The theory and phenomenology of perturbative QCD based jet quenching

The theory and phenomenology of perturbative QCD based jet quenching

Abstract

The study of the structure of strongly interacting dense matter via hard jets is reviewed. High momentum partons produced in hard collisions produce a shower of gluons prior to undergoing the non-perturbative process of hadronization. In the presence of a dense medium this shower is modified due to scattering of the various partons off the constituents in the medium. The modified pattern of the final detected hadrons is then a probe of the structure of the medium as perceived by the jet. Starting from the factorization paradigm developed for the case of particle collisions, we review the basic underlying theory of medium induced gluon radiation based on perturbative Quantum Chromo Dynamics (pQCD) and current experimental results from Deep Inelastic Scattering on large nuclei and high energy heavy-ion collisions, emphasizing how these results constrain our understanding of energy loss. This review contains introductions to the theory of radiative energy loss, elastic energy loss, and the corresponding experimental observables and issues. We close with a discussion of important calculations and measurements that need to be carried out to complete the description of jet modification at high energies at future high energy colliders.

Contents:

1 Introduction

During the past nine years, high energy heavy-ion collisions have been studied at the the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory (BNL). The three most striking findings in heavy-ion collisions at RHIC are the observation of a large azimuthal asymmetry in soft ( GeV) hadron production, denoted as elliptic flow; the observation of a large suppression of hadron yields at high transverse momentum () GeV and the finding that the yields and the elliptic flow at intermediate follow a scaling behaviour that separates baryons and mesons by a factor of 3/2. Each of these observations indicates that hot and dense strongly interacting matter is formed in collisions at RHIC, possibly a deconfined Quark-Gluon Plasma. Soon, the LHC will start colliding nuclei at much larger energies, which will provide critical tests of the current understanding of hot and dense QCD matter at RHIC.

In this review, we will concentrate solely on the suppression of high- hadron production and specifically its theoretical description within a perturbative QCD (pQCD) based formalism. In the collision of two heavy-ions at very high energies, most of the hard valence partons go through the collision un-deflected. The prevalent interaction is that between the softer “sea” partons. It is these soft interactions which lead to the formation of the hot and dense matter. Here “soft” means involving momentum scales of the order of . Occasionally the hard partons undergo hard scattering (at scales much larger than ) leading to the formation of two back-to-back hard partons with a large . High- hadrons originate in the fragmentation of such high partons after their escape from the medium. The hard partons lose energy through interactions with the hot and dense medium. High signatures are of special interest, because they are expected to be described by perturbative QCD due to the hard scale of the jets. At the LHC, it will be possible to reach GeV, i.e. a factor of 10 increase in the hard scale of the jets compared to RHIC. Jet physics based on perturbative QCD is therefore expected to play a central role in the study of hot QCD matter at the LHC.

While the development of the theoretical framework to describe parton energy loss was started a little less than 20 years ago [1], the basic observation that high- hadrons are suppressed was only made with the first RHIC data [2, 3]. Since then there has been significant progress in our understanding of parton energy loss. On the experimental side, the variety of measurements has greatly increased, now extending to a variety of di-hadron and heavy flavour measurements. The precision and momentum reach of these measurements is also much more extended compared to the first results from RHIC. Phenomenological calculations have become much more detailed, incorporating a variety of initial state effects such as shadowing, Cronin broadening [4] and more realistic models of the collision geometry based on hydrodynamical calculations constrained to describe the soft hadron spectrum. There are now four different theoretical schemes for energy loss calculations, which incorporate different physical assumptions regarding the scales involved and the microscopic structure of the medium. These approaches can be compared and contrasted to provide insight into the essential aspects of the dynamics of energy loss in heavy ion collisions.

In this review, we aim to provide a survey of the current state of the theoretical understanding of parton energy loss and the experimental tests of this understanding. Our objective is not to provide an exhaustive compilation of all experimental results and theoretical calculations that have been performed, but rather to provide an overview of the current understanding and methods which can serve as a ‘reading guide’ for the extensive literature on the subject. The theoretical part of the review introduces the pQCD based formalism of jet quenching as an extension of the standard factorized approach to hard processes in vacuum and then discusses the similarities and differences between the four different theoretical approaches to parton energy loss in some detail. In the experimental part of the review the emphasis is on a subset of observables which are, or can be, rigorously calculated within pQCD and focus on what has been learnt from these. This field is under active development, both theoretically and experimentally. On the theory side, the topic of most current interest is the development of Monte Carlo routines to address some of the shortcomings of the analytical approaches and to compute more exclusive jet based observables. On the experimental side, measurements have been extended to include photon-jet events and multi-hadron observables, including first attempts to reconstruct jets in heavy ion collisions at RHIC. In conclusion, we will discuss these developments and how they are expected to address some of the main open question about parton energy loss.

In the remainder of this chapter we provide a very brief introduction of how the factorized formalism arises in pQCD. In Chapter 2, we re-derive the multiple scattering induced single gluon emission cross section within this language. This is most straightforwardly achieved in a particular variant of what has come to be denoted as the higher twist approach. After this we relate our results to the derivations in other schemes. In Chapter 3, we describe the inclusion of multiple emissions. This is as yet a theoretically unsettled regime and we review the three different means by which multiple emissions have been included. This is followed by a direct comparison of the various formalisms, where the medium modified fragmentation function is calculated in an identical medium. In Chapter 4 we extend our formalism to include heavy flavor modification along with elastic loss and diffusion. In Chapter 5 we describe the phenomenological setup used to compute jet modification in a static large nucleus and in a dynamically evolving quark gluon plasma. In Chapter 6, we compare the results of recent calculations in the literature on single hadron, dihadron, photon triggered and heavy flavor observables to measurements in Deep-Inelastic Scattering (DIS) from HERA and those in high-energy nuclear collisions from RHIC. In Chapter 7, we provide an outlook to the LHC and forthcoming theory calculations.

1.1 Background

Quantum chromodynamics (QCD) is now the accepted theory of strong interactions. In spite of the large diversity of hadronic states observed in nature, the Lagrangian of QCD is rather simple, involving only two types of fundamental fields: quarks and gluons, interacting via an gauge theory [5]

(1)

where run from to representing the colors of the fundamental quarks of flavor and runs from to representing the colors of the adjoint gluon gauge field. For the purposes of this review will mostly be limited to 3, for the light quarks , , and , which will be treated as massless. The heavy flavors charm and bottom will be discussed separately and will have non-zero mass terms. The top quark will not be included in this review (thus ). The covariant derivative and the adjoint field strength have the usual definitions,

(2)

In the equation above, is the gluon four-vector potential with adjoint color , are the Gell-Mann matrices, is the completely antisymmetric tensor and is the strong interaction coupling constant. In this review, we will often refer to the fine structure constant also as the “coupling”. Which is meant, will be denoted using the appropriate symbol.

Like all renormalizable quantum field theories, the coupling depends on the renormalization scale , usually chosen as the relevant hard scale in the problem to minimize the effect of higher order contributions. Unlike the case of QED or weak interactions, QCD is asymptotically free, which implies that the coupling becomes weaker as the scale is raised [6, 7]. Solving the (one loop) renormalization group equation, and comparing with experimental data to obtain the initial condition, one obtains,

(3)

The above equation implies that at scales far above MeV, QCD should be weakly coupled and a perturbative expansion in terms of should become feasible. The applicability of this statement, of course, depends on the process in question. In the case of the total cross section in  annihilation, single hadron inclusive  annihilation, the total cross section in Deep Inelastic Scattering (DIS), single hadron inclusive DIS and the Drell-Yan effect, pQCD begins to become applicable beyond a GeV [8, 9, 10]. The quantity here refers to the relevant invariant mass that sets the hard scale in the problem. In all the three cases mentioned, this is the invariant mass squared of the intermediate photon. In the case of hadron-hadron scattering, one requires a specific hard momentum transfer process, such as the production of a high transverse momentum (high ) particle. The hard interaction in this case is mediated by the strong force (meaning the higher order corrections are of a different type) and such processes seem to need a minimum GeV ( GeV) for pQCD to be applicable [11]. In all cases mentioned above the applicability of pQCD refers to the agreement of calculations based on pQCD and experimental data.

With the diminishing of momentum transfers, the strong coupling fine structure constant continues to grow and confines all particles which carry color charge within composite color singlet hadrons. While one may still construct an effective Lagrangian using the underlying symmetries of QCD and carry out a perturbative analysis, such theories will not constitute any part of this review. All interactions with a momentum scale below a GeV will be considered as non-perturbative and will usually be contained in a non-perturbative distribution such as a fragmentation function, a parton distribution function or an in-medium transport coefficient. The separation of the non-perturbative part from the hard perturbative part of the calculation will be discussed in the next subsection.

1.2 pQCD and factorization

The ability to apply pQCD to describe a particular process should not be confused with the application of perturbative expansions in other gauge theories such as QED, the electro-weak theory or even in low energy effective theories of QCD such as chiral perturbation theory. In those theories, the incoming and outgoing asymptotic states in a scattering event carry the quantum numbers of the fields in the respective Lagrangian densities. While the QCD Lagrangian is cast in terms of quarks and gluons and by pQCD we do mean a diagrammatic expansion involving those fields, asymptotic states in strong interactions, due to confinement, are never quarks or gluons but composite hadrons. The ability to apply pQCD thus means simply the ability to isolate a section of the interaction which can be systematically described using a perturbative expansion in involving quarks and gluons, from the remaining part of the process which is non-perturbative.

In all the reactions mentioned above where pQCD is applicable, there exist sub-processes over a range of energy scales up to the hard scale . Most of these cannot be described using pQCD and need to be separated from those sub-processes which involve hard scales. The technical machinery which demonstrates this separation order-by-order in the coupling constant is called “factorization” [12, 13, 14]. The result of factorization is usually stated as a theorem with corrections power suppressed at very large . We illustrate an example of this theorem for the case of a high energy - collision leading to the formation of a high hadron. Assume the collision is in the center of mass frame with each proton carrying a momentum . The physical picture of this process is one where a hard parton in one of the incoming nucleons, carrying a light-cone momentum fraction () scatters of a hard parton in the other nucleon with light-cone momentum fraction () and produces two back-to-back partonic jets. Light-cone momenta are defined as and . The hadronization products of one of these jets () will yield the high hadron carrying a momentum fraction of the original jet’s momentum. The factorization theorem for the differential cross section (in and rapidity ) states that the process can be expressed as [13, 15],

(4)

where is the parton distribution function (PDF) to find a hard parton with a momentum fraction () in the incoming proton, is the hard partonic cross section (with a partonic Mandelstam variable ) and is the fragmentation function (FF), the distribution of hadrons with a momentum fraction produced in the hadronization of the outgoing hard parton. The second term on the indicates corrections to the factorization theorem that are suppressed by powers of the hard scale . The quantity represents the scale of soft processes in the collision. This contribution becomes negligible as . Even in this asymptotic limit, the only term in the equation above that can be completely calculated within pQCD is the hard partonic cross section; the PDFs and the FF are non-perturbative objects which represent physics at softer scales.

The primary utility of factorization is based on the lack of interference between the hard and the soft scale, i.e., the PDFs and the FF (as well as the partonic cross section) are all squares of amplitudes. Eq. (4) contains only a convolution in two parameters () between these probabilities. To compute the hard cross section one proceeds order by order and calculates all the amplitudes that may contribute to the process of two incoming partons scattering to outgoing partons [ for Leading Order (LO), for Next-to-Leading Order (NLO)]. These are then summed and squared to obtain . The non-perturbative quantities or cannot be calculated from first principles, however they have well defined operator expressions on the light-cone, e.g., the PDF to find a quark with momentum fraction , in a proton state traveling with a large light-cone momentum , is defined as,

(5)

Given the operator expression, one may calculate higher order corrections. These tend to have collinear divergences which are absorbed into a renormalization of the operator product (or non-perturbative expectation values in the case of the FF). Such a redefinition introduces a scale dependence of the expectation. Divergent contributions up to this scale, denoted as the factorization scale , are absorbed into the definition of the PDF. If is large compared to , the change of the PDF and the FF can be calculated by means of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [16, 17, 18],

(6)

In the equation above, is the splitting function which represents the probability for a hard quark to radiate a gluon and still retain a momentum fraction . The homogeneous DGLAP equation, as expressed in Eq. (6) applies only to the non-singlet quark distribution []. The singlet distributions () have couplings that mix the quark and gluon distributions.

Equation (6) is a differential equation and thus requires an initial condition for its solution. This requires the experimental measurement of the non-perturbative distribution at one value of the scale. Herein lies the other advantage of factorization: Once the non-perturbative distributions are factorized from the hard cross section, they are independent of the process and become universal functions in the sense that they may be given an identical definition and measured in a completely different process. In the case of totally inclusive DIS, the factorization theorem yields,

(7)

The operator expression for the PDF as well as its evolution equation, derived in this case, is identical to that in Eq. (4). Thus, the PDF measured in DIS (7), may be directly substituted in Eq. (4) to compute single particle inclusive cross section in - collisions.

To summarize, the applicability of pQCD to vacuum processes, such as in Eqs. (4,7), consists of the ability to calculate the hard partonic cross section order by order, as well as to compute the scale dependence of the non-perturbative distributions. While these non-perturbative distributions need to be measured at one scale in experiment, they have rigorous operator definitions which are identical in all processes where they can be factorized and thus are universal functions. Even though the discussion above was focused on the PDF, an almost identical factorization theorem and DGLAP evolution equation may be written down for the FF. Most of the remaining review will deal with the vacuum and in-medium modification of the FF.

1.3 Hard jets in Semi-inclusive DIS on a nucleus and heavy-ion collisions

The preceding subsection briefly described the factorized formalism of high energy pQCD as applied to hard processes. The underlying assumption of factorization has been rigorously proven in all the examples mentioned above. In this subsection, we outline the extension of this factorized formalism to the modification of hard jets in a dense medium. Unlike the case of hard processes in vacuum, the factorization theorems which are the underlying assumptions in these calculations, have not been proven to the same degree of rigor. So far there have only been a handful of attempts in this direction both in pQCD [19, 20, 21] as well as in an effective-field theory approach [22]. These factorization theorems will not be discussed further; we will assume their applicability. In this review we will focus on the application of the factorized theory to jet propagation in dense extended media and the ensuing phenomenology.

Imagine the single-inclusive DIS of a hard lepton on a proton in the Breit frame. This is illustrated in Fig. 1. The virtual photon with large negative light-cone momentum strikes a hard quark which carries a momentum fraction of the Lorentz contracted proton with large positive light-cone momentum . The quark is turned around and then exits the proton. Being considerably off-shell from the hard collision, it begins to shower gluons. The initial radiations have large transverse momentum (due to the large virtuality), and thus have shorter formation times:

(8)

Where is the momentum fraction of the parent parton carried away by the radiated gluon. Later radiations (those with larger formation times) have smaller transverse momentum. While not illustrated in the figure, the radiated gluons are also off-shell and tend to radiate gluons with smaller off-shellness. Eventually the local virtualities of the partons is so low that one cannot apply a perturbative partonic picture and hadronization begins to set in. As a result, the collection of hard collinear partons turns into a collimated jet of hadrons. The leading hadron in this picture is the one with the largest longitudinal momentum and, given some form of localized momentum conservation, is the result of hadronization of the highest longitudinal momentum part or the largest formation time part of the jet.

Figure 1: DIS on a nucleon leading to the formation of a quark jet which showers gluons in vacuum leading to the formation of a collimated jet of hadrons.

The picture in - is somewhat similar except that one has at least 2 back-to-back jets being produced. Both of these jets are produced considerably off-shell and tend to lose this virtuality by successive emissions. Unlike the case of DIS, jets which arise from the produced hard partons in large - collisions have a large momentum transverse to the incoming hard partons. We will always denote the momentum in this direction as , differentiating it from the momentum of gluons transverse to the produced jet which will be indicated with a , as in . The factorized formula to obtain the cross section of single hadrons is given by the first term on the right hand side of Eq. (4). At leading order, this formula simplifies to,

(9)

The reader will note that we have chosen all factorization and renormalization scales to be which is the hard scale in the problem. At leading order all hard scales that appear in the calculation are equivalent; in most cases one simply picks .

We now consider the process of DIS on a large nucleus. This will often be referred to as A-DIS (A referring to a nucleus). In the Breit frame, one may neglect the soft interactions between the various nucleons as these occur over a long time scale. The nucleus may then be modeled as a weakly interacting gas of nucleons traveling in the positive light-cone direction. The virtual photon strikes a hard quark in one of the nucleons and turns it backwards as in the case of the DIS on a single proton. The quark then propagates through the nucleons directly behind the one that is struck, as illustrated in Fig. 2, where we draw the struck quark as propagating outside the line of nucleons for clarity. The quark is virtual at production and radiates a shower of gluons with progressively longer formation times, similar to the case of DIS on a proton. In this case however, both the quark and the radiated gluons tend to scatter off the soft gluon field in the nucleons. This is indicated by the zig-zag lines. Note, the zig-zag lines do not indicate a pomeron or double gluon exchange, but rather are single-gluon exchanges which are distinct from the gluons in the shower of the jet. Diffractive exchanges with the nucleons may be neglected in the case of very large nuclei. This will be justified in the next chapter.

The multiple scattering of the partons in the shower changes their momentum distributions and as a result the final hadronization pattern is modified. In the case of a jet produced in a heavy ion collision, the picture is qualitatively similar except for the absence of nucleons. The different components of the jet now scatter off the quark-gluon substructure of the degrees of freedom in the hot deconfined matter. The leading effect of the multiple scattering on the shower profile is a broadening of the distribution in transverse momentum (recently this has been experimentally observed in cold nuclear matter [23]). Besides transferring transverse momentum, the medium also exchanges energy and longitudinal momentum with the jet (in the language of light-cone momentum, both and components are exchanged between the jet and the medium). While energy and momentum exchange (including transverse momentum) in the right proportion may cause minimal change in the virtuality of a given jet parton, arbitrary momentum exchanges may noticeably change the virtuality leading to induced radiation. In the case that the energy (and longitudinal momentum) of the jet partons far exceeds that of any constituent in the medium, such exchanges lead to a depletion of the light-cone momentum in the forward part of the jet. This is often referred to as radiative energy loss and results in a suppressed yield of leading particles [24, 25, 26, 27, 28, 29, 30]. This suppression has been quantitatively established in both DIS on a large nucleus [31] and in jets in heavy ion collisions [2, 3]. The primary difference between the two cases is the fact that a heavy-ion collision is not a static environment and evolves rapidly with time and then finally disintegrates into a cascade of hadrons.

Figure 2: DIS on a large nucleus. A quark in a nucleon on the front side is struck and then propagates through the nucleons behind the struck nucleon. In the process both the quark and the ensuing shower gluons scatter off the soft glue field inside the nucleons. This modifies the shower pattern.

The calculation of the modification of the hadronization pattern due to the multiple scattering induced radiation will be the subject of the next two chapters. Until recently, experiments have only been able to measure the modification of the yield of the leading hadron or the correlation between the leading and next-to-leading hadrons. The basic theoretical object required to describe this aspect of jet modification is called the medium modified fragmentation function. Along with the perturbative calculation of the evolution of the fragmentation function in vacuum, one computes the change in the fragmentation function due to the broadening and stimulated emission that occurs as the jet passes through matter. The medium modified fragmentation function is now a function of not just the momentum fraction and the scale , but also of the energy of the jet and the distance travelled in the medium. While it has become conventional to call this a “medium modified” fragmentation function, it should be pointed out that in all pQCD based calculations, the fragmentation is assumed to occur once the jet has escaped the medium.

We should point out that while this review will describe jet modification in dense matter as an extension of the factorized processes in vacuum pQCD [this is often referred to as the Higher-Twist (HT) scheme [32, 33, 34]], there are other approaches to this problem. An entirely orthogonal approach is that based on finite temperature field theory based on the work of Arnold, Moore, Yaffe and collaborators [35, 36, 37, 38, 39, 40]. Referred to as the AMY scheme, this formalism considers the effective Hard Thermal Loop (HTL) effective theory of dense matter and considers the hard jet to have the same virtuality or mass scale as a thermal plasma particle but with energy . Following this, one identifies and resums the collinear enhanced contributions emanating from the scattering and induced radiation off the hard parton. The entire calculation is carried out at the scale of the temperature which is assumed to be large () so that the effective coupling is small []. Another approach developed by Armesto, Salgado, Wiedemann and collaborators models the medium as a series of Debye-screened, heavy, colored scattering centers. In this approach, referred to as the ASW scheme, the hard parton radiates a virtual gluon which is then progressively brought on shell by a large number of soft scatterings off these heavy centers [41, 42, 43, 44, 45]. Yet another approach developed by Gyulassy, Levai, Vitev and their collaborators considers the same medium as ASW, however both the hard parton and the radiated gluon undergo a few but hard interactions with the centers leading to the emission of the gluon [46, 47, 48, 49, 50, 51]. After the calculation of the single gluon emission kernel, the AMY scheme uses rate equations to incorporate multiple emissions whereas the GLV and the ASW use a Poisson emission Ansatz.

The comparison of the yield of leading particles between the case with and without a medium is a measure of the properties of the medium as felt by the jet. In DIS, one measures the fragmentation function to produce a hadron with a momentum fraction of the original quark momentum immediately after being struck. For the case where the virtual photon carries a momentum, and the incoming struck quark has , the outgoing quark has a momentum . Thus the hadron momentum is . The fragmentation function for DIS on a proton (nucleus ) and the ratio R are defined as,

(10)

In the equation above, is the differential DIS cross section off a proton (nucleus ) integrated over a limited range of and energy imparted from the electron . In order to compare with theoretical calculations, the fragmentation function in the case of DIS on a large nucleus should be identified with the medium modified fragmentation function .

In the case of - or heavy-ion collisions, the jets span a range of momentum depending on the momenta of the two partons that undergo the hard scattering. In this case one cannot isolate separate bins in the final state momentum fraction and thus cannot measure the fragmentation function directly. Instead one measures the ratio of the binary scaled differential yield to produce a high hadron in a heavy-ion collision to that in a - collision referred to as the nuclear modification factor . The may be measured both differentially as a function of angle with the reaction plane and the transverse distance between the centers of the colliding nuclei, the impact parameter , or integrated. The angle integrated [also integrated over a small range of impact parameter ( to )], is defined as

(11)

where, is the mean number of binary nucleon nucleon encounters per ion-ion collision in the range of impact parameters chosen. The invariant yield in a heavy-ion collision and in a - collision are related to the the invariant differential cross section by the relation, , where the total cross section for a heavy-ion collision may be estimated as the geometrical cross section in that range of impact parameter selected. The relation between and the nuclear density profile will be discussed in Chapter 5.

The calculation of the nuclear cross section to produce a hard hadron may be expressed as the convolution of the nuclear PDFs, the hard partonic cross sections and the medium modified fragmentation function. The jets may be produced at various locations in the hot matter and propagate in any direction in the transverse plane. The location of the production point and the direction determine the extent and intensity of the medium as felt by the jet. One thus needs to integrate over all allowed production points and directions. This procedure will be described in Chapter 5.

1.4 Medium transport properties: what can be learnt from jets

As described in the preceding subsections, a jet is essentially a collection of high momentum particles which are somewhat collimated in a given direction. In the partonic part of the jet, these are virtual partons. To make this explicit we ascribe a virtuality to the hard jet, where describes the scale of the hard interaction which produces the jet. In effective field theory approaches to hard jets such as that of Soft Collinear Effective Theory (SCET) [52, 53] one introduces a small parameter with ; collinear radiations have a transverse momentum of and the virtuality of the jet may then be surmised as . Thus ; the two terminologies will often be used interchangeably in this review.

As the jet passes through matter each of these particles will scatter off the various constituents that it encounters. In the Breit frame (or infinite momentum frame) the nucleus has a large boost in the direction opposed to that of the jet. As we pointed out earlier, one may consider each nucleon within the nucleus to be traveling in almost straight lines independent of each other. At sufficiently high energies, the “large-” partons in the nucleons may also be considered to be traveling in straight lines independently of each other (see Fig. 3). The hard virtual partons in the jet with virtuality will resolve this sub-structure down to transverse sizes of order . The hard partons in the jet will scatter of these partons by exchanging gluons with transverse momenta . As a result, the hard partons in the jet will undergo a transverse diffusion as they propagate through the extended matter.

Figure 3: A single Coulomb (or Glauber) gluon interaction between a hard jet and a proton in the nucleus.

Along with the exchanged transverse momentum there may also be a certain amount of negative light-cone momentum () which may be exchanged between the partons in the jet and those from the medium. In this review, we will refer to this as “elastic energy loss” or “drag”, even though the incoming parton from the medium is not on-shell, is not a quasi-particle of the medium and may not go back on shell after the scattering. This is primarily done to distinguish this type of light-cone momentum loss from the light-cone momentum loss that occurs due to radiation. This is somewhat different from what is traditionally referred to as elastic energy loss, which involves a certain energy and momentum transfer to an in-medium quasi-particle which remains as one quasi-particle after the interaction, but with larger energy. The remaining component of the transferred momentum is completely determined by insisting on the criterion that the jet parton virtuality does not change too much after the momentum transfer. Thus if the original off-shellness is , then after the transfer, one obtains the virtuality as,

(12)

Thus, for we obtain that the virtuality of the parton will remain more or less unchanged if . For cases where exceeds this value, the hard parton from the jet is taken further off-shell leading to an induced radiation.

Treating the case of momentum change of the hard parton due to induced radiation separately, we obtain three separate components of momentum being exchanged between the jet and the medium: two components of transverse momentum and one component of negative light-cone momentum. At every exchange there is a distribution in transverse and light-cone momentum being imparted between the jet and the medium. Given the large number of exchanges, we invoke the Gaussian approximation, i.e., we approximate these distributions to be Gaussian and consider only the mean and the variance. The Gaussian approximation based on the central limit theorem is not completely unjustified. Each parton interacts multiple times with the medium and the shower distributions contain several hard partons. Along with this, one should also consider that except for the highest (or highest energy) hadrons, each bin in hadron momentum (or momentum fraction) contains several events that need to be summed over.

Given cylindrical symmetry around the jet axis, on may further argue that the imparted , and the mean of the transverse momentum distribution is vanishing. Thus the first transport coefficient may be defined as the variance of the distribution of imparted transverse momentum per unit length traversed. This is referred to as ,

(13)

Where is the total transverse momentum gained in traversing a length . In the case of negative light-cone momentum exchange, the mean of the distribution yields the drag per unit length (), whereas the variance is the fluctuation in light cone momentum transfer per unit length ():

(14)

Given these two quantities, the Gaussian distribution of the negative light-cone momentum transfer is completely specified.

Thus the modification of hard jets will reveal at most these three quantities. Due to the large number of scatterings that need to be summed over in order to obtain measurements with small error bars, access to higher moments in these distributions is somewhat limited. These transport coefficients can in principle be calculated given a model of the medium, e.g., in the asymptotically high temperature limit, the system can be described in the weakly coupled quasi-particle picture afforded by leading order Hard Thermal Loop (HTL) effective theory. This derivation will be described in some detail in the remaining review. Even in this case, the value of the transport coefficients are unknown unless the value of the coupling is specified. The relevant coupling is the in-medium coupling between thermal quasi-particles which is a priori unknown and is therefore set by fitting calculations to at least one data point. It should be pointed out that if the medium is determined to not be weakly coupled, the HTL formulae cannot be applied and some other model will have to be used. The value of the transport coefficient, in principle, will have to be evolved up to the scale of the hard jet prior to use in a jet quenching calculation.

1.5 Other approaches and medium response observables

In this review, we will primarily address the question of how a hard jet can be used as a weakly coupled probe to study the properties of a dense QCD medium. As such we will solely be interested in how different properties of the medium, codified as a set of transport coefficients will modify the shower pattern of the jet. As mentioned above, throughout this review, we will solely describe the the modification of the partonic portion of the jet shower, i.e., that part which may be computed using pQCD. This is often referred to in the literature as partonic energy loss to distinguish it from an alternate mechanism of hadronic energy loss. In this alternate hadronic energy loss scenario, one assumes that the jet hadronizes within a very short distance (of the order of fm) and produces a shower of hadrons. These hadrons then multiply scatter and lose forward momentum in the dense medium which is also assumed to be hadronic [54, 55].

When applied to heavy-ion collisions, these calculations assume a minimal partonic phase, if any. While such theories have experienced a degree of success in the description of jets produced in DIS on a large nucleus [31], they have performed less favorably in comparison to RHIC data [56]. A slightly more successful variant has been the model based on pre-hadronic absorption [57, 58, 59] which tend to model the absorption of the QCD string prior to hadronization. No doubt, all three mechanisms will be present in any description of jet modification where one attempts to capture the fate of all the hadrons which materialize from the shower, in media of arbitrary length. However, in sufficiently short media, and for considerably energetic jets, one may assume that the leading hadrons have hadronized outside the medium (these statements will be made more quantitative in the ensuing chapters). The yield of these leading hadrons may then be described by a factorized vacuum fragmentation function convoluted with the medium modified distribution of hard partons which have escaped the medium. The modification of the distribution of these few hard partons may then be calculated using pQCD with a minimal amount of non-perturbative input in the form of transport coefficients. The vacuum fragmentation functions are obtained from experiments and are thus well known. Thus the study of this subset of observables involves only one set of unknowns: the transport coefficients of the medium. It is in this very restricted sense that pQCD based jet modification may be used as a probe of the medium. This review will be focused on this very restricted set of observables.

As the medium modifies the jet, the jet in turn modifies the medium. There is a growing consensus that part of the energy radiated by a jet is deposited in the medium and this modifies the evolution of the soft medium at that location [60, 61]. Given the “supersonic” velocity of the hard jet, this may lead to the production of a Mach Cone [62] which trails the hard jet and may be observable as an excess in the hadronic yield at a large angle to the associated away side jet in the final soft hadronic spectra that is associated with a hard jet trigger [63, 64].

While the response of a soft medium to a hard jet may not be completely calculable in pQCD, in recent years an alternate theory based on the Anti-DeSitter space Conformal Field Theory (AdS/CFT) conjecture [65] has been used to compute both the drag experienced by a heavy quark [66, 67] and the Mach cone left in a wake of of a hard jet [68, 69]. Unlike calculations based on pQCD, these theories assume that the hard jet is strongly coupled with the dense medium. In the absence of a pQCD based picture it is not clear if the final hadronization process or the initial production process may be consistently factorized from the energy loss calculation. The three topics of (pre-)hadronic energy loss, medium response to a hard jet and the alternate theories of energy loss based on the AdS/CFT conjecture will not be covered in this review. For the reader interested in such topics we recommend these other excellent reviews: Refs. [70, 71, 72, 73].

2 Scattering induced single gluon radiation.

In order to compute the energy loss of hard jets and the medium modified fragmentation function, one needs to compute a series of multiple scattering and multiple emission diagrams. In all formalisms, the means to achieve this is to first compute the single gluon emission kernel due to multiple scattering. This is then iterated to include the effect of multiple emissions. The methodology of iteration is somewhat related to the approximations made in the underlying calculation of the single gluon emission kernel. In this chapter, we will describe this calculation in some detail. The iteration of the kernel will be dealt with in the next chapter. As stated in the introduction, our description of the single gluon emission kernel will follow that of the higher-twist approach. The differences with the other approaches will be pointed out at the end of this chapter.

We present the formalism in the Breit frame. In the case of DIS on a large nucleus this is characterized by the frame where the absolute magnitude of the () and () components of the photon momentum and the large () component of the initial momentum of the struck quark are equal:

(15)

is the momentum of a proton. The struck quark has a final () momentum and travels in the negative direction. It now scatters off the gluons in the nucleons which follow the struck nucleon. The picture is similar to that in Fig. 3.

We will present the calculations in negative light-cone gauge . This makes the discussion of gluons emitted collinear to the outgoing quark particularly simple. The first step is to quantify the magnitude of the different components of the gluons being exchanged between the outgoing quark and the incoming nucleons. The final outgoing quark has a negative light-cone momentum of . Being close to on-shell, it has a virtuality , which is built up from some combination of a and a small transverse momentum .

2.1 DIS on a proton and vacuum radiation

To familiarize the reader with the notation, we compute the simple process of vacuum gluon radiation from a quark struck by a hard photon. This constitutes the primary contribution to the scale evolution of fragmentation functions in vacuum. With this process we will also demonstrate an elementary example of factorization at leading order, where we factorize the parton distribution function (PDF) from the hard cross section. The basic quantity to be computed is the differential cross section for an electron with an initial momentum to scatter off a proton (with momentum ) with final electron momentum and producing an outgoing quark with momentum and a gluon with momentum :

(16)

The complete cross section for the process is represented by Fig. 4. The rectangular blob at the bottom of the diagram represents the contents of the proton after the hard quark is struck by the photon and removed from the proton.

The matrix element that needs to be evaluated may be represented as

(17)

The state represents an arbitrary hadronic state. For the case of interest, represents a state with at least a quark and a gluon endowed with the requisite momenta and an outgoing electron with momentum . At lowest order is simply the vacuum state. In the equation above, represents the interaction Hamiltonian in the interaction picture for QCD and QED. The eventual evaluation of the exponent will involve two orders of the electro-magnetic (EM) interaction and all orders of the strong interaction. For the case of one gluon emission in this section we will only have to expand the exponent to one order of the strong interaction (and two orders of the EM interaction) i.e.,

(18)

In the equation above, is the wave function of the lepton, while is that for a quark. The restriction to only two orders of the EM interaction, constrains the calculation to the one photon approximation. Squaring the matrix element one may calculate the cross section for this process. It may be straightforwardly demonstrated, in the one photon limit, that the process can be decomposed into a purely leptonic part and a partonic (or hadronic) part connected by a single photon propagator. The leptonic part will involve the trace over the Dirac matrix structure of the electron momenta; we will simply write this down without proof. The hadronic part will be somewhat more complicated as it will contain both a perturbative component describing the scattering of a quark off the virtual photon followed by the emission of a hard gluon and a non-perturbative component which describes the probability to find a quark with a particular momentum fraction in the incoming nucleon. This component will be described in some detail.

Figure 4: DIS on a nucleon leading to the formation of a quark and a radiated gluon. Left panel represents the complete diagram whereas the right panel represents the hadronic tensor. The dangling photon lines do not represent photon propagators, but rather the scattering of the quark off a photon. The photon propagator is not contained in the hadronic tensor, see text for details.

The quadruple differential cross section for the process in Eq. (16), in terms of can be decomposed, as stated above, into a purely leptonic part and a hadronic part in the one photon exchange approximation. This is symbolically represented as,

(19)

where and are the leptonic and hadronic tensors respectively. The Mandelstam variable . Each of the factors of and represents a photon propagator, one from the amplitude and one from the complex conjugate.

The leptonic tensor has the straightforward definition, , while the hadronic tensor, at leading order in the strong coupling, may be decomposed as

(20)

This expression requires some explanation. The interaction terms in are those contained in the QCD [Eq. (1)] and QED Lagrangians. These are composed of solely quarks, gluons, leptons and photons. While we have, somewhat artificially, constrained the final state to be a quark and a gluon, the initial state is a proton. In the Breit frame, the high energy proton can be approximated as a weakly interacting gas of collinear quarks and gluons. The distribution of these partons in bins of light-cone momentum fraction depends on the transverse size of these partons (i.e., resolution of the probe). In this high energy limit, the projection of the product of and its complex conjugate () along the large momentum direction may be viewed as an annihilation and creation of a near on-shell quark in the proton’s wave function. The Fourier transformation of the expectation of this operator product is referred to as the parton distribution function (PDF),

(21)

Unlike the parton distribution function in Eq. (21) above, the function that appears in Eq. (20) is not yet on the light-cone. Note that in the equation above only appears. This happens after the invocation of the high energy or collinear approximation. The incoming parton is assumed to be endowed with very high forward momentum with negligible transverse momentum . Within the kinematics chosen, the incoming virtual photon also has no transverse momentum. As a result, the produced final state parton also has a vanishingly small transverse momentum (i.e., with a distribution that may be approximated as up to corrections suppressed by powers of the hard scale). In this limit, the leading spin projection of the pieces which represent the initial state and final state may be taken. The factors,

(22)

are used to obtain the spin projections along the leading momenta of the outgoing state and the incoming state. See Refs. [34, 74] for precise details of how this is done.

The other function in Eq. (20) is which represents the physics of the final state after the hard scattering with the photon. The superscript on the operator implies that the quark undergoes no scattering in the final state. Taking the leading projection in Dirac matrix structure we obtain

(23)

In the equation above, is the sum over the product of polarization vectors of the radiated gluon with momentum . In light cone gauge (with ) this is given as,

(24)

To evaluate the above Feynman integral one integrates over the internal locations , which will yield the momentum conserving delta functions. This will set . Then one approximates the numerators of the fermion propagators as . The denominators may be expressed as where . The sets . The other -function over sets where . The remaining steps involve carrying out the simplifications associated with the matrices and carrying out the integrations over the momenta that do not appear in the integrand. This yields

(25)

The resulting phase factors constrain to the light cone and convert it into the PDF. The factor is the splitting function which expresses the probability for the hard quark to radiate a gluon. Re-incorporating this expression back into Eq. (20), we obtain the differential hadronic tensor for a quark in a proton, struck by a hard virtual photon, to radiate a gluon with momentum and itself have a momentum :

(26)

The above result should be compared with the simpler result of DIS without any radiation in the final state. The derivation is similar to that presented for the case of a produced radiation, we here simply state the result (referring the interested reader to Ref. [33]). The differential hadronic tensor for a quark in a proton, struck by a hard photon, to have a momentum is

(27)

In the equation above, is the quark PDF defined in Eq. (21). Note that the arguments in Eqs.(26,27) are slightly different. They differ by the quantity . This should be understood as the ratio of the off-shellness of the outgoing quark to , i.e.,

(28)

The momentum component is the extra ()-component of the momentum that must be brought in by the incoming quark such that the quark after being struck should be off-shell enough to radiate the gluon with momentum []. We work in the limit where a collinear jet is produced in the hard interaction, so that the off-shellness is very small compared to the hard scale :

(29)

As mentioned in the introduction, this ratio of the off-shellness of the produced quark to is used to qualify the small parameter , i.e., . In this limit of transverse momenta, if . In either case, this small correction to in Eq. (26) can be neglected.

In both equations  (26,27), the distribution of the final outgoing quark involves a -function. This is meant to be a simplification. A better expression is to use a narrow, normalized Gaussian distribution. The width is of order , i.e., small compared to the hard scale but still perturbatively large. The width of this outgoing quark distribution is related to the transverse momentum and off-shellness of the incoming quark; this controls the virtuality of outgoing quark and thus the virtualities involved in all final state processes. This also controls the scale at which the coupling constant is evaluated in the case of final state gluon emissions.

In the limit of small , the ratio of the cross sections for both processes, computed using Eq. (19), assumes a rather simple form. Integrating out the transverse momentum of the produced quark gives the differential number of gluons radiated,

(30)

In the subsequent section, we will derive the cross section for a hard quark to radiate a single gluon while undergoing multiple scattering in the medium. This will involve modifying a related version of Eqs. (26,27,30).

We close with a comment regarding Eq. (30). The reader will note that the expression for the differential number of gluons in possesses both an infra-red [ as ] and a collinear singularity . In the calculation of the final hadronic distribution, one convolutes this gluon distribution and the related final quark distribution with a fragmentation function as shown below in Eq. (31). In this convolution one also has to include the effect of virtual corrections, i.e., instances where the leading parton radiated a gluon and then reabsorbed it. Such diagrams also contain an infrared and collinear divergence. In the convoluted expression to obtain the yield of hadrons, the infrared singularity cancels between real and virtual diagrams. The real splitting function is replaced with the well known -functions [18]. The collinear divergence, however remains even after the inclusion of virtual corrections. The source of this divergence are gluons with , and thus those with formation times . In a factorized approach, such long distance effects are absorbed into a renormalization of the final fragmentation function to produce hadrons. In the case of single hadron inclusive DIS, absorbing gluons with transverse momenta up to the final factorization scale yields the scale dependent fragmentation function [] to produce a hadron with momentum fraction . Including the effect of multiple emissions yields the DGLAP evolution equation for the fragmentation function:

(31)

In the in-medium version of Eq. (30), Landau-Pomeranchuck-Migdal (LPM) interference tends to cancel the collinear divergence making the differential yield of gluons finite at . As a result, one obtains a finite energy of gluons emitted on integrating over . Two of the formalisms use this value to construct an iterative formalism, thus ignoring the fact that a fraction of the number of gluons are produced much later in time and should not be included in the calculation. This will be further discussed in Chapter 3.

2.2 Multiple scattering induced single gluon radiation

In this section, we compute the cross section for single gluon emission induced by multiple scattering. Iterating this process and convoluting with the fragmentation function will produce the medium modified fragmentation function. We will approximate each scattering to only transfer a small transverse momentum and a ()-component between the jet and the medium partons. There can also be an exchange of a small ()-component; this leads to elastic energy loss and will be discussed separately in Chapter 4.

Multiple scattering without gluon radiation

We begin by considering the case of a hard quark produced in a DIS on a large nucleus, which produces a hard quark propagating through the nucleus encountering multiple scattering without radiation. The virtual photon strikes a hard quark in one of the nucleons and turns it back towards the nucleus. The quark then propagates through the nucleus without radiating. In this process the hard quark scatters times in the amplitude and in the complex conjugate. This is represented by the diagram in the left panel of Fig. 5. The diagram represents a quark in one of the incoming nucleons with momentum , being struck with the photon with momentum . The outgoing quark has momentum . After encountering the scattering, its momentum is .

In the case of the higher-twist scheme, these gluon lines represent the gluon field at the point at which scattering takes place. Thus there is no meaning to crossing of gluon lines. The sources of the gluon lines are the nucleons in the nucleus (or rather the partons within these nucleons). The only assumption made regarding the gluon momenta are simple scaling relations regarding the momentum of the exchanged gluons and the magnitude of the corresponding components of the vector potentials. This is somewhat different in the other schemes. Under the assumption that the gluons transfer a transverse momentum of , and the incoming and outgoing quark lines remain close to on-shell, the component is constrained with the leading contribution given by the equation,

(32)

Ascribing the same criteria to the hard quark or gluon in the nucleons from which the exchanged gluon originates, implies that (where here is the momentum of some hard parton) also scales as . Exchanged gluons with transverse momenta much larger than their longitudinal components are referred to as Glauber gluons or Coulomb gluons. If the ()-component were to become larger than , it would drive the jet parton to go off-shell and radiate a hard gluon. If the component were to become larger, this will lead to energy loss from the jet parton but the parton in the nucleon will then go off shell and may radiate. As jet quenching measurements do not concern themselves with the fate of the target, the latter sort of momentum transfer is often referred to as elastic energy loss for the jet.

Figure 5: Left panel: DIS on a nucleus leading to the formation of a quark jet which which is constrained to propagate through the nucleus without radiating. Its only interactions are scatterings. Right panel: The dominant length enhancement arises from nested scattering diagrams, where the blobs represent individual nucleons, see text for details.

Given these approximations we may now compute the scaling of the different components of the gauge field. To remind the reader we are calculating in the Breit frame where the hard quark moves in the direction with a large light cone momentum and the valence quarks inside the nucleon are moving in the direction with a large . We use the linear response formula to ascertain the power counting of the field. Suppressing the color superscripts we obtain,

(33)

In the equation above, is the gluon propagator and at leading order in the light cone gauge is given as,

(34)

In Eq. (33), is the current of partons in the target which generates the gluon field. The fermionic operator may be decomposed as,

(35)

The scaling of the fermionic operator depends on the range of momentum which are selected from the in-state by the annihilation operator. Note that this influences both the scaling of the annihilation operator as well as the bispinor . The power counting of the annihilation operator may be surmised from the standard anti-commutation relation, The power counting of the bispinor can be obtained from the completeness relation, . Substituting the equation for the current in Eq. (33), and integrating out , we obtain,

(36)

If the incoming and out going momenta and scale as collinear momenta in the ()-direction, i.e., , then we get, , as one of the momenta will involve the large ()-component and the remaining are the small transverse components. Thus the annihilation (and creation) operator scales as . Also in the spin sum and thus ; one can check that the projects out the large components in and in the expression . We also institute the Glauber condition that , and .

Using these scaling relations we correctly find that the bispinor scales as