Doubly Heavy Baryons Expanded in $1/m_Q$

TL;DR

This paper develops a method to calculate the mass spectra and wave functions of doubly heavy baryons using a $1/m_Q$ expansion and a semirelativistic Hamiltonian, providing a foundation for further detailed studies.

Contribution

It introduces a formulation based on the Foldy-Wouthuysen-Tani transformation to expand baryon properties in $1/m_Q$, linking relativistic and nonrelativistic approaches.

Findings

Derived a $1/m_Q$ expansion for baryon masses and wave functions.

Examined the $ ho$ mode wave function with nonrelativistic models.

Proposed a comprehensive framework for doubly heavy baryon analysis.

Abstract

Starting from the semirelativistic Hamiltonian for a doubly heavy baryon system ($QQq$) with Coulomb and linear confining scalar potentials, and operating the naive Foldy-Wouthuysen-Tani transformation on two heavy quarks, I construct a formulation how to calculate mass spectra and wave functions of doubly heavy baryons. Based on our formulation, their masses and wave functions are expanded in $1/m_Q$ with a heavy quark mass $m_Q$ and the $\lambda$ mode lowest-order equation is examined carefully to obtain a complete set of eigenfunctions. I claim that the $\rho$ mode wave function should be given by other nonrelativistic methods, e.g., the Godfrey-Isgur model to give a complete wave function for a doubly heavy baryon.

Equations

$$\vec{\rho }=\frac{1}{\sqrt{2}}\left({\vec{r}}_1-{\vec{r}}_2\right),\vec{\lambda }=\frac{1}{\sqrt{6}}\left({\vec{r}}_1+{\vec{r}}_2-2{\vec{r}}_3\right).$$

$${\tilde{H}}_0=-{\vec{p}}_{{\lambda }^{\prime }}\cdot {\vec{\alpha }}_3+m_q{\beta }_3+2V({\lambda }^{\prime })+2{\beta }_{3}S({\lambda }^{\prime }),{\tilde{H}}_0{\psi }_0=E_0{\psi }_0,$$

$$y_{jm}^k(\Omega )$$

$$y_{jm}^{-k}(\Omega )$$

$$\mathrm{when}l=k\pm \frac{1}{2},\mathrm{then}k=\pm \left(j+\frac{1}{2}\right),$$

$${\psi }_{jm}^k$$

$$\displaystyle\left({\begin{array}[]{*{20}{c}}{{m_{q}}+2S+2V}&{-{\partial_{r}}+\frac{k}{r}}\\ {{\partial_{r}}+\frac{k}{r}}&{-{m_{q}}-2S+2V}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{u_{k}}(r)}\\ {{v_{k}}(r)}\end{array}}\right)=E_{0}^{k}\left({\begin{array}[]{*{20}{c}}{{u_{k}}(r)}\\ {{v_{k}}(r)}\end{array}}\right).,$$

$$\left(s_{\rho }=0,L_{\rho }=1,3,\cdots ,P_{\rho }=-\right),$$

$$\left(s_{\rho }=1,L_{\rho }=0,2,\cdots ,P_{\rho }=+\right).$$

$$Ref.[2]$$

$$Ref.[3]$$

$$\mathrm{Ours}$$

$${\mathcal{L}}_{\mathrm{int}}$$

$$(\varphi {)}_{ij}$$

$$\displaystyle\left({\begin{array}[]{*{20}{c}}{{\pi^{0}}}\\ \eta\end{array}}\right)$$

$${\Omega }_{cc}{\left(3/2^{+}\right)}^{+}(ccs)\to {\Omega }_{cc}{\left(1/2^{-}\right)}^{+}(ccs)+{\pi }^0.$$

$${\Omega }_{cc}{\left(3/2^{+}\right)}^{+}(ccs)\to {\Xi }_{cc}{\left(1/2^{-}\right)}^{++}(ccu)+K^{-}.$$

$$\displaystyle\left({\begin{array}[]{*{20}{c}}{(1S1p)1/{2^{-}}}\\ {(1S1p)1/{2^{{\rm{{}^{\prime}}}-}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{\cos\theta}&{-\sin\theta}\\ {\sin\theta}&{\cos\theta}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{(1S1p)1/{2^{-}}(j_{\lambda}=1/2)}\\ {(1S1p)1/{2^{-}}(j_{\lambda}=3/2)}\end{array}}\right).$$

Full text

heavy baryon, heavy quark symmetry, spectrum

1 Introduction

Heavy quark symmetry can be respected in hadrons in which at least one heavy quark is included and other quarks are light quarks. The simplest case is a heavy-light meson and the next will be a doubly heavy baryon $QQq$ because compared with $Qqq$ baryon, there is only one light quark involved. Treating two heavy quark is rather easier compared with two light quarks. The heavy-light meson has been extensively studied by us in Ref. [1] and also by many groups (see Ref. [2] for a review).

In this article, we will describe how to treat doubly heavy baryons using the method adopted in the former paper [1] where heavy-light mesons have been studied. Starting from the semirelativistic Hamiltonian for a doubly heavy baryon system ($QQq$) with Coulomb and linear confining scalar potentials, and operating the naive Foldy-Wouthuysen-Tani transformation on the heavy quarks, I expand all the physical quantities, Hamiltonian $H$, wave function $\psi$, and energy eigenvalue $E$, in $1/m_{Q}$ with heavy quark mass $m_{Q}$. However, as for the so-called $\rho$ mode wave function, we need to use other methods to calculate. Godfrey-Isgur relativised potential model is one choice. This is because by using the FWT transformation, we cannot obtain matrix structure for the $\rho$ mode Hamiltonian so that we cannot distinguish a couple of spin states of a heavy diquark.

2 Formulation

Using heaviness of $c$ or $b$ quarks compared with light quarks, $u,d,$ and $s$, we apply the method developed in Ref. [1] to a doubly heavy baryon. In Ref. [1], we have used the Foldy-Wouthuysen-Tani (FWT) transformation to expand the system in $1/m_{Q}$. Since there are two heavy quarks in a doubly heavy baryon, we need to operate two kinds of the FWT transformation on $QQq$.

Instead of using the Cartesian coordinates $\vec{r}_{1},\vec{r}_{2}$, and $\vec{r}_{3}$, we use the Jacobi relative coordinates $\vec{\lambda}$ and $\rho$. If we regard quarks 1 and 2 are heavy with mass $m_{1}=m_{2}=m_{Q}$ and quark 3 is light with $m_{3}=m_{q}$, the Jacobi coordinate system can be simplified as,

[TABLE]

After expansion, we obtain the lowest order eigenvalue equation for the $\lambda$ mode wave function as,

[TABLE]

where $\lambda^{\prime}=\lambda/\sqrt{6}$, one-gluon exchange potential $V(r)=-2\alpha_{s}/(3r)$, and confining linear potential $S(r)=r/a^{2}+b$. This equation expresses nothing but the one for interaction between a light quark and doubly heavy diquark, which is automatically derived from our formulation.

An angular part of a solution to Eq. (2) is explicitly given as follows. We define the following eigenfaunctions for Eq. (2),

[TABLE]

where $Y_{j}^{m}$ are spherical hamonics, and $k=j+1/2$. Here is a relation between $k$, $l$, and $j$ as

[TABLE]

and $k$ is an eigenvalue of an operator $K=-\beta_{q}\left(\vec{\Sigma}_{q}\cdot\vec{L}_{\lambda}+1\right)$. Then a general solution to Eq. (2) is given by

[TABLE]

where functions $u_{k}(r)$ and $v_{k}(r)$ satisfy the following eigenvalue equation:

[TABLE]

which can be solved like in Ref. [1] using the variational method.

3 Physics related to heavy-light systems

1. Relation among $L_{\rho}$, $s_{\rho}$, and state symbols

Because the total wave function of a diquark should be antisymmetric for two heavy quarks, there is a relation between $L_{\rho}$ and $s_{\rho}$. First, I should mention that a diquark with the same two heavy quarks, $c$ or $b$, is flavor symmetric and color antisymmetric. As for a combination of two heavy quarks, if a diquark has spin $s_{\rho}=0$, i.e., two heavy quarks have opposite spin directions, a spin wave function is antisymmetric. On the other hand, if a diquark has spin $s_{\rho}=1$, a spin wave function is symmetric. When we denote spin as $s_{\rho}$, angular momentum as $L_{\rho}$, and parity as $P_{\rho}$ for a $\rho$ mode diquark, we have the following combinations,

[TABLE]

As you can see from these combinations, when the value of $L_{\rho}$ is given, the value of $s_{\rho}$ becomes unique. Together with these quantum numbers, we need to consider quantum numbers coming from the $\lambda$ mode, i.e., principal quantum number $n_{\lambda}$, angular momentum $l_{\lambda}$, and parity $P_{\lambda}=(-)^{l_{\lambda}}$.

What kind of symbol should be adopted is an interesting problem. We list the three kinds of them.

[TABLE]

where $P=(-)^{L_{\rho}+L_{\lambda}}$, $k=\pm(j_{\rho}+1)$, and the $\rho$ mode principal quantum number $N_{\rho}$. Ref. [4] and ours are equivalent but Ref. [3] lacks information on $\rho$ mode quantum numbers.

2. Threshold behaviors

Let us consider the similarity of doubly heavy baryons to heavy-light mesons, especially to $D_{s}(0^{+},1^{+})$, which have very narrow widths. This is described as follows: Assume that $SU(3)$ light meson and quark interaction is given by

[TABLE]

Then, since $s$ quark ($i,j=3$) inside of $D_{s}$ couples to $\phi_{8}$ which is mixed with pion ($\pi^{0}$), heavy-light meson can decay into another heavy-light meson + $\pi^{0}$ with a small mixing parameter $\epsilon=1.0\times 10^{-2}$.

If $M(D_{s}(0^{+})>M(D(0^{-}))+M(K)$, then we would expect a broard decay width of the $D_{s}(0^{+})$. However, what we have found is the oppsite situation so that the decay width of this state becomes very narrow because the allowed decay channel is $D_{s}(0^{+})\to D_{s}(0^{-})+\pi$ which occurs through very small $\pi^{0}-\eta$ coupling.

A similar process to this in doubly heavy baryons is given by, if $M(\Omega_{cc}({3}/{2}^{+}))<M(\Xi_{cc}({1}/{2}^{+}))+M(K)$,

[TABLE]

if $M(\Omega_{cc}({3}/{2}^{+}))>M(\Xi_{cc}({1}/{2}^{+}))+M(K)$, we expect a broad decay width due to the existence of the process,

[TABLE]

where $K$ has mass of 494-498 MeV, the quark content of $\Xi_{cc}\left({1}/{2}^{+}\right)^{+}$ is $ccd$, while that of $\Xi_{cc}\left({1}/{2}^{+}\right)^{++}$ is $ccu$. Doubly heavy baryons have excitations in $\rho$ mode, i.e., the first excitation is given by $L_{\rho}L_{\lambda}=P_{\rho}S_{\lambda}$ with a total angular momentum $J^{P}=3/2^{+}$ and the ground states are given by $S_{\rho}S_{\lambda}$ with $J^{P}=1/2^{-}$.

2. Mixing angles in heavy-quark symmetry

In principle,there could be mixing between staes with the same quantum number $J^{P}$. For instance, there may be mixing between states with $J^{P}=1/2^{-}$, e.g., ${(1S1p)1/{2^{-}}(j_{\lambda}=1/2)}$, ${(1S1p)1/{2^{-}}(j_{\lambda}=3/2)}$, and ${(1P1s)1/{2^{-}}(j_{\lambda}=1/2)}$. We have checked whether there is mixing, e.g., between ${(1S1p)1/{2^{-}}(j_{\lambda}=1/2)}$ and ${(1P1s)1/{2^{-}}(j_{\lambda}=1/2)}$. It turns out that as long as we consider matrix elements with interactions of spin-spin and $LS$ coupling, we obtain no mixing between them, i.e., there is mixing between the states with the same $L_{\lambda}$ and different $L_{\rho}$ but no mixing between $S_{\rho}P_{\lambda}$ and $P_{\rho}S_{\lambda}$. Because we use the heavy quark symmetry, we obtain mixing angles between certain states à la heavy-light mesons [5]. One example is given by,

[TABLE]

with $\theta=\arctan(1/\sqrt{2})=35.3^{\circ}$ in the heavy quark limit.

4 Summary

In this article, I have described how one can obtain the $\lambda$ mode wave function with an explicit angular part. Other physical quantities, good quantum number $K$, state symbols, possible threshold behaviors, and mixing angles among states with the same quantum numbers, have been discussed.

Acknowledgement

I would like to thank Qi-Fang L${\ddot{\rm u}}$ of Huan Normal University for helpful discussions.