2011 QCD and High Energy Interactions - Rencontres de Moriond ...

Nonlocal Condensate Model for QCD Sum Rules
Ron-Chou Hsieh
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
We include effects of nonlocal quark condensates into QCD sum rules (QSR) via the Källén-
Lehmann (KL) representation for a dressed fermion propagator. Applying our formalism
to the pion form factor as an example, QSR results are in good agreement with data for
momentum transfer squared up to Q 2 ≈ 10 GeV 2 .
1 Introduction
It is known that the unique feature of the asymptotic freedom and the crucial concepts of infrared
safety and factorization make the perturbative quantum chromodynamics (PQCD) method powerful
for studying QCD processes. Despite of the remarkable success on high energy hard processes,
its applicability at moderate energies is limited. Hence, one needs to seek the aid of
non-perturbative approaches, which include lattice QCD, QCD sum rules (QSR) and instanton
gas model, etc. The basic idea of QSR is to construct a correlator which relates a quantity we
are interested in to an operator-product expansion under the assumption of the quark-hadron
duality.
2 Pion decay constant
The quark-hadron duality is implemented via a dispersion relation, in which an unknown parameter
s0 is introduced. The Borel transformation is then applied to determine the value of
s0. We first calculate the pion decay constant as an example, which is defined as
⟨Ω|ησ(y)|π(p)⟩ = ifπpσe −ip·y , ησ(y) = ū(y)γσγ5d(y), (1)
with |Ω⟩ representing the exact QCD vacuum. The quark-hadron duality leads to the dispersion
relation
with
(2π)f 2 πδ(P 2 − m 2 π) = 1
∫ s0
2πi 4m2 ds
q
ImΠ2(P 2 )
s − P 2 . (2)
Π2(P 2 ) =
Πµν(P 2 ) =
1
3P 4
[
∫
4P µ P ν Πµν(P 2 ) − P 2 g µν Πµν(P 2 ]
)
d 4 xe −iP ·x [
⟨Ω|T ηµ(x)η † ]
ν(0)
,
|Ω⟩. (3)