Heavy quarkonia mass-splittings in QCD: test of the 1/m -expansion and estimates of < α s G 2 > and α s
Abstract
New double ratios of exponential moments of different two-point functions, which are less sensitive to the heavy quark mass and to the continuum effects than the commonly used ratio of moments, are presented for a more refined analysis of the mass-splittings between the different heavy quarkonia states. We show that at the
c
and
b
quark mass scales the
1/m
-expansion does not converge for these quarkonia channels, while a connection of our mass and width formulae, with the potential model ones is done. Using the present value of the QCD coupling
α
s
, we deduce the value:
<
α
s
G
2
>=(7.5±2.5)×
10
−2
GeV
4
of the gluon condensate from
M
ψ
−
M
η
c
and
M
χ
b
−
M
Υ
, which we compare with the ones from different fits of the heavy and light quark channels. We also find that
M
χ
c
(
P
1
1
)
−
M
χ
c
(
P
3
1
)
is gouverned by the radiative corrections and gives
α
s
(1.3 GeV) = 0.64
+0.36
−0.18
for 4 flavours, implying
α
s
(
M
Z
)=0.127±0.011
. Our predictions for the splittings of different heavy quarkonia states are summarized in Table 2, where, in particular, we find
M
Υ
−
M
η
b
≈
63
−29
+51
MeV implying the possible observation of the
η
b
in the
Υ
-radiative decay.