Baryonic strings on a lattice

Nuclear Physics B267 (1986) 531-538 © North-Holland Publishing Company

BARYONIC STRINGS ON A LATFICE

R. SOMMER and J. WOSIEKl Physics Department, University of Wuppertal, Gauss-Str. 20, D-5600 Wuppertal I, F.R. Germany

Received 6 September 1985 Interaction of three static quarks on a lattice is studied using the Monte Carlo technique. The, overall additive description of the potential is satisfactory. However, some regularities of the data suggests the onset of three-body effects.

1. Introduction In an earlier paper we have suggested measuring b a r y o n i c loops in order to determine the energy o f the three static quarks in the singlet channel o f SU(3) lattice gauge theory [1]. In this way one can p r o b e the new features of the confining potential, not available in the q~l configuration. In ref. [1] we have concentrated on the c o r r e s p o n d e n c e with the mesonic channel: configurations with two quarks situated close to each other and the third one moving "far a w a y " (3-6 lattice units at fl = 6.0) were investigated. The resulting string tension was f o u n d to be consistent with the m e s o n i c one. This is expected from the equal slopes o f the corresponding Regge trajectories. In this p a p e r we report on a high statistics measurement for the more general three q u a r k configurations. The main goal o f this study is to decide whether the confining potential is o f two- or three-body character. Recent measurements of the qcl potential [2, 3] provide evidence for the existence o f the so called Liischer term [4]. To~ether with the non-zero value for the string tension, this directly suggests the effective description o f the nonabelian dynamics by fluctuating strings. For the b a r y o n i c case the general expectation is that, at least classica|ly, the strings meet at the so called Toricelli point which minimizes the sum o f the distances to the three sources [5, 6]. In this case one should see the genuine three-body interaction a m o n g the static quarks in the b a r y o n i c channel. We will be looking for such an effect here. In sect. 2 we define b a r y o n i c loops and discuss the prescription used for revealing the t h r e e - b o d y effects. Sect. 3 contains our results. Finally, the conclusions are given in sect. 4. 1 Research fellow of the Alexander yon Humboldt-Foundation. On leave from the Jagellonian University, Cracow, Poland. Address after August 31st: Institute of Physics, Jagellonian University, ul. Reymonta 4, Cracow, Poland. 531

532

R. Sommer, J. Wosiek / Baryonic strings on a lattice

T

r2 Fig. 1. The baryonic loop.

2. Bawonic loops and the three-body force The baryonic loop is defined as follows (cf. fig. 1)

~ ( c , , c2, c3) = ~(e,~ke,mnU,,(COgjm(C2) Uko(C3)),

(1)

with U(C) = 1-Ie~c Ue. The three paths CA, A = 1, 2, 3 have to meet at the end points. In the static limit they can be described by their time extent T and the positions of the heavy quarks rA. The potential energy of this configuration is obtained from [7]

1

V(rl, r2, r3) = T~o~lim- - ~ log B ( r l , r2, r3, r ) .

(2)

In this limit the details of the space-like sections connecting the paths together are unimportant. In the following, only those configurations where all three quarks are located in one of the main planes of the lattice will be considered. Hence rA will denote two-dimensional vectors. The three-body interactions would give rise to a dependence of the force acting between quarks 2 and 3, say, on the position of quark 1. On the cubiq lattice this can be checked as follows. Consider an arc of the circle, centered at the site r~, which passes through several sites r2, r2, etc. (cf. fig. 2). Choose r3 # rl, r2, rE,. For s m a l l Jr2 - r2,1, the difference A(rl)

•

m_ V ( r l

•

' r2 ' r3 ) -

•

•

r3 )

r2, '

V(rl"

e

®

•

.

•

•

•

•

•

2 •

•

•

x

®

•

•

•

0

X

• 1~

®

.

•

•

•

•

.

2'

. ll 3

.

Fig. 2. Geometry of the configurations used to search for the three-body interactions.

(3)

R. Sommer, Z Wosiek / Baryonic strings on a lattice

533

Fig. 3. Classical configurationof the baryonic strings. is the lattice counterpart of the above mentioned force A ( r l ) ~ 0 V/tgR23[RIE,Rt3. NOW move quark 1 along the diagonal shown in fig. 2, keeping quarks 2 and 3 at their old positions. Any dependence of A on rl will signal the existence of a three-body interaction. A qualitative estimate of the size of this effect can be obtained from the classical string model. In this case [5, 6, 8] V = tr min {Ix x

rll + Ix-/'21

q - I x - r3l}.

(4)

The Toricelli point xT, which minimizes the sum in eq. (4), lies inside the triangle, 2 spanned by the three quarks when all angles of this triangle aA <~Tr. If one aA is bigger than ~Tr 2 then xT = rA (cf. fig. 3). One should be aware, however, of the well-known limitations of this model. We use it only to provide a rough estimate of the size of the effect we are looking for. On the other hand, the proposed signature of the three-body interactions is clearly model independent.

3. R e s u l t s

Altogether 24 distinct quark configurations were investigated, corresponding to a total of 80 baryonic loops. The average values of baryonic loops are quoted in table 1 for further reference. The displayed errors were corrected for sweep-to-sweep correlations as discussed in ref. [3]. Depending on the quark configuration, the loops were averaged over 20-56 SU(3) gauge configurations generated on a 164 lattice at/3 = 5.8 [3]. When possible, the multihit procedure [8] was applied to the time-like links. Three to five time extents, T = 3, 4, 5 (6, 7), were used. For every quark configuration, the energy was determined by fitting a single exponential in T, and directly, from the ratio of B's. In both cases we obtained consistent results for the energies. Table 2 summarizes the geometry of the quark configurations. They are divided into five groups (A-E) corresponding to various radii of the circle, along which

534

R. Sommer, J. Wosiek / Baryonic strings on a lattice

TABLE 1 Average values of the baryonic loops eq. (1) in units of 10 6. They have been measured on a 164 lattice at/3 = 5.8. For assignment of quark configurations cf. table 2

Co n f~i g' u 'r a~t i o n s ~ . .T. . ~ AI A2 A3 A4, A5, A6 B3, B4, B5 B6 C1 C2 C3, C4, C5 C6 D1, D2, D4, D6, E2 E4 E5 El0

E8 E6 E1 E9

E3, D3, D9 E7

Dll D12 D5, D7, D10 D8

3

4

5

2 991 (23) 6 895 (52) 11396(34) 25 657 (35) 32 061 (70) 137 275 (160) 1 078 (14) 1 984 (11) 2 925 (22) 4 819 (23) 1 166 (15) 1 690 (21) 4 069 (16) 5 973 (14) 11 161 (34) 14 173 (44) 1 209 (19) 3 286 (32) 12 062 (53) 27 301 (71) 5 951 (41) 21 296 (38) 7 529 (36) 10 250 (29)

769 (10) 2 170 (43) 3644(17) 9 549 (23) 12 703" (49) 62 898 (140) 256 (12) 482 (4) 765 (14) 1 358 (9) 261 (10) 410 (9) 1 094 (7) 2 090 (7) 3 554 (22) 4 796 (25) 271 (11) 877 (14) 3 906 (24) 10 237 (43) 1 759 (32) 7 479 (22) 2 275 (14) 3 245 (17)

220 (8) 650 (45) 1 186(10) 3 588 (12) 5 057 (41) 28 888 (105) 52 (6) 123 (4) 209 (9) 396 (5) 54(9) 104 (7) 301 (3) 299 (4) 1 152 (16) 1 644 (10) 76 (4) 239 (11) 1 287 (12) 3 865 (25) 533 (13) 2 689 (13) 705 (10) 1 055 (9)

6

7

1 381 (20)

497 (23)

13 277 (73)

6 087 (54)

76 (45)

- 5 0 (49)

923 (45)

304 (30)

TABLE 2 Space configuration of quarks for various groups of baryonic loops A

B

C

D

E

(0, 1), (1, 0) (1, - 1 )

(1,2),(2,1) (2, - 2 )

(1, 2), (2, 1) (1, - 1 )

(0, 2), (2, 0) (1, - 1 )

(-1, 2), (2, 1) (1,-1)

Group r2, r2, r ~

_--

-2 - 1

0 1

2 3

AI A3 A5

A2 A4 A6

B3 B5

B4 B6

C1 C3 C5

C2 C4 C6

D1 D3 D5 D7 D9 Dll

D2 D4 D6 D8 D10 D12

E1 E3 E5 E7 E9

E2 E4 E6 E8 El0

535

R. Sommer, J. Wosiek / Baryonic strings on a lattice

quark 2 moved, as well as the position of the third quark !"3. In each group r~ is varied in the range shown in table 2. For fixed r~ (and r3), two positions of the second quark, r2 and rE,, were used to determine A ( r l ) from eq. (3). Figs. 4 and 5 show our results for A(r~). On fig. 4 we present the Monte Carlo measurements together with the expectations from formula (4)*. In general, no statistically significant variation of A with r] was found. However, there seems to exist a systematic trend in our data** which, at least qualitatively, follows the predictions of the classical string model. Definitely the effect is smaller by at least one order of magnitude from the one expected from eq. (4). We did not push our measurements to higher accuracy. Before looking for such a tiny variation of A one should have a better understanding of other systematic effects arising both from the small violation of rotational symmetry and the finite T dependence [3].

A

B

A -.10

- .10

(,

15

I

J -.05

.10

-.05

.05 -2

0

-2

0

-2

I

I

I

I

I

E

".30 .2C

o

o

o

) i

-.20

I .1C

i

I

I

I

I

-2

0

2

-2

0

I

2 d

Fig. 4. Energy differences A ( r l ) as the function of the position of quark 1, r I = (d, d). Open circles are results of our calculation, full points show the prediction of eq. (4). * We have also applied the hamiltonian strong coupling expansion for the off-axis qq potential [10] to our case. It shows qualitatively the same effect as expected from eq. (4). ** Especially for configurations D and E. Note that for these configurations A can hardly be regarded as a derivative (cf. table 2). However, still, any variation of A with respect to r, proves the lack of additivity in V.

536

R. Sommer, J. Wosiek / Baryonic strings on a lattice

.2~.

E

2C A .15

t

.10

C

I 05

i

-2

[

i

i

-1

0

1

¢f¢

|

i

-2

-1

0

l

i

L

1

2

d

Fig. 5. The energy ditterences for all five groups of quark configurations. The solid lines show the result of the additive fit based on eqs. (5) and (6).

Assuming a constant A, with rl, let us now turn to the additive description of the three-quark potential. Note that A (r~) varies within a factor of 5 between various groups of quark configurations. This is emphasized in fig. 5, where A's for all five groups are shown together. The solid lines are the result of a joined fit of all 24 energies to the additive formula [1] Vqqq(gl2, Rl3, R23)=1 ~

Vqq(RAB)

(5)

A
with C_

Vqq(R) = ~ - + Co+ c+R,

(6)

where c_ = - 0 . 2 4 7 ,

Co=0.632,

c+=0.117.

It is readily seen that this simple form works rather well fitting a variety of quark

R. Sommer, J. Wosiek / Baryonic strings on a lattice

537

configurations, which result in quite different values of A. The parameters ci, i = +, 0, - , given by the fit, are consistent with those obtained from the qcl analysis [3]. The quality of the fit can be further improved by including a small violation of the rotational symmetry [3]. 4. Summary and conclusions The static potential of three quarks in the baryonic channel was studied with an eye on the genuine three-body interaction. It was found that the energy of heavy quarks is described rather well by the additive formula (5). The parameters of the qq interaction agree with those extracted from the q?:l data. Formula (5) is well known from the baryonic spectroscopy [11 ]. In our case, however, many configurations with three equally separated quarks were considered, as opposed to the quark-diquark setting which is typical for excited baryons. No pronounced three-body effect was found. If any three-body interactions exist, they are smaller by an order of magnitude than those expected from the classical string model. The trivial way out of this dilemma is to say that at distances - 0 . 5 fm (3-5 lattice units at /3 = 5.8) one is still in the perturbative regime of the continuum theory*, where hardly any three-body effects are expected. However, one does see the non-perturbative effects in the q~l interaction at such distances. Namely, the linear term is important and the Liischer term has the proper magnitude [2]. We see the same in the qq potential extracted from eq. (5). One would have expected then, that the scale at which the effective strings show up is the same in mesonic and baryonic channels. In particular, the three-body interaction generated by the strings meeting at the junction should also appear at similar distances. Our results, however, suggest an alternative picture. At intermediate distances ( - 0 . 5 fm, say) the confining strings develop between pairs o f quarks. When all quarks are far from each other compared to the proton size, the strings merge together giving the configuration shown in fig. 3. When only one source is shifted beyond 1 fm, the two strings collapse giving rise to the linear quark-diquark potential with a slope of one tr. The latter situation is also suggested by.eqs. (5), (6) (~r = c+). It is quite possible that we see the onset of this phenomenon in configurations D and E (the largest inter-quark distances considered). Though the data are statistically consistent with the constant A(rt), our points show the systematic tendency for a small curvature with a proper** concavity. The fact that the additive fit is far from perfect for these configurations also adds some power to this speculation. Before, however, one pushes further by increasing the statistics, other effects of the same order of magnitude, like finite time contamination a n d / o r rotational invariance should be investigated more thoroughly. * Provided the lattice scaling is satisfied at fl = 5.8. ** By "proper" we mean consistent with the formula (4).

538

R. Sommer, J. Wosiek / Baryonic strings on a lattice

Formally, more understanding of the three-string system is needed. Quantum fluctuations will certainly modify the simple form (4)*. With linearized NambuGoto action, they can be easily computed, but the resulting formula applies only when all the quarks are far from the junction. This requirement is hardly met in our cases. The full quantum solution of the system of three strings would be very welcome. From the lattice side, clearly new techniques such as the one suggested in ref. [11] are needed to allow for realistic measurements of large Wilson and baryonic loops. We would like to thank Prof. K. Schilling for reading the manuscript. We also thank B. Bunk for numerous discussions. References [1] R. Sommer and J. Wosiek, Phys. Lett. 149B (1984) 497 [2] M. Flensburg and C. Peterson, Lund preprint LU-TP-84; D. Barkai, K.J.M. Moriarty and C. Rebbi, Phys. Rev. D30 (1984) 1293; S. Otto and J.D. Stack, Phys. Rev. Lett. 52 (1984) 2328; (E) 53 (1984) 1028; A. Hasenfratz, P. Hasenfratz, U. Heller and F. Karsch, Z. Phys. C25 (1984) 191; Ph. de Forcrand, G. Schierholz, H. Schneider and M. Teper, DESY preprint 84-116 (1984); K.C. Bowler et al., Edinburgh preprint ITFA-85-07 [3] R. Sommer and K. Schilling, Wuppertal preprint WUB 85-6 [4] M. Liischer, Nucl. Phys. B180 [FS2] (1981) 317 [5] Th.W. Ruijgrok, Eur. J. Phys., 5 (1984) 21 [6] H.G. Dosch and V.F. Miiller, Nucl. Phys. Bl16 (1976) 470 [7] G. Marchesini and E. Onofri, Nuovo Cim. 65A (1981) 298 [8] X. Artru, Nucl. Phys. B85 (1975) 442 [9] G. Parisi, R. Petronzio and F. Rapuano, Phys. Lett. 128B (1983) 418 [10] J.B. Kogut, D.K. Sinclair, R.B. Pearson, J.L. Richardson and I. Shigemitsu, Phys. Rev. D23 (1981) 2945 [11] A. De Rujala, H.D. Georgi and S.L. Glashow, Phys. Rev. D30 (1975) 147; A.J. Wey and R.L. Kelly, Phys. Reports 96 (1983) 73 and references therein [12] J. Ambj0rn, Copenhagen preprint NBI-HE 85/07

* Especially at small distances.