# Spectrum of the heavy axial vector and mesons in thermal QCD

###### Abstract

Using the additional operators coming up at finite temperature, we calculate the masses and decay constants of the wave heavy axial-vector and quarkonia in the framework of thermal QCD sum rules. In the calculations, we take into account the perturbative two loop order corrections and nonperturbative effects up to the dimension four condensates. It is observed that the masses and decay constants almost remain unchanged with respect to the variation of the temperature up to , however after this point, the decay constants decrease sharply and approach approximately to zero at critical temperature. The decreasing in values of the masses is also considerable after .

###### pacs:

11.55.Hx, 14.40.Pq, 11.10.Wx## I Introduction

Investigation of the in medium properties of heavy mesons such as bottomonium () and charmonium () are of considerable interest for hadron physics to date. These quarkonia play an important role in obtaining information on the restoration of the spontaneously broken chiral symmetry in a nuclear medium and understanding quark gluon plasma (QGP) as a new phase of hadronic matter. Such investigations can also provide us with substantial knowledge on the nonperturbative QCD and interaction of quarks and gluons with QCD vacuum.

In the last twenty years, a lot of theoretical and experimental works have been devoted to study the behavior of the heavy mesons in medium. The suppression which can be considered as an indication of QGP Matsui () has been observed in heavy ion collisions in super proton synchrotron (SPS) at CERN and relativistic heavy ion collider (RHIC) at BNL. A plenty of theoretical works have also been dedicated to study the thermal behavior of hadronic parameters as well as QCD degrees of freedom (for some of them and discussion on the QGP phase see for instance K.Morita (); O.Kaczmarek (); Loewe (); Cheng (); Miller (); Nora (); Nora1 (); Aoki (); Aoki1 (); Borsanyi (); Borsanyi1 (); Aoki2 (); S.Mallik Mukherjee (); S.Mallik sarkar (); E.V.Veliev (); C.A.Dominguez (); C.A.Dominguez2 (); E.V.Veliev2 (); F. Klingl (); K.Morita2 (); E.V.Veliev3 (); E.V.Veliev4 (); E.V.Veliev5 (); Azizi ()).

Hadrons are formed in a region of energy very far from the perturbative region, hence to calculate their parameters we need to have some nonperturbative approaches. The QCD sum rules as one of the most attractive, applicable and powerful techniques has been in the focus of much attention during last 32 years. This approach at zero temperature proposed in Shifman () and have applied to many decay channels in this period giving results in a good consistency with the existing experimental data as well as lattice QCD calculations. This method then was extended to finite temperature QCD in Bochkarev (). There are two new aspects in this extension compared to the case at zero temperature E.V.Shuryak (); T.Hatsuda (); S.Mallik (), namely interaction of the particles in the medium with the currents requiring modification of the hadronic spectral density as well as the breakdown of the Lorentz invariance via the choice of reference frames. Because of residual symmetry at finite temperature, more operators with the same dimensions come out in the operator product expansion (OPE) compared to that of vacuum.

The purpose of this paper is to calculate the masses and decay constants of the wave heavy axial vector and mesons in the framework of the thermal QCD sum rules. In our calculations, we use thermal propagator containing new non-perturbative contributions appearing at finite temperature, and take into account the perturbative two-loop order corrections to the correlation function Shifman (); L.J.Reinders (). We use the expressions of the temperature-dependent energy-momentum tensor obtained via Chiral perturbation theory P.Gerber () and lattice QCD Cheng (); Miller () as well as temperature-dependent gluon condensates and continuum threshold to obtain the behavior of the masses and decay constants of these mesons in terms of temperature.

## Ii Thermal QCD Sum Rule for Wave Heavy Axial Vector quarkonia

In order to extract the sum rules for the masses and decay constants of the heavy axial vector and mesons at finite temperature, we start considering the following two-point thermal correlation function:

(1) |

where, with or is the interpolating current of heavy axial vector meson, is temperature and indicates the time ordering product. The thermal average of any operator is defined as

(2) |

where is the QCD Hamiltonian and .

According to the general philosophy of the QCD sum rules formalism, the above correlation function can be calculated in two different ways. Once, in terms of QCD degrees of freedom by the help of OPE called the theoretical or QCD side. The OPE incorporates the effects of the QCD vacuum through an infinite series of condensates of increasing mass dimensions. The second, in terms of hadronic parameters called the physical or phenomenological side. Matching then these two representations, we find sum rules for the physical observables under consideration. To suppress the contribution of the higher states and continuum, we apply Borel transformation as well as continuum subtractions. In the following, we calculate the correlation function in two aforesaid windows.

### ii.1 The phenomenological side

Technically, to obtain the physical or phenomenological side of the correlation function, we insert a complete set of intermediate hadronic states with the same quantum numbers as the interpolating current into the correlation function. After performing the four-integral over and isolating the ground state contribution, we get

(3) |

where the and are decay constant and mass of the heavy axial vector meson, respectively. The in the above equation stands for the contribution of the excited heavy axial vector states and continuum. In deriving the Eq. (3), we have defined the decay constant by the matrix element of the current between the vacuum and the mesonic state in the following manner:

(4) |

where is the four-polarization vector. We have also used the summation over polarization vectors as

(5) |

### ii.2 The QCD side

In QCD side, the correlation function is calculated in deep Euclidean region where via OPE where the short or perturbative and long distance or non-perturbative effects are separated, i.e.,

(6) |

The short distance contributions are calculated using the perturbation theory, while the long distance contributions are expressed in terms of the thermal expectation values of the quark and gluon condensates as well as thermal average of the energy density coming up at finite temperature.

In the rest frame of the medium for axial vector meson at rest, the correlation function in QCD side can be written in terms of the transverse and longitudinal components as

(7) |

where the functions, and are found in terms of the total correlation function as

(8) |

Here, we would like to mention that the transverse and longitudinal components are related to each other, hence it is enough to use one of them to obtain the thermal sum rules for the physical quantities under consideration. Here, we use the function for this aim. It can be shown that this function for the fixed values of the , can be written as S.Mallik Mukherjee ():

(9) |

where , and

(10) |

is the spectral density. We also should stress that the function receives contributions from both annihilation and scattering parts (for more information see E.V.Veliev5 ()). However, as we deal with the mesons containing quark and antiquark with the same masses, the scattering part gives zero and here we focus our attention to calculate only the annihilation part.

The thermal correlation function in QCD side is obtained from Eq. (1) contracting out all quark fields via Wick’s theorem. As a result, we obtain the following expression in terms of thermal heavy quarks propagators:

(11) |

In real time thermal field theory, the function can be expressed in matrix representation, the elements of which depend on only one analytic function. Hence calculation of the 11-component of such matrix is enough to get information on the dynamics of the corresponding two-point correlation function. The 11-component of the thermal quark propagator which is given as a sum of its vacuum expression and a term depending on the temperature is given as A.Das ():

(12) |

where is the Fermi distribution function and is the quark mass. Performing the integral over in the limit, we get the imaginary part of the as:

(13) |

where . After standard calculations, we get the following expression for the annihilation part of the spectral density:

(14) |

where, . As we previously mentioned, we take into account also the perturbative two-loop order correction to the spectral density. At zero temperature, it is given as Shifman (); L.J.Reinders ():

(15) | |||||

where we have set and the functions , , and are given as:

(16) |

To get the thermal version of the above two-loop order correction, we replace the strong coupling by its temperature dependent lattice improved version given in E.V.Veliev5 () ( for more details see also K.Morita (); O.Kaczmarek ()).

Our final task in this section is to calculate the nonperturbative part of the thermal correlation function. The nonperturbative part in our case can be written in terms of operators up to dimension four as:

(17) |

where, are as Wilson coefficients. As we also previously mentioned, at finite temperature the Lorentz invariance is broken by the choice of reference frame and new operators appear in the Wilson expansion above. The new four-dimension operator here is , where is the energy momentum tensor and is the four-velocity of the heat bath and it is introduced to restore Lorentz invariance formally in the thermal field theory. In the rest frame of the heat bath, we have which leads to . Note that in our calculations, we ignore the heavy quark condensate since it suppress by inverse powers of the heavy quark mass.

To proceed in calculation of the nonperturbative part, we use the nonperturbative part of the quark propagator in an external gluon field, in the Fock-Schwinger gauge, . In this gauge, the vacuum gluon field is written in terms of gluon field strength tensor in momentum space as follows:

(18) |

where is the gluon momentum.

Taking into account one and two gluon lines attached to the quark line as shown in Fig. 1, up to terms required for our calculations, the non-perturbative part of the temperature-dependent massive quark propagator is obtained as:

(19) | |||||

where is the traceless gluonic part of the energy-momentum tensor of the QCD.

Using the above expression and after straightforward but lengthy calculations, we get the following expression for the nonperturbative part:

(20) | |||||

where and .

### ii.3 Thermal Sum Rules for Physical Quantities

Now it is time to equate two different representations of the correlation function from physical and QCD sides and perform continuum subtraction to suppress the contribution of the higher states and continuum. As a result of this procedure we get the following sum rule including the temperature-dependent mass and decay constant:

(21) |

where is temperature-dependent continuum threshold and for simplicity, the temperature-dependent width of meson has been neglected. To further suppress the higher states and continuum contributions, we also apply the Borel transformation with respect to to both sides of the above sum rule. As a result we get,

(22) |

where the nonperturbative part in Borel scheme is obtained as:

(23) | |||||

Here is the Borel mass parameter. Considering Eq. (22), the mass squared of the heavy axial vector meson alone can be obtained as:

(24) |

where,

(25) |

and

(26) |

### ii.4 Numerical Results

To numerically analyze the sum rules for mass and decay constant, we use the following temperature-dependent continuum threshold C.A.Dominguez2 ():

(27) |

where with being critical temperature and is the continuum threshold at zero temperature. For the temperature-dependent gluon condensate we also use Cheng (); Miller ()

(28) |

For the thermal average of total energy density we use both results: i) obtained in lattice QCD Cheng (); Miller ():

(29) |

where this parametrization is valid only in the region . ii) obtained via chiral perturbation theory P.Gerber ():

(30) |

where is trace of the total energy momentum tensor and is pressure. They are given as:

where , and .

We also use the values , and for quarks masses and gluon condensate at zero temperature. Finally, we should find the working region for the continuum threshold at zero temperature () and Borel mass parameter () such that the physical observables are weakly depend on these parameters according to the standard criteria of the QCD sum rules. The continuum threshold, is not totally arbitrary and it is correlated to the energy of the first exited state of the heavy axial vector meson. Our numerical calculations lead to the intervals and for the and heavy axial mesons, respectively. The working region for the Borel mass parameter is calculated requiring that not only the contributions of the higher states and continuum are efficiently suppressed but also the contributions of the operators with higher dimensions are ignorable. We get the working regions and respectively for the and channels.

Using the above obtained working regions for auxiliary parameters together with the other inputs, we plot the dependence on the Borel parameter of the masses and decay constants of the heavy axial and quarkonia at zero temperature in Figs. (2-5). From these figures, we see that the results weakly depend on the auxiliary parameters in their working regions. The numerical results for the masses and decay constants of the heavy axial vector mesons under consideration are depicted in tables I and II. We also compare the obtained results with the experimental values in the same tables. From table I we see a good consistency of our results with the experimental data. The errors in the results of our work belong to the uncertainties in calculation of the working regions for auxiliary parameters as well as those coming from other inputs.

At the end of this section we would like to discuss the behavior of the decay constants and masses of the heavy axial quarkonia under consideration in terms of temperature. We depict the variations of these quantities versus temperature in figures (6-9). From these figures, we see that the masses and decay constants remain unchanged with the variation of temperature up to . After this point they start to decrease increasing the temperature. At deconfinement or critical temperature, the decay constants decrease about (73-78)%, while the masses are decreased about 4%, and 19% for and states, respectively. The sharp decreasing in the values of the decay constants near the deconfinement temperature can be considered as a signal for existing the QGP as the new phase of hadronic matter.

## Iii Acknowledgement

The authors are grateful to T. M. Aliev for useful discussions. This work has been supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under the research project No. 110T284 and research fund of Kocaeli University under grant No. 2011/029.

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