NNLO predictions for event shapes and jet rates in electron-positron annihilation

0.1 MB·English

by  Stefan Weinzierl

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NNLO predictions for event shapes and jet rates in
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- Stefan Weinzierl∗
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e UniversityofMainz
h E-mail: [email protected]
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v Thestrongcouplingconstantisafundamentalparameterofnature. Itcanbeextractedfromex-
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perimentsmeasuringthree-jeteventsinelectron-positronannihilation.Forthisextractionprecise
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2 theoreticalcalculations for jet rates and eventshapes are needed. In this talk I will discuss the
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NNLOcalculationfortheseobservables.
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RADCOR2009-9thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumField
TheorytoPhenomenology)
October25-302009
Ascona,Switzerland
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NNLOpredictionsforeventshapesandjetratesinelectron-positronannihilation StefanWeinzierl
1. The calculation
Theprocesse+e−→3jetsisofparticularinterestforthemeasurementofthestrongcoupling
a . Three-jeteventsarewellsuitedforthistaskbecausetheleadingterminaperturbative calcula-
s
tionofthree-jetobservables isalreadyproportional tothestrongcoupling. Forapreciseextraction
of the strong coupling one needs in addition to a precise measurement of three-jet observables in
the experiment a precise prediction for this process from theory. This implies the calculation of
higher order corrections. Theprocess e+e− →3jets has been been calculated recently at next-to-
next-to-leading order(NNLO)inQCD[1,2]. Themasterformulaforthecalculation ofathree-jet
observable atanelectron-positron collider is
hOi = 1 (cid:229) df O (p ,...,p ,q ,q ) (cid:229) |A |2, (1.1)
8s Z n n 1 n 1 2 n
n≥3 helicity
where q and q are the momenta of the initial-state particles and 1/(8s) corresponds to the flux
1 2
factorandtheaverageoverthespinsoftheinitialstateparticles. Theobservable hastobeinfrared
safe,inparticular thisimpliesthatinsingleanddoubleunresolved limitswemusthave
O (p ,...,p ,q ,q ) → O (p′,...,p′,q ,q ) forsingleunresolved limits,
4 1 4 1 2 3 1 3 1 2
O (p ,...,p ,q ,q ) → O (p′,...,p′,q ,q ) fordoubleunresolved limits. (1.2)
5 1 5 1 2 3 1 3 1 2
A is the amplitude with n final-state partons. At NNLO we need the following perturbative ex-
n
pansions oftheamplitudes:
|A |2= A(0) 2+2Re A(0)∗A(1) +2Re A(0)∗A(2) + A(1) 2,
3 3 3 3 3 3 3
(cid:12) (cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12) (cid:12)
|A |2=(cid:12)(cid:12)A(0)(cid:12)(cid:12)2+2Re A(0)∗A(1) , (cid:12)(cid:12) (cid:12)(cid:12)
4 4 4 4
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)2
|A |2=(cid:12)A(0)(cid:12) . (1.3)
5 5
(cid:12) (cid:12)
(cid:12) (cid:12)
HereA(l) denotesan(cid:12)amplit(cid:12)udewithnfinal-statepartonsandlloops. Wecanrewritesymbolically
n
theLO,NLOandNNLOcontribution as
hOiLO = O ds (0),
Z 3 3
hOiNLO = O ds (0)+ O ds (1),
Z 4 4 Z 3 3
hOiNNLO = O ds (0)+ O ds (1)+ O ds (2). (1.4)
Z 5 5 Z 4 4 Z 3 3
ThecomputationoftheNNLOcorrection fortheprocesse+e−→3jetsrequirestheknowledgeof
the amplitudes for the three-parton final state e+e− →q¯qg up to two-loops [3, 4], the amplitudes
ofthefour-parton finalstatese+e−→q¯qggande+e−→q¯qq¯′q′ uptoone-loop [5–8]andthefive-
partonfinalstatese+e−→q¯qgggande+e−→q¯qq¯′q′gattreelevel[9–11]. Themostcomplicated
amplitude is of course the two-loop amplitude. For the calculation of the two-loop amplitude
specialintegrationtechniqueshavebeeninvented[12–15]. Theanalyticresultcanbeexpressedin
terms of multiple polylogarithms, which in turn requires routines for the numerical evaluation of
thesefunctions [16,17].
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NNLOpredictionsforeventshapesandjetratesinelectron-positronannihilation StefanWeinzierl
2. Subtraction andslicing
Is is well known that the individual pieces in the NLO and in the NNLO contribution of
eq. (1.4) are infrared divergent. To render them finite, a mixture of subtraction and slicing is
employed. TheNNLOcontribution iswrittenas[18]
hOiNNLO = O ds (0)−O ◦da single−O ◦da (0,2)
Z (cid:16) 5 5 4 4 3 3 (cid:17)
+ O ds (1)+O ◦da single−O ◦da (1,1)
Z (cid:16) 4 4 4 4 3 3 (cid:17)
+ O ds (2)+O ◦da (0,2)+O ◦da (1,1) . (2.1)
Z (cid:16) 3 3 3 3 3 3 (cid:17)
da single is the NLO subtraction term for 4-parton configurations, da (0,2) and da (1,1) are generic
4 3 3
NNLOsubtraction terms,whichcanbefurtherdecomposed into
da (0,2) = da double+da almost+da soft−da iterated,
3 3 3 3 3
da (1,1) = da loop+da product−da almost−da soft+da iterated. (2.2)
3 3 3 3 3 3
In a hybrid scheme of subtraction and slicing the subtraction terms have to satisfy weaker condi-
tionsascompared toastrictsubtraction scheme. Itisjustrequiredthat
(a)theexplicit poles inthe dimensional regularisation parameter e inthesecond lineofeq. (2.1)
cancel after integration over unresolved phase spaces for each point of the resolved phase
space.
(b)thephasespacesingularitiesinthefirstandinthesecondlineofeq.(2.1)cancelafterazimuthal
averaging hasbeenperformed.
Point (b) allows the determination of the subtraction terms from spin-averaged matrix elements.
The subtraction terms can be found in [19–21]. The subtraction term da (0,2) without da soft
3 3
would approximate all singularities except a soft single unresolved singularity. The subtraction
term da soft takes care of this last piece [2, 22]. The azimuthal average is not performed in the
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Monte Carlo integration. Instead a slicing parameter h is introduced to regulate the phase space
singularities related to spin-dependent terms. It is important to note that there are no numerically
large contributions proportional to a power of lnh which cancel between the 5-, 4- or 3-parton
contributions. Eachcontribution itselfisindependent ofh inthelimith →0.
3. Monte Carlointegration
The integration over the phase space is performed numerically with Monte Carlo techniques.
Efficiency of the Monte Carlo integration is an important issue, especially for the first moments
of the event shape observables. Some of these moments receive sizable contributions from the
close-to-two-jet region. Inthe5-partonconfigurationthiscorrespondsto(almost)threeunresolved
partons. The generation of the phase space is done sequentially, starting from a 2-parton config-
uration. In each step an additional particle is inserted [21, 23]. In going from n partons to n+1
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NNLOpredictionsforeventshapesandjetratesinelectron-positronannihilation StefanWeinzierl
partons,then+1-partonphasespaceispartitionedintodifferentchannels. Withinonechannel,the
phasespaceisgenerated iteratively according to
df = df df (3.1)
n+1 n unresolvedi,j,k
Theindicesi, jandkindicate thatthenewparticle jisinserted betweenthehardradiatorsiandk.
Foreachchannelwerequirethattheproductofinvariantss s isthesmallestamongallconsidered
ij jk
channels. Fortheunresolved phasespacemeasurewehave
1 1 2p
s
df = ijk dx dx dj Q (1−x −x ) (3.2)
unresolvedi,j,k 32p 3Z 1Z 2Z 1 2
0 0 0
We are not interested in generating invariants smaller than (h s), these configurations will be re-
jectedbytheslicingprocedure. Insteadweareinterestedingeneratinginvariantswithvalueslarger
than(h s)withadistributionwhichmimicstheoneofatypicalmatrixelement. Wethereforegener-
atethe(n+1)-partonconfigurationfromthen-partonconfigurationbyusingthreerandomnumbers
u ,u ,u uniformlydistributed in[0,1]andbysetting
1 2 3
x =h u1, x =h u2 j =2p u . (3.3)
1 PS 2 PS 3
Thephase space parameter h isanadjustable parameter ofthe order oftheslicing parameter h .
PS
Theinvariants aredefinedas
s =x s , s =x s , s =(1−x −x )s . (3.4)
ij 1 ijk jk 2 ijk ik 1 2 ijk
Fromtheseinvariantsandthevalueofj wecanreconstructthefour-momentaofthe(n+1)-parton
configuration [24]. Theadditionalphasespaceweightduetotheinsertionofthe(n+1)-thparticle
is
1 s s
w = ij jk ln2h . (3.5)
16p 2 s PS
ijk
Note that the phase space weight compensates the typical eikonal factor s /(s s ) of a single
ijk ij jk
emission. As mentioned above, the full phase space is constructed iteratively from these single
emissions.
4. Numerical results
Fig. 1 shows the results for the Durham three jet rate and the thrust distribution at the LEP
I centre-of-mass energy Q2 =m with a (m )=0.118. The LO, NLO and NNLO predictions
Z s Z
are shown together withpthe experimental measured values from the Aleph experiment [25]. The
bandsgivetherangeforthetheoretical prediction obtained fromvaryingtherenormalisation scale
from m =Q/2to m =2Q. Notethat thetheory predictions inthese plotsare thepure perturbative
predictions. Powercorrections orsoftgluonresummation effectsarenotincluded intheseresults.
InarecentcalculationthelogarithmictermsoftheNNLOcoefficientofthethrustdistribution
havebeencalculated basedonsoft-collinear effectivetheory[26]:
dCt = 1 a ln5t +a ln4t +a ln3t +a ln2t +a lnt +a +O(t ) , t =1−T. (4.1)
dt t 5 4 3 2 1 0
(cid:2) (cid:3)
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NNLOpredictionsforeventshapesandjetratesinelectron-positronannihilation StefanWeinzierl
Durhamthree-jetrate Thrust
0.6
0.7 LO LO
NLO NLO
NNLO 0.5 NNLO
0.6 Alephdata Alephdata
0.5 0.4
jetrate 00..43 s−d1Ts−d1T() 0.3
0.2
0.2
0.1
0.1
0 0
1e-04 0.001 0.01 0.1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ycut 1−T
Figure 1: The scale variation of the Durham three jet rate and the thrust distribution at Q2 =m with
Z
a (m )=0.118. Thebandsgivethe rangefor the theoreticalpredictionobtainedfromvapryingthe renor-
s Z
malisationscalefromm =m /2tom =2m .
Z Z
NNLO NNLO NNLO
76000000 150000000 430500000000 hhh ===111000−−−579
300000
5000 0 250000
dC−−1T()1T()−d1T()432000000000 dC−−1T()1T()−d1T()---112-0505000000000000 dC−−1T()1T()−d1T()21150500000000000000000
0
1000 -25000 -50000
0 -30000 -100000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.01 0.1 1 1e-04 0.001 0.01 0.1 1
1−T 1−T 1−T
Figure2: AcomparisonoftheNNLOcoefficientofthethrustdistributionasobtainedfromthenumerical
programwiththelogarithmictermsobtainedfromSCET.
Thevaluesofthea ’sareforN =5
j f
a =−18.96, a =−207.4, a =−122.3, a =1488.3, a =−822.3, a =−683.4.
5 4 3 2 1 0
The logarithmic terms give a good description of the thrust distribution in the close-to-two jet
region. Theyarenot expected to giveanaccurate result in thehard region. Fig.2shows thecom-
parison oftheNNLOcoefficientofthethrustdistribution asobtained fromthenumerical program
witheq.(4.1). Intheleftplotoffig.2thex-axisshows(1−T)onalinearscale. Thiscorresponds
tothehardregion,wheretheNNLOresultfromthenumericalprogramisexpectedtogivethecor-
rect answer. Themiddle plot offig.2shows(1−T)on alogarithmic scale around (1−T)≈0.1.
Thiscorrespondstothepeakregionortheoverlapregion,wheretheperturbativeNNLOresultand
the one obtained from SCET agree. The right plot of fig. 2 shows (1−T) on a logarithmic scale
around (1−T)≈0.001. Thiscorresponds totheextremetwo-jet region, inwhichthelogarithmic
terms are dominant. In this region the results from the numerical program show a dependence on
theslicingparameter. Thenumericalresultsforh =10−5,h =10−7andh =10−9areplotted. For
smallervaluesofh theSCETresultisapproached.
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NNLOpredictionsforeventshapesandjetratesinelectron-positronannihilation StefanWeinzierl
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