#### Precise determination of \(\alpha _{S}(M_Z)\) from a global fit of energy–energy correlation to NNLO+NNLL predictions

Eur. Phys. J. C
Precise determination of αS( M Z ) from a global fit of energy-energy correlation to NNLO+NNLL predictions
Adam Kardos 2
Stefan Kluth 1
Gábor Somogyi 0
Zoltán Tulipánt 2
Andrii Verbytskyi 1
0 MTA-DE Particle Physics Research Group, University of Debrecen , PO Box 105, Debrecen 4010 , Hungary
1 Max-Planck-Institut für Physik , Föringer ring 6, 80805 Munich , Germany
2 Institute of Physics, University of Debrecen , PO Box 105, Debrecen 4010 , Hungary
We present a comparison of the computation of energy-energy correlation in e+e− collisions in the backto-back region at next-to-next-to-leading logarithmic accuracy matched with the next-to-next-to-leading order perturbative prediction to LEP, PEP, PETRA, SLC and TRISTAN data. With these predictions we perform an extraction of the strong coupling constant taking into account non-perturbative effects modelled with Monte Carlo event generators. The final result at NNLO+NNLL precision is αS(MZ ) = 0.11750 ± 0.00018(ex p.) ± 0.00102(hadr.) ± 0.00257(r en.) ± 0.00078(r es.).
1 Introduction
The strong interaction in the Standard Model (SM) is
described by quantum chromodynamics (QCD) [
1–4
]. The
theory successfully models the interactions between quarks
and gluons and is a source of numerous predictions.
Verifying the predictions of QCD is instrumental for searches for
physics beyond the SM at the LHC, since the reliable
prediction of SM processes as sources of backgrounds for searches
is essential.
Precision measurements of event shape distributions in
e+e− annihilation have provided detailed experimental tests
of QCD and remain one of the most precise tools used for
extracting the strong coupling αS from data [
5,6
]. Quantities
related to three-jet events are particularly well suited for this
task.
The state of the art for QCD for event shape observables
currently includes exact fixed-order next-to-next-to-leading
order (NNLO) corrections for the six standard three-jet event
shapes of thrust, heavy jet mass, total and wide jet
broadening, C -parameter and the two-to-three jet transition variable
y23 [
7–9
] as well as jet cone energy fraction [
9
], oblateness
and energy–energy correlation [
10
]. The numerical matrix
element integration codes described in the references allow
the straightforward computation of any suitable, i.e. collinear
and infrared safe event shape or jet observable.
However, fixed-order predictions have a limited
kinematical range of applicability. For small values of an event
shape observable y corresponding to events with
two-jetlike topologies the fixed-order predictions do not converge
well. This is due to terms where each power of the strong
coupling αSn is enhanced by a factor (ln y)n+1 (leading logs),
(ln y)n (next-to-leading logs) etc. For three-jet event shapes
such logarithmically enhanced terms can be resummed at
next-to-next-to-leading logarithmic (NNLL) accuracy [
11–
17
], i.e. up to terms ∼ (ln y)n−1. Resummation in
nextto-next-to-next-to-leading logarithmic (N3LL) accuracy has
been achieved for the C -parameter [
18
] and thrust [
19
]. A
prediction incorporating the complete perturbative
knowledge about the observable can be derived by matching the
fixed-order and resummed calculations.
For the frequently used event shapes of thrust, heavy jet
mass, total and wide jet broadening, C -parameter and y23,
NNLO predictions matched to NLL resummation were
presented in [
20
]. Predictions at NNLO matched to N3LL
resummation are also known for thrust [
12,19
] and the C -parameter
[18].
In this paper we consider the energy–energy
correlation (EEC) in e+e− annihilation and present NNLO
predictions matched to NNLL resummation for the back-to-back
region. EEC was the first event shape for which a complete
NNLL resummation was performed [
11
] while the
fixedorder NNLO corrections to this observable were computed
recently [
10
]. Moreover, EEC is the first event shape
observable for which an analytic fixed-order NLO correction was
computed [
21
].
The agreement between the predictions at NNLO+NNLL
accuracy and the measured data is still not perfect. The
discrepancy can be attributed mainly to non-perturbative
hadronization corrections. We extract these corrections from
data by comparison to state-of-the-art Monte Carlo
predictions and determine the value of the strong coupling by
comparing our results to measurements over a wide range of
centre-of-mass energies. Our analysis allows us to target the
highest precision of αS determination and we present the first
global fit of the strong coupling to EEC at NNLO+NNLL
accuracy. Our analysis also represents the first extraction of
αS based on Monte Carlo hadronization corrections obtained
from NLO Monte Carlo setups at NNLO+NNLL precision.
e+e− → hadrons, σ0. In massless QCD this normalization
cancels all electroweak coupling factors, and the dependence
on the collision energy enters only through αS(Q).
However, experiments measure the distribution normalized to the
total hadronic cross section, so physical predictions must be
normalized to σt. The distribution normalized to the total
hadronic cross section can be obtained from the expansion in
Eq. (2) through multiplying by σ0/σt. For massless quarks,
this ratio is independent of all electroweak couplings and
reads
σσ0t = 1 −
αS(Q)
2π
At +
αS(Q) 2
2π
At2 − Bt + O(αS3),
where
A¯(χ , x R ) = A(χ ),
B¯ (χ , x R ) = B(χ ) +
C¯ (χ , x R ) = C (χ ) +
β0 ln(x 2R ) − At B(χ )
1
2 β0 ln(x 2R ) − At A(χ ),
+
1
4 β1 ln(x 2R ) + 41 β02 ln2(x 2R ) − Atβ0 ln(x 2R )
+ At2 − Bt A(χ ),
while nf denotes the number of light quark flavours.
The renormalization scale dependence of the fixed-order
prediction can be restored using the renormalization group
equation for αS and one finds
1 dΣ (χ , μ)
σt d cos χ
f.o.
=
αS(μ) d A¯(χ , x R )
2π d cos χ
+
+
αS(μ) 2 d B¯ (χ , x R )
2π d cos χ
αS(μ) 3 dC¯ (χ , x R )
2π d cos χ
+ O(αS4),
(3)
with
and
Bt = CF
2 EEC distribution in perturbation theory
EEC is the normalized energy-weighted cross section defined
in terms of the angle between two particles i and j in an event
[
22
]:
1 dΣ (χ ) 1
σt d cos χ ≡ σt
i, j
EQiE2 j dσe+e−→ i j+X δ(cos χ − cos θi j )(,1)
where Ei and E j are the particle energies, Q is the
centre-ofmass energy, θi j = χ is the angle between the two particles
and σt is the total hadronic cross section. The back-to-back
region θi j → 180◦ corresponds to χ → π , while the
normalization ensures that the integral of the EEC distribution
from χ = 0◦ to χ = 180◦ is unity.1
2.1 Fixed-order and resummed calculations
The differential EEC distribution has been computed
numerically at NLO accuracy in perturbation theory some time ago
[
24–34
] and efforts towards obtaining an analytic result at this
order [
35,36
] have culminated in a complete calculation very
recently [21]. The NNLO prediction has also been obtained
in ref. [
10
] using the CoLoRFulNNLO method [
9,37,38
]. At
the default renormalization scale2 of μ = Q the fixed-order
prediction reads
1 dΣ (χ , Q)
σ0 d cos χ
f.o.
=
αS(Q) d A(χ )
2π d cos χ +
αS(Q) 2 d B(χ )
2π d cos χ
αS(Q) 3 dC (χ )
2π d cos χ + O(αS4), (2)
where A, B and C are the perturbative coefficients at LO,
NLO and NNLO, normalized to the LO cross section for
1 Refs. [
11
] and [
23
] use the opposite convention of θi j = 180◦ − χ
such that the back-to-back region corresponds to χ → 0◦. Here we use
θi j = χ throughout which agrees with the experimental convention.
2 We use the MS renormalization scheme throughout the paper.
Q2
q2
S(Q, b) = exp
−
Q2
b02/b2
dq2
q2
.
+ B(αS(q2))
The zeroth order Bessel function J0 in Eq. (4) and b0 = 2e−γE
in Eq. (5) have a kinematic origin. The functions A, B (not
to be confused with the fixed-order expansion coefficients
appearing in Eq. (2)) and H in Eqs. (4) and (5) are free of
logarithmic corrections and can be computed as perturbative
expansions in αS,
(5)
(6)
(7)
(8)
(9)
with x R = μ/Q. Finally, using three-loop running the scale
dependence of the strong coupling is given by
The large logarithmic corrections are exponentiated in the
Sudakov form factor,
αS(μ) = β40πt ⎣⎡ 1 − ββ21t ln t
0
Here, t = ln(μ2/Λ2QCD) and the βi are the MS-scheme
coefficients of the QCD beta function,
β0 =
11CA
3
−
4nf TR
3
,
β1 = 334 C A2 − 3 CATRnf − 4CF TRnf ,
20
The fixed-order perturbative predictions diverge for both
small and large values of χ , due to the presence of large
logarithmic contributions of infrared origin. Concentrating
on the back-to-back region χ → 180◦, these contributions
take the form αSn log2n−1 y, where
y = cos2 χ .
2
As y decreases, the logarithms become large and invalidate
the use of the fixed-order perturbative expansion. In order to
obtain a description of EEC in this limit, the logarithmic
contributions must be resummed to all orders. This resummation
has been computed at NNLL accuracy in Ref. [
11
]3 while in
Ref. [
40
] a factorization theorem for EEC was derived based
on soft-collinear effective theory which will allow to preform
the resummation at N3LL accuracy once the corresponding
NNLO jet function is computed. Since the complete jet
function is currently not available, we use the NNLL results and
formalism of Ref. [
11
] in the following. The resummed
prediction at the default scale of μ = Q can be written as
3 Note that the NNLL A(3) coefficient in Ref. [
11
] is incomplete. The
full coefficient has been derived in Ref. [
39
].
A(αS) =
B(αS) =
∞
n=1
∞
n=1
H (αS) = 1 +
4π
αS n A(n),
αS n B(n),
4π
∞
n=1
4π
αS n H (n).
Explicit expressions for the expansion coefficients (up to
NNLL accuracy) in our normalization conventions can be
found in Ref. [
23
].
It is possible to perform the q2 integration in Eq. (5)
analytically and the Sudakov form factor can be written as
S(Q, b) = exp[Lg1(aSβ0 L) + g2(aSβ0 L)
+ aSg3(aSβ0 L) + . . .],
where aS = αS(Q)/(4π ) and L = ln(Q2b2/b02) corresponds
to ln y at large b (the y 1 limit corresponds to Qb 1
through a Fourier transformation). Writing the Sudakov form
factor this way clearly shows that S(Q, b) depends on its
variables only through the dimensionless combination b Q.
The functions g1, g2 and g3 correspond to the LL, NLL and
NNLL contributions. Their explicit expressions can be found
in Refs. [
11,23
].
So far, we have not considered the dependence of the
resummed prediction on the renormalization scale. Besides
the replacement of αS(Q) by αS(μ) in Eqs. (4) and (9), the
resummation functions gi (λ) also acquire renormalization
scale dependence,
g1(λ, x R ) = g1(λ),
g2(λ, x R ) = g2(λ) + λ2g1(λ) ln(x 2R ),
The differential EEC distribution is easily recovered from
Σ (χ , μ),
1 dΣ (χ , μ)
σt dχ
1 d
= 1 − cos χ dχ
1
σt
Σ (χ , μ) .
The particular linear combination of moments introduced
in Eq. (11) has the property that the divergence of the
differential EEC distribution in the forward region (χ → 0)
is suppressed by the factor of (1 − cos χ ). Hence, in
contrast to EEC itself, the fixed-order cumulative coefficients
of Σ (χ , μ) can be computed reliably. Furthermore, one can
show that in massless QCD this cumulative distribution is
unity when χ = 180◦. Hence, we can integrate the
fixedorder differential distribution in Eq. (3) and use the unitarity
constraint Σ (π, μ)/σt = 1 to all orders in αS to fix the
constants of integration,
Σ (χ , μ)
f.o.
+
+
g3(λ, x R ) = g3(λ) +
β1 2
λ g1(λ) + β0λg2(λ) ln(x 2R )
β0
+ β20 λ3g1 (λ) + β0λ2g1(λ) ln2(x 2R ),
where the prime denotes differentiation with respect to λ.
The factorization between the constant and logarithmic
terms H (αS) and S(Q, b) in Eq. (4) also involves some
arbitrariness, since the argument of the large logarithm L can
always be rescaled as
L = ln(Q2b2/b02) = ln(x L2 Q2b2/b02) − ln(x L2 ),
provided that xL is independent of b and that xL = O(1)
when Qb 1. This arbitrariness is parametrized by xL ,
which plays a role in the resummed computation which is
analogous to the role played by the renormalization scale in
the fixed-order calculation. This rescaling of the logarithm
introduces some modifications of the resummed formulae
and the expansion coefficients in Eqs. (6)–(8). We find
A˜(n)(xL ) = A(n),
B˜ (n)(xL ) = B(n) − A(n) ln(x L2 ),
H˜ (1)(xL ) = H (1) − β0g2(0) ln(x L2 ) + β0g1(0) ln2(x L2 ),
while the Sudakov form factor in Eq. (9) is also modified as
follows
S(Q, b, x R , xL ) = exp
x R
xL
x R
xL
× L˜ g1 aSβ0 L˜ ,
+ g2 aSβ0 L˜ ,
+αSg3 aSβ0 L˜ ,
+ · · · ,
x R
xL
where L˜ = ln(x L2 Q2b2/b02).
1
σt
1
Σ (χ , μ) ≡ σt
2.2 Matching the fixed-order and resummed predictions
In order to obtain a prediction which is valid over a wide
kinematical range4 the fixed-order and resummed
calculations must be matched. Here we employ the log-R matching
scheme as worked out for EEC in Ref. [
23
], and limit
ourselves to recalling the final results.
In the log-R matching scheme for EEC we consider the
cumulative distribution
χ
1
= σt
0
y(χ)
0
dχ (1 − cos χ )
dy 2(1 − y )
dΣ (χ , μ)
dχ
dΣ (y , μ)
dy
4 We note that another resummation in the forward limit would be
required to describe EEC over the full angular range.
5 Note a misprint in Eq. (3.12) of Ref. [
23
] where an overall factor of
1/2 appears erroneously.
Moreover, starting from Eq. (4) and using the definition of
Σ , Eq. (11), we obtain the following expression for the
resummed prediction5:
Σ (χ , μ)
= H (αS(μ))
res.
∞
0
Q√y(1 − y) J1(b Q√y)
2y
+ b J2(b Q√y) S(Q, b)db,
(13)
where the Sudakov form factor S(Q, b) is the one given in
Eq. (9).
The final expression for the matched prediction was
derived in Ref. [
23
] and reads
1
σt
ln
Σ(χ , μ) = ln
1 1
H (αS(μ)) σt
1 1
− ln H (αS(μ)) σt
αS(μ)
+ 2π A¯(χ , μ)
+
+
αS(μ) 2
2π
αS(μ) 3
2π
B¯(χ , μ) − 21 A¯2(χ , μ)
C¯(χ , μ) − A¯(χ , μ)B¯(χ , μ)
(14)
αS(μ) 2
αS(μ) 3
2π
2π
+
+
+O(αS4).
A¯res.(χ , μ)
B¯res.(χ , μ)
C¯res.(χ , μ)
(15)
The expansion coefficients A¯res., B¯res. and C¯res. can be found
in Ref. [
23
].
Notice that the function H (αS) does not appear in
Eq. (14) at all. In the log-R matching scheme such
nonlogarithmically enhanced contributions should not be
exponentiated, instead these terms, as well as subdominant
logarithmic contributions, are all implicit in the unsubtracted
parts of the fixed-order coefficients A¯, B¯ and C¯ [
41
]. Thus
the log-R matched prediction can be computed without the
explicit knowledge of H (n).
Finally, we comment on our implementation of the
unitarity constraint Σ (π, μ)/σt = 1. It can be shown that this
constraint can be satisfied by modifying the resummation
formula in Eq. (4) such that in the kinematical limit y = 1 the
Sudakov form factor is unity. This may be achieved in several
ways and here we choose a very simple solution and modify
the resummation coefficients A˜(n) and B˜ (n) according to
A˜(n)(xL ) → A˜(n)(y, xL ) = A˜(n)(xL )(1 − y) p,
B˜ (n)(xL ) → B˜ (n)(y, xL ) = B˜ (n)(xL )(1 − y) p,
where p is a positive number.6 This modification is fully
legitimate since it does not modify the logarithmic structure
of the result and introduces only power-suppressed terms.
In practice, we set p = 1 and quantify the impact of this
modification by comparing the results to those obtained with
p = 2.
(16)
2.3 Finite b-quark mass corrections
The theoretical prediction presented above was computed in
massless QCD. However, the assumption of vanishing quark
masses is not fully justified, especially at lower energies,
where b-quark mass effects are relevant at the percent level
[
43
]. In order to take b-quark mass corrections into account,
we subtract the fraction of b-quark events, rb(Q) from the
massless result and add back the corresponding massive
con6 A modification similar in spirit was employed in Ref. [
42
] although
in the context of matching the fixed-order and resummed predictions
for transverse observables in Higgs hadroproduction.
is the fixed-order
tribution. Hence, we include mass effects directly at the level
of matched distributions,
less QCD as outlined above, while σ1t dΣdc(oχs,χQ) mNaNssLivOe∗ is the
fixed-order massive distribution. As the complete NNLO
correction to this distribution is currently unknown, we model
it by supplementing the massive NLO prediction of the
parton level Monte Carlo generator Zbb4 [
44
], with the NNLO
coefficient of the massless fixed-order result.
We define the fraction of b-quark events as the ratio of the
total b-quark production cross section divided by the total
hadronic cross section,
σmassive(e+e− → bb¯)
rb(Q) ≡ σmassive(e+e− → hadrons)
.
We evaluate the ratio of these cross sections at NNLO
accuracy (O(αS2) in the strong coupling) including the exact
bquark mass corrections at O(αS) and the leading mass terms
up to (m2b/Q2)2 at O(αS2) [
45
]. We note that the electroweak
coupling factors do not cancel in this ratio and the
summation over quark flavours has to be carried out explicitly when
computing σmassive(e+e− → hadrons).
Distributions for σ1t dΣdc(oχs,χQ) massive were generated for
each of the considered energies using a pole b-quark mass
of mb = 4.75 GeV, which is consistent with world average
estimations of pole mass 4.78 ± 0.06 GeV [
46
].
In order to assess the uncertainty associated to the
modelling of b-quark mass corrections, we have investigated two
alternative approaches for including them in our predictions.
In approach A Eq. (17) is modified to
i.e., we simply subtract the massless fixed-order NLO
prediction multiplied by the fraction rb(Q) of b-quark events
and add back the corresponding massive NLO distribution.
Approach B is defined in a way very similar to our baseline,
Eq. (17), but we do not include any NNLO corrections to the
massive distribution,
To extract the strong coupling the predictions described
above were confronted with the available data sets. Namely,
the data obtained in SLD [
47
], L3 [
48
], DELPHI [
49
], OPAL
[
50, 51
], TOPAZ [52], TASSO [
53
], JADE [
54
], MAC [
55
],
MARKII [
56
], CELLO [
57
] and PLUTO [
58
] experiments
were included. The information on used data is summarised
in Tab. 1.
The criteria to include the data were high precision of
differential distributions obtained with charged and neutral
final state particles in the full χ range, presence of
corrections for detector effects, correction for initial state photon
radiation and sufficient amount of supplementary
information. Therefore, data sets without supplementary
information [
59
], with large uncertainties [
60
], superseded datasets
[
61, 62
] and measurements unfolded only to charged particles
in the final state [63] are not included in the analysis.
The data sets selected for the extraction procedure have
high precision and the measurements from different
experiments performed at close energy points are consistent.7 This
justifies their use in the extraction procedure in a wide
centreof-mass energy interval, similarly to studies of thrust [
43
] and
C parameter [
64
] and allows us to target the highest precision
of αS determination with available theoretical predictions.
3.1 Monte Carlo generation setup
In a previous study [
23
] the non-perturbative effects for the
EEC distribution were modelled with an analytic approach.
In this paper the non-perturbative effects in the e+e− →
hadr ons process are modelled using state-of-the-art
particlelevel Monte Carlo (MC) generators. The non-perturbative
corrections of the energy–energy correlation distributions
were extracted as ratios of energy–energy correlation
distributions at hadron and parton level in the simulated samples.
In this study the MC generators SHERPA2.2.4 [
65
]8 and
Herwig7.1.1, [
66–68
] were used.
The e+e− → hadr ons MC samples were generated at
centre-of-mass energies √s = 14.0, 22.0, 29.0, 34.0, 43.5,
53.3, 59.5 and 91.2 GeV. In all cases, the simulation of
initial state radiation was disabled and generator settings were
defaults if the opposite is not stated explicitly. The value
of the strong coupling used for the hard process was set to
αS(MZ ) = 0.1181 [
46
].
The SHERPA2.2.4 samples were generated with the
MENLOPS method using the matrix element generators
AMEGIC [
69
], COMIX [
70
] and the GoSam [
71
] one-loop
library to produce matrix elements for e+e− → Z /γ →
2, 3, 4, 5 partons processes. The 2−parton final state
processes had NLO accuracy in perturbative QCD. The QCD
matrix elements were calculated assuming massive b-quarks.
The merging parameter Ycut was set to 10−2.75 1.778 ×
10−3.
To test the fragmentation and hadronization model
dependence, the events generated with SHERPA2.2.4 were
hadronized using the Lund string fragmentation model [
72
]
or the cluster fragmentation model [
73
]. The first setup is
labelled below as S L and the second as SC .
To assure proper fragmentation of heavy quarks and heavy
hadron decays the cluster fragmentation model was adjusted.
The value of SPLIT_LEADEXPONENT parameter was set to
1.0, the parameter M_DIQUARK_OFFSET was set to 0.55,
7 Some observed differences between the measurements performed at
√s = 91.2 GeV are not statistically significant once the systematical
uncertainties and correlations are taken into account.
8 Partially updated to version 2.2.5.
the production of charm and beauty baryons was enhanced
by factors 0.8 and 1.7.
For the cross-check of SHERPA2.2.4 samples, the
Herwig7.1.1 generator was used. The Herwig7.1.1
samples were generated with improved unitarized merging
[
74
] using the MadGraph5 [
75
] matrix element generator
and the GoSam [
71
] one-loop library to produce matrix
elements of the e+e− → Z /γ → 2, 3, 4, 5 partons processes.
The 2−parton final state processes again had NLO accuracy
in perturbative QCD and the matrix elements were calculated
assuming massive b-quarks. The merging parameter was set
to √s × 10−1.25 √s × 5.623 × 10−2. For the modelling of
the hadronization process the default implementation of the
cluster fragmentation model [
76
] was used. To improve the
modelling of beauty production at the lowest energies, the
b-quark nominal mass was changed from the default value
of 5.3 GeV to 5.1 GeV. This setup is labelled below as H M .
3.2 Estimation of hadronization effects from MC models
Estimation of hadronization corrections is an integral part of
comparing the parton-level QCD predictions to the data
measured on hadron (particle) level. Despite the fact that under
certain conditions the local parton-hadron duality leads to
close values of quantities on parton and hadron level, the
difference between them is not negligible and should be taken
into account in precise analyses. One way to do so is to apply
correction factors estimated from MC simulations to the
perturbative QCD prediction. The factors, called hadronization
corrections H/ P, are defined as ratios of the corresponding
quantities at parton level to the same quantities at hadron
level at every point of the considered distribution.
To obtain the EEC distributions, the generated MC
samples were processed in the same way as data (see e.g. Ref.
[
50
]), using partons before hadronization for parton-level
calculations and undecayed/stable particles for hadron-level
calculations. For the parton-level calculations the parton
energies were used as provided by the MC generators.
The predictions obtained with all setups describe the data
well for all ranges of χ with the exception of the regions
near χ = 0◦ and χ = 90◦, for all values of √s. For √s <
29 GeV the H M setup is sensitive to the b-quark mass and
the corresponding predictions are not reliable.
However, to assure an even better description of data, a
reweighting procedure was applied to the simulated samples.
The samples were reweighted at hadron level on an
event-byevent basis to describe the data and the corresponding event
weights were propagated to the parton level. The resulting
distributions are shown in Fig. 1 for the SHERPA2.2.4
setups and in Fig. 2 for the Herwig7.1.1 setup.
As the SHERPA2.2.4 setups give the most stable and
physically reliable predictions these are used in the analysis
for reference hadronization corrections (S L ) and for
systemOPAL, 91.2 GeV
SL hadrons
SC hadrons
SL partons
SC partons
TOPAZ, 59.5 GeV
SL hadrons
SC hadrons
SL partons
SC partons
JADE, 34 GeV
SL hadrons
SC hadrons
SL partons
SC partons
JADE, 22 GeV
SL hadrons
SC hadrons
SL partons
SC partons
MAC, 29 GeV
SL hadrons
SC hadrons
SL partons
SC partons
JADE, 14 GeV
SL hadrons
SC hadrons
SL partons
SC partons
0 20 40 60 80 100 120 140 160 180
χ◦
0 20 40 60 80 100 120 140 160 180
χ◦
TOPAZ, 53.3 GeV
HM hadrons
HM partons
TASSO, 43.5 GeV
HM hadrons
HM partons
0 20 40 60 80 100 120 140 160 180
χ◦
OPAL, 91.2 GeV
HM hadrons
HM partons
TOPAZ, 59.5 GeV
HM hadrons
HM partons
0 20 40 60 80 100 120 140 160 180
χ◦
atic studies (SC ). The corresponding hadronization
corrections together with parametrizations are shown in Fig. 3 for
reweighted samples.
3.3 Estimation of statistical correlations between
measurements from MC models
To perform an accurate extraction procedure, the available
data and uncertainties were examined and for every
measured set of data a covariance matrix was built. The procedure
consisted of multiple steps.
In the first step the systematic uncertainties were
recalculated and separated from statistical uncertainties when this
was possible. For the measurements with the uncertainties
rounded to one significant digit [
50, 53, 54
] the uncertainties
were expanded assuming maximal uncertainty before
rounding. The measurements of TASSO [
53
] were converted from
the dΣ/d cos χ form to dΣ/dχ using values of cos χ on the
bin edges.
For the data from TOPAZ [
52
] the systematic uncertainty
was calculated from an estimated relative systematic
uncertainty of ±4% [
52
].
Taking into account the uncertainties of αS extraction
analysis [
58
] the same was done for PLUTO [
58
] data. The
systematic uncertainty estimation of ±5% for TASSO [
53
] is
based on the upper limit of 10% for the total uncertainty
mentioned in the paper [
53
]. The systematic uncertainties
from DELPHI [
49
] and SLD [
47
] were used as provided in
the original papers.
For all remaining data sets the published combined
uncertainty was treated as statistical.
The measurements of Σ are provided in the original
publications without correlations between the individual points.
The correlation matrix was estimated from the Monte Carlo
samples in terms of Fisher correlation coefficients [
77, 78
].
Some of the obtained correlation matrices are shown in Fig. 4.
The obtained correlation coefficients are sizeable, up to
0.5 for the closest points, which highlights the importance
of properly taking into account the correlations between
measured points in the fits. The obtained correlation matrix
together with statistical uncertainties was used to build a
statistical covariance matrix for every data set.
To construct the systematic covariance matrix, the
systematic uncertainties from the original publications were used
with an assumption that these are positively correlated with
correlation coefficient ρ = 0.5 between closest points. The
correlations between the uncertainties of data from different
experiments or different beam energies were neglected. The
final covariance matrix used in the fit for every data set was
a sum of statistical and systematic covariance matrices.
0 20 40 60 80 100 120 140 160 180
χ◦
0 20 40 60 80 100 120 140 160 180
χ◦
3.4 Fit procedure and estimation of uncertainties
The strong coupling extraction procedure is based on the
comparison of data to the perturbative QCD prediction
combined with non-perturbative (hadronization) corrections. The
perturbative part of the predictions was calculated in every
bin as described in previous sections. To tame the statistical
fluctuations present in the obtained binned hadronization
correction distributions, these were parametrized with analytic
functions, expressed as a sum of polynomials of χ − 90◦.
The value of the fitted function at the bin centre was used as
the correction factor.
To find the optimal value of αS, the MINUIT2 [
79,80
]
program was used to minimize
χ 2(αS) =
χ 2(αS)data set,
data sets
where the χ 2(αS) value was calculated for each data set as
χ 2(αS) = ( D − P (αS))V −1( D − P (αS))T ,
with D standing for the vector of data points, P (αS) for the
vector of calculated predictions and V for the covariance
matrix for D. The default scale used in the fit procedure was
μ = Q = √s.
The fit ranges were chosen to avoid regions where
resummed predictions or hadronization correction
calculations are not reliable. The selected fit ranges were 117◦−165◦,
60◦−165◦ and 60◦−160◦. The uncertainty on the fit result
was estimated with the χ 2 + 1 criterion as implemented in
the MINUIT2 program. The results of the fits are given in
Tab. 2 for each fit range. In order to assess the impact of
the NNLO corrections, in Tab. 2 we also present the results
obtained using NLO+NNLL predictions. The NNLO
corrections affect the fit in a moderate but non-negligible way. The
obtained values of χ 2 divided by the number of degrees of
freedom in the fit are of order unity for all cases. The
corresponding distributions obtained from the fits for different
√s points are shown in Figs. 5, 6, 7 and 8.
OPAL,91.2 GeV
Z.Phys.C59,1
NNLO+NNLL+SL
NLO+NNLL+SL
SLD,91.2 GeV
Phys.Rev.D51,962
NNLO+NNLL+SL
NLO+NNLL+SL
The systematic uncertainties of the obtained results were
estimated with procedures used in previous studies [
81
]. To
estimate the bias of the obtained result caused by the absence
of higher-order terms in the perturbative predictions, the
renormalization scale variation procedure was performed. In
this procedure the fits were repeated, with variation of the
renormalization scale in the range between x R = 1/2 and
x R = 2.
The bias of hadronization model selection is studied using
the S L and SC setups of hadronization corrections, see results
in Fig. 9. The bias related to the ambiguity of
resummation scale choice was estimated by varying xL in the range
between xL = 1/2 and xL = 2. To estimate the bias related
to the ambiguity of our prescription implementing the
unitarity constraint in the resummed calculation (see Eq. (16)),
two values of p were used: p = 1 and p = 2. The
difference between results obtained with two options is
negligible. In all cases above the sizes of the biases were estimated
numerically as half of the difference between the maximal
and minimal αS value obtained in the corresponding set of
0.1
fits. To estimate the potential bias of the result caused by
imperfections of specific hadronization model and parton
shower model, the fits were repeated with hadronization
corrections obtained with all setups described in previous
subsections. The numerical value of the bias was obtained as half
of the difference between the αS values obtained using
nonta1.2
1.1
a
D
/1.0
ry0.9
eo0.80 20 40 60 80 100 120 140 160 180
h
T χ◦
Fig. 6 Fits of theory predictions to the data for √s = 34 − 53.3 GeV.
The used fit range is shown with thick line. For the ratio plot only the
uncertainties of the data are taken into account
Fig. 7 Fits of theory predictions to the data for √s = 22 − 29 GeV.
The used fit range is shown with thick line. For the ratio plot only the
uncertainties of the data are taken into account
perturbative corrections from Lund and cluster hadronization
models implemented in SHERPA2.2.4. From Fig. 9 it is
seen that the estimated biases are relatively independent and,
therefore combined in the final result as such.
Besides the estimations, several cross-checks of the
obtained results were performed. First, the datasets were
grouped according to their energies and fitted separately for
each energy. The results are shown in Fig. 10. There is no
visible trend for the fitted value of αS with energy in the S L
and SC setups. For the H M setup, the results of the fits are
not reliable below √s < 29 GeV due to the sensitivity of this
setup to the b-quark mass. In addition to the MC
hadroniza0.1
)0.140
Z
(M0.135
s
α0.130
Fig. 9 Dependence of fit results on the renormalization scale (upper
left), resummation scale (upper right), non-perturbative simulation
model (bottom left) choice and b mass corrections (bottom right). The
fit range for SL , SC and H M setups is 60◦−160◦. The fit range for the
An.DMW setup is 117◦−165◦
tion models the fits were also performed with the analytic
hadronization model of Dokshitzer, Marchesini and Webber
(DMW) [
82
]. In this setup, non-perturbative effects in EEC
were accounted for by multiplying the Sudakov form factor
by a correction of the form
)0.140
Z
(M0.135
s
α0.130
0.125
Here a1 and a2 are non-perturbative parameters that can be
related in the dispersive approach to certain moments α¯ q, p
of the strong coupling αS [
82
]. These moments are the fit
parameters of the analytic model. The results obtained from
the fits with this setup are listed in Tab. 2. They show a high
degree of dependence on the selected fit range, but are close
to results obtained with the Monte Carlo based hadronization
corrections in the range 117◦−165◦, see Tab. 2. Hence we
conclude that away from the back-to-back region, the analytic
model cannot fully account for hadronization effects.
4 Results and discussions
In this paper we presented the first combined analysis and
extraction of αS at NNLO+NNLL precision from energy–
energy correlation in electron-positron annihilation.
Moreover, our analysis is the first extraction of the strong coupling
based on Monte Carlo hadronization corrections obtained
from NLO Monte Carlo setups at NNLO+NNLL precision.
For the central value of the final result we quote the results
obtained from the fits with the S L hadronization model in the
range 60◦−160◦ with uncertainties and estimations of biases
obtained as described above.
At NNLO+NNLL accuracy we obtain the best fit value of
αS(MZ ) = 0.11750 ± 0.00018(ex p.) ± 0.00102(hadr.)
±0.00257(r en.) ± 0.00078(r es.).
In order to appreciate the impact of NNLO corrections, we
also quote the result of the fit at NLO+NNLL accuracy
We see that the inclusion of the NNLO corrections has a
moderate but non-negligible effect on the extracted value of
αS.
It has been explicitly checked that there are no
correlations between estimated biases, therefore, the combined
values with combined estimations of bias at NNLO+NNLL
accuracy are:
αS(MZ ) = 0.11750 ± 0.00287(comb.)
while in comparison, for NLO+NNLL accuracy we obtain:
αS(MZ ) = 0.12200 ± 0.00535(comb.).
The value obtained from the analysis in NNLO+NNLL
approximation is in agreement with the world average as of
2017 [
83
], however it is visibly lower than the results from
measurements performed for other e+e− observables using
NNLO perturbative QCD predictions and MC hadronization
models [
83
]. The estimated uncertainties are dominated by
the uncertainty on the theoretical predictions. The results
obtained in this study can be compared to those described in
the original publications with NLO+NLL precision as well
as the results obtained with analytic hadronization model in
the sister paper [
23
].
Acknowledgements We are grateful to Simon Plätzer and Ludovic
Scyboz for fruitful discussions about the calculation of NLO
predictions with Herwig7.1.1 and GoSam, to Pier Monni for stimulating
discussions on resummation of event shapes and to Carlo Oleari for
providing us the Zbb4 code. ZT was supported by the ÚNKP-17-3 New
National Excellence Program of the Ministry of Human Capacities of
Hungary. AK acknowledges financial support from the Premium
Postdoctoral Fellowship program of the Hungarian Academy of Sciences.
This work was supported by Grant K 125105 of the National Research,
Development and Innovation Fund in Hungary.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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