51

沼津工業高等専門学校研究報告
第 52 号，2018 年 1 月

Lattice QCD study of the strong force among the three quarks
Yoshiaki KOMA∗
We present results of the static three-quark potential in quantum chromodynamics obtained byusing lattice Monte-Carlo simulations. We investigate the three-quark potential of O(200) sets ofthe three-quark geometries including not only the cases that three quarks are put at the vertices ofacute, right, and obtuse triangles, but also the extreme cases such that three quarks are put in line.
We ﬁnd a clear evidence that the string tension of the three-quark potential is the same as that of
the quark-antiquark potential regardless of the geometry of the three quarks.

Key Words: Lattice QCD, Inter-quark potential, Strong force, Baryon
INTRODUCTION

three quarks. So far, the three-quark potential has

Although hadrons such as mesons and baryons

been computed by several groups [5–12] by evaluat-

are known to be composed of quarks, the underlying

ing the expectation value of the three-quark Wilson

structure is not yet completely understood. This isloop. However, the use of the Wilson loop may cause

due to diﬃculty of solving quantum chromodynam-a systematic eﬀect due to the presence of the junction

ics (QCD), a quantum gauge ﬁeld theory with SU(3)of spatial Wilson lines, especially when the temporalgroup symmetry, at the energy scale of hadrons whereextent of the Wilson loop is not large enough [13, 14].

QCD becomes nonperturbative. A possible approach

It is particularly worth noting that our results are freeis then to deﬁne QCD in four dimensional Euclideanfrom such a systematic eﬀect.

I.

space on a hypercubic discrete lattice and compute theQCD partition function by the Monte-Carlo method

II.

NUMERICAL PROCEDURES(using supercomputers), which is referred to as lattice
QCD simulations. This method allows us to compute

We consider QCD within the quenched approx-

expectation values of various physical quantities di-imation (corresponding to pure SU(3) lattice gauge

rectly without relying on any perturbative techniques.

theory) in four dimensions with the lattice volume

In this report, we present our recent results of theL3 × T and the lattice spacing a, and impose pe-lattice QCD simulations on the static three-quark po-

riodic boundary conditions in all space-time direc-

tential [1, 2], which is relevant to the understanding oftions. By preparing the three-link correlators, which

the baryon structure. Our method consists of comput-

is a tensor product of the time-like link variables

ing the expectation values of the correlation function

U4 (x) placed at x = (x0 , �x1 ), (x0 , �x2 ) and (x0 , �x3 ) forof three Polyakov loops (PLCF) at zero temperaturex0 = 0, a, ..., T − a,accurately by using the noise reduction method called
the multilevel algorithm [3, 4], and of extracting thepotential from them. We obtain the three-quark potential of O(200) sets of the three-quark geometries
including not only the cases that three quarks are put

T(x0 , �x1,2,3 )αβγδ�ζ≡ U4 (x0 , �x1 )αβ U4 (x0 , �x2 )γδ U4 (x0 , �x3 )�ζ ,(1)where Greek indices take the values from 1 to 3 inSU(3), we construct the three-quark PLCF asat the vertices of acute, right, and obtuse triangles,TrP (�x1 )TrP (�x2 )TrP (�x3 )but also the extreme cases such that three quarks= {T(0, �x1,2,3 ) · · · T(T − a, �x1,2,3 )}ααγγ�� . (2)are put in line. From the derivative of the potential with respect to distances among the three quarks,
we ﬁnd a clear evidence that the string tension of thethree-quark potential is the same as that of the quarkantiquark potential regardless of the geometry of theThe ground state potential can be extracted from theexpectation value of the PLCF with a large T as1(3)V3q = − ln�TrP (�x1 )TrP (�x2 )TrP (�x3 )� ,T
where we employ the multilevel algorithm [3, 4]. Notethat this algorithm allows us to compute the expecta-∗ Division of Liberal Arts / Physics grouption value accurately even from one gauge conﬁguration [13].

52

沼津工業高等専門学校研究報告

�x3 

�x3 �x2

 

 �x1



第 52 号(i) acute

�x3

�x2

 �x1�x3�x2 �x�x2 �x1(ii) right(iii) obtuse

�x3

 �x2

1


 �x

(iv) line(v) quark-diquark

1FIG. 1: The three-quark geometries investigated in our numerical simulations. The circles represent the spatial location ofquarks. Three Polyakov loops are put at the vertices of (i) acute, (ii) right, (iii) obtuse triangles, and are also put in (iv)line, and are put to be (v) the quark-diquark system.

The spatial locations of the Polyakov loops, �x1 ,the determination of the functional form of the po-�x2 , and �x3 correspond to those of three quarks in thetential is nothing but the ﬁnding of appropriate dis-

three-dimensional space, respectively. There are ﬁvetances that can capture the systematic behavior of thetypes of three-quark geometries: three quarks are put

potential. Such distances should be symmetric underat the vertices of acute (ACT), right (RGT), obtuse

the permutation of the quark positions.

(OBT) triangles, and are put in line (LIN) as shown inThe simplest distance is then given by the sumFig. 1. As a special case, two of three quarks are putof interquark distances in Eq. (5),at the same location, which corresponds to a quark-Δ = r 1 + r2 + r3 .

diquark system (QDQ). These three-quark geometries(6)

can be classiﬁed by the value of the maximum inner

Another possible distance is given by the minimal to-angle of a triangle

tal length of lines connecting the three quarks via theFermat-Torricelli point of a triangle,√
r12 + r22 + r32 + 4 3S
Y =,2θmax = max(θ1 , θ2 , θ3 )2)rmax (r12 + r22 + r32 − 2rmax, (4)= cos−1
2r1 r2 r3

where S is the area of the triangle given by Heron’s

where

r1 = |�x2 −�x3 |,r2 = |�x3 −�x1 |,(7)

formula,r3 = |�x1 −�x2 | , (5)Acute triangles satisfy θmax < 90◦ , which contain1
Δ(Δ − 2r1 )(Δ − 2r2 )(Δ − 2r3 ) .
(8)
4Note that the distance between the Fermat-Torricelli

equilateral and isosceles triangles in our study. Right

point and each vertex is

S=are interquark distances and rmax = max(r1 , r2 , r3 ).

triangles are the case θmax = 90 .
◦Obtuse trian-li = Y −gles are further classiﬁed into two types depending

1 2 4S(r + √ ) .
Y i3(9)on θmax , obtuse-narrow (OBTN) triangles for 90◦ <Two distances Δ and Y were often used to examineθmax < 120 and obtuse-wide (OBTW) triangles forthe behavior of the potential in the earlier studies.

120  θmax < 180 .

We also follow them in our analyses of the three-quark◦◦◦

In contrast to the classiﬁcation of the three-quark

potential. In terms of the minimal length of connected

geometries, the parametrization of the three-quark po-lines, Y is reduced totential is not straightforward. This is due to the factΛ = Δ − rmax

that the potential can depend not only on the loca-(10)when θmax  120◦ . It is then convenient to introducetion of three quarks, �x1 , �x2 , and �x3 , but also on thea combined distance of Y and Λ classiﬁed by θmax asY(θmax < 120◦ ).
(11)
Lstr =Λ(θmax  120◦ )structure of the ﬂux tube spanned among the threequarks, which is unknown a priori because of the nonperturbative feature of the QCD vacuum. Therefore,2

Lattice QCD study of the strong force among the three quarks

III.

53

NUMERICAL RESULTS

We carried out the lattice QCD simulations using the standard Wilson gauge action in SU(3) latticegauge theory at β = 6.00 on the 244 lattice. The
lattice spacing is a = 0.093 [fm], determined by theSommer scale r0 = 0.50 [fm] [15]. One Monte Carlo
update consisted of 1 heatbath and 5 over-relaxation
steps. The parameters for the multilevel algorithm are
Nsub = 6 (the number of sublattices), which corresponds to Ntsl = L/(aNsub ) = 4, and Niupd = 500000(the number of internal updates).
Firstly, we focus on the potential of the isosceles triangle geometries within ACT, where the twoof three quarks are placed at �x1 = (x, 0, 0) and�x2 = (0, x, 0), and the remaining third quark is placed
at �x3 = (0, 0, z) with z  x. In this case, Lstr isidentical to Y . The distance between the FermatTorricelli point and �x1 and �x2 , respectively, is the√same l1 = l2 = ( 6/3)x. Therefore, pulling thethird quark (changing z) with the ﬁxed ﬁrst and second quarks does not aﬀect the location of the FermatTorricelli point, which just aﬀects the increase of theenergy between the Fermat-Torricelli point and the√third quark, where l3 = z 2 + x2 /2 − x/ 6. We thencompute the derivative of the potential with respect
to Y for several ﬁxed values of x,V3q (x, z + δz) − V3q (x, z)
.
(12)
δY
In Fig. 2 (upper), we plot the result for one gauge=V3q

conﬁguration at β = 6.00 with the classiﬁcation interms of the distance between the ﬁrst and second√quarks, rmin = 2 x, where x/a = 1, 2, and 3 (inFIG. 2: The derivatives of the three-quark potential withrespect to Lstr for acute (upper), right (middle), and line(lower) geometries. The dotted line in each plot corresponds to the string tension of the quark-antiquark potential σqq̄ a2 = 0.0449 [1].

this case, δY = δl3 ). We ﬁnd that all the derivatives
behave quite similarly and approach a constant value
at long distance. Remarkably, the constant value isnothing but the string tension in the quark-antiquarksystem, σqq̄ a2 = 0.0449. Since the third quark is cho-

the remaining ﬁrst quark is placed at �x1 = (x, 0, 0)

sen arbitrarily among the three, this result also sup-with x  y. In this case, although the location of theports a picture of the Y -shaped ﬂux-tube formation.

Fermat-Torricelli point is slightly dependent on changing x, it becomes insensitive to x when x � y. The

This feature agrees with that was pointed out by Taka-

derivative is then deﬁned byhashi et al. [9, 10] based on the χ ﬁt to the potential
2data with the Y Ansatz.

=V3q

Secondly, we pay attention to the potentials of
RGT, where �x2 = (0, y, 0) and �x3 = (0, 0, 0), andV3q (x + δx, y) − V3q (x, y).
δY(13)In Fig. 2 (middle), we plot the result for the same one354

沼津工業高等専門学校研究報告第 52 号

again ﬁnd that all the derivatives approach the con-0.20
quark-diquark (quark-antiquark (0.15)

stant value, σqq̄ a2 = 0.0449, at long distance.

)

Finally, we look at the quark-diquark system. In

2a V'Fig. 3, we plot the derivatives of the quark-diquark
and quark-antiquark potentials with respect to the in-0.10

terquark distance r. In this case, we ﬁnd that bothdata overlap with each other (including ﬁnite volume0.05

eﬀects at large r), which means that the string tension
0.00
02468

1012

is exactly the same.

14IV.

r/aFIG. 3: The derivatives of the quark-diquark and quarkantiquark potentials with respect to r. The horizontal
dotted line corresponds to the string tension of the quarkantiquark potential σqq̄ a2 = 0.0449 [1].

By using the lattice QCD simulations, we havecomputed the static three-quark potential of O(200)sets of the three-quark geometries including not only
the cases that three quarks are put at the vertices ofgauge conﬁguration with the classiﬁcation in terms

acute, right, and obtuse triangles, but also the extremeof the distance between the second and third quarks,

cases such that three quarks are put in line. We havermin = y, where y/a = 1 ∼ 4. We ﬁnd that allused the PLCF as the three-quark source, which has

the derivatives approach the constant value, σqq̄ a2 =been quite important to obtain the result with less0.0449, at long distance: the tendency is quite thesystematic eﬀect. From the analysis of the derivative

same as that for ACT.

of the potential with respect to distances among theThirdly, we examine the potentials of LIN, wherethree quarks, we have found a clear evidence that the�x1 = (x1 , 0, 0), �x2 = (x2 , 0, 0), and �x3 = (0, 0, 0),string tension of the three-quark potential is the same

which is an extreme case that there is probably noas that of the quark-antiquark potential regardless ofchance to form a junction of the ﬂux tube. For a ﬁxed

the geometry of the three quarks.

distance rmin = x2 (0 < x2 < x1 /2), the derivative is

The author is grateful to M. Koma for fruit-

then deﬁned by
=V3q

V3q (x1 + δx1 , x2 ) − V3q (x1 , x2 ).
δx1

SUMMARY

ful collaboration. The lattice QCD simulations havebeen performed on the NEC SX8 at Research Center(14)

for Nuclear Physics (RCNP), Osaka University. ThisIn Fig. 2 (lower), we plot the result for the same one

work is partially supported by JSPS KAKENHI Grant

gauge conﬁguration as a function of Lstr = x1 . We

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