# Towards a Stringy Description for the $Q\bar{Q}q\bar{q}$-Quark System

Oleg Andreev  L.D. Landau Institute for Theoretical Physics, Kosygina 2,
119334 Moscow, Russia  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 made of one heavy and one light quark-antiquark pair, with the aim of
exploring the two lower-lying Born-Oppenheimer potentials as a function of a
separation of the heavy quark-antiquark pair. Our analysis reveals three
critical separations related to the processes of string reconnection, breaking
and junction annihilation. In particular, for the ground state potential only
the process of string reconnection matters. We find that a tetraquark state
makes the dominant contribution to the potential of the first excited state at
small separations, and this is the big difference with the
$QQ\bar{q}\bar{q}$-quark system where it does so to the ground state
potential. Another big difference is the emergence of the full diquark picture
$[Qq][\bar{Q}\bar{q}]$ rather than the partial picture $QQ[\bar{q}\bar{q}]$
for the tetraquark state. On the other hand, the scales of string junction
annihilation, below which the systems can be thought of mainly as the compact
tetraquarks, are very close for both cases and become almost the same if the
phenomenological rule $E_{\text{\tiny QQ}}=\frac{1}{2}E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$ holds. The same is also true for the screening
lengths whose values are in agreement with lattice QCD.

††preprint: LMU-ASC 11/22

##  I Introduction

Multiquark hadrons remain a challenge for both theory and experiment book .
One class of those includes four-quark hadrons made of one heavy and one light
quark-antiquark pair. The most known example are the $Z_{b}$ states discovered
by the Belle Collaboration belle , whose quark content is $b\bar{b}q\bar{q}$
with $q\in\\{u,d\\}$. Because of the large ratio of the quark masses, one of
the frameworks for dealing with such a class is the Born-Oppenheimer (B-O)
approximation borrowed from atomic and molecular physics bo .111For the
further development of these ideas in the context of QCD, see braat and
references therein. 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 $Q$ and $\bar{Q}$ sources. The spectrum
is then calculated by solving the Schrödinger equation in these potentials.

Lattice gauge theory is one of the basic tools for studying nonperturbative
phenomena in QCD and in particular the B-O potentials in the
$Q\bar{Q}q\bar{q}$-quark system AP ; SP0 . But it still has a significant
disadvantage in regards to that problem. The available data are limited that
makes it difficult to understand the physics behind them. On the other hand,
the gauge/string duality uaw provides new theoretical tools for studying
strongly coupled gauge theories, and therefore may be used as an alternative
method to gain important physical insights into this problem. Within this
framework, the string configurations for tetraquarks were qualitatively
discussed in a-3q0 ; coba . Making it more precise for the case of the
$Q\bar{Q}q\bar{q}$-quark system will be one of our goals.

This paper continues our study on the doubly heavy quark systems started in
a-QQq and QQqq . The rest of the paper 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 $Q\bar{Q}q\bar{q}$ system.
Among those we find the configurations relevant to the two lower-lying Born-
Oppenheimer potentials. In addition, we comment on some other configurations
and introduce three scales. These length scales characterize transitions
between different dominant configurations, and in fact, are related to three
types of interactions between strings: reconnection, breaking and junction
(baryon vertex) annihilation. We go on in Sec.IV to compare our results with
those in lattice QCD and to discuss the construction of the potentials. We
conclude in Sec.V with a few important points. To make the paper more self-
contained, additional technical details are included in the Appendices.

##  II Preliminaries

###  II.1 General procedure

For QCD with light quarks, the B-O potentials can be defined along the line 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 correspond to the eigenvalues of the
correlation matrix.

Now consider the string configurations for the $Q\bar{Q}q\bar{q}$-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. For a first orientation, consider the simplest
configurations. These are the mesonic configurations shown in Figures 1(a) and
1(b). Each consists of the valence quarks and antiquarks joined by the strings
and looks

Figure 1: Various string configurations for the $Q\bar{Q}q\bar{q}$ system.

like a pair of mesons. Here all the strings are in the ground state and the
light quark-antiquark pair has zero momentum. Further, we can assume that
other configurations are constructed by adding extra string junctions and
virtual (light) quark-antiquark pairs. Intuitively, it is clear that such a
procedure will result in configurations of higher energy. And so to some
extent, 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. The first one is obtained by adding two junctions that
leads to the tetraquark configuration of Figure 1(c). The other two are
obtained by adding a virtual pair. This is a simple modification of the meson
configurations which gives rise to configurations (d) and (e), each having the
three mesons.

Before going on, it is worth making a few comments. First, the spatial
positions of quarks are not significant for what follows. The only thing which
matters is a separation between the heavy quark-antiquark pair. Second, apart
from string junctions and virtual pairs other elementary excitations may be
involved, which however are not relevant here. For example, these are some
gluonic excitations: excited strings and glueball states, as sketched in
Figure 2.

Figure 2: Some other configurations. The wavy line denotes an excited string
and the closed string (circle) a glueball.

Finally, adding both elementary excitations together we can get a baryon
configuration as that of Figure 2(h).

The possible transitions between the configurations arise from three type of
string interactions: (a) reconnection, (b) breaking, and (c) junction
annihilation (creation), as sketched in Figure 3.

Figure 3: Some string interactions.

All those are part of the big picture of QCD strings XA . For each
interaction, one can introduce a notion of a critical separation (between the
heavy quark-antiquark pair) which characterizes the system.

###  II.2 Key features of a five-dimensional framework

In our study of the $Q\bar{Q}q\bar{q}$ system, we will use the formalism
recently developed in a series of papers a-strb2 ; a-QQq ; QQqq . Although we
illustrate it using one of the simplest AdS/QCD models, the formalism is
applicable to other models as well.

We consider a five-dimensional space with metric tensor

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

Such a space represents a deformation of the Euclidean $\text{AdS}_{5}$ space
of radius $R$, with a deformation parameter $\mathsf{s}$. There is a boundary
at $r=0$ and a soft wall at $r=1/\sqrt{\mathsf{s}}$. Two features make it
especially attractive: computational simplicity and phenomenological
applications. Among those, let us only mention the model of az1 for the heavy
quark potential which yields a very good fit to the lattice data white .222For
more examples, see a-hyb ; a-3qPRD .

To construct the configurations of Figures 1 and 2 in five dimensions, we need
the building blocks. 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 pair of string
junctions, nowadays called the baryon vertices, at which three strings meet.
In the AdS/CFT correspondence the baryon vertex is supposed to be a dynamic
object living in ten dimensions. 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.333Like in AdS/CFT, one expects
an analog of the Ramond-Ramond fields living on the internal space
$\mathbf{X}$. In the case of zero baryon chemical potential, it is natural to
take the action (2.3) for the antibaryon vertex so that
$S_{\bar{\text{v}}}=S_{\text{v}}$.

In addition to the background metric (2.1), we introduce a background scalar
field $\text{T}(r)$ which describes light quarks son . In the present context
those are at string endpoints in the interior of five-dimensional space. For
our purposes it will suffice to restrict ourselves to a single scalar field
(tachyon), as we are interested in the case of two light flavors of equal
mass.444The use of the term tachyon seems particularly appropriate in virtue
of instability of a QCD string and the worldsheet coupling to the tachyon. 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. In what follows, we consider only a
constant field $\text{T}_{0}$ and worldsheets whose boundaries are 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)

It is nothing else but the action of a point particle of mass ${\text{T}}_{0}$
at rest. Clearly, at zero baryon chemical potential the same action also
describes the light antiquarks, and hence $S_{\bar{\text{q}}}=S_{\text{q}}$.

##  III The $Q\bar{Q}q\bar{q}$-Quark System via Gauge/String Duality

Now we will begin our discussion of the $Q\bar{Q}q\bar{q}$ system in the five-
dimensional framework. We approach this problem from a hadro-quarkonium point
of view voloshin and hence think of the light quarks (antiquarks) as clouds.
So, it makes sense to speak about their average positions or, equivalently,
the centers of the clouds. The heavy quark and antiquark are thought of as
point-like objects inside the clouds. Our goal is to determine the potentials
as a function of the separation between the heavy quark-antiquark pair.

An intuitive way to see the string configurations in five dimensions is to
place the configurations of Figures 1 and 2 on the boundary of five-
dimensional space. A gravitational force pulls the light quarks, antiquarks
and strings into the interior, whereas the heavy quark and antiquark remain at
rest. We start by describing configurations (a) and (b). Then we add a pair of
baryon vertices or correspondingly a pair of light quarks to those to
construct configurations (c)-(e). This is a good starting point for
understanding the energies of the ground and first excited states, as we will
see soon.

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

First consider configuration (a). It can be interpreted as a pair of mesons.
One of which is a pion, whereas the other is a quarkonium state realized by
the static quark-antiquark pair connected by a string. For infinitely
separated mesons, the total energy is just the sum of their masses (rest
energies). It is surprising that this also holds for finite separations if one
takes the average over the pion (cloud) position AP ; SP0 . In what follows,
we consider the configurations with pions only in the sense of averaging over
all possible pion positions.

From the five-dimensional perspective, the configuration looks like that shown
in Figure 4(a). It consists of two

Figure 4: Configurations (a) and (b) in five dimensions. The heavy quark and
antiquark are separated by distance $\ell$.

parts: the string connecting the heavy quark-antiquark pair and the light
quark pair. The first was discussed in az1 , where the relation between the
string energy $E_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ and separation $\ell$
was described in parametric form.555For more details on this relation, see
Eq.(C.1) in Appendix C. The second, which can be interpreted as a pion, is
unknown in the literature. This part requires some explanation. In the static
limit the only possible configuration is that shown in the Figure. The
intuitive idea is that nothing prevents a string connecting the light quark-
antiquark pair from collapsing into a point. So, the action is twice the
action $S_{\text{q}}$. Explicitly,

$S=2\mathsf{g}\mathsf{n}T\,\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{2q}^{2}}}{r_{2q}}\,,$
(3.1)

with $\mathsf{g}=\frac{R^{2}}{2\pi\alpha^{\prime}}$,
$\mathsf{n}=\frac{\text{T}_{0}R}{\mathsf{g}}$, and $T=\int dt$. By varying it
with respect to $r_{2q}$, we get

$1-\mathsf{s}r_{2q}^{2}=0\,.$ (3.2)

This equation determines the position of the light quarks in the bulk. It has
a single solution which is $r_{2q}=1/\sqrt{\mathsf{s}}$. This implies that the
light quark pair is located on the soft wall. The rest energy is then

$E_{\text{\tiny q}\bar{\text{\tiny q}}}=2\mathsf{n}\sqrt{\mathsf{g}\sigma}\,.$
(3.3)

Here $\sigma$ is the physical string tension defined by (C.4). Finally, the
energy of configuration (a) will be

$E^{\text{(a)}}=E_{\text{\tiny Q}\bar{\text{\tiny Q}}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}}\,.$ (3.4)

Now consider configuration (b). It represents a pair of non-interacting heavy-
light mesons so that the total energy is just twice that of the meson

$E^{\text{(b)}}=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (3.5)

In the static limit $E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ was calculated in
a-strb1 with the result

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

Here the function ${\cal Q}$ is defined in Appendix A, $c$ is a normalization
constant, and $q=\mathsf{s}r_{q}^{2}$ is a solution to the equation

$\text{e}^{\frac{q}{2}}+\mathsf{n}(q-1)=0\,$ (3.7)

which is nothing else but the force balance equation in the $r$-direction. It
is obtained by varying the action $S=S_{\text{\tiny NG}}+S_{\text{q}}$ with
respect to $r_{q}$.

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

It is natural to expect that if the connected configuration contributes to the
ground state, or at least to the first excited state, its shape is dictated by
symmetry. If so, there are the two most symmetric case: 1. The light quarks
are in the middle between the heavy quarks. 2. The light quark (antiquark)
sits on top of the heavy quark (antiquark).666This is the diquark picture
often used to describe the four-quark systems. In our situation, as we will
see, the former takes place at small enough separations between the heavy
quark-antiquark pair, whereas the latter at larger separations.

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

For this case, the corresponding string configuration is depicted in Figure 5,
which shows that the light quarks are indeed in the middle between the heavy
ones. Here $r_{q}$ and $r_{v}$ are assumed to satisfy the following condition:
$r_{q}>r_{v}$.777It indeed holds for the parameter values we are using.

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

$S=\sum_{i=1}^{4}S_{\text{\tiny NG}}^{(i)}+2S_{\text{vert}}+2S_{\text{q}}\,.$
(3.8)

Figure 5: A static string configuration for small $\ell$. The light quark
(antiquark) and baryon vertices are placed on the $r$-axis, respectively at
$r=r_{q}$ and $r=r_{v}$. $\alpha$ indicates the tangent angle at the endpoint
of the first string.

Picking the static gauge $\xi^{1}=t$ and $\xi^{2}=r$ for the Nambu-Goto
actions, we can write the boundary conditions for $x(r)$ as follows:

$x^{(1,2)}(0)=\mp\frac{1}{2}\ell\,,\qquad
x^{(i)}(r_{v})=x^{(3,4)}(r_{q})=0\,.$ (3.9)

The action is then888We drop the subscript $(i)$ when it does not cause
confusion.

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

with $\mathsf{k}=\frac{\tau_{v}}{3\mathsf{g}}$ and
$\partial_{r}x=\frac{\partial x}{\partial r}$.

To find a stable configuration, one has to extremize $S$ with respect to the
functions $x(r)$ describing the string profiles and, in addition, with respect
to $r_{v}$ and $r_{q}$ describing the locations of the baryon vertices and
light quarks. As explained in Appendix B, varying with respect to $x^{(1)}$
yields

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,v)\,$ (3.11)

and the energy of the configuration can be written in the form

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

Here the functions ${\cal L}^{+}$ and ${\cal E}^{+}$ are defined in Appendix
A. In addition, we introduced the dimensionless variable
$v=\mathsf{s}r_{v}^{2}$. Varying the action with respect to $r_{q}$ leads to
Eq.(3.7) and with respect to $r_{v}$ to

$\sin\alpha-1-3\mathsf{k}(1+4v)\text{e}^{-3v}=0\,.$ (3.13)

Both are the force balance equations in the $r$-direction at $r=r_{q}$ and
$r=r_{v}$, respectively.

Finally, the energy of the configuration is given parametrically by
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}(v)$ and $\ell=\ell(v)$ with the parameter taking values on a subinterval
of the interval $[0,1]$. Importantly, the region around $v=0$ belongs to this
subinterval. This enables us to take the limit $v\rightarrow 0$ that results
in ${\cal L}^{+}\rightarrow 0$, as follows from (A.1). Thus, such a
configuration does describe small values of $\ell$. We will return to this
issue in later subsections.

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

A simple numerical analysis shows that $\ell$ is an increasing function of
$v$. Therefore, increasing the separation between the quark-antiquark pair
will lead to a situation where the vertices reach the light quarks. In this
case the configuration looks like two string meeting at a point-like defect
made of the vertices and light quarks. We can continue in this way until we
approach infinite separation. 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. A way
out is that for these values of $\ell$ there is another configuration in which
the vertices are spatially separated, as depicted in Figure 6. It can be
obtained from the configuration shown

Figure 6: A static string configuration for larger $\ell$. The forces exerted
on the baryon vertex are depicted by the arrows. $r_{0}$ is a turning point.

in Figure 5 first by splitting the baryon vertices and then by stretching a
string between them.

The configuration is governed by the action

$S=\sum_{i=1}^{5}S_{\text{\tiny NG}}^{(i)}+2S_{\text{vert}}+2S_{\text{q}}\,.$
(3.14)

In the static gauge the boundary conditions (3.9) are replaced by

$x^{(1,2)}(0)=\mp\frac{1}{2}\ell\,,\qquad x^{(1,2)}(r_{v})=\mp x_{v}\,,\qquad
x^{(3,4)}(r_{v})=x^{(3,4)}(r_{q})=\mp x_{v}\,,\qquad x^{(5)}(r_{v})=\pm
x_{v}\,,$ (3.15)

so that the action now reads

$S=\mathsf{g}T\biggl{(}\int_{0}^{r_{v}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+\int_{r_{v}}^{r_{q}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}+\int_{r_{v}}^{r_{0}}\frac{dr}{r^{2}}\,\text{e}^{\mathsf{s}r^{2}}\sqrt{1+(\partial_{r}x)^{2}}+3\mathsf{k}\,\frac{\text{e}^{-2\mathsf{s}r_{v}^{2}}}{r_{v}}+\mathsf{n}\frac{\text{e}^{\frac{1}{2}\mathsf{s}r_{q}^{2}}}{r_{q}}\,\biggr{)}+(x\rightarrow-x)\,.$
(3.16)

Here the integrals correspond respectively to the contributions of strings
(1), (3), and (5). Because of the reflection symmetry of the configuration,
only the $x<0$ part of the action is written explicitly in (3.16).

Varying the action with respect to $x_{v}$ gives999To derive it, one has to
take into account the boundary conditions.

$\cos\alpha_{1}-\cos\alpha_{5}=0\,.$ (3.17)

The equation has a trivial solution $\alpha_{1}=\alpha_{5}$ which implies that
strings (1) and (5) are smoothly glued together to form a single string.
Another consequence of this solution is that variation with respect to $r_{v}$
results in

$1+3\mathsf{k}(1+4v)\text{e}^{-3v}=0\,.$ (3.18)

It is worth making a couple of points on this equation. First, in a-QQq we
found that in the interval $(0,1)$ it has solutions 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 $v=\frac{1}{12}$ at
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$. Second, in our analysis we
have assumed that $v\leq q$. While this assumption does not hold for all
possible parameter values, it holds for the ones we have in mind to use.

Now it is straightforward to write down the energy of the configuration. It is

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}+2\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}{\cal Q}(q)-{\cal
Q}(v_{\star})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}+3\mathsf{k}\frac{\text{e}^{-2v_{\star}}}{\sqrt{v_{\star}}}\Bigr{)}\,,$
(3.19)

with $q$ a solution to Eq.(3.7) and $v_{\star}$ to Eq.(3.18). In the process,
we combined the contributions from strings (1), (2) and (5), to get
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ (see Appendix C). Then we performed
the remaining integrals corresponding to strings (3) and (4). An important
point, which applies to this expression, is that the parameter $v$ runs from
$v_{\star}$ to $1$. Thus, such a configuration does not exist for $\ell$
smaller than $\ell(v_{\star})$.

It is particularly useful to formally write (3.19) as follows:

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=E_{\text{\tiny Q}\bar{\text{\tiny Q}}}+2\bigl{(}E_{\text{\tiny
Qqq}}-E_{\text{\tiny Q}\bar{\text{\tiny q}}}\bigr{)}\,,$ (3.20)

where $E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ and $E_{\text{\tiny Qqq}}$ are
defined in (3.6) and (3.37). These formulas imply the occurrence of the
diquark picture for the $Q\bar{Q}q\bar{q}$ system. Furthermore, it is
noteworthy that the constant term in (3.20) is consistent with string
breaking.

The main conclusions of our analysis are twofold. First of all,
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$
is a piecewise function of $\ell$ or, in other words, the shape of the
configuration depends on the separation between the heavy quarks. Second, the
diquark picture occurs for $\ell\geq\ell(v_{\star})$.

####  III.2.3 Glueing the two pieces together

Now let us discuss the gluing of the two branches of $E_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$. For this, we
need to specify the model parameters. We will use one of the two parameter
sets suggested in a-strb1 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 (C.4) to its value in bulava gives
$\mathsf{g}=0.176$.101010Note that this value is smaller than the value
$\mathsf{g}=0.196$ obtained by fitting the lattice data for the heavy quark-
antiquark potential in white but the discrepancy between these two values is
not significant. 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 , this results in $\mathsf{n}=3.057$.

In principle, the value of $\mathsf{k}$ could be adjusted to fit the lattice
data for the three-quark potential, as done in a-3q 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
phenomenology111111Note 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-3q . However, both are out of
the range of allowed values for $\mathsf{k}$ as follows from the analysis of
Eq.(3.18). In this situation it seems natural to pick
$\mathsf{k}=-\frac{1}{4}\text{e}^{\frac{1}{4}}$ which is most close to those.

Having fixed the parameters, we can immediately perform some simple
calculations. From (3.7) and (3.18), we get $q=0.566$ and
$v_{\star}=\frac{1}{12}$. Thus our construction of the connected configuration
makes sense for these values as $q>v_{\star}$. With this value of $v$, we find
that $\ell(v_{\star})=0.106\,\text{fm}$. It is quite surprising that the
diquark picture emerges at such small separations.

Given the parametric equations we derived above, it is straightforward to plot
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$
vs $\ell$. The result is presented in Figure 7. We see that the two pieces of
the function $E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ are smoothly glued together at
$\ell=0.106\,\text{fm}$, as expected.

Figure 7: $E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ as a function of $\ell$. The dashed curve
corresponds to the configuration of Figure 5 with the center pion cloud
between the heavy quarks, whereas the solid curve to the configuration of
Figure 6 for which the diquark picture holds. Here and later,
$c=0.623\,\text{GeV}$.

Also we will see shortly that it does behave as $1/\ell$ for small $\ell$ and
linearly for large $\ell$.

####  III.2.4 The limiting cases

We are ready to analyze the behavior of $E_{\text{\tiny Q}\bar{\text{\tiny
Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$ for small and large $\ell$. We begin
with the case of small $\ell$. In that case, the relevant configuration is
that of Figure 5. The reason for this is that ${\cal L}^{+}$ (and equally
$\ell$) is an increasing function of $v$ which vanishes at $v=0$. Taking the
limit $v\rightarrow 0$ in Eqs.(3.11) and (3.12), we find

$\ell=\sqrt{\frac{v}{\mathsf{s}}}\Bigl{(}\ell_{0}+\ell_{1}v\Bigr{)}+o\bigl{(}v^{\frac{3}{2}}\bigr{)}\,,\qquad
E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=\mathsf{g}\sqrt{\frac{\mathsf{s}}{v}}\Bigl{(}E_{0}+E_{1}v\Bigr{)}+2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+o\bigl{(}v^{\frac{1}{2}}\bigr{)}\,.$ (3.21)

The coefficients are given by

$\displaystyle\ell_{0}=\frac{1}{2}\tau^{-\frac{1}{2}}B\bigl{(}\tau^{2};\tfrac{3}{4},\tfrac{1}{2}\bigr{)}\,,\qquad\ell_{1}=\frac{1}{2}\tau^{-\frac{3}{2}}\Bigl{(}\bigl{(}\tau-3(1+3\mathsf{k})\tfrac{\mathsf{k}}{\tau}\bigr{)}B\bigl{(}\tau^{2};\tfrac{3}{4},-\tfrac{1}{2}\bigr{)}-B\bigl{(}\tau^{2};\tfrac{5}{4},-\tfrac{1}{2}\bigr{)}\Bigr{)}\,,$
(3.22) $\displaystyle
E_{0}=2(1+3\mathsf{k})+\frac{1}{2}\tau^{\frac{1}{2}}B\bigl{(}\tau^{2};-\tfrac{1}{4},\tfrac{1}{2}\bigr{)}\,,\quad
E_{1}=\frac{3}{2}(1+2\mathsf{k})\Bigl{(}-\frac{12\mathsf{k}}{1+3\mathsf{k}}+\frac{\tau^{\frac{1}{2}}}{2+3\mathsf{k}}B\bigl{(}\tau^{2};\tfrac{3}{4},-\tfrac{1}{2}\bigr{)}-\frac{\tau^{-\frac{1}{2}}}{1+2\mathsf{k}}B\bigl{(}\tau^{2};-\tfrac{5}{4},-\tfrac{1}{2}\bigr{)}\Bigr{)}\,.$
(3.23)

Here $\tau=\sqrt{-3\mathsf{k}(3\mathsf{k}+2)}$ and $B(z;a,b)$ is the
incomplete beta function. This makes it possible to eliminate the parameter,
and thereby to get a simple result for the energy

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=-\frac{\alpha_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}}{\ell}+2E_{\text{\tiny Q}\bar{\text{\tiny
q}}}+\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}\,\ell+o(\ell)\,,$ (3.24)

with

$\alpha_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=-\mathsf{g}E_{0}\ell_{0}\,,\qquad\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}=\frac{1}{\ell_{0}}\Bigl{(}E_{1}+\frac{\ell_{1}}{\ell_{0}}E_{1}\Bigr{)}\mathsf{g}\mathsf{s}\,.$
(3.25)

For completeness, let us estimate these coefficients. Using (C.3)-(C.4), we
get $\alpha_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}/\alpha_{\text{\tiny Q}\bar{\text{\tiny Q}}}=0.939$,
$\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}/\sigma=1.406$, with $\mathsf{k}$ as before.

To find the behavior for large $\ell$, we combine Eq.(3.20) with Eq.(C.4). So,

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}(\ell)=\sigma\ell+2\bigl{(}E_{\text{\tiny Qqq}}-E_{\text{\tiny
Q}\bar{\text{\tiny q}}}-\mathsf{g}\sqrt{\mathsf{s}}\,I_{0}\bigr{)}+2c+o(1)\,.$
(3.26)

This is one more indication of the universality of the string tension
$\sigma$. It is the same in all the known cases: $Q\bar{Q}$ az1 , $QQq$ a-QQq
, $QQqq$ QQqq , and $QQQ$ a-3q0 .

###  III.3 More disconnected configurations

Consider configurations (d) and (e). These are obtained from configurations
(a) and (b) by adding a light quark-antiquark pair (pion). In this case, the
analogy with Figure 4 is clear: one needs to add $q\bar{q}$ at position
$r=r_{2q}$ in the bulk. The resulting configurations are shown schematically
in Figure 8.

Figure 8: Configurations (d) and (e) in five dimensions.

Although there are no calculations on the lattice for those, in the following
we will assume that by analogy with the case of configuration (a) adding a
pion, with averaging over its position, leads to an energy increase by
$E_{\text{\tiny q}\bar{\text{\tiny q}}}$. If so, then the energies can be read
from the corresponding expressions for configurations (a) and (b). We have

$E^{\text{(d)}}=E_{\text{\tiny Q}\bar{\text{\tiny Q}}}+2E_{\text{\tiny
q}\bar{\text{\tiny q}}}\,$ (3.27)

and

$E^{\text{(e)}}=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}}\,.$ (3.28)

Clearly, these formulas also hold for non-interacting mesons.

###  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 quark-antiquark pair. In Figure 9
we plot those for our parameter values. These plots show that the energies of
the

Figure 9: Various $E$ vs $\ell$ plots. The curves relevant for $V_{0}$ and
$V_{1}$ are depicted in solid black. Here we set $\mathsf{k}_{\text{d}}=2000$,
as in a-hyb .

ground state and first excited state are determined only by the contributions
from configurations (a)-(c) and (e): $V_{0}=\min\\{E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}+E_{\text{\tiny q}\bar{\text{\tiny q}}},2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}\\}$ and $V_{1}=\min\\{E_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}},2E_{\text{\tiny
Q}\bar{\text{\tiny q}}},E_{\text{\tiny Q}\bar{\text{\tiny Q}}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}},2E_{\text{\tiny Q}\bar{\text{\tiny q}}}+E_{\text{\tiny
q}\bar{\text{\tiny q}}}\\}$.121212We will discuss configurations (f) and (h)
shortly. First one has to take a minimum to get $V_{0}$ and then, with that in
mind, $V_{1}$.

An interesting consequence of the pattern of Figure 9 is the emergence of
three scales which separate different configurations, or in other words
different descriptions. The first of those is a scale which refers to the
process of string reconnection. It is $Q\bar{Q}+q\bar{q}\rightarrow
Q\bar{q}+q\bar{Q}$ 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 heavy quark-antiquark pair in a pion cloud
for smaller quark separations and, respectively, as a pair of heavy-light
meson for large ones. To make this more quantitative, we define a critical
separation distance $l_{\text{\tiny Qq}}$ by

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(l_{\text{\tiny
Qq}})+E_{q\bar{q}}=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (3.29)

If reconnection occurs at small $\ell$ (as in Figure 9), then by using (3.6)
and (C.2), one gets131313For future reference, we keep track of
$E_{\text{\tiny q}\bar{\text{\tiny q}}}$.

$l_{\text{\tiny
Qq}}=\frac{\mathsf{g}\sqrt{\mathsf{s}}}{\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}\Bigl{(}{\cal
Q}(q)+\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}-\frac{E_{\text{\tiny
q}\bar{\text{\tiny
q}}}}{2\mathsf{g}\sqrt{\mathsf{s}}}\Bigr{)}+\sqrt{\frac{\alpha_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}{\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny
Q}}}}+\frac{\mathsf{g}^{2}\mathsf{s}}{\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}^{2}}\Bigl{(}{\cal
Q}(q)+\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}-\frac{E_{\text{\tiny
q}\bar{\text{\tiny q}}}}{2\mathsf{g}\sqrt{\mathsf{s}}}\Bigr{)}^{2}}\,.$ (3.30)

It is interesting to estimate $l_{\text{\tiny Qq}}$. For the pion mass defined
by (3.3), the result is $l_{\text{\tiny Qq}}=0.219\,\text{fm}$.

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

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}(\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}})=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (3.31)

An important observation one can make from the plots of $E_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$ and
$2E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ is that
$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ is of order $0.2\,\text{fm}$. From this, it follows
that in (3.31) $E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ is defined by (3.20) (also see Figure 7). Since
$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ is small, the equation can be solved approximately
by using (C.2). So, we get

$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny
q}}}=\frac{\mathsf{g}\sqrt{\mathsf{s}}}{\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}\Bigl{(}{\cal
Q}(v_{\star})-3\mathsf{k}\frac{\text{e}^{-2v_{\star}}}{\sqrt{v_{\star}}}\Bigr{)}+\sqrt{\frac{\alpha_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}{\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny
Q}}}}+\frac{\mathsf{g}^{2}\mathsf{s}}{\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}^{2}}\Bigl{(}{\cal
Q}(v_{\star})-3\mathsf{k}\frac{\text{e}^{-2v_{\star}}}{\sqrt{v_{\star}}}\Bigr{)}^{2}}\,,$
(3.32)

where in the last step we used (3.6) and (3.37). It is worth noting that such
defined critical distance is finite and scheme independent. The normalization
constant $c$ drops out of (3.32). Moreover, it depends on $v_{\star}$, which
describes the position of the baryon vertices in the bulk, rather than $q$
which describes the position of the light quarks. This suggests that
$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ is indeed related to gluonic degrees of freedom, as
expected from annihilation of string junctions made of gluons.

Let us make a simple estimate of $\boldsymbol{\ell}_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$. For the
parameter values we use, it gives

$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}\approx 0.196\,\text{fm}\,.$ (3.33)

This values is only slightly above the value $\boldsymbol{\ell}_{\text{\tiny
QQ}\bar{\text{\tiny q}}\bar{\text{\tiny q}}}=0.184\,\text{fm}$ obtained for
the $QQ\bar{q}\bar{q}$ system in QQqq . Thus, our estimate provides further
evidence that unlike the process of string breaking, the process of string
junction annihilation takes place at relatively small scales, 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 making comparison
with the lattice and gaining some information about the screening lengths.

The third scale is due to string breaking which occurs at the level of the
first excited state. Here $Q\bar{Q}+q\bar{q}\rightarrow
Q\bar{q}+q\bar{Q}+q\bar{q}$, as seen from Figure 9. The corresponding equation
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell_{\text{\tiny Q}\bar{\text{\tiny
Q}}})+E_{\text{\tiny q}\bar{\text{\tiny q}}}=2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}$ reduces to
Eq.(C.5) whose solution at large $\ell$ is given by (C.6). Numerically, it is
$\ell_{\text{\tiny Q}\bar{\text{\tiny Q}}}=1.22\,\text{fm}$ bulava . We used
this value to fix the parameter $\mathsf{n}$ in subsection B.

###  III.5 Comments on configurations (f)-(h)

Let us briefly discuss the role of configurations (f)-(h). The five-
dimensional counterparts of configurations (f) and (h) are depicted in Figure
10. In the configuration of Figure 10(f) the excited string is in the
$\Sigma_{u}^{-}$ state that corresponds to the $\Sigma_{u}^{-}$ hybrid
potential for the

Figure 10: Configurations (f) and (h) in five dimensions.

$Q\bar{Q}$ system kuti . In the current model, such a potential was
constructed from two strings meeting at a defect $D$ in the bulk a-hyb . The
configuration in Figure 10(h) represents a pair of non-interacting heavy-light
baryons.

For case (f), we assume that similarly to configuration (a), its energy is the
sum of two energies

$E^{\text{(f)}}=E_{\Sigma_{u}^{-}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\,.$
(3.34)

Here $E_{\Sigma_{u}^{-}}$ is the hybrid potential and $E_{\text{\tiny
q}\bar{\text{\tiny q}}}$ is the pion mass defined as before. The potential is
given in parametric form by a-hyb

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,v)\,,\qquad
E_{\Sigma_{u}^{-}}=2\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}{\cal
E}^{+}(\alpha,v)+\mathsf{k}_{\text{d}}\frac{\text{e}^{-2v}}{\sqrt{v}}\,\Bigr{)}+2c\,,\qquad$
(3.35)

with $\sin\alpha=\mathsf{k}_{\text{d}}(1+4v)\text{e}^{-3v}$. The parameter $v$
takes values on the interval $[v_{0},v_{1}]$, where $v_{0}$ is a solution to
$\cos\alpha=0$ and $v_{1}$ to $\cos\alpha=v\,\text{e}^{1-v}$.
$\mathsf{k}_{\text{d}}$ is a model parameter which characterizes the defect.

It makes no sense to consider configuration (g) in detail. The basic reason
that it is irrelevant for our purpose is that even for the scalar glueball,
lattice calculations yield a glueball mass in the range $1.5-1.7\,\text{GeV}$
bulava2 that is larger than the physical pion mass and also $E_{\text{\tiny
q}\bar{\text{\tiny q}}}$, as we will see shortly.

For case (h), the energy is simply

$E^{\text{(h)}}=2E_{\text{\tiny Qqq}}\,.$ (3.36)

Here $E_{\text{\tiny Qqq}}$ is given by a-strb1

$E_{\text{\tiny Qqq}}=\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}2{\cal Q}(q)-{\cal
Q}(v_{\star})+2\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}+3\mathsf{k}\frac{\text{e}^{-2v_{\star}}}{\sqrt{v_{\star}}}\Bigr{)}+c\,,$
(3.37)

with $q$ a solution to (3.7) and $v_{\star}$ to (3.18).

Given the formulas we have just described, it is straightforward to plot the
corresponding energies as a function of the separation between the heavy
quark-antiquark pair. The result is presented in Figure 9. It is seen that
these configurations contribute neither to $V_{0}$ nor to $V_{1}$, as
expected.

To complete the story, let us look at the decay channel
$Q\bar{Q}q\bar{q}\rightarrow Qqq+\bar{Q}\bar{q}\bar{q}$. Here we define a
characteristic scale by

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}(\ell)=2E_{\text{\tiny Qqq}}\,.$ (3.38)

It is natural to expect that a solution to this equation is large enough. If
so, then using (3.20), we can reduce this equation to Eq.(C.5) whose solution
is known. It is given by $\ell_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ and
describes the process of string breaking. In the current case, this process
takes place in context of a string stretched between a diquark and an
antidiquark.

##  IV More on the potentials $V_{0}$ and $V_{1}$

###  IV.1 An issue of $E_{\text{\tiny q}\bar{\text{\tiny q}}}$

With the help of (3.3), it is straightforward to make a simple estimate of the
pion mass. Using the parameter values of Sec.III one finds that
$E_{\text{\tiny q}\bar{\text{\tiny q}}}=1.190\,\text{GeV}$. This value is
considerably larger than the value $280\,\text{MeV}$ in the lattice
calculations of bulava . The problem is that the current model is not suitable
for describing light hadrons as it was originally developed for hadrons with
heavy quarks. What this means in practice is that at least one quark is needed
to be placed on the boundary of five-dimensional space.

A possible way out is to treat $E_{\text{\tiny q}\bar{\text{\tiny q}}}$ as a
model parameter. For $E_{\text{\tiny q}\bar{\text{\tiny q}}}=280\,\text{MeV}$,
this does not cause an essential change of the plots of Figure 9, as
illustrated in Figure 11.

Figure 11: The $E$’s relevant for $V_{0}$ and $V_{1}$. Here $E_{\text{\tiny
q}\bar{\text{\tiny q}}}=280\,\text{MeV}$.

The most visible effect of change in $E_{\text{\tiny q}\bar{\text{\tiny q}}}$
is that string reconnection occurs at much larger $\ell$. Thus, in order to
solve the equation (3.29), one now has to use the asymptotic expansion (C.4).
A simple algebra yields

$\ell_{\text{\tiny
Qq}}=\frac{2\mathsf{g}\sqrt{\mathsf{s}}}{\sigma}\Bigl{(}Q(q)+\mathsf{n}\frac{\text{e}^{\frac{q}{2}}}{\sqrt{q}}+I_{0}-\frac{E_{\text{\tiny
q}\bar{\text{\tiny q}}}}{2\mathsf{g}\sqrt{\mathsf{s}}}\,\Bigr{)}\,.$ (4.1)

For $E_{\text{\tiny q}\bar{\text{\tiny q}}}=280\,\text{MeV}$,
$\ell_{\text{\tiny Qq}}=0.964\,\text{fm}$. For $E_{\text{\tiny
q}\bar{\text{\tiny q}}}=496\,\text{MeV}$, $\ell_{\text{\tiny
Qq}}=0.766\,\text{fm}$ that agrees with the lattice result AP .

An interesting conclusion can be drawn from the examples given: Our results
indicate that the following scale hierarchy is met

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

At this point, one may well ask what happens at the physical value of the pion
mass. We expect that at least the string junction annihilation distance
remains the smallest scale, namely $\boldsymbol{\ell}_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}<l_{\text{\tiny
Qq}}$, $\ell_{\text{\tiny Q}\bar{\text{\tiny Q}}}$. Of course, it is not the
whole answer to this question, but it’s certainly an important piece of it.

###  IV.2 Comparison with the lattice

The potentials for the $Q\bar{Q}q\bar{q}$ system have been studied in lattice
QCD AP ; SP0 . In this case those are extracted from the set of correlators.
The data limitation prevents one from being able to analyze the effects of
string breaking and reconnection but not the effect of string junction
annihilation. In particular, the data extracted from the correlator of meson
operators allow one to gain some insight into the potential $V_{1}$ in the
quark separation range $0.1\text{-}0.7\,\text{fm}$ AP . They are fitted by a
single algebraic expression

$V_{1}(\ell)=-\frac{\alpha}{\ell}\exp\Bigl{(}-\frac{\ell^{\,p}}{d^{\,p}}\,\Bigl{)}\,+2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}\,\,,$ (4.3)

with the parameters $\alpha$, $d$, and $p$.

Keeping in mind the fact that $\boldsymbol{\ell}_{\text{\tiny
Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$ is
$0.196\,\text{fm}$, we expect that the screening length $d$ is near this
value. Therefore we use (3.20) for $V_{1}$, and then solve for the unknown
coefficients in the limit of small $\ell$. So, we get

$\alpha=\alpha_{\text{\tiny Q}\bar{\text{\tiny Q}}}\,,\qquad
d=\sqrt{\frac{\alpha_{\text{\tiny Q}\bar{\text{\tiny
Q}}}}{\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny Q}}}}}\,,\qquad
p=2\,.$ (4.4)

The parameters are expressed in terms of the coefficients of the quark-
antiquark potential. We can make an estimate of the screening length. With our
parameter set, this gives $d=0.238\,\text{fm}$. The value is close to the
range of lattice QCD, where $d=0.16^{+0.05}_{-0.02}\,\text{fm}$ AP ; wagner .

What makes the lattice approach less attractive is that it is difficult to say
whether the system can be thought of as a compact tetraquark or as a pair of
mesons. As we have just seen, this question can be answered in the framework
of the string model. Now for completeness let us show how to construct the
potential $V_{1}$ in this range of quark separations. Consider the model
Hamiltonian

${\cal H}(\ell)=\begin{pmatrix}E_{\text{\tiny Q}\bar{\text{\tiny
Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}(\ell)&\Theta\\\
\Theta&2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\\\ \end{pmatrix}\,,$ (4.5)

where $\Theta$ describes the strength of the mixing between the compact
tetraquark state and two mesons. The potential is given by the smallest
eigenvalue of ${\cal H}$. Explicitly,

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

In principle, $\Theta$ can be computed from a correlator of the operators
corresponding to these two states, but currently this is out of our reach. So,
we treat $\Theta$ as a free parameter and find its value by the best fit of
our prediction to the parameterization suggested from lattice QCD. In Fig.12
we plot $V_{1}$ vs $\ell$. There are two things worth mentioning here.

Figure 12: The potential $V_{1}$ defined by the model Hamiltonian (4.5). Here
$\Theta=75\,\text{MeV}$. The dashed curve corresponds to the lattice
parameterization (4.3).

The first is a visible deviation between the solid and dashed curves on the
interval $0.08\lesssim\ell\lesssim 0.28\,\text{fm}$. In our opinion, it is
early to speak about the real reasons for this as the lattice data are very
limited. The second is a falloff at large $\ell$. It is power law for (4.6)
with $\Theta=const$, but exponential for (4.3). Of course, one can get the
desired exponential falloff by taking $\Theta$ as a Gaussian function with a
pick at $\ell=\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny
Q}}{\text{\tiny q}}\bar{\text{\tiny q}}}$ but it will require one addition
parameter (a Gaussian width).

###  IV.3 The potentials

Having understood the string configurations relevant to the ground and first
excited states of the $Q\bar{Q}q\bar{q}$ system, one can gain some insight
into their energies. One way for doing so is to consider a model Hamiltonian,
similar as it is used in lattice QCD to study the phenomenon of string
breaking FK . For the problem at hand it is

${\cal H}(\ell)=\begin{pmatrix}\,E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}(\ell)+E_{\text{\tiny q}\bar{\text{\tiny q}}}&{}&{}&{}&{}\,\\\
\,{}&2E_{\text{\tiny Q}\bar{\text{\tiny q}}}&{}&{}&\Theta_{ij}\,\\\
\,\Theta_{ij}&{}&E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}(\ell)&{}&{}\,\\\ \,{}&{}&{}&{}&2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}+E_{\text{\tiny q}\bar{\text{\tiny q}}}\,\\\
\end{pmatrix}\,,$ (4.7)

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 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 concretely how the
potentials look like. Nevertheless, we can learn from our experience in the
previous subsection (see also Appendix C) about the order of magnitude of the
$\Theta$’s near the intersection points. With the help of this, the picture
will then look more like what is shown in Figure 13. A

Figure 13: The potentials $V_{0}$ and $V_{1}$. The relevant $E$’s are shown in
dashed lines. Here $E_{\text{\tiny q}\bar{\text{\tiny q}}}=280\,\text{MeV}$.

compact tetraquark configuration contributes dominantly to the $V_{1}$
potential at small quark separations, as we have already seen in Figure 12.

##  V Concluding comments

(1) There is really a difference between the $Q\bar{Q}q\bar{q}$ and
$QQ\bar{q}\bar{q}$-quark systems. In the former, a compact tetraquark
structure shows up in $V_{1}$, while in the latter in $V_{0}$. The basic
reason is that a pair $Q\bar{Q}$ being a color singlet enables configuration
(a) which dominates at small heavy quark separations and hence contributes to
$V_{0}$. Nevertheless the estimate of the corresponding scales shows that the
value of $\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}$ is small and close to that of
$\boldsymbol{\ell}_{\text{\tiny QQ}\bar{\text{\tiny q}}\bar{\text{\tiny
q}}}\,$, see (3.33). Interestingly, these values become almost the same if the
phenomenological rule $E_{\text{\tiny QQ}}=\frac{1}{2}E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$ holds true.141414In the literature it is sometimes
called the Lipkin rule. To see this, let us first note that the equation
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny q}}\bar{\text{\tiny
q}}}(\ell)=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ reduces to
$\frac{1}{2}E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell)=2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}-E_{\text{\tiny Qqq}}$ as follows from Eq.(3.20). On
the other hand, the heavy quark-diquark symmetry implies that the equation
$E_{\text{\tiny QQ}\bar{\text{\tiny q}}\bar{\text{\tiny
q}}}(\ell)=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ for small $\ell$ becomes
$E_{\text{\tiny QQ}}(\ell)\approxeq 2E_{\text{\tiny Q}\bar{\text{\tiny
q}}}-E_{\text{\tiny Qqq}}$. Thus

$\boldsymbol{\ell}_{\text{\tiny Q}\bar{\text{\tiny Q}}{\text{\tiny
q}}\bar{\text{\tiny q}}}\approxeq\boldsymbol{\ell}_{\text{\tiny
QQ}\bar{\text{\tiny q}}\bar{\text{\tiny q}}}$ (5.1)

if the Lipkin rule holds. The same is true for the screening lengths. Those
are given respectively by $d=\sqrt{\frac{\alpha_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}{\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny
Q}}}}}$ and $d=\sqrt{\frac{\alpha_{\text{\tiny
QQ}}}{\boldsymbol{\sigma}_{\text{\tiny QQ}}}}$ QQqq . The $\alpha$’s and
$\boldsymbol{\sigma}$’s are the coefficients in the small $\ell$ expansions of
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ and $E_{\text{\tiny QQ}}$ (see
Eq.(C.2)). Clearly, rescaling of the coefficients by the factor 2 has no
effect on the $d$’s.

(2) An interesting experimental observation is that the $Z$ states are very
close to the corresponding $B\bar{B}$ thresholds ali . Moreover, the mass
differences with respect to the thresholds are positive.151515This requires a
caveat because the errors in the mass determination are not small enough. If
so, then there are no binding energies which bind the mesons in such states.
This could seem puzzling for one who thinks only in terms of $V_{0}$, but not
for one who also takes into consideration $V_{1}$ (see Figure 13). Thus, the
story may really and truly be about the four states contributing to the
construction of $V_{1}$, one of which is a compact tetraquark.

(3) The diquark picture emerged in the case of configuration (c) gives rise to
a naturally defined diquark mass. Using the $\ell$-independent part in
Eq.(3.19), we define it by161616For this definition to make sense, the
separation between diquarks must be larger than $\ell(v_{\star})$.

$m_{[Qq]}=m_{Q}+\mathsf{g}\sqrt{\mathsf{s}}\Bigl{(}{\cal Q}(q)-{\cal
Q}(v_{\star})+\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}+3\mathsf{k}\frac{\text{e}^{-2v_{\star}}}{\sqrt{v_{\star}}}\Bigr{)}\,,$
(5.2)

with $q$ and $v_{\star}$ as in (3.19). The square brackets mean that a diquark
has zero spin. It is interesting to see what results come out of this
definition and how those agree or disagree with phenomenological values. Some
are presented in Table 1.

Diquark |  this work |  DE |  Gi
---|---|---|---
$[cq]$ | 2.508 | 1.973 |  2.120
$[bq]$ | 5.418 | 5.359 |  5.513
Table 1:  Diquark masses (in GeV). The expression (5.2) is evaluated at
$m_{c}=1.27\,\text{GeV}$ and $m_{b}=4.18\,\text{GeV}$ PDG , with the other
parameters as in Sec.III. The results of DE and Gi are fits.

Although there is a visible discrepancy in the case of $m_{[cq]}$, making use
of (5.2) gives a very reasonable value of $m_{[bq]}$.

(4) Actually, our discussion has been so general that it makes sense with or
without spin. But the neglect of spin-dependent effects is not good in making
contact with the real world. We ran into trouble with $q\bar{q}$, where one
expects such effects, and with $m_{[cq]}$, where the $1/m_{c}$ factor seems
unable to completely suppress those. Also including spin will be crucial in
predicting the masses of spin one diquarks.

(5) Intuitively, one expects the probability of reconnection between the
strings connecting quarks in $Q\bar{Q}$ and $q\bar{q}$ to increase with
increasing a separation of the heavy quark-antiquark pair. But it is not clear
at what separation the probability reaches a maximum.171717It is natural to
expect that such a separation coincides with the critical separation
$l_{\text{\tiny Qq}}$ defined from the energy balance equation (3.29). Let us
assume that this happens at the maximum possible separation of the $Q\bar{Q}$

Figure 14: A heavy quark-antiquark pair inside a pion cloud of radius
$r_{\pi}$.

pair inside the pion cloud, as sketched in Figure 14. If so, then

$l_{\text{\tiny Qq}}=2r_{\pi}\,.$ (5.3)

For $m_{\pi}=265\,\text{MeV}$, which is quite close to the value in bulava ,
the (charge) pion radius is $\langle r_{\pi}^{2}\rangle=0.232\,\text{fm}^{2}$
pionR . So, we immediately obtain $l_{\text{\tiny Qq}}=0.963\,\text{fm}$.
Although there is no satisfactory explanation of why our assumption is
correct, it is interesting that it results in approximately the same value of
$l_{\text{\tiny Qq}}$ as Eq.(4.1) evaluated at $m_{\pi}=280\,\text{MeV}$.

Eq.(5.3), if taken literally, gives $l_{\text{\tiny Qq}}=1.318\,\text{fm}$ at
$\langle r_{\pi}^{2}\rangle=0.434\,\text{fm}^{2}$ quoted by Particle Data
Group PDG . This then implies that, in the real world, one might expect
$\ell_{\text{\tiny Q}\bar{\text{\tiny Q}}}<l_{\text{\tiny Qq}}$. So the
hierarchy of scales in (4.2) has to be modified, but we will omit this from
the discussion.

(6) The $Q\bar{Q}q\bar{q}$-quark system has a rich complexity of physics and a
number of unanswered, pressing questions. Making further progress in
theoretical understanding and applications to hadron phenomenology will
require a joint effort by the high-energy community. We hope that our study
provides a useful starting point for a stringy approach to this system.

###### Acknowledgements.

We would like to thank S. Dubovsky, A. Peters, J.-M. Richard, J. Sonnenschein,
and M. Wagner for useful communications and conversations. This work is
supported by Russian Science Foundation grant 20-12-00200 in association with
Steklov Mathematical Institute. We acknowledge the warm hospitality at the
Simons Center for Geometry and Physics at Stony Brook University and at the
Center for Cosmology and Particle Physics at New York University, where part
of this work was completed.

##  Appendix A Notation and definitions

In all Figures throughout the paper, heavy and light quarks (antiquarks) are
denoted by $Q\,(\bar{Q})$ and $q\,(\bar{q})$, and baryon (antibaryon) vertices
by $V\,(\bar{V})$. Strings are represented by smooth curves without self-
intersections. When not otherwise noted, we usually set light quarks
(antiquarks) at $r=r_{q}\,(r_{\bar{q}})$ and vertices at
$r=r_{v}\,(r_{\bar{v}})$. For convenience, we 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 on the interval $[0,1]$ and show how far from the soft-wall these
objects are.181818In these dimensionless units, the soft wall is located at
$1$. To classify the critical separations related to the string interactions
of Figure 3, the notation $l$ is used for (a), $\ell$ for (b), and
$\boldsymbol{\ell}$ for (c).

In order to present the formulas in a general simplified 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)

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

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

This function is singular at $x=0$ and $(0,1)$;

${\cal Q}(x)=\sqrt{\pi}\text{erfi}(\sqrt{x})-\frac{\text{e}^{x}}{\sqrt{x}}\,,$
(A.3)

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

${\cal Q}(x)=-\frac{1}{\sqrt{x}}+\sqrt{x}+O(x^{\frac{3}{2}})\,;$ (A.4)

${\cal
I}(x)=I_{0}-\int_{\sqrt{x}}^{1}\frac{du}{u^{2}}\text{e}^{u^{2}}\Bigl{[}1-u^{4}\text{e}^{2(1-u^{2})}\Bigr{]}^{\frac{1}{2}}\,\,,\qquad
I_{0}=\int_{0}^{1}\frac{du}{u^{2}}\Bigl{(}1+u^{2}-\text{e}^{u^{2}}\Bigl{[}1-u^{4}\text{e}^{2(1-u^{2})}\Bigr{]}^{\frac{1}{2}}\Bigr{)}\,,\qquad
0<x\leq 1\,.$ (A.5)

Numerically, $I_{0}\approx 0.751$.

##  Appendix B A static Nambu-Goto string with fixed endpoints

The purpose of this Appendix is to briefly describe some facts about a static
Nambu-Goto string in the curved geometry (2.1) that may be helpful for
understanding the string configurations of Sect.III. For our purposes, the
only

Figure 15: A static string stretched between two points. $\alpha$ is the
tangent angle. (a) $0<\alpha<\frac{\pi}{2}$. (b) $\alpha=\frac{\pi}{2}$.

cases we need to consider are presented in Figure 15. More details and further
results omitted in this Appendix can be found in a-3qPRD .

Consider a string stretched between the two fixed points $Q(0,0)$ and
$V(\ell,r_{v})$ in the $xr$-plane as shown in Figure 15(a). To get to the
desired formulas here as quickly as possible, we will take as in malda the
static gauge $\xi_{1}=t$ and $\xi_{2}=x$. In this case the string profile is
described by the function $r(x)$ subject to the boundary conditions

$r(0)=0\,,\qquad r(\ell)=r_{v}\,.$ (B.1)

The Nambu-Goto action takes the form

$S=T\mathsf{g}\int_{0}^{\ell}dx\,w(r)\sqrt{1+(\partial_{x}r)^{2}}\,,\qquad
w(r)=\frac{\text{e}^{\mathsf{s}r^{2}}}{r^{2}}\,.$ (B.2)

Here $\mathsf{g}=\frac{R^{2}}{2\pi\alpha^{\prime}}$, $T=\int dt$, and
$\partial_{x}r=\frac{\partial r}{\partial x}$. Since the integrand does not
explicitly depend on $x$, the corresponding Euler-Lagrange equation has the
first integral

$I=\frac{w(r)}{\sqrt{1+(\partial_{x}r)^{2}}}\,.$ (B.3)

After expressing $I$ in terms of $\alpha$ and $r_{v}$, we get the differential
equation $w(r_{v})\cos\alpha=w(r)/\sqrt{1+(\partial_{x}r)^{2}}$ which can be
integrated over the variables $x$ and $r$. So we find

$\ell=\frac{1}{\sqrt{\mathsf{s}}}{\cal L}^{+}(\alpha,v)\,,$ (B.4)

with $v=\mathsf{s}r_{v}^{2}$. The function ${\cal L}^{+}$ is as defined in
Appendix A.

To compute the energy of the string, we first reduce the integral over $x$ in
$S$ to that over $r$. This can be done by using the first integral (B.3).
Since the resulting integral is divergent at $r=0$, we regularize it by
imposing a cutoff $\epsilon$ such that $r\geq\epsilon$. Finally, the
regularized expression is given by

$E_{R}=\frac{S_{R}}{T}=\mathsf{g}\sqrt{\frac{\mathsf{s}}{v}}\int^{1}_{\sqrt{\tfrac{\mathsf{s}}{v}}\epsilon}\,\frac{du}{u^{2}}\,\text{e}^{vu^{2}}\Bigl{(}1-\cos^{2}{}\alpha\,v^{4}\,\text{e}^{2v(1-u^{2})}\Bigr{)}^{-\frac{1}{2}}\,.$
(B.5)

In the limit as $\epsilon\rightarrow 0$, it behaves like

$E_{R}=\frac{\mathsf{g}}{\epsilon}+E+O(\epsilon)\,.$ (B.6)

Subtracting the $\tfrac{1}{\epsilon}$ term and letting $\epsilon=0$, we get a
finite result

$E=\mathsf{g}\sqrt{\mathsf{s}}\,{\cal E}^{+}(\alpha,v)+c\,,$ (B.7)

where $c$ is a normalization constant and ${\cal E}^{+}$ is the function
defined in Appendix A.

At $\alpha=\frac{\pi}{2}$ the string becomes straight. In this case, the
expression (B.7) reduces to

$E=\mathsf{g}\sqrt{\mathsf{s}}{\cal Q}(v)+c\,,$ (B.8)

with the function ${\cal Q}$ given by (A.3). From this it follows that the
energy of the string shown in Figure 15(b) is simply

$E=\mathsf{g}\sqrt{\mathsf{s}}\bigl{(}{\cal Q}(\bar{v})-{\cal
Q}(v)\bigr{)}\,.$ (B.9)

Here $\bar{v}=\mathsf{s}r_{\bar{v}}^{2}$.

##  Appendix C Some details on the quark-antiquark potential

In this Appendix we give a brief summary of the basic results about the heavy
quark-antiquark potential (the ground state energy of a static quark-antiquark
pair) in the presence of two light flavors of equal mass. These are relevant
for our discussion in Sec.III. For standard explanations, see az1 ; a-strb1
whose conventions we follow, unless otherwise stated.

In the problem at hand there are two static string configurations: a connected
configuration like that at bottom of Figure 1(a) and a disconnected
configuration similar to that of Figure 1(b). In five dimensions the connected
configuration includes a string attached to the heavy quark sources on the
boundary of five-dimensional space (see Figure 4(a)). For a Nambu-Goto string
in the background geometry (2.1), the relation between the string energy and
quark separation along the $x$-axis is written in parametric form

$\ell=\frac{2}{\sqrt{\mathsf{s}}}{\cal L}^{+}(0,v)\,,\quad E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}=2\mathsf{g}\sqrt{\mathsf{s}}\,{\cal E}^{+}(0,v)+2c\,,$
(C.1)

where $c$ is a normalization constant and $v$ is a dimensionless parameter
running from $0$ to $1$. It is given by $v=\mathsf{s}r_{0}^{2}$, with $r_{0}$
a turning point az1 .

The behavior of $E_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ for small $\ell$ is
given by

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell)=-\frac{\alpha_{\text{\tiny
Q}\bar{\text{\tiny Q}}}}{\ell}+2c+\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}\ell+o(\ell)\,,$ (C.2)

with

$\alpha_{\text{\tiny Q}\bar{\text{\tiny
Q}}}=(2\pi)^{3}\Gamma^{-4}\bigl{(}\tfrac{1}{4}\bigr{)}\mathsf{g}\,,\qquad\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny
Q}}}=\frac{1}{2}(2\pi)^{-2}\Gamma^{4}\bigl{(}\tfrac{1}{4}\bigr{)}\mathsf{g}\mathsf{s}\,.$
(C.3)

On the other hand, for large $\ell$ it is

$E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}(\ell)=\sigma\ell-2\mathsf{g}\sqrt{\mathsf{s}}\,I_{0}+2c+o(1)\,,\qquad\text{with}\qquad\sigma=\text{e}\mathsf{g}\mathsf{s}\,.$
(C.4)

Here $\sigma$ is the physical string tension and $I_{0}$ is defined in
Appendix A. Note that the coefficients $\boldsymbol{\sigma}_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$ and $\sigma$ are not equal to each other.
Numerically, $\boldsymbol{\sigma}_{\text{\tiny Q}\bar{\text{\tiny
Q}}}/\sigma\approx 0.805$.

The five-dimensional counterpart of the disconnected configuration is shown in
Figure 4(b). Since the mesons are non-interacting, the energy is twice the
heavy-meson mass. The latter is given by Eq.(3.6).

As in lattice gauge theory bulava , the critical separation (string breaking
distance) is defined by equating the energies of the configurations

$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell_{\text{\tiny Q}\bar{\text{\tiny
Q}}})=2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\,.$ (C.5)

The physical meaning of such a distance is that it gives a condition for
determining which configuration is dominant in the ground state of the system
(see Figure 16). The equation simplifies for large quark separations, where
$E_{\text{\tiny Q}\bar{\text{\tiny Q}}}(\ell)$ is a linear function of
$\ell$.191919For the parameter values we use, this is true for $\ell\gtrsim
0.5\,\text{fm}$, whereas the string breaking distance is about $1\,\text{fm}$.
If so, then it follows from Eqs.(3.6) and (C.4) that the string breaking
distance is

$\ell_{\text{\tiny Q}\bar{\text{\tiny
Q}}}=\frac{2}{\text{e}\sqrt{\mathsf{s}}}\Bigl{(}{\cal
Q}(q)+\mathsf{n}\frac{\text{e}^{\frac{1}{2}q}}{\sqrt{q}}+I_{0}\Bigr{)}\,.$
(C.6)

Here $q$ is a solution to Eq.(3.7).

The potential is formally defined by requiring $V_{\text{\tiny
Q}\bar{\text{\tiny Q}}}=\min\bigl{(}E_{\text{\tiny QQq}},2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}\bigr{)}$.202020Conventionally, $V_{0}$ is used to
denote the potential (ground state energy), but we reserve this notation for
the $Q\bar{Q}q\bar{q}$ system. Thus it interpolates between $E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$ at small quark separations and $2E_{\text{\tiny
Q}\bar{\text{\tiny q}}}$ at large ones. The problem with this formal
definition is that it does not say precisely what happens at intermediate
quark separations. A way out would be to use the same mixing analysis as in
lattice gauge theory FK ; bulava . So, consider a model Hamiltonian of a two-
state system

${\cal H}(\ell)=\begin{pmatrix}E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}(\ell)&\Theta_{\text{\tiny Q}\bar{\text{\tiny Q}}}\\\ \Theta_{\text{\tiny
Q}\bar{\text{\tiny Q}}}&2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\\\
\end{pmatrix}\,,$ (C.7)

with $\Theta_{\text{\tiny Q}\bar{\text{\tiny Q}}}$ describing the mixing
between the two states. The potential is given by the smallest eigenvalue of
the model Hamiltonian. Explicitly,

$V_{\text{\tiny Q}\bar{\text{\tiny Q}}}=\frac{1}{2}\Bigl{(}E_{\text{\tiny
Q}\bar{\text{\tiny Q}}}+2E_{\text{\tiny Q}\bar{\text{\tiny
q}}}\Bigr{)}-\sqrt{\frac{1}{4}\Bigl{(}E_{\text{\tiny Q}\bar{\text{\tiny
Q}}}-2E_{\text{\tiny Q}\bar{\text{\tiny q}}}\Bigr{)}^{2}+\Theta_{\text{\tiny
Q}\bar{\text{\tiny Q}}}^{2}}\,.$ (C.8)

We conclude by giving a simple example of the potential to illustrate this
construction. For constant $\Theta_{\text{\tiny Q}\bar{\text{\tiny Q}}}$, the
potential is as shown in Figure 16.

Figure 16: The static quark-antiquark potential determined using the model
Hamiltonian (C.7). Here $\Theta_{\text{\tiny Q}\bar{\text{\tiny
Q}}}=47\,\text{MeV}$, as in bulava .

It asymptotically approaches $E_{\text{\tiny QQq}}$ as $\ell$ tends to zero
and $2E_{\text{\tiny Q}\bar{\text{\tiny q}}}$ as $\ell$ tends to infinity. The
transition between these two regimes occurs around $\ell=\ell_{\text{\tiny
Q}\bar{\text{\tiny Q}}}$, as expected.

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