Nuclear and Particle Physics

Nuclear and Particle Physics Proceedings 00 (2016) 1–4

Nuclear and Particle Physics

Proceedings

α s 2016

S. Bethke

a

Max-Planck-Institute of Physics, F¨ohringer Ring 6, 80805 Munich, Germany

Abstract

An update of measurements of the strong coupling α

and the determination of the world average value of α

(M

) is presented, resulting in

α

(M

) = 0.1181 ± 0.0011.

Keywords: strong coupling, alpha-s, Quantum Chromodynamics

Several new measurements of α

, the coupling strength of the strong interaction between quarks and gluons, became available since previous summaries were given at this conference series [1] and in the 2014 Review of Particle Properties [2]. In the fol- lowing, those new results which are used to determine the new world average value of α

, i.e. those that are based 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 demonstrating asymptotic freedom, i.e. the specific energy dependence of α

as predicted by Quantum Chromodynamics, even if being based on next-to-leading (NLO) perturbation theory only, will be reviewed.

This update with status of April 2016 is extracted from 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 presentation of theoretical and experimental issues concerning tests of Quantum Chromodynamics.

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

∗

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

Email address: bethke@mpp.mpg.de (S. Bethke)

• updated results from τ-decays [4] [5] [6], based on a re-analysis of ALEPH data and on complete N

LO 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 e ff ective field theory matched to NNLO perturbation theory [10],

• α

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-jet di ff erential cross section [13], and from energy- correlations [14], all in NLO QCD, plus one deter- mination in complete NNLO, from a measurement of the tt cross section at √

s = 7 TeV [15];

• and finally, an update of α

from a global fit to electroweak precision data [16], based on complete N

LO perturbation theory.

All measurements based on at least full NNLO

perturbation theory are summarised in figure 1, and

are ordered according to subclasses of τ-decays, lat-

tice results, structure functions, e

e

-annihilation, elec-

troweak precision fits and hadron colliders.

S. Bethke / Nuclear and Particle Physics Proceedings 00 (2016) 1–4 2

With the exception of lattice results, most results within their subclass are strongly correlated, however to an unknown degree, as they largely use similar data sets and / or theoretical predictions. The large scatter between many of these measurements, sometimes with only 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 selected results is taken as pre-average value for each subclass, and the unweighted average of the quoted uncertainties is 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 available in full NNLO, so that these measurements alone define the average value for their subclass. Note that more measurements of top-quark pair production at LHC are meanwhile available, indicating that - on average - a larger value of α

(M

) is likely to emerge in the future;

see also [17] and the presentation of T. Klijnsma at this conference [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 of α

( M

).

Subclass α

(M

)

τ-decays 0.1192 ± 0.0018

lattice QCD 0.1188 ± 0.0011 structure functions 0.1156 ± 0.0021 e

e

[jets & shps] 0.1169 ± 0.0034 hadron collider 0.1151

ew precision fits 0.1196 ± 0.0030

Assuming that the resulting pre-averages are largely independent of each other, the final world average value is determined as the weighted average of the pre- averaged values. An initial uncertainty of the central value is calculated treating the uncertainties of all in- put values as being uncorrelated and of Gaussian nature, and the overall χ

to the central value is determined. If the initial χ

is smaller than the number of degrees of freedom, an overall, a-priori unknown correlation co- e ffi cient is introduced and adjusted such that the total χ

/ d.o.f. equals unity. Applying this procedure to the values listed in table 1 results in the new world average of

α

(M

) = 0.1181 ± 0.0011 .

This value is in good agreement with that from

Figure 1: Summary of determinations of α

. The light-shaded bands

and 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 and

short-dashed line represent the new overall world average of α

.

S. Bethke / Nuclear and Particle Physics Proceedings 00 (2016) 1–4 3

2013/2014, which was α

(M

) = 0.1185 ± 0.0006, how- ever at a somewhat decreased central value and with an overall uncertainty that has almost doubled. These changes are mainly due to the following reasons:

• the uncertainty of the lattice result, now deter- mined as unweighted average of central values and errors, is more conservative than that used in the previous review, leading to a reduced relative weight of lattice results, and to a larger uncertainty of the new world average;

• the relatively low value of α

from the new sub- class of hadron collider results, which currently consists of only one measurement of the tt cross section at √

s = 7 TeV, and which appears to be

”lowish” if compared to later measurements at the same and at higher √

s [17, 18].

It may be instructive to review the history and devel- opments of ”world average” values of α

(M

), which is extracted from a variety of reviews [2, 19–27] and given in fig. 2. ”Quantum jumps” of central values and of the size of overall uncertainties can be identified with the advent of precision data from the LEP and the HERA colliders, with the availability of and restriction to mea- surements based on NNLO perturbative predictions, and the inclusion of results from unquenched lattice compu- tations.

Figure 2: History of world average values of α

(M

).

While there is still room for improved measurements and treatments of systematic uncertainties, the data and results, especially when including measurements which are available at NLO only, consistently demonstrate and

QCD α

(M

) = 0.1181 ± 0.0011

e.w. precision fits (N³LO)

0.1 0.2 0.3

α

(Q

)

1 10 100

Q [GeV]

Heavy Quarkonia (NLO) e⁺e^– jets & shapes (res. NNLO) DIS jets ^(NLO)

April 2016

τ decays (N³LO)

1000 (NLO

pp –> tt(NNLO)

(–) )

Figure 3: Summary of measurements of α

as a function of the energy scale Q. The respective degree of QCD perturbation theory used in the extraction of α

is indicated in brackets (NLO: next-to-leading order;

NNLO: next-to-next-to leading order; res. NNLO: NNLO matched with resummed next-to-leading logs; N

LO: next-to-NNLO).

prove asymptotic freedom and the running of α

, as pre- dicted by QCD, up to energies beyond 1 TeV, see fig- ure 3.

Acknowledgements

I wish to cordially thank S. Narison for his untiring and successful organisation of this pleasant conference, and G. Dissertori and G.P. Salam for e ffi cient collaboration in preparing the QCD review and its updates for the Par- ticle Data Group’s Review of Particle Physics.

References

[1] S. Bethke, Nucl.Phys.Proc.Suppl. 234 (2013) 229-234.

[2] S. Bethke, G. Dissertori and G.P. Salam in: K. A. Olive et al.

[PDG Collab.], Chin. Phys. C 38 (2014) 090001.

[3] S. Bethke, G. Dissertori and G.P. Salam in: C. Patrignani et al.

[PDG Collab.], Chin. Phys. C 40 (2016) 100001.

[4] M. Davier et al., Eur. Phys. J. C74, 2803 (2014), [arXiv:1312.1501 [hep-ph]].

[5] D. Boito et al., Phys. Rev. D91, 034003 (2015), [arXiv:1410.3528 [hep-ph]] .

[6] A. Pich, Prog. Part.Nucl. Phys. 75 (2014) 41, [arXiv:1310.7922 [hep-ph]].

[7] S. Aoki et al., Eur. Phys. J. C74, 2890 (2014), [arXiv:1109.1388 [hep-ph]].

[8] A. Bazavov et al., Phys. Rev. D90, 074038 (2014), [arXiv:1407.8437 [hep-ph]].

[9] L. A. Harland-Lang et al., [arXiv:1506.05682 [hep-ph]]

[10] A.H. Hoang et al., Phys. Rev. D91, 094018 (2015),

[arXiv:1501.04111 [hep-ph]],

S. Bethke / Nuclear and Particle Physics Proceedings 00 (2016) 1–4 4

[11] S. Chatrchyan et al. [CMS], Eur. Phys. J. C73, 10 (2013), [arXiv:1304.7498 [hep-ex]].

[12] V. Khachatryan et al. [CMS], Eur. Phys. J C75, 6 (2015), [arXiv:1410.6765 [hep-ex]].

[13] V. Khachatryan et al. [CMS], Eur. Phys. J. C75 5, 186 (2015) [arXiv:1412.1633 [hep-ex]].

[14] G. Aad et al. [ATLAS], Phys.Lett. B750 (2015) 427, [arXiv:1508.01579 [hep-ex]].

[15] S. Chatrchyan et al. [CMS], Phys. Lett. B728, 496 (2014) [arXiv:1307.1907 [hep-ex]]

[16] M. Baak et al. [Gfitter], Eur. Phys. J. C74 , 304660 (2014), [arXiv1407.3792 [hep-ph]].

[17] S. Bethke, G. Dissertori, T. Klijnsma and G.P. Salam, contribu- tion to the Workshop on High Precision α

Measurements: from LHC to FCC-ee, CERN, Geneva, October 12th-13th, 2015.

[18] S. Bethke, G. Dissertori, T. Klijnsma and G.P. Salam, contribu- tion to this conference.

[19] G. Altarelli, Ann.Rev.Nucl.Part.Sci. 39 (1989) 357.

[20] S. Bethke, contribution to QCD’94, Montpellier (France) 1994, Nucl.Phys.Proc.Suppl. 39BC (1995) 198.

[21] S. Bethke, IV

Int. Symp. on Radiative Corrections, Barcelona 1998, hep-ex/9812026.

[22] S. Bethke, J.Phys. G26 (2000) R27, [hep-ex/0004021].

[23] S. Bethke, 9

Intern. Conf. in Quantum Chromodynamics (QCD 02), Montpellier (France) 2002, Nucl.Phys.Proc.Suppl.

121 (2003) 74, [hep-ex/0211012].

[24] S. Bethke, Loops and Legs in Quantum Field Theory, Zinnovitz 2004, Nucl.Phys.Proc.Suppl. 135 (2004) 345-352, [hep-ex/0407021].

[25] S. Bethke, Prog.Part.Nucl.Phys. 58 (2007) 351, [hep-ex/

0606035].

[26] S. Bethke, Eur.Phys.J. C64 (2009) 689, [arXiv:0908.1135].

[27] S. Bethke, in: Workshop on Precision Measurements of α

, Mu-

nich 2011, arXiv:1110.0016.