EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
欧洲核子研究组织

CERN-EP/2000-023

7 February 2000 2000 年 2 月 7 日

Charged and Identified Particles
带电和鉴定的粒子

in the Hadronic Decay of
在强子衰变中

W Bosons and in $e^{+}{e}^{-}\to$qq from
W 玻色子和在 qq 中的

130 to 200 GeV 130 到 200 吉电子伏特

DELPHI Collaboration DELPHI 合作组

Abstract 摘要

Inclusive distributions of charged particles in hadronic W decays are experimentally investigated using the statistics collected by the DELPHI experiment at LEP during 1997, 1998 and 1999, at centre-of-mass energies from 183 to around 200 GeV. The possible effects of interconnection between the hadronic decays of two Ws are not observed.
利用 DELPHI 实验在 1997 年、1998 年和 1999 年在 LEP 收集的数据，研究了强子 W 衰变中带电粒子的分布情况，中心质心能量范围从 183 到约 200 GeV。未观察到两个 W 强子衰变之间的相互作用可能产生的影响。

Measurements of the average multiplicity for charged and identified particles in $\mathrm{q}\overline{\mathrm{q}}$ and WW events at centre-of-mass energies from 130 to 200 GeV and in W decays are presented.
在中心质能从 130 到 200 GeV 的 $\mathrm{q}\overline{\mathrm{q}}$ 和 WW 事件以及 W 衰变中，给出了带电和鉴别粒子的平均多重性测量结果。

The results on the average multiplicity of identified particles and on the position ${\xi }^{\ast }$ of the maximum of the ${\xi }_p=-\mathrm{l}\mathrm{o}\mathrm{g}(\frac{2\mathrm{p}}{\sqrt{\mathrm{s}}})$ distribution are compared with predictions of JETSET and MLLA calculations.
对于已鉴定粒子的平均多重性和 ${\xi }^{\ast }$ 分布最大值的位置，将其结果与 JETSET 和 MLLA 计算的预测进行比较。

The study of the production properties of charged and identified hadrons (${\pi }^{+}$, K${}^{+}$, K${}^0$, p and $\mathrm{\Lambda }$)1 in $\mathrm{q}\overline{\mathrm{q}}$ events at LEP 2 allows QCD based models to be tested through the comparison with LEP 1 data, in particular the validity of high-energy extrapolations of Monte Carlo models tuned at the ${\mathrm{Z}}^0$.
在 LEP 2 的 $\mathrm{q}\overline{\mathrm{q}}$ 事件中研究带电和鉴别强子（ ${\pi }^{+}$ ，K ${}^{+}$ ，K ${}^0$ ，p 和 $\mathrm{\Lambda }$ ）的产生特性，可以通过与 LEP 1 数据进行比较来测试基于 QCD 的模型，特别是在 ${\mathrm{Z}}^0$ 调整的蒙特卡罗模型的高能外推的有效性。

Footnote 1: Unless otherwise stated antiparticles are included as well.
脚注 1：除非另有说明，反粒子也包括在内。

In the case of WW events in which both W bosons decay hadronically, these kind of studies, besides providing checks for QCD-inspired models, are expected to give insights into possible correlations and/or final state interactions between the decay products of the two W bosons.
在两个 W 玻色子衰变为强子的 WW 事件中，这类研究除了提供对 QCD 启发模型的检验外，还有望揭示两个 W 玻色子衰变产物之间可能存在的相关性和/或末态相互作用的见解。

Hadron production in $e^{+}{e}^{-}$ and QCD
在 $e^{+}{e}^{-}$ 和量子色动力学中的强子产生

The way quarks and gluons transform into hadrons is complex and can not be completely described by QCD. In the Monte Carlo simulations (as in [1]), the hadronisation of a $\mathrm{q}\overline{\mathrm{q}}$ pair is split into 3 phases. In a first phase, gluon emission and parton branching of the original $\mathrm{q}\overline{\mathrm{q}}$ pair take place. It is believed that this phase can be described by perturbative QCD (most of the calculations have been performed in leading logarithmic approximation LLA). In a second phase, at a certain virtuality cut-off scale $Q_0$, where ${\alpha }_s(Q_0)$ is still small, quarks and gluons produced in the first phase are clustered in colour singlets and transform into mesons and baryons. Only phenomenological models, which need to be tuned to the data, are available to describe this stage of fragmentation; the models most frequently used in $e^{+}{e}^{-}$ annihilations are based on string and cluster fragmentation. In the third phase, the unstable states decay into hadrons which can be observed and identified in the detector. These models account correctly for most of the features of the $\mathrm{q}\overline{\mathrm{q}}$ events such as, for instance, the average multiplicity and inclusive momentum spectra.
夸克和胶子转化为强子的方式是复杂的，不能完全用量子色动力学（QCD）来描述。在蒙特卡洛模拟中（如[1]中所示），一个 $\mathrm{q}\overline{\mathrm{q}}$ 对的强子化过程被分为 3 个阶段。在第一阶段，原始 $\mathrm{q}\overline{\mathrm{q}}$ 对发生胶子辐射和部分子分支。人们认为这个阶段可以用微扰 QCD 来描述（大部分计算都是在领头对数近似 LLA 下进行的）。在第二阶段，当虚拟度截断尺度 $Q_0$ ，其中 ${\alpha }_s(Q_0)$ 仍然很小时，第一阶段产生的夸克和胶子被聚集成色单态，并转化为介子和重子。只有需要根据数据进行调整的现象学模型可用于描述这个碎裂阶段；在 $e^{+}{e}^{-}$ 湮灭中最常用的模型是基于弦和簇状碎裂。在第三阶段，不稳定态衰变为可以在探测器中观测和鉴别的强子。这些模型正确地解释了 $\mathrm{q}\overline{\mathrm{q}}$ 事件的大部分特征，例如平均多重性和包容性动量谱。

A different and purely analytical approach (see e.g. [2] and references therein) giving quantitative predictions of hadronic spectra are QCD calculations using the so-called Modified Leading Logarithmic Approximation (MLLA) under the assumption of Local Parton Hadron Duality (LPHD) [3, 4]. In this picture multi-hadron production is described by a parton cascade, and the virtuality cut-off $Q_0$ is lowered to values of the order of 100 MeV, comparable to the hadron masses; it is assumed that the results obtained for partons are proportional to the corresponding quantities for hadrons.
使用不同且纯粹分析的方法（参见例如[2]及其引用文献），在局域粒子强子对偶（LPHD）[3, 4]的假设下，通过使用所谓的修正领头对数近似（MLLA）的量子色动力学（QCD）计算，可以给出强子谱的定量预测。在这个模型中，多强子产生被描述为一个粒子级联，虚拟性截断 $Q_0$ 被降低到约 100 MeV 的量级，与强子质量相当；假设得到的粒子结果与相应的强子数量成比例。

The MLLA/LPHD predictions involve three parameters: an effective scale parameter ${\mathrm{\Lambda }}_{\mathrm{eff}}$, a virtuality cut-off $Q_0$ in the evolution of the parton cascade and an overall normalisation factor $K_{\mathrm{LPHD}}$. The momentum spectra of hadrons can be calculated as functions of the variable ${\xi }_p=-\ln (\frac{2p}{\sqrt{s}})$, with $p$ being the particle's momentum and $\sqrt{s}$ the centre-of-mass energy:
MLLA/LPHD 预测涉及三个参数：一个有效的尺度参数 ${\mathrm{\Lambda }}_{\mathrm{eff}}$ ，在部子级联演化中的虚拟截断 $Q_0$ 和一个整体归一化因子 $K_{\mathrm{LPHD}}$ 。强子的动量谱可以根据变量 ${\xi }_p=-\ln (\frac{2p}{\sqrt{s}})$ 计算，其中 $p$ 是粒子的动量， $\sqrt{s}$ 是质心能量。

$$\begin{array}{}\text{(1)}& \frac{dn}{d{\xi }_p}=K_{\mathrm{LPHD}}\cdot f({\xi }_p,X,\lambda )\end{array}$$

with 与

$$\begin{array}{}\text{(2)}& X=\log \frac{\sqrt{s}}{Q_0};\lambda =\log \frac{Q_0}{{\mathrm{\Lambda }}_{\mathrm{eff}}}.\end{array}$$

Due to uncertainties from higher order corrections ${\mathrm{\Lambda }}_{\mathrm{eff}}$ cannot be identified with ${\mathrm{\Lambda }}_{\overline{MS}}$. In equation (1), $n$ is the average multiplicity per bin of ${\xi }_p$, and the function $f$ has the form of a "hump-backed plateau". It can be approximated by a distorted Gaussian [5, 6]$$\begin{array}{}\text{(3)}& DG(\xi ;N,\bar{\xi },\sigma ,s_k,k)=\frac{N}{\sigma \sqrt{2\pi }}\exp (\frac{1}{8}k+\frac{1}{2}{s}_k\delta -\frac{1}{4}(2+k){\delta }^2+\frac{1}{6}{s}_k{\delta }^3+\frac{1}{24}k{\delta }^4),\end{array}$$
由于高阶修正的不确定性， ${\mathrm{\Lambda }}_{\mathrm{eff}}$ 不能与 ${\mathrm{\Lambda }}_{\overline{MS}}$ 确定。在方程（1）中， $n$ 是每个 ${\xi }_p$ 区间的平均多重性，而函数 $f$ 的形式类似于“驼峰状高原”。它可以用扭曲的高斯函数 [5, 6] 近似表示。 $$\begin{array}{}\text{(3)}& DG(\xi ;N,\bar{\xi },\sigma ,s_k,k)=\frac{N}{\sigma \sqrt{2\pi }}\exp (\frac{1}{8}k+\frac{1}{2}{s}_k\delta -\frac{1}{4}(2+k){\delta }^2+\frac{1}{6}{s}_k{\delta }^3+\frac{1}{24}k{\delta }^4),\end{array}$$

where $\delta =(\xi -\bar{\xi })/\sigma$, $\overline{\xi }$ is the mean of the distribution, $\sigma$ is the square root of its variance, $s_k$ its skewness and $k$ its kurtosis. For an ordinary Gaussian these last two parameters vanish. The mean, $\bar{\xi }$, coincides with the peak of the distribution, ${\xi }^{\ast }$, only up to next-to-leading order in ${\alpha }_s$.
其中 $\delta =(\xi -\bar{\xi })/\sigma$ ， $\overline{\xi }$ 是分布的均值， $\sigma$ 是方差的平方根， $s_k$ 是偏度， $k$ 是峰度。对于普通的高斯分布，这两个参数都为零。均值 $\bar{\xi }$ 与分布的峰值 ${\xi }^{\ast }$ 仅在 ${\alpha }_s$ 的次领先阶段相符。

To check the validity of the MLLA/LPHD approach, one can study the evolution of the position of the maximum, ${\xi }^{\ast }$, as a function of $\sqrt{s}$. In the context of MLLA/LPHD the dependence of ${\xi }^{\ast }$ on the centre-of-mass energy can be expressed as [2, 5]:
为了检验 MLLA/LPHD 方法的有效性，可以研究最大值 ${\xi }^{\ast }$ 随 $\sqrt{s}$ 的变化。在 MLLA/LPHD 的背景下， ${\xi }^{\ast }$ 对于质心能量的依赖可以表示为[2, 5]：

$$\begin{array}{}\text{(4)}& {\xi }^{\ast }=Y(\frac{1}{2}+\sqrt{C/Y}-C/Y)+F_h(\lambda ),\end{array}$$

where 在哪里

$$\begin{array}{}\text{(5)}& Y=\log \frac{\sqrt{s}/2}{{\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}}C={\left(\frac{11N_c/3+2n_f/(3N_c^2)}{4N_c}\right)}^2\cdot \left(\frac{N_c}{11N_c/3-2n_f/3}\right),\end{array}$$

with $N_c$ being the number of colours and $n_f$ the number of active quark flavours in the fragmentation process. The function $F_h(\lambda )$ depends on the hadron type through $\lambda =\log (Q_0/{\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}})$[2], and it can be approximated as $F_h(\lambda )=-1.46\lambda +0.207{\lambda }^2$ with an error of $\pm 0.06$.
$N_c$ 是颜色的数量， $n_f$ 是碎裂过程中活跃夸克味道的数量。函数 $F_h(\lambda )$ 通过 $\lambda =\log (Q_0/{\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}})$ [2]与强子类型相关，并且可以近似为 $F_h(\lambda )=-1.46\lambda +0.207{\lambda }^2$ ，误差为 $\pm 0.06$ 。

Interference and final state interactions in W decays
W 衰变中的干涉和末态相互作用

The possible presence of interference due to colour reconnection and Bose-Einstein correlations (see for example [7, 8, 9, 10, 11, 12] and [13, 14] for reviews) in hadronic decays of WW pairs may provide information on hadron formation at a time scale smaller than 1 fm/$c$. At the same time it can induce a systematic uncertainty on the W mass measurement in the 4-jet mode [13] comparable with the expected accuracy of the measurement at LEP 2.
由于色重联和玻色-爱因斯坦关联的可能干扰存在（参见例如[7, 8, 9, 10, 11, 12]和[13, 14]的综述），WW 对的强子衰变可能提供关于小于 1 fm/ $c$ 时间尺度下强子形成的信息。同时，它可能对 4 喷注模式下 W 质量测量产生系统性不确定性[13]，与 LEP 2 上预期的测量精度相当。

Interconnection can happen due to the fact that the lifetime of the W (${\tau }_W\simeq \hslash /{\mathrm{\Gamma }}_W\simeq 0.1$ fm/$c$) is an order of magnitude smaller than the typical hadronisation times. The interconnection between the products of the hadronic decays of different Ws in WW pair events can occur at several stages: (1) from colour rearrangement between the quarks coming from the primary branching, (2) due to gluon exchanges during the parton cascade, (3) in the mixing of identical pions or kaons due to Bose-Einstein correlations. The first two are QCD effects. They can mix the two colour singlets and produce hadrons which cannot be uniquely assigned to either W. The perturbative effects are colour suppressed and the possible shift is expected to be only about 5 MeV in the W mass [7].
互连是因为 W 的寿命（ ${\tau }_W\simeq \hslash /{\mathrm{\Gamma }}_W\simeq 0.1$ fm/ $c$ ）比典型的强子化时间小一个数量级。在 WW 对事件中，不同 W 的强子衰变产物之间的互连可以在几个阶段发生：（1）来自主要分支的夸克之间的颜色重排，（2）在部分级联期间由于胶子交换，（3）由于玻色-爱因斯坦关联而导致相同的π介子或 K 介子的混合。前两者是量子色动力学效应。它们可以混合两个色单态并产生不能唯一归属于任何 W 的强子。微扰效应受到颜色抑制，预计 W 质量只会发生约 5 MeV 的偏移[7]。

Non-perturbative effects need model calculations. Several models have been proposed (for reviews see for example [13] and [15]) and have already been included in the widely used event generators PYTHIA [1], ARIADNE [16] and HERWIG [17]. In these models the final state quarks after the parton shower can be rearranged to form colour singlets with probabilities which in some cases are free parameters. The shift on the W mass in these models is typically smaller than 50 MeV [18], but other observables are affected by colour rearrangement. Generally these models suggest a small effect on the total charged particle multiplicity, of the order of $-1\mathrm{\%}$ to $-2\mathrm{\%}$[15, 19, 14]. Dedicated detailed simulations of the response of the DELPHI detector to such models showed that this effect is substantially unaffected by the event selection criteria and by the detector performance. For identified heavy particles, such as K${}^{+}$ and p, the effects due to colour reconnection are expected to be stronger [20], but the experimental verification is complicated by losses in statistics. The same applies to the particle spectrum at low momentum [15]. Bose-Einstein Correlations could also slightly change the multiplicity for (4$q$) events in some models [12, 21].
非摄动效应需要模型计算。已经提出了几种模型（有关综述请参见[13]和[15]），并已经包含在广泛使用的事件生成器 PYTHIA [1]、ARIADNE [16]和 HERWIG [17]中。在这些模型中，部分子喷注后的最终态夸克可以重新排列，形成彩色单态，其概率在某些情况下是自由参数。这些模型中 W 质量的偏移通常小于 50 MeV [18]，但其他可观测量受到了彩色重新排列的影响。一般来说，这些模型对总带电粒子多重性的影响很小，大约在 $-1\mathrm{\%}$ 到 $-2\mathrm{\%}$ 的数量级上 [15, 19, 14]。对 DELPHI 探测器对这些模型的响应进行了专门的详细模拟，结果表明这种效应在事件选择标准和探测器性能的影响下基本不受影响。对于识别的重粒子，如 K ${}^{+}$ 和 p，由于统计损失，彩色重连接效应预计会更强 [20]。对于低动量下的粒子谱，情况也是如此 [15]。 Bose-Einstein 相关也可能在一些模型中略微改变(4 $q$ )事件的多重性[12, 21]。

The WW events allow a comparison of the characteristics of the W hadronic decays when both Ws decay in hadronic modes (referred to here as the (4$q$) mode) with the case in which only one W decays hadronically ((2$q$) mode). These characteristics should be the same in the absence of interference between the hadronic decay products from different W bosons.
WW 事件允许比较两个 W 玻色子同时以强子模式衰变（在此称为（4 $q$ ）模式）和只有一个 W 玻色子以强子模式衰变的情况（（2 $q$ ）模式）下 W 强子衰变的特征。在没有不同 W 玻色子的强子衰变产物之间的干涉的情况下，这些特征应该是相同的。

Previous experimental results based on the statistics collected by LEP experiments during 1997 (see [22, 23, 24] for reviews) did not indicate at that level of statistics the presence of interconnection or correlation effects.
之前基于 LEP 实验在 1997 年收集的统计数据的实验结果（参见[22, 23, 24]进行综述）并未表明在那个统计水平下存在相互连接或相关效应。

This paper presents measurements of:
本文介绍了以下测量结果：

• the charged particle multiplicities for the q$\overline{\mathrm{q}}$ events in the data sample collected by the DELPHI experiment at LEP during 1997, 1998 and 1999, at the centre-of-mass energies from 183, 189 and from 192 to 200 GeV respectively;
  DELPHI 实验在 1997 年、1998 年和 1999 年在 LEP 收集的数据样本中，对于中心质能量分别为 183、189 和 192 至 200 GeV 的 q $\overline{\mathrm{q}}$ 事件的带电粒子多重性
• the charged particle multiplicity and inclusive distributions for WW events at 183 and 189 GeV (multiplicity values at 189 GeV are expected to be slightly higher than that at 183 GeV due to increased phase space. However this effect is below the precision obtainable with the present data samples);
  183 和 189 GeV 的 WW 事件的带电粒子多重性和包容分布（由于相空间增加，189 GeV 时的多重性值预计略高于 183 GeV。然而，这种效应低于目前数据样本可获得的精度）。
• The average multiplicities for identified charged and neutral particles (${\pi }^{+}$, K${}^{+}$, K${}^0$, p and $\mathrm{\Lambda }$), and the position ${\xi }^{\ast }$ of the maximum of the ${\xi }_p$ distribution for identified particles in q$\overline{\mathrm{q}}$ events from 130 GeV to 189 GeV and in WW events at 189 GeV.
  在 130 GeV 到 189 GeV 的 q $\overline{\mathrm{q}}$ 事件中，以及在 189 GeV 的 WW 事件中，已鉴别的带电和中性粒子（ ${\pi }^{+}$ ，K ${}^{+}$ ，K ${}^0$ ，p 和 $\mathrm{\Lambda }$ ）的平均多重性，以及鉴别粒子的 ${\xi }_p$ 分布的最大位置 ${\xi }^{\ast }$ 。

2 Data Sample and Event Preselection at 183 and 189 GeV
2 个数据样本和 183 和 189 GeV 的事件预选

Data corresponding to total luminosities of 157.7 pb${}^{-1}$ (54.1 pb${}^{-1}$) at centre-of-mass energies around 189 (183) GeV, and data taken in 1999 corresponding to total luminosities of 25.8 pb${}^{-1}$, 77.4 pb${}^{-1}$ and 83.8 pb${}^{-1}$ at centre-of-mass energies around 192 GeV, 196 GeV and 200 GeV respectively were analysed. A description of the DELPHI detector can be found in reference [25]; its performance is discussed in reference [26].
对应于质心能量约为 189（183）GeV 的 157.7 pb ${}^{-1}$ （54.1 pb ${}^{-1}$ ）的总光度数据，以及 1999 年在质心能量约为 192 GeV、196 GeV 和 200 GeV 的 25.8 pb ${}^{-1}$ 、77.4 pb ${}^{-1}$ 和 83.8 pb ${}^{-1}$ 的总光度数据进行了分析。DELPHI 探测器的描述可在参考文献[25]中找到；其性能在参考文献[26]中讨论。

A preselection of hadronic events was made, requiring at least 6 charged particles and a total transverse energy of all the particles above 20% of the centre-of-mass energy $\sqrt{s}$. In the calculation of the energies $E$, all charged particles were assumed to have the pion mass. Charged particles were required to have momentum $p$ above 100 MeV/$c$ and below 1.5 times the beam energy, a relative error on the momentum measurement $\mathrm{\Delta }p/p<1$, angle $\theta$ with respect to the beam direction between 20${}^{\circ }$ and 160${}^{\circ }$, and a distance of closest approach to the interaction point less than 4 cm in the plane perpendicular to the beam axis (2 cm in the analyses of identified charged particles) and less than 4/$\sin \theta$ cm along the beam axis (2 cm in the analyses of identified charged particles).
进行了强子事例的预选，要求至少有 6 个带电粒子和所有粒子的总横向能量超过中心质能的 20% $\sqrt{s}$ 。在能量计算中 $E$ ，假设所有带电粒子的质量为π介子质量。要求带电粒子的动量 $p$ 在 100 MeV/ $c$ 以上且低于束流能量的 1.5 倍，动量测量的相对误差 $\mathrm{\Delta }p/p<1$ ，与束流方向的夹角 $\theta$ 在 20 ${}^{\circ }$ 到 160 ${}^{\circ }$ 之间，并且在垂直于束流轴的平面上与相互作用点的最近距离小于 4 cm（在鉴别带电粒子的分析中为 2 cm），并且沿着束流轴的距离小于 4/ $\sin \theta$ cm（在鉴别带电粒子的分析中为 2 cm）。

After the event selection charged particles were also required to have a track length of at least 30 cm, and in the charged identified particles analysis a momentum $p>200$ MeV/$c$.
事件选择后，要求带电粒子的轨迹长度至少为 30 厘米，在带电粒子鉴别分析中，动量 $p>200$ MeV/ $c$ 。

The influence of the detector on the analysis was studied with the full DELPHI simulation program, DELSIM [26]; events were generated with PYTHIA 5.7, using the JETSETfragmentation with Parton Shower (PS) [1] with parameters tuned to fit LEP 1 data from DELPHI [27]. The initial state for the WW 1999 sample was generated using EXCALIBUR version 1.08 [28]. The particles were followed through the detailed geometry of DELPHI with simulated digitizations in each detector. These data were processed with the same reconstruction and analysis programs as the real data.
使用完整的 DELPHI 模拟程序 DELSIM [26]研究了探测器对分析的影响；使用 PYTHIA 5.7 生成事件，使用经过调整以适应 DELPHI [27]的 LEP 1 数据的 JETSET 碎裂和 Parton Shower（PS）[1]。WW 1999 样本的初始状态使用 EXCALIBUR 版本 1.08 [28]生成。通过 DELPHI 的详细几何结构跟踪粒子，并在每个探测器中进行模拟数字化。这些数据使用与真实数据相同的重建和分析程序进行处理。

To check the ability of the simulation to model the efficiency for the reconstruction of charged particles, the samples collected at the ${\mathrm{Z}}^0$ pole during 1998 and 1997 were used. From these samples, by integrating the distribution of ${\xi }_E=-\ln (\frac{2E}{\sqrt{s}})$, where $E$ is the energy of the particle, corrected bin by bin using the simulation, the average charged particle multiplicities at the ${\mathrm{Z}}^0$ were measured. The values were found to be $20.93\pm 0.03(stat)$ and $20.60\pm 0.03(stat)$ respectively, in satisfactory agreement with the world average of $21.00\pm 0.13$[29]. The ratios of the world average value to the measured multiplicities at the ${\mathrm{Z}}^0$, respectively $1.0033\pm 0.0064$ and $1.0194\pm 0.0065$ from the ${\mathrm{Z}}^0$ data in 1998 and 1997, were used to correct the measured multiplicities at high energies in the respective years.
为了检查模拟能否准确模拟充电粒子重建的效率，我们使用了 1998 年和 1997 年在 ${\mathrm{Z}}^0$ 极收集的样本。通过对这些样本进行积分，得到了 ${\xi }_E=-\ln (\frac{2E}{\sqrt{s}})$ 的分布，其中 $E$ 是粒子的能量，通过使用模拟进行逐个 bin 的修正，测量了 ${\mathrm{Z}}^0$ 处的平均充电粒子多重性。测量结果分别为 $20.93\pm 0.03(stat)$ 和 $20.60\pm 0.03(stat)$ ，与世界平均值 $21.00\pm 0.13$ [29] 较好地吻合。将世界平均值与 ${\mathrm{Z}}^0$ 处测量的多重性之比，分别用于修正相应年份高能量处的测量多重性，即 $1.0033\pm 0.0064$ 和 $1.0194\pm 0.0065$ 。

The cross-section for $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}(\gamma )$ above the ${\mathrm{Z}}^0$ peak is dominated by radiative $\mathrm{q}\overline{\mathrm{q}}\gamma$ events; the initial state radiated photons (ISR photons) are generally aligned along the beam direction and not detected. In order to compute the hadronic centre-of-mass energy, $\sqrt{s^{\prime }}$, the procedure described in reference [30] was used. The procedure clusters the particles into jets using the DURHAM algorithm [31], excluding candidate ISR photons and using a $y_{cut}=0.002$. The reconstructed jets and additional ISR photons are then fitted with a three constraint fit (energy and transverse momentum, leaving free the $z$ component of the missing momentum). The hadronic centre-of-mass energy, $\sqrt{s^{\prime }}$, is the invariant mass of the jets using the fitted jet energies and directions.
上述 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}(\gamma )$ 峰的截面主要由辐射 $\mathrm{q}\overline{\mathrm{q}}\gamma$ 事件主导；初始状态辐射光子（ISR 光子）通常沿着束流方向对齐且不被探测到。为了计算强子质心能量 $\sqrt{s^{\prime }}$ ，使用了参考文献[30]中描述的方法。该方法使用 DURHAM 算法[31]将粒子聚类成喷注，排除候选 ISR 光子并使用 $y_{cut}=0.002$ 。然后，使用三个约束拟合重建的喷注和额外的 ISR 光子（能量和横向动量，自由留下缺失动量的 $z$ 分量）。强子质心能量 $\sqrt{s^{\prime }}$ 是使用拟合后的喷注能量和方向计算的喷注不变质量。

3 Analysis of charged particles in $\mathrm{q}\overline{\mathrm{q}}$ events
$\mathrm{q}\overline{\mathrm{q}}$ 事件中带电粒子的分析

Centre-of-mass energies of 183 and 189 GeV
183 和 189 GeV 的质心能量

Events with $\sqrt{s^{\prime }}/\sqrt{s}$ above 0.9 were used to compute the multiplicities. A total of 3444 (1297) hadronic events were selected from the data at 189 (183) GeV, by requiring that the multiplicity for charged particles was larger than 9, that the total transverse energy of the charged particles exceeded $0.2\sqrt{s}$, and that the narrow jet broadening [32] was smaller than 0.065. From the simulation it was calculated that the expected background coming from WW and ${\mathrm{Z}}^0$${\mathrm{Z}}^0$ decays was 432+52 (127+21) events. The contamination from double radiative returns to the ${\mathrm{Z}}^0$, within 10 GeV of the nominal ${\mathrm{Z}}^0$ mass, was estimated by simulation to be below 5%. Other contaminations (from ${\mathrm{Z}}^0$ee, $\mathrm{W}\mathrm{e}\nu$, $\gamma \gamma$ interactions and Bhabhas) are below 2% in total.
使用大于 0.9 的事件计算多重性。通过要求带电粒子的多重性大于 9，带电粒子的总横向能量超过 $0.2\sqrt{s}$ ，以及窄喷注展宽小于 0.065，从 189（183）GeV 的数据中选择了 3444（1297）个强子事件。从模拟中计算出来的预期背景来自 WW 和 ${\mathrm{Z}}^0$ ${\mathrm{Z}}^0$ 衰变，为 432+52（127+21）个事件。模拟估计，距离标称 ${\mathrm{Z}}^0$ 质量 10 GeV 以内的双辐射返回到 ${\mathrm{Z}}^0$ 的污染低于 5%。其他污染（来自 ${\mathrm{Z}}^0$ ee， $\mathrm{W}\mathrm{e}\nu$ ， $\gamma \gamma$ 相互作用和巴巴）总共低于 2%。

The average multiplicity of charged particles with $p>0.1$ GeV/$c$ measured in the selected events at 189 GeV (183 GeV), after subtraction of the WW and ${\mathrm{Z}}^0$${\mathrm{Z}}^0$ backgrounds estimated by simulation, was $24.58\pm 0.16(stat)$ ($23.96\pm 0.23(stat)$), to be compared to $24.52\pm 0.05(stat)$ ($24.30\pm 0.07(stat)$) in the $\mathrm{q}\overline{\mathrm{q}}$ PS simulation including detector effects. The dispersion (square root of the variance) of the multiplicity distribution in the data was $7.57\pm 0.11(stat)$ ($7.00\pm 0.16(stat)$), to be compared to the dispersion from the $\mathrm{q}\overline{\mathrm{q}}$ PS simulation of $7.24\pm 0.03$ ($7.20\pm 0.05(stat)$).
在 189 GeV（183 GeV）选定事件中测量的带电粒子的平均多重性，经过模拟估计的 WW 和 ${\mathrm{Z}}^0$ ${\mathrm{Z}}^0$ 背景的减法后，为 $24.58\pm 0.16(stat)$ （ $23.96\pm 0.23(stat)$ ），与包括探测器效应的 $\mathrm{q}\overline{\mathrm{q}}$ PS 模拟中的 $24.52\pm 0.05(stat)$ （ $24.30\pm 0.07(stat)$ ）进行比较。数据中多重性分布的离散度（方差的平方根）为 $7.57\pm 0.11(stat)$ （ $7.00\pm 0.16(stat)$ ），与 $\mathrm{q}\overline{\mathrm{q}}$ PS 模拟的离散度 $7.24\pm 0.03$ （ $7.20\pm 0.05(stat)$ ）进行比较。

Detector effects and selection biases were corrected for using a $\mathrm{q}\overline{\mathrm{q}}$ simulation from PYTHIA with the JETSET fragmentation tuned by DELPHI without inital state radiation. The corrected average charge multiplicity was found to be $<n>=27.37\pm 0.18(stat)$($<n>=26.56\pm 0.26(stat)$), and the dispersion was found to be $D=8.77\pm 0.13(stat)$ ($D=8.08\pm 0.19(stat)$).
探测器效应和选择偏差通过使用 PYTHIA 中由 DELPHI 调整的 JETSET 碎裂模拟进行校正。校正后的平均电荷多重性为 $<n>=27.37\pm 0.18(stat)$ （ $<n>=26.56\pm 0.26(stat)$ ），方差为 $D=8.77\pm 0.13(stat)$ （ $D=8.08\pm 0.19(stat)$ ）。

The average multiplicity was computed by integrating the ${\xi }_E$ distribution, since the detection efficiency depends mostly on the momentum of the particle, after correcting for detector effects bin by bin using the simulation. The ${\xi }_E$ distribution was integrated up to a value of 6.3, and the extrapolation to the region above this cut was based on the simulation at the generator level.
平均多重性是通过积分 ${\xi }_E$ 分布计算得出的，因为探测效率主要取决于粒子的动量，在使用模拟进行逐个区间的探测器效应校正后。 ${\xi }_E$ 分布被积分到 6.3 的值，对超过此截断值的区域的外推基于生成器级别的模拟。

After multiplying by the Z${}^0$ corrections factors from section 2, the following values were obtained:
经过乘以第 2 节中的 Z ${}^0$ 校正因子，得到以下数值：

$$\begin{array}{}\text{(6)}& ⟨n{⟩}_{189\mathrm{G}\mathrm{e}\mathrm{V}}=27.47\pm 0.18(stat)\pm 0.30(syst)\end{array}$$ $$D_{189\mathrm{G}\mathrm{e}\mathrm{V}}=8.77\pm 0.13(stat)\pm 0.11(syst)$$ (7) $$⟨n{⟩}_{183\mathrm{G}\mathrm{e}\mathrm{V}}=27.05\pm 0.27(stat)\pm 0.32(syst)$$ (8) $$\begin{array}{}\text{(9)}& D_{183\mathrm{G}\mathrm{e}\mathrm{V}}=8.08\pm 0.19(stat)\pm 0.14(syst).\end{array}$$
(7) (8)

These values include the products of the decays of particles with lifetime $\tau <{10}^{-9}$ s. The systematic errors were obtained by adding in quadrature:
这些数值包括寿命小于 $\tau <{10}^{-9}$ 秒的粒子衰变产生的产物。系统误差通过平方相加得到：

1. the propagated uncertainty of the average values in the Z${}^0$ correction factors, $\pm 0.18$ ($\pm 0.17$) for the multiplicity.
   在多重性的修正因子 $\pm 0.18$ （ $\pm 0.17$ ）中，平均值的传播不确定性。
2. the effect of the cuts for the reduction of the background. The value of the cut on the narrow jet broadening was varied from 0.045 to 0.085 in steps of 0.010, in order to estimate the systematic error associated with the procedure of removing the contribution from WW events. The new values for the average charged particle multiplicity and the dispersion were stable within these variations, and half of the difference between the extreme values, 0.07 and 0.07 (0.06 and 0.12) respectively, were added in quadrature to the systematic error. The effect of the uncertainty on the WW cross-section was found to be negligible.
   削减背景的效果。对于狭窄喷流展宽的削减值从 0.045 到 0.085 进行了变化，步长为 0.010，以估计消除 WW 事件贡献的程序所带来的系统误差。在这些变化中，平均带电粒子多重性和离散度的新值保持稳定，并且极端值之间的差异的一半，分别为 0.07 和 0.07（0.06 和 0.12），以平方和的方式添加到系统误差中。WW 截面的不确定性影响被发现是可以忽略的。
3. the uncertainty on the modelling of the detector response in the forward region. The analysis was repeated by varying the polar angle acceptance of charged particles from 10-170 degrees to 40-140 degrees, both in the high energy samples and in the computation of the Z${}^0$ correction factors. The spread of the different values obtained for the multiplicities and for the dispersions were found to be respectively 0.18 and 0.08 (0.23 and 0.03). The effect of the variation of other track selection criteria was found to be negligible, and the same applies to the higher centre-of-mass energies.
   在前向区域对探测器响应的建模存在不确定性。分析通过改变带电粒子的极角接受范围从 10-170 度到 40-140 度来重复进行，无论是在高能样本中还是在计算 Z ${}^0$ 校正因子时。得到的多重性和离散度的不同值的分散分别为 0.18 和 0.08（0.23 和 0.03）。其他轨迹选择标准的变化对结果影响可以忽略不计，同样适用于更高的质心能量。
4. the systematic errors due to the statistics of the simulated samples, 0.04 (0.06) for the multiplicity and 0.04 (0.06) for the dispersion.
   由于模拟样本的统计误差，多重性为 0.04（0.06），离散度为 0.04（0.06）。
5. the uncertainty on the calculation of the efficiency correction factors in the multiplicity. The values of the multiplicities, before applying the Z${}^0$ correction factors, were also estimated: * from the observed multiplicity distribution as 27.37 (26.56); * from the integral of the rapidity distribution (with respect to the thrust axis), $y_T=\frac{1}{2}\ln \frac{E+p_{||}}{E-p_{||}}$ ($p_{||}$ is the absolute value of the momentum component on the thrust axis) as 27.43 (26.59). Half of the differences between the maximum and the minimum values of the multiplicity calculated from the multiplicity distribution and from the integration of the $y_T$ and ${\xi }_E$ distributions, 0.03 in both cases, were added in quadrature to the systematic error.
   在多重性中计算效率修正因子的不确定性。在应用 Z ${}^0$ 修正因子之前，也估计了多重性的值：*从观察到的多重性分布中，为 27.37（26.56）；*从快度分布的积分（相对于推力轴）， $y_T=\frac{1}{2}\ln \frac{E+p_{||}}{E-p_{||}}$ （ $p_{||}$ 是推力轴上的动量分量的绝对值），为 27.43（26.59）。将从多重性分布和 $y_T$ 和 ${\xi }_E$ 分布的积分计算出的最大值和最小值之间的差异的一半，即 0.03，在两种情况下都以平方和的形式添加到系统误差中。
6. Half of the extrapolated multiplicity in the high-${\xi }_E$ region, 0.14 (0.12).
   高 ${\xi }_E$ 区域的外推多重性的一半为 0.14（0.12）。

As a cross-check, a simulated sample based on HERWIG plus DELSIM was also used to unfold the data; the results were consistent with those based on PYTHIA plus DELSIM within the statistical error associated to the size of the Monte Carlo sample.
作为交叉检验，还使用基于 HERWIG 加 DELSIM 的模拟样本对数据进行展开；结果与基于 PYTHIA 加 DELSIM 的结果一致，仅在统计误差范围内与蒙特卡洛样本的大小相关。

Centre-of-mass energies of 192 to 200 GeV
192 到 200 GeV 的质心能量

To check the ability of the simulation to model the efficiency for the reconstruction of charged particles, the sample collected at the ${\mathrm{Z}}^0$ calibration runs of 1999 was used, following the procedure described in section 2. The average charged particle multiplicity at the ${\mathrm{Z}}^0$ was measured to be $20.82\pm 0.03(stat)$, in satisfactory agreement with the world average. The ratio of the world average value to the measured multiplicity at the ${\mathrm{Z}}^0$, $1.0084\pm 0.0064$, was used to correct the measured multiplicities at centre-of-mass energies of 192 to 200 GeV.
为了检查模拟能否准确模拟重建带电粒子的效率，我们使用了 1999 年校准运行收集的样本，并按照第 2 节中描述的步骤进行了处理。在 ${\mathrm{Z}}^0$ 处，平均带电粒子多重性被测量为 $20.82\pm 0.03(stat)$ ，与世界平均值相当。世界平均值与 ${\mathrm{Z}}^0$ 处测量到的多重性之比，即 $1.0084\pm 0.0064$ ，被用来校正 192 至 200 GeV 质心能量下测量到的多重性。

For each of the energies a separate analysis was performed following the procedure described in the previous subsection.
对于每种能量，按照前一小节中描述的程序进行了单独的分析。

The number of events selected, the number of expected signal and background events, estimated with Monte Carlo simulation, and the measured multiplicities and dispersions are listed in Table 1 for the centre-of-mass energies of 192 to 200 GeV. The systematic errors were estimated as in 3.1; a breakdown is shown in the table (the numbering of the sources of systematic error corresponds to the one in the previous subsection).
在 192 到 200 GeV 的质心能量下，通过蒙特卡洛模拟估计，列出了所选事件的数量、预期信号和背景事件的数量，以及测量的多重性和离散度，详见表 1。系统误差的估计方法与 3.1 节相同；表中显示了误差来源的细分（系统误差来源的编号与前一小节中的编号相对应）。

These results were then combined at an average centre-of-mass energy of 200 GeV, according to the following procedure. First each result was rescaled to a centre-of-mass energy of 200 GeV, calculating the scaling factors from the simulation. Then a weighted
平均值被计算在 200GeV 的平均质心能量下，以下是具体步骤。首先，每个结果都被重新调整到 200GeV 的质心能量，缩放因子从模拟中计算得出。然后根据统计权重进行加权平均。

$$\text{Unknown environment 'table'}$$ Table 1: Number of events selected, number of events expected for the signal ($\mathrm{q}\overline{\mathrm{q}}\gamma$) and for the background (WW and ZZ), estimated from simulation, and the measured multiplicities and dispersions for the three different energies.
表 1：所选事件数量，信号（ $\mathrm{q}\overline{\mathrm{q}}\gamma$ ）和背景（WW 和 ZZ）的预期事件数量，根据模拟估计，以及三种不同能量下的测量多重性和离散度。

average was computed using the statistical error as a weight. The systematic error is taken as the weighted average of the systematic errors, increased (in quadrature) by the difference between the values obtained when rescaling and when not rescaling to 200 GeV. This gives:
平均值是使用统计误差作为权重计算的。系统误差被视为系统误差的加权平均值，通过将重新缩放和不重新缩放到 200 GeV 时的值之间的差异（按平方和）增加。这给出：

$$\begin{array}{}\text{(10)}& ⟨n{⟩}_{200\mathrm{G}\mathrm{e}\mathrm{V}}=27.58\pm 0.19(stat)\pm 0.45(syst)\end{array}$$ $$\begin{array}{}\text{(11)}& D_{200\mathrm{G}\mathrm{e}\mathrm{V}}=8.64\pm 0.13(stat)\pm 0.20(syst).\end{array}$$

As a cross-check, a simulated sample based on HERWIG plus DELSIM was also used to unfold the data; the results were consistent with those based on PYTHIA plus DELSIM within the statistical error associated to the size of the Monte Carlo sample.
作为交叉检验，还使用基于 HERWIG 加 DELSIM 的模拟样本对数据进行展开；结果与基于 PYTHIA 加 DELSIM 的结果一致，仅在统计误差范围内与蒙特卡洛样本的大小相关。

4 Classification of the WW Events and Charged Multiplicity Measurement
4 WW 事件的分类和带电多重性测量

About 4/9 of the WW events are $\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2{\mathrm{q}}_3{\overline{\mathrm{q}}}_4$ events. At threshold, their topology is that of two pairs of back-to-back jets, with no missing energy; the constrained invariant mass of two jet-jet systems is close to the W mass. Even at 183 and 189 GeV these characteristics allow a clean selection.
大约有 4/9 的 WW 事件是 $\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2{\mathrm{q}}_3{\overline{\mathrm{q}}}_4$ 事件。在阈值处，它们的拓扑结构是两对背对背的喷注，没有缺失能量；两个喷注-喷注系统的约束不变质量接近 W 玻色子的质量。即使在 183 和 189 GeV 的能量下，这些特征也能够进行清晰的选择。

Another 4/9 of the WW events are $\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2\ell \overline{\nu }$ events. At threshold, their topology is 2-jets back-to-back, with a lepton and missing energy opposite to it; the constrained invariant mass of the jet-jet system and of the lepton-missing energy system equals the W mass.
另外 4/9 的 WW 事件是 $\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2\ell \overline{\nu }$ 事件。在阈值处，它们的拓扑结构是两个背靠背的喷注，一个轻子和相反的缺失能量；喷注-喷注系统和轻子-缺失能量系统的约束不变质量等于 W 玻色子质量。

Fully Hadronic Channel ($\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2{\mathrm{q}}_3{\overline{\mathrm{q}}}_4$)
全强子通道（ $\mathrm{W}\mathrm{W}\to {\mathrm{q}}_1{\overline{\mathrm{q}}}_2{\mathrm{q}}_3{\overline{\mathrm{q}}}_4$ ）

Events with both Ws decaying into $\mathrm{q}\overline{\mathrm{q}}$ are characterised by high multiplicity, large visible energy, and tendency of the particles to be grouped in 4 jets. The background is dominated by $\mathrm{q}\overline{\mathrm{q}}(\gamma )$ events.
具有两个 W 玻色子衰变成 $\mathrm{q}\overline{\mathrm{q}}$ 的事件具有高多重性、大可见能量和粒子倾向于分组成 4 个喷注的特征。背景主要由 $\mathrm{q}\overline{\mathrm{q}}(\gamma )$ 事件主导。

The events were pre-selected by requiring at least 12 charged particles (with $p>100$ MeV/$c$), with a total transverse energy (charged plus neutral) above 20% of the centre-of-mass energy. To remove the radiative hadronic events, the effective hadronic centre-of-mass energy $\sqrt{s^{\prime }}$, computed as described in section 2, was required to be above 110 GeV.
事件是通过要求至少 12 个带电粒子（能量小于 $p>100$ MeV/ $c$ ）以及总横向能量（带电和中性粒子）超过中心质能的 20%来预先选择的。为了排除辐射强子事件，需要将有效强子质心能量 $\sqrt{s^{\prime }}$ ，如第 2 节所述计算的，要求大于 110 GeV。

The particles in the event were then clustered to 4 jets using the LUCLUS algorithm [1], and the events were kept if all jets had multiplicity (charged plus neutral) larger than 3. It was also required that the separation between the jets ($d_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}}$ value) be larger than 6 GeV/c. The combination of these two cuts removed most of the remaining semi-leptonic WW decays and the 2-jet and 3-jet events of the $\mathrm{q}\overline{\mathrm{q}}\gamma$ background.
事件中的粒子随后使用 LUCLUS 算法[1]聚类成 4 个喷注，并且只保留所有喷注的多重性（带电加中性）大于 3 的事件。还要求喷注之间的分离（ $d_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}}$ 值）大于 6 GeV/c。这两个截断的组合去除了大部分剩余的半轻子 WW 衰变以及 $\mathrm{q}\overline{\mathrm{q}}\gamma$ 背景的 2 喷注和 3 喷注事件。

A five constraint fit was applied, imposing energy and momentum conservation and the equality of two di-jet masses. Of the three fits obtained by permutation of the jets, the one with the smallest ${\chi }^2$ was selected. Events were accepted only if
应用了五个约束条件，强加能量和动量守恒以及两个双喷注质量相等。在对喷注进行排列组合得到的三个拟合结果中，选择 ${\chi }^2$ 最小的一个。只有满足条件的事件才被接受。

$$D_{\mathrm{s}\mathrm{e}\mathrm{l}}=\frac{E_{min}{\theta }_{min}}{E_{max}(E_{max}-E_{min})}>0.004\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{G}\mathrm{e}{\mathrm{V}}^{-1}$$

where $E_{min}$ and $E_{max}$ are respectively the smallest and the largest fitted jet energy, and ${\theta }_{min}$ is the smallest angle between the fitted jet directions. The details of the selection variable $D_{\mathrm{s}\mathrm{e}\mathrm{l}}$ can be found in [33]. The purity and the efficiency of the selected data sample from the 189 (183) GeV data were estimated using simulation to be about 76%and 80% (75% and 80%) respectively. The data sample consists of 1256 (427) events, where 1255 (422) were expected from the simulation. The expected background was subtracted bin by bin from the observed distributions, which were then corrected bin by bin using scaling factors computed from the simulation generated using PYTHIA with the JETSET fragmentation tuned by DELPHI (EXCALIBUR plus JETSET for the 1999 data) without initial state radiation.
其中 $E_{min}$ 和 $E_{max}$ 分别是最小和最大的拟合喷注能量， ${\theta }_{min}$ 是拟合喷注方向之间的最小角度。选择变量 $D_{\mathrm{s}\mathrm{e}\mathrm{l}}$ 的详细信息可以在[33]中找到。使用模拟估计了来自 189（183）GeV 数据的选定数据样本的纯度和效率分别为约 76%和 80%（75%和 80%）。数据样本包括 1256（427）个事件，其中 1255（422）个来自模拟。预期的背景从观测到的分布中逐个 bin 减去，然后使用从使用 PYTHIA 生成的模拟中计算得到的缩放因子逐个 bin 进行校正，其中使用了由 DELPHI 调整的 JETSET 碎裂和 1999 年数据的 EXCALIBUR 加 JETSET，没有考虑初始态辐射。

Finally, the average multiplicity of charged particles $⟨n^{(4q)}⟩$ was estimated by integrating the ${\xi }_E$ distribution up to a value of 6.3 (and estimated above this value with simulation) and multiplying by the Z${}^0$ correction factors from section 2. The following values were obtained:
最后，通过将 ${\xi }_E$ 分布积分至 6.3（并通过模拟估计超过该值的部分），并乘以来自第 2 节的 Z ${}^0$ 校正因子，估计得到了带电粒子 $⟨n^{(4q)}⟩$ 的平均多重性。得到以下数值：

$$\begin{array}{}\text{(12)}& ⟨n^{(4q)}{⟩}_{189\mathrm{G}\mathrm{e}\mathrm{V}}=39.12\pm 0.33(stat)\pm 0.36(syst)\end{array}$$ $$\begin{array}{}\text{(13)}& ⟨n^{(4q)}{⟩}_{183\mathrm{G}\mathrm{e}\mathrm{V}}=38.11\pm 0.57(stat)\pm 0.44(syst).\end{array}$$

The systematic errors account for:
系统误差占据：

1. The propagated uncertainty of the average values in the Z${}^0$ correction factors, $\pm 0.24$ ($\pm 0.24$).
   在 Z ${}^0$ 校正因子的平均值中传播的不确定性， $\pm 0.24$ （ $\pm 0.24$ ）。
2. The spread of the measured values from the reference values by redoing the analysis varying the selection criteria, 0.05 (0.11).
   通过重新分析并改变选择标准，参考值与测量值之间的差异为 0.05（0.11）。
3. Modelling of the detector in the forward region. The analysis was repeated by varying the polar angle acceptance of charged particles from 10-170 degrees to 40-140 degrees, both in the WW samples and in the computation of the Z${}^0$ correction factors. The spreads of the different measured values were found to be 0.01 (0.08).
   在前向区域对探测器进行建模。通过改变带电粒子的极角接受范围从 10-170 度到 40-140 度，对 WW 样本和 Z ${}^0$ 校正因子的计算进行了重复分析。不同测量值的扩散程度为 0.01（0.08）。
4. Limited statistics in the simulated sample 0.03 (0.05).
   模拟样本中的有限统计数据为 0.03（0.05）。
5. Variation of the q$\overline{\mathrm{q}}\gamma$ cross-sections within 5%: 0.01 (0.01).
   q 截面的变化在 5%以内：0.01 (0.01)。
6. Calculation of the correction factors. The value of $⟨n^{(4q)}⟩$, before applying the Z${}^0$ correction factors, was also estimated: * from the observed multiplicity distribution as 38.96 (37.39); * from the integral of the rapidity distribution (with respect to the thrust axis) as 39.16 (36.97). * from the integral of the $p_T$ distribution (with respect to the thrust axis) as 38.95 (37.42). Half of the difference between the maximum and the minimum value, 0.11 (0.23), was added in quadrature to the systematic error.
   校正因子的计算。在应用 Z 校正因子之前， $⟨n^{(4q)}⟩$ 的值也被估计为：*根据观察到的多重性分布为 38.96（37.39）；*根据快度分布的积分（相对于推力轴）为 39.16（36.97）；*根据 $p_T$ 分布的积分（相对于推力轴）为 38.95（37.42）。最大值和最小值之间的差值的一半，0.11（0.23），被平方加到系统误差中。
7. Uncertainty on the modelling of the 4-jets q$\overline{\mathrm{q}}$ background, 0.05 (0.14). The uncertainty on the modelling of this background is the sum in quadrature of two contributions: * Uncertainty on the modelling of the 4-jet rate. The agreement between data and simulation was studied in a sample of 4-jet events at the Z${}^0$, selected with the DURHAM algorithm for $y_{cut}$ ranging from 0.003 to 0.005. The rate of 4-jet events in the simulated sample was found to reproduce the data within 10%. The correction due to background subtraction was correspondingly varied by 10%, which gives an uncertainty of 0.00 (0.01).
   对于 4 喷注 q $\overline{\mathrm{q}}$ 背景的建模存在不确定性，为 0.05（0.14）。对于该背景建模的不确定性是两个因素的平方和：*对 4 喷注率建模的不确定性。在选择 DURHAM 算法的 Z ${}^0$ 4 喷注事件样本中，研究了数据和模拟之间的一致性，其中 $y_{cut}$ 范围从 0.003 到 0.005。在模拟样本中，4 喷注事件的率被发现在 10%的范围内与数据相符。相应地，由于背景减法的修正也变化了 10%，这给出了 0.00（0.01）的不确定性。

• Uncertainty on the multiplicity in 4-jet events. The average multiplicity in 4-jet events selected at the Z${}^0$ data in 1998 (1997), with the DURHAM algorithm for a value of $y_{cut}=0.005$, is larger by $0.9\mathrm{\%}\pm 0.3\mathrm{\%}(stat)$ ($2.0\mathrm{\%}\pm 0.5\mathrm{\%}(stat)$) than the corresponding value in the simulation. A shift by 0.9% (2.0%) in the multiplicity for 4-jet events induces a shift of 0.05 (0.14) on the value in Equation (12) (Equation (13)).
  1998 年（1997 年）在 Z ${}^0$ 数据中选择的 4 喷注事件中，使用 DURHAM 算法得到的平均多重性，比模拟中对应的数值大 $0.9\mathrm{\%}\pm 0.3\mathrm{\%}(stat)$ （ $2.0\mathrm{\%}\pm 0.5\mathrm{\%}(stat)$ ）。4 喷注事件中多重性的 0.9%（2.0%）的偏移，会导致方程（12）（方程（13））中的数值偏移 0.05（0.14）。

8. Half of the extrapolated multiplicity in the high-${\xi }_E$ region, 0.23 (0.20).
   高 ${\xi }_E$ 区域的外推多重性的一半为 0.23（0.20）。

The presence of interference between the jets coming from the different Ws could create subtle effects, such as to make the application of the fit imposing equal masses inadequate. For this reason a different four constraint fit was performed, leaving the di-jet masses free and imposing energy-momentum conservation. Of the three possible combinations of the four jets into WW pairs, the one with minimum mass difference was selected. No ${\chi }^2$ cut was imposed in this case. The average multiplicity obtained was again fully consistent (within the statistical error) with the one measured in the standard analysis.
不同 W 产生的喷流之间的干扰存在可能会产生微妙的影响，例如使得应用等质量的拟合不合适。因此，进行了不同的四个约束拟合，将双喷流质量保持自由，并强制能量动量守恒。在将四个喷流组合成 WW 对的三种可能组合中，选择了质量差最小的那一对。在这种情况下，没有施加 ${\chi }^2$ 截断。得到的平均多重性与标准分析中测量的多重性完全一致（在统计误差范围内）。

A simulated sample based on HERWIG plus DELSIM was also used to unfold the data; the results were consistent with those based on PYTHIA plus DELSIM within the statistical error associated to the size of the Monte Carlo sample.
基于 HERWIG 加 DELSIM 的模拟样本也被用来展开数据；结果与基于 PYTHIA 加 DELSIM 的结果一致，仅在统计误差范围内与蒙特卡洛样本的大小相关。

The distribution of the observed charged particle multiplicity in (4$q$) is shown in Figure 1c(1a).
在图 1c（1a）中显示了（4 $q$ ）中观测到的带电粒子多重性的分布。

The value of the corrected multiplicity in the low momentum range 0.1 to 1. GeV/$c$, where the interconnection effects are expected to be most important, was found to be 14.47$\pm$0.20 (13.67$\pm$0.34), where the errors are statistical only.
在低动量范围 0.1 到 1 GeV/ $c$ 内，修正的多重性的值被发现为 14.47 $\pm$ 0.20（13.67 $\pm$ 0.34），其中误差仅为统计误差。

After correcting for detector effects, the dispersion was found to be:
经过校正探测器效应后，发现色散为：

$$\begin{array}{}\text{(14)}& D_{189\mathrm{GeV}}^{(4q)}=8.72\pm 0.23(stat)\pm 0.11(syst)\end{array}$$ $$\begin{array}{}\text{(15)}& D_{183\mathrm{GeV}}^{(4q)}=8.53\pm 0.39(stat)\pm 0.16(syst).\end{array}$$

In the systematic error:
在系统误差中：

1. 0.10 (0.15) accounts for the spreads of the measured values from the reference value when varying the event selection criteria;
   0.10（0.15）是在改变事件选择标准时，测量值与参考值之间的差异的比例。
2. 0.03 (0.01) is due to the modelling of the detector in the forward region. The dispersions were also measured using only charged particles with polar angle between 40 and 140 degrees and the differences with respect to the reference value were considered in the systematic error;
   0.03（0.01）是由于前向区域探测器的建模。还使用仅在 40 至 140 度之间的带电粒子进行了色散测量，并将与参考值的差异考虑在系统误差中；
3. 0.03 (0.05) from the limited simulation statistics.
   0.03（0.05）来自有限的模拟统计数据。

Mixed Hadronic and Leptonic Final States (WW$\to$ q${}_1$q${}_2$$l\nu$)
混合强子和轻子的最终态（WW $\to$ q ${}_1$ q ${}_2$ $l\nu$ ）

Events in which one W decays into lepton plus neutrino and the other one into quark and antiquark are characterised by two hadronic jets, one energetic isolated charged lepton, and missing momentum resulting from the neutrino. The main backgrounds to these events are radiative q$\overline{\mathrm{q}}$ production and four-fermion final states containing two quarks and two oppositely charged leptons of the same flavour.
一个 W 粒子衰变成轻子加中微子，另一个 W 粒子衰变成夸克和反夸克的事件，其特征是两个强子喷注、一个高能孤立带电轻子和由中微子引起的缺失动量。这些事件的主要背景是辐射性 q $\overline{\mathrm{q}}$ 产生和包含两个夸克和两个相同味道的反向带电轻子的四费米子末态。

Events were selected by requiring seven or more charged particles, with a total energy (charged plus neutral) above 0.2$\sqrt{s}$ and a missing momentum larger than 0.1$\sqrt{s}$. Events in the q$\overline{\mathrm{q}}\gamma$ final state with ISR photons at small polar angles, which would be lost inside the beam pipe, were suppressed by requiring the polar angle of the missing momentum vector to satisfy $|\cos {\theta }_{miss}|<0.94$.
通过要求七个或更多带电粒子，总能量（带电加中性）大于 0.2 $\sqrt{s}$ ，以及缺失动量大于 0.1 $\sqrt{s}$ ，来选择事件。通过要求缺失动量矢量的极角满足 $|\cos {\theta }_{miss}|<0.94$ ，来抑制在小极角处具有 ISR 光子的 q $\overline{\mathrm{q}}\gamma$ 末态事件，这些光子会在束流管内丢失。

Including the missing momentum as an additional massless neutral particle (the candidate neutrino), the particles in the event were clustered to 4 jets using the DURHAM algorithm. The jet for which the fractional jet energy carried by the highest momentum charged particle was greatest was considered as the "lepton jet". The most energetic charged particle in the lepton jet was taken as the lepton candidate, and the event was
包括将缺失的动量作为额外的无质量中性粒子（候选中微子）考虑在内，使用 DURHAM 算法将事件中的粒子聚类为 4 个喷注。其中，由动量最高的带电粒子携带的喷注能量最大的喷注被视为“轻子喷注”。轻子喷注中能量最高的带电粒子被视为轻子候选粒子，并且事件被

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• from the observed multiplicity distribution as 19.53 (19.60);
  从观察到的多重性分布中，得出的数值为 19.53（19.60）。
• from the integral of the rapidity distribution (with respect to the thrust axis) as 19.30 (18.88).
  根据快度分布（相对于推力轴）的积分结果为 19.30（18.88）。
• from the integral of the $p_T$ distribution (with respect to the thrust axis) as 19.39 (19.36). Half of the difference between the maximum and the minimum value, 0.12 (0.36), was included in the systematic error.
  从 $p_T$ 分布的积分（相对于推力轴）为 19.39（19.36）。最大值和最小值之间的差值的一半，0.12（0.36），被包括在系统误差中。
• Half of the extrapolated multiplicity in the high-${\xi }_E$ region, 0.12 (0.10).
  高能区域的外推多重性的一半为 0.12（0.10）。

A simulated sample based on HERWIG plus DELSIM was also used to unfold the data; the results were consistent with those based on PYTHIA plus DELSIM within the statistical error associated to the size of the Monte Carlo sample.
基于 HERWIG 加 DELSIM 的模拟样本也被用来展开数据；结果与基于 PYTHIA 加 DELSIM 的结果一致，仅在统计误差范围内与蒙特卡洛样本的大小相关。

The distribution of the observed charged particle multiplicity in ($2q$) is shown in Figure 1d(1b).
在图 1d(1b)中显示了（ $2q$ ）中观测到的带电粒子多重性的分布。

The value of the corrected multiplicity in the low momentum range 0.1 to 1. GeV/$c$ was found to be 7.29$\pm$0.19 (7.15$\pm$0.28) (where the errors are statistical only).
在动量范围为 0.1 至 1. GeV/ $c$ 的修正多重性的值为 7.29 $\pm$ 0.19（7.15 $\pm$ 0.28）（其中误差仅为统计误差）。

After correcting for detector effects, the dispersions were found to be:
经过校正探测器效应后，发现色散为：

$$\begin{array}{}\text{(18)}& D_{189\mathrm{GeV}}^{(2q)}=6.49\pm 0.21(stat)\pm 0.43(syst)\end{array}$$ $$\begin{array}{}\text{(19)}& D_{183\mathrm{GeV}}^{(2q)}=6.51\pm 0.33(stat)\pm 0.25(syst).\end{array}$$

In the systematic error:
在系统误差中：

1. 0.14 (0.08) accounts for the variation of the cuts;
   0.14（0.08）解释了切割的变化。
2. 0.41 (0.23) accounts for the modelling of the detector in the forward region. The dispersions were also measured using only charged particles with polar angle between 40 and 140 degrees and the differences with respect to the reference value were considered in the systematic error;
   0.41（0.23）解释了前向区域探测器的建模。还使用仅带电粒子的极角在 40 到 140 度之间进行了测量，并将与参考值的差异考虑在系统误差中；
3. 0.03 (0.05) is due to the limited simulation statistics.
   0.03（0.05）是由于有限的模拟统计数据。

5 Analysis of Interconnection Effects from Charged Particle Multiplicity and Inclusive Distributions
从带电粒子多重性和全包络分布的相互连接效应的分析

Most models predict that, in case of colour reconnection, the ratio between the multiplicity in ($4q$) events and twice the multiplicity in ($2q$) events would be smaller than 1; the difference is expected to be at the percent level. It was measured:
大多数模型预测，在颜色重联的情况下，( $4q$ ) 事件中的多重性与( $2q$ ) 事件中的多重性的两倍之比将小于 1；预计差异将在百分之一的水平上。已经测量出来：

$$\begin{array}{}\text{(20)}& {\left(\frac{⟨n^{(4q)}⟩}{2⟨n^{(2q)}⟩}\right)}_{189\mathrm{GeV}}=1.004\pm 0.018(stat)\pm 0.014(syst)\end{array}$$ $$\begin{array}{}\text{(21)}& {\left(\frac{⟨n^{(4q)}⟩}{2⟨n^{(2q)}⟩}\right)}_{183\mathrm{GeV}}=0.963\pm 0.028(stat)\pm 0.015(syst).\end{array}$$

In the calculation of the systematic error on the ratio, the correlations between the sources of systematic error were taken into account. If the systematic errors are taken as uncorrelated, except for the errors on the ${\mathrm{Z}}^0$ correction factors and the modelling of the detector in the forward region, for which full correlation is assumed, a compatible value of $\pm 0.014$ ($\pm 0.022$) is obtained for the systematic error.
在计算比值的系统误差时，考虑了系统误差来源之间的相关性。如果将系统误差视为不相关，除了 ${\mathrm{Z}}^0$ 修正因子和正向区域探测器建模的误差，这些误差被认为是完全相关的，那么系统误差的兼容值为 $\pm 0.014$ （ $\pm 0.022$ ）。

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between the sources of systematic error were taken into account):
在考虑系统误差来源之间的差异时：

$${\left(\frac{D^{(4q)}}{\sqrt{2}{D}^{(2q)}}\right)}_{189\mathrm{GeV}}=0.95\pm 0.04(\mathrm{stat})\pm 0.07(\mathrm{syst})$$ $${\left(\frac{D^{(4q)}}{\sqrt{2}{D}^{(2q)}}\right)}_{183\mathrm{GeV}}=0.93\pm 0.06(\mathrm{stat})\pm 0.04(\mathrm{syst}).$$

Using as weights the inverse of the sum in quadrature of the statistical and systematic errors, one obtains a weighted average of
使用统计误差和系统误差的平方和的倒数作为权重，可以得到加权平均值

$$\begin{array}{}\text{(26)}& \left(\frac{D^{(4q)}}{\sqrt{2}{D}^{(2q)}}\right)=0.94\pm 0.03(stat)\pm 0.03(syst).\end{array}$$

In conclusion, no depletion of the multiplicity was observed in fully hadronic WW events with respect to twice the semileptonic events, at this statistical precision; a possible depletion at the percent level can however not be excluded.
总之，在这个统计精度下，与两倍半轻子事件相比，完全强子化的 WW 事件中没有观察到多重性的耗尽；然而，不能排除在百分比水平上可能存在的耗尽情况。

6 Identified Particles from $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$
从 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 中识别出的 6 个粒子

This section describes the results obtained for ${\pi }^{+}$, K${}^{+}$, K${}^0$, p and $\mathrm{\Lambda }$ with data recorded by DELPHI at LEP 2. After the description of the event selection for identified particles at energies up to 189 GeV, the additional criteria for hadron identification are described.
本节描述了在 LEP 2 上由 DELPHI 记录的数据得到的 ${\pi }^{+}$ 、K ${}^{+}$ 、K ${}^0$ 、p 和 $\mathrm{\Lambda }$ 的结果。在对能量高达 189 GeV 的已鉴别粒子的事件选择进行描述之后，还描述了用于强子鉴别的附加标准。

Event selection at 130 and 136 GeV
130 和 136 GeV 的事件选择

After the hadronic preselection described in Section 2, events with $\sqrt{s^{\prime }}>0.85\sqrt{s}$ were used for further analysis. Data recorded at these two energies were combined and are referred to as the 133 GeV sample. The total of $\sim 12$ pb${}^{-1}$ recorded by DELPHI yields 1387 events while 1406 are expected from simulation.
在第 2 节中描述的强子预选之后，使用具有 $\sqrt{s^{\prime }}>0.85\sqrt{s}$ 的事件进行进一步分析。记录在这两个能量上的数据被合并，并称为 133 GeV 样本。DELPHI 记录的总共 $\sim 12$ pb ${}^{-1}$ 产生了 1387 个事件，而模拟预计有 1406 个事件。

Event selection at 161 and 172 GeV
161 和 172 GeV 的事件选择

Selected events at 161 GeV were required to have a minimum of 8 and a maximum of 40 charged particles, $\sqrt{s^{\prime }}>0.85\sqrt{s}$ and a visible energy of at least 50% of $\sqrt{s}$. A cut was imposed on the polar angle $\theta$ of the thrust axis to select events well within the acceptance of the detector. It was found that a selection based on the narrow jet broadening, $B_N$, is effective in removing the WW events and minimises the bias introduced on the remaining event sample. At threshold about 30 WW events are expected. Selecting events with $B_N\le 0.12$ reduces this background by 50%.
选择在 161 GeV 处的事件要求至少有 8 个至多 40 个带电粒子， $\sqrt{s^{\prime }}>0.85\sqrt{s}$ 和至少 50%的可见能量。对于推力轴的极角 $\theta$ 施加了一个截断，以选择在探测器接受范围内的事件。发现基于狭窄喷注展宽 $B_N$ 的选择对于去除 WW 事件是有效的，并且最小化对剩余事件样本引入的偏差。在阈值处预计大约有 30 个 WW 事件。选择具有 $B_N\le 0.12$ 的事件可以将这个背景减少 50%。

Using this selection 342 events are expected from simulation, while 357 were selected from the data, with an estimated remaining WW background of 15 events.
使用这个选择，预计从模拟中得到 342 个事件，而从数据中选择了 357 个事件，估计剩余的 WW 背景为 15 个事件。

At 172 GeV, in addition to the criteria described above, events were required to have at most 38 charged particles and $B_N\le 0.1$. This leads to 267 selected events, with 264 expected from simulation, out of which 36 are WW background.
在 172 GeV 能量下，除了上述所描述的标准外，事件需要最多有 38 个带电粒子和 $B_N\le 0.1$ 。这导致了 267 个被选中的事件，其中 264 个来自模拟，其中 36 个是 WW 背景。

Event selection at 183 and 189 GeV
183 和 189 GeV 的事件选择

The event selection for charged identified particles at 189 (183) GeV follows very closely the procedure already described in sections 2 and 3. Events with $\sqrt{s^{\prime }}/\sqrt{s}$ above 0.9 and more than 9(8) charged particles with $p\ge 200$ MeV/$c$ were used. WW background was suppressed by demanding $B_N\le 0.1(0.08)$. A total of 3617(1122) events were selected with an expected background from WW and Z${}^0$Z${}^0$ of 789 (146) events.
189（183）GeV 的带电粒子事件选择非常接近已在第 2 和第 3 节中描述的过程。使用 $\sqrt{s^{\prime }}/\sqrt{s}$ 大于 0.9 且 $p\ge 200$ MeV/ $c$ 的 9（8）个以上带电粒子的事件。通过要求 $B_N\le 0.1(0.08)$ 来抑制 WW 背景。共选择了 3617（1122）个事件，预期的 WW 和 Z ${}^0$ Z ${}^0$ 背景为 789（146）个事件。

Selection of charged particles for identification
选择带电粒子进行鉴别

A further selection was applied to the charged particle sample to obtain well identifiable particles. Two different momentum regions were considered, above and below 0.7 GeV/$c$, which correspond respectively to the separation of samples identified solely by the ionization loss in the Time Projection Chamber (TPC) and by the Cerenkov detectors (RICH) and TPC together. Below 0.7 GeV/$c$ tighter cuts were applied, namely to eliminate secondary protons. There had to be at least 30 wire hits in the TPC associated with the track and the measured track length had be to larger than 100 cm. In addition it was required at least two associated VD layer hits in $r\varphi$ and an impact parameter in the $r\varphi$ plane of less then 0.1 cm. If there were less than two associated VD layers in $z$, the corresponding impact parameter had to be less than 1 cm, else less then 0.1 cm. Particles above 0.7 GeV/c were required to have a measured track length bigger than 30 cm and good RICH quality, i.e. presence of primary ionization in the veto regions. Only particles which were well contained in the barrel region of DELPHI ($|\cos (\theta )|\le 0.7$) were accepted.
进一步对带电粒子样本进行了筛选，以获得可辨识的粒子。考虑了两个不同的动量区域，分别是大于和小于 0.7 GeV/c，分别对应仅通过时间投影室（TPC）中的电离损失和 Cerenkov 探测器（RICH）与 TPC 的鉴别样本。在小于 0.7 GeV/c 的情况下，采用了更严格的筛选条件，即排除次级质子。TPC 中与轨迹相关联的导线击中数必须至少为 30 个，并且测量的轨迹长度必须大于 100 厘米。此外，还要求在 $r\varphi$ 中至少有两个相关联的 VD 层击中，并且在 $r\varphi$ 平面上的冲击参数小于 0.1 厘米。如果在 $z$ 中相关联的 VD 层少于两个，则相应的冲击参数必须小于 1 厘米，否则小于 0.1 厘米。要求大于 0.7 GeV/c 的粒子的测量轨迹长度大于 30 厘米，并且具有良好的 RICH 质量，即在否决区域中存在主要电离。只有在 DELPHI 的桶区域（ $|\cos (\theta )|\le 0.7$ ）中完全包含的粒子才被接受。

Analysis 分析

For an efficient identification of charged particles over the full momentum region, information from the ionization loss in the TPC ("dE/dx") and information from the DELPHI RICH detectors were combined, using dedicated software packages [26]. One package fine-tunes the Monte Carlo simulation concerning detector related effects (such as slight fluctuations in pressures and refractive indices, background arising from photon feedback, crosstalk between the MWPC readout strips, $\delta$-rays, track ionization photoelectrons, etc.), and another package derives identification likelihoods from the specific energy loss, the number of reconstructed photons and the mean reconstructed Cerenkov angles respectively. The likelihoods are then multiplied and rescaled to one. From these, a set of "tags" which indicate the likelihood for a particular mass hypothesis ($\pi$, K, and p) are derived. Throughout this analysis leptons were not separated from pions. Their contribution to the pion sample was subtracted using simulation.
为了在全动量范围内高效地识别带电粒子，使用了来自 TPC（“dE/dx”）的电离损失信息和 DELPHI RICH 探测器的信息，使用专用软件包[26]进行组合。一个软件包对与探测器相关的效应进行蒙特卡洛模拟的微调（如压力和折射率的轻微波动，光子反馈引起的背景，MWPC 读出条之间的串扰，γ射线，轨迹电离光电子等），另一个软件包从特定能量损失、重建光子数和平均重建切伦科夫角度中得出识别似然度。然后将这些似然度相乘并重新缩放为一。从中得出一组“标签”，指示特定质量假设（π、K 和 p）的似然度。在整个分析过程中，没有将轻子与π介子分开。使用模拟减去了它们对π子样本的贡献。

A matrix inversion formalism was used to calculate the true particle rates in the detector from the tagged rates. The $3\times 3$ efficiency matrix is defined by
使用矩阵求逆形式计算探测器中的真实粒子速率，从标记速率中得出。 $3\times 3$ 效率矩阵的定义如下：

$$\begin{array}{}\text{(27)}& {\mathcal{E}}_i^j=\frac{\text{it Number of type }i\text{ hadrons tagged as type }j\text{ hadrons}}{\text{it Number of type }i\text{ hadrons}},\end{array}$$

where type $i,j$ can be either of $\pi$,$K^{\pm }$,p($\bar{\text{p}}$). It establishes the connection between the true particles in the RICH/TPC and the tagged ones:
其中 type $i,j$ 可以是 $\pi$ 、 $K^{\pm }$ 或 p( $\bar{\text{p}}$ )。它建立了 RICH/TPC 中真实粒子与标记粒子之间的连接。

$$\begin{array}{}\text{(28)}& \left(\begin{array}{c}{N}_{\pi }^{meas}\\ N_K^{meas}\\ N_p^{meas}\end{array}\right)=\mathcal{E}\left(\begin{array}{c}{N}_{\pi }^{true}\\ N_K^{true}\\ N_p^{true}\end{array}\right)\end{array}$$

The inverse of the efficiency matrix works on the three sets of tagged particles in two ways. First a particle can have multiple tags, meaning that the information from the tagging is ambiguous. This is not unlikely because in this analysis the low statistics of the data samples force rather loose selection criteria to be applied. Secondly a particle can escape identification. Both effects can be corrected by this method. The average identification efficiency is approximately 85% for pions and 60% for kaons and protons, whereas the purities are approximately 85% and 60% respectively. They show a strong momentum dependence.
效率矩阵的逆矩阵对三组标记粒子有两种作用。首先，一个粒子可以有多个标记，意味着标记信息是模糊的。这并不罕见，因为在这个分析中，数据样本的低统计量迫使我们采用相对宽松的选择标准。其次，一个粒子可能无法被鉴别。这两种效应都可以通过这种方法进行修正。平均鉴别效率对于π介子约为 85%，对于 K 介子和质子约为 60%，而纯度分别约为 85%和 60%。它们显示出强烈的动量依赖性。

K${}^0$ and $\mathrm{\Lambda }$ candidates were reconstructed by their decay in flight into ${\pi }^{+}{\pi }^{-}$ and p${\pi }^{-}$ respectively. Secondary decays candidates, $V^0$, in the selected sample of hadronic events were found by considering all pairs of oppositely charged particles. The vertex defined by each pair was determined so that the ${\chi }^2$ of the hypothesis of a common vertex was minimised. The particles were then refitted to the common vertex. The selection criteria were the "standard" ones described in [26]. The average detection efficiency from this procedure is about 36% for the decay K${}^0\to {\pi }^{+}{\pi }^{-}$ and about 28% for the decay $\mathrm{\Lambda }\to$ p${\pi }^{-}$ in multi-hadronic events. The background under the invariant mass peaks was subtracted separately for each bin of $V^0$ momentum. The background was estimated from the data by linearly interpolating two sidebands in invariant mass:
K 和Λ候选粒子通过它们在飞行中的衰变重建，分别衰变为π和 p。在所选的强子事件样本中，通过考虑所有异号电荷粒子对来找到次级衰变候选粒子Λ。每对粒子定义的顶点被确定为使得共同顶点假设的卡方最小化。然后将粒子重新拟合到共同顶点。选择标准是[26]中描述的“标准”标准。从这个过程中得到的平均探测效率约为 36%（K 衰变）和 28%（Λp 衰变）多强子事件。不变质量峰下的背景分别在每个动量区间进行了减去。背景是通过在不变质量中线性插值两个侧带来估计的。

• between 0.40 and 0.45 GeV/c${}^2$ and between 0.55 and 0.60 GeV/c${}^2$ for the K${}^0$ ;
  在 K 粒子的动量范围内，0.40 至 0.45 GeV/c 之间 ${}^2$ ，以及 0.55 至 0.60 GeV/c 之间 ${}^2$ 。
• between 1.08 and 1.10 GeV/c${}^2$ and between 1.14 and 1.18 GeV/c${}^2$ for the $\mathrm{\Lambda }$.
  在 1.08 和 1.10 GeV/c 之间 ${}^2$ ，以及在 1.14 和 1.18 GeV/c 之间 ${}^2$ ，对于 $\mathrm{\Lambda }$ 。

Calibration of the efficiency matrix using Z${}^0$ data
使用 Z ${}^0$ 数据校准效率矩阵

The Z${}^0$ data recorded during each year for calibration were used to tune the above described matrix before applying it to high energy data. This is made possible by the fact that studies at the Z${}^0$ pole [34] established that the exclusive particle spectra are reproduced to a very high level of accuracy by the DELPHI-tuned version of the generators. Therefore deviations of the rates of tagged particles between data and simulation in this sample can be interpreted as detector effects. Comparison of the tag rates allows a validation of the efficiency matrix, which would be impossible to measure from the data due to the limited LEP 2 statistics. The matrix is corrected so that it reproduces the simulated rates, assuming that the correction factors are linear in the number of tagged hadrons. The discrepancies were found to be smaller than 4%. This is taken into account when calculating the systematic uncertainties.
在应用于高能数据之前，使用每年校准记录的 Z ${}^0$ 数据来调整上述描述的矩阵。这是因为在 Z ${}^0$ 极点的研究已经证实，DELPHI 调整版本的生成器能够非常准确地再现独立粒子谱。因此，在数据和模拟之间标记粒子的比率偏差可以解释为探测器效应。比较标记比率可以验证效率矩阵，由于 LEP 2 统计数据有限，无法从数据中测量。矩阵被校正以使其再现模拟比率，假设校正因子与标记强子数成线性关系。发现的差异小于 4%。在计算系统误差时考虑了这一点。

As the high energy events are recorded over a long time period, stability of the identification devices becomes a major concern. Variations in the refractive index or the drift velocity in the RICH detectors may significantly change the performance of the identification. To estimate the effect of these variations on the measurement, the radiative returns to the Z${}^0$ were used. Such events were selected among the events passing the hadronic preselection, by requiring in addition that they contained at least 8 charged particles, they had $\sqrt{s^{\prime }}<130$ GeV, and a total energy transverse to the beam axis of more than 30 GeV. Figure 8 shows the good agreement for differential cross-sections for this event sample which may be taken as an indication of the stability of the detector during the year at the few percent level.
由于高能事件在长时间内记录，识别设备的稳定性成为一个主要关注点。RICH 探测器中的折射率或漂移速度的变化可能会显著改变识别的性能。为了估计这些变化对测量的影响，使用了返回到 Z ${}^0$ 的辐射事件。这些事件是在通过强子预选的事件中选择的，要求它们至少包含 8 个带电粒子，它们具有 $\sqrt{s^{\prime }}<130$ GeV 的能量，并且横向于束轴的总能量超过 30 GeV。图 8 显示了这个事件样本的微分截面的良好一致性，可以作为探测器在一年期间稳定性的指示，误差在几个百分点水平。

${\xi }_{\mathrm{p}}$ distributions and average multiplicities
分布和平均重数

After background subtraction, the tagged particle fractions were unfolded using the calibrated matrix. The full covariance matrix was calculated for the tag rates using multinomial statistics. It was then propagated to the true rates of identified particles using the unfolding matrix.
经过背景减法后，使用校准矩阵对标记粒子的比例进行展开。使用多项式统计方法计算标记率的完整协方差矩阵。然后将其传播到使用展开矩阵确定的粒子的真实比例。

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They are mainly due to the uncertainties in the modelling of the detector response. In addition for the charged particles there are time dependent effects such as variations of the drift velocity in the RICH detectors.
主要是由于探测器响应模型的不确定性。此外，对于带电粒子，还存在时间相关的效应，如 RICH 探测器中漂移速度的变化。

The unfolding matrix was adjusted using the Z${}^0$-calibration data recorded in the beginning of data taking in 1996, 1997 and 1998, as well as from the peak period in 1995. This is described in section 6.6. For 1997 and 1998 data also radiative return Z${}^0$-events were used. These are better suited as they were recorded under the same conditions as the signal events.
展开的矩阵是使用 1996 年、1997 年和 1998 年数据采集开始时记录的 Z ${}^0$ -校准数据进行调整的，以及 1995 年的高峰期。这在第 6.6 节中有描述。对于 1997 年和 1998 年的数据，还使用了辐射回波 Z ${}^0$ -事件。这些更适合，因为它们是在与信号事件相同的条件下记录的。

Spectra were obtained using the original and the adjusted matrices. The difference between the results obtained was taken as the corresponding systematic uncertainty.
使用原始和调整后的矩阵获得了光谱。所得结果之间的差异被视为相应的系统误差。

The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.006, 0.076 and 0.136 for charged pions, kaons and protons respectively.
相对于质心能量的平均不确定性，对于带电π介子、K 介子和质子分别为 0.006、0.076 和 0.136。

A relative systematic uncertainty of 3% was estimated for K${}^0$[35] and 5% for $\mathrm{\Lambda }$[36].
K ${}^0$ 的相对系统误差估计为 3% [35]，而 $\mathrm{\Lambda }$ 的相对系统误差估计为 5% [36]。

Size of the subtracted WW background (where applicable).
减去的 WW 背景的大小（适用时）。

Variation of the selection criteria results in a 10% uncertainty on the fraction of the W contamination in the q$\overline{\mathrm{q}}$ sample. This corresponds to a 5% uncertainty in the W cross-section and the size of the W background has been varied accordingly.
选择标准的变化导致 q $\overline{\mathrm{q}}$ 样本中 W 污染的比例存在 10%的不确定性。这相当于 W 截面的 5%不确定性，W 背景的大小相应地进行了变化。

The maximal variation observed in the distributions has been taken as the corresponding systematic uncertainty. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.003, 0.010 and 0.003 for charged pions, kaons and protons respectively, and is 0.014 (0.028, negligible) and 0.032 (0.057) for K${}^0$ and $\mathrm{\Lambda }$, respectively, at centre of mass energies of 189 (183, 161) GeV.
在分布中观察到的最大变化被视为相应的系统误差。在质心能量上平均计算，带电π介子、K 介子和质子的相对误差分别为 0.003、0.010 和 0.003，而在质心能量为 189（183、161）GeV 时，K ${}^0$ 和 $\mathrm{\Lambda }$ 的相对误差分别为 0.014（0.028，可忽略）和 0.032（0.057）。

Particles with momenta below 0.2 GeV/c or above 50 GeV/c were not identified. Their contribution was extrapolated from the simulation. Half of the extrapolated multiplicity was added in quadrature to the systematic uncertainty. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.021, 0.003 and 0.001 for charged pions, kaons and protons respectively, while for K${}^0$ and $\mathrm{\Lambda }$ at 133 GeV, 183 GeV and 189 GeV, is 0.034, and for K${}^0$ at 161 GeV is 0.042.
动量低于 0.2 GeV/c 或高于 50 GeV/c 的粒子未被鉴定。它们的贡献是通过模拟进行外推的。外推得到的多重性的一半被平方加到系统误差中。在质心能量上平均计算，对于带电π介子、K 介子和质子，这种不确定性的相对大小分别为 0.021、0.003 和 0.001，而对于 133 GeV、183 GeV 和 189 GeV 的 K ${}^0$ 和 $\mathrm{\Lambda }$ ，为 0.034，对于 161 GeV 的 K ${}^0$ ，为 0.042。

For the pions this analysis relies on a subtraction of the lepton contamination using simulation.
对于π介子，该分析依赖于使用模拟进行的减去轻子污染。

An extra uncertainty of 10% of the simulation prediction for the total number of leptons is added. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.003.
总的来说，对于轻子总数的模拟预测，增加了 10%的额外不确定性。这种不确定性的相对大小，在质心能量平均值上为 0.003。

In Figure 13 the results are compared to the predictions from PYTHIA 5.7 and HERWIG 5.8. The results shown for energies below 133 GeV (open squares) were extracted from reference [29]. The models account for the measurements, with the exception of the HERWIG predictions for K${}^0$ and $\mathrm{\Lambda }$ and of the PYTHIA predictions for K${}^0$ at high energy which are above the measured values.
在图 13 中，将结果与 PYTHIA 5.7 和 HERWIG 5.8 的预测进行了比较。能量低于 133 GeV 的结果（空心方块）来自参考文献[29]。这些模型能够解释测量结果，但 HERWIG 对于 K ${}^0$ 和 $\mathrm{\Lambda }$ 的预测以及 PYTHIA 对于高能量下 K ${}^0$ 的预测超过了测量值。

${\xi }^{\ast }$ and its evolution
${\xi }^{\ast }$ 及其演变

An interesting aspect of the ${\xi }_p$-distribution is the evolution of its peak position ${\xi }^{\ast }$ with increasing centre-of-mass energy. It is determined by fitting a parametrisation of the distribution to the peak region.
${\xi }_p$ -分布的一个有趣方面是其峰值位置 ${\xi }^{\ast }$ 随着质心能量的增加而演变。它是通过将分布的参数化拟合到峰值区域来确定的。

One such parametrisation is the distorted Gaussian in equation (3). Another parametrisation is a standard Gaussian distribution. While being a more crude approximation, it facilitates the analysis in the case of limited statistics.
其中一种参数化是方程（3）中的扭曲高斯分布。另一种参数化是标准高斯分布。虽然这是一种更粗糙的近似，但在统计数据有限的情况下，它有助于分析。

Since equation (3) is expected to describe well only the peak region, the fit range has to be carefully chosen around the peak. Table 3 shows the results and the fit range,with the ${\chi }^2$ per degree of freedom for the gaussian fit, except for the ${\mathrm{K}}^0$ and $\mathrm{\Lambda }$ at 133 GeV where only the distorted gaussian fit was performed. The errors in the data are the sum in quadrature of the statistical and systematic uncertainties (generator values were extracted from samples of one million events generated with the DELPHI tuned versions of the programs).
由于方程（3）预计只能很好地描述峰值区域，因此拟合范围必须在峰值周围谨慎选择。表 3 显示了结果和拟合范围，高斯拟合的每个自由度的 ${\chi }^2$

The systematic uncertainty has the following contributions which were added in quadrature to the statistical uncertainty of the fit.
系统误差包括以下贡献，与拟合的统计误差按平方和相加。

1. Uncertainty of the background evaluation (above the W threshold). This source was evaluated as the maximal difference obtained by a variation of the WW background cross-section. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.005, 0.010, 0.002, 0.024 and 0.023 for charged pions, kaons, protons, ${\mathrm{K}}^0$ and $\mathrm{\Lambda }$ respectively.
   背景评估的不确定性（超过 W 阈值）。该来源被评估为通过变化 WW 背景截面获得的最大差异。在质心能量上平均的相对大小分别为 0.005、0.010、0.002、0.024 和 0.023，对于带电π介子、K 介子、质子、 ${\mathrm{K}}^0$ 和 $\mathrm{\Lambda }$ 。
2. Uncertainty due to the particle identification. The analysis was repeated using the calibrated matrices or changes by 3% (${\mathrm{K}}^0$) or 5% ($\mathrm{\Lambda }$) in the bin contents of the ${\xi }_p$ distribution, and the fit redone. The maximal differences in the position of the peak thus obtained were added in quadrature. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.011, 0.017, 0.002, 0.019 and 0.017 for charged pions, kaons, protons, ${\mathrm{K}}^0$ and $\mathrm{\Lambda }$ respectively.
   由于粒子鉴别的不确定性。使用校准矩阵或 ${\mathrm{K}}^0$ 中的 bin 内容变化 3％或 5％重复分析，并重新进行拟合。因此得到的峰值位置的最大差异按平方相加。在质心能量上平均计算，相对大小的不确定性分别为 0.011、0.017、0.002、0.019 和 0.017，对于带电π介子、K 介子、质子、 ${\mathrm{K}}^0$ 和 $\mathrm{\Lambda }$ 。
3. Stability of the fit and dependence on the fit range. To estimate this effect, which arises from the combination of the limited statistics, the resulting need to choose a coarse binning, and the choice of the fit range, systematic shifts have been imposed on the data by variation within the statistical uncertainty. One standard deviation has been added to the values left of the peak and one standard deviation has been subtracted from the values to its right and vice versa. The maximum variation is taken as the contribution to the systematic uncertainty. The relative size of this uncertainty, averaged over the centre-of-mass energies is 0.026, 0.112, 0.161, 0.163 and 0.147 for charged pions, kaons, protons, ${\mathrm{K}}^0$ and $\mathrm{\Lambda }$ respectively.
   拟合的稳定性和对拟合范围的依赖性。为了估计这种效应，由于有限的统计量、选择粗糙的分组和选择拟合范围的需要，数据被施加了系统性偏移，通过在统计不确定性内进行变化。在峰值左侧的值上加了一个标准偏差，峰值右侧的值减去一个标准偏差，反之亦然。最大变化被视为系统性不确定性的贡献。在质心能量上平均计算，相对于带电π介子、K 介子、质子、 ${\mathrm{K}}^0$ 和 $\mathrm{\Lambda }$ ，这种不确定性的相对大小分别为 0.026、0.112、0.161、0.163 和 0.147。

Figure 14 shows the ${\xi }^{\ast }$ values from the Gaussian fits as a function of the centre-of-mass energy. The data up to centre-of-mass energies of 91 GeV were taken from previous measurements [37]. The fits to expression (4), with $Y$ and $C=0.351$ defined as in equation (5) and $F_h$ and ${\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ taken as free parameters, were done separately for each particle type and are superimposed on the data points (lines). Figure 14 shows that (within the statistics of the data samples analysed) the fitted functions follow well the data points. This suggests that MLLA+LPHD gives a good description of the observed particle spectra. From table 3 it is shown that there is fair agreement between the data and the predictions from the generators (PYTHIA 5.7, HERWIG 5.8, and ARIADNE 4.8).
图 14 显示了高斯拟合的 ${\xi }^{\ast }$ 值随质心能量的变化。质心能量为 91 GeV 以下的数据来自先前的测量[37]。对于每种粒子类型，使用式（4）进行拟合，其中 $Y$ 和 $C=0.351$ 的定义如方程（5）所示， $F_h$ 和 ${\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ 作为自由参数。拟合结果与数据点（线）叠加在一起。图 14 显示（在分析的数据样本统计误差范围内），拟合函数很好地符合数据点。这表明 MLLA+LPHD 很好地描述了观测到的粒子谱。从表 3 可以看出，数据与生成器（PYTHIA 5.7、HERWIG 5.8 和 ARIADNE 4.8）的预测之间存在较好的一致性。

7 Identified hadrons in WW events
WW 事件中鉴定出的 7 种强子

The selections of (4$q$) and (2$q$) WW events in the analysis of identified hadrons in WW events at 189 GeV are similar to the procedures described in [38]. A feed forward neural network, trained with back-propagation on PYTHIA simulated events, was used to improve the separation of (4$q$) WW events from 2-fermion (mainly ${\mathrm{Z}}^0/\gamma \to \mathrm{q}\bar{\mathrm{q}}(\gamma )$) and 4-fermion background (mainly ${\mathrm{Z}}^0{\mathrm{Z}}^0$ events). The network input variables, the training
在对 189 GeV 的 WW 事件中鉴别强子的分析中，选择(4 $q$ )和(2 $q$ ) WW 事件的方法与[38]中描述的过程类似。使用了一个前馈神经网络，通过 PYTHIA 模拟事件进行反向传播训练，以改善(4 $q$ ) WW 事件与 2 费米子(主要是 ${\mathrm{Z}}^0/\gamma \to \mathrm{q}\bar{\mathrm{q}}(\gamma )$ )和 4 费米子背景(主要是 ${\mathrm{Z}}^0{\mathrm{Z}}^0$ 事件)的分离。网络的输入变量、训练

$$\text{Unknown environment 'tabular'}$$ \end{table} Table 4: Values for ${\xi }^{\ast }$ for ${\pi }^{\pm }$, ${\mathrm{K}}^{\pm }$ and protons from a Gaussian fit to the semi-leptonic W data at 189 GeV. The fit was made to the spectra in the W rest frame. The first error is statistical and the second reflects the total systematic uncertainties. Also indicated is the ${\chi }^2/n.d.f$.
表 4：对 189 GeV 半轻子 W 数据进行高斯拟合得到的 ${\pi }^{\pm }$ ， ${\mathrm{K}}^{\pm }$ 和质子的 ${\xi }^{\ast }$ 值。拟合是在 W 静止参考系中进行的。第一个误差是统计误差，第二个误差反映了总体系统误差。还指示了 ${\chi }^2/n.d.f$ 。

The ratios of the average multiplicities in (4q) events to twice the multiplicities in (2q) events for different hadron species, conservatively assuming uncorrelated errors, for the full momentum range and for momentum between 0.2 and 1.25 GeV/c are shown in table 6, and are compatible with unity. The systematic errors for the restricted momentum range were assumed to be proportional to the sum in quadrature of the systematic errors in the full momentum range, excluding the contributions from the extrapolation.
不同强子种类中，(4q)事件的平均多重性与(2q)事件的两倍多重性之比，在整个动量范围和动量在 0.2 至 1.25 GeV/c 之间的情况下，保守地假设误差不相关，如表 6 所示，并且与单位相符。对于受限动量范围的系统误差被假设与整个动量范围的系统误差的平方和成比例，不包括外推的贡献。

8 Summary and Discussion
8. 总结与讨论

The mean charged particle multiplicities $⟨n⟩$ and dispersions $D$ in $\mathrm{q}\overline{\mathrm{q}}$ events at the different centre-of-mass energies were measured to be:
在不同的质心能量下，测量得到的 $\mathrm{q}\overline{\mathrm{q}}$ 事件中的平均带电粒子多重性 $⟨n⟩$ 和离散度 $D$ 为：

$$\begin{array}{}\text{(29)}& ⟨n{⟩}_{183\mathrm{GeV}}=27.05\pm 0.27(stat)\pm 0.32(syst)\end{array}$$ $$D_{183\mathrm{GeV}}=8.08\pm 0.19(stat)\pm 0.14(syst)$$ (30) $$⟨n{⟩}_{189\mathrm{GeV}}=27.47\pm 0.18(stat)\pm 0.30(syst)$$ (31) $$D_{189\mathrm{GeV}}=8.77\pm 0.13(stat)\pm 0.11(syst)$$ (32) $$⟨n{⟩}_{200\mathrm{GeV}}=27.58\pm 0.19(stat)\pm 0.45(syst)$$ (33) $$\begin{array}{}\text{(34)}& D_{200\mathrm{GeV}}=8.64\pm 0.13(stat)\pm 0.20(syst).\end{array}$$

Figure 15 shows the value of the average charged particle multiplicity in $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ events at 183, 189 and 200 GeV compared with lower energy points from JADE [39], PLUTO [40], MARK II [41], TASSO [42], HRS [43], and AMY [44], with DELPHI results in $\mathrm{q}\overline{\mathrm{q}}\gamma$ events at the ${\mathrm{Z}}^0$[45], with the world average at the ${\mathrm{Z}}^0$[29], and with LEP results
图 15 显示了 183、189 和 200 GeV 能量下 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 事件中平均带电粒子多重性的数值，与 JADE [39]、PLUTO [40]、MARK II [41]、TASSO [42]、HRS [43]和 AMY [44]的低能量数据进行比较，以及 $\mathrm{q}\overline{\mathrm{q}}\gamma$ 事件中 DELPHI 的结果在 ${\mathrm{Z}}^0$ [45]，与 ${\mathrm{Z}}^0$ 的世界平均值和 LEP 的结果

$$\text{Unknown environment 'table'}$$ Table 6: Ratio of average multiplicities in (4q) events to twice the values in (2q) events for ${\pi }^{\pm }$, ${\mathrm{K}}^{\pm }$ and protons from WW events at 189 GeV, for different momentum ranges. The first uncertainty is statistical and the second is systematic.
表 6：在不同动量范围内，189 GeV 的 $$\text{Unknown environment 'table'}$$ ， ${\pi }^{\pm }$ 和质子的(4q)事件的平均多重性与(2q)事件中两倍数值的比率。第一个不确定性是统计误差，第二个是系统误差。

$$\text{Unknown environment 'table'}$$ Table 5: Average multiplicity for ${\pi }^{\pm }$, ${\mathrm{K}}^{\pm }$ and protons for WW events at 189 GeV. In the data the first uncertainty is statistical and the second is systematic.
表 5：189 GeV 下 WW 事件的 ${\pi }^{\pm }$ 、 ${\mathrm{K}}^{\pm }$ 和质子的平均多重性。在数据中，第一个不确定性是统计误差，第二个是系统误差。

at high energy [46; 47; 48; 49; 50; 51; 52; 53; 54]. The points from JADE, PLUTO and MARK II do not include the decay products of short lived K${}^0$ and $\mathrm{\Lambda }$. The value at $M_{{\mathrm{Z}}^0}$ has been lowered by 0.20 to account for the different proportion of $b\overline{b}$ and $c\overline{c}$ events at the ${\mathrm{Z}}^0$ with respect to the $e^{+}{e}^{-}\to {\gamma }^{\ast }\to \mathrm{q}\overline{\mathrm{q}}$[55]. Similarly, the values at 133, 161, 172, 183 and 189 GeV were lowered by 0.15, 0.12, 0.11, 0.08, 0.05 and 0.07 respectively. The QCD prediction for the charged particle multiplicity has been computed as a function of ${\alpha }_s$ including the resummation of leading (LLA) and next-to-leading (NLLA) corrections [56]:
在高能量下[46; 47; 48; 49; 50; 51; 52; 53; 54]。JADE、PLUTO 和 MARK II 的数据不包括短寿命 K ${}^0$ 和 $\mathrm{\Lambda }$ 的衰变产物。在 $M_{{\mathrm{Z}}^0}$ 处的值已经降低了 0.20，以考虑 ${\mathrm{Z}}^0$ 相对于 $e^{+}{e}^{-}\to {\gamma }^{\ast }\to \mathrm{q}\overline{\mathrm{q}}$ 的 $b\overline{b}$ 和 $c\overline{c}$ 事件的不同比例[55]。类似地，133、161、172、183 和 189 GeV 处的值分别降低了 0.15、0.12、0.11、0.08、0.05 和 0.07。带电粒子多重性的 QCD 预测已经根据 ${\alpha }_s$ 的函数计算，包括领先（LLA）和次领先（NLLA）修正的重新求和[56]。

$$\begin{array}{}\text{(35)}& ⟨n⟩(\sqrt{s})=a[{\alpha }_s(\sqrt{s}){]}^b{e}^{c/\sqrt{{\alpha }_s(\sqrt{s})}}[1+O(\sqrt{{\alpha }_s(\sqrt{s})})]\end{array}$$

where $s$ is the squared centre-of-mass energy and $a$ is a parameter (not calculable from perturbation theory) whose value has been fitted from the data. The constants $b=0.49$ and $c=2.27$ are predicted by theory [56] and ${\alpha }_s(\sqrt{s})$ is the strong coupling constant. The fitted curve to the data between 14 GeV and 200 GeV, excluding the results from JADE, PLUTO and MARK II, is plotted in Figure 15, corresponding to $a=0.045$ and ${\alpha }_s(m_{\mathrm{Z}})=0.112$. The multiplicity values are consistent with the QCD prediction on the multiplicity evolution with centre-of-mass energy.
其中 $s$ 是质心能量的平方， $a$ 是一个参数（无法从微扰理论计算得出），其值已通过数据拟合得到。常数 $b=0.49$ 和 $c=2.27$ 由理论[56]预测， ${\alpha }_s(\sqrt{s})$ 是强耦合常数。在 14 GeV 至 200 GeV 之间的数据拟合曲线，不包括 JADE、PLUTO 和 MARK II 的结果，如图 15 所示，对应于 $a=0.045$ 和 ${\alpha }_s(m_{\mathrm{Z}})=0.112$ 。多重性数值与质心能量的多重性演化的量子色动力学（QCD）预测一致。

The ratios of the average multiplicity to the dispersion measured at 183 GeV, 189 GeV and 200 GeV, $⟨n⟩/D=3.35\pm 0.11$, $⟨n⟩/D=3.14\pm 0.07$ and $⟨n⟩/D=3.19\pm 0.10$ (where the errors are the sum in quadrature of the statistical and of the systematic) respectively, are consistent with the weighted average from the measurements at lower centre-of-mass energies (3.13 $\pm$ 0.04), as can be seen in Figure 16. From Koba-Nielsen-Olesen scaling [57] this ratio is predicted to be energy-independent. The ratio measured is also consistent with the predictions of QCD including 1-loop Higher Order terms (H.O.) [58].
在 183 GeV、189 GeV 和 200 GeV 测量的平均多重性与离散度的比率 $⟨n⟩/D=3.35\pm 0.11$ 、 $⟨n⟩/D=3.14\pm 0.07$ 和 $⟨n⟩/D=3.19\pm 0.10$ （其中误差是统计误差和系统误差的平方和）分别与较低质心能量的测量结果的加权平均值（3.13 $\pm$ 0.04）一致，如图 16 所示。根据 Koba-Nielsen-Olesen 缩放[57]，预测这个比率与能量无关。测量得到的比率也与包括 1 环高阶项（H.O.）的 QCD 预测[58]一致。

For WW events the measured multiplicities in the fully hadronic channel are:
对于 WW 事件，在全强子通道中测得的多重性为：

$$\begin{array}{}\text{(36)}& ⟨n^{(4q)}{⟩}_{189\mathrm{G}\mathrm{e}\mathrm{V}}=39.12\pm 0.33(stat)\pm 0.36(syst)\end{array}$$ $$\begin{array}{}\text{(37)}& ⟨n^{(4q)}{⟩}_{183\mathrm{G}\mathrm{e}\mathrm{V}}=38.11\pm 0.57(stat)\pm 0.44(syst),\end{array}$$

while for the semileptonic channel they are:
而对于半轻子通道，它们是：

$$\begin{array}{}\text{(38)}& ⟨n^{(2q)}{⟩}_{189\mathrm{G}\mathrm{e}\mathrm{V}}=19.49\pm 0.31(stat)\pm 0.27(syst)\end{array}$$ $$\begin{array}{}\text{(39)}& ⟨n^{(2q)}{⟩}_{183\mathrm{G}\mathrm{e}\mathrm{V}}=19.78\pm 0.49(stat)\pm 0.43(syst).\end{array}$$

The PYTHIA Monte Carlo program with parameters tuned to the DELPHI data at LEP 1, predicts multiplicities of 38.2 and 19.1 for the fully hadronic and semileptonic events respectively.
PYTHIA 蒙特卡洛程序根据 LEP 1 上的 DELPHI 数据进行参数调整，预测完全强子和半轻子事件的多重性分别为 38.2 和 19.1。

A possible depletion of the multiplicity in fully hadronic WW events with respect to twice the semileptonic events, as predicted by most colour reconnection models, is not observed in the full momentum range, in agreement with the results from other LEP collaborations [59]:
与大多数颜色重联模型预测的完全强子 WW 事件的多重性相比，与半强子事件的两倍相比，可能出现的减少并未在整个动量范围内观察到，与其他 LEP 合作组织的结果一致[59]

$$\begin{array}{}\text{(40)}& \left(\frac{⟨n^{(4q)}⟩}{2⟨n^{(2q)}⟩}\right)=0.990\pm 0.015(stat)\pm 0.011(syst),\end{array}$$

nor in the momentum range $0.1<p<1.0$ GeV/$c$:
在动量范围 $0.1<p<1.0$ GeV/ $c$ 内也不是

$$\begin{array}{}\text{(41)}& \frac{⟨n^{(4q)}⟩}{2⟨n^{(2q)}⟩}{|}^{0.1<p<1GeV/c}=0.981\pm 0.024(stat)\pm 0.013(syst).\end{array}$$No significant difference is observed between the dispersion in fully hadronic events and $\sqrt{2}$ times the dispersion in semileptonic events:
在完全强子衰变事件和半轻子衰变事件中，观察到的离散度之间没有显著差异

$$\frac{D^{(4q)}}{\sqrt{2}{D}^{(2q)}}=0.94\pm 0.03(\text{stat})\pm 0.03(\text{syst}).$$

Assuming uncorrelated systematic errors for the two centre-of-mass energies, 183 and 189 GeV, the inverse of the sums in quadrature of the statistical and the systematic errors were used as weights to give the averages:
假设两个质心能量 183 和 189 GeV 的系统误差不相关，将统计误差和系统误差的平方和的倒数作为权重，计算平均值

$$\begin{array}{}\text{(42)}& ⟨n^{(W)}⟩=19.44\pm 0.13(stat)\pm 0.12(syst)\end{array}$$ $$\begin{array}{}\text{(43)}& D^{(W)}=6.20\pm 0.11(stat)\pm 0.06(syst),\end{array}$$

where the multiplicities and their errors in (4q) were divided by 2 and the dispersions and their errors in (4q) were divided by $\sqrt{2}$.
在(4q)中，多重性及其误差被除以 2，离散度及其误差被除以 $\sqrt{2}$ 。

The value of $⟨n^{(W)}⟩$ is plotted in Figure 15 at an energy value corresponding to the W mass, with an increase of 0.35 applied to account for the different proportion of events with a $b$ or a $c$ quark. The measurement lies on the same curve as the neutral current data. The value of $⟨n^{(W)}⟩/D^{(W)}=3.14\pm 0.07(stat+syst)$, plotted in Figure 16, is also consistent with the $e^{+}{e}^{-}\to {\gamma }^{\ast }\to \text{q}\overline{\text{q}}$ average.
在图 15 中， $⟨n^{(W)}⟩$ 的值以与 W 质量相对应的能量值绘制，增加了 0.35 以考虑具有 $b$ 或 $c$ 夸克的事件比例的差异。测量结果与中性电流数据的曲线相同。在图 16 中绘制的 $⟨n^{(W)}⟩/D^{(W)}=3.14\pm 0.07(stat+syst)$ 的值也与 $e^{+}{e}^{-}\to {\gamma }^{\ast }\to \text{q}\overline{\text{q}}$ 的平均值一致。

The production of ${\pi }^{+}$, K${}^{+}$, K${}^0$, p and $\mathrm{\Lambda }$ from q$\overline{\text{q}}$ and WW events at 189 GeV was also studied. The results on the average multiplicity of identified particles and on the position ${\xi }^{\ast }$ of the maximum of the ${\xi }_p=-\text{log}(\frac{2\text{p}}{\sqrt{\text{s}}})$ distribution were compared with predictions of PYTHIA and with calculations based on MLLA+LPHD approximations. Within their uncertainties the data are in good agreement with the prediction from the generator as well as with the predictions based on the analytical calculations in the MLLA framework.
还研究了在 189 GeV 的 q 和 WW 事件中产生的 ${\pi }^{+}$ 、K ${}^{+}$ 、K ${}^0$ 、p 和 $\mathrm{\Lambda }$ 的生产。对于已鉴别粒子的平均多重性和 ${\xi }_p=-\text{log}(\frac{2\text{p}}{\sqrt{\text{s}}})$ 分布最大值的位置 ${\xi }^{\ast }$ 的结果与 PYTHIA 的预测以及基于 MLLA+LPHD 近似的计算进行了比较。在其不确定性范围内，数据与生成器的预测以及基于 MLLA 框架的解析计算的预测相吻合。

The ratio of the multiplicities of identified heavy hadrons in (4q) events to twice those in (2q) events is compatible with unity, both for the full momentum range and momenta between 0.2 and 1.25 GeV/c:
在（4q）事件中，已鉴定的重型强子的多重性与（2q）事件中的两倍相比，比值与单位相符，无论是在全动量范围内还是在 0.2 至 1.25 GeV/c 的动量范围内。

$$\text{Math not terminated in text box}$$

The evolution for the ${\xi }^{\ast }$ for identified hadrons is in good agreement with the prediction from perturbative QCD (equation (4)). This underlines the applicability of MLLA/LPHD for the description of hadron production in $e^{+}{e}^{-}$ annihilation over the full LEP energy range.
鉴于鉴别强子的演化与微扰量子色动力学（方程（4））的预测非常吻合，这强调了 MLLA/LPHD 在描述 LEP 能量范围内的强子产生中的适用性。

Acknowledgements 致谢

We are greatly indebted to our technical collaborators, to the members of the CERN-SL Division for the excellent performance of the LEP collider, and to the funding agencies for their support in building and operating the DELPHI detector.
我们对我们的技术合作伙伴深表感谢，对 CERN-SL 部门的成员在 LEP 对撞机的出色表现表示感谢，并对资助机构在建设和运营 DELPHI 探测器方面的支持表示感谢。

We acknowledge in particular the support of
我们特别感谢的支持

Austrian Federal Ministry of Science and Traffics, GZ 616.364/2-III/2a/98,
奥地利联邦科学和交通部，GZ 616.364/2-III/2a/98

FNRS-FWO, Belgium, FNRS-FWO，比利时

FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil,
FINEP, CNPq, CAPES, FUJB 和 FAPERJ，巴西

Czech Ministry of Industry and Trade, GA CR 202/96/0450 and GA AVCR A1010521,
捷克工业和贸易部，GA CR 202/96/0450 和 GA AVCR A1010521

Danish Natural Research Council,Commission of the European Communities (DG XII),
丹麦自然研究委员会，欧洲委员会（DG XII）

Direction des Sciences de la Matiere, CEA, France,
法国原子能与替代能源委员会材料科学部门

Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie, Germany,
德国联邦教育、科学、研究和技术部

General Secretariat for Research and Technology, Greece,
希腊研究与技术总秘书处

National Science Foundation (NWO) and Foundation for Research on Matter (FOM),
国家科学基金会（NWO）和物质研究基金会（FOM）

The Netherlands, 荷兰

Norwegian Research Council,
挪威研究委员会

State Committee for Scientific Research, Poland, 2P03B06015, 2P03B1116 and SPUB/P03/178/98,
波兰国家科学研究委员会，2P03B06015，2P03B1116 和 SPUB/P03/178/98

JNICT-Junta Nacional de Investigacao Cientifica e Tecnologica, Portugal,
葡萄牙国家科学技术研究委员会

Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134,
斯洛伐克科学研究基金机构 MS SR，编号 95/5195/134

Ministry of Science and Technology of the Republic of Slovenia,
斯洛文尼亚共和国科学与技术部

CICYT, Spain, AEN96-1661 and AEN96-1681,
CICYT，西班牙，AEN96-1661 和 AEN96-1681

The Swedish Natural Science Research Council,
瑞典自然科学研究委员会

Particle Physics and Astronomy Research Council, UK,
英国粒子物理与天文研究委员会

Department of Energy, USA, DE-FG02-94ER40817.
能源部，美国，DE-FG02-94ER40817。

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Figure 1: Charged particle multiplicity distributions for (a) the ($4q$) events and (b) the ($2q$) events at 183 GeV, for (c) the ($4q$) events and (d) the ($2q$) events at 189 GeV. The error bars in the data represent the statistical errors. The shaded areas represent the background contribution; the histograms are the sum of the expected signal and background.
图 1：在 183 GeV 时，(a) ( $4q$ ) 事件和 (b) ( $2q$ ) 事件的带电粒子多重性分布，以及在 189 GeV 时，(c) ( $4q$ ) 事件和 (d) ( $2q$ ) 事件的带电粒子多重性分布。数据中的误差棒表示统计误差。阴影区域表示背景贡献；直方图是预期信号和背景的总和。

Figure 2: (a) Corrected momentum distributions of charged particles for ($4q$) events (closed circles) and ($2q$) events (open circles), compared to simulation without colour reconnection, at 189 GeV. The error bars in the data represent the statistical errors. The difference between ($4q$) and twice ($2q$) is shown in (b).
图 2：（a）189 GeV 时，带电粒子的修正动量分布，与没有颜色重联的模拟结果进行比较，闭合圆圈表示（ $4q$ ）事件，开放圆圈表示（ $2q$ ）事件。数据中的误差棒表示统计误差。图（b）显示了（ $4q$ ）与两倍（ $2q$ ）之间的差异。

Figure 3: (a) Corrected momentum distributions of charged particles for ($4q$) events (closed circles) and ($2q$) events (open circles), compared to simulation without colour reconnection, at 183 GeV. The difference between ($4q$) and twice ($2q$) is shown in (b).
图 3：（a）183 GeV 时，（ $4q$ ）事件（实心圆）和（ $2q$ ）事件（空心圆）的修正动量分布，与没有颜色重联的模拟进行比较。 （b）显示了（ $4q$ ）和两倍（ $2q$ ）之间的差异。

Figure 4: (a) Corrected ${\xi }_E$ distributions of charged particles for $(4q)$ events (closed circles) and $(2q)$ events (open circles), at 189 GeV, compared to simulation without colour reconnection. The error bars in the data represent the statistical errors. The difference between $(4q)$ and twice $(2q)$ is shown in (b).
图 4：（a）189 GeV 能量下，带电粒子的修正 ${\xi }_E$ 分布，与没有颜色重联的模拟结果进行比较，其中 $(4q)$ 事件（实心圆）和 $(2q)$ 事件（空心圆）。数据中的误差棒表示统计误差。图（b）显示了 $(4q)$ 与两倍 $(2q)$ 之间的差异。

Figure 5: (a) Corrected ${\xi }_E$ distributions of charged particles for $(4q)$ events (closed circles) and $(2q)$ events (open circles), at 183 GeV, compared to simulation without colour reconnection. The error bars in the data represent the statistical errors. The difference between $(4q)$ and twice $(2q)$ is shown in (b).
图 5：（a）183 GeV 能量下，带电粒子的修正 ${\xi }_E$ 分布，与没有颜色重联的模拟结果进行比较，闭合圆圈表示 $(4q)$ 事件，开放圆圈表示 $(2q)$ 事件。数据中的误差棒代表统计误差。图（b）显示了 $(4q)$ 与两倍 $(2q)$ 之间的差异。

Figure 6: (a) Corrected $p_T$ distributions of charged particles for ($4q$) events (closed circles) and ($2q$) events (open circles), compared to simulation without colour reconnection, at 189 GeV. The internal error bars in the data represent the statistical error and the external error bars represent the sum in quadrature of the statistical and systematic errors. The difference between ($4q$) and twice ($2q$) is shown in (b).
图 6：（a）189 GeV 能量下，（ $4q$ ）事件（实心圆）和（ $2q$ ）事件（空心圆）的修正 $p_T$ 带电粒子分布，与没有颜色重联的模拟结果进行比较。数据中的内部误差条代表统计误差，外部误差条代表统计误差和系统误差的平方和。图（b）显示了（ $4q$ ）与两倍（ $2q$ ）之间的差异。

Figure 7: (a) Corrected $p_T$ distributions of charged particles for ($4q$) events (closed circles) and ($2q$) events (open circles), compared to simulation without colour reconnection, at 183 GeV. The internal error bars in the data represent the statistical error and the external error bars represent the sum in quadrature of the statistical and systematic errors. The difference between ($4q$) and twice ($2q$) is shown in (b).
图 7：（a）183 GeV 能量下，（ $4q$ ）事件（实心圆）和（ $2q$ ）事件（空心圆）的修正 $p_T$ 带电粒子分布，与没有颜色重联的模拟结果进行比较。数据中的内部误差条代表统计误差，外部误差条代表统计误差和系统误差的平方和。图（b）显示了（ $4q$ ）与两倍（ $2q$ ）之间的差异。

Figure 8: Fractions of identified particles in radiative Z${}^0$ events from 189 GeV data. The data have been taken under the same conditions as the signal data. Data (points) are in good agreement with the prediction from JETSET (lines) including full detector simulation. The top line and points indicate the sum of the fractions after unfolding. The error bars represent only the statistical error.
图 8：来自 189 GeV 数据的辐射 Z ${}^0$ 事件中已鉴别粒子的比例。数据在与信号数据相同的条件下进行采集。数据（点）与包括完整探测器模拟的 JETSET 预测（线）非常吻合。顶部的线和点表示展开后的比例之和。误差棒仅表示统计误差。

Figure 9: ${\xi }_p$ distributions (efficiency corrected and background subtracted) for charged particles, pions, kaons and protons in q$\overline{\mathrm{q}}$ events at 189 GeV. Data (points) are compared to the prediction from JETSET (solid line). Only the statistical uncertainties are shown. The dashed dotted line shows a fit to equation (3).
图 9：在 189 GeV 的 q $\overline{\mathrm{q}}$ 事件中，带电粒子、π介子、K 介子和质子的 ${\xi }_p$ 分布（经过效率校正和背景减除）。数据（点）与 JETSET 的预测（实线）进行比较。只显示统计误差。虚线点线显示对方程（3）的拟合。

Figure 10: ${\xi }_p$ distributions (efficiency corrected and background subtracted) for charged particles, pions, kaons and protons in fully hadronic WW events at 189 GeV. Data (points) which have been boosted back to the W rest-frame, compared to the prediction from JETSET (solid line). Only the statistical uncertainties are shown. The dashed dotted line shows a fit to equation (3).
图 10：在 189 GeV 的全强子 WW 事例中，带电粒子、π介子、K 介子和质子的 ${\xi }_p$ 分布（经过效率校正和背景减除）。数据（点）已经提升回 W 静止参考系，并与 JETSET 的预测（实线）进行比较。只显示统计不确定性。虚线点线显示对方程（3）的拟合。

Figure 11: ${\xi }_p$ distributions (efficiency corrected and background subtracted) for charged particles, pions, kaons and protons in semileptonic WW events at 189 GeV. Data (points) which have been boosted back to the W rest-frame, compared to the prediction from JETSET (solid line). Only the statistical uncertainties are shown. The dashed dotted line shows a fit to equation (3).
图 11：在 189 GeV 的半轻子 WW 事例中，带电粒子、π介子、K 介子和质子的 ${\xi }_p$ 分布（经过效率校正和背景减除）。数据（点）已经提升回 W 静止参考系，并与 JETSET 的预测（实线）进行比较。只显示统计不确定性。虚线点线显示对方程（3）的拟合。

Figure 12: ${\xi }_p$ distributions (efficiency corrected and background subtracted) for neutral Kaons and neutral Lambdas in $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ at 183 GeV and 189 GeV respectively, for data (points), and simulation using JETSET (dashed-dotted histogram). The full curves show the fit of the data to a Gaussian and the dashed ones to a distorted Gaussian. The error bars represent only the statistical error.
图 12：中性 Kaons 和中性 Lambdas 在 183 GeV 和 189 GeV 的 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 中的 ${\xi }_p$ 分布（经过效率校正和背景减除），分别对应数据（点）和使用 JETSET 进行的模拟（虚线点划直方图）。实线曲线显示数据拟合为高斯分布，虚线曲线显示数据拟合为畸变高斯分布。误差棒仅表示统计误差。

Figure 13: Average multiplicity of K${}^{+}$ (a), K${}^0$ (b), p (c) and $\mathrm{\Lambda }$ (d) as function of the centre-of-mass energy. Black squares are DELPHI high energy data, open squares are measurements from a previous DELPHI publication [34], and open bullets are values taken from the PDG [29]. Simulations using PYTHIA tuned to DELPHI data (solid line) and HERWIG 5.8 (dashed line) are superimposed. The error bars represent the sum in quadrature of the statistical and the systematic uncertainties.
图 13：K ${}^{+}$ （a）、K ${}^0$ （b）、p（c）和 $\mathrm{\Lambda }$ （d）的平均多重性作为质心能量的函数。黑色方块是 DELPHI 高能数据，空心方块是来自之前 DELPHI 出版物的测量结果[34]，空心圆点是从 PDG[29]中获取的数值。使用调整为 DELPHI 数据的 PYTHIA 模拟（实线）和 HERWIG 5.8（虚线）进行叠加。误差棒表示统计和系统误差的平方和。

Figure 14: Evolution of ${\xi }^{\ast }$ in $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ with increasing centre-of-mass energy. This measurement is compared with previous measurements from DELPHI from references [34] and lower energy experiments [37]. The lines are fits to equation (4) for the hadrons listed in the figure. The error bars represent the sum in quadrature of the statistical and the systematic uncertainties. For better legibility the data points have been shifted from the nominal energy.
图 14：随着质心能量的增加， ${\xi }^{\ast }$ 在 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 中的演化。该测量结果与 DELPHI 的先前测量结果[34]和较低能量实验[37]进行了比较。曲线是根据图中列出的强子的方程（4）进行拟合。误差棒表示统计和系统误差的平方和。为了更好地阅读，数据点已从名义能量偏移。

Figure 15: Measured average charged particle multiplicity in $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ events as a function of centre-of-mass energy $\sqrt{s}$. DELPHI high energy results are compared with other experimental results and with a fit to a prediction from QCD in Next to Leading Order. The error bars represent the sum in quadrature of the statistical and the systematic uncertainties. Some points are slightly shifted on the abscissa for clarity. The average charged multiplicity in W decays is also shown at an energy corresponding to the W mass. The measurements have been corrected for the different proportions of $b\overline{b}$ and $c\overline{c}$ events at the various energies.
图 15：作为质心能量 $\sqrt{s}$ 函数的 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 事例中测量得到的平均带电粒子多重性。DELPHI 高能结果与其他实验结果以及 QCD 在次领导阶的预测拟合进行了比较。误差棒表示统计和系统误差的平方和。为了清晰起见，一些点在横坐标上稍微偏移。还显示了 W 衰变对应能量下的平均带电多重性。测量结果已经校正为不同能量下 $b\overline{b}$ 和 $c\overline{c}$ 事例的比例。

Figure 16: Ratio of the average charged particle multiplicity to the dispersion in $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ events at 183, 189 and 200 GeV, compared with lower energy measurements. The error bars represent the sum in quadrature of the statistical and the systematic uncertainties. Some points are slightly shifted on the abscissa for clarity. The ratio in W decays is also shown at an energy corresponding to the W mass. The straight solid and dotted lines represent the weighted average of the data points and its error. The dashed line represents the prediction from QCD (see text).
图 16：在 183、189 和 200 GeV 能量下，平均带电粒子多重性与 $e^{+}{e}^{-}\to \mathrm{q}\overline{\mathrm{q}}$ 事件中离散度的比值，与较低能量的测量结果进行比较。误差棒表示统计和系统误差的平方和。为了清晰起见，某些点在横坐标上稍微偏移。在对应于 W 质量的能量下，还显示了 W 衰变的比值。实线和虚线表示数据点的加权平均值及其误差。虚线表示 QCD 的预测（见正文）。