Lattice QCD static potentials of the meson-meson and tetraquark systems computed with both quenched and full QCD

TL;DR

This study uses lattice QCD simulations to analyze static potentials in meson-meson and tetraquark systems, comparing quenched and full QCD, and explores the transition between different quark configurations.

Contribution

It provides a detailed comparison of static potentials in tetraquark and meson systems using both quenched and dynamical lattice QCD, including analysis of color excitations and transition mechanisms.

Findings

Potential differences between quenched and full QCD.

Identification of transition behavior between tetraquark and meson states.

Quantitative fits with string flip-flop models.

Abstract

We revisit the static potential for the $Q Q \bar Q \bar Q$ system using SU(3) lattice simulations, studying both the colour singlets groundstate and first excited state. We consider geometries where the two static quarks and the two anti-quarks are at the corners of rectangles of different sizes. We analyse the transition between a tetraquark system and a two meson system with a two by two correlator matrix. We compare the potentials computed with quenched QCD and with dynamical quarks. We also compare our simulations with the results of previous studies and analyze quantitatively fits of our results with anzatse inspired in the string flip-flop model and in its possible colour excitations.

Tables

Table 1. Table 1: Normalized mean values of the Casimir invariant operators ⟨ Ψ | C i ​ j | Ψ ⟩ quantum-operator-product Ψ subscript 𝐶 𝑖 𝑗 Ψ \langle\Psi|C_{ij}|\Psi\rangle . The indices i 𝑖 i and j 𝑗 j refer to the quarks and anti-quarks.

$|\Psi ⟩$	$C_{12}$	$C_{13}$	$C_{14}$
$|{\mathrm{𝟏}}_{13}{\mathrm{𝟏}}_{24}⟩$	0	1	0
$|{\mathrm{𝟖}}_{13}{\mathrm{𝟖}}_{24}⟩$	1/4	-1/8	7/8
$|{\mathrm{𝟏}}_{14}{\mathrm{𝟏}}_{23}⟩$	0	0	1
$|{\mathrm{𝟖}}_{14}{\mathrm{𝟖}}_{23}⟩$	1/4	7/8	-1/8
$|{\overline{\mathrm{𝟑}}}_{12}{\mathrm{𝟑}}_{34}⟩$	1/2	1/4	1/4
$|{\mathrm{𝟔}}_{12}{\overline{\mathrm{𝟔}}}_{34}⟩$	-1/4	5/8	5/8

Table 2. Table 2: Fits of the static Q ​ Q ¯ 𝑄 ¯ 𝑄 Q\bar{Q} meson potential (Wilson loop) in quenched QCD, for different intervals r ∈ [ r min , r max ] 𝑟 subscript 𝑟 min subscript 𝑟 max r\in[r_{\text{min}},r_{\text{max}}] , to the Cornell potential model of Eq. ( 24 ).

$r_{\text{min}}$	$r_{\text{max}}$	${\chi }^2/\text{d.o.f.}$	$Ka$	$\gamma$	$\sigma a^2$	$a$ (fm)	$a^{-1}\left(\text{GeV}\right)$
5	12	0.98	0.6406(21)	0.3078(77)	0.02490(14)	0.0681	2.898
6	12	0.62	0.6382(49)	0.2987(199)	0.02506(29)	0.0681	2.898
5	11	1.08	0.6409(21)	0.3085(75)	0.02488(14)	0.0681	2.898
6	11	0.79	0.6385(55)	0.2996(224)	0.02504(34)	0.0681	2.898

Table 3. Table 3: Fits of the static Q ​ Q ¯ 𝑄 ¯ 𝑄 Q\bar{Q} meson potential (Wilson loop) in full QCD, for different intervals r ∈ [ r min , r max ] 𝑟 subscript 𝑟 min subscript 𝑟 max r\in[r_{\text{min}},r_{\text{max}}] , to the Cornell potential model of Eq. ( 24 ).

$r_{\text{min}}$	$r_{\text{max}}$	${\chi }^2/\text{d.o.f.}$	$Ka$	$\gamma$	$\sigma a^2$	$a$ (fm)	$a^{-1}\left(\text{GeV}\right)$
3	12	0.43	0.2995(23)	0.3625(49)	0.03092(25)	0.0775	2.546
4	12	0.50	0.3005(84)	0.3654(270)	0.03085(64)	0.0775	2.546
5	12	0.43	0.2931(169)	0.3386(638)	0.03129(109)	0.0772	2.546
3	11	0.19	0.3017(42)	0.3666(117)	0.03065(33)	0.0773	2.553
4	11	0.04	0.3063(25)	0.3799(76)	0.03030(21)	0.0772	2.546
5	11	0.05	0.3042(58)	0.3728(204)	0.03044(40)	0.0772	2.546

Table 4. Table 4: Fits of the quenched Q ​ Q ​ Q ¯ ​ Q ¯ 𝑄 𝑄 ¯ 𝑄 ¯ 𝑄 QQ\bar{Q}\bar{Q} anti-parallel alignment excited state potential to a Cornell ansatz.

$r_1$	$r_{\text{min}}$	$r_{\text{max}}$	${\chi }^2/\text{d.o.f.}$	$Ca$	$\gamma$	$\sigma a^2$	$\sigma /{\sigma }_{meson}$
6	8	12	1.30	1.183(60)	0.14(30)	0.0570(30)	2.28(12)
	9	12	0.11	1.252(54)	0.50(28)	0.0537(26)	2.16(10)
7	8	12	1.00	1.124(77)	0.24(37)	0.0584(39)	2.35(16)
	9	12	0.03	1.218(32)	0.23(17)	0.0537(15)	2.16(6)

Table 5. Table 5: Fits of the quenched data anti-parallel alignment ground state potential to the transition ansatz of Eqs. ( VI.1 ) and ( 31 ).

$r_{\text{min}}$	$r_{\text{max}}$	${\chi }^2/\text{d.o.f.}$	${\Delta }_{0}a$	$c{a}^2$	$\lambda a$
5	11	1.02	0.0335(16)	0.1547(89)	0.0309(37)
5	12	1.15	0.0335(16)	0.1548(90)	0.0309(38)
6	11	0.75	0.0362(49)	0.1644(127)	0.0363(86)
6	12	0.79	0.0363(49)	0.1643(127)	0.0364(86)

Table 6. Table 6: Fits of the quenched data anti-parallel alignment ground state potential to the transition ansatz of Eqs. ( VI.1 ) and ( 32 ).

$r_{\text{min}}$	$r_{\text{max}}$	${\chi }^2/\text{d.o.f.}$	${\Delta }_{0}a$	$c{a}^2$	$d{a}^2$
5	11	1.07	0.0285(10)	0.1934(144)	0.0016(3)
5	12	1.19	0.0285(10)	0.1938(147)	0.0016(3)
6	11	0.77	0.0294(33)	0.2275(330)	0.0018(7)
6	12	0.80	0.0295(33)	0.2278(332)	0.0018(7)

Table 7. Table 7: Data for the ground state potential in the anti-parallel geometry for the pure-gauge simulation. All values are in lattice units.

$r_{13}$	$r_{14}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{13}$	$r_{14}$	$V_0$	$t_i-t_f$	${\chi }_r^2$
4	4	1.3010(1)	9-16	0.94	7	7	1.51984(14)	6-10	0.52
	5	1.32119(8)	8-16	0.82		8	1.53675(17)	6-10	0.65
	6	1.32452(4)	5-16	0.62		9	1.54052(23)	6-16	0.48
	7	1.32547(5)	5-16	0.69		10	1.54154(31)	6-16	0.88
	8	1.32581(5)	5-16	0.53		11	1.54208(32)	6-16	1.06
	9	1.32594(5)	5-16	0.82		12	1.54220(36)	6-16	0.49
	10	1.32601(5)	5-16	0.76	8	8	1.58247(33)	6-16	0.96
	11	1.32608(6)	5-16	1.40		9	1.59850(38)	6-13	1.01
	12	1.32604(6)	5-10	0.85		10	1.60205(42)	6-16	0.38
5	5	1.38254(8)	5-11	0.92		11	1.60289(59)	6-16	1.09
	6	1.40149(9)	5-11	0.63		12	1.60232(13)	7-12	0.52
	7	1.40546(10)	5-16	0.70	9	9	1.64322(81)	6-16	1.02
	8	1.40657(11)	5-16	0.87		10	1.65809(101)	6-15	0.94
	9	1.40657(12)	5-16	1.13		11	1.65919(90)	7-16	0.76
	10	1.40716(12)	5-16	0.86		12	1.65961(95)	7-10	0.75
	11	1.40729(11)	5-16	0.64	10	10	1.70108(219)	7-16	1.00
	12	1.40729(12)	6-13	1.06		11	1.71397(156)	7-10	1.03
6	6	1.45412(15)	5-10	0.88		12	1.71573(183)	7-10	1.06
	7	1.47203(21)	5-16	1.00	11	11	1.75794(335)	7-10	1.26
	8	1.47596(26)	5-16	1.21		12	1.77037(216)	7-10	0.70
	9	1.47710(31)	5-16	1.34	12	12	1.81682(432)	7-10	0.74
	10	1.47760(31)	5-16	1.37
	11	1.47783(31)	5-16	1.20
	12	1.47763(13)	6-13	0.38

Table 8. Table 8: Data for the first excited state potential for the anti-parallel geometry and for the pure-gauge simulation. All values are in lattice units.

$r_{13}$	$r_{14}$	$V_1$	$t_i-t_f$	${\chi }_r^2$	$r_{13}$	$r_{14}$	$V_1$	$t_i-t_f$	${\chi }_r^2$
4	4	1.37342(8)	4-7	1.11	7	7	1.57714(10)	7-10	0.17
	5	1.43446(6)	4-6	0.41		8	1.62034(21)	7-16	0.23
	6	1.50427(8)	4-5	0.32		9	1.67517(27)	7-13	0.39
	7	-	-	-		10	1.73150(34)	7-13	0.26
	8	-	-	-		11	1.78729(32)	7-14	0.34
	9	-	-	-		12	1.84313(215)	7-15	0.23
	10	-	-	-	8	8	1.63344(67)	7-15	0.39
	11	-	-	-		9	1.67532(85)	7-14	0.48
	12	-	-	-		10	1.72029(29)	9-10	0.01
5	5	1.45092(12)	5-7	0.66		11	1.77764(267)	8-10	0.51
	6	1.50245(14)	5-7	0.57		12	1.83159(607)	8-15	0.51
	7	1.56502(18)	5-7	1.02	9	9	1.68826(48)	7-16	0.24
	8	1.62775(18)	5-6	0.40		10	1.72195(153)	9-11	0.20
	9	-	-	-		11	1.78195(74)	7-8	0.37
	10	-	-	-		12	1.83526(272)	7-12	0.57
	11	-	-	-	10	10	1.73977(76)	7-14	0.22
	12	-	-	-		11	1.78172(233)	7-12	0.72
6	6	1.51750(20)	6-14	1.12		12	1.83504(281)	7-14	0.41
	7	1.56389(24)	6-16	0.44	11	11	1.79142(126)	7-12	0.40
	8	1.62139(19)	6-8	0.64		12	1.83291(374)	7-12	0.49
	9	1.68025(11)	6-7	0.07	12	12	1.84085(34)	7-13	0.16
	10	1.73928(61)	6-7	1.03
	11	1.79777(48)	5-6	0.79
	12	1.85509(28)	5-6	0.17

Table 9. Table 9: Data for the ground state potential for the anti-parallel geometry and for the full QCD simulation. All values are in lattice units.

$r_{13}$	$r_{14}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{13}$	$r_{14}$	$V_0$	$t_i-t_f$	${\chi }_r^2$
3	3	0.5126(1)	6-10	0.07	6	6	0.8248(26)	6-8	0.62
	4	0.5397(2)	6-10	0.07		7	0.8464(12)	6-7	0.26
	5	0.5431(2)	6-10	0.24		8	0.8504(18)	6-8	0.79
	6	0.5438(1)	6-10	0.12		9	0.8503(25)	6-8	0.82
	7	0.5437(2)	6-10	0.07		10	0.8520(10)	6-7	0.18
	8	0.5438(2)	6-10	0.20		11	0.8531(18)	6-11	0.38
	9	0.5433(3)	6-10	0.34	7	7	0.9088(67)	6-10	0.60
	10	0.5434(3)	6-10	0.32		8	0.9293(72)	6-10	1.07
	11	0.5440(2)	6-10	0.39		9	0.9157(23)	7-8	0.12
4	4	0.6364(6)	6-10	0.50		10	0.9314(71)	6-8	1.12
	5	0.6620(3)	6-10	0.08		11	0.9342(10)	6-8	0.19
	6	0.6658(1)	6-10	0.02	8	8	0.9854(214)	6-12	0.83
	7	0.6654(9)	6-10	0.80		9	1.0023(159)	6-11	1.25
	8	0.6655(10)	6-10	0.66		10	1.0048(96)	6-10	0.70
	9	0.6654(9)	6-10	0.58		11	1.0004(82)	6-10	0.40
	10	0.6667(5)	6-10	0.32	9	9	1.0650(277)	6-10	1.01
	11	0.6677(4)	6-10	0.14		10	1.0734(178)	6-10	0.55
5	5	0.7382(9)	6-10	0.26		11	1.0805(26)	5-6	0.22
	6	0.7605(5)	6-10	0.29	10	10	1.1377(56)	5-6	0.64
	7	0.7621(14)	6-10	0.66		11	1.1491(18)	5-6	0.06
	8	0.7630(20)	6-10	1.09	11	11	1.1630(298)	6-9	0.14
	9	0.7642(11)	6-10	0.54
	10	0.7643(8)	6-10	0.28
	11	0.7664(16)	6-10	0.20

Table 10. Table 10: Data for the first excited state potential for the anti-parallel geometry and for the full QCD simulation. All values are in lattice units.

$r_{13}$	$r_{14}$	$V_1$	$t_i-t_f$	${\chi }_r^2$	$r_{13}$	$r_{14}$	$V_1$	$t_i-t_f$	${\chi }_r^2$
3	3	0.6039(2)	6-7	0.12	6	6	0.8988(38)	6-12	0.16
	4	0.7062(10)	6-7	0.91		7	0.9549(19)	6-15	0.69
	5	-	-	-		8	1.0233(101)	6-10	1.26
	6	-	-	-		9	1.0933(184)	6-11	0.34
	7	-	-	-		10	1.1690(8)	6-10	0.10
	8	-	-	-		11	1.2341(136)	6-9	0.25
	9	-	-	-	7	7	0.9704(33)	6-10	0.67
	10	-	-	-		8	1.0235(96)	6-10	0.81
	11	-	-	-		9	1.0952(79)	6-10	0.34
4	4	0.7234(3)	6-10	0.19		10	1.1683(170)	6-9	1.26
	5	0.7990(8)	6-13	0.12		11	1.2274(408)	6-9	0.31
	6	0.8960(22)	6-12	0.78	8	8	1.0482(35)	6-12	0.42
	7	0.9974(47)	6-7	1.20		9	1.1122(9)	6-8	0.11
	8	-	-	-		10	1.1956(30)	6-9	0.42
	9	-	-	-		11	1.2612(89)	6-9	0.06
	10	-	-	-	9	9	1.1219(15)	6-10	0.13
	11	-	-	-		10	1.1852(360)	6-9	0.92
5	5	0.8160(12)	6-14	0.27		11	1.2526(507)	6-11	0.26
	6	0.8821(16)	6-12	0.27	10	10	1.1621(476)	6-9	0.79
	7	0.9617(25)	6-12	0.63		11	1.2069(751)	6-9	1.46
	8	1.0409(16)	6-12	0.63	11	11	1.2052(998)	6-9	1.30
	9	1.1287(136)	6-8	0.49
	10	1.1992(362)	6-9	0.71
	11	1.2929(606)	6-8	0.88

Table 11. Table 11: Data for the ground state potential for the parallel geometry and for the pure-gauge simulation. All values are in lattice units.

$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$
4	4	1.31360(7)	9-16	0.90	7	4	1.32551(5)	7-16	0.64	10	4	1.32607(8)	7-14	1.09
	5	1.37514(19)	9-16	0.90		5	1.40571(11)	7-16	0.88		5	1.40713(10)	7-15	0.75
	6	1.41847(24)	9-14	0.78		6	1.47380(8)	7-16	0.18		6	1.47735(10)	6-16	0.23
	7	1.45346(28)	9-15	0.76		7	1.53399(6)	7-15	0.41		7	1.54129(20)	7-14	0.54
	8	1.48445(25)	9-15	0.28		8	1.57844(29)	7-15	1.10		8	1.60145(57)	7-15	0.99
	9	1.51359(33)	9-15	0.49		9	1.63355(32)	7-15	0.68		9	1.65887(79)	7-10	0.86
	10	1.54153(63)	9-12	0.49		10	1.67247(85)	7-14	0.66		10	1.71108(252)	8-10	1.37
	11	1.57062(87)	8-12	0.91		11	1.70545(208)	7-14	0.81		11	1.76163(373)	8-10	1.18
	12	1.59793(83)	8-12	0.84		12	1.73700(298)	7-13	1.03		12	1.81737(460)	7-13	0.67
5	4	1.32224(5)	5-16	0.38	8	4	1.32583(5)	6-16	0.67	11	4	1.32618(9)	8-12	1.02
	5	1.39610(12)	6-16	0.88		5	1.40657(8)	7-16	0.53		5	1.40724(10)	6-16	0.85
	6	1.45207(32)	8-16	1.07		6	1.47618(10)	7-16	0.34		6	1.47750(12)	6-16	0.19
	7	1.49576(37)	8-16	0.94		7	1.53861(10)	7-14	0.25		7	1.54185(33)	6-16	0.92
	8	1.53054(45)	9-13	1.10		8	1.59605(24)	7-12	0.18		8	1.60217(25)	7-10	0.32
	9	1.56251(81)	8-15	1.00		9	1.64883(59)	7-15	0.63		9	1.65978(75)	7-10	1.01
	10	1.59199(100)	8-15	1.05		10	1.69660(123)	7-14	0.89		10	1.71562(138)	7-10	0.98
	11	1.62032(89)	8-13	0.72		11	1.73760(192)	7-12	0.70		11	1.76992(238)	7-10	0.88
	12	1.64931(87)	7-15	0.46		12	1.76709(231)	8-12	0.21		12	1.82503(171)	7-15	0.67
6	4	1.32467(4)	5-16	0.39	9	4	1.32597(6)	6-15	0.85	12	4	1.32661(17)	10-16	0.85
	5	1.40329(12)	5-16	0.74		5	1.40698(9)	6-14	1.02		5	1.40731(12)	6-16	1.07
	6	1.46806(17)	6-16	0.49		6	1.47693(11)	6-14	0.17		6	1.47769(11)	6-14	0.26
	7	1.52250(34)	6-16	0.57		7	1.54083(22)	6-14	0.28		7	1.54211(33)	6-16	0.52
	8	1.56670(24)	7-16	0.32		8	1.59999(25)	7-16	0.72		8	1.60188(67)	7-16	1.11
	9	1.60289(89)	8-14	1.01		9	1.65665(13)	7-15	0.26		9	1.65957(131)	7-16	1.21
	10	1.63500(109)	8-11	0.58		10	1.70918(103)	7-14	0.84		10	1.71533(282)	7-16	1.08
	11	1.66453(89)	8-16	0.19		11	1.75771(215)	7-14	0.89		11	1.77108(299)	7-10	1.34
	12	1.69415(147)	8-15	0.74		12	1.80102(441)	7-15	0.93		12	1.82536(277)	7-13	0.42

Table 12. Table 12: Data for the first excited state potential and for the parallel geometry for the pure-gauge simulation. All values are in lattice units.

$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$
4	4	-	-	-	7	4	1.59960(18)	7-13	0.40	10	4	1.74590(101)	7-16	0.33
	5	-	-	-		5	1.61629(19)	7-14	0.39		5	1.75813(40)	7-16	0.32
	6	-	-	-		6	1.63726(37)	7-15	0.84		6	1.77430(162)	7-14	0.80
	7	-	-	-		7	1.66201(25)	7-12	0.39		7	1.79224(143)	7-13	0.54
	8	-	-	-		8	1.69130(104)	7-15	0.67		8	1.81258(123)	7-14	0.91
	9	-	-	-		9	1.72705(87)	7-10	0.91		9	1.83436(263)	7-15	0.91
	10	-	-	-		10	1.76824(107)	7-10	0.49		10	1.85436(263)	7-15	0.86
	11	-	-	-		11	1.81376(100)	7-14	0.75		11	1.87964(35)	7-16	0.63
	12	-	-	-		12	1.85802(299)	7-13	0.55		12	1.90681(118)	7-12	0.30
5	4	1.48851(16)	7-11	0.61	8	4	1.65264(44)	7-15	0.31	11	4	1.78291(307)	7-15	0.96
	5	-	-	-		5	1.66678(121)	6-16	0.76		5	1.79754(100)	7-14	1.27
	6	-	-	-		6	1.68246(25)	7-14	0.42		6	1.81565(156)	7-11	1.04
	7	-	-	-		7	1.70390(12)	7-15	0.38		7	1.83565(337)	7-14	0.70
	8	-	-	-		8	1.72841(33)	7-15	0.42		8	1.85526(365)	7-15	0.74
	9	-	-	-		9	1.75774(96)	7-15	0.65		9	1.87600(800)	7-15	0.60
	10	-	-	-		10	1.79002(155)	7-11	0.59		10	1.89616(84)	7-12	0.80
	11	-	-	-		11	1.82871(186)	7-13	0.40		11	1.91507(533)	7-12	0.84
	12	-	-	-		12	1.87212(329)	7-15	0.91		12	1.93991(618)	7-13	0.67
6	4	1.54514(37)	6-13	0.85	9	4	1.70183(27)	7-15	0.68	12	4	1.80257(107)	8-12	1.17
	5	1.56559(22)	7-14	0.59		5	1.71365(17)	7-16	0.17		5	1.83135(262)	7-14	0.46
	6	1.59216(34)	7-14	0.33		6	1.72868(41)	7-15	0.73		6	1.85330(267)	7-11	0.34
	7	1.62468(83)	7-15	0.89		7	1.74737(42)	7-15	0.36		7	1.87618(254)	7-13	0.39
	8	-	-	-		8	1.76888(84)	7-14	0.80		8	1.89751(246)	7-13	0.18
	9	-	-	-		9	1.79388(113)	7-14	0.86		9	1.91888(266)	7-13	0.27
	10	-	-	-		10	1.82044(154)	7-14	0.22		10	1.93594(431)	7-9	0.45
	11	-	-	-		11	1.85045(370)	7-14	0.51		11	1.95381(962)	7-15	0.89
	12	-	-	-		12	1.88700(724)	7-14	0.94		12	1.97689(190)	7-13	0.45

Table 13. Table 13: Data for the ground state potential and for the parallel geometry and for the full QCD simulation. All values are in lattice units.

$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_0$	$t_i-t_f$	${\chi }_r^2$
3	3	0.5280(4)	5-15	1.07	6	3	0.5432(3)	5-13	0.47	9	3	0.5435(3)	5-14	0.74
	4	0.6163(6)	7-16	0.29		4	0.6654(4)	5-10	0.29		4	0.6665(6)	5-15	0.59
	5	0.6718(11)	7-12	0.75		5	0.7630(6)	5-15	0.58		5	0.7661(13)	5-13	1.03
	6	0.7162(13)	7-12	0.29		6	0.8450(13)	5-13	0.87		6	0.8447(57)	7-13	1.01
	7	0.7553(14)	7-12	0.74		7	0.9143(27)	5-11	0.58		7	0.9331(30)	5-15	0.59
	8	0.7925(11)	7-10	0.09		8	0.9696(56)	5-15	0.92		8	1.0092(39)	5-11	0.66
	9	0.8259(34)	7-10	0.57		9	1.0157(75)	5-11	0.93		9	1.0813(52)	5-10	0.80
	10	0.8546(27)	7-9	0.59		10	1.0551(104)	5-13	1.05		10	1.1512(14)	5-6	0.12
	11	0.8882(27)	7-8	0.30		11	1.0939(77)	5-10	1.26		11	1.2129(148)	5-9	1.60
4	3	0.5396(2)	7-16	0.46	7	3	0.5434(3)	5-16	0.92	10	3	0.5433(4)	5-16	0.74
	4	0.6513(6)	7-16	0.43		4	0.6660(6)	5-13	0.64		4	0.6666(6)	5-16	1.07
	5	0.7283(15)	7-16	0.66		5	0.7652(9)	5-14	0.74		5	0.7658(16)	5-16	0.90
	6	0.7827(12)	6-16	0.91		6	0.8516(12)	5-13	0.76		6	0.8505(18)	6-12	0.46
	7	0.8264(16)	6-15	0.45		7	0.9281(17)	5-10	1.27		7	0.9237(53)	7-9	0.71
	8	08649(26)	6-14	0.39		8	0.9958(36)	5-8	1.07		8	1.0054(92)	6-15	0.74
	9	0.9006(38)	6-9	0.95		9	1.0531(84)	5-10	1.27		9	1.0766(164)	6-12	0.92
	10	0.9316(73)	6-14	0.79		10	1.0868(140)	6-11	0.76		10	1.0857(296)	7-9	0.47
	11	0.9638(88)	6-16	0.80		11	1.1255(125)	6-10	1.22		11	1.2039(334)	6-9	0.72
5	3	0.5425(3)	5-16	0.42	8	3	0.5437(2)	5-15	0.46	11	3	0.5431(4)	5-14	0.80
	4	0.6622(6)	5-16	0.63		4	0.6666(5)	5-16	0.83		4	0.6665(5)	5-12	0.63
	5	0.7528(12)	6-11	0.72		5	0.7664(9)	5-13	1.18		5	0.7657(2)	6-14	0.47
	6	0.8244(13)	6-12	0.25		6	0.8534(14)	5-15	0.89		6	0.8531(11)	6-14	0.45
	7	0.8778(27)	6-16	0.38		7	0.9336(9)	5-10	0.40		7	0.9318(13)	6-11	0.42
	8	0.9214(48)	6-16	0.36		8	1.0065(29)	5-11	0.69		8	1.0040(63)	6-11	0.31
	9	0.9575(70)	6-11	0.94		9	1.0652(128)	6-10	1.20		9	1.0490(319)	7-10	1.18
	10	0.9905(123)	6-9	2.34		10	1.1188(330)	6-10	0.96		10	1.1035(584)	7-11	0.81
	11	1.0301(84)	6-11	1.09		11	1.1548(428)	6-10	0.96		11	1.2040(147)	6-8	0.52

Table 14. Table 14: Data for the first excited state potential for the parallel geometry and for the full QCD simulation. All values are in lattice units.

$r_{12}$	$r_{13}$	$V_1$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_1$	$t_i-t_f$	${\chi }_r^2$	$r_{12}$	$r_{13}$	$V_1$	$t_i-t_f$	${\chi }_r^2$
3	3	0.6637(29)	8-12	1.14	6	3	0.9136(16)	5-15	5-15	9	3	1.0911(229)	6-11	0.99
	4	-	-	-		4	0.9313(20)	5-9	0.42		4	1.1233(40)	5-10	0.52
	5	-	-	-		5	0.9575(33)	5-11	0.89		5	1.1457(79)	5-11	0.62
	6	0.6054(299)	10-15	1.05		6	0.9926(42)	5-12	1.30		6	1.1692(194)	5-11	1.22
	7	0.5737(83)	9-16	0.66		7	1.0292(103)	6-9	0.85		7	1.1897(286)	5-11	2.08
	8	0.5398(41)	9-11	0.37		8	1.0472(105)	7-10	0.23		8	1.2112(244)	5-10	1.29
	9	0.5420(79)	9-10	0.41		9	1.1304(298)	6-11	1.35		9	1.2476(130)	5-10	0.55
	10	0.5399(22)	8-9	0.08		10	1.1892(126)	6-7	0.72		10	1.2876(108)	5-9	0.76
	11	0.5491(149)	8-9	1.33		11	1.2250(268)	6-10	0.74		11	1.3220(308)	5-9	0.60
4	3	0.7502(4)	7-8	0.10	7	3	0.9855(32)	5-12	0.67	10	3	1.1340(108)	6-11	0.69
	4	0.7863(45)	7-12	1.16		4	1.0004(30)	5-12	0.75		4	1.1560(137)	6-10	0.81
	5	0.8363(94)	7-15	1.47		5	1.0226(55)	5-11	0.62		5	1.1748(75)	6-8	0.63
	6	0.8490(292)	8-13	1.49		6	1.0401(140)	6-11	1.00		6	1.1807(221)	6-7	1.41
	7	0.8468(298)	8-13	0.94		7	1.0727(113)	6-9	0.50		7	1.1899(578)	6-10	1.21
	8	0.8285(299)	8-13	0.59		8	1.1070(23)	6-11	0.45		8	1.2202(592)	6-9	1.17
	9	0.8141(492)	8-12	1.36		9	1.1638(102)	5-9	1.01		9	1.3152(104)	5-6	0.91
	10	0.7706(714)	8-12	1.22		10	1.2227(44)	5-9	0.66		10	1.3472(243)	5-9	0.48
	11	0.6894(1127)	8-14	1.42		11	1.2788(115)	5-7	0.62		11	1.3764(32)	5-9	0.18
5	3	0.8360(7)	5-8	0.40	8	3	1.0504(73)	5-11	1.20	11	3	1.1707(7)	6-7	0.001
	4	0.8614(13)	5-8	1.20		4	1.0642(30)	5-12	0.24		4	1.1916(188)	6-9	0.22
	5	0.8946(36)	6-9	0.89		5	1.0850(77)	5-10	0.75		5	1.2134(375)	6-10	1.01
	6	0.9284(129)	7-10	1.28		6	1.1121(60)	5-7	0.97		6	1.2265(479)	6-10	0.46
	7	0.9813(179)	7-12	0.66		7	1.1154(113)	6-12	1.12		7	1.2458(381)	6-10	0.18
	8	1.0230(200)	7-13	1.03		8	1.1610(141)	5-10	0.99		8	1.2981(96)	6-8	0.32
	9	1.0636(251)	7-12	0.94		9	1.1988(67)	5-9	0.87		9	1.3944(490)	5-10	0.84
	10	1.1215(245)	6-9	0.81		10	1.2453(48)	5-10	0.34		10	1.4238(236)	5-8	0.44
	11	1.1400(60)	6-7	0.12		11	1.2913(190)	5-12	0.81		11	1.4508(480)	5-9	0.40

Equations

$$|1_{13}{1}_{24}⟩$$

$$|1_{14}{1}_{23}⟩$$

$$⟨1_{13}{1}_{24}|1_{14}{1}_{23}⟩=\frac{1}{3}.$$

$$|8_{13}{8}_{24}⟩=\frac{1}{4\sqrt{2}}{\lambda }_{ik}^a{\lambda }_{jl}^a|Q_i{Q}_j{\bar{Q}}_k{\bar{Q}}_l⟩,$$

$$|8_{14}{8}_{23}⟩=\frac{1}{4\sqrt{2}}{\lambda }_{il}^a{\lambda }_{jk}^a|Q_i{Q}_j{\bar{Q}}_k{\bar{Q}}_l⟩,$$

$$⟨8_{13}{8}_{24}|8_{13}{8}_{24}⟩=⟨8_{14}{8}_{23}|8_{14}{8}_{23}⟩=1.$$

$$|8_{13}{8}_{24}⟩$$

$$|8_{14}{8}_{23}⟩$$

$$⟨1_{13}{1}_{24}|8_{13}{8}_{24}⟩$$

$$⟨1_{14}{1}_{23}|8_{14}{8}_{23}⟩$$

$$|{\bar{3}}_{12}{3}_{34}⟩$$

$$|6_{12}{\bar{6}}_{34}⟩$$

$$P_{12}|{\bar{3}}_{12}{3}_{34}⟩$$

$$P_{12}|6_{12}{\bar{6}}_{34}⟩$$

$$|1_{13}{1}_{24}⟩=\sqrt{\frac{2}{3}}|6_{12}{\bar{6}}_{34}⟩+\frac{1}{\sqrt{3}}|{\bar{3}}_{12}{3}_{34}⟩,$$

$$|1_{14}{1}_{23}⟩=\sqrt{\frac{2}{3}}|6_{12}{\bar{6}}_{34}⟩-\frac{1}{\sqrt{3}}|{\bar{3}}_{12}{3}_{34}⟩,$$

$$V_{CS}=\sum _{i<j}{C}_{ij}\text{leavevmode}{V}_M,$$

$$V_{\Psi }=⟨\Psi |V_{CS}|\Psi ⟩.$$

$$V_{33}$$

$$r_{12}=|{\mathbf{x}}_1-{\mathbf{x}}_2|=|{\mathbf{x}}_3-{\mathbf{x}}_4|,$$

$$r_{13}=|{\mathbf{x}}_1-{\mathbf{x}}_3|=|{\mathbf{x}}_2-{\mathbf{x}}_4|.$$

$$r_{13}$$

$$r_{14}$$

$$M_{ij}$$

$$M_{ij}(t)a_j(t)={\lambda }_k(t)M_{ij}(t_0)a_j(t).$$

$$M_{eff}(t)$$

$${\mathcal{O}}_{13,24}$$

$${\mathcal{O}}_{14,23}$$

$$M=\left(\begin{array}[]{cc}W_{13}W_{24}&\frac{1}{3}W_{1324}\\ \frac{1}{3}W_{1423}&W_{14}W_{23}\end{array}\right)\ ,$$

$${\mathcal{O}}_{YY}=$$

$$M=\left(\begin{array}[]{cc}W_{YY}&\frac{1}{2\sqrt{3}}W_{YY,1324}\\ \frac{1}{2\sqrt{3}}W_{1324,YY}&W_{13}W_{24}\end{array}\right)\ .$$

$$V_M(r)=K-\frac{\gamma }{r}+\sigma r.$$

$$r_0^2\frac{d{V}_M}{dr}(r_0)=1.65,$$

$$V_0$$

$$V_{MM}$$

Full text

BLAH BLAH BLAH

I Introduction

Our current understanding of strong interaction phenomenology, being the hadron spectrum or the form factors associated to transitions between hadrons, relies on the description of the quark and gluon interaction within Quantum Chromodynamics. Despite the efforts of several decades, the non-perturbative nature of QCD still ensconce several properties of its fundamental particles. Indeed, we still do not understand the confinement mechanism, which prevents the observation of free quarks and gluons in nature, and still do not have a satisfactory answer why the experimentally  Olive:2016xmw confirmed hadrons are composed of three valence quarks or a pair of quark and an anti-quark.

QCD is a gauge theory and physical observables should be gauge invariant objects. Gauge invariance implies that only certain combinations of quarks and/or gluons can lead to observables particles. If one applies blindly such a simple rule, the observed hadrons are necessarily composite states involving multi-quarks and multi-gluon configurations. There is a priori no reason why states with other valence composition than mesons or baryons, called in general exotic states, should not be observed. Exotic states can be pure glue states (glueballs), multi-quark states (tetraquark, pentaquarks, etc) or hybrid states (mesons with a non-vanishing valence gluon content). Besides the hadron states compatible with the quark model, the particle data book  Olive:2016xmw also reports candidates for the different types of exotic states, see e.g. the reviews on pentaquarks and non-$q\overline{q}$ mesons. The masses of the experimental states listed as candidates to multi-quark/gluon hadrons cover the full range of energies of the particle spectrum. In particular the exotics with most observations are the tetraquarks.

In what concerns the experimental observation of exotic tetraquarks, the quarkonium sector of double-heavy tetraquarks including a $Q\bar{Q}$ pair is the most explored experimentally, see e.g. the recent reviews Briceno:2015rlt ; Lebed:2016hpi ; Esposito20171 . In particular, the charged $Z_{c}^{\pm}$ and $Z_{b}^{\pm}$ are crypto-exotic, but technically they can be regarded as essentially exotic tetraquarks if we neglect $c\bar{c}$ or $b\bar{b}$ annihilation. There are two $Z_{b}^{\pm}$ observed only by the collaboration BELLE at KEK Belle:2011aa , slightly above $B\ \bar{B}^{*}$ and $B^{*}\ \bar{B}^{*}$ thresholds, the $Z_{b}(10610)^{+}$ and $Z_{b}(10650)^{+}$. Their nature is possibly different from the two $Z_{c}(3940)^{\pm}$ and $Z_{c}(4430)^{\pm}$, whose mass is well above $DD$ threshold Choi:2007wga . The $Z_{c}^{\pm}$ has been observed with very high statistical significance and has received a series of experimental observations by the BELLE collaboration Liu:2013dau ; Chilikin:2014bkk , the Cleo-C collaboration Xiao:2013iha , the BESIII collaboration Ablikim:2013mio ; Ablikim:2013emm ; Ablikim:2013wzq ; Ablikim:2013xfr ; Ablikim:2014dxl and the LHCb collaboration Aaij:2014jqa . This family is possibly related to the closed-charm pentaquark recently observed at LHCb Aaij:2015tga . Notice that, using naïve Resonant Group Method calculations, in 2008, some of us predicted Cardoso:2008dd a partial decay width to $\pi\ J/\psi$ of the $Z_{c}(4430)^{-}$ consistent with the recently observed experimental valueAaij:2014jqa .

On the other hand, in what concerns lattice QCD simulations, the most promising exotic tetraquark sector is also double-heavy, but it has a pair of heavy quarks $QQ$ or antiquarks $\bar{Q}\bar{Q}$, and thus it differs from the quarkonium sector. Note that in lattice QCD, the study of exotics is presently even harder than in the laboratory, since the techniques and computer facilities necessary to study of resonances with many decay channels remain to be developed. Lattice QCD searched for evidence of a large tetraquark component in the closed-charm $Z_{c}(3940)^{-}$ candidate but this resonance is well above threshold, and Ref. Prelovsek:2014swa ; Leskovec:2014gxa concluded there is no robust lattice QCD evidence of a $Z_{c}^{\pm}$ tetraquark resonance. Lattice QCD also searched for the expected boundstate in light-light-antiheavy-antiheavy channels Ikeda:2013vwa ; Guerrieri:2014nxa . Using dynamical quarks, the only heavy quark presently accessible to Lattice QCD simulations is the charm quark. No evidence for boundstates in this possible family of tetraquarks, say for a $ud\bar{c}\bar{c}$ was found. Moreover the potentials between two mesons, each composed of a light quark and a static (or infinitely heavy) antiquark , have been computed in lattice QCD Wagner:2010ad ; Wagner:2011ev . A static antiquark constitutes a good approximation to a spin-averaged $\bar{b}$ bottom antiquark. The potential between the two light-static mesons can then be used, with the Born-Oppenheimer approximation Born:1927 , as a $B-B$ potential, where the higher order $1/m_{b}$ terms including the spin-tensor terms are neglected. From the potential of the channel with larger attraction, which occurs in the Isospin=0 and Spin=0 quark-quark system, the possible boundstates of the heavy antiquarks have been investigated with quantum mechanics techniques. Recently, this approach indeed found evidence for a tetraquark $ud\bar{b}\bar{b}$ boundstate Bicudo:2012qt ; Brown:2012tm , while no boundstates have been found for states where the heavy quarks are $\bar{c}\bar{b}$ or $\bar{c}\bar{c}$ (consistent with full lattice QCD computations Ikeda:2013vwa ; Guerrieri:2014nxa ) or where the light quarks are $\bar{s}\bar{s}$ or $\bar{c}\bar{c}$ Bicudo:2012qt ; Bicudo:2015kna ; Peters:2015tra ; Bicudo:2015vta ; Bicudo:2016ooe ; Bicudo:2016jwl ; Peters:2016isf . The $\bar{b}\bar{b}$ probability density in the only binding channel has also been computed in Ref. Bicudo:2012qt ; Bicudo:2015kna ; Peters:2015tra ; Bicudo:2015vta ; Bicudo:2016ooe ; Bicudo:2016jwl ; Peters:2016isf .

The quark models for tetraquarks with the most sophisticated description of confinement are the string flip-flop models. Clearly, tetraquarks are always coupled to meson-meson systems, and we must be able to address correctly the meson-meson interactions. The first quark models had confining two-body potentials proportional to the SU(3) colour Casimir invariant $\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\,V(r_{ij})$ suggested by the One-Gluon-Exchange type of potential. However this would lead to an additional Van der Waals potential $V_{\text{Van der Waals}}={V^{\prime}(r)\over r}\times T,$ where $T$ is a polarization tensor. The resulting Van der Waals Fishbane:1977ay ; Appelquist:1978rt ; Willey:1978fm ; Matsuyama:1978hf ; Gavela:1979zu ; Feinberg:1983zz force between mesons, or baryons would be extremely large and this is clearly not compatible with observations. The string flip-flop potential for the meson-meson interaction was developed in Refs. Miyazawa:1979vx ; Miyazawa:1980ft ; Oka:1984yx ; Oka:1985vg ; Karliner:2003dt , to solve the problem of the Van der Waals forces produced by the two-body confining potentials. The first considered string flip-flop potential was the one minimizing the energy of the possible two different meson-meson configurations, say $M_{13}\,M_{24}$ or $M_{14}\,M_{23}$. This removes the inter-meson potential, and thus solves the problem of the Van der Waals force. An upgrade of the string flip-flop potential includes a third possible configuration Carlson:1991zt , in the tetraquark channel, say $T_{12\,,\,34}$, where the four constituents are linked by a connected string Vijande:2007ix ; Vijande:2009xx . The three confining string configurations differ in the strings linking the quarks and antiquarks, this is illustrated in Fig. 1. When the diquarks $qq$ and $\bar{q}\bar{q}$ distances are small, the tetraquark configuration minimizes the string energy. When the quark-antiquark pairs $q\bar{q}$ and $q\bar{q}$ are close, the meson-meson configuration minimizes the string energy. With a triple string flip-flop potential, boundstates below the threshold for hadronic coupled channels have been found Beinker:1995qe ; Zouzou:1986qh ; Vijande:2007ix ; Vijande:2009xx ; Bicudo:2010mv ; Bicudo:2015bra . On the other hand, the string flip-flop potentials allow fully unitarized studies of resonances Lenz:1985jk ; Oka:1984yx ; Oka:1985vg ; Bicudo:2010mv ; Bicudo:2015bra . Analytical calculations with a double flip-flop harmonic oscillator potential, Lenz:1985jk , using the resonating group method again with a double flip-flop confining harmonic oscillator potential, Oka:1984yx ; Oka:1985vg , anfd with the triple string flip-flop potential Bicudo:2010mv ; Bicudo:2015bra have already predicted resonances and boundstates.

So far. the theoretical and experimental interpretations of the observed states that can possibly be exotics is not clear crystal and, certainly, a better understanding of the colour force helps to elucidate our present view of the hadronic spectrum. For heavy quark systems its dynamics can be represented by a potential which, in general, is a function of the geometry of the hadrons, of the spin orientation of its components and of the quark flavours. In the limit of infinite quark mass one can compute the so-called static potential using first principle lattice QCD techniques via the evaluation of Wilson loops. The static potential provides an important input to the modelling of hadrons and it gives a simple realisation of the confinement mechanism. Moreover it can be applied to study tetraquarks $QQ\bar{Q}\bar{Q}$ with two heavy quarks and two heavy antiquarks, see for instance a Dyson-Schwinger study in Ref. Heupel:2012ua , at the intersection of the two sectors most studied experimentally and theoretically.

The static potential has been computed using lattice QCD for mesons, tetraquark, pentaquarks and hybrid systems see Bali:2000gf ; Alexandrou:2004ak ; Bornyakov:2005kn ; Okiharu:2004ve ; Okiharu:2004wy ; Bicudo:2007xp ; Cardoso:2008sb . For a quark and an anti-quark system, the static potential $V_{Q\overline{Q}}$ is a landmark calculation in lattice QCD and it is used to set the scale of the simulations. $V_{Q\overline{Q}}$ has been computed both in the quenched theory and in full QCD with the lattice data being well described by a one-gluon exchange potential (a Coulomb like potential) at short distances and a linear rising function of the quark distances at large separations. The behaviour at large interquark distances provides a nice explanation of the confinement mechanism. Moreover, for other hadronic systems and for large separations of its constituents a similar pattern of the corresponding static potentials has been observed in lattice simulations, i.e. a linear rising potential which, once more, is a simple realisation of quark confinement.

In the current work we revisit the static potential for tetraquarks using lattice simulations. The static potential for tetraquarks was computed for the gauge group SU(3) and in the quenched approximation in Alexandrou:2004ak ; Bornyakov:2005kn ; Okiharu:2004ve . The hybrid potential defined and measured in Bicudo:2007xp can also be viewed as a particular limit of the tetraquark potential. Herein, of all the possible geometries for the $QQ\bar{Q}\bar{Q}$ system we consider the case where quarks and anti-quarks are at the corners of a rectangle, see Fig. 2, and recompute the static potential of the system both in the quenched approximation and in full QCD. We focus our analysis in the comparison of the quenched and full QCD and also in the transition between a tetraquark system and a two meson system. Thus we go beyond the triple string flip-flop paradigm of Fig. 1 and analyse, in the transition region, the mixing between the meson-meson and tetraquark string configurations. Moreover we explore not only the groundstate but also the first excited state.

The current work is organised as follows. In Sec. II we discuss the possible colour structures for a $QQ\bar{Q}\bar{Q}$ system and introduce the $Q\bar{Q}$ potentials used to compare the results of the static potentials for the tetraquark. In Sec. III we revisit the geometries used to compute the static potentials and discuss the expected configurations at large separations. In Sec. IV the method used to evaluate the static potentials is described. In Sec. V we report on the parameters used in the lattice simulations and how we set the scale of the simulations. The results for the static tetraquark potential for the two geometries are described in Sec VI. In Sec. VII we resume and conclude. In the Appendix, the reader can find various tables with all our numerical results.

II The Color structure of a $QQ\bar{Q}\bar{Q}$ system

The colour-spin-spatial wave function of a $QQ\bar{Q}\bar{Q}$ system has multiple combinations, relevant for the computation of static potentials. In this section, we analyse the possible colour wave functions associated with a tetraquark system.

The quarks belong to the fundamental $3$ representation of SU(3), while anti-quarks are in a $\overline{3}$ representation of the group. The space built from the direct product $3\otimes 3\otimes\overline{3}\otimes\overline{3}$ includes two independent colour singlet states.

In a $QQ\bar{Q}\bar{Q}$ system, quarks and anti-quarks can combine into colour singlet meson-like states, leading naturally to the two meson states,

[TABLE]

where only the colour indices are written explicitly and $\mathbf{1}_{ij}$ refers to the meson-like colour singlet state built combining quark $i$ and anti-quark $j$. The two colour singlet states in Eq. (1) are not orthogonal to each other and a straightforward algebra gives,

[TABLE]

Moreover, a quark and anti-quark pair, besides a colour singlet state, can also form a colour octet state. With two colour octets it is again possible to build a colour singlet state. For the $QQ\bar{Q}\bar{Q}$ system the colour singlet states built from the octets read,

[TABLE]

where the factors comply with the normalization condition,

[TABLE]

The colour octet-octet states in Eq. (3) can be written in terms of the meson-meson states defined in Eq. (1),

[TABLE]

A simple calculation shows that the colour octet states (3) are not orthogonal (in colour space) to each other. However, each of the octet-octet states is orthogonal to the corresponding meson-meson state, i.e.

[TABLE]

The states in Eqs. (1) and (3) do not represent all the possible colour singlet states that can be associated to a $QQ\bar{Q}\bar{Q}$ system. We can also consider diquark-antidiquark configurations. For the group SU(3) it follows that $\mathbf{3}\otimes\mathbf{3}=\bar{\mathbf{3}}\oplus\mathbf{6}$, $\bar{\mathbf{3}}\otimes\bar{\mathbf{3}}=\mathbf{3}\oplus\bar{\mathbf{6}}$ and the two colour singlet states belong to the space spanned by $\mathbf{3}\otimes\bar{\mathbf{3}}$ and $\mathbf{6}\otimes\bar{\mathbf{6}}$,

[TABLE]

The states in Eqs. (7) and (8) are orthogonal to each other in colour space, i.e. $\langle\bar{\mathbf{3}}_{12}\mathbf{3}_{34}|\mathbf{6}_{12}\mathbf{\bar{6}}_{34}\rangle=0$. Furthermore, they are eigenstates of the exchange operators of quarks or anti-quarks, and verify the following relations,

[TABLE]

where $P_{ij}$ is the exchange operator of (anti)quark $i$ with (anti)quark $j$. Eqs. (7) and (8) can be inverted, giving,

[TABLE]

which shows that the meson-meson states of Eq. (1) are not eigenstates of the quark and of the anti-quark exchange operators $P_{12}$ and $P_{34}$.

The static potential $V$ for a $QQ\bar{Q}\bar{Q}$ system is a complicated object which may involve, two, three and four body interactions. In general, $V$ also depends on the allowed quantum numbers of the constituents of the multiquark state. The static potential should allow, when combined with quantum mechanics, for the groundstates to be the ones of Fig. 1. For example, the static potential should allow for the formation of two-meson states when the quark-anti-quark distances are small compared to the quark-quark and anti-quark-anti-quark distances, or possibly for the formation of a tetraquark at other particular distances.

As an approximate model to understand the results of the lattice simulations for the static potential in terms of overlaps with the various colour singlets, one can consider the two-body potential given by the Casimir scaling,

[TABLE]

where $V_{M}$ is the mesonic static $Q\bar{Q}$ potential with $C_{ij}=\frac{\lambda_{i}^{a}\cdot\lambda_{j}^{a}}{-16/3}$ and compare the results of the simulations with the one of any of the colour singlet states and the Casimir potential given by,

[TABLE]

Note, for a two body system, the one gluon exchange predicts a static potential proportional to $\lambda^{a}_{i}\cdot\lambda^{a}_{j}$.

The expectation values $\langle\Psi|C_{ij}|\Psi\rangle$ for the possible colour singlet states associated to the $QQ\bar{Q}\bar{Q}$ system are reported in Table 1. These numbers are important to obtain a qualitative insight into the result of the simulations. For instance, if for a given state $C_{ij}<0$, we don’t expect that the lattice result would give us a strong attraction between the particles $i$ and $j$ and, therefore, one can expected significant deviations of the static potential relative to the potential $V_{CS}$ associated to the corresponding colour singlet state.

Moreover we consider as well the first excitation of the $QQ\bar{Q}\bar{Q}$, which also depends in the particular distances of the system. Based in the orthogonality conditions and in a crude Casimir scaling where $V_{M}$ would be a spatial independent potential, we would expect the pairs of colour singlet states, ($|1_{13}1_{24}\rangle$, $|8_{13}8_{24}\rangle$) , ($|1_{14}1_{23}\rangle$, $|8_{14}8_{23}\rangle$) and ($|\overline{3}_{12}3_{34}\rangle$, $|6_{13}\overline{6}_{24}\rangle$) to form possible (groundstate, first excited state) pairs. This already goes beyond the simple paradigm of Fig. 1.

Nevertheless, Eq. (11) is clearly an approximation, and our aim is to compute more rigorous potentials. Previous lattice studies Alexandrou:2004ak ; Bornyakov:2005kn ; Okiharu:2004ve ; Okiharu:2004wy show that the static potential for a tetraquark system is not described entirely by a function proportional to this potential. An example of such a kind of potentials is the two-meson potential,

[TABLE]

which we expect to saturate the ground state when the quark-quark and anti-quark-anti-quark distances are large.

III Geometrical Setup

We aim to measure the static potential for the $QQ\bar{Q}\bar{Q}$ system but also to investigate the transition between the tetraquark and a two meson state, and the transition between the two two-meson states. This computation within lattice QCD simulations requires choosing a particular geometrical setup of the quark system under investigation. In principle, one could choose any of the available geometrical configurations allowed by the hypercubic lattice. In order to study in detail the transitions between the different states, in the current work we opt for restricting our study to the case where the four particles are at the corners of a rectangle and look at two particular alignments. In the so-called parallel alignment, see Fig. 2 (left), the two quarks (anti-quarks) are at adjacent corners of the rectangle. In the anti-parallel alignment, see Fig. 2 (right), the quarks (anti-quarks) are at the opposite corners of the rectangle.

III.1 Parallel Alignment of Quarks

For this geometry, where the two quarks are at neighbour corners of the rectangle, we can describe the system via the intra-diquark distances,

[TABLE]

and the inter-diquark distances,

[TABLE]

Note that for both cases the second equality holds only due to the particular geometrical configuration considered.

If one assumes that quarks are confined within colourless states, this geometrical setup has two limits which allow to study the transition between a tetraquark state and a two meson system. Indeed, when $r_{12}\ll r_{13}$ one expects the ground state of the $QQ\bar{Q}\bar{Q}$ system to be that of a tetraquark, while for the opposite case, i.e. for $r_{13}\ll r_{12}$, one expects the system, i.e. its potential, to behave as a two meson system.

For this geometrical setup, in the evaluation of the static potential we consider the basis of operators shown in Fig. 3. They are associated with a tetraquark operator (left in the figure) and a two-meson operator (right in the figure), the two ground state configurations expected for this particular geometry.

III.2 Anti-parallel Alignment of Quarks

For the anti-parallel alignment of quarks described in Fig. 2 (right), we take as distance variables,

[TABLE]

where, again, the second equalities are valid due to the particular characteristics of the geometrical distribution of quarks and anti-quarks.

For this geometrical setup, one expects the ground state of the system when $r_{13}\ll r_{14}$ and $r_{14}\ll r_{13}$ to be dominated by the two possible independent two-meson states. For the computation of the static potential we use the basis of operators shown in Fig. 4 that are associated with the two two-meson operators.

IV Computing the Static Potential

For the computation of the static potential, including the groundstate and the first excited state, we rely on a basis of two operators $\mathcal{O}_{i}$ for each of the geometrical setups discussed in Sec. III. Defining the correlation matrix,

[TABLE]

where $\langle\cdots\rangle$ stands for vacuum expectation value, $c_{in}=\langle n|\mathcal{O}_{i}|0\rangle$ and $|n\rangle$ are the eigenstates of the Hamiltonian of the system, the determination of the potential requires the knowledge of the solutions of the generalized eigenvalue problem

[TABLE]

In our calculation, we assume that the creation of an excited state out of the vacuum occurs at $t=0$. From the generalized eigenvalues $\lambda_{k}$, the energy levels of system $V_{k}$ can be estimated from the plateaux on the effective mass given by,

[TABLE]

In practice, the effective mass plateaux are identified fitting to a constant both generalized eigenvalues. In this way, one is able to compute both the static potential for the ground state and the first excited state of the system.

As described above, the basis of operators chosen to compute $V$ depends on the geometry of the system and on the expected ground states. For the anti-parallel alignment, we use two meson-meson operators, while for the parallel alignment a meson-meson operator and a diquark-antidiquark operator, i.e. a $\bar{\mathbf{3}}_{12}\mathbf{3}_{34}$ colour configuration, are used to compute the correlation matrix.

In the case where the quarks are in the anti-parallel alignment the operators used to compute the potential are,

[TABLE]

where $L$ are Wilson lines connecting the quark. Its representation in terms of closed Wilson loops is given in Fig. 6. The corresponding correlation matrix reads,

[TABLE]

where $W_{i}$ are normalized mesonic Wilson loops W=\mbox{\frac{1}{3}Tr}[U].

On the other hand, for the parallel alignment the two operators we consider are,

[TABLE]

The closed Wilson loops associated to $\mathcal{O}_{YY}$ and $\mathcal{O}_{13,24}$ are represented in Fig. 5 and the corresponding correlation matrix is given by,

[TABLE]

V Lattice Setup

From the static potential we aim to understand the transition between possible configurations of a $QQ\bar{Q}\bar{Q}$ system. Furthermore, we also want to glimpse any possible differences due to the quark dynamics. Therefore, for the computation of $V_{k}$ we consider two different simulations.

Our quenched simulation uses an ensemble of 1199 configurations provided by the PtQCD collaboration Cardoso:2011xu ; Cardoso:2010di ; PtQCD , generated using the Wilson action in a $24^{3}\times 48$ lattice for a value of $\beta=6.2$. The quenched configurations were generated using GPU’s and a combination of Cabbibo-Marinari, pseudo-heatbath and over-relaxation algorithms, and computed in the GPU servers of the PtQCD collaboration.

Our full QCD simulation uses a Wilson fermion dynamical ensemble of 156 configurations generated in a $24^{3}\times 48$ lattice and a $\beta=5.6$. In the dynamical ensemble we take $\kappa=0.15825$ for hopping parameter, which corresponds to a pion mass of $m_{\pi}=383$ MeV. For Wilson fermions the deviations from continuum physics are of order $\mathcal{O}(a)$ in the lattice spacing and, therefore, one can expect relative large systematic errors. However, we expect that the static potential as measured from the full QCD simulation away from the physical point to be more realistic when compared to the quenched simulation. The full QCD configuration generation has been performed in the Centaurus cluster LCA using the Chroma library Edwards:2004sx . The Hybrid Monte Carlo integrator scheme has been tuned using the methods described in Clark:2011ir ; Kennedy:2012gk .

Then, with both the quenched and full QCD ensembles of configurations, we perform our correlation matrix computations at the PC cluster ANIMAL of the PtQCD collaboration.

The Wilson loops at large Euclidean time are decaying exponential functions of the static potential times the Euclidean time and, therefore, for large Euclidean times the Wilson loops are dominated by the statistical noise of the Monte Carlo. A reliable measurement of the static potential requires techniques which reduce the contribution of the noise to the correlation functions used in the evaluation of $V$.

The quality of the measurement of the effective masses depends strongly on the overlap with the ground state of the system. In order to improve the ground state overlap we applied 50 iterations of APE smearing Albanese:1987ds with $w=0.2$ to the spatial links in both configuration ensembles. Furthermore, for the quenched ensemble, to further improve the signal to noise ratio, we used the extended multihit technique Cardoso:2013lla . This procedure generalizes the multihit as described in Parisi:1983hm by fixing the $n^{th}$ neighbouring links instead of the first ones when performing the averages of the links. However, this technique has the inconvenient of changing the short distance behaviour of the correlators and, therefore, one should not consider the points with $r<r_{\text{min}}$. In previous studies with the multihit, $r_{\text{min}}=2$ was sufficient, but in our study we consider $r_{\text{min}}=4$. For the dynamical configurations the multihit technique can not be applied and, therefore, we resorted on hypercubic blocking Hasenfratz:2001hp with the parameters $\alpha_{1}=0.75$, $\alpha_{2}=0.60$ and $\alpha_{3}=0.30$ to improve the signal to noise ratio.

For the conversion into physical units we first evaluate Wilson loops to access the ground state meson static potential on a single axis. In this calculation, we use a variational basis built using four different smearing levels to access the ground state meson static potential. The lattice data for the static meson potential is then fitted to the Cornell potential functional form,

[TABLE]

The fits for different fitting ranges are reported in Tables 2 and 3 for the quenched and the dynamical ensembles, respectively. The fits allows for the evaluation of the physical scale associated to the two ensembles through the Sommer method Sommer:1993ce . Indeed, by demanding that,

[TABLE]

where $r_{0}=0.5$ fm, the lattice spacing $a$ is measured and we present it in Tables 2 and 3 for various fitting ranges. The results show that $a$ is fairly independent of the fitting intervals and, in the following, we take $a\simeq 0.0681$ fm for the quenched data ensemble and $a\simeq 0.0775$ fm for the dynamical data set. Our QCD lattice spacing is essentially similar to the one obtained with different techniques. It follows that the lattice volumes used in the simulation are $(1.63\,$fm$)^{3}\times 3.27\,$fm for the quenched case and $(1.86\,$fm$)^{3}\times 3.72\,$fm for the dynamical simulation. For completeness, in Fig. 7 we show the ground state meson potentials for the two ensembles in physical units.

VI Results for the static $QQ\bar{Q}\bar{Q}$ potential

In this section, we report on the results for the static potential with the two different geometries mentioned in Sec. III, and we apply fits with ansatze bases in the string flip-flop potential and in the Casimir scaling.

In Fig. 8, as an example, we show effective mass plots for the pure gauge simulation (left), full QCD simulation (right) and for the ground state (top) and first excited state (bottom) for a $QQ\bar{Q}\bar{Q}$ system in the antiparallel geometry. The red curves are the results of fitting the lattice data to measure the static potential. See the appendix for further details on the numerics. We consider the maximum number of points aligned in a horizontal line with acceptable $\chi^{2}/\text{d. o. f.}$. Because the noise reduction technique in the quenched simulation rejects the cases with source distances smaller than $4a$, we end up by accepting a few more results in the full QCD case than in the quenched case.

VI.1 The anti-parallel alignment

We start by analyzing the simpler case of the anti-parallel geometry, where the meson-meson systems are expected to have lower energies than the tetraquark system. Our results are plotted in Figs. 9 and 10. Clearly there are two different trends for $r_{13}<r_{14}$ and for $r_{13}>r_{14}$ and a transition, with mixing, at the point $r_{13}=r_{24}$. Moreover we compare in detail our results with different ansatze.

From the string flip-flop paradigm of Fig. 1 we expect the ground state of the system to be that of a two meson system when the distance between a quark and an anti-quark, i. e. $r_{13}$ or $r_{14}$, is much smaller than the quark-quark distance, i.e. $r_{12}$. Then, for sufficiently small $r_{13}$ and/or $r_{14}$ the potential of the ground state of the $QQ\bar{Q}\bar{Q}$ should reproduce the string flip-flop potential,

[TABLE]

where the two different meson -meson potentials are

[TABLE]

and $V_{M}$ is the ground state potential of a meson in Eq. (24). Previous lattice simulations Alexandrou:2004ak ; Bornyakov:2005kn ; Okiharu:2004ve confirm that $V_{0}$ is compatible with such a result. Deviations from Eq. (26) are expected at intermediate distances together with a smooth transition from one picture to the other, i.e. from the two meson state with valence content $Q_{1}\bar{Q}_{3}$ and $Q_{2}\bar{Q}_{4}$ to the two meson with valence content $Q_{1}\bar{Q}_{4}$ and $Q_{2}\bar{Q}_{3}$.

On the other hand, for the excited state, we have two possible scenarios. From the string-flip-flop, we would again expect, when the distance between quark and anti-quark a quark and an anti-quark, i. e. $r_{13}$ or $r_{14}$, is much smaller than the quark-quark distance, i.e. $r_{12}$, the system to be that of the next two meson system,

[TABLE]

However, given that the colour wavefunctions of the two mesonic states are not orthogonal, see Eq. (2), and Section II, possibly the excited state is not another mesonic state and, but instead is an octet state,

[TABLE]

where we estimate the colour octet potential assuming Casimir scaling, i.e. using the decomposition in Eq. (11) and the values reported on Tab. 1,

[TABLE]

Thus we have two different simple anzatse to interpret our results. The ground state potential $V_{0}$ and the first excited state potential $V_{1}$ for the quenched and dynamical ensembles are reported in Figs. 9 and 10, respectively, together with $V_{MM}$, $V_{MM^{\prime}}$ and the octet potentials $V_{88}$, $V_{88^{\prime}}$.

As the figures show, the ground state static potential $V_{0}$ as a function of $r_{14}$ is compatible with two two meson potentials for small and large values of $r_{14}$. Indeed, for all $r_{13}$, at small values of $r_{14}$ the static potential is compatible with $V_{MM}$, while for large $r_{14}$ $V_{0}$ becomes compatible with $V_{MM^{\prime}}$. We show in Table 4 good fits with the meson-meson potentials. This type of behaviour is well described by the string flip-flop potential $V_{ff}$.

In the transition region $r_{13}\sim r_{14}$ where also $V_{MM}\sim V_{MM^{\prime}}$, deviations of $V_{0}$ from $V_{MM}$ or $V_{MM^{\prime}}$ can be seen. The difference between the ground state potential and the sum of the two meson potentials in physical units is detailed in Fig. 11, and in particular the transition point $r_{12}=r_{13}$ is analysed in Fig. 12. The results for the quenched simulation are well described assuming an off diagonal term $\Delta$ in the correlation matrix, leading to the functional form,

[TABLE]

where we may have either,

[TABLE]

or,

[TABLE]

Eq. (VI.1) interpolates between the two potentials in flip-flop picture of a meson-meson.

The fits for the functional forms in Eqs. (31) and (32) are reported in Tables 5 and 6. In order to quantify the deviation from the two limits where the system behaves as a two meson system, we refer that the fits give a $\Delta(0.5\,\mbox{fm},0.5\,\mbox{fm})\simeq 60\,\mbox{MeV}$, a number to be compared with typical values for the meson potential which are of the order of GeV (see Fig. 7). This results shows that the corrections due to $\Delta$ to the flip-flop picture are small when the quarks and anti-quarks are in an anti-parallel geometry.

The full QCD simulation shows similar results to the quenched QCD simulation. However, the results for $V_{0}$ for the full QCD configurations are not described by the same type of functional form given in Eq. (VI.1) which reproduces the flip-flop potential at large distances. We found no window where the fits are stable and, therefore, conclude that the dynamical $V_{0}$ is not reproduced by Eq. (VI.1) with the deviations parametrised by either Eq. (31) or Eq. (32).

In what concerns the excited state potential $V_{1}$ there are clearly two different regimes for $r_{13}$ very different from $r_{14}$, but we are not able to find an analytic form compatible with the lattice data, neither for the quenched simulations nor for the full QCD simulations. In both Figs. 9 and 10, it is clear the static potential $V_{1}$ lies between the functional forms of Eq. (27) and Eq. (28). There are subtle differences between Fig. 9 and 10. In general, the full QCD case is closer to the octet expression of Eq. (28) than the quenched QCD case.

A fortiori, we are not able as well to find a good ansatze to fit $V_{1}$ in the transition region. For a detailed view of the differences for the quenched simulation in this region, see Fig. 13.

This observed behaviour for $V_{1}$ can be understood in terms of adjoint strings. When the quark-anti-quark inside the octets are close to each other, they can be seen externally as a gluon. Therefore, we have a single adjoint string with a tension of $\sigma_{A}=\frac{9}{4}\sigma$. On the other hand, when the quark and the anti-quark are pulled apart, the adjoint string tends to split into two fundamental strings, with a total string tension of $2\sigma$. The splitting of the adjoint string, gives a repulsive interaction between the quark-anti-quark pairs that form octets in the excited state. This is qualitatively consistent with the behaviour predicted by Casimir scaling, where the potential for a quark and an anti-quark in an octet corresponds to a repulsive interaction.

VI.1.1 Mixing angle

For the anti-parallel geometry and for the ground state potential the lattice results show that the tetraquark is essentially a two meson state. Therefore, one can write the most general ket describing the ground state $|u_{0}\rangle$ of a $QQ\bar{Q}\bar{Q}$ system as a linear combination of the available colourless states

[TABLE]

For a pure two-meson state, the mixing angle is either $\theta=\theta_{0}$, for $|\mathbf{1}_{13}\mathbf{1}_{24}\rangle$, or $\theta=-\theta_{0}$, for $|\mathbf{1}_{14}\mathbf{1}_{23}\rangle$, with $\theta_{0}=\tan^{-1}(1/\sqrt{2})$. For the general case, the angle $\theta$ can be estimated using the generalized eigenvectors obtained solving Eq. (18) with the following operators,

[TABLE]

The results for $\theta$ for the quenched simulation can be seen in Fig. 14. From the lattice data one can estimate a typical length, or broadness, associated to the transition between the two two-meson states. In the region when $|r_{13}-r_{14}|\lesssim d_{\text{trans}}$, the transition occurs and the groundstate is a mixing of the $MM$ and $MM^{\prime}$ states. We estimate the typical transition length from,

[TABLE]

For the quenched data, see Fig. 14, the derivative stays within $0.36/a$ and $0.42/a$ and, therefore, $d_{trans}\sim 0.16-0.19$ fm. For the dynamical simulation, see Fig. 15, the typical transition length is essentially the same and we find $d_{trans}\sim 0.16-0.20$ fm.

The lattice data for the mixing angle gives a vanishing angle for $r_{13}=r_{14}$. This means that the ground state for the anti-parallel alignment is given only by $|\mathbf{6}_{12}\bar{\mathbf{6}}_{34}\rangle$ and has no $|\bar{\mathbf{3}}_{12}\mathbf{3}_{34}\rangle$ component.

The results reported in Figs. 14 and 15 show that, in general, a $QQ\bar{Q}\bar{Q}$ system is in a mixture of two possible colour meson states and it approaches meson states as the distance between the pairs of quark-anti-quark is much smaller than the distance between quarks or anti-quarks.

VI.2 The parallel alignment

For this particular geometry, the static potential was investigated with lattice methods in Alexandrou:2004ak ; Okiharu:2004ve . For the ground state and in the limit where $r_{12}\ll r_{13}$, the authors found that the lattice data is compatible with the double-Y (or butterfly) potential,

[TABLE]

where $\gamma$ and $K$ are the estimates of the static meson potential and $\sigma$ is the fundamental string tension. For the geometry described on the right hand side of Fig. 2 and for $r_{13}>r_{12}/\sqrt{3}$ the butterfly potential simplifies into,

[TABLE]

Moreover, from the expression for the Casimir scaling potential given in (11) and using the results reported on Tab. 1 it is possible to define various types of potentials to be compared with the static potential computed from the lattice simulations.

The potential associated to the state where the quarks and anti-quarks are in triplet states leads to the so-called triplet-antitriplet or diquark-antidiquark potential,

[TABLE]

or in a form similar to (37),

[TABLE]

Similarly, the anti-sextet-sextet potential is given by

[TABLE]

and the octet-octet potential reads

[TABLE]

The lattice estimates for the ground state and first excited (whenever possible) potentials can been in Figs. 16 and 17 for the quenched and for the dynamical simulation, respectively. The data shows that for large quark-anti-quark distances, i.e. for large $r_{13}$, the static potentials are compatible with a linearly rising function of $r_{13}$. This result can be viewed has an indication that the fermions on a tetraquark system are confined particles.

For both the pure gauge and dynamical simulations and for small quark-anti-quark distances, i.e. for small $r_{13}$, and up to $r_{13}\leq r_{12}$ the ground state potential reproduces that of a two meson state $V_{MM}$. In this sense, one can claim that for sufficiently small quark-anti-quark distances the ground state of a $QQ\bar{Q}\bar{Q}$ system is a two meson state. For the excited potential, the pure gauge results are among the double-Y potential (36) and the octet-octet potential (41). However, for the dynamical results, the static potential seems to be closer to $V_{88}$ at smaller and large $r_{13}$ and closer to $V_{YY}$ as $r_{13}$ approaches $r_{12}$.

On the other hand for sufficiently large $r_{13}$, the ground state potential is essentially that of a diquark-antidiquark system $V_{33}$ and the system enters its tetraquark phase. Indeed, the ground potential is given by $2V_{M}$ for quark-anti-quarks distances up to $r_{13}=r_{12}$ and is just above $V_{YY}$ for distances $r_{13}\geq r_{12}+1$ in lattice units. These results suggests that, for this geometrical setup, the transition of a two meson state towards a tetraquark state occurs at $r_{13}\sim r_{12}+1$ (in lattice units).

In what concerns the dependence of $V_{0}$ on $r_{12}$, the lattice data suggests that the potential increases with the quark-quark distance and favours a $V_{0}\sim V_{YY}$ for sufficiently large $r_{12}$ as was also observed in Okiharu:2004ve ; Alexandrou:2004ak .

For the quark models with four-body tetraquark potentials, in particular the string flip-flop potential illustrated in Fig. 1 it is very important to quantify the deviation of $V_{0}$ from the $V_{YY}$ ansatz; and we have studied several ansatze for this difference. Clearly $V_{0}$ is more attractive than the tetraquark potential $V_{YY}$ of Eq. (36) reported by previous authors, and this favours the existence of tetraquarks.

Adding a negative constant (attractive) to the double-Y potential is not sufficient for a good fit of the lattice data for any of the sets of configurations. Adding a correction to the double-Y potential which is linear in the quark-quark distance,

[TABLE]

describes quite well the dynamical simulation data and a fit gives $\delta K=-0.12(3)\,\sqrt{\sigma}$, $\delta\sigma_{12}=-0.34(5)\,\sigma$, where $\sigma$ is the fundamental string tension, for a $\chi^{2}/d.o.f=0.46$ (see the tables on the appendix for details on the fits). The dynamical data for the deviations from $V_{YY}$ are also compatible with a Coulomb like correction

[TABLE]

for a $\delta K=-0.67(4)\,\sqrt{\sigma}$, $\delta\gamma_{12}=0.22(3)$ with a $\chi^{2}/d.o.f.=0.62$ (see appendix for details). Such a functional form is not compatible with the lattice data for the pure gauge case. A possible explanation could come from the difference in the statistics of both ensembles. Recall that the number of configurations for the pure gauge ensemble is about ten times larger than for the dynamical simulation and, therefore, the associated statistical errors are much smaller.

In what concerns the first excited potential $V_{1}$, the data for the pure gauge and for the dynamical fermion simulation follows slightly different patterns. In the quenched simulation and for $r_{13}<r_{12}$, the potential is close to $V_{T}$ and the behaviour for larger values of $r_{13}$ does not reproduces any of the potentials considered here. On the other hand, in the dynamical simulation $V_{1}$ for small and large values of $r_{13}$ is just below the data for anti-sextet-sextet potential

[TABLE]

which, for this geometry, is given by

[TABLE]

and, at intermediate distances where $r_{13}\sim r_{12}$, is compatible with the octet-octet potential,

[TABLE]

Further, at very small distances the potential seems to flatten for full QCD and the data also suggests a flattening or a small repulsive core. Note, for the quenched simulation smaller distances than 4 are not accessible, this short distance effect is not visible.

VII Summary and Discussion

In this work the static potential for a $QQ\bar{Q}\bar{Q}$ system was investigated using both quenched and Wilson Fermion full QCD simulations for two different geometric setups. The two geometries are designed to investigate sectors where dominantly meson-meson or tetraquark static potentials are expected.

The simulations show that whenever one distance is much larger than the other, the ground state potential and the first excited state potential are compatible with a linearly rising function of the distance between constituents, suggesting that quarks and anti-quarks are confined particles. For the distances studied, the quenched and full QCD results are qualitatively similar, and their subtle differences only become clearer when we compare the lattice data with anzatse inspired in the string flip-flop potential and in Casimir scaling.

For the anti-parallel geometry setup, the groundstate potential $V_{0}$ is approximately described by a sum of two two meson potentials, i.e. it is compatible with the string flip-flop type of potential. We take this result as an indication that the $QQ\bar{Q}\bar{Q}$ wave function is given by a superposition of two meson states and we compute the mixing angle, as a function of the quark-anti-quark distances, which caracterize such a quantum state. The mixing angle shows that the tetraquark system undergoes a transition from one of the meson states to the other configuration as the quark-antiquark distance increases, and the broadness of this transition has a typical length scale of $0.16-0.20$ fm. Moreover, for the quenched simulation, we found an analytical expression which describes well the lattice groundstate. The analytical expression is essentially a flip-flop type of potential with corrections, parametrized by $\Delta(r_{1},r_{2})$, which are typically $\lesssim 10$% than the sum of two two mesons potentials.

In what concerns the first excited potential $V_{1}$ in the anti-parallel geometry, the results show that for small enough quark-anti-quark distances the potential is just below one of the possible octet-octet potentials and approaches a two meson potential from above from large quark-anti-quark distances. This results for the excited potential can be interpreted in terms of and excited state including a combination of meson-meson and octet-octet states.

For the parallel geometry setup, the groundstate potential $V_{0}$ is compatible with a diquark-antidiquark potential for large quark-antiquark distances and a sum of two meson potentials for small separations. Moreover, the lattice data for the full QCD simulation is compatible with a butterfly type of potential with corrections that we are able to parametrize. For the quenched simulation we found no analytical expressions that are able to describe the data, but the trend is the same.

The interpretation of the first excited potential $V_{1}$ for the parallet geometry, in terms of possible colour configurations is not as compliant with models as in the anti-parallel geometry. It seems that $V_{1}$ for the full QCD simulation is just below the octet-octet from small quark-anti-quark distances and approaches again the octet-octet potential for at large distances. For the quenched simulation, the interpretation of $V_{1}$ in terms of colour components is not so clear, as the lattice data seems to point for a combination of different colour potentials.

Importantly for quark models with four-body tetraquark potentials, in particular for the string flip-flop potential illustrated in Fig. 1, we obtain a groundstate potential $V_{0}$ more attractive, by a difference of $-300$ to $-500$ MeV, than the butterfly potential reported by previous authors Alexandrou:2004ak ; Bornyakov:2005kn ; Okiharu:2004ve ; Okiharu:2004wy , and this favours the existence of tetraquarks.

As an outlook, it would be interesting to measure the static $QQ\bar{Q}\bar{Q}$ potentials for larger distances and for different geometries. We leave this for future studies.

Acknowledgements.

The authors are extremely grateful to Nuno Cardoso Cardoso:2011xu ; Cardoso:2010di for generating the ensemble of quenched configurations utilized in this work. The authors also acknowledge both the use of CPU and GPU servers of the collaboration PtQCD PtQCD , supported by NVIDIA, CFTP and FCT grant UID/FIS/00777/2013, and the Laboratory for Advanced Computing at University of Coimbra LCA for providing HPC resources that have contributed to the research results reported within this paper. M. C. is supported by FCT under the contract SFRH/BPD/73140/2010. P. J. S. acknowledges support by FCT under contracts SFRH/BPD/40998/2007 and SFRH/BPD/109971/2015.

Appendix A Tables of Results