To construct a theory with Lorentz symmetry one naturally recurs to a Lagrangian formulation. So is done for the CCQM which is an effective field approach where both, quark and hadronic fields occur. The quark-meson interaction term is written as where M represents the mesonic field, q the quark one, $g_M$ is their coupling and H.c. stands for the hermitian conjugate. The interpolating quark current $J_M$ is non-local and the integral over the positions $x_1$, $x_2$ of constituent quarks is weighted by a vertex function $F_M$. The symbol ${\Gamma }_M$ represents a combination of gamma matrices depending on the spin of M. For a scalar M one has ${\Gamma }_M=1$, for pseudoscalar ${\Gamma }_M={\gamma }^5$ and for a vector particle the expression is ${\Gamma }_M={\gamma }^{\mu }$. In the latter case the mesonic field has a Lorentz index too ($M_{\mu }$) and the indices are contracted.

The explicit form of $F_M$ is driven by two requirements. First we constrain the positions of quarks so as to make the hadron be situated in their barycenter. For this a delta function is introduced where the weights in its argument depend on the constituent quark masses $w_i=m_i/(m_1+m_2)$. Second, to manifestly respect the Lorentz symmetry, the remaining dependence is written as a function of the spacetime interval

Further steps in the construction of $F_M$ are done with respect to the computational convenience. ${\Phi }_M$ is assumed to have a Gaussian form in the momentum representation where ${\Lambda }_M$ is a free parameter of the model related to the meson M. The square of the momentum in the argument of the exponential becomes negative in the Euclidean region $k^2=-k_E^2$ which implies an appropriate fall-off behavior and removes ultraviolet divergences in Feynman diagrams. The question of other possible function forms of ${\Phi }_M$ was addressed in [38], where four different ansatzes were tested, each having a meaningful physical interpretation. The dependence of the results on the function form was found to be small.

Besides the hadron-related ${\Lambda }_M$, the CCQM comprises four "global" parameters: three constituent quark masses and one universal cutoff which plays a role in the quark confinement (as explained later). The values expressed in GeV are where we do not distinguish between the two light quarks and use the same mass for both. The values slightly changed in the past, they were few times [39,40] updated if significant new data become available. They were extracted by over-constrained global fits of the model on available experimental points.

The CCQM does not include gluons. The gluonic effects are effectively taken into account by the vertex function which is adjusted to describe data by tuning the free parameter it contains.

At last we have to mention that the CCQM is suitable for description of various multi-quark states including baryons [41,42] and tetraquarks [43]. In this text we focus on mesons, the approach is in other cases very similar: the interpolating quark current is constructed for a given number of quarks (more alternatives can be considered) and multiplied by the hadronic field to give the interaction Lagrangian.

The interaction of a meson is given by the Lagrangian (2): the meson fluctuates into its constituent quarks, these interact and afterwards combine back into a mesonic final state. Yet, (2) implies that both, quark and mesons, are elementary and this rises concerns about the double counting.

We address these questions by implementing the so-called compositeness condition [37,39,44] which originates in works [45,46,47] (see [48] for a review). The interaction of a meson through the creation of virtual quark states implies the mesonic field is dressed, i.e. its vertex and wave function need to be renormalized. This is reflected in the renormalization constant $Z_M$ which can be interpreted as the overlap between the physical state and the corresponding bare state. By requiring $Z_M=0$ we make this overlap vanish, i.e. the physical state does not contain bare state and can be regarded as a bound state. As a consequence the quarks exist only as virtual and quark degrees of freedom do not appear on the level of the physical state.

$Z_M$ is expressed in terms of the derivative of the meson mass operator ${\Pi }_M^{{}^{\prime }}$ (its scalar part for vector mesons) and at one-loop level (Figure 1) is given by for pseudoscalar and vector mesons respectively. The symbol $S_i$ denotes the quark propagator $S_i=1/(m_{q_i}-{\gamma }^{\mu }{k}_{\mu })$ and the differentiation is done using the identity $d\Pi /d{p}^2=(p^{\mu }d\Pi /d{p}^{\mu })/\left(2p^2\right)$.

To reach the equality (6) we profit from the up-to-now undetermined coupling constant $g_M$ and tune its value so that (6) is satisfied. As consequence, the coupling $g_M$ becomes the function of ${\Lambda }_M$. In this way we reduce the number of parameters of the model and increase its predictive power and stability. If the values ${\Lambda }_M$ and $g_M$ are unknown from previous studies, their determination is the first step in the application of the CCQM.

The CCQM was initially built without the quark confinement, but new data on heavy particles required its extension to situations where the hadron is heavier than the sum of its constituent quarks [37]. To prevent the decay into free quarks in such a scenario, we use a technique inspired by confined propagators. Here the propagators are written in the Schwinger representation and a cutoff is introduced in the upper integration limit. The propagator then becomes an entire function where the absence of singularities indicates the absence of a single quark asymptotic state. We adopt a modified version of this strategy and apply the cutoff to the whole structure F of the Feynman diagram. It can be formally written as and can be obtained by inserting the unity $1={\int }_0^{\infty }dt\delta (t-{\sum }_{i=1}^n{\alpha }_i)$ into the expression on the left hand side. The single cutoff (indicated by the arrow) in the t variable is done in the last step, the remaining integration variables are confined to an n dimensional simplex. After the cutoff is applied the integral becomes convergent for arbitrary values of the kinematic variables meaning that the quark thresholds are removed with quarks never being on the mass shell. The the cutoff value (5) is the same for all processes we study.

Radiative decays represent another important class of processes measured at heavy meson factories. For their description one has to include the interactions with photons into the CCQM [37,49]. Because of the non-local interaction Lagrangian this is not straightforward and requires a dedicated approach. Taking into the account quarks and scalar mesons, the free parts of the Lagrangian are treated in the usual way, i.e. the minimal substitution is used where $\psi \in \{q,M\}$ and $e_{\psi }$ is its electric charge in the units of the proton charge. One then gets

The compositeness condition formulated above however prevents a direct interaction of the dressed particle, i.e. the meson, with photons: the contributions from the photon-meson tree level diagram and analogous diagrams with self-energy insertions into the external mesonic line determine the renormalization constant Z and $Z=0$ implies they cancel. The interaction thus proceeds only through intermediate virtual states.

The gauging of the non-local interaction (2) is done in a manner similar to [50]. First one multiplies the quark fields in (2) by a gauge field exponential where P is the path connecting $x_i$ and x, the latter being the position of the meson. One can verify that the Lagrangian is invariant under the following gauge transformations here $f\left(x\right)$ is some scalar function. The apparent path-dependence of the definition (14) is not an actual one: in the perturbative expansion only derivatives of the path integral appear and these are path independent

The individual terms of the Lagrangian are generated by expanding the gauge field exponential by orders in $A^{\mu }$. At first order one has where $E_M$ is defined through its Fourier transform ${\tilde{E}}_M$: $(x_1-x,x_2-x,y-x)\stackrel{FT}{\leftrightarrow }(p_1,p_2,q)$

Symbol ${\tilde{\Phi }}_M^{{}^{\prime }}$ denotes the derivative with respect to the argument. In corresponding Feynman diagrams the photon is attached to the non-local vertex.

From the Lagrangian one derives the Feynman diagrams. Gaussian expressions in the vertex function (4) and in the Fock-Schwinger propagator (9) can be joined into a single exponent which takes a quadratic form in the loop momenta k. It can be formally written as $exp(a{k}^2+2rk+z)$, $a=a\left(\right\{\alpha \left\}\right)$, $r=r\left(\right\{\alpha \},\{p\left\}\right)$, where $\left\{\alpha \right\}$ denotes the set of Schwinger parameters and $\left\{p\right\}$ external momenta. The exponential is preceded by a polynomial P in loop momenta which originates from the trace of Dirac matrices (numerators of propagators). Since the powers of k can be generated by differentiation with respect to r, the loop momenta integration is formally written as

Using the substitution $u=k+r/a$, the argument of the exponential is transformed and the integration is performed in the Euclidean region as a simple Gaussian integral. Further, the differential operator and the r-dependent exponential can be interchanged which simplifies the action of the differential operator. One arrives to where F represents the whole structure of the Feynman diagram including (23). We use a FORM [51] code to treat symbolic expressions: besides computing traces we also us it to repeatedly perform chain rule application in (23) and arrive to an explicit formula with no differential operators appearing. The implementation of the infrared confinement as expressed by (10) is the last step before the numerical integration.

Large mass difference between heavy mesons and leptons implies, by phase-space arguments, small branching fractions of pure and radiative leptonic decays. Some of these are further suppressed by CKM elements or helicity. Thus for most leptonic decays only limits have been measured.

At the usual 95% confidence level a branching fraction measurement is available only for $B_s^0\to 2\mu$[52,53,54,55] and $B^{\pm }\to {\tau }^{\pm }{\nu }_{\tau }$[56,57,58,59]. If the criteria are loosened to (at least) one sigma significance, additional results can be cited: $B^0\to 2\mu$[52], $B^{+}\to {\mu }^{+}{\nu }_{\mu }$[60,61] and $B^{+}\to {\ell }^{+}{\nu }_{\ell }\gamma$[62]. The limits are settled [23] for $B^{+}\to e^{+}{\nu }_e$, $B^{+}\to e^{+}{\nu }_e\gamma$, $B^{+}\to {\mu }^{+}{\nu }_{\mu }\gamma$, $B^{+}\to {\mu }^{+}{\mu }^{-}{\mu }^{+}{\nu }_{\mu }$, $B^0\to e^{+}{e}^{-}$, $B^0\to e^{+}{e}^{-}\gamma$, $B^0\to {\mu }^{+}{\mu }^{-}\gamma$, $B^0\to {\mu }^{+}{\mu }^{-}{\mu }^{+}{\mu }^{-}$ , $B^0\to {\tau }^{+}{\tau }^{-}$, $B_s^0\to e^{+}{e}^{-}$, $B_s^0\to {\tau }^{+}{\tau }^{-}$ and $B_s^0\to {\mu }^{+}{\mu }^{-}{\mu }^{+}{\mu }^{-}$.

These experimental results motivate various analyses. Pure leptonic decays are considered as theoretically clean with the main source of uncertainty represented by the hadronic effects of the initial state, which are contained in the leptonic decay constant of the hadron. The neutrino production process corresponds, in the leading order, to the annihilation of the constituent quarks into a virtual W meson which subsequently decays. The branching fraction is given by where $G_F$ is the Fermi coupling constant, $V_{ij}$ the CKM matrix element and ${\tau }_P$ the lifetime of the particle P.

A general information about B leptonic decays is contained in several reviews. Besides [26], a more specific focus on processes with charged pseudoscalar mesons is given in [63] and a summary concerning specifically B decays (leptonic and semileptonic ) is provided in [64]. The existing theoretical approaches follow two directions. One focuses on the SM contributions at different precision levels, the other is concerned with NP beyond the SM.

Dilepton final states are produced at one loop through box and penguin diagrams. The cross section formula can be found e.g. in [65], equation (4.10). The leptonic decays constants of B (and D) mesons where determined in a model-independent way using lattice calculations in [66]. The SM treatment of dilepton decays includes the computation of three-loop QCD corrections [67], the evaluation of the electroweak contributions at the two-loop level [68] and further improvements of theoretical predictions reached by combining additional EM and strong corrections [69]. The authors of [70] investigated the effect of the virtual photon exchange from scales below the bottom-quark mass and found a dynamical enhancement of the amplitude at the 1% level. The soft-collinear effective theory approach was used in [71] to evaluate the power-enhanced leading-logarithmic QED corrections.

The radiative processes have the advantage of not being helicity suppressed at the price of one additional ${\alpha }_{\mathrm{EM}}$ factor. A larger number of results can be cited for radiative dilepton production. An evaluation within a constituent quark model was performed in [72] to estimate branching fractions, the same observables were predicted by the authors of [73,74] using the light-cone QCD sum rules and by those of [75] using the light-front model. Universal form factors related to the light cone wave function of the $B_s$ meson allowed to make estimates in [76]. Interesting results were given in [77], where it was shown that the gauge invariance and other considerations allow to significantly constrain the form factor behavior, and also in [78] where the authors have demonstrated that the non-perturbative hadronic effects largely cancel in amplitude ratios of pure leptonic and radiative decays. The impact of the light meson resonances on long-distance QCD effects was studied in [79]. In [80] the authors have identified the effective $B\to \mu \mu \gamma$ lifetime and a related CP-phase sensitive observable as appropriate quantities to study the existing B decay discrepancies.

Also for decays $B\to \gamma l{\nu }_l$ several studies can be cited. The work [81] was concerned with photon spectrum and the decay rates of the process. The authors of [82] used the HQET to predict form factors and in [83] the heavy-quark expansion and soft-collinear effective theory were applied to evaluate the soft-overlap contribution to the photon. The process was also studied in [84]. Here, assuming an energetic photon, the authors aimed to quantify the leading power-suppressed corrections in $1/E_{\gamma }$ and $1/m_b$ from higher-twist B-meson light-cone distribution amplitudes. The soft-collinear effective theory was the approach adopted in [85,86].

The recent publication [87] focused on four-body leptonic B decays: off-shell photon form factors were computed within the QCD factorization formalism and predictions for differential distribution of various observables were presented. Similar processes are addressed also in [88,89,90].

Although the most tensions with the SM are seen in the semileptonic sector, the pure leptonic decays are of a concern too. The summary papers [29,91] mention two tensions. Fist is related to the combined likelihood for $B^0$ and $B_s^0$ decays to ${\mu }^{+}{\mu }^{-}$ where the theory-measurement difference reaches $2.3\sigma$. The other concerns the branching fraction ratio for the $B_s^0\to {\mu }^{+}{\mu }^{-}$ reaction $R={\mathcal{B}}_{\exp }/{\mathcal{B}}_{\mathrm{SM}}$ which deviates from 1 by $2.4\sigma$. In [92] is the difference between the theory and the experiment for the dimuon $B_s$ decay quantified to be $2.2\sigma$.

The possible NP contributions are usually assessed by introducing new, beyond SM four-fermion contact operators and the corresponding Wilson coefficients. Once evaluated in the appropriate NP approach, it is possible to conclude about their effect on the theory-experiment discrepancy, see e.g. [93].

An overview of various flavor-violating extensions of the SM also with relation to $B\to \ell \ell$ decay was presented in [94]. In [95] the $B_s$ dimuon decay was considered and it was argued that the decay width difference between the light and heavy $B_s$ mass eigenstates is a well-suited observable for the detection of NP. The work [31] points to the ambiguity in choice of the NP operators that might play role in explaining the tensions in the B semileptonic decays. They show that this ambiguity can be lifted by analyzing the longitudinal polarization asymmetry of the muons in $B_s^{*}\to \mu \mu$. Various discrepancies in measured data are addressed in [96], among them also dimuon branching fractions. The attempt to explain them is based on lepton-flavored gauge extensions of the SM, a specific construction with a massive gauge boson $X_{\mu }$ and "muoquark" $S_3$ is presented. Several texts are interested in decays with tau lepton in the final state. In [97,98,99] these decays are studied in relation with various alternative scenarios of the Higgs boson model and in [100] they are analyzed in the context of non-universal left-right models.

Before reviewing other CCQM results on leptonic B decays we present in more details the evaluation of the branching fraction for $B_s\to {\ell }^{+}{\ell }^{-}\gamma$[101]. The computations are in many ways similar to those of other cases and provide an insight of how leptonic and radiative decays are treated within the CCQM. Since $B_s$ is the only hadron, we need to extend the set of parameters (5) only by one number, i.e. ${\Lambda }_{B_s}=2.05\mathrm{GeV}$ which was settled in previous works. Two explicit forms of the effective Hamiltonian (1) are considered where ${\sigma }_{\mu \nu }=i[{\gamma }_{\mu },{\gamma }_{\nu }]$ and $F^{\mu \nu }$ is the EM field tensor. In (26) the dilepton is produced from the weak b-s transition, Figure 2, in (27) the weak transition gives birth to a real photon, Figure 3. An additional set of diagrams depicted in Figure 4 is considered too, where the real photon is emitted as the final-state radiation (FSR).

The tilde notation in (26)(27) indicates the QCD quark mass (different from (5)) which is $\widetilde{m_b}=4.68\pm 0.03\mathrm{GeV}$[102]. The values of scale-dependent Wilson coefficients were determined in [103] at the matching scale ${\mu }_0=2m_W$ and run to the hadronic scale ${\mu }_b=4.8\mathrm{GeV}$. The effective operators are defined through the standard SM operators as follows where

The $\Omega$ function in $C_9^{\mathrm{eff}}$ parameterizes, in the standard Breit-Wigner form, the resonant contributions from $\Psi \left(1s\right)$ and $\Psi \left(2s\right)$ charmonia states.

Amplitudes given by the diagrams in Figure 2 and Figure 3, where the photon originates from the intermediate QCD-generated states, are labeled as structure dependent and can be described by four $B_s\to \gamma$ transition form factors (see e.g. [79]). Defining momenta as $B_s\left(p_1\right)\to \gamma \left(p_2\right){\ell }^{+}\left(k_{+}\right){\ell }^{-}\left(k_{-}\right)$, $q=p_1-p_2$ with $p_1^2=m_{B_s}^2$, $p_2^2=0$, ${ϵ}_2^{†}·p_2=0$ and $k_{\pm }^2=m_{\ell }^2$ one has where $ϵ$ is the polarization vector. Each of the four introduced form factors can be expressed as sum of contributions from particular Feynman graphs in Figure 2 and Figure 3. We have where "$q\gamma q$" superscript refers to a real photon emission from the quark line, "$bubble$" to the real photon emission from the non-local hadron-quark vertex and "$q(\overline{\ell }\ell )q$" corresponds to the virtual photon emission from the quark line.

The branch point at $q^2=4m_s^2$ corresponding to the virtual photon emission from the s quark (left in Figure 3) is situated well inside the accessible physical $q^2$ region. This leads to the appearance of light vector meson resonance which prevents us to compute the corresponding form factors within the CCQM. An approach inspired by [104] is adopted and a gauge-invariant vector-meson dominance model is used to express the form factors in question where $P=p_1+p_2$. With all this objects defined, one can write down the amplitude for the structure dependent part where $T_1^{\mu \alpha }=[g^{\mu \alpha }{p}_1{p}_2-{\left(p_1\right)}^{\alpha }{\left(p_2\right)}^{\mu }]$. The structure-independent bremsstrahlung (Figure 4) amplitude takes the form

Here $t={(p_2+k_{-})}^2$, $u={(p_2+k_{+})}^2$. To avoid infrared divergences in (35) a lower boundary on the photon energy has to be introduced $E_{\gamma }>E_{\gamma \min }$ set later, in numerical computations (Table 1), to $20\mathrm{MeV}$.

The differential branching fraction in t and $s\equiv q^2$ has a general expression where one sums over the polarization of photons and leptons, $4m_{\ell }^2\le s\le m_{B_s}^2$, $t_{-}\le t\le t_{+}$ with $t_{\pm }=m_{\ell }^2+(m_{B_s}^2-s)[1\pm \sqrt{1-4m_{\ell }^2/s}]/2$. The explicit formulas for double and single differential distributions we omit here because of their complexity, they are stated in Eqs. (32)-(38) of [101].

The form factors predicted by the CCQM model are shown in Figure 5. For $F_{TV/TA}$ form factors we present two scenarios: by including the VMD component (32) these form factors become complex and thus their norm is shown. Alternatively, they can be shown without the VMD component as real functions

The form factors were also compared to those determined in [104] with which they well agreed. The differential branching fractions shown as a function of dimensionless variable $\widehat{s}=q^2/m_{B_s}$ are, together with the branching fraction ratio depicted in Figure 6.

The total branching fractions for the three lepton flavors are presented in Table 1.

The numbers in brackets indicate the results of computations with long-distance contributions included (but we exclude the region of the two low lying charmonia $0.33\le \widehat{s}\le 0.55$), results without the long distance contributions correspond to $\kappa =0$ in (29). The comparison with theoretical predictions of other authors is shown in Table 2.

We identify the dominant error source of our results to be the uncertainty of the hadronic form factors and estimate the error on the branching fractions to reach 30%. One should remark that the resonant peaks induced by light $\varphi$ particles lead to significant enhancement of the branching fraction (≈15%).

In summary, in our SM computations within the CCQM we evaluated the hadronic transition form factors and radiative leptonic branching fractions of the $B_s$ meson. Our form factors are in a very good agreement with those presented in [104] and our branching fraction numbers for light leptons agree with [105]. For the tau lepton decay mode, where bremsstrahlung dominates, we agree with all other authors. Together, these results from various authors with us included, reflect our understanding of the SM description of the $B_s\to {\ell }^{+}{\ell }^{-}\gamma$ decay process and provide an estimate on the error of theoretical SM predictions, beyond which one can claim NP manifestations.

The CCQM was applied also to the leptonic decays $B\to {\ell }^{-}{\overline{\nu }}_{\ell }$[107] and $B_c^{-}\to \tau \overline{\nu }$ [108].

The work [107] provides a SM analysis of pure leptonic and semileptonic decays. Most of the results presented there concern the semileptonic processes, which have richer structure and significant hints for the NP. Yet the results for purely leptonic branching fractions were presented too

ℓ	e	$\mu$	$\tau$
$\mathcal{B}(B^{-}\to {\ell }^{-}{\overline{\nu }}_{\ell })$	$1.16\times {10}^{-11}$	$0.49\times {10}^{-6}$	$1.10\times {10}^{-4}$

.

The numbers are in good agreement with the experimental values for the tau lepton $(1.090\pm 0.24)\times {10}^{-4}$[23] and the muon $(0.53\pm 0.22)\times {10}^{-9}$[61], which became more precisely measured since then, and also with the experimental limit for the electron. The agreement with several theoretical prediction of other authors was shown too. Since the leptonic decay constants are crucial in the description of purely leptonic decays and carry all of the necessary non-perturbavite information, their values have also been listed for $B_{(s,c)}^{(*)}$ and $D_{\left(s\right)}^{(*)}$ mesons, see Table I of [107] .

In [108] possible NP contributions were evaluated for chosen leptonic and semileptonic decays. It was assumed that these contributions affect only the third generation of leptons and all neutrinos were considered as left-handed. New, beyond-SM four-fermion operators were introduced in the Hamiltonian (1) with ${\sigma }_{\mu \nu }=i[{\gamma }_{\mu },{\gamma }_{\nu }]$, $P_{L,R}=(1\mp {\gamma }_5)/2$ and $i\in \{L,R\}$ (left, right). The most of the text deals with semileptonic decays where the $R_{D^{(*)}}$ discrepancy is observed (41). The set of observables was extended to of which the first is meant to analyze the R anomaly also for the $b\to u$ transition and the two remaining concern the leptonic decays. The limits on the Wilson coefficients $C_{V_i,S_i,T_l}$ were extracted assuming that only one of them is dominant at a time (besides the SM ones). Including into the analysis also the leptonic observable $R_{\tau }^u$ (together with $R_{D^{(*)}}$) it was found that no $C_{S_R,S_L}$ values were allowed (within 2 $\sigma$) and for $C_{V_L,V_R,T_L}$ allowed regions were identified in the complex plane (1 of [108]). Further, the leptonic ${\overline{B}}_c^{-}$ branching fractions were evaluated within the SM, $\mathcal{B}({\overline{B}}_c^{-}\to \tau \overline{\nu })=2.85\times {10}^{-2}$, $\mathcal{B}({\overline{B}}_c^{-}\to \mu \overline{\nu })=1.18\times {10}^{-4}$ and observables (40) were predicted for the SM and NP scenarios. In the latter case the corresponding Wilson coefficient $C_i$ was varied (one at a time) in the allowed region of the complex plain and the impact on the observable was determined. For the leptonic $R_{\tau }^c$ variable the prediction stands

	$SM$	$C_{V_L}$	$C_{V_R}$	$C_{T_L}$
$R_{\tau }^c=$	$3.03$	$3.945\pm 0.735$	$3.925\pm 0.815$	$3.03$.

As summary one can say that, within the given scenario, the text translated existing experimental information into the constraints on NP Wilson coefficients. Contributions of some of them ($C_{S_R,S_L}$) were excluded and some ($C_{V_L,V_R,T_L}$) were constrained.

The experimental information on the semileptonic B decays is much larger than on the pure leptonic decays. The LHCb experiment alone published in the past 10 years more than 35 papers on this topic and the number further increases if other experiments (Belle, BaBar, Belle II) are taken into the account. The same is true for theoretical publications which are large in quantity. With the aim to provide an overview of the CCQM results, we restrain ourselves only to most significant experimental measurements and theoretical predictions of other authors.

The focus of the community is predominantly driven by the so-called flavor anomalies. They are often defined as ratios of branching fractions, the most prominent of them are

The first observable is sensitive to the $b\to s$ quark transition, the two remaining to $b\to c$. Other quantities measured in semileptonic decays of the B meson are listed for example in VII of [109]. In these and other observables deviations were seen (see e.g. Tab XVIII of [110] for a nice review) with some of them reaching up to 4$\sigma$, which is naturally interpreted as significant argument in favor of the NP (see e.g. [111] ). The most recent LHCb measurements nevertheless weaken some of these observations and imply that the discrepancy with the SM may not be so pronounced after all. In [112] the deviation of a correlated observables $R_D$ and $R_{D^{*}}$ from the SM prediction is $1.9\sigma$ and the results for $R_K$ and $R_{K^{*}}$ given in [113] are in agreement with the SM. However, if one includes also older measurements and measurements of different experiments, the situation seem not to be yet solved and discrepancy is still close to $3\sigma$ [114].

The LHCb detector was specifically designed for b physics and the experiment successfully reaches its purpose by being the most important source of the experimental information on b decays. The measurements of $B\to K^{*}{\ell }^{+}{\ell }^{-}$ were presented in works [115,116,117,118,119,120,121,122,123]. Two of them [120,123] study the lepton-flavor universality by measuring $R_{K^{*}}$, but with no significant deviations from the SM. Most of the remaining works are concerned with angular distributions: the coefficients (noted for a p-wave process as $F_L$, $A_{FB}$, $S_{3,\cdots ,9}$) in front of angular terms which appear in the decay width formula are combined into so-called optimized observables $P_i^{{(}^{\prime })}$, and here some significant tensions are seen (e.g. $3\sigma$ in $P_2$ for $q^2$ between 6 and 8 ${\mathrm{GeV}}^2$ [122]).

The semileptonic B decays with the K meson in the final state are addressed in [124,125,126]. The first publication is concerned with the angular distribution and the differential branching fraction, the two others focus more specifically on the lepton flavor universality question, with an observation of a $2.5\sigma$ deviation from the SM in $R_K$. This was however, as mentioned earlier, undermined by the recent measurement [113] where no longer the deviation is seen.

The process $B\to D^{*}{\ell }^{+}{\ell }^{-}$ was analyzed in [112,127,128,129] and no deviation of $R_{D^{*}}$ from the SM greater than $2\sigma$ was detected. The same is true for the $R_{J/\Psi }$ observable measured in [130]. The decay of the $B_s^0$ particle to $\varphi {\mu }^{+}{\mu }^{-}$ was studied in [131,132,133], where, in the last analysis, a disagreement with the SM prediction is observed in the differential branching fraction for $1{\mathrm{GeV}}^2\le q^2\le 6{\mathrm{GeV}}^2$ at the level of $3.6\sigma$.

Various other semileptonic B decays were measured at the LHCb which we do not mention here. An overview of the lepton flavor universality question in b decays at the LHCb was, as of 2022, given in [134].

An additional experimental information on the semileptonic B decays comes from BaBar measurements. Studies of the $B\to D^{(*)}\ell {\nu }_{\ell }$ process were presented in [135,136,137,138,139,140,141]. In the first three references the question of the lepton flavor universality is addressed ($\ell =\tau$) and the measurement of $R_D$ and $R_{D^{*}}$ performed. The authors claim a deviation of $2.0\sigma$ for $R_D$, $2.7\sigma$ for $R_{D^{*}}$ and $3.4\sigma$ for their combination. The four latter references present the measurement of the $|V_{cb}|$ element of the CKM matrix and the analysis of corresponding transition form factors.

The decays with the $K^{(*)}{\ell }^{+}{\ell }^{-}$ final state were addressed in [142,143,144,145,146,147]. The texts present the measurements of branching fractions, the $R_{K^{(*)}}$ observable, the isospin and CP asymmetries, the forward-backward angular asymmetry of the lepton pair and the $K^{*}$ longitudinal polarization (and others). Overall, the results are in an agreement with the SM expectations, the anomaly observed for isospin asymmetries in both K and $K^{*}$ channels in [144] was not later confirmed in [145].

The BaBar collaboration also published results on semileptonic B decays into light mesons $\pi$ and $\rho$[148,149]. Here the branching fractions and the $|V_{ub}|$ element were determined and also transition form factors were discussed.

Further, BaBar published results on semileptonic decays where hadronic state $X_s$ containing kaons was produced and measured corresponding branching fractions [150,151]. One can also mention the measurement of charmless semileptonic decays [152,153] and the measurement with the electron in the final state [154], all of which were used to establish the $|V_{ub}|$ value. In [155] the semileptonic decay with five particles in the final state $D^{(*)}{\pi }^{+}{\pi }^{-}\ell {\nu }_{\ell }$, was confirmed.

Important contribution to measurements of semileptonic B decays comes form the Belle and Belle II collaborations.

Analyses [156,157,158,159,160] investigate both D and $D^{*}$ decay channels (with $\tau$ and ${\nu }_{\tau }$). They measure branching fractions and ratios $R_{D^{(*)}}$, where they do not see significant deviations from the SM expectations. The last work focuses also on the extraction of parameters for the Caprini-Lellouch-Neubert form factor parameterization.

Specifically $D^{*}$-containing final states are addressed in [161,162,163,164,165,166]. Also here the objects of interest are the branching fractions and the $R_{D^{*}}$ observable and, again, no significant deviations from the SM are seen. Works [162,166] present, in addition, the measurement of the $|V_{cb}|$ matrix element and form factor analysis, in works [164,165] the $\tau$ lepton polarization is measured.

The references [167,168] focus on the $D\ell {\nu }_{\ell }$ final state. The first work is concerned with the branching fraction and form factors, in both works $|V_{cb}|$ is measured. Authors of [169] report on the first observation of $B\to {\overline{D}}_1\ell {\nu }_{\ell }$ decay and measure the branching fractions of $B\to {\overline{D}}^{(*)}\pi {\ell }^{+}{\nu }_{\ell }$ and $B\to {\overline{D}}^{(*)}{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell }$ processes.

Production of strange mesons in semileptonic B decays is studied in [170,171] for the K meson, in [172,173,174] for the $K^{*}$ meson and in [175] for both, K and $K^{*}$. Besides branching fractions and $R_{K^{(*)}}$ ratios, some of the works present also measurements of angular and polarization variables and the isospin asymmetry. In general all measured values agree well with the SM predictions, some tensions for the subset of the optimized angular observables $P_i$ were reported in [173].

Semileptonic decays to light mesons ($\pi$, $\rho$ and $\eta$) were described in [176,177,178,179], the works are mostly concerned with the branching fractions and the determination of the $|V_{ub}|$ element of the CKM matrix.

The Belle(II) collaboration also published articles on semileptonic B decays to a general hadronic state X containing the s quark, $X_s$[180,181], the u quark, $X_u$[182,183,184] and the c quark, $X_c$[185,186]. The main objects of interest were branching fractions, CKM elements $|V_{ub}|$ and $|V_{cb}|$ and first four moments of the lepton mass squared (for $X_c$). The question of the lepton flavor universality in semileptonic decays to a general hadronic state X was addressed in [187].

Other results from different experiments could be cited in the domain of semileptonic B decays, yet the measurements of the above-mentioned B-factories represent the most important data from both, the quantity and quality perspective.

The large number of theoretical works implies strong selection criteria which we base on the impact of the work with some preference for review and pedagogical texts. We have already mentioned nice reviews [26,27,29,64,110] which cover (also) the semileptonic B decays. Further survey papers are [188], where the SM theory and appropriate observables are presented, a pedagogically-written article [189], which focuses on the charged lepton flavour violation and also a generally-oriented texts [190,191]. One can in addition mention [192], in which B flavor anomalies are discussed and also similarly oriented recent text [193].

Reliable SM predictions are the starting point for assessing various anomalies. Already decades ago a quark potential model was used to make predictions for semileptonic B and D decays [194] with an update several years later [195]. Decays to $D^{(*)}$ mesons were addressed in [196], the analyticity and dispersion relations were used to produce parametrizations of the QCD form factors with small model dependence. The same authors later published QCD two-loop level computations [197] including lepton mass effect, higher resonances and heavy quark symmetry, which further improved the theoretical precision. The heavy quark spin symmetry was used in [198] to derive dispersive constraints on $B\to D^{(*)}$ form factors and implications for the determination of $|V_{cb}|$. Semileptonic decays to light mesons $\rho$, $\omega$, $K^{*}$ and $\varphi$ were discussed in [199] in the framework of light-cone sum rules, the authors claim $10\%$ precision at zero momentum transfer. The angular analysis of the process $\overline{B}\to \overline{K}{\ell }^{+}{\ell }^{-}$ was presented in [200]. The work is based on the QCD factorization and large recoil symmetry relations and besides angular coefficients it also gives a prediction of $R_K$ and explores the potential of the introduced observables to reach the NP. Taking into the consideration also the excited state $K^{*}$, the publication [201] is dedicated to the charm-loop effect. The results are derived using QCD light-cone sum rules and hadronic dispersion relations and the evaluated charm loop effect, which is claimed to reach up to 20% , is represented as a contribution to the $C_9$ Wilson coefficient. Lattice QCD was used in [202,203,204] to predict form factors and matrix elements for processes with $D^{(*)}$ mesons. In [205] were the lattice form factors used as input and allowed to determine CKM matrix elements, or, alternatively, constrain the real part of the Wilson coefficients $C_9$ and $C_{10}$. The CKM matrix was also the subject of the work [206], where $|V_{cb}|$ was extracted using the OPE, the expansion in powers of the heavy quark mass and constraints derived from the experimental values on the normalized lepton energy moments. A process with a vector meson particle production $B\to V{\ell }^{+}{\ell }^{-}$ was considered in [207] where the authors used light-cone sum rules to predict form factors. The paper [208] has a somewhat review character, it present three common form factor parameterizations, summarizes the data and the available lattice information (as of 2016) and gives a special emphasis on the unitarity constraints. Then it presents fits to experimental points and to the lattice numbers from which the results on $R_D$ and $|V_{cb}|$ are extracted. Radiative corrections to the $R_{K^{(*)}}$ observables are of a concern to the authors of [209], their thorough analysis indicates that these observables are indeed well suited to be a probe of NP. Similar questions related to the same observables are addressed in [210]. Still the same observables are, together with the angular observables $P_i$, discussed in a pedagogical way in [211] with special emphasis on the hadronic uncertainties. Coming back to D particles and works published within few years after the first measurements indicating a possible lepton-flavor violation, one can mention [212], where the coefficients of the Boyd-Grinstein-Lebed form factor parametrization were constrained by analyzing the form factor ratios and their uncertainties in the heavy quark limit. With this knowledge fits to experimental data were performed and $R_{D^{*}}$ computed. In [213] two different form factors parameterizations are used to predict $R_{D^{*}}$ and $|V_{cb}|$. The approach uses, besides data, inputs from the light cone sum rules and lattice and the relations between form factors as given by HQET. To mention more recent theoretical works, one can point to e.g. [214,215], where QED corrections and non-local matrix elements are discussed for B decays to dilepton and a kaon. The status of the $b\to c\tau \nu$ anomalies as of 2022 is summarized in [216], where the models for global fits are based mostly on the HQET and lattice results. The latter are also reviewed the 8 of [18].

The number of NP papers progressively grew as the evidence for tensions and anomalies became more and more convincing, with the first hints appearing at the beginning of the new millennium. Often, the NP is theoretically addressed by non-SM operators appearing in the effective Hamiltonian. So was done in [217], where the approach was applied to the $b\to s$ process. No strong claims were given there, but it was shown that the evaluated NP effects can reach up to 13% for $R_{K^{*}}$. The same effective-operator approach was applied in [218] to $b\to c$ transition and the impact of the NP to $B\to D^{*}\tau {\overline{\nu }}_{\tau }$ observables was evaluated. The authors demonstrated that it is significant, i.e. the sensitivity of the process is high enough for the NP to be detected. Effective operators were used also in [219], where, after the NP operator contributions were discussed, two leptoquark models were proposed to explain two out of three possible scenarios which lead to the observed $R_K$ value. Leptoquarks (vector and scalar, respectively) are also considered in [220,221], both works claim that their theory allows to simultaneously resolve discrepancies appearing in $b\to s$ and $b\to c$ transitions. Still leptoquarks, the authors of [222] investigate single leptoquark extensions of the SM with $1\mathrm{TeV}\lesssim m_{LQ}\lesssim 2\mathrm{TeV}$ with conclusion that no such scalar leptoquark can be, a vector particle is the only option. The work [223] uses scenarios with light right-handed neutrinos appearing in low-scale seesaw models as the NP framework for analyzing the lepton flavor violation. Among other results the authors propose observables, i.e. properly chosen branching fraction ratios, which could discriminate between supersymmetric (SUSY) and non-SUSY NP realizations. Further works which analyze the $R_K$ and $R_{K^{*}}$ anomalies are [224] and [225], the former assumes a composite Higgs model, the latter uses a two-Higgs-doublet model. At last, let us mention a set of more generally-oriented works [91,92,226,227,228,229] which focus mainly on $b\to s{\ell }^{+}{\ell }^{-}$ and which aim to provide model-independent or theoretically clean conclusions. By different approaches they investigate the space for NP parameters and most of them presents arguments in favor of some NP scenario.

The $B_s\to \varphi {\ell }^{+}{\ell }^{-}\gamma$ and $B_s\to \varphi \gamma$ decays were within the CCQM analyzed in [230]. The analysis was done in the light of the LHCb measurements [131,132], where the second one was recent at that time. The measurement focused on angular observabes and the branching fraction distribution and reported on a deviation from the SM in the latter exceeding $3\sigma$ for $1{\mathrm{GeV}}^2\le q^2\le 6{\mathrm{GeV}}^2$. Several years later two new measurements were performed. The work [231] addressed the angular distribution where no significant tensions with the SM were observed, [133] however confirmed the discrepancy from the previous branching fraction measurement. One may put this observation in relation with $R_K$ and $R_{K^{*}}$ anomalies, which also happen for the $b\to s$ transition, from where the motivation to study this process in more details.