Thermal behaviors of light unflavored tensor mesons in the framework of QCD sum rule

# Thermal behaviors of light unflavored tensor mesons in the framework of QCD sum rule

K. Azizi    A. Türkan    H. Sundu    E. Veli Veliev    E. Yazıcı Department of Physics, Doğuş University, Acıbadem-Kadıköy, 34722 Istanbul, Turkey
Department of Physics, Kocaeli University, 41380 Izmit, Turkey
(Presenter)enis.yazici@kocaeli.edu.tr
###### Abstract

In this study, we investigate the sensitivity of the masses and decay constants of the light and tensor mesons to the temperature using QCD sum rule approach. In our calculations, we take into account the additional operators appearing in operator product expansion at finite temperature. It is obtained that at deconfinement temperature the decay constants and masses decrease with amount of and compared to their vacuum values, respectively. Our results on the masses at zero temperature are consistent with the vacuum sum rules predictions as well as the experimental data.

## 1 Introduction

In recent years, many researchers are focused on heavy-ion collision experiments in order to deeply understand the hadronic dynamics [1, 2, 3, 4, 5, 6]. It is believed that a transition occurs from hadronic matter to Quark Gluon Plasma (QGP) phase around critical temperature . Therefore, thermal QCD calculations on the properties of hadrons can also be useful in understanding the phase diagram and other properties of strong interactions. In spite of considerable theoretical studies on properties of (pseudo)scalar and (axial)vector mesons at finite temperature (for instance see [7, 8, 9, 10, 11]), there are few theoretical studies on the thermal properties of the tensor mesons ( as an example see [12]).

In this article, we use the thermal QCD sum rule approach to investigate the sensitivity of the masses and decay constants of the light and tensor mesons to temperature (for the original work see [13]). In the literature, there are also few studies on the vacuum properties of tensor mesons using different models [14, 15, 16, 17, 18]. The QCD sum rule approach was firstly proposed by Shifman, Vainshtein and Zakharov [19] in vacuum as one of the most applicable tools to hadron physics. This method then was extended to finite temperature by Bochkarev and Shaposhnikov [20]. In this extension, the Wilson expansion and the quark-hadron duality approximation are valid, but the quark and gluon condensates are replaced by their thermal expectation expressions. In thermal QCD sum rule, due to the breaking of the Lorentz invariance some extra operators, which are expressed in terms of 4-vector velocity of the medium and the energy-momentum tensor [21, 22] are appeared in the Wilson expansion. Taking into account these new operators at finite temperature we obtain the sum rule for the light tensor mesons under consideration.

## 2 Thermal QCD sum rule for light mesons

In this section our aim is to obtain sum rules for the masses and the decay constants of the and tensor mesons. For this reason, we start with the following thermal correlation function:

 Πμν,αβ(q,T)=i∫d4xeiq⋅(x−y)⟨T[Jμν(x)¯Jαβ(y)]⟩∣y=0, (1)

where indicates the time ordering operator and is the interpolating current of the tensor mesons. This current for the mesons under consideration is given by

 Jf2μν(x) = i2√2[¯u(x)γμ\lx@stackrel↔Dν(x)u(x)+¯u(x)γν\lx@stackrel↔Dμ(x)u(x) (2) + ¯d(x)γμ\lx@stackrel↔Dν(x)d(x)+¯d(x)γν\lx@stackrel↔Dμ(x)d(x)],

and

 Ja2μν(x) = i2√2[¯u(x)γμ\lx@stackrel↔Dν(x)u(x)+¯u(x)γν\lx@stackrel↔Dμ(x)u(x) (3) − ¯d(x)γμ\lx@stackrel↔Dν(x)d(x)−¯d(x)γν\lx@stackrel↔Dμ(x)d(x)],

where is the four-derivative with respect to the space-time acting on the left and right, simultaneously. We will set after applying derivatives with respect to .

To obtain the thermal QCD sum rule for the tensor mesons we need to calculate the correlation function both in QCD and hadronic representations. To get the QCD side we use the operator product expansion (OPE) to separate the perturbative and non-perturbative contributions. The perturbative part in spectral representation is written as

 Πpert(q,T)=∫dsρ(s)s−q2, (4)

where is the spectral density and it is given by the imaginary part of the correlation function, . We use the perturbative parts of the light quark propagator to get the spectral density (see [13] for more details). It is obtained as

 ρf2(a2)(s)=(m2u+m2d)s32π2+3s2160π2. (5)

From a similar manner, using the non-perturbative parts of the quark propagator we obtain the non-perturbative contributions as:

 Πnon−pertf2(a2)=mdm20144q2⟨¯dd⟩+mum20144q2⟨¯uu⟩−2⟨uΘfu⟩(q⋅u)29q2, (6)

where is the fermionic part of the energy momentum tensor and is the four-velocity of the heat bath. According to the general idea of the QCD sum rules, after calculation of also the hadronic side of the correlator we match both the hadronic and OPE representations of the correlation function. As a result, we obtain the following sum rule:

 f2f2(a2)(T)m6f2(a2)(T)e−mf2(a2)(T)/M2=∫s0(T)(mu+md)2dsρf2(a2)(s)e−s/M2+^BΠnon−pertf2(a2), (7)

where denotes the Borel transformation with respect to , is the Borel mass parameter and is the temperature-dependent continuum threshold. It is given as

 (8)

where on the right hand side is the hadronic threshold at zero temperature.

## 3 Numerical Results

To obtain the values of the masses and decay constants we need to determine the working regions of two auxiliary parameters: the Borel mass parameter and the hadronic threshold at zero temperature. We choose the intervals and for the continuum thresholds in and channels, respectively. Also the working region of the Borel mass is taken as . In these intervals, the dependence of the results on the auxiliary parameters are relatively weak.

Using the working regions of the auxiliary parameters and other input values, we obtain that the masses and decay constants are well described by the following fit functions in terms of temperature:

 mf2(a2)(T)==AeαT+B, (9)

and

 ff2(a2)(T)=CeβT+D, (10)

where the temperature is in units of GeV. The parameters A, B, C, D, and are given in Table 1.

We obtain that the values of the masses and decay constants are stable until temperature , but after this point they start to decrease with altering the temperature (see [13] for more details). Also at deconfinement temperature, the decay constants and masses decrease with amount of and compared to their vacuum values, respectively.

The obtained results of masses and decay constants at zero temperature are , , and , which are compatible with the vacuum predictions [14, 15, 18] as well as the existing experimental data [23]. Our predictions on the temperature behaviors of decay constants and masses can be verified in the future experiments.

This work has been supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under the research projects 110T284 and 114F018.

## References

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