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--> --> -->In previous theoretical investigations of the hidden-charm four-quark states with strangeness, the compact tetraquark configuration
A hadronic molecule is composed of two color-singlet hadrons by exchanging light mesons. This is a very useful configuration to study the nature of some exotic XYZ states and pentaquark states [14-19]. Because
The paper is organized as follows. In Sec. II, we construct the interpolating currents for the
$\begin{aligned}[b](\mathbf{3} \!\otimes\! \overline{\mathbf{3}})_{[q \bar{Q}]} \!\otimes\!(\mathbf{3} \!\otimes\! \overline{\mathbf{3}})_{[Q \bar{q}]} & = \!(\mathbf{1} \!\oplus\! \mathbf{8})_{[q \bar{Q}]} \!\otimes\!(\mathbf{1} \!\oplus\! \mathbf{8})_{[Q \bar{q}]} \\&=\! (\mathbf{1} \!\otimes\! \mathbf{1}) \!\oplus\!(\mathbf{1} \!\otimes\! \mathbf{8}) \!\oplus\!(\mathbf{8} \!\otimes\! \mathbf{1}) \!\oplus\!(\mathbf{8} \!\otimes\! \mathbf{8}) \\ &=\! \mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{8} \!\oplus\!(\mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{1 0} \!\oplus\! \overline{\mathbf{1 0}} \!\oplus\! \mathbf{27})\, ,\\(\mathbf{3} \!\otimes\! {\mathbf{3}})_{[q Q]} \!\otimes\!(\overline{\mathbf{3}} \!\otimes\! \overline{\mathbf{3}})_{[\bar Q \bar{q}]} &=\! (\mathbf{6} \!\oplus\! \overline{\mathbf{3}})_{[q Q]} \!\otimes\!(\mathbf{3} \!\oplus\! \overline{\mathbf{6}})_{[\bar Q \bar{q}]} \\ &=\! (\mathbf{6} \!\otimes\! \overline{\mathbf{6}}) \!\oplus\!(\overline{\mathbf{3}} \!\otimes\! \mathbf{3}) \!\oplus\! (\mathbf{6} \!\otimes\! \mathbf{3}) \!\oplus\!(\overline{\mathbf{3}} \!\otimes\! \overline{\mathbf{6}}) \\ &=\! (\mathbf{1} \!\oplus\! \mathbf{8} \!\oplus\! \mathbf{27}) \!\oplus\!(\mathbf{1} \!\oplus \!\mathbf{8}) \!\oplus\! (\mathbf{8}\! \oplus\! \mathbf{1 0}) \!\oplus\! (\mathbf{8} \!\oplus\! \overline{\mathbf{1 0}})\, ,\end{aligned} $ ![]() | (1) |
$ \begin{aligned}[b]&J_{1} = (\bar{c}_{a} \gamma_{5} s_{a})(\bar{q}_{b} \gamma_{5} c_{b} )\, , \qquad\quad J^P = 0^+\, , \\&J_{2} = (\bar{c}_{a} \gamma_{\mu} s_{a})(\bar{q}_{b} \gamma^{\mu} c_{b} )\, , \qquad\quad J^P = 0^+\, ,\\&J_{1\mu} = (\bar{c}_{a} \gamma_{\mu} s_{a})(\bar{q}_{b}\gamma_{5} c_{b} )\, ,\qquad\;\; J^P = 1^+\, ,\\&J_{2\mu} = (\bar{c}_{a} \gamma_{5} s_{a})(\bar{q}_{b}\gamma_{\mu} c_{b} )\, ,\qquad\;\; J^P = 1^+ \, ,\\&J_{3\mu} = (\bar{c}_{a} \gamma^{\alpha} s_{a})( \bar{q}_{b}\sigma_{\alpha\mu} \gamma_{5}c_{b} )\, ,\quad J^P = 1^+\, ,\\&J_{4\mu} = (\bar{c}_{a} \sigma_{\alpha\mu}\gamma_{5} s_{a})( \bar{q}_{b} \gamma^{\alpha} c_{b} )\, , \quad J^P = 1^+\, ,\\&J_{\mu\nu} = (\bar{c}_{a} \gamma_{\mu} s_{a})( \bar{q}_{b}\gamma_{\nu}c_{b} )\, ,\qquad\;\; J^P = 2^+\, , \end{aligned}$ ![]() | (2) |
$ \begin{aligned}[b]&\eta_{1} = s_{a}^{T} C \gamma_{5} c_{b}\left(\bar{q}_{a} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma_{5} C \bar{c}_{a}^{T}\right)\, , \qquad\qquad J^P = 0^+\, , \\ &\eta_{2} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma^{\mu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\mu} C \bar{c}_{a}^{T}\right)\, , \qquad\quad\;\;\;\; J^P = 0^+\, ,\\&\eta_{1\mu} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma_{5} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 1^+\, ,\\&\eta_{2\mu} = s_{a}^{T} C \gamma_{5} c_{b}\left(\bar{q}_{a} \gamma^{\mu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\mu} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 1^+ \, ,\\ &\eta_{3\mu} = s_{a}^{T} C \gamma^{\alpha} c_{b}\left(\bar{q}_{a} \sigma_{\alpha\mu} \gamma_{5} C \bar{c}_{b}^{T}-\bar{q}_{b} \sigma_{\alpha\mu} \gamma_{5} C \bar{c}_{a}^{T}\right)\, ,\,\, \; J^P = 1^+\, ,\\&\eta_{4\mu} = s_{a}^{T} C \sigma_{\alpha\mu}\gamma_{5} c_{b}\left(\bar{q}_{a} \gamma^{\alpha} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\alpha} C \bar{c}_{a}^{T}\right)\, ,\quad\;\;\;\, J^P = 1^+ \, ,\\ &\eta_{\mu\nu} = s_{a}^{T} C \gamma_{\mu} c_{b}\left(\bar{q}_{a} \gamma^{\nu} C \bar{c}_{b}^{T}-\bar{q}_{b} \gamma^{\nu} C \bar{c}_{a}^{T}\right)\, ,\qquad\quad\;\;\; J^P = 2^+\, , \end{aligned} $ ![]() | (3) |
$ \begin{array}{l} \Pi\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J(x) J^{\dagger}(0)\right]\right| 0\right\rangle\, , \end{array} $ ![]() | (4) |
$ \begin{array}{l} \Pi_{\mu \nu}\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{\mu}(x) J_{\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array} $ ![]() | (5) |
$ \Pi_{\mu \nu}\left(p^{2}\right) = \left(\frac{p_{\mu} p_{\nu}}{p^{2}}-g_{\mu \nu}\right) \Pi_{1}\left(p^{2}\right)+\frac{p_{\mu} p_{\nu}}{p^{2}}\Pi_{0}\left(p^{2}\right)\, , $ ![]() | (6) |
$ \begin{array}{l} \Pi_{\mu \nu,\;\rho \sigma}\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{\mu\nu}(x) J_{\rho\sigma}^{\dagger}(0)\right]\right| 0\right\rangle\, , \end{array} $ ![]() | (7) |
$\Pi_{\mu \nu,\;\rho\sigma} \left(p^{2}\right) = \left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\right) \Pi_{2}\left(p^{2}\right)+\cdots \, , $ ![]() | (8) |
$\eta_{\mu\nu} = \frac{p_{\mu} p_{\nu}}{p^{2}}-g_{\mu \nu}, $ ![]() | (9) |
At the hadronic level, the correlation function can be described via the dispersion relation
$ \Pi\left(p^{2}\right) = \frac{\left(p^{2}\right)^{N}}{\pi} \int_{4m_{c}^{2}}^{\infty} \frac{{\rm{Im}} \Pi(s)}{s^{N}\left(s-p^{2}-{\rm i} \epsilon\right)} {\rm d} s+\sum\limits_{n = 0}^{N-1} b_{n}\left(p^{2}\right)^{n}\, , $ ![]() | (10) |
$\begin{aligned}[b] \rho (s) =& \frac{1}{\pi} \text{Im}\Pi(s) = f_{H}^{2}\delta(s-m_{H}^{2})\\&+\text{QCD continuum and higher states}\, , \end{aligned}$ ![]() | (11) |
$ \begin{aligned}[b] &\left\langle 0|J| H\right\rangle = f_{H}\, , \\& \left\langle 0\left|J_{\mu}\right| H\right\rangle = f_{H} \epsilon_{\mu}\, , \\& \left\langle 0\left|J_{\mu\nu}\right| H\right\rangle = f_{H} \epsilon_{\mu\nu} \end{aligned} $ ![]() | (12) |
We can calculate the correlation function
$ \begin{aligned}[b] {\rm i} S_{q}^{a b}(x) =& \frac{{\rm i} \delta^{a b}}{2 \pi^{2} x^{4}} \hat{x} +\frac{{\rm i}}{32 \pi^{2}} \frac{\lambda_{a b}^{n}}{2} g_{s} G_{\mu \nu}^{n} \frac{1}{x^{2}}\left(\sigma^{\mu \nu} \hat{x}+\hat{x} \sigma^{\mu \nu}\right) \\&-\frac{\delta^{a b} x^{2}}{12}\left\langle\bar{q} g_{s} \sigma \cdot G q\right\rangle -\frac{m_{q} \delta^{a b}}{4 \pi^{2} x^{2}} \\ &+\frac{{\rm i} \delta^{a b} m_{q}(\bar{q} q)}{48} \hat{x} -\frac{{\rm i} m_{q}\left\langle\bar{q} g_{s} \sigma \cdot G q\right) \delta^{a b} x^{2} \hat{x}}{1152}\, , \\ {\rm i} S_{Q}^{a b}(p) = & \frac{{\rm i} \delta^{a b}}{\hat{p}-m_{Q}} +\frac{{\rm i}}{4} g_{s} \frac{\lambda_{a b}^{n}}{2} G_{\mu \nu}^{n} \frac{\sigma^{\mu \nu}\left(\hat{p}+m_{Q}\right)+\left(\hat{p}+m_{Q}\right) \sigma^{\mu \nu}}{12} \\&+\frac{{\rm i} \delta^{a b}}{12}\left\langle g_{s}^{2} G G\right\rangle m_{Q} \frac{p^{2}+m_{Q} \hat{p}}{(p^{2}-m_{Q}^{2})^{4}}\, , \end{aligned} $ ![]() | (13) |
$ {\cal{L}}_{k}\left(s_{0}, M_{\rm B}^{2}\right) = f_{H}^{2} m_{H}^{2 k} {\rm e}^{-m_{H}^{2} / M_{\rm B}^{2}} = \int_{4m_{c}^{2}}^{s_{0}} {\rm d} s {\rm e}^{-s / M_{\rm B}^{2}} \rho(s) s^{k}\, , $ ![]() | (14) |
$ \begin{array}{l} m_{H}\left(s_{0}, M_{\rm B}^{2}\right) = \sqrt{\frac{{\cal{L}}_{1}\left(s_{0}, M_{\rm B}^{2}\right)}{{\cal{L}}_{0}\left(s_{0}, M_{\rm B}^{2}\right)}}\, , \end{array} $ ![]() | (15) |
$ \begin{aligned}[b]m_{u}(2 \;{\rm{GeV}}) = &\, (2.2_{-0.4}^{+0.5} ) \;{\rm{MeV}}\ , \\ m_{d}(2 \;{\rm{GeV}}) = &\, (4.7_{-0.3}^{+0.5}) \;{\rm{MeV}}\, ,\\ m_{q}(2\; {\rm{GeV}}) = &\, (3.5_{-0.2}^{+0.5}) \;{\rm{MeV}}\, ,\\ m_{s}(2 \;{\rm{GeV}}) = &\, (95_{-3}^{+9}) \;{\rm{MeV}}\, ,\\m_{c}\left(m_{c}\right) = &\, (1.275 _{-0.035}^{+0.025}) \;{\rm{GeV}}\, , \\m_{b}\left(m_{b}\right) = &\, (4.18 _{-0.03}^{+0.04}) \;{\rm{GeV}}\, , \\\langle\bar{q} q\rangle = &\, -(0.24 \pm 0.03)^{3} \;{\rm{GeV}}^{3}\, , \\\left\langle\bar{q} g_{s} \sigma \cdot G q\right\rangle = &\, - M_{0}^{2}\langle\bar{q} q\rangle\, ,\\ M_{0}^{2} = &\, (0.8 \pm 0.2) \;{\rm{GeV}}^{2}\, , \\\langle\bar{s} s\rangle /\langle\bar{q} q\rangle = &\, 0.8 \pm 0.1\, , \\\left\langle g_{s}^{2} G G\right\rangle = &\, (0.48\pm0.14) \;{\rm{GeV}}^{4}\, , \end{aligned} $ ![]() | (16) |
To establish a stable mass sum rule, one should find appropriate parameter working regions first, i.e, the continuum threshold
$ {\rm{PC}}\left(s_{0}, M_{\rm B}^{2}\right) = \frac{{\cal{L}}_{0}\left(s_{0}, M_{\rm B}^{2}\right)}{{\cal{L}}_{0}\left(\infty, M_{\rm B}^{2}\right)}\, , $ ![]() | (17) |
We use the
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2021/9/PIC/CPC-2021-0215-1.jpg)
$ {\frac{\Pi^{\langle\bar{q}q\rangle \langle\bar{q}g_{s}\sigma\cdot G q\rangle}(M_{\rm B}^{2},\infty)}{\Pi(M_{\rm B}^{2},\infty)}<10\% } \, , $ ![]() | (18) |
As mentioned above, the variation of the output hadron mass
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2021/9/PIC/CPC-2021-0215-2.jpg)
$ m_{\bar{D}_s^\ast D^\ast, \, 0^+} = 4.11\pm0.14\; \text{GeV}\, , $ ![]() | (19) |
System | Current | ![]() ![]() | ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() | ![]() ![]() | PC (%) |
![]() ![]() | ![]() ![]() | ![]() ![]() | 18.0 ± 2.0 | 1.6 ~ 3.6 | 3.74 ± 0.13 | 52.5 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 20.5 ± 2.0 | 2.6 ~ 3.4 | 4.11 ± 0.14 | 42.4 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 20.7 ± 2.0 | 2.1 ~ 2.5 | 3.99 ± 0.12 | 68.2 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 20.5 ± 2.0 | 2.1 ~ 2.5 | 3.97 ± 0.11 | 67.7 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 21.5 ± 2.0 | 2.8 ~ 3.6 | 4.22 ± 0.14 | 40.1 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 21.5 ± 2.0 | 2.8 ~ 3.6 | 4.22 ± 0.14 | 40.0 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 23.0 ± 2.0 | 2.8 ~ 4.3 | 4.34 ± 0.13 | 48.7 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 18.0 ± 2.0 | 2.1 ~ 3.1 | 3.84 ± 0.15 | 46.3 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 20.0 ± 2.0 | 2.6 ~ 3.2 | 4.13 ± 0.17 | 35.6 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 19.0 ± 2.0 | 2.5 ~ 3.3 | 3.98 ± 0.16 | 41.0 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 19.0 ± 2.0 | 2.5 ~ 3.3 | 3.97 ± 0.15 | 41.6 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 22.0 ± 2.0 | 2.9 ~ 3.6 | 4.28 ± 0.14 | 40.9 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 22.0 ± 2.0 | 2.9 ~ 3.6 | 4.28 ± 0.14 | 41.1 |
![]() ![]() | ![]() ![]() | ![]() ![]() | 23.0 ± 2.0 | 2.8 ~ 4.3 | 4.33 ± 0.13 | 46.4 |
Table1.Numerical results for the
In Table 1, the mass of the scalar
The masses obtained from the axial-vector molecular currents
$ \begin{array}{l} \Pi_{12\mu \nu}^M\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{1\mu}(x) J_{2\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array} $ ![]() | (20) |
We also study the
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2021/9/PIC/CPC-2021-0215-3.jpg)
$ \begin{array}{l} \Pi_{12\mu \nu}^T\left(p^{2}\right) = {\rm i} \int {\rm d}^{4} x {\rm e}^{{\rm i} p \cdot x}\left\langle 0\left|T\left[J_{1\mu}(x) J_{2\nu}^{\dagger}(0)\right]\right| 0\right\rangle\, . \end{array} $ ![]() | (21) |
One may wonder if the two-meson scattering states can contribute to the correlation functions in our calculations. In general, the interpolating currents can couple to all structures with the same quantum numbers, including resonances, two-meson scattering states, and continuum. Thus, these structures will give contributions to the correlation functions. However, it has been demonstrated that the two-meson scattering states cannot saturate the QCD sum rules, while only exotic four-quark states can saturate the QCD sum rules. Moreover, the contributions from the two-meson scattering states to the correlation functions are numerically negligible [43, 55].
Our results show that the masses of the axial-vector
Note added: Since we finished this work, the LHCb Collaboration has reported two new charged resonances,
According to the observation of the LHCb, the decay width of
$ \rho(s) = \rho^{0}(s)+\rho^{3}(s)+\rho^{4}(s)+\rho^{5}(s)+\rho^{6}(s)+\rho^{8}(s)\, , \tag{A1}$ ![]() | (22) |
1. Spectral densities for
![]() |
$ \rho_{J_{1}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, , $ ![]() |
$ \rho_{J_{1}}^{3a}(s) \!=\! - \frac{3\langle\bar{s}s\rangle}{128\pi^{4} } \!\!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \!\!\int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{2m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha\beta}\right]\, ,$ ![]() |
$ \rho_{J_{1}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{128\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{2m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,$ ![]() |
$ \rho_{J_{1}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (2m_{c}^{2}(\alpha+ \beta)-3 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$ \rho_{J_{1}}^{4b}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)(m_{c}^{2}(\alpha+ \beta)-2\alpha \beta s)\left(\frac{1}{\alpha^{2}\beta}+\frac{1}{\alpha\beta^{2}}\right)\, , $ ![]() |
$ \rho_{J_{1}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta) \left(\frac{1}{\beta}-\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)+\frac{2m_{c}m_{q}}{\beta}\right]\, , $ ![]() |
$ \rho_{J_{1}}^{5b}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta) \left(\frac{1}{\alpha}-\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)+\frac{2m_{c}m_{s}}{\alpha}\right]\, ,$ ![]() |
$ \rho_{J_{1}}^{5c}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{512 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-6 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{512 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-6 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{1}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{32 \pi^{2}}(2m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{1}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{1}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}-\frac{ 2\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{\alpha m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \alpha_{\rm min } = \frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{4 m_{c}^{2}}{s}},\; \; \; \alpha_{\rm max } = \frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{4 m_{c}^{2}}{s}},\; \; \; \beta_{\rm min} = \frac{\alpha m_{c}^{2}}{\alpha s-m_{c}^{2}},\; \; \; \; \beta_{\rm max} = 1-\alpha \, , $ ![]() |
$ \rho_{J_{2}}^{0a}(s) = \frac{3} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\, , $ ![]() |
$ \rho_{J_{2}}^{0b}(s) = - \frac{3m_{c}} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, , $ ![]() |
$ \rho_{J_{2}}^{3a}(s) = - \frac{3\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)+2\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{2}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)+2\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{2}}^{4}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (2m_{c}^{2}(\alpha+ \beta)-3 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$ \rho_{J_{2}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta}{\beta} +\frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2 m_{c}^{2}(\alpha+\beta)-3 s \alpha \beta }{\alpha}\, , $ ![]() |
$ \rho_{J_{2}}^{5b}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{128 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-6 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{128 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-6 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{2}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{2}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{16 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{2}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{16 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{1\mu}}^{0a}(s) = \frac{3} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$ \rho_{J_{1\mu}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}+\frac{(m_{c}^{2}(\alpha+\beta)-4\alpha \beta s)m_{q}}{\alpha^{2} \beta^{3}}\right)\, , $ ![]() |
$ \rho_{J_{1\mu}}^{3a}(s) = - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha^{2} \beta} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{1\mu}}^{3b}(s) = - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}} -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{1\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\Big(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\Big)\, , $ ![]() |
$ \rho_{J_{1\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha\beta^{2}} -\frac{(3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s)}{\alpha^{2}\beta}\right)\, , $ ![]() |
$ \rho_{J_{1\mu}}^{5a}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\alpha}\, , $ ![]() |
$ \rho_{J_{1\mu}}^{5b}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\beta}-\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)+\frac{2m_{c}m_{s}}{\beta}\right]\, , $ ![]() |
$ \rho_{J_{1\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{1\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{1\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{1\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{2\mu}}^{0a}(s) = \frac{3} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$ \rho_{J_{2\mu}}^{0b}(s) = - \frac{3m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \Big(\frac{(2m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)m_{q}}{\alpha^{2} \beta^{3}} +\frac{(m_{c}^{2}(\alpha+\beta)-4\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}\Big)\, , $ ![]() |
$ \rho_{J_{2\mu}}^{3a}(s) = - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{c}} { \alpha\beta^{2} } -\frac{4m_{c}^{2}m_{s}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{2\mu}}^{3b}(s) = - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{2(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha ^{2} \beta} -\frac{4m_{c}^{2}m_{q}+(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{2\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$ \rho_{J_{2\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2} \beta} -\frac{(3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s)}{\alpha\beta^{2} }\right)\, , $ ![]() |
$ \rho_{J_{2\mu}}^{5a}(s) = \frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s} {\beta}\, , $ ![]() |
$ \rho_{J_{2\mu}}^{5b}(s) = \frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s ) \left(\frac{1}{\alpha}-\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)+\frac{2m_{c}m_{q}}{\alpha}\right]\, , $ ![]() |
$ \rho_{J_{2\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{768\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{768 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{2\mu}}^{6a}(s) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+2m_{c}m_{s}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{2\mu}}^{6b}\left(M_{\rm B}^{2}\right) = -\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{2\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} -\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{3\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$\rho_{J_{3\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, , $ ![]() |
$ \rho_{J_{3\mu}}^{3a}(s) = \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{3\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \!\!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}}\! {\rm d} \alpha \!\!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1\!-\!\alpha-\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta}\!-\!\frac{12m_{c}^{2}m_{s}\! +\! 3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{3\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$\rho_{J_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha\beta^{2}} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha^{2}\beta}\right)\, , $ ![]() |
$ \rho_{J_{3\mu}}^{5a}(s) = -\frac{3 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \alpha \, , $ ![]() |
$ \rho_{J_{3\mu}}^{5b}(s) = \frac{ \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\alpha}+\frac{2(1-\alpha-\beta)}{\alpha^{2}}\right)-\frac{6m_{c}m_{s}}{\alpha}\right]\, , $ ![]() |
$ \rho_{J_{3\mu}}^{5c}(s) = \frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{3\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\left(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\right) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{4\mu}}^{0a}(s) = \frac{9} {4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$ \rho_{J_{4\mu}}^{0b}(s) = - \frac{9m_{c}} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{2} \beta^{3}}-\frac{ \alpha \beta sm_{q}}{\alpha^{3} \beta^{2}}\right)\, , $ ![]() |
$ \rho_{J_{4\mu}}^{3a}(s) = \frac{3\langle\bar{q}q\rangle}{256\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \alpha}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{4\mu}}^{3b}(s) \!=\! - \frac{3\langle\bar{s}s\rangle}{256\pi^{4} } \!\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} \!{\rm d} \alpha \!\int_{\beta_{\rm min}}^{\beta_{\rm max}} \!{\rm d} \beta (m_{c}^{2}(\alpha\!+\! \beta)\! -\! \alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\! +\! \beta)\! -\! 5\alpha \beta s)m_{c}} { \alpha \beta^{2}} \! -\! \frac{12m_{c}^{2}m_{q}\! +\! 3(m_{c}^{2}(\alpha\! +\! \beta)\! -\! 3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{J_{4\mu}}^{4a}(s) = \frac{3\langle g_{s}^{2} G G\rangle m_{c}^{2} }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$\rho_{J_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{4096 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta) (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{3m_{c}^{2}(\alpha+ \beta)-5\alpha \beta s}{\alpha^{2}\beta} -\frac{3(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)}{\alpha\beta^{2}}\right)\, , $ ![]() |
$ \rho_{J_{4\mu}}^{5a}(s) = -\frac{3 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta s \beta \, , $ ![]() |
$ \rho_{J_{4\mu}}^{5b}(s) = \frac{ \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{256 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left[(3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{3}{\beta}+\frac{2(1-\alpha-\beta)}{\beta^{2}}\right)-\frac{6m_{c}m_{q}}{\beta}\right]\, , $ ![]() |
$ \rho_{J_{4\mu}}^{5c}(s) = \frac{\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}+\frac{\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{4\mu}}^{6a}(s) = \frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{64 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{64 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 3(\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2 \langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{\mu\nu}}^{0a}(s) = -\frac{5} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\, $ ![]() |
$ \rho_{J_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {512 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, , $ ![]() |
$ \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3a}(s) = &- \frac{15\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}} \right. \\&\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, ,\end{aligned} $ ![]() |
$ \begin{aligned}[b] \rho_{J_{\mu\nu}}^{3b}(s) = & - \frac{15\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left. -\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned} $ ![]() |
$ \rho_{J_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, , $ ![]() |
$ \begin{aligned}[b] \rho_{J_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{2048 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left(\frac{1}{\alpha^{2}\beta}+\frac{1}{\alpha\beta^{2}}\right)\\ &\times\left((1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)-4(m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\right)\, , \end{aligned} $ ![]() |
$ \begin{aligned}[b] \rho_{J_{\mu\nu}}^{5a}(s) = &\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{128 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned} $ ![]() |
$ \rho_{J_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{256 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{J_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{32 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{J_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{16 \pi^{2}} \int_{0}^{1} {\rm{d}} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{J_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{32 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}} {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{s}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{q}}{\alpha^{2} \beta^{3}}\right)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{3a}(s) = \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{s}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, ,$ ![]() |
$ \rho_{\eta_{3\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\!+\!\beta)\!-\!\alpha \beta s)\left[\frac{2(1-\alpha\!-\!\beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{q}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{5a}(s) = \frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{5b}(s) = -\frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{\eta_{3\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{\eta_{3\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{\eta_{3\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+2\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{J_{4\mu}}^{0a}(s) = \frac{3} {1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}(m_{c}^{2}(\alpha+\beta)-5 \alpha \beta s)\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{0b}(s) = - \frac{3m_{c}} {256 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2} \left(\frac{(m_{c}^{2}(\alpha+\beta)-2\alpha \beta s)m_{q}}{\alpha^{3} \beta^{2}}-\frac{ \alpha \beta sm_{s}}{\alpha^{2} \beta^{3}}\right)\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{3a}(s) = \frac{\langle\bar{s}s\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{4(1-\alpha-\beta) s m_{c}} { \beta}+\frac{12m_{c}^{2}m_{q}+3(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{s}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{3b}(s) \!=\! - \frac{\langle\bar{q}q\rangle}{64\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha\! + \! \beta)\! - \!\alpha \beta s)\left[\frac{2(1\! -\! \alpha\! -\! \beta)(3m_{c}^{2}(\alpha\!+\!\beta)\!-\!5\alpha \beta s)m_{c}} { \alpha^{2} \beta} \!-\!\frac{12m_{c}^{2}m_{s}\!+\!3(m_{c}^{2}(\alpha\!+\!\beta)\!-\!3\alpha \beta s)m_{q}}{\alpha\beta}\right]\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{4a}(s) = \frac{\langle g_{s}^{2} G G\rangle m_{c}^{2} }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2} (m_{c}^{2}(\alpha+ \beta)-2 \alpha \beta s)\left(\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}\right)\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{4b}(s) = \frac{\langle g_{s}^{2} G G\rangle }{1024 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)s \left(3+\frac{4(1-\alpha-\beta)}{\beta}-\frac{3(1-\alpha-\beta)^{2}}{4\beta^{2}}\right)\, , $ ![]() |
$ \ \rho_{\eta_{4\mu}}^{5a}(s) = \frac{ \langle\bar{s}g_{s}\sigma\cdot Gs\rangle m_{c}}{192 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (3 m_{c}^{2}(\alpha+\beta)-4 s \alpha \beta) \left(\frac{1-\alpha+2\beta}{\alpha\beta}\right)\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{5b}(s) = -\frac{ \langle\bar{q}g_{s}\sigma\cdot Gq\rangle m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \left(1+5\alpha-\beta\right)\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{5c}(s) = \frac{\langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{256\pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{s}-9 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{\langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{256 \pi^{4}}\left(\left(s- m_{c}^{2}\right) m_{q}-9 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \rho_{\eta_{4\mu}}^{6a}(s) = \frac{3\langle\bar{q}q\rangle\langle\bar{s}s\rangle }{16 \pi^{2}}(4m_{c}^{2}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{\eta_{4\mu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle m_{c}^{3}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{s}}{1-\alpha}+\frac{m_{q}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{\eta_{4\mu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ m_{c}^{4}}{96 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{ 6(\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \bar{q} g_{s}\sigma\cdot Gq\rangle)}{(1-\alpha)^{2}M_{\rm B}^{2}} +\frac{2\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle+\langle\bar{s}s\rangle \langle\bar{q} g_{s}\sigma\cdot Gq\rangle}{(1-\alpha)m_{c}^{2}}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \rho_{\eta_{\mu\nu}}^{0a}(s) = -\frac{5} {768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{(1-\alpha-\beta)^{2}} { \alpha^{3} \beta^{3}}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{3}\left((\alpha+\beta+2)(m_{c}^{2}(\alpha+\beta)-\alpha \beta s) -3(m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)\right)\, $ ![]() |
$ \rho_{\eta_{\mu\nu}}^{0b}(s) = - \frac{15m_{c}} {384 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}(m_{c}^{2}(\alpha+\beta)-\alpha \beta s)^{2}(m_{c}^{2}(\alpha+\beta)-4 \alpha \beta s) \left(\frac{m_{s}}{\alpha^{2} \beta^{3}}+\frac{m_{q}}{\alpha^{3} \beta^{2}}\right)\, , $ ![]() |
$ \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3a}(s) = & - \frac{5\langle\bar{s}s\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha^{2} \beta}\right.\\ &\left.-\frac{2m_{c}^{2}m_{q}-\alpha \beta s m_{s}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{s}}{\alpha\beta}\right]\, , \end{aligned} $ ![]() |
$ \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{3b}(s) = & - \frac{5\langle\bar{q}q\rangle}{16\pi^{4} } \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+\beta)-\alpha \beta s)\left[\frac{(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-3\alpha \beta s)m_{c}} { \alpha \beta^{2}}\right.\\ &\left.-\frac{2m_{c}^{2}m_{s}-\alpha \beta s m_{q}+(1-\alpha-\beta)(m_{c}^{2}(\alpha+\beta)-s\alpha\beta)m_{q}}{\alpha\beta}\right]\, , \end{aligned} $ ![]() |
$ \rho_{\eta_{\mu\nu}}^{4a}(s) = \frac{5 \langle g_{s}^{2} G G\rangle m_{c}^{2} }{768 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (1-\alpha-\beta)^{2}\left[ (1-\alpha-\beta)(m_{c}^{2}(\alpha+ \beta)- \alpha \beta s)\left(\frac{1}{3\alpha^{3}}+\frac{1}{3\beta^{3}}\right) -\left(\frac{\beta s}{2\alpha^{2}}+\frac{\alpha s}{2\beta^{2}}\right)\right]\, , $ ![]() |
$ \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{4b}(s) =& \frac{5\langle g_{s}^{2} G G\rangle m_{c}^{2} }{12288 \pi^{6}}\int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta (m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)\left[(m_{c}^{2}(\alpha+ \beta)-3\alpha \beta s)\left(1+\frac{2(1-\alpha-\beta)^{2}}{\alpha\beta}\right) \right.\\&\left.+\frac{4(m_{c}^{2}(\alpha+ \beta)-\alpha \beta s)(1-\alpha-\beta)(\alpha+\beta)}{\alpha\beta^{2}}\right]\, , \end{aligned} $ ![]() |
$ \begin{aligned}[b] \rho_{\eta_{\mu\nu}}^{5a}(s) =&\frac{5 \langle\bar{s} g_{s}\sigma\cdot G s\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{s}}{\beta}\, ,\\ &+\frac{5 \langle\bar{q} g_{s}\sigma\cdot G q\rangle m_{c}}{96 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{3( m_{c}^{2}(\alpha+\beta)-2 \alpha \beta s)m_{c}+2( 2m_{c}^{2}(\alpha+\beta)-3 \alpha \beta s)m_{q}}{\alpha}\, , \end{aligned} $ ![]() |
$ \rho_{\eta_{\mu\nu}}^{5b}(s) = \frac{5\langle\bar{q} g_{s}\sigma\cdot G q\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{q}-30 m_{c}^{2} m_{s}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}} +\frac{5\langle\bar{s} g_{s}\sigma\cdot G s\rangle}{192 \pi^{4}}\left(\left(s-2 m_{c}^{2}\right) m_{s}-30 m_{c}^{2} m_{q}\right) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, ,$ ![]() |
$ \rho_{\eta_{\mu\nu}}^{5c}(s) = \frac{5 \left(\langle\bar{s} g_{s}\sigma\cdot G s\rangle+\langle\bar{q} g_{s}\sigma\cdot G q\rangle\right) m_{c}}{384 \pi^{4}} \int_{\alpha_{\rm min}}^{\alpha_{\rm max}} {\rm d} \alpha \int_{\beta_{\rm min}}^{\beta_{\rm max}} {\rm d} \beta \frac{7(m_{c}^{2}(\alpha+\beta)-6 \alpha \beta s)(\alpha+5(1-\alpha+\beta))}{\alpha\beta}\, , $ ![]() |
$ \rho_{\eta_{\mu\nu}}^{6a}(s) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle }{24 \pi^{2}}(4m_{c}^{2}+m_{c}m_{q}+m_{c}m_{s}) \sqrt{1-\frac{4 m_{c}^{2}}{s}}\, , $ ![]() |
$ \Pi_{\eta_{\mu\nu}}^{6b}\left(M_{\rm B}^{2}\right) = \frac{5\langle\bar{s}s\rangle\langle\bar{q}q\rangle m_{c}^{3}}{12 \pi^{2}} \int_{0}^{1} {\rm d} \alpha\Big(\frac{m_{q}}{1-\alpha}+\frac{m_{s}}{\alpha}\Big) {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |
$ \Pi_{\eta_{\mu\nu}}^{8}\left(M_{\rm B}^{2}\right) = \frac{ 5m_{c}^{4}}{24 \pi^{2}} \int_{0}^{1} {\rm d} \alpha \left[\frac{ \langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{(1-\alpha)^{2}M_{\rm B}^{2}}-\frac{\langle\bar{s}s\rangle \langle\bar{q}g_{s}\sigma\cdot Gq\rangle+\langle\bar{q}q\rangle \langle\bar{s}g_{s}\sigma\cdot Gs\rangle}{12\alpha}\right] {\rm e}^{\frac{-m_{c}^{2}} { \alpha(1-\alpha) M_{\rm B}^{2}}}\, , $ ![]() |