††thanks: Also on leave from L.D. Landau Institute for Theoretical Physics

#  $QQqq\bar{q}$ Quark System, Compact Pentaquark, and Gauge/String Duality

Oleg Andreev  Arnold Sommerfeld Center for Theoretical Physics, LMU-München,
Theresienstrasse 37, 80333 München, Germany

###### Abstract

For the case of two light flavors we propose the stringy description of the
system consisting of two heavy and three light quarks, with the aim of
exploring the quark organization inside the system and its low-lying Born-
Oppenheimer potentials as a function of the heavy quark separation. Our
analysis reveals several critical separations related to the processes of
string reconnection, breaking and junction annihilation. We find that a
compact pentaquark configuration makes the dominant contribution to the
potential of the first excited state at small separations, and for separations
larger than $0.1\,\text{fm}$, the antiquark-diquark-diquark structure emerges.
Moreover, it turns out that the length scale of string junction annihilation
is the same as that for the $QQ\bar{q}\bar{q}$ system. We also discuss the
relation to the potential of the $QQq$ system and some relations among the
masses of hadrons in the heavy quark limit.

††preprint: LMU-ASC 26/22

##  I Introduction

Since the first observation X38 of the charmonium-like peak $X(3872)$, more
than $60$ new hadrons have been observed at high statistical significance.
Most of those are potentially exotic states such as tetraquarks, pentaquarks,
hybrid mesons, and glueballs.111For more details, see the recent review
article lebed and the book book . More recently, sightings of pentaquarks
consisting of three light quarks and one $c\bar{c}$ pair have been reported
P43 . These are the example of $Q\bar{Q}qqq$ states. In general, there could
exist other doubly heavy pentaquark states which are of type $QQqq\bar{q}$.
But so far they have not been observed experimentally.

One way to deal with doubly heavy quark systems is as follows. Because of the
large ratio of the quark masses, the Born-Oppenheimer (B-O) approximation
borrowed from atomic and molecular physics bo seems justified.222For the
further development of these ideas in the context of QCD, see braat . In that
case the corresponding B-O potentials are defined as the energies of
stationary configurations of the gluon and light quark fields in the presence
of the static heavy quark sources. The hadron spectrum is then calculated by
solving the Schrödinger equation in these potentials.

Although lattice gauge theory is a well-established approach to non-
perturbative QCD, it still remains to be seen what it can and can’t do with
regard to the doubly heavy pentaquark systems. Meanwhile, the gauge/string
duality is a powerful way to understand gauge theories outside of the weak
coupling limit, and therefore may be used as an alternative approach to gain
important physical insights into this problem.333The book uaw gives a thorough
review of much of what is known about the gauge/string duality in relation to
QCD. Also within this approach, matters relating to pentaquarks have not been
discussed in the literature. Filling the latter gap is one of the aims of the
present paper.

The paper continues our study a-QQq ; a-QQqq ; a-QQbqqb on the doubly heavy
quark systems. It is organized as follows. We begin in Sec.II by recalling
some preliminary results and setting the framework. Then in Sec.III, we
construct and analyze a set of string configurations in five dimensions which
provide a dual description of the $QQqq\bar{q}$ system in the heavy quark
limit. Among those we find the configurations relevant to the two low-lying
B-O potentials. In the process, we introduce several length scales. These
characterize transitions between different dominant configurations, and in
fact, are related to different types of string interactions: reconnection,
breaking and junction (baryon vertex) annihilation. We go on in Sec.IV to
discuss further improvements and extensions. Finally, in Sec.V, we make a few
comments on some of the consequences of our findings. In particular, we
discuss some relations among the masses of doubly-heavy-light and heavy-light
hadrons. Appendix A contains notation and definitions. To make the paper more
self-contained, we give a brief review of the $QQq$ system in Appendix B.

##  II Preliminaries

###  II.1 A general procedure

In the presence of light quarks, the B-O potentials can be determined along
the lines of lattice QCD. This implies that a mixing analysis based on a
correlation matrix is needed FK . Its diagonal elements are given by the
energies of stationary configurations, whereas the off-diagonal ones describe
transitions between those configurations. The potentials are determined by the
eigenvalues of this matrix.

Now consider string configurations for the $QQqq\bar{q}$ quark system from the
standard viewpoint in four dimensions XA . We do so for $N_{f}=2$, two
dynamical flavors of equal mass, but the extension to $N_{f}=2+1$ is
straightforward. First let us look at the simplest disconnected configurations
only with the valence quarks. These are the basic configurations shown in
Figure 1. Each consists of the valence quarks and antiquark connected by the
strings and looks like a pair

Figure 1: Basic string configurations. Three strings may join at a point
called the string junction XA2 . Here and later, non-excited strings are
designated by straight lines.

of non-interacting hadrons.

To pursue this further, we assume that other configurations are constructed by
adding extra string junctions and virtual quark-antiquark pairs to the basic
configurations. Intuitively, it is clear that such a procedure will result in
configurations of higher energy. And so to some extent, the junctions and
pairs can be thought of as kinds of elementary excitations. It turns out that
for our purposes we would only need relatively simple configurations. Adding
two string junctions to the basic configurations results in the pentaquark
configurations of Figure 2.444Since they describe the genuine five-body
interactions of quarks, we call them the pentaquark configurations. As we will
see in Sec.III, such configurations make the dominant contribution to the
potential at small heavy quark separations. Because of this, we will add the
word ”compact” as a prefix. On the other hand, adding

Figure 2: Pentaquark configurations.

one $q\bar{q}$ pair results in the configurations of Figure 3. The
configurations (d) and (e) are simple

Figure 3: String configurations with one virtual pair.

modifications of the configurations (a) and (b), respectively. The
configuration (f) is obtained from those by quark exchange. It is noteworthy
that apart from the string junctions and virtual pairs other elementary
excitations may be involved. We return to this in Sec.V.

The possible transitions between the configurations arise from string
interactions. In Figure 4, we sketch four

Figure 4: Some string interactions: (a) reconnection (rearrangement); (b)
breaking; (b’) double breaking; (c) junction annihilation.

different types of interactions which will be discussed in what follows. This
is part of the big picture of QCD strings XA . Later we will introduce the
notion of a critical separation between the heavy quarks, which characterizes
each interaction, that will enable us to shed some light on the physics of QCD
strings.

###  II.2 A short account of the five-dimensional string model

In our study of the $QQqq\bar{q}$ system, we will use the formalism recently
developed in a-strb . Although we illustrate it by performing calculations in
one of the simplest AdS/QCD models, the formalism is general and, therefore,
adaptable to other models as well.

For the purposes of this paper, we consider a five-dimensional Euclidean space
with coordinates $t,x^{i},r$ and with the metric

$ds^{2}=\text{e}^{\mathsf{s}r^{2}}\frac{R^{2}}{r^{2}}\Bigl{(}dt^{2}+(dx^{i})^{2}+dr^{2}\Bigr{)}\,.$
(2.1)

Such a space represents a deformation of the Euclidean $\text{AdS}_{5}$ space
of radius $R$, with a deformation parameter $\mathsf{s}$. The boundary is at
$r=0$ and the so-called soft wall at $r=1/\sqrt{\mathsf{s}}$. Two features
make it especially attractive: relative computational simplicity and
phenomenological applications. Here let us just mention that the model of az1
yields a perfect fit to the lattice data obtained for the heavy quark
potential white .555For another perfect example, see a-3qPRD .

As for Feynman diagrams in field theory, we need the building blocks to
construct the string configurations of Figures 1-3 in five dimensions. The
first is a Nambu-Goto string governed by the action

$S_{\text{\tiny NG}}=\frac{1}{2\pi\alpha^{\prime}}\int
d^{2}\xi\,\sqrt{\gamma^{(2)}}\,.$ (2.2)

Here $\gamma$ is an induced metric, $\alpha^{\prime}$ is a string parameter,
and $\xi^{i}$ are world-sheet coordinates.

The second is a high-dimensional counterpart of the string junction called the
baryon vertex. In the AdS/CFT correspondence this vertex is supposed to be a
dynamic object. It is a five brane wrapped on an internal space $\mathbf{X}$
witten , and correspondingly the antibaryon vertex is an antibrane. Both
objects look point-like in five dimensions. In a-3qPRD it was observed that
the action for the baryon vertex, written in the static gauge,

$S_{\text{v}}=\tau_{v}\int dt\,\frac{\text{e}^{-2\mathsf{s}r^{2}}}{r}\,$ (2.3)

yields very satisfactory results, when compared to the lattice calculations of
the three-quark potential. Notice that $S_{\text{v}}$ is given by the
worldvolume of the brane if $\tau_{v}={\cal T}_{5}R\,\text{vol}(\mathbf{X})$,
with ${\cal T}_{5}$ a brane tension. Unlike AdS/CFT, we treat $\tau_{v}$ as a
free parameter to somehow account for $\alpha^{\prime}$-corrections as well as
possible impact of the other background fields.666Like in AdS/CFT, one expects
an analog of the Ramond-Ramond fields on $\mathbf{X}$. In the case of zero
baryon chemical potential, it is natural to take the action (2.3) also for the
antibaryon vertex so that $S_{\bar{\text{v}}}=S_{\text{v}}$.

Finally, following son , we introduce a background scalar field $\text{T}(r)$
which describes two light quarks of equal mass. In the present context those
are at string endpoints in the interior of five-dimensional space. The scalar
field couples to the worldsheet boundary as an open string tachyon
$S_{\text{q}}=\int d\tau e\,\text{T}$, where $\tau$ is a coordinate on the
boundary and $e$ is a boundary metric.777The use of the term tachyon also
seems particularly appropriate in virtue of instability of a QCD string. In
what follows, we consider only a constant field $\text{T}_{0}$ and worldsheets
whose boundaries are straight lines in the $t$ direction. In that case, the
action written in the static gauge is simply

$S_{\text{q}}=\text{T}_{0}R\int
dt\frac{\text{e}^{\frac{1}{2}\mathsf{s}r^{2}}}{r}\,.$ (2.4)

One can immediately recognize it as the action of a point particle of mass
${\text{T}}_{0}$ at rest.888For the parameter values used in this paper, the
masses of the light quarks can be found by fitting the string breaking
distance for the $Q\bar{Q}$ system to the lattice data of bulava . This leads
to the result $m_{u/d}=46.6\,\text{MeV}$ a-stb3q . Clearly, at zero baryon
chemical potential the same action also describes the light antiquarks, and
hence $S_{\bar{\text{q}}}=S_{\text{q}}$.

It is worth noting a visual analogy between tree level Feynman diagrams and
static string configurations. In the language of Feynman diagrams the above
building blocks play respectively the roles of propagators, vertices and
tadpoles.

##  III The $QQqq\bar{q}$-Quark System: string theory analysis in five
dimensions

We will now describe the $QQqq\bar{q}$ system. Our basic approach is as
follows. Following the hadro-quarkonium picture voloshin , we think of the
light quarks/antiquarks as clouds. Therefore, it only makes sense to speak
about the average positions of those or, equivalently, the centers of the
clouds. The heavy quarks are point-like objects inside the clouds. Our goal is
to determine the low-lying B-O potentials as a function of separation between
the heavy quarks.

We start our discussion with the basic string configurations, then continue
with the remaining ones, and finally end up with the potentials. A simple
intuitive way to see how a configuration looks like in five dimensions is to
place it on the boundary of five-dimensional space. A gravitational force
pulls the light quarks and strings into the interior, whereas the heavy
(static) quarks remain at rest. Mostly this helps, but some exceptions exist.
We will see shortly that the shape of several configurations changes with the
heavy quark separation that makes the problem more complicated.

###  III.1 The disconnected configurations (a) and (b)

First consider configuration (a). It can be interpreted as a pair of non-
interacting hadrons: a doubly heavy baryon and a pion. Clearly, the total
energy is just the sum of the rest energies of the baryon and pion if they are
infinitely far apart. It is quite surprising that such a factorization also
holds at finite separation if one takes the average over the pion (cloud)
position pion-factor . In what follows, we consider the configurations with
pions only in the sense of averaging over all possible pion positions.

In five dimensions, the configuration looks like that shown in Figure 5(a). It
consists of two parts. The lower part

Figure 5: Configurations (a) and (b) in five dimensions. The heavy quarks are
placed on the boundary at $r=0$ and separated by distance $\ell$. The light
quarks, baryon vertices and pion are at $r=r_{q}$, $r=r_{v}$ and $r=r_{2q}$,
respectively. Generically, the shape of configuration (a) changes with $\ell$.
Sketched here is the configuration for intermediate separations (see Fig.19).

corresponds to the $QQq$ system and the upper to the pion. The total energy is
the sum of the energies of each part

$E^{\text{(a)}}=E_{\text{\tiny QQq}}+E_{\text{\tiny q}\bar{\text{\tiny
q}}}\,.$ (3.1)

In the static limit, $E_{\text{\tiny QQq}}$ was computed in a-QQq . For
convenience, we include a brief description of the results in Appendix B.
$E_{\text{\tiny q}\bar{\text{\tiny q}}}$ was computed in a-QQbqqb with the
result

$E_{\text{\tiny q}\bar{\text{\tiny
q}}}=2\mathsf{n}\sqrt{\mathsf{g}\sigma}\,,\qquad\sigma=\text{e}\mathsf{g}\mathsf{s}\,.$
(3.2)

Here $\mathsf{g}=\frac{R^{2}}{2\pi\alpha^{\prime}}$,
$\mathsf{n}=\frac{\text{T}_{0}R}{\mathsf{g}}$, and $\sigma$ is the string
tension obtained in az1 from the study of the $Q\bar{Q}$ system at large pair
separation.

Now let us consider configuration (b). Again, the total energy is just the sum
of the rest energies. So,

$E^{\text{(b)}}=E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}\,.$ (3.3)

The first term is the rest energy of a heavy-light meson. In a-strb it was
shown that it can be written as

$E_{\text{\tiny Q}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}{\cal
Q}({\text{q}})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}\Bigr{)}+c\,,$
(3.4)

where the function ${\cal Q}$ is defined in Appendix A and $c$ is a
normalization constant. q is a solution to the equation

$\text{e}^{\frac{q}{2}}+\mathsf{n}(q-1)=0\,$ (3.5)

in the interval $[0,1]$. Here $q=\mathsf{s}r_{q}^{2}$. This equation is
nothing else but the force balance equation in the $r$-direction. It is
derived by varying the action $S=S_{\text{\tiny NG}}+S_{\text{q}}$ with
respect to $r_{q}$.

The second term represents the rest energy of a heavy-light baryon. It was
also computed in a-strb . In this case, one has

$E_{\text{\tiny Qqq}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}2{\cal
Q}({\text{q}})-{\cal
Q}({\text{v}})+2\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+3\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}\Bigr{)}+c\,,$
(3.6)

with v a solution to the equation

$1+3\mathsf{k}(1+4v)\text{e}^{-3v}=0\,$ (3.7)

in the interval $[0,1]$. Here $\mathsf{k}=\frac{\tau_{v}}{3\mathsf{g}}$ and
$v=\mathsf{s}r_{v}^{2}$. The above equation is the force balance equation in
the $r$-direction at $r=r_{v}$. It is derived by varying the action
$S=3S_{\text{\tiny NG}}+2S_{\text{q}}+S_{\text{v}}$ with respect to $r_{v}$.

We conclude our discussion of the basic configurations with some remarks.
First, in a-QQq it was shown that in the interval $[0,1]$ Eq.(3.7) has
solutions if and only if $\mathsf{k}$ is restricted to the range
$-\frac{\text{e}^{3}}{15}<\mathsf{k}\leq-\frac{1}{4}\text{e}^{\frac{1}{4}}$.
In particular, there exists a single solution ${\text{v}}=\frac{1}{12}$ at
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$. Second, the analysis of
configuration (b) assumes that ${\text{v}}\leq{\text{q}}$. Although this is
not true for all possible parameter values, it definitely is for those we use
to make predictions. Finally, the solutions q and v describe the light quarks
and baryon vertices and, as a consequence, are independent of the heavy quark
separation.

###  III.2 The connected configuration (c)

Now let us discuss the pentaquark configuration (c) in five dimensions. In
doing so, it is natural to suggest that if a configuration contributes to the
ground state, or at least to the first excited state, its shape is dictated by
symmetry. For the configuration at hand, there are the two most symmetric
cases: 1. All the light quarks are in the middle between the heavy quarks. 2.
The light quarks sit on top of each heavy quark and the antiquark is in the
middle.999This is in fact the antiquark-diquark-diquark scheme of maiani . We
will see that the former takes place at small enough separations between the
heavy quarks, whereas the latter at larger separations, $\ell\gtrsim
0.1\,\text{fm}$.

####  III.2.1 Small $\ell$

For small separations, the corresponding string configuration is depicted in
Figure 6. It is obtained by placing the

Figure 6: The pentaquark configuration (c) for very small $\ell$. The light
quarks and baryon vertices are placed on the $r$-axis. Here and later,
$\alpha$ indicates the tangent angle at the endpoint of string (1).

standard configuration of Figure 2(c) on the boundary of five-dimensional
space. The light quarks are indeed in the middle between the heavy ones. Here
$r_{q}$, $r_{v}$, and $r_{\bar{v}}$ are assumed to satisfy the following
condition: $r_{q}>r_{\bar{v}}>r_{v}$.101010As we will see shortly, it holds
for the parameter values we use in this work.

The total action is the sum of the Nambu-Goto actions plus the actions for the
vertices and light quarks

$S=\sum_{i=1}^{6}S_{\text{\tiny NG}}^{(i)}+3S_{\text{vert}}+3S_{\text{q}}\,.$
(3.8)

We pick the static gauge $\xi^{1}=t$ and $\xi^{2}=r$ for the Nambu-Goto
actions and consider the $x^{(i)}$’s as a function of $r$. Then the boundary
conditions take the form

$x^{(1;2)}(0)=\mp\frac{1}{2}\ell\,,\qquad
x^{(1,2,6)}(r_{v})=x^{(3,4,5,6)}(r_{\bar{v}})=x^{(3,4,5)}(r_{q})=0\,,$ (3.9)

and the action becomes111111We drop the subscript $(i)$ when it does not cause
confusion.

$S=\mathsf{g}T\biggl{(}2\int_{0}^{r_{v}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}\,\,+\int_{r_{v}}^{r_{\bar{v}}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+3\int_{r_{\bar{v}}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}}+6\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{\bar{v}}^{2}}}{r_{\bar{v}}}+3\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}\,\biggr{)}\,,$
(3.10)

where $\partial_{r}x=\frac{\partial x}{\partial r}$ and $x^{(3,4,5,6)}=const$.
The integrals represent the contributions of the strings, and the remaining
terms the contributions of the vertices and light quarks.

To find a stable configuration, one has to extremize the action with respect
to $x(r)$ describing the profiles of strings (1) and (2), and, in addition,
with respect to $r_{v}$, $r_{\bar{v}}$, and $r_{q}$ describing the locations
of the vertices and light quarks. As explained in Appendix B of a-stb3q ,
varying with respect to $x^{(1)}$ results in the following expression for the
separation distance and the energy of the configuration:

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,v)\,,$ (3.11)
$E^{\text{(c)}}=\frac{S}{T}=E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}2{\cal E}^{+}(\alpha,v)+3{\cal
Q}(q)-2{\cal Q}(\bar{v})-{\cal
Q}(v)+3\mathsf{k}\frac{\text{e}^{-2v}}{\sqrt{v}}+6\mathsf{k}\frac{\text{e}^{-2\bar{v}}}{\sqrt{\bar{v}}}+3\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}\biggr{)}+2c\,.$
(3.12)

We have also used the fact that
$\int_{a}^{b}\frac{dx}{x^{2}}\text{e}^{cx^{2}}=\sqrt{c}\bigl{(}{\cal
Q}(cb^{2})-{\cal Q}(ca^{2})\bigr{)}$. Here $v=\mathsf{s}r_{v}^{2}$,
$\bar{v}=\mathsf{s}r_{\bar{v}}^{2}$, and the functions ${\cal L}^{+}$ and
${\cal E}^{+}$ are defined in Appendix A.

It is easy to see that varying the action with respect to $r_{q}$ and
$r_{\bar{v}}$ leads to Eq.(3.5) and Eq.(3.7) (with $v$ replaced by $\bar{v}$).
On the other hand, varying with respect to $r_{v}$ leads to

$\sin\alpha=\frac{1}{2}\bigl{(}1+3\mathsf{k}(1+4v)\text{e}^{-3v}\bigr{)}\,.$
(3.13)

All these equations are nothing else but the force balance equations in the
$r$-direction.

Combining Eqs.(3.12)-(3.13) with Eq.(B.2), we can write the energy as

$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}=E_{\text{\tiny
QQq}}+2\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}{\cal Q}({\text{q}})-{\cal
Q}({\text{v}})+3\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}\biggr{)}\,.$
(3.14)

Here $E_{\text{\tiny QQq}}$ is the energy of the configuration shown in Figure
19(s). q and v are solutions respectively to Eqs.(3.5) and (3.7) in the
interval $[0,1]$. In four dimensions, this formula assumes a structure like
$QQ(qq\bar{q})$ with a color-triplet $(qq\bar{q})$.

Thus, the energy of the configuration is given in parametric form by
$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}=E_{\text{\tiny
QQqq}\bar{\text{\tiny q}}}(v)$ and $\ell=\ell(v)$. The parameter takes values
in the interval $[0,{\text{v}}]$.

####  III.2.2 Larger $\ell$

A simple numerical analysis shows that $\ell$ is an increasing function of $v$
vanishing at the origin.121212This is indeed the case for the parameter values
we use. Therefore, at some separation between the heavy quarks the baryon
vertex reaches the point $r=r_{\bar{v}}$ where it meets the other vertices, as
shown in Figure 7(a).131313For $\ell<\ell({\text{v}})$, this configuration has
higher energy than the configuration of Figure 6. Its four-dimensional
counterpart is sketched in Figure 2(c’). This defines the upper bound on the
range of $v$ in (3.11) and (3.12). We can continue with a $V\bar{V}V$ object
until we approach infinite separation.

Figure 7: Some other pentaquark configurations.

Why do we need to be concerned with such a configuration? The reason is that
from the string theory perspective it is natural to expect that the brane-
antibrane annihilation or, in other words, the string junction annihilation
occurs if the positions of the vertices coincide or close enough to each
other. This could lead to instability. However, there is a natural way out of
it: employing another configuration in which the vertices are spatially
separated, as depicted in Figure 8.141414Another reason is that it is most
symmetric. For comparison, see the configuration shown in Figure 7(b), which
does not have reflection symmetry.

Figure 8: The pentaquark configuration (c) for small $\ell$ (but larger than
$\ell({\text{v}}))$. It is labeled by (s), like the corresponding
configuration for the $QQq$-system (see Fig. 19). The forces exerted on the
baryon vertex are depicted by the arrows. $\alpha_{6}$ is the tangent angle at
the endpoint of string (6).

This configuration is governed by the action

$S=\sum_{i=1}^{7}S_{\text{\tiny NG}}^{(i)}+3S_{\text{vert}}+3S_{\text{q}}\,.$
(3.15)

In the static gauge the boundary conditions are

$x^{(1;2)}(0)=\mp\frac{\ell}{2}\,,\qquad
x^{(1,3,6;2,5,7)}(r_{v})=x(r_{q})^{(3;5)}=\mp x_{v}\,,\qquad
x^{(4,6,7)}(r_{v})=x^{(4)}(r_{q})=0\,,$ (3.16)

and the action reads

$\begin{split}S=&\mathsf{g}T\biggl{(}\int_{0}^{r_{v}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+\int_{r_{v}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+\int_{r_{v}}^{r_{\bar{v}}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}\,\biggr{)}+(x\rightarrow-x)\\\
+&\mathsf{g}T\biggl{(}\int_{r_{\bar{v}}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{\bar{v}}^{2}}}{r_{\bar{v}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}\,\biggr{)}\,.\end{split}$
(3.17)

Here we set $x^{(3,4)}=const$. Because of the reflection symmetry of the
configuration, only the $x<0$ part of the action is written explicitly in the
first line of (3.17). The integrals correspond respectively to the
contributions of strings (1), (3), and (6). The second line represents the
$x=0$ part of the action.

Given the action (3.17), it is straightforward to extremize it with respect to
the $x(r)$’s describing the profiles of strings and with respect to $x_{v}$,
$r_{v}$, $r_{\bar{v}}$, and $r_{q}$ describing the locations of the baryon
vertices and light quarks. We begin with $x_{v}$ and $r_{v}$. In this case,
the result can be conveniently written in a vector form as

$\mathbf{e}_{1}+\mathbf{e}_{3}+\mathbf{e}_{6}+\mathbf{f}_{v}=0\,,$ (3.18)

where $\mathbf{e}_{1}=\mathsf{g}w(r_{v})(-\cos\alpha,-\sin\alpha)$,
$\mathbf{e}_{3}=\mathsf{g}w(r_{v})(0,1)$,
$\mathbf{e}_{6}=\mathsf{g}w(r_{v})(\cos\alpha_{6},\sin\alpha_{6})$, and
$\mathbf{f}_{v}=(0,-3\mathsf{g}\mathsf{k}\,\partial_{r_{v}}\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}})$,
with $\alpha_{i}\leq\frac{\pi}{2}$. This is the force balance equation at the
position of the baryon vertex (see Fig. 8). Its $x$-component reduces simply
to

$\cos\alpha-\cos\alpha_{6}=0\,.$ (3.19)

The equation has a trivial solution $\alpha_{6}=\alpha$. ”Trivial” means that
in this case the strings (1) and (6) are smoothly glued together to form a
single string. The vertex in turn does not affect the string.151515In fact,
this is true only for $v={\text{v}}$. If so, then the $r$-component becomes
equivalent to Eq.(3.7). As a result, a significant simplification of the
expression (3.17) is achieved. Now it takes the form

$S=\mathsf{g}T\biggl{(}2\int_{0}^{r_{\bar{v}}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+\int_{r_{\bar{v}}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+2\int_{r_{v}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+6\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{\bar{v}}^{2}}}{r_{\bar{v}}}+3\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}\,\biggr{)}\,.$
(3.20)

Varying the action with respect to $r_{q}$ and $r_{\bar{v}}$ results in
Eqs.(3.5) and (3.13), with $v$ replaced by $\bar{v}$. The same reasoning that
led to (3.11) gives

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,\bar{v})\,.$ (3.21)

Similarly the obvious analog of Eq.(3.12) holds for the energy

$E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}2{\cal E}^{+}(\alpha,\bar{v})+3{\cal
Q}({\text{q}})-2{\cal Q}({\text{v}})-{\cal
Q}(\bar{v})+6\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+3\mathsf{k}\frac{\text{e}^{-2\bar{v}}}{\sqrt{\bar{v}}}+3\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}\biggr{)}+2c\,.$
(3.22)

Given this, we learn by using (B.2) that the same formula as in (3.14) is also
valid in the interval $[{\text{v}},{\text{q}}]$. An important point is that
now it assumes an antiquark-diquark-diquark $\bar{q}[Qq][Qq]$ structure. The
reason for the upper bound is that the antibaryon vertex goes up as the
separation between the heavy quarks is increased. This continues until it
reaches the light quark whose position is independent of the separation. So,
the string configuration becomes that shown in Figure 9(m).

Figure 9: Static configurations for intermediate (m) and large (l)
separations. The gray horizontal line in (l) represents the soft wall at
$r=1/\sqrt{\mathsf{s}}$. For string (1) a turning point is located at
$(x_{0},r_{0})$.

Again, a short numerical calculation shows that $\ell$ reaches its maximum
value at $v={\text{q}}$ but remains finite. Because of this, for larger
separations one needs to analyze the configuration of Figure 9(m). By
essentially the same arguments that we have given for the configuration of
Figure 8, the strings (1) and (2), which are stretched between the heavy
quarks and antibaryon vertex, are smooth. In that case, the action can be
obtained from (3.20) by simply dropping the contribution of string (4) out.
So,

$S=\mathsf{g}T\biggl{(}2\int_{0}^{r_{\bar{v}}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+2\int_{r_{v}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+6\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{\bar{v}}^{2}}}{r_{\bar{v}}}+2\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{\bar{v}}^{2}}}{r_{\bar{v}}}\,\biggr{)}\,.$
(3.23)

As before, varying the action with respect to $r_{q}$ leads to Eq.(3.5) and
with respect to $r_{v}$ to Eq.(3.7). But when one varies it with
$r_{\bar{v}}$, the resulting formula can be written as

$\sin\alpha=\frac{1}{2}\bigl{(}\mathsf{n}(1-\bar{v})\text{e}^{-\frac{1}{2}\bar{v}}+3\mathsf{k}(1+4\bar{v})\text{e}^{-3\bar{v}}\bigr{)}\,.$
(3.24)

Since $\alpha$ is non-negative by construction, this places an upper bound on
$\bar{v}$. It is defined by the equation $\alpha=0$, or equivalently, by

$\mathsf{n}(1-\bar{v})+3\mathsf{k}(1+4\bar{v})\text{e}^{-\frac{5}{2}\bar{v}}=0\,.$
(3.25)

We denote the solution of this equation in the interval $[0,1]$ as
${\text{v}}_{\text{\tiny 0}}$.

Clearly, Eq.(3.21) holds for the heavy quark separation. It follows then from
(3.23) that the energy of the configuration is

$E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}2{\cal E}^{+}(\alpha,\bar{v})+2{\cal
Q}({\text{q}})-2{\cal
Q}({\text{v}})+6\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+3\mathsf{k}\frac{\text{e}^{-2\bar{v}}}{\sqrt{\bar{v}}}+2\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\bar{v}}}{\sqrt{\bar{v}}}\biggr{)}+2c\,.$
(3.26)

It is straightforward now, using the formula (B.3), to see that the relation
between $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ and $E_{\text{\tiny QQq}}$
is valid in the interval $[{\text{q}},{\text{v}}_{\text{\tiny 0}}]$.

Since $\ell({\text{v}}_{\text{\tiny 0}})$ is finite, the question arises of
whether larger separations can be reached. The short answer to this question
is yes, they can. For this, one needs to consider a change of the sign of
$\alpha$. If so, then the configuration profile becomes convex near $x=0$, as
depicted in Figure 9(l). For some $\alpha$ the strings reach the soft wall
that corresponds to infinite separation between the heavy quarks.

The configuration is governed by the same action as before. The expressions
for the separation distance and energy are simply obtained by respectively
replacing ${\cal L}^{+}$ and ${\cal E}^{+}$ with ${\cal L}^{-}$ and ${\cal
E}^{-}$, as follows from the analysis in Appendix B of a-stb3q . So, we have

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{-}(\lambda,\bar{v})\,$ (3.27)

and

$E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}2{\cal E}^{-}(\lambda,\bar{v})+2{\cal
Q}({\text{q}})-2{\cal
Q}({\text{v}})+6\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+3\mathsf{k}\frac{\text{e}^{-2\bar{v}}}{\sqrt{\bar{v}}}+2\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\bar{v}}}{\sqrt{\bar{v}}}\biggr{)}+2c\,.$
(3.28)

The functions ${\cal L}^{-}$ and ${\cal E}^{-}$ are given in Appendix A. The
dimensionless variable $\lambda$ is defined by $\lambda=\mathsf{s}r_{0}^{2}$,
with $r_{0}=\max r(x)$ as shown in Figure 9(l). Importantly, $\lambda$ can be
expressed in terms of $\bar{v}$ as a-QQq

$\lambda=-\text{ProductLog}\biggl{[}-\bar{v}\text{e}^{-\bar{v}}\Bigl{(}1-\frac{1}{4}\Bigl{(}3\mathsf{k}(1+4\bar{v})\text{e}^{-3\bar{v}}+\mathsf{n}(1-\bar{v})\text{e}^{-\frac{1}{2}\bar{v}}\Bigr{)}^{2}\,\Bigr{)}^{-\frac{1}{2}}\,\biggr{]}\,.$
(3.29)

Here $\text{ProductLog}(z)$ is the principal solution for $w$ in
$z=w\,\text{e}^{w}$ wolfram .

The parameter $\bar{v}$ now varies from ${\text{v}}_{\text{\tiny 0}}$ to
${\text{v}}_{\text{\tiny 1}}$. The upper bound is found by solving the
equation $\lambda=1$, or equivalently the equation

$2\sqrt{1-\bar{v}^{2}\text{e}^{2(1-\bar{v})}}+3\mathsf{k}(1+4\bar{v})\text{e}^{-3\bar{v}}+\mathsf{n}(1-\bar{v})\text{e}^{-\frac{1}{2}\bar{v}}=0\,,$
(3.30)

in the interval $[0,1]$. The reasoning for this is that since ${\cal
L}^{-}(\lambda,\bar{v})$ is singular at $\lambda=1$, ${\text{v}}_{\text{\tiny
1}}$ corresponds to infinite separation between the heavy quarks.

We conclude the discussion of configuration (c) with a brief summary of our
analysis. First, $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ is a piecewise
function of $\ell$. The different parts of its domain are described by the
different string configurations. Second, for $\ell>\ell({\text{v}})$ our model
provides an explicit realization of the antiquark-diquark-diquark scheme of
the pentaquark maiani . In that case the formula (3.14) establishes the
relation between the energies of a compact pentaquark and a doubly heavy
baryon in the heavy quark limit.

####  III.2.3 The limiting cases

It is instructive to examine an asymptotic behavior of $E_{\text{\tiny
QQqq}\bar{\text{\tiny q}}}$ for small and large $\ell$. These are easy to
understand because the relation (3.14) between $E_{\text{\tiny
QQqq}\bar{\text{\tiny q}}}$ and $E_{\text{\tiny QQq}}$ is valid for all
separations.

The behavior for small $\ell$ can be read off from Eq.(B.5). We get exactly
what we expect from heavy quark-diquark symmetry wise , namely

$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}(\ell)=E_{\text{\tiny
QQ}}(\ell)+E_{\bar{\text{\tiny Q}}\text{\tiny qq}\bar{\text{\tiny q}}}\,.$
(3.31)

Here $E_{\text{\tiny QQ}}$ is the heavy quark-quark potential (in the
antitriplet channel) and $E_{\bar{\text{\tiny Q}}\text{\tiny
qq}\bar{\text{\tiny q}}}$ is the rest energy of a heavy-light tetraquark. At
zero baryon chemical potential $E_{\bar{\text{\tiny Q}}\text{\tiny
qq}\bar{\text{\tiny q}}}=E_{\text{\tiny Qq}\bar{\text{\tiny
q}}\bar{\text{\tiny q}}}$, where

$E_{\text{\tiny Qq}\bar{\text{\tiny q}}\bar{\text{\tiny
q}}}=3\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}{\cal
Q}({\text{q}})-\frac{2}{3}{\cal
Q}({\text{v}})+2\mathsf{k}\frac{\text{e}^{-2{{\text{v}}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}\,\biggr{)}+c\,,$
(3.32)

as described in a-QQqq .

Similarly, the behavior for large $\ell$ can be read off from Eq.(B.7). So, we
have

$E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\sigma\ell+2\mathsf{g}\sqrt{\mathsf{s}}\biggl{(}{\cal
Q}({\text{q}})-{\cal
Q}({\text{v}})+3\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}-I_{\text{\tiny
QQq}}\biggr{)}+2c+o(1)\,.$ (3.33)

Here $\sigma$ is the same string tension as in (3.2).

####  III.2.4 Gluing the pieces together

Now let us discuss the gluing of all the branches of $E_{\text{\tiny
QQqq}\bar{\text{\tiny q}}}(\ell)$. For this, we need to specify the model
parameters. Here we use one of the two parameter sets suggested in a-strb
which is mainly a result of fitting the lattice QCD data to the string model
we are considering. The value of $\mathsf{s}$ is fixed from the slope of the
Regge trajectory of $\rho(n)$ mesons in the soft wall model with the geometry
(2.1), and as a result, one gets $\mathsf{s}=0.45\,\text{GeV}^{2}$ a-q2 .
Then, fitting the value of the string tension $\sigma$ defined in (3.2) to its
value in bulava gives $\mathsf{g}=0.176$. The parameter $\mathsf{n}$ is
adjusted to reproduce the lattice result for the string breaking distance in
the $Q\bar{Q}$ system. With $\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny
Q}}}=1.22\,\text{fm}$ for the $u$ and $d$ quarks bulava , one gets
$\mathsf{n}=3.057$ a-strb .

In principle, the value of $\mathsf{k}$ could be adjusted to fit the lattice
data for the three-quark potential, as done in a-3qPRD for pure $SU(3)$ gauge
theory, but there are no lattice data available for QCD with two light quarks.
There are still two special options: $\mathsf{k}=-0.102$ motivated by
phenomenology161616Note that $\mathsf{k}=-0.102$ is a solution to the equation
$\alpha_{\text{\tiny QQ}}(\mathsf{k})=\frac{1}{2}\alpha_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$ which follows from the phenomenological rule
$E_{\text{\tiny QQ}}(\ell)=\frac{1}{2}E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}(\ell)$ in the limit $\ell\rightarrow 0$. and $\mathsf{k}=-0.087$ obtained
from the lattice data for pure gauge theory a-3qPRD . However, both are out of
the range of allowed values for $\mathsf{k}$ as follows from the analysis of
Eq.(3.7). In this situation it is reasonable to pick
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$, as the closest to those.

Having fixed the model parameters, we can immediately perform some simple but
important calculations. First let us check that ${\text{q}}>{\text{v}}$. That
is, our construction of the string configurations makes sense. From Eqs.(3.5)
and (3.7), we find that ${\text{q}}=0.566$ and ${\text{v}}=\frac{1}{12}$, as
desired. In addition, from (3.25) and (3.30), we get ${\text{v}}_{\text{\tiny
0}}=0.829$ and ${\text{v}}_{\text{\tiny 1}}=0.930$. Second, with this value of
$v$, we can immediately calculate the smallest separation between the heavy
quarks for the configuration shown in Figure 8. So,
$\ell({\text{v}})=0.106\,\text{fm}$. It is quite surprising that the
antiquark-diquark-diquark scheme of maiani arises at such small separations.

It is now straightforward to plot $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$
as a function of $\ell$. The result is shown in Figure 10. From this Figure it
is seen that all the pieces of the function

Figure 10: $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ vs $\ell$. The dashed
curve corresponds to the configuration of Figure 6, whereas the solid curve to
the configurations of Figures 8 and 9 for which the the antiquark-diquark-
diquark scheme holds. Here and later, $c=0.623\,\text{GeV}$.

are smoothly glued together. Moreover, $E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}(\ell)$ is with good approximation linear for separations larger than
$0.4\,\text{fm}$.

###  III.3 The disconnected configurations (d)-(f)

We start with configuration (d). It is obtained from configuration (a) by
adding a $q\bar{q}$ pair (pion). So, by analogy with Figure 5(a), we place the
pion at $r=r_{2q}$. The resulting configuration is shown schematically in
Figure 11(d).

Figure 11: Configurations (d), (e), and (f) in five dimensions. The heavy
quarks in (d) are placed at an intermediate separation distance (see Fig.19).

Although there are no calculations on the lattice for two pions, we will
assume that adding another pion and averaging over its position, leads to an
energy increase by $E_{\text{\tiny q}\bar{\text{\tiny q}}}$. If so, then the
energy is

$E^{\text{(d)}}=E_{\text{\tiny QQq}}+2E_{\text{\tiny q}\bar{\text{\tiny
q}}}\,,$ (3.34)

with $E_{\text{\tiny QQq}}$ described in Appendix B and $E_{\text{\tiny
q}\bar{\text{\tiny q}}}$ given by (3.2).

In a similar way, configuration (e) is obtained from configuration (b). It is
as shown in Figure 11 (e). By the same assumption that we have made previously
in our treatment of configuration (d), the energy is given by

$E^{\text{(e)}}=E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\,,$ (3.35)

where the rest energies were defined in Eqs. (3.2), (3.4), and (3.6).

From the five-dimensional perspective, configuration (f) looks like that shown
in Figure 11(f). It consists of the two heavy-light mesons which we have
discussed above and a three-quark state (nucleon). Such a state requires some
explanation. In the static limit it is natural to guess, by analogy with the
case of $q\bar{q}$, that all the strings connecting the quarks collapse into a
point as shown in the Figure.171717This still leaves another choice in which
the strings don’t collapse, but get stretched along the $r$-axis so that the
quarks and vertex are spatially separated. The problem with such a
configuration is that it is unstable at
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$. If so, then the corresponding
action is just the sum of the actions of the vertex and light quarks.
Explicitly,

$S=3\frac{\mathsf{g}T}{r_{3q}}\Bigl{(}\mathsf{k}\text{e}^{-2\mathsf{s}r_{3q}^{2}}+\mathsf{n}\text{e}^{\frac{1}{2}\mathsf{s}r_{3q}^{2}}\Bigr{)}\,.$
(3.36)

The force balance equation obtained by varying $r_{3q}$ in (3.36) is

$\mathsf{k}(1+4q_{3})+\mathsf{n}(1-q_{3})\text{e}^{\frac{5}{2}q_{3}}=0\,.$
(3.37)

Here $q_{3}=\mathsf{s}r_{3q}^{2}$. The nucleon rest energy is thus

$E_{\text{\tiny
qqq}}=3\mathsf{g}\sqrt{\frac{\mathsf{s}}{{\text{q}}_{3}}}\Bigl{(}\mathsf{k}\text{e}^{-2{\text{q}}_{3}}+\mathsf{n}\text{e}^{\frac{1}{2}{\text{q}}_{3}}\Bigr{)}\,,$
(3.38)

where ${\text{q}}_{3}$ is a solution to Eq.(3.37) in the interval
$[0,1]$.181818 Numerically, ${\text{q}}_{3}=0.953$ at
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$ and $\mathsf{n}=3.057$.

This configuration can be interpreted as a pair of mesons in a nucleon cloud.
Like in the case of configuration (e), we assume averaging over a nucleon
position and therefore expect that

$E^{\text{(f)}}=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
qqq}}\,,$ (3.39)

with $E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ given by Eq.(3.4).

###  III.4 What we have learned

It is instructive to see concretely how the energies of the configurations
depend on the separation between the heavy quarks and which configurations
contribute to the two low-lying B-O potentials. In Figure 12 we plot the $E$’s
for our

Figure 12: Various $E$ vs $\ell$ plots. The curves relevant for $V_{0}$ and
$V_{1}$ are depicted in solid black.

parameter values. These show that the energies of the ground and first excited
states are determined only by the contributions from configurations (a)-(c)
and (f). In other words, $V_{0}=\min\\{E_{\text{\tiny QQq}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{Qqq}\\}$ and
$V_{1}=\min\\{E_{\text{\tiny QQqq}\bar{\text{\tiny q}}},E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}},E_{\text{\tiny
QQq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}},2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny qqq}}\\}$.191919Note that in these
formulas, one has first to take a minimum to get $V_{0}$ and then, with that
in mind, $V_{1}$.

The essential feature of the pattern of Figure 12 is the emergence of three
length scales which separate different configurations, or in other words
different descriptions. This enables us to better understand how the quarks
are organized inside the $QQqq\bar{q}$ system.

The first is a scale which refers to the process of string reconnection. It
goes like this: $QQq+q\bar{q}\rightarrow Qqq+Q\bar{q}$ for $V_{0}$ and vice
versa for $V_{1}$, if $\ell$ is increased. In the case of $V_{0}$ the physical
meaning of this scale is that the system can be though of as a doubly heavy
baryon in a pion cloud for smaller quark separations and, respectively, as a
pair of heavy-light hadrons for larger ones. To make this more quantitative,
we define a critical separation distance $l_{\text{\tiny Qq}}$ by

$E_{\text{\tiny QQq}}(l_{\text{\tiny Qq}})+E_{q\bar{q}}=E_{\text{\tiny
Qqq}}+E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (3.40)

If reconnection occurs at small $\ell$ (as in Fig.12), then by using (B.5)
together with (3.2), (3.4) and (3.6), we get

$l_{\text{\tiny
Qq}}=\frac{\mathsf{g}\sqrt{\mathsf{s}}}{\boldsymbol{\sigma}_{\text{\tiny
QQ}}}\Bigl{(}{\cal Q}({\text{q}})-\frac{1}{2}{\cal
Q}({\text{v}})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}-\mathsf{n}\sqrt{\text{e}}+\frac{3}{2}\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}\Bigr{)}+\sqrt{\frac{\alpha_{\text{\tiny
QQ}}}{\boldsymbol{\sigma}_{\text{\tiny
QQ}}}+\frac{\mathsf{g}^{2}\mathsf{s}}{\boldsymbol{\sigma}_{\text{\tiny
QQ}}^{2}}\Bigl{(}{\cal Q}({\text{q}})-\frac{1}{2}{\cal
Q}({\text{v}})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}-\mathsf{n}\sqrt{\text{e}}+\frac{3}{2}\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}\Bigr{)}^{2}}\,.$
(3.41)

For our parameter values, $l_{\text{\tiny Qq}}\approx 0.203\,\text{fm}$. This
value is rather close to the value of $0.219\,\text{fm}$ obtained for
$l_{\text{\tiny Qq}}$ in the $Q\bar{Q}q\bar{q}$ system a-QQbqqb .

The second scale is related to the process of string junction annihilation
which occurs at the level of the first excited state. More specifically, it
goes like this: $QQqq\bar{q}\rightarrow Q\bar{q}+Qqq$, if $\ell$ is increased.
In this case we define a critical separation distance by

$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}(\boldsymbol{\ell}_{\text{\tiny
QQqq}\bar{\text{\tiny q}}})=E_{\text{\tiny Q}\bar{\text{\tiny
q}}}+E_{\text{\tiny Qqq}}\,.$ (3.42)

It implies that at the first excited level the system can be thought of mainly
as a compact pentaquark if $\ell\leq\boldsymbol{\ell}_{\text{\tiny
QQqq}\bar{\text{\tiny q}}}$. As seen from Figure 12,
$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ is of order
$0.2\,\text{fm}$. For this range of $\ell$ values, the function
$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}(\ell)$ is well approximated by
Eq.(3.31). Then a simple calculation shows that

$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\frac{\mathsf{g}\sqrt{\mathsf{s}}}{2\boldsymbol{\sigma}_{\text{\tiny
QQ}}}\Bigl{(}{\cal
Q}({\text{v}})-3\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}\Bigr{)}+\sqrt{\frac{\alpha_{\text{\tiny
QQ}}}{\boldsymbol{\sigma}_{\text{\tiny
QQ}}}+\frac{\mathsf{g}^{2}\mathsf{s}}{4\boldsymbol{\sigma}_{\text{\tiny
QQ}}^{2}}\Bigl{(}{\cal
Q}({\text{v}})-3\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}\Bigr{)}^{2}}\,,$
(3.43)

where in the last step we used (3.4) and (3.6). One important point is that
$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ is finite and
scheme independent ($c$ drops out). The other point is that it depends on v,
which describes the position of the baryon vertices in the bulk, rather than
on q which describes the position of the light quarks. This suggests that
$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ is related to
gluonic degrees of freedom, as expected from annihilation of string junctions
made of gluons. A simple calculation shows that

$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}\approx
0.184\,\text{fm}\,.$ (3.44)

This value is equal to the value obtained for the $QQ\bar{q}\bar{q}$ system in
a-QQqq and only slightly below the value $\boldsymbol{\ell}_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}\approx
0.196\,\text{fm}$ for the $Q\bar{Q}q\bar{q}$ system a-QQbqqb . Thus, our
estimate provides further evidence that in systems with two heavy quark
sources the process of string junction annihilation takes place at relatively
small heavy quark separations, of order $0.2\,\text{fm}$. Of course the
question arises of whether this scale is universal. We will return to this
question in Sec.V, after gaining some information about the screening lengths.

The third scale is associated to the transition: $QQq+q\bar{q}\rightarrow
2Q\bar{q}+qqq$ which also occurs at the level of the first excited state. Such
a transition can be subdivided into two elementary ones, as those of Figure 4.
The first is the process of string reconnection: $QQq+q\bar{q}\rightarrow
Qqq+Q\bar{q}$, and the second is the process of string breaking:
$Qqq+Q\bar{q}\rightarrow 2Q\bar{q}+qqq$. The corresponding equation for a
critical distance reads

$E_{\text{\tiny QQq}}(\ell_{\text{\tiny QQq}}^{(\text{\tiny q}\bar{\text{\tiny
q}})})+E_{\text{\tiny q}\bar{\text{\tiny q}}}=2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny qqq}}\,.$ (3.45)

Since this scale is large enough, as seen in Figure 12, the solution can be
found just by using the asymptotic expansion (B.7). So,

$\ell_{\text{\tiny QQq}}^{(\text{\tiny q}\bar{\text{\tiny
q}})}=\frac{2}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})+\mathsf{n}\Bigl{(}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}-\text{e}^{\frac{1}{2}}\Bigr{)}+\frac{3}{2\sqrt{{\text{q}}_{3}}}\Bigl{(}\mathsf{k}\text{e}^{-2{\text{q}}_{3}}+\mathsf{n}\text{e}^{\frac{1}{2}{\text{q}}_{3}}\Bigr{)}+{\cal
I}_{\text{\tiny QQq}}\biggr{)}\,,$ (3.46)

where we have used Eqs. (3.2), (3.4) and (3.38). A numerical calculation shows
that $\ell_{\text{\tiny QQq}}^{(\text{\tiny q}\bar{\text{\tiny q}})}\approx
1.22\,\text{fm}$. This value is surprisingly close to that found for the
string breaking distance in the $Q\bar{Q}$ system bulava .

##  IV More Detail On the Potentials

###  IV.1 An issue with $E_{\text{\tiny q}\bar{\text{\tiny q}}}$ and
$E_{\text{\tiny qqq}}$

With the help of (3.2) and (3.38), it is straightforward to estimate the rest
energies of the pion and nucleon. Using the parameter values of Sec.III, one
finds that $E_{\text{\tiny q}\bar{\text{\tiny q}}}=1.190\,\text{GeV}$ and
$E_{\text{\tiny qqq}}=1.769\,\text{GeV}$. These values are considerably larger
than the values of $280\,\text{MeV}$ and $1060\,\text{GeV}$ used in the
lattice calculations bulava .202020This is a reasonable value of the nucleon
mass in the case of two flavors because $E_{\text{\tiny
qqq}}=1060\,\text{GeV}$ at $E_{\text{\tiny q}\bar{\text{\tiny
q}}}=285\,\text{MeV}$ nucl .  The problem is that the current model is not
suitable for describing light hadrons as it was originally developed for
applications in the heavy quark limit. In the context of string theory on AdS-
like geometries, this means that at least one quark is needed to be placed on
the boundary of five-dimensional space.

A possible way out is to treat $E_{\text{\tiny q}\bar{\text{\tiny q}}}$ and
$E_{\text{\tiny qqq}}$ as model parameters. For $E_{\text{\tiny
q}\bar{\text{\tiny q}}}=280\,\text{MeV}$, the corresponding $E$’s are plotted
in Figure 13. There are two main differences with the plots of Figure 12.
First, the scale of $l_{\text{\tiny Qq}}$ is now large, about $1\,\text{fm}$.
Second, configuration (e), rather than (f), is relevant for determining the
potential $V_{1}$. So, $V_{1}=\min\\{E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}},E_{\text{\tiny QQq}}+E_{\text{\tiny q}\bar{\text{\tiny
q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\\}$. Note that there is no change
in the behavior of $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ and
$E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}}$. Thus, the value
of $\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ remains the
same and the compact pentaquark is expected at the level of the first excited
state.

Figure 13: Sketched here are the two low-lying B-O potentials of the
$QQqq\bar{q}$ system. The dashed lines indicate the relevant $E$’s.

.

Since $l_{\text{\tiny Qq}}$ is large, one can use the large $\ell$ asymptotic
for $E_{\text{\tiny QQq}}$ to solve Eq. (3.40). With the help of (B.7), one
finds that

$l_{\text{\tiny Qq}}=\frac{3}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})-\frac{1}{3}{\cal
Q}({\text{v}})+\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+\frac{2}{3}I_{\text{\tiny
QQq}}-\frac{E_{\text{\tiny q}\bar{\text{\tiny
q}}}}{3\mathsf{g}\sqrt{\mathsf{s}}}\biggr{)}\,.$ (4.1)

For $E_{\text{\tiny q}\bar{\text{\tiny q}}}=280\,\text{MeV}$, $l_{\text{\tiny
Qq}}\approx 1.001\,\text{fm}$. Interestingly, this value differs only by
$3.7\%$ from the value obtained within the $Q\bar{Q}q\bar{q}$ system a-QQbqqb
, where the string reconnection process goes like this:
$Q\bar{Q}+q\bar{q}\rightarrow Q\bar{q}+q\bar{Q}$.

Because of the relevance of configuration (e), the third scale is now related
to the process of string breaking: $QQq+q\bar{q}\rightarrow
Q\bar{q}+Qqq+q\bar{q}$, which obviously reduces to the process in the $QQq$
system, namely $QQq\rightarrow Q\bar{q}+Qqq$. The corresponding scale is given
by (B.11). Numerically, $\ell_{\text{\tiny QQq}}=1.257\,\text{fm}$ a-QQq .

Just like in the case of the $Q\bar{Q}q\bar{q}$ system, our results indicate
that the following scale hierarchy is met

$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}<l_{\text{\tiny
Qq}}<\ell_{\text{\tiny QQq}}\,.$ (4.2)

In other words, the scale of string junction annihilation is the smallest and
that of string breaking is the largest. One may ask what happens at the
physical value of the pion mass. We expect that
$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ remains the
smallest so that $\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}<l_{\text{\tiny Qq}}$, $\ell_{\text{\tiny QQq}}$. Of course, this is not
the whole answer to this question, but it’s certainly an important piece of
it.

Having understood the relevant configurations, one can gain some insight into
the two low-lying B-O potentials. A natural way for doing so is to consider a
model Hamiltonian similar to that used in lattice QCD to study the phenomenon
of string breaking FK . For the problem at hand it is

${\cal H}(\ell)=\begin{pmatrix}\,E_{\text{\tiny QQq}}(\ell)+E_{\text{\tiny
q}\bar{\text{\tiny q}}}&{}&{}&{}&{}\,\\\ \,{}&E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}}&{}&{}&\Theta_{ij}\,\\\
\,\Theta_{ij}&{}&E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}(\ell)&{}&{}\,\\\
\,{}&{}&{}&{}&E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\,\\\ \end{pmatrix}\,,$ (4.3)

where the off-diagonal elements describe the strength of mixing between the
four states (string configurations). The potentials of interest are the two
smallest eigenvalues of the matrix ${\cal H}$.

Unlike lattice QCD, where the Hamiltonian could in principle be extracted from
a correlation matrix, it is not clear how to compute the off-diagonal elements
within the string models. This makes it difficult to see precisely how the
potentials look like. Nevertheless, we can learn from our experience with the
other quark systems about the order of magnitude of the $\Theta$’s near the
transition points.212121For example, one can take these $\Theta$’s to be of
order $47\,\text{MeV}$, as in the case of the $Q\bar{Q}$ system bulava . With
the help of this, the picture will then look more like what is shown in Figure
13. The compact pentaquark configuration contributes dominantly to the
potential $V_{1}$ at heavy quark separations smaller than $0.2\,\text{fm}$.

###  IV.2 Simple approximations to $V_{1}$

For practical purposes, the parametric expressions for the potentials may look
somewhat cumbersome. On the other hand, a simple parametrization motivated by
lattice calculations was proposed for the $V_{0}$ potential of the
$QQ\bar{q}\bar{q}$ system in wagner . From a string point of view the
description includes two distinct configurations, one associated with a
compact tetraquark and the other with a pair of non-interacting mesons a-QQqq
. This is similar to what we have seen above for $V_{1}$. Indeed, at
separations $\ell\lesssim 0.6\,\text{fm}$, this potential is determined in
terms of the two string configurations, one of which corresponds to the
compact pentaquark. The physical reason for this similarity is, in fact, the
underlying heavy quark-diquark symmetry.222222We return to this in Sect.V.

Thus, it is reasonable, following wagner , to suggest that for $\ell\lesssim
0.6\,\text{fm}$

$V_{1}=-\frac{\alpha}{\ell}\exp\Bigl{\\{}-\frac{\ell^{\,p}}{d^{\,p}}\,\Bigl{\\}}\,+E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}}\,,$ (4.4)

with parameters $\alpha$, $d$, and $p$. To express these parameters in terms
of ours, we use the small $\ell$ expansion (3.31) and solve for the unknown
coefficients. As a result, except for the constant term, we get

$\alpha=\alpha_{\text{\tiny QQ}}\,,\qquad d=\sqrt{\frac{\alpha_{\text{\tiny
QQ}}}{\boldsymbol{\sigma}_{\text{\tiny QQ}}}}\,,\qquad p=2\,.$ (4.5)

It is no coincidence that they are the same as those of the $QQ\bar{q}\bar{q}$
a-QQqq system. This is a consequence of heavy quark-diquark symmetry. In
particular, a numerical calculation gives for the screening length $d\approx
0.200\,\text{fm}$.

For the given range of $\ell$, the model Hamiltonian is well approximated by a
$2\times 2$ submatrix of ${\cal H}$. Its diagonal elements are given by
$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ and $E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}}$, and an off-diagonal element by
$\Theta_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$. This leads to the following
expression for $V_{1}$:

$V_{1}=\frac{1}{2}\Bigl{(}E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}+E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}\Bigr{)}-\sqrt{\frac{1}{4}\Bigl{(}E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}-E_{\text{\tiny Q}\bar{\text{\tiny q}}}-E_{\text{\tiny
Qqq}}\Bigr{)}^{2}+\Theta_{\text{\tiny QQqq}\bar{\text{\tiny q}}}^{2}}\,.$
(4.6)

It is now interesting to compare it to the approximation (4.4). Figure 14
shows the graphs for both cases. We set

Figure 14: $V_{1}$ vs $\ell$ for $\ell\lesssim 0.6\,\text{fm}$. The solid
curve corresponds to (4.6), whereas the dashed and dotted curves to the
approximations (4.4) and (4.7).

$\Theta_{\text{\tiny QQqq}\bar{\text{\tiny q}}}=47\,\text{MeV}\,\text{GeV}$ to
have a small deviation between the curves.

There are two observations to be made here. The first is a falloff at large
$\ell$. It is power-law for (4.6), but exponential for (4.4).232323The
difference between these is slightly visible around $\ell=0.3\,\text{fm}$. The
reason for the power-law falloff is the choice of $\Theta=const$. In fact, one
can get the exponential falloff by taking $\Theta$ as a Gaussian function, but
then one additional parameter is required, a Gaussian width. The second is a
more visible deviation between the solid and dashed curves in the interval
$0.05\,\text{fm}\lesssim\ell\lesssim 0.23\,\text{fm}$. The appearance of this
deviation can be ascribed to the mismatch of the constant terms in the small
$\ell$ expansions of (4.4) and (4.6). The point is that the constant term
dictated by quark-diquark symmetry for (4.4) is equal to
$c+E_{\bar{\text{\tiny Q}}\text{\tiny qq}\bar{\text{\tiny q}}}$, with $c$
coming from the heavy quark-quark potential in (B.5). If so, then the improved
approximation to $V_{1}$ takes the form

$V_{1}=-\frac{\alpha_{\text{\tiny
QQ}}}{\ell}\exp\Bigl{\\{}-\frac{\boldsymbol{\sigma}_{\text{\tiny
QQ}}}{\alpha_{\text{\tiny QQ}}}\,\ell^{2}\Bigl{\\}}\,+c+E_{\bar{\text{\tiny
Q}}\text{\tiny qq}\bar{\text{\tiny q}}}\,.$ (4.7)

As seen from the Figure, now there is no visible deviation in the above
interval, but instead another deviation comes out for larger $\ell$. This is
so because of the difference between the constant terms in (4.4) and
(4.7).242424Numerically, $E_{\bar{\text{\tiny Q}}\text{\tiny
qq}\bar{\text{\tiny q}}}+c-E_{\text{\tiny Q}\bar{\text{\tiny
q}}}-E_{\text{\tiny QQq}}\approx 41\,\text{MeV}$.

The lesson to learn from this example is that a consistent parametrization
should be based on a function having different constant terms in its small and
large $\ell$ expansions, as for example in (4.6). Hopefully it will be
possible eventually to determine $V_{1}$ reliably by computer simulations.

###  IV.3 Another view on the $QQqq\bar{q}$ system

According to one of our assumptions made in Sec.II, the string configurations
corresponding to excited states are constructed by adding $q\bar{q}$ pairs to
the basic configurations corresponding to the ground state. This implies that
the $QQqq\bar{q}$ system can be thought of as a subsystem of the $QQq$ system
obtained by adding one pair.252525We did not adopt this viewpoint in Sec.III,
in part because it might have made the analysis more tedious. Here we briefly
explore such an idea using the results of the previous Section and Appendix B.
This will also enable us to show some of the subtleties of the $QQq$ system
and thereby extend the analysis of Ref.a-QQq .

We start with the ground state energy. As discussed in Appendix B, from the
string theory point of view, it is described in terms of the configurations of
Figure 18 and the corresponding potential is defined by
$V_{0}=\min\\{E_{\text{\tiny QQq}},E_{\text{\tiny Q}\bar{\text{\tiny
q}}}+E_{\text{\tiny Qqq}}\\}$. The transition between these two occurs because
of string breaking. The potential, which is the same as in Figure 20, is shown
in Figure 15 below.262626Like in Figure 13, the resulting plots of the
$V_{i}$’s require a caveat. Here too, we are not aware precisely of any
$\Theta_{ij}$.

Figure 15: Sketched here are the three low-lying B-O potentials of the $QQq$
system. Various $E$ vs $\ell$ plots are indicated by dashed curves and
horizontal lines. Here $E_{\text{\tiny q}\bar{\text{\tiny
q}}}=280\,\text{MeV}$ and $E_{\text{\tiny qqq}}=1060\,\text{GeV}$.

The energy of the first excited state is described in terms of four
configurations. These are the configurations of Figures 1, 3(e), and 18(a).
So, we have $V_{1}=\min\\{E_{\text{\tiny QQq}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}},E_{\text{\tiny QQq}},E_{\text{\tiny Q}\bar{\text{\tiny
q}}}+E_{\text{\tiny Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\\}$. As seen
from the Figure, for $\ell\lesssim 1.15\,\text{fm}$ the potential looks
similar to what is shown in Figure 13 for the ground state energy. Thus, the
transition at $\ell=1.001\,\text{fm}$ is due to string reconnection, with the
formula (4.1) remaining valid. For larger separations there is no similarity
between them. The transition at $\ell=1.257\,\text{fm}$ is now interpreted as
a result of fusion of two strings: $Q\bar{q}+Qqq\rightarrow QQq$. This process
is inverse to the process of string breaking, which is discussed in Appendix
B. Because of this, the formula (B.11) holds true. The final transition near
$\ell=1.51\,\text{fm}$ is an example of double string breaking:
$QQq\rightarrow Q\bar{q}+Qqq+q\bar{q}$, as sketched in Figure
4(b’).272727Another possibility could be that two different strings break
down. One of which is attached to a heavy quark, and the other to a light
quark. However, from the four dimensional perspective this doesn’t seem
possible because the light quark sits on top of a string junction so that the
corresponding string is shrunk into a point. By analogy with what we did for
the other transitions, we define a critical separation distance by

$E_{\text{\tiny QQq}}(\ell_{\text{\tiny QQq}}^{(2)})=E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny
q}}}\,.$ (4.8)

For large $\ell$, this equation can be easily solved by using the asymptotic
expression (B.7). The solution is given by

$\ell_{\text{\tiny
QQq}}^{(2)}=\frac{3}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})-\frac{1}{3}{\cal
Q}({\text{v}})+\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+\frac{2}{3}I_{\text{\tiny
QQq}}+\frac{E_{\text{\tiny q}\bar{\text{\tiny
q}}}}{3\mathsf{g}\sqrt{\mathsf{s}}}\biggr{)}\,.$ (4.9)

A simple calculation shows that for $E_{\text{\tiny q}\bar{\text{\tiny
q}}}=280\,\text{MeV}$, $\ell_{\text{\tiny QQq}}^{(2)}=1.513\,\text{fm}$.

Before proceeding further, we pause here to briefly discuss the process of
double string breaking in the $QQq$ and $Q\bar{Q}$ systems. In the $QQq$
system, we consider a string attached to a heavy quark so that the critical
separation distance is given by (4.9). It is straightforward to extend the
above analysis to the $Q\bar{Q}$ system. First, one has to consider the
process: $Q\bar{Q}\rightarrow Q\bar{q}+q\bar{Q}+q\bar{q}$, and then to define
a critical separation distance by

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell_{\text{\tiny Q}\bar{\text{\tiny
Q}}}^{(2)})=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}}\,.$ (4.10)

Using the large $\ell$ expansion for $E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}(\ell)$ of a-strb , one gets

$\ell_{\text{\tiny Q}\bar{\text{\tiny
Q}}}^{(2)}=\frac{2}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+I_{0}+\frac{E_{\text{\tiny
q}\bar{\text{\tiny q}}}}{2\mathsf{g}\sqrt{\mathsf{s}}}\biggr{)}\,.$ (4.11)

Here $I_{0}$ is defined in Appendix A. If, for instance, $E_{\text{\tiny
q}\bar{\text{\tiny q}}}=280\,\text{MeV}$, then $\ell_{\text{\tiny
Q}\bar{\text{\tiny Q}}}^{(2)}=1.476\,\text{fm}$. This value is very close
(within $2.4\%$) to that obtained above for the $QQq$ system. Interestingly, a
similar story also holds for the standard string breaking (by one quark pair)
distances, where the difference is about $2.9\%$ a-QQq .

The energy of the second excited state is described in terms of six
configurations. These are shown in Figures 1, 2, 3(e)-(f), and 18(a). We
therefore have $V_{2}=\min\\{E_{\text{\tiny QQqq}\bar{\text{\tiny
q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}},E_{\text{\tiny QQq}}+E_{\text{\tiny q}\bar{\text{\tiny
q}}},E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
Qqq}}+E_{\text{\tiny q}\bar{\text{\tiny q}}},E_{\text{\tiny
QQq}},2E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny qqq}}\\}$. As for
$V_{1}$, a quite similar story holds true for $V_{2}$. For $\ell\lesssim
1.35\,\text{fm}$ it looks similar to what is shown in Figure 13 for the energy
of the first excited state. So, the transition at $\ell=0.184\,\text{fm}$ is
due to the process of string junction annihilation, with the formula (3.43)
remaining valid. The next transition at $\ell=1.001\,\text{fm}$ is interpreted
as a result of string reconnection. In this case, the critical separation
distance is given by (4.1). The transition at $\ell=1.257\,\text{fm}$ is due
to string breaking, as discussed in Appendix B. The remaining transition at
$\ell=1.513\,\text{fm}$ can be interpreted as a result of fusion of two
strings: $Q\bar{q}+Qqq+q\bar{q}\rightarrow QQq$. Clearly, it is inverse to the
process of double string breaking so that the above formula can also be
applied to estimate the critical separation distance.

So far we have considered the domain of $V_{2}$, where the analysis is
straightforward and relies on the previous results. However, for larger
separations a new analysis is needed. The transition near
$\ell=1.65\,\text{fm}$ is another example of double string breaking:
$QQq\rightarrow 2Q\bar{q}+qqq$. A crucial difference from the first example is
that two strings break down. Both have a heavy quark at their endpoints. For
each of them, the process of string breaking is as sketched in Figure 4(b). In
this case we define a critical separation distance by

$E_{\text{\tiny QQq}}(\ell_{\text{\tiny QQq}}^{(2^{\prime})})=2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny qqq}}\,.$ (4.12)

As before, it is easy to find a solution for large $\ell$. It is

$\ell_{\text{\tiny
QQq}}^{(2^{\prime})}=\frac{2}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+I_{\text{\tiny
QQq}}+\frac{E_{\text{\tiny qqq}}}{2\mathsf{g}\sqrt{\mathsf{s}}}\biggr{)}\,.$
(4.13)

For $E_{\text{\tiny qqq}}=1060\,\text{GeV}$, a simple estimate gives
$\ell_{\text{\tiny QQq}}^{(2^{\prime})}=1.661\,\text{fm}$ that is smaller than
the value of $\ell_{\text{\tiny QQq}}^{(2)}$.

We conclude that in the $QQq$ system the pentaquark configuration makes the
dominant contribution to the potential of the second excited state at small
separations. This is quite similar to what happen in the $QQqq\bar{q}$ system
to the potential of the first excited state.

##  V Concluding comments

(i) An interesting relation between the energies of some connected
configurations can be deduced from heavy quark-diquark symmetry. Indeed, one
can express $E_{\text{\tiny QQ}}$ from Eq.(B.5) and then substitute it into
Eq.(3.31) to get

$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}(\ell)=E_{\text{\tiny
QQq}}(\ell)+E_{\text{\tiny Qq}\bar{\text{\tiny q}}\bar{\text{\tiny
q}}}-E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (5.1)

The above derivation holds for relatively small $\ell$. We have developed all
the necessary machinery to directly check if this relation holds for larger
values of $\ell$. To do so, it is convenient to think in terms of the $QQq$
system and to further restrict discussion to the potentials of the ground and
second excited states. As seen from Figure 15, $V_{0}$ can be well
approximated by $E_{\text{\tiny QQq}}$ up to separations of order
$1,2\,\text{fm}$, while $V_{2}$ by $E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$
up to separations of order $0.2\,\text{fm}$. The reason for the latter is the
flattening of $V_{2}$ near $\ell=0.2\,\text{fm}$ due to string junction
annihilation. For illustration, in Figure 16 we plot the potentials $V_{2}$

Figure 16: The potentials $V_{0}$, shifted by $E_{\text{\tiny
Qq}\bar{\text{\tiny q}}\bar{\text{\tiny q}}}-E_{\text{\tiny Q}\bar{\text{\tiny
q}}}$, and $V_{2}$. The part of $V_{2}$ which is well approximated by
$E_{\text{\tiny QQqq}\bar{\text{\tiny q}}}$ is shown in blue.

and $V_{0}$ shifted by the constant $E_{\text{\tiny Qq}\bar{\text{\tiny
q}}\bar{\text{\tiny q}}}-E_{\text{\tiny Q}\bar{\text{\tiny q}}}$. Obviously,
the refined version of (5.1) is

$V_{2}(\ell)=V_{0}(\ell)+E_{\text{\tiny Qq}\bar{\text{\tiny
q}}\bar{\text{\tiny q}}}-E_{\text{\tiny Q}\bar{\text{\tiny
q}}}\,,\qquad\text{if}\qquad\ell\lesssim 0.2\,\text{fm}\,.$ (5.2)

It is tempting to apply this relation in the heavy quark limit, where
contributions from the motion of the heavy quarks and spin interactions are
negligible, to derive a relation among the masses of heavy-light and doubly-
heavy-light hadrons. We will thus have

$m_{\text{\tiny QQqq}\bar{\text{\tiny q}}}-m_{\text{\tiny QQq}}=m_{\text{\tiny
Qq}\bar{\text{\tiny q}}\bar{\text{\tiny q}}}-m_{\text{\tiny Q}\bar{\text{\tiny
q}}}\,.$ (5.3)

It is important to realize, however, that because of the use of (5.2), the
doubly heavy hadrons are compact in sense of heavy quark separation.

Similar formulas arise in the case of the $QQ\bar{q}\bar{q}$ system a-QQqq .
Again, the phenomenon of string junction annihilation plays a pivotal role in
establishing the upper bound on $\ell$. In the heavy quark limit one can use
those to derive the relation among the masses of heavy-light and doubly-heavy-
light mesons and baryons as that of quigg . Moreover, by essentially the same
arguments that we have given above and in a-QQqq , one can show that similar
relations hold between the corresponding potentials of the $QQqq\bar{q}$ and
$QQ\bar{q}\bar{q}$ systems. In this case, the relation among masses
reads282828Alternatively, it can be obtained by combining (5.3) with the
relation of quigg .

$m_{\text{\tiny QQqq}\bar{\text{\tiny q}}}-m_{\text{\tiny QQ}\bar{\text{\tiny
q}}\bar{\text{\tiny q}}}=m_{\text{\tiny Qq}\bar{\text{\tiny
q}}\bar{\text{\tiny q}}}-m_{\text{\tiny Qqq}}\,.$ (5.4)

It is interesting to see what results come out of both relations and how those
agree or disagree with phenomenological values. For reasons of brevity, we
will not discuss all this here, but only make a crude estimate. Consider the
first relation. For the mass of the $QQqq\bar{q}$ state several values can be
found in the literature wang ; FG ; zhu . We use these as an input to estimate
the mass of $Qq\bar{q}\bar{q}$ and compare the output with the result of a
direct calculation in lu . Our results are presented in Table 1.

State |  wang |  FG |  zhu |  lu
---|---|---|---|---
$ccqq\bar{q}$ | 4.21 | 4.54 | 4.70 | $-$
$bbqq\bar{q}$ | 10.75 | 11.15 | 11.37 | $-$
$cq\bar{q}\bar{q}$ | 2.41 | 2.74 | 2.90 | $2.57$
$bq\bar{q}\bar{q}$ | 5.85 | 6.25 | 6.47 | $5.98$
Table 1:  Masses (in GeV) of $QQqq\bar{q}$ and $Qq\bar{q}\bar{q}$ states.

The values used here for the masses of $QQq$ are the same as the smallest
values in quigg , namely $m_{ccq}=3.66\,\text{GeV}$ and
$m_{bbq}=10.18\,\text{GeV}$. The masses of $Q\bar{q}$ are those of the $D^{0}$
and $B^{\pm}$ mesons PDG , $m_{c\bar{q}}=1.86\,\text{GeV}$ and
$m_{b\bar{q}}=5.28\,\text{GeV}$. Since the ranges of values are quite wide, it
is not surprising that both values of lu are inside those ranges. While it is
surprising that for $cq\bar{q}\bar{q}$ all the values don’t exceed
$2.90\,\text{GeV}$, which is the mass of the $X_{0}(2900)$ state whose flavor
content is $ud\bar{s}\bar{c}$.

(ii) The assumptions made in Sect.II are oversimplified for various reasons.
For one thing, excited strings, as one sketched in Figure 17(a), must be
included when building string configurations for excited states. These strings

Figure 17: Some gluonic excitations.

represent a kind of gluonic excitations of the system which have been
extensively studied in the literature, but only within the $Q\bar{Q}$ system.
Also, glueballs must be included. They are another kind of gluonic
excitations. The two simplest examples are sketched in Figures 17(b) and (b’).
The first is just a closed string, and the second involves a pair of vertices
connected by strings. Both kinds of gluonic excitations are natural from the
point of view of string theory in four dimensions XA . There is however a
novelty related to the description of the baryon vertex as a five-brane in ten
dimensions witten . Accordingly, brane excitations must be included. This
would lead to a bunch of excited vertices. One example of these is sketched in
Figure 17 (c), where excitation is due to an open string with endpoints on the
brane. One can think of this as a new kind of gluonic excitations. It would be
of great interest to find evidence for this kind of excitations by means of
lattice simulations.

(iii) As we have seen, at small heavy quark separations the pentaquark
configuration makes the dominant contribution to the first excited B-O
potential of the $QQqq\bar{q}$ system. This does not exclude the possibility
of its dominance at larger separations, but for higher excited potentials. The
key difference with the first case is that if one thinks in terms of the heavy
quark separation, the pentaquarks are compact (small) in the first case but
not in the others.

(iv) An interesting observation, which one can make from Eq.(3.43) and
Eq.(3.42) in a-QQqq , is that in the $QQqq\bar{q}$ and $QQ\bar{q}\bar{q}$
systems the critical separation distances characterizing the process of string
junction annihilation are equal to each other. So,

$\boldsymbol{\ell}_{\text{\tiny QQqq}\bar{\text{\tiny
q}}}=\boldsymbol{\ell}_{\text{\tiny QQ}\bar{\text{\tiny q}}\bar{\text{\tiny
q}}}\,.$ (5.5)

This is also true for the screening lengths, if one uses the Gaussian form for
the corresponding potentials (see Eq.(4.5) and Eq.(4.5) in a-QQqq ). At first
glance, the reason for this is that both systems possess heavy quark-diquark
symmetry. It is sufficient for guaranteeing the equal screening lengths, but
not the equal critical separation distances. There one needs some peculiar
relation among the rest energies of heavy-light hadrons. So we are left with
the two important questions: 1) Is the critical separation distance
$\boldsymbol{\ell}$ universal for doubly heavy quark systems? 2) Why is it so
small? Hopefully, lattice QCD will be able to make further progress in
addressing these questions in the near future.

###### Acknowledgements.

We would like to thank I.Ya. Aref’eva, J.-M. Richard, and M. Wagner for useful
communications and conversations. This work was supported in part by Russian
Science Foundation grant 20-12-00200.

##  Appendix A Notation

In all Figures throughout the paper, heavy and light quarks (antiquarks) are
denoted by $Q$ and $q\,(\bar{q})$, and baryon (antibaryon) vertices by
$V\,(\bar{V})$. Straight lines represent non-excited strings. Light quarks
(antiquarks) are set at $r=r_{q}\,(r_{\bar{q}})$ and vertices at
$r=r_{v}\,(r_{\bar{v}})$, unless otherwise indicated. It is convenient to
introduce dimensionless variables: $q=\mathsf{s}r_{q}^{2}$,
$\bar{q}=\mathsf{s}r_{\bar{q}}^{2}$, $v=\mathsf{s}r_{v}^{2}$, and
$\bar{v}=\mathsf{s}r_{\bar{v}}^{2}$. They take values between $0$ and $1$, and
show how far from the soft-wall these objects are.292929The soft wall in such
units is located at $1$. To classify the critical separations related to the
string interactions of Figure 4, the notation $l$ is used for (a), $\ell$ for
(b-b’), and $\boldsymbol{\ell}$ for (c).

To present the resulting formulas in a compact form, we use the set of basic
functions a-stb3q :

${\cal
L}^{+}(\alpha,x)=\cos\alpha\sqrt{x}\int^{1}_{0}du\,u^{2}\,\text{e}^{x(1-u^{2})}\Bigl{[}1-\cos^{2}{}\alpha\,u^{4}\text{e}^{2x(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}\,,\qquad
0\leq\alpha\leq\frac{\pi}{2}\,,\qquad 0\leq x\leq 1\,.$ (A.1)

${\cal L}^{+}$ is a non-negative function which vanishes if
$\alpha=\frac{\pi}{2}$ or $x=0$, and has a singular point at $(0,1)$;

${\cal
L}^{-}(y,x)=\sqrt{y}\biggl{(}\,\int^{1}_{0}du\,u^{2}\,\text{e}^{y(1-u^{2})}\Bigl{[}1-u^{4}\,\text{e}^{2y(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}+\int^{1}_{\sqrt{\frac{x}{y}}}du\,u^{2}\,\text{e}^{y(1-u^{2})}\Bigl{[}1-u^{4}\,\text{e}^{2y(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}\,\biggr{)}\,,\quad
0\leq x\leq y\leq 1\,.$ (A.2)

${\cal L}^{-}$ is also a non-negative function. It vanishes at the origin and
becomes singular at $y=1$. There is a simple relation between ${\cal L}^{+}$
and ${\cal L}^{-}$, namely ${\cal L}^{+}(0,x)={\cal L}^{-}(x,x)$;

${\cal
E}^{+}(\alpha,x)=\frac{1}{\sqrt{x}}\int^{1}_{0}\,\frac{du}{u^{2}}\,\biggl{(}\text{e}^{xu^{2}}\Bigl{[}1-\cos^{2}{}\alpha\,u^{4}\text{e}^{2x(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}-1-u^{2}\biggr{)}\,,\qquad
0\leq\alpha\leq\frac{\pi}{2}\,,\qquad 0\leq x\leq 1\,.$ (A.3)

${\cal E}^{+}$ is singular at $x=0$ and $(0,1)$;

${\cal
E}^{-}(y,x)=\frac{1}{\sqrt{y}}\biggl{(}\int^{1}_{0}\,\frac{du}{u^{2}}\,\Bigl{(}\text{e}^{yu^{2}}\Bigl{[}1-u^{4}\,\text{e}^{2y(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}-1-u^{2}\Bigr{)}+\int^{1}_{\sqrt{\frac{x}{y}}}\,\frac{du}{u^{2}}\,\text{e}^{yu^{2}}\Bigl{[}1-u^{4}\,\text{e}^{2y(1-u^{2})}\Bigr{]}^{-\frac{1}{2}}\biggr{)}\,,\,\,\,0\leq
x\leq y\leq 1\,.$ (A.4)

${\cal E}^{-}$ is singular at $(0,0)$ and $y=1$. Just like for the ${\cal
L}$’s, one also has ${\cal E}^{+}(0,x)={\cal E}^{-}(x,x)$;

${\cal Q}(x)=\sqrt{\pi}\text{erfi}(\sqrt{x})-\frac{\text{e}^{x}}{\sqrt{x}}\,,$
(A.5)

which is the special case of ${\cal E}^{+}$ with $\alpha=\frac{\pi}{2}$. Here
$\text{erfi}(x)$ is the imaginary error function. A useful fact is that its
small-$x$ behavior is given by

${\cal Q}(x)=-\frac{1}{\sqrt{x}}+\sqrt{x}+O(x^{\frac{3}{2}})\,;$ (A.6)

${\cal
I}(x)=I_{0}-\int_{\sqrt{x}}^{1}\frac{du}{u^{2}}\text{e}^{u^{2}}\Bigl{[}1-u^{4}\text{e}^{2(1-u^{2})}\Bigr{]}^{\frac{1}{2}}\,,\qquad
I_{0}=\int_{0}^{1}\frac{du}{u^{2}}\Bigl{(}1+u^{2}-\text{e}^{u^{2}}\Bigl{[}1-u^{4}\text{e}^{2(1-u^{2})}\Bigr{]}^{\frac{1}{2}}\Bigr{)}\,,\qquad
0<x\leq 1\,.$ (A.7)

Numerically, $I_{0}=0.751$.

##  Appendix B The $QQq$ system

We will give a brief review of the string construction for the $QQq$ system
proposed in a-QQq , whose conventions we follow. We are limited here to the
potential $V_{0}$ corresponding to the ground state.

From the point of view of four dimensional string models XA , the only
relevant string configurations are those shown in Figure 18.

Figure 18: String configurations contributing to the potential $V_{0}$ of the
$QQq$ system.

The first consists of the valence quarks joined by strings. The strings meet
at the string junction. This is a usual string picture for doubly heavy
baryons. The second is obtained by adding a virtual pair $q\bar{q}$ to the
first configuration. It is the same as that of Figure 1 (b) which describes
two non-interacting hadrons: $Qqq$ and $Q\bar{q}$.303030Another disconnected
configuration consisting of $QQq$ and $q\bar{q}$ doesn’t contribute to the
ground state. The reason that the second configuration contributes to the
ground state is that for large $\ell$ its energy is of order $1$, whereas the
energy of the first configuration is of order $\ell$. The transition between
the two different regimes corresponds to the baryon decay

$QQq\rightarrow Qqq\,+\,Q\bar{q}\,.$ (B.1)

In the language of string theory, such a decay can be interpreted as string
breaking. One of the strings attached to the heavy quarks breaks down.

Now consider these configurations in the five-dimensional framework. We begin
with the connected configuration of Figure 18(a). The important point here is
that the shape of the configuration changes with the increase of heavy quark
separation. As a result, the single string configuration in four dimensions is
replaced by three different configurations in five dimensions, as those of
Figure 19.

Figure 19: Three types of connected configurations for the $QQq$ system in
five dimensions. $\alpha$ denotes the tangent angle of the left string. The
gray horizontal line in (l) represents the soft wall at
$r=1/\sqrt{\mathsf{s}}$.

For small $\ell$ the corresponding configuration is labeled by (s). In this
case, the total action is the sum of the Nambu-Goto actions for the three
fundamental strings plus the actions for the baryon vertex and background
scalar. The relation between the energy and heavy quark separation is written
in parametric form

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,v)\,,\qquad
E_{\text{\tiny QQq}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}2{\cal
E}^{+}(\alpha,v)+{\cal Q}({\text{q}})-{\cal
Q}(v)+3\mathsf{k}\frac{\text{e}^{-2v}}{\sqrt{v}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}\Bigr{)}+2c\,,$
(B.2)

with the parameter $v$ varying from $0$ to q. The latter is a solution to
Eq.(3.5) in the interval $[0,1]$. The functions ${\cal L}^{+}$ and ${\cal
E}^{+}$ are as defined in Appendix A. $c$ is a normalization constant. The
tangent angle $\alpha$ can be expressed in terms of the parameter $v$ by using
the force balance equation at $r=r_{v}$. The result is given by Eq.(3.13).

Since $\ell$ is an increasing function of $v$, increasing $\ell$ leads to a
situation where the vertex reaches the position of the light quark. In this
case the configuration looks like that of Figure 19(m). It differs from the
first only by the absence of the string stretched between the vertex and light
quark so that the quark sits on top of the vertex. Hence the distance $\ell$
is expressed in terms of $v$ and $\alpha$ by the same formula as before, only
for another parameter range, whereas the energy by

$E_{\text{\tiny QQq}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}2{\cal
E}^{+}(\alpha,v)+\frac{1}{{\sqrt{v}}}\bigl{(}3\mathsf{k}\text{e}^{-2v}+\mathsf{n}\text{e}^{\frac{1}{2}v}\bigr{)}\Bigr{)}+2c\,.$
(B.3)

Clearly, it can be obtained from (B.2) by formally setting ${\text{q}}=v$. In
this case, $\alpha$ is expressed in terms of $v$ by Eq.(3.24), with $\bar{v}$
replaced by $v$. By construction, the tangent angle must be non-negative. This
condition allows one to find a range for $v$. It is given by
$[{\text{q}},{\text{v}}_{\text{\tiny 0}}]$, where ${\text{v}}_{\text{\tiny
0}}$ is a solution to Eq.(3.25) (with $\bar{v}$ replaced by $v$). The meaning
of the upper bound is that $\alpha({\text{v}}_{\text{\tiny 0}})=0$. In other
words, there is no cusp at the string endpoints at $x=0$.

From the expression (B.2), it follows that $\ell$ remains finite at
$v={\text{v}}_{\text{\tiny 0}}$. The question now is how to reach larger
values of $\ell$. The answer is to consider negative values of $\alpha$. In
that case, the configuration profile becomes convex near $x=0$, as shown in
Figure 19(l). The desired result is obtained from the previous one by
replacing ${\cal L}^{+}$ and ${\cal E}^{+}$ with ${\cal L}^{-}$ and ${\cal
E}^{-}$. So,

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{-}(\lambda,v)\,,\qquad
E_{\text{\tiny QQq}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}2{\cal
E}^{-}(\lambda,v)+\frac{1}{\sqrt{v}}\bigl{(}3\mathsf{k}\text{e}^{-2v}+\mathsf{n}\text{e}^{\frac{1}{2}v}\bigr{)}\Bigr{)}+2c\,.$
(B.4)

Here the parameter $v$ goes from ${\text{v}}_{\text{\tiny 0}}$ to
${\text{v}}_{\text{\tiny 1}}$, where ${\text{v}}_{\text{\tiny 1}}$ is a
solution to Eq.(3.30) in the interval $[0,1]$. $\lambda$ is the function of
$v$ given by Eq.(3.29). The strings approach the soft wall as
$\lambda\rightarrow 1$. This places the upper bound on $v$.

A summary of the above discussion is as follows. The energy of the connected
configuration as a function of the heavy quark separation is given in
parametric form by the two piecewise functions $E_{\text{\tiny
QQq}}=E_{\text{\tiny QQq}}(v)$ and $\ell=\ell(v)$.

For future reference, we give a brief description of $E_{\text{\tiny
QQq}}(\ell)$ for small and large $\ell$. It behaves for $\ell\rightarrow 0$ as

$E_{\text{\tiny QQq}}(\ell)=E_{\text{\tiny QQ}}(\ell)+E_{\text{\tiny
q}\bar{\text{\tiny Q}}}+o(\ell)\,,\qquad\text{with}\qquad E_{\text{\tiny
QQ}}=-\frac{\alpha_{\text{\tiny QQ}}}{\ell}+c+\boldsymbol{\sigma}_{\text{\tiny
QQ}}\ell\,.$ (B.5)

$E_{\text{\tiny q}\bar{\text{\tiny Q}}}$ is equal to $E_{\text{\tiny
Q}\bar{\text{\tiny q}}}$ defined in (3.4) and the coefficients are given by

$\alpha_{\text{\tiny
QQq}}=-l_{0}E_{0}\mathsf{g}\,,\qquad\boldsymbol{\sigma}_{\text{\tiny
QQq}}=\frac{1}{l_{0}}\Bigl{(}E_{1}+\frac{l_{1}}{l_{0}}E_{0}\Bigr{)}\mathsf{g}\mathsf{s}\,,$
(B.6)

where
$l_{0}=\frac{1}{2}\xi^{-\frac{1}{2}}B\bigl{(}\xi^{2};\tfrac{3}{4},\tfrac{1}{2}\bigr{)}$,
$l_{1}=\frac{1}{2}\xi^{-\frac{3}{2}}\bigl{[}\bigl{(}2\xi+\frac{3}{4}\frac{\mathsf{k}-1}{\xi}\bigr{)}B\bigl{(}\xi^{2};\tfrac{3}{4},-\tfrac{1}{2}\bigr{)}-B\bigl{(}\xi^{2};\tfrac{5}{4},-\tfrac{1}{2}\bigr{)}\bigr{]}$,
$E_{0}=1+3\mathsf{k}+\frac{1}{2}\xi^{\frac{1}{2}}B\bigl{(}\xi^{2};-\tfrac{1}{4},\tfrac{1}{2}\bigr{)}$,
and
$E_{1}=\xi\,l_{1}-1-6\mathsf{k}+\frac{1}{2}\xi^{-\frac{1}{2}}B\bigl{(}\xi^{2};\tfrac{1}{4},\tfrac{1}{2}\bigr{)}$.
Here $B(z;a,b)$ is the incomplete beta function and
$\xi=\frac{\sqrt{3}}{2}(1-2\mathsf{k}-3\mathsf{k}^{2})^{\frac{1}{2}}$. Thus
the model we are considering has the desired property of factorization,
expected from heavy quark-diquark symmetry wise .

$E_{\text{\tiny QQq}}$ behaves for $\ell\rightarrow\infty$ as

$E_{\text{\tiny QQq}}=\sigma\ell-2\mathsf{g}\sqrt{\mathsf{s}}I_{\text{\tiny
QQq}}+2c+o(1)\,,\qquad\text{with}\qquad I_{\text{\tiny QQq}}={\cal
I}({\text{v}}_{\text{\tiny
1}})-\frac{3\mathsf{k}\text{e}^{-2{\text{v}}_{\text{\tiny
1}}}++\mathsf{n}\text{e}^{\frac{1}{2}{\text{v}}_{\text{\tiny
1}}}}{2\sqrt{{\text{v}}_{\text{\tiny 1}}}}\,$ (B.7)

and the same string tension $\sigma$ as in (3.2). The function ${\cal I}$ is
defined in Appendix A.

A five-dimensional counterpart of the disconnected configuration of Figure
18(b) is shown schematically in Figure 5(b). It describes the non-interacting
hadrons and, therefore, the total energy is the sum of the rest energies of
the hadrons. Explicitly, it is given by Eq.(3.3).

Like in lattice QCD, the potential $V_{0}$ is given by the smallest eigenvalue
of a model Hamiltonian

${\cal H}(\ell)=\begin{pmatrix}E_{\text{\tiny QQq}}(\ell)&\Theta_{\text{\tiny
QQq}}\\\ \Theta_{\text{\tiny QQq}}&E_{\text{\tiny Qqq}}+E_{\text{\tiny
Q}\bar{\text{\tiny q}}}\\\ \end{pmatrix}\,,$ (B.8)

with $\Theta_{\text{\tiny QQq}}$ describing the mixing between the two states.
Explicitly,

$V_{0}=\frac{1}{2}\Bigl{(}E_{\text{\tiny QQq}}+E_{\text{\tiny
Qqq}}+E_{\text{\tiny Q}\bar{\text{\tiny
q}}}\Bigr{)}-\sqrt{\frac{1}{4}\Bigl{(}E_{\text{\tiny QQq}}-E_{\text{\tiny
Qqq}}-E_{\text{\tiny Q}\bar{\text{\tiny q}}}\Bigr{)}^{2}+\Theta_{\text{\tiny
QQq}}^{2}}\,.$ (B.9)

It is instructive to give an example of this potential. For the same parameter
values as in Figure 10, it is shown in Figure 20.

Figure 20: The potential $V_{0}$ of the $QQq$ system. Here
$\Theta_{\text{\tiny QQq}}=47\,\text{MeV}$.

The potential asymptotically approaches $E_{\text{\tiny QQq}}$ as $\ell$ tends
to zero and $E_{\text{\tiny Qqq}}+E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ as
$\ell$ tends to infinity. The transition between these two regimes occurs
around $\ell=\ell_{\text{\tiny QQq}}$ which is a solution to the equation

$E_{\text{\tiny QQq}}(\ell_{\text{\tiny QQq}})=E_{\text{\tiny
Qqq}}+E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (B.10)

This equation simplifies drastically at large $\ell$, where the phenomenon of
string breaking is expected to occur. Combining (B.7) with (3.3), one gets

$\ell_{\text{\tiny QQq}}=\frac{3}{\text{e}\sqrt{\mathsf{s}}}\biggl{(}{\cal
Q}({\text{q}})-\frac{1}{3}{\cal
Q}({\text{v}})+\mathsf{k}\frac{\text{e}^{-2{\text{v}}}}{\sqrt{{\text{v}}}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}{\text{q}}}}{\sqrt{{\text{q}}}}+\frac{2}{3}{\cal
I}_{\text{\tiny QQq}}\biggr{)}\,.$ (B.11)

For the parameter values of Sec.III, a simple estimate gives
$\ell_{\text{\tiny QQq}}=1.257\,\text{fm}$.

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