Nuclear Physics A497 (1989) 229c-234~ North-Holland, Amsterdam

229c

PERTURBATIVE QCD IN EXCLUSIVE PROCESSES

Nathan ISGUR

Department of Physics, University of Toronto, Toronto, Ontario, Canada

Since Llewellyn Smith and I first sounded a warning [l] s g ainst the use of perturbative QCD for exclusive processes [Z-5], there have been many developments in both theory and experiment. We have therefore re-examined the issues involved. In this talk I will summarize our results: a detailed discussion is given elsewhere [6].

1)

2)

3)

The factors which prompted this re-examination include: Reanalyses ]7-191 of the perturbative calculation, which reveal a sign error that re- verses, for a given nucleon wave function, the predicted sign of the

G& . Sum rule calculations 191, supported by lattice Monte Carlo calculations [II, 121, which find distribution amplitudes of the type apparently needed to explain both the sign and magnitude of the nucleon magnetic form factors, and the magnitude of the pion electric form factor F,, , in terms of perturbative &CD. On the other hand, direct sum rule calculations 113-161 of the pion and nucleon form factors find, in agreement with our earlier conclusion (11, that these form factors are dominated by non-perturbative effects at accessible values of Q’. New data for exclusive hadronic [17,18] and gamma-gamma (191 reactions, which gen- erally disagree with the predictions of perturbative QCD.

We concentrate on form factors, since they constitute the one case in which advocates of perturbative QCD claim unmitigated success for their calculations. Our main point is that, as anticipated in [I], these calculations cannot be believed since they involve an illegitimate use of perturbation theory. On the other hand, we find [l,S] that non- perturbative effects can easily explain the data.

As before [I], we start by observing that according to perturbative QCD

Q4G& -a;j (11

where j, which has dimensions rnr , varies logarithmically with Q” and would be expected to be of the order of some typical hadronic mass scale such as < pg >‘. This guess is borne out by simple models [I] which lead to values of GP,, that are about two orders of magnitude less than the value of about I GeV4 observed for Q” between 3 and 31 GeV’. Simple models combined with perturbative QCD also underestimate the pion form factor, by a factor of about three.

We now explain how asymmetric wave functions apparentiy generate perturbative contributions to form factors much greater than the order of magnitude estimate above and why this enhancement is an artefact due to the illegitimate use of perturbation theory. We consider first the pion form factor which is given asymptotically by

0375-9474/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

230~ N. Isgur / Perturbative QCD in exclusive processes

where the distribution amplitude 4, is related to the S = 0, L, = 0 component of the (infinite momentum or light-cone) pion q-q wave function $9 by

where z is the quark momentum fraction in the infinite momentum frame, $z satisfies

P,” being the associated q-6 probabihty, and we have used the symmetry of the wave function for 5 = z1 - z2 = 1 - .r in writing (2). In ref. 111 we found that with the asymptotic form @“,” = Azfl - z), or other forms suggested by simple models with probability peaked at x = l/2, eqn.(2) cannot explain the data unless A corresponds to an impossibly large value of < pt >. However, sum rules suggest [Q] a distribution 4:” = Bs(l - 2)(1 - 2s)’ which concentrates probability near z = 0 and z = 1 and generates a much bigger form factor than 4;’ for a given < ps > because of the factor l/(z’zf in (2).

Eqn.(Z) is obtained by convoluting the wave functions for the incoming and outgoing pion with a hard scattering amplitude TH which, to leading order in l/Q’, can be calculated perturbatively as a power series in as for Q” + co with x and z’ fixed. To leading order

in as, T, is given by the one gluon exchange contributions shown in fig.1. It is intuitively clear that the momentum transfer flowing through the gluons, which is given by Z$ZZQ’

for the first two diagrams, and z; z1 Q” for the second two, must be large in order for perturbation theory to be applicable. If this condition is not satisfied, we anticipate two problems -

a) the perturbative expansion of the leading twist contribution to TH may not be under

control (using Q’ as the scale, one expects corrections of order ty, (Q2)ln(z));

b) there may be important higher twist contributions.

The first problem is dealt with in [20] and 1211 (where the work of [20] is extended and corrected) where it is suggested that the perturbative expansion may be unreliable unless a scale very much less than Q’ is used and Q” is very large. We are concerned with the

second problem which we find to be even more serious.

An illustrative model of higher twist contributions is given by including an effective &on mass mp and using l/(s’zQ* + m,2) for the propagators in fig.1; in such a model the higher twist contributions are proportional to rni /(z’~Q”)_ The message is that ZZ’&* must be large compared to the typical hadronic mass scale in order for perturbation theory to be applicable. This condition is not satisfied in calculations based on 4:” at available

9”. To show just how misleading perturbation theory is when combined with this asym-

metric wave function, we follow two procedures - a) we completely exclude the regions where perturbation theory is a priori unreliable by

including factors B(s’rQ2 - mf,,,) in the calculation; b) we include a gluon mass m, when calculating the amplitudes corresponding to fig.1.

In both cases the difference between the results obtained with these procedures and the asymptotic result is a higher twist effect. The results are given in Table 1 for several

N. Isgur / Perfurbative QCD in exclusive processes 231c

Fig. 1 : The hard scattering amplitude which controls the asymptotic behaviour of the pion form factor; p(p’) is the four-momentum of the initial (final) pion and z1 -t-z2 = z: +s; = 1.

procedure : I I @I

rnki, or rni : 0.25 1.0 0.25 1.0

(GeV’) !

Q2(GeV2) = 1 2 0 10 0

2 8 1 16 6

4 16 2 23 10

8 27 8 32 16

Table 1. The percentage of the fuI1 contribution of 4,“” to Eqn.(2) which remains after regulating higher twist effects.

232~ N. Isgur / Perturbative QCD in exclusive processes

plausible values* of rnkin and rnz for Q2 in the range where perturbative QCD has been

claimed [S] to be applicable to the existing data. These results suggest that when 4,“” is used, the asymptotic formula (2) overestimates the perturbative contribution by about an order of magnitude. The perturbative contribution exceeds the higher twist effects only for c = m~i,/Q2 or rni/Q’ less than about 0.01 (see Ref. 6).

Exactly analogous considerations apply in the case of nucleon form factors. If per- turbation theory is used uncritically, amplitudes which concentrate probability at small z lead to large form factors because the hard scattering amplitude contains factors l/z. However, it is not legitimate to use perturbation theory unless z;z~.Q* is large, and this

condition is not satisfied unless Q” is enormous. Once again we quantify the degree to which perturbation theory is misleading a) by introducing theta functions which exclude the regions where perturbation theory is a priori unreliable (for details see [6]), and b) in- cluding a gluon mass in the calculation of the hard scattering amplitude. The suppressions that result for GP, (the effects on G; are equally devastating) are given in Table 2. We see that in this case the asymptotic formula for GP, , when used with the distribution am- plitude $,“” of Ref. [q], overestimates the perturbative contribution by about two orders of magnitude. The perturbative contribution exceeds the higher twist effects only for s less than about 3~10~~ (see Ref.6); this value of c corresponds to an m:,,, or rni of about 0.003 GeV’ at Q’ = 10 Gel” or to a Q” of about 1000 GeV2 for ml,.,, or rni equal to 0.25 GeV’ !

Having argued that perturbative QCD cannot explain the observed form factors, we must ask what mechanism is responsible. The answer given by us earlier [l], and discussed in great detail in [6] for both “intuitive” and asymmetric (CZ) wave functions, is that overlap of the “soft” components of the wave functions for the initial and final particles generates a higher twist contribution which, although very model dependent, has the same magnitude as the data in all cases.

procedure : (4 0))

main or rni : 0.25 1.0 0.25 1.0

(GeVz)

Q2(GeVZ) = 5 1 0 3 1 10 1 0 4 2 25 1 1 7 3

Table 2: The percentage of the full contribution of 4,“” to the asymptotic formula for GP, which remains after regulating higher twist effects.

* It is presumably the scale of glueball masses that controls higher twist effects in the t-channel, so that it is probably safe to set mki, or rni 2 1GeV’. However, we would not consider it ridiculous to assume a value as small as 0.25GeV2 given that perturbative QCD is not totally misleading in deep inelastic scattering for Q” - 1GeV2.

N. Isgur f Perturbative QCD in exclusive processes 233~

Finally we must ask whether our conclusions preclude the use of perturbation theory at available Q” in other processes. As far as inclusive processes such as deep inelastic scattering or the Drell Yan production of massive mu-pairs is concerned, the answer is no[6]. In these cases the expansion parameter is Q” (except for z -+ 1) and the leading order terms, which are of order 1, dominate the higher twist contributions, which are of order m2/Q2, for Q” above a few GeVZ. In the case of form factors, the leading order terms are of order aP,, with p = 1 for the pion and p = 2 for the nucleon, which is to

be compared to higher twist terms of order m2/@ where l/@ is greater than l/Q’ by factors such as < (l/z) >; it follows immediately that we would expect much higher Q* to be needed to test form factors. In the case of other exclusive processes, more powers of QS are involved in general and the situation is even worse.

We conclude that, very unfortunately, the many elegant tests of perturbative QCD in exclusive processes that have been proposed are irrelevant at experimentally accessible values of momentum transfer.

Acknowledgements

This work was begun while N.I. was a visitor at the Department of Theoretical Physics, Cxford University. He is grateful to the Department and to the Science and Engineering Research Council for making this visit possible. He would also like to acknowledge the ho:pitality of the Theory Division at CERN, where it was completed.

This research was funded in part by grants from the Science and Engineering Council of the UK and the Natural Sciences and Engineering Research Council of Canada.

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