Photons production in heavy-ion collisions as a signal of deconfinement phase

Abstract

The photon production due to conversion of two gluons into a photon, $gg\rightarrow \gamma$, in the presence of the background gauge fields is studied within the specific mean-field approach to QCD vacuum. In this approach, mean field in the confinement phase is represented by the statistical ensemble of almost everywhere homogeneous abelian (anti-)self-dual gluon configurations, while the deconfined phase can be characterized by the purely chromomagnetic fields. The probability of gluon conversion of two gluons into a photon vanishes in the confinement phase due to the randomness of the background field configurations. The anisotropic strong electromagnetic field, generated in the collision of relativistic heavy ions, serves as a catalyst for deconfinement with the appearance of an anisotropic purely chromomagnetic mean field. Respectively, deconfined phase is characterized by nonzero probability of the conversion of two gluons into a photon with strongly anisotropic angular distribution.

Appendix

The amplitude in Eq. (11) consists of 32 terms. Below an explicit form of the tensor structures and corresponding form factors are listed.

Set of tensors $\mathcal {F}^{l}_{\mu \nu \rho }(p,k)$ includes

$$\begin{aligned} \mathcal {F}^{1}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }f_{\beta \nu }f_{\lambda \rho }p_{\perp }^{\alpha }p_{\perp }^{\beta }p_{\perp }^{\lambda },~\\{} & {} \mathcal {F}^{2}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }f_{\beta \nu }f_{\lambda \rho }p_{\perp }^{\alpha }p_{\perp }^{\beta }k_{\perp }^{\lambda },~\\{} & {} \mathcal {F}^{3}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }f_{\beta \rho }f_{\lambda \nu }p_{\perp }^{\alpha }p_{\perp }^{\beta }k_{\perp }^{\lambda }\\ \mathcal {F}^{4}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }f_{\beta \nu }f_{\lambda \rho }p_{\perp }^{\alpha }k_{\perp }^{\beta }k_{\perp }^{\lambda },~\\{} & {} \mathcal {F}^{5}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }f_{\beta \rho }f_{\lambda \mu }p_{\perp }^{\alpha }p_{\perp }^{\beta }k_{\perp }^{\lambda },~\\{} & {} \mathcal {F}^{6}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }f_{\beta \mu }f_{\lambda \rho }p_{\perp }^{\alpha }k_{\perp }^{\beta }k_{\perp }^{\lambda },\\ \mathcal {F}^{7}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \rho }f_{\beta \mu }f_{\lambda \nu }p_{\perp }^{\alpha }k_{\perp }^{\beta }k_{\perp }^{\lambda },~\\{} & {} \mathcal {F}^{8}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }f_{\beta \nu }f_{\lambda \rho }k_{\perp }^{\alpha }k_{\perp }^{\beta }k_{\perp }^{\lambda },\\ \mathcal {F}^{9}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }p_{\perp }^{\alpha }\delta _{\nu \rho }p_{\perp }^2,~\\{} & {} \mathcal {F}^{10}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }p_{\perp }^{\alpha }\delta _{\nu \rho }p_{\perp }k_{\perp }, ~\\{} & {} \mathcal {F}^{11}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }p_{\perp }^{\alpha }\delta _{\nu \rho }k_{\perp }^2,\\ \mathcal {F}^{12}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }p_{\perp }^{\alpha }\delta ^{||}_{\nu \rho },~\\ \mathcal {F}^{13}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \nu }p_{\perp }^{\alpha }\delta _{\mu \rho }p_{\perp }^2, ~\\{} & {} \mathcal {F}^{14}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }p_{\perp }^{\alpha }\delta _{\mu \rho }p_{\perp }k_{\perp },~\\{} & {} \mathcal {F}^{15}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }p_{\perp }^{\alpha }\delta _{\mu \rho }k_{\perp }^2,\\ \mathcal {F}^{16}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \nu }p_{\perp }^{\alpha }\delta ^{||}_{\mu \rho },~\\ \mathcal {F}^{17}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \rho }p_{\perp }^{\alpha }\delta _{\mu \nu }p_{\perp }^2,~\\{} & {} \mathcal {F}^{18}_{\mu \nu \rho }(p,k)=f_{\alpha \rho }p_{\perp }^{\alpha }\delta _{\mu \nu }p_{\perp }k_{\perp },~\\{} & {} \mathcal {F}^{19}_{\mu \nu \rho }(p,k)=f_{\alpha \rho }p_{\perp }^{\alpha }\delta _{\mu \nu }k_{\perp }^2,\\ \mathcal {F}^{20}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \rho }p_{\perp }^{\alpha }\delta ^{||}_{\mu \nu },~\\ \mathcal {F}^{21}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }k_{\perp }^{\alpha }\delta _{\nu \rho }p_{\perp }^2,~\\{} & {} \mathcal {F}^{22}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }k_{\perp }^{\alpha }\delta _{\nu \rho }p_{\perp }k_{\perp },~\\{} & {} \mathcal {F}^{23}_{\mu \nu \rho }(p,k)=f_{\alpha \mu }k_{\perp }^{\alpha }\delta _{\nu \rho }k_{\perp }^2,\\ \mathcal {F}^{24}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \mu }k_{\perp }^{\alpha }\delta ^{||}_{\nu \rho },~\\ \mathcal {F}^{25}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \nu }k_{\perp }^{\alpha }\delta _{\mu \rho }p_{\perp }^2,~\\{} & {} \mathcal {F}^{26}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }k_{\perp }^{\alpha }\delta _{\mu \rho }p_{\perp }k_{\perp },~\\{} & {} \mathcal {F}^{27}_{\mu \nu \rho }(p,k)=f_{\alpha \nu }k_{\perp }^{\alpha }\delta _{\mu \rho }k_{\perp }^2\\ \mathcal {F}^{28}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \nu }k_{\perp }^{\alpha }\delta ^{||}_{\mu \rho },~\\ \mathcal {F}^{29}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \rho }k_{\perp }^{\alpha }\delta _{\mu \nu }p_{\perp }^2,~\\{} & {} \mathcal {F}^{30}_{\mu \nu \rho }(p,k)=f_{\alpha \rho }k_{\perp }^{\alpha }\delta _{\mu \nu }p_{\perp }k_{\perp },~\\{} & {} \mathcal {F}^{31}_{\mu \nu \rho }(p,k)=f_{\alpha \rho }k_{\perp }^{\alpha }\delta _{\mu \nu }k_{\perp }^2,\\ \mathcal {F}^{32}_{\mu \nu \rho }(p,k)= & {} f_{\alpha \rho }k_{\perp }^{\alpha }\delta ^{||}_{\mu \nu }, \end{aligned}$$

where $\delta ^{||}_{\alpha \beta }=\textrm{diag}(0,0,1,1)$ - Kronecker symbol in the longitudinal space. Form factors $F^l(p,k)$ have the following representation

$$\begin{aligned}{} & {} F^l(p,k)= \frac{2\sqrt{B}}{\pi }\sum _f q_f \textrm{Tr}_{\hat{n}} |\hat{n}|^3\\{} & {} \quad \int ^{\infty }_{0}\frac{ds_{1}ds_{2}ds_{3}}{\left( t_1+t_2+t_3\right) } \frac{\xi _1\xi _2\xi _3}{\left( \xi _1+\xi _2+\xi _3+\xi _1\xi _2\xi _3\right) } ~\mathcal {P}^l(s_1,s_2,s_3)\\{} & {} \quad \times \exp \left\{ -p^{2}_{||}\phi _{1}(s_{1},s_{2},s_{3})-p_{||}k_{||}\phi _{2}(s_{1},s_{2},s_{3})-k^{2}_{||}\phi _{3}(s_{1},s_{2},s_{3}) \right. \\{} & {} \quad -p^{2}_{\perp }\phi _{4}(s_{1},s_{2},s_{3})-p_{\perp }k_{\perp }\phi _{5}(s_{1},s_{2},s_{3})\\{} & {} \quad \left. -k^{2}_{\perp }\phi _{6}(s_{1},s_{2},s_{3})-m^{2}_f(s_{1}+s_{2}+s_{3})\right\} , \\{} & {} t_j=B\left| \hat{n}\right| s_j, \ \ \xi _j=\frac{2t_j}{t_j\coth (t_j)+1} \\{} & {} \phi _1=\frac{t_1\left( t_2+t_3\right) }{t_1+t_2+t_3}, ~ \phi _2=\frac{2t_1t_2}{t_1+t_2+t_3},~ \phi _3=\frac{t_2\left( t_1+t_3\right) }{t_1+t_2+t_3},\\{} & {} \phi _4=\frac{\xi _3\left( \xi _1+\xi _2\right) }{\xi _1+\xi _2+\xi _3+\xi _1\xi _2\xi _3}, \ \phi _5= \frac{2\xi _2\xi _3}{\xi _1+\xi _2+\xi _3+\xi _1\xi _2\xi _3}, ~\\{} & {} \quad \phi _6=\frac{\xi _2\left( \xi _1+\xi _3\right) }{\xi _1+\xi _2+\xi _3+\xi _1\xi _2\xi _3}, \end{aligned}$$

where $\mathcal {P}^l(s_1,s_2,s_3)$ - are the rational functions

$$\begin{aligned} \mathcal {P}^1(s_1,s_2,s_3) = \frac{\begin{array}{c} 32\xi _1^2 X \big [ \coth (t_1)(C_{12}+4) +4D_{23} \big ] \\ \big [ X Y_{13}(Y_{23} +\xi _3) + Y_{13}^2Y_{23} + X^2\xi _3 \big ]\end{array} }{Y_{13}\left( 2Y + X\right) ^3 }, \end{aligned}$$

$$\begin{aligned} \mathcal {P}^2(s_1,s_2,s_3){} & {} = \Big ( 16\xi _1 X\left[ \coth (t_1)\left( C_{12}+4\right) +4D_{23} \right] \\{} & {} \quad \left[ X \left( 2\xi _1Y_{23} + \xi _3 \left( Y_{23}+\xi _3 \right) \right) + Y_{13}\left( \xi _1\left( Y_{23}+\xi _2 \right) \right. \right. \\{} & {} \quad \left. \left. + \xi _3Y_{23} \right) \right. \\{} & {} \quad \left. + X^2\xi _3 \right] \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) ,\\ \mathcal {P}^3(s_1,s_2,s_3)\!{} & {} =\! \Big ( 8\xi _1 X \Big [ X \Big (\coth (t_1)\big [2\xi _1(C_{23}\xi _2\!+\!4\xi _2\!+\!\xi _3 ) \\{} & {} \quad +\xi _3(-2C_{23}\xi _2-7\xi _2+2\xi _3 ) \big ] + D_{23} \big [2\xi _1(4\xi _2+\xi _3) \\{} & {} \quad + \xi _3(2\xi _3-7\xi _2) \big ] \Big ) \\{} & {} \quad +Y_{13}\Big (\coth (t_1)\big [\xi _1(2C_{23}\xi _2+8\xi _2+\xi _3) \\{} & {} \quad -Y_{23}(2C_{23}\xi _2+8\xi _2-\xi _3)\big ] \\{} & {} \quad +D_{23}\big [\xi _1(8\xi _2+\xi _3)-8\xi _2^2-7X_{23}+\xi _3^2 \big ] \Big ) \\{} & {} \quad + X^2\xi _3D \Big ] \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3\right) ,\\ \mathcal {P}^4(s_1,s_2,s_3){} & {} = -\Big ( 8 X\xi _2 \Big [ \Big ( \coth (t_1) \big [ -2\xi _1^2(C_{23}+4)\\{} & {} \quad +2X_{12}(C_{23}+4)-X_{13}(2C_{23}\!+\!7)\!+\!\xi _3Y_{23} \big ] \\{} & {} \quad \!-\! D_{23} \big [ 8\xi _1^2\!+\!\xi _1(7\xi _3\!-\!8\xi _2)\!-\!\xi _3Y_{23} \big ] \Big )\! \!\\{} & {} \quad +X \big ( \coth (t_1)[\xi _3-2\xi _1(C_{23}+4)]\\{} & {} \quad -D_{23}[8\xi _1-\xi _3] \big ) \Big ] \Big ) \Big / \left( \left( Y + X\right) ^3 \right) ,\\ \mathcal {P}^5(s_1,s_2,s_3){} & {} = \Big (8\xi _1 X \Big [ X \big (2\coth (t_1)\xi _1[\xi _2(C_{23}+4)\\{} & {} \quad +\xi _3(2C_{23}+7)] \\{} & {} \quad +\coth (t_1)\xi _3[2\xi _3(2C_{23}+7)-\xi _2] + D_{23}(8X_{12}\\{} & {} \quad +14X_{13}-X_{23}+14\xi _3^2) \big ) \\{} & {} \quad +Y_{13} \big (\coth (t_1)\xi _1[2\xi _2(C_{23}+4)+\xi _3(2C_{23}+7)] \\{} & {} \quad -\coth (t_1)Y_{23}[2\xi _2(C_{23}+4)-\xi _3(2C_{23}+7)] \\{} & {} \quad + D_{23}(8X_{12}+7X_{13}-8\xi _2^2-X_{23}+7\xi _3^2) \big ) \\{} & {} \quad +X^2\xi _3[\coth (t_1)(2C_{23}+7)\\{} & {} \quad +7D_{23}] \Big ] \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned} \mathcal {P}^6(s_1,s_2,s_3){} & {} = -\Big ( 8X\xi _2 \big ( \coth (t_1)[-2\xi _1^2(C_{23}+4)\\{} & {} \quad +2X_{12}(C_{23}+4) \\{} & {} \quad + \xi _3(2C_{23}+7)Y_{23}-X_{13}] - D_{23}[8\xi _1^2\\{} & {} \quad +\xi _1(\xi _3-8\xi _2)-7\xi _3Y_{23}] \big ) \\{} & {} \quad + X \big [-2\coth (t_1)\xi _1(C_{23}+4)+\coth (t_1)\\{} & {} \quad \xi _3(2C_{23}+7)\\{} & {} \quad - D_{23}(8\xi _1-7\xi _3) \big ] \Big ) \Big / \left( \left( Y + X\right) ^3\right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^7(s_1,s_2,s_3) = - \frac{16 X\xi _2 \big [\coth (t_1)(C_{23}+4)+4D_{23} \big ] \big [ \xi _1^2(Y_{23}+\xi _2)+\xi _3Y_{23} + X\xi _3 \big ]}{ \left( Y + X\right) ^3 }, \\{} & {} \mathcal {P}^8(s_1,s_2,s_3)= -\frac{ 32 X\xi _2 Y_{13} \big (\coth (t_1)(C_{23}+4)+4D_{23} \big ) }{\left( Y + X\right) ^3 },\\{} & {} \mathcal {P}^9(s_1,s_2,s_3) = - \frac{ \xi _1 \big [\coth (t_1)(C_{23}+4)+4D_{23} \big ] \big [X Y_{13}(Y_{23}+\xi _3) + Y_{13}^2Y_{23} + X^2\xi _3 \big ] }{ Y_{13}^2\left( Y + X\right) ^3}, \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{10}(s_1,s_2,s_3)= -\Big ( 8X\xi _1\big (Y_{13}+X\big ) \big [\big (\coth (t_1)[\xi _1(2C_{23}\xi _2\\{} & {} \quad + 8\xi _2+\xi _3)+\xi _3Y_{23}] \\{} & {} \quad + D_{23}[\xi _1(8\xi _2+\xi _3)+\xi _3Y_{23}] \big ) + X\xi _3Y \big ] \Big ) \Big / \left( Y_{13} \left( Y + X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{11}(s_1,s_2,s_3)= \\{} & {} \quad - \Big ( 8X\xi _2 \big [ \big ( \coth (t_1) [\xi _1^2(C_{23}+4)+X_{13}(2C_{23}+7)\\{} & {} \quad +\xi _3(C_{23}+3)Y_{23}] + D_{23}[4\xi _1^2+7X_{13}+3\xi _3Y_{23}] \big ) \\{} & {} \quad +X \big ( \coth (t_1)\xi _1(C_{23}+4)+\coth (t_1)\xi _3(C_{23}+3)\\{} & {} \quad +D_{23}(4\xi _1+3\xi _3) \big ) \big ] \Big ) \Big / \left( \left( Y + X\right) ^3 \right) , \\{} & {} \mathcal {P}^{12}(s_1,s_2,s_3)= \Big ( 4X \big [ Y_{13}[\coth (t_1)(\xi _1(4C_{23}+15)\\{} & {} \quad -\xi _2-\xi _3 +D_{23}(15\xi _1-\xi _2-\xi _3)] \\{} & {} \quad - X [\coth (t_1)(\xi _3-\xi _1(4C_{23}+15)) \\{} & {} \quad -D_{23}(15\xi _1-\xi _3)] \big ] \Big ) \Big / \left( Y_{13}\left( Y +X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned} \mathcal {P}^{13}(s_1,s_2,s_3) = -\frac{\begin{array}{c} 8X\xi _1 \big [ \coth (t_1)(C_{23}+4)+4D_{23} \big ]\\ \big [ XY_{13}(Y_{23}+\xi _3) + ^4Y_{13}^2Y_{23}+^4X\xi _3 \big ] \end{array}}{Y_{13}\left( Y + X\right) ^3 }, \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{14}(s_1,s_2,s_3)= -\Big ( 8 X\xi _1 \big (Y_{13}+X \big ) \big (\coth (t_1)[2X_{12}(C_{23}+4)\\{} & {} \quad +X_{13}(2C_{23}+7)+\xi _3(2C_{23}+7)Y_{23}]\\{} & {} \quad + D_{23}[\xi _1(8\xi _2+7\xi _3)+7\xi _3Y_{23}] + X\xi _3[\coth (t_1)(2C_{23}+7)\\{} & {} \quad +7D_{23}] \big ) \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) ,\\{} & {} \mathcal {P}^{15}(s_1,s_2,s_3) \\{} & {} \quad = - \Big ( 8X\xi _2 \big [ \big ( \coth (t_1) [\xi _1^2(C_{23}+4)-\xi _3(C_{23}+3)Y_{23}\\{} & {} \quad +X_{13}] + D_{23}[4\xi _1^2+X_{13}-3\xi _3Y_{23}] \big ) \\{} & {} \quad +X \big ( \coth (t_1)\xi _1(C_{23}+4)-\coth (t_1)\xi _3(C_{23}+3)\\{} & {} \quad +D_{23}(4\xi _1-3\xi _3) \big ) \big ] \Big ) \Big / \left( \left( Y + X\right) ^3 \right) ,\\{} & {} \mathcal {P}^{16}(s_1,s_2,s_3)\!=\! \Big ( 4X \big [ Y_{13}[\coth (t_1)(\xi _1(2C_{23}\!+\!9) \\{} & {} \quad -(2C_{23}+7)Y_{23})+D_{23}(9\xi _1-7Y_{23})] \\{} & {} \quad - X [-\coth (t_1)\xi _1(2C_{23}+9)+\coth (t_1)\xi _3(2C_{23}+7) \\{} & {} \quad - D_{23}(9\xi _1-7\xi _3)] \big ] \Big ) \Big / \left( Y_{13}\left( Y +X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{17}(s_1,s_2,s_3) = -\frac{ 8X\xi _1\big (4D_{23} + \coth (t_1)(4 + C_{23})\big )\big (X^2\xi _1 + Y_{13}^2Y_{23} + XY_{13}(Y_{23} + \xi _3) \big ) }{BY_{13}^2(X + Y)^3},\\{} & {} \mathcal {P}^{18}(s_1,s_2,s_3) = - \frac{16X\xi _2 \big [ \coth (t_1)(C_{23}+4)+4D_{23} \big ] \big [Y_{13}Y_{23} +X\xi _3 \big ]}{Y_{13}\left( Y + X\right) ^3 },\\{} & {} \mathcal {P}^{19}(s_1,s_2,s_3)= \frac{ 8X\xi _2 \big (\coth (t_1)(C_{23}+4)+4D_{23} \big ) \big (\xi _1^2+2\xi _1Y_{23}+\xi _3Y_{23} + XY_{13} \big ) }{\left( Y + X\right) ^3 },\\{} & {} \mathcal {P}^{20}(s_1,s_2,s_3) = \frac{8 \big (\coth (t_1) (C_{23}+4)+4D_{23}\big ) \big (Y_{13}(\xi _1-Y_{23})+ X(\xi _1-\xi _3)\big )}{(Y_{13} \left( Y + X\right) ^2)}, \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{21}(s_1,s_2,s_3)\\{} & {} \quad = -\Big ( 8X_{12}\xi _1 \big [X\xi _3\big (\coth (t_1)(2\xi _1(C_{23}+3)+2C_{23}\xi _3\\{} & {} \qquad -\xi _2+6\xi _3)+ D_{23}(6\xi _1-\xi _2+6\xi _3)\big ) \\{} & {} \qquad +\xi _3Y_{13}\big (\coth (t_1)(\xi _3(C_{23}+3)Y_{13}-\xi _2^2(C_{23}+4)-X_{23})\\{} & {} \qquad - D_{23}(-3\xi _3Y_{13}+4\xi _2^2+X_{23})\big ) \\{} & {} \qquad + X^2\xi _3^2(\coth (t_1)(C_{23}+3)+3D_{23}) \big ] \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{22}(s_1,s_2,s_3)= \Big ( 8 X\xi _2 \big ( \coth (t_1)[2X_{12}(C_{23}+4)\\{} & {} \quad +X_{13}(2C_{23}+7)+\xi _3(2C_{23}+7)Y_{23}]\\{} & {} \quad + D_{23}[\xi _1(8\xi _2+7\xi _3)+7\xi _3Y_{23}] + X\xi _3[\coth (t_1)(2C_{23}+7)\\{} & {} \quad +7D_{23}] \big ) \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{23}(s_1,s_2,s_3)=-\frac{1}{4}\mathcal {P}^{8}(s_1,s_2,s_3),\\{} & {} \mathcal {P}^{24}(s_1,s_2,s_3)= \Big ( 4 X\big [\coth (t_1)(2C_{23}\xi _1-2C_{23}\xi _2+2C_{23}\xi _3\\{} & {} \quad +7\xi _1-9\xi _2+7\xi _3) + D_{23}(7\xi _1-9\xi _2+7\xi _3) \\{} & {} \quad +X(\coth (t_1)(2C_{23}+7)+7D_{23}) \big ] \Big ) \Big / \left( Y +X\right) ^2,\\{} & {} \mathcal {P}^{25}(s_1,s_2,s_3)= -\Big ( 8X_{12}\xi _1 \big [X\xi _3\big (\coth (t_1)(2\xi _1(C_{23}+3)\\{} & {} \quad +2C_{23}\xi _2+2C_{23}\xi _3+7\xi _2+6\xi _3)\\{} & {} \quad +D_{23}(6\xi _1+7\xi _2+6\xi _3)\big )+ \xi _3Y_{13}\big (\coth (t_1)(\xi _3(C_{23}+3)Y_{13}\\{} & {} \quad +\xi _2^2(C_{23}+4)-X_{23}(2C_{23}+7)) \\{} & {} \quad +D_{23}(3\xi _3Y_{13}+4\xi _2^2+ 7X_{23})\big )+ X^2\xi _3^2(\coth (t_1)(C_{23}+3)\\{} & {} \quad +3D_{23}) \big ] \Big ) \Big / \left( Y_{13}\left( Y + X\right) ^3 \right) , \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{26}(s_1,s_2,s_3) = \frac{\begin{array}{c} 8X \big (\coth (t_1)[\xi _1(2C_{23}\xi _2+8\xi _2+\xi _3)+\xi _3Y_{23}]\\ + D_{23}[\xi _1(8\xi _2+\xi _3)+\xi _3Y_{23}]+X\xi _3D \big )\end{array} }{ \left( Y + X\right) ^3 }, \\{} & {} \mathcal {P}^{27}(s_1,s_2,s_3)=-\frac{1}{4}\mathcal {P}^{8}(s_1,s_2,s_3), \\ \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{28}(s_1,s_2,s_3)\\{} & {} \quad = \frac{\begin{array}{c}4X \big (\coth (t_1)[-4C_{23}\xi _2+\xi _1-15\xi _2+\xi _3]\\ +D_{23}[Y_{13}-15\xi _2] + XD \big )\end{array}}{\left( Y + X\right) ^3 }, \\{} & {} \mathcal {P}^{29}(s_1,s_2,s_3)\\{} & {} \quad = - \frac{\begin{array}{c} \xi _1 \big [\coth (t_1)(C_{23}+4)+4D_{23} \big ]\\ \big [X Y_{13}Y_{23} +Y_{13}(\xi _1(Y_{23}+\xi _2)+Y_{23}^2) + X^2\xi _3 \big ]\end{array} }{Y_{13}^2\left( Y + X\right) ^3}, \end{aligned}$$

$$\begin{aligned}{} & {} \mathcal {P}^{30}(s_1,s_2,s_3)\\{} & {} \quad = -\frac{16 X\xi _1\xi _2 [\coth (t_1)(C_{23}+4)+4D_{23}][Y_{13}+D]}{\left( Y + X\right) ^3 },\\{} & {} \mathcal {P}^{31}(s_1,s_2,s_3)=-\frac{1}{4}\mathcal {P}^{8}(s_1,s_2,s_3),\\{} & {} \mathcal {P}^{32}(s_1,s_2,s_3)\\{} & {} \quad = \frac{8 X \big (\coth (t_1)(C_{23}+4)+4D_{23}\big ) \big (Y_{13}-\xi _2 +X\big ) }{\left( Y + X\right) ^3 }, \end{aligned}$$

where the following notations is used

$$\begin{aligned} X= & {} \xi _1\xi _2\xi _3,~ X_{ij}=\xi _i\xi _j,~ Y=\xi _1+\xi _2+\xi _3,~ Y_{ij}=\xi _i+\xi _j, \\ C= & {} \coth (t_1)\coth (t_2)\coth (t_3),~ C_{ij}=\coth (t_i)\coth (t_j), \\ D= & {} \coth (t_1)+\coth (t_2)+\coth (t_3),~ D_{ij}=\coth (t_i)+\coth (t_j). \end{aligned}$$