Gauge-invariant field strength correlations in QCD at zero and non-zero temperature

B ELSEVIER

Nuclear Physics B 483 (1997) 371-382

Gauge-invariant field strength correlations in QCD at zero and non-zero temperature A. Di Giacomo a, E. Meggiolaro a'l, H. Panagopoulos b a Dipartimento di Fisica, Universit~idi Pisa, and INFN, Sezione di Pisa, 1-56100 Pisa, Italy b Department of Natural Sciences, University of Cyprus, 1678 Nicosia, Cyprus Received 1 July 1996; accepted 11 October 1996

Abstract We determine, by numerical simulations on a lattice, the gauge-invariant two-point correlation functions of the gauge field strengths in the QCD vacuum at zero temperature, down to a distance of 0.1 fm. We also study the behaviour of the gauge-invariant field correlators in the theory at finite temperature, across the deconfinement phase transition.

PACS: 12.38.Gc; 11.10.Wx;

1. Introduction The gauge-invariant t w o - p o i n t correlators o f the field strengths in the Q C D v a c u u m are defined as 7 9 ~ p , ~ ( x ) = (OITr {Gup(X) S ( x , O ) G ~ ( O ) S t (x, O) } I0),

(1.1)

a and T a are the generators of the colour gauge group in the where Gup = gT a Gup f u n d a m e n t a l representation. Moreover, in Eq. ( 1 , 1 ) ,

S ( x , 0) = P e x p

i

dt x ~ A ~ ( x t )

,

(1.2)

* Partially supported by MURST (Italian Ministry of the University and of Scientific and Technological Research) and by the EC contract CHEX-CT92-0051. i Postal address: Enrico Meggiolaro, Dipartimento di Fisica, Universit~ di Pisa, Piazza Torricelli 2, 1-56100 Pisa, ltaly. E-mail: [email protected] 0550-3213/97/$17.00 Copyright I~) 1997 Elsevier Science B.V. All rights reserved PII S0550-32 13 ( 9 6 ) 0 0 5 8 0 - 9

372

A. Di Giacomo et aL/Nuclear Physics B 483 (1997) 371-382

with A~, = gTaA~u, is the Schwinger phase operator needed to parallel transport the tensor G~,(0) to the point x. These field-strength correlators play an important role in hadron physics. In the spectrum of heavy QQ bound states, they govern the effect of the gluon condensate on the level splittings [1-3]. They are the basic quantities in models of stochastic confinement of colour [4-6] and in the description of high-energy hadron scattering [7-10]. A numerical determination of the correlators on lattice (with gauge group SU(3)) already exists, in the range of physical distances between 0.4 and 1 fm [ 11 ]. In that range D~o,~,~ falls off exponentially D u p , ~ ( x ) ,.~ e x p ( - I x ] / • ) ,

(1.3)

with a correlation length A "~ 0.22 fm [ 1 1 ]. What makes the determination of the correlators possible on the lattice, with a reasonable computing power, is the idea [ 12,13] of removing the effects of short-range fluctuations on large distance correlators by a local c o o l i n g procedure. Freezing the links of QCD configurations one after the other, damps very rapidly the modes of short wavelength, but requires a number n of cooling steps proportional to the square of the distance d in lattice units to affect modes of wavelength d: n "~ k d 2 .

(1.4)

Cooling is a kind of diffusion process. If d is sufficiently large, there will be a range of values of n in which lattice artefacts due to short-range fluctuations have been removed, without touching the physics at distance d; by lattice a r t e f a c t s we mean statistical fluctuations and renormalization effects from lattice to continuum. This removal will show up as a plateau in the dependence of the correlators on n. This was the technique successfully used in Ref. [ 11 ]. There, the range of distances explored was from 3-4 up to 7-8 lattice spacings at fl ~ 6 (/3 = 6/gZ), which means approximately from 0.4 up to l fm in physical distance. The lattice size was 164. In Section 2 we discuss new results obtained on a 324 lattice, at/3 between 6.6 and 7.2: at these values of/3 the lattice size is still bigger than 1 fm, and therefore safe from infrared artefacts, but d = 3, 4 lattice spacings now correspond to physical distances of about O. 1 fm. Since what matters to our cooling procedure is the distance in lattice units, we obtain in this way a determination of the correlators at distances down to 0.1 fm. All of the above concerns the theory at zero temperature. In Section 3 we go further and determine the behaviour of the correlators at finite temperature for the pure-gauge theory with SU(3) colour group and in particular we study their behaviour across the deconfining phase transition. The motivations to do that stem from Refs. [ 14-16].

A. Di Giacomo et al./Nuclear Physics B 483 (1997) 371-382

373

2. C o m p u t a t i o n s and results at T = 0

The most general form of the correlator compatible with the invariances of the system at zero temperature is [4-6] 791~p,vo'( X ) = (guvgpa~ -- gtxo'gpv ) [ D ( x 2) + 791 (x2)]

+(x~x~gp~ - xt, x~gpv + xpx,~&,v - xpxvgu,~)

•791 (X2)

Ox2

(2.1)

79 and ~)1 are invariant functions of x 2. We work in the Euclidean region. We can define a 7911(x 2) and a D ± ( x 2) as follows. We go to a reference frame in which x ~ is parallel to one of the coordinate axes, s a y / , = 0. Then 1

2 ODl

3

7911=-- -~ Z 79oi,oi( x ) -- 79 "~- 791 -q- x ~ i=1 3

1 D l_ ~- -~ . ~ .

,

Dij, i j ( X ) = D "q- D 1 .

(2.2)

/<.1=1 L On the lattice we can define a lattice operator Dtw,~,,, which is proportional to Dup,,,~, in the continuum limit, i.e. when the lattice spacing a --+ 0. Since the lattice analogue of the field strength is the plaquette Huo (n) (the parallel transport along an elementary L square of the lattice, lying on the/.tp-plane), Duo,~r will be defined as

Dcup.~( da ) = ~N 1 { (Tr[//~w(n + da)S(n + da, n)S2~(n)S t (n + da, n)1}} , (2.3) where ~ stands for real part and the lattice operator/2~,~(n) is given by 1

S2ucr(n) =//~t~(n) - / T r y ( n ) - 7Tr[//~a(n) - / / v a ( n ) ] 3 --

(2.4)

da is a line parallel to one of the coordinate axes, with integer length d in units of the lattice spacing a, joining the two sites to be correlated; S is the parallel transport along this line. The inclusion of the operator ½Or,, (in place of simply putting//~,~) on the right-hand side of Eq. (2.3) ensures that the disconnected part and the singlet part of the correlator are left out. In particular, the subtraction of the singlet part (which is anyway a small contribution of order (.9(a 12) ) means that we are indeed taking the correlation of two operators with the quantum numbers of a colour octet. In the naive continuum limit (a ---* 0) we have that

79~p.wr( da) a~oa479~zp,vcr(da) + (.9(a6) . Making use of the definition (2.1) we can also write, in the same limit,

(2.5)

374

A. Di Giacomo et al./Nuclear Physics B 483 (1997) 371-382

....

I

O. 0 0 0 2

, .... #

Z l z S R S l z

! ....

!

#

a ~ s ~ a s ~ m

O. 0 0 0 0

-0. 0 0 0 2

X

-o. 0004

z

I,,,,I,,,,I,,,,I,,,,I 0

5

i0

15 COOLING

20 STEP

Fig. 1. A typical behaviour of D~ (crosses; d = 6, ,8 = 6.6, lattice 324) and of "DLL (triangles; d = 12, /3 = 6.6, lattice 324) during cooling. ~ a4791i ( d 2 a 2) D (da) a--.O

+ O(a6) ,

D~_(da) a~oa479_L( d2a 2) 4- O(a6)

.

(2.6)

Eqs. (2.5) and (2.6) come from a formal expansion of the operator, and are expected to be modified, when the expectation value is computed, by lattice artefacts, i.e. by effects due to the ultraviolet cutoff. These effects can be estimated in perturbation theory and subtracted [ 17]. Instead we remove them by freezing the quantum fluctuations at the scale of the lattice spacing. After cooling, D~ and D~ are expected to obey Eq. (2.6). We shall omit the details of our cooling procedure, which have been described in several papers (see Refs. [ 12,13] and references therein). We only remark that it is a local procedure, which affects correlations at distances that grow with the number of cooling steps as in a diffusion process. We then expect that, if the distance at which we observe the correlation is sufficiently large, lattice artefacts are frozen by cooling long before the correlation is affected: this produces a plateau in the correlation versus cooling step. Our data are the values of the correlation at the plateau; the error is the typical statistical error at the plateau, plus a systematic error which is estimated as the difference between neighbouring points at the plateau (whenever the plateau is not long enough, looking more like an extremum). Typically the global error is three times larger than the statistical error. The typical behaviour of 7)~ and of D L along cooling is shown in Fig. 1. We have measured the correlations on a 324 lattice at distances ranging from 3 to 14 lattice spacings and at fl = 6.6, 6.8, 7.0, 7.2. From renormalization group arguments,

A. Di Giacomo et al./Nuclear Physics B 483 (1997) 371-382 1 a(fl) = ~-[f(fl),

375 (2.7)

where AL is the fundamental constant of QCD in the lattice renormalization scheme: its value, extracted from the string tension [ 18], turns out to be about 5 MeV. At large enough/3, f ( / 3 ) is given by the usual two-loop expression

/ 8

\ 51/121

_ 47t.2/3 ~

for gauge group SU(3) and in the absence of quarks. At sufficiently large /3 one also expects that

~_~LD_I_

f2(fl)

,

(2.9)

where f ( / 3 ) is given by Eq. (2.8) and terms of higher order in a are negligible. In Figs. 2 and 3 we plot respectively D ~ f ( f l ) -4 and D ~ f ( f l ) -4 v e r s u s dphys = (dIAL) f(fl). In these figures we have also plotted the values of the correlators obtained in Ref. [ 11 ], corresponding to physical distances dphys /> 0.4 fro. We have applied a best fit to all of these data with the functions D ( x 2) = A e x p

-

+ =-=.,exp

D l ( x z) = B e x p

-

+ ;---rT.,exp

\ aa)

\ &)

, .

(2.10)

We have obtained the following results: A A4 -~3.3 x 108, 1

1

AA~-- A----~18--2 '

B - - ~- 0.7 x 108 A4

1

a _ 0.69

b-~ 0.46,

1

Aa -- A L 9 4 '

(2.11)

with x2/Nd.o.f. ~ 1.7. The continuum lines in Figs. 2 and 3 have been obtained using the parameters of this best fit. With the value of A/. determined from the string tension [ 18 ] we obtain /~A "~

0.22 f m ,

,~<, ~- 0.43 fm.

(2.12)

The correlation length AA, which enters the non-perturbative exponential terms of D and Dr, as well as the magnitude of the coefficients A and B, are compatible with the values obtained in Ref. [ 11 ]. We have also tried a different fit, in which D1 is described by a purely perturbative-like term, while D is still the sum of a non-perturbative exponential term plus a purely perturbative-like term. In other words, we have fixed B = 0 and ,~a = 0 in the parametrization (2.10) for 7) and Dl:

376

A. Di Giacomo et a l . / N u c l e a r Physics B 483 (1997) 371-382

/'"f 10

....

I ....

I ....

I ....

TM

1 O*

101

1Or

,,,I,,,,I,,,,I,,,,I,,,,I 0

0.2

0.4

0.6

0.8

1 FM

Fig. 2. The function :D~If ( f l ) - 4 versus physical distance (in fermi units). Crosses correspond to fl = 6.6, triangles to fl = 6.8, hexagons to fl = 7.0, diamonds to fl = 7.2; crossed circles correspond to the data of Ref. [11]. The line is the curve for 7:)11 obtained from the best fit of Eqs. (2.10) and (2.11). a

79(x 2) = A e x p D 1 (X 2) =

-

+ ixl-- ,

b

(2.13)

ixl 4.

We have thus obtained A A---~L___2.7 x 108

1 1 AA -- AL 183

a

0.4,

b

0.3

(2.14)

The value of x2/Nj.o.f, is again acceptable (about 2) ; the slope of the non-perturbative exponential term is practically unchanged. The behaviour for DII and D ± obtained using the parameters from this best fit is close to the continuum lines reported in Figs. 2 and 3. We stress again that we have been able to observe terms proportional to 1/Ixl 4 in the correlations because we have worked at larger values of/3, where the distance between two points (far enough in lattice units so that the correlation is not modified by cooling before lattice artefacts are eliminated) is small compared with 1 fm in physical units. A larger lattice (324 ) has been necessary to avoid infrared artefacts.

A. Di Giacomo et al./Nuclear Physics B 483 (1997) 371-382

'"1

....

377

I""1''"1''"

10 '1

1 0 '°

--

1 O"

10 I

10 ~

I I I I I [ I I I I I I , [ , I I I I I I I , I 0

0.2

0.4

0.6

0.8

1 FNI

Fig. 3. The function D~f(/3) -4 versus physical distance (in fermi units). The symbols are the same as in Fig. 2. The line is the curve for 79± obtained from the best fit of Eqs. (2.10) and (2.11).

3. Computations and results at finite T To simulate the system at finite temperature, a lattice is used of spatial extent N~ >> Nr, Nr being the temporal extent, with periodic boundary conditions. The temperature T corresponding to a given value o f / 3 = 6 / g 2 is given by

1

N~.a(/3) = ~,

(3.1)

where a(/3) is the lattice spacing, whose expression in terms o f / 3 is given by Eqs. (2.7) and (2.8). The gauge-invariant two-point correlators of the field strengths in the QCD vacuum are defined in Eq. (1.1). At zero temperature, that is on a symmetric lattice N~ = NT, they are expressed in terms of two independent invariant functions of x 2, called 79(x 2) and 791 (x2), as shown in Eq. (2.1) [ 4 - 6 ] . At finite temperature, the 0 ( 4 ) space-time symmetry is broken down to the spatial 0 ( 3 ) symmetry and in principle the bilocal correlators are now expressed in terms of five independent functions [ 14-16] ; two of them are needed to describe the electric-electric correlations

378

A. Di Giacomo et al./Nuclear Physics B 483 (1997)371-382 (OITr { E i ( x ) S ( x , y ) E k ( y ) S t ( x , y) } 10) 20Df] 8Of = ~ik DE + D f + u4 0--~-]j + blil,I k 6~U2 ,

(3.2)

where Ei = Gi4 is the electric field operator and u u = x u - yu [u 2 = (x - y)2]. We are considering the Euclidean theory. Two further functions are needed for the magnetic-magnetic correlations

[DS + D f

{OITr{Bi(x)S(x,y)Bk(y)St(x,y)}lO)=Sik

+

1

a u Z j --UiUk ou2 '

(3.3) where Bk = 7eokGi 1 j is the magnetic field operator. Finally, one more function is necessary to describe the mixed electric-magnetic correlations: {0jTr { E i ( x ) S ( x , y)Bk ( y ) S t (x, y) } 10)

1

=

ODBE

---~13iknU n OU4

(3.4)

In Eqs. (3.2), (3.3) and (3.4), the five quantities D e, D~, D ~, D~ and D~ E are all functions of u 2, due to rotational invariance, and of u 2, due to time-reversal invariance. From the conclusions of Refs. [ 14-16], one expects that D e is related to the (temporal) string tension and should have a drop just above the deconfinement critical temperature To In other words, D E is expected to be the order parameter of the confinement; on the contrary, D~ does not contribute to the area law of the temporal Wilson loop. Similarly, D B is related to the spatial string tension [ 14,15], while D~ does not contribute to the area law of the spatial Wilson loop. We have determined the following four quantities: "Off(u2,0) ~ ~)E(u2,0) --~ O f ( u 2 , 0 )

+ u2 az~f (u2, o) art2

79~(u 2, 0) = DE(u z, 0) + 7 ~ ( u 2 , 0) Dff(u2,0) ~ "~B(u2,0) --F ~IB(u2,0) "D~_(u2,0) ~ "DB(u2,0) -q--~)B(u2,0 )

2 0DB

2

+ u -~5-uz(u ,0), (3.5)

by measuring appropriate linear superpositions of the correlators (3.2) and (3.3) at equal times (u4 = 0), on a 163 x 4 lattice (so that, in our notation, Nr = 4). The critical temperature Tc for such a lattice corresponds to tic -~ 5.69. The behaviour of DI~ and Dff_ is shown in Figs. 4 and 5 respectively, on three-dimensional plots versus T/Tc and the physical distance in the range 0.4 up to 1 fm. Due to the logarithmic scale, the errors are comparable with the size of the symbols and the lines connecting the points are drawn as a guide to the eye. A clear drop is observed for DI~ and D~ across the phase transition, as expected. The analogous behaviour for DI~ and Dff_ is shown in Figs. 6 and 7. No drop is visible across the transition for the magnetic correlations.

A. Di Giacomo et aL /Nuclear Physics B 483 (1997) 371-382

379

corr_l: ara_E

le+08

I :

le+07 i

le+06

, , . ,

,

:

'

'

'

'

i

~

i

i

i

i

,"

'

ii

'

i i

i

'1

1

~ !h , ~ I~ " ~i ',I ."~ I 095 I ii'l ~ ~ "i I *'1 ." I • • " ~Z_.,.,. v. 4--,-"1 ............................ , ......... , .................... ":-'-'-'~ 1 • j ./q,-- - -r-:.'--,-- --~i' . . . . . . . . . ;~ ' . . . . . . . . / . . . . . . . 7 " " . . . . . . . . . " ' . . . . . . . . •. ,*. . . . . . . S " ~' t 1.05

10000o /-i---y

....... :-:,~ . . . . . . . . 7 ........ J

0.4

0.5

0.6

..... f ....... / " ....... ;,. . . . . . . . -~,~"

0.7fm

0.8

0.9

T/T(:

1.1

1.1

1

Fig. 4. The quantity ~ fEl / A ~4 (defined by the first Eq. (3.5)) versus T/Tc and the physical distance (in fro). corr_p ~rp_E

i i

i

le+08

', i

!

I i

i

, ,

,

I

J

i

i

i:,, i

t i

le+07

i

,,'

i

i i

'

,

i J t

i

, i

' le+06

i

III

i i

''

~ '

I JL

i

,

' ' "

'

,

' '

Pt i

g

i

i

i

d

, ,"' i i,

,

i q i

i

"' i i

t

:

ii

, i

ii J

i

I

lit

'

I

,, i

i

i i

~,' ',

*r ,

J t

i i

i i i

,i

Ii i I

'

'

i

i

' ", i

'

i i

i

.~ i

i i

i

, ''':' ~,, ~

, i i

t~ ii ip

i

i

i

i

~

I i

I I

i

i t p

i

i i

i i

,

i

,

i

,' . . . ., .

',

'

i

i i i

i

,

I i i

, i

i

,i

I,, ' I i

i

i i

i

,i

i i

i

i

!/ ; l0 .|t : ,., 'z :.. i' ,.: ,'i.',; ,,.-i ,; . , i ,../ 0.95 ~---r'1"~,~'d'"'4-:,1"," ~,'"':,~"1-~"Y"~'"t -h"• . . . . . r"" ",1~. . . . . . "'~'~('-'"':1 / I . . . . . . . . . . . 1 - - ~ ' . . . . . . ;--;,'4 . . . . . . ,~/ . . . . . . . , ~ . . . . . -:--,,-~ . . . . . . -,~. . . . . . . . . , , ' 7 " . 1 . 0 5

TEe

100000 / - i - - - / ..... :-7 ........ V'....... 2( ..... /'I ...... 2"--,----) ...... --7:7" 1.t 0.4

0.5

0.6

0.7fm

0.8

0.9

1

Fig. 5. The quantity 79~L/A4 (defined by the second Eq. (3.5)) versus fm).

1.1

T/Tc and the physical distance (in

Finally, w e have also measured the mixed electric-magnetic correlator of Eq. (3.4). When computed at equal times (u4 = 0 ) , this correlator turns out to be zero both at zero temperature and at finite temperature. This is because the function D1BE at the right-hand side of Eq. (3.4) depends on u~, as a consequence of the invariance of the theory under time-reversal. We have verified this directly on the lattice. Then, we have measured the same correlator (3.4) for u4 = 1 lattice spacing and lul = 3 , 4 , 5 , 6 , 7 , 8 lattice spacings. The results of these measurements are shown in Fig. 8.

380

A. Di Giacomo et aL/Nuclear Physics B 483 (1997) 371-382 c o r r _ l araB

I, ii' ,, ,, j

le+08

J

I

i

' i

i

',

i

i i

le+07

'

] e ~

i

i

Ii i

I

t

Ii

~

j

o

=l

"'

' ,

i

i

i

i

,.

i

Jl

i

iii

i

, :l,ll'

i I

' '

i

i

' :.ill

i

' ,

'

i

i

i

i

i

,

i

Ii

i

iI

i

i

I

i

I

i

i

ii

i

~ '.-', iI ;,:'

;:i.;

i

i

i

iI

;~

;l,i

i

i

',1~,;

i

i

i

i

i

i

;,-;

/~?:`!~i~.)~.~`~-~;'`~!~!~-~;r'.~.~:~.~.:-.~.~`

IO0000V

t/--, . . . . . .. . . . . . ,--~, . . . . . . . ,-.' . . . . . . . . ~' . . . . . . . . " ~ . . . . . . . / - - , . . . . . , . . . . . . . ' ~ ' ," '/ /1 / ' t ' ~ 0.4 0.5 0.6 0.7fm 0.8 0.9 1 1.1

Fig. 6. The quantity 7 9 1m1 / A 4L

(defined

by the third

Eq.

(3.5))

versus

T/Tc a n d

.-,:-/

t/

0.95

1705

I 1

T/Tc

""

the physical

distance

(in

fm).

corr_I erpB

1 e+08

~i 0.4

0.5

0.6

0.7fm

i:11: 0.8

0.9

1

1.1

Fig. 7. The quantity "Dj_/A B 4L (defined by the fourth Eq. (3.5)) versus T/Tc and the physical distance (in fm). Our results can be summarized as follows: (i) In the confined phase (T < To), until very near to the temperature of deconfinement, the correlators, both the electric-electric type (3.2) and the magnetic-magnetic type (3.3), are nearly equal to the correlators at zero temperature: in other words, D e_~D B-~DandD E-~D~'~D] forT
381

A. Di Giacomo et a l . / N u c l e a r Physics B 483 (1997) 3 7 1 - 3 8 2 corr tt

BE

1e+08

le+07

', ',

I I =

t ii Ii i

i

ii:, ,

le+06

,i 'I i

/

."

i

:iE ', , I i

',

i

",

•

~

,i

,

,.'

i

i

i

.:

'

.i

;

'.

~

;.-)

/ ....... .~........ :':" ...... .~'"""--" ....... ","........ ; ' " ...... ,-'---"", ---,"-¢ 1

0.95

./.----::7.'~7--1;........ 7 ...... - / ....... -t ........ ,.......... ........... : ' ~ . 1.05 100000

/ 0.4

"t" . . . . . . . / 0.5

0.6

. . . . . . . ~ . . . . . . . ~" . . . . . . . 7 " . . . . . . . ~," . . . . O.7fm

0.8

0.9

1

7"7"

T/rc

1.1

1.1

Fig. 8. The quantity - ( 1 / 2 ) ( [ul/A 4) (o~DBE/au4), appearing at the right-hand side of Eq. (3.4), for u4 = 1 lattice spacing (and lu[ = 3,4, 5, 6, 7, 8 lattice spacings), versus T/Tc and the physical spatial distance lul (in

fm). Dff_ (u 2 , O) - D~ (u 2, O) = - u 2 a79~ (u2 ' O) 3u 2

(3.6)

between the two quantities reported in Figs. 5 and 4, respectively, one finds that the quantity 79~ does not show any drop across the phase transition at Tc. So the clear drop seen in the quantities ~DI~ and 79~_ across Tc is entirely due to a drop of 79e alone. This result confirms the conclusion of Refs. [ 14-16], where 79e was related to the (temporal) string tension ~re. Similarly, one can look at the difference _

= -u

-ff~-u2(u2, 0)

(3.7)

between the quantities reported in Figs. 7 and 6, respectively. One thus finds that 792 does not show any drop across the transition and, in addition, it is nearly equal to 79~ (79 2 _~ 79~). From the fact that the quantities 791~ and 79~_ stay almost unchanged (or even show a slight increase) across Tc, we conclude that the same must be true also for 79B. It was shown in Refs. [ 14,15] that 79B is related to the spatial string tension o's. Recent lattice results [ 19,20] indicate that O's is almost constant around Tc and increases for T >>,2Tc: this fact is in good agreement with the behaviour that we have found for 79B.

Acknowledgements This work was done using the CRAY T3D of the CINECA Inter University Computing Centre (Bologna, Italy). We would like to thank the CINECA for having put the CRAY T3D at our disposal and for the kind and highly qualified technical assistance. We thank Gtinther Dosch and Yuri Simonov for many useful discussions.

382

A. Di Giacomo et al./Nuclear Physics B 483 (1997) 371-382

References [ 11 I21 131 14] [5] [61 [71 [81 [91 [101 Ill] [ 121 I 131 [14] [15] 116] [171 [181 [19] [201

D. Gromes, Phys. Lett. B 115 (1982) 482. M. Campostrini, A. Di Giacomo and S. Olejnik, Z. Phys. C 31 (1986) 577. Yu.A. Simonov, S. Titard and EJ. Yndurain, Phys. Lett. B 354 (1995) 435. H.G. Dosch, Phys. Lett. B 190 (1987) 177. H.G. Dosch and Yu.A. Simonov, Phys. Lett. B 205 (1988) 339. Yu.A. Simonov, Nucl. Phys. B 324 (1989) 67. O. Nachtmann and A. Reiter, Z. Phys. C 24 (1984) 283. P.V. Landshoff and O. Nachtmann, Z. Phys. C 35 (1987) 405. A. Krfimer and H.G. Dosch, Phys. Lett. B 252 (1990) 669. H.G. Dosch, E. Ferreira and A. Krtimer, Phys. Rev. D 50 (1994) 1992. A. Di Giacomo and H. Panagopoulos, Phys. Lett. B 285 (1992) 133. M. Campostrini, A. Di Giacomo, M. Maggiore, H. Panagopoulos and E. Vicari, Phys. Lett. B 225 (1989) 403. A. Di Giacomo, M. Maggiore and S. Olejnik, Phys. Lett. B 236 (1990) 199; Nucl. Phys. B 347 (1990) 441. Yu.A. Simonov, JETP Lett. 54 (1991) 249. Yu.A. Simonov, JETP Lett. 55 (1992) 627; Yad. Fiz. 58 (1995) 357. Yu.A. Simonov and E.L. Gubankova, Phys. Lett. B 360 (1995) 93. M. Campostrini, A. Di Giacomo and G. Mussardo, Z. Phys. C 25 (1984) 173. C. Michael and M. Teper, Nucl. Phys. B 305 (1988) 453. G.S. Bali et al., Phys. Rev. Lett. 71 (1993) 3059. E. Laermann, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 120.