Nuclear Physics B256 (1985) 717-726 fc) North-Holland Publishing Company

THE SPIN-DEPENDENT FORCES BETWEEN HEAVY QUARKS IN LATI'ICE GAUGE THEORIES

Massimo CAMPOSTRINI

Scuola Normale Superiore, Pisa and INFN, Sezione di Pisa, 1-56100 Pisa, Italy

Received 13 November 1984

We discuss the possibility of obtaining the spin-dependent forces between heavy quarks from a lattice Monte Carlo computation. The quantities to be measured are various combinations of Wilson loops with two plaquette insertions.

We present a Monte Carlo computation with the U(I) gauge group that reproduces well the perturbative results for the spin-orbit and spin-spin interactions, and suggests that an accurate SU(3) calculation is indeed feasible and meaningful.

1. Introduction

One of the most interesting problems in the study of QCD is the calculation of the mass spectrum and various properties of the bound states of the theory (like wave functions, magnetic moments and so on).

Lattice Monte Carlo methods are presently the most powerful tool in this analysis [1]; now we have reached the capability of performing the necessary calculations for a pure gauge theory on what is believed to be a large enough lattice [2], while for the much more demanding full QCD we must wait for more powerful computers.

The study of the bound states of heavy quarks offers considerable simplifications: these particles can be considered in the first approximation as non-relativistic and a consistent 1/m expansion is possible [3]. The effective hamiltonian for these systems can be obtained from Monte Carlo calculations without explicitly introducing heavy quarks [4]. By using the pure SU(3) gauge theory we obtain an approximation equivalent to the "quenching" for light quarks [5], but numerically much less demanding.

When light quarks are introduced into the simulation (with large enough lattice and statistics) this method could take care of the heavy quarks with a small extra numerical effort.

It also gives more detailed information than a "brute force" (purely numerical) calculation, like the fermion propagator computation needed to study light-quark bound states.

717

718 M. Campostrim / Spin-dependent forces

2. The spin-dependent potentials

Eichten and Feinberg [3] developed the 1/m expansion to find the spin-dependent part of the quark-antiquark system effective hamiltonian*. Starting from the full QCD lagrangian, at the zero order one obtains the familiar result

H'°) = E P~ ~--~m + V(R) , (2.1)

V ( R ) = - lim l i o g ( W ( R , T ) ) , (2.2) 1

where R is the quark-antiquark separation, W(R, T) the rectangular Wilson loop R x T and + , - are the quark and the antiquark respectively.

There is no first-order term and the spin-dependent part of the second-order term is

H~2)=S4"L++S_'L_ 1 ( d V 9 d V l ] S , .L_+S_ .L+ 1 d~ 2m 2 R - ~ + " d-R ] - m 2 R dR

l [(s+.k)(s . i¢)-~s+.s ]v~(n)- l-L-s .s v~(R) (2.3) m2 - . 3m 2 ~ ,

where L is the angular momentum, S the spin operator of each particle and

~'Jk k k dl?"t( R ' T ) d R f )rdt fordt ' ( t ' - t )(g2B'(O't)Ej(O't ' )) w, (2.4)

~'Jkk~df/2(R'T)dR fordt f~ cdt'(t'-~T)(g2B'(R't)E'(O't'))w'- (2.5)

( f~,R,- ~8,j)P'3(R,T) +~,,("4( R, r ) = f0?dt f r d t ' (g2B,(R,t)Bs(O,,')) w ,

(2.6)

where

(~) w = (e{ ~'w( R, r ) ) ) (2.7)

is the expectation value of ~? in the presence of a Wilson loop (P is the path-order- ing operator) and

1 ( / , (R,T) / (W(R T)) (2.8) V , ( R ) = lim ~ , .

* A detailed determinat ion and discussion of the following formulae is also found in ref. [6].

M. Campostrmi / Spin-dependent forces 719

The spin-dependent potential V 4 is not independent of the others; in fact

V4( R ) = 2V2V2 ( R ). (2.9)

The first-order perturbation theory gives for the V,

Ot v = v =0, ,

V 3 = - 3 C ~ 3 , C = C2(O ). (2.10)

Another contribution at the same order that must be considered is the virtual annihilation term, but of course it can be reliably calculated perturbatively; its spin-dependent part is

H2an)= 2~'ot C,(G)S+. S_33(R) (2.11) • ~ I 2 ~

For the U(1) theory, C2(G ) = 1 and the previous formulae reproduce the usual result for the positronium fine and hyperfine structure.

3. Introducing a lattice

To obtain the four I~, from QCD only, without postulating a particular model, we can perform a Monte Carlo simulation on a lattice. The measure of the i~, allows the calculation of the fine and hyperfine structure of the quarkonia, improving the results of [4].

Of course, there is a lot of freedom in the choice of lattice operators that give the spin-dependent potentials in the continuum limit. Our choice is to sum over the maximum number of loops of the appropriate form, but excluding combinations that would enhance finite-size effects.

In the following, we will take 1 as the R and 4 as the T direction, to fix the ideas. Starting from the plaquette operator

G ~ ( x ) = G ( x ) V ~ ( x + i , ) V £ ( x + b)V~'(~,), (3.1)

we define

- v ; . ( x ) ] .

that gives in the continuum limit

(3.2)

~v( x ) = ga2F~( x ) + O(a6) . (3.3)

720 M. Campostrmi / Spin-dependent force.s

I ~-i co) 3,~c-t) l~t 3,jc-Y) 3~/c-~-;)

Fig. 1. The four ~'~,. in contact with a side of the Wilson loop.

We define also

T

~ ( x ) = Y'~ ?~,(x, t ) , (3.4) t = 0

r

#.Ax)= E (t- ~r)G(x.t). (3.5) t = 0

To calculate the two spin-spin potentials, we average the mean value of two parallel space-like plaquettes in the presence of a fixed Wilson loop over the four positions with a corner in contact with the loop at a given time (see fig. 1), with the constraint of having separation R in direction 1: setting R = ] R, we define

% ( R , T) = ~ {[,%(0) + ~ 2 ( - 2 ) 1 1 % ( R ) + ~Jn2(R- 2)]

+ [ ~ l 1 2 ( - ] ) + ~ 1 2 ( - i - 2 ) ] [ - ~ 1 2 ( R - i ) + } i 1 2 ( R - ] - 2 ) ] } , ( 3 . 6 )

and the same for ~.3 2 (changing 2 with 3) and

~.'B,(R, T) = ~ [ .~23(0) + ~t23(- 2) + ~23(-:~) + ![23(-2 - :3)]

×[.%(R)+~2~(R-~)+~2~(R-$)+q~(R-2-$)]. (3.7)

The lattice analogues of (2.6) and (2.7) are then

V3(R, T) = (~.~'~1 - ~(t:~2 + ~-~3)) w, (3.8)

~'4(R, T) = ~ ~'~ (3.9) ( ,t~ + ?L~ 2 + ':~3) w ,

where (t'~)w is defined like in (2.7) (including the path ordering) and the Wilson loop has 0 as origin.

M. Campostrmt / Spin-dependent forces 721

For the two spin-orbit potentials, we choose the definitions

~';2(R, T) = ~[~2,(0) + ~ 2 , ( - 2 ) ]

× [ S t 2 ( R ) + ~ q t 2 ( R - 2 ) + ~ t 2 ( R - i ) + ! ~ , z ( R - ' l - 2 ) ] , (3.10)

~

and the same for ~';3 (changing 2 with 3) and for ~2.3 (changing fl with ~,i ). Finally, the lattice analogues of (2.4) and (2.5) are

dlT](R, T) - . dR = (( b';2 + ~';3 - ~';~ - ~';3)1 g =°) w" (3.11)

dlT"2(R, T ) dR (&2 + ~';3) w" (3.12)

We have verified by perturbative methods that no divergences are present in the I/, at the leading order.

To obtain the ~ from the I~,, it is better to use the formula

V , ( R , T ) = ~(R,T+I)/(W(R,T+ 1)) - ~(R,T)/(W(R,T)), (3.13)

~ ( R ) = lim V,(R,T), (3.14) T - ~ o c

rather than (2.8), for the same reason that one uses the XS to calculate the force between two static sources: the new definition reduces very much the dependence on T [2,7].

4. Monte Carlo simulation

To study the possibility of a Monte Carlo simulation to get precise values of the spin-dependent potentials for the gauge group SU(3), we have performed a simula- tion with the gauge group U(1).

This group has the great advantage of not requiring path ordering; we can therefore calculate for every configuration the sums ~q and ~i independently for each site and then multiply them by the relative Wilson loop, with a big computational simplification. Moreover, we can compare the results with the known continuum perturbation theory to check the reliability of the method.

We have used a 16 × 123 lattice with periodic boundary conditions and the gauge group Z(128), that approximates very well U(I) up to large fl, as was shown in [8]. We have chosen the value fl = 2.5, far enough from the phase transition and with an a = l/(4¢rfl) not too small.

722 M. Campostrmi / Spin-dependent fi~rces

The FORTRAN program updates the configurations with a standard Metropolis 3-hit algorithm. Every 6 iterations, the following operations are performed:

(i) For every site of the lattice, the six ~,, are computed and stored. (ii) The transport factors for length up to 6 in the three space directions are

computed and stored. (iii) Finally, the Wilson loops and the four operators I?, are computed in the

following way. For each time slice, in a loop over time length T of the loops up to 8, the !~s, the ~{~s and the transport factors in time direction are computed; if T is bigger than 2 they are immediately used together with the pre-calculated quantities to compute the loops for every space length R up to 6 and averaging over the three space directions 1, 2 and 3.

Since the lattice is too big to fit into the computer memory, the link variables, the ','~ and the transport factors are kept on direct-access disk files, divided into time slices and (for the latter) also divided in the three space directions.

The CPU time required on an IBM 3081 is 6.3 sec for each upgrading and 85.2 sec for each measure; measuring only the Wilson loops with one side in the time direction with the same code requires 17.9 sec.

After the 500 sweeps discarded to ensure thermalization, 1200 sweeps are used to obtain 200 almost statistically independent measures of the Wilson loops and of the I~,.

5. Analysis of the data and results

The analysis is performed in analogy with [2]: the values of the five loops are averaged in blocks of 4 measures (and so of 24 sweeps), that are used to compute the values, the statistical errors and the cross-correlations of the I~, and the Wilson loops.

The values and the errors for V,(R, T) and

x ( R , T) = - l o g (W(R + 1, T + 1 ) ) ( W ( R : T ) ) (5.1) (W(R,T+ I) ) (W(R+ I r ) )

are then given by

( i F ) 2 = _ _ N-1 1 y'~,~ 0~, A,=(A,> OA,CgF A,_(A,)d(A,A,) ' (5.2)

where F is V, or X and is given by (3.13) or (5.1), A~ is the appropriate I?,, or W and

d( A,Aj) = ( A,A,) - ( A,)( A,). (5.3)

We have checked that the block-block correlation is negligible and that bigger block lengths give similar results.

n~

-I

6 "o

0 C

O

ci.

E"

-I

0

o 0 0 C

) C

:)

I:1

I~:

0 --

.M

k ~ II

(D

e

-

PO

I - m

!

N

6 (2)

0

0 0

o 0

C)

0 0

724 M. Campostrim / Spin-dependent fin'ces

0 0 0

- . 0 0 2

- . 0 0 4

- 0 0 6

- 0 0 8

i L h ~ _ i L i L

2

, - . . . . . " r ~ - -

0 2 4 6

R

Fig. 4. The spin-orbi t potenl ia l d l ,~/d R as a funct ion of R.

Note that, since F is linear in ~ and the relative error on W is always small (less than 0.5%), the errors calculated in this way are reliable also when AV, > V,.

Since the T-dependence of these quantities is negligible (except for the smallest T) we have computed V,(R) and x(R) as averages over T = 5.6,7 of V,(R, T) and x(R,T), with the appropriate statistical weight; they are shown in figs. 2-6, compared with the perturbative results (2.10).

The resulting V, are in good agreement with (2.10), and their statistical errors are similar to those of X.

We can therefore conclude that an accurate Monte Carlo computation of the spin-dependent potentials for the gauge group SU(3) is possible and not much more demanding than the computation of the force between static sources. In fact, in the latter case, most of the CPU time is spent in the upgradings [2], while in the present case the upgrading procedure and the Wilson-loop computation arc made very fast by the use of a discrete group, that cannot also be used for the computation of the V.

Since the results for U(1) are encouraging, the physically interesting case of SU(3) will be the subject of a subsequent work [9 I.

,000

-.002

- 004

-.006

-.008

i

0

’ R ‘4 6

Fig. 5. The tensorial spin-spin potential V3 as a function of R.

I

.O

-. 1

- 2

-.3

-.4

_-

t

A

‘1 0

2 R 4 6

Fig. 6. The scalar spin-spin potential V., as a function of R.

725

726 M. ('ampostrim / .~ptn.ch'pendent [orce~

W e w i s h to t h a n k C. R e b b i for h a v i n g s u g g e s t e d the s u b j e c t of the p r e s e n t w o r k

a n d for h i s c o n s t a n t in t e res t , a n d G. Pa f f u t i for he lp fu l d i s c u s s i o n s .

P a r t o f t h i s w o r k was d o n e at the B r o o k h a v e n N a t i o n a l L a b o r a t o r y a n d s u p p o r t e d

b y t h e D e l l a R i c c i a F o u n d a t i o n . T h e c o m p u t a t i o n was p e r f o r m e d a t the C e n t r o

N a z i o n a l e di C a l c o l o E l e t t r o n i c o in Pisa, w h o s e c o l l a b o r a t i o n is a c k n o w l e d g e d .

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