Manifestations of minimum-bias dijets in high-energy nuclear collisions
Thomas A. Trainor

































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    åßp_t⟨ p_t ⟩p̅_t2̌v_2y_tm_tη_Δϕ_ΔΔρ / √(ρ_ref)y_t × y_tp_t × p_terfdE_t/dηE_t⟨ E_t ⟩⟨ e_t ⟩p-pn_chVersion 1.2CENPA 354290, University of Washington, Seattle, Washington 98195
    Dijets observed near midrapidity in high-energy nuclear collisions result from large-angle scattering of low-x partons (gluons) within projectile hadrons as a signature manifestation of QCD. Within the same collisions it has been claimed that hydrodynamic flows (radial, elliptic and “higher harmonic” flows) carried by a dense QCD medium or quark-gluon plasma (QGP) dominate the observed hadronic final state. The flow-QGP narrative is imposed   a priori on primary particle data, and of all possible analysis methods  a subset A that seems to support that narrative is preferred. The present study explores an alternative minimum-bias (MB) jet narrative – quantitative correspondence of MB dijet manifestations in the hadronic final state with measured  isolated jet properties. The latter incorporates a different set of methods B that emerge from inductive study of primary particle data without  a priori assumptions. The  resulting system of methods and data manifestations is represented by a two-component (soft + hard) model (TCM) of hadron production. A survey of methods reveals that type A tends to discard substantial information carried by primary particle data whereas type B retains almost all information in both primary particle data from nuclear collisions and from isolated jets. The main goal of the present study is a review of MB dijet  contributions to high-energy collisions in small and large systems relative to measured isolated-jet properties. Representative analysis methods from types A and B are compared in the context of MB jet manifestations. This study suggests that at least some data features commonly attributed to flows actually result from MB dijets and thereby challenges the flow-QGP narrative.
    12.38.Qk, 13.87.Fh, 25.75.Ag, 25.75.Bh, 25.75.Ld, 25.75.NqManifestations of minimum-bias dijets in high-energy nuclear collisions
    
        Thomas A. Trainor
        December 30, 2023
    ========================================================================
    
    
     
    
    § INTRODUCTION
    
    
    This study reviews manifestations of minimum-bias (MB) dijets in the hadronic final state of high-energy nuclear collisions in the context of claimed collectivity (flows) in the same systems. There is widespread belief that hydrodynamic flows play a dominant role in high-energy collisions wherein a dense medium (quark-gluon plasma or QGP) is formed that supports a collective velocity field manifested by features of hadron distributions <cit.>. However, it is possible that at least some data features attributed to flows may relate to MB dijets <cit.>. In order to clarify such ambiguities MB dijets should be understood in ,  and å collisions in relation to eventwise-reconstructed (isolated) jets derived independently from  and  collisions over a broad range of collision energies. That is the main goal of this study.
    
    Flows and QGP  are intimately related by the assumption that a dense, strongly-interacting medium developed during nucleus-nucleus (å) collisions should respond to initial-state energy- and matter-density gradients by developing a velocity field (various flows) whose consequences may be observed in the hadronic final state <cit.>. Observation of flow manifestations and a causal relation to å initial-state geometry is sought  a priori and interpreted to imply that a QGP has been established <cit.>.
    
    Flow-QGP claims then rely on assignment of certain data features to flows and may refer to others as “nonflow” without further elaboration <cit.>. The flow-QGP narrative forms the basis for other data analysis and interpretation, and MB jet contributions may be minimized by preferred analysis methods and interpretation strategies. 
    
    The flow narrative is related to data by an assumption that most hadrons emerge from “freezeout” of a locally-thermalized, flowing dense medium <cit.>.  Hadron  is divided into several intervals with specific physical interpretations: thermalization and flows for p_t < 2 GeV/c <cit.>, high- jet phenomena for p_t > 5 GeV/c <cit.> and an intermediate region where production mechanisms are debated <cit.>. Such assumptions provide a preferred context for analysis and interpretation of high-energy data.
    
    For example, transverse-momentum  spectra may be fitted with a monolithic model function interpreted to reflect a thermodynamic context and to measure radial flow <cit.>. Two-dimensional (2D) angular correlations are projected onto periodic 1D azimuth ϕ and represented by Fourier series wherein each Fourier term is interpreted to represent a type of transverse flow <cit.>. Charge and  fluctuations are addressed with statistical measures motivated by thermodynamic assumptions including some degree of local thermalization of a bulk medium. Intensive ratios or  ratios of ratios are preferred over extensive measures of collision observables such as integrated charge multiplicity  or integrated P_t within some angular domain <cit.>. Jet contributions are acknowledged only within restricted  intervals including a small fraction of all hadrons, and  assumed jet properties are based on conjecture rather than actual jet measurements <cit.>.
    
    In contrast, the properties of eventwise-reconstructed dijets, their fragment momentum distributions (fragmentation functions or FFs) <cit.> and jet (leading-parton) energy spectra <cit.> measured over thirty years predict quantitatively certain manifestations of MB dijets that appear in high-energy collision data <cit.>. Predictions from  isolated-jet measurements [as opposed to perturbative QCD (pQCD) theory with its limitations] are inconsistent with much of the flow narrative and its presumed basis in measurement as discussed below.
    
    In  this study I examine the process of measure design in several critical areas and compare competing analysis methods. I demonstrate how measured properties of isolated jets predict certain data features from nuclear collisions corresponding to MB dijets and how analysis methods motivated by the flow narrative may lead to attribution of the same features to flows. I show how a two-component (soft + hard) model (TCM) of hadron production in high-energy collisions emerges naturally from inductive analysis of  spectrum and correlation data, is not imposed  a priori, and how the TCM is manifested in other contexts. A number of examples are presented. I conclude that when data features are reexamined in the context of isolated-jet measurements little substantial evidence remains to support the flow narrative.
    
    
    This article is arranged as follows:
    Section <ref> discusses preferred analysis methods.
    Section <ref> introduces a two-component spectrum model for  collisions.
    Section <ref> reviews the measured properties of reconstructed jets.
    Section <ref> describes MB jet contributions to  spectra.
    Section <ref> presents MB jet contributions to å spectra.
    Section <ref> reviews MB jet contributions to two-particle charge correlations.
    Section <ref> presents MB jet contributions to  fluctuations and  angular correlations.
    Sections <ref> and <ref>  present discussion and summary.
    
    
    
     
    
    § PREFERRED ANALYSIS METHODS
    
    
    For some aspects of data analysis alternative methods may give significantly different results and support different physical interpretations. The overall interpretation of high-energy nuclear collisions then depends on a sequence of method choices. How should such choices be made to establish an overall result that best reflects reality? One possible criterion is the fraction of information carried by primary particle data that is retained by a method for hypothesis testing. MB dijet manifestations in nuclear collisions compared to measured properties of isolated jets may provide a basis for evaluation.
    
    
    
     §.§ Primary and secondary observables
    
    
    All analysis methods are based on primary particle data. A charged-particle detector (e.g. time-projection chamber) determines the primary hadron single-particle (SP) observables in high-energy nuclear collisions: transverse momentum , pseudorapidity η, azimuth angle ϕ, charge sign and possibly hadron species via particle ID, in which case η→ y_z (longitudinal rapidity) and  p_t → m_t = √(p_t^2 + m_h^2) (transverse mass) with m_h a hadron mass. Transverse rapidity y_t ≡ln[(p_t + m_t)/m_h] provides superior visual access to SP spectrum structure at lower . For unidentified hadrons the pion mass may be assumed. SP densities are defined on  as  spectra and on (η,ϕ) as angular densities. 
    
    Two-particle (pair) densities defined on 6D momentum space (p_t1,η_1,ϕ_1,p_t2,η_2,ϕ_2) or a subspace may reveal certain pair correlations identified with physical mechanisms. 2D pair densities on (x_1,x_2) may be  projected by averaging onto difference variables x_Δ = x_1 - x_2 to obtain a  joint angular autocorrelation on the reduced space (p_t1,p_t2,η_Δ,ϕ_Δ) <cit.> which can be further reduced by integration over  bins or the entire  acceptance.  In discussing 2D angular correlations it is convenient to separate azimuth difference ϕ_Δ into two intervals:  same-side (SS, |ϕ_Δ| < π / 2) and  away-side (AS, |ϕ_Δ - π| < π / 2).
    
    Charge multiplicity and particle  may be integrated over some angular acceptance (Δη,Δϕ) or multiple bins within an angular acceptance to obtain  and P_t as  extensive eventwise random variables (RVs) whose fluctuations may be of interest <cit.>. Uncorrected (observed) charge multiplicities denoted by n̂_ch are relevant to Poisson statistics, for instance in determining  void probabilities defined below.  then denotes corrected values. 
    
    Secondary observables may be defined as combinations of primary observables, for instance eventwise mean ⟨ p_t ⟩ = P_t / n_ch as an  intensive RV <cit.>, event-ensemble-mean p̅_t = P̅_t / n̅_ch characterizing a collision system <cit.>, spectrum ratio R_AA as the ratio of a central å  spectrum to a  spectrum <cit.>, and v_2(p_t) also as a ratio of distinct hadron spectra <cit.>. Fluctuation and pair-correlation measures (variances and covariances) may be combined with other statistics in sums, differences or ratios to define secondary statistics. SP and pair momentum spaces may be partitioned (possibly based on  a priori assumptions), for instance defining certain  intervals within which specific physical mechanisms are expected to dominate collision data according to some narrative. 
    
    
    
     §.§ Competing analysis methods
    
    
    Analysis methods are generally not unique. A specific combination of methods contributing to a published analysis may comprise a subset of available methods determined by a sequence of choices among alternatives, possibly guided by a preferred narrative.  In a given context (e.g. SP spectra or pair angular correlations) alternative selections may lead to significantly different physical interpretations of collision data. 
    
    For instance, each of extensive measures P_t and n_ch or their ensemble means P̅_t and n̅_ch may reveal certain data trends inconsistent with a temperature hypothesis but supporting an alternative hypothesis whereas fluctuations of intensive eventwise ⟨ p_t ⟩ or systematics of ensemble p̅_t may be seen as reflecting local temperature variations of a conjectured bulk medium. Critical extensive trends may be suppressed by cancellations within intensive ratios.
    
    Hadron pair correlations from high-energy nuclear collisions projected onto 1D azimuth exhibit strong nonuniformities (literally “azimuthal anisotropy”) that may originate from several physical mechanisms including MB dijets. However, the term “anisotropic flow” is commonly interpreted as synonymous with azimuthal anisotropy <cit.>. Any distribution on periodic azimuth, no matter what its physical origins, can be described exactly by a Fourier series (FS). The assertion “The second Fourier coefficient of the azimuthal asymmetry [i.e. anisotropy] is called elliptic flow” <cit.> reflects a common assumption. Alternative modeling of azimuth distributions may favor a different physical interpretation.
    
    This study emphasizes manifestations of MB dijets from high-energy nuclear collisions within several contexts (e.g. yields, spectra, correlations, statistical fluctuations) and their relation to selection of specific analysis methods: How are MB dijets, consistent with measured isolated-jet properties, revealed or concealed by method choices, and what criteria would insure conscious and unbiased choices that lead to meaningful interpretations?
    
    
    
     
    
    § TWO-COMPONENT SPECTRUM MODEL
    
    
    In Ref. <cit.> a detailed analysis of  spectra was applied to ten multiplicity classes of 200 GeV  collisions. No  a priori assumptions about spectrum structure were imposed. The main goal was to understand systematic variation of spectrum shape with  in terms of algebraic models  inferred from data alone: given available spectrum data what is the  most efficient algebraic description? For reasons given in Sec. <ref> this analysis is presented in terms of transverse rapidity .
    
    
    
     §.§ Spectrum data and soft component
    
    
    Figure <ref> (left) shows  spectra for ten  multiplicity classes normalized by soft-component multiplicity n_s (points) and displaced upward from each other by successive factors 40 relative to the lowest spectrum (the terms “soft” and “hard” are interpreted below) <cit.>. Empirically, all spectra are observed to coincide at lower  if normalized by n_s ≈ n_ch - α n_ch^2 for some α≈ 0.01. The definition of n_s in terms of  is refined further in Sec. <ref>.
    
    
    
    
    Figure <ref> (right) shows  running integrals of the ten normalized spectra in the left panel. The spectra have been extrapolated to y_t = 0 (note that  spectra are nearly constant at lower ). The running integral is a means to enhance a long-wavelength signal (spectrum shape) over short-wavelength (statistical) noise. Different behavior is observed in each of  intervals A, B and C (spanning the detector acceptance). In interval A the integrals approximately coincide. In interval B the integrals diverge substantially. In interval C the integrals are nearly constant, and those constant values increase approximately linearly with . The limiting case for n_s→ 0 is described by function N_0(y_t) (dash-dotted curve) defined below. Those results indicate that the main  dependence lies within interval B as the running integral of a  peaked spectrum component with amplitude ∝ n_ch.
    
    The limiting case  N_0(y_t) is modeled by the running integral of a unit-normal Lévy distribution on  <cit.>Ŝ_0(m_t)    =   A(T_0,n_0)/[1 + (m_t - m_h) / n_0 T_0]^n_0,
    
    where T_0 ≈ 145 MeV controls the function mainly in interval A and n_0 ≈ 12.8 controls the function mainly in interval C. The Jacobian factor from  to  is p_t m_t / y_t. The resulting Ŝ_0(y_t) model inferred directly from data trends can be subtracted to reveal the peaked spectrum (hard) component residing mainly within interval B.
    
    
    
     §.§ Spectrum hard component
    
    
    Figure <ref> (left) shows the normalized spectrum data in Fig. <ref> (left) with fixed model Ŝ_0(y_t) inferred from Eq. (<ref>) subtracted to reveal peaked distributions (points and dashed curves) centered on interval B with amplitudes increasing approximately  ∝ n_ch. With a few exceptions discussed below the distributions are well described by a two-parameter Gaussian function (solid curves). That result suggests that the peaked  hard componentH(y_t,n_ch) has the factorized form H(y_t,n_ch) / n_s ∝ n_chĤ_0(y_t).
    
    
    
    
    Figure <ref> (right) shows data in the left panel rescaled by soft-component multiplicity n_s (rather than ) (solid) compared with a fixed Gaussian model in the form αĤ_0(y_t) with centroid y̅_t ≈ 2.7 and width σ_y_t≈ 0.45 (dashed) and with coefficient α≈ 0.006 determined by the data-model comparison. In summary, a TCM for   spectra is inferred inductively from  spectrum data alone as
    dn_ch/y_t dy_t   =    S_pp(y_t,n_ch) + H_pp(y_t,n_ch)
    
       =    n_s(n_ch)  Ŝ_0(y_t) + n_h(n_ch) Ĥ_0(y_t),
    
    where  and  trends have been factorized separately for the two spectrum components. There are two exceptions: (a) Significant systematic deviations from Ĥ_0(y_t) are observed for the lowest  classes. (b) Smaller systematic deviations are also observed for all  classes near the upper limit of the  acceptance. Both exceptions were reconsidered in Ref. <cit.> and incorporated into an extended TCM describing spectrum data over the full range of collision energies (see Sec. <ref>).
    
    More-detailed analysis <cit.> shows that the relation n_h = α n_s^2 is required by data, with α≈ 0.006 for 200 GeV  collisions <cit.> (and see Fig. <ref>, left). Adding the condition n_ch = n_s + n_h defines n_s(n_ch) and n_h(n_ch) in terms of  corrected total multiplicity  (as opposed to  detectedn̂_ch≈ n_ch/2). The resulting SP spectrum TCM with  fixed parameters and functional forms describes  spectra accurately over an  interval corresponding to 10-fold increase in the soft component and  100-fold increase in the hard (dijet) component. There was no requirement for physical interpretation of the TCM components as the model was inferred from spectrum data.
    
    
    
     §.§ Alternative spectrum models
    
    
    The TCM derived inductively from spectrum data requires five fixed parameters to describe  spectra for any event multiplicity within a large interval <cit.>. Alternative spectrum models could be proposed with free parameters determined for each event class by fits to data. One such model is the so-called “power-law” (n) or Tsallis (q) distribution that is similar in form to Eq. (<ref>)
    
    P(x_t)    =   A/[1 + x_t / n T]^n,
    
    where x_t is p_t <cit.> or m_t <cit.> and n ↔ 1/(q-1).
    
    Figure <ref> shows the result of fitting the spectrum data from Fig. <ref> (left) with Eq. (<ref>). On the left are fit residuals for ten multiplicity classes  in units of statistical uncertainty for each  value. That format differs from more-conventional presentations in terms of data/model ratios that strongly suppress residuals at lower . Especially for lower event multiplicities those fits should be rejected given standard criteria. On the right  are values for parameter n derived from fitting Eq. (<ref>) to spectrum data (solid points) and to the TCM described above (open points) compared to the fixed value n_0 = 12.8 for the TCM itself. The variation descends from values exceeding the TCM soft-component value for lower  to values consistent with the jet-related TCM hard component for larger  as dijet production increases ∝n̂_ch^2 per Ref. <cit.>.
    
    
    
    
    It could be argued that the TCM for  spectra is simply one of several competing spectrum models and that the power-law model should be preferred as requiring only two parameters compared to five for the TCM. However,  the TCM is not a simple fitting exercise applied to a single data spectrum. Ref. <cit.> established that a spectrum TCM inferred inductively from the  dependence of  spectra without any  a priori assumptions is  necessary to describe consistently an ensemble of high-statistics  spectra over a large  range (e.g. factor 10). Separation of  spectrum  data into two components does not depend on an imposed model. Specific model functions were introduced only  after resolution of data spectra into two components based on   trends.
    
    In contrast, what emerges from power-law fits is twenty parameter values to describe ten multiplicity classes as opposed to five for the TCM. The large penalty for poor fits in Figure <ref> (left)  is even more determining. The dramatic variation of power-law parameter n in the right panel has no  a priori explanation, whereas in a TCM context the variation occurs because an inappropriate (soft) model attempts to accommodate quadratic increase of the jet-related (hard) spectrum component with .
    
    
    
     
    
    § EVENTWISE-RECONSTRUCTED JETS
    
    
    Jets may be reconstructed eventwise from final-state hadrons within the full hadron momentum space including scalar momentum and angular correlations relative to an inferred leading-parton four momentum. This section emphasizes scalar-momentum dependence of reconstructed jets and jet fragments. A joint density distribution on parton (jet) energy E_jet and scalar fragment momentum p_frag can be factorized as P(E_jet,p_frag) = P(E_jet) P(p_frag|E_jet), where conditional distribution P(p_frag|E_jet) → D_p^h(p_frag|E_jet) is a  fragmentation function (FF) for parton type p fragmenting to hadron type h, and P(E_jet) → dσ_j/dE_jet is the jet energy spectrum for a given dijet source (e.g.  collisions). In this section simple and accurate parametrizations of isolated-jet data for  and  collision systems are based on logarithmic rapidity variables, with E_jet↔ p_t to accommodate some conventional notation.
    
    
    
     §.§ p-p jet (scattered-parton) energy spectra
    
    
    A QCD-related energy dependence is typically of the form log(Q/Q_0) where Q_0 represents some characteristic energy scale. A description of fragmentation functions in terms of rapidity variable y = ln[(p  + E) / m_h] ≈ln(2p/m_h) as in  Ref. <cit.> is presented in the next subsection.
    
    A jet spectrum near midrapidity for  collision energy √(s) can be written in terms of a jet “rapidity” by
     
    p_t d^2 σ_j/dp_t dη   =   d^2σ_j/dy_max dη,
     
    where  y_max≡ln(2 E_jet / m_π) was first defined in Ref. <cit.> in connection with fragmentation functions as summarized in the next subsection and E_jet→ p_t as noted above.
    
    Study of jet-related yields, spectra and angular correlations  in 200 GeV  collisions reveals that the jet-related (hard-component) density dn_h/dη (and presumably jet production dn_j/dη) scales with the soft-component density as dn_h/dη∝ (dn_s / dη)^2. 
    Given that relation and  dn_s / dη∝log(s/s_0) ≡ 2Δ y_b (with √(s_0)≈ 10 GeV) near midrapidity <cit.> the number of MB dijets (dominated by lowest-energy jets) appearing near midrapidity in  collisions should vary with collision energy as <cit.>.
    d^2σ_j/dy_max dη   ∝   Δ y_b^2   near some E_min.
    
    
    Kinematic constraints impose the upper limit 2E_jet < √(s) (y_max < y_b) with y_b = ln(√(s) / m_π). Evidence from jet <cit.> and SP-hadron <cit.> spectra suggests a lower limit  E_jet > E_min (y_max > y_min) with E_min≈ 3 GeV. A normalized  jet rapidity variable can then be defined by
    
    u    =   y_max - y_min/y_b - y_min =  log(E_jet / E_min)/log(√(s) / 2 E_min)∈ [0,1].
    
    
    Figure <ref> (left) shows ISR and Spp̄S jet spectrum data (points) with the jet spectra rescaled vertically by factor (Δ y_b)^2 and parton rapidity y_max rescaled horizontally to u assuming E_min≈ 3 GeV. 
    
    All spectrum data for  collision energies below 1 TeV fall on the common locus 0.15 exp(-u^2/2σ_u^2) (solid curve). 
    
    The parametrized parton spectrum conditional on  beam energy is then 
     d^2σ_j/dy_max dη   =    0.026 Δ y_b^2  1/√(2πσ^2_u) e^-u^2 / 2 σ^2_u,
     
    where 0.026/√(2πσ^2_u) = 0.15 and σ_u ≈ 1/7 is determined empirically from the  data. All jet production over nine decades is then represented by parameters √(s_0)≈ 10 GeV, E_min≈ 3 GeV and σ_u ≈ 1/7. Endpoints √(s_0) and E_min are closely related by kinematic constraints on fragmentation to charged hadrons from low-x gluons <cit.>.
    
    
        
    
    Figure <ref> (right) shows ISR and Spp̄S spectrum data from the left panel plotted vs  p_t ↔ E_jet in a conventional log-log format. The curve for each beam energy is defined by Eq. (<ref>). The dotted curve corresponds to √(s) = 630 GeV <cit.>. The model curves extend to u = 0.9 corresponding to partons with momentum fraction x = 2E_jet / √(s)≈ 2/3 beyond which the  collision-energy constraint should strongly influence the spectra.
    
    
    
     §.§ Fragmentation functions
    
    
    Dijet formation depends on parton energy scale Q = 2E_jet with rapidity y = ln[(E + p)/m_π] (for an unidentified hadron fragment with scalar momentum p) and maximum rapidity y_max as defined below Eq. (<ref>) to describe  FFs with D(y|y_max) = 2dn_ch,j/dy, the fragment rapidity density per dijet.  
    
    The FF parametrization is D(y|y_max) = 2 n_ch,j(y_max) β(u;p,q)/y_max, where β(u;p,q) is a unit-normal (on u) beta distribution, u = (y - y_min) / (y_max - y_min) ∈ [0,1] is a normalized fragment rapidity, and parameters p and q (for each parton-hadron combination) are nearly constant over a large jet energy interval <cit.>. The total fragment multiplicity 2 n_ch,j(y_max) (from two jets) is inferred from the shape of β(u;p,q) (and therefore parameters p and q) via parton (jet) energy conservation. 
     
     Figure <ref> (left) shows measured FFs (points) for three dijet energies <cit.> extending down to very low fragment momentum (less than 100 MeV/c). When plotted on fragment rapidity y the FFs show a self-similar evolution with maximum rapidity y_max. The solid curves show the corresponding FF parametrization developed in Ref. <cit.> based on the beta distribution as noted above.
     
     
      
     
     Figure <ref> (right) shows the self-similar data in the left panel rescaled to unit integral and plotted on scaled fragment rapidity u with y_min≈ 0.35 (p ≈ 50 MeV/c). The solid curves are corresponding beta distributions with parameters p and q nearly constant over a large jet energy interval. The simple two-parameter description is accurate to a few percent within the E_jet interval 3 GeV (y_max≈ 3.75) to 100 GeV (y_max≈ 7.25) <cit.>.  FF data for light-quark and gluon jets are parametrized separately but the parametrizations for gluon and light-quark jets converge near E_jet = 3 GeV.
     
    All minimum-bias jet fragment production can be described with a few universal parameters via introduction of logarithmic rapidities.
    
    
    
    FFs for isolated dijets derived from  collisions are quite different from FFs derived from  or  collisions as noted below. Minor differences might arise from alternative jet reconstruction algorithms, but larger observed differences suggest that the concept of universality may not apply to FFs from distinct collision systems.
    
    Figure <ref> (left) shows FFs for ten dijet energies from 78 to 573 GeV inferred from 1.8 TeV  collisions (points) using eventwise jet reconstruction <cit.>.  The solid curves are a parametrization. Comparison with the   FF data in Fig. <ref> (left) (e.g. dashed curves in this panel for 2E_jet =  6 and 91 GeV) reveals that a substantial portion of  dijet FFs at lower fragment momenta may be missing from reconstructed  FFs. This comparison is dominated by quark jets, but quark and gluon FFs converge near E_jet≈ 3 GeV where most MB jets appear.
    
    
     
    
    Figure <ref> (right) shows the ratio of  FF data in the left panel to the  FF parametrization for each jet energy (points), revealing systematic differences. The solid curve is tanh[(y-1.5)/1.7] which describes measured   FFs relative to  FFs for dijet energies below 70 GeV. FF parametrizations for quark and gluon jets used for  collisions in the present study are the  parametrizations of Ref. <cit.> and Fig. <ref> multiplied by the same tanh factor for both quark and gluon FFs.
    
    
    
     
    
    § JET CONTRIBUTIONS TO P-P SPECTRA
    
    
    The MB dijet contribution to  spectra and other momentum-space measures should consist of all hadron fragments from all dijets emerging from a given collision system and appearing within a certain angular and  acceptance. Given the jet spectrum and FF parametrizations in the previous section the resulting MB fragment distribution can be obtained from a convolution integral.
    
    
    
     §.§ Minimum-bias fragment distributions
    
    
    The midrapidity η density of MB dijets from non-single-diffractive (NSD)  collision is estimated by
    
    f_NSD   =   1/σ_NSDdσ_j/dη  ≈ 0.028 at 200 GeV
    given  σ_NSD≈ 36.5 mb <cit.> and dσ_j/dη≈ 1 mb <cit.> at that energy. The ensemble-mean  fragment distribution for MB dijets is defined by the convolution integral
    D̅_u(y)    ≈   1/dσ_j/dη∫_0^∞  dy_max  D_pp(y|y_max) d^2σ_j/dy_maxdη,  
    where subscript u denotes unidentified-hadron fragments.
    Given that a spectrum hard component H(y_t) represents hadron fragments from MB dijets it can be expressed as y_t H(y_t) ≈ϵ  f_NSDD̅_u(y), where ϵ(Δη,Δη_4π) ∈ [0.5,1] is the average fraction of a dijet appearing in detector acceptance Δη compared to effective 4π acceptance Δη_4π (which depends on collision energy). At 200 GeV ϵ≈ 0.6 within detector acceptance Δη = 2. 
    
    The spectrum hard component so defined represents the fragment contribution from MB  scattered parton pairs into acceptance Δη. It is assumed that for midrapidity jets y_t ≈ y (i.e. collinearity) except for low-momentum fragments (e.g. p_t < 0.5 GeV/c).
    
    Fig. <ref> (left) shows a surface plot of the Eq. (<ref>) integrand—D_pp(y|y_max)  d^2σ_j/dy_max dη—incorporating  FFs from Fig. <ref> (left) and the 200 GeV jet  (→ y_max) spectrum from Fig. <ref> (right).   The  FFs are bounded below by y_min≈ 1.5 (p ≈ 0.3 GeV/c). The jet-spectrum effective lower bound is E_min≈ 3 GeV. The z axis is logarithmic to show all distribution structure.
    
    
    
    
    Fig. <ref> (right) shows the corresponding mean fragment distribution ϵD̅_u(y) as a projection (dashed) described by Eq. (<ref>) and compared to hard-component data from 200 GeV NSD  collisions (solid points <cit.>) in the form y_t H(y_t) / f_NSD corresponding to their relation in the text just below Eq. (<ref>).  The open boxes are 200 GeV  jet-spectrum data from Ref. <cit.>. The dash-dotted curve S_p is Eq. (<ref>) reduced by factor 2/3 so D̅_u(y) from Eq. (<ref>) (dashed) best accommodates the  SP spectrum data (solid points). 
    
    The apparent difference between calculated and measured jet spectra falls within the systematic uncertainties of Ref. <cit.>. On the other hand the FF parametrization from Fig. <ref> extrapolated down to E_jet < 10 GeV (y_max < 5) may overestimate the fragment yield there substantially. In any case the measured spectrum  hard component from 200 GeV NSD  collisions is quantitatively consistent with a dijet contribution derived from eventwise-reconstructed jets <cit.>. Given the spectrum hard-component mode near p_t = 1 GeV/c this comparison also establishes that a parton (jet) spectrum effective lower limit E_min = 3.0 ± 0.2 GeV is required by 200 GeV  spectrum data. Extrapolating the (≈ power-law) jet spectrum significantly below 3 GeV (e.g. as in some Monte Carlos) would result in a large overestimate of H(y_t) for fragment momenta below 1 GeV/c.
    
    
    
     §.§ Measured spectrum hard components
    
    
    The comparison in Fig. <ref> provides convincing evidence from a specific 200 GeV NSD  SP spectrum that its TCM hard component does indeed represent a MB jet fragment distribution. A more-recent study reveals what can be learned from  and collision-energy dependence.
    
    Figure <ref> (left) shows the  evolution of a revised TCM spectrum hard-component model as derived in Ref. <cit.> based on recent high-statistics spectrum data from Ref. <cit.>. The variation below the hard-component mode near y_t = 2.7 is already apparent in the early data of Ref. <cit.> shown in Fig. <ref> (right). Whereas the model shape was initially held fixed independent of  to retain simplicity, the revised TCM of Ref. <cit.> accommodates all significant variation of spectrum data. Shape variations below and above the mode are evidently tightly correlated.  Interpreted in a jet-related context the revised model suggests that as larger event multiplicities are required the underlying jet spectrum is biased to more-energetic jets with larger fragment multiplicities.
    
    It is important to note  that while the   TCM hard component was initially modeled by a simple Gaussian on  <cit.> subsequent detailed analysis of  and å spectrum data combined <cit.> revealed that an exponential tail for the Gaussian on  is required for the hard-component model. The corresponding power-law trend on  required by data for 200 GeV  collisions is then ≈ 1/p_t^7 as indicated by the dashed line in Figure <ref> (left). The power-law exponent evolution with collision energy in Figure <ref> (right)  <cit.> is compatible with jet spectrum measurements <cit.>. In contrast, the “power law” of soft component Ŝ_0(m_t) corresponding to ≈ 1/p_t^13 at 200 GeV appears to be unrelated to jet physics.
    
    
     
    
    Figure <ref> (right) shows spectrum hard components for a range of  collision energies from SPS to top LHC energies based on the analysis in Ref. <cit.>. 
    
    The main variation is reduction of the power-law exponent describing the distribution high- tail (note dashed lines at right) which can be compared with the jet-spectrum evolution shown in Fig. <ref> (right).  The close correspondence between TCM spectrum hard components and jet spectra provides additional support for interpretation of the  hard component as a MB jet fragment distribution.
    
    
    
     §.§ Hadron yields and p̅_t vs p-p multiplicity
    
    
    TCM analysis of differential spectrum structure as described above can be supplemented by statistical measures, e.g. integrated yields n_x or mean angular densities ρ̅_x within some angular acceptance and ensemble-mean  . Given TCM soft and hard components inferred from differential  spectra the integrated multiplicities n_s and n_h and ensemble means p̅_ts and p̅_th can be computed. Figure <ref> (right) shows an initial comparison from Ref. <cit.> suggesting a quadratic relation between n_h and n_ch. A study of recent high-statistics  spectrum data from Ref. <cit.> establishes more accurate relations.
    
    Figure <ref> (left) shows ratio n_h / n_s vs soft-component mean density ρ̅_s = n_s / Δη for ten multiplicity classes <cit.>. n_h is the integral of hard-component H(y_t) appearing differentially in Fig. <ref> (left) and as a running integral in Fig. <ref> (right) (excesses above unity at right). The linear trend for n_h/n_s confirms the relation ρ̅_h ∝ρ̅_s^2 <cit.>. Soft-component density ρ̅_s may be interpreted as a proxy for the density of low-x gluons released from projectile nucleons in a  collision.  spectrum data then reveal that the number of midrapidity dijets ∝ρ̅_h varies quadratically with number of participant gluons. But in an eikonal collision model the number of gluon-gluon binary collisions should vary as the dashed curve representing ρ̅_h ∝ρ̅_s^4/3 as for the Glauber model of å collisions. The  data appear inconsistent with the eikonal model.
    
    
    
    
    Figure <ref> (right) shows   vs soft-component mean density ρ̅_s well described by a constant plus linear term. The solid line is a TCM for that quantity derived from the spectrum TCM of Eq. (<ref>). Given the evidence in  this section one may conclude that  variation with ρ̅_s is determined entirely by  the jet-related hard component.
    
    
    
    
    
    § JET CONTRIBUTIONS TO A-A SPECTRA
    
    
    The  TCM provides an important reference for å collisions. It is reasonable to expect more-peripheral å collisions to be described by the same basic model elements modulo the Glauber model of å collision geometry – dependence on number of participant nucleons N_part and  binary collisions N_bin, with ν≡ 2N_bin / N_part as the mean number of binary collisions per participant. However, it has been established that jets are strongly modified in more-central å collisions. The TCM hard-component model should then be altered to accommodate such changes.
    
    
    
     §.§ Identified-hadron spectrum evolution
    
    
    The TCM for  spectra can be extended to describe data from å collisions (and identified hadrons) as in Ref. <cit.>. The TCM reference for å collisions corresponding to Glauber linear superposition (GLS) of isolated  collisions is based on the  result in Eq. (<ref>) evaluated for inelastic  collisions as represented by
    ρ̅_0(y_t)    ≈    S_NN(y_t) + H_NN(y_t).
    
    By hypothesis the TCM soft component in å collisions should scale with N_part/2 and the hard component should scale with N_bin leading to the å spectrum TCM
    ρ̅_0(y_t,b)    ≈    (N_part / 2) S_NN(y_t) + N_bin H_AA(y_t,b)
    
    2/N_partρ̅_0(y_t,b)    ≈    S_NN(y_t) + ν H_AA(y_t,b),
    
    where the soft component is assumed to be invariant with å centrality but the jet-related hard component may vary substantially and is then a principal object of study.
    
    
     
    
    Figure <ref> shows identified-pion and -proton spectra (solid curves) for five centrality classes of 200 GeV  collisions plotted vs rapidity y_t(π) with pion mass assumed <cit.>. The rapidity variable is used in this case simply as a logarithmic momentum variable y_t ≈ln(2 p_t / m_π) but with well-defined zero. That choice is explained below. Also plotted with the pion spectra is the unidentified-hadron spectrum for 200 GeV NSD  collisions (points) <cit.>. The TCM soft components S_NNx (dotted) are defined as the limits of normalized spectrum data as N_part→ 0, equivalent to the definition for  collisions. The pion soft component is consistent with the  soft component inferred from unidentified hadrons. The pion hard component H_NN is also consistent with the  analysis. The proton soft and hard components have the same algebraic structure, but model parameters are adjusted to accommodate peripheral  data as described below.  The dash-dotted curves are GLS reference spectra for ν = 1 and 6 (å limiting cases).
    
    Figure <ref> shows data hard components for identified pions and protons (solid) from five centrality classes of 200 GeV  collisions in the form ν H_AA per Eq. (<ref>). Also plotted are the hard component for unidentified hadrons from 200 GeV NSD  collisions (points) and hard-component models H_NNx (dashed). The dotted curves are GLS references corresponding to the five centrality classes assuming that H_AA = H_NN (i.e. linear scaling with factor ν). Relative to those reference curves the data exhibit substantial suppression at higher  for more-central collisions as inferred from conventional ratio measure R_AA <cit.>. However, substantial  enhancement at lower  is a new feature not revealed by R_AA data.
    
    
     
    
    The structure of the proton hard component is a surprise. The mode for  (≈) collisions appears near p_t = 1 GeV/c as for pions but the peak width is substantially less, and there is a significant difference in the “power-law” slope at high  as described below. However, it is most notable that hard-component maximum values are  essentially the same for protons and pions in contrast to the soft components. The similarity of two hard components on  is the main motivation for adopting y_t(π) ≈ln(2 p_t / m_π) as the independent variable <cit.>.
    
    
    
     §.§ Spectrum ratios for identified hadrons
    
    
    Figure <ref> shows ratios r_AA = H_AA / H_NN for pions and protons from five centrality classes of 200 GeV  collisions (curves of several line styles). Also plotted are  hard-component data in ratio to the pion H_NN reference (points). Several features are notable. The peripheral pion and proton data for 60-80% central collisions indicate no jet modification (r_AA = 1), a result consistent with the observation from jet-related 2D angular correlations that below a  sharp transition (ST) in jet characteristics near 50% of the total cross section jets remain unmodified in 200 GeV  collisions <cit.>.
    
    
     
    
    Above the ST there is increasing suppression at higher , with a saturation value ≈ 0.2 for both pions and protons in central  collisions as observed with conventional ratio parameter R_AA. However, because R_AA as defined is a ratio of entire spectra  including soft components the evolution of jet-related H_AA below p_t = 3 GeV/c (y_t ≈ 3.75) is visually inaccessible. In contrast, ratio r_AA including only hard components reveals large enhancements of jet-related hadron yields at lower  tightly correlated on centrality with suppressions at higher . But whereas enhancement for pions extends below 0.5 GeV/c (y_t = 2) enhancement for protons peaks near 2.5 GeV/c (y_t ≈ 3.6), and below p_t = 1 GeV/c the proton data for all centralities remain consistent with the  reference (r_AA = 1) within the detector acceptance. 
    
    These jet-related hard-component trends have important consequences for other (e.g. ratio) measures and for the flow narrative. In particular, whereas conjectured radial flow should boost all hadron species to higher  proportional to hadron mass the trends in Fig. <ref> show that while protons appear boosted to higher  relative to the  hard-component mode pions move to  lower.
    
    Figure <ref> (left) shows a conventional ratio comparison of proton and pion spectra – the proton-to-pion ratio – for two centralities of 200 GeV  collisions (points) <cit.>. The  curves are derived from a TCM for  spectra including hard-component modification in more-central collisions <cit.> that describes the spectrum evolution in Fig. <ref>. Those curves were not fitted to the ratio data.
    
    
     
    
    Figure <ref> (right) shows the curves in the left panel replotted on  to improve visibility of the low- region. Since the TCM soft and hard components contributing to those spectrum ratios are known their ratios can be plotted separately (dotted and dash-dotted curves respectively). The soft component by hypothesis does not vary with å centrality. The hard-component ratio for peripheral å (and therefore ) collisions is unity at the mode ( = 2.7, p_t ≈ 1 GeV/c). It falls off on either side of the mode because of the peak width difference but increases for larger  because the proton power-law exponent is smaller (the spectrum high- tail is harder), all consistent with the discussion of Fig. <ref> above.
    
    The hard-component ratio maximum for central collisions has a substantially larger value and the mode on  shifts up to 3 GeV/c. Protons are strongly enhanced relative to pions above the  mode but pions are strongly enhanced relative to protons below the mode, a feature concealed by the conventional R_AA ratio and plotting format. Comparison with the solid curve matching the ratio data reveals that the data peak is dominated by the jet-related hard component. In contrast, the ratio peak for peripheral collisions is dominated by the soft component; the hard component only influences the peripheral ratio above 7 GeV/c.  Note that the ratio of hard to soft hadron production in  collisions increases with centrality according to parameter ν with no jet modification (5-fold increase) and by an additional factor 3 due to jet modification <cit.>. Thus, from peripheral to central  collisions the hard/soft ratio increases by about 3 × 5 = 15-fold. Although peaks in the left panel appear similar and suggest a common mechanism this TCM analysis reveals that they represent distinct soft and hard hadron production mechanisms. Comparisons of spectrum ratio data with in-vacuum  FFs and nonjet (soft) recombination or coalescence hadronization models as in Ref. <cit.> are likely misleading.
    
    
    
     §.§ Hadron yields and p̅_t vs A-A centrality
    
    
    Just as for  collisions differential  spectrum structure for å collisions can be supplemented by trends for integrated spectrum yields and  mean values. For instance, hadron yields vs  multiplicity were discussed in Sec. <ref> where observed dijet production appears inconsistent with the eikonal approximation. The centrality  dependence of -integral hadron yields (or mean angular densities) in å collisions is similarly of interest.
    
    Figure <ref> (left) shows integrated unidentified-hadron densities in the form (2/N_part) dn_ch/dη (solid points) reconstructed from spectrum data in Fig. <ref> <cit.>. Due to multiplicity fluctuations the most-central point of such a trend is typically high by an amount controlled by the detector angular acceptance: the excess is less for a larger acceptance <cit.>.  The open point is a reference value from NSD  collisions. A simple TCM with fixed constant x = 0.095 from Ref. <cit.> is represented by the dash-dotted line. A color-glass condensate (CGC) trend approximated by 0.9 ln(8ν) is shown as the dashed curve <cit.>. The hatched region labeled GLS is a  reference TCM extrapolation from the  TCM of Sec. <ref> assuming no jet modification in  collisions. The solid curve is a TCM with varying x based on dijet angular correlations <cit.> combined with the spectrum TCM from Ref. <cit.> that models jet modification in terms of spectrum hard components as in Fig. <ref> (right) below. The hatched band labeled ST marks the “sharp transition” in jet-related angular-correlation characteristics reported in Ref. <cit.> (see Sec. <ref>). Quantitative correspondence of those results with yield data from spectra provides additional compelling evidence that the TCM spectrum hard component remains jet-related in all å collisions <cit.>.
    
    
    
    
    Figure <ref> (right) shows  hadron production vs centrality for 2.76 TeV  collisions (inverted triangles)  and a reference value for NSD  collisions (upright triangle) <cit.>. The solid curve is the solid curve for 200 GeV in the left panel extrapolated to higher energy by two modifications. The soft component S_NN is multiplied by 1.87 = log(2760/10) / log(200/10) motivated by the observed soft-component energy trend ∝log(√(s_NN) / 10 GeV) <cit.>. The hard component is then multiplied by another factor 1.87 corresponding to n_h ∝ n_s^2 for  collisions <cit.>.  Whereas the centrality evolution of jet correlation structure for 200 GeV  collisions is consistent with a ST near ν = 3 (50% fractional cross section) <cit.> the  data suggest that the ST at higher energy may have shifted down to ν≈ 2. Otherwise the same TCM describes data well at two widely-spaced energies depending only on a log(s/s_0) QCD energy trend.
    
    Figure <ref> (left) shows ensemble-mean  data from 5 TeV  and 2.76 TeV  collisions (points) vs charge density ρ̅_0 = n_ch / Δη <cit.>. The solid and dotted curves are TCMs for the respective collision systems <cit.>.  p̅_ts' ≈ 0.51 GeV/c (hatched band) is the soft-component value corresponding to a -acceptance lower limit near 0.2 GeV/c, whereas p̅_ts≈ 0.38 GeV/c is a universal feature of any  spectrum extrapolated down to p_t = 0 as in Fig. <ref> (right).  The  GLS reference (dashed curve) reflects the eikonal approximation assumed for the å Glauber model and no jet modification.  The dash-dotted curve is the TCM for 5 TeV  data consistent with Fig. <ref> (right) for 200 GeV data and inconsistent with the eikonal approximation. The  data (open points) suggest a smooth transition from a non-eikonal  trend to eikonal å trend with increasing  (centrality).
    
    
    
    
    Figure <ref> (right) shows evolution of the measured TCM spectrum hard component for 200 GeV  collisions from peripheral (open points) to central (solid points) collisions. The corresponding dashed and solid curves are derived in Ref. <cit.> from jet measurements. The dash-dotted curve is a GLS reference (no jet modification) for central .  It seems likely that spectrum hard-component evolution in 2.76 TeV  collisions is similar but with some quantitative differences.
    
    According to the TCM for  the GLS trend  for 2.76 TeV  in the left panel should be a weighted average of soft-component  value p̅'_ts≈ 0.5 GeV/c (for p_t > 0.2 GeV/c) and hard-component value p̅_th≈ 1.7 GeV/c (for 2.76 TeV  collisions <cit.>), the average increasing with increasing fraction of jet-related hard-component hadrons. The  data trend differs from GLS by two consequences of jet modification: (a) the hard-component yield increases by factor 3 and (b) the hard-component mode decreases substantially from 1.7 GeV/c, as illustrated in Figure <ref> (right) and Fig. <ref> (right). The combination results in a net increase of data p̅_t over GLS.
    
    
    
     §.§ Alternative spectrum models
    
    
    Section <ref> presents an alternative to the  spectrum TCM in the form of a “power-law” or Tsallis distribution approximating the TCM soft component alone. The fit results indicate poor fit quality, and the large parameter variations are not physically interpretable. The blast-wave model is a popular alternative for å spectra in which deviations from some reference spectrum are assumed to result from radial flow of a bulk medium.
    
    The assumed reference is usually a Maxwell-Boltzmann (M-B) exponential <cit.>, but a power-law or Tsallis distribution has been adopted in recent studies <cit.>. The model parameters include slope parameter T and mean radial speed ⟨β_t ⟩.
    
    Blast-wave fits may be restricted to smaller  intervals lying below 2 GeV/c under the assumption that such intervals exclude jet contributions and that any spectrum variation is then determined by flows: “In central Au+Au collisions the flattening of the spectra [below 2 GeV/c] is likely dominated by collective transverse radial flow, developed due to the large pressure buildup in the early stage of heavy-ion collisions” <cit.>.
    
    Figure <ref> (left) illustrates an early radial-flow analysis of spectrum data from 19 GeV ß collisions (solid points) <cit.>. The corresponding M-B distribution is shown by the dash-dotted line. A  spectrum for 17 GeV (open points) and a TCM soft-component Lévy distribution (dashed) with universal T_0 = 145 GeV and with exponent n_0 = 17 adjusted to accommodate the S-S data are shown for comparison (n_0 = 27 describes the  spectrum <cit.>). Deviation of the ß data from the M-B reference is interpreted to represent radial flow with ⟨β_t ⟩≈ 0.25 (β_s ≈ 0.5 is the maximum for a radial β_t distribution). The same fit model applied to a 200 GeV  spectrum returns a similar ⟨β_t ⟩ value as noted below. 
    
    
     
    
    Figure <ref> (right) shows published ⟨β_t ⟩ values derived from fits to 62 GeV  (open points) and 200 GeV (solid points)  spectra for several collision centralities <cit.> plotted vs Glauber centrality parameter ν. The location of the ST inferred from jet-related angular correlations <cit.> is indicated by the hatched band. To the left of that point  collisions are effectively transparent linear superpositions (GLS) of  collisions <cit.>, but to the right of that point jet structure shows substantial modification (“jet quenching”). It is notable that the inferred ⟨β_t ⟩ data are not zero in the å transparency interval (or for  collisions). The  ⟨β_t ⟩ data instead increase  more rapidly in a centrality interval where rescattering is less likely based on jet data but increase  less rapidly in an interval where jet modification is substantial suggesting copious rescattering is more likely. Results in Fig. <ref> (right) interpreted to indicate radial flow are in direct conflict with the results in Fig. <ref> consistent with measured jet properties. In the blast-wave model jet-related spectrum structure described by the TCM hard component is in effect reassigned to radial flow.
    
    
    
     
    
    § JETS AND ANGULAR CORRELATIONS
    
    
    Angular correlation methods are introduced briefly in Sec. <ref>. A  measured pair density ρ(x_1,x_2) can be compared with some reference ρ_ref(x_1,x_2) = ρ̅_0(x_1)ρ̅_0(x_2) ≈ρ_mix(x_1,x_2) to define a  correlated-pair density
    Δρ(p_t1,p_t2,η_Δ,ϕ_Δ) ≡ρ_ref (ρ' / ρ_mix - 1),
    
    where ρ' is an  uncorrected pair density and ρ_mix is a mixed-pair reference also uncorrected.  The correlated-pair density can be normalized to form a  per-particle measure Δρ / √(ρ_ref)≡√(ρ_ref)(ρ' / ρ_mix - 1) →ρ̅_0(ρ' / ρ_mix - 1) or a  per-pair measure Δρ / ρ_ref =(ρ' / ρ_mix - 1) <cit.>.
    
    
    
    
     §.§ A TCM for two-particle correlations
    
    
    In addition to angular correlations on (η_Δ,ϕ_Δ) two-particle correlations can be studied on transverse rapidity as (y_t1,y_t2) → y_t × y_t where they are directly comparable with SP spectra on  and the -spectrum TCM.
    
    Figure <ref> (left) shows  correlations from 200 GeV NSD  collisions for p_t ∈ [0.15,6] GeV/c <cit.>. The two peaked features are identified as TCM soft and hard components as follows. The lower- peak falls mainly below 0.5 GeV/c (y_t < 2) and consists exclusively of unlike-sign (US) pairs. Corresponding angular correlations consist of a narrow 1D peak on η_Δ centered at the origin. That combination suggests longitudinal fragmentation of low-x gluons to charge-neutral hadron pairs nearby on η, consistent with TCM spectrum soft component S_pp(y_t). The higher- peak falls mainly above 0.5 GeV/c with mode near p_t = 1 GeV/c (y_t ≈ 2.7) corresponding to SP spectrum hard component H_pp(y_t) and with charge structure (LS or US) depending on angular constraints.
    
    
    
    
    Figure <ref> (right) shows angular correlations for the same collision system with the condition p_t ≈ 0.6 GeV/c, i.e. near the lower boundary of the  hard component in the left panel. Despite the low hadron momentum the observed angular correlations exhibit structure expected for jets: a SS 2D peak representing  intra jet  correlations and an AS 1D peak representing   inter jet (back-to-back jet) correlations. The SS peak is dominated by US pairs while the AS peak shows no preference, consistent with fragmentation of back-to-back  charge-neutral gluons.
    
    
    
     §.§ 2D angular correlations and model fits
    
    
    The general structure of 2D angular correlations was established for  collisions in Refs. <cit.> and for  collisions in Ref. <cit.>. Quantitative discrimination was achieved among jets, a  nonjet (NJ) azimuth quadrupole and Bose-Einstein correlations (BEC). 2D angular correlations on (η_Δ,ϕ_Δ) have a simple structure  modeled by a few 1D and 2D functions <cit.>. 
    
    The six-element fit model of Ref. <cit.> includes eleven model parameters but describes more than 150 data degrees of freedom for typical 25× 25-bin data histograms on (η_Δ,ϕ_Δ). Model parameters are thus strongly constrained. The NJ quadrupole component of angular correlations can be extracted accurately via such  fits. For per-particle 2D angular correlations as Δρ / √(ρ_ref) the NJ quadrupole is represented by A_Q{2D}≡ρ̅_0 v_2^2{2D} <cit.>.
    
    Figure <ref> shows 2D angular correlations from two of seven multiplicity classes of 200 GeV  collisions (index values n = 1, 6, dn_ch/dη≈ 1.8, 15, for left and right panels respectively) <cit.>. Based on 2D model fits contributions from a soft component (1D peak on η_Δ) and BEC (narrow 2D peak at the origin) have been subtracted. What remains is a jet contribution (broad SS 2D peak at the origin and AS 1D peak on azimuth) and a NJ quadrupole contribution manifested in the right panel by increased curvature of the AS 1D peak, reduced curvature of the SS background for |η_Δ| > 1 and  apparent narrowing of the SS 2D peak on azimuth. Systematic variations of several correlation components are presented in Ref. <cit.>. The main message of such studies is that MB dijets dominate  correlation structure.
    
    
     
    
    
    
    Figure <ref> shows 2D angular correlations from the most peripheral (left) and most central (right) 200 GeV  collisions. The statistical errors for those histograms are about 4.5 times larger than for the high-statistics  data in Fig. <ref>. The same six-element 2D fit model was applied to the  data <cit.>. The peripheral  data are approximately equivalent to NSD  data similar to Figure <ref> (left) (but before subtraction of two model elements). The NJ quadrupole  in those panels is negligible compared to both the jet-related structure (in both panels) and the soft component (1D peak on η_Δ at the origin in the left panel). Note the narrower BEC 2D peak atop the broader jet-related SS 2D peak in each panel.
    
    Figure <ref> shows fitted parameter values vs Glauber centrality parameter ν for 2D model fits to 200 GeV  data as in Fig. <ref> including (a) the SS 2D peak amplitude A_1, (b) the SS 2D peak η_Δ width σ_η_Δ and (c) the AS 1D peak amplitude A_D <cit.>. The data systematics reveal two intervals on ν with markedly different behavior: (i) variation of three parameters consistent with Glauber linear superposition (GLS) for ν < 3 equivalent to å  transparency and (ii) large increases in the rate of variation for ν > 3 where ν≈ 3 corresponds to a factional cross section σ / σ_0 ≈ 0.5. The rapid change from one trend to the other is characterized as a “sharp transition” or ST in Ref. <cit.>. Note that the SS peak amplitudes in panel (a) for 62 GeV coincide for those for 200 GeV when rescaled by factor 1/0.63 representing a ln(√(s) / 10 GeV) energy trend for jet-related structure that  persists above the ST <cit.>. The data in panels (b) and (c) are not rescaled. The close correspondence among three correlation parameters strongly suggests that MB dijets remain the dominant source of correlation structure in  collisions even for the most-central collisions, although jets are substantially modified there as demonstrated by spectrum data (e.g. Fig. <ref>).
    
    
    
     §.§ Trigger-associated 1D correlation analysis
    
    
    Data and analysis methods presented above correspond to MB dijets with no conditions imposed on jet structure. Model fits to 2D angular correlations integrated over the entire  acceptance arguably extract almost all angular-correlation information carried by primary particle data. Alternative methods (a) impose  trigger-associated conditions on data, (b) are typically confined to 1D projections onto azimuth and (c) subtract model-dependent backgrounds to arrive at  nominal jet-related structure. Two examples are considered below.
    
    
     
    
    Figure <ref> shows a highly-cited (over 700 citations) trigger-associated analysis of 200 GeV  azimuth correlations compared to  and  data <cit.>. Trigger-associated  conditions are p_t,trig∈ [4,6] GeV/c and p_t,assoc∈ [2 GeV/c,p_t,trig],  admitting only a tiny fraction of all jet fragments observed in spectrum hard components (as in Fig. <ref>). In panel (b) backgrounds have been subtracted including a v_2 contribution (for  data)  based on published v_2 data that may include a jet contribution in the form of “nonflow.” The principal message is that the “away-side jet” is suppressed (disappears) in central  collisions. The suppression is attributed to absorption of jets in a dense medium.
    
    Indications of some form of jet modification (e.g. reduction of the  AS peak amplitude) are clearly apparent but the full implications are not clear. The AS peak “disappearance” suggested by  Fig. <ref> is consistent (within statistics) with the factor-5 high- reduction in SP spectra indicated in  Fig. <ref>, but there is no information about changes in jet structure (possible  enhancements) at lower  as in Fig. <ref>. Imposition of a high- trigger on one jet may tend to bias its partner jet to softer fragmentation – fewer higher- fragments but more lower- fragments – independent of any medium effects. One cannot then conclude from such biased data that the AS jet has “disappeared.” More can be learned by relaxing the associated-particle  cut.
    
    Figure <ref> (left) shows a similar trigger-associated analysis of 200 GeV  azimuth correlations with the associated-particle  condition extended down to the detector-acceptance lower bound: p_t,assoc∈ [0.15 GeV/c,p_t,trig] <cit.>. The  result (solid points) is compared with  data treated similarly (open points). 
    
    A combinatoric background to be subtracted from “raw” data is defined by published v_2 data and a ZYAM ( zero yield at minimum) principle based on the  ad hoc assumption that SS and AS jet peaks  never overlap on azimuth. The data minimum value after background subtraction is then defined as the zero for both  and  data. 
    
    Consequences of the updated analysis are two-fold: (a) the AS peak remains substantial, has not “disappeared” as reported in Ref. <cit.>, but (b) the AS peak for central  collisions appears to be softened and broadened compared to individual  () collisions – interpreted to signal “progressive equilibration” of the AS jet in the medium and “thermalization within the m
    An alternative treatment of the same basic correlation data leads to different conclusions. The  data are fitted with a 1D projection of the 2D fit model from Sec. <ref> described in Refs. <cit.> including a 1D Gaussian for the SS peak (dash-dotted), a dipole term for the AS peak (dashed) and  a quadrupole term that should correspond to v_2 data (dotted). The inferred  negative value for 2v_2^2 is notable. The fit to  data is the bold solid curve in Figure <ref> (left) that describes those data within statistical uncertainties. The fit parameters then estimate the actual background subtracted in Ref. <cit.>. The ZYAM offset is 1.37 (lower dash-dotted line) and the assumed background v_2^2 value is |2 v_2^2| ≈ 0.43 (amplitude of dotted curve). The  nonjet quadrupole amplitude for 0-5% central 200 GeV  collisions inferred from model fits to 2D angular correlations is essentially zero (upper limit consistent with data uncertainties) <cit.>. The ZYAM |2 v_2^2| ≈ 0.43 value can then be compared with the quadrupole component of the SS jet peak 2 v_2^2{2}≈ 0.25 × 3.36 = 0.84 <cit.> and an assumption that the “true” background value is v^2_2 = (v^2_2{2} + v^2_2{4})/2 ≈ v^2_2{2}/2 [Ref. <cit.>, Eqs. (9) and (10)] so the  jet-related value for central  is 2v_2^2 ≈ 0.42. It is then likely that the 2v_2^2 value adopted for the ZYAM subtraction is dominated by the jet-related SS peak Fourier decomposition <cit.>. In essence, the quadrupole component of the SS jet peak is subtracted from raw data to produce a distorted result.
    
    
     
    
    Figure <ref> (right) shows the same  data with the ZYAM background subtraction in effect reversed. The data are described within uncertainties (solid) by SS Gaussian (dash-dotted) plus AS dipole (dashed). The  data (open points) have been treated similarly except with no quadrupole term and are also described within uncertainties by the same basic model (bold dotted). Aside from the amplitude differences the  SS peak is  broader (σ_ϕ≈ 0.7) than the  SS peak  (σ_ϕ≈ 0.5) (consistent with 2D correlation trends reported in Ref. <cit.>). The  and  AS peaks are described by the same dipole model implying equivalent large widths. The peak amplitudes indicate that the number of jet fragments  per trigger is about 70% larger in central  than in  collisions, consistent with the lower- enhancements evident in Fig. <ref>. The SS and AS jet peaks  strongly overlap in all collision systems (except for very high  cuts) consistent with 2D model fits from Ref. <cit.> and as illustrated in Figs. <ref> and <ref>.  Jet-related correlation analysis on 1D azimuth based on ZYAM background subtraction is thus strongly inconsistent with data properties derived from other contexts.
    
    
    
     §.§ Bayesian analysis of azimuth correlations
    
    
    It could be argued that the results in the previous subsection depend on a chosen fit model and are therefore arbitrary. Bayesian analysis provides neutral criteria for comparison of data models based on competition between goodness of fit (e.g. the χ^2 measure) and cost of model complexity (based on information parameter I) as measured by  evidenceE with -2ln(E) = χ^2 + 2I <cit.>.
    
    
    
    
    Figure <ref> shows 2D angular correlation data from 0-5% central 200 GeV  collisions as reported in Ref. <cit.> projected onto 1D azimuth difference ϕ_Δ. A calculated distribution integral has been subtracted from the data. The dashed curve is a fit to data with the same 1D model used in  Figure <ref>. A fit with a Fourier-series (FS) model would achieve the same apparent result given a sufficient number of terms. Which model should be preferred?
    
    
    
    
    Figure <ref> shows  negative log evidence-2LE ≡ -2ln(E) values for several competing models, where Bayesian evidence E measures the competition between goodness of fit (better fit increases the evidence) and cost of model complexity (more complexity decreases the evidence). With increasing model complexity (e.g. number of parameters) improved fit quality could reduce χ^2 but at the cost of increasing information I. The entry labeled “Model” (solid square) consisting of SS Gaussian and AS dipole as applied to the data in Fig. <ref> achieves the minimum log-evidence value (largest evidence). Adding an m = 2 quadrupole term (solid diamond) increases -2LE ( decreases evidence E) because the additional model parameter does not achieve a compensating improvement in fit quality (consistent with results from Refs. <cit.>). The same is true for other additions (m = 3 sextupole and m = 4 octupole terms).
    
    The -2LE trend for a Fourier cosine-series (FS) model (solid dots) achieves a minimum at four elements (including an m = 3 sextupole term) and then increases monotonically. In all cases the FS model (commonly interpreted to represent flows) is strongly rejected by Bayesian analysis compared to the simple two-peaked model (consistent with MB dijet production as in Fig. <ref>). In Ref. <cit.> the large difference is traced to the  predictivity of a model. The term in the negative log evidence that increases with model complexity is information I gained by a model upon acquisition of new data.[Information can be described as a logarithmic measure of volume reduction. In this case “volume” refers to some part of a fit-model parameter space. Information I compares the volume allowed before data (prior) to that allowed after data (posterior) <cit.>.] A fixed model or one with few parameters may gain little or no information from the addition of new data and is thus highly predictive, might be falsified by new data. Predictivity and falsifiability are equivalent concepts. In contrast a Fourier series on periodic azimuth has no predictivity, can describe any data distribution. Thus, introduction of new data adds substantial information to the FS model in the form of parameter adjustments that lead to corresponding increase in the negative log evidence. In effect, the two-peak model predicts that any azimuth distribution from high-energy nuclear collisions should include peaks at 0 (SS) and π (AS),  and the SS peak should be substantially narrower than the AS peak (as expected for MB dijets). New data should modify only the SS peak width and the peak amplitudes. Any data with different features would falsify that model but would remain well described by a FS model.
    
    
    
     §.§ Predicting trigger-associated correlations
    
    
    Trigger-associated (TA) analysis of jet structure in 1D azimuth distributions, such as summarized in Sec. <ref>,  follows precedents established at lower energies during a period when the jet concept was not well established and detector technology was quite limited <cit.>. With much-higher collision energies, established jet phenomenology, higher-statistics  and å data and much-improved particle detectors the analysis of full pair-momentum space is both possible and necessary to access all information carried by particle data. Correlation data with and without  conditions should be combined quantitatively with SP-spectrum and yield data to provide the strongest possible challenge to competing theories and better inductive understanding of underlying mechanisms.
    
    This subsection summarizes a study in Ref. <cit.> where the pair density on asymmetric TA rapidity space (y_t,trig,y_t,assoc) for 200 GeV  collisions is predicted based on jet data and the TCM for  SP spectra. When symmetrized the distribution on  (y_t,trig,y_t,assoc) can be compared with the distribution on  in Fig. <ref> (left). 
    In a MB TA analysis all  collision events and all hadrons within a collision are accepted for analysis. A “trigger particle” is the single hadron in each collision with  the highest ; all other hadrons are “associated.” Some fraction of all trigger hadrons may be related to jets (the leading  detected hadron in a jet serving as proxy for the leading parton), and some fraction of all associated hadrons may be fragments from a triggered jet or its back-to-back partner jet. The TA pair distribution on  (y_t,trig,y_t,assoc) resulting from such conditions includes soft-soft (SS), soft-hard (SH) and hard-hard (HH) pair combinations. It is assumed that the HH contribution is amenable to prediction based on measured jet properties. A full analysis involves two parts: (a) predict the HH contribution to MB TA correlations <cit.> and (b) extract the HH contribution from measured MB TA correlations for direct comparison based on the SP spectrum TCM <cit.>. The full derivations  are summarized schematically below.
    
    (a) The HH component of TA correlations can be predicted from measured conditional FFs D_u(y|y_max) and jet (parton) spectrum Ŝ_p(y_max) = (1/σ_j) d^2σ_j/dy_max/dη introduced in Sec. <ref> <cit.>. FF distribution D_u(y|y_max) is first decomposed into a trigger component Ŝ_t(y_trig|y_max) and an associated component D_a(y_assoc|y_max) based on void probability G_t(y|y_max). The void probability is defined as the Poisson probability that no fragment appears for y > y_trig based on the FF integral over that interval, which is just the probability that a fragment at y = y_trig is a trigger particle for given y_max. Trigger spectrum Ŝ_t(y_trig) is then obtained by convoluting Ŝ_t(y_trig|y_max) with the measured jet spectrum Ŝ_p(y_max). An intermediate conditional spectrum is obtained from Bayes' theorem as Ŝ_p(y_max|y_trig) = Ŝ_t(y_trig|y_max)Ŝ_p(y_max) / Ŝ_t(y_trig). The associated spectrum D_a(y_assoc|y_max) is the complement of Ŝ_t(y_trig|y_max) in D_u(y|y_max). The associated fragment distribution D_a(y_assoc|y_trig) is obtained by convoluting D_a(y_assoc|y_max) with Ŝ_p(y_max|y_trig), thus eliminating jet rapidity y_max. The TA HH distribution  per jet derived from eventwise-reconstructed jet data is then
    
    F_at(y_assoc,y_trig)    =   Ŝ_t(y_trig) D_a(y_assoc|y_trig).
    
    
    Figure <ref> (left) shows HH distribution F_at(y_assoc,y_trig) for 200 GeV  collisions based on the measured  FFs and 200 GeV jet spectrum summarized in Sec. <ref>. The mode on y_trig corresponds to p_t ≈ 1.2 GeV, and the mode on y_assoc corresponds to p_t ≈ 0.6 GeV/c. Data from eventwise-reconstructed jets thus predict that the  great majority of jet-related TA pairs appear near 1 GeV/c.
    
    
     
    
    Figure <ref> (right) shows 1D projections D_a(y_assoc) (solid) and Ŝ_t(y_trig) (dash-dotted). The dashed curve shows D_a(y_assoc) with F_at(y_assoc,y_trig) extrapolated to large y_trig (beyond the 2D plot boundaries at left).
    
    (b) Isolation of hard component HH from correlation data requires a TA TCM based on the SP spectrum TCM in Sec. <ref> <cit.>. The TA TCM assumes that hadrons from soft and hard components of the SP spectrum are  uncorrelated in pairs. The TA TCM is factorized in the form F_at(y_ta,y_tt) = T̂(y_tt) A(y_ta|y_tt), the product of a unit-integral trigger spectrum and a conditional associated-particle distribution. The derivation distinguishes  soft events (no jet within acceptance) from hard events (at least one jet within acceptance) with probabilities P_s and P_h depending on event multiplicity  based on measured  dijet systematics <cit.>. Because  dijets ∝ n_s^2, multiple dijets may appear within acceptance Δη for higher  multiplicities. The trigger spectrum is then
    T̂(y_tt;n_ch)    =     P_s(n_ch) T̂_s(y_tt) + P_h(n_ch) T̂_h(y_tt),
    
    where each T̂_x is defined by a corresponding void probability based on the appropriate TCM spectrum model – with or without a spectrum hard component. The TCM for the full TA distribution is then
    
    F_at(y_ta,y_tt;n_ch)    =    P_s T̂_s A_ss + P_h T̂_h (A_hs + A_hh)  
    that can be compared with a measured F_at TA distribution. Each column of A_xy(y_ta|y_tt) on y_ta is a SP spectrum TCM soft (+ hard where applicable) component set to zero above trigger rapidity y_tt (comprising the void) as the condition, each normalized to the associated-particle number n̂_ch - 1 for that event class. The object of analysis for comparison with jet correlations is the  per jet hard component of hard events A^*_hh. To obtain that quantity from TA data the appropriate nonjet elements of the TA TCM are subtracted from the TA data. A = F_ta / T̂ is first obtained from TA data using the TCM trigger spectrum. The associated hard component HH is then isolated by subtracting SS and HS TCM elements
    
    P_h R_h A_hh = A - P_s R_s A_ss - P_h R_h A_hs,
    
    where R_x ≡T̂_x / T̂. Two more steps are required to obtain A^*_hh (correlations from trigger jet only) from A_hh (correlations from soft and hard triggers and from both trigger jet and partner jet if present) as described in Ref. <cit.>.
    
    Figure <ref> (left) shows the  per jet hard component of TA correlations F_at in the form T̂_h y_ta A^*_hh for multiplicity class n = 5 from 200 GeV  collisions. That result can be compared directly with the prediction in Fig. <ref> (left) derived from eventwise-reconstructed jet data. The z-axis limits are the same for the two plots. There is quantitative correspondence except near the low- boundary.
    
    
     
    
    Figure <ref> (right) shows a projection of the left panel onto y_ta (solid points) that can be compared directly with projection D_a(y_assoc) (mean associated fragment distribution) from Fig. <ref> (right) (solid and dashed curves). Also plotted is the hard component from 200 GeV NSD  collisions (open points) in the  per jet form y_t,assoc H / f [also plotted in Fig. <ref> (right)] where f is the dijet η density per  collision. For both SP spectra as in Fig. <ref> and more-complex TA correlations as in Figs. <ref> and <ref> there is a quantitative correspondence between eventwise-reconstructed jets and MB dijet manifestations in  spectra and correlations. The correspondence extends down to at least p_t = 0.3 GeV/c, and most jet fragments (> 90%) appear  below 2 GeV/c (y_t ≈ 3.3).
    
    It is notable that trigger-associated  cuts applied to data as in Fig. <ref> from Ref. <cit.> or Fig. <ref> from Ref. <cit.> correspond respectively to rectangles on (y_t,assoc,y_t,trig) in Fig. <ref> (left) bounded by ([3.3,y_t,trig],[4,4.5]) and ([1,y_t,trig],[4,4.5]) for the two cases. Such cuts include only a tiny fraction of correlated pairs and exclude the dominant TA peak near y_t = 2.7 (p_t ≈ 1 GeV/c). It has been argued that since pQCD calculations are valid only for “high-” fragments and energetic partons high- cuts are required for valid jet analysis. But the validity of pQCD calculations is not relevant to comparisons between MB jet phenomena and  measured jet properties (FFs and jet spectra) as above.
    
    
     
    
    § JETS AND P_T FLUCTUATIONS
    
    
    When the √(s_NN) = 17 GeV  program commenced at  the CERN super proton synchrotron it was expected that certain  eventwise fluctuation measures might reflect thermodynamic properties of a quark-gluon plasma. In particular, eventwise mean- fluctuations might serve as a proxy for temperature fluctuations within a locally-thermalized plasma <cit.>. Exceptional fluctuations (i.e. deviations from independent particle emission) might signal proximity to a QCD phase boundary and reveal its nature. That concept continues to provide the principal context for recent fluctuation studies <cit.> but does not include possible contributions to  fluctuations from MB jets that are considered below.
    
    
    
     §.§ Fluctuation measures
    
    
    Fluctuation measure definitions described below follow those presented in Ref. <cit.>. n̅_ch and p̅_t are event-ensemble means averaged over a detector acceptance. To reduce notation complexity n_ch↔ n where there is no ambiguity. Symbols Δσ^2_x denote  per-particle variance differences that are negligible in case of central-limit conditions (CLT) wherein data are comprised of (a) independent samples from (b) a fixed parent distribution <cit.>. Deviations from the CLT may represent significant two-particle correlations. Extensive RVs n_ch and P_t represent eventwise integrals over some angular domain: a full detector acceptance or smaller bins within an acceptance as described in Ref. <cit.>. Overlines or bars represent ensemble means whereas angle brackets represent eventwise means. The charge-multiplicity variance is
    σ^2_n = (n_ch - n̅_ch)^2≡n̅_ch(1 + Δσ^2_n), where Δσ^2_n represents a non-Poisson contribution from correlations.
    
    Most measurements of “mean- fluctuations” are based on one of two statistical measures with different interpretations. If the emphasis is on eventwise mean  represented by  intensive ratio ⟨ p_t ⟩ = P_t / n_ch as a proxy for local temperature the conventional fluctuation measure is
    σ^2_⟨ p_t ⟩ = (⟨ p_t ⟩ - p̅_t)^2≈ (σ^2_p_t + Δσ^2_⟨ p_t ⟩) / n̅_ch. Alternatively, a system based on  extensive quantities is conditional measure σ^2_P_t|n = (P_t - n_chp̅_t)^2 = n̅_ch (σ^2_p_t + Δσ^2_P_t|n) where in each case Δσ^2_x represents a non-CLT contribution from two-particle correlations and/or dynamical fluctuations that may shed light on collision mechanisms. 
    
    In Ref. <cit.> several statistical measures are compared in reference to  fluctuation measurements reported in Ref. <cit.>. An extensive differential measure is denoted by B̅ = σ^2_P_t|n - n̅_chσ^2_p_t with associated per-particle measure Δσ^2_P_t|n = B̅ / n̅_ch, both negligible for CLT conditions. In Ref. <cit.> the preferred intensive measure is C ≡B̅ / n_ch(n_ch-1)≈B̅ / n̅_ch^2.  fluctuations are then reported in terms of the r.m.s. quantity √(C) / p̅_t ≈√(B̅ / P̅_t^2). One motivation for that construction may be that C ≈σ^2_⟨ p_t ⟩ - σ^2_p_t / n̅_ch, so √(C) / p̅_t may be interpreted as a  fractional r.m.s. ⟨ p_t ⟩ fluctuation measure. However, that choice is based on several questionable assumptions, in particular that a local temperature is a relevant concept and that  non thermodynamic sources (e.g. MB dijets) do not dominate collision dynamics. Per-pair measure C tends to decrease strongly with increasing system size (i.e. å collision centrality) other things being equal, which might suggest that thermalization increases with å centrality. In contrast, fluctuations of extensive measures n_ch and P_t can (and do) exhibit informative TCM trends (strong increases with å centrality) that are concealed by the intensive ratio ⟨ p_t ⟩ = P_t / n_ch.
    
    
    
     §.§ A-A fluctuation data at 200 GeV and 2.76 TeV
    
    
    Figure <ref> (left) shows  fluctuation data for 2.76 TeV  collisions from  Ref. <cit.> in the form √(C) / p̅_t (which decreases monotonically with increasing centrality) transformed to (2/N_part) B̅ rather than B̅ / n̅_ch (a per-particle measure in terms of initial-state participant nucleons rather than final-state charged hadrons) plotted vs mean participant pathlength ν. The trend for more-peripheral collisions is consistent with a TCM GLS trend expected for dijet production in transparent å collisions (dashed line). In more-central collisions the data significantly exceed the GLS trend consistent with increased fragment yields from jet modification as noted for 200 GeV  collisions in Fig. <ref> (a) and (c).
    
    
    
    
    Figure <ref> (right) shows comparable TCM results for 200 GeV  collisions reported in Ref. <cit.>. The general trend is similar but with reduced overall amplitude as expected given the log(s/s_0) collision-energy dependence of dijet production <cit.>. The dash-dotted curves are the GLS expectations extrapolated from isolated  collisions. The dashed lines relate to production from “wounded” projectile nucleons after a first  collision.
    
    
    
     §.§ Inversion of fluctuation scaling at 200 GeV
    
    
    Fluctuation data for a given collision system depend strongly (and possibly nonmonotonically) on the detector acceptance or bin size of a particular analysis as demonstrated in Ref. <cit.>. Results from different experiments are therefore not simply comparable. However, measurements of the scale (bin size) dependence of fluctuations are “portable” and may be inverted to infer underlying angular correlations as first demonstrated in Ref. <cit.>.
    
    Figure <ref> (a) and (b) show  angular correlations from peripheral and central 200 GeV  collisions inferred from inversion of fluctuation scale dependence <cit.>. Model elements representing an AS 1D peak and nonjet quadrupole have been subtracted to isolate the SS 2D peak structure. The structure is equivalent to that in Figs. <ref> and <ref>, and those results appear to confirm that  fluctuations are dominated by MB dijets. However, the complex inversion procedure could be questioned.
    
    
    
    
    Figure <ref> (c) and (d) show  angular correlations from the same collision systems determined directly by pair counting, not by inversion of fluctuation scaling. The  results are in quantitative agreement with (a) and (b) confirming that dijets are the dominant source of  fluctuations in high-energy nuclear collisions. There are significant differences due to reduced angular resolution of the inversion process, but the equivalence is clearly apparent.  fluctuation data compel the conclusion that a local temperature is not relevant  for  fluctuation measurements and their interpretation in high-energy nuclear collisions, and direct correlation measurements by pair counting supply equivalent information more accurately.
    
    
    
     
    
    § DISCUSSION
    
    
    As noted in the introduction, interpretation of high-energy collision data from more-central å collisions near midrapidity tends to follow one of two themes: (a) a flowing bulk medium identified as a QGP with unique properties or (b) a combination of two or three hadron production mechanisms including dijet production, albeit with jet modification increasing with centrality. Theme (a) assumes the existence of flows and local thermalization  a priori and prefers methods and data interpretations that favor a flow narrative. Theme (b) assumes dijet production as the principal manifestation of QCD in high-energy nuclear collisions and uses measurements of eventwise-reconstructed (isolated) jets in elementary collisions to form expectations for other jet manifestations in  and å collisions. It is not unreasonable to pursue both, but theme (b) should be engaged at least as thoroughly as (a) to insure a balanced scientific outcome.
    
    To pursue theme (b) properly requires direct and quantitative comparisons of all available information on isolated jets in elementary collisions with all available information on MB dijets in nuclear collisions over the largest possible range of collision systems.  Comparisons should be based on accurate isolation of jet-related hard components from other contributions to yields, spectra, correlations and fluctuations combined with extensive measures that preserve TCM trends, as demonstrated by various examples and cited references in the present study.
    
    
    
     §.§ Universality of the TCM
    
    
    The TCM represents a conceptually simple idea – two complementary mechanisms dominate hadron production near midrapidity consisting of projectile-nucleon dissociation (soft) and MB dijet production (hard). The TCM concept was first related to RHIC data fifteen years ago <cit.>. Elaboration of the TCM has proceeded in a number of subsequent publications (e.g. Refs. <cit.>). The TCM hard component in hadron yields, spectra, fluctuations and correlations from ,  and å collisions corresponds quantitatively to eventwise-reconstructed jet properties over a large range of  multiplicity, å centrality, collision energies and hadron momenta. The general TCM pattern for  extensive quantities X (e.g. n_ch or P_t) is
    
    (1 /ρ̅_s) X    =     X_s + αρ̅_s X_h   p-p
    
    or
    
    (2/N_part) X    =    X_s + ν X_h   A-A,
    
    where ρ̅_s is by hypothesis a proxy for the number of participant low-x partons (gluons) in  collisions and N_part is the number of participant nucleons in å collisions according to the Glauber model of that collision system. When left-hand-side quantities are plotted vs ρ̅_s () or ν (å) the TCM signature is a constant (soft component) plus an approximately linear increase (jet-related hard component).  Those trends are illustrated for X = integrated charge  in Fig. <ref> (left) () or Fig. <ref> (å), for integrated P_t in Fig. <ref> (right) (), for P_t variance difference B̅ in Fig. <ref> (å) and for number of correlated hadron pairs Δρ in Fig. <ref> (a) and (c). 
    Intensive ratios  tend to conceal jet-related TCM trends by partial cancellations.
    
    
    
     §.§ Phenomenology of MB dijets
    
    
    For MB analysis as described in this study all collision events and all hadrons within each collision are accepted for analysis. No conditions are imposed (except those that define a detector acceptance). All hadron fragments from dijets that enter the detector acceptance are therefore retained within the primary particle data. If properties of the MB dijet population for a given collision system are known (i.e. measured) the detected fragment system should correspond quantitatively to predictions derived from jet data as a  critical test for any jet interpretations of data. An example is provided in  Sec. <ref>.
    
    Section <ref> summarizes a comprehensive description of properties of isolated jets in terms of FFs and jet (scattered-parton) energy spectra represented by simple and accurate parametrizations for an array of collision systems <cit.>. Those parametrizations permit quantitative comparisons with MB fragment data from nuclear collisions. For some analysis methods (e.g. those associated with the TCM) the comparisons demonstrate accurate correspondence (e.g. Refs. <cit.>). For other methods (some of those associated with the flow narrative) there is no clear pattern (e.g. Refs. <cit.>).
    
    Descriptions of dijet contributions in terms of pQCD theory cannot represent MB dijets. Due to its inevitable limitations pQCD can represent less than 10% of jet fragments <cit.>. Next-to-leading-order (NLO) predictions of  spectrum structure are usually constrained to p_t > 2 GeV/c with large systematic uncertainties near the lower bound (see the next subsection), whereas MB fragment distributions peak well below that point (Figs. <ref> and <ref>).
    
    Biased representations of fragment data distributions that rely on  model-dependent background subtraction,  ad hoc cuts or intensive (spectrum or statistics) ratios represent only a distorted fraction of the MB fragment population. Comparisons of  such biased data to eventwise-reconstructed jet properties is likely misleading and cannot test jet-related interpretations of data components.
     
    
    
     §.§ Comparisons with NLO p_t spectrum predictions
    
    
    A substantial effort has been expended to compare NLO pQCD theory predictions of inclusive hadron production with measured hadron  spectra <cit.>. The form of such predictions is represented schematically by
    d^2σ/dp_t dη   =   ∫dz/z∬ dx_1 dx_2 f_p_1^h_1(x_1) f_p_2^h_2(x_2) 
    
       ×   d^2σ̂(x_1,x_2,p̂_t)/dp̂_t dη D_p_3^h_3(z),
    
    a convolution of projectile-hadron parton distribution functions (PDFs) f_p^h(x), QCD parton scattering cross section d^2σ̂(x_1,x_2,p̂_t)/dp̂_t dη and scattered-parton FF D_p^h(z). It is argued that comparisons of NLO predictions to inclusive spectra may test the role of jets in nuclear collisions, but there are problems with that argument.
    
    Such NLO predictions (e.g. Ref. <cit.>) assume that almost all hadrons from  collisions are the result of binary parton-parton collisions. But that assumption conflicts with the TCM inferred from 200 GeV  collisions (Sec. <ref>) wherein most hadrons emerge from a (soft) process that does not scale with a number of binary collisions and is interpreted to represent dissociation of single projectile nucleons as the result of a soft  interaction. Instead, Eq. (<ref>) corresponds to Eq. (<ref>) describing (within a constant factor) the  spectrum hard component.
    
    [Note that p̂_t is here the scattered-parton (jet) momentum,  is a fragment momentum, z = p_t / p̂_t, dz/z = dln(z) = dy_max for given , y_max≡ln(2p̂_t / m_π) and ln(1/z) = y_max - y = ξ (conventional FF parameter). The assumption that FFs D(z) depend only on z or y_max - y is contradicted by  FF data as in Fig. <ref>, and  FFs have a universal form on u ≈ y/y_max <cit.>.]
    Comparisons between NLO predictions and isolated spectrum hard components might provide a useful test of MB dijet production, but there are further issues.
    
    Because of limitations on the applicability of perturbative methods NLO predictions are typically restricted to hadron p_t > 2 GeV/c (approximating p_t ≫ 1 GeV/c). But examples above and cited references demonstrate that most jet fragments (> 90%) appear in p_t < 2 GeV/c, and that fraction plays a dominant role in yields, spectra, fluctuations and correlations from  and å collisions.
    
    An alternate (perhaps the main) purpose for comparison of NLO predictions to hadron spectra is tests of QCD factorization and FF universality and optimizing FF parametrizations from  global fits to ,  and  or  data. A recent review stipulates that NLO predictions for jet spectra combining PDFs with pQCD parton-parton cross sections describe the spectrum data well <cit.>. The major remaining uncertainty lies with current FFs, especially gluon-to-unidentified-hadron FFs whereby hadron spectra are typically overpredicted by a factor 2. It is proposed to reduce such errors by refinement of FFs through improved global fits to data.
    
    However, one can question such a strategy, especially the assumption of FF universality as it is conventionally invoked. Section <ref> illustrates the large discrepancies between light-quark FFs derived from  collisions and those inferred from  collisions. The differences lie well outside systematic uncertainties.  light-quark FFs used to describe the  spectrum hard component  already produce a large overprediction <cit.>. But low- hadrons that dominate the spectrum hard component near midrapidity arise mainly from low-x gluons. If  gluon FFs are applied instead the overprediction is worse due to the increased (and softer) hadron yield from gluon jets. The approximate agreement between jet systematics and spectrum hard components as in Fig. <ref> is achieved only by using  light-quark FFs <cit.>. The result suggests that there are substantial differences in FFs depending on the jet environment in  collisions. Strong jet modification in å collisions is a notable result of the RHIC experimental program confirmed at the LHC. But jets appear to be modified significantly already in  collisions relative to  collisions. As a result, attempts to optimize “universal” FFs via global fits to multiple systems may produce a strongly-biased result.
    
    Another issue for NLO comparisons is the structure of spectrum hard components below 5 GeV/c (e.g. Fig. <ref>). The maximum near 1 GeV for pions and protons <cit.> (and therefore possibly kaons and other hadrons) is probably outside the scope of pQCD. The evolution of jet-related structure with å centrality as in Sec. <ref> is certainly so, consistent with the conclusion that “...only the region above p_T≈ 10 GeV/c of these charged-hadron data, with theoretical scale uncertainties below ±20%, should be included in forthcoming global fits of parton-to-hadron fragmentation functions” <cit.>. However, the effectiveness of such global fits may still be questioned as noted above.
    
    Three areas may then be distinguished: (a) comparison of measured isolated-jet characteristics with various MB jet manifestations in high-energy nuclear-collision data as described in the present study, (b) comparisons of PDF and pQCD parton-spectrum combinations with measured jet spectra as one test of QCD factorization and (c) improved measurement of scattered-parton FFs in elementary collisions with the understanding that FFs may depend on collision context – e.g.  vs  – as an extension of jet modification in å collisions.
    
    
    
     §.§ Methods that misidentify or distort MB dijets
    
    
    Section <ref> provides an example of analysis methods that tend to minimize and/or distort MB dijet manifestations in nuclear-collision data. Other methods tend to reveal contributions from MB dijets consistent with isolated-jet measurements as described in this study. A principal difference is the amount of information carried by primary particle data that is retained by a method and may be utilized for direct comparison with isolated jets.
    
    Examples are provided below for integrated  or P_t vs  () or centrality (å), SP  spectra vs  or centrality,  correlations, 2D angular correlations,  fluctuations and choice of plot formats.
    
    Integrated charge  from å collisions may be plotted as (2/N_part) n_ch vs ν over the complete å centrality range to reveal a TCM trend with strong MB dijet contribution as in Fig. <ref>, varying from a GLS trend extrapolated from  for more-peripheral å collisions to a trend reflecting strong jet modification for more-central collisions. Alternatively, the same per-participant quantity may be plotted vs N_part over a limited centrality interval (e.g. top 40% as in Ref. <cit.>), and the  GLS extrapolation is then effectively concealed. Alternative nonjet hypotheses may seem to describe the more-central data but could be falsified by more-peripheral data (e.g. the CGC dashed trend in Fig. <ref>, left).
    
    A full analysis of  spectra vs   or å centrality over a large  acceptance may reveal all available information in spectrum data through inductive study as in Refs. <cit.> leading to or consistent with the TCM. Alternatively, spectra for single collision systems fitted over limited  intervals with  a priori model functions may appear to support nonjet interpretations (e.g. radial flow as in Ref. <cit.>). Studies that seem to emphasize jets by applying “high-” cuts may actually discard almost all evidence of MB jets (more than 90% of MB jet fragments).
    
    Possible two-particle correlation studies include  correlations as in Fig. <ref> (left) for various collision systems and several combinations of pair angular acceptance, and 2D angular correlations for various  conditions including full pair  acceptance <cit.>. The result in Fig. <ref> (left) provides compelling evidence for the TCM, is directly comparable to the SP  spectrum TCM, and the hard-component peak centered near (2.7,2.7) corresponds directly to jet-related angular correlations as in Fig. <ref> (right). It is notable that  correlations remain largely unexplored despite the essential information they convey.
    
    2D angular correlations for a number of collision systems have been studied and characterized by a universal 2D fit model <cit.>. The results are quantitatively compatible with TCM results from other analysis including MB jet spectrum manifestations <cit.>. An important feature of 2D angular correlations is resolution of the jet-related SS 2D peak from other structure that may or may not be jet related. In contrast, projection of 2D angular correlations onto 1D azimuth discards information relating to η dependence and reduces the ability to distinguish jet-related structures from others. 
    
    In particular, the NJ quadrupole cannot be uniquely distinguished from the quadrupole component of the jet-related SS 2D peak. Some fraction of the latter may then be attributed to “elliptic flow”v_2 leading to distortion of nominal jet structure following ZYAM background subtraction <cit.> or attribution of all MB jet structure to flows including “triangular” <cit.> and “higher harmonic” <cit.> flows <cit.>.
    [While some data features attributed to flows may represent MB dijet manifestations the NJ quadrupole is most likely a distinct  nonjet phenomenon with a unique physical mechanism <cit.>.]
    
    While extensive quantities reveal clear TCM trends (as in this study) intensive ratios of such quantities discard essential information in primary data by canceling data trends required to interpret collision data, especially MB jet manifestations. Examples include spectrum ratio R_AA that obscures contributions from MB jets below p_t ≈ 3 GeV/c including more than 99% of MB jet fragments,  eventwise mean ⟨ p_t ⟩ = P_t / n_ch and corresponding ensemble mean p̅_t that partially obscure TCM trends in the extensive numerators and denominators, and ratio v_2(p_t) including NJ quadrupole spectrum (numerator) and SP hadron spectrum (denominator) that may represent different hadron populations confused by the v_2 ratio <cit.>. Extensive fluctuation measures (e.g. variance differences such as Δσ^2_P_t|n) convey almost all information on underlying correlations including MB jet manifestations, whereas intensive ratios of statistical quantities (or  ratios of ratios such as √(C̅) / p̅_t) tend to suppress jet manifestations and may favor interpretations within a thermodynamic context (see Sec. <ref> for notation).
    
    The choice of plot format, including independent-variable choice, may further suppress essential information. The choice of participant number N_part as å centrality measure over mean participant pathlength ν visually suppresses the more-peripheral part of centrality dependence, especially GLS trends extrapolated from  collisions that provide a valuable reference for å collisions. The choice of linear  over rapidity  [or at least log(p_t)] visually suppresses the low- region where most MB jet fragments appear. Semilog plots showing model curves passing through spectrum data points are often misleading; highly significant deviations may appear much smaller than the plotted points (as are the statistical uncertainties). Plots of spectrum data/model  ratios are similarly misleading because deviations at lower  may be strongly suppressed. Fit quality is tested only by comparing [data - model]  differences in ratio to statistical uncertainties, as demonstrated in Fig. <ref> (left).
    
    
    
     §.§ Consequences for physical interpretation
    
    
    Section <ref> notes that a specific analysis may rely on sequential selection from several alternative methods. A given combination of methods may then contribute to support of a preferred narrative whereas a different combination might falsify that narrative. A possible criterion for optimum selection is maximized use of the information carried by primary particle data. Another is insistence on consistent application and interpretation across a range of collision systems and measured quantities.
    
    The present study demonstrates that one can assemble a combination of methods providing clear evidence for manifestations of MB dijets in all collision systems. Measure choices are guided by the principle that almost all available information from nuclear collision data is compared with almost all available information from isolated jets. No information is intentionally discarded. Measured MB jet manifestations then correspond quantitatively and consistently to properties of isolated jets established by separate experiments. Recognition of MB jet manifestations as such then precludes much of the evidence advanced to support flow interpretations.
    
    In contrast, alternative combinations of methods can be assembled that minimize MB jet manifestations and appear to support a flow narrative to describe high-energy nuclear collisions (possibly even in small systems). However, as argued and demonstrated with examples in the present study, such preferred methods tend to discard major fractions of the information in primary particle data – in the form of projections to lower dimensions,  cuts based on a preferred narrative, intensive ratios and nonoptimal plotting formats – and discard information in isolated jet data by invoking pQCD as an imposed intermediary context (a sort of filter). Established jet physics is thereby largely excluded from descriptions of high-energy nuclear collisions in favor of a flow narrative.
    
    
    
     
    
    § SUMMARY
    
    
    Primary particle data from high-energy nuclear collisions may be processed with alternative classes of analysis methods that seem to support one of two narratives: (a) collisions are dominated by flows carried by a dense medium (quark-gluon plasma or QGP) or (b) collisions are dominated near midrapidity by two hadron production mechanisms consisting of projectile-nucleon dissociation and minimum-bias (MB) dijet production.
    
    Because a data analysis system leading to physical interpretations typically relies on multiple selections from among several possible methods there is no unique method combination. Results of the present study suggest that flow-QGP interpretations tend to arise from a certain class A of analysis methods  imposed  a priori within a narrative context that exhibit substantial  information discard. Interpretations based on MB dijets are favored by an alternative class B inferred inductively from primary particle data that retain most information.
    
    This article presents a detailed study of measured manifestations of MB dijets in high-energy nuclear collisions and quantitative comparisons of those manifestations with measured properties of eventwise-reconstructed (isolated) jets. The study considers jet manifestations in yields, spectra, correlations and fluctuations from a range of collision systems. The MB dijet system is then used to evaluate properties of alternative analysis methods and the quality of support for the two competing narratives.
    
    The two-component (soft + hard) model (TCM) of hadron production near midrapidity provides a context for MB dijets and narrative (b). The TCM as applied in this work was derived inductively from the  dependence of   spectra. The soft and hard components of yields, spectra, correlations and fluctuations are observed to evolve independently with  charge multiplicity or å centrality. For each form of primary data the TCM hard component is quantitatively related to measured isolated-jet properties. For instance, evolution of the   spectrum hard component with collision energy tracks quantitatively with measured isolated-jet spectra. Evolution of hadron yields with å centrality and collision energy follows simple TCM trends on centrality and a QCD trend log(s / s_0) on collision energy with √(s_0)≈ 10 GeV. The observed TCM simplicity applies to methods based on  extensive variables such as  and P_t integrated over some acceptance and supports narrative (b).
    
    Alternative methods that appear to support narrative (a) rely on  intensive ratios (or ratios of ratios),  a priori imposed fit models, projections from higher- to lower-dimensional spaces, imposed  cuts and subtraction of  ad hoc backgrounds that tend to discard essential information carried by primary particle data and may then lead to biased and distorted results. Those conclusions are based on manifestations of MB dijets using such methods in quantitative comparison to the measured properties of isolated jets. A number of examples are provided in this study. The overall results suggest that at least some data features interpreted to arise from flows actually represent MB dijet manifestations.
    
    I conclude that the TCM for high-energy nuclear collisions emerges naturally from inductive analysis of yield, spectrum and correlation data, is not imposed  a priori. The TCM hard component agrees quantitatively with the measured properties of isolated jets. Statistical measures based on extensive variables retain substantially more of the information carried by primary particle data. When various data features are reexamined in the context of isolated-jet measurements little substantial evidence remains to support a flow narrative. And any description of high-energy nuclear collisions that omits a clear, quantitative description of MB dijets consistent across all measures and collision systems may be questioned.
    
    
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