Comparing the same-side "ridge" in CMS p-p angular correlations to RHIC p-p data

0.65 MB·English

by  Thomas A. Trainor

Version 1.5
Comparing the same-side “ridge” in CMS p-p angular correlations to RHIC p-p data
Thomas A. Trainor and David T. Kettler
CENPA 354290, University of Washington, Seattle, WA 98195
(Dated: January 7, 2011)
The CMS collaboration hasrecently reported theappearance of a same-side “ridge” structure in
two-particleangularcorrelations from 7TeVp-pcollisions. Theridgeinp-pcollisions at 7TeVhas
beencomparedtoaridgestructureinmore-centralAu-Aucollisionsat0.2TeVinterpretedbysome
asevidenceforadense,flowingQCDmedium. Inthisstudywemakeadetailedcomparisonbetween
0.2 TeV p-p correlations and the CMS results. We find that 7 TeV minimum-bias jet correlations
are remarkably similar to those at 0.2 TeV, even to the details of the same-side peak geometry.
Extrapolation of azimuth quadrupole systematics from 0.2 TeV suggests that the same-side ridge
at 7 TeV is a manifestation of theazimuth quadrupolewith amplitude enhanced byapplied cuts.
1
PACSnumbers: 12.38.Qk,13.87.Fh,25.75.Ag,25.75.Bh,25.75.Ld,25.75.Nq
1
0
2
I. INTRODUCTION II. CMS CORRELATION ANALYSIS

n
a
J The CMS collaboration has recently released the first The CMS analysis is directly related to correlation
6 study of angular correlations from p-p collisions at the analysis previously carried out at the RHIC. In order to
Large Hadron Collider (LHC) [1]. A notable result is distinguishwhatistrulynovelattheLHCitisimportant

] the appearanceof a same-side “ridge”structure possibly to establish the quantitative relationships among differ-
h
related to the expected jet peak at the angular origin ent analysis methods, and the p-p phenomenology that
p
(intrajet correlations). The same-side ridge observed in has emerged from previous work at lower energies.
-
p p-p collisions at 7 TeV has been compared to a ridge
e structure observed in more-central Au-Au collisions at
h
the Relativistic Heavy Ion Collider (RHIC) [2], inter- A. CMS analysis method
[
preted by some as evidence for a dense, flowing QCD

2 medium[3,4]. Thepossiblecorrespondencesuggeststhat TheCMSanalysisisbasedonnormalizedpairdensities
v adensemediummayalsoforminp-pcollisionsat7TeV.
8 WhiletheCMSresultisintriguingonecanaskwhetherit 1 d2Nsignal
4 doessignalnovelphysicsinp-pcollisionsatLHCenergies SN = N(N 1) d∆ηd∆φ (1)
0
−
3 or whether it is simply an extrapolation of p-p phenom- 1 d2Nbkd
0. ena previously observed at 0.2 TeV. BN = N2d∆ηd∆φ,
Inthepresentstudywemakequantitativecomparisons
1
0 between CMS results at 7 TeV and correlationmeasure- where for example Nsignal is a pair number, to be dis-
1 ments at 0.2 TeV. The generalstrategy is to extrapolate tinguished from N, the number of (charged) particles in
: measured 0.2 TeV p-p correlations to 7 TeV via the en-
v the angular acceptance. Pair densities SN and BN are
ergydependence establishedbelow 0.2TeVandthen de-
Xi terminewhatotheralterationsoftheextrapolatedcorre- equivalenttopairnumbersnˆab definedinRef.[5,6],with
sibling-to-mixed pair ratio rˆ = nˆ /nˆ for 2D
ab ab,sib ab,mix
ar lbaottiohnmsitnruimctuumre-bairaesrdeaqtuairaenddtowidtehsccruibtseatphpeli7edT.eV data, bins (a,b) on difference variables η∆ = η1 − η2 (pseu-
dorapidity) and φ∆ = φ1 φ2 (azimuth). The CMS
We address a central question: Is the CMS “ridge” a correlationmeasure is −
trulynovelmanifestationat7TeV,suggestinganomalous
physics in p-p collisions previously encountered in RHIC SN
R(∆η,∆φ) = ( N 1) 1 (2)
heavy ion collisions at 0.2 TeV? Or is the unexpected (cid:28) h i− (cid:18)B − (cid:19)(cid:29)
N N
structure simply explained as an azimuth quadrupole = ( N 1)( rˆ 1),
component consistent with RHIC quadrupole systemat- h i− h i−
ics extrapolated to 7 TeV and CMS cut conditions? with the correspondence to measure N ( rˆ 1) of
h i h i −
The paper is arranged as follows. We briefly review Refs. [5, 6] established in the second line. The corre-
the CMS angular correlation results. We then describe lationmeasuredefinedinEq.(2)is directly proportional
general correlation analysis methods applied to nuclear to the specific detector angular acceptance.
collisions atthe RHIC. We review the systematics of an- Event multiplicity selection strongly influences spec-
gular correlations from 0.2 TeV p-p collisions. We then trum and correlation structure in p-p collisions, mainly
report quantitative A-B comparisons between 0.2 TeV by biasing the jet frequency per p-p collision [7]. In the
p-p correlations extrapolated to 7 TeV and the CMS re- CMS analysis p -integral angular correlations were ob-
t
sults. We conclude that the CMS “ridge” is consistent tained for minimum-bias data and for a cut on “offline
with RHIC correlation data suitably extrapolated. tracks” N > 110 (p > 0.4 GeV/c, η < 2.4) which
trk t
| |
2
corresponds to Ntrk = 118. The corresponding cor- B. Correlations on yt yt
h i ×
rected angular density is dN /dη 40. The NSD den-
ch
≈
sity is dNch/dη ≈ 5.8, with hNtrki ≈ 16 [8]. The ratio 2D correlations on pt or transverse rapidity yt =
of cut-selected to minimum-bias correctedmultiplicity is ln[(p +m )/m ] (m for unidentified hadrons)are com-
t t π π
then about 7. pt-differential correlations were also ob- plementaryto4Dangularcorrelationsin6Dtwo-particle
tained for specific pt bins. pt cuts should modify jet cor- momentum space. yt is preferred for visualizing correla-
relations and possibly the nonjet azimuth quadrupole. tionstructureontransversemomentum. Similartoangu-
larcorrelations,y y correlationscanbedefinedforLS
t t
×
and US charge combinations but also for same-side (SS,
B. CMS correlation measurements
φ∆ <π/2) and away-side (AS, π/2< φ∆ < π) subre-
| | | |
gionsofangularcorrelations. Differentcorrelationmech-
The main results are briefly summarized here. The anismscanbedistinguishedinthefoursubspaces[15,16].
correlation histograms are discussed in more detail in
Sec. V. The CMS study concluded that jet correlations
are enhanced in high-multiplicity collisions. The away- C. Correlation measures
side ridge (interjet correlations)is flat for η <1.5, con-
| |
sistentwithSTARmeasurementswithintheTPCaccep- Forourinitial angularcorrelationanalysiswe adopted
tance η < 1 [5, 9]. A CMS breakdown of p-p angular N ( rˆ 1) as the correlation measure [5, 6]. Sib-
| | h chi h i −
correlation structure at the LHC can be compared with ling and mixed pair numbers are normalized to unit in-
200 GeV p-p phenomenology described in Sec. IV. tegral, and pair ratio rˆ is averaged over kinematic bins
Several pt cut intervals were defined. For pt ∈ [1,3] (e.g. multiplicity, pt, vertex position). That per-particle
GeV/c and high-multiplicity cut a same-side “ridge” is measure is an improvement over conventional per-pair
assumedtobeassociatedwiththesame-side2Djetpeak. correlation function C r or r 1 ∆ρ/ρ (∆ρ
ref
↔ h i h i− →
No corresponding ridge is observed in PYTHIA data. is the correlated-pair density and ρ is the reference-
ref
The pt and Nch dependence of the same-side ridge is or mixed-pair density). The per-particle measure elimi-
studied via ZYAM subtraction, usually applied to jet nates a trivial 1/N trend common to all per-pair mea-
ch
studies with 1D dihadron correlations [10]. The same- sures [11]. However, that extensive measure is propor-
side ridge does not depend on charge combinations. A tionaltothespecificdetectorangularacceptance∆η∆φ.
similarstructureisobservedforcorrelatedγsfromπ0 de- The corresponding intensive correlation measure is a
cay. Observation of a same-side ridge in p-p collisions at form of Pearson’s normalized covariance [11],
7TeVandpossibleimplicationsinrelationtoasame-side
∆ρ N
“ridge” reported in more-centralRHIC Au-Au collisions = h i ( rˆ 1) ρ0( rˆ 1) (3)
are the featured results of the CMS analysis. √ρref ∆η∆φ h i− → h i−
assuming single-particle density ρ0 uniform within the
angularacceptanceandfactorizationofthereferenceden-
III. ANALYSIS METHOD FOR THIS STUDY sity, ρref ρ20. ∆ρ/√ρref is invariant under combina-
≈
tion of uncorrelated parts, therefore should not change
We reviewtechnicalaspectsofSTARcorrelationanal- with linear superposition of N-N collisions. The measure
ysis applied to nuclear collisions at the RHIC. Method is independent of angular acceptance if the underlying
details are provided in Refs. [5, 9, 11–16]. physical mechanisms are uniform across the acceptance.
A. Angular correlations on (η∆,φ∆) D. A-B comparisons
Two-particle angular correlations are defined on 4D Histograms for this study were binned as 25 28 on
×
momentum subspace (η1,η2,φ1,φ2). In acceptance in- (η∆,φ∆) to match the CMS binning. Detailed compar-
tervals where correlation structure is invariant on mean isons of contour lines between data and model functions
position (e.g. ηΣ = η1 + η2) angular correlations can then permit quantitative inference of model parameters
be projected by averaging onto difference variables (e.g. from the CMS data, typically accurate to 10% for the
η∆ = η1 η2) without loss of information to form an- simple structures in p-p correlations. The CMS color
−
gular autocorrelations [6, 11]. The 2D subspace (η∆,φ∆) palette includes 20 colors, whereas this study employs
is then visualized. The notation x∆ rather than ∆x for over 50 colors, causing significant differences in shading
difference variablesis adoptedto conformto mathemati- insomeregions. Anattemptwasmadetoinsurethatthe
calnotationconventionsandto reserve∆x as a measure intervals spanned by vertical scales are nearly the same
ofthe detector acceptanceonparameterx. Angular cor- for model and CMS data in each comparison, although
relations can be formed separately for like-sign(LS) and the offsets may differ due to analysis details. By care-
unlike-sign (US) charge combinations, as well as for the ful A-B comparisons a good approximation to direct χ2
charge-independent (CI = LS + US) combination [5, 6]. model fits to histogram data can be achieved.
3
IV. 0.2 TEV p-p ANGULAR CORRELATIONS C. Correlation model functions
The soft component of angular correlations is mod-
p-pandAu-AuangularcorrelationsatRHIChavebeen
extensively studied by the STAR collaboration [5, 6, 9, eled by a 1D Gaussian on η∆ with r.m.s. width approxi-
12–16]. Those results provide an essential reference for mately1. Themodelisassumedtobeuniformonφ∆ for
simplicity. However, there are indications that the soft
the CMS p-p measurements at 7 TeV.
componentis suppressednearφ∆ =0 asa resultof local
transverse-momentumconservation. Thesoftcomponent
falls within p < 0.5 GeV/c. Its amplitude decreases to
t
zerowithincreasingcentralityinA-Acollisions. Thesoft
A. Two-component angular correlations componentisexclusivelyUSpairs,reflectinglocalcharge
conservation during projectile-nucleon fragmentation.
The correlation hard component has two parts, a SS
Spectra and correlations in nuclear collisions can be
2D peak at the origin and an AS 1D peak on azimuth
decomposed (near mid-rapidity) into soft and hard com-
ponents, denoting respectively longitudinal fragmenta- uniformonη∆ (withintheSTARTPCacceptance). The
SS jet peak (intrajet correlations) is well modeled by
tion (mainly diffractive dissociation) of projectile nu-
a 2D Gaussian. The SS peak is also characterized as
cleons and transverse fragmentation of large-angle scat-
tered partons [7, 15–17]. The hard-component fraction ρ0(b)j2(η∆,φ∆,b)withinangularacceptance(∆η,∆φ)=
(2,2π)—the STAR TPC acceptance [18]. The corre-
amounts to a few percent in minimum-bias (NSD) p-p
sponding CMS acceptance is (4.8,2π). Except for p-p
collisions [7] but increasesto about 1/3of the final-state
and most-peripheral A-A collisions the AS peak is con-
yield in central Au-Au collisions [18].
venientlymodeledasanASdipolebasedoncos(φ∆ π).
−
The combined model function [5, 9], including the az-
imuth quadrupole term cos(2φ∆) required to describe
minimum-bias A-A angular correlations,is
B. Minijet phenomenology
∆ρ = A0+A1e−12(cid:26)(cid:18)σφφ∆∆(cid:19)2+(cid:16)σηη∆∆(cid:17)2(cid:27)+A2e−12(cid:16)ησ∆2(cid:17)2
Minijets playakeyrolein RHIC collisions[17,19–21]. √ρref
ThehistoryofminijetsdatesfromtheUA1observationin + AD [1+cos(φ∆ π)]/2+AQ2cos(2φ∆), (4)
−
1985ofjetproductionat√s=200GeVfollowingpQCD
where a narrow 2D exponential describing quantum cor-
cross-sectionpredictionsdownto5GeV(backgroundcor-
relations and electron pairs from γ conversions has been
rectedto 3-4 GeV) [22]. Becauseof the partonspectrum
omitted for clarity. The dipole term in Eq.(4) is defined
structure (power law with lower cutoff near 3 GeV) the
differently from that in Ref. [9].
minimum-bias parton (jet) spectrum is dominated by 3
For p-p collisions the AS dipole term (interjet corre-
GeV jets [20]. Thus, correlations from 3 GeV minijets
lations) may be replaced by a 1D Gaussian on azimuth
and from minimum-bias jets are essentially equivalent.
centered at π with periodic image at π [10]. The total
Theinterplayofcorrelationson(η∆,φ∆)andyt ytfor quadrupoleinferredfromEq.(4)then−includesthenonjet
×
SS/AS angular subregions and LS/US charge combina- quadrupole and a possible quadrupole contribution from
tions distinguishes soft- and hard-componentcorrelation the AS jet peak. For minimum-bias A-A collisions the
structure [15, 16]. For instance, SS-US yt yt corre- latter is small, since for r.m.s. widths greater than 1.2
×
lations show clear intrajet structure extending down to the higher Fourier components of the AS periodic peak
0.3 GeV/c with mode at yt =2.7 (pt 1 GeV/c), corre- array(includingthequadrupoleterm)approachzero[10].
∼
spondingclosely(whenprojected)tothep-ppt spectrum The minimum-bias SS 2D jet peak in p-p collisions is
hard component first revealed in Ref. [7]. The SS-LS stronglyelongatedintheazimuthdirection,withapprox-
combination is dominated by HBT correlations appear- imate3:2aspectratio[15,16]. Thestrongφelongationin
ing below 0.5 GeV/c, with negligible jet contribution. p-pcollisionscontrastswith strongη elongationinmore-
AS-LS and AS-US correlations contain identical in- central Au-Au collisions, with 3:1 aspect ratio [5, 9].
terjet contributions appearing above 0.7 GeV/c, also
with mode at y = 2.7 (p 1 GeV/c). The absence
t t
of charge dependence for AS∼jet structure is consistent D. 200 GeV p-p model parameters
with scatteredpartons(mainly gluons)havingno charge
correlation. The AS-US combination also shows a soft- Figure 1 shows model results for 200 GeV NSD p-p
componentcontributionbelow0.5GeV/c. Cutsony y collisions for the STAR intensive measure (left panel)
t t
×
inturnisolatecorrespondingelementsofangularcorrela- and for the CMS extensive measure including the an-
tions. The combination provides a clear picture of local gular acceptance (right panel), the latter as in Ref. [5].
chargeconservationintwoorthogonalfragmentationpro- ResultsobtaineddirectlyfromNSDp-pcollisions[15,16]
cessespluslike-signquantumcorrelations(HBT)[15,16]. areconsistentwithperipheralA-Acollisionsextrapolated
4
to N-N collisions [9], providing a cross check of methods F. Collision-energy dependence
anddata consistency. The two casesarethus equivalent.
We observe that minijet correlations in the form
ρ0(b)j2(η∆,φ∆,b) [23, 24] and the azimuth quadrupole
(a) (b) in the form ρ0(b)v22(b) [12, 25] (next subsection) scale
ρref00.1.15 ρref 4 wGietVh,ewnehregryea√psp0roxi1m3a.5teGlyeaVs.loWge(√thseNrNef/o√resd0)efibneelow200
∆ρ√ / 0.005 ∆ρ√ / 02 R(√sNN) = l≈og(√sNN/13.5 GeV)/log(200/13.5)(5)
φ φ
∆ ∆
∆η 4 4 ∆η 4 4 to represent collision-energy scaling of jets and
φ∆ 2 0 2η ∆ φ∆ 2 0 2η ∆ quadrupolerelativeto200GeV[12]. Assumingthe same
0 -2 0 -2 scalingup to 7 TeVimplies that both amplitudes should
-4 -4 be larger by factor log(7000/13.5)/log(200/13.5) 2.3
≈
at 7 TeV compared to 200 GeV.
FIG. 1: (Color online) (a) Model angular correlations repre-
The symbol R introduced in the CMS analysis should
senting200GeVNSDp-pcollisions[15,16]. (b)Thesameplot
be distinguished from energy scaling factor R(√s) in
rescaled bythe CMS angular acceptance ∆η∆φ=4.8 2π.
× Eq.(5)definedinRef.[12]inconnectionwithv2analysis.
The measured 200 GeV N-N correlation parameters
are summarized in the first row of Table I. The 2D G. Azimuth quadrupole systematics
Gaussian parameters [9] are A1 =0.058, ση∆ =0.64 and
σφ∆ = 0.9. The amplitude of the 1D Gaussian on η∆ is The azimuth quadrupole, with form cos(2φ∆), is con-
A2 =0.023, with σ2 1. The AS azimuth dipole ampli- ventionallyinterpretedinA-Acollisionsas“ellipticflow,”
≈
tude is A 0.046 (twice the value reported in Ref. [9]
D aconjecturedresponsetoearlydevelopmentoflargepres-
≈
becauseofthedifferentdipoledefinition). Whenmodeled
sure gradients in non-centralA-A collisions [26]. v2 data
as a 1D Gaussianthe AS peak amplitude remains 0.046,
inferredfromfitsto2Dangularcorrelationsanddenoted
but with r.m.s. width 1 implying a significant jet-
≈ v2 2D ,whichaccuratelyexcludecontributionsfromjets
related quadrupole component. The nonjet quadrupole { }
(“nonflow”),revealsystematicbehaviorinconsistentwith
amplitude extrapolated from Au-Au centrality depen-
hydroexpectations,suggestinganalternativeinterpreta-
dence is A 0.0005. The coefficients of the dipole
Q tion in terms of interacting gluonic fields [12, 25]. Al-
∼
and nonjet quadrupole sinusoids are A /2 = 0.023 and
D thoughtheazimuthquadrupoleisnotusuallyconsidered
2A = 0.001 respectively. The sinusoid coefficients are
Q relevant to p-p collisions it may explain the CMS ridge
important for the curvature discussion in Sec. VIF.
manifestation, as demonstrated by this analysis.
An analysis of pt-integral v2 2D for Au-Au collisions
{ }
at 62 and 200 GeV, combined with SPS v2 EP data
{ }
E. Multiplicity dependence of the hard component at 17 GeV, led to the following simple relation which
describes v2 2D data from 13.5 to 200 GeV [12],
{ }
colTlihsieonpts-sspceacletsruimn ahmarpdlictoumdepo(nreelnattiivne2t0o0tGheeVsoNfStDcopm-p- √∆ρρr[2ef] =ρ0(b)v22{2D}(b)=0.0045R(√sNN)ǫ2optnbin.(6)
ponent) nearly linearly with N as N increases by a
ch ch
factor ten relative to the NSD value [7]. The spectrum For 200 GeV NSD p-p collisions n = 1 and ǫ
bin opt
hardcomponentisquantitativelyconsistentwithminijet 0.3, yielding ρ0v22 0.0005. For the CMS analy≈-
≈
correlations[18]andwithpQCD-calculatedfragmentdis- sis additional factors (defined above) lead to 2R
Q
tributions[20]. Thesingle-particlehard-componentyield R(7 TeV)∆η∆φ2ρ0v22 = 2.3 30 2 0.0005 = 0.0≡7
× × ×
increases by a factor 50 as the particle multiplicity near asthe predicted quadrupoleamplitude forminimum-bias
mid-rapidity increases ten-fold. Equivalently, the jet fre- p-pangularcorrelationsconsistentwithCMSmeasureR.
quency per p-p collision within one unit of η increases The role of p-p collision centrality is of significant in-
from 2% to nearly 100%. The jet frequency saturates at terest at the LHC [27]. Requiring increased multiplicity
one jet (pair) per collision for large event multiplicities. shouldbiasthep-pcollisiongeometrytomore-centralcol-
Correspondingjetangularcorrelationsscaleasfollows. lisions. The product ǫ2 (b)n (b) in Eq. (6) increases
opt bin
Themeanjetfragmentmultiplicity( 2.5,dominatedby rapidly with increasing centrality in Au-Au collisions to
∼
3 GeV jets) does not change significantly with N (the a maximum for mid-central collisions (50-fold increase),
ch
number of correlatedpairs per jet is then fixed), but the beyond which the rapidly decreasing eccentricity dom-
jet frequency increases by a factor 50 with 10 increase inates [12]. The centrality trend for A-A collisions is
×
in N , rising to one jet per event. Jet correlationsmea- independentofcollisionenergyandabsolutesystemsize,
ch
sured by ρ0j2 then scale up as 10 (50/102) = 5, since suggesting that with increasing multiplicity (centrality)
pairratioj2 representscorrelatedp×airs/referencepairs. ρ0v22 2D for p-p collisions also increases substantially.
{ }
5
V. 7 TEV p-p ANGULAR CORRELATIONS Nonjet and jet-related quadrupoles (in that order) are
shownin both the 2A andthe 2R columns ofTable I.
Q Q
Wecompareenergy-scaled0.2TeVangularcorrelation
results to CMS 7 TeVresults for minimum-bias andcut-
selected data. The inferred parameters for two energies
and several cut conditions are summarized in Table I. B. High-multiplicity cut
Selecting high-multiplicity p-p collisions biases the jet
A. Minimum-bias angular correlations frequency per event to larger values [7]. The increased
hardscatteringcanbeinterpretedasaresultofincreased
For the extrapolation we assume the simplest case: p-p centrality induced by the multiplicity cut. Figure 3
log(√s) scaling observed below 200 GeV continues up (leftpanel)showsthemodelfunctioninFig.2(leftpanel)
to 7 TeV. We then scale all STAR minimum-bias with the following changes: (i) The jet structure (SS 2D
200 GeV angular correlation structure as X(7 TeV) = peak, AS ridge)is scaledup by factor 3, (ii) the SS peak
[R(7 TeV)2π∆η] X(0.2 TeV), with R(7 TeV) = 2.3 and φ width is reduced from 0.9 to 0.65 and the η width
∆η = 4.8. The overall scaling factor is therefore from 0.64 to 0.58, and (iii) the 1D η Gaussian is elim-
[2.3 30] 70. TheresultingmodelinFig.2(leftpanel) inated. The quadrupole component for this figure (not
× ≈
compareswellwithCMSminimum-biasdataintheright visible in this plotting format) is unchanged from Fig. 2
panel, but only if the extrapolated AS ridge amplitude (minimum-bias RQ = 0.07). The result compares well
is reduced by factor 0.6 (compare with Fig. 1). The AS with CMS data in the right panel, with the exception of
amplitude reduction may be a consequence of the larger the narrow contribution from quantum correlations and
4π η acceptance at 7 TeV compared to 200 GeV [20]. electron pairs at the peak not included in the model.
The same reduction factor is retained for all other com-
parisons. The largeφelongation(3:2)ofthe SS2Dpeak
observedin 200 GeV p-p collisions [16] persists in 7 TeV
(a)
collisions,asdoesthe1DGaussianonη∆ associatedwith
longitudinal projectile-nucleon fragmentation [15].
√ρref 2300
(a) φ∆ρ / 100
∆
√ρref 71.505 ∆η φ4∆ 2 0 2η ∆4
∆ρ / 2.05 0 -4 -2
φ
∆
∆η 4 4 FIG. 3: (Color online) (a) Model function from Fig. 2 (left
φ2 2η panel) multiplied by factor 3, with SS peak narrowed on η∆
∆ 0 -2 0 ∆ and with 1D η∆ Gaussian removed. (b) CMS pt-integral an-
-4 gularcorrelationsforhigh-multiplicity7TeVp-pcollisions[1].
FIG. 2: (Color online) (a) The function from Fig. 1 (left
panel) rescaled by energy-dependent factor R(7TeV) = 2.3. Figure 4 shows the results in Fig. 3 with the verti-
(b) CMS angular correlations for minimum-bias 7 TeV p-p cal axis range reduced by a factor 5 to enhance small-
collisions [1]. amplitude details. Correspondence of the SS jet peaks
near the base is evident. The AS ridge shows significant
TheactualASpeakmodelinFig.2(leftpanel)isa1D reductionin the data at largerη∆ (lowerpanel) which is
notincludedinthemodelfunction(upperpanel)inferred
Gaussian with σ 1. The quadrupole component of
φ∆ ≈ within the STAR TPC acceptance.
the Gaussian is then nominally 2A 0.125 A [10].
Q D
≈ ×
However, for the purpose of the curvature discussion in The nonjet quadrupole amplitude is increased by fac-
Sec. VIF we represent the Gaussian as a combination tor 6 for this figure (relative to the minimum-bias value)
of only AS dipole and quadrupole terms. Requiring an to 2R = 0.42. The larger quadrupole amplitude is
Q,NJ
equivalent net curvature at the angular origin φ∆ = 0 required to change the SS curvature in η∆ > 2 from
| |
defines 2A 0.05A . The jet-related quadrupole concave upward to slightly concave downward (compare
Q D
≈
then also represents curvature contributions from higher with Fig. 3 – left panel). Thus, the 6-fold quadrupole
Fourier components, mainly the sextupole. We there- increase is already required by CMS data prior to impo-
fore define jet-related quadrupole 2AQ,J = 0.05 AD sitionofpt cuts. Thecurvatureonazimuthatφ∆ =0for
×
and nonjet quadrupole AQ,NJ, with total quadrupole η∆ > 2 is the sole basis for identification of a “ridge.”
| |
A =A +A inferredfrommodelfitswithEq.(4). The curvature issue is further discussed in Sec. VIF.
Q Q,NJ Q,J
6
(a) (a)
3
ef 6 ef
r r
ρ ρ 2
4
√ √
/ 2 / 1

ρ ρ 0
0
∆ ∆

φ φ
∆ ∆
η η
4 4
∆ 4 ∆ 4
2 2
φ 2 η φ 2 η
∆ 0 ∆ ∆ 0 ∆
0 -2 0 -2
-4 -4
FIG. 4: (Color online) Histograms from Fig. 3 rescaled by FIG. 5: (Color online) (a) Model function from Fig. 4 (up-
factor5. (a)ModelfunctionfromFig.3(leftpanel)butwith per panel) with the same quadrupole amplitude but with jet
quadrupole amplitude increased by factor 6 to RQ = 0.42. correlation amplitudes and peak widths reduced (see text).
(b) CMS pt-integral correlations for high-multiplicity cut [1]. (b) CMS correlations for high-multiplicity cut and pt [1,3]
∈
GeV/c [1].
C. High-multiplicity plus high-pt cuts
multiplicity cut alone—2R = 0.42 increased by fac-
Q,NJ
With high-p cuts we expect to eliminate some frac- tor 6 from minimum-bias 0.07. The comparisonto CMS
t
tion of the jet contribution, to eliminate any remaining datainthelowerpanelshowsgoodagreement,especially
evidence of the soft component and possibly to change near the base of the SS 2D peak.
thequadrupoleamplitude. Figure5(upperpanel)shows The nonjet quadrupole remains the same in Figs. 4
the model in Fig. 4 (upper panel) with the following and 5, but the jet structure in Fig. 5 is reduced in am-
changes: (i) The φ width of the SS 2D peak is fur- plitude byfactor2-3relativetothe quadrupolebythe p
t
ther reduced from 0.65 to 0.55 leading to a symmetric cuts. BecausetheverticalsensitivityinFig.5isdoubled,
SS peak, (ii) The SS peak amplitude is reduced by fac- because the nonjet quadrupole remains enhanced by at
tor 1/3 and AS ridge by factor 1/2 compared to the least factor 6, and because the jet structure (particulary
high-multiplicity cut alone (consequence of reduced p theASridge)isstronglyreducedinamplitudebyp cuts
t t
acceptance for jet-correlated pairs) and (iii) the nonjet the SS “ridge” structure becomes visually apparent as a
quadrupole amplitude remains the same as for the high- change in sign of the azimuth curvature (Sec. VIF).
7
TABLE I: p-p correlation systematics. Entries represent systematics inferred from RHIC data at 0.2 TeV and parameters
inferred by modeling CMS angular correlation histograms at 7 TeV from Ref. [1]. The left columns indicate collision energy
and cut conditions, including minimum-bias (MB) data. The AX represent parameters from STAR per-particle analysis with
anintensivecorrelationmeasure. TheRX representcorrespondingCMSdataincludingextensiveacceptancefactor2π∆η 30.
≈
The two entries in the2AQ and 2RQ columns correspond tononjet and jet-related (AS peak) quadrupoles respectively.
√s (TeV) Condition 2AQ AD/2 ASS 16AQ/AD 2RQ RD/2 RSS ση σφ
0.2 MB 0.001+.0023 0.023 0.058 0.57 0.03+0.069 0.69 1.74 0.64 0.9
7 MB 0.0023+0.0032 0.032 0.133 0.69 0.07+0.096 0.96 4.0 0.64 0.9
7 n cut 0.014+0.0095 0.095 0.39 0.99 0.42+0.29 2.9 12.0 0.58 0.65
ch
7 pt, nch cuts 0.014+0.0047 0.047 0.13 1.59 0.42+0.14 1.45 4.0 0.55 0.55
VI. DISCUSSION The CMS high-multiplicity cut produces a 7-fold in-
crease over the NSD angular density (dN /dη = 40 vs
ch
5.8). In a study of multiplicity dependence of 200 GeV
We observe that minimum-bias jet structure at the
p-p p spectra [7] the multiplicity variation included a
LHCisremarkablysimilartothatatRHIC,simplyscaled t
10-fold increase (dN /dη = 25 vs 2.5). The maximum
upbyalog(√s)energy-dependencefactor. Possiblenov- ch
particledensityat200GeVis thus60%ofthemaximum
elty arises with application of multiplicity and p cuts.
t
at 7 TeV—less than a factor two difference. RHIC and
Does the unanticipated “ridge” structure imply novel
LHC p-p multiplicity trends are directly comparable.
physics in LHC p-p collisions, or is it also an extrapola-
tionofphenomenaobservedpreviouslyatlowerenergies?
C. p-p multiplicity trends
A. Summary of inferred model parameters
From the CMS data we conclude that both A
Q
(quadrupole) and A (jets) increase with increasing p-p
TableIpresentsresultsfromextrapolationofRHICp-p D
multiplicity. Aresuchtrendsreasonablegivenourknowl-
data to 7 TeV and from modeling CMS data histograms
edge of p-p collisions at 0.2 TeV? We have two sources
forseveralcutconditions. Uncertaintiesaretypicaly10%
of information about p-p multiplicity trends at 0.2 TeV:
except for the 0.2 TeV p-p nonjet quadrupole estimate
1) direct observations of 0.2 TeVp-p systematics and 2)
which is an upper limit. The first row summarizes re-
arguments by analogy with Au-Au collision centrality.
sults for p-p collisions at 0.2 TeV (jet parameters) or
a. p-p multiplicity systematics at 0.2 TeV Se-
extrapolated from Au-Au centrality trends (quadrupole
lection of high-multiplicity p-p events is expected to
parameter). STAR(A )andCMS(R )parametersare
X X
biastowardmore-centralcollisions[27]. Two-component
related by R =2π∆ηA 30A , with ∆η =4.8.
X X X
≈ analysisofp spectrarevealsthatwithaten-foldincrease
In the second row the SS peak and quadrupole ampli- t
ineventmultiplicitythejetfrequencyin200GeVp-pcol-
tudes are scaled up with energy by factor R(√s) = 2.3
lisions increases by a factor 50. The increase saturates
inferred from jet and quadrupole energy systematics be-
when the jet probability nears 100% (Sec. IVE) [7]. Jet
low 0.2 TeV. AS peak amplitude A increases by factor
D correlationmeasurements at 200 GeV confirm that mul-
0.6 2.3 to match the structures observed in Fig. 2. In
× tiplicitycutsincreasethejetfrequencypercollision. The
the third row jet amplitudes with applied multiplicity
per-particle jet correlation amplitude increases by up to
cuts are further increasedby factor 3 and quadrupole by
a factor 5, but the mean fragment multiplicity per jet
factor 6 to match the structures observed in Fig. 4.
is not significantly altered (the multiplicity cut does not
In the fourth row the results of p cuts applied to
t bias minimum-bias parton fragmentation) [15, 16].
CMS data corresponding to Fig. 5 are presented. The
In Sec. VB we conclude from CMS data that applica-
quadrupoleamplitudedoesnotchangesignificantly. The
tionofahigh-multiplicitycutincreasesthejetcorrelation
SS 2D peak amplitude A is reduced by factor 1/3 and
SS amplitude by factor 3. At 7 TeV the minimum-bias jet
the AS dipole amplitude A is reduced by factor 1/2.
D frequency increases by factor R(7 TeV) = 2.3 to almost
5%. The maximum frequency increase is then 50/2.3
= 22. With a 7-fold increase in event multiplicity we
B. Particle densities in p-p collisions thereforeexpectthecorrelationamplitudetoincreaseby
factor 7(22/72) 3, which we observe in the CMS data.
≈
The R(√s) trend defined in Eq. (5) and inferred b. Au-Aucentralitysystematics TheCMSdata
from angular correlation energy systematics below 200 (SS curvature) suggest a factor-6 increase in the nonjet
GeV [12, 24] also describes NSD particle densities above quadrupole with high-multiplicity cut. Is that consis-
200 GeV. Given the NSD value dN /dη = 2.5 at 200 tent with the increase in jet correlations? The nonjet
ch
GeV,factorR(√s)predictsthevalue5.75at7TeV,con- azimuthquadrupoleinA-Acollisionsbelow200GeVde-
sistent with the recent CMS measurement 5.8 [8]. pendsonlyoncollisionenergyandinitialgeometryinthe
∼
8
form b/b0 (the geometry of intersecting spheres), not on tude for like-sign and unlike-sign charge pairs [6]. In
absolutesystemsize. Thetwotrendsarefactorized,asin minimum-bias p-p collisions at 7 TeV the quadrupole is
Eq.(6)[12]. Wearguebyanalogythattheincreaseinp-p not visually apparent but may be determined by model
centralitywhichleads to increasedjet productionshould fits to 2D histogram data. With the imposition of
produce a comparable or greater quadrupole increase. multiplicity and p cuts to CMS data the extrapolated
t
Thenonjetquadrupoleamplitudeisexpressedinterms quadrupole can become visually apparent as a SS ridge.
of A-A centrality parameters by 2AQ = ρ0(b)v22 The correspondence between quadrupole properties and
n ǫ2 . TheASridge(jets)scaleswithA-Acentralitya∝s theSSridgeinCMSdatastronglysuggeststhattheridge
bin opt
AD/2 ρ0(b)j2 ν/(1+x(ν 1)) ν. TheGlauberpa- is in fact a manifestation of the azimuth quadrupole.
ramete∝rs are rela∝ted by n −ν4 a≈nd n ν3. Thus, InterpretationoftheSSridgeat7TeVasamanifesta-
bin part
A /A should scale with ce∼ntrality as ν3. ∼In p-p colli- tion of the azimuth quadrupole does not imply that “el-
Q D
sions ν can be interpreted as a thickness measure or in- lipticflow”playsaroleinp-pcollisions. Instead,appear-
teractionlengthoftwooverlappingspheres. Aquadrople ance of the quadrupole in elementarycollisions is consis-
increasewithmultiplicity twice the jetincreaseis consis- tentwithevidenceagainstahydrodynamicinterpretation
tent with measured p-p and Au-Au centrality trends. in more-central Au-Au collisions [17, 21, 28]. Recently-
measuredsystematictrends[12–14]suggestthatthenon-
jet quadrupole is a novel QCD phenomenon [25].
D. Jet correlations response to pt cuts
F. Azimuth curvatures and “ridge” phenomena
Minimum-bias SS peak properties for 7 TeV jet an-
gular correlations are quantitatively similar to those at
0.2 TeV. In p-p correlations on y y at the lower en- Mathematically,the“ridge”observedinCMSp-pdata
t t
×
ergy [15, 16] SS jet correlations extend down to pt = 0.3 resultsfromacompetitionbetweentwocurvaturesonφ∆
GeV/c, with mode at 1 GeV/c . Nearly half the SS- within η∆ >2(whichexcludesmostoftheSS2Dpeak).
| |
correlatedpairsappear below the mode. Incontrast,AS A ridge appears when azimuth structure near φ∆ = 0
correlations are cut off near 0.7 GeV/c due to initial- becomessignificantlyconcavedownward(definedasneg-
state k effects. Thus, a smaller fraction of AS pairs ative net curvature). In minimum-bias p-p collisions we
t
appears below 1 GeV/c. In Sec. VC we observed that observe only positive curvatures in that region. A small
with p [1,3] GeV/c cuts imposed the 7 TeV SS peak relative change in correlation amplitudes may result in
t
∈
amplitude is reduced by factor 1/3 and the AS ridge is thequalitativeappearanceordisappearanceofa“ridge.”
reduced by factor 1/2,consistent with 0.2 TeV jet struc- In the context of the present analysis the dominant
ture on yt yt. With the high-multiplicity cut the SS structures within η∆ > 2 are the AS dipole sinu-
× | |
peak azimuth width decreases from 0.9 to 0.65. With soid cos(φ∆ π) and the azimuth quadrupole sinusoid
−
the additional high-pt cut the azimuth width is further cos(2φ∆). The curvature of cos(2φ∆) is four times the
reduced to 0.55 leading to a symmetric SS 2D jet peak. curvature of cos(φ∆ π) (and with opposite sign) at
−
φ∆ = 0. The net curvature is then determined by the
coefficients of the two sinusoids—A /2 for the dipole
D
E. The azimuth quadrupole in p-p collisions and 2AQ for the quadrupole. Zero net curvature cor-
responds to 4 2A = A /2 or 16A /A = 1. That
Q D Q D
×
The nonjet quadrupole in minimum-bias p-p collisions ratiois includedin Table I. Aridge(negative curvature)
at 200 GeV is ill-defined because of a fitting ambiguity. is observed if the ratio is significantly greater than one.
The total η∆-independent quadrupole in p-p 2D angular As revealed in Table I the combination of pt and mul-
correlations within the STAR TPC includes a possible tiplicity cuts increases the nonjet quadrupole amplitude
nonjet quadrupole and the second Fourier component of byatleastafactor4relativetotheASjetpeak,changing
the AS jet peak, the latter depending on the actual AS the curvature sign and producing an apparent SS ridge.
peak width [10]. If the AS peak is modeled as a dipole In effect, the azimuth curvature functions as a compara-
(width π/2) then the fitted quadrupole component tor, switching states from valley to ridge as one ampli-
serves a∼s an upper limit on the nonjet quadrupole. tude changes value relative to another. A quantitative
The nonjet quadrupole in p-p collisions is better de- change is thereby transformed into a qualitative change,
termined by extrapolation from peripheral Au-Au colli- interpreted as the emergence of a novel phenomenon.
sions. The nonjet quadrupole in Au-Au collisions at and
below 200GeV has the simple parametrizationshownin
Eq. (6) [12], quite different from minijets which undergo G. Is the “ridge” at RHIC relevant to p-p at LHC?
a sharptransitiononcentrality[9]. The parametrization
includes a log(√s) factor which predicts the quadrupole Two aspects of the so-called “ridge” phenomenon at
for minimum-bias p-p collisions at 7 TeV. RHIC can be distinguished: (i) η elongation of a mono-
Like the CMSridge,the 200GeVazimuthquadrupole lithic SS jet peak well described by a single 2D Gaus-
is insensitive to charge combination, with equal ampli- sian [5, 9] and (ii) claimed development of a separate
9
ridge-likestructurebeneathasymmetric2Djetpeak[2]. TeV minimum-bias p-p collisions. Midrapidity correla-
Item (i), well established for untriggered(no p cuts) jet tionamplitudesandmultiplicitydensitiesincreasebyfac-
t
correlations and for some combinations of p cuts, has tor R(√s = 7 TeV) = 2.3 over those measured at 0.2
t
been referred to as a “soft ridge,” although there is no TeV. The minimum-bias correlation structure is other-
separate ridge per se. Item (ii) is inferred from other wise quite similar within present measurement accuracy.
combinations of p cuts (“triggered” analysis). Response to multiplicity and p cuts is also consistent
t t
Elongation of the minimum-bias SS peak (i) under- withtrendsat0.2TeV.Thechangeinjet-likecorrelation
goes a sharp transition on centrality in RHIC A-A colli- amplitudes and same-side 2D peak widths is consistent
sions (both Au-Au and Cu-Cu collisions) [9]. For more- with STAR correlation analysis at 0.2 TeV.
peripheralandp-pcollisionstheSSpeakisstronglyelon-
The single novel manifestation at the LHC is the ap-
gatedintheφdirection(3:2)[16]. Formore-centralcolli-
pearance of a same-side “ridge” for certain multiplicity
sions the SS peak transitions to strong elongation in the
and p cuts. It is conjectured that the ridge structure in
t
η direction (3:1) [5, 9]. The transition occurs within a
7 TeV p-p collisions may be similar to that observed in
small centrality interval. The mechanism is unknown.
more-centralAu-Au collisionsat0.2TeV, interpretedby
VariationofSS jet peak structure with p cuts is com-
t some to indicate formation of a dense QCD medium.
plex, depending on the details of fragmentation for dif-
However,extrapolationofazimuthquadrupolesystem-
ferentcharge-signcombinations,fragmentp sandhadron
t atics measured in Au-Au collisions at 0.2 TeV to 7 TeV
species. Such details are currently poorly explored. As-
p-p collisions predicts a significant quadrupole ampli-
signmentofcertainaspectsofpeakstructuretoadistinct
tude in minimum-bias p-p collisions. Although the ex-
ridge phenomenon for some p cuts is questionable.
t
act consequences for the quadrupole of multiplicity and
For the CMS data we observe that the SS 2D peak η
p cuts applied in the CMS analysis cannot be predicted
widthat7TeVhasthesamevalueasthatat0.2TeVand t
quantitatively,a six-foldincreaseoverthe minimum-bias
isreducedslightlywithappliedcuts. Thus,ηbroadening,
quadrupole prediction fully accounts for the same-side
item (i) observed at RHIC, is not present. The azimuth
ridge structure in 7 GeV p-p collisions. A modest in-
width varies with cuts just as in 0.2 TeV p-p collisions.
crease in the quadrupole amplitude (enhanced by multi-
The appearanceofaSS “ridge”in CMSdatafor some
plicity cuts) relative to the away-side ridge (reduced by
cutcombinationsisassociatedwithitem(ii)above,given
p cuts) can reverse the sign of the curvature near the
the apparentsimilarity. However,the absenceofitem(i) t
azimuth origin, resulting in an apparent ridge structure.
andconsistencywithknownazimuthquadrupolesystem-
atics makes interpretation(ii) unlikely. There is no indi- We concludethatthe azimuthquadrupoleapparentin
cation from present data that the CMS ridge is directly more-centralAu-AucollisionsatRHICisdirectlyvisual-
associatedwiththeSS2Dpeak. Amoredefinitiveresolu- izedinp-pangularcorrelationsattheLHCasacurvature
tionmaycomewithsystematicstudyofPb-Pbcentrality reversal resulting from larger collision energy combined
systematics in upcoming LHC heavy ion runs. with kinematic cuts. LHC p-p data offer the possibility
tostudythesystematicsoftheazimuthquadrupoleinel-
ementarycollisions. Modelfitscanbe appliedasincited
VII. SUMMARY references to increase sensitivity to nonjet quadrupole
structure. By such means the azimuth quadrupole may
be better understood as a QCD phenomenon.
Direct A-B comparisons between CMS 2D angu-
lar correlations from 7 TeV p-p collisions and STAR We greatly appreciate extensive discussions on two-
parametrizationsof 0.2 TeV collisions with identical his- particlecorrelationsinnuclearcollisionsoveranumberof
togram formats provide a quantitative model descrip- years with Lanny Ray, Jeff Porter and Duncan Prindle.
tion of the CMS data. The model fits reveal that there This workwassupportedinpartby the Office ofScience
is remarkably little difference between 0.2 TeV and 7 of the U.S. DOE under grant DE-FG03-97ER41020.
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