Doubly Heavy Baryons Expanded in $1/m_Q$

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.


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 ψ, and energy eigenvalue E, in 1/m Q with heavy quark mass m Q . However, as for the so-called ρ 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 ρ mode Hamiltonian so that we cannot distinguish a couple of spin states of a heavy diquark.

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 r 1 , r 2 , and r 3 , we use the Jacobi relative coordinates λ and ρ. 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, After expansion, we obtain the lowest order eigenvalue equation for the λ mode wave function as,H where λ ′ = λ/ √ 6, one-gluon exchange potential V(r) = −2α 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), where Y m j are spherical hamonics, and k = j + 1/2. Here is a relation between k, l, and j as and k is an eigenvalue of an operator K = −β q Σ q · L λ + 1 . Then a general solution to Eq. (2) is given by where functions u k (r) and v k (r) satisfy the following eigenvalue equation: which can be solved like in Ref. [1] using the variational method.
3. Physics related to heavy-light systems 1) Relation among L ρ , s ρ , and state symbols Because the total wave function of a diquark should be antisymmetric for two heavy quarks, there is a relation between L ρ and s ρ . 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 ρ = 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 ρ = 1, a spin wave function is symmetric. When we denote spin as s ρ , angular momentum as L ρ , and parity as P ρ for a ρ mode diquark, we have the following combinations, s ρ = 1, L ρ = 0, 2, · · · , P ρ = + .
As you can see from these combinations, when the value of L ρ is given, the value of s ρ becomes unique. Together with these quantum numbers, we need to consider quantum numbers coming from the λ mode, i.e., principal quantum number n λ , angular momentum l λ , and parity P λ = (−) l λ . What kind of symbol should be adopted is an interesting problem. We list the three kinds of them.
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 S U(3) light meson and quark interaction is given by Then, since s quark (i, j = 3) inside of D s couples to φ 8 which is mixed with pion (π 0 ), heavy-light meson can decay into another heavy-light meson + π 0 with a small mixing parameter ǫ = 1.0 × 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 + ) → D s (0 − ) + π which occurs through very small π 0 − η coupling.

Summary
In this article, I have described how one can obtain the λ 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.