Collectivity and manifestations of minimum-bias jets in high-energy nuclear collisions

Collectivity, as interpreted to mean flow of a dense medium in high-energy A-A collisions described by hydrodynamics, has been attributed to smaller collision systems -- p-A and even p-p collisions -- based on recent analysis of LHC data. However, alternative methods reveal that some data features attributed to flows are actually manifestations of minimum-bias (MB) jets. In this presentation I review the differential structure of single-particle $p_t$ spectra from SPS to LHC energies in the context of a two-component (soft + hard) model (TCM) of hadron production. I relate the spectrum hard component to measured properties of isolated jets. I use the spectrum TCM to predict accurately the systematics of ensemble-mean $\bar p_t$ in p-p, p-A and A-A collision systems over a large energy interval. Detailed comparisons of the TCM with spectrum and correlation data suggest that MB jets play a dominant role in hadron production near midrapidity. Claimed flow phenomena are better explained as jet manifestations agreeing quantitatively with measured jet properties.


Introduction
Collectivity (interpreted to represent hydrodynamic flows [1]) has been attributed recently to smaller collision systems [2] based on certain correlation phenomena conventionally attributed to flows in larger A-A systems (e.g. v 2 data). But the term "collectivity" simply represents any correlated collection: correlations ⇒ collectivity. For instance, dijet production represents a collective phenomenon that arguably dominates high-energy nuclear collisions. Several hadron production mechanisms may contribute to observed correlations. It should be our task to identify those mechanisms by comprehensive data analysis based on a variety of methods. The goal is then to extract all information from available particle data and interpret that information based on recognized physics principles. In this study measuredp t trends for three collision systems are compared to corresponding p t spectra and measured jet properties.

Two-component (soft + hard) model or TCM for A-B collisions
The TCM is a comprehensive and accurate model of hadron production in A-B collisions near midrapidity inferred inductively from p t spectra [3,4] and angular correlations [5][6][7]. For instance, intensive charge densityρ 0 can be separated into two components:ρ 0 =ρ s +ρ h (soft + a e-mail: ttrainor99@gmail.com arXiv:1709.10229v2 [hep-ph] 4 Oct 2017 The Journal's name hard). Based on comprehensive data analysis (e.g. [5,8,9]) the two components have been interpreted physically as follows: Soft component SC results from projectile-nucleon dissociation to participant low-x gluons ∼ρ s ≡ n s /∆η ∝ log( √ s/10 GeV). Hard component HC results from large-angle scattering of participant gluons which then fragment to a minimum-bias (MB) ensemble of dijets. Measured SC and HC (yields, spectra and correlations) are related in p-p collisions viaρ h ≡ n h /∆η ≈ αρ 2 s (noneikonal trend) with α ≈ O(0.01). An eikonal trend (e.g. as in the Glauber model of A-A collisions) would instead followρ h ∝ρ 4/3 s [10]. The p-pρ h ≈ αρ 2 s trend is illustrated in Fig. 1 (a) as n h /n s ≈ 0.005ρ s (within ∆η = 1) [3]. A TCM for hadron production in A-B collisions follows suite but requires additional geometry parameters defined within the Glauber model of collision geometry based on the eikonal approximation. N part /2 is the number of participant nucleon pairs, N bin is the number of N -N binary collisions, and ν ≡ 2N bin /N part is the mean number of binary collisions per participant pair. For instance, the event-ensemble mean of extensive total P t (p t integrated within some acceptance ∆η) can be represented in TCM form asP t =P ts +P th or P t = (N part /2)n sN NptsN N + N bin n hN NpthN N ;P t /n s =p ts + x(n s )ν(n s )p thN N (n s ), (1) where x ≡ρ hN N /ρ sN N = n hN N /n sN N and the TCM incorporates factorization of N-N lowx gluon participants (noneikonal) and A-B nucleon participants (eikonal). Note thatp t SC p tsN N →p ts ≈ 0.4 GeV/c is observed to be universal for N -N collisions within A-B collisions.
3 Minimum-bias jet manifestations in p-p p t spectra and correlations Quantitative correspondence between measured jet properties and single-particle p t spectra has been demonstrated previously for 200 GeV p-p and Au-Au spectra down to 0.5 GeV/c [8]. The demonstration relies on the observation that jet spectra and fragmentation functions (FFs) have simple universal forms (Gaussian and beta distribution) when plotted on rapidity variables. Some details are reviewed in this section (angular correlations are described below). (d) Figure 1. (a) Hard/soft ratio vs soft densityρs. (b) Jet pt spectra for several energies from the ISR and SppS. Curves through data are from Ref. [9]. (c) Fragmentation functions from p-p collisions described by a simple parametrization from Ref. [9]. (d) Spectrum hard component for 200 GeV NSD p-p collisions (solid points) compared to a fragment distribution as predicted by Eq. (2) (dashed). Figure 1 (b) shows jet spectra from the ISR (43 and 63 GeV points) and SppS (remaining data points). The curves are derived from a universal parametrization of jet spectra applicable up to 13 TeV [9]. Panel (c) shows jet FFs from p-p collisions obtained by the CDF collaboration plotted on rapidity variable y = ln[(E + p)/m π ]. The FFs show self-similar variation with jet energy making a simple parametrization possible. Schematically, the fragment distribution (FD) P (p) describing the contribution to a p t spectrum from MB dijets (spectrum HC) is the convolution of jet spectrum P (E) for a given p-p collision energy and FF ensemble P (p|E) with E min ≈ 3 GeV established by comparisons with hadron spectra [8,11]. In panel (d) the dashed curve is P (d). The solid points are the spectrum HC from 200 GeV NSD p-p collisions [3]. The open boxes are the 200 GeV UA1 jet spectrum appearing in panel (b). It is notable that the majority of jet fragments appear near 1 GeV/c. If E min is reduced significantly below 3 GeV a large overestimate of the measured FD contribution below 1 GeV/c results, since the jet spectrum scales as dσ jet /dp t ∼ 1/p 6 t near that energy. Two-particle angular correlations have been studied for 200 GeV p-p collisions [5][6][7] and Au-Au collisions [12]    GeV p-p collisions with no special p t cuts. The main features are a SS (same-side on azimuth) 2D peak representing individual jets and an AS (away-side) 1D peak on φ ∆ representing backto-back jet pairs. Superposed on the broader SS peak is a narrower 2D contribution from Bose-Einstein correlations. Just visible is a small SC contribution -a narrow 1D peak on η ∆ . Figure 2 (b-d) shows systematic variation with n ch or n s of individual correlation amplitudes obtained by 2D model fits to correlation data [5]. The linear per-hadron trends in panels (b,c) indicate that the correlation hard component (dijets) follows a noneikonal trend: jet-correlated pairs ∝ρ 2 s as observed for the p-p spectrum HC. Panel (d) shows a quadratic trend onρ s for nonjet quadrupole component A Q indicating that quadrupole correlated pairs are ∝ρ 3 s . That trend is actually consistent with the quadrupole trend observed for A-A collisions. Ifρ s ∼ low-x participant gluons andρ 2 s ∼ participant-gluon binary collisions then ρ 3 s ∼ N part N bin for p-p collisions. The centrality trend observed for the nonjet quadrupole (ρ 2 0 v 2 2 ) in Au-Au collisions is N part N bin 2 opt [13]. Absence of a 2 opt factor for p-p collisions is consistent with the noneikonalρ h ∝ρ 2 s trend: centrality is not relevant for p-p collisions.
The Journal's name 4 p-p p t spectrum TCM hard components The illustrations above pertain to 200 GeV RHIC data as a testbed for MB dijet manifestations in spectra and angular correlations. In preparation forp t analysis the full collision-energy and charge-multiplicity dependence of the spectrum hard component should be understood. Figure 3 (a) shows the TCM hard components for p-p p t spectra as a function of collision energy from √ s = 17 GeV to 13 TeV (curves of various line styles) [4] compared to data for two energies (points) [3,14]. The TCM spectrum HC model is simple and relies on a few log( √ s) terms. The description of spectrum data is in all cases within systematic uncertainties of the data. Spectrum hard components show significant dependence on event multiplicity n ch (e.g. multiple fine lines for the 200 GeV model). For thep t study fiducial hard-componentp th0 values in panel (b) are inferred from a symmetric spectrum HC model (forρ s ≈ 2ρ sN SD ) [15].    [5]. The n ch dependence of the shape has been known for more than ten years [3] and is attributed to bias of the underlying jet energy spectrum in response to the n ch condition. The TCM is modified to accommodate those changes as follows: The hard-component model is a Gaussian plus exponential tail. The transition from Gaussian to exponential is determined by slope matching. The Gaussian widths are treated separately below and above the mode with widths σ yt− and σ yt+ respectively. The exponential constant q is a separate parameter. Figure 3 (d) shows evolution of spectrum HC model parameters σ yt+ and q with n ch at 200 GeV and 13 TeV. The trend for the width below the mode σ yt− is available in Ref. [4]. For 200 GeV the parameter trends are tightly constrained by spectrum data over the full n ch interval relevant top t data. For 13 TeV the spectrum data span a more-limited n ch interval. The 2/q trend (upper solid curve) has been extended by fittingp t data in the present study.
Thep t analysis of Ref. [16] produced uncorrectedp t values, where the prime denotes the effect on spectra of a lower-p t acceptance cut at p t,min near 0.15 GeV/c. The p t acceptance cut strongly affects thep t SC but has negligible effect on the HC which is small below 0.5 GeV/c. The relation isp ts =p ts /ξ where ξ is the SC efficiency ≈ 0.78 for an effective p t cutoff near 0.17 GeV/c. For universalp ts ≈ 0.4 GeV/c uncorrectedp ts ≈ 0.51 ± 0.2 GeV/c. The TCM forp t data from p-p collisions is constructed as follows [15]: The uncorrected yield n ch within angular acceptance ∆η relative to the corrected soft component n s is n ch /n s = ξ +x(n s ) with ξ = 0.76 -0.80 (depending on the effective p t,min ). For noneikonal p-p collisions is predicted from jet characteristics [4]. The TCM for uncorrectedp t is the first of wherep ts ≈ 0.4 GeV/c and p th (n s , √ s) is the new information derived from p-pp t data.    Figure 4 (b) shows correctedP t /n s data (points) according to the second of Eqs. (3). In that format the TCM is a straight line with slope α( √ s)p th0 predicted from p t spectra and jet data. Figure 4 (c) showsp th (n s , √ s) data inferred from the uncorrectedp t data according to Eqs. (3). The horizontal lines are thep th0 ( √ s) values appearing in Fig. 3 (b). Clearly evident are substantial variations with n s orρ s . However, such variations are expected based on previous spectrum studies as in Refs. [3,4]. Figure 4 (d) showsp th (n s ) values for 200 GeV and 7 TeV (points) compared to a TCM including HC n ch dependence according to Fig. 3  (d). The correspondence between TCM andp t data is within data uncertainties. Accurate correspondence between p-pp t data and the TCM compels the conclusion that there is a direct, quantitative correspondence among measured isolated-jet properties, p t spectrum hard components andp t data. p-p collisions are dominated by MB dijets and are noneikonal.

p-Pbp t TCM analysis
For a TCM description of p-Pb collisions [15] the composite structure of the Pb nucleus must be incorporated by adding mean participant pathlength ν(n s ) ≡ 2N bin /N part as a geometry measure, so for instance n ch /n s → ξ + x(n s )ν(n s ). The soft-component charge density factorizes asρ s ≡ (N part /2)ρ sN N (n s ), similarlyρ h ≡ N binρhN N (n s ) and hard/soft ratio x(n s ) →ρ hN N (n s )/ρ sN N (n s ) ≈ αρ sN N (n s ) averaged over N -N collisions as for p-p collisions. Combining those relations gives N part (n s )/2 = αρ s /x(n s ), and for p-A collisions N part ≡ N bin + 1. Thus, if a model for x(n s ) is specified then N part , N bin and ν are also defined. As described below, x(n s ) for more-peripheral p-A collisions is based on the noneikonal p-p trend and is then extrapolated to more-central collisions with N part /2 > 1 based onp t data. The TCM for p-Pbp t data is similar to Eqs. (3) but with additional ν(n s ) factors p t ≡P t n ch ≈p ts + x(n s )ν(n s )p thN N (n s ) ξ + x(n s )ν(n s ) ;P t (n s ) n s =p ts + x(n s )ν(n s )p thN N (n s ) assumingp thN N (n s ) ≈p th0 (p-p values) for p-Pb collisions (no jet modification in p-A).  shows the corresponding ν(n s ) trend (solid curve) defined by x(n s ) as noted above. The dash-dotted curve is a Pb-Pb ν(n s ) trend for comparison. Figure 5 (c) shows uncorrectedp t data from Ref. [16] (points) withp ts ≈ 0.525 GeV/c. The solid curve is the TCM defined by Eq. (4) (first) withρ s0 = 15 and m 0 = 0.1 adjusted to accommodatep t data andp thN N (n s ) →p th0 = 1.30 GeV/c from Fig. 3 (b). The dashed curve is the p-p TCM for 5 TeV. Figure 5 (d) shows the same features for correctedP t /n s data with TCM described by Eq. (4) (second). This p-Pb exercise illustrates the transition from noneikonal p-p or p-N trend to eikonal p-A trend with increasing multiplicity. The value of N part /2 remains near one (p-N only) up to the transition point but then slowly increases as greater p-A centrality becomes competitive to deliver larger overall charge multiplicity.

Pb-Pbp t TCM analysis
For the Pb-Pbp t TCM published yield measurements and a Glauber model based on the eikonal approximation can be used to determined some TCM model elements. Parameter x(n s ) can be derived from per-participant-pair charge-density data based on the TCM Parameter ν(n s ) can be derived from the Glauber model in the form ν(n s ) ≈ (N part /2) 1/3 [10] with N part /2 ≈ρ s /ρ sN N as in the previous section andρ sN N ≈ρ sN SD in both cases. Figure 6 (a) shows per-participant-pair yield data from Ref. [17] (points). The dashed line is a Glauber linear superposition (GLS) model representing no jet modification. The upper bold dotted curve shows the expected effect of multiplicity fluctuations for more-central A-A collisions [18]. The solid curve is the TCM in Eq. (5) with x(ν) defined below. Panel (b) shows solution of Eq. (5) for parameter x(ν) (points) assumingρ sN N →ρ sN SD ≈ 4.6 [4]. A sharp transition (ST), first observed for Au-Au jet-related 2D angular correlations [12], representing evidence of jet modification in the yield trend [8] is evident. The bold dotted curve is a simple TCM expression for x(ν) that starts from the NSD p-p value x pp ≈ 0.045, increases rapidly through the ST and maintains a saturation value ≈ 0.142 for more-central collisions, approximately threefold increase over the NSD value. The dashed curve is the equivalent for 200 GeV Au-Au collisions. The intervalρ s = 10 -40 is not well-defined a priori by yield data so a TCM expression is required that includes an adjustable x pp [15] x with ST near ν = 2.3.    Fig. 6 (c,d). The dashed curve is the corresponding TCM for p-p data. The dash-dotted curve is a GLS model assuming no jet modification. Figure 7 (b) shows product x(n s )p thN N (n s ) ≈P thN N /n sN N (open squares) obtained from the data in panel (a) according to Eqs. (4) given the ν(n s ) model in Fig. 6 (c). The open circles are corresponding p-p data. The solid curve is the Pb-Pb TCM and the dashed curve is the p-p TCM. Note that the ST is clearly evident in this more-differential plot format, but  The solid curve through thep thN N (n s ) data, representing new information obtained from Pb-Pbp t data, is a parametrization that forms the third element of the Pb-Pbp t TCM [15].

Comment on Glauber-model analysis of p-Pb collision centrality
The TCM for p-Pbp t data described in Sec. 6 and Ref. [15] can be compared with a recent Glauber analysis of 5 TeV p-Pb centrality in Ref. [19]. For the Glauber analysis it is assumed that integrated charge multiplicity is proportional to Glauber N part : "[n ch ] at mid-rapidity scales linearly with [N part ]." That assumption is equivalent to assuming that a probability (i.e. event-frequency) distribution on charge multiplicity -dP/dn ch -is equivalent to a fractional cross-section distribution on N part -(1/σ 0 )dσ/dN part -inferred from a Glauber Monte Carlo. Figure 8 (a) shows an event-frequency distribution (histogram) on VOA amplitude (charge multiplicity within a pseudorapidity interval in the Pb hemisphere) from Ref. [19]. The assumption N part ∝ n ch is equivalent to dP/dn ch ∼ (1/σ 0 )dσ/dN part as noted above, and the fractional cross sections in the form 100(1 − σ/σ 0 ) noted in the panel and apparently obtained from a running integral of the histogram seem to be consistent with that statement. Figure 8 (b) shows Glauber estimates of (2/N part )dn ch /dη from Ref. [19] (solid points) compared to the TCM equivalent for p-p (dashed) and p-Pb (solid) collisions from Ref. [15] as reported above. The large discrepancy is apparent. Given N part and N bin from the Glauber analysis corresponding values of ν are plotted in panel (c) (solid dots). The TMC equivalent is the solid curve, and the Glauber trend for Pb-Pb collisions is shown as the dashed curve. That ν for p-Pb collisions might anywhere greatly exceed that for Pb-Pb collisions is notable.   [19] reports 0-5% central p-Pb collisions forρ 0 ≈ 45 the p-Pbp t data from Ref. [16] (same collaboration) extend toρ 0 ≈ 120, nearly three times larger. The bold dotted curve suggests a more realistic trend for b/b 0 .
The basic problem with the analysis in Ref. [19] is the initial assumption N part ∝ n ch which is strongly violated for p-Pb collisions. Just as for p-p collisions in Sec. 5 initial increase of the n ch condition is more easily met in p-Pb collisions by higher multiplicities in individual p-N collisions than by increasing p-Pb centrality. N part /2 ≈ 1 persists untilρ s0 ≈ 15, three times the NSD value. Only then does N part /2 begin to increase above 1 significantly. Below that point dσ/dn ch ≈ 0 and p-Pb "centrality" remains fixed at 100%. The correct relation is where dσ/dN part may be obtained from a Glauber Monte Carlo but the Jacobian dN part /dn ch is determined by the p-Pbp t TCM as described above. As noted, the Jacobian factor is essentially zero over a substantial n ch interval below transition pointρ s0 in Sec. 6. Based on the assumption in Ref. [19] the Jacobian would have a fixed value near 2/n chN SD for all n ch .

Summary
Recently, claims have emerged that hydrodynamic flows must play a role in smaller collision systems (p-A and even p-p) based on observation of the same phenomena in those systems attributed to flows in more-central A-A systems. However, the argument could be reversed to conclude that if phenomena appear in collision systems so small and low-density that flows arising from particle rescattering are impossible then the same phenomena observed in more-central A-A may not represent flows. Recentp t data may contribute to a resolution. The two-component (soft + hard) model (TCM) of hadron production near midrapidity provides an accurate and comprehensive description of yield, spectrum and correlation data applicable to any A-B collision system. While one or more other mechanisms may contribute to hadron production in minor ways (e.g. a nonjet quadrupole component) projectile-nucleon dissociation (soft) and dijet production (hard) appear as the dominant processes in all cases.
From TCM analysis ofp t data described above for three successive collision systems the following may be concluded: (a) p-pp th (n s , √ s) hard-component trends agree with spectrum HC evolution and measured MB dijet properties. (b) p-p dijet production is noneikonal, p-p centrality is not relevant. (c) The p-Pbp th (n s ) hard component establishes factorization of A-B Glauber and N -N noneikonal trends. (d) p-Pbp t data confirm that MB dijets dominatē p t (n s ) variation. (e) In effect, MB dijets probe the centrality evolution of p-Pb collisions with n ch . (f) Pb-Pbp t data confirm that the Glauber model effectively describes more-central A-A collisions, but peripheral collisions accurately follow p-p (i.e. N -N ) trends. (g) The Pb-Pb p thN N (n s ) trend confirms that jets are modified quantitatively above the ST, but jets still dominate spectrum and correlation structure in more-central Pb-Pb collisions.
Thus, the likelihood that flows play any role in small collision systems appears negligible. Claimed novel phenomena may be better explained either as due to MB dijets or as a related QCD process (e.g. multipole radiation). The same argument may be extended even to the most-central A-A collisions at the highest energies. Before novelty is claimed the contribution of MB dijets and related QCD processes to data trends must first be well understood.