Properties of spin-1/2 heavy baryons at nonzero temperature

One of the most attractive subjects in particle physics is to investigate the spectroscopic properties of hadrons at finite temperatures. Such studies provide us with a better understanding of the perturbative and non-perturbative natures of QCD at hot mediums. They will also help us in analyses of data provided by future heavy ion collision experiments aiming to investigate the hadronic properties and possible phase transitions to quark gluon plasma (QGP) at finite temperatures and densities. In the last two decades, there have been significant experimental and theoretical studies on single heavy baryons in vacuum. Roughly, all single heavy baryons containing a heavy $b$ or $c$ quark have been successfully observed PDG . The investigation of these baryons at a medium with nonzero temperature is a very prominent research subject now and it will be in agenda of different experimental and theoretical studies.

Single heavy baryons $(Qq_{1}q_{2})$ are composite particles made of one heavy ($Q=b$ or $c$) and two light quarks ($q_{1,2}=u,d$ or $s$). These particles belong to either antitriplet of flavor antisymmetric state $\overline{\textbf{3}}$ or sextet of flavor symmetric state 6. It is well known that total spin-parity of the ground state single heavy baryons in sextet state is either $J^{P}=\frac{1}{2}^{+}$ or $J^{P}=\frac{3}{2}^{+}$ while spin-parity of the single heavy baryons in antitriplet state is only $J^{P}=\frac{1}{2}^{+}$. In this paper, we study the spectroscopic parameters of the spin-1/2 heavy bottom/charmed baryons both in antitriplet and sextet representations, whose members are shown in Table 1.

In the literature, a lot of theoretical studies on the investigation of the spin-1/2 heavy baryons in vacuum have been performed using different phenomenological approaches such as the quark model Roberts ; Karliner , quark potential model Capstick , heavy quark effective theory (HQET) Dai ; Liu ; Korner , chiral perturbation theory Savage , Feynman-Hellman theorem Roncaglia , hypercentral approach Ghalenovi1 ; Ghalenovi2 ; Patel1 ; Patel2 , lattice QCD simulation Brown ; Mathur ; Lewis ; Bahtiyar , relativistic (constituent) quark model Ebert1 ; Ebert2 ; Ebert3 ; Migura ; Gerasyuta1 ; Gerasyuta2 ; Gerasyuta3 ; Garcilazo , chiral quark-soliton model Kim , symmetry-preserving treatment of a vector$\times$vector contact interaction model Yin , QCD sum rules Shuryak ; Bagan ; Azizi ; Wang1 ; Wang2 ; Wang3 ; Zhang1 ; Zhang2 ; Agaev1 ; Agaev2 ; Agaev3 ; Agaev4 , etc. As we also previously mentioned, to better understand the hot and dense QCD matter created in relativistic heavy-ion collision experiments such as Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory’s (BNL) and the Large Hadron Collider (LHC) at the European Organization for Nuclear Research’s (CERN), the investigation of effects of temperature on spectroscopic properties of hadrons at nonzero temperature is needed. Such investigations can also help us to improve our understanding of phase transition, quark-gluon deconfinement and chiral symmetry restoration. By increasing in the temperature, a transition or chiral crossover Aoki ; MCheng from the hadronic phase to QGP phase may be occurred. Lattice QCD calculations show that thepseudocritical temperature is $T_{pc}\approx 155MeV$ for chiral crossover Bhattacharya ; Bazavov2 to QGP.

One of the most applicable and powerful phenomenological tools that can be used to analyze the spectroscopic properties of hadrons at nonzero temperature is the thermal QCD sum rule method (TQCDSR). This method is the thermal version Bochkarev of the QCD sum rule, firstly introduced by Shifman, Vainshtein and Zakharov for mesons at zero temperature Shifman and then applied to baryons in vacuum by Ioffe Ioffe . In thermal version, some additional operators appear in the operator product expansion (OPE/Wilson expansion) due to the breaking of the Lorentz invariance and vacuum expectation values are replaced by their thermal forms at finite temperature. The essential objective of this study is to extend our previous work on the thermal properties of the spin-3/2 heavy baryons at nonzero temperature AziziTurkan and investigate the shifts on the mass and residue of the spin-1/2 heavy $\Lambda_{Q}$, $\Xi_{Q}$, $\Sigma_{Q}$, $\Xi_{Q}^{{}^{\prime}}$ and $\Omega_{Q}$ baryons with respect to temperature using TQCDSR. In our calculations, we take account the extra operators arising from the OPE at nonzero temperature and use the thermal quark, gluon and mixed condensates up to dimension eight as well as the temperature-dependent energy-momentum tensor.

This study is structured as follows: In Sec. II, we derive the TQCDSR for masses and residues of the spin-1/2 heavy baryons at nonzero temperature. In Sec. III, we present the numerical analysis of the obtained sum rules for the physical parameters and a comparison of our results at zero temperature with those existing in the literature. The last section is devoted to both the summary of the results and our concluding remarks.

The aim of this section is to obtain TQCDSR for the masses and residues of the spin-1/2 heavy $\Lambda_{Q}$, $\Xi_{Q}$, $\Sigma_{Q}$, $\Xi_{Q}^{{}^{\prime}}$ and $\Omega_{Q}$ baryons at nonzero temperature. For this purpose, we start our calculations by considering the following two-point thermal correlation function:

$$\Pi(q,T)=i\int d^{4}x~{}e^{iq\cdot x}\langle\Psi|\mathcal{T}\{J_{B_{Q}}(x)\bar{J}_{B_{Q}}(0)\}|\Psi\rangle,$$

(1)

where $q$ denotes the four-momentum of the considered spin-1/2 heavy baryon (${B_{Q}}$), $\Psi$ indicates the ground state of the hot medium and $\mathcal{T}$ is the time-ordering operator. $J_{B_{Q}}(x)$ is the interpolating current of ${B_{Q}}$ baryon, which couples to both the positive and negative parities. It is represented by the following expressions for anti-triplet ($\overline{\textbf{3}}$) and sextet (6) baryons Bagan :

	$\displaystyle J_{\overline{\textbf{3}}}$	$\displaystyle=$	$\displaystyle\dfrac{1}{\sqrt{6}}\epsilon_{abc}\sum_{l=1}^{2}\ \Big{\{}2\Big{(}q_{1}^{T,a}(x)C\Gamma_{1}^{l}q_{2}^{b}(x)\Big{)}\Gamma_{2}^{l}Q^{c}(x)$		(2)
		$\displaystyle+$	$\displaystyle\Big{(}q_{1}^{T,a}(x)C\Gamma_{1}^{l}Q^{b}(x)\Big{)}\Gamma_{2}^{l}q_{2}^{c}(x)$
		$\displaystyle+$	$\displaystyle\Big{(}Q^{T,a}(x)C\Gamma_{1}^{l}q_{2}^{b}(x)\Big{)}\Gamma_{2}^{l}q_{1}^{c}(x)\Big{\}},$

	$\displaystyle J_{\textbf{6}}$	$\displaystyle=$	$\displaystyle-\dfrac{1}{\sqrt{2}}\epsilon_{abc}\sum_{l=1}^{2}\ \Big{\{}\Big{(}q_{1}^{T,a}(x)C\Gamma_{1}^{l}Q^{b}(x)\Big{)}\Gamma_{2}^{l}q_{2}^{c}(x)$		(3)
		$\displaystyle-$	$\displaystyle\Big{(}Q^{T,a}(x)C\Gamma_{1}^{l}q_{2}^{b}(x)\Big{)}\Gamma_{2}^{l}q_{1}^{c}(x)\Big{\}},$

where $\epsilon_{abc}$ is the Levi-Civita tensor with color indices $a,b,c$ , $C$ is the charge conjugation operator, $\Gamma_{1}^{1}=1$, $\Gamma_{1}^{2}=\Gamma_{2}^{1}=\gamma^{5}$, and $\Gamma_{2}^{2}=t$ in which $t$ denotes an arbitrary mixing parameter with $t=-1$ corresponds to the famous Ioffe currents that we consider in the present study. As we previously noted, $q_{1}$ and $q_{2}$ stand for light quarks and $Q$ for heavy quark field and they are given in Table 1 for all considered ${B_{Q}}$ baryons. Thus, considering Eq. (2) and Eq. (3), the interpolating currents for each state can be written as

	$\displaystyle J_{\Xi_{Q}^{-(0)}}$	$\displaystyle=$	$\displaystyle-\dfrac{1}{\sqrt{6}}\epsilon_{abc}\sum_{l=1}^{2}\ \Big{\{}2\Big{(}d^{T,a}(x)C\Gamma_{1}^{l}s^{b}(x)\Big{)}\Gamma_{2}^{l}Q^{c}(x)$
		$\displaystyle+$	$\displaystyle\Big{(}d^{T,a}(x)C\Gamma_{1}^{l}Q^{b}(x)\Big{)}\Gamma_{2}^{l}s^{c}(x)$
		$\displaystyle+$	$\displaystyle\Big{(}Q^{T,a}(x)C\Gamma_{1}^{l}s^{b}(x)\Big{)}\Gamma_{2}^{l}d^{c}(x)\Big{\}},$
	$\displaystyle J_{\Xi_{Q}^{0(+)}}$	$\displaystyle=$	$\displaystyle J_{\Xi_{Q}^{-(0)}}(d\rightarrow u),$
	$\displaystyle J_{\Lambda_{Q}^{0(+)}}$	$\displaystyle=$	$\displaystyle J_{\Xi_{Q}^{-(0)}}(d\rightarrow u,s\rightarrow d),$
	$\displaystyle J_{\Sigma_{Q}^{0(+)}}$	$\displaystyle=$	$\displaystyle-\dfrac{1}{\sqrt{2}}\epsilon_{abc}\sum_{l=1}^{2}\ \Big{\{}\Big{(}u^{T,a}(x)C\Gamma_{1}^{l}Q^{b}(x)\Big{)}\Gamma_{2}^{l}d^{c}(x)$
		$\displaystyle-$	$\displaystyle\Big{(}Q^{T,a}(x)C\Gamma_{1}^{l}d^{b}(x)\Big{)}\Gamma_{2}^{l}u^{c}(x)\Big{\}},$
	$\displaystyle J_{\Xi_{Q}^{-(0)^{\prime}}}$	$\displaystyle=$	$\displaystyle J_{\Sigma_{Q}^{0(+)}}(u\rightarrow d,d\rightarrow s),$
	$\displaystyle J_{\Xi_{Q}^{0(+)^{\prime}}}$	$\displaystyle=$	$\displaystyle J_{\Xi_{Q}^{-(0)^{\prime}}}(d\rightarrow u),$
	$\displaystyle J_{\Sigma_{Q}^{-(0)}}$	$\displaystyle=$	$\displaystyle J_{\Xi_{Q}^{-(0)^{\prime}}}(s\rightarrow d),$
	$\displaystyle J_{\Sigma_{Q}^{+(++)}}$	$\displaystyle=$	$\displaystyle J_{\Sigma_{Q}^{-(0)}}(d\rightarrow u),$
	$\displaystyle J_{\Omega_{Q}^{-(0)}}$	$\displaystyle=$	$\displaystyle J_{\Sigma_{Q}^{+(++)}}(u\rightarrow s).$		(4)

According to the standard philosophy of the QCD sum rule method, the aforementioned thermal correlation function is evaluated in two basic ways: i) On the hadronic side, it is calculated in terms of hadronic parameters such as the temperature-dependent mass and residue of hadron. ii) On the QCD side, it is calculated in terms of temperature-dependent QCD degrees of freedom in the deep Euclidean region with the help of OPE. Then, matching the coefficients of the selected structures from both sides in momentum space, via the dispersion relation, the QCD sum rules for the spectroscopic parameters of the ${B_{Q}}$ at nonzero temperature are obtained. In the final step, to suppress the contributions of the higher states and continuum in obtained sum rules, Borel transformation and continuum subtraction are applied to both sides of these equalities. One may first apply the Borel transformation and continuum subtraction, then match the coefficients of the selected structures from both sides.

The thermal correlation function in the hadronic side is obtained by inserting the full set of hadronic states having the same quantum numbers as the related interpolating current into Eq. (1). After performing the integration over four-$x$, the thermal correlation function for the hadronic side can be written as

			$\displaystyle\Pi^{Had.}(q,T)$
		$\displaystyle=$	$\displaystyle-\frac{{\langle}\Psi|J_{B_{Q}}(0)|B_{Q}^{+}(q,s){\rangle}{\langle}B_{Q}^{+}(q,s)|J^{{\dagger}}_{B_{Q}}(0)|\Psi{\rangle}}{q^{2}-m_{B_{Q}^{+}}^{2}(T)}$
		$\displaystyle-$	$\displaystyle\frac{{\langle}\Psi|J_{B_{Q}}(0)|B_{Q}^{-}(q,s){\rangle}{\langle}B_{Q}^{-}(q,s)|J^{{\dagger}}_{B_{Q}}(0)|\Psi{\rangle}}{q^{2}-m_{B_{Q}^{-}}^{2}(T)}$
		$\displaystyle+$	$\displaystyle\mbox{...},$

where $|B_{Q}^{+}(q,s)\rangle$ and $|B_{Q}^{-}(q,s)\rangle$ are spin-1/2 single heavy baryon states with the positive and negative parity, respectively, dots stand for the contributions of the higher states and continuum and $m_{B_{Q}^{\pm}}(T)$ is the temperature-dependent mass of $B_{Q}^{\pm}$. The matrix elements ${\langle}\Psi|J_{B_{Q}}(0)|B_{Q}^{\pm}(q,s){\rangle}$ for $B_{Q}^{\pm}$ are defined as

	$\displaystyle{\langle}\Psi|J_{B_{Q}}(0)|B_{Q}^{+}(q,s){\rangle}$	$\displaystyle=$	$\displaystyle\lambda_{B_{Q}^{+}}(T)u_{B_{Q}^{+}}(q,s),$
	$\displaystyle{\langle}\Psi|J_{B_{Q}}(0)|B_{Q}^{-}(q,s){\rangle}$	$\displaystyle=$	$\displaystyle\lambda_{B_{Q}^{-}}(T)\gamma_{5}u_{B_{Q}^{-}}(q,s),$		(6)

where $\lambda_{B_{Q}^{\pm}}(T)$ is the temperature-dependent residue of $B_{Q}^{\pm}$ and $u_{B_{Q}^{\pm}}(q,s)$ is Dirac spinor of spin $s$ and the four-momentum $q$. To proceed, we insert Eq. (II) into Eq. (II) and perform summation over spins of $B_{Q}^{\pm}$. The hadronic side of thermal correlation function in its final form is decomposed in terms of different structures as

	$\displaystyle\Pi^{Had.}(q,T)$	$\displaystyle=$	$\displaystyle-\frac{\lambda^{2}_{B_{Q}^{+}}(T)(\!\not\!{q}+m_{B_{Q}^{+}}(T))}{q^{2}-m^{2}_{B_{Q}^{+}}(T)}$		(7)
		$\displaystyle-$	$\displaystyle\frac{\lambda^{2}_{B_{Q}^{-}}(T)(-\!\not\!{q}+m_{B_{Q}^{-}}(T))}{q^{2}-m^{2}_{B_{Q}^{-}}(T)}+\cdots.$

This correlation function can be written in terms of the structures $\!\not\!{q}$ and $I$ as

$\displaystyle\Pi^{Had.}(q,T)$

$\displaystyle=$

$\displaystyle\Pi_{1}^{Had.}(T)\!\not\!{q}+\Pi_{2}^{Had.}(T)I+\cdots,$

where $\Pi^{Had.}_{1(2)}(T)$, as the coefficients of the selected Lorentz structures, in Borel scheme are obtained as

	$\displaystyle\hat{B}\Pi_{1}^{Had.}(T)$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(9)
		$\displaystyle-$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}},$

and

	$\displaystyle\hat{B}\Pi_{2}^{Had.}(T)$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)m_{B_{Q}^{+}}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(10)
		$\displaystyle+$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)m_{B_{Q}^{-}}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}},$

where $M^{2}$ is the Borel parameter to be determined in next section.

Now, we have to evaluate the QCD side of the thermal correlation function in terms of the quark-gluon degrees of freedom in the deep Euclidean region. For this aim, we insert the related interpolating current of ${B_{Q}}$ given in Eq. (II) into Eq. (1) and contract all light and heavy quark fields using the Wick theorem. Thus, the most general form of the thermal correlation function on the QCD side in terms of thermal light (heavy) quark propagators $S_{q(Q)}^{ij}(x)$ for $\overline{\textbf{3}}$ and 6 baryons are obtained as

	$\displaystyle\Pi_{\overline{\textbf{3}}}^{QCD}(q,T)$	$\displaystyle=$	$\displaystyle\frac{i}{6}\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\int d^{4}xe^{iq\cdot x}$		(11)
		$\displaystyle\times$	$\displaystyle\sum_{l=1}^{2}\sum_{k=1}^{2}\ \left\{\Gamma^{l}_{2}\Big{(}2S^{ca^{\prime}}_{Q}(x)A^{k}_{1}\widetilde{S}^{ab^{\prime}}_{q_{1}}(x)\Gamma^{l}_{1}S^{bc^{\prime}}_{q_{2}}(x)\Gamma^{k}_{2}\right.$
		$\displaystyle+$	$\displaystyle 2S^{cb^{\prime}}_{Q}(x)\Gamma^{k}_{1}\widetilde{S}^{ba^{\prime}}_{q_{2}}(x)\Gamma^{l}_{1}S^{ac^{\prime}}_{q_{1}}(x)\Gamma^{k}_{2}$
		$\displaystyle-$	$\displaystyle S^{ca^{\prime}}_{q_{2}}(x)\Gamma^{k}_{2}\widetilde{S}^{bb^{\prime}}_{Q}(x)\Gamma^{l}_{1}S^{ac^{\prime}}_{q_{1}}(x)\Gamma^{k}_{1}$
		$\displaystyle-$	$\displaystyle 2S^{ca^{\prime}}_{q_{2}}(x)\Gamma^{k}_{2}\widetilde{S}^{ab^{\prime}}_{q_{1}}(x)\Gamma^{l}_{1}S^{bc^{\prime}}_{Q}(x)\Gamma^{k}_{1}$
		$\displaystyle-$	$\displaystyle S^{cb^{\prime}}_{q_{1}}(x)\Gamma^{k}_{2}\widetilde{S}^{aa^{\prime}}_{Q}(x)\Gamma^{l}_{1}S^{bc^{\prime}}_{q_{2}}(x)\Gamma^{k}_{1}$
		$\displaystyle-$	$\displaystyle 2S^{cb^{\prime}}_{q_{1}}(x)\Gamma^{k}_{2}\widetilde{S}^{ba^{\prime}}_{q_{2}}(x)\Gamma^{l}_{1}S^{ac^{\prime}}_{Q}(x)\Gamma^{k}_{1}\Big{)}$
		$\displaystyle-$	$\displaystyle\Gamma^{k}_{2}\Big{(}S^{cc^{\prime}}_{q_{1}}(x)\Gamma^{l}_{2}Tr[\Gamma^{k}_{1}S^{ab^{\prime}}_{Q}(x)\Gamma^{l}_{1}\widetilde{S}^{ba^{\prime}}_{q_{2}}(x)]$
		$\displaystyle+$	$\displaystyle S^{cc^{\prime}}_{q_{2}}(x)\Gamma^{l}_{2}Tr[\Gamma^{k}_{1}S^{ab^{\prime}}_{q_{1}}(x)\Gamma^{l}_{1}\widetilde{S}^{ba^{\prime}}_{Q}(x)]$
		$\displaystyle+$	$\displaystyle 4S^{cc^{\prime}}_{Q}(x)\Gamma^{l}_{2}Tr[\Gamma^{k}_{1}S^{ab^{\prime}}_{q_{1}}(x)\Gamma^{l}_{1}\widetilde{S}^{ba^{\prime}}_{q_{2}}(x)]\Big{)}\left.\right\},$

	$\displaystyle\Pi_{\textbf{6}}^{QCD}(q,T)$	$\displaystyle=$	$\displaystyle-\frac{i}{2}\epsilon_{abc}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\int d^{4}xe^{iq\cdot x}$		(12)
		$\displaystyle\times$	$\displaystyle\sum_{l=1}^{2}\sum_{k=1}^{2}\ \left\{\Gamma^{l}_{2}\Big{(}S^{ca^{\prime}}_{q_{1}}(x)A^{k}_{1}\widetilde{S}^{ab^{\prime}}_{Q}(x)\Gamma^{l}_{1}S^{bc^{\prime}}_{q_{2}}(x)\right.$
		$\displaystyle+$	$\displaystyle S^{cb^{\prime}}_{q_{2}}(x)\Gamma^{k}_{1}\widetilde{S}^{ba^{\prime}}_{Q}(x)\Gamma^{l}_{1}S^{ac^{\prime}}_{q_{1}}(x)\Big{)}\Gamma^{k}_{2}$
		$\displaystyle+$	$\displaystyle\Gamma^{k}_{2}\Big{(}S^{cc^{\prime}}_{q_{1}}(x)\Gamma^{l}_{2}Tr[\Gamma^{l}_{1}\widetilde{S}^{aa^{\prime}}_{Q}(x)\Gamma^{k}_{1}S^{bb^{\prime}}_{q_{2}}(x)]$
		$\displaystyle+$	$\displaystyle S^{cc^{\prime}}_{q_{2}}(x)\Gamma^{l}_{2}Tr[\Gamma^{l}_{1}\widetilde{S}^{aa^{\prime}}_{q_{1}}(x)\Gamma^{k}_{1}S^{bb^{\prime}}_{Q}(x)]\Big{)}\left.\right\},$

where $\widetilde{S}^{ij}_{q(Q)}=CS^{ijT}_{q(Q)}C$. To proceed, thermal light (heavy) quark propagators $S_{q(Q)}^{ij}(x)$ in coordinate space are needed, which are used as (see also Azizi1 ; Azizi2 ; Prop_C )

	$\displaystyle S_{q}^{ij}(x)$	$\displaystyle=$	$\displaystyle i\frac{\!\not\!{x}}{2\pi^{2}x^{4}}\delta_{ij}-\frac{m_{q}}{4\pi^{2}x^{2}}\delta_{ij}-\frac{\langle\bar{q}q\rangle_{T}}{12}\delta_{ij}$
		$\displaystyle-$	$\displaystyle\frac{x^{2}}{192}m_{0}^{2}\langle\bar{q}q\rangle_{T}\Big{[}1-i\frac{m_{q}}{6}\!\not\!{x}\Big{]}\delta_{ij}$
		$\displaystyle+$	$\displaystyle\frac{i}{3}\Big{[}\!\not\!{x}\Big{(}\frac{m_{q}}{16}\langle\bar{q}q\rangle_{T}-\frac{1}{12}\langle u^{\mu}\Theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{)}$
		$\displaystyle+$	$\displaystyle\frac{1}{3}\Big{(}u\cdot x\!\not\!{u}\langle u^{\mu}\Theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{)}\Big{]}\delta_{ij}$
		$\displaystyle-$	$\displaystyle\frac{ig_{s}\lambda_{A}^{ij}}{32\pi^{2}x^{2}}G_{\mu\nu}^{A}\Big{(}\!\not\!{x}\sigma^{\mu\nu}+\sigma^{\mu\nu}\!\not\!{x}\Big{)}$
		$\displaystyle-$	$\displaystyle i\frac{x^{2}\!\not\!{x}g_{s}^{2}\langle\bar{q}q\rangle_{T}^{2}}{7776}\delta_{ij}-\frac{x^{4}\langle\bar{q}q\rangle_{T}\langle g_{s}^{2}G^{2}\rangle_{T}}{27648}+...~{},$

	$\displaystyle S_{Q}^{ij}(x)$	$\displaystyle=$	$\displaystyle i\int\frac{d^{4}ke^{-ik\cdot x}}{(2\pi)^{4}}\left(\frac{\!\not\!{k}+m_{Q}}{k^{2}-m_{Q}^{2}}\delta_{ij}\right.$		(14)
		$\displaystyle-$	$\displaystyle\frac{g_{s}G^{\alpha\beta}_{ij}}{4}\frac{\sigma^{\alpha\beta}(\!\not\!{k}+m_{Q})+(\!\not\!{k}+m_{Q})\sigma^{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
		$\displaystyle+$	$\displaystyle\frac{m_{Q}}{12}\frac{k^{2}+m_{Q}\!\not\!{k}}{(k^{2}-m_{Q}^{2})^{4}}\langle g_{s}^{2}G^{2}\rangle_{T}\delta_{ij}+\cdots\Bigg{)}.$

Here, $m_{q(Q)}$ is the light (heavy) quark mass, $\langle\bar{q}q\rangle_{T}$ is the thermal quark condensate, $\langle g_{s}^{2}G^{2}\rangle_{T}$ is thermal gluon condensate, $m_{0}^{2}\langle\bar{q}q\rangle_{T}=\langle\bar{q}g_{s}\sigma Gq\rangle_{T}$ is the thermal mixed condensate. The new operators, emerging in OPE, appear to restore the Lorentz invariance broken out by the choice of the thermal rest frame at nonzero temperature. They are expressed in terms of fermionic and gluonic parts of the energy momentum tensor $\Theta^{f,g}_{\mu\nu}$ $(\langle u^{\mu}\Theta^{f,g}_{\mu\nu}u^{\nu}\rangle=\langle u\Theta^{f,g}u\rangle=\langle\Theta^{f,g}_{00}\rangle=\langle\Theta^{f,g}\rangle)$ and the four-vector velocity of the medium, $u^{\mu}$. To this end, $u^{\mu}=(1,0,0,0)$ is chosen which leads to $u^{2}=1$. In the rest frame of the heat bath, $q\cdot u=q_{0}$, with $q_{0}$ being the energy of quasi particle, $q_{0}=E_{\vec{q}}=(|\vec{q}|^{2}+m^{2})^{1/2}$, in the mass-shell condition. At $\vec{q}=0$ limit it is the mass of the particle. In the light quark propagator, the fermionic part of the energy momentum tensor, $\langle u^{\mu}\Theta_{\mu\nu}^{f}u^{\nu}\rangle$, is seen explicitly whereas the gluonic part, $\langle u^{\lambda}{\Theta}^{g}_{\lambda\sigma}u^{\sigma}\rangle$, appears in the trace of two-gluon field strength tensor in the heat bath Mallik , i.e.

	$\displaystyle\langle Tr^{c}G_{\alpha\beta}G_{\mu\nu}\rangle$	$\displaystyle=$	$\displaystyle\frac{1}{24}(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu})\langle G^{2}\rangle_{T}$		(15)
		$\displaystyle+$	$\displaystyle\frac{1}{6}\Big{[}g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}-2(u_{\alpha}u_{\mu}g_{\beta\nu}$
		$\displaystyle-$	$\displaystyle u_{\alpha}u_{\nu}g_{\beta\mu}-u_{\beta}u_{\mu}g_{\alpha\nu}+u_{\beta}u_{\nu}g_{\alpha\mu})\Big{]}$
		$\displaystyle\times$	$\displaystyle\langle u^{\lambda}{\Theta}^{g}_{\lambda\sigma}u^{\sigma}\rangle.$

The QCD side of the correlation function can be written similar to hadronic side in terms of different Lorentz structures as

$\displaystyle\Pi^{QCD}(q,T)$

$\displaystyle=$

$\displaystyle\Pi_{1}^{QCD}(T)\!\not\!{q}+\Pi_{2}^{QCD}(T)I,$

where $\Pi^{QCD}_{1(2)}(T)$ are the coefficients of the selected Lorentz structures and they contain both the perturbative and non-perturbative contributions. The perturbative and some non-perturbative parts are written in terms of the dispersion integrals in the present study. Thus,

$$\Pi^{QCD}_{1(2)}(T)=\int_{s_{min}}^{\infty}ds\dfrac{\rho^{{QCD}}_{1(2)}(s,T)}{s-q^{2}}+\Gamma_{1(2)}^{QCD}(T),$$

(17)

where $s_{min}=(m_{q_{1}}+m_{q_{2}}+m_{Q})^{2}$, $\rho^{QCD}_{1(2)}(s,T)$ are the spectral densities and $\Gamma_{1(2)}^{QCD}(T)$ stand for the remaining non-perturbative contributions that are calculated directly by applying the Borel transformation. The related spectral densities are defined as

$$\rho^{QCD}_{1(2)}(s,T)=\frac{1}{\pi}\mathrm{Im}[\Pi^{QCD}_{1(2)}(T)].$$

(18)

After Borel transformation and continuum subtraction we get

$\displaystyle\hat{B}\Pi_{1(2)}^{QCD}(T)=\int_{s_{min}}^{s_{0}(T)}ds\rho_{1(2)}^{{QCD}}(s,T)e^{-s/M^{2}}+\hat{B}\Gamma_{1(2)}^{QCD}(T),$

where $s_{0}(T)$ is the temperature-dependent continuum threshold. The main task in this part is to find the expressions for $\rho^{QCD}_{1(2)}(s,T)$ and $\hat{B}\Gamma_{1(2)}^{QCD}(T)$. They are obtained after inserting the explicit forms of the heavy and light quark propagators into the QCD side of the thermal correlation function given in Eqs. (11) and (12) for $\overline{\textbf{3}}$ and 6 baryons with spin-1/2, performing the Fourier integral to go to the momentum space and applying the steps above to get the perturbative and non-perturbative parts. As a result, the explicit forms of the functions $\rho^{QCD}_{1(2)}(s,T)$ and $\hat{B}\Gamma_{1(2)}^{QCD}(T)$ are obtained. Because of obtained expressions are quite lengthy, we present the expressions of $\rho^{QCD}_{1}(s,T)$ and $\hat{B}\Gamma_{1}^{QCD}(T)$ for only $\Sigma_{b}^{0}$ baryon as an example in the Appendix.

Finally, we match the coefficients of the selected structures from the hadronic and QCD sides of the correlation function and find the sum rules:

	$\displaystyle\hat{B}\Pi_{1}^{QCD}(T)$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(20)
		$\displaystyle-$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}},$

and

	$\displaystyle\hat{B}\Pi_{2}^{QCD}(T)$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)m_{B_{Q}^{+}}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(21)
		$\displaystyle+$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)m_{B_{Q}^{-}}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}}.$

In order to obtain the four unknowns, $m_{B_{Q}^{+}}$, $m_{B_{Q}^{-}}$, $\lambda_{B_{Q}^{+}}$ and $\lambda_{B_{Q}^{-}}$, two more equations which can be achieved by applying derivatives with respect to $\frac{d}{d(-\frac{1}{M^{2}})}$ to both sides of Eqs. (20) and (21) are needed. Therefore, we get

	$\displaystyle\frac{d\Pi_{1}^{QCD}(T)}{d(-1/M^{2})}$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)m_{B_{Q}^{+}}^{2}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(22)
		$\displaystyle-$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)m_{B_{Q}^{-}}^{2}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}},$

	$\displaystyle\frac{d\Pi_{2}^{QCD}(T)}{d(-1/M^{2})}$	$\displaystyle=$	$\displaystyle\lambda^{2}_{B_{Q}^{+}}(T)m_{B_{Q}^{+}}^{3}(T)e^{-m_{B_{Q}^{+}}^{2}(T)/M^{2}}$		(23)
		$\displaystyle+$	$\displaystyle\lambda^{2}_{B_{Q}^{-}}(T)m_{B_{Q}^{-}}^{3}(T)e^{-m_{B_{Q}^{-}}^{2}(T)/M^{2}},$

By simultaneously solving the above four equations, the temperature-dependent masses and residues for the positive and negative parity spin-1/2 heavy baryons are obtained in terms of $s_{0}(T)$, $M^{2}$, QCD degrees of freedom and other inputs. For the masses, as examples, we get

$\displaystyle m_{B_{Q}^{+}}=\dfrac{(\alpha_{4}\alpha_{1}-\alpha_{3}\alpha_{2}+\sqrt{4\alpha_{3}^{3}\alpha_{1}+\alpha_{4}^{2}\alpha_{1}^{2}-6\alpha_{3}\alpha_{4}\alpha_{1}\alpha_{2}-3\alpha_{3}^{2}\alpha_{2}^{2}+4\alpha_{4}\alpha_{2}^{3}})}{2\alpha_{3}\alpha_{1}-2\alpha_{2}^{2}},$

(24)

$\displaystyle m_{B_{Q}^{-}}=\dfrac{(-\alpha_{4}\alpha_{1}+\alpha_{3}\alpha_{2}+\sqrt{4\alpha_{3}^{3}\alpha_{1}+\alpha_{4}^{2}\alpha_{1}^{2}-6\alpha_{3}\alpha_{4}\alpha_{1}\alpha_{2}-3\alpha_{3}^{2}\alpha_{2}^{2}+4\alpha_{4}\alpha_{2}^{3}})}{2\alpha_{3}\alpha_{1}-2\alpha_{2}^{2}},$

(25)

where

$\displaystyle\alpha_{1}=\hat{B}\Pi_{1}^{QCD}(T),\alpha_{2}=\hat{B}\Pi_{2}^{QCD}(T),\alpha_{3}=\frac{d\Pi_{1}^{QCD}(T)}{d(-1/M^{2})},\alpha_{4}=\frac{d\Pi_{2}^{QCD}(T)}{d(-1/M^{2})}.$

(26)

Similar results are obtained for the temperature-dependent residues.

In this section, we perform the numerical analyses of the sum rules for the masses and residues of the spin-1/2 heavy $\Lambda_{Q}$, $\Xi_{Q}$, $\Sigma_{Q}$, $\Xi_{Q}^{{}^{\prime}}$ and $\Omega_{Q}$ baryons at nonzero temperature. For this aim, firstly we use the numerical values of some input parameters collected in Table 2 in our calculations.