A Proposal to the ‘12 vs. 32 Puzzle’

Abstract

We reconsider the semileptonic decays of $B\to D_1^{(\prime )}l{\overline{\nu }}_l$. The previous theoretical calculations predict a significantly smaller rate for the semileptonic decay of B to $D_1^{\prime }(J_l=\frac{1}{2})$ compared with that to the $D_1(J_l=\frac{3}{2})$, which is not consistent with the current experimental data. This conflict is known as the so-called ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’. In this work, we propose a simple scheme to fix this problem, where we suppose the strong eigenstates $D_1^{(\prime )}$ that do not coincide with the eigenstate of the weak interaction, since no experimental results show that the weak and the strong interactions have to share the same eigenstates. Within the framework of this tentative scheme, meson B first weakly decays to the weak eigenstates $D_{\alpha \left(\beta \right)}$ and then the latter are detected as the $D_1^{(\prime )}$ by the strong decay products $D^{*}\pi$. We predict that there exist two new particles $D_{\alpha \left(\beta \right)}$ with $J^P=1^{+}$, which were not previously identified. The good performance of the new scheme in describing the experimental data may hint at new symmetry in the weak decays of $B_q$ to $1^{+}$ heavy–light mesons. To test the scheme proposed here, we suggest an experiment to detect the difference in the invariant mass spectra of $D_1$ that is reconstructed from the B weak decay and from the strong decay products.

1. Introduction

In the heavy quark limit, the $J^P=1^{+}$ heavy–light mesons, such as $B_1$, $B_{s1}$, $D_1$, and $D_{s1}$, that contain a doublet: one with the light quark total angular momentum $J_l=\frac{1}{2}$ and the other with $J_l=\frac{3}{2}$. Since the $|J_l=\frac{3}{2}\rangle$ state mainly decays through a D-wave barrier, it has a narrow width, while the $|J_l=\frac{1}{2}\rangle$ state usually decays through a S-wave way, and its width is much broader. Notice that both of the two $1^{+}$ charmed-strange mesons $D_{s1}\left(2460\right)$ and $D_{s1}\left(2536\right)$ are narrow, which seems contradictory with the above analysis. However, this may be caused by the low mass of $D_{s1}\left(2460\right)$, and, hence, it cannot strongly decay to the $D^{*}K$ channel as with the $D_{s1}\left(2536\right)$. The $J^P=1^{+}$$\left(c\overline{u}\right)$ state with $J_l=\frac{3}{2}$ is usually labeled as the $D_1$, and that with $J_l=\frac{1}{2}$ is labeled as $D_1^{\prime }$. The theoretical calculations of the semileptonic decays rates of B to the $|J_l=\frac{1}{2}\rangle$ state yield a much smaller value than that of B to the $|J_l=\frac{3}{2}\rangle$ state. However, this theoretical prediction is not supported by current experimental data. This is the famous ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ [1,2,3,4], which is more clearly showed in Table 1. The theoretical predictions of the branching fraction for the decay $B\to D_1^{\prime }l{\overline{\nu }}_l$ are generally one order less than that for $B\to D_{1}l{\overline{\nu }}_l$. The significant discrepancies between the theoretical predictions and the experimental results have been discussed in several works [5,6,7,8,9,10,11,12]. Most of these previous theoretical results are derived in the heavy quark limit and the corrections might be large. It is expected that the $1/m_c$ corrections induce a large mixing between $|J_l=\frac{3}{2}\rangle$ and $|J_l=\frac{1}{2}\rangle$ state, which could soften this puzzle [9]. However, since the traditional theoretical results for the strong decays of $D_1$ and $D_1^{\prime }$ are consistent with the experimental data [13,14,15,16], introducing a large mixing angle would inevitably change the strong decay calculation results. A dilemma arises here. In a previous work [17], we found that the puzzle can not be overcome by adding only relativistic corrections, but it can be partly explained in a small special range of the string parameter $\lambda$. However, the mixing angle used leads to an inconsistent mass order between $D_1$ and $D_1^{\prime }$ according to the latest experimental data for the mass of $D_1\left(2430\right)$ [18]. The absence of a satisfactory understanding and explanations may suggest the existence of a special scheme in this type of decay.

On the other hand, the BESIII collaboration reported the experimental observations of semileptonic decays $D^{+}\to {\overline{K}}_1\left(1270\right)\overline{l}{\nu }_l$ in 2019 [19] and the $D^0\to K_1\left(1270\right)e^{+}{\nu }_e$ in 2021 [20], where the measured branching fractions were $2.3\times {10}^{-3}$ and $1.1\times {10}^{-3}$, respectively. The semileptonic decays of $D\to K_1$ are quite similar with the problem we discussed above. From the experimental results, if the branching fractions $D\to K_1\left(1400\right)$ were comparable with the $D\to K_1\left(1270\right)$, it would be possible to detect the semileptonic decays of D to the broader $K_1$ system. This can help us answer the question as to whether the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ just happened accidentally or has generally existed in such decays involving the unnatural parity mesons, which occupy the spin-parity $J^P=1^{+},2^{-},3^{+},\cdots$.

Table 1. Branching fraction ($\times {10}^{-3}$) of semileptonic decays $B_{\left(s\right)}\to D_{\left(s\right)1}^{(\prime )}l{\overline{\nu }}_l$$(l=e$ or $\mu )$. The results labeled ‘BS’ are calculated according to the traditional method by using the Bethe–Salpeter (BS) wave functions; ‘Ansatz’ are calculated within the new scheme proposed here. The theoretical uncertainties in our results are calculated by varying the mixing angle ${(\theta \pm 5)}^{\circ }$. Since the PDG data only gives the fraction of the cascade decay $B\to D_1^{(\prime )}(D_1^{(\prime )}\to {\overline{D}}^{*0}{\pi }^{-})l\overline{\nu }$, we assumed the branching fraction $\mathcal{B}(D_1^{(\prime )}\to D^{*0}{\pi }^{-})=2/3$; and $\mathcal{B}\left(B_s\to D_{s1}l\overline{\nu }\right)$ is determined by $\left(2.94·\frac{1+0.85}{0.85}\right)\times {10}^{-3}$. The last line denotes the summation of the branching fractions $\mathcal{B}(B^{-}\to D_{1}l{\overline{\nu }}_l)$ and $\mathcal{B}(B^{-}\to D_1^{\prime }l{\overline{\nu }}_l)$.

Decay	Ansatz	BS	[5]	[21]	[22]	[23]	[24]	PDG [25]
$B^{-}\to D_{1}l{\overline{\nu }}_l$	${5.06}_{+0.38}^{-0.40}$	${7.82}_{+0.16}^{-0.28}$	3.0–5.0	$7.04$	$6.3$	$3.85$	-	$4.54\pm 0.3$
$B^{-}\to D_1^{\prime }l{\overline{\nu }}_l$	${3.46}_{-0.38}^{+0.40}$	${0.64}_{-0.16}^{+0.27}$	0.0–0.7	$0.45$	$0.9$	$1.98$	-	$4.05\pm 0.9$
$B_s\to D_{s1}l{\overline{\nu }}_l$	${6.03}_{+0.43}^{-0.46}$	${8.44}_{+0.20}^{-0.32}$	-	-	$10.6$	$4.77$	$8.4\pm 0.9$	$5.66\pm 1.52$
$B_s\to D_{s1}^{\prime }l{\overline{\nu }}_l$	${4.21}_{-0.43}^{+0.46}$	${1.36}_{-0.25}^{+0.28}$	-	-	$1.8$	$1.74$–$5.7$	$1.9\pm 0.2$	-
$B^{-}\to D_1^{(\prime )}l{\overline{\nu }}_l$	8.42	8.46	3.0–5.7	$7.49$	$7.2$	$5.83$	-	$8.54$

Table 1. Branching fraction ($\times {10}^{-3}$) of semileptonic decays $B_{\left(s\right)}\to D_{\left(s\right)1}^{(\prime )}l{\overline{\nu }}_l$$(l=e$ or $\mu )$. The results labeled ‘BS’ are calculated according to the traditional method by using the Bethe–Salpeter (BS) wave functions; ‘Ansatz’ are calculated within the new scheme proposed here. The theoretical uncertainties in our results are calculated by varying the mixing angle ${(\theta \pm 5)}^{\circ }$. Since the PDG data only gives the fraction of the cascade decay $B\to D_1^{(\prime )}(D_1^{(\prime )}\to {\overline{D}}^{*0}{\pi }^{-})l\overline{\nu }$, we assumed the branching fraction $\mathcal{B}(D_1^{(\prime )}\to D^{*0}{\pi }^{-})=2/3$; and $\mathcal{B}\left(B_s\to D_{s1}l\overline{\nu }\right)$ is determined by $\left(2.94·\frac{1+0.85}{0.85}\right)\times {10}^{-3}$. The last line denotes the summation of the branching fractions $\mathcal{B}(B^{-}\to D_{1}l{\overline{\nu }}_l)$ and $\mathcal{B}(B^{-}\to D_1^{\prime }l{\overline{\nu }}_l)$.

Decay	Ansatz	BS	[5]	[21]	[22]	[23]	[24]	PDG [25]
$B^{-}\to D_{1}l{\overline{\nu }}_l$	${5.06}_{+0.38}^{-0.40}$	${7.82}_{+0.16}^{-0.28}$	3.0–5.0	$7.04$	$6.3$	$3.85$	-	$4.54\pm 0.3$
$B^{-}\to D_1^{\prime }l{\overline{\nu }}_l$	${3.46}_{-0.38}^{+0.40}$	${0.64}_{-0.16}^{+0.27}$	0.0–0.7	$0.45$	$0.9$	$1.98$	-	$4.05\pm 0.9$
$B_s\to D_{s1}l{\overline{\nu }}_l$	${6.03}_{+0.43}^{-0.46}$	${8.44}_{+0.20}^{-0.32}$	-	-	$10.6$	$4.77$	$8.4\pm 0.9$	$5.66\pm 1.52$
$B_s\to D_{s1}^{\prime }l{\overline{\nu }}_l$	${4.21}_{-0.43}^{+0.46}$	${1.36}_{-0.25}^{+0.28}$	-	-	$1.8$	$1.74$–$5.7$	$1.9\pm 0.2$	-
$B^{-}\to D_1^{(\prime )}l{\overline{\nu }}_l$	8.42	8.46	3.0–5.7	$7.49$	$7.2$	$5.83$	-	$8.54$

This paper is organized as follows. In Section 2, we try to explore the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ by re-examining the related weak decay process and the final strong products in experimental measurements, and then we give our proposal for the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ and the related discussions. Finally we give a brief summary and outlook of this work. A brief review of the calculation methods and some numerical details used are all collected in the Appendix A.

2. Revisitinf the $\mathit{B}\mathbf{\to }{\mathit{D}}_{\mathbf{1}}^{\mathbf{(}\mathbf{\prime }\mathbf{)}}\mathit{l}{\overline{\mathit{\nu }}}_{\mathit{l}}$ Decays

In this section, we try to deal with the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’. It should be pointed out that the proposed scheme and our main conclusion here are not sensitive or dependent on the specific quark model or calculation methods introduced in the Appendix. We start by rechecking the experimental measurements first.

2.1. The Weak and Strong Eigenstates

The effective Hamiltonian responsible for the semileptonic decays of B to $1^{+}$ charmed mesons can be expressed at the hadronic level as follows:

$$\begin{array}{c}{H}_{\mathrm{eff}}=\frac{G_F}{\sqrt{2}}{V}_{cb}\left(\overline{l}{\Gamma }_{\nu }{\nu }_l\right)\mathrm{tr}\left({\overline{D}}_{\alpha }{\Gamma }^{\nu }B+{\overline{D}}_{\beta }{\Gamma }^{\nu }B\right),\end{array}$$

(1)

where B and $D_{\alpha \left(\beta \right)}$ here denote the fields of the corresponding mesons, and will they also be used to denote the corresponding mesons without causing confusion; ${\Gamma }^{\nu }={\gamma }^5(1-{\gamma }^{\nu })$ represents the Dirac structure of a weak vertex; and l and ${\nu }_l$ represent the fields of the lepton and the related neutrino, respectively. The two weak eigenstates of the $1^{+}$ charmed mesons are generally denoted as $D_{\alpha }$ and $D_{\beta }$.

First, we re-examined the previous theoretical calculations and experimental measurements on the decays of $B\to D_1^{(\prime )}l{\overline{\nu }}_l$. In these processes, $D_1$ or $D_1^{\prime }$ is regarded as a particle participating in both the semileptonic weak decays and the strong decays to the $D^{*}\pi$, which are then detected by the detectors to reconstruct the $D_1^{(\prime )}$. Namely, in the previous studies, the weak and strong decays of the D meson system share the same eigenstates. Here, the weak eigenstates $D_{\alpha \left(\beta \right)}$ refer to the direct hadron products in the semileptonic decay of the B meson, which are the eigenstates of the above Hamiltonian, and the strong eigenstates denote the involved charmed mesons, which can directly strongly decay to $D^{*}\pi$.

For the later one, namely, the strong decays to $D^{*}\pi$, there is no controversy. The angular momentum $J_l$ is conserved in the infinite mass limit, and hence the $|J_l=\frac{3}{2}\rangle$ and $|J_l=\frac{1}{2}\rangle$ states, instead of $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$, are the strong eigenstates of the QCD Hamiltonian in this limit. In the nonrelativistic and heavy quark limit, with the help of the Clebsch–Gordan coefficients, the states $|J_l=\frac{3}{2}\rangle$ and $|J_l=\frac{1}{2}\rangle$ can be decomposed on the basis of $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$ (see Appendix B for detailed calculations).

$$\left(\begin{array}{c}|J_l=\frac{3}{2}\rangle \\ |J_l=\frac{1}{2}\rangle \end{array}\right)=\left[\begin{array}{cc}+\frac{\sqrt{2}}{\sqrt{3}}& \frac{1}{\sqrt{3}}\\ -\frac{1}{\sqrt{3}}& \frac{\sqrt{2}}{\sqrt{3}}\end{array}\right]\left(\begin{array}{c}|{}^1{P}_1\rangle \\ |{}^3{P}_1\rangle \end{array}\right)=\left[\begin{array}{cc}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{array}\right]\left(\begin{array}{c}|{}^1{P}_1\rangle \\ |{}^3{P}_1\rangle \end{array}\right),$$

(2)

which corresponds to a counter-clock rotation with rotation angle $\alpha =-{35.3}^{\circ }$. The most recent data shows that $M_{D_1^{\prime }}=2.412\mathrm{GeV}$ [18] is slightly lighter than $M_{D_1}=2.421\mathrm{GeV}$, which is opposite to the previous situation. Then from the above equation, it is easy to see that the $|{}^1{P}_1\rangle$ state should correspond to the higher mass state compared with the $|{}^3{P}_1\rangle$ state. The recent data is also consistent with the relevant mass relationship in charmonia and bottomonia systems, namely, $h_{c\left(b\right)}\left(1P\right)$ is heavier than ${\chi }_{c\left(b\right)1}\left(1P\right)$. In our experiments, $D_1$ and $D_1^{\prime }$ were reconstructed in the $D^{*}\pi$ invariant mass spectrum as the strong decay eigenstates. Then throughout this work, we took the states $|J_l=\frac{3}{2}\rangle$ and $|J_l=\frac{1}{2}\rangle$ as the two strong eigenstates $D_1$ and $D_1^{\prime }$, respectively, which is also supported by the consistence between the previous theoretical calculations and the experimental data for the strong decays of $D_1^{(\prime )}$ [13,14,15,16,26,27]. Namely, even if $D_1$ and $D_1^{\prime }$ were the mixing states of $|J_l=\frac{3}{2}\rangle$ and $|J_l=\frac{1}{2}\rangle$, respectively, the mixing effects should be quite small and would not influence the main discussion in this work.

However, for the former one, namely, the semileptonic weak decays of B, we could not ensure that if the B weakly decays to the $D_1$ and $D_1^{\prime }$ directly or it first decays to some other states, which are the mixtures of $D_1$ and $D_1^{\prime }$, and the latter two are just the final states we detected. This feature reminded us to review a very similar example: the neutral kaons ${\overline{K}}^0$ and $K^0$, as well as $K_{\mathrm{L}}^0$ and $K_{\mathrm{S}}^0$. The neutral kaons are typically produced by the strong interactions as the strong eigenstates ${\overline{K}}^0$ and $K^0$, which are the superposition states as follows [28]:

$$\left(\begin{array}{c}|{\overline{K}}_0\rangle \\ |K_0\rangle \end{array}\right)=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]\left(\begin{array}{c}|K_{\mathrm{L}}^0\rangle \\ |K_{\mathrm{S}}^0\rangle \end{array}\right).$$

(3)

Then, these neutral kaons decay by the weak interactions to the weak eigenstates $K_{\mathrm{L}}^0$ and $K_{\mathrm{S}}^0$ with different lifetimes, where the $CP$ is conserved. If the similar case happens in the $1^{+}$ charmed mesons, it may be responsible for the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’.

The neutral kaons hinted us to introduce the weak eigenstates $D_{\alpha }$ and $D_{\beta }$, which can be generally expressed as the mixtures of the strong eigenstates $D_1$ and $D_1^{\prime }$:

$$\left(\begin{array}{c}{D}_{\alpha }\\ D_{\beta }\end{array}\right)=\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]\left(\begin{array}{c}{D}_1\\ D_1^{\prime }\end{array}\right),\mathrm{or}\left(\begin{array}{c}{D}_1\\ D_1^{\prime }\end{array}\right)=\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]\left(\begin{array}{c}{D}_{\alpha }\\ D_{\beta }\end{array}\right),$$

(4)

where $D_1$ and $D_1^{\prime }$ decay by the strong interaction via the strong eigenstates, and the $J_l$ is conserved under the heavy quark spin symmetry, which is similar to $K_{\mathrm{L}}^0$ and $K_{\mathrm{S}}^0$ conserving the $CP$. The symbol $\theta$ denotes the corresponding mixing angle between the weak and the strong eigenstates. Then, if the weak and the strong decays share the same eigenstates, we have $\theta =0$, which is trivial and the standard treatment to this problem, but it was not established in experiments. In general, the weak eigenstate $D_{\alpha \left(\beta \right)}$ may be different from the strong eigenstate $D_1^{(\prime )}$. Then, the assumed $D_{\alpha \left(\beta \right)}$ would be a more general description to the $1^{+}$ charmed meson involved in the semileptonic decay of B. Notice that this treatment in Equation (4) can naturally recover the standard calculation when $\theta =0$, since then the $D_{\alpha }$ and $D_{\beta }$ are exactly the same with the traditional $D_1$ and $D_1^{\prime }$, respectively. In this work, we are trying to discuss whether there is any possibility that the mixing angle $\theta$ is not equal to 0. Upon comparing Equation (4) with Equation (3), we find that $D_{\alpha \left(\beta \right)}$ corresponds to the ${\overline{K}}^0\left(K^0\right)$, while $D_1^{(\prime )}$ corresponds to the $K_{\mathrm{L}\left(\mathrm{S}\right)}^0$. The only difference is that the kaons are generated through strong interactions, but they undergo weak decays, while the charmed mesons are produced via weak interactions, but they undergo strong decays.

Let us examine the actual effects in the measurement of branching fraction between the proposal here and the traditional treatment in more detail. The semileptonic weak decay widths of B→$D_{\alpha \left(\beta \right)}l{\overline{\nu }}_l$ are expressed as follows:

$$\begin{array}{c}\Gamma \left(D_{\alpha }\right)={\left|\mathcal{A}(B\to D_{\alpha }l{\overline{\nu }}_l)\right|}^2,\\ \Gamma \left(D_{\beta }\right)={\left|\mathcal{A}(B\to D_{\beta }l{\overline{\nu }}_l)\right|}^2,\end{array}$$

(5)

where $\mathcal{A}$ denotes the corresponding decay amplitude, and the universal phase space integral is omitted for simplicity. In any one process of above decays, B either directly decays to a $D_{\alpha }$ or a $D_{\beta }$ but not to the superposition state of the two. Since the weak eigenstates $D_{\alpha \left(\beta \right)}$ were assumed to be the real physical states, the two processes were consequently different and distinguishable. We could determine which one was actually taken from the lineshapes of the invariant mass spectrum of the produced charmed meson, for the masses and widths of $D_{\alpha }$ and $D_{\beta }$ were different. On the other hand, what we really detected in experiments was that the $D_1$ and $D_1^{\prime }$ are, in fact, their strong decay products. $D_{\alpha }$ and $D_{\beta }$ are related to $D_1$ and $D_1^{\prime }$ by Equation (4). Then, by combining Equations (4) and (5), we obtain the widths for B decaying to $D_1$ and $D_1^{\prime }$:

$$\begin{array}{cc}\hfill \Gamma \left(D_1\right)& =c^2\Gamma \left(D_{\alpha }\right)+s^2\Gamma \left(D_{\beta }\right),\hfill \\ \hfill \Gamma \left(D_1^{\prime }\right)& =s^2\Gamma \left(D_{\alpha }\right)+c^2\Gamma \left(D_{\beta }\right),\hfill \end{array}$$

(6)

where $c\left(s\right)$ denotes the $cos\theta (sin\theta )$ for simplicity. Notice here that the weak eigenstates $D_{\alpha }$ and $D_{\beta }$ are two different physical states, but they are not the virtual intermediate particles. In a semileptonic decay of B, the hadronic product could only be a definite $D_{\alpha }$ or a definite $D_{\beta }$. Then, if we start with a produced $D_{\alpha }$, we will have a probability of $c^2$ to detect a $D_1$ with a long lifetime (${\Gamma }_{D_1}\sim 31\mathrm{MeV}$) and a probability of $s^2$ to detect a $D_1^{\prime }$ with a quite short lifetime (${\Gamma }_{D_1^{\prime }}\sim 314\mathrm{MeV}$) and vice versa for a $D_{\beta }$. Then, we would sum over the decay widths instead of the invariant amplitudes.

On the other hand, in the traditional theoretical calculations, namely, states $D_1$ and $D_1^{\prime }$ are taken as the direct participants of the weak decays, the corresponding results are as follows:

$$\begin{array}{cc}\hfill \Gamma (B\to D_{1}l{\overline{\nu }}_l)& =|\mathcal{A}\left(D_1\right){|}^2={|c\mathcal{A}\left(D_{\alpha }\right)-s\mathcal{A}\left(D_{\beta }\right)|}^2=c^2\Gamma \left(D_{\alpha }\right)+s^2\Gamma \left(D_{\beta }\right)-2sc{\Gamma }_{\mathrm{In}},\hfill \\ \hfill \Gamma (B\to D_1^{\prime }l{\overline{\nu }}_l)& =|\mathcal{A}\left(D_1^{\prime }\right){|}^2={|s\mathcal{A}\left(D_{\alpha }\right)+c\mathcal{A}\left(D_{\beta }\right)|}^2=s^2\Gamma \left(D_{\alpha }\right)+c^2\Gamma \left(D_{\beta }\right)+2sc{\Gamma }_{\mathrm{In}},\hfill \end{array}$$

(7)

where ${\Gamma }_{\mathrm{In}}=\mathcal{A}\left(D_{\alpha }\right){\mathcal{A}}^{*}\left(D_{\beta }\right)+{\mathcal{A}}^{*}\left(D_{\alpha }\right)\mathcal{A}\left(D_{\beta }\right)$ denotes the interference part between $D_{\alpha }$ and $D_{\beta }$. Accordingly, by using the BS wave functions and the Mandelstam formalism, the calculated numerical results are listed in Table 1 and labeled as ‘BS’. It is obvious that, in the traditional calculations, the semileptonic B decays had a substantially smaller rate to the $J_l^{P_l}={\frac{1}{2}}^{+}$ doublet than to the $J_l^{P_l}={\frac{3}{2}}^{+}$ doublet, which was consistent with other theoretical calculations but contrary to the experimental data labeled as ‘PDG’ in Table 1.

Comparing the traditional theoretical results Equation (7) with Equation (6), we found that the difference came from the interference parts, which may be responsible for the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’. Any difference between the weak eigenstates $D_{\alpha \left(\beta \right)}$ and the strong ones $D_1^{(\prime )}$ would cause this kind of interference.

2.2. Test of the New Scheme

To test the assumption proposed here, we provided an experimental proposal. Since the assumed weak eigenstate $D_{\alpha \left(\beta \right)}$ might be different from the strong eigenstate $D_1$, the invariant mass spectra of the charmed mesons would also be different when reconstructed from the weak decay and the strong decay. Namely, we reconstructed the relatively narrow $D_1$ from both the weak B decay process and its strong decay products:

$$\begin{array}{c}{M}_{\mathrm{w}}^2={(P-p_l-p_{\nu })}^2,\end{array}$$

(8)

$$\begin{array}{c}{M}_{\mathrm{s}}^2={(p_{D^{*}}+p_{\pi })}^2,\end{array}$$

(9)

where $M_{\mathrm{w}}$ and $M_{\mathrm{s}}$ denote the invariant masses of the long lifetime $D_1$ meson reconstructed from the B weak decays and from its strong decay products, respectively. Then, the difference (in both the mass peak and width) between $M_{\mathrm{w}}^2$ and $M_{\mathrm{s}}^2$ was used to deny or verify our proposed assumption. If the assumed physical states $D_{\alpha \left(\beta \right)}$ did not exist, the detected properties of $M_{\mathrm{s}}^2$ and $M_{\mathrm{w}}^2$ would be exactly the same in experiments. Otherwise, the two new resonances for $D_{\alpha \left(\beta \right)}$ would exist in the weak decays as the weak eigenstates. Moreover, since both $D_{\alpha }$ and $D_{\beta }$ can be detected as $D_1$, there may exist two charmed peak structures in the invariant mass spectrum $M_{\mathrm{w}}^2$ with the reconstructed final charmed products labeled as $D_1$. However, since the detection of the neutrino is quite difficult in experiments, an alternative choice may be detecting the above difference in the corresponding nonleptonic decay channels of B, namely, replacing the lepton pair with a light-charged meson. Under the factorization assumptions, the nonleptonic decays would be quite similar with the semileptonic one, and then we can expect them to share the same physics concerned here.

In addition to the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ itself, there are also some other ways to verify our results. It is obvious that, unless the mixing angle $\theta =0$, these kinds of discrepancies would generally exist in all weak decay modes that involve the mesons with the unnatural parity $J^P=1^{+},2^{-},\cdots$. Namely, these kinds of discrepancies would happen generally when the weak and strong decay eigenstates are different. The further experimental information on the $B_c$ to $B_{\left(s\right)1}^{(\prime )}$ or $D_1^{(\prime )}$ could also test our assumption proposed here. Our scheme in Equation (5) also predicted that the branching fractions of $B_c$ to the primed $B_{1\left(s\right)}$ and $D_1$ would be comparable with those of the unprimed ones, while, in the traditional calculations, the fractions of the primed ones would be negligible compared with the unprimed ones.

In addition, notice that the sum of the two results in Equations (6) and (7) are equal when ignoring the small difference in phase space:

$$\Gamma \left(D_1\right)+\Gamma \left(D_1^{\prime }\right)=\Gamma \left(D_{\alpha }\right)+\Gamma \left(D_{\beta }\right).$$

(10)

Namely, the traditional calculations can obtain the right results for the total widths of B to $D_1$ and $D_1^{\prime }$, regardless of whether the weak and strong decays share the same eigenstates. This can then be used as a first check on our thoughts proposed here. The sum of the branching fractions for B to $D_1$ and $D_1^{\prime }$ are listed in the last line of Table 1, which shows a satisfactory consistence between the theoretical predictions and the experimental data.

2.3. Determination of the Mixing Angle $\theta$

In fact, we have already given the reason for the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’. In our proposal, this puzzle is presumed to be caused by the difference between the weak and strong eigenstates. To finally fix the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’, we just calculated the decay widths and then fit them to the data to obtain the mixing angle $\theta$. In this work, we calculated the decay branching fractions by Equation (5) combined with Equation (6) when the mixing angle $\theta$ varied from $-{90}^{\circ }$ to ${90}^{\circ }$. The obtained results of B to $D_1^{(\prime )}$ are represented in Figure 1, where we also show the experimental data (labeled as ‘PDG’ in Figure 1) using the circle and square, respectively.

The values of the theta angle that could reproduce the experimental data were $-{49.0}^{\circ }$, $-{31.3}^{\circ }$, ${41.0}^{\circ }$, and ${58.7}^{\circ }$, as shown in Figure 1. Notice that the mixing angle $\theta$ was defined in the basis of $D_1\left(2420\right)$ and $D_1\left(2430\right)$. By combining Equation (4) with Equation (2), we could also express the weak eigenstates $D_{\alpha \left(\beta \right)}$ on the basis of $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$ as follows:

$$\left(\begin{array}{c}{D}_{\alpha }\\ D_{\beta }\end{array}\right)=\left[\begin{array}{cc}cos\vartheta & -sin\vartheta \\ sin\vartheta & cos\vartheta \end{array}\right]\left(\begin{array}{c}|{}^1{P}_1\rangle \\ |{}^3{P}_1\rangle \end{array}\right),$$

(11)

where the mixing angle $\vartheta =\theta +\alpha$, and $\alpha =-{35.3}^{\circ }$ is obtained in the heavy quark limit (see Appendix B). Then, the possible $\vartheta$ is $-{84.3}^{\circ }$, $-{66.6}^{\circ }$, ${5.7}^{\circ }$, or ${23.4}^{\circ }$. It is interesting to see that $\vartheta =-{84.3}^{\circ }$ and ${5.7}^{\circ }$ are, in fact, equivalent if we interchange the states $D_{\alpha }$ and $D_{\beta }$ and then add a global minus sign to $D_{\beta }$. Also notice that adding a global minus sign to $D_{\alpha \left(\beta \right)}$ would not affect the physics. Similarly, the mixing angles $\vartheta =-{66.6}^{\circ }$ and ${23.4}^{\circ }$ are also equivalent. Namely, the two redundant mixing angles can be eliminated by a proper definition to the two weak eigenstates. Now, we let $D_{\alpha }$ always denote the higher mass one by definition, and then the possible mixing angles, which can recover the experimental data, are left to be $\vartheta ={5.7}^{\circ }$ and ${23.4}^{\circ }$. It can be verified that, under these two mixing angles, the obtained branching fractions are $4.5\times {10}^{-3}$ and $4.0\times {10}^{-3}$ for the semilepontic decays of $B\to D_1\left(2420\right)$ and $B\to D_1\left(2430\right)$, respectively, which agree well with the experimental data.

Namely, we fixed the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ by assuming $D_{\alpha \left(\beta \right)}$ as the real weak eigenstate and then calculating the corresponding mixing angles. On the other hand, it is easy to see that one of the mixing angles $\vartheta ={5.7}^{\circ }$ was quite close to $0^{\circ }$, which corresponded to the pure $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$. By considering the experimental uncertainty and the theoretical errors, a natural ansatz of $|D_{\alpha }\rangle$ and $|D_{\beta }\rangle$ is $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$, respectively. Namely, the mixing angle $\vartheta$ in Equation (11) may be $0^{\circ }$. Roughly speaking, above obtained numerical results further hinted us to assume that the states $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$, while not being the $D_1$ and $D_1^{\prime }$, might be the real weak eigenstates $D_{\alpha }$ and $D_{\beta }$, respectively, in the semileptonic decay of a B meson. Additionally, $D_{\alpha }$ can then strongly decay to $D^{*}\pi$, either by $D_1$ with a long lifetime or by $D_1^{\prime }$ with a much shorter lifetime. It should be pointed out here that the assumed particle $D_{\alpha \left(\beta \right)}$ was not the virtual intermediate particle but a real physical state. Notice that, in the traditional studies of the $1^{+}$ open flavored mesons [29,30,31], the $|{}^3{P}_1\rangle$ and $|{}^1{P}_1\rangle$ were just taken as the theoretical pure states used to obtain the physical of the $1^{+}$ mesons by ${}^1{P}_1-{}^3{P}_1$ mixing.

Under this ansatz, the obtained branching fractions are $\mathcal{B}\left(D_{\alpha }\right)=6.65\times {10}^{-3}$ and $\mathcal{B}\left(D_{\beta }\right)=1.86\times {10}^{-3}$. Then, it is easy to find that the branching fractions of $B^{-}\to D_1\left(D_1^{\prime }\right)l{\overline{\nu }}_l$ are $5.1\times {10}^{-3}$ and $3.5\times {10}^{-3}$, which are also consistent with the experimental results and labeled as ‘Ansatz’ in Table 1 for comparison. From the Table 1, we can see that the new scheme could resolve the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’, and the calculations agreed with the data pretty well, even without any fine-tuning to the mixing angle $\alpha$. The theoretical errors were calculated by varying $\theta$ by $\pm 5^{\circ }$ to see the dependence on the mixing angle.

3. Summary and Outlook

In this work, we reconsidered the branching fractions of B to the $J^P=1^{+}$ doublet $D_1$ and $D_1^{\prime }$. To resolve the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’, we proposed that $D_1$ and $D_1^{\prime }$ may not be the eigenstates in such semileptonic weak decays but may only be the eigenstates of the strong decays in the final detection, while the latter case has been well established in both experiments and theoretical calculations. The real weak eigenstates $D_{\alpha }$ and $D_{\beta }$ can be expressed as the superposition states of $D_1$ and $D_1^{\prime }$. The B meson first weakly decayed to a $D_{\alpha }$ ($D_{\beta }$), and then it was detected as a $D_1$ or $D_1^{\prime }$ by the corresponding strong decay products. By fitting to the experimental data, we found that two mixing angles $\theta ={41.0}^{\circ }$ and ${58.7}^{\circ }$ (or equivalently, $\vartheta ={5.7}^{\circ }$ and ${23.4}^{\circ }$ under the $|{}^1{P}_1\rangle$ and $|{}^3{P}_1\rangle$ basis, see Equation (11) well described the experimental data and then resolved this puzzle.

To test this assumption here, we proposed an experiment to detect the difference between $M_{\mathrm{w}}^2$ and $M_{\mathrm{s}}^2$, namely, the invariant mass distributions of the long lifetime $D_1$ mesons reconstructed from the B weak decays and from the strong decay products respectively. We also predicted that similar situation would occur in the semileptonic decays of $B_c$ to other $1^{+}$ heavy–light mesons, such as $B_{s1}^{(\prime )}$, $B_{q1}^{(\prime )}$, and ${\overline{D}}_1^{(\prime )}$, etc. Namely, we predicted comparable branching fractions for $B_c$ weakly decaying to the two $1^{+}$ heavy–light mesons, which can also be used to test the scheme proposed here. Our scheme proposed here may also be tested in the similar processes, such as $D\to K_1^{(\prime )}l{\overline{\nu }}_l$, in the very near future experiments.

A Further Ansatz and Discussions

In addition to the above discussions, the obtained numerical results hinted us to make an interesting ansatz that the real weak eigenstates $D_{\alpha }$ and $D_{\beta }$ may be the states represented by the wave functions $\psi {(}^1{P}_1)$ and $\psi {(}^3{P}_1)$, respectively. Under this ansatz, the theoretical predictions could also agree with the experimental data pretty well. The ansatz hints that B could only directly weakly decay to the $J^P=1^{+}$$\left(c\overline{q}\right)$ systems represented by the wave functions with the definite behaviors (1 or $-1$) under charge conjugation transformation, but not mixtures of the two, and then the produced states would be detected as $D_1$ or $D_1^{\prime }$. If this ansatz is correct, this phenomena should also appear in the weak decays, $B_c\to D_1(B_1,B_{s1})l{\nu }_l$, $B_s\to D_{s1}^{(\prime )}l{\nu }_l$ etc, or in the unnatural parity $J^P=2^{-}$ mesons, which means we need to reconsider all the weak decays involving the unnatural parity mesons.

Within the new scheme, even without knowing the exclusive decay widths of B to the two weak eigenstates $D_{\alpha }$ and $D_{\beta }$, we can estimate the corresponding ratio for the decay width of $B\to D_{1}l{\overline{\nu }}_l$ over that of $B\to D_1^{\prime }l{\overline{\nu }}_l$. From Equation (6), this ratio can be expressed as follows:

$$\begin{array}{c}R(D_1,D_1^{\prime })\equiv \frac{\Gamma (B\to D_{1}l{\overline{\nu }}_l)}{\Gamma (B\to D_1^{\prime }l{\overline{\nu }}_l)}=\frac{1+{tan}^2\theta R(D_{\alpha },D_{\beta })}{{tan}^2\theta +R(D_{\alpha },D_{\beta })},\end{array}$$

(12)

where we define the ratio $R(D_{\alpha },D_{\beta })\equiv \Gamma (B\to D_{\alpha }l{\overline{\nu }}_l)/\Gamma (B\to D_{\beta }l{\overline{\nu }}_l)$, and $R(D_{\alpha },D_{\beta })$ can vary from 0 to ∞. Then, it is easy to see that the ratio locates in the range of ${tan}^2\theta$ to $1/{tan}^2\theta$, and it only depends on the mixing angle $\theta$. In the heavy quark limit, the mixing angle $\theta$ was predicted to be equal to $\alpha =-{35.23}^{\circ }$ (see Appendix B). Then, we obtained that the ratio $R(D_1,D_1^{\prime })$ was located in the range of $\frac{1}{2}$ to 2, namely, in the order of one, which also well described the experimental data.

The above discussion is even more applicable to the $J^P=1^{+}$ bottomed mesons where the heavy quark limit approximation works better than the charmed ones. Namely, we can also predict that the branching fractions of $B_c$ to the primed $B_{1\left(s\right)}$ and $D_1$ are in the same order with the unprimed ones, while, in the traditional calculations, the fractions of the primed ones have been shown to be negligible compared with the unprimed ones. For example, within this framework, we predicted that the branching fraction of $D\to K_1\left(1400\right)\overline{l}\nu$ were comparable (∼60%) with that to the $K_1\left(1270\right)$, while, in the traditional calculations, the former one was shown to be one or two orders less than the latter [32,33,34]. We also suggested the experiments to measure the branching fraction of $D\to K_1\left(1400\right)\overline{l}\nu$, which should have at least two effects, to see if the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’ happens in the D decay, and to check if the weak and strong decays share the same eigenstates.

The good performance of this ansatz hinted that there may exist some more deeper physical constraints or symmetry requirements in the weak decays involving $1^{+}$ heavy–light mesons, which restrict the wave functions of the weak decay eigenstates produced in B mesons to have certain forms. Finally, it should be pointed out that we proposed the weak eigenstates $D_{\alpha \left(\beta \right)}$ to resolve the ‘$\frac{1}{2}$ vs. $\frac{3}{2}$ puzzle’, while the ansatz $\vartheta =0$ is not necessary, but it is interesting and intriguing.