Table Of ContentGluon and quark jet multiplicities at N3LO+NNLL
P. Bolzonia, B. A. Kniehla, A.V. Kotikova,b
aII. Institut fu¨r Theoretische Physik, Universita¨t Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
bBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Wepresentanewapproachtoconsiderandincludeboththeperturbativeandthenon-perturbative
contributions to the multiplicities of gluon and quark jets. Thanks to this new method, we have
included for the first time new contributions to these quantities obtaining next-to-next-to-leading-
logarithmic(NNLL)resummedformulae. Ouranalyticexpressionsdependontwonon-perturbative
parameters with a clear and simple physical interpretation. A global fit of these two quantities
shows how our results solve a longstanding discrepancy in thetheoretical description of thedata.
2
1 PACSnumbers: 12.38.Cy,12.39.St,13.66.Bc,13.87.Fh
0
2
Collisions of particles and nuclei at high energies usu- coupledgluon-singletsystemwhoseevolutioninthescale
t
c ally produce many hadrons. In Quantum Chromody- µ2 is governed in QCD by the DGLAP equations:
O
namics (QCD) their production is due to the interac-
0 tions of quarks and gluons and to test it as a theory µ2 d Ds = Pqq Pgq Ds . (1)
3 of strong interactions, the transition from a description dµ2 Dg Pqg Pgg Dg
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
based in terms of quarks and gluons to the hadrons ob-
] served in experiments is always needed. The production ThetimelikesplittingfunctionsPij canbecomputedper-
h
of hadrons is a typical process where non-perturbative turbatively in the strong coupling constant:
p
- phenomenaareinvolved. However,the hypothesis ofLo-
hep cdaisltPriabruttoino-nHsaadrreonsimDupalylitryen(LorPmHaDli)zeadssiunmtehsethhaadtrpoanritzoan- Pij(ω,as)= ∞ aks+1Pi(jk)(ω), as = 4απs, i,j =g,q,
k=0
[ tion process without changing their shape [1], allowing X (2)
perturbativeQCDtomakepredictions. Thesimplestob- whereω =N 1withN beingtheusualMellinconjugate
2
v servables of this kind are gluon and quark multiplicities variabletothe−fractionoflongitudinalmomentumx. The
4 nh g and nh s which represent the number of hadrons functions P(k)(ω) with k =0,1,2 appearing in Eq.(2) in
h i h i ij
1 produced in a gluon and a quark jet respectively. In
the MS scheme can be found in Refs. [10–12] through
9 the framework of the generating-functional approach in
5 NNLO and in Refs. [9, 13, 14] through the NNLL.
themodifiedleadinglogarithmicapproximation(MLLA)
. In general it is not possible to diagonalize Eq.(1) be-
9 [2], several studies of the multiplicities have been per-
0 cause the contributions to the splitting function matrix
formed [3–5]. In such studies, the ratio r = n / n
2 h hig h his do not commute at different orders. It is, therefore, con-
is at least10%higher than the data orit has a slope too
1 venient(seee.g.Ref.[15])tointroduceanewbasis,called
: small. Good agreement with the data has been achieved
v “plus-minus” basis where the LO splitting matrix is di-
in Ref. [6] where recoileffects are included. Nevertheless
i agonalwith eigenvaluesP(0) andP(0). We define sucha
X in Ref. [6] a constantoffset to be fitted to the quark and ++ −−
changeofbasisaccordingtothefollowingtransformation
r gluon multiplicities has been introduced, while the au-
a thors of Ref. [7] suggested that other, better motivated of the gluon and the singlet fragmentation functions in
Mellin space:
possibilities should be studied.
In this Letter, we study such a possibility inspired by
D+(ω,µ2) = (1 α )D (ω,µ2) ǫ D (ω,µ2),
the new formalism that has recently been proposed in 0 − ω s 0 − ω g 0
Ref. [8]. Thanks to very recent new results in small-x D−(ω,µ20)) = αωDs(ω,µ20)+ǫωDg(ω,µ20), (3)
timelikeresummationobtainedinRef.[9],weareableto
where
reach the NNLL accuracy level. A purely perturbative
and analytic prediction has been already attempted in
Ref.[7]uptothe thirdorderintheexpansionparameter α = Pq(q0)(ω)−P+(0+)(ω), ǫ = Pg(q0)(ω) .
√αs i.e. α3s/2, where paradoxically the quark multiplic- ω P−(0−)(ω)−P+(0+)(ω) ω P−(0−)(ω)−P+(0+)(ω)
ity and the ratio are not well described even if the be- (4)
havior of the perturbative expansion is very good. Our The general solution to Eq.(1) can be formally written
new resummed results that we present here are a gener- as
alization of what was obtained in Ref. [7] and represent
also a solution to this apparent paradox. D(µ2)=T exp µ2 dµ¯2P(µ¯2) D(µ2), (5)
WeconsiderthestandardMellin-spacemomentsofthe µ2( Zµ20 µ¯2 ) 0
2
where T denotes the path ordering with respect to µ2 WenotethatD˜± differfromD± startingathigherorders
µ2 a a
and [15]. Inthefollowing,wecollecttheresummedformulae.
Detailsofthecalculationwillbepresentedelsewhere[16].
D+
D = . (6) After the resummation is perfomed for P in Eq.(8)
D− ±±
(cid:18) (cid:19) thankstotheresultsobtainedinRefs.[9,17,18],wefind
Now making the following ansatz: for the first Mellin moment (ω =0) at NNLL:
µ2 dµ¯2 PNNLL(ω =0) = γ (1 K γ +K γ2+O(γ3)),
Tµ2(expZµ20 µ¯2 P(µ¯2))= P+N+NLL(ω =0) = 08nf−TRC1F0a +2O(0a2), 0 (16)
µ2 dµ¯2 −− − 3CA s s
=Z−1(µ2)exp PD(µ¯2) Z(µ2), (7)
"Zµ20 µ¯2 # 0 where
where γ PLL(ω =0)= 2a C , (17)
0 ≡ ++ s A
P (ω) 0
PD(ω)= ++ (8) p
0 P (ω) and
−−
(cid:18) (cid:19)
is the all-order diagonal part of the splitting matrix in 1 nfTR 2CF
K = 11+4 1 ,
1
the “plus-minus” basis and Z is a matrix in the same 12 C − C
(cid:18) A (cid:18) A (cid:19)(cid:19)
basis with a perturbative expansion of the form: 1 n T C
f R F
K = 1193 576ζ 56 5+2
2 2
Z(µ2)=1+as(µ2)Z(1)+O(a2s), (9) 288(cid:18) − − CA (cid:18) CA(cid:19)(cid:19)
n2T2 C C2
we obtain that +16 f R 1+4 F 12 F , (18)
C2 C − C2
A (cid:18) A A(cid:19)
D (ω,µ2) D+(ω,µ2)+D−(ω,µ2); a=s,g, (10)
a ≡ a a with ζ = π2/6. Now we can perform the integration in
2
where D+(ω,µ2) evolves like a “plus” component, Eq.(12) up to the NNLL to obtain that
a
D−(ω,µ2) evolves like a “minus” component, and
a
TNNLL(Q2)
Da±(ω,µ2)=D˜a±(ω,µ20)Tˆ±(ω,µ2,µ20)Ha±(ω,µ2). (11) Tˆ±NNLL(0,Q2,Q20)= T±NNLL(Q2), (19)
± 0
Here Tˆ (ω,µ2,µ2) is a renoramlization group exponent 4C
± 0 TNNLL(Q2)=exp A 1+
which is given by + β γ0(Q2)
0
n h
as(µ2) da¯ + b 2C K a (Q2) a (Q2) d+, (20)
Tˆ (ω,µ2,µ2)=exp s P (ω,a¯ ) , (12) 1− A 2 s s
± 0 "Zas(µ20) β(a¯s) ±± s # TNN(cid:16)LL(Q2)= a(cid:17)(Q2) di−o,(cid:16) (cid:17) (21)
− s
with
(cid:16) (cid:17)
∂ where
β(a (µ2)) µ2 a (µ2)
s ≡ ∂µ2 s
8n T C 2C K
f R F A 1
= β a2(µ2) β a3(µ2)+O(a4). (13) b1 =β1/β0, d− = , d+ = . (22)
− 0 s − 1 s s 3CAβ0 β0
We recall that
We are now ready to define the avarage multiplicities
11 4 in our formalism:
β = C n T ,
0 A f R
3 − 3
34 20 n (Q2) D (0,Q2)=D+(0,Q2)+D−(0,Q2), (23)
β1 = 3 CA2 − 3 CAnfTR−4CFnfTR, (14) h h ia ≡ a a a
with a = s,g for the quark and gluon multiplicities re-
where C =3, C =4/3 and T =1/2 in QCD and n
A F R f spectively. From Eqs.(11,15) we have that
isthenumberofactiveflavors. InEq.(11)H±(ω,µ2)are
a
perturbativefunctionscontainingoff-diagonaltermsofP
D+(0,Q2) α H+(ω,Q2)
beyond the LO and the normalizationfactors D˜±(ω,µ2) g = lim ω g r (Q2), (24)
satisfy the following conditions: a 0 Ds+(0,Q2) −ω→0 ǫω Hs+(ω,Q2) ≡ +
D˜+(ω,µ2) = αω D˜+(ω,µ2); and
g 0 −ǫ s 0
ω
D˜g−(ω,µ20) = 1−ǫωαω D˜s−(ω,µ20). (15) DDsg−−((00,,QQ22)) =ωli→m01−ǫωαω HHsg−−((ωω,,QQ22)) ≡r−(Q2). (25)
3
Using these definitions, it is convenient to write for the 35
gluon and quark multiplicities in general:
30
à
hhnnhh(+(QQD2˜2)s−)iis+(g0=D=,˜Q−DD˜20˜(r)g+0g++r,(−((Q00Q(,,2QQ2Q))T220ˆ20)))rTˆTTeˆˆ−sr++rr(eee0sss(,((0Q00,,,2QQQ,2Q22,,,2QQQ)H202020)))H−HH(s−gg0++(,((0Q00,,,2QQQ).222)()),26) <nh> 11220505 àìàæìàæàææàìôææìæææáàççæìæòòìôæàææìæìôæææíæíæíííòòííí õõáææàààóóó æõóàõ çìæíáôòà AMTTTODJæàAMPOAPEADSCAPLYRESPLAKOHZ-æàII æàæà æ
s 0 − 0 s ó ALEPH
For the coefficients of the renormalization group expo- 5 ààà æàõ CHL3LRESO
nents, we clearly have the following simple relations at 0
0 50 100 150 200
the lowest order in a
s Q2
r (Q2)=C /C ; r (Q2)=0;
+ A F − FIG.1: Gluonandquarkmultiplicitiesfitscomparedtothedata.
H±(0,Q2)=1; D˜±(0,Q2)=D±(0,Q2),(27) ThegraydashedlineistheLO+NNLLresult,theorangesolidline
s a 0 a 0
istheN3LOapprox+NNLLresultandthereddottedlineisthefit
witha=s,g. Onewouldliketoincludehigher-ordercor- withfourconstantcoefficients. Theorangebandcorrespondstothe
rections to Eq.(27). However, this is highly non-trivial estimatederrorofthefittedparametersintheN3LOapprox+NNLL
case.
because the general perturbative structures of the func-
tions H± and Z , whose knowledge is required for
a ±∓,a
theresummation,arenotknown. Fortunatly,generalas-
sumptions and approximations can be made to improve 1.8
them. Firstly,itisawellknownfactthatthe“plus”com- á áá á
ponents by themselves represent the dominant contribu- 1.6 á
tionsforboththe gluonandthe quarkmultiplicities (see á æææ æ æ æààæ æ æ ææææ
e.g. Refs. [19,20]). Secondly, Eq.(25) tells us thatDg− is r 1.4 ææ ææ ææ ææàææ æ æ DELPHI
suppressed with respect to Ds−, because αω ∼1+O(ω). ææôôææææ òòæàà à OPAL
These twofacts suggestus that to keepr−(Q2)=0 even 1.2 ææ ì TASSO
at higher orders should still represent a good approxi- ç à ò ALEPH
mation. Then we notice that higher-order corrections to 1.0 ç ìì à çô CHLRESO
D˜±(0,Q2)andH±(0,Q2)justrepresentaredefinitionof á CDF
a 0 a
Da±(0,Q20) apart from running coupling effects starting 0.80 10 20 30 40 50 60 70
at order a2. Therefore we assume that these corrections Q2
s
canbeneglected. Nowwecanfinallydiscusshigher-order
corrections to r (Q2), which represents the ratio of the FIG.2: Glun-quarkmultiplicityratiopredictioncomparedtodata.
+
The gray solid upper line is the prediction of Ref. [7], the others
pure “plus” components. Accordingly, we can intepret
areasinFig.1
theresultinEq.(5)ofRef.[7]ashigher-ordercorrections
to Eq.(24). This interpretation is explicitly confirmed
up to order a in Chapter 7 of Ref. [2], where also the
s
for the multiplicities and
samesetofequationsusedinthe computationofRef.[7]
are obtained. Further arguments to support it and its
n (Q2)
scheme dependence will be discussed in Ref. [16]. We r(Q2) h h ig (30)
denote the approximation in which Eqs.(19,27) are used ≡ nh(Q2) s
h i
as LO+NNLL and the one in which r+(Q2) in Eq.(27) r+(Q2)
= ,
is replaced by the result of Eq.(5) in Ref. [7] up to order
aal3sl/y2aagsoNo3dLaOpapprporxoxim+aNtioNnLwLi.llTbheasthtohwisnlbaesltoown.e iIsnabcottuh- (cid:20)1+ rr++((QQ220))(cid:16)Ds(D0,gQ(200),Qr+20()Q20) −1(cid:17)TTˆˆ+−rreess((00,,QQ22,,QQ2020))(cid:21)
approximations considered we can summarize the main
for the gluon-quark multiplicity ratio. Eqs.(28,29) de-
theoretical result of this Letter in the following way pend only on two parameters, D (0,Q2) and D (0,Q2),
g 0 s 0
n (Q2) =D (0,Q2)Tˆres(0,Q2,Q2), (28) with a simple physical interpretation: they are just the
h h ig g 0 + 0 gluon and the quark multiplicities at the arbitrary scale
n (Q2) =D (0,Q2)Tˆ+res(0,Q2,Q20) Q0.
h h is g 0 r+(Q2) Wehaveperformedaglobalfitofourresummedformu-
D (0,Q2) lae, Eqs.(28,29), to the experimental data to extract the
+ D (0,Q2) g 0 Tˆres(0,Q2,Q2), (29)
(cid:20) s 0 − r+(Q20) (cid:21) − 0 values of Dg(0,Q20) and Ds(0,Q20). With Q0 = 50GeV,
4
the result of the fit is given by a fit and have a simple physical meaning because they
just represent the gluon and the quark multiplicity at a
D (0,Q2) = 24.31 0.85; 90% C.L.
g 0 ± certainarbitraryscaleQ0. Wehopethatadditionalmea-
Ds(0,Q20) = 15.49±0.90; 90% C.L., (31) surements of these observables will come from the LHC
to test our results on a much wider energy range.
in the LO+NNLL case and by
We kindly thank A. Vogt and G. Kramer for valu-
Dg(0,Q20) = 24.02±0.36; 90% C.L. able discussions. The work of A.V.K. was supported
D (0,Q2) = 15.83 0.37; 90% C.L., (32) in part by the Russian Foundation for Basic Research
s 0 ±
RFBR through Grant No. 11–02–01454–a. This work
in the N3LO +NNLL case in agreement with the
approx was supported in part by the German Federal Ministry
experimentalvalueswithintheerrors. However,the90%
for Education and Research BMBF through Grant No.
C.L.errorintheN3LO +NNLLcaseismuchsmaller
approx 05H12GUE, by the German Research Foundation DFG
reflecting a much better fit to the data at all energies. throughthe CollaborativeResearchCentreNo.676Par-
Indeed, per degree of freedom we obtain χ2 = 18.09 in ticles, Strings and the Early Universe—The Structure of
the LO+NNLL case, while we have χ2 = 3.71 in the Matter and Space Time, and by the Helmholtz Associa-
N3LOapprox+NNLL case. In our analysis, we have used tionHGFthroughtheHelmholtzAllianceHa101Physics
the NLO solution for the running coupling according to at the Terascale.
Eq.(13) with α (M ) = 0.118 and n = 5. We have
s Z f
checked that varying the arbitrary scale Q2 does not
0
change the resulting value of χ2 as expected and that
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