P H Y S I C A L R E V I E W D V O L U M E 1 9 , N U M B E R 8 1 5 A P R I L 1 9 7 9

Spin-orbit terms of the heavy-quark potential in the presence of instantons

Carlos Aragiio de Carvalho Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540

and Departamento de ~isisica, ~ont i j fc ia Uniuersidade Catblica do Rio de Janeiro, Brasil (Received 21 August 1978)

Spin-orbit (3.f) terms of the heavy-quark potential are calculated taking into account the effect of instantons in the dilute-gas approximation. It is shown that these terms can be expressed by means of differential operators acting on a certain potential which is not related to the spin-independent potential of Callan et al . A rough estimate of the terms is also presented.

I. INTRODUCTION

In a recent paper Callan et ul.' have calculated contributions to the heavy-quark potential in the presence of instantons using the dilute-gas approx- imation (DGA).2 Their approach consisted of ob- taining the Green's function f o r fermions in a n ex- ternal non-Abelian gauge field through a system- atic expansion in powers of m"' of the correspond- ing Dirac Hamiltonian. Such an expansion, pro- vided by a Foldy-Wouthuysen transformation, allows one to wri te the Hamiltonian a s H =HI +Hz, where to 0 ( l /mz)

with A, =A: ra/2, B, = + ~ , , ~ F ; ~ 7 " / 2 , and Di i s the covariant derivative. One can then use s tandard nonrelativistic Feynman path-integral methods to a r r i v e a t the propagator. The calculation was performed for the one-instanton c a s e and then used to obtain the time-evolved quark wave func- tion and, f r o m that, an effective Hamiltonian f o r the quark-antiquark system. Integrating this r e - sul t over positions and sca le s i z e s of instantons in the DGA with density D@) leads then to the final resul t .

P a r t of the t e r m s presen t in the effective Ham- iltonian, in fact mos t of them, can be expressed a s the resul t of differential opera tors acting on the spin-independent heavy-quark potential (V). These t e r m s have been thoroughly analyzed in Ref. 1 and a r e essent ial ly of two types: "spin-spinJ' t e r m s and "spin-orbit" t e rms . There a r e , however, t e r m s which a r e not related to the spin-independ- ent potential, although they can be written in t e r m s of another potential function 7. They a r e spin-orbit (a'. z) t e r m s , identical in f o r m to s o m e that were already encountered in Ref. 1, although

s m a l l e r . The purpose of this paper is to present a detailed derivation of the new t e r m s and give a quantitative est imate of the i r importance.

11. THE PROPAGATOR

In order to obtain the one-instanton fermion propagator it i s useful to look a t the Euclidean A, = O gauge t i m e his tory of a quark that s t a r t s in a t ime region wkere there i s no gauge field (re- gion I: t < - T , A =O), then interacts with a n SU(2) instanton in region I1 (- T < t < T ) and finally reaches region I11 ( t > T ) where there i s only a pure gauge field (x =~u$)?u-' (x) , U(2) zexp{i?i?. (2- ?)/[(x'- E')' + P ' ] ~ ' ~ ) , with if and p the position and sca le s ize of the instanton). The resul t fo r regions I and I11 i s quite s imple, a s shown in Ref. 1. (It can a l so be obtained a s the limit of the resu l t f o r region I1 a s * T -*m.) In region I1 however we mus t make use of the ex- pansion in powers of m-' mentioned in the Intro- duction to wr i te the propagator ~ ( 2 , T; 7, - T ) a s

where S, i s the spin-independent propagator of H I . It can be obtained f r o m the path-integral expres- sion

Since m i s large, the integral i s dominated by the straight-line path between x' and x" and can be written

19 - S P I N - O R B I T T E R M S O F T H E H E A V Y - Q U A R K P O T E N T I A L I N . . . 2503

sl(g2, t2;zl, 4) where So i s the free-particle Schrijdinger propa- gator and the path-ordered integral from ji, to Z2

{ ( c2 dy'. x)) , i s taken over the straight-line path. The caleula-

=So(Z2, t2;<, t , ) P exp ie I tion of the exponential i s s impler if we go to the

"standard" gauge where we can use A",2qaPuxv/ (2.2) e (x2 + p2), TO do this we use

where

- 1' {T + tan-l[t / ( 3 + u(Z,t)=exp[-i@(Z,t)f*?] and $(x',t)=- ( 3 + ~ 2 ) l ' ~ 2

Then, denoting the left-hand side of (2.3) simply by P we have (the antisymmetry of the q symbol makes the integral Abelian)

where we have parametrized the straight line a s F(t ) =% +?(t- t ,) and expanded the result in powers of G = (z2 - zl)/(t2 - t1)[0(m-I)], keeping only f irst-order te rms. If we now define

and expand the exponentid in (2.4), we get

where I is the identity in SU(2). Expression (2.6) can be compactly written a s

The f i r s t se t of square brackets of Eq. (2.7) has already been dealt with in Ref. 1. The second one (after being converted back to the A, = O gauge) can be shown not to contribute to the f i r s t t e rm sf (2.1) a s we take iT - im (this limit will be implicit throughout). However, it does give a contribution to the integral of the second term. Let us call this second t e rm SPeIt and incorporate to i t the SU(2) matrix U-'(Z) a s was done in Ref. 1 s o a s to obtain the net effect of the instanton on the propagation. Then from (2.1) i t follows that

where we have used So(%, T; F, - ~) - -6 '~ ' g - F) and the exponentials a r e calculated in the A, = O gauge. By inserting UG, t ) ~ - ' ( z , t ) in the spaces between the square brackets of Eq. (2.8) and remembering that U(F, - T)- I we can use the "standard" gauge result (2.7) for the exponentials. The only new contribution to order (m?) comes from taking the ?.g t e rm of Hz, the t e r m involving the t ime derivatives in (2.7), and the identity te rm, s o that the new par t S,,,, can be written

U ( ) ( ; y ' - ) = dl I d '2 [V1 (%)(? - i) dU(ill ) ] [L G * (%, t )] U-I (i, t )6 (x' - z')6 (5 - y') - T dt 2 m

where we have dropped the t ime argument of U whenever we meant it t o be T. Using 6 ( z - y') to do the space integration and integrating by parts in t ime, taking advantage o f the vacuum boundary conditions, we obtain

+ + e d15 U ~ ( ) ~ ~ ( X T , - T ) = - - 2 ~n ( ? ) 6 ( - y ' ) 1 ( y ' ) J T d t U ( F , t ) B . ; i i - F 1 ( j , t ) .

- T (2.10)

In both (2.9) and (2.10) the field g is in the standard gauge. Since we know the explicit forms of U and E, we can calculate the integral to get

At this point it is important to notice that so far we have used a parametrization of the straight-line path which privileges one of its end points. W e could have used y'(t) =it2 +?(t- t,) instead. Indeed, if we want to avoid problems with operator ordering later on, we must use some symmetric way of parametrizing the path. The simplest thing to do is to average the results with each type of parametrization. This i s equivalent to using the parametrization y'(t) =i+?(t -zO), with z , =$(xl +x,),. Expression (2.7) can then be rewritten as

P = (1- 7 7 a p u ~ a ~ l i a u)u-1(Z)7 t,)U(ZI, t l ) ,

where 8,' a/azU and t, =xy. Expression (2.11) [ i f we make use of the identity GS, = ( l / m ) $ , ~ , = - ( l / ~ z ) \ ~ , ] will then be replaced by

where we distinguish between the gradient operator 5 acting on the 6 function and the ? which only acts on the S U ( 2 ) matrices.

Now we are ready to use the symmetrized kernel (2.12) to propagate the quark wave function $. Thus,

Integrating and using - i 5 ~ 5 (the momentum operator on the wave function), we can finally write

where repeated indices means summation and the symbol { )+ i s being used to represent the anticommuta- tor of the operators inside.

Ill. THE EFFECTIVE HAMILTONIAN IN THE DGA

The effective quark-antiquark Hamiltonian in the one-instanton case follows immediately f rom (2.14). I f we define the wave function for the qq system as PalN1, a2%(sl,Z2) with a , 0,s being color, spin, and space indices, respectively, the e f f ec t of the instanton on P is given by the tensor product of single quark expressions :

The effective Hamiltonian is h, such that - =G - 9. Since we are only interezted in color-singlet states we have toperform atraceover color variables. For an instantonlocated at r of scale s izep, there-

19 - S P I N - O R B I T T E R M S O F T H E H E A V Y - Q U A R K P O T E N T I A L I N ... 2505

sul t i s [for ~ ~ ( 3 1 1

We mus t now integrate over positions and s i z e s of the instantons with the density D@) charac te r i s t i c of the DGA. I t can be shown that going through the s a m e s teps with a n anti-instanton instead of a n instanton will s imply double the resul t of the integration. Therefore, the effective Hamiltonian in the DGA is , up to o r d e r n?-',

where the subscr ip t s 1 and 2 r e f e r to quark and antiquark and w is defined a s

u -* T ~ [ U ( Z ~ - ? ) r 1 g 2 - ?) - 11.

We can rewri te (3.2) a s

Doing the integration in d 3~ by p a r t s we obtain

- 2 H = - ---- 5m12 1 $ ~ ( ~ * i j k ~ I j [ ~ l i { ~ l n , ~ : k } + -{pin, '::kit ( I - 2, 1 (3.5)

where X ik and Yi: a r e defined a s follows:

Expanding the integrand in Y and using the propert ies of the E symbol, we get

- 2 H z - - - - 1

5m12 J ~ b k , j k 'li [vli{~ln,xkk}+ - p b m , ~ : a } + ] + (1 - 2 ) ,

where the quantity 2;: is defined a s

Note that both X i , and 2:: a r e dimensionless quantities (the supersc r ip t s = 1 , 2 r e f e r s to quark and anti- quark). It can be shown that their tensor s t ruc ture is

where Hsx ' , - sl and the derivation of (3.9) together with the definitions of 2, X ( * , , and X ,,) can be found in the Appendix, These a r e s c a l a r functions that depend only on the "radialJJ coordinate x / p = /$ - zll /p. F r o m (3.7) and (3.9) we may wri te

which c a n b e wri t ten a s

+ * with L 4 x ' x G l and L 2 a - &?x62. Since the functions inside the anticommutator depend only on the radial

coordinate, we simply write

The integration over scale s izes is to be performed from ze ro up to a cutoff s ize p, a s discussed in Ref. 1. At this stage we may call this contribution to the heavy-quark potential Hk to distinguish it from the other spin-orbit te rms in Ref. 1. Its final form is

IV. THE ASYMPTOTIC BEHAVIOR OF W In order to have an idea of the order of magnitude of H:, when compared with other t e rms in the heavy-

quark potential, it is useful to look a t the limits x<<p and x>>p which correspond to instantons of s ize much bigger and much smal ler than the quark-antiquark separation. To do s o one must calculate the asymptotic behavior of the functions 2, X,,,, and X,,, defined in the Appendix. This i s accomplished by making use of the form w [see (3.3) and Appendix] in these limits:

(ii) x >>p 2 n (Z/2 + F) (Z/2 - F) - , +oc Go/x)"), y>>p; wll--j (5) l i / 2 + ? 1 3 l Z / ~ - r \

where we have distinguished between "self-energy" (w,) and "interaction" (w,, ) t e rms in the large-x/p limit, the total contribution being given by the sum o = w , +w,,. The variable 5 is just x'*F//x'/ /?I. After some calculation one ar r ives a t

A rough sketch of a plausible graph for is presented in Fig. 1.

V. CONCLUSIONS

The te rm that we have calculated (Hi,) adds a 16% correction to a similar one already present in H,, of Ref. 1. There we had

(we note that our definition of 2 i s the negative of the one used in Ref. 1). Acting with the derivatives on V we get the t e rms of interest:

Xd $ ~ ( p ) f [v] + others ,

where we have used * O C dp

v(x) = 2 J - D ( P ) W(X/P) . 0 p2 FIG. 1. A plausible graph for % ( x / p ) .

19 - S P I N - O R B I T T E R M S O F T H E H E A V Y - Q U A R K P O T E N T I A L I N . . . 2507

Defining Z sx /p ,

Comparison with (3.13) shows that the relative importance of the new term is given by the ratio z I ~ ( z ) / ' 2w1(2), with Wf -d~/d~. The asymptotic behavior of 2 ~ ( 2 ) / 2 i s known to be'

At 2 - 0 (where in both cases we have the maximum contribution) the ratio i s 16O/c. As Z becomes large the relative importance of the new term goes to zero like 1/Z.

In ei ther case we see that the instantons will induce a coupling of the spin of each quark with i t s own angular momentum whenever the distance between the quarks i s small compared with the instanton scale size. This coupling vanishes rapidly a s we increase the separation between the quarks.

ACKNOWLEDGMENTS

I thank Professor Curt is G. Callan Jr. for suggesting this problem. This work was supported by CAPES, Research Agency of the Brazilian Ministry of Education.

APPENDIX

In order to calculate X:, and 2:; let us introduce the variables $n;, +cl and x'=$ - ?'. Choosing the orig- in of our coordinate system at if, we obtain

This particular choice allows u s to go from quark (q) to antiquark (&) by simply letting 2--x'. Then, re- writing the expression for w (Ref. 1) in t e rms of the new variables we get

- (2/2 + P) . (x'/2 - P) n lk /2+F1 n lZ /2 -F1 lx'/2+Fll?/2- ? I s in ( [(2/2 + P)'+ $ p J 2 ) s ~ ' ( [ ( ~ / z - F)2 + P2P/2

This expression i s invariant under the change 2--2. It can be expressed a s a function of x/p, y/p, and 5 =z SF/ 1 % / I ? I with the property

W ( X / P , ~ / P , 5 ) = O ( X / P , ~ / P , - 5 ) . (A3

We can then write

where Z(x, p) i s a sca lar function independent of the particle type since w i s an even function of 5 and J d 3 r =JOm2n./2dy j - :d<. Thus the only t e rms from the curly brackets to give a nonvanishing contribution a r e those even in <. Choosing s = 1 we can then write

Analogous reasoning allows us to write

2508 C A R 1 , O S A R A G A O D F C A R V A L H O 19 -

where again X , , , and X,,, a re s ca l a r functions independent of particle type. Their values can be calculated from

'c. G, Callan, R. Dashen, D. J. G r o s s , F. Wilczek, 'c. G. Callan, R. Dashen, and D. J. G r o s s , Phys. Rev. and A. Zee, Phys, Rev. D 18, 4684 (1978). D g, 2717 (1978).