Pion form factor in the NLC QCD SR approach
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ELEMENTARY PARTICLES AND FIELDS Theory
Pion Form Factor in the NLC QCD SR Approach* A. P. Bakulev1)** , A. V. Pimikov1)*** , and N. G. Stefanis2)**** Received September 23, 2009
Abstract—We present results of a calculation of the electromagnetic pion form factor within the framework of QCD sum rules with nonlocal condensates and using a perturbative spectral density which includes O(αs ) contributions. DOI: 10.1134/S1063778810060189
1. INTRODUCTION
where the asymptotic pion DA has the form
An archetypical example of a QCD (hadronic) observable is the pion form factor, which is typical for a hard-scattering process obeying the factorization theorem [1, 2]. Consequently, at asymptotically large Q2 it can be cast in terms of a scale-dependent pion distribution amplitude (DA) [3] of leading twist two, ϕπ (x, Q2 ), convoluted with the hard-scattering amplitude of the process which contains the large external scale Q2 : 8παs (Q2 )fπ2 π 2 2 (Q ) , I −1 9Q2 1 ϕπ (x, Q2 ) π 2 dx, I−1 (Q ) = x
Fπpert (Q2 ) =
(1)
0
where fπ is the pion decay constant. The nonperturbative input—the pion DA ϕπ (x, μ2 )—can be expressed as an expansion over Gegenbauer polynomials ⎡ × ⎣1 +
ϕπ (x, μ2 ) = ϕas (x) n≥1
⎤
(2)
a2n (μ2 )C2n (2x − 1)⎦ , 3/2
⎡
π (μ2 ) = 3 ⎣1 + I−1
⎤ a2n (μ2 )⎦ ,
n≥1 ∗
The text was submitted by the authors in English. Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia. 2) ¨ Theoretische Physik II, Ruhr-Universitat ¨ Institut fur Bochum, Germany. ** E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] 1)
ϕas (x) = 6x(1 − x),
(3)
while the scale dependence of the coefficients a2n (μ2 ) is controlled by the Efremov–Radyushkin–Brodsky–Lepage (ERBL) evolution equation [1, 2]. At the one-loop level and at asymptotically large pert 2 Q , the pion form factor simplifies to Fπ (Q2 ) = 2 2 2 8παs (Q )fπ /Q . The onset of the asymptotic regime cannot be determined precisely; estimates [4, 5] show that this transition scale is of the order of 100 GeV2 . On the other hand, at intermediate momentum transfers 1 ≤ Q2 ≤ 20 GeV2 , the situation is more complicated because of the interplay of perturbative and nonperturbative effects. The latter effects are contained in a nonfactorizable part—called the soft contribution—so that one has to take it into account using some nonperturbative concepts, e.g., the method of QCD sum rules (SR) [6–9], the local quark–hadron duality (LD) approach [6, 10], and others. Note in this context that, describing the pion form factor within the three-point QCD SR approach [6– 8], the shape of the pion DA becomes irrelevant. This considerably reduces the inherent theoretical uncertainty of the method. The same applies to the LD approach, but the latter contains an additional uncertainty related to the continuum threshold parameter s0 (Q2 ) for intermediate and large values of Q2 —see for a discussion in [5]. However, the standard QCD SR’s [6–8] are plagued by instabilities arising at Q2 3 GeV2 , which are induced by
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