1. Introduction
As a key prediction of QCD, the doubly heavy baryons are composed of two heavy quarks and one light quark. The SELEX collaboration found the ${{\rm{\Xi }}}_{{cc}}^{+}$ baryon with the mass ${m}_{{{\rm{\Xi }}}_{{cc}}^{+}}=3518\pm 3\,\mathrm{MeV}$ [1, 2]. However, this result was not confirmed in other collaborations [3–5]. The LHCb collaboration found the ${{\rm{\Xi }}}_{{cc}}^{++}$ baryon with the mass ${m}_{{{\rm{\Xi }}}_{{cc}}^{++}}\,=3621.55\,\pm \,0.23\,\pm \,0.30\,\mathrm{MeV}$ [6, 7]. However, people did not find experimental evidence on the other doubly heavy baryons.
Nowadays, people widely studied the masses of the doubly heavy baryons. In the quark model, the authors studied the masses of the doubly heavy baryons in different potential models [8–10], the masses of the doubly heavy baryons were studied in a chromomagnetic interaction model [11], and the authors used a constituent quark model to study the masses of the doubly heavy baryons [12, 13]. In lattice QCD, the authors studied the masses of the doubly charmed baryons [14–17], and the masses of the doubly bottom baryons and the charmed-bottom baryons were studied [14, 18–21]. The lattice QCD simulations predicted the masses of the doubly heavy baryons at the unphysical point, so the authors performed the chiral extrapolations [14, 20]. Recently, the lattice QCD simulations predicted the masses of the doubly heavy baryons near the physical point [16, 17]. Other methods were involved, such as Bethe–Salpeter equation [22, 23], contact interaction models [24, 25], and QCD sum rules [26–28].
Chiral perturbation theory (ChPT) is an effective field theory of QCD [29, 30]. ChPT is used in the meson sector [31, 32], people make the expansion in terms of the momentum, but the power counting scheme is broken in the baryon sector due to the nonzero baryon mass in the chiral limit [33, 34]. People proposed many schemes to solve this issue, such as heavy baryon chiral perturbation theory (HBChPT) [35], infrared chiral perturbation theory (IRChPT) [36, 37], and extended on mass shell chiral perturbation theory (EOMSChPT) [38]. In HBChPT, people make the expansion in terms of the inverse baryon mass, and the power counting scheme is restored. Later, quenched chiral perturbation theory (QChPT) and partially quenched chiral perturbation theory (PQChPT) were proposed [39–41].
For the doubly heavy baryons, the heavy quarks are static, and the light quark governs the chiral dynamics. Nowadays, people widely studied the chiral corrections to the masses of the doubly heavy baryons. The authors used the Lagrangians with the heavy quark-diquark symmetry to study the masses of the doubly heavy baryons [42] and extended this result to QChPT and PQChPT [43]. The authors studied the masses of the doubly charmed baryons up to ${ \mathcal O }\left({p}^{4}\right)$ in HBChPT [44]. The masses of the doubly charmed baryons were studied up to ${ \mathcal O }\left({p}^{3}\right)$ in EOMSChPT [45], and this result was extended up to ${ \mathcal O }\left({p}^{4}\right)$ [46]. In [44–46], the authors showed the numerical results up to ${ \mathcal O }\left({p}^{3}\right)$.
Besides the doubly charmed baryons, the chiral corrections to the masses of the doubly bottom baryons and the charmed-bottom baryons are an important topic. We study the masses of the doubly bottom baryons and the charmed-bottom baryons up to ${ \mathcal O }\left({p}^{3}\right)$ in HBChPT. We show the numerical results for the masses of the doubly bottom baryons and the charmed-bottom baryons up to ${ \mathcal O }\left({p}^{3}\right)$.
Our work is organized as follows. We introduce the Lagrangians of the doubly bottom baryons in section 2 . We derive the mass formulas of the doubly bottom baryons in section 3 . We show the numerical results for the masses of the doubly bottom baryons and the charmed-bottom baryons in section 4 . A summary is given in section 5 .
2. The Lagrangians of the doubly bottom baryons
In ChPT, the Lagrangians of the pseudoscalar mesons and the doubly bottom baryons were constructed [44, 47]. The ${ \mathcal O }\left({p}^{1}\right)$ Lagrangian reads
$\begin{eqnarray}{{ \mathcal L }}_{{MB}}^{\left(1\right)}=\bar{\psi }\left({\rm{i}}\rlap{/}{D}-{m}_{0}+\displaystyle \frac{{g}_{A}}{2}\rlap{/}{u}{\gamma }^{5}\right)\psi ,\end{eqnarray}$
where m0 is the mass of the doubly bottom baryons in the chiral limit, and gA is the axial vector charge of the doubly bottom baryons. The doubly bottom baryon field reads $\begin{eqnarray}\psi =\left(\begin{array}{c}{{\rm{\Xi }}}_{{bb}}^{0}\\ {{\rm{\Xi }}}_{{bb}}^{-}\\ {{\rm{\Omega }}}_{{bb}}^{-}\end{array}\right).\end{eqnarray}$
For the building blocks, $\begin{eqnarray}u=\exp \left(\displaystyle \frac{{\rm{i}}\phi }{2{F}_{P}}\right),\end{eqnarray}$
$\begin{eqnarray}{u}_{\mu }=\displaystyle \frac{{\rm{i}}}{2}\left({u}^{\dagger }{\partial }_{\mu }u-u{\partial }_{\mu }{u}^{\dagger }\right),\end{eqnarray}$
$\begin{eqnarray}{D}_{\mu }={\partial }_{\mu }+\displaystyle \frac{1}{2}\left({u}^{\dagger }{\partial }_{\mu }u+u{\partial }_{\mu }{u}^{\dagger }\right),\end{eqnarray}$
where FP are the decay constants of the pseudoscalar mesons. We take the experimental values of FP [48], Fπ = 92 MeV, FK = 113 MeV, Fη = 116 MeV. The pseudoscalar meson field reads $\begin{eqnarray}\phi =\left(\begin{array}{ccc}{\pi }^{0}+\displaystyle \frac{\sqrt{3}}{3}\eta & \sqrt{2}{\pi }^{+} & \sqrt{2}{K}^{+}\\ \sqrt{2}{\pi }^{-} & -{\pi }^{0}+\displaystyle \frac{\sqrt{3}}{3}\eta & \sqrt{2}{K}^{0}\\ \sqrt{2}{K}^{-} & \sqrt{2}{\bar{K}}^{0} & -\displaystyle \frac{2\sqrt{3}}{3}\eta \end{array}\right).\end{eqnarray}$
The ${ \mathcal O }\left({p}^{2}\right)$ and ${ \mathcal O }\left({p}^{3}\right)$ Lagrangians read $\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{{MB}}^{\left(2\right)}=\bar{\psi }\left[{c}_{1}\left\langle {\chi }_{+}\right\rangle +{c}_{2}{u}^{2}+{c}_{3}\left\langle {u}^{2}\right\rangle \right.\\ \quad +\left({c}_{4}\left\{{u}^{\mu },{u}^{\nu }\right\}\left\{{D}_{\mu },{D}_{\nu }\right\}+{\rm{H}}.{\rm{c}}.\right)\\ \quad +\left({c}_{5}\left\langle {u}^{\mu }{u}^{\nu }\right\rangle \left\{{D}_{\mu },{D}_{\nu }\right\}+{\rm{H}}.{\rm{c}}.\right)\\ \quad \left.+{\rm{i}}{c}_{6}\left[{u}^{\mu },{u}^{\nu }\right]{\sigma }_{\mu \nu }+{c}_{7}{\tilde{\chi }}_{+}\right]\psi ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{{MB}}^{\left(3\right)} & = & \bar{\psi }\left({d}_{1}\rlap{/}{u}{\gamma }^{5}\left\langle {\chi }_{+}\right\rangle +{d}_{2}\left\{{u}^{\mu },{\tilde{\chi }}_{+}\right\}{\gamma }_{\mu }{\gamma }^{5}\right.\\ & & \left.+{d}_{3}\left\langle {u}^{\mu }{\tilde{\chi }}_{+}\right\rangle {\gamma }_{\mu }{\gamma }^{5}+\cdots \right)\psi ,\end{array}\end{eqnarray}$
where ci and dj are the coupling constants. For the building blocks, $\begin{eqnarray}\chi =2{B}_{0}\mathrm{diag}\left({m}_{u},{m}_{d},{m}_{s}\right),\end{eqnarray}$
$\begin{eqnarray}{\chi }_{+}={u}^{\dagger }\chi {u}^{\dagger }+u{\chi }^{\dagger }u.\end{eqnarray}$
$\left\langle {\chi }_{+}\right\rangle $ is the trace of χ+, and ${\tilde{\chi }}_{+}={\chi }_{+}-\tfrac{1}{3}\left\langle {\chi }_{+}\right\rangle $ is traceless. In the isospin limit, mu = md, we use ${m}_{\pi }^{2}=2{B}_{0}{m}_{u/d}$, ${m}_{K}^{2}={B}_{0}\left({m}_{u/d}+{m}_{s}\right)$, ${m}_{\eta }^{2}=\tfrac{2}{3}{B}_{0}\left({m}_{u/d}+2{m}_{s}\right)$, so $\chi \,=\mathrm{diag}\left({m}_{\pi }^{2},{m}_{\pi }^{2},-2{m}_{K}^{2}+3{m}_{\eta }^{2}\right)$.In HBChPT, the doubly bottom baryon field is separated into two parts,
$\begin{eqnarray}\psi =\exp \left(-{\rm{i}}{m}_{0}v\cdot x\right)\left(H+h\right),\end{eqnarray}$
where H and h are the light and heavy fields. The heavy baryon Lagrangians of the pseudoscalar mesons and the doubly bottom baryons were constructed [44, 47]. The ${ \mathcal O }\left({p}^{1}\right)$ heavy baryon Lagrangian reads $\begin{eqnarray}{\hat{{ \mathcal L }}}_{{MB}}^{\left(1\right)}=\bar{H}\left({\rm{i}}v\cdot D+{g}_{A}S\cdot u\right)H,\end{eqnarray}$
where ${S}^{\mu }=\tfrac{{\rm{i}}}{2}{\gamma }^{5}{\sigma }^{\mu \nu }{v}_{\nu }$ is the spin operator. The ${ \mathcal O }\left({p}^{2}\right)$ and ${ \mathcal O }\left({p}^{3}\right)$ heavy baryon Lagrangians read $\begin{eqnarray}\begin{array}{l}{\hat{{ \mathcal L }}}_{{MB}}^{\left(2\right)}=\bar{H}\left[{c}_{1}\left\langle {\chi }_{+}\right\rangle +{c}_{2}{u}^{2}+{c}_{3}\left\langle {u}^{2}\right\rangle -8{m}_{0}^{2}{c}_{4}{\left(v\cdot u\right)}^{2}\right.\\ \quad -\ 4{m}_{0}^{2}{c}_{5}\left\langle {\left(v\cdot u\right)}^{2}\right\rangle +2{c}_{6}\left[{S}_{\mu },{S}_{\nu }\right]\left[{u}^{\mu },{u}^{\nu }\right]+{c}_{7}{\tilde{\chi }}_{+}\\ \quad \left.+\ \displaystyle \frac{2}{{m}_{0}}{\left(S\cdot D\right)}^{2}-\displaystyle \frac{{\rm{i}}{g}_{A}}{2{m}_{0}}\left\{S\cdot D,v\cdot u\right\}-\displaystyle \frac{{g}_{A}^{2}}{8{m}_{0}}{\left(v\cdot u\right)}^{2}\right]H,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\hat{{ \mathcal L }}}_{{MB}}^{\left(3\right)} & = & \bar{H}\left(-2{d}_{1}S\cdot u\left\langle {\chi }_{+}\right\rangle -2{d}_{2}\left\{S\cdot u,{\tilde{\chi }}_{+}\right\}\right.\\ & & \left.-2{d}_{3}\left\langle S\cdot u{\tilde{\chi }}_{+}\right\rangle -\displaystyle \frac{{\rm{i}}}{{m}_{0}^{2}}S\cdot {Dv}\cdot {DS}\cdot D+\cdots \right)H.\end{array}\end{eqnarray}$
3. The mass formulas of the doubly bottom baryons
In HBChPT, with the variables of the self-energy, η = v · p − mB and $\xi ={\left(p-{m}_{B}v\right)}^{2}$, the Feynman propagator of the doubly bottom baryons reads
$\begin{eqnarray}{\rm{i}}{G}=\displaystyle \frac{{\rm{i}}}{v\cdot p-{m}_{0}-{{\rm{\Sigma }}}_{B}\left(\eta ,\xi \right)}=\displaystyle \frac{{\rm{i}}{Z}_{N}}{\eta -{Z}_{N}{\tilde{{\rm{\Sigma }}}}_{B}\left(\eta ,\xi \right)},\end{eqnarray}$
where ${{\rm{\Sigma }}}_{B}\left(\eta ,\xi \right)={{\rm{\Sigma }}}_{B}\left(0,0\right)+\eta {{\rm{\Sigma }}}_{B}^{{\prime} }\left(0,0\right)+{\tilde{{\rm{\Sigma }}}}_{B}\left(\eta ,\xi \right)$ is the self-energy, and ${Z}_{N}=\tfrac{1}{1-{{\rm{\Sigma }}}_{B}^{{\prime} }\left(0,0\right)}$ is the wave function renormalization constant. Thus, the mass formulas of the doubly bottom baryons read $\begin{eqnarray}{m}_{B}={m}_{0}+{{\rm{\Sigma }}}_{B}\left(0,0\right).\end{eqnarray}$
We show the Feynman diagrams contributing to the self energies of the doubly bottom baryons in figure 1. Equations (13 ) and (14 ) contribute to the tree vertices in figures 1(a) and 1(d), and equation (12 ) contributes to the loop vertices in figures 1(b) and (c). In HBChPT, the chiral order of the Feynman diagrams reads
$\begin{eqnarray}{D}_{\chi }=4{N}_{L}-2{I}_{M}-{I}_{B}+\sum _{n}{{nN}}_{n},\end{eqnarray}$
where NL is the number of loops, IM and IB are the number of internal pseudoscalar meson and doubly bottom baryon lines, and Nn is the number of vertices contributed by the ${ \mathcal O }\left({p}^{n}\right)$ heavy baryon Lagrangians. Thus, Dχ = 2 in figure 1(a), and Dχ = 3 in figure 1(b)–(d).
Figure 1. The Feynman diagrams contributing to the self-energies of the doubly bottom baryons. The dashed and solid lines are the pseudoscalar mesons and the doubly bottom baryons. The squares with the numbers n are the vertices contributed by the ${ \mathcal O }\left({p}^{n}\right)$ heavy baryon Lagrangians. |
Figures 1(a) and (c) contribute to the ${ \mathcal O }\left({p}^{2}\right)$ tree and ${ \mathcal O }\left({p}^{3}\right)$ loop masses, and figures 1(b) and (d) contribute to the zero masses. After the derivations, equation (16 ) reads18 ) reads
$\begin{eqnarray}{m}_{B}={m}_{0}-{c}_{1}\left\langle {\chi }_{+}\right\rangle -{c}_{7}{\tilde{\chi }}_{+}-\displaystyle \frac{{g}_{A}^{2}}{128\pi }\sum _{P}\displaystyle \frac{{C}_{{PB}}{m}_{P}^{3}}{{F}_{P}^{2}},\end{eqnarray}$
where CPB are the Clebsch–Gordan coefficients of the pseudoscalar mesons and the doubly bottom baryons, and mP are the masses of the pseudoscalar mesons. We show the values of CPB in table 1. We take the experimental values of mP [48], mπ = 140 MeV, mK = 494 MeV, mη = 550 MeV. After the organizations, equation ( $\begin{eqnarray}\begin{array}{l}{m}_{{{\rm{\Xi }}}_{{bb}}}={m}_{0}-2{c}_{1}\left(2{m}_{\pi }^{2}-2{m}_{K}^{2}+3{m}_{\eta }^{2}\right)\\ \quad -\displaystyle \frac{2}{3}{c}_{7}\left({m}_{\pi }^{2}+2{m}_{K}^{2}-3{m}_{\eta }^{2}\right)\\ \quad -\displaystyle \frac{{g}_{A}^{2}}{384\pi }\left(\displaystyle \frac{9{m}_{\pi }^{3}}{{F}_{\pi }^{2}}+\displaystyle \frac{6{m}_{K}^{3}}{{F}_{K}^{2}}+\displaystyle \frac{{m}_{\eta }^{3}}{{F}_{\eta }^{2}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{m}_{{{\rm{\Omega }}}_{{bb}}}={m}_{0}-2{c}_{1}\left(2{m}_{\pi }^{2}-2{m}_{K}^{2}+3{m}_{\eta }^{2}\right)\\ \quad +\displaystyle \frac{4}{3}{c}_{7}\left({m}_{\pi }^{2}+2{m}_{K}^{2}-3{m}_{\eta }^{2}\right)\\ \quad -\displaystyle \frac{{g}_{A}^{2}}{96\pi }\left(\displaystyle \frac{3{m}_{K}^{3}}{{F}_{K}^{2}}+\displaystyle \frac{{m}_{\eta }^{3}}{{F}_{\eta }^{2}}\right).\end{array}\end{eqnarray}$
Table 1. The values of CPB. |
| CπB | CKB | CηB | |
|---|---|---|---|
| Ξbb | 3 | 2 | $\tfrac{1}{3}$ |
| Ωbb | 0 | 4 | $\tfrac{4}{3}$ |
4. Numerical results
There are the unknown low energy constants, m0, c1, c7, and gA in equations (19 ) and (20 ). Due to the limited experimental data, we determine the unknown low energy constants by some theoretical information.
After the lattice QCD simulations, the authors performed the chiral extrapolations for the subtracted masses ${E}_{X}^{\left(\mathrm{sub}\right)}$, rather than the full masses EX of the doubly bottom baryons in [14],
$\begin{eqnarray}{E}_{X}^{\left(\mathrm{sub}\right)}={E}_{X}-\displaystyle \frac{{n}_{c}}{2}{\overline{E}}_{c\overline{c}}-\displaystyle \frac{{n}_{b}}{2}{\overline{E}}_{b\overline{b}},\end{eqnarray}$
where nc and nb are the number of c and b quarks in the doubly bottom baryons, and ${\overline{E}}_{c\overline{c}}$ and ${\overline{E}}_{b\overline{b}}$ are the spin average charmonium and bottomonium masses. The chiral extrapolated values of ${E}_{X}^{\left(\mathrm{sub}\right)}$ were added to the experimental values of $\tfrac{{n}_{c}}{2}{\overline{E}}_{c\overline{c}}+\tfrac{{n}_{b}}{2}{\overline{E}}_{b\overline{b}}$, ${\overline{E}}_{c\overline{c}}=3069\,\mathrm{MeV}$ and ${\overline{E}}_{b\overline{b}}=9445\,\mathrm{MeV}$ [48], and the full masses of the doubly bottom baryons were obtained.In the chiral limit, the masses of the doubly bottom baryons are m0. We assume only m0 has the information of the heavy quark mass dependence. From the lattice QCD schemes in [14], $\tfrac{{n}_{c}}{2}{\overline{E}}_{c\overline{c}}+\tfrac{{n}_{b}}{2}{\overline{E}}_{b\overline{b}}$ only depend on the heavy quark masses. We absorb $\tfrac{{n}_{c}}{2}{\overline{E}}_{c\overline{c}}+\tfrac{{n}_{b}}{2}{\overline{E}}_{b\overline{b}}$ into m0.
With equations (19 ) and (20 ), we perform the chiral extrapolations for the lattice QCD data of ${E}_{X}^{\left(\mathrm{sub}\right)}$. We show the lattice QCD data in table 2. The lattice QCD data of ${E}_{X}^{\left(\mathrm{sub}\right)}$ are added to the experimental values of $\tfrac{{n}_{c}}{2}{\overline{E}}_{c\overline{c}}+\tfrac{{n}_{b}}{2}{\overline{E}}_{b\overline{b}}$. The π and ηs meson masses are used to tune the u/d and s quark masses [50, 51]. We use ${m}_{{\eta }_{s}}^{2}=2{B}_{0}{m}_{s}$, so the K and η meson masses read
$\begin{eqnarray}{m}_{K}=\sqrt{\displaystyle \frac{1}{2}\left({m}_{\pi }^{2}+{m}_{{\eta }_{s}}^{2}\right)},\end{eqnarray}$
$\begin{eqnarray}{m}_{\eta }=\sqrt{\displaystyle \frac{1}{3}\left({m}_{\pi }^{2}+2{m}_{{\eta }_{s}}^{2}\right)}.\end{eqnarray}$
We obtain the results of m0, c1, and c7, and predict the masses of the doubly bottom baryons, as shown in table 3.Table 2. The lattice QCD data in [14]. All the physical quantities are in MeV. |
| mπ | ${m}_{{\eta }_{s}}$ | ${m}_{{{\rm{\Xi }}}_{{bb}}}$ | ${m}_{{{\rm{\Omega }}}_{{bb}}}$ | ${m}_{{{\rm{\Xi }}}_{\{{cb}\}}}$ | ${m}_{{{\rm{\Omega }}}_{\{{cb}\}}}$ | ${m}_{{{\rm{\Xi }}}_{[{cb}]}}$ | ${m}_{{{\rm{\Omega }}}_{[{cb}]}}$ | |
|---|---|---|---|---|---|---|---|---|
| I | 245 | 761 | $10227\left(13\right)$ | — | $7016\left(14\right)$ | — | $6969\left(13\right)$ | — |
| II | 270 | 761 | $10219\left(14\right)$ | — | $7026\left(14\right)$ | — | $6980\left(13\right)$ | — |
| III | 336 | 761 | $10224\left(13\right)$ | $10297\left(13\right)$ | $7027\left(13\right)$ | $7104\left(13\right)$ | $6982\left(13\right)$ | $7065\left(13\right)$ |
Table 3. The results of m0, c1, and c7 with the errors from the lattice QCD data, and the masses of the doubly bottom baryons and the charmed-bottom baryons with the errors from m0, c1, c7, and gA. |
| ${m}_{0}\left(\mathrm{MeV}\right)$ | ${c}_{1}\left({\mathrm{GeV}}^{-1}\right)$ | ${c}_{7}\left({\mathrm{GeV}}^{-1}\right)$ | ${m}_{{{\rm{\Xi }}}_{{QQ}}}\left(\mathrm{MeV}\right)$ | ${m}_{{{\rm{\Omega }}}_{{QQ}}}\left(\mathrm{MeV}\right)$ | ${\chi }_{{\rm{d}}.{\rm{o}}.{\rm{f}}.}^{2}$ | |
|---|---|---|---|---|---|---|
| Ξbb/Ωbb | $10264\left(53\right)$ | $-0.015\left(36\right)$ | $-0.104\left(20\right)$ | $10235\left(63\right)$ | $10299\left(64\right)$ | 0.182 |
| Ξ{cb}/Ω{cb} | $7000\left(51\right)$ | $-0.059\left(34\right)$ | $-0.108\left(20\right)$ | $7010\left(60\right)$ | $7078\left(61\right)$ | 0.161 |
| Ξ[cb]/Ω[cb] | $6854\left(48\right)$ | $-0.326\left(32\right)$ | $-0.323\left(20\right)$ | $6930\left(63\right)$ | $7017\left(84\right)$ | 0.148 |
The mass formulas of the doubly bottom baryons and the charmed-bottom baryons are the same due to the heavy quark symmetry. For the charmed-bottom baryons, the heavy quarks are regarded as the spin symmetric and spin antisymmetric diquarks and form different triplets with the light quark. gA was determined in the quark model [49], ${g}_{A}\left(\{{cb}\}q\right)\,=-0.50\left(5\right)$ and ${g}_{A}\left([{cb}]q\right)=1.51\left(15\right)$. With the same lattice QCD schemes as the doubly bottom baryons, we obtain the results of m0, c1, and c7, and predict the masses of the charmed-bottom baryons, as shown in table 3.
We show the ${ \mathcal O }\left({p}^{2}\right)$ tree and ${ \mathcal O }\left({p}^{3}\right)$ loop masses of the doubly bottom baryons and the charmed-bottom baryons in table 4. The mass differences of the nonstrange and the strange baryons are determined in equations (19 ) and (20 ). There are two reasons. First, the u/d and s quark mass difference is involved. Second, the interactions of the pseudoscalar mesons and the doubly bottom baryons and the charmed-bottom baryons are involved. We show the mass curves of the doubly bottom baryons and the charmed-bottom baryons as functions of ${m}_{\pi }^{2}$, and the mass points from our results and the lattice QCD data in figure 2. We show our results and other theoretical results for the masses of the doubly bottom baryons and the charmed-bottom baryons in table 5.
Figure 2. The mass curves of the doubly bottom baryons and the charmed-bottom baryons as functions of ${m}_{\pi }^{2}$, and the mass points from our results and the lattice QCD data. |
Table 4. The ${ \mathcal O }\left({p}^{2}\right)$ tree and ${ \mathcal O }\left({p}^{3}\right)$ loop masses of the doubly bottom baryons and the charmed-bottom baryons. All the physical quantities are in MeV. |
| m0 | ${ \mathcal O }\left({p}^{2}\right)$ tree | ${ \mathcal O }\left({p}^{3}\right)$ loop | mB | |
|---|---|---|---|---|
| Ξbb | 10 264 | −14 | −15 | 10 235 |
| Ωbb | 10 264 | 69 | −34 | 10 299 |
| Ξ{cb} | 7000 | 25 | −15 | 7010 |
| Ω{cb} | 7000 | 112 | −34 | 7078 |
| Ξ[cb] | 6854 | 213 | −137 | 6930 |
| Ω[cb] | 6854 | 471 | −308 | 7017 |
Table 5. Our results and other theoretical results for the masses of the doubly bottom baryons and the charmed-bottom baryons. All the physical quantities are in MeV. |
| Cases | ${m}_{{{\rm{\Xi }}}_{{bb}}}$ | ${m}_{{{\rm{\Omega }}}_{{bb}}}$ | ${m}_{{{\rm{\Xi }}}_{\{{cb}\}}}$ | ${m}_{{{\rm{\Omega }}}_{\{{cb}\}}}$ | ${m}_{{{\rm{\Xi }}}_{[{cb}]}}$ | ${m}_{{{\rm{\Omega }}}_{[{cb}]}}$ | |
|---|---|---|---|---|---|---|---|
| HBChPT | Our results | $10235\left(63\right)$ | $10299\left(64\right)$ | $7010\left(60\right)$ | $7078\left(61\right)$ | $6930\left(63\right)$ | $7017\left(84\right)$ |
| Quark models | [8, 9] | $10162\left(12\right)$ | $10208\left(18\right)$ | $6933\left(12\right)$ | $6984\left(19\right)$ | $6914\left(13\right)$ | $6968\left(19\right)$ |
| [10] | 10210 | 10319 | — | — | — | — | |
| [11] | $10168.9\left(9.2\right)$ | $10259.0\left(15.5\right)$ | $6947.9\left(6.9\right)$ | $7047.0\left(9.3\right)$ | $6922.3\left(6.9\right)$ | $7010.7\left(9.3\right)$ | |
| [12, 13] | 9716 | 10870 | 6628 | 7329 | 6628 | 7329 | |
| Lattice QCD | [14] | $10143\left(30\right)\left(23\right)$ | $10273\left(27\right)\left(20\right)$ | $6959\left(36\right)\left(28\right)$ | $7032\left(28\right)\left(20\right)$ | $6943\left(33\right)\left(28\right)$ | $6998\left(27\right)\left(20\right)$ |
| [18] | $10127\left(13\right)\left({}_{26}^{12}\right)$ | $10225\left(9\right)\left({}_{13}^{12}\right)$ | — | — | — | — | |
| [19] | $10267\left(44\right)\left(61\right)$ | $10356\left(34\right)\left(68\right)$ | $6937\left(105\right)\left(46\right)$ | $7077\left(109\right)\left(47\right)$ | $6887\left(103\right)\left(46\right)$ | $7039\left(108\right)\left(47\right)$ | |
| [20] | — | — | $6966\left(23\right)\left(14\right)$ | $7045\left(16\right)\left(13\right)$ | $6945\left(22\right)\left(14\right)$ | $6994\left(15\right)\left(13\right)$ | |
| [21] | $10091\left(17\right)$ | $10190\left(17\right)$ | $6843\left(19\right)$ | $6946\left(17\right)$ | $6787\left(12\right)$ | $6893\left(16\right)$ | |
| QCD sum rules | [27] | $10220\left({}_{70}^{70}\right)$ | $10330\left({}_{80}^{70}\right)$ | — | — | — | — |
| [28] | — | — | $6930\left(70\right)$ | $7040\left(80\right)$ | $6890\left(80\right)$ | $7010\left(80\right)$ |
5. Summary
We study the masses of the doubly bottom baryons and the charmed-bottom baryons up to ${ \mathcal O }\left({p}^{3}\right)$ in HBChPT. There are unknown low energy constants, m0, c1, c7, and gA. gA was determined in the quark model. For the doubly bottom baryons and the charmed-bottom baryons, with the same lattice QCD schemes, we fit the lattice QCD data to determine m0, c1, and c7. We show the numerical results for the masses of the doubly bottom baryons and the charmed-bottom baryons up to ${ \mathcal O }\left({p}^{3}\right)$.
The properties of the doubly heavy baryons are worth exploring, which can help people to understand the mechanism of the baryon spectrum. With the discovery of the ${{\rm{\Xi }}}_{{cc}}^{++}$ baryon, we hope for experimental evidence on the other doubly heavy baryons. Our numerical results may be useful for future experiments.