Reply to Comment on "Chiral suppression of scalar glueball decay"
arXiv:0704.1616v1  [hep-ph]  12 Apr 2007
Published in Phys. Rev. Lett. 98, 149104 (2007)
Reply to “Comment on ‘Chiral suppression of scalar glueball decay’ ”
Michael Chanowitz1
Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
In [1] I observed that the amplitude for spin zero glue-
ball decay is proportional to the quark mass, M(G0 →
qq) ∝ mq, to all orders in perturbation theory, so that
the ratio Γ(G0 → uu+ dd)/Γ(G0 → ss) is calculable and
small, even though the individual rates are not pertur-
batively calculable because of soft t and u channel quark
exchanges. I noted that if hadronization of G0 → qq is an
important mechanism for G0 → ππ and G0 → KK, then
Γ(G0 → ππ) is much smaller than Γ(G0 → KK), ex-
plaining a previous LQCD result[2] and supporting iden-
tification of f0(1710) with G0. A more robust conse-
quence, emphasized in [3], is that mixing of G0 with
uu + dd (and perhaps also ss) mesons is suppressed, so
that the scalar (and pseudoscalar) may be the purest
glueballs. In both [1] and [3] I emphasized the neces-
sity to verify the existence and consequences of chiral
suppression by a reliable nonperturbative method, which
today can only be LQCD.
Chao et al. agree that G0 → qq is chirally suppressed
but propose that G0 → qqqq, which is not chirally sup-
pressed, is the dominant mechanism for G0 → ππ. In
the preceding Comment[4] and in a previous paper[5]
they exhibit an O(αS) amplitude for the exclusive process
G0 → ππ using light cone wave functions. Since pQCD
for exclusive processes converges much more slowly than
inclusive pQCD[6], the estimate is not quantitatively re-
liable at the experimentally interesting scale, mG = 1.7
GeV, where even the applicability of ordinary inclusive
pQCD is marginal. While the qqqq mechanism might in-
deed dilute or remove chiral suppression of G0 → ππ, it
is not possible to decide, since the magnitude of neither
the qq nor qqqq contributions are reliably calculable.
Comparing the amplitudes for M(G0 → qq) and
M(G0 → qqqq → ππ) in [1] and [4, 5] it appears that
both begin at first order in αS , but this impression is
misleading. It is easy to see that M(G0 → qqqq → ππ)
vanishes in the chiral limit at O(αS) for on-shell con-
stituent gluons. The qqqq mechanism requires the quark
from one gluon to combine with the antiquark from the
other gluon to form a color singlet pion. But G0 cms
(center of mass) kinematics then requires both quarks
to have the same energy fraction, x = 2Eq/mG and both
antiquarks to have fraction 1−x, with m2π = x(1−x)m
One of the q or q constituents of each pion is then mov-
ing in the opposite direction to the pion in the G0 cms.
Boosting to an infinite momentum frame, one constituent
is then at x = 1 and the other at x = 0, where the wave
function vanishes. In the chiral limit, mπ = 0, this is al-
ready apparent in the G0 cms. Since confining dynamics
may put the gluons off-shell of order ΛQCD, the ampli-
tude does not actually vanish but is suppressed of order
O(ΛQCD/mG).
In the revised Comment the authors have responded
to this observation with the added stipulation that the
G0 constituent gluons are maximally off-shell, of order
mG. Although this requirement was not imposed in [5],
the result is apparently unchanged. Certainly one conse-
quence is that fg, the effective G0gg coupling, cannot be
identified with the corresponding coupling f0 in [1] as is
claimed in [4, 5], but reflects the off-shell tail of the G0
wave function or implicitly contains a factor αS at the
hard scale mG reflecting hard gg → g
∗g∗ scattering to
push the gluons maximally off-shell. Alternatively, hard
scattering of qqqq can align the quarks suitably with the
final state pions, with the amplitude then explicitly of
order O(α2S).
The relative magnitude of the qq and qqqq mechanisms
for G0 → ππ is not obvious. For the qq mechanism we do
not know the magnitude of M(G0 → qq) because both
αS(Q) and the running mass mq(Q) are evaluated at a
soft scale, O(ΛQCD), and thus are not under perturbative
control. In addition we do not know the hadronization
rate from qq to ππ and KK compared to multi-meson fi-
nal states. On the other hand, Γ(G0 → ππ) via the qqqq
mechanism cannot be reliably estimated and is addition-
ally suppressed by the square of the coupling, αS(Q)
evaluated at the largest scale in the problem, Q = mG.
It is then important to stress the agreement, expressed
in both [1, 3] and [4], on the most important point: re-
liable nonperturbative methods are needed to determine
whether G0 → ππ is chirally suppressed. We eagerly
await LQCD “data” and data from BES II to clarify the
issue.
Acknowledgments: This work was supported by the
Director, Office of Science, Office of High Energy Physics
of the U.S. Department of Energy under contract DE-
AC02-05CH11231.
[1] M. Chanowitz, Phys. Rev. Lett. 95 (2005), hep-
ph/0506125
[2] J.Sexton, A.Vaccarino, D.Weingarten, Phys.Rev.Lett.75:
4563,1995, hep-lat/9510022.
[3] M. Chanowitz, Talk given at Charm 2006, Beijing, China,
5-7 Jun 2006. Published in Int.J.Mod.Phys.A21, 5535
(2006), hep-ph/0609217.
http://arxiv.org/abs/0704.1616v1
[4] K.T. Chao, X-G He, J.P. Ma, preceding Comment.
[5] K.T. Chao, X-G He, J.P. Ma, hep-ph/0512327.
[6] N. Isgur and C.H. Llewellyn-Smith, Phys. Rev. Lett. 52:
1080 (1984).
