Comment on "Chiral Suppression of Scalar Glueball Decay"

TL;DR

This paper provides a critical commentary on the claims made in the original work regarding the suppression of scalar glueball decay due to chiral symmetry effects.

Contribution

It offers an analysis that questions or clarifies the original conclusions about chiral suppression in scalar glueball decay.

Findings

Reassesses the evidence for chiral suppression

Highlights potential theoretical inconsistencies

Suggests alternative interpretations of decay mechanisms

Abstract

Comment on ``Chiral Suppression of Scalar Glueball Decay''

Equations

$${\mathcal{T}}_{q\bar{q}}=\bar{u}(p_1){\gamma }_{\mu }{A}^{\mu }v(p_2).$$

$${\mathcal{T}}_{\pi \pi }=-{\alpha }_s{f}_g\frac{8\pi }{9}{f}_{\pi }^2{\int }_0^{1}d{u}_{1}d{u}_2{\varphi }_{{\pi }^{+}}(u_1){\varphi }_{{\pi }^{-}}(u_2)$$

$$\times \left(\frac{1}{u_1{u}_2}+\frac{1}{(1-u_1)(1-u_2)}\right)\left[1+\mathcal{O}({\alpha }_s,\lambda /M_{G_s})\right],$$

Full text

In a recent letter, based on an effective Lagrangian, ChanowitzChan showed that in the limit that the mass $m_{q}$ of a light quark $q$ goes to zero, the decay amplitude for a scalar glueball $G_{s}$ decaying into $q\bar{q}$ goes to zero, and conjectured further that this chiral suppression also occurs at the hadron level for $G_{s}$ decays into $\pi\pi,KK$ with the ratio of these two branching ratios to be of the order $\mathcal{O}(m^{2}_{u,d}/m^{2}_{s})$ for finite quark masses. Here we show that the decay $G_{s}\to q\bar{q}$ 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 $q\bar{q}$ pair through a multi-gluon annihilation process. The decay amplitude for $G_{s}\to q(p_{1})\bar{q}(p_{2})$ can be written as a product of a spinor pair $\bar{u}(p_{1})$ and $v(p_{2})$ with a product of any number of $\gamma$ matrices sandwiched between the spinors. Because vector-like coupling in QCD, for $m_{q}=0$ the number of the $\gamma$-matrices is an odd number which can always be reduced to one $\gamma$-matrix. Therefore the amplitude can be written as:

[TABLE]

Lorentz covariance of the amplitude then dictates $A^{\mu}(p_{1},p_{2})$ to be of the form $a_{1}p_{1}^{\mu}+a_{2}p_{2}^{\mu}$. Therefore in the chiral limit $m_{q}=0$, ${\mathcal{T}}_{q\bar{q}}=0$. The result also applies to a pseudoscalar glueball decays into a $q\bar{q}$ pair.

To study whether there is a chiral suppression in $G_{s}\to\pi\pi,KK$ or not, we work with an effective Lagrangian, $L_{s}=f_{g}G^{a,\mu\nu}G^{a}_{\ \mu\nu}G_{s}$, as in Chan , and employ QCD factorizationBrLe to calculate the amplitude $\mathcal{T}_{\pi\pi}$ for $G_{s}\to\pi^{+}\pi^{-}$. 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 $M_{G_{s}}$. A direct calculation gives:

[TABLE]

where $\phi_{\pi}$ is normalized as $\int du\phi_{\pi}(u)=1$. $u_{i}(i=1,2)$ is the momentum fraction carried by the anti-quark in the meson. In the above, $\lambda$ can be any soft scale, such as quark mass, $\Lambda_{QCD}$ and $m_{\pi}$. Clearly, ${\mathcal{T}}_{\pi\pi}$ is not zero in the chiral limit $m_{q}=0$.

The amplitude for $G_{s}\to K^{+}K^{-}$ decay can be obtained by replacing quantities related to $\pi$ by those related to $K$ correspondingly. We would obtain, $R={B(G_{s}\to\pi\pi)/B(G_{s}\to KK)}\approx{f_{\pi}^{4}/f_{K}^{4}}=0.48$, 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) calculation the decay of $G_{s}\to\pi\pi,KK$ is related to the coupling of $G_{s}$ to two pairs of $q\bar{q}$ compared with conjectured by Chanowitz in Chan , where it is assumed that $G_{s}$ just couples to one $q\bar{q}$ 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 direct two quark hadronization into $\pi\pi$ and $KK$. If the pQCD contribution dominates, the result of $R\approx{f_{\pi}^{4}/f_{K}^{4}}$ can be obtained without the assumption of the effective Lagrangian. Because glueball is a pure gluon state, the amplitude of the decay $G_{s}\to\pi^{+}\pi^{-}$ can always be written with QCD factorization as $T_{\pi\pi}=f_{\pi}^{2}H_{g}\otimes\phi_{\pi^{+}}\otimes\phi_{\pi^{-}}$, where the higher-twist effects related to $\pi$’s are neglected and $H_{g}$ consists of some perturbative coefficient functions and some quantities related to the structure of $G_{s}$. Although $H_{g}$ is unknown, one can easily find the result of $R\approx{f_{\pi}^{4}/f_{K}^{4}}$.

The $f_{0}(1710)$ is a candidate for scalar glueball. Early measurement obtained $R\leq 0.11$PDG , and a larger one by BESbes $R=0.41^{+0.11}_{-0.17}$ recently. It is interesting to notice that the later is consistent with our result and may favor that the $f_{0}(1710)$ is a gluebal. However one should remember that the prediction $R\approx{f_{\pi}^{4}/f_{K}^{4}}$ can have substantial non-perturbative corrections and there may be further complication by mixing effects of a glueball with $q\bar{q}$ states. A more detailed study can be found in CHM .

Acknowledgments: This work was supported in part by grants from NSC and NNSFC (No 10421503).