# Once More on the Applicability of Perturbative QCD to Elastic Form Factors Page: 4 of 6

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analyze to what extent one can justify the assumptions that the hadronic wave

functions at accessible energies are very broad, humpy functions of CZ type, but, at

the same time, the soft nonperturbative contributions to the hadronic form factors

rapidly vanish with Q2.

The standard argument in favor of the CZ wave functions is that the are dictated

by the QCD sum rules. In particular, the moments ((N) . ft P!L) (N d( of the

pion wave function are given by the SR [8]

r N M _ 3 _' 4M M ) a (GG) l6 wa ,(Qq)a( l + 4N) (8}'

f {f } 4i r( + N +}N + 3) + 12tMs + 81 M+ ( + 4N) (8)

where a47 is the effective threshold characterising the position of higher states and

M' is-the so-called Borel parameter. The values of (() are extracted from the sum

rule using the requirement of the best agreement between its left- and right-hand

sides. The N = 0 case was considered in the pioneering SVZ paper: if one takes

s4 . 0.7 GeV', the right-hand side is fairly constant for all M= > 0.6GeV', with the

output value f, 130 MeV, in a good agreement with experiment The correlation

between the s and f, values (obtained from a fitting procedure) is well reproduced

by the local duality relation f; s/(4x') that follows from the SR in the formal

M' -. oo limit.

The same strategy was used by CZ for the higher moments N = 2 and N = 4.

Note, however, that the nonperturbative terms in their sum rule have a completely

different N-dependence compared to the perturbative one: the perturbative term

decreases like 1/N' for higher moments while the condensate terms are either con-

stant or even increasing with N. Thus, the effective scale in the channel (settled by

the ratio of the condensate terms to the perturbative one) substantially increases

for higher N: s2)= 2as, a4 3sv, etc. Again, the fitted values of ((N)} are well

reproduced by the local duality relation:

N [N)

' x'f; (N + 1)(N 4 3) ( ")

As a result, the value ((}eCZ = 0.43 found by CZ is by factor 2 larger than ((_)"

1/5. Such a large value can be attributed only to a wave function concentrated in

the region 1(1~ t 1. This is how the CZ wave function was extracted from the QCD

sum rules. A crucial implicit assumption in this derivation is that one can neglect

higher power corrections, and it is suffcient to take into account only the lowest

condensates.

The soft contribution to the pion form factor can be also estimated within the

framework of the QCD sum rules. The relevant sum ruler ,. , a : e

fIF,(Q) - x dot /24 p 5(ai, saq')exp ( 2 s

+a,{G0) 16 ra(q)' / 2Q '

+ I2rrM' 81 M4\3 M'j,in fact, has a striking similarity to the pion wave function sum rule (just take

Q2/M' - N): the perturbative term vanishes like 1/(Q2)2 for large Q', while the

(q)- and (GG)-terms are constant or linearly increasing with Q', - even the numer-

ical values of the coefficients are almost identical The ratios of the nonperturbative

terms to the perturbative term grow with Q' and, as a result, the parameter a5 I

straightforwardly extracted from the SR (9) increases with Q2. In particular, for

Q' = 1 GeV' the fitting gives the so-value very close to that obtained from the

SR for f,: i~49''- 0eV') 0.7 GeV2, just demonstrating the self-consistency

of the physical interpretation of so and of the whole SR approach. However, for

Q' = 6GeV', a formal fitting gives the value as(Q' = 6GeV') t 1.5GeV' - by

factor of 2 larger than aF 17]. This is precisely the same effect that produced the

growth of a4N7 and ((N) in the CZ SR. Using the local duality relation for the pion

form factor(10)

s 1 + 6so/Q' _

N" ftQ ) ~ 4af l {- + - ,Q=one can see that one gets larger Fdwi(Q2) if one takes larger ao. It should be

emphasized that the soft contribution, estimated by local duality with a constant

duality interval sa 0.70eV', is sufficiently large to describe existing data li as

increases with Q', then the soft contribution is larger than the data 117[.

Main lesson is that, within the QCD sum rule approach, the form of the pion

wave function and the magnitude of the soft nonperturbative contribution to the

pion form factor are strongly correlated the sum rules for ((N) and for F,(Q') have

an essentially identical structure (with N .-. Q'), and it is impossible to increase {{'}

compared to ()'" and at the same time get a rapidly decreasing soft contribution

In full accordance with the results obtained (in a wave function formalism) by lsgur

and Llewellyn Smith, a big perturbative contribution is always accompanied by a

big nonperturbative term, and the basic assumption of the pQCD scenario fails A

pragmatic observation is that taking a = conast . 4'jf, one gets F"01i close to

data and a small one-gluon-exchange contribution, the addition of which improves

the agreement with data. The same statement is true for the nucleons: taking the

nucleon duality parameter Sv = cost a 2.3GeV' (the numerical value extracted

from the QCD sum rule for the nucleon mass 13]) in the local duality formula for

the proton magnetic form factorGJ(Q') = 3 T2 - l {(472 - 1)(T' - 1) + (4T2 - 3)T T= - 1 } ,

(11)

(where T = 1 + Q'/2S) one gets a curve describing the data in a wide region

30eV' < Q' 20GeV=. Asymptotically, this soft contribution goes lke 0(a/Q }

One may ask, however what is wrong with the original CZ-derivation that

produced a broad wave function? This sum rule, taken at face value, definitely

requires a drastic increase of 47) for N = 2, 4, .... To better understand the nature

of the approximations made, it is instructive to rewrite the sum rule (8) for the wave(9)

6

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Radyushkin, Anatoly. Once More on the Applicability of Perturbative QCD to Elastic Form Factors, report, June 1, 1992; [Newport News, Virginia]. (https://digital.library.unt.edu/ark:/67531/metadc930585/m1/4/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.