A Precise Determination of $\alpha_s$ from the C-parameter Distribution

4 MB·English

by  André H. Hoang

UWTHPH 2015-1
MIT–CTP 4630
LPN14-128
A Precise Determination of α from the C-parameter Distribution
s
Andr´e H. Hoang,1,2 Daniel W. Kolodrubetz,3 Vicent Mateu,1 and Iain W. Stewart3
1University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien, Austria
2Erwin Schro¨dinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Vienna, Austria
3Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
We present a global fit for α (m ), analyzing the available C-parameter data measured at
s Z
center-of-mass energies between Q = 35 and 207GeV. The experimental data is compared to a
N3LL(cid:48) + O(α3) + Ω theoretical prediction (up to the missing four-loop cusp anomalous dimen-
s 1
sion), which includes power corrections coming from a field theoretical nonperturbative soft func-
tion. The dominant hadronic parameter is its first moment Ω , which is defined in a scheme which
1
eliminates the O(Λ ) renormalon ambiguity. The resummation region plays a dominant role
QCD
5 in the C-parameter spectrum, and in this region a fit for α (m ) and Ω is sufficient. We find
s Z 1
1 α (m ) = 0.1123±0.0015 and Ω = 0.421±0.063GeV with χ2/dof = 0.988 for 404 bins of data.
s Z 1
0 TheseresultsagreewiththepredictionofuniversalityforΩ betweenthrustandC-parameterwithin
1
2 1-σ.

y
a
M I. INTRODUCTION which include higher-order resummation and corrections
up to O(α3) [3–12].
s
7 In order to study Quantum Chromodynamics (QCD) Several previous high-precision studies which deter-
2 accurately in the high-energy regime, it is useful to ex- mine αs(mZ) [4, 5, 9–11] focus on the event shape called
ploit the wealth of data from previous e+e− colliders thrust [13],
]
h suchasLEP.Herethefinalstatescomingfromtheunder-
(cid:32) (cid:80) (cid:33)
p lying partons created in the collisions appear as boosted |(cid:126)n·p(cid:126) |
τ = 1−T = min 1− i i , (1)
p- and collimated groups of hadrons known as jets. Event (cid:126)n (cid:80)j|p(cid:126)j|
shapes have proven to be very successful to study these
e
h collisions quantitatively. They combine the energy and
where (cid:126)n is called the thrust axis and it follows from the
[ momentaofallofthemeasuredhadronsintoaninfrared-
above equation that 0 ≤ τ ≤ 1/2. Another event shape,
andcollinear-safeparameterwhichdescribesthegeomet-
2 known as C-parameter [14, 15], can be written as:
ric properties of the whole event by a single variable dis-
v
11 tnriicbeutthioeno.reDtiucaeltporothpeeirrtigelso,bmalakniantguriet epvoesnsitbslehatpoeosbhtaavine C = 3 (cid:80)i,j|p(cid:126)i||p(cid:126)j|sin2θij , (2)
1 very accurate theoretical predictions using QCD. Most 2 ((cid:80) |p(cid:126) |)2
i i
4 e+e− event shape variables quantify how well the event
0
resembles the situation of two narrow back-to-back jets, where θ gives the angle between particles i and j. It
. ij
1 calleddijets,byvanishinginthislimit. Becausethedijet is straightforward to show that 0 ≤ C ≤ 1. In a pre-
0 limit involves restrictions that only allow collinear and vious paper [12] we computed the C-parameter distribu-
5 soft degrees of freedom for the final-state radiation, such tion with a resummation of large logarithms at N3LL(cid:48)
1 QCD predictions involve a number of theoretical aspects accuracy, including fixed-order terms up to O(α3) and
: s
v that go beyond the calculation of higher-order pertur- hadronizationeffectsusingafield-theoreticnonperturba-
i bative loop corrections. These include factorization, to tive soft function. These results were achieved by using
X
systematically account for perturbative and nonpertur- theSoftCollinearEffectiveTheory(SCET)[16–20]. Our
r
a bative contributions, and the resummation of large log- results for C are valid in all three of the peak, tail, and
arithmic corrections by renormalization group evolution. far-tail regions of the distribution, and are the most ac-
Comparisonsofpredictionsforeventshapeswithexperi- curate predictions available in the literature, having a
mental data thus provide non-trivial tests of the dynam- perturbative uncertainty of (cid:39)3% at Q=m for the re-
Z
ics of QCD. gionrelevantforα (m )andΩ fits. Thesameaccuracy
s Z 1
Due to the high sensitivity of event shapes to jets was previously achieved for thrust, where the remaining
induced by gluon radiation they are an excellent tool perturbative uncertainty in the τ distribution is (cid:39)2% in
to measure the strong coupling α . For more inclusive this region [9]. In this paper we make use of these new
s
hadronic cross sections (like e+e− → hadrons) the α C-parametertheoreticalresults[12]tocarryoutaglobal
s
dependence is subleading because it only occurs in cor- fit to all available data, comparing the results with the
rections to a leading order term, while for event shapes analogous global fit for thrust [9] where appropriate.
the α dependence is a leading-order effect. For this rea- Sincebothτ andC vanishinthedijetlimit,itisworth-
s
son, the study of event shapes for determining α has while to contrast them in order to anticipate differences
s
a long history in the literature (see the review [1] and that will appear in the analysis. Differences between C
the workshop proceedings [2]), including recent analyses and τ include the following:
2
a) calculating τ requires identifying the thrust axis wepresenthere,weprovideanotherevent-shapedetermi-
with a minimization procedure, while C does not nation of α (m ) with a high level of precision. We will
s Z
involve a minimization; alsosimultaneouslyexaminetheleadingpowercorrection
to the distribution, which should be universal between
b) τ has a single sum over particles while C has a
thrust and C-parameter.
double sum;
Thispaperisorganizedasfollows. InSec.IIwereview
the theoretical calculation of the C-parameter cross sec-
c) the size of the nonperturbative region, where the
tion,presentedinmoredetailinRef.[12]. Thedetailson
entire shape function is important, is larger for C
theexperimentaldataandfitprocedureusedinouranal-
compared to τ due to an enhancement by a factor
ysisaregiveninSecs.IIIandIV.InSec.Vwepresentthe
of 3π/2;
results of our fit for α (m ) and the first moment of the
s Z
d) the resummation region for C is larger than that nonperturbative soft function, ΩC. Fits which include
1
for thrust since the logarithms appear as ln(C/6) hadron-mass effects are discussed in Sec. VE. In Sec. VI
compared to ln(τ), which increases the range of C we make predictions for the peak and far-tail regions of
values that are useful for α fits; the distribution, which are not used in our fit, and com-
s
parethoseregionstoexperimentaldata. Theuniversality
e) fixed-order predictions for the thrust cross sec- of Ω is discussed in Sec VII, where we compare our re-
1
tion are smooth across the threshold where non- sults with the previous fit done using thrust in Ref. [9].
planar events first contribute, τ = 1/3, while the Finally, Sec. VIII contains our conclusions. We also in-
fixed-order C-parameter cross section has an inte- clude three appendices. The first, Appendix A, contains
grablesingularityatthisthreshold,Cshoulder =3/4. the formulae needed to calculate the profile functions,
The singularity for C comes from the fact that thesecondAppendixBcontainsresultsthatsupportour
the leading-order distribution is not continuous at choice for the definition of the renormalon free Ω pa-
1
C = Cshoulder. (The C-distribution can be made rameter to use for C; and the third compares fit results
smooth here using LL resummation [21].) for thrust with our profiles and those from Ref. [9].
A key similarity is that in the dijet limit (C, τ (cid:28)1) the
partonic cross sections for thrust and C-parameter are
related up to NLL by using τ =C /6 [22]. This II. THEORY REVIEW
NLL NLL
relation quantifies several qualitative relations between
C and τ in the dijet region. Until a few years ago, the theoretical uncertainties re-
Recent higher-order event-shape analyses [4, 9–11] lated to perturbative contributions as well as hadronic
have found values of α (m ) significantly lower than the power corrections were still larger than the experimental
s Z
world average of α (m ) = 0.1185±0.0006 [23] which uncertainties. The situation on the theory side has dra-
s Z
is dominated by the lattice QCD determination [24]. matically changed with the calculation of O(α3) correc-
s
This includes the determination carried out for thrust tions[3,7,32–35],andthepioneeringuseofSCETtoob-
at N3LL(cid:48)+O(α3) in Ref.[9]1, which is also consistent tainhigher-orderperturbativecorrectionsinRef.[5],and
s
with analyses at N2LL+O(α3) [11, 25] which consider to obtain accurate predictions for the full spectrum and
s
the resummation of logs at one lower order. In Ref. [26] a precision description of power corrections in Ref. [9].
a framework for a numerical code with N2LL precision Inthissectionwereviewthetheoreticalworkbehindour
formanye+e− eventshapeswasfound, whichcouldalso calculation of dσ/dC, presented in Ref.[12].
be utilized for α fits in the future. In Ref. [9] it was SCET separates the physics occurring at the different
s
pointed out that including a proper fit to power correc- energy and momentum scales relevant to the underlying
tions for thrust causes a significant negative shift to the jetswhosepropertiesarecharacterizedbyaneventshape.
value obtained for α (m ), and this was also confirmed For the C-parameter distribution the relevant scales are:
s Z
by subsequent analyses [11]. Recent results for α (m ) (i) the hard scale µ , which is related to the production
s Z H
fromτ decays[27], DISdata[28], thestaticpotentialfor of partons at short distances and is of the order of the
(cid:112)
quarks [29], as well as global PDF fits [30, 31] also find center-of-mass energy Q; (ii) the jet scale µ ∼Q C/6,
J
values below the world average. With the new analysis which governs the formation and evolution of the jets;
and (iii) the soft scale µ ∼ QC/6, which is the scale of
S
large-anglesoftradiation. Allthreescalesarewidelysep-
aratedinthedijetregionC (cid:28)1,wheremostoftheevents
1 Note that results at N3LL require the currently unknown QCD occur and where the distribution is maximal. There is
four-loopcuspanomalous,butconservativeestimatesshowthat one function associated to each one of these scales in
this has a negligible impact on the perturbative uncertainties. the factorization theorem that describes the dominant
Results at N3LL(cid:48) also technically require the unknown 3-loop
contribution of the C-parameter distribution in the dijet
non-logarithmic constants for the jet and soft functions which
are also varied when determining our uncertainties, but these limit: (i) the hard function H (the modulus squared of
parameters turn out to only impact the peak region which is the QCD-to-SCET matching coefficient), which is com-
outsidetherangeofourαs(mZ)fitsintheresummationregion. mon to all dijet event-shape factorization theorems; (ii)
3
the jet function J (given by matrix elements of quark A. Factorized Cross Section Formula
τ
fields with collinear Wilson lines), which is common for
C-parameter [12], thrust [9] and Heavy-Jet-Mass (ρ) [8]; In order to understand the perturbative components
and (iii) the soft function SC(cid:101) (defined by a vacuum ma- of the C-parameter cross section we make use of the
trixelementofpurelysoftWilsonlines),whichingeneral C-parameterfactorizationformula. Tomaketheconnec-
dependsonthespecificformoftheeventshape. Whereas tion to thrust simpler we will often use functions defined
the former two are perturbative (µ ,µ (cid:29) Λ ), per-
H J QCD with the variable C(cid:101) = C/6. For the perturbative cross
mitting the calculation of the hard and jet functions as
section we find [12]:
an expansion in powers of α , the soft function has per-
s
ttuurrbbaattiivvee ccoornrtercibtiuotnison(sµSth(cid:29)at ΛneQeCdDt)oasbewealclcaosunntoendpfeorr- σ1 ddσˆCs = Q6H(Q,µ)(cid:90) dsJτ(s,µ)SC(cid:101)(cid:18)Q6C − Qs,µ(cid:19). (4)
(µS >∼ ΛQCD). Renormalization evolution between the 0
three scales µH, µJ, and µS sums up large logarithms Here Jτ is the thrust jet function, obtained by the con-
to all orders in perturbation theory. It turns out that volutionofthetwohemispherejetfunctions,J =J⊗J.
τ
the soft function anomalous dimension for C and τ are J describesthecollinearradiationinthedirectionofthe
τ
identical [12], providing a connection between these two two jets. Its definition and expression up to O(α3)[44–
s
eventshapesateveryorderofperturbationtheory. Only 46] can be found in Refs. [5, 9]. The three-loop non-
corrections related to non-logarithmic terms in their soft logarithmic coefficient of this jet function, j , is not
3
functions, and the associated towers of logarithms, differ known, and we vary it in our scans. The anomalous
between C and τ. dimension of J is known at three loops, and can be ob-
τ
tained from Ref. [47].
The soft function can be further factorized into a par-
tonic soft function Sˆ , calculable in perturbation theory, ThehardfactorH containsshort-distanceQCDeffects
C(cid:101) and is obtained from the Wilson coefficient of the QCD-
and a nonperturbative shape function F , which has to
C SCET matching of the vector and axial-vector currents.
be obtained from fits to data. In the strict MS scheme
The hard function is the same for all event shapes for
this factorization was achieved in Refs. [36, 37]. (Ana-
massless quarks, and its expression up to O(α3) [48–54],
lyticpowercorrectionsfortheC-parameterdistributions s
can be found in Ref. [9]. The full anomalous dimension
havealsobeenstudiedinotherschemesandframeworks,
of H is known at three loops, O(α3) [49, 51, 55].
see e.g. Refs. [38–40].) s
The soft function S describes wide-angle soft radia-
C(cid:101)
The treatment of hadronic power corrections greatly tion between the two jets. It is defined as
simplifies in the tail of the distribution, defined by
(cid:18) (cid:19)
QC (cid:29)3πΛQCD, where the shape function can be ex- S ((cid:96),µ)= 1 (cid:104)0|trYTY δ (cid:96)− QC(cid:98) Y†Y∗ |0(cid:105), (5)
panded in an OPE. Here the leading power correction is C(cid:101) N n¯ n 6 n n¯
C
parametrizedbyΩC,thefirstmomentoftheshapefunc-
1
tion. The main effect of this leading power correction where C(cid:98) is an operator whose eigenvalues on physical
is a shift of the cross section, dσˆ(C)→dσˆ(C − ΩC1/Q). states correspond to the value of C-parameter for that
Interestingly, this matrix element is related to the corre-
state: C(cid:98)|X(cid:105) = C(X)|X(cid:105). Since the hard and jet func-
sponding one in thrust by
tions are the same as for thrust, the anomalous dimen-
sion of the soft function has to coincide with the anoma-
lous dimension of the thrust soft function. Hence one
Ωτ ΩC only needs to determine the non-logarithmic terms of
1 = 1 ≡Ω . (3)
2 3π 1 the C-parameter soft function. In Ref. [12] we computed
it analytically at one loop, sC(cid:101) = −π2C /6, and used
1 F
EVENT2 to numerically determine the two-loop non-
This relation was first derived using dispersive models
logarithmic coefficient sC(cid:101), with the result
with a single soft-gluon approximation in Ref. [40]. The 2
equalitycanactuallybederivedtoallordersinQCDjust
sC(cid:101) =−43.2 ± 1.0. (6)
using quantum field theory [41], but ignoring hadron- 2
mass effects [42]. These hadron-mass effects can also
The three-loop non-logarithmic coefficient of the
be formulated purely with quantum field theory oper-
C-parameter soft function, sC(cid:101), is currently not known,
ators [43]. Although they may in general give large cor- 3
andweestimateitwithaPad´eapproximation,assigning
rections, hadron-mass effects turn out to violate Eq. (3)
a very conservative uncertainty. We vary this constant
at only the 2% level, which is well below the 15% level
determinationofΩ1thatwewillachievehere. Whenpre- inourscananalysis. Theprecisedefinitionsofj3 andsC2(cid:101)
senting the results of our fits, we parametrize the power as well as sC(cid:101) can be found in Eqs. (A10) and (A12) of
3
correction using Ω defined in Eq. (3) to ease compari- Ref. [12].
1
son with our previous analysis which determined this Ω In Eq. (4) the hard, jet and soft functions are evalu-
1
based on thrust [9, 10]. ated at a common scale µ. If fixed-order expressions are
4
usedforthesefunctions,thenthereisnoscalechoicethat Oneimportantpropertyofthisshapefunctionisthatits
simultaneously minimizes the logarithms for these three firstmomentencodestheleadingpowercorrectiontoour
functions. Onecaninsteadrenormalization-groupevolve cross section. In the MS scheme this moment is given by
(cid:112)
from µ to the respective scales µ ∼ Q, µ ∼ Q C/6
H J (cid:90)
and µ ∼QC/6 at which the logs in each of H, J , and C
S τ Ω ≡ dkkF (k). (11)
S are minimized, and only use fixed-order expressions 1 C
C(cid:101)
forthesefunctionsatthesescales. Inthisway, largelogs
Up to the normalization factors shown in Eq. (3) we
ofratiosofthescalesaresummedupintherenormaliza-
tion group evolution kernels U , Uτ, and Uτ: expect approximate universality between the Ω1 for
H J S C-parameter and thrust. For the calculation of the cross
1 dσˆ Q section,theshapefunctionisexpandedinacompleteba-
s = H(Q,µ )U (Q,µ ,µ)× (7)
σ dC 6 H H H sisoffunctionsobtainedbyanappropriateinfinite-range
0
(cid:90) mapping of the Legendre polynomials [37], with the co-
dsds(cid:48)Jτ(s,µJ)UJτ(s−s(cid:48),µ,µJ) efficients chosen to maintain the first moment. For fur-
ther details on the implementation of the shape function
(cid:90) (cid:18)QC s (cid:19)
dkUSτ(k,µ,µS)SC(cid:101) 6 − Q −k,µS . rfoernoCrm-paalroanmbeyteurssineegRtheef.R[1g2a]p. sWcheemreem[6o0v,e6a1n],Ow(hΛicQhCiDn)-
troduces a subtraction scale R into our formula, as well
The evolution kernels U , Uτ and Uτ resum the as the gap parameter ∆¯ and the perturbative scheme-
H J S
large logarithms, ln(C/6), and explicit expressions can change gap parameter δ(R,µS). Here, δ(R,µS) is given
be found in Ref. [12]. The only unknown piece in our byaperturbativeseriesinαs(µS)whosemassdimension
resummation of logarithms at N3LL order is the small is set by an overall factor of R, and which also contains
contribution from the four-loop cusp anomalous dimen- ln(µS/R) factors. The convolution with the shape func-
sion, Γcusp, which we estimate using a Pad´e approxima- tion now becomes,
3
tion and conservatively vary in our analysis.
While Eq. (7) gives the part of the cross section that dσ =(cid:90) dpe−3πδ(RQ,µs)∂∂C dσˆ(cid:16)C− p(cid:17) (12)
is singular and non-integrable as C →0, we also need to dC dC Q
include the integrable or nonsingular contribution. This ×F (cid:0)p−3π∆¯(R,µ )(cid:1).
C S
can be written as
The final component of our cross section is properly
1 dσˆ α (µ )
ns = s ns f (C) (8) accounting for hadron-mass effects following Ref. [43].
σ0 dC 2π 1 Hadron-mass effects induce an additional series of
(cid:18)α (µ )(cid:19)2(cid:20) (cid:16)µ (cid:17) (cid:21) large perturbative logarithms which start at NLL,
+ s2πns f2(C)+β0ln Qns f1(C) αsklnk(µS/ΛQCD), and also break the exact universality
between ΩC and Ωτ given in Eq. (3). These effects are
(cid:18)α (µ )(cid:19)3(cid:26) (cid:16)µ (cid:17) 1 1
+ s ns f (C)+2β ln ns f (C) accountedfor byincludingdependenceonthetransverse
2π 3 0 Q 2 velocity, r ≡ √ p⊥ , in the nonperturbative matrix
(cid:20)β (cid:16)µ (cid:17) (cid:16)µ (cid:17)(cid:21) (cid:27) p2⊥+m2H
+ 21 ln Qns +β02ln2 Qns f1(C) eplaermtiecnutlsar(,hienret,hmeHRggaivpesscthheemneonth-zeerfiorshtamdroomnemntasosf).thIne
+ O(α4). shape function is actually given by
s
(cid:90)
The functions f1(C), f2(C), and f3(C) were determined dkkF (cid:0)k−3π∆¯(R,µ )(cid:1) = ΩC(R,µ ) (13)
in Ref. [12] using the fixed-order results at O(α1,2,3) [3, C S 1 S
s
7, 32–35, 56–59]. The nonsingular cross section dσˆ /dC (cid:90) 1
ns
is independent of the renormalization scale µ order by =3π drgC(r)Ω1(R,µS,r).
order, and therefore we evaluate these pieces at the non- 0
singularscaleµ ,andvarythisscaletoestimatehigher-
ns In the MS scheme the definition accounting for hadron-
order perturbative nonsingular corrections. The scale mass effects is the same as Eq. (13), but one sets ∆¯ =0
variation of µ will be discussed further in Section IIB.
ns andremovesRasanargumentfortheseparameters. Ac-
The full partonic cross section is then given by
counting for both the the MS running due to hadron
massesandtheR-evolutionrunningintheRgapscheme,
1 dσˆ 1 dσˆ 1 dσˆ
= s + ns . (9) the evolution of the integrand on the right-hand side of
σ dC σ dC σ dC
0 0 0 Eq. (13) is given by,
Nonperturbative effects are included by convolving
Eq. (9) with a shape function: g (r)Ω (R,µ,r)=g (r)(cid:20) αs(µ) (cid:21)γˆ1(r)Ω (R ,µ ,r)
C 1 C α (µ ) 1 ∆ ∆
1 dσ (cid:90) 1 dσˆ(cid:16) p(cid:17) s ∆
σ dC = dpσ dC C− Q FC(p). (10) +∆diff(R∆,R,µ∆,µ,r), (14)
0 0
5
where R and µ give the initial scales where the func- C-independent formula
∆ ∆
tionΩ (R ,µ ,r)isdefined. Theperturbativeevolution
1 ∆ ∆
kernel ∆diff gives the R and µ running for each value of µH = eHQ, (16)
r. The function g encodes the event-shape dependence
C
where e is a parameter that we vary from 0.5 to 2 in
of the hadron-mass effects and γˆ gives the solution to H
1
order to account for theory uncertainties.
the one-loop RGE for Ω with hadron masses derived
1
The soft scale has different functional dependence in
in Ref. [43]. Since the two- and three-loop r-dependent
the three regions of Eq. (15), and hence depends on the
anomalous dimensions are unknown, we do not treat the
following parameters:
logs generated by hadron-mass effects to the same level
of precision. When hadron-mass effects are accounted
µ =µ (C,µ ,r ,µ ,t ,t ,t ,t ). (17)
forwealwayssumtheassociatedlogarithmsatNLL.An S S 0 s H 0 1 2 s
analogous formula to Eq. (14) also holds for the thrust
Here,µ controlstheinterceptofthesoftscaleatC =0,
parameter Ωτ. 0
1 t is near the boundary of the purely nonperturbative
0
Combining all of these elements gives the complete
region and t controls the end of this transition, where
1
cross section. Note that we can resum to any order
theresummationregionbegins. Thetransitionfromnon-
up to N3LL(cid:48) and can choose to include or leave out the
perturbative to perturbative is Q dependent, so we use
shapefunction,renormalonsubtractionandhadron-mass
the Q-independent parameters n ≡ t Q/(1GeV) and
0 0
effects. This flexibility allows us to see how the analysis
n ≡ t Q/(1GeV). In the resummation region the pa-
1 1
changes when we take into account each of these addi-
rameter r determines the slope of the soft scale rela-
s
tional physical considerations and enables us to test how
tive to the canonical resummation region scaling, with
robust the fits are to various changes in the theoretical
µ = r µ C/6. The parameter t controls where the
S s H 2
treatment.
transition occurs between the resummation and fixed-
order regions and t sets the value of C where the renor-
s
malization scales all become equal as required in the
B. Profile Functions fixed-order region. For the jet scale we have the depen-
dence
Inordertosmoothlytransitionbetweenthenonpertur-
bative, resummation, and fixed-order regions we make µJ =µJ(µH,µS(C),eJ), (18)
use of profile functions µ (C) for the renormalization
i
wheree isaparameterthatisvariedinourtheoryscans
scalesµ ,µ (C),µ (C),R(C),andµ (C). Inthethree J
H J S ns
toslightlymodifythenaturalrelationbetweenthescales.
C regions, therequirementsonthescaleswhichproperly
Theexactfunctionalformforµ andµ inEqs.(17)and
deal with large logarithms, nonperturbative effects, and S J
(18) is given in Appendix A.
the cancellations between singular and nonsingular con-
To avoid large logarithms in the soft function subtrac-
tributions in the fixed-order region are
tionsδ,thescaleR(C)ischosentobethesameasµ (C)
S
1) nonperturbative: C <∼3πΛQCD in the resummation and fixed-order regions. In the non-
perturbative region we need R(C) < µ (C) to obtain
(cid:112) S
µH ∼Q, µJ ∼ ΛQCDQ, µS∼R∼ΛQCD, an O(αs) subtraction that stabilizes the soft function in
this region (removing unphysical negative dips that ap-
2) resummation: 3πΛ (cid:28)C <0.75 (15)
QCD pear in the MS scheme). This introduces an additional
(cid:114)
C QC parameter R ≡R(C =0). Therefore we have
µ ∼Q, µ ∼Q , µ ∼R∼ (cid:29)Λ , 0
H J 6 S 6 QCD
R=R(µ (C),R ). (19)
3) fixed-order: C >0.75 S 0
µns =µH =µJ =µS =R∼Q(cid:29)ΛQCD. The exact functional form for R is also given in Ap-
pendix A.
In addition we take the fixed-order nonsingular scale For the nonsingular scale µ , we use the variations
ns
µ ∼ µ in the nonperturbative and resummation re-
ns H

sgtioranisn.ts,Oaunrd pprroovfiildeefuconncttiinounosuµsia(Cnd)ssmatoiosftyh ttrhaensesitcioonns- 21(cid:2)µH(C)+µJ(C)(cid:3), ns = 1
µ (C)= µ , n = 0 . (20)
between these C regions. The resummed perturbative ns H s
cross section is independent of O(1) variations in all 1(cid:2)3µ (C)−µ (C)(cid:3), n =−1
2 H J s
renormalization scales order by order in the logarith-
micresummation. Thereforethedependenceonparame- Here the three choices vary µ in a manner that allows
ns
tersappearingintheprofilefunctionsgetssystematically it to have some independence from µ in the resumma-
H
smalleraswegotohigherorders,andtheirvariationpro- tion and nonperturbative regions, while still being equal
vides us with a method of assessing perturbative uncer- µ in the fixed-order region (where µ = µ ). These
H J H
tainties. variations of µ probe the higher-order fixed-order un-
ns
For the hard renormalization scale we use the certaintyinthenonsingularcrosssectioncontribution. In
6
parameter default value range of values parameter default value range of values
µ 1.1GeV - µ 1.1GeV -
0 0
R 0.7GeV - R 0.7GeV -
0 0
n 12 10 to 16 n 2 1.5 to 2.5
0 0
n 25 22 to 28 n 10 8.5 to 11.5
1 1
t 0.67 0.64 to 0.7 t 0.25 0.225 to 0.275
2 2
t 0.83 0.8 to 0.86 t 0.4 0.375 to 0.425
s s
r 2 1.78 to 2.26 r 2 1.77 to 2.26
s s
e 0 −0.5 to 0.5 e 0 −1.5 to 1.5
J J
e 1 0.5 to 2.0 e 1 0.5 to 2.0
H H
n 0 −1, 0, 1 n 0 −1, 0, 1
s s
Γcusp 1553.06 −1553.06 to +4659.18 j 0 −3000 to +3000
3 3
sC(cid:101) −43.2 −44.2 to −42.2 sτ 0 −500 to +500
2 3
j 0 −3000 to +3000 (cid:15) 0 −1, 0, 1
3 2
sC(cid:101) 0 −500 to +500 (cid:15) 0 −1, 0, 1
3 3
(cid:15)low 0 −1, 0, 1
2 TABLE II. Thrust theory parameters relevant for estimating
(cid:15)h2igh 0 −1, 0, 1 thetheoryuncertainty,theirdefaultvaluesandrangeofvalues
(cid:15)low 0 −1, 0, 1 used for the theory scan during the fit procedure. The last
3
two parameters control the statistical errors induced by fit
(cid:15)high 0 −1, 0, 1
3 functions used in the non-singular terms at O(α2) ((cid:15) ) and
s 2
O(α3) ((cid:15) ); see Sec. E of Ref. [9].
TABLE I. C-parameter theory parameters relevant for esti- s 3
matingthetheoryuncertainty,theirdefaultvaluesandrange
of values used for the theory scan during the fit procedure.
Thelastfourparameterscontrolthestatisticalerrorsinduced vertical arrow on the hard scale indicates the overall
byfitfunctionsusedinthenon-singulartermsatO(αs2)((cid:15)l2ow up/down variation, which causes a variation to all the
and(cid:15)h2igh)andO(αs3)((cid:15)l3owand(cid:15)h3igh)intheregionbelow((cid:15)l2ow scales. Also shown (as gray dashed lines) are plots of
and(cid:15)l3ow)andabove((cid:15)h2igh and(cid:15)h3igh)theshoulder;seeSec. V the canonical soft scale QC/6 and canonical jet scale
of Ref. [12]. (cid:112)
Q C/6. In the resummation region, these correspond
fairly well with the profile functions, indicating that in
100 Μ this region our analysis will avoid large logarithms. As
H
Μ
i discussed in detail in Ref. [12], to improve the conver-
80 Q=91.2GeV genceoftheC-parametercrosssectionswetaker =2as
s
ourdefaultslopeparameter,explainingwhyoursoftand
60 jet scales are larger than the canonical values in Fig. 1
(by a factor that does not induce further large logs). In
40 ΜJ ΜJ=Q C6 the analysis of Sec. V, we will see how varying each of
these profile parameters affects our final fit results.
C
20 ΜS ΜS=Q6 For the numerical analyses carried out in this work
we have created within our collaboration two completely
0
independent codes. One code within Mathematica [62]
0.0 0.2 0.4 0.6 0.8 1.0 implements the theoretical expressions exactly as given
C
inRef.[12],andanothercodeisbasedontheoreticalfor-
mulaeinFourierspaceandrealizedasafastFortran[63]
FIG.1. Bandsfortheprofilefunctionsfortherenormalization
code suitable for parallelized runs on computer clusters.
scalesµ ,µ (C),µ (C)whenvaryingtheprofileparameters.
H J S
These two codes agree for the C-parameter distribution
at the level of 10−6.
We will also repeat the thrust fits of Ref. [9], imple-
thefixed-orderregionthevariationofµ aloneprecisely menting the same type of profile functions used here.
H
reproduces the standard fixed-order scale variation. These profiles have several advantages over those in
Thedetailsofthevariationsoftheprofilefunctionpa- Ref. [9], including a free variable for the slope, a flat
rameters used to assess uncertainties are given in Tab. I. nonperturbative region, and parameters whose impact
The plot in Fig. 1 shows how the scales vary with the is much more confined to one of the three regions in
changes to our C-parameter profile parameters. The Eq. (15). For the thrust profiles we redefine r →6r ,
s s
7
which eliminates the factors of 6 in Eqs. (A1) and (A4). 3ΠgCHrLW1HRD,ΜD,rL
This way, the canonical choice of slope is r =1 for both W1CHRD,ΜDL
s 4
C-parameterandthrust. Weuser =2asourdefaultfor
s Θ=Π
thrustaswell, againtoimprovetheperturbativeconver- Θ=3Π 4
gence, as discussed in Ref. [12]. The profile parameters 4
3
forthrustandtheirvariationsaresummarizedinTab.II. Θ=0
Thesechoicescreateprofilesthatareverysimilartothose
used in Ref. [9]. The new fit results for thrust are fully 2
compatible with those of Ref. [9] in the resummation re-
gionusedfortheα fits. Additionally, theygiveabetter Θ=Π
s 2
description in the nonperturbative region which is out- 1
side of our fit range.
0
0.0 0.2 0.4 0.6 0.8 1.0
r
C. Hadron-mass effects
FIG. 2. Plots of the r dependence of g (r)Ω (R ,µ ,r) for
In Ref. [43] it was shown that hadron-mass effects, C 1 ∆ ∆
different values of θ(R ,µ ). We normalize to ΩC(R ,µ ),
apart from breaking the universality properties of the ∆ ∆ 1 ∆ ∆
since it is simply an overall multiplicative factor.
leading power correction for various event shapes, also
induce a nontrivial running. Since these are single loga-
rithms, they start at NLL order. In Ref. [43] the corre-
sponding leading anomalous dimension was determined, leading order by
which yields the NLL resummation of larger logs be-
(cid:20)α (µ )(cid:21)γˆ1(r)
tween the scales µS and ΛQCD for a large set of event Ω (µ ,r)=Ω (µ ,r) s S , (22)
shapes. The pieces necessary for a higher-order resum- 1 S 1 0 αs(µ0)
mation have not yet been computed. One might be wor-
ried that accounting for only the NLL running for Ω1 in withγˆ1(r)= 2βC0A ln(1−r2). Thecorrespondingevolution
an expression as accurate as N3LL in cross-section logs formula for the Rgap scheme is considerably more com-
plex, as shown by the form displayed in Eq. (14) above.
could be inadequate, or that it could leave significant
Ensuring that the renormalon is not reintroduced by the
perturbative uncertainties. However, one should recall
renormalization group evolution requires an additional
thatthehadronicparameterΩ itselfisacorrection,and
1
evolution in the scale R, so ∆diff(R ,R,µ ,µ,r) con-
hence it is valid to account for the related resummation ∆ ∆
tains evolution in both the µ and R scales. Also here we
with less precision. In this section we show that the Ω
1
have two reference scales µ and R to specify the ini-
evolution at NLL order suffices for the precision of our ∆ ∆
N3LL(cid:48)+O(α3)analysis. Indeed, itturnsoutthattheef- tial parameter Ω1(R∆,µ∆,r). The full formula for ∆diff
s
is given in Eq. (67) of Ref. [12].
fectofthehadronmassrunningonthefitoutcomeisvery
Note that the renormalization group evolution is a
small as compared to other uncertainties, and therefore
function of r and takes place independently for each r,
can be safely neglected.
buttherequiredresultforC-parameterorthrustrequires
For our C-parameter analysis the implementation of
an integral over r. Due to this integration the functional
hadronmassrunningeffectshasbeenexplainedatlength
form that we assume for the initial condition Ω (r,µ )
in Ref. [12], and we only summarize here the most rele- 1 0
or Ω (R ,µ ,r) gets entangled with the perturbative
vant aspects needed to understand the fit results. In the 1 ∆ ∆
resummation.
MS scheme the leading power correction can be written
With current constraints on the r dependence, and
as an integral of a universal hadron function, Ω (µ,r),
1
with the lack of even more precise experimental data to
common to all event shapes
probethisissue,amodel-independentformulation(likea
(cid:90) 1 completefunctionalbasisfortherdependenceattheref-
e
Ω (µ) = c drg (r)Ω (µ,r), (21) erence scales, R and µ ) is not feasible. To implement
1 e e 1 ∆ ∆
0 this running we have therefore tested several models for
ther dependenceinRef.[12], andfoundthatgenerically
where r is the transverse velocity, e denotes a specific
the experimental data is sensitive to the normalization
event shape, c is a calculable normalization factor, and
e
which specifies Ω , but not to the detailed form used for
g (r) is an event-shape-dependent function encoding the 1
e
the r dependence as long as it satisfies several reason-
hadron-masseffects. Thefunctionsg (r)areknownana-
e
able constraints. Therefore for the fits performed here
lytically, and specific examples can be found in Ref. [43].
we simply adopt the default form from Ref. [12],
For C-parameter c = 3π, while for thrust c = 2. For
C τ
the simple case of the MS scheme the running between Ω (R ,µ ,r)=[a(R ,µ )f (r)+b(R ,µ )f (r)]2,
1 ∆ ∆ ∆ ∆ a ∆ ∆ b
the initial reference scale µ where the universal hadron
function is specified, and t0he soft scale µS, is given at fa(r)=3.510e−1−r2r2, (23)
8
fb(r)=13.585e−12−rr22 −21.687 e−14−rr22. (cid:52)=(cid:22)(cid:24)(cid:42)(cid:72)(cid:57)
DW
(cid:19)(cid:17)(cid:19)(cid:22) (cid:20)
ThismodelensuresthatΩ (R ,µ ,r)isalwayspositive DΑ
1 ∆ ∆ (cid:19)(cid:17)(cid:19)(cid:21) (cid:86)
definite and smoothly goes to zero at the r =1 endpoint
(where the ratio of the hadron mass to p goes to zero). (cid:19)(cid:17)(cid:19)(cid:20) DW
The functions f form an orthonormal bTasis upon inte- DC(cid:71)Σ (cid:21)
a,b Σ (cid:19)
gration with g (r), which yields the following relation: (cid:71)(cid:38)
C -
(cid:19)(cid:17)(cid:19)(cid:20)
ΩC1(R∆,µ∆)=3π[a(R∆,µ∆)2+b(R∆,µ∆)2], (24) -(cid:19)(cid:17)(cid:19)(cid:21)
-
(cid:19)(cid:17)(cid:19)(cid:22)
which determines the normalization. We also define
(cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26)
(cid:18)b(R ,µ )(cid:19) (cid:38)
θ(R∆,µ∆)≡arctan a(R∆,µ∆) , (25) (a)
∆ ∆
=
which was chosen to have an effect orthogonal to the (cid:52) (cid:28)(cid:20)(cid:17)(cid:21)(cid:42)(cid:72)(cid:57)
(cid:19)(cid:17)(cid:19)(cid:20)(cid:19)(cid:19)
more relevant parameter ΩC(R ,µ ). By examining DW
1 ∆ ∆ (cid:19)(cid:17)(cid:19)(cid:19)(cid:26)(cid:24) DΑ (cid:20)
our ability to simultaneously measure ΩC(R ,µ ) and (cid:86)
1 ∆ ∆
(cid:19)(cid:17)(cid:19)(cid:19)(cid:24)(cid:19)
θ(R ,µ ) we have a means to test for the impact that
∆ ∆
the initial r dependence has on our fits. As we can see C Σ (cid:19)(cid:17)(cid:19)(cid:19)(cid:21)(cid:24) DW
D (cid:71) (cid:21)
in Fig. 2, our model captures different behavior for the r Σ (cid:19)
(cid:71)(cid:38)
dependenceofΩ (R ,µ ,r)bychoosingdifferentvalues -
1 ∆ ∆ (cid:19)(cid:17)(cid:19)(cid:19)(cid:21)(cid:24)
of θ(R∆,µ∆). Over the interval r ∈ [0,1], all the curves -(cid:19)(cid:17)(cid:19)(cid:19)(cid:24)(cid:19)
in this figure are normalized so that they integrate to 1.
-
(cid:19)(cid:17)(cid:19)(cid:19)(cid:26)(cid:24)
(cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26)
III. EXPERIMENTAL DATA (cid:38)
(b)
Data on the C-parameter cross section are given by =
(cid:52) (cid:21)(cid:19)(cid:25)(cid:42)(cid:72)(cid:57)
several experiments for a range of Q from 35 to 207
DΑ
GeV. We use data from ALEPH2 with Q = {91.2, 133, (cid:19)(cid:17)(cid:19)(cid:19)(cid:23) (cid:86)
161, 172, 183, 189, 200, 206} GeV [64], DELPHI with DW
(cid:20)
Q = {45, 66, 76, 89.5, 91.2, 93, 133, 161, 172, 183, (cid:19)(cid:17)(cid:19)(cid:19)(cid:21)
189, 192, 196, 200, 202, 205, 207} GeV [65–68], JADE C Σ DW
D (cid:71) (cid:21)
with Q = {35, 44} GeV [69], L3 with Q = {91.2, 130.1, Σ (cid:19)
(cid:71)(cid:38)
136.1,161.3,172.3,182.8,188.6,194.4,200.2,206.2}GeV
-
[70, 71], OPAL with Q = {91, 133, 177, 197} GeV [72], (cid:19)(cid:17)(cid:19)(cid:19)(cid:21)
and SLD with Q = 91.2 GeV [73]. As each of these
-
datasets is given in binned form, our cross section in (cid:19)(cid:17)(cid:19)(cid:19)(cid:23)
Eq. (12) is integrated over each bin before being com-
(cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26)
pared to the data. The default range on C used in fit- (cid:38)
ting the data is 25 GeV/Q ≤ C ≤ 0.7. A lower limit (c)
of 25 GeV/Q eliminates the peak region where higher
nonperturbative moments ΩC become important. The FIG. 3. Difference between the default cross section and the
n>1
upper limit is chosen to be 0.7 in order to avoid the far- cross section varying only one parameter. We vary αs(mZ)
tail region as well as the Sudakov shoulder at C = 0.75. by ±0.001 (solid red), 2Ω1 by ±0.1 (dashed blue) and ΩC2
by ±0.5 (dotted green). The three plots correspond to three
Any bin that contains one of the end points of our range
different center of mass energies: (a) Q = 35GeV, (b) Q =
(C = 25GeV/Q or 0.7) is included if more than half of
91.2GeV, (c) Q=206GeV.
that bin lies within the range. Using the default range
anddatasetsgivesatotalof404bins. Asafurthertestof
thestabilityofouranalysis,boththisC-parameterrange
and the selection of datasets is varied in the numerical
analysis contained in Sec. V.
In our fitting procedure, we consider both the sta-
2 The ALEPH dataset with Q = 91.2GeV has two systematic
tistical and systematic experimental uncertainties. The
uncertainties for each bin. The second of these uncertainties
is treated as correlated while the first one is treated as an un- statistical uncertainties can be treated as independent
correlated uncertainty and simply added in quadrature to the between bins. The systematic experimental uncertain-
statisticaluncertainty. tiescomefromvarioussourcesandfulldocumentationof
9
their correlations are not available, so dealing with them fully contains all 500 of the central-fit values by first cre-
inourχ2analysisismorecomplicated,andwehavetouse atingtheconvexenvelopethatcontainsallofthesepoints
a correlation model. For this purpose we follow the LEP within it. Then, we find the equation for the ellipse that
QCDworkinggroup[64,72]andusetheminimaloverlap best fits the points on the envelope, with the additional
model. Within one C-parameter dataset, which consists restrictions that all values lie within the ellipse and its
of various C-parameter bins at one Q value for one ex- center is the average of the maximum and minimum val-
periment, we take for the bin i, bin j off-diagonal entry ues in each direction. This ellipse determines the per-
oftheexperimentalcovariancematrix[min(∆sys,∆sys)]2. turbative theoretical uncertainty, which turns out to be
i j
Here ∆sys are the quoted systematic uncertainties of the the dominant uncertainty in our fit results. In our final
i,j
bins i and j. Within each dataset, this model implies results the perturbative and experimental uncertainties
a positive correlation of systematic uncertainties. In ad- are added in quadrature. This procedure is similar to
dition to this default model choice, we also do the fits that discussed in the Appendix of Ref. [10].
assuming uncorrelated systematic uncertainties, in order
to test whether the minimal overlap model introduces
anybias. SeeSec.VBformoredetailsonthecorrelation V. RESULTS
matrix.
In this section we discuss the results from our global
analysis. We split the presentation into several subsec-
IV. FIT PROCEDURE tions. In Sec. VA we discuss the impact that resumma-
tionandtheinclusionofpowercorrectionshaveonthefit
Inordertoaccuratelydeterminebothα (m )andthe results. In Sec. VB we present the analysis which yields
s Z
leading power correction in the same fit, it is important theperturbativeuncertaintyindetail,cross-checkingour
to perform a global analysis, that is, simultaneously fit- methodbyanalyzingtheorder-by-orderconvergence. We
tingC-spectraforawiderangeofcenter-of-massenergies also analyze the impact of removing the renormalon. In
Q. For each Q, effects on the cross sections induced by Sec. VC we discuss the experimental uncertainties ob-
changesinα (m )canbepartlycompensatedbychanges tained from the fit. Section VD discusses the impact
s Z
in Ω , resulting in a fairly strong degeneracy. This is re- that varying the theory parameters one by one has on
1
solved by the global fit, just as in the thrust analysis of the best-fit points, allowing us to determine which pa-
Ref. [9]. Fig. 3 shows the difference between the theoret- rameters dominate the theoretical uncertainty. The im-
icalpredictionforthecrosssectionatthreedifferentval- pact of hadron-mass resummation is discussed in detail
ues of Q, when α (m ) or Ω are varied by ±0.001 and in Sec. VE. We examine the effects of changing the de-
s Z 1
±0.05 GeV, respectively. It is clear that the potential faultdatasetinSec.VF.Thefinalfitresultsarecollected
degeneracyintheseparametersisbrokenbyhavingdata in Sec. VG. When indicating the perturbative precision,
at multiple Q values. In Fig. 3 we also vary the higher- and whether or not the power correction Ω1 is included
orderpowercorrectionparameterΩC,whichclearlyhasa andatwhatlevelofprecision, weusethefollowingnota-
2
much smaller effect than the dominant power correction tion:
parameter Ω .
1
O(αk) fixed order up to O(αk)
To carry out a fit to the experimental data we fix s s
the profile and theory parameters to the values shown NkLL(cid:48)+O(αk) perturbative resummation
s
in Tab. I. The default values are used for our primary
NkLL(cid:48)+O(αk)+Ω MS scheme for Ω
theory cross section. We integrate the resulting theo- s 1 1
retical distribution over the same C-parameter bins as NkLL(cid:48)+O(αk)+Ω (R,µ) Rgap scheme for Ω
s 1 1
those available experimentally, and construct a χ2 func-
NkLL(cid:48)+O(αk)+Ω (R,µ,r) Rgap scheme with
tionwiththeuncorrelatedstatisticalexperimentaluncer- s 1
tainties and correlated systematic uncertainties. This χ2 hadron masses for Ω1.
is a function of α (m ) and Ω , and is very accurately
s Z 1
describedbyaquadraticnearitsglobalminimum,which
thereforedeterminesthecentralvaluesandexperimental
uncertainties. The value of Ω and its associated uncer- A. Impact of Resummation and Power Corrections
1
taintiesencodethedominanthadronizationeffectaswell
asthedominantresidualuncertaintyfromhadronization. In Fig. 4 we show α (m ) extracted from fits to the
s Z
To obtain the perturbative theoretical uncertainty we tail of the C-parameter distribution including sequential
considertherangeofvaluesshownforthetheoryparame- improvements to the treatment of perturbative and non-
tersinTab.I.Treatingeachoftheseasaflatdistribution, perturbative components of our code, using the highest
werandomlygeneratevaluesforeachoftheseparameters perturbative accuracy at each stage. The sequence from
and then repeat the fit described above with the new χ2 lefttorightshowsthefitresultsusing: O(α3)fixed-order
s
function. This random sampling and fit is then repeated results only, adding N3LL resummation, adding the Ω
1
500 times. We then construct the minimum ellipse that power correction, adding renormalon subtractions and
10
Α Hm L from global C-parameter tail fits
s Z
0.135 Belowerrorbars&± ® perturbativeerror
ΑsHmZL OHΑs3Lfixed-order Allerrors:ΑsHmZL=0.1123±0.0015
0.1317±0.0052
0.130
0.125
+N3LL'summation +PowerCorrection
0.120 0.1219±0.0028 0.1117±0.0016
+R-scheme +hadronmasseffects
0.1123±0.0014 0.1119±0.0013
0.115
0.110
FIG.4. Theevolutionofthevalueofα (m )addingcomponentsofthecalculation. Anadditional∼8%uncertaintyfromnot
s Z
including power corrections is not included in the two left points.
using the Rgap power correction parameter Ω (R ,µ ), tractions, and the inclusion of the log-resummation ef-
1 ∆ ∆
and adding hadron-mass effects. These same results to- fectsassociatedtothehadron-masseffects. Bothofthese
gether with the corresponding χ2/dof are also collected corrections have a fairly small impact on the determi-
in Tab. III. The fit with only fixed-order O(α3) results nation of α (m ), shifting the central value by +0.5%
s s Z
has a relatively large χ2/dof and also its central value and −0.3% respectively. Since adding the −0.3% shift
has the largest value of α (m ). Including the resum- fromthehadronmasscorrectionsinquadraturewiththe
s Z
mation of large logarithms decreases the central α (m ) (cid:39) 1.2% perturbative uncertainty does not change the
s Z
by 8% and also decreases the perturbative uncertainty overall uncertainty we will use the R-scheme determi-
by ∼50%. Due to this smaller perturbative uncertainty nation for our main result. This avoids the need to fully
it becomes clear that the theoretical cross section has discuss the extra fit parameter θ(R ,µ ) that appears
∆ ∆
a different slope than the data, which can be seen, for when hadron masses are included. Further discussion
example, at Q = m for 0.27 < C < 0.35. This leads of the experimental uncertainties and the perturbative
Z
to the increase in the χ2/dof for the “N3LL(cid:48) no power uncertainty from the random scan are given below in
corr.” fit, and makes it quite obvious that power correc- Secs. VB and VD, and a more detailed discussion of
tions are needed. When the power correction parameter the impact of hadron-mass resummation is given below
Ω is included in the fit, shown by the third entry in in Sec. VE.
1
Tab. III and the result just to the right of the vertical ThevaluesofΩ obtainedfromthefitsdiscussedabove
1
dashed line in Fig. 4, the χ2/dof becomes 1.004 and this can be directly compared to the Ω power correction ob-
1
issue is resolved. Furthermore, a reduction by ∼ 50% tained from the thrust distribution. Values for Ω from
1
is achieved for the perturbative uncertainty in α (m ). theC-parameterfitsaregivenbelowinSecs.VBandVD
s Z
This reduction makes sense since some of the perturba- andthecomparisonwiththrustisconsideredinSec.VII.
tive uncertainty of the cross section is now absorbed in
Ω , and a much better fit is achieved for any of the vari-
1
ations associated to estimating higher-order corrections. B. Perturbative Uncertainty from the Scan
TheadditionofΩ alsocausedthefitvalueofα (m )to
1 s Z
dropbyanother8%,consistentwithourexpectationsfor
Toexaminetherobustnessofourmethodofdetermin-
theimpactofpowercorrectionsandtheestimatemadein
ingtheperturbativeuncertaintybytherandomscan, we
Ref. [12]. Note that the error bars of the first two purely
considertheconvergenceandoverlapoftheresultsatdif-
perturbative determinations, shown at the left-hand side
ferent perturbative orders. Figure 5 shows the spread of
of the vertical thick dashed line in Fig. 4 and in the last
best-fitvaluesatNLL(cid:48),N2LL(cid:48)andN3LL(cid:48). Theupperleft
two entries in Tab. III, do not include the ∼ 8% uncer-
panel, Fig. 5(a), shows results from fits performed in the
tainties associated with the lack of power corrections.
Rgap scheme, which implements a renormalon subtrac-
The remaining corrections we consider are the use of tion for Ω , and the upper right-panel, Fig. 5(b), shows
1
the R-scheme for Ω which includes the renormalon sub- results in the MS scheme without renormalon subtrac-
1