Revisiting $\Xi_{Q}-\Xi_{Q}^{\prime}$ mixing in QCD sum rules

The semileptonic decays of hadrons are of great significance for extracting CKM matrix elements and testing the standard model. Recently, the semileptonic decay $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ is measured by the Belle collaboration Belle:2021crz

$${\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(1.31\pm 0.39)\%,$$

(1)

while the Lattice QCD prediction in Ref. Zhang:2021oja is

$${\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(2.38\pm 0.44)\%.$$

(2)

Our preliminary calculation based on QCD sum rules in Ref. Zhao:2021sje even gives a larger result

$${\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(3.4\pm 0.7)\%.$$

(3)

It can be seen that there exist one tension between experimental data and theoretical predictions.

In Refs. He:2021qnc ; Geng:2022xfz ; Geng:2022yxb ; Ke:2022gxm , the authors suggested that this puzzle can be resolved by considering $\Xi_{c}-\Xi_{c}^{\prime}$ mixing on theoretical side. If so, one would expect that there exists a sizable $\Xi_{c}-\Xi_{c}^{\prime}$ mixing angle. Some efforts have been made in this direction. Early in 2010, a QCD sum rules analysis was performed, and the authors arrived at $\theta_{c}=5.5\degree\pm 1.8\degree$ Aliev:2010ra . In Ref. Matsui:2020wcc , this mixing angle is obtained as $|\theta_{c}|=8.12\degree\pm 0.80\degree$ in heavy quark effective theory. In Ref. Liu:2023feb , the result of Lattice QCD shows that this mixing angle is equal to $1.2\degree\pm 0.1\degree$. More theoretical predictions can be found in Table 2.

One can see that, large differences exist among different theoretical predictions. In this work, we intend to perform a new QCD sum rules analysis. First of all, it is necessary to figure out the concepts of flavor eigenstates and mass eigenstates. The flavor eigenstates are defined as follows

	$\displaystyle\Xi_{Q}^{\bar{3}}$	$\displaystyle=\frac{1}{\sqrt{2}}(qs-sq)Q,$
	$\displaystyle\Xi_{Q}^{6}$	$\displaystyle=\frac{1}{\sqrt{2}}(qs+sq)Q$		(4)

with $Q=c,b$ and $q=u,d$. Eqs. (4) are of course the classification of quark model, where $\Xi_{Q}^{\bar{3}}$ belongs to the SU(3) flavor antitriplet, and $\Xi_{Q}^{6}$ belongs to the sextet, as indicated by their notations. The two light quarks are usually considered to form a scalar diquark and an axial-vector diquark in $\Xi_{Q}^{\bar{3}}$ and $\Xi_{Q}^{6}$, respectively. In reality, the physical mass eigenstates $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$ are the mixing of flavor eigenstates

$$\left(\begin{array}[]{c}|\Xi_{Q}\rangle\\ |\Xi_{Q}^{\prime}\rangle\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}|\Xi_{Q}^{\bar{3}}\rangle\\ |\Xi_{Q}^{6}\rangle\end{array}\right).$$

(5)

Although there already exists a QCD sum rules analysis in Ref. Aliev:2010ra , while in this work, we will highlight the following points:

• •

  New definitions (see Eq. (9)) of interpolating currents are adopted. These definitions have been proved in a quark model Zhao:2023yuk , and are considered to be possibly better definitions of interpolating currents for baryons.

• •

  We attempt to reveal the nature of mixing. Through detailed calculation, one can clearly see that the gluon exchange involving the two light quarks plays a crucial role in flavor mixing. It is gluon exchange that can change the spin of the system of two light quarks.

• •

  In the heavy quark limit, the spin of the system of two light quarks is a good quantum number, therefore, the mixing angle between $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$ should be zero. Our calculation results show such a trend.

The rest of this article is arranged as follows. In Sec. II, QCD sum rules analysis is performed, contributions from up to dimension-6 four-quark operators are considered. In Sec. III, numerical results are shown and are compared with other predictions in the literature. We conclude this article in the lat section.

The mass sum rule for $\Xi_{Q}^{(\prime)}$ can be obtained by considering the following two-point correlation function

$$\Pi^{(\prime)}(p)=i\int d^{4}xe^{ip\cdot x}\langle 0|T\{J^{(\prime)}(x)\bar{J}^{(\prime)}(0)\}|0\rangle,$$

(6)

where $J^{(\prime)}$ stands for the interpolating current of the mass eigenstate $\Xi_{Q}^{(\prime)}$. It is natural to expect that $\bar{J}^{(\prime)}$ creates only $\Xi_{Q}^{(\prime)}$ and not the other one, and in this sense, the following two correlation functions should be zero

	$\displaystyle i\int d^{4}xe^{ip\cdot x}\langle 0|T\{J(x)\bar{J}^{\prime}(0)\}|0\rangle$	$\displaystyle=0,$
	$\displaystyle i\int d^{4}xe^{ip\cdot x}\langle 0|T\{J^{\prime}(x)\bar{J}(0)\}|0\rangle$	$\displaystyle=0.$		(7)

$J$ and $J^{\prime}$ are linear combinations of $J_{0}$ and $J_{1}$ – the interpolating currents of flavor eigenstates $\Xi_{Q}^{\bar{3}}$ and $\Xi_{Q}^{6}$:

$$\left(\begin{array}[]{c}J\\ J^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}J_{0}\\ J_{1}\end{array}\right),$$

(8)

which is a counterpart of Eq. (5). However, it should be noted that since there is no exact one-to-one correspondence between the interpolating current and the hadron state, the quark-hadron duality ansatz is actually implicit in Eq. (8). In this work, $J_{0,1}$ are given by

	$\displaystyle J_{0}$	$\displaystyle=\epsilon_{abc}[q_{a}^{T}C\gamma_{5}(1+\not{v})s_{b}]Q_{c},$
	$\displaystyle J_{1}$	$\displaystyle=\epsilon_{abc}[q_{a}^{T}C(\gamma^{\mu}-v^{\mu})(1+\not{v})s_{b}]\frac{1}{\sqrt{3}}\gamma_{\mu}\gamma_{5}Q_{c},$		(9)

where $a,b,c$ are color indices, and $v^{\mu}\equiv p^{\mu}/\sqrt{p^{2}}$ is the 4-velocity of baryon. As mentioned in the Introduction, these new definitions have been proved in a quark model, and are possibly better definitions of interpolating currents for baryons.

It can be seen from Eqs. (6) and (7) that, we need to calculate the following 4 correlation functions

$$\Pi_{ij}(p)=i\int d^{4}xe^{ip\cdot x}\langle 0|T\{J_{i}(x)\bar{J}_{j}(0)\}|0\rangle$$

(10)

with $i,j=0,1$.

From Eq. (6), one can obtain the mass sum rules for $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$

	$\displaystyle\Pi=\Pi_{00}\cos^{2}\theta+\Pi_{11}\sin^{2}\theta+\Pi_{01}\sin 2\theta$	,		(11)
	$\displaystyle\Pi^{\prime}=\Pi_{11}\cos^{2}\theta+\Pi_{00}\sin^{2}\theta-\Pi_{01}\sin 2\theta$	.		(12)

As explicit calculation has shown, $\Pi_{01}=\Pi_{10}$, then from Eq. (7), one can arrive at

$$\Pi_{01}\cos 2\theta+(\Pi_{11}-\Pi_{00})\frac{1}{2}\sin 2\theta=0,$$

(13)

or

$$\tan 2\theta=\frac{2\ \Pi_{01}}{\Pi_{00}-\Pi_{11}}.$$

(14)

One can easily check that the above description is equivalent to the following matrix diagonalization formula

$$O\Pi O^{-1}=\Pi_{{\rm diag}}$$

(15)

with

$$\Pi=\left(\begin{array}[]{cc}\Pi_{00}&\Pi_{01}\\ \Pi_{01}&\Pi_{11}\end{array}\right),\quad O=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}\right),\quad\Pi_{{\rm diag}}=\left(\begin{array}[]{cc}\Pi&0\\ 0&\Pi^{\prime}\end{array}\right).$$

(16)

One important note. From Eq. (14), one can see that, we had better normalize the two interpolating currents in Eq. (9) to a same factor, and we have indeed done that. Therefore, in this work, $\Pi_{00}$, $\Pi_{11}$, and $\Pi_{01}$ are on an equal footing, and we can explicitly compare their respective contributions from the same dimensions at the QCD level, see below.

In this work, we calculate the 4 correlation functions in Eq. (10), considering the contributions from perturbative term (dim-0), quark condensate (dim-3), gluon condensate (dim-4), quark-gluon condensate (dim-5), and four-quark condensate (dim-6), as can be seen in Fig. 1. The analytical results are listed in Appendix A. Through detailed calculation, one can clearly see that:

• •

  For $\Pi_{01}$, it turns out that, only 4 diagrams are nonzero–they are dim-4(a,b) and dim-5(a,c). The physical meaning of $\Pi_{01}$ is: it provides the absolute possibility for the diquark to transition from $0^{+}$ to $1^{+}$, or vice versa. As far as we are concerned, the mixing between $\Xi_{Q}^{\bar{3}}$ and $\Xi_{Q}^{6}$ originates from that the two light quarks exchange gluons with the background fields in vacuum, and with the heavy quark $Q$.

• •

  For $\Pi_{00}$ and $\Pi_{11}$, dim-0,3,6, and dim-4(d,e,f) are respectively equal to each other, so they do not contribute to the denominator $\Pi_{00}-\Pi_{11}$ in Eq. (14). Only dim-4(a,b,c) and dim-5(a,b,c,d) contribute to $\Pi_{00}-\Pi_{11}$. The physical meaning of $\Pi_{00}-\Pi_{11}$ is: it measures the difference, or the “gap” between $\Xi_{Q}^{\bar{3}}$ and $\Xi_{Q}^{6}$; The larger the difference, the less likely the two flavor eigenstates are to mix.

The mixing angle formula in Eq. (14) is of course our main research object. However, the corresponding QCD sum rules are very different from the traditional ones: it does not have the hadron-level representation. For this point, try to consider the hadron-level representation of $\Pi_{01}$. That is, Eq. (14) has only a representation at the QCD level. The continuum threshold parameter $\sqrt{s_{0}}$ and Borel parameters $T^{2}$ cannot determined by the methods commonly used in the literature. However, note that a reasonable threshold parameter for Eq. (14) should lie between those of $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$. Naturally, in the following, we present the mass sum rule of $\Xi_{Q}^{(\prime)}$.

The following parameters are adopted ParticleDataGroup:2022pth :

	$\displaystyle m_{c}(m_{c})=1.27\pm 0.02\ {\rm GeV},\quad m_{s}(2\ {\rm GeV})=0.093\pm 0.009\ {\rm GeV},$
	$\displaystyle m_{b}(m_{b})=4.18\pm 0.03\ {\rm GeV}.$		(25)

The condensate parameters are taken as Colangelo:2000dp : $\langle\bar{q}q\rangle(1\ {\rm GeV})=-(0.24\pm 0.01\ {\rm GeV})^{3}$, $\langle\bar{s}s\rangle=(0.8\pm 0.2)\langle\bar{q}q\rangle$, and $\langle g_{s}^{2}G^{2}\rangle=(0.47\pm 0.14)\ {\rm GeV}^{4}$, and $\langle\bar{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$ and $\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$ with $m_{0}^{2}=(0.8\pm 0.2)\ {\rm GeV}^{2}$. The renormalization scale is taken as $\mu_{c}=1\sim 3\ {\rm{\rm GeV}}$, and $\mu_{b}=3\sim 6\ {\rm{\rm GeV}}$, from which, one can estimate the dependence of calculation results on the energy scale.

Following similar steps in Refs. Zhao:2020mod ; Zhao:2021sje , one can arrive at the optimal parameter selections for continuum thresholds $\sqrt{s_{0}}$ and Borel parameters $T^{2}$ for $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$. The corresponding results can be found in Fig. 2 and Table 1. Some comments are given in order.

• •

  As expected in Ref. Zhao:2023yuk , the pole residues of $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$ are almost equal when the interpolating currents in Eq. (9) are used.

• •

  The continuum threshold and Borel parameter at the minimum point in Fig. 2 are selected as the optimal parameters, as can be seen in Table 1. These optimal parameters correspond to the experimental value of the baryon mass.

• •

  As can be seen in Table 1, the optimal parameter selection satisfies: $\sqrt{s_{0}}$ is about 0.5 GeV higher than the corresponding baryon mass, and $T^{2}\sim{\cal O}(m_{H}^{2})$ with $m_{H}$ the baryon mass.

• •

  As can be seen in Fig. 2, the dependence of pole residues on the Borel parameters is weak, while they are sensitive to changes in energy scales. The latter leads to the main source of error.

For the sum rule in Eq. (14), considering the continuum threshold should lie between those of $\Xi_{Q}$ and $\Xi_{Q}^{\prime}$, and assuming $T^{2}\sim{\cal O}(m_{H}^{2})$, we choose the following parameters:

• •

  For $\theta_{c}$, when $\mu=m_{c}$, $\sqrt{s_{0}}=2.98\ {\rm GeV}$, and when $\mu=3\ {\rm GeV}$, $\sqrt{s_{0}}=3.05\ {\rm GeV}$; the Borel parameters $T^{2}\in[6,14]\ {\rm GeV}^{2}$.

• •

  For $\theta_{b}$, when $\mu=m_{b}$, $\sqrt{s_{0}}=6.33\ {\rm GeV}$, and when $\mu=6\ {\rm GeV}$, $\sqrt{s_{0}}=6.38\ {\rm GeV}$; the Borel parameters $T^{2}\in[30,70]\ {\rm GeV}^{2}$.

Our main results are shown in Fig. 3, and the corresponding central values and error estimates are:

• •

  $\theta_{c}=(1.3\pm 0.1)\degree$ from the first sum rule, and $\theta_{c}=(2.0\pm 0.8)\degree$ from the second sum rule;

• •

  $\theta_{b}=(0.31\pm 0.03)\degree$ from the first sum rule, and $\theta_{b}=(0.32\pm 0.02)\degree$ from the second sum rule.

Here the first and second sum rules respectively refer to those from the coefficients of $\not{p}$ and constant terms, since all the $\Pi_{ij}$, at the QCD level, can be computed like

$$\Pi^{{\rm QCD}}(p)=A(p^{2})\not{p}+B(p^{2}).$$

(26)

In Table 2, we compare our results with others in the literature. It can be seen that, our result for $\theta_{c}$ is consistent with that of Lattice QCD in Ref. Liu:2023feb if the uncertainty is taken into account.

There is a tension between the recent Belle’s measurement and Lattice QCD calculation for the branching ratio of semileptonic decay ${\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})$. Some people proposed that it is possible to resolve this puzzle by considering the $\Xi_{Q}-\Xi_{Q}^{\prime}$ mixing. Following this suggestion, we investigate the $\Xi_{Q}-\Xi_{Q}^{\prime}$ mixing using QCD sum rules in this work. Contributions from up to dimension-6 four-quark operators are considered. However, it turns out that only dimension-4 and dimension-5 operators contribute, which reveals the non-perturbative nature of mixing. Especially we notice that only the diagrams with the two light quarks participating in gluon exchange contribute to the mixing. Contributions from three-gluon condensate, and radiative corrections in Fig. 4 may be sizable and deserve further investigation. We leave these more detailed consideration for future works.

Our results show that the mixing angle $\theta_{c}$ is very small, and is consistent with the most recent Lattice QCD calculation result within error. Such a small mixing angle seems unlikely to resolve the tension between experimental measurement and Lattice QCD calculation for the semileptonic decay $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$. We have to draw the conclusion that the tension is still there.

Finally, it is worth pointing out that Ref. Xing:2022phq recently proposed a method for measuring the mixing angle in experiment, which is helpful to further clarify the issue of the $\Xi_{Q}-\Xi_{Q}^{\prime}$ mixing.

The authors are grateful to Profs. Yue-Long Shen, Wei Wang, Zhi-Gang Wang, and Drs. Hang Liu, Zhi-Peng Xing for valuable discussions. This work is supported in part by National Natural Science Foundation of China under Grant No. 12065020.