Quark number scaling of hadronic $p_T$ spectra and constituent quark degree of freedom in $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV

We show that the experimental data of $p_T$ spectra of identified hadrons released recently by ALICE collaboration for $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV exhibit a distinct universal behavior --- the quark number scaling. We further show that the scaling is a direct consequence of quark (re-)combination mechanism of hadronization and can be regarded as a strong indication of the existence of the underlying source with constituent quark degree of freedom for the production of hadrons in $p$-Pb collisions at such high energies. We make also predictions for production of other hadrons.

Introduction -The striking features observed recently by ALICE and CMS collaborations for high multiplicity events at Large Hadron Collider (LHC) such as long range angular correlations [1,2], flow-like patterns [3], enhanced strangeness [4,5] and baryon to mesons ratios at soft transverse momenta [6,7] have attracted many discussions [8][9][10][11][12][13][14][15][16][17]. A core problem is whether Quark Gluon Plasma (QGP) is also formed in such small system in pp and p-Pb collisions. At the same time, a series of measurements of transverse momentum p T spectra have also been carried out and high accuracy data have been obtained [4,18,19] even for decuplet hyperons such as Ω − , Ξ * and Σ * and vector mesons such as φ and K * in p-Pb collisions at √ s NN = 5.02 TeV. Because the decay influence is almost negligible, behaviors of such hadrons are usually believed as carrying more direct information from hadronization. It is thus of particular interest to see whether such data [4,18,19] show any regularities that may lead to deeper insights into reaction mechanism.
Quark number scaling of p T spectra in p-Pb collisions at LHC energies -For all the decuplet hyperons and vector mesons, we see in particular that Ω − and φ are composed of only strange quarks (antiquarks). Besides the s-quark momentum distribution, their momentum spectra should be solely determined by the hadronization mechanism. Indeed, by looking at the midrapidity data on p T spectra of Ω − and φ [4,18], we see a very distinct feature. If we divide p T h by the number n c of constituent quark(s) and/or antiquark(s), i.e. p T h /3 for Ω − and p T h /2 for φ, and compare the inverse quark number 1/n c power of p T spectra, i.e. f 1/3 Ω − and f 1/2 φ , with each other, we see that they are parallel to each other. [Here, f h (p T h ) = dN h /d p T h dy is the p T spectrum of h at midrapidities.] This can be seen clearly in Fig. 1 where we re-normalize the data by a constant so that they just fall on one line. More precisely, we see that the data exhibit the following regularity where κ φ,Ω is a constant independent of p T . In other words, both f Ω − and f φ are given by a single f s (p T ) as where κ Ω and κ φ are constants, and κ φ,Ω = κ 1/3 Ω /κ 1/2 φ . We call this property the "quark number scaling" because it is similar to that of the elliptic flow of identified hadrons observed in relativistic heavy ion collisions [20][21][22].
Such a simple scaling behavior is consistent with that observed in AA [23] but is surprising for pA collisions. We therefore continue to examine the data [18,19] for other hadrons such as Ξ * 0 and K * 0 . Unfortunately, the results show that such simple scaling behavior is slightly violated. However, if we introduce a modification factor r ≈ 2/3, i.e., take p T h /(2 + r) instead of p T h /3 for Ξ * 0 and p T h /(1 + r) instead of p T h /2 for K * 0 , and divide that for Ξ * 0 by that for K * 0 , the result is again parallel to f s (p T ) obtained from f Ω − and f φ . More precisely, we obtain where κ φ,K * ,Ξ * is a constant. In panels (b), (d), (e) of Fig. 1, we show results obtained this way as a function of p T . We see that the scaling behavior is quite impressive. QCM and constituent quark degree of freedom -We show that the scaling behavior given by Eqs. (1)(2)(3)(4) and demonstrated in Fig. 1 is a direct consequence of the quark (re-)combination mechanism (QCM) [22,[24][25][26][27][28][29][30][31][32][33] for quarks and antiquarks with independent momentum distributions.
The situation for hadrons composed of quark(s) and/or antiquark(s) of only one flavor such as Ω − and φ discussed above is very simple. For consistency, we start with the general formulae. As formulated explicitly in e.g. [30], in general, in QCM, for a baryon B j composed of q 1 q 2 q 3 and a meson M j composed of q 1q2 , we have where f q 1 q 2 q 3 (p 1 , p 2 , p 3 ) is the joint momentum distribution for q 1 , q 2 and q 3 ; and R B j (p 1 , p 2 , p 3 ; p B ) is the combination  Insets show the data of p T spectra of these hadrons taken from [4,18,19].
function that is the probability for a given q 1 q 2 q 3 with momenta p 1 , p 2 and p 3 to combine into a baryon B j with momentum p B ; and similar for mesons. If we assume independent distributions of quarks and/or antiquarks, we have f q 1q2 (p 1 , p 2 ) = f q 1 (p 1 ) fq 2 (p 2 ).
Suppose the combination takes place mainly for quark and/or antiquark that takes a given fraction of momentum of the hadron, i.e., we obtain Now, for the simplest case, i.e. for hadrons such as Ω − and φ that are composed of only strange quark(s) and/or antiquark(s), we obtain immediately the results given by Eqs. (2) and (3) from Eqs. (11) and (12) respectively if we take f s (p T ) = fs(p T ). The combination takes place for three squarks with the same p T h /3 to form a Ω − with p T h and s ands with p T h /2 to form a φ with p T h .
For combinations of quark(s) and/or antiquark(s) with different flavors such as Ξ * and K * , the result given by Eq. (4) is actually a direct consequence of combination of equal transverse velocity. We recall that the velocity is v = p/E = p/γm. Equal velocity implies p i = γvm i ∝ m i that leads to We denote x u /x s = x d /x s = m u /m s = r and we obtain This leads immediately to Eq. (4) and r ≈ 2/3 if we take m s = 500MeV and m u = m d = 330MeV. Here, we take f u (p T ) = f d (p T ) = fū(p T ) = fd(p T ) for the midrapidity region at LHC. We see clearly that the quark number scaling exhibited by the hadronic p T -spectra in p-Pb collisions is a direct consequence of QCM of quarks and antiquarks with independent momentum distributions. The combination takes place among quarks and/or antiquarks with the same transverse velocity. Furthermore, we obtain also the following direct results.
(1) We see that f s (p T ) in Eq. (2) or (3) is nothing else but the p T spectrum of the strange quarks and antiquarks. We can easily extract the p T spectra of the constituent quarks and antiquarks at hadronization from the data [4,18,19].
Inspired by the Lévy-Tsallis function [34] for p T spectra of hadrons, we use the following form to parameterize the p Tdistribution for quarks where N q is the normalization constant and we use a superscript (n) to denote the normalized p T -distribution. By using the data of φ [18] and Eq. (3), we fix the parameters n s and c s for strange quarks. For u and d quarks, we use data of K * 0 [18] and Eq. (15). The obtained results for these parameters in different multiplicity classes are shown in Table I [35]. We see in particular that n q decreases with decreasing centrality and n s is larger than n u . As an example, we plot f (n) s (p T ) and f (n) u (p T ) in 20-40% multiplicity class in Fig. 2. We see that the obtained p T -spectrum for strange quarks is harder than that for u or d quarks for p T less than 3 GeV. We also plot the ratio between them where we see it raises with p T and seems to reach the maximum at p T around 3 GeV. These behaviors are similar to those obtained in AA collisions at RHIC and LHC energies [23,36,37].
(2) By applying Eqs. (11) and (12) to other hadrons such as ∆ and ρ, we obtain e.g., and other similar results that can be tested by future experiments.
(3) We note that the constants κ B j and κ M j can be obtained from Eqs. (11) and (12) as where N h is the average yield of h, N q i is the average number of q i ; and N q 1 q 2 q 3 and N q 1q2 are determined by the normalization conditions respectively. We see that besides the normalization constant that depends on the shape of f (n) q (p T ), the constant κ h is determined by the average yield of hadron and average numbers of quarks and/or antiquarks.
We recall that, in QCM, roughly speaking, if we take the approximation that the probability for a qq to form a meson or a qqq to form a baryon is flavor independent, the relative average yields of hadrons are well determined with a few parameters. This was formulated explicitly in e.g. [30], where we found [38] where C M j is the probability for a produced meson to be M j if it is of flavor q 1q2 and similar for C B j , they are determined by the vector to pseudo-scalar meson ratio R V/P and the decuplet to octet baryon ratio R D/O respectively; N iter is number of iteration for q 1 q 2 q 3 ; N M and N B are average numbers of mesons and baryons and N B / N M ≈ 1/12 in QCM; p q i is the probability for a quark q to take flavor q i , p u : p d : p s = 1 : 1 : λ and λ is the strangeness suppression