Event Shape Variables at NLLA+NNLO

0.16 MB·

by  G. Luisoni

Event Shape Variables at NLLA+NNLO
∗
Gionata Luisoni
Institut fu¨r Theoretische Physik,Universit¨at Zu¨rich,
CH-8057 Zu¨rich, Switzerland
9
0 In this talk [1] we report work on the matching of the next-to-leading logarithmic ap-
0 proximation(NLLA)ontothefixednext-to-next-to-leadingorder(NNLO)calculations
2 foreventshapedistributionsinelectron-positronannihilation. Furthermorewepresent
preliminaryresultsonthedeterminationofthestrongcouplingconstantobtainedusing
n
a NLLA+NNLOpredictions and ALEPH data.
J

6 1 Introduction
2
Thereactionofe+e−annihilationintothreejetshasplayedhistoricallyaveryprominentrole
]
h forphenomenology. Itpermittedforexamplethediscoveryofthegluonandthemeasurement
p of its properties and allows also a precise determination of the strong coupling constant α ,
s
- since the deviation from two-jet configurations is proportionalto it. Not only jet rates, but
p
e also the shape of the single events can be studied in a systematic fashion. The so-called
h event shape observables became very popular mainly because they are well suited both for
[ experimental measurement and for theoretical description since many of them are infrared

1 andcollinearsafe. Themainideabehindeventshapevariablesistoparameterizetheenergy-
v momentum flow of an event, such that one can smoothly describe its shape passing from
3 pencil-like two-jet configurations, which are a limiting case in event shapes, up to multijet
5
final states. At LEP a set of six different event shape observables were measured in great
9
detail: thrust T (which is substituted here by τ = 1 −T), heavy jet mass ρ, wide and
3
. totaljet broadeningBW andBT, C-parameterandtwo-to-three-jettransitionparameterin
1
the Durham algorithm y . The definitions of these variables, which we denote collectively
0 3
as y in the following, are summarized in [2]. The two-jet limit of each variable is y → 0.
9
0 Until very recently, the theoretical state-of-the-art description of event shape distributions
: was based on the matching of the NLLA [3] onto the NLO [4, 5] calculation. Using these
v
i predictions the largestcontributionto the errorin the determinationof the strong coupling
X
constant came from theoreticalscale uncertainties. Recently the NNLO correctionsbecame
r available. Using this new results we computed the matching of the resummed NLLA onto
a
the fixed order NNLO.
2 Fixed order and resummed calculations
At NNLO the integrated fixed order differential cross section
1 y dσ(x,Q,µ)
R(y,Q,µ) ≡ dx,
σ Z dx
had 0
is given by
R(y,Q,µ) = 1+ α¯ (µ)A(y) + α¯2(µ)B(y,x ) + α¯3(µ)C(y,x ) ,
s s µ s µ
∗OnbehalfofG.Dissertori,A.Gehrmann-DeRidder,T.Gehrmann,E.W.N.Glover,G.HeinrichandH.
Stenzel.
LCWS/ILC2008
where α¯ =α /(2π) and x =µ/Q.
s s µ
Table 1 shows the relevant contributions for the
computationofthethreecoefficientfunctionsA, LO γ∗ → qq¯g tree level
B andC. Thecarefulsubtractionofrealandvir- NLO γ∗ → qq¯g one loop
tual divergences is done using the antenna for- γ∗ → qq¯gg tree level
malismandimplementedinanumericalintegra- γ∗ → qq¯qq¯ tree level
tion program. Recently an inconsistency in the NNLO γ∗ → qq¯g two loop
treatment of large-angle soft radiation was dis- γ∗ → qq¯gg one loop
covered [8]. This was corrected (erratum to [6]) γ∗ → qq¯qq¯ one loop
and it results in numerically minor changes to γ∗ → qq¯qq¯g tree level
the NNLO coefficients in the kinematical region γ∗ → qq¯ggg tree level
of phenomenological interest here. The correc-
tions turn out to be significant only in the deep Table 1: Contributions order by order.
two-jet region, e.g. (1 −T) < 0.05 (figure 1).
Approaching the two-jet region the infrared logarithms in the coefficient functions become
T 0.5 T 0.5
d NNLO new d NLLA+NNLO new
σ/ σ/
d 0.4 NNLO old d 0.4 NLLA+NNLO old
σT) 1/ had 0.3 σT) 1/ had 0.3 Qαs =(M MZ)Z = 0.1189
1- Q = MZ 1-
( 0.2 α (M) = 0.1189 ( 0.2
s Z
0.1 0.1
0 0
10-2 10-1 10-2 10-1
1-T 1-T
Figure 1: Comparison between old and corrected distributions for τ. In the fixed order
distribution (left) a small difference is visible in the far infrared region, in the matched
distribution (right) the curves are equal since the resummation becomes dominant in the
infrared region.
large spoiling the convergence of the series expansion. The main contribution in this case
comes from the highest powers of the logarithms which have to be resummed to all orders.
For suitable observables resummation leads to exponentiation. At NLLA the resummed
expression is given by
R(y,Q,µ) = (1+C α¯ ) e(Lg1(αsL)+g2(αsL)) ,
1 s
where the function g (α L) contains all leading-logarithms (LL), g (α L) all next-to-
1 s 2 s
leading-logarithms (NLL) and µ = Q is used. Terms beyond NLL have been consistently
omitted. The resummationfunctions g (α L)andg (α L)canbe expandedas powerseries
1 s 2 s
in α¯ L:
s
Lg (α L) = G α¯ L2+G α¯2L3+G α¯3L4+... (LL),
1 s 12 s 23 s 34 s
g (α L) = G α¯ L+G α¯2L2+G α¯3L3+... (NLL). (1)
2 s 11 s 22 s 33 s
Table 2 shows the logarithmic terms present up to the third order in perturbation theory.
At the fixed order level the LL are terms of the form αnLn+1, the NLL those which go like
s
LCWS/ILC2008
α¯ A(y) α¯ L α¯ L2
s s s
α¯2B(y,x ) α¯2L α¯2L2 α¯2L3 α¯2L4
s µ s s s s
α¯3C (y,x ) α¯3L α¯3L2 α¯3L3 α¯3L4 α¯3L5 α¯3L6
s µ s s s s s s
Table 2: Powers of the logarithms present at different orders in perturbation theory. The
colorhighlightsthedifferentordersinresummation: LL(red)andNLL(blue). Thetermsin
greenare containedin the LL andNLL contributions andexponentiate trivially with them.
αnLn, and so on. Notice that this can be read off the expansion (1) of the exponentiated
s
resummation functions.
Closedanalyticformsforthefunctionsg (α L)andg (α L)areavailableforτ andρ[9],
1 s 2 s
B and B [10, 11], C [12] and Y [13], and are collected in the appendix of [16]. Recently
W T 3
also g (α L) and g (α L) were computed for τ using effective field theory methods [15].
3 s 4 s
3 Matching of fixed order and resummed calculations
To obtain a reliable description of the event shape distributions over a wide range in y,
it is mandatory to combine fixed order and resummed predictions. The two predictions
have to be matched in a way that avoids the double counting of terms present in both.
A number of different matching procedures have been proposed in the literature, see for
example [2] for a review. In the so-calledR-matching scheme, the two expressions for R(y)
are matched. We computed the matching in the so-called ln R-matching [3] since in this
particularscheme,allmatchingcoefficientscanbeextractedanalyticallyfromtheresummed
calculation. The ln R-matching at NLO is described in detail in [3]. In the ln R-matching
scheme, the NLLA+NNLO expression is
ln(R(y,α )) = Lg (α L) + g (α L)+ α¯ A(y)−G L−G L2 +
s 1 s 2 s S 11 12
1 (cid:0) (cid:1)
+α¯2 B(y)− A2(y)−G L2−G L3
S(cid:18) 2 22 23 (cid:19)
1
+α¯3 C(y)−A(y)B(y)+ A3(y)−G L3−G L4 . (2)
S(cid:18) 3 33 34 (cid:19)
Thematchingcoefficientsappearinginthisexpressioncanbeobtainedfrom(1)andarelisted
in[16]. Toensurethevanishingofthematchedexpressionatthekinematicalboundaryy
max
a further shift of the logarithm is made [2].
The renormalisation scale dependence of (2) is given by making the following replace-
ments:
α → α (µ),
s s
B(y) → B(y,µ)=2β lnx A(y)+B(y) ,
0 µ
C(y) → C(y,µ)=(2β lnx )2A(y)+2 lnx [2β B(y)+2β A(y)]+C(y) ,
0 µ µ 0 1
β
g (α L) → g α L,µ2 =g (α L)+ 0 (α L)2 g′ (α L) lnx ,
2 s 2 s 2 s π s 1 s µ
(cid:0) (cid:1)
G → G (µ)=G + 2β G lnx ,
22 22 22 0 12 µ
G → G (µ)=G + 4β G lnx .
33 33 33 0 23 µ
LCWS/ILC2008
′
Intheabove,g denotesthederivativeofg withrespecttoitsargument. TheLOcoefficient
1 1
A and the LL resummation function g , as well as the matching coefficients G remain
1 ii+1
independent on µ.
4 Discussion of the matched distribution
For the resulting plots of the matched distributions we refer to [16]. The most striking
observation is that the difference between NLLA+NNLO and NNLO is largely restricted
to the two-jet region, while NLLA+NLO and NLO differ in normalisation throughout the
full kinematical range. This behavior may serve as a first indication for the numerical
smallness of corrections beyond NNLO in the three-jet region. In the approach to the two-
jet region, the NLLA+NLO and NLLA+NNLO predictions agree by construction, since
the matching suppresses any fixed order terms. Although not so visible on these plots,
the difference between NLLA+NNLO and NLLA+NLO is only moderate in the three-jet
region. The renormalisation scale uncertainty in the three-jet region is reduced by 20-40%
between NLLA+NLO and NLLA+NNLO. This effect is due to the smaller renormalization
scale dependence of the NNLO contributions. It is also important to observe that the scale
dependence remains the same and is larger in the two-jet region, because the resummed
calculations atNLLA take into accountonly the one-looprunning ofthe coupling constant.
This has important consequences in the determination of α and we will comment more on
s
this in the next section.
The description of the hadron-level data improves between parton-level NLLA+NLO
andparton-levelNLLA+NNLO,especiallyinthe three-jetregion. The behaviorinthe two-
jet region is described better by the resummed predictions than by the fixed order NNLO,
although the agreement is far from perfect. This discrepancy can in part be attributed to
missing higher order logarithmic corrections and in part to non-perturbative corrections,
which become large in the approach to the two-jet limit.
Therightplotinfigure1showsthattheinconsistencyinthetreatmentofthelarge-angle
soft radiation does not affect the matched prediction since the infrared region is dominated
by the resummation.
5 Determination of the strong coupling constant
After the extraction of α using only the NNLO distributions and the experimental data of
s
ALEPH [14], a new extraction of α using the new matched results was performed using
s
JADE data [18]. The improvement in the error coming from the inclusion of resummed
calculation is not as drammatic as passing from NLO to NLLA+NLO calculations. As
already anticipated, this is due to the fact that the NNLO coefficients compensate the two-
looprenormalizationscalevariation,whereastheNLLApartonlycompensatestheone-loop
variation. A more natural way of matching would be the consider NNLLA and NNLO, but
the NNLLA function g is by now only known for τ. A new determination of α using
3 s
ALEPH data is in progress. The analysiswill follow the lines of the previous determination
using pure NNLO predictions with a few improvements.
LCWS/ILC2008
6 Outlook
The matching of NLLA and NNLO has improved the theoretical prediction of event shape
distributions, but further improvement is possible by including the NNLL corrections into
the calculations. These corrections are known only for τ, where higher order logarithmic
correctionshavebeencomputed[15]usingsoft-collineareffectivetheory(SCET).Fromthese
calculationsonecanextractthe functionsg (α L)andg (α L). Thenextsteptowardsthe
3 s 4 s
further improvement in the extraction of α from event shape distributions could be to
s
compute them for all six observables mentioned here. As shown in [15] the subleading
logarithmic corrections can also account for about half of the discrepancy between parton-
level theoretical predictions and hadron-levelexperimental data.
Improvements can also come from non-perturbative corrections. A very recent non-
perturbativestudyforτ usingalow-scaleeffectivecoupling[17]showsthatnon-perturbative
1/Qpowercorrectionscauseashiftinthedistributions,whichcanaccountforanimportant
partofthe differencebetweenparton-leveldistributionsandhadron-levelexperimentaldata
discussed in the previous section.
Acknowledgements
We wish to thank the Swiss NationalScience Foundation (SNF) which supported this work
under contract 200020-117602.
References
[1] Presentation:
http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=73&sessionId=18&confId=2628
[2] R.W.L.Jones,M.Ford,G.P.Salam,H.StenzelandD.Wicke,JHEP0312(2003)007[hep-ph/0312016].
[3] S.Catani,L.Trentadue, G.TurnockandB.R.Webber,Nucl.Phys.B407(1993)3.
[4] R.K.Ellis,D.A.RossandA.E.Terrano,Nucl.Phys.B178(1981)421.
[5] Z.KunsztandP.Nason,inZ Physics at LEP 1,CERNYellowReport89-08,Vol.1,p.373;
W.T.GieleandE.W.N.Glover,Phys.Rev.D46(1992) 1980;
S.CataniandM.H.Seymour,Phys.Lett. B378(1996) 287[hep-ph/9602277].
[6] A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover and G. Heinrich, JHEP 0711 (2007) 058
[arXiv:0710.0346].
[7] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, JHEP 0712 (2007) 094
[arXiv:0711.4711].
[8] S.Weinzierl,Phys.Rev.Lett. 101(2008) 162001[arXiv:0807.3241].
[9] S.Catani,G.Turnock,B.R.Webber andL.Trentadue, Phys.Lett. B263(1991)491.
[10] S.Catani,G.TurnockandB.R.Webber,Phys.Lett. B295(1992)269.
[11] Y.L.Dokshitzer,A.Lucenti,G.MarchesiniandG.P.Salam,JHEP9801(1998)011[hep-ph/9801324].
[12] S.CataniandB.R.Webber,Phys.Lett. B427(1998)377[hep-ph/9801350];
E.GardiandL.Magnea,JHEP0308(2003)030[hep-ph/0306094].
[13] A.Banfi,G.P.SalamandG.Zanderighi,JHEP0201(2002) 018[hep-ph/0112156].
[14] G. Dissertori, A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, G. Heinrich and H. Stenzel,
JHEP0802(2008)040[arXiv:0712.0327].
[15] T.BecherandM.D.Schwartz, JHEP0807(2008) 034[arXiv:0803.0342].
[16] T.Gehrmann,G.LuisoniandH.Stenzel, Phys.Lett.B664(2008) 265[arXiv:0803.0695].
[17] R.A.DavisonandB.R.Webber,Eur.Phys.J.C59(2009) 13[arXiv:0809.3326].
[18] S.Bethke, S.Kluth,C.Pahl,J.SchieckandtheJADECollaboration,[arXiv:0810.1389].
LCWS/ILC2008