You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    ------------------------------------------------------------------------

    We evaluate static properties and semileptonic decays for the ground state of doubly heavy $\Xi,\,\Xi',\,\Xi^*$ and $\Omega,\,\Omega',\,\Omega^*$ baryons. Working in the framework of a nonrelativistic quark model, we solve the three–body problem by means of a variational ansatz made possible by heavy quark spin symmetry constraints. To check the dependence of our results on the inter-quark interaction we use five different quark-quark potentials that include a confining term plus Coulomb and hyperfine terms coming from one–gluon exchange. Our results for static properties (masses, charge radii and magnetic moments ) are, with a few exceptions for the magnetic moments, in good agreement with a previous Faddeev calculation. Our much simpler wave functions are used to evaluate semileptonic decays of doubly heavy $\Xi,\Xi'(J=1/2)$ and $\Omega,
    \Omega'(J=1/2)$ baryons. Our results for the decay widths are in good agreement with calculations done within a relativistic quark model in the quark–diquark approximation.
author:
- 'C. Albertus'
- 'E. Hernández'
- 'J. Nieves'
- 'J. M. Verde-Velasco'
title: ' Static properties and semileptonic decays of doubly heavy baryons in a nonrelativistic quark model. '
---

[^1]

Introduction
============

[cccccccc]{}Baryon &    $S$    &    $J^P$    &     $I$    &    $S_{ h}^\pi$    & Quark content

\
& & & & &\
$\Xi_{cc}$& 0 &$\frac12^+$& $\frac12$ &$1^+$&$ccl$\
$\Xi^*_{cc}$ & 0 &$\frac32^+$&$\frac12$ &$1^+$&$ccl$\
$\Omega_{cc}$ & $-1$ &$\frac12^+$& 0 &$1^+$&$ccs$\
$\Omega^*_{cc}$ & $-1$ &$\frac32^+$&$0$&$1^+$&$ccs$\
\
$\Xi_{bb}$& 0 &$\frac12^+$& $\frac12$ &$1^+$&$bbl$\
$\Xi^*_{bb}$ & 0 &$\frac32^+$&$\frac12$ &$1^+$&$bbl$\
$\Omega_{bb}$ & $-1$ &$\frac12^+$& 0 &$1^+$&$bbs$\
$\Omega^*_{bb}$ & $-1$ &$\frac32^+$&$0$&$1^+$&$bbs$\
\
$\Xi'_{bc}$& 0 &$\frac12^+$& $\frac12$ &$0^+$&$bcl$\
$\Xi_{bc}$ & 0 &$\frac12^+$&$\frac12$ &$1^+$&$bcl$\
$\Xi^*_{bc}$& 0 &$\frac32^+$& $\frac12$ &$1^+$&$bcl$\
$\Omega'_{bc}$ & $-1$ &$\frac12^+$& 0 &$0^+$&$bcs$\
$\Omega_{bc}$ & $-1$ &$\frac12^+$& 0 &$1^+$&$bcs$\
$\Omega^*_{bc}$ & $-1$ &$\frac32^+$&$0$&$1^+$&$bcs$\

The subject of doubly heavy baryons has been attracting attention for a long time. Magnetic moments of doubly charmed baryons were evaluated back in the 70’s by Lichtenberg [@lichtenberg77] within a nonrelativistic approach. The infinite heavy quark mass limit was already used in the 90’s to relate the spectrum of doubly heavy baryons to the one of mesons with a single heavy quark [@savage90], or to analyze their semileptonic decay [@white91]. A factor of two error in the hyperfine splittings of Ref. [@savage90] have been recently noticed by the potential nonrelativistic QCD (pNRQCD) calculation of Ref. [@brambilla05].

On the experimental side the SELEX Collaboration claimed evidence for the $\Xi^+_{cc}$ baryon, in the $\Lambda_c^+K^-\pi^+$ and $pD^+K^-$ decay modes, with a mass of $M_{\Xi^+_{cc}}=3519\pm 1\ \mathrm{MeV/c^2}$ [@mattson02]. Those results were challenged by a theoretical analysis [@kili] which claimed the observed events by the SELEX Collaboration could be explained without the involvement of doubly charmed baryons. Other experimental collaborations like FOCUS [@focus03], [*BABAR*]{}  [@babar06] and BELLE [@lesiak06] have found no evidence for doubly charmed baryons so far. At present the $\Xi^+_{cc}$ has only a one star status and it is not listed in the particle summary table [@pdg06].

In hadrons with a heavy quark and working in the infinite heavy quark mass limit the dynamics of the light degrees of freedom becomes independent of the heavy quark flavor and spin. This is known as heavy quark symmetry (HQS) [@hqs1; @hqs2; @hqs3; @hqs4]. This symmetry was developed into an effective theory (HQET) [@georgi90] that allowed a systematic, order by order, evaluation of corrections in inverse powers of the heavy quark masses. Unfortunately ordinary HQS can not be applied directly to hadrons containing two heavy quarks as the kinetic energy term needed in those systems to regulate infrared divergences breaks heavy flavor symmetry [@thacker91]. For those hadrons the symmetry that survives is heavy quark spin symmetry (HQSS) [@jenkins93], which amounts to the decoupling of the heavy quark spins in the infinite heavy quark mass limit. In that limit one can consider the total spin of the two heavy quark subsystem ($S_h$) to be well defined. In this work we shall assume this is a good approximation for the actual heavy quark masses. This approximation, which is the only one related to the infinite heavy quark mass limit that we shall use, will certainly simplify the solution of the baryon three–quark problem. Recently the authors of Ref. [@brambilla05] have developed and effective theory (pNRQCD) suitable to describe baryons with two and three heavy quarks.

Solving the three–body problem is not an easy task and here we shall do it by means of a variational approach. The approach, with obvious changes, was already applied with good results in the study of baryons with one heavy quark [@conrado04]. This method, that leads to simple and manageable wave functions, is made possible by the simplifications introduced in the problem by the fact that we can consider $S_h$ to be well defined. We shall consider several simple phenomenological quark–quark interactions [@BD81; @SS94; @Si96] which free parameters have been adjusted in the meson sector and are thus free of three–body ambiguities. The use of different interactions will allow us to estimate part of the theoretical uncertainties affecting our calculation. Uncertainties related to the nonrelativistic baryon states that we use are difficult to estimate. We are aware of the limitations of a non-relativistic approach to describe light quark physics. However, for the kind of study performed in this work these are likely not as relevant as in other contexts. We study the semileptonic decays of double heavy baryons, in which the light quark is merely an spectator (heavy-to-heavy transitions). Indeed the lack of a proper relativistic treatment did prevent us to study semileptonic transitions of the type $b \to
   u$ of a greater phenomenological interest. One should notice however that at least part of the relativistic effects not explicitly taken into account are included in an effective way in the parameters of the interaction which had been fitted to experimental data. We think this explains why the nonrelativistic quark model is so successful phenomenologically even in the presence of light quarks.

Our simple variational calculation reproduces the results for static properties obtained in Ref. [@Si96] by solving more involved Faddeev type equations. Our method has the advantage that we provide explicit and manageable wave functions that can be used to evaluate further observables. Static properties like masses and magnetic moments of doubly heavy baryons have also been studied in other models. Masses have been calculated in the relativistic quark model assuming a light quark heavy diquark structure [@ebert02], the potential approach and sum rules of QCD [@kiselev02], the nonperturbative QCD approach [@narodetskii02], the Bethe–Salpeter equation applied to the light quark heavy diquark [@tong00], the nonrelativistic quark model with harmonic oscillator potential [@itoh00] or with the use of QCD derived potentials [@vijande04; @gershtein00], the relativistic quasi–potential quark model [@ebert97], with the use of the Feynman-Hellman theorem and semi-empirical mass formulas within the framework of a nonrelativistic constituent quark model [@roncaglia95], in effective field theories [@korner94; @brambilla05], or in lattice nonrelativistic QCD [@mathur02]. There are also lattice QCD determinations  [@khan00; @lewis01; @flynn03]. Similarly, magnetic moments have been evaluated in a nonrelativistic approach [@lichtenberg77], in the relativistic three–quark model [@faessler06], the relativistic quark model using different forms of the relativistic kinematics [@bruno04], in the skyrmion model [@oh91], in the Dirac equation formalism [@jena86], or using the MIT bag model [@bose80].

We shall further use our manageable wave functions to study semileptonic decays of doubly heavy , $J=1/2$, baryons. We shall evaluate form factors, decay widths and angular asymmetry parameters. Previous calculations of semileptonic decay widths have been done in different relativistic quark model approaches  [@ebert04; @faessler01; @guo98], or with the use of HQET [@sanchis95].

The paper is organized as follows. In Sect. \[sec:3bp\] we study the Hamiltonian of the system (Subsect. \[subsec:h\]) and briefly introduce the different inter-quark interactions used in this work (Subsect. \[subsec:qqi\]). The variational wave functions are discussed in Sect. \[sec:vwf\]. In Sect. \[sec:sp\] we present results for the static properties: masses (Subsect. \[subsec:masses\]), charge densities and radii (Subsect. \[subsec:chmdr\]), and magnetic moments (Subsect. \[subsec:mm\]). Semileptonic decays are analyzed in Sect. \[sec:sd\]. After the presentation of general formulas, in Subsect. \[subsec:ff\] we relate the form factors to matrix elements and show how the latter ones are evaluated within our model. In Subsect. \[subsec:results\] we present our results for the form factors, differential and total semileptonic decay widths, and angular asymmetry parameters. The findings of this work are summarized in Sect. \[sec:summary\]. The paper also includes three appendices: in appendix  \[app:ihqml\] we analyze the infinite heavy quark mass limit of our radial wave functions. In appendix \[app:fg123\] we relate the form factors for semileptonic decay to the two basic integrals in terms of which all of them can be obtained. Finally, in appendix \[app:integrals\] we give explicit expressions for those basic integrals.

In Table \[tab:summ\] we summarize the quantum numbers of the doubly heavy baryons considered in this study[^2].

Three Body Problem {#sec:3bp}
==================

Intrinsic Hamiltonian {#subsec:h}
---------------------

In the Laboratory (LAB) frame (see Fig. \[fig:coor\]), the Hamiltonian ($H$) of the three quark ($h_1,\,h_2,\,q$, where $h_1,\,h_2=c,b$ and $q=l(u,\,d),\,s$) system reads: $$\begin{aligned}
H&=&  \sum_{j=h_1,\,h_2,\,q} \left (m_j
-\frac{\stackrel{\rightarrow}{\nabla}\stackrel{}{^2}_{\hspace{-.1cm}\vec x_j}}{2m_j}\right ) +
V_{h_1h_2} + V_{h_1q}+V_{h_2 q }\end{aligned}$$ where $m_{h_1},\,m_{h_2},\,m_q$ are the quark masses and the quark-quark interaction terms $V_{jk}$ depend on the quark spin-flavor quantum numbers and the quark coordinates ($\vec{x}_{h_1},
\vec{x}_{h_2},\, \vec{x}_q$ for the $h_1,\,h_2,\,q$ quarks respectively). To separate the Center of Mass (CM) free motion, we go to the light quark frame ($\vec{R},\vec{r}_1,\vec{r}_2$), $$\begin{aligned}
\vec{R}&=&\frac{m_{h_1}\vec{x}_{h_1} + m_{h_2}\vec{x}_{h_2} + m_q \vec{x}_q}
{m_{h_1}+m_{h_2}+m_q} \nonumber \\
\vec{r}_1 & = & \vec{x}_{h_1} - \vec{x}_q \nonumber \\
\vec{r}_2 & = & \vec{x}_{h_2} - \vec{x}_q \end{aligned}$$ where $\vec{R}$ and $\vec{r}_1,\, \vec{r}_2$ are the CM position in the LAB frame and the relative positions of the $h_1,\,h_2$ heavy quarks with respect to the light quark $q$. The Hamiltonian now reads $$\begin{aligned}
H&=&
-\frac{\stackrel{\rightarrow}{\nabla}
\stackrel{}{^2}_{\hspace{-.1cm}\vec{R}}}{2 \overline M} +
H^{\rm int} \end{aligned}$$ $$\begin{aligned}
 H^{\rm
int}&=&\overline M+\sum_{j=1,2}\,H_j^{sp}+V_{h_1h_2}(\vec r_1-\vec r_2,\, spin)-
\frac{\stackrel{\rightarrow}{\nabla}_1\cdot\stackrel{\rightarrow}{\nabla}_2}
{m_q}
\nonumber\\
H_j^{sp}&=&-\frac{\stackrel{\rightarrow}{\nabla}\stackrel{}{^2}_{\hspace{-0.2cm}j}
}{2\mu_j}+V_{h_jq}(\vec r_j,\, spin),\ j=1,2
\label{eq:hint}\end{aligned}$$ where $\overline M = m_{h_1}+m_{h_2}+m_{q}$, $\mu_{j} = \left (
1/m_{h_j} + 1/m_q\right)^{-1}$ and $\stackrel{\rightarrow}{\nabla}_{j}
= \partial/\partial_{\vec{r}_j},\ j=1,2$. The intrinsic Hamiltonian $H^{\rm int}$ describes the dynamics of the baryon. Apart from the sum of the quark masses $\overline M$, it consists of the sum of two single particle Hamiltonian $H^{sp}_j$, each of them describing the dynamics of a heavy-light quark system, plus the heavy–heavy interaction term, including the Hughes-Eckart term ($\stackrel{\rightarrow}{\nabla}_1\cdot
\stackrel{\rightarrow}{\nabla}_2$). We will use a variational approach to solve it.

Quark–Quark Interactions {#subsec:qqi}
------------------------

We have examined five different interactions, one suggested by Bhaduri and collaborators [@BD81] (BD) and four suggested by B. Silvestre-Brac and C. Semay [@Si96; @SS94] (AL1, AL2, AP1 y AP2). All of them contain a confinement term, plus Coulomb and hyperfine terms coming from one-gluon exchange, and differ from each other in the form factors used for the hyperfine terms, the power of the confining term or the use of a form factor in the one gluon exchange Coulomb potential. All free parameters in the potentials had been adjusted to reproduce the light ($\pi$, $\rho$, $K$, $K^*$, etc.) and heavy-light ($D$, $D^*$, $B$, $B^*$, etc.) meson spectra[^3]. All details on the above interactions can be found in Refs. [@BD81; @Si96; @SS94].

These interactions were also used in Ref. [@Si96] to obtain, within a Faddeev calculation, the spectrum and static properties of heavy baryons. Our simpler variational method will not only give equally good results for the observables analyzed in [@Si96], but it will also provide us with easy to handle wave functions that can be used to evaluate other observables.

Variational Wave Functions {#sec:vwf}
==========================

For the above mentioned interactions, we have that both the total spin and the internal orbital angular momentum given as $$\begin{aligned}
{\vec
S} &=& \left ( {\vec \sigma}_{h_1} + {\vec \sigma}_{h_2} +
{\vec \sigma}_q \right )/2\nonumber\\
{\vec L} &=& {\vec l}_1 + {\vec l}_2, \qquad {\rm with}~~ {\vec
l}_j=-{\rm i}~~{\vec r}_j \times {\stackrel{\rightarrow}{\nabla}}_j, \quad j=1,2\end{aligned}$$ commute with the intrinsic Hamiltonian and are thus well defined. We are interested in the ground state of baryons with total angular momentum $J=1/2,\,3/2$ so that we can assume the orbital angular momentum of the baryons to be $L=0$. This implies that the spatial wave function can only depend on the relative distances $r_1$, $r_2$ and $r_{12}=|\vec{r}_1-\vec{r}_2|$. Furthermore when the heavy quark mass is infinity ($m_h \to \infty$), the total spin of the heavy degrees of freedom, $\vec{S}_{\rm heavy} = \left ( {\vec \sigma}_{h_1}
+ {\vec \sigma}_{h_2} \right )/2$, commutes with the intrinsic Hamiltonian, since the spin–spin terms in the potentials vanish in this limit. We can then assume the spin of the heavy degrees of freedom to be well defined.

With these simplifications we have used the following intrinsic wave functions in our variational approach[^4]

-   [*$\Xi_{h_1h_2},\,\Omega_{h_1h_2}-$type baryons:* ]{} $$\begin{aligned}
    |\Xi_{h_1h_2},\,\Omega_{h_1h_2}; J=\frac12, M_J  \rangle  &=& 
    \sum_{M_{S_h}M_{S_q}} ( 1\frac12\frac12|M_{S_h}M_{S_q}M_J)\  |h_1h_2;1M_{S_h}
     \rangle \otimes |q;\frac12 M_{S_q}\rangle \nonumber\\
    &&\hspace{3cm}\times\Psi_{h_1h_2}^{\Xi,\,\Omega}(r_1,r_2,r_{12})\end{aligned}$$ where $M_J$ is the third component of the baryon total angular momentum while $|h_1h_2;S_h,M_{S_h}\rangle$ and $|q;\frac12
    M_{S_q}\rangle $ represent spin states of the $h_1h_2$ subsystem and the light quark respectively. $(j_1j_2j|m_1m_2m)$ is a Clebsch-Gordan coefficient. For $h_1=h_2$ we need $\Psi_{h_1h_1}^{\Xi,\,\Omega}(r_1,r_2,r_{12})
    =\Psi_{h_1h_1}^{\Xi,\,\Omega}(r_2,r_1,r_{12})$ to guarantee a complete symmetry of the wave function under the exchange of the two heavy quarks.

-   [*$\Xi^*_{h_1h_2},\,\Omega^*_{h_1h_2}-$type baryons:* ]{} $$\begin{aligned}
    |\Xi^*_{h_1h_2},\,\Omega^*_{h_1h_2}; J=\frac32, M_J  \rangle  &=& 
    \sum_{M_{S_h}M_{S_q}} ( 1\frac12\frac32|M_{S_h}M_{S_q}M_J)\  |h_1h_2;1M_{S_h}
     \rangle \otimes |q;\frac12 M_{S_q}\rangle \nonumber\\
    &&\hspace{3cm}\times\Psi_{h_1h_2}^{\Xi^*,\,\Omega^*}(r_1,r_2,r_{12})\end{aligned}$$ Similarly to the case before for $h_1=h_2$ we need $\Psi_{h_1h_1}
    ^{\Xi^*,\,
    \Omega^*}(r_1,r_2,r_{12})
    =\Psi_{h_1h_1}^{\Xi^*,\,\Omega^*}(r_2,r_1,r_{12})$.

-   [*$\Xi'_{h_1h_2},\,\Omega'_{h_1h_2}-$type baryons:* ]{} $$\begin{aligned}
    |\Xi'_{h_1h_2},\,\Omega'_{h_1h_2}; J=\frac12, M_J  \rangle  &=& 
      |h_1h_2;00 \rangle \otimes |q;\frac12 M_{J}\rangle \nonumber\\
    &&\hspace{3cm}\times\Psi_{h_1h_2}^{\Xi',\,\Omega'}(r_1,r_2,r_{12})\end{aligned}$$ In this case $h_1\ne h_2$ and we do not need the orbital part to have a definite symmetry under the exchange of the two quarks.

The spatial wave functions $\Psi(r_1,r_2,r_{12})$ in the above expressions will be determined by the variational principle: $\delta
\langle B | H^{\rm int}| B \rangle = 0$. For simplicity, we shall assume a Jastrow–type functional form[^5]:

$$\begin{aligned}
\Psi_{h_1h_2}^{B} (r_1,r_2,r_{12}) &=& N\, F^{B}(r_{12})\,
\phi_{h_1q}(r_1)\,\phi_{h_2q}(r_2)
\label{eq:wf}\end{aligned}$$

where $N$ is a constant, which is determined from normalization $$\begin{aligned}
1=\int d^3r_1 \int d^3 r_2\ \left |\Psi_{h_1h_2}^{B}(r_1,r_2,r_{12})
\right |^2 = 8\pi^2 \int_0^{+\infty}dr_1~r_1^2~ \int_0^{+\infty}dr_2~r_2^2
\int_{-1}^{+1} d\mu~\left |\Psi_{h_1h_2}^{B}(r_1,r_2,r_{12})
\right |^2 \end{aligned}$$ with $\mu$ being the cosine of the angle between the vectors $\vec
{r}_1$ and $\vec {r}_2$ ($r_{12}=(~r_1^2+r_2^2-2 r_1 r_2
\mu)^{1/2}$).

The functions $\phi_{h_1q}$ and $\phi_{h_2q}$ will be taken as the $S-$wave ground states $\varphi_j(r_j)$ of the single particle Hamiltonians $H^{sp}_{j}$ of Eq. (\[eq:hint\]) modified at large distances. $$\begin{aligned}
\phi_{h_jq} (r_j) &=& (1+\alpha_j\,r_j)\,\varphi_j(r_j),\ \ j=1,2
\label{eq:onebody}\end{aligned}$$ The heavy–heavy correlation function $F^{B}$ will be given by a linear combination of gaussians[^6] $$\begin{aligned}
F^{B}(r_{12}) &=& 
\sum_{j=1}^4 a_j e^{-b_j^2(r_{12}+d_j)^2},\quad a_1=1 
\label{eq:f12}\end{aligned}$$ The value of one of the $a_j$ parameters can be absorbed into the normalization constant $N$, so that we fix $a_1=1$. The rest of the variational parameters are determined by the variational condition $\delta
\langle B | H^{\rm int}| B \rangle = 0$.

Although it is not self-evident from the functional form assumed, our variational wave functions are consistent with the infinite heavy quark mass limit, reducing in that limit to the product of the internal wave function for the heavy diquark times the wave function for the relative motion of the light quark with respect to a pointlike heavy diquark. All this is discussed in appendix \[app:ihqml\].

Static Properties {#sec:sp}
=================

Masses {#subsec:masses}
------

The mass of the baryon is simply given by the expectation value of the intrinsic Hamiltonian. Our results (VAR) appear in Table \[tab:xiomegamasses\] where we compare them with the ones obtained in Ref. [@Si96] with the use of the same inter-quark interactions but within a Faddeev approach (FAD). For that purpose we have eliminated from the latter a small three-body force contribution of the type $V_{123}= {\rm constant}/m_{h_1}m_{h_2}m_q$ that was also included in the evaluation of Ref. [@Si96]. We will show their full results in the following tables. Whenever comparison is possible we find an excellent agreement between the two calculations. In some cases the variational masses are even lower than the Faddeev ones. Besides we give predictions for states that were not considered in the study of Ref. [@Si96].

[lcccccc]{} & & AL1 & AL2 &AP1 &AP2 &BD\

$\Xi_{cc}$ &VAR& 3612 & 3619 & 3629 & 3630 & 3639\
&FAD [@Si96]& 3609 & 3616 & 3625 & 3628 & 3633\
\
$\Xi_{cc}^*$ &VAR& 3706 & 3715 & 3722 & 3729 & 3722\
\
$ \Xi_{bb} $ &VAR& 10197 & 10180 & 10207 & 10179 & 10202\
&FAD [@Si96]& 10194 & 10175 & 10204 & 10176 & 10197\
\
$\Xi_{bb}^*$ &VAR& 10236 & 10219 & 10245 & 10219 & 10235\
\
$\Xi_{bc}$ &VAR& 6919 & 6912 & 6933 & 6917 & 6936\
&FAD [@Si96]& 6916 & 6913 & 6928 & 6907 & 6934\
\
$\Xi_{bc}'$ &VAR& 6948 & 6942 & 6957 & 6944 & 6965\
\
$\Xi_{bc}^*$ &VAR& 6986 & 6981 & 7000 & 6987 & 6993\

[lc|ccccc]{} & & AL1 & AL2 &AP1 &AP2 &BD\

$\Omega_{cc}$ &VAR& 3702 & 3718 & 3711 & 3710 & 3743\
&FAD [@Si96]& 3711 & 3718 & 3710 & 3709 & 3741\
\
$\Omega_{cc}^*$ &VAR& 3783 & 3802 & 3800 & 3802 & 3805\
\
$\Omega_{bb}$ &VAR& 10260 & 10249 & 10259 &10226 &10274\
&FAD [@Si96]& 10267 & 10246 & 10258 &10224 &10271\
\
$\Omega_{bb}^*$ &VAR& 10297 & 10287 & 10301 &10269 &10302\
\
$\Omega_{bc}$ &VAR& 6986 & 6986 & 6990 & 6969 & 7013\
&FAD [@Si96]& 7003 & 6996 & 6996 & 6971 & 7023\
\
$\Omega_{bc}'$ &VAR& 7009 & 7010 & 7011 & 6994 & 7033\
\
$\Omega_{bc}^*$ &VAR& 7046 & 7047 & 7055 & 7037 & 7057\

In Tables \[tab:ximassesdiffmod\] and \[tab:omegamassesdiffmod\] we compare our results with other theoretical calculations[^7]. Our central values correspond to the results obtained with the AL1 potential, while the errors quoted take into account the variation when using different potentials. The same presentation is used for the results obtained in Ref. [@Si96] for which we now show their full values including the contribution of the three-body force. All calculations give similar results that vary within a few per cent. From the experimental point of view the SELEX Collaboration [@mattson02] has recently measured the value of $M_{\Xi_{cc}}$. This experimental value is 100 MeV smaller that our result. On account of what has been said in the introduction, one should take this experimental value with due caution. Note also that in Ref. [@mattson02] the systematic error is not given. There are also different lattice determinations for baryons with two equal heavy quarks [@khan00; @lewis01; @flynn03]. Our results agree within errors with the lattice data for baryons with two $c$ quarks, while they are roughly 100MeV below lattice results for doubly $b$-quark baryons. The best overall agreement with lattice data available so far is achieved in the calculation of Ref. [@roncaglia95] where they use the Feynman-Hellmann theorem and semiempirical mass formulas in the framework of a nonrelativistic quark model but without the use of an explicit Hamiltonian. Within full dynamical calculations, ours and the relativistic calculations in Refs. [@ebert02; @tong00; @ebert97] have the best overall agreement with lattice data.

                       This work              [@Si96]         [@ebert02]   [@kiselev02]   [@narodetskii02]   [@tong00]   [@itoh00]   [@vijande04]   [@gershtein00]   [@ebert97]     [@roncaglia95]      [@korner94]   [@mathur02]
  -------------- --------------------- --------------------- ------------ -------------- ------------------ ----------- ----------- -------------- ---------------- ------------ --------------------- ------------- --------------
  $\Xi_{cc}$         $3612^{+17}$          $3607^{+24}$          3620          3480             3690           3740        3646          3524            3478           3660          $3660\pm70$          3610       $3588\pm72$
  $\Xi^*_{cc}$       $3706^{+23}$                                3727          3610                            3860        3733          3548            3610           3810          $3740\pm80$          3680      
  $\Xi_{bb}$      $10197^{+10}_{-17}$   $10194^{+10}_{-19}$     10202         10090            10160           10300                                    10093          10230         $10340\pm100$                   
  $\Xi^*_{bb}$    $10236^{+9}_{-17}$                            10237         10130                            10340                                    10133          10280         $10370\pm100$                   
  $\Xi_{bc}$       $6919^{+17}_{-7}$     $6915^{+17}_{-9}$       6933          6820             6960           7010                                      6820           6950      $6965\pm90^\dagger$                 $6840\pm236$
  $\Xi'_{bc}$      $6948^{+17}_{-6}$                             6963          6850                            7070                                      6850           7000      $7065\pm90^\dagger$                
  $\Xi^*_{bc}$     $6986^{+14}_{-5}$                             6980          6900                            7100                                      6900           7020          $7060\pm90$                    

  : First panel: doubly heavy $\Xi$ masses in MeV as obtained in different models. Our central values, and the ones of Ref. [@Si96], have been evaluated with the AL1 potential. Second panel: we compare our results with the experimental value for $M_{\Xi_{cc}}$ measured by the SELEX Collaboration [@mattson02] (Note the cautions that appear on this experimental mass in the introduction), and lattice results from Refs. [@khan00; @lewis01; @flynn03]. Entries with $^\dagger$ should be taken with due caution (see footnote 6). []{data-label="tab:ximassesdiffmod"}

                       This work        Exp. [@mattson02]   Latt. [@khan00]   Latt. [@lewis01]   Latt. [@flynn03]  
  -------------- --------------------- ------------------- ----------------- ------------------ ------------------ --
  $\Xi_{cc}$         $3612^{+17}$          $3519\pm1$                           $3605\pm23$        $3549\pm 95$    
  $\Xi^*_{cc}$       $3706^{+23}$                                               $3685\pm 23$       $3641\pm97$     
  $\Xi_{bb}$      $10197^{+10}_{-17}$                        $10314\pm47$                                          
  $\Xi^*_{bb}$    $10236^{+9}_{-17}$                         $10333\pm 55$                                         

  : First panel: doubly heavy $\Xi$ masses in MeV as obtained in different models. Our central values, and the ones of Ref. [@Si96], have been evaluated with the AL1 potential. Second panel: we compare our results with the experimental value for $M_{\Xi_{cc}}$ measured by the SELEX Collaboration [@mattson02] (Note the cautions that appear on this experimental mass in the introduction), and lattice results from Refs. [@khan00; @lewis01; @flynn03]. Entries with $^\dagger$ should be taken with due caution (see footnote 6). []{data-label="tab:ximassesdiffmod"}

                          This work             [@Si96]         [@ebert02]   [@kiselev02]   [@narodetskii02]   [@tong00]   [@itoh00]   [@gershtein00]   [@ebert97]     [@roncaglia95]      [@korner94]   [@mathur02]
  ----------------- --------------------- -------------------- ------------ -------------- ------------------ ----------- ----------- ---------------- ------------ --------------------- ------------- --------------
  $\Omega_{cc}$         $3702^{+41}$       $3710^{+29}_{-2}$       3778          3590             3860           3760        3749           3590           3760          $3740\pm70$          3710       $3698\pm65$
  $\Omega^*_{cc}$       $3783^{+22}$                               3872          3690                            3900        3826           3690           3890          $3820\pm80$          3760      
  $\Omega_{bb}$      $10260^{+14}_{-34}$   $10267^{+4}_{-43}$     10359         10180            10340           10340                     10180          10320         $10370\pm100$                   
  $\Omega^*_{bb}$    $10297^{+5}_{-28}$                           10389         10200                            10380                     10200          10360         $10400\pm100$                   
  $\Omega_{bc}$      $6986^{+27}_{-17}$    $7003^{+20}_{-32}$      7088          6910             7130           7050                       6910           7050      $7045\pm90^\dagger$                 $6954\pm225$
  $\Omega'_{bc}$     $7009^{+24}_{-15}$                            7116          6930                            7110                       6930           7090      $7105\pm90^\dagger$                
  $\Omega^*_{bc}$     $7046^{+11}_{-9}$                            7130          6990                            7130                       6990           7110          $7120\pm90$                    

  : Same as Table \[tab:ximassesdiffmod\] for doubly heavy $\Omega$ baryons.[]{data-label="tab:omegamassesdiffmod"}

                          This work                   Latt. [@khan00]                Latt. [@lewis01]      Latt. [@flynn03]
  ----------------- --------------------- --------------------------------------- ----------------------- ------------------
  $\Omega_{cc}$         $3702^{+41}$                                               $3733\pm9^{+7}_{-38}$     $3663\pm 97$
  $\Omega^*_{cc}$       $3783^{+22}$                                               $3801\pm9^{+3}_{-34}$     $3734\pm 98$
  $\Omega_{bb}$      $10260^{+14}_{-34}$   $10365\pm40^{-11}_{+12}$$^{+16}_{-0}$                          
  $\Omega^*_{bb}$    $10297^{+5}_{-28}$     $10383\pm39^{-8}_{+8}$$^{+12}_{-0}$                           

  : Same as Table \[tab:ximassesdiffmod\] for doubly heavy $\Omega$ baryons.[]{data-label="tab:omegamassesdiffmod"}

There are also independent determinations of mass splittings in lattice QCD [@khan00; @lewis01; @flynn03], nonrelativistic lattice QCD [@mathur02] and pNRQCD [@brambilla05]. In Table \[tab:ms\] we compare those results to the ones obtained in the present calculation and in other models.

                                            This work       [@ebert02]   [@kiselev02]   [@tong00]   [@gershtein00]   [@ebert97]   [@brambilla05]   [@mathur02]   Latt. [@khan00]   Latt. [@lewis01]   Latt. [@flynn03]
  ------------------------------------- ------------------ ------------ -------------- ----------- ---------------- ------------ ---------------- ------------- ----------------- ------------------ ------------------
  $M_{\Xi^*_{cc}}-M_{\Xi_{cc}}$          $94^{+5}_{-11}$       107           130           120           132            150         $120\pm40$      $70\pm13$                         $80\pm11$          $87\pm 19$
  $M_{\Xi^*_{bb}}-M_{\Xi_{bb}}$           $39^{+1}_{-6}$        35            40           40             40             50          $34\pm4$       $20\pm7$        $20\pm6$                         
  $M_{\Xi^*_{bc}}-M_{\Xi_{bc}}$          $67^{+3}_{-10}$        47            80           90             80             70                         $43\pm11$                                        
  $M_{\Xi'_{bc}}-M_{\Xi_{bc}}$            $29^{+1}_{-5}$        30            30           60             30             50                          $9\pm7$                                         
  $M_{\Omega^*_{cc}}-M_{\Omega_{cc}}$    $81^{+11}_{-19}$       94           100           140           100            130                         $63\pm9$                           $68\pm7$          $67\pm 16$
  $M_{\Omega^*_{bb}}-M_{\Omega_{bb}}$     $37^{+6}_{-9}$        30            20           40             20             40                         $19\pm5$        $20\pm5$                         
  $M_{\Omega^*_{bc}}-M_{\Omega_{bc}}$    $60^{+8}_{-16}$        42            80           80             80             60                         $39\pm8$                                         
  $M_{\Omega'_{bc}}-M_{\Omega_{bc}}$      $23^{+2}_{-3}$        28            20           60             20             40                          $9\pm6$                                         

  : Mass splittings in MeV for doubly heavy $\Xi$ and $\Omega$ baryons. Our central values have been obtained with the AL1 potential. Entries with a $^\dagger$ should be taken with due caution (see footnote 6).[]{data-label="tab:ms"}

Our central results evaluated with the AL1 potential are larger than the ones obtained in lattice QCD [@khan00; @lewis01; @flynn03] and lattice nonrelativistic QCD [@mathur02]. The agreement is better when we use the BD potential of Ref. [@BD81] for which we always get the lowest results. Similar results are obtained by the relativistic calculation of Ref. [@ebert02], whereas for the relativistic calculations in Refs. [@kiselev02; @tong00; @ebert97] and the nonrelativistic one in Ref. [@gershtein00] the agreement with lattice QCD and nonrelativistic lattice QCD data worsens. As for the calculation in Ref. [@roncaglia95] we do not quote their results due to the large theoretical errors involved.

Charge densities and radii {#subsec:chmdr}
--------------------------

The baryon charge density at the point $P$ (coordinate vector $\vec{r}$ in the CM frame, see Fig. \[fig:coor\]) is given by: $$\begin{aligned}
\rho_e^{B} (\vec{r}\,) 
& = & \int d^3 r_1 d^3 r_2\ \Big |
\Psi_{h_1h_2}^{B}(r_1,r_2,r_{12}) \Big |^2 \left \{ e_{h_1} \delta^3
(\vec{r}-\vec{y}_{h_1}) + e_{h_2} \delta^3 (\vec{r}-\vec{y}_{h_2}) + e_{q}
\delta^3 (\vec{r}-\vec{y}_q) \right \} \nonumber \\ 
& \equiv & \rho_e^{B} (\vec{r}\,)\big|_{h_1} 
+ \rho_e^{B}(\vec{r}\,)\big|_{h_2}  +
\rho_e^{B}(\vec{r}\,)\big|_{q} \label{eq:dens}\end{aligned}$$ where $e_{h_1,\,h_2,\,q}$ are the quark charges in proton charge units $e$, and from Fig. \[fig:coor\] we have[^8] $\vec{y}_{h_1}=\vec{y}_q+\vec{r}_1$, $\vec{y}_{h_2}=\vec{y}_q+\vec{r}_2$ and $\vec{y}_q= - \left
(m_{h_1}\vec{r}_1+m_{h_2}\vec{r}_2 \right)/\overline M$. Since our $L=0$ wave functions only depend on scalars ($r_1,r_2$ and $r_{12}$) the charge density is spherically symmetric ($\rho_e^{B} (\vec{r}\,) =
\rho_e^{B} (|\vec{r}~|)$).

The charge form factor is defined as usual $$\begin{aligned}
{\cal F}_e^{B} (\vec{q}~) = \int d^3 r\ e^{{\rm
i}\vec{q}\cdot\vec{r}}\rho_e^{B} (r)\label{eq:fe}\end{aligned}$$ and it only depends on $|\vec{q}~|$. Its value at $\vec{q}=\vec 0$ gives the baryon charge in units of the proton charge.

The charge mean square radii are defined $$\langle r^2 \rangle_e^{B} = 
\int d^3 r\ r^2 \rho_e^{B} (r) = 
4\pi \int_0^{+\infty}dr\ r^4 \rho_e^{B} (r) 
\label{eq:r2q}$$

\

In Figs. \[fig:cffxi\], \[fig:cffomega\] and \[fig:cffbc\] we show the charge form factors for $\Xi_{cc},\, \Xi_{bb}$, $\Omega_{cc},\,\Omega_{bb}$ and $\Xi_{bc},\,\Omega_{bc}$ baryons. In each case similar results are obtained for the star and prime excitations. We show the calculations with both the AL1 potential of Refs. [@Si96; @SS94] and the BD potential of Ref. [@BD81]. The differences between the two calculations are minor in most cases.

In Table \[tab:xiomegacr\] we show the charge mean square radii. With the exceptions of the $\Xi_{bc}^0$ and $\Omega_{bc}^0$, we find good agreement with the results obtained in Ref. [@Si96] within a Faddeev calculation. The possible presence of a $S_h=0$ contribution in the wave functions of Ref. [@Si96] could be the possible explanation for this discrepancy. We also compare with the results obtained, for a few states, in Ref. [@bruno04] with the use a relativistic quark model in the instant form. The agreement is bad in this case.

[lccc]{} &This work & [@Si96]&[@bruno04]\

$\Xi_{cc}^+$ &$-0.030^{+0.003}_{-0.016}$ &$-0.038^{+0.004}_{-0.016}$&$0.1$\
$\Xi_{cc}^{++}$ & $0.298^{+0.034}_{-0.028}$ &$0.315^{+0.035}_{-0.030}$&$1.3$\
\
$\Xi_{cc}^{*+}$ &$-0.042^{+0.007}_{-0.019}$& &\
$\Xi_{cc}^{*++}$ &$0.341^{+0.041}_{-0.042}$&&\
\
$ \Xi_{bb}^0 $ &$0.221^{+0.033}_{-0.025}$& $0.242^{+0.035}_{-0.027}$&\
$ \Xi_{bb}^- $ &$-0.133^{+0.014}_{-0.016}$& $-0.143^{+0.006}_{-0.018}$&\
\
$\Xi_{bb}^{*-}$ &$-0.142^{+0.018}_{-0.018}$& &\
\
$\Xi_{bb}^{*0}$ &$0.238^{+0.035}_{-0.031}$& &\
\
$\Xi_{bc}^0$ &$-0.057^{+0.006}_{-0.013}$& $-0.072^{+0.008}_{-0.017}$&\
$\Xi_{bc}^+$ &$0.279^{+0.026}_{-0.031}$& $0.306^{+0.035}_{-0.011}$&\
\
$\Xi_{bc}'^0$ &$-0.065^{+0.010}_{-0.015}$& &\
$\Xi_{bc}'^+$ &$0.283^{+0.036}_{-0.025}$& &\
\
$\Xi_{bc}^{*0}$ &$-0.065^{+0.010}_{-0.018}$& &\
$\Xi_{bc}^{*+}$ &$0.305^{+0.031}_{-0.039}$&&\

[lccc]{} &This work &[@Si96]&[@bruno04]\

$\Omega_{cc}^+$ &$0.013^{+0.001}_{-0.002}$&$0.009_{-0.003}$&$0.2$\
\
$\Omega_{cc}^{*+}$ &$0.009^{+0.001}_{-0.002}$& &\
\
$\Omega_{bb}^-$ &$-0.086^{+0.008}_{-0.001}$& $-0.090^{+0.007}_{-0.002}$&\
\
$\Omega_{bb}^{*-}$ &$-0.092^{+0.011}_{-0.001}$& &\
\
$\Omega_{bc}^0$ &$-0.016_{+0.003}$& $-0.025^{+0.002}_{-0.003}$&\
\
$\Omega_{bc}'^0$ &$-0.019^{+0.003}_{-0.003}$&&\
\
$\Omega_{bc}^{*0}$ &$-0.021^{+0.004}_{-0.002}$& &\

Magnetic moments {#subsec:mm}
----------------

The orbital part of the magnetic moment is defined in terms of the velocities $\vec{v}$ of the quarks, with respect to the position of the CM, and it reads $$\begin{aligned}
{\mu}^{B} &=& \int  d^3 r_1 d^3 r_2 \left(\Psi_{h_1h_2}^{B}(r_1,r_2,r_{12})\right)^* \left \{
\frac{e_{h_1}}{2m_{h_1}} (\vec{y}_{h_1} \times m_{h_1}\vec{v}_{{h_1}})_z \right.\nonumber \\
&+&\left. \frac{e_{h_2}}{2m_{h_2}} (\vec{y}_{h_2} \times
m_{h_2}\vec{v}_{h_2})_z + 
\frac{e_{q}}{2m_{q}} (\vec{y}_q \times m_{q}
\vec{v}_{y_q})_z \right \}\Psi_{h_1h_2}^{B}(r_1,r_2,r_{12})\end{aligned}$$ with[^9] $m_{h_1}\vec{v}_{h_1} = - i \vec{\nabla}_1$, $m_{h_2}\vec{v}_{h_2} = - i \vec{\nabla}_2$ and $m_q\vec{v}_{q}
= i \left ( \vec{\nabla}_1 + \vec{\nabla}_2 \right)$. Since our orbital wave function has $L=0$, the orbital magnetic moment vanishes. The magnetic moment of the baryon is then entirely given by the spin contribution. $$\begin{aligned}
\langle B;\, J,\,M_J=J|\ \frac{e_{h_1}}{2m_{h_1}} (\vec{\sigma}_{h_1})_z
+ \frac{e_{h_2}}{2m_{h_2}} (\vec{\sigma}_{h_2})_z  + 
\frac{e_{q}}{2m_{q}} (\vec{\sigma}_q)_z\ | B;\,J,\, M_J=J\rangle\end{aligned}$$ Those matrix elements are trivially evaluated with the results $$\begin{aligned}
\Xi_{h1h_2},\ \Omega_{h1h_2}&\longrightarrow&
\frac{2}{3}\left( \frac{e_{h_1}}{2m_{h_1}}+\frac{e_{h_2}}{2m_{h_2}}-\frac12
\ \frac{e_{q}}{2m_{q}} \right)\nonumber\\
\Xi^*_{h1h_2},\ \Omega^*_{h1h_2}&\longrightarrow&
 \frac{e_{h_1}}{2m_{h_1}}+\frac{e_{h_2}}{2m_{h_2}}+
\frac{e_{q}}{2m_{q}} \nonumber\\
\Xi'_{h1h_2},\ \Omega'_{h1h_2}&\longrightarrow&
 \frac{e_{q}}{2m_{q}}\end{aligned}$$

In Table \[tab:xiomegamm\] we give our numerical results. Our central values, as the ones obtained in Ref. [@Si96] within a Faddeev approach, have been evaluated with the use of the AL1 potential. When compared to the values obtained in Ref. [@Si96] we find very good agreement with just a few exceptions ($\Xi_{bc}^0,\,\Xi_{bc}^+,\,\Omega_{bc}^0$). The discrepancy for the latter baryons may come from a possible non negligible $S_h=0$ contribution to their wave functions in the calculation of Ref. [@Si96]. In our case we have fixed $S_h=1$ which we think is a very good approximation since in the limit of infinite heavy quark masses the spin of the heavy quark degrees of freedom is well defined.

We also compare our results to the ones obtained in Refs.[@lichtenberg77; @faessler06; @bruno04; @oh91; @jena86; @bose80] using different approaches. The differences between different calculations are in some cases large. Being $L=0$ a good approximation the magnetic moments are essentially determined by the spin contribution of the quarks. With $m_b\gg m_u,\,m_d,\,m_s$, the contribution from the $b$ quarks is negligible compared to the one of the light quark. This is also true to a lesser extent for the $c$ quark.

[lcccccccc]{} & This work& [@Si96]&[@lichtenberg77]&[@faessler06]&[@bruno04]&[@oh91] &[@jena86]&[@bose80]\

$\Xi_{cc}^+$ &$0.785^{+0.050}_{-0.030}$ &$0.784^{+0.050}_{-0.029}$&0.806 &0.72&$0.72$&$0.89\sim0.98$&$0.778\sim0.790$&0.86\
$\Xi_{cc}^{++}$&$-0.208^{+0.035}_{-0.086}$ &$-0.206^{+0.034}_{-0.086}$&$-0.124$&0.13&$-0.10$&$-0.47$&$-0.172\sim-0.154$&0.17\
\
$\Xi_{cc}^{*+}$ &$-0.311^{+0.052}_{-0.130}$ &$$ &$-0.186$& &&$-1.17\sim-0.98$&&0.20\
$\Xi_{cc}^{*++}$&$2.67^{+0.27}_{-0.15}$ &$$ &2.60 &&&$3.16\sim3.18$&&2.54\
\
$ \Xi_{bb}^0 $ &$-0.742^{+0.044}_{-0.091}$&$-0.742^{+0.044}_{-0.092}$&&$-0.53$&&&$-0.726\sim-0.705$&0.61\
$ \Xi_{bb}^- $ &$0.251^{+0.045}_{-0.021}$ &$0.251^{+0.046}_{-0.021}$ &&0.18 &&&$0.226\sim0.236$&0.14\
\
$\Xi_{bb}^{*0}$ &$1.87^{+0.27}_{-0.13}$ &$$ && &&&&1.37\
$\Xi_{bb}^{*-}$ &$-1.11^{+0.06}_{-0.14}$ &$$ &&& &&&$-0.95$\
\
$\Xi_{bc}^0$ &$0.518^{+0.048}_{-0.020}$&$0.058^{+0.059}_{-0.054}$&& 0.42 &&&&\
$\Xi_{bc}^+$ &$-0.475^{+0.040}_{-0.088}$ &$-0.198^{+0.057}_{-0.056}$ &&$-0.12$ &&&&\
\
$\Xi_{bc}'^0$ &$-0.993^{+0.065}_{-0.137}$ &$$ &&$-0.76$&&& $-0.385\sim-0.366$ &\
$\Xi_{bc}'^+$ &$1.99^{+0.27}_{-0.13}$ &$$ && 1.52 & && $1.50\sim1.54$ &\
\
$\Xi_{bc}^{*0}$ &$-0.712^{+0.059}_{-0.133}$ &$$ &&&&& &$-0.39$\
$\Xi_{bc}^{*+}$ &$2.27^{+0.27}_{-0.14}$ &$$ &&&&& &2.04\

\

[lcccccccc]{} & This work&[@Si96]&[@lichtenberg77]&[@faessler06]&[@bruno04]&[@oh91] & [@jena86]&[@bose80]\

$\Omega_{cc}^+$ &$0.635^{+0.012}_{-0.015}$ &$0.635^{+0.011}_{-0.015}$ &0.688&0.67 &$0.72$&$0.59\sim0.64$&$0.657\sim0.663$&0.84\
\
$\Omega_{cc}^{*+}$ &$0.139^{+0.009}_{-0.017}$ &$$ &0.167& &&$-0.20\sim0.03$&&0.39\
\
$\Omega_{bb}^-$ &$0.101^{+0.007}_{-0.007}$ &$0.101^{+0.007}_{-0.006}$ &&0.04& && $0.105\sim0.108$&0.084\
\
$\Omega_{bb}^{*-}$ &$-0.662^{+0.022}_{-0.024}$ &$$& & &&&&$-1.28$\
\
$\Omega_{bc}^0$ &$0.368^{+0.010}_{-0.011}$&$0.009^{+0.038}_{-0.029}$ &&0.45&&&&\
\
$\Omega_{bc}'^0$ &$-0.542^{+0.021}_{-0.024}$ &$$& &$-0.61$&&& $-0.130\sim-0.125$ &\
\
$\Omega_{bc}^{*0}$ &$-0.261^{+0.015}_{-0.021}$ &$$& &&&&&$-0.22$\

Semileptonic decay {#sec:sd}
==================

In this section we shall use the wave functions obtained with the variational method to study different doubly $B(1/2^+)\to B'(1/2^+)$ baryon semileptonic decays involving a $b\to c $ transition at the quark level.

The differential decay width reads $${\rm d}\Gamma= 
8 |V_{cb}|^2 m_{B'} G_F^{\,2}  
 \frac{d^3p^\prime}{(2\pi)^32E^\prime_{B'} }
\frac{d^3k}{(2\pi)^32E_{\bar \nu_l} } \frac{d^3k^\prime}{(2\pi)^32E^\prime_{l}
}  (2\pi)^4 \delta^4(p-p^\prime-k-k^\prime)\ {\cal
L}^{\alpha\beta}(k,k')
{\cal H}_{\alpha\beta}(p,p')$$ where $|V_{cb}|$ is the modulus of the corresponding Cabibbo–Kobayashi–Maskawa matrix element, $m_{B'}$ is the mass of the final baryon, $G_F= 1.16637(1)\times
10^{-11}$MeV$^{-2}$[@pdg06] is the Fermi decay constant, $p$, $p'$, $k$ and $k'$ are the four-momenta of the initial baryon, final baryon, final anti-neutrino and final lepton respectively, and ${\cal L}$ and ${\cal H}$ are the lepton and hadron tensors.

The lepton tensor is given as $$\begin{aligned}
{\cal L}^{\mu\sigma}(k,k')&=& k'^\mu k^\sigma +k'^\sigma k^\mu
- g^{\mu\sigma} k\cdot k^\prime + {\rm i}
\epsilon^{\mu\sigma\alpha\beta}k'_{\alpha}k_\beta \label{eq:lep}\end{aligned}$$ where we use the convention $\epsilon^{0123}=-1$, $g^{\mu\mu}=(+,-,-,-)$.

The hadron tensor is given as $$\begin{aligned}
{\cal H}_{\mu\sigma}(p,p') &=& \frac12 \sum_{r,r'}  
 \left\langle B', r'\
\vec{p}^{\,\prime}\left|\,
\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)\right| B, r\ \vec{p}   \right\rangle 
\ \left\langle B', r'\ 
\vec{p}^{\,\prime}\left|\,\overline \Psi^c(0)\gamma_\sigma(I-\gamma_5)
\Psi^b(0) \right|  B, r\ \vec{p} \right\rangle^*
\label{eq:wmunu}\end{aligned}$$ with $\left|B, r\ \vec p\right\rangle\, (\left|B', r'\
\vec{p}\,'\right\rangle)$ representing the initial (final) baryon with three–momentum $\vec p$ ($\vec{p}\,'$) and spin third component $r$ ($r'$). The baryon states are normalized such that $\langle r\
\vec{p}\, |\, r' \ \vec{p}^{\,\prime} \rangle = (2\pi)^3 (E(\vec
p\,)/m)\,\delta_{rr'}\, \delta^3(\vec{p}-\vec{p}^{\,\prime})$. The hadron matrix elements can be parametrized in terms of six form factors as $$\begin{aligned}
\left\langle B', r'\ \vec{p}^{\,\prime}\left|\,\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)
 \right| B, r\ \vec{p}
\right\rangle& =& {\bar u}^{B'}_{r'}(\vec{p}^{\,\prime})\Big\{
\gamma_\mu\left(F_1(w)-\gamma_5 G_1(w)\right)+ v_\mu\left(F_2(w)-\gamma_5
G_2(w)\right)\nonumber\\
&&\hspace{1.5cm}+v^\prime_\mu\left(F_3(w)-\gamma_5 G_3(w)
\right)\Big\}u^{B}_r(\vec{p}\,) \label{eq:def_ff}\end{aligned}$$ where $u^{B,B'}$ are dimensionless Dirac spinors, normalized as ${\bar
u} u = 1$, and $v_\mu = p_\mu/m_{B}$ ($v^\prime_\mu =
p^\prime_\mu/m_{B'}$) is the four velocity of the initial $B$ (final $B'$) baryon. The form factors are functions of the velocity transfer $w=v\cdot v^\prime$ or equivalently of the four momentum transfer ($q=p-p'$) square $q^2= m_{B}^2 + m_{B'}^2 - 2m_{B}m_{B'}w$. In the decay $w$ ranges from $w=1$, corresponding to zero recoil of the final baryon, to a maximum value given by $w=w_{\rm max}= (m_{B}^2 +
m_{B'}^2)/(2m_{B}m_{B'})$ which depends on the transition.

Neglecting lepton masses, we have for the differential decay rates from transversely $(\Gamma_T)$ and longitudinally $(\Gamma_L)$ polarized $W$’s (the total width is $\Gamma=\Gamma_L+\Gamma_T$) [@korner92] $$\begin{aligned}
\frac{{\rm d}\Gamma_T}{{\rm d}w}&=&
\frac{G^2_F |V_{cb}|^2}{12\pi^3}m_{B'}^3\sqrt{w^2-1}\,q^2 
\Big\{ (w-1)|F_1(w)
|^2+(w+1)|G_1(w)|^2 \Big\} \nonumber\\
&&\nonumber\\
\frac{{\rm d}\Gamma_L}{{\rm d}w}&=&
\frac{G^2_F |V_{cb}|^2}{24\pi^3}m_{B'}^3\sqrt{w^2-1}
\Big\{(w-1)|{\cal F}^V(w)|^2 + (w+1)|{\cal
F}^A(w)|^2  \Big\} \nonumber\\
&&\nonumber\\
 {\cal F}^{V,A}(w) &=& \Big[ (m_{B}\pm m_{B'}) F_1^{V,A}(w) +
(1\pm w)\left(m_{B'} F_2^{V,A}(w)+m_{B}
F_3^{V,A}(w)\right)\Big],\nonumber\\&& \quad F_j^V \equiv F_j(w) , ~ F_j^A \equiv
G_j(w),~ j=1,2,3\nonumber\\
 \label{eq:dg}\end{aligned}$$

One can also evaluate the polar angle distribution [@korner92]: $$\frac{{\rm d}^2\Gamma}{{\rm d}w\,{\rm d}\cos\theta} = \frac38
\left(\frac{{\rm d}\Gamma_T}{{\rm d}w} +
 2\frac{{\rm d}\Gamma_L}{{\rm d}w} \right)\Big\{1+2\alpha^\prime\cos\theta +
 \alpha^{\prime\prime} \cos^2\theta \Big\}
\label{eq:asymmetry1}$$ where $\theta$ is the angle between $\vec{k}^\prime$ and $\vec{p}^{\,\prime}$ measured in the off–shell $W$ rest frame, and $\alpha^\prime$ and $\alpha^{\prime\prime}$ are asymmetry parameters given by $$\begin{aligned}
\alpha^\prime &=&  - \frac{G^2_F
    |V_{cb}|^2}{6\pi^3} 
{m_{B'}^3}\frac{
    q^2\,(w^2-1)\,F_1(w)G_1(w)}{{\rm d}\Gamma_T/{{\rm d}w}+
    2\,{\rm d}\Gamma_L/{{\rm d}w}}  \nonumber\\&&\nonumber\\
\alpha^{\prime\prime} &=& \frac{{\rm d}\Gamma_T/{{\rm d}w}  
   - 2\,{\rm d}\Gamma_L/{{\rm d}w}}
   {{\rm d}\Gamma_T/{{\rm d}w}+
    2\,{\rm d}\Gamma_L/{{\rm d}w}} \label{eq:asymmetry2}\end{aligned}$$ These asymmetry parameters are functions of the velocity transfer $w$ and on averaging over $w$ the numerators and denominators are integrated separately and thus we have $$\begin{aligned}
\langle \alpha' \rangle &=& - \frac{G^2_F |V_{cb}|^2}{6\pi^3}
\frac{m_{B'}^3}{\Gamma_T}\frac{ \int_1^{w_{\rm max}}
q^2\,(w^2-1)\,F_1(w)G_1(w) {\rm d }w}{1+2R_{L/T}} 
 ,
 \quad \langle \alpha^{\prime\prime} \rangle =
\frac{1-2R_{L/T}
 } {1+2R_{L/T}},  \qquad R_{L/T} = \frac{\Gamma_L}{\Gamma_T} \label{eq:rlt} 
\label{eq:averageasymmetry}\end{aligned}$$

Form factors {#subsec:ff}
------------

To obtain the form factors we have to evaluate the matrix elements $$\left\langle B', r'\ \vec{p}^{\,\prime}\left|\,\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)
 \right| B, r\ \vec{p}
\right\rangle$$ which in our model are given by $$\sqrt{\frac{E_{B}(\vec p\,)}{m_{B}}}\ \sqrt{\frac{E_{B'}(\vec
p\,')\,}{m_{B'}}}\
{}_{\stackrel{}{\stackrel{}{NR}}}\left\langle B', r'\ \vec{p}^{\,\prime}\left|\,\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)
 \right| B, r\ \vec{p}
\right\rangle_{NR}$$ where the suffix “$NR$” denotes our nonrelativistic states and the factors $\sqrt{E/m}$ take into account the different normalization. We shall work in the initial baryon rest frame so that $\vec p=\vec
0,\,\vec p\,'=-\vec q$, and take $\vec q$ in the positive $z$ direction. Furthermore we shall use the spectator approximation. Having all this in mind we have in momentum space $$\begin{aligned}
&&\hspace {-1.5cm}\sqrt{\frac{E_{B'}(-\vec q\,)}{m_{B'}}}\
{}_{\stackrel{}{\stackrel{}{NR}}}\left\langle B', r'\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)
 \right| B, r\ \vec{0}
\right\rangle_{NR}\nonumber\\
&&\hspace{1.cm}=\sqrt{2}\sqrt{\frac{E_{B'}(-\vec q\,)}{m_{B'}}}\ \sum_{s_1}\sum_{s_2} \left(\frac12\frac12 S_h
\bigg|s_1, s_2-s_1, s_2\right) \left(S_h\frac12\frac12
\bigg|s_2, r-s_2, r\right)\nonumber\\
&&\hspace{3.35cm}\times\left(\frac12\frac12 S'_h
\bigg|r'-r+s_1, s_2-s_1, r'-r+s_2\right) \left(S'_h\frac12\frac12
\bigg|r'-r+s_2, r-s_2, r'\right)\nonumber\\
&&\hspace{3.35cm}\times\int d\,^3q_1\, d\,^3q_2\, 
\left(\Phi^{B'}_{c\,h_2}(\vec q_1-\frac{m_{h_2}+m_q}{\overline M\,'}\,\vec q,
\,\vec{q}_2+\frac{m_{h_2}}{\overline M\,'}\,\vec q\,)\right)^*\ 
\Phi^B_{b\,h_2}(\vec{q}_1,\vec{q}_2)
\nonumber\\
&&\hspace{3.35cm}\times\sqrt{\frac{m_b}{E_b(\vec{q}_1)}}\sqrt{\frac{m_c}
{E_c(\vec{q}_1-\vec q\,)}}\ \bar{u}^c_{r'-r+s_1}(\vec{q}_1-\vec q\,)
\gamma_\mu(I-\gamma_5)\,u^b_{s_1}(\vec{q}_1)\end{aligned}$$ where $\Phi^B_{b\,h_2}(\vec{q}_1,\vec{q}_2)$ ($\Phi^{B'}_{c\,h_2}(\vec{q}_1,\vec{q}_2)$) is the Fourier transform of the coordinate space wave function $\Psi^B_{b\,h_2}(r_1,r_2,r_{12})$ ($\Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})$) with $\vec q_1,\,\vec q_2$ being the conjugate momenta to the space variables $\vec r_1,\,\vec
r_2$. The factor of two comes from the fact that: i) for $bc-$baryon decays, the charm quark resulting from the $b\to c$ transition could be either the particle 1 or the particle 2 in the final $cc$ baryon, while ii) for $bb-$baryon decays, there exist two equal contributions resulting for the decay of each of the two bottom quarks of the initial baryon.

The actual calculation is done in coordinate space where we have $$\begin{aligned}
&&\hspace {-1.5cm}\sqrt{\frac{E_{B'}(-\vec q\,)}{m_{B'}}}\
{}_{\stackrel{}{\stackrel{}{NR}}}\left\langle B', r'\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_\mu(I-\gamma_5)\Psi^b(0)
 \right| B, r\ \vec{0}
\right\rangle_{NR}\nonumber\\
&&\hspace{1.cm}=\sqrt{2}\sqrt{\frac{E_{B'}(-\vec q\,)}{m_{B'}}}\ \sum_{s_1}\sum_{s_2} \left(\frac12\frac12 S_h
\bigg|s_1, s_2-s_1, s_2\right) \left(S_h\frac12\frac12
\bigg|s_2, r-s_2, r\right)\nonumber\\
&&\hspace{3.35cm}\times\left(\frac12\frac12 S'_h
\bigg|r'-r+s_1, s_2-s_1, r'-r+s_2\right) \left(S'_h\frac12\frac12
\bigg|r'-r+s_2, r-s_2, r'\right)\nonumber\\
&&\hspace{3.35cm}\times\int d\,^3r_1\, d\,^3r_2\, 
\Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})
\ e^{i\frac{m_{h_2}}{\overline M\,'}\,\vec
q\cdot\vec r_2}\,
e^{-i\frac{m_{h_2}+m_q}{\overline M\,'}\,\vec
q\cdot\vec r_1}
\nonumber\\
&&\hspace{3.35cm}\times\sqrt{\frac{m_b}{E_b(\vec{l}\ )}}\sqrt{\frac{m_c}
{E_c(\vec{l}-\vec q\,)}}\ \bar{u}^c_{r'-r+s_1}(\vec{l}-\vec q\,)
\gamma_\mu(I-\gamma_5)\,u^b_{s_1}(\vec{l}\ )\ \Psi^B_{b\,h_2}(r_1,r_2,r_{12})
\label{eq:coor}\end{aligned}$$ where $\vec l=-i\stackrel{\rightarrow}{\nabla}_1$ represents an internal momentum which is much smaller than the heavy quark masses $m_b,\,m_c$. On the other hand $|\vec q\,|$ can be large[^10]. Thus, to evaluate the above expression we have made use of an expansion in $\vec l$, introduced in Ref. [@albertus05], where second order terms in $\vec l$ are neglected, while all orders in $|\vec q\,|$ are kept. For instance $E_c(\vec l-\vec q\,)$ is approximated by $E_c(\vec l-\vec q\,)\approx
E_c(\vec q\,)\times (1-\vec l\cdot\vec q\,/E_c^2(\vec q\,))$ with $E_c(\vec q\,)=\sqrt{m_c^2+\vec q\,^2}$.

The three vector and three axial form factors can be extracted from the set of equations[^11] $$\begin{aligned}
\left\langle B',1/2\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_1\Psi^b(0)
 \right| B, -1/2\ \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}
\,\frac{|\vec q\,|}{E_{B'}(-\vec q\, )+m_{B'}}\ 
F_1(|\vec q\,|)\nonumber\\
\left\langle B', 1/2\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_3\Psi^b(0)
 \right| B, 1/2\ \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}\ 
|\vec q\,|\,\left(
\frac{F_1(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}+\frac{F_3(|\vec q\,|)}{m_{B'}}\right)\nonumber\\
\left\langle B', 1/2\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_0\Psi^b(0)
 \right| B, 1/2 \ \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}\left(
F_1(|\vec q\,|)+F_2(|\vec q\,|)+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,F_3(|\vec q\,|)\right)\nonumber\\
\label{eq:f123}\end{aligned}$$ for the vector form factors and $$\begin{aligned}
\left\langle B',1/2\  -\vec{q}\left|\,\overline \Psi^c(0)\gamma_1\gamma_5\Psi^b(0)
 \right| B, -1/2\ \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}
\ (-G_1(|\vec q\,|))\nonumber\\
\left\langle B', 1/2\ -\vec{q}\left|\,\overline \Psi^c(0)\gamma_3\gamma_5\Psi^b(0)
 \right| B, 1/2\ \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}\  \
\left(-G_1(|\vec q\,|)+\frac{|\vec q\,|^2\,G_3(|\vec q\,|)}{m_{B'}(E_{B'}(-\vec q\,
)+m_{B'})}\right)\nonumber\\
\left\langle B',1/2\  -\vec{q}\left|\,\overline \Psi^c(0)\gamma_0\gamma_5\Psi^b(0)
 \right| B,1/2\  \vec{0}
\right\rangle&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2m_{B'}}}\,
\frac{|\vec q\,|}{E_{B'}(-\vec q\, )+m_{B'}}\bigg(
-G_1(|\vec q\,|)+G_2(|\vec q\,|)\nonumber\\
&&\hspace{5.75cm}+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,G_3(|\vec q\,|)\bigg)\nonumber\\
\label{eq:g123}\end{aligned}$$ for the axial ones. All the left hand side terms can be evaluated using Eq.(\[eq:coor\]) with the approximation mentioned above.

For each transition there are only two different coordinate space integrals from which all different matrix elements can be evaluated. Those integrals are $$\begin{aligned}
{\cal I}^{B'B}(|\vec q\,|)&=&\int\, d\,^3r_1\, d\,^3r_2\, 
e^{i\frac{m_{h_2}}{\overline M\,'}\,\vec
q\cdot\vec r_2}\,
e^{-i\frac{m_{h_2}+m_q}{\overline M\,'}\,\vec
q\cdot\vec r_1}\ \left[\Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})\right]^*\ \Psi^{B}_{b\,h_2}(r_1,r_2,r_{12}) \nonumber\\
{\cal K}^{B'B}(|\vec q\,|)&=&\frac{1}{|\vec q\,|^2}\int\, d\,^3r_1\, d\,^3r_2\, 
e^{i\frac{m_{h_2}}{\overline M\,'}\,\vec
q\cdot\vec r_2}\,
e^{-i\frac{m_{h_2}+m_q}{\overline M\,'}\,\vec
q\cdot\vec r_1}\ \left[\Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})\right]^*\ 
\vec l\cdot\vec q\ \Psi^{B}_{b\,h_2}(r_1,r_2,r_{12})
\label{eq:integrales}\end{aligned}$$ In appendix \[app:fg123\] we relate the form factors to the integrals ${\cal I}^{B'B}(|\vec q\,|)$ and ${\cal K}^{B'B}(|\vec
 q\,|)$ for the different $S_h,\,S'_h$ combinations, while in appendix \[app:integrals\] we give the actual expressions we use to evaluate those integrals.

### Current conservation

In the limit $m_b=m_c$ and for $B'=B$ (and thus $S_h=S'_h$) vector current conservation provides a relation among the vector $F_2$ and $F_3$ form factors, namely $$F_2(w)=F_3(w)
\label{eq:cc}$$ On the other hand, if the two quarks in the current had the same flavour, vector current conservation would fix the normalization of the zeroth component of the vector current matrix element at $w=1$, since this just counts the number of heavy quarks, two, in that case. For quarks with distinct flavours, but still in the limit $m_b=m_c$, the forward vector matrix element has an extra Clebsch-Gordan factor of $1/\sqrt2$, and thus we have $$F_1(1)+F_2(1)+F_3(1)=\sqrt{2}
\label{eq:baryonnumber}$$ In this limiting situation the integrals ${\cal I}^{BB}(|\vec q\,|)$ and ${\cal K}^{BB}(|\vec q\,|)$ are related by[^12]$${\cal K}^{BB}(|\vec q\,|)=\frac{m_{h_2}+m_q}{2\overline M}\
{\cal I}^{BB}(|\vec q\,|)$$ Besides one has that ${\cal I}^{BB}(0)=1$.

Using now the relations in Eq.(\[eq:appf123\]) in appendix \[app:fg123\] we see that our model satisfies the constraint in Eq.(\[eq:baryonnumber\]) exactly. On the other hand we have a small violation of vector current conservation due to binding effects. For instance, and again using the relations in Eq.(\[eq:appf123\]), we obtain for $w=1$ $$F_2(1)=F_3(1)+\sqrt{2}(1-\frac{m_B}{\overline M})$$ which shows that current conservation is violated by a term proportional to the binding energy of the baryon divided by the sum of the masses of its constituents. This violation disappears in the infinite heavy quark mass limit. This deficiency is avoided in most calculations by the neglect of binding effects between the heavy diquark and the light quark[^13]. Improvements on vector current conservation would require at minimum the introduction of two–body currents [@buchmann94], going thus beyond the spectator approximation, that we have not considered in this analysis.

Note also that, for transitions that do not conserve the spin of the heavy quark subsystem $S_h$ (i.e. $\Xi'^0_{bc} \to \Xi^+_{cc} l \bar\nu_l)$ we have in the $m_b=m_c$ limit and at zero recoil that $$F_1(1)+F_2(1)+F_3(1)=0$$ due to the orthogonality of the initial and final baryon wave–functions.

Results {#subsec:results}
-------

In Fig. \[fig:ffxi\] we show the form factors for $\Xi$ decays evaluated with the AL1 potential. Variations when using a different potential are at the level a few per cent at most. The results for doubly heavy $\Omega$ decays are almost identical to the corresponding ones for doubly heavy $\Xi$ decays. The fact that we have two heavy quarks and that the light one acts as a spectator makes the results almost independent of the light quark mass.

In Fig. \[fig:dgdwxi\] we show now our results for the differential $d\Gamma_T/dw$, $d\Gamma_L/dw$ and $d\Gamma/dw$ decay widths evaluated with the AL1 and BD potentials. The differences between the results obtained with the two inter-quark interactions could reach 30% for some transitions and for some regions of $w$. As a consequence of the apparent SU(3) symmetry in the form factors we also find that the results for doubly heavy $\Xi$ and $\Omega$ decays (not shown) are very close to each other. This apparent SU(3) symmetry goes over to the integrated decay widths and asymmetry parameters.

\

In Table \[tab:semi\] we give our results for the semileptonic decay width (transverse $\Gamma_T$, longitudinal $\Gamma_L$ and total $\Gamma$) for the different processes under study. Our central values have been evaluated using the AL1 potential while the errors show the variations when changing the interaction. The biggest variations appear for the BD potential for which one obtains results which are larger by $7\sim12\% 
$. In Table \[tab:gamma\] we compare our results to the ones calculated in different models. For that purpose we need a value for $|V_{cb}|$ for which we take $|V_{cb}|=0.0413$. Our results are in reasonable agreement with the ones in Ref. [@ebert04] where they use a relativistic quark model evaluated in the quark-diquark approximation[^14]. The result for $\Gamma(\Xi_{bc}\to\Xi_{cc})$ obtained in Ref. [@sanchis95] using HQET is a factor of 4 larger than ours[^15]. We can not directly compare our result for the latter transition with the one published in Ref. [@faessler01], the reason being that in this reference the $\Xi_{bc}$ state is defined differently, with the $c$ and the light quark coupled to total spin 1. Had they defined the state with the $bc$ pair coupled to spin 1 they would had obtained roughly a factor of 4 larger result [@gutsche08], in reasonable agreement with our calculation or the one by Ebert [*et al.*]{}[^16]. Finally in Ref. [@guo98], where they use the Bethe–Salpeter equation applied to a quark-diquark system, they obtain much larger results for all transitions.

[lccc]{} &$\Gamma_T$ & $\Gamma_L$& $\Gamma$\
\
$\Xi_{bb}\to\Xi_{bc}\,l\bar\nu_l$ & $0.49^{+0.5}_{-0.01}$ & $0.64^{+0.10}_{-0.02} $ &$ 1.13^{+0.15}_{-0.03}$\
\
$\Xi_{bc}\to\Xi_{cc}\,l\bar\nu_l$ & $0.58^{+0.04}_{-0.01}$ &$ 0.93^{+0.11}_{-0.01}$ &$ 1.51^{+0.15}_{-0.02} $\
\
$\Xi_{bb}\to\Xi_{bc}'\,l\bar\nu_l$ & $0.37^{+0.04}_{-0.01}$ & $0.26^{+0.04}_{-0.01} $ & $0.62 ^{+0.08}_{-0.02}$\
\
$\Xi_{bc}'\to\Xi_{cc}\,l\bar\nu_l$ & $0.45^{+0.03}_{-0.01}$ & $0.35^{+0.03}_{-0.01} $ & $0.80^{+0.06}_{-0.02}$\

[lccc]{} &$\Gamma_T$ & $\Gamma_L$& $\Gamma$\
\
$\Omega_{bb}\to\Omega_{bc}\,l\bar\nu_l$ & $0.53^{+0.04}_{-0.01}$& $0.73^{+0.08}_{-0.01}$ & $1.26^{+0.12}_{-0.01}$\
\
$\Omega_{bc}\to\Omega_{cc}\,l\bar\nu_l$ &$ 0.58^{+0.03}$ &$0.94^{+0.09}$ &$1.52^{+0.12}$\
\
$\Omega_{bb}\to\Omega_{bc}'\,l\bar\nu_l$ & $0.40^{+0.04}$ & $0.29^{+0.04} $ &$0.68^{+0.08}$\
\
$\Omega_{bc}'\to\Omega_{cc}\,l\bar\nu_l$ & $0.45^{+0.03}$ & $0.35^{+0.03}$&$0.80^{+0.05} $\

[lccccc]{} &This work &[@ebert04]$^\dag$&[@faessler01]$^\ddag$&[@guo98]&[@sanchis95]$^\S$\
\
$\Gamma(\Xi_{bb}\to\Xi_{bc}\,l\bar\nu_l)$ &$ 1.92^{+0.25}_{-0.05}$& 1.63&&28.5&\
\
$\Gamma(\Xi_{bc}\to\Xi_{cc}\,l\bar\nu_l)$ &$ 2.57^{+0.26}_{-0.03} $ & 2.30&0.79&8.93&8.0\
\
$\Gamma(\Xi_{bb}\to\Xi_{bc}'\,l\bar\nu_l)$ & $1.06 ^{+0.13}_{-0.03}$ &0.82&&4.28&\
\
$\Gamma(\Xi_{bc}'\to\Xi_{cc}\,l\bar\nu_l)$ & $1.36^{+0.10}_{-0.03}$ & 0.88&&7.76&\

[lccc]{} &This work &[@ebert04]$^\dag$\
\
$\Gamma(\Omega_{bb}\to\Omega_{bc}\,l\bar\nu_l)$ & $2.14^{+0.20}_{-0.02}$& 1.70\
\
$\Gamma(\Omega_{bc}\to\Omega_{cc}\,l\bar\nu_l)$ &$2.59^{+0.20}$&2.48\
\
$\Gamma(\Omega_{bb}\to\Omega_{bc}'\,l\bar\nu_l)$ &$1.16^{+0.13}$& 0.83\
\
$\Gamma(\Omega_{bc}'\to\Omega_{cc}\,l\bar\nu_l)$ &$1.36^{+0.09} $ &0.95\

In Table \[tab:asimetrias\] we compile our results for the average angular asymmetries $\alpha'$ and $\alpha''$, as well as the $R_{L/T}$ ratio, introduced in Eq.(\[eq:averageasymmetry\]). The central values have been obtained with the AL1 potential. Being all quantities ratios the variation when changing the inter-quark interaction are in most cases small.

[lccc]{} & $\left<\alpha'\right>$ & $\left<\alpha''\right>$ &$R_{L/T}$\
\
$\Xi_{bb}\to\Xi_{bc}\,l\bar\nu_l$ &$-0.13^{+0.01}$&$-0.45_{-0.02}$& $1.33^{+0.06}_{-0.01}$\
\
$\Xi_{bc}\to\Xi_{cc}\,l\bar\nu_l$ &$-0.12^{+0.01}$&$-0.53_{-0.01}$& $1.62^{+0.07}$\
\
$\Xi_{bb}\to\Xi_{bc}'\,l\bar\nu_l$ & $-0.19$ & $-0.17_{-0.01}$ & $0.71^{+0.01}_{-0.01}$\
\
$\Xi_{bc}'\to\Xi_{cc}\,l\bar\nu_l$ & $-0.19$ & $-0.23_{-0.01}$ & $0.79^{+0.02}$\

[lccc]{} & $\left<\alpha'\right>$ & $\left<\alpha''\right>$ &$R_{L/T}$\
\
$\Omega_{bb}\to\Omega_{bc}\,l\bar\nu_l$ &$ -0.13^{+0.01}$ & $-0.47_{-0.01}$&$1.37^{+0.06}$\
\
$\Omega_{bc}\to\Omega_{cc}\,l\bar\nu_l$ & $-0.12^{+0.01}$ &$-0.53_{-0.01}$ &$1.63^{+0.06}$\
\
$\Omega_{bb}\to\Omega_{bc}'\,l\bar\nu_l$ & $-0.19_{-0.01}$ & $-0.18_{-0.01}$&$0.72^{+0.02}$\
\
$\Omega_{bc}'\to\Omega_{cc}\,l\bar\nu_l$ & $-0.19$ & $-0.23_{-0.01}$ & $0.79^{+0.02}$\

Summary {#sec:summary}
=======

We have evaluated static properties and semileptonic decays for the ground state of doubly heavy $\Xi$ and $\Omega$ baryons. The calculations have been done in the framework of a nonrelativistic quark model with the use of five different inter-quark interactions. The use of different quark-quark potentials allows us to obtain an estimation of the theoretical uncertainties related to the quark-quark interaction. In order to build our wave functions we have made use of the constraints imposed by the infinite heavy quark mass limit. In this limit the spin–spin interactions vanish and the total spin of the two heavy quarks is well defined. With this approximation we have used a simple variational approach, with Jastrow type orbital wave functions, to solve the involved three-body problem.

Among the static properties, our results for the masses are in very good agreement with previous results obtained with the same inter-quark interactions but within a more complicated Faddeev approach [@Si96]. In some cases we even get lower, and thus better, masses[^17]. We have calculated mass densities and charge densities (charge form factors) finding that the corresponding mean square radii are again in good agreement with the Faddeev calculation of Ref. [@Si96]. We have also evaluated magnetic moments. Being the total orbital angular momentum of the baryon $L=0$, the magnetic moments come from the spin contributions alone. With the exception of $\Xi_{bc}^0$, $\Xi_{bc}^+$ and $\Omega_{bc}^0$ we agree perfectly with the Faddeev calculation in Ref. [@Si96]. For the magnetic moments of $\Xi_{bc}^0$, $\Xi_{bc}^+$ and $\Omega_{bc}^0$ the discrepancies between the two calculation are very large. The origin might be attributed to the presence of a non–negligible $S_h=0$ component in the wave functions of Ref. [@Si96]. In our case we have $S_h=1$ which we think is a good approximation based on the infinite heavy quark mass limit. This assertion seems to be corroborated by the results obtained in the relativistic calculation of Ref. [@faessler06], at least for the $\Xi_{bc}^0$ and $\Omega_{bc}^0$ cases.

We have used our wave functions to study the semileptonic decay of doubly $\Xi(J=1/2)$ and $\Omega(J=1/2)$ baryons. We have worked in the spectator approximation with one–body currents alone. In the $m_b=m_c$ case and for $B=B'$ baryons we have checked that our model satisfies baryon number conservation. On the other hand we have a small vector current violation by an amount given by the binding energy over the mass of the baryon, violation that disappears in the infinite heavy quark mass limit. Small vector current violations due to binding effects seems to be present in most calculations to date. With this model we have evaluated form factors, asymmetry parameters, differential decay widths and total decay widths. Our results for the latter are in reasonable agreement with the ones obtained in Ref. [@ebert04] using a relativistic quark model in the quark–diquark approximation, while they are much smaller than the ones obtained in Ref. [@guo98] by means of the Bethe–Salpeter equation applied to a quark-diquark system.

For the weak form factors the results exhibit an apparent SU(3) symmetry when going from $\Xi$ to $\Omega$ baryons. This is due to the fact that we have two heavy quarks and the light one acts as a spectator in the weak transition. This apparent symmetry appears also in the decay widths and asymmetry parameters. On the other hand SU(3) violating effects are clearly visible in some static quantities like the charge form factors and radii, and the magnetic moments, that depend strongly on the light quark charge and/or mass.\

This research was supported by DGI and FEDER funds, under contracts FIS2005-00810, BFM2003-00856 and FPA2004-05616, by Junta de Andalucía and Junta de Castilla y León under contracts FQM0225 and SA104/04, and it is part of the EU integrated infrastructure initiative Hadron Physics Project under contract number RII3-CT-2004-506078. J. M. V.-V. acknowledges an E.P.I.F. contract with the University of Salamanca.

Infinite heavy quark mass limit of our variational wave functions {#app:ihqml}
=================================================================

In the infinite heavy quark mass limit the wave function of the system should look like the one for a “meson” composed of a light quark and a heavy diquark. The two heavy quarks bind into a $\bar 3$ color source diquark that to the light degrees of freedom appears to be pointlike [@white91]. In our model the pointlike nature of the heavy diquark comes about through the one-gluon exchange Coulomb potential which binds the two heavy quarks into a distance[^18] $$\begin{aligned}
r_{h_1h_2}\propto\frac{1}{\mu_{h_1h_2}};\ 
\mu_{h_1h_2}=\frac{m_{h_1}m_{h_2}}{m_{h_1}+m_{h_2}}\end{aligned}$$ that tends to zero if both quark masses go to infinity.

For our wave functions we can define the probability distribution $P_{h1h2}$ for the two heavy quarks to be found at a distance $r_{h_1h_2}$ $$P_{h1h2}\,(r_{h_1h_2}) =\int d^3r_1\int d^3r_2\ \delta(r_{12}-r_{h_1h_2})
\left|\Psi^B_{h_1h_2}(r_2,r_2,r_{12})
\right|^2
\label{eq:ph1h2}$$ In Fig. \[fig:p12\], we show the $P_{h1h2}$ probability distributions for the $\Xi_{cc},\,\Xi_{bc},\, \Xi_{bb}$ and $\Omega_{cc},\,\Omega_{bc},\, \Omega_{bb}$ baryons evaluated for the AL1 potential.

We see how the maximum moves, as expected, to lower $r_{h_1h_2}$ values as the quark masses increase.

On the other hand the relative wave function for the light quark with respect to the heavy quark subsystem should tend to the one of a light quark relative to a pointlike diquark. This limit is not evident in the coordinates we work as one has first to separate the diquark internal orbital wave function from the total one. To see that this limiting situation is in fact reached within our variational ansatz let us do the following: introduce the new set of coordinates $$\begin{aligned}
\vec R &=&\frac{m_{h_1}\vec x_{h_1}+m_{h_2}\vec x_{h_2}+m_{q}\vec x_{q}}
{m_{h_1}+m_{h_2}+m_{q}}\nonumber\\
\vec r_{12}&=&\vec x_{h_1}-\vec x_{h_2}\nonumber\\
\vec r_q&=&\frac{m_{h_1}\vec x_{h_1}+m_{h_2}\vec x_{h_2}}{m_{h_1}+m_{h_2}}-\vec
x_q=\frac{m_{h_1}\vec r_{1}+m_{h_2}\vec r_{2}}{m_{h_1}+m_{h_2}}\end{aligned}$$ in terms of which the Hamiltonian now reads $$\begin{aligned}
H&=&
-\frac{\stackrel{\rightarrow}{\nabla}
\stackrel{}{^2}_{\hspace{-.1cm}\vec{R}}}{2 \overline M} +
H^{\rm int} \nonumber\\
 H^{\rm
int}&=&\overline M+H_{h1h_2}+H_{qh_1h_2}\end{aligned}$$ where $$\begin{aligned}
H_{h1h_2}&=&-\frac{\stackrel{\rightarrow}{\nabla}_{12}^2}{2\mu_{h_1h_2}}
+V_{h_1h_2}(\vec r_{12},\, spin)\nonumber\\
 H_{qh_1h_2}&=&-\frac12\left(\frac{1}{m_{h_1}+m_{h_2}}+\frac{1}{m_q}\right)
 \stackrel{\rightarrow}{\nabla}_{q}^2
+V_{h_1q}(\vec r_q+\frac{m_{h_2}}{m_{h_1}+m_{h_2}}\vec r_{12},\,
spin)\nonumber\\
&&\hspace{4.55cm}+V_{h_2q}(\vec r_q-\frac{m_{h_1}}{m_{h_1}+m_{h_2}}\vec r_{12},\, spin)\end{aligned}$$ with $\stackrel{\rightarrow}{\nabla}_{12}=\partial /\partial_{\vec r_{12}}$ and $\stackrel{\rightarrow}{\nabla}_{q}=\partial /\partial_{\vec r_q}$. Defining now $$\begin{aligned}
 H^0_{qh_1h_2}&=&-\frac12\left(\frac{1}{m_{h_1}+m_{h_2}}+\frac{1}{m_q}\right)
 \stackrel{\rightarrow}{\nabla}_{q}^2
+V_{h_1q}(\vec r_q,\,
spin)+V_{h_2q}(\vec r_q,\, spin)\end{aligned}$$ one would have $$\begin{aligned}
H^{\mathrm{int}}=H_{h_1h_2}+H^0_{qh_1h_2}+(H_{qh_1h_2}-H^0_{qh_1h_2})\end{aligned}$$ $H_{h_1h_2}$ is the Hamiltonian for the relative movement of the two heavy quarks while $H^0_{qh_1h_2}$ is the Hamiltonian for the relative movement of the light quark with respect to a pointlike heavy diquark where the two heavy quarks are located in their center of mass. Both movements are coupled through the term $(H_{qh_1h_2}-H^0_{qh_1h_2})$. If the heavy quark masses get arbitrarily large the average $r_{12}$ value tends to zero and thus one can neglect the effect of $H_{qh_1h_2}-H^0_{qh_1h_2}$. In that limiting situation the light and heavy quark degrees of freedom decouple completely and the internal Hamiltonian reduces, as it should, to the sum of the Hamiltonians $H_{h_1h_2}$, corresponding to the relative movement of the two heavy quarks, and $H^0_{qh_1h_2}$, corresponding to the relative movement of the light quark with respect to the pointlike heavy diquark subsystem.

As to the wave function it should reduce in that limit to the product $\Phi_{h_1h_2}(r_{12})\cdot \Phi^0_{qh_1h_2}(r_q)$, being $\Phi_{h_1h_2}(r_{12}),\, \Phi^0_{qh_1h_2}(r_q)$ the ground state wave functions for $H_{h_1h_2},\,H^0_{qh_1h_2}$ respectively. To check that we reach that limiting situation we have evaluated the projection ${\cal P}$ of our variational wave functions onto $\Phi_{h_1h_2}(r_{12})\cdot \Phi^0_{qh_1h_2}(r_q)$ evaluated with the actual heavy quark masses. This projection is given by[^19] $${\cal P}=\int d^3r_1 \int d^3r_2 \left(\Psi^B_{h_1h_2}(r_1,r_2,r_{12}\right)^*
\Phi_{h_1h_2}(r_{12})\ \Phi^0_{qh_1h_2}(r_q)
\label{eq:project}$$ and the values for $|{\cal P}|^2$ that we obtain for the $\Xi_{cc},\,\Xi_{bc},\, \Xi_{bb}$ and $\Omega_{cc},\,\Omega_{bc},\, \Omega_{bb}$ baryons using the AL1 potential are given in Table \[tab:project\]. We see how $|{\cal P}|^2$ increases with increasing heavy quark masses indicating that for very high heavy quark masses the total wave function tends to the one of a light quark relative to a pointlike diquark times the diquark internal wave function.

                    $\Xi_{cc}$   $\Xi_{bc}$   $\Xi_{bb}$   $\Omega_{cc}$   $\Omega_{bc}$   $\Omega_{bb}$
  ---------------- ------------ ------------ ------------ --------------- --------------- ---------------
  $|{\cal P}|^2$      0.974        0.975        0.991          0.959           0.966           0.984

  : Absolute value square of the ${\cal P}$ projection coefficient defined in Eq. (\[eq:project\])[]{data-label="tab:project"}

In summary, our variational wave functions respect the infinite heavy quark mass limit.

We note by passing that the product $\Phi_{h_1h_2}(r_{12})\cdot \Phi^0_{qh_1h_2}(r_q)$ corrected by a correlation function in the variable $\vec r_{12}\cdot \vec r_q$ would have been a good variational orbital wave function. For instance, a calculation using the AL1 potential and an orbital wave function given just by that product $\Phi_{h_1h_2}(r_{12})\cdot \Phi^0_{qh_1h_2}(r_q)$ gives for the expectation value of $H^{\mathrm{int}}$ for the $\Xi_{cc},\,\Xi_{bc},\,\Xi_{bb}$ baryons $$\begin{aligned}
 \left.\langle H^{\mathrm{int}}\rangle\right|^{AL1}_{\Xi_{cc}}=3640\,\mathrm{MeV}; \
 \left.\langle H^{\mathrm{int}}\rangle\right|^{AL1}_{\Xi_{bc}}=6943\,\mathrm{MeV};\
 \left.\langle H^{\mathrm{int}}\rangle\right|^{AL1}_{\Xi_{bb}}=10198\,\mathrm{MeV}
 \end{aligned}$$ which are respectively 28MeV, 24MeV, and 1MeV larger, and therefore worse (see footnote 13), than our best values in Table \[tab:xiomegamasses\]. The correlation function would clearly be needed, being its role more important for the “$cc$” and “$bc$” systems. The improvement in going from a “$cc$” to a “$bc$ system” is not as good as one would naively expect, the reason being that, considering $\vec r_{12}$ to be a small quantity, $H_{qh_1h_2}-H^0_{qh_1h_2}$ is first order in $\vec r_{12}$ for the “$bc$” system while, for symmetry reasons, it is necessarily of second order for the “$cc$” and “$bb$” ones. In any case one sees how the expectation values obtained with this simple ansatz get closer to our best values in Table \[tab:xiomegamasses\] as the heavy quark masses increase.

Form factors in terms of the ${\cal I}^{B'B}(|\vec q\,|)$ and ${\cal K}^{B'B}(|\vec q\,|)$ integrals {#app:fg123}
====================================================================================================

In this appendix we relate the vector $F_1,\,F_2,\,F_3$ and axial $G_1,\,G_2,\,G_3$ form factors, that we evaluate in the center of mass of the decaying baryon, to the integrals ${\cal I}^{B'B}(|\vec q\,|),
\,{\cal K}^{B'B}(|\vec q\,|)$ defined in Eq.(\[eq:integrales\]). To simplify the expressions it is convenient to introduce $$\begin{aligned}
\widehat F_j(|\vec q\,|)&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2E_{B'}(-\vec q\, )}}
\,\sqrt{\frac{2E_{c}(\vec q\, )}{E_{c}(\vec q\, )+m_{c}}}\ F_j(|\vec q\,|)\ \ ,\ j=1,2,3\nonumber\\
\widehat G_j(|\vec q\,|)&=&\sqrt{\frac{E_{B'}(-\vec q\, )+m_{B'}}{2E_{B'}(-\vec q\, )}}
\,\sqrt{\frac{2E_{c}(\vec q\, )}{E_{c}(\vec q\, )+m_{c}}}\ G_j(|\vec q\,|)\ \ ,\ j=1,2,3\nonumber\\\end{aligned}$$

-   Cases $S_h=1,\,S'_h=0$ or $S_h=0,\,S'_h=1$: $$\begin{aligned}
    \frac{\widehat F_1(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}\ 
    &=&-\sqrt\frac{2}{3}\,
    \left(\frac{{\cal I}^{B'B}(|\vec q\,|)}{E_{c}(\vec q\, )+m_c}
    -\frac{{\cal K}^{B'B}(|\vec q\,|)}{2}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}-\frac{1}{m_b}
    \right)
    \right)
    \nonumber\\
    \frac{\widehat F_1(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}+\frac{\widehat F_3(|\vec q\,|)}{m_{B'}}&=&0\nonumber\\
    \widehat F_1(|\vec q\,|)+\widehat F_2(|\vec q\,|)+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,\widehat F_3(|\vec q\,|)&=&0\nonumber\\
    \label{eq:appf123s10}\end{aligned}$$ where $E_c(\vec q\,)=\sqrt{m_c^2+\vec q^{\,2}}$. From the above expressions we have that $F_2=F_3$. $$\begin{aligned}
    -\widehat G_1(|\vec q\,|)&=&\sqrt\frac{2}{3}\,
    \left(\,{\cal I}^{B'B}(|\vec q\,|)+\frac{|\vec q\,|^2\,{\cal K}^{B'B}(|\vec q\,|)}{2\left(E_{c}(\vec q\, )+m_c\right)}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}+\frac{1}{m_b}
    \right)
    \right)\nonumber\\
    \left(-\widehat G_1(|\vec q\,|)+\frac{|\vec q\,|^2\,\widehat G_3(|\vec q\,|)}{m_{B'}(E_{B'}(-\vec q\,
    )+m_{B'})}\right)&=&\sqrt\frac{2}{3}\,
    \left(\,{\cal I}^{B'B}(|\vec q\,|)+\frac{|\vec q\,|^2\,{\cal K}^{B'B}(|\vec q\,|)}{2\left(E_{c}(\vec q\, )+m_c\right)}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}-\frac{1}{m_b}
    \right)
    \right)\nonumber\\
    \frac{-\widehat G_1(|\vec q\,|)+\widehat G_2(|\vec q\,|)+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,
    \widehat G_3(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}&=&\sqrt\frac{2}{3}\,
    \left(\,\frac{{\cal I}^{B'B}(|\vec q\,|)}{E_{c}(\vec q\, )+m_c}
    -\frac{{\cal K}^{B'B}(|\vec q\,|)}{2}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}+\frac{1}{m_b}
    \right)
    \right)\nonumber\\ 
    $$

-   Case $S_h=1,\,S'_h=1$

    $$\begin{aligned}
    \frac{\widehat F_1(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}\ 
    &=&\frac{4}{3\sqrt2}\,
    \left(\frac{{\cal I}^{B'B}(|\vec q\,|)}{E_{c}(\vec q\, )+m_c}
    -\frac{{\cal K}^{B'B}(|\vec q\,|)}{2}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}-\frac{1}{m_b}
    \right)
    \right)
    \nonumber\\
    \frac{\widehat F_1(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}+
    \frac{\widehat F_3(|\vec q\,|)}{m_{B'}}&=&
    \sqrt2\,
    \left(\frac{{\cal I}^{B'B}(|\vec q\,|)}{E_{c}(\vec q\, )+m_c}
    -\frac{{\cal K}^{B'B}(|\vec q\,|)}{2}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}+\frac{1}{m_b}
    \right)
    \right)
    \nonumber\\
    \widehat F_1(|\vec q\,|)+\widehat F_2(|\vec q\,|)+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,\widehat F_3(|\vec q\,|)&=&
    \sqrt2\,
    \left(\,{\cal I}^{B'B}(|\vec q\,|)
    +\frac{|\vec q\,|^2\,{\cal K}^{B'B}(|\vec q\,|)}
    {2\left(E_{c}(\vec q\, )+m_c\right)}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}-\frac{1}{m_b}
    \right)
    \right)
    \nonumber\\
    \label{eq:appf123}\end{aligned}$$

    $$\begin{aligned}
    \widehat G_1(|\vec q\,|)&=&\frac{4}{3\sqrt2}\,
    \left(\,{\cal I}^{B'B}(|\vec q\,|)+\frac{|\vec q\,|^2\,{\cal K}^{B'B}(|\vec q\,|)}
    {2\left(E_{c}(\vec q\, )+m_c\right)}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}+\frac{1}{m_b}
    \right)
    \right)\nonumber\\
    \left(-\widehat G_1(|\vec q\,|)+\frac{|\vec q\,|^2\,\widehat G_3(|\vec q\,|)}{m_{B'}(E_{B'}(-\vec q\,
    )+m_{B'})}\right)&=&-\frac{4}{3\sqrt2}\,
    \left(\,{\cal I}^{B'B}(|\vec q\,|)+\frac{|\vec q\,|^2\,{\cal K}^{B'B}(|\vec q\,|)}
    {2\left(E_{c}(\vec q\, )+m_c\right)}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}-\frac{1}{m_b}
    \right)
    \right)\nonumber\\
    \frac{-\widehat G_1(|\vec q\,|)+\widehat G_2(|\vec q\,|)+\frac{E_{B'}(-\vec q\, )}{m_{B'}}\,
    \widehat G_3(|\vec q\,|)}{E_{B'}(-\vec q\, )+m_{B'}}&=&-\frac{4}{3\sqrt2}\,
    \left(\,\frac{{\cal I}^{B'B}(|\vec q\,|)}{E_{c}(\vec q\, )+m_c}
    -\frac{{\cal K}^{B'B}(|\vec q\,|)}{2}
    \left(\frac{m_c}{E^2_{c}(\vec q\, )}+\frac{1}{m_b}
    \right)
    \right)\nonumber\\ 
    $$

${\cal I}^{B'B}(|\vec q\,|)$ and ${\cal K}^{B'B}(|\vec q\,|)$ integrals {#app:integrals}
=======================================================================

To evaluate ${\cal I}^{B'B}(|\vec q\,|)$ and ${\cal K}^{B'B}(|\vec q\,|)$ we use a partial wave expansion of the orbital wave functions: $$\begin{aligned}
\Psi^{B}_{b\,h_2}(r_1,r_2,r_{12})=\sum_{l=0}^\infty f_l^{B}(r_1,r_2) P_l
(\mu)\nonumber\\
\Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})=\sum_{l=0}^\infty f_l^{B'}(r_1,r_2) P_l (\mu)\end{aligned}$$ where $\mu$ is the cosine of the angle made by $\vec r_1$ and $\vec r_2$ and $P_{l}(\mu)$ is a Legendre polynomial of rank $l$. The radial functions $f_l^{B}(r_1,r_2),\,f_l^{B'}(r_1,r_2),\,$ are evaluated as $$\begin{aligned}
f_l^{B}(r_1,r_2)&=&\frac{2l+1}{2}\int_{-1}^{+1}d\mu\
P_l(\mu)\, \Psi^{B}_{b\,h_2}(r_1,r_2,r_{12})\nonumber\\
f_{l}^{B'}(r_1,r_2)&=&\frac{2l+1}{2}\int_{-1}^{+1}d\mu\
P_{l}(\mu)\, \Psi^{B'}_{c\,h_2}(r_1,r_2,r_{12})\end{aligned}$$ In terms of those we have

-   ${\cal I}^{B'B}(|\vec q\,|)$ $$\begin{aligned}
    {\cal I}^{B'B}(|\vec q\,|)&=&16\pi^2\sum_l\sum_{l'}\sum_{l''}
    \left(ll'l''\left|\right.000\right)^2
    \int_0^\infty d\,r_1\, r_1^2\ j_{l''}(\,\frac{m_{h_2}+m_q}{\overline M'} 
    |\vec q\,|r_1)
    \int_0^\infty d\,r_2\, r_2^2\ j_{l''}(\,\frac{m_{h_2}}{\overline M'} 
    |\vec q\,|r_2)\nonumber\\
    &&\hspace{5cm}\times f_{l'}^{B'}(r_1,r_2)\,f_{l}^{B}(r_1,r_2)\end{aligned}$$ with $j's$ being spherical Bessel functions.

-   ${\cal K}^{B'B}(|\vec q\,|)$ $$\begin{aligned}
    {\cal K}^{B'B}(|\vec q\,|)&=&-\frac{16\pi^2}{\sqrt3\,|\vec q\,|}
    \sum_l\sum_{l'}\sum_{l''}\sum_{l'''}\sum_L
    (-1)^{(l{''}+l{'''}+1)/2}\sqrt{(2L+1)(2l''+1)(2l'''+1)}\nonumber\\&&
    \times\left(ll'l''\left|\right.000\right)\,\left(l''l'''1\left|\right.000\right)
    \,\left(l'Ll'''\left|\right.000\right)\,W(l''l'1L:ll''')\nonumber\\&& 
    \times\int_0^\infty d\,r_1\, r_1^2\ j_{l'''}(\,\frac{m_{h_2}+m_q}{\overline M'} 
    |\vec q\,|r_1)
    \int_0^\infty d\,r_2\, r_2^2\ j_{l''}(\,\frac{m_{h_2}}{\overline M'} 
    |\vec q\,|r_2)\
     f_{l'}^{B'}(r_1,r_2)\,\Omega_L\,f_{l}^{B}(r_1,r_2)\hspace{1cm}\end{aligned}$$ with $W(l''l'1L;l,l''')$ being a Racah coefficient and $\Omega_L$ the differential operator[^20] $$\begin{aligned}
    \Omega_{L=l+1}&=&-\sqrt{\frac{l+1}{2l+1}}\left(
    \frac{\partial}{\partial r_1}-\frac{l}{r_1}\right)\nonumber\\
    \Omega_{L=l-1}&=&\ \sqrt{\frac{l}{2l+1}}\left(
    \frac{\partial}{\partial r_1}+\frac{l+1}{r_1}\right)\end{aligned}$$

For the actual evaluation we restrict the $l,\,l'$ values to $l,\,l'=0,\cdots,6$

[99]{}

D. B. Lichtenberg, Phys. Rev. D 15, 345 (1977).

M.J. Savage and M.B. Wise, Phys. Lett. B 248, 177 (1990).

M.J. White and M.J. Savage, Phys. Lett. B 271, 410 (1991). N. Brambilla, A. Vairo, and T. Rösch, Phys. Rev. D 72, 034021 (2005). M. Mattson [*et al.*]{} (SELEX) Collaboration, Phys. Rev. Lett. 89, 112001 (2002) ç. V.V. Kiselev and A.K. Likhoded, hep-ph/0208231. S.P. Ratti (FOCUS Collaboaration), Nuc. Phys. B (Proc. Suppl.) 115, 33 (2003). See also\
. B. Aubert [*et al.*]{} ([*BABAR*]{} Collaboration), Phys. Rev. D 74, 011103 (2006). T. Lesiak (BELLE Collaboration), hep-ex/0605047 to appear in the Proceedings of Moriond QCD 2006.

W.-M- Yao [*et al.*]{} (Particle Data Group), J. Phys. G 33, 1 (2006) S. Nussinov and W. Wetzel, Phys. Rev. D 36, 130 (1987). M.A. Shifman and M.B. Voloshin, Sov. J. Nucl. Phys. 45, 292 (1987) (Yad. Fiz. 45, 463 (1987)) . H.D. Politzer and M.B. Wise, Phys. Lett. B 206, 681 (1988); 208, 504 (1988). N. Isgur and M.B. Wise, Phys. Lett. B 232, 113 (1989); 237, 527 (1990).

H Georgi, Phys. Lett. B 240, 447 (1990).

B.A. Thacker and G.P. Lepage, Phys. Rev. D43, 196 (1991).

E. Jenkins, M. Luke. A.V. Manohar, and M.J. Savage, Nucl. Phys. B 390, 463 (1993).

C. Albertus, J.E. Amaro, E. Hernández, and J. Nieves, Nucl. Phys. A 740, 333 (2004).

R. K. Bhaduri, L.E. Cohler, Y. Nogami, Nuovo Cim. A 65, 376 (1981).

C. Semay and B. Silvestre-Brac, Z. Phys. C 61, 271 (1994).

B. Silvestre-Brac, Few-Body Systems 20, 1 (1996).

D. Ebert, R.N. Faustov, V.O. Galkin, and A.P. Martynenko, Phys. Rev. D 66, 014008 (2002).

V.V. Kiselev and A.K. Likhoded, Phys. Usp. 45, 455 (2002) (Usp. Fiz. Nauk 172, 497 (2002)). arXiv:hep-ph/0103169.

I.M. Narodetskii and M.A. Trusov, Phys. At. Nucl. 65, 917 (2002) (Yad. Fiz. 65, 944 (2002)).

S.-P. Tong, Y.-B. Ding, X.-H. Guo, H.-Y. Jin, X.-Q. Li, P.-N. Shen, and R. Zhang, Phys. Rev. D 62, 054024 (2000).

C. Itoh, T. Minamikawa, K. Miura, and T. Watanabe, Phys. Rev. D 61, 057502 (2000).

J. Vijande, H. Garcilazo, A. Valcarce, and F. Fernández, Phys. Rev. D 70, 054022 (2004). S.S. Gershtein, V.V. Kiselev, A.K. Likhoded, and A.I. Onishchenko, Phys. Rev. D 62, 054021 (2000).

D. Ebert, R.N. Faustov, V.O. Galkin, A.P. Martynenko, and V.A. Saleev, Z. Phys. C 76, 111 (1997).

R. Roncaglia, D. B. Lichtenberg, and E. Predazzi, Phys. Rev. D 52, 1722 (1995); R. Roncaglia, A. Dzierba, D. B. Lichtenberg, and E. Predazzi, Phys. Rev D 51, 1248 (1995).

J.G. Körner, M. Krämer, and D. Pirjol, Prog. Part. Nucl. Phys. 33, 787 (1994). N. Mathur, R. Lewis, and R.M. Woloshyn, Phys. Rev. D 66, 014502 (2002).

A. Ali Khan [*et al.*]{}, Phys. Rev. D 62, 054505 (2000). R. Lewis, N. Mathur, and R.M. Woloshyn, Phys. Rev D 64, 094509 (2001). J.M. Flynn, F. Mescia, and A.S.B. Tariq (UKQCD Collaboration), JHEP 0307, 066 (2003).

A. Faessler, Th. Gutsche, M.A. Ivanov, J.G. Körner, V.E. Lyubovitskij, D. Nicmorus, and K. Pumsa-ard, Phys. Rev. D 73, 094013 (2006).

B. Juliá-Díaz and D.O. Riska, Nucl. Phys. A 739, 69 (2004).

Y. Oh, D.-P. Min, M. Rho, and N. Scoccola, Nucl. Phys. A 534, 493 (1991). S.N. Jena and D.P. Rath, Phys. Rev. D 34, 196 (1986).

S.K. Bose and L.P. Singh, Phys. Rev. D 22, 773 (1980).

D. Ebert, R.N. Faustov, V.O. Galkin, and A.P. Martynenko, Phys. Rev. D 70, 014018 (2004). Erratum to be published.

A. Faessler, Th. Gutsche, M.A. Ivanov, J. G. Körner, and V.E. Lyubovitdkij, Phys. Lett. B 518, 55 (2001).

X.-H. Guo, H.-Y. Jin, and X.-Q. Li, Phys. Rev. D 58, 114007 (1998).

M. A. Sanchis-Lozano, Nucl. Phys. B 440, 251 (1995).

J.G. Körner and M. Krämer, Phys. Lett. B 275, 495 (1992).

C. Albertus, E. Hernández, and J. Nieves, Phys. Rev. D 71, 014012 (2005).

A. Buchmman, E. Hernández, and K. Yazaki, Nucl. Phys. A 569, 661 (1994).

D. Ebert, R.N. Faustov, and V.O. Galkin, private communication.

M. A. Sanchis-Lozano, private communication.

Th. Gutsche, private communication.

[^1]: Present address: School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom.

[^2]: Note that the definitions of $\Xi_{bc}$ and $\Xi'_{bc}$ are interchanged in some references, with $\Xi_{bc}$ having $S_h=0$ and $\Xi'_{bc}$ having $S_h=1$. The same applies to $\Omega_{bc}$ and $\Omega'_{bc}$. In tables we always quote the results corresponding to the convention we use.

[^3]: To get the quark–quark interaction starting from a quark-antiquark one the usual $V_{ij}^{qq} = V_{ij}^{q \bar q}/2$ prescription, coming from a $\vec{\lambda}_i\vec{\lambda}_j$ color dependence ($\vec{\lambda}$ are the Gell-Mann matrices) of the whole potential, has been assumed.

[^4]: We omit the antisymmetric color wave function and the plane wave for the center of mass motion which are common to all cases.

[^5]: A similar form lead to very good results in the case of baryons with a single heavy quark [@conrado04].

[^6]: Note that $F^{B}$ should vanish at large distances because of the confinement potential. The confinement potential is also responsible for the non–vanishing values of the parameters $\alpha_j,\ j=1,2$ in Eq. (\[eq:onebody\]).

[^7]: Note in Ref. [@roncaglia95] the $\Xi_{bc},\,\Xi'_{bc}$ and $\Omega_{bc},\,\Omega'_{bc}$ baryons are defined such that the total spin of the light $q$ quark and the heavy $c$ quark are well defined, being 0 for $\Xi_{bc},\,\Omega_{bc}$ and 1 for $\Xi'_{bc},\,\Omega'_{bc}$. They are thus linear combinations of our states. The different spin functions are related by $$\begin{aligned}
    \left(|qc;0\rangle\otimes|b;\frac12\rangle\right)^{J=1/2}=
    \frac12\left(|bc;0\rangle\otimes|q;\frac12\rangle   \right)^{J=1/2}
    -\frac{\sqrt3}{2}\left(|bc;1\rangle\otimes|q;\frac12\rangle  
    \right)^{J=1/2}\nonumber\\
    \left(|qc;1\rangle\otimes|b;\frac12\rangle\right)^{J=1/2}=
    -\frac{\sqrt3}{2}\left(|bc;0\rangle\otimes|q;\frac12\rangle   \right)^{J=1/2}
    -\frac{1}{2}\left(|bc;1\rangle\otimes|q;\frac12\rangle  
    \right)^{J=1/2}\end{aligned}$$ In order to extract their predictions for the $\Xi_{bc},\,\Xi'_{bc}$ and $\Omega_{bc},\,\Omega'_{bc}$ baryons with total spin of the two heavy quarks well defined, we have assumed that the above relations, but with coefficients square, are also valid for the masses. Note this may be incorrect as we are neglecting a possible non negligible interference contribution.

[^8]: There exists the obvious restriction $m_{h_1}\vec{y}_{h_1}+m_{h_2}\vec{y}_{h_2}+m_{q}\vec{y}_q=\vec 0$.

[^9]: Note that the classical kinetic energy has a term on $\vec{v}_{h_1}\cdot\vec{v}_{h_2}$ and then the operator $m_{h_1} \vec{v}_{h_1}$ is not proportional to $-i\stackrel{\to}{\nabla}_{y_{h_1}}$, but it is rather given by $m_{h_1} \vec{v}_{h_1}={(\overline M-m_{h_1})}/{\overline M}\cdot
    (-i\stackrel{\to}{\nabla}_{y_{h_1}}) 
    -{m_{h_2}}/{\overline M}\cdot(-i\stackrel{\to}{\nabla}_{y_{h_2}})
    =(-i\stackrel{\to}{\nabla}_1)$. Similarly $m_{h_2} \vec{v}_{h_2}=(-i\stackrel{\to}{\nabla}_2)$.

[^10]: At $q^2=0$ one has $|\vec q\,|=(m_B^2-m_{B'}^2) /2m_B$ which is $\approx
    m_B/3$ for the transitions under study.

[^11]: Remember $\vec q$ is in the $z$ direction. Notice also that, for $\vec p=\vec 0$, $w$ is just a function of $|\vec q\,|$.

[^12]: One just has to integrate by parts in the ${\cal K}^{BB}(|\vec q\,|)$ expression

[^13]: In Refs. [@ebert04; @guo98] the currents are constructed at the diquark level. At the baryon level, vector current is conserved because light quark-heavy diquark binding effects are neglected. The calculation in Ref. [@sanchis95] uses the infinite heavy quark mass limit thus canceling binding effects.

[^14]: Note we have divided by a factor 2 the results originally published in Ref. [@ebert04] as there, and as we initially did, the authors forgot a symmetry factor $1/\sqrt2$ for the case of diquarks with two equal quarks [@ebert08].

[^15]: Note we have multiplied by a factor 2 the result originally published in Ref. [@sanchis95] as the author overlooked a normalization factor $\sqrt2$ [@sanchis08].

[^16]: Note the actual physical states would be an admixture of what we call $\Xi_{bc}\,\Xi'_{bc}$, and similarly in the $\Omega$ case. A measurement of the decay width could give information on the admixtures.

[^17]: For a given Hamiltonian a variational mass is an upper bound of the true ground state mass.

[^18]: The relation can only be approximate due to confinement and the interaction with the light quark.

[^19]: Note $r_q$ can be expressed in terms of $r_1,r_2,r_{12}$ and that $d^3r_1d^3r_2=d^3r_{12}d^3r_q$

[^20]: Note that the Racah and Clebsch-Gordan coefficients restrict $L$ to the two possible values $L=l\pm1$.