Thermal behaviors of exotic \(Z_{c}\) (3900) state in tetraquark approach

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The discovery of the charged \(Z_{c}\)(3900) exotic resonance with the quantum numbers \(J^{P}\)= \(1^{+}\) has crucial consequences for comprehending the nature of multi-quark hadrons. In this work, the meson-current coupling constant and mass of \(Z_{c}\) state are calculated using the Thermal QCD Sum Rules method. We suppose that the positively charged state \(Z_{c}\) has a quark content \({\bar{c}}cu{\bar{d}}\) and use tetraquark current with the corresponding quantum numbers. We evaluate the two-point correlation function taking into account the non-perturbative condensates up to six-dimension. We found the hot medium contributions to hadronic parameters of the \(Z_{c}\) state and exhibit that its meson-current coupling constant and mass are unsusceptible to the change of temperature in low temperature region, but these parameters significantly change in the vicinity of phase transition region. At critical temperature, the meson current-coupling constant reaches approximately to 10% of its vacuum value, while the mass decreases about 28%.

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Authors thank Kocaeli University for the partial financial support through Grant no. BAP 2018/070.

Author information

Correspondence to Hayriye Sundu.

Appendix A: The two-point thermal spectral density

Appendix A: The two-point thermal spectral density

The results of the spectral densities obtained in the calculations are presented in this section. Collecting them as perturbative and nonperturbative parts, we obtain that,

$$\begin{aligned} \rho ^{\mathrm {QCD}}(s,T)=\rho ^{\mathrm {pert.}}(s)+\sum _{k=3}^{6}\rho _{k}(s,T), \end{aligned}$$

where \(\rho _{k}(s,T)\) denote the nonperturbative contributions:

$$\begin{aligned} \rho _\mathrm{{pert.}}(s)= & {} \frac{1}{3072\pi ^6}\int _{0}^{1}\nonumber \\&\mathrm{{d}}z\int _{0}^{1-z}\mathrm{{d}}w\frac{wz}{ht^8} \Big \lbrace \left( hswz-m_c^2t(w+z)\right) ^2 (-7hswz+m_c^2t(w+z)) \nonumber \\&\times \,(-5hswz+3m_c^2t(w+z))\Big \rbrace \theta [L], \end{aligned}$$
$$\begin{aligned} \rho _{3}(s,T)= & {} \frac{1}{192\pi ^4}\int _{0}^{1}\mathrm{{d}}z\int _{0}^{1-z} \frac{{\text {d}}w}{t^6}\Big \lbrace \langle {\bar{q}}q\rangle \Big [21h^2m_cs^2tw^2z^2(w+z)\nonumber \\&-\,30hm_c^3st^2wz(w+z)^2+9m_c^5t^3(w+z)^3\Big ]\nonumber \\&+ \,{\langle \Theta ^f_{00}\rangle }\Big [-240h^3s^2w^3z^3\nonumber \\&+\,208h^2m_c^2stw^2z^2(w+z)-16m_c^4t^2(w+z)^2\Big ]\Big \rbrace \theta [L], \end{aligned}$$
$$\begin{aligned} \rho _{4}(s,T)= & {} -\frac{1}{18432\pi ^{6}} \int _{0}^{1}\mathrm{{d}}z\int _{0}^{1-z}\frac{{\text {d}}w}{h^2t^6} \Big \langle \frac{g^2G^2}{\pi }\Big \rangle \Big \lbrace -60h^4s^2w^3z^3(w+z)\nonumber \\&+\,m_c^4t^2(w+z)(12(-1+w)^3 \nonumber \\&\times \,w^2(-1+2w)+3(-1+w)w(-8+w(46\nonumber \\&+\,w(-75+(37+2\pi )w)))z+2(6+w(-81 \nonumber \\&+\, w(258+w(-299+(116+3\pi )w))))z^2\nonumber \\&+\,(-1+w)(60+w(-303+295w))z^3+2(54-3 \nonumber \\&\times \, (56+\pi )w+(116+3\pi )w^2)z^4\nonumber \\&+\,3(-28+(37+2\pi )w)z^5+24z^6) -hm_c^2stwz(12(-1+w)^3 \nonumber \\&+\,w^2(-1+2w)+(-1+w)w(-24+w(138+w(-177\nonumber \\&+\,(63+16\pi )w)))z+2(6+w(-81\nonumber \\&\times \, w((210+w(-155+4(5+2\pi )w))))z^2\nonumber \\&+\,(-60+w(315+w(-310+7w)))z^3+4(27+2(5\nonumber \\&\times \, (-6+w)+2\pi (-1+w))w)z^4\nonumber \\&+\,(-84+(63+16\pi )w)z^5+24z^6) \Big \rbrace \theta [L]\nonumber \\&+\, \frac{1}{4608}\int _{0}^{1}\mathrm{{d}}z\int _{0}^{1-z}\frac{{\text {d}}w}{h^2t^6}\langle \Theta ^f_{00}\rangle \Big \lbrace \Big [-30h^4s^2 w^3z^3(w+z)\nonumber \\&+\,hm_c^2twz(4swz(3(-1+w)^2w \nonumber \\&\times \,(-2+9w)+2(-1+w)(3+w(-33+50w))z\nonumber \\&+\,(39+w(-166+145w))z^2+20(-3+5w)z^3 \nonumber \\&+\, 27z^4) +3s(4(-1+w)^3w^2(-1+2w)\nonumber \\&+\,(-1+w)^2w(8+w(-42+23w))z+2(-1+w)(-2+w\nonumber \\&\times \,(27+w(-51+8w)))z^2\nonumber \\&+\,(-20+(-1+w)w(-115+3w))z^3\nonumber \\&+\,4(9-22w+4w^2)z^4+(-28 \nonumber \\&+\, 23w)z^5+8z^6))-3m_c^4t^2 \Big ((8w^7\nonumber \\&+\,4(-1+z)^3z^3(-1+2z)+7w^6(-4+7z)\nonumber \\&+\,w(-1+z)^2z^2(12\nonumber \\&\times \, (z (-54+49z))+ w^5(36+z(-152+129z))\nonumber \\&+\,w^2(-1+z)^2z(12+z(-92+129z))+ w^4\nonumber \\&\times \,(-20 z(169+z(-350+199z)))\nonumber \\&+\,w^3(4+z(-78+z(325+z(-452+199z))))\Big )\Big ] \Big \rbrace \theta [L], \end{aligned}$$
$$\begin{aligned} \rho _{5}(s,T)= & {} -\frac{m_cm_0^2\langle {\overline{q}}q\rangle }{128\pi ^4} \int _{0}^{1}\mathrm{{d}}z\nonumber \\&\int _{0}^{1-z}\mathrm{{d}}w\frac{wzh(w+z) }{t^5} (w+z) \Big [20hswz-9m_c^2(w^3+2w \nonumber \\&\times \, (-1+z)z+(-1+z)z^2+w^2(-1+2z)) \Big ] \theta [L], \end{aligned}$$
$$\begin{aligned} \rho _{6}(s,T)= & {} \frac{1}{108\pi ^2}\int _{0}^{1}\mathrm{{d}}z\int _{0}^{1-z}\mathrm{{d}}w\Big [ 9m_c^2\langle {\overline{q}}q\rangle ^2-\nonumber \\&3m_c\langle {\overline{q}}q\rangle \langle \Theta ^f_{00}\rangle +20z \langle \Theta ^f_{00}\rangle ^2 \Big ]\theta [L]. \end{aligned}$$


$$\begin{aligned} L= & {} \frac{1}{t^2} \Big [\Big (m_c^2(w^3+w^2(2z-1)+(z^2+2wz)(z-1))-hswz \Big ) (w-1) \Big ], \nonumber \\ t= & {} w^2+(z-1)(w+z),\ h=w+z-1 \end{aligned}$$

and \(\theta \) is step function.

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Işık, İ.B., Sundu, H. & Veliev, E.V. Thermal behaviors of exotic \(Z_{c}\) (3900) state in tetraquark approach. Eur. Phys. J. Plus 135, 48 (2020) doi:10.1140/epjp/s13360-019-00026-x

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