Decay properties of $P$-wave heavy baryons accompanied by vector mesons within light-cone sum rules

We use the method of light-cone sum rules to study decay properties of $P$-wave bottom baryons belonging to the $SU(3)$ flavor $\mathbf{6}_F$ representation. In Ref.~\cite{Cui:2019dzj} we have studied their mass spectrum and pionic decays, and found that the $\Sigma_{b}(6097)$ and $\Xi_{b}(6227)$ can be well interpreted as $P$-wave bottom baryons of $J^P = 3/2^-$. In this paper we further study their decays into ground-state bottom baryons and vector mesons. We propose to search for a new state $\Xi_b({5/2}^-)$, that is the $J^P = 5/2^-$ partner state of the $\Xi_{b}(6227)$, in the $\Xi_b({5/2}^-) \to \Xi_b^{*}\rho \to \Xi_b^{*}\pi\pi$ decay process. Its mass is $12 \pm 5$~MeV larger than that of the $\Xi_{b}(6227)$.

In Refs. [57,58] we have systematically applied the method of QCD sum rules [52,53] to study P-wave heavy baryons within the heavy quark effective theory (HQET) [54][55][56], where we systematically constructed all the P-wave heavy baryon interpolating fields, and applied them to study the mass spectrum of P-wave heavy baryons. Later in Ref. [59] we further studied their decay properties using light-cone sum rules, including: -S-wave decays of flavor3 F P-wave heavy baryons into ground-state heavy baryons and pseudoscalar mesons; -S-wave decays of flavor 6 F P-wave heavy baryons into ground-state heavy baryons and pseudoscalar mesons; -S-wave decays of flavor3 F P-wave heavy baryons into ground-state heavy baryons and vector mesons.
Very quickly, one notices that in order to make a complete study of P-wave heavy baryons, we still need to study:

Decay properties of P-wave bottom baryons
At the beginning let us briefly introduce our notations. A Pwave bottom baryon (bqq) consists of one bottom quark (b) and two light quarks (qq). Its orbital excitation can be either between the two light quarks (l ρ = 1) or between the bottom quark and the two-light-quark system (l λ = 1), so there are ρ-type bottom baryons (l ρ = 1 and l λ = 0) and λ -type ones (l ρ = 0 and l λ = 1). Altogether its internal symmetries are as follows: -The color structure of the two light quarks is antisymmetric (3 C ). -The SU(3) flavor structure of the two light quarks is either antisymmetric (3 F ) or symmetric (6 F ). -The spin structure of the two light quarks is either antisymmetric (s l ≡ s qq = 0) or symmetric (s l = 1). -The orbital structure of the two light quarks is either antisymmetric (l ρ = 1) or symmetric (l ρ = 0). -Due to the Pauli principle, the total symmetry of the two light quarks is antisymmetric.
According to the above symmetries, one can categorize the P-wave bottom baryons into eight baryon multiplets, as shown in Fig. 1. We denote these multiplets as [F(flavor), j l , s l , ρ/λ ], with j l the total angular momentum of the light components ( j l = l λ ⊗ l ρ ⊗ s l ). Every multiplet contains one or two bottom baryons, whose total angular momenta are j = j l ⊗ s b = | j l ± 1/2|, with s b the spin of the bottom quark. Especially, the heavy quark effective theory tells that the bottom baryons inside the same doublet with j = j l − 1/2 and j = j l + 1/2 have similar masses.
In this section we investigate S-wave decays of flavor 6 F P-wave bottom baryons into ground-state bottom baryons and vector mesons. To do this we use the method of light-cone sum rules within HQET, and investigate the following decay channels (the coefficients at right hand sides are isospin factors): Fig. 1 Categorization of P-wave bottom baryons. Taken from Ref. [1].
We can calculate their decay widths through the following Lagrangians , and V µ denotes the P-wave bottom baryon, ground-state bottom baryon, and vector meson, respectively.
As an example, we study the S-wave decay of the In this subsection we study the S-wave decay of the To do this we consider the following three-point correlation function: where J 1/2,−,Σ − b ,1,0,ρ and J Λ 0 b are the interpolating fields cou- We refer to Refs. [57,94], where we systematically constructed all the Sand P-wave heavy baryon interpolating fields. In the above expressions h v (x) is the heavy quark field; k = k +q, with k , k, and q the momenta of the Σ − b (1/2 − ), Λ 0 b , and ρ − , respectively; ω = v · k and ω = v · k . Note that the definitions of ω and ω in the present study are the same as those used in Refs. [1,59], but different from those used in Refs. [57,58].
At the hadronic level, we write G At the quark and gluon level, we calculate G After Wick rotations and making double Borel transformation with the variables ω and ω to be T 1 and T 2 , we obtain .
x k k! . Explicit forms of the light-cone distribution amplitudes contained in the above expression can be found in Refs. [78][79][80][81][82][83][84][85], and we work at the renormalization scale 2 GeV for the parameters involved. More sum rule examples can be found in Appendix B.
In the present study we work at the symmetric point We use the following values for the bottom quark mass and various quark and gluon condensates [2,[86][87][88][89][90][91][92][93]: b ρ − only depends on two free parameters, the threshold value ω c and the Borel mass T . We choose ω c = 1. 485GeV to be the average of the threshold values of the Σ b (1/2 − ) and Λ 0 b mass sum rules (see Appendix A and Ref. [1] for the parameters of the Σ b (1/2 − ) and Λ 0 b ), and extract the coupling constant g

+0.10
−0.11 , where the uncertainties are due to the Borel mass, the parameters of the Λ 0 b , the parameters of the Σ − b (1/2 − ), the lightcone distribution amplitudes of vector mesons [78][79][80][81][82][83][84][85], and various quark masses and condensates listed in Eq. (35), respectively. Besides these statistical uncertainties, there is another (theoretical) uncertainty, which comes from the scale dependence. In the present study we do not consider this, and simply work at the renormalization scale 2 GeV, since M 2 GeV. However, it is useful to give a rough estimation on this. Since the largest uncertainty comes from the light-cone distribution amplitudes of vector mesons, we choose the values for the parameters contained in these amplitudes to be at the renormalization scale 1 GeV (see Tables 1 and 2 of Ref. [85]), and redo the above calculations to obtain: Hence, the scale dependence leads to a significant uncertainty, and the total uncertainty of our results can be even larger. For completeness, we show g Fig. 2(a) as a function of the Borel mass T , and find that it only slightly depends on the Borel mass, where the working region for T has been evaluated in the Σ b (1/2 − ) mass sum rules [1,[57][58][59] to be 0.31 GeV < T < 0.34 GeV (we also summarize this in Table 5). Note that the definitions of ω and ω in this paper are the same as those used in Refs. [1,59], but different from those used in Refs. [57,58], so the Borel windows used in this paper are also the same/similar as those used in Refs. [1,59], but just about half of those used in Refs. [57,58].
The two-body decay is kinematically allowed, whose decay amplitude is Here 0 denotes the initial state Σ − b (1/2 − ); 4 denotes the intermediate state ρ − ; 1, 2 and 3 denote the finial states π − , π 0 and Λ 0 b , respectively. This amplitude can be used to further calculate its decay width +13.6 −7.5 keV . In the following subsections we apply the same procedures to separately study the four bottom baryon multiplets,

The bottom baryon doublet
The bottom baryon doublet [6 F , 1, 0, ρ] consists of six members: We use the method of light-cone sum rules within HQET to study their decays into ground-state bottom baryons and vector mesons.
There are altogether twenty-four non-vanishing decay channels, whose coupling constants are extracted to be Then we compute the three-body decay widths, which are kinematically allowed: We summarize these results in Table 1. For completeness, we also show the coupling constants as functions of the Borel mass T in Fig. 2.

The bottom baryon singlet
The bottom baryon doublet [6 F , 1, 0, ρ] consists of three members: We use the method of light-cone sum rules within HQET to study their decays into ground-state bottom baryons and vector mesons.
There are altogether eight non-vanishing decay channels, whose coupling constants are extracted to be Then we compute the three-body decay widths, which are kinematically allowed: We summarize these results in Table 2. For completeness, we also show the coupling constants as functions of the Borel mass T in Fig. 3. There are altogether twenty-four non-vanishing decay channels, whose coupling constants are extracted to be
Then we compute the three-body decay widths, which are kinematically allowed: We summarize these results in Table 3. For completeness, we also show the coupling constants as functions of the Borel mass T in Fig. 4. There are altogether twelve non-vanishing decay channels, whose coupling constants are extracted to be (h) Fig. 4 The coupling constants (a) g  Table 5.
Then we compute the three-body decay widths, which are kinematically allowed: 0.14 +0.19 −0.12 keV 0.14 +0.19 −0.12 keV We summarize these results in Table 4. For completeness, we also show the coupling constants as functions of the Borel mass T in Fig. 5.

Summary and Discussions
To summarize this paper, we have used the method of lightcone sum rules within heavy quark effective theory to study decay properties of P-wave bottom baryons belonging to the flavor 6 F representation. We have studied their S-wave decays into ground-state bottom baryons and vector mesons. The possible decay channels are given in Eqs. , and the extracted decay widths are listed in Tables 1, 2 In Ref. [1] we have studied the mass spectrum and pionic decay properties of the Σ b (6097) and Ξ b (6227) [7,8]. Our results suggest that they can be well interpreted as Pwave bottom baryons of J P = 3/2 − , belonging to the bottom baryon doublet [6 F , 2, 1, λ ]. This doublet contains altogether six bottom baryons, In the present study we further investigate their S-wave decays into ground-state bottom baryons and vector mesons, and extract: Hence, these three J P = 5/2 − states are probably quite narrow, because their S-wave decays into ground-state bottom baryons and pseudoscalar mesons can not happen, and widths of the following D-wave decays are also calculated to be zero in Ref. [1]: We suggest the LHCb and Belle/Belle-II experiments to search for these three narrow states. Especially, we propose to search for the Ξ b (5/2 − ), that is the ππ decay process. Its mass is 12 ± 5 MeV larger than that of the Ξ b (6227).
To end this work, we note that in the present study we have studied S-wave decays of flavor 6 F P-wave heavy baryons into ground-state heavy baryons and vector mesons, which is actually a complement to Ref. [59], where we studied Swave decays of flavor3 F P-wave heavy baryons into groundstate heavy baryons together with pseudoscalar and vector mesons, and S-wave decays of flavor 6 F P-wave heavy baryons into ground-state heavy baryons together with pseudoscalar mesons. To make a complete QCD sum rule studies of Pwave heavy baryons within HQET, we still need to systematically study their D-wave and radiative decay properties, which is currently under investigation.
(d) Fig. 5 The coupling constants (a) g  Table 5. Their QCD sum rule parameters can be found in Refs. [1,59,94]. We list masses of P-wave bottom baryons used in the present study, taken from the LHCb experiments [7,8] as well as our previous QCD sum rule studies [1,57,58]: Their QCD sum rule parameters can be found in Table 5.

Appendix B: Sum rule equations
In this appendix we show several examples of sum rule equations, which are used to extract S-wave decays of P-wave