Comment on "Chiral Suppression of Scalar Glueball Decay"
Comment on “Chiral Suppression of Scalar Glueball Decay”
Kuang-Ta Chao1, Xiao-Gang He2, and Jian-Ping Ma3,1
Department of Physics, Peking University, Beijing
Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei
Institute Of Theoretical Physics, Academia Sinica, Beijing
PACS numbers: PACS numbers: 12.39.Mk, 12.38.Bx
In a recent letter, based on an effective Lagrangian,
Chanowitz[1] showed that in the limit that the mass mq
of a light quark q goes to zero, the decay amplitude for
a scalar glueball Gs decaying into qq̄ goes to zero, and
conjectured further that this chiral suppression also oc-
curs at the hadron level for Gs decays into ππ,KK with
the ratio of these two branching ratios to be of the or-
der O(m2u,d/m
s) for finite quark masses. Here we show
that the decay Gs → qq̄ is forbidden in the chiral limit
in QCD without assumptions. More essentially, we show
that this chiral suppression may be spoiled and may not
materialize itself at the hadron level.
A glueball here is assumed to be a pure gluonic state.
It decays into a qq̄ pair through a multi-gluon annihila-
tion process. The decay amplitude for Gs → q(p1)q̄(p2)
can be written as a product of a spinor pair ū(p1) and
v(p2) with a product of any number of γ matrices sand-
wiched between the spinors. Because vector-like coupling
in QCD, for mq = 0 the number of the γ-matrices is
an odd number which can always be reduced to one γ-
matrix. Therefore the amplitude can be written as:
Tqq̄ = ū(p1)γµA
µv(p2).
Lorentz covariance of the amplitude then dictates
Aµ(p1, p2) to be of the form a1p
+ a2p
. Therefore in
the chiral limit mq = 0, Tqq̄ = 0. The result also applies
to a pseudoscalar glueball decays into a qq̄ pair.
To study whether there is a chiral suppression in
Gs → ππ,KK or not, we work with an effective La-
grangian, Ls = fgG
a,µνGaµνGs, as in [1], and employ
QCD factorization[2] to calculate the amplitude Tππ for
Gs → π
+π−. To the leading twist-2 order, there are two
diagrams with the two gluons splitting into two quarks
and two anti-quarks, and then form two pions. The two
gluons are off-shell by the scale at order of MGs . A direct
calculation gives:
Tππ = −αsfg
du1du2φπ+(u1)φπ−(u2)
(1 − u1)(1 − u2)
[1 +O(αs, λ/MGs)] ,
where φπ is normalized as
duφπ(u) = 1. ui(i = 1, 2) is
the momentum fraction carried by the anti-quark in the
meson. In the above, λ can be any soft scale, such as
quark mass, ΛQCD and mπ. Clearly, Tππ is not zero in
the chiral limit mq = 0.
The amplitude for Gs → K
+K− decay can be ob-
tained by replacing quantities related to π by those re-
lated to K correspondingly. We would obtain, R =
B(Gs → ππ)/B(Gs → KK) ≈ f
K = 0.48, which is
substantially different from 1. This suppression is much
milder compared with the one at the quark level. This is
due to the fact that in perturbative QCD (pQCD) calcu-
lation the decay of Gs → ππ,KK is related to the cou-
pling of Gs to two pairs of qq̄ compared with conjectured
by Chanowitz in [1], where it is assumed that Gs just cou-
ples to one qq̄ pair. We should point out that whether
the chiral suppression at quark level can be realized still
waits for better non-perturbative calculation for the di-
rect two quark hadronization into ππ and KK. If the
pQCD contribution dominates, the result of R ≈ f4π/f
can be obtained without the assumption of the effective
Lagrangian. Because glueball is a pure gluon state, the
amplitude of the decay Gs → π
+π− can always be writ-
ten with QCD factorization as Tππ = f
πHg ⊗φπ+ ⊗φπ− ,
where the higher-twist effects related to π’s are neglected
andHg consists of some perturbative coefficient functions
and some quantities related to the structure of Gs. Al-
though Hg is unknown, one can easily find the result of
R ≈ f4π/f
The f0(1710) is a candidate for scalar glueball. Early
measurement obtained R ≤ 0.11[3], and a larger one by
BES[4] R = 0.41+0.11
−0.17 recently. It is interesting to notice
that the later is consistent with our result and may fa-
vor that the f0(1710) is a gluebal. However one should
remember that the prediction R ≈ f4π/f
K can have sub-
stantial non-perturbative corrections and there may be
further complication by mixing effects of a glueball with
qq̄ states. A more detailed study can be found in [5].
Acknowledgments: This work was supported in part
by grants from NSC and NNSFC (No 10421503).
[1] M.S. Chanowitz, Phys. Rev. Lett. 95, 172001(2005) .
[2] S.J. Brodsky and G.P. Lepage, Phys. Rev. D24,
2848(1981), G.P. Lepage and S.J. Broadsky, Phys. ReV.
D22, 2157(1980).
[3] W.-.M Yao et al., (Particle Data Group), J. Phys. G33,
1(2006).
http://arxiv.org/abs/0704.1061v1
[4] M. Ablikim et al. (BES Collaboration), Phys. Lett. B642,
441(2006).
[5] K.T. Chao, Xiao-Gang He and J.P. Ma, hep-ph/0512327.
http://arxiv.org/abs/hep-ph/0512327
