348 S. Narison / Physics Letters B 738 (2014) 346–360
2.3. Tachyonic gluon mass and estimate of larger order PT-terms
The tachyonic gluon mass λ of dimension two has been intro-
duced
in [42,43] and appears naturally in most holographic QCD
models [44]. Its contribution is “dual” to the uncalculated higher
order terms of the PT series [45] and disappears for long PT se-
ries
like in the case of lattice calculations [46], but should remain
when only few terms of the PT series are calculated like in the
case studied here. Its contribution reads [43]:
L
P
5
(τ )
tach
=−
3
2π
2
(m
u
+m
q
)
2
a
s
λ
2
τ
−1
. (11)
Its value has been estimated from e
+
e
−
[28,47] and τ -decay [29]
data:
a
s
λ
2
=−(0.07 ±0.03) GeV
2
. (12)
2.4. The instanton contribution
The inclusion of this contribution into the operator product ex-
pansion
(OPE) is not clear and controversial [19–21]. In addition,
an analogous contribution might lead to some contradiction to the
OPE in the scalar channel [22]. Therefore, we shall consider the
sum rule including the instanton contribution as an alternative ap-
proach.
For our purpose, we parametrize this contribution as in
[19,20], where its corresponding contribution to the Laplace sum
rule reads:
L
P
5
(τ )
inst
=
3
8π
2
(m
u
+m
q
)
2
τ
−3
ρ
2
c
e
−r
c
K
0
(r
c
) + K
1
(r
c
)
, (13)
where K
i
is the Bessel–Mac-Donald function; r
c
≡ ρ
2
c
/(2τ ) and
ρ
c
=(1.89 ±0.11) GeV
−1
[48] is the instanton radius.
2.5. Duality violation
Some eventual additional contribution from duality violation
(DV) [49] could also be considered. However, as the LSR use the
OPE in the Euclidean region where the DV effect is exponen-
tially
suppressed, one may safely neglect such contribution in the
present analysis.
7
2.6. The QCD input parameters
There are several estimates of the QCD input parameters in the
current literature using different approaches and sometimes dis-
agree
each others. For a self-consistency, we shall work in this
paper with the input parameters given in Table 1 obtained using
the same approach (Laplace or/and τ -decay-like sum rule) as the
one used here and within the same criterion of stability (mini-
mum,
maximum, inflexion point or plateau in τ and t
c
).
–
ˆ
m
q
and
ˆ
μ
q
are RGI invariant mass and condensates which are
related to the corresponding running parameters as [2]:
m
q
(τ ) =
ˆ
m
q
(−β
1
a
s
)
−2/β
1
(1 +ρ
m
)
¯
qq(τ ) =−
ˆ
μ
3
q
(−β
1
a
s
)
2/β
1
/(1 +ρ
m
)
¯
qGq(τ) =−M
2
0
ˆ
μ
3
q
(−β
1
a
s
)
1/3β
1
/(1 +ρ
m
), (14)
where β
1
=−(1/2)(11 − 2n
f
/3) is the first coefficient of the QCD
β-function for n
f
-flavors. ρ
m
is the QCD correction which reads to
7
We thank the 2nd referee for this suggestion and for different provocative com-
ments
leading to the improvements of the final manuscript.
Table 1
Input
parameters: the value of
ˆ
μ
q
has been obtained from the running masses eval-
uated
at 2GeV: (m
u
+m
d
) =7.9(6) MeV [6,50]. Some other predictions and related
references can be found in [51]; ρ denotes the deviation on the estimate of the
four-quark condensate from vacuum saturation. The error on Γ
K
is a guessed con-
servative
estimate.
Parameters Values Ref.
Λ(n
f
=3)(353 ± 15) MeV [30,52]
ˆ
m
s
(0.114 ± 0.021) GeV [6,30,50,53]
ˆ
μ
d
(253 ± 6) MeV [50,53]
κ ≡
¯
ss/
¯
dd (0.74
+0.34
−0.12
) [6,54]
−a
s
λ
2
(7 ± 3) ×10
−2
GeV
2
[29,47]
α
s
G
2
(7.0 ±2.6) ×10
−2
GeV
4
[48]
M
2
0
(0.8 ± 0.2) GeV
2
[38–40]
ρα
s
¯
qq
2
(5.8 ± 1.8) ×10
−4
GeV
6
[27,38,41]
ρ
c
(1.89 ± 0.11) GeV
−1
[48]
Γ
π
(0.4 ± 0.2) GeV [51]
Γ
K
(0.25 ± 0.05) GeV [51]
N4LO accuracy for n
f
=3 [6,55]:
ρ
m
=0.8951a
s
+1.3715a
2
s
+0.1478a
3
s
, (15)
where a
s
≡ α
s
/π is the QCD running coupling. The value of
ˆ
m
s
quoted in Table 1 will serve as an initial value for the m
s
cor-
rections
in the PT expression of the kaon correlator. It will be
re-extracted by iteration in the estimate of m
s
from the kaon sum
rule where one obtains a convergence of the obvious iteration pro-
cedure
after two iterations.
– The value of the μ
q
RGI condensate used in Table 1 comes
from the value (m
u
+ m
d
) = (7.9 ± 0.6) MeV evaluated at 2GeV
from [50] after the use of the GMOR relation:
2m
2
π
f
2
π
=−(m
u
+m
d
)
¯
uu +
¯
dd, (16)
where f
π
=(92.23 ±0.14) MeV [63].
–
The value of the gluon condensate used here comes from
recent charmonium sum rules. Since SVZ, several determinations
of the gluon condensates exist in the literature [20,28,29,32,41,47,
56–60].
The quoted error is about 2 times the original error for
making this value compatible with the SVZ original value and char-
monium
analysis in [20] commented in [48].
8
– We use the value of the four-quark condensate obtained from
e
+
e
−
and VV +AA τ -decay [27–29,41] data and from light baryons
sum rules [38] where a deviation from the vacuum saturation by
a factor ρ (4.2 ± 1.3) has been obtained if one evaluates
¯
dd
from μ
d
given in Table 1 at M
τ
where the four-quark condensate
has been extracted (for a conservative result, we have multiplied
the original error in [27] by a factor 2). Similar conclusions have
been derived from FESR [56] and more recently from the VV–AA
component of τ -decay data [60,61]. We assume that a similar de-
viation
holds in the pseudoscalar channels. We shall see again later
on that the error induced by this contribution on our estimate is
relatively negligible.
–
We use the value of the SU(3) breaking parameter κ ≡
¯
ss/
¯
dd from [54] which agrees with the ones obtained from light
baryons [38] and from kaon and scalar [7,10,12] sum rules recently
reviewed in [6] but more accurate. For a conservative estimate we
have enlarged the original error by a generous factor 4 and the
upper value for recovering the central value κ =1.08 from recent
lattice calculations [62].
8
The sets of FESR in [56] tend give large values of the condensates which are in
conflict with the ones from LSR and τ -like sum rules using similar e
+
e
−
data [27,
28,41,47] and
previous charmonium analysis [3,20,32,48]. They will not be consid-
ered
here. However, as shall see explicitly later on, the effects of α
s
G
2
and of the
tachyonic gluon mass used in this paper are relatively small in the present analysis.
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