Inclusive production of small radius jets in heavy-ion collisions

0.48 MB·

by  Zhong-Bo Kang

LA-UR-17-20230
Inclusive production of small radius jets in heavy-ion collisions
Zhong-Bo Kang,1,2,3 Felix Ringer,3 and Ivan Vitev3
1Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA
2Mani L. Bhaumik Institute for Theoretical Physics,
University of California, Los Angeles, CA 90095, USA
3Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
We develop a new formalism to describe the inclusive production of small radius jets in heavy-
ion collisions, which is consistent with jet calculations in the simpler proton-proton system. Only
at next-to-leading order (NLO) and beyond, the jet radius parameter R and the jet algorithm
dependence of the jet cross section can be studied and a meaningful comparison to experimental
measurements is possible. We are able to consistently achieve NLO accuracy by making use of the
recently developed semi-inclusive jet functions within Soft Collinear Effective Theory (SCET). In
addition, single logarithms of the jet size parameter αnlnnR are resummed to next-to-leading log-
s
7 arithmic (NLL ) accuracy. The medium modified semi-inclusive jet functions are obtained within
R
1 theframeworkofSCETwithGlaubergluonsthatdescribetheinteractionofjetswiththemedium.
0 We present numerical results for the suppression of inclusive jet cross sections in heavy ion colli-
2 sions at the LHC and the formalism developed here can be extended directly to corresponding jet
n substructure observables.
a
J
I. INTRODUCTION surements. In addition, parton distribution functions
0
(PDFs) are fitted at NLO, typically for larger values of
2
In heavy-ion collisions at RHIC and LHC, a quark- R ∼ 0.7. For the analyses of heavy-ion collisions, how-
] ever, the jet size parameter is typically chosen to be rel-
h gluonplasma(QGP)canbecreatedandstudiedbyusing
atively small, R ∼ 0.2−0.4, in order to minimize fluc-
p bothhardandsoftprobes[1]. Jetsareproducedinhard-
tuations in the heavy-ion background. The perturbative
- scattering events and constitute one of most frequently
p series exhibits a single logarithmic structure αnlnnR to
studied examples of hard probes in heavy-ion collisions. s
e all orders in the QCD strong coupling constant, which
h Jets traverse the hot and dense QCD medium and are
have to be resummed to render the convergence of the
[ identified as energetic and collimated sprays of particles
perturbative calculations.
in the detectors. Examination of their properties can,
1
therefore,provideinformationabouttheQGP.TheLHC Such a lnR-resummation has recently been achieved
v
experimentalcollaborationsCMS[2],ATLAS[3]andAL- for proton-proton collisions to next-to-leading logarith-
9
3 ICE [4] have provided precise data sets for the inclusive mic (NLLR) accuracy [9] within the framework of Soft
8 production of jets in both proton-proton and heavy-ion Collinear Effective Theory (SCET) [10–13]. See [14–
5 collisions. In heavy-ion collisions jets are modified, or 17] for related work along these lines. It was also
0 quenched, duetotheinteractionwiththeQCDmedium. demonstrated that the cross section for inclusive jets
1. Most commonly, the quenching of jet production yields can be factorized into convolution products of PDFs,
0 is studied using the nuclear modification factor R , hard functions and so-called semi-inclusive jet functions
AA
7 whichisgivenbytheratiooftherespectivecrosssection Ji(z,ωJR,µ) (siJFs) [9]. The siJFs describe the forma-
1 in heavy-ion collisions normalized by the corresponding tion of a jet with energy ωJ and jet parameter R orig-
:
v proton-proton baseline. In order to reliably extract in- inating from a parent parton i at scale µ. They sat-
i formationabouttheQGPfromtheavailabledatasets,it isfythesametimelikeDGLAPequationsthatgovernthe
X
is important that the experimentally achieved precision scale evolution of fragmentation functions. By solving
ar is matched with corresponding theoretical calculations. the DGLAP equations, the resummation of logarithms
This is precisely what we are going to address in this lnR can be achieved. For values of R in the range of
work. 0.2−0.4, fixed NLO calculations fail to describe the ex-
perimentaldata[2]. WhentheresummationoflnRterms
The identification of jets relies on a jet algorithm that
is included, good agreement can be achieved, as we will
specifies when particles are clustered together into the
show below. The need of lnR resummation to describe
same jet. Typical algorithms used by the experimental
the proton-proton baseline makes this the ideal starting
analyses involve for example the anti-k and the cone
T
point to also study inclusive jet production in heavy-ion
algorithm [5, 6]. In addition, jets are defined by the jet
collisions. Inthiswork,weextendourearliercalculations
parameter R which represents the size of the identified
for proton-proton collisions to heavy-ion collisions. See
jets, see e.g. [7, 8] for more details. The first non-trivial
forexample[18–25]forearlierworkonthedescriptionof
order in the perturbative expansion of the cross section
jets in heavy-ion collisions.
where these specifications play a role is next-to-leading
order (NLO) in QCD. Therefore, full NLO is the min- We address the medium modification within the effec-
imally required perturbative order allowing meaningful tivefieldtheory(EFT)frameworkofSCETwithGlauber
comparisons between theory and the experimental mea- gluons which is generally denoted by SCET [26, 27].
G
2
(A) (B) (C) (D) (E)
FIG.1: Feynmandiagramsforquarkinitiatedjets. (A)istheleading-ordercontributionand(B)-(E)aretherelevantdiagrams
at NLO: (B) both partons are inside the jet, (C) virtual correction, (D) and (E) only one parton is inside the jet. The dashed
lines correspond to a collinear quarks and the curly lines to collinear gluons within SCET.
The interaction of collinear quarks and gluons with the collinear splitting functions obtained within SCET . In
G
hot and dense QCD medium can be described via the Section III, we present numerical results for the nuclear
exchange of Glauber gluons. Within SCET , the rel- modification factor R and we compare to recent data
G AA
evant interaction terms are included at the level of the from the LHC. Finally, we draw our conclusions in Sec-
Lagrangian. By making use of the collinear sector of the tion IV.
corresponding EFT, the full collinear in-medium split-
tingfunctionshavebeenderivedinthepastyearstofirst
order in opacity [28–31]. When finite quark masses are II. THEORETICAL FRAMEWORK
neglected,thein-mediumsplittingfunctionsaregivenby
the vacuum splitting functions times a modification fac- We start by summarizing the main results of [9] for
tor that depends on the properties of the medium. The inclusive jet production in proton-proton collisions. We
opacity expansion for the medium interactions is analo- then outline how this framework can be extended to the
gous to the traditional Gyulassy-Levai-Vitev (GLV) ap- heavy-ion case.
proach to parton energy loss in the QCD medium [32].
At first order in opacity, an average number of uncorre-
latedinteractionswiththemediumistakenintoaccount. A. Proton-proton collisions
Higherordersintheopacityexpansioncorrespondtocor-
relations between the interactions which are yet to be Thefactorizedformofthedoubledifferentialcrosssec-
calculated and are neglected in this work. In the tradi- tionforinclusivejetswithagiventransversemomentum
tional GLV approach, all radiated gluons in the splitting p and rapidity η is given by
T
processes are approximated to be soft. Within SCET ,
G
one can go beyond this approximation and obtain full dσpp→jetX (cid:88)
= f ⊗f ⊗Hc ⊗J . (1)
controlofthecollineardynamicsofsplittingprocessesin dp dη a b ab c
T
a,b,c
themedium. Forexample, thein-mediumsplittingfunc-
tions have already been successfully applied to describe Here, we suppressed the arguments of the various func-
themodificationoflighthadrons[33,34]aswellasheavy tions for better readability. See [9] for more details. The
flavor mesons [31] in heavy-ion collisions. symbols⊗denoteconvolutionproductsandwearesum-
In this work, we derive an analogous treatment of the mingoverallrelevantpartonicchannels. Thef denote
a,b
in-mediumeffectsforinclusivejetsbydefiningin-medium the PDFs, Hc are hard-functions and the J are the si-
ab c
siJFs. Inthevacuum,thesiJFscanbewrittenintermsof JFs. The hard-functions are evaluated up to NLO and
collinear vacuum splitting functions. In the medium, we were shown [9] to be the same as the hard-functions for
needtoincludeadditionalcontributionstothesiJFsthat inclusive hadron production pp→hX, see [35, 36]. Note
can be expressed in terms of collinear in-medium split- that Eq. (1) is a factorization of purely hard-collinear
ting functions derived from SCETG. In general, both dynamics, i.e. no soft function is needed. The siJFs
radiative and collisional energy loss can play a role in are perturbatively calculable functions that describe the
modifying jet production in the medium. In this work, formation of the observed jet originating from a parent
weconcentrateonthehigh-pT jetsandthusonlyconsider parton.
radiativeenergyloss,andwillleavecollisionalenergyloss If both Hc and J are expanded to NLO, we get back
ab c
forfuturepublications. Inaddition,weincludeColdNu- to the standard NLO results for inclusive jets as derived
clear Matter (CNM) effects. in [8, 37, 38]. Within the framework developed in [9], we
The remainder of this paper is organized as follows. can go beyond the fixed order approach. Using the siJFs
In Section II, we recall the basic framework for inclu- in Eq. (1) represents an additional final state factoriza-
sive jet production in proton-proton collisions for small- tion. As pointed out in [14, 15], fixed order jet cross sec-
R jets. We outline a consistent extension of the jet tions can have a vanishing, unphysical scale dependence.
cross section to heavy-ion collisions using the in-medium This problem is overcome by using the factorized form
3
of the cross section in Eq. (1), where the interpretation equations from this characteristic scale to the hard scale
of the QCD scale uncertainty as a measure of missing µ∼p . InsectionIII,wepresentsomenumericalresults
T
higherordercorrectionsisrestored. See[9]fornumerical at NLO+NLL accuracy showing the impact of the lnR
R
results concerning the scale dependence. resummation for narrow jets.
In [9], the siJFs J (z,ω R,µ) were calculated to NLO Notethatin(2),wechosetoexpresstheresultinterms
i J
from their operator definition within SCET. We have of the initiating parton energy ω = ω /z. This conven-
J
z = ω /ω, where ω (ω) denotes the jet (initiating par- tion differs from the one chosen in [9]. The different
J J
ton) energy. Here, we summarize only the results for convention results in an additional term ∼−2lnz in the
quark initiated jets to keep the discussion short. Up to second line in of Eq. (2). In principle, ω is the rele-
J
NLO,onehastoconsiderthediagramspresentedinFig.1 vant external quantity as it is related to the observed jet
for jets that are initiated by an outgoing quark. (A) cor- transverse momentum ω = 2p coshη. However, the
J T
respondstotheleading-order(LO)diagram. ToLO,one underlying structure of the siJFs becomes more appar-
findsJ(0)(z,ω R,µ)=δ(1−z). AtNLO,onehastocon- ent when writing the result in terms of the energy ω of
q J
sider the contributions (B)-(E). Here (B) corresponds to the initiating parton. This will be particularly relevant
a splitting process, where both final state partons are for deriving the in-medium siJFs below. Also note that
in the jet. (C) is a virtual correction and (D), (E) are the sum rule for the siJFs associated with momentum
the contributions, where one of the final state partons conservation of the initiating parton i,
exits the jet. All contributions (B)-(E) can be written
(cid:90) 1
in terms of integrals over the quark-to-quark or quark-
dzzJ (z,ωR,µ)=1, (4)
i
to-gluon LO Altarelli-Parisi splitting functions [39]. We 0
makeuseofthisfactbelowwhenderivingthein-medium
(anti-k algorithm)onlyholdswhenexpressingthesiJFs
siJFs. For completeness, we present here the result for T
in terms of the parton energy ω instead of the jet energy
the quark siJF in dimensional regularization up to NLO
ω , see also [17].
for the anti-k algorithm J
T
J (z,ωR,µ)=δ(1−z)
q
B. Heavy-ion collisions
α (cid:18)1 (cid:18) µ2 (cid:19) (cid:19)
+ s +ln −2lnz [P (z)+P (z)]
2π (cid:15) ω2tan2(R/2) qq gq Wenowturntothecrosssectionforinclusivejetspro-
duced in heavy-ion collisions. First, we note that the
(cid:40) (cid:34) (cid:18) (cid:19) (cid:35)
− αs C 2(1+z2) ln(1−z) +(1−z) QGP is only present in the final state after the hard-
2π F 1−z scatteringevent. Therefore,itissufficienttomodifyonly
+
thesiJFswhichcapturetheformationoftheobservedjet.
(cid:18)13 2π2(cid:19) (cid:41) Second, all Feynman diagrams shown in Fig. 1 that are
−δ(1−z)CF − +2Pgq(z)ln(1−z)+CFz , relevant for the vacuum also appear in the medium cal-
2 3
culation. In other words, in the heavy ion collisions, the
(2)
siJFs obtained in proton-proton collisions are modified
as
whereP (z)andP (z)aretheusualLOAltarelli-Parisi
qq gq
splitting functions. The remaining 1/(cid:15) pole is a UV pole J →Jvac+Jmed, (5)
i i i
which is removed by renormalization. The associated
RG equations turn out to be the same DGLAP evolu- where Jvac are the vacuum contributions, and Jmed are
i i
tion equations that are also satisfied by fragmentation the in-medium siJFs that take into account medium in-
functions, which describe the transition of a final state teractions.
parton into a specific observed hadron. In other words, The Feynman diagrams that contribute to Jmed can
i
the DGLAP equations for the siJFs read be obtained from the corresponding vacuum ones shown
in Fig. 1. As an example, see Fig. 2 for the relevant
µ d J = αs (cid:88)P ⊗J , (3) so-called single-Born (SB) diagrams Amed for the case
dµ i 2π ji j SB
j that both partons remain inside the jet (B). In addition,
we need to calculate double-Born (DB) diagrams Amed
DB
where we omitted again the arguments of the involved where two interactions with the medium are considered.
functions. These evolution equations can be solved The relevant diagrams are not shown here explicitly. In
in Mellin moment space using the methods developed order to obtain a physical in-medium cross section, we
in [40, 41]. By solving the DGLAP equations, we ob- schematically need to calculate the combination
tain the evolved siJFs as they appear in the factoriza-
tion theorem in Eq. (1). The resummation of terms |Amed|2+2Re(cid:8)Amed×Avac(cid:9) , (6)
SB DB
∼ lnR is achieved by choosing µ ∼ ω tan(R/2) ≈ p R
J T
in Eq. (2) which eliminates terms ∼lnR in the fixed or- where Avac denotes the vacuum diagrams as shown in
derresult. WethenevolvethesiJFsthroughtheDGLAP Fig. 1. See [27, 31] for a more detailed discussion.
4
As mentioned above, in the calculation of the vacuum
siJFs Jvac, all contributions (B)-(E) in Fig. 1 can be ex-
i
pressed in terms of integrals over the LO real emission
splitting functions P (z,q ). For example, for a quark-
ji ⊥
to-quark splitting, we have
α 1+x2 1
s
P (x,q )= C , (7)
qq ⊥ π F 1−x q
⊥
where we include explicitly a 1/q (its transverse mo-
⊥
FIG. 2: Single-Born diagrams contributing to the medium
mentumdependence)relativetotheAltarelliParisisplit-
siJF Jmed where both partons are inside the jet, cf. Fig. 1
ting functions used above. See Ref. [9] for more de- q
(B).ThedottedlinesrepresenttheinteractionwiththeQCD
tails. In order to obtain the corresponding in-medium
medium via Glauber gluon exchange.
siJFs Jmed, the vacuum splitting functions P (z,q )
i ji ⊥
are replaced with the collinear in-medium splitting func-
tions Pmed(z,q ) derived from SCET . With all quark
ji ⊥ G associated with diagram (B), where both partons are in-
massessettozero,onefindsthattheyhavethefollowing
side the jet, as the following integral over the quark-to-
structure
quark splitting function [9, 42]
Pjmied(z,q⊥)=Pji(z,q⊥)fji(z,q⊥;β) (8) (cid:90) 1 (cid:90) x(1−x)ωtan(R/2)
(B)=δ(1−z) dx dq P (x,q ).
⊥ qq ⊥
where the characteristics (properties) of the medium are 0 0
collectivelydenotedbyβ [33]. Thefunctionsf (z,q ;β) (9)
ji ⊥
describe the modification of the vacuum splitting func- Note that here we can use the energy of the initiating
tion due to the presence of the QCD medium. parton or the jet as the delta function enforces z = 1 or
TheSCETGsplittingfunctionscanbepartlyevaluated equivalentlyω =ωJ. Wewouldliketopointoutthatthe
analytically. TheexplicitformoftherelevantSCET in- result for (B) by itself is divergent both in the vacuum
G
medium splitting functions for massless partons can be and in the medium. This can be most easily seen for the
found for example in [28]. Eventually, they have to be vacuumcaseusingdimensionalregularization,whereone
integratedalsooverthesizeofthemediumandthetrans- finds both double poles 1/(cid:15)2 and single poles 1/(cid:15) [9].
verse momentum transfer that is acquired due to the In order to evaluate the in-medium result numeri-
medium interactions. These integrations can only be cally, we first have to appropriately combine the result
evaluated numerically, as they depend on the specific in Eq. (9) with the other contributions (C)-(E). We con-
properties of the medium. Therefore, we have to find tinue with the virtual correction shown in Fig. 1 (C).
awaytoevaluatethein-mediumsiJFsJmed numerically, Following [43], this contribution can be written as
i
suchthatalldivergencesthatappearonlyattheinterme-
(cid:90) 1 (cid:90) µ
diate steps of the calculation cancel. For the calculation
(C)=−δ(1−z) dx dq P (x,q ), (10)
⊥ qq ⊥
of the in-medium siJFs, we choose to work in a cut-off
0 0
scheme rather than dimensional regularization in order
to facilitate the numerical evaluation. A single regula- where we introduced the UV cut-off µ as the upper inte-
tor µ cutting off UV divergences is sufficient as there is gration boundary of the q⊥ integral. Both contributions
onlyoneremainingUVdivergenceoncethecontributions (B) and (C) are ∼δ(1−z) and we can directly combine
from all diagrams are taken into account. This can also them as
be seen from the vacuum result of the quark siJF in di-
(cid:90) 1 (cid:90) µ
mensional regularization in Eq. (2), where there is only (B)+(C)=δ(1−z) dx dq P (x,q ).
⊥ qq ⊥
one 1/(cid:15) UV pole left. Note that in dimensional regu- 0 x(1−x)ωtan(R/2)
larization, the virtual correction as shown in Fig. 1 (C) (11)
leads to a scaleless integral. Effectively, it only changes Thecontributionfromthediagram(D),wherethegluon
IR poles into UV poles. However, in order to work out exits the jet can be written as
the in-medium result numerically, we have to explicitly
(cid:90) µ
take this contribution into account.
(D)= dq P (z,q ), (12)
⊥ qq ⊥
Wenowpresenttheexpressionsfortheindividualcon-
z(1−z)ωtan(R/2)
tributions (B)-(E) to the quark siJF written in terms
of integrals over splitting functions. Here, we generally where symmetry of the lower integration boundary with
write the splitting functions as P (z,q ). For Jvac, we respect to z ↔ 1−z is obtained by using the energy ω
ji ⊥ i
refertothevacuumones,seee.g. Eq.(7),andforJmedto insteadofω . Notethatthecontribution(D)inEq.(12)
i J
the in-medium SCET splitting functions as in Eq. (8). by itself is divergent for z →1 as P (z,q )∼1/(1−z).
G qq ⊥
The two cases eventually will be summed over as indi- Similarly,theresultfor(B)+(C)inEq.(11)isdivergent
cated in Eq. (5). Following [9], we can write the result by itself. However, it can be seen immediately that the
5
1.5
R=0.2 R=0.3 R=0.4
O
L 1
N
σ
d
/
es 0.5
r
dσ anti-kT,√s=2.76TeV
η <2
| |
0
100 200 300 100 200 300 100 200 300
p
T
FIG. 3: Ratio of the resummed NLO+NLL inclusive jet cross section and the fixed NLO result in proton-proton collisions.
R √
ThekinematicsarechosenasintheCMSanalysisof[2]: s=2.76TeV,|η|<2andtheobservedjetsarereconstructedusing
the anti-k algorithm. The jet size parameter is chosen as R=0.2, 0.3, 0.4 in the three panels from left to right.
T
tworesultshaveaverysimilarstructureandtheycanbe the medium modification of hadron and jet production
combined together by introducing a plus distribution: yieldsinheavy-ioncollisionsessentiallyonthesamefoot-
ing. The result for the gluon siJF can be obtained in an
(cid:34) (cid:35)
(cid:90) µ
analogous way. However, there are some subtleties when
(B)+(C)+(D)= dq P (z,q ) ,
⊥ qq ⊥ introducing the plus prescription. The same issues were
z(1−z)ωtan(R/2)
+ discussed in details in [31] in the context of heavy flavor
where the plus distribution is defined as usual via production in heavy-ion collisions.
Finally,followingtheadditivestructureofthevacuum
(cid:90) 1 (cid:90) 1
and in-medium siJFs described in Eq. (5), we obtain the
dzf(z)[g(z)] ≡ dz(f(z)−f(1))g(z). (13)
+
following structure of the inclusive jet production cross
0 0
section in heavy-ion (PbPb) collisions
Apparently such a combination written in this form is
finite for z →1. dσjet =dσjet,vac+dσjet,med. (16)
Finally, for the case that the gluon makes the jet and PbPb pp PbPb
the quark is outside of the jet, we have
Here the first term is given in Eq. (1). The second term
(cid:90) µ is given by
(E)= dq P (z,q ). (14)
⊥ gq ⊥
z(1−z)ωtan(R/2) dσjet,med = (cid:88) σ(0)⊗Jmed, (17)
PbPb i i
The splitting function Pgq(x,q⊥) describes the quark-to- i=q,q¯,g
gluon splitting process and can be obtained from Eq. (7)
(0)
bysubstitutingx→1−x. Diagram(E)isfinitebyitself where σ is the LO production cross section for quarks
i
for z → 1. When summing over all diagrams at NLO and gluons since Jmed is formally already NLO. Ideally,
i
(B)-(E), we can write the result as one would like to evolve the in-medium siJFs also using
the DGLAP evolution equations in (3). However, in this
(cid:34) (cid:35)
(cid:90) µ workwelimitourselvestoperformingthelnRresumma-
Jmed,(1)(z,ωR,µ)= dq P (z,q )
q ⊥ qq ⊥ tiononlyforthefirsttermin(16)andweaddthesecond
z(1−z)ωtan(R/2)
+ term consistently at NLO.
(cid:90) µ
+ dq P (z,q ). (15)
⊥ gq ⊥
z(1−z)ωtan(R/2)
III. NUMERICAL RESULTS
The result for Jmed written in this form is finite for z →
q
1 and we only need the upper UV cut-off µ, which is In this section, we present numerical results for in-
suitable for numerical implementations and integrations. clusive jet production at NLO+NLL accuracy in both
R
The structure of the result in Eq. (15) is completely proton-proton and heavy-ion collisions using the frame-
analogous to the sum of real and virtual NLO correc- work outlined in section II. For all numerical results
tions to the partonic calculation of fragmentation func- presented in this section, we use the CT14 NLO set of
tions (FFs), see [31, 43]. Note that for jet production PDFs [44].
the sum of diagrams (B) + (C) in Eq. (11) is analogous We start by considering inclusive small-R jet produc-
to the virtual NLO correction for FFs. On the other tion in proton-proton collisions at the LHC. In [9], we
hand, the results from diagrams (D) and (E) involving presented a more detailed study of the effects of lnR
radiation outside of the jet corresponds to the real cor- resummation and we also considered the scale depen-
rection for FFs. This close analogy allows us to treat dence of the cross section. Only very recently, the CMS
6
collaboration presented precise measurements of small- impact of possible CNM effects and by comparing our
R jets [2] where the jet radius parameter was chosen as results to the ATLAS data of [3] in Fig. 4. The ATLAS
R = 0.2, 0.3, 0.4. The jets are reconstructed using the data for the nuclear modification factor R was taken
√ √ AA
anti-k algorithm at s=2.76 TeV and |η|<2. It was at s =2.76 TeV and 0−10% centrality for |η|<2.
T NN
shownin[2]thatstandardNLOcalculationsarenotable The jets were reconstructed using the anti-k clustering
T
todescribethedata. Thediscrepancybecomeslargerfor algorithm with a jet size parameter of R = 0.4. In this
small R and low p . One has to keep in mind that mod- figure,thestatisticalandsystematicerrorsoftheATLAS
T
ern sets of PDFs are also fitted to inclusive jet spectra data were added in quadrature. We leave the coupling
whichconstrain, inparticular, thegluonPDFatlargex. strength g between the jet and the medium as a param-
However, the jet size parameter R is typically relatively eter that can eventually be determined by comparing to
large (∼ 0.7) for the data sets included in these fits. In data. We choose g = 2.1 as a central value and we ob-
Fig. 3, we show results when the resummation of lnR tain a band as shown in Fig. 4 by varying that value of
terms is included. We choose the same kinematics as in g by ±0.1. Note that the coupling g, and the associ-
the CMS analysis and plot the ratio of the NLO+NLL ated α = g2/(4π), correspond to the vertices shown in
R s
resultsnormalizedbytheNLOresultforthethreediffer- Fig.2betweentheGlaubergluons(dottedlines)andthe
ent values of R. By comparing with the results shown collinear partons. The green band in Fig. 4 shows our
in [2], we find that the discrepancy between the the- SCET results at NLO+NLL accuracy without CNM
G R
ory calculation at NLO and the data for small-R jets effects. We emphasize again that resummed accuracy is
iswelldescribedwhenlnRresummationisincluded. We achieved for the vacuum contribution. The medium con-
conclude that lnR resummation is necessary to describe tribution is consistently included at NLO. The hatched
small-RjetdataattheLHC.Itis,therefore,desirableto red band shows the same results but with CNM effects.
work with this proton-proton baseline calculation when We obtain a good description of the ATLAS data once
considering the modification of inclusive jet spectra in CNM effects are included. The initial state energy loss
heavy-ion collisions. considered here corresponds to a momentum exchange
scale of µ =0.35 GeV. See [45] for more details. As
CNM
1.4 η <2.0,g=2.1 0.1 w/CNM mentioned above, we do not include collisional energy
|√|sNN=2.76TeV± w/AoTCLNAMS loss in this work. Eventually, it is important to clearly
1.2 centrality0 10% disentangle the numerical size of the different contribu-
−
R=0.4 tions of radiative and collisional energy loss mechanisms
1
as well as CNM effects. One possibility is to consider
A0.8 jet substructure observables where CNM effects are ex-
A
R pected to only play a marginal role [23]. More detailed
0.6
studies along these lines will be left for future work.
0.4 Finally, wecompareourcalculationstotheCMSmea-
surement of the nuclear modification factor R [2] in
AA
0.2
Fig. 5. Similar to the ATLAS measurement, the data
SCETG,NLO+NLLR √
0 was taken at sNN = 2.76 TeV for |η| < 2 using the
50 100 150 200 250 300 350 400 anti-k algorithm. We show the results for both central
T
pT collisions (0−10% centrality) on the left side as well as
formid-peripheralcollisions(30−50%centrality)onthe
FIG. 4: The nuclear modification factor R for heavy-ion
√ AA
collisionsattheLHCfor s =2.76TeV,|η|<2,R=0.4, rightsideofFig.5. Theresultsfordifferentvaluesofthe
NN
anti-k jets and 0−10% centrality. The SCET results at jet radius parameter are all shown in one plot: R = 0.2
T G
NLO+NLL are shown without CNM effects (green band) (green), R = 0.3 (blue) and R = 0.4 (red). Again, sys-
R
and with CNM effects (hatched red band). The coupling tematic and statistical errors are added in quadrature.
strength between the jet and the QCD medium is chosen as We show our SCET based results at NLO+NLL ac-
G R
g=2.1±0.1. WecomparetotheATLASdataof[3]. Statis- curacy for the different values of R using the same color
tical and systematic errors are added in quadrature.
coding. We only show our results with CNM effects and
we choose the coupling of the jet and the medium as
Having established a framework that can describe the g =2.1±0.1 as in Fig. 4. Again, we find that our calcu-
proton-protonbaseline,wenowturntonumericalresults lations describe the data well for both centrality regions
for inclusive jet production in heavy-ion collisions. We within the experimental error bars.
consider the nuclear modification factor R which is
AA
defined as
dσPbPb→jetX IV. CONCLUSIONS
R = , (18)
AA (cid:104)N (cid:105)dσpp→jetX
coll
In summary, we developed a new framework to de-
where (cid:104)N (cid:105) is the average number of binary nucleon- scribe the inclusive production of small-R jets in heavy-
coll
nucleon collisions. We start by assessing the numerical ioncollisions. Forsmall-Rjets,theallorderresummation
7
1.4 |η|<2.0,g=2.1±0.1 SCETGRR==00..43 1.4 |η|<2.0,g=2.1±0.1 SCETGRR==00..43
√sNN=2.76TeV R=0.2 √sNN=2.76TeV R=0.2
1.2 centrality0−10% CMSR=0.4 1.2 centrality30−50% CMSR=0.4
R=0.3 R=0.3
R=0.2 R=0.2
1 1
A0.8 A0.8
A A
R R
0.6 0.6
0.4 0.4
0.2 0.2
NLO+NLLR,w/CNM NLO+NLLR,w/CNM
0 0
100 150 200 250 300 100 150 200 250 300
pT pT
FIG. 5: Comparison of the SCET results at NLO+NLL with the CMS data of [2] for R =0.2 (green), R =0.3 (blue) and
G R √
R =0.4 (red). Statistical and systematic errors are added in quadrature. We have s =2.76 TeV, |η|<2 and we choose
NN
the coupling constant of the medium to the jet as g = 2.1±0.1. On the left side, we show the results for central collisions
(0−10% centrality) and on the right side for mid-peripheral collisions (30−50% centrality).
of single logarithms of the jet radius parameter αnlnnR servables in heavy-ion collisions in the future. (Semi-)
s
has to be taken into account. We consistently calculated inclusivejetsubstructuremeasurementsperformedonan
both the proton-proton and the heavy-ion jet cross sec- inclusively measured jet in heavy-ion collisions can now
tion at NLO+NLL accuracy. This new results was en- be calculated consistently with the proton-proton base-
R
abled by the use of the recently developed semi-inclusive line. See for example [46, 47].
jet functions. For the heavy-ion jet cross section, we
introduced in-medium semi-inclusive jet functions anal-
Acknowledgments
ogously to the vacuum ones. We calculated the QCD
mediumcontributionusingthecollinearin-mediumsplit- We would like to thank Yang-Ting Chien, Raghav
tingkernelsderivedwithinSCET tofirstorderinopac- Elayavalli, Wouter Waalewijn and Hongxi Xing for help-
G
ity. We found good agreement with recent experimental ful discussions. This research is supported by the US
measurementsfromtheLHCforthenuclearmodification Department of Energy, Office of Science under Contract
factor R of inclusive jets. The calculation outlined in No. DE-AC52-06NA25396andbytheDOEEarlyCareer
AA
this work sets the stage to address jet substructure ob- Program under Grant No. 2012LANL7033.
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