# Medium-induced gluon emission via transverse and longitudinal scattering in dense nuclear matter

###### Abstract

We study the medium-induced gluon emission from a hard quark jet traversing the dense nuclear matter within the framework of deep inelastic scattering off a large nucleus. We extend the previous work and compute the single gluon emission spectrum including both transverse and longitudinal momentum exchanges between the hard jet parton and the medium constituents. On the other hand, with only transverse scattering and using static scattering centers for the traversed medium, our induced gluon emission spectrum in the soft gluon limit reduces to the Gyulassy-Levai-Vitev one-rescattering-one-emission formula.

## I Introduction

The study of parton energy loss and jet quenching has been regarded as a very useful tool to probe the properties of the quark-gluon plasma (QGP) produced in ultra-relativisitic heavy-ion collisions at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) Wang:1991xy (); Qin:2015srf (); Blaizot:2015lma (); Majumder:2010qh (). After they are produced from early stage hard collisions, high transverse momentum partonic jets propagate through the dense nuclear medium and interact with the medium constituents via binary elastic and radiative inelastic collisions before fragmenting into hadrons. Jet-medium interaction not only changes the energy of the leading parton, but also modifies the internal structure of the jets, such as the distribution of jet momentum among different jet constituents. The picture of jet-medium interaction and parton energy loss has been confirmed by many experimental observations at RHIC and the LHC, such as the significant suppression of high transverse momentum hadron and jet production in nucleus-nucleus collisions as compared to the expectations from independent nucleon-nucleon collisions Adcox:2001jp (); Adler:2002xw (); Aamodt:2010jd (); CMS:2012aa (); Abelev:2012hxa (); Aad:2015wga (); Aad:2010bu (); Chatrchyan:2011sx (); Chatrchyan:2012gt (); Aad:2014bxa ().

Several theoretical schemes have been founded to study the radiative component of energy loss experienced by the hard jet partons propagating through the dense nuclear medium, such as Baier-Dokshitzer-Mueller-Peigne-Schiff-Zakharov (BDMPS-Z) Baier:1996kr (); Baier:1996sk (); Baier:1998kq (); Zakharov:1996fv (); Zakharov:1997uu (), Gyulassy, Levai and Vitev (GLV) Gyulassy:1999zd (); Gyulassy:2000fs (); Gyulassy:2000er (), Armesto-Salgado-Wiedemann (ASW) Wiedemann:2000za (); Wiedemann:2000tf (); Armesto:2003jh (), Arnold-Moore-Yaffe (AMY) Arnold:2001ba (); Arnold:2002ja () and higher twist (HT) Guo:2000nz (); Wang:2001ifa (); Majumder:2009ge () formalisms. For a more detailed comparison of different jet energy loss schemes, the reader is referred to Ref. Armesto:2011ht () and references therein. There have also been many literatures studying the effect of binary elastic collisions between the hard partons and the medium constituents Bjorken:1982tu (); Braaten:1991we (); Djordjevic:2006tw (); Qin:2007rn (). Also, extensive studies have been performed for various jet quenching observables, such as the suppression of large transverse momentum hadron production Bass:2008rv (); Armesto:2009zi (); Burke:2013yra (); Cao:2017hhk (), the nuclear modification of back-to-back dihadron and photon-hadron pair correlations Zhang:2007ja (); Majumder:2004pt (); Qin:2009bk (); Renk:2008xq (); Chen:2016vem (); Chen:2017zte (), and the full jet modification Qin:2010mn (); CasalderreySolana:2010eh (); He:2011pd (); Young:2011qx (); Dai:2012am (); Zapp:2012ak (); CasalderreySolana:2012ef (); Wang:2013cia (); Blaizot:2013hx (); Chang:2016gjp (); Chen:2016cof (); Mehtar-Tani:2016aco (); Casalderrey-Solana:2016jvj (); Tachibana:2017syd (); Cao:2017zih (); KunnawalkamElayavalli:2017hxo (); Luo:2018pto (); Chen:2018fqu (). By comparing phenomological jet modification calculations to experimental measurements, one may obtain the quantitative values of jet transport coefficients, such as the transverse momentum broadening rate and the longitudinal momentum loss rate Burke:2013yra (), from which a lot of information about the dense nuclear medium traversed by the hard jets can be inferred.

While the studies of jet quenching in heavy-ion collisions have already entered the quantiative era, there still exist many theoretical uncertainties in detailed calculations of the effects caused by jet-medium interaction. For example, most current studies of inelastic radiative contribution mainly focus on the gluon emission induced by the transverse momentum exchange between the propagating hard jet partons and the dense nuclear matter. However, when a hard parton interacts with the traversed dense medium, both transverse and longitudinal momenta are exchanged between them Majumder:2008zg (); Qin:2012fua (); Abir:2014sxa (). While there have been many studies on the longitudinal momentum exchange (loss) experienced by the propagating jet partons, the main focus was on the evaluation of purely collisional energy loss either from the leading parton Wicks:2005gt (); Djordjevic:2006tw (); Qin:2007rn (); Schenke:2009ik () or by the shower partons of the full jet Qin:2009uh (); Neufeld:2009ep (); Qin:2010mn (); Qin:2012gp (); Chang:2016gjp (); Tachibana:2017syd (). In Refs. Qin:2014mya (); Zhang:2016avg (); Abir:2015hta (), the effect of longitudinal momentum transfer between the hard parton and the medium constituents on the medium-induced emission vertex has been studied.

In this work, we study the medium-induced gluon emission from a hard quark jet which scatters off the medium constituents during the propagation through the dense nuclear medium, within the framework of deep-inelastic scattering (DIS). We extend the HT radiative energy loss approach Guo:2000nz (); Wang:2001ifa () and include the contributions from both transverse and longitudinal momentum exchanges to the gluon emission vertex. It is also an extension of Refs. Qin:2014mya (); Zhang:2016avg () which study the medium-induced photon emission from longitudinal and transverse scatterings. Here we derive a closed formula for the medium-induced single gluon emission spectrum with the inclusion of the contributions from both transverse and longitudinal momentum transfers between the hard parton and the medium constituents. We further show that if one neglects the longitudinal momentum transfer and only considers the transverse scattering, our medium-induced gluon emission spectrum in the soft gluon limit can reduce to the GLV one-rescattering-one-emission formula.

The paper is organized as follows. In Sec. II, we provide some details for the derivation of the medium-induced gluon emission spectrum for a quark jet parton traversing the dense nuclear medium. Other details may be found in the Appendix. Sec. III contains our summary.

## Ii Medium-induced gluon emission

Here we study the gluon emission from a hard jet parton in dense nuclear matter in the framework of deep inelastic scattering (DIS) off a large nucleus. We consider the following DIS process:

(1) |

where and are the momenta of the incoming and outgoing leptons, and is the momentum of the incoming nucleus, with being the momentum carried by each nucleon in the nucleus, and are the momenta of the produced hard quark and the radiated gluon. Here the light-cone notation are used for four-vectors, e.g., , and . In the Briet frame, the virtual photon carries a momentum , with being the Bjorken fraction variable and the invariant mass of the virtual photon.

The differential cross section the lepton production from the above DIS process can be expressed as follows:

(2) |

Here is the electromagnetic coupling and is the center-of-mass energy of the lepton-nucleon collision system. The leptonic tensor is given as:

(3) |

The hadronic tensor is expressed as:

(4) | |||||

where denotes initial state of the incoming nucleus A, and represent the final hadronic (or partonic) states, with running over all possible final states except the outgoing hard quark jet and the emitted gluon. is the hadron electromagnetic current for a quark of flavor and the electric charge (in units of the electric charge ). In our study, the focus is the hadronic tensor which contains the detailed information about the final state interaction between the struck hard quark and the traversed dense nuclear medium.

Figure 1 shows the gluon emission process in semi-inclusive DIS at leading twist level. It represents the process that a hard quark is first excited by the virtual photon from one nucleon of the nucleus, then radiates a gluon and exits the medium without further interaction. Here we use the light-cone gauge, , with . The sum of the gluon polarization in this light-cone gauge is given by:

(5) |

In the limit of very high energy and collinear emission, one may neglect the () component of the incoming quark field operators and factor out the one-nucleon state from nucleus state. After some straightforward calculation, the contribution to the hadronic tensor at leading twist can be obtained as follows:

(6) | |||||

Here , denotes the probability of finding a nucleon state with a momentum inside the nucleus , is the strong coupling, and and are the fractions of the forward momentum and transverse momentum carried by the radiated gluon with respect to the parent quark. For convenience, the momentum fraction is also defined. The quark parton distribution function is defined as:

(7) |

where is the fraction of the forward momentum carried by the quark from the nucleon. The leading order quark-to-gluon (photon) splitting function is given by:

(8) |

Note the color factor for quark to gluon splitting vertex is factored out. Therefore, the differential single gluon emission spectrum at leading order (without rescattering with the medium constituents) is given as:

(9) |

We now study the medium modification on the above gluon emission spectrum due to the re-scattering with the medium constituents. We will compute the medium-induced gluon emission spectrum from a hard quark jet traversing the dense nuclear medium in the DIS framework. In this work, we use the following power counting scheme and notations: for the hardest momentum scale, and for a small dimensionless parameter. Considering the scattering of a nearly on-shell projectile parton carrying a momentum off a nearly on-shell target parton carrying a momentum , the exchanged gluons is then the standard Glauber gluon carrying a momentum Idilbi:2008vm (). If the target parton is allowed to be off shell, the longitudinal momentum component of the exchanged gluon may be the same order as the transverse components; such type of gluons is often referred to as the longitudinal-Glauber gluon which carries a momentum Qin:2012fua (). In this work, we will investigate the influence of both transverse and longitudinal momentum transfers on the single gluon emission from a hard quark jet which interacts with the constituents of the dense nuclear medium.

Figure 2 shows one of central-cut diagrams that contributes to the hadronic tensor at the twist-four level. It describes the process with a single rescattering on the radiated gluon in both the amplitude and the complex conjugate. The other 20 diagrams at the twist-four level are presented in Appendix. In this section, we provide the detailed calculation of the hadronic tensor and the medium-induced gluon emission spectrum for Figure 2. The calculations for the other 20 diagrams are completely analogous and their main results are listed in Appendix.

In Figure 2, a hard virtual photon carrying a momentum strikes a quark from the nucleus at the location ( in the complex conjugate). The incoming quark from the nucleus carries a momentum ( in the complex conjugate). The struck quark carries a momentum ( in the complex conjugate) and emits a gluon with a momentum ( in the complex conjugate). The emitted gluon then scatters off the gluon field at the location ( in the complex conjugate) and picks up a momentum ( in the complex conjugate). The final radiated gluon and the final outgoing quark carry the momenta and , respectively. The contribution to the hadronic tensor from Figure 2 can be written as follows:

(10) | |||||

Here, ( in complex conjugate) is the three-gluon vertex,

(11) |

where is the anti-symmetric structure constant of group, and

(12) |

( in complex conjugate) is the gluon propagator,

(13) |

To simplify the hadronic tensor , one first isolates the phase factors associated with the gluon insertions: . After carrying out the integration over and , we obtain two functions which may be used to integrate over the momenta and , rendering the relations: and . From the momentum conservation at each vertex, we can obtain the following relations among various momenta in Figure 2:

(14) |

Re-introducing the momentum and changing the integration variables and , the phase factor can be expressed as: .

In this study, we work in the very high energy and collinear emission limit, and the dominant component of the rescattered gluon field is the forward component. In such limit, one may factor out one-nucleon state from the nucleus state and ignore the component of the quark field operators,

(15) |

The gluon propagators together with the three-gluon vertices in the hard trace part may be simplified as:

(16) |

With the above simplification, the hadronic tensor may be written as follows:

(17) | |||||

We now look at the internal propagators and external lines. For the quark propagator before the gluon emission,

(18) |

where we have defined the momentum factions,

(19) |

For the internal gluon propagator,

(20) |

where we have defined the momentum factions,

(21) |

For the final outgoing quark, the on-shell condition gives:

(22) |

where we have defined the momentum factions,

(23) |

Combining the internal quark and gluon lines with the final outgoing quark,

(24) | |||||

where

(25) |

The trace part of the hadronic tensor [the last line of Eq. (17)] can be simplified as:

(26) |

With the above simplifications, the hadronic tensor now reads:

(27) | |||||

where for convenience, we have used the three-vector notations for momentum and coordinate: and , and .

Now we perform the integration over the momentum fractions , , and . Using the on-shell condition for the outgoing quark and the overall momentum conservation , we can carry out the integration over ,

(28) |

The remaining integration over may be performed with a counterclockwise semicircle in the upper half of the complex plane:

(29) |

where we have defined the variable for convenience. The integration over the momentum fraction is completely analogous. After performing the integration over the quark lines and the gluon lines, we obtain the hadronic tensor as: