$B \rightarrow T$ transition form factors in light-cone sum rules

TL;DR

This paper calculates $B$-meson to tensor meson transition form factors using light-cone sum rules, including higher-twist corrections, and explores their implications for specific decay processes.

Contribution

It introduces the first analytical inclusion of higher-twist corrections up to twist four in $B$ to tensor meson form factors using light-cone sum rules.

Findings

Higher-twist effects significantly influence the form factors.

Results impact predictions for ${B} o K_2^{*} ext{(gamma, semileptonic, dileptonic)}$ decays.

Provides new theoretical inputs for flavor physics analyses.

Abstract

We present a new calculation of the semileptonic tree-level and flavor-changing neutral current form factors describing $B$-meson transitions to tensor mesons $T=D_2^*,K_2^*,a_2,f_2$ ($J^{P}=2^{+}$). We employ the QCD Light-Cone Sum Rules approach with $B$-meson distribution amplitudes. We go beyond the leading-twist accuracy and provide analytically, for the first time, higher-twist corrections for the two-particle contributions up to twist four terms. We observe that the impact of higher twist terms to the sum rules is noticeable. We study the phenomenological implications of our results on the radiative ${B} \to K_2^{*}\gamma$ and semileptonic ${B} \to D_2^* \ell {\bar \nu}_\ell$, ${B} \to K_2^{*}\ell^+\ell^-$ decays.

Tables

Table 1. Table 1: In this table, σ ρ ​ { q } ≡ σ ρ ​ α ​ q α superscript 𝜎 𝜌 𝑞 superscript 𝜎 𝜌 𝛼 subscript 𝑞 𝛼 \sigma^{\rho\{q\}}\equiv\sigma^{\rho\alpha}q_{\alpha} , and the position-space covariant derivative D ν superscript 𝐷 𝜈 D^{\nu} is defined in Eq. ( 10 ).

Transition	$j_{\text{int}}^{\mu \nu }$	$j_{\text{weak}}^{\rho }$	Form factor
${\overline{B}}^0\to D_2^{\ast }{}_{}{}^{+}$	$\frac{i}{2}\overline{d}[{\gamma }^{\mu }\overleftrightarrow{D^{\nu }}+\mu \leftrightarrow \nu ]c$	$\overline{c}{\gamma }^{\rho }{h}_v$	${\tilde{V}}^{B\to D_2^{\ast }}$
		$\overline{c}{\gamma }^{\rho }{\gamma }_5{h}_v$	${\tilde{A}}_0^{B\to D_2^{\ast }},{\tilde{A}}_1^{B\to D_2^{\ast }},{\tilde{A}}_2^{B\to D_2^{\ast }}$
		$\overline{c}{\sigma }^{\rho \left\{q\right\}}{h}_v$	${\tilde{T}}_1^{B\to D_2^{\ast }}$
		$\overline{c}{\sigma }^{\rho \left\{q\right\}}{\gamma }_5{h}_v$	${\tilde{T}}_2^{B\to D_2^{\ast }},{\tilde{T}}_3^{B\to D_2^{\ast }}$
${\overline{B}}^0\to K_2^{\ast }{}_{}{}^0$	$\frac{i}{2}\overline{d}[{\gamma }^{\mu }\overleftrightarrow{D^{\nu }}+\mu \leftrightarrow \nu ]s$	$\overline{s}{\gamma }^{\rho }{h}_v$	${\tilde{V}}^{B\to K_2^{\ast }}$
		$\overline{s}{\gamma }^{\rho }{\gamma }_5{h}_v$	${\tilde{A}}_0^{B\to K_2^{\ast }},{\tilde{A}}_1^{B\to K_2^{\ast }},{\tilde{A}}_2^{B\to K_2^{\ast }}$
		$\overline{s}{\sigma }^{\rho \left\{q\right\}}{h}_v$	${\tilde{T}}_1^{B\to K_2^{\ast }}$
		$\overline{s}{\sigma }^{\rho \left\{q\right\}}{\gamma }_5{h}_v$	${\tilde{T}}_2^{B\to K_2^{\ast }},{\tilde{T}}_3^{B\to K_2^{\ast }}$
${\overline{B}}^0\to a_2^{+}$	$\frac{i}{2}\overline{d}[{\gamma }^{\mu }\overleftrightarrow{D^{\nu }}+\mu \leftrightarrow \nu ]u$	$\overline{u}{\gamma }^{\rho }{h}_v$	${\tilde{V}}^{B\to a_2}$
		$\overline{u}{\gamma }^{\rho }{\gamma }_5{h}_v$	${\tilde{A}}_0^{B\to a_2},{\tilde{A}}_1^{B\to a_2},{\tilde{A}}_2^{B\to a_2}$
		$\overline{u}{\sigma }^{\rho \left\{q\right\}}{h}_v$	${\tilde{T}}_1^{B\to a_2}$
		$\overline{u}{\sigma }^{\rho \left\{q\right\}}{\gamma }_5{h}_v$	${\tilde{T}}_2^{B\to a_2},{\tilde{T}}_3^{B\to a_2}$
$B\to f_2^0$	$\frac{i}{2\sqrt{2}}\overline{u}[{\gamma }^{\mu }\overleftrightarrow{D^{\nu }}+\mu \leftrightarrow \nu ]u$ + $u\leftrightarrow d$	$\overline{u}\left(\overline{d}\right){\gamma }^{\rho }{h}_v$	${\tilde{V}}^{B\to f_2}$
		$\overline{u}\left(\overline{d}\right){\gamma }^{\rho }{\gamma }_5{h}_v$	${\tilde{A}}_0^{B\to f_2},{\tilde{A}}_1^{B\to f_2},{\tilde{A}}_2^{B\to f_2}$
		$\overline{u}\left(\overline{d}\right){\sigma }^{\rho \left\{q\right\}}{h}_v$	${\tilde{T}}_1^{B\to f_2}$
		$\overline{u}\left(\overline{d}\right){\sigma }^{\rho \left\{q\right\}}{\gamma }_5{h}_v$	${\tilde{T}}_2^{B\to f_2},{\tilde{T}}_3^{B\to f_2}$

Table 2. Table 2: Tensor mesons’ decay constants used in our numerical results.

Tensor meson	$f_2^0$	$a_2$	$K_2^{\ast }$	$D_2^{\ast }$
$f_T$	$0.040$ Aliev and Shifman (1982)	$0.0406\pm 0.0023$ Cheng et al. (2010)	$0.050\pm 0.002$ Aliev et al. (2010)	$0.0185\pm 0.0020$ Wang and Di (2014)

Table 3. Table 3: Results for the fit parameters α n F superscript subscript 𝛼 𝑛 𝐹 \alpha_{n}^{F} by fitting our LCSRs results for B → T → 𝐵 𝑇 B\to T form factors to Eq. ( 26 ).

Form Factor	${\alpha }_0$	${\alpha }_1$
$B\to D_2^{\ast }$
${\tilde{V}}^{B\to D_2^{\ast }}$	${1.45}_{-0.35}^{+0.31}$	$-{8.63}_{-1.26}^{+1.99}$
${\tilde{A}}_0^{B\to D_2^{\ast }}$	${1.13}_{-0.17}^{+0.11}$	$-{6.25}_{-0.17}^{+0.77}$
${\tilde{A}}_1^{B\to D_2^{\ast }}$	${1.10}_{-0.23}^{+0.21}$	$-{4.57}_{-0.16}^{+0.75}$
${\tilde{A}}_2^{B\to D_2^{\ast }}$	${1.03}_{-0.33}^{+0.38}$	$-{6.16}_{-2.17}^{+2.13}$
${\tilde{T}}_1^{B\to D_2^{\ast }}$	${1.15}_{-0.25}^{+0.23}$	$-{6.67}_{-0.83}^{+1.34}$
${\tilde{T}}_2^{B\to D_2^{\ast }}$	${1.15}_{-0.25}^{+0.23}$	$-{3.86}_{-0.22}^{+0.55}$
${\tilde{T}}_3^{B\to D_2^{\ast }}$	${0.44}_{-0.18}^{+0.23}$	$-{2.18}_{-1.27}^{+0.98}$
$B\to K_2^{\ast }$
${\tilde{V}}^{B\to K_2^{\ast }}$	${0.22}_{-0.08}^{+0.11}$	$-{0.90}_{-0.50}^{+0.37}$
${\tilde{A}}_0^{B\to K_2^{\ast }}$	${0.30}_{-0.05}^{+0.06}$	$-{1.23}_{-0.23}^{+0.23}$
${\tilde{A}}_1^{B\to K_2^{\ast }}$	${0.19}_{-0.07}^{+0.09}$	$-{0.46}_{-0.25}^{+0.19}$
${\tilde{A}}_2^{B\to K_2^{\ast }}$	${0.11}_{-0.06}^{+0.05}$	$-{0.40}_{-0.16}^{+0.23}$
${\tilde{T}}_1^{B\to K_2^{\ast }}$	${0.19}_{-0.06}^{+0.09}$	$-{0.75}_{-0.38}^{+0.28}$
${\tilde{T}}_2^{B\to K_2^{\ast }}$	${0.19}_{-0.06}^{+0.09}$	$-{0.17}_{-0.12}^{+0.09}$
${\tilde{T}}_3^{B\to K_2^{\ast }}$	${0.09}_{-0.04}^{+0.06}$	$-{0.27}_{-0.22}^{+0.15}$
$B\to a_2^{+}$
${\tilde{V}}^{B\to a_2^{+}}$	${0.18}_{-0.07}^{+0.12}$	$-{0.70}_{-0.52}^{+0.31}$
${\tilde{A}}_0^{B\to a_2^{+}}$	${0.30}_{-0.05}^{+0.06}$	$-{1.21}_{-0.26}^{+0.22}$
${\tilde{A}}_1^{B\to a_2^{+}}$	${0.16}_{-0.05}^{+0.09}$	$-{0.33}_{-0.26}^{+0.15}$
${\tilde{A}}_2^{B\to a_2^{+}}$	${0.07}_{-0.03}^{+0.08}$	$-{0.15}_{-0.31}^{+0.06}$
${\tilde{T}}_1^{B\to a_2^{+}}$	${0.15}_{-0.05}^{+0.09}$	$-{0.59}_{-0.39}^{+0.23}$
${\tilde{T}}_2^{B\to a_2^{+}}$	${0.15}_{-0.05}^{+0.09}$	$-{0.10}_{-0.12}^{+0.07}$
${\tilde{T}}_3^{B\to a_2^{+}}$	${0.07}_{-0.03}^{+0.06}$	$-{0.19}_{-0.22}^{+0.12}$
$B\to f_2$
${\tilde{V}}^{B\to f_2}$	${0.11}_{-0.05}^{+0.07}$	$-{0.42}_{-0.32}^{+0.20}$
${\tilde{A}}_0^{B\to f_2}$	${0.20}_{-0.04}^{+0.04}$	$-{0.80}_{-0.17}^{+0.15}$
${\tilde{A}}_1^{B\to f_2}$	${0.10}_{-0.04}^{+0.06}$	$-{0.20}_{-0.16}^{+0.10}$
${\tilde{A}}_2^{B\to f_2}$	${0.04}_{-0.01}^{+0.05}$	$-{0.07}_{-0.20}^{+0.02}$
${\tilde{T}}_1^{B\to f_2}$	${0.10}_{-0.04}^{+0.05}$	$-{0.36}_{-0.24}^{+0.15}$
${\tilde{T}}_2^{B\to f_2}$	${0.10}_{-0.04}^{+0.05}$	$-{0.06}_{-0.08}^{+0.05}$
${\tilde{T}}_3^{B\to f_2}$	${0.04}_{-0.02}^{+0.04}$	$-{0.11}_{-0.13}^{+0.08}$

Table 4. Table 4: Comparison of our form factor results at q 2 = 0 superscript 𝑞 2 0 q^{2}=0 with existing results in the literature.

Form Factor	This work	Literature
${\tilde{V}}^{B\to D_2^{\ast }}\left(0\right)$	${1.45}_{-0.35}^{+0.31}$	$-0.41\pm 0.12$ Azizi et al. (2013)
${\tilde{A}}_0^{B\to D_2^{\ast }}\left(0\right)$	${1.13}_{-0.17}^{+0.11}$	$-0.12\pm 0.33$ Azizi et al. (2013)
${\tilde{A}}_1^{B\to D_2^{\ast }}\left(0\right)$	${1.10}_{-0.23}^{+0.21}$	$0.37\pm 0.10$ Azizi et al. (2013)
${\tilde{A}}_2^{B\to D_2^{\ast }}\left(0\right)$	${1.03}_{-0.33}^{+0.38}$	$1.23\pm 0.41$ Azizi et al. (2013)
${\tilde{T}}_1^{B\to D_2^{\ast }}\left(0\right)$	${1.15}_{-0.25}^{+0.23}$	—
${\tilde{T}}_2^{B\to D_2^{\ast }}\left(0\right)$	${1.15}_{-0.25}^{+0.23}$	—
${\tilde{T}}_3^{B\to D_2^{\ast }}\left(0\right)$	${0.44}_{-0.18}^{+0.23}$	—

Equations

$$f_2\equiv \frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})\cos \delta +s\bar{s}\sin \delta ,f_2^{\prime }\equiv -s\bar{s}\cos \delta +\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})\sin \delta ,$$

$$⟨T(k,\epsilon )|{\bar{q}}_1{\gamma }^{\rho }b|B(p)⟩$$

$$⟨T(k,\epsilon )|{\bar{q}}_1{\gamma }^{\rho }{\gamma }_{5}b|B(p)⟩$$

$$\displaystyle\qquad\qquad-q^{\rho}q^{\beta}\frac{2m_{T}}{q^{2}}\left({\tilde{A}}_{3}-{\tilde{A}}_{0}\right)\bigg{]}\,,$$

$$⟨T(k,\epsilon )|{\bar{q}}_1{\sigma }^{\rho \alpha }{q}_{\alpha }b|B(p)⟩$$

$$⟨T(k,\epsilon )|{\bar{q}}_1{\sigma }^{\rho \alpha }{q}_{\alpha }{\gamma }_{5}b|B(p)⟩$$

$$\displaystyle\qquad\qquad+q^{\beta}\left(q^{\rho}-\frac{q^{2}}{m_{B}^{2}-m_{T}^{2}}(p+k)^{\rho}\right){\tilde{T}}_{3}^{B\to T}\bigg{]}\,,$$

$${\tilde{A}}_3^{B\to T}=\frac{m_B+m_T}{2m_T}{\tilde{A}}_1^{B\to T}-\frac{m_B-m_T}{2m_T}{\tilde{A}}_2^{B\to T}.$$

$${\tilde{A}}_0^{B\to T}(q^2=0)$$

$${\tilde{T}}_1^{B\to T}(q^2=0)={\tilde{T}}_2^{B\to T}(q^2=0).$$

$${\Pi }^{\mu \nu \rho }(q,k)\equiv i\int {\text{d}}^{4}x{e}^{ik\cdot x}⟨0|\mathcal{T}\{j_{\text{int}}^{\mu \nu }(x),j_{\text{weak}}^{\rho }(0)\}|{\bar{B}}_{q_2}(q+k)⟩$$

$$\begin{gathered} j_{\text{int}}^{\mu \nu }=\frac{i}{2}{\bar{q}}_2(x)\left[{\gamma }^{\mu }\overleftrightarrow{D^{\nu }}+\mu \leftrightarrow \nu \right]q_1(x);\overleftrightarrow{D^{\nu }}=\frac{1}{2}\left[\overrightarrow{D^{\nu }}-\overleftarrow{D^{\nu }}\right], \\ \text{vspace}0.15cm \end{gathered}$$

$${\Pi }_{\text{OPE}}^{\mu \nu \rho }(q,k)=\int {\text{d}}^{4}x\int \frac{{\text{d}}^4{p}^{\prime }}{(2\pi {)}^4}{e}^{i(k-p^{\prime })\cdot x}{\left[{\Gamma }_2^{\mu \nu }\frac{/p^{\prime }+m_{q_1}}{m_{q_1}^2-p^{\prime 2}}{\Gamma }_1^{\rho }\right]}_{\alpha \beta }\times ⟨0|{\bar{q}}_2^{\alpha }(x)h_v^{\beta }(0)|{\bar{B}}_{q_2}(v)⟩,$$

$${\Pi }_{\text{had}}^{\mu \nu \rho }(q,k)=\frac{⟨0|j_{\text{int}}^{\mu \nu }(x)|T(k)⟩⟨T(k)|j_{\text{weak}}^{\rho }(0)|{\bar{B}}_{q_2}(q+k)⟩}{m_T^2-k^2}+{\int }_{s_{\text{thr.}}^h}^{\infty }\text{d}s\frac{{\rho }_{\text{had}}^{\mu \nu \rho }(s)}{s-k^2}.$$

$$⟨0|j_{\text{int}}^{\mu \nu }|T(k,\epsilon )⟩={\epsilon }^{\mu \nu }{m}_T^3{f}_T.$$

$${\epsilon }_{\mu \nu }(k){\epsilon }_{\alpha \beta }^{\ast }(k)=\frac{1}{2}{\kappa }_{\mu \alpha }{\kappa }_{\nu \beta }+\frac{1}{2}{\kappa }_{\mu \beta }{\kappa }_{\nu \alpha }-\frac{1}{3}{\kappa }_{\mu \nu }{\kappa }_{\alpha \beta };{\kappa }_{\mu \nu }=-g_{\mu \nu }+\frac{k_{\mu }{k}_{\nu }}{m_T^2}.$$

$${\Pi }_{\text{(V-A)}}^{\mu \nu \rho ,BT}(q,k)$$

$$+{ϵ}_{\mu \rho \alpha \beta }{q}^{\nu }{q}^{\beta }{k}^{\alpha }{\Pi }_{\mathrm{OPE}}^{(3,BT)}(q^2,k^2)+q^{\nu }{g}^{\mu \rho }{\Pi }_{\mathrm{OPE}}^{(4,BT)}(q^2,k^2)+\text{...},$$

$${\Pi }_{\text{Tensor}}^{\mu \nu \rho ,BT}(q,k)$$

$$+{ϵ}_{\mu \rho \alpha \beta }{q}^{\nu }{q}^{\beta }{k}^{\alpha }{\Pi }_{\mathrm{OPE}}^{(7,BT)}(q^2,k^2)+\text{...},$$

$$F^{B\to T}=$$

$$\displaystyle-\Bigg{[}\frac{(-1)^{n-1}}{(n-1)!}e^{(-s(\sigma,q^{2})+m^{2}_{T})/M^{2}}\sum_{j=1}^{n-1}\frac{1}{(M^{2})^{n-j-1}}\frac{1}{s^{\prime}}\left(\frac{\text{d}}{\text{d}\sigma}\frac{1}{s^{\prime}}\right)^{j-1}I_{n}^{(F)}\Bigg{]}_{\sigma=\sigma_{0}}\Bigg{\}}\,,$$

$$s(\sigma ,q^2)=\sigma m_B^2+\frac{m_{q_1}^2-\sigma q^2}{\bar{\sigma }},s^{\prime }(\sigma ,q^2)=\frac{\text{d}s(\sigma ,q^2)}{\text{d}\sigma },\text{with}\bar{\sigma }=1-\sigma .$$

$${\left(\frac{\text{d}}{\text{d}\sigma }\frac{1}{s^{\prime }}\right)}^{n}I(\sigma )\to \left(\frac{\text{d}}{\text{d}\sigma }\frac{1}{s^{\prime }}\left(\frac{\text{d}}{\text{d}\sigma }\frac{1}{s^{\prime }}\dots I(\sigma )\right)\right).$$

$${\sigma }_0=\frac{s_0+m_B^2-q^2-\sqrt{4(m_{q_1}^2-s_0)m_B^2+(m_B^2+s_0-q^2{)}^2}}{2m_B^2},$$

$$I_n^{(F,\text{2p})}(\sigma ,q^2)$$

$${\tilde{A}}_0^{B\to T}$$

$${\tilde{T}}_2^{B\to T}$$

$${\tilde{T}}_3^{B\to T}$$

$$m_{f_2}=1275.5\pm 0.8\text{MeV},m_{a_2}=1318.3_{-0.6}^{+0.5}\text{MeV},$$

$$m_{K_2^{\ast }}=1425.6\pm 1.5\text{MeV},m_{D_2^{\ast }}=2465.4\pm 1.3\text{MeV},$$

$$m_B=5279.55\pm 0.26\text{MeV}.$$

$$m_b(m_b)$$

$$m_s(1\text{GeV})$$

$${\lambda }_B=460\pm 110\text{MeV}\text{cite[cite]@@bibrefAuthors Phrase1YearPhrase2Braun:2003wx@@citephrase(@@citephrase)},{\lambda }_E^2=0.03\pm 0.02{\text{GeV}}^2\text{cite[cite]@@bibrefAuthors Phrase1YearPhrase2Nishikawa:2011qk@@citephrase(@@citephrase)},{\lambda }_H^2=0.06\pm 0.03{\text{GeV}}^2\text{cite[cite]@@bibrefAuthors Phrase1YearPhrase2Nishikawa:2011qk@@citephrase(@@citephrase)}.$$

$$7.2{\text{GeV}}^2$$

$$2.7{\text{GeV}}^2$$

$$2.3{\text{GeV}}^2$$

Full text

B $\to$ tensor meson transition form factors, light-cone QCD sum rules

††preprint: TUM-HEP-1207/19

I Introduction

$B$-meson decays represent a promising area for checking the gauge structure of the Standard Model (SM), looking for physics beyond it, as well as precise determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.

Interest to the $B$ meson decays increases considerably after a number of measurements that deviate from the respective Standard Model predictions. These results are observed in two types of decays:

1. (1)

Decays due to the flavor-changing neutral currents: $b\to s\mu^{+}\mu^{-}$. Discrepancies with the SM predictions are obtained in several observables in $B\to K^{*}\mu^{+}\mu^{-}$ Aaij et al. (2013a, 2016); Abdesselam et al. (2016a); Aaboud et al. (2018); Sirunyan et al. (2018); Khodjamirian et al. (2010); Bobeth et al. (2018) and $B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij et al. (2013b); Geng and Liu (2003); Erkol and Turan (2002); Yilmaz (2008); Chang and Gao (2011) as well as in the measurements of $R_{K(K^{*})}=B(B\to K^{(*)}\mu^{+}\mu^{-})/B(B\to K^{(*)}e^{+}e^{-})$ Aaij et al. (2017, 2014); Bordone et al. (2016). 2. (2)

The charged current $b\to cl\nu$ transitions that take place at tree-level in SM. Tension between theory predictions and experimental data has been observed in the ratios $R_{D^{(*)}}=B(B\to D^{(*)}\tau\bar{\nu}_{\tau})/B(B\to D^{(*)}\ell{\bar{\nu}}_{\ell})$ ($\ell=e,\mu$) Huschle et al. (2015); Lees et al. (2013); Abdesselam et al. (2016b); Amhis et al. (2017) as well as $R_{J\psi}=B(B_{c}\to J/\psi\tau\bar{\nu}_{\tau})/B(B_{c}\to J/\psi\mu\bar{\nu}_{\mu})$ Aaij et al. (2018); Dutta and Bhol (2017).

If these results are confirmed by the forthcoming experiments, it will be an unambiguous discovery of existence of new physics (NP).

With respect to these experimental observations one expects that if NP exists at the quark-level $b\to c$ transition, then such discrepancies should also be seen in $B$-meson transitions to tensor mesons in addition to $B$ decays to pseudo-scalar or vector-mesons111Tests of lepton flavor universality (LFU) regarding the quark-level $b\to c$ transition are not restricted to mesonic decays. For a very recent analysis of the baryonic counter part decay $\Lambda_{b}\to\Lambda_{c}(\to\Lambda\pi)\ell\bar{\nu}$, we refer the reader to see Böer et al. (2019)..

In regard to seeking NP effects, the $B$-meson decays to tensor mesons have the following advantage: tensor mesons have additional polarizations compared to the vector mesons and therefore this could provide additional kinematical quantities that are sensitive to the existence of NP. As a result, $B$-meson decays to tensor mesons could provide a complementary platform to search for new helicity structures, that deviate from the SM ones.

The main ingredients of $B\to T$ transitions are the relevant form factors. In this work, the form factors of $B\to T$ transitions are calculated within the light-cone QCD sum rules (LCSRs) Balitsky et al. (1989); Chernyak and Zhitnitsky (1990) (for reviews see e.g. Colangelo and Khodjamirian (2000)) using $B$-meson Light-Cone Distribution Amplitudes (LCDAs). Note that the light-cone sum rules have successfully been applied to a wide range of problems of hadron physics. The recent applications of LCSRs with heavy meson and heavy baryon distribution amplitudes are discussed in detail in many works (see for example Khodjamirian et al. (2007); Faller et al. (2009); Wang and Shen (2015, 2016); Cheng et al. (2017); Gubernari et al. (2019); Gao et al. (2019); Cheng and Shen (2019); Descotes-Genon et al. (2019) and references therein).

It should be noted that the $B\to T$($J^{P}=2^{+}$) form factors have previously been calculated by several groups using various methods Khosravi and Sadeghi (2016); Bernlochner et al. (2018); Azizi et al. (2013); Emmerich et al. (2018); Yang (2011); Wang (2011a, b); Sharma and Verma (2010); Cheng and Chua (2010a); Cheng et al. (2004); Cheng and Chua (2010b); Safir (2001); Scora and Isgur (1995); Charles et al. (1999); Ebert et al. (2001); Datta et al. (2008). For example, the $B\to f_{2}(1270)$ form factors have recently been calculated in Emmerich et al. (2018) within the LCSRs framework using the $f_{2}(1270)$ meson DAs. Also, for the light tensor meson final states $B\to f_{2},a_{2},K_{2}^{*},f_{2}^{{}^{\prime}}$ the form factor calculation has been carried out previously by Yang (2011) within LCSRs employing tensor-meson DAs, and in Wang (2011a) using perturbative QCD approach. Within three-point QCD sum-rule approach, a sub-set of the relevant form factors under consideration was estimated in Ref. Khosravi and Sadeghi (2016) for $B\to f_{2},a_{2},K_{2}^{*}$ transitions, and in Ref. Azizi et al. (2013) for $B\to D_{2}^{*}$. The LCSRs analysis carried out in Ref. Wang (2011b) computes the relevant form factors for $B\to f_{2},a_{2},K_{2}^{*}$ transitions considering only the $\phi_{+},\bar{\phi}$ $B$-LCDAs with vanishing virtual quark masses regarding the $f_{2},a_{2}$ final states. Our analysis here extends previous works by providing new results for the full set of $B\to T$($D_{2}^{*},K_{2}^{*},a_{2},f_{2}$) transition form factors up to and including twist-four accuracy of $B$-LCSRs as well as takes into account the finite virtual quark mass effects in the results of the form factors. Moreover, we provided results for the tensor form factors in $B\to D_{2}^{*}$ transitions for the first time.

We should further mention that the tensor isosinglet final state $f_{2}(1270)$ considered in this study, in principle, possesses a mixing pattern with the other isosinglet tensor meson of the same quantum numbers $f_{2}^{\prime}(1525)$ in the form

[TABLE]

where the mixing angle $\delta$ has been found to be small indicating that $f_{2}$ could be considered nearly as a pure $\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$ state ($\sim 98.2\%$) while $f_{2}^{\prime}$ is nearly a pure $s\bar{s}$ state (for details, see Refs. Li et al. (2001) and Olive et al. (2014)). We will therefore assume no mixing between $f_{2}$ with $f_{2}^{\prime}$ when studying the $B\to f_{2}$ form factors in this paper (see e.g. Emmerich et al. (2018); Khosravi and Sadeghi (2016) for similar assumptions in regard to analyses of $B\to f_{2}$ form factors).

The outline of our paper is as follows. In Sec. II, the LCSRs for the relevant form factors are derived. Sec. III is devoted to the numerical analysis of the sum rules obtained in Sec. II. In Sec. IV, we study the phenomenological implications of our form factor results on the radiative ${B}\to K_{2}^{*}\gamma$ and semileptonic ${B}\to D_{2}^{*}\ell{\bar{\nu}}_{\ell}$, ${B}\to K_{2}^{*}\ell^{+}\ell^{-}$ decays within the context of SM. Sec. V contains a summary of our findings. Last, in Appendix A we collect the two-particle DAs of the $B$-meson and in Appendix B we present analytical expressions for the coefficient functions needed for the determination of the relevant form factors.

II Theoretical Framework and Analytical Results for the Form Factors

In general, the $B\to T$ transitions, where $T=D_{2}^{*},K_{2}^{*},a_{2},f_{2}$, can be described by seven form factors ${\tilde{A}}_{0}^{B\to T}$, ${\tilde{A}}_{1}^{B\to T}$, ${\tilde{A}}_{2}^{B\to T}$, ${\tilde{V}}^{B\to T}$, ${\tilde{T}}_{1}^{B\to T}$, ${\tilde{T}}_{2}^{B\to T}$, and ${\tilde{T}}_{3}^{B\to T}$, which are defined in analogy to the $B\to V$ case222Note that for the case of semileptonic tree-level transitions ${B}\to D_{2}^{*}\ell{\bar{\nu}}_{\ell}$, ${\tilde{T}}_{1}^{B\to D_{2}^{*}}$,${\tilde{T}}_{2}^{B\to D_{2}^{*}}$ and ${\tilde{T}}_{3}^{B\to D_{2}^{*}}$ form factors would only be induced by possible NP tensor type operators, which are absent in the SM.:

[TABLE]

where $\varepsilon$ represents the polarization of the final state tensor meson with shorthand notation $\varepsilon_{\beta}=\varepsilon_{\beta\alpha}q^{\alpha}/m_{B}$, and we use $\epsilon_{0123}=+1$. The polarization tensor $\varepsilon_{\beta\alpha}$ is symmetric in its indices and satisfies $\varepsilon_{\beta\alpha}(k)k^{\alpha}=0$. Throughout, $k$ and $p$ represent the final-state tensor meson’s and the $B$-meson’s momentum, respectively, with $q^{2}=(p-k)^{2}$ being the momentum transfer squared.

${\tilde{A}}_{3}^{B\to T}$ is superfluous, because it is correlated with ${\tilde{A}}_{1}^{B\to T}$ and ${\tilde{A}}_{2}^{B\to T}$ form factors as

[TABLE]

The unphysical singularities of the matrix elements defined in Eq. (3) at $q^{2}=0$ are removed by

[TABLE]

Besides, one has the following identity using algebraic relations between $\sigma^{\mu\nu}$ and $\sigma^{\mu\nu}\gamma_{5}$:

[TABLE]

Our starting point is the correlation function

[TABLE]

of two quark currents $j_{\text{int}}^{\mu\nu}=\bar{q}_{2}(x)\Gamma_{2}^{\mu\nu}q_{1}(x)$ and $j_{\text{weak}}^{\rho}(0)=\bar{q}_{1}(0)\Gamma_{1}^{\rho}h_{v}(0)$, where $h_{v}$ denotes the Heavy Quark Effective Theory field instead of a $b$-quark. The spin structures of $\Gamma_{1,2}$ together with various choices of quark flavors $q_{1}$ and $q_{2}$ for the form factors extracted in this paper are given in Table 1.

The interpolating current for tensor mesons (with valence quark content $q_{1}{\bar{q}}_{2}$) is, in general, given by

[TABLE]

where $\overrightarrow{D^{\nu}}=\overrightarrow{\partial^{\nu}}-ig\frac{\lambda^{a}}{2}A_{a}^{\nu}(x)$ and $\overleftarrow{D^{\nu}}=\overleftarrow{\partial^{\nu}}+ig\frac{\lambda^{a}}{2}A_{a}^{\nu}(x)$ with $\partial^{\nu}=\frac{\partial}{\partial x_{\nu}}$. When regard to two-particle contributions, which we are interested in this work, it suffices to take the first terms in the covariant derivatives, because the second terms involving the fields $A^{\nu}(x)$ only contribute to three-particle effects333In a recent comprehensive work with $B$-LCDAs Gubernari et al. (2019) for $B\to P,V$ transitions it has been shown that compared to two-particle contributions, the relative impact of the three-particle contributions to the form-factors is only at percent level or less (for details see Gubernari et al. (2019)). The same conclusion was also drawn e.g. in Ref. Wang (2011b) for $B\to T$ transitions. We therefore feel safe to neglect such effects in the present analysis..

The higher Fock state contributions to the correlation function arise when expanding the position-space virtual-quark $q_{1}$ propagator in $x^{2}$ near the light-cone $x^{2}\simeq 0$. In the present work, we focus on the two-particle contributions, while higher Fock state contributions are beyond our current scope. We summarize the two-particle Operator-Product-Expansion (OPE) contributions as

[TABLE]

where $p^{\prime}=k-l$, and $l$ describes the momentum of the spectator quark inside the $B$-meson with $\alpha$,$\beta$ being spinor indices. In Eq. (11), the $B$-meson to vacuum matrix elements are non-perturbative objects that are expressed in terms of $B$-meson LCDAs, whose explicit definitions are relegated to Appendix A.

The hadronic correlator $\Pi^{\mu\nu\rho}$ reads:

[TABLE]

The decay constant $f_{T}$ is defined via444Note that this definition implies $f_{T}$ to be dimensionless.:

[TABLE]

The spin sum for the tensor mesons is given by:

[TABLE]

The form-factors are extracted by matching independent Lorentz structures appearing in both correlators $\Pi_{\text{OPE}}^{\mu\nu\rho}(q,k)$ and $\Pi_{\text{had}}^{\mu\nu\rho}(q,k)$.

For the particular choice of the weak currents as in Table 1, the correlator $\Pi_{\text{OPE}}^{\mu\nu\rho}(q,k)$ can be split as:

[TABLE]

where the ellipsis stand for terms involving other Lorentz structures. The extraction of the $B\rightarrow T$ form factors is then achieved as follows:

• •

${\tilde{V}}$: we considered terms with Lorentz-structure $\epsilon_{\mu\rho\alpha\beta}q^{\nu}q^{\beta}k^{\alpha}$ in Eq. (II).

• •

${\tilde{A}}_{1}$: we considered terms with Lorentz-structure $q^{\nu}g^{\mu\rho}$ in Eq. (II).

• •

${\tilde{A}}_{2}$: we considered terms with Lorentz-structure $k^{\rho}q^{\mu}q^{\nu}$ in Eq. (II).

• •

$({\tilde{A}}_{3}-{\tilde{A}}_{0})$: in this case, the form factors ${\tilde{A}}_{0},{\tilde{A}}_{2}$ and ${\tilde{A}}_{3}$ possess some common Lorentz structures. Hence, we define a combined term as ${\tilde{A}}_{023}=\frac{{\tilde{A}}_{2}}{m_{B}+m_{T}}+\frac{2m_{T}({\tilde{A}}_{3}-{\tilde{A}}_{0})}{q^{2}}$, and then extract ${\tilde{A}}_{023}$ by considering terms with Lorentz-structure $q^{\rho}q^{\mu}q^{\nu}$ in Eq. (II).

• •

${\tilde{T}}_{1}$: we considered terms with Lorentz-structure $\epsilon_{\mu\rho\alpha\beta}q^{\nu}q^{\beta}k^{\alpha}$ in Eq. (16).

• •

${\tilde{T}}_{2}$ and ${\tilde{T}}_{3}$: in this case, the form factors ${\tilde{T}}_{2}$ and ${\tilde{T}}_{3}$ possess some common Lorentz structures. We, therefore, define the combination terms ${\tilde{T}}_{23A},{\tilde{T}}_{23B}$ as in Eq. (22)-Eq. (23), from which we then extract ${\tilde{T}}_{23A}({\tilde{T}}_{23B})$ by considering terms with Lorentz-structure $k^{\rho}q^{\mu}q^{\nu}(q^{\rho}q^{\mu}q^{\nu})$ in Eq. (16).

The choice of these structures is dictated by the fact that they contain contributions coming purely from tensor mesons.

Following the formulation introduced in Ref. Gubernari et al. (2019), we write down the sum rule for all the $B\to T$ form factors in a form of a master-formula555Our results in this work provide additional ingredients to the master-formula introduced in Ref. Gubernari et al. (2019) to also include the $B\to T$ form factors at the same twist accuracy of the $B$-LCDAs. For details on the derivation of this formula we refer the reader to see Ref. Gubernari et al. (2019). as:

[TABLE]

where

[TABLE]

In Eq. (17), $\chi=\sqrt{2}\,\left(\chi=1\right)$ for the light unflavored states $f_{2}^{0},a_{2}^{0}$ (for other states), and the differential operator is understood to act as

[TABLE]

Using the first relation in Eq. (18) one obtains

[TABLE]

where $s_{0}$ is an effective threshold parameter to be determined and supplied as an input.

The two-particle LCDAs appear in the definitions of the functions $I_{n}^{(F)}$ Gubernari et al. (2019):

[TABLE]

with $\sigma=\omega/m_{B}$ in Eq. (20). The analytical expressions for the normalization factors $K^{(F)}$ together with the matching coefficients $C^{(F,\psi_{\text{2p}})}$ for the considered $B\to T$ transition form factors are relegated to Appendix B.

We provide results for $F={\tilde{V}}^{B\to T},{\tilde{A}}_{1}^{B\to T},{\tilde{A}}_{2}^{B\to T},{\tilde{A}}_{30}^{B\to T},{\tilde{T}}_{1}^{B\to T},{\tilde{T}}_{23A}^{B\to T}$, and ${\tilde{T}}_{23B}^{B\to T}$. The remainder of the form factors ${\tilde{A}}_{0}^{B\to T},{\tilde{T}}_{2}^{B\to T}$ and ${\tilde{T}}_{3}^{B\to T}$ are then simply obtained using

[TABLE]

Further, we give our results for generic final state tensor meson $T(q_{1}\bar{q}_{2})$, where $q_{1}=c,s,u$  ($q_{1}=u(d)$) for ${D_{2}^{*}}^{+},{K_{2}^{*}}^{0},a_{2}^{+}$ ($f_{2}^{0}$), respectively666The theoretical approach presented in this work, together with our form factor results, is generic and can also be readily applied to other tensor mesons with $J^{P}=2^{+}$ by making obvious replacements..

The analytical results presented in this work for $C^{(F^{B\to T},\psi_{\text{2p}})}_{n}$ coefficients of Eq. (20) constitute the first complete results up to twist four accuracy of $B$-LCDAs for the two-particle Fock state contributions to the correlation function. As a result, our present $B\to T$ form factor results improve upon previous results in the literature.

At this stage, a remark on our form factor results is in order. We compared our analytical results related to the two-particle contributions at the leading-twist limit to those of Ref. Wang (2011b). We observe the followings: first, we see that the surface-term contributions777Surface-terms arise after performing continuum subtraction. We observed and stress that the numerical impact of the surface terms on the form factor results could be sizable. given in Eq. (17) of our paper have not been taken into account in the work of Wang (2011b). Nonetheless, when we still compare our analytical results to Wang (2011b), after also dropping the mentioned surface-term effects in our results, we then reproduce the analytical results for their form factors called $V,A_{1},T_{1},{\tilde{A}}_{3}$ and ${\tilde{T}}_{3}$. Next, for ${\tilde{A}}_{2}$ of Ref. Wang (2011b) we reproduce their results for $\phi_{+}$ terms, while for the ${\bar{\phi}}$ terms we have a disagreement. Last, for the ${T}_{2}$ form factor of Ref. Wang (2011b) we have a complete disagreement. The disagreement in the ${T}_{2}$ form factor of Ref. Wang (2011b) is particularly interesting because while in our case the condition ${\tilde{T}}_{1}^{B\to T}(0)={\tilde{T}}_{2}^{B\to T}(0)$ is exactly fulfilled (as required by equation-of-motion conditions), the analytical results given in Eqs. 20–21 of Ref. Wang (2011b) (arXiv v6) for these two form factors seem not to satisfy this condition.

III Numerical Illustrations

III.1 Input

In this section we collect the input parameters used in our numerical estimates. We use up-to-date input parameters.

The meson masses entering our numerics are quoted from the latest PDG averages Tanabashi et al. (2018):

[TABLE]

Moreover, the quark masses $m_{q_{1}}$($q_{1}=c,s,u(d)$) appearing in the $C^{(F,\psi_{\text{2p}})}_{n}$ coefficients of Appendix B together with ${b}$-quark mass are defined in $\overline{\text{MS}}$ scheme, for which we use Tanabashi et al. (2018)

[TABLE]

$B$-meson decay constant is taken from the currently most precise lattice-QCD analysis $f_{B}=189.4\pm 1.4\,\text{MeV}$ Bazavov et al. (2018), on the other side the tensor mesons’ decay constants used in our numerical results are quoted in Table 2.

For the non-perturbative parameters entering the explicit expressions of $B$-LCDAs we use:

[TABLE]

III.2 LCSRs Results

We obtained results for the full set of $B\to T$ form factors within LCSRs up to $q^{2}=0$ $\text{GeV}^{2}$. Our LCSRs results involve, besides other input, free parameters introduced by the method; the continuum threshold $s_{0}$ and Borel mass parameter $M^{2}$, which we determine by fulfilling some physical criteria. First, the working interval of the Borel parameter $M^{2}$ is determined following a standard criteria, i.e demanding that both the power corrections and the continuum contributions in the sum rules should be suppressed. Next, the working region of the continuum threshold is determined by defining so-called first-moments for each form factor and respective final state by differentiating the OPE correlator with respect to $-1/M^{2}$ and normalizing it to itself. These first-moments are then expected to give the mass squares $m_{T}^{2}$ of the respective final state mesons. Imposing $\pm 5\%$ uncertainty on the mass of each final state tensor meson, we were then able to find validity window for $s_{0}$ too.

Based on these discussions, we determined the following working regions for $s_{0}$ and $M^{2}$ for the considered transitions:

[TABLE]

With these working regions for $M^{2}$ and $s_{0}$’s, the smallness of the sub-leading twist-4 contributions compared to the leading twist-2 ones as well as the suppression of higher state contributions are satisfied simultaneously.

In Fig. 1, we illustrate the Borel parameter dependence of all form factors for $B\to D_{2}^{*}$ transition at $q^{2}=0$ based on our LCSRs results including higher twist contributions. Within the chosen interval for $M^{2}$, it is seen that the form factors posses a very mild dependence on $M^{2}$. Similar behavior holds at other negative $q^{2}$ values and for the other final states ($K_{2}^{*},a_{2},f_{2}$) too in their respective $M^{2}$ ranges.

III.3 Parametrization of the form-factors

After determining the best fit intervals of the threshold and the Borel parameters from Sec. III.2, we are now in a position to extrapolate our LCSRs results to the physical region where the phenomenology of the considered $B\to T$ transitions take place. As mentioned in Sec. III.2 we truncate our LCSRs results at $q^{2}=0$ $\text{GeV}^{2}$ for all the tensor meson final states. The extrapolation from the calculated LCSRs input points ($q^{2}\lesssim 0\,\text{GeV}^{2}$) to larger $q^{2}$ values888For instance, the upper $q^{2}$ limit in the case of semi-leptonic decays is $q_{\text{max}}^{2}=(m_{B}-m_{T})^{2}$. is then achieved by parametrizing each of the form factors in a simple pole form with $z$-expansion999We observed that the $B\to T$ transition form factors under consideration, are well fitted by the fit-function of Eq. (26) to first order in $z$. as Bharucha et al. (2016):

[TABLE]

where $z(s)\equiv\frac{\sqrt{t_{+}-s}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-s}+\sqrt{t_{+}-t_{0}}}$, $t_{\pm}=(m_{B}\pm m_{T})^{2}$ and $t_{0}\equiv t_{+}\left(1-\sqrt{1-t_{-}/t_{+}}\right)$. In Eq. (26), $\alpha_{0,1}^{F}$ are the fit parameters that are constrained and presented in Table 3 for each form factor and final state transition separately. Beside, $m_{R,F}$ quantities describe the mass of the resonances associated with the quantum numbers of the respective form factor $F$, whose values can be found in Ref. Gubernari et al. (2019) (for details see Table 5 of Gubernari et al. (2019) and references therein). Note that the kinematical conditions given in Eqs. (7)–(8) impose the following relations among the fit parameters

[TABLE]

which are respected in our numerical results presented in Table 3.

The uncertainties in the values of the form factors of Table 3 are due to the variation of various input parameters involved in the LCSR calculation. In particular, the non-perturbative parameters $\lambda_{B},\,\lambda_{H}^{2},\,\lambda_{E}^{2}$ of $B$-LCDAs together with the continuum threshold $s_{0}$ are mostly responsible for these errors.

In order to estimate the uncertainties of the results presented in this work, such as the form factors, decay rates etc., we followed a Monte Carlo based analysis as performed e.g. in Refs. Leinweber (1997); Erkol and Oka (2008). For this analysis, randomly selected data sets of thousands of data points are generated for any input parameter and its given uncertainty. This led us to determine the mean and corresponding standard deviations of our results.

III.4 Illustrations

The $q^{2}$ dependence of the complete set of $B\to T$ form factors is depicted in Figs. 2,3,4 and 5. In these plots, comparing the leading-twist central results (empty red-circles) with the corresponding new results including twist-four terms (dotted-blue curves) we see that the relative impact of the calculated higher twist terms for the two-particle contributions could be sizable101010For the $B\to T$ form factors under consideration, in the charmed case the relative impact of the calculated higher-twist terms is observed to be relatively less significant when compared to light final state transitions. In our opinion, this could mainly be related to the presence of the heavy mass scale $m_{c}$ in the problem. (in particular for light tensor meson transitions) and therefore should be included in the estimations of the form factors. The magnitude of the central values of the form factors based on the leading-twist terms, is observed to decrease due to the calculated higher twist terms.

In Table 4, we have also compared our present results for the $B\to T$ form factors at $q^{2}=0$ with existing results in literature. Regarding the comparison of $B\to D_{2}^{*}$ form factors with Ref. Azizi et al. (2013), we normalized their results to obtain dimensionless form factors (as in our case), and extracted their value for ${\tilde{A}}_{0}^{B\to D_{2}^{*}}(0)$ using Eq. (6) and Eq. (7). We observe that our numerical results for ${\tilde{V}}^{B\to D^{*}_{2}},{\tilde{A}}_{0}^{B\to D^{*}_{2}},{\tilde{A}}_{1}^{B\to D^{*}_{2}},{\tilde{A}}_{2}^{B\to D^{*}_{2}}$ at $q^{2}=0$, given in the top-left pane of Table 4, severely differ111111A remark on this point is in order. Our definition for the ${\tilde{A}}_{3}-{\tilde{A}}_{0}$ form factor is related to the form factor $b_{-}$ of Ref. Azizi et al. (2013) (arXiv v3) in the following way: $(2m_{T}/q^{2})({\tilde{A}}_{3}-{\tilde{A}}_{0})/m_{B}=-b_{-}$. At $q^{2}=0$, ${\tilde{A}}_{3}-{\tilde{A}}_{0}$ should exactly be zero according to the equation-of-motion condition given in Eq. (7) of our paper. However, the $b_{-}(q^{2}=0)$ form factor of Ref. Azizi et al. (2013) is seen to differ from zero (see Table 2 of the mentioned reference), in explicit violation of this condition. from the corresponding values quoted in Ref. Azizi et al. (2013), which use three-point sum rules. On the other side, concerning the light tensor transition form factors, our numerical results are in agreement with some of the existing results in the literature, which use various calculation methods.

IV Phenomenological analyses

In this section, using our new results for the relevant form factors we give SM predictions for some selected observables. We considered the decay channels $B\to D_{2}^{*}\ell\bar{\nu}$, $B\to K_{2}^{*}\gamma$ and $B\to K_{2}^{*}\ell^{+}\ell^{-}$.

IV.1 SM prediction for $B\to D_{2}^{*}\ell\bar{\nu}$

For the $D_{2}^{*}(2460)$ mode, currently experimental data only on the decay chain ${\cal{B}}(B\rightarrow D_{2}^{*}\ell\overline{\nu}){\cal{B}}(D_{2}^{*}\rightarrow D\pi)$ is available,

[TABLE]

where $\ell=e$ or $\mu$. Despite the current progress in collider physics, there are no data available on $B\to D_{2}^{*}\ell\bar{\nu}$ decays yet. On the other side, regarding the vector $D^{*}$ and the pseudoscalar $D$ modes although the most recent measurements for the ratios ${\cal{B}}({B}\rightarrow D^{(*)}\tau\overline{\nu}_{\tau})/{\cal{B}}({B}\rightarrow D^{(*)}\ell\overline{\nu}_{\ell})$ from the Belle collaboration Abdesselam et al. (2019) alone are compatible with the corresponding SM predictions, when combined with previous experiments the tension between theory and experiment stays around $3.1\sigma$ Amhis et al. (2017), which indicates a violation of lepton flavor universality.

The size of LFU-violation in ${\cal{B}}({B}\rightarrow D_{2}^{*}\tau\overline{\nu}_{\tau})/{\cal{B}}({B}\rightarrow D_{2}^{*}\ell\overline{\nu}_{\ell})$ can therefore be further tested for the charmed tensor meson $D_{2}^{*}(2460)$ case. Since the matrix elements given in Eqs. (2)–(3) are the only ones relevant to $B\to D_{2}^{*}\ell\bar{\nu}$ decays in SM, the differential decay widths of these channels are obtained in terms of ${\tilde{V}}^{B\to D_{2}^{*}}$, ${\tilde{A}}_{0}^{B\to D_{2}^{*}}$, ${\tilde{A}}_{1}^{B\to D_{2}^{*}}$ and ${\tilde{A}}_{2}^{B\to D_{2}^{*}}$ form factors as Wang et al. (2009); Azizi et al. (2013)

[TABLE]

where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$ is the Källén function. We presented the $q^{2}$ dependence of $B\to D_{2}^{*}\ell\bar{\nu}$ form factors up to and including twist-four accuracy in Table 3. Using these results together with the input parameters $G_{F}=1.167\times 10^{-5}\,{\rm GeV}^{-2}$ and $V_{cb}=0.0405$ Olive et al. (2014), we obtain the following predictions

[TABLE]

and

[TABLE]

The variance of our predictions from those of Ref. Azizi et al. (2013) is due to aforementioned discrepancy in the estimation of the form factors (see Table 4).

IV.2 SM predicition for $B\to K_{2}^{*}\gamma$

We continue with a phenomenological analysis on exclusive rare radiative decay of $B$ meson to radially excited tensor meson $K_{2}^{*}(1430)$. The branching ratio of this radiative mode has been measured by several experiments:

[TABLE]

which gives the PDG average of $(1.24\pm 0.24)\times 10^{-5}$ Tanabashi et al. (2018). In the SM, $B\to K_{2}^{*}\gamma$ decay is governed by the electromagnetic dipole operator $\mathcal{O}_{7}$, and its matrix elements between initial $B$ and final $K_{2}^{*}$ states are given in Eqs. (4)–(5). The exclusive decay rate of emission of a real photon ($q^{2}=0$) depends only on the form factor ${\tilde{T}}_{1}^{B\to K_{2}^{*}}$ and is given by Ebert et al. (2000)

[TABLE]

where $V_{ij}$ are the CKM matrix elements, $\alpha$ is the fine-structure constant and $C_{7}(m_{b})$ is the Wilson coefficient associated with $\mathcal{O}_{7}$. Since the inclusive radiative decay $B\to X_{s}\gamma$ is accurately measured by several experiments Koppenburg et al. (2004); Aubert et al. (2006), it is more convenient121212Considering this ratio, one avoids most of the parametric uncertainties. to consider the ratio of exclusive to inclusive branching ratios Ebert et al. (2000)

[TABLE]

where the world average of the inclusive decay is given by the Heavy Flavour Averaging Group Amhis et al. (2017) as ${\cal{B}}(B\to X_{s}\gamma)=(3.32\pm 0.16)\times 10^{-4}$, which is compatible with the theoretical estimate Misiak et al. (2007).

We determine the experimental ratio $R^{\text{exp}}_{K_{2}^{*}}$ by normalizing the experimental world averages of the corresponding decays. Similarly, using the value of the form factor ${\tilde{T}}_{1}^{B\to K_{2}^{*}}$ from Table 3 we obtain our SM prediction for $R^{\text{SM}}_{K_{2}^{*}}$. They read

[TABLE]

which are in agreement within the quoted error budget.

IV.3 SM prediction for $B\to K_{2}^{*}\ell^{+}\ell^{-}$

In the standard model, the effective Hamiltonian governing $B\to K_{2}^{*}\ell^{+}\ell^{-}$ decay is

[TABLE]

with $\mathcal{O}_{i}({\mu})$ being the effective operators and $C_{i}({\mu})$ the respective Wilson coefficients at the renormalization scale $\mu$. Among the ten operators in Eq. (38), $\mathcal{O}_{7}$, $\mathcal{O}_{9}$ and $\mathcal{O}_{10}$

[TABLE]

are the only ones contributing to $B\to K_{2}^{*}\ell^{+}\ell^{-}$. The related Wilson coefficients are discussed thoroughly in the literature (for details, see e.g. Buras and Munz (1995); Buras (2011); Hurth and Nakao (2010) and references therein). In terms of the Wilson coefficients and the form factors defined in Eqs. (2)–(5), the general expression of the differential decay width for $B\to K_{2}^{*}\ell^{+}\ell^{-}$ can be written as Junaid et al. (2012):

[TABLE]

where $\lambda\equiv\lambda(m_{B}^{2},m_{K_{2}^{\ast}}^{2},q^{2})$, and the individual quantities $\mathcal{F}_{i}$ read

[TABLE]

The new input parameters entering the decay rate prediction here are taken as $V_{tb}=0.77^{+0.18}_{-0.24}$ Olive et al. (2014), $V_{ts}=0.0406\pm 0.0027$ Olive et al. (2014), $C_{7}^{eff}(m_{b})=-0.306$ Asatrian et al. (2017), $C_{9}^{eff}(m_{b})=4.344$ Buras and Munz (1995); Dag et al. (2011) and $C_{10}=-4.669$ Buras and Munz (1995); Dag et al. (2011). Using the calculated LCSR results for the form factors we obtain

[TABLE]

Our predictions are compatible with the references given within the error budget. Furthermore, in analogy to Eq. (33) we also give our prediction for the LFU ratio:

[TABLE]

As a final remark before summary, we would like to stress that the results presented in this work include only factorizable contributions and non-factorizable (non-local $c\bar{c}$-loop) effects are not taken into account in this work. Analysis of such non-factorizable contributions lies beyond the scope of this paper and we plan to come back to discuss this point separately in the future.

V Conclusion

The study of semileptonic $B$-meson decays involving tensor mesons can provide additional information on physics beyond the Standard Model due to the rich polarization structure of the tensor mesons. In connection to that we calculated the $B\to D_{2}^{*},K_{2}^{*},a_{2},f_{2}$ ($J^{P}=2^{+}$) transition form factors within light-cone sum rules using $B$-meson distribution amplitudes, including the twist-four terms. We find that the calculated higher-twist terms have a noticeable impact on the sum rules. Using the obtained results for the form factors we estimate the decay rates of $B\to D_{2}^{*}\ell\bar{\nu}$, $B\to K_{2}^{*}\gamma$ and $B\to K_{2}^{*}\ell^{+}\ell^{-}$ in the SM. Our results indicate that these decays can be within reach for LHCb and Belle experiments in the near future.

Acknowledgements.

The work of A.K. is supported by the DFG within the Emmy Noether Programme under Grant No. DY-130/1-1 and the DFG Collaborative Research Center 110 ”Symmetries and the emergence of structure in QCD.” H.D. acknowledges support through the Scientific and Technological Research Council of Turkey (TUBITAK) Grant No. BIDEP-2219. A.O. thanks the Physics Department at the Technical University of Munich for the hospitality at the early stage of this work. A.K. is thankful to Danny van Dyk and Javier Virto for helpful discussions on the topic.

Appendix A Distribution Amplitudes of the $B$-meson

The two-particle momentum-space projector can be expressed in terms of $B$-LCDAs (up to twist-four) as

[TABLE]

where $v_{\mu}$ is the four-velocity of the $B$-meson, and $\partial_{\mu}\equiv\partial/\partial l^{\mu}$ with $l^{\mu}=\omega v^{\mu}$ in the two-particle case. The above momentum-space derivatives are understood to act on the hard-scattering kernel of Eq. (11). Moreover, we abbreviate

[TABLE]

In our numerical estimates for the form factors we follow the local duality model131313The model we employ in this work corresponds to model II A of Ref. Braun et al. (2017). proposed in Ref. Braun et al. (2017) for the two-particle $B$-LCDAs $\phi_{+}$, $\phi_{-}$, and $g_{+}$. The explicit expressions for $\phi_{+}$, $\phi_{-}$, and $g_{+}$ in this model are given in Eqs. 5.22–5.23 of Ref. Braun et al. (2017).

For $g_{-}$ no model expression is available yet; we therefore use the Wandzura-Wilczek (WW) approximation

[TABLE]

In the local duality model considered in this work, Eq. (48) explicitly yields:

[TABLE]

where $\theta(x)$ is the heavy-side step function.

The parameters $\lambda_{E}^{2},\lambda_{H}^{2}$ and $\lambda_{B}$ appearing in the explicit expressions of $B$-LCDAs are provided as input in Sec. III.1.

Appendix B $B\to T$ coefficients of Eq. (17) from two-particle contributions

B.1 $K^{(F)}$ factors of Eq. (17)

The pre-factors appearing in Eq. (17) read:

[TABLE]

B.2 $C^{(F,\psi_{\text{2p}})}_{n}$ coefficients of Eq. (20)

We collect here the (non-vanishing) coefficients appearing in Eq. (20).

For ${\tilde{V}}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{A}}_{1}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{A}}_{2}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{A}}_{30}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{T}}_{1}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{T}}_{23A}^{B\to T}$ we obtain:

[TABLE]

For ${\tilde{T}}_{23B}^{B\to T}$ we obtain:

[TABLE]

where shorthand notations $\hat{\sigma}_{1}=m_{q_{1}}^{2}(1-2\sigma)-m_{B}m_{q_{1}}\sigma\bar{\sigma}+\sigma(m_{B}^{2}\bar{\sigma}^{2}+q^{2}(2\sigma-1))$ and $\hat{\sigma}_{2}=2\bar{\sigma}^{2}m_{B}^{2}-\bar{\sigma}m_{B}m_{q_{1}}-2q^{2}+4\sigma q^{2}+m_{q_{1}}^{2}$ are introduced for simplicity. The ${\tilde{T}}_{23A}^{B\to T}$ and ${\tilde{T}}_{23B}^{B\to T}$ form factors are defined via Eq. (22) and Eq. (23).