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Heavy quarkonium decay in the presence of instantons
Volume 85B, number 2,3
PHYSICS LETTERS
13 August 1979
HEAVY QUARKONIUM DECAY IN THE PRESENCE OF INSTANTONS W. BERNREUTHER, N. MARINESCU and M.G. SCHMIDT Institut fiir Theoretische Physik der Universitiit, D-6900 Heidelberg, West Germany Received 25 May 1979
The instanton contribution to the on-sheU quark-antiquark annihilation amplitude is investigated by means of the dilute gas approximation in the limit of large quark masses. We find that the instanton correction - evaluated with an instanton scale size cut off Pc - to the decay rate of pseudoscalar states is not necessarily positive and decreases with the inverse of the quark mass squared. There is no instanton contribution to the decay of vector states.
Quantum chromodynamics (QCD) possesses a rich vacuum structure due to instantons [ 1 - 4 ] (and other field configurations [5] with non-trivial topological quantum number), but up to now the phenomenological success of QCD as a theory of strong interactions is based on its asymptotic freedom property, i.e. is connected with its perturbative vacuum properties. Yet there is substantiated hope that important problems like quark confinement/ hadronization and the spontaneous breaking of chiral symmetry can be solved in a satisfactory way by better understanding the role of the dense i n s t a n t o n - m e r o n plasma [6,7]. In the vicinity of quarks, however, the analog gas composed o f instantons is expected to be dilute [6,7], and effects o f the non-trivial vacuum structure can be estimated by means of the so-called dilute gas approximation [8]. Discussions in the literature of such effects include instanton induced s p i n - s p i n and s p i n - o r b i t forces in'heavy quarkonium [9] and instanton contributions to a hadr [8,10] and to deep inelastic e l e c t r o n - n u c l e o n scattering e+e [ 1 1] ; but at energies o f a few GeV these non-perturbative corrections to the latter process turn out to be fairly smaller than the perturbative QCD contributions. In this note we consider instanton corrections to a process for which the perturbative contribution is suppressed by powers o f %: the decay rate of heavy quarkonium into hadrons. The total decay is related to the imaginary part o f the forward q u a r k - a n t i q u a r k annihilation amplitude (fig. 1) via the optical theorem. Of course, bound state wave functions are not reliably calculable from first principles yet, but they can be eliminated b y considering branching ratios only. Actually the application of the optical theorem to processes with instanton interactions is indispensable, in contrast to the situation in perturbative QCD. The optical theorem is valid for hadronic intermediate states which constitute asymptotic states. In perturbative QCD hadronic asymptotic states are replaced by quarks and gluons hence an equivalent proof of the theorem can be formulated by using quark and gluon intermediate states, in agreement with the Cutkowsky cutting rules. In the presence of a (dilute) instanton gas, however, this is no longer true because quarks (and gluons) no longer form asymptotic states. Therefore the calculation of, say, a quark proq
~
- - ~ - 258
+
crossedgraph
Ib~
Fig. 1. The forward quark-antiquark annihilation amplitude, (a) instanton contribution, (b) lowest order perturbative contribution. The unshaded blob depicts the bound-state wave function.
Volume 85B, number 2,3
PHYSICS LETTERS
13 August 1979
duction amplitude is problematic * ~. Use o f the optical theorem avoids this difficulty. We are thus led to evaluate the euclidean spin-averaged colour singlet amplitude represented in fig. la in the dilute gas approximation. As essential ingredient we need the heavy quark propagator in the presence of an (anti-)instanton field A ~ ( x - z) o f scale size p located at z. Only the propagator for massless quarks is explicitly known and it can be represented in terms of the massless scalar propagator [12]. This is also possible in the case o f massive quarks. Using the (anti-)self-duality property of the (anti-)instanton field [12] one can show that S±(x,y) = (~± - im)A±(x,y)
{1 - (~9 +- - i m ) A ± ( x , y ) ( I p ± + ira)}
+.
2
'
(1)
where S+-(x, y ) , A-+(x, y ) are, respectively, the spinor and scalar propagator in an (anti-)instanton field and D~ = Ou - iA~ denotes the covariant derivative. A +- is not known in closed form but we can expand it in a perturbation series (see fig. 2): A+=
1 1 = -D2+m 2 _~2+m2
+
1 m2 ( - 2 i A +-u~u - A [+A [ ~+) _ ~ 2 1+ m 2 + .... _82+
(2)
The advantage o f this particular expansion of S -+ will become clear below. There is no obviously small parameter available to expand about as in ordinary perturbation theory, b u t we will discuss those terms in the series (1) which are the relevant ones in the particular process we are considering in the limit of large quark masses. Translation invariance for the amplitude shown in fig. l a is restored after integrating over the instanton position z and we end up with Feynman-type amplitudes with the instanton field at z = 0. Introducing a colour G-parity, G c = C exp (i n~2), the (anti-)instanton field has G c = +1 ; hence only q u a r k - a n t i quark bound states with positive G c parity, i.e., pseudoscalar states like r/ce ( 0 - ) but not vector states like ffce (1--), receive an instanton contribution to their hadronic decay rates. The calculation simplifies when we consider the heavy bound-state quarks to be at rest, which should be reasonable for S-wave states (but this is not an essential restriction). The projection factor resulting from the pseudoscalar combination u,~+ - u ~ t is 23,5(~/4P4 - m), when rotated to euclidean space and the non-perturbative contribution to the amplitude (fig. 1a) without bound-state wave functions is given to first order in the SUc(3)-instanton density D+-(O) by A(p, -p) =
D±(,o) Tr [275(3'4P4 - m ) S t r ( P , - P ) ] Tr [275(74P4 - m ) ~ r ( - P , P ) ] ,
.
(3)
¢1 In particular, when interference terms between the perturbative and (anti-)instanton sectors are absent, the cross section would be of second order in the instanton density - in disagreement with the calculations of the cross section via the optical theorem which leads to a result being of first order in the instanton density.
I I "1-
-
+
{a]
t
I
(d)
ae
i
+
(e)
~.
(b)
{c)
+
(f}
(g)
Fig. 2. Perturbation expansion of the massive scalar propagator in the presence of an (anti-)instanton field. Dashed lines and wiggly lines represent, respe'ctively, 2iA~au and A ~ A ~ vertices. 259
Volume 85B, number 2,3
PHYSICS LETTERS
13 August t 979
where ,2 D+-(19) = gT6(8zr2/ff2(p))6exp(-Srr2/~2(p)) 1
l-I
(1.3mio),
87r2/ff2(19)=(ll-~Nf)ln(1/19t.£),
(4)
i=u,d,s
and Str denotes the Fourier transform of
Str(X'Y) = (~x + im)S+(x,Y)(~y + i m ) .
(5)
Inserting formula (1) into eq. (3) only the 3'5 part of eq. (1) survives and eq. (3) becomes
A (p,-p) =
f _
n+(19)Tr
[ A t r ( P , - p ) ] Tr [ A ~ ( - p , p ) ] ,
(6)
19o
where Atr(P, --p) is the Fourier transform of the truncated scalar propagator: AtrCx,y ) = (_-~2 + m2)A+(x,y)(_~2
+ m2).
(7)
By using eq. (1) one completely avoids the complicated Dirac algebra which would result from a perturbative expansion analogous to eq. (2) of S_+. What are the relevant terms in the expansion (1) (see fig. 2) of Tr A+r(p, - p ) in the limit of large quark masses? By performing the colour trace the terms depicted in figs. 2b, 2e, 2fvanish. The most elementary contribution to Tr Atr is the AuA u vertex shown in fig. 2a. In the regular gauge the instanton has the form
A;(x) = ~'a~#vXv/(x2 + 192), and one obtains with euclidean momentum p = -Tr
(8)
(P4, O)
fA~A~exp (2 ipx)d4x = - 1 2 +72p2 [K0(2P 4 p) + (2/p 4 P)K1 (2P4 p ) .
(9)
For heavy on-shell quarks the vertex (9) behaves like
--6027r5/2(pp4)-l/2ex p ( - 2 p p 4 ) .
(10)
After having established the high mass behaviour of the first term in the expansion of Tr Atr we will investigate the convergence of the Born series. The first step is to show that the terms containing 2iAu3 u vertices are suppressed at least by a power of limp compared to (10). To this aim let us first consider the term represented in fig. 2c which after contraction of the ~auv symbols becomes const. (iP4) 2
fd4xf d4x'A0(x - x ' ) e x p [iP4(X 4 +x'4)]f(x,x' ),
f(x,x')=x'x'/(x 2 + P2)(x'2 +192). (11) = 1 (x - x ' ) at a value ~l/rn, so that for practical
The free propagator h0(x - x ' ) cuts off the integration over x _ purposes the function f(x, x') can be expanded in a Taylor series in x _ , which after integration over x _ leads to a power series in 1/m. However, the Fourier tran¢orm of the Taylor coefficients may bring powers of p4 in the numerator and may so compensate the factor 1/m~ This is the case for the terms containing x~ only. Thus the leading behaviour in 1/m (or 1/P4) is obtained by putting x = x ' in f(x, x'). Using this approximation and integrating over x 4, x~ one finds that (11) has the behaviour 4zr3(iP4)2 f 0
r4dr 1 oo dk4 exp [-(IP4 - k4[ + [P4 + k4l)(r 2 + 192)1/2] r 2 + 192 27r f k24 + m 2
(12)
--~
,2 Note that for large rnp the renormalized quark determinant in a background (anti-)instanton field is 1 + O(1/(mp)2). (See ref. [8] .) For the values of the light quark masses we take m s = 0.5 GeV, m d = (1/20)ms, m u = (1/40)m s. (See, e.g., ref. [10].) 260
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PHYSICS LETTERS
13 August 1979
The integration over k 4 can be written as 1 [~ ~--~nexp [-2(r2 + p2)l/2p4]
+ 1 f~ 0
+p2)l/2+p4-im
dz ln z/(r2 /92)1/2 -z/( r2 + + P4 + lm
t
(13)
.
This means that for large P4 and m, expression (12) is dominated by
(p2/mpx/~4O) exp (-2P4P),
(14)
(In Minkowski space, P4 -+ iP0, the term 2c behaves up to a logarithm also like (14).) An analysis similar to the one just outlined shows that the asymptotic behaviour (14) is the dominant one for any term containing a vertex 2iA~ 8~. Such contributions to eq. (6) can thus be neglected in the limit of large rap. We are left with a Born series for the truncated scalar propagator (7) withAuA ~ vertices only. Its Fourier transform is the solution of the integral equation + -+ d 4x f e x p (iP4x4)Atr(x , y ) exp (ik4Y4) d4x d4y = - f e x p [i(p 4 + k4)x4] A/~A/~(x) (15) -
fexp(iP4x4) Atr (X, z)A 0 (Z -- y ) A ; A ~ ( v )
exp (ik4Y4) d4xd4y d4z.
Arguments similar to those used in the estimate of the leading order of graph 2d allow us to p u t y = z in AuAu(y ). The resulting integral equation is: +~ dk~ Ax(p 4 , - k 4) = lx(P4 + k4) + ~ - _ " k~2 + m 2 Ax(P4, - k 4 ) I x (k 4 - k~)
(16)
where
Ix(q4) = - f A~A~(x)exp(iq4x 4) dx 4 ~ Tclq41 exp(-lqala)
, c = -anp2 ,
a = (x 2 +p2)1/2 ,
"~+ Ax(P4, - k 4 ) = f d x a d Y 4 d 3 y exp (iPaX4) exp(ik4Y 4) A~r(X + ,y).
(17)
(18)
It is convenient to rewrite eq. (16) for the function
M±(P4 ' - k 4 ) = °t2exp [t~lP4 + k41] ~'x(P4' -k4)"
(19)
Eq. (16)then reads
f
M±(P4, _k4) = clP4 + k41 + 2net c2 ~
. . . -k4)K(k4, . dk4M-(P4, k4).
(19')
The kernel K(k'4, k4) has now the following form
K(k' 4, k 4) = [Ik 4 - k'4l/(k'42 + m2)] exp [-a{[p4 + k~l + Ik~ - k41 - IP4 + k41)] •
(20)
It can be easily seen that in the limit of large P4 only the integration domain in which the exponent in (20) vanishes is important, i.e., - P 4 < k~ < k 4 and k 4 < k~ < -P4" As a consequence the leading term ofM-*(p4, - k 4 ) satisfies the following integral equation ( - P 4 ~< k4 ~
Volume 85B, number 2,3
PHYSICS LETTERS
13 August 1979
f dkr4M±(P4,-lc'4)(k~ 4-k~4). k4
M±(P4, - k 4 )
= c (P4 + k4) +- c 27rot2
--P4
(21)
k'4Z+m2
For convenience let us discuss eq. (21) for complex P4 = m exp i~o and k 4 such that K = we deduce that the leading part of the nth Born term is
cn+l M;n)(P4,-P4):(2rtt~2) n P4
k4/P4 is real. From
+1 t~1 Kn-1 d• n f dK1 f dt¢2 "'" f -1 -1 -I
(1 - KI)(K 1 -- K2) ... 0 % -- K n - 1 ) ( 1 + K4) X (exp ( - 2 i ¢ ) + •2)(exp (-2i~0) + K2) ... (exp (-2kp) + K2) '
(22)
Consider eq. (22) for ~0= 0. Since the numerator of the integrand is bounded by g
(n)(P4,_P4)<~(27rl~2nn_)n_l{2]np 4 ?1 .dK n ±
eq.(21)
(2/n)n it follows that (23)
m
~n]n~.-1
K2+l
'
i.e., the series converges rapidly in euclidean space. In Minkowski space (~0 = i n) the Born series still converges, although less rapidly than in euclidean space• This follows from the observation that
If_ln-1 (t¢n--Kn-1)(1 +Kn) dtCn [<~ Kn-I 2
/On
--
1
+ 1.
(24)
By successively using the bound (24) in the multiple integral (22) one obtains M÷
~< Icn+ll
12P4[,
[ (n)(p4,-p4)I (2rra2)n where Ic/2na21 <~-~. a Thus the Born series for Ax(P4 , - P 4 ) is absolutely convergent in Minkowski space and 2 bounded by the geometric series (2cP4/a)exp (-2aP4)Zn=O(-~)n . •
.
oo
(25)
3
In order to estimate the behavxour of Tr A~-r(P4, - P 4 ) for large P4 = im we take into account the first three terms of its Born series with AuAt, vertices only, Tr Str(P4, - P 4 ) =
E1 f d3x~+Ax(p4, - P 4 ) ~ f d3x -e-L-2na2 +[~2not 2]-c]2[2- ~
ff~)] 2cp4 Vexp (--2°tP4)'
(26)
Integrating over x we find the same asymptotic behaviour as given in (10), multiplied by a factor of about 2. To obtain the on-shell quark-antiquark annihilation amplitude T = (1/i)A let us insert (10), multiplied by a factor 2, into eq. (6). In the vicinity of heavy quarks large scale size instantons are suppressed [6,7] ; thus the integral over/9 should be cut off at a "critical value" Pc" Since D (/9) ~/91.1- (2/3)Nf(1 n 0)6/93 increases very rapidly with /9, the logarithm can be approximated by a constant. (For definiteness, we shall evaluate the density for four quark flavours.) We now assume that the naive continuation to Minkowski space can be made• The imaginary part of T arises from the analytic structure of (10) and is given by * s ,a Calculating with an instanton in the singular gauge [5], the leading behaviour for large mp of Im T turns out to be the same as (27). 262
Volume 85B, number 2,3
Im T ~
PHYSICS LETTERS
4mPc 367r3 25 r x 34/3 sin x dx (mPc)2 -jf(Oc) J (4mPc)34/~
13 August 1979
,
(27)
where f(Pc) is the space-time fraction occupied by instantons and anti-instantons, Pc
f(Pe) =
rr2
f
(28)
0 Strictly speaking, our formulae do not hold for the integration region x ~< 1 in (27). This region can be investigated with the formulae for S -+ and D(p), valid for small mp [8]. But it gives a negligible contribution to Im T so it can be included. The integral in (27) can be approximated by - cos 4mOc when 4mPc/> 5 ~r. At 4mPc = 37r it has a minimum o f - 0 . 5 , and it is O(10 - 2 ) when 4mPc <. 2ft. For a numerical estimate let us compare Im T with the imaginary part of the amplitude given by the QCD graphs 1b, which (in the same normalization) is given by 2rra 2 . Their ratio is ImT _ 2rro~
187r2 25 f(Pc) ~ mpc x34/3sin x dx (mOo)2
3
°t2 0d
(29)
(4mPc) 34/3
Recently it has been argued [6,7] that inside of heavy quark-antiquark bound states the space-time fraction f(Oc) is about 10 -3 and the critical instanton size Pc is of the order of magnitude of the inverse o-meson mass. This means, say, for charm and bottom quark bound states 2 <~ mPc <~ 5. Finally, setting c~2 = 0.1, as is usually assumed in calculations with heavy quark bound states, one sees that the ratio (29) is of order one. It is remarkable that the instanton contribution is not positive definite. We argued before that the instanton term by itself is not the absolute square of a production amplitude (quarks and gluons are not asymptotic states any more). Thus this fact does not lead to a conceptual difficulty. Of course, the total (i.e. perturbative and nonperturbative) contribution to the decay rate should be positive definite - in agreement with the optical theorem with hadronic intermediate states. The introduction of a cut-off Pc connected with hadronization [6,7] seems very appealing to us. However, it should be mentioned that one can evaluate eq. (6) alternatively as follows. For large P4, the behaviour of Tr ~tr (9) gives rise to a rapid convergence of the o-integral in euclidean space. Hence eq. (6) can be integrated up to infinity. After having rescaled P by x = 2P4P, eq. (6) reads (Nf = 4) A
2(24rr2)2 640
(-~)
6
l'3mi
la 34/3 f
n s ( - - - Z ) (2-~4) i:-u,d,
~o ( d x ln-Tff]2p4~6x31/3K2(x),0
(30)
0
and the leading behaviour of (30) for large P4 is A ~ (24;:)2 225/3 256 [ F ( ~ ) ] 4 l-I ~l'3mi~.CP ]34/3[ln2P4'~6 (~-) V ( ~ ) i=u,d,s \-~---]kfff-44I ~ .--ff--]
.
(31)
The imaginary part of (31), continued to Minkowski space, is totally different in structure from the one obtained in (27). This demonstrates that the result in Minkowski space depends drastically on the approximation made in the euclidean region - which is necessary because of our limited knowledge about hadronization (dilute gas approximation). Note that the numerical value of the imaginary part of (31) depends drastically on p/2P4. E . g . , when setting 2m = 3 G e V - a value where the dilute gas approximation starts to break down because the dominant x in integral (30) corresponds to O = 1 GeV -1 - Im A varies in the range 0.3 to 30 when p varies between 0.3 and 0.7 GeV, respectively. Our crude estimates show that it is conceivable for the instanton contribution to heavy quarkonium decay to 263
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be sizable over a broad range of quark masses. It is interesting that only C = +1 states, in particular the pseudoscalars, obtain a contribution. Unfortunately the evidence for the existence of the pseudoscalar states r/c and rl'C and their hadronic decays is poor [ 13]. Usually one identifies the state rio with the resonance at 2.8 GeV. The hadronic width for tic predicted by perturbative QCD is larger than the experimental upper bound for the width of that resonance. This discrepancy could be removed by negative instanton contributions. In this context it seems worthwhile to note that instantons could distort the two-jet-like distribution of the heavy meson decay products as predicted by perturbative QCD. But a truly quantitative discussion of these effects is premature as long as a more precise control over the scale size cut-off pc and the boundary of the dilute instanton phase is lacking. We would like to thank M. Bace for discussions and collaboration on related problems.
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11 ] [12] [13]
264
A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B (1975) 85. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 18; Phys. Rev. D14 (1976) 3432, Erratum; D18 (1978) 2199. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. C. Callan, R. Dashen and D. Gross, Phys. Lett. 63B (1976) 334. C. CaUan, R. Dashen and D. Gross, Phys. Rev. D17 (1978) 2717. C. Callan, R. Dashen and D. Gross, Phys. Lett. 78B (1978) 307, and Princeton preprint (1978). E. Shuryak, Phys. Lett. 79B (1978) 135. N. Andrei and D. Gross, Phys. Rev. D18 (1978) 468. C. Callan, R. Dashen, D. Gross, A. Wilczek and A. Zee, Princeton preprint (1978). L. Beaulieu, J. Ellis, M.K. Gaillard and W. Zakrzewski, Phys. Lett. 77B (1978) 290. L. Beaulieu, J. Ellis, M.K. Gaillard and W. Zakrzewski, Phys. Lett. 81B (1979) 41. L.S. Brown, R.D. Carlitz, D. Craemer and C. Lee, Phys. Rev. D17 (1978) 1583. E. Bloom, talk given at the Rencontre de Moriond (March 1979).
PHYSICS LETTERS
13 August 1979
HEAVY QUARKONIUM DECAY IN THE PRESENCE OF INSTANTONS W. BERNREUTHER, N. MARINESCU and M.G. SCHMIDT Institut fiir Theoretische Physik der Universitiit, D-6900 Heidelberg, West Germany Received 25 May 1979
The instanton contribution to the on-sheU quark-antiquark annihilation amplitude is investigated by means of the dilute gas approximation in the limit of large quark masses. We find that the instanton correction - evaluated with an instanton scale size cut off Pc - to the decay rate of pseudoscalar states is not necessarily positive and decreases with the inverse of the quark mass squared. There is no instanton contribution to the decay of vector states.
Quantum chromodynamics (QCD) possesses a rich vacuum structure due to instantons [ 1 - 4 ] (and other field configurations [5] with non-trivial topological quantum number), but up to now the phenomenological success of QCD as a theory of strong interactions is based on its asymptotic freedom property, i.e. is connected with its perturbative vacuum properties. Yet there is substantiated hope that important problems like quark confinement/ hadronization and the spontaneous breaking of chiral symmetry can be solved in a satisfactory way by better understanding the role of the dense i n s t a n t o n - m e r o n plasma [6,7]. In the vicinity of quarks, however, the analog gas composed o f instantons is expected to be dilute [6,7], and effects o f the non-trivial vacuum structure can be estimated by means of the so-called dilute gas approximation [8]. Discussions in the literature of such effects include instanton induced s p i n - s p i n and s p i n - o r b i t forces in'heavy quarkonium [9] and instanton contributions to a hadr [8,10] and to deep inelastic e l e c t r o n - n u c l e o n scattering e+e [ 1 1] ; but at energies o f a few GeV these non-perturbative corrections to the latter process turn out to be fairly smaller than the perturbative QCD contributions. In this note we consider instanton corrections to a process for which the perturbative contribution is suppressed by powers o f %: the decay rate of heavy quarkonium into hadrons. The total decay is related to the imaginary part o f the forward q u a r k - a n t i q u a r k annihilation amplitude (fig. 1) via the optical theorem. Of course, bound state wave functions are not reliably calculable from first principles yet, but they can be eliminated b y considering branching ratios only. Actually the application of the optical theorem to processes with instanton interactions is indispensable, in contrast to the situation in perturbative QCD. The optical theorem is valid for hadronic intermediate states which constitute asymptotic states. In perturbative QCD hadronic asymptotic states are replaced by quarks and gluons hence an equivalent proof of the theorem can be formulated by using quark and gluon intermediate states, in agreement with the Cutkowsky cutting rules. In the presence of a (dilute) instanton gas, however, this is no longer true because quarks (and gluons) no longer form asymptotic states. Therefore the calculation of, say, a quark proq
~
- - ~ - 258
+
crossedgraph
Ib~
Fig. 1. The forward quark-antiquark annihilation amplitude, (a) instanton contribution, (b) lowest order perturbative contribution. The unshaded blob depicts the bound-state wave function.
Volume 85B, number 2,3
PHYSICS LETTERS
13 August 1979
duction amplitude is problematic * ~. Use o f the optical theorem avoids this difficulty. We are thus led to evaluate the euclidean spin-averaged colour singlet amplitude represented in fig. la in the dilute gas approximation. As essential ingredient we need the heavy quark propagator in the presence of an (anti-)instanton field A ~ ( x - z) o f scale size p located at z. Only the propagator for massless quarks is explicitly known and it can be represented in terms of the massless scalar propagator [12]. This is also possible in the case o f massive quarks. Using the (anti-)self-duality property of the (anti-)instanton field [12] one can show that S±(x,y) = (~± - im)A±(x,y)
{1 - (~9 +- - i m ) A ± ( x , y ) ( I p ± + ira)}
+.
2
'
(1)
where S+-(x, y ) , A-+(x, y ) are, respectively, the spinor and scalar propagator in an (anti-)instanton field and D~ = Ou - iA~ denotes the covariant derivative. A +- is not known in closed form but we can expand it in a perturbation series (see fig. 2): A+=
1 1 = -D2+m 2 _~2+m2
+
1 m2 ( - 2 i A +-u~u - A [+A [ ~+) _ ~ 2 1+ m 2 + .... _82+
(2)
The advantage o f this particular expansion of S -+ will become clear below. There is no obviously small parameter available to expand about as in ordinary perturbation theory, b u t we will discuss those terms in the series (1) which are the relevant ones in the particular process we are considering in the limit of large quark masses. Translation invariance for the amplitude shown in fig. l a is restored after integrating over the instanton position z and we end up with Feynman-type amplitudes with the instanton field at z = 0. Introducing a colour G-parity, G c = C exp (i n~2), the (anti-)instanton field has G c = +1 ; hence only q u a r k - a n t i quark bound states with positive G c parity, i.e., pseudoscalar states like r/ce ( 0 - ) but not vector states like ffce (1--), receive an instanton contribution to their hadronic decay rates. The calculation simplifies when we consider the heavy bound-state quarks to be at rest, which should be reasonable for S-wave states (but this is not an essential restriction). The projection factor resulting from the pseudoscalar combination u,~+ - u ~ t is 23,5(~/4P4 - m), when rotated to euclidean space and the non-perturbative contribution to the amplitude (fig. 1a) without bound-state wave functions is given to first order in the SUc(3)-instanton density D+-(O) by A(p, -p) =
D±(,o) Tr [275(3'4P4 - m ) S t r ( P , - P ) ] Tr [275(74P4 - m ) ~ r ( - P , P ) ] ,
.
(3)
¢1 In particular, when interference terms between the perturbative and (anti-)instanton sectors are absent, the cross section would be of second order in the instanton density - in disagreement with the calculations of the cross section via the optical theorem which leads to a result being of first order in the instanton density.
I I "1-
-
+
{a]
t
I
(d)
ae
i
+
(e)
~.
(b)
{c)
+
(f}
(g)
Fig. 2. Perturbation expansion of the massive scalar propagator in the presence of an (anti-)instanton field. Dashed lines and wiggly lines represent, respe'ctively, 2iA~au and A ~ A ~ vertices. 259
Volume 85B, number 2,3
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13 August t 979
where ,2 D+-(19) = gT6(8zr2/ff2(p))6exp(-Srr2/~2(p)) 1
l-I
(1.3mio),
87r2/ff2(19)=(ll-~Nf)ln(1/19t.£),
(4)
i=u,d,s
and Str denotes the Fourier transform of
Str(X'Y) = (~x + im)S+(x,Y)(~y + i m ) .
(5)
Inserting formula (1) into eq. (3) only the 3'5 part of eq. (1) survives and eq. (3) becomes
A (p,-p) =
f _
n+(19)Tr
[ A t r ( P , - p ) ] Tr [ A ~ ( - p , p ) ] ,
(6)
19o
where Atr(P, --p) is the Fourier transform of the truncated scalar propagator: AtrCx,y ) = (_-~2 + m2)A+(x,y)(_~2
+ m2).
(7)
By using eq. (1) one completely avoids the complicated Dirac algebra which would result from a perturbative expansion analogous to eq. (2) of S_+. What are the relevant terms in the expansion (1) (see fig. 2) of Tr A+r(p, - p ) in the limit of large quark masses? By performing the colour trace the terms depicted in figs. 2b, 2e, 2fvanish. The most elementary contribution to Tr Atr is the AuA u vertex shown in fig. 2a. In the regular gauge the instanton has the form
A;(x) = ~'a~#vXv/(x2 + 192), and one obtains with euclidean momentum p = -Tr
(8)
(P4, O)
fA~A~exp (2 ipx)d4x = - 1 2 +72p2 [K0(2P 4 p) + (2/p 4 P)K1 (2P4 p ) .
(9)
For heavy on-shell quarks the vertex (9) behaves like
--6027r5/2(pp4)-l/2ex p ( - 2 p p 4 ) .
(10)
After having established the high mass behaviour of the first term in the expansion of Tr Atr we will investigate the convergence of the Born series. The first step is to show that the terms containing 2iAu3 u vertices are suppressed at least by a power of limp compared to (10). To this aim let us first consider the term represented in fig. 2c which after contraction of the ~auv symbols becomes const. (iP4) 2
fd4xf d4x'A0(x - x ' ) e x p [iP4(X 4 +x'4)]f(x,x' ),
f(x,x')=x'x'/(x 2 + P2)(x'2 +192). (11) = 1 (x - x ' ) at a value ~l/rn, so that for practical
The free propagator h0(x - x ' ) cuts off the integration over x _ purposes the function f(x, x') can be expanded in a Taylor series in x _ , which after integration over x _ leads to a power series in 1/m. However, the Fourier tran¢orm of the Taylor coefficients may bring powers of p4 in the numerator and may so compensate the factor 1/m~ This is the case for the terms containing x~ only. Thus the leading behaviour in 1/m (or 1/P4) is obtained by putting x = x ' in f(x, x'). Using this approximation and integrating over x 4, x~ one finds that (11) has the behaviour 4zr3(iP4)2 f 0
r4dr 1 oo dk4 exp [-(IP4 - k4[ + [P4 + k4l)(r 2 + 192)1/2] r 2 + 192 27r f k24 + m 2
(12)
--~
,2 Note that for large rnp the renormalized quark determinant in a background (anti-)instanton field is 1 + O(1/(mp)2). (See ref. [8] .) For the values of the light quark masses we take m s = 0.5 GeV, m d = (1/20)ms, m u = (1/40)m s. (See, e.g., ref. [10].) 260
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The integration over k 4 can be written as 1 [~ ~--~nexp [-2(r2 + p2)l/2p4]
+ 1 f~ 0
+p2)l/2+p4-im
dz ln z/(r2 /92)1/2 -z/( r2 + + P4 + lm
t
(13)
.
This means that for large P4 and m, expression (12) is dominated by
(p2/mpx/~4O) exp (-2P4P),
(14)
(In Minkowski space, P4 -+ iP0, the term 2c behaves up to a logarithm also like (14).) An analysis similar to the one just outlined shows that the asymptotic behaviour (14) is the dominant one for any term containing a vertex 2iA~ 8~. Such contributions to eq. (6) can thus be neglected in the limit of large rap. We are left with a Born series for the truncated scalar propagator (7) withAuA ~ vertices only. Its Fourier transform is the solution of the integral equation + -+ d 4x f e x p (iP4x4)Atr(x , y ) exp (ik4Y4) d4x d4y = - f e x p [i(p 4 + k4)x4] A/~A/~(x) (15) -
fexp(iP4x4) Atr (X, z)A 0 (Z -- y ) A ; A ~ ( v )
exp (ik4Y4) d4xd4y d4z.
Arguments similar to those used in the estimate of the leading order of graph 2d allow us to p u t y = z in AuAu(y ). The resulting integral equation is: +~ dk~ Ax(p 4 , - k 4) = lx(P4 + k4) + ~ - _ " k~2 + m 2 Ax(P4, - k 4 ) I x (k 4 - k~)
(16)
where
Ix(q4) = - f A~A~(x)exp(iq4x 4) dx 4 ~ Tclq41 exp(-lqala)
, c = -anp2 ,
a = (x 2 +p2)1/2 ,
"~+ Ax(P4, - k 4 ) = f d x a d Y 4 d 3 y exp (iPaX4) exp(ik4Y 4) A~r(X + ,y).
(17)
(18)
It is convenient to rewrite eq. (16) for the function
M±(P4 ' - k 4 ) = °t2exp [t~lP4 + k41] ~'x(P4' -k4)"
(19)
Eq. (16)then reads
f
M±(P4, _k4) = clP4 + k41 + 2net c2 ~
. . . -k4)K(k4, . dk4M-(P4, k4).
(19')
The kernel K(k'4, k4) has now the following form
K(k' 4, k 4) = [Ik 4 - k'4l/(k'42 + m2)] exp [-a{[p4 + k~l + Ik~ - k41 - IP4 + k41)] •
(20)
It can be easily seen that in the limit of large P4 only the integration domain in which the exponent in (20) vanishes is important, i.e., - P 4 < k~ < k 4 and k 4 < k~ < -P4" As a consequence the leading term ofM-*(p4, - k 4 ) satisfies the following integral equation ( - P 4 ~< k4 ~
Volume 85B, number 2,3
PHYSICS LETTERS
13 August 1979
f dkr4M±(P4,-lc'4)(k~ 4-k~4). k4
M±(P4, - k 4 )
= c (P4 + k4) +- c 27rot2
--P4
(21)
k'4Z+m2
For convenience let us discuss eq. (21) for complex P4 = m exp i~o and k 4 such that K = we deduce that the leading part of the nth Born term is
cn+l M;n)(P4,-P4):(2rtt~2) n P4
k4/P4 is real. From
+1 t~1 Kn-1 d• n f dK1 f dt¢2 "'" f -1 -1 -I
(1 - KI)(K 1 -- K2) ... 0 % -- K n - 1 ) ( 1 + K4) X (exp ( - 2 i ¢ ) + •2)(exp (-2i~0) + K2) ... (exp (-2kp) + K2) '
(22)
Consider eq. (22) for ~0= 0. Since the numerator of the integrand is bounded by g
(n)(P4,_P4)<~(27rl~2nn_)n_l{2]np 4 ?1 .dK n ±
eq.(21)
(2/n)n it follows that (23)
m
~n]n~.-1
K2+l
'
i.e., the series converges rapidly in euclidean space. In Minkowski space (~0 = i n) the Born series still converges, although less rapidly than in euclidean space• This follows from the observation that
If_ln-1 (t¢n--Kn-1)(1 +Kn) dtCn [<~ Kn-I 2
/On
--
1
+ 1.
(24)
By successively using the bound (24) in the multiple integral (22) one obtains M÷
~< Icn+ll
12P4[,
[ (n)(p4,-p4)I (2rra2)n where Ic/2na21 <~-~. a Thus the Born series for Ax(P4 , - P 4 ) is absolutely convergent in Minkowski space and 2 bounded by the geometric series (2cP4/a)exp (-2aP4)Zn=O(-~)n . •
.
oo
(25)
3
In order to estimate the behavxour of Tr A~-r(P4, - P 4 ) for large P4 = im we take into account the first three terms of its Born series with AuAt, vertices only, Tr Str(P4, - P 4 ) =
E1 f d3x~+Ax(p4, - P 4 ) ~ f d3x -e-L-2na2 +[~2not 2]-c]2[2- ~
ff~)] 2cp4 Vexp (--2°tP4)'
(26)
Integrating over x we find the same asymptotic behaviour as given in (10), multiplied by a factor of about 2. To obtain the on-shell quark-antiquark annihilation amplitude T = (1/i)A let us insert (10), multiplied by a factor 2, into eq. (6). In the vicinity of heavy quarks large scale size instantons are suppressed [6,7] ; thus the integral over/9 should be cut off at a "critical value" Pc" Since D (/9) ~/91.1- (2/3)Nf(1 n 0)6/93 increases very rapidly with /9, the logarithm can be approximated by a constant. (For definiteness, we shall evaluate the density for four quark flavours.) We now assume that the naive continuation to Minkowski space can be made• The imaginary part of T arises from the analytic structure of (10) and is given by * s ,a Calculating with an instanton in the singular gauge [5], the leading behaviour for large mp of Im T turns out to be the same as (27). 262
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Im T ~
PHYSICS LETTERS
4mPc 367r3 25 r x 34/3 sin x dx (mPc)2 -jf(Oc) J (4mPc)34/~
13 August 1979
,
(27)
where f(Pc) is the space-time fraction occupied by instantons and anti-instantons, Pc
f(Pe) =
rr2
f
(28)
0 Strictly speaking, our formulae do not hold for the integration region x ~< 1 in (27). This region can be investigated with the formulae for S -+ and D(p), valid for small mp [8]. But it gives a negligible contribution to Im T so it can be included. The integral in (27) can be approximated by - cos 4mOc when 4mPc/> 5 ~r. At 4mPc = 37r it has a minimum o f - 0 . 5 , and it is O(10 - 2 ) when 4mPc <. 2ft. For a numerical estimate let us compare Im T with the imaginary part of the amplitude given by the QCD graphs 1b, which (in the same normalization) is given by 2rra 2 . Their ratio is ImT _ 2rro~
187r2 25 f(Pc) ~ mpc x34/3sin x dx (mOo)2
3
°t2 0d
(29)
(4mPc) 34/3
Recently it has been argued [6,7] that inside of heavy quark-antiquark bound states the space-time fraction f(Oc) is about 10 -3 and the critical instanton size Pc is of the order of magnitude of the inverse o-meson mass. This means, say, for charm and bottom quark bound states 2 <~ mPc <~ 5. Finally, setting c~2 = 0.1, as is usually assumed in calculations with heavy quark bound states, one sees that the ratio (29) is of order one. It is remarkable that the instanton contribution is not positive definite. We argued before that the instanton term by itself is not the absolute square of a production amplitude (quarks and gluons are not asymptotic states any more). Thus this fact does not lead to a conceptual difficulty. Of course, the total (i.e. perturbative and nonperturbative) contribution to the decay rate should be positive definite - in agreement with the optical theorem with hadronic intermediate states. The introduction of a cut-off Pc connected with hadronization [6,7] seems very appealing to us. However, it should be mentioned that one can evaluate eq. (6) alternatively as follows. For large P4, the behaviour of Tr ~tr (9) gives rise to a rapid convergence of the o-integral in euclidean space. Hence eq. (6) can be integrated up to infinity. After having rescaled P by x = 2P4P, eq. (6) reads (Nf = 4) A
2(24rr2)2 640
(-~)
6
l'3mi
la 34/3 f
n s ( - - - Z ) (2-~4) i:-u,d,
~o ( d x ln-Tff]2p4~6x31/3K2(x),0
(30)
0
and the leading behaviour of (30) for large P4 is A ~ (24;:)2 225/3 256 [ F ( ~ ) ] 4 l-I ~l'3mi~.CP ]34/3[ln2P4'~6 (~-) V ( ~ ) i=u,d,s \-~---]kfff-44I ~ .--ff--]
.
(31)
The imaginary part of (31), continued to Minkowski space, is totally different in structure from the one obtained in (27). This demonstrates that the result in Minkowski space depends drastically on the approximation made in the euclidean region - which is necessary because of our limited knowledge about hadronization (dilute gas approximation). Note that the numerical value of the imaginary part of (31) depends drastically on p/2P4. E . g . , when setting 2m = 3 G e V - a value where the dilute gas approximation starts to break down because the dominant x in integral (30) corresponds to O = 1 GeV -1 - Im A varies in the range 0.3 to 30 when p varies between 0.3 and 0.7 GeV, respectively. Our crude estimates show that it is conceivable for the instanton contribution to heavy quarkonium decay to 263
Volume 85B, number 2,3
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be sizable over a broad range of quark masses. It is interesting that only C = +1 states, in particular the pseudoscalars, obtain a contribution. Unfortunately the evidence for the existence of the pseudoscalar states r/c and rl'C and their hadronic decays is poor [ 13]. Usually one identifies the state rio with the resonance at 2.8 GeV. The hadronic width for tic predicted by perturbative QCD is larger than the experimental upper bound for the width of that resonance. This discrepancy could be removed by negative instanton contributions. In this context it seems worthwhile to note that instantons could distort the two-jet-like distribution of the heavy meson decay products as predicted by perturbative QCD. But a truly quantitative discussion of these effects is premature as long as a more precise control over the scale size cut-off pc and the boundary of the dilute instanton phase is lacking. We would like to thank M. Bace for discussions and collaboration on related problems.
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11 ] [12] [13]
264
A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B (1975) 85. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 18; Phys. Rev. D14 (1976) 3432, Erratum; D18 (1978) 2199. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. C. Callan, R. Dashen and D. Gross, Phys. Lett. 63B (1976) 334. C. CaUan, R. Dashen and D. Gross, Phys. Rev. D17 (1978) 2717. C. Callan, R. Dashen and D. Gross, Phys. Lett. 78B (1978) 307, and Princeton preprint (1978). E. Shuryak, Phys. Lett. 79B (1978) 135. N. Andrei and D. Gross, Phys. Rev. D18 (1978) 468. C. Callan, R. Dashen, D. Gross, A. Wilczek and A. Zee, Princeton preprint (1978). L. Beaulieu, J. Ellis, M.K. Gaillard and W. Zakrzewski, Phys. Lett. 77B (1978) 290. L. Beaulieu, J. Ellis, M.K. Gaillard and W. Zakrzewski, Phys. Lett. 81B (1979) 41. L.S. Brown, R.D. Carlitz, D. Craemer and C. Lee, Phys. Rev. D17 (1978) 1583. E. Bloom, talk given at the Rencontre de Moriond (March 1979).