SI-HEP-2023-25, P3H-23-087; RBI-ThPhys-2023-37
A guide to the QCD light-cone sum rules
for $b$-quark decays

The $B\to\pi$ form factors, generated in SM by the weak $b\to u$ transition, were among the first applications of LCSRs. The vector and scalar form factors¹¹1We use standard definitions of these form factors, see, e.g., Ref. [18]. $f^{+}_{B\pi}$ and $f^{0}_{B\pi}$ determine the weak semileptonic decay $\bar{B}^{0}\to\pi^{+}\ell\nu_{\ell}$. To describe the rare $B\to\pi\ell^{+}\ell^{-}$ decay, generated by the flavour-changing neutral current (FCNC) $b\to d$ transition, one needs, in addition to $f^{+,0}_{B\pi}$, also a form factor $f_{B\pi}^{T}(q^{2})$ of the tensor $b\to d$ quark current.

In the rest of this subsection, we briefly recall the derivation of LCSR for the most important form factor $f^{+}_{B\pi}$. The underlying correlator is defined as a vacuum-to-pion matrix element of the time-ordered product of two currents:

	$\displaystyle F_{\mu}(p,q)$	$\displaystyle=$	$\displaystyle i\int d^{4}x~{}e^{iq\cdot x}\langle\pi^{+}(p)|T\left\{\bar{u}(x)\gamma_{\mu}b(x),m_{b}\bar{b}(0)i\gamma_{5}d(0)\right\}|0\rangle$		(1)
		$\displaystyle=$	$\displaystyle F((p+q)^{2},q^{2})\,p_{\mu}+\widetilde{F}((p+q)^{2},q^{2})\,q_{\mu}\,,$

where $q$ and $(p+q)$ are, respectively, the momenta of the $b\to u$ weak current and of the $B$-meson interpolating current. Only the first invariant amplitude $F$ in the Lorentz-decomposition is relevant for $f^{+}_{B\pi}$. The amplitude $\widetilde{F}$ is used to obtain a second LCSR, so that a linear combination of the two sum rules yields the scalar form factor. The tensor form factor $f_{B\pi}^{T}$ is accessed by replacing the vector weak current in the correlator by the tensor $b\to d$ transition current. Ability to access different form factors by varying currents or invariant amplitudes in the correlator reveals flexibility and universality of the LCSR method.

The correlator in Eq. (1) is calculated at the external momenta squared far below the open $b$-flavour thresholds, that is, at

$$(p+q)^{2}\ll m_{b}^{2},~{}~{}q^{2}\ll m_{b}^{2}.$$

(2)

These conditions induce a strongly oscillating exponent in the correlator, retaining only the region near the light-cone $x^{2}\simeq 0$ in the four-coordinate integral. Hence, the product of currents can be expanded near $x^{2}=0$. In particular, a highly virtual b-quark is replaced with a light-cone expansion [19] of its propagator. Contraction of $b$-quark fields generates a perturbatively calculable kernel which is factorized from the remaining long-distance part. The resulting expression for the invariant amplitude $F$ (or similarly for $\tilde{F})$ has the following schematic form:

	$\displaystyle F((p+q)^{2},q^{2})$	$\displaystyle=$	$\displaystyle i\int d^{4}x\,e^{iqx}\Bigg{\{}\big{[}S_{0}(x^{2},m_{b},\mu)+\alpha_{s}S_{1}(x^{2},m_{b},\mu)+\dots\big{]}$		(3)
		$\displaystyle\otimes$	$\displaystyle\langle\pi(p)\mid\bar{u}(x)\Gamma d(0)\!\mid\!0\rangle_{|_{\mu}}+\int_{0}^{1}dv~{}\big{[}\tilde{S}_{0}(x^{2},m_{b},\mu,v)+\dots\big{]}$
		$\displaystyle\otimes$	$\displaystyle\langle\pi(p)\mid\bar{u}(x)G(vx)\tilde{\Gamma}d(0)\}\mid 0\rangle_{|_{\mu}}+\dots\Bigg{\}}\,,$

where $S_{0}$ and $S_{1}$ are the LO and NLO parts of the perturbative kernel. They correspond, respectively, to the free $b$-quark propagator and $O(\alpha_{s})$ perturbative gluon corrections. This kernel is convoluted with the vacuum-to-pion matrix element of the operator formed by a product of the $\bar{u}$ and $d$ quark fields at a near-light-cone separation, with a generic Dirac structure $\Gamma$. The term with $\tilde{S}_{0}$ corresponds to a low-virtuality gluon emitted at a light-cone separation from the $b$-quark propagator and absorbed, together with a quark-antiquark pair, in the final pion state. The ellipsis in Eq. (3) denotes higher-order terms, e.g., the NNLO, $O(\alpha^{2}_{s})$ corrections to $S_{0}$, NLO corrections to $\tilde{S}_{0}$ or the terms with two quarks and two antiquarks entering the vacuum-to-pion matrix element. The diagrams and detailed expressions for this correlator at NLO are given in Ref. [18], (see also the introductory review [15] for a description of derivation at the LO level).

The scale $\mu$ indicated in Eq. (3) corresponds to a separation between the short-distance and long-distance parts in this convolution. The optimal choice is $\mu\sim\sqrt{\Lambda m_{b}}$, where an intermediate scale $\Lambda\sim$ 1 GeV is parametrically larger than $\Lambda_{\rm QCD}$, but at the same time does not scale with the heavy mass $m_{b}$. This choice guarantees that the average interval $|x^{2}|\sim 1/\mu^{2}$, between the emission points of the light-quark and gluon fields in the vacuum-to-pion matrix elements, can still be considered small. The subsequent transition of light constituents into an on-shell pion state includes all nonperturbative effects at energy-momenta below $\mu$.

The formula (3) reveals a general structure of the light-cone OPE, emerging after expanding the correlator in terms of (i) quark-gluon coupling and (ii) multiplicity of light-quark and gluon fields entering the vacuum-to-pion matrix elements. The possibility to retain the lowest terms in the expansions (i) and (ii) is based on the suppression, respectively, due to extra powers of $\alpha_{s}$ (for which the scale $\mu$ is a natural choice) and due to inverse powers of $\mu$.

The complete OPE of the correlator outlined in Eq. (3) is in fact more involved. Within each vacuum-to-pion matrix element, an additional expansion emerges after expressing these matrix elements in terms of the pion light-cone DAs with growing twists, starting from the lowest twist-2 DA:

$\displaystyle\langle\pi(p)|\bar{u}(x)[x,0]\gamma_{\mu}\gamma_{5}d(0)|0\rangle_{x^{2}\sim 0}=-ip_{\mu}f_{\pi}\int_{0}^{1}du\,e^{iup\cdot x}\varphi_{\pi}(u)+O(x^{2})+\dots$

(4)

Here $u$ and $1-u$ are the fractions of the pion momentum $p$ distributed among the two partons (quark and antiquark) in the pion, and $[x,0]$ is the gauge factor. Two similar vacuum-to-pion matrix elements emerging in the light-cone expansion of the correlator contain (instead of $\gamma_{\mu}\gamma_{5}$) the Dirac-matrices $\Gamma=i\gamma_{5}$ and $\Gamma=\sigma_{\mu\nu}$, yielding, respectively, two additional DAs of the next-to-lowest twist-3. In addition, the $O(x^{2})$ terms in Eq.(4) generate two more two-particle (quark-antiquark) DAs of twist-4. The second hadronic matrix element in Eq. (3) is expressed via three-particle (quark-antiquark-gluon) pion DAs, starting from twist 3. Currently, altogether five two-particle DAs with $t=2,3,4$ and five three-particle DAs with $t=3,4$ are taken into account in the OPE. Their expressions, normalization parameters and other important features, such as the relations between two and three-particle DAs due to QCD equations of motion, can be found in the most updated form in Ref. [20]. A short summary of these DAs and convenient defining formulas for the two- and three-particle vacuum-to-pion matrix elements are also given in Appendix A of Ref. [18].

After substitution of DAs and the coordinate integration, the OPE (3) for the invariant amplitude $F$ transforms into a sum of separate DA contributions. In a compact generic form we have:

	$\displaystyle F_{OPE}((p+q)^{2},q^{2})=\sum\limits_{m=1}^{\infty}\int{\cal D}u_{\{m\}}~{}\sum\limits_{t\geq 2}\Bigg{[}T_{0}^{(m,t)}\big{(}(p+q)^{2},q^{2},u_{1},..u_{m},\bar{m}_{b},\mu\big{)}$
	$\displaystyle+\,\alpha_{s}T_{1}^{(m,t)}\big{(}(p+q)^{2},q^{2},u_{1},..u_{m},\bar{m}_{b},\mu\big{)}+O(\alpha_{s}^{2})\Bigg{]}\phi_{\pi}^{(m,t)}(u_{1},u_{2},..u_{m},\mu)\,,$		(5)

where $\phi_{\pi}^{(m,t)}$ is a pion DA with the light-parton multiplicity $m$ and twist $t$. This DA depends on the pion momentum fractions $u_{1},u_{2},...u_{m}$, and we use the notation ${\cal D}u_{\{m\}}=\displaystyle\int\left(\prod\limits_{i=1}^{m}du_{i}\right)\,\delta\left(1-\sum\limits_{i=1}^{m}u_{i}\right)$ for the integration element. In the above expression, the twist-2 DA defined in Eq. (4) corresponds to $\phi^{(2,2)}(u,1-u)=f_{\pi}\varphi_{\pi}(u)$. A nontrivial feature of the light-cone OPE is the factorization proven at NLO for the leading power twist-2 terms, so that the $\mu$ dependence of the perturbative kernel $T_{0}^{(2,2)}+\alpha_{s}T_{1}^{(2,2)}$ is compensated by the perturbative evolution of $\phi_{\pi}^{(2,2)}$, well known as the ERBL evolution [10, 12]. Also the choice of the $\overline{\rm MS}$ scheme for the $b$-quark mass $\overline{m}_{b}$ is justified for a correlator with a highly-virtual $b$-quark. The currently known terms in Eq.(5) are listed in Table 1 with references to the papers in which they have been computed and where further important details can be found.

Finally, one more expansion implicitly present in the OPE (5) concerns the dependence of each DA on the fractions $u_{i}$ of the pion momenta shared between light degrees of freedom. Here the formalism of conformal partial-wave expansion in QCD is used. Comprehensive reviews on that subject can be found , e.g. in Refs. [26], [27]. The conformal expansion represents a given pion DA in a form of series in orthogonal polynomials. For the twist-2 DA it is the well familiar series in Gegenbauer polynomials with multiplicatively renormalizable coefficients:

$$\varphi_{\pi}(u,\mu)=6u(1-u)\Big{(}1+a_{2}^{\pi}(\mu)C_{2}^{3/2}(2u-1)+a_{4}^{\pi}(\mu)C_{4}^{3/2}(2u-1)+\dots\Big{)}\,.$$

(6)

The polynomial expansion for this and other DAs, is usually taken up to the second conformal partial wave, that is, retaining only $a^{\pi}_{2}$ and $a^{\pi}_{4}$ in the above formula. A usual motivation is that the anomalous dimensions of polynomial coefficients grow with $n$, suppressing terms with larger $n$, so that at sufficiently large scale $\mu$, the DA (6) is not far from its asymptotic form, which is $6u(1-u)$. The Gegenbauer moments in $\varphi_{\pi}$ and similar polynomial coefficients for the higher-twist pion DAs, all taken at a reference scale $\mu=1.0$ GeV, represent universal, process independent inputs for the OPE. Various methods to determine these parameters include lattice QCD computation and two-point QCD sum rules. Another useful strategy applied to determine the shape of the lowest twist-2 pion DA is to fit to their measured values the pion transition form factor $\gamma\gamma^{*}\to\pi^{0}$ and the pion electromagnetic form factor, both calculated from LCSRs (see, respectively, e.g., Refs. [28, 29, 30] and Refs. [31, 32]).

Having at hand the OPE result (5) for the invariant amplitude as an analytical function of the variable $(p+q)^{2}$ at fixed $q^{2}$, one then applies the standard tools of the conventional QCD sum rule method:

$\bullet$ equating this result to the hadronic dispersion relation in the variable $(p+q)^{2}$:

$$F^{OPE}((p+q)^{2},q^{2})=\frac{2f_{B}m_{B}^{2}f^{+}_{B\pi}(q^{2})}{m_{B}^{2}-(p+q)^{2}}+...,$$

(7)

where the ellipsis indicates all heavier states with the $B$-meson quantum numbers, starting from the lowest threshold at $m_{B^{*}}+m_{\pi}$. Note that the hadronic dispersion relation is valid at any $(p+q)^{2}$, whereas the relation (7) is only used in the region (2) of the OPE validity;

$\bullet$ using quark-hadron semi-local duality approximation:

$\displaystyle\int\limits_{(m_{B^{*}}+m_{\pi})^{2}}^{\infty}\!\!\!ds\,\frac{\mbox{Im}F(s,q^{2})}{s-(p+q)^{2}}=\int\limits_{s_{0}^{B}}^{\infty}ds\,\frac{[\mbox{Im}F(s,q^{2})]_{OPE}}{s-(p+q)^{2}}\,,$

(8)

where the effective channel-specific threshold $s_{0}^{B}$ is introduced;

$\bullet$ subtracting the heavier state contributions from Eq. (7) with the help of Eq. (8) and applying the Borel transform $(p+q)^{2}\to M^{2}$.

The final form of LCSR is then obtained:

$$m_{B}^{2}f_{B}f_{B\pi}^{+}(q^{2})e^{-m_{B}^{2}/M^{2}}=\int\limits_{m_{b}^{2}}^{s_{0}^{B}}ds\,e^{-s/M^{2}}[\mbox{Im}F(s,q^{2})]_{\rm OPE}\,.$$

(9)

This sum rule has three different sets of input parameters, ordered according to their universality. The first one includes the $b$-quark mass and $\alpha_{s}$, both determined with great accuracy. The second set contains universal parameters of the pion DAs, including their normalizations and polynomial coefficients. Finally, the third set includes parameters specific for the $B$-meson channel: (a) the ranges of the scale $\mu$ and Borel parameter $M$; (b) the decay constant of $B$-meson $f_{B}$, that can be taken from lattice QCD average or, with a larger uncertainty, from the two-point QCD sum rule calculation (see e.g., Ref. [33]); and (c) the threshold $s_{0}^{B}$ which is usually extracted from a derivative sum rule obtained from LCSR. The latter procedure was systematically used in more recent analyses (see e.g., Ref. [34]).

The latest numerical results for the $B\to\pi$ form factors at $q^{2}=0$ are given in the next subsection, in Table 3, together with the form factors of all other $B$ transitions to light pseudoscalar mesons. The standard $z$-expansion (here preferred is the BCL version suggested in Ref. [35]) allows one to extrapolate the $B\to\pi$ form factors from the region of LCSR validity, typically at $q^{2}\leq 12-15$ GeV² to larger $q^{2}$, that is, to the low recoil region of the pion, where this extrapolation can be compared with the lattice QCD predictions ²²2 For brevity, we do not quote for comparison the lattice QCD results here, they can be found in the updated and averaged form in Ref. [36], see also the review on lattice QCD applications to $b$ quark physics in this volume..

The uncertainties quoted in Table 3 are parametric and they are usually estimated in quadratures. A more advanced Bayesian analysis of LCSRs was initiated in Ref. [34]. There are however not many possibilities left to substantially decrease these uncertainties in the future. Indeed, the still missing parts of OPE such as the NLO corrections to the nonasymptotic twist-3 terms and to the twist-4 terms are expected to be very small. Also the factorizable twist-5, 6 contributions estimated in Ref. [25] turned out to be negligible. On the other hand, there is still a relatively large uncertainty in the parameters of the pion twist-2 DA. Further efforts are desirable to improve the knowledge of this key element of LCSRs, combining all available methods.

The most elusive uncertainty of systematic origin in LCSRs remains the one related to the application of quark-hadron duality. The attempts in the literature to quantify this uncertainty by varying the threshold parameter $s_{0}^{B}$ with the Borel scale or in any other way are only remedies, because these analyses still use duality as a basic assumption. In the future, one has to find alternative methods to estimate the integrated hadronic spectral density of higher states in the sum rules.

As an indirect way to assess the actual accuracy of LCSR predictions, we suggest to increase the accuracy of the $q^{2}$-shape in the measured differential (binned) width of the $B\to\pi\ell\nu_{\ell}$ decays. This observable, being directly proportional to the squared shape of the form factor $f^{+}_{B\pi}(q^{2})$, provides a direct test of a QCD method, independent of the value of CKM parameter $V_{ub}$.

Finally, let us mention that straightforward byproducts of LCSRs with pion DAs considered in this subsection are the analogous sum rules for the $D\to\pi$ form factors. Switching to the charmed sector is straightforward, and demands only a replacement of $b$ quark by the $c$ quark in the correlator and a corresponding adjustment of all channel-specific inputs. The last LCSR calculation of these form factors in Ref. [37] deserves an update, e.g., including the twist 5,6 terms in the OPE. A comparison with experimental data on $D\to\pi\ell\nu_{\ell}$ semileptonic decay will allow one to tune the universal input parameters of LCSRs and to further increase the accuracy of the $B\to\pi$ form factor determination.

The $B\to\pi$ form factor $f^{+}_{B\pi}(q^{2})$ at large $q^{2}$ (near the zero recoil of the pion), where the $B^{\ast}$-pole dominates, is determined by the strong $B^{*}B\pi$ coupling. It is defined as the invariant constant parametrizing the hadronic matrix element

$$\langle B^{*}(q)\pi(p)|B(p+q)\rangle=-g_{B^{*}\!B\pi}\,p^{\mu}\epsilon_{\mu}^{(B^{*})}\,,$$

(10)

where $\epsilon_{\mu}^{(B^{*})}$ is the polarization vector of $B^{*}$ meson. The coupling $g_{B^{*}\!B\pi}$ is not measurable, since the $B^{\ast}\to B\pi$ decay is kinematically forbidden, contrary to its analog in charm sector, the $D^{\ast}\to D\pi$ decay.

As originally suggested in Ref. [6], the coupling $g_{B^{*}B\pi}$ can be obtained from a LCSR, considering the same correlator (1) and using its OPE. For the hadronic representation of the invariant function $F(q^{2},(p+q)^{2})$ a double dispersion relation in both variables $p^{2}$ and $(p+q)^{2}$ should then be used:

$\displaystyle F(q^{2}\!,(p\!+\!q)^{2})=\frac{m_{B}^{2}m_{B^{*}}f_{B}f_{B^{*}}\,g_{B^{*}B\pi}}{(m_{B}^{2}\!-\!(p+q)^{2})(m_{B^{*}}^{2}\!-\!q^{2})}\!+\!\frac{1}{\pi^{2}}\!\!\int\!\!\!\!\int\!\!ds_{2}ds_{1}\frac{{\rm Im}_{s_{1}}{\rm Im}_{s_{2}}F(s_{1},s_{2})}{(s_{2}\!-\!(p+q)^{2})(s_{1}\!-\!q^{2})}\,,\hskip 14.22636pt$

(11)

where the lowest double-pole term contains the $B^{*}B\pi$ coupling multiplied by the decay constants of pseudoscalar ($f_{B}$) and vector ($f_{B^{*}}$) bottom mesons. The duality approximation has to be defined for a two-dimensional region in the $\{s_{1},s_{2}\}$ plane, adding an uncertainty, related to the freedom to choose the shape of that region, whereas the parametric accuracy of the OPE is the same as in the LCSRs for $B\to\pi$ form factors. Instead of a single Borel transform, as in Eq. (9), the double Borel transform is then performed, removing all subtraction terms that are not shown in Eq. (11) for brevity, including single-variable dispersion integrals. Due to the approximate mass degeneracy of $B^{*}$ and $B$ mesons, usually the two Borel parameters, $q^{2}\to M_{1}^{2}$ and $(p+q)^{2}\to M_{2}^{2}$, are taken equal, $M_{1}^{2}=M_{2}^{2}=2M^{2}$.

The strong coupling $g_{B^{\ast}B\pi}$ is then extracted from the sum rule:

$\displaystyle f_{B}\,f_{B^{*}}\,\,g_{B^{*}B\pi}=\frac{1}{m_{B}^{2}m_{B^{*}}}e^{\frac{m_{B}^{2}+m_{B^{*}}^{2}}{2M^{2}}}\!\!\!\!\int\limits^{\Sigma(s_{0})}\!\!\!\!\!\!\int\!ds_{2}ds_{1}e^{-\frac{s_{2}+s_{1}}{2M^{2}}}\frac{1}{\pi^{2}}{\rm Im}_{s_{1}}{\rm Im}_{s_{2}}F_{OPE}(s_{1},s_{2})\,,$

(12)

where $\Sigma_{0}(s_{0})$ indicates the duality region parametrized with the effective threshold $s_{0}$.

In Ref. [6] the LO result for this sum rule at twist-4 accuracy was obtained. The NLO correction to the twist-2 contribution was computed in Ref. [38]. The most recent and substantially improved analysis of the LCSR (12) for the $B^{*}B\pi$ coupling and (replacing $b\to c$ quark in the correlator) for the $D^{*}D\pi$ coupling is in Ref. [39], where also the NLO twist-3 contributions were calculated.

In Table 2, the results obtained in Ref. [39] for the $B^{\ast}B\pi$ strong coupling are displayed for two choices of the decay constants $f_{B}$ and $f_{B}^{\ast}$: from two-point sum rules, and from lattice QCD. Also, two models for the leading twist-2 pion DA are considered. Additional details of this analysis and references relevant for the choice of the input parameters can be found in Ref. [39].

It is important to stress that the coupling $g_{B^{*}B\pi}$ is calculated from LCSR at a finite $b$-quark mass. Hence, replacing $b\to c$ in the sum rule provides also the charmed meson coupling $g_{D^{*}D\pi}$. The infinitely heavy-quark limit of this coupling, known as the static coupling $\hat{g}$, and serving as a key parameter in the Heavy-Meson Chiral Perturbation Theory (HM$\chi$PT), can also be obtained from the same LCSR. In Ref. [39] this limit was estimated, together with the inverse heavy mass correction combining the couplings for both heavy mesons and fitting them to the parametrization:

$$g_{H^{*}H\pi}=\frac{2m_{H}\,\hat{g}}{f_{\pi}}\left(1+\frac{\delta}{m_{H}}\right)\,,~{}~{}(H=D,B)\,.$$

(13)

The results are given in Table 2.

Analogously to the $B\to\pi$ form factors, the other $B_{(s)}\to P$ form factors ($P=K,\eta,\eta^{\prime}$) can be obtained from LCSRs. These form factors are of a particular interest, for an alternative $V_{ub}$ determination, for various rare $B_{(s)}\to P\ell^{+}\ell^{-}$ decays, and also for testing the factorization approximation in nonleptonic $B_{(s)}$ decays.

The initial correlator for a $B_{(s)}\to K$ transition is obtained from Eq.(1), replacing the pion state with the kaon state and making necessary changes in the interpolation and transition quark currents. It is also important that LCSRs allow for a complete account of $SU(3)_{fl}$-breaking effects, originating from a nonvanishing $s$-quark mass and revealing themselves in differences between the kaon and pion DAs, starting from the ratio $f_{K}/f_{\pi}$ as well as in the ratios of other hadronic parameters entering the sum rules, e.g., $m_{B_{s}}/m_{B}$ and $f_{B_{s}}/f_{B}$.

The lowest twist-2 DA of a kaon has an expansion in Gegenbauer polynomials similar to Eq. (6), but including also the odd moments $a_{1,3,...}^{K}$, in contrast to the pion DA, where due to the $G$-parity conservation, the odd moments vanish. The set of higher twist kaon DAs are worked out in Ref. [20].

The $B_{(s)}\to K$ form factors were calculated from LCSRs first in Ref. [40] and more recently in Ref. [41], with the same accuracy as for $B\to\pi$ form factors discussed in the previous subsection. The calculation of the $B_{(s)}\to\eta,\eta^{\prime}$ transition form factors is somewhat more complicated due to the $\eta-\eta^{\prime}$ mixing and a related $U(1)_{A}$ QCD anomaly contribution to that mixing. The first calculation of the $f^{+}_{B\eta}$ transition form factors at NLO level for the leading twist-2 was done in Ref. [42]. This calculation was further improved in Ref. [43], where also the $B\to\eta^{\prime}$ transition was considered. The $U(1)_{A}$ anomaly induces, in addition to flavour-singlet quark-antiquark DA the two-gluon DA which contributes to the $B\to\eta,\eta^{\prime}$ transitions, at NLO level, and has to be taken into account. This introduces additional uncertainty in the calculation since the coefficients in the Gegenbauer expansion of the twist-2 gluon DA are not known. The most recent application of LCSRs to all the $B_{(s)}\to\eta,\eta^{\prime}$ form factors (as well as to the $D_{(s)}\to\eta,\eta^{\prime}$ form factors), at NLO and including two-gluon DAs is in Ref. [44] (see also Ref. [45]). The results for all form factors of the $B$-meson transitions to pseudoscalar mesons at $q^{2}=0$ are summarized in Table 3.

Extension of the LCSR method to the $B\to V$ form factors, where $V=\rho,\omega,\phi,K^{*}$, demands a vacuum-to-$V$ correlator which otherwise has the same structure as the vacuum-to-$\pi$ correlator in Eq. (1). Correspondingly, the OPE is obtained in terms of vector meson distribution amplitudes. Importantly, these DAs are defined neglecting the widths of vector mesons. As an example, one of the two twist-2 DAs of $\rho$-meson is defined as:

$$\langle\rho(p)|\bar{u}(x)\sigma_{\mu\nu}[x,0]d(0)|0\rangle=-if_{\rho}^{\perp}\big{(}\epsilon^{*(\rho)}_{\mu}p_{\nu}-p_{\mu}\epsilon^{*(\rho)}_{\nu}\big{)}\int\limits_{0}^{1}du\,e^{iup\cdot x}\phi_{\perp}^{(\rho)}(u)\,,$$

(14)

where $\epsilon^{(\rho)}$ is the polarization vector of $\rho$. The DA $\phi_{\perp}^{(\rho)}$ corresponds to the transversely polarized $\rho$-meson and $f_{\rho}^{\perp}$ is the transverse decay constant. Note that the shape of this DA is also determined by the Gegenbauer polynomial expansion. A comprehensive analysis of the twist-3 and twist-4 DAs was done, respectively, in Refs. [48] and [49].

The very first LCSRs for $B\to V$ form factors were derived in Ref. [7], where the radiative $B\to V\gamma$ decays were considered. The first application of this method to the semileptonic $B\to\rho\ell\nu$ decay was done, at LO level, in Refs. [50, 8], where also the advantages of the LCSR approach over the three-point QCD sum rules for the calculation of the heavy-to-light transition form factors were discussed in detail. A major update including the NLO twist-2 contributions and extending the method to almost all $B\to V$ channels was made in Ref. [51]. After that, in Ref. [52] the analysis was improved by adding to OPE the NLO twist-3 terms and obtaining LCSRs also for the $B\to\omega$ transition. The most recent update of LCSRs with vector meson DAs for all $B_{(s)}\to V$ form factors, and with OPE based on the results of Ref. [52], can be found in Ref. [53]. In the latter paper, the form factors are also extrapolated to the whole semileptonic decay region after fitting them to the z-expansion. Their results for the $B\to V$ transition form factors at $q^{2}=0$ are summarized in Table 4.

The LCSRs for $B\to\rho,K^{*}$ form factors with the vector meson DAs considered in the previous subsection are derived neglecting the total widths of vector mesons. This is certainly a poor approximation for such a broad resonances as $\rho(770)$ and $K^{*}(892)$. A more comprehensive approach is to consider the $B\to 2\pi$ and $B\to K\pi$ transitions, where $\rho$ and $K^{*}$ resonances are only a part, albeit dominant, of the dimeson $2\pi$ and $K\pi$ states, respectively. The phenomenology of $B\to 2\pi$ form factors was studied in Ref. [54] were one can find all necessary definitions (see also Ref. [55]).

Here, as an example, we consider the $\bar{B}^{0}\to\pi^{+}\pi^{0}\ell\nu_{\ell}$ semileptonic transition in which the $\bar{B}^{0}\to\rho^{+}$ form factors are the resonance parts of the $\bar{B}^{0}\to\pi^{+}\pi^{0}$ form factors. To avoid lengthy formulas, we take only the vector part of the weak $b\to u$ current, which yields a single form factor defined as:

$$i\langle\pi^{+}(k_{1})\pi^{0}(k_{2})|\bar{u}\gamma^{\mu}b|\bar{B}^{0}(p)\rangle=-F_{\perp}(q^{2},k^{2},\zeta)\,\frac{4}{\sqrt{k^{2}\lambda_{B}}}\,i\epsilon^{\mu\alpha\beta\gamma}\,q_{\alpha}\,k_{1\beta}\,k_{2\gamma}\,,$$

(15)

where $k^{2}=(k_{1}+k_{2})^{2}$ is the squared invariant mass of the dipion state and $\zeta$ is an additional angular variable. The $B\to 2\pi$ form factor is then expanded in partial waves corresponding to angular momenta $\ell=1,3,5...$ of the dipion state, yielding a series in Legendre polynomials in the angular variable. Note that the partial waves with $\ell=0,2,4,...$ are forbidden for the $\pi^{+}\pi^{0}$ state on symmetry grounds. The $B\to\rho$ form factor contributes only to the $\ell=1$ component $F_{\perp}^{(\ell=1)}(q^{2},k^{2})$ of the $B\to 2\pi$ form factor via dispersion relation in the variable $p^{2}$ at fixed $q^{2}$:

$$\frac{\sqrt{3}F_{\perp}^{(\ell=1)}(q^{2},k^{2})}{\sqrt{k^{2}}\sqrt{\lambda_{B}}}=\frac{g_{\rho\pi\pi}}{m_{\rho}^{2}-k^{2}-im_{\rho}\Gamma_{\rho}(k^{2})}\frac{V^{B\to\rho}(q^{2})}{m_{B}+m_{\rho}}+...\,,$$

(16)

where $g_{\rho\pi\pi}$ is the strong $\rho\pi\pi$ coupling. In the above relation, the energy-dependent width is inserted in the Breit-Wigner formula and contributions of excited states, such as the $\rho(1450)$ resonance, are indicated by the ellipsis. Although dispersion relation by itself is model-independent and follows from analyticity and unitarity principles, a certain model-dependence is unavoidable in the resonance term. In any case, this relation is the only realistic possibility to extract the $B\to\rho$ form factor from the $B\to 2\pi$ form factor, taking into account not only the $\rho$ width effect, but also the nonresonant background. The latter emerges from the continuum and excited state contributions hidden under the ellipsis in Eq. (16).

To access the $B\to 2\pi$ form factors directly, the method of LCSRs, resembling the one presented in the previous subsections, was suggested in Ref. [56] (see also Ref. [57]). A correlator similar to Eq. (1) was used in which, instead of a single meson state, there is an on-shell state of two pions with a variable invariant mass squared $k^{2}\geq 4m_{\pi}^{2}$. Applying light-cone OPE yields an expression with a structure resembling Eq. (3). But in this case, the perturbative kernels with a virtual $b$-quark are convoluted with the vacuum-to-dipion matrix elements of quark-antiquark or quark-antiquark-gluon operators. These matrix elements are parametrized in terms of a set of new objects, the dipion DAs. The latter were introduced and used much earlier [58, 59, 60, 61, 62], to describe hard exclusive processes with dimeson states, such as $\gamma^{*}\gamma\to 2\pi$.

The dipion DAs are also classified by their twist and by the multiplicity of quark and gluon fields. Currently, only the most important two-particle (quark-antiquark) DAs of twist-2 are available. One of them is defined as:

$$\langle\pi^{+}(k_{1})\pi^{0}(k_{2})|\bar{u}(x)\gamma_{\mu}[x,0]d(0)|0\rangle=-\sqrt{2}k_{\mu}\int\limits_{0}^{1}due^{iu(k\cdot x)}\Phi^{I=1}_{\parallel}(u,\zeta,k^{2})\,,$$

(17)

and the second one denoted as $\Phi^{I=1}_{\perp}$ has a $\sigma_{\mu\nu}$ Dirac structure between quark fields. The index $I=1$ reflects the isospin of the $\pi^{+}\pi^{0}$ state. Full definitions and many important properties of these DAs can be found in Ref. [61]. Both DAs undergo a double expansion in Legendre polynomials – i.e. in partial waves of the dipion state – and in Gegenbauer polynomials. The latter expansion reflects the momentum distribution between quark and antiquark and has the same form as Eq. (6). The coefficients of this double expansion replace Gegenbauer moments $a_{2n}$ in Eq. (6) and are complex valued functions of $k^{2}$, with the phase reflecting strong rescattering of the final-state pions. The local limit $x\to 0$ of the matrix element (17) in the isospin symmetry limit, is proportional to the pion electromagnetic form factor in the timelike region which is well measured. However, for the second twist-2 DA this normalization coincides with the timelike form factor of the tensor current which is not directly measurable. A calculation of this hadronic parameter together with a few lowest Gegenbauer functions for both dipion DAs was only performed at small $k^{2}$ in Ref. [63], employing the instanton vacuum model of QCD.

The OPE of the correlator for the $\bar{B}\to\pi^{+}\pi^{0}$ form factors was obtained in Refs. [56, 57] with a twist-2 accuracy and in the LO. This expansion is valid at sufficiently small dipion invariant masses, $k^{2}\ll m_{b}^{2}$ and, simultaneously, in the large recoil region $q^{2}\ll m_{b}^{2}$. The rest of the LCSR derivation is essentially the same as for the sum rules with a single light-meson DAs discussed in Section 2.1. In particular, the same hadronic dispersion relation and duality in the channel of the $B$-meson interpolating current are used. In the resulting LCSRs, the partial wave components $F_{\perp}^{(\ell)}(q^{2},k^{2})$ with $\ell=1,3,..$ are separated from each other. As shown in Ref. [56] in detail, the sum rule for $F_{\perp}^{(\ell=1)}$, together with its analogs for the $B\to 2\pi$ form factors of the axial weak current, determine the proportion of the $B\to\rho$ channel in the general $B\to 2\pi$ transition. In addition, the ratios of the form factors with $\ell>1$ with respect to the lowest one with $\ell=1$ were estimated.

The method of LCSRs with dipion DAs has a considerable potential for further improvement. To increase the precision, one needs detailed studies of the twist expansion for dipion DAs, including the three-particle (quark-antiquark-gluon) DAs. On the other hand, a better knowledge of Gegenbaeur functions for the leading twist-2 DA is necessary. A possibility to gain some information on these universal functions from the $D\to 2\pi\ell\nu$ decays (using the $b\to c$ replacement in the LCSRs) is currently being studied [64]. In the future, in order to extend this method to other important form factors, DAs for the different states of two light pseudoscalar mesons should also be studied. Most important are the dipion states with the spin-parities $J^{P}=0^{+},2^{+}$ relevant for $B\to\pi^{+}\pi^{-}$ form factors, as well as the $K\pi$ and $K\bar{K}$ states needed for the $B\to K^{*}$ and $B\to\phi$ form factors, respectively. In this respect, let us mention an earlier paper [65] where the form factors of $B$ transitions to the scalar dimeson ($K\pi$ and $2\pi$) were obtained from LCSRs.

The $B$-meson transition form factors obtained from the LCSRs with light-meson DAs are mainly used to determine the modulus of the CKM matrix element $V_{ub}$ from the data on exclusive semileptonic $b\to u\ell\nu_{\ell}$ decays, predominantly using the $B\to\pi\ell\nu_{\ell}$ decay, but also the $B_{s}\to K\ell\nu_{\ell}$, $B\to\rho\ell\nu_{\ell}$ and $B\to\omega\ell\nu_{\ell}$ decays.

One way to extract $|V_{ub}|$ is to use the LCSR result for the $B\to\pi$ form factor $f^{+}_{B\pi}(q^{2})$ and integrate the predicted differential $B\to\pi\ell\nu_{\ell}$ width ($\ell=e,\mu$) over the LCSR validity region $0<q^{2}<q^{2}_{max}$ (see e.g. Refs. [34, 41]). The other way is to extrapolate the form factor up to the zero recoil point of the pion , $q^{2}=(m_{B}-m_{\pi})^{2}$ using the $z$-parametrization, e.g. the one in Ref. [35]. A combined fit to both LCSR and lattice QCD predictions can also be performed, to achieve the theoretically most accurate form factor in the whole semileptonic region of $q^{2}$.

The recent determination of $|V_{ub}|$ using the combined approach to the $B\to\pi$ form factors was performed in Ref. [46]. An independent extraction of $|V_{ub}|$ from $B\to\rho(\omega)\ell\nu_{\ell}$ decays was performed in Ref. [66] using the LCSR form factors from Ref. [53]. In Ref. [67], the channels with vector mesons were added to the combined analysis of the $B\to\pi\ell\nu_{\ell}$ decay, employing an advanced statistical tool [68]. Separate process-specific $|V_{ub}|$ values were obtained: $|V_{ub}|_{B\to\pi}=(3.79\pm 0.15)\cdot 10^{-3}$,  $|V_{ub}|_{B\to\rho}=(2.92^{+0.28}_{-0.25})\cdot 10^{-3}$, $|V_{ub}|_{B\to\omega}=(3.00^{+0.38}_{-0.32})\cdot 10^{-3}$, with an overall average:

$$|V_{ub}|=(3.59^{+0.13}_{-0.12})\cdot 10^{-3}\,.$$

(18)

The observed difference between the $|V_{ub}|$ values extracted using $B\to\pi$ and $B\to\rho(\omega)$ form factors demands further investigation, in particular, an update of LCSRs for $B\to V$ form factors is desirable. The point is that the twist-3 NLO contributions to LCSRs for $B\to V$ form factors are not available in analytical form and should be completely recalculated and reassessed [69]. It would also be useful to quantify the nonresonant background for $B\to\rho(\omega)\ell\nu_{\ell}$ decays, along the lines presented in Ref. [56] as already discussed in Section 2.3.

Returning to the CKM matrix elements, quite recently, the combined analysis of the $B_{s}\to K$ form factors was done in Ref. [47] and used to extract the ratio $|V_{ub}/V_{cb}|$ from the data on the $B_{s}\to K\ell\nu_{\ell}$ and $B_{s}\to D_{s}\ell\nu_{\ell}$ decays. A similar analysis was also done in Ref. [70].

The $B\to K^{*}$ form factors obtained from LCSRs were also extensively used in the exploration of various observables in the $B\to K^{*}\ell^{+}\ell^{-}$ and $B\to K^{*}\gamma$ FCNC decays. Already in Ref. [71] and after that in Ref. [53] these observables were extended from SM to various models of new physics. An independent determination of Wolfenstein parameters of CKM matrix from a combination of observables in $B_{(s)}\to P\ell^{+}\ell^{-}$ decays was suggested in Ref. [41].

In the current tests of the lepton flavour universality (LFU) in semileptonic $B$ decays, the LCSR results for the form factors, combined with the lattice QCD results, were also employed. The main goal was to obtain predictions of LFU ratios, such as $R_{\pi}$ in Ref. [46] or $R_{\rho}$ and $R_{\omega}$ in Ref. [66], where also a possible influence of new physics on the polarizations and asymmetries in $B\to(\rho,\omega)\ell\nu_{\ell}$ decays was examined. Note that for LFU tests involving $B\to P\tau\nu_{\tau}$ decays, the scalar form factors $f_{BP}^{0}(q^{2})$ are essential. The latter ones, as well as the tensor form factors $f_{BP}^{T}(q^{2})$, are not always available from the lattice QCD, enhancing even more the importance of their LCSR calculations.

Based on the form factors extracted from LCSRs, in Ref. [67] also the effects of possible new physics were analysed in the framework of the Weak Effective Theory (WET). Similarly, the LCSR results for $B\to\pi,\rho,\omega$ form factors from Refs [18] and [53] were used in Ref. [72] to examine new physics interpretation in the Standard Model Effective Field Theory (SMEFT).

The underlying idea of this version of LCSRs suggested in Ref. [73] and developed further in Ref. [74] was to swap the meson state and interpolating current in the initial correlator. E.g., for the $B\to\pi$ transition one has to consider, instead of Eq. (1), the $B$-to-vacuum correlator with the pion interpolating current:

$$F_{\mu\nu}^{(B)}(p,q)=i\int d^{4}x~{}e^{ip\cdot x}\langle 0|T\left\{\bar{d}(x)\gamma_{\mu}\gamma_{5}u(x),\bar{u}(0)\gamma_{\nu}b(0)\right\}|\bar{B}(p+q)\rangle\,.$$

(19)

This opens up a possibility to obtain form factors of $B$-meson transitions into any light or charmed meson by just switching from one interpolating current to another, and, correspondingly, applying quark-hadron duality in that channel. There is no need to introduce a separate set of DAs for a particular light meson. New nonperturbative objects that emerge in the OPE of the correlator (19), after contracting the virtual $u$-quark fields, are the universal DAs of $B$ meson. However, these DAs can only be systematically defined in the infinite heavy-quark mass limit, in the framework of heavy-quark effective theory (HQET). Hence, we have to replace the $B$-meson state in the correlator by a state $B_{v}$ with a definite velocity $v=(p+q)/m_{B}$, and the $b$-quark field by an effective HQET field $h_{v}$.

The leading twist-2 and subleading twist-3 $B$-meson DAs ³³3Note that the concept of twist for the $B$ meson DAs introduced and explained in detail in Ref. [75] differs from the one for the light-meson DAs., denoted, respectively as $\phi_{+}^{B}$ and $\phi_{-}^{B}$ have been defined originally in Ref. [76] (see also Ref. [77] and the review [78]):

	$\displaystyle\langle 0|\bar{d}_{\beta}(x)[x,\!0]h_{v,\,\alpha}(0)|\bar{B}_{v}\rangle$
	$\displaystyle\hskip 5.69046pt\!=\!-\frac{if_{B}m_{B}}{4}\!\!\!\int\limits_{0}^{\infty}\!\!d\omega\,e^{-i\omega v\cdot x}\left[(1\!+\!/\!\!\!v)\!\left\{\!\phi^{B}_{+}(\omega)\!-\!\frac{\phi_{+}^{B}(\omega)\!-\!\phi_{-}^{B}(\omega)}{2v\cdot x}/\!\!\!x\!\right\}\!\gamma_{5}\right]_{\alpha\beta},$		(20)

where the normalization constant is taken equal to the physical $B$-meson decay constant (as usually done in LCSRs at LO considered in this subsection). The HQET distribution amplitudes also serve as an indispensable ingredient for the theory descriptions of exclusive $B$-meson decay matrix elements in the QCD factorization framework [79].

The LCSR derivation starting from the HQET limit of the correlator (19) basically repeats the procedure described in Section 2. Hadronic dispersion relation of the correlator in the pion channel (in the invariant variable $p^{2}$) is now used, and one has to isolate the pion pole contribution with the help of duality approximation, introducing the effective threshold $s_{0}^{\pi}$. The resulting sum rule for the $B\to\pi$ vector form factor has a surprisingly simple expression in the lowest twist-3 approximation, after Borel transformation $p^{2}\to M^{2}$:

$\displaystyle f^{+}_{B\pi}(0)=\frac{f_{B}}{f_{\pi}\,m_{B}}\int\limits_{0}^{s_{0}^{\pi}}dse^{-s/M^{2}}\phi^{B}_{-}(s/m_{B})\,.$

(21)

The r.h.s. of this sum rule becomes significantly more involved after adding higher-twist (power suppressed) contributions, including those from the quark-antiquark-gluon B-meson DAs. The first calculations for $B\to\pi$ and $B\to K,\rho,K^{*}$ form factors in Ref. [74] were superseded by more recent results for these form factors in Refs. [80] and [81]. In both analyses the set of higher-twist DAs worked out in Ref. [75] was taken into account. Note that there is an important difference in the achieved accuracy: in Ref. [80] both two- and three-particle DAs were included (the latter up to twist-six level) whereas in Ref. [81] only the two-particle twist-five contributions were taken into account, which is not consistent from the point of view of HQET equations of motion [75] (see Refs. [82, 83] for further discussions).

The LCSRs described in this section were obtained in the LO, that is, at the zeroth order in $\alpha_{s}$, albeit with the corrections beyond the leading power (LP). Therefore, renormalization and scale dependence for $B$-meson DAs, with their specific evolution discovered in Ref. [84] and discussed in some detail in the next subsection are not fully used here. As a result, the accuracy of OPE in these sum rules is still less than for the LCSRs with light-meson DAs. Also the key parameter of leading-twist DA – the inverse moment $1/\lambda_{B}=\int_{0}^{\infty}d\omega\phi_{+}^{B}(\omega)/\omega$ – is not yet determined with a sufficient accuracy, in particular, this moment is not yet accessible in lattice QCD (see, however, [85] for interesting discussions in this respect). The input interval of this parameter employed in LCSRs for $B_{(s)}$ meson is usually taken from the two-point QCD sum rules worked out in [86] (see also Ref. [87] and other independent estimates in [88, 89]). The behaviour of the DAs at $\omega\to 0$ is model-independent [75]. The actual form of these DAs at large $\omega$ is unimportant at tree level, since in the sum rules such as Eq. (21) the duality interval cuts off the region $\omega>s_{0}^{\pi}/m_{B}$. One popular choice of $\omega$-dependence is the exponential model from Ref. [76] (see also Refs.  [84, 90, 91] ).

The two additional parameters determining all higher twist DAs in this model are $\lambda_{B}$ and $\lambda_{H}$, parametrizing the quark-antiquark-gluon $B$-to-vacuum matrix elements in HQET. Their estimates from independent two-point sum rules [92, 93] still yield large uncertainty intervals.

Numerically, the form factors of $B$ meson transitions to light mesons obtained from the LCSRs with $B$ meson DAs (see the Tables in the next subsection) agree with the ones calculated from the sum rules with light-meson DAs, but only within their larger uncertainties. Apart from narrowing down the uncertainty intervals of the inputs, there are two important questions to be addressed for a further improvement of this version of LCSRs. The first question, already raised in Ref. [74], is the size of inverse heavy-quark mass effects which are implicitly neglected when one starts from the correlation function (19) in full QCD with an on-shell $B$-meson state and then takes the HQET limit of that state. A pragmatic argument in favour of smallness of such correction is a relatively good agreement between form factors obtained with both LCSR methods with light and $B$-meson DAs. The second question is the size of NLO, $O(\alpha_{s})$ corrections to the correlator (19). The answer is found using an alternative formulation of this method to be discussed below.

Finally, we would like to comment on the possibility to extend the method of LCSR with $B$-meson DAs to $D$-meson semileptonic form factors, introducing the $D$-meson DAs. This however implies using HQET for the $c$-quark and $D$-meson, hence, considerably limiting the accuracy in phenomenological applications to the $D$-meson decays.

The method of LCSRs with $B$-meson DAs was independently formulated [94, 95] in the framework of soft-collinear effective theory (SCET) (see [96] where this theory is introduced in the context of heavy-to-light transitions). To derive the SCET sum rules for our standard example – the $B\to\pi$ form factors – the following definition of the correlator is used (see e.g. [88]):

	$\displaystyle{\cal F}_{\mu}^{(B)}(n\cdot p,\bar{n}\cdot p)=\int d^{4}x\,e^{ip\cdot x}\langle 0|{\rm T}\left\{\bar{d}(x)\,\not{n}\gamma_{5}\,u(x),\,\bar{u}(0)\,\gamma_{\mu}\,b(0)\right\}|\bar{B}(p_{B})\rangle$
	$\displaystyle={\cal F}^{(B)}(n\cdot p,\bar{n}\cdot p)\,n_{\mu}+\tilde{\cal F}^{(B)}(n\cdot p,\bar{n}\cdot p)\,\bar{n}_{\mu}\,,$		(22)

where in the $B$-meson rest frame the two light-cone vectors $n_{\mu}$ and $\bar{n}_{\mu}$ are introduced, such that $n\cdot v=\bar{n}\cdot v=1$, $v_{\perp}=0$ and $n\cdot\bar{n}=2$, and a power-counting scheme for the four-momentum of the pion interpolation current is employed:

$\displaystyle n\cdot p\sim{\cal O}(m_{b})\,,\qquad\bar{n}\cdot p\sim{\cal O}(\Lambda_{\rm QCD})\,.$

(23)

In this scheme, a generic momentum $P_{\mu}\equiv(n\cdot P,\bar{n}\cdot P,P_{\perp})$ is split into the three different momentum modes: hard (h), hard-collinear (hc) and soft (s), with the scaling behavior, respectively, $P_{h,\,\mu}\sim{\cal O}(1,1,1)$, $P_{hc,\,\mu}\sim{\cal O}(1,\lambda,\lambda^{1/2})$, $P_{s,\,\mu}\sim{\cal O}(\lambda,\lambda,\lambda)$. The expansion parameter $\lambda$ scales as $\Lambda/m_{b}$ where $\Lambda$ is a typical hadronic scale. The momentum transfer $q$ in the large and intermediate recoil region of the pion (accessible to LCSRs) belongs to the hard or hard-collinear mode.