Gluon emission at small longitudinal momenta in the QCD effective action approach
Abstract
In the framework of the QCD effective action the vertices of gluon emission in interaction of reggeons are studied in the limit of small longitudinal momenta of the emitted gluon. It is found that the vertices drastically simplify in this limit, so that the gluon becomes emitted from a single reggeon coupled to the projectile and target via multireggeon vertices. The contribution from this kinematical region is studied for double and \(2 \times 2\) elementary collisions inside the composite projectile and target.
1 Introduction
One of the main observables in highenergy collisions of heavy nuclei is the inclusive crosssection for production of secondaries. In the perturbative approach it reduces to production of gluons, which subsequently transform into observed secondary hadrons. The study of gluon production in the central rapidity region with transverse momenta much smaller than the longitudinal momenta of the colliding particles (”Regge kinematics”) has a long history, starting from the pioneer work on the production of minijets from the BFKL pomeron [1]. Later this problem was studied in the framework of the dipole picture for the inclusive crosssection in deepinelastic scattering (DIS) on the heavy nucleus [2], where it was shown that the inclusive crosssection was related to the socalled unintegrated gluon density in the nucleus. Still later in the formalism of reggeized gluons (”BFKLBartels” or the BFKLB framework [3, 4, 5]) it was demonstrated that the same crosssection consists of a sum of two contributions coming from the BFKL pomeron and the cut triple pomeron vertex [6]. In collisions of two heavy nuclei (”AB collisions”) the situation is not so straightforward with the results obtained only in the framework of the JIMWLK formalism (see [7] and the references therein). However, the lack of connection with the actual gluon production in a collision of composite targets and the absence of the confirmation in the BFKLB framework leave certain open questions, which are waiting for their solution.
In the BFKLB framework gluon production is based on the vertices obtained by the dispersive technique which uses multiple discontinuities at poles corresponding to intermediate particles. Such vertices depend only on the transversal momenta of gluons and reggeons. In this form the relation to the scattering on composites is poorly understood. To describe it one rather has to study the vertices with the dependence on all 4momenta included, both transversal and longitudinal. Such vertices can be found by means of the Lipatov Effective Action (LEA) for QCD [8], which introduces the reggeons as independent dynamical variables and describes their interaction with the gluons apart from the standard QCD action. The simplest vertex for gluon emission from a single reggeon \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RG}}\) was constructed in the original BFKL paper [3, 4]. In our previous papers we have found higher vertices for gluon production in the interaction of one and two reggeons, \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\), \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRRG}}\) and \(\Gamma _{\mathrm{RR}\rightarrow \mathrm{RRG}}\). They are quite complicated and the derivation of the full inclusive crosssection in hA and especially AA collisions seems to require an extraordinary effort, also taking into account that apart from the contribution from the vertices one has to calculate numerous contributions from rescattering.
One has to take into account that LEA only describes the interaction of gluons and reggeons at a given rapidity y (or rather within a finite interval of rapidities \(\Delta y\)). Slices of the total rapidity region separated by large rapidity intervals interact via the exchange of reggeons. So one has to divide the total rapidity into different number of large intervals connecting different slices of effective action and one obtains different diagrams made of effective vertices and reggeon propagators. Neglecting the restriction imposed on the width of each slice may lead to divergencies in the integrals over rapidities in the loop integrals. The practical realization of this picture was first achieved by the separation from the whole yintegration parts of small intervals \(\Delta y\) in the calculation of the NLO BFKL kernel in [9] where clusters of real gluons were produced in the intermediate state. Later it was found that in loop calculations the introduction of such a direct slicing in rapidity by \(\Delta y\) was inconvenient. Instead a different method to avoid divergencies due to the limited validity of the LEA vertices was developed based on the socalled tilted Wilson lines (or tilted lightcone variables). Within this method all different divergencies coming from the loop integrations in vertices and propagators cancel [10, 11, 12, 13].
Vertices studied in this paper refer to a fixed value of rapidity. They do not contain internal integrations nor any divergencies. So their calculation can be safely done within the rules of LEA for a given rapidity, without any cutoff \(\Delta y\) or tilted lightcone technique. We shall find that when the longitudinal momenta of the emitted gluon become small our vertices acquire a simple limiting form, which actually corresponds to quite different diagrams of the LEA containing the internal reggeon exchange. These limiting diagrams do not appear in LEA normally and exist only as limits of the standard contributions. However, their simple structure greatly simplifies the calculation of the contribution to the crosssection in the limiting domain of momenta of the emitted particles.
2 Vertices \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\) and \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRRG}}\) at small \(p_\)
We are going to study the vertex in the limiting region \(p_<<r_{1,2}\) (region of small \(p_\)). In this case \(r_{1}\simeq r_{2}\), so that the “” momenta of the two outgoing reggeons are large and opposite. The ’’ momentum comes along one of the outgoing reggeons and goes back along the other.
3 Vertex RR\(\rightarrow \) RRG at small \(p_\pm \)
The vertex RR\(\rightarrow \) RRG in the general kinematics was derived in [18]. It is much more complicated than the vertices R\(\rightarrow \) RRG and R\(\rightarrow \) RRRG.
We shall demonstrate that at small \(p_\pm \) the same reduction holds for this vertex as for \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\) and \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRRG}}\) discussed in the previous section, which drastically simplifies the vertices expressing them via the simple BFKL vertex \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RG}}\) and multigluon vertices. This reduction is illustrated in Fig. 11.
2. Figure 3b.
3. Figure 3c.
4 The inclusive crosssections in hA and AA collisions at small \(p_\pm \)
4.1 hA collisions
As we have seen the vertices for gluon production in the interaction of reggeons drastically simplify in the region of small longitudinal momenta of the produced gluon. At first sight it promises to facilitate calculation of the physical observables, such as inclusive crosssections in the collision of composite particles, e.g. deuterons. However, we shall discover that while such a facilitation certainly takes place, the resulting crosssections vanish in this region.
We also do not show explicitly evolution of the pomerons attached to the projectile and target (which is well known and standardly realized by the BFKL equation) nor the actual coupling to colorless scattering centers in the nucleus (in fact nucleons). One has to take into account that for the heavy nucleus the two centers have to refer to different nucleons. Otherwise the contribution is down by \(A^{1/3}\), assumed large. For clarity we show some diagrams with the nuclear target explicitly indicated in Fig. 7. Several cuts in Figs. 5 and 6 imply that the sum over the contribution of each cut should be taken. Our aim is to study these contributions in the limit when the longitudinal momentum \(p_\) of the created gluon is much smaller than the ””momentum \(\epsilon \) transferred to the target, which is the only dimensional variable after the integration over the longitudinal momenta of the reggeons.
Leaving the discussion of the rescattering contribution to the end of this section we concentrate here on the diagrams with the reduced vertices \(\Gamma \) in Fig. 8. Since the pomeron vanishes when the two reggeons are located at the same point, all diagrams in which the two final reggeons in \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\) or \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRRG}}\) are coupled to the same target vanish. So in the domain \(p_<<\epsilon \) the inclusive crosssection coming from the cut triple pomeron vertex will be given exclusively by diagram 3 in Fig. 8. which corresponds to squaring the vertex \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\).
Twice the imaginary part of the highenergy amplitude H will be given by the square of the two production amplitudes, each containing the vertex \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\). It contains color, energetic, transverse momentum and numerical factors.
Later we shall see that the diagrams with rescattering also give no contribution in this domain.
4.2 AB collisions
Due to AGK cancelations [14] we expect that contributions from emission from the cut pomerons cancel and the final result comes exclusively from the cut 4pomeron interaction vertex, including improper terms corresponding to its disconnected part, corresponding to rescattering in our language. Apart from rescattering and the suppressed evolution each diagram becomes a convolution of two amplitudes for production of the observed gluon in the transition from 1, 2 or 3 initial reggeons into 1, 2 or 3 final reggeons. Then graphically H corresponds to the cut diagrams shown in Fig. 10. We do not show the different ways in which the reggeons may be coupled to the two projectiles and targets. They again follow the pattern illustrated in Fig. 4.
As we have previously shown in the domain (13), the vertices \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRG}}\), \(\Gamma _{\mathrm{R}\rightarrow \mathrm{RRRG}}\) and \(\Gamma _{RR\rightarrow RRG}\) degenerate into the simple expressions (11), (12) and (33), respectively. Unfortunately we do not know explicit expressions for the production amplitudes \(\Gamma _{\mathrm{RR}\rightarrow \mathrm{RRRG}}\) and \(\Gamma _{\mathrm{RRR}\rightarrow \mathrm{RRRG}}\). However, in the spirit of the QCD effective action and comparing with cases with smaller numbers of reggeons we firmly believe that also for them a similar reduction takes place. This reduction is graphically shown in the lower part of Fig. 11.
As we have seen in the kinematics \(p_\pm \rightarrow 0\) vertex \(\Gamma _{\mathrm{RR}\rightarrow \mathrm{RRG}}\) is simplified to (33). The imaginary part of H will be obtained by squaring this expression together with the relevant energetic, transverse momenta, numerical and color factors.
4.3 Rescattering
The fact that the diagrams with rescattering do not contribute to the crosssection at small \(p_\pm \) follows directly from the number L of longitudinal integrations and dimensional considerations.
In the region \(p_<<\epsilon \) the only remaining variable is \(\epsilon \). So in the case of no integration the result has to be \(\propto 1/\epsilon \) and in the case of \(L=1\) it has to be \(\propto \delta (\epsilon )\). As we have already mentioned the last case lies outside the assumed kinematics. If the amplitude is \(\propto 1/\epsilon \) it gives zero in (34) as it is odd in \(\epsilon \). Calculations also show that in this case the contribution to the amplitude is real and its imaginary part is zero. So rescattering amplitudes do not contribute to the inclusive hA crosssection either.
 1.

\(M=8:\ \ (0,14,2),\ \ (1,12,0),\ \ (2,10,2)\);
 2.

\(M=9:\ \ (1,14,1),\ \ (2,12,1)\);
 3.

\(M=10:\ \ (2,14,0),\ \ (3,12,2)\ (\mathrm{only}\ M_1=M_2=5)\);
 4.

\(M=11:\ \ (3,14,1),\ \ (\mathrm{only}\ M_1=6,M_2=5)\);
 5.

\(M=12:\ \ (4,14,2)\ (\mathrm{only}\ M_1=M_2=6)\).
Inspecting these data we first find our main diagram with \(M=8\) and no rescattering. It contains two longitudinal integrations as we have previously seen. We also see that some diagrams with many rescatterings contain no integrations but one or two \(\delta \)functions as factors. The final dimension of the longitudinal integral is, however, \(2\) in all cases.
In the domain (13) the result of the longitudinal integration should be a function of \(\epsilon \) and \(\lambda \) which is Lorenzinvariant and of dimension \(2\). The only candidates are terms proportional to \(1/\epsilon \lambda \), \(\delta (\epsilon )/\lambda \), \(\delta (\lambda )/\epsilon \) and \(\delta (\epsilon )\delta (\lambda )\). In all these cases no contribution to the crosssection follows in the domain (13) either because of the oddness in \(\epsilon \) or \(\lambda \) or because of violating the kinematics.
5 Conclusions
The bulk of our paper is devoted to the study of the vertex for transition RR\(\rightarrow \)RRG in the special kinematical region where the longitudinal momentum of the emitted gluon is much smaller than the longitudinal momenta of participating reggeons. We were able to show that the vertex drastically simplifies, so that emission proceeds from a single intermediate reggeon connected to the participants via two 3reggeon vertices. This also reestablishes the role of the 3reggeon vertex, absent in many cases due to signature conservation but appearing under certain kinematical conditions. It also indicates the general rule for gluon emission in the interaction of reggeons when the longitudinal momenta of the emitted gluon turn out to be much smaller (larger) than those of the reggeons. Under this condition the emission vertex drastically simplifies from a very complicated general expression to a simple and physically clear form. First examples of such simplification were already mentioned in [16]. Here we found that it remains valid also for the highly complicated vertex \(\Gamma _{\mathrm{RR}\rightarrow \mathrm{RRG}}\). We firmly believe that this phenomenon holds also for vertices with any number of incoming and outgoing reggeons.
Our results have a certain significance in the general theory of interacting reggeons within the effective action approach. They refer to the use of the effective action for calculating not only the vertices at a given rapidity but also the amplitudes for physical processes at large overall rapidity. Then according to the idea of effective action one has to divide the total rapidity interval in slices of definite intermediate rapidities, which are then connected by reggeon exchanges. It is initially assumed that the vertices determined by the effective action are working only within a given rapidity slice, so that the set of diagrams describing the amplitude depends on the number of divisions of the total rapidity (resolution in rapidity). Our results show that this situation is different: the set of diagrams actually does not depend on the resolution. Taking the lowest resolution and using the appropriate set of diagrams for the intermediate rapidity one automatically obtains a different set of diagrams appropriate for higher resolution when one considers the limiting expressions for the initial resolution. In this sense we prove the independence of the slicing of the whole rapidity into partial intervals as supposed to be true in the original derivation of LEA.
Our proof is not complete and does not cover all possible cases. It is based on the vertices which have been explicitly calculated earlier. In fact the vertices become very complicated with the growth of the number of reggeons and emitted particles. However, we firmly believe that the result obtained for the considered comparatively simple vertices is true in more complicated cases.
As an application we considered the contribution of the vertices in the limiting cases of higher rapidity resolution to the calculation of the inclusive crosssection for gluon production. One finds that this contribution is zero. In fact this result trivially follows from two circumstances. First, in the integration over longitudinal momenta of the reggeons at small \(p_\pm \) their order of magnitude automatically reduces to the transferred momenta \(\epsilon \) and \(\lambda \). Then the condition of smallness of \(p_\pm \) relative to longitudinal momenta of the reggeons transforms into smallness relative to \(\epsilon \) and \(\lambda \). Second, dimensional considerations restrict the final dependence of the longitudinal integrals over “” components to either \(1/\epsilon \) or \(\delta (\epsilon )\) and over “+” components to either \(1/\lambda \) or \(\delta (\lambda )\). Then vanishing of the contribution in the domain (13) immediately follows.
We do not exclude cases that the obtained properties of the vertices at limiting values of gluon momenta may have other less trivial applications. We are going to search for such applications in our future study.
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