高精度强耦合常数αs的全球喷流速率拟合结果

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本文档探讨了对量子色动力学（QCD）基本常数αs的高精度测定，通过对e+e-碰撞实验数据的深入分析，特别是利用达勒姆聚类算法进行N3LO（Next-to-Next-to-Leading Order）精度的理论预测和NNLL（Next-to-Next-to-Leading Logarithmic）修正。研究集中在LEP和PETRA对撞机收集的大量数据上，这些数据覆盖的能量范围从35 GeV到207 GeV，这是一个关键的能量区间，因为在这个范围内，粒子物理实验能够提供关于强相互作用的重要信息。研究团队通过全球拟合的方式，对两次和三次喷射速率进行精确测量，并考虑了不同观测值之间的相关性。这一方法的优势在于能够充分利用数据的多维度信息，从而减小统计和系统性误差的影响，提高αs值的精度。他们使用的预测模型具有高度的准确性和稳定性，使得微扰不确定性成为次要因素，而强耦合带来的不确定性则成为主要考量。结果显示，通过对Z玻色子质量附近的喷射率进行分析，得出αs(MZ) = 0.11881±0.00063 (实验误差) ±0.00101 (hadronic误差) ±0.00045 (重整化误差) ±0.00034 (残留误差)。这个结果与当前世界平均值相当，展示了该方法在精确测量标准模型参数方面的潜力。此外，文章发表于《Journal of High Energy Physics》(JHEP) 2019年8月号，是开放获取资源，表明研究成果对学术界开放，便于同行评审和后续研究。论文作者来自德国马克斯·普朗克物理研究所、英国苏塞克斯大学、匈牙利德布勒森大学、CERN理论部、马尔堡大学和埃托沃什洛朗德大学等多个机构，他们的合作反映了国际粒子物理研究领域的协同努力。总结来说，这篇论文的核心贡献是提供了一个基于精确理论计算的αs值，这是通过细致的实验数据分析和高阶理论校正得出的，对于理解QCD的性质和标准模型的精确性测试具有重要意义。

JHEP08(2019)129

contribution computed at ﬁxed order. Hence, we include mass eﬀects directly at the level

of the ﬁnal distributions according to the formulae

(y) = R

LO+NNLL

(y)

(1 − r

(Q)) + r

(Q)R

NNLO

(y)

6=0

(y) = R

NNLO

(y)

(1 − r

(Q)) + r

(Q)R

NLO

(y)

6=0

. (2.7)

The latter quantities are obtained by combining the total cross section at NNLO in-

cluding mass corrections as obtained from ref. [35], and the three- and four-jet rate O(α

)

predictions as computed with the Zbb4 program [33, 36]. The strong coupling used in the

prediction of Zbb4 is then converted into a 5-ﬂavour coupling by means of the matching

relation for α

) [37, 38].

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,

(Q) ≡

6=0

−

→ b

6=0

−

→ hadrons)

. (2.8)

We evaluate the ratio of these cross sections to approximate O(α

) according to ref. [35].

For the bottom quark we used a pole mass of m

= 4.78 GeV, which is consistent with

the corresponding world average [39].

3 Determination of α

To extract the strong coupling we compare the theory predictions described above to the

available data, taking into account the non-perturbative corrections from Monte Carlo

(MC) models, as described in detail in the following.

3.1 Data sets

We select experimental data that satisfy the following basic requirements: (i) measurements

are obtained with both charged and neutral ﬁnal state particles, (ii) corrections for detector

eﬀects have been taken into account, and (iii) corrections for initial-state QED radiation

have been taken into account. We found that the data from refs. [40–48] satisfy these

basic criteria.

However, the measurements of ref. [41] and ref. [42] are superseded by the measure-

ments in ref. [43] and hence not included in our analysis. The analyses in ref. [44] was

excluded as the provided combined uncertainties are much lower than those from later

reﬁned analyses and are close to expected statistical uncertainties estimated from earlier

analysis in ref. [49]. We also excluded the measurements from ref. [45] as these contain

explicit corrections for the contributions of the process e

−

→ b

b from MC simulations

and cannot be treated in the same way as the data without such corrections.

From the selected analyses we also excluded the measurements at

√

s = 172 GeV from

ref. [46] as the background subtraction procedure performed with data of limited statistics

has biased the measurement, e.g. introduced a non-monotonous behaviour of R

(y), see

ref. [50] for details. For similar reasons we exclude data from ref. [47]. The sum of the rates

– 5 –

JHEP08(2019)129

Data MC

Experiment

√

s, Events

GeV GeV

OPAL [43] 91.2(91.2) 91.2 1508031

OPAL [43] 189.0(189.0) 189 3300

OPAL [43] 183.0(183.0) 183 1082

OPAL [43] 172.0(172.0) 172 224

OPAL [43] 161.0(161.0) 161 281

OPAL [43] 130.0–136.0(133.0) 133 630

L3 [46] 201.5–209.1(206.2) 206 4146

L3 [46] 199.2–203.8(200.2) 200 2456

L3 [46] 191.4–196.0(194.4) 194 2403

L3 [46] 188.4–189.9(188.6) 189 4479

L3 [46] 180.8–184.2(182.8) 183 1500

L3 [46] 161.2–164.7(161.3) 161 424

L3 [46] 135.9–140.1(136.1) 136 414

L3 [46] 129.9–130.4(130.1) 130 556

JADE [43] 43.4–44.3(43.7) 44 4110

JADE [43] 34.5–35.5(34.9) 35 29514

ALEPH [48] 91.2(91.2) 91.2 3600000

ALEPH [48] 206.0(206.0) 206 3578

ALEPH [48] 189.0(189.0) 189 3578

ALEPH [48] 183.0(183.0) 183 1319

ALEPH [48] 172.0(172.0) 172 257

ALEPH [48] 161.0(161.0) 161 319

ALEPH [48] 133.0(133.0) 133 806

Table 1. Data used for the determination of α

in this work. The ranges of collision energies, their

weighted average value (in brackets) and the number of events for each experiment are given as

quoted in the original publications.

for the data sets in ref. [48] deviates from unity, and the largest deviation reaches almost

0.03 for the data set at

√

s = 200 GeV. For this reason we excluded this data from the ﬁts

as well.

We summarise the information on the selected data sets in table 1. For some data

sets the uncertainties were updated before the ﬁt, as follows. We added in quadrature

the two available systematic uncertainties for the measurements at

√

s = 91 GeV from

ref. [48]. The data from JADE [43] does not include systematic uncertainties related to

the choice of MC samples used for the calculations of detector corrections. Later studies

with the same data [45] indicated that such uncertainties can be at least as large as the

statistical uncertainties. Therefore, in this case we added a relative uncertainty of 1.5% as

an estimation of missing systematic uncertainties.

The measurements of the jet rates selected for the analysis are provided in the original

publications without correlations between the individual points and without decomposition

of the total systematic uncertainties. To perform an accurate extraction procedure, we ex-

amined the available data and uncertainties and built a covariance matrix for all measured

sets of data. This procedure consists of the following steps.

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