Saturation momentum scale extracted from semi-inclusive transverse   spectra in high-energy pp collisions

Takeshi Osada; Takuya Kumaoka

arXiv:1904.10823·hep-ph·October 2, 2019

Saturation momentum scale extracted from semi-inclusive transverse spectra in high-energy pp collisions

Takeshi Osada, Takuya Kumaoka

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TL;DR

This paper demonstrates that semi-inclusive transverse momentum spectra in high-energy proton-proton collisions exhibit geometric scaling when using an energy-dependent saturation momentum, unifying data across different energies and explaining fluctuation properties.

Contribution

It introduces an energy-dependent saturation momentum scale that enables geometric scaling of semi-inclusive spectra and fluctuation measures in high-energy pp collisions.

Findings

01

Semi-inclusive spectra at different energies overlap on a universal curve.

02

Particle density and mean transverse momentum scale with the saturation momentum.

03

The model explains the scaling of transverse momentum fluctuations.

Abstract

Geometric scaling is well confirmed for transverse momentum distributions observed in proton-proton collisions at LHC energies. We introduced multiplicity dependence on a saturation momentum of the geometrical scaling, assuming the scaling holds for semi-inclusive distributions as well as for inclusive distributions. The saturation momentum is usually given by Bjorken's $x$ variable, but redefinition of the scaling variable can make the saturation momentum a function of collision energy $W$ . We treat the energy as a free parameter (denoted $W^{*}$ to distinguish it from $W$ ) and associate the energy-dependent saturation momentum $Q_{sat} (W^{*})$ with particle number density. By using $Q_{sat} (W^{*})$ for a scaling variable $τ$ , we show semi-inclusive distributions can be geometrically scaled. i.e., all semi-inclusive spectra observed at $W$ =0.90, 2.76 and 7.00 TeV overlap one…

Tables2

Table 1. Table 1: The values of x 0 W ∗ superscript 𝑥 0 superscript 𝑊 x^{0}W^{*} , Q sat subscript 𝑄 sat Q_{\rm sat} , R T subscript 𝑅 T R_{\rm T} , and minimum chi-squared χ min 2 subscript superscript 𝜒 2 min \chi^{2}_{\rm min} obtained from the fitting to the semi-inclusive transverse momentum distribution observed by ALICE Aamodt:2010my with multiplicity class n acc subscript 𝑛 acc n_{\rm acc} = 3, 7, 17 (accepted number of charged particles per inelastic event in the range | η | < 0.8 𝜂 0.8 |\eta|<0.8 ) and by CMS Chatrchyan:2012qb with multiplicity class n tracks subscript 𝑛 tracks n_{\rm tracks} = 40, 63, 75, 98, 120 and 131(average number of true tracks multiplicity in the range | η | < 2.4 𝜂 2.4 |\eta|<2.4 ). For assignment from n acc subscript 𝑛 acc n_{\rm acc} to ⟨ n ch ⟩ delimited-⟨⟩ subscript 𝑛 ch \langle n_{\rm ch}\rangle in ALICE data, we use results presented in Table 2 of Ref. Aamodt:2010my . The particle densities d n ch / d y 𝑑 subscript 𝑛 ch 𝑑 𝑦 dn_{\rm ch}/dy at central rapidity region, are estimated by ⟨ n ch ⟩ / Δ η delimited-⟨⟩ subscript 𝑛 ch Δ 𝜂 \langle n_{\rm ch}\rangle/\Delta\eta for simplicity. X − b + a subscript superscript 𝑋 𝑎 𝑏 {X}^{+a}_{-b} denotes X ( = x 0 W ∗ , Q sat , R T ) X(=x_{0}W^{*},Q_{\rm sat},R_{\rm T}) giving χ min 2 subscript superscript 𝜒 2 min \chi^{2}_{\rm min} and a 𝑎 a , b 𝑏 b mean a boundary X − b ≦ X ≦ X + a 𝑋 𝑏 𝑋 𝑋 𝑎 X-b\leqq X\leqq X+a giving χ 2 = 1.5 χ min 2 superscript 𝜒 2 1.5 subscript superscript 𝜒 2 min \chi^{2}=1.5~{}\chi^{2}_{\rm min} .

$\sqrt{s}$ (TeV)	$n_{acc}$	$⟨ n_{ch} ⟩ / Δ η$	$x_{0} W^{*}$ [GeV]	$Q_{sat}$ [GeV/ $c$ ]	$R_{T}$ [fm]	$χ_{\min}^{2}$ /dof
0.90	3	4.8/1.6	0.18 ${}_{- 0.09}^{+ 0.16}$	0.84 ${}_{- 0.06}^{+ 0.06}$	0.63 ${}_{- 0.07}^{+ 0.09}$	97.5/33
0.90	7	10.0/1.6	1.39 ${}_{- 0.43}^{+ 0.60}$	1.03 ${}_{- 0.04}^{+ 0.04}$	0.69 ${}_{- 0.04}^{+ 0.05}$	20.5/33
0.90	17	22.5/1.6	9.03 ${}_{- 4.32}^{+ 7.70}$	1.24 ${}_{- 0.08}^{+ 0.08}$	0.82 ${}_{- 0.07}^{+ 0.08}$	23.2/33
$\sqrt{s}$ (TeV)	$n_{tracks}$	$⟨ n_{ch} ⟩ / Δ η$	$x_{0} W^{*}$ [GeV]	$Q_{sat}$ [GeV/ $c$ ]	$R_{T}$ [fm]	$χ_{\min}^{2}$ /dof
0.90	40	$40 / 4.8$	0.77 ${}_{- 0.14}^{+ 0.16}$	0.97 ${}_{- 0.02}^{+ 0.02}$	0.84 ${}_{- 0.02}^{+ 0.04}$	10.5/18
0.90	63	$63 / 4.8$	1.45 ${}_{- 0.39}^{+ 0.53}$	1.04 ${}_{- 0.03}^{+ 0.03}$	0.98 ${}_{- 0.04}^{+ 0.05}$	21.8/18
0.90	75	$75 / 4.8$	1.72 ${}_{- 0.45}^{+ 0.22}$	1.06 ${}_{- 0.03}^{+ 0.04}$	1.05 ${}_{- 0.05}^{+ 0.05}$	17.6/18
2.76	40	$40 / 4.8$	1.05 ${}_{- 0.24}^{+ 0.33}$	1.00 ${}_{- 0.03}^{+ 0.03}$	0.81 ${}_{- 0.03}^{+ 0.04}$	19.8/18
2.76	63	$63 / 4.8$	2.23 ${}_{- 0.80}^{+ 1.15}$	1.08 ${}_{- 0.05}^{+ 0.05}$	0.93 ${}_{- 0.05}^{+ 0.07}$	41.0/18
2.76	75	$75 / 4.8$	2.92 ${}_{- 1.17}^{+ 1.79}$	1.11 ${}_{- 0.06}^{+ 0.05}$	0.99 ${}_{- 0.06}^{+ 0.07}$	46.4/18
2.76	98	$98 / 4.8$	3.94 ${}_{- 1.54}^{+ 2.24}$	1.15 ${}_{- 0.05}^{+ 0.05}$	1.10 ${}_{- 0.06}^{+ 0.07}$	32.4/18
7.00	40	$40 / 4.8$	1.06 ${}_{- 0.19}^{+ 0.35}$	1.01 ${}_{- 0.02}^{+ 0.03}$	0.81 ${}_{- 0.03}^{+ 0.03}$	19.1/18
7.00	63	$63 / 4.8$	2.47 ${}_{- 0.88}^{+ 1.25}$	1.09 ${}_{- 0.05}^{+ 0.05}$	0.92 ${}_{- 0.05}^{+ 0.07}$	41.1/18
7.00	75	$75 / 4.8$	3.24 ${}_{- 1.17}^{+ 1.71}$	1.12 ${}_{- 0.05}^{+ 0.05}$	0.98 ${}_{- 0.05}^{+ 0.07}$	40.9/18
7.00	98	$98 / 4.8$	4.95 ${}_{- 2.00}^{+ 3.20}$	1.17 ${}_{- 0.06}^{+ 0.06}$	1.07 ${}_{- 0.07}^{+ 0.07}$	47.1/18
7.00	120	$120 / 4.8$	6.25 ${}_{- 2.17}^{+ 3.92}$	1.20 ${}_{- 0.05}^{+ 0.06}$	1.17 ${}_{- 0.07}^{+ 0.07}$	30.1/18
7.00	131	$131 / 4.8$	8.25 ${}_{- 3.25}^{+ 5.67}$	1.23 ${}_{- 0.06}^{+ 0.07}$	1.18 ${}_{- 0.07}^{+ 0.07}$	35.6/18

Table 2. Table 2: The best fit values of the parameters σ 𝜎 \sigma and n 𝑛 n to the experimental data on the event-by-event fluctuation of mean p T subscript 𝑝 T p_{\rm T} observed by ALICE Collaboration Abelev:2014ckr and results of evaluation of κ 2 subscript 𝜅 2 \kappa_{2} by Eq.( 21 ).

$\sqrt{s}$ [TeV]	$σ$	$n$	$χ_{\min}^{2}$ /dof	$κ_{2}$
0.90	1.14	$- 0.71$	4.93/23	1.79
2.76	1.01	$- 0.73$	14.0/45	1.14
7.00	1.01	$- 0.81$	20.4/64	1.14

Equations45

Q_{s} (x) \equiv Q_{0} (\frac{x}{x _{0}})^{- λ /2} .

Q_{s} (x) \equiv Q_{0} (\frac{x}{x _{0}})^{- λ /2} .

Q_{s} (\frac{p _{T}}{W}) = Q_{0} \cdot (x_{0} W)^{\frac{λ}{2}} p_{T}^{- \frac{λ}{2}},

Q_{s} (\frac{p _{T}}{W}) = Q_{0} \cdot (x_{0} W)^{\frac{λ}{2}} p_{T}^{- \frac{λ}{2}},

τ = (\frac{p _{T}}{Q _{sat} ( W )})^{2 + λ},

τ = (\frac{p _{T}}{Q _{sat} ( W )})^{2 + λ},

Q_{sat} (W) \equiv Q_{0} (\frac{x _{0} W}{Q _{0}})^{\frac{λ}{λ + 2}}

\frac{1}{S _{T}} \frac{d ^{2} N _{ch}}{d p _{T}^{2} d y} = F (τ),

⟨ \frac{d N _{ch}}{d y} ⟩ \propto S_{T} Q_{sat}^{2} (W),

⟨ \frac{d N _{ch}}{d y} ⟩ \propto S_{T} Q_{sat}^{2} (W),

\frac{1}{S _{T}^{*}} \frac{d ^{2} n _{ch}}{d p _{T}^{2} d y} = F (τ),

\frac{1}{S _{T}^{*}} \frac{d ^{2} n _{ch}}{d p _{T}^{2} d y} = F (τ),

τ = (\frac{p _{T}}{Q _{sat} ( W ^{*} )})^{2 + λ} .

F (τ) = [1 + (q - 1) \frac{τ ^{1/ (λ + 2)}}{κ}]^{- 1/ (q - 1)},

F (τ) = [1 + (q - 1) \frac{τ ^{1/ (λ + 2)}}{κ}]^{- 1/ (q - 1)},

\frac{1}{2 π p _{T}} \frac{d ^{2} N _{ch}}{d p _{T} d y} = C [1 + (q - 1) \frac{p _{T}}{T _{eff}}]^{- 1/ (q - 1)} .

\frac{1}{2 π p _{T}} \frac{d ^{2} N _{ch}}{d p _{T} d y} = C [1 + (q - 1) \frac{p _{T}}{T _{eff}}]^{- 1/ (q - 1)} .

p_{T} = Q_{0} (\frac{x _{0} W}{Q _{0}})^{\frac{λ}{λ + 2}} τ^{\frac{1}{2 + λ}},

p_{T} = Q_{0} (\frac{x _{0} W}{Q _{0}})^{\frac{λ}{λ + 2}} τ^{\frac{1}{2 + λ}},

T_{eff} = κ Q_{sat} .

T_{eff} = κ Q_{sat} .

⟨ \frac{d N _{ch}}{d y} ⟩

⟨ \frac{d N _{ch}}{d y} ⟩

\frac{1}{S _{T}^{*}} \frac{1}{2 π p _{T}} \frac{d ^{2} n _{ch}}{d p _{T} d y} = F (τ),

\frac{1}{S _{T}^{*}} \frac{1}{2 π p _{T}} \frac{d ^{2} n _{ch}}{d p _{T} d y} = F (τ),

τ^{1/ (2 + λ)} = \frac{p _{T}}{Q _{sat} ( W ^{*} )} .

τ^{1/ (2 + λ)} = \frac{p _{T}}{Q _{sat} ( W ^{*} )} .

\frac{d n _{ch}}{d y} \propto S_{T}^{*} Q_{sat}^{2} (W^{*}) .

\frac{d n _{ch}}{d y} \propto S_{T}^{*} Q_{sat}^{2} (W^{*}) .

\frac{W ^{*}}{W}

\frac{W ^{*}}{W}

\frac{Q _{sat}}{[ GeV / c ]}

\frac{Q _{sat}}{[ GeV / c ]}

\frac{R _{T}}{[ fm ]}

C_{m}

C_{m}

\displaystyle C({\bf p}_{\rm T_{1}},{\bf p}_{\rm T_{2}})\equiv{\frac{d^{4}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{1}}d{\bf p}^{2}_{\rm T_{2}}}}\Bigg{/}\frac{d^{2}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{1}}}\frac{d^{2}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{2}}}.

\displaystyle C({\bf p}_{\rm T_{1}},{\bf p}_{\rm T_{2}})\equiv{\frac{d^{4}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{1}}d{\bf p}^{2}_{\rm T_{2}}}}\Bigg{/}\frac{d^{2}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{1}}}\frac{d^{2}n_{\rm ch}}{d{\bf p}^{2}_{\rm T_{2}}}.

C_{GFT} (p_{T_{1}}, p_{T_{2}}) = 1 + \frac{κ _{2}}{S _{T} Q _{sat}^{2}},

C_{GFT} (p_{T_{1}}, p_{T_{2}}) = 1 + \frac{κ _{2}}{S _{T} Q _{sat}^{2}},

C (p_{T_{1}}, p_{T_{2}}) =

C (p_{T_{1}}, p_{T_{2}}) =

\displaystyle\qquad 1+\Big{(}S^{*}_{\rm T}~{}[\kappa Q_{\rm sat}]^{2}\Big{)}^{n}\exp\left(-\frac{({\bf p}_{\rm T_{1}}-{\bf p}_{\rm T_{2}})^{2}}{\sigma[\kappa Q_{\rm sat}]^{2}}\right),\quad

κ_{2} \sim \frac{1}{κ ^{2}} exp (- \frac{4}{σ}) .

κ_{2} \sim \frac{1}{κ ^{2}} exp (- \frac{4}{σ}) .

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Full text

Saturation momentum scale extracted from semi-inclusive transverse spectra in high-energy pp collisions

Takeshi Osada

[email protected]

Department of Physics, Faculty of Liberal Arts and Sciences,

Tokyo City University, Tamazutsumi 1-28-1, Setagaya-ku, Tokyo 158-8557, Japan

Takuya Kumaoka

Department of Physics, Shinshu University, Matsumoto 390-8621, Japan

Abstract

Geometric scaling is well confirmed for transverse momentum distributions observed in proton-proton collisions at LHC energies. We introduced multiplicity dependence on a saturation momentum of the geometrical scaling, assuming the scaling holds for semi-inclusive distributions as well as for inclusive distributions. The saturation momentum is usually given by Bjorken’s $x$ variable, but redefinition of the scaling variable can make the saturation momentum a function of collision energy $W$ . We treat the energy as a free parameter (denoted $W^{*}$ to distinguish it from $W$ ) and associate the energy-dependent saturation momentum $Q_{\rm sat}(W^{*})$ with particle number density. By using $Q_{\rm sat}(W^{*})$ for a scaling variable $\tau$ , we show semi-inclusive distributions can be geometrically scaled. i.e., all semi-inclusive spectra observed at $W$ =0.90, 2.76 and 7.00 TeV overlap one universal function. The particle density dependences of mean transverse momentum $\langle p_{\rm T}\rangle$ for LHC energies scales in terms of $Q_{\rm sat}(W^{*})$ . Furthermore, our model explains a scaling property of event-by-event $p_{\rm T}$ fluctuation measure $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ at LHC energies for pp collisions, where $C_{m}$ is two-particle transverse momentum correlator. Our analysis of the $p_{\rm T}$ fluctuation makes possible to evaluate a non-perturbative coefficient of the gluon correlation function.

pacs:

13.75.Cs, 24.60.Ky, 25.75.Gz

I Introduction

Studies of small collision systems in high multiplicity events is attracting considerable interest Preghenella:2018moc because of the collective phenomena which attribute to the formation of a strongly-interacting collectively-expanding quark-gluon medium Khachatryan:2010gv ; Khachatryan:2016txc ; Dusling:2015gta ; Nagle:2018nvi . A remarkable similarity has been observed between strange particles production in pp collisions and that in Pb-Pb collisions, suggesting the possibility of deconfined QCD phase formation in small systems ALICE:2017jyt . In such pp collisions, the charged particle pseudo-rapidity density rises as a power of energy Kharzeev:2000ph ; McLerran:2010ex , which can be explained by the theory of gluon saturation Blaizot:1987nc ; Gribov:1984tu . Recombination of gluons Mueller:1985wy in high particle number density state causes the saturation, and the gluon distribution function ceases growing from some intrinsic scale of the transverse momentum $Q_{s}$ Kharzeev:2001gp . The Color Glass Condensate (CGC) Kovchegov:2012mbw ; Iancu:2002xk ; Blaizot:2004px is an effective theory to describe saturated gluons with small $x$ as classical color fields radiated by color sources at higher rapidity. The existence of $Q_{s}$ which separates the degree of freedom into fast frozen color sources and slow dynamical color fields Gelis:2010nm is the underlying assumption of the effective theory. The scaling of the limiting fragmentation curves Stasto:2011zza is one of the crucial pieces of evidence for the picture of the CGC JalilianMarian:2002wq ; McLerran:2004fg .

Another experimental evidence of CGC hypothesis is a geometrical scaling Iancu:2003jg ; Iancu:2002tr (GS) confirmed originally in results on total $\gamma^{*}$ p cross section Stasto:2000er . A term of the ‘geometrical’ of this GS comes from that survival probability of a color dipole Tribedy:2010ab ; Iancu:2003ge ; Gotsman:2019vrv is determined by the geometric relationship between the dipole size and the saturation radius given by $Q_{s}^{-1}(x)$ Gelis:2010nm ; Munier:2003vc , where $x$ is a Bjorken variable. In this article, since we will deal with multi-particle production in the central rapidity region of high-energy pp collisions, we have $x=p_{\rm T}/W$ , where $p_{\rm T}$ and $W$ are transverse momentum and colliding energy of the incident proton, respectively. With $x_{0}$ , $Q_{0}$ and $\lambda$ are constants (see, Sec.II for details), the saturation momentum is given by GolecBiernat:1998js

[TABLE]

If such momentum is the only scale that controls $p_{\rm T}$ distribution, it should exhibit GS behavior; i.e., when one normalizes inclusive transverse momentum spectra observed with an appropriate constant $S_{\rm T}$ (interpreted later as reaction effective transverse cross-sectional area), the data points lies on a characteristic curve ${\cal F}(\tau)$ which is only depends on the scaling variable $\tau\equiv p^{2}_{\rm T}/Q^{2}_{\rm sat}$ and the curve does not depend on $W$ . In particular, the scaling property has been vigorously studied for pp collisions obtained at the Large Hadron Collider (LHC) energies Praszalowicz:2015dta ; Praszalowicz:2013fsa ; Praszalowicz:2011tc ; Praszalowicz:2011rm and GS is observed in single inclusive distributions of charged hadrons Praszalowicz:2015dta and and recently observed direct photons from heavy-ion collisions Khachatryan:2019uqn . Since $Q_{s}(x)$ includes $p_{\rm T}$ dependence via Bjorken $x$ ; i.e.,

[TABLE]

we unify the terms of $p_{\rm T}$ contained in $\tau$ and redefine the rest that depends on $W$ as an energy dependent saturation momentum $Q_{\rm sat}(W)$ Kharzeev:2004if . Namely, the scaling variable can be rewritten as

[TABLE]

Under an assumption that a local parton hadron duality Azimov:1984np as a hadronization model is appropriate, the particle density at the central rapidity region ( $y\approx 0$ ) relates to $Q_{\rm sat}(W)$ as follows:

[TABLE]

where $\langle\cdots\rangle$ denotes the average over single inclusive distribution (or over minimum bias events). Since the particle number density is known to increases gradually with collision energy $W$ , we expect $Q_{\rm sat}(W)$ to also increases gradually with $W$ .

Let us suppose that GS holds not only for inclusive distributions but also for the semi-inclusive distributions, i.e., inclusive distribution with fixed multiplicity or limited multiplicity class Kanki:1988tz 111In this paper, the inclusive spectra is denoted by $d^{2}N_{\rm ch}/dp_{\rm T}dy$ , and the semi-inclusive spectra is denoted by $d^{2}n_{\rm ch}/dp_{\rm T}dy$ to distinguish it from the inclusive one.. For the semi-inclusive spectra $\frac{d^{2}n_{\rm ch}}{dp_{\rm T}^{2}dy}$ , as the case of inclusive one, we assume that there exists a saturation momentum for the spectrum classified by multiplicity as well and we propose to represent it by effective energy $W^{*}$ ; i.e.,

[TABLE]

It should be noted here that the universal function ${\cal F}$ in Eq.(5a) is the same as that in Eq.(3c). Here, $S^{*}_{\rm T}$ and $W^{*}$ are determined to reproduce the spectrum obtained by the experiment. In particular, this $W^{*}$ is a fit parameter introduced replacing the actual beam energy $W$ in eq.(3b). Hence, we intend to check whether GS found in inclusive distribution is restored even in semi-inclusive distribution.

It may be appropriate to give some explanations for $W^{*}$ here. As discussed in detail later in Sec.II, the energy-dependent saturation momentum $Q_{\rm sat}$ gives a typical scale of transverse momentum $p_{\rm T}$ . That is, $Q_{\rm sat}$ is the solution $p_{\rm T}$ of an equation $Q_{s}(p_{\rm T}/W)=p_{\rm T}$ for each colliding energy $W$ . Because $Q_{\rm sat}$ itself is a scale of transverse momentum, the inverse of it is a typical transverse size scale of saturated gluons. Hence as seen in Eq.(4), the ratio of effective interaction cross sectional area $S_{\rm T}$ to the cross-sectional area per gluon $Q_{\rm sat}^{-2}$ governs the mean charged particle density of the inclusive distribution. On the other hand, for semi-inclusive collisions classified by multiplicity, $S^{*}_{\rm T}$ and $Q_{\rm sat}(W^{*})$ should be related to each other by the constraint of the fixed multiplicity. We will discuss the relation between $Q_{\rm sat}(W^{*})$ and $S_{\rm T}^{*}$ of the semi-inclusive distribution in some detail in Sec.III and also comment on the physical meaning of $W^{*}$ .

This article is organized as follows. In the following Sec.II, we briefly review GS hypothesis and we confirm that it holds well for inclusive transverse spectra observed in pp collisions at LHC energies. Then, we determine the universal function of GS used throughout this article. In Sec.III, the effective energy $W^{*}$ is determined from the semi-inclusive transverse spectra. By using the scaling variables with $Q_{\rm sat}(W^{*})$ , we show that the transverse momentum spectra observed in the different multiplicity classes at the different collision energies scale to the universal function found in Sec.II. We also show that the multiplicity dependence of the mean transverse momentum scales with $Q_{\rm sat}(W^{*})$ . Furthermore, we analyze the scaling behavior of a normalized fluctuation measure of transverse momentum and consider it as a result of the correlation between particles generated from color flux tubes. We close with Sec.IV containing the summary and some concluding remarks.

II GS for inclusive $p_{\rm T}$ distribution

The transverse momentum spectra of various energies for pp collisions never scale with variable $p_{\rm T}$ because their intensities and slopes depend on the colliding energy $W$ . However, for high energy collisions in which the number of soft gluons inside the proton saturates, the transverse momentum spectrum depends only on a scaling variable defined by Eq.(3a) with Eq.(3b). Let us examine the quantitative difference between $Q_{s}(x)$ and $Q_{\rm sat}(W)$ at LHC energies. We show them as a function of $p_{\rm T}$ in Fig.1 for the case of $\lambda=0.22$ , $x_{0}=1.0\times 10^{-3}$ , $Q_{0}=1.0$ GeV/ $c$ McLerran:2014apa , and we will fix the values from now on.

Since $Q_{s}$ is less dependent on $p_{\rm T}$ for $p_{\rm T}\gtrsim 0.5$ GeV/ $c$ , one may use $Q_{\rm sat}(W)$ instead of $Q_{s}$ as a typical momentum scale. The values of $Q_{\rm sat}(W)$ obtained from inclusive $p_{\rm T}$ spectra at energy $W=$ 0.90, 2.76, 7.00 TeV are 0.99, 1.11 and 1.21 GeV/ $c$ , respectively. As shown in Fig.2, experimental data observed by ALICE Abelev:2013ala and CMS Collaboration Chatrchyan:2012qb suggests the validity of GS especially for $\tau^{1/(2+\lambda)}\lesssim 10$ . The curve emerging from a plot of the $p_{\rm T}$ spectra with using the scaling variable $\tau$ can be fitted well by the so-called Tsallis type function Rybczynski:2012pn ; Rybczynski:2012vj ;

[TABLE]

where the non-extensive parameter $q$ =1.134 and $\kappa$ =0.1293 are used. The effective cross-sectional area in Eq.(3c) is determined as $S_{\rm T}=$ 22.66 GeV*-2*. In this way, the transverse momentum distribution indeed exhibits GS behavior for pp collisions in the LHC energies.

It seems to be appropriate to shortly comment on the energy-dependent saturation momentum $Q_{\rm sat}$ and an effective temperature $T_{\rm eff}$ (or a slope parameter) Praszalowicz:2013fsa ; McLerran:2013una here. In case of Tsallis-type distribution function, $T_{\rm eff}$ can be defined as

[TABLE]

Here, one may interpret the constant $C$ as $S_{\rm T}$ . Since the transverse spectra experimentally observed exhibits good GS behavior, the effective temperature $T_{\rm eff}$ must have energy dependence to cancel the energy dependence of

[TABLE]

which is obtained from Eq.(3a). Hence, the property of the GS determined the energy dependence of $T_{\rm eff}$ and that $T_{\rm eff}$ should be proportional to $Q_{\rm sat}$ Praszalowicz:2013fsa . Substituting Eq.(8) into Eq.(7) yields the expression of the universal function of Eq.(6) in the case of

[TABLE]

The gluon saturation is physics of the intermediate energy scale $Q_{\rm sat}$ , while GS observed in the final state is physics of energy scale $T_{\rm eff}$ which is much lower than $Q_{\rm sat}$ . Therefore, the parameter $\kappa$ in Eq.(9) (or equivalently Eq.(6)) may have a physical meaning of a linkage between two energy scales of $Q_{\rm sat}$ and $T_{\rm eff}$ . Before closing Sec.II, let us check $S_{\rm T}$ and $Q_{\rm sat}$ obtained here. By integrating Eq.(7), we obtain the average multiplicity densityOsada:2017oxe ;

[TABLE]

which gives 3.68, 4.63, and 5.50 for $W=$ 0.90, 2.76 and 7.00 TeV, respectively. These values should be compared with values obtained by experiments Adam:2015gka i.e., 3.75 ${}^{+0.06}_{-0.05}$ , 4.76 ${}^{+0.08}_{-0.07}$ , 5.98 ${}^{+0.09}_{-0.07}$ for $\sqrt{s}=$ 0.90, 2.76 and 7.00 TeV, respectively.

III GS for semi-inclusive $p_{\rm T}$ distribution

III.1 Extraction of saturation momentum scale

In this Section, we will extract the multiplicity dependence of saturation momentum $Q_{\rm sat}$ from the semi-inclusive spectrum observed. Our central assumption is that the semi-inclusive $p_{\rm T}$ distribution scales to the same universal function ${\cal F}(\tau)$ as the inclusive one (i.e., Eq.(6) with $q=1.134$ , $\kappa=$ 0.1293 and $\lambda=$ 0.22), providing that the appropriate $Q_{\rm sat}(W^{*})$ is used. Since $S_{\rm T}$ in Eq.(3c) now depends on the multiplicity, we require GS for the semi-inclusive spectra as shown by Eq.(5a) with (5b) in Sec.I;

[TABLE]

and

[TABLE]

These two parameters, $W^{*}$ and $S^{*}_{\rm T}$ , are determined by fitting to the experimental data on the semi-inclusive $p_{\rm T}$ distribution. Note that, in this case, Eq.(4) should be modified as

[TABLE]

Since the universal function in Eq.(5a) is the same as that in Eq.(3c), the proportionality constants in Eqs.(4) and (11) are equal. Therefore, using Eq.(3b), the ratio of $W^{*}$ to $W$ is given by

[TABLE]

where $s^{*}_{\rm T}\equiv S^{*}_{\rm T}/\frac{dn_{\rm ch}}{dy}$ . As can be seen from Eq.(15), $W^{*}$ and $S^{*}_{\rm T}$ are not independent parameters. Hence, whether $W^{*}/W$ becomes larger or smaller than unity depends on whether a cross-sectional area per gluon $s^{*}_{\rm T}$ in the semi-inclusive distribution is larger or smaller than that in the inclusive distribution. Even if $W^{*}$ has a value greater than $W$ , it does not mean an unphysical situation.

In order to determine the multiplicity dependence of $W^{*}$ in Eq.(5b), we fit Eq.(5a) to $p_{\rm T}$ spectra at energy 0.90 TeV for the accepted number of charged particles $n_{\rm acc}=$ 3, 7 and 17 observed ALICE Collaboration Aamodt:2010my and at energy 0.90, 2.76 and 7.00 TeV for the average track multiplicity $n_{\rm tracks}=$ 40, 63, 75, 98, 120 and 131 observed CMS Collaboration Chatrchyan:2012qb . Figure 3 and 4 show the results of fitting with $S^{*}_{\rm T}{\cal F}$ to ALICE and CMS data, respectively.

Besides, Table 1 shows the values of $W^{*}$ (multiplied by $x_{0}$ ) and effective radius $R_{\rm T}\equiv\sqrt{S^{*}_{\rm T}/\pi}$ obtained by the fit. Table 1 also shows the value of $Q_{\rm sat}(W^{*})$ and the minimum value of $\chi^{2}$ (denoting by $\chi^{2}_{\rm min}$ ) in each fitting.

As shown in Fig.5, we confirm that the semi-inclusive transverse momentum spectra depicted in Figs.3 and 4 scale in terms of the scaling variable $\tau$ of Eq.(5b). Note that the solid curve (the universal function ${\cal F}$ ) in Fig.5 is exactly the same as that obtained in the inclusive distribution in Fig.1.

We also show $Q_{\rm sat}$ and $R_{\rm T}$ as function of $dn_{\rm ch}/dy$ in Fig.6. It is found that $Q_{\rm sat}(W^{*})$ and $R_{\rm T}$ are proportional to $\left(dn_{\rm ch}/dy\right)^{1/6}$ and $\left(dn_{\rm ch}/dy\right)^{1/3}$ , respectively. The curves depicted by broken lines in the left panel (for $Q_{\rm sat}$ ) and the right panel (for $R_{\rm T}$ ) of Fig.6 are given by

[TABLE]

These $dn_{\rm ch}/dy$ dependencies are consistent with Eq.(11) when $dn_{\rm ch}/dy$ is sufficiently large and the constant term can be ignored. Here, it is interesting to find a particle number density $dn_{\rm ch}/dy$ to give $W^{*}(dn_{\rm ch}/dy)=W$ . Using Eqs.(15) and (16), we can evaluate $s^{*}_{\rm T}$ that satisfies $W^{*}/W=1$ . For simplicity, ignoring the constant term of Eq.(16), we obtain $W^{*}=W$ when $dn_{\rm ch}/dy=$ 10.4, 20.5 and 27.4 for 0.90, 2.76 and 7.00 TeV, respectively. In fact, for CMS event classes with $n_{\rm tracks}$ = 63, 98 and 131 in $|\Delta\eta|<2.4$ at $W=$ 0.90, 2.76 and 7.00 TeV, respectively, it can be read from Table 1 that $W^{*}>W$ is realized.

III.2 Mean transverse momentum

Next, we turn our attention to the average transverse momentum $\langle p_{\rm T}\rangle$ obtained from the semi-inclusive distributions. The energy-dependent saturation momentum $Q_{\rm sat}(W^{*})$ should be proportional to $\langle p_{\rm T}\rangle$ in GS framework Osada:2017oxe . As seen in the left panel of Fig.7, $dn_{\rm ch}/dy$ dependences of $\langle p_{\rm T}\rangle$ at 0.90, 2.76 and 7.00 TeV observed by ALICE Abelev:2013bla and CMS Chatrchyan:2012qb do not show scaling behavior in terms of $dn_{\rm ch}/dy$ . However, since GS holds for the semi-inclusive distributions, one expects that $\langle p_{\rm T}\rangle$ is linearly proportional to $Q_{\rm sat}(W^{*})$ and those data lie on a straight line regardless of the colliding energy $W$ . Figure 7 shows results of the conversion of the $dn_{\rm ch}/dy$ dependence of $\langle p_{\rm T}\rangle$ on the left side panel into the dependence of $Q_{\rm sat}(W^{*})$ on the right side panel. The difference in scaling curves between ALICE and CMS seems to be due to that in acceptance employed in each observation. Thus, the behavior of GS is observed not only in the inclusive distributions but also in the semi-inclusive distributions in high energy pp collisions.

III.3 Normalized fluctuation measure of transverse momentum

A prominent scaling behavior emerges in event-by-event mean $p_{\rm T}$ fluctuations in pp collisions at LHC energies Abelev:2014ckr ; Heckel:2015swa ; Stefan:2011es . In our previous work Osada:2017oxe , we studied it focusing only on the energy $W=$ 0.90 TeV, and we did not discuss the GS behavior by extending the analysis to other energies. In this article, we analyze data on transverse momentum fluctuations observed at $\sqrt{s}=$ 0.90, 2.76, 7.00 TeV using $Q_{\rm sat}(W^{*})$ and $S^{*}_{\rm T}$ without changing the basic idea of the model proposed in Ref.Osada:2017oxe . The fluctuation measure is essentially a two-particle distribution as defined below,

[TABLE]

where $m=\frac{dn_{\rm ch}}{d\eta}\times|\Delta\eta|$ is the multiplicity in the pseudo-rapidity window $|\Delta\eta|$ . Since the universal function of GS is essentially one particle distribution, two-particle correlation function Gavin:2008ev as shown below is required to obtain the two-particle distribution in Eq.(17);

[TABLE]

It is known that a gluon two-particle correlation function takes the following simple geometrical form in the CGC / Glasma framework Dumitru:2008wn ; Lappi:2009xa ; Lappi:2010cp ,

[TABLE]

where $\kappa_{2}$ is a non-perturbative constant, and the evaluation of this constant is a challenging problem in theoretical physics. On the other hand, we consider an extreme model in which the correlation in momentum space between gluons is inherited to that between hadrons in the final state. Since the transverse size of color flux tubes stretching between the receding protons is expected to be of order in $1/Q_{\rm sat}$ , one may write the following correlation function commonly found in Bose-Einstein correlation (BEC) analysis:

[TABLE]

where $n$ and $\sigma$ are model parameters. Here, $\kappa$ is the parameter that appears in the universal function Eq.(6) which connects intermediate energy scale $Q_{\rm sat}$ and hadronization energy scale $T_{\rm eff}$ . Since $\kappa$ and $Q_{\rm sat}$ always appear together in the inclusive distribution, there must also be such property in the two particle distribution in Eq.(20). Note also that the term $S^{*}_{\rm T}Q^{2}_{\rm sat}$ in Eq.(20) is proportional to the number of flux tubes Tribedy:2010ab , especially when $n=-1$ , it can be interpreted as chaoticity of the BEC effectOsada:2017oxe . Another parameter $\sigma$ is for adjusting the size of the flux tube. When $\sigma\approx 1$ , it means that the size of the color flux tube is expanded by about $1/\kappa\approx 7.7$ times in the transverse direction and the source size scale is the inverse of the temperature of the system $\sim 1/T_{\rm eff}$ .

As seen in Fig.8, ALICE observed normalized fluctuation measure $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ at $W=$ 0.90, 2.76 and 7.00 TeV, and they found almost no energy dependence in them. Our model based on GS easily explain the reason why the measure $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ hardly depends on the collision energy: i.e., By noting that $p_{\rm T}=Q_{\rm sat}\tau^{1/(2+\lambda)}$ , $\langle p_{\rm T}\rangle\propto Q_{\rm sat}$ , and $m\propto S^{*}_{\rm T}Q_{\rm sat}^{2}$ , one can represent the measure as a function of the scaling variable $\tau$ except for the term $S^{*}_{\rm T}Q_{\rm sat}^{2}$ in the correlation function Eq.(20). However, as shown by Eqs.(16) and (16), the energy dependence of both $Q_{\rm sat}$ and $S^{*}_{\rm T}$ are considerably small. Moreover, recall that $S^{*}_{\rm T}Q_{\rm sat}^{2}$ is the number of color flux tubes. Since the gluon in the incident proton is saturated regardless of the energy, it is natural that the energy dependence of this factor is small. Therefore, it is explained that $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ is almost independently of the colliding energy $W$ in our model. The fit results to the experimental data by Eq.(20) are shown by solid lines in Fig.8. We also show values of the parameter both $\sigma$ and $n$ giving $\chi^{2}_{\rm min}$ in Table 2. The values of $n$ obtained by the fits are from $-0.71$ to $-0.81$ , which are larger than $-1$ , but Eq.(20) can be compared with the Eq.(19) in the Glasma framework. Evaluating the typical momentum scale of BEC as $|{\bf p}_{\rm T_{1}}-{\bf p}_{\rm T_{2}}|^{2}\sim[2\kappa Q_{\rm sat}]^{2}\approx[200{\rm MeV}]^{2}$ , the comparison leads us to a rough estimation of $\kappa_{2}$ as the following;

[TABLE]

Table 2 also shows the values of $\kappa_{2}$ evaluated by Eq.(21). Since there are considerable variations in the extracted values of $\kappa_{2}$ from experimental data based on the Glasma framework, its value is not known to be as accurate as an order of 1 Lappi:2009xa . It is interesting to note that the values of $\kappa_{2}$ extracted from our model are comparable to the estimation by the Glasma framework, although the picture for particle correlation of each other is different.

IV Summary and concluding remarks

In this article, we have phenomenologically investigated multiplicity dependence on the gluon saturation momentum in high energy pp collisions. This result makes it possible to classify events by energy-dependent saturation momentum $Q_{\rm sat}(W^{*})$ , which in turn can provide a new research approach to high energy multi-particle production.

If the local parton-hadron duality hypothesis is correct, $Q_{\rm sat}(W^{*})$ must link to observables in the final state of the charged hadrons. In order to extract $Q_{\rm sat}(W^{*})$ that governs the multiplicity of the final states, we assumed the semi-inclusive transverse momentum spectra exhibit geometrical scaling behavior independently of its fixed multiplicity and its colliding energy. Furthermore, the universal function is assumed to be the same as that of the inclusive distribution. Through the effective energy $W^{*}$ defined by Eq.(5b), we determined $Q_{\rm sat}(W^{*})$ for the semi-inclusive distributions. We have shown that the transverse momentums distribution of various multiplicity class at $\sqrt{s}$ =0.90, 2.76 and 7.00 TeV do scale in terms of the scaling variable $\tau^{1/(2+\lambda)}=p_{\rm T}/Q_{\rm sat}(W^{*})$ . We have also confirmed that $Q_{\rm sat}(W^{*})$ dependence on the average transverse momentum also scales to a linear function of $Q_{\rm sat}(W^{*})$ , which is consistent with the behavior expected from GS.

It is meaningful to note on works by Korus and Mrówczyński Korus:2001fv ; Mrowczynski:2004cg and to compare with the model we have proposed. Korus and Mrówczyński have introduced a multiplicity-dependent temperature and related the nontrivial behavior of fluctuations in the transverse momentum to that in the multiplicity distribution. In our model, on the other hand, the energy-dependent saturation momentum $Q_{\rm sat}(W^{*})$ is related to the multiplicity of the final state via the effective energy $W^{*}$ and is also related to the temperature evaluated from the semi-inclusive spectra by Eq.(9). About fluctuation of transverse momentum, Korus and Mrówczyński argue that the reason for it is that the fluctuation in the multiplicity distribution is almost independent of energy. In fact, the normalized $q$ -moment values of $C_{2},C_{3},C_{4}$ for the multiplicity distribution in the central rapidity region $|\eta|<0.5$ are almost independent of the collision energy Adam:2015gka ; Khachatryan:2010nk . On the other hand, in our model, the reason why there is almost no dependence on collision energy in the fluctuation measure $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ is that the energy dependences on $Q_{\rm sat}(W^{*})$ and $S_{\rm T}$ are considerably small (see, Fig.6) in addition to the fact that the semi-inclusive transverse momentum spectrum shows the behavior of geometrical scaling.

In this article, we thought that the two-particle Bose-Einstein correlation between identical gluons produced from color flux tubes could explain the experimental results of the fluctuation measure. The measure $\sqrt{C_{m}}/\langle p_{\rm T}\rangle$ can be fitted by Eq.(20) nicely, in which the correlation between gluons is considered to remain between charged particles after hadronization. Comparing Eq.(19) with our model Eq.(20) we can estimate the value of the non-perturbative constant of the gluon correlation function $\kappa_{2}$ . If a typical value for $|{\bf p}_{\rm T_{1}}-{\bf p}_{\rm T_{2}}|$ in Eq.(20) as 200 MeV/ $c$ is adopted, one obtain $\kappa_{2}=1.4-1.8$ . It is interesting to extract $Q_{\rm sat}$ from other reaction such as pA McLerran:2015lta and A-AAndres:2012ma collision and to discuss the relationship between the fluctuation of multiplicity and that of the saturation momentum. However, we plan to investigate those issues at some other opportunity.

Acknowledgements.

We acknowledge stimulating discussions with Grzegorz Wilk concerning the $q$ -scaling in high-energy production processes and the effective energy. We would also like to thank the anonymous referee for thorough comments which have greatly improved the manuscript.

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Saturation momentum scale extracted from semi-inclusive transverse spectra in high-energy pp collisions

Abstract

pacs:

I Introduction

II GS for inclusive pTp_{\rm T}pT​ distribution

III GS for semi-inclusive pTp_{\rm T}pT​ distribution

III.1 Extraction of saturation momentum scale

III.2 Mean transverse momentum

III.3 Normalized fluctuation measure of transverse momentum

IV Summary and concluding remarks

Acknowledgements.

II GS for inclusive $p_{\rm T}$ distribution

III GS for semi-inclusive $p_{\rm T}$ distribution