Quantum effects in the quark-antiquark potential due to instantons

Volume 78B, number 2,3

PHYSICS LETTERS

25 September 1978

QUANTUM EFFECTS IN THE Q U A R K - A N T I Q U A R K POTENTIAL DUE TO INSTANTONS Herbert LEVINE 1

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08540, USA Received 22 June 1978

The effects of instantons on the static quark potential are investigated beyond leading order in g2. Qualitative results are obtained in two limits of interest: large and small quark-antiquark separation. The results have some bearing on both phenomenology and the question of quark confinement.

Recently it has been argued that instantons are responsible for much of the dynamics of QCD [1 ]. Their effect on the static quark interaction has been investigated [2] and may account for some of the structure of phenomenological charmonium potentials. In this letter, we consider quantum corrections to this interaction, and derive severn qualitative features of interest. In particular, it is shown that the instanton contribution to the potential is nonsingular at short distances and of order 1/R at large separations. Within the dilute gas approximation [1 ], the full effect of multiple instanton configurations may be taken into account by calculating the potential due to one instanton at spatial position x I and size p, V(R; xi, p), and integrating over position and scale with appropriate density factors. Inasmuch as instantons are nonsingular field configurations, we do not expect quantum fluctuations to dramatically change the picture obtained at the classical level. Nevertheless, it is still useful to show this explicitly and set up the formalism which allows one in principle to do a complete numerical evaluation of the effects of these fluctuations. Following ref. [2], the potential *1 is given by

V(Xl _ x2 ; xi ' p) = _ 1 Tr [ s - l ( x 1 - xi, e~, _c~) S(x2 _ xi ' ~, _~a) _ 1 ],

(1)

where

S(x 1 - xi, tl, t2) = exp [-iTr x- (x 1 - xi) / [(x 1 - xi)2 + 1921 1/2] S1 (x 1 _ xI ' tl ' t2).

(2)

S 1 is the static limit of the fermion propagator in the presence of the instanton, given by AF(Xl'X2' t l ' t 2 ) MF_. 2 6(Xl --X 2) SI(X 1 --X I, t 1, t2)-

(3)

(We will work in SU(2) but extension to SU(3) is straightforward.) From the non-relativistic path integral, we obtain 1 Supported in Part by National Science Foundation Predoctoral Fellowship Program. :1:1What is actually being calculated is the energy of the quark-antiquark pair. It is not correct in general to substitute this expression into a non-relativistic Schr6dinger equation and solve for the spectra. This can only be justified if the energy can be interpreted as an effectively instantaneous interaction (a Bethe-Salpeter kernel). The problem is discussed in several places [3] and will not be considered here. 235

Volume 78B, number 2,3

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25 September 1978 Xl

2XF(Xl ,X2, tl ' t 2 ) ~ AO(Xl,X2, tl, t 2 ) U - I ( x 1 - x i , t l ) P f e x p

ig f

A dy,] U(x 2 - x I, t2),

(4)

X2

where A 0F is the free fermion propagator, A u is the full gauge field given by At ~ =Aag 7a/2,

A a# = 2flav(x _ xi)v/g(( x _ xI )2 + 02) + 6A a

(5)

U(x 1 - x I, t) = exp [ i , - ( x I - x i ) (rr/2 + tan -1 [t/[p2 + (x 1 _ xi)2 ] 1/21 )1.

(6)

(In the limit t -+ 0% U reduces to the mapping of 3,space onto the SU(2) group. This mapping characterizes the instanton in A 0 = 0 gauge.) We are now in a position to evaluate the potential in the static limit. In this limit &o is highly peaked around x 1 = x 2 and we may calculate S 1 from eq. (4), and substitute it into eqs. (1,2) to obtain V. Neglecting all finite velocity terms we obtain, up to O(g 2)

V(x 1 - x 2 ; x I, P), = - I Tr [U(x I - x I, oo) u - l ( x 2 _ xi ' oo) _ 1]

(7a)

co

_ 1 ig f

dt 6Aao(x2 - x I, t) Tr [U(x 1 - xi, °° ) u - l ( x 2

- xi, °°) U(x 2 - x I, t) 1 "ra U(x 2 - x I, t)]

__oo

dt 6Aao(xl

+ ~ ig f

f

- x I , t ) { za U - 1 ( x I _ x i '

t) U(x 1 - x I , co) U -1 (x 2 _ xi,

dt

--oo

f

dt --co

(7b)

dt'Tr[U(x 1 - x i , ° ° ) u--l(x 2 - x i , ° ° ) U(x 2 - x i , t)½rau-l(x2 - x I , t)

- - ~

× U(x 2 - x i , t')½ r b U - I ( x 2 - x I, ~)] 6A~(x 1 - x I, t') 6Abo(Xl --x I, t) + (x 1 -+x 2, U-+ U - I ) _1g2

oo)]

t

0o

+¼g2

- x I , t) Tr [U(x I

f

dt'Tr[U(Xl-Xi, t ) l r a u - l ( x l - x i ,

t) U ( x l - x i , ° ° ) u - l ( x 2 - x i

(7c)

,°°)

--oo

X U(X 2 ~ X i , t ' ) ½ T b U - I ( x 2 - x I, t)l

6A~(x 1 - - X I, t) 6Abo(X2 - - X I, t').

(7d)

Although this expression looks complicated, it is actually simple to understand. To lowest order, the q u a r k antiquark pair simply propagates in the background instanton field. To O(g), either the quark or the antiquark interacts with a gluon emitted by the vacuum, while to O(g2), a gluon is emitted and reabsorbed by either its emit. ter, or by the other particle. In fact, these are the same terms one would have written down in a perturbative approach. The only real change is that all our propagators are those in the background field. We must now evaluate this expression by calculating the zero, one and two point function of the fluctuation 8A~). Normally one point functions (the vacuum expectation values of quantum fields) are not interesting; they average to zero upon integration over different color orientations of the instanton. However, here, the trace over color is taken only after multiplying by the fermion propagators and the one ~oint terms survive. Since the one point function must be O(g), we obtain an additionai contribution to the O(g ) term of the potential. The situation is summarized in figs. l a - d , where the four different types of interaction, corresponding to expressions ( 7 a d), are displayed. 236

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I

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25 September 1978 I

I I

I

I

I I

I

I I I

I I

I I I

EXCHANGE

ULAbblUAL

I

I I I

"LOOP"

I I L

"SELF-ENERGY"

Fig. 1. Diagrammatic representation of eq. (7).

The origin of the term (7b) can be understood more clearly by the following argument. Physically, we expect the entire contribution of instantons to be manifested in the changes of the fermion and gluon propagators. The instanton corrections to the fermion propagators have been discussed many times [4] and are included here by means of the U's and the "self-energy" term, (7c). When we attempt to calculate the gluon propagator, we must evaluate (schematically),

(A(z) A(z ')>=fed (~A) (ACl(z) + 6A(z)) (ACl(z ') + ~A(z ')) e ~8~r2/g2 A(gAcl + g6A) × exp [-~A .2 (2) (ACl) 6A

+g(6A)3 +g2(6A)4 ].

To O(1/g2), we obtain just the expectation value of a product of classical fields and there are no 1/g terms. At O(1) we have three types of contributions. First, the "classical" ~piecemust be multiplied by the two loop density of instantons. Also, we have the term which is just [12(2J(ACl)] -1. This is the object calculated by Brown et al. [5] and recently improved by Levine and Yaffe [6]. The last type of term comes about from the cross terms A(z) 8A(z'), which, when the self-interactions of the gluon are taken into account will contain an O(1) piece. This piece will not vanish upon global gauge averaging. The calculation of the two loop density of instantons is a highly non-trivial computation and has not yet been done. However, we expect and will therefore assume that aside from putting in the correct renormalization of the coupling constant, it has no effect on the shape of the q-Ct potential. We shall therefore restrict our attention to the diagrams in figs. lb, c, d and investigate first the short distance limit of eq. (7). As shown by Callan et al. [2], the leading behavior of the "classical" O(1) potential is

V(R) "

constant R2 f

~do D(O),

(9)

where DO9) is the instanton density. The only way quantum corrections can be important as if they are singular in the limit R -+ 0. Otherwise, their magnitude would always be sifnificantly smaller than this piece and could be ignored. One quantum correction is just the 1/R Coulomb force. We wish to ch~ck whether the 1/R piece changes in any way or if a logarithmic singularity could possibly result. This question is of some importance due to the recent suggestion [7] that a logarithmic potential may be useful in fitting charmonium spectra. For this calculation, we set x 1 = x 2 = 0, in all fermion propagators, noting that this makes an error of O(R 2) and hence cannot affect the singularity structure of V(R). Doing this we immediately see that there is no contribu. tion from the "loop" term, (7b). This is because the product of U's reduces to 1, and we are left with Tr ~'a = 0. Similarly, we can neglect the "self-energy" piece because it can never be singular in the R ~ 0 limit. We must only consider (7d), the exchange term. Explicitly, we must evaluate g2 /

dt f dt' G~bo(-xi-R/2, t;-xi +R/2, t')

V(R;xI, o) = - - T _

o~

_

oo

X Tr [exp(-i'c- x I qb(t, t')) 1 za exp(i x- x I qb(t, t')) 1 r b ],

(lo) 237

Volume 78B, number 2,3

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25 September 1978

where q~(t, t') = tan -1 [t/(x 2 + p2)112] _ tan-1 it,/(x2 + p2)1/21 ;

(11)

taking the trace gives

g2f 4

dt

_co

~

ab

dt' G0O ( - x I - R / 2 , t ; - x I + R/2, t )

--oo

X [sab cos 2qb(t, t') + e abe 2i, a sin 2aS(t, t') + 2Y~Io?ibsin2qb(t, t')].

(12)

As explained in ref. [6], the matrix inverse Guab (x, y) is only well-defined if we specify localized constraints to fix the zero mode contribution. Doing this leads to the equation

Guv ab(x ab(x 'Y)" -- gau,i(x)Nijfd4z~,yHCb(z t" ' Y ") - H uv~ av" ' y ) + g~,,(x)

f d 4 z H t~': ac(x 'z) f~,i(z) c b (N T )i]gu,](y)


(13)

where Huv=(6

v 6 e + r l .av ~ )a

-+ +D ( l I D2)2 D~

(14)

is the propagator of Brown [4], g(i) is the set of zero modes, f(i) some localized constraint functions, and

(N-1)ij = f f~u,i(z) ga,] (z) d4z.

(15)

The singularity structure of G is not effected by terms in eq. (13) other than the first, and we are free to consider the effect o f H ~ b in eq. (12). It is also clear that after we integrate along the time direction, it is only terms that were singular as both x - y and t 1 - t 2 -+ 0 that can lead to singularities at x = y . In this limit we can explicitly calculate g g b . From eq. (14), we have

Hab=(~ #v pv ~oep~+~l.w %~)D fW(x,z) W(z,y)d4zD~,

(16)

where the gauge fixing parameter has been set to 1, and W, the isospin 1 scalar propagator is given in non-singular gauge by c 2 +x - z) + 2~7~v a ~Xa b xuzvxxza 6 ab (/94 + 2px • Z + 2(X "z) 2 =X2Z 2) + 2e abc ~vxUzv(p

wab(x, z) =

4rr2 (p 2 + x2)(p2+z 2) (x - z) 2

(17)

Doing the convolution integral for small values of x - y , yields H ab (X, y ) 12

8 ab 8 #v

2e abe ~Te~(x - y ) a ((x +y)/2)t 3 6uu

4rr 2 (x - y)2

47r2 (x - y)2 (p:2 + ((x + y)/2) 2)

eabc c

r~uv In/ p2(x _ y ) 2 - 2 . 2 (O2 +((x +y)/2) 2) ~O2-+~x~YY))2)2) 2 ] ' 238

(18)

Volume 78B, number 2,3

PHYSICS LETTERS

25 September 1978

plus terms finite as x - y -+ 0. Substibuting eq. (18) into eq. (15) and recognizing that • is odd under the interchange of t and t', we obtain (aside from the Coulomb term)

V(R;xi,p)= g 2 f°° at f= .

.

.

dt'

[

sin2qb(t't') + - -

( t - t ' ) s i n 2 d g ( t , t' )

: (t-

.

)]

+ t )12)

(19)

"

Since cb ~ (t - t')lx I for small t - t', we see that eq. (19) is finite as R -+ 0. This therefore demonstrates the desired result, that quantum effects do not significantly change the results of the classical analysis of the short distance limit. In principle, one could keep all the terms of eq. (13) and evaluate the magnitude of the quantum correction. Then after integrating * ~ over a scale dependent density one would have an expression for the short distance limit of the energy of the quark-antiquark state. The long distance limit of the potential bears on the question of what happens when we attempt to liberate a quark. Qualitatively [1 ], it appears that at the classical level instantons do not confine quarks, that is, they do not lead to a linearly rising potential barrier at large distances. Recently it was suggested [81 that at the O(g 2) level, gluon exchange may generate such a confining force. Unfortunately, this will turn out to be incorrect. Specifically, it will now be shown that the leading behavior of V(R) is a correction to the 1/R tail, that is, a coupling constant renormalization. As shown by Levine and Yaffe, the asymptotic behavior of the full propagator in singular gauge is

Gl~v(x y) ~

6,u 6ab/4~r2(x _ y)2 + O(p2) ~ O(1/x2),

(20)

and - 1 Guu t7£/ = 6#,/4frZ(x - y ) 2 + O(p4) Yr G -=-~

(21)

The first term in eq. (21) is just the free propagator G 0. Consider first substituting this expression into the "exchange" term, (7d). In singular gauge,

U(x, t) = 1 + 7ri(x "x p2/x3)f(x, O,

(22)

where the actual form o f f is irrelevant. By simple power counting and by the vanishing of 6abTr r a r b r c, we are led to an expression of the form

E(R) "~ (" dp D(p) 04 V(R), J p5

where

V(R) = aiR + O(p2[R3).

(23)

Secondly, it is easy to see that the contribution of the "loop" will also be at least order p4. There will have to be at least one U expanded out to order p2 to avoid the vanishing of the trace and the loop itself vanishes as p2. (Dimensionally, the loop integral which defines 6A 0 goes as (length) -1 .) Schematically, it is of the form

Goa(X, Y) Fa(Y) d4y,

(24)

where Fa(y) is some function determined by the self-interaction of the gluons. Since Ks(v) converges rapidly as y -+ 0% this integral must vanish like O(1/x 2) at large x. Since everything depends on p2, this requires a behavior 4:2 Although it appears that the contribution calculated so far would diverge if integrated over xi, when all the other terms are added, V(xi) ~ 1/x~ at large x I. Alternatively, we could have done the calculation in singular gauge and all parts of V(xi) would have converged separately. 239

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,,~p2/X3 +a.

By the same argument as above, this will lead to a 1/R potential. Finally, there is the "self-energy". The first stage at which a dependence on R is introduced is when we consider, for example, multiplying the leading non-trivial term of the anti-quark propagator by a quark propagator with a self-energy correction. This term is just of the form (see (7c)),

g2 -4-

~

oo

f dt f --~

dt'Tr[rarbu-l(x 2-xI,°°)]

ab (x I - x i , G00

t;x 2 - x i , t').

(25)

--oo

Using eq: (22) with

f(x,

o~

V(R;xi,p)= .f

. .dt.

oo f

oo) = 1, we obtain, aside from a trivial R independent term,

p2eabC(x2 _ X l ) e dt' [(~x2-~i) 2 + - ~

2 G ab 00(Xl-xI't;x2-xI't)"

(26)

Again by using eq. (20) and dimensional analysis, it is easy to see that this too leads to a 1/R form in E(R). It is important to realize why one seems to recover a different answer [8] by considering the propagator H~ ) instead of G~ub. It can be readily shown that the asymptotic behavior of H is

H ~ G O+ O(p2/x3y, p2/xy3).

(27)

By previous considerations this is expected to and indeed does lead to a linear potential if only the exchange term is considered. However, here the "loop" term will also have a O2 piece and, although this has not yet been demonstrated explicitly, can and presumably does cancel the linear piece of the other term. In summary, it has been shown that the O(g 2) quantum fluctuations will not modify the qualitative picture of quark-antiquark forces in QCD. There remains the purely technical problem of calculating the magnitude of these effects. This will be relevant if and when real calculations of hadronic structure are attempted but that requires a much better understanding of the confinement mechanism that exists at present. It is my pleasure to acknowledge the advice and encouragement of Roger Dashen and the many stimulating conversations I have had on this topic with Larry Yaffe. I thank David Laughton for a critical reading of the manuscript. 4:3 Another argument to this effect is that in the p ~ 0 limit, the instanton becomes the zero field configuration for which the 10op diagram vanishes identically. This is not completely valid and, in fact, fails if the propagator being considered is H instead of G.

References [ 1] C. Callan, R. Dashen and D. Gross, Towards a theory of the strong interaction, I.A.S. preprint COO-2220-115, to be published in Phys. Rev. D. [2] C. CaUan, R. Dashen, D. Gross, F. Wilczek and A. Zee, The effect of instantons on the heavy quark potential, I.A.S. preprint COO-2220-132 (1978). [3] T. Applequist, M. Dine and I. Muzinich, Phys. Lett. 69B (1977) 231; F. Feinberg, Hamiltonian formulation of non-abelian gauge fields and non-relativistic bound states, Phys. Rev. D, to be published. [4] N. Andrei and D. Gross, The effect of instantons on the short distance structure of hadronic currents, Princeton Univ. preptint; E. Mottola, Phys. Rev. D17 (1978) 1102. [5] L.S. Brown, R. Carlitz, D. Creamer and C. Lee, Phys. Rev. D17 (1978) 1583: [6] H. Levine and L.G. Yaffe, Gluon propagators in the presence of instantons, in preparation. [7] C. Quigg and J.L. Rosner, Phys. Lett. 71B (1977) 153. [8] A. Duncan, Instanton effects on bound-state dynamics, Columbia Univ. preprint (1978). 240