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

We then apply the Borel transformation 1 a
ˆB(P 2 → M 2 )F (P 2 ) = 1
∫
dP
2πi C
2
M 2 e−P 2 /M 2
F (P 2 ), (4)
where C denotes any contour that encloses the branch cuts in the P 2 plane from zero to infinity.
Under the above transformation, Eq. (2) becomes
f 2 π = − M 2em2π/M 2 ∫ s0
2π
4m 2 q
ds ˆ B(P 2 → M 2 [
ImΠ2(P
)
2 )
s − P 2
]
. (5)
The duality interval s0 is determined by the requirement that fπ is least sensitive to the Borel
mass M in the fπ − M plot. In Fig.1(a) we present the M dependence of fπ for three different
s0 inputs. It indicates that for s0 = 0.69 GeV 2 , we obtain an almost M-independent behavior
for about M > 2 GeV, which implies the value of fπ ≃ 0.132 GeV.
3 Nonlocal condensate model
With the Wick contraction, we express the following matrix element as
[
⟨Ω|T ηµ(x)η † ]
ν(0)
|Ω⟩ = −⟨[iS u F (0 − x)]γµγ5[iS d F (x − 0)]γνγ5⟩
+ ⟨: ū(x)γµγ5[iS d F (x − 0)]γνγ5u(0) :⟩
+ ⟨: ¯ d(0)γνγ5[iS u F (0 − x)]γµγ5d(x) :⟩
+ ⟨: ū(x)γµγ5d(x) ¯ d(0)γνγ5u(0) :⟩. (6)
where the normal ordering terms, representing non-perturbative effects, are usually dropped for
trivial vacuum. We assume that these effects can be absorbed into a dressed fermion propagator,
[
⟨Ω|T ηµ(x)η † ]
ν(0) |Ω⟩ = −⟨[iS u (0 − x)]γµγ5[iS d (x − 0)]γνγ5⟩. (7)
The KL representation for a dressed propagator of the quark q is written as 3 ,
∫
⟨Ω|T[q(z)q(0)]|Ω⟩ = i
=
d4 ∫
k
∞
2 /kρq1
e−ik·z dµ
(2π) 4
0
(µ2 ) + ρ q
2 (µ2 )
k2 − µ 2 + iϵ
∫ (
∞ z2 ds exp
0 4 s
) ∫ ∞
dµ
0
2 (
exp − µ2
) [i/z
s 2 sρq1
(µ2 ) + ρ q
2 (µ2 ]
) ,(8)
1
16π 2
where the spectral density functions ρ q
1,2 (µ2 ) describe the glutinous medium effect, and µ is
an effective mass. The KL representation can be deemed as a superposition of free quark
propagators for all mass eigenstates with the weights ρ q
1,2 (µ2 ). We then decompose the above
matrix element into the perturbative and non-perturbative pieces
⟨Ω|T[q(z)q(0)]|Ω⟩ ≡ iZSF (z, mq) + ⟨Ω| : q(z)q(0) : |Ω⟩, (9)
respectively, with Z being a renormalization constant, SF (z, mq) being the quark propagator
in perturbation theory, mq
contribution from large µ
being the quark mass. The non-perturbative piece collects the
2 ,
⟨Ω| : q(z)q(0) : |Ω⟩ = 1
16π2 ∫ ∞
ds exp
0
(
z2 4 s
) ∫ ∞
µ 2 dµ
c
2 (
exp − µ2
) [i/z
s 2 sρq 1 (µ2 ) + ρ q
2 (µ2 ]
) . (10)
a The Borel transformation can be defined in different ways 2 . In this work we adopt the integral representation.