# Multiparticle Production at Mid-Rapidity in the Color-Glass Condensate

###### Abstract

In this paper, we compute a number of cross sections for the production of multiple particles at mid-rapidity in the semi-dilute / dense regime of the color-glass condensate (CGC) effective field theory. In particular, we present new results for the production of two quark-antiquark pairs (whether the same or different flavors) and for the production of one quark-antiquark pair and a gluon. We also demonstrate the existence of a simple mapping which transforms the cross section to produce a quark-antiquark pair into the corresponding cross section to produce a gluon, which we use to obtain various results and to cross-check them against the literature. We also discuss hadronization effects in the heavy flavor sector, writing explicit expressions for the production of various combinations of and mesons, mesons, and light hadrons. The various multiparticle cross sections presented here contain a wealth of information and can be used to study heavy flavor production, charge-dependent correlations, and “collective” flow phenomena arising from initial-state dynamics.

###### pacs:

## I Introduction

Correlations in the production of multiple soft or semi-hard particles in the mid-rapidity region of hadronic collisions are important probes of novel phenomena in quantum chromodynamics (QCD). Whether in proton-proton (pp), proton-nucleus (pA), or heavy-ion (AA) collisions, multiparticle production reflects the many-body correlations generated by QCD. In pp collisions, such correlations may be produced by quantum evolution through Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution Dokshitzer (1977); Gribov and Lipatov (1972); Altarelli and Parisi (1977) or by a range of higher-order corrections to the hard part (see, e.g. Ellis et al. (1979); Curci et al. (1980)) or small- evolution, including linear Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution Kuraev et al. (1977); Balitsky and Lipatov (1978), nonlinear Balitsky-Kovchegov (BK) evolution Kovchegov (1999); Balitsky (1996) and Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution Jalilian-Marian et al. (1997, 1998); Iancu et al. (2001). The resulting correlations are sensitive probes of the perturbative hard vertex and of the strongly-ordered emission structure of the evolution equations. In pA collisions, these higher-order and evolution corrections are augmented by a new set of dynamical correlations arising from the enhancement of multiple scattering in the high charge densities of the heavy nucleus, characterized by the color-glass condensate (CGC) effective field theory (see e.g. Kovchegov and Levin (2012) and references therein). The resulting correlations are sensitive probes of the multiple scattering dynamics, including the significant effects of Bose enhancement in the strong gluon fields Altinoluk et al. (2015a, 2018a). Finally, in AA collisions (as well as potentially in high-multiplicity pp and pA collisions), all these initial state correlations are modified and complemented by the final-state dynamics of a strongly-coupled quark-gluon plasma (QGP) phase.

A detailed characterization of multiparticle production in the strong color fields of the CGC is especially important in trying to differentiate intial-state effects from the final-state dynamics of the QGP, where strongly-coupled interactions lead to substantial many-body correlations among soft and semi-hard particles. The canonical measures of this collective flow are the cumulants of azimuthal anisotropies Luzum and Petersen (2014), with correlations among increasing numbers of particles reflected in higher values of the cumulants. Other landmark properties believed to be possible in the QGP phase include the onset of novel transport mechanisms associated with the axial anomaly: the chiral magnetic effect, the chiral separation effect, the chiral vortical effect, and the chiral magnetic wave ^{1}^{1}1For a review on chiral magnetic and vortical effects in high-energy nuclear collisions we refer to the reader to Ref. Kharzeev et al. (2016).. Signatures for all of these novel chiral dynamics are encoded in multiparticle correlations, often charge dependent, such as the same-sign and opposite-sign correlators and Sirunyan et al. (2018). For all of these critical signatures of the quark-gluon plasma, it is essential to disentangle the “background” contributions coming from initial-state mechanisms to better quantify the properties of the QGP and improve the chances of discovering such novel anomalous dynamics. Accordingly, a substantial effort has been made in recent years to compute multiparticle production in the CGC framework.

The purest realization of the CGC formalism is in the “dilute / dense” framework, in which density-enhanced effects of the “dilute projectile” are kept only to lowest order, while density-enhanced corrections in the “dense target” are resummed to all orders. Few-particle production has been studied in the dilute / dense framework from the earliest days of the CGC formalism, starting with the inclusive single-gluon production cross section Kovchegov and Mueller (1998); Blaizot et al. (2004a); Kovchegov and Tuchin (2002); Dumitru and McLerran (2002); Kopeliovich et al. (1999); Jalilian-Marian and Kovchegov (2006) at mid-rapidity and followed shortly thereafter by the inclusive cross section of a single pair via gluon pair production Levin et al. (1991); Blaizot et al. (2004b); Kovchegov and Tuchin (2006); Gelis and Venugopalan (2004); Fujii et al. (2006); Gelis et al. (2006, 2005); Gelis and Tanji (2016); Tanji and Berges (2018). Corrections to these production channels were also considered in the form of small- evolution corrections Kovchegov and Tuchin (2006); Fujii et al. (2006). However, a detailed computation of higher multiparticle production cross sections in the dilute / dense framework becomes increasingly difficult due to the proliferation of ways another soft particle could be radiated from a pre-existing one.

A significant step toward overcoming this barrier was made through the development of the “semi-dilute / dense” framework Kovchegov and Wertepny (2013). This regime is designed to fill the gap between the dilute / dense regime, in which the projectile charge density is kept only to lowest order, and the dense / dense regime, where both projectile and target densities must be simultaneously resummed to all orders. The semi-dilute / dense framework is appropriate for “heavy-light ion collisions” intermediate to, say, pPb and PbPb collisions. For a collision between one light ion and one heavy ion, such as CuAu collisions, it is possible to construct a regime in which the large target density is resummed to all orders while corrections from the projectile density are calculated order by order in perturbation theory. In the semi-dilute / dense framework, higher-order corrections which are enhanced by the projectile density are more important than genuine quantum corrections. Formally, for a dense target nucleus with nucleons and a semi-dilute projectile nucleus with nucleons, the semi-dilute / dense regime can be quantified by the hierarchy of scales

(1a) | |||

(1b) |

or equivalently, in term of the saturation momenta and of the projectile and target respectively,

(2) |

With the help of the semi-dilute / dense framework, a number of significant steps have been taken in recent years toward the calculation of genuine multiparticle production in the CGC framework. The key simplification that makes this possible in the semi-dilute / dense framework is that the independent emission of new soft particles from the high-density projectile becomes dominant over emission from the pre-existing system of soft particles. As such, the first observable computed in the semi-dilute / dense framework was the production cross section for two soft gluons Kovchegov and Wertepny (2013); Kovner and Lublinsky (2013). A similar effort was made toward determining the production cross section for two quarks coming from separate pairs, with a partial calculation having been performed in Ref. Altinoluk et al. (2016a) emphasizing the new role played by Fermi-Dirac quantum statistics among the two pairs. This calculation later formed the basis of the partial calculation of the cross section for two quark / antiquark pairs plus a gluon, with the intent of studying the CGC contribution to the same-sign correlators Kovner et al. (2017). It should be emphasized that in this important calculation Kovner et al. (2017), only correlations generated at the level of the wave functions were taken into account, without including the effects of multiple scattering that translate these wave functions into actual production cross sections. And very recently, a new attempt has been made to extend these calculations to the third order in the projectile charge density through the computation of the triple-gluon production cross section Altinoluk et al. (2018b). Other notable developments in soft multiparticle production include the identification of Bose enhancement as a driving mechanism of the Ridge Altinoluk et al. (2015a) in double-gluon production Altinoluk et al. (2015a, 2018a), the calculation of the soft double-photon cross section Kovner and Rezaeian (2017a), and the realization that soft double-pair production can be used to probe the gluon Wigner distribution with Weizsäcker-Williams gauge structure Boussarie et al. (2018). A variant of the semi-dilute / dense power counting can be found in the form of the lowest-order “glasma graph” calculations, which have been used to calculate multiparticle correlations such as the triple-gluon cross section Dusling et al. (2009).

Other important developments in the calculation of multiparticle production in the CGC formalism have emphasized production in the forward regime, where the “hybrid factorization” framework makes it possible to rigorously relate the particle production cross sections to collinear parton distribution functions in the (semi-)dilute projectile, dressed with the effects of multiple scattering in the dense target Dumitru et al. (2006); Altinoluk and Kovner (2011); Altinoluk et al. (2015b, 2016b). In this approach, observables such as forward double valence-quark production cross sections Kovner and Rezaeian (2017b), forward triple valence-quark production cross sections Kovner and Rezaeian (2018), and forward valence-quark + photon + gluon production Altinoluk et al. (2018c) have been calculated. Similar studies of quadruple valence-quark production cross sections have also been considered in a “parton model” description Dusling et al. (2018a, b) without the benefit of an underlying hybrid factorization. Other recent work on the subject also includes the demonstration Kovchegov and Skokov (2018) that two-gluon correlations can break the “accidental” back-to-back symmetry which occurs at lowest order and related phenomenology Mace et al. (2018a, b). And finally, in a recent work Martinez et al. (2018), we have considered single- and double-pair production and in coordinate space as a means of initializing spatial corrections of conserved charges in the quark-gluon plasma.

In this paper, our primary goal is to systematically extend the calculation of multiparticle production at mid-rapidity in the semi-dilute / dense framework to higher orders. One of the key results we will derive here for the first time is the complete expression at exact for the double-pair production cross section in momentum space, as written in Eqs. (IV.1), (IV.1.1), and (IV.1.2). This expression significantly generalizes the result obtained in Ref. Altinoluk et al. (2016a) by including contributions that were intentionally omitted there, by working with exact , and by keeping the multiple scattering corrections to all orders. In a key conceptual development, we show in Eq. (21) that it is possible to map the amplitude (and therefore, the cross section) for producing pairs into the corresponding quantities for producing gluons. Thus, we are able to directly map the double-pair cross section into the corresponding cross section to produce a quark / antiquark pair and a gluon. This expression, as written in Eq. (IV.2), is also a new result. We also perform a number of validations of this gluonic mapping, verifying explicitly that it correctly reproduces the known results for single- and double-gluon production from the literature.

While the preceding results all reflect the final-state production of multiple partons, they also open the door to a substantial program of computing hadronic-level observables derived from them. By convoluting the partonic-level results with the appropriate fragmentation functions or projection operators and long-distance matrix elements, we can translate these partonic-level cross sections to full hadronic cross sections. The resulting hadronic observables can be used to rigorously study the correlations among same-sign and opposite-sign charged hadrons, open and hidden heavy-flavor hadrons, heavy-flavor vs light hadrons, and more. The phenomenology based on these hadronic obserables will provide critical new insight into initial-state mechanisms for collective flow, quarkonium correlations, and charge-dependent correlations which form the background to anomalous chiral dynamics in the QGP.

This paper is organized as follows. In Sec. II we construct the scattering amplitudes for the production of soft particles in momentum space, starting with the quark/antiquark production amplitude in Sec. II.1 and deriving the mapping to the gluon production amplitude in Sec. II.2. Then in Sec. III we compute the production cross section for a single pair in Sec. III.1 and map it in Sec. III.2 to the well-known gluon production cross section to validate the gluonic mapping. Then in Sec. IV we proceed to calculate the new cross sections for the production of two sets of soft particles: double pair production in Sec. IV.1, mixed production in Sec. IV.2, and double gluon production in Sec. IV.3. The successful cross-check against the double-gluon production cross section in Sec. IV.3 represents another validation of the gluonic mapping derived in Sec. III.2. In Sec. V we utilize the techniques enumerated in Ref. Ma et al. (2018) to translate our partonic-level cross sections into hadronic cross sections for the production of open and hidden heavy flavor as an illustration of how to straightforwardly apply the results derived here to hadronic observables. Finally, we conclude in Sec. VI by reiterating the primary new results and exploring the many opportunities for phenomenological applications and further theoretical development which this work provides. In Appendix A we provide details of the Gaussian color averaging used for the (semi-)dilute projectile, and in Appendix B we formulate some useful algebraic properties of Wilson lines in momentum space.

Throughout this paper, we denote longitudinal momenta in light-front coordinates and transverse vectors by with magnitudes . Different authors use different conventions for the light-front metric ; we will use , but it is also common to encounter .

## Ii Production Amplitudes for (Anti)Quark Pairs and Gluons

### ii.1 Quark / Antiquark Pair Production Amplitude

The amplitude to radiate a soft quark/antiquark pair at mid-rapidity has been derived many times in the literature Blaizot et al. (2004b); Kovchegov and Tuchin (2006). In the notation of our previous work Martinez et al. (2018) as illustrated in Fig. 1, we denote the light-front wave functions Brodsky et al. (1998); Lepage and Brodsky (1980) to radiate a soft pair as corresponding to the various time orderings of the scattering in the target fields. The term corresponds to scattering after the pair is created, to scattering after the gluon is emitted but before the pair is created, and to scattering before the pair is created. The three wave functions are not all independent, but satisfy , and the explicit expressions are given by

(3a) | ||||

(3b) | ||||

(3c) |

where is the fraction of the pair longitudinal momentum carried by the quark and is the center-of-mass transverse momentum of the pair. In (3), we have omitted the explicit dependence of the wave functions on the momentum fraction for brevity. Note also that, in comparison to Eqs. (21) of Martinez et al. (2018), we have removed a factor of the coupling from the definition of the wave functions. This corresponds to absorbing this coupling constant into the scale defined in (74) characterizing the sources of soft gluons.

In terms of these wave functions, the single-pair amplitude summed over all time orderings is given in coordinate space by (see Eqs. (30 - 31) of Martinez et al. (2018))

(4) |

where, as labeled in Fig. 1, , , and are the final-state positions of the quark, antiquark, and valence quark, respectively, and is the center-of-mass position of the pair (equal to the gluon position). The scattering of partons in the color fields of the target are described by Wilson lines in the fundamental or adjoint representations,

(5a) | ||||

(5b) |

where we work in the light cone gauge. With (II.1) written this way, the Wilson line associated with the valence quark will always cancel against a corresponding one in the complex-conjugate amplitude.

It is convenient to translate the specific model of the projectile as a distribution of valence quarks into a generic continuous charge density. This can be accomplished by introducing the quantity , which loosely corresponds to the wave function of a color source in the projectile at position which radiates a soft gluon with color . We can translate from the valence quark model of the projectile to the continuous color charge density by effectively replacing . (For another discussion of the translation between discrete and continuous charge distributions, see e.g. Kovchegov and Skokov (2018).) With this change of notation, we can Fourier transform the buildling block (II.1) into momentum space to obtain

(6) |

where the momenta of the final-state quark and antiquark are and , respectively. Note that the Fourier factor for the valence quark cancels because its position is the same in the initial and final states under the eikonal approximation.
^{2}^{2}2At first glance, the amplitude (II.1) may appear to be problematic, because it contains an impact over impact parameters of the source at the amplitude level, leading to two such impact parameter integrals in the cross section. This is true; however, when averaged over color states of the projectile as in (74), the correlator of two ’s possesses a delta function which sets these two positions equal. Thus the continuous charge distribution leads to one integral over per source at the cross section level, as with the model of discrete valence quarks.

Inserting the inverse transformation of the wave functions (3), Wilson lines, and source density

(7a) | ||||

(7b) | ||||

(7c) | ||||

(7d) |

gives

(8) |

with for brevity. We can combine all three diagrams by redefining the dummy integration variables, obtaining the compact form

(9) |

where the differences among the three diagrams are all encoded in the combined wave function

(10) |

With this expression, it is easy to do the manipulations over all diagrams at once and particularly to study their color structure, since the Wilson lines enter in exactly the same form for all diagrams. As such, when we construct cross sections for the production of multiple pairs, we will only have to perform one calculation per diagrammatic topology, rather than having to repeat the calculation for many possible time orderings. These various topologies will correspond to different ways to contract the diagrams, including both the color matrix and the wave function , which is a matrix in the spin space of the pair.

### ii.2 Gluon Production Amplitude

In comparison with (II.1) for the production amplitude of a soft pair in coordinate space, the corresponding amplitude to emit a soft gluon is illustrated in Fig. 2 and is given by

(11) |

where is the light-front wave function

(12a) | ||||

(12b) |

to radiate a soft gluon from a valence quark projectile. Here and are the spin states of the valence quark before and after gluon emission, and is the spin of the emitted gluon. When we write the trace over the square of these wave functions, we mean the averaging over the quantum numbers of the initial state, together with a sum over the quantum numbers of the final state:

(13a) | ||||

(13b) |

The first term of (11) corresponds to the shockwave passing through the gluon, the second term corresponds to the shockwave passing through the valence quark before the gluon is emitted, and we have used the fact that the wave functions for the two time orderings differ by a minus sign (similar to for the quark pair case (3)).

As we did in Sec. II.1, we can rewrite the second time ordering so that the valence quark scattering looks the same as the first one, and we can convert to the continuous charge density to write

(14) |

Comparing the gluon amplitude (14) with the pair amplitude (9), we note that the gluon has a specified color in the final state, resulting in the pair-like structure being contracted with to form a trace. Squaring the gluon amplitude gives

(15) |

where we denote the averaging over color fields of the projectile and target by and , respectively. By we denote a trace over color indices and by we denote a trace over the spin states of the wave function which averages over spins in the initial state and sums over spins in the final state.

If we use the Fierz identity over the repeated color index , we can combine the two traces into one, with the -suppressed term in the Fierz identity vanishing exactly by the Wilson line identity (80). This gives

(16) |

which has the same Wilson line structure as we would obtain by squaring the pair amplitude (9). Thus we can, without loss of generality, replace the gluon amplitude (14) with the equivalent expression

(17) |

that has the same same form as the pair amplitude (9).

This similar structure appears to suggest a possible mapping between the pair amplitude (9) and the gluon amplitude (17). Comparing the two expressions, we see that in the limit , the Wilson line structure of the pair amplitude (9) can be cast in the same form as the gluon amplitude (17):

(18) |

The change of variables followed by makes the comparison with (17) explicit,

(19) |

Thus we see that by mapping the pair wave function

(20) |

and setting the final-state quark and antiquark to have equal momenta, , we can map the pair amplitude (9) onto the gluon amplitude (17):

(21) |

Thus any cross section calculated for the production of multiple pairs via (9) can be mapped onto the cross section to instead produce gluons via (20) and (21). The existence of this mapping allows us to efficiently compute multiparticle production at mid rapidity, by first computing the production of various pairs and then mapping them systematically back to gluons. Aside from (21), the only other modifications to the cross section will be a change in the prefactor of to reflect the changed number of final-state particles and the exclusion of quark entanglement in the pair which has been mapped to a gluon. We will use this strategy in Sec. III.2 to obtain the single-inclusive gluon cross section from the pair cross section and in Secs. IV.2 and IV.3 to obtain the and cross sections from the double-pair cross section.

## Iii Cross Sections for Single Pairs and Gluons

### iii.1 Cross Section for Single-Pair Production

The inclusive cross section to produce a single soft pair with quark (antiquark) transverse momentum () and rapidity () is simply related to the square of the amplitude (9):

(22) |

where and . To help keep the notation compact let us introduce the following notation for the differentials:

(23a) | ||||

(23b) | ||||

(23c) |

We will also often exclude the underlines for the many transverse vectors when it is clear from context that they refer to -vectors rather than -vectors. Then squaring (9) and performing the averaging as in Appendix A, we straightforwardly obtain

(24) |

as illustrated in Fig. 3. Using the Gaussian averaging of the projectile with the assumption of Locality from (74), we obtain

(25) |

where the second argument of vanishes in this case because the eikonal Wilson lines preserve the plus momentum. This feature will in general be violated when producing multiple pairs. The color trace is straightforward to simplify using using the Fierz identity, yielding the compact expression

(26) |

where the fundamental dipole and quadrupole operators are defined in (72).

Comparing this expression with the corresponding ones (32) and (37) from Ref. Martinez et al. (2018) in coordinate space, we see that Eq. (III.1) is far more compact. This is largely because the condensed amplitude (9) in momentum space combines the two time orderings “1” and “2” into a single form, allowing us to perform a single calculation for all time orderings, rather than requiring a sum over all the distinct time orderings for the pair emission. The single-pair cross section is then immediately given by combing Eqs. (III.1) and (22).

### iii.2 Cross Section for Single-Gluon Production

As a cross-check and to illustrate the “gluonic mapping” of a pair (21), let us map (III.1) onto the cross section for single-inclusive gluon production. Applying (21) to (III.1) gives

(27) |

where we changed the prefactor from (22) to to reflect the fact that there is now only one particle tagged in the final state. Changing variables to and gives