## 65.5 Other long distance dominated modes

The rates and spectra for the decays $K^+ \to \pi^+\ell^+\ell^-$ ($\ell = e$ or $\mu$) have been measured [119–124] and are successfully described with a parameterization inspired by chiral perturbation theory but with large corrections beyond leading order [96, 97] (see [102, 125] for a review). The most recent NA62 result shows good agreement between the electron and muon mode form factors [126]. NA62 has also obtained a first measurement of the forward-backward asymmetry in the muon mode [126].

The mode $K^+ \to \pi^+\pi^0 e^+e^-$, analyzed by NA48/2 [127], is also dominated by long distance physics but it has been argued that measuring asymmetries can provide information on the short distance components [128]. The current status of these modes is discussed in [129]. Contributions to $K^+ \to \pi^+e^+e^-e^+e^-$ beyond the double Dalitz decay in $K^+ \to \pi^+\pi^0$ are considered in [130], and have been measured by NA62 [131].

The decay $K^+ \to \pi^+\gamma\gamma$ can be predicted in terms of one unknown parameter to leading order in $\chi^2$PT, dubbed $\hat{c}$, resulting in a correlation between the rate and the diphoton mass spectrum [132]. Certain important corrections at the next order are also known [133]. The rate and spectrum were first measured by E787 [134], followed by NA48/2 [135] and NA62 [136]. The most recent measurement from NA62 [137], based on a sample of over 4000 events, finds $B(K^+ \to \pi^+\gamma\gamma) = (9.73 \pm 0.17 \pm 0.08) \times 10^{-7}$ and a precise value for $\hat{c}$.

Much information has been recorded by KTeV and NA48 on the rates and spectrum for the Dalitz pair conversion modes $K_L \to \ell^+\ell^-\ell^+\ell^-$ [138, 139], and $K_L \to \ell^+\ell^-\ell^+\ell^-$ for $\ell, \ell' = e$ or $\mu$ [22, 140]. LHCb has recently placed limits on $K_{L,S} \to \mu^+\mu^-\mu^+\mu^-$ [141]. All these results are used to test hadronic models and should eventually help unravel the underlying physics in $K_L \to \mu^+\mu^-$ [87, 92, 142].

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## 66. CPT Invariance Tests in Neutral Kaon Decay

Revised August 2023 by M. Antonelli (INFN, Frascati), G. D'Ambrosio (INFN, Napoli) and M.S. Sozzi (Pisa U.).

CPT theorem is based on three assumptions: quantum field theory, locality, and Lorentz invariance, and thus it is a fundamental probe of our basic understanding of particle physics. Strangeness oscillation in $K^0-\bar{K}^0$ system, described by the equation

$$i \frac{d}{dt} \left[ \frac{K^0}{\bar{K}^0} \right] = [M - i\Gamma/2] \left[ \frac{K^0}{\bar{K}^0} \right],$$

where $M$ and $\Gamma$ are hermitian matrices (see PDG review [1], references [2,3], and KLOE paper [4] for notations and previous literature), allows a very accurate test of CPT symmetry; indeed since CPT requires $M_{11} = M_{22}$ and $\Gamma_{11} = \Gamma_{22}$, the mass and width eigenstates, $K_{S,L}$, have a CPT-violating piece, $\delta$, in addition to the usual CPT-conserving parameter $\epsilon$:

$$K_{S,L} = \frac{1}{\sqrt{2(1 + |\epsilon_{S,L}|^2)}} \left[ (1 + \epsilon_{S,L}) K^0 \pm (1 - \epsilon_{S,L}) \bar{K}^0 \right] \\ \epsilon_{S,L} = \frac{-i\Im(M_{12}) - \frac{1}{2}\Im(\Gamma_{12}) \mp \frac{1}{2} [M_{11} - M_{22} - \frac{i}{2}(\Gamma_{11} - \Gamma_{22})]}{m_L - m_S + i(\Gamma_S - \Gamma_L)/2} \\ \equiv \epsilon \pm \delta. \qquad (66.1)$$

Using the phase convention $\Im(\Gamma_{12}) = 0$, we determine the phase of $\epsilon$ to be $\phi_{SW} = \arctan \frac{2(m_L - m_S)}{\Gamma_S - \Gamma_L}$. Imposing unitarity to an arbitrary combination of $K^0$ and $\bar{K}^0$ wave functions, we obtain the Bell-Steinberger relation [5] connecting CP and CPT violation in the mass matrix to CP and CPT violation in the decay; in fact, neglecting $O(\epsilon)$ corrections to the coefficient of the CPT-violating parameter, $\delta$, we can write [4]

$$\left[ \frac{\Gamma_S + \Gamma_L}{\Gamma_S - \Gamma_L} + i \tan \phi_{SW} \right] \left[ \frac{\Re(\epsilon)}{1 + |\epsilon|^2} - i \Im(\delta) \right] = \frac{1}{\Gamma_S - \Gamma_L} \sum_f A_L(f) A_S^*(f), \qquad (66.2)$$

where $A_{L,S}(f) \equiv A(K_{L,S} \rightarrow f)$. We stress that this relation is phase-convention-independent. The advantage of the neutral kaon system is that only a few decay modes give significant contributions to the r.h.s. in Eq. (66.2); in fact, defining for the hadronic modes

$$\alpha_i \equiv \frac{1}{\Gamma_S} \langle A_L(i) A_S^*(i) \rangle = \eta_i \mathcal{B}(K_S \rightarrow i), \\ i = \pi^0 \pi^0, \pi^+ \pi^-(\gamma), 3\pi^0, \pi^0 \pi^+ \pi^-(\gamma), \qquad (66.3)$$

the recent data from CPLEAR, KLOE, KTeV, and NA48 have led to the following determinations (the analysis described in Ref. [4] has been updated by using the recent measurements of $K_L$ branching ratios from KTeV [6, 7], NA48 [8, 9], the results described in the CP violation in $K_L$ decays minireview, and the KLOE result [10])

$$\begin{aligned} \alpha_{\pi+\pi-} &= ((1.121 \pm 0.010) + i(1.061 \pm 0.010)) \times 10^{-3}, \\ \alpha_{\pi^0\pi^0} &= ((0.493 \pm 0.005) + i(0.471 \pm 0.005)) \times 10^{-3}, \\ \alpha_{\pi+\pi-\pi^0} &= ((0 \pm 2) + i(0 \pm 2)) \times 10^{-6}, \\ |\alpha_{\pi^0\pi^0\pi^0}| &< 1.5 \times 10^{-6} \text{ at 95\% CL.} \end{aligned} \qquad (66.4)$$

The semileptonic contribution to the right-handed side of Eq. (66.2) requires the determination of several observables: we

define [2,3]

$$\begin{aligned} A(K^0 \rightarrow \pi^- l^+ \nu) &= A_0(1 - y), \\ A(K^0 \rightarrow \pi^+ l^- \nu) &= A_0^*(1 + y^*)(x_+ - x_-)^*, \\ A(\bar{K}^0 \rightarrow \pi^+ l^- \nu) &= A_0^*(1 + y^*), \\ A(\bar{K}^0 \rightarrow \pi^- l^+ \nu) &= A_0(1 - y)(x_+ + x_-), \end{aligned} \qquad (66.5)$$

where $x_+(-x_-)$ describes the violation of the $\Delta S = \Delta Q$ rule in CPT-conserving (violating) decay amplitudes, and $y$ parameterizes CPT violation for $\Delta S = \Delta Q$ transitions. Taking advantage of their tagged $K^0(\bar{K}^0)$ beams, CPLEAR has measured $\Im(x_+)$, $\Re(x_-)$, $\Im(\delta)$, and $\Re(\delta)$ [11]. These determinations have been improved in Ref. [4] by including the information $A_S - A_L = 4[\Re(\delta) + \Re(x_-)]$ (valid at first order in the small parameters), where $A_{L,S}$ are the $K_L$ and $K_S$ semileptonic charge asymmetries, respectively, from the PDG [12] and the new KLOE semileptonic measurement [13]. Here we are also including the T-violating asymmetry measurement from CPLEAR [14] with a finer binning than appearing in the published article.

<table><caption>**Table 66.1:** Values, errors, and correlation coefficients for $\Re(\delta)$, $\Im(\delta)$, $\Re(x_+)$, $\Im(x_+)$, and $A_S + A_L$ obtained from a combined fit, including KLOE [4, 13] and CPLEAR [14].</caption><thead><tr><th></th><th>value</th><th>Correlations coefficients</th></tr></thead><tbody><tr><td>Re(δ)</td><td>(4.3 ± 2.7) × 10<sup>-4</sup></td><td>1</td></tr><tr><td>I(δ)</td><td>(-0.9 ± 0.6) × 10<sup>-2</sup></td><td>-0.40 1</td></tr><tr><td>Re(x<sub>-</sub>)</td><td>(-0.22 ± 0.10) × 10<sup>-2</sup></td><td>-0.14 -0.30 1</td></tr><tr><td>I<sub>+</sub>(x<sub>+</sub>)</td><td>(0.06 ± 0.19) × 10<sup>-2</sup></td><td>-0.12 -0.02 0.34 1</td></tr><tr><td>A<sub>S</sub> + A<sub>L</sub></td><td>(-0.23 ± 0.38) × 10<sup>-2</sup></td><td>-0.12 -0.29 0.94 0.18 1</td></tr></tbody></table>

The value $A_S + A_L$ in Table 66.1 can be directly included in the semileptonic contributions to the Bell Steinberger relations in Eq. (66.2)

$$\begin{aligned} \sum_{\pi \ell \nu} \langle A_L(\pi \ell \nu) A_S^*(\pi \ell \nu) \rangle &= 2\Gamma(K_L \rightarrow \pi \ell \nu)(\Re(\epsilon) - \Re(y) - i(\Im(x_+) + \Im(\delta))) \\ &= 2\Gamma(K_L \rightarrow \pi \ell \nu)((A_S + A_L)/4 - i(\Im(x_+) + \Im(\delta))). \end{aligned} \qquad (66.6)$$

Defining

$$\alpha_{\pi \ell \nu} \equiv \frac{1}{\Gamma_S} \sum_{\pi \ell \nu} \langle A_L(\pi \ell \nu) A_S^*(\pi \ell \nu) \rangle + 2i \frac{\tau_{K_S}}{\tau_{K_L}} \mathcal{B}(K_L \rightarrow \pi \ell \nu) \Im(\delta), \qquad (66.7)$$

we find:

$$\alpha_{\pi \ell \nu} = ((-0.1 \pm 0.2) + i(-0.1 \pm 0.5)) \times 10^{-5}. \qquad (66.8)$$

<table><caption>**Table 66.2:** Summary of results: values, errors, and correlation coefficients for $\Re(\epsilon)$, $\Im(\delta)$, $\Re(\delta)$, and $\Re(x_-)$.</caption><thead><tr><th></th><th>value</th><th>Correlations coefficients</th></tr></thead><tbody><tr><td>Re(ε)</td><td>(161.2 ± 0.5) × 10<sup>-5</sup></td><td>+1</td></tr><tr><td>I(δ)</td><td>(-0.3 ± 1.4) × 10<sup>-5</sup></td><td>+0.08 1</td></tr><tr><td>Re(δ)</td><td>(2.6 ± 2.5) × 10<sup>-4</sup></td><td>+0.00 -0.05 1</td></tr><tr><td>I(x<sub>-</sub>)</td><td>(-2.7 ± 1.0) × 10<sup>-3</sup></td><td>+0.05 0.13 -0.30 1</td></tr></tbody></table>

The analysis of semileptonic decay asymmetries implicitly assumes Lepton Flavour Universality (LFU) of any effect violating CPT or $\Delta S = \Delta Q$, through the use of a single set of $x, y$ parameters, consistently with the availability of experimental information on $A_S$ for the electron mode only. The explicit LFU assumption in the input BR measurements has been lifted exploiting the measurement of $BR(K_S \rightarrow \pi \mu \nu)$ Ref. [15], with no effect on the numerical results.
Inserting the values of the $\alpha$ parameters into Eq. (66.2), we find

$$ \begin{aligned} \Re(\epsilon) &= (161.2 \pm 0.5) \times 10^{-5}, \\ \Im(\delta) &= (-0.3 \pm 1.4) \times 10^{-5}. \end{aligned} \qquad (66.9) $$

The complete information on Eq. (66.9) is given in Table 66.2.

<figure><img src="image_327.png" /><figcaption>**Figure 66.1:** Top: allowed region at 68% and 95% C.L. in the $\Re(\epsilon)$, $\Im(\delta)$ plane. Bottom: allowed region at 68% and 95% C.L. in the $\Delta M$, $\Delta\Gamma$ plane.</figcaption></figure>

Now the agreement with CPT conservation, $\Im(\delta) = \Re(\delta) = \Re(x_-) = 0$, is at 18% C.L.

The allowed region in the $\Re(\epsilon) - \Im(\delta)$ plane at 68% CL and 95% C.L. is shown in the top panel of Fig. 66.1.

The process giving the largest contribution to the size of the allowed region is $K_L \to \pi^+\pi^-$, through the uncertainty on $\phi_{+-}$.

The limits on $\Im(\delta)$ and $\Re(\delta)$ can be used to constrain the $K^0 - \bar{K}^0$ mass and width difference

$$ \delta = \frac{i(m_{K^0} - m_{\bar{K}^0}) + \frac{1}{2}(\Gamma_{K^0} - \Gamma_{\bar{K}^0})}{\Gamma_S - \Gamma_L} \cos \phi_{SW} e^{i\phi_{SW}} [1 + \mathcal{O}(\epsilon)]. $$

The allowed region in the $\Delta M = (m_{K^0} - m_{\bar{K}^0})$, $\Delta\Gamma = (\Gamma_{K^0} - \Gamma_{\bar{K}^0})$ plane is shown in the bottom panel of Fig. 66.1. As a result, we improve on the previous limits (see for instance, P. Bloch in Ref. [12]) and in the limit $\Gamma_{K^0} - \Gamma_{\bar{K}^0} = 0$ we obtain

$$ -4.0 \times 10^{-19} \text{ GeV} < m_{K^0} - m_{\bar{K}^0} < 4.0 \times 10^{-19} \text{ GeV} \text{ at 95 \% C.L.} $$

References

[1] See the "CP Violation in Meson Decays," in this Review.

[2] L. Maiani, "CP And CPT Violation in Neutral Kaon Decays," L. Maiani, G. Pancheri, and N. Paver, *The Second DAΦNE Physics Handbook*, Vol. 1,2.

[3] G. D'Ambrosio, G. Isidori and A. Pugliese, in "2nd DAΦNE Physics Handbook:63-96," 63-96 (1994), [hep-ph/9411389], URL http://preprints.cern.ch/cgi-bin/setlink?base=preprint&categ=cern&id=th-7504-94.

[4] G. D'Ambrosio and G. Isidori (KLOE), JHEP 12, 011 (2006), [hep-ex/0610034].

[5] J. S. Bell and J. Steinberger, in "Wolfenstein, L. (eq.): CP violation, 42-57. (In Oxford International Symposium Conference on Elementary Particles)," 195-208, 221-222 (1966), (See Book Index).

[6] T. Alexopoulos et al. (KTeV), Phys. Rev. D70, 092006 (2004), [hep-ex/0406002].

[7] E. Abouzaid et al. (KTeV), Phys. Rev. D83, 092001 (2011), [arXiv:1011.0127].

[8] A. Lai et al. (NA48), Phys. Lett. B645, 26 (2007), [hep-ex/0611052]; A. Lai et al. (NA48), Phys. Lett. B602, 41 (2004), [hep-ex/0410059].

[9] We thank G. Isidori and M. Palutan for their contribution to the original analysis [4] performed with KLOE data.

[10] D. Babusci et al. (KLOE), Phys. Lett. B723, 54 (2013), [arXiv:1301.7623].

[11] A. Angelopoulos et al. (CPLEAR), Phys. Lett. B444, 52 (1998).

[12] W. M. Yao et al. (Particle Data Group), J. Phys. G33, 1 (2006).

[13] A. Anastasi et al. (KLOE-2), JHEP 09, 021 (2018), [arXiv:1806.08654].

[14] P. Bloch, M. Fidecaro, private communication of the data in a finer binning format; A. Angelopoulos et al. (CPLEAR), Phys. Lett. B444, 43 (1998).

[15] D. Babusci et al. (KLOE-2), Phys. Lett. B 804, 135378 (2020), [arXiv:1912.05990].
## 67. $V_{ud}$, $V_{us}$ the Cabibbo Angle, and CKM Unitarity

Revised March 2024 by E. Blucher (Chicago U.), G. D'Ambrosio (INFN, Napoli) and W.J. Marciano (BNL).

The Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] three-generation quark mixing matrix written in terms of the Wolfenstein parameters ($\lambda, A, \rho, \eta$) [3] nicely illustrates the orthonormality constraint of unitarity, as well as the central role played by $\lambda$.

$$ V_{CKM} = \begin{pmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{pmatrix} = \begin{pmatrix} 1 - \lambda^2/2 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1 - \lambda^2/2 & A\lambda^2 \\ A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1 \end{pmatrix} + \mathcal{O}(\lambda^4). \quad (67.1) $$

That cornerstone parameter is a carryover from the two-generation Cabibbo angle, $\lambda = \sin(\theta_{\text{Cabibbo}}) = V_{us}$. Its value is an important component in tests of CKM unitarity.

Up until 2003, the precise value of $\lambda$ was controversial, with kaon decays (specifically $K \to \pi e \nu$) branching fractions suggesting [4] $\lambda \simeq 0.220$, while indirect determinations via $V_{ud}$ obtained from nuclear $\beta$-decays combined with unitarity preferred a somewhat larger $\lambda \simeq 0.225 - 0.230$. This difference implied a 2 - 2.5 sigma deviation from the first row unitarity requirement

$$ |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1, \quad (67.2) $$

a possible hint [5, 6] of new physics effects. Below, we describe the current status of $V_{ud}$ and $V_{us}$, and their implication for the unitarity test in Eq. (67.2). (Since $|V_{ub}|^2 \simeq 1.7 \times 10^{-5}$ is negligibly small, it is ignored in this discussion.) Eq. (67.2) is currently the most stringent test of unitarity in the CKM matrix. As we shall see, it is again showing signs of a possible 2 to 3 sigma inconsistency.

### 67.1 $V_{ud}$

Precise values of $V_{ud}$ have been obtained from superallowed nuclear, neutron, and pion beta decays. Currently, the best determination of $V_{ud}$ comes from analysis of a set of 15 precisely measured superallowed nuclear beta-decays [5, 6] ($0^+ \to 0^+$ transitions). Measuring their half-lives, $t$, and $Q$ values gives the decay rate factors, $f$, which lead to a precise determination of $V_{ud}$ via [7–12]. Based on several decades of dedicated studies, Hardy and Towner recently updated the average of the half-life, $\bar{\mathcal{F}}t$, [6]

$$ \begin{aligned} |V_{ud}|^2 &= \frac{K}{2G_F^2(1 + \Delta_R^V)\bar{\mathcal{F}}t} = \frac{2912.95 \pm 0.54}{\bar{\mathcal{F}}t} \\ &= 0.94815 \pm 0.00060 \quad \text{for } \Delta_R^V = 2.454(19)\%, \quad (67.3) \end{aligned} $$

where $K/(\hbar c)^6 = 2\pi^3\hbar \ln 2/(m_e c^2)^5 = 8120.27648(26) \times 10^{-10} \text{GeV}^{-4} s$, $\Delta_R^V$ denotes the so-called inner or universal electroweak radiative corrections (RC) to superallowed nuclear beta decays. A dispersion relation (DR) calculational approach [13] to quantum loop corrections, specifically the gamma-W box diagram, gives $\Delta_R^V = 0.02467(22)$. Actually the authors of Ref. [14] have combined the different theoretical calculations leading to $\Delta_R^V = 0.02467(27)$ and

$$ V_{ud} = 0.97367(11)_{\text{exp}}(13)_{\Delta_R^V}(27)_{\text{NS}}(32)_{\text{total}} \text{ (superallowed)}. \quad (67.4) $$

As a result of its small uncertainty and more rigorous theoretical footing, we use this value below. A somewhat different approach [15] found $\Delta_R^V = 0.02426(32)$. These recent values are roughly consistent. Both are larger than the 2018 PDG value of 0.02361(38). Implications and possible nuclear physics modifications of those studies are still under scrutiny [16, 17]. Note that an additional uncertainty (54)$_{\text{NS}}$ from nuclear structure (NS) [17] has been recently included [6] in that master formula. This recent determination of $V_{ud}$ has shifted significantly down compared to the 2018 PDG value of 0.97420(21). In addition, NS uncertainties are now the dominant contribution to the overall uncertainty.

Taken at face value, along with current $V_{us}$ determinations (see subsection 67.2), the reduced $V_{ud}$ would seem to violate the first row unitarity requirement, and thus suggest the presence of deviations from unitarity or unaccounted uncertainties.

Measurements of the neutron lifetime, $\tau_n$, and the ratio of axial-vector/vector couplings, $g_A \equiv G_A/G_V$, via neutron decay asymmetries combined with the inner radiative corrections can also be used to determine $V_{ud}$ via the precise formula [14, 18]

$$ |V_{ud}|^2 = \frac{5024.7 \text{ s}}{\tau_n(1 + 3g_A)(1 + \Delta_R)}, \quad (67.5) $$

where $\Delta_R = 0.03983(27)$ represents radiative corrections [14]. We present the $V_{ud}$ results using either the current published world averages [19] or the best measured values for $\lambda$ [20] and for $\tau_n$ [21]. The average leads to

$$ \begin{aligned} \tau_n^{\text{ave}} &= 879.4(6) \text{ s (1.6 PDG scale factor)} \\ g_A^{\text{ave}} &= 1.2756(13), (2.6 \text{ PDG scale factor}), \end{aligned} \quad (67.6) $$

$$ |V_{ud}^{n, \text{PDG}}| = 0.97441(28)_{\tau_n}(82)_{\lambda}(13)_{\Delta_R}(88)_{\text{TOT}}, \quad (67.7) $$

while the best values lead to

$$ |V_{ud}^{n, \text{best}}| = 0.97413(20)_{\tau_n}(35)_{\lambda}(13)_{\Delta_R}(42)_{\text{TOT}}, \quad (67.8) $$

Both central values are similar to the superallowed nuclear beta decay result reported above. Reconciliation with CKM unitarity suggests a shorter neutron lifetime near 878.5 s or a smaller $g_A$. We note that the most precise recent neutron lifetime update reported [21], $\tau_n = 877.75(34)$ s, has an uncertainty about half as big as the average given in Eq. 67.6. It corresponds to $|V_{ud}| = 0.9746(8)$, a value more in keeping with unitarity expectations; but too large an uncertainty from $g_A$ to be meaningful. Future neutron studies [22] are expected to resolve any current inconsistencies and further reduce the uncertainties in $g_A$ and $\tau_n$ making them a potentially better way to determine $V_{ud}$ without the nuclear physics uncertainties.

The PIBETA experiment at PSI measured the very small ($\mathcal{O}(10^{-8})$) branching ratio for $\pi^+ \to \pi^0 e^+ \nu_e$ with about $\pm 0.6\%$ precision. Its result gives [23]

$$ |V_{ud}| = 0.9739(27) \left[ \frac{BR(\pi^+ \to e^+ \nu_e(\gamma))}{1.2325 \times 10^{-4}} \right]^{\frac{1}{2}}, \quad (67.9) $$

which is normalized using the very precisely measured $BR(\pi^+ \to e^+ \nu_e(\gamma)) = 1.2325(23) \times 10^{-4}$ [7], rather than the theoretical branching ratio of $1.2350(2) \times 10^{-4}$, which if used, would increase $|V_{ud}|$ to 0.9749(27). Theoretical uncertainties in pion beta decay are very small [24], leaving open more than an order of magnitude improvement of its experimental branching ratio before theory uncertainties become a problem. Although challenging, improved measurements of pion beta decay currently under discussion would allow this decay mode to compete with superallowed beta decays and future neutron decay efforts for the most precise direct $|V_{ud}|$ determination.

### 67.2 $V_{us}$

$|V_{us}|$ may be directly obtained from kaon decays, hyperon decays, and tau decays. Early determinations most often used $K\ell 3$ decays:

$$ \Gamma_{K\ell 3} = \frac{G_F^2 M_K^5}{192\pi^3} S_{EW} (1 + \delta_K^\ell + \delta_{SU2}) C^2 |V_{us}|^2 f_+^2(0) I_K^\ell. \quad (67.10) $$

Here, $\ell$ refers to either $e$ or $\mu$, $G_F$ is the Fermi constant, $M_K$ is the kaon mass, $S_{EW}$ is the short-distance radiative correction, $\delta_K^\ell$ is the mode-dependent long-distance radiative correction, $f_+(0)$ is the calculated form factor at zero momentum transfer for the $\ell\nu$ system, and $I_K^\ell$ is the phase-space integral, which depends on measured semileptonic form factors. For charged kaon decays, $\delta_{SU2}$ is the deviation from one of the ratio of $f_+(0)$ for the charged to neutral kaon decay; it is zero for the neutral kaon. $C^2$ is 1 (1/2) for neutral (charged) kaon decays. Most early determinations of
$|V_{us}|$ were based solely on $K \to \pi e \nu$ decays; $K \to \pi \mu \nu$ decays were not used because of large uncertainties in $f_K^\mu$. The experimental measurements are the semileptonic decay widths (based on the semileptonic branching fractions and lifetime) and form factors (allowing calculation of the phase space integrals). Theory is needed for $S_{EW}$, $\delta_K^\ell$, $\delta_{SU2}$, and $f_+(0)$.

Many measurements during the last 20 years have resulted in a shift in $|V_{us}|$. Most importantly, the $K \to \pi e \nu$ branching fractions are significantly different than much earlier PDG averages, probably as a result of inadequate treatment of radiation in older experiments. This effect was first observed by BNL E865 [25] in the charged kaon system and then by KTeV [26, 27] in the neutral kaon system; subsequent measurements were made by KLOE [28–31], NA48 [32–34], and ISTRA+ [35]. Current averages (e.g., by the PDG [36] or Flavianet [37]) of the semileptonic branching fractions are based only on recent, high-statistics experiments where the treatment of radiation is clear. In addition to measurements of branching fractions, new measurements of lifetimes [38] and form factors [39–43], have resulted in improved precision for all of the experimental inputs to $|V_{us}|$. Precise measurements of form factors for $K_{\mu 3}$ decay make it possible to use both semileptonic decay modes to extract $V_{us}$.

Following the analysis of Moulson [18, 44] and the Flavianet group [37, 45, 46], along with recent improvements in the QED radiative corrections [47], one finds [48], after including the isospin violating effect, $\delta_{SU2}$, the values of $|V_{us}|f_+(0)$ in Table 67.1. The average of these measurements, including correlation effects [18, 48–50] gives [14]

$$f_+(0)|V_{us}| = 0.21656(35) \quad \text{FLAVIANET} \qquad (67.11)$$

where the errors correspond to kaon experimental parameters and radiative corrections, respectively, leading, as we shall see, to $|V_{us}| = 0.22330(35)_{\text{exp}}(39)_{\text{LAT}}(8)_{\text{RC+IB}}(53)_{\text{TOT}}$.

Lattice QCD calculations of $f_+(0)$ have been carried out for 2, 2+1, and 2+1+1 quark flavors and range from about 0.96 to 0.97. Here, we illustrate recent FLAG (2020) updated averages [51] for 2+1 and 2+1+1 flavors:

$$ \begin{aligned} f_+(0) &= 0.9677(27) \quad N_f = 2+1 \\ f_+(0) &= 0.9698(17) \quad N_f = 2+1+1 \end{aligned} \qquad (67.12) $$

Updating the result for $N_f = 2+1+1$ we obtain for $V_{us}$ from Eq. (67.11) and Eq. (67.12),

$$ \begin{aligned} |V_{us}| &= 0.2238(4)_{\text{exp+RC}}(6)_{\text{lattice}} (N_f = 2+1, K_{\ell 3} \text{ decays}) \\ &= 0.22330(4)_{\text{exp+RC}}(4)_{\text{lattice}} (N_f = 2+1+1, K_{\ell 3} \text{ decays}) \end{aligned} \qquad (67.13) $$

based on ref. [18] and ref. [48]

<table><caption>**Table 67.1:** $|V_{us}|f_+(0)$ from $K_{\ell 3}$, based on ref. [18, 48]</caption><thead><tr><th>Decay Mode</th><th>|V<sub>us</sub>|f<sub>+(0)</sub></th></tr></thead><tbody><tr><td>K<sup>±</sup>e3</td><td>0.2169(6)</td></tr><tr><td>K<sup>±</sup>μ3</td><td>0.2168(10)</td></tr><tr><td>K<sub>L</sub>e3</td><td>0.2162(5)</td></tr><tr><td>K<sub>L</sub>μ3</td><td>0.2165(6)</td></tr><tr><td>K<sub>S</sub>e3</td><td>0.2169(8)</td></tr><tr><td>K<sub>S</sub>μ3</td><td>0.2125(47)</td></tr><tr><td>Average (including correlation effects [18, 48])</td><td>0.21656(35)</td></tr></tbody></table>

A value of $V_{us}$ can also be obtained from a comparison of the radiative inclusive decay rates for $K \to \mu \nu(\gamma)$ and $\pi \to \mu \nu(\gamma)$ combined with a lattice gauge theory calculation of $f_{K+}/f_{\pi+}$ via

$$ \frac{|V_{us}|f_{K+}}{|V_{ud}|f_{\pi+}} = 0.23871(20) \left[ \frac{\Gamma(K \to \mu\nu(\gamma))}{\Gamma(\pi \to \mu\nu(\gamma))} \right]^{\frac{1}{2}} \qquad (67.14) $$

with the small error coming from electroweak radiative corrections [52–54]; these corrections were confirmed by direct lattice calculation of the kaon and pion leptonic decay rates [53, 54].

A recent fit [14, 18] gives

$$ \frac{|V_{us}|f_{K+}}{|V_{ud}|f_{\pi+}} = 0.27679(28)_{\text{BR}}(20)_{\text{corr.}} \qquad (67.15) $$

Employing the FLAG [51] lattice QCD averages for the isospin broken decay constants

$$ \frac{f_{K+}}{f_{\pi+}} = 1.1978(22) \quad N_f = 2+1+1 \qquad (67.16) $$

leading to

$$ \frac{|V_{us}|}{|V_{ud}|} = 0.23108(51) \quad (N_f = 2+1+1, K_{\mu 2} \text{ decays}). \qquad (67.17) $$

Actually we can average the Cabibbo angle determinations from kaons: from $K_{\ell 3}$ decays in eqs. 67.11, 67.12 and from $K_{\mu 2}$ in eqs. 67.15, 67.16 with $N_f = 2+1+1$. Assuming 20% correlation between $(|V_{us}|/V_{ud}|f_K/f_\pi)$ and $(f_K/f_\pi)$ we obtain

$$ V_{us} = 0.22431(85) \quad S = 2.5. \qquad (67.18) $$

The result shown in Eq. 67.18 is what we use when we consider the first row test of CKM unitarity. The quoted uncertainty includes a scale factor of 2.5 to account for the discrepancies between the values of $V_{us}$ obtained from $K_{\ell 3}$ and $K_{\mu 2}$ decays.

This is the current status of the Cabibbo Angle determination from the kaon sector, and theoretical and experimental progress is expected to clarify the situation soon.

It should be mentioned that hyperon decay fits suggest [55]

$$ |V_{us}| = 0.2250(27) \quad (\text{Hyperon Decays}) \qquad (67.19) $$

modulo SU(3) breaking effects that could shift that value up or down. We note that a representative effort [56] that incorporates SU(3) breaking found $V_{us} = 0.226(5)$. Strangeness changing tau decays, averaging both inclusive and exclusive measurements, give [57, 58]

$$ |V_{us}| = 0.2207(14) \quad (\text{Tau Decays}), \qquad (67.20) $$

which differs by about 2 sigma from the kaon determination discussed above, and would, if combined with $V_{ud}$ from super-allowed beta decays, lead to a 4 sigma deviation from unitarity. This discrepancy results mainly from the inclusive tau decay results that rely on Finite Energy Sum Rule techniques and assumptions, as well as experimental uncertainties. Recent investigation of that approach suggests a larger value for $V_{us}$, which is more in accord with other determinations [59].

Employing the values of $V_{ud}$ and $V_{us}$ with an error scale factor of 2 from Eq. (67.4) and the average obtained after rescaling the errors from mEq. 67.18 leads to the unitarity consistency check

$$ |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9983(6)(4), \qquad (67.21) $$

where the first error is the uncertainty from $|V_{ud}|^2$ and the second error is the uncertainty from the average $|V_{us}|^2$ in Eq. 67.18. One finds about an overall 2 sigma deviation from unitarity. (The deviation increases to 3 sigma if nuclear structure uncertainties are ignored.) That deviation could be due a problem with $|V_{ud}|$ theory (RC or NS), the lattice determination of $f_+(0)$ or new physics.

## 67.3 CKM Unitarity Constraints

The current 2 sigma experimental disagreement with unitarity, $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9983(7)$, still provides strong confirmation of Standard Model radiative corrections (which range between 3-4% depending on the nucleus used) at a high significance level [60]. In addition, it implies constraints on “New Physics” effects at both the tree and quantum loop levels. Those effects could be in the form of contributions to nuclear beta decays, K decays and/or muon decays, with the last of these providing normalization via the muon lifetime [61], which is used to obtain the Fermi constant, $G_\mu = 1.1663787(6) \times 10^{-5} \text{GeV}^{-2}$.

In the following examples, we illustrate the implications of CKM unitarity for (1) exotic muon decays [62] (beyond ordinary
muon decay $\mu^+ \to e^+\nu_e\bar{\nu}_\mu$) and (2) new heavy quark mixing $V_{uD}$ [63]. Other examples in the literature [64, 65] include $Z_\chi$ boson quantum loop effects, supersymmetry, leptoquarks, compositeness etc.

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# 68. CP Violation in $K_L^0$ Decays

Revised August 2023 by C.-J. Lin (LBNL).

The symmetries C (particle-antiparticle interchange) and P (space inversion) hold for strong and electromagnetic interactions. After the discovery of large C and P violation in the weak interactions, it appeared that the product CP was a good symmetry. In 1964 CP violation was observed in $K^0$ decays at a level given by the parameter $\epsilon \approx 2.3 \times 10^{-3}$.

A unified treatment of CP violation in K, D, B, and $B_s$ mesons is given in “CP Violation in Meson Decays” by D. Kirkby and Y. Nir in this Review. A more detailed review including a thorough discussion of the experimental techniques used to determine CP violation parameters is given in a book by K. Kleinknecht [1]. Here we give a concise summary of the formalism needed to define the parameters of CP violation in $K_L$ decays, and a description of our fits for the best values of these parameters.

## 68.1 Formalism for CP violation in Kaon decay

CP violation has been observed in the semi-leptonic decays $K_L^0 \to \pi^\mp \ell^\pm \nu$, and in the nonleptonic decay $K_L^0 \to 2\pi$. The experimental numbers that have been measured are

$$A_L = \frac{\Gamma(K_L^0 \to \pi^- \ell^+ \nu) - \Gamma(K_L^0 \to \pi^+ \ell^- \nu)}{\Gamma(K_L^0 \to \pi^- \ell^+ \nu) + \Gamma(K_L^0 \to \pi^+ \ell^- \nu)} \quad (68.1a)$$

$$\begin{aligned} \eta_{+-} &= A(K_L^0 \to \pi^+ \pi^-) / A(K_S^0 \to \pi^+ \pi^-) \\ &= |\eta_{+-}| e^{i\phi_{+-}} \end{aligned} \quad (68.1b)$$

$$\begin{aligned} \eta_{00} &= A(K_L^0 \to \pi^0 \pi^0) / A(K_S^0 \to \pi^0 \pi^0) \\ &= |\eta_{00}| e^{i\phi_{00}}. \end{aligned} \quad (68.1c)$$

CP violation can occur either in the $K^0-\bar{K}^0$ mixing or in the decay amplitudes. Assuming CPT invariance, the mass eigenstates of the $K^0-\bar{K}^0$ system can be written

$$|K_S\rangle = p|K^0\rangle + q|\bar{K}^0\rangle, \quad |K_L\rangle = p|K^0\rangle - q|\bar{K}^0\rangle. \quad (68.2)$$

If CP invariance held, we would have $q = p$ so that $K_S$ would be CP-even and $K_L$ CP-odd. (We define $|\bar{K}^0\rangle$ as CP $|K^0\rangle$). CP violation in $K^0-\bar{K}^0$ mixing is then given by the parameter $\epsilon$ where

$$\frac{p}{q} = \frac{(1 + \tilde{c})}{(1 - \tilde{c})}. \quad (68.3)$$

CP violation can also occur in the decay amplitudes

$$A(K^0 \to \pi\pi(I)) = A_I e^{i\delta_I}, \quad A(\bar{K}^0 \to \pi\pi(I)) = A_I^* e^{i\delta_I}, \quad (68.4)$$

where I is the isospin of $\pi\pi$, $\delta_I$ is the final-state phase shift, and $A_I$ would be real if CP invariance held. The CP-violating observables are usually expressed in terms of $\epsilon$ and $\epsilon'$ defined by

$$\eta_{+-} = \epsilon + \epsilon', \quad \eta_{00} = \epsilon - 2\epsilon'. \quad (68.5a)$$

One can then show [2]

$$\epsilon = \tilde{\epsilon} + i \left( \operatorname{Im} A_0 / \operatorname{Re} A_0 \right), \quad (68.5b)$$

$$\sqrt{2}\epsilon' = ie^{i(\delta_2 - \delta_0)}(\operatorname{Re} A_2 / \operatorname{Re} A_0) \left( \operatorname{Im} A_2 / \operatorname{Re} A_2 - \operatorname{Im} A_0 / \operatorname{Re} A_0 \right), \quad (68.5c)$$

$$A_L = 2\operatorname{Re}\epsilon/(1 + |\epsilon|^2) \approx 2\operatorname{Re}\epsilon. \quad (68.5d)$$

In Eq. (68.5a), small corrections [3] of order $\epsilon' \times \operatorname{Re}(A_2/A_0)$ are neglected, and Eq. (68.5d) assumes the $\Delta S = \Delta Q$ rule.

The quantities $\operatorname{Im} A_0$, $\operatorname{Im} A_2$, and $\operatorname{Im} \tilde{\epsilon}$ depend on the choice of phase convention, since one can change the phases of $K^0$ and $\bar{K}^0$ by a transformation of the strange quark state $|s\rangle \to |s\rangle e^{i\alpha}$; of course, observables are unchanged. It is possible by a choice of phase convention to set $\operatorname{Im} A_0$ or $\operatorname{Im} A_2$ or $\operatorname{Im} \tilde{\epsilon}$ to zero, but none of these is zero with the usual phase conventions in the Standard Model. The choice $\operatorname{Im} A_0 = 0$ is called the Wu-Yang phase convention [4], in which case $\epsilon = \tilde{\epsilon}$. The value of $\epsilon'$ is independent of phase convention, and a nonzero value demonstrates CP violation

in the decay amplitudes, referred to as direct CP violation. The possibility that direct CP violation is essentially zero, and that CP violation occurs only in the mixing matrix, was referred to as the superweak theory [5].

By applying CPT invariance and unitarity the phase of $\epsilon$ is given approximately by

$$\phi_{\epsilon} \approx \tan^{-1} \frac{2(m_{K_L} - m_{K_S})}{\Gamma_{K_S} - \Gamma_{K_L}} \approx 43.52 \pm 0.05^{\circ}, \quad (68.6a)$$

while Eq. (68.5c) gives the phase of $\epsilon'$ to be

$$\phi_{\epsilon'} = \delta_2 - \delta_0 + \frac{\pi}{2} \approx 42.3 \pm 1.5^{\circ}, \quad (68.6b)$$

where the numerical value is based on an analysis of $\pi-\pi$ scattering using chiral perturbation theory [6]. The approximation in Eq. (68.6a) depends on the assumption that direct CP violation is very small in all $K^0$ decays. This is expected to be good to a few tenths of a degree, as indicated by the small value of $\epsilon'$ and of $\eta_{+-0}$ and $\eta_{000}$, the CP-violation parameters in the decays $K_S \to \pi^+\pi^-\pi^0$ [7], and $K_S \to \pi^0\pi^0\pi^0$ [8]. The relation in Eq. (68.6a) is exact in the superweak theory, so this is sometimes called the superweak-phase $\phi_{SW}$. An important point for the analysis is that $\cos(\phi_{\epsilon'} - \phi_{\epsilon}) \simeq 1$. The consequence is that only two real quantities need be measured, the magnitude of $\epsilon$ and the value of $(\epsilon'/\epsilon)$, including its sign. The measured quantity $|\eta_{00}/\eta_{+-}|^2$ is very close to unity so that we can write

$$|\eta_{00}/\eta_{+-}|^2 \approx 1 - 6\operatorname{Re}(\epsilon'/\epsilon) \approx 1 - 6\epsilon'/\epsilon, \quad (68.7a)$$

$$\operatorname{Re}(\epsilon'/\epsilon) \approx \frac{1}{3} (1 - |\eta_{00}/\eta_{+-}|). \quad (68.7b)$$

From the experimental measurements in this edition of the Review, and the fits discussed in the next section, one finds

$$|\epsilon| = (2.228 \pm 0.011) \times 10^{-3}, \quad (68.8a)$$

$$\phi_{\epsilon} = (43.5 \pm 0.5)^{\circ}, \quad (68.8b)$$

$$\operatorname{Re}(\epsilon'/\epsilon) \approx \epsilon'/\epsilon = (1.66 \pm 0.23) \times 10^{-3}, \quad (68.8c)$$

$$\phi_{+-} = (43.4 \pm 0.5)^{\circ}, \quad (68.8d)$$

$$\phi_{00} - \phi_{+-} = (0.34 \pm 0.32)^{\circ}, \quad (68.8e)$$

$$A_L = (3.32 \pm 0.06) \times 10^{-3}. \quad (68.8f)$$

Direct CP violation, as indicated by $\epsilon'/\epsilon$, is expected in the Standard Model. However, the numerical value cannot be reliably predicted because of theoretical uncertainties [9]. The value of $A_L$ agrees with Eq. (68.5d). The values of $\phi_{+-}$ and $\phi_{00} - \phi_{+-}$ are used to set limits on CPT violation [see “Tests of Conservation Laws”].

## 68.2 Fits for $K_L^0$ CP-violation parameters

In recent years, $K_L^0$ CP-violation experiments have improved our knowledge of CP-violation parameters,

and their consistency with the expectations of CPT invariance and unitarity. To determine the best values of the CP-violation parameters in $K_L^0 \to \pi^+\pi^-$ and $\pi^0\pi^0$ decay, we make two types of fits, one for the phases $\phi_{+-}$ and $\phi_{00}$ jointly with $\Delta m$ and $\tau_S$, and the other for the amplitudes $|\eta_{+-}|$ and $|\eta_{00}|$ jointly with the $K_L^0 \to \pi\pi$ branching fractions.

### 68.2.1 Fits to $\phi_{+-}$, $\phi_{00}$, $\Delta\phi$, $\Delta m$, and $\tau_S$ data

These are joint fits to the data on $\phi_{+-}$, $\phi_{00}$, the phase difference $\Delta\phi = \phi_{00} - \phi_{+-}$, the $K_L^0 - K_S^0$ mass difference $\Delta m$, and the $K_S^0$ mean life $\tau_S$, including the effects of correlations.

Measurements of $\phi_{+-}$ and $\phi_{00}$ are highly correlated with $\Delta m$ and $\tau_S$. Some measurements of $\tau_S$ are correlated with $\Delta m$. The correlations are given in the footnotes of the $\phi_{+-}$ and $\phi_{00}$ sections of the $K_L^0$ Listings, and the $\tau_S$ section of the $K_S^0$ Listings.

In most cases, the correlations are quoted as 100%, with the value and error of $\phi_{+-}$ or $\phi_{00}$ given at a fixed value of $\Delta m$ and $\tau_S$, with additional terms specifying the dependence of the value on $\Delta m$ and $\tau_S$. These cases lead to diagonal bands in Figs. 68.1
<figure><img src="image_328.png" /><figcaption>**Figure 68.1:** $\phi_{+-}$ vs $\Delta m$ for experiments which do not assume CPT invariance. $\Delta m$ measurements appear as vertical bands spanning $\Delta m \pm 1\sigma$, cut near the top and bottom to aid the eye. Most $\phi_{+-}$ measurements appear as diagonal bands spanning $\phi_{+-} \pm \sigma_{\phi}$. Data are labeled by letters: “b”-FNAL KTeV, “c”-CERN CPLEAR, “d”-FNAL E773, “e”-FNAL E731, “f”-CERN, “g”-CERN NA31, and are cited in Table 68.1. The narrow band “j” shows $\phi_{SW}$. The ellipse “a” shows the $\chi^2 = 1$ contour of the fit result.</figcaption></figure>

and 68.2. The KTeV experiment [10] quotes its results as values of $\Delta m$, $\tau_S$, $\phi_\epsilon$, Re($\epsilon'/\epsilon$), and Im($\epsilon'/\epsilon$) with correlations, leading to the ellipses labeled “b.” The correlations for the KTeV measurements are given in the Im($\epsilon'/\epsilon$) section of the $K_L^0$ Listings. For small $|\epsilon'/\epsilon|$, $\phi_{+-} \approx \phi_\epsilon + \text{Im}(\epsilon'/\epsilon)$.

<table><caption>**Table 68.1:** References, Document ID's, and sources corresponding to the letter labels in the figures. The data are given in the $\phi_{+-}$ and $\Delta m$ sections of the $K_L$ Listings, and the $\tau_S$ section of the $K_S$ Listings.</caption><thead><tr><th>Label</th><th>Source</th><th>PDG Document ID</th><th>Ref.</th></tr></thead><tbody><tr><td>a</td><td>This Review</td><td>OUR FIT</td><td></td></tr><tr><td>b</td><td>FNAL KTeV</td><td>ABOUZAID 11</td><td>[10]</td></tr><tr><td>c</td><td>CERN CPLEAR</td><td>APOSTOLAKIS 99C</td><td>[11]</td></tr><tr><td>d</td><td>FNAL E773</td><td>SCHWINGENHEUER 95</td><td>[12]</td></tr><tr><td>e</td><td>FNAL E731</td><td>GIBBONS 93,93C</td><td>[13], [14]</td></tr><tr><td>f</td><td>CERN</td><td>GEWENIGER 74B,74C</td><td>[15], [16]</td></tr><tr><td>g</td><td>CERN NA31</td><td>CAROSI 90</td><td>[17]</td></tr><tr><td>h</td><td>CERN NA48</td><td>LAI 02C</td><td>[18]</td></tr><tr><td>i</td><td>CERN NA31</td><td>BERTANZA 97</td><td>[19]</td></tr><tr><td>j</td><td>This Review</td><td>SUPERWEAK 16</td><td></td></tr></tbody></table>

The data on $\tau_S$, $\Delta m$, and $\phi_{+-}$ shown in Figs. 68.1 and 68.2 are combined with data on $\phi_{00}$ and $\phi_{00}-\phi_{+-}$ in two fits, one without assuming CPT, and the other with this assumption. The results without assuming CPT are shown as ellipses labeled “a.” These ellipses are seen to be in good agreement with the superweak phase

$$ \phi_{SW} = \tan^{-1} \left( \frac{2\Delta m}{\Delta\Gamma} \right) = \tan^{-1} \left( \frac{2\Delta m \tau_S \tau_L}{\hbar(\tau_L - \tau_S)} \right) . \quad (68.9) $$

In Figs. 68.1 and 68.2, $\phi_{SW}$ is shown as narrow bands labeled “j.”

Table 68.2 column 2, “Fit w/o CPT,” gives the resulting fitted parameters, while Table 68.3 gives the correlation matrix for this fit. The white ellipses labeled “a” in Figure 68.1 and 68.2 are the $\chi^2 = 1$ contours for this fit.

For experiments which have dependencies on unseen fit parameters, that is, parameters other than those shown on the x or y

<figure><img src="image_329.png" /><figcaption>**Figure 68.2:** $\phi_{+-}$ vs $\tau_S$. $\tau_S$ measurements appear as vertical bands spanning $\tau_S \pm 1\sigma$, some of which are cut near the top and bottom to aid the eye. Most $\phi_{+-}$ measurements appear as diagonal or horizontal bands spanning $\phi_{+-} \pm \sigma_{\phi}$. Data are labeled by letters: “b”-FNAL KTeV, “c”-CERN CPLEAR, “d”-FNAL E773, “e”-FNAL E731, “f”-CERN, “g”-CERN NA31, “h”-CERN NA48, “i”-CERN NA31, and are cited in Table 68.1. The narrow band “j” shows $\phi_{SW}$. The ellipse “a” shows the fit result's $\chi^2 = 1$ contour.</figcaption></figure>

axis of the figure, their band positions are evaluated using the fit results and their band widths include the fitted uncertainty in the unseen parameters. This is also true for the $\phi_{SW}$ bands.

If CPT invariance and unitarity are assumed, then by Eq. (68.6a), the phase of $\epsilon$ is constrained to be approximately equal to

$$ \phi_{SW} = (43.50258 \pm 0.00021)^{\circ} + 54.1(\Delta m - 0.5289)^{\circ} + 32.0(\tau_S - 0.89564) \quad (68.10) $$

where we have linearized the $\Delta m$ and $\tau_S$ dependence of Eq. (68.9). The error $\pm 0.00021$ is due to the uncertainty in $\tau_L$. Here $\Delta m$ has units $10^{10} \hbar s^{-1}$ and $\tau_S$ has units $10^{-10} \text{ s}$.

If in addition we use the observation that Re($\epsilon'/\epsilon$) $\ll 1$ and $\cos(\phi_{\epsilon'} - \phi_\epsilon) \simeq 1$, as well as the numerical value of $\phi_{\epsilon'}$ given in Eq. (68.6b), then Eqs. 68.5a, which are sketched in Fig 68.3, lead to the constraint

$$ \begin{align} \phi_{00} - \phi_{+-} & \approx -3 \operatorname{Im} \left( \frac{\epsilon'}{\epsilon} \right) \nonumber \\ & \approx -3 \operatorname{Re} \left( \frac{\epsilon'}{\epsilon} \right) \tan(\phi_{\epsilon'} - \phi_\epsilon) \nonumber \\ & \approx 0.006^{\circ} \pm 0.008^{\circ}, \tag{68.11} \end{align} $$

so that $\phi_{+-} \approx \phi_{00} \approx \phi_\epsilon \approx \phi_{SW}$.

<table><caption>**Table 68.2:** Fit results for $\phi_{+-}$, $\Delta m$, $\tau_S$, $\phi_{00}$, $\Delta\phi = \phi_{00} - \phi_{+-}$, and $\phi_\epsilon$ without and with the CPT assumption</caption><thead><tr><th>Quantity (units)</th><th>Fit w/o CPT</th><th>Fit w/ CPT</th></tr></thead><tbody><tr><td>φ<sub>+-</sub>(°)</td><td>43.4 ± 0.5 (S=1.2)</td><td>43.51 ± 0.05 (S=1.2)</td></tr><tr><td>Δm(10<sup>10</sup>ħ s<sup>-1</sup>)</td><td>0.5289 ± 0.0010</td><td>0.5293 ± 0.0009 (S=1.3)</td></tr><tr><td>τ<sub>S</sub>(10<sup>-10</sup>s)</td><td>0.89564 ± 0.00033</td><td>0.8954 ± 0.0004 (S=1.1)</td></tr><tr><td>φ<sub>00</sub>(°)</td><td>43.7 ± 0.6 (S=1.2)</td><td>43.52 ± 0.05 (S=1.3)</td></tr><tr><td>Δφ(°)</td><td>0.34 ± 0.32</td><td>0.006 ± 0.014 (S=1.7)</td></tr><tr><td>φ<sub>ε</sub>(°)</td><td>43.5 ± 0.5 (S=1.3)</td><td>43.52 ± 0.05 (S=1.2)</td></tr><tr><td>χ<sup>2</sup></td><td>16.4</td><td>20.0</td></tr><tr><td># Deg. Free.</td><td>14</td><td>16</td></tr></tbody></table>
In the fit assuming CPT, we constrain $\phi_\epsilon = \phi_{SW}$ using the linear expression in Eq. (68.10), and constrain $\phi_{00} - \phi_{+-}$ using Eq. (68.11). These constraints are inserted into the Listings with the Document ID of SUPERWEAK 16. Some additional data for which the authors assumed CPT are added to this fit or substitute for other less precise data for which the authors did not make this assumption. See the Listings for details.

<figure><img src="image_330.png" /><figcaption>**Figure 68.3:** Sketch of Eqs. 68.5a. Not to scale</figcaption></figure>

The results of this fit are shown in Table 68.2, column 3, “Fit w/CPT,” and the correlation matrix is shown in Table 68.4. The $\Delta m$ precision is improved by the CPT assumption.

<table><caption>**Table 68.3:** Correlation matrix for the results of the fit without the CPT assumption</caption><thead><tr><th></th><th>φ<sub>+-</sub></th><th>Δm</th><th>τ<sub>S</sub></th><th>φ<sub>00</sub></th><th>Δφ</th><th>φ<sub>ε</sub></th></tr></thead><tbody><tr><th>φ<sub>+-</sub></th><td>1.000</td><td>0.596</td><td>-0.488</td><td>0.827</td><td>-0.040</td><td>0.976</td></tr><tr><th>Δm</th><td>0.596</td><td>1.000</td><td>-0.572</td><td>0.487</td><td>-0.035</td><td>0.580</td></tr><tr><th>τ<sub>S</sub></th><td>-0.488</td><td>-0.572</td><td>1.000</td><td>-0.423</td><td>-0.014</td><td>-0.484</td></tr><tr><th>φ<sub>00</sub></th><td>0.827</td><td>0.487</td><td>-0.423</td><td>1.000</td><td>0.529</td><td>0.929</td></tr><tr><th>Δφ</th><td>-0.040</td><td>-0.035</td><td>-0.014</td><td>0.529</td><td>1.000</td><td>0.178</td></tr><tr><th>φ<sub>ε</sub></th><td>0.976</td><td>0.580</td><td>-0.484</td><td>0.929</td><td>0.178</td><td>1.000</td></tr></tbody></table>

<table><caption>**Table 68.4:** Correlation matrix for the results of the fit with the CPT assumption</caption><thead><tr><th></th><th>φ<sub>+-</sub></th><th>Δm</th><th>τ<sub>S</sub></th><th>φ<sub>00</sub></th><th>Δφ</th><th>φ<sub>ε</sub></th></tr></thead><tbody><tr><th>φ<sub>+-</sub></th><td>1.000</td><td>0.972</td><td>-0.311</td><td>0.957</td><td>-0.105</td><td>0.995</td></tr><tr><th>Δm</th><td>0.972</td><td>1.000</td><td>-0.509</td><td>0.958</td><td>-0.007</td><td>0.977</td></tr><tr><th>τ<sub>S</sub></th><td>-0.311</td><td>-0.509</td><td>1.000</td><td>-0.306</td><td>0.004</td><td>-0.312</td></tr><tr><th>φ<sub>00</sub></th><td>0.957</td><td>0.958</td><td>-0.306</td><td>1.000</td><td>0.189</td><td>0.981</td></tr><tr><th>Δφ</th><td>-0.105</td><td>-0.007</td><td>0.004</td><td>0.189</td><td>1.000</td><td>-0.006</td></tr><tr><th>φ<sub>ε</sub></th><td>0.995</td><td>0.977</td><td>-0.312</td><td>0.981</td><td>-0.006</td><td>1.000</td></tr></tbody></table>

### 68.2.2 Fits for $\epsilon'/\epsilon$, $|\eta_{+-}|$, $|\eta_{00}|$, and $B(K_L \to \pi\pi)$

We list measurements of $|\eta_{+-}|$, $|\eta_{00}|$, $|\eta_{00}/\eta_{+-}|$, and $\epsilon'/\epsilon$. Independent information on $|\eta_{+-}|$ and $|\eta_{00}|$ can be obtained from measurements of the $K_L^0$ and $K_S^0$ lifetimes ($\tau_L$, $\tau_S$), and branching ratios (B) to $\pi\pi$, using the relations

$$|\eta_{+-}| = \left[ \frac{B(K_L^0 \to \pi^+\pi^-)}{\tau_L} \frac{\tau_S}{B(K_S^0 \to \pi^+\pi^-)} \right]^{1/2}, \quad (68.12a)$$

$$|\eta_{00}| = \left[ \frac{B(K_L^0 \to \pi^0 \pi^0)}{\tau_L} \frac{\tau_S}{B(K_S^0 \to \pi^0 \pi^0)} \right]^{1/2} . \quad (68.12b)$$

For historical reasons, the branching ratio fits and the CP-violation fits are done separately, but we want to include the

influence of $|\eta_{+-}|$, $|\eta_{00}|$, $|\eta_{00}/\eta_{+-}|$, and $\epsilon'/\epsilon$ measurements on $B(K_L^0 \to \pi^+\pi^-)$ and $B(K_L^0 \to \pi^0\pi^0)$ and vice versa. We approximate a global fit to all of these measurements by first performing two independent fits: 1) BRFIT, a fit to the $K_L^0$ branching ratios, rates, and mean life, and 2) ETAFIT, a fit to the $|\eta_{+-}|$, $|\eta_{00}|$, $|\eta_{+-}/\eta_{00}|$, and $\epsilon'/\epsilon$ measurements. The results from fit 1, along with the $K_S^0$ values from this edition, are used to compute values of $|\eta_{+-}|$ and $|\eta_{00}|$, which are included as measurements in the $|\eta_{00}|$ and $|\eta_{+-}|$ sections with a document ID of BRFIT 16. Thus, the fit values of $|\eta_{+-}|$ and $|\eta_{00}|$ given in this edition include both the direct measurements and the results from the branching ratio fit.

The process is reversed in order to include the direct $|\eta|$ measurements in the branching ratio fit. The results from fit 2 above (before including BRFIT 16 values) are used along with the $K_L^0$ and $K_S^0$ mean lives and the $K_S^0 \to \pi\pi$ branching fractions to compute the $K_L^0$ branching ratio $\Gamma(K_L^0 \to \pi^0\pi^0)/\Gamma(K_L^0 \to \pi^+\pi^-)$. This branching ratio value is included as a measurement in the branching ratio section with a document ID of ETAFIT 16. Thus, the $K_L^0$ branching ratio fit values in this edition include the results of the direct measurement of $|\eta_{00}/\eta_{+-}|$ and $\epsilon'/\epsilon$. Most individual measurements of $|\eta_{+-}|$ and $|\eta_{00}|$ enter our fits directly via the corresponding measurements of $\Gamma(K_L^0 \to \pi^+\pi^-)/\Gamma(\text{total})$ and $\Gamma(K_L^0 \to \pi^0\pi^0)/\Gamma(\text{total})$, and those that do not have too large errors to have any influence on the fitted values of these branching ratios. A more detailed discussion of these fits is given in the 1990 edition of this Review [20].

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# 69. Review of Multibody Charm Analyses

Revised August 2023 by J. Rademacker (Bristol U.); written with D.M. Asner (BNL)

## 69.1 Overview

The study of multibody charm decays is a vibrant field, with vast and fast-increasing datasets, new developments in theory and experimental technique, and implications reaching beyond charm. This review is structured as follows: Sec. 69.2 summarizes key aspects of amplitude models that describe multibody charm decays, their application to data, and theory progress; Sec. 69.3 reviews the special role of charm threshold data in model-independent approaches; and Sec. 69.4 describes applications of multi-body charm analyses, focusing on Charge-Parity (CP) violation, charm mixing and the role of threshold data in this context. In Sec. 69.5, we conclude.

## 69.2 Kinematics & Models

The differential decay rate to a point s = (s₁, ..., sₙ) in n dimensional phase space can be expressed as

$$d\Gamma = |\mathcal{M}(s)|^2 \left| \frac{\partial^n \phi}{\partial(s_1 \dots s_n)} \right| d^n s \quad (69.1)$$

where $|\partial^n\phi/\partial(s_1 \dots s_n)|$ represents the density of states at s, and $\mathcal{M}$ the matrix element for the decay at that point in phase space, which is 2, 5, 8, ... dimensional for D decays to 3, 4, 5, ... spinless particles. Additional parameters are required to fully describe decays involving particles with non-zero spin in the initial or final state.

For the important case of D decays to three pseudoscalars, the decay kinematics can be represented in a two-dimensional Dalitz plot [1]. This is usually parameterized in terms of $s_{12} \equiv (p_1+p_2)^2$ and $s_{23} \equiv (p_2+p_3)^2$, where $p_1, p_2$, and $p_3$ are the four-momenta of the final-state particles. In terms of these variables, phase-space density is constant across the kinematically allowed region, so that any structure seen in the Dalitz plot is a direct consequence of the dynamics encoded in $|\mathcal{M}|^2$. Note that decays to four or more particles cannot be unambiguously described in terms of analogously-defined variables $s_{ij}$ and $s_{ijk}$, which are all parity-even. Parity-odd kinematics are absent in three-body decays because the final-state momenta are confined to a plane; the use of parity-odd observables in four body decays is discussed in Sec. 69.4.4.

In the widely-used isobar approach, the matrix element $\mathcal{M}$ is modeled as a sum of interfering decay amplitudes, each proceeding through resonant two-body decays [2–6]. See [2, 7, 8] for a review of resonance phenomenology. In most analyses, each resonance is described by a relativistic Breit-Wigner [9] or Flatté [10] lineshape, and the model includes a non-resonant term with a constant phase and magnitude. This approach has well-known theoretical limitations, such as the violation of unitarity and analyticity, which can break the relationship between magnitude and phase across phase space. This motivates the use of more sophisticated descriptions, especially for broad, overlapping resonances (frequently found in S-wave components) where these limitations are particularly problematic. In charm analyses, these approaches have included the K-matrix [11] which respects two-body unitarity; the use of LASS scattering data [12]; dispersive methods [13–16]; multi-meson models using chiral Lagrangians [17–20]; and methods based on chiral unitarity [21–24]; QCD factorization [25–28]; and quasi model-independent parameterizations that use generic lineshapes, with minimal theory input and many free parameters, for a subset of resonances [29–37].

An important example for a multibody charm decay is $D^0 \to K_S^0 \pi^+ \pi^-$, which is a key channel in CP violation and charm-mixing analyses. The latest analysis, published jointly by BaBar and Belle [38], describes the Dalitz plot with 1.1M $D^0 \to K_S^0 \pi^+ \pi^-$ events with 18 resonant components, including four doubly Cabibbo suppressed ones. The model uses a K-matrix description for the $\pi\pi$ S-wave based on [39] and input from LASS scattering data [12, 40] for the $K\pi$ S-wave, with no need to add a non-resonant component to describe the data. These data were re-analyzed by [25] in a QCD factorization framework, using line-shape parameterizations for the S [41, 42] and P wave [14] con-

tributions that preserve two-body unitarity and analyticity. The analyses give compatible results for the components they share.

The field of charm amplitude analyses remains very active. Publications since the last update of this review two years ago include Dalitz plot analyses of $D^0 \to K_S^0 \pi^+ \pi^-$, $D^0 \to K_S^0 \pi^+ \pi^-$, $D_S^+ \to K_S^0 K_S^0 \pi^+$, $D_S^+ \to K^+ \pi^- \pi^+$, $D_S^+ \to \pi^+ \pi^0 \eta'$, and $D_S^+ \to \pi^+ \pi^0 \pi^0$ by BESIII [43–48]; $D_S^+ \to \pi^+ \pi^- \pi^+$ by LHCb [36] and BESIII [35]; and $D^+ \to \pi^+ \pi^- \pi^+$ by LHCb [37]. LHCb studied the amplitude structure of $\Lambda_c \to pK^- \pi^+$ [49] and BESIII that of $\Lambda_c \to \Lambda \pi^+ \pi^0$ [50]. BESIII continued its series of four-body amplitude analyses with $D^+ \to K_S^0 \pi^+ \pi^0 \pi^0$ and $D_S^+ \to K^+ \pi^- \pi^+ \pi^0$ [51, 52], and performed a five-body amplitude analysis of $D_S^+ \to K^+ K^- \pi^+ \pi^+ \pi^-$ [53]. These analyses provide valuable insight into hadronic dynamics and have wider impact beyond that. The amplitude structure of $\Lambda_c \to pK^- \pi^+$ for example is found to have a high sensitivity to the $\Lambda_c$ polarization, which can prove valuable in polarization measurements of $\Lambda_b$ baryons decaying to $\Lambda_c$ [49, 54]. The importance of charm amplitude analyses for CP violation and charm mixing is further discussed below. Comparing the $D^0 \to K_L^0 \pi^+ \pi^-$ amplitude structure with that of $D^0 \to K_S^0 \pi^+ \pi^-$ reveals U-spin breaking effects [43]. The $D_S^+$ and $D^+ \to \pi^+ \pi^- \pi^+$ analyses mentioned above, as well an earlier one by BaBar [31], analyse the $\pi^+ \pi^-$ S-wave using a quasi-model-independent approach. They observe substantial differences between the $\pi^+ \pi^+$ S-wave components observed in the two decays, and between the associated phase shift seen in either compared to that observed in scattering data [55–57]. These observations suggest that broad scalar meson resonances in these decays might be generated from meson-meson rescattering [37]; more broadly, they emphasize the importance of final state interactions in multibody decays.

Noteworthy is the increasing sophistication of recent amplitude analyses, most of which go substantially beyond the isobar model with Breit Wigner and Flatté lineshapes. However, with the notable exceptions of [15, 16, 23, 24, 58, 59], they remain within the isobar framework which describes the decay as a series of two-body processes, ignores long-range hadronic effects such as rescattering, and does not respect three (or four)-body unitarity and analyticity.

Several groups work on improved models. Dispersive techniques, which respect three-body unitarity and analyticity, have been applied to regions of the $D^+ \to K^- \pi^+ \pi^+$ and $D^+ \to K_S^0 \pi^0 \pi^+$ Dalitz plots below the $\eta'K$ threshold [15, 16], where they provide a good description of the data with fewer fit parameters than the isobar approach. Reference [58] uses a unitary coupled channel approach to describe $D^+ \to K^- \pi^+ \pi^+$, which has no restrictions on the kinematic range, but requires additional parameters to describe the Dalitz plot above the $\eta'K$ threshold. Using an effective chiral Lagrangian, the authors of reference [19] provide a description of the annihilation contribution to the decay amplitude which respects three-body unitarity. The approach provides a good description of LHCb's $D^+ \to K^+ K^- K^+$ data, with fewer parameters than an equivalent isobar model [59]. The same channel has more recently been re-analyzed by the authors of [23], who argue that the internal emission diagram should dominate and use a chiral unitarity-based approach to achieve a reasonable description of the data with two free parameters.

Limitations in the theoretical description of interfering resonances are the leading source of systematic uncertainty in many analyses. In some cases, the model uncertainty can be removed through model-independent methods, often relying on input from the charm threshold, as discussed in Sec. 69.3. The authors of [60] expand the scope and applicability of the quasi model-independent approach in amplitude fits. At the same time, increasingly sophisticated models are being developed, and applied to data. The authors of [20] and [61] provide valuable practical frameworks for sophisticated amplitude analyses.
## 69.3 Model Independent Methods and the Charm Threshold

Precision measurements of the CP-violation phase $\gamma/\phi_3$ using $B^- \to DK^-$, $D \to K_S^0\pi^+\pi^-$ and related decay modes, as well as precision measurements of charm-mixing parameters in decay modes such as $D^0 \to K_S^0\pi^+\pi^-$, require input on the relative decay amplitudes of $D^0$ and $\bar{D}^0$. In the above examples, $K_S^0\pi^+\pi^-$ can be replaced with other final state accessible to both $D^0$ and $\bar{D}^0$ [62–70]. While the magnitudes of the decay amplitudes can be measured precisely using the vast charm samples at the B factories and LHCb, obtaining the relative phases requires either amplitude models with reliable phase motion, or model-independent approaches. More details on the measurements of $\gamma/\phi_3$ and charm mixing can be found in sections 69.4.1 and 69.4.2.

Model-independent measurements of relative phases between $D^0$ and $\bar{D}^0$ decay amplitudes rely on interference effects in the decays of well-defined coherent superpositions of $D^0$ and $\bar{D}^0$. These are accessible at the charm threshold [65, 68, 70–75], where CLEO-c and BESIII operate. There, quantum-correlated D-meson pairs are produced, which have opposite CP and flavour content. For example if one (the tag) is identified as a CP eigenstate through its decay (e.g. to $K^+K^-$ or $K_S\pi^0$) the other D meson is an eigenstate state with the opposite CP eigenvalue. The decay rate of a CP eigenstate $\frac{1}{\sqrt{2}}(D^0 \pm \bar{D}^0)$ is proportional to $|A_D|^2 + |\bar{A}_D|^2 \pm 2|A_D||\bar{A}_D|\cos(\delta_D)$, where $A_D$ and $\bar{A}_D$ are the $D^0$ and $\bar{D}^0$ decay amplitudes, and $\delta_D$ is their sought-after phase difference. Other quantum correlations give access to $\sin(\delta_D)$. For multibody decays, $A_D$, $\bar{A}_D$ and $\delta_D$ vary across phase space. Model-independent phase information is measured either integrated over the entire phase space of the decay, or in sub-regions/bins. The results can be expressed in terms of one complex parameter $\mathcal{Z} = Re^{-i\delta} = c + i\bar{c}$ per pair of CP-conjugate bins; c and $\bar{c}$ are the average of $\cos(\delta_D)$ and $\sin(\delta_D)$ over that region of phase space. Amplitude models can be used to optimize the binning for sensitivity to $\gamma/\phi_3$, without introducing a model-dependent bias in the result [76]; novel unbinned approaches have the potential to further increase the precision on $\gamma/\phi_3$ in future analyses [77–79].

CLEO-c data have been analyzed to provide binned $\mathcal{Z}$ for the self-conjugate decays $D^0 \to K_{S,L}\pi^+\pi^-$, $K_{S,L}K^+K^-$, $\pi^+\pi^-\pi^+\pi^-$, and $K_S^0\pi^-\pi^+\pi^0$ and phase space-integrated values for $D^0$, $\bar{D}^0 \to K_S^0K^+\pi^-$, $K^+\pi^-\pi^0$ and $K^+\pi^-\pi^+\pi^+$ [80–85]. BESIII significantly improved the precision for the $K_S^0\pi^+\pi^-$, $K_S^0K^+K^-$, $K^+\pi^-\pi^0$ and $K^+\pi^-\pi^+\pi^-$ final states [86–89]. CLEO-c and BESIII $K^+\pi^-\pi^+\pi^-$ data were also analysed in four bins of five-dimensional phase space [89, 90] with a binning based on [33].

For self-conjugate decays such as $D^0 \to \pi^+\pi^-\pi^0$, one can define the CP-even fraction $F_+$ which is +1 for a CP-even eigenstate and 0 for a CP-odd one [73]. $F_+$ is related to $\text{Re}(\mathcal{Z}) = c$, defined for a single CP-conjugate bin pair. For the purpose of $\gamma/\phi_3$ measurements, this information allows the decay to be treated equivalently to a two-body decay, where $\gamma/\phi_3$ is determined from relative decay rates in multiple decay modes, without the need to resolve the phase-space distribution. In the mathematical expressions for the decay rates, a factor $(2F_+ - 1)$ multiplies the $\gamma/\phi_3$-sensitive term, leading to best sensitivity for $F_+ = 1$ or $F_+ = 0$. $D^0 \to \pi^+\pi^-\pi^0$ is found to be, with $F_+ = (97.3 \pm 1.7)\%$, compatible with being completely CP-even (CLEO-c data); a recent BESIII analysis of $D^0 \to K_{L,S}\pi^+\pi^-$ finds $F_+ = (35.3 \pm 0.6 \pm 1.4)\%$ and $(55.6 \pm 0.6 \pm 1.2)\%$, respectively; $D^0 \to \pi^+\pi^-\pi^+\pi^-$ (BESIII, CLEO-c data), $D^0 \to K^+K^-\pi^+\pi^-$ (BESIII), $D^0 \to K^+K^-\pi^0$ (CLEO-c data), are all predominantly CP even with $F_+$ ranging from 73% to 77%; $D^0 \to K_S^0\pi^+\pi^-\pi^0$ (BESIII, CLEO-c data), is predominantly CP odd with $F_+ = (23.5 \pm 10)\%$ [43, 74, 82, 83, 91–93]. In cases where results from both BESIII and CLEO-c data exist, BESIII's more precise results are quoted.

Comparing measured parameters like $F_+$ and $\mathcal{Z}$ with those predicted from amplitude models provides a test of the models' phase motion. Model-predicted values for $F_+$ in $D^0 \to K_{L,S}\pi^+\pi^-$ agree, with $(35.3 \pm 0.6 \pm 1.4)\%$ and $(55.6 \pm 0.6 \pm 1.2)\%$ [38, 43] well

with the measured values given above; as does the measured $F_+$ for $D^0 \to \pi^+\pi^-\pi^+\pi^-$, $F_+ = 0.769 \pm 0.023$ [92], with $0.729 \pm 0.020$ calculated from [32]; and the coherence factor $|\mathcal{Z}| = R$ measured in $D^0 \to K^\pm\pi^+\pi^\pm\pi^\mp$, $0.44^{+0.10}_{-0.09}$ [85, 94], with $0.459 \pm 0.025$ predicted by [33]. The latest bin-wise comparisons between model and data for $D^0 \to K_{L,S}\pi^+\pi^-$, $K^\pm\pi^+\pi^\pm\pi^\mp$, $K^\pm\pi^+\pi^0$, $K_S^0K^+K^-$, and $\pi^+\pi^-\pi^+\pi^-$ can be found in [43, 82, 86–89, 89] and generally show good agreement. These results are a welcome surprise given the preceding discussion on the theoretical shortcomings of amplitude models.

## 69.4 Applications of multibody charm analyses for CP violation and mixing

Amplitude analyses provide sensitivity to both relative magnitudes and phases of the interfering decay amplitudes. It is especially the sensitivity to phases that makes amplitude analyses such a uniquely powerful tool for studying a wide range of phenomena. Here we concentrate on their critical role in CP violation and mixing measurements. Properties of light-meson resonances determined in D amplitude analyses are reported in the light-unflavored-meson section of this Review.

The closely related topics of multibody charm decays in measurements of CP violation in beauty decays to charm and charm mixing will be discussed in turn, followed by a review of searches for time-integrated CP violation in multibody charm decays.

### 69.4.1 CP violation in decays of Beauty to Charm

Neutral D mesons originating from $B^- \to DK^-$ (here denoted as $D_{B^-}$) are a superposition of $D^0$ and $\bar{D}^0$ with a relative phase that depends on the CKM unitarity triangle parameter $\gamma/\phi_3$,

$$D_{B^{-}} \propto D^{0} + r_{B} e^{i(\delta_{B^{-}} - \gamma)} \bar{D}^{0},$$

where $\delta_B$ is a CP conserving strong phase, and $r_B \sim 0.1$ is the magnitude of the ratio of the $B^- \to D^0K^-$ and $B^- \to D^0K^-$ decay amplitudes. In the corresponding CP-conjugate expression, $\gamma/\phi_3$ changes sign. The amplitude analysis of the subsequent $D_{B^\pm}$ decay to a state accessible to both $D^0$ and $\bar{D}^0$ allows the measurement of $\gamma/\phi_3$ [62–69]. The method generalizes to similar B hadron decays, such as $B^0 \to DK^{*0}$, and, albeit with reduced sensitivity, to decays where the kaon is replaced by a pion.

Measurements based on this technique have been reported by BaBar [95, 96], Belle [97–99], Belle II [100], and LHCb [101–114]. Since it relies on $D^0 - \bar{D}^0$ interference, the phase differences between the relevant $D^0$ and $\bar{D}^0$ decay amplitudes are required in order to extract $\gamma/\phi_3$. Because the theoretical shortcomings of amplitude models discussed above make their phases unreliable, most recent $\gamma/\phi_3$ measurements use amplitude model-independent approaches to obtain this information [98–105, 107–114]. These depend critically on information from the charm threshold described in Sec. 69.3. Charm mixing also results in a (time-dependent) $D^0 - \bar{D}^0$ superposition, that can be used to measure the relevant phase information as input to $\gamma/\phi_3$ measurements. This method is particularly powerful in doubly Cabibbo-suppressed decays such as $D^0 \to K^+\pi^-\pi^+\pi^-$, and when used in combination with threshold data [115, 116]. Under some circumstances, with large data sets, the relevant strong phases and $\gamma/\phi_3$ can be extracted simultaneously without external input, for example in simultaneous analysis of the $B^0 \to DK^+\pi^-$ Dalitz plot and that of the subsequent $D \to K_S^0\pi^+\pi^-$ decay [117]. However, the global effort to measure $\gamma/\phi_3$ to sub-degree precision will continue to rely critically on input from the charm threshold and BESIII's increasing datasets [118].

The most precise individual $\gamma/\phi_3$ measurement with an uncertainty of $\sim 5^\circ$ results from LHCb's amplitude model-independent study of $D_{B^-} \to K_S^0\pi^+\pi^-$ and $D_{B^-} \to K_S^0K^+K^-$ [119], using input from the charm threshold [80, 87]. LHCb's analysis of $D_{B^-} \to K^+\pi^-\pi^+\pi^-$ decays leads to the second most precise individual $\gamma/\phi_3$ measurement with $\gamma/\phi_3 = 55.8^{+6.0}_{-5.8}^\circ \pm 0.6^{+6.7}_{-4.3}^\circ$ [113]. Here, the first uncertainty is due to $B^\pm \to DK^\pm$ statistics, the second systematic, and the third due to the uncertainty on charm-threshold input. This illustrates the importance of charm-threshold data, and the need for the analysis of larger threshold datasets to keep pace with the sharp increase in $B^\pm \to DK^\pm$ data
expected at the upgraded LHCb detector and Belle II. Further improvements in this mode are expected from including information on relative phases between $D^0$ and $\bar{D}^0$ amplitudes obtained from charm mixing [85, 94, 116].

The current world average on $\gamma/\phi_3$ is dominated by LHCb who finds, combining its results with input from CLEO-c and BESIII in an amplitude-model independent approach, $\gamma = 63.8^{+3.5}_{-3.7}^\circ$ [111, 120]. Belle II, which is expected to reach excellent precision in $\gamma/\phi_3$, recently released its first $\gamma/\phi_3$ result, based on Belle, Belle II and charm threshold data [100].

The interference between mixing and decay in $B^0 \to D^0 h^0$ with $h^0 = \pi^0, \eta, \omega$ provides sensitivity to $\beta$, which can be extracted from the Dalitz plot of the subsequent $D^0 \to K_S^0 \pi^+ \pi^-$ decay [38, 121–124]. The combined BaBar/Belle analysis based on this technique resolved the ambiguity in $\beta$ present in other measurements [38].

Further details on CP violation in beauty and charm can be found in [125, 126].

### 69.4.2 Charm Mixing and CP violation

Time-dependent amplitude analyses in decays to final states that are accessible to both $D^0$ and $\bar{D}^0$ have unique sensitivity to mixing parameters. A Dalitz plot analysis of a self-conjugate final state, such as $K_S^0 \pi^+ \pi^-$ and $K_S^0 K^+ K^-$, allows a direct measurement of $x$ and $y$, the normalized mass and width difference of the $D^0 - \bar{D}^0$ system's mass eigenstates [127]. This is in contrast to decays like $D^0 \to K^+ \pi^-$ which only provide access to decay-specific parameters $x'^2, y'$.

The phase differences between $D^0$ and $\bar{D}^0$ amplitudes across the Dalitz plot affect these measurements in the same way as those of $\gamma/\phi_3$ in $B^+ \to DK^+$, and can be taken into account in an amplitude model-independent way using the same charm threshold results [65, 70–75, 125]. This approach recently resulted in the first observation of a non-zero mass difference between the neutral charm mass eigenstates by LHCb [128], using a model-independent analysis of $D^0 \to K_S^0 \pi^+ \pi^-$. The “bin flip” technique applied in these measurements exploits that CP-conjugate regions of the $D^0 \to K_S^0 \pi^+ \pi^-$ Dalitz plot have near-identical experimental efficiencies [129]. This measurement is also sensitive to CP violation in mixing and in the interference between mixing and decay, which is discussed further in [125, 126].

### 69.4.3 Everything, Everywhere, All at Once

The discussion above shows how closely related the measurements of $\gamma/\phi_3$, charm mixing, and the analysis of charm threshold data are, due to the shared charm parameters. While above, the emphasis was on the importance of charm as input to $\gamma/\phi_3$, and threshold data as input to charm mixing and $\gamma/\phi_3$, these relationships are in fact omnidirectional. This is demonstrated by LHCb's combined $\gamma/\phi_3$ fit to $\gamma/\phi_3$-sensitive beauty decays to charm such as $B^- \to DK^-$, charm mixing, and charm threshold data. This combined analysis has a notable effect on charm parameters due to the input from beauty decays to charm [111].

### 69.4.4 Searches for time-integrated CPV in charm

The recent observation of direct CP violation in LHCb's analysis of $D^0 \to \pi^+ \pi^-$ and $D^0 \to K^+ K^-$ decays [130, 131], and the open question as to its Standard Model or beyond the Standard Model nature, add renewed interest in the search for CP violation in other decays.

Multibody decays, with their rich structure and varying strong phase across phase space, could potentially have particularly high sensitivity to CP violation. Comparing the results of amplitude fits for CP-conjugate decay modes provides a measure of CP violation, as for example done for $D^0 \to K_S^0 K^\pm \pi^\mp$, $D^0 \to K^+ K^- \pi^+ \pi^-$, and $D^0 \to \pi^+ \pi^- \pi^+ \pi^-$ using LHCb and CLEO-c data [32, 132–134]. A widely-used amplitude model-independent technique to search for local CP violation in multi-body phase space is based on performing a $\chi^2$ comparison of CP-conjugate phase-space distributions. This method was pioneered by BaBar [135] and developed further in [136–138] with results reported by BaBar [135, 139] and LHCb in $D^\pm \to K^+ K^- \pi^\pm$ [140, 141], CDF in $D^0 \to K_S^0 \pi^+ \pi^-$ [142], and LHCb in $D_{(s)}^+ \to K^+ K^- K^+$, $D^+ \to \pi^- \pi^+ \pi^+$, $D^0 \to K^+ K^- \pi^+ \pi^-$,

$D^0 \to \pi^+ \pi^- \pi^+ \pi^-$, and $\Xi_c^+ \to p K^- \pi^+$ [138, 143–145]. Unbinned methods can increase sensitivity [146, 147] and have been applied by LHCb to $D^+ \to \pi^- \pi^+ \pi^+$, $D^0 \to \pi^+ \pi^- \pi^0$, $D^0 \to \pi^+ \pi^- \pi^+ \pi^-$, and $\Xi_c^+ \to p K^- \pi^+$ [144, 145, 148–150].

An alternative model-independent approach is based on observables in four body decays that are odd under motion reversal (“naïve T”) [151–159], which is equivalent to P for scalar particles [159]. One such observable is $C_T = \bar{p}_2 \cdot (\bar{p}_3 \times \bar{p}_4)$ where $\bar{p}_i$ are the decay products' three momenta in the decay's center of mass frame. Comparing the P violating asymmetry $A_T \equiv \frac{\Gamma(C_T>0)-\Gamma(C_T<0)}{\Gamma(C_T>0)+\Gamma(C_T<0)}$ with its C-conjugate provides sensitivity to CP violation. Additional variables are proposed in [159], which also provides an excellent review of this topic. Searches for CP violation using P-odd variables have been carried out by BaBar, Belle, FOCUS, and LHCb in $D^0 \to K^+ K^- \pi^+ \pi^-$ [160–163], $D^+ \to K^+ K_S^0 \pi^+ \pi^-$, $D_S^+ \to K^+ K_S^0 \pi^+ \pi^-$ [164], $D^0 \to K_S^0 \pi^+ \pi^- \pi^0$ [165], and $D^0 \to \pi^+ \pi^- \pi^+ \pi^-$ [149].

Recently, LHCb performed the first full angular analysis of the rare decays $D^0 \to \pi^+ \pi^- \mu^+ \mu^-$ and $D^0 \to K^+ K^- \mu^+ \mu^-$. From this, several observables with sensitivity to physics beyond the Standard Model are constructed, including CP-violating ones. The results are in agreement with Standard-Model predictions [166, 167].

All results of analyses mentioned in this section are compatible with CP conservation. The observation of CP violation in two body charm decays [130, 131], and the vast data samples about to be collected, provide grounds for optimism that this may change in the foreseeable future.

## 69.5 Summary

Multibody charm decays offer a rich phenomenology, including unique sensitivity to CP violation and charm mixing. This is a highly dynamic field with many new results (some of which we presented here) and rapidly increasing, high quality datasets. These datasets constitute a huge opportunity, but also a challenge to improve the theoretical descriptions of soft hadronic effects in multibody decays. For some measurements, model-independent methods, many relying on input from the charm threshold, provide a way of removing model-induced uncertainties. At the same time, substantial progress in the theoretical description of multi-body decays is being made.

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## 70. $D^0-\bar{D}^0$ Mixing

Revised September 2023 by D.M. Asner (BNL) and A.J. Schwartz (U. of Cincinnati).

The formalism for $D^0-\bar{D}^0$ mixing is closely related to that for CP violation; for further details on the latter, see the note “CP Violation in the Quark Sector” in this Review. The time evolution of the $D^0-\bar{D}^0$ system is described by the Schrödinger equation

$$i \frac{\partial}{\partial t} \left( \frac{D^0(t)}{\bar{D}^0(t)} \right) = \left( \mathbf{M} - \frac{i}{2} \Gamma \right) \left( \frac{D^0(t)}{\bar{D}^0(t)} \right), \quad (70.1)$$

where the **M** and **Γ** matrices are Hermitian, and CPT invariance requires that $M_{11} = M_{22} \equiv M$ and $\Gamma_{11} = \Gamma_{22} \equiv \Gamma$. The off-diagonal elements of **M** and **Γ** are referred to as the dispersive and absorptive parts, respectively, of the mixing. The mass eigenstates $D_1$ and $D_2$ of the Hamiltonian $\mathbf{M} - i\mathbf{\Gamma}/2$ are defined as

$$|D_{1,2}\rangle \equiv p|D^0\rangle \pm q|\bar{D}^0\rangle, \quad (70.2)$$

where normalization imposes $|p|^2 + |q|^2 = 1$. If $p = \pm q$, then the mass eigenstates are CP eigenstates and CP is conserved in mixing. Our phase convention is $CP|D^0\rangle = -|\bar{D}^0\rangle$ and Arg($q/p$) $\in [-\pi/2, \pi/2]$, which imply that, in the absence of CP violation in mixing, $D_2$ is CP-even and $D_1$ is CP-odd.

The eigenvalues of $\mathbf{M} - i\mathbf{\Gamma}/2$ are

$$\omega_{1,2} = \left( M - \frac{i}{2} \Gamma \right) \pm \frac{q}{p} \left( M_{12} - \frac{i}{2} \Gamma_{12} \right) \equiv m_{1,2} - \frac{i}{2} \Gamma_{1,2}, \quad (70.3)$$

where $m_{1,2}$ and $\Gamma_{1,2}$ are real and correspond to the masses and decay widths, respectively, of the $D_{1,2}$ mass eigenstates. As the trace $\Gamma_{11} + \Gamma_{22} = 2\Gamma$ is unchanged by diagonalizing $\mathbf{\Gamma}$, $\Gamma$ must equal $(\Gamma_1 + \Gamma_2)/2$, the mean decay width. Solving for the eigenstates of the eigenvalues yields

$$\left(\frac{q}{p}\right)^2 = \frac{M_{12}^* - \frac{i}{2}\Gamma_{12}^*}{M_{12} - \frac{i}{2}\Gamma_{12}}. \quad (70.4)$$

If CP is conserved in mixing, then $(q/p)^2 = 1$ and $M_{12}$ and $\Gamma_{12}$ must be real. In this case, the difference in eigenvalues is $\Delta m \equiv m_2 - m_1 = 2M_{12}$ and $\Delta \Gamma \equiv \Gamma_2 - \Gamma_1 = 2\Gamma_{12}$. The signs of $\Delta m$ and $\Delta \Gamma$ are difficult to predict from theory and thus must be determined experimentally.

We define dimensionless mixing parameters $x$ and $y$ as

$$x \equiv \frac{\Delta m}{\Gamma} \quad (70.5)$$

$$y \equiv \frac{\Delta \Gamma}{2\Gamma}. \quad (70.6)$$

These parameters are measured in several ways. The most precise values are obtained by measuring the time dependence of $D^0$ decays. For all methods, the initial flavor of the $D^0$ or $\bar{D}^0$ when produced must be determined. The most common method used for this is to reconstruct $D^{*+} \to D^0\pi^+$ or $D^{*-} \to \bar{D}^0\pi^-$ decays; the charge of the accompanying pion (which has low momentum in the lab frame and is often referred to as the “soft” pion) determines the flavor of the neutral D. BABAR and LHCb have also identified the flavor of the neutral D by reconstructing the semileptonic decays $B^+ \to \bar{D}^0\ell^+\nu$, $B^0 \to D^{*+}\ell^+\nu$, $B^- \to D^0\ell^-\nu$, and $\bar{B}^0 \to D^{*+}\ell^-\nu$; in this case the charge of the accompanying lepton determines the D flavor. Both experiments have used both tags together to select “double-tagged” $B \to D^{*\pm}\ell^+\nu$, $D^{*\pm} \to (D^0, \bar{D}^0)\pi^\pm$ decays, which have especially high purity. At $e^+e^- \to \psi(3770) \to D^0\bar{D}^0$, there is relatively little background and the purity of opposite-side tagging is equivalent to that achieved using $D^{*\pm}$ decays. However, BESIII operates at a symmetric $e^+e^-$ collider, and the $\bar{D}D$ pairs are produced almost at rest in the lab frame. As a consequence, the D's do

At BESIII, where $\bar{D}D$ pairs are produced near their threshold via $e^+e^- \to \psi(3770) \to D^0\bar{D}^0$, there is relatively little background and the purity of opposite-side tagging is equivalent to that achieved using $D^{*\pm}$ decays. However, BESIII operates at a symmetric $e^+e^-$ collider, and the $\bar{D}D$ pairs are produced almost at rest in the lab frame. As a consequence, the D's do

not travel any appreciable distance before decaying, and time-dependent analyses are not possible. To overcome this, measurements of mixing at BESIII utilize the quantum coherence of the initial $\psi(3770) \to D^0\bar{D}^0$ state and time-integrated measurements [1–5].

### 70.1 Time-Dependent Analyses

Our notation is as follows: Cabibbo-favored (CF) decay amplitudes are denoted $\bar{A}_f \equiv \langle f | H | \bar{D}^0 \rangle$ and $A_{\bar{f}} \equiv \langle \bar{f} | H | D^0 \rangle$; i.e., the final state is $f = K^+\ell^-\nu$, $K^+\pi^-$, $K^+\pi^-\pi^0$, etc. Doubly-Cabibbo-suppressed (DCS) decay amplitudes are denoted $A_f \equiv \langle f | H | D^0 \rangle$ and $\bar{A}_{\bar{f}} \equiv \langle \bar{f} | H | \bar{D}^0 \rangle$. When discussing decay rates, we neglect constant phase-space factors, which divide out in ratios of rates to common final states.

Starting from a pure $|D^0\rangle$ or $|\bar{D}^0\rangle$ state at $t = 0$, the time-dependent decay rates to “wrong-sign” final states can be written

$$r(t) \equiv \left| \langle f | H | D^0(t) \rangle \right|^2 = \left| \bar{A}_f \right|^2 \left| \frac{q}{p} \right|^2 \left| g_+(t) \lambda_f^{-1} + g_-(t) \right|^2 \quad (70.7)$$

$$\bar{r}(t) \equiv \left| \langle \bar{f} | H | \bar{D}^0(t) \rangle \right|^2 = \left| A_{\bar{f}} \right|^2 \left| \frac{p}{q} \right|^2 \left| g_+(t) \lambda_{\bar{f}} + g_-(t) \right|^2, \quad (70.8)$$

where

$$\lambda_f \equiv \frac{q}{p} \frac{\bar{A}_f}{A_f}, \quad \lambda_{\bar{f}} \equiv \frac{q}{p} \frac{\bar{A}_{\bar{f}}}{A_{\bar{f}}}, \quad (70.9)$$

and

$$g_{\pm}(t) = \frac{1}{2} (e^{-i\omega_1 t} \pm e^{-i\omega_2 t}). \quad (70.10)$$

A change in convention for the relative phase of $D^0$ and $\bar{D}^0$ would cancel between $q/p$ and $\bar{A}_f/A_f$ or $\bar{A}_{\bar{f}}/A_{\bar{f}}$, leaving $\lambda_f$ and $\lambda_{\bar{f}}$ unchanged. For multibody final states, these equations apply separately to each point in phase-space. Integrating over regions of phase-space can lead to enhanced sensitivity to CP violation; see the discussion below on multibody decays and the note “Review of Multibody Charm Analyses* in this Review. As the mixing parameters $x$ and $y$ are very small, $r(t)$ and $\bar{r}(t)$ are usually expanded to second order in $x$ and $y$.

### 70.2 Semileptonic decays

Consider the final state $f = K^+\ell^-\bar{\nu}_\ell$, where $A_f = \bar{A}_{\bar{f}} = 0$ is an excellent approximation in the Standard Model. The final state $f$ is accessible from a $D^0$ only via mixing,$^1$ and the decay rate is

$$r(t) = \left| \bar{A}_f \right|^2 \left| \frac{q}{p} \right|^2 |g_-(t)|^2 \approx \left| \bar{A}_f \right|^2 \left| \frac{q}{p} \right|^2 \left( \frac{x^2 + y^2}{4} \right) (\Gamma t)^2 e^{-\Gamma t}. \quad (70.11)$$

For $\bar{r}(t)$, $q/p$ is replaced by $p/q$. In the Standard Model, CP violation in charm mixing is small and $|q/p| \approx 1$. In the limit of CP conservation, $r(t) = \bar{r}(t)$, and the time-integrated mixed decay rate relative to the time-integrated unmixed decay rate for semileptonic decays is

$$\frac{\int_0^\infty r(t) dt}{\int_0^\infty |\bar{A}_f|^2 e^{-\Gamma t} dt} = \frac{x^2 + y^2}{2} \equiv R_M. \quad (70.12)$$

Table 70.1 summarizes results for $R_M$ from semileptonic decays. The world average from the Heavy Flavor Averaging Group (HFLAV) [6] is $R_M = (1.30 \pm 2.69) \times 10^{-4}$.

<footnote>$^1$There exists a doubly Cabibbo-suppressed amplitude in which the c and $\bar{u}$ quarks exchange a W, and then the resulting d quark (from c) decays semileptonically. We neglect this second-order process.</footnote>
<table><caption>**Table 70.2:** Results for $R$, $R_D$, and $A_D$ as measured using $D^0 \to K^\pm\pi^\mp$ decays. When a single uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The measurements with an asterisk (*) have been superseded and thus are not included in the HFLAV global fit (Section 70.7). The measurements with a dagger (†) are not included in the HFLAV global fit due to much poorer precision.</caption><thead><tr><th>Year</th><th>Experiment</th><th>R (×10<sup>-3</sup>)</th><th>R<sub>D</sub> (×10<sup>-3</sup>)</th><th>A<sub>D</sub> (%)</th></tr></thead><tbody><tr><td>2018</td><td>LHCb (5.0 fb<sup>-1</sup> D* tag) [16]</td><td>—</td><td>3.454±0.031</td><td>-0.01±0.91</td></tr><tr><td>2017</td><td>LHCb (3.0 fb<sup>-1</sup> B+D*<br>double tag) [17]</td><td>—</td><td>3.48±0.10</td><td>-3.15±3.31</td></tr><tr><td>2014</td><td>Belle (976 fb<sup>-1</sup>) [18]</td><td>3.86±0.06</td><td>3.53±0.13</td><td>—</td></tr><tr><td>2013</td><td>CDF (9.6 fb<sup>-1</sup>) [19]</td><td>4.30±0.05</td><td>3.51±0.35</td><td>—</td></tr><tr><td>2007</td><td>BABAR (384 fb<sup>-1</sup>) [20]</td><td>3.53±0.08±0.04</td><td>3.03±0.16±0.10</td><td>-2.1±5.2±1.5</td></tr><tr><td colspan="2"><b>HFLAV Fit Result [15]</b></td><td><b>3.434 ± 0.019</b></td><td><b>-0.70 ± 0.36</b></td><td></td></tr><tr><td>2013b*</td><td>LHCb (3.0 fb<sup>-1</sup> D* tag) [21]</td><td>—</td><td>3.568±0.066</td><td>-0.7±1.9</td></tr><tr><td>2013a*</td><td>LHCb (1.0 fb<sup>-1</sup>) [22]</td><td>4.25±0.04</td><td>3.52±0.15</td><td>—</td></tr><tr><td>2008*</td><td>CDF (1.5 fb<sup>-1</sup>) [23]</td><td>4.15±0.10</td><td>3.04±0.55</td><td>—</td></tr><tr><td>2006*</td><td>Belle (400 fb<sup>-1</sup>) [24]</td><td>3.77±0.08±0.05</td><td>3.64±0.18</td><td>2.3±4.7</td></tr><tr><td>2005†</td><td>FOCUS (234 evts) [25]</td><td>4.29<sup>+0.63</sup><sub>-0.61</sub>±0.27</td><td>5.17<sup>+1.47</sup><sub>-1.58</sub>±0.76</td><td>13<sup>+33</sup><sub>-25</sub>±10</td></tr><tr><td>2000†</td><td>CLEO (9.0 fb<sup>-1</sup>) [26]</td><td>3.32<sup>+0.63</sup><sub>-0.65</sub>±0.40</td><td>4.8±1.2±0.4</td><td>-1<sup>+16</sup><sub>-17</sub>±1</td></tr><tr><td>1998†</td><td>E791 (5643 evts) [27]</td><td>6.8<sup>+3.4</sup><sub>-3.3</sub>±0.7</td><td>—</td><td>—</td></tr></tbody></table>

separately fitting the $D^0 \to K^+\pi^-$ and $\bar{D}^0 \to K^-\pi^+$ event samples.

Extraction of the mixing parameters $x$ and $y$ from measurements of $x'$ and $y'$ requires knowledge of the strong phase difference $\delta_{K\pi}$. This can be determined from the decay rates of $D_\pm \to K^+\pi^-$, where $D_+(D_-)$ denotes the CP-even (CP-odd) eigenstate. Since $|D_\pm\rangle = (|D^0\rangle \mp |\bar{D}^0\rangle)/\sqrt{2}$,

$$ \sqrt{2} A(D_{\pm} \to K^{+} \pi^{-}) = A(D^{0} \to K^{+} \pi^{-}) \mp A(\bar{D}^{0} \to K^{+} \pi^{-}). \quad (70.26) $$

Squaring this amplitude and using Eq. (70.13) yields the relation

$$ \cos \delta_{K\pi} = \frac{|A(D_+ \to K^+\pi^-)|^2 - |A(D_- \to K^+\pi^-)|^2}{2|A(D^0 \to K^+\pi^-)||A(\bar{D}^0 \to K^+\pi^-)|\quad (70.27)} $$

Measuring the right-hand side is possible if one can identify pure $D_+$, $D_-$, $D^0$, and $\bar{D}^0$ initial states. This is accomplished at CLEOc and BESIII utilizing the processes $e^+e^- \to \psi(3770) \to \bar{D}^0D^0 \to (f_{CP})(K^+\pi^-)$, or $\psi(3770) \to \bar{D}^0D^0 \to (f_{\bar{D}^0})(K^+\pi^-)$, where $f_{CP}$ denotes a CP-specific final state, and $f_{\bar{D}^0}$ denotes a $\bar{D}^0$-flavor-specific final state. In the first case, quantum coherence and CP symmetry ensures that the $K^+\pi^-$ state originates from a neutral $D$ with CP opposite that of $f_{CP}$. In the second case, at the time when the $\bar{D}^0$ decays, the opposite side is $D^0$. However, it can potentially mix to $\bar{D}^0$ before decaying to $K^+\pi^-$, and this introduces some dependence on the mixing parameters $x$ and $y$. This dependence is seen explicitly in the observable

$$ A_{K\pi}^{CP} \equiv \frac{|A(D_{-} \to K^{-}\pi^{+})|^2 - |A(D_{+} \to K^{-}\pi^{+})|^2}{|A(D_{-} \to K^{-}\pi^{+})|^2 + |A(D_{+} \to K^{-}\pi^{+})|^2} \quad (70.28) $$

To lowest order in the mixing parameters [28],

$$ A_{K\pi}^{CP} = \frac{2\sqrt{R_D} \cos \delta_{K\pi} + y}{1 + R}, \quad (70.29) $$

where $R$ is defined in Eq. (70.25). Such measurements are discussed in Section 70.5.

### 70.3.1 Wrong-sign decays to multibody final states

For multibody final states, Eqs. (70.13)-(70.25) apply to each point in phase-space.<fnref n="3" /> Although $x$ and $y$ do not vary across phase-space, knowledge of the resonant substructure is needed to determine the strong phase difference $\delta$ from point to point to extract $x$

and $y$. Alternatively, experimental knowledge of the strong phase difference between $D^0$ and $\bar{D}^0$ decay amplitudes across phase space [29] allow one to determine $x$ and $y$ independent of a decay model of resonant substructure. This phase information can be measured at the charm threshold, where CLEO-c and BESIII took data.

A time-dependent analysis at BABAR [30, 31] of $D^0 \to K^+\pi^-\pi^0$ decays, relative to CF $\bar{D}^0 \to K^+\pi^-\pi^0$ decays, determined the strong phase variation across the Dalitz plot and reported $x'' = (2.61^{+0.57}_{-0.68}\pm0.39)\%$ and $y'' = (-0.06^{+0.55}_{-0.64}\pm0.34)\%$. These mixing parameters are defined as

$$ x'' = x \cos \delta_{K\pi\pi 0} + y \sin \delta_{K\pi\pi 0} \quad (70.30) $$

$$ y'' = y \cos \delta_{K\pi\pi 0} - x \sin \delta_{K\pi\pi 0}, \quad (70.31) $$

in analogy with $x'$, $y'$, and $\delta_{K\pi}$ of Eqs. (70.20) and (70.21). Here, $\delta_{K\pi\pi 0}$ is the strong phase difference between the amplitudes $A(D^0 \to K^+\rho^-)$ and $A(\bar{D}^0 \to K^+\rho^-)$. The phase difference $\delta_{K\pi\pi 0}$ can be determined in a manner similar to that for $\delta_{K\pi}$: by using Eq. (70.27) and quantum-correlated measurements of the branching fractions $B(D_+ \to K^+\rho^-)$, $B(D_- \to K^+\rho^-)$, $B(D^0 \to K^+\rho^-)$, and $B(\bar{D}^0 \to K^+\rho^-)$ in $e^+e^- \to \psi(3770)$ events.

For the decay modes $D^0$ and $\bar{D}^0 \to K^+\pi^-\pi^+\pi^-$, Belle measured $R = (0.324 \pm 0.008 \pm 0.007)\%$ [32]. Subsequently, a phase-space-integrated analysis from LHCb [33] measured the product of a “coherence factor” $R_D^{K3\pi}$ and the strong-phase-rotated mixing parameter $y''_{K3\pi}$. This measurement resulted in an observation of charm mixing with $8.2\sigma$ significance.

Both the sign and magnitude of $x$ and $y$ without strong phases entering or sign ambiguity can be determined by measuring the time-dependent resonant substructure of multibody $D^0$ decays to self-conjugate final states [34, 35]. For such decays, e.g., $D^0 \to K_S^0\pi^+\pi^-$, the DCS and CF decay amplitudes populate the same Dalitz plot, which allows for direct measurement of the strong phase difference. Belle [35, 36], BABAR [37], and CLEO [38] have measured the overall phase difference between $D^0 \to K^*(892)^-\pi^+$ and $D^0 \to K^*(892)^+\pi^-$ to be [173.9 ± 0.7 (stat. only)]°, [177.6 ± 1.1 (stat. only)]°, and [189 ± 10 ± 3 ± 5]°, respectively. These results are close to the 180° expected from Cabibbo factors, i.e., the relative minus sign between $V_{cs}V_{cd}$ and $V_{us}V_{cd}$; thus they indicate a small strong phase. Four LHCb measurements [39–42] of $x, y$ using $D^0 \to K_S^0\pi^+\pi^-$ decays are decay-model independent, as the model of resonances in the intermediate state is replaced by strong-phase measurements from CLEO-c [43] and BESIII [44]. Table 70.4 summarizes results from time-dependent analyses of self-conjugate multibody final states. World average values for the measurements listed are given later, as a result of the HFLAV global fit.

With regard to resonant substructure in $D^0 \to K_S^0\pi^+\pi^-$ decays, Belle [35, 36] measured the relative strong phase (statistical errors only) and the ratio $R$ (central values only) of the DCS fit

<footnote n="3">³However, if the decay amplitudes involve the third generation, then $A_f/\bar{A}_f$ and $\bar{A}_f/A_f$ in Eq. (70.13) also have weak phase differences. However, this phase difference is estimated to be negligible for the sensitivity of current and foreseeable future experimental measurements.</footnote>
<table><caption>**Table 70.3:** Results for $x'^2$ and $y'$, as measured using $D^0 \to K^\pm\pi^\mp$ decays. When a single uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The measurements with an asterisk (*) have been superseded and thus are not included in the HFLAV global fit. The measurements with a dagger (†) are not included in the HFLAV global fit due to much poorer precision. All confidence limits and intervals correspond to 95% C.L. The Belle 2006 results restrict $x'^2$ to the physical region. The BABAR confidence intervals are obtained from the fit, whereas Belle uses a Feldman-Cousins method, and CDF uses a Bayesian method.</caption><thead><tr><th rowspan="2">Year</th><th rowspan="2">Experiment</th><th colspan="2">No CP violation</th><th colspan="2">Allowing for CP violation</th></tr><tr><th>x'^2 (×10<sup>-3</sup>)</th><th>y' (%)</th><th>x'^2 (×10<sup>-3</sup>)</th><th>y' (%)</th></tr></thead><tbody><tr><td>2018</td><td>LHCb (5.0 fb<sup>-1</sup> <i>D</i><sup>*</sup> tag) [16]</td><td>0.039±0.027</td><td>0.528±0.052</td><td>{ <i>D</i><sup>0</sup>: 0.061±0.037<br> <i>D</i><sup>0</sup>: 0.016±0.039}</td><td>0.501±0.074<br>0.554±0.074</td></tr><tr><td>2017</td><td>LHCb (3.0 fb<sup>-1</sup> <i>B</i>+<i>D</i><sup>*</sup> double tag) [17]</td><td>0.028±0.310</td><td>0.46±0.37</td><td>{ <i>D</i><sup>0</sup>: -0.019±0.447<br> <i>D</i><sup>0</sup>: 0.079±0.433}</td><td>0.581±0.526<br>0.332±0.523</td></tr><tr><td>2014</td><td>Belle (976 fb<sup>-1</sup>) [18]</td><td>0.09±0.22</td><td>0.46±0.34</td><td>—</td><td>—</td></tr><tr><td>2013</td><td>CDF (9.6 fb<sup>-1</sup>) [19]</td><td>0.08±0.18</td><td>0.43±0.43</td><td>—</td><td>—</td></tr><tr><td>2007</td><td>BABAR (384 fb<sup>-1</sup>) [20]</td><td>-0.22±0.37</td><td>0.97±0.54</td><td>{ <i>D</i><sup>0</sup>: -0.24±0.52<br> <i>D</i><sup>0</sup>: -0.20±0.50}</td><td>0.98±0.78<br>0.96±0.75</td></tr><tr><td>2006</td><td>Belle (400 fb<sup>-1</sup>) [24]</td><td>(0.18<sup>+0.21</sup><sub>-0.23</sub>)<sup>*</sup></td><td>(0.06<sup>+0.40</sup><sub>-0.39</sub>)<sup>*</sup></td><td>&lt; 0.72</td><td>-2.8 &lt; y' &lt; 2.1</td></tr><tr><td>2013b*</td><td>LHCb (3.0 fb<sup>-1</sup> <i>D</i><sup>*</sup> tag) [21]</td><td>0.055±0.049</td><td>0.48±0.10</td><td>{ <i>D</i><sup>0</sup>: 0.049±0.070<br> <i>D</i><sup>0</sup>: 0.060±0.068}</td><td>0.51±0.14<br>0.45±0.14</td></tr><tr><td>2013a*</td><td>LHCb (1.0 fb<sup>-1</sup>) [22]</td><td>-0.09±0.13</td><td>0.72±0.24</td><td>—</td><td>—</td></tr><tr><td>2008*</td><td>CDF (1.5 fb<sup>-1</sup>) [23]</td><td>-0.12±0.35</td><td>0.85±0.76</td><td>—</td><td>—</td></tr><tr><td>2005†</td><td>FOCUS (234 evts) [25]</td><td>&lt; 8.3</td><td>-7.2 &lt; y' &lt; 4.1</td><td>&lt; 8.0</td><td>-11.2 &lt; y' &lt; 6.7</td></tr><tr><td>2000†</td><td>CLEO (9.0 fb<sup>-1</sup>) [26]</td><td>0.00±0.23</td><td>-2.3<sup>+1.3</sup><sub>-1.4</sub></td><td>0.00±0.23</td><td>-2.5<sup>+1.4</sup><sub>-1.6</sub></td></tr><tr><td>1998†</td><td>E791 (5643 evts) [27]</td><td>&lt; 17</td><td>&lt; 13</td><td>—</td><td>—</td></tr></tbody></table>

<table><caption>**Table 70.4:** Results from time-dependent multibody analyses. The errors are statistical, systematic, and, when a third error is listed, due to the decay-model, respectively. The measurement with an asterisk (*) has been superseded and thus is not included in the HFLAV global fit. The measurement with a dagger (†) is not included in the HFLAV global fit due to poorer precision. The 2019 LHCb result utilizes strong-phase measurements from CLEO-c [43] and thus is decay-model independent. The 2019–2023 LHCb fits determine CP-violating parameters $\Delta x$ and $\Delta y$; the translation of these parameters to $|q/p|$ and $\phi$ is given in Ref. [45].</caption><thead><tr><th rowspan="2">Year</th><th rowspan="2">Experiment</th><th colspan="3">No CP Violation</th></tr><tr><th>Final state(s)</th><th><i>x</i> (×10<sup>-3</sup>)</th><th><i>y</i> (×10<sup>-3</sup>)</th></tr></thead><tbody><tr><td>2023</td><td>LHCb (5.4 fb<sup>-1</sup> <i>B</i> tag) [42]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>4.29 ± 1.48 ± 0.26</td><td>12.61 ± 3.12 ± 0.83</td></tr><tr><td>2021</td><td>LHCb (5.4 fb<sup>-1</sup> <i>D</i><sup>*</sup> tag) [41]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>3.97 ± 0.46 ± 0.29</td><td>4.59 ± 1.20 ± 0.85</td></tr><tr><td>2019</td><td>LHCb (3.0 fb<sup>-1</sup> <i>B</i>, <i>D</i><sup>*</sup> tags) [40]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>2.7 ± 1.6 ± 0.4</td><td>7.4 ± 3.6 ± 1.1</td></tr><tr><td>2016</td><td>LHCb (1.0 fb<sup>-1</sup> <i>D</i><sup>*</sup> tag) [39]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>-8.6 ± 5.3 ± 1.7</td><td>0.3 ± 4.6 ± 1.3</td></tr><tr><td>2016</td><td>BABAR (468 fb<sup>-1</sup>) [46]</td><td>π<sup>+</sup>π<sup>-</sup>π<sup>0</sup></td><td>15 ± 12 ± 6</td><td>2 ± 9 ± 5</td></tr><tr><td>2014</td><td>Belle (921 fb<sup>-1</sup>) [36]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>5.6 ± 1.9<sup>+0.3</sup><sub>-0.9</sub> ± 0.6</td><td>3.0 ± 1.5<sup>+0.4</sup><sub>-0.5</sub> ± 0.3</td></tr><tr><td>2010</td><td>BABAR (469 fb<sup>-1</sup>) [37]</td><td>{ <i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup><br><i>K</i><sub>S</sub><sup>0</sup> <i>K</i><sup>+</sup><i>K</i><sup>-</sup></td><td>1.6 ± 2.3 ± 1.2 ± 0.8</td><td>5.7 ± 2.0 ± 1.3 ± 0.7</td></tr><tr><td>2007*</td><td>Belle (540 fb<sup>-1</sup>) [35]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>8.0 ± 2.9<sup>+0.9</sup><sub>-0.7</sub> ± 1.0</td><td>3.3 ± 2.4<sup>+0.8</sup><sub>-1.2</sub> ± 0.6</td></tr><tr><td>2005†</td><td>CLEO (9.0 fb<sup>-1</sup>) [34]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>19<sup>+32</sup><sub>-33</sub> ± 4 ± 4</td><td>-14 ± 24 ± 8 ± 4</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">Year</th><th rowspan="2">Experiment</th><th colspan="3">With CP Violation</th></tr><tr><th>Final state(s)</th><th>|<i>q</i>/<i>p</i>|</th><th><i>&phi;</i></th></tr></thead><tbody><tr><td>2023</td><td>LHCb (5.4 fb<sup>-1</sup>) [42]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>{ Δ<i>x</i> × 10<sup>3</sup> =<br>-0.77 ± 0.93 ± 0.28<br>0.996 ± 0.052<br>Δ<i>x</i> × 10<sup>3</sup> =<br>-0.27 ± 0.18 ± 0.01<br>1.05<sup>+0.22</sup><sub>-0.17</sub><br>Δ<i>x</i> × 10<sup>3</sup> =<br>-0.53 ± 0.70 ± 0.22</td><td>{ Δ<i>y</i> × 10<sup>3</sup> =<br>3.01 ± 1.92 ± 0.26<br>(3.2<sup>+2.7</sup><sub>-2.9</sub>)°<br>Δ<i>y</i> × 10<sup>3</sup> =<br>0.20 ± 0.36 ± 0.13<br>(-5.2<sup>+6.3</sup><sub>-9.2</sub>)°<br>Δ<i>y</i> × 10<sup>3</sup> =<br>0.6 ± 1.6 ± 0.3</td></tr><tr><td>2021</td><td>LHCb (5.4 fb<sup>-1</sup>) [41]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td></td><td></td></tr><tr><td>2019</td><td>LHCb (3.0 fb<sup>-1</sup>) [40]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td></td><td></td></tr><tr><td>2014</td><td>Belle (921 fb<sup>-1</sup>) [36]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>0.90<sup>+0.16</sup><sub>-0.15</sub> ± 0.05 ± 0.06<br>-0.16 ± 0.04 ± 0.05</td><td>(-6 ± 11 ± 3<sup>+3</sup><sub>-4</sub>)°</td></tr><tr><td>2007*‡</td><td>Belle (540 fb<sup>-1</sup>) [35]</td><td><i>K</i><sub>S</sub><sup>0</sup> π<sup>+</sup>π<sup>-</sup></td><td>0.86<sup>+0.30</sup><sub>-0.29</sub> ± 0.06<br>-0.29 ± 0.03</td><td>(-14<sup>+16</sup><sub>-18</sub> ± 5<sup>+2</sup><sub>-3</sub>)°</td></tr></tbody></table>

fraction relative to the CF fit fraction for five excited K states: $K^*(892)^+\pi^-$, $K_0^*(1430)^+\pi^-$, $K_2^*(1430)^+\pi^-$, $K^*(1410)^+\pi^-$, and $K^*(1680)^+\pi^-$. Similarly, BABAR [37, 47, 48] reported central values of R for $K^*(892)^+\pi^-$, $K_0^*(1430)^+\pi^-$, and $K_2^*(1430)^+\pi^-$. The systematic uncertainties on R are not evaluated. Large differences in R are observed among these final states, which indicates significant hadronic effects.

## 70.4 Decays to CP Eigenstates

When the final state $f$ is a CP eigenstate, there is no distinction between $f$ and $\bar{f}$. Thus $A_f = A_{\bar{f}}$ and $\bar{A}_f = \bar{A}_f$. We denote final states with CP eigenvalues $\pm 1$ by $f_\pm$. Decays to CP eigenstates proceed mainly via singly Cabibbo-suppressed amplitudes. Such amplitudes can contain internal loops and thus involve the third quark generation; in this manner a weak phase would appear in

<footnote>‡This result allows for all types of CP violation and is superseded by Ref. [36], which assumes no direct CP violation in CF or DCS decays.</footnote>
<table><caption>**Table 70.5:** Results for $y_{CP}$ and $A_Γ$ from $D^0$ decays to CP eigenstates. When a single uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The measurements with an asterisk (*) have been superseded.</caption><thead><tr><th>Year</th><th>Experiment</th><th>Final state(s)</th><th>y<sub>CP</sub> (%)</th><th>A<sub>Γ</sub> (×10<sup>-3</sup>)</th></tr></thead><tbody><tr><td>2022<sup>‡</sup></td><td>LHCb (6 fb<sup>-1</sup> D* tag) [49]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>0.696±0.026±0.013</td><td>—</td></tr><tr><td>2021</td><td>LHCb (8.4 fb<sup>-1</sup> B, D* tags) [50]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.10±0.11±0.03</td></tr><tr><td>2021</td><td>LHCb (6 fb<sup>-1</sup> D* tag) [50]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.27±0.13±0.03</td></tr><tr><td>2021</td><td>LHCb (6 fb<sup>-1</sup> D* tag) [50]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>0.23±0.15±0.03</td></tr><tr><td>2021</td><td>LHCb (6 fb<sup>-1</sup> D* tag) [50]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.40±0.28±0.04</td></tr><tr><td>2020</td><td>Belle (976 fb<sup>-1</sup>) [51]</td><td>K<sub>S</sub><sup>0</sup> ω</td><td>0.96±0.91<sup>+0.64</sup><sub>-0.62</sub></td><td>—</td></tr><tr><td>2019</td><td>LHCb (3 fb<sup>-1</sup> B tag) [52]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>0.57±0.13±0.09</td><td>—</td></tr><tr><td>2017</td><td>LHCb (3 fb<sup>-1</sup> D* tag) [53]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>-0.13±0.28±0.10</td></tr><tr><td>2017</td><td>LHCb (3 fb<sup>-1</sup> D* tag) [53]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>-0.30±0.32±0.10</td></tr><tr><td>2017</td><td>LHCb (3 fb<sup>-1</sup> D* tag) [53]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.46±0.58±0.12</td></tr><tr><td>2016</td><td>Belle (976 fb<sup>-1</sup>) [54]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>1.11±0.22±0.09</td><td>-0.3±2.0±0.7</td></tr><tr><td>2015</td><td>LHCb (3 fb<sup>-1</sup> B tag) [55]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>-1.25±0.73</td></tr><tr><td>2015</td><td>LHCb (3 fb<sup>-1</sup> B tag) [55]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>-1.34±0.77<sup>+0.26</sup><sub>-0.34</sub></td></tr><tr><td>2015</td><td>LHCb (3 fb<sup>-1</sup> B tag) [55]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>-0.92±1.45<sup>+0.25</sup><sub>-0.33</sub></td></tr><tr><td>2015</td><td>BES III (2.9 fb<sup>-1</sup>) [56]</td><td>K<sup>+</sup>K<sup>-</sup>, π<sup>+</sup>π<sup>-</sup><br>K<sub>S</sub><sup>0</sup> π<sup>0</sup>, K<sub>S</sub><sup>0</sup> π<sup>0</sup>π<sup>0</sup><br>K<sub>S</sub><sup>0</sup> η, K<sub>S</sub><sup>0</sup> ω</td><td>-2.0±1.3±0.7</td><td>—</td></tr><tr><td>2014</td><td>CDF (9.7 fb<sup>-1</sup>) [57]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>-1.2±1.2</td></tr><tr><td>2014</td><td>CDF (9.7 fb<sup>-1</sup>) [57]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>-1.9±1.5±0.4</td></tr><tr><td>2014</td><td>CDF (9.7 fb<sup>-1</sup>) [57]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>-0.1±1.8±0.3</td></tr><tr><td>2012</td><td>BABAR (468 fb<sup>-1</sup>) [58]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>0.72±0.18±0.12</td><td>0.9±2.6±0.6</td></tr><tr><td>2009</td><td>Belle (673 fb<sup>-1</sup>) [59]</td><td>K<sub>S</sub><sup>0</sup> K<sup>+</sup>K<sup>-</sup></td><td>0.11±0.61±0.52</td><td>—</td></tr><tr><td>2002</td><td>CLEO (9.0 fb<sup>-1</sup>) [60]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>-1.2±2.5±1.4</td><td>—</td></tr><tr><td>2000</td><td>FOCUS (1 × 10<sup>6</sup> evts) [61]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>3.42±1.39±0.74</td><td>—</td></tr><tr><td>1999</td><td>E791 (2 × 10<sup>10</sup> evts) [62]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>0.73±2.89±1.03</td><td>—</td></tr><tr><td colspan="2"><strong>HFLAV Average [63]</strong></td><td></td><td><strong>0.719 ± 0.113</strong></td><td><strong>0.089 ± 0.113</strong></td></tr><tr><td>2020*</td><td>LHCb (5.4 fb<sup>-1</sup> B tag) [64]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>-0.43±0.36±0.05</td></tr><tr><td>2020*</td><td>LHCb (5.4 fb<sup>-1</sup> B tag) [64]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.22±0.70±0.08</td></tr><tr><td>2013*</td><td>LHCb (1.0 fb<sup>-1</sup> D* tag) [65]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>—</td><td>-0.35±0.62±0.12</td></tr><tr><td>2013*</td><td>LHCb (1.0 fb<sup>-1</sup> D* tag) [65]</td><td>π<sup>+</sup>π<sup>-</sup></td><td>—</td><td>0.33±1.06±0.14</td></tr><tr><td>2011*§</td><td>LHCb (29 pb<sup>-1</sup> D* tag) [66]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>0.55±0.63±0.41</td><td>-5.9±5.9±2.1</td></tr><tr><td>2009*</td><td>BABAR (384 fb<sup>-1</sup>) [67]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>1.16±0.22±0.18</td><td>—</td></tr><tr><td>2008*</td><td>BABAR (384 fb<sup>-1</sup>) [68]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>1.03±0.33±0.19</td><td>2.6±3.6±0.8</td></tr><tr><td>2007*</td><td>Belle (540 fb<sup>-1</sup>) [69]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>1.31±0.32±0.25</td><td>0.1±3.0±1.5</td></tr><tr><td>2003*</td><td>BABAR (91 fb<sup>-1</sup>) [70]</td><td>K<sup>+</sup>K<sup>-</sup> + π<sup>+</sup>π<sup>-</sup></td><td>0.8±0.4<sup>+0.5</sup><sub>-0.4</sub></td><td>—</td></tr><tr><td>2001*</td><td>Belle (23.4 fb<sup>-1</sup>) [71]</td><td>K<sup>+</sup>K<sup>-</sup></td><td>-0.5±1.0<sup>+0.7</sup><sub>-0.8</sub></td><td>—</td></tr></tbody></table>

the decay amplitude, leading to direct CP violation. However, such internal loop amplitudes are suppressed, and the presence of a weak phase is often neglected.

The mixing parameter $y$ may be measured by comparing the rate for $D^0$ decays to CP eigenstates such as $K^+K^-$ with the rate to flavor eigenstates such as $K^-\pi^+$ [14]. If decays to $K^+K^-$ have a shorter effective lifetime than those to $K^-\pi^+$, then $\Gamma_+ > \Gamma_-$, or, since CP violation is very small, $\Gamma_2 > \Gamma_1$ and $y$ is positive. For small mixing ($x, y \ll 1$), the decay rates for $D^0 \to f_\pm$ and $\bar{D}^0 \to f_\pm$ have an approximately exponential time dependence:

$$r_{\pm}(t) \propto \exp(-\Gamma_{\pm}t) \qquad (70.32)$$

$$\bar{r}_{\pm}(t) \propto \exp(-\bar{\Gamma}_{\pm}t), \qquad (70.33)$$

where the effective decay widths are given by

$$\Gamma_{\pm} = \Gamma \left( 1 \pm \left| \frac{q}{p} \right| (y \cos \phi - x \sin \phi) \right) \qquad (70.34)$$

$$\bar{\Gamma}_{\pm} = \Gamma \left( 1 \pm \left| \frac{p}{q} \right| (y \cos \phi + x \sin \phi) \right). \qquad (70.35)$$

Thus, the effective decay rate to a CP eigenstate combining equal numbers of $D^0$ and $\bar{D}^0$ decays (e.g., an untagged sample with no production asymmetry) is

$$r_{\pm}(t) + \bar{r}_{\pm}(t) \propto e^{-(1 \pm y_{CP})\Gamma t}, \qquad (70.36)$$

where

$$y_{CP} = \frac{1}{2} \left( \left| \frac{q}{p} \right| + \left| \frac{p}{q} \right| \right) y \cos \phi - \frac{1}{2} \left( \left| \frac{q}{p} \right| - \left| \frac{p}{q} \right| \right) x \sin \phi \qquad (70.37)$$

$$\approx y \cos \phi - A_M x \sin \phi. \qquad (70.38)$$

If CP is conserved, $y_{CP} = y$. Most measurements of $y_{CP}$ have used $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$ decays, which are CP-even, measured relative to $D^0 \to K^-\pi^+$. Belle measured $y_{CP}$ also using $D^0 \to K_S^0 \omega$ decays [51], which are CP-odd, and $D^0 \to K_S^0 K^+K^-$ decays [59], which are dominated by the CP-odd final state $K_S^0 \phi$. Table 70.5 summarizes the current status of measurements.

In addition to $y_{CP}$, Belle [54], BABAR [58], CDF [57], and LHCb [50, 66] have reported measurements of the decay-rate asymmetry for CP-even final states:

$$A_{\Gamma} \equiv \frac{\Gamma_{+} - \bar{\Gamma}_{+}}{\Gamma_{+} + \bar{\Gamma}_{+}} = \frac{(1/\tau_{+}) - (1/\bar{\tau}_{+})}{(1/\tau_{+}) + (1/\bar{\tau}_{+})} = \frac{\bar{\tau}_{+} - \tau_{+}}{\bar{\tau}_{+} + \tau_{+}} \qquad (70.39)$$

$$\approx \frac{1}{2} \left( \left| \frac{q}{p} \right| - \left| \frac{p}{q} \right| \right) y \cos \phi - \frac{1}{2} \left( \left| \frac{q}{p} \right| + \left| \frac{p}{q} \right| \right) x \sin \phi \qquad (70.40)$$

$$\approx A_M y \cos \phi - x \sin \phi. \qquad (70.41)$$

<footnote>‡This measurement has sufficient precision that $y_{CP}$ for the normalization channel $D^0 \to K^-\pi^+$ must be accounted for. Thus, the measurement is $y_{CP}(h^+h^-) - y_{CP}(K^-\pi^+)$. HFLAV accounts for this small correction in their global fit.</footnote>

<footnote>§This result for $y_{CP}$ is not superseded, but it is not included in the HFLAV average due to having some correlations with the result of Ref. [52] but much worse precision.</footnote>
If CP is conserved, $A_Γ = 0$.

There is a contribution to Eq. (70.41) from direct CP violation, i.e., $|\bar{A}_f/A_f| \neq 1$ [72, 73]. For $f = K^+K^-$ and $\pi^+\pi^-$, this contribution can be estimated from measurements of $A_{CP}(K^+K^-)$ and $A_{CP}(\pi^+\pi^-)$ (see below) and is much smaller than the current uncertainty on $A_M$; thus we neglect it here. We note that, when averaging $A_Γ$ measurements over $K^+K^-$ and $\pi^+\pi^-$ final states, the contribution from direct CP violation cancels, as it has the same magnitude but opposite signs for $K^+K^-$ and $\pi^+\pi^-$ due to U-spin symmetry [73].

The asymmetry $A_Γ$ is an asymmetry in the full decay widths. An asymmetry in partial widths is referred to as $A_{CP}$ and is final-state dependent:

$$A_{CP} \equiv \frac{\Gamma(D^0 \to f) - \Gamma(\bar{D}^0 \to \bar{f})}{\Gamma(D^0 \to f) + \Gamma(\bar{D}^0 \to \bar{f})} \quad (70.42)$$

Unlike $A_Γ$, which is measured by fitting decay time distributions, $A_{CP}$ is measured by fitting for signal yields and (aside from acceptance effects) does not require measuring decay times. For neutral $D$ decays, $A_{CP}$ receives contributions from both direct (in the decay amplitudes) and indirect (due to mixing) processes: $A_{CP}(D^0 \to f) = A_{CP}^f + A_{CP}^{\text{indirect}}$. The latter indirect contribution depends on the mixing parameters $x$ and $y$:

$$A_{CP}^{\text{indirect}} = \frac{1}{2} \left( \left| \frac{q}{p} \right| + \left| \frac{p}{q} \right| \right) x \sin \phi - \frac{1}{2} \left( \left| \frac{q}{p} \right| - \left| \frac{p}{q} \right| \right) y \cos \phi \\ = -A_Γ. \qquad (70.43)$$

Numerous measurements of $A_{CP}$ for decays to CP eigenstates are listed in this Review [74]. Table 70.6 summarizes the current status of measurements of the difference in $A_{CP}$ for $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$ decays: $\Delta A_{CP} \equiv A_{CP}(K^+K^-) - A_{CP}(\pi^+\pi^-)$. Within the Standard Model, $A_{CP}^{KK} \approx -A_{CP}^{\pi\pi}$ [75], and $\Delta A_{CP}$ essentially doubles any direct CP violation present. The difference is also advantageous experimentally, as several systematic uncertainties cancel. As $A_{CP}^{\text{indirect}}$ is independent of final state, it subtracts out of $\Delta A_{CP}$. However, at hadron experiments such as LHCb, there is a difference in efficiencies between $K^+K^-$ and $\pi^+\pi^-$ such that $\langle t \rangle_{KK} \neq \langle t \rangle_{\pi\pi}$, i.e., the mean decay times slightly differ. This difference leads to a small contribution to $\Delta A_{CP}$ from $A_{CP}^{\text{indirect}}$ [72]. The most recent $\Delta A_{CP}$ result from LHCb [76], based on 8.9 fb$^{-1}$ of data, differs from zero with a statistical significance of 5.3$\sigma$. Thus, this measurement constitutes the first observation of CP violation in charm decays. These CP asymmetries are included in HFLAV's global fit for charm mixing parameters discussed below.

<table><caption>**Table 70.6:** Results for the difference in time-integrated CP asymmetries $\Delta A_{CP}$ between $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$ decays. When a single uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The measurements with an asterisk (*) have been either superseded or combined with subsequent results and thus are not included in the HFLAV global fit.</caption><thead><tr><th>Year</th><th>Experiment</th><th>ΔA<sub>CP</sub> (×10<sup>-3</sup>)</th></tr></thead><tbody><tr><td>2019</td><td>LHCb (8.9 fb<sup>-1</sup> B, D* tags) [76]</td><td>-1.54 ± 0.29</td></tr><tr><td>2013</td><td>CDF (9.7 fb<sup>-1</sup> D* tag) [77]</td><td>-6.2 ± 2.1 ± 1.0</td></tr><tr><td>2008</td><td>BABAR (386 fb<sup>-1</sup>) [78]</td><td>2.4 ± 6.2 ± 2.6</td></tr><tr><td>2008</td><td>Belle (540 fb<sup>-1</sup>) [79]</td><td>-8.6 ± 6.0 ± 0.7</td></tr><tr><td>2016*</td><td>LHCb (3.0 fb<sup>-1</sup> D* tag) [80]</td><td>-1.0 ± 0.8 ± 0.3</td></tr><tr><td>2014*</td><td>LHCb (3.0 fb<sup>-1</sup> B tag) [81]</td><td>1.4 ± 1.6 ± 0.8</td></tr><tr><td>2013*</td><td>LHCb (1.0 fb<sup>-1</sup> B tag) [82]</td><td>4.9 ± 3.0 ± 1.4</td></tr><tr><td>2012*</td><td>LHCb (0.62 fb<sup>-1</sup> D* tag) [83]</td><td>-8.2 ± 2.1 ± 1.1</td></tr><tr><td>2012‡</td><td>Belle (976 fb<sup>-1</sup>) [84]</td><td>-8.7 ± 4.1 ± 0.6</td></tr></tbody></table>

## 70.5 Quantum-correlated $D^0\bar{D}^0$ Analyses

Measurements of $R_D$, $\cos\delta_{K\pi}$, $\sin\delta_{K\pi}$, $x$, and $y$ can be obtained from a combined fit to time-integrated yields of single-tagged (ST) and double-tagged (DT) $D^0\bar{D}^0$ events produced at the $\psi(3770)$ resonance. Single-tagged events are those in which either the $D^0$ or $\bar{D}^0$ decay is reconstructed (identified), and the other neutral $D$ decays generically. Double-tagged events are those in which both the $D^0$ and $\bar{D}^0$ decays are identified. Due to quantum correlations, the decay of a $D^0$, $\bar{D}^0$, $D_+$, or $D_-$ projects the other neutral $D$ into a state $\bar{D}^0$, $D^0$, $D_-$, or $D_+$, respectively. The CP-specific $D_-$ and $D_+$ decays (or, neglecting CP violation, $D_1$ and $D_2$ decays) include interference between $D^0$ and $\bar{D}^0$ amplitudes, and this provides sensitivity to $R_D$ and $\cos\delta_{K\pi}$. The flavor-specific $D^0$ and $\bar{D}^0$ decays include interference between $D_1$ and $D_2$ amplitudes, and this provides sensitivity to $x$ and $y$. For details of this method, see Refs. [1–5].

BESIII has reported results using $2.9 \text{ fb}^{-1}$ of $e^{+}e^{-} \to \psi(3770)$ data, where the quantum-correlated $D^0\bar{D}^0$ pairs are produced in a $C = -1$ state. They measure $y_{CP} = (-2.0 \pm 1.3 \pm 0.7)\%$ [56] from DT yields using a CP-eigenstate tag for one $D$ and a flavor-specific semileptonic tag for the other; and they measure $A_{K\pi}^{CP} = (13.2 \pm 1.1 \pm 0.7)\%$ [85] from DT yields using a CP tag for one $D$ and a $K^{\pm}\pi^{\mp}$ tag for the other. For $y_{CP}$, the CP eigenstates used are $K^-K^+(f_+)$, $\pi^+\pi^-(f_+)$, $K_S^0\pi^0\pi^0(f_+)$, $K_S^0\pi^0(f_-)$, $K_S^0\eta(f_-)$, and $K_S^0\omega(f_-)$. For $A_{K\pi}^{CP}$, seven additional CP eigenstates are included: $\pi^0\pi^0(f_+)$, $K_S^0\eta'(f_-)$, $K_S^0\phi(f_-)$, $K_L^0\pi^0(f_+)$, $K_L^0\omega(f_+)$, $K_L^0\pi^0\pi^0(f_-)$, and $\pi^+\pi^-\pi^0$ (mixed CP). Using Eq. (70.29) and external inputs for the CP-even fraction of $D^0 \to \pi^+\pi^-\pi^0$ (from Ref. [86]) and values of $R_D$ and $y$ (from HFLAV [87]), BESIII obtains $\delta_{K\pi} = (7.6^{+10.4}_{-11.6})^{\circ}$ [85].

CLEO-c has reported results using $0.82 \text{ fb}^{-1}$ of $e^{+}e^{-} \to \psi(3770)$ data [88–90]. The values for $y$, $R_M$, $\cos\delta_{K\pi}$, and $\sin\delta_{K\pi}$ are obtained from a combined fit to the ST (hadronic only) and DT yields. The DT yields include events in which one $D$ is reconstructed in a hadronic mode and the other $D$ is partially reconstructed in flavor-specific $D \to K^{\pm}e^{\pm}\nu$ and $D \to K^{\pm}\mu^{\pm}\nu$ modes. The CLEO-c analysis obtains $\cos\delta_{K\pi} = 0.81^{+0.22}_{-0.18} \pm 0.07$ and $\sin\delta_{K\pi} = -0.01 \pm 0.41 \pm 0.04$. These fits allow $\cos\delta_{K\pi}$ and $\sin\delta_{K\pi}$ (and also $x^2$) to be unphysical. Constraining $\cos\delta_{K\pi}$ and $\sin\delta_{K\pi}$ to the physical range $[-1, +1]$ (i.e., interpreting $\delta_{K\pi}$ as an angle) and also using external inputs for $x$, $y$, and $y_{CP}$ from HFLAV 2012 [91], CLEO-c obtains $\delta_{K\pi} = (18^{+11}_{-17})^{\circ}$ [90].

## 70.6 Summary of Experimental Results

The first evidence for $D^0\bar{D}^0$ mixing was obtained in 2007 by Belle [69] and BABAR [20]. These results were confirmed by CDF [92] and, much later, by LHCb [22]. There are now numerous measurements of $D^0\bar{D}^0$ mixing with various levels of sensitivity. For $D^0 \to K^+\pi^-$ decays, LHCb [21, 22], CDF [19], and Belle [18] each exclude the no-mixing hypothesis by more than five standard deviations. LHCb [33] reported the observation of charm mixing in $D^0 \to K^+\pi^-\pi^+\pi^-$ decays with 8.2$\sigma$ significance. However, the strong phase difference for this decay is not known, and thus the mixing parameters $x$ and $y$ cannot be extracted. The most precise measurements of $x$ and $y$ are obtained from a time-dependent Dalitz plot analysis of $D^0 \to K_S^0\pi^+\pi^-$ decays. This method was developed at CLEO [93] and subsequently used at Belle [35, 36] and BABAR [37] with much higher statistics. BABAR has applied this method also to $D^0 \to K_S^0K^+K^-$ decays [37]. It has recently been used by LHCb with very high statistics [41, 42] to obtain the most precise values of $x$ and $y$ to date. This measurement resulted in the first observation ($> 5\sigma$ significance) of dispersive mixing ($x \neq 0$).

The experimental measurements establish that $D^0$ and $\bar{D}^0$ mesons mix. This mixing is presumably dominated by long-distance amplitudes, which are difficult to calculate. Under the assumption that the observed mixing is due entirely to non-Standard Model processes, significant constraints on new physics models can be obtained [94]. A significant limitation to interpreting charm mixing in terms of new physics is the theoretical uncertainty on Standard Model predictions [95, 96]. We note that the HFLAV global fit results for $x$ and $y$ (see below) indicate that

<footnote>‡This preliminary result was not published and thus is not included in the HFLAV global fit.</footnote>
<table><caption>**Table 70.7:** HFLAV global fit results (see text) [15].</caption><thead><tr><th rowspan="2">Parameter</th><th>No CP violation</th><th>No subleading amplitudes for indirect CPV</th><th>No subleading amplitudes in CF+DCS decays</th><th>All CP violation allowed</th><th>95% C.L. Interval (CPV allowed)</th></tr><tr><th>Fit (a)</th><th>Fit (b)</th><th>Fit (c)</th><th>Fit (d)</th><th></th></tr></thead><tbody><tr><td>x (%)</td><td>0.434<sup>+0.126</sup><sub>-0.139</sub></td><td>0.407 ± 0.044</td><td>—</td><td>0.407 ± 0.044</td><td>[0.320, 0.493]</td></tr><tr><td>y (%)</td><td>0.646 ± 0.024</td><td>0.643<sup>+0.024</sup><sub>-0.023</sub></td><td>—</td><td>0.645<sup>+0.024</sup><sub>-0.023</sub></td><td>[0.600, 0.692]</td></tr><tr><td>δ<sub>Kπ</sub> (°)</td><td>11.5<sup>+3.6</sup><sub>-3.7</sub></td><td>11.3<sup>+3.6</sup><sub>-3.8</sub></td><td>10.9<sup>+3.6</sup><sub>-3.8</sub></td><td>11.4<sup>+3.5</sup><sub>-3.8</sub></td><td>[3.7, 18.2]</td></tr><tr><td>R<sub>D</sub> (%)</td><td>0.344 ± 0.002</td><td>0.344 ± 0.002</td><td>0.344 ± 0.002</td><td>0.344 ± 0.002</td><td>[0.340, 0.347]</td></tr><tr><td>A<sub>D</sub> (%)</td><td>—</td><td>—</td><td>—</td><td>-0.77 ± 0.35</td><td>[-1.46, -0.08]</td></tr><tr><td>|q/p|</td><td>—</td><td>1.005 ± 0.007</td><td>—</td><td>0.994<sup>+0.016</sup><sub>-0.015</sub></td><td>[0.96, 1.02]</td></tr><tr><td>φ (°)</td><td>—</td><td>-0.19 ± 0.26</td><td>—</td><td>-2.6<sup>+1.1</sup><sub>-1.2</sub></td><td>[-4.88, -0.37]</td></tr><tr><td>δ<sub>Kππ</sub> (°)</td><td>23.1<sup>+22.8</sup><sub>-23.2</sub></td><td>24.5<sup>+21.4</sup><sub>-22.5</sub></td><td>24.4<sup>+21.5</sup><sub>-22.5</sub></td><td>24.6<sup>+21.4</sup><sub>-22.5</sub></td><td>[-20.1, 65.7]</td></tr><tr><td>A<sub>π</sub> (%)</td><td>—</td><td>0.202 ± 0.059</td><td>0.200 ± 0.059</td><td>0.218 ± 0.060</td><td>[0.10, 0.34]</td></tr><tr><td>A<sub>K</sub> (%)</td><td>—</td><td>0.045 ± 0.053</td><td>0.043 ± 0.053</td><td>0.062 ± 0.054</td><td>[-0.04, 0.17]</td></tr><tr><td>x<sub>12</sub> (%)</td><td>—</td><td>0.407 ± 0.044</td><td>0.407 ± 0.044</td><td>—</td><td>[0.321, 0.494]</td></tr><tr><td>y<sub>12</sub> (%)</td><td>—</td><td>0.643<sup>+0.024</sup><sub>-0.023</sub></td><td>0.641<sup>+0.024</sup><sub>-0.023</sub></td><td>—</td><td>[0.595, 0.687]</td></tr><tr><td>φ<sub>12</sub> (°)</td><td>—</td><td>0.65<sup>+0.92</sup><sub>-0.90</sub></td><td>—</td><td>—</td><td>[-1.13, 2.49]</td></tr><tr><td>φ<sub>2</sub><sup>M</sup> (°)</td><td>—</td><td>—</td><td>0.48 ± 0.92</td><td>—</td><td>[-1.35, 2.32]</td></tr><tr><td>φ<sub>2</sub><sup>T</sup> (°)</td><td>—</td><td>—</td><td>2.40<sup>+1.55</sup><sub>-1.54</sub></td><td>—</td><td>[-0.61, 5.45]</td></tr><tr><td>χ<sup>2</sup>/d.o.f.</td><td>100.6/(63-5)</td><td>68.9/(63-8)</td><td>66.5/(63-9)</td><td>65.4/(63-10)</td><td></td></tr></tbody><tfoot><tr><td></td><td colspan="2">= 1.74</td><td colspan="2">= 1.23</td><td colspan="2">= 1.23</td></tr></tfoot></table>

<figure><img src="image_331.png" /><figcaption>**Figure 70.1:** Two-dimensional $1\sigma-5\sigma$ contours for $(x, y)$ (left) and for $(|q/p|, \text{Arg}(q/p))$ (right) as obtained by HFLAV [15]. In the right plot, the black dot denotes the no-CP-violation point $(0, 0)$.</figcaption></figure>

charm mixing is at the upper end of the range of these predictions. The current situation would benefit from knowledge of the strong phase difference $\delta_{K\pi\pi}$ between $D^0 \to K^+\pi^-\pi^0$ and $D^0 \to K^+\pi^-\pi^0$ decays, and similarly the strong phase difference $\delta_{K\pi\pi\pi}$ between $D^0 \to K^+\pi^-\pi^+\pi^-$ and $D^0 \to K^+\pi^-\pi^+\pi^-$ decays. Such knowledge would allow one to extract $x$ and $y$ from measurements of $(x''^2, y'')$ and $(x''_{K3\pi}, y''_{K3\pi})$ for these three- and four-body decays.

With regard to CP violation, by combining four separate measurements from two data sets totalling 8.9 fb$^{-1}$ of data, LHCb observed CP violation in $D$ decays for the first time [76]. The observable measured is the difference $\Delta A_{CP} = A_{CP}(K^+K^-) - A_{CP}(\pi^+\pi^-)$, which is dominated by direct CP violation, i.e., the potential contributions from indirect CP violation mostly cancel. The amount of direct CP violation observed is small: $\Delta A_{CP} = (-0.154 \pm 0.029)\%$. A theory calculation indicates this value is consistent with Standard Model expectations [97]; however, new physics contributions cannot be excluded. A subsequent LHCb measurement of $A_{CP}(K^+K^-) = (0.068 \pm 0.056)\%$ differs from zero by only $1.2\sigma$ [98]; thus, the $\Delta A_{CP}$ result is interpreted

as indicating direct CP violation in $D^0 \to \pi^+\pi^-$ decays.

## 70.7 HFLAV Global Fit for Charm Mixing Parameters

The Heavy Flavor Averaging Group (HFLAV) performs global fits to all relevant mixing measurements to obtain world average values for the following parameters: mixing parameters $x$ and $y$; strong phase differences $\delta_{K\pi}$ and $\delta_{K\pi\pi 0}$; the ratio $R_D$ of $D^0 \to K^+\pi^-$ and $D^0 \to K^+\pi^-$ partial widths; direct CP-violating parameters $A_D(K^+\pi^-)$, $A_{CP}^{K\pi} \equiv A_K$, and $A_{CP}^{\pi\pi} \equiv A_\pi$; and indirect CP-violating parameters $|q/p|$, $\text{Arg}(q/p) \equiv \phi$, $\phi_{12}$, $\phi_2^M$, and $\phi_2^T$. Four separate fits are performed: (a) assuming no CP violation; (b) assuming no subleading amplitudes for indirect CP violation; (c) assuming no subleading amplitudes in Cabibbo-favored and doubly Cabibbo-suppressed decays; and (d) allowing for all CP violation. Fits (b) and (c) correspond to theory expectations [73,99,100], with the latter being less restrictive. For fit (b), four fitted parameters are reduced to three using the relationship $\tan \phi = (x/y) \cdot (1 - |q/p|^2)/(1 + |q/p|^2)$ [99-101]; alternatively, one can fit for the three parameters $x_{12} \equiv 2|M_{12}|/\Gamma$,
$y_{12} \equiv |\Gamma_{12}|/\Gamma$, and $\phi_{12} \equiv \text{Arg}(M_{12}/\Gamma_{12})$, from which $x, y, |q/p|$, and $\phi$ can be derived. Details of these fits are given in Ref. [87].

The fits use Belle, BABAR, CDF, and LHCb measurements of $D^0 \to K^(*)\ell^-\bar{\nu}$, $K^+K^-$, $\pi^+\pi^-$, $K^+\pi^-$, $K^+\pi^-\pi^0$, $K^+\pi^-\pi^+\pi^-$, $K_S^0\pi^+\pi^-$, $K_S^0K^+K^-$, and $\pi^+\pi^-\pi^0$ decays, as well as CLEO-c and BESIII measurements of $\cos\delta$, $\sin\delta$, and $A_{CP}(K^+\pi^-)$ obtained from quantum-correlated branching fractions measured in $e^+e^- \to \psi(3770) \to D^0\bar{D}^0$ reactions. Correlations among observables are taken into account by using the error matrices provided by the experiments. Three observables input to the fit are themselves world average values calculated by HFLAV: $R_M = (x^2 + y^2)/2$ from $D^0 \to K^(*)\ell^-\bar{\nu}$ decays (Table 70.1), and $y_{CP}$ and $A_\Gamma$ from $D^0 \to f_{CP}$ decays (Table 70.5). A measurement by LHCb of $R_M$ using $D^0 \to K^+\pi^-\pi^+\pi^-$ decays is input separately.

The results of the fit as of September, 2023 are listed in Table 70.7. Confidence contours in the two dimensions $(x, y)$ and $\langle|q/p|, \phi\rangle$ resulting from the fit are plotted in Fig. 70.1. These contours are obtained by allowing, for any point in the two-dimensional plane, all other fitted parameters to take their preferred values. The $1\sigma-5\sigma$ boundaries drawn are the loci of points in which the $\chi^2$ has risen above the minimum by 2.30, 6.18, 11.83, 19.33, and 28.67 units. The fit excludes the no-mixing point $x=y=0$ at more than $11.5\sigma$. The fit is consistent with CP conservation ($|q/p| = 1, \phi = 0$) at the $2\sigma$ level. The $\chi^2$ of the all-CP-violation-allowed fit is 65.4 for 63–10 = 53 degrees of freedom, which is satisfactory. One-dimensional likelihood functions for parameters are obtained by allowing, for any value of the parameter, all other fitted parameters to take their preferred values. The resulting likelihood functions give central values, 68.3% C.L. intervals, and 95% C.L. intervals as listed in Table 70.7.

From the results of the HFLAV global fit, we conclude the following: (1) Since CP violation is small and $y_{CP}$ is positive, the CP-even state is shorter-lived, as in the $K^0\bar{K}^0$ system. (2) Since $x$ is positive, the CP-even state is heavier, unlike in the $K^0\bar{K}^0$ system. (3) The strong phase difference $\delta_{K\pi}$ is small but probably nonzero: the 95% C.L. interval is $3.7^\circ < \delta_{K\pi} < 18.2^\circ$. (4) While direct CP violation has been observed in $D$ decays, there is still no evidence for indirect CP violation, i.e., $|q/p| \neq 1$ or $\phi \neq 0$. Observing such CP violation at the current level of sensitivity would indicate new physics.

## 70.8 Future Data

Current results are based primarily upon CLEO-c (0.82 fb⁻¹ of $e^+e^- \to \psi(3770)$ data), Belle and BABAR (~1.4 ab⁻¹ of $e^+e^- \to \Upsilon(4S)$ data), CDF (9.6 fb⁻¹ of $p\bar{p}$ collision data at $\sqrt{s} = 1.96$ TeV), and LHCb Runs 1 and 2 (3.0 fb⁻¹ + 5.9 fb⁻¹ of $p\bar{p}$ collision data at $\sqrt{s} = 7, 8, 13$ TeV).

BESIII has accumulated 8 fb⁻¹ of $e^+e^- \to \psi(3770)$ data and plans to collect a total of 20 fb⁻¹ by 2025. These data should provide strong phase measurements that would enable improved model-independent determinations of mixing parameters from Belle II and LHCb. In 2019, Belle II began accumulating 50 ab⁻¹ of $e^+e^- \to \Upsilon(4S)$ data [102], which is expected to take approximately ten years to collect. At LHCb, Run 2 was completed in 2018, Run 3 is now in progress, and Run 4 is planned for 2027–30 [103]. The goal for Runs 3 + 4 is to accumulate an additional 50 fb⁻¹ of $p\bar{p}$ data at $\sqrt{s} \approx 14$ TeV [104]. These data, along with the large $e^+e^-$ dataset from Belle II, should provide more precise measurements of $D^0\bar{D}^0$ mixing and direct CP violation, and might possibly uncover indirect CP violation in the neutral $D$ system.

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# 71. $D_s^+$ Branching Fractions

Revised September 2021 by J.L. Rosner (Chicago U.) and C.G. Wohl (LBNL).

Figure 71.1 shows a partial breakdown of the $D_s^+$ branching fractions. The rest of this note is about how the figure was constructed. The values shown make heavy use of CLEO measurements of inclusive branching fractions [1]. For references to other data cited in the following, see the Listings. An addendum updates branching fractions to two-body final states reported by the BESIII Collaboration in 2020 [2], and the conclusion stresses modes still to be identified.

## 71.1 Modes with leptons

The bottom $(18.0 \pm 1.0)\%$ of Fig. 71.1 shows the fractions for the modes that include leptons. The measured $K^0e^+\nu_e$ and $K^*0e^+\nu_e$ fractions have been doubled to take account of the corresponding $\mu^+\nu_\mu$ fractions. The sum of the exclusive $Xe^+\nu_e$ fractions is $(6.0 \pm 0.3)\%$, consistent with an inclusive semileptonic measurement of $(6.5 \pm 0.4)\%$. There seems to be little missing here.

## 71.2 Inclusive hadronic $K\bar{K}$ fractions

The Cabibbo-favored $c \to s$ decay in $D_s^+$ decay produces a final state with both an $s$ and an $\bar{s}$; and thus modes with a $K\bar{K}$ pair or with an $\eta$, $\omega$, $\eta'$, or $\phi$ predominate (as may already be seen in Fig. 71.1 in the semileptonic fractions). We consider the $K\bar{K}$ modes first. A complete picture of the exclusive $K\bar{K}$ charge modes is not yet possible, because branching fractions for many of those modes have not yet been measured. However, CLEO has measured the inclusive $K^+$, $K^-$, $K_S^0$, $K^+K^-$, $K^+K_S^0$, $K^-K_S^0$, and $2K_S^0$ fractions (these include modes with leptons) [1]. And each of these inclusive fractions with a $K_S^0$ is equal to the corresponding fraction with a $K_S^0$: $f(K^+K_S^0) = f(K^+K_S^0)$, $f(2K_S^0) = f(2K_S^0)$, etc. Therefore, of all-inclusive fractions pairing a $K^+$, $K_S^0$, or $K_L^0$ with a $K^-$, $K_S^0$, or $K_L^0$, we know all but $f(K_S^0K_L^0)$.

We can get that fraction. The total $K_S^0$ fraction is

$$f(K_S^0) = f(K^+K_S^0) + f(K^-K_S^0) + 2f(2K_S^0) + f(K_S^0K_L^0) + f(\text{single } K_S^0),$$

where $f(\text{single } K_S^0)$ is the sum of the branching fractions for modes such as $K_S^0\pi^+2\pi^0$ with a $K_S^0$ and no second $K$. The $K_S^0\pi^+2\pi^0$ mode is in fact the only unmeasured single-$K_S^0$ mode (throughout, we shall assume that fractions for modes with a $K$ or $K\bar{K}$ and more than three pions are negligible), and we shall take its fraction to be the same as for the $K_S^02\pi^+\pi^-$ mode, $(0.30 \pm 0.11)\%$. Any reasonable deviation from this value would be too small to matter much in the following. Adding the several small single-$K_S^0$ branching fractions, including those from semileptonic modes, we get $f(\text{single } K_S^0) = (1.7 \pm 0.2)\%$. Using this, we have:

$$\begin{aligned} f(K_S^0K_L^0) &= f(K_S^0) - f(K^+K_S^0) - f(K^-K_S^0) \\ &\quad - 2f(2K_S^0) - f(\text{single } K_S^0) \\ &= (19.0 \pm 1.1) - (5.8 \pm 0.5) - (1.9 \pm 0.4) \\ &\quad - 2 \times (1.7 \pm 0.3) - (1.7 \pm 0.2) \\ &= (6.2 \pm 1.4)\%. \end{aligned}$$

Here and below we treat the errors as uncorrelated, although often they are not. However, our main aim is to get numbers for Fig. 71.1; errors are secondary.

There is a check on our result: The $\phi$ inclusive branching fraction is $(15.7 \pm 1.0)\%$, of which 34%, or $(5.34 \pm 0.34)\%$ of $D_s^+$ decays, produces a $K_S^0K_L^0$. Our $f(K_S^0K_L^0) = (6.2 \pm 1.4)\%$ has to be at least this large—and it is, within the sizable error.

We now have all the inclusive $K\bar{K}$ fractions. We use $f(K^+K\bar{K}^0) = 2f(K^+K_S^0)$, and likewise for $f(K^-K^0)$. For $K^+K^-$ and $K_S^0K_L^0$, we subtract off the contributions from $\phi\ell^+\nu$ decay to

get the purely hadronic $K\bar{K}$ inclusive fractions:

$$\begin{aligned} f(K^+K^-, \text{hadronic}) &= (15.8 \pm 0.7) - (2.1 \pm 0.3) \\ &= (13.7 \pm 0.8)\% \end{aligned}$$
$$\begin{aligned} f(K^+K\bar{K}^0, \text{hadronic}) &= (11.6 \pm 1.0)\% \\ f(K^-K^0, \text{hadronic}) &= (3.8 \pm 0.8)\% \end{aligned}$$
$$\begin{aligned} f(2K_S^0 + 2K_L^0, \text{hadronic}) &= (3.4 \pm 0.6)\% \\ f(K_S^0K_L^0, \text{hadronic}) &= (6.2 \pm 1.4) - (1.5 \pm 0.2) \\ &= (4.7 \pm 1.4)\% \end{aligned}$$

The fractions are shown in Fig. 71.1. They total $(37.2 \pm 2.2)\%$ of $D_s^+$ decays.

<figure><img src="image_332.png" /><figcaption>Figure 71.1: A partial breakdown of $D_s^+$ branching fractions. The hadronic bins in the left column show inclusive fractions. Shading within a bin shows how much of the inclusive fraction is not yet accounted for by adding up all the relevant exclusive fractions. The inclusive hadronic $\phi$ fraction is spread over three bins, in proportion to its decay fractions into $K^+K^-$, $K_S^0K_L^0$, and no-$K\bar{K}$ modes.</figcaption></figure>
We can add more information to the figure by summing up measured branching fractions for exclusive modes within each bin:

**$K^+K^-$ modes**—The sum of measured $K^+K^-\pi^+$, $K^+K^-\pi^+\pi^0$, and $K^+K^-2\pi^+\pi^-$ branching fractions is (12.6 ± 0.6)%. That leaves (1.1 ± 1.0)% for the $K^+K^-\pi^+2\pi^0$ mode, which is the only other $K^+K^-$ mode with three or fewer pions. In Fig. 71.1, this unmeasured part of the $K^+K^-$ bin is shaded.

**$K^+\bar{K}^0$ modes**—Two times the sum of the measured $K^+K_S^0$, $K^+K_S^0\pi^0$, and $K^+K_S^0\pi^+\pi^-$ branching fractions is (8.0 ± 0.5)%. This leaves (3.6 ± 1.1)% for the unmeasured $K^+\bar{K}^0$ modes (there are three such modes with three or fewer pions). This is shaded in the figure.

**$K^-K^0$ modes**—Twice the $K^-K_S^02\pi^+$ fraction is (3.4 ± 0.2)%, which leaves about (0.4 ± 0.8)% for $K^-K^02\pi^+\pi^0$, the only other $K^-K^0$ mode with three or fewer pions.

**$2K_S^0+2K_L^0$ modes**—The $2K_S^0\pi^+$ and $2K_S^02\pi^+\pi^-$ fractions sum to (0.86 ± 0.07)%; this times two (for the corresponding $2K_L^0$ modes) is (1.72 ± 0.14)%. This leaves about (1.7 ± 0.7)% for other $2K_S^0+2K_L^0$ modes.

**$K_S^0K_L^0$ modes**—Most of the $K_S^0K_L^0$ fraction is accounted for by $\phi$ decays (see below).

## 71.3 Inclusive hadronic $\eta, \omega, \eta',$ and $\phi$ fractions

These are easier. We start with the inclusive branching fractions, and then, to avoid double counting, subtract: (1) fractions for modes with leptons; (2) $\eta$ mesons that are included in the inclusive $\eta'$ fraction; and (3) $K^+K^-$ and $K_S^0K_L^0$ from $\phi$ decays:

$$f(\eta \text{ hadronic}) = f(\eta \text{ inclusive}) - 0.65 f(\eta' \text{ inclusive}) - f(\eta \ell^+ \nu) = (18.5 \pm 3.0)\%$$

$$f(\omega \text{ hadronic}) = f(\omega \text{ inclusive}) - 0.026 f(\eta' \text{ inclusive}) = (5.8 \pm 1.4)\%$$

$$f(\eta' \text{ hadronic}) = f(\eta' \text{ inclusive}) - f(\eta' \ell^+ \nu) = (8.5 \pm 1.5)\%$$

$$f(\phi \text{ hadronic}, \not\to K\bar{K}) = 0.17 [f(\phi \text{ inclusive}) - f(\phi \ell^+ \nu)] = (1.9 \pm 0.2)\%.$$

The factors 0.65, 0.026, and 0.17 are the $\eta' \to \eta, \eta' \to \omega$, and $\phi \to K\bar{K}$ branching fractions. Figure 71.1 shows the results; the sum is (34.7 ± 3.6)%, which is about equal to the hadronic $K\bar{K}$ total.

Note that the bin marked $\phi$ near the top of Fig. 71.1 includes neither the $\phi\ell^+\nu$ decays nor the 83% of other $\phi$ decays that produce a $K\bar{K}$ pair. There is twice as much $\phi$ in the $K_S^0K_L^0$ bin, and nearly three times as much in the $K^+K^-$ bin. These contributions are indicated in those bins.

Again, we can show how much of each bin is accounted for by measured exclusive branching fractions: $\eta$ modes—The sum of $\eta\pi^+$, $\eta\pi^+\pi^0$ (nearly all $\eta\rho^+$), and $\eta K^+$ branching fractions is (11.1 ± 1.2)%, which leaves a good part of the inclusive hadronic $\eta$ fraction, (18.5 ± 3.0)%, to be accounted for. This is shaded in the figure. $\omega$ modes—The sum of $\omega\pi^+$, $\omega\pi^+\pi^0$, and $\omega2\pi^+\pi^-$ fractions is (4.6 ± 0.9)%, which is nearly as large as the inclusive hadronic $\omega$ fraction, (5.8 ± 1.4)%. $\eta'$ modes—The sum of $\eta'\pi^+$, $\eta'\rho^+$, and $\eta'K^+$ fractions is (9.9 ± 1.5)%, which is larger than but not in serious disagreement with the inclusive hadronic $\eta'$ fraction, (8.5 ± 1.5)%.

## 71.4 Cabibbo-suppressed modes

The sum of the fractions for modes with a $K\bar{K}$, $\eta, \omega, \eta'$, or leptons is (89.9 ± 4.4)%. The remaining (10.1 ± 4.4)% is to

Cabibbo-suppressed modes, mainly single-$K^+$ pions and multiple-pion modes (see below). However, it should be noted that some small parts of the modes already discussed are Cabibbo-suppressed. For example, the (1.1 ± 0.2)% of $D_s^0$ decays to $K^0\ell\nu$ or $K^0\ell\nu$ is already in the $X\ell\nu$ bin in Fig. 71.1. And the inclusive measurements of $\eta, \omega$, and $\eta'$ fractions do not distinguish between (and therefore include both) Cabibbo-allowed and -suppressed modes. We shall not try to make a separation here.

**$K^0 + \text{pions}$**—Above, we found that $f(\text{single } K_S^0) = (1.7 \pm 0.2)\%$. Subtracting semileptonic fractions with a $K_S^0$ leaves (1.3 ± 0.2)%. The hadronic single-$K^0$ fraction is twice this, about (2.6 ± 0.4)%. The sum of measured $K^0\pi^+$, $K^0\pi^+\pi^0$, and $K^02\pi^+\pi^-$ fractions is (1.8 ± 0.3)%, about two-thirds as much.

**$K^+ + \text{pions}$**—The $K^+\pi^0$ and $K^+\pi^+\pi^-$ fractions sum to (0.72 ± 0.05)%. The total $K^+$ fraction wanted here is probably in the 1-to-2% range.

**Multi-pions**—The $2\pi^+\pi^-, \pi^+2\pi^0$, and $3\pi^+2\pi^-$ fractions total (2.5 ± 0.2)%. Modes not measured might double this.

The sum of the actually measured fractions is, including the semileptonics, (4.9 ± 0.3)%. The error on our Cabibbo-suppressed total, (10.1 ± 4.4)% is too large to know how much we might be missing.

## 71.5 Addendum: Improved precision in some measurements

The Table compares measurements of branching fractions to pairs of two pseudoscalar mesons used in our 2019 update of the 2018 PDG review with values available in 2021 [2]. They have not changed much but are much better known.

<table><thead><tr><th>Final state</th><th>2019 PDG avg or fit (%)</th><th>New BESIII entry in 2021 PDG (%)</th><th>2021 PDG avg or fit (%)</th></tr></thead><tbody><tr><td>$K^+\eta'$</td><td>0.18±0.06</td><td>0.268±0.025</td><td>0.265±0.025</td></tr><tr><td>$K^+\eta$</td><td>0.177±0.035</td><td>0.162±0.012</td><td>0.160±0.011</td></tr><tr><td>$\eta\pi^+$</td><td>1.70±0.09</td><td>1.741±0.063</td><td>1.68±0.10</td></tr><tr><td>$K^+K_S^0$</td><td>1.50±0.05</td><td>1.502±0.055</td><td>1.46±0.04</td></tr><tr><td>$K_S^0\pi^+$</td><td>0.122±0.006</td><td>0.1109±0.005</td><td>0.110±0.005</td></tr><tr><td>$K^+\pi^0$</td><td>0.063±0.021</td><td>0.0748±0.0057</td><td>0.074±0.0057</td></tr></tbody></table>

Other recent BESIII additions include the measurements $B(D_s \to \omega\pi^+, \omega K^+) = (1.77 \pm 0.32 \pm 0.13, 0.87 \pm 0.24 \pm 0.08) \times 10^{-3}$ [3] and $B(D_s \to K_{S,L}K^+) = (1.425 \pm 0.038 \pm 0.031, 1.485 \pm 0.039 \pm 0.046)\%$ [4].

## 71.6 Concluding remarks

The shaded bands in the figure imply that about eight percent of the total $D_s$ decay rate is into $\eta X$, with $X$ still to be accounted for. Unidentified Cabibbo-suppressed modes and the modes $K^+\bar{K}^0X$ and $(2K_S^0 + 2K_L^0)X$ respectively correspond to ~ (5, 3, 2)% of the total. However, one can consider the vast majority of $D_s$ decays to be represented by exclusive final states.

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## 72. Leptonic Decays of Charged Pseudoscalar Mesons

Updated Aug. 2023 by R. A. Briere (Carnegie Mellon U.); written with J. L. Rosner (Chicago U.), S. L. Stone (Syracuse U.) and R. Van de Water (FNAL)

### 72.1 Introduction

This review updates one in Ref. [1]. Extensive use is made of results of the Flavor Lattice Averaging group (FLAG 21 and the 2023 online update), Ref. [2,3].

Charged mesons formed from a quark and antiquark can decay to a lepton-neutrino pair when these objects annihilate via a virtual W boson. Figure 72.1 illustrates this process for the purely leptonic decay of a $D^+$ meson.

<figure><img src="image_333.png" /><figcaption>Figure 72.1: The annihilation process for pure $D^+$ leptonic decays in the standard model.</figcaption></figure>

Similar quark-antiquark annihilations via a virtual $W^+$ to the $\ell^+\nu$ final states occur for the $\pi^+, K^+, D_s^+$, and $B^+$ mesons. (Whenever pseudoscalar-meson charges are specified in this article, use of the charge-conjugate particles and corresponding decays are also implied.) Let P be any of these pseudoscalar mesons. To lowest order, the decay width is

$$ \Gamma^{(0)}(P \to \ell\nu) = \frac{G_F^2}{8\pi} f_P^2 m_\ell^2 M_P \left(1 - \frac{m_\ell^2}{M_P^2}\right)^2 |V_{q1q2}|^2 . \quad (72.1) $$

Here $M_P$ is the P mass, $m_\ell$ is the $\ell$ mass, $V_{q1q2}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element between the quarks $q_1\bar{q}_2$ in P, and $G_F$ is the Fermi coupling constant. The decay constant $f_P$ is proportional to the matrix element of the axial current between the one-P-meson state and the vacuum:

$$ \langle 0|\bar{q}_1\gamma_\mu\gamma_5 q_2|P(p)\rangle = ip_\mu f_P, \quad (72.2) $$

and can be thought of as proportional to the “wave function overlap” of the quark and antiquark. In this article, we use the convention in which $f_\pi \approx 130$ MeV. For brevity, we will often denote the purely leptonic decay width in Eq. (72.1) by $\Gamma^{(0)}$.

The decay of $P^\pm$ starts with a spin-0 meson, and ends up with a left-handed neutrino or right-handed antineutrino. By angular momentum conservation, the $\ell^\pm$ must then also be left-handed or right-handed, respectively. In the $m_\ell = 0$ limit, the decay is forbidden, and can only occur as a result of the finite $\ell$ mass. This helicity suppression is the origin of the $m_\ell^2$ factor in the decay width.

Experimentally, it is difficult to isolate events in which there are only a lepton and neutrino in the final state from those with a lepton, neutrino, and soft photon. Thus, radiative contributions must be removed from the experimental measurements a *posteriori* to obtain $\Gamma^{(0)}$. The radiative contributions can be broken into three pieces: the short-distance contribution to leptonic and semileptonic decays mediated by a $W^\pm$ boson that accounts for electroweak corrections not included in the definition of $G_F$, the long-distance internal bremsstrahlung (IB) contribution, and the contribution from photon emission that depends upon the hadron’s structure. The universal electroweak correction was calculated at $O(\alpha)$ by Sirlin [4], and increases the purely leptonic decay rate by ~1.8–2.2% depending on the decaying meson. The $O(\alpha)$ IB contribution was calculated by Kinoshita [5], and again is universal for all leptonic decays at this order. Numerically, this universal long-distance contribution lowers the purely leptonic decay rate by ~0.4–2.4%, where the correction is smallest for pions and largest for $D_{(s)}$ mesons. The structure-dependent

contributions have been estimated within various effective theories to increase the purely leptonic rate by one to a few percent [6–11].

In this review, we treat the radiative corrections differently for the light, charm, and bottom meson systems for several reasons. First, the experimental uncertainties on the decay widths vary substantially. Thus, while the inclusion of radiative corrections is essential for the pion, kaon, and $D_{(s)}$-meson decay widths, which have been measured to (sub)-percent precision, radiative corrections can be neglected (for now) for $B \to \tau\nu$ decay. Second, the photons are treated differently on the experimental side for the different decay processes. For pions and kaons, the experimental measurements of $\Gamma_{P\ell 2[\gamma]}$ are fully inclusive, while for $D_{(s)}$ mesons, the experiments impose cuts on the energy of any neutral cluster deposited in the calorimeter, which reduce the soft-photon background substantially. Some experiments also remove the QED bremsstrahlung in the leading-logarithmic approximation using the PHOTOS Monte-Carlo generator [12]. Third, the theoretical knowledge of the structure-dependent corrections varies for each meson system.

Once radiative corrections have been accounted for, measurements of purely leptonic decay branching fractions and lifetimes allow an experimental determination of the product $|V_{q1q2}| f_P$. If the decay constant $f_P$ is known to sufficient precision from theory, one can obtain the corresponding CKM element within the standard model. If, on the other hand, one takes the value of $|V_{q1q2}|$ assuming CKM unitarity, one can infer an “experimental measurement” of the decay constant that can then be compared with theory.

The importance of measuring $\Gamma(P \to \ell\nu)$ depends on the particle being considered. Leptonic decays of charged pseudoscalar mesons occur at tree level within the standard model. Thus one does not expect large new-physics contributions to measurements of $\Gamma(P \to \ell\nu)$ for the lighter mesons $P = \pi^+, K^+, \text{and these processes, in principle, provide clean standard-model determinations of } |V_{ud}| \text{ and } |V_{us}|$. The situation is different for leptonic decays of charm and bottom mesons. The presence of new heavy particles such as charged Higgs bosons or leptoquarks could lead to observable effects in $\Gamma(P \to \ell\nu)$ for $P = D_{(s)}^+$, $B^+$ [13–17]. Thus, the determination of $|V_{ub}|$ from $B^+ \to \tau\nu$ decay, in particular, should be considered a probe of new physics. More generally, the ratio of leptonic decays to $\tau\nu$ over $\mu\nu$ final states probes lepton universality [13, 18].

The determinations of CKM elements from leptonic decays of charged pseudoscalar mesons provide complementary information to those from other decay processes. The decay $P \to \ell\nu$ proceeds in the standard model via the axial-vector current $\bar{q}_1\gamma_\mu\gamma_5 q_2$, whereas semileptonic pseudoscalar meson decays $P_1 \to P_2\ell\nu$ proceed via the vector current $\bar{q}_1\gamma_\mu q_2$. Thus the comparison of determinations of $|V_{q1q2}|$ from leptonic and semileptonic decays tests the $V-A$ structure of the standard-model electroweak charged-current interaction. More generally, a small right-handed admixture to the standard-model weak current would lead to discrepancies between $|V_{q1q2}|$ obtained from leptonic pseudoscalar-meson decays, exclusive semileptonic pseudoscalar-meson decays, exclusive semileptonic baryon decays, and inclusive semileptonic decays [19, 20].

Both measurements of the decay rates $\Gamma(P \to \ell\nu)$ and theoretical calculations of the decay constants $f_P$ for $P = \pi^+, K^+, D_{(s)}^+$ from numerical lattice-QCD calculations are now quite precise. As a result, the elements of the first row of the CKM matrix $|V_{ud}|$ and $|V_{us}|$ can be obtained to sub-percent precision from $\pi^+ \to \ell\nu$ and $K^+ \to \ell\nu$, where the limiting error is from continuum theory radiative corrections. The elements of the second row of the CKM matrix $|V_{cd(s)}|$ can be obtained from leptonic decays of charmed pseudoscalar mesons to few-percent precision, where here the limiting error is from experiment. These enable stringent tests of the unitarity of the first and second rows of the CKM matrix.

This review is organized as follows. Because the experimental and theoretical issues associated with measurements of pions and kaons, charmed mesons, and bottom mesons differ, we discuss each one separately. We begin with the pion and kaon sys-