Comment on "Finding the $0^{--}$ Glueball" [arXiv:1408.3995]

TL;DR

This paper provides a critical commentary on a previous publication about the potential existence of a specific exotic glueball state, discussing its implications and the validity of the original claims.

Contribution

It offers an analysis and critique of the methods and conclusions presented in the original paper on the $0^{--}$ glueball candidate.

Findings

Highlights potential issues in the original analysis

Questions the interpretation of experimental data

Suggests alternative theoretical perspectives

Abstract

A Comment on the Letter by C. F. Qiao and L. Tang, Phys. Rev. Lett. 113, 221601 (2014) [arXiv:1408.3995]

Equations

$$\frac{1}{\pi }{\int }_0^{s_0}\text{Im}{\Pi }^{\text{(QCD)}}(s)e^{-s\tau }ds$$

$${\Pi }^{\text{QCD,LO}}=\frac{487}{2^6{3}^{3}11\cdot 13\pi }{\alpha }_S^3{Q}^{12}\ln (Q^2/{\mu }^2),$$

$$\frac{1}{\pi }\text{Im}{\Pi }^{\text{QCD,LO}}(s)=-\frac{487}{2^6{3}^{3}11\cdot 13\pi }{\alpha }_S^3{s}^6,$$

$$j_{0^{--}}^A(x)=g_s^3{d}^{abc}[g_{\alpha \beta }^t{\tilde{G}}_{\mu \nu }^a(x)][{\partial }_{\alpha }{\partial }_{\beta }{G}_{\nu \rho }^b(x)][G_{\rho \mu }^c(x)],$$

$$g_{\alpha \beta }^t\equiv g_{\alpha \beta }-{\partial }_{\alpha }{\partial }_{\beta }/{\partial }^2.$$

$$\Pi (Q^2)=i\int d^{4}x{e}^{iqx}⟨Tj(x)j^{†}(0)⟩,$$

Full text

Comment on “Finding the $0^{--}$ Glueball”

Alexandr Pimikov

pimikov@mail.ru

Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

Dubna, Moscow Region, 141980 Russia

  

Hee-Jung Lee

Department of Physics Education, Chungbuk National University, Cheongju, Chungbuk 361-763, Korea

  

Nikolai Kochelev

kochelev@theor.jinr.ru

Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

Dubna, Moscow Region, 141980 Russia

Glueball, oddball, QCD sum rules, condensates

††preprint: …

Comment on “Finding the $0^{--}$ Glueball”

In the Letter Qiao:2014vva the authors explore the mass of the exotic three-gluon $0^{--}$ glueball within the QCD sum rules (SR) approach Shifman:1978bx . Even though the Letter presents a relevant discussion on the advantages of this state for experimental observation, the value obtained for the glueball mass is wrong, since the SRs used are inconsistent. Indeed, combining Eq.(11) through Eq.(15) of Qiao:2014vva one gets the master SR for the imaginary part of the correlator of the glueball interpolating currents in the following form:

[TABLE]

where $M_{0}$ is glueball mass, $f_{G}$ is the decay constant, and $\tau$ is so-called Borel parameter. The major contribution to the correlator $\Pi^{\text{QCD}}$ of the A,B,C,D currents, presented in Eq.(1)-Eq.(4) of  Qiao:2014vva , comes from the Leading Order (LO) part of Operator Product Expansion for the glueball correlator (see Eq(6) and Eq.(7) in Qiao:2014vva ). They obtained the LO part of the correlator as

[TABLE]

where $\alpha_{S}$ is the coupling constant, and $\mu$ is the momentum scale. It is easy to check that the theoretical (left) side in Eq. (1) is negative because

[TABLE]

while the phenomenological (right) part is positive.

Therefore, the SR obtained in Qiao:2014vva is inconsistent. We found that the negative sign of the theoretical part of SR calculated in Qiao:2014vva is related to the specific structure of the interpolating currents used. For instance, the A current has the following form

[TABLE]

where

[TABLE]

This specific current induces additional nonphysical poles in the correlator coming from the second term of Eq. (5), which finally leads to the negative imaginary part of the correlator, Eq. (3). Note that the correlator does not depend on the phase of the current by the definition of the correlators Narison:2002pw :

[TABLE]

where Hermitian conjugation $\dagger$ is often omitted considering Hermitian currents. The above shortcomings lead to the conclusion that the currents, presented in Qiao:2014vva , do not couple to gluonic bound states and should not be considered in glueball studies.

Recently a new interpolating current for the exotic three-gluon $0^{--}$ glueball state has been proposed. This current leads to consistent SRs and gives a mass for the glueball of 6.3 GeV Pimikov:2016pag .

We would like to thank V. Vento for stimulating discussions and useful remarks. This work has been supported by Chinese Academy of Sciences President’s International Fellowship Initiative (Grant No. 2013T2J0011 and 2016PM053). The work by H.J.L. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education under Grants No. 2013R1A1A2009695. This work was also supported by the Heisenberg–Landau Program (Grant 2016), the Russian Foundation for Basic Research under Grants No. 15-52-04023.

Alexandr Pimikov1,2,∗, Hee-Jung Lee3 and Nikolai Kochelev1,2,†

1Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China

2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

Dubna, Moscow Region, 141980 Russia

3Department of Physics Education, Chungbuk National University, Cheongju, Chungbuk 361-763, Korea

• pimikov@mail.ru

† kochelev@theor.jinr.ru