analysis is also possible. The Z, like all unstable particles, is associated with a complex pole in the S matrix. The pole position is process-independent and gauge-invariant. The mass, $\overline{M}_Z$, and width, $\overline{\Gamma}_Z$, can be defined in terms of the pole in the energy plane via [11–14]

$$ \bar{s} = \overline{M}_Z^2 - i \overline{M}_Z \overline{\Gamma}_Z \qquad (55.8) $$

leading to the relations

$$ \begin{aligned} \overline{M}_Z &= M_Z / \sqrt{1 + \Gamma_Z^2 / M_Z^2} \\ &\approx M_Z - 34.1 \text{ MeV} \end{aligned} \qquad (55.9) $$

$$ \begin{aligned} \overline{\Gamma}_Z &= \Gamma_Z / \sqrt{1 + \Gamma_Z^2 / M_Z^2} \\ &\approx \Gamma_Z - 0.9 \text{ MeV.} \end{aligned} \qquad (55.10) $$

The LEP collaborations [15] have analyzed their data using the S-matrix approach as defined in Eq. (55.8), in addition to the conventional one. They observe a downward shift in the Z mass as expected.

## 55.3 Handling the large-angle $e^+e^-$ final state

Unlike other $f\bar{f}$ decay final states of the Z, the $e^+e^-$ final state has a contribution not only from the s-channel but also from the t-channel and s-t interference. The full amplitude is not amenable to fast calculation, which is essential if one has to carry out minimization fits within reasonable computer time. The usual procedure is to calculate the non-s channel part of the cross section separately using the Standard Model programs ALIBABA [16] or TOPAZ0 [17], with the measured value of $M_{top}$, and $M_{Higgs} = 150$ GeV, and add it to the s-channel cross section calculated as for other channels. This leads to two additional sources of error in the analysis: firstly, the theoretical calculation in ALIBABA itself is known to be accurate to $\sim 0.5\%$, and secondly, there is uncertainty due to the error on $M_{top}$ and the unknown value of $M_{Higgs}$ (100–1000 GeV). These errors are propagated into the analysis by including them in the systematic error on the $e^+e^-$ final state. As these errors are common to the four LEP experiments, this is taken into account when performing the LEP average.

## 55.4 Errors due to uncertainty in LEP energy determination

The systematic errors related to the LEP energy measurement, see [18–23], can be classified as:

* The absolute energy scale error;

* Energy-point-to-energy-point errors due to the nonlinear response of the magnets to the exciting currents;

* Energy-point-to-energy-point errors due to possible higher-order effects in the relationship between the dipole field and beam energy;

* Energy reproducibility errors due to various unknown uncertainties in temperatures, tidal effects, corrector settings, RF status, etc.

Precise energy calibration was done outside normal data-taking using the resonant depolarization technique. Run-time energies were determined every 10 minutes by measuring the relevant machine parameters and using a model which takes into account all the known effects, including leakage currents produced by trains in the Geneva area and the tidal effects due to gravitational forces of the Sun and the Moon. The LEP Energy Working Group has provided a covariance matrix from the determination of LEP energies for the different running periods during 1993–1995 [18].

## 55.5 Choice of fit parameters

The LEP Collaborations have chosen the following primary set of parameters for fitting:

$$ M_Z, \Gamma_Z, \sigma_{\text{hadron}}^0, R(\text{lepton}), A_{\text{FB}}^{(0,\ell)}, $$

where

$$ \begin{aligned} R(\text{lepton}) &= \Gamma(\text{hadrons})/\Gamma(\text{lepton}), \sigma_{\text{hadron}}^0 \\ &= 12\pi\Gamma(e^+e^-)\Gamma(\text{hadrons})/M_Z^2\Gamma_Z^2. \end{aligned} $$

With a knowledge of these fitted parameters and their covariance matrix, any other parameter can be derived. The main advantage of these parameters is that they form a physics motivated set of parameters with much reduced correlations.

Thus, the most general fit carried out to cross section and asymmetry data determines the **nine parameters**: $M_Z$, $\Gamma_Z$, $\sigma_{\text{hadron}}^0$, $R(e)$, $R(\mu)$, $R(\tau)$, $A_{\text{FB}}^{(0,e)}$, $A_{\text{FB}}^{(0,\mu)}$, $A_{\text{FB}}^{(0,\tau)}$. Assumption of lepton universality leads to a **five-parameter fit** determining $M_Z$, $\Gamma_Z$, $\sigma_{\text{hadron}}^0$, $R(\text{lepton})$, $A_{\text{FB}}^{(0,\ell)}$.

## 55.6 Combining results from LEP and SLC experiments

With a steady increase in statistics over the years and improved understanding of the common systematic errors between LEP experiments, the procedures for combining results have evolved continuously [24]. The Line Shape Sub-group of the LEP Electroweak Working Group investigated the effects of these common errors, and devised a combination procedure for the precise determination of the Z parameters from LEP experiments. Using these procedures, this note also gives the results after combining the final parameter sets from the four experiments, and these are the results quoted as the fit results in the Z listings below. Transformation of variables leads to values of derived parameters like partial decay widths and branching ratios to hadrons and leptons. Finally, transforming the LEP combined nine parameter set to $(M_Z, \Gamma_Z, \sigma_{\text{hadron}}^0, g_A^f, g_V^f, f=e, \mu, \tau)$ using the average values of lepton asymmetry parameters $(A_e, A_\mu, A_\tau)$ as constraints, leads to the best fitted values of the vector and axial-vector couplings $(g_V, g_A)$ of the charged leptons to the Z.

Brief remarks on the handling of common errors and their magnitudes are given below. The identified common errors are those coming from

(a) LEP energy-calibration uncertainties, and

(b) the theoretical uncertainties in (i) the luminosity determination using small angle Bhabha scattering, (ii) estimating the non-s channel contribution to large angle Bhabha scattering, (iii) the calculation of QED radiative effects, and (iv) the parametrization of the cross section in terms of the parameter set used.

## 55.7 Common LEP energy errors

All the collaborations incorporate in their fit the full LEP energy error matrix as provided by the LEP energy group for their intersection region [18]. The effect of these errors is separated out from that of other errors by carrying out fits with energy errors scaled up and down by $\sim 10\%$ and redoing the fits. From the observed changes in the overall error matrix, the covariance matrix of the common energy errors is determined. Common LEP energy errors lead to uncertainties on $M_Z$, $\Gamma_Z$, and $\sigma_{\text{hadron}}^0$ of 1.7, 1.2 MeV, and 0.011 nb, respectively.

## 55.8 Common luminosity errors

BHLUMI 4.04 [25] is used by all LEP collaborations for small-angle Bhabha scattering leading to a common uncertainty in their measured cross sections of 0.061% [26]. BHLUMI does not include a correction for production of light fermion pairs. OPAL explicitly corrects for this effect and reduces their luminosity uncertainty to 0.054%, which is taken fully correlated with the other experiments. The other three experiments among themselves have a common uncertainty of 0.061%.

## 55.9 Common non-s channel uncertainties

The same standard model programs ALIBABA [16] and TOPAZ0 [17] are used to calculate the non-s channel contribution to the large angle Bhabha scattering [27]. As this contribution is a function of the Z mass, which itself is a variable in the fit, it is parameterized as a function of $M_Z$ by each collaboration to properly track this contribution as $M_Z$ varies in the fit. The common errors on $R_e$ and $A_{\text{FB}}^{(0,e)}$ are 0.024 and 0.0014 respectively, and are correlated between them.

## 55.10 Common theoretical uncertainties: QED

There are large initial-state photon and fermion pair radiation effects near the Z resonance, for which the best currently available evaluations include contributions up to $O(\alpha^3)$. To estimate the
remaining uncertainties, different schemes are incorporated in the standard model programs ZFITTER [5], TOPAZ0 [17], and MIZA [28]. Comparing the different options leads to error estimates of 0.3 and 0.2 MeV on $M_Z$ and $\Gamma_Z$ respectively, and of 0.02% on $\sigma^o_{\text{hadron}}$.

## 55.11 Common theoretical uncertainties: parametrization of lineshape and asymmetries

To estimate uncertainties arising from ambiguities in the model-independent parametrization of the differential cross-section near the Z resonance, results from TOPAZ0 and ZFITTER were compared by using ZFITTER to fit the cross sections and asymmetries calculated using TOPAZ0. The resulting uncertainties on $M_Z$, $\Gamma_Z$, $\sigma^o_{\text{hadron}}$, $R(\text{lepton})$, and $A_{FB}^{(0,\ell)}$ are 0.1 MeV, 0.1 MeV, 0.001 nb, 0.004, and 0.0001 respectively.

Thus, the overall theoretical errors on $M_Z$, $\Gamma_Z$, $\sigma^o_{\text{hadron}}$ are 0.3 MeV, 0.2 MeV, and 0.008 nb respectively; on each $R(\text{lepton})$ is 0.004 and on each $A_{FB}^{(0,\ell)}$ is 0.0001. Within the set of three $R(\text{lepton})$'s and the set of three $A_{FB}^{(0,\ell)}$'s, the respective errors are fully correlated.

All the theory-related errors mentioned above utilize Standard Model programs which need the Higgs mass and running electromagnetic coupling constant as inputs; uncertainties on these inputs will also lead to common errors. All LEP collaborations used the same set of inputs for Standard Model calculations: $M_Z = 91.187$ GeV, the Fermi constant $G_F = (1.16637 \pm 0.00001) \times 10^{-5}$ GeV$^{-2}$ [29], $\alpha^{(5)}(M_Z) = 1/128.877 \pm 0.090$ [30], $\alpha_s(M_Z) = 0.119$ [31], $M_{\text{top}} = 174.3 \pm 5.1$ GeV [31] and $M_{\text{Higgs}} = 150$ GeV. The only observable effect, on $M_Z$, is due to the variation of $M_{\text{Higgs}}$ between 100–1000 GeV (due to the variation of the $\gamma/Z$ interference term which is taken from the Standard Model): $M_Z$ changes by +0.23 MeV per unit change in $\log_{10} M_{\text{Higgs}}/\text{GeV}$, which is not an error but a correction to be applied once $M_{\text{Higgs}}$ is determined. The effect is much smaller than the error on $M_Z$ (±2.1 MeV).

## 55.12 Methodology of combining the LEP experimental results

The LEP experimental results actually used for combination are slightly modified from those published by the experiments (which are given in the Listings below). This has been done in order to facilitate the procedure by making the inputs more consistent. These modified results are given explicitly in [24]. The main differences compared to the published results are (a) consistent use of ZFITTER 6.23 and TOPAZ0 (the published ALEPH results used ZFITTER 6.10); (b) use of the combined energy-error matrix, which makes a difference of 0.1 MeV on the $M_Z$ and $\Gamma_Z$ for L3 only as at that intersection the RF modeling uncertainties are the largest.

Thus, nine-parameter sets from all four experiments with their covariance matrices are used together with all the common errors correlations. A grand covariance matrix, $V$, is constructed and a combined nine-parameter set is obtained by minimizing $\chi^2 = \Delta^T V^{-1} \Delta$, where $\Delta$ is the vector of residuals of the combined parameter set to the results of individual experiments. Imposing lepton universality in the combination results in the combined five-parameter set.

## 55.13 Study of $Z \to b\bar{b}$ and $Z \to c\bar{c}$

In the sector of c- and b-physics, the LEP experiments have measured the ratios of partial widths $R_b = \Gamma(Z \to b\bar{b})/\Gamma(Z \to \text{hadrons})$, and $R_c = \Gamma(Z \to c\bar{c})/\Gamma(Z \to \text{hadrons})$, and the forward-backward (charge) asymmetries $A_{FB}^{b\bar{b}}$ and $A_{FB}^{c\bar{c}}$. The SLD experiment at SLC has measured the ratios $R_c$ and $R_b$ and, utilizing the polarization of the electron beam, was able to obtain the final state coupling parameters $A_b$ and $A_c$ from a measurement of the left-right forward-backward asymmetry of b- and c-quarks. The high precision measurement of $R_c$ at SLD was made possible owing to the small beam size and very stable beam spot at SLC, coupled with a highly precise CCD pixel detector. Several of the analyses have also determined other quantities, in particular the semileptonic branching ratios, $B(b \to \ell^-)$, $B(b \to c \to \ell^+)$, and $B(c \to \ell^+)$, the average time-integrated $B^0\bar{B}^0$ mixing parameter

$\bar{\chi}$ and the probabilities for a c-quark to fragment into a $D^+$, a $D_s$, a $D^{*+}$, or a charmed baryon. The latter measurements do not concern properties of the Z boson, and hence they do not appear in the Listing below. However, for completeness, we will report at the end of this minireview their values as obtained fitting the data contained in the Z section. All these quantities are correlated with the electroweak parameters, and since the mixture of b hadrons is different from the one at the $\Upsilon(4S)$, their values might differ from those measured at the $\Upsilon(4S)$.

All the above quantities are correlated to each other since:

* Several analyses (for example the lepton fits) determine more than one parameter simultaneously;

* Some of the electroweak parameters depend explicitly on the values of other parameters (for example $R_b$ depends on $R_c$);

* Common tagging and analysis techniques produce common systematic uncertainties.

The LEP Electroweak Heavy Flavour Working Group has developed [32] a procedure for combining the measurements taking into account known sources of correlation. The combining procedure determines fourteen parameters: the six parameters of interest in the electroweak sector, $R_b$, $R_c$, $A_{FB}^{b\bar{b}}$, $A_{FB}^{c\bar{c}}$, $A_b$ and $A_c$ and, in addition, $B(b \to \ell^-)$, $B(b \to c \to \ell^+)$, $B(c \to \ell^+)$, $\bar{\chi}$, $f(D^+)$, $f(D_s)$, $f(c_{\text{baryon}})$ and $P(c \to D^{*+})$ $\times$ $B(D^{*+} \to \pi^+ D^0)$, to take into account their correlations with the electroweak parameters. Before the fit both the peak and off-peak asymmetries are translated to the common energy $\sqrt{s} = 91.26$ GeV using the predicted energy-dependence from ZFITTER [5].

## 55.14 Summary of the measurements and of the various kinds of analysis

The measurements of $R_b$ and $R_c$ fall into two classes. In the first, named single-tag measurement, a method for selecting b and c events is applied and the number of tagged events is counted. A second technique, named double-tag measurement, has the advantage that the tagging efficiency is directly derived from the data thereby reducing the systematic error on the measurement.

The measurements in the b- and c-sector can be essentially grouped in the following categories:

* Lifetime (and lepton) double-tagging measurements of $R_b$. These are the most precise measurements of $R_b$ and obviously dominate the combined result. The main sources of systematics come from the charm contamination and from estimating the hemisphere b-tagging efficiency correlation;

* Analyses with $D/D^{*\pm}$ to measure $R_c$. These measurements make use of several different tagging techniques (inclusive/exclusive double tag, exclusive double tag, reconstruction of all weakly decaying charmed states) and no assumptions are made on the energy-dependence of charm fragmentation;

* A measurement of $R_c$ using single leptons and assuming $B(b \to c \to \ell^+)$;

* Lepton fits which use hadronic events with one or more leptons in the final state to measure the asymmetries $A_{FB}^{b\bar{b}}$ and $A_{FB}^{c\bar{c}}$. Each analysis usually gives several other electroweak parameters. The dominant sources of systematics are due to lepton identification, to other semileptonic branching ratios and to the modeling of the semileptonic decay;

* Measurements of $A_{FB}^{b\bar{b}}$ using lifetime tagged events with a hemisphere charge measurement. These measurements dominate the combined result;

* Analyses with $D/D^{*\pm}$ to measure $A_{FB}^{c\bar{c}}$ or simultaneously $A_{FB}^{b\bar{b}}$ and $A_{FB}^{c\bar{c}}$;

* Measurements of $A_b$ and $A_c$ from SLD, using several tagging methods (lepton, kaon, $D/D^*$, and vertex mass). These quantities are directly extracted from a measurement of the left-right forward-backward asymmetry in $c\bar{c}$ and $b\bar{b}$ production using a polarized electron beam.

## 55.15 Averaging procedure

All the measurements are provided by the LEP and SLD Collaborations in the form of tables with a detailed breakdown of
the systematic errors of each measurement and its dependence on other electroweak parameters.

The averaging proceeds via the following steps:

* Define and propagate a consistent set of external inputs such as branching ratios, hadron lifetimes, fragmentation models etc. All the measurements are checked to ensure that all use a common set of assumptions (for instance, since the QCD corrections for the forward-backward asymmetries are strongly dependent on the experimental conditions, the data are corrected before combining);

* Form the full (statistical and systematic) covariance matrix of the measurements. The systematic correlations between different analyses are calculated from the detailed error breakdown in the measurement tables. The correlations relating several measurements made by the same analysis are also used;

* Take into account any explicit dependence of a measurement on the other electroweak parameters. As an example of this dependence, we illustrate the case of the double-tag measurement of $R_b$, where c-quarks constitute the main background. The normalization of the charm contribution is not usually fixed by the data and the measurement of $R_b$ depends on the assumed value of $R_c$, which can be written as:

$$R_b = R_b^{\text{meas}} + a(R_c) \frac{(R_c - R_c^{\text{used}})}{R_c}, \quad (55.11)$$

where $R_b^{\text{meas}}$ is the result of the analysis which assumed a value of $R_c = R_c^{\text{used}}$ and $a(R_c)$ is the constant which gives the dependence on $R_c$;

* Perform a $\chi^2$ minimization with respect to the combined electroweak parameters.

After the fit the average peak asymmetries $A_{FB}^{cc}$ and $A_{FB}^{b\bar{b}}$ are corrected for the energy shift from 91.26 GeV to $M_Z$ and for QED (initial state radiation), $\gamma$ exchange, and $\gamma Z$ interference effects, to obtain the corresponding pole asymmetries $A_{FB}^{0,c}$ and $A_{FB}^{0,b}$.

This averaging procedure, using the fourteen parameters described above, and applied to the data contained in the Z particle listing below, gives the following results (where the last 8 parameters do not depend directly on the Z):

$$R_b^0 = 0.21629 \pm 0.00066$$

$$R_c^0 = 0.1721 \pm 0.0030$$

$$A_{FB}^{0,b} = 0.0992 \pm 0.0016$$

$$A_{FB}^{0,c} = 0.0707 \pm 0.0035$$

$$A_b = 0.923 \pm 0.020$$

$$A_c = 0.670 \pm 0.027$$

$$B(b \to \ell^-) = 0.1071 \pm 0.0022$$

$$B(b \to c \to \ell^+) = 0.0801 \pm 0.0018$$

$$B(c \to \ell^+) = 0.0969 \pm 0.0031$$

$$\bar{\chi} = 0.1250 \pm 0.0039$$

$$f(D^+) = 0.235 \pm 0.016$$

$$f(D_s) = 0.126 \pm 0.026$$

$$f(c_{\text{baryon}}) = 0.093 \pm 0.022$$

$$P(c \to D^{*+}) \times B(D^{*+} \to \pi^+ D^0) = 0.1622 \pm 0.0048 \quad (55.12)$$

Among the non-electroweak observables, the B semileptonic branching fraction $B(b \to \ell^-)$ is of special interest, since the dominant error source on this quantity is the dependence on the semileptonic decay model for $b \to \ell^-$, with $\Delta B(b \to \ell^-)_{b \to \ell^-_{\text{model}}}$ = 0.0012. Extensive studies have been made to understand the size of this error. Among the electroweak quantities, the quark asymmetries with leptons depend also on the semileptonic decay model, while the asymmetries using other

methods usually do not. The fit implicitly requires that the different methods give consistent results and this effectively constrains the decay model, and thus reduces in principle the error from this source in the fit result.

To obtain a conservative estimate of the modelling error, the above fit has been repeated removing all asymmetry measurements. The results of the fit on B-decay related observables are [24]: $B(b \to \ell^-) = 0.1069 \pm 0.0022$, with $\Delta B(b \to \ell^-)_{b \to \ell^-_{\text{model}}}$ = 0.0013, $B(b \to c \to \ell^+) = 0.0802 \pm 0.0019$ and $\bar{\chi} = 0.1259 \pm 0.0042$.

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56. Muon Anomalous Magnetic Moment

Revised August 2023 by A. Höcker (CERN) and W.J. Marciano (BNL).

The Dirac equation predicts a muon magnetic moment, $\vec{M} = g_\mu \frac{e}{2m_\mu} \vec{S}$, with gyromagnetic ratio $g_\mu = 2$. Quantum loop effects lead to a small calculable deviation from $g_\mu = 2$, parameterized by the magnetic anomaly<fnref n="1" />

$$a_{\mu} \equiv \frac{g_{\mu} - 2}{2}. \qquad (56.1)$$

That quantity can be accurately measured and, within the Standard Model (SM) framework, precisely predicted. Hence, comparison of experiment and theory tests the SM at its quantum loop level. A deviation in $a_\mu^{\text{exp}}$ from the SM expectation would signal effects of new physics, with current sensitivity reaching up to mass scales of $\mathcal{O}(\text{TeV})$ [1]. For recent thorough muon $g-2$ reviews, see e.g. Refs. [2–4].

The E821 experiment at Brookhaven National Lab (BNL) studied during the years 1997–2001 the precession of $\mu^+$ and $\mu^-$ in a constant external magnetic field as they circulated in a confining storage ring. It found<fnref n="2" /> [6]

$$\begin{aligned} a_{\mu^+}^{\text{exp,BNL}} &= 116\,592\,040(60)(50) \times 10^{-11}, \\ a_{\mu^-}^{\text{exp,BNL}} &= 116\,592\,150(80)(30) \times 10^{-11}, \end{aligned} \qquad (56.2)$$

where the first errors are statistical and the second systematic. Assuming CPT invariance and taking into account correlations between systematic uncertainties, one finds for their average [5,6]

$$a_{\mu}^{\text{exp,BNL}} = 116\,592\,089(54)(33) \times 10^{-11}. \qquad (56.3)$$

These results represent about a factor of 14 improvement over the classic CERN experiments of the 1970s [7].

Further improvement of the measurement by a factor of four by setting up the E821 storage ring at the Fermilab National Accelerator Laboratory (FNAL), and utilizing a cleaner and more intense muon beam and improved detectors [8] is in progress. After six runs, the experiment has completed data taking in 2023. A first analysis with positive muons based on a fraction of the data collected in 2018 was released in 2021 confirming the BNL result with similar precision [9]. A second analysis released in 2023 was based on a four times larger positive muon data sample collected in 2019 and 2020, and benefitted from instrumental improvements and reduced systematic uncertainties [10]. The result is dominated by statistical uncertainty and consistent with but more precise than the previous one. Their combination yields [10]

$$a_{\mu^+}^{\text{exp,FNAL}} = 116\,592\,055(24) \times 10^{-11}. \qquad (56.4)$$

The ongoing analysis of the remaining 2021–2023 dataset is expected to further reduce the uncertainty by a factor of two.

The FNAL result agrees with the BNL measurement and has improved precision. Their average assuming CPT invariance reads [10]

$$a_{\mu}^{\text{exp}} = 116\,592\,059(22) \times 10^{-11}, \qquad (56.5)$$

providing a relative precision of 0.19 parts per million.

Another muon $g-2$ experiment with similar sensitivity but using an alternative zero-electric-field technique with a low-emittance and low-momentum muon beam is currently under construction at J-PARC in Japan [11].

The SM prediction $a_\mu^{\text{SM}}$ is generally divided into three parts (see Fig. 56.1 for representative Feynman diagrams)

$$a_{\mu}^{\text{SM}} = a_{\mu}^{\text{QED}} + a_{\mu}^{\text{EW}} + a_{\mu}^{\text{Had}}. \qquad (56.6)$$

<figure><img src="image_305.png" /><figcaption>Figure 56.1: Representative diagrams contributing to $a_\mu^{\text{SM}}$. From left to right: first order QED (Schwinger term), lowest-order weak, lowest-order hadronic.</figcaption></figure>

In the following discussion we use the numerical estimates provided by the Muon $g-2$ Theory Initiative White Paper [4].

The QED part includes all photonic and leptonic $(e, \mu, \tau)$ loops starting with the classic $\alpha/2\pi$ Schwinger contribution [12]. It has been computed through five loops [4, 13, 14]

$$\begin{aligned} a_{\mu}^{\text{QED}} = \frac{\alpha}{2\pi} &+ 0.765\,857\,420(13) \left(\frac{\alpha}{\pi}\right)^2 + 24.050\,509\,85(23) \left(\frac{\alpha}{\pi}\right)^3 \\ &+ 130.8782(60) \left(\frac{\alpha}{\pi}\right)^4 + 751.0(9) \left(\frac{\alpha}{\pi}\right)^5 + \dots \end{aligned} \qquad (56.7)$$

with little change in the coefficients since our last update of this review. Employing $\alpha^{-1} = 137.035\,999\,046(27)$, obtained from the precise measurements of $h/m_{\text{Cs}}$ [15], the Rydberg constant, and $m_{\text{Cs}}/m_e$ leads to<fnref n="3" /> [13]

$$a_{\mu}^{\text{QED}} = 116\,584\,718.93(0.10) \times 10^{-11}, \qquad (56.8)$$

where the small error results mainly from the uncertainties in the estimate of the six loop contribution and in $\alpha$.

Loop contributions involving heavy $W^\pm$, $Z$ or Higgs particles are collectively labeled as $a_\mu^{\text{EW}}$. They are suppressed by at least a factor of $(\alpha/\pi) \cdot (m_\mu^2/m_W^2) \simeq 4 \times 10^{-9}$. At 1-loop order [19]

$$\begin{aligned} a_{\mu}^{\text{EW}}[\text{1-loop}] = & \frac{G_{\mu}m_{\mu}^2}{8\sqrt{2}\pi^2} \left[ \frac{5}{3} + \frac{1}{3} \left(1 - 4\sin^2\theta_{\text{W}}\right)^2 + \mathcal{O}\left(\frac{m_{\mu}^2}{M_W^2}\right) + \mathcal{O}\left(\frac{m_{\mu}^2}{M_H^2}\right) \right] \\ = & 194.8 \times 10^{-11}, \end{aligned} \qquad (56.9)$$

for $\sin^2\theta_W \equiv 1 - M_W^2/M_Z^2 \simeq 0.223$, and where $G_\mu \simeq 1.166 \times 10^{-5} \text{ GeV}^{-2}$ is the Fermi coupling constant. Two-loop corrections are relatively large and negative [20]. For a Higgs boson mass of 125 GeV it amounts to $a_\mu^{\text{EW}}[\text{2-loop}] = -41.2(1.0) \times 10^{-11}$ [20], where the uncertainty stems from quark triangle loops. The 3-loop leading logarithms are negligible, $\mathcal{O}(10^{-12})$ [20, 21]. Overall one finds

$$a_{\mu}^{\text{EW}} = 153.6(1.0) \times 10^{-11}. \qquad (56.10)$$

A recent complete 2-loop numerical evaluation of the electroweak correction [22] when adjusted for appropriate light quark masses confirmed the result in Eq. (56.10).

Hadronic (quark and gluon) loop contributions to $a_\mu^{\text{SM}}$ give rise to its main uncertainty. One traditionally relies on a data-driven dispersion relation approach to evaluate the lowest-order $\mathcal{O}(\alpha^2)$ hadronic vacuum polarization contribution $a_\mu^{\text{Had}}[\text{LO}]$ from corresponding cross section measurements [23]

$$a_{\mu}^{\text{Had}}[\text{LO}] = \frac{1}{3} \left(\frac{\alpha}{\pi}\right)^2 \int_{m_{\pi}^2}^{\infty} ds \frac{K(s)}{s} R^{(0)}(s), \qquad (56.11)$$

<footnote n="1">¹Also referred to as anomalous magnetic moment despite being dimensionless.</footnote>

<footnote n="2">²The results reported by the experiment have been updated in Eqs. (56.2) and (56.3) to the newest value for the absolute muon-to-proton magnetic ratio $\lambda = 3.183\,345\,142(71)$ [5]. The change induced in $a_\mu^{\text{exp}}$ with respect to the value of $\lambda = 3.183\,345\,39(10)$ used in Ref. [6] amounts to $+10 \times 10^{-11}$.</footnote>

<footnote n="3">³A recent measurement using Rb-87 atoms [16] resulted in $\alpha^{-1} = 137.035\,999\,206(11)$, which exhibits a larger than $5\sigma$ discrepancy with the Cs-133 based result. Using this value in Eq. (56.7) leads to a reduction of $a_\mu^{\text{QED}}$ by $0.14 \times 10^{-11}$, which is larger than the quoted uncertainty in Eq. (56.8), but still negligible compared to other uncertainties affecting $a_\mu^{\text{SM}}$. This discrepancy impacts, however, the SM prediction of the magnetic moment of the electron [14, 17], which differs by respectively $-3.9\sigma$ and $+2.1\sigma$ from the most recent experimental value [18], depending on whether the Cs-133 or Rb-87 based $\alpha$ value is used.</footnote>
where $K(s)$ is a QED kernel function [24], and where $R^{(0)}(s)$ denotes the ratio of the bare<fnref n="4" /> cross section for $e^+e^-$ annihilation into hadrons to the pointlike muon-pair cross section at center-of-mass energy $\sqrt{s}$. The function $K(s) \sim 1/s$ in Eq. (56.11) emphasizes the low-energy part of the integral so that $a_\mu^{\text{Had}}[\text{LO}]$ is dominated by the $\rho(770) \to \pi^+\pi^-$ resonance.

The analysis of Eq. (56.11) results in the representative value [4, 25–30]

$$a_{\mu}^{\text{Had}}[\text{LO}] = 6931(40) \times 10^{-11}, \qquad (56.12)$$

whose error is dominated by systematic uncertainties in the $e^+e^- \to$ hadrons cross-section data, and also includes a small uncertainty due to perturbative QCD, which is used at intermediate and large energies in the dispersion integral to predict the contribution from the quark-antiquark continuum. The experimental precision of Eq. (56.12) is limited by a discrepancy between the most precise $\pi^+\pi^-$ data from the BABAR and KLOE experiments [27, 29]. Recent preliminary $\pi^+\pi^-$ data from the CMD-3 experiment [31] give a significantly larger contribution to $a_\mu^{\text{Had}}[\text{LO}]$ but disagree with essentially all other datasets. New or updated measurements are expected from several experiments during the next years.

Alternatively, one can use precise vector spectral functions from $\tau \to \nu_\tau$ + hadrons decays [32] that can be related to isovector $e^+e^- \to$ hadrons cross sections by isospin symmetry. Analyses replaced $e^+e^-$ data in the two-pion and four-pion channels with the corresponding isospin- transformed $\tau$ data, and applied isospin-violating corrections [33]. Owing to the progress in the precision of the $e^+e^-$ data, the $\tau$ data are now less precise and less reliable due to additional theoretical uncertainties, so that recent $a_\mu^{\text{Had}}[\text{LO}]$ evaluations ignored them.

Progress has been achieved on the evaluation of $a_\mu^{\text{Had}}[\text{LO}]$ using lattice QCD, which allows to directly compute the real part of the two-point correlation function without invoking the resonances occurring on the imaginary axis [34–36]. The required sub-percent precision represents a challenge requiring to master systematic and extrapolation uncertainties. A combination of recent lattice QCD evaluations [37, 38] gives [4]

$$a_{\mu}^{\text{Had}}[\text{LO}] = 7116(184) \times 10^{-11}, \qquad (56.13)$$

with an uncertainty that is not yet competitive with the data-driven approach. Another recent lattice QCD calculation with much reduced uncertainty found [39]

$$a_{\mu}^{\text{Had}}[\text{LO}] = 7075(55) \times 10^{-11}, \qquad (56.14)$$

whose error is dominated by systematic uncertainties. It exhibits a $2.1\sigma$ tension with the data-driven result, growing up to $3.7\sigma$ for the modified observable $a_{\mu,\text{win}}$ that has favourable properties for computation on the lattice<fnref n="5" /> [38, 39, 41]. Because of the complexity and challenges of such a calculation, it is critical that it is confronted with independent lattice results of comparable precision [38, 41, 42].

Higher order hadronic loop contributions are also obtained from dispersion relations using the same $e^+e^- \to$ hadrons data but with different integration kernels [43]. Recent evaluations found $a_\mu^{\text{Had,Disp}}[\text{NLO}] = (-98.3 \pm 0.7) \times 10^{-11}$ [4, 30] and $a_\mu^{\text{Had,Disp}}[\text{NNLO}] = (12.4 \pm 0.1) \times 10^{-11}$ [44]. In the case of hadronic light-by-light scattering contributions, which enter at NLO, recent studies [3, 4, 45, 46] based on short-distance QCD,

<figure><img src="image_306.png" /><figcaption>Figure 56.2: Compilation of recent results for $a_\mu$ (in units of $10^{-11}$), subtracted by the central value of the experimental average (56.3). The light blue vertical band indicates the total experimental uncertainty. The SM predictions shown differ only in the value used for the $a_\mu^{\text{Had}}[\text{LO}]$ contribution. They are taken from: WP 2020 [4] (Eq. (56.16), also indicated by the light grey vertical band), DHMZ 2019 [29], KNT 2019 [30], all three based on the data-driven (DD) $a_\mu^{\text{Had}}[\text{LO}]$ evaluation, and the lattice QCD (LQ) based evaluations WP 2020 [4, 37, 38] and BMW 2021 [39]. Note that the quoted errors in the figure do not include the uncertainty on the subtracted experimental value. To obtain for each theory calculation a result equivalent to Eq. (56.17), the errors from theory and experiment must be added in quadrature.</figcaption></figure>

dispersion relations and lattice QCD calculations lead to the improved prediction $a_\mu^{\text{Had,LBL}}[\text{NLO}] = 92(18) \times 10^{-11}$ [4].<fnref n="6" /> One thus finds for the sum of the three higher order hadronic terms

$$a_{\mu}^{\text{Had}}[\text{N}(\text{N})\text{LO}] = 6(18) \times 10^{-11}, \qquad (56.15)$$

where the error is dominated by the hadronic light-by-light scattering uncertainty.

Adding Eqs. (56.8), (56.10), (56.12) and (56.15) gives the representative SM prediction

$$a_{\mu}^{\text{SM}} = 116591810(1)(40)(18) \times 10^{-11}, \qquad (56.16)$$

whose errors are due to the electroweak, lowest-order hadronic, and higher-order hadronic contributions, respectively. The difference between experiment and theory

$$\Delta a_{\mu} = a_{\mu}^{\text{exp}} - a_{\mu}^{\text{SM}} = 249(22)(43) \times 10^{-11}, \qquad (56.17)$$

where the errors are from experiment and theory prediction (with all errors combined in quadrature, taking into account a small anticorrelation [4] between the data-driven lowest and higher order contributions), exhibits a discrepancy of $5.2\sigma$.<fnref n="7" /> The new FNAL measurement has increased the discrepancy with the SM prediction from previously $3.7\sigma$ (BNL only) and $4.2\sigma$ (including the first FNAL result), respectively. Despite the large significance, the situation remains inconclusive due to the discrepancies in the hadronic cross section data underlying the dispersive $a_\mu^{\text{Had}}[\text{LO}]$ determination.

Figure 56.2 represents the difference between experimental and predicted $a_\mu$ values based on the most recent data-driven and lattice QCD based evaluations of $a_\mu^{\text{Had}}[\text{LO}]$.

A leading candidate explanation for the deviation observed in $\Delta a_\mu$ has been supersymmetry particle loop contributions. Such a scenario is quite natural, since generically, supersymmetric models predict [1] an additional contribution to $a_\mu^{\text{SM}}$

$$a_{\mu}^{\text{SUSY}} \approx \pm 130 \times 10^{-11} \cdot \left( \frac{100 \text{ GeV}}{m_{\text{SUSY}}} \right)^2 \tan\beta, \qquad (56.18)$$

<footnote n="4">⁴The bare cross section is defined as the measured cross section corrected for initial-state radiation, electron-vertex loop contributions and vacuum-polarization effects in the photon propagator. However, QED effects in the hadron vertex and final state, as photon radiation, are included.</footnote>

<footnote n="5">⁵A dispersive approach similar to that for $a_\mu^{\text{Had}}[\text{LO}]$ is used to determine the hadronic contribution to the running of the electromagnetic fine structure constant at the Z-boson mass $\Delta\alpha_{\text{had}}(m_Z)$, which is a critical ingredient of the global electroweak fit. One may therefore ask if the goodness of that fit would break down were $a_\mu^{\text{Had}}[\text{LO}]$ in Eq. (56.12) be brought in compatibility with the result (56.14). However, the difference in the relevant integration kernels (whereas 73% of the contribution to $a_\mu^{\text{Had}}[\text{LO}]$ stems from the $\pi^+\pi^-$ channel, it is only 13% in the case of $\Delta\alpha_{\text{had}}(m_Z)$) allows to absorb large effects in $a_\mu^{\text{Had}}[\text{LO}]$ without creating a significant tension in the electroweak fit. Detailed discussions are available in the literature [40].</footnote>

<footnote n="6">⁶A very recent lattice QCD based calculation of $a_\mu^{\text{Had,LBL}}[\text{NLO}]$ [47], not included in [4], gives a consistent result with [4, 46] using a different approach.</footnote>

<footnote n="7">⁷The probabilistic interpretation of the discrepancy requires caution due to the systematic nature of the $a_\mu^{\text{SM}}$ uncertainty.</footnote>
where $m_{SUSY}$ is a representative supersymmetric mass scale, $\tan\beta \simeq 3-40$ a potential enhancement factor, and $\pm 1$ corresponds to the sign of the $\mu$ term in the supersymmetric Lagrangian. Supersymmetric particles in the mass range 100–500 GeV could be the source of the deviation $\Delta a_{\mu}$. If so, those particles should be directly observable at the Large Hadron Collider at CERN. So far, there is however no direct evidence in support of the supersymmetry interpretation.

New physics effects [1] other than supersymmetry could also explain a non-vanishing $\Delta a_{\mu}$. A popular scenario involves the “dark photon”, a relatively light hypothetical vector boson from the dark matter sector that couples to our world of particle physics through mixing with the ordinary photon [48–50]. As a result, it couples to ordinary charged particles with strength $\varepsilon \cdot e$ and gives rise to an additional muon magnetic anomaly contribution

$$a_{\mu}^{\text{dark photon}} = \frac{\alpha}{2\pi} \varepsilon^2 F(m_V/m_{\mu}), \quad (56.19)$$

where $F(x) = \int_0^1 2z(1-z)^2/[(1-z)^2 + x^2z] dz$. For values of $\varepsilon \sim 1-2 \times 10^{-3}$ and $m_V \sim 10-100$ MeV, the dark photon, which was originally motivated by cosmology, can provide a viable solution to the muon $g-2$ discrepancy. However, experimental constraints appear to rule out the simplest model in which the dark photon has equal couplings to electrons and muons and decays primarily to $e^+e^-$ pairs [51] or invisible dark particles [52] that give rise to missing event energy. One can expand the dark photon scenario into a more complete theory in which several new particles contribute to the muon $g-2$. Direct searches for a dark photon, therefore, continue to be well motivated, but with primary guidance coming from astrophysics [53].

Recent popular solutions to the muon anomaly discrepancy have also focused on loop contributions induced by relatively light scalar or pseudo scalar particle appendages from physics beyond the SM [54, 55].

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# 57. Muon Decay Parameters

Revised August 2023 by W. Fetscher (ETH Zurich) and H.-J. Gerber (ETH Zurich).

## 57.1 Introduction:

All measurements in direct muon decay, $\mu^- \to e^- + 2$ neutrals, and its inverse, $\nu_\mu + e^- \to \mu^- + \text{neutral}$, are successfully described by the “V-A interaction,” which is a particular case of a local, derivative-free, lepton-number-conserving, four-fermion interaction [1]. As shown below, within this framework, the Standard Model assumptions, such as the V-A form and the nature of the neutrals ($\nu_\mu$ and $\bar{\nu}_e$), and hence the doublet assignments ($\nu_e e^-)_L$ and ($\nu_\mu \mu^-)_L$, have been determined from experiments [2, 3]. All considerations on muon decay are valid for the leptonic tau decays $\tau^- \to \ell^- + \nu_\tau + \bar{\nu}_\ell$ with the replacements $m_\mu \to m_\tau$, $m_e \to m_{e\ell}$, $\ell^- = e^-, \mu^-$.

## 57.2 Parameters:

The differential decay probability to obtain an $e^\pm$ with (reduced) energy between $x$ and $x + dy$, emitted in the direction $\hat{x}_3$ at an angle between $\vartheta$ and $\vartheta + d\vartheta$ with respect to the muon polarization vector $P_\mu$, and with its spin parallel to the arbitrary direction $\hat{\zeta}$, neglecting radiative corrections, is given by

$$ \frac{d^2\Gamma}{dx d\cos\vartheta} = \frac{m_\mu}{4\pi^3} W_{e\mu}^4 G_F^2 \sqrt{x^2 - x_{20}^2} \times (F_{\text{IS}}(x) \pm P_\mu \cos\vartheta F_{\text{AS}}(x)) \times \left[1 + \hat{\zeta} \cdot \mathbf{P}_e(x, \vartheta)\right]. \quad (57.1) $$

Here, $W_{e\mu} = \max(E_e) = (m_\mu^2 + m_e^2)/2m_\mu$ is the maximum $e^\pm$ energy, $x = E_e/W_{e\mu}$ is the reduced energy, $x_0 = m_e/W_{e\mu} = 9.67 \times 10^{-3}$, and $P_\mu = |P_\mu|$ is the degree of muon polarization. $\hat{\zeta}$ is the direction in which a perfect polarization-sensitive electron detector is most sensitive. The isotropic part of the spectrum, $F_{\text{IS}}(x)$, the anisotropic part $F_{\text{AS}}(x)$, and the electron polarization, $P_e(x, \vartheta)$, may be parametrized by the Michel parameter $\rho$ [1], by $\eta$ [4], by $\xi$ and $\delta$ [5, 6], etc. These are bilinear combinations of the coupling constants $g_{\epsilon\mu}^\gamma$, which occur in the matrix element (given below).

If the masses of the neutrinos as well as $x_0^2$ are neglected, the energy and angular distribution of the electron in the rest frame of a muon ($\mu^\pm$) measured by a polarization insensitive detector, is given by

$$ \frac{d^2\Gamma}{dx d\cos\vartheta} \sim x^2 \cdot \left\{ 3(1-x) + \frac{2\rho}{3}(4x-3) + 3\eta x_0(1-x) \pm P_\mu \cdot \xi \cdot \cos\vartheta \left[1 - x + \frac{2\delta}{3}(4x-3)\right] \right\}. \quad (57.2) $$

Here, $\vartheta$ is the angle between the electron momentum and the muon spin, and $x \equiv 2E_e/m_\mu$. For the Standard Model coupling, we obtain $\rho = \xi\delta = 3/4$, $\xi = 1$, $\eta = 0$ and the differential decay rate is

$$ \frac{d^2\Gamma}{dx d\cos\vartheta} = \frac{G_F^2 m_\mu^5}{192\pi^3} [3 - 2x \pm P_\mu \cos\vartheta(2x-1)] \cdot x^2. \quad (57.3) $$

The coefficient in front of the square bracket is the total decay rate.

If only the neutrino masses are neglected, and if the $e^\pm$ polarization is detected, then the functions in Eq. (57.1) become

$$ F_{\text{IS}}(x) = x(1-x) + \frac{2}{9}\rho(4x^2 - 3x - x_0^2) + \eta x_0(1-x) \quad (57.4a) $$

$$ F_{\text{AS}}(x) = \frac{1}{3}\xi\sqrt{x^2 - x_0^2} \times \left[ (1-x) + \frac{2}{3}\delta \left\{ (4x-3) + \left(\sqrt{1-x_0^2}-1\right) \right\} \right] \quad (57.4b) $$

$$ P_e(x, \vartheta) = P_{T1} \cdot \hat{x}_1 + P_{T2} \cdot \hat{x}_2 + P_L \cdot \hat{x}_3 \quad (57.4c) $$

Here $\hat{x}_1, \hat{x}_2$, and $\hat{x}_3$ are orthogonal unit vectors defined as follows:

$$ \frac{\hat{x}_3}{|\hat{x}_3 \times P_\mu|} = \hat{x}_2 \quad \text{is transverse to } p_e \text{ and perpendicular to} \\ \text{“the decay plane”} \\ \hat{x}_2 \times \hat{x}_3 = \hat{x}_1 \quad \text{is transverse to } p_e \text{ and in the decay plane} $$

The components of $P_e$ then are given by

$$ P_{T1}(x, \vartheta) = P_{\mu} \sin \vartheta \cdot F_{T1}(x) / \{F_{\text{IS}}(x) \pm P_{\mu} \cos \vartheta \cdot F_{\text{AS}}(x)\} \quad (57.5a) $$

$$ P_{T2}(x, \vartheta) = P_{\mu} \sin \vartheta \cdot F_{T2}(x) / \{F_{\text{IS}}(x) \pm P_{\mu} \cos \vartheta \cdot F_{\text{AS}}(x)\} \quad (57.5b) $$

$$ P_L(x, \vartheta) = \frac{\pm F_{\text{IP}}(x) + P_\mu \cos \vartheta \cdot F_{\text{AP}}(x)}{F_{\text{IS}}(x) \pm P_\mu \cos \vartheta \cdot F_{\text{AS}}(x)}, \quad (57.5c) $$

where

$$ F_{T1}(x) = \frac{1}{12} \left\{ -2 \left[ \xi'' + 12 \left( \rho - \frac{3}{4} \right) \right] (1-x)x_0 - 3\eta (x^2 - x_0^2) + \eta'' \left( -3x^2 + 4x - x_0^2 \right) \right\} \quad (57.6a) $$

$$ F_{T2}(x) = \frac{1}{3} \sqrt{x^2 - x_0^2} \cdot \left\{ 3 \frac{\alpha'}{A} (1-x) + 2 \frac{\beta'}{A} \sqrt{1-x_0^2} \right\} \quad (57.6b) $$

$$ F_{\text{IP}}(x) = \frac{1}{54} \sqrt{x^2 - x_0^2} \cdot \left\{ 9\xi' \left(-2x + 2 + \sqrt{1-x_0^2}\right) + 4\xi \left(\delta - \frac{3}{4}\right) \left(4x - 4 + \sqrt{1-x_0^2}\right) \right\} \quad (57.6c) $$

$$ F_{\text{AP}}(x) = \frac{1}{6} \left\{ \xi'' (2x^2 - x - x_0^2) + 4 \left( \rho - \frac{3}{4} \right) (4x^2 - 3x - x_0^2) + 2 \eta'' (1-x) x_0 \right\}. \quad (57.6d) $$

For the experimental values of the parameters $\rho$, $\xi$, $\xi'$, $\xi''$, $\delta$, $\eta$, $\eta''$, $\alpha/A$, $\beta/A$, $\alpha'/A$, $\beta'/A$, $\alpha'/A$, $\beta'/A$ (and $\eta = (\alpha - 2\beta)/2A$), as defined by Kinoshita and Sirlin [5, 6]. They serve as a model-independent summary of all possible measurements on the decay electron (see Listings below). The relations between the two sets of parameters are

$$ \rho - \frac{3}{4} = \frac{3}{4} (-a + 2c)/A \quad (57.7a) $$

$$ \eta = (\alpha - 2\beta)/A \quad (57.7b) $$

$$ \eta'' = (3\alpha + 2\beta)/A \quad (57.7c) $$

$$ \delta - \frac{3}{4} = \frac{9}{4} \cdot \frac{(a' - 2c')/A}{1 - [a + 3a' + 4(b + b') + 6c - 14c']/A} \quad (57.7d) $$

$$ 1 - \xi \frac{\delta}{\rho} = 4 \frac{[(b + b') + 2(c - c')] / A}{1 - (a - 2c) / A} \quad (57.7e) $$

$$ 1 - \xi = \left[ (a + a') + 4(b + b') + 6(c + c') \right] / A \quad (57.7f) $$

$$ 1 - \xi'' = (-2a + 20c)/A \quad (57.7g) $$

$$ \text{with} \quad A = a + 4b + 6c \quad (57.7h) $$

The differential decay probability to obtain a left-handed $\nu_e$ with (reduced) energy between $y$ and $y + dy$, neglecting radiative corrections as well as the masses of the electron and of the neutrinos, is given by [7]

$$ \frac{d\Gamma}{dy} = \frac{m_{\mu}^{5} G_{F}^{2}}{16 \pi^{3}} \cdot Q_{L}^{\nu_{e}} \cdot y^{2} \left\{ (1 - y) - \omega_{L} \cdot \left(y - \frac{3}{4}\right) \right\}. \quad (57.8) $$
Here, $y = 2 E_{\nu_e}/m_\mu$. $Q_L^{\nu_e}$ and $\omega_L$ are parameters. $\omega_L$ is the neutrino analog of the spectral shape parameter $\rho$ of Michel. Since in the Standard Model, $Q_L^{\nu_e} = 1$, $\omega_L = 0$, the measurement of $d\Gamma/dy$ has allowed a null-test of the Standard Model (see Listings below).

<table><caption>**Table 57.1:** Coupling constants $g_{\epsilon\mu}^{\gamma}$ and some combinations of them. Ninety-percent confidence level experimental limits. The limits on $|g_{LL}^S|$ and $|g_{LL}^V|$ are from [8–10], and the others from a general analysis of muon decay measurements. Top three rows: [11], fourth row: [12], next three rows: [13], last row: [14]. The experimental uncertainty on the muon polarization in pion decay is included. Note that, by definition, $|g_{\epsilon\mu}^S| \le 2$, $|g_{\epsilon\mu}^V| \le 1$ and $|g_{\epsilon\mu}^T| \le 1/\sqrt{3}$.</caption><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>$|g_{RR}^S| < 0.035$</td><td>$|g_{RR}^V| < 0.017$</td><td>$|g_{RR}^T| \equiv 0$</td></tr><tr><td>$|g_{LR}^S| < 0.050$</td><td>$|g_{LR}^V| < 0.023$</td><td>$|g_{LR}^T| < 0.015$</td></tr><tr><td>$|g_{RL}^S| < 0.420$</td><td>$|g_{RL}^V| < 0.105$</td><td>$|g_{RL}^T| < 0.105$</td></tr><tr><td>$|g_{LL}^S| < 0.550$</td><td>$|g_{LL}^V| > 0.960$</td><td>$|g_{LL}^T| \equiv 0$</td></tr><tr><td>$|g_{LR}^S + 6 g_{LR}^T| < 0.143$</td><td>$|g_{RL}^S + 6 g_{RL}^T| < 0.418$</td><td></td></tr><tr><td>$|g_{LR}^S + 2 g_{LR}^T| < 0.108$</td><td>$|g_{RL}^S + 2 g_{RL}^T| < 0.417$</td><td></td></tr><tr><td>$|g_{LR}^S - 2 g_{LR}^T| < 0.070$</td><td>$|g_{RL}^S - 2 g_{RL}^T| < 0.418$</td><td></td></tr><tr><td colspan="3">$Q_{RR} + Q_{LR} < 8.2 \times 10^{-4}$</td></tr></tbody></table>

## 57.3 Matrix element:

All results in direct muon decay (energy spectra of the electron and of the neutrinos, polarizations, and angular distributions), and in inverse muon decay (the reaction cross section) at energies well below $m_W c^2$, may be parametrized in terms of amplitudes $g_{\epsilon\mu}^{\gamma}$ and the Fermi coupling constant $G_F$, using the matrix element

$$ \frac{4G_F}{\sqrt{2}} \sum_{\substack{\gamma=S,V,T \\ \epsilon, \mu=R,L}} g_{\epsilon\mu}^{\gamma} \langle \bar{e}_{\epsilon} | \Gamma^{\gamma} | (\nu_{\epsilon})_n \rangle \langle \bar{\nu}_{\mu} | m | \Gamma_{\gamma} | \mu_{\mu} \rangle \quad (57.9) $$

We use the notation of Fetscher et al. [2], who in turn use the sign conventions and definitions of Scheck [15]. Here, $\gamma = S, V, T$ indicates a scalar, vector, or tensor interaction; and $\epsilon, \mu = R, L$ indicate a right- or left-handed chirality of the electron or muon. The chiralities $n$ and $m$ of the $\nu_e$ and $\bar{\nu}_\mu$ are then determined by the values of $\gamma, \epsilon$, and $\mu$. The particles are represented by fields of definite chirality [16].

As shown by Langacker and London [17], explicit lepton-number nonconservation still leads to a matrix element equivalent to Eq. (57.9). They conclude that it is not possible, even in principle, to test lepton-number conservation in (leptonic) muon decay if the final neutrinos are massless and are not observed.

The ten complex amplitudes $g_{\epsilon\mu}^{\gamma}$ ($g_{RR}^T$ and $g_{LL}^T$ are identically zero) and $G_F$ constitute 19 independent (real) parameters to be determined by experiment. The Standard Model interaction corresponds to one single amplitude $g_{LL}^V$ being unity and all the others being zero.

The (direct) muon decay experiments are compatible with an arbitrary mix of the scalar and vector amplitudes $g_{LL}^S$ and $g_{LL}^V$ – in the extreme even with purely scalar $g_{LL}^S = 2$, $g_{LL}^V = 0$. The decision in favour of the Standard Model comes from the quantitative observation of inverse muon decay, which would be forbidden for pure $g_{LL}^S$ [2].

## 57.4 Experimental determination of V-A:

In order to determine the amplitudes $g_{\epsilon\mu}^{\gamma}$ uniquely from experiment, the following set of equations, where the left-hand sides

represent experimental results, has to be solved.

$$ a = 16 (|g_{RL}^V|^2 + |g_{LR}^V|^2) + |g_{RL}^S + 6 g_{RL}^T|^2 + |g_{LR}^S + 6 g_{LR}^T|^2 \quad (57.10a) $$

$$ a' = 16 (|g_{RL}^V|^2 - |g_{LR}^V|^2) + |g_{RL}^S + 6 g_{RL}^T|^2 - |g_{LR}^S + 6 g_{LR}^T|^2 \quad (57.10b) $$

$$ \alpha = 8 \operatorname{Re} \{ g_{RL}^V (g_{LR}^{S*} + 6 g_{LR}^{T*}) + g_{LR}^V (g_{RL}^{S*} + 6 g_{RL}^{T*}) \} \quad (57.10c) $$

$$ \alpha' = 8 \operatorname{Im} \{ -g_{RL}^V (g_{LR}^{S*} + 6 g_{LR}^{T*}) + g_{LR}^V (g_{RL}^{S*} + 6 g_{RL}^{T*}) \} \quad (57.10d) $$

$$ b = 4 (|g_{RR}^V|^2 + |g_{LL}^V|^2) + |g_{RR}^S|^2 + |g_{LL}^S|^2 \quad (57.10e) $$

$$ b' = 4 (|g_{RR}^V|^2 - |g_{LL}^V|^2) + |g_{RR}^S|^2 - |g_{LL}^S|^2 \quad (57.10f) $$

$$ c = \frac{1}{2} \{ |g_{RL}^S - 2 g_{RL}^T|^2 + |g_{LR}^S - 2 g_{LR}^T|^2 \} \quad (57.10g) $$

$$ c' = \frac{1}{2} \{ |g_{RL}^S - 2 g_{RL}^T|^2 - |g_{LR}^S - 2 g_{LR}^T|^2 \} \quad (57.10h) $$

and

$$ Q_L^{\nu_e} = 1 - \left\{ \frac{1}{4} |g_{LR}^S|^2 + \frac{1}{4} |g_{LL}^S|^2 + |g_{RR}^V|^2 + |g_{RL}^V|^2 + 3 |g_{LR}^T|^2 \right\} $$

$$ \omega_L = \frac{3}{4} \frac{|g_{RR}^S|^2 + 4 |g_{LR}^V|^2 + |g_{RL}^S + 2 g_{LR}^T|^2}{|g_{RL}^S|^2 + |g_{RR}^S|^2 + 4 |g_{LL}^V|^2 + 4 |g_{LR}^V|^2 + 12 |g_{RL}^T|^2} \quad (57.10i) $$

It has been noted earlier by C. Jarlskog [18], that certain experiments observing the decay electron are especially informative if they yield the V-A values. The complete solution is now found as follows. Fetscher et al. [2] introduced four probabilities $Q_{\epsilon\mu}(\epsilon, \mu = R, L)$ for the decay of a $\mu$-handed muon into an $\epsilon$-handed electron, and showed that there exist upper bounds on $Q_{RR}$, $Q_{LR}$, and $Q_{RL}$, and a lower bound on $Q_{LL}$. These probabilities are given in terms of the $g_{\epsilon\mu}^{\gamma}$'s by

$$ Q_{\epsilon\mu} = \frac{1}{4} |g_{\epsilon\mu}^S|^2 + |g_{\epsilon\mu}^V|^2 + 3 (1 - \delta_{\epsilon\mu}) |g_{\epsilon\mu}^T|^2 \quad (57.11) $$

where $\delta_{\epsilon\mu} = 1$ for $\epsilon = \mu$, and $\delta_{\epsilon\mu} = 0$ for $\epsilon \neq \mu$. They are related to the parameters $a, b, c, a', b'$, and $c'$ by

$$ Q_{RR} = 2 (b + b') / A \quad (57.12a) $$

$$ Q_{LR} = [(a - a') + 6 (c - c')] / 2A \quad (57.12b) $$

$$ Q_{RL} = [(a + a') + 6 (c + c')] / 2A \quad (57.12c) $$

$$ Q_{LL} = 2 (b - b') / A \quad (57.12d) $$

with $A = 16$. In the Standard Model, $Q_{LL} = 1$ and the others are zero.

Since the upper bounds on $Q_{RR}$, $Q_{LR}$, and $Q_{RL}$ are found to be small, and since the helicity of the $\nu_\mu$ in pion decay is known from experiment [19,20] to very high precision to be $-1$ [21], the cross section $S$ of inverse muon decay, normalized to the V-A value, yields [2]

$$ |g_{LL}^S|^2 \le 4(1 - S) \quad (57.13a) $$

and

$$ |g_{LL}^V|^2 = S. \quad (57.13b) $$

Thus the Standard Model assumption of a pure V-A leptonic charged weak interaction of $e$ and $\mu$ is derived (within errors) from experiments at energies far below the mass of the $W^\pm$: Eq. (57.13 b) gives a lower limit for V-A, and Eqs. (57.12 a, b, c)
and (57.13 a) give upper limits for the other four-fermion inter-
actions. The existence of such upper limits may also be seen
from $Q_{RR}+Q_{RL} = (1-\xi')/2$ ($e^+$ longitudinal polarization) and
$Q_{RR}+Q_{LR} = \frac{1}{2}(1+\xi/3-16\xi\delta/9)$ (decay asymmetry). Ta-
ble 57.1 gives the current experimental limits on the magnitudes
of the $g_{\epsilon\mu}^{\gamma}$'s. More stringent limits on the six coupling constants
$g_{LR}^{S}, g_{LR}^{V}, g_{LR}^{T}, g_{RL}^{S}, g_{RL}^{V}$, and $g_{RL}^{T}$ have been derived from upper
limits on the neutrino mass [22]. Limits on the “charge retention”
coordinates, as used in the older literature (e.g., Ref. [23]), are
given by Burkard et al. [24].

References

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[2] W. Fetscher, H. J. Gerber and K. F. Johnson, Phys. Lett. **B173**, 102 (1986).

[3] P. Langacker, Comments Nucl. Part. Phys. **19**, 1, 1 (1989).

[4] C. Bouchiat and L. Michel, Phys. Rev. **106**, 170 (1957).

[5] T. Kinoshita and A. Sirlin, Phys. Rev. **107**, 593 (1957).

[6] T. Kinoshita and A. Sirlin, Phys. Rev. **108**, 844 (1957).

[7] W. Fetscher, Phys. Rev. **D49**, 5945 (1994).

[8] S. R. Mishra et al., Phys. Lett. **B252**, 170 (1990).

[9] S. R. Mishra, private communication.

[10] P. Vilain et al. (CHARM-II), Phys. Lett. **B364**, 121 (1995).

[11] A. Hillairet et al. (TWIST), Phys. Rev. D85, 092013 (2012),
[arXiv:1112.3606].

[12] R. P. MacDonald et al. (TWIST), Phys. Rev. D78, 032010 (2008), [arXiv:0807.1125].

[13] C. A. Gagliardi, R. E. Tribble and N. J. Williams, Phys. Rev. D72, 073002 (2005), [hep-ph/0509069].

[14] R. Bayes et al. (TWIST), Phys. Rev. Lett. 106, 041804 (2011).

[15] F. Scheck, in *Electroweak and Strong Interactions* (Springer Verlag, 1996).

[16] K. Mursula and F. Scheck, Nucl. Phys. **B253**, 189 (1985).

[17] P. Langacker and D. London, Phys. Rev. D39, 266 (1989).

[18] C. Jarlskog, Nucl. Phys. **75**, 659 (1966).

[19] A. Jodidio et al., Phys. Rev. D34, 1967 (1986), [Erratum: Phys. Rev.D37,237(1988)].

[20] L. P. Roesch et al., Helv. Phys. Acta **55**, 74 (1982).

[21] W. Fetscher, Phys. Lett. **140B**, 117 (1984).

[22] G. Prezeau and A. Kurylov, Phys. Rev. Lett. **95**, 101802 (2005), [hep-ph/0409193].

[23] S. E. Derenzo, Phys. Rev. **181**, 1854 (1969).

[24] H. Burkard et al., Phys. Lett. **160B**, 343 (1985).
# 58. τ Branching Fractions

Revised April 2024 by Sw. Banerjee (Louisville U.) and A. Lusiani (SNS, Pisa; INFN, Pisa).

## 58.1 τ Branching Fractions

The τ Listings contains 252 entries that correspond to either a τ partial decay fraction into a specific decay mode (branching fraction) or a ratio of two τ partial decay fractions (branching ratio). Experimental measurements provide values for 148 of these quantities, upper limits for 67 branching fractions to Lepton Family number, Lepton number, or Baryon number violating modes, and 37 additional upper limits for other modes. A total of 129 τ branching fractions and branching ratios are determined with a fit of 170 measurements. 85 quantities have at least one measurement in the fit.

## 58.2 The constrained fit to τ branching fractions

The τ branching fractions fit uses the reported values, uncertainties and statistical correlations of the τ branching fractions and branching ratios measurements. Asymmetric uncertainties are symmetrized as $σ_{\text{symm}}^2 = (σ_+^2 + σ_-^2)/2$. If only a few measurements are correlated, the correlation coefficients are listed in the footnote for each measurement (see for example $\Gamma(\text{particle}^- \ge 0 \text{ neutrals} \ge 0 K^0 \nu_\tau)$ ("1-prong"))/\Gamma_{\text{total}}$. If a large number of measurements are correlated, then the full correlation matrix is listed in the footnote to the measurement that first appears in the τ Listings. Footnotes to the other measurements refer to the first one. For example, the large correlation matrices for the branching fraction or ratio measurements contained in Refs. [1] [2] are listed in Footnotes to the $\Gamma(e^-\bar{e}\nu_\tau)/\Gamma_{\text{total}}$ and $\Gamma(h^-\nu_\tau)/\Gamma_{\text{total}}$ measurements respectively. Additionally, the most precise experimental inputs are treated according to how they depend on external parameters on the basis of their documentation [3]. The τ measurements may depend on parameters such as the τ pair production cross-section in $e^+e^-$ annihilations at the $\Upsilon(4S)$ peak. In some cases, measurements reported in different papers by the same collaboration may depend on common parameters like the estimate of the integrated luminosity or of particle identification efficiencies. For all the significant detected dependencies, the τ measurements and their uncertainties are updated to account for the updated values of the external parameters. The dependencies on common systematic effects are also determined in size and sign, and all the common systematic dependencies of different measurements are used together with the published statistical and systematic uncertainties and correlations in order to compute an updated all-inclusive variance and covariance matrix of the experimental inputs of the fit.

The fit procedure parameters correspond to all measured τ branching fractions and ratios, to some non-measured branching fractions and ratios like for instance $\mathcal{B}(\tau^- \to \pi^- K_L^0 K_L^0 \nu_\tau)$ and to one nuisance variable. When discussing the fit results in the following, the fit $\chi^2$, the number of degrees of freedom, the residuals and pulls all refer to the subset of fit parameters that correspond to τ branching fractions and ratios, excluding nuisance variables. The fit parameters are optimized while respecting relations described by a series of constraint equations. All the experimental inputs and all the constraint equations are reported in the τ Listings section that follows this review. In some cases, constraints describe approximate relations that nevertheless hold within the present experimental precision. For instance, the constraint $\mathcal{B}(\tau^- \to K^- K^- K^+ \nu_\tau) = \mathcal{B}(\tau^- \to K^- \phi \nu_\tau) \times \mathcal{B}(\phi \to K^+ K^-)$ is justified within the current experimental evidence. The constraint equations between the τ branching fractions and ratios include quantities other than τ branching fractions and branching ratios, like for instance the branching fractions of the $\eta$ and $\omega$ mesons. We neglect the uncertainties on these values for all quantities except $\mathcal{B}(a_1^- \to \pi^- \gamma)$, whose value and uncertainty were estimated by ALEPH [1] to be $(0.21 \pm 0.08) \cdot 10^{-2}$, relying on a measurement of $\Gamma(a_1^- \to \pi^- \gamma)$ [4]. This quantity is included in the fit parameters as nuisance variable, with a $\chi^2$ term corresponding to its estimate. We assume that $\mathcal{B}(\tau^- \to a_1^- \nu_\tau) = \mathcal{B}(\tau^- \to \pi^- \pi^+ \pi^- \nu_\tau \text{ (ex. }K^0, \omega\text{)}) + \mathcal{B}(\tau^- \to \pi^- 2\pi^0 \nu_\tau \text{ (ex. }K^0\text{)}) + \mathcal{B}(\tau^- \to a_1^-(\pi^- \gamma) \nu_\tau)$, neglecting the observed but negligible branching fractions to other modes, and that

$\mathcal{B}(\tau^- \to a_1^-(\pi^- \gamma) \nu_\tau) = \mathcal{B}(\tau^- \to a_1^- \nu_\tau) \cdot \mathcal{B}(a_1^- \to \pi^- \gamma)$. The values of all other quantities in the constraint equations are taken from the 2023 edition of the Review of Particle Physics.

In the fit, uncertainty scale factors are applied to the published uncertainties of measurements only if significant inconsistency between different measurements remain after accounting for all relevant uncertainties and correlations. When performing the fit with no scale factors, the two measurements of $\mathcal{B}(\tau^- \to K^- K^- K^+ \nu_\tau)$ have pulls exceeding 5 $\sigma$ from the fit values. There are 170 pulls, one per measurement. They are partially correlated, and the effective number of independent pulls is equal to the number of degrees of freedom of the fit, 125. The probability of getting pulls equal or larger than either one of the two very large pulls in a sample of 125 is smaller that the probability of a 3 $\sigma$ deviation for a Normal variable. Therefore, it has been decided to apply an uncertainty scale factor of 5.4 on all measurements of $\mathcal{B}(\tau^- \to K^- K^- K^+ \nu_\tau)$ (one by BaBar and one by Belle). The scale factor has been computed according to the standard PDG procedure. After applying the scale factor, the pull distribution of the measurements in figure 58.1 is reasonably Normal and the pull probability distribution in figure 58.2 is reasonably flat.

<figure><img src="image_307.png" /><figcaption>Figure 58.1: Pulls of individual measurements against the respective fitted quantity.</figcaption></figure>

<figure><img src="image_308.png" /><figcaption>Figure 58.2: Probability of individual measurement pulls against the respective fitted quantity.</figcaption></figure>

Considering only the residuals with respect to τ branching fractions and ratios measurements, the constrained fit has a $\chi^2$ of 135 for 125 degrees of freedom, corresponding to a $\chi^2$ probability of 26.0%. We use 170 measurements and 84 constraints on the branching fractions and ratios to determine 129 quantities, consisting of 112 branching fractions and 17 branching ratios. The constraints include the unitarity constraint on the sum of all the exclusive τ decay modes, $B_{\text{all}} = 1$. If the unitarity constraint is released, the fit result for $B_{\text{all}}$ is consistent with unitarity with $1 - B_{\text{all}} = (0.07 \pm 0.11)\%$.

For the convenience of summarizing the fit results, we list in the following the values and uncertainties for a set of 46 "basis" decay modes, from which all remaining branching fractions and ratios
can be obtained using the constraints. The basis decay modes are not intended to sum up to 1. Since some basis quantities represent multiple branching fractions that are related by constraint equations, the unitarity constraint corresponds to a linear combination of branching fractions, with the coefficients listed in the following. The coefficients of $B(\tau^{-} \rightarrow \mu^{-}\bar{\nu}_{\mu}\nu_{\tau})$ (ex. $K^{0}$)) and $B(\tau^{-} \rightarrow \pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ (ex. $K^{0}$, $\omega$)) depend on the fitted value of the nuisance variable, $B(a_{1}^{-} \rightarrow \pi^{-}\gamma) = 0.0022 \pm 0.0008$. The correlation matrix between the basis modes is reported in the $\tau$ Listings.

<table><thead><tr><th>decay mode</th><th>fit result</th><th>coeff.</th></tr></thead><tbody><tr><td>B(τ<sup>-</sup> → μ<sup>-</sup>ν<sub>μ</sub>ν<sub>τ</sub>)</td><td>0.1739 ± 0.0004</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → e<sup>-</sup>ν<sub>e</sub>ν<sub>τ</sub>)</td><td>0.1782 ± 0.0004</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>ν<sub>τ</sub>)</td><td>0.1082 ± 0.0005</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>ν<sub>τ</sub>)</td><td>0.00696 ± 0.00010</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>0.2549 ± 0.0009</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>0.00433 ± 0.00015</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>2π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>0.0926 ± 0.0010</td><td>1.0022</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>2π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(0.65 ± 0.22) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>3π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>0.0104 ± 0.0007</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>3π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, η))</td><td>(0.48 ± 0.21) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → h<sup>-</sup>4π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, η))</td><td>0.0011 ± 0.0004</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>ν<sub>τ</sub>)</td><td>0.00838 ± 0.00014</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>K<sup>0</sup>ν<sub>τ</sub>)</td><td>0.001486 ± 0.000034</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>0.00382 ± 0.00013</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>K<sup>0</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>0.00150 ± 0.00007</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>2π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(0.26 ± 0.23) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>K<sup>0</sup>ν<sub>τ</sub>)</td><td>(235 ± 6) · 10<sup>-6</sup></td><td>2.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>K<sup>0</sup>ν<sub>τ</sub>)</td><td>0.00108 ± 0.00024</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>K<sup>0</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>(18.2 ± 2.1) · 10<sup>-6</sup></td><td>2.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>K<sup>0</sup>K<sup>0</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>(0.32 ± 0.12) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>0</sup>h<sup>-</sup>h<sup>-</sup>h<sup>+</sup>h<sup>+</sup>ν<sub>τ</sub>)</td><td>(0.25 ± 0.20) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>π<sup>+</sup>π<sup>-</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω))</td><td>0.0899 ± 0.0005</td><td>1.0022</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>π<sup>+</sup>π<sup>-</sup>π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω))</td><td>0.0274 ± 0.0007</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → h<sup>-</sup>h<sup>-</sup>h<sup>+</sup>h<sup>+</sup>2π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω, η))</td><td>0.0010 ± 0.0004</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>-</sup>K<sup>+</sup>ν<sub>τ</sub>)</td><td>0.001435 ± 0.000027</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>-</sup>K<sup>+</sup>π<sup>0</sup>ν<sub>τ</sub>)</td><td>(61 ± 18) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>π<sup>0</sup>ην<sub>τ</sub>)</td><td>0.00139 ± 0.00007</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>ην<sub>τ</sub>)</td><td>(155 ± 8) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>π<sup>0</sup>ην<sub>τ</sub>)</td><td>(48 ± 12) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>K<sup>0</sup>ην<sub>τ</sub>)</td><td>(94 ± 15) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>π<sup>+</sup>π<sup>-</sup>ην<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(220 ± 13) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>ων<sub>τ</sub>)</td><td>(0.41 ± 0.09) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → h<sup>-</sup>π<sup>0</sup>ων<sub>τ</sub>)</td><td>0.0041 ± 0.0004</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>φν<sub>τ</sub>)</td><td>(44 ± 16) · 10<sup>-6</sup></td><td>0.8300</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>ων<sub>τ</sub>)</td><td>0.0195 ± 0.0006</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>π<sup>-</sup>π<sup>+</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω))</td><td>0.00293 ± 0.00007</td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>π<sup>-</sup>π<sup>+</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω, η))</td><td>(0.39 ± 0.14) · 10<sup>-3</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>2π<sup>0</sup>ων<sub>τ</sub>)</td><td>(72 ± 16) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → 2π<sup>-</sup>π<sup>+</sup>3π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, η, ω, f<sub>1</sub>))</td><td>(14 ± 27) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → 3π<sup>-</sup>2π<sup>+</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, ω, f<sub>1</sub>))</td><td>(775 ± 30) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>2π<sup>+</sup>2π<sup>+</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(0.6 ± 1.2) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → 2π<sup>-</sup>π<sup>+</sup>ων<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(84 ± 6) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → 3π<sup>-</sup>2π<sup>+</sup>π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>, η, ω, f<sub>1</sub>))</td><td>(38 ± 9) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → K<sup>-</sup>2π<sup>+</sup>2π<sup>+</sup>π<sup>0</sup>ν<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(1.1 ± 0.6) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>f<sub>1</sub>(2π<sup>-</sup>2π<sup>+</sup>)ν<sub>τ</sub>)</td><td>(52 ± 4) · 10<sup>-6</sup></td><td>1.0000</td></tr><tr><td>B(τ<sup>-</sup> → π<sup>-</sup>2π<sup>0</sup>ην<sub>τ</sub> (ex. K<sup>0</sup>))</td><td>(0.19 ± 0.04) · 10<sup>-3</sup></td><td>1.0000</td></tr></tbody></table>

In defining the fit constraints and in selecting the modes that sum up to one we made some assumptions and choices. We assume that some channels, like $\tau^{-} \rightarrow \pi^{-}K^{+}\pi^{-} \ge 0\pi^{0}\nu_{\tau}$ and $\tau^{-} \rightarrow \pi^{+}K^{-}K^{-} \ge 0\pi^{0}\nu_{\tau}$, have negligible branching fractions as expected from the Standard Model, even if the experimental limits for these branching fractions are not very stringent. The 95% confidence level upper limits are $B(\tau^{-} \rightarrow \pi^{-}K^{+}\pi^{-} \ge 0\pi^{0}\nu_{\tau}) < 0.25\%$ and $B(\tau^{-} \rightarrow \pi^{+}K^{-}K^{-} \ge 0\pi^{0}\nu_{\tau}) < 0.09\%$, values not so different from measured branching fractions for allowed 3-prong modes containing charged kaons. For decays to final states containing one neutral kaon we assume that the branching fraction with the $K_{L}^{0}$ are the same as the corresponding one with a $K_{S}^{0}$. On decays with two neutral kaons we assume that the branching fractions with $K_{L}^{0}K_{L}^{0}$ are the same as the ones with $K_{S}^{0}K_{S}^{0}$.

## 58.3 BaBar and Belle measure on average lower branching fractions and ratios.

We compare the BaBar and Belle measurements with the results of a fit where all the B-factories measurements have been excluded. We restrict the comparison to the measurements that are used in the fit, omitting two measurements that are superseeded with other results, and one measurement that is fully correlated with other measurements. We find that BaBar and Belle measure on average lower $\tau$ branching fractions and ratios than the other experiments. Figures 58.3 and 58.4 show histograms of the 25 pulls of the differences between B-factory measurements and the respective non-B-factory fit results. The average pull between the two sets of measurements is -0.8$\sigma$ (-0.7$\sigma$ for the 14 Belle measurements and -0.9$\sigma$ for the 11 BaBar measurements).

<figure><img src="image_309.png" /><figcaption>Figure 58.3: Distribution of the normalized difference between 11 measurements of branching fractions and ratios published by the BaBar collaboration and the respective averages computed using only non-B-factory measurements.</figcaption></figure>

<figure><img src="image_310.png" /><figcaption>Figure 58.4: Distribution of the normalized difference between 14 measurements of branching fractions and ratios published by the Belle collaboration and the respective averages computed using only non-B-factory measurements.</figcaption></figure>

## 58.4 Overconsistency of Leptonic Branching Fraction Measurements.

As observed in the previous editions of this review, measurements of the leptonic branching fractions are more consistent with each other than expected from the quoted uncertainties on the individual measurements. The $\chi^{2}$/n.d.o.f. are 0.34/4 for $B_{e}$ and 0.08/4 for $B_{\mu}$. The probability of getting a smaller $\chi^{2}$ is 1.3% for $B_{e}$ and 0.08% for $B_{\mu}$.

## 58.5 Technical implementation of the fit

The fit computes a set of quantities denoted with $q_{i}$ by minimizing a $\chi^{2}$ while respecting a series of equality constraints on the $q_{i}$. The quantities $q_{i}$ represent $\tau$ branching fractions and branching ratios, and nuisance variables. The fit minimization procedure is equivalent to choosing a set of basis fit variables, using functions of these basis variables to predict measurements, and determining these basis variables by minimizing the measurements' $\chi^{2}$. The $\chi^{2}$ is computed using the measurements $m_{i}$ and their covariance matrix $E_{ij}$ as $\chi^{2} = (m_{i} - A_{ik}q_{k})^{T} E_{ij}^{-1} (m_{j} - A_{jl}q_{l})$, where the model
matrix $A_{ij}$ is used to get the vector of the predicted measurements $m_i^l$ from the vector of the fit parameters $q_j$ as $m_i^l = A_{ij}q_j$. There is one fit variable for each of the modes that have measurements, therefore $A_{ij} = 1$ when $q_j$ corresponds to the $\tau$ branching fraction or branching ratio of the measurement $m_i$, and $A_{ij} = 0$ otherwise. The constraints are equations involving the fit parameters. The fit does not impose limitations on the functional form of the constraints. In summary, the fit requires:

$$ \min [\chi^2(q_k)] = \min [(m_i - A_{ik}q_k)^t E_{ij}^{-1}(m_j - A_{jl}q_l)], \quad (58.1) $$

$$ \text{subjected to} \quad f_r(q_s) - c_r = 0, \quad (58.2) $$

where the left term of Eq. 58.2 defines the constraint expressions. Using the method of Lagrange multipliers, a set of equations is obtained by taking the derivatives with respect to the fitted quantities $q_k$ and the Lagrange multipliers $\lambda_r$ of the sum of the $\chi^2$ and the constraint expressions multiplied by the Lagrange multipliers $\lambda_r$, one for each constraint:

$$ \begin{aligned} \min & \left[ (A_{ik}q_k - m_i)^t E_{ij}^{-1} (A_{jl}q_l - m_j) + 2\lambda_r (f_r(q_s) - c_r) \right] = \\ & = \min \left[ \tilde{\chi}^2(q_k, \lambda_r) \right], \\ & \left(\partial/\partial q_k, \partial/\partial \lambda_r\right) \left[ \tilde{\chi}^2(q_k, \lambda_r) \right] = 0. \end{aligned} \quad (58.3) $$

Eq. 58.3 defines a set of equations for the vector of the unknowns $(q_k, \lambda_r)$, some of which may be non-linear, in case of non-linear constraints. An iterative minimization procedure approximates at each step the non-linear constraint expressions by their first order Taylor expansion around the current values of the fitted quantities, $\bar{q}_s$:

$$ f_r(q_s) - c_r = f_r(\bar{q}_s) + \left. \frac{\partial f_r(q_s)}{\partial q_s} \right|_{\bar{q}_s} (q_s - \bar{q}_s) - c_r, $$

which can be written as

$$ B_{rs}q_s - c'_r, $$

where $c'_r$ are the resulting constant known terms, independent of $q_s$ at first order. After linearization, the differentiation by $q_k$ and $\lambda_r$ is trivial and leads to a set of linear equations

$$ A_{ki}^t E_{ij}^{-1} A_{jl} q_l + B_{kr}^t \lambda_r = A_{ki}^t E_{ij}^{-1} m_j, \quad (58.4) $$

$$ B_{rs}q_s = c'_r, \quad (58.5) $$

which can be expressed as

$$ F_{ij}u_j = v_i, \quad (58.6) $$

where $u_j = (q_k, \lambda_r)$ and $v_i$ is the vector of the known constant terms running over the index $k$ and then $r$ in the right terms of Eq. 58.4 and Eq. 58.5, respectively. Solving the equation set in Eq. 58.6 by matrix inversion gives the fitted quantities and their variance and covariance matrix, using the measurements and their variance and covariance matrix. The fit procedure starts by computing the linear approximation of the non-linear constraint expressions around the quantities seed values. With an iterative procedure, the unknowns are updated at each step by solving the equations and the equations are then linearized around the updated values, until the variation of the fitted unknowns is reduced below a numerically small threshold.

### References

[1] S. Schael et al. (ALEPH), Phys. Rept. 421, 191 (2005), [hep-ex/0506072].

[2] J. Abdallah et al. (DELPHI), Eur. Phys. J. C46, 1 (2006), [hep-ex/0603044].

[3] D. Asner et al. (Heavy Flavor Averaging Group) (2010), [arXiv:1010.1589].

[4] M. Zielinski et al., Phys. Rev. Lett. 52, 1195 (1984).
## 59. τ-Lepton Decay Parameters

Revised February 2024 by A. Stahl (RWTH Aachen U.).

The purpose of the measurements of the decay parameters (also known as Michel parameters) of the τ is to determine the structure (spin and chirality) of the current mediating its decays.

### 59.1 Leptonic Decays:

The Michel parameters are extracted from the energy spectrum of the charged daughter lepton $ℓ = e, μ$ in the decays $τ → ℓν_ℓν_τ$. Ignoring radiative corrections, neglecting terms of order $(m_ℓ/m_τ)^2$ and $(m_τ/\sqrt{s})^2$, and setting the neutrino masses to zero, the spectrum in the laboratory frame reads

$$ \frac{d\Gamma}{dx} = \frac{G_{\tau\ell}^2 m_\tau^5}{192 \pi^3} \times \left\{ f_0(x) + \rho f_1(x) + \eta \frac{m_\ell}{m_\tau} f_2(x) - P_\tau[\xi g_1(x) + \xi \delta g_2(x)] \right\} \quad (59.1) $$

with

$$ f_0(x) = 2 - 6x^2 + 4x^3 \quad (59.2) $$

$$ f_1(x) = -\frac{4}{9} + 4x^2 - \frac{32}{9}x^3 \quad g_1(x) = -\frac{2}{3} + 4x - 6x^2 + \frac{8}{3}x^3 $$

$$ f_2(x) = 12(1-x)^2 \quad g_2(x) = \frac{4}{9} - \frac{16}{3}x + 12x^2 - \frac{64}{9}x^3. $$

The quantity $x$ is the fractional energy of the daughter lepton $ℓ$, i.e., $x = E_ℓ/E_{ℓ,max} \approx E_ℓ/(\sqrt{s}/2)$ and $P_τ$ is the polarization of the tau leptons. The integrated decay width is given by

$$ \Gamma = \frac{G_{\tau\ell}^2 m_\tau^5}{192 \pi^3} \left( 1 + 4\eta \frac{m_\ell}{m_\tau} \right) . \quad (59.3) $$

The situation is similar to muon decays $μ → eν_eν_μ$. The generalized matrix element with the couplings $g_{eμ}^γ$ and their relations to the Michel parameters $ρ, η, ξ$, and $δ$ have been described in the “Note on Muon Decay Parameters”. The Standard Model expectations are 3/4, 0, 1, and 3/4, respectively. For more details, see [1].

More information can be extracted from the decays, if the polarization of the daughter lepton is measured. Belle presented a first result (see [2]). A new parameter $ξ'$ is introduced. Its relation to the couplings is given by (see [3]):

$$ \xi' = 1 - 2 \sum_{\omega=L,R} \left( \frac{1}{4} |g_{R\omega}^S|^2 + |g_{R\omega}^V|^2 \right) + 3 |g_{RL}^T|^2 . \quad (59.4) $$

The new measurement improves the limit on $g_{RL}^T$ by about a factor of two.

Additional Michel parameters can be defined in radiative decays $τ → ℓν_ℓν_τγ$, see [4]. A first measurement is presented in [5]. The experimental precision does not provide a significant impact on the knowledge about the couplings, yet.

### 59.2 Hadronic Decays:

In the case of hadronic decays $τ → hν_τ$, with $h = π, ρ$, or $a_1$, the ansatz is restricted to purely vectorial currents. The matrix element is

$$ \frac{G_{\tau h}}{\sqrt{2}} \sum_{\lambda=R,L} g_{\lambda} \langle \bar{\Psi}_{\omega}(\nu_{\tau}) | \gamma^{\mu} | \Psi_{\lambda}(\tau) \rangle J_{\mu}^{h} \quad (59.5) $$

with the hadronic current $J_μ^h$. The neutrino chirality $ω$ is uniquely determined from $λ$. The spectrum depends only on a single parameter $ξ_h$

$$ \frac{d^n \Gamma}{dx_1 dx_2 \dots dx_n} = f(\vec{x}) + \xi_h P_\tau g(\vec{x}) , \quad (59.6) $$

with $f$ and $g$ being channel-dependent functions of the n observables $\vec{x} = (x_1, x_2, \dots, x_n)$ (see [6]). The parameter $ξ_h$ is related to the couplings through

$$ \xi_h = |g_L|^2 - |g_R|^2 . \quad (59.7) $$

$ξ_h$ is the negative of the chirality of the τ neutrino in these decays. In the Standard Model, $ξ_h = 1$. Also included in the Data Listings for $ξ_h$ are measurements of the neutrino helicity which coincide with $ξ_h$, if the neutrino is massless (ASNER 00 [7], ACKERSTAFF 97R [8], AKERS 95P [9], ALBRECHT 93C [10], and ALBRECHT 90I [11]).

### 59.3 Combination of Measurements:

The individual measurements are combined, taking into account the correlations between the parameters. In a first fit, universality between the two leptonic decays, and between all hadronic decays, is assumed. A second fit is made without these assumptions. The results of the two fits are provided as OUR FIT in the Data Listings below in the tables whose title includes “(e or mu)” or “(all hadronic modes)”, and “(e)”, “(mu)” etc., respectively. The measurements show good agreement with the Standard Model. The $χ^2$ values with respect to the Standard model predictions are 24.1 for 41 degrees of freedom and 26.8 for 56 degrees of freedom, respectively. The correlations are reduced through this combination to less than 20%, with the exception of $ρ$ and $η$ which are correlated by +23%, for the fit with universality and by +70% for $τ → μν_μν_τ$.

### 59.4 Model-independent Analysis:

From the Michel parameters, limits can be derived on the couplings $g_{eλ}^γ$ without further model assumptions. In the Standard model $g_{eL}^γ = 1$ (leptonic decays), and $g_L = 1$ (hadronic decays) and all other couplings vanish. First, the partial decay widths have to be compared to the Standard Model predictions to derive limits on the normalization of the couplings $A_x = G_{\tau x}^2/G_F^2$ with Fermi’s constant $G_F$:

$$ \begin{aligned} A_e &= 1.0022 \pm 0.0041, \\ A_\mu &= 0.983 \pm 0.017, \\ A_\pi &= 0.9957 \pm 0.0049. \end{aligned} \quad (59.8) $$

Then limits on the couplings (95% CL) can be extracted (see [12] and [13]). Without the assumption of universality, the limits given in Table 59.1 are derived.

<table><caption>**Table 59.1:** Coupling constants $g_{eμ}^γ$. 95% confidence level experimental limits. The limits include the quoted values of $A_e, A_μ$, and $A_π$ and assume $A_ρ = A_{a_1} = 1$.</caption><tbody><tr><th colspan="3" style="text-align: left;">τ → eν<sub>e</sub>ν<sub>τ</sub></th></tr><tr><td>|g<sup>S</sup><sub>RR</sub>| &lt; 0.70</td><td>|g<sup>V</sup><sub>RR</sub>| &lt; 0.17</td><td>|g<sup>T</sup><sub>RR</sub>| ≡ 0</td></tr><tr><td>|g<sup>S</sup><sub>LR</sub>| &lt; 0.99</td><td>|g<sup>V</sup><sub>LR</sub>| &lt; 0.13</td><td>|g<sup>T</sup><sub>LR</sub>| &lt; 0.082</td></tr><tr><td>|g<sup>S</sup><sub>RL</sub>| &lt; 2.01</td><td>|g<sup>V</sup><sub>RL</sub>| &lt; 0.52</td><td>|g<sup>T</sup><sub>RL</sub>| &lt; 0.51</td></tr><tr><td>|g<sup>S</sup><sub>LL</sub>| &lt; 2.01</td><td>|g<sup>V</sup><sub>LL</sub>| &lt; 1.004</td><td>|g<sup>T</sup><sub>LL</sub>| ≡ 0</td></tr><tr><th colspan="3" style="text-align: left;">τ → μν<sub>μ</sub>ν<sub>τ</sub></th></tr><tr><td>|g<sup>S</sup><sub>RR</sub>| &lt; 0.72</td><td>|g<sup>V</sup><sub>RR</sub>| &lt; 0.18</td><td>|g<sup>T</sup><sub>RR</sub>| ≡ 0</td></tr><tr><td>|g<sup>S</sup><sub>LR</sub>| &lt; 0.95</td><td>|g<sup>V</sup><sub>LR</sub>| &lt; 0.12</td><td>|g<sup>T</sup><sub>LR</sub>| &lt; 0.079</td></tr><tr><td>|g<sup>S</sup><sub>RL</sub>| &lt; 2.01</td><td>|g<sup>V</sup><sub>RL</sub>| &lt; 0.52</td><td>|g<sup>T</sup><sub>RL</sub>| &lt; 0.27</td></tr><tr><td>|g<sup>S</sup><sub>LL</sub>| &lt; 2.01</td><td>|g<sup>V</sup><sub>LL</sub>| &lt; 1.006</td><td>|g<sup>T</sup><sub>LL</sub>| ≡ 0</td></tr><tr><th colspan="3" style="text-align: left;">τ → πν<sub>τ</sub></th></tr><tr><td>|g<sup>V</sup><sub>R</sub>| &lt; 0.15</td><td>|g<sup>V</sup><sub>L</sub>| &gt; 0.984</td><td></td></tr><tr><th colspan="3" style="text-align: left;">τ → ρν<sub>τ</sub></th></tr><tr><td>|g<sup>V</sup><sub>R</sub>| &lt; 0.10</td><td>|g<sup>V</sup><sub>L</sub>| &gt; 0.995</td><td></td></tr><tr><th colspan="3" style="text-align: left;">τ → a<sub>1</sub>ν<sub>τ</sub></th></tr><tr><td>|g<sup>V</sup><sub>R</sub>| &lt; 0.16</td><td>|g<sup>V</sup><sub>L</sub>| &gt; 0.987</td><td></td></tr></tbody></table>
**59.5 Model-dependent Interpretation:**

More stringent limits can be derived assuming specific models. For example, in the framework of a two Higgs doublet model, the measurements correspond to a limit of $m_{H^±} > 2.0$ GeV × tan $β$ on the mass of the charged Higgs boson, or a limit of 500 GeV on the mass of the second W boson in left-right symmetric models for arbitrary mixing (both 95% CL). See [13] and [14].

**References**

[1] F. Scheck, Phys. Rept. **44**, 187 (1978); W. Fetscher and H.J. Gerber in *Precision Tests of the Standard Model*, edited by P. Langacker, World Scientific, 1993; A. Stahl, *Physics with τ Leptons*, Springer Tracts in Modern Physics, **160**, 1, (2000).

[2] D. Bodrov et al. (Belle), Phys. Rev. D **108**, 1, 012003 (2023), [arXiv:2303.10574].

[3] D. Bodrov and P. Pakhlov, JHEP **10**, 035 (2022), [arXiv:2203.12743].

[4] A. Arbuzov and T. Kopylova, JHEP 109 (2016).

[5] N. Shimizu et al. (Belle), Prog. Theo. Exp. Phys. **2**, 023C01 (2018).

[6] M. Davier et al., Phys. Lett. B **306**, 411 (1993).

[7] D. Asner et al. (CLEO), Phys. Rev. D **61**, 012002 (2000), [hep-ex/9902022].

[8] K. Ackerstaff et al. (OPAL), Z. Phys. C **75**, 593 (1997).

[9] R. Akers et al. (OPAL), Z. Phys. C **67**, 45 (1995).

[10] H. Albrecht et al. (ARGUS), Z. Phys. C **58**, 61 (1993).

[11] H. Albrecht et al. (ARGUS), Phys. Lett. B **250**, 164 (1990).

[12] K. Ackerstaff et al. (OPAL), Eur. Phys. J. C **8**, 3 (1999), [hep-ex/9808016].

[13] A. Stahl, Nucl. Phys. B Proc. Suppl. **76**, 173 (1999).

[14] M.-T. Dova, J. Swain and L. Taylor, Phys. Rev. D **58**, 015005 (1998), [hep-ph/9712384]; T. Hebbeker and W. Lohmann, Z. Phys. C **74**, 399 (1997); A. Pich and J. P. Silva, Phys. Rev. D **52**, 4006 (1995), [hep-ph/9505327].
## 60. Quark Masses

Revised August 2023 by R.M. Barnett (LBNL), L.P. Lellouch (CNRS & Aix-Marseille U.) and A.V. Manohar (UC San Diego).

### 60.1 Introduction

This note discusses some of the theoretical issues relevant for the determination of quark masses, which are fundamental parameters of the Standard Model of particle physics. Unlike the leptons, quarks are confined inside hadrons and are not observed as physical particles. Quark masses therefore cannot be measured directly, but must be determined indirectly through their influence on hadronic properties. Although one often speaks loosely of quark masses as one would of the mass of the electron or muon, any quantitative statement about the value of a quark mass must make careful reference to the particular theoretical framework that is used to define it. It is important to keep this scheme dependence in mind when using the quark mass values tabulated in the data listings.

Historically, the first determinations of quark masses were performed using quark models. These are usually called constituent quark masses and are of order 350 MeV for the u and d quarks. Constituent quark masses model the effects of dynamical chiral symmetry breaking discussed below, and are not directly related to the quark mass parameters $m_q$ of the QCD Lagrangian of Eq. (60.1). The resulting masses only make sense in the limited context of a particular quark model, and cannot be related to the quark mass parameters, $m_q$, of the Standard Model. In order to discuss quark masses at a fundamental level, definitions based on quantum field theory must be used, and the purpose of this note is to discuss these definitions and the corresponding determinations of the values of the masses.

### 60.2 Mass parameters and the QCD Lagrangian

The QCD [1] Lagrangian is

$$ \mathcal{L} = \sum_{q=u,d,s,...,t} \bar{q} (i \not D - m_q) q - \frac{1}{2} \text{tr} G_{\mu\nu} G^{\mu\nu}, \quad (60.1) $$

where the sum runs over the quark flavors u, d, s, c, b and t. $\not D = (\partial_\mu - igA_\mu)\gamma^\mu$ is the gauge covariant derivative, $A_\mu$ is the SU(3)-valued gluon field, $G_{\mu\nu} = \frac{i}{g}[D_\mu, D_\nu]$ is the gluon field strength, $m_q$ is the mass parameter of quark flavor q, and q is the quark Dirac field. After renormalization, the QCD Lagrangian Eq. (60.1) gives finite values for physical quantities, such as scattering amplitudes. Renormalization is a procedure that invokes a subtraction scheme to render the amplitudes finite, and requires the introduction of a dimensionful scale parameter $\mu$. The mass parameters in the QCD Lagrangian Eq. (60.1) depend on the renormalization scheme used to define the theory, and also on the scale parameter $\mu$. The most commonly used renormalization scheme for QCD perturbation theory is the $\overline{\text{MS}}$ scheme.

The QCD Lagrangian has a chiral symmetry in the limit that the quark masses vanish. This symmetry is spontaneously broken by dynamical chiral symmetry breaking, and explicitly broken by the quark masses. The nonperturbative scale of dynamical chiral symmetry breaking, $\Lambda_\chi$, is around 1 GeV [2]. It is conventional to call quarks heavy if $m_q > \Lambda_\chi$, so that explicit chiral symmetry breaking dominates (c, b, and t quarks are heavy), and light if $m_q < \Lambda_\chi$, so that spontaneous chiral symmetry breaking dominates (the u and d are light and the s is considered to be light when using $SU(3)_L \times SU(3)_R$ chiral perturbation theory). The determination of light- and heavy-quark masses is considered separately in Sec. 60.5 and Sec. 60.6 below.

At high energies or short distances, nonperturbative effects, such as chiral symmetry breaking, become small and one can, in principle, determine quark masses by analyzing mass-dependent effects using QCD perturbation theory. Such computations are conventionally performed using the $\overline{\text{MS}}$ scheme at a scale $\mu \gg \Lambda_\chi$, and give the $\overline{\text{MS}}$ “running” mass $\overline{m}(\mu)$. We use the $\overline{\text{MS}}$ scheme when reporting quark masses; one can readily convert these values into other schemes using perturbation theory.

The $\mu$ dependence of $\overline{m}(\mu)$ at short distances can be calculated

using the renormalization group (RG) equation,

$$ \mu^2 \frac{d\overline{m}(\mu)}{d\mu^2} = -\gamma(\overline{\alpha}_s(\mu)) \overline{m}(\mu), \quad (60.2) $$

where $\gamma$ is the anomalous dimension which is now known to four-loop order in perturbation theory [3] [4]. $\overline{\alpha}_s$ is the coupling constant [1] in the $\overline{\text{MS}}$ scheme. Defining the expansion coefficients $\gamma_r$ by

$$ \gamma(\overline{\alpha}_s) \equiv \sum_{r=1}^{\infty} \gamma_r \left( \frac{\overline{\alpha}_s}{4\pi} \right)^r, \quad (60.3) $$

the first four coefficients are given by

$$ \begin{aligned} \gamma_1 &= 4, \\ \gamma_2 &= \frac{202}{3} - \frac{20N_L}{9}, \\ \gamma_3 &= 1249 + \left(-\frac{2216}{27} - \frac{160}{3}\zeta(3)\right) N_L - \frac{140}{81}N_L^2, \\ \gamma_4 &= \frac{4603055}{162} + \frac{135680}{27}\zeta(3) - 8800\zeta(5) \\ &\quad + \left(-\frac{91723}{27} - \frac{34192}{9}\zeta(3) + 880\zeta(4) + \frac{18400}{9}\zeta(5)\right) N_L \\ &\quad + \left(\frac{5242}{243} + \frac{800}{9}\zeta(3) - \frac{160}{3}\zeta(4)\right) N_L^2 \\ &\quad + \left(-\frac{332}{243} + \frac{64}{27}\zeta(3)\right) N_L^3, \end{aligned} $$

where $N_L$ is the number of active light quark flavors at the scale $\mu$, i.e. flavors with masses $\le \mu$, and $\zeta$ is the Riemann zeta function ($\zeta(3) \simeq 1.2020569$, $\zeta(4) \simeq 1.0823232$, and $\zeta(5) \simeq 1.0369278$). Eq. (60.2) must be solved in conjunction with the RG equation for $\overline{\alpha}_s(\mu)$ given in [1]. In addition, as the renormalization scale crosses quark mass thresholds one needs to match the scale dependence of $\overline{m}$ below and above the threshold. There are finite threshold corrections; the necessary formulae can be found in Ref. [5].

### 60.3 Lattice QCD

Ab initio lattice QCD calculations of the fundamental parameters of QCD, i.e. the coupling constant and quark masses (except for the top-quark mass), has become a precision science (see also the sections on Lattice Quantum Chromodynamics and on Quantum Chromodynamics in this Review). Here we only briefly recall those features which are required for the determination of quark masses. For more details on lattice QCD, please see the section on Lattice Quantum Chromodynamics.

The inputs into lattice calculations are bare, dimensionless versions of the fundamental parameters of QCD: the bare strong coupling and the bare quark masses in lattice units, i.e. multiplied by the lattice spacing $a$, the distance between neighboring points of the lattice. The physical values of the quark masses are determined by computing as many physical quantities as there are parameters, including one for the gauge coupling. In practice, the computations are performed for multiple lattice spacings, determined by the bare coupling via dimensional transmutation, and a number of bare quark masses. After renormalizing these masses and converting them to physical units, the resulting renormalized masses are interpolated to their physical values. This is achieved by requiring that the computed physical quantities reproduce their measured values as the $a \to 0$ limit to our continuum world is taken. Such a tuning of fundamental parameters is required in any regularization of QCD: lattice QCD is simply a tool to perform QCD calculations in the low-energy, nonperturbative domain.

As discussed in the previous paragraph, one must renormalize the bare quark masses of lattice QCD simulations. The values of these bare masses refer to a particular discretization of QCD with the lattice spacing as the ultraviolet cut-off. In order for the resulting physical masses to be useful in phenomenological applications, it is necessary to give their values in some standard
renormalization scheme such as $\overline{\text{MS}}$. Provided that both the ultraviolet cut-off $a^{-1}$ and the renormalization scale $\mu$ are much greater than $\Lambda_{\text{QCD}}$, the bare masses can be renormalized in perturbation theory. However, in order to avoid uncertainties due to the unknown higher-order coefficients in lattice perturbation theory, most results obtained recently use *nonperturbative renormalization*. This procedure relates the bare masses to those defined in renormalization schemes which can be realized directly in lattice and continuum QCD, e.g. those obtained from quark and gluon Green functions at specified momenta in the Landau gauge [6] or those defined using finite-volume techniques and the Schrödinger functional [7], but not $\overline{\text{MS}}$ that is only defined for dimensional regularization. These methods require $\mu \gg \Lambda_{\text{QCD}}$ so that unwanted (nonperturbative) corrections proportional to inverse powers of $\mu$, which appear in some approaches, remain small corrections that can be identified and removed. This condition is also necessary so that matching to other schemes can be performed reliably in perturbation theory. Moreover these methods require $a^{-1} \gg \mu$ so that cutoff effects are small enough to be extrapolated away when the continuum limit is taken. The conversion to the $\overline{\text{MS}}$ scheme is then performed using continuum perturbation theory, which is more readily obtained to higher orders and is usually better behaved than its lattice counterpart.

It is important to note that the final renormalized values for the quark masses must be the same in the continuum limit for any valid discretization of QCD and for any sensible choice of the physical quantities used for calibration, as long as the calculation is performed with a sufficient number of sea-quark flavors and with the relevant electromagnetic and strong-isospin breaking corrections for the claimed precision. It is also worth noting that issues surrounding the renormalization of quark masses disappear when considering pairwise ratios of these masses in QCD alone. Indeed, if the same renormalization scheme and scale are implemented, also for the definition of QCD in the absence or electromagnetism,<fnref n="1" /> QCD renormalization factors are identical for all quark flavors, and these factors therefore cancel exactly in quark-mass ratio.<fnref n="2" /> In particular, this means that these ratios are scheme and scale independent up to possible QED corrections. Moreover these ratios may suffer less from the uncertainties in the determination of the lattice scale, in particular in cases where the quantities used to fix the two quarks masses depend on these masses in a similar way, because these ratios are dimensionless. Thus, quark-mass ratios are often determined with significantly higher precision using lattice QCD than are the individual masses.

The determination of quark masses using lattice simulations is well established and the current emphasis is on the reduction and control of uncertainties. With better methods, improved algorithms and access to more powerful computing resources, the precision of the results has improved significantly in recent years. Vacuum polarization effects are included with $N_f = 2, 2+1$ or $N_f = 2+1+1$ flavors of sea quarks. The number 2 here indicates that the up and down quarks are degenerate. The first $+1$ corresponds to the inclusion of strange sea-quark effects and the second $+1$ to those of the charm. Simulations with $2+1$ and $2+1+1$ flavors represent controlled approximations to physical QCD at the low energies considered for quark mass determinations, up to corrections of $O((\Lambda_{\text{QCD}}/m_c)^2/N_c)$ and $O((\Lambda_{\text{QCD}}/m_b)^2/N_c)$, respectively. This is not the case for simulations with $N_f = 2$ or in which vacuum polarization effects are treated as a mean field (this is the so-called *quenched approximation*) and results obtained in those frameworks will not enter the discussion here.

Particularly pleasing is the observation that different formulations of lattice QCD and different approaches, with different systematic uncertainties, yield results which are largely consistent with each other. This gives us broad confidence in the estimates of the systematic errors. As the precision of the results approaches (or even exceeds in some cases) 1%, isospin breaking effects, including electromagnetic corrections need to be considered and this

is beginning to be done as will be discussed below. In particular, a reliable estimate of these effects is required for determining the individual $u$ and $d$ quark masses.

Members of the lattice QCD community have organized a Flavour Lattice Averaging Group (FLAG) which, against stated quality criteria, critically reviews quantities computed in lattice QCD that are relevant to flavor physics and presents its view of the current status of the results. This includes the determination of quark masses. The latest edition reviews lattice results that appeared in print before April 30th 2021 [8]. Since that deadline and the writing of the previous edition of the present review, the only new results on quark masses that have been published are those of Ref. [9].

## 60.4 Averaging procedure for lattice QCD results

In this review we present summary numbers for each quark mass which are solely based on averages of lattice results. Those summary numbers differ from those presented in the particle listings, for three reasons. The first is that they do not include non-lattice estimates of the quark masses. The second is that they may differ slightly in the lattice results retained, as made explicit here and in the listings. The third difference comes from the averaging procedures, which are described in the listings and below. In the listings, summary numbers are obtained from averages of the individual quark-mass estimates, weighted by their total errors. Moreover, the resulting one-standard-deviation errors are rescaled by the usual $\sqrt{\chi^2/dof}$ factor, which is further multiplied by 1.645 to obtain the uncertainty corresponding to a 90% confidence interval. Nevertheless, the averages presented here and in the listings are fully consistent, with comparable uncertainties, signalling the robustness of the two combination procedure.

The results on which the lattice summary numbers presented here are those used in the FLAG reviews, supplemented by any new calculations that satisfy their selection criteria. The methodology that we use is also close to that of FLAG, but here we make an attempt to separate uncertainties into their statistical and systematic components, to provide some idea of what the relative weight of each is for different quark masses. In some cases, also, slightly different choices are also made. Thus, agreement between the results obtained is a further sign of the robustness of the combination procedures used here, as well as in FLAG.

We now lay out our averaging procedure. For each of $m_{ud}$, $m_s$ or $m_s/m_{ud}$, and for each of $N_f = 2+1$ or $N_f = 2+1+1$, we perform averages of the relevant, individual lattice results with weights that include the sum in quadrature of the statistical and systematic uncertainties. The weights also account for correlations by assuming that they are 100% on any uncertainty components that may be correlated. This procedure yields a central value and a total error for each of the quantities of interest. The total uncertainty is rescaled by the usual factor of $\sqrt{\chi^2/dof}$ if that factor is larger than 1, where $\chi^2$ is the one corresponding to the weighted average procedure. To isolate the statistical component of the uncertainty on any given average, like for the total uncertainty we use the standard error formula for correlated data, this time only taking into account the statistical uncertainties on the different collaborations' results. If the total error has been rescaled, this statistical uncertainty is multiplied by the same factor. The candidate systematic error on an average is then obtained by subtracting in quadrature the statistical uncertainty from the total one. For each quantity, the final systematic error on the average is the largest of two uncertainties: the candidate systematic error and the minimum of the systematic uncertainties on the individual lattice determinations. This guarantees that the final systematic error is no smaller than the one of any given calculation. We then repeat this whole procedure to average the $N_f = 2+1$ and $N_f = 2+1+1$ averages, for reasons discussed below. In cases where the error separation is not explicit in a particular collaboration's result, we assume that the quoted error is statistical and take, as the minimum of systematic error, the smallest non-vanishing one among those of the other individual results. While this procedure may change the balance between statistical and systematic errors, it does change the average (which

<footnote n="1">¹Since QCD without QED is not the real world, it makes a difference, for example, which hadron mass is used to tune a quark mass and the QED correction will differ for different choices.</footnote>

<footnote n="2">²The same remains true in the presence of QED if the two quarks in the ratio have the same charge.</footnote>
is weighted by the total errors of the individual entries and the correlations between them) and does not decrease the total error. When quantities involving $m_c$ or $m_b$ are averaged, we consider only the averages of $N_f = 2+1+1$ calculations, for reasons also explained below.

The values of the individual results used in the averages obtained below can be found in the quark properties listings of this Review.

## 60.5 Light quarks

In this section we review the determination of the masses of the light quarks u, d and s from lattice simulations and then discuss the consequences of the approximate chiral symmetry.

### 60.5.1 Lattice QCD results

The most reliable determinations of the strange quark mass, $m_s$, and of the up and down quark masses, $m_u$ and $m_d$, are obtained from lattice simulations. This is reflected in part by the uncertainties of individual determinations of these masses given in this Review's listings of quark properties: lattice results typically have smaller error bars than phenomenological extractions. As explained in the previous section, the simulations are generally performed in the isospin-symmetric limit of QCD, i.e. with degenerate up and down quarks ($m_u = m_d$) and the electromagnetic coupling $\alpha$ set to zero. Thus, it is initially $m_{ud} = (m_u + m_d)/2$ which is obtained from the computations. The numerical values of these masses depend on the choice of renormalization scale and scheme.<fnref n="3" />

Below we discuss the derivation of $m_u$ and $m_d$ separately, but we start by briefly presenting our estimates for $m_{ud}$, $m_s$ and their ratio, using the latest lattice results. FLAG [8] bases its summary numbers on references that the authors consider to have the most reliable estimates of systematic uncertainties, i.e. [12–18] for $N_f = 2+1$ and [19–24] for $N_f = 2+1+1$ flavors of sea quarks. Note that the $N_f = 2+1+1$ results of Ref. [9] did not make the FLAG deadline. In the estimates given below, we use the fact that Ref. [9] supersedes Ref. [22].

Applying the averaging procedure described in the previous section and the results of Refs. [12–18], for $N_f = 2+1$ we obtain $\bar{m}_{ud} = (3.382 \pm 0.026 \pm 0.040[\pm 0.050])$ MeV, $\bar{m}_s = (92.24 \pm 0.24 \pm 1.00[\pm 1.03])$ MeV and $(\bar{m}_s/\bar{m}_{ud}) = 27.42 \pm 0.05 \pm 0.12[\pm 0.12]$. These numbers are $\bar{m}_{ud} = (3.421 \pm 0.035 \pm 0.049[\pm 0.060])$ MeV, $\bar{m}_s = (92.93 \pm 0.51 \pm 0.37[\pm 0.063])$ MeV and $(\bar{m}_s/\bar{m}_{ud}) = 27.25 \pm 0.03 \pm 0.07[\pm 0.08]$ using the latest $N_f = 2+1+1$ calculations [9, 19–21, 23, 24]. The masses are given in the $\overline{\text{MS}}$ scheme at a renormalization scale of 2 GeV. In these averages, the first error is statistical, the second systematic and the third one in brackets is the sum in quadrature of the first two. Also, QED corrections have been accounted for only to the extent that they have been removed from the hadron mass inputs phenomenologically.

A few comments are in order concerning the compatibility of the results which enter these averages. We begin with the case of $N_f = 2+1$. For $m_{ud}$ the results show a good degree of compatibility with a $\chi^2$/dof = 5.7/4 and the final systematic uncertainty is given by the one of Ref. [17]. The level of compatibility is similar in the case of $m_s$ and the final systematic uncertainty is also given by the one of Ref. [17]. For $m_s/m_{ud}$ the results are even more compatible and here the final systematic error is derived from the averaging procedure. In the case of $N_f = 2+1+1$, there are only two results for $m_{ud}$ [9, 20] and their level of compatibility is poor, with $\chi^2$/dof = 6.4/1. This is due to the value of Ref. [9], $m_{ud} = (3.636 \pm 0.066 \pm 0.059)$ MeV, which is significantly larger than all other determinations, including those obtained in $N_f = 2+1$ determinations.

The scenario repeats for $m_s$, but is diluted by the fact that there are four results [9, 19–21] which enter the average. However, compatibility is recovered for $m_s/m_{ud}$, suggesting a problem in the determination of the renormalization constants or the lattice spacing in at least one of the calculations. Since the different collaborations use different formulations of lattice QCD, the (relatively) small variations of the results between the groups provides important information about the reliability of the estimates. Furthermore, the conservative nature of our method for estimating uncertainties is confirmed by the excellent agreement of the averages obtained for $N_f = 2+1$ and $N_f = 2+1+1$ calculations.

Despite being reported in the $\overline{\text{MS}}$ scheme at a renormalization scale of 2 GeV, the schemes in simulations with $2+1$ and $2+1+1$ sea-quark flavors actually differ. This is because the former are renormalized with $N_L = 3$ and the latter with $N_L = 4$. Thus for a comparison one should convert the results to the same scheme. That is not the case for $(m_s/m_{ud})$, where renormalization factors cancel in the absence of electromagnetic corrections. The conversion of the $N_f = 2+1$ results to the $N_L = 4$ scheme can be performed, for instance, by running them down to the charm threshold in the $N_L = 3$ theory, matching the results to the $N_L = 4$ theory at that scale, and running them back up to 2 GeV with four active flavors. Such a conversion leads to an upward shift in the values of the quark masses of only around 0.3%, well within the quoted uncertainties on the quark masses themselves. Quark-mass ratios are not concerned by this upward shift, because it is identical for the numerator and denominator. There are also nonperturbative contributions to this procedure, of $O((\Lambda_{\text{QCD}}/m_c)^2/N_c) \sim 2\%$. However lattice QCD studies of these effects have shown that they are typically an order of magnitude smaller on hadronic quantities or on $\Lambda_{\text{QCD}}$ [25, 26]. Given that the total errors on the $N_f = 2+1$ results for $\bar{m}_{ud}$ and $\bar{m}_s$ are larger than 1%, we consider these matching effects negligible. In addition, for $m_{ud}$ in particular there are six $N_f = 2+1$ calculations in which all sources of uncertainty have been accounted for, while there are only two for $N_f = 2+1+1$. Thus, we average the averages from the two frameworks in the same way as above (see Sec. 60.4), yielding as a final lattice QCD estimate in the $\overline{\text{MS}}$ scheme at $\mu = 2$ GeV in the $N_L = 4$ theory:

$$ \bar{m}_{ud} = (3.397 \pm 0.021 \pm 0.040[\pm 0.045]) \text{ MeV}, \quad (60.4) $$

$$ \bar{m}_s = (92.74 \pm 0.22 \pm 0.49[\pm 0.54]) \text{ MeV}, \quad (60.5) $$

and

$$ r_s \equiv \frac{\bar{m}_s}{\bar{m}_{ud}} = 27.30 \pm 0.03 \pm 0.07[\pm 0.08]. \quad (60.6) $$

In these averages, the first error is statistical, the second systematic and the third one in brackets is the sum in quadrature of the first two. QED corrections have been accounted for only to the extent that they have been removed from the hadron mass inputs phenomenologically. It is worth noting that the result for $r_s$ in Eq. (60.6) is identical to the ratio of $m_s$ and $m_{ud}$ from Eqs. (60.4) and (60.5), within the four digits for $r_s$ given in Eq. (60.6), albeit with a significantly smaller total uncertainty.

To obtain the individual values of $\bar{m}_u$ and $\bar{m}_d$ requires the introduction of isospin breaking effects, including electromagnetism. This is now being done completely using lattice field theory, albeit neglecting electromagnetic effects in the sea in most cases (for an exception, see Ref. [27] which includes a calculation of the valence and sea QED contributions to $\Delta M_K^2 = M_{K^0}^2 - M_{K^+}^2$ that is critical for determining $\bar{m}_d - \bar{m}_u$). The effect of neglecting QED contributions from the sea on the u and d quark masses has been estimated in Ref. [28] to induce a contribution to the uncertainty that ranges from about 3% in $\bar{m}_u/\bar{m}_d$ to less than 1% in $\bar{m}_d$. FLAG has also reviewed results for these masses [8]. Again, they separate results obtained from $N_f = 2+1$ and $N_f = 2+1+1$ simulations. For the former, they use the results of Ref. [28] and for the latter, those of Refs. [20, 29, 30].

To obtain our estimates, we proceed as for the light quark masses above. For $N_f = 2+1$, our estimates are the results

<footnote n="3">³ If QED is not included, then quark masses also depend on the conditions used to match isospin-symmetric QCD to the low energy effective theory of the Standard Model, which includes QCD and QED with $m_u \neq m_d$. Those matching conditions, which define a matching scheme, are quite similar in the calculations reported on below. Moreover, they have been designed to minimize the impact of isospin-breaking effects on isospin-symmetric combinations of quark masses, such $m_{ud}$ and $m_s$ here, and $m_c$ and $m_b$ below. Thus, as does FLAG [8], we generally assume that any remaining differences between schemes are within the quoted errors. Such an assumption is certainly true for $\bar{m}_c(\bar{m}_c)$ [10], where QED effects have been studied explicitly. On the other hand, in the case of $m_b/m_c$, where the effects of QED have also been studied explicitly [11] and where uncertainties are less than 0.3%, these effects are no longer negligible.</footnote>
of the only calculation in which all sources of uncertainty are ac-
counted for [28]. These are $\overline{m}_u = (2.27 \pm 0.06 \pm 0.06[\pm 0.09])$ MeV,
\overline{m}_d = (4.67 \pm 0.06 \pm 0.06[\pm 0.09])$ MeV and $\overline{m}_u/\overline{m}_d = (0.485 \pm 0.011 \pm 0.016[\pm 0.020])$. For $N_f = 2 + 1 + 1$, they rely on
Refs. [20, 29, 30]. Our estimates are $\overline{m}_u = (2.14 \pm 0.04 \pm 0.07[\pm 0.08])$ MeV,
\overline{m}_d = (4.70 \pm 0.03 \pm 0.04[\pm 0.05])$ MeV and
$\overline{m}_u/\overline{m}_d = (0.460 \pm 0.008 \pm 0.018[\pm 0.020])$. These masses are
given the the $\overline{\text{MS}}$ scheme at 2 GeV in the $N_L = 3$ and 4 theories,
respectively. Given the size of the uncertainties on these aver-
ages, tiny corrections due to QED effects in the renormalization
constants have been neglected.

In obtaining the $N_f = 2 + 1 + 1$ averages, there is a significant $\chi^2/1 = 4.7$ tension between the two results for $m_u$ [20, 29] and a $\chi^2/1 = 3.4$ one between those for $m_u/m_d$ [29, 30]. These tensions are accounted for by the usual rescaling of uncertainties. However, agreement is found between the two results for $m_d$ [20, 29]. Moreover, at the level of our conservative uncertainties, agreement is found with the $N_f = 2 + 1$ results.

Again, given the small number of results and the few percent
uncertainties, we average the $N_f = 2 + 1$ and $N_f = 2 + 1 + 1$
results, as explained in Sec. 60.4, to obtain our final estimates:

$$
\overline{m}_u = (2.20 \pm 0.04 \pm 0.06[\pm 0.07]) \text{ MeV}, \quad (60.7)
$$

$$
\overline{m}_d = (4.69 \pm 0.03 \pm 0.04[\pm 0.05]) \text{ MeV}, \quad (60.8)
$$

$$
r \equiv \frac{\overline{m}_u}{\overline{m}_d} = (0.473 \pm 0.007 \pm 0.016[\pm 0.017]), \quad (60.9)
$$

where the three uncertainties have the same meaning as described
after Eq. 60.6. *r* in Eq. (60.9) is the combination of direct lat-
tice computations of that ratio in which some systematics can-
cel. The averages are given in the MS scheme at 2 GeV in the
*N*<sub>L</sub> = 4 theory and, as above, tiny corrections due to QED effects
in the renormalization constants have been neglected. Within one
standard deviation it agrees with the ratio of *m*<sub>u</sub> over *m*<sub>d</sub> from
Eqs. (60.7) and (60.8).

It is also worth noting that $m_u$ differs from zero by more than 30 times the quoted error, making a scenario in which $m_u = 0$ very unlikely. This is important because there would be no strong CP problem if $m_u$ were to vanish.

60.5.2 Chiral Perturbation Theory

For light quarks, one can use the techniques of chiral perturba-
tion theory [31–33] to extract quark mass ratios. The mass term
for light quarks in the QCD Lagrangian is

$$
\overline{\Psi} M \Psi = \overline{\Psi}_L M \Psi_R + \overline{\Psi}_R M^\dagger \Psi_L, \quad (60.10)
$$

where *M* is the light quark mass matrix,

$$
M = \begin{pmatrix} m_u & 0 & 0 \\ 0 & m_d & 0 \\ 0 & 0 & m_s \end{pmatrix}, \quad (60.11)
$$

$\Psi = (u, d, s)$, and $L$ and $R$ are the left- and right-chiral com-
ponents of $\Psi$ given by $\Psi_{L,R} = P_{L,R}\Psi$, $P_L = (1 - \gamma_5)/2$,
$P_R = (1 + \gamma_5)/2$. The mass term is the only term in the QCD
Lagrangian that mixes left- and right-handed quarks. In the limit
$M \to 0$, there is an independent SU(3)×U(1) flavor symmetry for
the left- and right-handed quarks. The vector U(1) symmetry is
baryon number; the axial U(1) symmetry of the classical theory is
broken in the quantum theory due to the anomaly. The remaining
$G_\chi = \text{SU}(3)_L \times \text{SU}(3)_R$ chiral symmetry of the QCD Lagrangian
is spontaneously broken to SU(3)$_V$, which, in the limit $M \to 0$,
leads to eight massless Goldstone bosons, the $\pi$'s, $K$'s, and $\eta$.

The symmetry $G_\chi$ is only an approximate symmetry, since it
is explicitly broken by the quark mass matrix $M$. The Goldstone
bosons acquire masses which can be computed in a systematic
expansion in $M$, in terms of low-energy constants, which are un-
known nonperturbative parameters of the effective theory, and
are not fixed by the symmetries. One treats the quark mass ma-
trix $M$ as a uniform, external field that transforms under $G_\chi$ as
$M \to LMR^\dagger$, where $\Psi_L \to L\Psi_L$ and $\Psi_R \to R\Psi_R$ are the SU(3)$_L$
and SU(3)$_R$ transformations, and writes down the most general

Lagrangian invariant under $G_\chi$. Then one sets $M$ to its given constant value Eq. (60.11), which implements the symmetry breaking.
To first order in $M$ one finds that [34]

$$
\begin{align*}
m_{\pi^0}^2 &= B(m_u + m_d), \\
m_{\pi^\pm}^2 &= B(m_u + m_d) + \Delta_{\text{em}}, \\
m_{K^0}^2 &= m_{K^0}^2 = B(m_d + m_s), \\
m_{K^\pm}^2 &= B(m_u + m_s) + \Delta_{\text{em}}, \\
m_\eta^2 &= \frac{1}{3} B(m_u + m_d + 4m_s), \tag{60.12}
\end{align*}
$$

with two unknown constants *B*, related to the light quark con-
densate, and $\Delta_{em}$, the electromagnetic mass difference. From
Eq. (60.12), one can determine the quark mass ratios [34]

$$
\begin{align}
\frac{m_u}{m_d} &= \frac{2m_{\pi^0}^2 - m_{\pi^+}^2 + m_{K^+}^2 - m_{K^0}^2}{m_{K^0}^2 - m_{K^+}^2 + m_{\pi^+}^2} = 0.56, \notag \\
\frac{m_s}{m_d} &= \frac{m_{K^0}^2 + m_{K^+}^2 - m_{\pi^+}^2}{m_{K^0}^2 + m_{\pi^+}^2 - m_{K^+}^2} = 20.2, \tag{60.13}
\end{align}
$$

to lowest order in chiral perturbation theory, with an error which
will be estimated below. Since the mass ratios extracted using
chiral perturbation theory use the symmetry transformation prop-
erty of M under the chiral symmetry $G_\chi$, it is important to use a
renormalization scheme for QCD that does not change this trans-
formation law. Any mass-independent subtraction scheme such
as $\overline{\text{MS}}$ is suitable. The ratios of quark masses are scale indepen-
dent in such a scheme (up to electromagnetic corrections), and
Eq. (60.13) can be taken to be the ratio of $\overline{\text{MS}}$ masses. Chi-
ral perturbation theory cannot determine the overall scale of the
quark masses, since it uses only the symmetry properties of M,
and any multiple of M has the same $G_\chi$ transformation law as
M.

Chiral perturbation theory is a systematic expansion in powers of the light quark masses. The typical expansion parameter is $m_K^2/\Lambda_\chi^2 \sim 0.25$ if one uses SU(3) chiral symmetry, and $m_\pi^2/\Lambda_\chi^2 \sim 0.02$ if instead one uses SU(2) chiral symmetry. Electromagnetic effects at the few percent level also break SU(2) and SU(3) symmetry. The mass formulae Eq. (60.12) were derived using SU(3) chiral symmetry, and are expected to have approximately a 25% uncertainty due to second order corrections. This estimate of the uncertainty yields results consistent with the lattice results summarized in Eq. (60.5)-(60.6).

There is a subtlety which arises when one tries to determine
quark mass ratios at second order in chiral perturbation theory.
The second order quark mass term [35]

$$
(M^\dagger)^{-1} \det M^\dagger \quad (60.14)
$$

(which can be generated by instantons) transforms in the same
way under $G_\chi$ as $M$. Chiral perturbation theory cannot distin-
guish between $M$ and $(M^\dagger)^{-1} \det M^\dagger$; one can make the replace-
ment $M \to M(\lambda) = M + \lambda M(M^\dagger M)^{-1} \det M^\dagger$ in the chiral La-
grangian,

$$
\begin{align}
M(\lambda) &= \operatorname{diag}(m_u(\lambda), m_d(\lambda), m_s(\lambda)) \tag{60.15} \\
&= \operatorname{diag}(m_u + \lambda m_d m_s, m_d + \lambda m_u m_s, m_s + \lambda m_u m_d), \nonumber
\end{align}
$$

and leave all observables unchanged.

The combination

$$
\left(\frac{m_u}{m_d}\right)^2 + \frac{1}{Q^2} \left(\frac{m_s}{m_d}\right)^2 = 1 \quad (60.16)
$$

where

$$
Q^2 = \frac{m_s^2 - m_{ud}^2}{m_d^2 - m_u^2}, \quad m_{ud} = \frac{1}{2}(m_u + m_d),
$$

is insensitive to the transformation in Eq. (60.15). Eq. (60.16)
gives an ellipse in the $m_u/m_d-m_s/m_d$ plane. The ellipse is well-
determined by chiral perturbation theory, but the location on the
ellipse, and the absolute normalization of the quark masses, have larger uncertainties.

A leading-order result for $Q$ in SU(3) chiral perturbation theory can be derived using Eq. (60.12) and the values for the relevant meson masses. This result actually holds to next-to-leading order, yielding $Q^{\text{NLO}}=24.3$. Phenomenologically, the preferred way to determine $Q$ is from $\eta \to 3\pi$ decay, giving the smaller value $Q = 22.1(7)$ [36]. Lattice QCD collaborations have also reported determinations of $Q$. Using $N_f = 2+1$ simulations, Ref. [28] obtains $Q = 23.4 \pm 0.6$ and Ref. [29] determines $Q = 23.8 \pm 1.1$ with $N_f = 2+1+1$ simulations, results which are fully compatible. These results are also compatible with $Q = 23.4 \pm 0.5$, obtained from $Q^2 = (r_s^2 - 1)(1+r)/[4(1-r)]$ and from our lattice averages for $r_s$ and $r$ from Eqs. (60.6) and (60.9), respectively. Given the size of the uncertainties on the lattice results for $Q$, it is safe to neglect effects due to the definition of isosymmetric QCD (see footnote 3). On the whole, the lattice results are larger than the one from phenomenology [36]. This difference could be due to surprisingly large corrections to the NLO substitution of meson masses by $Q$ in the phenomenological determination, as suggested by the authors of [36].

The absolute normalization of the quark masses cannot be determined using chiral perturbation theory. Other methods, such as lattice simulations discussed above, or spectral function sum rules [37, 38] for hadronic correlation functions reviewed next, are necessary.

### 60.5.3 Sum rules

Sum rule methods have been used extensively to determine quark masses and for illustration we briefly discuss here their application to hadronic $\tau$ decays [39]. Other applications involve very similar techniques.

<figure><img src="image_311.png" /><figcaption>**Figure 60.1:** The analytic structure of $\Pi(s)$ in the complex $s$-plane. The contours $C_1$ and $C_2$ are the integration contours discussed in the text, and the integral over the closed contour $C_1+C_2$ vanishes. $m_\tau^2$ has not been drawn to scale; $m_\tau^2 \sim 40(4m_\pi^2)$.</figcaption></figure>

The experimentally measured quantity is $R_\tau$,

$$ \frac{dR_{\tau}}{ds} = \frac{d\Gamma/ds (\tau^{-} \rightarrow \text{hadrons} + \nu_{\tau}(\gamma))}{\Gamma(\tau^{-} \rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}(\gamma))} \qquad (60.17) $$

the hadronic invariant mass spectrum in semihadronic $\tau$ decay, normalized to the leptonic $\tau$ decay rate. It is useful to define $q$ as the total momentum of the hadronic final state, so $s = q^2$ is the hadronic invariant mass. The total hadronic $\tau$ decay rate $R_\tau$ is then given by integrating $dR_\tau/ds$ over the kinematically allowed range $0 \le s \le M_\tau^2$.

$R_\tau$ can be written as

$$ R_{\tau} = 12\pi \int_{0}^{M_{\tau}^{2}} \frac{ds}{M_{\tau}^{2}} \left(1 - \frac{s}{M_{\tau}^{2}}\right)^{2} \times \left[ \left(1 + 2\frac{s}{M_{\tau}^{2}}\right) \text{Im} \Pi^{T}(s) + \text{Im} \Pi^{L}(s) \right] \quad (60.18) $$

where the hadronic spectral functions $\Pi^{L,T}$ are defined from the time-ordered correlation function of two weak currents ($j^\mu(x)$ and

$j^\nu(0)$) by

$$ \Pi^{\mu\nu}(q) = i \int d^4 x \, e^{iq \cdot x} \langle 0 | T (j^\mu(x) j^\nu(0)^\dagger) | 0 \rangle, \qquad (60.19) $$

$$ \Pi^{\mu\nu}(q) = (-g^{\mu\nu} + q^{\mu}q^{\nu}) \Pi^{T}(s) + q^{\mu}q^{\nu}\Pi^{L}(s), \qquad (60.20) $$

and the decomposition Eq. (60.19) is the most general possible structure consistent with Lorentz invariance.

By the optical theorem, the imaginary part of $\Pi^{\mu\nu}$ is proportional to the total cross-section for the current to produce all possible states. A detailed analysis including the phase space factors leads to Eq. (60.18). The spectral functions $\Pi^{L,T}(s)$ are analytic in the complex $s$ plane, with singularities along the real axis. There is an isolated pole at $s = m_\pi^2$, and single- and multi-particle singularities for $s \ge 4m_\pi^2$, the two-particle threshold. The discontinuity along the real axis is $\Pi^{L,T}(s+i0^+) - \Pi^{L,T}(s-i0^+) = 2i\text{Im} \Pi^{L,T}(s)$. As a result, Eq. (60.18) can be rewritten with the replacement $\text{Im} \Pi^{L,T}(s) \to -i\Pi^{L,T}(s)/2$, and the integration being over the contour $C_1$. Finally, the contour $C_1$ can be deformed to $-C_2$ without crossing any singularities, and so leaving the integral unchanged, i.e. the integral over the closed contour $C_1+C_2$ vanishes. One can derive a series of sum rules analogous to Eq. (60.18) by weighting the differential $\tau$ hadronic decay rate by different powers of the hadronic invariant mass [40],

$$ R_{\tau}^{kl} = \int_{0}^{M_{\tau}^{2}} ds \left(1 - \frac{s}{M_{\tau}^{2}}\right)^{k} \left(\frac{s}{M_{\tau}^{2}}\right)^{l} \frac{dR_{\tau}}{ds} . \qquad (60.21) $$

This leads to the final form of the sum rule(s),

$$ R_{\tau}^{kl} = -6\pi i \int_{C_2} \frac{ds}{M_{\tau}^2} \left(1 - \frac{s}{M_{\tau}^2}\right)^{2+k} \left(\frac{s}{M_{\tau}^2}\right)^l \times \left[ \left(1 + 2\frac{s}{M_{\tau}^2}\right) \Pi^T(s) + \Pi^L(s) \right] . \qquad (60.22) $$

The manipulations so far are completely rigorous and exact, relying only on the general analytic structure of quantum field theory. The left-hand side of the sum rule Eq. (60.22) is obtained from experiment. The right hand-side can be computed for $s$ far away from any physical cuts using the operator product expansion (OPE) for the time-ordered product of currents in Eq. (60.18), and QCD perturbation theory. The OPE is an expansion in a series of local operators, and is an expansion about the $q \to \infty$ limit. It gives $\Pi^{L,T}(s)$ as an expansion in powers of $\alpha_s(s)$ and $\Lambda_{\text{QCD}}^2/s$, and is valid when $s$ is far (in units of $\Lambda_{\text{QCD}}^2$) from any singularities in the complex $s$-plane.

The OPE gives $\Pi^{L,T}(s)$ as a series in $\alpha_s$, quark masses, and various nonperturbative vacuum matrix elements. By computing $\Pi^{L,T}(s)$ theoretically, and comparing with the experimental values of $R_\tau^{kl}$, one determines various parameters such as $\alpha_s$ and the quark masses. The theoretical uncertainties in using Eq. (60.22) arise from neglected higher order corrections (both perturbative and nonperturbative), and because the OPE is no longer valid near the real axis, where $\Pi^{L,T}$ have singularities. The contribution of neglected higher order corrections can be estimated as for any other perturbative computation. The error due to the failure of the OPE is more difficult to estimate. In Eq. (60.22), the OPE fails on the endpoints of $C_2$ that touch the real axis at $s = M_\tau^2$. The weight factor $(1-s/M_\tau^2)$ in Eq. (60.22) vanishes at this point, so the importance of the endpoint can be reduced by choosing larger values of $k$.

Light quark masses are often determined using QCD sum rules [38], which are similar to the $\tau$ sum rules. One takes the correlator of two light-quark-bilinear operators (e.g. an axial vector current), as in Eq. (60.18), and computes their Laplace transforms or moments

$$ \mathcal{L}_n(\tau) = \int_0^\infty ds \, s^n e^{-\tau s} \text{Im} \Pi(s), \\ \mathcal{M}_n(Q^2) = \int_0^\infty \frac{ds}{(s+Q^2)^n} \text{Im} \Pi(s) $$
to get Laplace or moment sum rules, respectively. The quark masses are extracted by comparing the theoretical and experimental values of $L_n(\tau)$ and $M_n(Q^2)$. Considerable theoretical effort has gone into optimizing $n$ and $Q^2$ to improve the precision of the resulting light quark masses.

## 60.6 Heavy quarks

### 60.6.1 Continuum approaches and results

For heavy quark physics one can exploit the fact that $m_Q \gg \Lambda_{QCD}$ to construct effective theories ($m_Q$ is the mass of the heavy quark $Q$). The masses and decay rates of hadrons containing a single heavy quark, such as the $B$ and $D$ mesons can be determined using the heavy quark effective theory (HQET) [41]. The theoretical calculations involve radiative corrections computed in perturbation theory with an expansion in $\alpha_s(m_Q)$ and nonperturbative corrections with an expansion in powers of $\Lambda_{QCD}/m_Q$. Due to the asymptotic nature of the QCD perturbation series, the two kinds of corrections are intimately related; an example of this are renormalon effects in the perturbative expansion which are associated with nonperturbative corrections.

Systems containing two heavy quarks such as the $\Upsilon$ or $J/\Psi$ are treated using non-relativistic QCD (NRQCD) [42]. The typical momentum and energy transfers in these systems are $\alpha_s m_Q$, and $\alpha_s^2 m_Q$, respectively, so these bound states are sensitive to scales much smaller than $m_Q$. However, smeared observables, such as the cross-section for $e^+e^- \to \bar{b}b$ averaged over some range of $s$ that includes several bound state energy levels, are better behaved and only sensitive to scales near $m_Q$. For this reason, most determinations of the $c, b$ quark masses using perturbative calculations compare smeared observables with experiment [43–45]. The method is similar to that outlined for $\tau$ decays. The current correlator in Eq. (60.18) is the electromagnetic current, and the experimental data is the value of $R(s)$ in the threshold region for $e^+e^- \to Q\bar{Q}$. The theoretical values for the moments are computed using renormalization group improved calculations in non-relativistic QCD.

There are many continuum extractions of the $c$ and $b$ quark masses, some with quoted errors of 10 MeV or smaller. There are systematic effects of comparable size, which are typically not included in these error estimates. Reference [46], for example, shows that even though the error estimate of $m_c$ using the rapid convergence of the $\alpha_s$ perturbation series is only a few MeV, the central value of $m_c$ can differ by a much larger amount depending on which algorithm (all of which are formally equally good) is used to determine $m_c$ from the data. This leads to a systematic error from perturbation theory of around 20 MeV for the $c$ quark and 25 MeV for the $b$ quark. Electromagnetic effects, which also are important at this precision, are often not included. For this reason, we inflate the errors on the continuum extractions of $m_c$ and $m_b$. The average values of $m_c$ and $m_b$ from continuum determinations are (see Sec. 60.7 for the 1S scheme)

$$ \begin{aligned} \bar{m}_c(\bar{m}_c) &= (1.280 \pm 0.025)\text{GeV}, \\ \bar{m}_b(\bar{m}_b) &= (4.18 \pm 0.03)\text{GeV}, \quad m_b^{1S} = (4.65 \pm 0.03)\text{GeV}. \end{aligned} $$

### 60.6.2 Lattice approaches and results

Lattice QCD simulations of a heavy quark $Q$, described in a relativistic fermion formulation, lead to potentially large discretization errors which are powers of $am_Q$ (modulated by logarithms); the leading power depends on the formulation of lattice QCD being used and in most cases is quadratic. Clearly those errors can be reduced by performing simulations at smaller lattice spacings $a$, but also by using improved discretizations of the theory, in which the leading discretization errors are systematically eliminated. Recently, with more powerful computing resources, better algorithms and improved discretizations of QCD, it has become possible to perform simulations with quarks more massive than the charm and even up to the $b$ [11], decreasing or eliminating the extrapolation which has to be performed to reach the $m_b$. It is worth noting that computations using a relativistic fermion formulation for the valence charm with $N_f = 2+1$ sea-quark flavors, or with $N_f = 2+1+1$ flavors and a relativistic $b$, suffer from partial-quenching effects. This is because the heavy valence

quarks are absent from the sea. While these effects are not expected to be large in $N_f = 2+1+1$ computations, they should be investigated when quoting results below the percent level in the case of $N_f = 2+1$ sea-quark flavors. For that reason we will omit heavy-quark results obtained with $N_f = 2+1$ sea-quark flavors from our averages and, more generally focus on $N_f = 2+1+1$ results. Note that these partial-quenching effects are accounted for systematically in lattice calculations in which the heavy valence quarks are described by HQET or NRQCD.

Traditionally the charm quark mass is obtained by tuning its bare, simulation input value to reproduce the physical mass of charmonium mesons or of the $D$, $D_s$ mesons (requiring a more precise tuning of the light quark masses and the computation of larger QED effects in some cases). This mass can then be renormalized and matched to the $\overline{\text{MS}}$ scheme using the methods discussed for the light quarks.

An alternative approach for obtaining the $\overline{\text{MS}}$ mass was proposed in [47]. Euclidean-time moments of pseudoscalar, two-point functions of $c\bar{c}$ quark-bilinear operators can readily be computed on the lattice and extrapolated to the continuum limit where they can be compared to perturbative calculations of the same quantities at 4-loop order. In this way, both the strong coupling constant and the charm quark mass can be determined with remarkably small errors. As this approach uses the same perturbative expressions for two-point correlators as the continuum determinations discussed in Sec. 60.6.1, it suffers from similar perturbation-theory, systematic errors. FLAG [8] has reviewed lattice determinations of the charm-quark mass obtained using both approaches. For their $N_f = 2+1$ average they retain the results [13, 48–50], with [51] being published after their deadline. As stated above, we focus here on $N_f = 2+1+1$ results. For these, FLAG uses the results of Refs. [10, 19, 20, 22, 52].

Since the FLAG review, Refs. [22, 52] have been superseded by Ref. [9]. In that reference, only $\bar{m}_c(3\text{GeV})$ is given and we multiply it by the running factor $0.7739 \pm 0.0060$ from $\mu = 3\text{GeV}$ to $\bar{m}_c$ [8]. With this substitution and using the combination method described in Sec. 60.4, we obtain:

$$ \bar{m}_c(\bar{m}_c) = (1.275 \pm 0.004 \pm 0.008[\pm 0.009])\text{GeV} \quad (60.23) $$

where the three uncertainties have the same meaning as described after Eq. (60.6). This result is in good agreement with the continuum result given above, but with a smaller error.

It is worth noting that three [10, 19, 20] of the four results entering this average agree, while the other [9] is over 2 standard deviations larger than the average. This is taken into account by stretching errors in the usual way, but the total uncertainty on the lattice average of $\bar{m}_c$ will not significantly be reduced until this discrepancy is resolved. It should also be noted that [10] includes QED directly in the lattice calculation, albeit only for valence quarks, while some of the other computations account for these effects using phenomenology. For $\bar{m}_c(\bar{m}_c)$ considered above, these effects are below the permil [10] and are therefore neglected here.

Historically, the main approach to controlling the discretization errors in lattice studies of b-quark physics was to perform simulations of effective theories such as HQET and NRQCD. This remains an important technique, both in its own right and in providing additional information for extrapolations from lower masses to the bottom region. Using effective theories, $m_b$ is obtained from what is essentially a computation of the difference of $M_{H_b} - m_b$, where $M_{H_b}$ is the mass of a hadron $H_b$ containing a b quark. The relative error on $m_b$ is therefore much smaller than that for $M_{H_b} - m_b$. The principal systematic errors are the matching of the effective theories to QCD and the presence of power divergences in $a^{-1}$ in the $1/m_b$ corrections which have to be subtracted numerically. A procedure for performing these subtractions fully nonperturbatively was proposed and implemented for the first time in [53].

The most recent $N_f = 2+1+1$ lattice QCD determinations of the b quark mass rely on a variety of approaches, including Euclidean-time moments of correlation functions with [54] or without NRQCD [19] and interpolations [55, 56] (using results from HQET simulations) or extrapolations [11, 20] from above the
<figure><img src="image_312.png" /><figcaption>**Figure 60.2:** The allowed region (shown in white) for up quark and down quark masses renormalized in the $\overline{\text{MS}}$ scheme at 2 GeV, determined from the allowed ranges of mass parameters (see Fig. 60.3). The parameter $(m_u + m_d)/2$ yields the two downward-sloping white lines, while $m_u/m_d$ yields the two rising white lines originating at (0, 0). The white region is the remaining allowed parameter space after applying the limits on $m_u$, $m_d$, $(m_u + m_d)/2$ and $m_u/m_d$, where those limits are 90% confidence levels including the scale factors shown in the ideograms in the Data Listings.</figcaption></figure>

charm mass to that of the b. One calculation [11] even includes simulations performed with a heavy quark whose mass is very near that of the b quark. The overall agreement of the results obtained using these very different approaches, which have different systematic errors, is a confirmation that the various groups control these uncertainties. As the range of heavy-quark masses which can be used in numerical simulations increases, the extrapolation or interpolation of results to the physical b mass, possibly including some obtained directly at that mass, are becoming ever more reliable (see e.g. Refs. [11, 20]).

Using the results from Refs. [11, 19, 20, 55, 56], converting those given in the $N_L = 4$ theory to the $N_L = 5$ one and applying the averaging method described in Sec. 60.4, we obtained the estimate:

$$ \bar{m}_b(\bar{m}_b) = (4.196 \pm 0.009 \pm 0.009[\pm 0.012]) \text{ GeV}, \quad (60.24) $$

where the three uncertainties have the same meaning as in the estimates given above for light-quark masses.

Among the results entering this average, only one [11] accounts for electromagnetic effects. Using Refs. [10, 11], it is straightforward to show that QED effects on the value of $\bar{m}_b$ at $\mu = 3$ GeV,

and thus at $\mu = \bar{m}_b$, are smaller than one permil in the separation scheme of those two references. Assuming that the same is true for the other results, we neglect these corrections here.

The lattice result of Eq. (60.24) is compatible with the average value of continuum results, but with a significantly smaller uncertainty. Note that FLAG [8] also provides an average of $N_f = 2+1$ results, taken from [13, 50]. It is about one combined, standard deviations lower than the result quoted above.

As explained in Sec. 60.3, ratios of quark masses can have significantly smaller errors than the individual masses if they are determined within the same lattice QCD framework. This led HPQCD to leverage their precise determination of $m_c$ [47] to determine $m_s$ and $m_{ud}$ [57], through a precise computation of $m_c/m_s$ [57] and of $m_s/m_{ud}$ [58]. The $N_f = 2+1$ calculation [57] was updated using $N_f = 2+1+1$ simulations in [19]. The ratio $m_c/m_s$ was also computed in Refs [48, 59] with $N_f = 2+1$ simulations and in Refs. [20, 22] with $N_f = 2+1+1$ ones. On the basis of those references, FLAG [8] gives average results from $N_f = 2+1$ and $N_f = 2+1+1$ simulations. Here we focus on $N_f = 2+1+1$ results and we replace the results of Ref. [22] by the collaboration's update [9]. Applying to the modern $N_f = 2+1+1$ computations of
<figure><img src="image_313.png" /><figcaption>Figure 60.3: The allowed regions (shown in white) for quark masses renormalized in the $\overline{\text{MS}}$ scheme at 2 GeV. The points (taken from the Data Listings) are in chronological order with the more recent measurements shown at the top. The white regions indicate values allowed at the 90% confidence level including the scale factors shown in the ideograms in the Data Listings.</figcaption></figure>

Refs. [9, 19, 20] the combination procedure described in Sec. 60.4, we obtain the following lattice average:

$$ \overline{m}_c/\overline{m}_s = 11.769 \pm 0.021 \pm 0.028[\pm 0.035], $$

where the three uncertainties have the some meaning as described after Eq. 60.6. The average is dominated by Ref. [20] that has a substantially smaller error than the other results. The result of Ref. [9] is a little more that 1 standard deviation below that average and the one of Ref. [19] close to $2\sigma$ above, for a total $\chi^2$/dof = 5.0/2 and a rescaling factor of ~ 1.6 in the statistical and systematic errors.

The above ratio is renormalization scheme and scale independent, in the absence of electromagnetic corrections. None of the calculations above estimates the size of those corrections. Nevertheless, they are expected to still be smaller than the total uncertainty quoted above in a standard scheme for defining QCD in the isospin limit.

The ratio $m_b/m_c$ has also been computed on the lattice. The two $N_f = 2 + 1$ results which satisfy FLAG's criteria are [13, 50].

As explained above, we focus on $N_f = 2 + 1 + 1$ calculations [11, 20, 55]. Ref. [11] is the only calculation which accounts for QED directly in the lattice simulations, albeit only for valence quarks, and thus must specify the renormalization scheme and scale of its result, which it chooses to be $\overline{\text{MS}}$ at 3 GeV. Here we correct the results of Refs. [20, 55] by the QED factor 1.0017 obtained in Ref. [11], and ascribe to it a 100% systematic uncertainty with 100% correlation. This means that we assume that the differences due to the separation schemes used in each calculation lies within that error. Averaging the resulting numbers, as prescribed in Sec. 60.4, yields

$$ \overline{m}_b/\overline{m}_c = 4.584 \pm 0.006 \pm 0.010[\pm 0.012], $$

where exceptionally the bar indicates that the masses are renormalized in the $\overline{\text{MS}}$ at 3 GeV in the four-flavor theory. Here a scale factor of $\sqrt{\chi^2}$/dof = 1.30 has been applied to the error bar. Indeed, [55] contributes 3.3 to the total $\chi^2$.
### 60.6.3 Warnings concerning the use of the pole mass

For an observable particle such as the electron, the position of the pole in the propagator is the definition of its mass. In QCD this definition of the quark mass is known as the pole mass. It is known that the on-shell quark propagator has no infrared divergences in perturbation theory [60,61], so this provides a perturbative definition of the quark mass. However, the pole mass cannot be used to arbitrarily high accuracy because of nonperturbative infrared effects in QCD. In fact the full quark propagator has no pole because the quarks are confined, so that the pole mass cannot be defined outside of perturbation theory. The relation between the pole mass $m_Q$ and the $\overline{\text{MS}}$ mass $\overline{m}_Q$, used throughout this review, is known to three loops [62–65]

$$ m_Q = \overline{m}_Q(\overline{m}_Q) \left\{ 1 + \frac{4\bar{\alpha}_s(\overline{m}_Q)}{3\pi} \right. \quad (60.25) \\ \left. + \left[ -1.0414 \sum_q \left( 1 - \frac{4}{3} \frac{\overline{m}_q}{\overline{m}_Q} \right) + 13.4434 \right] \left[ \frac{\bar{\alpha}_s(\overline{m}_Q)}{\pi} \right]^2 \right. \\ \left. + \left[ 0.6527 N_L^2 - 26.655 N_L + 190.595 \right] \left[ \frac{\bar{\alpha}_s(\overline{m}_Q)}{\pi} \right]^3 \right\}, $$

where $\bar{\alpha}_s(\mu)$ is the strong interaction coupling constants in the $\overline{\text{MS}}$ scheme, and the sum over $q$ extends over the $N_L$ flavors lighter than $Q$. The complete mass dependence of the $\alpha_s^2$ term can be found in [62]; the mass dependence of the $\alpha_s^3$ term is not known. For the b-quark, Eq. (60.25) reads

$$ m_b = \overline{m}_b(\overline{m}_b) [1 + 0.10 + 0.05 + 0.03], \quad (60.26) $$

where the contributions from the different orders in $\alpha_s$ are shown explicitly. The two and three loop corrections are comparable in size and have the same sign as the one loop term. This is a signal of the asymptotic nature of the perturbation series (there is a renormalon in the pole mass [66]). Such a badly behaved perturbation expansion can be avoided by directly extracting, from data, the mass defined in the $\overline{\text{MS}}$ (used in this review) or other short-distance schemes (see below), without invoking the pole mass as an intermediate step.

## 60.7 Numerical values and caveats

The quark masses in the particle data listings have been obtained by using a wide variety of methods. Each method involves its own set of approximations and uncertainties. In most cases, the errors are an estimate of the size of neglected higher-order corrections or other uncertainties. The expansion parameters for some of the approximations are not very small (for example, they are $m_K^2/\Lambda_\chi^2 \sim 0.25$ for the SU(3) chiral expansion and $\Lambda_{QCD}/m_b \sim 0.1$ for the heavy-quark expansion), so an unexpectedly large coefficient in a neglected higher-order term could significantly alter the results. Thus, before using a particular result, it is important to understand the possible limitations of the approach used to obtain it. It is also important to note that the quark mass values can be significantly different in the different schemes.

We have specified all masses in the $\overline{\text{MS}}$ scheme. For light quarks, the renormalization scale has been chosen to be $\mu = 2$ GeV. Quoting these masses at smaller values of $\mu$, where perturbative corrections become significantly larger, would introduce unnecessary uncertainties in the results. In fact, as lattice calculations, performed on finer and finer lattices, allow to determine quark masses, fully nonperturbatively, at larger and larger values of $\mu$, it may become advantageous to quote quark mass results at renormalization scales above 2 GeV, where perturbative uncertainties are smaller.

The heavy quark masses obtained using HQET, QCD sum rules, or lattice gauge theory are consistent with each other if they are all converted into the same scheme and scale. For these quarks it is conventional to choose the renormalization scale equal to the quark mass, so we have quoted $\overline{m}_Q(\mu)$ at $\mu = \overline{m}_Q$ for the c and b quarks. Given the small size of the charm quark mass, in the future it may become advantageous to quote its value at larger values of $\mu$ so as not to introduce unnecessary perturbative uncertainties (see discussion above). Analyses of inclusive

$B$ meson decays have shown that other mass definitions lead to a better behaved perturbation series than for the $\overline{\text{MS}}$ mass, and hence to more accurate mass values [67–70]. Thus, we have chosen to also give values for one of these, the b quark mass in the 1S scheme [67, 68]. Other schemes that have been proposed are the PS-scheme [69], the kinetic scheme [71] and, most recently, the minimal renormalon-subtracted mass (MRS) [70] used in the lattice calculation [20].

If necessary, we have converted values in the original papers to our chosen scheme using two-loop formulæ. It is important to realize that our conversions introduce significant additional errors. In converting to the $\overline{\text{MS}}$ b-quark mass, for example, the three-loop conversions from the 1S and pole masses give values about 35 MeV and 135 MeV lower than the two-loop conversions. The uncertainty in $\alpha_s(M_Z) = 0.1179 \pm 0.0010$ [1] gives an uncertainty of $\pm 9$ MeV and $\pm 21$ MeV respectively in the same conversions. We have not added these additional errors when we do our conversions. The $\alpha_s$ value in the conversion is correlated with the $\alpha_s$ value used in determining the quark mass, so the conversion error is not a simple additional error on the quark mass.

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# 61. Top Quark

Revised September 2023 by T.M. Liss (City Coll. of New York), F. Maltoni (CP3 U. catholique de Louvain; Bologna U.) and A. Quadt (Göttingen U.).

## 61.1 Introduction and SM theory overview

In the Standard Model (SM), the left-handed top quark is the $Q = 2/3$, $T_3 = +1/2$ member of the weak-isospin doublet containing the bottom quark, while the right-handed top is an $SU(2)_L$ singlet (see, e.g., the review "Electroweak Model and Constraints on New Physics" here). Its phenomenology is driven by its large mass, $m_t \simeq v/\sqrt{2}$, where $v = 246$ GeV is the vacuum expectation value of the Higgs field, i.e., it is the only quark whose Yukawa coupling to the Higgs boson is of order unity. As it is heavier than the W boson, it is the only quark that decays semi-weakly into a real W boson and a b quark. This results in a lifetime that is shorter than the time that is needed for strong interactions to modify its properties or to bind it into a hadron. In this respect, the top is the only quark that, in its brief life, behaves as a free one. Because of its large mass and Yukawa coupling to the Higgs boson, it also gives important contributions (via loops) to Standard Model precision observables measured at lower scales.

For these reasons, the top quark provides a unique laboratory to test our understanding of matter and fundamental interactions at the ElectroWeak (EW) symmetry-breaking scale and beyond. In addition, since it is abundantly produced at the Large Hadron Collider (LHC), a precise knowledge of its properties (mass, couplings, production cross sections, decay branching ratios, etc.) can be achieved. This review provides a concise discussion of the experimental and theoretical issues involved in the determination of the top-quark properties.

### 61.1.1 Properties

#### 61.1.1.1 Mass

The top quark mass determines to a large extent the unique phenomenology of this quark. In the SM quark masses are derived parameters from the Yukawa coupling and the vacuum expectation value. At the loop level, with the Higgs boson discovery and per mil measurement of its mass, the values of the W-boson and top quark mass are correlated, so that their precise determinations provides a strong test of the SM (see Section 10 "Electroweak Model and constraints on new physics" of this Review). At present there is some tension at the $1.7\sigma$ level, between the indirect top quark (pole) mass determination from electroweak precision data (176.12±1.9 GeV) and the combination of direct measurements at the LHC that yields 172.52±0.14(stat.) ± 0.30(syst.) GeV [1].

The top mass value is also critical in the issue of vacuum stability in the SM [2–4]. At high scales, the Higgs quartic coupling $\lambda$ evolves to increasingly small values as $m_t$ grows. Above $m_t = 171$ GeV, i.e., very close to the most precise measurements, $\lambda$ becomes negative at the Planck scale leading to a meta-stable electroweak vacuum, while for slightly larger values, $m_t > 176$ GeV the electroweak vacuum would become unstable. Current top-quark measurements therefore point to a Higgs quartic coupling which is nearly vanishing at the Planck scale. While being quite suggestive, in absence of a clear UV picture this argument allows only to conclude that comparison between our best SM predictions and the data does not imply new physics below the Planck scale (see Section 11.2.3 of "Status of Higgs Boson Physics" in this Review).

Given its importance and the large samples of top quarks produced at the LHC, the top mass determination has been one of the most important goals in the precision measurement campaign of the ATLAS and CMS collaborations. The basic methodology rests on the idea of fitting $m_t$-dependent kinematic distributions to fully-exclusive (Monte Carlo) predictions, via the full or partial reconstruction of the system of the $t$ and $\bar{t}$ decay products. These are called "direct measurements" and aim at a target accuracy in the permil range. With an absolute uncertainty of order or better than $\Lambda_{QCD}$, however, a clear relation between the extracted mass and a well defined quantum field theory parameter of the underlying theory has become necessary.

Several mass parameter definitions exist with a precise field-

theoretical meaning, that can be organised in two broad classes, the long-distance ones, such as the "pole mass", and short-distance masses, such as $\overline{\text{MS}}$ mass. A mass definition, the MSR mass [5], exists that interpolates between the two. The one measured through direct measurements is sometimes referred to as "Monte Carlo" mass. Several other observables have been put forward that could be expressed in terms of well-defined masses and then overcome the problems of the direct measurements, yet with a target accuracy not quite competitive with the direct determination.

A top mass cannot be defined solely in terms of the mass of the system of its decay products: the top quark is a colored object, therefore it cannot be an asymptotic state of the theory and no final-state hadronic system can be unambiguously associated with it. On the other hand, final state observables such as those arising from its decay, can be related to the top mass through computations, which have perturbative as well as non-perturbative components, as Monte Carlo generators.

From a purely theoretical (calculational) point of view, the top mass parameter is defined within a given renormalization scheme, since divergent perturbative corrections arise order by order in perturbation theory, and need to be subtracted. The pole mass scheme prescribes to subtract the divergent mass corrections so that the pole in the quark propagator remains fixed order by order in perturbation theory. The $\overline{\text{MS}}$ scheme prescribes to employ dimensional regularization and subtract the pure $1/\epsilon$ pole in the divergent mass correction. In doing so, the corresponding mass becomes scale dependent. In this scheme the pole in the top propagator receives corrections at all orders in perturbation theory. This scheme has the advantage that it makes the relation between the mass and the Yukawa coupling straightforward in the SM and is independent from non-perturbative corrections.

The relation between the top pole mass $m_t^{\text{pole}}$ and $m_t^{\overline{\text{MS}}}$ up to four loops reads [6]

$$m_t^{\text{pole}} = m_t^{\overline{\text{MS}}}(m_t) [1 + 0.4244 \alpha_s + 0.8345 \alpha_s^2 + 2.375 \alpha_s^3 + (8.49 \pm 0.25) \alpha_s^4], \quad (61.1)$$

which for $\alpha_s = \alpha_s^{(6)}(m) = 0.1088$ and $m_t^{\text{pole}} = 172.5$ GeV gives $m_t^{\overline{\text{MS}}}(m_t) = 162.69 \pm 0.006$ GeV. The two definitions lead to perturbatively equivalent theories: a perturbative expression of the pole mass in terms of the $\overline{m}^{\overline{\text{MS}}}$ mass can be translated to a physical result in the pole mass scheme into the corresponding physical result in the $\overline{\text{MS}}$ scheme. However, the relation above has only a perturbative meaning, as the two masses differ by non-perturbative, long-distance effects. The difference between the top pole mass and the mass extracted in direct measurements, $\Delta m_{\text{MC}}$, arises due to non-perturbative effects, that are currently modelled by shower Monte Carlo programs. It is expected to be of order $Q_0 \cdot \alpha_s(Q_0)$ with $Q_0 \sim 1$ GeV and estimated of order 0.5 GeV. An overview on the on-going efforts to precisely define and to estimate the uncertainties of the top mass can be found in Refs. [7, 8].

#### 61.1.1.2 Couplings

The SM couplings involving top quarks are of two types: gauge couplings, which are universal, and Yukawa couplings, which instead depend on the generation. Following the notation of Sections 9 and 10 of this Review, after EWSB and in the mass eigenstate basis, top quark interactions read

$$ \begin{aligned} \mathcal{L}_t^{\text{SM}} ={}& \bar{\psi}_t [i\not\partial - m_t] \psi_t - g_s \bar{\psi}_t \gamma^\mu t^C \psi_t A_\mu^C - eQ_t \bar{\psi}_t \gamma^\mu \psi_t A_\mu \\ & - \frac{g}{2\sqrt{2}} \bar{\psi} \gamma^\mu (1 - \gamma^5) (T^+ W_\mu^+ + T^- W_\mu^-) \Psi \\ & - \frac{g}{2 \cos \theta_W} \bar{\psi}_t \gamma^\mu (g_V^t - g_A^t \gamma^5) \psi_t Z_\mu \frac{m_t}{v} H \bar{\psi}_t \psi_t, \end{aligned} \quad (61.2) $$

where in the first line we include the coupling to the $SU(3)$ field $A_\mu$, and $\Psi = (t, b')$ with $b' = V_{tb}b + V_{ts}s + V_{td}d$. There are no neutral flavor changing interactions in the SM at tree level.

In absence of evidence for physics beyond the SM at the weak scale or below, the Standard Model Effective Field Theory (SMEFT) [9–11] provides a simple and consistent framework
<table><caption>**Table 61.1:** List of SMEFT operators at dimension six, assuming $U(2)_q \times U(2)_u \times U(2)_d$ flavor symmetry. For more details on notation and conventions, see reference in the text.</caption><thead><tr><th>Operator</th><th>Field content</th><th>Operator</th><th>Field content</th></tr></thead><tbody><tr><td colspan="4" style="text-align:center; font-weight:bold;">Four-quark</td></tr><tr><td>O<sup>1(iijkl)</sup><sub>qq</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>q<sub>j</sub>)(q&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>q<sub>l</sub>)</td><td>O<sup>1(iijkl)</sup><sub>lq</sub></td><td>(l&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>l<sub>j</sub>)(q&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>q<sub>l</sub>)</td></tr><tr><td>O<sup>3(iijkl)</sup><sub>qq</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>&tau;<sup>I</sup>q<sub>j</sub>)(q&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>&tau;<sup>I</sup>q<sub>l</sub>)</td><td>O<sup>3(iijkl)</sup><sub>lq</sub></td><td>(l&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>&tau;<sup>I</sup>l<sub>j</sub>)(q&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>&tau;<sup>I</sup>q<sub>l</sub>)</td></tr><tr><td>O<sup>1(iijkl)</sup><sub>qu</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>q<sub>j</sub>)(u&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>u<sub>l</sub>)</td><td>O<sup>(ijkl)</sup><sub>lu</sub></td><td>(l&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>l<sub>j</sub>)(u&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>u<sub>l</sub>)</td></tr><tr><td>O<sup>8(iijkl)</sup><sub>qu</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>T<sup>A</sup>q<sub>j</sub>)(u&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>T<sup>A</sup>u<sub>l</sub>)</td><td>O<sup>eq</sup><sub>lu</sub></td><td>(e&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>e<sub>j</sub>)(q&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>q<sub>l</sub>)</td></tr><tr><td>O<sup>1(iijkl)</sup><sub>qd</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>q<sub>j</sub>)(d&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>d<sub>l</sub>)</td><td>O<sup>(ijkl)</sup><sub>eu</sub></td><td>(e&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>e<sub>j</sub>)(u&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>u<sub>l</sub>)</td></tr><tr><td>O<sup>8(iijkl)</sup><sub>qd</sub></td><td>(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>T<sup>A</sup>q<sub>j</sub>)(d&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>T<sup>A</sup>d<sub>l</sub>)</td><td>‡O<sup>1(iijkl)</sup><sub>lequ</sub></td><td>(l&#773;<sub>i</sub>e<sub>j</sub>) &epsilon; (q&#773;<sub>k</sub>u<sub>l</sub>)</td></tr><tr><td>O<sup>(ijkl)</sup><sub>uu</sub></td><td>(u&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>u<sub>j</sub>)(u&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>u<sub>l</sub>)</td><td>‡O<sup>3(iijkl)</sup><sub>lequ</sub></td><td>(l&#773;<sub>i</sub>&sigma;<sup>&mu;&nu;</sup>e<sub>j</sub>) &epsilon; (q&#773;<sub>k</sub>&sigma;<sub>&mu;&nu;</sub>u<sub>l</sub>)</td></tr><tr><td>O<sup>1(iijkl)</sup><sub>ud</sub></td><td>(u&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>u<sub>j</sub>)(d&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>d<sub>l</sub>)</td><td>‡O<sup>(ijkl)</sup><sub>ledq</sub></td><td>(l&#773;<sub>i</sub>e<sub>j</sub>)(d&#773;<sub>k</sub>q<sub>l</sub>)</td></tr><tr><td>O<sup>8(iijkl)</sup><sub>ud</sub></td><td>(u&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>T<sup>A</sup>u<sub>j</sub>)(d&#773;<sub>k</sub>&gamma;<sub>&mu;</sub>T<sup>A</sup>d<sub>l</sub>)</td><td></td><td></td></tr><tr><td>‡O<sup>quqd</sup><sub>u(iijkl)</sub></td><td>(q&#773;<sub>i</sub>u<sub>j</sub>) &epsilon; (q&#773;<sub>k</sub>d<sub>l</sub>)</td><td></td><td></td></tr><tr><td>‡O<sup>8(iijkl)</sup><sub>quqd</sub></td><td>(q&#773;<sub>i</sub>T<sup>A</sup>u<sub>j</sub>) &epsilon; (q&#773;<sub>k</sub>T<sup>A</sup>d<sub>l</sub>)</td><td></td><td></td></tr><tr><td colspan="4" style="text-align:center; font-weight:bold;">Two-quark operators</td></tr><tr><td></td><td></td><td>Baryon- and lepton-number-violating</td><td></td></tr><tr><td>‡O<sup>(ij)</sup><sub>u&phi;</sub></td><td>q&#773;<sub>i</sub>u<sub>j</sub>&phi; (&phi;<sup>&dagger;</sup>&phi;)</td><td>‡O<sup>(ijkl)</sup><sub>qqu</sub></td><td>(q&#773;<sup>c</sup><sub>i&alpha;</sub>&epsilon;q<sub>j&beta;</sub>)(u&#773;<sup>c</sup><sub>k&gamma;</sub>e<sub>l</sub>) &epsilon;<sup>&alpha;&beta;&gamma;</sup></td></tr><tr><td>O<sup>1(ij)</sup><sub>u&phi;</sub></td><td>(&phi;<sup>&dagger;</sup>D&#773;<sub>&mu;</sub>&phi;)(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>q<sub>j</sub>)</td><td>‡O<sup>1(iijkl)</sup><sub>qq</sub></td><td>(q&#773;<sup>c</sup><sub>i&alpha;</sub>&epsilon;q<sub>j&beta;</sub>)(q&#773;<sup>c</sup><sub>k&gamma;</sub>e<sub>l</sub>) &epsilon;<sup>&alpha;&beta;&gamma;</sup></td></tr><tr><td>O<sup>3(ij)</sup><sub>u&phi;</sub></td><td>(&phi;<sup>&dagger;</sup>D&#773;<sub>&mu;</sub><sup>I</sup>&phi;)(q&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>&tau;<sup>I</sup>q<sub>j</sub>)</td><td>‡O<sup>3(iijkl)</sup><sub>qqq</sub></td><td>(q&#773;<sup>c</sup><sub>i&alpha;</sub>&tau;<sup>I</sup>&epsilon;q<sub>j&beta;</sub>)(q&#773;<sup>c</sup><sub>k&gamma;</sub>&tau;<sup>I</sup>e<sub>l</sub>) &epsilon;<sup>&alpha;&beta;&gamma;</sup></td></tr><tr><td>O<sup>(ij)</sup><sub>u&phi;</sub></td><td>(&phi;<sup>&dagger;</sup>D&#773;<sub>&mu;</sub>&phi;)(u&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>u<sub>j</sub>)</td><td>‡O<sup>(ijkl)</sup><sub>duu</sub></td><td>(d&#773;<sup>c</sup><sub>i&alpha;</sub>u<sub>j&beta;</sub>)(u&#773;<sup>c</sup><sub>k&gamma;</sub>e<sub>l</sub>) &epsilon;<sup>&alpha;&beta;&gamma;</sup></td></tr><tr><td>‡O<sup>(ij)</sup><sub>u&phi;</sub><sup>ud</sup></td><td>(&phi;<sup>&dagger;</sup>iD<sub>&mu;</sub>&phi;)(u&#773;<sub>i</sub>&gamma;<sup>&mu;</sup>d<sub>j</sub>)</td><td></td><td></td></tr><tr><td>‡O<sup>(ij)</sup><sub>uW</sub></td><td>(q&#773;<sub>i</sub>&sigma;<sup>&mu;&nu;</sup>&tau;<sup>I</sup>u<sub>j</sub>) &phi;W<sup>I</sup><sub>&mu;&nu;</sub></td><td></td><td></td></tr><tr><td>‡O<sup>(ij)</sup><sub>dW</sub></td><td>(q&#773;<sub>i</sub>&sigma;<sup>&mu;&nu;</sup>&tau;<sup>I</sup>d<sub>j</sub>) &phi;W<sup>I</sup><sub>&mu;&nu;</sub></td><td></td><td></td></tr><tr><td>‡O<sup>(ij)</sup><sub>uB</sub></td><td>(q&#773;<sub>i</sub>&sigma;<sup>&mu;&nu;</sup>u<sub>j</sub>) &phi;B<sub>&mu;&nu;</sub></td><td></td><td></td></tr><tr><td>‡O<sup>(ij)</sup><sub>uG</sub></td><td>(q&#773;<sub>i</sub>&sigma;<sup>&mu;&nu;</sup>T<sup>A</sup>u<sub>j</sub>) &phi;G<sup>A</sup><sub>&mu;&nu;</sub></td><td></td><td></td></tr></tbody></table>

to systematically parameterise possible deviations from the SM predictions in the interactions among the known particles, using minimal theoretical assumptions. It amounts to extend the Lagrangian of the SM, by all higher-dimensional operators that respect the gauge symmetry

$$ \mathcal{L}^{\text{SMEFT}} = \mathcal{L}^{\text{SM}} + \sum_{D>4} \sum_{i}^{N_D} \frac{C_{i}^{(D)} O_{i}^{(D)}}{\Lambda^{D-4}}, \quad (61.3) $$

where $D$ is the dimension of the operator $O_i^{(D)}$ and $\Lambda$ provides an upper bound for the scale of new physics. An EFT model is generally characterised by power counting rules that identify a hierarchy among operators. In the case of the SMEFT, a minimal approach is considered where the large scale $\Lambda$ provides a universal suppression factor for the higher dimensional operators.

The currently adopted parametrization for SMEFT interpretations of top quark measurements relies on the Warsaw basis of gauge-invariant dimension-six operators [11] and it is presented in Tab. 61.1 as detailed in Ref. [12] (see [13, 14] for early discussions of top-quark related operators). For convenience, often specific degrees of freedom are identified from combinations of Warsaw-basis operator coefficients aligned with the directions of the EFT parameter space which appear in given processes, in interferences with SM amplitudes, and in top-quark interactions with some of the gauge boson mass eigenstates. Model implementations are available for tree-level and even one-loop Monte Carlo simulations.

The definitions of the SMEFT operators can be organised in four categories: Four-quark, two-quark, two-quark-two-lepton, and baryon-lepton-number violating operators. The overwhelming number of four-fermion operators is tamed by adopting simplifying assumptions about beyond-the-standard-model flavor structures. A baseline flavor scenario in the quark sector and motivated by the minimal flavor violation (MFV) ansatz [15–17] corresponds to imposing a $U(2)_q \times U(2)_u \times U(2)_d$ symmetry among the first two generations. In this case the following numbers of degrees of freedom are produced for the operators of each category of field content:

four heavy quarks 11 + 2 CPV

two light and two heavy quarks 14

two heavy quarks and bosons 9 + 6 CPV

two heavy quarks and two leptons (8 + 3 CPV) × 3 lepton flavors

where we counted separately CP-conserving and CP-violating (CPV) parameters. They are collected in Table 61.1. Other less restricted scenarios, such as that obtained by imposing $U(2)_{q+u+d}$ symmetry featuring additional 10 + 10 CPV degrees of freedom, or more restricted ones, such as top-phile scenario where it is assumed that new physics couples dominantly to the left-handed doublet and right-handed up-type quark singlet of the third generation as well as to bosons, featuring only 19+6 (CPV) degrees of freedom, are often considered. It is also customary to analyse top-quark flavor-changing neutral currents (FCNCs) separately as, at the tree level, they enter only quadratically. More details can be found in Ref. [12].

### 61.1.2 Decay

As other unstable elementary particles, the lifetime of the top quark, and therefore its width, is perturbatively calculable within the SM. With a mass above the Wb threshold, and $|V_{tb}| \gg |V_{td}|$, $|V_{ts}|$, the decay width of the top quark is expected to be dominated by the two-body channel $t \to Wb$. Neglecting terms of order $m_b^2/m_t^2$, $\alpha_s^2$, and $(\alpha_s/\pi)M_W^2/m_t^2$, the width predicted in the SM at NLO is [18]:

$$ \Gamma_t = \frac{G_F m_t^3}{8\pi\sqrt{2}} \left(1 - \frac{M_W^2}{m_t^2}\right)^2 \left(1 + 2 \frac{M_W^2}{m_t^2}\right) \left[1 - \frac{2\alpha_s}{3\pi} \left(\frac{2\pi^2}{3} - \frac{5}{2}\right)\right], \quad (61.4) $$

where $m_t$ refers to the top-quark pole mass. The order $\alpha_s^2$ QCD corrections to $\Gamma_t$ as well as the EW NLO corrections are known [19, 20], thereby improving the overall theoretical accuracy to better than 1%. As a result, between $m_t = 170$ GeV and 175 GeV the width changes from 1.258 GeV to 1.394 GeV, with a linear dependence $(\alpha_s(M_Z) = 0.1179$ and $m_W = 80.377$ GeV). At the reference value of $m_t = 172.5$ GeV, $\Gamma_t = 1.326$ GeV.

With its correspondingly short lifetime of about $0.5 \times 10^{-24}$ s, the top quark is expected to decay before top-flavored hadrons
or $\bar{t}\bar{t}$-quarkonium-bound states can form [21]. In fact, since the decay time is close to the would-be-resonance binding time, a peak will be visible in $e^{+}e^{-}$ scattering at the $\bar{t}\bar{t}$ threshold [22] and it is in principle present (yet very difficult to measure) in hadron collisions too [23, 24].

As mentioned above, flavor changing neutral interactions are allowed starting at one loop. The branching ratios to charm final state are estimated to be [25, 26]:

$$ \text{Br}^{\text{SM}}(t \to gc) = 5 \cdot 10^{-12}, \quad \text{Br}^{\text{SM}}(t \to gu) = 4 \cdot 10^{-14}, $$

$$ \text{Br}^{\text{SM}}(t \to \gamma c) = 5 \cdot 10^{-14}, \quad \text{Br}^{\text{SM}}(t \to \gamma u) = 4 \cdot 10^{-16}, $$

$$ \text{Br}^{\text{SM}}(t \to Zc) = 1 \cdot 10^{-14}, \quad \text{Br}^{\text{SM}}(t \to Zu) = 7 \cdot 10^{-17}, $$

$$ \text{Br}^{\text{SM}}(t \to Hc) = 3 \cdot 10^{-15}, \quad \text{Br}^{\text{SM}}(t \to Hu) = 2 \cdot 10^{-17}. $$

### 61.1.3 Production

#### 61.1.3.1 Pair production

In hadron collisions, top quarks are produced dominantly in pairs through the processes $q\bar{q} \to t\bar{t}$ and $gg \to \bar{t}\bar{t}$, at leading order in QCD. At the Tevatron ($p\bar{p}$ at 1.96 TeV) approximately 85% of the production cross section is from $q\bar{q}$ annihilation, with the remainder from gluon-gluon fusion. Conversely, at the LHC about 90% (80%) of $t\bar{t}$ production is from gluon-gluon fusion at $\sqrt{s} = 13$ TeV ($\sqrt{s} = 7$ TeV).

Predictions for the top-quark production total cross sections are available at next-to-next-to-leading order (NNLO) [27, 28], also including next-to-next-to-leading-log (NNLL) soft gluon resummation. Assuming a top-quark mass of 173.3 GeV, close to the Tevatron + LHC combination [29], the resulting theoretical prediction of the top-quark pair cross-section at NNLO+NNLL accuracy at the Tevatron at $\sqrt{s} = 1.96$ TeV is $\sigma_{t\bar{t}} = 7.16^{+0.11}_{-0.12} + 0.17$ pb where the first uncertainty is from scale dependence and the second from parton distribution functions. At the LHC, assuming a top-quark mass of 172.5 GeV the cross sections are: $\sigma_{t\bar{t}} = 179.6^{+4.8}_{-6.2} + 6.1$ pb at $\sqrt{s} = 7$ TeV, $\sigma_{t\bar{t}} = 256.0^{+6.7}_{-8.9} + 8.0$ pb at $\sqrt{s} = 8$ TeV, $\sigma_{t\bar{t}} = 833.9^{+20.5}_{-30.0} + 21.0$ pb at $\sqrt{s} = 13$ TeV, $\sigma_{t\bar{t}} = 923.6^{+22.6}_{-33.4} + 22.8$ pb at $\sqrt{s} = 13.6$ TeV, and $\sigma_{t\bar{t}} = 985.7^{+24.1}_{-35.7} + 24.1$ pb at $\sqrt{s} = 14$ TeV [27], where the first uncertainty is from scale dependence and the second from parton distribution functions and $\alpha_s$.

The identification of $t\bar{t}$ events at a collider has to take into account that top quarks decay at microscopic scales and only their decay products can be detected. The final states for the leading pair-production process can be divided into three classes:

$$ \text{A. } t\bar{t} \to W^+ b W^- \bar{b} \to q\bar{q}' b q'' \bar{q}''\bar{b}, \quad (45.7\%) $$

$$ \text{B. } t\bar{t} \to W^+ b W^- \bar{b} \to q\bar{q}' b \ell^- \bar{\nu}_\ell \bar{b} + \ell^+ \nu_\ell b q'' \bar{q}''\bar{b}, \quad (43.8\%) $$

$$ \text{C. } t\bar{t} \to W^+ b W^- \bar{b} \to \ell^+ \nu_\ell b \ell^- \bar{\nu}_\ell' \bar{b}, \quad (10.5\%) $$

The quarks in the final state evolve into jets of hadrons. A, B, and C are referred to as the all-hadronic, lepton+jets ($\ell$+jets), and dilepton ($\ell\ell$) channels, respectively. Their relative contributions, including hadronic corrections, are given in parentheses assuming lepton universality. While $\ell$ in the above processes refers to $e$, $\mu$, or $\tau$, most of the analyses distinguish the $e$ and $\mu$ from the $\tau$ channel, which is more difficult to reconstruct. Therefore, in what follows, we will use $\ell$ to refer to $e$ or $\mu$, unless otherwise noted. Here, typically leptonic decays of $\tau$ are included. In addition to the quarks resulting from the top-quark decays, extra QCD radiation (quarks and gluons) from the colored particles in the event can lead to extra jets.

The number of jets reconstructed in the detectors depends on the decay kinematics, as well as on the algorithm for reconstructing jets used by the analysis. Information on the transverse momenta, $p_T$ of the neutrinos is obtained from the imbalance in transverse momentum measured in each event, the missing $p_T$, whose magnitude is also called missing transverse energy, $E_T$.

#### 61.1.3.2 Single top production

Electroweak single top-quark production mechanisms, namely from $q\bar{q}' \to t\bar{b}$ [30], $qb \to q't$ [31], mediated by virtual s-channel and t-channel W-bosons, and Wt-associated production, through $bg \to W^-t$, lead to somewhat smaller cross sections. For example, t-channel production, while suppressed by the weak coupling with respect to the strong pair production, is kinematically

enhanced, resulting in a sizeable cross section both at Tevatron and LHC energies. At the Tevatron, the t- and s-channel cross sections for top quarks are identical to those for antitop quarks, while at the LHC they are not, due to the charge-asymmetric initial state. NNLO cross sections for t-channel single top-quark production ($t + \bar{t}$) are calculated for $m_t = 173.2$ GeV to be $2.08^{+0.04}_{-0.03} + 0.08$ pb in $p\bar{p}$ collisions at $\sqrt{s} = 1.96$ TeV, where the first uncertainty is from scale dependence and the second from parton distribution functions. [32]. A calculation at NNLO accuracy for the t-channel cross section at the LHC has first appeared in [33], superseded by more recent calculations [32, 34] which predict ($m_t = 172.5$ GeV): $\sigma_{t+\bar{t}} = 30.3^{+0.4}_{-0.3} + 0.6$ pb at $\sqrt{s} = 5.02$ TeV, $\sigma_{t+\bar{t}} = 63.7^{+0.9}_{-0.5} + 1.1$ pb at $\sqrt{s} = 7$ TeV, $\sigma_{t+\bar{t}} = 84.3^{+1.1}_{-0.7} + 1.4$ pb at $\sqrt{s} = 8$ TeV, $\sigma_{t+\bar{t}} = 214.2^{+2.4}_{-1.7} + 3.3$ pb at $\sqrt{s} = 13$ TeV, $\sigma_{t+\bar{t}} = 232.2^{+2.6}_{-1.7} + 3.4$ pb at $\sqrt{s} = 13.6$ TeV, and $\sigma_{t+\bar{t}} = 244.5^{+2.7}_{-2.0} + 3.5$ pb at $\sqrt{s} = 14$ TeV, where the first uncertainty is from scale dependence and the second from parton distribution functions and $\alpha_s$. The corresponding fraction of top quarks are 67%, 65%, 65%, 63%, 62%, and 62%. For the s-channel, NNLO approximated calculations yield $1.03^{+0.05}_{-0.05}$ pb for the Tevatron [35]. An NNLO calculation gives ($t + \bar{t}$), $3.00^{+0.03}_{-0.03}$ pb, $3.61^{+0.03}_{-0.03}$ pb, $6.84^{+0.06}_{-0.03}$ pb, and $7.25^{+0.06}_{-0.04}$ pb, for $\sqrt{s} = 7, 8, 13, 13.6$ TeV at the LHC, respectively, where the uncertainty is from scale dependence only. While negligible at the Tevatron, at LHC energies Wt-associated production becomes relevant. At $\sqrt{s} = 7, 8, 13, 13.6, 14$ TeV, an NLO+NLL calculation gives ($t + \bar{t}$), $17.1^{+0.4}_{-0.3} + 0.7$ pb, $24.4^{+0.6}_{-0.5} + 0.9$ pb, $79.3^{+1.9}_{-1.8} + 2.2$ pb, $87.9^{+2.0}_{-1.9} + 2.3$ pb, and $93.8^{+2.2}_{-2.1} + 2.5$ pb, respectively, where the first uncertainty is from scale dependence and the second from parton distribution functions and $\alpha_s$. In this process, an equal proportion of top and anti-top quarks is foreseen. [36].

Assuming $|V_{tb}| \gg |V_{td}|, |V_{ts}|$ (see the Section “The CKM Quark-Mixing Matrix” in this Review for more information), the cross sections for single top production are proportional to $|V_{tb}|^2$, and no extra hypothesis is needed on the number of quark families or on the unitarity of the CKM matrix in extracting $|V_{tb}|$. Separate measurements of the s- and t-channel processes provide sensitivity to physics beyond the Standard Model [37].

The identification of top quarks in the electroweak single top channel is much more difficult than in the QCD $t\bar{t}$ channel, due to a less distinctive signature and significantly larger backgrounds, mostly due to $t\bar{t}$ and W+jets production.

#### 61.1.3.3 Monte Carlo predictions

Fully exclusive predictions via Monte Carlo generators for the $t\bar{t}$ and single top production processes at NLO accuracy in QCD, including top-quark decays and possibly off-shell effects are available [38, 39] through the MC@NLO [40] and POWHEG [41] methods. Recently, the first Monte Carlo implementation of the NNLO QCD computation has become available [42].

#### 61.1.3.4 Associated production

Besides fully inclusive QCD or EW top-quark production, more exclusive final states can be accessed at hadron colliders, whose cross sections are typically much smaller, yet can provide key information on the properties of the top quark. For all relevant final states (e.g., $t\bar{t}V$, $t\bar{t}VV$ with $V = \gamma, W, Z, t\bar{t}H$, $t\bar{t}+jets$, $t\bar{b}\bar{b}$, $t\bar{t}\bar{t}\bar{t}$) automatic or semi-automatic predictions at NLO accuracy both in QCD and EW expansions, also in the form of event generators are available (see the review “Monte Carlo event generators” for more information). Results for total cross sections at NLO+NLL in QCD and NLO EW, are available for $t\bar{t}Z, t\bar{t}W$, $t\bar{t}H$ [43] and for $t\bar{t}\bar{t}$ [44]. Recently, approximate NNLO QCD predictions for and $t\bar{t}H$ and $t\bar{t}W$ have become available [45, 46].

## 61.2 Top-quark and precision SM tests

Since the discovery of the top quark, direct measurements of $t\bar{t}$ production have been made at seven center-of-mass energies in pp or $p\bar{p}$ and in $pPb$ or $PbPb$ collisions, providing stringent tests of QCD and electroweak theory including parton distribution functions, the strong coupling $\alpha_s$ and the dependence on the top-quark mass $m_{top}$. The first measurements were made in Run I at the Tevatron at $\sqrt{s} = 1.8$ TeV. In Run II at the Tevatron