−q1)−(k3−q3)|R ⎧ ⎪ ⎩ 1 2F|(k2−q2)+(k3−q3)|R F|(k1−q1)+(k2−q2)| +F|k1−k2|RF|(k1−q1)−(k3−q3)|R⎫ ⎪ ⎭. (80) The first term in this equation is a contribution to the forward Bose enhancement of the gluons k1−q1and k3−q3together with a contribution to the backward Bose enhancement of the gluons k2−q2and k3−q3as well as the gluons k1−q1 and k2−q2in the projectile wave function. The second term in Eq. (79) is a contribution to forward Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function together with a contribution to forward HBT of the gluons k1and k2. The identification of the symmetry partners of I(2) 1tr,2can be performed in a straight forward way by adopting the same procedure. (c) The third term that originates from the singletrace contribution, I(2) 1tr,3is defined in Eq. (A38) and its symmetry partners are given in Eq. (A37). This term has also four pieces and the first one is proportional to μ2k1−q1,k2+q1 ⎧ ⎪ ⎩μ2k3+q2,−q2−k1μ2−q3−k2,q3−k3 +1 2μ2k3+q2,−q3−k2μ2−q2−k1,q3−k3⎫ ⎪ ⎭ +1 4μ2k1−q1,q3−k3μ2k2+q1,−q2−k1 μ2k3+q2,−q3−k2 ∝F|k1+k2|R⎧ ⎪ ⎩F|k1−k3|RF|k2+k3|R +1 2F2|(k3−q3)−(k2−q2)|R⎫ ⎪ ⎭ +1 4F|(k1−q1)−(k3−q3)|R F|(k1−q1)−(k2−q2)|R F|(k2−q2)−(k3−q3)|R.(81) The first term in this equation is a contribution to the backward HBT of the gluons k1and k2 as well as the gluons k2and k3together with a contribution to forward HBT of the gluons k1 and k3. The second term in Eq. (81) is a contri- 123 600 Page 18 of 27 Eur. Phys. J. C (2019) 79 :600 bution to backward HBT of the gluons k1and k2together with a contribution to forward Bose enhancement of the gluons k2−q2and k3−q3 in the projectile wave function. The third term is a contribution to forward Bose enhancement of the all three gluons k1−q1,k2−q2and k3−q3in the projectile wave function. The second piece of I(2) 1tr,3is proportional to ⎧ ⎪ ⎩μ2k1−q1,k3−q3μ2k2+q1,q3−k1 μ2q2−k2,−k3−q2 +μ2k1−q1,q2−k2μ2k2+q1,−k3−q2 μ2k3−q3,q3−k1⎫ ⎪ ⎭ ∝⎧ ⎪ ⎩F|k2+k3|RF2|(k1−q1)+(k3−q3)|R +F|k1−k3|RF2|(k1−q1)−(k2−q2)|R⎫ ⎪ ⎭. (82) The first term here is a contribution to backward HBT of the gluons k2and k3together with a contribution to backward Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function. The second term in this equation is a contribution to forward HBT of the gluons k1and k3together with a contribution to forward Bose enhancement of the gluons k1−q1and k2−q2in the projectile wave function. The third piece of I(2) 1tr,3is proportional to μ2k1−q3,k3−q1μ2k2+q3,−k3−q2 μ2q1−k1,q2−k2 ∝F|(k1−q1)+(k3−q3)|R F|(k2−q2)−(k3−q3)|R F|(k1−q1)+(k2−q2)|R.(83) This term is a contribution to backward Bose enhancement of the gluons k1−q1and k3− q3as well as the gluons k1−q1and k2−q2 in the projectile wave function together with a contribution to forward Bose enhancement of the gluons k2−q2and k3−q3in the projectile wave function. The last piece of I(2) 1tr,3is proportional to μ2k1+q2,−q1−k2 μ2k2−q2,k3−q3μ2q3−k1,q1−k3 ∝F|(k1−q1)−(k2−q2)|R F|(k2−q2)+(k3−q3)|R F|(k1−q1)+(k3−q3)|R.(84) This term is a contribution to the backward Bose enhancement of the gluons k1−q1and k3− q3as well as the gluons k2−q2and k3−q3 in the projectile wave function together with a contribution to the forward Bose enhancement of the gluons k1−q1and k2−q2in the projectile wave function. The symmetry partners of I(2) 1tr,3 can be identified in a similar manner. (d) The last term that originates from the singletrace contribution, I(2) 1tr,4, is defined in Eq. (A41) with its symmetry partners given in Eq. (A40). This term has four pieces and the first one is proportional to μ2k1−q1,q1−k2⎧ ⎪ ⎩ 1 2μ2k2−q3,k3−q2 μ2q2−k1,q3−k3 +μ2k2−q3,q3−k3μ2k3−q2,q2−k1⎫ ⎪ ⎭ +1 4μ2k1−q1,q3−k3μ2k2−q3,k3−q2 μ2q2−k1,q1−k2 ∝F|k1−k2|R⎧ ⎪ ⎩ 1 2F2|(k2−q2)+(k3−q3)|R +F|k2−k3|RF|k1−k3|R⎫ ⎪ ⎭ +1 4F|(k1−q1)−(k3−q3)|R F|(k2−q2)+(k3−q3)|R F|(k1−q1)+(k2−q2)|R.(85) The first term in this equation is a contribution to the forward HBT of the gluons k1and k2together with a contribution to the backward Bose enhancement of the gluons k2−q2and k3−q3in the projectile wave function. The second term in Eq. (85) is a contribution to forward HBT of the three gluons k1,k2and k3.The last term is a contribution the backward Bose enhancement of the gluons k1−q1and k2−q2 as well as the gluons k2−q2and k3−q3together with a contribution to the forward Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function. The second piece of I(2) 1tr,4is proportional to ⎧ ⎪ ⎩μ2k1−q1,k2−q2μ2k3−q3,q3−k1 μ2q1−k2,q2−k3 +μ2k1−q1,k3−q3μ2k2−q2,q2−k3 μ2q3−k1,q1−k2⎫ ⎪ ⎭ 123 Eur. Phys. J. C (2019) 79 :600 Page 19 of 27 600 ∝⎧ ⎪ ⎩F2|(k1−q1)+(k2−q2)|R F|k1−k3|R +F2|(k1−q1)+(k3−q3)|R F|k2−k3|R⎫ ⎪ ⎭.(86) The first term in this equation is a contribution to the forward HBT of the gluons k1and k3together with a contribution to the backward Bose enhancement of the gluons k1−q1and k2−q2in the projectile wave function. The second term in Eq. (86) is a contribution to the forward HBT of the gluons k1and k2together with a contribution to the backward Bose enhancement of the gluons k1−q1and k3−q3in the projectile wave function. The third piece of I(2) 1tr,4 is proportional to μ2k1−q2,k2−q1μ2k3−q3,q2−k2 μ2q3−k1,q1−k3 ∝F|(k1−q1)+(k2−q2)|RF|(k3−q3) −(k2−q2)|RF|(k3−q3)+(k1−q1)|R. (87) This term is a contribution to the backward Bose enhancement of the gluons k1−q1and k2−q2as well as the gluons k1−q1and k3−q3together with a contribution to forward Bose enhancement of the gluons k2−q2and k3−q3in the projectile wave function. The last piece of I(2) 1tr,4 is proportional to μ2k1−q3,k3−q1μ2k2−q2,q1−k1 μ2q3−k2,q2−k3 ∝F|(k1−q1)+(k3−q3)|RF|(k1−q1) −(k2−q2)|RF|(k2−q2)+(k3−q3)|R. (88) This term is a contribution to the backward Bose enhancement of the gluons k1−q1and k3− q3as well as the gluons k2−q2and k3−q3 together with a contribution to the forward Bose enhancement of the gluons k1−q1and k2−q2. The symmetry partners to I(2) 1tr,4can be identified in a similar way. 4 Discussion and outlook To conclude, we have derived the non-eikonal Lipatov vertex that takes into account the finite longitudinal width of the target to all orders. This result was conjectured in [12]after considering the first two corrections to the eikonal limit of the Lipatov vertex coming from the non-eikonal expansion of the gluon propagagor obtained in [5,11]. However, here we have presented a different derivation from first principles. Then, we have used the non-eikonal Lipatov vertex to study the single, double and triple inclusive gluon production cross sections in p-A collisions at mid pseudorapidity. Our results are valid for dilute-dilute collisions since we consider the dilute target limit which, for double and triple inclusive gluon production, corresponds to the original Glasma graph calculation with the exception that we take into account the non-eikonal corrections due to the finite longitudinal thickness of the target. In the single inclusive gluon production cross section, we have shown that the non-eikonal corrections are encoded in function GNE 1(k−,λ +)that is defined in Eq. (33) with k− being the light cone energy of the produced gluon and λ+ the colour correlation length along the longitudinal direction in the target. On the one hand, in the limit of (k−λ+)→ 0, our result reproduces the well known eikonal expression which is often referred to as the kt-factorized formula in the CGC. On the other hand, by expanding our result to second order in (k−λ+), we recover the result calculated in [12]. Our numerical results show that in the kinematic region where the non-eikonal effects are expected to be sizeable, the relative importance of the non-eikonal corrections can vary from 2 to 15% with respect to the eikonal result. This shows that, depending on the kinematic region that one is interested in, the non-eikonal effects might very well be sizable. We have also used the non-eikonal Lipatov vertex to calculate the double inclusive gluon production cross section for dilute-dilute scattering. Adopting the same strategy that was introduced in [58], we have identified the terms that contribute to uncorrelated production, those that are responsible for Bose enhancement of the gluons in the projectile and in the target wave functions, and the terms that contribute to HBT interference effects. Our results agree with the results in [58]uptotheNccounting of the target Bose enhancement and part of the projectile Bose enhancement terms. However, it is known that this difference is a consequence of the fact that some aspects of Nccounting are different in the dilute and dense limits [58,80,81]. Moreover, including the non-eikonal corrections in the double inclusive gluon production cross section has a direct consequence. On top of the function GNE 1(k− 1;λ+)that also exists in the single inclusive case, a new function GNE 2(k− 1,k− 2;L+), defined in Eq. (46), appears which also encodes non-eikonal effects. The partners of the terms that contain GNE 2(k− 1,k− 2;L+), obtained via (k2→−k2),also appear in the double inclusive gluon production cross section but they are accompanied by GNE 2(k− 1,−k− 2;L+). However, in some specific kinematic regions, namely when k− 1∼k− 2, GNE 2(k− 1,k− 2;L+)GNE 2(k− 1,−k− 2;L+)which creates an 123 600 Page 20 of 27 Eur. Phys. J. C (2019) 79 :600 asymmetry. We would like to emphasize that this asymmetry is absent in the eikonal limit. One can immediately realize that this asymmetry created by the non-eikonal corrections in the double inclusive gluon production indeed mimics the asymmetry between the forward and backward peaks in the ridge observed in the two particle correlations. The consequences of this asymmetry are illustrated in Fig. 7and in Fig. 8. This is one of the most striking results of our current study. A dedicated study of two particle correlations and azimuthal harmonics with non-eikonal corrections is left for a forthcoming work [78]. Finally, we have also considered the non-eikonal triple inclusive gluon production cross section in the dilute target limit. We have identified all the terms that appear in the final result. Compared to the work performed in [58], the main difference – apart from non-eikonal corrections that we have included in our study – is that we have included all terms while only the leading Ncones were considered in [58]. This difference is again due to the fact that Nccounting is different in the dilute and dense regimes. In our study, we have identified the terms that correlate all three gluons which originate from three-trace or double-trace contributions, which were absent in [58] since they are suppressed in powers of Ncin the dense target limit and therefore discarded there. Moreover, the non-eikonal effects enter through two new functions G3(k− 1,k− 2,k− 3;L+)and G4(k− 1,k− 2,k− 3;L+) that are defined in Eqs. (A8) and (A9) respectively, on top of the functions G1(k−;λ+)and G2(k− 1,k− 2;L+)that already appeared in the double inclusive case. Obviously, in the limit of the vanishing L+these functions become one and provide the eikonal limit of the triple inclusive gluon production cross section in the dilute target limit. Acknowledgements We thank Raju Venugopalan for comments on the first version of this manuscript. TA expresses his gratitude to Instituto Galego de Física de Altas Enerxías for support and hospitality when part of this work was done. PA and NA are supported by Ministerio de Ciencia e Innovación of Spain under projects FPA2014- 58293-C2-1-P, FPA2017-83814-P and Unidad de Excelencia María de Maetzu under project MDM-2016-0692, by Consellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia under project ED431C 2017/07, and by FEDER. The work of TA is supported by Grant no. 2017/26/M/ST2/01074 of the National Science Centre, Poland. This work has been performed in the framework of COST Action European Cooperation in Science and Technology CA15213 “Theory of hot matter and relativistic heavy-ion collisions” (THOR). Data Availability Statement This manuscript has no associated data or the data will not be deposited [Authors’ comment: This work is theoretical and there is no associated data. The results shown in Figs. 5, 6,7and 8can be obtained from the formulae in the manuscript, please contact the corresponding author in case of any problem.] 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: Details of the calculation of the triple inclusive gluon cross section beyond the eikonal approximation As in the case of single and double inclusive gluon production, we first take the dilute target limit which corresponds to the expansion of the Wilson lines in powers of the background field of the target, Eq. (24). Then the triple inclusive gluon production cross section reads dσ d2k1dη1d2k2dη2d2k3dη3dilute =(4π)3α3 s z1¯z1z2¯z2z3¯z3 eik1·(z1−¯z1)+ik2·(z2−¯z2)+ik3·(z3−¯z3) x1x2x3y1y2y3×Ai(x1−z1)Ai(¯z1−y1)Aj(x2−z2) Aj(¯z2−y2)Ak(x3−z3)Ak(¯z3−y3) ρa1 x1ρa2 x2ρa3 x3ρb1 y1ρb2 y2ρb3 y3P ×g6dx+ 1dx+ 2dx+ 3dx+ 4dx+ 5dx+ 6 d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 d2q5 (2π)2 d2q6 (2π)2 Tc1Tc2a1b1Tc3Tc4a2b2Tc5Tc6a3b3 ×A− c1(x+ 1,q1)A− c2(x+ 2,q2)A− c3(x+ 3,q3) ×A− c4(x+ 4,q4)A− c5(x+ 6,q6)T e−iq1·z1−e−iq1·x1eiq2·¯z1−eiq2·y1 ×e−iq3·z2−e−iq3·x2eiq4·¯z2−eiq4·y2 e−iq5·z3−e−iq5·x3eiq6·¯z3−eiq6·y3.(A1) In the calculation of the single and double inclusive gluon production cross section, we performed the averaging over the colour charge densities of the projectile first. However, it can also be left for further stages of the calculation for convenience since the expressions for the triple inclusive gluon production are longer. Therefore, we leave it for later and perform the integrals over the transverse coordinates which yields dσ d2k1dη1d2k2dη2d2k3dη3dilute =(4π)3α3 sg6 dx+ 1dx+ 2dx+ 3dx+ 4dx+ 5dx+ 6 d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 d2q5 (2π)2 d2q6 (2π)2 123 Eur. Phys. J. C (2019) 79 :600 Page 21 of 27 600 ×A− c1(x+ 1,q1)A− c2(x+ 2,q2)A− c3(x+ 3,q3) ×A− c4(x+ 4,q4)A− c5(x+ 6,q6)T ρa1 k1−q1ρa2 k2−q3ρa3 k3−q5ρb1 q2−k1ρb2 q4−k2ρb3 q6−k3P ×Tc1Tc2a1b1Tc3Tc4a2b2Tc5Tc6a3b3 Li(k1,q1)Li(k1,q2)Lj(k2,q3)Lj(k2,q4) Lk(k3,q5)Lk(k3,q6), (A2) where Li(k,q)is the eikonal Lipatov vertex defined in Eq. (19). At this point, we can incorporate the non-eikonal effects for the triple inclusive gluon production cross section. As discussed earlier, these effects are taken into account by exchanging each eikonal Lipatov vertex in Eq. (A2) with the corresponding non-eikonal Lipatov vertex given in Eq. (20), and using Eq. (29) for the correlator of two target fields. After exchanging each eikonal Lipatov vertex with the corresponding non-eikonal one, the dilute target limit of the non-eikonal triple inclusive gluon production cross section reads dσ d2k1dη1d2k2dη2d2k3dη3 NE dilute =(4π)3α3 sg6d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2 d2q4 (2π)2 d2q5 (2π)2 d2q6 (2π)2 dx+ 1dx+ 2dx+ 3dx+ 4dx+ 5dx+ 6 ×eik− 1(x+ 1−x+ 2)+ik− 2(x+ 3−x+ 4)+ik3(x+ 5−x+ 6)A− c1(x+ 1,q1) A− c2(x+ 2,q2)A− c3(x+ 3,q3)A− c4(x+ 4,q4)A− c5(x+ 6,q6)T ×ρa1 k1−q1ρa2 k2−q3ρa3 k3−q5ρb1 q2−k1ρb2 q4−k2ρb3 q6−k3P Tc1Tc2a1b1Tc3Tc4a2b2Tc5Tc6a3b3×Li(k1,q1)Li(k1,q2) ×Lj(k2,q3)Lj(k2,q4)Lk(k3,q5)Lk(k3,q6). (A3) Let us now consider the averaging over the colour fields of the target. As in the case of the double inclusive gluon production, the average over six colour fields of the target can be factorized into all possible Wick contractions: "A− 1A− 2A− 3A− 4A− 5A− 6#T="A− 1A− 2#T "A− 3A− 4#T"A− 5A− 6#T+"A− 3A− 5#T"A− 4A− 6#T +"A− 3A− 6#T"A− 4A− 5#T+"A− 1A− 3#T "A− 2A− 4#T"A− 5A− 6#T+"A− 2A− 5#T"A− 4A− 6#T+"A− 2A− 6#T "A− 4A− 5#T+"A− 1A− 4#T "A− 2A− 3#T"A− 5A− 6#T+"A− 2A− 5#T"A− 3A− 6#T+"A− 2A− 6#T "A− 3A− 5#T+"A− 1A− 5#T "A− 2A− 3#T"A− 4A− 6#T+"A− 2A− 4#T"A− 3A− 6#T+"A− 2A− 6#T "A− 3A− 4#T+"A− 1A− 6#T "A− 2A− 3#T"A− 4A− 5#T+"A− 2A− 4#T "A− 3A− 5#T+"A− 2A− 5#T"A− 3A− 4#T,(A4) where we have introduced a shorthand notation for the target fields A− i≡A− ci(x+ i,qi)for convenience. The target fields are originating from the expansion of the Wilson line in the amplitude (complex conjugate amplitude) when the subscript iis odd (even). With this shorthand notation, the correlator of two target fields defined in Eq. (29), can be written in the most convenient way as A− iA− jT=n(x+ i)1 2λ+λ+−x+ i−x+ jij,(A5) where ij is defined as ij =δcicj(2π)2δ(2)qi+(−1)i+jqja(qi) 2.(A6) Note that Eq. (A3) can now be integrated over the longitudinal coordinates. After plugging the factorized expression for averaging of the colour fields of the target given in Eq. (A4) into Eq. (A3), the longitudinal coordinate dependent part of the dilute target limit of the non-eikonal triple inclusive gluon production cross section can be written as dx+ 1dx+ 2dx+ 3dx+ 4dx+ 5dx+ 6 eik− 1(x+ 1−x+ 2)+ik− 2(x+ 3−x+ 4)+ik3(x+ 5−x+ 6) "A− 1A− 2A− 3A− 4A− 5A− 6#T =GNE 1(k− 1;λ+)GNE 1(k− 2;λ+)GNE 1(k− 3;λ+) 123456 +12GNE 2(k− 2,−k− 3;L+)3546 +GNE 2(k− 2,k− 3;L+)3645 +34GNE 2(k− 1,−k− 3;L+)1526 +GNE 2(k− 1,k− 3;L+)1625 +56GNE 2(k− 1,−k− 2;L+)1324 +GNE 2(k− 1,k− 2;L+)1423 +GNE 3(k− 1,k− 2,k− 3;L+)132546 +162435 +GNE 3(k− 2,k− 1,k− 3;L+)132645 +152436 +GNE 3(k− 1,k− 3,k− 2;L+)142635 +152346 +GNE 4(k− 1,k− 2,k− 3;L+)142536 +162345, (A7) 123 600 Page 22 of 27 Eur. Phys. J. C (2019) 79 :600 where the functions GNE 1(k− i;λ+)and GNE 2(k− i,k− j;L+)are the functions that account for non-eikonal effects and they are defined in Eqs. (33) and (46), respectively. Moreover, for the triple inclusive gluon production the longitudinal coordinate integral produces two new functions GNE 3(k− i,k− j,k− k;L+) and GNE 4(k− i,k− j,k− k;L+)that also account for the noneikonal effects and read GNE 3(k− 1,k− 2,k− 3;L+) =2−sin (k+ 1+k+ 2)L++sin (k− 1−k− 3)L++sin (k− 2+k− 3)L+ (k+ 1+k+ 2)L+(k− 1−k− 3)L+(k− 2+k− 3)L+(A8) and GNE 4(k− 1,k− 2,k− 3;L+) = sin (k− 1−k− 2) 2L+sin (k− 1−k− 3) 2L+sin (k− 2−k− 3) 2L+ (k− 1−k− 2) 2L+(k− 1−k− 3) 2L+(k− 2−k− 3) 2L+. (A9) Both functions go to 1 when we consider the shockwave (eikonal) limit L+→0. We can now substitute Eq. (A7) into the dilute target limit of the non-eikonal triple inclusive gluon production cross section given in Eq. (A3). By using the definition of ij given in Eq. (A6) and integrating over the three transverse momenta, we get dσ d2k1dη1d2k2dη2d2k3dη3 NE dilute =(4π)3α3 sg6 d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2a(q1) 2a(q2) 2a(q3) 2 ×GNE 1(k− 1,λ +)GNE 1(k− 2,λ +)GNE 1(k− 3,λ +) ×C3 Aρa k1−q1ρb k2−q2ρc k3−q3ρa q1−k1ρb q2−k2ρc q3−k3P Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2)Lk(k3,q3)Lk(k3,q3) +⎧ ⎪ ⎩GNE 2(k− 1,k− 2;L+)CA(TaTb)a1b1(TbTa)a2b2δa3b3 Li(k1,q1)Li(k1,q2)Lj(k2,q1)Lj(k2,q2)Lk(k3,q3)Lk(k3,q3) ×ρa1 k1−q1ρa2 k2−q2ρa3 k3−q3ρb1 −k1+q2ρb2 −k2+q1ρb3 −k3+q3P +k2→−k2+k1↔k3+k2↔k3⎫ ⎪ ⎭ +⎧ ⎪ ⎩GNE 3(k− 1,k− 2,k− 3;L+)(TaTb)a1b1(TaTc)a2b2(TbTc)a3b3Li(k1,q1) Li(k1,q2)Lj(k2,−q1)Lj(k2,q3)Lk(k3,q2)Lk(k3,−q3) ×ρa1 k1−q1ρa2 k2+q1ρa3 k3−q2ρb1 −k1+q2ρb2 −k2+q3ρb3 −k3−q3P+k3→−k3 +k1↔k3+k2↔k3⎫ ⎪ ⎭ +⎧ ⎪ ⎩GNE 4(k− 1,k− 2,k− 3;L+)(TaTb)a1b1(TcTa)a2b2(TbTc)a3b3Li(k1,q1) Li(k1,q2)Lj(k2,q1)Lj(k2,q3)Lk(k3,q2)Lk(k3,q3) ×ρa1 k1−q1ρa2 k2−q3ρa3 k3−q2ρb1 −k1+q2ρb2 −k2+q1ρb3 −k3+q3P+k2↔k3⎫ ⎪ ⎭, (A10) where we remind the notation k≡(k−,k). Our next order of business is to perform the averaging over the projectile colour charge densities. As in the previous subsections, we adopt the generalized MV model for the average of two projectile colour charge densities and write down all possible Wick contractions of their products. Then, the average of six generic projectile colour charge densities can be written ρa1 k1ρa2 k2ρa3 k3ρb1 p1ρb2 p2ρb3 p3P="ρa1 k1ρb1 p1#"ρa2 k2ρb2 p2#"ρa3 k3ρb3 p3# +"ρa1 k1ρb1 p1#"ρa2 k2ρa3 k3#"ρb2 p2ρb3 p3#+"ρa2 k2ρb3 p3#"ρa3 k3ρb2 p2# +"ρa2 k2ρb2 p2#"ρa1 k1ρa3 k3#"ρb1 p1ρb3 p3# +"ρa1 k1ρb3 p3#"ρa3 k3ρb1 p1#+"ρa3 k3ρb3 p3#"ρa1 k1ρa2 k2#"ρb1 p1ρb2 p2# +"ρa1 k1ρb2 p2#"ρa2 k2ρb1 p1# +"ρa1 k1ρa2 k2#"ρa3 k3ρb1 p1#"ρb2 p2ρb3 p3#+"ρa3 k3ρb2 p2#"ρb1 p1ρb3 p3# +"ρa2 k2ρa3 k3#"ρa1 k1ρb2 p2#"ρb1 p1ρb3 p3#+"ρa1 k1ρb3 p3#"ρb1 p1ρb2 p2# +"ρa2 k2ρb1 p1#"ρa1 k1ρa3 k3#"ρb2 p2ρb3 p3#+"ρa1 k1ρb3 p3#"ρa3 k3ρb2 p2# +"ρa2 k2ρb3 p3#"ρa1 k1ρb2 p2#"ρa3 k3ρb1 p1#+"ρa1 k1ρa3 k3#"ρb1 p1ρb2 p2#, (A11) where the two projectile colour charge correlator is given by Eq. (26). One can use Eq. (A11) in order to perform the projectile colour charge density averaging in Eq. (A10). The resulting expression consists of three distinct parts: a term with a single trace, a term with double trace and a term with three traces of the colour generators (these terms are the analogue of three-dipole, dipole-quadrupole and sextuple contributions in [58] for the dilute-dense set up). Therefore, we write the dilute target limit of the non-eikonal triple inclusive gluon production cross section as sum of those three contributions: dσ d2k1dη1d2k2dη2d2k3dη3 NE dilute =dσ(3tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute +dσ(2tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute +dσ(1tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute .(A12) 123 Eur. Phys. J. C (2019) 79 :600 Page 23 of 27 600 Let us now write down the explicit expressions for each of these three contributions starting from the the three-trace one: dσ(3tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute =(4π)3α3 sg6C3 A(N2 c−1)3 ×d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2a(q1) 2a(q2) 2a(q3) 2 ×GNE 1(k− 1;λ+)GNE 1(k− 2;λ+)GNE 1(k− 3;λ+) ×I(0) 3tr +1 (N2 c−1)I(1) 3tr +1 (N2 c−1)2I(2) 3tr,1+I(2) 3tr,2, (A13) where I(0) 3tr =μ2k1−q1,q1−k1μ2k2−q2,q2−k2 ×μ2k3−q3,q3−k3 ×Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) ×Lk(k3,q3)Lk(k3,q3). (A14) For O1/(N2 c−1)terms, we have introduced the following compact notation I(1) 3tr =˜ I(1) 3tr +k2→−k2+k1↔k3+k2↔k3 (A15) with ˜ I(1) 3tr =GNE 2(k− 1,k− 2;L+)μ 2k1−q1,q2−k1 ×μ2k2−q2,q1−k2μ2k3−q3,q3−k3 ×Li(k1,q1)Li(k1,q2)Lj(k2,q1)Lj(k2,q2) ×Lk(k3,q3)Lk(k3,q3). (A16) A similar compact notation has been adopted for the O1/(N2 c−1)2terms in Eq. (A12): I(2) 3tr,1=˜ I(2) 3tr,1+k3→−k3+k1↔k3+k2↔k3 (A17) with ˜ I(2) 3tr,1=GNE 3(k− 1,k− 2,k− 3;L+)μ 2k1−q1,q2−k1 ×μ2k2+q1,q3−k2μ2k3−q2,−q3−k3 ×Li(k1,q1)Li(k1,q2)Lj(k2,−q1)Lj(k2,q3) ×Lk(k3,q2)Lk(k3,−q3), (A18) and I(2) 3tr,2=˜ I(2) 3tr,2+k2↔k3(A19) with ˜ I(2) 3tr,2=GNE 4(k− 1,k− 2,k− 3;L+)μ 2k1−q1,q2−k1 ×μ2k2−q3,q1−k2μ2k3−q2,q3−k3 ×Li(k1,q1)Li(k1,q2)Lj(k2,q1)Lj(k2,q3) ×Lk(k3,q2)Lk(k3,q3). (A20) The double-trace contribution to the dilute target limit of the non-eikonal triple inclusive gluon production cross section can be organized in a similar way: dσ(2tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute =(4π)3α3 sg6C3 A(N2 c−1)2 ×d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)2a(q1) 2a(q2) 2a(q3) 2 ×GNE 1(k− 1,λ +)GNE 1(k− 2;λ+)GNE 1(k− 3;λ+) ×I(1) 2tr,1+I(1) 2tr,2+1 (N2 c−1)I(2) 2tr,1+I(2) 2tr,2+I(2) 2tr,3. (A21) Similar compact notations can be adopted for each term in the double-trace contribution. Let us start with the O(1)terms: I(1) 2tr,1=˜ I(1) 2tr,1 +k2→−k2+k1↔k2+k1↔k3,(A22) with ˜ I(1) 2tr,1=μ2k1−q1,q1−k1μ2k2−q2,q3−k3 ×μ2k3−q3,q2−k2 ×Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) ×Lk(k3,q3)Lk(k3,q3), (A23) and I(1) 2tr,2=˜ I(1) 2tr,2+k2→−k2+k1↔k3+k2↔k3 (A24) with ˜ I(1) 2tr,2=GNE 2(k− 1,k− 2;L+)μ 2k3−q3,q3−k3Li(k1,q1) ×Li(k1,q2)Lj(k2,q1)Lj(k2,q2)Lk(k3,q3)Lk(k3,q3) ×μ2k1−q1,q1−k2μ2k2−q2,q2−k1 +1 2μ2k1−q1,k2−q2μ2q2−k1,q1−k2. (A25) 123 600 Page 24 of 27 Eur. Phys. J. C (2019) 79 :600 O1/(N2 c−1)terms in the double-trace contribution can be written in a similar manner. The first term reads I(2) 2tr,1=˜ I(2) 2tr,1+k2→−k2+k1↔k3+k2↔k3, (A26) with ˜ I(2) 2tr,1=GNE 2(k− 1,k− 2;L+)μ 2k1−q1,q2−k1 ×Li(k1,q1)Li(k1,q2)Lj(k2,q1) ×Lj(k2,q2)Lk(k3,q3)Lk(k3,q3) ×μ2k2−q2,q3−k3μ2k3−q2,q1−k2 +μ2k2−q2,k3−q3μ2q1−k2,q3−k3. (A27) The second term can be written as I(2) 2tr,2=˜ I(2) 2tr,2+k3→−k3+k1↔k3+k2↔k3, (A28) with ˜ I(2) 2tr,2=GNE 3(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,−q2) ×Lj(k2,−q1)Lj(k2,−q3)Lk(k3,−q2)Lk(k3,q3) ×μ2k1−q1,−q2−k1⎧ ⎪ ⎩ 1 2μ2k2+q1,q3−k3 ×μ2k3+q2,−q3−k2+μ2k2+q1,k3+q2 ×μ2−q3−k2,q3−k3⎫ ⎪ ⎭ +GNE 3(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q3) ×Lj(k2,−q1)Lj(k2,q2)Lk(k3,q3)Lk(k3,−q2) ×μ2k2+q1,q2−k2⎧ ⎪ ⎩μ2k1−q1,−k3−q2 ×μ2q3−k1,k3−q3 +1 2μ2k1−q1,k3−q3μ2q3−k1,−k3−q2⎫ ⎪ ⎭ +GNE 3(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,−q2) ×Lj(k2,−q1)Lj(k2,q3)Lk(k3,−q2)Lk(k3,−q3) ×μ2k3+q2,−k3−q3⎧ ⎪ ⎩ 1 2μ2k1−q1,q3−k2 ×μ2−q2−k1,k2+q1 +μ2k1−q1,k2+q1μ2−q2−k1,q3−k2⎫ ⎪ ⎭. (A29) Finally, the last term can be written as I(2) 2tr,3=˜ I(2) 2tr,3+k2↔k3(A30) with ˜ I(2) 2tr,3=GNE 4(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q2) ×Lj(k2,q1)Lj(k2,q3)Lk(k3,q2)Lk(k3,q3) ×μ2k1−q1,q2−k1⎧ ⎪ ⎩μ2k2−q3,q3−k3 ×μ2k3−q2,q1−k2+1 2μ2k2−q3,k3−q2 ×μ2q1−k2,q3−k3⎫ ⎪ ⎭ +μ2k3−q2,q3−k3⎧ ⎪ ⎩μ2k1−q1,q1−k2 ×μ2q2−k1,k2−q3+1 2μ2k1−q1,k2−q3 ×μ2q2−k1,q1−k2⎫ ⎪ ⎭ +GNE 4(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q2) ×Lj(k2,q2)Lj(k2,q3)Lk(k3,q1)Lk(k3,q3) ×μ2k2−q3,q2−k2⎧ ⎪ ⎩μ2k1−q2,q3−k3 ×μ2q1−k1,k3−q1 +1 2μ2k1−q2,k3−q1μ2q1−k1,q3−k3⎫ ⎪ ⎭. (A31) The last contribution to the dilute target limit of the noneikonal triple inclusive gluon production cross section that we need to consider is the single-trace contribution which can be organized as follows: dσ(1tr) d2k1dη1d2k2dη2d2k3dη3 NE dilute =(4π)3α3 sg6C3 A(N2 c−1) ×d2q1 (2π)2 d2q2 (2π)2 d2q3 (2π)3a(q1) 2a(q2) 2a(q3) 2 ×GNE 1(k− 1;λ+)GNE 1(k− 2;λ+)GNE 1(k− 3;L+) ×I(2) 1tr,1+I(2) 1tr,2+I(2) 1tr,3+I(2) 1tr,4.(A32) The first term in the single-trace contribution can be written as I(2) 1tr,1=˜ I(2) 1tr,1+k2→−k2+k2↔k3(A33) with ˜ I(2) 1tr,1=Li(k1,q1)Li(k1,q1)Lj(k2,q2)Lj(k2,q2) ×Lk(k3,q3)Lk(k3,q3) ×μ2k1−q1,k2−q2⎧ ⎪ ⎩μ2k3−q3,q1−k1 μ2q2−k2,q3−k3+μ2k3−q3,q2−k2 123 Eur. Phys. J. C (2019) 79 :600 Page 25 of 27 600 μ2q1−k1+q1,q3−k3⎫ ⎪ ⎭.(A34) In a similar manner, the second term in the single-trace contribution can be written as I(2) 1tr,2=˜ I(2) 1tr,2+k2→−k2+k1↔k3+k2↔k3 (A35) with ˜ I(2) 1tr,2=GNE 2(k− 1,k− 2;L+)Li(k1,q1)Li(k1,q2) ×Lj(k2,q1)Lj(k2,q2)Lk(k3,q3)Lk(k3,q3) ×μ2k1−q2,k2−q1⎧ ⎪ ⎩ 1 2μ2k3−q3,q1−k1 ×μ2q2−k2,q3−k3+1 2μ2k3−q3,q2−k2 ×μ2q1−k1,q3−k3⎫ ⎪ ⎭ +μ2k1−q1,k3−q3⎧ ⎪ ⎩μ2k2−q2,q2−k1 ×μ2q1−k2,q3−k3+1 2μ2k2−q2,q3−k3 ×μ2q2−k1,q1−k2⎫ ⎪ ⎭ +μ2k1−q1,q1−k2⎧ ⎪ ⎩μ2k2−q2,k3−q3 ×μ2q2−k1,q3−k3+μ2k2−q2,q3−k3 ×μ2k3−q3,q2−k1⎫ ⎪ ⎭ +μ2k1−q1,q3−k3⎧ ⎪ ⎩ 1 2μ2k2−q2,k3−q3 ×μ2q2−k1,q1−k2+μ2k2−q2,q2−k1 ×μ2k3−q3,q1−k2⎫ ⎪ ⎭.(A36) The third term in the single-trace contribution reads I(2) 1tr,3=˜ I(2) 1tr,3+k3→−k3+k1↔k3+k2↔k3 (A37) with ˜ I(2) 1tr,3=G3(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,−q2) ×Lj(k2,−q1)Lj(k2,−q3)Lk(k3,−q2)Lk(k3,q3) ×μ2k1−q1,k2+q1⎧ ⎪ ⎩μ2k3+q2,−q2−k1 ×μ2−q3−k2,q3−k3 +1 2μ2k3+q2,−q3−k2μ2−q2−k1,q3−k3⎫ ⎪ ⎭(A38) +1 4μ2k1−q1,q3−k3μ2k2+q1,−q2−k1 ×μ2k3+q2,−q3−k2 +G3(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q3) ×Lj(k2,−q1)Lj(k2,q2)Lk(k3,q3)Lk(k3,−q2) ×1 2μ2k1−q1,k3−q3μ2k2+q1,q3−k1 ×μ2q2−k2,−k3−q2 +μ2k1−q1,q2−k2μ2k2+q1,−k3−q2 ×μ2k3−q3,q3−k1 +G3(k− 1,k− 2,k− 3;L+)Li(k1,q3)Li(k1,q1) ×Lj(k2,−q3)Lj(k2,q2)Lk(k3,q1)Lk(k3,−q2) ×1 4μ2k1−q3,k3−q1 ×μ2k2+q3,−k3−q2μ2q1−k1,q2−k2 +G3(k− 1,k− 2,k− 3;L+)Li(k1,−q2)Li(k1,q3) ×Lj(k2,q2)Lj(k2,−q1)Lk(k3,q3)Lk(k3,q1) ×1 4μ2k1+q2,−q1−k2μ2k2−q2,k3−q3 ×μ2q3−k1,q1−k3.(A39) Finally, the last term in the single-trace contribution can be written as I(2) 1tr,4=˜ I(2) 1tr,4+k2↔k3(A40) with ˜ I(2) 1tr,4=GNE 4(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q2) ×Lj(k2,q1)Lj(k2,q3)Lk(k3,q2)Lk(k3,q3) ×μ2k1−q1,q1−k2⎧ ⎪ ⎩ 1 2μ2k2−q3,k3−q2 ×μ2q2−k1,q3−k3+μ2k2−q3,q3−k3 ×μ2k3−q2,q2−k1⎫ ⎪ ⎭ +1 4μ2k1−q1,q3−k3μ2k2−q3,k3−q2 ×μ2q2−k1,q1−k2 +GNE 4(k− 1,k− 2,k− 3;L+)Li(k1,q1)Li(k1,q3) ×Lj(k2,q1)Lj(k2,q2)Lk(k3,q2)Lk(k3,q3) ×1 2μ2k1−q1,k2−q2μ2k3−q3,q3−k1 ×μ2q1−k2,q2−k3 +μ2k1−q1,k3−q3μ2k2−q2,q2−k3 123