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

The lower bound for the integration variable µ 2 is usually set to the multi-particle threshold m 2 γ
in the KL representation. In this work we have modified it into 4
µ 2 ⎧
⎪⎨
c =
⎪⎩
cs, s > m2 m
γ
2 γ, s ≤ m2 γ
, (11)
where the free parameter c of order unity will be fixed later. This modification respects the
multi-particle threshold, and at the same time guarantees a finite integral in Eq. (10).
We obtain the dressed propagator
S q (p) =
with the definitions
/p + mq
p 2 − m 2 q
− 1
2 i(γα /pγ βGαβ − mqγαGαβγβ) (p2 − m2 q) 2
− παs⟨G 2 αβ ⟩mq/p(mq + /p)
(p 2 − m 2 q) 4
Î q
1,2f(µ) ≡
∫ ∞
m 2 γ
[
+ /p Îq 1
] exp[c(p2 − µ 2 )/µ 2 ]
+ Îq 2
p2 − µ 2 , (12)
dµ 2 ρ q
1,2 (µ2 )f(µ). (13)
The second and third terms on the right hand side of Eq. (12) arise from the background gluon
field 5,6 , and the forth term comes from the nonlocal quark condensates with the integrations
over µ 2 and s being exchanged in Eq. (10). We define the distribution functions
fs(s) =
fv(s) =
−3
4π2 ⟨qq⟩
3
2π2 ⟨qq⟩
∫ ∞
µ 2 c
∫ ∞
and parameterize the spectral density functions as
µ 2 c
dµ 2 e −µ2 /s ρ q
2 (µ2 ), (14)
dµ 2 e −µ2 /s sρ q
1 (µ2 ), (15)
ρ q
1 (µ2 ) = N1 exp(−aµ 2 )/µ, ρ q
2 (µ2 ) = N2 exp(−aµ 2 ). (16)
The choice of µ 2 c in Eq. (11) then renders the integral in Eq. (14)
fs(s) ∝
s
1 + as exp(−µ2 c/s − aµ 2 c), (17)
exhibit the limiting behaviors exp(−m 2 γ/s) at small s and exp(−acs) at large s, consistent with
exp(−m 2 γ/s) and the exponential ansatz exp(−σqs) postulated in the literature 7,8 , respectively.
The threshold mass mγ is expected to take a value of order of the constituent quark mass 9 , and
set to mγ ∼ 0.36 GeV below.
Comparing the Taylor expansion of the nonlocal quark condensates 2,8 with Eq. (10), we
have the constraints
∫ ∞
0
fs(s)ds = 1,
∫ ∞
0
sfs(s)ds = 1
2 (λ2 q − m 2 q),
∫ ∞
0
fv(s)ds = mq, (18)
which determine the free parameters a, N1 and N2 in Eq. (16), given values of λq and mq. The
average virtuality λq and the lower bound c in Eq. (11), being not known with certainty, are
fixed by fits to the data of the pion decay constant fπ = 0.1307. In Fig. 1(b) we display the
allowed values of c and λq as a curve in the c-λq plane. We adopt λq = 0.75 GeV and c = 0.3,
and choose the light quark masses mu = 4.2 MeV and md = 7.5 MeV. We then solve for the free
parameters a, N1 and N2 from the constraints in Eq. (18), whose results are listed in Table 1.
The opposite signs of N1 and N2 imply the violation of positivity, which can be interpreted as a
manifestation of confinement 10 . The fixed parameters are then employed to calculate the pion
form factor, and the results are shown in Fig. 1(c). It is obvious that our results are consistent
with the experimental data for Q 2 > 1GeV 2 , the region where QSR are applicable.