The string tension from lattice QCD

Volume

139B, number

4

PHYSICS

17 May 1984

LETTERS

THE STRING TENSION FROM LATTICE QCD N.A. CAMPBELL, C. MICHAEL and P.E.L. RAKOW Department ofAppliedMathematicsand Received

31 January

Theoretical Physics, University of Liverpool, PO Box 147, Liverpool L69 3BX, UK

1984

We discuss the extraction of the interquark crepancies between different methods.

potential

1. Introduction. Euclidean lattice techniques have been very successful in exploring the non-perturbative features of quantum chromodynamics. The pureglue sector is best understood. The potential between static colour sources is found to be consistent with a linear increase with increasing separation at large separation. The coefficient of this linear rise, the string tension K, can be related to quantities measured from the hadron spectrum and so may be used to set the mass scale AI_ of lattice QCD . Recently evidence has been presented [l] using a new analysis of lattice data that the lattice string tension was a/A, = 84 f 4, considerably smaller than the previously accepted value [2-41 a/AL = 125-170. This reduction in @/A, would have the effect of increasing AL and hence results for glueball masses, the deconfinement temperature, etc., would be increased too. In this letter we shall reconsider the various methods available to extract the string tension from lattice measurements. As well as the conventional Creutz [5] method involving ratiosof large rectangular Wilson loops and the new method referred to above [ 11, we shall also use a variational method [6]. Careful analysis shows that these methods do agree in practice. The root of the apparent discrepancy is traced and the easiest way to assign this discrepancy is due to a neglect of finite temperature effects. Such effects would clearly not be included in a perturbative analysis of corrections to the extraction of the string ten*’

In terms of the lattice spacing a at couplinggZ(P ALa = (llg2/16n2)~51’121exp(-8~*/llg2).

288

= 6/g’)

and string tension

in lattice

QCD and we resolve the apparent

dis-

sion [7]. A more appropriate analysis of such corrections is from a hadronic model - the spectrum of excited states of a string. This has already been investigated by lattice techniques [6] and so provides the required information. 2. Wilson loops and potentials. (a) The classic argument is that the average value W of an R X T rectangular Wilson loop (see fig. la) is related to the potential V(R) between static colour sources on a lattice of spacing a by

(1) where V,,(R) differs from V(R) by an a-dependent constant describing the self energy of the static sources. The conventional Creutz ratio x arises from using eq. (1) for two large T-values and then taking the difference between V,(R) and V,(R -a) to give the string tension K at large R

Fig. 1. Wilson loops: (a) Rectangular, (b) correlating paths of length R separated by time T, (c) correlation of Polyakov loops that encircle the periodic time direction.

0.370-2693/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 139B, number 4

PHYSICS LETTERS

e x p ( - K a ) = Lt exp [-aVo(r ) +aVo(R - a ) ] R-~ =

Lt

W ( R , T, a)

. W ( R - a, T - a, a)

T--, = W(R, T - a, a)

W(R - a, T, a)

R-""

-

Lt

(2)

×(R,T,a).

r--* oo R-~

In practice the measurable Wilson loops averages W become very small for large R and T and are then swamped by the noise in the Monte Carlo calculation. So the key to using these relationships is in estimating accurately the corrections to them for finite R and T. Consider first the extraction o f Vo(R ) from Wilson loop data. The behaviour of an R × T rectangular loop is controlled by the covariant eigenmodes of the transfer matrix [6] - physically these can be thought o f as the excited states o f a string with ends flexed at distance R apart and which propagates for time T. Then

paths between the fixed ends at separation R. The advantage o f this method is that many eigenvalues can be extracted from relatively small values of T/a. For example with T = a, 2a, 3a and 3 independent path combinations it would be possible to extract 4 eigenvalues whereas a study of rectangular loops o f length up to T = 8a would be needed to obtain the same informa~tion in principle. Since loops with large values o f T/a have small average values, the errors are relatively large and so accurate numerical calculation o f their averages is very time consuming. Conversely the variational method measures a larger number o f Wilson loops (the overlaps between path Pi at T = 0 and path P / a t T) at relatively small T/a values and so enables very accurate extraction o f the ground state string potential Vo(R ) and o f the excited state spectrum. Analysis o f the discrete symmetries o f these paths [6] shows that only excited states with the same symmetry (Alg) as a straight line path will contribute to eqs. (3) and (4). Roughly the results [6] are that

R[VI(R ) - V0(R)] -~ 1 - 2 .

oo

W(R, T, a) = ~

17 May 1984

ca exp [ - Vc~(R)T ] ,

(3)

ee=0

where a = 0 is the ground state and a / > 1 are excited states. The positivity o f the transfer matrix ensures that c a / > 0. Previous work [6] and general arguments indicate that the lowest excited states will be discrete although a continuum of ground state string plus glueballs will exist. So the corrections due to the first excited state are of the form

W(R, T,a) = c o exp [ - Vo(R)T ]

(5)

so that if c 0 -~ c 1, the contribution to eq. (4) from ths excited state is important if T ~
× {1 + (Cl/cO)ex p [ - ( V 1 - V0)T ] + ...) = CoCOr]a) + c1~.1T[a) + ....

(4)

C(R, T, a) = ~ exp [-Va(R)T],

(6)

a=0

A fit o f the latter form can be used to extract

Vo(R ) and VI(R ) in principle [8]. A more sophisticated way to disentangle this spectrum Vc,(R ) at fixed R is discussed next. In passing let us note that W(R, T, a) for rectangular loops can be thought o f as either a string o f energy Vo(R ) lasting for time T or a string o f energy V0(T) lasting for time R. The more appropriate description is that with the excited state contributions minimized - which is R < T as we shall see. (b) the variational method [6] involves Wilson loops as illustrated in fig. lb and uses a trial basis o f

where the essential simplification is that the overlap factor c a of eq. (3) is now equal to one since the situation persists for all T. Of course the sum over a now runs over all excited states o f any symmetry - so that additional contributions o f symmetry Eu [6] etc., will now appear. This approach is rather similar to considering R × Trectangular loops. The advantage is that the normalization c a is fLxed. One disadvantage is that to vary Tmeans a different lattice simulation rather than a different measurement o f the same lattice. 289

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PHYSICS LETTERS

A different viewpoint o f the same configuration (fig. lc) is obtained by considering this correlation as the energy E 0 o f an endless string of length Twhich propagates for time R. Then taking account o f the geometrical factor for 3 space dimensions, one has Lt C(R, T, a) = (d/R) exp [-Eo(T)R] .

17 May 1984

oEgT)

(7)

R--~

Here d represents the overlap between a straight loop of length T and the ground-state endless string eigenmode, and from positivity d > 0. Now this eigenmode is similar to a glueball in the sense that it is a closed loop excitation o f the vacuum. Thus, as for the glueball, the rest mass o f the system can be extracted by using a sum over spatial positions to project out zero momentum. Since the straight loop o f length T is already symmetric in the T-direction, a sum over the remaining two directions is necessary. Then the correlation o f two suchloops at spatial separation (Ax, Ay, T) and at time separation R is

T, a) =

Ax ~,y

C(R, Xx, Ay, T, a)

=dexp[-Eo(T)R ]

f o r R ~ oo

(8)

This is indeed the quantity evaluated by Prisi et al., in their new method of determining the string tension [ 1 ]. They evaluate the string tension K by assuming, for R > 2,

C?(R, T,a) = d e x p ( - K T R ) , which is equivalent to assuming that

Eo(T ) = K T .

(9)

At first sight this appears reasonable since an endless string o f zero length should have zero energy - there are no end effects. But in practice for a lattice o f sufficiently small size N in the T-direction there will be a first order transition to a deconfining phase. In this gluon plasma phase, the correlation C between two Polyakov loops will be given by an inverse power law rather than an exponential because o f the zero mass deconfined gluon modes. Thus

Eo(T ) = 0

forT
(10)

This situation is exhibited in fig. 2. Clearly their extracted string tension Kef f = Eo(T)/T will be smaller than the true string tension given by the asymptotic 290

/

/

/

/

J

Fig. 2. Expected form of the energy Eo(T) of a closed path that encircles the periodic time direction in a TX ~3 lattice. The finite temperature transition at T =N~3on a NI3 × 003 lattice is shown. The naive estimate that E0(T) = KeffT leads to Keff < K where K is the asymptotic slope of Eo(T ).

slope of E 0 ( T ) with T. In the case o f ref. [1], T = 10a wasused with/5 = 6/g 2 = 6 where N = 5.5a [8] .Thus very large corrections ensue. This finite temperature effect highlights the fact that E (T) cannot be linear with T and some extra contribution such as c/T is needed for a fuller description. [9] :

3. Results. We concentrate on t3 = 6/g 2 = 6 to compare results on the potential Vo(R ) from several different methods. (a) Rectangular Wilson loops [10]. From the tabulated values o f R × T Wilson loop averages with R, T~< 6a evaluated [2] from 50 X 124 lattices, we extract Vo(R ) using fits at each R for T 1> a using eq. (4) with two eigenmodes (ground state and first excited state). The resulting aVo(R ) is plotted in fig. 3. (b) Variational method [6]. A 124 lattice is again used and several are constructed using a discrete set approximation to SU(3) [11] in the Monte Carlo algorithm. This speeds up the computation somewhat and reduces storage enormously. Then a basis o f three types o f path with ends R apart is used: (i) a straight path, (ii) a sidewise excursion o f one spacing for the whole length and (iii) a sidewise excursion for the central R/a - 2 spacings only (see fig. lb). We evaluate the Wilson loop averages for all combinations o f paths at the ends and for R = 3a-7a and T <~3a. For the ground state symmetry (Alg) there are three independent path combinations and the relevant combinations

Volume 139B, n u m b e r 4

PHYSICS LETTERS

aVo(R) / /

1.2

10

0.8

0.6

®/

/ /

/

/ I

0.4

o/

0

I

i

i

i

i

i

i

i

2

3

Z,

5

6

7

8

9

i

~

10 11

I

)

12

R/o Fig. 3. The static colour source potential aVo(R) at separation R/a o n a lattice at ~ = 6. Points are from (a) rectangular Wilson loops (o), (b) variational analysis o f paths (u) and (c) Polyakov loop correlations (X). The straight line has slope Ka 2 = 0.06. The curve is from a fit (ref. [10]) using a string model with Ka 2 = 0.05. In physical units at/3 = 6, a ~- 0 .56 GeV-1.

of Wilson loop averages (with i, j ~< 3) are given by

Wi/(R, T,a)= ~ c~c i /" exp [-Va(R)T].

(11)

An upper limit for Vo(R ) can be obtained from a variational method. In fig. 3, the upper limits shown come from applying this variational technique to T = 3a and 2a but using the eigenmodes determined from T = 2a and a to reduce errors [12]. As welt as upper limits, it is possible to fit directly to eq. (11) and using two eigenmodes for T = 3a and 2a gives the points shown in fig. 3. The difference between the upper limit and the fitted value gives an estimate of the error. As expected, this method agrees with the analysis of rectangular loops above and, moreover, it is parti-

17 May 1984

cularly economical of computer time. A discussion of the excited state spectrum will be presented elsewhere [131. (c) Polyakov loop correlation [ 1 ]. Recently data on C (Az, T, a), the correlation between two Polyakov loops ~z aFart (for ~z = 2 a - 1 8 a and T = 10a), have been presented [1] from 103 × 20 lattices. In order to extract Vo(R ) from this data, we use eq. (6) and note that the contribution of excited states will be negligible for these values o f ~z and T. The sum over the relative transverse positions (Ax, A y ) o f the Polyakov loops implies that there is a sum over separations R = [(£tx') 2 + (Ay)2 + (~z)2] 1/2 for each Az. Assuming restoration of rotational symmetry, this can be taken into account by assuming a dependence for Vo(R ) near_R =/~ where/~ is the average value of R. In practice R ~ zSz + 2a, and the output Vo(R) is insensitive to this dependence. A further complication is that "image" Polyakov loops arise due to the periodic boundary conditions used. The most straightforward way to take them into account is to sum over periodic copies of one of the loops. To be specific we express C ( ~ z , T, a) = exp [ - V0(/~)a ]

× ~ e x p { - a [ V o ( R ) - V0(/~)]),

(12)

and /~ = ~ R

exp

[ - a V o ( R ) ] / ~ exp [-aVo(R)],

where the sum runs over z2~x,Ay = 0, +-a, ... +~o and Dz = zXz, ~z -+ 20a, ... with R 2 = (zXx)2 + (2xy) 2 + (Dz) 2. Choosing a linear form, Vo(R ) - Vo(R) = K(R - R), then allows V0(/~ ) to be extracted from the data on C. A new value of K can then be obtained from the slope o f V0(R ) versus k and this procedure converges to a consistent solution for Vo(R) which is shown in fig. 3. The resulting slope Ka 2 is larger, at about 0.053 (for a linear fit), than the value 0.039 claimed in ref. [1] from fitting C (zXz, T, a) = exp(-KTzXz). The discrepancy can be traced in eq. (12) to the sum over ~x, Ay 4= 0 which enhances the correlation relatively more for larger R. An alternative way to see the source o f the discrepancy is from the picture of endless loops of length Tpropagating for time R, as discussed in section 2c, where the finite 291

Volume 139B, number 4

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17 May 1984

one determines a = 0.56 -+ 0.05 GeV -1 . Thus Vo(R ) has been determined out to R = 6.7 GeV -1 at/3 = 6. At other/3 values Vo(R ) has only been determined to smaller R-values. At/3 = 5.7 the string tension [4], assuming scaling, is steeper at x/K'/A t -~ 133 compared to the/3 = 6 value o f x/K'/A L = 105 --- 10. Data at/3 = 5.4 confirm that the scaling region for the string tension seems to be reached from above for SU(3).

temperature phase transition will be important - see fig. 2. When full account is taken o f the relation between and V0(/~ ) the resulting data on V0(/~') is in excellent agreement with the other two methods. Any contribution from excited states would reduce V0(/~ ) preferentially at large/~. This could explain the somewhat smaller slope observed at the largest values o f , ~ although our estimates suggest that this effect would be small compared to the errors.

References

4. Discussion. Our main conclusion is that, at/3 = 6, three different methods o f extracting the static colour source potential Vo(R ) are in agreement. Thus the string tension K must be in agreement too. So we have resolved previous apparent discrepancies by a careful analysis. It is still not straightforward to extract the string tension K to compare with the value needed to describe the high-spin light-quark meson spectrum. This is because the data on J = 3, 4 mesons correspond naively to a rotating string o f length R -~ 10 GeV -1 (-~ 18a at/3 = 6). Thus Vo(R ) is needed at larger R values than currently measured accurately in lattice simulations - this implies an extrapolation in R. This extrapolation is made uncertain b y the presence o f coulombic terms (1/R terms) in Vo(R ). Some sample fits are shown in fig. 3. Overall one can only claim that Ka 2 at/3 = 6 is in the region 0.06 -+ 0.01. Then since X/~ = 0.44 GeV

[ 1] G. Parisi, R. Petronzio and F. Rapuano, Phys. Lett. 128B (1983) 418. [ 2 ] D. Barkai, M. Creutz and K.J.M. Moriarty, BNL preprint BNL-34027 (1983). [3] M. Fukugita, T. Kaneko and A. Ukawa, KEK preprint TH-63 (1983). [4] F. Gutbrod, P. Hasenfratz, Z. Kunszt and I. Montvay, Phys. Lett. 128B (1983) 415. [5 ] M. Creutz, Phys. Rev. D21 (1980) 2308. [6] L.A. Gfiffiths, C. Michael and P.E.L. Rakow, Phys. Lett. 129B (1983) 351. [7] G.Curciand R.Petronzio, Phys. Lett. 132B (1983) 133. [8] T. ~elik, J. Engels and H. Satz, Phys. Lett. 129B (1983) 323. [9] P.E.L. Rakow, Liverpool preprint LTH 111 (1984). [10] J.D. Stack, Phys. Rev. D27 (1983) 412. [11] P. Lisboa and C. Michael, Phys. Lett. 113B (1982) 303. [12] K. Ishikawa, M. Teper and G. Schierholz, Phys. Lett. l16B (1982) 429. [13] N.A. Campbell, L.A. Griffiths, C. Michael and P.E.L. Rakow, in preparation.

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