Strings of diquark-quark (QQ)Q baryonbefore phase transition

A. S. Bakry,1 X. Chen,1 M. A. Deliyergiyev,2, 1, ∗ A. A. Galal,3

A. M. Khalaf,3 M. KhalilA William,4, 5, 6, 3 and P. Ming Zhang7

1Institute of Modern Physics, Chinese Academy of Sciences, Gansu 730000, China2Institute of Physics, The Jan Kochanowski University in Kielce, 25-406, Poland

3Department of Physics, Al Azhar University, Cairo 11651, Egypt4Department of Mathematics, Bergische Universitat Wuppertal, 42097 Germany

5Department of Physics, University of Ferrara, Ferrara 44121, Italy6Research and computing center, The Cyprus Institute, Nicosia 2121, Cyprus

7San Yat-sen University, Guangzhou 510275, China(Dated: January 31, 2020)

We explore the limit at which effective baryonic Y-string model of the junction approaches themesonic stringlike behavior. We calculate and compare the numerical values of the static potentialand energy-density correlators of diquark-quark (QQ)Q and quark-antiquark (QQ) configurations.The gauge model is pure Yang-Mills SU(3) lattice gauge theory at coupling β = 6.0 and finitetemperature. The diquark set up is approximated as two-quarks confined within a sphere of radius0.1 fm. The lattice data of the potential and energy show that the string binding the diquark-quark(QQ)Q configuration exhibits an identical behavior to the quark-antiquark (QQ) confining string.However, with the temperature increase to a small enough neighborhood of the critical point Tc,the gluonic symmetries between the two systems do not manifest neither at short nor intermediatedistance scales. The comparison between the Polyakov loop correlators and the second momentof the action-density correlators for both system shows significant splitting. This suggests thatsubsisted baryonic decouplet states overlap with the mesonic spectrum. The baryonic junction’smodel for the potential and profile returns good fit to the numerical lattice data of the diquark-quark (QQ)Q arrangement. However, near the critical point the mesonic string displays fits withlarge deviations compared to fits of the corresponding (QQ) data.

PACS numbers: 12.38.Gc, 12.38.Aw, 14.70.Dj

I. INTRODUCTION

In the ordinary quark model, baryons are regarded asthe bound states of the three quarks with two relativecoordinates as the expressions of the internal space-timedegrees of freedom. On the other hand, in the quark-diquark model two quarks out of the three, are gluedtogether to form a boson (diquark) system, and the re-maining third quark is assumed to revolve around thisboson.

The notion of diquark is as old as the quark model it-self. Gell-Mann introduced the notion of the diquark in[1], and soon afterward constituent quark-diquark modelsfor baryons were developed by Ida and Kobayashi [2] andLichtenberg et al. [3–5] also studying, within the model,the electromagnetic properties of the baryons. Later thediquark model has been invoked to explain many phe-nomena of strong interactions [6–13].Interest in the properties of diquark in hadronic systemswas recently renewed, since they may play an importantrole in the existence of exotic states [14–19].

Relationship between strings and gauge field the-ory [20] of hadrons is a surviving speculation [21, 22].

∗maksym.deliyergiyev@ujk.edu.pl

The interaction of two oppositely (magnetically) chargedmonopoles by means of a massive gauge field (mass µ)leads to the Nambu-Nielsen-Susskind-Goto string [23]with point masses at its ends in the limit µ→∞ [24]. Ithas also been observed that a suitable limit of a non-abelian gauge theory resembles the dual string model[25].

Thus, it can be assumed that in a certain approxima-tion a meson can be regarded as two point-quarks con-fined to each other by a string, and a baryon as threepoint-quarks confined to each other by three strings. Ap-propriate internal quantum numbers can be assigned bytreating the point-quarks as Dirac fields confined to worldlines [26, 27]. The remarkable result obtained in thisway is that this 1 + 1 dimensional string model theory ofhadron dynamics is equivalent to QCD [28].

Now, we have theoretical developments related tohadronic string models on both the mesonic and baryonicconfigurations [29–33]. These studies have investigatedthe mass of diquarks [34–38], and, more recently, the na-ture of diquark correlations[39]. However, the investiga-tion of the properties of the quark-antiquark (QQ) versusthe properties of the quark-diquark (Q−QQ) configura-tions effective bosonic strings has not fully addressed onthe lattice and in extreme conditions on particular.

Recently we have reported the formation of thehadronic Y-strings systems in static baryonic configura-

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tions on the lattice at finite T [40–42]. In that workisosceles triangle three-quark configurations with baselengths greater than A = 0.6 fm were investigated. Fortheses configurations a formation of three Gaussian-likeflux tubes has been observed even by adopting Polyakovloops operators which are string-less operators in the spa-tial directions.

In this work we extend our study of three quark sys-tems to the case where two quarks form a tightly boundstate or a diquark, which in turn interacts with a thirdquark to form a baryon [40–42].

In SU(3) color group, a diquark, two-quarks in closevicinity to each other, transforms according to the con-jugate representation [3]. The scalar-diquark channel isattractive in the spin singlet rather than the spin tripletof the axial vector diquark. Hence low-lying diquarkslie within the conjugate representation [3] and acquires(+) parity and belong to the color [3], and thereof, sharecommon properties to the antiquark.

Owing to the formation of flux tubes of the same en-ergy density and transverse size, the long range confiningforce put forth to the linearly rising slop of the quark-antiquark potential [43, 44]. The target of this investi-gation is to search this interesting limit In lattice QCDif the quark-diquark flux tubes [40–42, 45–53, 53–57] athigh temperature with similar physical characteristics.

The paper is organized as follows: We review the freebosonic string theory for the baryon and meson in Sec. II.The simulations setup and lattice measurements opera-tors will be described in Sec. III. In Sec. IV we mea-sure and compare the numerical data of the potentialof the QQ and the (QQ)Q systems. The string modelsimplications for each of these systems are discussed andcompared with the numerical data at several lattice pa-rameters. We present the action density analysis of boththe QQ and the (QQ)Q systems in Sec. V. The bary-onic profile of the junction and meson strings and thebroadening aspects the Monte-Carlo data is scrutinizedat several transverse planes. In the last section Sec. VIconcluding remarks are drawn.

II. EFFECTIVE BOSONIC STRING MODEL

It is an old conjecture that a pure Yang-Mills (YM)vacuum allowing for the formation of stable string-likeobjects which confine static color charges and gives riseto the linearly rising potential. In the dual supercon-ductor model of the QCD vacuum, the QCD vacuumsqueezes the color fields into a confining string dual tothe Abrikosov line by the dual Meissner effect [58, 59].

An idealized string-like system of flux tubes transmit-ting the strongly interacting forces between the colorsources was proposed [22, 60]. The formation of stringlikedefects is not a peculiar to QCD [20, 62–75], and arises inmany strongly interacting systems such as vortices in su-perfluids [76], flux tubes in superconductors [77], vorticesin Bose Einstein condensates [78], Nielsen-Olesen vortices

of field theory [79], and cosmic strings [80]. The physicalparameters of each of the relevant strongly interactingsystem fix the properties of this stringlike object. How-ever, measurable effects of quantum fluctuations presentin certain phases of the model.

In a mesonic string picture [81–86], the classical so-lution itself breaks the translational invariance of (YM)vacuum and leads to the generation of massless Gold-stone modes [87]. The quantum fluctuations give rise tothe Luscher subleading correction [60, 61] to the linearlyrising potential, on distance scales larger than the intrin-sic thickness of the flux tube 1/Tc [88] at zero tempera-ture in the so-called rough phase of lattice gauge theories(LGT).

The roughening transition proceeds with the decreaseof the coupling constants and in this phase, the flux-tube admits a collective co-ordinate description with alogarithmic increase [89] with the string length.

This has been found in consistency with very preciselattice measurements of the the QQ potential for colorsource separation commencing from distance R = 0.4fm [60]. These predictions have been verified as well inother confining gauge models [61, 88, 93, 94].

At sufficiently high temperature, the equations of mo-tion of a Nambu-Goto type bosonic strings are indicatinga linear growth in the tube’s width if solved [95] for thewidth of the action density at the middle plane betweentwo quarks. This prediction has been also verified inLGT by studying configurations with a static quark andanti-quark pair [55, 95–99] at large separations and nearthe deconfinement point.

It is widely accepted that the Y-shaped string is the rel-evant picture of the baryonic flux tubes to the IR regionof the non-Abelian gauge and amounts to three squeezedflux tubes that meet at a junction [31–33]. Static colorcharges corresponding to a multiquark system may in-duce an intricate stringy system in the QCD vacuumincluding the formation of multijunction systems [100–104]. In tetraquark and pentaquark systems, the analy-sis of color fields shows evidence on the formation of amulti-Y-type shaped flux tube [101, 102].

The flux tube junction points are compatible withFermat-Steiner points minimizing the total length of theflux-tube. The Y-string configuration can be derivedperturbatively [105–110] and from the strong couplingapproximation and is consistent with the dual supercon-ducting picture of QC. The Y-ansatz describes the lead-ing string effect and can be successful for parametrisationthe large distance lattice data of the confining poten-tial [31, 32] at zero temperature.

In a static baryon, the Y-string system is expected [31]to be a stable configuration in the IR region of the the-ory. This system accounts for three strings originatingfrom a node to the static three quarks. A theoreticaldevelopment discussing the effects of the quantum fluc-tuations of the Y-string on the 3Q potential has beenreported [31]. The calculations of the Casimir energy inthis model have indicated a geometrical Luscher-like sub-

3

leading term for the Y-(3Q) potential [32]. The Luscher-like term is a gauge-group-independent and depends onlyon the geometry of the three static charges.

The simulation of 3-Potts model showed a 3Q poten-tial [32] consistent with the Luscher-like correction γ/LYto the linearly rising Y-ansatz, and hence the applicabil-ity of the Y-baryonic string picture at large distances.

We summarize the motivation to discuss an effectiveY-string model versus the lattice data at high temper-ature in the following main points: The linear growthproperty of the confining flux-tube at high temperaturehas been verified on the lattice [55, 95–99], no substan-tial changes [111] in the nature of the confining thin-tubes between a quark-antiquark pair on large distancescales, the Y-model seems consistent with lattice datacorresponding to the confining potential at T = 0 [32].The expectations that the observed features of the glu-onic distribution may arise as a result of the vibration ofthis underlying Y-shaped string system.

The string model assumptions of the effective descrip-tion of the tube with a collective coordinate referringto the underlying thin-string are working at high tem-perature [57]. This is of particular relevance when dis-cussing a Y-shaped baryonic string model [31–33] toscrutinize the large distance ∆ baryonic flux arrange-ment [112, 113].

At high temperature, the thermal behavior of thefree-string manifests at source separation distance scaleslarger than what one would expect normally in the zerotemperature regime R ≈ 0.8 fm [55]. Even so, the inclu-sion of higher-order string’s self-interactions [98, 99, 114–128] improves the match between the lattice data and thestring model.

A. String phenomenology of quark potential

In the Y-string ansatz [129, 130, 133–135], it is as-sumed that the quarks are connected by three stringsthat meet at a junction (Fig. 2). The classical config-uration corresponds to the minimal area of the stringworld sheets. Each string’s world-sheet (blade) consistsof a static quark line and the world-line of the fluctuat-ing [31, 33, 136, 137] junction Fig. 1.

The parameter s and t (time) label the position onstring world-sheet (blade) i. The position of the junctionis given by s = Li + ηi.φ(t). The transverse fluctuationsξi(t, s) vanish at the location of the quarks (s = 0), andare periodic in the time t, with period 1/LT (see Fig. 1)and is the temporal extent governing the inverse temper-ature.

The simplest choice for the string action S is theNambu–Goto (NG) action which is proportional to thesurface area

S[X] = σ

∫dζ1

∫dζ2√g, (1)

FIG. 1: World sheet traced by one of the strings up to thejunction position.

where gαβ is the two dimensional induced metric on theblade world sheet embedded in the background R4,

gαβ =∂X

∂ζα· ∂X∂ζβ

, (α, β = 1, 2),

g = det(gαβ).

The vector Xµ(ζ1, ζ2) maps the region C ⊂ R2 into R4.Gauge fixing is required for the path integrals involvingthe string partition functions to be well defined with re-spect to Weyl and re-parametrization invariance. Thephysical gauge, X1 = ζ1, X

4 = ζ2 would restrict thestring fluctuations to transverse directions to C. In thequantum level, Weyl invariance is broken in 4 dimen-sions, however, the anomaly is known to vanish at largedistances [138]. The leading-order contribution

SNG`o [X] = σ0A+

σ0

2

∫dζ2

(∂X

∂ζα· ∂X∂ζα

), (2)

In mesonic string The Casimir energy is extracted fromthe string partition function as

V (R, T ) = − 1

LTlog(Z(R, T )). (3)

The partition function of the NG model in the physicalgauge is a functional integrals over all the world sheetconfigurations swept by the string

Z(R, T ) =

∫C[DX] exp(−S(X)). (4)

For a periodic boundary condition along the time direc-tion such that

X(ζ0 = 0, ζ1) = X(ζ0 = LT , ζ1), (5)

4

with an extent equals to the inverse of the temperatureLT = 1

T and Dirichlet boundary condition at the sourcesposition given by

X(ζ0, ζ1 = 0) = X(ζ0, ζ1 = R) = 0, (6)

the path integral Eq. (4) and Eq. (3) yields the staticpotential for the leading order contribution Eq. (2) ofthe NG action SNG`o The eigenfunctions are given by

φmn = e2πi(mR+ n

LT

), (7)

and eigenvalues of −4 are given by

Γnm =

(2πn

LT

)2

+

(2πm

R

)2

. (8)

The determinant of the Laplacian after ζ function regu-larization [114] reads

Det (−4) = (q124

∞∏n=1

(1− qn))2, (9)

where q = e2πτ and τ = LT2R is the modular parameter

of the cylinder. The partition function and the staticpotential are given by

Z(NG)`o = e−σRT−µ(T )[Det (−4)]−

(d−2)2 , (10)

V NG`o (R, T ) = σ0R+ (d− 2)T log η (τ) + µT, (11)

µ is a UV-cutoff and η is the Dedekind η function definedon the real axis as

η(τ) = q124

∞∏n=1

(1− qn); (12)

The second term on the right hand side encompasses theLuscher term of the interquark potential. This term sig-nifies a universal quantum effect which is a character-istic of the CFT in the infrared free-string limit and isindependent of the interaction terms of the correspond-ing effective theory. One can extract the string tensiondependency on temperature from the slop of the linearterms in R.

Considering the modular transform of the Eq. (11) τ →1/τ and taking the limit of long string, the renormalizedstring tension [139, 140] to leading order is given by

σ(T ) = σ0 −π(d− 2)

6T 2 +O(T 4). (13)

In a three blade baryonic system, the transverse fluctu-ations X⊥ = ξµ(t, s) vanish at the location of the quarks(s = 0), and are periodic in the time t, with period LT ,that is, Dirichlet boundary condition in addition to the

boundary condition from the continuity of the transversefluctuations ξi(t, s)

ξi(t, Li + ηi · φ(t)) = φ⊥i(t) , (14)

where ηi are spatial unit vectors in the direction ofthe strings such that Σiηi = 0. The NG action aftergauge-fixing and expanding around the equilibrium con-figuration yields

SFluct = σLY LT +σ

2

∑i,j

∫Θi

d2ζ∂ξi∂ζj· ∂ξi∂ζj

, (15)

where, LY =∑i Li above denotes the total string

length. In this model [31, 33], the junction is assumedto acquire a self-energy term m. This results in an addi-tional boundary term to NG action

S = SFluct + SBoundary,

with a static energy and a kinetic energy terms of junc-tion defined as

SBoundary =

(mLT +

m

2

∫ LT

0

dt |φ|2),

respectively.The system’s partition function then reads

Z = e−(σLY +m)LT

∫Dφ exp

(−m

2

∫dt |φ|2

) 3∏i=1

Zi(φ),

(16)where Zi(φ) denotes the partition function for the fluc-tuations of a given blade that is bounded by the junctionworldline φ(t):

Zi(φ) =

∫φ

Dξi exp

(−σ

2

∫|∂ξi|2

). (17)

The Y-string may be described by the Nambu-Gototype string action with Dirichlet boundary condition atthe static color sources. In this formalism, the action isproportional to the total area of the three blade worldsheet system swept by the fluctuating world lines of thestrings, in addition to a boundary term which accountsfor the junction fluctuations. This Y-string system [31]has a mixed Dirichlet-Neumann condition [141] and thestring’s partition function Zi(φ) in D dimensions [31] isgiven by

Zi(φ) = e−σ2

∫|∂ξmin,i|2 |det(−4Θi)|−(D−2)/2 , (18)

where ξmin,i is the minimal-area solution for a given junc-tion configuration φ(t), and 4Θi denotes the Laplacianacting on the domain (blade) Θi.

5

(a)The three and two bladeworldsheet systems swept by the

fluctuating world lines of thegluonic strings of the baryon and

meson.

(b)The domain Θi is conformally mapped into a rectangleL′i × LT , the first order fluctuations are convoluted with a

smoothing scalar ψ

FIG. 2: The action is proportional to the total area of thethree blade world sheet system swept by the fluctuating worldlines of the strings Y-string system [31] has a mixed Dirichlet-Neumann condition [141] .

Jahn and de Forcrand [31, 32] calculated the Casimirenergy for the baryonic potential V3Q. This was done byevaluating the determinant of the Laplacian in Eq. (18)by conformally mapping the resulting domains to rectan-gles. This results in an expression for the determinant interms of the Dedekind η function and a Gaussian func-tion of the junction fluctuations, ei · φwn , which in 4D isgiven by

det(−4Θi) =

η2

(iLT2Li

)exp

(− 1

12π

∑wn

w3n coth(wnLi)|ei · φwn |2

),

with wn = 2πn/LT , where LT = 1/T is the time extentof the blade, and Li are the lengths of the strings. Thesum over all eigen-energies would result in a Luscher-likecorrection to the V3Q potential at zero temperature [31].The baryonic potential V3Q then reads

V3Q(Li) = σLY + V‖ + 2V⊥ +O(L−2i ), (19)

with

V‖(Li) =−∑i

1

2LTη

(iLT2Li

)+

∑w=0

1

LTln

1

3

∑i<j

coth(wLi) coth(wLj)

,(20)

for the in-plane component and

V⊥(Li) =−∑i

1

2LTη

(iLT2Li

)

+∑w=0

1

LTln

[1

3

∑i

coth(wLi)

],

(21)

is the potential component due to the perpendicular fluc-tuations such that LY ≡ L1 + L2 + L3 is the minimal ofthe sum of the lengths of the three strings. The corre-sponding mesonic limit would read

V⊥ =− 1

2LTη

(iLT2L1

)− 1

2LTη

(iLT2L2

)+∑w=0

1

LTln

[1

2coth(wL1) + coth(wL2)

].

(22)

The quark anti-quark (QQ) potential is then

VQQ = σ(L1 + L2) +1

LTln

[η

(iLT

2(L1 + L2)

)], (23)

which is in agreement with the mesonic string poten-tial (26). Expressing the sum in Eq. (19) in terms ofDedekind η functions, the potential in the 3Q channelwould read

V3Q = σLY −γ

LTln

[η

(iLT2LY

)], (24)

where γ is a geometrical factor that can be evaluatednumerically by solving Eq. (19).

B. String’s Width profile

The mean-square width of the string is defined as,

w2(ξ; τ) = 〈X2(ξ; τ) 〉,

=

∫C [DX]X2 exp(−S[X])∫C [DX] exp(−S[X])

, (25)

6

(a)

3 4 5 6 7 8 9 10 11 12

-8.0

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

l3a-1

A=10a A=8a A=6a A=4a

(b)

FIG. 3: a) Schematic diagram showing the configuration ofthe Y string relative to the quark source positions. The junc-tions locus is fixed at Fermat point xf . The isosceles base isdenoted as A = 2d, the Fermat point is xF = A/2

√3. b) The

geometrical factor γ of the log term in Eq. (24). The solidlines correspond to γ as a function of the third string lengthL3.

ξ = (ξ1, iξ2) is a complex parametrisation of the worldsheet, such that ξ1 ∈ [−R/2, R/2], ξ2 ∈ [−L/2, L/2], withτ = L

R is the modular parameter of the cylinder, and L isthe temporal extent governing the inverse temperature.

The expectation value of the mean square width wouldthen read,

w2(ξ1, τ) =1

2πσlog(

R

R0) +

1

2πσlog∣∣ θ2(π ξ1/R; τ))

θ′1(0; τ)

∣∣.(26)

We calculate the string’s thickness at the junction po-sition. In the following, we follow the same proce-dures of the calculations presented in Refs. [31, 33],however, taking into account a convoluted fluctuationsφ→

∫∞−∞ φ(τ)ψ(t− τ)dτ to incorporate thermal effects.

The calculation of the corresponding partition functionEq.(16) requires evaluating the integral over the minimalarea swept due to perpendicular fluctuations φ, and thedeterminant of the Laplacian.

Conformally mapping the string’s blade a to a rectan-

gle [31] see Fig. 2

fi(z) = z +1√T

∑w 6=0

ηi · φwψ(wLi)

sinh(wLi)ewz. (27)

The minimal area solution for a fixed junction config-uration

ξmin,i =1√T

∑w

φw,ziψwsinh(ws)

sinh(wLi)eiwt, (28)

taking into account that the minimal-area solution fora fixed position of the junction, ξmin,i(t, s), is harmonicand satisfies the boundary conditions

4ξmin,i = 0 , ξmin,i(t, Li + ηi · φ(t)) = φzi(t) . (29)

The integral in Eq. (25) would then read

∫Θi

d2 ζ∑i

∂ξmin,i

∂ζi· ∂ξmin,i

∂ζi(30)

=∑w

w coth(wLi)|φw,zi|2ψ2w.

The determinant in Eq. (18) is obtained by mappingthe domain Θi conformally to a rectangle L′i×LT . Takinginto account the change in the Laplacian [33, 60]. Usingthe above conformal map Eq. (27), we obtain to leadingorder

lndet(−4Θ)

det(−4Θ)=

1

12π

∑w

w3|ηi · φw|2 coth(wLi)ψ(w),

(31)

Further conformally mapping the above into a circleand making use of (31), the determinant of the Laplacianwith respect to the blade a would then read

det(−4Θi) = η2

(iLT2L′i

)× exp

(− 1

12π

∑w

w3 coth(wLi)|ηi · φw|2ψ2w

).(32)

with η(τ) is the Dedekind function.The thickness of the string at the junction can be cal-

culated [33] taking the expectation value of φ2

〈φ2〉 =

∫Dφφ2 ψ2

we−S∫

Dφe−S. (33)

The above second moment of the junction can be de-composed into perpendicular z and parallel (in-plane) xyfluctuations

〈φ2〉 = 〈φz2〉+ 〈φxy2〉 =Iz,2Iz,0

+Ixy,2Ixy,0

, (34)

7

where

Iz,2 =

∫Dφzφ

2z exp

− 1

2

∑w

mw2 + σw∑a

coth(wLa)

×|φw,z|2, (35)

Ixy,2 =

∫Dφφ2 exp

∑w

−1

2

(mw2 + σw

∑i

coth(wLi))

×|φw|2 + |φw,x|2Qx + |φw,y|2Qy + |φw,y|2Qy

+2 ((φw,x.φw,y))Qxy

], (36)

with Qx, Qy and Qxy defined as in Eq. (43). Orthogonal-izing the fluctuations for parallel fluctuations the abovemoments would then read

Ix,2 =

∫Dφx φ

2x exp

∑w

−1

2

(F (w) +Gx(w)

)|φw,x|2

,

Iy,2 =

∫Dφyφ

2y exp

∑w

−1

2

(F (w) +Gy(w)

)|φw,y|2

,

Iz,2 =

∫Dφzφ

2z exp

− 1

2

∑w

R(w)|φw,z|2.

F (w), and G(w) are defined as

F (w) = Qx,w +Qy,w, (37)

G(w) = (Q2xy,w + (Qx,w −Qy,w)2)1/2,

R(w) = mw2 + σw∑i

coth(wLi)ψ(w,Li).

Solving for the above Gaussian integrals

〈φ2x〉 =

Ix,2Ix,0

=2

LT

∑w>0

1

F (w)−G(w),

〈φ2y〉 =

Iy,2Iy,0

=2

LT

∑w>0

1

F (w) +G(w),

〈φ2z〉 =

Iz,2Iz,0

=2

LT

∑w>0

1

R(w), (38)

with w = 2πn/LT .

This expression converges for modular parametersclose to 1, and contains in addition to the logarithmicdivergence term a correction term that encodes the de-pendence of the width at different transverse planes onthe modular parameter of the cylinder. At finite temper-ature, this term is contributing to the width at all theplanes.

〈φ2z〉 =

2

LT

∑w>0

1

mw2 + σw∑i coth(wLi)ψ(w,Li)

(39)

and the string parallel (in-plane) fluctuations

〈φ2x〉 =

2

LT

∑w>0

1

Qx,w +Qy,w −√Q2xy,w + (Qx,w −Qy,w)2

〈φ2y〉 =

2

LT

∑w>0

1

Qx,w +Qy,w +√Q2xy,w + (Qx,w −Qy,w)2

,

(40)

where w = 2πn/LT , the period of the transverse fluctua-tions is 1/LT , Li is the string length on the i’s world-sheet(blade). It is more convenient for our further discussionof the in-plane fluctuations on the lattice to consider theabove rotated decoupled form, the smoothing ψ(w,Li) isgiven by

ψ(wn, Li) =−kwn

2σcoth(wnLi)−

1− 12n

2coth(wnLi)(2Liχ(τi) + 1

2Liχ(τi)− 1

)2n−1

.

(41)

Qx, Qy, Qxy are defined in Ref.[33, 42] where Qx, Qy andQxy are defined [33] as

Qx =

(kw2 + σw

∑i

coth(wLi)ψ(w,Li)

)+

(σ

2w +

w3

12π

)[∑i

η2i,x coth(wLi)ψ(w,Li)

], (42)

Qy =

(kw2 + σw

∑i

coth(wLi)ψ(w,Li)

)+

(σ

2w +

w3

12π

)[∑i

η2i,y coth(wLi)ψ(w,Li)

],

Qxy =

(σ

2w +

w3

12π

)[∑i

ηi,xηi,y coth(wLi)ψ(w,Li)

].

8

III. LATTICE MEASUREMENTS

A. Potential operators

A transfer matrix interpretation to the Polyakov loopscorrelator allow to obtain the 3Q static potential V3Q

with the center symmetry preserving operator

〈P3Q〉 = 〈P(~r1)P(~r2)P(~r3)〉,= exp(−V3Q(~r1, ~r2, ~r3;T )), (43)

where the Polyakov loop is given by

P (~r) =1

3Tr

[Nt∏nt=1

Uµ=4(~r, nt)

]. (44)

In the context of the Polyakov loop method we havecontributions of temperature-dependent effects to thefree energy of a system of three static charges coupledto a heatbath. In the following we describe the latticearrangement and parameters used to extract the abovecorrelator Eq. (43) at finite temperature.

For the study of the 3Q potential V3Q, we are interestedin the large-distance behavior which is directly relatedto the properties of the confining force rather than theshort-distance behavior [142] that is well described by thetwo-body Coulomb-type potential.

In this section we extract the long range potential forthe quark-antiquark [139, 143] and quark-diquark flux-tubes at finite temperature close to critical point. At thezero temperature according to QCD, the effective poten-tials should have the same slope [144, 145], correspondingto the flux tubes having the same energy density. Theeffective potential is obtained from the Polyakov loops inthe standard manner: The static mesonic state is con-structed by means of a pair of Polyakov loops

〈PQQ(~r1, ~r2)〉 = 〈P (~r1)P †(~r2)〉 (45)

where the Polyakov loop on an Euclidean lattice of sizeN3s × Nt is defined as a product of gauge field variables

Uµ=4(~ri, nt):

P (~ri) =1

3Tr

[Nt∏nt=1

Uµ=4(~ri, nt)

], (46)

which corresponds to three Polyakov lines all in the sametime direction, the vectors ~r define the positions of thequarks.

At fixed temperature T , Assuming the transfer ma-trix in interpretation is preserved as justified in Ref.[55],the Monte Carlo evaluation of the temperature depen-dent quark–antiquark potential at each R is calculated

through the Polyakov loop correlators as

〈PQQ(0, R)〉 = 〈P (0)P †(R)〉 (47)

=

∫d[U ]P (0)P †(R) exp(−Sw)

= e−1T VQQ(R,T ), (48)

where Sw is the plaquette action and T is the physicaltemperature.

The infinitely heavy quark state is constructed bymeans of Polyakov loop correlators. In the confinementphase, for pure SU(3) gauge configurations, the correla-tors respect the center symmetry transformation

Uµ=4(x, nt = 1) = CUµ=4(x, nt = 1), (49)

where center C of the gauge group SU(3) is all the el-ements z such that zgz−1 = g, with g ∈ SU(3) orz = exp(2πl/3) ∈ Z(3) with l = 0, 1, 2. The form ofthe center symmetry preserving baryonic correlators isthen

〈P3Q(~r1, ~r2, ~r3)〉 → 〈P (~r1)P (~r2)P (~r3)〉= 〈e2iπlP (~r1)P (~r2)P (~r3)〉= 〈P (~r1)P (~r2)P (~r3)〉. (50)

The quark–diquark potential can be identified via athree-loop correlator from Eq.(50) as

〈P3Q〉 = 〈P (~r1)P (~r2)P (~r3)〉

= e−1T V3Q(~r1,~r2,~r3) (51)

B. Energy-density operators

After construction of the gauge-invariant color-averaged quark states, we evaluate a gauge-invariant ac-tion density operator S(~ρ, t), that characterize the colorfield, at the spatial coordinate ~ρ of the three dimensionaltorus corresponding to each Euclidean time slice, nt. Themeasurements are repeated for each time slice and thenaveraged,

S(~ρ) =1

Nt

Nt∑nt=1

S(~ρ, t). (52)

The action density operator is calculated via a highly-improvedO(a4) three-loop improved lattice field strengthtensor [146]. The symmetry of the quark positions wasalso exploited to gain a reduction in the statistical uncer-tainties. The flux distribution has been averaged aroundall the symmetry planes of a given quark configuration.

A dimensionless scalar field that characterizes the glu-onic field can be defined as

C(~ρ;~r1, ~r2, ~r3) =〈P3Q(~r1, ~r2, ~r3)S(~ρ)〉〈P3Q(~r1, ~r2, ~r3)〉 〈S(~ρ)〉

, (53)

9

for baryonic systems,

C(~ρ;~r1, ~r2) = 1−〈PQQ(~r1, ~r2)S(~ρ)〉〈PQQ(~r1, ~r2)〉〈S(~ρ)〉

, (54)

for mesonic systems, where 〈......〉 denotes averagingover configurations and lattice symmetries, the vectors~ri define the positions of the quarks and the vector ~ρ de-notes position of the flux probe with respect to someorigin. The Polyakov loop correlators, for the heavybaryonic and mesonic state are defined as in Eq.(50) andEq.(45) respectively.

In the case of the quark-aniquark state cluster decom-position of the operators leads to C −→ 0 away from thequark space. In the quark-diquark state due to cluster de-composition of the operators, C should approach a valueC ' 1 away from the interquark space. For noise reduc-tion, we make use of translational invariance by comput-ing the correlation on every node of the lattice, averag-ing the results over the volume of the three-dimensionaltorus, in addition to the averaging the action measure-ments taken at each time slice in Eq.(52).

C. Lattice parameters and Monte-Carlo updates

Measurements are taken on a set of SU(3) pure gaugeconfigurations. The configurations are generated usingthe standard Wilson gauge action on two lattices of aspatial volume of 363 and temporal extents of Nt = 8and Nt = 10 for a coupling value β = 6.00. These tem-poral extents correspond to temperatures T/Tc = 0.9and T/Tc = 0.8, respectively.The lattice observable usedto reproduce the value of the lattice spacing of a = 0.1fm is the string tension σ = 440 MeV [90–92].

The Monte Carlo updates are implemented with apseudo-heatbath algorithm [147] using Fabricius-Haanand Kennedy-Pendelton (FHKP) [148, 149] updating.The operator S(r) is taken as a one-loop Wilson’s opera-tor. The first set consists of 500 configurations. Eachupdate step consists of one heat bath and five over-relaxations. The measurements are taken on 500 config-urations separated with 2000 updating sweeps. For eachconfiguration considered a sweep consisting of a heat bathand four overrelaxation steps is applied on the entire lat-tice four times n = 4.

The gauge configurations were generated using thestandard Wilson gauge action. The two lattices em-ployed in this investigation are of a typical spatial sizeof 3.63fm3. Performing the simulations on large enoughlattice sizes would be beneficial to gain high statisticsin a gauge-independent manner and also minimizing themirror effects and correlations across the boundaries asa by-product [72, 150].

The SU(3) gluonic gauge configurations has been gen-erated employing a pseudo-heatbath algorithm [148, 149]updating the corresponding three SU(2) subgroup ele-ments [147]. Each update step consists of one heat bath

and 5 micro-canonical reflections. We chose to performour analysis with lattices as fine as a = 0.1 fm by adopt-ing a coupling of value β = 6.00, with temporal extentsof Nt = 8, and Nt = 10 slices, which correspond to tem-peratures T/Tc ' 0.9, and T/Tc ' 0.8, respectively.

We perform a set of measurements nsub = 20 sepa-rated by 70 sweeps of updates. Each set of measurementsis taken following 2000 updating sweeps. These sub-measurements are binned together in evaluating Eq. (53).The total measurements taken on 500 bins. In this in-vestigation, we have taken 10,000 measurement at eachtemperature. The measurements are taken on hierarchi-cally generated configurations.

These gauge configurations have been spatiallysmeared according to a stout-link smearing [56, 151, 152].The values of the smearing parameters used are 0.25 andρ = 0.06 as we describe below in detail.

We consider four data sets corresponding to four equa-tions of evaluating Polyakov lines correlators Eq. andEq. (46) for the meson and baryon with the action-density correlations Eq. (54) and Eq. (53), respectively.The statistical uncertainties are estimated using the jack-knife method [153].

D. Cooling method

An ultraviolet filtering (UV) step precedes our mea-surements of the action density distribution throughoutthe lattice. The UV-filtering of the gauge configurationssuppresses the short distance quantum fluctuations of thevacuum and is beneficial in attaining a good signal tonoise ratio in the correlations Eq. (53). This involves alocal action reduction by smearing the gauge links of thewhole 4 dimensional lattice.

Smoothing the gauge fields complements our use oflattice symmetries to gain noise reduction in our mea-surement setup. It was shown, that the effective stringphysics in the heavy meson are independent of the UVfluctuations at large source separation [55]. Moreover, itwas demonstrated that the lattice data compares favor-ably with the predictions of the free string model withincreasing the levels of gauge field smoothing at the inter-mediate source separation distance at high temperatureswhere the free string picture is known to poorly describethe flux tube width profile (See also a detailed discussionon the effects of smearing in [154]).

Variant to [155, 156] where the Cabbibo-Marinari cool-ing [147] has been employed, we have chosen to smooththe gauge field by an over-improved stout-link smearingalgorithm [151]. Filtering methods such as this are alsoregularly used in calculations of physical observable toimprove overlap with low energy states. Since we areapproximating space-time by a 4-D lattice, the appliedapproximations to the continuum gluonic action will con-tain unavoidable discretisation errors. These errors canhave a negative effect on the topological objects presentin the gauge field being studied, in our case – strings.

10

FIG. 4: Geometry of the three well-separated quarks onthe nutshell. The large spheres represent the motion of thediffused field of characteristic smearing radius of Rs centeredat the quarks (small spheres).

These errors can be tamed by the introduction of a tun-able parameter ε into the action [157].

Beginning with the sum of the two staples touchingUµ(x) which reside in the µ− ν plane

Σµν = Uν(x+ µ)U†µ(x+ ν)U†ν (x)

+ U†ν (x+ µ− ν)U†µ(x− ν)Uν(x− ν)(55)

define the weighted sum of the perpendicular stapleswhich begin at lattice site x and terminate at neighboringsite x+ µ:

Cµ(x) =∑ν

ν 6=µ

ρµνΣ†µν(x)

(56)

which defines the standard stout-link smearing [158].The weights ρµν are tunable real parameters. Then thematrix Qµ(x), defined in SU(3) by

Qµ(x) =i

2

(Ω†µ(x)− Ωµ(x)

)− i

6Tr(Ω†µ(x)− Ωµ(x)

)(57)

with

Ωµ(x) = Cµ(x)U†µ(x) (58)

is by definition Hermitian and traceless, and henceeiQµ(x) ∈ SU(3). The new smeared link is then defined

Uµ(x) = eiQµ(x)Uµ(x) (59)

where the smearing parameter ρsm is unchanged.In this work, we use a value of ρ = 0.06 in the standard

stout-link cooling which is roughly equivalent in terms of

UV filtering to ε = −0.25 and ρ = 0.06 of the improvedalgorithm [151]. We have considered measurements onthree sets of ensembles after the corresponding coolingsweeps nsw = 40, 60, and 80.

On Fig.4 we show this quark-diquark system with thecorresponding cooled spheres. The link fuzzing Eq. (59)procedure is analogous to the Brownian motion of a dif-fused sharp field [130]. The diffuse field is Gaussian dis-tributed [130] through a sphere centered at positionr whose evolution with a smearing time τ , in a four-dimensional smearing scheme [56, 130] is given by

G(r; τ) =1

(4πDτ)3/2exp

[−r · r4Dτ

], (60)

with D describing the diffuseness of the field. The dif-fused field characteristic radius is defined as

Rs =

∫d3rG(r, τ)r2∫d3rG(r, τ)

,

= a√ρcnsw.

(61)

The proportionality constant c provides the scaling be-tween nsw in the improved stout-link smearing definedin Eq. (59) with respect to APE smearing [152, 159]).For instance, a careful calibration [56] yields a value ofc = 6.15 [56]for ρ = 0.06, that is, nsw is roughly twicethat of the corresponding sweeps in APE smearing [159]of parameter α = 0.7.

A string condensate that resembles the free Nambu-Goto NG string is induced in QCD vacuum at colorsource separations in the intermediate separation regionwith the gradual filtering out of the UV-fluctuations. Theflux tube, measured as a correlation between the mesonicoperator and the vacuum action density, is found to ex-hibit a broadening pattern and a transverse structuresimilar to the free-bosonic string.

Since one of our targets in this investigation is to ex-tract and draw a comparison between the physics of theQQ and (QQ)Q systems. The effective bosonic stringmodels are appropriate and systematic framework tohandle this quest. The application of the cooling to thegluonic gauge configuration is, thereof, consistent andjustified from a methodological point of view.

IV. DIQUARK-QUARK POTENTIAL

We evaluate the correlator Eq. (43) to extractthe heavy quarks potential for a planar 3Q ar-rangement corresponding to isosceles triangles of basewidth A and height R as illustrated in the schematicFigs. 3(a).However, Fig 3(b) shows the numerical val-ues of the geometrical factor γ as a function of trianglelength L3 for isosceles configurations L1 = L2 evaluatedby solving Eq. (19) for the Y-string potential.

The measured numerical values of the potentialEq. (43) are depicted in Fig. 6 and Fig.7 for isosceles

11

(QQ)Q triangle corresponding to bases of width, A = 0.2fm, at the two considered temperatures T/Tc = 0.8 andT/Tc = 0.9. The potential data for each base are ex-tracted in accord to Eq. (24) by varying the height of thetriangle, R, for each fixed base, as shown in Fig. 3. Thenumerical values of the potential have been subtractedfrom their values at isosceles height R = 1.2 fm.

The gauge configurations have been UV filteredthrough the above prescribed cooling algorithm. Thelattice data consist of four data sets with the gradual in-crease of the number of sweeps all over the lattice. Thatis, nsw = 20, 40, 60 and 80 sweeps. The link-fuzzing pro-ceedure creates overlapping regions (see Fig. 4) betweenthe two quark Q1 and Q2 constituting the diquark systemat the base.

Nevertheless, when we draw comparison between theQQ system and the (QQ)Q diquark-quark system at bothconsidered time extent of the lattice. We find identicalbehavior of the two systems with respect to each other ifthe same number of smearing sweeps is applied to gaugelinks.

Figure. 5 illustrates the measured potential at T/Tc =0.9 for both the bosonic and the fermionic systems at thedepicted number of smearing sweeps. The increase in thenumber of smearing sweeps on the diquark-quark baryonshould not affect the response of the QCD vacuum whichindicates the splitting of both system from the identicalgluonic structure after increase in the temperature.

FIG. 5: The lattice data of the QQ potential at T/Tc = 0.9and the corresponding 3Q potential of a planar 3Q isoscelestriangle with a base length A = 0.2 fm with measurementstaken at various smearing sweeps. The solid lines correspondto the best fits to the string model Eq. (11).

We find the diquark-quark configuration to exhibitan identical behavior to the mesonic string for the po-tential for temperature near the end of QCD plateauT/Tc = 0.8. In the vicinity of the deconfinement point;however, We observe significantly different numerical val-ues for Polyakov loop correlators corresponding to eachsystem. The symmetry in the confining potential withthe meson does not manifest at both short and interme-

-6

-5

-4

-3

-2

-1

0

1

2 4 6 8 10 12 14

Va

Ra-1

β = 6.0,T/Tc=0.8QQ,80

Q-QQ,80

FIG. 6: Compares of the static potential of quark-antiquarkQQ and diquark-quark (QQ)Q configuration of base lengthA = 0.2 fm at T/Tc = 0.8 and β = 6.0.

diate distance scales (Fig.7).The same assertion on the effects of the number of

cooling sweeps holds true at lower temperature T/Tc =0.8 where both the bosonic quark-antiquark system QQand and fermionic (QQ)Q diquark-quark have identicalconfining potential. We observe this symmetry betweenboth system at all considered number of smearing sweeps.

Formula Eq. (24) sums up the contributions to the 3Qpotential resulting from both the in-plane and the per-pendicular fluctuations as discussed above. Before fittingthis string potential to the lattice data, the Y-stringslength has to be minimized, i.e., with the node’s posi-tion at Fermat point of the 3Q triangular configuration.The position of the Fermat point of the planar isoscelesarrangement can easily be shown, see Fig. 3, to lie at

xf =A

2√

3.

The three-quark system exhibits a ∆-shaped energy-density distribution [42]. The ∆ ansatz for the potentialmay be relevant to the system under scrutiny [41]. The∆ potential is given by

V3Q(~r1, ~r2, ~r3) = −1

2AQQ

∑i<j

1

|~ri − ~rj |+

1

2σQQ

∑i<j

|~ri−~rj |,

(62)with AQQ signifying the strength of the OGE Coulom-bic term derived from perturbative QCD (see Ref. [130,142]). The value of the string tension is taken to the halfthat of the mesonic string.

To throughly gain insight into the behavior of thestrings binding the constituent quarks we draw a com-parison between the ∆-ansatz paramterization Eq. (62)

12

TABLE I: The returned χ2dof(x) for fits of the lattice data to (QQ)Q with diquark diameter A = 0.2 fm and 3Q isosceles of

width A = 0.4 fm and A = 0.6 fm at T/Tc = 0.9 . The string tension is fixed to its value returned from the mesonic stringmodel Eq. (11). The fits are compare Eq. (24) for the Y-string model (with QQ string tension σ0a

2 = 0.0365) and the ∆-ansatzEq. (62) (with QQ string tension σ(T )a2 = 0.032).

Fit Range R ∈ [Rm, RM ] [4− 7] [4− 8] [5− 8] [5− 12] [7− 11] [8− 11]Fit Parameters χ2 µ χ2 µ χ2 µ χ µ χ2 µ χ2 µ

A = 2aY-string Model 6.29 -4.62 12.65 -4.57 6.49 -4.51 47.99 -4.32 11.48 -4.18 5.30 -4.11

∆-ansatz 13.29 -5.07 24.69 -5.01 11.35 -4.92 69.96 -4.69 14.35 -4.56 6.28 -4.48A = 4a

Y-string Model 4.48 -3.36 7.54 -3.37 2.89 -3.39 1.99 -3.37 9.27 -3.45 5.64 -3.48∆-ansatz 1.74 -4.42 2.50 -4.41 1.13 -4.39 1.13 -4.39 2.18 -4.39 1.89 -4.40A = 6a

Y-string model 6.18 -3.92 15.12 -3.87 9.03 -3.84 117.0 -3.53 34.71 -3.55 18.49 -3.53∆-ansatz 21.10 -5.65 45.01 -5.56 23.30 -5.49 257.19 -5.04 71.13 –5.07 37.25 -5.04

TABLE II: Lists the returned values of the χ2dof for fits of the lattice data to the string model formula Eq. (24), the fits are for

isosceles three-quark configurations of base width A = 2 a,A = 4 a and A = 6 a at two temperatures.

Fit range R ∈ [3− 11] R ∈ [4− 11] R ∈ [5− 11]Fit Parameters χ2 σ0a

2 χ2 σ0a2 χ2 σ0a

2

(a)T/Tc = 0.8

A = 2a 5.40 0.041 3.04 0.045 0.39 0.045A = 4a 13.87 0.39 4.21 0.040 7.68 0.044A = 6a 13.97 0.39 6.54 0.0415 5.04 0.0425

(b)T/Tc = 0.9

A = 2a 6.05 0.045(1) 5.53 0.045(1) 4.15 0.045(1)A = 4a 6.85 0.0335(5) 4.76 0.0335(5) 4.45 0.0335(5)A = 6a 8.25 0.0360(5) 5.86 0.0360(5) 5.65 0.0360(5)

-3

-2.5

-2

-1.5

-1

-0.5

0

2 4 6 8 10 12

Va

Ra-1

β=6.0,T/Tc=0.9

Q--Q,nsw=80

(QQ)Q,nsw=80

FIG. 7: Compares of the static potential of quark-antiquarkQQ and diquark-quark (QQ)Q configuration of base lengthA = 0.2 fm at T/Tc = 0.9 and β = 6.0.

and the Y-string model with a fluctuating junction. Thecomparison between the fit behavior of either Eq. (62)and Eq. (24) may divulge the effects of the junction fluc-tuations.

Considering the highest temperature T/Tc = 0.9, Ta-ble I summarizes the returned values of χ2 from the re-sultant fits of the thee-quark 3Q potential to the Y-stringmodel formula Eq. (24) and ∆ model Eq. (62). The ul-traviolet (UV) cutoff parameter µ is the only fit param-eter used. However; the zero temperature string tensionis fixed to its value, σ0a

2 = 0.365, which is returnedfrom the best fit to the free NG mesonic string potentialEq. (11). The resultant χ2 of the fits the ∆-ansatz [130–132, 142] to 3Q potential are enlisted as well.

For the fixed value of the string tension the fit returngood χ2 for either Q3 sources separations only over a lim-ited fit range including either few points at small sourceseparation or large separation, that is, R ∈ [0.4, 0.7] or[0.9, 1.1] fm. Figure 8, for example, depicts the potentialdata with fits to Eq. (24) corresponding to a diquark-quark configuration (QQ)Q of diameter A = 0.2 fm.

We plot in Fig. 9 the diquark-quark (QQ)Q potential,with the (QQ) diquark diameter is A = 0.2 fm, as thequark is pulled distance R apart. The lines correspond tothe fit of the Y-string formula of Eq. (24) and ∆-ansatzEq. (62) over either of the interval R ∈ [0.4, 0.7] fm orR ∈ [0.8, 1.1] fm.

13

-4

-3

-2

-1

0

1

4 6 8 10 12

V a

R a-1

(QQ)Q, T/Tc=0.9

∆-Baryon;σa2=0.0365

Y-Baryon;σa2=0.0365

SU(3) LGT, β=6.0

(a)

-4

-3

-2

-1

0

1

4 6 8 10 12

V a

R a-1

(QQ)Q, T/Tc=0.9

∆-Baryon;σa2=0.0365

Y-Baryon;σa2=0.0365

SU(3) LGT, β=6.0

(b)

FIG. 8: The diquark-quark (QQ)Q potential as the quark is pulled apart R, the diquark diameter is A = 0.2 fm. The linescorrespond to the fit of the Y-string formula of Eq. (24) and ∆-ansatz Eq. (62) (a) Fit over interval R ∈ [0.4, 0.7] fm; however,in (b) the fit interval range is R ∈ [0.8, 1.1] fm.

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

5 6 7 8 9 10 11 12

V a

R a-1

β = 6.0

T/Tc=0.8

T/Tc=0.9

(a)

-3

-2.5

-2

-1.5

-1

-0.5

0

4 6 8 10 12

V a

R a-1

β = 6.0,T/Tc=0.9

Lattice

∆-model

Y-model

(b)

FIG. 9: The 3Q potential versus the height length of isosce-les triangular configuration of the planar 3Q with bases oflength A = 0.8 fm. (a) The lines correspond to the 3Q po-tential according to the best fits to the string picture formulaof Eq. (24) at T/Tc = 0.8 and T/Tc = 0.9. (b) The linescorrespond to fits to Eq. (24) and a linear Y-ansatz [130, 142]for fit range R ∈ [5, 12].

The returned χ2 in Table. I and the plots in Fig. 9show subtle differences between the two potential ansatzcorresponding to the Y and ∆ models. The triangularconfiguration of the diquark-quark base width A = 0.2fm is small enough that both types approach similar fitbehavior.

TABLE III: The retrieved fit parameters of free mesonic stringpotential Eq. (11) to the lattice data of (QQ)Q with a diquarkdiameter A = 0.2 fm and (3Q) with A = 2a, 4a and A = 6a.

Fit Range R ∈ [4− 11] R ∈ [5− 11]χ2 σ0a

2 χ2 σ0a2

T/Tc = 0.8

A = 2a27.22 0.0445 5.95 0.0445

A = 4a18.54 0.035 5.16 0.0378

A = 6a16.35 0.036 8.28 0.041

T/Tc = 0.9

A = 2a4.53 0.0455(5) 1.15 0.0445(5)

A = 4a6.68 0.0296(1) 4.12 0.0325(5)

A = 6a7.85 0.032 6.15 0.0345(5)

Let us consider 3Q configuration with large separationdistance between the two quarks Q1 and Q2 to system-atically compare the behavior of different models. Withthe increase of the width of the base of the isosceles toA = 0.8 fm, Fig. 9 shows the 3Q potential versus theheight length of isosceles triangular configuration.

In Figure. 9-(a) the lines correspond to the 3Q poten-tial in accord to the best fits to the string picture formulaof Eq. (24) at the two considered temperatures. The Y-string model fits well to the lattice data considering stringtension value returned from the mesonic fits.

14

-4

-3

-2

-1

0

1

2 3 4 5 6 7 8 9 10 11

V a

R a-1

T/Tc=0.9, A=2a

Baryon-σa2=0.0445

Meson-σa2=0.0365

SU(3) LGT, β=6.0

(a)T/Tc = 0.9

-4

-3

-2

-1

0

1

2 3 4 5 6 7 8 9 10 11

V a

R a-1

T/Tc=0.8, A=2a

Baryon-σa2=0.0445

Meson-σa2=0.0445

SU(3) LGT, β=6.0

(b)T/Tc = 0.8

FIG. 10: The (QQ)Q diquark-quark potential versus theheight length of configuration of the planar quark system withdiquark diameter A = 0.2 fm. The lines compares the fit ofEq. (24) for the baryonic potential and that of the mesonicstring Eq. (11) over range R ∈ [9, 11].

However, in Fig. 9-(b) the lines correspond to fits toEq. 24 and linear confining Y-model with a coulombterm [41, 130, 142] for fit over the range R ∈ [5, 12].

At large distances and temperatures [41, 42] the Y-string assumes a higher energy content and broadeningprofile. This can be read as well from the value of the χ2

corresponding to the ∆-parametrization Eq. (62) if thesame fit range is considered. The returned χ2 values arerelatively smaller and closer to the corresponding valuesof the Y-string model Eq. (24). A consistent physical re-alization of these observations is that the fluctuations ofthe node within the static baryon broaden largely enough

to give similar effects to a hypothetical ∆-string circum-venting the 3Q triangle.

Assuming the string potential of the Y-string junctionEq. (24) approaches the mesonic string limit of the cor-responding QQ pair Eq. (11), then a fit of the potentialto the mesonic string potential returns the χ2 and σ0a

2

shown in Table. III for the diquark-quark data at bothtemperatures.

The diquark-quark string at T/Tc = 0.9 retrieves astring tension that is significantly different than the cor-responding QQ at the same temperature. In Table.I apoor fit with high value of χ2 is retrieved if the QQ stringtension σ0a

2 = 0.0365 is used as input for the string ten-sion in formula Eq. (24) for the baryonic Y-string poten-tial.

Fig. 10-(a) compares both fits to the diquark-quarkpotential data for mesonic string Eq. (11) and the poten-tial from the baryonic junction model at the relativelylow temperature T/Tc = 0.8. It is remarkable that bothstring models fit well to the lattice data of the diquark-quark system and return the same value of the stringtension σ0a

2 = 0.045. The value coincides with that ofthe QQ system at the same temperature.

With the temperature increases closer to the criti-cal point T/Tc = 0.9 Fig 10-(b) shows the fit of themesonic string potential, with fixed value of σ0a

2 =0.036, Eq. (13). A drastic deviation from the latticedata of diquark-quark system is evident. The fits of thebaryonic string model, however, are returning a valuethat is surprisingly matches that of the string tension atT = 0 [160].

These may be relevant to the interaction of the junc-tion with the strings and the two quarks at the base, Q1

and Q2. These are higher order effects that is beyondthe free string approximations Eq. (11) and Eq.(24) andought to be addressed considering other ansatz such asthose encompassing the boundary, rigid and self interac-tions [57]. Nevertheless, the fit to the baryonic Y-stringmodel with a fixed value of the σ0a

2 = 0.0365 is shownin Fig. 8 which still reflects an improved paramterizationbehavior than the mesonic counterpart in Fig. 10(a).

At T/Tc = 0.9 the fits of (QQ)Q diquark-quark data,with diameter A = 0.2 fm, over distances R ≥ 0.5 fmto the mesonic string (Table. III) returns a value of theσ0a

2 = 0.045 that is dramatically different from the thatof the corresponding QQ system σ0a

2 = 0.0365.In summary, the gluonic string binding diquark-quark

(QQ)Q configuration shows identical behavior as quark-antiquark QQ string up to temperature T/Tc = 0.8.Within the uncertainty of measurements, the Polyakovloop correlators and string tension of both system arethe same.

The baryonic string model approaches the mesonicstring potential for quark source separation R ≥ 0.5fm. Within a smaller neighborhood of the critical pointT/Tc = 0.9, the increase in temperature breaks the sym-metry into the bosonic and fermionic arrangments. TheY-baryonic string model and ∆-ansatz show no signifi-

15

cant difference in the fit of the diquark-quark data, bothpotential models returns good χ2, however, with stringtension value that is significantly different from the QQ.

V. DIQUARK-QUARK (QQ)Q ENERGYPROFILE

A. Vacuum’s Action-density

The simulation setup has been described in section(V).The lattice operator which characterizes the gluonic fieldC is usually taken as the correlation between the vac-uum lattice action-density S(~ρ, t) operator, and a gauge-invariant operator representing the heavy baryon state,that is, three Polyakov lines Eq. (53). The action-densityoperator is calculated through an O(a4) improved latticeversion of the continuum field-strength tensor given by

F Impµν =

2∑i=1

wi C(i,i)µν +

4∑i=3

wi C(1,i−1)µν + w5C

(3,3)µν , (63)

where C(i,j) is a combination of Wilson loop terms cor-responding to loops with lattice extent i used to con-struct the clover term and wi are the correspondingweights [146].

The reconstructed action density

S(~ρ) = β∑µ>ν

1

2Tr(F Imp

µν )2 (64)

is accordingly measured on 80 sweeps of stout-link smear-ing. For convenience, we consider the complementarydistribution C′ = 1− C in the following. The correlationfunction Eq.(53) is found C′(~ρ) = 0 away from the quarkposition.

In general, the action density distribution is non-uniformly distributed. The distribution C′(~ρ(x, y, z =0)) = 0 has an action density maximal curve along themiddle line ~ρ(x, y = 0, z = 0) between the two quarksQ1,2. With the increase of source Q3 separation, thepeak point along the maximal curve C′(~ρ(x, y = 0, z = 0))shows only subtle movement [52, 154].

In Fig.14 we show examples of the expulsion of vac-uum fluctuations and the formation of flux-tubes for ourquark-diquark and quark-antiquark configurations.

Different possible components of the field-strength ten-sor in Eq.(63) can separately measure the chromo-electricand magnetic components of the flux. The action density,however, is related to the chromo-electromagnetic fieldsvia 1

2 (E2−B2) and is the quantity of direct relevance tothe comparison with both the string perpendicular fluc-tuations

The width of the flux tube may be then estimatedthrough fitting the density distribution C(~ρ), [Eq. (3)], ineach selected transverse plane ~ρ = (xi, y, z0) to a Gaus-sian [88, 89]. To estimate the mean-square width of the

gluonic action density along the planes transverse to thequark-diquark axis, we choose a double Gaussian func-tion of the same amplitude, A, and mean value µ = 0

G(r, θ; z) = A(e−r2/σ2

1 + e−r2/σ2

2 ), (65)

for baryonic and meson systems [161], respectively, withthe color charge position ~ri and flux probe vector ~ρ. TheQQ and (QQ)Q static potential is extracted from the log-arithm [161] of Polyakov loop correlators P2Q(~r1, ~r2) andP3Q(~r1, ~r2, ~r3), respectively. The Diquark-quark (QQQ)is constructed as an isoscele triangle of two quark at thebase (diquark) separated by two lattice spacing.

Assuming that σ1 = σ2 the above form corresponds tothe standard Gaussian distribution, σ1 6= σ2 correspondsto unconstrained form. The fits of the double Gaussianform return acceptable values of χ2 at the intermediatedistances. Good χ2 values are returned as well whenfitting the action density profile to a convolution of theGaussian with an exponential [42, 162].

The fits to the action density distribution C(r, θ, y =R/2) at the center of the tube Eq. (53), R/2, is shownfor Diquark-quark separation distances R = 0.6, 0.9 fmand R = 1.2 fm in Fig 11 and Fig. 12 at temperaturesT/Tc ≈ 0.8 and temperatures T/Tc ≈ 0.9, respectively.The plots depicts the fits for both σ1 = σ2 and σ1 6= σ2

of Eq.(65).We can gain more insight into the properties of the flux

tubes by examining their profiles close to the quark. Westudy the values of the correlators C3Q(~ρ) and CQQ(~y)where ~ρ = (x, y, 0) is constrained to the plane of thecolor sources. The quark is at the position (x, 0, 0) whereR = x varies from 1 to 12 lattice steps. Color source sep-aration R ≥ 1.2 fm are excluded due to correlations fromthe other side of the lattice due to periodic boundary.

In Appendix. A, we show in Fig. 25 the action densityin 2D-plane for source separations R = 0.6, 0.9, 1.2 and1.4 fm and temperature T/Tc ≈ 0.8. The panels com-pares the quark-diquark system with bases length,thedistance between Q1 and Q2 quarks, A = 0.2 fm andA = 0.4 fm for the in-plane and perpendicular fluctua-tions

Similarly, Fig. 26-of Appendix. A- exhibits the samepanels; however, at the highest temperature T/Tc = 0.9.Surface plot of the Flux density in the quark-plane forsource separations R = 0.5, 0.9 and 1.2 fm at the sametemperature T/Tc ≈ 0.9. The flux density is shown inthe projection plane yx for the distance between Q1 andQ2 quarks A = 0.2 fm is shown in Fig. 27 and for A = 0.4fm in Fig. 28.

First we examine the longitudinal profiles of bothquark-diquark and quark-antiquark flux-tubes along theline (~ρ) = (x, 0, 0) in Fig. 15. As expected, the vacuumexpulsion close to the diquark is stronger than in thevicinity of the antiquark. However, near the quark thetwo flux-tubes show very similar profiles.

Next we examine the transverse profiles along a lineorthogonal to the midpoint of the flux tube, ie. along(x/2, y, 0) for x even, or along ((x+ 1)/2, y, 0) for x odd.

16

y0 5 10 15 20 25 30 35

(y)

3QC

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

=6distanceQQ-Q

=9distanceQQ-Q

=12distanceQQ-Q

2σ=1σ

2σ≠1σ

=6.0βLattice data: T/Tc=0.8, YX-projection

=DOF

/N2χ 2.30914

width = 13.8 ± 0.06922

=DOF

/N2χ 2.71292

width = 15.27 ± 0.3473

=DOF

/N2χ 1.11139

width = 18.09 ± 1.024

(a)yx-plane, A = 2d = 0.2 fm

y0 5 10 15 20 25 30 35

(y)

3QC

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

=6distanceQQ-Q

=9distanceQQ-Q

=12distanceQQ-Q

2σ=1σ

2σ≠1σ

=6.0βLattice data: T/Tc=0.8, YX-projection

=DOF

/N2χ 1.78439

width = 14.66 ± 0.133

=DOF

/N2χ 0.908588

width = 16.53 ± 0.3362

=DOF

/N2χ 0.271063

width = 17.26 ± 1.38

(b)yx-plane, A = 2d = 0.4 fm

FIG. 11: The action density distribution C(r, θ, y = R/2) at the center of the tube Eq. (53), R/2, for source separations R =0.6, 0.9, 1.2 fm and temperature T/Tc ≈ 0.8. The red dotted lines correspond to fits with the standard Gaussian distribution,assuming σ1 = σ2 in Eq.(65), the blue solid lines correspond to unconstrained form, σ1 6= σ2. The distance between Q1 andQ2 quarks denoted as A = 2d, see Fig.4 for details. Transverse profiles for quark-diquark system denoted as yx-plane, thelongitudinal flux tube profiles denoted as yx-plane.

y0 5 10 15 20 25 30 35

(y)

3QC

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

=6distanceQQ-Q

=9distanceQQ-Q

=12distanceQQ-Q

2σ=1σ

2σ≠1σ

=6.0βLattice data: T/Tc=0.9, YX-projection

=DOF

/N2χ 0.952934

width = 22.99 ± 0.1215

=DOF

/N2χ 3.36809

width = 28.86 ± 0.3047

=DOF

/N2χ 2.57544

width = 38.06 ± 4.219

(a)yx-plane, A = 2d = 0.2 fm

y0 5 10 15 20 25 30 35

(y)

3QC

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

=6distanceQQ-Q

=9distanceQQ-Q

=12distanceQQ-Q

2σ=1σ

2σ≠1σ

=6.0βLattice data: T/Tc=0.9, YX-projection

=DOF

/N2χ 5.06309

width = 25.92 ± 0.1923

=DOF

/N2χ 3.72472

width = 32.25 ± 0.5002

=DOF

/N2χ 1.79722

width = 38.06 ± 0.5155

(b)yx-plane, A = 2d = 0.4 fm

FIG. 12: Same as Fig.11; however, the temperature scale isset to T/Tc = 0.9.

FIG. 13: The action density Eq.(54) of meson QQ at T/Tc =0.8 and color source separation R = 1.0 fm measured afternsw = 60.

In Fig. 14 we show profiles of both quark-diquark andquark-antiquark flux-tubes for x = 12. We find that aslong as x is larger than one third of the total length ofthe quark-antiquark system, the transverse profiles areclose to identical.

Table IV and V the transverse profile of the flux-tubeis fitted well by a double Gaussian function C(x; y) =

1−A(e(−y2/σ21 +e(−y2/σ2

2)). The fit enables us to estimatethe mean-square width/second moment of the flux-tubeas well as its depth/amplitude.

W 2(x) =

∫C(r;x)r2 dA∫C(r;x) dA

. (66)

The fit parameters of the flux-tube are given in Ta-ble IV and V, the fits show that for long enough fluxtubes the transverse profiles of quark-diquark and quark-antiquark flux-tubes are statistically identical at bothtemperatures.

17

Q1

Q2

Q3

RA

x

y

(0,0,0)

(a)Baryon 3Q with base A = 0.6 fm (b)Baryon (QQ)Q; base A = 0.2 fm

FIG. 14: The baryonic action density Eq. (53) with the decrease of the two quark (QQ) separation at the planar-triangle basefrom A = 6a to A = 2a and the formation of quark-antiquark QQ gluonic field at T/Tc = 0.8

C(y

)

0

0.05

0.1

0.15

0.2

0.25

0.3

y-axis10 12 14 16 18 20 22 24

z-ax

is

0

1

2

3

4

5

6 Lattice data

R=7=6.0β

T/Tc=0.9

(a)

C(y

)

0.2−

0.1−

0

0.1

0.2

0.3

0.4

y-axis10 12 14 16 18 20 22 24

z-ax

is

0

1

2

3

4

5

6 Lattice data

R=7=6.0β

T/Tc=0.9

(b)

C(y

)

0.2−

0.1−

0

0.1

0.2

0.3

0.4

y-axis10 12 14 16 18 20 22 24

z-ax

is

0

1

2

3

4

5

6 Lattice data

R=7=6.0β

T/Tc=0.9

(c)

FIG. 15: The in-plane action density Eq. (54) for meson QQ in (a) and baryon (QQ)Q with isosceles height R = 0.7 fm andbase length A = 0.2 fm in (b) and A = 0.4 fm in (c) at T/Tc = 0.9 and R = 0.7 fm, nsw = 60.

In the vicinity of the deconfinement point T/Tc = 0.9,the flux density corresponding to each system shows verysimilar profiles near the quark. However, the vacuumexpulsion close to the diquark is stronger than that inthe proximity of the antiquark as can be seen from theamplitude profile in Table. IV.

The perpendicular and in-plane fluctuation’s width ofthe Q(QQ) system exhibit cylindrical symmetry; even so,the string profile is apparently not identical to the QQsystem. The coincidence with the mesonic string doesnot manifest neither at small nor intermediate separa-tion regions. Figures 16 and 17 consecutively exhibit thedifference in the mean square width at both the middleplanes and first four lattice slices from the diquark sys-tem.

This could be an indication that the string is energeticenough to induce a splitting towards the baryonic-likestructure [41, 42]. Indeed we observe a significant changein the numerical values of the Polyakov correlators of(QQ)Q system for R < 1 fm with the increase of thetemperature as depicted in Fig. 7. In the previous sectionwe found a consonant measurement of the string tension,which encodes the energy per unit length of the string,that is significantly higher than the corresponding value

of the mesonic string.Apart the broadening aspects at the two temperatures,

one would like also to compare the curvatures which maybe related to the form factors of (QQ)Q and QQ systems.For each color source separation we calculate the differ-ence at each adjacent plane from the diquark and quarkfrom the mean-square width from the middle plane inbetween it.

Figure 18 compares the change in the mean-squarewidth with respect to the middle plane, δW 2(x) =W 2(x)−W 2(R/2), for quark source separation R = 0.8fm at T/Tc = 0.9. In the mesonic width difference QQ isshown in Fig. 18(a) and the diquark-quark (QQ)Q withdiameter A = 2a is depicted in Fig. 18(b).

For QQ system, the oscillations of a free NG stringfixed at the ends by Dirichlet boundaries traces out anonuniform width profile with a geometrical curved finestructure. This is in consistency with the free NG stringat source separations R ≥ 1.0 fm at the highest temper-ature T/Tc = 0.9. In the intermediate region the latticedata is better described with self-interaction rigid mod-els [45] as shown in Fig. 18-(a).

In contrast to the QQ, the diquark-quark (QQ)Q sys-tem Fig. 18-(a) exhibits asymmetry along the tube. That

18

FIG. 16: Compares the in-plane and perpendicular mean-square width at T/Tc = 0.9 for (QQ)Q and QQ systems at the middleplane after nsw = 40 for diquark base of diameter A = 2a.

plane x = 1 x = 2 x = 3 x = 4n = R/a A w2a−2 χ2

dof A w2a−2 χ2dof A w2a−2 χ2

dof A w2a−2 χ2dof

R=

4a QQ 0.0654(1) 17.75(5) 4.3 0.0711(1) 17.27(5) 4.40 0.0654(1) 17.75(5) 4.3

(QQ)Qxy 0.0837(2) 24.92(7) 5.8 0.0864(2) 24.59(7) 5.74 0.0810(2) 25.03(7) 5.84(QQ)Qxz 0.0833(2) 24.93(6) 8.13 0.0862(2) 24.63(6) 8.57 0.0810(2) 25.03(7) 10.51

R=

5a QQ 0.0777(2) 18.75(6) 3.98 0.0894(2) 17.97(6) 4.03 0.0894(2) 17.97(5) 4.3 0.0777(2) 18.75(6) 3.98

(QQ)Q‖ 0.1(2) 27.22(15.012) 239.03 0.1064(2) 24.20(7) 4.87 0.1053(2) 24.24(7) 5.10 0.0951(2) 24.71(7) 5.33(QQ)Q⊥ 0.0986(2) 24.81(7) 7.09 0.1062(2) 24.27(7) 6.83 0.1052(2) 24.23(7) 7.68 0.0952(2) 24.83(7) 9.45

R=

6a QQ 0.0842(2) 20.28 3.27 0.1003(2) 19.31(7) 3.2 0.1062(3) 19.04(6) 3.1 0.0843(2) 19.3(6) 3.21

(QQ)Q‖ 0.1097(3) 24.93(9) 2.98 0.1212(3) 24.67(9) 3.51 0.1245(3) 24.57(9) 3.86 0.1190(3) 24.67(8) 4.30(QQ)Q⊥ 0.1092(3) 25.47(8) 5.10 0.1209(3) 24.82(8) 5.19 0.1243(3) 24.58(8) 5.17 0.1190(3) 24.73(8) 5.98

R=

7a QQ 0.0867(3) 22.1(2) 2.5 0.1048(3) 21.04(8) 2.47 0.1147(4) 20.64(9) 2.3 0.0867(3) 20.64(9) 2.3

(QQ)Q‖ 0.1161(3) 25.7(1) 1.6 0.1306(4) 25.6(1) 2.15 0.1449(6) 26.7(1) 1.44 0.1367(5) 25.5(1) 3.02(QQ)Q⊥ 0.1156(3) 26.6(1) 5.20 0.1303(4) 25.9(1) 4.02 0.1448(6) 27.0(1) 2.35 0.1366(5) 25.5(1) 3.48

R=

8a QQ 0.9(2) 24.1(1) 1.94 0.1049(4) 23.03(1) 1.91 0.1165(5) 22.6(1) 1.71 0.1206(5) 22.5(1) 1.62

(QQ)Q‖ 0.1195(4) 26.5(1) 0.74 0.1354(5) 26.6(1) 1.10 0.1449(6) 26.7(1) 1.44 0.1476(6) 26.8(1) 1.83(QQ)Q⊥ 0.1190(4) 27.9(1) 4.67 0.1351(5) 27.3(1) 3.32 0.1448(6) 27.0(1) 2.35 0.1475(7) 26.9(1) 2.00

R=

9a QQ 0.0856(4) 26.2(2) 1.38 0.1025(5) 25.2(2) 1.43 0.1140(6) 24.9(2) 1.30 0.1199(7) 24.9(2) 1.17

(QQ)Q‖ 0.1209(5) 27.4(2) 0.33 0.1368(6) 27.6(2) 0.49 0.1474(8) 28.0(2) 0.48 0.1524(8) 28.3(2) 0.96(QQ)Q⊥ 0.1204(5) 29.2(2) 4.11 0.1366(7) 28.8(2) 2.92 0.1473(8) 28.6(2) 1.79 0.1523(9) 28.6(2) 1.21

R=

10a QQ 0.0840(5) 28.26 0.91 0.0992(6) 27.6(2) 0.98 0.1094(8) 27.6(2) 0.91 0.1151(9) 27.8(2) 0.83

(QQ)Q‖ 0.1210(6) 28.3(2) 0.17 0.1361(8) 28.6(2) 0.20 0.1464(9) 29.1(1) 0.20 0.152(1) 29.8(2) 0.46(QQ)Q⊥ 0.1205(7) 30.4(2) 3.27 0.1359(8) 30.1(2) 2.52 0.146(1) 30.2(2) 1.5 0.152(1) 30.4(2) 0.80

R=

11a QQ 0.0821(7) 30.3 0.57 0.0953(8) 29.9(3) 0.6 0.1038(9) 30.3(3) 0.55 0.1084(1) 31.1(3) 0.50

(QQ)Q‖ 0.1205(8) 29.5(3) 0.09 0.134(1) 29.6(3) 0.09 0.143(1) 30.2(3) 0.14 0.149(1) 31.2(3) 0.22(QQ)Q⊥ 0.1199(8) 31.5(3) 2.21 0.134(1) 31.3(1) 1.95 0.143(1) 31.6(3) 1.27 0.149(1) 32.2(3) 0.61

R=

12 QQ 0.0799(8) 32.1(4) 0.34 0.0910(9) 32.0(4) 0.34 0.097(1) 33.0(4) 0.29 0.101(1) 34.6(4) 0.24

(QQ)Q‖ 0.119(1) 31.2(4) 0.08 0.131(1) 30.9(4) 0.06 0.138(1) 31.3(4) 0.07 0.143(2) 32.4(4) 0.12(QQ)Qperp 0.119(1) 32.8(4) 1.21 0.131(1) 32.5(4) 1.28 0.138(1) 33.0(4) 0.97 0.143(2) 33.9(4) 0.49

TABLE IV: The mean-square width W 2 and amplitude of the in-plane and perpendicular action-density measured at thecorresponding planes using fits to a double-Gaussian ansatz Eq. (65) at temperature T/Tc = 0.9 and nsw = 40.

is the width decreases near the diquark and increases nearthe quark source. The second observation is that thewidth difference is around 50% smaller than the mesoniccounter part. This is another instance that shows thatboth profiles are nonidentical near the critical point.

Near the end of QCD plateau (see Fig. 14), T/Tc = 0.8,in the vicinity of the quark, the action density exhibits

cylindrical symmetry with no measurable difference be-tween the transverse profiles of the diquark-quark flux-tubes and the quark-antiquark flux-tubes (See for com-parison Ref. [42]). We find the energy profiles of (QQ)Qto be very similar considering middle plane as depictedin Fig. 19-(a). The same assertion on the broadening ofthe profile at planes other than the middle, see Fig. 20,

19

]-1 distance x-axis, [RaQQ-Q

4 5 6 7 8 9 10 11 12 13 14

]-2 a2

Str

ing

wid

th, [

w

10

15

20

25

30

35

40 Q-QQ: YX-plane [A=0.2]

Q-QQ: YX-plane [A=0.4]

Q-QQ: ZX-plane [A=0.2]

Q-QQ: ZX-plane [A=0.4]

QQ

Plane ID=1

=6.0β

T/Tc=0.9

]-1 distance x-axis, [RaQQ-Q

4 5 6 7 8 9 10 11 12 13 14

]-2 a2

Str

ing

wid

th, [

w

10

15

20

25

30

35

40

Plane ID=2=6.0β

T/Tc=0.9

]-1 distance x-axis, [RaQQ-Q

4 5 6 7 8 9 10 11 12 13 14

]-2 a2

Str

ing

wid

th, [

w

10

15

20

25

30

35

40

Plane ID=3=6.0β

T/Tc=0.9

]-1 distance x-axis, [RaQQ-Q4 5 6 7 8 9 10 11 12 13 14

]-2 a2

Str

ing

wid

th, [

w

10

15

20

25

30

35

40

Plane ID=4=6.0β

T/Tc=0.9

FIG. 17: The in-plane and Perpendicular-plane mean-squarewidth at T/Tc = 0.9 for (QQ)Q and QQ systems at Planesx = 1, 2, 3 and x = 4, nsw40.

holds as well.

It is important to mention that we observe no signifi-cant effect of the number of UV filtering sweeps nsw onthe observed symmetry or asymmetry between both sys-tems. Figure. 19-(b) is an illustration for measurementstaken at the middle planes considering two cooling levels.

Indeed, these results are compatible with our analysisin the previous section of the Polyakov loop correlatorwhich assumes same values for the diquark-quark (QQ)Qand the mesonic QQ string or the same values of thestring tension. The findings that string tension [41] whichis observed also recently employing the three Polyakovloop analysis at T = 0 [160] (This instance elaborate onthe close analogy between the analysis at end of QCDplateau [41, 42], in addition to using the Wilson loopoverlap formalism [144] at T = 0.

Plane ID0.5− 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

]-2 a2

wδS

trin

g w

idth

diff

eren

ce, [

6−

4−

2−

0

2

4Lattice data

LO2W∆

NLO2W∆

=8-1 separtion RaQQ=6.0β

T/Tc=0.9

(a)QQ-Meson

PlaneID0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

]­2

a2

wδ

Str

ing w

idth

diffe

rence, [

5−

0

5

10

15

20LDATA: Gauss+C

LDATA: Gauss+Gauss

LDATA: Gauss+Gauss+C

=8­1 separtion RaQQ­Q YX­projection=6.0β

T/Tc=0.9 base: A=0.2 fmQQ

40 sweeps

(b)(QQ)Q-Baryon

FIG. 18: The change in the mean-square width with respectto the middle plane at T/Tc = 0.9 for (a)-(QQ)Q with baseA = 2a and (b)-QQ systems with nsw = 40, the quark sourceseparation R = 0.8 fm.

(a)

(b)

FIG. 19: The mean-square width of the string in the middleplane for QQ and (QQ)Q of isosceles base A = 0.2, 0.4 fm“in-plane and Perpendicular-plane” at T/Tc = 0.8.

20

FIG. 20: The width of the string in the plane, z = 2, 3, 4 forQQ and (QQ)Q of isosceles base A = 0.2, 0.4 fm. In-planeand perpendicular mean-square width at T/Tc = 0.8.

B. Broadening of effective strings

In this section our target is to divulge the character ofbroadening of the flux tube. The mean-square width iscompared to the corresponding string model formulas inEqs (39), Eq. (40) and (26) for both mesonic and baryonicstring, respectively.

We focus on examining the fit behavior at the firstfour consecutive plane in the vicinity of the diquark asthe the third quark Q3 is pulled apart. This ought toreveal whether the baryonic junction has nontrivial sig-natures on the broadening profile at the correspondingquark configuration.

As we proceed in the fit with the action density dataone have to fix the geometrical form of the Y stringsconfiguration before hand. The junction position is fixedto Steiner point where the sum of the length of the threestrings is minimal. Figure 3 depicts the classical solutionof the Y-string’s configuration with respect to the quarkpositions with the junction at Fermat point xf of the

0

50

100

150

200

250

4 5 6 7 8

χ2

Ra-1

β=6.0, T/Tc=0.8, A=2aPlane x=1Plane x=2Plane x=3Plane x=4

(a)A = 2a

0

20

40

60

80

100

120

140

4 5 6 7 8

χ2

Ra-1

β=6.0, T/Tc=0.9, A=4aPlane x=1Plane x=2Plane x=3Plane x=4

(b)A = 4a

FIG. 21: The returned χ2 for fits to the baryonic string modelfor the in-plane fluctuations Eq. (40) at the first four planeswith Fig.(a)Diquark diameter A = 2a and (b)Baryon withbase A = 4a.

isosceles triangular configuration.

Similar to our analysis regarding the confining force inthe last section We focus on the flux tubes action den-sity due to the 3Q planar configuration correspondingto isosceles triangles, however, with the diquark-quark(QQ)Q bases of length A = 2a, A = 4a only consid-ered. This planar quark setup is convenient to simplifythe study of a baryonic junction on a lattice structureas the position of the Fermat point depends only on thebase length which is fixed as the third quark position Rvaries with the same locus of the Fermat point.

In Tables IV and V enlisted are the two components

21

0

50

100

150

200

250

4 5 6 7 8

χ2

Ra-1

β=6.0, T/Tc=0.8, A=2aPlane x=1Plane x=2Plane x=3Plane x=4

FIG. 22: The returned χ2 for fits to the mesonic string modelfor the in-plane fluctuations Eq. (26) at the first four planesat T/Tc = 0.9.

plane A w2a−2 χ2dof

R=

4a QQ 0.0752(1) 14.21(5) 2.29

(QQ)Q‖ 0.0937(2) 14.57(5) 1.60(QQ)Q⊥ 0.0936(2) 14.33(6) 1.30

R=

5a QQ 0.1066(2) 14.51(5) 0.51

(QQ)Q‖ 0.1171(2) 12.89(1) 15.91(QQ)Q⊥ 0.1183(3) 14.73(7) 1.22

R=

6a QQ 0.1332(3) 15.11 0.23

(QQ)Q‖ 0.1416(3) 15.35(7) 1.02(QQ)Q⊥ 0.1415(4) 15.17(8) 0.99

R=

7a QQ 0.1530(5) 15.89(9) 0.07

(QQ)Q‖ 0.1607(5) 15.9(1) 1.15(QQ)Q⊥ 0.1606(5) 16.0(1) 0.99

R=

8a QQ 0.1721(9) 16.8(1) 0.02

(QQ)Q‖ 0.1768(9) 16.4(2) 0.73(QQ)Q⊥ 0.1768(7) 16.6(1) 0.89

R=

9a QQ 0.183(2) 18.0(2) 0.005

(QQ)Q‖ 0.188(1) 17.0(2) 0.85(QQ)Q⊥ 0.1879(9) 17.8(2) 1.14

R=

10a QQ 0.193(2) 19.6(3) 0.003

(QQ)Q‖ 0.196(2) 17.6(4) 0.57(QQ)Q⊥ 0.196(1) 18.8(2) 1.14

TABLE V: The mean-square width W 2 and amplitude ofthe in-plane and perpendicular action-density measured atthe middle plane R/2 using fits to a double-Gaussian ansatzEq. (65) at temperature T/Tc = 0.8 and nsw = 60.

of the width at the first four transverse planes x = 1to x = 4 as the third color source Q3 is pulled apart toseparation R (see Figs. 16, 17, 19 and 20).

The Y-string implies in-plane and perpendicular-planemean-square width for the junction fluctuations given byEq. (39) and Eq. (40), respectively. Since the junctionsfluctuations are nonlocal, the fit behavior ought to be

examined at each selected transverse plane to the tubesmeasured widths.

At the highest temperature T/Tc = 0.9, TableVI andTableVII summarize the returned values of χ2(x) fromresultant fits to the two width components (in-planeand perpendicular-Table IV) for the indicated separa-tion range R ∈ [a, b] at four consecutive transverse planesx = 1 to x = 4.

In general, the fits in TableVI and TableVII showstrong dependency on the fit range with high values ofχ2(x) when including points at small Q3 source separa-tions or long fit intervals. However, the values of χ2(x)rapidly decrease when excluding those points at shortdistance separations.

Nevertheless, It is apparent that best fits are for planesone to two lattice spacings from the base, that is x = 1and x = 2. In Ref. [42], where we have carried out thesimilar analysis however with large base length, it hasbeen observed that at planes closer to the junction’s meanposition the corresponding χ2 is minimized. That is someplanes may exhibit more contributions of the junctionsfluctuations.

Figure 21 depicts the change in the residuals at thefirst four planes from the diquark with minimal at theplanes x = 1 and x = 2 which are near Fermat pointof the configuration. Generally speaking, the existenceof particular planes at which the above indicated bestmatches with the formulas of Eq. (39) and Eq. (40) sug-gests the junction effects manifest at the highest temper-ature T/Tc = 0.9.

At farther planes from the diquark the effects of thejunction diminishes gradually. One may expect, accord-ingly, that with the increase of the base length results inreduction of the residuals which can be by examining thevalues in Tables VI and VII. Figure 21 plots χ2(x) forselected fit regions at planes x = 1 to x = 4.

As mentioned above, the plane at which we obtain theminimal in χ2(x) is sensitive to the length of the baseof the triangular isosceles quark configuration. In thecase of the small base the two strings of the Y shape aremore close in space and self-interactions can cause largerdeviations to be observed. This is also indication that thejunction effects are more pronounced when the systemmove towards the ordinary baryonic configuration.

Table VIII summarizes the returned χ2(x) from thefits to the mesonic string mean-square width given byEq. (26) to the in-plane fluctuations W 2

y (x) of thediquark-quark (QQ)Q enlisted in Table IV. The fits re-turn higher values of the residuals χ2(x) compared tothe corresponding fits for baryonic string Tables.VI, Ta-ble. VII. This can readily be seen by inspecting Fig. 22which shows significantly higher values of χ2 at the firstfour considered planes.

At the same temperature T/Tc = 0.9 We plot both fitsthe baryonic string Eq. (40) and mesonic string Eq. (26)in Fig.23. The fit interval corresponds to R ∈ [6, 11].Remarkably, the lattice data for the diquark configura-tion A = 2a poorly matches the QQ mesonic string. This

22

6

8

10

12

14

16

18

20

5 6 7 8 9 10 11

W2 a

-2

R a-1

T/Tc=0.9, A=2aBaryon-StringMeson-StringLGT, ID:X=1

(a)Plane x = 1

6

8

10

12

14

16

18

20

5 6 7 8 9 10 11

W2 a

-2

R a-1

T/Tc=0.9, A=2aBaryon-StringMeson-StringLGT, ID:X=2

(b)Plane x = 2

FIG. 23: The data correspond to the mean-square width of the action density in the 3Q plane at T/Tc = 0.9 at plane x = 1from the diquark, the lines are the fit of baryonic string Eq. (40) for the in-plane fluctuation of the junction and mesonic stringEq. (26).

TABLE VI: The returned values of the χ2dof(x) corresponding to fits of the in-plane width W 2

y (x) of the action density at eachplane x to the string model formula Eq.(40), the fits are for isosceles triangle quark configuration of base A = 2 a and A = 4 aat T/Tc = 0.9 after nsw = 40 sweeps.

(a)A = 2aFit range 4-8 4-9 4-10 4-12 5-9 5-10 5-12 6-9 6-10 6-12 7-12 8-12χ2(1) 15.7467 27.0586 39.7006 76.2242 5.1557 8.18385 21.888 0.694732 1.73399 9.67608 4.48314 2.1474χ2(2) 36.6744 58.1804 78.4037 116.516 10.6706 16.1961 29.815 1.28484 2.48185 7.21768 2.07623 0.781562χ2(3) 54.4166 88.878 121.68 172.52 17.6709 27.2391 44.2826 2.41245 4.59691 9.50593 1.80113 0.3457χ2(4) 15.16 31.21 49.12 80.01 4.56 9.10 18.51 1.24

(b)A = 4aFit range 4-8 4-9 4-10 4-12 5-9 5-10 5-12 6-9 6-10 6-12 7-12 8-12χ2(1) 16.728 22.3562 28.3737 47.6369 19.3351 22.4557 33.4371 0.216339 0.237681 2.60272 2.44289 1.23397χ2(2) 11.9139 18.9942 26.0145 42.4123 3.49132 5.41441 11.7304 0.398579 0.819422 3.296 1.20461 0.541966χ2(3) 19.2606 31.8565 43.9118 68.6434 6.7706 10.3598 19.3688 0.897718 1.68898 4.59129 1.12771 0.360786χ2(4) 6.21 12.85 19.99 38.56 1.94 3.72 9.82 0.46

TABLE VII: Same as Table VI; however the values of the χ2dof are returned from the fits of formula Eq. (39) to the perpendicular

width of the action density W 2(x) and nsw = 40 sweeps.

(a)A = 2aFit range 4-8 4-9 4-10 4-12 5-9 5-10 5-12 6-9 6-10 6-12 7-12 8-12χ2(1) 43.6488 70.643 95.8211 139.296 14.0156 21.01 35.5021 1.76644 3.20865 7.46718 1.55846 0.451334χ2(2) 52.935 89.9815 125.779 182.096 19.3283 29.9856 48.5269 2.85559 5.28193 10.3058 1.68293 0.254988χ2(3) 65.7849 113.944 163.981 244.385 24.7828 40.5754 68.2562 3.94029 7.97634 15.9491 2.90368 0.363801χ2(4) 16.59 35.86 59.61 105.76 5.62 12.0726.94 11.88 1.56

(b)A = 4aFit range 4-8 4-9 4-10 4-12 5-9 5-10 5-12 6-9 6-10 6-12 7-12 8-12χ2(1) 17.0601 27.9284 38.6724 59.6416 5.46255 8.43977 15.7773 0.670376 1.3058 3.72269 0.997622 0.352228χ2(2) 20.45 34.4918 48.0338 70.617 7.53652 11.5326 19.0597 1.02559 1.8569 3.85362 0.650788 0.134616χ2(3) 25.7955 44.7738 63.8724 94.1709 10.0837 15.9441 25.8421 1.51211 2.86368 5.38821 0.806941 0.083399χ2(4) 6.82 14.71 23.92 40.32 2.32 4.71 9.50 0.65

23

5

6

7

8

9

10

11

12

13

14

4 5 6 7 8 9 10 11 12

W2 a

-2

R a-1

T/Tc=0.9, A=2aBaryon-StringMeson-StringLGT, ID:X=1

(a)Plane x = 1

5

6

7

8

9

10

11

12

13

14

4 5 6 7 8 9 10 11 12

W2 a

-2

R a-1

T/Tc=0.9, A=2aBaryon-StringMeson-StringLGT, ID:X=2

(b)Plane x = 2

FIG. 24: Same as Fig. 23; however, the data and fits of mean-square width are taken at temperature T/Tc = 0.8.

TABLE VIII: The returned values of the χ2(x) correspondingto fits of the in-plane width W 2

y (x) to the mesonic string for-mula given by Eq. (26) at the corresponding plane x, the fitsare for isosceles triangle quark configuration of base A = 2 aand A = 4 a at T/Tc = 0.9.

(a)A = 2aFit range 4-9 5-10 4-12 5-12 6-12 7-12 8-12χ2(1) 29.9 46.6 99.6 99.4 91.3 57.3 32.1χ2(2) 65.0 91.7 156.7 155.5 120.3 69.6 33.5χ2(3) 93.3 117.4 203.0 193.5 156.0 91.2 41.9χ2(4) —- 122.38 —- 215.3 195.3 123.9 59.9

(b)A = 4aFit range 4-9 5-10 4-12 5-12 6-12 7-12 8-12χ2(1) 23.1 30.8 58.5 56.8 56.8 21.9 13.1χ2(2) 26.2 38.4 69.26 66.7 47.6 27.0 13.5χ2(3) 33.6 50.7 88.40 88.3 67.3 38.9 18.6χ2(4) —- 52.4 —- 105.6 92.8 59.6 30.2

rationalize that the junction effects may survive this tem-perature.

On the other-hand, at the lower temperature T/Tc =0.8 we enlisted in Table IX the returned χ2(x) from thefits to the in-plane fluctuations W 2

y (x) of the diquark-quark (QQ)Q. The two panels in the table compares the

fits for both the mesonic and baryonic strings given byEq. (26) and Eq. (40), respectively.

The fits return good values of χ2(x) at the first fourconsidered planes compared to the corresponding fits forbaryonic string Tables.VI. This is reflected in the plots ofFig. 24 which compare the resultant fits of both strings attemperature T/Tc = 0.8. The diminish of the deviations(Fig. 23) of the lattice data from the mesonic string’swidth profile is palpable.

The Y-string model at the depicted planes x = 1 andx = 2 the in-plane fluctuations of the diquark-quarkflux-tube return good χ2(x) base lengths A = 2a (seeFig. 24). The diquark-quark and QQ similarity manifestobviously. This points to the subtle effects of thejunction as the Y-string approaches the limit of itsmesonic counterpart at this temperature.

In summary the analysis of the mean-square width ofthe energy profile indicates the following points; we de-tect diquark-quark (QQ)Q baryon equivalent to quark-antiquark QQ system at temperatures up to the end ofthe QCD plateau. However, with the decrease of thelattice time slices Nt = 8 at β = 6.0 which brings thetemperature to T/Tc = 0.9, the diquark-quark (QQ)Qbaryon exhibits baryonic-like aspects consistent with theGoldstone modes of Y-bosonic string. The correspondingprofile QQ quark-antiquark system for source sepationsup to R ≤ 0.8 fm.

The gluonic meson limit of diquark-quark baryon withthe considered lattice parameters and diquark diame-ters [144, 160] can be a good approximation for low-lyingstates. The propagation in the Euclidean time of Wilsonloop in addition to the APE link-blocking in [144] hasbeen sufficient for the excited decuplet baryonic statesto decay yielding an optimal overlap with the mesonicground state, which is evidently not the case at very hightemperatures.

Otherwise, to draw a conclusive remarks on the rough-ness of the approximation at extreme conditions such assmall neighborhoods of TC or higher states of excitedspectrum [93, 164, 165], a careful analysis consideringsmaller lattice spacings, even higher temperatures andstrong magnetic fields [166–169] ought to be addressed.

24

TABLE IX: The returned values of the χ2dof(x) corresponding to fits of the in-plane width W 2

y (x) of the action density at eachplane x to the string model formula Eq. (40), the fits are for isosceles triangle quark configuration of base A = 2 a and A = 4 aat T/Tc = 0.8 , nsw = 60 sweeps.

Baryon(a)A = 2a (b)A = 4aFit range 4-8 4-9 4-10 5-10 4-12 5-12 6-12 4-8 4-9 4-10 5-9 4-12 5-12 6-12χ2(1) 3.10 1.60 1.61 6.98 13.90 7.00 1.93 0.83 1.02 1.10 1.02 1.18 1.18 0.51χ2(2) 4.12 4.89 5.20 0.39 5.65 0.45 7.17 1.60 2.04 2.29 1.86 2.93 2.82 1.73χ2(3) 9.51 11.33 12.5 1.92 14.43 2.61 0.40 2.81 3.31 3.63 2.35 4.31 3.48 1.84χ2(4) — — — 2.10 — 3.09 0.71 — — — 1.37 1.41 0.25

Meson(a)A = 2a (b)A = 4aFit range 4-8 5-8 6-9 4-12 5-12 6-12 7-12 4-8 5-8 6-9 4-12 5-12 6-12 7-12χ2(1) 2.1 1.96 0.83 2.33 2.20 0.87 0.19 0.68 0.75 0.55 0.79 0.58 0.30 0.30χ2(2) 58.26 1.14 0.22 58.76 1.22 6.15 1.15 202.31 16.7 1.98 218.47 20.05 2.46 0.40χ2(3) 1215.1 97.31 6.98 1249.5 103.93 16.99 7.40 2520.96 243.0 31.7 2679.5 295.1 39.17 7.13χ2(4) — 640.11 84.68 — 698.68 87.03 8.85 — 737.8 144.1 — 827.7 162.95 14.53

VI. CONCLUSION AND PROSPECT

In this work, we inspect the symmetry between thegluon flux-tubes for the quark-antiquark (QQ) and threequark systems at finite temperature. We approximatethe baryonic quark-diquark (QQ)Q configuration by con-structing the two quark at small separation distance ofat least 0.2 fm.

We measure the potential and the energy-density pro-file of the diquark-quark (QQ)Q and quark-antiquarkQQsystem. The Polyakov loop and the action-density cor-relator return identical values at temperatures up to theend of the QCD plateau. However, these symmetries donot manifest with the increase of the temperature closeto the deconfinement point. The splitting between thebosonic and fermionic arrangements in the gluonic char-acteristics shows up to quark source separations R ≤ 0.8fm.

The baryonic string model approaches the free mesonicstring for the fits of the numerical data of the (QQ)Qpotential with both returning the same value of the stringtension. In the neighborhood of the critical point T/Tc =0.9, the fit of the potential from Y-baryonic string modelreturns good χ2, however, with a string tension valueσ0a

2 that is significantly different from the corresponding(QQ) arrangement. In addition we detect no significantdifference in the fit behavior of the diquark-quark whena ∆-ansatz for the potential is employed.

Similarly, the analysis of the mean-square width ofthe energy profile indicates baryonic-like aspects consis-tent with the Goldstone modes of Y-bosonic string atT/Tc = 0.9 at all considered transverse planes. Themesonic string profile displays large deviations from thediquark-quark data at planes close to the diquark sys-tem. At the lower temperature the diquark-quark (QQ)Qbaryon exhibits a broadening profile consistent with both

the mesonic and the Y-string model with the same stringtension as the quark-antiquark (QQ) string.

These findings limit the validity that, in the quenchedapproximation, the diquark-quark (QQ)Q precisely sharemany properties in common with the quark-antiquark(QQ) only to the plateau region of QCD [144, 160]; oth-erwise, excited baryonic states seem to manifest aroundsmall neighborhoods of the QCD critical point.

It would be interesting to reiterate the presented calcu-lations with the employment of finer lattices consideringhigher temperatures or dynamical quarks. One wouldquestion the meson-baryon symmetries for the excitedspectrum and in the presence of strong magnetic fields.These analysis is likely to be of a considerable interestto phenomenological models of hadron structure and willbe discussed in a future work.

Acknowledgments

The authors thank the Yukawa Institute for The-oretical Physics, Kyoto University. Discussions dur-ing the YITP workshop YITP-T-14-03 on Hadronsand Hadron Interactions in QCD” were useful in com-pleting this work. We thank Philippe de Forcrandfor suggesting the investigation of the mesonic aspectsof the (3Q) systems. This work has been fundedby the Chinese Academy of Sciences President’s In-ternational Fellowship Initiative grants No.2015PM062and No.2016PM043, the Polish National Science Cen-ter (NCN) grant 2016/23/B/ST2/00692, the Recruit-ment Program of Foreign Experts, NSFC grants(Nos. 11035006, 11175215, 11175220) and the HundredTalent Program of the Chinese Academy of Sciences(Y101020BR0).

25

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VII. APPENDIX A

29

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.8

(a)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.8

(b)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.8

(c)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.8

(d)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.8

(e)yx-plane,A = 0.4 fmy)(

3QC

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.8

(f)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.8

(g)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.8

(h)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.8

(i)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.8

(j)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.8

(k)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.8

(l)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.8

(m)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.8

(n)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.8

(o)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.8

(p)zx-plane,A = 0.4 fm

(y)

2QC

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.8

(q)QQ-system: R = 0.6 fm

(y)

2QC

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.8

(r)QQ-system: R = 0.9 fm

(y)

2QC

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.8

(s)QQ-system: R = 1.2 fm

(y)

2QC

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.8

(t)QQ-system: R = 1.4 fm

FIG. 25: Flux tubes in 2D-plane for source separations R = 0.6, 0.9, 1.2, 1.4 fm (from top to bottom) and temperatureT/Tc ≈ 0.8. Panels on the top correspond to the quark-diquark system (baryon), panels on the very bottom – to the quark-antiquark system (meson). For the baryon system the flux tube is shown in two projection planes yx(a–h) and zx(i–p), andfor the two values of the distance between Q1 and Q2 quarks A = 0.2, 0.4 fm, see Fig.4 for details.

30

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.9

(a)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.9

(b)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.9

(c)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.9

(d)yx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.9

(e)yx-plane,A = 0.4 fmy)(

3QC

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.9

(f)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.9

(g)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance y-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.9

(h)yx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.9

(i)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.9

(j)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.9

(k)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.9

(l)zx-plane,A = 0.2 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.9

(m)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.9

(n)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.9

(o)zx-plane,A = 0.4 fm

y)(3Q

C

0.2−

0.1−

0

0.1

0.2

0.3

0.4

]-1QQ distance z-axis, [Ra8 10 12 14 16 18 20 22 24 26 28

]-1

Q-Q

Q d

ista

nce

x-ax

is, [

Ra

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.9

(p)zx-plane,A = 0.4 fm

(y)

2QC

0

0.05

0.1

0.15

0.2

0.25

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 Lattice data

R=6=6.0β

T/Tc=0.9

(q)QQ-system: R = 0.6 fm

(y)

2QC

0

0.05

0.1

0.15

0.2

0.25

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

1

2

3

4

5

6

7

8 Lattice data

R=9=6.0β

T/Tc=0.9

(r)QQ-system: R = 0.9 fm

(y)

2QC

0

0.05

0.1

0.15

0.2

0.25

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

2

4

6

8

10

Lattice data

R=12=6.0β

T/Tc=0.9

(s)QQ-system: R = 1.2 fm

(y)

2QC

0

0.05

0.1

0.15

0.2

0.25

y-axis8 10 12 14 16 18 20 22 24 26 28

z-ax

is

0

2

4

6

8

10

12Lattice data

R=14=6.0β

T/Tc=0.9

(t)QQ-system: R = 1.4 fm

FIG. 26: Flux tubes in 2D-plane for source separations R = 0.6, 0.9, 1.2, 1.4 fm (from top to bottom) and temperatureT/Tc ≈ 0.9. Panels on the top correspond to the quark-diquark system (baryon), panels on the very bottom – to the quark-antiquark system (meson). For the baryon system the flux tube is shown in two projection planes yx(a–h) and zx(i–p), andfor the two values of the distance between Q1 and Q2 quarks A = 0.2, 0.4 fm, see Fig.4 for details.

31

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=5

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=5

(a)yx-plane,A = 0.2 fm

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=9

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=9

(b)yx-plane,A = 0.4 fm

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=12

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=12

(c)yx-plane,A = 0.4 fm

FIG. 27: Surface plot of the Flux density in the quark-plane for source separations R = 0.5, 0.9, 1.2 fm (from top to bottom)and temperature T/Tc ≈ 0.9. The flux density is shown in the projection plane yx for the distance between Q1 and Q2 quarksA = 0.2 fm, see Fig.4 for the geometrical configuration.

32

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=5

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=5

(a)yx-plane,A = 0.2 fm

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=9

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=9

(b)yx-plane,A = 0.4 fm

]-1

QQ distance y-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-YX

R=12

=6.0β

T/Tc=0.8

]-1

QQ distance z-axis, [Ra

05

1015

2025

3035

]-1

Q-Q

Q distance x-axis, [R

a

0

2

4

6

8

10

12

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y)(3Q

C

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Action Density in plane-ZX

R=12

(c)yx-plane,A = 0.4 fm

FIG. 28: Surface plot of the Flux density in the quark-plane for source separations R = 0.5, 0.9, 1.2 fm (from top to bottom)and temperature T/Tc ≈ 0.9. The flux density is shown in the projection plane yx for the distance between Q1 and Q2 quarksA = 0.4 fm, see Fig.4 for the geometrical configuration.