Gluon overlap factors extracted from Monte Carlo lattice calculations

Volume 258, n u m b e r 3,4

PHYSICS LETTERS B

11 April 1991

Gluon overlap factors extracted from Monte Carlo lattice calculations O. M o r i m a t s u Instttute for Nuclear Study, Umverstty of Tolcvo, Tokyo 188, Japan A.M. Green 2 Research Institute for Theoretical Physics, University of Helsmkt, SF-O0170 Helsmkt, Fmland and J. P a t o n 3 Department of Theoretical Physics, Umversltv of Oxford, Oxford OXI 3NP, UK Received 21 D e c e m b e r 1990, revised m a n u s c r i p t received 4 February 1991

A strategy is proposed for extracting gluon field overlap factors from Monte Carlo lattice calculatmns involving static ( q q ) (qdl) configurations This ~s d e m o n s t r a t e d by m e a n s of an existing lattice calculation that restricts the quarks and a n t l q u a r k s to a rectangular geometry The results are m q u a h t a t i v e agreement with weak coupling at short distances, but more definite conclusions could not be m a d e at this stage, since the a p p r o p r i a t e M o n t e Carlo calculations are - as yet - not suffioently precise.

1. Introduction In two recent papers [ 1,2 ] m e s o n - m e s o n scattermg has been formulated in an adtabatic basts for QCD in which " m e s o n " states [M (q,CtjG,:)) have quarks and antlquarks at fixed points and G,: represents the ground state of the gluon field h a m i l t o n l a n for these positions of q u a r k - a n t t q u a r k sources. Variational wave functions for states containing two quarks and two anUquarks are then made m terms of I A ) = IM ( q j c l 2 G t ~ ) M ( q 2 ( 1 4 G 2 a )

) ,

IB) = [M (qlft4Gaa)M(q2dI3G2~)) ,

( 1.1 )

as shown m fig. 1 In ref. [2] a third channel, with a different gluon configuration, was also introduced. But th~s has only a very m m o r effect on the main issue of interest here. Therefore, in this discussion the Bltnet address M O R I M A T U @ J P N U T I N S 2 Bltnet address G R E E N @ F I N U H C B 3 B~tnet address J A N E T % " P A T O N @ V 1 P H O X AC U K "

11 t a)

b)

Fig 1 The two-gluon configurations A and B o f r e f s [ 1,2] (a) The two meson state I A ) =M(q~ChG~3)M (q2q4G24)), (b) The two meson state I B ) = I M (ql~14G 14) M (q2q3G23) ) • =~ q u a r k × = ant~quark

formahsm will be illustrated only in the two channel case. In this varlattonal basts there are three dtstinct contributions to the scattertng, each of which preserves the two extreme limits of weak and strong coupling. ( 1 ) Non-orthogonahty effect. The states A and B are not orthogonal. In the weak coupling limit the offdiagonal matrix elements of the overlap matrix N have the form

0 3 7 0 - 2 6 9 3 / 9 1 / $ 03 50 © 1991 - Elsevier Science P u b h s h e r s B V ( N o r t h - H o l l a n d )

257

PHYSIC‘S LEl’TEKS

Volume 25X. numhcr 3.4

I I April

H

teal. which

IS coulomblc

at long distances. clcmcnts

= ‘/‘ 1

=(M(q,ql)M(q,qJ)IM(q,q,)

whereas.

In the strong

- as polntcd

(1 2)

coupling

Ilmlt

[ 3 1.

out In ref.

the quarks

ample.

IS Icro

result

lntcractlons

m a matrix

the off-diagonal

1’

matrix

I’. where.

elements

bcfor cx-

(weak

llmlt)

(strong

coupling

teals I’,, arc replaced

.

and

when

). The

lntcrquark

side of the second form

are close

1-O

limit

by the full

also on the right-hand

(1.6)

and antlquarks

coupling

( I .6 ) The complete

are

as

/.A)

+I when all quarks

togcthcr

are all far apart

( 2 ) D~w(Y /~~~ouc~/o~~. The gluon tween

1B)

(A

whcrc

and linear

the off-diagonal

.

(BI l/‘l.A) =,/< BI I’(wcak)

xM(qzS1) > = \ >

dlstanccs

hand.

I ‘arc wrlttcn

of .Y and

(HI:\)

at short

On the other

1991

proposed

they

potcn-

potcnttal equation

In

I IS,

for :I’ and

therefore. (AI

I’iB)

=(M((~,q,G,)M(q,qJCi?)lH,,lM(q,q,(i,) xM(q,cl~G,))

(1.3)

In the weak coupling

I has the form

llmlt

I ‘(weak )

In rcl’. mcnts whcrc

[ I].

[4],

on the basis of strong f IS parametrized

CTIS the string

tension

as a free paramctcr where

I’,, IS the Interquark

gluon-eschange strong

where

(OGE).

coupling

I ‘(strong)

=

potential

0

energy since

vo1L.e static

quarks.

for

duced I’=
given

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>.

the

A and B arc nonmatrix

[ I.?]

for the

for

wll

dIscusston

details. only

In-

wll not enter.

parametrizations

matrices

between above.

the

strong

In the same form the only

dlffercncc

potcntlal.

by lust replacing

the whole

and

weak

clcmcnts

introand

coupling of N and

m the two cxtrcmc they

practically Ilmlt.

as /, =exp( down

wcrc

I’,, b> the full lntcrquark

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Intcrpopotcn-

meson-meson -/&‘

all

of

form

system.

check

value

prlnclplc.

InteractIon

such

this

form

with

as algcbralc

matrix.

out that Monte

of the above

.A strategy

kzl

regime

that can bc done

for the pure gauge system

the valIdIt\:

Carlo

lattice

can be used to

approach

and also to

Idea that IS csscntlally is outllncd from

cutpres-

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In a separable

to point

I.C. the

In the transitional

for the scattering

[ 2 1.

In ref.

parametrized

of the SIX links

s~rnpl~c~t~cs

suggest the I‘orm off-an in r-cl: (41

to gcncrate

interaction.

of,/ leads to lntcgrals

11s attendant

calculations

~fcu IS > 0. I

too weak

~11th P=kr7/6

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WC here wish

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the average

seems to bc a t!,plcal

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ent In a (qq)(qq)

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reverts

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InteractIon

argu-currS).

CYIS treated

If CYIS chosen

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Therefore.

and antiquarks

This latter

were

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The diagonal

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see refs.

this term

quarks

mainly

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In refs

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bounded

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there IS also a non-diagonal

llowever.

In the

=. I GcV/fm.

and S ts the mlnlmal

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then the model

(I,,AfL?J)

( 3) Krncv/c P/Ic,:~J~ /cm. kinetic

hand.

0

(L’l,+1’2.0

II,, is now a linear

orthogonal

the other

by onc-

1‘has the form

llmlt

(

On

- given

coupling

as f, =cxp(

for

“cxpcrlmental

glven

cxtractlng. data“

In

Volume 258, n u m b e r 3,4

meaning

Monte

Carlo

PHYSICS LETTERS B

lattice

calculations

for

(qCl) (qq) states. In the Monte Carlo lattice calculations for the pure gauge system the ground state energies o f the gluon field hamfltonlan are obtained for given q u a r k - a n t i quark sources. If the models proposed in refs. [ 1,2 ] a r e v a l i d , the lowest potential elgenvalues 2 o b t a m e d by solving det ( V - 2 N ) = 0, where N and V are gaven by eqs (1.7), should be c o m p a r e d with the corresponding ground state energies in lattice calculations.

2. Results At the present rime, almost the only data available for the above strategy is that o f ref. [ 5 ]. However, it should be noted that they are still far from being ideal for this purpose. In partacular, in a d d i t i o n to the statistical errors explicitly referred to m the following comparison, there as also a systematic error due to their use o f a small time-lattice i.e. the results do not correspond to the zero temperature h i l t . This could well mean that, even if we could obtain f it might differ to some extent from the zero t e m p e r a t u r e one In v~ew o f this, few definite statements can be drawn from the comparison with the present data These should be reserved until a calculation better than that o f r e f [ 5 ] has been carried out and m which the above objecnons have been confronted. In spite o f this the authors believe that the following qualitative mvesn g a n o n is worth making. In ref. [ 5 ] the two types of rectangular configuranons for (qCt) ( q q ) states shown in fig. 2 are consadered: ( a ) shows the antlparallel ( A P ) c o m b i n a t i o n as the two associated meson configuranons A and B and ( b ) shows the corresponding parallel ( P ) comblnanons. In this way, as a function o f d and m for

k A

B

a)

A

B

b)

Fig 2 (a) Annparallel (AP) combmanons glwng mesons of "'length" L=da and a d~stance apart ofR=ma A and B refer to the two meson configuranons of fig 1 (b) Parallel (P) comblnanons and the correspondmg meson configurations A, B

I 1 April 1991

d = 0 , 1, 2 and l ~ < m ~ 8 , a (qq)(qCl) potential V(d, m) was extracted, m umts o f 1/a, where a is the lattice spacing. Also in this reference, a Monte Carlo latrice calculation for qCl configuranons showed that, for a quark and antiquark a distance ra apart, ogo

av(r) = a a 2 r - - - - vo,

(2.1)

r

where aa2=0.088_+0.003, ao=0.30+001 and Vo= 1.04_+0.01, consistent with other d e t e r m m a tions. In eq (1.7) for the matrix V, the v,j are identified with the v(r) o f e q . (2 1 ) As can be seen from fig. 3 in ref. [5], the only values o f (d, m ) that lead to significantly non-zero potentials V(d, m ) are for (d, m ) = (1, 1), (2, 1 ) a n d (2, 2). A comparison is now m a d e between V(d, m ) , the (qcl) (qCl) potential o f ref. [ 5 ], and the lowest potential eigenvalues 2 (d, m ), i.e V(d, m ) <~ I?(d, m )

=2(d, rn)- [v13(d)+v2~(d)] .

(2 2)

Here the second term simply removes the energies o f the two mesons as R ( m ) ~ and ensures 17(d, rn) -~0 in this limit. Results are shown in table 1 for f = 1, 0.9 and 0.5 as the gluon field overlap factor defined by eq. ( 1.6 ) Thas selectaon o f f ' s is chosen since: (a) The value f = 1 corresponds to no cut-down due to the overlap and so maxamlzes the absolute value of the potentml I?(d, m ) . ( b ) F r o m the above defimtlons off~,2 for a ~ 0 . 1 , k~½ the value o f f - - 0.9 occurs roughly at (d, m) = (2, 2). It is therefore selected as a typical value in the transitional regime for the range of (d, m) o f interest. For this estimate, a lattice spacing of 0.15 fm ~s used as m ref. [ 5 ]. (c) The numbers for f = 0.5 are quoted smce they a p p r o x i m a t e l y correspond to the choaces a and k ~ 1 which are representatives of strong cut-down factors. Further results are shown m fig. 3 for the two extremes f = 1 and f = 0 , but treating R as a continuous variable. Several poants should be noted. ( 1 ) All the numbers are in quahtatlve agreement with the " d a t a " of ref. [5] p r o v l d l n g f ~ 1. Thas implies that the range o f (d, m ) stays m the weak coupling regime. 259

Volume 258, number 3,4

PHYSICS LETTERS B

1 l April 1991

Table 1 Column 1 the values of (d, m) corresponding to significantly non-zero potenuals m ref [5] Column 2 a representauve selectmn of gluon field overlap factors/defined by eq ( 1 6) Columns 3 and 4 the (qq) (qq) potenUal a V(d, m ) defined by eq (2 2 ) for the two quark configuratmns, anUparallel (AP) and parallel (P), shown in figs 2a and 2b, respecuvely Columns 5 and 6 the corresponding potenual values read from fig 3 o f r e f [5] (dm )

(1,1)

(2,1)

(2,2)

/

a V(d, m )

"'Experimental" data

AP

P

AP

10 09 05

-00621 - 0 0574 - 0 0355

-00311 - 0 0246 - 0 00710

-0

I0 09 05

- 0 477 -0477 - 0 476

- 0 224 -0187 - 0 0734

-048+001

10 09 05

- 0 0584 - 0 0539 - 0 0333

- 0 0292 - 0 0231 - 0 00668

(2) Even though there is quahtatlve agreement, m detail the results are not what was expected, since some of the "data" from ref. [5] appear to be larger In magnttude than the above numbers even for the case w~th f = 1 It is not clear whether this should be taken too seriously at this stage m view of the present accuracy of Monte Carlo lattice calculations. (3) In the (d, m ) = ( 2 , 1) case, the constancy of the anttparallel results with appreciable magnitude, asfdecreases, is an artifice of the definition of 17(d, m) by eq. (2.2), since asymptotically the state correspondmg to ( 1, 2) - B m fig. 2a - has a lower energy than the one corresponding to (2, 1 ) - the state The most part of 12(2, 1 ), thus, comes from the difference of asymptotic quark-antxquark potentials m two states, i e . (t,12~-~/)23) - ( u 1 3 - ] - v 2 4 ) . After removing thts "trivial" part, the rest becomes - 0 0012 equal to that for ( 1, 2) A stmllar problem does not arise with the parallel results, since there state B always has a higher energy because the q u a r k - a n t i quark distances are always larger In the state B - as seen m fig. 2b - and the lnterquark potential is monotonically increasing. This suggests that the antlparallel configurations with d > tn are less useful in the present strategy for extracting f 's. (4) One definite conclusion that can already be seen from fig 3 Is that even for f= 1 - l e. no cut down due to exphclt gluon effects - there is still a rapid decrease in the mteractlon as R increases In ref [5] 260

P

-005±001

1 ±001

-027±001

~-008+005

thts Is interpreted as due to colour charge shielding. However, the above calculation shows that th~s rapid decrease ts already present when using the conventional two-body potential ofeq. (2.1). This is one of the reasons why tt did not prove possible to extract f from the results ofref. [ 5 ] with their present accuracy. Several questtons can be asked concerning the stablhty and completeness of the above analysts and also its feastblhty with the present data from ref. [ 5 ]. ( 1 ) Since the qq potential ofeq. (2.1) plays a crucial role, it was checked that variations of the parameters o~a2 and e~o within the quoted error bars did not make significant changes. For example, using a a 2 = - 0 091 and c%=0.31 changed I?(2, 1 ) from - 0 . 1 8 7 to - 0 . 1 9 4 1 e. an increase of less than 5%. (2) As discussed in ref [ 2 ], exotic q2q2 states can be introduced However, these make a rather small contribution at the values of (d, m ) of interest here. Thts ~s because, m the weak coupling hmlt, states A and B form a complete set for descrtbmg the scatterrag. If the Monte Carlo calculations were extended for the positions of q u a r k - a n t i q u a r k sources beyond the weak couphng regime, to where an e x o t t c q2Ct2 state has an energy which ~s either the lowest or comparable to the lowest, it would probably be necessary to also include these exotic states in the complete analysis.

Volume 258, number 3,4

PHYSICS LETTERS B

aC/(d=l,R) 08

15 R/a

0

)

//

-005

/i/I p ,,/

-0 10 f = 0 "/'1

/'%" f = 1

AP

aV (d =2,R) 05 I

10 i

~

I

~

I

15 i

i

i

~

]

20 L

i

i

I

/

R/a

I

11 April 1991

t r a c t m g f t h e g l u o n field o v e r l a p . I n t h e a p p h c a U o n to t h e d a t a c u r r e n t l y a v a i l a b l e , t h e results are m q u a l i t a t i v e a g r e e m e n t w i t h w e a k c o u p l i n g at s h o r t distances. H o w e v e r , a m o r e d e f i n i t e e x t r a c t i o n c o u l d n o t b e a c h i e v e d at present, s i n c e t h i s d a t a , in effect, is so f a r o n l y r e s t r i c t e d to s h o r t d i s t a n c e s a n d also suffers f r o m t h e use o f a s m a l l U m e - l a t t l c e as d i s c u s s e d earher. T h e r e f o r e , t h e r e is a g o o d case for p e r f o r m i n g m o r e M o n t e C a r l o latUce c a l c u l a U o n s m t h e q2C12 syst e m u s i n g l a m c e s t h a t are a b l e to yield (qCl) (qCt) pot e n t i a l s V(d, m ) t h a t are s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o a n d yet n o t in t h e w e a k c o u p l i n g regime. Bec a u s e o f t h e dlfficulUes m e n t i o n e d e a r h e r , for n u m e r i c a l r e a s o n s it is p r o b a b l y b e t t e r to c o n c e n t r a t e such calculations on quark configurations which a v o i d i n c l u d i n g a large " t n w a l " c o m p o n e n t as ~s p r e s e n t in t h e a n U p a r a l l e l case w i t h d > m

Acknowledgement -05-

P/

//""

/ "

/

O n e o f t h e a u t h o r s ( O . M . ) w i s h e s to t h a n k t h e Res e a r c h I n s t i t u t e for T h e o r e t i c a l Physics, H e l s m k i , for t h e i r h o s p i t a l i t y d u r i n g w h i c h txme m o s t o f t h i s w o r k w a s c a r r i e d out.

-10-

b Fig 3 The potential ~2(d, m), m units of a, defined by eq (2 2), for (a) d = 1 and (b) d = 2 and with continuous values for R/a Solid lines: the antlparallel (AP) combmauon of (qcl)(qCl) m fig 2a with f=0, 1 Dashed line the parallel (P) combination m fig 2b with f= 1, the f = 0 potential being always zero In (a) the two points at R/a = 1 were read from fig. 3 of ref [ 5 ] and correspond to the AP(O ) and P( × ) combinations In (b) the point ([]) at R/a = 2 as read from ref [ 5 ] cannot d]stlngmsh between AP and P

3. Conclusion In t h e a b o v e , a s t r a t e g y h a s b e e n o u t l i n e d f o r ex-

References [ 1 ] c Alexandrou, T Karaplpens and O Monmatsu, Nucl Phys A 518 (1990) 723. [2 ] B Masud, J Paton, A M Green and G Q Llu, A model for quark exchange m chromodynamlcs illustrated by a description of meson-meson scattering, Helslnk~ preprmt HU-TFT-90-67, Nucl Phys A 527 (1991), to appear [3] D Robson, Phys Rev D 35 (1986) 1029, G A Miller, Phys Rev D 37 (1988)2431, Phys Rev C 39 (1989) 1563 [4] O Monmatsu, Nucl Phys A 505 (1989) 655 [ 5 ] S Ohta, M. Fukugna and A Ukawa, Phys Lett B 173 (1986) 15

261