ρ-meson longitudinal leading-twist distribution amplitude revisited and theD→ρsemileptonic decay

Theρ-meson leading-twist longitudinal distribution amplitude (DA) is a key input parameter when investigating related exclusive processes including theρmeson. Charmed semileptonic decay processes can provide a clear platform for researching theρ-meson DA. In charmed factories, the BESIII collaboration reported new results for the semileptonic decay $ D^0\to \rho^-\mu^+\nu_\mu $ in 2021 [1]. In 2019, the BESIII collaboration provided improved measurements of $ D\to\rho e^+\nu_e $ [2]. The CLEO collaboration provided measurements of the form factors in the decays $ D^{0/+}\to \rho^{-/0}e^+\nu_e $ in 2013 [3], and their previous measurements can be found in [4,5]. It is known that transition form factors (TFFs) are key components of semileptonic $ D \to \rho \ell^+ \nu_\ell $ decays in the standard model. Therefore, accurate TFFs are very important for theoretical groups and experimental collaborations. Theoretically, the $ D\to\rho $ TFFs can be treated using different approaches, such as the 3-point sum rule (3PSR) [6], heavy quark effective field theory (HQEFT) [7,8], relativistic harmonic oscillator potential model (RHOPM) [9], quark model (QM) [10,11], light-front quark model (LFQM) [12,13], light-cone sum rule (LCSR) [14−16], covariant confined quark model (CCQM) [17], heavy meson and chiral symmetry theory (HMχT) [18], and lattice QCD (LQCD) [19,20]. The LCSR approach is based on operator production expansion near the light-cone, and all the non-perturbative dynamics are parameterized into light-cone DAs (LCDAs), which is suitable for calculating the heavy to light transition. In this study, we take the LCSR approach to calculate the $ D\to\rho $ TFFs with a left-handed current ¹ . Thus, the key task is obtaining an accurateρ-meson longitudinal twist-2 DA.

Theoretically, theρ-meson longitudinal leading-twist DA can be studied using various methods, such as QCD sum rules (SRs) [21−26], LQCD [27−30], the AdS/QCD holographic method [31−34], extraction from experimental data [35−38], the light-front quark model (LFQM) [39,40], Dyson-Schwinger equations (DSEs) [41], large momentum effective theory (LMET) [42], instanton vacuum [43,44], and other models [45,46]. The QCDSR in the framework of background field theory (BFT) is an effective approach in calculating light and heavy meson DAs [47]. In early 2016, we preliminarily studied theρ-meson longitudinal leading-twist DA [48], in which the first two order nonzeroξ-moments and Gegenbauer moments were obtained, and the behavior of $ \phi_{2;\rho}^\|(x,\mu) $ described by the light-cone harmonic oscillator (LCHO) model was determined [49].

In 2021, we proposed a new research scheme for the QCDSR study on pionic leading-twist DAs, in which a new SR formula forξ-moments was proposed after considering that the SR for the zerothξ-moment cannot be normalized in the entire Borel region [50]. It enabled us to calculate more higher-orderξ-moments. Furthermore, the behavior of the pion DA $ \phi_{2;\pi}(x,\mu) $ could be determined by fitting sufficientξ-moments with the least squares method. Subsequently, this scheme was used to study pseudoscalar $ \eta^{(\prime)} $ -meson and kaon leading-twist DAs [51,52] andD-meson twist-2,3 DAs [53], axial vector $ a_1(1260) $ -meson twist-2 longitudinal DAs [54], and scalar $ a_0(980) $ and $ K_0^\ast(1430) $ -meson leading-twist DAs [55,56]. Inspired by this, we restudy theρ-meson leading-twist longitudinal DA $ \phi_{2;\rho}^\|(x,\mu) $ by adopting the research scheme proposed in Ref. [50].

The remainder of this paper are organized as follows. In Sec. II, we present the calculations for theξ-moments of theρ-meson leading-twist DA, the $ D\to \rho $ TFFs, and the semileptonic decays $ D\to\rho\ell^+\nu_\ell $ . In Sec. III, we present the numerical results and discussions on theξ-moments, $ D\to \rho $ TFFs, and $ D\to \rho\ell^+\nu_\ell $ decay widths and branching ratios. Section IV is contains a summary.

To derive the sum rules ofρ-meson leading-twist longitudinal DAξ-moments, we adopt the following correlation function (also known as the correlator):

where the current $J_n(x) = \bar{d} {\not {z}} ({\rm i}z\cdot {\mathop D\limits^ \leftrightarrow})^n u(x) $ with $ z^2 = 0 $ . After taking the standard QCDSR calculation procedure within the framework of BFT [50], we obtain an expression for $ \langle\xi^n\rangle_{2;\rho}^\| \times \langle\xi^0\rangle_{2;\rho}^\| $ ,

where $ m_\rho $ and $ f_\rho $ are theρ-meson mass and decay constant, respectively, $ s_\rho $ is the continuum threshold, $ m_u $ and $ m_d $ are the current quark masses of theuanddquarks, respectively,Mis the Borel parameter, $ \langle\bar{q}q\rangle $ with $ q = u (d) $ is the double-quark condensate, $ \langle\alpha_sG^2\rangle $ is the double-gluon condensate, $ \langle g_s\bar{q}\sigma TGq\rangle $ is the quark-gluon mix condensate, $ \langle g_s^3fG^3\rangle $ is the triple-gluon condensate, and $ \langle g_s\bar{q}q \rangle^2 $ and $ \langle g_s^2\bar{q}q \rangle^2 $ are the four-quark condensates, respectively. In addition, $ \kappa = \langle\bar{s}s\rangle / \langle\bar{q}q\rangle $ with the doublesquark condensate $ \langle\bar{s}s\rangle $ . By taking $ n = 0 $ for Eq. (2), the SR of the zeroth-orderξ-moment $ \langle\xi^0\rangle_{2;\rho}^\| $ can be obtained as

Eq. (3) indicates that the zeroth-orderξ-moment $ \langle\xi^0\rangle_{2;\rho}^{\|} $ in Eq. (2) cannot be normalized in the entire Borel parameter region. The main reason for this is because the contributions from vacuum condensates larger than dimension-six are truncated normally. As discussed in Ref. [50], a more accurate and reasonable SR for thenth-orderξ-moment $ \langle\xi^n\rangle_{2;\rho}^\| $ should be

However, to describe the behavior of theρ-meson leading-twist longitudinal DA, we take the following DSE model for $ \phi_{2;\rho}^\parallel(x,\mu) $ [57,58]:

where $ \alpha_- = \alpha - 1/2 $ , and $ \mathcal{N} = 4^\alpha \Gamma(\alpha + 1) / [\sqrt{\pi} \Gamma(\alpha + 1/2)] $ is the normalization constant. In this notation, the Gegenbauer polynomial series is considered the most accurate form for describing the meson DA. Unfortunately, one cannot obtain all Gegenbauer moments in principle. Therefore, a truncated form of the Gegenbauer polynomial series (TF model) is typically used to approximately describe the behavior of the meson DA in the literature. However, the TF model is not an ideal model for describing the behavior of meson DAs, because it truncates at the lowest Gegenbauer coefficient. In fact, the DSE model is still a truncated form of the Gegenbauer polynomial series. The difference between these two models is that the DSE model is based on the $ {C_n^\alpha} $ -basis, whereas the TF model is based on the $ {C_n^{3/2}} $ -basis. In addition, the factor $ [x(1-x)]^{\alpha_-} $ in the DSE model can adjust and compensate for the impact caused by truncation to a certain extent. Another advantage of the DSE model is that, as mentioned in Ref. [58], it can reduce the introduced spurious oscillations.

Next, to obtain the $ D\to\rho $ TFFs, we can take the following correlation function:

where $j_D^{L} (x)= {\rm i} \bar q_2(x)(1 - \gamma_5)c(x)$ is the left-handed current. As we know, there are fifteen DAs for a vector meson up to twist-4 accuracy, and the left-handed chiral current can reduce the uncertainties from chiral-odd vector meson DAs with $ \delta^{0,2} $ -order and leave the chiral-even with a $ \delta^{1,3} $ -order meson. The relationships are listed inTable 1. In this table, except $ j_D^{L} (x) $ , the current $ j_D^{R} (x) $ respects the right-handed current with the expression $j_D^{R} (x)= {\rm i} \bar q_2(x) \times (1 + \gamma_5)c(x)$ , as researched in our previous study [15]. The parameter $ \delta \simeq m_\rho/m_c\sim 52\ $ % [59,60]. Meanwhile, the chiral current approach has been considered in other studies [61−66], which improves the predictions of the LCSR approach.

		twist-2	twist-3	twist-4
	$ \delta^0 $	$ \phi_{2;\rho}^\bot $
$ j_D^{L} (x) $	$ \delta^2 $		$ \phi_{3;\rho}^\|, \psi_{3;\rho}^\|, \Phi_{3;\rho}^\bot $	$ \phi_{4;\rho}^\bot,\psi_{4;\rho}^\bot,\Psi_{4;\rho}^\bot, \widetilde{\Psi} _{4;\rho}^\bot $
	$ \delta^1 $	$ \phi_{2;\rho}^\| $	$ \phi_{3;\rho}^\bot, \psi_{3;\rho}^\bot, \Phi_{3;\rho}^\|,\tilde\Phi_{3;\rho }^\bot $
$ j_D^{R} (x) $	$ \delta^3 $			$ \phi_{4;\rho}^\|,\psi_{4;\rho}^\| $

Table 1.ρ-meson DAs with different twist-structures up to $ \delta^3 $ , where $ \delta \simeq m_\rho/m_c $ .

Based on the procedures of the LCSR approach, we obtain the expressions of the $ D\to \rho $ TFFs $ A_{1,2}(q^2) $ and $ V(q^2) $ LCSRs with the left-handed current in the correlator. The analytic formulas are similar to the $ B\to\rho $ TFFs of our previous study [45], which replace the input parameter of theB-meson with aD-meson, such as $ m_B\to m_D $ , $ f_B \to f_D $ , $ m_b \to m_c $ . The detailed expressions can be written as follows:

where $ s(\varrho)=[m_b^2-\bar \varrho(q^2-\varrho m_\rho^2)]/\varrho $ with $ \bar \varrho = 1 - \varrho $ ( $ \varrho $ representsuorX), and $ X=a_1 + v a_3 $ . $ f_\rho^\| $ represents theρ-meson decay constant, and $ c(u,s_0)=u s_0 - m_b^2 + \bar u q^2 - u \bar u m_\rho^2 $ . $ \Theta(c(\varrho,s_0)) $ is the usual step function, and the definitions for $ \widetilde\Theta (c(u,s_0)) $ , $ \widetilde{\widetilde\Theta}(c(u,s_0)) $ , and $ \widetilde{\widetilde{\widetilde\Theta}}(c(u,s_0)) $ can be found in our previous study [45]. The combinedρ-meson DA $ A_\rho^\| (x), B_\rho^\| (x) $ , and $ C_\rho^\| (x) $ have the expressions

which originate from the definition of the chiral-even two-particle DAs [60]. From Eqs. (7), (8), and (9), we can see that only $ \delta^{1,3} $ -order DAs have contributions to TFFs with a left-chiral correlation function. Conversely, $ \delta^{0,2} $ -order DAs do not have contributions, which also indicates that transverse twist-2 DAs have no contribution to $ A_{1,2}(q^2) $ and $ V(q^2) $ . Furthermore, the twist-3 DAs $ \psi_{3;\rho}^\bot(x,\mu) $ , $ \phi_{3;\rho}^\bot(x,\mu) $ , and $ A_\rho^\| (x,\mu) $ can be related to $ \phi_{2;\rho}^\| $ directly using the Wandzura-Wilczek approximation [67,68],