Ridge Formation Induced by Jets in pp Collisions at 7 TeV
aa r X i v : . [ h e p - ph ] F e b Ridge Formation Induced by Jets in pp Collisions at 7 TeV
Rudolph C. Hwa and C. B. Yang , Institute of Theoretical Science and Department of PhysicsUniversity of Oregon, Eugene, OR 97403-5203, USA Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P. R. China (Dated: November 10, 2018)An interpretation of the ridge phenomenon found in pp collisions at 7 TeV is given in terms ofenhancement of soft partons due to energy loss of semihard jets. A description of ridge formationin nuclear collisions can directly be extended to pp collisions, since hydrodynamics is not used,and azimuthal anisotropy is generated by semihard scattering. The observed ridge structure isthen understood as a manifestation of soft-soft transverse correlation induced by semihard partonswithout long-range longitudinal correlation. Both the p T and multiplicity dependencies are wellreproduced. Some predictions are made about other observables. PACS numbers: 25.75.Gz, 13.85.Ni
I. INTRODUCTION
The observation of ridge structure in two-particle cor-relation in pp collisions at 7 TeV by the CMS Collabo-ration at Large Hadron Collider (LHC) [1] has openedup the question of whether it has a similar origin as thatalready found at Relativistic Heavy-Ion Collider (RHIC)in Au-Au collisions at 0.2 TeV [2–6]. A great deal isknown about the ridge in heavy-ion collisions, since vari-ous experiments have studied two-particle (with or with-out trigger) and three-particle correlations. The domi-nant theme is that the ridge exhibits the effect of highor intermediate- p T jets on a dense medium. If the phe-nomenon seen at LHC reveals similar features upon fur-ther investigation, it would imply that soft partons ofhigh density can be created in pp collisions and can af-fect the passage of hard partons through them. If not, anew mechanism needs to be found. Various theoreticalspeculations have been advanced with varying degrees ofattention to the specifics of the CMS data [7–9]. In thisarticle we propose a model that is an extension of ourpast interpretation of the ridge phenomena in the RHICdata, but is particularly suitable for pp collisions at LHC,since the dynamical origin is jet production rather thanhydrodynamics. We have a simple formula that can re-produce the CMS data quantitatively with the use of twoparameters that can clearly describe the physics involved.The most direct approach to the study of ridges isto consider only events selected by triggers with p trig T inan intermediate p T range, as first reported by Putschke[3, 10]. The dependence of the ridge yield on centrality innuclear collisions indicates that the ridge is formed whenthere is a jet in a dense medium. Having an exponen-tial behavior in p assoc T at values less than p trig T suggeststhat the ridge particles are related to the soft partons,but they have an inverse slope larger than that of theinclusive distribution, implying an enhancement effect ofthe jet [3, 11]. If triggers are not used as in the study ofautocorrelation, ridges are also observed at | ∆ η | > pp collisions at LHC we can- not presume the existence of a dense medium of partons,which is a possibility we leave open. However, we canand shall assume that ridge formation is due to high- orintermediate- p T jets, whether or not the jets are detectedby triggers. Our goal is to study the properties of cor-relation generated by semihard jets. It should be notedthat there are models in which the ridge phenomenon canoccur without jets, such as in Refs. [9, 12–15].In the hadronization model based on Refs. [11, 16] theridge component (due to the recombination of thermalpartons) manifests the effect of the semihard parton onthe medium. The soft partons have exponential depen-dence on the transverse momentum k T , whose inverseslope is T in the absence of semihard partons. For theridge component the inverse slope is increased to T ′ > T due to the enhancement of the thermal motion of the softpartons caused by the energy loss of the semihard par-ton that passes through the medium in the vicinity [17].That is soft-semihard correlation, which we shall applyto even pp collisions where the notion of thermal partonsmay be questionable. It is known empirically that thereexists an exponential peak at small p T at LHC [18–20];that is sufficient for us to refer to the underlying par-tons as soft, the recombination of which gives the low- p T hadrons.In Sec. II we give a short summary of our past work onridges with emphasis on the distinction between trans-verse and longitudinal correlations. It is significant tonote that the data on ridge reported by PHOBOS [4] donot imply the existence of long-range longitudinal corre-lation upon closer examination. In Sec. III the transversecorrelation is extended to | ∆ η | > pp collisions is then carried out in Sec. IV. In the last sec-tion we give the conclusion along with some predictions. II. TRANSVERSE AND LONGITUDINALCORRELATIONS
Longitudinal correlation has been the primary concernof most theoretical studies on ridges [12–14, 21–23]. Theobservation by PHOBOS [4] that | ∆ η | can be as largeas 4 has led to the conclusion that there is empiricalevidence for long-range correlation, which is an inherentproperty of flux-tube models. There are, however, twoother aspects about the ridge structure that one shouldalso consider in addition to the large-∆ η aspect of thePHOBOS data. One is A : the property of ridge in thesmall ∆ η limit, and the other is B : the question of howlarge should ∆ η be in order for the correlation to beregarded as long-range. We comment on them in thecontext of what have been observed at RHIC as a preludeto our discussion about the ridge found at LHC. A. Transverse Correlation.
At midrapidity dihadron cor-relations in the azimuthal angles have been studied indetail at RHIC; in particular, the dependence of the az-imuthal correlation on the trigger angle φ s relative tothe reaction plane reveals features that are importantabout ridge formation [24–27]. Any model on the originof ridges at | ∆ η | > | ∆ η | <
1, sinceall observed ridge structure have common behavior in ∆ φ throughout the ∆ η range. The ridge yield as a function of φ s has been studied in a model where the angular correla-tion between the trigger and local flow direction is limited[28]. It is found that a Gaussian width of σ ∼ .
34 canreproduce the data [24, 26, 27]. The model suggests thatthermal activities of the soft partons in the vicinity ofthe trajectory of the semihard parton (i.e., within a coneof angular range of σ ) are enhanced by the energy lossof the latter to the medium. Those enhanced thermalpartons hadronize into the ridge particles that rise abovethe background. That is transverse correlation betweenthe soft and semihard partons, the only type that can bestudied when | ∆ η | is restricted to <
1. After finding sat-isfactory explanation of the azimuthal correlation in thedata this way for triggered events, the natural questionto follow is how such correlation influences the single-particle distribution when triggers are not used. Semi-hard partons can be pervasive if their k T is around 3GeV/c or lower. It is found that the semihard-soft trans-verse correlation can give rise to a significant azimuthalanisotropy [17, 29], and that v ( p T , N part ) can be quanti-tatively reproduced as a consequence of the ridge effect ininclusive distribution [30]. This will become a key inputin our discussion below where the nature of the transversecorrelation will be made explicit. B. Longitudinal Correlation.
At first sight of the PHO-BOS data on the ∆ η range of the ridge distribution [4],anyone having some familiarity with multiparticle pro-duction is likely to regard | η − η | ∼ η and ridgeparticle at η . However, to quantify the notion of corre-lation range it is important to compare it to the η -range of the single-particle distribution. A recent study showsthat the ridge distribution in ∆ η , denoted by dN chR /d ∆ η ,can be related empirically to the single-particle distribu-tion, dN ch /dη , by using the two relevant sets of PHOBOSdata only [4, 31] without any theoretical input [32]. Thatphenomenological relationship dN chR d ∆ η ∝ Z . dη dN ch dη (cid:12)(cid:12)(cid:12)(cid:12) η = η +∆ η (1)involves a shift in η of the charge hadron and an inte-gration over the trigger η , and shows that the range ofcorrelation in ∆ η is no more than the range of the inclu-sive distribution apart from the smearing of the triggeracceptance, which lengthens the ∆ η range by 1.5. Theimplication is that there is no long-range longitudinal cor-relation. Any successful model of ridge formation shouldbe able to explain the simple relationship shown in Eq.(1). In Ref. [32] an interpretation of that relationship isgiven in terms of transverse correlation that we discussin more detail in the next section. III. RIDGE AT | ∆ η | > The phenomenological verification of Eq. (1) directsone’s attention to the origin of ridge formation withoutintrinsic longitudinal correlation at large ∆ η . From allthat have been learned experimentally about the ridges,there is no indication that such structure can be foundin the absence of any jet. Even in autocorrelation stud-ies where no triggers are used, ridges are found in thekinematical region where minijets are detected [2]. Ourapproach is therefore to start with jet-induced transversecorrelation at | ∆ η | < η sep-aration, in contrast to other studies where long-rangelongitudinal correlation at low p T exists without jetsand then a large- p T parton is introduced to define the∆ η range. The approach we adopt was actually advo-cated even before the discovery of ridge was reported byPutschke [10] at a time when the phenomenon was re-garded as the pedestal lying under the jet peak [33, 34].Now, with more data and model analyses of the trans-verse correlation at hand, the extension to large ∆ η canbe done with more definiteness.To be more specific, let us consider the ridge found byCMS at LHC, where only charged particles with | η | < . p T < p L is less than 22 GeV/c, soFeynman x F is < . × − at √ s = 7 TeV, and thecorresponding partons that recombine have even lower x values. Those are soft wee partons deep in the sea,whose correlations can be strongly influenced by fluctua-tions. Suppose that a semihard scattering occurs in a pp collision at 7 TeV and sends a parton to the η ≈ k T in the 5-10 GeV/c range,which we shall regard as intermediate at LHC. Whateverthe medium effect on it may be, it can lead to a clusterof hadrons with limited range in η and φ [1]. It cannotdirectly cause the production of an associated particle at η = 2 . p L of that particle can exceed 20 GeV/c,hence forbidden by energy conservation. Any particleproduced outside the jet peak carries longitudinal mo-mentum that is driven by the initial partons (right- orleft-movers) of the incident protons. In the conventionalparton model it is assumed that there are no significantlongitudinal constraints on those initial partons [35, 36].We add, however, that their transverse momentum distri-bution can be affected by the semihard scattering beforethey recede from one another. At early time the right-and left-movers need not be arranged as in Hubble ex-pansion, i.e., a right-moving parton may be located onthe left side of the region of uncertainty, and vice-versa;hence, those initial partons can be sensitive the passageof the semi-hard parton across their ways. The quantumfluctuations that generate the transverse k T distributionof the forward (or backward) moving partons may be en-hanced by the energy loss of the semihard parton. Morespecifically, let exp( − k T /T ) represent the k T distributionin the absence of semihard scattering; then our asser-tion is that the distribution changes to exp( − k T /T ′ ) with T ′ > T in the presence of semihard scattering, providedthat the affected partons are in the vicinity of the semi-hard parton trajectory in the transverse plane, i.e., ∆ φ is limited on the near side. Furthermore, such a changeoccurs for all partons independent of their longitudinalmomenta up to x ∼ − , say. This enhancement is inessence the transverse correlation discussed in Sec. II.A,but now the semihard parton at η ≈ T to T ′ at all η in the limited region | η | < . | η | < . η , = ± .
4, resulting in | ∆ η | = | η − η | as large as 4.8. Hereafter, η and η do not refer totrigger and associated particles, respectively, but to anytwo particles whose correlation is measured by CMS. Thesemihard parton may be anywhere in between ± .
4. Thehuge jet peak observed in Ref. [1] corresponds to particlesthat are produced by thermal-shower recombination andtherefore must be close in η to the semihard parton, butthe peak distribution in ∆ η does not indicate where it is.The flat ridge distribution that lies below the jet peakonly reveals the response of the medium in terms of en-hanced thermal partons without any information aboutthe locations of the shower partons. The ridge particleshave transverse distribution that is characterized by thesame inverse slope T ′ as for the enhanced soft partons.That is a property of recombination [16, 30]. No explicitlongitudinal correlation has been put in.In order to describe pion and proton production in thesame formalism of recombination of thermal partons atlow p T , it is shown that the replacement of p T by E T , where E ( p T ) = ( m h + p T ) / − m h , h = π or p , is sufficientto account for the mass effect and that the inclusive ridgedistribution can reproduce v h ( E T , N part ) at low E T [30].Being the difference between the enhanced distributionand the background, that ridge distribution is R ( p T ) = R ( e − E T /T ′ − e − E T /T ) (2)for nuclear collisions. It is the soft response to the semi-hard partons. We will apply the same description to pp collision below. The difference ∆ T = T ′ − T is a measureof the magnitude of the influence by semihard scatteringwithout which there is no ridge. IV. RIDGE YIELD IN pp COLLISION AT LHC
We now focus on the ridge yield measured byCMS. Let the single-particle distribution be ρ ( p T , η ) = dN/p T dηdp T , which will be abbreviated by ρ ( i ) forthe i th particle, so that two-particle distribution is de-noted by ρ (1 , C (1 ,
2) = ρ (1 , − ρ (1) ρ (2) . The measure for ridgeused by CMS is R CMS (1 ,
2) =
N C (1 , /ρ (1) ρ (2) , (3)where N is the number of charged particles in a multi-plicity bin. In more detail the quantities in Eq. (3) areaveraged over bins of p T , so Ref. [1] exhibits R CMS ( p T , ∆ η, ∆ φ ) = N Q i =1 , hR [ p T ] dp T i p T i i C (1 , Q i =1 , hR [ p T ] dp T i p T i ρ ( i ) i (4)where [ p T ] denotes the range of integration from p T − . p T + 0 . η -∆ φ distribution. A projection of it onto ∆ φ is done byintegrating | ∆ η | over the range 2.0 to 4.8. The associatedyield in the ridge is then determined by integrating over arange of ∆ φ around 0 where R CMS is above its minimum,i.e., Y R ( p T , N ) = Z R d ∆ φ Z ± . ± d ∆ η R CMS ( p T , ∆ η, ∆ φ ) . (5)This measure of the ridge yield is given for 4 bins of p T and N each [1]. The data points are shown in Fig. 1.What is remarkable about the data is that Y R is verysmall for both 0 . < p T < < p T < < p T < p T behavior is so drastically differenton the two sides of 1 GeV/c. The increase of Y R with N is not surprising, especially if one has in mind that jetsare connected with the ridge phenomenon.Our explanation of the p T and N dependencies of Y R is very simple, based on what has already been discussed.We assume no longitudinal correlation, as in [32], which T <1 (GeV/c)CMS Y R T <2 (GeV/c) T <3 (GeV/c) T <4 (GeV/c) N FIG. 1: Ridge yield vs multiplicity N for 4 bins of p T . Dataare from Ref. [1], and lines are from model calculation. can explain the PHOBOS data [4]. Thus the only con-tribution to C (1 ,
2) is from transverse correlation thatgives rise to the ridge distribution given in Eq. (2) as an η -independent response to the semihard jet at any η jet .We therefore write C (1 ,
2) = R (1) R (2) . (6)This is a very unconventional description of correlationthat we are proposing, since one usually expects an un-factorizable form for correlation. The two particles at η and η are correlated because their p T distributions areboth enhanced by the jet. R (1) and R (2) are indepen-dent responses, so they enter into C (1 ,
2) as factorizedproducts. We emphasize that Eq. (6) is a correlation be-tween two soft particles, each of which being correlatedtransversely to the unobserved jet as described by Eq.(2). An analogy for this is the adage that rising tideraises all boats — even though, we add, there are no in-trinsic horizontal correlations among the boats. PuttingEq. (6) in (4) and (5) we obtain Y R ( p T , N ) = cN Y i =1 " R [ p T ] dp T i p T i R ( p T i , N ) R [ p T ] dp T i p T i ρ ( p T i ) , (7)where c is an adjustable parameter that depends on theexperiment. This is an explicit formula that enables usto do phenomenological analysis.The single-particle distribution for | η | < . ρ ( p T ) = ρ (1 + E T nT ) − n (8)with T = 0 .
145 GeV/c and n = 6 .
6. The average p T found from the above fit is h p T i = 0 .
545 GeV/c.We use Eq. (8) in (7) and fit the data in Fig. 1 with twoparameters (apart from normalization), which we chooseto be T and β , where∆ TT = β ln N, ∆ T = T ′ − T. (9) This dependence on N is reasonable, since at higher N there is higher probability for jet production and hencelarger ∆ T , which is in the exponent in Eq. (2). The resultof the fit is shown by the solid lines in Fig. 1 for T = 0 .
294 GeV and β = 0 . . (10)Evidently, our model reproduces the data very well for all p T and N bins. Y R ( p T , N ) is small at small p T because R ( p T ) in Eq. (2) is suppressed as p T →
0. The reason forthat is discussed below. Y R ( p T , N ) is also small at large p T ; that is due both to the exponential suppression of R ( p T ) and the power-law decrease of ρ ( p T ) at high p T .The increase with N that is most pronounced in the 1
22 MeV/c for 4 < p trig T < R ( p T ) must vanish as p T → φ contains all the φ dependence ofthe inclusive distribution [17, 37]. In that approach whichhas been worked out in more detail recently in [30], it isshown without using hydrodynamics that v (referred toas elliptic flow in hydro description) can be reproducedat all centralities, provided that R ( p T ) → p T because v ( p T ) →
0. Since the azimuthal behavior isdetermined primarily by the initial geometry of the col-lision system [17, 29, 37], such an approach may well beapplicable to pp collisions, for which the validity of hy-drodynamics used for nuclear collisions is doubtful. Theorigin of the φ dependence in the geometrical approachis the anisotropy of semihard emission when the initialconfiguration is almond-shaped. Similarly, it is reason-able to consider the initial configuration in pp collisionsalso, when the impact parameter is non-zero, and we ex-pect significant φ anisotropy in the produced particles.The Tsallis distribution in Eq. (8) has the propertyof a power-law behavior at large p T , but an exponentialbehavior, exp( − E T /T ), at low p T . It is then of interestto note the difference between the values of T and T ,the latter being twice larger than the former. It mayappear as being inconsistent; however, the average h p T i of exp( − E T /T ) is 0.6 GeV/c, only 10% higher than thatfor Eq. (8). Thus different parametrizations of the E T distribution give essentially the same physical quantity.Eq. (8) is a fit of the CMS data [19] that emphasizes the p − nT behavior at high p T , while Eq. (2) is a theoreticalmodel of the ridge distribution at low p T . V. CONCLUSION
We have given an interpretation of the ridge phe-nomenon in pp collisions in terms of soft partons on whichvery little is known. By drawing on what we do knowabout the soft partons in nuclear collisions, we are ledto the implication that a dense medium can be createdeven in pp collisions at 7 TeV. The primary input in ourapproach to explaining the observed ridge yield is theassertion that the correlation is of the factorizable form R (1) R (2), where R ( i ) is the response of the i th soft parti-cle to the unobserved jet, so that two independent trans-verse correlations of the semihard-soft type can lead to anet soft-soft correlation in C (1 , pp collisions at7 TeV suggests that (a) the medium can be responsive tosemihard jets, (b) there can be azimuthal anisotropy, (c)the p T spectrum in the ridge is harder than that of theinclusive, and (d) that hadronization is by recombination.None of the above rely on the validity of hydrodynamicsfor pp collisions, or the existence of intrinsic long-rangelongitudinal correlation, and all of them can be checkedby further experimental measurements. The last itemcannot be checked directly, but one of its consequencesis that the p/π ratio can be large, which is a propertyof all recombination/coalescence models [39]. We expectthe p/π ratio in the ridge to increase with p T at low p T in pp collisions at 7 TeV, although the rate of thatincrease depends on the soft parton density, on which we have insufficient knowledge to predict. A ratio largerthan 0.2 cannot be explained by fragmentation. Thusthe experimental determination of the p/π ratio in theridge will be very interesting and should provide furtherinsight on the structure and origin of the ridge.The basic issue that the observation of a ridge by CMShas opened up is whether a system of high density softpartons can be created in pp collisions. The systemmay be too small for the applicability of hydrodynam-ics, but azimuthal anisotropy can nevertheless exist forsmall systems in non-central collisions, so consequenceson φ asymmetry should be measurable, as the ridge struc-ture on the near side demonstrates. Our consideration ofridge formation as being generated by semihard jets ap-plies to both hadronic and nuclear collisions. 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