SUM RULES AND PION FORM FACTOR IN QCD

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SUM RULES AND PION FORM FACTOR IN QCD

V. Nesterenko, A. Radyushkin, A. Efremov

To cite this version:

V. Nesterenko, A. Radyushkin, A. Efremov. SUM RULES AND PION FORM FACTOR IN QCD.

Journal de Physique Colloques, 1982, 43 (C3), pp.C3-242-C3-245. �10.1051/jphyscol:1982348�. �jpa-

00221902�

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JOURNAL DE PHYSIQUE

CoZZoque C3, suppZe'ment m n o 12, Tome 43, de'cembre 2982 page C3-242

SUM RULES AND P I O N FORM FACTOR I N QCD

V.A. Nesterenko and A.V. Radyushkin

Presented by A.V. Efremov

Laboratory ^ofZ'heoretieaZ Physics, JINR, Eubna, USSR

The discovery that perturbative QCD methods can be applied to stu- dy the asymptotic behaviour of the pion electromagnetic form factor

(see, e.g., refs./l,2/ and the reviews / 3 / ) has been an important step in the development of the'theory of hard processes.The pion form factor for all orders and for all logarithms in perturbation theory was shown to have the factorized form:

where ^{Q 2}=-q2

,

p is a renormalization parameter, 6 is the pion wave function and E is the amplitude of the short-distance (SD) sub-

?recess q&* + q'q' (Fig. 1 )

a)(X

I %- -

_"PC

^I

_YP,^@*(Y)

A A

p, "2

Fig. 1

As it is now generally accepted /4-6/, the best choice for p is in the region where the higher order PT corrections are small. The 1 - l~op~yorrection to E have been calculated by Dittes and Radyushkin /7/

.

Its magnitude depends on the wave function +(x, p ) . It was shown /9/, that asymptotically +(x, p) + 6fTx(l-x) as p +

- ^.

However, for p h I GeV the wave function should be very broad for nearly-massless strongly in- teracting quarks. Thus; if one parametrizes +(x) simply as +(x)

-

^xr(l-x):

then one should expect that for pion r < < I (say, r z0.1

-

0.3). As a result, for the naive choice p = Q and small r the correction is huge:

X , A sixilar calculation was performed by Field et al. / 3 / , but

their results for some diagrams are incorrect.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982348

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A . V . Efremov C3-243

To reduce it, one should take _{p 2}= p

Ept

t Q 2 e x p ( - 3 / 2 r ) ( 5 0.007 Q 2 )

.

This means that for Q 2 ( 10 Gev2 the exchanged gluon (Fig. la) has vir- tuality much smaller than 1 G~v'. In such a situation it is, of course, misleading to rely on the asymptotic freedom.

First attempt to find the way out was a QCD inspired model /10,4/

with the result that for moderately large Q 2 the ?ion EM form factor is dominated by the Feynman mechanism /11/. The aim of this report is a brief description of a new approach /12,13,14/ that enables one to analyse the behaviour of 3 , ( € J 2 ) for moderately large ~ 2 .

Technically our analysis is based on the QCD sum rule approach /15/. We consider the three-point amplitude

(see for-notation Fig. Ib) where J p is the electromagnetic current and j a = ~ Y ~ Y , U is the axial one: < 0 1 f a (0)lP > = ifspa ,

where IP> is the I-pion state and f , = 133 MeV the pion decay constant.

The invariant amplitudes T i (corresponding to various structures present in T,B

!

^{depend on}³variables: p ; , p q 2 = ( P - p 2 ) 2. Owing to the asymptotic freedom one may calculate T

, &I\,

^{p g}^,^{q 2}⁾ ^{in the}

Euclidean region P

y ,

^{P $}

,

^q ⁽-1 G ~ v ~ . To extract hhe desired infor- mation about the physical states, we use the dispersion relation

and apply the Bore1 procedure /15/ in

p: and p ; : T i ( p : , p g , 6 1 2 ) - @ ( ~ f , ~ ~ ~ , ~ ~ ) .

For the pion form factor analysis the most important is the invariant amplitude Q

,

related to the structure Pa PBP

,

where P = g l c p 2 . Neglecting quark masses ( 210 MeV) we calculated the contribution of the diagram Ib into the p density:

In the real world p ( s l , s 2 . q 2 ) differs, of course, from po ( s l , s 2 , q 2 ) especially for small s l , s , . In particular, it contains the pion term p , ( s l , s 2 , q 2 ) = n 2 f ~ ~ n ( ~ 2 ) ~ ( s 1 - )m: 6 ( s z

-

):m

. .

The basic idea of the QCD sum rule approach /15/ is that the difference @

- a o

is mainly due to power corrections*) ( 1 / ~ ~ ) ~ generated by nonzero vacuum expectation values of quark and gluon operators. Taking into account the contributions proportional to < C ; , G ; > and a , < q q > (cf

.

^ref.^/I^{5 / )} we arrive at the following representat~on for the pion form factor

-

6

where ( a S / n ) < G G > = 0.012 G ~ v ~ , a,<\p2=1 .83.10 GeV2 /15/ and s o is the

"effective threshold" characterizing the region where one can substi-

X, To treat initial and final states on equal footing we take henceforth M1 = M 2 = FA.

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C3-244 JOURNAL DE PHYSIQUE

tute p(sl, s2, q2) by its f ree-quark value P ~ ( s ~ , s ~ , q2)

.

It is natural to define that the "true" so is that for which the region of weak sensitivity to variations of unphysical parameter M is the most broad. For Q

'

⁼1-3 Gev2 this gives so= 0.6-0.8 Gev2 respecti- vely (Fig, 2) which is in good agreement with the value s o=0.7 G ~ ex- V ~ tracted from the analykis /15/.

To get a reliable result for Yn(Q2)one should choose M~ SO as to reduce to a minimum both the power corrections (blowing up as M ~ + o ) and

the contribution of states with sl, s2 > so. In our case, however, the situation IS more com- plicated than in /15/, mainly because the free -quark term in eq. (2) behaves like l/C14 for l a r g e ~ ~ w h i l e the power corrections grow with

CI2. In particular to reduce the power corrections to 30 % for Q2 = 2 Gev2 one should take

~ ~ Gev2. The contribution of higher states 2 2 (estimated by use of the free-quark density) 0 2 L 6 is then =I00

.

However, it is the difference

M'(G~v') between the true higher state contribution and its free-quark approximation that matters, Fiq. 2 which may well be even zero for an appropriate choice of so. As is seen from Fig.3, in the region 1<Q2<4 Gev2 our theoretical curve (broken line) agrees with experimental data /16/

within 10-20% accuracy.

Taking M' sufficiently large one can control the power corrections for arbirarily high Q' values. The most radical way out is to take M2 =-

.

Then the power corrections are absent altogether, and if one assumes that the "effective threshold" s

,,

is Q2-indepen- dent, one obtains the finite energy sum rule

0 1.0 2.0 3.0 L.0

q2

(Gev2) (3) is nothing else but a duality rela- :?on between the pion and free-quark contri- Fig. 3 butions (cf. /17/ and references bherein).

Using ( 1 ) one can reduce eq. (3 ) to

For so = 4n2f

2

= 0.7 Gev2 this formula is in excellent agreement with experimental data (fig. 3, solid line). The agreement indicates that the local duality between the pion and free-quark contributions is indeed a very good approximation.

In principle, apart from power corrections one must take into account also higher perturbative corrections due to diagrams like that shown in Fig. Ic. However, their contribution is damped by thea /n=0.1 factor with respect to that of Fig. Ib. Our estimates show that for Q 2

5

^{10 G} ^~ these diagrams produce only 10% (positive) ^V ^~ correction to 3,(Q2). Thus, the celebrated one-gluon-exchange contribution is of little importance for available Q2. However, the situation changes drastically in the asymptotic Q 2 +

-

region, because Fig. Ic gives in this region 1/Q2 -contribution (corresponding to the asymptotic QCD analysis /I-3/) exceeding the 1/Q4 -contributions of Fig. Ib for sufficiently large Q2.

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A.V. Efremov

References

1. A.V.Radyushkin, JINR preprint P2-10717 (June, 1977);

A.V.Efremov and A.V.Radyushkin. Phys-Lett. 94B, 2451 (1980) and references therein.

2. S ; J . ~ r o d s k ~ and G.P.Lepage. Phys-Rev.,

E ,

157 (1980).

3. V.L.Chernyak. Proc. XV LINP Winter School, Leningrad (1980),1, p. 65;

A.H.Mueller. Phys.Reports,

2,

237 (1951).

4. A.V.Efremov and A.V.Radyushkin, JINX preprint E2-30-521 (1980j, submitted to Madison Conference.

5. P.M.Stevenson. Phys-Rev.,

GI

2916 (1981).

6. W-Celmaster and D.Sivers. Phys.Rev.

G,

227 (1981).

7. F.-11.Dittes and A.V.Radyushkin. Yad-Fiz., 34, 529 (1951).

8. R.D.Fheld et. al. Nucl.Phys.,

e,

429 (1-1).

9. A.V.Efremov and A.V.Radyushkin, JINR preprint E2-11983, (October, 1978)

.

10. A.V.Radyushkin. JINR report P2-80-687 (1990); see also Proc.

of the Symp. on Particle Physics, Visegrad, Hungary (September, 1981), p. 79.

11. R.P.Feynman. Photon-Hadron Interactions. Benjamin (1972).

12. V.A.Nesterenko and A.V.Radyushkin. Pis ma v JETF,

2 ,

395 (1932).

13. V.A.Nesterenko and A-V-Radyushkin. JINR preprint E2-82-204 (1932), to be published in Phys.Lett. B.

14. B.L.Ioffe and A.V.Smilga. Preprint ITEP-27 (1982).

15. M.A.Shifman, A.I.Vainshtein and V.I.Zakharov, Nucl-Phys.,

m,

447 (1 979)

.

16. C.Bebek et al. Phys.Rev.

m,

1693 (1978).

17. N.V.Krasnikov and A.A.Pivovarov. Report NBI-HZ-81-39 (Kopenhagen