Weak decays of doubly heavy baryons: <inline-formula><tex-math id="M1-3">\begin{document}${{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="CPC-2020-0577_M1-3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="CPC-2020-0577_M1-3.png"/></alternatives></inline-formula>

Run-Hui Li; Juan-Juan Hou; Bei He; Ya-Ru Wang

doi:10.1088/1674-1137/abe0bc

Abstract：

The discovery of

$\Xi_{cc}^{++}$ has inspired new interest in studying doubly heavy baryons. In this study, the weak decays of a doubly charmed baryon

${\cal B}_{cc}$ to a light baryon

${\cal B}$ and a charm meson

$D^{(*)}$ (either a pseudoscalar or a vector one) are calculated. Following our previous work, we calculate the short distance contributions under the factorization hypothesis, whereas the long distance contributions are modeled as the final state interactions, which are calculated with the one particle exchange model. We find that the

${\cal B}_{cc}\to {\cal B} D^{*}$ decays' branching ratios are obviously larger, as they receive contributions of more polarization states. Among the decays that we investigate, the following have the largest branching fractions:

${\cal BR}(\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{*+}) \in [0.46 \%, 3.33 \%]$ estimated with

$\tau_{\Xi_{cc}^{++}} = 256$ fs;

${\cal BR}(\Xi_{cc}^{+}\rightarrow\Lambda D^{*+}) \in [0.38 \%, 2.63 \%]$ and

${\cal BR}(\Xi_{cc}^{+}\rightarrow\Sigma^{0} D^{*+}) \in [0.45 \%, 3.16 \%]$ with

$\tau_{\Xi_{cc}^+} = 45$ fs; and

${\cal BR}(\Omega_{cc}^{+}\rightarrow \Xi^{0} D^{*+}) \in [0.27 \%, 1.03 \%]$ ,

${\cal BR}(\Omega_{cc}^{+}\rightarrow\Xi^{0} D^{+}) \in [0.07 \%, 0.44 \%]$ , and

${\cal BR}(\Omega_{cc}^{+}\rightarrow\Sigma^{0} D^{*+}) \in [0.06 \%, 0.45 \%]$ with

$\tau_{\Omega_{cc}^+} = 75$ fs. By comparing the decay widths of pure color commensurate channels with those of pure bow-tie ones, we find that the bow-tie mechanism plays an important role in charm decays.

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I. INTRODUCTION

Studies on doubly heavy baryons that contain two heavy constituent quarks (c or b quark) have been conducted for a long time. They are predicted by the quark model and allowed by the quantum chrodynamics theory. Physicists believe in their existence, even though they have not been found in experiments. The SELEX collaboration announced the discovery of $\Xi_{cc}^+$ in 2002 and 2005 [1, 2]. However, the reported production is large, and the lifetime that was measured is very long. Their results do not agree with the theoretical predictions and have not been confirmed by other experiments. In 2017, the LHCb collaboration declared the discovery of $\Xi_{cc}^{++}$ via $\Xi_{cc}^{++}\to \Lambda_c^+ K^-\pi^+\pi^+$ with $m_{\Xi_{cc}^{++}} = 3.621$ GeV [3]. In 2018, the LHCb collaboration measured its lifetime as $256$ fs [4] and confirmed the discovery via $\Xi_{cc}^{++}\to \Xi_c^+\pi^+$ [5]. $\Xi_{cc}^{++}$ is the first doubly heavy baryon to be discovered in experiments with properties that agree with the theoretical expectations. Its discovery is meaningful to the study of the hadron spectrum and baryon decays. Physicists have already conducted substantial research on the spectrum of doubly heavy baryons. However, the determination of a proper framework to study their weak decays is a very challenging task. Numerous studies have been conducted on this topic [6-24], and the form factors as well as the semileptonic decays of a doubly heavy baryon to a singly heavy baryon have been studied under various frameworks. However, few systematic methods are available to deal with even two body nonleptonic decays, which is essential for guiding new particle discoveries, understanding the dynamics of strong interactions, and testing the standard model precisely.

In 2017, we applied final state interactions (FSIs) to baryon decays at the charm scale to estimate the branching fractions of two body nonleptonic weak decays of doubly charmed baryons [25]. We suggested two golden discovery channels of $\Xi_{cc}^{++}$ , which, as mentioned above, were adopted by the LHCb collaboration and aided in the discovery of the $\Xi_{cc}^{++}$ particle. The discovery inspired the research of doubly heavy baryons and further questions are posed: which are the golden discovery channels of the other doubly charmed baryons and what else can we find in the decays of doubly charmed baryons? To answer these questions, further research on the weak decays of doubly heavy baryons is required. In our previous work, we calculated the decays of a doubly charmed baryon to a singly charmed baryon and a light vector meson. We also investigated the possibility of these decays as potential discovery channels [26]. After discovering $\Xi_{cc}^{++}$ , measuring its lifetime, and confirming the discovery with another decay, the LHCb collaboration also focused on the weak decays of $\Xi_{cc}^{++}$ with a charm meson in the final state [27]. Motivated by these theoretical questions and experimental efforts, we study the two body nonleptonic decays of a doubly charmed baryon ${\cal B}_{cc}\to {\cal B} D^{(*)}$ , where ${\cal B}_{cc}$ represents a doubly charmed baryon, ${\cal B}$ denotes a light baryon, and $D^{(*)}$ is either a pseudoscalar or vector charm meson.

There are many interesting physics to be explored in baryon decays. For example, the CP violations have already been observed in K, B, and D meson decays but have not been observed in baryon decays. Theoretical progress in this topic is slow because it is challenging to calculate the dynamics. No systematic factorization method has been established thus far, even for two body nonleptonic decays. In general, the contributions in two body nonleptonic baryon decays can be topologically classified into several types: T, C, E, and B [28]. In b baryon decays, the E and B contributions are numerically small [29]. In the charm sector, the situation differs, and the E and B contributions may become important [30]. The study of these decays will aid in understanding the dynamics of baryon decays at the charm scale.

The remainder of this paper is organized as follows. In Section II, the phenomenological framework is introduced, the contributions in these decays are discussed, and the analytical expressions are presented. Section III presents several inputs, tables of our results, and discussions. A summary is provided in Section IV. Owing to space limitations, we list all of the expressions of the amplitudes in Appendix B, whereas the strong couplings are presented in Appendix C.

II. THEORETICAL FRAMEWORK AND ANALYTICAL CALCULATIONS

IV. SUMMARY

The discovery of $\Xi_{cc}^{++}$ in 2017 has inspired interest in studying doubly charmed baryons. Among all of the related topics, how to calculate their weak decays is a meaningful and challenging one, which can provide valuable suggestions for experimental research as well as aid in understanding the dynamics of baryon decays. In our previous work, we applied the model of FSIs to baryon decays and realized the estimation of two body nonleptonic decays of charm baryons.

In this study, we calculated the decays of a doubly charmed baryon to a light baryon and a charm meson. In the same decay mode, ${\cal B}_{cc} \to {\cal B} D$ or ${\cal B}_{cc} \to {\cal B} D^*$ , the CKM favored, singly CKM suppressed, and doubly CKM suppressed decays fall into a hierarchy naturally. The ${\cal B}_{cc} \to {\cal B} D^*$ decays tends to have larger branching ratios or decay widths because they have more polarization states. Moreover, $\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{*+}$ has the largest branching ratio in $\Xi_{cc}^{++} \to {\cal B} D^{(*)}$ decays, which lies in the range of $(0.46\sim 3.33) \%$ . The two largest branching ratios in the $\Xi_{cc}^{+} \to {\cal B} D^{(*)}$ mode are ${\cal BR}(\Xi_{cc}^{+}\rightarrow\Lambda D^{*+}) \in [0.38 \%, 2.63 \%]$ and ${\cal BR}(\Xi_{cc}^{+}\rightarrow\Sigma^{0} D^{*+}) \in [0.45 \%, 3.16 \%]$ , which are estimated with $\tau_{\Xi_{cc}^+} = 45$ fs. For the $\Omega_{cc}^{+} \to {\cal B} D^{(*)}$ mode, ${\cal BR}(\Omega_{cc}^{+}\rightarrow\Xi^{0} D^{*+}) \in [0.27 \%, 1.03 \%]$ , ${\cal BR}(\Omega_{cc}^{+}\rightarrow\Xi^{0} D^{+}) \in [0.07 \%, 0.44 \%]$ , and ${\cal BR}(\Omega_{cc}^{+}\rightarrow\Sigma^{0} D^{*+}) \in [0.06 \%, 0.45 \%]$ are the three largest ones, and they are calculated with $\tau_{\Omega_{cc}^+} = 75$ fs.

By comparing the decay widths of pure color commensurate processes with those of pure bow-tie processes, we found that the bow-tie mechanism also plays an important role in charm decays.

ACKNOWLEDGEMENTS

We would like to thank Z.-T. Zou, F.-S. Yu, and F.-K. Guo for helpful discussions.

APPENDIX A: EXPRESSIONS OF DIAGRAMS (C)-(I) IN FIG. 3

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$f_{\pi}$	$f_{K}$	$f_{\eta_8}$	$f_{\eta_1}$	$f_{\rho}$	$f_{K^*}$	$f_{\omega}$	$f_{\phi}$
130	156	163	152	216	217	195	233

Channel	${\cal BR}(10^{-3})$	CKM	Channel	${\cal BR}(10^{-3})$	CKM
$\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{+}$	$2.98_{-2.02}^{+3.16}$	CF	$\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{*+}$	$16.06_{-10.50}^{+17.28}$	CF
$\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{+}_{s}$	$0.17_{-0.12}^{+0.18}$	SCS	$\Xi_{cc}^{++}\rightarrow\Sigma^{+}D^{*+}_{s}$	$2.68_{-1.71}^{+2.64}$	SCS
$\Xi_{cc}^{++}\rightarrow p D^{+}$	$0.16_{-0.11}^{+0.18}$	SCS	$\Xi_{cc}^{++}\rightarrow p D^{*+}$	$2.96_{-2.06}^{+3.38}$	SCS
$\Xi_{cc}^{++}\rightarrow p D^{+}_{s}$	$0.01_{-0.00}^{+0.02}$	DCS	$\Xi_{cc}^{++}\rightarrow p D^{*+}_{s}$	$0.11_{-0.07}^{+0.13}$	DCS

Weak decays of doubly heavy baryons: ${{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}$

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Weak decays of doubly heavy baryons: ${{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}$

Corresponding author: Run-Hui Li, lirh@imu.edu.cn

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A. Theoretical framework

B. Calculation of weak vertices

C. Rescattering at long distance

D. Analytical expressions for diagrams

目录

1.	Tousi, M.S., Azizi, K. Properties of doubly heavy spin- 12 baryons: The ground and excited states[J]. Physical Review D, 2024, 109(5): 054005. doi: 10.1103/PhysRevD.109.054005
2.	Chen, H.-X., Chen, W., Liu, X. et al. An updated review of the new hadron states[J]. Reports on Progress in Physics, 2023, 86(2): 026201. doi: 10.1088/1361-6633/aca3b6

Channel	$\Gamma/{\rm GeV}$	CKM	Contributions	Channel	$\Gamma/{\rm GeV}$	CKM	Contributions
$\Xi_{cc}^{+}\rightarrow\Sigma^{0} D^{+}$	$(5.93_{-4.05}^{+6.31})*10^{-15}$	CF	$C^\prime$ B	$\Xi_{cc}^{+}\rightarrow\Lambda D^{*+}$	$(1.82_{-1.26}^{+2.03})*10^{-13}$	CF	$C^\prime$ B
$\Xi_{cc}^{+}\rightarrow\Lambda D^{+}$	$(5.84_{-3.98}^{+6.16})*10^{-15}$	CF	$C^\prime$ B	$\Xi_{cc}^{+}\rightarrow\Sigma^{0} D^{*+}$	$(2.17 _{-1.51}^{+2.45})*10^{-13}$	CF	$C^\prime$ B
$\Xi_{cc}^{+}\rightarrow\Sigma^{+} D^{0}$	$(1.23 _{-0.77 }^{+1.24 })*10^{-15}$	CF	B	$\Xi_{cc}^{+}\rightarrow\Sigma^{+} D^{*0}$	$(6.77 _{-4.46}^{+7.37})*10^{-14}$	CF	B
$\Xi_{cc}^{+}\rightarrow\Xi^{0} D_{s}^{+}$	$(4.52 _{-3.49}^{+5.22})*10^{-16}$	CF	B	$\Xi_{cc}^{+}\rightarrow\Xi^{0} D_{s}^{*+}$	$(2.52 _{-1.48 }^{+2.23})*10^{-14}$	CF	B
$\Xi_{cc}^{+}\rightarrow p D^{0}$	$(1.85 _{-1.27}^{+2.02})*10^{-15}$	SCS	B	$\Xi_{cc}^{+}\rightarrow\Sigma^{0} D_{s}^{*+}$	$(1.15_{-0.85}^{+1.41})*10^{-14}$	SCS	$C^\prime$ B
$\Xi_{cc}^{+}\rightarrow\Lambda D_{s}^{+}$	$(3.00_{-2.00}^{+2.93})*10^{-16}$	SCS	$C^\prime$ B	$\Xi_{cc}^{+}\rightarrow\Lambda D_{s}^{*+}$	$(1.58_{-1.09}^{+1.73})*10^{-14}$	SCS	$C^\prime$ B
$\Xi_{cc}^{+}\rightarrow n D^{+}$	$(1.59 _{-1.13}^{+1.87})*10^{-16}$	SCS	$C^\prime$ B	$\Xi_{cc}^{+}\rightarrow n D^{*+}$	$(1.04 _{-0.87 }^{+1.41})*10^{-15}$	SCS	$C^\prime$ B
$\Xi_{cc}^{+}\rightarrow\Sigma^{0} D_{s}^{+}$	$(2.85_{-1.78}^{+2.72})*10^{-16}$	SCS	$C^\prime$ B	$\Xi_{cc}^{+}\rightarrow p D^{*0}$	$(9.46 _{-6.49}^{+10.20 })*10^{-15}$	SCS	B
$\Xi_{cc}^{+}\rightarrow n D_{s}^{+}$	$(3.26 _{-2.41}^{+3.88})*10^{-17}$	DCS	$C^\prime$	$\Xi_{cc}^{+}\rightarrow n D_{s}^{*+}$	$(1.47 _{-1.00}^{+1.57})*10^{-16}$	DCS	$C^\prime$

Vertex	$f_{1}$	$f_{2}$	Vertex	$f_{1}$	$f_{2}$	Vertex	$f_{1}$	$f_{2}$
$p\rightarrow\ n\rho^{+}$	−2.40	32.95	$\Lambda\rightarrow\Sigma^{-}\rho^{+}$	2.00	12.30	$\Sigma^{0}\rightarrow\Sigma^{+}\rho^{+}$	7.20	−25.00
$\Sigma^{+}\rightarrow\Lambda\rho^{+}$	2.00	12.30	$\Sigma^{+}\rightarrow p\overline{K}^{*0}$	5.66	−1.70	$\Sigma^{+}\rightarrow\Xi^{0}K^{*+}$	−2.26	37.05
$\Sigma^{+}\rightarrow\Sigma^{+}\phi$	−6.00	2.50	$p\rightarrow\Sigma^{0}K^{*+}$	4.00	−1.20	$p\rightarrow\Lambda K^{*+}$	5.10	−28.00
$p\rightarrow\Sigma^{+}K^{*0}$	5.66	−1.70	$\Sigma^{0}\rightarrow\Sigma^{-}\rho^{+}$	−7.20	25.00	$\Sigma^{0}\rightarrow n\overline{K}^{*0}$	−4.00	1.20
$\Lambda\rightarrow n\overline{K}^{*0}$	5.10	−28.00	$\Sigma^{0}\rightarrow\Xi^{-}K^{*+}$	−1.60	26.20	$\Sigma^{0}\rightarrow\Sigma^{0}\phi$	−6.00	2.50
$\Sigma^{0}\rightarrow\Sigma^{0}\omega$	4.30	−1.10	$\Lambda\rightarrow\Sigma^{0}\rho^{0}$	1.90	11.90	$p\rightarrow p\rho^{0}$	−2.50	22.20
$\Lambda\rightarrow\Xi^{0}K^{*0}$	−6.00	17.10	$\Lambda\rightarrow\Xi^{-}K^{*+}$	−6.00	17.10	$\Lambda\rightarrow\Lambda\phi$	−5.30	24.60
$n\rightarrow\Sigma^{0}K^{*0}$	−4.00	1.20	$\Lambda\rightarrow\Lambda\omega$	−7.10	8.70	$n\rightarrow\Sigma^{-}K^{*+}$	5.66	−1.70
$n\rightarrow\Lambda K^{*0}$	5.10	−28.00	$\Xi^{-}\rightarrow\Sigma^{+}\overline{K}^{*0}$	−2.26	37.05	$n\rightarrow n\rho^{0}$	2.50	−22.20
$\Xi^{0}\rightarrow\Lambda\overline{K}^{*0}$	4.45	−27.54	$\Xi^{0}\rightarrow\Xi^{-}\rho^{+}$	6.08	−1.56	$\Xi^{0}\rightarrow\Sigma^{0}\overline{K}^{*0}$	1.60	−26.20
$\Xi^{0}\rightarrow\Xi^{0}\phi$	−9.50	32.30	$\Sigma^{0}\rightarrow\Xi^{0}K^{*0}$	1.60	−26.20

Vertex	g	Vertex	g	f
$\Lambda_{c}\rightarrow N{D}_{q}$	4.82	$\Lambda_{c}\rightarrow N{D}_{q}^{*}$	−5.80	3.60
$\Sigma_{c}\rightarrow N{D}_{q}$	3.78	$\Sigma_{c}\rightarrow N{D}_{q}^{*}$	11.21	4.64

Weak decays of doubly heavy baryons: Bcc→BD(∗){{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}

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Weak decays of doubly heavy baryons: Bcc→BD(∗){{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}

Corresponding author: Run-Hui Li, lirh@imu.edu.cn

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A. Theoretical framework

B. Calculation of weak vertices

C. Rescattering at long distance

D. Analytical expressions for diagrams

目录

Weak decays of doubly heavy baryons: ${{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}$

Weak decays of doubly heavy baryons: ${{\cal{B}}_{{cc}}\to {\cal{B}} D^{(*)}}$