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--> --> --> $ {\cal A} = \sum\limits_{p,X} C_{p,X} {\cal M}_{p,X}, $ | (1) |
For convenience, we can define a map for each primitive amplitude from the helicity state
$ f_p: H \mapsto X, {\; s.t.\; } {\mathbb P}_H {\cal M}_{p,X} \neq 0, $ | (2) |
$ \bar u(k_1) {\not\!\! q} v(k_2), $ | (3) |
$ f_p(+-) = LR, \quad f_p(-+) = RL. $ | (4) |
$ \bar u(k_1) {\not\!\! q} v(k_2), $ | (5) |
$ f_p(--) = f_p(+-) = LR, \quad f_p(++) = f_p(-+) = RL. $ | (6) |
Therefore we can obtain
$ {\cal M}_{p,X} = \sum\limits_H {\mathbb P}_H {\cal M}_{p,X} = \sum\limits_{H = f_p^{-1}(X)} \delta_{X,f_p(H)}{\cal M}_{p,H}, $ | (7) |
Consequently, the amplitude
$ {\cal A} = \sum\limits_{p,H}C_{p,X = f(H)} ( {\mathbb P}_H {\cal M}_{p} ). $ | (8) |
$ {\cal A}_H = \sum\limits_{p}C_{p,X = f(H)} ( {\mathbb P}_H {\cal M}_{p} ). $ | (9) |
$ {\cal M}_p = \sum\limits_{i} d_{i,p} F_i. $ | (10) |
$ M_{ij}\equiv F_iF_j^\dagger. $ | (11) |
$ d_{i,p} = \sum\limits_{j} (M^{-1})_{ij}{\cal M}_p F_j^\dagger. $ | (12) |
$ {\cal A} = \sum\limits_{i,H} c_{i,H}({\mathbb P}_{H} F_i), $ | (13) |
$ c_{i,H} = \sum\limits_{p}d_{i,p}C_{p,X = f(H)}. $ | (14) |
$ k = k_0+\dfrac{m^2_k}{2 k_{0}\cdot k_{r}}k_{r}, $ | (15) |
$ \begin{split} u^+(k,m_k)& = \left|k_{0} \right \rangle+\dfrac{m_k}{\left[k_{0}k_{r}\right]}\left|k_{r}\right],\\ u^-(k,m_k)& = \left|k_{0} \right ]+\dfrac{m_k}{\left\langle k_{0}k_{r}\right\rangle}\left|k_{r}\right\rangle,\\ v^+(k,m_k)& = \left|k_{0} \right ]-\dfrac{m_k}{\left\langle k_{0}k_{r}\right\rangle}\left|k_{r}\right\rangle,\\ v^-(k,m_k)& = \left|k_{0} \right \rangle-\dfrac{m_k}{\left[k_{0}k_{r}\right]}\left|k_{r}\right]. \end{split} $ | (16) |
$ \begin{split} u(k,m_k)& = u_{0}(k_0)+ u_{r}(k_{r}),\\ v(k,m_k)& = v_{0}(k_0)+ v_{r}(k_{r}). \end{split} $ | (17) |
$ \begin{split} u^+_{0}(k_0) &\equiv \left|k_{0} \right \rangle,\quad u^+_{r}(k_{r}) \equiv \dfrac{m_k}{\left[k_{0}k_{r}\right]}\left|k_{r}\right],\\ u^-_{0}(k_0) &\equiv \left|k_{0} \right ],\quad u^-_{r}(k_{r}) \equiv \dfrac{m_k}{\left\langle k_{0}k_{r}\right\rangle}\left|k_{r}\right\rangle,\\ v^+_{0}(k_0) &\equiv \left|k_{0} \right ],\quad v^+_{r}(k_{r}) \equiv -\dfrac{m_k}{\left\langle k_{0}k_{r}\right\rangle}\left|k_{r}\right\rangle,\\ v^-_{0}(k_0) &\equiv \left|k_{0} \right \rangle,\quad v^-_{r}(k_{r}) \equiv -\dfrac{m_k}{\left[k_{0}k_{r}\right]}\left|k_{r}\right]. \end{split} $ | (18) |
$ \begin{split} u^+_{0}(k_0) \bar{u}^+_{0}(k_0) + u^-_{0}(k_0) \bar{u}^-_{0}(k_0) & = {{\not\! k}_0},\\ v^+_{0}(k_0) \bar{v}^+_{0}(k_0) + v^-_{0}(k_0) \bar{v}^-_{0}(k_0) & = {{\not\! k}_0},\\ u^+_{r}(k_r) \bar{u}^+_{r}(k_r) + u^-_{r}(k_r) \bar{u}^-_{r}(k_r) & = \dfrac{m^2_k}{2 k_0 \cdot k_r} {{\not\! k}_r}, \\ v^+_{r}(k_r) \bar{v}^+_{r}(k_r) + v^-_{r}(k_r) \bar{v}^-_{r}(k_r) & = \dfrac{m^2_k}{2 k_0 \cdot k_r} {{\not\! k}_r}, \end{split} $ |
$ \begin{split} u^+_{0}(k_0) \bar{u}^+_{r}(k_r) + u^-_{0}(k_0) \bar{u}^-_{r}(k_r) & = \dfrac{1}{m_k}{{\not\! k}_0}{{\not\! k}},\\ u^+_{r}(k_r) \bar{u}^+_{0}(k_0) + u^-_{r}(k_r) \bar{u}^-_{0}(k_0) & = \dfrac{1}{m_k}{{\not\! k}}{{\not\! k}_0},\\ v^+_{0}(k_0) \bar{v}^+_{r}(k_r) + v^-_{0}(k_0) \bar{v}^-_{r}(k_r) & = -\dfrac{1}{m_k}{{\not\! k}_0}{{\not\! k}},\\ v^+_{r}(k_r) \bar{v}^+_{0}(k_0) + v^-_{r}(k_r) \bar{v}^-_{0}(k_0) & = -\dfrac{1}{m_k}{{\not\! k}}{{\not\! k}_0}. \end{split} $ | (19) |
$ \begin{split} {{\not\! k}}u_{0}(k_0) & = m_k\, u_{r}(k_r),\quad {{\not\! k}}u_{r}(k_r) = m_k\, u_{0}(k_0),\\ {{\not\! k}}v_{0}(k_0) & = -m_k\, v_{r}(k_r),\quad {{\not\! k}}v_{r}(k_r) = -m_k\, v_{0}(k_0). \end{split} $ | (20) |
Figure1. One-loop triangle diagram for a virtual Z boson decaying into a top-quark pair.
The relevant amplitude can be written as
$ {\cal A} = \int {\rm d}^D q \dfrac{N(q,k_1,k_2,k_3)}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }, $ | (21) |
$ \begin{split} {\cal D}_1 & = q^2 -m_t^2,\\ {\cal D}_2 & = (q-k_1)^2 -m_t^2,\\ {\cal D}_3 & = (q-k_3)^2. \end{split} $ | (22) |
$ \begin{split} N_R(q,k_1,k_2,k_3) = & -g^2_s \, \bar{u}(k_2,m_t)\gamma^a({{\not\! k}_1}-{\not\!\! q}+m_t)\\&\times{\not\! {\varepsilon}}P_R({\not \!\!q}+m_t)\gamma^a{v}(k_3,m_t). \end{split} $ | (23) |
$ \begin{split} {v}({k_3,m_t}) & = {v}_0(k_{30})+{v}_r(k_{3r}),\\ k_3& = k_{30}+\dfrac{m^2_t}{2k_{30}\cdot k_{3r}}k_{3r}. \end{split} $ | (24) |
$ \begin{split} s_1 &\equiv k_{3r}\cdot k_{30}, \\ s_2 &\equiv k_{3r}\cdot k_{1}. \end{split} $ | (25) |
$ \begin{split} {\cal M}_1 & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not\! {\varepsilon}} v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{2} & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not\! {\varepsilon}} v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{3} & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not\! {\varepsilon}}{\not\!\! q}v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{4} & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not\! {\varepsilon}}{\not\!\! q}v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{5}& = \int {\rm d}^D q \dfrac{(k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{6}& = \int {\rm d}^D q \dfrac{(k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{7}& = \int {\rm d}^D q \dfrac{(k_3\cdot\varepsilon)\bar{u}(k_2,m_t){\not\!\! q}v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{8}& = \int {\rm d}^D q \dfrac{(k_3\cdot\varepsilon)\bar{u}(k_2,m_t){\not\!\! q}v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{9}& = \int {\rm d}^D q \dfrac{(q\cdot\varepsilon)\bar{u}(k_2,m_t)v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{10}& = \int {\rm d}^D q \dfrac{(q\cdot\varepsilon)\bar{u}(k_2,m_t)v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{11}& = \int {\rm d}^D q \dfrac{(q\cdot\varepsilon)\bar{u}(k_2,m_t){\not\!\! q}v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{12}& = \int {\rm d}^D q \dfrac{(q\cdot\varepsilon)\bar{u}(k_2,m_t){\not\!\! q}v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }. \end{split} $ | (26) |
$ \begin{split} F_{1} & = \bar{u}(k_2,m_t){\not \!\!{\varepsilon}} v_{0}(k_{30}),\\ F_{2} & = \bar{u}(k_2,m_t){\not\!\! {\varepsilon}} v_{r}(k_{3r}),\\ F_{3} & = (k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{0}(k_{30}),\\ F_{4} & = (k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{r}(k_{3r}). \end{split} $ | (27) |
$ \begin{align} {\cal M}_{p} = d_{1,p}F_{1}+d_{2,p}F_{2}+d_{3,p}F_{3}+d_{4,p}F_{4}. \end{align} $ | (28) |
$ \begin{split} {\cal M}_{3} & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not \!\!{\varepsilon}}{\not\!\! q}v_{0}(k_{30})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 },\\ {\cal M}_{4} & = \int {\rm d}^D q \dfrac{\bar{u}(k_2,m_t){\not \!\!{\varepsilon}}{\not\!\! q}v_{r}(k_{3r})}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }. \end{split} $ | (29) |
$ \begin{split} {{\not\! k}_3}v_{0}(k_{30}) = -m_t v_{r}(k_{3r}),\quad {{\not\! k}_3}v_{r}(k_{3r}) = -m_t v_{0}(k_{30}), \end{split} $ | (30) |
$ d_{3,1} = d_{4,2},\quad d_{3,2} = d_{4,1},\quad d_{3,3} = d_{4,4},\quad d_{3,4} = d_{4,3}. $ | (31) |
$ \begin{split} {k_{3r}} \cdot {{q}} = &\dfrac{2}{Q^4-4 {m^2_t} {Q^2}}\Big \{({k_2}\cdot {{q}} ) ({Q^2} {s_1}-2 {m^2_t} {s_2})\\ &+({k_3}\cdot {{q}} )({Q^2} ({s_2}-{s_1})-2 {m^2_t} {s_2})\Big\}. \end{split} $ | (32) |
$ \begin{split} {u}({k_2,m_t}) & = {u}_0(k_{20})+{u}_r(k_{2r}),\\ k_2& = k_{20}+\dfrac{m^2_t}{2k_{20}\cdot k_{2r}}k_{2r}. \end{split} $ | (33) |
$ \begin{split} {\mathbb P}_{-+} F_1 & = \left\langle k_{20}|{\not\!\! {\varepsilon}}|k_{30}\right],\\ {\mathbb P}_{++} F_1 & = \dfrac{m_t}{\left\langle k_{2r}|k_{20}\right\rangle}\left\langle k_{2r}|{\not\!\! {\varepsilon}}|k_{30}\right],\\ {\mathbb P}_{++} F_2 & = -\dfrac{m_t}{\left\langle k_{30}|k_{3r}\right\rangle} \left[k_{20}|{\not\!\! {\varepsilon}}|k_{3r}\right\rangle,\\ {\mathbb P}_{-+} F_2 & = -\dfrac{m_t^2}{\left[k_{2r}|k_{20}\right]\left\langle k_{30}|k_{3r}\right\rangle} \left[k_{2r}|{\not\!\! {\varepsilon}}|k_{3r}\right\rangle,\\ {\mathbb P}_{++} F_3 & = (k_3\cdot\varepsilon)\left[k_{20}|k_{30}\right],\\ {\mathbb P}_{-+} F_3 & = (k_3\cdot\varepsilon)\dfrac{m_t}{\left[k_{2r}|k_{20}\right]}\left[k_{2r}|k_{30}\right],\\ {\mathbb P}_{-+} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t}{\left\langle k_{30}|k_{3r}\right\rangle} \left\langle k_{20}|k_{3r}\right\rangle,\\ {\mathbb P}_{++} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t^2}{\left\langle k_{2r}|k_{20}\right\rangle\left\langle k_{30}|k_{3r}\right\rangle}\left\langle k_{2r}|k_{3r}\right\rangle,\\ {\mathbb P}_{+-} F_1 & = \left[k_{20}|{\not \!\!{\varepsilon}}| k_{30} \right\rangle ,\\ {\mathbb P}_{--} F_1 & = \dfrac{m_t}{\left[ k_{2r}|k_{20}\right]} \left[k_{2r}|{\not\!\! {\varepsilon}} |k_{30}\right\rangle,\\ {\mathbb P}_{--} F_2 & = -\dfrac{m_t}{\left[ k_{30}|k_{3r}\right]} \left\langle k_{20}|{\not\!\! {\varepsilon}}|k_{3r}\right],\\ {\mathbb P}_{+-} F_2 & = -\dfrac{m_t^2}{\left\langle k_{2r}|k_{20}\right\rangle \left[ k_{30}|k_{3r}\right]} \left\langle k_{2r}|{\not \!\!{\varepsilon}}| k_{3r}\right],\\ {\mathbb P}_{--} F_3 & = (k_3\cdot\varepsilon)\left\langle k_{20}|k_{30} \right\rangle,\\ {\mathbb P}_{+-} F_3 & = (k_3\cdot\varepsilon)\dfrac{m_t}{\left\langle k_{2r}|k_{20}\right\rangle} \left\langle k_{2r}|k_{30} \right\rangle,\\ {\mathbb P}_{+-} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t}{\left[ k_{30}|k_{3r}\right]} \left[ k_{20}|k_{3r}\right],\\ {\mathbb P}_{--} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t^2}{\left[ k_{2r}|k_{20}\right]\left[ k_{30}|k_{3r}\right]} \left[ k_{2r}|k_{3r}\right]. \end{split} $ | (34) |
$ \begin{align} c_{1,-+} = &c_{1,++} = c_{2,--} = c_{2,+-} \\ = &\dfrac{-g^2_s}{Q^4-4 m^2_{t} Q^2} \int {\rm d}^D q \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }\big\{8 m^4_{t} Q^2-2 m^2_{t} Q^4+\left(8 (D-4) m^4_{t}-4 m^2_{t} Q^2\right) (k_2 \cdot q )+(8 (D-4) m^4_{t}-4 (D-5) m^2_{t} Q^2)(k_3 \cdot q )\big\},\\ \\ c_{2,++} = &c_{2,-+} = c_{1,+-} = c_{1,--} \\ = & \dfrac{-g^2_s}{Q^4-4 m^2_{t} Q^2} \int {\rm d}^D q \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }\big\{(D-4) m^2_{t} Q^2 \left(Q^2-4 m^2_{t}\right)+(8 (D-4) m^4_{t}-4 (D-1)m^2_{t} Q^2+4 Q^4)(k_2 \cdot q )\\&+\left(8 (D-4) m^4_{t}+4 (7-2 D) m^2_{t} Q^2+2 (D-4) Q^4\right) (k_3 \cdot q )+\left(16 m^2_{t}-8 Q^2\right)(k_2 \cdot q ) (k_3 \cdot q )\\&+8 m^2_{t} (k_2 \cdot q )^2+8 m^2_{t} (k_3 \cdot q )^2 -(D-4) Q^2\left(Q^2-4 m^2_{t}\right) q^2\big\},\\ \\ c_{3,++} = & c_{3,-+} = c_{4,+-} = c_{4,--} \\ = & \dfrac{-g^2_s}{\left(Q^3-4 m^2_t Q\right)^2}\int {\rm d}^D q \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }\big\{2(D-4) m_{t} \left(Q^3-4 m^2_{t} Q\right)^2+(-32 (D-4) m_{t}^5+8 \,D\, m^3_{t} Q^2-8 m_{t} Q^4)(k_2 \cdot q )\\&-4 m_{t} \left(4 m^2_{t}-Q^2\right) \left(2 (D-4) m^2_{t}-(D-6) Q^2\right) (k_3 \cdot q )+\left(8\, D \,m_{t} Q^2-32 m^3_{t}\right)(k_2 \cdot q ) (k_3 \cdot q )\\&+\left(16 (D-3) m^3_{t}-8 (D-2) m_{t} Q^2\right) (k_2 \cdot q )^2-16 (D-1) m^3_{t} (k_3 \cdot q)^2-4 m_{t} Q^2\left(Q^2-4 m^2_{t}\right) q^2\big\},\\ \\ c_{4,-+} = & c_{4,++} = c_{3,--} = c_{3,+-} \\ = & \dfrac{-g^2_s}{\left(Q^3-4 m^2_t Q\right)^2}\int {\rm d}^D q \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 }\big\{-8 (D-4) m^3_{t} \left(4 m^2_{t}-Q^2\right) (k_2 \cdot q )-4 (D-4) m_{t} (8 m^4_{t}-6 m^2_{t} Q^2 +Q^4) (k_3 \cdot q )\\&+\left(8\,D\,m_{t} Q^2-32 m^3_{t}\right) (k_2 \cdot q ) (k_3 \cdot q )-16 (D-1) m^3_{t}(k_2 \cdot q )^2 +\left(16 (D-3) m^3_{t}-8 (D-2) m_{t} Q^2\right) (k_3 \cdot q )^2\\&-4 m_{t} Q^2 \left(Q^2 -4 m^2_{t}\right)q^2\big\}. \end{align} $ | (35) |
Figure2. Two-loop triangle diagram for a virtual Z boson decaying into a top-quark pair.
$ {\cal A} = \int {\rm d}^D q_1 {\rm d}^D q_2 \dfrac{N(q_1,q_2,k_1,k_2,k_3)}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 {\cal D}_4 {\cal D}_5 {\cal D}_6 }, $ | (36) |
$ \begin{split} {\cal D}_1 & = (q_2+k_2)^2 -m_t^2,\\ {\cal D}_2 & = (q_1)^2 - m_t^2,\\ {\cal D}_3 & = (q_1-k_1)^2 ,\\ {\cal D}_4 & = (q_2-k_3)^2 - m_t^2,\\ {\cal D}_5 & = (q_2)^2,\\ {\cal D}_6 & = (q_1-q_2-k_2)^2. \end{split} $ | (37) |
$ {\cal D}_7 = (q_1-k_2)^2. $ | (38) |
$ \begin{split} N_R(&q_1,q_2,k_1,k_2,k_3) = g^4_s \bar{u}(k_2)\gamma^a({{\not\!\! q}_2}+{{\not\!\! k}_2}+m_t)\gamma^b({{\not\!\! q}_1}+m_t)\\ &\times {\not\!\! {\varepsilon}}P_R({{\not \!\!k}_1}-{{\not\!\! q}_1}+m_t) \gamma^b ({{\not\!\! k}_3}-{{\not \!\!q}_2}+m_t)\gamma^a v(k_3). \end{split} $ | (39) |
$ \begin{split} s_1 \equiv k_{3r}\cdot k_{30},\quad s_2 \equiv k_{3r}\cdot k_{1} . \end{split} $ | (40) |
$ \begin{split} F_{1} & = \bar{u}(k_2,m_t){\not\!\! {\varepsilon}} v_{0}(k_{30}),\quad F_{2} = \bar{u}(k_2,m_t){\not\!\! {\varepsilon}} v_{r}(k_{3r}),\\ F_{3} & = (k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{0}(k_{30}),\\ F_{4} & = (k_3\cdot\varepsilon)\bar{u}(k_2,m_t)v_{r}(k_{3r}). \end{split} $ | (41) |
$ \begin{split} {k_{3r}} \cdot {{q_1}} = & \dfrac{2}{Q^4-4 {m^2_t} Q^2} \{({k_2}\cdot {{q_1}} ) ({Q^2} {s_1}-2 {m^2_t} {s_2})\\ &+({k_3}\cdot {{q_1}} )({Q^2} ({s_2}-{s_1})-2 {m^2_t} {s_2})\}, \end{split} $ | (42) |
$ \begin{split} \\ {k_{3r}} \cdot {{q_2}} = & \dfrac{2}{Q^4-4 {m^2_t} Q^2} \{({k_2}\cdot {{q_2}} ) ({Q^2} {s_1}-2 {m^2_t} {s_2})\\ &+({k_3}\cdot {{q_2}} )({Q^2} ({s_2}-{s_1})-2 {m^2_t} {s_2})\}. \end{split} $ | (43) |
$ \begin{split} {\mathbb P}_{-+} F_1 & = \left\langle k_{20}|{\not \!\! {\varepsilon}}|k_{30}\right],\\ {\mathbb P}_{++} F_1 & = \dfrac{m_t}{\left\langle k_{2r}|k_{20}\right\rangle}\left\langle k_{2r}|{\not \!\! {\varepsilon}}|k_{30}\right],\\ {\mathbb P}_{++} F_2 & = -\dfrac{m_t}{\left\langle k_{30}|k_{3r}\right\rangle} \left[k_{20}|{\not \!\! {\varepsilon}}|k_{3r}\right\rangle,\\ {\mathbb P}_{-+} F_2 & = -\dfrac{m_t^2}{\left[k_{2r}|k_{20}\right]\left\langle k_{30}|k_{3r}\right\rangle} \left[k_{2r}|{\not \!\! {\varepsilon}}|k_{3r}\right\rangle,\\ {\mathbb P}_{++} F_3 & = (k_3\cdot\varepsilon)\left[k_{20}|k_{30}\right],\\ {\mathbb P}_{-+} F_3 & = (k_3\cdot\varepsilon)\dfrac{m_t}{\left[k_{2r}|k_{20}\right]}\left[k_{2r}|k_{30}\right], \end{split} $ |
$ \begin{split} {\mathbb P}_{-+} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t}{\left\langle k_{30}|k_{3r}\right\rangle} \left\langle k_{20}|k_{3r}\right\rangle,\\ {\mathbb P}_{++} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t^2}{\left\langle k_{2r}|k_{20}\right\rangle\left\langle k_{30}|k_{3r}\right\rangle}\left\langle k_{2r}|k_{3r}\right\rangle,\\ {\mathbb P}_{+-} F_1 & = \left[k_{20}|{\not \!\! {\varepsilon}}| k_{30} \right\rangle ,\\ {\mathbb P}_{--} F_1 & = \dfrac{m_t}{\left[ k_{2r}|k_{20}\right]} \left[k_{2r}|{\not \!\! {\varepsilon}} |k_{30}\right\rangle,\\ {\mathbb P}_{--} F_2 & = -\dfrac{m_t}{\left[ k_{30}|k_{3r}\right]} \left\langle k_{20}|{\not \!\! {\varepsilon}}|k_{3r}\right],\\ {\mathbb P}_{+-} F_2 & = -\dfrac{m_t^2}{\left\langle k_{2r}|k_{20}\right\rangle \left[ k_{30}|k_{3r}\right]} \left\langle k_{2r}|{\not \!\! {\varepsilon}}| k_{3r}\right],\\ {\mathbb P}_{--} F_3 & = (k_3\cdot\varepsilon)\left\langle k_{20}|k_{30} \right\rangle,\\ {\mathbb P}_{+-} F_3 & = (k_3\cdot\varepsilon)\dfrac{m_t}{\left\langle k_{2r}|k_{20}\right\rangle} \left\langle k_{2r}|k_{30} \right\rangle,\\ {\mathbb P}_{+-} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t}{\left[ k_{30}|k_{3r}\right]} \left[ k_{20}|k_{3r}\right],\\ {\mathbb P}_{--} F_4 & = -(k_3\cdot\varepsilon) \dfrac{m_t^2}{\left[ k_{2r}|k_{20}\right]\left[ k_{30}|k_{3r}\right]} \left[ k_{2r}|k_{3r}\right]. \end{split} $ | (44) |
$ \begin{split} x_1 & = q_1 \cdot k_2,\quad x_2 = q_1 \cdot k_3,\\ x_3 & = q_2 \cdot k_2,\quad x_4 = q_2 \cdot k_3,\\ x_5 & = q_1^2,\quad x_6 = q_2^2,\quad x_7 = q_1 \cdot q_2. \end{split} $ | (45) |
$ \begin{split} c_{1,-+} = &c_{1,++} = c_{2,--} = c_{2,+-} = \dfrac{2g^4_s {m^2_t}}{(D-2) {Q^2} \left(Q^2-4 {m^2_t}\right)}\int {\rm d}^D q_1 {\rm d}^D q_2 \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 {\cal D}_4 {\cal D}_5 {\cal D}_6 }\\& \times \Big\{ -16 (D-2) m^2_{t} x_1^2 +8 (D-2) x_4 x_1^2 -16 (D-2) m^2_{t} x_2^2 -4 \left(D^3-16 D^2+80 D-124\right) x_1 x_4^2 \\& +4 \left(D^3-12 D^2+44 D-52\right) m^2_{t} x_4^2 -4 (D-3) (D-2) Q^2 \left(Q^2-4 m^2_{t}\right) x_1 -4 (D-2) Q^2 \left(Q^2-4 m^2_{t}\right) x_2 \\& -2 Q^2 \left(2 D^2-19 D+38\right) \left(Q^2-4 m^2_{t}\right) x_7 +2 (D-4)(D-2) \left(Q^2-4 m^2_{t}\right) Q^2 x_5\\& +(D-4) \left(D^2-6 D+10\right) \left(Q^2-4 m^2_{t}\right) Q^2 x_6 +4 (D-4) \left((D-4)^2 m^2_{t}-(D-5) Q^2\right) x_1 x_6 \\& +2 (D-4) \left(2 (D-4)^2 m^2_{t}-\left(D^2-10 D+26\right) Q^2\right) x_2 x_6 +2 (D-2) \left(Q^2-4 m^2_{t}\right) Q^2 \left((D-2) m^2_{t}-Q^2\right) \\& + \left(4 \left(D^3-12 D^2+44 D-52\right) m^2_{t} -2 (D-4)^2 (D-2) Q^2\right)x_3^2 +4 \left(D^3-14 D^2+64 D-92\right) x_2 x_3^2 \\& +16 (D-2) \left(Q^2-2 m^2_{t}\right) x_1 x_2 -8 (D-2) x_1 x_2 x_3 +8 (D-5) (D-2) x_2^2 x_3 +2 (D-4) (D-2) \left(2 (D-4) m^2_{t}-Q^2\right) x_3 x_5 \\& +2 (D-2) \left(2 (D-4) (D-2) m^4_{t}+\left(-4 D^2+25 D-42\right) Q^2 m^2_{t}+\left(D^2-7 D+14\right) Q^4\right) x_3 \\& +\left(4 (D-4) (D-2) Q^2-8 \left(2 D^2-19 D+38\right) m^2_{t}\right) x_1 x_3 \\& +\left(8 \left(D^3-12 D^2+51 D-70\right) m^2_{t} -4 \left(D^3-11 D^2+43 D-58\right) Q^2\right) x_2 x_3 \\& +2 (D\!-\!4) (D\!-\!2) \left(2 (D\!-\!4) m^2_{t}-(D\!-\!5) Q^2\right) x_4 x_5 +2 (D-2) \left(2 (D-4) (D-2) m^4_{t}-\left(D^2-9 D+14\right) Q^2 m^2_{t}-2 Q^4\right) x_4 \\& +\left(4 \left(D^3-10 D^2+25 D-10\right) Q^2-8 (D-3) \left(D^2-5 D-2\right) m^2_{t}\right) x_1 x_4 \\& +\left(8 (D-5) (D-2) Q^2 -8 \left(2 D^2-19 D+38\right) m^2_{t}\right) x_2 x_4 -8 (D-5) (D-2) x_1 x_2 x_4 \\& +\left(8 \left(D^3-12 D^2+44 D-52\right) m^2_{t} -2 \left(D^3-14 D^2+56 D-72\right) Q^2\right) x_3 x_4 -4 \left(D^3-14 D^2+64 D-92\right) x_1 x_3 x_4 \\& +4 \left(D^3\!-\!16 D^2\!+\!80 D\!-\!124\right) x_2 x_3 x_4 \!+\!4 (D\!-\!2) \left(2 (D-4) m^2_{t}\!-\!Q^2\right) x_1 x_7 +4 (D-2) \left(2 (D-4) m^2_{t}-(D-5) Q^2\right) x_2 x_7 \\& -8 (D-3) \left((D-4)^2 m^2_{t}-(D-5) Q^2\right) x_3 x_7-4 (D-3) \left(2 (D-4)^2 m^2_{t}-\left(D^2-10 D+26\right) Q^2\right) x_4 x_7 \Big\}, \end{split} $ | (46) |
$ \begin{split} c_{2,++} = &c_{2,-+} = c_{1,+-} = c_{1,--} = \dfrac{g^4_s}{(D-2)Q^2(Q^2-4m^2_t)}\int {\rm d}^D q_1 {\rm d}^D q_2 \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 {\cal D}_4 {\cal D}_5 {\cal D}_6 }\\ &\times\Big\{ 8 (D-4)^2 (D-3) m^4_{t} x_3^2 +4 (D-2) \left(2 (D-4) (D-2) m^2_{t}+(14-3 D) Q^2\right) m^4_{t} x_4 -16 (D-2)\left(2 m^2_{t}-Q^2\right) m^2_{t}x_2^2 \\& -8 (D-2)^2 m^2_{t} x_3^2 x_5 -8 (D-2)^2 m^2_{t} x_4^2 x_5 -8 (D-2)^2 m^2_{t} x_1^2 x_6 -8 (D-2)^2 m^2_{t} x_2^2 x_6 -(D-4)^3 Q^2 m^2_{t} \left(4 m^2_{t}-Q^2\right) x_6 \\& +16 (D-2) \left(Q^2-2 m^2_{t}\right) m^2_{t}x_1^2 -2 (D-2) Q^2 m^2_{t} \left(4 m^2_{t}-Q^2\right) \left(2 (D-2) m^2_{t}-(D-4) Q^2\right)\\& +4 (D-4)^2 \left(2 (D-3) m^2_{t}-(D-2) Q^2\right) m^2_{t}x_4^2\\& +4 (D-2) \left(2 (D-4) (D-2) m^4_{t}-(D-3) (D-2) Q^2 m^2_{t}+(D-4) Q^4\right)m^2_{t} x_3 \\& +4 (D-4)^2 m^2_{t} \left(4 (D-3) m^2_{t}-(D-4) Q^2\right) x_3 x_4 +16 (D-2)^2 m^2_{t} x_1 x_3 x_7 +16 (D-2)^2 m^2_{t} x_2 x_4 x_7 \\& -16 (D-2) \left(3 m^2_{t}-Q^2\right) x_3 x_2^2 +4 (D-2)^2 \left(Q^2-4 m^2_{t}\right) Q^2 x_7^2 -2 (D-4) (D-2) Q^2 \left(8 m^4_{t}-6 Q^2 m^2_{t}+Q^4\right) x_5\\& -8 (D-2)^2 \left(2 m^2_{t}-Q^2\right) x_3 x_4 x_5 +(D-6) (D-2)^2 \left(Q^2-4 m^2_{t}\right) Q^2 x_5 x_6\\& +2 (D-4)\left(4 (D-4)^2 m^4_{t} +2 \left(D^2-6 D+14\right) Q^2 m^2_{t}-\left(D^2-8 D+20\right) Q^4\right) x_1 x_6\\& +8 (D-4) \left((D-4)^2 m^4_{t} +(3 D-11) Q^2 m^2_{t}-(D-4) Q^4\right) x_2 x_6 -8 (D-2)^2 \left(2 m^2_{t}-Q^2\right) x_1 x_2 x_6 \\& +\left(4 \left(D^3-14 D^2+68 D-104\right) Q^2-8 \left(D^3-16 D^2+80 D-124\right) m^2_{t}\right) x_1 x_4^2 \\& +4 (D-2) Q^2 \left(4 m^2_{t}-Q^2\right) \left(2 (D-3) m^2_{t}-(D-4) Q^2\right) x_1\\& + \left(8 \left(D^3-14 D^2+64 D-92\right) m^2_{t}-4 (D-4) \left(D^2-12 D+28\right) Q^2\right) x_2 x_3^2 +8 (D-2) Q^2 \left(4 m^4_{t}-5 Q^2 m^2_{t}+Q^4\right) x_2 \\& -16 (D-2) \left(Q^2-2 m^2_{t}\right)^2 x_1 x_2 -16 (D-2) \left((D-1) m^2_{t}-Q^2\right) x_1 x_2 x_3 \\& +4 (D-4) (D-2) \left(2 (D-4) m^4_{t} -(D-1) Q^2 m^2_{t}+Q^4\right) x_3 x_5 \\& +\left(-16 \left(3 D^2-21 D+38\right) m^4_{t}+16 (3 D-8) (D-4) Q^2 m^2_{t} -8 (D-4) (D-2) Q^4\right) x_1 x_3 \\& +\left(16 \left(D^3-13 D^2+53 D-70\right) m^4_{t}+8 \left(5 D^2-35 D+66\right) Q^2 m^2_{t} -8 \left(D^2-7 D+14\right) Q^4\right) x_2 x_3 \\& -16 (D-2) \left(Q^2-(D-1) m^2_{t}\right) x_1^2 x_4 +4 (D-4) (D-2) \left(2 (D-4) m^4_{t}+3 Q^2 m^2_{t}-Q^4\right) x_4 x_5 \\& -8 \left(2 \left(D^3-7 D^2+11 D+6\right) m^4_{t}+\left(-2 D^2+23 D-54\right) Q^2 m^2_{t}+(10-3 D) Q^4\right) x_1 x_4 \\& +\left(-16 \left(3 D^2-21 D+38\right) m^4_{t}+8 \left(D^2-16 D+36\right) Q^2 m^2_{t} +16 (D-2) Q^4\right) x_2 x_4 +16 (D-2) \left(3 m^2_{t}-Q^2\right) x_1 x_2 x_4 \\& +\left(4 (D-4) \left(D^2-12 D+28\right) Q^2 -8 \left(D^3-14 D^2+64 D-92\right) m^2_{t}\right) x_1 x_3 x_4 \\& +\left(8 \left(D^3-16 D^2+80 D-124\right) m^2_{t} -4 \left(D^3-14 D^2+68 D-104\right) Q^2\right) x_2 x_3 x_4 \\& +4 \left((D-6) (D-3) Q^6+(7 D-22) (5-D) m^2_{t} Q^4 +4 \left(3 D^2-21 D+38\right) m^4_{t} Q^2\right) x_7 \\& +8 (D-2) \left(2 (D-4) m^4_{t}-(D-1) Q^2 m^2_{t}+Q^4\right) x_1 x_7 +8 (D-2) \left(2 (D-4) m^4_{t}+3 Q^2 m^2_{t}-Q^4\right) x_2 x_7 \\& +2 \left(-8 (D-4)^2 (D-3) m^4_{t} +4 \left(3 D^2-20 D+34\right) Q^2 m^2_{t}+(D-4) \left(D^2-12 D+28\right) Q^4\right) x_3 x_7 \\& +8 (D-2)^2 \left(2 m^2_{t}-Q^2\right)x_2 x_3 x_7 +8 (D-2)^2 \left(2 m^2_{t}-Q^2\right) x_1 x_4 x_7 \\& -2 \left(8 (D-4)^2 (D-3) m^4_{t}-4 \left(D^3-12 D^2+52 D-74\right) Q^2 m^2_{t} +\left(D^3-14 D^2+68 D-104\right) Q^4\right) x_4 x_7 \Big\}, \end{split} $ | (47) |
$ \begin{split} c_{3,++} = & c_{3,-+} = c_{4,+-} = c_{4,--} = \dfrac{4m_t g^4_s}{(D-2)(Q^3-4m^2_tQ)^2}\int {\rm d}^D q_1 {\rm d}^D q_2 \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 {\cal D}_4 {\cal D}_5 {\cal D}_6 }\\ &\times\Big\{ -2 (D-2)^2 \left(Q^2-4 m^2_t\right) Q^2 x_7^2 +4 (D-1)(D-2)^2 m^2_t x_5 x_4^2 +2 (D-2)^2 \left((D-2) Q^2-2 (D-3) m^2_t\right) x_5 x_3^2 \\& -2 (D-2)^2 \left(D Q^2-4 m^2_t\right) x_3 x_4 x_5 +4 (D-1) (D-2)^2 m^2_t x_6 x_2^2 +2 (D-2)^2 \left(Q^2-4 m^2_t\right) Q^2 x_5 x_6 \\& +2 (D-2)^2 \left((D-2) Q^2-2 (D-3) m^2_t\right) x_6 x_1^2 -2 (D-2)^2 \left(D Q^2-4 m^2_t\right) x_1 x_2 x_6 -4 (D-1)(D-2)^2 \left(2 m^2_t-Q^2\right) x_2 x_3 x_7 \\ & +4 (D-2)^2 \left(2 (D-3) m^2_t-(D-2) Q^2\right) x_1 x_3 x_7 +4 (D-2)^2 \left(2 (D-3) m^2_t+Q^2\right) x_1 x_4 x_7 -8 (D-1)(D-2)^2 m^2_t x_2 x_4 x_7 \end{split} $ |
$ \begin{split} & +2 (D-3)(D-2) \left(Q^3-4 m^2_t Q\right)^2 x_5 -2 (D-2) \left(4 m^2_t-Q^2\right) \left(2 \left(D^2-6 D+10\right) m^2_t-(D-3) D Q^2\right) x_3 x_5\\ & -4 (D-2)\left(4 m^2_t-Q^2\right) \left(\left(D^2-6 D+10\right) m^2_t+(D-3) Q^2\right) x_4 x_5 +8 (D-2) m^2_t \left(4 m^2_t-Q^2\right) x_2^2 \\& +4 (D-2) \left(4 m^2_t-Q^2\right) \left(2 m^2_t-(D-2) Q^2\right) x_1^2 -2 (D-4)(D-2) Q^2 \left(8 m^4_t-6 Q^2 m^2_t+Q^4\right) x_1 \\& +4 (D-2) \left(Q^2-4 m^2_t\right) Q^2 \left((D+4) m^2_t-2 Q^2\right) x_2 +4 (D-2)\left(4 m^2_t-Q^2\right) \left(4 m^2_t+(D-4) Q^2\right) x_1 x_2 \\& -4 (D-4)(D-2)(D-1) \left(2 m^2_t-Q^2\right) x_1 x_2 x_3 +2 (D-2) m^2_t \left(4 m^2_t-Q^2\right) \left(2 (D-2) m^2_t+(D-4)^2 Q^2\right) x_3 \\& +4 (D-2) \left(2 \left(D^2-5 D+12\right) m^2_t -\left(D^2-5 D+8\right) Q^2\right) x_3 x_2^2 +4 (D-4) (D-2) (D-1) \left(2 m^2_t-Q^2\right) x_4 x_1^2\\& +2 (D-2) m^2_t \left(8 (D-2) m^4_t-2 \left(2 D^2-13 D+26\right) Q^2 m^2_t+\left(D^2-7 D+14\right) Q^4\right) x_4 \\& -4 (D-2) \left(2 \left(D^2-5 D+12\right) m^2_t-\left(D^2-5 D+8\right) Q^2\right) x_1 x_2 x_4 -2 (D-2)\left(4 m^2_t-Q^2\right) \left(2 (D-4) m^2_t +(D-3) D Q^2\right) x_1 x_7 \\& -2 (D-2)\left(8 (D-4) m^4_t+2 \left(-2 D^2+5 D+4\right) Q^2 m^2_t+(D-3) D Q^4\right) x_2 x_7 \\& -4 \left(4 \left(5 D^2-34 D+58\right) m^2_t+\left(D^3-12 D^2+50 D-68\right) Q^2\right) x_1 x_4^2 -\left(3 D^2-20 D+36\right) \left(4 m^2_t-Q^2\right) m^2_t Q^2x_6\\& +\left(-8 (D-4)^3 m^4_t+6 \left(D^3-4 D^2+16\right) Q^2 m^2_t-\left(D^3-24 D+56\right) Q^4\right) x_1 x_6\\& +\left(-8 (D-4)^3 m^4_t+2 \left(D^3-28 D^2+144 D-224\right) Q^2 m^2_t+8 \left(D^2-6 D+10\right) Q^4\right) x_2 x_6 \\& + \left(4 \left(D^3-3 D^2-8 D+28\right) m^4_t+2 (D-4)^2 (D-2) Q^2 m^2_t\right) x_3^2 \\& + \left(4 (D-2) \left(D^2-6 D+10\right) m^2_t Q^2-4 \left(D^3-9 D^2+32 D-44\right) m^4_t\right) x_4^2 \\& + \left(32 (D-3) (2 D-7) m^2_t -4 \left(D^3-3 D^2-10 D+32\right) Q^2\right) x_2x_3^2 \\& -4 \left(2 \left(D^3-15 D^2+76 D-116\right) m^4_t+(5 D-12) (D-6) Q^2 m^2_t+2 (D-2) Q^4\right) x_1 x_3 \\& +\left(-8 (D-3) \left(D^2-24 D+60\right) m^4_t-4 \left(D^3+18 D^2-114 D+172\right) Q^2 m^2_t+2 \left(D^3+D^2-22 D+40\right) Q^4\right) x_2 x_3 \\& +\left(8 \left(D^3-5 D^2-12 D+52\right) m^4_t+ 8 \left(D^3-11 D^2+47 D-70\right) Q^2 m^2_t-2 (D-4) \left(D^2-7 D+14\right) Q^4\right) x_1 x_4 \\& +\left(8 \left(D^3+7 D^2-68 D+116\right) m^4_t-4 \left(D^3+D^2-46 D+88\right) Q^2 m^2_t-16 (D-2) Q^4\right) x_2 x_4 \\& -2 \left(3 D^2-20 D+36\right) \left(D Q^2-4 m^2_t\right)m^2_t x_3 x_4 +\left(4 \left(D^3-3 D^2-10 D+32\right) Q^2 -32 (D-3) (2 D-7) m^2_t\right) x_1 x_3 x_4 \\& +4 \left(4 \left(5 D^2-34 D+58\right) m^2_t+\left(D^3-12 D^2+50 D-68\right) Q^2\right) x_2 x_3 x_4 \\& +\left((D-6) (D-4) (D-3) Q^6+2 \left(-4 D^3+51 D^2-212 D+284\right) m^2_t Q^4 +8 \left(2 D^3-25 D^2+104 D-140\right) m^4_t Q^2\right) x_7 \\& +2 (D-4) \left(4 m^2_t-Q^2\right) \left(2 (D-4) (D-3) m^2_t +(10-3 D) Q^2\right) x_3 x_7 \\& +2 \left(8 (D-4)^2 (D-3) m^4_t+2 \left(-3 D^3+35 D^2-140 D+184\right) Q^2 m^2_t +\left(D^3-12 D^2+50 D-68\right) Q^4\right) x_4 x_7 \Big\}, \end{split} $ | (48) |
$ \begin{split} c_{4,-+} = & c_{4,++} = c_{3,--} = c_{3,+-} = \dfrac{2m_tg^4_s}{(D-2)(Q^3-4m^2_tQ)^2}\int {\rm d}^D q_1 {\rm d}^D q_2 \dfrac{1}{ {\cal D}_1 {\cal D}_2 {\cal D}_3 {\cal D}_4 {\cal D}_5 {\cal D}_6 }\\&\times \Big\{-4 (D-2)^2 \left(Q^2-4 m^2_t\right) Q^2 x_7^2 +8 (D-1) x_3^2 m^2_t x_5 (D-2)^2 +4 (D-2)^2 \left((D-2) Q^2-2 (D-3) m^2_t\right)x_4^2 x_5 \\& -4 (D-2)^2 \left(D Q^2-4 m^2_t\right) x_3 x_4 x_5 +8 (D-1) (D-2)^2 m^2_t x_6 x_1^2 +4 (D-2)^2 \left(Q^2-4 m^2_t\right) Q^2 x_5 x_6 \\& +4 (D-2)^2 \left((D-2) Q^2-2 (D-3) m^2_t\right) x_6 x_2^2-4 (D-2)^2 \left(D Q^2-4 m^2_t\right) x_1 x_2 x_6 -16 (D-1)(D-2)^2 m^2_t x_1 x_3 x_7 \\& +8 (D-2)^2 \left(2 (D-3) m^2_t+Q^2\right) x_2 x_3 x_7 -8 (D-1) (D-2)^2 \left(2 m^2_t-Q^2\right) x_1 x_4 x_7 \\& +8 (D-2)^2 \left(2 (D-3) m^2_t-(D-2) Q^2\right) x_2 x_4 x_7 -8 (D-4)(D-2) \left(2 (D+1) m^2_t-Q^2\right) x_3 x_2^2 \\& -4 (D-2)\left(Q^3-4 m^2_t Q\right)^2 x_5 +16 (D-2) m^2_t \left(4 m^2_t-Q^2\right) x_1^2 +8 (D-2)\left(4 m^2_t-Q^2\right) \left(2 m^2_t+(D-2) Q^2\right) x_2^2 \\& +2 (D-2)\left(Q^3-4 m^2_t Q\right)^2 \left(2 (D-2) m^2_t-(D-4) Q^2\right) -4 (D-2)\left(Q^2-4 m^2_t\right) \left(D \left(Q^2-2 m^2_t\right)-2 Q^2\right)Q^2 x_1 \\& +8 (D-2)\left(4 m^2_t-Q^2\right) \left((3 D-8) m^2_t-(D-3) Q^2\right)Q^2 x_2 -8 (D-2)\left(4 m^2_t-Q^2\right) \left(D Q^2-4 m^2_t\right) x_1 x_2\\ & +8 (D-2)\left(4 m^2_t-Q^2\right) \left(2 (D-3) m^2_t-Q^2\right) x_3 x_5 \end{split} $ |
$ \begin{split} & -4 (D-2)\left(4 m^2_t-Q^2\right) \left(2 (D-3) (D-2) m^4_t -(D-2) (D-1) Q^2 m^2_t+(D-4) Q^4\right) x_3 \\& +8 (D-2)\left(2 \left(D^2-3 D+4\right) m^2_t-D Q^2\right) x_1 x_2 x_3 +4 (D-2)\left(4 m^2_t-Q^2\right) \left(4 (D-3) m^2_t-(D-4) Q^2\right) x_4 x_5 \\& +8 (D-2) \left(D Q^2-2 \left(D^2-3 D+4\right) m^2_t\right) x_4 x_1^2 \\& -4 (D-2)\left(4 m^2_t-Q^2\right) \left(2 (D-3) (D-2) m^4_t +2 (D-2) Q^2 m^2_t-(D-4) Q^4\right) x_4 \\& +8 (D-4)(D-2) \left(2 (D+1) m^2_t-Q^2\right) x_1 x_2 x_4 -4 (D-2) \left(4 m^2_t-Q^2\right) \left(2 (D-4) m^2_t-D Q^2\right) x_1 x_7 \\& -4 (D-2)\left(4 m^2_t-Q^2\right) \left(2 (D-4) m^2_t+D Q^2\right) x_2 x_7 \\& +\left((D-4)^2 (D-2) Q^6+2 \left(-4 D^3+43 D^2-148 D+164\right) m^2_t Q^4 +8 \left(2 D^3-23 D^2+84 D-100\right) m^4_t Q^2\right) x_6 \\& +\left(-16 (D-4)^3 m^4_t-4 \left(D^3-4 D^2+24 D-64\right) Q^2 m^2_t+2 \left(D^3-8 D^2+36 D-64\right) Q^4\right) x_1 x_6 \\& -4 \left(4 (D-4)^3 m^4_t+\left(-5 D^3+44 D^2-168 D+240\right) Q^2 m^2_t+\left(D^3-8 D^2+30 D-44\right) Q^4\right) x_2 x_6 \\& + \left(16 (D-2) \left(D^2-7 D+13\right) m^2_t Q^2 -8 \left(5 D^3-49 D^2+160 D-172\right) m^4_t\right)x_3^2 \\& +\left(12 (D-4)^2 (D-2) m^2_t Q^2-8 (D-3) \left(3 D^2-28 D+52\right) m^4_t\right)x_4^2 \\& + \left(32 (D-3) \left(D^2-8 D+22\right) m^2_t-16 \left(D^3-9 D^2+31 D-38\right) Q^2\right) x_1 x_4^2 \\& -8 \left(4 \left(D^3-10 D^2+38 D-50\right) m^2_t+(3 D-10) (D-4) Q^2\right) x_2 x_3^2 \\& -4 \left(4 \left(3 D^3-23 D^2+76 D-100\right) m^4_t-4 \left(D^3-4 D^2+11 D-26\right) Q^2 m^2_t+(3 D-10) D Q^4\right) x_2 x_3 \\& +8 \left(2 \left(D^3-5 D^2-4 D+36\right) m^4_t+\left(-D^3+D^2+14 D-32\right) Q^2 m^2_t+(D-2)^2 Q^4\right) x_1 x_3 \\& +\left(16 \left(3 D^3-25 D^2+60 D-28\right) m^4_t-8 \left(D^3-12 D^2+22 D+20\right) Q^2 m^2_t-4 \left(D^2+2 D-16\right) Q^4\right) x_1 x_4 \\& -8 \left(2 (D-3) \left(D^2+12\right) m^4_t+(3 D-8) (D-6) Q^2 m^2_t-(D-4) (D-2) Q^4\right) x_2 x_4 \\& +\left(-16 \left(4 D^3-43 D^2+148 D-164\right) m^4_t+4 \left(9 D^3-100 D^2+348 D-384\right) Q^2 m^2_t-8 (D-4)^2 (D-2) Q^4\right) x_3 x_4 \\& +8 \left(4 \left(D^3-10 D^2+38 D-50\right) m^2_t+(3 D-10) (D-4) Q^2\right) x_1 x_3 x_4 \\& +\left(16 \left(D^3-9 D^2+31 D-38\right) Q^2-32 (D-3) \left(D^2-8 D+22\right) m^2_t\right) x_2 x_3 x_4 \\& -2 \left((D-6) (D-4) Q^6-2 \left(3 D^2-36 D+92\right) m^2_t Q^4+8 \left(D^2-16 D+44\right) m^4_t Q^2\right) x_7 \\& +4 (D-4) \left(4 m^2_t-Q^2\right) \left(2 (D-4) (D-3) m^2_t+(10-3 D) Q^2\right) x_3 x_7 \\& +4 \left(8 (D-4)^2 (D-3) m^4_t+2 \left(-3 D^3+35 D^2-140 D+184\right) Q^2 m^2_t+\left(D^3-12 D^2+50 D-68\right) Q^4\right) x_4 x_7\Big\}. \end{split} $ | (49) |
$ I_{a_1,a_2,a_3,a_4,a_5,a_6,a_7} \equiv \int \dfrac{{\rm d}^D q_1 {\rm d}^D q_2}{ {\cal D}^{a_1}_1 {\cal D}^{a_2}_2 {\cal D}^{a_3}_3 {\cal D}^{a_4}_4 {\cal D}^{a_5}_5 {\cal D}^{a_6}_6 {\cal D}^{a_7}_7 }. $ | (50) |
$ \begin{split} c_{1,-+} = &c_{1,++} = c_{2,--} = c_{2,+-} = \dfrac{g_s^4}{547285587552000}(-349758480361050 I_{0,0,0,1,1,1,1}+12411033570133 I_{0,0,1,1,1,0,0}\\&+561600032236650 I_{0,1,0,0,0,1,0}+54203486007512020 I_{0,1,0,1,1,1,0}-2524949213666400 I_{0,1,0,1,1,1,1}\\&-37669046330192400 I_{0,1,1,0,0,1,0}+1725030658739520 I_{0,1,1,1,1,0,0}-4392957485551487040 I_{0,1,1,1,1,1,0}\\&-43861242480409840 I_{0,2,0,1,1,1,0}-29063980357164172800 I_{0,2,1,1,1,1,0}+1620333018035519040 I_{1,1,0,1,1,1,1}\\&-72885095952847200 I_{1,1,1,1,1,0,0}+11596870539170956800 I_{2,1,0,1,1,1,1}), \end{split} $ | (51) |
$ \begin{split} c_{2,++} = &c_{2,-+} = c_{1,+-} = c_{1,--} =\dfrac{g_s^4}{16965853214112000}(8269804372235850 I_{0,0,0,1,1,1,1}+128265951369391 I_{0,0,1,1,1,0,0}\\&-91088086096643130 I_{0,1,0,0,0,1,0}-649848399059280020 I_{0,1,0,1,1,1,0}+1445530903043788800 I_{0,1,0,1,1,1,1}\\&+3309765421122010800 I_{0,1,1,0,0,1,0}-48712048971181200 I_{0,1,1,1,1,0,0}+100038303369820946880 I_{0,1,1,1,1,1,0}\\&+2255145332468200880 I_{0,2,0,1,1,1,0}+558971258163723993600 I_{0,2,1,1,1,1,0}-351050113244960902080 I_{1,1,0,1,1,1,1}\\&+4171217340297348000 I_{1,1,1,1,1,0,0}-1701371669315481561600 I_{2,1,0,1,1,1,1}), \end{split} $ | (52) |
$ \begin{split} \\ c_{3,++} = &c_{3,-+} = c_{4,+-} = c_{4,--} =\dfrac{m_tg_s^4}{169658532141120000}(15697729816313400 I_{0,0,0,1,1,1,1}-465105777241369 I_{0,0,1,1,1,0,0}\\&+158565732883624440 I_{0,1,0,0,0,1,0}-576650937419554060 I_{0,1,0,1,1,1,0}-2324636412144847200 I_{0,1,0,1,1,1,1}\\&-5066094845578723200 I_{0,1,1,0,0,1,0}-86512591540544400 I_{0,1,1,1,1,0,0}-14868254094936246720 I_{0,1,1,1,1,1,0}\\&-1977030323227905200 I_{0,2,0,1,1,1,0}-159507262360225382400 I_{0,2,1,1,1,1,0}+591796539388769901120 I_{1,1,0,1,1,1,1}\\&+544504397058468000 I_{1,1,1,1,1,0,0}+2749018573751407718400 I_{2,1,0,1,1,1,1}), \end{split} $ | (53) |
$ \begin{split} \\ c_{4,-+} = &c_{4,++} = c_{3,--} = c_{3,+-} =\dfrac{m_tg_s^4}{18850948015680000}(2930558305311000 I_{0,0,0,1,1,1,1}-155537250759781 I_{0,0,1,1,1,0,0}\\&+20515397403643560 I_{0,1,0,0,0,1,0}-89892274740750940 I_{0,1,0,1,1,1,0}-259544532715984800 I_{0,1,0,1,1,1,1}\\&-575810327185300800 I_{0,1,1,0,0,1,0}-12234451069011600 I_{0,1,1,1,1,0,0}+16492851980585536320 I_{0,1,1,1,1,1,0}\\&-210958059931545200 I_{0,2,0,1,1,1,0}+108826317247815014400 I_{0,2,1,1,1,1,0}+86665228257872911680 I_{1,1,0,1,1,1,1}\\&-54851022351180000 I_{1,1,1,1,1,0,0}+401116455907129497600 I_{2,1,0,1,1,1,1}). \end{split} $ | (54) |