Inclusive Ridge Distributions in Heavy-Ion Collisions

0.22 MB·English

by  Rudolph C. Hwa

Inclusive Ridge Distributions in Heavy-Ion Collisions
Rudolph C. Hwa1 and Lilin Zhu2
1Institute of Theoretical Science and Department of Physics
University of Oregon, Eugene, OR 97403-5203, USA
2Department of Physics, Sichuan University, Chengdu 610064, P. R. China
(Dated: January 10, 2011)
The formation of ridges induced by semihard scattering in nuclear collisions is included in the
descriptionofsingle-particledistributionsforbothpionandprotonatlowtransversemomenta. The
ridge component is characterized by an azimuthal dependent factor that is derived in the study of
the ridge structure in two-particle correlation distributions involving triggers. It is shown that the
inclusive ridge can reproduce the observed data on v2(pT) if the base component underlying the
ridge has no azimuthal dependence. A common description of pion and proton spectra is given
1
in the recombination model that can smoothly join the low- and intermediate-p regions. All the
1 T
important properties of single-particle distributions in those regions can be satisfactorily described
0
in this approach.
2

n PACSnumbers: 25.75.-q,25.75.Dw
a
J

6 I. INTRODUCTION gered by jets there is a ridge phenomenon in the struc-
ture of associated particles with narrow ∆φ (azimuthal

] angle relative to that of the trigger) and extended ∆η
h As the data on single-particle distributions of iden-
(pseudorapidity relative to the trigger). Such a struc-
t tified hadrons produced in heavy-ion collisions become
- ture should be present in the inclusive distribution even
l more abundant and precise [1–7], more demands are put
c iftriggersarenotusedto selectthe jet events. When k
u on theoretical models to reproduce them. It is gener- T
is low enough so that minijets are copiously produced,
n ally recognized that in Au-Au collisions at √sNN =200 the correspondingeffect onthe φ anisotropycanbecome
[ GeV at the Relativistic Heavy-Ion Collider (RHIC) the
dominant, rendering the consideration of pressure gra-
low transverse-momentum (p ) region (p < 2 GeV/c)
1 T T dients along different φ direction unreliable if semihard
is well described by hydrodynamics [8] and the high-
v scatterings are ignored. In this paper we give specific
4 pT region (pT > 6 GeV/c) by perturbative QCD [9], attention to the ridge contribution to the single-particle
3 both subjects being reviewed recently in Ref. [10]. In
distributionsinthelow-p region. Itisinthissensethat
3 the intermediate region (2 < p < 6 GeV/c) neither T
1 approaches work very well. WhaTt stands out in that re- we use the terminology: inclusive ridge distribution.
.
1 gionarethelargebaryon/mesonratioandquark-number
0 scaling (QNS), which give empirical support to the re-
Another area of concern is the variation of the p de-
1 combination/coalescence models [11–16]. The connec- T
pendence as the focus is moved to the low-p region,
1 T
tion between the intermediate- and high-p regions is
: T wherepionandprotonappearempirically to havediffer-
v smooth, since the dominance of shower-shower recombi-
ent behaviors. In the parton recombination model the
i nation is equivalent to parton fragmentation. The tran-
X hadrons should have the same inverse slope as that of
sition across the lower p boundary at p 2 GeV/c is
r T T ∼ the coalescing quarks if the hadrons are formed by re-
a notsosmoothbecauseofthedifferenceinthecontinuum combination of the thermal partons, but because of the
descriptioninhydrodynamicsandthepartondescription
difference in the meson and baryon wave functions, the
in hadronization. Our aim in this article is to extend
net p distributions turn out to be different. This line
T
our previous considerations [17, 18] to the lower-p re-
T of analysis takes into account the quark degree of free-
gionandto describe ina self-consistentwayboth the p
T dom just before hadronization, which is overlooked by
and azimuthal φ behaviorsof pions and protons without
the fluid description of the flow effect. The burden is to
explicit reliance on hydrodynamics.
showthatthedataonv (p )canbereproducedforboth
2 T
One specific point that motivates our study is related pionandprotonatlowp withoutthe hydrodescription
T
to the question of what happens to the initial system of elliptic flow. That is indeed what we shall show for
within 1 fm/c after collision. Semihard partons cre- various centralities.
ated within 1 fm from the surface will have already left
the initial overlap region before thermalization is com-
plete. There are many of them with parton transverse- We confine our consideration in this paper to the
momentum k 2-3 GeV/c even at RHIC, let alone at physicsatmidrapidity. Atlargerηthereareotherissues,
T
∼
theLargeHadronCollider(LHC).Theyareminijetsthat such as large p/π ratio [6] and large ∆η distribution of
cancauseazimuthalanisotropy,notaccountedforbycon- triggered ridge [7], which have been examined in Refs.
ventionalhydrodynamics. Itisknownthatineventstrig- [19, 20], and will not be further considered here.
2
II. SINGLE-PARTICLE DISTRIBUTION WITH collisions the almond-shaped initial configuration leads
RIDGE to φ anisotropy. The conventional description in terms
of hydrodynamics relates the momentum anisotropy to
We begin with a recapitulation of our description of the variation of pressure gradient at early times upon
single-particle distribution [11, 17, 18]. At low p we equilibration [21]. The success in obtaining the large
T
consider only the recombination of thermal partons, so v2 as observed gives credibility to the approach. We
the pion and proton spectra at y =0 are given by adopt an alternative approach and justify our point of
view on the basis that we can also reproduce the em-
p0ddNpπ = 2 dqqiT(qi) Rπ(q1,q2,pT), (1) opffireicrianlgv2a,samsowotehschoanllneschtoiwon. wFiuthrththeremionrtee,rmaesiddieatferopmT
T Z i=1(cid:20) i (cid:21) region by the inclusion of TS recombination, our ap-
Y
dNp 3 dq proach describes also the effect of semihard scattering
p0 = i (qi) p(q1,q2,q3,pT), (2) on the soft sector. The ridge phenomenon that we at-
dp q T R
T Z i=1(cid:20) i (cid:21) tribute to that effect can be with trigger [20, 22, 23] or
Y
without trigger [14, 17, 18, 24]. Although data on the
where (q ) is the thermal distribution of the quark (or
T i ridge structure must necessarily make use of triggers in
antiquark) with momentum q , and h is the recombi-
i R ordertodistinguishitfrombackground[25–28],inclusive
nation function (RF) for h = π or p. On the assump-
distribution must include ridges along with background.
tion that collinear quarks make the dominant contribu-
Thus theoretically a single-particle distribution should
tion to the coalescence process (so that the integrals are
have a ridge component in the soft sector due to unde-
one-dimensionalforeachquarkalongthedirectionofthe
tectedsemihardorhardpartons. Thatcomponenthas φ
hadron), the RFs are
dependencethatcanbecalculatedfromgeometricalcon-
sideration[18],andhasbeenshowntobeconsistentwith
2
q q q
π(q ,q ,p) = 1 2δ i 1 , (3) the dependence of the ridge yieldin two-particlecorrela-
R 1 2 p2 i=1 p − ! tiononthetriggerangleφs relativetothe reactionplane
X [27].
3
p(q1,q2,q3,p) = f q1,q2,q3 δ qi 1 (4) Letususeρh1(pT,φ,b)todenotethesingle-particledis-
R (cid:18) p p p (cid:19) i=1 p − ! tributionofhadronhproducedatmid-rapidityinheavy-
X ion collision at impact parameter b, i.e.
where the details of f(q /p) that depends on the proton
i
wavefunctionaregivenin[11],andneednotberepeated dNh
ρh(p ,φ,b)= (N ), (8)
here. The main point to be made here is that if the 1 T p dp dφ part
T T
thermal distribution (q ) has the canonical invariant
i
form T where Npart is the number of participants related to b
in a known way through Glauber description of nuclear
(q)=qdNq =Cqe−q/T, (5) collision [29]. At low pT let ρh1 be separated into two
T dq components
thenthe δ-functionsinthe RFsrequirethatdNh/pTdpT ρh1(pT,φ,b)=Bh(pT,b)+Rh(pT,φ,b), (9)
has the common exponential factor, exp( p /T), for
T
−
bothh=πandp. Theprefactorsaredifferent;wesimply whereBh(p ,b)isreferredtoasBase,nottobeconfused
T
write down the results obtained previously withthebulkthatisusuallydeterminedinhydrodynam-
ics;thisisachangefromearliernomenclature[18],where
dNπ
= e−pT/T, (6) the use of “bulk” did lead to some misunderstanding.
π
pTdpT N Our emphasis here is that B(pT,b) is independent of φ.
dNp p2 In our approachwe regardthe semihard partons created
= T e−pT/T, m =(p2 +m2)1/2, (7)
p dp Npm T T p near the surface, and directed outward, give rise to all
T T T
the φ dependence of the medium before equilibrium is
where C2 and C3, and C has the dimen- established; the recoil partons being directed inward are
π p
sion(GNeV)−∝1. NotethNatth∝efactorp2/m intheproton absorbedand randomized. The componentexpressedby
T T
spectrum (that must be present for dimensional reason) Rh(p ,φ,b) is referred to as ridge on the basis of its φ
T
causes the p/π ratio to vanish as p 0 on the one dependence discussed below. The Bh(p ,b) component
T T
→
hand,buttobecome large,asp increases,onthe other. consists of all the soft and semihard partons that are
T
When p exceeds 2 GeV/c, shower partons become im- farther away from the surface and are unable to lead to
T
portantandtheabovedescriptionmustbesupplemented hadronswithdistinctiveφdependence. Thusthesepara-
by thermal-shower (TS) recombination that limits the tion between Bh(p ,b) and Rh(p ,φ,b) relies primarily
T T
increase of the p/π ratio to a maximum of about 1 [11]. ontheφdependencethattheridgecomponentpossesses.
We restrict our consideration to p < 2 GeV/c, but In Ref. [18] we have given an extended derivation of
T
nowbroadenittoincludeφdependence. Fornon-central whatthatφdependenceis. ItisembodiedinS(φ,b)that
3
is the segment of the surface through which a semihard where
partoncanbeemittedtocontributetoaridgeparticleat
φ. From the geometry of the initial ellipse (with width R¯0(pT)=e−pT/T e−pT/TB =e−pT/TB(epT/T˜ 1) (20)
− −
w and height h that depend on b) and from the angular 1 1 1 ∆T
= = , ∆T =T T . (21)
constraintbetween the semihardparton and ridge parti- T˜ TB − T TBT − B
cle prescribed by a Gaussian width σ determined earlier
in treating the ridge formation for nuclear density not There are two undetermined inverse-slopes: TB and T,
too low [23], it is found that common for both π and p. They are for single-particle
inclusive distributions, so only T is directly observable.
S(φ,b)=h[E(θ2,α) E(θ1,α)], (10) We postpone phenomenology to a later section. In ridge
−
analysis using triggered events for two-particle correla-
where E(θ ,α) is the elliptic integral of the second kind
i tion the two corresponding inverse slopes are separately
with α=1 w2/h2 and
− measured[26]. Here,however,wearedealingwithsingle-
θi =tan−1 h tanφi , φ1 =φ σ, φ2 =φ+σ,(11) hpaasrtitcoleddoiswtriitbhutriiodngse.sTahneddtihffeeirrenecffeecbtetowneetnheTBφ adnisdtrTi-
w −
(cid:18) (cid:19) bution. Thus we expect ∆T to be related to azimuthal
for φ π/2, and an analytic continuation of it for φ > asymmetry, a topic we next turn to.
i 2
≤
π/2. Thus S(φ,b) is completely calculable for any given
b, and Rh(p ,φ,b) is proportional to it.
T
We can now rewrite Eq. (9) unambiguously as III. QUADRUPOLE MOMENTS OF φ
ASYMMETRY
S(φ,b)
ρh(p ,φ,b)=Bh(p ,b)+ R¯h(p ,b) (12)
1 T T S¯(b) T
This topic is usually referred to as elliptic flow, a ter-
minologythatisrootedinhydrodynamics. Sincewehave
where
not used hydro in the previous section, it is more appro-
π/2
S¯(b)=(2/π) dφS(φ,b) (13) priatetousethe unbiasedlanguageinitiatedinRef.[30],
and call it azimuthal quadrupole. It is the familiar v
Z0 2
that is defined by
and R¯h(p ,b) is a similar average of Rh(p ,φ,b). Ac-
T T
cording to Eqs. (6) and (7) the inclusive distributions 2πdφcos2φρh(p ,φ,b)
ρe¯xh1p(p(T,pbT)/sTh)o,uflodr hsh=areπtahnedcpo,mams foonr qexuparoknse.ntTiahlatfadcotoers v2h(pT,b)=hcos2φihρ1 = R0 02πdφρh1(p1T,φT,b) .(22)
−
not take into consideration the enhancement of pions at
Using Eqs. (12) - (21) yields R
very small p due to resonance decay. We account for it
T
by a phenomenological term u(p ,b), and write
T [2R¯(p ,b)/πS¯(p ,b)] π/2dφcos2φS(φ,b)
ρ¯π1(pT,b) = Nπ(b)[1+u(pT,b)]e−pT/T, (14) v2h(pT,b) = T B(pTT,b)+RR0¯(pT,b)
p2 cos2φ
ρ¯p(p ,b) = (b) T e−pT/T, (15) = h iS , (23)
1 T Np m Z−1(p )+1
T T
wherethe resonanceeffect onthe protonis neglectedbe- where
cause ofbaryon-numberconservation. These expressions
are for the left-hand side of Eq. (12) after φ averaging. 2/π π/2
cos2φ = dφcos2φS(φ,b), (24)
ThebasetermBh(pT,b)ontherightsideisthesoftcom- h iS S¯(b)Z0
ponent without the contribution from semihard scatter-
Z(p ) = epT/T˜ 1. (25)
ing near the surface and should have the same common T −
structureasinEqs.(6)and(7)duetothermalpartonre-
Theseequationsareremarkableinthatthebdependence
combination, except that the inverse slope is lowerwith-
resides entirely in Eq. (24) and the p dependence en-
out the enhancement by the energy loss from the semi- T
tirely in Eq. (25); furthermore, there is no explicit de-
hard partons. We can therefore write
pendence on the hadron type nor the resonance term
Bπ(pT,b) = π(b)[1+u(pT,b)]e−pT/TB, (16) represented by u(pT,b). As we have noted at the end
N of the preceding section, T can be determined by the
p2
Bp(pT,b) = p(b) T e−pT/TB, (17) pT spectra, but TB is not directly observable. However,
N mT the quadrupole is measurable, so it can constrain T˜ and
where TB <T. It then follows that thereforeTB. Inshort,thetwoparametersT andTB can
be fixed by fitting the data on ρ¯h(p ,b) and vh(p ,b).
R¯π(pT,b) = π(b)[1+u(pT,b)]R¯0(pT) (18) Withoutusingamodeltodesc1ribeTtheevolu2tionTofthe
N
p2 dense medium, it is clearthat we cannotpredict the val-
R¯p(p ,b) = (b) T R¯ (p ), (19)
T Np mT 0 T uesofT andTB. However,ouraimistodiscoverhowfar
4
onecangowithoutusingsuchamodel. NeitherT norT
B 0.8
depend on φ. Yet non-trivial vh(p ,b) can be obtained
2 T
because of the presence of the ridge term in Eq. (12).
If phenomenology turns out to support this interpreta- 0.6
S
tion of azimuthal asymmetry, as we shall do in the next >
φ
section, then the ridges induced by undetected semihard 2 0.4
partonsplayamoreimportantroleingivingrisetotheφ s
o
dependence in inclusive single-particle distribution than c
<
hydro expansion that is based on assuming equilibration 0.2
to be completely at a later time without semihard scat- (a)
tering.
0
From Eqs. (10) and (24) we can calculate cos2φ 0 0.5 1 1.5 2
h iS b/R
and obtain its dependence on b. For the initial elliptical
A
configuration the width and height are
0.3
w =1 b/2, h=(1 b2/4)1/2, (26) p =1.58 GeV/c (×0.7)
− − T
where all lengths are in units of the nuclear radius RA. 0.25 p =0.975 GeV/c
Setting the Gaussianwidth σ betweenthe azimuthalan- T
gleφ1 ofthesemihardpartonandφ2 oftheridgeparticle 0.2 p =0.525 GeV/c (×1.83)
T
to be σ = 0.33 [23], we determine cos2φ as shown in
Fig. 1(a). h iS 20.15 <cos2φ> (×0.25)
v
S
According to Eq. (23) cos2φ contains all the b de-
pendence of v2h(pT,b) forhany pTiSin the soft region. To 0.1
check how realistic that is phenomenologically, we show
first in Fig. 1(b) the data on v2h(pT,Npart) for three pT 0.05 (b)
values from Ref. [2], but shifted vertically so that they
agree with the data for p = 0.975 GeV/c for most of
T 0
large N . The diamond and square points are slightly 0 50 100 150 200 250 300 350 400
part
shifted horizontally to spread out the overlappingpoints N
part
for the sake of visual distinguishability. The fact that
their dependencies on N are so nearly identical is re-
part
FIG. 1: (Color online) (a) Average of cos2φ weighted by
markablein itself. The solidline is areproductionofthe
S(φ,b) vs impact parameter in units of R . (b) Common
curve in Fig. 1(a) but plotted in terms of N , and re- A
part dependence of v2h(pT,b) on Npart for various pT, shifted ver-
duced in normalization by a factor 0.25 to facilitate the
tically for comparison. The diamond and square points are
comparison with the data points. For N > 100 the
part horizontally shifted slightly from the points in circles to aid
lineagreeswiththedataonv2verywell,thusprovingthe visualization. The solid line is from hcos2φi shown in (a),
S
factorizability of pT and b dependencies of Eq. (23). For butrescaledandplottedintermsofNpart. Thedataarefrom
N < 100, corresponding to b/R > 1.3 or centrality Ref. [2].
part A
>40%, there is disagreement which is expected because
the density is too low in peripheral collisions to justify
thesimpleformulainEq.(23). Adensity-dependentcor- Our first task is to determine the inverse slope T that
rectionisconsideredinRef.[18],butwillnotberepeated is sharedby T(q),ρ¯π1(pT,b)andρ¯p1(pT,b). Since the nor-
here. Ourfocus inthis paperis onthe inclusive ridge,so malizationfactorsinEqs.(5),(14)and(15)havenotyet
we proceed to phenomenology on the basis that the for- been specified, we consider first a particular centrality,
malism given above is valid for central and mid-central 20-30%, and fit the pT dependence of the proton spec-
collisions at Npart > 100. To have a compact analytic trum for pT <2 GeV/c, as shown in Fig. 2, and obtain
expression for S(φ,b) as given in Eqs. (10) and (11) to
T =0.283 GeV. (27)
summarizethe φdependence isnotonlyeconomical,but
also provides a succinct feature to distinguish the ridge Note that the one-parameter fit (apart from normaliza-
from the base components in Eq. (12). tion) is very good compared to the data from Ref. [1].
It demonstrates that the proton is produced in that p
T
range by thermal partons and that the flattening of the
IV. PHENOMENOLOGY spectrumatlowp isduetotheprefactorp2/m arising
T T T
from the proton wave function.
We now determine the parameters in our model Having determined T, we next consider the pion spec-
throughphenomenology. Asuccessin fitting allthe rele- trum ρ¯π(p ,b). According to Eq. (14) it has the same
1 T
vant data can givesupport to our approachthat empha- exponential factor as does ρ¯p(p ,b), but has also an ad-
1 T
sizes issues not considered in the standard model [31]. ditional factor [1+u(p ,b)] due to resonance decay. We
T
5
−2] 102 PHENIX proton 102 PHENIX pion
)
V/c 20−30% 2] 20−30%
e −)
y [(G 100 eV/c 100
d G
T
p ( no resonance
pdT10−2 T=0.283 GeV πρ [| 110−2 resonance only
p/
N with resonance
2
d
10−4 10−4
0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.6 1 1.4 1.8 2.2
p (GeV/c) p (GeV/c)
T T
FIG. 2: Proton spectrum at y ≈ 0 averaged over φ (hence, FIG. 3: Pion spectrum showing e−pT/T by the dashed line,
no 1/2π factor) at 20-30% centrality. The solid line is a fit and the resonance contribution by the dash-dotted line. The
of the data by Eqs. (15) and (27) with free adjustment of sum is in solid line. The data are from Ref. [1].
normalization. The data are from Ref. [1].
Eq. (25) so as to account for the mass effect, i.e.,
show in Fig. 3 the data from PHENIX [1] on the pion
distribution for 20-30% centrality; the exp( pT/T) fac- Z(p )=eET(pT)/T˜ 1, (30)
− T
tor is shown by the dashed line, the normalizationbeing −
adjusted to fit (and to be discussed later). For pT > 1 where T˜ is as given in Eq. (21). In Fig. 4 is shown the
GeV/c they agree very well, demonstrating the validity data from Ref. [2] when vh is plotted against E for 20-
of the common T. For p <1 GeV/c there is resonance 2 T
T 30% centrality. We fit the data points for both h = π
contribution to the pion spectrum which we cannot pre-
and p by Eqs. (23) and (30) with the choice
dict. Thus we fit the low-p region by the addition of a
T
rteersmponedxpin(g−tpoT/TTr=),0s.h1o7w4nGbeVy.thTehedasushm-ddoetpteicdtelidneb,yctohre- TB =0.253±0.003 GeV, (31)
r
solidlineagreeswiththedataperfectly. Thepointofthis
0.2
exerciseismainlytoshowthatthecommonexp( p /T)
T
behavioris validforpionasforproton,butthe r−ealityof STAR π
resonance contribution for pT < 1 GeV/c obscures that 0.15 20−30% p
commonality. Convertingtheresonancetermtotheform
given in Eq. (14) we write
0.1
2
u(p ,b)=u (b)e−pT/T0, (28) v
T 0
whereT =0.45GeVandu =3.416for20-30%central- 0.05
0 0
ity. We do not regardthis u term as a fundamental part
of our model; we attach the factor [1+u(p ,b)] to all
T 0
expressionsof the pion distributions, as in Eqs. (16) and
(18). Of more significance is the role that T has played 0 0.2 0.4 0.6 0.8 1
in the phenomenology,and so far T has played no role. E (GeV)
B T
T is not directly related to any observable spec-
B
trum, since it describes the p dependence of the base
Bh(p ,b) that lies under the rTidge. The important con- FIG.4: (Coloronline)v2h forh=π andp. Theshadedregion
T corresponds to T = 0.253±0.003 GeV. The data are from
B
cept we advance here is that it is φ independent, and
Ref. [2].
that Rh(p ,φ,b) carries all the φ dependence. Thus we
T
turn to vh(p ,b) in Eq. (23) and examine its p depen-
2 T T which is representedby the shaded regionin Fig. 4. The
dence for both h = π and p. In order to emphasize the
upperboundaryofthatregionisforT =0.25GeVthat
universality between π and p, we consider vh versus the B
2 fits the pionv almostperfectly, andthe lowerboundary
transverse kinetic energy E , for E <0.8 GeV, where 2
T T is for T = 0.256 GeV that fits well the proton v . It is
B 2
evident that v is very sensitive to T due to the expo-
E (p )=m (p ) m . (29) 2 B
T T T T h
− nentialfactorinEq.(30),yetthedatasupportacommon
We adopt the ansatz that p is to be replaced by E in valueforT towithin1-2%deviationforpionandproton
T T B
6
production. One cannot expect an accuracy better than Having obtained the correct centrality dependence of
thatinthe universalityofvh forh=π andp. Weregard ˇ that is calculable, we now consider the centrality de-
2
this result to be remarkable, since the normalization of pendenceoftheinclusivespectraρ¯h(p ,b). Wenotethat
1 T
vh is fixed by Eq. (23) without freedom of adjustment. the unknown normalization factors (b) and (b) in
2 Nπ Np
Notethatwehavenotusedanymoreparametersbesides Eqs. (14) and (15) never enter into the calculation of ˇ
T andT toaccomplishthis,whichisafittingprocedure because of cancellation, but for ρ¯h(p ,b) they must be
B 1 T
not more elaborate than the hydro approach where the reckoned with. As remarked after Eq. (7), (b) and
π
initial condition and viscosity are adjusted. (b) are proportional to C2 and C3, respecNtively, due
p
N
So far we have concentrated on 20-30% centrality to qq¯ and qqq recombination. The magnitude C of the
partly because we want to separate the p and φ de- thermal partons depends on b in a way that cannot be
T
pendencies from the issue of centrality dependence, and reliablycalculated. Byphenomenologyonthe pionspec-
partly because vh(p ,b) is large at 20-30%centrality for trum it was previously estimated for p > 1.2 GeV/c
2 T T
low p . To extend our consideration to other centrali- [18], but that is inadequate for our purpose here; more-
T
ties, we fix T and T at the values obtained in Eqs. (27) over, (b) and (b) have different statistical factors
B π p
N N
and(31) sothat Z(p )is no longeradjustable. The cen- that can depend on b because of resonances. We give
T
trality dependence of ˇ is then examined using Eq. (23). heredirectparametrizationsofthe normalizationfactors
Figure 5 shows the results for different centrality bins in terms of N
part
for both h = π and p. The shaded regions due to the
uncertainty in Eq. (31) become narrower in more cen- (N ) = 0.516N1.05, (32)
Nπ part part
tral collisions. The agreement with data from STAR [2] (N ) = 0.149N1.18, (33)
is evidently very good. Since there has been no more Np part part
u (N ) = 2.8+0.003N . (34)
adjustment of free parameters to achieve that, we find 0 part part
substantial support from Fig. 5 for our view that the
The parameters are determined by fitting the centrality
φ dependence arises entirely from the ridge component
dependence to be shown, but the essence of our predic-
in the inclusive distribution. This raisesserious question
tionis inp andφdependenciesthatarenotadjustable.
onwhetherviscoushydrodynamicsistheonlyacceptable T
Using the above in Eqs. (14) and (15) we obtain the
description of heavy-ioncollisions, if the reproductionof
curves in Fig. 6 (a) pion and (b) proton for three cen-
ˇ is the primary criterion for the success of a model.
trality bins. They agree with the data from PHENIX[1]
verywelloverawiderangeoflowp . Inallthosecurves
T
0.15
T is kept fixed at 0.283 GeV, thus reaffirming our point
(a) 0−5% π (b) 5−10% that both pions and protons are produced by the same
0.1 0.1
p setofthermalpartonsdespitetheapparentdifferencesin
STAR
the shapes of their p dependencies.
T
0.05 0.05
0 0
2 V. INCLUSIVE RIDGE DISTRIBUTION
v
0.2 0.4 0.6 0.8 01 0.2 0.4 0.6 0.8 1
(c) 10−20% (d) 20−30%
0.1 0.1 Itisnowopportuneforustorevisitthetwo-component
description of the single-particle distribution and focus
0.05 0.05 on the ridge component, in particular. As stated explic-
itlyinEq.(12),theφdependenceseparatestheBh(p ,b)
T
0 and R¯h(p ,b) components, the former being described
T
by Eqs. (16) and (17), the latter by Eqs. (18) and (19).
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
E (GeV) Upon averagingover φ, we have
T
ρ¯h(p ,b)=Bh(p ,b)+R¯h(p ,b). (35)
1 T T T
FIG. 5: (Color online) Same as in Fig. 4 for four centrality
bins. Since the exponentialfactorsare the same for h=π and
p, let us consider only the pion distribution specifically.
Itispossibletofurtherimprovetheagreementbetween In Fig. 7 we show B and R components by dashed and
the values of vh for pion and proton in Fig. 5 if those dash-dotted lines, respectively, for (a) 0-5% and (b) 20-
2
figures are replotted in accordance to the idea of quark 30%. It is in those figures that we exhibit the basic dif-
numberscaling(QNS),i.e.,vh/n vsE /n ,wheren is ference between our description of inclusive spectra and
2 h T h h
thenumberofconstituentquarksinhadronh[16,32]. As those of others. Inclusive ridge represented by R is al-
wehaveconsideredQNSanditsbreakingintherecombi- ways present in the single-particle distribution whether
nation model before [17], we do not revisit that problem or not an experiment chooses to do correlationmeasure-
here,especiallysinceourmaingoaltousev toconstrain ment to examine the ridge. Semihard scattering is un-
2
T has already been accomplished. avoidable in any nuclear collisions at high energy. Its
B
7
4
10
−2] 0−5% 102 PHENIX pion
)
V/c 20−30% 2]
Ge 102 40−50% −c) 0
( / 10
dy [ eV
T G B
dpT100 π [(110−2 R
p ρ|
π/ (a) pion B+R (a) 0−5%
N
2
d
10−2 10−4
0.2 0.6 1 1.4 1.8 2.2 0.2 0.6 1 1.4 1.8 2.2
p (GeV/c) p (GeV/c)
T T
−2] 102 0−5% 102 PHENIX pion
)
c 20−30%
V/ 2]
e −
G 40−50% c) 0
dy [( 100 eV/ 10
T G B
pdpT πρ [(| 110−2 R
pN/ 10−2 (b) proton B+R (b) 20−30%
2
d
−4
10
0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.6 1 1.4 1.8 2.2
p (GeV/c) p (GeV/c)
T T
FIG.6: Inclusivespectraatthreecentralitiesfor(a)pionand FIG. 7: Inclusivedistributions for pion showing thebase (B)
(b) proton. The data are from Ref. [1]. componentbydashedlineandridge(R)componentbydash-
dotted line for (a) 0-5% and (b) 20-30%. The solid line is
theirsum. The data are from Ref. [1].
effect on soft partons is therefore also unavoidable. We
quantify the effect by the R component which is deter-
curve for p >1 GeV/c by
T
mined by the azimuthal anisotropy that is well repro-
duced in Fig. 5. Here in Fig. 7 we see it rising above the R¯ (p ) R e−pT/T′, T′ =0.326 GeV. (36)
0 T 0
φ-independentbaseB componentwhenp ishigherthan ≈
T
1.4 GeV/c. It is a consequence of the recombination of Thus the ridge distribution is harder than the inclusive
enhancedthermalpartons. ForpT >3GeV/cinaddition distribution characterized by T = 0.283 GeV. This is a
to the inclusive ridge the jet component of the semihard property that is known from triggered ridges [26], but
partons themselves manifests in the spectra in the form now it is for untriggered inclusive ridge.
of thermal-shower recombination that characterizes the The enhancement of T′ over T is an important point
intermediate-pT region. Thus we have a smooth transi- to note. Physically, it means that the ridge is a conse-
tion from low- to intermediate-pT regionsby recognizing quence of the passage of semihard partons through the
the importance of the inclusive ridge component. medium, whose energy losses enhance the thermal par-
It is observed that the dash-dotted lines in Fig. 7 are tons in the vicinities of the trajectories. We know that
not exactly straight because the ridge component is not the enhancement factor is Z(p ), which has the neces-
T
exponentialinp . However,forp >1GeV/c,R¯π(p ,b) sary p dependence to render vh(p ,b) to be in good
T T T T 2 T
can be wellapproximatedby pure exponential. In Fig. 8 agreement with the quadrupole data. In particular, the
we show by the solid line the p dependence of R¯ (p ), property that Z(p ) 0 as p 0 is essential to guar-
T 0 T T T
definedinEq.(20);itisthepartoftheridgedistributions antee that vh(p ,b) → 0 in the→same limit. The effect
R¯h(p ,b)inEqs.(18)and(19)thatiscommonforh=π of Z(p ) at l2argTer p →is to increase T to T′. Although
T T T B
and p andis independent of b. From the values ofT and Z(p ) increases exponentially, its net effect on R¯ (p )
T 0 T
T thatwenowknow,wehaveT˜ =2.39GeV.The(red) is suppressed by e−pT/TB. The effective inverse slope T′
B
dashed line is a straight-line approximation of the solid for p > 1 GeV/c is larger than T of the inclusive by
T
8
−1 Similar consideration has also been used in the explana-
10
tionofthe ridgestructure foundatLHC [34, 35]. Inthis
paper we have presented the most detailed quantitative
]
2
− analysis of the RHIC data in the formalism of inclusive
)
c −2 ridge that sets the foundation for the ridges at ∆η >0.
V/ 10 | |
e
G
(
[ VI. CONCLUSION

)T10−3 Ridge
p
(
π 0 exp(−p /T’) Our study of inclusive ridge distributions has consoli-
|R
T dated earlier exploratory work with firm phenomenolog-
−4 ical support, and therefore succeeded in extending the
10
0 0.5 1 1.5 2 hadronization formalism from intermediate-p region to
T
p (GeV/c) below 2 GeV/c, exposing thereby an aspect of physics
T
that has not been included in other approaches. The ef-
fect of semihard scattering on soft partons is accounted
FIG.8: (Color online) ThepT dependenceof R¯0(pT)defined for by the ridge component whose azimuthal behavior is
in Eq. (20), represented by the solid line. The (red) dashed
totallycharacterizedbyS(φ,b);itisacalculablequantity
line is a straight-line approximation for p > 1 GeV/c, ex-
T
bounding the surface segment through which semihard
pressed by Eq.(36).
partonscancontributetotheformationofaridgeparticle
at φ. In an earlier paper [18] we showed the connection
betweenS(φ,b)andthedependenceofthetriggeredridge
43MeV,roughlythesameaswhatPutschkereportedon
yield on the φ of the trigger angle relative to the reac-
s
the firstdiscoveryofridge,where the triggeredridge has
tion plane. Now, we have exhibited the central role that
′
T larger than that of the inclusive by 45 MeV [33].
S(φ,b) plays in determining the azimuthal quadrupole
There are, however, subtle differences between trig- vh(p ,b) of inclusive distributions. Thus the inclusive
2 T
gered and untriggered ridges. Experimentally, it is nec- ridgedistributionthat we haveadvancedin this seriesof
essarytodocorrelationmeasurementstolearnaboutthe workservesasabridgebetweenthesingle-particledistri-
properties of the ridge, which is extracted by a subtrac- bution and the two-particle correlation. Since semihard
′
tion scheme. The inverse slope T can be compared to partons are copiously produced before thermalization is
theinclusiveT ofthebackground. Forsingle-particledis- complete,itisanaspectofphysicsthatshouldnotbeig-
tribution the only measurable inverse slope is T for the nored. The success in fitting vh(p ,b)for all centraland
2 T
inclusive spectrum. Theoretically, we assert that ridges mid-central collisions and for both π and p by one pa-
do notdisappearjust because triggersarenotused. The rameterT thereforeleadstoclaimofrelevanceasmuch
B
inverseslope T′ for R¯0(pT) cannotbe measureddirectly. as viscous hydro does.
It is larger than both T and TB because R¯0(pT) is the Another attribute of our approachis to unify the pro-
difference between the two exponentials for ρ¯1 and B, ductionofpionsandprotonsinonehadronizationscheme
represented by the middle term in Eq. (20), which van- basedontherecombinationofenhancedthermalpartons
ishes as pT 0. A physicallymore sensible way to com- so that their spectra have the same inverse slope T de-
→ ′
pare the various inverse slopes is to recognize that T is spiteapparentdifferencesinthelow-p data. Thatsame
T
significantly larger than TB because of the enhancement schemewhenextendedtopT 3GeV/cexplainsreadily
effect due to semihard scattering, and that T is the ef- theobservedlargep/πratio. ∼Sincetheridgecomponents
fective slope of the inclusive distribution, B+R¯, that is inourformalismarethesameforπ andp,wecanpredict
′
measurable and is between TB and T . that the p/π ratio is also large in the triggered ridge as
Althoughourconcerninthispaperhasbeenrestricted in the inclusive.
to the midrapidity region, the physics of inclusive ridge The property about ridge that most investigators are
can be extended to non-vanishing pseudo-rapidity η. In concerned about is the large ∆η range found in correla-
Ref. [20] a phenomenological relationship is found be- tion experiments. That is an aspect of the problem that
tween the triggered ridge distribution in ∆η and the in- has been addressed in Ref. [20]. Our focus in this paper
clusive distribution in η with the implication that there is on the hidden aspectof the ridge that is not easilyde-
is nolong-rangelongitudinalcorrelation. However,there tected, but is pervasive because it is in the inclusive dis-
can be transverse correlationdue to transverse broaden- tribution. Phenomenologicalsuccess found in this paper
ing of forward (or backward) soft partons as they move puts the idea onsolid footing. If the conceptof inclusive
throughtheconicalvicinityofthesemihardpartons. The ridge is important at RHIC, then its relevance at LHC
enhancement of the thermal partons due to energy loss will be predominant. Since the structure of vh(p ,b) ex-
2 T
is just as we have described in this paper. Indeed, the pressed in terms of S(φ,b) in Eq. (23) is independent of
term representing the enhanced p distribution in Ref. initial density, viscosity, or even collision energy, except
T
[20] is essentially identical to that expressed in Eq. (20). T˜,wewouldexpectv measuredatLHCtobeessentially
2
9
similar to what is shown in Fig. 5 for both π and p. A and by the Scientific Research Foundation for Young
preliminarylookatthedatafromALICE[36]leadsusto Teachers, Sichuan University under No. 2010SCU11090
believe that such an expectation may not be unrealistic. andKeyLaboratoryofQuarkandLeptonPhysicsunder
Grant No. QLPL2009P01.
Acknowledgment
Thisworkwassupported,inpart,bytheU.S.Depart-
ment of Energy under Grant No. DE-FG02-96ER40972
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