Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

PERTURBATIVE QCD IN EXCLUSIVE PROCESSES

Nathan I S G U R CERN, CH-1211 Geneva 23, Switzerland

and

C.H. LLEWELLYN SMITH Department qf Theoretical Physics, 1 Keble Road, Oxford OXI 3NP. UK

Received 29 June 1988; revised manuscript received 23 September 1988

We have re-examined our arguments against the dominance of perlurbative QCD over soft, non-perturbative effects in exclu- sive processes at currently available Q2 We find that claims that our earlier objections can be overcome are based on the illegiti- mate use of perturbative QCD, and that the perturbative contributions are much smaller than soft contributions (which are capable of explaining the data) in the simple test cases of the pion and nucleon electromagnetic form factors. We see no reason why our results should not generalize, and conclude that there is no justification for the continued application of perturbative QCD to exclusive processes.

Since we first sounded a warning [ 1 ] against the use of per turba t ive Q C D for exclusive processes [2 - 5 ] there have been many deve lopments in both the- ory and exper iment . We have therefore re-examined the issues involved. In this let ter we summar ise our results: a deta i led discussion is given elsewhere [ 6 ].

The factors which p rompted this re -examinat ion include:

( 1 ) Reanalyses [ 7 -10 ] o f the per turba t ive calcu- lat ion, which reveal a sign error that reverses, for a given nucleon wavefunction, the predicted sign of the nucleon magnetic form factors. This implies that only asymmetr ic and very sub-asymptot ic wavefunct ions have any hope of describing the data for GPm and G%.

(2 ) Sum rule calculat ions [9 ], suppor ted by lat- tice Monte Carlo calculat ions [11,12] , which find distr ibution ampli tudes o f the type apparent ly needed to explain both the sign and magni tude of the nu- cleon magnet ic form factors, and the magni tude of the pion electric form factor b~, in terms of pertur-

ba t ive QCD. On the other hand, direct sum rule cal-

culat ions [ 13-16 ] o f the pion and nucleon form fac-

tors find, in agreement with our earl ier conclusion

[ ! ], that these form factors are domina ted by non- per turba t ive effects at accessible values of Q2.

(3) New data for exclusive hadronic [ 17,18 ] and

g a m m a - g a m m a [ 19 ] reactions, which generally dis-

agree with the predic t ions o fpe r tu rba t ive QCD.

We concentrate on form factors, since they consti-

tute the one case in which advocates of per turba t ive Q C D claim unmit igated success for their calcula-

tions. Our main point is that, as ant ic ipated in ref.

[ 1 ], these calculations cannot be bel ieved since they involve an illegitimate use of perturbat ion theory. On

the other hand, we find [ 1,6 ] that non-per turbat ive effects can easily explain the data.

As before [ i ], we start by observing that according to per turbat ive QCD

Q4G~, ~ ce~ f , ( 1 )

Permanent address: Department of Physics, University of To- ronto. Toronto, Ontario, Canada M5S IA7.

where f which has d imensions m 4, varies logarith- mical ly with Q2 and would be expected to be of the

0370-2693 /89 /$ 03.50 © Elsevier Science Publishers B.V. ( Nor th -Hol l and Physics Publishing Divis ion )

535

Volume 217, number 4 PHYS1CS LETTERS B 2 February 1989

order of some typical hadronic mass scale such as ( p 2 ) 2. This guess is borne out by simple models [ 1 ] which lead to values of G~ that are about two orders of magnitude less than the value of about 1 GeV 4 ob- served for Q2 between 3 and 31 GeV 2. Simple models combined with perturbative QCD also underesti- mate the pion form factor, by a factor of about three.

We now explain how asymmetric wavefunctions apparently generate perturbative contributions to form factors much greater than the order of magni- tude 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

1 1

F~(Q 2) - 3rc2 dx' dx 0* )0~(x) x , x Q 2 , (2) 0 0

where the distribution amplitude #~ is related to the S= 0, Lz= 0 component of the (infinite momentum or light-cone) pion q-cl wavefunction q/o by

0~(x) = f d~P~ ~,o (x, pT), (3)

where x is the quark momentum fraction in the infi- nite momentum frame, q/o satisfies

f dx d Zp,r 2 o I~°(x, PT) I - P q c l , (4)

p o being the associated q-~l probability, and we have used the symmetry of the wavefunction for x = xj ~-~ x2= 1 - x in writing (2). In ref. [ i ] we found that with the asymptotic form q~as = A x ( 1 - x ) , or other forms suggested by simple models with probability

peaked at x = ½, eq. (2) cannot explain the data un- less A corresponds to an impossibly large value of (p2x). However, sum rules suggest [9 ] a distribution

cz = Bx ( 1 - x) ( 1 - 2x) 2 which concentrates prob- ability near x = 0 and x = 1 and generates a much big- ger form factor than ~ for a given ( p 2 ) because of the factor 1 / x ' x in (2).

Eq. (2) is obtained by convoluting the wavefunc- tions for the incoming and outgoing pion with a hard scattering amplitude Tn which, to leading order in 1/Q2, can be calculated perturbatively as a power se- ries in as for Q2-~oo with x and x ' fixed. To leading order in O~s, TH 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 x'2x2Q 2 for the first two diagrams and x'~x~ Q2 for the second two, must be large in order for perturbation theory to be applica- ble. I f 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 Q2 as the scale, one expects corrections of order c~(Q 2) In(x) 1;

(b) there may be important higher twist contributions.

The first problem is dealt with in ref. [20] and in ref. [21] (where the work of ref. [20] is extended and corrected) where it is suggested that the pertur- bative expansion may be unreliable unless a scale very much less than Q2 is used and Q2 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

p' 4 p'

xl P xz P

J o o o t g

m t l | l l

4-

I I

4-

I

Fig. l. 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 x~ +x2 =x'~ +x~ = I.

536

Volume 217, number 4 PHYSICS LETTERS B

Table 1 The percentage of the full contribution of ~c,z to eq. (2) which remains after regulating higher twist effects.

2 February 1989

Q2 (GeV z) Procedure (a) Procedure (b)

m~,,° or m~ m~,i, or m~ m~in or m 2 m~,,i~ or m~ = 0.25 GeV e = 1.0 GeV 2 = 0.25 GeV 2 = 1.0 GeV 2

1 2 0 I0 0 2 8 1 16 6 4 16 2 23 10 8 27 8 32 16

is given by including an effective gluon mass mg and using 1/(x'xQ2+m~ ) for the propagators in fig. 1; in such a model the higher twist contributions are proportional to m2/y'xQ 2. The message is that xx' Q2 must be large compared to the typical hadronic mass scale in order for perturbation theory to be applica- ble. This condition is not satisfied in calculations based on 0~ "z at available Q2.

To show just how misleading perturbation theory is when combined with this asymmetric wavefunc- tion, we follow two procedures:

(a) we completely exclude the regions where per- turbation theory is a priori unreliable by including factors , 2 O(x xQ -m~li, ) in the calculation;

(b) we include a gluon mass mg when calculating the amplitudes corresponding to fig. 1.

In both cases the difference between the results ob- tained with these procedures and the asymptotic re- sult is a higher twist effect. The results are given in table 1 for several plausible values ~1 of 2 mini n and mg for Q2 in the range where perturbative QCD has been claimed [ 5 ] to be applicable to the existing data. These results suggest that when g}cz is used, the asymptotic formula (2) overestimates the perturba- tire contribution by about an order of magnitude. The perturbative contribution exceeds the higher twist ef- fects only for ~_=m2mm/Q 2 o r m g / Q 2 less than about 0.01 (seeref. [6]) .

Exactly analogous considerations apply in the case of nucleon form factors. If perturbation theory is used uncritically, amplitudes which concentrate probabil-

~ It is presumably the scale ofglueball masses that controls higher twist effects in the t-channel, so that it is probably safe to set m~, , or m~ > 1 GeV a. However, we would not consider it ri- diculous to assume a value as small as 0.25 GeV 2 given that perturbative Q C D is not totally misleading in deep inelastic scattering for Q2 ~ 1 GeV 2.

ity at small x lead to large form factors because the hard scattering amplitude contains factors 1/x. How- ever, it is not legitimate to use perturbation theory unless x,x~Q 2 is large, and this condition is not sat- isfied 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 unreli- able (for details see ref. [6]) , and (b) including a gluon mass in the calculation of the hard scattering amplitude. The suppressions that result for G, p, (the effects on Gp, are equally devastating) are given in table 2. We see that in this case the asymptotic for- mula for GPm, when used with the distribution ampli- tude ¢~pCZ of ref. [9 ], overestimates the perturbative contribution by about two orders of magnitude. The perturbative contribution exceeds the higher twist ef- fects only for e less than about 3 X I 0 -4 (see ref. [ 6 ] ); this value ofc corresponds to an rn~in or m~ of about 0.003 GeV 2 at Q2 = 10 GeV 2 or to a Q2 of about i 000 GeV 2 for m~,i, or m~ equal to 0.25 GeV2!

Having argued that perturbative QCD cannot ex- plain the observed form factors, we must ask what mechanism is responsible. The answer given by us earlier [ 1 ], and discussed in great detail in ref. [6] for both "intuitive" and asymmetric (CZ) wave- functions, is that overlap of the "soft" components of the wavefunctions 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.

Finally we must ask whether our conclusions pre- clude the use of perturbation theory at available Q2 in other processes. As far as inclusive processes such as deep inelastic scattering or the Drell-Yan produc- tion of massive mu-pairs is concerned, the answer is no [6]. In these cases the expansion parameter is Q2

537

Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

Table 2 The percentage of the full contribution of O cz to the asymptotic formula for G~, which remains after regulating higher twist effects.

Q2 (GeV 2) Procedure (a) Procedure (b)

rrt21in or m 2 mZi, or m~ 2 rn~li, or m 2 rnZmi, or rn 2 = 0.25 GeV 2 -- 1.0 GeV 2 = 0.25 GeV 2 = 1.0 GeV 2

5 1 0 3 1 10 l 0 4 2 25 1 1 7 3

( excep t for x ~ 1 ) and the l ead ing o rde r te rms , wh ich

are o f o rde r 1, d o m i n a t e the h igher twist con t r ibu- t ions, which are o f o rde r m 2 / Q 2, for Q2 above a few

G e V 2. In the case o f f o r m factors , the l ead ing o rde r

te rms are o f o rder ozg, wi th p = 1 for the p ion and p = 2

for the nuc leon , which is to be c o m p a r e d to h igher twist t e rms o f o rde r m2/O~ 2, where 1 / 0 2 is greater

than 1/Q 2 by factors such as ( l / x ) ; it fo l lows im-

med ia t e ly that we wou ld expec t m u c h h igher Q2 to

be needed to test f o r m factors. In the case o f o the r

exc lus ive processes , m o r e power s o f OLs are i n v o l v e d

in genera l and the s i tua t ion is even worse.

We conc lude that , ve ry unfor tuna te ly , the m a n y el-

egant tests o f pe r tu rba t ive Q C D in exclus ive pro-

cesses that have been p r o p o s e d are i r r e l evan t at ex-

per imenta l ly accessible va lues o f m o m e n t u m transfer.

This work was begun whi le N.I . was a v i s i to r at the

D e p a r t m e n t o f Theore t i ca l Physics , O x f o r d U n i v e r -

sity. He is grateful to the D e p a r t m e n t and to the Sci-

ence and Eng inee r ing Resea rch Counc i l for m a k i n g

this v i s i t possible , He wou ld also l ike to acknowledge

the hospi ta l i ty o f the T h e o r y D i v i s i o n at C E R N ,

where it was comple ted . Th is research was f u n d e d in

par t by grants f r o m the Science and Eng inee r ing

Counc i l o f the U K and the Na tu r a l Sc iences and En-

g ineer ing Resea rch Counc i l o f Canada .

R e f e r e n c e s

[1] N. lsgur and C.H, Llewellyn Smith, Phys. Rev. Lett. 52 (1984) 1080.

[2] G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 43 (1979) 246; A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245; A. Duncan and A.H. Mueller, Phys. Lett. B 90 (1980) 159; Phys. Rev. D 21 (1980) 1626.

538

[ 3 ] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22 (1980) 2157. [4] G.P. Lepage, S.J. Brodsky, T. Huang and P.B. MacKenzie,

in: Particles and fields 2, eds. A.Z. Capri and A.N. Kamal (Plenum, New York, 1983) p. 83; S.J. Brodsky, T. Huang and G.P. Lepage, in: Particles and fields 2, eds. A.Z. Capri and A.N. Kamal (Plenum, New York, 1983) p. 143; S.J. Brodsky and G.P. Lepage, Phys. Rev. D 24 ( 1981 ) 1808; G.R. Farrar, E. Maina and F. Neri, Nucl. Phys. B 259 ( 1985 ) 702.

[ 5 ] S.J. Brodsky, in: Intern. Syrup. on Medium-energy physics (Beijing, China, 1987); SLAC-PUB-4387 ( 1987 ).

[6] N. Isgur and C.H. Llewellyn Smith, CERN preprint TH- 5013/88.

[7 ] I.G. Aznauryan, S.V. Esaybegyan and N.L. Ter-lsaakyan, Phys. Lett. B 90 (1980) 151;B 92 (1980) 371(E).

[8] V.M. Belyaev and B.L. Ioffe, Soy. Phys. JETP 56 (1982) 493.

[ 9 ] V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112 ( 1984 ) 174; Nucl. Phys. B 246 (1984) 52.

[ 10] N. lsgur and C.H. Llewellyn Smith, unpublished ( 1983 ). [ 11 ] A.S. Kronfeld and D.M. Photiadis, Phys. Rev. D 31 ( 1985 )

2939. [ 12 ] S. Gottlieb and A.S. Kronfeld, Phys. Rev. Lett. 55 ( 1985 )

2531; Phys. Rev. D 33 (1986) 227. [ 13 ] B.L. Ioffe and A.V. Smilga, Phys. Lett. B 114 (1982) 353;

Nucl. Phys. B 216 (1983) 373. [ 14 ] V.A. Nesterenko and A.V. Radyushkin, Phys. Lett.'B 115

(1982) 410; Sov. Phys. JETP Lett. 35 (1982) 488. [ 15] V.A. Nesterenko and A.V. Radyushkin, Phys. Lett. B 128

(1983) 439; Yad. Fiz. 39 (1984) 1287. [ 16] A.V. Radyushkin, Acta Phys. Pol. B 15 (1984) 403. [ 17 ] P.R. Cameron et al., Phys. Rev. D 32 ( 1985 ) 3070, [ 18 ] S. Heppelmann et al., Phys. Rev. Lett. 55 ( 1985 ) 1824. [19] H. Aihara et al., Phys. Rev. D 36 (1987) 3506; Phys. Rev.

Lett. 57 (1986) 404; M. Althoffet al., Phys. Lett. B 130 (1983) 449.

[20] R.D. Field et al., Nucl. Phys. B 186 ( 1981 ) 429. [ 21 ] F.M. Dittes and A.V. Radushkin, Yad. Fiz. 34 ( 1981 ) 529;

Phys. Lett. B 134 (1984) 359; M.H. Sarmadi, Phys. Lett. B 143 (1984) 471; S.V. Mikhailov and A.V. Radyushkin, Nucl. Phys. B 254 (1985) 89; B 273 (1986) 297; A.V. Radyushkin and R.S. Khalmuradov, Yad. Fiz. 42 (1985) 458.