1 Introduction

Bottom quarks can be produced at the high energy colliders, such as LHC and Tevatron, and then hadronized into B mesons and b-baryons. The probability of a bottom quark fragments into a certain weakly decaying b-hadron is called the fragmentation fractions, i.e. \(f_{u,d,s}\equiv \mathcal {B}(b\rightarrow B^-,\overline{B}^0, \overline{B}_s^0)\), \(f_{\Lambda _b}\equiv \mathcal {B}(b\rightarrow \Lambda _b^0)\), \(f_{\Xi _b}\equiv \mathcal {B}(b\rightarrow \Xi _b^0, \Xi _b^-)\) and \(f_{\Omega _b}\equiv \mathcal {B}(b\rightarrow \Omega _b^-)\). As non-perturbative effects, the fragmentation fractions can only be determined by experiments in some phenomenological approaches.

Table 1 List of measurements related to the fragmentation fraction ratio \(f_{\Xi _b}/f_{\Lambda _b}\)
Full size table

The B-meson fragmentation fractions have been measured by LEP, Tevatron and LHC with a relatively high precision [1, 2]. However, the current understanding of b-baryon productions is still a challenge. The total fragmentation fraction of b-baryon is the sum of all the weakly-decaying b-baryons,

$$\begin{aligned} f_\mathrm{baryon}=f_{\Lambda _b}\left( 1+2{f_{\Xi _b}\over f_{\Lambda _b}}+{f_{\Omega _b}\over f_{\Lambda _b}}\right) =f_{\Lambda _b}(1+\delta ), \end{aligned}$$
(1)

where the isospin symmetry is assumed as \(f_{\Xi _b^-}=f_{\Xi _b^0}=f_{\Xi _b}\), and \(\delta =2{f_{\Xi _b}\over f_{\Lambda _b}}+{f_{\Omega _b^-}\over f_{\Lambda _b}}\) is the correction from \(f_{\Lambda _b}\) to \(f_\mathrm{baryon}\). The averages of the total baryon production fractions are [2]

$$\begin{aligned} f_\mathrm{baryon}=\left\{ \begin{aligned}&0.084\pm 0.011,\quad \text {at LEP,}\\&0.196\pm 0.046,\quad \text {at Tevatron}, \end{aligned} \right. \end{aligned}$$
(2)

which are inconsistent with each other, and of large uncertainties.

The total fraction of b-baryons has not been determined by LHCb because of its lack of measurements on \(\Xi _{b}^{0,-}\) and \(\Omega _{b}^{-}\). It has been found that the ratio \(f_{\Lambda _{b}}/f_{d}\) depends on the \(p_\mathrm{T}\) of the final states [3,4,5,6]. At LHCb, the kinematic averaging ratio is [5]

$$\begin{aligned} {f_{\Lambda _{b}}\over f_{u}+f_{d}}\Big |_\mathrm{LHCb}=0.240\pm 0.022. \end{aligned}$$
(3)

It is required for the information of \(f_{\Xi _{b}}\) and \(f_{\Omega _{b}}\)to determine the other fragmentation fractions at LHCb due to the constraint of

$$\begin{aligned} f_{u}+f_{d}+f_{s}+f_\mathrm{baryon}=1. \end{aligned}$$
(4)

Since the production of \(\Omega _{b}^{-}\) is suppressed compared to those of \(\Xi _{b}^{0,-}\) by the production of an additional strange quark, the determination of \(f_{\Xi _b}/f_{\Lambda _b}\) is essential to understand the productions of b-baryons and B mesons.

So far, only Refs. [7, 8] have predicted the ratio \(f_{\Xi _b}/f_{\Lambda _b}\), both based on the processes of \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) and \(\Lambda _b^0\rightarrow J/\psi \Lambda \) with the data given by CDF and D0. At LHCb, the productions of \(\Lambda _{b}\) and \(\Xi _{b}\) have been measured by the heavy-flavor-conserving process of \(\Xi _{b}^{-}\rightarrow \Lambda _{b}^{0}\pi ^{-}\) [9], and the charm-baryon involving decays of \(\Xi _b^0\rightarrow \Xi _c^+\pi ^-\) and \(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-\) via \(\Xi _c^+/\Lambda _c^+\rightarrow pK^-\pi ^+\) [10]. All the results are listed in Table 1. The production with the charm-baryon involving method is of the most high precision. The ratio \(f_{\Xi _b}/f_{\Lambda _b}\) can be obtained as long as the related branching fractions are determined. Among them, the absolute branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) has never been measured [1], thus is of the largest ambiguity. In this work, we determine \(f_{\Xi _b}/f_{\Lambda _b}\) with \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)\) obtained under the U-spin symmetry.

This article is organized as follows. In Sect. 2, we introduce the status of \(f_{\Xi _b}/f_{\Lambda _b}\). The branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) and \(f_{\Xi _b}/f_{\Lambda _b}\) are obtained in Sects. 3 and 4, respectively. Section 5 is the summary.

2 Status of \(f_{\Xi _b}/f_{\Lambda _b}\)

In some literatures, it is usually assumed that the difference between the productions of \(\Xi _b\) and \(\Lambda _b\) is from the strange quark and up or down quarks [10, 12],

$$\begin{aligned} {f_{\Xi _b}\over f_{\Lambda _b}}\simeq {f_s\over f_u},\quad \text {or}\quad {f_{\Xi _b}\over f_{\Lambda _b}}\simeq 0.2. \end{aligned}$$
(5)

However, since the fragmentation fractions are non-perturbative effects, they can only be extracted from experimental data. In this section, we introduce the status of \(f_{\Xi _b}/f_{\Lambda _b}\) by means of the relevant measurements.

2.1 \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) v.s. \(\Lambda _b^0\rightarrow J/\psi \Lambda \)

So far, the only theoretical analysis on \(f_{\Xi _b}/f_{\Lambda _b}\) are performed in Refs. [7, 8] based on the experimental data of \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) and \(\Lambda _b^0\rightarrow J/\psi \Lambda \). In Ref. [1], the relevant results averaging the measurements by CDF and D0 [13,14,15,16], are given as

$$\begin{aligned} \begin{aligned}&f_{\Lambda _b}\cdot \mathcal {B}(\Lambda _b^0\rightarrow J/\psi \Lambda )=(5.8\pm 0.8)\times 10^{-5}, \\&f_{\Xi _b}\cdot \mathcal {B}(\Xi _b^-\rightarrow J/\psi \Xi ^-)=(1.02_{-0.21}^{+0.26})\times 10^{-5}. \end{aligned} \end{aligned}$$
(6)

The fragmentation fraction ratio of \(f_{\Xi _b}/f_{\Lambda _b}\) can be obtained unless the ratio of branching fractions of \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) and \(\Lambda _b^0\rightarrow J/\psi \Lambda \) is known.

Both \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) and \(\Lambda _b^0\rightarrow J/\psi \Lambda \) are the \(b\rightarrow c \bar{c} s\) transitions with the spectators of \((ds-sd)/\sqrt{2}\) and \((ud-du)/\sqrt{2}\), respectively. Therefore, the two processes are related to each other under the flavor SU(3) symmetry. The width relation of

$$\begin{aligned} \Gamma (\Lambda _b^{0}\rightarrow J/\Psi \Lambda )=\frac{2}{3}\Gamma (\Xi _b^-\rightarrow J/\Psi \Xi ^-), \end{aligned}$$
(7)

is given by Voloshin [7]. Using the experimental data in (6), the ratio of the fragmentation fractions can then be obtained as [7]

$$\begin{aligned} \frac{f_{\Xi _b}}{f_{\Lambda _b}}=0.11\pm 0.03. \end{aligned}$$
(8)

Hsiao et ac express the decay amplitudes of \(\Xi _b^-\rightarrow J/\psi \Xi ^-\) and \(\Lambda _b^0\rightarrow J/\psi \Lambda \) in the factorization approach [8]. They relate the form factors of b-baryon to light baryon octet transitions based on the SU(3) symmetry, and obtain the ratio of branching fractions \(\mathcal {B}(\Xi _b^-\rightarrow J/\psi \Xi ^-)/\mathcal {B}(\Lambda _b^0\rightarrow J/\psi \Lambda )=1.63\pm 0.04\). Utilizing the data in Eq. (6), the authors give a result similar to Eq. (8),

$$\begin{aligned} \frac{f_{\Xi _b}}{f_{\Lambda _b}}=0.108\pm 0.034. \end{aligned}$$
(9)

2.2 Heavy-flavor-conserving decay \(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-\)

The LHCb collaboration has observed the first heavy-flavor-conserving \(\Delta S=1\) hadronic weak decay \(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-\) [9], with

$$\begin{aligned} \frac{f_{\Xi _b}}{f_{\Lambda _b}}\cdot \mathcal {B}(\Xi _b^-\rightarrow \Lambda _b^0 \pi ^-)=(5.7\pm 1.8_{-0.9}^{+0.8})\times 10^{-4}. \end{aligned}$$
(10)

In Ref. [9], \(f_{\Xi _b}/f_{\Lambda _b}\) is assumed to be bounded between 0.1 and 0.3, and then obtain the branching fraction of \(\Xi _b^-\rightarrow \Lambda _b^0 \pi ^-\) lie in the range from \((0.57 \,\pm \, 0.21)\%\) to \((0.19 \,\pm \, 0.07)\%\). On the contrary, the fragmentation ratio \(f_{\Xi _b}/f_{\Lambda _b}\) can be obtained if \(\mathcal {B}(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-)\) is determined.

In Ref. [11], the branching fraction of \(\Xi _b^-\rightarrow \Lambda _b^0 \pi ^-\) is calculated in the MIT bag model and the diquark model,

$$\begin{aligned} \mathcal {B}(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-)=\left\{ \begin{aligned}&2.0\times 10^{-3},\qquad \text {MIT bag model,} \\&6.9\times 10^{-3},\qquad \text {diquark model.} \end{aligned} \right. \end{aligned}$$
(11)

Subsequently, we can obtain the ratio of fragmentation fractions according to Eq. (10) as,

$$\begin{aligned} \frac{f_{\Xi _b^-}}{f_{\Lambda _b^0}}=&\,0.29\pm 0.10,\qquad \qquad \text {MIT bag model,} \end{aligned}$$
(12)
$$\begin{aligned} \frac{f_{\Xi _b^-}}{f_{\Lambda _b^0}}=&\,0.08\pm 0.03,\qquad \qquad \text {diquark model.} \end{aligned}$$
(13)

2.3 \(\Xi _b^0\rightarrow \Xi _c^+\pi ^-\) v.s. \(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-\) via \(\Xi _c^+/\Lambda _c^+\rightarrow pK^-\pi ^+\)

In the above two methods, the experimental measurements are of large uncertainties, as seen in Eqs. (6) and (10). In the decay of \(\Xi _{b}^{-}\rightarrow J/\Psi \Xi ^{-}\), the efficiency of reconstruction of \(\Xi ^{-}\) with \(\Xi ^{-}\rightarrow \Lambda \pi ^{-}\) and \(\Lambda \rightarrow p \pi ^{-}\), is very small in the hadron colliders [13, 14]. On the other hand, the branching fraction of \(\Xi _{b}^{-}\rightarrow \Lambda _{b}^{0}\pi ^{-}\) is expected to be very small.

Compared to the above processes, the relative production ratio between \(\Xi _b^0\rightarrow \Xi _c^+\pi ^-\) and \(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-\) has been measured by LHCb with much higher precision [10],

$$\begin{aligned}&\frac{f_{\Xi _b^0}}{f_{\Lambda _b^0}}\cdot \frac{\mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+ \pi ^-)}{\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)}\cdot \frac{\mathcal {B} (\Xi _c^+\rightarrow pK^-\pi ^+)}{\mathcal {B}(\Lambda _c^+\rightarrow pK^-\pi ^+)}\nonumber \\&\quad =(1.88\pm 0.04\pm 0.03)\times 10^{-2}. \end{aligned}$$
(14)

As long as the branching fractions of the relevant b- and c-baryon decays are known, Eq. (14) could provide a good determination of \(f_{\Xi _b}/f_{\Lambda _b}\). In Ref. [10], with naively expected values of \(\mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+\pi ^-)/\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)\approx 1\) and \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)/\mathcal {B}(\Lambda _c^+\rightarrow pK^-\pi ^+)\approx 0.1\), it can be obtained that \(f_{\Xi _b}/f_{\Lambda _b}\approx 0.2\).

The branching fraction of \(\Xi _b^0\rightarrow \Xi _c^+\pi ^-\) has not been directly measured in experiment. \(\mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+\pi ^-)\) and \(\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)\) are equal to each other in the heavy quark limit and the flavor SU(3) symmetry. In literatures, only Refs. [17] and [18] calculate both the branching fractions of \(\Xi _b^0\rightarrow \Xi _c^+\pi ^-\) and \(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-\). With the transition form factors in the non-relativistic quark model, the ratio of branching fractions involving the factorizable contribution can be obtained in [17]:

$$\begin{aligned} \frac{\mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+\pi ^-)}{\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)}&=\frac{\Gamma (\Xi _b^0\rightarrow \Xi _c^+\pi ^-)}{\Gamma (\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)}\nonumber \\&=\frac{0.33a_1^2\times 10^{10}s^{-1}}{0.31a_1^2\times 10^{10}s^{-1}}=1.07, \end{aligned}$$
(15)

where the difference in the lifetimes is neglected since \(\tau (\Xi _b^0)/\tau (\Lambda _b^0)=1.006\pm 0.021\), and \(a_{1}=C_{1}+C_{2}/3\) is the effective Wilson coefficient. The deviation from unity results from the mass difference between \(m_{\Xi _{b}}+m_{\Xi _{c}}\) and \(m_{\Lambda _{b}}+m_{\Lambda _{c}}\), i.e. the SU(3) breaking effect. In the soft-collinear effective theory, the non-factorizable contributions from the color-commensurate and the W-exchange diagrams are suppressed by \(\mathcal {O}(\Lambda _\mathrm{QCD}/m_{b})\) [19]. In Ref. [18] in a relativistic three-quark model, it is found that the non-factorizable contributions amount up to 30% of the factorizable ones, with the ratio of \( \mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+\pi ^-)/\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+\pi ^-)=1.25. \) Therefore, even without a reliable study in a QCD-based approach, it can still be expected that the deviation of the ratio from unity is under control.

The absolute branching fraction of \(\Lambda _c^+\rightarrow pK^-\pi ^+\) has been well measured by Belle and BESIII [20, 21], with \(\mathcal {B}(\Lambda _c^+\rightarrow pK^-\pi ^+)=(6.35\pm 0.33)\%\) [1]. However, branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) is of large ambiguity. The ratio of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)/\mathcal {B}(\Lambda _c^+\rightarrow pK^-\pi ^+)\approx 0.1\) used in [10], is only naively assumed by the Cabibbo factor. In the next section, we will obtain the branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) via U-spin analysis, and then determine \(f_{\Xi _b}/f_{\Lambda _b}\).

3 Branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\)

The understanding of charmed baryon decays are still of high deficiency both in theory and in experiment. So far, there has not been any measurement on the absolute branching fraction of \(\Xi _{c}^{0,+}\) decays [1]. The ratio of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)/\mathcal {B}(\Xi _c^+\rightarrow \Xi ^-\pi ^+\pi ^+)\) has been measured to be \(0.21\pm 0.04\) [1, 22, 23]. But it is still unknown for the absolute branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\).

It is first found in Ref. [24] that \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)=(2.2\pm 0.8)\%\) can be obtained from the measured ratio of \(\mathcal {B}(\Xi _c^+\rightarrow p\overline{K}^{*0})/\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)=0.54\pm 0.10\) [22], and the U-spin relation between \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) and \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\). We show the U-spin analysis in detail in the present work.

The decays of \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) and \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\) are both singly Cabibbo-suppressed modes, with the transition of \(c\rightarrow (s\bar{s}- d\bar{d})u\) where the minus sign between \(s\bar{s}\) and \(d\bar{d}\) comes from the Cabibbo–Kobayashi–Maskawa matrix elements, \(V_{cd}^{*}V_{ud}=-V_{cs}^{*}V_{us}\). Note that the U-spin doublets are \((|d\rangle , |s\rangle )\) and \((|\bar{s}\rangle , - |\bar{d}\rangle )\). The effective Hamiltonian of \(c\rightarrow (s\bar{s}- d\bar{d})u\) changes the U-spin by \(\Delta U=1\), \(\Delta U_{3}=0\), i.e. \(|\mathcal {H}_\mathrm{eff}\rangle =\sqrt{2}|1,0\rangle \). \(\Xi _c^+\) and \(\Lambda _c^+\) form a U-spin doublet of \((\Lambda _{c}^{+}, \Xi _{c}^{+})\). We have

$$\begin{aligned} \mathcal {H}_\mathrm{eff}|\Xi _{c}^{+}\rangle&=\sqrt{2}\left| 1,0;{1\over 2},-{1\over 2}\right\rangle ={2\over \sqrt{3}}\left| {3\over 2},-{1\over 2}\right\rangle +\sqrt{2\over 3}\left| {1\over 2},-{1\over 2}\right\rangle , \end{aligned}$$
(16)
$$\begin{aligned} \mathcal {H}_\mathrm{eff}|\Lambda _{c}^{+}\rangle&=\sqrt{2}\left| 1,0;{1\over 2},{1\over 2}\right\rangle ={2\over \sqrt{3}}\left| {3\over 2},{1\over 2}\right\rangle -\sqrt{2\over 3}\left| {1\over 2},{1\over 2}\right\rangle . \end{aligned}$$
(17)

The U-spin representations of the \(|p\overline{K}^{*0}\rangle \) and \(|\Sigma ^+K^{*0}\rangle \) states are

$$\begin{aligned}&\big |p\overline{K}^{*0}\big \rangle =\left| {1\over 2},{1\over 2};1,-1\right\rangle ={1\over \sqrt{3}}\left| {3\over 2},-{1\over 2}\right\rangle -\sqrt{2\over 3}\left| {1\over 2}, -{1\over 2}\right\rangle , \end{aligned}$$
(18)
$$\begin{aligned}&\left| \Sigma ^+K^{*0}\right\rangle =\left| {1\over 2},-{1\over 2};1,1\right\rangle ={1\over \sqrt{3}}\left| {3\over 2},{1\over 2}\right\rangle +\sqrt{2\over 3}\left| {1\over 2}, {1\over 2}\right\rangle . \end{aligned}$$
(19)

The decay amplitudes are then

$$\begin{aligned} \mathcal {A}(\Xi _c^+\rightarrow p\overline{K}^{*0}) =&\,\big \langle p\overline{K}^{*0}\big |\mathcal {H}_\mathrm{eff}\big |\Xi _c^+\big \rangle ={2\over 3}A_{3/2}-{2\over 3}A_{1/2}, \end{aligned}$$
(20)
$$\begin{aligned} \mathcal {A}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}) =&\,\big \langle \Sigma ^+K^{*0}\big |\mathcal {H}_\mathrm{eff}\big |\Lambda _c^+\big \rangle ={2\over 3}A_{3/2}-{2\over 3}A_{1/2}, \end{aligned}$$
(21)

where \(A_{3/2}\) and \(A_{1/2}\) are the amplitudes of U-spin of 3 / 2 and 1 / 2, respectively. It is clear that the amplitudes satisfy

$$\begin{aligned} \mathcal {A}(\Xi _c^+\rightarrow p\overline{K}^{*0})=\mathcal {A}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}). \end{aligned}$$
(22)

This relation can also be seen from the topological diagrams in Fig. 1.

Fig. 1
figure 1

The topological diagrams of \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) and \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\). The top and bottom diagrams are color-commensurate and W-exchange ones, respectively

Full size image

According to the relation in Eq. (22), the branching ratio of \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) can be obtained by

$$\begin{aligned} \mathcal {B}(\Xi _{c}^{+}\rightarrow p\overline{K}^{*0})&=\frac{m_{\Lambda _c^+}^2}{m_{\Xi _c^+}^2}\cdot \frac{\tau _{\Xi _c^+}}{\tau _{\Lambda _c^+}}\cdot \frac{|p_c(m_{\Xi _c},m_{p},m_{K^{*}})|}{|p_c(m_{\Lambda _c},m_{\Sigma },m_{K^{*}})|}\nonumber \\&\quad \cdot \mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}), \end{aligned}$$
(23)

where \(|p_c(M,m_{1},m_{2})|={\sqrt{[M^2-(m_1+m_2)^2][M^2-(m_1-m_2)^2]}}/{2M}\). The data of masses and lifetimes are taken from PDG [1]: \(m_{\Xi _{c}}=2468\)MeV, \(m_{\Lambda _{c}}=2286\) MeV, \(m_{p}=938\) MeV, \(m_{\Sigma }=1189\) MeV, \(m_{K^{*}}=892\) MeV, \(\tau _{\Xi _{c}^{+}}=(4.42\pm 0.26)\times 10^{-13}\) s, \(\tau _{\Lambda _{c}^{+}}=(2.00\pm 0.06)\times 10^{-13}\) s. Besides, \(\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+\pi ^{+}\pi ^{-})=(4.57\pm 0.29)\%\) [1, 21], and the branching ratios are [22, 25]

$$\begin{aligned} {\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0})\over \mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+\pi ^{+}\pi ^{-})}&=0.078\pm 0.022, \end{aligned}$$
(24)
$$\begin{aligned} {\mathcal {B}(\Xi _c^+\rightarrow p\overline{K}^{*0})\over \mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)}&=0.54\pm 0.10 . \end{aligned}$$
(25)

Then we can obtain

$$\begin{aligned} \mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)=(2.2\pm 0.8)\%. \end{aligned}$$
(26)

The uncertainty is dominated by the ratios of branching fractions of \(\Lambda _{c}^{+}\) and \(\Xi _{c}^{+}\) decays in Eqs. (24) and (25).

The central value of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)\) at the order of percent, is larger than the typical order of \(10^{-3}\) of the ordinary singly Cabibbo-suppressed processes, such as \(\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0})=(3.6\pm 1.0)\times 10^{-3}\). This can be clarified by Eq. (23). Firstly, the lifetime of \(\Xi _{c}^{+}\) is two times larger than that of \(\Lambda _{c}^{+}\). Secondly, the phase space of \(\Xi _{c}^{+}\rightarrow p\overline{K}^{*0}\) is larger than that of \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\) by another factor of two, i.e. \(|p_{c}(m_{\Xi _c},m_{p},m_{K^{*}})|=828\) MeV and \(|p_c(m_{\Lambda _c},m_{\Sigma },m_{K^{*}})|=470\) MeV. Due to the larger lifetime and phase space, the branching fraction of \(\Xi _{c}^{+}\rightarrow p\overline{K}^{*0}\) is then at the order of percent, \((1.2\pm 0.4)\%\).

The understanding of the dynamics of charmed baryon decays is still a challenge at the current stage. Recent theoretical studies are mostly based on the flavor SU(3) analysis [26,27,28,29,30] and the current algebra [31]. They have not yet been applied to the singly Cabibbo-suppressed charmed baryon decays into a light baryon and a vector meson. Therefore, it is not available to estimate the U-spin breaking effects in the above analysis of Eq. (26). In the D-meson decays, the U-spin breaking effects, or say the SU(3) breaking effects, are mainly from the transition form factors and decay constants in the factorizable amplitudes, the difference between \(u\bar{u}\), \(d\bar{d}\) and \(s\bar{s}\) produced from vacuum in the W-exchange and W-annihilation amplitudes, and the Glauber strong phase with pion in the non-factorizable contributions [32, 33]. In Fig. 1, both amplitudes in the \(\Xi _{c}^{+}\rightarrow p\overline{K}^{*0}\) and \(\Lambda _{c}^{+}\rightarrow \Sigma ^{+}K^{*0}\) decay are non-factorizable. The vacuum production of \(d\bar{d}\) and \(s\bar{s}\) in the W-exchange diagrams would be a main source of U-spin breaking. In the modes involving a vector meson and a pseudoscalar meson in the final states of D-meson decays, the difference between \(d\bar{d}\) and \(s\bar{s}\) production in the W-exchange diagrams can be seen from \(\chi _{d}^{E}e^{i \phi _{d}^{E}}=(0.49\pm 0.03)e^{i (92\pm 4)^{\circ }}\) and \(\chi _{s}^{E}e^{i \phi _{s}^{E}}=(0.54\pm 0.03)e^{i (128\pm 5)^{\circ }}\) [34] where \(\chi \) and \(\phi \) are the magnitude and strong phase of the non-perturbative parameters in the W-exchange diagrams, and the subscripts d and s denotes the quark flavor of \(q\bar{q}\) produced from the vacuum. It can be found that the U-spin breaking effects are not large in \(D\rightarrow VP\) modes. The W-exchange diagrams in charmed baryon decays are similar to the ones in charmed meson decays, with an additional spectator quark. It can be expected that the U-spin breaking effects between \(\Xi _{c}^{+}\rightarrow p\overline{K}^{*0}\) and \(\Lambda _{c}^{+}\rightarrow \Sigma ^{+}K^{*0}\) would not be large, and thus the prediction of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)=(2.2\pm 0.8)\%\) would be under control.

The process of \(\Xi _c^+\rightarrow pK^-\pi ^+\) with all the charged final particles is widely used to study the properties of, or to search for some heavier baryons. The mass and lifetime of \(\Xi _{b}^{0}\) are measured with the most high precision via \(\Xi _{b}^{0}\rightarrow \Xi _{c}^{+}\pi ^{-}\), \(\Xi _c^+\rightarrow pK^-\pi ^+\) [10]. New states of \(\Xi _{b}^{\prime -}({1\over 2}^{+})\) and \(\Xi _{b}^{*-}({3\over 2}^{+})\) are observed in the \(\Xi _{b}^{0}\pi ^{-}\) spectrum with \(\Xi _{b}^{0}\rightarrow \Xi _{c}^{+}\pi ^{-}\), \(\Xi _c^+\rightarrow pK^-\pi ^+\) [35]. Five new \(\Omega _{c}^{0}\) resonances are observed in the final states of \(\Xi _{c}^{+}K^{-}\) with \(\Xi _c^+\rightarrow pK^-\pi ^+\) [36]. It is suggested to search for the doubly charmed baryons in the decay of \(\Xi _{cc}^{++}\rightarrow \Xi _{c}^{+}\pi ^{+}\) with \(\Xi _c^+\rightarrow pK^-\pi ^+\) [24, 37]. In this work, the ratio of fragmentation fractions \(f_{\Xi _b}/f_{\Lambda _b}\) can be obtained as long as the branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) is determined by Eq. (14).

4 \(f_{\Xi _b}/f_{\Lambda _b}\)and its implications

Utilizing the prediction of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)\) in Eq. (26), the measured value of \(\mathcal {B}(\Lambda _c^+\rightarrow pK^-\pi ^+)=(6.35\pm 0.33)\%\)[1] and the reasonable theoretical ratio \(\mathcal {B}(\Xi _b^0\rightarrow \Xi _c^+\pi ^-)/\mathcal {B}(\Lambda _b^0\rightarrow \Lambda _c^+ \pi ^-)\approx 1\), the ratio of the fragmentation fraction for b quark into \(\Xi _b^0\) and \(\Lambda _b^0\) can be obtained from Eq. (14) as

$$\begin{aligned} \frac{f_{\Xi _b^0}}{f_{\Lambda _b^0}}=0.054\pm 0.020. \end{aligned}$$
(27)

The uncertainty is mainly from the branching fraction of \(\Xi _c^+\rightarrow pK^-\pi ^+\) in Eq. (26). This result of \(f_{\Xi _b}/f_{\Lambda _b}\) is much smaller than the naive estimation of \(f_{s}/f_{u}\) or 0.2 in Eq. (5), and the MIT bag model for the branching fraction of \(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-\) in Eq. (12). The central value of our result in Eq. (27) is one half of those obtained via \(\Xi _{b}^{-}(\Lambda _{b}^{0})\rightarrow J/\Psi \Xi ^{-}(\Lambda )\) in Eqs. (8) and (9). Only the prediction in the diquark model for \(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-\) in Eq. (13) is consistent with our result within the uncertainties, while the central value is larger as well.

The total b-baryon fraction can be obtained by the ratio \(f_{\Xi _b}/f_{\Lambda _b}\) in Eq. (27). The production of \(\Omega _{b}^{-}\) is doubly suppressed by two strange quarks, estimated as 15% of the \(\Xi _{b}\) [38]. It is smaller than the error of \(f_{\Xi _b}/f_{\Lambda _b}\) in Eq. (27), and thus can be neglected in the total fraction of b-baryons. We then have

$$\begin{aligned} f_\mathrm{baryon}&=f_{\Lambda _b}+2f_{\Xi _b}+f_{\Omega _b} \approx f_{\Lambda _b}+2f_{\Xi _b}\nonumber \\&= (1.11\pm 0.04)f_{\Lambda _b}. \end{aligned}$$
(28)

It is equivalent that the correction \(\delta \) in Eq. (1) is \(\delta =0.11\pm 0.04\), which is smaller than the estimation of \(\delta =0.25\pm 0.10\) in Ref. [38].

With the result of \(f_{\Xi _b}/f_{\Lambda _b}\) in Eq. (27), the branching fraction of \(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-\) can be determined from Eq. (10),

$$\begin{aligned} \mathcal {B}(\Xi _b^-\rightarrow \Lambda _b^0\pi ^-)=(1.06\pm 0.54)\%. \end{aligned}$$
(29)

This is consistent with the diquark model, but larger than the MIT bag model, seen in Eq. (11).

The precision of our result of \(f_{\Xi _b}/f_{\Lambda _b}\) in Eq. (27) can be significantly improved by the measurements of \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) and \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\) at LHCb, BESIII and Belle II. The large uncertainty of \(\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)\) in Eq. (26), inducing the major uncertainty of \(f_{\Xi _b}/f_{\Lambda _b}\), is dominated by two ratios of branching fractions: \(\mathcal {B}(\Xi _c^+\rightarrow p\overline{K}^{*0})/\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)=0.54\pm 0.10\) by FOCUS in 2001 [22] and \(\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0})/\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+\pi ^{+}\pi ^{-})=0.078\pm 0.022\) measured by FOCUS in 2002 [25]. For the former, a more precise measurement can be performed by LHCb with partial wave analysis. At LHCb with the data of 3.3 fb\(^{-1}\), there are already \(1\times 10^{6}\) events of \(\Xi _c^+\rightarrow pK^-\pi ^+\) [36], which is four orders of magnitude larger than 200 events in Ref. [22]. The latter can be improved by the BESIII or Belle(II) experiments, which have recently performed a dozen measurements of \(\Lambda _{c}^{+}\) decays [20, 21, 39,40,41,42,43,44,45,46], especially the observation of some singly or doubly Cabibbo-suppressed processes [44,45,46]. With the updated measurements of \(\Xi _c^+\rightarrow p\overline{K}^{*0}\) and \(\Lambda _c^+\rightarrow \Sigma ^+K^{*0}\) in the near future, \(f_{\Xi _b}/f_{\Lambda _b}\) could be of higher precision.

5 Summary

In this work, we study the ratio of fragmentation fractions \(f_{\Xi _b}/f_{\Lambda _b}\) with the data of \(\Xi _{b}^{0}\rightarrow \Xi _{c}^{+}\pi ^{-}\) and \(\Lambda _{b}^{0}\rightarrow \Lambda _{c}^{+}\pi ^{-}\), \(\Xi _{c}^{+}/\Lambda _{c}^{+}\rightarrow p K^{-}\pi ^{+}\) at LHCb, which is the most precise measurement related to the fragmentations of \(\Xi _{b}\) and \(\Lambda _{b}\), seen in Table 1. The least known branching fraction of \(\Xi _{c}^{+}\rightarrow p K^{-}\pi ^{+}\) is obtained under the U-spin symmetry, \(\mathcal {B}(\Xi _{c}^{+}\rightarrow p K^{-}\pi ^{+})=(2.2\pm 0.8)\%\). The ratio \(f_{\Xi _b}/f_{\Lambda _b}\) is then determined to be \(f_{\Xi _b}/f_{\Lambda _b}\)=\(0.054\pm 0.020\). This is the first analysis of \(f_{\Xi _b}/f_{\Lambda _b}\) using the LHCb data. It helps to understand the production of b-baryons. To improve the precision, we suggest to measure the ratios of branching fractions \(\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+K^{*0})/\mathcal {B}(\Lambda _c^+\rightarrow \Sigma ^+\pi ^{+}\pi ^{-})\) and \(\mathcal {B}(\Xi _c^+\rightarrow p\overline{K}^{*0})/\mathcal {B}(\Xi _c^+\rightarrow pK^-\pi ^+)\) at BESIII, Belle(II) and LHCb using the current data set or in the near future.