HADRON CORRELATIONS IN JETS AND RIDGES THROUGH PARTON RECOMBINATION

Hadron Correlations in Jets and Ridges Through Parton Recombination

RUDOLPH C. HWA
Abstract

Hadron correlations in jets, ridges and opposite dijets at all above 2 GeV/c are discussed. Since abundant data are available from RHIC at intermediate , a reliable hadronization scheme at that range is necessary in order to relate the semihard partonic processes to the observables. The recombination model is therefore first reviewed for that purpose. Final-state interaction is shown to be important for the Cronin effect, large B/M ratio and forward production. The effect of semihard partons on the medium is then discussed with particular emphasis on the formation of ridge with or without trigger. Azimuthal anisotropy can result from ridges without early thermalization. Dynamical path length distribution is derived for any centrality. Dihadron correlations in jets on the same or opposite side are shown to reveal detail properties of trigger and antitrigger biases with the inference that tangential jets dominate the dijets accessible to observation.

1 Introduction

Among the many properties of the dense medium that have been studied at RHIC, the nature of jet-medium interaction has become the subject of particular current interest. Jet quenching, proposed as a means to reveal the effect of the hot medium produced in heavy-ion collisions on the hard parton traversing that medium, has been confirmed by experiments and has thereby been referred to as a piece of strong evidence for the medium being a deconfined plasma of quarks and gluons. By the time of Quark Matter 2006 the frontier topic has moved beyond the suppression of single-particle distribution at high and into the correlation of hadrons on both the near side and the away side of jets. The data on dihadron and trihadron correlations are currently analyzed for low and intermediate , so the characteristic of hydrodynamical flow is involved in its interplay with semihard partons propagating through the medium. The physics issues are therefore broadened from the medium effects on jets to include also the effect of jets on medium. Theoretical studies of those problems can no longer be restricted to perturbative QCD that is reliable only at high or to hydrodynamics that is relevant only at low . In the absence of any theory based on first principles that is suitable for intermediate , phenomenological modeling is thus inevitable. A sample of some of the papers published before 2008 are given in Refs. [?-?].

Hadron correlation at intermediate involves essentially every complication that can be listed in heavy-ion collisions. First, there is the characteristic of the medium created. Then there is the hard or semihard scattering that generates partons propagating through it. The interaction of those partons with the medium not only results in the degradation of the parton momentum, but also gives rise to ridges in association with trigger jets and to broad structure on the away side. Those features observed are in the correlations among hadrons, so hadronization is an unavoidable subprocess that stands between the partonic subprocess and the detected hadrons. Any realistic model must deal with all aspects of the various subprocesses involved. Without an accurate description of hadronization, observed data on hadron correlation cannot be reliably related to the partonic origin of ridges and jet structure.

It is generally accepted that fragmentation is the hadronization subprocess at high , as in lepton-initiated processes. At intermediate and lower recombination or coalescence subprocess (ReCo) in heavy-ion collisions has been found to be more relevant. Despite differences in detail, the three formulations of ReCo are physically very similar. In Refs. [?, ?] the descriptions are 3-dimensional and treat recombination and fragmentation as independent additive components of hadronization. In Ref. [?] the formulation is 1D on the basis that acollinear partons have low probability of coalescence, and is simple enough to incorporate fragmentation as a component of recombination (of shower partons) so that there is a smooth transition from low to high . Since the discussions on jet-medium interaction in the main part of this review are based largely on the formalism developed in Ref. [?] that emphasizes the role of shower partons at intermediate , the background of the subject of recombination along that line is first summarized along with an outline of how non-trivial recombination functions are determined. Some questions raised by critics, concerning such topics as entropy and how partons are turned into constituent quarks, are addressed. More importantly, how shower partons are determined is discussed.

Large baryon-to-meson ratio observed in heavy-ion collisions is a signature of ReCo, since the physical reason for it to be higher than in fragmentation is the same in all three formulations. The discussion here that follows the formulation of the recombination model (RM) by Hwa-Yang should not be taken to imply less significance of the other two, but only the limits of the scope of this review. Considerable space is given to the topics of the Cronin effect (to correct a prevailing misconception) and to forward production at low and intermediate in Sec. 3. The large B/M ratio observed at forward production cements the validity of recombination so that one can move on to the main topic of jet-medium interaction.

The two aspects of the jet-medium interaction, namely, the effect of jets on the medium and that of the medium on jets, are discussed in Secs. 4 and 6, respectively. In between those two sections we insert a section on azimuthal anisotropy because semihard jets can affect what is conventionally referred to as elliptic flow at low and also because ridge formation can depend on the trigger azimuth at intermediate . Much theoretical attention has been given in the past year to the phenomena of ridges on the near side and double hump on the away side of triggers at intermediate .- Our aim here is not to review the various approaches of those studies, but to give an overview of what has been accomplished on these topics in the RM. The focus is necessary in order to cover a range of problems that depend on a reliable description of the hadronization subprocess. This review is complementary to the one given recently by Majumder, which emphasizes the region of much higher than what is considered here, so that factorized fragmentation can be applied.

Due to space limitations this review cannot go into the mathematical details of either the basic formalism or the specific problems. Adequate referencing is provided to guide the interested reader to the original papers where details can be found. The discussions will mainly be qualitative, thus rendering an opportunity to describe the motivations, assumptions and physical ideas that underlie the model calculations. For example, the shower parton is an important ingredient in this approach that interpolates between what are soft (thermal-thermal) and hard (shower-shower), but we have neither space nor inclination to revisit the precise scheme in which the shower-parton distributions are derived from the fragmentation functions. The concept of thermal-shower recombination and its application to intermediate- physics are more important than the numerical details. Similarly, we emphasize the role that the ridges play (without triggers) in the inclusive distributions of single particles because of the pervasiveness of semihard scattering, the discussion of which can only be phenomenological.

Attempts are made to distinguish our approach from conceptions and interpretations that are generally regarded as conventional wisdom. Some examples of what is conventional are: (1) Cronin effect is due to initial-state transverse broadening; (2) large B/M ratio is anomalous; (3) azimuthal anisotropy is due to asymmetric high pressure gradient at early time; (4) recombination implies quark number scaling (QNS) of ; (5) dijets probe the medium interior. In each case evidences are given to support an alternative interpretation. In (4), it is the other way around: QNS confirms recombination but the breaking of QNS does not imply the failure of recombination. Other topics are more current, so no standard views have been developed yet. Indeed, there exist a wide variety of approaches to jet-medium interaction, and what is described here is only one among many possibilities.

2 Hadronization by Recombination

2.1 A historical perspective

In the 70s when inclusive cross sections were beginning to be measured in hadronic processes the only theoretical scheme to treat hadronization was fragmentation for lepton-initiated processes for which the interaction of quark were known to be the basic subprocess responsible for multiparticle production. The same fragmentation process was applied also to the production of high- particles in hadronic collisions. Local parton-hadron duality was also invoked as a way to avoid focusing on the issue of hadronization. In dual parton model where color strings are stretched between quarks and diquarks, the fragmentation functions (FFs) are attached to the ends of the strings to materialize the partons to hadrons, even if one of the ends is a diquark. However, at low in collisions quarks are not isolated objects in the parton model since there are gluons and wee-partons at small , so the justification for the confinement of color flux to a narrow string is less cogent than at high . A more physically realistic description of hadronization seemed wanting.

The first serious alternative to fragmentation against the prevalent scheme for hadronization was the suggestion that pion production at low in collisions can be treated by recombination. The simple equation that describes it is

(1)

where is the distribution, taken to be the product of the and distributions already known at the time among the parton distributions of a proton. The recombination function (RF) contains the momentum conserving with a multiplicative factor that is constrained by the counting rule developed for quarks in hadrons. That simple treatment of hadronization turned out to produce results that agreed with the existing data very well.

The next important step in solidifying the treatment of recombination is the detailed study of the RF. If RF is circumscribed by the characteristics of the wave function of the hadron formed, then it should be related to the time-reversed process of describing the structure of that hadron. In dealing with that relationship it also becomes clear that the distinction between partons and constituent quarks must be recognized and then bridged — a problem that has puzzled some users of the RM even in recent years. Since hadron structure is the basis for RF, it became essential to have a description of the constituents of a hadron in a way that interpolates between the hadronic scale and the partonic scale. It is in the context of filling that need that the concept of valons was proposed.

The origin of the notion of constituent quarks (CQs) is rooted in solving the bound-state problem of hadrons. However, in describing the structure of a nucleon in deep inelastic scattering the role of CQs seems to be totally absent in the structure functions , such as . The two descriptions are not merely due to the difference in reference frames, CQs being in the rest frame, the partons in a high-momentum frame. Also important is that the bound state is a problem at the hadronic scale, i.e. low , while deep inelastic scattering is at high . The two aspects of the problem can be connected by the introduction of valons as the dressed valence quarks, i.e., each being a valence quark with its cloud of gluons and sea quarks which can be resolved only by high- probes. At low the internal structure of a valon cannot be resolved, so a valon becomes what a CQ would be in the momentum-fraction variable in an infinite-momentum frame. Thus the valon distribution in a hadron is the wave-function squared of the CQs, whose structure functions are described by pQCD at high . Note that the usual description of -evolution by DGLAP has no prescription within the theory for the boundary condition at low . That distribution at low is precisely what the valon distribution specifies. In summary, the structure function of a hadron is a convolution of the valon distribution and the structure function of a valon

(2)

where is the momentum fraction (not rapidity) of a valon in the hadron . The first description of the properties of is given in [?, ?], derived from the early data . More recent determination of is described in Ref. [?] where more modern parton distribution functions have been used.

is the single-valon inclusive distribution in hadron , and is the appropriate integral of the exclusive distribution, for pion and for proton. More specifically, is the absolute square of the pion wave function in the infinite-momentum frame. Once we have that, it is trivial to get the RF for pion (i.e., by complex conjugation), since it is the time-reversed process. Thus for pion and proton, we have

(3)
(4)

where the factors on the RHS are due to the fact that the RFs are invariant distributions defined in the phase space element , whereas are non-invariant defined in , as seen in (2.1). The exclusive distribution contains the momentum conserving . For pion there is nothing else, but for other hadrons the prefactors are given in Refs. [?, ?].

Having determined the RF, the natural question next is how partons turn into valons before recombination in a scattering process. Let us suppose that we can calculate the multi-parton distribution for a and moving in the same direction, whether at low or high . If their momentum vectors are not parallel, with relative transverse momentum larger than the inverse hadronic size, then the probability of recombination is negligible. Relative longitudinal momentum need not be small, since the RF allows for the variation in the momentum fractions, just as the partons in a hadron can have various momentum fractions. Now, as the and move out of the interaction region, they may undergo color mutation by soft gluon radiation as well as dress themselves with gluon emission and reabsorption with the possibility of creating virtual pairs, none of which can be made precise without a high probe. The net effect is that given enough time before hadronization the quarks convert themselves to valons with essentially the same original momenta, assuming that the energy loss in vacuum due to soft gluon radiation is negligible (even though color mutation is not negligible). For that reason we may simply write as multiplicative factors, as done in (2.1), while treating as the distribution of partons and as the RF of valons. The detail of this is explained in Ref. [?].

The question of entropy conservation has been raised at times, especially by those with experience in nuclear physics. In elementary processes, such as , unlike a nuclear process , the color degree of freedom is important. Since a pion is colorless, the and that recombine must have opposite color. If they do not, they cannot travel in vacuum without dragging a color flux tube behind them. The most energy-efficient way for them to evolve is to emit soft gluons thereby mutating their color charges until the pair becomes colorless and recombine. Such soft processes leave behind color degrees of freedom from the system whose entropy is consequently not conserved. It is therefore pointless to pursue the question of entropy conservation in recombination, since the problem is uncalculable and puts no constraint on the kinematics of the formation of hadrons. Besides, the entropy principle should not be applied locally. A global consideration must recognize that the bulk volume is increasing during the hadronization process, and thus this compensates any decrease of local entropy density.

After the extensive discussion given above on the RF, we have come to the point of being able to assert that the main issue about recombination is the determination of the multi-parton distribution, such as in (2.1), of the quarks that recombine. Related to that is the question about the role of gluons which have to hadronize also. By moving the focus to the distributions of partons that hadronize, the investigation can then concentrate on the more relevant issues in heavy-ion collisions concerning the effect of the nuclear medium.

2.2 Shower Partons

At low in the forward direction the partons that recombine are closely related to the low- partons in the projectile. It is a subject to be discussed in a following section. At intermediate and high the partons are divided into two types: thermal (T) and shower (S). The former contains the medium effect; the latter is due to semihard and hard scattered partons. The consideration of shower partons is a unique feature of our approach to recombination, which is empowered by the possibility to include fragmentation process as SS or SSS recombination. The jet-medium interaction is taken into account at the hadronization stage by TS recombination, although at an earlier stage the energy loss of the partons before emerging from the medium is another effect of the interaction that is, of course, also important. A quantitative theoretical study of that energy loss in realistic heavy-ion collisions at fixed centrality cannot be carried out and compared with data without a reliable description of hadronization. At intermediate there is no evidence that fragmentation is applicable because the baryon/meson ratio would be too small, as we shall describe in the next section.

The fragmentation function (FF), , is a phenomenological quantity whose evolution is calculable in pQCD; however, at some low before evolution the distribution in is parametrized by fitting the data. With that reality in mind it is reasonable to consider an alternative way of treating the FF, one that builds in more dynamical content by regarding fragmentation as a recombination process. That is, if we replace the LHS of Eq. (2.1) by the invariant function , then the corresponding two-parton distribution in the integrand on the RHS is the product distributions of two shower partons in a jet initiated by a parton of type . To be specific, consider, for example, the fragmentation of gluon to pion

(5)

where the and distributions in a shower initiated by the gluon are the same, but their momentum-fraction dependencies are such that if one is a leading quark, the other has to be from the remainder of the parton pool. With being a phenomenological input, it is possible to solve (2.2) numerically to obtain . It has been shown in Ref. [?] that there are enough FFs known from analyzing leptonic processes to render feasible the determination of various shower parton distributions (SPDs), which are denoted collectively by with and , where can be either or . If in the initiating hard parton is an quark, it is treated as . That is not the case if is in the produced shower. The parameterization of has the form

(6)

where the dependence of , , etc., on and are given in a Table in Ref. [?]. Those parameters were determined from fitting the FFs at GeV, and have been used for all hadronization processes without further consideration of their dependence on . It should be recognized that those shower partons are not to be identified with the ones due to gluon radiation at very high virtuality calculable in pQCD, which is not applicable for the description of hadronization at low virtuality.

To sum up, in the conventional approach the FF is treated as a black box with a parton going in and a hadron going out, whereas in the RM we open up the black box and treat the outgoing hadron as the product of recombination of shower partons, whose distributions are to be determined from the FFs. Once the SPDs are known, one can then consider the possibility that a shower parton may recombine with a thermal parton in the vicinity of a jet and thus give a more complete description of hadronization at intermediate- region, especially in the case of nuclear collisions.

The SPDs parametrized by (2.2), being derived from the meson FFs, open up the question of what happens if, instead of , three quarks in the shower recombine, e.g., , or . A self-consistent scheme of hadronization would have to demand that the formation of baryons is a possibility in a fragmentation process and that the SPDs already determined should give an unambiguous prediction of what baryon FFs are. The calculation has been carried out in Ref. [?] where the results for and in gluon jets are in good agreement with data without the use of any adjustable parameters. To be able to relate meson and baryon FFs is an attribute of our formalism for hadronization that has not been achieved in other theoretical approaches, and provides further evidence that the SPDs are reliable for use at the hadronization scale.

2.3 Parton distributions before recombination

In the study of shower partons in a jet we have assumed the validity of the approximation that the fragmentation process is essentially one dimensional. One may question whether the recombination process in a nuclear collision for a hadron produced at high may necessitate a 3D consideration, since two different length scales seem to be involved, one being that of the hadron produced, the other being the size of nuclear medium at hadronization time. Indeed, recombination schemes formulated in 3D have been proposed, and various groups have independently found satisfactory results that are similar to one another.

The essence of recombination is, however, not in the 2D transverse plane normal to the direction of hadron momentum because if the coalescing parton momenta are not roughly parallel, then the relative momentum would have a large component in that transverse plane. If that component is larger than the inverse of the hadron size, then the two (or three) partons cannot recombine. Thus partons from regions of the nuclear medium that are far apart cannot form a hadron, rendering the concern over different length scales in the problem inessential. Only collinear partons emanating from the same region of the dense medium can recombine. For that reason the 1D formulation of recombination is adequate, as simple as expressed in (2.1). If one asks why the relative momentum can be large in the hadron direction, but not transverse to it, the answer lies in the foundation of the parton model where the momentum fraction can vary from 0 to 1, while the transverse parton momentum is limited to (hadron radius). The RFs in (2.1) and (2.1) are related to the 1D wave function in that framework.

Having justified the 1D formulation of recombination, let us now focus on the distributions of the recombining partons at low , and later at high . Since pQCD cannot be applied to multiparticle production at low , our consideration of the problem is based on Feynman’s parton model, which was originally proposed for hadron production at low . In a collision there are valence and sea quarks and gluons whose -distributions at low are known. Without hard scattering their momenta carry them forward, and they must hadronize in the fragmentation region of the initial proton. RM has provided a quantitative treatment of the single-hadron inclusive distribution in not only for , but also for all realistic hadronic collisions. What is to be remarked here is how gluons hadronize. In an incident proton as in other hadrons, the gluons carry about half the momentum of the host. Since gluons cannot hadronize by themselves, but can virtually turn to pairs in the sea, we require that all gluons be converted to the sea quarks (thus saturating the sea) before recombination. This idea was originally suggested by Duke and Taylor, and was implemented quantitatively in the valon model in Refs. [?, ?]. The simplest way to achieve that is to increase the normalization of the sea quarks without changing their distributions so that the total momentum of the valence (unchanged) and sea quarks (enhanced) exhausts the initial momentum of the hadron without any left over for gluons. With the thus obtained, the use of (2.1) results in an inclusive distribution that agrees with data in both normalization and spectrum. Using the appropriate valon distributions of pion and kaon, the success extends beyond to , , and in hadronic collisions at low . In nuclear collisions there is the additional complication arising from momentum degradation when partons traverse nuclear medium. It is a subject that will be brought up in Sec. 3.4.

When is not small, then there has to be a semihard or hard scattering at the partonic level so that a parton with GeV/c has to be created. In that case shower partons are developed in addition to the thermal partons, so the partons before recombination can be separated into the following types: TT+TS+SS for mesons and TTT+TTS+TSS+SSS for baryons. The thermal partons have mainly GeV/c. If one has a reliable scheme to calculate the thermal partons, then their distributions can, of course, be used in the recombination equation. It does not mean that hydrodynamics is a necessary input in the RM. In collisions, for instance, hydrodynamics is not reliable, yet the Cronin effect can be understood in the RM for both proton and pion production without associating the effect with initial-state scattering — a departure from the conventional thinking that will be discussed in the next section. In most applications reviewed here, the distributions of thermal partons are determined from fitting the data at low , and are then used in the RM to describe the behavior of hadrons at GeV/c. When we consider correlation at a later section, careful attention will be given to the enhancement of thermal partons due to the energy loss of a semihard or hard parton passing through the nuclear medium. It is only in the framework of a reliable hadronization scheme can one learn from the detected hadrons the nature of jet-medium interaction, as aspired in jet tomography.

3 Large Baryon/Meson Ratios

3.1 Intermediate in heavy-ion collisions

A well-known signature of the RM is that the baryon/meson (B/M) ratio is large — larger than what is customarily expected in fragmentation. The ratio of the FFs, i.e., , is at most 0.2 at , and is much lower at other values of . However, for inclusive distributions in heavy-ion collisions at RHIC the ratio is as large as at GeV/c, as shown in Fig. 1. Thus hadronization at intermediate cannot be due to parton fragmentation. Three groups (TAM, Duke and Oregon) have studied the problem in the Recombination/Coalescence (ReCo) model and found large in agreement with the data. The underlying reason that is common in all versions of ReCo is that for and at the same the three quarks that form the has average momentum , while the and that form the has . Since parton distributions are suppressed severely at increasing , there are more quarks at than at , so the formation of proton is not at a disadvantage compared to that of a pion despite the difference in the RFs. For either hadron the recombination process is at an advantage over fragmentation because of the addivity of momenta. Fragmentation suffers from two penalties: first, the initiating parton must have a momentum higher than , and second, the FFs are suppressed at any momentum fraction, more for proton than for pion. Thus the yield from parton fragmentation is lower compared to that from parton recombination at intermediate , even apart from the issue of B/M ratio. When faced with the question why baryon production is so efficient, the proponents of pion fragmentation regard it as an anomaly. Despite efforts to explain the enhancement in terms of baryon junction, the program has not been successful in establishing it as a viable mechanism for the formation of baryons. From the point of view of ReCo there is nothing anomalous.

Fig. 1: Comparison of baryon/meson ratio from STAR data to theoretical curves from ReCo. Left panel is from Ref. [?]; right panel is from Ref. [?].

A simple way to understand the dependence of is to consider the 1D formulation of ReCo given in Ref. [?], where the invariant distributions of meson and baryon production are expressed as

(7)
(8)

in which all quarks are collinear with the hadron momentum . We assume that the rapidity is , so the transverse momenta are the only essential variables, for which the subscripts of all momenta are therefore omitted, for brevity. Mass effect at low renders the approximation poor and the 1D description inadequate. However, in order to gain a transparent picture analytically, let us ignore those complications and assume provisionally that all hadrons are massless. Then the experimental observation of exponential behavior of the distribution of pions at low , i.e., , implies that the thermal partons behave as

(9)

where has dimension (GeV), and given in (2.1) is dimensionless. When thermal partons dominate and , the multiparton distributions can be written as products: and , respectively. It is then clear from the dimensionlessness of the quantities in (3.1) and (3.1) that with the proton distribution having dependence, as opposed to the pion distribution being , the ratio has the property

(10)

so long as thermal recombination dominates. This linear rise with is the behavior seen in Fig. 1, although the mass effect of proton makes it less trivial in . Nevertheless, this simple feature is embodied in the more detailed computation until shower partons become important for GeV/c.

From the above analysis which should apply to any baryon and meson, it follows that the ratios and should also increase with in a way similar to . Such behaviors have indeed been observed by STAR, as have been obtained in theoretical calculation. Taken altogether, it means that without TS and TTS recombination the B/M ratios would continue to rise with . But the data all show that the ratios peak at around GeV/c. In the RM the bend-over is due to the increase of the TS component of the meson earlier than the TTS component of the baryon, since two thermal partons in the latter have more weight than the single thermal parton in the former. The shower parton distribution in heavy-ion collisions is a convolution of the hard parton distribution and the distribution derived from FF, discussed in Sec. 2.2, i.e.,

(11)

is the transverse-momentum distribution of hard parton at midrapidity and contains the shadowing effect of the parton distribution in nuclear collisions. A simple parametrization of it is given in Ref. [?] as follows

(12)

where and are tabulated for each parton type for nuclear collisions at RHIC and LHC. The parameter is the average suppression factor that can be related to the nuclear modification factor , and was denoted by in Ref. [?] and other references thereafter. Since has a power-law dependence on , so does on in contrast to the exponential behavior of the thermal partons, . This upward bending of relative to is the beginning of the dominance of TS and TTS components over TT and TTT components, resulting in a peak in the B/M ratio at around GeV/c. Detailed descriptions of these calculations are given in Ref. [?, ?].

We add here that the effort made to consider the shower partons before recombination is motivated by our concern that a hard parton with high virtuality cannot hadronize by coalescing with a soft parton with low virtuality. The introduction of shower partons is our way to bring the effects of hard scattering to the hadronization scale. At the same time the formalism does not exclude fragmentation by a hard parton, since SS and SSS recombination at high are equivalent to fragmentation but in a language that has dynamical content at the hadronization scale.

One could ask how the RM can be applied reliably in the intermediate- region before the shower partons were introduced. The approach adopted in Ref. [?] does not involve the determination of the hard parton distribution by perturbative calculation, but uses the pion data as input to extract the parton distribution at the hadronization scale at all in the framework of the RM. It is on the basis of the extracted parton distribution (which must in hindsight contain the shower partons) that the proton inclusive distribution is calculated. Thus the procedure is self-consistent. The result is that the ratio is large at GeV/c in agreement with data; furthermore, it was a prediction that the ratio would decrease as increases beyond 3 GeV/c, as confirmed later by data shown in Fig. 1.

3.2 Cronin effect

The conventional explanation of the Cronin effect, i.e., the enhancement of hadron spectra at intermediate in collisions with increasing nuclear size, is that it is due to multiple scattering of projectile partons as they propagate through the target nucleus, thus acquiring transverse momenta, and that a moderately large- parton hadronizes by fragmentation. The emphasis has been on the transverse broadening of the parton in the initial-state interaction (ISI) and not on the final-state interaction (FSI). In fact, the Cronin effect has become synonymous to ISI effect in certain circles. However, that line of interpretation ignores another part of the original discovery where the dependence of hadrons produced in collisions, when parameterized as

(13)

has the property that for all measured. That experimental result alone is sufficient to invalidate the application of fragmentation to the hadronization process, since if the dependence in (3.2) arises mainly from the ISI, where the multiply-scattered parton picks up it , then the transverse broadening of that parton should have no knowledge of whether the parton would hadronize into a proton or a pion, so should be independent of the hadron type .

A modern version of the Cronin effect is given in terms of the central-to-peripheral nuclear modification factor for collisions at midrapidity

(14)

where and denote central and peripheral, respectively, and is the average number of inelastic collisions. If hadronization is by fragmentation, which is a factorizable subprocess, the FFs for any given should cancel in the ratio of (3.2), so should be independent of . However, the data show that for all GeV/c when 0-20 % and 60-90 % centralities. See Fig. 2. Clearly ISI is not able to explain this phenomenon, which strongly suggests the medium-dependence of hadronization. The data further indicate that the dependence of peaks at GeV/c for both and , reminiscent of the ratio at fixed centrality in collisions although the ratio for collisions is distinctly different.

Hadron production at intermediate and in collisions can be treated in the RM in a similar way as for collisions. Although no hot and dense medium is produced in a collision, so thermal partons are not generated in the same sense as in collisions, nevertheless soft partons are present to give rise to the low- hadrons. For notational uniformity we continue to refer to them as thermal partons. We apply the same formalism developed in Ref. [?] to the problem and consider the TT+TS+SS contributions to production (TTT+TTS+TSS+SSS for ). The thermal distribution is determined by fitting the spectra at GeV/c for each centrality; the shower-parton distribution is calculated as before but without nuclear suppression. Unlike the dense thermal system created in collisions, the distribution in this case is weaker; its parameters and (inverse slope) that correspond to the ones in Eq. (3.1) are smaller. Furthermore, decreases with increasing peripherality, while remains unchanged at 0.21 GeV/c. Thus thermal-shower recombination becomes important at GeV/c, which is earlier than in collisions. As a consequence, becomes at GeV/c. That is the Cronin effect, but not due to ISI. The same situation occurs for proton production, only stronger. The calculated results for the inclusive distributions of both and agree well with data at all centralities, hence also and . Fig. 2 shows for 0-20 % and 60-90 % in collisions for and ; the lines are the results obtained in the RM. The reason for can again be traced to 3-quark recombination for and only 2 quarks for . When is large, fragmentation dominates (i.e. SS and SSS), and both approach 1, since FFs cancel and the yields are normalized by . No exotic mechanism need be invoked to explain the production process. FSI alone is sufficient to provide the underlying physics for the Cronin effect.

Fig. 2: Central-to-peripheral ratios for the production of pion (left panel) and proton (right panel) in collisions. Data are from Ref. [?] and lines are from Refs. [?, ?].

3.3 Forward production in collisions

Hadron production at forward rapidities in collisions was regarded as a fertile ground for exposing the physics of ISI, especially saturation physics, since the nuclear effect in the deuteron fragmentation region was thought to cause minimal FSI. It was further thought that the difference in nuclear media for the side and the side () would lead to backward-forward asymmetry in particle yield in such a way as to reveal a transition in basic physics from multiple scattering in ISI for to gluon saturation for . The observation by BRAHMS that decreases with increasing was regarded as an indication supporting that view. That line of thinking, however, assumes that FSI is invariant under changes in so that any dependence on observed is a direct signal from ISI. Such an assumption is inconsistent with the result of a study of forward production in collisions in the RM, where both and are shown to be well reproduced by considering FSI only. Any inference on ISI from the data must first perform a subtraction of the effect of FSI, and just as in the case of the Cronin effect there is essentially nothing left after the subtraction.

In Sec. 3.2 the Cronin effect at midrapidity is considered. The extension to along the same line involves no new physics. However, it is necessary to determine the dependencies of the soft and hard parton spectra at various centralities. For the soft partons, use is made of the data on to modify the normalization of already determined at . For the hard partons, modified parametrizations of their distributions are obtained from leading order minijet calculations using the CTEQ5 pdf and the EKS98 shadowing. A notable feature of the result is that falls rapidly with as increases, especially near the kinematical boundary GeV/c and . Thus TS and SS components are negligible compared to TT at large for any and any centrality, even though the TT component is exponentially suppressed. In central collisions there is the additional suppression due to momentum degraduation of the forward partons going through the nuclear medium of the target . Putting the various features together leads to the ratio shown in Fig. 3(a), where the data are from Ref. [?] and the curves from the calculation in Ref. [?]. It is evident that the decrease of at GeV/c as increased from to is well reproduced in the RM. Only one new parameter is introduced to describe the centrality and dependence of the inverse slope of the soft partons, but no new physics has been added. The suppression of at is due mainly to the reduction of the density of soft partons in the forward direction, where hard partons are suppressed.

Extending the consideration to the backward region and using the same extrapolated to , the backward/forward ratio of the yield can be calculated. For corresponding to the data of STAR at and 0-20% centrality, the calculated result on for is shown by the solid line in Fig. 3(b). While it agrees with the data very well for GeV/c, it is noticably lower than the data for all charged particles for GeV/c. However, more recent data on for , shown in the inset of Fig. 3(b), exhibit excellent agreement with the same theoretical curve that should be regarded as a prediction.

Fig. 3: (a) for forward production. Data are from Ref. [?] and lines from Ref. [?]. (b) Back/Forward ratio for that shows agreement of theoretical result with preliminary data for GeV/c but not for higher , but later data (inset) show agreement for GeV/c also.

The fact that is for all measured may be regarded as a proof against initial transverse broadening of partons, since forward partons of have more nuclear matter of to go through than the backward partons of . Thus if ISI is responsible for the acquisition of of the final-state hadrons, then should be . The data clearly indicate otherwise.

3.4 Forward production in collisions

Theoretical study of hadron production in the forward direction in heavy-ion collision is a difficult problem for several reasons. The parton momentum distribution at low and large momentum fraction in nuclear collisions is hard to determine, especially when momentum degradation that accounts for what is called “baryon stopping” cannot be ignored. Furthermore, degradation of high-momentum partons in the nuclear medium implies the regeneration of soft partons at lower ; that is hard to treat also. The use of data as input to constrain unknown parameters is unavoidable; however, existent data have their own limitations. Measurement at fixed cannot be used to provide information on dependence unless is known. Measurement of both and has been limited to charged hadrons that cannot easily be separated into baryons and mesons. For these various reasons forward production in collisions has not been an active area of theoretical investigation. However, there are gross features at large that suggest important physics at play and deserve explanation.

PHOBOS data show that particles are detected at where is the shifted pseudorapidity defined by . It is significant because it suggests that if is not too small, it corresponds to , where . Instead of violation of momentum conservation, the interpretations in the RM is that a proton can be produced in the region, if three quarks from three different nucleons in the projectile nucleus, each with , recombine to form a nucleon with . That kinematical region is referred to as transfragmentation region (TFR), which is not accessible, if hadronization is by fragmentation. The theoretical calculation in the RM involves an unknown parameter, , which quantifies the degree of momentum degradation of low- partons, in the forward direction. For in a reasonable range, not only can nucleons be produced continuously across the boundary, but also can ratio attain an amazingly large value.

BRAHMS has determined the distribution of all charged particles at . For GeV/c, the corresponding values of for pion and proton are, respectively, 0.4 and 0.54. Taking the preliminary value of the ratio at 0.05 into account, it is possible to estimate the value of and then calculate the distribution of . The ratio was predicted to be at GeV/c. However, at QM2008 the more recent data on was reported to have a lower value at 0.02 and on a higher value at at GeV/c. Those new data prompted a reexamination of the problem in the RM; with appropriate changes in the treatment of degradation, regeneration and transverse momentum, the very large ratio can be understood.

Since , and production at large depends sensitively on and distributions, which in turn depend strongly on the dynamical process of momentum degradation and soft-parton regeneration (the parameterization of which requires phenomenological inputs), the procedure in Ref. [?] is to use and as input in order to determine and then calculate the distributions of the hadrons. At fixed the and distributions are related. It turns out that the result on the distribution leads to a large contribution to the distribution of shown by the dashed line in Fig. 4(a). The additional enhancement shown by the solid line arises from the mass dependence of the inverse slopes due to flow. While the ratio is insensitive to the absolute normalizations of the yields, the inclusive distribution of all charged particles is not. In Fig. 4(b) is shown the good agreement between the calculated result and the data in both normalization and shape with no extra parameters beyond already fixed.

Fig. 4: (a) Proton/pion ratio in forward production at showing agreement between data and solid line from the RM; the dashed line is the contribution from the longitudinal components at fixed . (b) Comparison of the distribution of charged particles at from BRAHMS with calculated result from the RM.

It should be noted that the ratio, shown in Fig. 4(a), is extremely large at and modest GeV/c. The underlying physics is clearly the suppression of at medium and the enhancement of due to recombination, where the (valence) quarks are from three different nucleons in the projectile. No other hadronization mechanisms are known to be able to reproduce the data on the large at large .

3.5 Recombination of adjacent jets at LHC

So far we have considered only the physics at RHIC energies and the recombination of thermal and shower partons, either between them or among themselves. At RHIC high- jets are rare, so the shower partons are from one jet at most in an event. At LHC, however, high- jets are copiously produced for GeV/c. When the jet density is high, the recombination of shower partons in neighboring jets becomes more probable and can make a significant contribution to the spectra of hadrons in the GeV/c range, high by RHIC standard, but intermediate at LHC. If that turns out to be true, then a remarkable signature is predicted and is easily measurable: the ratio will be huge, perhaps as high as 20.

If a hard parton of momentum is produced, shower partons in its jet with momenta are limited by the constraint , so that the recombination of those shower partons can produce a hadron with momentum not exceeding . However, if there are two adjacent jets with hard-parton momenta and , then to form a hadron at from shower partons in those two jets, neither nor need to be larger than , so the rate of such a process would be higher. Furthermore, to form a proton at the shower parton can be lower than those for pion formation at the same , so can be even lower. Thus in 2-jet recombination can be much higher than the ratio in 1-jet fragmentation.

The probability for 2-jet recombination, however, also depends on the overlap of jet cones, since the coalescing shower partons must be nearly collinear. That overlap decreases with increasing , so there is a suppression factor in the SS or SSS recombination integral that depends on the widths of the jet cones. Using some reasonable estimates on all the factors involved, it is found that can be between 5 and 20 in the range GeV/c, decreasing with increasing . Although exact numbers are unreliable, the approximate value of is about 2 orders of magnitude higher than what is expected in the usual scenario of fragmentation from single hard partons.

The origin of the large at LHC discussed above is basically the same as that for forward production in collisions at RHIC. In both cases it is the multi-source supply of the recombining partons that enhances the proton production. At large at LHC there are more than one jet going in the same direction; at large at RHIC there are more than one nucleon going in the forward direction. In the latter case we already have data supporting our view that should be large as shown in Fig. 3(a). It would be surprising that our prediction of large at LHC turns out to be untrue.

4 Ridgeology – Phenomenology of Ridges

In the previous section the topics of discussion have been exclusively on the single-particle distributions in various regions of phase space. Everywhere it is found that the B/M ratio is large when is in the intermediate range. We now consider two-particle correlations, on which there is a wealth of data as a result of the general consensus in both the experimental and theoretical communities that more can be learned about the dense medium when one studies the system’s effect on (and response to) penetrating probes. The strong interaction between energetic partons and the medium they traverse, resulting in jet quenching, is the underlying physics that can be revealed in the jet tomography program. To calibrate the medium effect theoretically, it is necessary to have a reliable framework in which to do calculation from first principles, and that is perturbative QCD. Although many studies in pQCD have been carried out to learn about the modification of jets in dense medium in various approximation schemes, they are mainly concerned with the effect of the medium on jets at high , and the results can only be compared with data on single-particle distributions, such as . The response of the medium to the passage of hard partons is not what can be calculated in pQCD, since it involves soft physics. That is, however, the physical origin of most of the characteristics in the correlation data. An understanding of that response is one of the objectives of studying correlations. Without the reliable theory to describe correlation, especially at low to intermediate where abundant data exist, it becomes necessary to use phenomenological models to relate various features of correlation. When all the features can consistently be explained in the framework of a model, then one may feel that a few parameters are a small price to pay for the elucidation of the jet-medium interaction.

On two-particle correlation the most active area in recent years has been the use of triggers at intermediate or high to select a restricted class of events and the observation of associated particles at various values of and relative to the trigger. Among the new features found, the discovery of ridges on the near side has stimulated extensive interest and activities. We review in this section only those aspects in which recombination plays an important role, which in turn makes inferences on the origin of the ridges. We start with a summary of the experimental facts.

4.1 Experimental features of ridges

The distribution of particles associated with a trigger at intermediate exhibits a peak at small and sitting on top of a ridge that has a wide range in , where and are, respectively, the differences of and of the associated particle from those of the trigger. A 2D correlation function in () first shown by Putschke at QM06 is reproduced here in Fig. 5(a). STAR has been able to separate the ridge from the peak , where refers to Jet, although both are features associated with jets. The structure shown in Fig. 5(a) is for GeV/c and GeV/c in central collisions. The ridge yield integrated over and decreases with decreasing , until it vanishes at the lowest corresponding to collisions, so depends strongly on the nuclear medium. That is not the case with . On the other hand, is also strongly correlated to jet production, since the ridge yield is insensitive to . Thus the ridge is a manifestation of jet-medium interaction.

Putschke further showed that the ridge yield is exponential in its dependence on and that the slope in the semi-log plot is essentially independent of . That is shown by the solid lines in Fig. 5(b). The inverse slope parameterized by is slightly higher than of the inclusive distribution, also shown in that figure. Since the range in that figure is between 2 and 4 GeV/c, we know from single-particle distribution that the shape of the inclusive spectrum is at the transition from pure exponential on the low side to power-law behavior on the high side. The last data point at GeV/c being above the straight line is an indication of that. Thus the value of the pure exponential part for the bulk is lower than what that straight line suggests. The exponential behavior of should be taken to mean that the particles in the ridge are emitted from a thermal source. Usually thermal partons are regarded as begin uncorrelated. In the case of they are all correlated to the semihard parton that initiates the jet. We thus interpret the observed characteristics as indicating that the ridge is from a thermal source at , enhanced by the energy lost by the semihard parton transversing the medium at .

Fig. 5: (a) Jet structure from Ref. [?] for charged particles associated with a trigger. (b) Dependence of ridge yield on for various .

The B/M ratio of particles in the ridge is found to be even higher than the same ratio of the inclusive distributions in collisions at 200 GeV. More specifically, in for GeV/c and is about 1 at GeV/c. In contrast, that ratio in is more than 5 times lower. There is indication that the ratio in the ridge is just as large. As discussed in Sec. 3, it is hard to find any way to explain the large B/M ratio outside the framework of recombination. Since the exponential behavior in implies the hadronization of thermal partons, the application of recombination very naturally gives rise to large B/M ratio, as we have seen in Sec. 3.1.

Putting together all the experimental features discussed above, we can construct a coherent picture of the dynamical origin of the and components of the jet structure, although no part of it can be rigorously proved for lack of a calculationally effective theory of soft physics. There are several stages of the dynamical process.

(a)

A hard or semihard scattering takes place in the medium resulting in a parton directed outward in the transverse plane at midrapidity. Because of energy loss to the medium, those originating in the interior are not able to transverse the medium as effectively as those created near the surface. That leads to trigger bias.

(b)

Whatever the nature of the jet-medium interaction is, the energy lost from the semihard parton goes to the enhancement of the thermal energy of the partons in the near vicinity of the passing trajectory. Those enhanced thermal partons are swept by the local collective movement outward whether or not the flow can be described by equilibrated hydrodynamics initially.

(c)

Since the initial scattering takes place at , which is the pseudorapidity range of the trigger acceptance, the shower (S) partons associated with the jet are restricted to the same range of . However, the enhanced thermal partons that interact strongly with the medium can be carried by the high- initial partons that they encounter on the way out and be boosted to higher . Thus the distribution of the enhanced thermal partons is elongated in , but not in because the expansion of the bulk system is in longitudinal and radial directions, not in the azimuthal direction. Consequently, the hadronization of the enhanced thermal partons has the shape of a ridge.

(d)

In terms of recombination the ridge is formed by TT and TTT recombination, while the peak is formed largely by TS and TTS (or TSS) recombination, and possibly also by fragmentation (SS and SSS), depending on and centrality. Since the component involves S, it is restricted to a narrow cone in and .

An initial attempt to incorporate all these properties in the RM was made in Ref. [?] before the ridge data were reported in QM06. By the time of QM08 ridgeology has become an intensely studied subject, as evidence by the talks in Ref. [?].

4.2 Recombination of enchanced thermal partons

Although the properties of ridges described in the above subsection are derived from events with triggers, it should be recognized that ridges are present with or without triggers. That is because the ridges are induced by semihard scattering which can take place whether or not a hadron in a chosen range is used to select a subset of events. Experimentally, it is known that the peak and ridge structure is seen in auto-correlation where no triggers are used. The implication is that the ridge hadrons are pervasive and are always present in the single-particle spectra.

Hard scattering of partons can occur at all virtuality , with increasing probability at lower and lower . When the parton is GeV/c, the rate of such semihard scattering can be high, while the time scale involved is low enough ( fm/c) to be sensitive to the initial spatial configuration of the collision system. Thus for noncentral collisions there can be nontrivial dependence, which we shall discuss in Sec. 5. Hadron formation does not take place until much later, so it is important to bear in mind the two time scales involved in ridgeology. Ridges are the hadronization products of enhanced thermal partons at late time, which are stimulated by semihard parton created at early time. In the absence of a theoretical framework to calculate the degree of enhancement due to energy loss, we extract the characteristics of the thermal distributions from the data. Although hydrodynamics may be a valid description of the collective flow after local thermal equilibrium is established, it does not take semihard scattering into consideration and assumes fast thermaliztion without firmly grounded justification. If the semihard scattering occurs in the interior of the dense medium, the energy of the scattered parton is dissipated in the medium and contributes to the thermalization of the bulk (). That process may take some time to complete. If the semihard scattering occurs near the surface of the medium, its effect can be detected as in these events selected by a trigger with the trigger direction not far from the local flow direction, a point to be discussed in more detail later in Sec. 4.4. Inclusive distribution averages over all events without triggers, including all manifestation of hard and semihard scatterings; hence, it is the sum of . Since is associated with the shower partons , we identify with the recombination of TS+SS for the mesons and TTS+TSS+SSS for the baryons, leaving TT+TTT for . Thus the exponential behavior of the thermal partons is revealed in the exponential behavior of in , for which we emphasize the inclusion of the ridge contribution to the inclusive distribution.

In noncentral collisions the ridges are not produced uniformly throughout all azimuth, so that averages over all has varying proportions of and contributions depending on centrality. To be certain that we can get a measure of the contribution independent of , we focus on only the most central collisions in this and the next subsections. Continuing to use the notation for the transverse momentum of the semihard parton at the point of creation in the medium, for that at the point of exit from the medium, and for the hadron outside, we have for thermal partons the distribution given in (3.1) just before recombination. Our first point to stress here is that the inverse slope in (3.1) includes the effects of both and . Putting that expression into (3.1) where one takes

(15)

and being more explicit with the RF for pion in (2.1), i.e.,

(16)

one obtains

(17)

although in 2004 no one was aware of the existence of ridges. From the data for identified hadrons, one can fit the distribution for 0-5% centrality in the range GeV/c and get GeV/c. This value is slightly lower than the one given in Ref. [?] which takes the slope of the inclusive distribution in the range GeV/c. Ref. [?] provides data for and also, which have the same value of as above for GeV/c, thus confirming that the exponential behaviors of the hadronic spectra can be traced to the common thermal distribution in (3.1) through recombination. At lower the spectra for and deviate from exponential behavior because of mass effect, which can largely be taken into account by using instead of , where

(18)

being the hadron rest mass. Thus we write for all hadrons

(19)

where is a constant for pion, but for proton where is a numerical factor that arises from the wave functions (valon distribution) of the proton. Note that the inverse slope is now denoted by , since the data show dependence on hadron type when the distributions are plotted as functions of . Furthermore, is found to depend on centrality, which is a feature that can be understood in the RM as being due to the non-factorizability of the thermal parton distributions of at very peripheral collisions where the density of thermal partons is low. For central collisions, GeV. We summarize the empirical results for and as follows:

(20)
(21)

where denotes % centrality, e. g., for 10 %. We shall hereafter use to denote the inverse slope in for , and for that in for only, i.e.,

(22)

It is hard to find data that describes the bulk contribution only, since the effect of semihard scattering cannot easily be filtered out. Indeed, as , there is no operational way without using trigger to distinguish from all inclusive. For that reason the prefactor in (4.2) is the same as that in (4.2). In events with trigger above a threshold momentum, semihard partons with lower momenta than that threshold can contribute to ; it becomes a part of the background, which is experimentally treated as . Thus the only meaningful way to isolate quantitatively is by use of correlation, while accepting the difficulty of separating and outside the momentum ranges where the correlated particles are measured. Another way of stating that attitude is to accept the experimental paradigm of regarding the mixed events as a measure of the background (hence, by definition, the bulk), and treating as only that associated with a trigger. Our cautionary point to make is that such a background can contain untriggered ridges. In practice, one can take the difference between (4.2) and (4.2) and identify it as the ridge yield

(23)

where

(24)

If , then the quantity in the square bracket makes a small correction to the behavior, and one can determine from the data. The only data available that address the ridge distribution are in Ref. [?] where the associated particles are in the range GeV/c, exhibiting an approximately exponential behavior. It is shown in Ref. [?] by using the data for trigger momentum in the range GeV/c that with MeV in (4.2) the ridge distributions can be well fitted. The expression for in (4.2) has no explicit dependence on , as is roughly the case with the data. It does have strong dependence on , which is in (4.2). Experimental exploration of the lower region would provide further validation that (4.2) needs. The physics basis for that distribution is the recombination of thermal partons given in (3.1).

4.3 Trigger from the ridge

We have discussed above the observation of ridge in triggered events, but to have a trigger from the ridge seems to put the horse behind the cart. There must be a phenomenological motivation for that role reversal.

Let us start with the single-pion inclusive distribution that shows an exponential decrease in followed by a power-law behavior. The boundary between the two regions is at GeV/c. We have associated the exponential region to TT recombination and the power-law region to TS+SS. We have also discussed the contribution to T from the enhanced thermal partons arising from the medium response to semihard partons. In order to be able to investigate the TT component better without the interference from the shower contribution so that one can examine the components cleanly, it would be desirable to be able to push the TS+SS components out of the way. That is not possible with the light and quarks, but not impossible with the quark, since the heavier quark is suppressed in hard scattering.

If one observes the hadrons formed from only the quarks, either or , one finds exponential behavior at all measured, which in the case extends to as high as GeV/c.- The absence of any indication of up-bending of the distributions clearly suggests that the source of the quarks is thermal in nature and that no shower partons participate in the formation of and . That problem is studied in Ref. [?] along with and production. Indeed, the data can be well reproduced by TT for , TTT for , TT+TS for , and TTT+TTS+TSS for .

Since quarks in S make insignificant contribution to production for GeV/c, and since thermal partons are uncorrelated, it is reasonable to expect that the observed has no correlated particles. It was therefore predicted