NNLO QCD calculations at work: determining S in e e ...Adam Kardos ! University of Debrecen ! in collaboration with Gábor Somogyi and Zoltán Tulipánt ! Appeared as: arXiv:1708.04093

Adam Kardos !

University of Debrecen !

in collaboration with Gábor Somogyi and Zoltán Tulipánt

!Appeared as: arXiv:1708.04093

(accepted for publication in EPJC)

NNLO QCD calculations at work: determining 𝛼S in e+e- annihilation

Theoretical Physics Department, ELTE 25th, October, 2017

2

Introduction

Introduction

•𝛼S is one fundamental parameter of the standard model

3

⟹ its most precise determination is mandatory

With the exception of lattice results, most resultswithin their subclass are strongly correlated, howeverto an unknown degree, as they largely use similar datasets and/or theoretical predictions. The large scatterbetween many of these measurements, sometimes withonly marginal or no agreement within the given errors,indicate the presence of additional systematic uncer-tainties from theory or caused by details of the anal-yses. Therefor the unweighted average of all selectedresults is taken as pre-average value for each subclass,and the unweighted average of the quoted uncertaintiesis assigned to be the respective overall error of this pre-average.

For the subclasses of hadron collider results and elec-troweak precision fits, only one result each is availablein full NNLO, so that these measurements alone definethe average value for their subclass. Note that moremeasurements of top-quark pair production at LHC aremeanwhile available, indicating that - on average - alarger value of αs(M2Z) is likely to emerge in the future;see also [17] and the presentation of T. Klijnsma at thisconference [18]. The resulting subclass averages are in-dicated in figure 1, and are summarized in table 1.

Table 1: Pre-average values of subclasses of measurements ofαs(M2Z).

Subclass αs(M2Z)τ-decays 0.1192 ± 0.0018lattice QCD 0.1188 ± 0.0011structure functions 0.1156 ± 0.0021e+e− [jets & shps] 0.1169 ± 0.0034hadron collider 0.1151 + 0.0028− 0.0027ew precision fits 0.1196 ± 0.0030

Assuming that the resulting pre-averages are largelyindependent of each other, the final world averagevalue is determined as the weighted average of the pre-averaged values. An initial uncertainty of the centralvalue is calculated treating the uncertainties of all in-put values as being uncorrelated and of Gaussian nature,and the overall χ2 to the central value is determined. Ifthe initial χ2 is smaller than the number of degrees offreedom, an overall, a-priori unknown correlation co-efficient is introduced and adjusted such that the totalχ2/d.o.f. equals unity. Applying this procedure to thevalues listed in table 1 results in the new world averageof

αs(M2Z) = 0.1181 ± 0.0011 .This value is in good agreement with that from

Figure 1: Summary of determinations of αs. The light-shaded bandsand long-dashed vertical lines indicate the pre-average values as ex-plained in the text and as listed in table 1; the dark-shaded band andshort-dashed line represent the new overall world average of αs.

S. Bethke / Nuclear and Particle Physics Proceedings 282–284 (2017) 149–152150

[S. Bethke, ’17]

•World average from various observables:from lattice to hadron collisions

•Most of the measurements were done in e+e-

•World average: ↵S(MZ) = 0.1181± 0.0011

•e+e- is still not fully exploited:- new observables- new calculations- new tools to estimate non-perturbative effects

•LHC measurements are just coming up…

Introduction

•𝛼S determination not only needs precise measurements but theoretical predictions with equally high quality

4

αs 2016

S. Bethkea

aMax-Planck-Institute of Physics, Föhringer Ring 6, 80805 Munich, Germany

Abstract

An update of measurements of the strong coupling αs and the determination of the world average value of αs(M2Z)is presented, resulting in

αs(M2Z) = 0.1181 ± 0.0011.

Keywords: strong coupling, alpha-s, Quantum Chromodynamics

Several new measurements of αs, the couplingstrength of the strong interaction between quarks andgluons, became available since previous summarieswere given at this conference series [1] and in the2014 Review of Particle Properties [2]. In the fol-lowing, those new results which are used to determinethe new world average value of αs, i.e. those that arebased on at least complete next-to-next-to-leading or-der (NNLO) perturbation theory, are published in peer-reviewed journals and contain complete estimates of ex-perimental and systematic uncertainties, will be sum-marised. Also results which are used for demonstratingasymptotic freedom, i.e. the specific energy dependenceof αs as predicted by Quantum Chromodynamics, evenif being based on next-to-leading (NLO) perturbationtheory only, will be reviewed.

This update with status of April 2016 is extractedfrom the most recent version of the Review of Parti-cle Properties [3]; see this reference and [2] for a com-plete list of references, and for a detailed presentationof theoretical and experimental issues concerning testsof Quantum Chromodynamics.

The newest and most actual entries satisfying thequality criteria given above are:

∗Talk given at 19th International Conference in Quantum Chromo-dynamics (QCD 16), 4 - 8 July 2016, Montpellier - F

Email address: [email protected] (S. Bethke)

• updated results from τ-decays [4] [5] [6], basedon a re-analysis of ALEPH data and on completeN3LO perturbation theory,

• more results from unquenched lattice calculations,[7][8],

• further results from world data on structure func-tions, in NNLO QCD [9],

• from e+e− hadronic event shapes (C-parameter)in soft collinear effective field theory matched toNNLO perturbation theory [10],

• αs determinations at LHC, from data on the ra-tio of inclusive 3-jet to 2-jet cross sections [11],from inclusive jet production [12], from the 3-jetdifferential cross section [13], and from energy-correlations [14], all in NLO QCD, plus one deter-mination in complete NNLO, from a measurementof the tt cross section at

√s = 7 TeV [15];

• and finally, an update of αs from a global fit toelectroweak precision data [16], based on completeN3LO perturbation theory.

All measurements based on at least full NNLOperturbation theory are summarised in figure 1, andare ordered according to subclasses of τ-decays, lat-tice results, structure functions, e+e−-annihilation, elec-troweak precision fits and hadron colliders.

Available online at www.sciencedirect.com

Nuclear and Particle Physics Proceedings 282–284 (2017) 149–152

2405-6014/© 2016 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/nppp

http://dx.doi.org/10.1016/j.nuclphysbps.2016.12.028

•The new 𝛼S value has to meet certain criteria to end up contributing to the world average:

•We need at least Next-to-next-to-leading order (NNLO) accuracy in the theoretical calculation

luckily this can be fulfilled…

[S. Bethke, ’17]

5

Nuts and bolts of QCD perturbation theory

QCD perturbation theory

•We are always interested in cross sections, e.g. for a jet function J

6

•Treat 𝛼S as the perturbation parameter, thus expand in powers of 𝛼S:

�[J ] = �LO[J ] + �NLO[J ] + �NNLO[J ] + . . .

•To be calculable J has to be IR safe/finite:arbitrary number of soft and/or collinear partons should not change the value of J

•Beyond LO singularities appear in each order in different contributions but vanish when contributions are combined due to the KLN theorem.

�NLO[J ] = �NLOm+1[J ] + �NLOm [J ] =

Z

m+1d�Rm+1Jm+1 +

Z

md�VmJm

one more final state parton one more loop(kinematic singularity)(𝜖 poles)

cancellation


•There would be no problem if we could calculate everything analytically

•This is not possible, need something numerically integrable (with Monte-Carlo techniques)

•One possible remedy is subtraction

I = lim✏!0

✓Z 1

0

dx

x

1�✏ f(x)�1

✏

f(0)

◆

•As 𝜖→0 the sum is finite, terms are separately divergent•Find a g(x) function which mimics f(x) as x→0:

I = lim✏!0

Z 1

0

dx

x

1�✏ [f(x)� g(x)] + lim✏!0

✓Z 1

0

dx

x

1�✏ g(x)�1

✏

f(0)

◆

•g(x) has to be analytically integrable…

subtraction integrated subtraction

7


•At NLO the subtraction looks like:�NLO[J ] = �NLOm+1[J ] + �

NLOm [J ]

�NLOm+1[J ] =

Z

m+1d�Rm+1Jm+1

�NLOm [J ] =

Z

md�VmJm

•

•

8



NLOm [J ]

�NLOm [J ] =

Z

md�VmJm

•

•

�NLOm+1[J ] =

Z

m+1

hd�Rm+1Jm+1 � d�

R,Am+1Jm

i

d=4

•Kinematic singularities emerging in single parton emissions in the m+1 parton contribution regularized by the subtractions

8



NLOm [J ]

•

•

�NLOm+1[J ] =

Z

m+1

hd�Rm+1Jm+1 � d�

R,Am+1Jm

i

d=4

�NLOm [J ] =

Z

m

d�VmJm +

Z

1d�R,Am+1Jm

�

d=4

•Kinematic singularities emerging in single parton emissions in the m+1 parton contribution regularized by the subtractions

•Explicit 𝜖 poles coming from the loop integral are cancelled by those 𝜖 poles arising from the integrated subtraction terms

8


•At NNLO the philosophy is the same but the subtractions are more elaborate:

�NNLO[J ] = �NNLOm+2 [J ] + �NNLOm+1 [J ] + �

NNLOm [J ]

•

•

•

�NNLOm+2 [J ] =

Z

m+2d�RRm+2Jm+2

�NNLOm+1 [J ] =

Z

m+1d�RVm+1Jm+1

�NNLOm [J ] =

Z

md�VVm Jm

9




NNLOm [J ]

•

•

•

�NNLOm+1 [J ] =

Z

m+1d�RVm+1Jm+1

�NNLOm [J ] =

Z

md�VVm Jm

�NNLOm+2 [J ] =

Z

m+2

n

d�RRm+2Jm+2�d�RR,A2m+2 Jm�

h

d�RR,A1m+2 Jm+1�d�RR,A12m+2 Jm

io

d=4

9

• Kinematic singularities in single and double parton emissions in the m+2 parton contribution regularized by the subtractions




NNLOm [J ]

•

•

• �NNLOm [J ] =Z

md�VVm Jm

�NNLOm+2 [J ] =

Z

m+2

n


h


io

d=4

�NNLOm+1 [J ] =

Z

m+1

(✓d�RVm+1+

Z

1d�RR,A1m+2

◆Jm+1�

"d�RV,A1m+1 +

✓Z

1d�RR,A1m+2

◆A1#Jm

)

d=4

9


• Kinematic singularities due to single emissions and pole cancellations with integrated subtraction terms in the m+1 parton line




NNLOm [J ]

•

•

•

�NNLOm+2 [J ] =

Z

m+2

n


h


io

d=4

�NNLOm+1 [J ] =

Z

m+1

(✓d�RVm+1+

Z

1d�RR,A1m+2

◆Jm+1�

"d�RV,A1m+1 +

✓Z

1d�RR,A1m+2

◆A1#Jm

)

d=4

�NNLOm [J ] =

Z

m

(d�VVm +

Z

2

hd�RR,A2m+2 �d�

RR,A12m+2

i+

Z

1

"d�RV,A1m+1 +

✓Z

1d�RR,A1m+2

◆A1#)

d=4

Jm

9


• Kinematic singularities due to single emissions and pole cancellations with integrated subtraction terms in the m+1 parton line

• explicit poles cancelled across the two-loop and integrated subtraction terms


•The philosophy of local subtraction was used in the construction of the CoLoRFuLNNLO method (Del Duca, Somogyi and Trocsanyi)

10

•It is derived from first principles: factorization and limit behavior of QCD amplitudes

•It defines fully local subtractions: beneficial for good numerical properties

•Derived for colorless initial states (electron-pozitron annihilation)•The generalization for colored initial states (hadron-hadron [LHC]

and hadron-lepton [DIS] collisions) is on the way•The theory was well laid down but needed a powerful numerical

framework…

11

MCCSM

MCCSM

•MCCSM stands for Monte-Carlo for the CoLoRFuL Subtraction Method

12

•Fortran90 program•Fully automatic and general capable of calculating the NNLO QCD

corrections for any process at electron-pozitron annihilation•More than 80k lines of code excluding the matrix elements•Flexible, user friendly and equipped with many features•Tested in 2- and 3-jet production•Already used to obtain new observables:

- Oblateness and Energy-energy correlation

- Jet-cone energy fraction[Del Duca, Duhr, AK, Somogyi et al. PRL 117 (2016) no.15, 152004]

[Del Duca, Duhr, AK, Somogyi et al. PRD94 (2016) no.7, 074019]

MCCSM

13

MCCSM

13

MCCSM

14

MCCSM

14

MCCSM

15

MCCSM

15

MCCSM

16

MCCSM

16

MCCSM

17

18

Resummation 101

Resummation

19

0.1

0.2

0.3

0.4

0.5

⌧ �d� d⌧

ALEPH dataGGGH

LONLONNLO

⇠R 2 [0.5, 2]⇠R 2 [0.5, 2]⇠R 2 [0.5, 2]

pQ2 = 91.2GeV

↵s(Q2) = 0.118

0.951.01.05

GGGH

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45⌧

GGGH

0.951.01.05

SW

SW

There are regions of phase space where perturbation theory breaks down:

𝛼S ≪ 1 but logs can become large𝛼S log(…) ≈1 ⟹ all orders become important in the perturbative series!

𝜏=1-Thrust distribution in e+ e- → 3jets

Indeed, the LO is:1

�

d�

d⌧

⌧!0⇡ ↵S2⇡

CF4

⌧log ⌧

This is clearly divergent, to have a meaningful prediction in the small 𝜏 region have to sum up all orders!

Resummation

20

Let’s stick to the 1-Thrust distribution, its perturbative expansion:d�

d⌧⇠ ↵S

�A+ ↵SB + ↵2SC + . . .

Keeping only singular terms order by order:d�

d⌧⇠ 1

⌧

1X

n=1

2n�1X

m=0

↵nS logm ⌧

or writing out explicitly order by order and grouping accordingly:d�

d⌧⇠ 1

⌧

⇢↵S

�log ⌧ + 1

�+

+↵2S�log

3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+↵3S�log

5 ⌧ + log4 ⌧+ log3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+ . . .

�

Resummation

21

d�

d⌧⇠ 1

⌧

⇢↵S

�log ⌧ + 1

�+

+↵2S�log

3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+↵3S�log

5 ⌧ + log4 ⌧+ log3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+ . . .

�

Fixed order calculation:

Resummation:d�

d⌧⇠ 1

⌧

⇢↵S

�log ⌧ + 1

�+

+↵2S�log

3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+↵3S�log

5 ⌧ + log4 ⌧+ log3 ⌧ + log2 ⌧+ log ⌧ + 1�+

+ . . .

�

LO

NLONNLO

Resummation

22

Formally the resumed formula for an observable y:1

�

d�

dy= H(↵S(µ))

Z 1

0db b J0(b y)S(Q, b)

S(Q, b) = exp

(�Z Q2

b20/b2

dq2

q2

A(↵S(q

2)) log

Q2

q2+B(↵S(q

2))

�)

A(↵S) =1X

n=1

⇣↵S4⇡

⌘nA(n), B(↵S) =

1X

n=1

⇣↵S4⇡

⌘nB(n), H(↵S) = 1 +

1X

n=1

⇣↵S4⇡

⌘nH(n)

with:

and

(Q is the CM energy)To carry out resummation at a given order have to determine the various A(n), B(n) and H(n) terms such as:•for LL accuracy: A(1)•for NLL accuracy: A(1), A(2), B(1) ( and H(1))•for NNLL accuracy: A(1), A(2), A(3), B(1), B(2) ( , H(1) and H(2))

Resummation

23

•The resumed result contains some logs exponentiated providing meaningful prediction in some parts of phase space

•Since it only contains part of the singular terms the resumed prediction is not at all accurate in the whole available phase space⟹ when logs are not essential have to use the fixed-order prediction•To have good precision over the whole available phase space we need to

match the resumed prediction with the fixed-order one1

�

d�

dy6=

1

�

d�

dy

�

res.

+

1

�

d�

dy

�

f.o.

•Even the fixed order contains logs ⟹ these have to be subtracted:1

�

d�

dy=

1

�

d�

dy

�

res.

+

1

�

d�

dy

�

f.o.

�⇢

1

�

d�

dy

�

res.

��f.o.

•This is called the (naive) R matching scheme

Resummation

24

•If the resumed prediction is not precise enough (not enough Ns) not all the logs are cancelled in the fixed-order calculation⟹non-resumed logs in the fixed order spoil physicality for small y values

•To ensure physicality in the widest range use a different prescription:•Define a cumulative observable:

R(y) =1

�

Z y

0dy0

d�(y0)

dy0

•with the constraint of: R(ymax

) = 1

•Note also that:dR(y)

dy=

1

�

d�(y)

dy

Resummation

25

•The fixed-order expansion of R is:

[R(y)]f.o. = 1 +

↵S

2⇡Ā(y) +

⇣↵S

2⇡

⌘2

B̄(y) +⇣↵

S

2⇡

⌘3

C̄(y) +O(↵4S

)

•The resumed formula takes the form of:[R(y)]res. =

�1 + C1↵S + C2↵

2S + . . .

�exp [Lg1(↵SL) + g2(↵SL) + ↵Sg3(↵SL) + . . . ] +O(↵Sy)

L stands for the essential log

gn(↵SL) =1X

i=1

Gi,i+2�n⇣↵S2⇡

⌘iLi+2�n

Resummation

26

•Taking the log of both and expanding in terms of 𝛼S:

log[R(y)]f.o. =

↵S

2⇡¯A(y) +

⇣↵S

2⇡

⌘2

¯B(y)� 1

2

¯A2(y)�

+

⇣↵S

2⇡

⌘3

¯C(y)� ¯A(y) ¯B(y) + 1

3

¯A3(y)�+O(↵4

S

)

log[R(y)]res. = Lg1(↵SL) + g2(↵SL) + ↵Sg3(↵SL)

+ ↵SC1 + ↵2S

✓C2 � 1

2

C21

◆+ ↵3S

✓C3 � C1C2 + 1

3

C31

◆+O(↵4S)

•The terms coming from non-exponentiated contributions in the resumed prediction can be replaced by those coming from the fixed-order one minus those logs which are available in the resumed prediction.

Resummation

27

•Hence the master formula for log-R matching:logR(y) = Lg1(↵SL) + g2(↵SL) + ↵Sg3(↵SL)

+

↵S2⇡

⇥¯A(y)�G11L�G12L2

⇤

+

⇣↵S2⇡

⌘2 ¯B(y)� 1

2

¯A2(y)�G21L�G22L2 �G23L3�

+

⇣↵S2⇡

⌘3 ¯C(y)� ¯A(y) ¯B(y) + 1

3

¯A3(y)�G32L2 �G33L3 �G34L4�

+O(↵4S)In the above only those logs were subtracted which are present at NNLL accuracy

28

Energy-energy correlation @ NNLO + NNLL

Energy-Energy Correlation (EEC)

29

Observable Resummation NNLO NNLO + N

The “6” event shapes

[Chien et al. ’10], [Abbate et al. ’10], [Becher et al. ’12], [Banfi et al. ’14],

[Hoang et al. ’14],…

[Gehrmann-De Ridder et al. ’07], [Weinzierl ’09],

[Del Duca et al. ’16][Gehrmann et al. ’08]

Jet rate [Banfi et al. ’16][Gehrmann-De Ridder et al. ’07],

[Weinzierl ’09], [Del Duca et al. ’16]

[Banfi et al. ’16]

Oblateness [Banfi et al. ’14] [Del Duca et al. ’16]

JCEF [Del Duca et al. ’16]

EEC [de Florian et al. ’04] [Del Duca et al. ’16]

…

For 𝛼S we need an observable measured precisely enough and for which accurate prediction can be obtained

EEC

30

EEC is the normalized energy-weighted cross section defined in terms of the angle between two particles i and j in the event:

1

�t

d⌃(�)

d cos�⌘ 1

�t

Z X

i,j

EiEjQ2

d�e+e�!ij+X�(cos�+ cos ✓ij)

Back-to-back region corresponds to 𝜒 = 0Available resummation corresponds to the back-to-back region ( 𝜒⟶ 0 ).

5

10�1

2

5

1

2

1/�td⌃

/d�

[1/rad

]

0 20 40 60 80 100 120 140 160 180� [deg]

Q

OPAL

[OPAL, ‘93]

Why EEC?

31









•In e+ e- collisions only global event shapes and jet rates were used for 𝛼S extraction (so far).

•EEC is based upon two-particle correlations.

•At Q = MZ the high(er) hadronization corr.s may affect differently this type of observable.

[S. Bethke, ’17]

Why EEC?

31









•In e+ e- collisions only global event shapes and jet rates were used for 𝛼S extraction (so far).

•EEC is based upon two-particle correlations.









•At Q = MZ the high(er) hadronization corr.s may affect differently this type of observable.

[S. Bethke, ’17]

Why EEC?

32

•EEC was measured in several experiments at various energies:Exp.

ps (h

psi) [GeV] Events

SLD 91.2(91.2) 6476OPAL 91.2(91.2) 336247OPAL 91.2(91.2) 128032L3 91.2(91.2) 169700

DELPHI 91.2(91.2) 120600TOPAZ 59.0� 60.0(59.5) 540TOPAZ 52.0� 55.0(53.3) 745TASSO 38.4� 46.8(43.5) 6434TASSO 32.0� 35.2(34.0) 52118PLUTO 34.6(34.6) 6964JADE 29.0� 36.0(34.0) 12719CELLO 34.0(34.0) 2600MARKII 29.0(29.0) 5024MARKII 29.0(29.0) 13829MAC 29.0(29.0) 65000TASSO 21.0� 23.0(22.0) 1913JADE 22.0(22.0) 1399CELLO 22.0(22.0) 2000TASSO 12.4� 14.4(14.0) 2704JADE 22.0(22.0) 2112

EEC at Fixed-Order

33

Perturbative expansion of EEC up to NNLO in terms of 𝛼S:1

�t

d⌃(�, µ)

d cos�

�

f.o.

=

↵S

(µ)

2⇡

d

¯A(�, µ)

d cos�+

✓↵S

(µ)

2⇡

◆2

d

¯B(�, µ)

d cos�+

✓↵S

(µ)

2⇡

◆3

d

¯C(�, µ)

d cos�+O(↵4

S

)

Comparison of EEC at various orders to OPAL data

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]Q↵S(Q)

LONLONNLOOPAL

0.75

1.0

1.25

ratio

0 20 40 60 80 100 120 140 160 180

� [deg]

EEC with Resummation

34

The resummation of EEC @ NNLL was done by de Florian and Grazzini [de Florian et al. ’04]

1

�t

d⌃(�, µ)

d cos�

�

res.

=

Q2

8

H(↵S(µ))

Z 1

0db b J0(bQ

py)S(Q, b)

Large logs are exponentiated via the already seen Sudakov form factor:

S(Q, b) = exp

(�Z Q2

b20/b2

dq2

q2

A(↵S(q

2)) log

Q2

q2+B(↵S(q

2))

�)

A(↵S) =1X

n=1

⇣↵S4⇡

⌘nA(n), B(↵S) =

1X

n=1

⇣↵S4⇡

⌘nB(n), H(↵S) = 1 +

1X

n=1

⇣↵S4⇡

⌘nH(n)

To retain NNLL accuracy we need: A(1), A(2), A(3), B(1), B(2), ( H(1) , H(2))

with y = sin2�

2

EEC with Resummation

35

Resumed prediction vs. data

5

10�1

2

5

1

2

1/�td⌃

/d�

[1/rad

]

0 10 20 30 40 50 60� [deg]

Q↵S(Q)

LLNLLNNLLOPAL

Resumed prediction describes the Sudakov peak well, starts to deviate from data at moderate angles ⟹ have to match w/ fixed-order!

Matching

36

•In case of EEC only NNLL resummation is available ⟹ R scheme cannot be used

•Modified R scheme can be used: determining the coefficient of the missing logs from the fixed-order calculation

•Fitting of the coefficients is challenging at NNLO

•When missing logs are present the log-R scheme is much more natural to use

•To be able to use the log-R scheme we have to define a cumulative observable

log-R matching for EEC

37

To define the cumulative distribution have to get rid of the divergence at large 𝜒:1

�te⌃(�, µ) ⌘ 1

�t

Z �

0d�0(1 + cos�0)

d⌃(�0, µ)

d�0=

1

�t

Z y(�)

0d�0 2(1� y0)d⌃(y

0, µ)

dy0

From the cumulative distribution the original one can be easily obtained:

1

�t

d⌃(�, µ)

d�=

1

1 + cos�

d

d�

1

�te⌃(�, µ)

�

In massless QCD: 1�t

e⌃(�max

, µ) = 1

naive R and log-R matching with NLO

38

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]

Q↵S(Q)

NLONNLL+NLO (R)NNLL+NLO (log-R)

0.95

1.0

1.05

ratio

0 20 40 60 80 100 120

� [deg]

Matched EEC predictions using naive R and log-R matching with fixed-order NLO, bands: renorm. uncertainty 𝜇R ∈ [ Q/2 , 2Q ]

log-R matching with NNLO

39

Matched EEC prediction using log-R matching with fixed-order NNLO, band: renorm. uncertainty 𝜇R ∈ [ Q/2 , 2Q ]

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]

Q↵S(Q)

NNLONNLL+NNLO

0.95

1.0

1.05

ratio

0 20 40 60 80 100 120

� [deg]

log-R matching with NNLO

40

Matched EEC predictions using log-R scheme with N(N)LO, bands: renorm. uncertainty 𝜇R ∈ [ Q/2 , 2Q ]

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]

Q↵S(Q)

NNLL+NLONNLL+NNLO

0.9

1.0

1.1

ratio

0 20 40 60 80 100 120

� [deg]

41

Phenomenology

Phenomenology

42

Considered Q = MZ = 91.2 GeV

Comparisons are made against OPAL and SLD data

Due to the low Q value non-perturbative corrections are sizable:1) Comparisons are made with N(N)LO + NNLL 2) N(N)LO + NNLL + analytic model for NP effects

Dokshitzer, et al., ‘99

𝛼S(MZ) value is extracted from various setups

Fits without NP effects

43

Fit rangeNNLL+NLO (R) NNLL+NLO (log-R) NNLL+NNLO (log-R)↵S(MZ) �2/d.o.f. ↵S(MZ) �2/d.o.f. ↵S(MZ) �2/d.o.f.

0� < � < 63� 0.133± 0.001 1.96 0.131± 0.003 1.21 0.129± 0.003 4.1315� < � < 63� 0.132± 0.001 0.59 0.131± 0.003 0.54 0.128± 0.003 1.5815� < � < 120� 0.135± 0.002 3.96 0.134± 0.004 5.12 0.127± 0.003 1.12

•Fit ranges were selected to comply with de Florian et al.

•Large hadronization corrections around Sudakov peak⟹ large 𝜒2/d.o.f. & 𝛼S value

•Excluding the peak region the quality of fit only improves if NNLO is used in the matching

•R-matched NNLL+NLO results are in agreement with de Florian et al. slight change in value only due to difference in A(3) used

Fits with NP effects

44

To incorporate non-perturbative effects the analytical model of Dokshitzer, Marchesini and Webber (’99) was used:

The Sudakov form factor is multiplied by:SNP = e

� 12a1b2

(1� 2a2b)where a1 and a2 are free parameters of the NP model

⟹ the 𝛼S fit becomes a three-parameter fit for 𝛼S, a1, a2

Fit range is selected to be between 0° and 63°

NNLL+NLO+NP

45

NNLL+NLO+NP predictions for EEC fitted to OPAL and SLD data

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]

Q↵S(Q) = 0.134 (R)↵S(Q) = 0.128 (log-R)

OPALSLD

NNLL+NLO+NP (R)NNLL+NLO+NP (log-R)

0.95

1.0

1.05

ratio

0 10 20 30 40 50 60

� [deg]

Note the clear difference in shape between fit and data!

NNLL+NNLO+NP

46

NNLL+NNLO+NP prediction for EEC fitted to OPAL and SLD dataShape is described by prediction! One more `N’ was needed in fixed-order!

5

10

�1

2

5

1

2

1/�

td⌃/d

�[1/rad

]

Q↵S(Q)

OPALSLD

NNLL+NNLO+NP

0.95

1.0

1.05

ratio

0 10 20 30 40 50 60

� [deg]

𝛼S values from fits using NP effects

47

•NNLL+NLO (R):

•NNLL+NLO (log-R):

•NNLL+NNLO:

↵S(MZ)= 0.134+0.001�0.009 , a1 = 1.55

+4.26�1.54 GeV

2 , a2 = �0.13+0.50�0.05 GeV ,�2/d.o.f. = 0.81

↵S(MZ)= 0.128+0.002�0.006 , a1 = 1.17

+1.46�0.29 GeV

2 , a2 = 0.13+0.14�0.09 GeV ,

�2/d.o.f. = 0.85

↵S(MZ)= 0.121+0.001�0.003 , a1 = 2.47

+0.48�2.38 GeV

2 , a2 = 0.31+0.27�0.05 GeV .

�2/d.o.f. = 1.18

48

Outlook & Conclusions

Outlook

49

•The used NP model was very naive•NP effects can be approximated with other methods:

- NLO QCD calculations can be matched with parton shower- Through parton shower different hadronization models are usable:‣Cluster fragmentation model [Winter, Krauss and Soff, ’04]‣Lund string fragmentation model [Sjostrand, Mrenna and Skands, ’06]

•𝛼S will be obtained using all data ever taken for EEC (see slide no. 32)•In order to correctly assess uncertainties we teamed up with

experimentalists

•The study is underway, to appear soon   [AK, Kluth, Somogyi, Tulipant and Verbytskyi]

Conclusions

•NNLL+NNLO matching was performed for EEC

50

•log-R matching scheme was worked out for EEC•𝛼S was obtained from fits to OPAL and SLD data both at NNLL+NLO and

NNLL+NNLO accuracy at parton level and taking into account non-perturbative correction via an analytical model

•Found that adding an extra `N’ in the fixed-order prediction is crucial to get good quality fits

•To be able to use naive R matching one more `N’ is needed in the resummed formula

•Our best fit is:

•A precise fit of 𝛼S is on the way using modern tools to determine hadronization corrections

↵S(MZ)= 0.121+0.001�0.003 ,�

2/d.o.f. = 1.18

51

Thank you for your attention!

Documents

NNLO QCD calculations at work: determining S in e e ...Adam Kardos ! University of Debrecen ! in collaboration with Gábor Somogyi and Zoltán Tulipánt ! Appeared as: arXiv:1708.04093