tem in Sec. 72.2. First, in Sec. 72.2.1, we review current measurements of the experimental decay rates. We provide tables of branching-ratio measurements and determinations of the product $|V_{ud(s)}|f_{\pi^+(K^+)}$, as well as average values for these quantities including correlations and other effects needed to combine results. Then, in Sec. 72.2.2, we summarize the status of theoretical calculations of the decay constants. We provide tables of recent lattice-QCD results for $f_{\pi^+}$, $f_{K^+}$, and their ratio from calculations including dynamical $u, d, s$, and (in some cases $c$) quarks, along with averages including correlations and strong SU(2)-isospin breaking corrections as needed. We next discuss the charmed meson system in Sec. 72.3, again reviewing current experimental rate measurements in Sec. 72.3.1 and theoretical decay-constant calculations in Sec. 72.3.2. Last, we discuss the bottom meson system in Sec. 72.4, following the same organization as the two previous sections. For almost all of the decay constants presented in Secs. 72.2.2, 72.3.2, and 72.4.2, we take as our preferred values the four-flavor averages from the FLAG 21 review [2] as updated online in 2023 [3]. This 2023 web update contains only one relevant new result since the 2021 version, on $f_{K^+}/f_{\pi^+}$ from ETM 21 [21].

After having established the status of both experimental measurements and theoretical calculations of leptonic charged pseudoscalar-meson decays, we discuss some implications for phenomenology in Sec. 72.5. For each process discussed in Secs. 72.2-72.4, we combine the average $B(P \to \ell\nu)$ with the decay constant $f_P$ to infer the associated CKM matrix element. We then compare these results with determinations of the same CKM elements from other processes. We also use the CKM elements obtained from leptonic decays to test the unitarity of the first and second rows of the CKM matrix. Further, as in previous reviews, we combine the experimental $B(P \to \ell\nu)$s with the associated CKM elements obtained from CKM unitarity to infer “experimental” values for the decay constants. The comparison of these values with theory provides a test of lattice and other QCD approaches, assuming that new-physics contributions to these processes are not significant.

## 72.2 Pions and kaons

### 72.2.1 Experimental rate measurements

Experimental rate measurements of pion and kaon leptonic decays are fully radiation inclusive. Following Refs. [22, 23], and references therein, we combine the $O(\alpha)$ radiative corrections to the purely leptonic rate as follows:

$$ \Gamma(P \to \ell\nu[\gamma]) = \Gamma^{(0)}(P \to \ell\nu) \left[ 1 + \frac{\alpha}{\pi} C_P \right], \quad (72.3) $$

where $P = \pi, K$. The full expressions for $C_\pi$ and $C_K$ are given in Eq. (114) of Ref. [8]. In addition to the universal short- [4] and long-distance [5] corrections, $C_\pi$ and $C_K$ include hadronic-structure dependent contributions [6] through $O(\alpha p^4)$ in chiral perturbation theory (χPT), where $p$ is the pion or kaon momentum. The inclusion of radiative corrections to the purely leptonic rates is numerically important given the level of precision achieved on the experimental measurements of the $\pi^\pm \to \mu^\pm\nu$ and $K^\pm \to \mu^\pm\nu$ decay widths.

We evaluate $\delta_P \equiv (\alpha/\pi)C_P$ using experimentally-measured meson and lepton masses and coupling constants from the Particle Data Group [24], and taking the low-energy constants (LECs) that parametrize the hadronic contributions from Refs. [8, 25, 26]. Because the finite non-logarithmic parts of the LECs were estimated within the large-$N_c$ approximation assuming that contributions from the lowest-lying resonances dominate, we conservatively assign a 100% uncertainty to the LECs, which leads to a $\pm 0.9$ error in $C_{\pi,K}$.<fnref n="1" /> We obtain the following correction factors to the indi-

vidual charged pion and kaon decay widths:

$$ \delta_{\pi} = 0.0176(21) \quad \text{and} \quad \delta_{K} = 0.0107(21). \qquad (72.4) $$

The error on the ratio of kaon-to-pion leptonic decay widths is under better theoretical control because the hadronic contributions from low-energy constants estimated within the large-$N_c$ framework cancel at lowest order in the chiral expansion. For the ratio, we use the correction factor

$$ \delta_{K/\pi} = -0.0069(17), \qquad (72.5) $$

where we take the estimated error due to higher-order corrections in the chiral expansion from Ref. [28].

There have been no new measurements of the pion leptonic decay rate since the review of Ref. [29]. The sum of branching fractions for $\pi^- \to \mu^-\bar{\nu}$ and $\pi^- \to \mu^-\bar{\nu}\gamma$ is 99.98770(4)% [1]. Together with the lifetime 26.033(5) ns [1] this implies $\Gamma(\pi^- \to \mu^-\bar{\nu}[\gamma]) = 3.8408(7) \times 10^7$ s$^{-1}$. We then subtract the estimated radiative correction factor $\delta_\pi$ in Eq. (72.4) to obtain the purely leptonic rate $\Gamma^{(0)}(\pi^- \to \mu^-\bar{\nu})$. Using this rate and the masses from the 2022' PDG review [1] in Eq. (72.1) gives

$$ f_{\pi+} |V_{ud}| = (127.13 \pm 0.01 \pm 0.13) \text{ MeV}, \qquad (72.6) $$

where the errors are from the experimental rate measurement and the radiative correction factor, respectively.

The uncertainty on $f_{\pi^-}|V_{ud}|$ is dominated by that from the theoretical estimate of the hadronic structure-dependent radiative corrections. The first direct lattice-QCD calculation of the radiative corrections to the pion and kaon leptonic decay rates was performed by the RM123-Soton Collaboration [11]. The results for both $\delta_\pi = 0.0153(19)$ and $\delta_K = 0.0088(9)$, which are given in the Gasser-Rusetsky-Scimemi scheme for separating strong and electromagnetic effects [30], are compatible with our chiral-perturbation-theory estimates above and have smaller quoted uncertainties, especially for $\delta_K$. While independent confirmation of these results is needed, they demonstrate a promising approach for reducing the theoretical uncertainties on the pion and kaon leptonic decay rates in the future.

The world average for the $K \to \mu\nu$ decay rate was previously obtained from a global fit of several kaon-decay branching ratios and lifetime measurements by the FlaviaNet Working Group on Kaon Decays in 2014 [41]; we have now updated to the PDG [1]. The radiation-inclusive branching ratio $B(K^+ \to \mu^+\nu[\gamma]) = 63.56(11)%$ and lifetime $\tau_{K^\pm} = 12.380(20)$ ns are now used; this is a very small change. These measurements imply $\Gamma(K^+ \to \mu^+\nu[\gamma]) = 5.134(12) \times 10^7$ s$^{-1}$. As before, we subtract $\delta_K$ in Eq. (72.4) from the radiation-inclusive decay width to obtain $\Gamma^{(0)}(K^+ \to \mu^+\nu)$. We then use Eq. (72.1) to obtain

$$ f_{K^+}|V_{us}| = (35.083 \pm 0.042 \pm 0.037) \text{ MeV}, \qquad (72.7) $$

where the errors are from the experimental rate measurement and the radiative correction factor, respectively.

Short-distance radiative corrections cancel in the ratio of pion-to-kaon decay rates [42]:

$$ \frac{\Gamma_{K_{\ell 2[\gamma]}}}{\Gamma_{\pi \ell 2[\gamma]}} = \frac{|V_{us}| f_{K-}^2}{|V_{ud}|^2 f_{\pi-}^2} \frac{m_K (1 - m_\ell^2/m_K^2)^2}{m_\pi (1 - m_\ell^2/m_\pi^2)^2} (1 + \delta_{K/\pi}), \quad (72.8) $$

where $\delta_{K/\pi}$ is given in Eq. (72.5). The left-hand side of Eq. (72.8) is 1.3367(32), which implies

$$ \frac{|V_{us}| f_{K+}}{|V_{ud}| f_{\pi+}} = 0.27599 \pm 0.00033 \pm 0.00024, \qquad (72.9) $$

where the first uncertainty is from the branching fractions and the second is from $\delta_{K/\pi}$. Here, the estimated error on the hadronic structure-dependent radiative corrections is commensurate with the experimental error.

<footnote n="1">¹This uncertainty on $C_{\pi,K}$ is smaller than the error estimated by Marciano and Sirlin in Ref. [27], which predates the calculations of the hadronic-structure contributions in Refs. [6, 8, 25, 26]. The hadronic LECs incorporate the large short-distance electroweak logarithm discussed in Ref. [27], and their dependence on the chiral renormalization scale cancels the scale-dependence induced by chiral loops, thereby removing the dominant scale uncertainty of the Marciano-Sirlin analysis [27].</footnote>
<table><caption>**Table 72.1:** Recent published lattice-QCD results for $f_{\pi+}$, $f_{K+}$, and their ratio. The upper and lower panels show (2+1+1)-flavor and (2+1)-flavor determinations, respectively. When two errors are shown, they are statistical and systematic, respectively. Results for $f_{\pi}$ and $f_{K}$ in the isospin symmetric limit $m_{u} = m_{d}$ are noted with a $\ddagger$; they are corrected for isospin breaking via Eq. (72.13) before computing the averages. The 2+1+1 FLAG 23 averages are dominated by the Fermilab/MILC values.</caption><thead><tr><th>Reference</th><th>N<sub>f</sub></th><th>f<sub>π+</sub> (MeV)</th><th>f<sub>K+</sub> (MeV)</th><th>f<sub>K+</sub>/f<sub>π+</sub></th></tr></thead><tbody><tr><td>ETM 21 [21]</td><td>2+1+1</td><td>—</td><td>—</td><td>1.1957(44)(7)</td></tr><tr><td>CalLat 20 [31]</td><td>2+1+1</td><td>—</td><td>—</td><td>1.1942(32)(31)</td></tr><tr><td>Fermilab/MILC 17 [32] *</td><td>2+1+1</td><td>—</td><td>—</td><td>1.1950(15)(<sup>+6</sup><sub>-18</sub>)</td></tr><tr><td>ETM 14 [33] *</td><td>2+1+1</td><td>—</td><td>154.4(1.5)(1.3)</td><td>1.184(12)(11)</td></tr><tr><td>Fermilab/MILC 14 [10] * †</td><td>2+1+1</td><td>—</td><td>155.92(13)(<sup>+42</sup><sub>-34</sub>)</td><td>1.1956(10)(<sup>+26</sup><sub>-18</sub>) † ‡</td></tr><tr><td>HPQCD 13 [34] *</td><td>2+1+1</td><td>—</td><td>155.37(20)(27)</td><td>1.1916(15)(16)</td></tr><tr><td>FLAG 23 average [3]</td><td>2+1+1</td><td>—</td><td>155.7(3)</td><td>1.1934(19)</td></tr><tr><td>QCDSF/UKQCD 16 [35]</td><td>2+1</td><td>—</td><td>—</td><td>1.190(10)(13)</td></tr><tr><td>BMW 16 [36]</td><td>2+1</td><td>—</td><td>—</td><td>1.178(10)(26)</td></tr><tr><td>RBC/UKQCD 14 [37]‡</td><td>2+1</td><td>130.19(89)</td><td>155.51(83)</td><td>1.1945(45)</td></tr><tr><td>MILC 10 [38]</td><td>2+1</td><td>129.2(0.4)(1.4)</td><td>156.1(4)(<sup>+6</sup><sub>-9</sub>)</td><td>1.197(2)(<sup>+3</sup><sub>-7</sub>)</td></tr><tr><td>BMW 10 [39]‡</td><td>2+1</td><td>—</td><td>—</td><td>1.192(7)(6)</td></tr><tr><td>HPQCD/UKQCD 07 [40]‡</td><td>2+1</td><td>132(2)</td><td>157(2)</td><td>1.189(2)(7)</td></tr><tr><td>FLAG 23 average [3]</td><td>2+1</td><td>130.2(8)</td><td>155.7(7)</td><td>1.1917(37)</td></tr></tbody></table>

In summary, the main experimental results pertaining to charged pion and kaon leptonic decays are

$$|V_{us}|f_{\pi+} = (127.13 \pm 0.01 \pm 0.13) \text{ MeV}, \quad (72.10)$$

$$|V_{us}|f_{K+} = (35.083 \pm 0.042 \pm 0.037) \text{ MeV}, \quad (72.11)$$

$$\frac{|V_{us}|f_{K+}}{|V_{ud}|f_{\pi+}} = 0.27599 \pm 0.00033 \pm 0.00024, \quad (72.12)$$

where the errors are from the experimental uncertainties in the branching fractions and the theoretical uncertainties in the radiative correction factors $\delta_P$, respectively. These values contain small changes from earlier reviews [29, 43, 44].

### 72.2.2 Theoretical decay-constant calculations

Table 72.1 presents recent published results for the charged pion and kaon decay constants and their ratio from numerical lattice-QCD calculations with three ($N_f = 2 + 1$) or four flavors ($N_f = 2 + 1 + 1$) of dynamical quarks. The uncertainties on both the individual decay constants and their ratio are at the sub-percent level. The SU(3)-breaking ratio $f_{K+}/f_{\pi+}$ can be obtained with especially small errors because statistical errors associated with the Monte Carlo calculations are correlated between the numerator and denominator, as are some systematics. The results in Table 72.1 were obtained using several independent sets of gauge-field configurations, and a variety of lattice fermion actions that are sensitive to different systematic uncertainties.<fnref n="2" /> Thus, the good agreement between them indicates that the lattice-QCD uncertainties are controlled and the associated error estimates are reliable.<fnref n="3" />

Table 72.1 also shows the three- and four-flavor averages for the pion and kaon decay constants and their ratio from the FLAG 23 review [2, 3] in the lines labeled “FLAG 23 average.” There is no four-flavor average for the pion decay constant in Table 72.1 because all of the four-flavor calculations use the quantity $f_{\pi+} = 130.41(20)$ MeV [24] as an input to fix the absolute lattice scale needed to convert from lattice-spacing units to GeV [10, 33, 34].

All of the results in Table 72.1 were obtained using isospin-symmetric gauge-field configurations, i.e., the dynamical up and

down quarks have the same mass. Fortunately, however, the dominant effect of strong-isospin breaking is easily included in lattice-QCD calculations as follows. Because the up-down mass difference $\Delta m_{ud} \equiv (m_u - m_d) \sim -2.5$ MeV [50, 52] is much less than typical hadronic scales, the strong-isospin breaking corrections to physical observables can be systematically expanded in the small parameter $\delta m_{ud} \equiv \Delta m_{ud}/\Lambda_{QCD}$. The leading strong-isospin-breaking corrections to pseudoscalar-meson decay constants arise from the light valence quarks in the initial- and final-state hadrons. (See, e.g., the discussion in Ref. [53], for a detailed discussion of isospin-breaking effects in pion and kaon observables.) Thus, to include the effect of non-degenerate up- and down-quark masses, most recent lattice-QCD calculations of $f_{\pi+}$ and $f_{K+}$ evaluate the masses of the valence quarks in the pion at the physical $m_u$ and $m_d$, and the mass of the valence light quark in the kaon at the physical $m_u$. This procedure yields a correction to the kaon decay constant below 0.5%. Consequently, strong-isospin breaking corrections from the light sea-quark masses — which are suppressed by an additional power of $\delta m_{ud}$ — can be neglected given present uncertainties.

Some earlier lattice-QCD calculations, however, only provide the decay constants and their ratio in the SU(2) isospin-symmetric limit [37, 39, 40]. The Flavour Lattice Averaging Group corrects these results for strong-isospin breaking using chiral perturbation theory before including them in the averages. The leading strong-isospin-breaking corrections to the pion and kaon decay constants in $\chi$PT can be parametrized as [28, 60]

$$f_{\pi+} = f_{\pi}, \quad f_{K+} = f_{K} \sqrt{1 + \delta_{\text{SU}(2)}}, \quad (72.13)$$

where $f_{\pi}$ and $f_{K}$ denote the values of the decay constants in the isospin-symmetric limit. The pion decay constant does not receive corrections linear in $m_u - m_d$ because of the G-parity symmetry<fnref n="4" /> of the pion triplet, so at first order the $\delta m_{ud}$ expansion, strong-isospin breaking corrections are characterized by a single parameter, $\delta_{\text{SU}(2)}$. Next-to-leading order $\chi$PT yields numerical values for $\delta_{\text{SU}(2)}$ of approximately $-0.004$. Recent direct lattice-QCD calculations of $\delta_{\text{SU}(2)}$ give larger values of around $-0.005$ to $-0.008$ [11, 32–34, 53, 61], but further studies are needed. Thus, to be conservative, FLAG includes an uncertainty of 100% on the $\chi$PT estimate for $\delta_{\text{SU}(2)}$ when correcting those decay-constant values that are quoted in the isospin-symmetric limit.

<footnote>* PDG 2014 value of $f_{\pi+} = 130.41(21)$ MeV used to set absolute lattice scale.</footnote>

<footnote>† Superseded by $f_{K+}/f_{\pi+}$ from Fermilab/MILC 17.</footnote>

<footnote n="2">²See the PDG mini-review on “Lattice Quantum Chromodynamics” [45–47] for a general review of numerical lattice-QCD calculations. Details on the different methods used in modern lattice-QCD calculations are provided in Appendix A of the FLAG “Review[s] of lattice results concerning low energy particle physics” [2, 48–50].</footnote>

<footnote n="3">³See the review by Kronfeld [51] for a summary of the large body of evidence validating the methods employed in modern lattice-QCD calculations.</footnote>

<footnote n="4">⁴This is a combination of charge conjugation, C, and an $e^{i\pi I_2}$ rotation in isospin space that maps the triplet onto itself.</footnote>
<table><caption>**Table 72.2:** Experimental results for $B(D^+ \to \mu^+\nu[\gamma])$, $B(D^+ \to \tau^+\nu[\gamma])$, and $|V_{cd}|f_{D^+}$. The systematic errors on the inferred values of $|V_{cd}|f_{D^+}$ include those from the $D^+$ lifetime and mass. The error from radiative corrections is only included in the entries labeled "our average." In the second CLEO line the $\tau^+$ decays to $\mu^+\nu_\mu\bar{\nu}_\tau$.</caption><thead><tr><th>Experiment</th><th>Mode</th><th><i>B</i></th><th>|<i>V</i><sub>cd</sub>|<i>f</i><sub>D<sup>+</sup></sub> (MeV)</th></tr></thead><tbody><tr><td>CLEO-c [54, 55]</td><td>μ<sup>+</sup>ν</td><td>(3.93 ± 0.35 ± 0.09) × 10<sup>-4</sup></td><td>46.80 ± 2.08 ± 0.55</td></tr><tr><td>CLEO-c [54, 55]</td><td>μ<sup>+</sup>ν + τ<sup>+</sup>ν</td><td>(3.82 ± 0.32 ± 0.09) × 10<sup>-4</sup></td><td>46.14 ± 1.93 ± 0.56</td></tr><tr><td>BESIII [56]</td><td>μ<sup>+</sup>ν</td><td>(3.71 ± 0.19 ± 0.06) × 10<sup>-4</sup></td><td>45.47 ± 1.16 ± 0.38</td></tr><tr><td>Our average</td><td>Lines 2+3</td><td>(3.74 ± 0.17) × 10<sup>-4</sup></td><td>45.65 ± 1.19</td></tr><tr><td>CLEO-c [57, 58]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>&lt; 1.2 × 10<sup>-3</sup></td><td></td></tr><tr><td>BESIII [59]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>(1.20 ± 0.24 ± 0.12) × 10<sup>-3</sup></td><td>50.08 ± 5.62</td></tr><tr><td>Our average</td><td>μ<sup>+</sup>ν + τ<sup>+</sup>ν</td><td></td><td>45.82 ± 1.10</td></tr></tbody></table>

The errors on the decay-constant results in Table 72.1 obtained from $(2+1)$-flavor lattice-QCD calculations do not include an estimate of the systematic uncertainty from the omission of charm sea quarks in the calculation. Consequently, when the uncertainty on the $(2+1+1)$-flavor FLAG average is comparable to or better than that on the $(2+1)$-flavor FLAG average, we simply use the four-flavor average as our preferred value. This is not possible, however, for the pion decay constant. To account for this, we first estimate the systematic uncertainty on pseudoscalar-meson decay constants associated with the omission of charm sea quarks. We then add this estimate in quadrature to the quoted error on the $(2+1)$-flavor FLAG average for $f_{\pi^+}$ to obtain our preferred value.

The error introduced by omitting charm sea quarks can be roughly estimated by expanding the charm-quark determinant in powers of $1/m_c$ [62]; the resulting leading contribution is of order $\alpha_s (\Lambda_{QCD}/2m_c)^2$ [63]. Taking the $\overline{\text{MS}}$ values $\overline{m}_c(\overline{m}_c) = 1.275$ GeV, $\overline{\Lambda}_{QCD} \sim 340$ MeV from FLAG [48], and $\overline{\alpha}_s(\overline{m}_c) \sim 0.4$, leads to an estimate of about 0.7% for the contribution to the decay constants from charm sea quarks. We can compare this power-counting estimate of charm sea-quark contributions with the observed differences between the $(2+1)$- and $(2+1+1)$-flavor lattice-QCD averages for kaon, $D_{(s)}$-meson, and $B_{(s)}$-decay constants in Tables 72.1, 72.4, and 72.6. Looking at Table 72.1, the three- and four-flavor averages for $f_{K^+}$ agree to much better than our simple power-counting estimate. Inspection of Tables 72.4 and 72.6 shows, however, that charm sea-quark effects of this size are still allowed for both $D_{(s)}$-meson and $B_{(s)}$-meson decay constants.

Our final preferred theoretical values for the charged pion and kaon decay constants are

$$f_{\pi^+} = 130.2(1.2) \text{ MeV}, \quad f_{K^+} = 155.7(3) \text{ MeV}, \quad \frac{f_{K^+}}{f_{\pi^+}} = 1.193(2), \qquad (72.14)$$

where $f_{K^+}$ and $f_{K^+}/f_{\pi^+}$ are simply the four-flavor FLAG 2023 averages [2, 3], and $f_{\pi^+}$ is the three-flavor FLAG 2023 average with the error increased by the estimated 0.7% charm sea-quark contribution.

## 72.3 Charmed mesons

### 72.3.1 Experimental rate measurements

Measurements have been made of the branching fractions for $D^+$ and $D_s^+$ mesons decaying to both $\mu^+\nu$ and $\tau^+\nu$ final states. The CLEO-c and BESIII experiments have made measurements of $D^+$ decays using $e^+e^-$ collisions at the $\psi(3770)$ resonant energy where $D^-D^+$ pairs are copiously produced. They fully reconstruct one of the $D$ mesons; for concreteness, we will take this to be the $D^-$. Counting the number of these events provides the normalization for the branching fraction measurement. The experimental analyses then proceed by identifying a candidate $\mu^+$ and forming the missing-mass squared, $MM^2 = (E_{CM} - E_{D^-})^2 - (\vec{p}_{CM} - \vec{p}_{D^-} - \vec{p}_{\mu^+})^2$, where $E_{CM}$ and $p_{CM}$ are the center-of-mass energy (which is known) and momentum (which equals zero in $e^+e^-$ collisions). A peak at zero $MM^2$ implies the existence of a missing neutrino, and hence the $\mu^+\nu$ decay of the $D^+$. CLEO-c does not explicitly identify the muon, so their data consists of a

combination of $\mu^+\nu$ and $\tau^+\nu$, $\tau^+ \to \pi^+\nu$ events. This permits them to do two fits: in one they fit for the individual components, and in the other they fix the ratio of $\tau^+\nu/\mu^+\nu$ events to be that given by the standard-model expectation. Thus, the former measurement should be used for new-physics searches, and the latter for standard-model predictions. Our average uses the fixed-ratio value; this ratio is implied by Eq. 72.1 which is necessary to extract $|V_{cd}|f_{D^+}$.

Table 72.2 shows the available measurements of $D^+ \to \mu^+\nu$, an upper limit on $D^+ \to \tau^+\nu$ from CLEO-c, and the first measurement of this decay from BESIII. To extract the values of $|V_{cd}|f_{D^+}$ via Eq. (72.1), we use [1] the mass $m_{D^+} = 1.86966(5)$ GeV and the lifetime $\tau_{D^+} = 1.033(5)$ ps, and apply radiative corrections as described below. For calculating the average $\mu^+\nu$ number, we use the CLEO-c result from $\mu^+\nu + \tau^+\nu$. Since the last review the $D^+$ lifetime is updated and the propagated uncertainties on $|V_{cd}|f_{D^+}$ from the last measurement [59] are corrected.

To obtain the purely leptonic rates $\Gamma^{(0)}(D^+ \to \mu^+(\tau^+)\nu)$, we subtract the radiative contributions as in Sec. 72.2.1, but use numerical values for the corrections appropriate for $D$ mesons. First, we reduce both the $\mu^+\nu$ and $\tau^+\nu$ branching fractions in Table 72.2 by 1.8%, which is the universal short-distance electroweak contribution of Sirlin [4] evaluated using the $D$-meson mass for the factorization scale. We do not adjust the experimental rates by the universal long-distance correction [5]. This is because QED bremsstrahlung contributions have already been subtracted at leading-log order from the measurements in Table 72.2 using Monte-Carlo estimates computed with PHOTOS [12]. The $\mu^+\nu$ rates should also be reduced by the 1% estimate of the structure-dependent contributions from Dobrescu and Kronfeld [17]. This correction accounts for tree-level radiative processes in which the $D$ meson decays into a real photon and an off-shell vector meson, which subsequently decays weakly to a charged lepton and neutrino. It is estimated using Eq. (12) of Burdman et al. [7] with the CLEO-c cut on the photon energy from Ref. [74], which is typical of all the measurements. We do not need to apply the structure-dependent correction to the $\mu^+\nu$ branching fractions in Table 72.2, however, because the experiments have already included it in their quoted results. Therefore, in summary, we reduce both the $D^+ \to \mu^+\nu$ and the $D^+ \to \tau^+\nu$ rates by 1.8% to account for radiative corrections. It is worth noting, however, that the universal long-distance electromagnetic contribution estimated for point-like charged mesons by Kinoshita [5], which we are not including because IB contributions are already subtracted from the measurements via PHOTOS, would increase both rates by about 2.5%.

We now discuss the $D_s^+$ decay process; there is no change since our previous review [1]. Measurements of $D_s^+$ leptonic decay branching fractions by several groups are listed in Table 72.3. We exclude older values obtained by normalizing to $D_s^+$ decay modes that are not well-defined. Many measurements, for example, used the $\phi\pi^+$ mode. This decay is a subset of the $D_s^+ \to K^+K^-\pi^+$ channel which has interferences from other modes populating the $K^+K^-$ mass region near the $\phi$, the most prominent of which is the $f_0(980)$. Thus, the extraction of the effective $\phi\pi^+$ rate is sensitive to the mass resolution of the experiment and the cuts used
<table><caption>**Table 72.3:** Experimental results for $B(D_s^+ \to \mu^+\nu[\gamma])$, $B(D_s^+ \to \tau^+\nu[\gamma])$, and $|V_{cs}|f_{D_s^+}$. The systematic errors on the inferred values of $|V_{cs}|f_{D_s^+}$ include those from the $D_s^+$ lifetime and mass. The entries labeled “our average” take into account correlations between systematic errors common to the experiments, and also include errors from radiative corrections.</caption><thead><tr><th>Experiment</th><th>Mode</th><th>B(%)</th><th>|V<sub>cs</sub>|f<sub>D<sub>s</sub><sup>+</sup></sub> (MeV)</th></tr></thead><tbody><tr><td>CLEO-c [57, 58]</td><td>μ<sup>+</sup>ν</td><td>0.565 ± 0.045 ± 0.017</td><td>247.6 ± 9.9 ± 4.1</td></tr><tr><td>BaBar* [65]</td><td>μ<sup>+</sup>ν</td><td>0.602 ± 0.038 ± 0.034</td><td>254.3 ± 8.0 ± 7.4</td></tr><tr><td>Belle [66]</td><td>μ<sup>+</sup>ν</td><td>0.531 ± 0.028 ± 0.020</td><td>238.8 ± 6.3 ± 4.8</td></tr><tr><td>BESIII [67]</td><td>μ<sup>+</sup>ν</td><td>0.517 ± 0.075 ± 0.021</td><td>238.9 ± 17.3 ± 4.9</td></tr><tr><td>BESIII [68]</td><td>μ<sup>+</sup>ν</td><td>0.535 ± 0.013 ± 0.016</td><td>240.9 ± 2.9 ± 4.0</td></tr><tr><td>Our average</td><td>μ<sup>+</sup>ν</td><td>0.543 ± 0.017</td><td>242.6 ± 4.0</td></tr><tr><td>CLEO-c [57, 58]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>6.42 ± 0.81 ± 0.18</td><td>267.3 ± 16.9 ± 4.2</td></tr><tr><td>CLEO-c [69]</td><td>τ<sup>+</sup>ν (ρ<sup>+</sup>ν̄)</td><td>5.52 ± 0.57 ± 0.21</td><td>247.9 ± 12.8 ± 5.0</td></tr><tr><td>CLEO-c [70, 71]</td><td>τ<sup>+</sup>ν (e<sup>+</sup>ν̄ν̄)</td><td>5.30 ± 0.47 ± 0.22</td><td>242.9 ± 10.8 ± 5.3</td></tr><tr><td>BaBar [65]</td><td>τ<sup>+</sup>ν (e<sup>+</sup>(μ<sup>+</sup>)ν̄ν̄)</td><td>5.00 ± 0.35 ± 0.49</td><td>236.9 ± 8.3 ± 11.7</td></tr><tr><td>Belle [66]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>6.04 ± 0.43<sup>+0.46</sup><sub>-0.40</sub></td><td>260.3 ± 9.3<sup>+10.1</sup><sub>-8.8</sub></td></tr><tr><td>Belle [66]</td><td>τ<sup>+</sup>ν (e<sup>+</sup>ν̄ν̄)</td><td>5.37 ± 0.33<sup>+0.35</sup><sub>-0.31</sub></td><td>244.5 ± 7.5<sup>+8.2</sup><sub>-7.4</sub></td></tr><tr><td>Belle [66]</td><td>τ<sup>+</sup>ν (μ<sup>+</sup>ν̄ν̄)</td><td>5.86 ± 0.37<sup>+0.34</sup><sub>-0.59</sub></td><td>255.4 ± 8.0<sup>+7.6</sup><sub>-13.1</sub></td></tr><tr><td>BESIII [67]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>3.28 ± 1.83 ± 0.37</td><td>193 ± 54 ± 11</td></tr><tr><td>BESIII [68]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>ν̄)</td><td>5.21 ± 0.25 ± 0.17</td><td>240.8 ± 5.8 ± 4.3</td></tr><tr><td>BESIII [72]</td><td>τ<sup>+</sup>ν (e<sup>+</sup>ν<sub>e</sub>ν̄)</td><td>5.27 ± 0.10 ± 0.12</td><td>242.2 ± 2.3 ± 3.2</td></tr><tr><td>BESIII [73]</td><td>τ<sup>+</sup>ν (π<sup>+</sup>π<sup>0</sup>ν̄)</td><td>5.29 ± 0.25 ± 0.20</td><td>242.7 ± 5.7 ± 4.0</td></tr><tr><td>Our average</td><td>τ<sup>+</sup>ν</td><td>5.33 ± 0.13</td><td>244.0 ± 3.1</td></tr><tr><td>Our average</td><td>μ<sup>+</sup>ν + τ<sup>+</sup>ν</td><td></td><td>243.5 ± 2.7</td></tr></tbody></table>

to define the $\phi$ mass region [75].<fnref n="5" />

To find $D_s$ decays in the $\mu^+\nu$ signal channels, the experiments rely on fully reconstructing all of the final state particles except for the neutrino and use a missing-mass technique to infer the existence of the neutrino. CLEO and BESIII use $e^+e^- \to D_s D_s^*$ collisions at 4170 MeV, while BaBar and Belle use $e^+e^- \to D K n \pi D_s^*$ collisions at energies near the $\Upsilon(4S)$. CLEO and BESIII do a similar analysis as was done for the $D^+$ above. BaBar and Belle do a similar $MM^2$ calculation by using the reconstructed hadrons, the photon from the $D_s^+$ decay and a detected $\mu^+$. To get the normalization they do a $MM^2$ fit without the $\mu^+$ and use the signal at the $D_s^+$ mass squared to determine the total $D_s^+$ yield.

When selecting the $\tau^+ \to \pi^+\bar{\nu}$ and $\tau^+ \to \rho^+\bar{\nu}$ decay modes, CLEO uses both the calculation of the missing-mass and the fact that there should be no extra energy in the event beyond that deposited by the measured tagged $D_s$ and the $\tau^+$ decay products. The $\tau^+ \to e^+\nu\bar{\nu}$ mode, however, uses only extra energy. BaBar and Belle also use the extra energy to discriminate signal from background in their $\tau^+\nu$ measurements. BESIII uses $\tau^+ \to \pi^+\bar{\nu}$ decays, where they calculate the $MM^2$ and discriminate against $\mu^+$ from $D_s^+ \to \mu^+\nu$ decays, and more recently $\tau^+ \to e^+\nu\bar{\nu}$ and $\tau^+ \to \pi^+\pi^0\bar{\nu}$ decays.

When extracting $|V_{cs}|f_{D_s^+}$ via Eq. (72.1), we first apply the $-1.8\%$ universal electroweak correction [4] to all of the $\mu^+\nu$ and $\tau^+\nu$ branching fractions in Table 72.3; this is the same as for $D^+$ mesons. We also decrease the BaBar and Belle $\mu^+\nu$ branching fractions by the 1% structure-dependent correction [17]. This correction was already included in CLEO and BESIII results for the $\mu^+\nu$ branching fractions in Table 72.3. We use the masses and lifetimes $m_{D_s^+} = 1.96834(7)$ GeV, $m_{\tau^+} = 1.77686(12)$ GeV, and $\tau_{D_s^+} = 0.504(4)$ ps [43]. The inferred values for $f_{D_s^+}|V_{cs}|$ are in good agreement for the $\mu^+\nu$ and $\tau^+\nu$ decay modes.

It is clear from the discussion of radiative corrections in this section that they are less well understood theoretically for $D^+$ and

$D_s^+$ meson decays than for pions and kaons. We therefore assign a 1.4% systematic uncertainty to the purely leptonic decay rates, which is half the size of the applied radiative corrections. This translates to a 0.7% error on the products of the decay constant times CKM matrix element. Putting everything together, the main experimental results pertaining to charmed meson leptonic decays are (see the bottom lines of Tables 72.2 and 72.3):

$$|V_{cd}|f_{D^+} = 45.82 \pm 1.05 \pm 0.32 = 45.82 \pm 1.10 \text{ MeV}(72.15)$$

$$|V_{cs}|f_{D_s^+} = 243.5 \pm 2.1 \pm 1.7 = 243.5 \pm 2.7 \text{ MeV}, \quad (72.16)$$

where the errors are from the measured branching fractions and the applied radiative corrections, respectively.

### 72.3.2 Theoretical decay-constant calculations

Table 72.4 presents recent theoretical calculations of charmed heavy-light meson decay constants and their ratio in the isospin-symmetric limit $m_u = m_d$. (As in Sec. 72.2.2, we denote the physical $D^+$-meson decay constant by $f_{D^+}$, and use $f_D$ for the isospin-symmetric value.) The upper two panels show results from lattice-QCD calculations with three ($N_f = 2 + 1$) or four flavors ($N_f = 2 + 1 + 1$) of dynamical quarks. Although there are fewer available results than for the pion and kaon sector, both $f_D$ and $f_{D_s}$ have been obtained using multiple sets of gauge-field configurations with different lattice fermion actions, providing independent confirmation. For comparison, the bottom panel of Table 72.4 shows QCD-model calculations of the $D$- and $D_s$-meson decay constants for which uncertainty estimates are provided. The lattice and non-lattice results agree, but numerical lattice-QCD calculations have now reached significantly greater precision than other approaches.

The lattice-QCD decay-constant results in Table 72.4 were all obtained using isospin-symmetric gauge-field configurations. As discussed in Sec. 72.2.2, however, the leading strong-isospin breaking corrections to heavy-light pseudoscalar-meson decay constants can be accounted for by using the physical down (or up) quark in the $D$ (or $B$) meson. Strong-isospin breaking corrections to heavy-strange meson decay constants are roughly an order-of-magnitude smaller because there are no light valence quarks involved. Re-

<footnote>*We do not use a previous unpublished BaBar result from a subsample of data that uses a different technique for obtaining the branching fraction normalization [64].</footnote>

<footnote n="5">⁵We have not included the BaBar result for $B(D_s^+ \to \mu^+\nu)$ reported in Ref. [76] because this measurement determined the ratio of the leptonic decay rate to the hadronic decay rate $\Gamma(D_s^+ \to \ell^+\nu)/\Gamma(D_s^+ \to \phi\pi^+)$.</footnote>
<table><caption>**Table 72.4:** Recent theoretical determinations of $f_D$, $f_{D_s}$, and their ratio in the isospin-symmetric limit. The upper panels show results from lattice-QCD calculations with (2 + 1 + 1) and (2 + 1) dynamical quark flavors, respectively. Statistical and systematic errors are quoted separately. The bottom panel shows estimates from QCD sum rules (QCD SR). These are not used to obtain our preferred decay-constant values. The 2+1+1 averages are dominated by the Fermilab/MILC values.</caption><thead><tr><th>Reference</th><th>Method</th><th>N<sub>f</sub></th><th>f<sub>D</sub> (MeV)</th><th>f<sub>D<sub>s</sub></sub> (MeV)</th><th>f<sub>D<sub>s</sub></sub>/f<sub>D</sub></th></tr></thead><tbody><tr><td>Fermilab/MILC 17 [32]</td><td>LQCD</td><td>2+1+1</td><td>212.1(0.3)(0.5)</td><td>249.9(0.3)(0.3)</td><td>1.1782(0.6)(15)*</td></tr><tr><td>ETM 14 [33]</td><td>LQCD</td><td>2+1+1</td><td>207.4(3.7)(0.9)</td><td>247.2(3.9)(1.4)</td><td>1.192(19)(11)</td></tr><tr><td>FLAG 23 average [3]</td><td>LQCD</td><td>2+1+1</td><td>212.0(0.7)</td><td>249.9(0.5)</td><td>1.1783(16)</td></tr><tr><td>χQCD 20A [77]</td><td>LQCD</td><td>2+1</td><td>213(5)</td><td>249(7)</td><td>1.16(3)</td></tr><tr><td>RBC/UKQCD 18A [78]<sup>†</sup></td><td>LQCD</td><td>2+1</td><td>—</td><td>—</td><td>1.1740(51)(68)</td></tr><tr><td>RBC/UKQCD 17 [79]</td><td>LQCD</td><td>2+1</td><td>208.7(2.8)(+2.1/-1.8)</td><td>246.4(1.3)(+1.3/-1.3)</td><td>1.1667(77)(+57/-43)</td></tr><tr><td>χQCD 14 [80]</td><td>LQCD</td><td>2+1</td><td>—</td><td>254(2)(4)</td><td>—</td></tr><tr><td>HPQCD 12 [81]</td><td>LQCD</td><td>2+1</td><td>208.3(1.0)(3.3)</td><td>—</td><td>1.187(4)(12)</td></tr><tr><td>Fermilab/MILC 11 [82]</td><td>LQCD</td><td>2+1</td><td>218.9(9.2)(6.6)</td><td>260.1(8.9)(6.1)</td><td>1.188(14)(21)</td></tr><tr><td>HPQCD 10 [83]</td><td>LQCD</td><td>2+1</td><td>—</td><td>248.0(1.4)(2.1)</td><td>—</td></tr><tr><td>FLAG 23 average [3]</td><td>LQCD</td><td>2+1</td><td>209.0(2.4)</td><td>248.0(1.6)</td><td>1.174(7)</td></tr><tr><td>Pullin 21 [84]</td><td>QCD SR</td><td></td><td>190(15)</td><td>226(17)</td><td>1.19(7)</td></tr><tr><td>Wang 15 [85]<sup>‡</sup></td><td>QCD SR</td><td></td><td>208(10)</td><td>240(10)</td><td>1.15(6)</td></tr><tr><td>Gelhausen 13 [86]</td><td>QCD SR</td><td></td><td>201(+12/-13)</td><td>238(+13/-23)</td><td>1.15(+0.04/-0.05)</td></tr><tr><td>Narison 12 [87]</td><td>QCD SR</td><td></td><td>204(6)</td><td>246(6)</td><td>1.21(4)</td></tr><tr><td>Lucha 11 [88]</td><td>QCD SR</td><td></td><td>206.2(8.9)</td><td>245.3(16.3)</td><td>1.193(26)</td></tr></tbody></table>

cently, the Fermilab Lattice and MILC Collaborations used this approach to calculate directly the dominant strong-isospin breaking corrections to both $f_D$ and $f_B$, finding [32]

$$ \begin{aligned} f_{D+} - f_D &= +0.58(1)_{\text{stat}}(7)_{\text{syst}}(1)_{\text{EM scheme}} \text{MeV}, \\ f_{B+} - f_B &= -0.53(5)_{\text{stat}}(7)_{\text{syst}}(0)_{\text{EM scheme}} \text{MeV}. \end{aligned} \quad (72.17) $$

These results agree with independent estimates of the strong-isospin-breaking corrections to heavy-light meson decay constants from QCD sum rules [89]. Combined with the determinations of $f_D$ and $f_B$ from the same work, Eq. (72.17) implies that the corrections to the SU(3)-flavor breaking ratios are

$$ \frac{f_{D_s}}{f_{D+}} = \frac{f_{D_s}}{f_D} (1 - 0.0027(3)), \quad \frac{f_{B_s}}{f_{B+}} = \frac{f_{B_s}}{f_B} (1 + 0.0028(5)). \quad (72.18) $$

These estimated strong-isospin-breaking corrections to $f_D$ and $f_{D_s}/f_D$ above are commensurate with the uncertainties on the (2+1+1)-flavor FLAG averages in Table 72.4. Consequently, it is important to account for isospin-breaking effects before combining the theoretical decay constants with the corresponding experimental decay rates.

To obtain the charged $D^+$-meson decay constant, we apply the correction in Eq. (72.17) to the (2+1+1)-flavor 2021 FLAG average for the D-meson decay constant in the isospin-symmetric limit. Similarly we use Eq. (72.18) to correct the (2+1+1)-flavor 2021 FLAG average for $f_{D_s}/f_D$. We take the four-flavor FLAG 2021 average for $f_{D_s}$ directly. Our final preferred theoretical values for the charmed pseudoscalar-meson decay constants are

$$ \begin{aligned} f_{D+} &= 212.0(7) \text{ MeV}, \\ f_{D_s} &= 249.9(5) \text{ MeV}, \\ \frac{f_{D_s}}{f_{D+}} &= 1.1783(16). \end{aligned} \quad (72.19) $$

## 72.4 Bottom mesons

### 72.4.1 Experimental rate measurements

The Belle and BaBar collaborations have found evidence for $B^- \to \tau^- \bar{\nu}$ decay in $e^+ e^- \to B^- B^+$ collisions at the $\Upsilon(4S)$ energy. The analysis relies on reconstructing a hadronic or semileptonic B decay tag, finding a $\tau$ candidate in the remaining track and photon candidates, and examining the extra energy in the event which should be close to zero for a real $\tau^-$ decay to $e^- \nu \bar{\nu}$ or $\mu^- \nu \bar{\nu}$ opposite a $B^+$ tag. The results are listed in Table 72.5. These are unchanged since the previous review [1].

<table><caption>**Table 72.5:** Experimental results for $B(B^- \to \tau^- \bar{\nu})$ and $|V_{ub}|f_{B+}$. To extract the values of $|V_{ub}|f_{B+}$ via Eq. (72.1), we use the PDG values of the $B^+$ lifetime of 1.638±0.004 ps, and the $\tau^+$ and $B^+$ masses of 1.77686 and 5.27934 GeV, respectively.</caption><thead><tr><th>Experiment</th><th>Tag</th><th>B (units of 10<sup>-4</sup>)</th><th>|V<sub>ub</sub>|f<sub>B+</sub> (MeV)</th></tr></thead><tbody><tr><td>Belle [90]</td><td>Hadronic</td><td>0.72<sup>+0.27</sup><sub>-0.25</sub> ± 0.11</td><td></td></tr><tr><td>Belle [91]</td><td>Semileptonic</td><td>1.25 ± 0.28 ± 0.27</td><td></td></tr><tr><td>Belle [91]</td><td>Average</td><td>0.91 ± 0.22</td><td>0.72 ± 0.09</td></tr><tr><td>BaBar [92]</td><td>Hadronic</td><td>1.83<sup>+0.53</sup><sub>-0.49</sub> ± 0.24</td><td></td></tr><tr><td>BaBar [93]</td><td>Semileptonic</td><td>1.7 ± 0.8 ± 0.2</td><td></td></tr><tr><td>BaBar [92]</td><td>Average</td><td>1.79 ± 0.48</td><td>1.01 ± 0.14</td></tr><tr><td colspan="3">Our average</td><td>1.06 ± 0.20</td><td>0.77 ± 0.07</td></tr></tbody></table>

Because there are large backgrounds under the signals for these measurements, as well as substantial systematic errors, the significances of the individual results are still below the $5\sigma$ discovery threshold. Belle quotes $4.6\sigma$ for their combined hadronic and semileptonic tags, while BaBar quotes $3.3\sigma$ and $2.3\sigma$, for hadronic and semileptonic tags. Greater precision is necessary to determine if any effects beyond the standard model are present.

We do not correct the measured branching ratios in Table 72.5 for radiative corrections because the experimental uncertainties are so large. The relative radiative corrections are expected to be larger, however, for $B \to \mu\nu$ leptonic decays because the corrections are no longer helicity suppressed [9], and may be a significant fraction of the purely leptonic rate. More theoretical work is needed to understand radiative corrections to leptonic B decays in anticipation of future measurements with greater precision, and of new decay channels.

### 72.4.2 Theoretical decay-constant calculations

Table 72.6 presents recent theoretical calculations of bottom heavy-light meson decay constants and their ratio in the isospin-symmetric limit $m_u = m_d$. The upper two panels show results from lattice-QCD calculations with three ($N_f = 2 + 1$) or four flavors ($N_f = 2 + 1 + 1$) of dynamical quarks. For all decay constants, calculations using different gauge-field configurations, light-quark actions, and b-quark actions provide independent confirmation. For comparison, the bottom panel of Table 72.6 shows QCD-model calculations of the B- and $B_s$-meson decay constants

<footnote>*Ref. [32] provides values for $f_D$ and $f_{D_s}$ in the isospin-symmetric limit, but not for their ratio. Here we infer the central value from those of the individual decay constants, and take the statistical and systematic errors to be the same as for the physical ratio $f_{D_s}/f_{D+}$.</footnote>

<footnote>†Slight difference from preliminary value quoted in 2019 FLAG review.</footnote>

<footnote>‡Obtained using $m_c^{\text{MS}}$; results using $m_c^{\text{pole}}$ are also given in the paper.</footnote>
<table><caption>**Table 72.6:** Recent theoretical determinations of $f_B$, $f_{B_s}$, and their ratio in the isospin-symmetric limit. The upper panels show results from lattice-QCD calculations with (2 + 1 + 1) and (2 + 1) dynamical quark flavors, respectively. When available, statistical and systematic errors are quoted separately. The bottom panel shows estimates from the relativistic potential model (RPM) and QCD sum rules (QCD SR), which are not used to obtain our preferred decay-constant values. The 2+1+1 FLAG 23 averages are dominated by the Fermilab/MILC values.</caption><thead><tr><th>Reference</th><th>Method</th><th>N<sub>f</sub></th><th>f<sub>B</sub> (MeV)</th><th>f<sub>B<sub>s</sub></sub> (MeV)</th><th>f<sub>B<sub>s</sub></sub>/f<sub>B</sub></th></tr></thead><tbody><tr><td>Fermilab/MILC 17 [32]</td><td>LQCD</td><td>2+1+1</td><td>189.9(1.4)</td><td>230.7(1.2)</td><td>1.2146(49)</td></tr><tr><td>HPQCD 17 [94]*</td><td>LQCD</td><td>2+1+1</td><td>190(4)</td><td>229(5)</td><td>1.206(5)</td></tr><tr><td>ETM 16 [95]</td><td>LQCD</td><td>2+1+1</td><td>193(6)</td><td>229(5)</td><td>1.184(25)</td></tr><tr><td>HPQCD 13 [96]</td><td>LQCD</td><td>2+1+1</td><td>186(4)</td><td>224(5)</td><td>1.205(7)</td></tr><tr><td>FLAG 23 average [3]</td><td>LQCD</td><td>2+1+1</td><td>190.0(1.3)</td><td>230.3(1.3)</td><td>1.209(5)</td></tr><tr><td>Aoki 14 [97]<sup>†</sup></td><td>LQCD</td><td>2+1</td><td>218.8(6.5)(30.8)</td><td>263.5(4.8)(36.7)</td><td>1.193(20)(44)</td></tr><tr><td>RBC/UKQCD 14 [98]<sup>‡</sup></td><td>LQCD</td><td>2+1</td><td>195.6(6.4)(13.3)</td><td>235.4(5.2)(11.1)</td><td>1.223(14)(70)</td></tr><tr><td>HPQCD 12 [99]</td><td>LQCD</td><td>2+1</td><td>191(1)(8)</td><td>228(3)(10)</td><td>1.188(12)(13)</td></tr><tr><td>HPQCD 12 [99]</td><td>LQCD</td><td>2+1</td><td>189(3)(3)*</td><td>—</td><td>—</td></tr><tr><td>HPQCD 11 [100]</td><td>LQCD</td><td>2+1</td><td>—</td><td>225(3)(3)</td><td>—</td></tr><tr><td>Fermilab/MILC 11 [82]</td><td>LQCD</td><td>2+1</td><td>196.9(5.5)(7.0)</td><td>242.0(5.1)(8.0)</td><td>1.229(13)(23)</td></tr><tr><td>FLAG 23 average [3]</td><td>LQCD</td><td>2+1</td><td>192.0(4.3)</td><td>228.4(3.7)</td><td>1.201(16)</td></tr><tr><td>Pullin 21 [84]</td><td>QCD SR</td><td></td><td>192<sup>+20</sup><sub>-19</sub></td><td>225<sup>+21</sup><sub>-20</sub></td><td>1.17(7)</td></tr><tr><td>Sun 16 [101]<sup>§</sup></td><td>RPM</td><td></td><td>219(15)</td><td>266(19)</td><td>1.21(9)</td></tr><tr><td>Wang 15 [85]<sup>§</sup></td><td>QCD SR</td><td></td><td>194(15)</td><td>231(16)</td><td>1.19(10)</td></tr><tr><td>Baker 13 [102]</td><td>QCD SR</td><td></td><td>186(14)</td><td>222 (12)</td><td>1.19(4)</td></tr><tr><td>Lucha 13 [103]</td><td>QCD SR</td><td></td><td>192.0(14.6)</td><td>228.0(19.8)</td><td>1.184(24)</td></tr><tr><td>Gelhausen 13 [86]</td><td>QCD SR</td><td></td><td>207<sup>+17</sup><sub>-9</sub></td><td>242<sup>+17</sup><sub>-12</sub></td><td>1.17<sup>+3</sup><sub>-4</sub></td></tr><tr><td>Narison 12 [87]</td><td>QCD SR</td><td></td><td>206(7)</td><td>234(5)</td><td>1.14(3)</td></tr></tbody></table>

*Average of new HPQCD 17 and previous HPQCD 13 results allowing for correlations between the two sets of results.

†Obtained with static b quarks (i.e., $m_b \to \infty$).

‡Ref. [98] does not provide results in the isospin-symmetric limit, so we show $f_{B+}$ and $f_{B_s}/f_{B+}$ for this work.

§Obtained using $m_b^{\text{MS}}$; results using $m_b^{\text{pole}}$ are also given in the paper.

for which uncertainty estimates are provided. These are consistent with the lattice values, but with much larger uncertainties.

The lattice-QCD decay-constant results in Table 72.6 were all obtained using isospin-symmetric gauge-field configurations. Some calculations, however, account for the dominant effect of strong-isospin-breaking by using the correct value for the valence light-quark mass in the B meson ($m_u$ for $f_{B+}$ and $m_d$ for $f_{B^0}$). Early estimates of the strong-isospin-breaking correction obtained $f_{B^+} - f_B \sim -2$ MeV [96, 98], which would be significant given the present lattice-QCD uncertainties. It turns out, however, that these calculations inadvertently introduced a spurious sea-quark contribution, and therefore overestimated the size of the effect. A more recent calculation by the Fermilab/MILC Collaboration finds very little evidence for isospin violation ($f_{B^+} - f_B \sim -0.5$ MeV) [32], which is more than two times smaller than the total uncertainties on present lattice-QCD calculations. For this reason, we quote isospin averages in the current review. Our preferred theoretical values for the bottom pseudoscalar-meson decay constants are

$$ 
\begin{aligned}
f_B &= 190.0(1.3) \text{ MeV}, \\
f_{B_s} &= 230.3(1.3) \text{ MeV}, \\
\frac{f_{B_s}}{f_B} &= 1.209(5),
\end{aligned} 
\qquad (72.20) $$

which are simply the $N_f = 2 + 1 + 1$ FLAG 2023 averages [3]. Because the uncertainties on the three-flavor results in Table 72.6 are substantially larger than those on the four-flavor results, including them in the average leaves the central values unchanged, and decreases the errors only slightly.

## 72.5 Phenomenological implications

### 72.5.1 $|V_{ud}|, |V_{us}|$, and status of first-row unitarity

Using the average values for $f_{\pi^+}|V_{ud}|, f_{K^+}|V_{us}|$, and their ratio from Eqs. (72.10)–(72.12) and for $f_{\pi^+}$, $f_{K^+}$, and their ratio from Eq. (72.14), we obtain the following determinations of the CKM matrix elements $|V_{ud}|, |V_{us}|$, and their ratio from leptonic decays

within the standard model:

$$ 
\begin{aligned}
|V_{ud}| &= 0.9763(1)(90)(10), & |V_{us}| &= 0.2253(3)(4)(2), \\
|V_{us}| &= 0.2313(3)(4)(2), & |V_{ud}| &= 0.2253(3)(4)(2),
\end{aligned} 
\qquad (72.21) $$

where the errors are from the experimental branching fraction(s), the pseudoscalar decay constant(s), and radiative corrections, respectively. These results enable a test of the unitarity of the first row of the CKM matrix from leptonic decays alone (the contribution from $|V_{ub}|$ is negligible). Using the values of $|V_{ud}|$ and $|V_{us}|$ from Eq. (72.21), we find

$$ |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 - 1 = 0.004(18), \qquad (72.22) $$

which is consistent with three-generation unitarity at the few-percent level.

The determinations of $|V_{ud}|$ and $|V_{us}|$ from leptonic decays in Eq. (72.21) can be compared to those obtained from other processes. The result above for $|V_{ud}|$ agrees with the determination from super-allowed $\beta$-decay (a previous value of $|V_{ud}| = 0.97420(21)$ [104] has been updated to $|V_{ud}| = 0.97373(31)$ [105]) but has a considerably larger error, primarily due to the uncertainty in the theoretical determination of $f_{\pi^+}$.

The CKM element $|V_{us}|$ can be determined from semileptonic $K^+ \to \pi^0 \ell^+ \nu$ decay. Here experimental measurements provide a value for the product $f_{+}^{K\pi}(0)|V_{us}|$, where $f_{+}^{K\pi}(0)$ is the form factor at zero four-momentum transfer between the initial state kaon and the final state pion. We quote a recent analysis by Seng et al. [106] which finds $|V_{us}|f_{+}^{K\pi}(0) = 0.21635(38)(3)$, with the first error denoting experiment and the second reflecting uncertainty in higher order corrections. Choosing the 2021 2+1+1-flavor FLAG average for $f_{+}(0)^{K\pi} = 0.9698(17)$ [2] based on the calculations of ETM [107] and Fermilab/MILC [108] gives $|V_{us}| = 0.22309(39)_{\text{exp}}(39)_{\text{LQCD}}(3)_{\text{HO}}$ from semileptonic decay. The determinations of $|V_{us}|$ from leptonic and semileptonic kaon decays are both quite precise, with comparable errors, but the central values differ by $2.8\sigma$.
Finally, the combination of the ratio $|V_{us}|/|V_{ud}|$ from leptonic decays [Eq. (72.21)] with $|V_{ud}|$ from $\beta$ decay implies an alternative determination of $|V_{us}| = 0.2252(5)$ which agrees with the value from leptonic kaon decay, but again disagrees with the semileptonic-decay result at the $2.8\sigma$ level.

Given the roughly $3\sigma$ tension between $|V_{us}|$ from leptonic and semileptonic kaon decays, it is important to scrutinize the uncertainties on the theoretical and experimental inputs to $|V_{us}|$ and other elements of the first row of the CKM matrix. Seng et al. have introduced a new approach for calculating radiative corrections to neutron and nuclear beta decays using dispersion relations [109–112]. These calculations imply a value of $|V_{ud}| = 0.97395(23)$, consistent with the updated analysis of Ref. [105]. An independent calculation of the radiative corrections by Czarnecki and Marciano using QCD sum rules yields similar results [113]. Using this value of $|V_{ud}|$ with the determination of $|V_{us}|$ from leptonic kaon decays in Eq. (72.21), we obtain $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 - 1 = -0.0007(5)$, again consistent with first-row unitarity.

Last, we combine the experimental measurement of $f_{\pi^+}|V_{ud}|$ in Eq. (72.10) with $|V_{ud}|$ from super-allowed $\beta$-decay [105] to infer an “experimental” value for the pion decay constant:

$$f_{\pi^-}^{\text{"exp}} = 130.56(2)(4)(13) \text{ MeV}, \qquad (72.23)$$

where the uncertainties are from the errors on $\Gamma$, $|V_{ud}|$, and higher-order corrections, respectively. Many recent (2+1+1)-flavor lattice-QCD calculations use this quantity to set the overall physical scale in their simulations, e.g., Refs. [10, 32–34]. Conversely, comparing $f_{\pi^-}^{\text{"exp}}$ with the 2021 FLAG (2+1)-flavor average $f_{\pi^+} = 130.2(8)$ MeV, which only includes lattice-QCD results that employ other observables to set the scale [35–40], provides a test of lattice-QCD methods. The values are in good agreement within present uncertainties. We do not quote an “experimental” value for the kaon decay constant because the value of $|V_{us}|$ is less clear given the approximately $3\sigma$ tension between the values of $|V_{us}|$ obtained from leptonic and semileptonic kaon decays.

### 72.5.2 $|V_{cd}|, |V_{cs}|$, and status of second-row unitarity

Using the average values for $|V_{cd}|f_{D^+}$ and $|V_{cs}|f_{D_s^+}$ from Eqs. (72.15) and (72.16), and for $f_{D^+}$ and $f_{D_s^+}$ from Eq. (72.19), we obtain the following determinations of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ from leptonic decays within the standard model:

$$|V_{cd}| = 0.2161(50)(15)(7) \quad \text{and} \quad |V_{cs}| = 0.9744(84)(16)(20), \qquad (72.24)$$

where the errors are from the measured branching fractions, radiative corrections, and decay constants, respectively. These results enable a test of the unitarity of the second row of the CKM matrix. Taking $|V_{cb}| = 40.8(1.4) \times 10^{-3}$ [1], we obtain

$$|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 - 1 = -0.002(22), \qquad (72.25)$$

in agreement with three-generation unitarity.

The uncertainty on $|V_{cd}|$ in Eq. (72.24) is limited by the measurement error on the $D^+ \to \mu^+\nu$ decay rate. For $|V_{cs}|$, however, the experimental and radiative-correction errors are commensurate. The value of $|V_{cs}|$ from leptonic $D_s$ decays decreased substantially from the value of 1.007(17) in an earlier version of this review [29, 114], and is now below unity as expected in the three-generation CKM framework. This change is due to our more consistent treatment of the radiative corrections, which lower the purely leptonic decay rates for $D_s^+ \to \mu^+\nu$ and $D_s^+ \to \tau^+\nu$. We emphasize, however, that we have taken a generous uncertainty on these estimates, and that more theoretical work is needed to really pin down the sizes of the radiative corrections to $D_{(s)}$-meson leptonic decays.

The CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ can also be obtained from semileptonic $D^+ \to \pi^0\ell^+\nu$ and $D_s^+ \to K^0\ell^+\nu$ decays, respectively. Here experimental measurements determine the product of the form factor times the CKM element, and theory provides the value for the form factor at zero four-momentum transfer between the initial $D_{(s)}$ meson and the final pion or kaon.

The latest experimental averages (2023, with data as of 2021) from the Heavy Flavor Averaging Group (HFLAV) are $f_{+}^{D\pi}(0)|V_{cd}| = 0.1426(18)$ and $f_{+}^{DK}(0)|V_{cs}| = 0.7180(33)$ [115]. There were not enough published lattice-QCD calculations of the zero-momentum D-meson semileptonic form factors with $N_f \ge 3$ to permit an average by the FLAG Collaboration; instead, we summarize the 2+1+1 flavor results here. The newest calculation, from the Fermilab Lattice and MILC Collaborations [116] appeared after the FLAG 2023 online update [3]. They report $f_{+}^{D\pi}(0) = 0.6300(51)$ and $f_{+}^{DK}(0) = 0.7452(31)$; these results are the most precise and include both decay modes. For the decay $D \to K\ell\nu$, the HPQCD Collaboration [117] calculates $f_{+}^{DK}(0) = 0.7380(44)$, about $1.3\sigma$ lower. An update from HPQCD [118] gives $f_{+}^{DK}(0) = 0.7441(40)$, in agreement with Fermilab-MILC. Note that the two HPQCD results and Fermilab-MILC all share MILC gauge configurations and are hence not independent. The ETM Collaboration [119], has results for both decays, obtaining $f_{+}^{D\pi}(0) = 0.612(35)$ and $f_{+}^{DK}(0) = 0.765(31)$, which are consistent but less precise. A naive combination of the HFLAV $f(0)|V|$ products and the new Fermilab-MILC results gives results for $|V_{cd}|$ and $|V_{cs}|$ consistent with the leptonic decays in Eq. (72.24). However, as with the leptonic case, one must exercise some care with corrections; this is beyond the scope of the present review. We note that Ref. [116] presents a detailed discussion of CKM element extractions using their LQCD results, including consideration of QED, electroweak and isospin corrections; a comparison of leptonic and semileptonic results and unitarity tests are also included.

We can combine the experimental measurements of $f_{D^+}|V_{cd}|$ and $f_{D_s^+}|V_{cs}|$ from Tables 72.2 and 72.3 with $|V_{cd}| = 0.22486(67)$ and $|V_{cs}| = 0.97349(16)$ from the PDG 2022 global unitarity-triangle analysis [1] to infer “experimental” values for the decay constants within the standard model. We take the CKM elements from the global fit because they are based on many input quantities, thereby reducing the sensitivity to any one outlying measurement or calculation. We obtain for the decay constants

$$ \begin{aligned} f_{D^+}^{\text{"exp}} &= 203.8(4.7)(0.6)(1.4) \text{ MeV}, \\ f_{D_s^+}^{\text{"exp}} &= 250.1(2.2)(0.04)(1.8) \text{ MeV}, \\ (f_{D_s^+}/f_{D^+})^{\text{"exp}} &= 1.228(30)(4)(9). \end{aligned} \qquad (72.26) $$

where the uncertainties are from the errors on $\Gamma^{(0)}$, CKM matrix elements, and radiative corrections, respectively. Here, the third uncertainty on the ratio is taken as 0.7%, the same as on the first two results, due to presumed partial cancellation. Possible further cancellation is not very relevant given the current statistical uncertainty. The “experimental” values $f_{D^+} (f_{D_s^+}/f_{D^+})$ are consistent within $1\sigma$ with the (2+1+1)-flavor lattice-QCD averages in Eq. (72.19). The CKM matrix element $|V_{cd}|$ is approximately equal to $|V_{us}|$, as first shown by Wolfenstein [120, 121]. Thus, resolving the inconsistencies between determinations of $|V_{us}|$ from leptonic and semileptonic decays discussed in Sec. 72.5.1 may benefit from the constraints observed here.

Last, we can test lepton-flavor universality in charm meson decays by checking the following relationship derived from Eq. (72.1):

$$ \frac{\Gamma(D_s^+ \to \tau^+\nu)}{\Gamma(D_s^+ \to \mu^+\nu)} = \frac{m_\tau^2 (1 - m_\tau^2 / M_{D_s}^2)^2}{m_\mu^2 (1 - m_\mu^2 / M_{D_s}^2)^2} = 9.75, \qquad (72.27) $$

where the uncertainties from the masses are negligible to the number of digits quoted. The measured ratio of $\tau^+\nu$ to $\mu^+\nu$ rates is $9.82 \pm 0.40$, consistent with this standard-model expectation.

### 72.5.3 $|V_{ub}|$ and other applications

Using the average value for $|V_{ub}|f_{B^+}$ from Table 72.5, and for $f_{B^+}$ from Eq. (72.20), we obtain the following determination of the CKM matrix element $|V_{ub}|$ from leptonic decays within the standard model:

$$ |V_{ub}| = 4.05(37)(3) \times 10^{-3}, \qquad (72.28) $$
where the errors are from experiment and theory, respectively. One should bear in mind when interpreting Eq. (72.28) that none of the experimental measurements of the branching fractions that enter the average for $|V_{ub}|f_{B+}$ have individually reached the $5\sigma$ discovery level (see Sec. 72.4.1). Further, decays involving the third generation of quarks and leptons may be particularly sensitive to new physics associated with electroweak symmetry breaking due to their larger masses [13,15], so Eq. (72.28) is more likely to be influenced by new physics than the determinations of the elements of the first and second rows of the CKM matrix in the previous sections.

The CKM element $|V_{ub}|$ can also be obtained from semileptonic $B$-meson decays. A long-standing $2-3\sigma$ tension between the determinations of $|V_{ub}|$ from exclusive $B \to \pi\ell\nu$ decay and from inclusive $B \to X_u\ell\nu$ decay has diminished somewhat; here, $X_u$ denotes all hadronic states populated by the $b \to u$ weak transition [24, 122–126]. The currently most precise determination of $|V_{ub}|^{\text{excl}} = 3.73(14) \times 10^{-3}$ is obtained from a joint z-fit by FLAG [2] of the vector and scalar form factors $f_{+}^{B\pi}(q^2)$ and $f_{0}^{B\pi}(q^2)$ calculated in (2+1)-flavor lattice QCD [127–129] and experimental measurements of the differential decay rate from BaBar [130, 131] and Belle [132, 133]. This is consistent with the value of $|V_{ub}|^{\text{excl}} = 3.70(16) \times 10^{-3}$ obtained in the 2022 PDG review on semileptonic determinations of $V_{cb}, V_{ub}$ [134]. On the other hand, the inclusive determination in that same review, obtained using the theoretical frameworks in Refs. [135–137], is $|V_{ub}|^{\text{incl}} = 4.13(26) \times 10^{-3}$ [1, 134]. This value is significantly lower than the $4.49(28) \times 10^{-3}$ value from the corresponding PDG 2018 review [134], reducing the exclusive-inclusive tension to less than $2\sigma$. The value of $|V_{ub}|$ from leptonic $B \to \tau\nu$ decay in Eq. (72.28) lies between these inclusive and exclusive determinations, and is compatible (within large uncertainties) with both.

Given the large uncertainties on the experimental measurements of $B(B^- \to \tau^-\bar{\nu})$, and the lingering tension between $|V_{ub}|$ obtained from inclusive and exclusive semileptonic $B$ decays, we do not present an “experimental” value of the decay constant $f_{B+}$.

## 72.6 Concluding remarks

The increasing precision of measurements of leptonic decays of charged pseudoscalar mesons has begun to require control over radiative corrections at a level superior to that presented here. Improvements will be called for both in theory and in experiment (for example, by BESIII and Belle II).

Unitarity of the CKM matrix is in reasonable shape. However, there is a lingering almost $3\sigma$ tension between the values of $V_{us}$ determined by semileptonic and leptonic kaon decays. It will be interesting to follow experimental and theoretical developments on this quantity.

## Acknowledgements

I thank J. L. Rosner for useful discussions on this review. Previous assistance from C. Davies, S. Gottlieb, S. Hashimoto, A. Kronfeld, H.-L. Ma, W. Marciano, and S. Stone is also acknowledged.

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# 73. Production and Decay of b-flavored Hadrons

Revised March 2024 by P. Eerola (Research Council of Finland), M. Kreps (Warwick U.) and Y. Kwon (Yonsei U., Seoul).

The b quark belongs to the third generation of quarks and is the weak-doublet partner of the t quark. The existence of the third-generation quark doublet was proposed in 1973 by Kobayashi and Maskawa [1] in their model of the quark mixing matrix ("CKM" matrix), and confirmed four years later by the first observation of a $b\bar{b}$ meson [2]. In the KM model, CP violation is explained within the Standard Model (SM) by an irreducible phase of the 3 × 3 unitary matrix. The regular pattern of the three lepton and quark families is one of the most intriguing puzzles in particle physics. The existence of families gives rise to many of the free parameters in the SM, including the fermion masses, and the elements of the CKM matrix.

Since the b quark is the lighter element of the third-generation quark doublet, the decays of b-flavored hadrons occur via generation-changing processes through CKM matrix. Because of this, and the fact that the CKM matrix is close to a 3 × 3 unit matrix, many interesting features such as loop and box diagrams, flavor oscillations, as well as large CP asymmetries, can be observed in the weak decays of b-flavored hadrons.

The CKM matrix is parameterized by three real parameters and one complex phase. This complex phase is the source of CP violation in B meson decays in the Standard Model. A crucial milestone was the first observation of CP violation in the B meson system in 2001, by the BaBar [3] and Belle [4] collaborations. They measured a large value for the parameter $\sin 2\beta$ (= $\sin 2\phi_1$) [5], almost four decades after the discovery of a small CP asymmetry in neutral kaons. A more detailed discussion of the CKM matrix and CP violation can be found elsewhere in this Review [6,7].

The structure of this mini-review is organized as follows. After a discussion of b-quark production and current results on spectroscopy, we discuss lifetimes of b-flavored hadrons. We then discuss some basic properties of B-meson decays, followed by summaries of dominant hadronic, rare hadronic, and electroweak penguin decays of B-mesons. There are separate mini-reviews for $B^0$-$\bar{B}^0$ mixing [8] and the extraction of the CKM matrix elements $V_{cb}$ and $V_{ub}$ from B-meson decays [9] in this Review.

## 73.1 Production and spectroscopy

The bound states of a $\bar{b}$ antiquark and a u, d, s, or c quark are referred to as the $B_u(B^+)$, $B_d(B^0)$, $B_s(B^0)$, and $B_c(B^+)$ mesons, respectively. The $B_c^+$ is the heaviest of the ground-state b-flavored mesons, and the most difficult to produce: it was observed for the first time in the semileptonic mode by CDF in 1998 [10], but its mass was accurately determined only in 2006, from the fully reconstructed mode $B_c^+ \to J/\psi\pi^+$ [11]. Many exclusive decay channels can now be used for the accurate mass measurements, given the large statistics available at the LHC. Currently the most precise measurement is made by LHCb and yields $m(B_c^+) = 6274.47 \pm 0.27 \pm 0.17$ MeV/c$^2$ [12], combining $B_c^+ \to J/\psi\pi^+$, $B_c^+ \to J/\psi\pi^+\pi^-\pi^+$, $B_c^+ \to J/\psi p\bar{p}\pi^+$, $B_c^+ \to J/\psi D_s^+$, $B_c^+ \to J/\psi D^0 K^+$ and $B_c^+ \to B_s^0\pi^+$ decay modes.

The first excited meson is called the $B^*$ meson, while $B^{**}$ is the generic name for the four orbitally excited ($L=1$) B-meson states that correspond to the P-wave mesons in the charm system, $D^{**}$. Excited states of the $B_s^0$ meson are similarly named $B_s^*$ and $B_s^{**}$.

Of the possible bound $\bar{b}b$ states, the $\Upsilon(nS)$ and $\chi_{bJ}(nP)$ states are well studied. The pseudoscalar ground state $\eta_b$ has been observed for the first time by BaBar [13] indirectly through the decay $\Upsilon(3S) \to \gamma\eta_b$, and then confirmed by Babar in $\Upsilon(2S)$ decays [14] and CLEO in $\Upsilon(3S)$ decays [15]. The most accurate mass and width measurements come now from Belle, using decays $\Upsilon(5S) \to h_b(1P)\pi^+\pi^-$, $h_b(1P) \to \gamma\eta_b(1S)$ [16], $\Upsilon(4S) \to \eta h_b(1P)$, $h_b(1P) \to \gamma\eta_b(1S)$ [17], and $\Upsilon(2S) \to \gamma\eta_b(1S)$ [18]. Belle has also reported first evidence for the $\eta_b(2S)$ in the $h_b(2P) \to \eta_b(2S)\gamma$ transition [16]. In addition, Belle has observed $T_{b\bar{b}1}(10610)$ (was $Z_b(10610)$) and $T_{b\bar{b}1}(10650)$ (was $Z_b(10650)$) states in the processes $\Upsilon(5S) \to \Upsilon(nS)\pi^+\pi^-$ ($n=1,2,3$) and $\Upsilon(5S) \to h_b(mP)\pi^+\pi^-$ ($m=1,2$) [19]. These $T_{b\bar{b}1}$ states are

observed to decay to $\Upsilon(nS)\pi^\pm$ and $h_b(mP)\pi^\pm$, hence electrically charged and do not belong to ordinary $q\bar{q}$ mesons. For classification and naming of these and other states, see Ref. [20].

Experimental studies of b decays have been performed in $e^+e^-$ collisions at the $\Upsilon(4S)$ (ARGUS, CLEO, Belle, BaBar) and $\Upsilon(5S)$ (CLEO, Belle) resonances. The $e^+e^- \to b\bar{b}$ production cross-section at the $\Upsilon(4S)$ ($\Upsilon(5S)$) resonance is about 1.1 nb (0.3 nb). The full data samples of BaBar and Belle are 560 fb$^{-1}$ and 1020 fb$^{-1}$, respectively, of which 433 fb$^{-1}$ and 710 fb$^{-1}$ are at the $\Upsilon(4S)$ resonance. Since the $\Upsilon(4S)$ decays dominantly to a pair of B mesons (either $B^+B^-$ or $B^0\bar{B}^0$), a precise knowledge of the energy-momentum of one B meson (the 'tagging B') enables deducing the properties of the other B (the 'signal B'). This property has been exploited by both BaBar and Belle, in particular, to measure inclusive decay modes as well as final states with missing neutrinos. The Belle II experiment at SuperKEKB has started recording data in 2019, and the experiment has so far collected about 428 fb$^{-1}$ of data by summer 2022, when their long shutdown 1 started. Of collected data, 363 fb$^{-1}$ is at the $\Upsilon(4S)$ resonance. At the Z resonance (SLC, LEP) all species of b-flavored hadrons could be studied for the first time. The $e^+e^- \to b\bar{b}$ production cross-section at the Z resonance is about 6.6 nb.

High-energy $p\bar{p}$ (Tevatron) and $p\bar{p}$ collisions (LHC) produce b-flavored hadrons of all species with large cross-sections. At the Tevatron ($\sqrt{s} = 1.96$ TeV) the visible cross section $\sigma(p\bar{p} \to bX, |\eta| < 1)$ is about 30 $\mu$b. CDF and D0 experiments at the Tevatron have accumulated by the end of their running about 10 fb$^{-1}$ each.

At the LHC $p\bar{p}$ collider at $\sqrt{s} = 7-13$ TeV, the visible b-hadron cross section at the LHCb experiment with pseudorapidity acceptance $2 < \eta < 5$ has been measured to be $\sim 72 \mu$b at 7 TeV and $\sim 144 \mu$b at 13 TeV [21] (cross section at 13 TeV corrected in Erratum). LHCb has collected about 1 fb$^{-1}$ at 7 TeV, 2 fb$^{-1}$ at 8 TeV, and close to 5.9 fb$^{-1}$ at 13 TeV during LHC Runs 1 and 2. CMS and ATLAS have collected each about 5 fb$^{-1}$ of data at $\sqrt{s} = 7, 20$ fb$^{-1}$ at 8 TeV and about 150 fb$^{-1}$ at 13 TeV during LHC Runs 1 and 2. The latest LHC Run 3 at 13.6 TeV started in summer 2022, with upgraded detectors. By the time of the writing (March 2024), LHC has delivered about 70 fb$^{-1}$ to ATLAS and CMS experiments, and about 300 pb$^{-1}$ to the LHCb experiment with major detector upgrades.

In hadron collisions, production happens as $b\bar{b}$ pairs via leading order flavor creation or higher order processes such as gluon-splitting. Single b-quarks can be produced by flavor excitation. The total b-production cross section is an interesting test of our understanding of leading and higher order QCD processes. With a wealth of measurements at LHC and at Tevatron (see Ref. [21] and references therein), and improved calculations [22], there is a reasonable agreement between measurements and predictions.

Each quark of a $b\bar{b}$ pair produced in hadron collisions hadronizes separately and incoherently from the other, but it is still possible to obtain a statistical indication of the charge of a produced $b/\bar{b}$ quark ("flavor tag" or "charge tag") from the accompanying particles produced in the hadronization process, or from the decay products of the other quark. The momentum spectrum of produced b-quarks typically peaks near the b-quark mass, and extends to much higher momenta, dropping by about a decade for every ten GeV. Typical decay lengths are of the order of a centimeter at 13 TeV $p\bar{p}$ collisions; the resolution for the decay vertex must be more precise than this to resolve the fast oscillations of $B_s^0$ mesons.

In $e^+e^-$ colliders, since the B mesons are very slow in the $\Upsilon(4S)$ rest frame, asymmetric beam energies are used to boost the decay products to allow time-dependent measurements that are crucial for the study of CP violation. At KEKB, the boost was $\beta\gamma = 0.43$, while PEP-II used a slightly larger boost, $\beta\gamma = 0.55$. The typical B-meson decay length is dilated from $\approx 20 \ \mu m$ to $\approx 200 \ \mu m$. At SuperKEKB the boost is lower, $\beta\gamma = 0.28$, which puts more demanding requirements on the track reconstruction precision at Belle II to reach a resolution in decay time measurements similar to Belle. The two B mesons produced in $\Upsilon(4S)$ decay are in
a coherent quantum state, which makes it easier than in hadron collisions to infer the charge state of one B meson from observation of the other; however, the coherence also requires determination of the decay time of both mesons, rather than just one, in order to perform time-dependent CP-violation measurements. For $B_s^0$, which can be produced at $\Upsilon(5S)$ the situation is less favourable, as boost is not high enough to provide sufficient time resolution to resolve the fast $B_s^0$ oscillations.

For the measurement of branching fractions, the initial composition of the data sample must be known. The $\Upsilon(4S)$ resonance decays predominantly to $B^0\bar{B}^0$ and $B^+B^-$; the current experimental upper limit for non-$B\bar{B}$ decays of the $\Upsilon(4S)$ is less than 4% at the 95% confidence level (CL) [23]. The observed modes of this category are decays to lower $\Upsilon$ states and a pion pair, $\eta$, or $\eta'$, measured branching fractions being of order $10^{-4} - 10^{-5}$ [24], and decays to $h_b(1P)\eta$ with branching fraction of order $10^{-3}$ [17].

The ratio $f_+/f_0$ of the fractions of charged to neutral B productions from $\Upsilon(4S)$ decays has been measured by CLEO, BaBar, and Belle in various ways. They typically use pairs of isospin-related decays of $B^+$ and $B^0$, such that it can be assumed that $\Gamma(B^+ \to x^+) = \Gamma(B^0 \to x^0)$. In this way, the ratio of the number of events observed in these modes is proportional to $(f_+\tau_+)/(f_0\tau_0)$ [25]. BaBar has also performed an independent measurement of $f_0$ with a different method that does not require isospin symmetry or the value of the lifetime ratio, based on the number of events with one or two reconstructed $B^0 \to D^*\ell^+\nu$ decays [26]. The combined result, from the current average of $\tau_+/\tau_0$, is $f_+/f_0 = 1.058 \pm 0.024$ [27]. The result is consistent within $2.4\sigma$ with equal production of $B^+B^-$ and $B^0\bar{B}^0$ pairs, and we assume $f_+/f_0 = 1$ in this mini-review except where explicitly stated otherwise. This assumption is also supported by the near equality of the $B^+$ and $B^0$ masses: our fit yields $m(B^0) = 5279.72 \pm 0.08$ MeV/c$^2$, $m(B^+) = 5279.41 \pm 0.07$ MeV/c$^2$, and $m(B^0) - m(B^+) = 0.31 \pm 0.05$ MeV/c$^2$.

Data collected at the $\Upsilon(5S)$ resonance gave CLEO, Belle and BaBar access to $B_s^0$ decays. In $\Upsilon(5S)$ decays there are seven possible final states including a pair of non-strange B mesons and 0, 1 or 2 pions, and three final states with a pair of strange B mesons ($B_s^{*0}\bar{B}_s^{*0}$, $B_s^{*0}\bar{B}_s^0$, and $B_s^0\bar{B}_s^0$). The fraction of events with a pair of $B_s^0$ mesons over the total number of events with a pair of b-flavored hadrons has been measured to be $f_s[\Upsilon(5S)] = 0.199^{+0.030}_{-0.029}$ [28], of which 88% is $B_s^{*0}\bar{B}_s^{*0}$ events. However, the small boost of $B_s^0$ mesons produced in this way prevents resolution of their fast oscillations for time-dependent measurements; these are only accessible in hadron collisions (or at the Z peak).

In high-energy collisions, the produced b or $\bar{b}$ quarks can hadronize with different probabilities into the full spectrum of b-hadrons, either in their ground or excited states. The hadronization does not have to be identical in $p\bar{p}$ or $pp$ collisions and in Z decay, because of the different momentum distributions of the b-quark in these processes; the sample used in the $p\bar{p}$ measurements has momenta close to the b mass, rather than $m_Z/2$ in Z decay. The available data from Tevatron and LHC show that the production fractions $f_d$, $f_u$, $f_s$, and $f_{baryon}$ of $B^0$, $B^+$, $B_s^0$, and baryons, respectively, of weakly decaying b hadrons depend on the kinematics of the produced b hadron. Recently LHCb experiment found evidence for dependence of b-quark hadronization on multiplicity in $pp$ collisions [29]. The production fractions of b hadrons are discussed in more detail in the $B^0 - \bar{B}^0$ mixing section in this Review [8].

Excited B-meson states have been thoroughly studied by CLEO, LEP, CUSB, D0 and CDF (an admixture of B mesons) and LHCb ($B^{*-+}$-meson). The current world average of the $B^{*-B}$ mass difference is $45.21 \pm 0.21$ MeV/c$^2$. Excited $B_s^*-\text{meson}$ states have observed in $\Upsilon(5S)$ decays by CUSB, CLEO and Belle.

For orbitally excited $B_{(s)}$ meson states, with relative angular momentum L=1 of the two quarks, there exist four states $(J, j_q) = (0, 1/2)$, $(1, 1/2)$, $(1, 3/2)$, $(2, 3/2)$, where $j_q$ is the total angular momentum of the light u, d or s quark and J is the total angular momentum of the B meson. These states are collectively called as $B_{(s)}^{**}$ mesons. The $j_q = 1/2$ states are named $B_{(s)0}^*$ ($J = 0$) and $B_{(s)1}^*$ ($J = 1$) mesons, while the states with

$j_q = 3/2$ are named $B_{(s)1}$ ($J = 1$) and $B_{(s)2}^*$ ($J = 2$) mesons. The states with $j_q = 1/2$ can decay through an S-wave transition and are expected to have a large width, but the $j_q = 3/2$ states are narrow D-wave decays. Evidence for $B^{**}$ production has been initially obtained at LEP as a broad $B\pi$ resonance [30] or a $B^+K^-$ enhancement [31]. Detailed results have been obtained for the narrow states $B_1(5721)^{0,+}$ and $B_2(5747)^{0,+}$ at the Tevatron and by LHCb, and clear enhancements compatible with the higher mass states $B_J(5840)^{0,+}$ and $B_J(5970)^{0,+}$ have been observed [32]. Also the narrow $B_s^{**}$ states $B_{s1}(5830)^0$ and $B_{s2}(5840)^0$ have been measured at the CDF [32], LHCb [33], and CMS [34].

Excited states of $B_c^+$ mesons will provide important information about the strong potential. A $B_c^+\pi^+\pi^-$ resonance has been observed for the first time by ATLAS [35]. The mass of the resonance has been measured precisely by CMS and LHCb as $6871.2 \pm 1.0$ MeV/c$^2$ [36]. The resonance may be interpreted as the second S-wave state of the $B_c^+$ meson, $B_c^+(2S)$, but the quantum numbers are to be confirmed.

Baryon states containing a b quark are labeled according to the same scheme used for non-b baryons, with the addition of a b subscript [20]. The first observed b baryon was the $\Lambda_b^0$ (quark composition ud b). Thanks to the large samples accumulated at the Tevatron and specially at the LHC many new b baryons have been found. The masses of all these new baryons have been measured to a precision of a few MeV/c$^2$, and found to be in agreement with predictions from Heavy Quark Effective Theory (HQET).

Clear signals of four strongly-decaying baryon states, $\Sigma_b^+$, $\Sigma_b^{*+}$ (uub), $\Sigma_b^-$, $\Sigma_b^{*-}$ (ddb) have been obtained by CDF [37] and LHCb [38]. LHCb has also observed two new mass peaks in the $\Lambda_b^0\pi^\pm$ systems, consistent with single resonances and named as $\Sigma_b^\pm$ (6097) [38]. The nature of these resonances is, however, not yet clear. The isodublet of strange b baryons $\Xi_b^0$ (usb) and $\Xi_b^\pm$ (dsb) has been observed by CDF and D0 [39]. Masses, lifetimes, and branching ratios have been accurately measured by LHCb [40-42] and CDF [43]. LHCb has also measured several parameters sensitive to P and CP violation [42,44]. Other observed $\Xi_b$ baryons are spin-3/2 states $\Xi_b(5945)^0$ ($\Xi_b^{*0}$) [45, 46], $\Xi_b^*(5955)^-$ [46, 47] and $\Xi_b(6100)^-$ [46, 48], a spin-1/2 state $\Xi_b^*(5935)^-$ [46, 47], and resonance states $\Xi_b^-$ (6227) [41, 49], $\Xi_b^0(6327)$ [50] and $\Xi_b^0(6333)$ [50]. The doubly-strange bottom baryon $\Omega_b^-$ has been observed first by D0 and CDF [51]. Mass and mean life have been measured precisely by LHCb [52] and CDF [43]. LHCb has also observed four excited $\Omega_b^-$ states $\Omega_b^-(6316)$, $\Omega_b^-(6330)$, $\Omega_b^-(6340)$ and $\Omega_b^-(6350)$ [53].

The so-called exotic states have raised a lot of interest recently. While many exotic states were seen in the charm sector, in bottom sector there are fewer seen. The D0 Collaboration claimed a narrow state X(5568) decaying into a $B_s^0\pi^\pm$ final state [54]. While this would be an interesting addition to the observed states as the first exotic state with constituent quarks with four different flavours (b, s, u, d), analysis by LHCb yields negative result [55]. Also CMS finds no such a state [56].

## 73.2 Lifetimes

Precise lifetimes are key in extracting the weak parameters that are important for understanding the role of the CKM matrix in CP violation, such as the determination of $V_{cb}$ and $B_s^0 - \bar{B}_s^0$ mixing parameters. In the naive spectator model, the heavy-flavored hadrons can decay only via the external spectator mechanism, and thus, the lifetimes of all mesons and baryons containing b quarks would be equal. Non-spectator effects, such as the interference between contributing amplitudes, modify this simple picture and give rise to a lifetime hierarchy for b-flavored hadrons similar to the one in the charm sector. However, since the lifetime differences are expected to scale as $1/m_Q^2$, where $m_Q$ is the mass of the heavy quark, the variations in the b system are expected to be only 10% or less [57, 58]. We expect:

$$ \tau(B^+) \ge \tau(B^0) \approx \tau(B_s^0) > \tau(\Lambda_b^0) \gg \tau(B_c^+) . \quad (73.1) $$
For the $B_c^+$, both quarks decay weakly, so the lifetime is much shorter.

Measurements of the lifetimes of the different b-flavored hadrons thus provide a means to determine the importance of non-spectator mechanisms in the b sector. Availability of large samples of fully-reconstructed decays of different b-hadron species has resulted in precise measurements with small statistical and systematic uncertainties (~1%). The world averages given in Table 73.1 have been determined by the Heavy Flavor Averaging Group (HFLAV) [59].

<table><caption>**Table 73.1:** Summary of the world-average b-hadron lifetime measurements. For the $B_s^0$ lifetimes, see text below.</caption><thead><tr><th>Particle</th><th>Lifetime [ps]</th></tr></thead><tbody><tr><td>$B^+$</td><td>1.638 ± 0.004</td></tr><tr><td>$B^0$</td><td>1.517 ± 0.004</td></tr><tr><td>$B_s^0$</td><td>1.520 ± 0.005</td></tr><tr><td>$B^0_s$</td><td>1.429 ± 0.006</td></tr><tr><td>$B^0_sL$</td><td>1.623 ± 0.008</td></tr><tr><td>$B^+_c$</td><td>0.510 ± 0.009</td></tr><tr><td>$\Lambda_b^0$</td><td>1.471 ± 0.009</td></tr><tr><td>$\Xi_b^-$</td><td>1.572 ± 0.040</td></tr><tr><td>$\Xi_b^0$</td><td>1.480 ± 0.030</td></tr><tr><td>$\Omega_b^-$</td><td>1.64<sup>+0.18</sup><sub>-0.17</sub></td></tr></tbody></table>

The $B_s^0$ lifetime in Table 73.1 is defined as $1/\Gamma_s$, where $\Gamma_s$ is the average width of the light (L) and heavy (H) mass eigenstates, $(\Gamma_L + \Gamma_H)/2$. In the absence of CP violation, the light (heavy) $B_s^0$ mass eigenstate is the CP-even (CP-odd) eigenstate. Thus, the lifetime of the light (heavy) mass eigenstate can be measured from CP-even (odd) final states. The lifetimes can also be obtained from time-dependent angular analysis of $B_s^0 \to J/\psi\phi$ decays.

The short $B_c^+$ lifetime is in good agreement with predictions [60]. With large samples of $B_c^+$ mesons at the LHC precision on the lifetimes can still improve. The measurement using semileptonic decays gives $\tau_{B_c^+} = 0.509 \pm 0.008 \pm 0.012$ ps [61] while using decays $B_c^+ \to J/\psi\pi^+$ yields $\tau_{B_c^+} = 0.5134 \pm 0.0110 \pm 0.0057$ ps [62]. Each of these is more precise than the combination of all previous experiments.

The recent $\Lambda_b^0$ lifetime measurements from LHC experiments and CDF are precise and favour lifetime close to the lifetime of $B^0$ meson, in agreement with theory.

For precision comparisons with theory, lifetime ratios are more sensitive. Experimentally it is found [59]:

$$ \begin{aligned} \frac{\tau_{B^+}}{\tau_{B^0}} &= 1.076 \pm 0.004, & \frac{\tau_{B_s^0}}{\tau_{B^0}} &= 1.002 \pm 0.004, \\ \frac{\tau_{\Lambda_b^0}}{\tau_{B^0}} &= 0.969 \pm 0.006, \end{aligned} $$

while recent Heavy Quark Expansion (HQE) predictions give [58]:

$$ \begin{aligned} \frac{\tau_{B^+}}{\tau_{B^0}} &= 1.04^{+0.05}_{-0.01} \pm 0.02 \pm 0.01, & \frac{\tau_{B_s^0}}{\tau_{B^0}} &= 1.001 \pm 0.002, \\ \frac{\tau_{\Lambda_b^0}}{\tau_{B^0}} &= 0.935 \pm 0.054. \end{aligned} $$

The ratio of $B^+$ to $B^0$ lifetimes has a precision of better than 1%, and is significantly different from 1.0, in agreement with predictions [57]. The ratio of $B_s^0$ to $B^0$ lifetimes is expected to be very close to 1.0.

For a detailed discussion on neutral $B^0$ and $B_s^0$ oscillation and relevant CP violation measurements see Ref. [8].

### 73.3 Features of decays

The ground states of b-flavored hadrons decay via weak interactions. In most decays of the b-flavored hadrons, where the b-quark is accompanied by lighter partner quarks (d, u, s, or c), the decay modes are well described by the decay of the b quark (spectator

model) [63]. The dominant decay mode of a b quark is $b \to cW^*-$ (referred to as a “tree” or “spectator” decay), where the virtual W materializes either into a pair of leptons $\ell\bar{\nu}$ (“semileptonic decay”), or into a pair of quarks which then hadronizes. The transition $b \to u$ is suppressed by $|V_{ub}/V_{cb}|^2 \sim (0.1)^2$ relative to $b \to c$ transitions. The decays in which the spectator quark combines with one of the quarks from $W^*$ to form one of the final state hadrons are suppressed by a factor $\sim (1/3)^2$, because the colors of the two quarks from different sources must match (“color-suppression”).

Semileptonic B decays $B \to X_c\ell\nu$ and $B \to X_u\ell\nu$ provide an excellent way to measure the magnitude of the CKM elements $|V_{cb}|$ and $|V_{ub}|$ respectively, because the strong interaction effects are much simplified due to the two leptons in the final state. Both exclusive and inclusive decays can be used with dominant uncertainties being complementary. For exclusive decay analysis, knowledge of the form factors for the exclusive hadronic system $X_{c(u)}$ is required. For inclusive analysis, it is usually necessary to restrict the available phase-space of the decay products to suppress backgrounds; subsequently uncertainties are introduced in the extrapolation to the full phase-space. Moreover, restriction to a small corner of the phase-space may result in breakdown of the operator-product expansion scheme, thus making theoretical calculations unreliable. A more detailed discussion of B semileptonic decays and the extraction of $|V_{cb}|$ and $|V_{ub}|$ is given elsewhere in this Review [9]. While traditionally $B^0$ and $B^+$ decays were used, over time also other B hadron studies became available. Most notably, determination of $|V_{ub}|$ using $\Lambda_b^0 \to p\mu^-\bar{\nu}_\mu$ decays by LHCb [64] was more precise than expected. Besides, there have been measurements of inclusive semileptonic decay rates of $B_s^0$ [65] and $B_c^+$ [66] mesons. One of the latest additions in this area is the observation of the $B^0 \to K^-\mu^+\nu_\mu$ decays by LHCb using only a fraction of their available data [67].

On the other hand, hadronic B decays are complicated because of strong interaction effects caused by the surrounding cloud of light quarks and gluons. While this complicates the extraction of CKM matrix elements, it also provides a great opportunity to study perturbative and non-perturbative QCD, hadronization, and Final State Interaction (FSI) effects.

Many aspects of B decays can be understood through the Heavy Quark Effective Theory (HQET) [68]. This has been particularly successful for semileptonic decays. For further discussion of HQET, see for instance Ref. [69]. For hadronic decays, one typically uses effective Hamiltonian calculations that rely on a perturbative expansion with Wilson coefficients. In addition, some form of the factorization hypothesis is commonly used, where, in analogy with semileptonic decays, two-body hadronic decays of B mesons are expressed as the product of two independent hadronic currents, one describing the formation of a charm meson (in case of the dominant $b \to cW^*-\bar{\nu}_s$ decays), and the other describing the hadronization of the remaining $\bar{u}\bar{d}$ (or $c\bar{s}$) system from the virtual $W^-$. Qualitatively, for B decays with a large energy release, e.g. $b \to uW^*-\bar{\nu}_s$ transitions, the $\bar{u}\bar{d}$ pair (produced as a color singlet) travels fast enough to leave the interaction region without influencing the meson containing the spectator quark. This is known to work well for the dominant spectator decays [70]. There are several common implementations of these ideas for hadronic B decays, the most common of which are QCD factorization (QCDF) [71], perturbative QCD (pQCD) [72], and soft collinear effective theory (SCET) [73].

The transitions $b \to s$ and $b \to d$ are flavor-changing neutral-current (FCNC) processes. Although they are not allowed in the SM as a tree-process, they can occur via loop diagrams (denoted “penguin” decays). The rates for $b \to s$ penguin decays are comparable to the CKM-suppressed $b \to u$ tree processes. Pure-penguin decays were first established by the observation of $B \to K^*(892)\gamma$ [74]. Penguin processes involving $b \to d$ transitions are further suppressed by CKM, and have been observed for $B \to (\rho/\omega)\gamma$ decays [75, 76]. LHCb has observed a $b \to d$ penguin transition in the $B^+ \to \pi^+\mu^+\mu^-$ mode and measured its branching fraction to be $(1.83 \pm 0.24 \pm 0.05) \times 10^{-8}$ [77].

Other decay processes discussed in this Review include W-exchange (a W is exchanged between initial-state quarks), penguin annihilation (the gluon from a penguin loop attaches to
the spectator quark, similar to an exchange diagram), and pure-
annihilation (the initial quarks annihilate to a virtual W, which
then decays). Some observed decay modes such as B$^{0} \to D_{s}^{-} K^{+}$,
may be interpreted as evidence of a W-exchange process [78]. The
evidence for the purely leptonic decay B$^{+} \to \tau^{+}\nu$ from Belle [79]
and BaBar [80] is the first sign of a pure annihilation decay. The
average branching fraction is (1.09 $\pm$ 0.24) $\times$ 10$^{-4}$, which is some-
what larger than, though consistent with, the value expected in
the SM. A substantial region of parameter space of charged Higgs
mass vs. $\tan\beta$ is excluded by the measurements of this mode.
A dedicated discussion of purely leptonic decays of charged pseu-
doscalar mesons is given elsewhere in this Review [81].

73.4 Dominant hadronic decays

Most of the hadronic B decays involve $b \to c$ transition at the
quark level, resulting in a charmed hadron or charmonium in the
final state. Other types of hadronic decays are very rare and will
be discussed separately in the next section. The experimental re-
sults on hadronic B decays have steadily improved over the years,
and the measurements have reached sufficient precision to chal-
lenge our understanding of the dynamics of these decays. With
good particle detection and hadron identification capabilities of
B-factory detectors, a substantial fraction (roughly on the order
of a few per mill) of hadronic B decay events can be fully recon-
structed. In particular, good performances for detecting $\pi^0$ and
other neutral particles helped Belle and BaBar to make compre-
hensive measurements of the decays $\bar{B}^0 \to D^{(*)0}h^0$ [82], where
$h^0$ stands for light neutral mesons such as $\pi^0$, $\eta^{(\prime)}$, $\rho^0$, $\omega$. The
measurements are being complemented by LHCb, in decays like
$\bar{B}^0 \to D^0\pi^+\pi^-$ [83], where no neutral particles reconstruction is
needed. These decays proceed through color-suppressed diagrams,
hence they provide useful tests on the factorization models.

Because of the kinematic constraint of $\Upsilon(4S) \to B\bar{B}$, the energy sum of the final-state particles of a B meson decay is always equal to one half of the total energy in the center of mass frame. As a result, the two variables, $\Delta E$ (energy difference) and $M_B$ (B candidate mass with a beam-energy constraint) are very effective for reducing combinatorial background both from $\Upsilon(4S)$ and $e^+e^- \to q\bar{q}$ continuum events. In particular, the energy-constraint in $M_B$ improves the signal resolution by almost an order of magnitude.

The kinematically clean environment of B meson decays provides an excellent opportunity to search for new states. For instance, quark-level $b \to c\bar{c}s$ decays have been used to search for new charmonium and charm-strange mesons and study their properties in detail. While narrow charm-strange states $D_{s0}^*(2317)$ [84] and $D_{s1}(2460)$ [85] were discovered by BaBar and CLEO, respectively, the properties of these new states were revealed by studying the B meson decays, $B \to DD_{s0}^*(2317)$ and $B \to DD_{s1}(2460)$ by Belle [86] and BaBar [87]. Another example is Dalitz plot analysis of decay $B_s^0 \to \bar{D}^0 K^- \pi^+$ in which the decay to spin-3 resonance was observed for the first time [88]. One of the most significant improvements in the past decade is establishment of decays $B^+ \to D^0 K^+$ and $B^+ \to D^0 K^*(892)^+$ with $D^0$ decaying to final state common to $D^0$ and $\bar{D}^0$ [89]. This allows direct determination of CKM angle $\gamma(= \phi_3)$ to be 65.9$^{+3.3}_{-3.5}$.

Information on $B_s^0$, $B_c^+$ and $\Lambda_b^0$ decays have been remarkably
improved with recent studies of large samples from LHCb. Notice-
able additions in $B_s$ include decay modes to $D_s^{(*)+} D_s^{(*)-}$, $\bar{D}^0 K^0$,
and $J/\psi \bar{K}^*(892)^0$. The $B_s^0 \to D_s^{(*)+} D_s^{(*)-}$ decays were first ob-
served by CDF [90], followed by Belle [91]. LHCb has improved
the precision with $B(B_s^0 \to D_s^{(*)+} D_s^{(*)-}) = (3.07 \pm 0.22 \pm 0.33)\%$
[92], which suggests that $B_s^0 \to D_s^{(*)+} D_s^{(*)-}$ decays do not sat-
urate the CP-even modes of the $B_s$ decays. The $B_s^0 \to \bar{D}^0 K^0$
decay occurs mostly via a color-suppressed tree diagram, and has
a small theoretical uncertainty in the SM, thus this mode can
significantly improve the determination of the CP-violation angle
$\phi_s$. LHCb has observed this decay and the branching fraction
is $(4.3 \pm 0.5 \pm 0.7) \times 10^{-4}$ [93]. The $B_s^0 \to J/\psi \bar{K}^*(892)^0$ decay
can be used to constrain the penguin pollution in determining $\phi_s$.
LHCb has updated the branching fraction and measured the CP
asymmetries of this decay, thereby constraining the penguin pol-

lution in $\phi_s$ [94], although a much more stringent constraint on
penguin pollution can come from $B^0 \to J/\psi p^0$ which has been
observed by BaBar [95] and LHCb [96]. The $B_c^+ \to B_s^0 \pi^+$ decay
is unique as the only observed mode of b-flavored hadron decays
where the partner quark decays (c in this case) while the b quark
remains a spectator. LHCb has observed this mode [97] and mea-
sured $B(B_c^+ \to B_s^0 \pi^+)/B(B_c^+ \to J/\psi \pi^+) = (91 \pm 10 \pm 8 \pm 3)$
[98]. In addition, LHCb [99] and ATLAS [100] have measured
$B_c^+ \to J/\psi D_s^{(*)+}$, which, by comparing with $B_c^+ \to B_s^0 \pi^+$, pro-
vides a ratio of exclusive $b \to c$ and $c \to s$ decays of $B_c^+$. For
$\Lambda_b^0 \to \Lambda_c^+ \pi^+ \pi^- \pi^-$ [101], not only the total rate is measured,
but also structure involving decays through excited $\Lambda_c$ and $\Sigma_c$
baryons.

In addition, a variety of exotic particles that do not fit the con-
ventional meson spectroscopy have been discovered in B decays.
Belle found the X(3872) state by studying $B^+ \to J/\psi \pi^+ \pi^- K^+$
[102], which was confirmed by CDF [103], D0 [104] and BaBar
[105]. Production of X(3872) has been studied by the LHC ex-
periments, LHCb [106], CMS [107] and ATLAS [108].

A charged charmonium-like state X(4430)$^{\pm}$ that decays to $\psi(2S)\pi^{\pm}$ was observed by Belle in $B \to \psi(2S)K\pi^{\pm}$ [109]. Since it is charged, it could not be an ordinary charmonium state. A high-statistics study by LHCb confirmed the existence of the X(4430)$^{\pm}$ in decays $B \to \psi(2S)K\pi^{\pm}$ [110], demonstrated its resonance character by studying the phase motion, unambiguously determined its spin-parity, and saw evidence for another state. In a Dalitz plot analysis of $\bar{B}^0 \to J/\psi K^- \pi^+$ [111], Belle has found another state, labeled as X(4200)$^+$ in this Review, adding to the list of exotic charged charmonium-like states. In an amplitude analysis of the decay $\Lambda_b^0 \to J/\psi p K^-$, LHCb observed exotic structures, labeled as $P_c(4380)^+$ and $P_c(4450)^+$ in this Review, in the $J/\psi p$ channel [112]. The subsequent analysis with significantly increased statistics observed additional state and resolved the peak at 4450 MeV/c² as being due to the two states close in the mass [113]. The structure in the $J/\psi p$ channel was also seen in the $B_s^0 \to J/\psi p \bar{p}$ decays [114] and in the $J/\psi \Lambda$ channel in $B^+ \to J/\psi \Lambda p$ and $\Xi_b^- \to J/\psi \Lambda K^-$ decays [115]. They are referred to as charmonium-pentaquark states. More detailed discussions of exotic meson-like states and pentaquarks are given elsewhere in this Review [116].

73.5 Rare hadronic decays

All B-meson decays that do not occur through the $b \to c$ transition are usually called rare B decays. These include both semileptonic and hadronic $b \to u$ decays that are suppressed at leading order by the small CKM matrix element $V_{ub}$, as well as higher-order $b \to s(d)$ processes such as electroweak and gluonic penguin decays. In this section, we review hadronic rare B decays, while electroweak penguin decays and others are discussed in the next.

Charmless B meson decays into two-body hadronic final states such as $B \to \pi\pi$ and $K\pi$ are experimentally clean, and provide good opportunities to probe new physics and search for indirect and direct CP violations. Since the final state particles in these decays tend to have larger momenta than average B decay products, the event environment is cleaner than for $b \to c$ decays. Branching fractions are typically around $10^{-5}$. Over the past decade, many such modes have been observed not only by $e^+e^-$ collider experiments such as BaBar and Belle, but also by hadron collider experiments such as CDF ($\bar{p}\bar{p}$) and LHCb ($p\bar{p}$). In the latter cases, huge data samples of the modes with all charged final-state particles have been reconstructed by triggering on the impact parameter of the charged tracks. This has also allowed observation of charmless decays of the $B_s$, in final states such as $\phi\phi$ [117], $K^+K^-$ [118,119], and $K^-\pi^+$ [119,120], and of charmless decays of the $\Lambda_b^0$ baryon [120]. The large samples available at LHCb experiment allow to perform also time-dependent CP violation measurements [121]. Charmless $B_s$ modes are related to corresponding $B^0$ modes by U-spin symmetry, and are determined by similar amplitudes. Combining the observables from $B_s^0$ and $B^0$ modes is a further way of eliminating hadronic uncertainties and extracting relevant CKM information [122].

Because of relatively high-momenta for final state particles, the
dominant source of background in $e^{+}e^{-}$ collisions is $q\bar{q}$ contin-
uum events; sophisticated background suppression techniques ex-
ploiting event shape variables are essential for these analyses. In
hadron collisions, the dominant background comes from QCD or
partially reconstructed heavy flavors, and is similarly suppressed
by a combination of kinematic and isolation requirements. The
results are in general consistent among the experiments.

Most rare decay modes including $B^0 \to K^+\pi^-$ have contri-
butions from both $b \to u$ tree and $b \to sg$ penguin processes.
If the size of the two contributions are comparable, the interfer-
ence between them may result in direct CP violation, seen exper-
imentally as a charge asymmetry in the decay rate measurement.
BaBar [123], Belle [124], CDF [118], and LHCb [121, 125] have
measured the direct CP violating asymmetry in $B^0 \to K^+\pi^-$ de-
cays. Direct CP violation has been observed in this decay with a
significance of more than $5\sigma$. The world average value of the asym-
metry is now rather precise, $A_{CP}(K^+\pi^-) = -0.0834 \pm 0.0032$.
The CP asymmetry in $B^+ \to K^+\pi^0$ mode has been measured by
BaBar [126], *Belle [124] and LHCb [127] with the average value
$A_{CP}(K^+\pi^0) = 0.030 \pm 0.013$. These two asymmetries differ signif-
icantly, in contrast to a naive expectation based on simplified pic-
ture in the SM. For more detailed tests, there are sum rules [128]
that relate the decay rates and decay-rate asymmetries between
the four $K\pi$ charge states. A crucial ingredient of the sum rule
test is $A_{CP}(K^0\pi^0)$. Currently, measured values are reported by
both BaBar [129] and Belle [130]. Using the $A_{CP}(K^0\pi^0)$ value of
Ref. [130], Belle reports the sum rule test result that is consistent
with zero within $1.9\sigma$ [124]. With the future improvements via
Belle II and upgraded LHCb, the measurements are expected to
become precise enough to shape a definite conclusion. The CP
asymmetry in the $\pi^+K^-$ mode has also been measured in $B_s^0$
decays, by CDF [118] and LHCb [121, 125]. The combined value is
$A_{CP}(B_s^0 \to \pi^+K^-) = 0.224 \pm 0.012$.

In addition to $B_{(s)} \to K\pi$ modes, significant ($> 3\sigma$) non-zero CP asymmetries have been measured in several other rare decay modes: $A_{CP}(B^+ \to \rho^0 K^+) = 0.37 \pm 0.10$ [131], $A_{CP}(B^+ \to \eta K^+) = -0.37 \pm 0.08$ [132], $A_{CP}(B^0 \to \eta K^{*0}) = 0.19 \pm 0.05$ [133], and $A_{CP}(B^+ \to f_2(1270) K^+) = -0.68^{+0.19}_{-0.17}$ [131]. In at least the first two cases, a large direct CP violation might be expected since the penguin amplitude is suppressed so the tree and penguin amplitudes may have comparable magnitudes. There are also measurements by LHCb of CP asymmetries in several 3-body modes: $A_{CP}(B^+ \to \pi^+\pi^-\pi^+) = 0.057 \pm 0.013$, $A_{CP}(B^+ \to K^+\pi^-\pi^+) = 0.027 \pm 0.008$, $A_{CP}(B^+ \to K^+K^-\pi^+) = -0.122 \pm 0.0021$, and $A_{CP}(B^+ \to K^+K^-K^+) = -0.033 \pm 0.008$ [134]. Many of these analyses now include Dalitz plot treatments with many intermediate resonances.

BaBar [135] and Belle [124, 136] have observed the decays
$B^+ \to \bar{K}^0 K^+$ and $B^0 \to K^0 \bar{K}^0$. The world-average branching
fractions are $B(B^0 \to K^0 \bar{K}^0) = (1.21 \pm 0.16) \times 10^{-6}$ and $B(B^+ \to$
$\bar{K}^0 K^+) = (1.31 \pm 0.17) \times 10^{-6}$. These are the first observa-
tions of hadronic $b \to d$ transitions, with significance bigger than
$5\sigma$ for all four measurements. CP asymmetries have been mea-
sured for these modes, but with large errors. LHCb has observed
$B^0 \to K^+ K^-$ mode which occurs via a weak-annihilation process
and is the rarest hadronic B-meson decay thus far observed, with
$B(B^0 \to K^+ K^-) = (7.80 \pm 1.52) \times 10^{-8}$ [137]. $B_s^0 \to K^+ K^-$
decay mode, which occurs mostly via $b \to s$ penguin process, has
been observed by Belle [138], CDF [139] and LHCb [119]. The av-
erage branching fraction is $B(B_s^0 \to K^+ K^-) = (26.6 \pm 2.2) \times 10^{-6}$.
Belle has also observed $B_s^0 \to K^0 \bar{K}^0$ [140] which also occurs
via $b \to s$ penguin transition in the SM. This was recently
confirmed by LHCb [141]. The average branching fraction is
$(1.76 \pm 0.31) \times 10^{-5}$.

The decay $B^0 \to \pi^+\pi^-$ can be used to extract the CKM angle $\alpha$ (for details see elsewhere in this Review [142]). This is complicated by the presence of significant contributions from penguin diagrams. An isospin analysis [143] can be used to untangle the penguin complications. The decay $B^0 \to \pi^0\pi^0$ is crucial in this analysis. Both BaBar and Belle have observed $B^0 \to \pi^0\pi^0$, with a mild tension in the measured branching fractions: $(1.83 \pm 0.25) \times 10^{-6}$ for BaBar [123] and $(1.31 \pm 0.26) \times 10^{-6}$ for Belle [144]. It turns out that the amount of penguin pollution in the $B \to \pi\pi$ system is

rather large. In the past few years, measurements in the $B^0 \to \rho\rho$
system have produced more precise values of $\alpha$, since penguin am-
plitudes are generally smaller for decays with vector mesons. An
important ingredient in the analysis is the $B^0 \to \rho^0\rho^0$ branch-
ing fraction. The average of measurements from BaBar [145] and
Belle [146] yields a branching fraction of $(0.96 \pm 0.15) \times 10^{-6}$. This
is only 3% of the $\rho^+\rho^-$ branching fraction, much smaller than the
corresponding ratio $(\gtrsim 20\%)$ in the $\pi\pi$ system.

Since $B \to \rho\rho$ has two vector mesons in the final state, the CP eigenvalue of the final state depends on the longitudinal polarization fraction $f_L$ for the decay. Therefore, a measurement of $f_L$ is needed to extract the CKM angle $\alpha$. Both BaBar and Belle have measured $f_L$ for the decays $\rho^+\rho^-$ [147] and $\rho^+\rho^0$ [148] and in both cases the measurements show $f_L > 0.9$, making a complete angular analysis unnecessary. In $B^0 \to \rho^0\rho^0$, $f_L$ is measured by BaBar [145], Belle [146] and LHCb [149], with the average value being $0.71^{+0.08}_{-0.09}$.

By analyzing the angular distributions of the B decays to two
vector mesons, we can learn a lot about both weak- and strong-
interaction dynamics in B decays. Decays that are penguin-
dominated surprisingly have values of $f_L$ near 0.5. The list of such
decays has now grown to include $B \to \phi K^*(892)$, $B \to \rho K^*(892)$,
and $B \to \omega K^*(892)$. The reasons for this "polarization puzzle" are
not fully understood. A detailed description of the angular anal-
ysis of B decays to two vector mesons can be found in a separate
mini-review [150] in this Review.

## 73.6 Electroweak penguin decays

Electroweak penguin decays are one-loop FCNC decays pro-
ceeding through penguin or box Feynman diagrams with final
state including real photon or pair of leptons. Such decays
were first observed by CLEO experiment when it observed decay
$B \to K^*(892)\gamma$ [74]. Since then significant amount of experi-
mental information was obtained. Branching fractions for these
decays are $10^{-5}$ or less, which makes them excellent candidates
for searches for new physics beyond SM. Often several observables
are available, which allows for stringent tests of the SM.

Starting with radiative decays, experimentally easiest to study
are exclusive decays with a fully reconstructed final state. The
best studied decay in this class is $B \to K^*(892)\gamma$ seen by CLEO,
Belle, BaBar experiments [151, 152] with world average branch-
ing fraction $B(B^0 \to K^*(892)^0\gamma) = (41.8 \pm 2.5) \times 10^{-6}$. Decays
through several other kaon resonances such as $B \to K_1(1270)\gamma$,
$K_2^*(1430)\gamma$, etc. were studied at B-factories [153, 154]. It is worth
to mention decay $B^+ \to K^+\pi^+\pi^-\gamma$ for which besides measure-
ments of the branching fraction [154, 155] one can also use the
angular distribution to access photon polarization. Such a mea-
surement was done by the LHCb experiment, which was able to
clearly demonstrate that the photon in $B^+ \to K^+\pi^+\pi^-\gamma$ decay
is polarized [156]. Unfortunately given non-trivial hadronic struc-
ture, more work is needed before turning this into test of the SM.
The exclusive radiative decays $B_s^0 \to \phi\gamma$ was seen by the Belle
and LHCb experiments [157] with an average branching fraction
of $(3.4 \pm 0.4) \times 10^{-5}$ and more recently also decay $\Lambda_b \to \Lambda\gamma$ was
observed by LHCb [158].

Compared to $b \to s\gamma$, the $b \to d\gamma$ transitions such as $B \to \rho\gamma$,
are suppressed by the CKM elements ratio $|V_{td}/V_{ts}|^2$. Both Belle
and BaBar have observed these decays [75, 76]. The world average
$B(B \to (\rho, \omega)\gamma) = (1.30 \pm 0.23) \times 10^{-6}$. This can be used to
calculate $|V_{td}/V_{ts}|$ [159]; the measured values are $0.195^{+0.025}_{-0.024}$ from
Belle [75] and $0.233^{+0.033}_{-0.032}$ from BaBar [76].

The observed radiative penguin branching fractions can con-
strain a large class of SM extensions [160]. However, due to the
uncertainties in the hadronization, only the inclusive $b \to s\gamma$ rate
can be reliably compared with theoretical calculations. This rate
can be measured from the endpoint of the inclusive photon spec-
trum in B decay. By combining the measurements of $B \to X_s\gamma$
from the CLEO, BaBar, and Belle experiments [161, 162], HFLAV
obtains the average: $B(B \to X_s\gamma) = (3.49 \pm 0.19) \times 10^{-4}$ [59]
for $E_\gamma \gtrsim 1.6$ GeV, averaging over $B^+$ and $B^0$. Consistent but
less precise results have been reported by ALEPH for inclusive
$b$-hadrons produced at the Z, which includes also contribution
from $B_s^0$ and $\Lambda_b^0$ hadrons. Using the sum of seven exclusive final
states, the BaBar experiment measured the branching fraction of inclusive $b \to d\gamma$ decays to be $(9.2 \pm 2.0 \pm 2.3) \times 10^{-6}$ [163]. The measured branching fraction can be compared to theoretical calculations. Recent calculations of $B(b \to s\gamma)$ at NNLO level predict for the $E_\gamma \ge 1.6$ GeV values of $(3.36 \pm 0.23) \times 10^{-4}$ for $b \to s\gamma$ and $(1.68 \pm 0.17) \times 10^{-5}$ for $b \to d\gamma$ decays [164].

The CP asymmetry in $b \to s\gamma$ is extensively studied theoretically both in the SM and beyond [165]. According to the SM, the CP asymmetry in $b \to s\gamma$ is smaller than 1%, but some non-SM models allow significantly larger CP asymmetry ($\sim 10\%$) without altering the branching fraction. The current world average is $A_{CP} = 0.015 \pm 0.011$, again dominated by BaBar and Belle [166, 167]. In addition to the CP asymmetry, BaBar and Belle also measured the isospin asymmetry $\Delta_{0-} = -0.005 \pm 0.020$ in $b \to s\gamma$ measured using sum of exclusive decays [166, 168]. An alternative measurement using full reconstruction of the companion $B$ in the hadronic decay modes yields a consistent, but less precise result [169]. Both Belle and BaBar experiments measured the isospin asymmetry in exclusive $B \to K^*(892)\gamma$ decay with average of $6.3 \pm 1.7\%$ [152] and therefore providing evidence for the non-zero isospin asymmetry.

In addition, experiments have measured the inclusive photon energy spectrum for $b \to s\gamma$, and by analyzing the shape of the spectrum they obtain the first and second moments for photon energies. Belle has measured these moments covering the widest range in the photon energy $(1.7 < E_\gamma < 2.8$ GeV) [162]. The measurement by BaBar has slightly smaller range with lower limit at 1.8 GeV [170]. These results can be used to extract non-perturbative HQET parameters that are needed for precise determination of the CKM matrix element $V_{ub}$ (see further discussion elsewhere in this Review [171]).

Additional information on FCNC processes can be obtained from $b \to s\ell^+\ell^-$ decays. These processes are studied as a function of dilepton invariant mass squared, $q^2$. Different $q^2$ regions are sensitive to different physics. Starting at the very low $q^2$ decays exhibit sensitivity to the same physics as the radiative decays. Then for the $q^2$ in region 1.1 to 6.0 GeV$^2$/c$^4$ the SM and new physics have best chance to compete. At the high $q^2$ above the $\psi(2S)$ mass, the interference of SM and new physics is to some extend complementary to that in lower $q^2$. Regions around $J/\psi$ and $\psi(2S)$ is normally excluded from measurements as these are dominated by the $b \to c$ transitions to charmonia. For exclusive decays, theory predictions require calculations of hadronic form factors. With current theory predictions, the most useful are measurements within the $q^2$ regions 1.1 to 6.0 GeV$^2$/c$^4$ and from 16.0 GeV$^2$/c$^4$ up to the kinematic limit. From this reason in the listing we provide results mainly in those two regions.

Similar as for radiative decays, also for the $b \to s\ell^+\ell^-$ decays the inclusive measurements provide some benefits. Both Belle and BaBar performed such measurement without reconstructing hadronic part exclusively and measure a branching fraction of $(5.8 \pm 1.3) \times 10^{-6}$ [172]. Unfortunately this measurement is not trivially possible at hadron colliders and also does not easily allow the angular distributions of the decay products to be exploited. One alternative is to extract information on the inclusive decay as sum of exclusive decays. Such a measurement was performed by Belle [173], but in this case the difficulty lies in extrapolation for the missing hadronic states.

Turning to the exclusive decays, the initial measurements performed by B-factories typically averaged between charged and neutral $B$ mesons as well as between $e^+e^-$ and $\mu^+\mu^-$ finals states. The experiments CDF, LHCb, ATLAS and CMS are much better suited for the $\mu^+\mu^-$ finals states compared to the $e^+e^-$ finals states. As such most measurements at hadron colliders are done only with $\mu^+\mu^-$ pairs and by separating charged and neutral $B$ mesons. Recently, however, with much increased statistics, LHCb measured several final states with $e^+e^-$, to make a series of tests of $e/\mu$ universality in $B \to K^*(\ell^+\ell^-)\ell^+\ell^-$ decays. At hadron colliders other $b$ hadrons are produced and as such CDF and LHCb experiments did observe also $B_s^0 \to \phi\mu^+\mu^-$ [174, 175], $\Lambda_b^0 \to \Lambda\mu^+\mu^-$ [174, 176] and $\Lambda_b^0 \to pK^-\mu^+\mu^-$ decays [177]. The averages of the total branching fractions integrated over whole $q^2$ regions are $(5.6 \pm 0.6) \times 10^{-7}$ for $B^+ \to K^+e^+e^-$, $(4.53 \pm 0.35) \times 10^{-7}$ for

$B^+ \to K^+\mu^+\mu^-$, $(1.03^{+0.19}_{-0.17}) \times 10^{-6}$ for $B^0 \to K^*(892)^0e^+e^-$ and $(0.94 \pm 0.05) \times 10^{-6}$ for $B^0 \to K^*(892)^0\mu^+\mu^-$ decays [178–182]. The total branching fractions for $B_s^0 \to \phi\mu^+\mu^-$ and $\Lambda_b^0 \to \Lambda\mu^+\mu^-$ decays are $(8.4 \pm 0.4) \times 10^{-7}$ [174, 175, 183] and $(1.08 \pm 0.28) \times 10^{-6}$ [174, 176] respectively. With increased precision of $B^0 \to K^*(892)^0\ell^+\ell^-$ decay, there is a question on what fraction of the seen branching fraction is due to the $K^*(892)^0$ resonance and what fraction is due to the $K\pi$ in s-wave. This has been studied by LHCb which found that the $K\pi$ in s-wave fraction varies between 1% and about 10% depending on the $q^2$ region [182]. It should be noted, that for all relevant $B$ meson decays the branching fractions so far studied are consistently below the SM expectation.

In the $b \to s\ell^+\ell^-$ decays angular distributions offer rich source of information. The full angular analysis was performed for decays $B^+ \to K^+\ell^+\ell^-$, $B^0 \to K^0\ell^+\ell^-$, $B^+ \to K^*(892)^0\ell^+\ell^-$, $B^0 \to K^*(892)^0\ell^+\ell^-$, $B_s^0 \to \phi\mu^+\mu^-$ and $\Lambda_b^0 \to \Lambda\mu^+\mu^-$ decays [184–192]. An attempt to increase sensitivity to the NP was made by constructing observables, which have reduced theory uncertainties and measurements of these are done. Most notably the observable called $P'_5$ [193] shows a discrepancy with the SM in the $q^2$ region which is highly sensitive to new physics [186–188]. Measurements of the CP asymmetries [177, 178, 181, 194, 195] and the isospin asymmetry [179–181, 194] were also performed. All these measurements are well consistent with the small $A_{CP}$ and small isospin asymmetry expected in the SM [196]. With statistics available at the LHC, the measurement of phase difference between long- and short-distance contribution in $B^+ \to K^+\mu^+\mu^-$ decays became possible [197].

With the data samples available at LHC, the lepton universality in $b \to s\ell^+\ell^-$ can be tested. While in the SM decays to electron-positron and muon pairs are expected to be same up to small corrections due to the different masses of leptons, in extensions of the SM this does not have to hold. The angular analysis of $B^0 \to K^*(892)^0e^+e^-$ decays was performed by LHCb at low dilepton invariant masses [198] and Belle in several regions over whole $q^2$ range [188]. The LHCb measurement yields the most stringent constraint on the photon polarization. The result on lepton universality test which over past few years attracted most attention is the ratio of branching fractions between $B^+ \to K^+\mu^+\mu^-$ and $B^+ \to K^+e^+e^-$ and between $B^0 \to K^*(892)^0\mu^+\mu^-$ and $B^0 \to K^*(892)^0e^+e^-$ decays [179, 199]. The measurements by LHCb showed mild discrepancy from the SM, with significance of $3.1\sigma$ for $B^+ \to K^+\ell^+\ell^-$ and about $2.4\sigma$ for $B^0 \to K^*(892)^0\ell^+\ell^-$. The latest analysis in Ref. [200] identified previously missed hadronic misidentification background. With better handling of such background the results are now consistent with the SM. LHCb experiment performed similar test with $\Lambda_b^0 \to pK^-\ell^+\ell^-$ decays [201], $B^0 \to K_S^0\ell^+\ell^-$ and $B^+ \to K^*(892)^0\ell^+\ell^-$ [202].

While $b \to d\ell^+\ell^-$ decays are further suppressed, they recently became accessible. Signals were observed for $B^+ \to \pi^+\mu^+\mu^-$ [77], $B^0 \to \pi^+\pi^-\mu^+\mu^-$ [203] and $\Lambda_b^0 \to p\pi^-\mu^+\mu^-$ [204] decays. Search for the $B_s^0 \to \bar{K}^*(892)^0\mu^+\mu^-$ at LHCb is complicated by large background from $B^0 \to K^*(892)^0\mu^+\mu^-$ and current significance for the decay is $3.4\sigma$ [205]. The total branching fractions are only quantities measured and these are about $2 \times 10^{-8}$ for the meson decays and about $7 \times 10^{-8}$ for the $\Lambda_b^0$ decay.

A closely related process is $B \to X_s\nu\bar{\nu}$. Since the neutrinos are not detected, the final state is a strange hadron system $X_s$ plus missing energy-momentum. Depending on $X_s$, the SM branching fraction is $\mathcal{O}(10^{-6})$. New physics effects beyond SM, e.g. those from dark sector models can greatly enhance the yield of $X_s$ plus missing energy. BaBar [206], Belle [207], and Belle II [208] have searched for these decays and determined the upper limits in the range $\mathcal{O}(10^{-5})$.

Finally the decays $B_s^0 \to e^+e^-$ and $\mu^+\mu^-$ are interesting since they only proceed at second order in weak interactions in the SM, but may have large contributions from supersymmetric loops, proportional to $(\tan\beta)^6$. First limits were published more than 30 years ago and since then experiments at Tevatron, B-factories and LHC gradually improved those and effectively ex-
cluded whole models of new physics and significantly constrained allowed parameter space of others. For the decays to $\mu^+\mu^-$, Tevatron experiments pushed the limits down to roughly factor of 5-10 above the SM expectation [209]. The long journey in the search for these decays culminated in 2012, when first evidence for $B_s^0 \to \mu^+\mu^-$ decay was seen [210]. Subsequently, LHC experiments ATLAS [211], CMS [212] and LHCb [213] observed statistically significant signal for $B_s^0 \to \mu^+\mu^-$ decay. The average branching fraction is found to be $(3.01 \pm 0.435) \times 10^{-9}$. In experiments at hadron colliders searches for $B^0 \to \mu^+\mu^-$ decays are performed at the same time. The $B(B^0 \to \mu^+\mu^-)$ is extracted in simultaneously with $B(B_s^0 \to \mu^+\mu^-)$ and is found to be $(0.07_{-0.11}^{+0.13}) \times 10^{-9}$. The limits for the $e^+e^-$ modes are: $< 9.4 \times 10^{-9}$ and $< 2.5 \times 10^{-9}$, respectively, for $B_s^0$ and $B^0$ [214]. The searches for decays to $\tau^+\tau^-$ are more challenging with current best limits of $B(B^0 \to \tau^+\tau^-) < 2.1 \times 10^{-3}$ and $B(B_s^0 \to \tau^+\tau^-) < 6.8 \times 10^{-3}$ at 95% C.L. [215]. All existing measurements of $B^0$ and $B_s^0$ decays to same flavour dilepton pair is consistent with SM expectation [216]. With $B_s^0 \to \mu^+\mu^-$ decay observed, it was suggested that the effective lifetime is useful further test of the decay [217]. Attempt was made by LHCb and CMS experiments, but its precision is not yet sufficient to provide test of the SM [212, 213]. It will take couple of years until interesting precision is reached. The searches were also performed for lepton flavour violating decays to two leptons with best limits in $e^{\pm}\mu^{\mp}$ channel, where limits are $< 1.3 \times 10^{-9}$ for $B^0$ and $< 6.3 \times 10^{-9}$ for $B_s^0$, at 95% confidence level [218].

Several theory groups performed global analysis of electroweak decays concluding that significant tension between data and SM is present [219]. The tension can be relieved by new physics beyond SM. For more detailed reviews see e.g. Ref. [220].

## 73.7 Summary and Outlook

The study of $B$ mesons continues to be one of the most productive fields in particle physics. With the two asymmetric $B$-factory experiments Belle and BaBar, we now have a combined data sample of well over $1 \text{ ab}^{-1}$. CP violation has been firmly established in many decays of $B$ mesons. Evidence for direct CP violation has been observed. Many rare decays resulting from hadronic $b \to u$ transitions and $b \to s(d)$ penguin decays have been observed, and the emerging pattern is still full of surprises. Despite the remarkable successes of the $B$-factory experiments, many fundamental questions in the flavor sector remain unanswered.

At Fermilab, CDF and D0 each has accumulated about $10 \text{ fb}^{-1}$, which is the equivalent of about $10^{12} \text{ b-hadrons}$ produced. In spite of the low trigger efficiency of hadronic experiments, a selection of modes have been reconstructed in large quantities, giving a start to a program of studies on $B_s$ and $b$-flavored baryons, in which a first major step has been the determination of the $B_s$ oscillation frequency.

As Tevatron and $B$-factories finished their data taking few year ago, the experiments at the LHC have become very active. LHCb has collected about $1 \text{ fb}^{-1}$ at 7 TeV, $2 \text{ fb}^{-1}$ at 8 TeV, and close to $5.9 \text{ fb}^{-1}$ at 13 TeV during LHC Runs 1 and 2. CMS and ATLAS have collected each about $5 \text{ fb}^{-1}$ of data at $\sqrt{s} = 7$ TeV, $20 \text{ fb}^{-1}$ at 8 TeV and about $150 \text{ fb}^{-1}$ at 13 TeV during LHC Runs 1 and 2. The latest LHC Run 3 at 13.6 TeV started in summer 2022, with upgraded detectors. By the time of the writing (summer 2023), LHC has delivered about $70 \text{ fb}^{-1}$ to ATLAS and CMS experiments. LHCb, which is dedicated to the studies of $b$- and $c$-hadrons, has a data sample that is for many decays larger than the sum of all previous experiments. With it, we are entering to regime of precision physics even for many rare decays, which allows much more detailed measurements.

The Belle II experiment at the SuperKEKB has started recording data in 2019 and has by summer 2022 collected about 428 fb$^{-1}$ of data, when their long shutdown 1 started. The aim to increase sample to $\sim 50 \text{ ab}^{-1}$ will make it possible to explore the indirect evidence of new physics beyond the SM in the heavy-flavor particles $(b, c, \text{ and } \tau)$, in a way that is complementary to the LHC. The LHCb Collaboration is commissioning upgrade of its detector and is planning on nominal running from 2024. The aim of the upgrade was to increase flexibility of the trigger, which will

allow about a factor of five increase in instantaneous luminosity and of about a factor of two in efficiencies on triggering on purely hadronic decays. The plan is to integrate about $50 \text{ fb}^{-1}$ of data during LHC runs 3 and 4.

These experiments promise a rich spectrum of rare and precise measurements that have the potential to fundamentally affecting our understanding of the SM and CP-violating phenomena.

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## 74. Polarization in B Decays

Revised July 2023 by A.V. Gritsan (Johns Hopkins U.).

We review the notation used in polarization measurements in particle production and decay, with a particular emphasis on the B decays and the CP-violating observables in polarization measurements. We look at several examples of vector-vector and vector-tensor B meson decays, while more details about the theory and experimental results in B decays can be found in a separate mini-review [1] in this Review.

Figure 74.1 illustrates angular observables in an example of the sequential process $ab \to X \to P_1 P_2 \to (p_{11}p_{12})(p_{21}p_{22})$ [2]. The angular distributions are of particular interest because they are sensitive to spin correlations and reveal properties of particles and their interactions, such as quantum numbers and couplings. In the case of a spin-zero particle X, such as B meson or a Higgs boson, there are no spin correlations in the production mechanism and the decay chain is to be analyzed. The angular distribution of decay products can be expressed as a function of three helicity angles which describe the alignment of the particles in the decay chain. The analyzer of the B-daughter polarization is normally chosen for two-body decays, as the direction of the daughters in the center-of-mass of the parent (e.g., $\rho \to 2\pi$) [3], and for three-body decays as the normal to the decay plane (e.g., $\omega \to 3\pi$) [4]. An equivalent set of transversity angles is sometimes used in polarization analyses [5]. The differential decay width depends on complex amplitudes $A_{\lambda_1\lambda_2}$, corresponding to the X-daughter helicity states $\lambda_i$.

<figure><img src="image_334.png" /><figcaption>**Figure 74.1:** Definition of the production and helicity angles in the sequential process $ab \to X \to P_1 P_2 \to (p_{11}p_{12})(p_{21}p_{22})$. The three helicity angles include $\theta_1$ and $\theta_2$, defined in the rest frame of the two daughters $P_1$ and $P_2$, and $\Phi$, defined in the X frame as the angle between the two decay planes. The two production angles $\theta^*$ and $\Psi$ are defined in the X frame, where $\Psi$ is the angle between the production plane and the average of the two decay planes.</figcaption></figure>

In the case of a spin-zero B-meson decay, its daughter helicities are constrained to $\lambda_1 = \lambda_2 = \lambda$. Therefore we simplify amplitude notation as $A_\lambda$. Moreover, most B-decay polarization analyses are limited to the case when the spin of one of the B-meson daughters is 1. In that case, there are only three independent amplitudes corresponding to $\lambda = 0$ or $\pm 1$ [6], where the last two can be expressed in terms of parity-even and parity-odd amplitudes $A_{\parallel, \pm} = (A_{+1} \pm A_{-1})/\sqrt{2}$. The overall decay amplitude involves three complex terms proportional to the above amplitudes and the Wigner $d$ functions of helicity angles. The exact angular dependence would depend on the quantum numbers of the B-meson daughters and of their decay products, and can be found in the literature [6,7]. When both B-meson daughters are tensor mesons and the smaller of the two daughter spins is $J_1 > 1$, this formalism

can be easily extended by introducing the parity-even and parity-odd amplitudes of higher order $A_{\parallel n, \perp n} = (A_{+n} \pm A_{-n})/\sqrt{2}$, with $1 < n \le J_1$, while the general angular parameterization may be found in Ref. [7]. However, we limit the following discussion to $J_1 = 1$. The differential decay rate would involve six real quantities $\alpha_i$, including interference terms,

$$ \frac{d\Gamma}{\Gamma d\cos\theta_1 d\cos\theta_2 d\Phi} = \sum_i \alpha_i f_i (\cos\theta_1, \cos\theta_2, \Phi), \quad (74.1) $$

where each $f_i (\cos\theta_1, \cos\theta_2, \Phi)$ has unique angular dependence specific to particle quantum numbers, and the $\alpha_i$ parameters are defined as:

$$ \alpha_1 = \frac{|A_0|^2}{\Sigma|A_\lambda|^2} = f_L, \quad (74.2) $$

$$ \alpha_2 = \frac{|A_{\parallel}|^2 + |A_{\perp}|^2}{\Sigma|A_{\lambda}|^2} = (1 - f_L), \quad (74.3) $$

$$ \alpha_3 = \frac{|A_{\parallel}|^2 - |A_{\perp}|^2}{\Sigma|A_{\lambda}|^2} = (1 - f_L - 2f_{\perp}), \quad (74.4) $$

$$ \alpha_4 = \frac{\Im(A_{\perp} A_{\parallel}^*)}{\Sigma|A_{\lambda}|^2} = \sqrt{f_{\perp}(1-f_L-f_{\perp})} \sin(\phi_{\perp} - \phi_{\parallel}), \quad (74.5) $$

$$ \alpha_5 = \frac{\Re(A_{\parallel} A_0^*)}{\Sigma|A_{\lambda}|^2} = \sqrt{f_L (1 - f_L - f_{\perp})} \cos(\phi_{\parallel}), \quad (74.6) $$

$$ \alpha_6 = \frac{\Im(A_{\perp} A_0^*)}{\Sigma|A_{\lambda}|^2} = \sqrt{f_{\perp} f_L} \sin(\phi_{\perp}), \quad (74.7) $$

where the amplitudes have been expressed with the help of polarization parameters $f_L$, $f_\perp$, $\phi_\parallel$, and $\phi_\perp$ defined in Table 74.1. Note that the terms proportional to $\Re(e^{A_\perp A_0^*})$, $\Im(e^{A_\parallel A_0^*})$, and $\Re(e^{A_\perp A_0^*})$ are absent in Eqs. (2-7). However, these terms may appear for some three-body decays of a B-meson daughter, see Ref. [7].

<table><caption>**Table 74.1:** Rate, polarization, and CP-asymmetry parameters defined for the B-meson decays to mesons with non-zero spin. Numerical examples are shown for the average of the $B^0 \to \phi K^*(892)^0$ decay measurements obtained from BABAR [8], Belle [9], and LHCb [10]. The first six parameters are defined under the assumption of no CP violation in decay, while they are averaged between the $\bar{B}$ and B parameters in general. The last six parameters involve differences between the $\bar{B}$ and B meson decay parameters. The phase convention $\delta_0$ is chosen with respect to a single $A_{00}$ amplitude from a reference B decay mode, which is $B^0 \to \phi K_0^*(1430)^0$ for numerical results.</caption><thead><tr><th>parameter</th><th>definition</th><th>average</th></tr></thead><tbody><tr><td>B</td><td>Γ/Γ<sub>total</sub></td><td>(10.1<sup>+0.6</sup><sub>-0.5</sub>) × 10<sup>-6</sup></td></tr><tr><td>f<sub>L</sub></td><td>|A<sub>0</sub>|<sup>2</sup>/Σ|A<sub>λ</sub>|<sup>2</sup></td><td>0.497 ± 0.017</td></tr><tr><td>f<sub>⊥</sub></td><td>|A<sub>⊥</sub>|<sup>2</sup>/Σ|A<sub>λ</sub>|<sup>2</sup></td><td>0.225 ± 0.015</td></tr><tr><td>φ<sub>‖</sub> − π</td><td>arg(A<sub>‖</sub>/A<sub>0</sub>) − π</td><td>−0.712 ± 0.058</td></tr><tr><td>φ<sub>⊥</sub> − π</td><td>arg(A<sub>⊥</sub>/A<sub>0</sub>) − π</td><td>−0.615 ± 0.056</td></tr><tr><td>δ<sub>0</sub> − π</td><td>arg(A<sub>00</sub>/A<sub>0</sub>) − π</td><td>−0.26 ± 0.10</td></tr><tr><td>A<sub>CP</sub></td><td>(Γ − Γ)/(Γ + Γ)</td><td>−0.003 ± 0.038</td></tr><tr><td>A<sup>0</sup><sub>CP</sub></td><td>(f&#773;<sub>L</sub> − f<sub>L</sub>)/(f&#773;<sub>L</sub> + f<sub>L</sub>)</td><td>−0.007 ± 0.030</td></tr><tr><td>A<sub>⊥</sub><sup>CP</sup></td><td>(f&#773;<sub>⊥</sub> − f<sub>⊥</sub>)/(f&#773;<sub>⊥</sub> + f<sub>⊥</sub>)</td><td>−0.014 ± 0.057</td></tr><tr><td>Δφ<sub>‖</sub></td><td>(φ<sub>‖</sub> − φ<sub>‖</sub>)/2</td><td>+0.051 ± 0.053</td></tr><tr><td>Δφ<sub>⊥</sub></td><td>(φ&#773;<sub>⊥</sub> − φ<sub>⊥</sub> − π)/2</td><td>+0.075 ± 0.050</td></tr><tr><td>Δδ<sub>0</sub></td><td>(δ<sub>0</sub> − δ<sub>0</sub>)/2</td><td>+0.13 ± 0.08</td></tr></tbody></table>

Overall, six real parameters describe three complex amplitudes $A_0$, $A_\parallel$, and $A_\perp$. These could be chosen to be the four polarization
parameters $f_L$, $f_⊥$, $\phi_‖$, and $\phi_⊥$, one overall size normalization, such as decay rate $\Gamma$, or branching fraction $B$, and one overall phase $\delta_0$. The phase convention is arbitrary for an isolated $B$ decay mode. However, for several $B$ decays, the relative phase could produce meaningful and observable effects through interference with other $B$ decays with the same final states, such as for $B \to V K_J^*$ with $J = 0, 1, 2, 3, 4, ...$ The phase could be referenced to the single $B \to V K_0^*$ amplitude $A_{00}$ in such a case, as shown in Table 74.1. Here $V$ stands for any spin-one vector meson.

Moreover, CP violation can be tested in the angular distribution of the decay as the difference between the $B$ and $\bar{B}$. Each of the six real parameters describing the three complex amplitudes would have a counterpart CP-asymmetry term, corresponding to three direct-CP asymmetries in three amplitudes, and three CP-violating phase differences, equivalent to the phase measurements from the mixing-induced CP asymmetries in the time evolution of $B$-decays [1]. In Table 74.1 and Ref. [11], these are chosen to be the direct-CP asymmetries in the overall decay rate $A_{CP}$, in the $f_L$ fraction $A_{CP}^0$, and in the $f_⊥$ fraction $A_{CP}^⊥$, and three weak phase differences:

$$ \Delta\phi_‖ = \frac{1}{2}\arg(\bar{A}_‖A_0/A_‖\bar{A}_0), \quad (74.8) $$

$$ \Delta\phi_⊥ = \frac{1}{2}\arg(\bar{A}_⊥A_0/A_⊥\bar{A}_0) - \frac{\pi}{2}, \quad (74.9) $$

$$ \Delta\delta_0 = \frac{1}{2}\arg(\bar{A}_{00}A_0/A_{00}\bar{A}_0). \quad (74.10) $$

The $\pi/2$ term in Eq. (74.8) reflects the fact that $A_⊥$ and $\bar{A}_⊥$ differ in phase by $\pi$ if CP is conserved. The two parameters $\Delta\phi_‖$ and $\Delta\phi_⊥$ are equivalent to triple-product asymmetries constructed from the vectors describing the decay angular distribution [12]. The CP-violating phase difference in the reference decay mode [11] is, in the Wolfenstein CKM quark-mixing phase convention,

$$ \Delta\phi_0 = \frac{1}{2}\arg(A_{00}/\bar{A}_{00}). \quad (74.11) $$

This can be measured only together with the mixing-induced phase difference for some of the neutral $B$-meson decays similar to other mixing-induced CP asymmetry measurements [1].

It may not always be possible to have a phase-reference decay mode which would define $\delta_0$ and $\Delta\delta_0$ parameters. In that case, it may be possible to define the phase difference directly similarly to Eq. (74.11):

$$ \Delta\phi_0 = \frac{1}{2}\arg(A_0/\bar{A}_0). \quad (74.12) $$

One can measure the angles of the CKM unitarity triangle, assuming Standard Model contributions to the $\Delta\phi_0$ and $B$-mixing phases. Examples include measurements of $\beta = \phi_1$ with $B \to J/\psi K^*$ and $\alpha = \phi_2$ with $B \to \rho\rho$.

Most of the $B$ decays that arise from tree-level $b \to c$ transitions have the amplitude hierarchy $|A_0| > |A_+| > |A_-|$ which is expected from analyses based on quark-helicity conservation [13]. The larger the mass of the vector-meson daughters, the weaker the inequality. The $B$ meson decays to heavy vector particles with charm, such as $B \to J/\psi K^*$, $\psi(2S)K^*$, $\chi_{c1}K^*$, $D^*\rho$, $D^*K^*$, $D^*D^*$, and $D^*D_s^*$, show a substantial fraction of the amplitudes corresponding to transverse polarization of the vector mesons ($A_{\pm 1}$), in agreement with the factorization prediction. The detailed amplitude analysis of the $B \to J/\psi K^*$ decays has been performed by the BABAR [14], Belle [15], CDF [16], CLEO [17], D0 [18], and LHCb [19] collaborations. Most analyses are performed under the assumption of the absence of direct CP violation. The parameter values are given in the particle listing of this Review. The difference between the strong phases $\phi_‖$ and $\phi_⊥$ deviates significantly from zero. The measurements [14, 15] of CP-violating terms similar to those in $B \to \varphi K^*$ [8] shown in Table 74.1 are consistent with zero.

In addition, the mixing-induced CP-violating asymmetry is measured in the $B^0 \to J/\psi K^{*0}$ decay [1, 14, 15] where angular analysis allows one to separate CP-eigenstate amplitudes. This allows one to resolve the sign ambiguity of the $\cos 2\beta$ ($\cos 2\phi_1$) term that appears in the time-dependent angular distribution due

to interference of parity-even and parity-odd terms. This analysis relies on the knowledge of discrete ambiguities in the strong phases $\phi_‖$ and $\phi_⊥$, as discussed below. The BABAR experiment used a method based on the dependence on the $K\pi$ invariant mass of the interference between the $S$- and $P$-waves to resolve the discrete ambiguity in the determination of the strong phases ($\phi_‖, \phi_⊥$) in $B \to J/\psi K^*$ decays [14]. The result is in agreement with the amplitude hierarchy expectation [13]. The CDF [20], D0 [21], LHCb [22], and ATLAS [23] experiments have studied the $B_s^0 \to J/\psi(K^+K^-), J/\psi(\pi^+\pi^-), \psi(K^+\pi^-)$ decays and provided the lifetime, polarization, and phase measurements.

The amplitude hierarchy $|A_0| \gg |A_+| \gg |A_-|$ was expected in $B$ decays to light vector particles in both penguin transitions [24, 25] and tree-level transitions [13]. There is confirmation by the BABAR and Belle experiments of predominantly longitudinal polarization in the tree-level $b \to u$ transition, such as $B^0 \to \rho^+\rho^-$ [26], $B^+ \to \rho^0\rho^+$ [27], and $B^+ \to \omega\rho^+$ [28]; this is consistent with the analysis of the quark helicity conservation [13]. Because the longitudinal amplitude dominates the decay, a detailed amplitude analysis is not possible with current $B$ samples, and limits on the transverse amplitude fraction are obtained. The small branching fractions of $B^0 \to \rho^0\rho^0, \omega\rho^0, \omega\omega$ [28–31] indicate that $b \to d$ penguin pollution is small in the charmless, strangeless vector-vector $B$ decays. There is a measurement of large longitudinal polarization in $B^0 \to \rho^0\rho^0$ [29–31] decays. The fraction of transverse polarization is large in decays to heavier mesons such as $B^0 \to a_1(1260)^+a_1(1260)^-$ [32].

The interest in the polarization and CP-asymmetry measurements in penguin transition, such as $b \to s$ decays $B \to \varphi K^*$, $\rho K^*$, $\omega K^*$, or $B_s^0 \to \varphi\varphi, K^*K^*$, and $b \to d$ decay $B \to K^*\bar{K}^*$, is motivated by their potential sensitivity to physics beyond the Standard Model. The decay amplitudes for $B \to \varphi K^*$ have been measured by the BABAR, Belle, and LHCb experiments [9–11, 33, 34]. The fractions of longitudinal polarization are $f_L = 0.50 \pm 0.05$ for the $B^+ \to \varphi K^{*+}$ decay and $f_L = 0.497 \pm 0.017$ for the $B^0 \to \varphi K^{*0}$ decay. These indicate significant departure from the naive expectation of predominant longitudinal polarization, suggesting other contributions to the decay amplitude, previously neglected, either within the Standard Model, such as penguin annihilation [35] or QCD rescattering [36], or from physics beyond the Standard Model [37]. The complete set of twelve amplitude parameters measured in the $B^0 \to \varphi K^{*0}$ decay is given in Table 74.1. Several other parameters could be constructed from the above twelve parameters, as suggested in Ref. [38].

The discrete ambiguity in the phase ($\phi_‖, \phi_⊥, \Delta\phi_‖, \Delta\phi_⊥$) measurements has been resolved by BABAR in favor of $|A_+| \gg |A_-|$ through interference between the $S$- and $P$-waves of $K\pi$. The search for vector-tensor and vector-axialvector $B \to \varphi K_J^{(*)}$ decays with $J = 1, 2, 3, 4$ revealed a large fraction of longitudinal polarization in the decay $B \to \varphi K_2^{(*)}(1430)$ with $f_L = 0.90^{+0.06}_{-0.07}$ [11, 39], but large contribution of transverse amplitude in $B \to \varphi K_1(1270)$ with $f_L = 0.46^{+0.13}_{-0.15}$ [40].

Like $B \to \varphi K^*$, the decays $B \to \rho K^*$ and $B \to \omega K^*$ may be sensitive to New Physics. Measurements of the longitudinal polarization fraction in $B \to \rho K^*$ [41] and in both vector-vector and vector-tensor final states of $B \to \omega K_J^*$ [28] by BABAR and Belle reveal a large fraction of transverse polarization, indicating an anomaly similar to $B \to \varphi K^*$ except for a different pattern in vector-tensor final states. An angular analysis of the $B^0 \to \rho^0 K^{*0}$ decay mode by LHCb [42] provides much higher precision and indicates remarkably small longitudinal polarization fraction and a significant direct CP asymmetry observed in angular distributions of $B \to VV$ decays for the first time. A large transverse polarization is also observed in the $B_s^0 \to \varphi\varphi$ decay by CDF [43] and LHCb [44], $B_s^0 \to K^{*0}K^{*0}$ decays by LHCb [45], and $B_s^0 \to \varphi K^{*0}$ decays by LHCb [46]. At the same time, measurement of the polarization in the $b \to d$ penguin decays $B \to K^*\bar{K}^*$ indicates a large fraction of longitudinal polarization [45, 47]. The LHCb experiment has also provided the very first polarization results on the tensor-tensor, as well as vector-tensor, decays of the $B_s^0$ meson in the $(K\pi)(K\pi)$ final state [45]. The polarization pattern in penguin-dominated $B$-meson decays is not fully understood [35–37].
The three-body semileptonic B-meson decays, such as $B \to V\ell_1\ell_2$, share many features with the two-body $B \to VV$ decays. Their differential decay width can be parameterized with the two helicity angles defined in the V and $(\ell_1\ell_2)$ frames and with the azimuthal angle, as defined in Fig. 74.1. However, since the $(\ell_1\ell_2)$ pair does not come from an on-shell particle, the angular distribution is unique to each point in the dilepton mass $m_{\ell\ell}$ spectrum. The polarization measurements as a function of $m_{\ell\ell}$ provide complementary information on physics beyond the Standard Model, as discussed for $B \to K^*\ell^+\ell^-$ and $B_s \to \phi\ell^+\ell^-\ell^-$ decays in Ref. [48]. The data in these modes have been analyzed by the BABAR, Belle, CDF, CMS, ATLAS, and LHCb experiments [49–54].

The examples of the angular distributions and observables in $B \to K^*\ell^+\ell^-$ are discussed in Ref. [48]. Two angular observables have been measured in this decay in certain ranges of the dilepton mass $m_{\ell\ell}$. One parameter is the fraction of longitudinal polarization $F_L$, which is determined by the $K^*$ angular distribution and is similar to $f_L$ defined for exclusive two-body decays. The other parameter is the forward-backward asymmetry of the lepton pair $A_{FB}$, which is the asymmetry of the decay rate with positive and negative values of $\cos\theta_1$. A complete set of observables and angular terms has been adopted by the LHCb collaboration [54] following Ref. [48] with the $F_L$, $A_{FB}$, and $S_3-S_9$ coefficients in the angular distributions. Additional set of optimized observables $P_i^{(l)}$ is derived from those, for example $P_2 = 2A_{FB}/(3-3F_L)$ and $P_5' = S_5/\sqrt{F_L(1-F_L)}$. These observables have the advantage that the leading form-factor uncertainties cancel. There have been hints of deviations from SM in the measurement of $P_5'$ and lepton flavor universality [49–54].

In summary, there has been considerable interest in the polarization measurements of B-meson decays because they reveal both weak- and strong-interaction dynamics [35–37,55]. New measurements will further elucidate the pattern of spin alignment measurements in rare B decays, and further test the Standard Model and strong interaction dynamics, including the non-factorizable contributions to the B-decay amplitudes.

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## 75. B⁰-B⁰ Mixing

Revised March 2024 by O. Schneider (EPFL).

There are two neutral $B^0-\bar{B}^0$ meson systems, $B_d^0-\bar{B}_d^0$ and $B_s^0-\bar{B}_s^0$ (generically denoted $B_q^0-\bar{B}_q^0$, $q=s,d$), which exhibit particle-antiparticle mixing [1]. This mixing phenomenon is described in Ref. [2]. In the following, we adopt the notation introduced in Ref. [2], and assume CPT conservation throughout. In each system, the light (L) and heavy (H) mass eigenstates,

$$|B_{L,H}\rangle = p|B_q^0\rangle \pm q|\bar{B}_q^0\rangle, \quad (75.1)$$

have a mass difference $\Delta m_q = m_H - m_L > 0$, a total decay width difference $\Delta\Gamma_q = \Gamma_L - \Gamma_H$ and an average decay width $\Gamma_q = (\Gamma_L + \Gamma_H)/2$. In the absence of CP violation in the mixing, $|q/p| = 1$, the differences are given by $\Delta m_q = 2|M_{12}|$ and $|\Delta\Gamma_q| = 2|\Gamma_{12}|$, where $M_{12}$ and $\Gamma_{12}$ are the off-diagonal elements of the mass and decay matrices [2]. The evolution of a pure $|B_q^0\rangle$ or $|\bar{B}_q^0\rangle$ state at $t=0$ is given by

$$|B_q^0(t)\rangle = g_+(t)|B_q^0\rangle + \frac{q}{p}g_-(t)|\bar{B}_q^0\rangle, \quad (75.2)$$

$$|\bar{B}_q^0(t)\rangle = g_+(t)|\bar{B}_q^0\rangle + \frac{p}{q}g_-(t)|B_q^0\rangle, \quad (75.3)$$

which means that the flavor states remain unchanged (+) or oscillate into each other (-) with time-dependent probabilities proportional to

$$|g_{\pm}(t)|^2 = \frac{e^{-\Gamma_q t}}{2} \left[ \cosh\left(\frac{\Delta\Gamma_q}{2} t\right) \pm \cos(\Delta m_q t) \right]. \quad (75.4)$$

In the absence of CP violation, the time-integrated mixing probability $\int |g_-(t)|^2 dt / (\int |g_-(t)|^2 dt + \int |g_+(t)|^2 dt)$ is given by

$$\chi_q = \frac{x_q^2 + y_q^2}{2(x_q^2 + 1)}, \quad \text{where} \quad x_q = \frac{\Delta m_q}{\Gamma_q}, \quad y_q = \frac{\Delta \Gamma_q}{2\Gamma_q}. \quad (75.5)$$

### 75.1 Standard Model predictions and phenomenology

In the Standard Model, the transitions $B_q^0 \to \bar{B}_q^0$ and $\bar{B}_q^0 \to B_q^0$ are due to the weak interaction. They are described, at the lowest order, by box diagrams involving two W bosons and two up-type quarks (see Fig. 75.1), as is the case for $K^0-\bar{K}^0$ mixing. However, the long range interactions arising from intermediate virtual states are negligible for the neutral B meson systems, because the large B mass is off the region of hadronic resonances. The calculation of the dispersive and absorptive parts of the box diagrams yields the following predictions for the off-diagonal element of the mass and decay matrices [3],

$$M_{12} = - \frac{G_F^2 m_W^2 \eta_B m_{B_q} B_{B_q} f_{B_q}^2}{12\pi^2} S_0(m_t^2/m_W^2) (V_{tq}^* V_{tb})^2, \quad (75.6)$$

$$\Gamma_{12} = \frac{G_F^2 m_b^2 \eta'_B m_{B_q} B_{B_q} f_{B_q}^2}{8\pi} \times \left[ (V_{tq}^* V_{tb})^2 + V_{tq}^* V_{tb} V_{cq}^* V_{cb} \mathcal{O}\left(\frac{m_c^2}{m_b^2}\right) + (V_{cq}^* V_{cb})^2 \mathcal{O}\left(\frac{m_c^4}{m_b^4}\right) \right], \quad (75.7)$$

where $G_F$ is the Fermi constant, $m_W$ the W boson mass, and $m_i$ the mass of quark i; $m_{B_q}$, $f_{B_q}$ and $B_{B_q}$ are the $B_q^0$ mass, weak decay constant and bag parameter, respectively. The known function $S_0(x_t)$ can be approximated very well by $0.784 x_t^{0.76}$ [4], and $V_{ij}$ are the elements of the CKM matrix [5]. The QCD corrections $\eta_B$ and $\eta'_B$ are of order unity. The only non-negligible contributions to $M_{12}$ are from box diagrams involving two top quarks. The phases of $M_{12}$ and $\Gamma_{12}$ satisfy

$$\phi_M - \phi_\Gamma = \pi + \mathcal{O}\left(\frac{m_c^2}{m_b^2}\right), \quad (75.8)$$

<figure><img src="image_335.png" /><figcaption>**Figure 75.1:** Dominant box diagrams for the $B_q^0 \to \bar{B}_q^0$ transitions ($q=d$ or $s$). Similar diagrams exist where one or both $t$ quarks are replaced with $c$ or $u$ quarks.</figcaption></figure>

implying that the mass eigenstates have mass and width differences of opposite signs. This means that, like in the $K^0-\bar{K}^0$ system, the heavy state is expected to have a smaller decay width than that of the light state: $\Gamma_H < \Gamma_L$. Hence, $\Delta\Gamma_q = \Gamma_L - \Gamma_H$ is expected to be positive in the Standard Model.

Furthermore, the quantity

$$\left| \frac{\Gamma_{12}}{M_{12}} \right| \approx \frac{3\pi}{2} \frac{m_b^2}{m_W^2} \frac{1}{S_0(m_t^2/m_W^2)} \sim \mathcal{O}\left(\frac{m_b^2}{m_t^2}\right) \quad (75.9)$$

is small, and a power expansion of $|q/p|^2$ yields

$$\left|\frac{q}{p}\right|^2 = 1 + \left|\frac{\Gamma_{12}}{M_{12}}\right| \sin(\phi_M - \phi_\Gamma) + \mathcal{O}\left(\left|\frac{\Gamma_{12}}{M_{12}}\right|^2\right). \quad (75.10)$$

Therefore, considering both Eqs. (75.8) and (75.9), the CP-violating parameter

$$1 - \left|\frac{q}{p}\right|^2 \approx \operatorname{Im}\left(\frac{\Gamma_{12}}{M_{12}}\right) \quad (75.11)$$

is expected to be very small: $\sim \mathcal{O}(10^{-3})$ for the $B_d^0-\bar{B}_d^0$ system and $\lesssim \mathcal{O}(10^{-4})$ for the $B_s^0-\bar{B}_s^0$ system [6].

In the approximation of negligible CP violation in mixing, the ratio $\Delta\Gamma_q/\Delta m_q$ is equal to the small quantity $|\Gamma_{12}/M_{12}|$ of Eq. (75.9); it is hence independent of CKM matrix elements, i.e., the same for the $B_d^0-\bar{B}_d^0$ and $B_s^0-\bar{B}_s^0$ systems. Calculations [7] yield $\sim 5 \times 10^{-3}$ with a $\sim 15\%$ uncertainty. Given the published experimental knowledge [8] on the mixing parameter $x_q$

$$\left\{ \begin{array}{ll} x_d = 0.7697 \pm 0.0035 & (B_d^0-\bar{B}_d^0 \text{ system}) \\ x_s = 26.99 \pm 0.09 & (B_s^0-\bar{B}_s^0 \text{ system}) \end{array} \right. , \quad (75.12)$$

the Standard Model thus predicts that $\Delta\Gamma_d/\Gamma_d$ is very small (below 1%), but $\Delta\Gamma_s/\Gamma_s$ considerably larger ($\sim 10\%$). These width differences are caused by the existence of final states to which both the $B_q^0$ and $\bar{B}_q^0$ mesons can decay. Such decays involve $b \to c\bar{c}q$ quark-level transitions, which are Cabibbo-suppressed if $q=d$ and Cabibbo-allowed if $q=s$.

A complete set of Standard Model predictions for all mixing parameters in both the $B_d^0-\bar{B}_d^0$ and $B_s^0-\bar{B}_s^0$ systems can be found in Ref. [7].