---
abstract: 'We perform the detailed study of the tetraquark (4Q) potential $V_{\rm 4Q}$ for various QQ-$\rm \bar{Q}\bar{Q}$ systems in SU(3) lattice QCD with $\beta=6.0$ and $16^3 \times 32$ at the quenched level.  For about 200 different patterns of 4Q systems, $V_{\rm 4Q}$ is extracted from the 4Q Wilson loop in 300 gauge configurations, with the smearing method to enhance the ground-state component.  We calculate $V_{\rm 4Q}$ for planar, twisted, asymmetric, and large-size 4Q configurations, respectively.  Here, the calculation for large-size 4Q configurations is done by identifying $16^2 \times 32$ as the spatial size and 16 as the temporal one, and the long-distance confinement force is particularly analyzed in terms of the flux-tube picture.  When QQ and $\rm \bar{Q}\bar{Q}$ are well separated, $V_{\rm 4Q}$ is found to be expressed as the sum of the one-gluon-exchange Coulomb term and multi-Y type linear term based on the flux-tube picture.  When the nearest quark and antiquark pair is spatially close, the system is described as a “two-meson" state.  We observe a flux-tube recombination called as “flip-flop" between the connected 4Q state and the “two-meson" state around the level-crossing point.  This leads to infrared screening of the long-range color forces between (anti)quarks belonging to different mesons, and results in the absence of the color van der Waals force between two mesons.' 
author:
- Fumiko Okiharu
- Hideo Suganuma
- 'Toru T. Takahashi'
title: 'Detailed analysis of the tetraquark potential and flip-flop in SU(3) lattice QCD'
---

Introduction {#sec1}
============

The inter-quark force is one of the elementary quantities for the study of the multi-quark system in the quark model.  Since the first application of lattice QCD simulations was done for the inter-quark potential between a quark and an antiquark using the Wilson loop in 1979 [@C7980], the study of the inter-quark force has been one of the important issues in lattice QCD [@R97].  Actually, in hadron physics, the inter-quark force can be regarded as an elementary quantity to connect “the quark world" to “the hadron world", and plays an important role to hadron properties. 

In addition to the $\rm Q\bar Q$ potential [@C7980; @R97; @APE87; @BS93], recent lattice QCD studies clarify the three-quark (3Q) potential [@TMNS01; @TSNM02; @TS0304; @SIT04], which is responsible to the baryon structure.  In fact, our group recently studied the 3Q potential $V_{\rm 3Q}$ in detail with lattice QCD, and clarified that it obeys the Coulomb plus Y-type linear potential [@TMNS01; @TSNM02; @TS0304; @SIT04].  However, no one knows the inter-quark force from QCD in the exotic multi-quark system such as tetraquark mesons (QQ-$\bar {\rm Q}\bar{\rm Q}$), pentaquark baryons (4Q-$\bar{\rm Q}$), dibaryons (6Q) and so on. 

In these years, various candidates of multi-quark hadrons have been experimentally observed.  $\Theta^+$(1540) [@LEPS; @DIANA; @CLAS; @SAPHIR], $\Xi^{--}(1862)$ [@H1] and $\Theta_c(3099)$ [@NA49] are considered to be pentaquark (5Q) states [@DPP97; @Z04] because of their exotic quantum numbers, although some high-energy experiments report null results[@null].  $X$(3872) [@BelleX; @CDF; @D0; @BABARX] and $D_s(2317)$ [@BABARDs; @BelleDs] are expected to be tetraquark (4Q) states [@CG03; @CP04; @T03; @PS04; @W04; @BK04; @BG04; @ELQ04; @S04] from the consideration of their mass, narrow decay width and decay mode. 

These discoveries of multi-quark hadrons are expected to reveal new aspects of hadron physics, especially for the inter-quark force such as the quark confinement force, the color-magnetic interaction and the diquark correlation [@JW03].  According to these experimental results, it is desired to investigate the inter-quark force in the multi-quark system directly based on QCD [@GLPM96; @PGM99; @OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p; @AK05], which would present the proper Hamiltonian for the quark-model calculation of multi-quarks [@WI82; @SR03; @KMN04]. 

As for the 4Q candidates, $X$(3872) was discovered in the process $B^+ \rightarrow K^++X(3872) \rightarrow K^++\pi^-\pi^+J/\psi$ at KEK(Belle) in 2003 [@BelleX], and its existence was confirmed by Fermilab (CDF[@CDF], D0[@D0]) and SLAC(BABAR)[@BABARX].  $D_s$(2317) was also found in $B$-$\bar{B}$ reaction at $\Upsilon(4S)$ resonance at SLAC(BABAR) [@BABARDs] and consecutively at KEK(Belle) [@BelleDs] in 2003.  As the unusual features of $X$(3872), its mass is rather close to the threshold of $D^0(c\bar{u})$ and $\bar{D}^{0*}(u\bar{c})$, and its decay width is very narrow as $\Gamma <$ 2.3MeV (90 % C.L.).  These facts seem to indicate that $X$(3872) is a 4Q state [@CG03; @CP04] or a molecular state of $D^0(c\bar{u})$ and $\bar{D}^{0*}(u\bar{c})$ [@T03; @PS04; @W04; @BK04] rather than an excited-state of a $c\bar c$ system [@BG04; @ELQ04; @S04].  Also, $D_s$(2317) cannot be regarded as the simple meson of $c\bar{s}$, but is conjectured to be a 4Q state from the similar reasons on the mass and the narrow decay width. 

Also in the light quark sector, the possibility of 4Q hadrons has been pointed out.  For instance, Jaffe proposed in 1977 that the light scalar nonet including $f_0$(980) and $a_0$(980) can be interpreted as $\rm QQ\bar{Q}\bar{Q}$ rather than $\rm Q\bar{Q}$ [@J77].  Since then, many studies of the scalar nonet have been done in terms of the 4Q state [@BFSS99; @AJ00]. 

As an analytical guiding model for the multi-quark system, the flux-tube picture [@N74; @KS75; @CKP83; @CNN79; @IP8385; @P84; @O85; @LLMRSY86] has been investigated for the structure and the reaction of hadrons, and is supported by recent lattice QCD studies [@TMNS01; @TSNM02; @PGM99; @OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p; @IBSS03].  In this picture, valence quarks are linked by the color flux-tube as a quasi-one-dimensional object.  The flux-tube has a large string tension of about $\sigma \simeq 0.89$ GeV/fm, and therefore its length is to be minimized.  For the multi-quark system, this picture predicts an interesting phenomenon of the “flip-flop", i.e., a recombination of the flux-tube configuration so as to minimize the total length of the flux-tube in accordance with the change of the quark location [@O85; @LLMRSY86].  This process is important not only for the structure of multi-quark systems but also for the reaction process of hadrons. 

In this paper, we study the tetraquark (4Q) potential, i.e., the interaction between quarks in the 4Q system directly from QCD by using SU(3) lattice QCD at the quenched level, and investigate the hypothetical flux-tube picture for the multi-quark system and the flip-flop in terms of QCD.  Here, the lattice QCD Monte Carlo simulation is the first-principle calculation of QCD and is considered as the only reliable method for nonperturbative QCD at present.  We note that lattice QCD is also a useful method to select out the correct picture for nonperturbative QCD in the low-energy region through the comparison with the lattice results. 

The organization of this paper is as follows.  In Sect. II, after a brief review on the lattice studies of static quark potentials, we present a theoretical form for the 4Q potential based on the flux-tube picture.  In Sect. III, we present the formalism for the 4Q Wilson loop and the 4Q potential.  The lattice QCD results are shown in Sect. IV.  In Sect. VI, we compare the lattice QCD results with the theoretical form, and discuss the flux-tube picture and the flip-flop.  Sect. VI is devoted for summary and concluding remarks. 

Theoretical consideration for the 4Q potential {#theoreticalAnsatz}
==============================================

$\bf Q\bar Q$, 3Q and 5Q potentials
-----------------------------------

To begin with, we give a theoretical consideration for the multi-quark potential.  From a lot of lattice QCD studies [@C7980; @R97; @APE87; @BS93; @TMNS01; @TSNM02], the static Q$\rm \bar{Q}$ potential is known to be well described by $$\begin{aligned}
V_{\rm Q\bar{Q}}
&=&-\frac{A_{\rm Q\bar{Q}}}{r}
             +\sigma_{\rm Q\bar{Q}}r+C_{\rm Q\bar{Q}},
\label{QQbar}\end{aligned}$$ where $r$ denotes the distance between the quark and the antiquark.  The first term is considered to be the Coulomb term due to the one-gluon exchange (OGE) and $A_{\rm Q\bar{Q}}$ denotes the Coulomb coefficient.  The second term is the linear confinement term with the string tension, $\sigma_{\rm Q\bar{Q}} \simeq$ 0.89 GeV/fm. 

From the recent detailed studies in lattice QCD with ($\beta$=5.7, $12^3 \times 24$), ($\beta$=5.8, $16^3 \times 32$), ($\beta$=6.0, $16^3 \times 32$), and ($\beta$=6.2, $24^4$) [@TMNS01; @TSNM02; @TS0304; @SIT04; @STOI04a; @STOI04b; @SOTI04; @SIOT04], the 3Q potential is clarified to be the sum of the OGE Coulomb term and the Y-type linear confinement term as $$\begin{aligned}
V_{\rm 3Q}=-A_{\rm 3Q}\sum_{i<j}\frac{1}{|{\bf r}_i-{\bf r}_j|}
             +\sigma_{\rm 3Q}L_{\rm min}+C_{\rm 3Q}.\end{aligned}$$ Here, $L_{\rm min}$ denotes the minimal value of the total flux-tube length, which corresponds to the Y-shaped flux-tube linking the three quarks.  In fact, the lattice data of the 3Q potential, $V_{\rm 3Q}^{\rm latt}$, can be well reproduced with only three parameters, $A_{\rm 3Q}$, $\sigma_{\rm 3Q}$ and $C_{\rm 3Q}$.

To demonstrate the validity of the Y-Ansatz, we show in Fig.1 the lattice QCD data of the 3Q confinement potential $V_{\rm 3Q}^{\rm conf}$, i.e., the 3Q potential subtracted by its Coulomb and constant parts, $$\begin{aligned}
V_{\rm 3Q}^{\rm conf} \equiv V_{\rm 3Q}^{\rm latt}-\left\{-A_{\rm 3Q}\sum_{i<j}\frac{1}{|{\bf r}_i-{\bf r}_j|}+C_{\rm 3Q}\right\},\end{aligned}$$ plotted against Y-shaped flux-tube length $L_{\rm min}$, at $\beta$=5.8, 6.0 and 6.2 in the lattice unit.  For each $\beta$, clear linear correspondence is found between 3Q confinement potential $V_{\rm 3Q}^{\rm conf}$ and $L_{\rm min}$ as $V_{\rm 3Q}^{\rm conf} \simeq \sigma_{\rm 3Q} L_{\rm min}$, which indicates the Y-Ansatz for the 3Q potential [@STOI04b; @SOTI04; @SIOT04]. 

![  The lattice QCD result for the 3Q confinement potential $V_{\rm 3Q}^{\rm conf}$, i.e., the 3Q potential subtracted by its Coulomb and constant parts, plotted against Y-shaped flux-tube length $L_{\rm min}$ at $\beta$=5.8, 6.0 and 6.2 in the lattice unit.  The clear linear correspondence between 3Q confinement potential $V_{\rm 3Q}^{\rm conf}$ and $L_{\rm min}$ indicates the Y-Ansatz for the 3Q potential. ]( Fig1.eps){height="7cm"}

\[Fig1\]

This lattice QCD result strongly supports the flux-tube picture for baryons, and the Y-type flux-tube formation is actually observed in lattice QCD through the direct measurement of the action density in the presence of spatially-fixed three quarks [@STOI04a; @STOI04b; @SOTI04; @SIOT04; @IBSS03].  The Y-Ansatz for the 3Q system is also supported by recent further lattice QCD studies [@BIMPS04; @BS04] and analytical studies [@KS03; @C04]. 

As for the relation between $V_{\rm Q\bar Q}$ and $V_{\rm 3Q}$, we have found the OGE result $A_{\rm 3Q} \simeq \frac12 A_{\rm Q\bar Q}$ and the universality of the string tension $\sigma_{\rm 3Q} \simeq \sigma_{\rm Q\bar Q}$, which also supports the flux-tube picture [@N74; @KS75; @CKP83; @CNN79; @IP8385; @P84; @O85; @LLMRSY86] and the strong-coupling expansion scheme [@KS75; @CKP83]. 

Very recently, the 5Q potential is also studied in lattice QCD [@OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p; @AK05].  It is well described by the OGE Coulomb plus multi-Y type linear potential [@OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p].  With the minimal length $L_{\rm min}$ of the flux-tube, the 5Q potential can be well described as $$\begin{aligned}
&&V_{\rm 5Q}
=\frac{g^2}{4\pi} \sum_{i<j} \frac{T^a_i T^a_j}{|{\bf r}_i-{\bf r}_j|}+\sigma_{\rm 5Q} L_{\rm min}+C_{\rm 5Q} \nonumber \\
&=&-A_{\rm 5Q}\{ ( \frac1{r_{12}}  + \frac1{r_{34}}) 
+\frac12(\frac1{r_{15}} +\frac1{r_{25}} +\frac1{r_{35}} +\frac1{r_{45}}) \nonumber \\
&+&\frac14(\frac1{r_{13}} +\frac1{r_{14}} +\frac1{r_{23}} +\frac1{r_{24}}) \}
+\sigma_{\rm 5Q} L_{\rm min}+C_{\rm 5Q},
\label{V5Qth}\end{aligned}$$ with $(A_{\rm 5Q}, \sigma_{\rm 5Q})$ fixed to be $(A_{\rm 3Q}, \sigma_{\rm 3Q})$ following the OGE result and the universality of the string tension.  This lattice result also supports the flux-tube picture. 

Theoretical Ansätze for the 4Q potential
----------------------------------------

Now, we investigate the theoretical form of the tetraquark (4Q) potential $V_{\rm 4Q}$ with respect to the flux-tube picture, which seems workable for $\rm Q\bar Q$ mesons and baryons.  For the argument of the low-lying 4Q states, we consider the 4Q state of ((QQ)$_{\bar{3}}$-($\rm \bar{Q}\bar{Q}$)$_3$)${_1}$ as shown in Fig.2.  Here, (QQ)$_{\bar{3}}$ denotes that two quarks form the $\rm \bar{\bf 3}$ representation of the color SU(3).  The meaning of ($\rm \bar{Q}\bar{Q}$)$_3$ is similar.  By combining (QQ)$_{\bar{\bf 3}}$ with ($\rm \bar{Q}\bar{Q}$)$_3$, the 4Q system can be constructed as a color-singlet state.  We note that another possible 4Q system of ((QQ)$_6$-$(\bar{\rm Q}\bar{\rm Q}$)$_{\bar{6}})_1$ is expected to be a highly-excited state, since the attractive (repulsive) force acts between quarks, when the QQ cluster belongs to $\bar{\bf 3}$ (${\bf 6}$) representation in a perturbative sense, which leads to the $\bar{\bf 3}$ diquark picture [@JW03]. 

![ The tetraquark (QQ-$\rm \bar{Q}\bar{Q}$) state.  The QQ ($\rm \bar{Q}\bar{Q}$) cluster belongs to $\bar{\bf 3}$ ([**3**]{}) representation of the color SU(3). []{ data-label="Fig2"}](Fig2.eps){width="5cm"}

In the flux-tube picture, the flux-tube is formed so as to minimize the total flux-tube length of the system for the low-lying state.  For the low-lying 4Q system, there are two candidates for the flux-tube configuration according to the 4Q location.  One is the connected flux-tube system where all quarks and antiquarks are connected with the single flux-tube as shown in Fig.3.  The other is the disconnected flux-tubes corresponding to a “two-meson" state as shown in Fig.4.  Note that the 4Q state of ((QQ)$_{\bar{3}}$-($\rm \bar{Q}\bar{Q}$)$_3$)${_1}$ generally includes such a “two-meson" state of $(\rm Q \bar Q)_1 (\rm Q \bar Q)_1$ as shown in Appendix A. For each case, we consider below the theoretical form of the tetraquark potential $V_{\rm 4Q}$.

![ The connected tetraquark system.  All quarks and antiquarks are connected with the single flux-tube, which is double-Y-shaped. []{ data-label="Fig3"}](Fig3.eps){height="4cm"}

![ The disconnected tetraquark system, which corresponds to a “two-meson" state. []{ data-label="Fig4"}](Fig4.eps){height="4cm"}

### OGE plus multi-Y Ansatz for the connected 4Q system. 

For the connected 4Q system, we propose the “OGE plus multi-Y Ansatz" as a theoretical form of $V_{\rm 4Q}$ from the viewpoint of the flux-tube picture.  This type of the flux-tube configuration is plausible when the distance between QQ and $\rm \bar Q\bar Q$ is enough long compared with the size of these clusters.  For such a system, all quarks and antiquarks are linked by the connected double-Y-shaped flux-tube as shown in Fig.3, and the 4Q potential $V_{\rm 4Q}$ is described by the OGE Coulomb plus multi-Y linear potential $V_{\rm c4Q}$ as $$\begin{aligned}
V_{\rm 4Q}
&=&\frac{g^2}{4\pi}\sum_{i<j}\frac{T^a_i T^a_j}{|r_i-r_j|}
     +\sigma_{\rm 4Q}L_{\rm min}+C_{\rm 4Q} \nonumber \\
&=&-A_{\rm 4Q}\{(\frac{1}{r_{12}}+\frac{1}{r_{34}})
    +\frac{1}{2}(\frac{1}{r_{13}}+\frac{1}{r_{14}} \nonumber \\
& &             +\frac{1}{r_{23}}+\frac{1}{r_{24}}) \}
   +\sigma_{\rm 4Q}L_{\rm min}+C_{\rm 4Q} \equiv V_{\rm c4Q}
\label{Vc4Q}\end{aligned}$$ with $r_{ij} \equiv |{\bf r}_i-{\bf r}_j|$ and $L_{\rm min}$ being the minimal value of the total flux-tube length.  Here, ${\bf r}_i$ denotes the location of $i$th (anti)quark in Fig.3. 

The first term describes the Coulomb term due to the OGE process.  Note that there appear two kinds of Coulomb coefficients ($A_{\rm 4Q}$, $\frac12 A_{\rm 4Q}$) in the 4Q system, while only one Coulomb coefficient, $A_{\rm Q\bar {\rm Q}}$ or $A_{\rm 3Q}$, appears in the Q$\bar{\rm Q}$ or the 3Q system.  In this definition, the Coulomb coefficient $A_{\rm 4Q}$ is expected to satisfy $A_{\rm 4Q} \simeq \frac{1}{2}A_{\rm Q\bar{Q}}$ as the OGE results.  The brief derivation of the OGE Coulomb terms is shown in Appendix A.

The second term is the linear confinement term with the string tension $\sigma_{\rm 4Q}$, which is expected to satisfy $\sigma_{\rm 4Q} \simeq \sigma_{\rm Q \bar Q} \simeq$0.89 GeV/fm as the universality of the string tension.  Similar to the 3Q and the 5Q systems, the Y-type junction appears in this case, and $L_{\rm min}$ is expressed by the length of the double-Y-shaped flux-tube as shown in Fig.3. 

In the extreme case, e.g., $r_{12}=r_{34} \gg r_{13}=r_{24}$, the lowest connected 4Q system takes an X-shaped flux-tube, although the energy of such system is larger than that of the two-meson state in most cases. 

### The two-meson Ansatz for the disconnected 4Q system

For the disconnected 4Q system corresponding to the “two-meson" state as shown in Fig.4, we adopt the “two-meson Ansatz" for $V_{\rm 4Q}$. Such a flux-tube configuration is plausible when the nearest quark and antiquark pair is spatially close and the system can be regarded as the “two-meson state" rather than an inseparable 4Q state.  For such a system as shown in Fig.4, the total potential $V_{\rm 4Q}$ for the 4Q system would be approximated to be the sum of two Q$\rm \bar{Q}$ potentials in Eq.(\[QQbar\]) as $$\begin{aligned}
V_{\rm 4Q}
&=&V_{\rm Q\bar{Q}}(r_{13}) + V_{\rm Q\bar{Q}}(r_{24}) \nonumber \\
&=&-A_{\rm Q\bar{Q}}(\frac{1}{r_{13}}+\frac{1}{r_{24}})
           +\sigma_{\rm Q\bar{Q}}(r_{13}+r_{24}) +2C_{\rm Q\bar{Q}} \nonumber \\
&\equiv& V_{\rm 2Q\bar Q},
\label{V2QQbar}\end{aligned}$$ assuming that the inter-meson force is subdominant. 

The 4Q potential form and the flip-flop
---------------------------------------

For the lowest 4Q system, the 4Q potential $V_{\rm 4Q}$ would be expressed with lower energy of the connected 4Q system or the two-meson system, $$\begin{aligned}
V_{\rm 4Q}={\rm min}(V_{\rm c4Q}, V_{2\rm Q\bar Q}). 
\label{4Qth}\end{aligned}$$

As a physical consequence of Eq.(\[4Qth\]) based on the flux-tube picture, there can occur a physical transition between the connected 4Q state and the two-meson state according to the change of the 4Q location.  This phenomenon occurs through the recombination of the flux-tube and is called as the “flip-flop". ( A popular usage of the “flip-flop" may be for the simple flux-tube recombination between two-meson states.  We here use this term as the general flux-tube recombination.) 

The flip-flop is important for the properties of 4Q states especially for the decay process into two mesons.  In addition, the flux-tube recombination between two-meson states can be realized through the two successive flip-flops between the two-meson state and the connected 4Q state.  Therefore, this type of the flip-flop is important also for the reaction mechanism between two mesons. 

Formalism for the 4Q potential in lattice QCD {#formalism}
=============================================

In this section, we present the formalism to extract the static 4Q potential.  Similar to the derivation of the Q$\rm\bar{Q}$ potential from the Wilson loop, the 4Q potential $V_{\rm 4Q}$ can be derived from the gauge-invariant 4Q Wilson loop $W_{\rm 4Q}$ as shown in Fig.5 [@STOI04b; @OST04p; @AK05]. 

![ The tetraquark (4Q) Wilson loop for the calculation of the 4Q potential $V_{\rm 4Q}$. The contours $M_i$ ($i$=1,2) are line-like, and $L_i,R_i$ ($i$=1,2) are staple-like.  The 4Q gauge-invariant state is generated at $t=0$ and is annihilated at $t=T$. The four quarks ($\rm QQ \bar Q \bar Q$) are spatially fixed in $\rm \bf R^3$ for $0<t<T$.[]{data-label="Fig5"}](Fig5.eps){width="7cm"}

The 4Q Wilson loop is defined by $$\begin{aligned}
W_{\rm 4Q} \equiv 
\frac{1}{3} {\rm tr} 
(\tilde{M}_1 \tilde{L}_{12} \tilde{M}_2 \tilde{R}_{12}).\end{aligned}$$ $\tilde{M}_i$, $\tilde{L_i}$,$\tilde{R_i}$ ($i$=1,2) are given by $$\begin{aligned}
\tilde{M}_i,\tilde{L_i}, \tilde{R_i}
\equiv 
P \exp{\{ig \int_{M_i,L_i,R_i}dx^\mu A_\mu (x)\}}\in \rm{SU(3)_c}.\end{aligned}$$ Here, $\tilde{L}_{12}$ and $\tilde{R}_{12}$ are defined by $$\begin{aligned}
\tilde{L}_{12}^{a'a} 
\equiv \frac{1}{2}\epsilon^{abc}\epsilon^{a'b'c'}
L_1^{bb'}L_2^{cc'}, \\
\tilde{R}_{12}^{a'a} 
\equiv \frac{1}{2}\epsilon^{abc}\epsilon^{a'b'c'}
R_1^{bb'}R_2^{cc'}.\end{aligned}$$ The ground-state 4Q potential $V_{\rm 4Q}$ is extracted as $$\begin{aligned}
V_{\rm 4Q}=-\lim_{T \rightarrow \infty} 
\frac{1}{T} \ln \langle W_{\rm 4Q} \rangle.\end{aligned}$$

In general, the 4Q Wilson loop $\langle W_{\rm 4Q} \rangle$ contains excited-state contributions, and is expressed as $$\begin{aligned}
\langle W_{4Q} \rangle = \sum_{n=0}^\infty C_n e^{-V_n T},\end{aligned}$$ where $V_0$ denotes the ground-state 4Q potential $V_{\rm 4Q}$ and $V_{n}$ $(n=1,2,3..)$ the $n$th excited-state potential.  In principle, $V_{\rm 4Q}$ can be obtained from the behavior of $\langle W_{\rm 4Q} \rangle$ at the large $T$ region where the ground-state contribution becomes dominant.  In the practical simulation, however, the accurate measurement of $V_{\rm 4Q}$ is not easy for large $T$, since $\langle W_{\rm 4Q} \rangle$ decreases exponentially with $T$.

To extract the ground-state potential $V_{\rm 4Q}$ in lattice QCD, we adopt the gauge-covariant smearing method [@R97; @APE87; @BS93; @TMNS01; @TSNM02; @TS0304] to enhance the ground-state component of the 4Q state in the 4Q Wilson loop.  The smearing is known to be a powerful method for the accurate measurement of the Q-$\bar {\rm Q}$ [@R97; @APE87; @BS93] and the 3Q potentials [@TMNS01; @TSNM02; @TS0304], and is expressed as the iterative replacement of the spatial link variables $U_i(s)$ ($i$=1,2,3) by the obscured link variables $\bar U_i(s)\in {\rm SU}(3)_c$ which maximizes ${\rm Re} \,{\rm tr} \,\,\{\bar U_i^{\dagger}(s) V_i(s)\}$ with $$\begin{aligned}
V_i(s)\equiv 
\alpha U_i(s)+\sum_{j \ne i} \sum_{\pm} 
\{ U_{\pm j}(s)U_i(s\pm \hat j)U_{\pm j}^\dagger (s+\hat i) \}~~~ 
\label{smearing}\end{aligned}$$ with the simplified notation of $U_{-j}\equiv U^\dagger_{j}(s-\hat j)$. We here adopt the smearing parameter $\alpha=2.3$ and the iteration number $N_{\rm smr}=30$, which enhance the ground-state component in the 4Q Wilson loop at $\beta$=6.0 in most cases. ( See the next section.) 

Lattice QCD results for the 4Q potential {#results}
========================================

The lattice QCD simulations are performed with the standard plaquette action at $\beta=6.0$ on the $16^3 \times 32$ lattice at the quenched level.  The lattice spacing $a$ is estimated as $a \simeq$ 0.104fm, which leads to the string tension $\sigma_{\rm Q\bar Q} \simeq (427{\rm MeV})^2$ in the Q$\rm \bar Q$ potential, using the numerical relation $\sigma_{\rm Q\bar Q} \simeq 0.0506a^{-2}$ obtained from the fitting analysis on the on-axis data of the Q$\bar {\rm Q}$ potential in lattice QCD at $\beta=6.0$ [@TSNM02; @SIT04].  The gauge configurations are taken every 500 sweeps after 5000 sweeps using the pseudo-heat-bath algorism.  We use 300 configurations for the 4Q potential simulation.  For the estimation of the statistical error of the lattice data, we adopt the jack-knife error estimate. 

On the $16^3 \times 32$ lattice, we investigate the typical configuration of 4Q systems as shown in Figs.6 and 7.  In Fig.6, the 4Q system has a planar structure.  In Fig.7, the 4Q system has a twisted (three-dimension) structure.  In particular, we analyze in detail the symmetric planar and twisted 4Q configurations with $d_1=d_2=d_3=d_4\equiv d$, although more general asymmetric 4Q configurations with various $(d_1,d_2,d_3,d_4)$ are also investigated. 

For the 4Q configurations with $h \le 8$, we identify $16^3$ as the spatial size and 32 as the temporal one.  On the other hand, the calculation for the large-size 4Q configurations with $h > 8$ is performed by identifying $16^2 \times 32$ as the spatial size and 16 as the temporal one.  In both cases, we use corresponding translational and rotational symmetries on lattices for the calculation of $\langle W_{\rm 4Q}\rangle$.

![ A planar configuration of the tetraquark system.  $\rm Q_1 Q_2$ is parallel to $\rm \bar{Q}_3 \bar{Q}_4$, and $\rm H_1 H_2$ is perpendicular to $\rm Q_1 Q_2$ and $\rm \bar{Q}_3 \bar{Q}_4$. We call the cases with $d_1=d_2=d_3=d_4\equiv d$ as “symmetric planar 4Q configurations".[]{data-label="Fig6"}](Fig6.eps){width="5cm"}

![ A twisted configuration of the tetraquark system.  $\rm Q_1 Q_2$ is perpendicular to $\rm \bar{Q}_3 \bar{Q}_4$, and $\rm H_1 H_2$ is perpendicular to $\rm Q_1 Q_2$ and $\rm \bar{Q}_3 \bar{Q}_4$. We call the cases with $d_1=d_2=d_3=d_4\equiv d$ as “symmetric twisted 4Q configurations".[]{data-label="Fig7"}](Fig7.eps){width="5cm"}

For these types of 4Q configurations, we construct the 4Q Wilson loop $W_{\rm 4Q}$ with the junctions locating at ${\rm H}_1$ and ${\rm H}_2$, and calculate the 4Q potential $V_{\rm 4Q}$ from $\langle W_{\rm 4Q}\rangle$ using the smearing method. 

For the suitable choice of the smearing parameter $\alpha$ and the iteration number $N_{\rm smr}$ in Eq.(\[smearing\]), we perform some numerical tests with various values of $\alpha$ and $N_{\rm smr}$, and finally adopt $\alpha=2.3$ and $N_{\rm smr}=30$, which are found to enhance the ground-state component in the 4Q Wilson loop at $\beta$=6.0 in most cases.  For the demonstration, we show in Fig.8 a typical example of the ground-state-overlap quantity $$\begin{aligned}
C_0 \equiv \langle W_{\rm 4Q}(T)\rangle^{T+1}/\langle W_{\rm 4Q}(T+1)\rangle^{T}\end{aligned}$$ plotted against the iteration number $N_{\rm smr}$ at $\alpha=2.3$ for the symmetric planar 4Q configuration with $(d,h)=(1,5)$. Here, the ground-state-overlap quantity $C$ indicates the magnitude of the ground-state component [@TMNS01; @TSNM02], and is found to take a large value close to unity around $N_{\rm smr}=30$. (Note here that the enhancement of the ground-state component is the aim of the smearing, and hence any approximate optimization is applicable as long as the ground-state overlap is enough large.) 

![ A typical example of the ground-state-overlap quantity $C_0 \equiv \langle W_{\rm 4Q} (T) \rangle^{T+1}/\langle W_{\rm 4Q}(T+1) \rangle^{T}$ plotted against the iteration number $N_{\rm smr}$ at $\alpha=2.3$ for the symmetric planar 4Q configuration with $(d,h)=(1,5)$. $C_0$ takes a large value close to unity around $N_{\rm smr}=30$. []{data-label="Fig8"}](Fig8.eps){height="5.5cm"}

Due to the smearing, the ground-state component is largely enhanced in most cases, and therefore the 4Q Wilson loop $\langle W_{\rm 4Q} \rangle$ composed with the smeared link-variable exhibits a single-exponential behavior as $\langle W_{\rm 4Q} \rangle \simeq e^{-V_{\rm 4Q}T}$ even for a small value of $T$. Then, for each 4Q configuration, we extract $V_{\rm 4Q}$ from the least squares fit with the single-exponential form $$\langle W_{\rm 4Q}\rangle =\bar{C}e^{-V_{\rm 4Q}T}
\label{expfit}$$ in the range of $T_{\rm min}\leq T\leq T_{\rm max}$ listed in Table I-VI.  The prefactor $\bar C$ physically means the ground-state overlap, and $\bar C\simeq 1$ corresponds to the realization of the quasi-ground-state.  Here, we choose the fit range of $T$ such that the stability of the “effective mass” $$\begin{aligned}
V(T)\equiv \ln\{\langle W_{\rm 4Q}(T) \rangle /
\langle W_{\rm 4Q}(T+1)\rangle\}\end{aligned}$$ is observed in the range of $T_{\rm min}\leq T\leq T_{\rm max}-1$.

To see how excited-state contamination is removed in this calculation, we show in Fig.9 several effective-mass plots, $V(T)$ v.s. $T$, for planar and twisted 4Q configurations at small, intermediate and large distances, respectively.  Owing to the smearing, the effective mass $V(T)$ seems to be stable even for small $T$. To show the quality of the single-exponential fit for $\langle W_{\rm 4Q}\rangle$ as in Eq.(\[expfit\]), we list the chi-square per degree of freedom, $\chi^2/N_{\rm DF}$, for each fit in Table I-VI.  For most cases, $\chi^2/N_{\rm DF}$ takes a small value less than unity, and the fitting seems to be plausible. ( We note that the errors listed in Table I-VI are the statistical ones, and there are some systematical errors in the lattice QCD calculation.  For instance, when $T_{\rm min}=2$ is adopted for large 4Q systems as $(d,h)$=(3,16), the systematical error originating from the fit-range choice seems to be several times larger than the statistical error.) 

![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9a.eps "fig:"){width="4.2cm"} ![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9b.eps "fig:"){width="4.2cm"} ![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9c.eps "fig:"){width="4.2cm"} ![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9d.eps "fig:"){width="4.2cm"} ![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9e.eps "fig:"){width="4.2cm"} ![  The effective-mass plots, $V(T)$ v.s. $T$, for several 4Q configurations at small, intermediate and large distances: the symmetric planar and twisted 4Q configurations with $(d,h)$=(1,2), $(d,h)$=(2,7), and $(d,h)$=(3,16).  The fit range of $T_{\rm min} \le T \le T_{\rm max}-1$, which is regarded as the plateau region, is indicated by the vertical dashed lines.  The solid horizontal line denotes the final fit of $V_{\rm 4Q}$ from the least square fit with the single-exponential form. []{ data-label="Fig9"}](Fig9f.eps "fig:"){width="4.2cm"}

In this way, we calculate the tetraquark potential $V_{\rm 4Q}$ for various 4Q system, [*i.e.,*]{} planar, twisted, asymmetric, and large-size 4Q configurations, respectively.  We summarize in Table I-VI the lattice QCD results for $V_{\rm 4Q}$ together with the ground-state overlap $\bar C$, the fit range of $T$, $\chi^2/N_{\rm DF}$, the minimal flux-tube length $L_{\rm min}^{\rm c4Q}$ for the connected 4Q configuration, and the theoretical Ansätze $V_{\rm c4Q}^{\rm th}$ and $V_{\rm 2Q\bar Q}^{\rm th}$ presented in Sect. II. 

1.   Table I and II show $V_{\rm 4Q}$ for the symmetric planar 4Q configurations as shown in Fig.6 with $d_1=d_2=d_3=d_4\equiv d$. $V_{\rm 4Q}$ is shown in terms of $d$ and $h$.

2.   Table III and IV show $V_{\rm 4Q}$ for the symmetric twisted 4Q configurations as shown in Fig.7 with $d_1=d_2=d_3=d_4\equiv d$. $V_{\rm 4Q}$ is shown in terms of $d$ and $h$.

3.   Table V shows $V_{\rm 4Q}$ for the asymmetric planar 4Q configurations as shown in Fig.6 with various $(d_1,d_2,d_3,d_4)$ for $h$=8. 

4.   Table VI shows $V_{\rm 4Q}$ for the asymmetric twisted 4Q configurations as shown in Fig.7 with various $(d_1,d_2,d_3,d_4)$ for $h$=8. 

Thus, we obtain the tetraquark potential $V_{\rm 4Q}$ for about 200 different patterns of 4Q systems. 

   ($d,h$)   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
    (1,1)     0.8590(45)    0.4970(135)             6-11                     0.140                   4.47                     1.1293                      0.8224
    (1,2)     1.2124(31)    0.9049(111)              4-9                     0.710                   5.46                     1.2560                      1.2004
    (1,3)     1.3218(17)    0.9572(48)               3-9                     0.195                   6.46                     1.3352                      1.3939
    (1,4)     1.3938(11)    0.9627(19)               2-8                     0.755                   7.46                     1.3999                      1.5412
    (1,5)     1.4531(13)    0.9556(22)               2-8                     0.633                   8.46                     1.4579                      1.6701
    (1,6)     1.5051(33)    0.9354(95)               3-9                     0.072                   9.46                     1.5123                      1.7897
    (1,7)     1.5644(19)    0.9434(32)               2-7                     0.336                   10.46                    1.5643                      1.9041
    (1,8)     1.6184(20)    0.9375(36)               2-8                     0.124                   11.46                    1.6150                      2.0152
    (1,9)     1.6706(22)    0.9295(35)               2-4                     0.119                   12.46                    1.6646                      2.1241
   (1,10)     1.7269(28)    0.9286(48)               2-7                     0.184                   13.46                    1.7136                      2.2314
   (1,11)     1.7774(30)    0.9177(48)               2-4                     0.001                   14.46                    1.7620                      2.3377
   (1,12)     1.8234(37)    0.8983(62)               2-7                     1.094                   15.46                    1.8100                      2.4431
   (1,13)     1.8797(44)    0.8979(74)               2-7                     0.351                   16.46                    1.8577                      2.5478
   (1,14)     1.9285(49)    0.8838(78)               2-6                     0.899                   17.46                    1.9052                      2.6521
   (1,15)     1.9787(55)    0.8722(91)               2-7                     0.361                   18.46                    1.9525                      2.7559
   (1,16)     2.0365(62)    0.8741(101)              2-6                     0.702                   19.46                    1.9996                      2.8594
    (2,1)     0.8263(46)    0.3324(93)              6-13                     0.624                   8.25                     1.3992                      0.8224
    (2,2)    1.2633(183)    0.5646(514)              5-9                     0.727                   8.94                     1.5022                      1.2004
    (2,3)    1.4592(111)    0.7457(328)              4-7                     0.119                   9.93                     1.5734                      1.3939
    (2,4)     1.5950(17)    0.9223(29)               2-7                     0.643                   10.93                    1.6340                      1.5412
    (2,5)     1.6619(21)    0.9286(36)               2-6                     0.779                   11.93                    1.6896                      1.6701
    (2,6)     1.7215(23)    0.9285(41)               2-7                     0.653                   12.93                    1.7426                      1.7897
    (2,7)     1.7791(29)    0.9283(51)               2-6                     0.089                   13.93                    1.7938                      1.9041
    (2,8)     1.8279(31)    0.9121(54)               2-6                     0.631                   14.93                    1.8439                      2.0152
    (2,9)     1.8827(34)    0.9101(57)               2-7                     0.111                   15.93                    1.8932                      2.1241
   (2,10)     1.9366(38)    0.9049(64)               2-6                     0.108                   16.93                    1.9419                      2.2314
   (2,11)     1.9917(45)    0.9026(73)               2-5                     0.785                   17.93                    1.9902                      2.3377
   (2,12)     2.0419(49)    0.8911(82)               2-5                     0.215                   18.93                    2.0381                      2.4431
   (2,13)     2.0954(55)    0.8856(90)               2-6                     0.445                   19.93                    2.0857                      2.5478
   (2,14)     2.1387(61)    0.8619(101)              2-5                     1.272                   20.93                    2.1331                      2.6521
   (2,15)     2.1990(67)    0.8683(111)              2-5                     1.103                   21.93                    2.1804                      2.7559
   (2,16)     2.2482(78)    0.8567(122)              2-5                     0.687                   22.93                    2.2274                      2.8594

   ($d,h$)   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
    (3,1)     0.8281(20)    0.3242(33)              5-12                     0.502                   12.17                    1.6129                      0.8224
    (3,2)    1.2143(228)    0.3390(387)              5-9                     0.237                   12.65                    1.7043                      1.2004
    (3,3)    1.5031(168)    0.5540(370)              4-7                     2.112                   13.42                    1.7636                      1.3939
    (3,4)     1.6992(68)    0.7751(158)              3-7                     0.654                   14.39                    1.8213                      1.5412
    (3,5)     1.8185(31)    0.8797(53)               2-6                     0.446                   15.39                    1.8756                      1.6701
    (3,6)     1.8896(35)    0.8922(60)               2-6                     0.299                   16.39                    1.9275                      1.7897
    (3,7)     1.9459(37)    0.8849(63)               2-4                     1.708                   17.39                    1.9780                      1.9041
    (3,8)     2.0016(45)    0.8808(76)               2-5                     0.933                   18.39                    2.0276                      2.0152
    (3,9)     2.0544(45)    0.8739(73)               2-4                     0.003                   19.39                    2.0766                      2.1241
   (3,10)     2.1123(52)    0.8748(85)               2-6                     0.116                   20.39                    2.1250                      2.2314
   (3,11)     2.1658(57)    0.8698(92)               2-7                     0.241                   21.39                    2.1731                      2.3377
   (3,12)     2.2234(65)    0.8720(105)              2-5                     0.373                   22.39                    2.2208                      2.4431
   (3,13)     2.2698(73)    0.8548(118)              2-5                     0.168                   23.39                    2.2683                      2.5478
   (3,14)     2.3192(88)    0.8418(143)              2-5                     0.417                   24.39                    2.3157                      2.6521
   (3,15)     2.3843(94)    0.8564(154)              2-6                     0.653                   25.39                    2.3628                      2.7559
   (3,16)    2.4393(112)    0.8530(185)              2-5                     0.469                   26.39                    2.4098                      2.8594
    (4,1)     0.8228(21)    0.3069(34)              5-12                     0.586                   16.12                    1.8119                      0.8224
    (4,2)     1.2510(77)    0.3541(110)              4-9                     0.528                   16.49                    1.8975                      1.2004
    (4,3)    1.4643(247)    0.3390(336)              4-7                     0.247                   17.09                    1.9482                      1.3939
    (4,4)    1.7452(112)    0.5783(194)              3-7                     0.910                   17.89                    1.9972                      1.5412
    (4,5)    1.9147(159)    0.7137(337)              3-5                     1.694                   18.86                    2.0493                      1.6701
    (4,6)    2.0168(230)    0.7591(518)              3-7                     0.425                   19.86                    2.1007                      1.7897
    (4,7)     2.1167(49)    0.8435(84)               2-4                     0.041                   20.86                    2.1507                      1.9041
    (4,8)     2.1712(56)    0.8341(89)               2-4                     0.436                   21.86                    2.2000                      2.0152
    (4,9)     2.2444(62)    0.8611(102)              2-6                     0.363                   22.86                    2.2486                      2.1241
   (4,10)     2.2970(84)    0.8505(138)              2-5                     1.626                   23.86                    2.2968                      2.2314
   (4,11)     2.3430(84)    0.8332(133)              2-6                     0.514                   24.86                    2.3447                      2.3377
   (4,12)     2.3976(96)    0.8309(149)              2-5                     0.544                   25.86                    2.3923                      2.4431
   (4,13)    2.4503(109)    0.8225(168)              2-4                     0.030                   26.86                    2.4397                      2.5478
   (4,14)    2.5043(123)    0.8178(194)              2-4                     0.070                   27.86                    2.4869                      2.6521
   (4,16)    2.6037(177)    0.7952(276)              2-4                     2.015                   29.86                    2.5809                      2.8594

   ($d,h$)   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
    (1,1)     1.1779(06)    0.9695(11)               2-8                     0.695                   4.47                     1.1693                      1.1305
    (1,2)     1.2577(06)    0.9687(11)               2-5                     0.009                   5.46                     1.2611                      1.2967
    (1,3)     1.3311(08)    0.9676(15)               2-8                     0.465                   6.46                     1.3362                      1.4435
    (1,4)     1.3960(11)    0.9642(20)               2-6                     0.676                   7.46                     1.4002                      1.5737
    (1,5)     1.4532(13)    0.9546(23)               2-8                     0.546                   8.46                     1.4580                      1.6941
    (1,6)     1.5100(35)    0.9497(99)               3-8                     0.221                   9.46                     1.5123                      1.8088
    (1,7)     1.5661(18)    0.9472(32)               2-7                     0.305                   10.46                    1.5644                      1.9200
    (1,8)     1.6177(21)    0.9357(37)               2-7                     0.311                   11.46                    1.6150                      2.0288
    (1,9)     1.6712(24)    0.9300(41)               2-7                     0.416                   12.46                    1.6646                      2.1360
   (1,10)     1.7144(68)    0.8956(180)              3-5                     0.006                   13.46                    1.7136                      2.2421
   (1,11)     1.7751(32)    0.9134(53)               2-5                     0.890                   14.46                    1.7620                      2.3472
   (1,12)     1.8302(38)    0.9109(65)               2-6                     0.039                   15.46                    1.8100                      2.4518
   (1,13)     1.8778(45)    0.8946(76)               2-6                     0.689                   16.46                    1.8577                      2.5558
   (1,14)     1.9306(49)    0.8879(79)               2-6                     0.543                   17.46                    1.9052                      2.6595
   (1,15)     1.9860(56)    0.8856(90)               2-6                     0.649                   18.46                    1.9525                      2.7628
   (1,16)     2.0378(62)    0.8771(99)               2-5                     0.100                   19.46                    1.9996                      2.8658
    (2,1)     1.4571(27)    0.9244(72)               3-6                     0.130                   8.25                     1.4778                      1.3939
    (2,2)     1.5027(32)    0.9296(89)               3-8                     0.131                   8.94                     1.5221                      1.4656
    (2,3)     1.5613(14)    0.9445(24)               2-4                     0.033                   9.93                     1.5800                      1.5578
    (2,4)     1.6152(18)    0.9388(31)               2-7                     0.339                   10.93                    1.6365                      1.6576
    (2,5)     1.6713(18)    0.9367(31)               2-4                     0.499                   11.93                    1.6907                      1.7598
    (2,6)     1.7249(22)    0.9305(38)               2-4                     0.266                   12.93                    1.7431                      1.8626
    (2,7)     1.7780(27)    0.9241(47)               2-7                     0.439                   13.93                    1.7941                      1.9655
    (2,8)     1.8305(29)    0.9162(50)               2-4                     0.044                   14.93                    1.8441                      2.0683
    (2,9)     1.8752(35)    0.8945(58)               2-6                     2.241                   15.93                    1.8933                      2.1708
   (2,10)     1.9335(40)    0.8985(68)               2-6                     0.989                   16.93                    1.9420                      2.2732
   (2,11)     1.9882(45)    0.8961(74)               2-6                     0.776                   17.93                    1.9902                      2.3755
   (2,12)     2.0295(51)    0.8681(82)               2-5                     1.912                   18.93                    2.0381                      2.4776
   (2,13)     2.0933(56)    0.8818(92)               2-4                     0.042                   19.93                    2.0857                      2.5796
   (2,14)     2.1470(62)    0.8768(102)              2-5                     0.015                   20.93                    2.1331                      2.6815
   (2,15)     2.1910(70)    0.8552(115)              2-7                     0.874                   21.93                    2.1804                      2.7833
   (2,16)     2.2315(76)    0.8272(119)              2-5                     0.884                   22.93                    2.2274                      2.8850

   ($d,h$)   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
    (3,1)     1.6641(20)    0.9086(34)               2-8                     1.052                   12.17                    1.7093                      1.5889
    (3,2)     1.6980(21)    0.9095(35)               2-4                     0.227                   12.65                    1.7359                      1.6314
    (3,3)     1.7444(25)    0.9098(43)               2-6                     0.649                   13.42                    1.7769                      1.6941
    (3,4)     1.7960(26)    0.9085(46)               2-5                     0.278                   14.39                    1.8275                      1.7700
    (3,5)     1.8473(31)    0.9017(53)               2-5                     0.747                   15.39                    1.8787                      1.8540
    (3,6)     1.9015(36)    0.8995(60)               2-6                     0.151                   16.39                    1.9292                      1.9431
    (3,7)     1.9563(39)    0.8969(66)               2-6                     0.334                   17.39                    1.9790                      2.0355
    (3,8)     2.0077(46)    0.8888(75)               2-6                     0.216                   18.39                    2.0282                      2.1301
    (3,9)     2.0609(47)    0.8838(77)               2-6                     0.067                   19.39                    2.0769                      2.2261
   (3,10)     2.1146(52)    0.8787(86)               2-5                     0.121                   20.39                    2.1252                      2.3232
   (3,11)     2.1695(56)    0.8763(91)               2-4                     0.104                   21.39                    2.1732                      2.4210
   (3,12)     2.2284(65)    0.8805(109)              2-6                     0.053                   22.39                    2.2209                      2.5194
   (3,13)     2.2684(69)    0.8513(112)              2-6                     0.676                   23.39                    2.2684                      2.6182
   (3,14)     2.3303(82)    0.8597(133)              2-4                     0.373                   24.39                    2.3157                      2.7174
   (3,15)     2.3725(96)    0.8357(151)              2-5                     0.631                   25.39                    2.3629                      2.8168
   (3,16)    2.4051(112)    0.7968(168)              2-6                     0.378                   26.39                    2.4099                      2.9165
    (4,1)     1.8453(33)    0.8666(56)               2-6                     0.389                   16.12                    1.9179                      1.7598
    (4,2)    1.8396(124)    0.7745(285)              3-6                     1.225                   16.49                    1.9368                      1.7897
    (4,3)    1.8832(135)    0.7758(315)              3-6                     1.204                   17.09                    1.9671                      1.8363
    (4,4)     1.9745(44)    0.8706(71)               2-4                     0.193                   17.89                    2.0072                      1.8960
    (4,5)     2.0271(47)    0.8670(79)               2-6                     0.321                   18.86                    2.0549                      1.9655
    (4,6)     2.0749(52)    0.8536(85)               2-5                     0.226                   19.86                    2.1040                      2.0422
    (4,7)     2.1306(52)    0.8522(86)               2-4                     0.543                   20.86                    2.1528                      2.1241
    (4,8)     2.1932(66)    0.8640(108)              2-6                     0.250                   21.86                    2.2012                      2.2099
    (4,9)     2.2409(62)    0.8510(101)              2-5                     0.801                   22.86                    2.2494                      2.2985
   (4,10)     2.2867(66)    0.8331(107)              2-4                     0.060                   23.86                    2.2973                      2.3893
   (4,11)     2.3324(87)    0.8152(135)              2-4                     0.680                   24.86                    2.3450                      2.4818
   (4,12)    2.3951(107)    0.8247(166)              2-5                     0.199                   25.86                    2.3925                      2.5756
   (4,13)    2.4477(110)    0.8173(174)              2-6                     2.138                   26.86                    2.4398                      2.6705
   (4,14)    2.4910(128)    0.7958(196)              2-5                     0.053                   27.86                    2.4870                      2.7662
   (4,15)    2.5625(154)    0.8210(238)              2-6                     0.904                   28.86                    2.5341                      2.8626
   (4,16)    2.5995(155)    0.7897(238)              2-5                     0.099                   29.86                    2.5810                      2.9596

   $(d_1,d_2,d_3,d_4)$   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------------------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
        (0,1,0,1)         1.4227(18)    0.9368(31)              2-10                     0.610                   9.73                     1.3983                      2.0152
        (0,1,1,0)         1.4243(17)    0.9377(29)              2-10                     0.265                   9.78                     1.4008                      2.0220
        (0,1,1,1)         1.5215(19)    0.9385(34)               2-8                     0.544                   10.61                    1.5073                      2.0186
        (0,1,2,0)         1.5170(39)    0.9054(103)              3-7                     0.424                   10.70                    1.5118                      2.0321
        (0,1,1,2)         1.5689(43)    0.8991(115)              3-8                     0.596                   11.46                    1.5697                      2.0220
        (0,1,2,1)         1.5830(21)    0.9353(36)               2-7                     0.155                   11.51                    1.5718                      2.0287
        (0,1,3,0)         1.5926(21)    0.9289(36)               2-7                     0.086                   11.64                    1.5781                      2.0483
        (0,2,1,1)         1.6211(21)    0.9300(38)               2-7                     0.388                   11.51                    1.6171                      2.0220
        (0,2,2,0)         1.6301(22)    0.9310(40)               2-7                     0.199                   11.64                    1.6234                      2.0422
        (0,1,1,3)         1.6281(22)    0.9200(38)               2-8                     0.687                   12.34                    1.6219                      2.0321
        (0,1,2,2)         1.6231(49)    0.9144(132)              3-8                     0.221                   12.34                    1.6219                      2.0321
        (0,1,3,1)         1.6326(23)    0.9253(40)               2-7                     0.758                   12.42                    1.6258                      2.0449
        (0,2,1,2)         1.6809(22)    0.9262(39)               2-6                     0.779                   12.34                    1.6784                      2.0186
        (0,2,2,1)         1.6834(23)    0.9273(42)               2-8                     0.762                   12.42                    1.6824                      2.0321
        (0,2,3,0)         1.6934(23)    0.9208(39)               2-5                     0.470                   12.58                    1.6902                      2.0584
        (1,1,1,2)         1.6784(23)    0.9324(41)               2-7                     0.510                   12.34                    1.6784                      2.0186
        (0,1,2,3)         1.6668(61)    0.9034(165)              3-6                     0.309                   13.20                    1.6687                      2.0422
        (0,1,3,2)         1.6732(25)    0.9199(45)               2-6                     0.502                   13.23                    1.6705                      2.0483
        (0,2,1,3)         1.7270(25)    0.9125(45)               2-6                     0.683                   13.20                    1.7296                      2.0220
        (0,2,3,1)         1.7349(25)    0.9210(43)               2-7                     0.323                   13.35                    1.7370                      2.0483
        (0,3,1,2)         1.7441(26)    0.9177(47)               2-7                     0.369                   13.23                    1.7426                      2.0220
        (0,3,2,1)         1.7473(26)    0.9191(46)               2-7                     0.503                   13.35                    1.7482                      2.0422
        (0,3,3,0)         1.7604(25)    0.9172(45)               2-7                     0.166                   13.53                    1.7573                      2.0747
        (1,1,1,3)         1.7295(26)    0.9259(45)               2-7                     0.880                   13.23                    1.7314                      2.0287
        (1,1,2,2)         1.7218(24)    0.9221(45)               2-6                     0.307                   13.20                    1.7296                      2.0220
        (1,2,1,2)         1.7398(24)    0.9311(44)               2-6                     0.254                   13.20                    1.7408                      2.0152
        (1,2,2,1)         1.7299(66)    0.9013(176)              3-9                     0.298                   13.23                    1.7426                      2.0220
        (0,1,3,3)         1.7159(29)    0.9127(49)               2-6                     0.559                   14.07                    1.7142                      2.0584
        (0,2,3,2)         1.7754(29)    0.9167(49)               2-9                     0.893                   14.14                    1.7807                      2.0449
        (0,3,3,1)         1.8006(28)    0.9162(49)               2-6                     1.013                   14.28                    1.8033                      2.0584
        (1,1,2,3)         1.7687(27)    0.9179(47)               2-9                     0.558                   14.07                    1.7772                      2.0321
        (1,2,2,2)         1.7866(27)    0.9268(49)               2-6                     0.192                   14.07                    1.7928                      2.0186
        (1,2,3,1)         1.7900(28)    0.9208(49)               2-6                     0.436                   14.14                    1.7963                      2.0321
        (1,2,2,3)         1.8273(31)    0.9109(55)               2-6                     0.881                   14.93                    1.8395                      2.0220
        (1,2,3,2)         1.8311(30)    0.9170(54)               2-7                     0.284                   14.96                    1.8412                      2.0287
        (1,3,3,1)         1.8437(32)    0.9191(57)               2-6                     0.449                   15.06                    1.8506                      2.0422
        (1,3,3,2)         1.8845(35)    0.9154(59)               2-6                     0.173                   15.87                    1.8946                      2.0321
        (2,2,2,3)         1.8774(34)    0.9131(60)               2-7                     0.197                   15.80                    1.8914                      2.0186
        (2,3,2,3)         1.9213(37)    0.9050(63)               2-6                     0.030                   16.66                    1.9380                      2.0152
        (2,3,3,3)         1.9648(39)    0.8987(65)               2-6                     0.603                   17.53                    1.9832                      2.0186

   $(d_1,d_2,d_3,d_4)$   $V_{\rm 4Q}$    $\bar C$     $T_{\rm min}-T_{\rm max}$   $\chi^2/N_{\rm DF}$   $L_{\rm min}^{\rm c4Q}$   $V_{\rm c4Q}^{\rm th}$   $V_{\rm 2Q\bar{Q}}^{\rm th}$
  --------------------- -------------- ------------- --------------------------- --------------------- ------------------------- ------------------------ ------------------------------
        (0,1,0,1)         1.4179(34)    0.9197(92)               3-8                     0.857                   9.76                     1.3997                      2.0220
        (0,1,1,1)         1.5239(18)    0.9435(34)              2-10                     0.590                   10.61                    1.5074                      2.0254
        (0,1,0,2)        1.5152(129)    0.8961(453)              4-9                     0.173                   10.66                    1.5100                      2.0320
        (0,1,1,2)         1.5819(47)    0.9340(132)              3-7                     0.262                   11.49                    1.5709                      2.0354
        (0,1,0,3)         1.5896(21)    0.9257(37)               2-9                     0.525                   11.58                    1.5756                      2.0481
        (0,2,1,1)         1.6248(20)    0.9373(36)               2-5                     0.889                   11.52                    1.6176                      2.0354
        (0,2,0,2)         1.6259(22)    0.9263(38)               2-7                     0.367                   11.57                    1.6202                      2.0417
        (0,1,1,3)         1.6279(53)    0.9144(143)              3-7                     0.538                   12.39                    1.6241                      2.0515
        (0,1,2,2)         1.6252(50)    0.9204(140)              3-9                     0.070                   12.34                    1.6219                      2.0455
        (0,2,1,2)         1.6813(24)    0.9250(42)               2-8                     0.770                   12.39                    1.6811                      2.0452
        (0,2,0,3)         1.6923(23)    0.9233(38)               2-7                     0.545                   12.49                    1.6858                      2.0575
        (1,1,1,2)         1.6790(22)    0.9334(39)               2-7                     0.652                   12.34                    1.6785                      2.0388
        (0,1,2,3)         1.6724(24)    0.9192(43)               2-8                     0.221                   13.22                    1.6696                      2.0616
        (0,2,1,3)         1.7335(25)    0.9213(45)               2-7                     0.040                   13.29                    1.7343                      2.0609
        (0,3,1,2)         1.7462(25)    0.9187(45)               2-7                     0.323                   13.31                    1.7467                      2.0609
        (0,3,0,3)         1.7567(27)    0.9166(47)               2-6                     0.557                   13.41                    1.7514                      2.0726
        (1,1,1,3)         1.7293(25)    0.9252(45)               2-7                     0.392                   13.24                    1.7317                      2.0549
        (1,1,2,2)         1.7220(24)    0.9222(45)               2-6                     0.420                   13.20                    1.7296                      2.0488
        (1,2,1,2)         1.7388(27)    0.9280(46)               2-6                     0.721                   13.22                    1.7420                      2.0485
        (0,1,3,3)         1.7198(27)    0.9199(47)               2-8                     0.746                   14.08                    1.7141                      2.0778
        (0,2,2,3)         1.7734(28)    0.9138(49)               2-7                     0.497                   14.12                    1.7798                      2.0710
        (0,3,1,3)         1.7979(28)    0.9145(49)               2-6                     0.096                   14.21                    1.7999                      2.0761
        (1,1,2,3)         1.7687(29)    0.9174(53)               2-8                     0.832                   14.07                    1.7772                      2.0649
        (1,2,2,2)         1.7840(28)    0.9210(49)               2-7                     0.820                   14.07                    1.7930                      2.0585
        (1,2,1,3)         1.7890(30)    0.9201(51)               2-7                     1.063                   14.12                    1.7952                      2.0643
        (1,2,2,3)         1.8312(30)    0.9169(53)               2-6                     0.095                   14.95                    1.8407                      2.0743
        (1,3,1,3)         1.8393(31)    0.9128(54)               2-7                     0.474                   15.02                    1.8484                      2.0794
        (1,3,2,3)         1.8853(34)    0.9164(58)               2-6                     1.208                   15.85                    1.8939                      2.0894
        (2,2,2,3)         1.8807(33)    0.9180(55)               2-4                     0.032                   15.80                    1.8917                      2.0840
        (2,3,2,3)         1.9171(36)    0.8943(64)               2-5                     0.564                   16.68                    1.9394                      2.0992
        (2,3,3,3)         1.9653(40)    0.8968(69)               2-6                     0.128                   17.54                    1.9838                      2.1149

Discussions {#discussion}
===========

Comparison with theoretical Ansätze
-----------------------------------

In this section, we compare the lattice QCD results of the 4Q potential $V_{\rm 4Q}$ with the theoretical Ansätze presented in Sect. II, i.e., the OGE plus multi-Y Ansatz (\[Vc4Q\]) and the two-meson Ansatz (\[V2QQbar\]). 

For the OGE plus multi-Y Ansatz (\[Vc4Q\]), we set the parameters ($A_{\rm 4Q}, \sigma_{\rm 4Q}$) to be ($A_{\rm 3Q}, \sigma_{\rm 3Q}$) in the 3Q potential $V_{\rm 3Q}$ in Ref.[@TSNM02], i.e., $A_{\rm 4Q}=A_{\rm 3Q} \simeq$ 0.1366, $\sigma_{\rm 4Q}=\sigma_{\rm 3Q} \simeq$ 0.0460$a^{-2}$. Note that there are no adjustable parameters except for an irrelevant constant $C_{\rm 4Q} \simeq$ 1.2579$a^{-1}$.

For the two-meson Ansatz (\[V2QQbar\]), we adopt the lattice result for the Q$\rm \bar{Q}$ potential $V_{\rm Q \bar Q}$ in Ref.[@TSNM02], i.e., $A_{\rm Q\bar{Q}} \simeq$ 0.2768, $\sigma_{\rm Q\bar{Q}} \simeq$ 0.0506$a^{-2}$, $C_{\rm Q\bar{Q}} \simeq$ 0.6374$a^{-1}$. Then, there are no adjustable parameters also for the two-meson Ansatz. 

We demonstrate the two Ansätze for the symmetric planar 4Q configurations as shown in Fig.6 with $d_1=d_2=d_3=d_4 \equiv d$. In this case, the OGE Coulomb plus multi-Y Ansatz for the connected 4Q system reads $$\begin{aligned}
V_{\rm c4Q}(d,h)&=&-A_{\rm 4Q}\left(\frac{1}{d}+\frac{1}{h}+\frac{1}{\sqrt{h^2+4d^2}}\right) \nonumber \\
   &+&\sigma_{\rm 4Q}L_{\rm min} +C_{\rm 4Q}.  
\label{Vc4Qpl}\end{aligned}$$ In the case of $h \ge \frac{2}{\sqrt{3}}d$, the lowest connected 4Q system takes the double-Y-shaped flux-tube, and the minimal value of the total flux-tube length is expressed as $$\begin{aligned}
L_{\rm min}=h+2\sqrt{3} d.\end{aligned}$$ In the case of $h \le \frac{2}{\sqrt{3}}d$, the lowest connected 4Q system takes the X-shaped flux-tube with $L_{\rm min}=2\sqrt{h^2+4d^2}$, although it must be unstable against the decay into two mesons.  On the other hand, the two-meson Ansatz reads $$\begin{aligned}
V_{\rm 2Q\bar Q}(h)=2 \times V_{\rm Q\bar Q}(h), 
\label{V2QQbarpl}\end{aligned}$$ which is independent of $d$.

We show in Fig.10(a) the lattice QCD results of the 4Q potential $V_{\rm 4Q}$ for symmetric planar 4Q configurations [@STOI04b; @OST04p; @AK05] with $d=1-4$. The symbols denote lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz (\[Vc4Q\]), and the dashed-dotted curve the two-meson Ansatz (\[V2QQbar\]). 

For large value of $h$ compared with $d$, the lattice data seem to coincide with the OGE Coulomb plus multi-Y Ansatz [@STOI04b; @OST04p].  On the other hand, for small $h$, the lattice data tend to agree with the two-meson Ansatz and seem independent of $d$ [@STOI04b; @OST04p].  These tendencies were also observed in a recent lattice work by other group [@AK05].  This would correspond to the transition from the connected 4Q state into the two-meson state as $h$ decreases, as will be discussed in the next subsection. 

Here, we comment on the transition in terms of the ground-state overlap $\bar C$ listed in Table I. For large $h$, the ground-state overlap $\bar C$ is almost unity, which implies the realization of the quasi-ground-state in the present calculation with the smeared 4Q Wilson loop based on the connected 4Q configuration.  For small $h$, however, $\bar C$ tends to be small, and hence, for accurate measurements, we have to take relatively large values of $T$ as the fit range.  This would indicate that the ground-state configuration is largely different from the connected 4Q configuration for small $h$. (In other words, it may be nontrivial to obtain the result indicating the two-meson state for small $h$ from the 4Q Wilson loop based on the connected 4Q configuration.) 

![ The tetraquark (4Q) potential $V_{\rm 4Q}$: (a) for symmetric planar 4Q configurations as shown in Fig.6; (b) for symmetric twisted 4Q configurations as shown in Fig.7.  The symbols denote the lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz, and the dashed-dotted curve the two-meson Ansatz. []{ data-label="Fig10"}](Fig10a.eps "fig:"){width="6.5cm"} ![ The tetraquark (4Q) potential $V_{\rm 4Q}$: (a) for symmetric planar 4Q configurations as shown in Fig.6; (b) for symmetric twisted 4Q configurations as shown in Fig.7.  The symbols denote the lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz, and the dashed-dotted curve the two-meson Ansatz. []{ data-label="Fig10"}](Fig10b.eps "fig:"){width="6.5cm"}

Next, we investigate the twisted 4Q configuration [@STOI04b; @OST04p] as shown in Fig.7.  We show in Fig.10(b) the lattice QCD results of the 4Q potential for symmetric twisted 4Q configurations with $d=1-4$. The symbols denote lattice QCD results for each $d$, and the curves describe the theoretical form of the OGE plus multi-Y Ansatz. 

The lattice data seem to agree with the OGE plus multi-Y Ansatz in the wide region of $h$ [@STOI04b; @OST04p].  In the twisted 4Q configuration, the distance between the nearest quark and antiquark cannot take a smaller value than the “inter-diquark distance" $h$, and therefore $V_{\rm c4Q}$ is smaller than $V_{\rm 2Q\bar Q}$ in most cases except for extreme configurations as $d>h$. (See Fig.10(b).)  Then, different from the planar case, it is not easy to make the transition from the connected 4Q state into the two-meson state for the twisted case.  Also from the lattice data, the ground-state overlap $\bar C$ is found to be almost unity for all twisted 4Q configurations, which indicates that the ground state resembles a connected 4Q state.  In other words, the twisted 4Q configuration seems to be rather stable against the transition into the two $\rm Q\bar Q$ mesons, which may indicate a stability of the “twisted structure" or the “tetrahedral structure" of the 4Q system. 

We also investigate more general asymmetric 4Q configurations with various $(d_1,d_2,d_3,d_4)$ for both planar and twisted cases, as shown in Table V and VI.  Also for the asymmetric planar and twisted 4Q configurations, $V_{\rm 4Q}$ seems to be well described with the OGE plus multi-Y Ansatz in the case of $h > \frac{2}{\sqrt{3}}d_i$ $(i=1,2,3,4)$. Note here that some 4Q configurations are physically equivalent, e.g., the planar cases with $(d_1,d_2,d_3,d_4)$=(1,1,1,2) and (0,2,1,2), although the corresponding smeared 4Q Wilson loops are different.  For such cases, the lattice QCD results are found to be almost the same.  In fact, the extracted lattice results are almost independent of the way how the 4Q Wilson loop is constructed, as long as the spatial locations of the static four quarks are the same.  This indicates that the ground-state contribution is properly extracted in the present calculation. 

As the conclusion, the OGE plus multi-Y Ansatz well describes the 4Q potential $V_{\rm 4Q}$, when QQ and $\rm \bar{Q}\bar{Q}$ are well separated, e.g., the “inter-diquark distance" $h$ is large in comparison with the “diquark size" $d$. On the other hand, when the nearest quark and antiquark pair is spatially close, the system is described as a “two-meson" state. 

Together with the previous studies [@TMNS01; @TSNM02; @OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p] for the inter-quark potentials in lattice QCD, we have found the universality of the string tension as $$\begin{aligned}
\sigma_{\rm Q\bar{Q}} \simeq \sigma_{\rm 3Q} \simeq 
\sigma_{\rm 4Q} \simeq \sigma_{\rm 5Q} \simeq (420{\rm MeV})^2,\end{aligned}$$ and the OGE result for the Coulomb coefficient as $$\begin{aligned}
A_{\rm Q\bar{Q}} \simeq 2A_{\rm 3Q} \simeq 
2A_{\rm 4Q} \simeq 2A_{\rm 5Q} \simeq 0.27.\end{aligned}$$ In particular, these lattice QCD studies [@OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p] indicate a fairly good agreement among $\sigma_{\rm 3Q}$, $\sigma_{\rm 4Q}$ and $\sigma_{\rm 5Q}$, which seem to be slightly smaller than $\sigma_{\rm Q\bar Q}$. (As an interesting possibility, the numerical similarity among $\sigma_{\rm 3Q}$, $\sigma_{\rm 4Q}$ and $\sigma_{\rm 5Q}$ may reflect the similar structure of the Y-type flux-tube in the multi-quark systems.)  The universality of the string tension observed in our lattice QCD studies [@TMNS01; @TSNM02; @OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p] seems to be consistent with the hypothetical flux-tube picture [@N74; @KS75; @CKP83; @CNN79; @IP8385; @P84; @O85; @LLMRSY86] or the strong-coupling expansion scheme [@KS75; @CKP83], although strong-coupling QCD does not have a continuum limit and is far from real QCD.  As for the irrelevant constant, $C_{\rm Q\bar Q}$, $C_{\rm 3Q}$, $C_{\rm 4Q}$ and $C_{\rm 5Q}$ are non-scaling unphysical quantities appearing in the lattice regularization, and we find an approximate relation as $$\begin{aligned}
\frac{C_{\rm Q\bar Q}}{2} \simeq \frac{C_{\rm 3Q}}{3} 
\simeq \frac{C_{\rm 4Q} }{4} \simeq \frac{C_{\rm 5Q} }{5} \simeq 0.32 a^{-1}\end{aligned}$$ in lattice QCD [@TMNS01; @TSNM02; @OST04; @STOI04a; @STOI04b; @SOTI04; @SIOT04; @OST04p]. 

The quark confinement force in 4Q systems
-----------------------------------------

While the short-distance OGE Coulomb force can be understood with perturbative QCD, the long-distance confinement force is a typical nonperturbative quantity and highly nontrivial particularly for multi-quark systems.  To specify the long-distance property of $V_{\rm 4Q}$ is important to clarify the confinement mechanism from a wide viewpoint including multi-quarks, and it also leads to a proper quark-model Hamiltonian to describe multi-quark systems.  Therefore, we perform a further analysis for the long-distance force in 4Q systems. 

To clarify the long-distance force in the 4Q system, we plot $V_{\rm 4Q}$ against $L_{\rm min}^{\rm c4Q}$ for planar and twisted 4Q configurations in Figs.11(a) and (b), respectively.  Here, $L_{\rm min}^{\rm c4Q}$ denotes the minimal flux-tube length for the connected 4Q system.  In both planar and twisted cases, for large $L_{\rm min}^{\rm c4Q}$, $V_{\rm 4Q}$ approaches a linearly arising function of $L_{\rm min}^{\rm c4Q}$.

![  The 4Q potential $V_{\rm 4Q}$ plotted against $L_{\rm min}^{\rm c4Q}$: (a) for planar 4Q configurations; (b) for twisted 4Q configurations.  $L_{\rm min}^{\rm c4Q}$ denotes the minimal flux-tube length for the connected 4Q configuration.  We plot all the lattice QCD data of $V_{\rm 4Q}$ for the symmetric case with $d=1$ (solid circles), $d=2$ (solid squares), $d=3$ (open triangles) and $d=4$ (crosses) together with the asymmetric case (open diamonds). []{ data-label="Fig11"}](Fig11a.eps "fig:"){width="6cm"} ![  The 4Q potential $V_{\rm 4Q}$ plotted against $L_{\rm min}^{\rm c4Q}$: (a) for planar 4Q configurations; (b) for twisted 4Q configurations.  $L_{\rm min}^{\rm c4Q}$ denotes the minimal flux-tube length for the connected 4Q configuration.  We plot all the lattice QCD data of $V_{\rm 4Q}$ for the symmetric case with $d=1$ (solid circles), $d=2$ (solid squares), $d=3$ (open triangles) and $d=4$ (crosses) together with the asymmetric case (open diamonds). []{ data-label="Fig11"}](Fig11b.eps "fig:"){width="6cm"}

To single out the long-distance confinement force, we consider the 4Q potential subtracted by the Coulomb part.  Here, we subtract the OGE Coulomb part $V_{\rm c4Q}^{\rm Coul}$ of $V_{\rm c4Q}$ in Eq.(\[Vc4Q\]) for the connected 4Q system, with the Coulomb coefficient $A_{\rm 4Q}$ fixed to be $A_{\rm 3Q}$ in the 3Q potential $V_{\rm 3Q}$ in Ref.[@TSNM02].  We plot $V_{\rm 4Q}-V_{\rm c4Q}^{\rm Coul}$ against $L_{\rm min}^{\rm c4Q}$ for planar and twisted 4Q configurations in Figs.12 (a) and (b), respectively.  For the planar 4Q system, $V_{\rm 4Q}-V_{\rm c4Q}^{\rm Coul}$ approaches $\sigma_{\rm 4Q} L_{\rm min}^{\rm c4Q}+C_{\rm 4Q}$ except for a small $h$ region, where the flip-flop into a two-meson state can take place.  For the twisted 4Q system, we observe remarkable agreement between the lattice QCD data of $V_{\rm 4Q}-V_{\rm c4Q}^{\rm Coul}$ and $\sigma_{\rm 4Q} L_{\rm min}^{\rm c4Q}+C_{\rm 4Q}$ for the wide region of $L_{\rm min}$, which corresponds that the flip-flop into the two-meson state does not occur in most twisted 4Q configurations. 

![  The 4Q potential subtracted by the OGE Coulomb part of the connected 4Q system, $V_{\rm 4Q}-V_{\rm c4Q}^{\rm Coul}$, plotted against $L_{\rm min}^{\rm c4Q}$: (a) for planar 4Q configurations; (b) for twisted 4Q configurations.  We plot all the lattice QCD data including asymmetric cases.  The meaning of the symbols is the same as in Fig.11.  The solid line denotes $\sigma_{\rm 4Q} L_{\rm min}^{\rm c4Q}+C_{\rm 4Q}$. []{data-label="Fig12"}](Fig12a.eps "fig:"){width="6cm"} ![  The 4Q potential subtracted by the OGE Coulomb part of the connected 4Q system, $V_{\rm 4Q}-V_{\rm c4Q}^{\rm Coul}$, plotted against $L_{\rm min}^{\rm c4Q}$: (a) for planar 4Q configurations; (b) for twisted 4Q configurations.  We plot all the lattice QCD data including asymmetric cases.  The meaning of the symbols is the same as in Fig.11.  The solid line denotes $\sigma_{\rm 4Q} L_{\rm min}^{\rm c4Q}+C_{\rm 4Q}$. []{data-label="Fig12"}](Fig12b.eps "fig:"){width="6cm"}

Thus, the confinement potential in the 4Q system as shown in Fig.2 is proportional to $L_{\rm min}$, which indicates that the quark confinement force is genuinely 4-body and the flux tube is multi-Y shaped. 

The flip-flop, the level crossing, and absence of the color van der Waals force
-------------------------------------------------------------------------------

Finally, we investigate the flip-flop between the connected 4Q state and the two-meson state.  Since the flux-tube changes its shape so as to have the minimal length, the multi-Y type flux-tube is expected to change into a two-meson state for small $h$.

This type of the flip-flop is physically important for the properties of 4Q states especially for their decay process into two mesons.  Note also that, in the flux-tube picture, the meson-meson reaction is described by the flux-tube recombination between the two mesons, and this process can be realized through the two successive flip-flops between the two-meson state and the connected 4Q state.  Therefore, this type of the flip-flop is important also for the reaction mechanism between two mesons. 

As a clear signal of the flip-flop, we again show the 4Q potential $V_{\rm 4Q}$ for the symmetric planar 4Q configuration with $d$=1,2,3 separately in Fig.13.  The solid curve denotes $V_{\rm c4Q}$ for the OGE plus multi-Y Ansatz, and the dashed-dotted curve $V_{\rm 2Q\bar Q}=2V_{\rm Q\bar Q}(h)$ for the two-meson Ansatz.  For large $h$, $V_{\rm 4Q}$ coincides with the energy $V_{\rm c4Q}$ of the connected 4Q system.  For small $h$, $V_{\rm 4Q}$ coincides with the energy $2V_{\rm Q\bar Q}$ of the “two-meson" system composed of two flux-tubes.  In the intermediate region of $h$, one can observe the cross over from one Ansatz to another. 

Thus, in these particular cases, we can observe a clear evidence of the flip-flop as $$\begin{aligned}
V_{\rm 4Q}(d,h) \simeq {\rm min}(V_{\rm c4Q}(d,h), 2V_{\rm Q\bar Q}(h)),\end{aligned}$$ which indicates the transition between the connected 4Q state and the two-meson state around the level-crossing point where these two systems are degenerate as $V_{\rm c4Q}(d,h)=2V_{\rm Q\bar Q}(h)$. This result also supports the flux-tube picture even for the 4Q system. 

The present lattice QCD results on the “flip-flop" leads to infrared screening and disappearance of the long-range color interactions, i.e., the confining force and the OGE Coulomb force, between (anti)quarks belonging to different “mesons".  This physically results in the absence of the tree-level color van der Waals force between two mesons [@AF78; @GYOPRS79; @FS79]. 

![ The typical lattice QCD results for the flip-flop between the connected 4Q state and the two-meson state for the symmetric planar 4Q configuration with fixed $d$. The symbols denote lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz, and the dashed-dotted curve the two-meson Ansatz. []{ data-label="Fig13"}](Fig13a.eps "fig:"){width="6cm"} ![ The typical lattice QCD results for the flip-flop between the connected 4Q state and the two-meson state for the symmetric planar 4Q configuration with fixed $d$. The symbols denote lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz, and the dashed-dotted curve the two-meson Ansatz. []{ data-label="Fig13"}](Fig13b.eps "fig:"){width="6cm"} ![ The typical lattice QCD results for the flip-flop between the connected 4Q state and the two-meson state for the symmetric planar 4Q configuration with fixed $d$. The symbols denote lattice QCD results.  The curves describe the theoretical form: the solid curve denotes the OGE plus multi-Y Ansatz, and the dashed-dotted curve the two-meson Ansatz. []{ data-label="Fig13"}](Fig13c.eps "fig:"){width="6cm"}

Summary and concluding remarks {#summary}
==============================

We have performed the detailed study of the tetraquark (4Q) potential $V_{\rm 4Q}$ for various QQ-$\rm \bar{Q}\bar{Q}$ systems in SU(3) lattice QCD with $\beta=6.0$ and $16^3 \times 32$ at the quenched level.  For about 200 different patterns of 4Q systems, we have extracted $V_{\rm 4Q}$ from the 4Q Wilson loop in 300 gauge configurations, with the smearing method to enhance the ground-state component.  We have calculated $V_{\rm 4Q}$ for planar, twisted, asymmetric, and large-size 4Q configurations, respectively.  The calculation for large-size 4Q configurations has been done by identifying $16^2 \times 32$ as the spatial size and 16 as the temporal one, and the long-distance confinement force has been particularly analyzed in terms of the flux-tube picture. 

When QQ and $\rm \bar{Q}\bar{Q}$ are well separated, $V_{\rm 4Q}$ is found to be expressed as the sum of the one-gluon-exchange Coulomb term and multi-Y type linear term based on the flux-tube picture.  In this case, all the four quarks are linked by the connected double-Y-shaped flux-tube, where the total flux-tube length is minimized.  On the other hand, when the nearest quark and antiquark pair is spatially close, the system is described as a “two-meson" state rather than the connected 4Q state. 

We have observed a flux-tube recombination called as “flip-flop" between the connected 4Q state and the “two-meson" state around the level-crossing point.  This “flip-flop" leads to infrared screening of the long-range color interactions between (anti)quarks belonging to different mesons, and results in the absence of the tree-level color van der Waals force between two mesons. 

As a next step, it is interesting to investigate the transition in terms of the level crossing between the connected 4Q state and the two-meson state through the diagonalization of the correlation matrix with various 4Q states [@GLPM96].  Through the investigation of the excited-state levels of the 4Q system, a realistic picture for the reaction mechanism between two mesons may be obtained. 

The dynamical quark effect for the flux-tube picture and the flip-flop would be also an interesting subject.  In this context, the string-breaking effect may cause more complicated variation of the transition between single $\rm Q\bar Q$ meson and the multi-quark system. 

In any case, recent lattice QCD studies begin to shed light on the realistic picture in hadron physics and to reveal even the properties of the multi-quark system. 

H.S. was supported in part by a Grant for Scientific Research (No.16540236) from the Ministry of Education, Culture, Science and Technology, Japan.  T.T.T. was supported by the Japan Society for the Promotion of Science.  The lattice QCD Monte Carlo simulations have been performed on NEC-SX5 at Osaka University. 

OGE Coulomb terms in the 4Q potential
=====================================

In the Appendix, we briefly show the derivation of the OGE Coulomb terms in the 4Q potential in Eq.(\[Vc4Q\]) by calculating $\langle {\rm T}| T_i^aT_j^a|{\rm T} \rangle$ for the tetraqurak (4Q) state $\rm |T \rangle$.

In the quark picture, the 4Q state $\rm |T \rangle$ corresponding to Fig.2 is expressed as $$\begin{aligned}
|{\rm T} \rangle 
&=& {\rm |(Q_1Q_2)_{\bar{3}}(\bar{Q}_3\bar{Q}_4)_3 \rangle_1} \nonumber \\
&=& \frac{1}{2\sqrt{3}}
\epsilon^{abc}\epsilon^{ab'c'}
|{\rm Q}_1^b {\rm Q}_2^c \bar{\rm Q}_3^{b'} \bar{\rm Q}_4^{c'} \rangle,
\label{A-one}\end{aligned}$$ where the indices denote the SU(3) color indices of (anti)quarks.  Here, $\rm |T\rangle $ is normalized as $$\begin{aligned}
\langle {\rm T} | {\rm T} \rangle = \frac{1}{12}
\epsilon^{abc}\epsilon^{ab'c'}\epsilon^{def}\epsilon^{de'f'}=1.\end{aligned}$$

The color matrix factor $T_i^a T_j^a$ in the OGE process can be expressed with the Casimir operator $C_2(R)$ as $$\begin{aligned}
T_i^a T_j^a 
&=& \frac{1}{2}\{ (T_i^a+T_j^b)^2-(T_i^a)^2-(T_j^a)^2 \} \nonumber\\
&=& \frac{1}{2}\{ C_2(R_{i+j})-C_2(R_i)-C_2(R_j) \} \nonumber\\
&=& \frac{1}{2}C_2(R_{i+j})-\frac{4}{3},\end{aligned}$$ where $R_{i+j}$ denotes the total SU(3) color representation of the $(i+j)$ system.  Here, $C_2(R_i)=C_2(R_j)=\frac43$ has been used for each (anti)quark belonging to ${\bf 3}$ ($\bar{\bf 3}$). 

In this 4Q system, the two quarks, $\rm Q_1$ and $\rm Q_2$, form the $\bar {\bf 3}$ representation, i.e., $C_2(R_{1+2})=C_2(\bar {\bf 3})=\frac{4}{3}$, and then one gets $\langle {\rm T}|T_1^a T_2^a|{\rm T}\rangle=-\frac{2}{3}$. This type of the Coulomb coefficient between two quarks is the same as that in the 3Q system.  Similarly, one gets $\langle {\rm T}|T_3^a T_4^a|{\rm T}\rangle=-\frac{2}{3}$ for the two antiquarks, $\bar {\rm Q}_3$ and $\bar {\rm Q}_4$.

Next, we consider the Coulomb interaction between the quark and the antiquark.  Owing to the symmetry, we only have to investigate the interaction between $\rm Q_1$ and $\rm \bar Q_3$. To this end, we rewrite the 4Q state $|\rm T\rangle$ in terms of the irreducible representation fo the $\rm Q_1$+$\rm \bar Q_3$ system.  Since $\rm Q_1$ and $\rm \bar Q_3$ can form the singlet ([**1**]{}) or the octet ([**8**]{}) representation, the 4Q state $\rm |T\rangle$ can be rewritten as $$\begin{aligned}
| {\rm T} \rangle 
&=& C_1 |({\rm Q_1\bar{Q}_3})_1({\rm Q_2\bar{Q}_4})_1 \rangle \nonumber \\
&+& C_8 |({\rm Q_1\bar{Q}_3})_8({\rm Q_2\bar{Q}_4})_8 \rangle_1 \end{aligned}$$ with appropriate constants $C_1$ and $C_8$ satisfying $$\begin{aligned}
|C_1|^2+|C_8|^2=1.\end{aligned}$$ After some calculation, one finds $$\begin{aligned}
|{\bf 11}\rangle &\equiv&
|({\rm Q_1\bar{Q}_3})_1({\rm Q_2\bar{Q}_4})_1 \rangle =
\frac{1}{3}|{\rm Q}_1^a{\rm Q}_2^b \bar{\rm Q}_3^b \bar{\rm Q}_4^b \rangle, 
\label{A-six}
\\
|{\bf 88}\rangle &\equiv&
|({\rm Q_1\bar{Q}_3})_8({\rm Q_2\bar{Q}_4})_8 \rangle_1 \nonumber \\
&=& \frac{1}{2\sqrt{2}}
\{|{\rm Q}_1^a {\rm Q}_2^b \bar{\rm Q}_3^b \bar{\rm Q}_4^a\rangle 
-\frac{1}{3}|{\rm Q}_1^a {\rm Q}_2^b \bar{\rm Q}_3^a \bar{\rm Q}_4^b \rangle \},~~~
\label{A-seven}\end{aligned}$$ which satisfy the orthonormal condition, $$\begin{aligned}
\langle{\bf 11}|{\bf 11}\rangle=\langle{\bf 88}|{\bf 88}\rangle=1, \ \langle{\bf 11}|{\bf 88}\rangle=0.  \end{aligned}$$ Using Eqs.(\[A-one\]), (\[A-six\]) and (\[A-seven\]), $C_1$ and $C_8$ can be obtained as $$\begin{aligned}
C_1&=&\langle {\bf 11} |{\rm T} \rangle \nonumber \\
&=&\frac{1}{6\sqrt{3}}
\epsilon^{abc}\epsilon^{ab'c'}
\langle {\rm Q}_1^d{\rm Q}_2^e \bar{\rm Q}_3^d \bar{\rm Q}_4^e |
{\rm Q}_1^b {\rm Q}_2^c \bar{\rm Q}_3^{b'} \bar{\rm Q}_4^{c'}\rangle \nonumber \\
&=&\frac{1}{6\sqrt{3}}
\epsilon^{abc}\epsilon^{abc}=\frac{1}{\sqrt{3}}, \end{aligned}$$ $$\begin{aligned}
C_8&=&\langle {\bf 88} |{\rm T} \rangle \nonumber \\
&=&\frac{1}{4\sqrt{6}}
\epsilon^{abc}\epsilon^{ab'c'}
\langle
{\rm Q}_1^d {\rm Q}_2^e \bar{\rm Q}_3^e \bar{\rm Q}_4^d|
{\rm Q}_1^b {\rm Q}_2^c \bar{\rm Q}_3^{b'} \bar{\rm Q}_4^{c'}\rangle \nonumber \\
&-&\frac{1}{12\sqrt{6}}
\epsilon^{abc}\epsilon^{ab'c'}
\langle 
{\rm Q}_1^d {\rm Q}_2^e \bar{\rm Q}_3^d \bar{\rm Q}_4^e|
{\rm Q}_1^b {\rm Q}_2^c \bar{\rm Q}_3^{b'} \bar{\rm Q}_4^{c'}\rangle \nonumber \\
&=&\frac{1}{4\sqrt{6}}\epsilon^{abc}\epsilon^{acb}
-\frac{1}{12\sqrt{6}}\epsilon^{abc}\epsilon^{abc}
=-\sqrt{\frac23}.\end{aligned}$$ Then, using Eq.(\[A-one\]), $C_2({\bf 1})=0$ and $C_2({\bf 8})=3$, we get $$\begin{aligned}
\langle {\rm T} | T_1^a T_3^a | {\rm T} \rangle 
&=&   |C_1|^2 \langle {\bf 11} | T_1^a T_3^a | {\bf 11} \rangle
    + |C_8|^2 \langle {\bf 88} | T_1^a T_3^a | {\bf 88} \rangle \nonumber\\
&=& |C_1|^2 \{ \frac12 C_2({\bf 1})-\frac43 \} +|C_8|^2 \{ \frac12 C_2({\bf 8})-\frac43 \} \nonumber \\
&=& -\frac{1}{3}.\end{aligned}$$

In this way, for the Coulomb interaction in the 4Q system as shown in Fig.3, we obtain $$\begin{aligned}
\langle {\rm T} | T_1^a T_2^a | {\rm T} \rangle&=&\langle {\rm T} | T_3^a T_4^a | {\rm T} \rangle= -\frac{2}{3}, \\
\langle {\rm T} | T_1^a T_3^a | {\rm T} \rangle&=&\langle {\rm T} | T_1^a T_4^a | {\rm T} \rangle
=\langle {\rm T} | T_2^a T_3^a | {\rm T} \rangle \nonumber \\
&=&\langle {\rm T} | T_2^a T_4^a | {\rm T} \rangle= -\frac{1}{3},\end{aligned}$$ and derive the Coulomb terms in Eq.(\[Vc4Q\]). 

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