-string three-quark potential

Smooth crossover transition from the�-string to the Y-string three-quark potential

V. Dmitrasinovic

Vinca Institute of Nuclear Sciences, lab 010, P.O. Box 522, 11001 Beograd, Serbia

Toru Sato

Department of Physics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan

Milovan Suvakov

Institute of Physics, Pregrevica 118, Zemun, P.O. Box 57, 11080 Beograd, Serbia(Received 24 June 2009; published 3 September 2009)

We comment on the assertion made by Caselle et al. [M. Caselle, G. Delfino, P. Grinza, O. Jahn, and

N. Magnoli, J. Stat. Mech. (2006) P008.] that the confining (string) potential for three quarks ‘‘makes a

smooth crossover transition from the �-string to the Y-string configuration at interquark distances of

around 0.8 fm’’. We study the functional dependence of the three-quark confining potentials due to a

Y-string, and the � string and show that they have different symmetries, which lead to different constants

of the motion (i.e. they belong to different ‘‘universality classes’’ in the parlance of the theory of phase

transitions). This means that there is no ‘‘smooth crossover’’ between the two, when their string tensions

are identical, except at the vanishing hyper-radius. We also comment on a certain two-body potential

approximation to the Y-string potential.

DOI: 10.1103/PhysRevD.80.054501 PACS numbers: 12.38.Gc, 11.15.Ha, 11.27.+d, 12.39.Pn

I. INTRODUCTION

The so-called Y-junction string three-quark potential,defined by

VY ¼ �minx0

X3i¼1

jxi � x0j (1)

has long been advertised [1,2] as the natural approximationto the flux tube confinement mechanism, that is allegedlyactive in QCD. Lattice investigations, Refs. [3,4], however,contradict each other in their attempts to distinguish be-tween the Y-string, Fig. 1, and the �-string potential, seeFig. 2,

V� ¼ �X3

i<j¼1

jxi � xjj; (2)

which, in turn, is indistinguishable from the sum of threelinear two-body potentials. One may therefore view thepresent lattice results as inconclusive and await the nextgeneration of lattice calculations [5]. Another point of viewheld among some lattice QCD practitioners [8] is that thereshould be a smooth crossover from the � to the Y-potentialat interquark distances of around 0.8 fm. This opinion isbased on certain similarities between the Potts model andlattice QCD which were made more precise in Ref. [9]. Itwas not stated in Ref. [9], however, how exactly this cross-over should be implemented, nor what they meant by‘‘interquark distances’’.

In the course of our studies of the (difference betweenthe) Y-string and the �-string potentials [10], we havetaken this assertion at face value and tried to devise a

‘‘smooth crossover’’, i.e. to make a smooth interpolationbetween these two potentials, that is as simple as possible.What we found is simple enough to state: there can be nosmooth crossover interpolation between these two poten-tials; there must be always a discontinuity in some variable(s). This fact may perhaps even be simply understood onthe basis of the different topologies of the two configura-tions, see the discussion below, but the proof which weshow (in some detail) is complicated by the (technical)requirements of the S3 permutation symmetry.We shall address the above questions one after another

and for this reason we divide the paper in four sections. InSec. II, we define the potentials that we use, in the secondSec. III, we show an analytic proof of incompatibility of �and Y-strings, and finally the third Sec. IV, addressesapproximations to the string potentials that can be usedto extract results from the lattice or to be used in the

x0

x3

Y

x0

3x-

||

x2

x1

2π3

FIG. 1 (color online). Three-quark Y-junction string potential.

PHYSICAL REVIEW D 80, 054501 (2009)

1550-7998=2009=80(5)=054501(10) 054501-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.054501

constituent quark model. The final Sec. V contains a sum-mary of our results and the discussion.

II. THREE-BODY POTENTIALS

Any reasonable static, spin-independent three-bodypotential must be: (1) translation-invariant, which meansthat it must depend only on the two linearly independentrelative coordinates, which we call ð�;�Þ, but not onthe center-of-mass coordinate; (2) rotation-invariant,which means that it may depend only on the three scalarproducts of the relative coordinates �2, �2, ð� � �Þ; and(3) permutation-invariant, which means that it may dependonly on certain combinations, yet to be determined, of theabove three scalar products of the relative coordinates.

We shall show that there are (precisely) three indepen-dent permutation-symmetric functions/variables of therelative coordinates, that are related to simple geometri-cal/physical properties (the moment of inertia, the area,and the perimeter of the triangle) which clearly distin-guishes them, and that the potential’s dependence on anyone of them, in particular, carries dynamical consequences.

Then we show that the (central, or three-body part of, forprecise definition see Sec. II A 2 below) Y-string potentialdepends on only two (the moment of inertia and the area ofthe triangle, but not on the perimeter) of these three vari-ables in most geometrical configurations; whereas the�-string potential depends only on the perimeter.

A. Derivation of the �- and Y-string potentials

1. Derivation of the �-string potentials

The �-string potential

V� ¼ �X3

i<j¼1

jxi � xjj ¼ 2�s; (3)

is proportional to the perimeter of the triangle 2s ¼ ðaþbþ cÞ, where a ¼ AB, b ¼ BC, and c ¼ CA are the threesides of the triangle and A, B, and C are positions of thequarks. When written in this form, the potential is mani-festly translation-, rotation-, and permutation-invariant.

2. Derivation of the Y-string potential

Three strings (‘‘flux tubes’’) merge at the point x0,which is chosen such that the sum of their lengths l3q ¼lY is minimized

lY ¼ minx0

X3i¼1

jxi � x0j: (4)

If all the angles in the triangle are less than 120�, then theequilibrium Y-junction position is the so-called Toricelli(or Fermat, or Steiner) point of classical geometry, I ¼ x0

in Fig. 1 that has the property that the straight linesemanating from the junction point x0 and leading to thequarks (‘‘strings’’) all form an angle 2�=3 at (see Fig. 1).The corresponding ‘‘three-string length lY is

lY ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2ðAB2 þ BC2 þ CA2Þ þ 2

ffiffiffi3

pArea4 ABC

sif ∡A;∡B;∡C � 120�; (5)

where a ¼ AB, b ¼ BC, and c ¼ CA are the three sides of the triangle. Here one can see that the ‘‘three-string’’ potentialEq. (5) depends on the ‘‘harmonic oscillator’’ variable (a2 þ b2 þ c2), which is permutation-symmetric and proportionalto the moment of inertia (divided by the quark mass) of this triangle [11], and on the triangle area

Area 4 ABC ¼ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðABþ BCþ CAÞð�ABþ BCþ CAÞðAB� BCþ CAÞðABþ BC� CAÞ

p¼ 4ða; b; cÞ

¼ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ða2b2 þ b2c2 þ b2c2Þ � a4 � b4 � c4

q; (6)

which is also permutation-symmetric.This form of the Y-string potential, when expressed in

terms of triangle sides ða; b; cÞ only, exhibits its permuta-tion symmetry S3, but hides the hidden/implicit angulardependence of the potential, and potential interdependen-cies on other permutation-symmetric variables: The tri-

angle area can be written using Heron’s formula

Area 4 ABC ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisðs� aÞðs� bÞðs� cÞp

; (7)

as a function of the triangle’s semiperimeter s ¼ 12 ðaþ

bþ cÞ and the three sides a, b, and c.

x3

∆

x2

x1

x-

||

x 1

2

FIG. 2 (color online). Three-quark �-shape string potential.

DMITRASINOVIC, SATO, AND SUVAKOV PHYSICAL REVIEW D 80, 054501 (2009)

054501-2

It should be intuitively clear, however, that the perimeterand the area of the triangle are two independent propertiesof the triangle. Moreover, all three variables have nonzerodimensionality, which seems to imply that there are threedifferent measures of ‘‘interquark distances’’. The tasknow becomes to find the most suitable independent coor-dinates/variables for an interpolation.

If one of the angles within the triangle equals or exceedsthe value 2�=3 ¼ 120�, however, the corresponding vertexof the triangle is the junction point (although the Toricellipoint can be constructed in that case, as well, but generallylies outside of the triangle and need not coincide with thevertex) and the minimal two-string length is

lV ¼ ðaþ cÞ if � ¼ ∡A � 120�; cos� ¼ 1

2acða2 þ c2 � b2Þ � � 1

2(8)

¼ ðaþ bÞ if � ¼ ∡B � 120�; cos� ¼ 1

2abða2 þ b2 � c2Þ � � 1

2(9)

¼ ðbþ cÞ if � ¼ ∡C � 120�; cos� ¼ 1

2bcðb2 þ c2 � a2Þ � � 1

2(10)

where a ¼ AB; b ¼ BC; c ¼ CA: (11)

Each one of these three expressions explicitly violates thepermutation symmetry (because of the ‘‘missing piece ofstring’’), but taken in totality they maintain it, in the sensethat no one vertex is different from the others when itsangle exceeds 2�=3 ¼ 120�.

B. String potentials in terms of relative coordinates

The three sides of a triangle are not be the most usefulcoordinates for practical calculations, however. Below, weshall express these potentials in terms of conventionalthree-body Jacobi relative coordinates �, �

� 12 ¼ 1ffiffiffi2

p ðx1 � x2Þ; (12)

� 12 ¼ 1ffiffiffi6

p ðx1 þ x2 � 2x3Þ; (13)

x CM ¼ 1

3ðx1 þ x2 þ x3Þ; (14)

which obscures the permutation symmetry, however (seeSec. III C). From now on, we always specialize to the pair(12) and drop the 12 index everywhere, so that the Jacobicoordinates in our problem will be denoted by � (instead of�12) and � (instead of �12).

1. String potentials in terms of hyper-sphericalcoordinates

We receive help here in the form of hyper-sphericalcoordinates: instead of the moduli � and � of the twoJacobi vectors �, �, shown in Fig. 3, the hyper-sphericalcoordinates introduce the hyper-radius, which ispermutation-symmetric:

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

q; (15)

as the only variable with dimension of length, the hyper-angle � through the polar transformation

� ¼ R sin�; � ¼ R cos� with 0 � � � �=2:

(16)

and the (physical) angle between � and �: cos ¼ � ��=ð��Þ. The boundary in the � vs plane between theregions in which the two- and the three-string potentials arevalid is determined by Eqs. (17). There are three suchboundaries, determined by the three (in)equalities, thatmerge continuously one into another at two ‘‘contactpoints’’ and one line, see Fig. 4.

26

x3

x1

2x

ρ2

λ

H

FIG. 3 (color online). Two three-body Jacobi coordinates �, �

that define the ‘‘hyper-radial’’ string lengthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij�j2 þ j�j2p

.

SMOOTH CROSSOVER TRANSITION FROM THE �- . . . PHYSICAL REVIEW D 80, 054501 (2009)

054501-3

cot�1ðÞ ¼ �1ffiffiffi3

pcosþ sin

;

cot�2ðÞ ¼ 1ffiffiffi3

pcos� sin

;

cot�3ðÞ ¼ 1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5� 2cos2� 2j sinj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� cos2

pq:

(17)

The two hyper-angles ð�; Þ describe the shape of thetriangle, so the (z ¼ cos2�, x ¼ cos) plane (square) maybe termed the ‘‘shape-space’’. Manifestly, for each angle there is another configuration with angle �� that de-scribes a ‘‘similar’’ triangle geometry that is a mirrorimage of the other. For this reason any three-body potentialdefined by geometric variables must be symmetric underreflections across the ¼ �=2 axis. In Figs. 5 and 6 weshow contour plots of the Y-string and the �-string poten-tials, respectively, in the ‘‘shape-space’’ plane, i.e. as func-tions of z ¼ cos2� (vertical axis) and x ¼ cos (horizontalaxis), where this symmetry is plain to behold. We see thatthe functional forms of these two potentials are different:even their symmetries, obvious to the naked eye, are differ-ent—one is symmetric only under reflections w.r.t verticalaxis, see Fig. 5, whereas the other is also symmetric underreflections w.r.t. the diagonals, as well as the vertical andhorizontal axes, see Fig. 6. Below we shall show that theY-string potential has a continuous O(2) dynamical sym-metry, that is the source of the ‘‘extra symmetry’’ visible inFig. 6.

Because of their manifestly different symmetries, thetwo potentials cannot coincide in the whole ðz; xÞ planethat describes the shapes of the triangles. The intersectionof the two potentials may, at best, yield a curve in this planeof admissible triangle configurations, which is but onereal continuum R out of a double real continuum R� R,

0 25 50 75 100 125 150 1750

10

20

30

40

50

60

70

o , o

FIG. 4. The boundary in the � vs plane, between the regionsin which the two- and the three-string potentials are appropriate,see Eqs. (17). The three-string potential holds in the centralregion, whereas the two-string potential holds in the ‘‘corners’’of this parallelogram.

FIG. 6. Contour plot of the Y-string potential as a function ofthe cosines of the two hyper-angles z ¼ cos2� (vertical axis) andx ¼ cos (horizontal axis) at any fixed value of the hyper-radiusR. The darker regions indicate a smaller value of the potential.Note, however, that this potential is not valid in the whole regionof ‘‘shape-space’’ depicted here: the ‘‘two-body’’ potential isvalid beyond the boundary shown in Fig. 11.

FIG. 5. Contour plot of the �-string potential as a function ofthe cosines of the two hyper-angles z ¼ cos2� (vertical axis) andx ¼ cos (horizontal axis) at any fixed value of the hyper-radiusR. The darker regions indicate a smaller value of the potential.The reflection symmetry about the x ¼ 0 axis should be visibleto the naked eye.


054501-4

which is measure-zero compared with the disallowedconfigurations.

A simple numerical exercise, see Sec. II C 1 below,shows however that even that much does not happen, i.e.there is absolutely no intersection of the two potentialswhen their string tensions �� ¼ �Y are equal, except atvanishing hyper-radius R ¼ 0, i.e. when the triangleshrinks to a point. That constitutes the proof of our con-tention that there is no smooth transition from the � to theY-string potential at nonzero R � 0 and equal string ten-sions �� ¼ �Y .

C. Difference of string potentials

1. Difference of string potentials as a function ofhyper-angles at equal string tensions �� ¼ �Y

The (normalized) difference of the � and the Y-stringpotentials (with �R factored out) is shown in Fig. 7 as afunction of the cosine of the hyper-angle cos2�, at thefixed values of cos ¼ 0, �1. Note that the difference isalways (substantially) larger than zero, i.e. that there are nozeros on the real line. The same holds as a function of cos.This shows that there is no continuous connection betweenthe � and the Y-string potentials at nonzero R � 0 andequal string tensions �� ¼ �Y .

One may try and change the string tensions �� Y , inwhich case there is an intersection of the two surfaces, butthat is not the problem that we started with. We shall returnto this new scenario below. Yet, even in that scenario therecan be a transition from one to another type of string onlyin a limited subspace of shapes.

2. Difference of string potentials with unequal stringtensions �� Y

One may readily circumvent the latter condition andthen find some small overlap of the two-string potentials,see Figs. 8 and 9. Yet, even then the overlap region is (only)a continuous curve in the ðx; zÞ plane, i.e. it covers anegligibly small (‘‘measure-zero’’) ‘‘number’’/set of al-

FIG. 7 (color online). The difference of the � and the Y-stringpotentials divided by the product of the string tension and thehyper-radius ðV� � VYÞ=ð�RÞ, as a function of the cosine of thehyper-angle z ¼ cos2�, at fixed values of x ¼ cos ¼ �1 (redlong dashes) and x ¼ cos ¼ 0 (blue short dashes). Note that thedifference is always (substantially) larger than unity, let alonezero.

FIG. 8 (color online). The difference of the � and the Y-stringpotentials divided by the product of the string tension and thehyper-radius ðV� � VYÞ=ð�RÞ, as a function of the cosine of thehyper-angle z ¼ cos2�, at two fixed values of x ¼ cos ¼ 0, 1and �Y=�� ¼ ffiffiffi

3p ¼ 1:732. Note that the difference vanishes at

only one point: x ¼ 0 ¼ z.

1 0.5 0 0.5 1

1

0.5

0

0.5

1

x

z

FIG. 9 (color online). The locus of vanishing difference of the� and the Y-string potentials (blue dashes) as a function of thecosine of the hyper-angle z ¼ cos2�, and x ¼ cos, at�Y=�� ¼ 1:96. Note that this whole curve lies outside the rangeof validity (continuous black line) of the three-body potential VY ,in which all of the angles are smaller than 120�.


054501-5

lowed triangular configurations (‘‘shapes’’) as comparedwith the extent of all possible such sets, described by thecomplete ‘‘shape-space’’ ðx; zÞ plane. Note, moreover, thatthis overlap lies within the ‘‘allowed region’’ of shape-space only in a limited range of values of the ratio

�Y=�� 2 ð ffiffiffi3

p; 1:86Þ ¼ ð1:732; 1:86Þ. For higher values

of this ratio the overlap curve lies partially, or completely(for �Y=�� > 1:96) outside the range of validity (continu-ous black line in Fig. 9) of the three-body potential VY , sothis solution is essentially irrelevant [12].

III. ANALYTIC PROOF OF INCOMPATIBILITY OF� AND Y-STRINGS

This was just a numerical proof which did not exposethe deeper underlying reasons for this incompatibility: the(in)dependence of these two strings on differentpermutation-symmetric variables. For an analytic proofof incompatibility we turn to the study of the permutationsymmetry in the three-body potential.

A. Permutation symmetry properties of the Jacobicoordinates and of the potential

The Jacobi relative coordinate vectors ð�; �Þ furnish atwo-dimensional irreducible representation of the S3 per-mutation symmetry group. For example, let Pij be the

(two-body) ij-th particle permutation operator (transposi-tion); then

P12� ! �� (18a)

P12� ! � (18b)

P13� ! 1

2��

ffiffiffi3

p2

� (18c)

P13� ! �ffiffiffi3

p2

�� 1

2�: (18d)

Here we see that the S3 permutation symmetry implies aninvariance (of the potential) under two rotations, throughinteger multiples of 2�

3 ¼ 120�, and three reflections in theð�;�Þ plane [13]. One special case of such a permutation isthe P12 transposition, which is a reflection that reverses thesign of cos ! � cos, which is equivalent to the alreadymentioned (geometrical) mirror symmetry.Starting from ð�;�Þ one can construct one symmetric V,

and two antisymmetric vectors: A, W.

V ¼ �ð�2 � �2Þ � 2�ð� � �Þ (19)

A ¼ �ð�2 � �2Þ þ 2�ð� � �Þ (20)

W ¼ �� : (21)

The ‘‘lengths’’ (norms) of vectors V ¼ jVj, A ¼ jAj,W ¼jWj are invariant under the quark permutations. The hyper-radius squared R2 is the fourth permutation-invariant sca-lar. Of course, we expect only three out of fourpermutation-invariant scalars to be (nonlinearly) indepen-dent. The nonlinear relationship reads

V2 þ A2 ¼ R2½R4 � 4W2�; (22)

so the third permutation-symmetric variable may be takenas (V2 � A2). Any (reasonable) confining three-body po-tential must be permutation-symmetric, so it must be afunction of (only) R, W and (V2 � A2). As all three ofthese variables have nonzero dimensions, one might thinkthat each one represents a potentially new definition of the‘‘interquark distance’’; that is not the case, as can be seenwhen one changes to hyper-spherical coordinates.

B. String potentials in terms of symmetrizedhyper-spherical coordinates

The three permutation-symmetric hyper-spherical vari-ables are R and

W ¼ 1

2R2 sin2� sin (23)

V2 � A2 ¼ R6 cos2�ðcos22�� 3ðsin2� cosÞ2Þ: (24)

When the interaction potential does not depend func-tionally on all three permutation-symmetric variables, thenadditional dynamical symmetries appear: e.g. the ‘‘cen-tral’’ part of the string potential Eq. (5),

1 0.5 0 0.5 1

1

0.5

0

0.5

1

x

z

FIG. 10 (color online). The locus of vanishing difference of the� and the Y-string potentials (blue dashes) as a function of thecosine of the hyper-angle z ¼ cos2�, and x ¼ cos, at�Y=�� ¼ 1:86. Note that this whole curve lies inside the rangeof validity (continuous black line) of the three-body potential, inwhich all of the angles are smaller than 120�.


054501-6

VY ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2R2ð1þ sin2� sinÞ

s¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2ðR2 þ 2WÞ

s; (25)

being a function of only W and R, i.e. not a function of(V2 � A2), is invariant under arbitrary rotations (not justthrough integer multiples of 2�

3 ¼ 120�) in the ð�;�Þplane. That leads to a new integral-of-motion G ¼� � p� � � � p�, associated with the dynamical symmetry

(Lie) groupOð2Þ, see Sec. below, which is not conserved inthe case of the � string potential. Further, when the poten-tial depends only on R, the dynamical symmetry is ex-tended to the Oð6Þ Lie group.

The �-string potential

V� ¼ �X3

i<j¼1

jxi � xjj

¼ �

� ffiffiffi2

p�þ

ffiffiffi3

2

s ��þ �ffiffiffi3

p��þ

ffiffiffi3

2

s �� ffiffiffi3

p��

(26)

can be expressed in terms of ðV;AÞ: just solve Eqs. (19)and (20) for ð�;�Þ as functions of ðV;AÞ and insert theresults

� ¼ ðVð�2 � �2Þ þ 2Að� � �ÞÞðR2 � ð2WÞ2Þ�2 (27)

� ¼ ðAð�2 � �2Þ � 2Vð� � �ÞÞðR2 � ð2WÞ2Þ�2 (28)

i.e.

� ¼ ðV cos2�þAðsin2� cosÞÞ�1�

�2W

R2

�2��2

(29)

� ¼ ðA cos2�� Vðsin2� cosÞÞ�1�

�2W

R2

�2��2

(30)

into Eq. (26) which shows that

V� ¼ V�ðR;W; ðV2 � A2ÞÞ (31)

is a function of all three permutation-symmetric three-bodyvariables, however. So, we see that in general the two-string potentials depend on different symmetric three-bodyvariables, and thus cannot be smoothly connected, exceptperhaps in special cases/geometries, where the variable(V2 � A2), and/or other variable(s) vanish.

C. Admissible region(s) for a ‘‘smooth crossover’’ oftwo strings

Thus, we need to solve

V2 � A2 ¼ R6 cos2�ðcos22�� 3ðsin2� cosÞ2Þ ¼ 0:

(32)

There are two ‘‘trivial’’ solutions: 1) R ¼ 0; 2) cos2� ¼ 0;

and a family of nontrivial solutions to 3) cos22��3ðsin2� cosÞ2 ¼ 0, i.e. cot2� ¼ � ffiffiffi

3p j cosj, which de-

termines the admissible ‘‘smooth crossover’’ region (theblue long-dashed curve) in Fig. 11. Thus, we can see thatout of the double-continuum (‘‘plane’’) of possible geo-metric configurations at a given ‘‘interquark distance’’ R, a‘‘smooth crossover’’ is potentially admissible, though notguaranteed, only on a single continuum (the blue dashedcurves and the vertical axis at x ¼ 0 in Fig. 11). Now, tomake this an actual ‘‘smooth crossover’’ region, rather thanmerely an admissible one, one must find the intersection ofthis curve and the locus of zeros of V� � VY , that wesearched for in Sec. II C 1, to no avail.This fact, together with our numerical results from

Sec. II C 1, are sufficient proof of our claim that even alongthis curve the crossover is never smooth. This means that,in order to provide a continuous transition from the� to theY-string configuration at any finite-R crossover point, in-cluding the R ¼ 0:8 fm one, the string tension � must bevariable and have a discontinuity in at least one(permutation-symmetric) variable. All-in-all, this showsthat with constant string tension �Y ¼ �� there can beno smooth transition from the � to the Y-string configura-tion in any geometry. Even when �Y=�� 2 ð1:732; 1:86Þthere are only three geometric configurations where asmooth transition can be accomplished, see the discussionbelow and Fig. 13.

D. Dynamical symmetry of the Y-string potential

As stated above, independence of the variable (V2 � A2)leads to the invariance under ‘‘generalized rotation’’

1 0.5 0 0.5 11

0.5

0

0.5

1

x cos

z cos2

FIG. 11 (color online). The boundary in the � vs plane (solidline), between the regions in which one of the angles is largerthan 120�, and the curves along which the functional identity ofthe � and the Y-string potentials is admissible (the blue long-dashed curve).


054501-7

� ¼ "� � ¼ �"�: (33)

in the six-dimensional hyper-space and thus leads to thenew integral-of-motion G ¼ � � p� � � � p�, associated

with the dynamical symmetry (Lie) group Oð2Þ that is asubgroup of the full Oð6Þ Lie group. In certain cases thenew integral-of-motion G can be integrated and the result-ing holonomic constraint can be used to eliminate onedegree of freedom [15]. In the case of the� string potentialthis G is not an integral-of-motion, i.e. it is not constant intime, however, due to the absence of the O(2) symmetry ofthis potential [15]. The dynamical O(2) transformationrules Eqs. (33) can be applied to the coordinates ðz; xÞ asfollows:

x ¼ 2"ð1� x2Þ zffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p z ¼ �2"xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p:

(34)

Note that these equations are nonlinear in ðx; zÞ implying anontrivial transformation of the ‘‘shape-space’’ under thisnew dynamical O(2) symmetry. Only near the originðx; zÞ ¼ ð0; 0Þ does this transformation look like an infini-tesimal rotation; at the edges ðx; zÞ ¼ ð�1;�1Þ it eithervanishes, or diverges. One can find another set of scalar

variables ðxffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p; zÞ in which the dynamical O(2) sym-

metry transformation rules Eqs. (33) are linear, but at theprice of introducing imaginary parts of the potential, atleast in some regions of the ‘‘shape-space’’. A detailedstudy of this symmetry is a task for the future [15].

IV. APPROXIMATIONS TO THE STRINGPOTENTIALS

A. String approximations to the lattice: the compositestring

One possible way to fit the lattice results is, perhaps, tohave a ‘‘composite Y-string’’ that contains a ‘‘core tri-angle’’ of variable size (proportional to, yet smaller thanthe quark triangle) instead of the Y-junction point, seeFig. 12. In that case, however, one may not talk about a

definite crossover interquark distance/hyper-radius, andone would still have contributions from both kinds of stringat all distances. Note, however, that this prescription isprecisely equivalent to the sum of two strings with unequaltensions: for a given (fixed) value of parameter � 2 ð0; 1Þ(coefficient of proportionality) one has

Vcomposite ¼ �VY þ ð1� �ÞV�

¼ �Yminx0

X3i¼1

jxi � x0j þ ��

X3i<j

jxi � xjj: (35)

There is at least one good reason for the unequal stringtensions, viz. the different color factors associated with thequadratic and the cubic Casimir operators of SU(3), c.f.Ref. [16]. It appears ‘‘natural’’ to associate the cubicCasimir color factor with the Y-string and the quadraticCasimir color factor with the � string. As there is no(reliable) way of dealing with color nonsinglets on thelattice (see, however the work of Saito et al. [17]) here-tofore there was little hope of extracting these factors (or atleast their ratio) from the lattice. This Ansatz allows, atleast in principle, the extraction of the ratio of the Y-stringand the �-string tensions �Y=��.Note that in this case the smooth crossover from the� to

the Y-string is not entirely out of the question, albeit it isstill severely restricted in the shape-space. The intersectionof the curve defined by Eq. (32) and the locus of zeros ofV� � VY , Figs. 9 and 10, yields the allowed crossover

x0

x3x2

x1

2π3

FIG. 12 (color online). The new three-quark Y-junction stringpotential.

1 0.5 0 0.5 11

0.5

0

0.5

1

x cos

z cos2

FIG. 13 (color online). The boundary in the � vs plane (solidline), between the regions in which all of the angles are smallerthan 120� and the ones where they are not, the locus of zeros ofV� � VY at �Y=�� ¼ 1:86 (the blue short-dashed curve), andthe lines along which identity of the� and the Y-string potentialsis admissible (the blue long-dashed curve).


054501-8

points/configurations, of which there are at most six (out ofa double-continuum, see Fig. 13), and that only when�Y=�� 2 ð1:732; 1:86Þ. One obtains a region of admis-sible values of � from the admissible values of �Y ¼ ��and �� ¼ ð1� �Þ�. Then �Y=�� ¼ �=ð1� �Þ 2ð1:732; 1:86Þ, implies � 2 ð0:634; 0:650Þ. If this turns outtoo restrictive for the actual lattice results, then one mayeven introduce a hyper-radially dependent coefficient �ðRÞthat peaks at some nonzero R, i.e. a not-quite-linearlyrising two-body confining potential.

In order to facilitate this separation of the Y-string fromthe �-string, the hyper-angular dependence of the three-body potential ought to be expressed in terms of the new

variables x0 ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

pand z0 ¼ z. Then the Y-string

component is manifested through the sole dependence on�2 ¼ ðx02 þ z02Þ ¼ ðV2 þ A2ÞR�6 ¼ ð1� ð2W

R2 Þ2Þ, whereasthe �-string is manifested through the dependence of the

potential on the new hyper-angle � ¼ tan�1ðx0z0Þ, within theconfines of the ‘‘central potential’’ boundary (defined byEqs. (17) in terms of ‘‘old’’ variables ð�; Þ). If both theY-string and the �-string components are present in thelattice three-body potential, their ratio can be disentangled

by measuring the ratio of the � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix02 þ z02

pto the � ¼

tan�1ðx0z0Þ dependencies, again within the confines of the

‘‘central potential’’ boundary, because the �-string alsodepends on �.Before closing this subsection, we ought to point out that

another, perhaps similar in spirit, way to fit the latticeresults has been devised and applied in Ref. [3]. In that‘‘generalized Y-string Ansatz’’, a ‘‘core circle’’ of a defi-nite radius has been assumed around the Y-junction point,see Fig. 14 in Ref. [3]. This Ansatz is rather difficult todescribe analytically in terms of hyper-spherical coordi-nates, so its validity would be difficult to ascertain on thelattice. The ‘‘composite string’’ Ansatz, on the other hand,can be readily recognized by its dependence on the ‘‘thirdvariable’’ which may be chosen either as ðV2 � A2ÞR�6 ¼zðz2 � 3x2ð1� z2ÞÞ ¼ R6z0ðz02 � 3x02Þ, or as �.

B. Optimal two-body approximation to the centralY-string potential

In this light one may try using a linear combination ofthe Y-string potential and a (linearly) rising two-bodypotential, perhaps with a variable string tension in theconstituent quark model calculations. The Y-string poten-tial has been known for the difficulty of implementation inthe Schrodinger equation, which has only recently beensolved systematically, see [10,18], so the authors ofRefs. [19,20] tried to approximate the central Y-stringpotential VY with a two-body V� plus possibly a one-body potential VCM, that are easier to deal with numeri-

FIG. 14. Contour plot of the difference between the genuineY-string potential VY , shown in Fig. 6, and the combined CM-and �-string potentials 1

2 ½VCM þ 1=ffiffiffi3

pV�� as a function of the

cosine of the hyper-angle z ¼ cos2�, and x ¼ cos. Note thatthe overall (quasirotational) symmetry of this figure is the sameas that of Fig. 6, except in the corners and near the edges of thesquare, where the difference is the largest. The darker regionsindicate a larger value of the difference of the potentials.

FIG. 15. Contour plot of the difference between the genuineY-string potential VY and the combined CM- and �-stringpotential 1

2 ½VCM þ V�� used in Ref. [20] as a function of the

cosine of the hyper-angle z ¼ cos2�, and x ¼ cos. Note thatthe overall (quasirotational) symmetry of this figure is broken ascompared with that of Fig. 6, due to the missing factor 1=

ffiffiffi3

p.

The difference is the largest in the darkest regions.


054501-9

cally. Such approximations may still be valuable in calcu-lations of multiquark states. Here we show that one par-ticular linear combination of V� and VCM

Vcomb ¼ 1

2

�VCM þ 1ffiffiffi

3p V�

�

¼ �

2

�X3i¼1

jxi � xCMj þ 1ffiffiffi3

p X3i<j

jxi � xjj�;

¼ �

� ffiffiffi2

3

sð�þ �Þ þ

ffiffiffi1

2

s ��þ �ffiffiffi3

p��þ

�� ffiffiffi3

p��

þ��þ �ffiffiffi

3p

��þ�� ffiffiffi

3p

��

(36)

assumes the highest degree of dynamical symmetry pos-sible with these variables and a linear hyper-radial depen-dence, viz. the symmetry under the exchange � $ �, thatexceeds the usual S3 permutation symmetry.

That new symmetry does not amount to the exact O(2)symmetry of the central Y-string potential VY , as yet, but isnumerically sufficiently close to it in the region of appli-cability, see Fig. 14, for most practical purposes, like thatof the quark model calculations. In Fig. 15, we show thedifference between the combined CM- and �-string poten-tial 1

2 ½VCM þ V�� used in Ref. [20]. Note the conspicuous

broadening of the contour in the lower half-plane (the pear-shape of the contours) and consequently the absence of the

continuous ‘‘quasirotational’’ symmetry in this figure,which goes to show that this approximation to the genuineY-string potential does not have the correct symmetry,

which in turn is a consequence of the missing factor 1=ffiffiffi3

p.

V. SUMMARYAND DISCUSSION

In summary, we have studied the functional dependenceof the three-quark confining potential due to a Y-string, andthe �-string. We have found fundamentally different re-sults for these two kinds of strings, which lead to differentconstants of the motion (‘‘universality classes’’). Thismeans that there can be no smooth crossover between thetwo, except at the vanishing hyper-radius R, which isphysically singular and mathematically not very meaning-ful. Perhaps, this should be no surprise, as the two stringshave different topologies: one separates the plane into twodisjoint parts, whereas the other one does not.

ACKNOWLEDGMENTS

We wish to thank the referee of our previous paper,Ref. [10] for drawing our attention to Refs. [8,9]. One ofus (V.D.) wishes to thank Professor S. Fajfer for herhospitality at the Institute Jozef Stefan, Ljubljana, wherethis work was started and to Professors H. Toki and A.Hosaka for their hospitality at RCNP, Osaka University,where it was continued.

[1] X. Artru, Nucl. Phys. B85, 442 (1975).[2] H. G. Dosch and V. Mueller, Nucl. Phys. B116, 470

(1976).[3] T. T. Takahashi, H. Matsufuru, Y. Nemoto, and H.

Suganuma, Phys. Rev. Lett. 86, 18 (2001); Phys. Rev. D65, 114509 (2002).

[4] C. Alexandrou, P. De Forcrand, and A. Tsapalis, Phys.Rev. D 65, 054503 (2002).

[5] For more recent lattice results see Refs. [6,7].[6] H. Iida, N. Sakumichi, and H. Suganuma,

arXiv:0810.1115.[7] V. G. Bornyakov et al. (DIK Collaboration), Phys. Rev. D

70, 054506 (2004).[8] Ph. de Forcrand and O. Jahn, Nucl. Phys. A755, 475

(2005).[9] M. Caselle, G. Delfino, P. Grinza, O. Jahn, and N.

Magnoli, J. Stat. Mech. (2006) P008.[10] V. Dmitrasinovic, T. Sato, and M. Suvakov, Eur. Phys. J. C

62, 383 (2009); see also Proceedings of the Mini-Workshop ‘‘Few-Quark states and the Continuum’’, BledWorkshops in Physics, Vol. 9, No. 9, edited by B. Golli, M.Rosina, and S. Sirca (DMFA - Zaloznistvo, Ljubljana,Slovenia, 2008), pp. 30–35.

[11] The (trace of the tensor of) moment of inertiaI ¼ P3

i¼1 mjxi � xCMj2 ¼ mða2 þ b2 þ c2Þ ¼ 3mR2 ¼3mð�2

12 þ �212Þ ¼ 3mð�2

23 þ �223Þ ¼ 3mð�2

31 þ �231Þ,

where xCM is the radius vector of the center-of-mass.[12] One should have rather compared with the two-body

potential Eqs. (8)–(10), which is not permutation-symmetric, however.

[13] These properties are perhaps most easily shown when thevectors are expressed in terms of Simonov’s [14] complexvector z ¼ �þ i�.

[14] Yu. A. Simonov, Yad. Fiz. 3, 630 (1966) [Sov. J. Nucl.Phys. 3, 461 (1966)].

[15] V. Dmitrasinovic and M. Suvakov (unpublished).[16] V. Dmitrasinovic, Phys. Lett. B 499, 135 (2001); Phys.

Rev. D 67, 114007 (2003).[17] A. Nakamura and T. Saito, Phys. Lett. B 621, 171 (2005).[18] I.M. Narodetskii, C. Semay, and A. I. Veselov, Eur. Phys.

J. C 55, 403 (2008).[19] M. Fabre de la Ripelle and M. Lassaut, Few-Body Syst.

23, 75 (1998).[20] B. Silvestre-Brac, C. Semay, I.M. Narodetskii, and A. I.

Veselov, Eur. Phys. J. C 32, 385 (2003); I.M. Narodetskiiand M.A. Trusov, Phys. At. Nucl. 67, 762 (2004).


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