D*Dρ and B*Bρ strong couplings in light-cone sum rules

$\begin{array}{l} \langle\, \rho(p,\eta^*)\,H(q)\,|\,H^*(p+q,\varepsilon)\, \rangle = -g_{H^*H\rho}\, \epsilon_{pq\eta\varepsilon}\,, \end{array}$

$\begin{array}{l} g_{D^*D\rho}\equiv g_{D^{*-}\bar{D}^0\rho^-} = -\sqrt{2}\,g_{D^{*-}D^-\rho^0}\,. \end{array}$

$\begin{split} \Pi_\mu(p,q) = &\int {\rm d}^4x\,{\rm e}^{-{\rm i}(p+q)\cdot x}\langle\,\rho^-(p,\eta^*)\,|\,T\big\{\bar{d}(x)\,\gamma_{\mu \perp}\, Q(x),\\&\,\bar{Q}(0)\,\gamma_5\,u(0)\big\}\,|\,0\rangle\,, \end{split}$

$\begin{array}{l} p^{\mu} = \dfrac{n\cdot p}{2}\bar{n}_{\mu}\,. \end{array}$

$\begin{split} &\langle\,H^*(p+q,\varepsilon^*)\,|\,\bar{d}\,\gamma^\mu \,Q\,|\,0\,\rangle = f_{H^*}\,m_{H^*}\varepsilon^{*\mu}\,,\\& \langle\,0\,|\,\bar{Q}\,\gamma_5\,u\,|\,H(p)\,\rangle = -{\rm i}\,f_H\,\frac{m_H^2}{m_Q}\,, \end{split}$

$\begin{split} \Pi_\mu^{\rm had}(p,q) =& \frac{g_{H^*H\rho}\,f_{H^*}\,f_H}{[m_{H^*}^2-(p+q)^2-i\,0]\,[m_H^{2}-q^2-i\,0]}\, \frac{m_H^2\,m_{H^*}}{m_Q}\,\epsilon_{\mu pq\eta} \\ &+\iint_{\Sigma}\frac{\rho^h(s,s')\,{\rm d}s\,{\rm d}s'}{\left[s'-(p+q)^2\right](s-q^2)}+\cdots\,, \end{split}$

$\begin{split} \Pi_\mu^{\rm had}(p,q) =& \frac{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}{m_Q}\,g_{H^*H\rho}\, {\rm e}^{-\frac{m_H^2+m_{H^*}^2}{M^2}} \epsilon_{\mu pq \eta} \\&+\iint_{\Sigma}\,{\rm d}s\,{\rm d}s'\,{\rm e}^{-\frac{s+s'}{M^2}}\,\rho^h(s,s')\,. \end{split}$

$\begin{split} \Pi_\mu^{{\rm LT},(0)}(p,\,q) =& -\frac{\rm i}{2}\frac{\bar{n}\cdot q}{u\,q^2+\bar{u}\,(p+q)^2-u\,\bar{u}\,m_\rho^2-m_Q^2}\, \\&\times\bar{q}(u\,p)\,\gamma_{\mu\,\perp}\,\not\!\!{n}\,\gamma_5\,q(\bar{u}\,p)\\ =& -\frac{\rm i}{2}\frac{\bar{n}\cdot q}{u\,q^2+\bar{u}\,(p+q)^2-m_Q^2}\,\\&\times \bar{q}(u\,p)\,\gamma_{\mu\perp}\,\not\!\!{n}\,\gamma_5\,q(\bar{u}\,p)\\&+{\cal{O}}\left(\frac{\Lambda^2_{\rm QCD}}{m_Q^2}\right)\,. \end{split}$

$\begin{split} \Pi_{\mu}^{{\rm LT},(0)}(p,\,q)\!=\! -\!f_\rho^T(\mu)\,\epsilon_{\mu p q\eta}\!\int_0^1\!\! {\rm d}u\,\phi_\perp(u,\mu)\,\dfrac{1}{u\,q^2\!+\!\bar{u}\,(p\!+\!q)^2\!-\!m_Q^2}\,. \end{split}$

$\begin{split} \Pi_{\mu}^{{\rm LT},(0)}(p,\,q) =& -f_\rho^T(\mu)\,\epsilon_{\mu p q\eta}\iint {\rm d}s\,{\rm d}s'\,\\&\times\frac{\rho^{\rm LT,(0)}(s,s')}{[s-q^2]\,[s'-(p+q)^2]}\,, \end{split}$

$\begin{split} \rho^{{\rm LT},(0)}(s,s') = \frac{1}{\pi^2}\,{\rm Im}_{s'}\,{\rm Im}_{s} \int_0^1 {\rm d} u\,\frac{\phi_\perp(u,\mu)}{u\,s+\bar{u}\,s'-m_Q^2}\,. \end{split}$

$\begin{array}{l} \phi_\perp(u,\mu) = 6\,u\,\bar{u}\,\displaystyle\sum\limits_{n = 0}^{\infty}a_n^\perp(\mu)\,C_n^{3/2}(2u-1) = \displaystyle\sum\limits_{k = 1}^\infty b_k\,u^k\,, \end{array}$

$\begin{split} \rho^{{\rm LT},(0)}(t,v) =& \displaystyle\sum\limits_{k = 1}^\infty b_k\,\frac{1}{\pi^2}\,{\rm Im}_{s'}\,{\rm Im}_{s}\int_0^1 {\rm d}u\,\frac{u^k}{u\,s+\bar{u}\,s'-m_Q^2} \\ =& \displaystyle\sum\limits_{k = 1}^\infty b_k\,\frac{(-1)^{k+1}}{k!}\,\frac{1}{2^{k+1}\,t}\,\left(\frac{m_Q^2}{t}-\bar{v}\right)^k\, \delta^{(k)}\left(v-\frac{1}{2}\right)\,\\&\times\theta(v\,t-m_Q^2)\,. \end{split}$

$\begin{split} g^{{\rm LT},(0)} =& \dfrac{m_Q}{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}\,f_\rho^T(\mu)\,\frac{M^2}{2}\, \phi_\perp\Big(\frac{1}{2},\mu\Big)\, \\&\times\Big[{\rm e}^{\textstyle\frac{m_H^2+m_{H^*}^2-2m_Q^2}{M^2}}-{\rm e}^{\textstyle\frac{m_H^2+m_{H^*}^2-2s_0}{M^2}} \Big]\,. \end{split}$

$\begin{split} g^{{\rm LT}} =& -\frac{m_Q}{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}\, {\rm e}^{\frac{m_H^2+m_{H^*}^2}{M^2}}\, f_\rho^T(\mu)\,\\&\times \Big[-\frac{M^2}{2}\,\phi_\perp\Big(\frac{1}{2},\mu\Big)\,\big( {\rm e}^{-\frac{2m_Q^2}{M^2}} -{\rm e}^{-\frac{2s_0}{M^2}}\big) + {\cal{F}}^{{\rm LT},(1)}\Big]\,, \end{split}$

$\begin{split} {\cal{F}}^{{\rm LT},(1)} =& \dfrac{\alpha_s\,C_F}{4\,\pi}\,\Big\{m_Q^2\int_{0}^{2\hat{s}_0-2}{\rm d}\sigma\, {\rm e}^{-\frac{\sigma+2}{\hat{M}^2}}\,g(\sigma) \\&+\Delta\,g(\hat{M}^2,m_Q^2)\Big\}\,, \end{split}$

$\begin{split} g(\sigma) =& 3\,{\rm Li}_2(-\sigma)+3\,{\rm Li}_2(-\sigma-1)-6\,{\rm Li}_2\left(-\frac{\sigma}{2}\right) -3\,\ln\frac{\sigma}{2}\ln\frac{\sigma+2}{2}\\&+3\,\ln(\sigma+1) \,\ln(\sigma+2) -\frac{6\,(\sigma+1)^2}{(\sigma+2)^3}\,\ln (\sigma+1) \\&+ \frac{3\,(7\,\sigma^3+50\,\sigma^2+100\,\sigma+64)}{4\,(\sigma+2)^3}\,\ln\frac{\sigma}{2} \\&+ \frac{3}{2}\,\ln\frac{\sigma+2}{2} +\frac{3\,(11\,\sigma^2+28\,\sigma+24)}{8\,(\sigma+2)^2}\\&-3\,\ln\frac{\mu^2}{m_Q^2} -\frac{9}{4}\,\ln\frac{\nu^2}{\mu^2}-\frac{\pi^2}{4}\,, \\ & \Delta\,g(\hat{M}^2,m_Q^2) = 3\,\left(4+3\,\ln\frac{\mu^2}{m_Q^2}\right)\, m_Q^2\,{\rm e}^{-\frac{2m_Q^2}{M^2}}\,. \end{split}$

$\begin{split} \frac{\rm d}{{\rm d}\ln\mu}m_Q(\mu) =& -6\frac{\alpha_sC_F}{4\pi}m_Q(\mu)\,, \\ \frac{\rm d}{{\rm d}\ln\mu}f_\rho^T(\mu) =& -2\frac{\alpha_sC_F}{4\pi}f_\rho^T(\mu)\,, \end{split}$

$\begin{array}{l} \dfrac{\rm d}{{\rm d}\ln\mu}g^{\rm LT} = 0+{\cal{O}}(\alpha_s^2)\,, \end{array}$

$\begin{split} &\langle\,0\,|\,T\{\bar{Q}(x),\,Q(0)\}\,|\,0\,\rangle \\ &\quad \supset {\rm i}\,g_s\int\frac{{\rm d}^4k}{(2\pi)^4}\,{\rm e}^{-{\rm i}\,k\cdot x}\int_0^1 {\rm d} u \,\bigg[ {u \, x_{\mu} \over k^2-m_Q^2 } \, G^{\mu \nu}(u\, x) \, \gamma_{\nu} \\&\quad- {\not{\!k} + m_Q \over 2\,(k^2-m_Q^2)^2} \, G^{\mu \nu}(u\, x) \, \sigma_{\mu \nu}\bigg]\,. \end{split}$

$\begin{split} \Pi_{\mu}^{\rm 2P, HT}(p,q) = &- \frac{1}{4}\,\epsilon_{\mu pq\eta}\, \int_0^1 {\rm d} u\,\\& \left\{ \frac{-2\,m_Q\,f_\rho\,m_\rho\,g_\perp^{(a)}(u)+f_\rho^T(\mu)\,m_\rho^2\,\mathbb{A}_T(u)} {[u\,q^2+\bar{u}\,(p+q)^2-m_Q^2]^2}\right.\\&+ \left.\frac{-2\,m_Q^2\,f_\rho^T(\mu)\,m_\rho^2\,\mathbb{A}_T(u)}{[u\,q^2+\bar{u}\,(p+q)^2-m_Q^2]^3}\right\}\,. \end{split}$

$\begin{split} \Pi_\mu^{\rm LT,NLP}(p,\,q) =& -\frac{\rm i}{2}\,\bar{n}\cdot q\,\frac{u\,\bar{u}\,m_\rho^2}{[u\,q^2+\bar{u}\,(p+q)^2-m_Q^2]^2}\, \\&\times\bar{q}(u\,p)\,\gamma_{\mu\perp}\,\not{\!n}\,\gamma_5\,q(\bar{u}\,p) \\ =& -f_\rho^T\,\epsilon_{\mu pq\eta}\,\int_0^1 {\rm d} u\, \phi_\perp(u,\mu)\,\\&\times \frac{u\,\bar{u}\,m_\rho^2}{[u\,q^2+\bar{u}\,(p+q)^2-m_Q^2]^2}\,. \end{split}$

$\begin{split} g^{{\rm 2P}} =& -\frac{m_Q}{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}\,{\rm e}^{\frac{m_H^2+m_{H^*}^2-2m_Q^2}{M^2}}\,\frac{m_\rho}{2} \, \Big[\frac{m_\rho}{2}\,f_\rho^T(\mu)\,\phi_\perp\Big(\frac{1}{2},\,\mu\Big) \\ & -m_Q\,f_\rho\,g_\perp^{(a)}\Big(\frac{1}{2},\,\mu\Big) +m_\rho\,f_\rho^T(\mu)\,\Big(\frac{1}{2}+\frac{m_Q^2}{M^2}\Big)\, \mathbb{A}_T\Big(\frac{1}{2},\,\mu\Big)\Big]\,. \end{split}$

$\begin{split} \Pi_{\mu}^{{\rm3P}}(p,q) =& \; {\rm i}\,g_s\int {\rm d}^4x\int \frac{{\rm d}^4k}{(2\pi)^4}\,{\rm e}^{-{\rm i}(k+p+q)\cdot x}\\&\times\int_0^1 {\rm d} u \,\langle\, \rho^-(p,\eta^*)\,|\,\bar{d}(x)\, \gamma_{\mu \perp}\, \Big[\frac{u\,x_\alpha}{k^2-m_Q^2}\,\gamma_\beta \\& -\frac{\not\!{k}+m_Q}{2\,(k^2-m_Q^2)^2}\,\sigma_{\alpha\beta}\Big]\, G^{\alpha\beta}(ux)\,\gamma_5\,q(0)\,|\,0\rangle\,. \end{split}$

$\begin{split} g^{{\rm3P}} =& -\frac{m_Q}{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}\,{\rm e}^{\frac{m_H^2+m_{H^*}^2-2m_Q^2}{M^2}}\,f_\rho^T(\mu)\,m_\rho^2\, \\&\times\Big[ \Big(\frac{21}{8}-4\ln2\Big)\,\tilde{t}_{10}(\mu)-\frac{15}{8}\,s_{00}(\mu)\Big]\,. \end{split}$

$\begin{split} g_{H^*H\rho} =& -\frac{m_Q}{f_H\,f_{H^*}\,m_H^2\,m_{H^*}}\,{\rm e}^{\frac{m_H^2+m_{H^*}^2-2m_Q^2}{M^2}}\,\\&\times \bigg\{ f_\rho^T(\mu)\Big[ -\frac{M^2}{2}\,\Big(1-{\rm e}^{-\frac{2s_0-2m_Q^2}{M^2}}\Big)+\frac{m_\rho^2}{4} \Big]\, \phi_\perp\Big(\frac{1}{2},\mu\Big)\\ &+f_\rho^T(\mu)\,{\cal{F}}^{{\rm LT},(1)}\,{\rm e}^{\frac{2m_Q^2}{M^2}} -\frac{m_Q}{2}\,m_\rho\,f_\rho\,g_\perp^{(a)}\Big(\frac{1}{2},\mu\Big) \\& +m_\rho^2\,f_\rho^T(\mu)\, \Big[\frac{1}{2}\, \Big(\frac{1}{2}+\frac{m_Q^2}{M^2}\Big)\, \mathbb{A}_T\Big(\frac{1}{2},\,\mu\Big)\\&+ \Big(\frac{21}{8}-4\ln2\Big)\,\tilde{t}_{10}(\mu)-\frac{15}{8}\,s_{00}(\mu)\Big]\bigg\}\,, \end{split}$

$\begin{split} f_\rho^T(\mu)\,a_n^\perp(\mu) =& \Big[ E_{T, n}^{\rm NLO}(\mu, \mu_0)+\frac{\alpha_s(\mu)}{4\pi} \,\sum\limits_{k = 0}^{n-2}\, E_{T, n}^{\rm LO}(\mu,\mu_0)\,\\&\times d_{T, n}^{k}(\mu,\mu_0) \Big]\,f_\rho^T(\mu_0)\,a_n^\perp(\mu_0)\,, \end{split}$

$\begin{split}& \left\{ \begin{array}{l} s_0 = 6.0\pm0.5 \; {\rm GeV}^2\,,\\ M^2 = 4.5\pm1.0 \; {\rm GeV}^2\, , \end{array} \right. {\rm for} \; D^*\,D\,\rho\,;\\& \left\{ \begin{array}{l} s_0 = 34.0\pm1.0 \; {\rm GeV}^2\,,\\ M^2 = 18.0\pm3.0 \; {\rm GeV}^2\, , \end{array} \right. {\rm for}\; B^*\,B\,\rho\,, \end{split}$

$\begin{split} &\langle\,\rho^-(p,\eta^*)\,|\,\bar{d}\,\gamma_\mu\,b\,|\,B^-(p+q)\,\rangle = \frac{2}{m_B+m_\rho}\,V(q^2)\,\epsilon_{\mu\eta pq}\,, \\& \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}\,i\,\sigma_{\mu\nu}\,q^\nu\,b\,|\,B^-(p+q)\,\rangle = -2\,T_1(q^2)\,\epsilon_{\mu\eta pq}\,. \end{split}$

$\begin{split} F_i(q^2) = \frac{r_1^i}{1-q^2/m_{B^*}^2}+\int_{(m_B+m_\rho)^2}^\infty \frac{\rho(s)}{s-q^2-{\rm i}\epsilon}\,, \end{split}$

$\begin{split} r_1^V = &\lim_{q^2\to m_{B^*}^2} \big(1-q^2/m_{B^*}^2\big)\,V(q^2) = \frac{m_B+m_\rho}{2\,m_B^*}\,f_{B^*}\,g_{B^*B\rho}\,, \\ r_1^{T_1} =& \lim_{q^2\to m_{B^*}^2} \big(1-q^2/m_{B^*}^2\big)\,T_1(q^2) = \frac{1}{2}\,f_{B^*}^T\,g_{B^*B\rho}\,, \end{split}$

$\begin{array}{l} \langle\,0\,|\,\bar{b}\,\sigma_{\mu\nu}\,q\,|\,B^*(q,\epsilon)\,\rangle = {\rm i}\,f_{B^*}^T\,(\epsilon_\mu\,q_\nu-\epsilon_\nu\,q_\mu)\,. \end{array}$

$\begin{array}{l} {\cal{L}}_V = {\rm i}\,\lambda\,{\rm Tr}[{\cal{H}}_b\,\sigma^{\mu\nu}\,F_{\mu\nu}(\rho)_{ba}\,\bar{{\cal{H}}}_a]\,, \end{array}$

$\begin{array}{l} F_{\mu\nu}(\rho) = \partial_\mu\, \rho_\nu-\partial_\nu\, \rho_\mu +[\rho_\mu,\,\rho_\nu]\,, \quad \rho_\mu = {\rm i}\,\dfrac{g_V}{\sqrt{2}}\,\hat{\rho}_\mu\,, \end{array}$

$\begin{array}{l} \lambda = \dfrac{\sqrt{2}}{4}\,\dfrac{1}{g_V}\,g_{H^*H\rho}\,. \end{array}$

$\tag{A1}\begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,\gamma_\mu\,u(0)\,|\,0\,\rangle = \; f_\rho\,m_\rho\,\bigg\{\frac{\eta^*\cdot x}{p\cdot x}\,p_\mu\int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,\big[\phi_\parallel(u)+\frac{m_\rho^2\,x^2}{16}\,\mathbb{A}(u)\big] +\eta^*_{\perp\mu}\int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,g_\perp^{(v)}(u)-\frac{1}{2}\,x_\mu\,\frac{\eta^*\cdot x}{(p\cdot x)^2}\,m_\rho^2 \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,g_3(u)\bigg\}\,, \end{split}$

$\tag{A2} \begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,\gamma_\mu\,\gamma_5\,u(0)\,|\,0\,\rangle = \frac{1}{4}\,f_\rho\,m_\rho\,\epsilon_{\mu\eta px} \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,g_\perp^{(a)}(u)\,, \end{split}$

$\tag{A3} \begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,\sigma_{\mu\nu}\,u(0)\,|\,0\,\rangle =& -{\rm i}\,f_\rho^T\,\bigg\{ (\eta^*_{\perp\mu}\,p_\nu-\eta^*_{\perp\nu}\,p_\mu) \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\, \big[\phi_\perp(u)+\frac{m_\rho^2\,x^2}{16}\,\mathbb{A}_T(u)\big] +(p_\mu\,x_\nu-p_\nu\,x_\mu)\,\frac{\eta^*\cdot x}{(p\cdot x)^2}\,m_\rho^2 \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,h_\parallel^{(t)}(u) \\ &+\frac{1}{2}\,\frac{\eta^*_{\perp\mu}\,x_\nu-\eta^*_{\perp\nu}\,x_\mu}{p\cdot x}\,m_\rho^2 \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,h_3(u) \bigg\}\,, \end{split}$

$\tag{A4} \begin{split} \langle\,\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,u(0)\,|\,0\,\rangle = -\frac{\rm i}{2}\,f_\rho^T\,(\eta^*\cdot x) \,m_\rho^2 \int_0^1{\rm d}u\,{\rm e}^{{\rm i}\,u\,p\cdot x}\,h_\parallel^{(s)}(u)\,, \end{split}$

$\tag{A5}\begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,g_s\,\tilde{G}_{\mu\nu}(vx)\,\gamma_\alpha\,\gamma_5\,u(0)\,|\,0\,\rangle =& -f_\rho\,m_\rho\,p_\alpha\,\big[p_\nu\, \eta^*_{\perp\mu}-p_\mu\,\eta^*_{\perp\nu}\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,{\cal{A}}(\alpha) -f_\rho\,m_\rho^3\,\frac{\eta^*\cdot x}{p\cdot x}\,\big[p_\mu\,g^\perp_{\alpha\nu}-p_\nu\,g^\perp_{\alpha\mu}\big] \\ &\times\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,\tilde{\Phi}(\alpha) -f_\rho\,m_\rho^3\,\frac{\eta^*\cdot x}{(p\cdot x)^2}\, p_\alpha\, \big[p_\mu\,x_\nu-p_\nu\,x_\mu\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\, \tilde{\Psi}(\alpha)\,, \end{split}$

$\tag{A6} \begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,g_s\,G_{\mu\nu}(vx)\,i\,\gamma_\alpha\,u(0)\,|\,0\,\rangle =& -f_\rho\,m_\rho\,p_\alpha\,\big[p_\nu\, \eta^*_{\perp\mu}-p_\mu\,\eta^*_{\perp\nu}\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,{\cal{V}}(\alpha) -f_\rho\,m_\rho^3\,\frac{\eta^*\cdot x}{p\cdot x}\,\big[p_\mu\,g^\perp_{\alpha\nu}-p_\nu\,g^\perp_{\alpha\mu}\big] \\ &\times\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,\Phi(\alpha) -f_\rho\,m_\rho^3\,\frac{\eta^*\cdot x}{(p\cdot x)^2}\, p_\alpha\, \big[p_\mu\,x_\nu-p_\nu\,x_\mu\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,\Psi(\alpha)\,, \end{split}$

$\tag{A7} \begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,g_s\,G_{\mu\nu}(vx)\,\sigma_{\alpha\beta}\,u(0)\,|\,0\,\rangle = & f^T_\rho\,m_\rho^2\,\frac{\eta^*\cdot x}{2\,p\cdot x} \,\big[ p_\alpha\,p_\mu\,g^\perp_{\beta\nu}-p_\beta\,p_\mu\,g^\perp_{\alpha\nu}-(\mu\leftrightarrow \nu) \big] \int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,{\cal{T}}(\alpha) +f^T_\rho\,m_\rho^2\,\big[ p_\alpha\,\eta^*_{\perp\mu}\,g^\perp_{\beta\nu}-p_\beta\,\eta^*_{\perp\mu} \,g^\perp_{\alpha\nu}-(\mu\leftrightarrow \nu) \big]\\&\times \int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,T_1^{(4)}(\alpha) +f^T_\rho\,m_\rho^2\,\big[ p_\mu\,\eta^*_{\perp\alpha}\,g^\perp_{\beta\nu}-p_\mu\,\eta^*_{\perp\beta} \,g^\perp_{\alpha\nu} -(\mu\leftrightarrow \nu) \big] \int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,T_2^{(4)}(\alpha)\\& +f^T_\rho\,m_\rho^2\,\frac{(p_\mu\,x_\nu-p_\nu\,x_\mu)\,(p_\alpha\,\eta^*_{\perp\beta}-p_\beta\,\eta^*_{\perp\alpha})}{p\cdot x} \int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,T_3^{(4)}(\alpha)\\& +f^T_\rho\,m_\rho^2\,\frac{(p_\alpha\,x_\beta-p_\beta\,x_\alpha)\,(p_\mu\,\eta^*_{\perp\nu}-p_\nu\,\eta^*_{\perp\mu})}{p\cdot x} \int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,T_4^{(4)}(\alpha)\,, \\[-13pt]\end{split}$

$\tag{A8} \begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,g_s\,G_{\mu\nu}(vx)\,u(0)\,|\,0\,\rangle = -{\rm i}\,f^T_\rho\,m_\rho^2\,\big[\eta^*_{\perp\mu}\,p_\nu-\eta^*_{\perp\nu}\,p_\mu\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,S(\alpha)\,, \end{split}$

$\tag{A9}\begin{split} \langle\,\rho^-(p,\eta^*)\,|\,\bar{d}(x)\,{\rm i}\,g_s\,\tilde{G}_{\mu\nu}(vx)\,\gamma_5\,u(0)\,|\,0\,\rangle = {\rm i}\,f^T_\rho\,m_\rho^2\,\big[\eta^*_{\perp\mu}\,p_\nu-\eta^*_{\perp\nu}\,p_\mu\big]\int[D\alpha]\,{\rm e}^{{\rm i}\,(\alpha_q+v\,\alpha_g)\,p\cdot x}\,\tilde{S}(\alpha)\,, \end{split}$

$\tag{A10} \begin{split} \widetilde{G}_{\alpha \beta} = \frac{1}{2} \, \epsilon_{\alpha \beta \rho \tau } \, G^{\rho \tau} \,. \end{split}$

$\tag{B1} \begin{split} g_\perp^{(a)}(u) = 6\,u\,\bar{u}\,\Big\{1+ \Big[\frac{1}{4}\,a_2^\parallel +\frac{5}{3}\,\zeta_3\,\Big(1-\frac{3}{16}\,\omega_3^A +\frac{9}{16}\,\omega_3^V\Big)\Big] \,(5\,\xi^2-1) \Big\}\,, \end{split}$

$\tag{B2} \begin{split} \mathbb{A}_T(u) =& 30\,u^2\,\bar{u}^2\,\Big[\frac{2}{5}\,\Big(1+\frac{2}{7}\,a_2^\perp+ \frac{10}{3}\,\zeta_4^T- \frac{20}{3}\,\tilde{\zeta}_4^T\Big)\\&+\Big(\frac{3}{35}\,a_2^\perp+\frac{1}{40}\,\zeta_3\,\omega_3^T\Big)\,C_2^{5/2}(\xi)\Big] \\& -\Big[\frac{18}{11}\,a_2^\perp-\frac{3}{2}\,\zeta_3\,\omega_3^T+ \frac{126}{55}\,\langle\langle Q^{(1)}\rangle\rangle \\&+\frac{70}{11}\,\langle\langle Q^{(3)}\rangle\rangle\Big]\, \big[u\,\bar{u}\,(2+13\,u\,\bar{u}) \\& +2\,u^3\,(10-15\,u+6\,u^2)\,\ln u \\&+2\,\bar{u}^3\,(10-15\,\bar{u}+6\,\bar{u}^2)\,\ln \bar{u}\big]\,. \end{split}$

$\begin{split} S(\alpha_i) =& \; 30\,\alpha_g^2\,\Big\{ s_{00}\,(1-\alpha_g)+s_{10}\, \Big[ \alpha_g\,(1-\alpha_g)-\frac{3}{2}\,(\alpha_{\bar{q}}^2+ \alpha_q^2) \Big] \\ & +s_{01}\, \big[ \alpha_g\,(1-\alpha_g)-6\,\alpha_{\bar{q}}\,\alpha_q \big]\Big\}\,, \\ \tilde{S}(\alpha_i) =& \; 30\,\alpha_g^2\,\Big\{ \tilde{s}_{00}\,(1-\alpha_g)+\tilde{s}_{10}\, \Big[ \alpha_g\,(1-\alpha_g)-\frac{3}{2}\,(\alpha_{\bar{q}}^2+ \alpha_q^2) \Big] \\ & +\tilde{s}_{01}\, \big[ \alpha_g\,(1-\alpha_g)-6\,\alpha_{\bar{q}}\,\alpha_q \big]\Big\}\,, \\ T_1^{(4)}(\alpha_i) =& \; 120\,t_{10}\,(\alpha_{\bar{q}}-\alpha_q)\, \alpha_{\bar{q}}\,\alpha_q\,\alpha_g\,, \\ T_2^{(4)}(\alpha_i) =& -30\,\alpha_g^2\,(\alpha_{\bar{q}}-\alpha_q)\, \Big[\tilde{s}_{00}+\frac{1}{2}\, \tilde{s}_{10}\, (5\,\alpha_g-3)+\tilde{s}_{01}\,\alpha_g\Big]\,, \\\end{split}$

$\tag{B3} \begin{split} T_3^{(4)}(\alpha_i) =& -120\,\tilde{t}_{10}\,(\alpha_{\bar{q}}-\alpha_q)\, \alpha_{\bar{q}}\,\alpha_q\,\alpha_g\,, \\ T_4^{(4)}(\alpha_i) =& \; 30\,\alpha_g^2\,(\alpha_{\bar{q}}-\alpha_q)\, \Big[s_{00}+\frac{1}{2}\, s_{10}\, (5\,\alpha_g-3) +s_{01}\,\alpha_g\Big]\,. \end{split}$

$\tag{B4} \begin{split} s_{00} =& \zeta_4^T\,\quad \tilde{s}_{00} = \tilde{\zeta}_4^T\,, \\ s_{10} =& -\frac{3}{22}\,a_2^\perp-\frac{1}{8}\,\zeta_3\,\omega_3^T+\frac{28}{55}\,\langle\langle Q^{(1)}\rangle\rangle +\frac{7}{11}\,\langle\langle Q^{(3)}\rangle\rangle+ \frac{14}{3}\,\langle\langle Q^{(5)}\rangle\rangle\,, \\ \tilde{s}_{10} =& \frac{3}{22}\,a_2^\perp-\frac{1}{8}\,\zeta_3\,\omega_3^T-\frac{28}{55}\,\langle\langle Q^{(1)}\rangle\rangle -\frac{7}{11}\,\langle\langle Q^{(3)}\rangle\rangle+ \frac{14}{3}\,\langle\langle Q^{(5)}\rangle\rangle\,, \\ s_{01} =& \frac{3}{44}\,a_2^\perp+\frac{1}{8}\,\zeta_3\,\omega_3^T+\frac{49}{110}\,\langle\langle Q^{(1)}\rangle\rangle -\frac{7}{22}\,\langle\langle Q^{(3)}\rangle\rangle+ \frac{7}{3}\,\langle\langle Q^{(5)}\rangle\rangle\,, \\ \tilde{s}_{01} =& -\frac{3}{44}\,a_2^\perp+\frac{1}{8}\,\zeta_3\,\omega_3^T-\frac{49}{110}\,\langle\langle Q^{(1)}\rangle\rangle +\frac{7}{22}\,\langle\langle Q^{(3)}\rangle\rangle+ \frac{7}{3}\,\langle\langle Q^{(5)}\rangle\rangle\,, \\ t_{10} =& -\frac{9}{44}\,a_2^\perp-\frac{3}{16}\,\zeta_3\,\omega_3^T-\frac{63}{220}\,\langle\langle Q^{(1)}\rangle\rangle +\frac{119}{44}\,\langle\langle Q^{(3)}\rangle\rangle\,, \\ \tilde{t}_{10} =& \frac{9}{44}\,a_2^\perp-\frac{3}{16}\,\zeta_3\,\omega_3^T+\frac{63}{220}\,\langle\langle Q^{(1)}\rangle\rangle +\frac{35}{44}\,\langle\langle Q^{(3)}\rangle\rangle\,. \end{split}$

$\tag{B5}\begin{split} a_{2}^{\parallel }(\mu )=&{{L}^{\frac{25}{6}{{C}_{F}}/{{\beta }_{0}}}}a_{2}^{\parallel }({{\mu }_{0}}), \\{{\zeta }_{3}}(\mu )=&{{L}^{\left( -\frac{1}{3}{{C}_{F}}+3{{C}_{A}} \right)/{{\beta }_{0}}}}{{\zeta }_{3}}({{\mu }_{0}}), \\ \omega _{3}^{T}(\mu )=&{{L}^{\left( \frac{25}{6}{{C}_{F}}-2{{C}_{A}} \right)/{{\beta }_{0}}}}\omega _{3}^{T}({{\mu }_{0}}), \\ (\zeta _{4}^{T}+\tilde{\zeta }_{4}^{T})(\mu )=&{{L}^{\left( 3{{C}_{A}}-\frac{8}{3}{{C}_{F}} \right)/{{\beta }_{0}}}}(\zeta _{4}^{T}+\tilde{\zeta }_{4}^{T})({{\mu }_{0}}), \\ (\zeta _{4}^{T}-\tilde{\zeta }_{4}^{T})(\mu )=&{{L}^{\left( 4{{C}_{A}}-4{{C}_{F}} \right)/{{\beta }_{0}}}}(\zeta _{4}^{T}-\tilde{\zeta }_{4}^{T})({{\mu }_{0}}), \\ \langle \langle {{Q}^{(1)}}\rangle \rangle (\mu )=&{{L}^{\left( -4{{C}_{F}}+\frac{11}{2}{{C}_{A}} \right)/{{\beta }_{0}}}}\langle \langle {{Q}^{(1)}}\rangle \rangle ({{\mu }_{0}}), \\ \langle \langle {{Q}^{(3)}}\rangle \rangle (\mu )=&{{L}^{\frac{10}{3}{{C}_{F}}/{{\beta }_{0}}}}\langle \langle {{Q}^{(3)}}\rangle \rangle ({{\mu }_{0}}), \\\langle \langle {{Q}^{(5)}}\rangle \rangle (\mu )=&{{L}^{\left( -\frac{5}{3}{{C}_{F}}+5{{C}_{A}} \right)/{{\beta }_{0}}}}\langle \langle {{Q}^{(5)}}\rangle \rangle ({{\mu }_{0}}), \\ \end{split}$

$\tag{B6} \begin{split} \left( \begin{array}{c} \omega^{V}_3(\mu) - \omega^{A}_3(\mu) \\ \omega^{V}_3(\mu) + \omega^{A}_3(\mu) \end{array} \right) = L^{\Gamma_{\omega} / \beta_0} \, \left( \begin{array}{c} \omega^{V}_3(\mu_0) - \omega^{A}_3(\mu_0) \\ \omega^{V}_3(\mu_0) + \omega^{A}_3(\mu_0) \end{array} \right) \,, \end{split}$

$\tag{B7} \begin{split} \Gamma_{\omega} = \left( \begin{array}{c} 3 \, C_F - {2 \over 3} \, C_A \qquad {2 \over 3} \, C_F - {2 \over 3} \, C_A \\ {5 \over 3} \, C_F - {4 \over 3} \, C_A \qquad {1 \over 2} \, C_F + C_A \end{array} \right)\,. \end{split}$