A two-component model for identified-hadron p t spectra from 5 TeV p-Pb collisions
vversion 2.4 * A two-component model for identified-hadron p t spectra from 5 TeV p-Pb collisions Thomas A. Trainor
CENPA 354290, University of Washington, Seattle, WA 98195 (Dated: January 14, 2020)In preparation for the heavy ion program at the relativistic heavy ion collider (RHIC) d -Aucollisions were designated as a control experiment for possible discovery of a quark-gluon plasma(QGP) in more-central Au-Au collisions, and contrasting results from the two systems seemed tosupport such a discovery. In contrast, recent results ( p t -spectrum and angular-correlation features)from p -Pb collisions at the large hadron collider (LHC) have been interpreted to support claims ofhydrodynamic flows and QGP formation even in small collision systems. The present study addressessuch claims via a two-component (soft + hard) model (TCM) of identified-hadron (PID) p t spectrafrom 5 TeV p -Pb collisions. p -Pb centrality is adopted from a previous study of ensemble-mean ¯ p t data from the same system. p -Pb p t spectra for pions, kaons, protons and Lambdas are described bythe TCM within their point-to-point uncertainties. Invariance of the TCM hard component vs p -Pbcentrality indicates that jet formation remains unchanged in p -Pb collisions relative to p - p collisions,and radial-flow contributions to p t spectra are negligible. These p -Pb TCM results have implicationsfor interpretation of similar data features from A-A collisions in terms of QGP formation. PACS numbers: 12.38.Qk, 13.87.Fh, 25.75.Ag, 25.75.Bh, 25.75.Ld, 25.75.Nq
I. INTRODUCTION
It is conventionally asserted that conditions arisingin more-central high-energy nucleus-nucleus (A-A) col-lisions (high temperatures and densities) are sufficientto achieve deconfinement of colored quarks and gluonsfrom nucleons to form a quark-gluon plasma (QGP) [1].But interpretation of certain data manifestations fromA-A collisions to confirm QGP formation relies on es-tablishment of control experiments involving low-density p - p and p -A collision systems where a QGP is unlikely:“The interpretation of heavy-ion results depends cru-cially on the comparison with results from smaller col-lision systems such as proton-proton (pp) or proton-nucleus (pA)” [2]. Initial results from the relativisticheavy ion collider (RHIC) seemed to confirm those expec-tations by the presence (in Au-Au) vs absence (in d -Au)of jet quenching in spectra and angular correlations [3].However, claims have emerged, based mainly on datafrom the large hadron collider (LHC), that evidence for“collectivity” (conventionally interpreted to mean hydro-dynamic flows) has been observed in p -A data [4], andsome even conclude that collectivity is evident in p - p col-lisions [5], thus negating the intended role of p -A or d -A and p - p collisions as experimental controls. The re-vised conclusion has emerged that QGP and flows mustbe universal phenomena for all high-energy nuclear colli-sions [6]. However, it has been recognized that such inter-pretations, and the implication that there may be no den-sity threshold for QGP formation, pose a central problemfor interpretation of high-energy particle data [7].The new experimental evidence is derived from two-particle angular correlations and from p t spectra. 2D an-gular correlations on ( η, φ ) from high-charge-multiplicity n ch p - p collisions exhibit a same-side (on azimuth) sin- gle “ridge” (maximum at φ = 0) extending over a largepseudorapidity η interval [5]. A so-called “double ridge”structure (with maxima at φ = 0 , π ) has also been ob-served in p -Pb data [4]. The latter is formally the samequadrupole structure cos(2 φ ) associated with elliptic flowparameter v in A-A collisions. The ridge structures havebeen interpreted by some as flow manifestations [6].Evidence from p t spectra relates to apparent indica-tions of radial flow in the form of “flattening” or “hard-ening” of spectra (increase of slope parameter T ) withincreasing n ch or centrality. The changes are more pro-nounced for higher-mass hadrons. Fits to spectra witha blast-wave model return parameters T kin and ¯ β t , thelatter interpreted as a measure of radial flow [8]. The gen-eral trends with n ch and hadron mass are similar to thoseencountered in A-A collisions where they are interpretedto indicate radial flow increasing with A-A centrality [9].However, such a reversal of the control function of p - p and p -A systems is questionable. It is reasonable to main-tain that if a phenomenon is observed in low-density p - p or p -A collisions it is unlikely to demonstrate formation ofa dense medium in A-A collisions and should not be inter-preted as such. Alternative analysis methods applied tosmall-system data have lead to interpretations based onminimum-bias (MB) jets rather than flows, casting doubton flow interpretations inferred from the same data [10].Abandonment of the control function of p - p and p -A or d -A data should therefore be reconsidered.High-energy nuclear data exhibit a basic property: cer-tain composite structures require a two-component (soft+ hard) model (TCM) of hadron production as demon-strated initially for p - p collisions [11, 12]. The TCM hassince been applied successfully to Au-Au [13, 14] and p -Pb [15, 16] collisions. As a composite production modelthe TCM is apparently required by spectrum [17] andcorrelation [11] data for all A-B collision systems. a r X i v : . [ h e p - ph ] J a n In a previous study [16] a TCM for spectra andensemble-mean p t or ¯ p t data from 5 TeV p -Pb collisionswas formulated based on certain assumptions: (a) hadronproduction near midrapidity proceeds via two distinctmechanisms (i.e., the TCM), (b) one mechanism (hardcomponent) represents MB jets and (c) jet production isunmodified in p -A collisions relative to isolated p - p col-lisions. The p -Pb TCM describes ¯ p t [16] and spectrumdata (present study) within their published uncertaintiesand requires only minor modification of the p - p TCM.In the present study the p -Pb TCM is extended toidentified-hadron (PID) p t spectra to provide the most-differential possible test of the TCM and its conclusionsfor p - p and p -A collision systems: almost all hadron pro-duction arises from the two mechanisms represented bythe TCM – longitudinal projectile-nucleon dissociation(soft) and transverse MB dijet production (hard) – asmanifested in yields, spectra and two-particle correla-tions. Differential p t spectra for four species of identi-fied hadrons from seven centrality classes of 5 TeV p -Pbcollisions are analyzed. The TCM description of data isexhaustive and allows no room for flow interpretations.This article is arranged as follows: Section II presentsPID p t spectra from 5 TeV p -Pb collisions. Section IIIcompares alternative data descriptions. Section IV de-scribes a p t spectrum TCM for composite A-B collisions.Section V derives a TCM for ¯ p t data and centrality pa-rameters for p -Pb collisions. Section VI presents a TCMfor PID p t spectra from 5 TeV p -Pb collisions. Sec-tion VII describes a TCM for PID ¯ p t data. Section VIIIdiscusses PID spectrum ratios derived from TCM spectrain Sec. VI. Section IX reviews systematic uncertainties.Sections X and XI present discussion and summary. Ap-pendix A describes a TCM for ¯ p t data from p - p collisions. II. 5 TeV p-Pb PID SPECTRUM DATA
The identified-hadron spectrum data obtained fromRef. [2] for the present analysis were produced by theALICE collaboration at the LHC. The event samplefor charged hadrons is 12.5 million non-single-diffractive(NSD) collisions and for neutral hadrons 25 million NSDcollisions. Collision events are divided into seven charge-multiplicity n ch or p -Pb centrality classes. Correspond-ing estimated centrality parameters from Ref. [2] areshown in Table I. A more detailed analysis of p -Pb cen-trality including estimates of systematic biases for dif-ferent methods is reported in Ref. [18]. Hadron speciesinclude charged pions π ± , charged kaons K ± , K-zeroshorts K S , protons p, ¯ p and Lambdas Λ , ¯Λ. Spectrafor charged vs neutral kaons and particles vs antiparti-cles are reported to be statistically equivalent. A. p-Pb PID spectrum data
Figure 1 shows PID spectrum data from Ref. [2](points) in a conventional semilog plotting format vs lin-ear hadron p t . The curves are TCM parametrizationsderived in Sec. VI. The spectra for panels (a) and (c-f)have been scaled by powers of 2 according to 2 n − where n ∈ [1 ,
7] is the centrality class index and n = 7 is mostcentral. Panel (b) shows pion spectra with no such scal-ing, the variation due solely to the different p -Pb central-ity classes. The elevation of pion data (points) in (a) and(b) above TCM (solid curves) at higher p t correspondsto the hard-component excess for pions in Fig. 4 (right).It is notable that the maximum spectrum variation cor-responds to the most peripheral n ch classes n ∈ [1 , p -Pb centrality varies most slowly (seeTable II). The chosen plot format limits visual accessto differential spectrum features, especially as they varywith hadron species and p -Pb centrality and especially atlower p t where most jet fragments appear [12, 19]. Com-pare with corresponding figures in Sec. VI B.The baryon data in panels (e) and (f) are expected tocorrespond closely [compare solid (Lambda) and dashed(proton) TCM curves in panel (f)]. The differences therearise from the mass difference (soft component) and thehard-component centroid difference (see Table III). How-ever, the proton data in panel (e) fall well below theproton TCM expectation [solid curves, same as dashedcurves in panel (f)]. To characterize the discrepancy pre-cisely the proton TCM was modified (dash-dotted curves)to include an additional suppression factor ≈ . p t and is said to besimilar to that observed in A-A collisions [20, 21]. Itis suggested that commonalities between p -Pb data andthose from Pb-Pb collisions imply the presence of collec-tive flow in p -Pb collisions: “In heavy-ion [A-A] collisions,the flattening of transverse momentum distribution andits mass ordering find their natural explanation in the col-lective radial expansion of the system [emphasis added].”Several parametrizations of PID spectra were exploredaccording to Ref. [2], of which the so-called blast-wave(BW) model is said to “give the best description of thedata over the full p T range.” The BW model assumes “alocally-thermalized medium, expanding collectively witha common velocity field....” It is acknowledged that “theactual values of the [BW] fit parameters depend substan-tially on the fit range [on p t ].” The chosen fit ranges“...have been defined according to the available data atlow p t and based on the agreement with the data at high p t ” [emphasis added]. It is concluded from BW fits to p -Pb spectrum data that results are “consistent with thepresence of radial flow in p -Pb collisions.” It is furthernoted that “a larger [inferred] radial velocity in p -Pb [vsPb-Pb] collisions has been suggested as a consequence of -4 -3 -2 -1 t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] -4 -3 -2 -1 t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] (a) (b) -4 -3 -2 -1 p t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] -7 -6 -5 -4 -3 -2 -1 p t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] (c) (d) -5 -4 -3 -2 -1
110 0 1 2 3 4 p t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] -7 -6 -5 -4 -3 -2 -1
110 0 2 4 6 8 p t (GeV/c) ( / p p t ) d n c h / dp t d y [( G e V / c ) - ] (e) (f ) FIG. 1: p t spectra for identified hadrons from 5 TeV p -Pb [2]:(a) pions, (b) pions without multiplicative factors, (c) chargedkaons, (d) neutral kaons, (e) protons, (f) Lambdas. Solidcurves represent the PID spectrum TCM from Sec. VI. Thedashed curves in (f) repeat the proton solid curves in (e). Thedash-dotted curves in (e) are described just below Fig. 6. stronger radial gradients ” [emphasis added].However, Ref. [2] includes the following disclaimer inits Sec. 4: ”Other processes not related to hydrodynamiccollectivity could also be responsible for the observed re-sults” and goes on to cite BW model fits to p - p spectrafrom the PYTHIA Monte Carlo with and without colorreconnection (CR). The fit results with CR are observedto be similar to those from p -Pb and Pb-Pb spectra. It isconcluded that “This generator study shows that otherfinal state mechanisms, such as color reconnection, canmimic the effects of radial flow.”Reference [2] concludes that p -Pb PID spectra “rep-resent a crucial set of constraints for the modeling ofproton-lead collisions at the LHC. The transverse mo-mentum distributions show a clear evolution with mul-tiplicity, similar to the pattern observed in high-energypp and heavy-ion collisions, where in the latter case theeffect is usually attributed to collective radial expansion [emphasis added]. Models incorporating final state ef-fects give a better description of the data.” B. p-Pb Glauber-model centrality parameters
Table I shows centrality parameters for 5 TeV p -Pbcollisions from Table 2 of Ref. [18] nominally correspond-ing to the spectra in Fig. 1. The ¯ ρ = n ch / ∆ η chargedensities are measured quantities inferred from Fig. 16of that reference, whereas the centrality parameters areinferred from a Glauber model Monte Carlo [18]. Specif-ically, charge density distributions in Fig. 16 of Ref. [2]were averaged over | η lab | < .
5. The results in the fifthcolumn agree with those in Table 1 of Ref. [2] within thedata uncertainties presented in that table.
TABLE I: V0A Glauber parameters for 5 TeV p -Pb collisionsare from Table 2 and ¯ ρ = n ch / ∆ η densities are derived fromFig. 16, both in Ref. [18]. The event sample is ≈ NSD. Com-parable charge densities are reported in Table 1 of Ref. [2].centrality (%) b (fm) N part N bin n ch / ∆ η To interpret the PID p t spectra from Ref. [2] prop-erly centralities and geometry parameters should be es-timated as accurately as possible. Reference [18] exam-ines several centrality estimation methods identified bydetector designations (e.g. V0A, V0C, ZNA) and empha-sizes estimates based on V0A, a large- η detector on thePb-going side. A detailed analysis of centrality biases isincluded in that study. The spectrum data from Ref. [2]that form the basis for the present analysis are based onthe V0A method. However, the V0A centrality determi-nation (corresponding to primed quantities in Table II)is not utilized in the present study. Instead, an inde-pendent p -Pb centrality determination based on TCMdescriptions of p -Pb ¯ p t data as reported in Ref. [15] (cor-responding to unprimed quantities in Table II) is utilized.The large differences between V0A and TCM geometriesexceed the systematic biases estimated in Ref. [18] (seeSec. V D). III. ALTERNATIVE INTERPRETATIONS
The conclusions in this paper contradict popular inter-pretations of collision data, especially that QGP may beformed in possibly all collision systems as demonstratedby experimental and theoretical evidence for flows and jetquenching in a dense QCD medium. This section reviewsevidence seen as supporting the flow/QGP paradigm andevidence that appears to contradict such conclusions inorder to provide a balanced context for the present study.
A. Collectivity in A-A collision systems
The terms “collectivity” and “collective phenomena”are commonly understood to represent data features fromhigh-energy nuclear collisions interpreted as manifesta-tions of transverse expansion in the form of hydrody-namic flow(s). Perceived evidence for strong expansionof a bulk medium in heavy-ion (A-A) collisions has beenbased on PID p t spectra, azimuth correlations, Bose-Einstein correlations and high- p t suppression (jet quench-ing). The evidence as of 2004 and summarizing the firstyears of RHIC operation was reported in so-called whitepapers [22–25]. The data as analyzed were interpreted toindicate that a QGP is formed and undergoes nearly-idealhydrodynamic expansion (a nearly perfect liquid) [1].Evidence for transverse flows is derived mainly from“blast-wave” fits to p t spectra to infer radial flow [8, 26]and from Fourier-series fits to azimuth angular correla-tions to infer elliptic flow ( v ) [27] and “higher harmon-ics” ( v , etc.) [28]. Aside from blast-wave model fits indi-cations in p t spectra of “hardening” (increased slope pa-rameter) with collision centrality and with hadron massare seen as indicators of radial flow. Azimuthal asym-metries (i.e. any azimuth correlation structure) are inter-preted to indicate azimuthal modulations of transverseflow, with elliptic flow ( v ) being most prominent. Datafeatures and conjectured physical phenomena are oftentreated as synonymous. The higher harmonics, as modu-lations of transverse flow, are seen as arising from fluctu-ations in the initial-state (IS) collision geometry [29, 30].An important element in such arguments is a correla-tion feature denoted by “the ridge,” a peak at the originon azimuth difference extending symmetrically over large η difference intervals (“long-range” correlations). Theridge in Au-Au collisions has been associated with jetstructure resulting from certain p t cuts in which the 2Djet peak apparently develops tails extending over sub-stantial η intervals [31–33]. The ridge in that case isattributed to bulk matter as opposed to jets based on ob-served large baryon/meson ratios. Note that if no p t cutsare applied the 2D jet peak itself is observed to broadensubstantially on η for more-central Au-Au collisions [14].In Ref. [29] the concept of triangularity and triangu-lar flow is introduced, in which conjectured fluctuationsof initial-state geometry (triangularity) are transformedto observed triangular flow via hydrodynamic evolutionof a bulk medium. The concept is then generalized to“higher harmonics” in which all “long-range” (on η ) az-imuth structure (i.e. the ridge) is a flow manifestationrepresented by a Fourier series with amplitudes v n [30].Interpretation of v n data as representing flow of a densemedium in A-A is conventionally justified by data com- parisons with viscous-hydro theory descriptions invokinga small viscosity ( η/s ratio) (e.g. Ref. [34]). Based onthe apparent success of many such comparisons with anassortment of v n data, formation of a low-viscosity QGPin A-A collisions is considered to be broadly accepted [6]. B. Collectivity in small collision systems
In light of the sequence of developments respondingto correlation data from RHIC Au-Au collisions the firstobservation of a “ridge” in 7 TeV p - p collisions at theLHC [35] was very surprising. Given interpretation ofthe ridge feature in A-A collisions as indicating collectiveflow of a dense bulk medium the p - p result suggested thepossibility of collectivity in the smallest collision systemfor some imposed conditions (high charge multiplicity,certain p t cuts). A follow-up study of p -Pb collisionsrevealed a similar ridge structure with larger amplitudes,comparable to those observed in A-A collisions [4].More generally, p - p and p -A p t spectra have trendssimilar to those found in A-A collisions, including “hard-ening” (increased slope parameter) increasing with n ch orcentrality and with hadron mass (relevant to radial flow).Equivalently, ensemble-mean p t also increases with cen-trality or n ch faster for more-massive hadrons. v and v data for p -Pb are similar to those for Pb-Pb, and massordering observed in PID v ( p t ) data for A-A collisionsis also observed for p -Pb data [36]. The v ( p t ) mass or-dering is interpreted as further evidence for radial flow.Just as for A-A collision systems description of relevantdata by hydro models is seen as confirming a QGP/flowinterpretation of p -A and even p - p data. In Ref. [6], com-menting on novel collective phenomena in small collisionsystems, the situation is summarized by the statement“...it is possible to describe all characteristic featuresmeasured in p-p and p/d/ He-A collisions with modelsbased on the collective response to an initial state geom-etry. In particular hydrodynamic models can reproducethe azimuthal anisotropies of charged hadrons v n , themass splitting of the mean transverse momentum and v for identified particles and the HBT radii.” However, thereport warns that “Depending on the assumptions madeabout how the initial shape of the system is generated, final results can vary dramatically [emphasis added].” C. Responses to claims for collectivity
The evidence and arguments summarized above over-look a number of issues that present a strong challengeto the flow/QGP paradigm. In general, much of theinformation carried by basic particle data tends to besuppressed or ignored, by data selection and by specificchoices of analysis methods and variables. Focus is main-tained on certain collision centralities and not others.Spectrum models are applied to selected p t intervals andnot others. 2D angular correlations are projected onto1D azimuth for certain key analyses, thereby discardingmuch information. p t cuts are applied as biases to angu-lar correlations based on a priori assumptions that maybe invalid. Some characterizations of data features arequalitative rather then quantitative, such as “hardening”of spectra and “mass ordering” for PID v ( p t ) data. Theill-defined term “ridge” is applied to more than one datafeature leading to confusion. Extensive quantities are in-voked (e.g. N track based on some arbitrary η acceptance)when intensive counterparts (e.g. a mean charge density)would facilitate clearer comparisons among A-B systems.More specifically, a growing body of negative evidenceis ignored. In A-A collisions the centrality trend for jetmodification (“quenching”) as revealed by modeling of2D angular correlations [14] corresponds to the trend for p t spectra [13] but is very different from the trend for v data also inferred by modeling of 2D angular corre-lations [37]. A close correspondence should be expectedif both trends are related to a common flowing densemedium. In fact, the v trend on centrality is generallyinconsistent with hydro expectations. The blast-wavemodel conventionally used to infer radial flow from spec-tra is an inferior spectrum model applied to limited p t intervals, in some cases determined solely by the fit qual-ity [2]. The contrast with an accurate and generalizedspectrum model with no p t limits is noted in Sec. VI.The use of Fourier series alone to model 1D azimuthcorrelations is strongly rejected by Bayesian analysis ofmodel quality [38]. For 200 GeV Au-Au collisions amodel consisting of a narrow Gaussian with cos( φ ) andcos(2 φ ) terms only is strongly preferred on the basis of how little information is acquired by a model upon en-countering new data. For example, with increasing dataacquisition the Fourier model requires additional termsto maintain a low χ whereas the model with a Gaussiandoes not. In effect, “higher harmonics” in the Fourier se-ries are substituted for a fixed two-parameter Gaussian.In more-central A-A collisions, and for certain p t cuts,the 1D Gaussian on azimuth includes projected contri-butions from the same-side “ridge” that has not beeneffectively ruled out as an aspect of jet modification.The “mass ordering” of PID v ( p t ) data is said toconfirm the presence of transverse expansion of a densemedium, but differential analysis of such data providesmuch more information. The same data plotted on trans-verse rapidity y t (with proper mass for each hadronspecies) reveal a common zero intercept correspond-ing quantitatively to a particle-source boost distribu-tion [39, 40]. But the effective boost distribution is con-sistent with a single value, not the broad distributionexpected from Hubble expansion of a bulk medium. Fur-ther analysis isolates a “quadrupole spectrum” associatedwith the cos(2 φ ) correlation feature that is quite differentfrom what is inferred for the great majority of hadrons.In p - p and p -A collisions misuse of the term “ridge”has caused confusion since it is applied to at least twodifferent phenomena. The ridge observed in 7 TeV p - p collisions [35] is actually one lobe of a quadrupole cos(2 φ ) correlation. The same-side lobe is easily visible as a ridgebecause the curvature of that lobe is opposite in sign tothe background, whereas superposition of the second lobeat π as an “away-side” ridge increases the magnitude ofthe like-sign curvature there which is usually overlooked.In Ref. [6] the p - p ridge is described as “seen” only forhigh event multiplicities, is “not present” in minimum-bias events and is not well understood. However, the sys-tematics (e.g. n ch dependence) of the nonjet quadrupolefor 200 GeV p - p collisions have been accurately deter-mined via 2D model fits [11] and differ strongly fromany reasonable hydro expectation. The same trend isfollowed from lowest to highest p - p n ch correspondingto a thousand-fold increase in the associated number ofcorrelated pairs. The factorized p - p quadrupole trend isformally equivalent to the trend for Au-Au collisions [41].The nonjet quadrupole has also been observed in 5 TeV p -Pb collisions [4] where the quadrupole cos(2 φ ) featurehas been isolated via a background subtraction but isdescribed as a “double ridge.” It is notable that the“ridge” first observed in more-central Au-Au collisionsas a centrality-dependent elongation of the same-side 2Djet peak [14] projecting to a narrow 1D Gaussian on az-imuth is distinct from the nonjet quadrupole cos(2 φ ) fea-ture that has been described as a “ridge” in p - p collisionsor “double ridge” in p -Pb collisions but is correctly iden-tified as a cylindrical quadrupole structure in A-A data. p -Pb PID p t spectra are described as exhibiting “hard-ening” with increasing n ch or centrality and with hadronmass [2]. The p -Pb trend is seen as matching a similartrend in PID spectra from A-A collisions that is inter-preted as an indicator for “collectivity” in the form ofradial flow. A similar argument is applied to ensemble-mean p t data [42] which simply reflect measured p t spec-tra. Those specific conjectures motivated the present p -Pb spectrum study: Hadron mass- and centrality-dependent “hardening” and blast-wave model fits to PIDspectra are interpreted to demonstrate radial flow uponwhich the ridge and associated higher harmonics are as-sumed to be modulations. If the same data features canbe identified with confidence as arising from a nonflowmechanism (e.g. jets) then the flow conjecture is unlikely.Hydrodynamic modeling of A-A collisions is reviewedin Ref. [43]. The agreement between data and theory isapparently very good as illustrated in Ref. [44]. The hy-brid model utilized in the latter reference consists of com-ponents IP-Glasma + Music where IP-Glasma [45] mod-els initial conditions and MUSIC [46] models viscous hy-dro evolution. The hydro model utilized for a recent anal-ysis reported in Ref. [47], comparisons with v n { EP } datafrom small asymmetric x -Au collisions, is based on theSONIC hydro model. According to Refs. [48, 49] SONICcombines Monte-Carlo Glauber initial conditions witha 2+1 viscous hydrodynamics evolution and hadronic-cascade afterburner. As noted, Ref. [6] warns that suchmodels are typically very sensitive to initial conditions.Although hydro models may appear to describe se-lected data quite precisely there are several major issues:(a) Sensitivity to initial conditions (IC): If a modelis very sensitive to some of its parameters then for pa-rameter values varying across some a priori reasonableintervals the model may disagree strongly with data. Toachieve good agreement with data the parameters mustthen be confined to a small volume within the parameterspace that corresponds to the data. But that is simplya data-fitting procedure. As noted in connection withBayesian model evaluation [38], a model that acquiresmuch information from new data is disfavored comparedto a model that acquires little (i.e. a predictive model).(b) Validity of some IC estimators: Some estimatorsmay be questioned, especially for modeling small asym-metric collision systems. For example, a conventionalGlauber Monte Carlo applied to 5 TeV p -Pb collisionsproduces strongly-biased estimates for N part , N bin andcollision centrality [15] (primed numbers in Table II).For p -Pb collisions assigned to 0-5% centrality via MonteCarlo Glauber the mean charge density is 45 [18] whereashigh-statistics ensemble-mean p t (¯ p t ) data for the samecollision system extend out to charge density 115 [50].The number of participants estimated by the GlauberMC for more-central collisions is roughly 3 times largerthan what is consistent with a TCM description of the ¯ p t data (unprimed numbers in Table II). Note that Ref. [18]includes a detailed study of possible biases from severalestimation methods, but the difference between Glauberand TCM estimates substantially exceeds such biases.If the IP-Glasma estimator is used the IC geometry de-pends strongly on the projectile proton transverse struc-ture, especially its fluctuations. Strong fluctuations ofthe IC geometry are considered essential to generate“higher harmonics” and the ridge structure(s). However,spectrum and correlation data from p - p collisions and ¯ p t data from p -Pb collisions imply that transverse geometryis not relevant for nucleon-nucleon collisions. Any N-Ncollision appears to achieve full overlap, and simultaneousmultiple collisions are excluded. Those principles emergefrom the p - p rate of dijet production [11, 12] and fromanalysis of p -Pb geometry [15, 16, 51].(c) Superiority of alternative data descriptions: Fig-ure 4 of Ref. [44] (ALICE data) or Fig. 20 of Ref. [6](ATLAS data) show IP-Glasma + MUSIC calculationscompared to v n { } ( b ) data from 2.76 TeV Pb-Pb colli-sions. Again the agreement appears to be very good.However, the ALICE v n data have been previously de-scribed within data uncertainties by the combination ofa nonjet quadrupole (inconsistent with hydro) and mul-tipoles that are Fourier components of the same-side 2Djet peak as demonstrated in Figs. 17 (200 GeV Au-Au)and 18 (2.76 TeV Pb-Pb) of Ref. [52]. v { } is knownto have a strong jet contribution despite a cut on η ac-ceptance intended to exclude “nonflow” imposed by theALICE analysis. And v { } and higher coefficients areconsistent with the jet peak alone as the source, i.e. areFourier components of a narrow Gaussian. The full cen-trality dependence of “higher harmonics” is exactly asexpected from jets, including jet modification (quench- ing) in more-central collisions as reported in Ref. [14]. D. Conclusions
The material in this section includes only a sampling ofan abundance of apparent evidence both for and againstthe flow/QGP paradigm. Nevertheless, some criticalissues raised above remain unresolved after nearly tenyears. As noted, recent claims for flow/QGP appearing insmall asymmetric collision systems and possibly even in p - p collisions have been recognized as presenting a majorpuzzle for the nuclear physics community. Any evidencethat might resolve the puzzle should be welcomed.The combination of data features interpreted to indi-cate transverse flow(s) and jet modification (quenching)in A-A collisions have been accepted as demonstratingformation of a dense flowing medium or QGP. It is rea-sonable to apply the same criteria to any collision systemin which QGP formation is claimed. The present studyreports arguably the most accurate PID spectrum analy-sis to date, applied in this case to p -Pb data. The intentis to test for the presence simultaneously of both radialflow and jet modification to the statistical limits of avail-able spectrum data. In the event of a null result claimsof QGP in small systems should be strongly questioned. IV. p-Pb SPECTRUM TCM
The TCM for p - p and p -Pb collisions utilized in thisstudy is the product of phenomenological analysis ofdata from a variety of collision systems and data for-mats [11, 12, 16, 17]. As such it does not representimposition of a priori physical models but does assumeapproximate linear superposition of p -N collisions within p -Pb collisions consistent with no significant jet modifi-cation. Physical interpretations of TCM soft and hardcomponents have been derived a posteriori by comparinginferred TCM characteristics with other relevant mea-surements [13, 19], in particular measured MB jet char-acteristics [53, 54]. Development of the TCM contrastswith data models based on a priori physical assumptionssuch as PYTHIA [55] and the BW model [8]. It is no-table that the TCM does not result from fits to individ-ual spectra (or other data formats), which would requiremany parameter values. The few TCM parameters arerequired to have simple log( √ s ) trends on collision en-ergy and simple extrapolations from p - p trends. A. Spectrum TCM for unidentified hadrons
The p t or y t spectrum TCM is by definition the sumof soft and hard components with details inferred fromdata (e.g. Ref. [12]). For p - p collisions¯ ρ ( y t ; n ch ) ≈ ¯ ρ s ( n ch ) ˆ S ( y t ) + ¯ ρ h ( n ch ) ˆ H ( y t ) , (1)where n ch is an event-class index, and factorization ofthe dependences on y t and n ch is a central feature ofthe spectrum TCM inferred from 200 GeV p - p spectrumdata in Ref. [12]. The motivation for transverse rapid-ity y ti ≡ ln[( p t + m ti ) /m i ] (applied to hadron species i )is described in Sec. IV B. The y t integral of Eq. (1) is¯ ρ = n ch / ∆ η = ¯ ρ s + ¯ ρ h , a sum of soft and hard chargedensities. ˆ S ( y t ) and ˆ H ( y t ) are unit-normal modelfunctions approximately independent of n ch , and thecentrally-important relation ¯ ρ h ≈ α ¯ ρ s with α ≈ O (0 . p - p spectrum data [11, 12, 17].For composite A-B collisions the spectrum TCM is gen-eralized to¯ ρ ( y t ; n ch ) ≈ N part ρ sNN ˆ S ( y t ) + N bin ¯ ρ hNN ˆ H ( y t ) , (2)which includes a further factorization of charge densi-ties ¯ ρ x = n x / ∆ η into A-B Glauber geometry param-eters N part (number of nucleon participants N) and N bin (N-N binary collisions) and mean charge densi-ties ¯ ρ xNN per N-N pair averaged over all N-N interac-tions within the A-B system. For A-B collisions ¯ ρ s =[ N part ( n s ) / ρ sNN ( n s ) is a factorized soft-componentdensity and ¯ ρ h ( n s ) = N bin ( n s )¯ ρ hNN ( n s ) is a factorizedhard-component density.Integrating Eq. (2) over y t the mean charge density is¯ ρ = N part ρ sNN ( n s ) + N bin ¯ ρ hNN ( n s ) (3)¯ ρ ¯ ρ s = n ch n s = 1 + x ( n s ) ν ( n s ) , where the hard/soft ratio is x ( n s ) ≡ ¯ ρ hNN / ¯ ρ sNN and themean number of binary collisions per participant pair is ν ( n s ) ≡ N bin /N part . If the p t acceptance is limited bya low- p t cutoff¯ ρ (cid:48) ¯ ρ s = n (cid:48) ch n s = ξ + x ( n s ) ν ( n s ) , (4)where ξ ≤ S ( p t ) admitted by a low- p t acceptance cut p t,cut , and primes indicate correspondinguncorrected (biased) quantities. It is assumed that a typ-ical p t,cut is below the effective ˆ H ( y t ) lower limit.To obtain details of model functions and other aspectsof the TCM the measured hadron spectra are normalizedby charge-density soft component ¯ ρ s . Normalized spectrathen have the form¯ ρ ( y t ; n ch )¯ ρ s = ˆ S ( y t ) + x ( n s ) ν ( n s ) ˆ H ( y t ) , (5)where n s is the soft component of event-class index n ch integrated within some η acceptance ∆ η . For A-B colli-sions x ( n s ) is generally inferred from data. For p - p colli-sions x ( n s ) ≡ ¯ ρ h / ¯ ρ s ≈ α ¯ ρ s is inferred with α ≈ O (0 . p - p collision energies [17]. For p -Acollisions x ( n s ) ≈ α ¯ ρ sNN is assumed by analogy with p - p collisions, and other p -Pb TCM elements are in turn defined in terms of x ( n s ). For unidentified hadrons thenormalization factor in Eq. (5) is1¯ ρ s = 1¯ ρ sNN N part / x ( n s ) ν ( n s )¯ ρ ( n s ) . (6) B. Spectrum TCM model functions
Given normalized spectrum data as in Eq. (5) and thetrend x ( n s ) ∼ n s ∼ n ch the spectrum soft component isdefined as the asymptotic limit of normalized data spec-tra as n ch goes to zero. Hard components of data spectraare then defined as complementary to soft components.The data soft component for a specific hadron species i is typically well described by a L´evy distribution on m ti = (cid:112) p t + m i . The unit-integral soft-componentmodel is ˆ S i ( m ti ) = A [1 + ( m ti − m i ) /nT ] n , (7)where m ti is the transverse mass-energy for hadrons i ofmass m i , n is the L´evy exponent, T is the slope param-eter and coefficient A is determined by the unit-integralcondition. Reference parameter values for unidentifiedhadrons from 5 TeV p - p collisions reported in Ref. [17]are ( T, n ) ≈ (145 MeV , . T, n )for each species of identified hadrons as in Table III aredetermined from p -Pb spectrum data as described below.The unit-integral hard-component model is a Gaussianon y tπ ≡ ln(( p t + m tπ ) /m π ) (as explained below) withexponential (on y t ) or power-law (on p t ) tail for larger y t ˆ H ( y t ) ≈ A exp (cid:26) − ( y t − ¯ y t ) σ y t (cid:27) near mode ¯ y t (8) ∝ exp( − qy t ) for larger y t – the tail , where the transition from Gaussian to exponential on y t is determined by slope matching [19]. The ˆ H tail den-sity on p t varies approximately as power law 1 /p q +2 t . Co-efficient A is determined by the unit-integral condition.Model parameters (¯ y t , σ y t , q ) for identified hadrons as inTable III are also derived from p -Pb spectrum data.All spectra are plotted vs pion rapidity y tπ with pionmass assumed. The motivation is comparison of spec-trum hard components assumed to arise from a com-mon underlying jet spectrum on p t , in which case y tπ serves simply as a logarithmic measure of hadron p t withwell-defined zero. ˆ S ( m ti ) in Eq. (7) is converted toˆ S ( y tπ ) via the Jacobian factor m tπ p t /y tπ , and ˆ H ( y t )in Eq. (8) is always defined on y tπ as noted. For uniden-tified hadrons a pion mass is assumed. In general, plot-ting spectra on a logarithmic rapidity variable permitssuperior access to important low- p t structure where the majority of jet fragments appear . In what follows, hadronspecies index i may be suppressed for simplicity. V. p-Pb MEAN-p t TCM
Appendix A describes a TCM for ¯ p t data from p - p collisions which provides a context for p -Pb ¯ p t analysis.With the dominant role of MB jets established for p - p ( p -N, N-N) collisions and elements of the p - p ¯ p t TCMintroduced the p -Pb ¯ p t TCM is presented here in moredetail. p -Pb ¯ p t data can in turn be used to infer p -Pbcentrality parameters with improved accuracy [16]. A. TCM for p-Pb Mean-pt vs n ch Given the TCM for p -A or A-B p t spectra ¯ ρ ( p t ) asdescribed in the previous section the associated TCM for¯ p t vs n ch data is simply determined [16]. The ensemble-mean total p t for unidentified hadrons integrated over all y t and within some angular acceptance ∆ η is¯ P t = ∆ η (cid:90) ∞ dp t p t ¯ ρ ( p t ) = ¯ P ts + ¯ P th (9)= N part n sNN ( n s )¯ p tsNN + N bin n hNN ( n s )¯ p thNN , where ¯ p tsNN and ¯ p thNN are determined by model func-tions ˆ S ( y t ) and ˆ H ( y t ). Data indicate that ¯ p tsNN → ¯ p ts ≈ .
40 GeV/c is a universal quantity (for unidenti-fied hadrons) corresponding to spectrum slope parameter T ≈
145 MeV [17]. A mean- p t expression based on theTCM (with n s = n sNN N part /
2) has the simple form¯ P t n s = ¯ p ts + x ( n s ) ν ( n s ) ¯ p thNN ( n s ) . (10)In general, ¯ p thNN ( n s ) may depend on the imposed mul-tiplicity condition ¯ n ch [17]. However, for this analysis itis assumed that ¯ p thNN ( n s ) → ¯ p th fixed. If the p t in-tegral in Eq. (9) does not extend down to zero becauseof limited p t acceptance (e.g. termination at some p t,cut )the expression is modified. The corresponding TCM for uncorrected conventional ratio ¯ p (cid:48) t with p t,cut > P (cid:48) t n (cid:48) ch ≡ ¯ p (cid:48) t ≈ ¯ p ts + x ( n s ) ν ( n s ) ¯ p th ξ + x ( n s ) ν ( n s ) , (11)where ξ is the fraction of the p t spectrum soft componentincluded by acceptance cut p t,cut , and that cut does notaffect the hard component. The lower limit for ¯ p (cid:48) t is ¯ p (cid:48) ts ≡ ¯ p ts /ξ with ξ ≈ .
75 for a p t,cut ≈ .
15 GeV/c [16].
B. Centrality parameter x ( n s ) model Formulation of a TCM for p -Pb data requires accuratedetermination of centrality parameter N part = N bin + 1which in turn requires an expression for x ( n s ) from whichother model parameters may be derived. An expressionfor x ( n s ) can be established by generalizing from x ( n s ) ≈ α ¯ ρ s for p - p collisions. The relation x ( n s ) ≈ α ¯ ρ sNN ( n s ) then defines ¯ ρ sNN ( n s ) and N part ( n s ) / α ¯ ρ s /x ( n s )from which ν ( n s ) = 2 N bin ( n s ) /N part ( n s ) follows. Pa-rameter n s is the independent variable for the model.In an analysis of ¯ p t vs n ch data from 5 TeV p -Pb col-lisions a simple algebraic expression for x ( n s ), as an ex-trapolation of the p - p α ¯ ρ s trend, is found to describe¯ p t data accurately [16]. For p -Pb data the evolution offactors x ( n s ) ν ( n s ) from strictly p - p –like to alternativebehavior is observed near a transition point ¯ ρ s , but¯ p thNN ( n s ) → ¯ p th is assumed to maintain a fixed p - p ( p -N) value in the p -A system (i.e. no jet modificationper supporting evidence in Sec. VI). The derivation fol-lows.Figure 2 (left) shows a model for x ( n s ) expressed as x ( n s ) = α { [1 / ¯ ρ s ] n + [1 /f ( n s )] n } /n , (12)where f ( n s ) = ¯ ρ s + m (¯ ρ s − ¯ ρ s ). Below a transitionpoint at ¯ ρ s , x ( n s ) ≈ α ¯ ρ s as for p - p collisions (dashedline). Above the transition x ( n s ) still increases linearlybut with reduced slope controlled by parameter m < n controls the transition width.The horizontal dotted line and vertical hatched band es-timate values of x ( n s ) and ¯ ρ s for NSD p - p collisions. n s / Dh x ( n s ) a r s p-N p-PbNSD NSD5 TeV n s / Dh n ( n s ) p-Np-PbNSD Pb-Pb5 TeV2.76 TeV FIG. 2: Left: Evolution of TCM hard/soft ratio parameter x ( n s ) with mean soft charge density ¯ ρ s = n s / ∆ η followinga linear p -N ( p - p ) trend (dashed) for lower multiplicities anda trend with ten-fold reduced slope for higher multiplicities(dotted) to describe p -Pb ¯ p t data. Right: Mean participantpath length ν ≡ N bin /N part vs ¯ ρ s (solid) as determined bythe x ( n s ) trend in the left panel (see text). A ν trend forPb-Pb collisions (dash-dotted) is included for comparison. Figure 2 (right) shows ν ≡ N bin /N part for p -Pb data(solid curve) based on N part ( n s ) / α ¯ ρ s /x ( n s ) and N bin = N part − x ( n s ) as described in the leftpanel (solid curve). The dash-dotted curve indicates a ν ∼ ( N part / / trend for Pb-Pb collisions for com-parison, consistent with the eikonal approximation as-sumed for the A-A Glauber model. For Pb-Pb collisions The transition arises from competition between two probabilitydistribution, not between physical mechanisms. See Sec. IX B. ν ∈ [1 ,
8] whereas for p -Pb ν ∈ [1 , p t TCM is compared with p -Pb ¯ p t data in Fig. 3. C. ¯p t TCM for p-Pb collisions vs data
Figure 3 (left) shows uncorrected ¯ p (cid:48) t data for 106 mil-lion 5 TeV p -Pb collisions vs corrected n ch (open boxes)from Ref. [42]. The dashed curve is the TCM for 5 TeV p - p collisions given by Eq. (A3) with α = 0 . p ts ≈ . p th = 1 . ξ = 0 .
73 [16]. The solidcurve through points is the TCM described by Eqs. (11)and (12) with parameters α = 0 . p th = 1 . p - p collisions (assumingno jet modification). Parameters ¯ ρ s ≈ ρ sNSD ≈
15 and m ≈ .
10 are adjusted to accommodate the p -Pb data.Exponent n = 5 affects the TCM only near ¯ ρ s . Soliddots and dash-dotted curve represent a ¯ p t trend impliedby a Glauber analysis of p -Pb centrality [18], with N part and N bin taken from Table I and with x = 0 .
06, ¯ p ts = 0 . p th = 1 . p - p values. n ch / Dh ¯ p t ¢ ( G e V / c ) p-p TCM5 TeV p-Pb¯p ts ¢ NSD p-Pb Glauber n s / Dh ( n c h ¢ / n s ) ¯ p t ¢ ( G e V / c ) NSD ¯p ts p-p 5 TeV p-Pb FIG. 3: Left: Uncorrected ensemble-mean ¯ p t data from 5TeV p -Pb collisions (open squares) vs corrected charge density¯ ρ = n ch / ∆ η from Ref. [42]. Solid and dashed curves areTCM data descriptions from Ref. [16]. Solid dots are derivedfrom a Glauber analysis [18]. Right: Curves and data in theleft panel transformed by factor ¯ ρ (cid:48) / ¯ ρ s = n (cid:48) ch /n s from Eq. (4). Figure 3 (right) shows data in the left panel convertedto ( n (cid:48) ch /n s ) ¯ p (cid:48) t ≈ ¯ P t /n s by factor ξ + x ( n s ) ν ( n s ) as inEq. (4). The dashed line is the TCM for 5 TeV p - p col-lisions defined by Eq. (A4). The solid curve is the p -PbTCM defined by Eq. (10) corresponding also to the solidcurve in the left panel. Transforming data from left toright panels requires an estimate of n s for the ¯ p (cid:48) t data toevaluate the required conversion factor ξ + x ( n s ) ν ( n s ).The map n s → n ch for the TCM from Eq. (3) (secondline) is inverted via linear interpolation to provide themap n ch → n s for data. Figure 3 demonstrates via ¯ p t data that the p -Pb TCM for unidentified-hadron p t spec-tra and their centrality evolution is quite accurate. D. p-Pb geometry inferred from non-PID data
With the TCM relations derived in the previous sub-section it is possible to associate with any measured p -Pbcharge density ¯ ρ = n ch / ∆ η a complete set of TCM spec-trum and geometry parameters. For each value of ¯ ρ thecorresponding ¯ ρ s or n s is obtained by inverting Eq. (3)(second line). x ( n s ) is obtained from Eq. (12) with pa-rameters N part ( n s ) and ν ( n s ) as described below Fig. 2.Table II presents geometry parameters for 5 TeV p -Pb collisions inferred from a Glauber-model analysis inRef. [18] (primed values) compared to TCM values fromthe analysis as in Ref. [15] and as described above (un-primed values). The large differences between Glauberand TCM values are explained in Ref. [15]. The primedquantities (also see Table I) based on a Glauber MonteCarlo study result from assumptions inconsistent with ¯ p t data. Large differences in fractional cross section σ/σ and binary-collision number N bin are especially notable.Charge densities ¯ ρ are derived directly from data andcorrectly characterize the seven centrality classes, but thecentralities reported in Ref. [18] may be questioned. TABLE II: Nominal (primed [18]) and TCM (unprimed [15])fractional cross sections and Glauber parameters, midrapid-ity charge density ¯ ρ , N-N soft component ¯ ρ sNN and TCMhard/soft ratio x ( n s ) used for 5 TeV p -Pb PID spectrum data. σ (cid:48) /σ σ/σ N (cid:48) bin N bin ν (cid:48) ν ¯ ρ ¯ ρ sNN x ( n s )0.025 0.15 14.7 3.20 1.87 1.52 44.6 16.6 0.1880.075 0.24 13.0 2.59 1.86 1.43 35.9 15.9 0.1800.15 0.37 11.7 2.16 1.84 1.37 30.0 15.2 0.1720.30 0.58 9.4 1.70 1.80 1.26 23.0 14.1 0.1590.50 0.80 6.42 1.31 1.73 1.13 15.8 12.1 0.1370.70 0.95 3.81 1.07 1.58 1.03 9.7 8.7 0.0980.90 0.99 1.94 1.00 1.32 1.00 4.4 4.2 0.047 Note that aside from the experimentally-determined ¯ ρ values all TCM (unprimed) parameters in Table II aredetermined by three numbers: α , ¯ ρ s and m . α is de-fined for all p - p collision energies by Eq. (15) of Ref. [17]based on measured jet properties and is not adjusted forindividual collision systems or hadron species. ¯ ρ s and m are inferred by comparing p - p and p -Pb ¯ p t data as inSec. V B. That parameter combination represents a tran-sition from individual peripheral p -N collisions at lower n ch with N part = 2 to increase of N part above 2 for highermultiplicities, which in turn depends on the relation be-tween the p - p probability distribution on n ch and the p -Pb cross-section distribution on N part (Sec. IX B).The TCM (unprimed) geometry parameters in Ta-ble II, derived from p -Pb p t spectrum and ¯ p t data forunidentified hadrons, are assumed to be valid for eachidentified-hadron species and are used unchanged to pro-cess PID spectrum data below. However, certain addi-tions to the spectrum TCM of Sec. (IV A) are requiredto accommodate PID data as described next.0 VI. p-Pb PID SPECTRUM TCM
PID spectrum data from Sec. II are analyzed to developa TCM for each hadron species. Spectrum hard compo-nents for the individual hadron species are isolated andcompared to e + - e − fragmentation functions for identifiedhadrons to support inference of jet-related origins. A. Spectrum TCM for identified hadrons
To establish a TCM for p -Pb PID p t spectra it is as-sumed that (a) N-N parameters α , ¯ ρ sNN and ¯ ρ hNN havebeen inferred from unidentified-hadron data and (b) ge-ometry parameters N part ( n s ), N bin ( n s ) are a commonproperty (i.e. centrality) of p -Pb collisions independent ofdetected hadron species. The required TCM parametersinferred from previous ensemble-mean ¯ p t data analysis ofthe same centrality classes for 5 TeV p -Pb collisions [16]are presented in Table II.Given the p -Pb spectrum TCM for unidentified-hadronspectra in Eq. (2) a corresponding TCM for identifiedhadrons can be generated by assuming that each hadronspecies i comprises certain fractions of soft and hardTCM components denoted by z si and z hi (both ≤ ρ i ( y t ) ≈ N part z si ¯ ρ sNN ˆ S i ( y t ) + N bin z hi ¯ ρ hNN ˆ H i ( y t )¯ ρ i ( y t )¯ ρ si = ˆ S i ( y t ) + ( z hi /z si ) x ( n s ) ν ( n s ) ˆ H i ( y t ) , (13)where unit-integral model functions ˆ S i ( y t ) and ˆ H i ( y t )may depend on hadron species i . For identified hadronsof species i the normalization factor 1 / ¯ ρ si in the secondline follows the form of Eq. (6) but can be re-expressedin terms of 1 / ¯ ρ s for unidentified hadrons already inferred1¯ ρ si = 1 + ( z hi /z si ) x ( n s ) ν ( n s )¯ ρ i ≡ z i ¯ ρ (14) ≈ (cid:26) z hi /z si ) x ( n s ) ν ( n s )1 + x ( n s ) ν ( n s ) (cid:27) z i · ρ s . For each hadron species i ratio z hi /z si is first adjustedto achieve coincidence of all seven normalized spectra as y t →
0. Parameter z i is then adjusted to match thoserescaled spectra to unit-normal ˆ S i ( y t ), also as y t → S i ( y t ) and ˆ H i ( y t ) mustbe determined for each hadron species; however a closerelation to unidentified-hadron models is expected. Asnoted in Sec. IV B ˆ S i ( y t ) is first defined on proper m ti for a given hadron species i and then transformed to y tπ .ˆ H i ( y t ) is defined on y tπ in all cases. B. p-Pb differential PID spectrum data
In the figures below, PID p t spectra from Sec. II arereplotted in left panels in the normalized form of Eq. (13) (second line) and compared to TCM soft componentsˆ S i ( y t ) (bold dotted curves). For each hadron speciespublished spectra are plotted on y tπ as a logarithmic rep-resentation of p t with well-defined zero since the under-lying jet (parton) p t spectrum determines the spectrumhard component [19]. Data spectra on p t are transformedto densities on y tπ via Jacobian factor m tπ p t /y tπ where m tπ = p t + m π and y tπ = ln[( m tπ + p t ) /m π ]. The plotboundaries for full spectra (left) and for spectrum hardcomponents (right) are maintained consistent among theseveral hadron species to facilitate comparisons. The ex-ception is for kaon data where the left panels are extendeddown to zero to accommodate the K S data.Figure 4 (left) shows identified-pion spectra from Fig. 1(a). The published spectra have been multiplied by 2 π to be consistent with the η densities used in this study.The spectra are then normalized by soft-component den-sity ¯ ρ si as defined in Eq. (14) with TCM parametervalues reported in Tables II, III and IV. The normal-ized spectra X ( y t ) can then be compared with spectrumsoft-component model ˆ S ( y tπ ) shown as the bold dottedcurve: a L´evy distribution defined on m tπ with parame-ters T = 145 MeV and n = 8 . y tπ thatalso describes unidentified hadrons from 5 TeV p - p colli-sions as reported in Ref. [17]. The solid line labeled BWmarks the p t ( y tπ ) interval over which a BW model fitwas imposed as reported in Ref. [2]. -3 -2 -1 y t p X ( y t ) = [ / N p a r t ( n s )] r ( y t ) / r s NN S -3 -2 -1 y t p [ X ( y t ) - S ( y t )] / x ( n s ) n ( n s ) Ae -4yt H FIG. 4: Left: Identified-pion spectra for 5 TeV p -Pb colli-sions from Ref. [2] transformed to y t with Jacobian m t p t /y t and normalized by TCM quantities in Table II (7 thinnercurves of several styles). ˆ S ( y t ) (bold dotted) is the soft-component model. Right: Difference X ( y t ) − ˆ S ( y t ) normal-ized by x ( n s ) ν ( n s ) ≈ α ¯ ρ sNN ν ( n s ) using TCM values from Ta-ble II reported in Ref. [16] (6 thinner curves, most-peripheralcurve is omitted). The bold dashed curve is hard-componentmodel ˆ H ( y t ) with exponential tail. Figure 4 (right) shows difference X ( y t ) − ˆ S ( y t ) nor-malized by ( z hi /z si ) x ( n s ) ν ( n s ) using TCM values as re-ported in Tables II and III. The result should be directlycomparable to the p - p spectrum hard-component modelin the form ˆ H ( y t ) per Eq. (13). The bold dashed curveis ˆ H ( y t ) with model parameters (¯ y t , σ y t , q ) for pions asin Table III. The dotted line is a reference to verify thatthe model is properly normalized. Any deviations fromˆ H ( y t ) in the right panels are the “fit” residuals for the1model, but the model is highly constrained with only afew adjustable parameters. There is substantial uncer-tainty in the data hard component for the first n ch class,so those data are omitted to improve access to the othercentrality classes. It is evident that the pion hard com-ponents are systematically above the model, by ≈ K + + K − and neutral K S spectra from Fig. 1 (c) and (d) processed in the samemanner as for charged pions. The K S and K ± spectraare consistent within data uncertainties as reported inRef. [2]. The TCM model functions are therefore con-strained to be the same for all kaons. Whereas the K ± data are quite limited the K S data subtend the spectac-ular interval p t ∈ [0 ,
7] GeV/c. K S data below y tπ ≈ . ≈ . fixed soft compo-nent independent of p -Pb centrality contributes in thatinterval, and a L´evy distribution on m tK describes thedata well. Actual data points (open circles) for the low-est and highest ¯ n ch classes are shown. A usable estimatefor ˆ H ( y t ) obtained down to y t = 1 . p t ≈ . -3 -2 -1 y t p X ( y t ) = [ / N p a r t ( b )] r ( y t ) / r s NN S -3 -2 -1 y t p [ r ( y t ) / r s - S ( y t )] / x ( b ) n ( b ) H (a) (b) -3 -2 -1 y t p X ( y t ) = [ / N p a r t ( b )] r ( y t ) / r s NN S S0 BW 10 -3 -2 -1 y t p [ X ( y t ) - S ( y t )] / x ( n s ) n ( n s ) Ae -4.0yt H (c) (d) FIG. 5: Identified-kaon spectra for 5 TeV p -Pb collisionsfrom Ref. [2]: (a), (b) charged kaons K ± ; (c), (d) neutralkaons K S . The panel descriptions are otherwise as for pions.In panel (c) individual data points (open circles) are providedfor the most-peripheral and most-central event classes. Figure 6 shows proton p + ¯ p and Lambda Λ+ ¯Λ spectrafrom Fig. 1 (e) and (f) processed in the same manner asfor charged pions. Given the complementary p t coverageof the two species the proton data were used to determineˆ S ( y t ) at lower y t and the Lambda data were used todetermine ˆ H ( y t ) at higher y t . -3 -2 -1 y t p X ( y t ) = [ / N p a r t ( b )] r ( y t ) / r s NN S -3 -2 -1 y t p [ X ( y t ) - S ( y t )] / x ( n s ) n ( n s )
200 GeVAu-Au Ae -4.8yt Be -5.0yt H (a) (b) -3 -2 -1 y t p X ( y t ) = [ / N p a r t ( b )] r ( y t ) / r s NN S -3 -2 -1 y t p [ r ( y t ) / r s - S ( y t )] / x ( b ) n ( b ) H (c) (d) FIG. 6: (a), (b) Identified-proton p + ¯ p spectra for 5 TeV p -Pb collisions from Ref. [2]; (c), (d) identified Lambda Λ + ¯Λspectra. The panel descriptions are otherwise as for pions. In(b) and (d) the dotted curves follow shifts on y t of the hardcomponents with increasing n ch or centrality. The dashedcurves are defined by values in Table III. In (b) the dottedcurves are suppressed relative to the nominal ˆ H ( y t ) model(dashed) to accommodate the proton data (see text). The Lambda data are well described out to 7 GeV/cbut, as noted in Sec. II regarding Fig. 1 (e), the pro-ton hard components in panel (b) appear to be stronglybiased relative to the expected ˆ H ( y t ). Proton soft com-ponents appear consistent with the TCM prediction [re-fer to data in Fig. 1 (e) compared to the TCM solidcurves]. In order to accommodate the proton hard com-ponents ˆ H ( y t ) is multiplied by function 1 − a exp {− [( y t − ¯ y t ) /σ y t,corr ] / } with ¯ y t = 2 . σ y t,corr = 0 .
85 and a = 0 .
40 which suppresses the hard component over alimited y t interval . The result is the dotted curves in (b)corresponding to dash-dotted curves in Fig. 1 (e).In each of the right panels of Fig. 6 hard-componentmodel ˆ H ( y t ) appears as multiple curves (dotted)that accommodate systematic shifts of hard-componentmodes to higher y t with increasing n ch or p -Pb centrality.The shift is clearly apparent for baryons but no shift wasrequired for meson data in this study. Hard-componentmodel ¯ H ( y t ) is modified by ¯ y t → ¯ y t + δ n where for n ∈ [1 , δ n = ( n − δ with δ = 0.03 for protonsand 0.015 for Lambdas. Consequences for PID spectrumratios are discussed in Sec. VIII.In contrast to BW fits to the same PID spectrum dataas described in Ref. [2] the TCM spectrum descriptionsabove do not rely on selecting some limited p t or y t rangebased on agreement with data (see intervals labeled BW2in Figs. 4, 5 and 6) as discussed in Sec. IX E. The TCMsystem is accurate, exhaustive and predictive. C. p-Pb TCM PID spectrum parameters
Table III shows TCM model parameters for hard com-ponent ˆ H ( y t ) (first three) and soft component ˆ S ( y t )(last two). Hard-component model parameters varyslowly but significantly with hadron species. Centroids¯ y t shift to larger y t with increasing hadron mass. Widths σ y t are substantially larger for mesons than for baryons.Only K s and Λ data extend to sufficiently high p t todetermine exponent q which is substantially larger forbaryons than for mesons. The combined centroid, widthand exponent trends result in near coincidence amongthe several models for larger y t . Hard components frommultiple hadron species all point to a common underlyingparton spectrum as demonstrated in Fig. 7. TABLE III: TCM model parameters for unidentified hadrons h from Ref. [17] and for identified hadrons from 5 TeV p -Pb collisions from this study: hard-component parameters(¯ y t , σ y t , q ) and soft-component parameters ( T, n ). Numberswithout uncertainties are adopted from a comparable hadronspecies with greater accuracy.¯ y t σ y t q T (MeV) nh . ± .
03 0 . ± .
03 3 . ± . ± . ± . π ± . ± .
03 0 . ± .
03 4 . ± ± . ± . K ± .
65 0 .
58 4 . K s . ± .
03 0 . ± .
02 4 . ± . ± ± p . ± .
02 0 .
47 4 . ±
10 14 ±
4Λ 2 . ± .
02 0 . ± .
03 4 . ± . Soft-component model parameter T ≈
145 MeV for pi-ons is consistent with that for unidentified hadrons foundto be universal over all A-B collision systems and colli-sion energies [17]. The values for higher-mass hadronsare substantially larger. L´evy exponent n ≈ . √ s/
10 GeV) energy depen-dence [17]. Soft-component exponent n values for more-massive hadrons are not well-defined because the hard-component fraction is much larger than for pions. Vary-ing n then has little impact on the overall spectra.Figure 7 (left) shows hard-component model functionsˆ H ( y tπ ) for four hadron species (pions, kaons, protons,Lambdas). The TCM spectrum hard component has pre-viously been interpreted as a fragment distribution aris-ing from MB dijets [19]. These similar model shapes fur-ther support that interpretation. With increasing hadronmass the lower- p t tails and distribution modes move tothe right, but the models for protons and Lambdas dropfaster on the higher- p t side so the high- p t intercepts co-incide, consistent with a common underlying parton p t spectrum having a lower bound near 3 GeV -3 -2 -1 y t p [ r ( y t ) / r s - S ( y t )] / x ( b ) n ( b ) p K p, L -2 -1 -2 -1 x p dn c h / d x p pionskaonsprotons91 GeV e + -e - FIG. 7: Left: Hard-component models ˆ H ( y t ) from Figs. 4,5 and 6. Models for charged and neutral kaons are assumedidentical consistent with spectrum data. Right: e + - e − frag-mentation functions for identified pions, kaons and protonsfrom unidentified partons (mainly light quarks) from Ref. [54]. Figure 7 (right) shows identified-hadron fragmentationfunctions (FFs) from unidentified (light) partons [Fig. 7(left) of Ref. [54]]. A shift to lower fragment momenta forpions is expected based on those data. The main trend –pion FFs are softer than kaon FFs are softer than protonFFs – is consistent with the left panel. Two conclusionsemerge from PID p t spectra: (a) For all p -Pb centralitiesthere is no apparent jet modification, no “jet quenching.”(b) The MB jet contribution dominates baryon produc-tion, which is far greater than expected from the statis-tical model, leading to large values for baryon z h /z s .Table IV shows PID parameters z and z h /z s for fivehadron species that are determined from spectrum dataas fixed values independent of centrality. The choiceto hold z and z h /z s fixed rather than z s and z h sep-arately arises from PID spectrum data structure as fol-lows: Given the TCM expression in Eq. (13) the correctnormalization 1 / ¯ ρ si should result in data spectra coin-cident with ˆ S ( y t ) as y t → z si . Empirically, therequired TCM condition is met by holding z hi /z si and z i fixed as described in the previous subsection. TABLE IV: TCM model parameters for identified hadronsfrom 5 TeV p -Pb collisions. Numbers without uncertaintiesare adopted from a comparable hadron species with greateraccuracy. Parameters ¯ p ts and ¯ p th are determined by modelfunctions ˆ S ( y t ) and ˆ H ( y t ) with parameters from Table III. h represents results for unidentified hadrons. z z h /z s ¯ p ts (GeV/c) ¯ p th (GeV/c) h ≡ ≡ . ± .
02 1 . ± . π ± . ± .
02 0 . ± .
05 0 . ± .
02 1 . ± . K ± . ± .
01 2 . ± . .
60 1 . K s . ± .
005 3 . ± . . ± .
02 1 . ± . p . ± .
005 7 . ± . ± .
02 1 . ± .
03Λ 0 . ± .
005 7 . . ± .
02 1 . ± . Separate fractions z s and z h may be derived from fixed3model parameters z h /z s and z via the relation z s = 1 + x ( n s ) ν ( n s )1 + ( z h /z s ) x ( n s ) ν ( n s ) z , (15)and since z and z h /z s are held fixed z s and z h mustthen be centrality dependent per x ( n s ) ν ( n s ). As a conse-quence, for hadron species 1 and 2 (with 2 more massive) z x z x ∝ z h /z s ) x ( x s ) ν ( n s )1 + ( z h /z s ) x ( x s ) ν ( n s ) , (16)where x = s or h . For increasing p -Pb centrality those ra-tios must then decrease according to parameter values inTable IV. If TCM model functions are held fixed indepen-dent of p -Pb n ch or centrality then according to Eq. (13)(first line) the spectrum ratio of two hadron species mustdecrease with increasing centrality. If that trend is notobserved the assumption of fixed model functions shouldbe questioned, as discussed in Sec. VIII. VII. p-Pb PID ENSEMBLE-MEAN ¯p t The TCM for PID spectra in the previous section maybe tested by comparison with measured PID ¯ p t values.A TCM for ensemble-mean ¯ p t for unidentified hadronsis described in Sec. V. In this section the ¯ p t TCM isgeneralized to describe identified hadrons. Just as forPID spectra it is assumed that all hadron species sharecommon p -Pb geometry parameters x ( n s ) and ν ( n s ).The ensemble-mean total p t for identified hadrons ofspecies i integrated over some angular acceptance ∆ η is¯ P ti = ∆ η (cid:90) ∞ dp t p t ¯ ρ i ( p t ) = ¯ P tsi + ¯ P thi (17)= N part z si n sNN ¯ p tsNNi + N bin z hi n hNN ¯ p th i . As for unidentified hadrons it is assumed that ¯ p tsNNi → ¯ p tsi is a universal quantity for each hadron species. Anensemble-mean ¯ p t expression based on the TCM (with n si = z si n sNN N part /
2) then has the simple form¯ P ti n si = ¯ p tsi + ( z hi /z si ) x ( n s ) ν ( n s ) ¯ p th i . (18)The corresponding TCM for conventional ratio ¯ p ti is¯ P ti ¯ n chi = ¯ p ti ≈ ¯ p tsi + ( z hi /z si ) x ( n s ) ν ( n s ) ¯ p th i z hi /z si ) x ( n s ) ν ( n s ) , (19)assuming that the p t integral extends down to p t = 0 (byTCM extrapolation of ¯ ρ i data). The lower limit for ¯ p ti isthen ¯ p tsi . If the p t acceptance has lower bound p t,cut > → ξ i above similar to Eq. (11).Based on results in the previous section ratio z hi /z si for each hadron species is held fixed independent of cen-trality or n ch using values from Table IV. That tablealso includes values for ¯ p tsi and ¯ p th i derived from model functions ˆ S i ( y t ) and ˆ H i ( y t ) respectively as defined byparameters in Table III. The centrality parameters, in-dependent of hadron species, are taken from Table II.Figure 8 shows ¯ p t vs n ch data from Ref. [2] (solidpoints) compared to the TCM described by Eq. (19)(solid curves). The dash-dotted curve for protons isEq. (19) with x ( n s ) reduced by factor 0.6 to represent thebias effect in Fig. 6 (b). Both versions assume fixed TCMmodel functions. The open circles are ¯ p t data for uniden-tified hadrons from Ref. [42] with corresponding TCMfrom Ref. [16] (dashed). The dotted curve representsa MC trend derived from Ref. [42] (Fig. 3). The opensquares represent a prediction in Ref. [16] for unidenti-fied hadrons based on the Glauber p -Pb centrality anal-ysis from Ref. [18]. All data and curves are correctedto full p t acceptance. The same z h /z s = 3 . p t = 0. Point-to-point uncertainties are muchsmaller. Deviations of proton and Lambda data relativeto corresponding TCM curves vary smoothly from lesserto greater, consistent with shifts on y t of data and model(dotted) hard components in Fig. 6 (b) and (d). n ch / Dh ¯ p t ( G e V / c ) pionshadronskaonsprotonsLambdashadrons¯p ts MCGlauber
FIG. 8: Corrected ¯ p t vs n ch data from Ref. [2] (solid points)with TCM described by Eq. (19) (solid curves). K S data (in-verted triangles) are displaced slightly to the right of the K ± data. The proton solid curve is an expectation from TCMsystematics. The dash-dotted curve represents apparent sup-pression of the proton hard component (see text). The opencircles are unidentified-hadron data from Ref. [42] with corre-sponding TCM (dashed) from Ref. [16]. ¯ p t values implied bya Glauber-model analysis in Ref. [18] are represented by opensquares. The dotted curve MC is derived from Ref. [42]. Thehatched bands indicate ¯ p ts values from Table IV. Several conclusions can be drawn from the results in4Fig. 8: (a) The strong increase in ¯ p t values with n ch corresponds to the quadratic relation between soft andhard TCM components in p - p or N-N collisions ¯ ρ hNN ≈ α ¯ ρ sNN which in turn follows the quadratic trend for p - p dijet production as described in Ref. [53]. (b) The strongincrease in ¯ p t values with hadron mass correspond tothe properties of parton fragmentation to jets (i.e. fol-lows the spectrum hard component associated with jets).(c) Whereas PID ¯ p t data from Ref. [2] (12.5 or 25 mil-lion events) extend only to ¯ ρ = n ch / ∆ η ≈
45 (0-5%central p -Pb collisions reported in Ref. [18]) the p -Pb ¯ p t data for unidentified hadrons extend to ¯ ρ ≈
116 accord-ing to Ref. [42] from the same collaboration (106 millionevents). (d) ¯ p t evolution implied by the Glauber-modelanalysis of Ref. [18] (open boxes) or Monte Carlos basedon a Glauber model of p - p collisions (dotted curve, e.g.PYTHIA) deviate strongly from the p -Pb data. VIII. p-Pb PID SPECTRUM RATIOS
Figure 9 shows spectrum ratios for (a) 2 K S / ( π + + π − ),(b)Λ /K S and (c) ( p + ¯ p ) / ( π + + π − ) for two p -Pb central-ities, 0-5% and 60-80% (as inferred from the Glauberanalysis of Ref. [18]). Panel (d) is discussed below. Thesolid curves are derived from TCM solid curves in Fig. 1that describe spectrum data well (except for protons).The dash-dotted curves for protons in panel (c) are de-termined using the biased dash-dotted curves in Fig. 1(e). The dashed curves are derived from the TCM withhard components omitted. Those three panels can becompared with data in Fig. 2 (left panels) of Ref. [2]. Theagreement between TCM and data (referring also to pro-ton data vs dash-dotted curves) is generally good with noTCM parameter adjustment. However, baryon data de-scription does require relaxing the assumption that TCMmodel function ˆ H ( y t ) is independent of n ch or centrality.In Sec. VI C the nominal TCM prediction for spectrumratios is uniform (on p t ) decrease with increasing p -Pbcentrality assuming TCM model functions independentof centrality, as illustrated in Fig. 9 (d) where the ra-tios are generated with no hard-component shift. Thehatched band in panel (d) is an estimate of ratio reduc-tion determined by Eq. (16), where the ratio value 0.66 isderived from Table values 0.8 and 7 for z h /z s and valuesfor parameters x and ν of 0.19 and 1.52 for 0-5% and 0.10and 1.03 for 60-80%. The upper solid curve is the ratioof lower solid curves and is consistent with that estimate.However, variation of baryon/meson data ratios withincreasing centrality in panels (b) and (c) changes fromdecreasing to increasing with increasing p t , as noted inthe top panel of Fig. 3 in Ref. [2]. Differential analysisof spectrum data as in Sec. VI B provides deeper insightinto PID spectrum evolution with p -Pb centrality.Spectrum-ratio centrality trends actually result fromtwo effects working in opposition: (a) common reductionof fractions z s and z h with increasing centrality as aboveand (b) the effect of baryon hard components shifting to t (GeV/c) K S / ( p + + p- ) t (GeV/c) L / K S (a) (b) p t (GeV/c) p + + p - / p + + p- t (GeV/c) p + + p - / p + + p- (c) (d) FIG. 9: Spectrum ratios for three combinations of identifiedhadrons from 5 TeV p -Pb collisions as derived from the cor-responding TCMs. Dashed curves represent soft componentsonly. Other curves represent soft + hard. In panels (c) and(d) the solid curves correspond to expected proton spectraand the dash-dotted curves represent suppression of the hardcomponent to accommodate spectrum data from Ref. [2]. Inpanel (d) the TCM hard-component position is fixed whereasin panel (c) the proton hard component shifts to greater y t with increasing centrality following data as shown in Fig. 6(b). There is no corresponding shift for pions or kaons. higher p t . Spectrum ratios tend to decrease uniformlyon p t with increasing n ch or centrality because of thetrend for decreasing fractions z s and z h , assuming modelfunctions (and their data equivalents) do not vary withcentrality. The z s and z h trends are determined by spec-trum data below 0.5 GeV/c where the hard-componentcontribution is negligible. The nearly-uniform (on p t )decrease is illustrated in Fig. 9 (d).In Fig. 6 (b) and (d) spectrum hard components forprotons and Lambdas shift significantly to higher y t withincreasing n ch , and the TCM equivalents are shifted toaccommodate the data. The difference between panels(c) and (d) in Fig. 9 is consistent with the spectrum hardcomponent for protons shifting to substantially higher y t with increasing n ch while no significant shift for pions isobserved in Fig. 4 (or for kaons in Fig. 5). The effectsof such shifts for unidentified hadrons were observed al-ready for p - p collisions in Refs. [16, 17]. The contrast ofeffects in the lower panels of Fig. 9 should be larger for p - p collisions where all of the n ch variation contributesto quadratic increase of dijet production, with resultingsubstantial bias of ¯ p t trends as demonstrated in Ref. [16].Reference [2] interprets ratio data in the context ofPb-Pb spectrum ratios (right panels in Fig. 2 of Ref. [2])5as follows: Again arguing by analogy there is “significantenhancement [of spectrum ratios] at intermediate p T ∼ p -Pb collisions the spectrum-ratio centrality trend [e.g. panel (c) solid or dash-dottedcurves] results from a baryon hard component shifting tohigher p t while a meson hard component exhibits negligi-ble shift. In contrast, for Pb-Pb collisions the dominantvariation with centrality is the meson (e.g. pion) hardcomponent shifting to lower p t while the baryon (e.g. pro-ton) hard component shifts only slightly to higher p t as demonstrated in Ref. [13]. These TCM results thendemonstrate that evolution of spectrum ratios as in Fig. 9is dominated in any A+B collision system by MB jet pro-duction, differently for different hadron species. IX. SYSTEMATIC UNCERTAINTIES
Uncertainties for p -Pb collision-geometry determina-tion, TCM spectrum model functions and accuracy ofspectrum models for identified-hadron data are discussedin the context of the TCM as a lossless data-compressionstrategy with a small number of degrees of freedom. A. TCM degrees of freedom
In contrast to a typical MC model with tens of pa-rameters readjusted (tuned) to each individual collisionsystem the TCM includes only a few parameters appliedself-consistently to a broad array of collision systems withlittle or no individual adjustment. The TCM can be seenas a form of data compression: a large number of colli-sion systems and data formats is represented by a smallnumber of tightly-constrained parameters. The result isa simple global data model with predictive power.For p - p spectra the parameters are hard/soft ratio α and y t model parameters ( T, n ; ¯ y t , σ y t , q ). The energyand n ch systematics of y t model parameters are describedin Ref. [17] covering a span from 17 GeV to 13 TeV. Forunidentified hadrons T ≈
145 MeV is universal, ¯ y t and σ y t variations are small or negligible and n and q varyas log( √ s ) as expected for QCD processes. The energydependence of α depends on measured jet properties andis also described in Ref. [17]. For p -Pb collisions addi-tional model parameters ¯ ρ s ( p -N– p -Pb transition point)and m [ x ( n s ) slope reduction factor] are introduced toaccommodate ¯ p t data [16]. For p - p and p -A collisions jetformation is assumed to be unmodified: The spectrum hard component is then approximately invariant on n ch in agreement with data. Spectrum and ¯ p t data are typi-cally described within point-to-point uncertainties.Within a composite TCM, A-B centrality is factorizedfrom N-N ( p - p ) densities which are factorized from p t or y t dependence leading to a simple model with largely-independent degrees of freedom that can be evaluatedaccurately. In contrast, within a one-component modelsuch as a QGP/flow model (all soft) or PYTHIA (allhard) multiple physical mechanisms may not be properlydistinguished leading to complexity, misinterpretationsand substantially increased parameter uncertainties.For the present identified-hadron study two additionalparameters are introduced, density fractions z s and z h asthe combinations z h /z s and z , and the TCM y t modelparameters are determined individually for each hadronspecies. However, the combinations z h /z s and z are con-strained to be independent of centrality and y t , and cen-trality parameters are maintained independent of hadronspecies, thereby achieving factorization of the TCM.In terms of degrees of freedom the geometry parame-ters in Table II depend on only three parameters, α ( √ s ),¯ ρ s and m . The last two are derived from p -Pb ¯ p t datain Ref. [16] but the first is determined by a simple log( √ s )function derived from p - p data in Ref. [17]. Parameters inTable III include those for unidentified hadrons h derivedfrom p - p data as in Refs. [11, 19] and for PID data. Forthe latter there are three lines (pions, kaons, baryons) offive parameters each for a total of 15. However, the hard-component parameters are closely correlated with mea-sured PID fragmentation functions as in Fig. 7. For theparameters in Table IV there are again three lines (pions,kaons, baryons) of two parameters each, z and z h /z s , fora total of 6 that are newly derived from the present studyand can be contrasted with the corresponding many p t -dependent PID parameter values represented by Fig. 9 ofRef. [2]. Because of correlations among PID parametersthe actual number of degrees of freedom in the 5 TeV p -Pb PID TCM is substantially less than 15 + 6 + 2 = 23. B. p-Pb geometry estimation
As noted in Sec. V D accurate centrality determina-tion is essential to establish a TCM for any A-B collisionsystem. Entries in Table II reveal major discrepancies be-tween Glauber (primed) and TCM (unprimed) centralityparameters suggesting large uncertainties in p -Pb geom-etry estimation. However, available evidence indicatesthat the TCM version is quite accurate as argued here.The TCM for 7 TeV p - p collisions provides a self-consistent description of yields, spectra and two-particlecorrelations over an n ch range corresponding to 100-foldincrease in dijet production for ¯ ρ increasing to more thanten times its NSD value 6 [11, 12]. Quantitative relationsamong jets, 2D angular correlations and spectrum hardcomponents have been established. A central element ofthe TCM is the quadratic relation ¯ ρ h ≈ α ¯ ρ s between jet6production per ¯ ρ h and low- x participant gluons per ¯ ρ s .Figure 3 (left) demonstrates close agreement betweenthe TCM description for p -Pb collisions (solid curve) and¯ p t data (open squares). As noted above, the p -Pb ¯ p t trend coincides with that for p - p collisions (dashed) up to¯ ρ ≈
20 or four times the 5 TeV NSD value, implying that p -Pb collisions within that interval are nearly equivalentto single peripheral p -N collisions (with increasing n ch )or that N bin ≈
1. In contrast, the Glauber trend for N (cid:48) bin in Table II increases in the same interval to greater than6 and coincides with an implicit assumption that all p -Ncollisions retain the same mean n ch and other properties.If that were literally true the result would be the solidpoints in the left panel deviating greatly from p -Pb data,with the restriction ¯ ρ <
50 for 0-5% central collisions.The TCM for charge densities from A-B systems is2 N part ¯ ρ = ¯ ρ sNN ( n s ) + ν ( n s )¯ ρ hNN ( n s ) , (20)and in relation to the Glauber analysis of Ref. [18] threecases can be considered: (a) N part ∝ ¯ ρ per Ref. [18], (b)¯ ρ sNN and ¯ ρ hNN remain fixed but ν varies, and (c) thefull TCM with all components is inferred from ¯ p t data.Figure 10 (left) shows Glauber parameter N part / p -Pb collisions from Ref. [18] (solid dots, primedvalues in Table II) and corrected Glauber values fromRef. [15] (open circles). The dashed curve is N part / ρ / . ρ is 5.0). The solid curveis the full TCM described in Sec. V. The short hatchedbands indicate limits on two models based on the GlauberMC simulation (near ¯ ρ = 50) and on ¯ p t data (near 120). n ch / Dh N p a r t / TCMGlauberNSDNSD n ch / Dh r NN = ( / N p a r t ) r NSDGlauberNSD 5 TeV p-Pbp-p TCMp-Pb TCM datalimit
FIG. 10: Left: Participant-pair number N part / ρ from a Glauber study of 5 TeV p -Pb collisions inRef. [18] (points) and from a TCM that describes ¯ p t datareported in Ref. [15] (solid). The dashed curve is N part / ρ / .
5. The hatched bands denote effective limits for thetwo trends. Right: Hadron production per participant pair:trends from the Glauber study in Ref. [18] (dash-dotted), fromthe p -Pb ¯ p t TCM (solid) and from the p - p TCM (dashed).
Figure 10 (right) shows ¯ ρ NN ≡ (2 /N part )¯ ρ (vs ¯ ρ )which, for the Glauber-model analysis of Ref. [18], isassumed constant near the NSD value 5 as in case (a)(dash-dotted). In contrast, the p -Pb TCM trend of case(c) follows that for p - p (dashed) with N part / ≈ p t data. Above a transition point (¯ ρ s ≈
15) ¯ ρ NN continues to increaselinearly but with reduced slope ( m ≈ .
1) as N part in-creases above 2. Near the limit of ¯ p t data at ¯ ρ ≈ N part ≈
8) ¯ ρ NN ≈
30 is just half the value 60reached by isolated 7 TeV p - p collisions for ¯ p t data as inRef. [42] and App. A but implies compared to the NSD value ≈ N part and ¯ ρ NN with n ch condition (or p -Pb cen-trality) depend on the relation between probability distri-butions on those parameters. The assumption N part ∝ ¯ ρ in Ref. [18] (points in the left panel) implies that the dis-tribution on N part (as simulated by the Glauber MC) ismuch broader than that on ¯ ρ NN , so with increasing n ch variation of N part is rapid while variation of ¯ ρ NN is neg-ligible. The TCM result implies that the distribution on N part must be much narrower than that on ¯ ρ NN .A study in Reference [15] concludes from ¯ p t data thatthe distribution on N part is indeed quite narrow com-pared to a p - p probability distribution on n ch , and muchnarrower than the Glauber MC result. For smaller n ch the cases (a) and (b) relating to Eq. (20) are clearly ex-cluded by the coincidence of p - p and p -Pb data in Fig. 3(left); N part must remain near 2 for lower n ch , contradict-ing the Glauber MC. Above the transition the TCM andGlauber model represent limiting cases for N part , but ¯ p t data also strongly favor the TCM distribution there.A follow-up study in Ref. [51] offers an explanation:The Glauber approach with eikonal approximation esti-mates the number of p -N geometric encounters duringprojectile passage through a target nucleus. Each suchencounter is then assumed to be an actual p -N collision,with multiple simultaneous collisions not only possiblebut likely. The result is the Glauber trends in Fig. 10.Reference [51] suggests that simultaneous p -N collisionsare excluded, in which case the maximum N part value for p -Pb is near 8 as for the TCM trends in Fig. 10. Exclu-sion of multiple simultaneous collisions is also consistentwith the quadratic relation ¯ ρ h ≈ α ¯ ρ s for p - p collisionsthat implies full overlap for any actual p -N collision .The consequence for uncertainty estimation is thatbased on ¯ p t data the actual trend for N part vs ¯ ρ must beclose to the TCM trend in Fig. 10 (left) as a lower limit.For more-peripheral collisions the uncertainty is negligi-ble due to the constraints of ¯ p t data. For more-centralcollisions the possibility of additional mechanisms for ¯ p t variation (jet modification, flows) could increase uncer-tainties, but indications from spectrum analysis (e.g. thepresent study) exclude a flow contribution. Observedspectrum hard-component trends indicate that jet pro-duction is effectively unmodified in p -Pb collisions.7 C. TCM model functions
Parameter values for PID TCM model functions on y t are shown in Table III. Coverage or acceptance on y t for various hadron species varies greatly, leading to quitedifferent uncertainties. For pions q is poorly determinedbecause the spectrum terminates at 3 GeV/c near thetransition point from Gaussian to exponential tail. Gaus-sian parameters ¯ y t and σ y t are better determined and areconsistent with the trend for unidentified hadrons fromRef. [17], as are soft-component parameters ( T, n ).The charged-kaon data are similarly restricted on y t and cannot therefore compete with the neutral-kaon datawith their large acceptance p t ∈ [0 ,
7] GeV/c. Since spec-tra for the two species are reported to be statisticallyequivalent where they overlap (confirmed in the presentstudy) the parameters for charged kaons K ± are simplycopied from those for neutral kaons K S . The uncertaintyfor soft-component parameter n is large because the hardcomponent makes a large contribution to spectra, thusreducing sensitivity to the tail of the L´evy distribution.Because protons and Lambdas have similar massestheir spectra are expected to be similar in form. Their y t coverage is complementary in that Lambda data alsoextend to 7 GeV/c but proton data extend to lower p t .Those differences are reflected in the parameter uncer-tainties. ( T, n ) are determined by proton data whereas(¯ y t , σ y t ) are determined by Lambda data. The baryonhigh- y t tails drop much faster than those for mesons so q is substantially less certain (and larger) for baryons.In Sec. II A it is observed that the proton TCM doesnot describe spectrum data properly. The differenceis shown in Fig. 6 (b) and problems seem confined tothe spectrum hard component. As noted in Sec. IX Athe TCM is constrained such that parameters are notadjusted arbitrarily for individual collision systems orhadron species. The proton TCM provides a predic-tion from which the data deviate strongly. The devia-tion source is not evident but simple tracking inefficiencyseems unlikely: The soft component is unaffected [seeFig. 1 (e) below 1 GeV/c], and the suppression is exactlycentered on the hard-component peak [see Fig. 6 (b)]. D. Accuracy of PID spectrum parameters
Given the PID spectrum y t model parameters in Ta-ble III those related to species abundances for soft andhard components are presented in Table IV. As notedabove, those parameters are determined by spectrumproperties for y t → z h /z s . The accuracy of the parameter values is then re-flected in TCM descriptions of spectrum and ¯ p t data.Roughly speaking, z h /z s and z parameter values canbe determined to about 10% by comparing data to TCMsoft components as in Figs. 4 (left), 5 (c) and 6 (a). The accuracy of the resulting TCM is then demonstrated inthe right panels of those figures where, except for protons,the inferred data hard components agree in amplitudewith the unit-normal TCM models within the same 10%.Table IV also includes values for soft and hard en-semble means ¯ p tsi and ¯ p th i derived from TCM modelfunctions ˆ S i ( y t ) and ˆ H i ( y t ) according to parameter val-ues in Table III. When combined with hadron-species-independent geometry parameters x ( n s ) and ν ( n s ) inEq. (19) predictions for ensemble-mean ¯ p ti are produced.Figure 8 shows ¯ p t vs n ch data for pions, kaons, protonsand Lambdas from Ref. [2] (solid points) vs TCM trendsfrom Eq. (19) (solid curves). The agreement for mesons iswell within point-to-point uncertainties. The large errorbars for charged kaons are associated with extrapolationof limited spectrum data to y t → K ± and K S spectra are statistically equivalent thereal extrapolation uncertainty is negligible [see Fig. 5 (c)],and that is reflected in the close correspondence of dataand TCM(s) for the two kaon species.The situation with baryons is markedly different. Ex-trapolation uncertainties for protons should be much lessthan for Lambdas because of the lower spectrum cutoff[see Fig. 6 (a) and (c)], but proton hard components ap-pear to be systematically suppressed by about 40% onlynear the mode [see Fig. 6 (b)] as represented by the dash-dotted curve in Fig. 8. The baryon solid curves in Fig. 8are determined by Eq. (19) with fixed hard componentsand no suppression, but the systematic centroid shifts inFig. 6 (b) and (d) lead to displacement of ¯ p t data relativeto TCM trend from lower to higher with increasing n ch .The proton discrepancy illustrates the predictive powerof the TCM: it is not adjusted to accommodate individualcases. The TCM serves as a fixed reference system appli-cable to any A-B collision system. Anomalous behaviorcan then be detected and characterized accurately. E. Comparison with blast-wave model fits
In Ref. [2] the BW spectrum model applied to p -Pbcollisions is described as giving “the best description ofthe data over the full p T range,” referring to individualBW fits to each hadron species and centrality class, wherethe “full p T range” depends strongly on specific hadronspecies (see Fig. 1). However, BW parameters ( T kin , ¯ β t )for each centrality class as reported in Table 5 of Ref. [2]are obtained from simultaneous fits to all hadron species.For the simultaneous fits the actual p t range is deter-mined by the low- p t acceptance limit for each species asabove but by “agreement with the data at high p T ” (i.e.the p t upper limits are chosen based on fit quality – seehorizontal lines labeled BW in Figs. 4, 5 and 6). But χ is a measure of “agreement with the data,” so the χ val-ues in Table 5 simply reflect a process of data selection,are not indicative of the likelihood of the fit model. Notethat χ values for seven centrality classes are all less than1 with mean value 0.45, whereas the mean of the χ dis-8tribution per degree of freedom should be 1, with roughlyhalf the values expected to be above 1 for an acceptabledata description. The χ values in Table 5 then likelyreflect a strong bias resulting from data selection via p t cuts. Reference [2] offers the following comment on itsTable 5: “Positive and negative variations of the param-eters using the different [ p t ] fit ranges...are also reported.Variations of the fit range lead to large shifts ( ∼ χ is greatest for the most-peripheralcentrality class where the TCM soft component (whatshould be described by a hydro model if such a modelwere appropriate) is by far the dominant component com-pared to the jet contribution. For the most-central col-lisions (at least according to the Glauber model), wherejets clearly dominate, χ is lowest. That combination offeatures suggests that the BW model is unlikely to reflectactual hadron production mechanisms in p -Pb collisions.In contrast, the TCM is required to describe all avail-able data self-consistently with minimal parametrization.Section VI B illustrates the description quality. The devi-ations for proton data in Fig. 6 (b) emphasize the impor-tance of a fixed model in revealing data-model anomalies. X. DISCUSSION
As summarized in Sec. II, Ref. [2] interprets PID spec-trum data to suggest that “collectivity” (e.g. radial flow)is manifested in p -Pb collisions. The suggestion is basedon certain terminology, preferred analysis methods andargument by analogy. If correct, p -A or d -A data, ini-tially assumed to serve as a control for QGP discovery inA-A collisions, would no longer serve that purpose. p -Pb PID spectra are said to “flatten” or become“harder” with increasing n ch or p -Pb centrality, i.e. spec-trum slopes for lower p t are observed to decrease. Theslope changes exhibit “mass ordering,” i.e. the effects in-crease with increasing hadron mass. It is then argued byanalogy that since similar effects are observed in Pb-Pbcollisions, and since such effects observed in A-A colli-sions are conventionally interpreted to arise from radialflow as the “natural explanation,” the cause in p -Pb colli-sions must also be “collective” radial flow. In this sectionsuch arguments are confronted with alternative evidencederived in part from the present PID p -Pb study. A. Evidence for and against flows from p t spectra Evidence for “collectivity” (flows) has been reportedpreviously for p -Pb collisions at the LHC, including ap-parent indications of hydrodynamic flows [6, 56, 57].In A-A collisions the presence of a flowing dense QCDmedium has been associated with “jet quenching” [58].In Sec. II BW fits to PID spectrum data as reported inRef. [2] are briefly summarized. It is observed that BWparameters ¯ β t and T kin have similar values for p -Pb and Pb-Pb collisions. Those results are then interpreted as“consistent with the presence of radial flow in p -Pb col-lisions.” It is further noted that “a larger radial velocityin p -Pb collisions has been suggested as a consequence ofstronger radial gradients,” albeit within a smaller system.However, p -Pb PID spectrum data as presented in theTCM formats of Sec. VI B contradict such claims. Softcomponent ˆ S ( y t ) (left panels) for a range of p -Pb n ch remains consistent with isolated p - p collisions and witha common energy dependence extending over three or-ders of magnitude in √ s . The inferred spectrum hardcomponents (right panels) are approximately indepen-dent of p -Pb n ch and consistent with jet properties in-ferred from isolated p - p collisions over a large energy in-terval [17]. There is no significant evidence for radialflow (e.g. boosted y t spectra) or for modification of par-ton fragmentation to jets (i.e. FF modification [19]). Theargument for larger gradients in smaller systems impliesthat central densities must remain similar for all collisionsystems, but central densities must scale with participantnumber which should be proportional to system size.As mentioned in Sec. II, Ref. [2] offers a disclaimerabout radial flow interpretations based on BW fits tospectrum data. Given results from BW fits applied toPYTHIA p - p spectra it is concluded that the PYTHIACR mechanism produces results that “...can mimic theeffects of radial flow.” The single disclaimer is juxta-posed with a number of positive statements regardingradial flow based on BW fits and on argument by anal-ogy with A-A spectra, as summarized in Sec. II, and isnot mentioned in the paper summary. A more importantomission is lack of acknowledgement of the contributionof minimum-bias jets to spectrum data with a peak near p t = 1 GeV/c that lies well within any BW fitting inter-val (see left panels in Sec. VI B). In the context of BWfits the large jet contribution can certainly mimic the ef-fects of radial flow, and thus demonstrates that the BWmodel is inappropriate for p t spectrum description. B. ¯p t trends vs flow interpretations Figure 8 shows PID ¯ p t data reported in Ref. [2] (solidpoints) and corresponding TCM trends (solid curves) for5 TeV p -Pb collisions. ¯ p t data certainly reflect evolving p t spectrum structure, but with greater statistical pre-cision due to integration. It is notable that the mostrapid increase of ¯ p t occurs for lowest n ch . For unidenti-fied hadrons from 5 TeV p - p and p -Pb collisions the two¯ p t trends are identical up to ¯ ρ ≈
20 (i.e. four times theNSD value 5) and it is likely that PID data for p - p colli-sions also agree with those for p -Pb within that interval.¯ p t values do increase strongly with hadron mass, but forlarger n ch the overall ¯ p t trend is strong decrease withincreasing system size from p - p to p -Pb to Pb-Pb [16].Several questions arise concerning flow interpretationsof ¯ p t data: In the context of a flowing dense mediumwhere is the transition (e.g. on n ch ) from free-flying par-9ticles (no inertial confinement) to rescattering within adense medium (confinement increasing with densities andsystem volume)? For instance, why doesn’t ¯ p t increase more rapidly for larger n ch or more rapidly in Pb-Pb(than in p - p ) where inertial confinement is more likely?In a flow context it is expected that ¯ p t increases morerapidly for more-massive hadrons because within a com-mon velocity field particle momentum should increasewith hadron mass. But that implies soft components of p t spectra should be boosted to higher y t proportional tohadron mass. Why is a common boost not observed forthe PID spectra in Sec. 8 [ ˆ S ( y t ) shapes do depend onmass, but all proceed from y t = 0, not a boosted value]?TCM analysis of p t spectra and ¯ p t data provide an-swers. Based on differential spectrum structure as shownin Sec. VI B ¯ p t increases with n ch and hadron mass be-cause of the large MB dijet contribution to hadron pro-duction, while soft components remain invariant . Morejet-related hadrons are produced but with the same hard-component y t distributions, and there are copious jet-related baryons (large z h /z s values in Table IV) as de-termined experimentally. There is no indication of a flowcomponent in PID spectra. The inverse hierarchy p - p >p -Pb > Pb-Pb for ¯ p t data is explained in terms of typi-cal ¯ ρ sNN values for the three collision systems that havethe same ordering and the relation ¯ ρ hNN ∝ ¯ ρ sNN thatdetermines MB jet production with, at least in p - p and p -A systems, no jet modification. C. Preferred analysis methods
Certain analysis methods, statistical measures andplotting formats have been conventionally preferred overa range of alternatives in the analysis of high-energy par-ticle data. p t spectra are typically modeled and inter-preted with the implicit assumption that they are mono-lithic (a single component), represented then by a sin-gle functional form. Examples include the “power law”model for p - p spectra from the S p ¯ p S (jets plus soft pro-cess) [59], Tsallis model (similar to the TCM soft com-ponent) [60] and the blast-wave model for A-A spectra(locally-thermalized flowing medium) [8]. Other choicesinclude linear p t as an independent variable rather than alogarithmic scale and the use of spectrum ratios to eval-uate fit quality and relations among hadron species.In Ref. [2] the BW spectrum model is applied to indi-vidual BW fits for each hadron species and p -Pb central-ity class. The BW results are described as “...the bestdescriptions of data over the full p T range” (presumablythe data ranges subtended by the various species). It isnoted that for equivalent ¯ ρ values ¯ β t for p -Pb is “sig-nificantly higher” than for Pb-Pb (e.g. compare Figs. 11and 14 of Ref. [16]), leading to speculation that “strongerradial gradients” may arise in p -Pb collisions. The sameargument applied to p - p ¯ p t data must conclude that suchgradients are highest in the smallest collision system .Among alternative possibilities are treatment of spec- tra as possibly composite, plotting spectra vs logarith-mic variable y t and differential study of individual spec-tra rather than their ratios, the goal being to extract asmuch information as possible from particle data. Com-parison of the same data as plotted in Sec. II vs Sec. VI Billustrates the major differences in visually accessible in-formation derived from alternative plotting formats.The TCM results in Sec. VI B provide a very differentpicture of mechanisms relevant to p -Pb collisions. In-stead of a monolithic model function with BW parame-ters varying strongly with collision conditions the TCMsoft and hard model functions are approximately invari-ant with p -Pb centrality, greatly simplifying the model.Because the TCM hard component is quantitatively re-lated to measured jet properties the ¯ p t trends attributedto radial flow in the BW context are instead related tojet physics in the TCM context. (The logarithmic y t ra-pidity variable allows greatly improved access to low- p t data where the great majority of jet fragments resides.)That jet production per final-state hadron (e.g. TCM ra-tio ¯ ρ h / ¯ ρ s ) might be largest in the smallest collision sys-tem (hence driving up ¯ p t values) is easily explained interms of conventional jet physics and the quadratic rela-tion ¯ ρ h ∝ ¯ ρ s consistent with measured jet cross sections . p -Pb centrality determination based on the Glaubermodel [18], as opposed to direct inference from ¯ p t data [15], strongly limits information available from par-ticle data. In Fig. 8 PID ¯ p t data from Ref. [2] extendonly to ¯ ρ ≈
45, assigned by the Glauber analysis to 0-5% centrality, whereas high-statistics ¯ p t data extend to¯ ρ ≈
116 corresponding to in dijet pro-duction. Claims for collectivity or flows in p -Pb colli-sions could be much better tested with PID spectra fromhigher charge densities and associated jet production. D. Argument by analogy
Reference [2], arguing by analogy, suggests that com-monalities between p -Pb spectrum and ¯ p t data and thosefrom Pb-Pb collisions imply the presence of collectiveflow in p -Pb collisions: “In heavy-ion collisions, the flat-tening of transverse momentum distribution and its massordering [i.e. relating to variation with n ch ] find their nat-ural explanation in the collective radial expansion of thesystem.” and “The [ p -Pb] transverse momentum distri-butions show a clear evolution with multiplicity, similarto the pattern observed in high-energy pp and heavy-ioncollisions, where in the latter case the effect is usuallyattributed to collective radial expansion.”That a particular data characteristic might appear inPb-Pb and p -Pb data does not require similar causes,and attribution of certain trends in A-A collisions to flowphenomena or quark recombination may be questioned.Spectrum and yield ratios tend to discard a substantialfraction of the information carried by particle data [10]and therefore cannot be relied on for definitive modeltests. The same comments apply to fit models: Applica-0tion of a flow-related spectrum model to spectrum datadoes not demonstrate the presence of flow in the corre-sponding collision system, and differential spectrum de-tails (e.g. the TCM applied to PID hadron spectra as inthe present study) may exclude a flow interpretation. E. Abandoning control experiments
In preparation for initial RHIC operations data from d -Au collisions were promoted as a control experiment forpossible discovery of QGP in more-central Au-Au colli-sions (e.g. Ref. [61]). Whereas a phase transition to aQGP was expected to require large energy and particledensities d -Au collisions should represent “cold nuclearmatter” and no transition. The apparent contrast be-tween Au-Au and d -Au data (e.g. apparent presence orabsence of jet quenching) was then cited as confirmingdiscovery [3]. But recent observations in p -Pb [4] andeven p - p [5] data from the LHC of certain characteristics(e.g. one or more ridge structures, p t spectrum “harden-ing” with increased centrality or hadron mass) conven-tionally associated with “collective” behavior (i.e. flows)in A-A collisions have led to speculation that flows andQGP might be possible even in the smallest systems, al-beit corresponding to higher energies and charge densi-ties [6].Such reasoning betrays the function of a control exper-iment. Theoretical understanding of QCD and a conjec-tured phase transition led to a prediction: QGP may bepossible in the largest A-A collision systems but not pos-sible in low-density smaller systems. If phenomenon X isobserved in A-A but not in p -A it is a candidate QGPindicator (e.g. elliptic flow, jet quenching). But if X isobserved to a significant extent in both large and smallcollision systems it is unlikely to be associated with QGPformation and flows according to the original theoreticalexpectation . Concluding that QGP must be a universalmanifestation even in small low-density systems betraysthe role of the control experiment: In essence, theory ismodified to accommodate data and is then not falsifiable.For example, a nonjet quadrupole feature [i.e. v ∼ cos(2 φ )] is accurately inferred for 2D angular correlationson ( η, φ ) from 200 GeV p - p collisions [11]. Arguing byanalogy with the v interpretation in A-A collisions onemight then conclude that elliptic flow must appear inthe smallest collision system. But p - p collisions are in-tended to serve as a control experiment for claims of jetquenching and flows in A-A collisions, for instance in theform of spectrum ratio R AA . In fact, systematics of thequadrupole component in p - p (e.g. its n ch dependence)make a flow interpretation unlikely there [11], castingdoubt on flow interpretations of v data in A-A collisions.Similarly, interpretation (again by analogy) of certain p t spectrum properties in terms of radial flow in p -Pbcollisions (as a nominal control experiment) are falsi-fied by the PID spectrum analysis of the present study.Significant radial flow (i.e. corresponding to a boosted hadron source) should result in substantial suppressionof spectra near zero momentum to complement enhance-ment (“hardening”) at higher momentum. However, thespectra from Ref. [2] are accurately and exhaustively de-scribed by a TCM including an invariant soft component(i.e. consistent with a L´evy distribution) and a hard com-ponent quantitatively related to jet production. That isparticularly evident for the K data in Fig. 5 (c).This p -Pb result is consistent with previous analysis of200 GeV Au-Au collisions [13] which concluded ”Since1/3 of the hadrons originate effectively from rapidly-moving sources (parton fragmentation), the significanceof statistical-model spectrum measures (chemical poten-tials, decoupling temperatures, (cid:104) p t (cid:105) s, [ ¯ β t ]) attributed toan expanding bulk medium can be strongly questioned.Upon close examination of pion and proton spectra no[evidence for] identifiable radial flow is apparent.”Control experiments based on differential analysis ofall available information reveal that certain data featuresappearing in A-A and even p -A or d -A collisions, previ-ously interpreted in terms of flows and QGP, are morelikely manifestations of MB jets [10].Problems raised by claims for collectivity in small sys-tems (i.e. control experiments) have been recognized [7]:“With decreasing system size, one may expect a transi-tion from collective behavior to free streaming on scaleswhere the mean free path...of medium [QGP, perfectliquid] constituents becomes comparable to the systemsize.... Experimentally, however, no indications for theexistence of such an onset of collective behavior withsystem size have been identified so far [see (a) below].Signatures of collectivity such as higher-cumulant flowharmonics [see (b) below] display a remarkably weak de-pendence on system size. Understanding the apparentabsence of a minimal scale for collective behavior is acentral problem of the field for which qualitatively differ-ent solutions remain to be explored in more detail....” (a)Experimental evidence for a “sharp transition” (on cen-trality) of jet modification within Au-Au collisions (fromno jet quenching to strong jet quenching) has been es-tablished previously [14], whereas no jet modification isobserved in p - p [11] or p -Pb (the present study) collisions,and (b) emphasis on what are denoted “flow [higher] har-monics” and their interpretation as indicators of “collec-tivity” (flows) may be strongly questioned [52, 62]. Thepresent TCM study indicates no significant evidence forradial flow in p -Pb PID hadron spectra (see Sec. X A). An illustration from recent hydro theory descriptions may befound in Fig. 16 (right) of Ref. [41]. The hydro curve (bold solid)is strongly suppressed at lower p t as the model accommodatesthe strong jet contribution at higher p t . F. Collectivity inferred from angular correlations
While the present study focuses on p -Pb p t spectrumproperties for PID hadrons, arguments for manifestationsof “collectivity” or flows in small systems have mainlybeen based on one or two “ridge” features in 2D angularcorrelations [4, 5]. However, the n ch trends for thosefeatures rule out a flow interpretation as argued here.In Ref. [11] 2D angular correlations from high-statistics200 GeV p - p collisions are studied. The TCM for n ch de-pendence of p t spectra is first confirmed and correspondsto 100-fold increase of dijet production over the n ch in-terval spanned by the p - p data. Model fits to 2D angularcorrelations then reveal three components: (a) the TCMsoft component manifested as a 1D peak on pseudora-pidity η centered at η = 0 (charge-neutral pairs only);(b) the TCM hard component consisting of a same-side( φ = 0) 2D peak on ( η, φ ) centered also at η = 0 and anaway-side ( φ = π ) 1D peak on φ ; and (c) a quadrupolecos(2 φ ) component on φ representing a third component.Component (c) should be distinguished from a contribu-tion to the total quadrupole moment from the 2D jetpeak and is thus referred to as the nonjet quadrupole.The three correlation components have distinct depen-dences on charge-density soft component ¯ ρ s employed asthe system control parameter. As for p t spectra the soft-component amplitude (a) varies as ∝ ¯ ρ s and the hard-component amplitude (b) varies as ∝ ¯ ρ s , where “ampli-tude” here refers to the number of correlated pairs asso-ciated with a given component. Further analysis revealsthat the nonjet quadrupole component (c) varies as ∝ ¯ ρ s .Those trends are followed precisely over the full n ch range of data. Thus, as dijet production increases 100-fold the quadrupole amplitude increases 1000-fold andbecomes clearly visible in 2D angular correlations for thelarger n ch values. As a result of the increasing two -lobed(maxima at 0 and π ) quadrupole contribution the same-side ( φ = 0) curvature on φ (excluding the 2D jet peak) changes sign to become the apparently-single “ridge”usually referred to [5], while the away-side ( φ = π ) curva-ture increases in magnitude although that aspect is typi-cally ignored. The full two-lobe quadrupole structure forLHC data was only recently inferred from p -Pb data [4],based on visual examination of 2D angular correlationsrather than precision model fits as in Ref. [11].The observed ∝ ¯ ρ s trend for the p - p nonjet quadrupoleleads to several important conclusions: (a) Jet-relatedand quadrupole trends are clearly distinct. (b) The cubictrend is followed precisely over the full n ch range for datadown to small charge densities. A flowing dense medium(collectivity) interpretation is therefore quite unlikely.(c) The cubic trend can be recast as ∝ N part N bin interms of number of participant low- x gluons N part ∝ ¯ ρ s and their binary collisions to produce dijets N bin ∝ ¯ ρ s .The ∝ N part N bin quadrupole trend for p - p collisionscan be compared with the ∝ N part N bin (cid:15) opt trend forAu-Au collisions as determined in Ref. [37], where in thelatter case N part and N bin refer to number of nucleon N participants and N-N binary collisions. As for p - p col-lisions the same n ch trend is followed precisely by thenonjet quadrupole over the complete centrality range forAu-Au collisions. The same trend is observed for LHCPb-Pb collisions [37]. There is no significant correlationwith the strongly-varying modification of jet formation(jet quenching) [14] attributed to energy loss in a densemedium, again making a flow interpretation unlikely.For p - p collisions there is no eccentricity dependence(i.e. no (cid:15) opt factor as appears for A-A collisions [37]).That observation is consistent with a conclusion from p t spectrum data that centrality plays no role in p - p collisions [11, 12, 63, 64]. Each p - p (or N-N) collisionrequires full transverse overlap [51]. It is possible thatthe ∝ N part N bin ∝ ¯ ρ s trend for the p - p quadrupole cor-responds to a three-gluon interaction resulting in QCDquadrupole radiation [65]. The ∝ N part N bin (cid:15) opt trend forA-A collisions then suggests that the same three-gluon in-teraction may be responsible as well, but that the overallA-A geometry must be eccentric to produce a significant net quadrupole component from the composite system.The present TCM study of PID p t spectra from p -Pbcollisions confirms that hadron production is dominatedby TCM soft and hard components. Evidence from 2Dangular correlations reveals that the nonjet quadrupole isa distinct third component and must therefore be “car-ried” by a small minority (i.e. few percent) of hadronseven for large n ch . That conclusion is consistent witha study of A-A quadrupole trends [37] but inconsis-tent with a hydro/QGP narrative in which the nonjetquadrupole is carried by almost all hadrons and mosthadrons emerge from a flowing dense medium. In anycase, evidence from 2D angular correlations corroboratesconclusions derived from p t spectrum data as in thepresent study: collectivity (i.e. flows) in small collisionsystems is unlikely, and the same features interpreted toindicate collectivity or flows in small systems may alsobe misinterpreted as such in more-central A-A collisions. XI. SUMMARY
Certain data features from small collision systems ( p - p and p -Pb collisions) at the large hadron collider (LHC),e.g. properties of p t spectrum and 2D angular correlationdata, have been interpreted recently as demonstratingthe presence of “collectivity” (hydrodynamic flows) evenfor low charge densities. Such interpretations are basedon arguments by analogy, that similar data features inA-A collisions are interpreted to indicate formation of aflowing dense medium or quark-gluon plasma (QGP).Such findings reverse the intended role of small col-lision systems as control experiments based on the as-sumption that for low-enough particle and energy den-sities free particle streaming should be inconsistent witha hydrodynamic flow description. Claimed evidence forQGP formation in A-A should then not appear in a con-trol collision system representing “cold nuclear matter.”2The inference of collectivity in small systems impliesthat there is no threshold point for the onset of collectiv-ity (and possible QGP formation). Absence of a densitythreshold for collectivity or flows has been recognized asa central problem for interpretation of high-energy parti-cle data. One suggested approach is development of newtheoretical mechanisms to account for the lack of such atransition, but an alternative approach is reexaminationof conventional interpretations of certain data featuresas demonstrating collectivity, thus challenging the basicconcept of collectivity or flows in any collision system.To that end the present study is a differential analysisof identified-hadron (PID) p t spectra from 5 TeV p -Pbcollisions for seven centrality or n ch classes. The PIDspectra are described by a two-component (soft + hard)model (TCM) of hadron production mechanisms. Thesoft component is associated with longitudinal projectile-nucleon dissociation while the hard component is associ-ated with large-angle scattering of low- x partons (gluons)to form jets. p -Pb centrality is adopted from a previousTCM study of p -Pb ensemble-mean ¯ p t data for uniden-tified hadrons. Questions to be addressed include: (a)Is spectrum evolution with n ch consistent with expec-tations for effects of radial flow (soft)? and (b) Is jetformation modified by parton interactions with a denseflowing medium (hard)? In a previous study it was ob-served that ¯ p t data for peripheral p -Pb are equivalent tothose for p - p over a significant n ch interval, thus provid-ing information on both systems from the p -Pb data.The main analysis results are as follows: (a) The PIDspectrum TCM provides an accurate description of allspectrum and ¯ p t data with one exception – protons –where the spectrum hard component is found to be sup-pressed by about 40%. (b) The TCM provides access tounprecedented details of spectrum structure and system-atic variation with n ch . (c) Spectrum variations with n ch and hadron mass that have been attributed to radial floware actually jet manifestations represented by the spec-trum hard component. (d) Approximate invariance ofPID spectrum hard components with n ch demonstratesthat the jet component of hadron production remains un-modified over a large range of hadron and jet densities.Precise distinction between TCM soft and hard compo-nents is based on a quadratic hard-vs-soft relation that isalready evident for measured eventwise-reconstructed-jetcross sections. PID spectrum soft components, includingthe majority of produced hadrons, retain fixed shapesindependent of n ch or p -Pb centrality and show no evi-dence for a common source boost that would indicate ra-dial flow. The dominant contribution to the TCM hardcomponent is from lowest-energy jets near 3 GeV whichshould be most sensitive to a dense flowing medium if oneexisted. PID spectrum data indicate that p -Pb collisionsfor any n ch are linear superpositions of p -N collisions.It could be argued that appearance of a ridge or ridgesin 2D angular correlations from small systems may stillindicate some form of medium formation and collectiv-ity. Angular correlations are outside the scope of the present study, but evidence from other studies suggeststhat the presence of a nonjet quadrupole in p - p and p -Acollisions is the manifestation of a QCD process basedon direct few-gluon interactions, not medium formation.Observation of a ridge structure in p -Pb collisions wherespectrum structure precludes a dense medium may serveto buttress such an interpretation. The role of small sys-tems as control experiments would then be restored. Appendix A: ¯p t TCM for p-p collisions
The present study of PID hadron spectra from 5 TeV p -Pb collisions relies critically on p -Pb centrality deter-mination inferred from ¯ p t data for unidentified hadronsas reported in Ref. [15]. This appendix briefly reviewsbasic ¯ p t TCM analysis for elementary p - p collisions.Reference [50] established that a TCM for ratio ¯ p t =¯ P t /n ch constitutes a good description of LHC data from p - p and Pb-Pb collisions at several energies and providedhints as to the mechanism of p -Pb ¯ p t variation – evolv-ing from a p - p or p -N trend for smaller n ch to a quan-titatively different but similar trend above a transitionpoint. Reference [17] presented a detailed TCM anal-ysis of p - p p t spectra for a range of energies from 17GeV to 13 TeV. Soft component ˆ S ( m t , √ s ) varies weaklywith energy and not at all with n ch , but hard compo-nent ˆ H ( y t , n s , √ s ) varies strongly with energy (consis-tent with variation of the underlying jet spectrum) andsignificantly with n ch (as established with 200 GeV spec-tra). Those new spectrum results were then incorporatedwithin a revised ¯ p t analysis in Ref. [16], summarized for p - p here and for p -Pb in Sec. V C.Quantities ¯ p th ( n s , √ s ), α ( √ s ) and an effective detec-tor acceptance ratio ξ are used to update results fromRef. [50]. The TCM for charge densities averaged oversome angular acceptance ∆ η (e.g. 0.6 for Ref. [42]) is¯ ρ = ¯ ρ s + ¯ ρ h (A1)= ¯ ρ s [1 + x ( n s )] , ¯ ρ (cid:48) ¯ ρ s = n (cid:48) ch n s = ξ + x ( n s ) , where x ( n s ) ≡ ¯ ρ h / ¯ ρ s ≈ α ¯ ρ s is the ratio of hard-component to soft-component yields [12] and α ( √ s ) isdefined in Ref. [17]. Primes denote uncorrected quan-tities. The TCM for ensemble-mean total p t integratedover some angular acceptance ∆ η from p - p collisions forgiven ( n ch , √ s ) is expressed as¯ P t = ¯ P ts + ¯ P th (A2)= n s ¯ p ts + n h ¯ p th . The conventional intensive ratio of extensive quantities¯ P (cid:48) t n (cid:48) ch ≡ ¯ p (cid:48) t ≈ ¯ p ts + x ( n s )¯ p th ( n s ) ξ + x ( n s ) (A3)3(assuming ¯ P (cid:48) t ≈ ¯ P t [16]) in effect partially cancels MBdijet manifestations represented by ratio x ( n s ). The al-ternative ratio n (cid:48) ch n s ¯ p (cid:48) t ≈ ¯ P t n s = ¯ p ts + x ( n s )¯ p th ( n s ) (A4)= ¯ p ts + α ( √ s ) ¯ ρ s ¯ p th ( n s , √ s )preserves the simplicity of Eq. (A2) and provides a con-venient basis for testing the TCM hypothesis precisely.Figure 11 (left) shows ¯ p t data for four p - p collision ener-gies from the RHIC (solid triangles [12]), the Sp¯pS (openboxes [66]) and the LHC (upper points [42]) increasingmonotonically with charge density ¯ ρ = n ch / ∆ η . Thelower points and curves correspond to full p t acceptance.For acceptance extending down to zero ( ξ = 1), ¯ p (cid:48) t → ¯ p t in Eq. (A3) should vary between the universal lower limit¯ p ts ≈ . n ch = 0) and ¯ p th ( n ch → ∞ ) as lim-iting cases. For a lower p t cut p t,cut > p (cid:48) ts = ¯ p ts /ξ (dotted lines) and the data are systemati-cally shifted upward (upper points and curves). The solidcurves represent the p - p ¯ p t TCM from Ref. [16]. Notethat the 7 TeV ¯ p t data extend to ¯ ρ ≈
10 ¯ ρ NSD ≈ p - p collision events. n ch / Dh ¯ p t ¢ ( G e V / c ) p-p ts ¯p ts ¢ n s / Dh ( n c h ¢ / n s ) ¯ p t ¢ ( G e V / c ) p-p ts FIG. 11: Left: ¯ p t vs n ch for several collision energies. Theupper group of points from Ref. [42] are derived from particledata with a lower p t cutoff. The lower 900 GeV data fromRef. [66] and 200 GeV data from Ref. [12] are extrapolated tozero p t . Right: Data from the left panel multiplied by factor n (cid:48) ch /n s that removes the jet contribution and the effect of thelow- p t cut on the soft component from the denominator of ¯ p t . Figure 11 (right) shows data on the left transformedvia Eq. (A4) to ( n (cid:48) ch /n s )¯ p (cid:48) t ≈ ¯ P t /n s (points). The TCMcurves undergo the same transformation and the slopes ofthe resulting straight lines are α ( √ s )¯ p th ( √ s ). The datadeviate significantly from the straight-line TCM becauseof systematic variation with n ch of the p t spectrum hard-component shape as reported in Refs. [16, 17]. However,those details are beyond the scope of this spectrum study.The success of the p - p ¯ p t TCM confirms that variationof p - p ¯ p t is dominated by jet fragments from large-angle-scattered low- x gluons. The hard yield or angular den-sity ¯ ρ h ≈ α ( √ s ) ¯ ρ s represents the dijet fragment densitydetermined precisely by soft component ¯ ρ s . The p t spec-trum TCM hard component and underlying jet energyspectrum evolve according to the same rules [53]. Thequadratic relation ¯ ρ h ∝ ¯ ρ s implies that p - p collisions are noneikonal (compared to the eikonal trend ¯ ρ h ∝ ¯ ρ / s ).The quadratic trend (each participant gluon in one pro-ton can interact with any participant gluon in the partnerproton) implies that p - p collisions with large n ch are veryjetty. Reference [16] demonstrates a direct connectionbetween ¯ p t hard component ¯ p th ( n s ), p t spectrum hardcomponent H ( p t , n s ) [17] and jet spectra as in Ref. 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