New estimate of the chromomagnetic dipole moment of quarks in the standard model

Abstract

A new estimate of the one loop contributions of the standard model to the chromomagnetic dipole moment (CMDM) $\hat{\mu }_q(q^2)$ of quarks is presented with the aim to address a few disagreements arising in previous calculations. The most general case with arbitrary $q^2$ is considered and analytical results are obtained in terms of Feynman parameter integrals and Passarino-Veltman scalar functions, which are then expressed in terms of closed form functions when possible. It is found that while the QCD contribution to the static CMDM ($q^2=0$) is infrared divergent, which agrees with previous evaluations and stems from the fact that this quantity has no sense in perturbative QCD, the off-shell CMDM ($q^2\ne 0$) is finite and gauge independent, which is verified by performing the calculation for arbitrary gauge parameter $\xi$ via both a renormalizable linear $R_\xi$ gauge and the background field method. It is thus argued that the off-shell $\hat{\mu }_q(q^2)$ can represent a valid observable quantity. For the numerical analysis we consider the region 30 GeV$<\Vert q\Vert<$ 1000 GeV and analyze the behavior of $\hat{\mu }_q(q^2)$ for all the standard model quarks. It is found that the CMDM of light quarks is considerably smaller than that of the top quark as it is directly proportional to the quark mass. In the considered energy interval, both the real and imaginary parts of $\hat{\mu }_t(q^2)$ are of the order of $10^{-2}-10^{-3}$, with the largest contribution arising from the QCD induced diagrams, though around the threshold $q^2=4m_t^2$ there are also important contributions from diagrams with Z gauge boson and Higgs boson exchange.

Analytic results for the loop functions

We now present the results for the loop functions appearing in the contributions to the CMDM of quarks discussed in Sect. 2 in term of Feynman parameter integrals, Passarino-Veltman scalar functions, and closed form functions. For sake of completeness we also include the results for $q^2=0$.

1.1 Feynman parameter integrals

We note that this calculation was done via the unitary gauge as the result are gauge independent, which was explicitly verified via the Passarino-Veltman reduction scheme. Therefore, in the EW sector we only computed by this method the Feynman diagrams (a) through (c) of Fig. 3.

We introduce the definition $r_{a,b}=m_a/m_b$ and present the loop functions for the QCD contributions to the CMDM of quarks. Feynman diagram 2(a) yields the loop function [Eq. (9)]:

$$\begin{aligned} \mathcal {F}^{\mathrm{QCD}_1}_{q}\left( q^2\right) =m_q^2\int _0^1\int _0^{1-u}\frac{ (u-1) u}{ m_q^2 (u-1)^2+q^2 v (u+v-1)}\mathrm{d}v\mathrm{d}u, \end{aligned}$$

(31)

whereas Feynman diagram 2(b) gives [Eq. (14)]:

$$\begin{aligned} \mathcal {F}^{\mathrm{QCD}_2}_{q}\left( q^2\right) =m_q^2\int _0^1\int _0^{1-u} \frac{(u-1)u}{ m_q^2 u^2 - q^2 v (1- u - v)}\mathrm{d}v\mathrm{d}u, \end{aligned}$$

(32)

which for $q^2=0$ reduces to

$$\begin{aligned} \mathcal {F}^{\mathrm{QCD}_2}_q(0)=\int ^1_0 \frac{ (1-u)^2 }{ u}\mathrm{d}u. \end{aligned}$$

(33)

As far as the EW contributions to the CMDM of quarks are concerned, the Feynman diagram with photon exchange of Fig. 3a gives [Eq. (17)]:

$$\begin{aligned} \mathcal {F}_{q}^A\left( q^2\right) =m_q^2\int _0^1\int _0^{1-u}\frac{ (u-1) u}{m_q^2 (u-1)^2+q^2 v (u+v-1)}\mathrm{d}v\mathrm{d}u, \end{aligned}$$

(34)

whereas the Z boson exchange diagram gives [Eq. (23)]

$$\begin{aligned} \mathcal {A}_q^Z\left( q^2\right)= & {} \int ^1_0\int ^{1-u}_0 \frac{\mathrm{d}u\mathrm{d}v}{\varDelta _Z}\Bigg ( (3 u-1) \varDelta _Z \log \left( \frac{\varDelta _Z}{\mu ^2}\right) \nonumber \\&+2 \left( m_q^2 (u-1)^3-2 m_Z^2 u\right) +2 q^2 u v (u+v-1) \Bigg ), \end{aligned}$$

(35)

and

$$\begin{aligned} \mathcal {V}_q^Z\left( q^2\right) =-\int ^1_0\int ^{1-u}_0 \frac{\mathrm{d}u\mathrm{d}v}{\varDelta _Z} m_Z^2 (u-1) u, \end{aligned}$$

(36)

where $\varDelta _Z=m_q^2 (u-1)^2+m_Z^2 u+q^2 v(u+v-1)$ and $\mu$ is the scale of dimensional regularization, which cancels out after integration.

For $q^2=0$, the last two loop functions give:

$$\begin{aligned} \mathcal {A}_q^Z(0)=\int _0^1\mathrm{d}u\frac{ u \left( 2 u^2+r_{Z,q}^2 (u-4) (u-1)\right) }{ r_{Z.q}^2 (u-1)- u^2}, \end{aligned}$$

(37)

and

$$\begin{aligned} \mathcal {V}_q^Z(0)=\int _0^1\mathrm{d}u\frac{ {g_V^q}^2 r_{Z,q}^2 (u-1) u^2}{ r_{Z.q}^2 (u-1)- u^2}. \end{aligned}$$

(38)

As for the W boson exchange contribution of diagram 3(b), it is given by [Eq. (28)]:

$$\begin{aligned} \mathcal {F}_{qq'}^W\left( q^2\right)&=\int ^1_0\int ^{1-u}_0 \frac{\mathrm{d}u\mathrm{d}v}{\varDelta _W}\Bigg ( -(1-3 u) \log \left( \frac{\varDelta _W}{\mu ^2}\right) \varDelta _W\nonumber \\&\quad -2m_{q^\prime }^2(u-1)^2+u\big (2m_q^2(u-1)^2\nonumber \\&\quad -m_W^2(u+3)+2 q^2 v(u+y-1) \big )\Bigg ), \end{aligned}$$

(39)

with the following result for $q^2=0$:

$$\begin{aligned} \mathcal {F}_{qq'}^W(0)=\int _0^1\frac{ u \left[ u \left( (u-1)+r_{q^\prime , q}^2 (u+1)\right) +2 r_{W , q}^2 (u-2) (u-1)\right] }{r_{W , q}^2 (u-1)-u \left( (u-1)+r_{q^\prime , q}^2\right) }\mathrm{d}u, \end{aligned}$$

(40)

where $\varDelta _W= u \left( m_{q}^2 (u-1)+m_W^2\right) -m_{q^\prime }^2 (u-1)+q^2 v(u+v-1)$. Again this calculation was done via the unitary gauge.

Finally, for the Higgs contribution [Eq. (30)] we obtain:

$$\begin{aligned} \mathcal {F}_q^h\left( q^2\right) =m_q^2\int _0^1\int _0^{1-u}\frac{ (u^2-1) }{m_q^2 (u-1)^2+u m_h^2 + q^2 v(u+v-1)} \mathrm{d}v \mathrm{d}u, \end{aligned}$$

(41)

which for $q^2=0$ reduces to

$$\begin{aligned} \mathcal {F}_q^h(0)=-\int ^1_0 \mathrm{d}u \frac{(1+u)(1-u)^2}{(1-u)^2+ur_{h,q}^2}. \end{aligned}$$

(42)

1.2 Passarino-Veltman results

We now present the above results in terms of Passarino-Veltman scalar integrals, where we use the standard notation for the two- and three-point scalar functions. Our calculation was done via a renormalizable linear $R_\xi$ gauge and the BFM to verify that the dependence on the $\xi$ gauge parameter drops out. The loop functions are thus gauge independent and read

$$\begin{aligned} {\mathcal {F}}_q^{\mathrm{QCD}_1}\left( q^2\right) =\frac{m_q^2 }{\eta ^2(\Vert q\Vert ,m_q)}\Big \{\mathbf {B}_0(0;m_q,m_q)-\mathbf {B}_0\left( q^2;m_q,m_q\right) +2\Big \}, \end{aligned}$$

(43)

where we define $\eta (x,y)=\sqrt{x^2-4y^2}$. Also

$$\begin{aligned} {\mathcal {F}}_q^{\mathrm{QCD}_2}\left( q^2\right)&=\frac{m_q^2}{\eta ^4(\Vert q\Vert ,m_q)} \Big \{\left( 8 m_q^2+q^2\right) \left( \mathbf {B}_0(0;m_q,m_q)-\mathbf {B}_0\left( q^2;0,0\right) \right) \nonumber \\&\quad -6 m_q^2 \left( q^2 \mathbf {C}_0\left( m_q^2,m_q^2,q^2;0,m_q,0\right) -4\right) \Big \}, \end{aligned}$$

(44)

$$\begin{aligned} \mathcal {F}_q^{\mathrm{QCD}_2}(0)&=\frac{1}{2}\Big \{\mathbf {B}_0(0;m_q,m_q)+3\Big \}, \end{aligned}$$

(45)

$$\begin{aligned} \mathcal {F}_q^{A}\left( q^2\right)&=\frac{m_q^2}{\eta ^2(\Vert q\Vert ,m_q)}\Big \{\mathbf {B}_0 \left( 0;m_q,m_q\right) -\mathbf {B}_0\left( q^2;m_q,m_q\right) +2\Big \}, \end{aligned}$$

(46)

$$\begin{aligned} \mathcal {A}_q^{Z}\left( q^2\right)&=\frac{m_Z^2}{m_q^2\eta ^4(\Vert q\Vert ,m_q)} \Bigg \{\Big (q^2 \left( m_Z^2-8 m_q^2\right) \nonumber \\&\quad +10 m_q^2 \left( 2 m_q^2-m_Z^2\right) \Big ) \mathbf {B}_0\left( m_q^2;m_q,m_Z\right) \nonumber \\&\quad +\frac{m_q^2}{m_Z^2} \left( q^2 \left( 9 m_Z^2-2 m_q^2\right) + \left( 8 m_q^4-24 m_q^2 m_Z^2+6 m_Z^4\right) \right) \mathbf {B}_0\left( q^2;m_q,m_q\right) \nonumber \\&\quad + \frac{\left( 2 m_q^2+m_Z^2\right) \eta ^2(\Vert q\Vert ,m_q)}{m_Z^2}\left( m_q^2\mathbf {B}_0(0;m_q,m_q)-m_Z^2 \mathbf {B}_0(0;m_Z,m_Z)\right) \nonumber \\&\quad +2\Big (2 q^2 \left( 3 m_Z^2-7 m_q^2\right) +2 q^4 \nonumber \\&\quad +3\left( 8 m_q^4-6 m_q^2 m_Z^2+m_Z^4\right) \Big )m_q^2 \mathbf {C}_0\left( m_q^2,m_q^2,q^2;m_q,m_Z,m_q\right) \nonumber \\&\quad +\frac{(4m_q^4-m_Z^4) \eta ^2(\Vert q\Vert ,m_q)}{m_Z^2}\Bigg \}, \end{aligned}$$

(47)

$$\begin{aligned} \mathcal {V}_q^{Z}\left( q^2\right)&=\frac{m_Z^2}{m_q^2\eta ^4(\Vert q\Vert ,m_q)} \Bigg \{\Big (q^2 \left( m_Z^2-2 m_q^2\right) \nonumber \\&\quad +2 m_q^2 \left( 4 m_q^2-5 m_Z^2\right) \Big ) \mathbf {B}_0\left( m_q^2;m_q,m_Z\right) \nonumber \\&\quad + m_q^2\Big (2 \left( 3 m_Z^2-2 m_q^2\right) + q^2 \Big )\mathbf {B}_0\left( q^2;m_q,m_q\right) \nonumber \\&\quad +\eta ^2(\Vert q\Vert ,m_q)\left( m_q^2\mathbf {B}_0(0;m_q,m_q)-m_Z^2 \mathbf {B}_0(0;m_Z,m_Z)\right) \nonumber \\&\quad +\Big (4 q^2 +2 \left( 3 m_Z^2-8 m_q^2\right) \Big ) m_q^2 m_Z^2\mathbf {C}_0\left( m_q^2,m_q^2,q^2;m_q,m_Z,m_q\right) \nonumber \\&\quad +(2m_q^2-m_Z^2)\eta ^2(\Vert q\Vert ,m_q)\Bigg \}, \end{aligned}$$

(48)

$$\begin{aligned} \mathcal {A}_q^{Z}(0)&=\frac{m_Z^2}{8m_q^4}\Bigg \{\Big (10 m_q^2-5 m_Z^2\Big ) \mathbf {B}_0\left( m_q^2;m_q,m_Z\right) +\Big (3 m_Z^2 -14 m_q^2 \Big )\mathbf {B}_0(0;m_q,m_q)\nonumber \\&\quad +2\Big (2 m_q^2 + m_Z^2 \Big )\mathbf {B}_0(0;m_Z,m_Z)\nonumber \\&\quad +3\Big (m_Z^4 -6 m_q^2 m_Z^2 +8 m_q^4 \Big ) \mathbf {C}_0\left( m_q^2,m_q^2,0;m_q,m_Z,m_q\right) \nonumber \\&\quad +\frac{2}{m_Z^2} \left( m_Z^4-4m_q^4\right) \Bigg \}, \end{aligned}$$

(49)

$$\begin{aligned} \mathcal {V}_q^{Z}(0)&=\frac{m_Z^2}{8m_q^4} \Bigg \{\Big (4 m_q^2 -5 m_Z^2\Big ) \mathbf {B}_0\left( m_q^2;m_q,m_Z\right) +\Big (3 m_Z^2-4 m_q^2\Big ) \mathbf {B}_0(0;m_q,m_q)\nonumber \\&\quad +2 m_Z^2 \mathbf {B}_0(0;m_Z,m_Z) +\Big (3 m_Z^2 -8 m_q^2\Big )m_Z^2 \mathbf {C}_0\left( m_q^2,m_q^2,0;m_q,m_Z,m_q\right) \nonumber \\&\quad +2\left( m_Z^2-2 m_q^2\right) \Bigg \}, \end{aligned}$$

(50)

$$\begin{aligned} \mathcal {F}_{qq'}^{W}\left( q^2\right)&=\frac{1}{m_q^2\eta ^4(\Vert q\Vert ,m_q)} \Big \{ \eta ^2(\Vert q\Vert ,m_q) \left( m_q^2+m_{q^\prime }^2+2 m_W^2\right) \left( m_{q^\prime }^2\mathbf {B}_0(0;m_{q^\prime },m_{q^\prime })\right. \nonumber \\&\quad -\left. m_W^2\mathbf {B}_0(0;m_W,m_W)\right) \nonumber \\&\quad -\Big [q^2 \left( m_q^4+m_q^2 \left( 9 m_W^2-2 m_{q^\prime }^2\right) +m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \nonumber \\&\quad +2 m_q^2 \Big (m_q^4+m_q^2 \left( 4 m_{q^\prime }^2-9 m_W^2\right) \nonumber \\&\quad -5 \left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \Big )\Big ] \mathbf {B}_0\left( m_q^2;m_{q^\prime },m_W\right) \nonumber \\&\quad +m_q^2 \Big [q^2 \left( m_q^2-3 m_{q^\prime }^2+10 m_W^2\right) \nonumber \\&\quad +2 \left( m_q^4+m_q^2 \left( 6 m_{q^\prime }^2-11 m_W^2\right) -3 \left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \right) \Big ]\nonumber \\&\quad \mathbf {B}_0\left( q^2;m_{q^\prime },m_{q^\prime }\right) \nonumber \\&\quad -2 m_q^2 \Big [m_q^2\left( m_q^4-m_q^2 \left( 5 m_{q^\prime }^2+12 m_W^2\right) +\left( 7 m_{q^\prime }^4-12 m_{q^\prime }^2 m_W^2+17 m_W^4\right) \right) \nonumber \\&\quad -3\left( m_{q^\prime }^6-3 m_{q^\prime }^2 m_W^4+2 m_W^6\right) \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\quad -q^2 \left( m_q^4-2 m_q^2 \left( m_{q^\prime }^2+4 m_W^2\right) +m_{q^\prime }^4-6 m_{q^\prime }^2 m_W^2+8 m_W^4\right) \nonumber \\&\quad -2 m_W^2 \left( q^2\right) ^2\Big ] \mathbf {C}_0\left( m_q^2,m_q^2,q^2;m_{q^\prime },m_W,m_{q^\prime }\right) \nonumber \\&\quad +\eta ^2(\Vert q\Vert ,m_q) \left( m_q^2+m_{q^\prime }^2-m_W^2\right) \left( m_q^2+m_{q^\prime }^2+2 m_W^2\right) \Big \}, \end{aligned}$$

(51)

$$\begin{aligned} \mathcal {F}_{qq'}^{W}(0)&=\frac{1}{ 8m_{q}^4}\Big \{2 m_W^2 \left( m_{q}^2+m_{q^\prime }^2+2 m_W^2\right) \mathbf {B}_0(0;m_W,m_W)\nonumber \\&\quad +\Big [m_{q}^4+m_{q}^2 \left( 4 m_{q^\prime }^2-11 m_W^2\right) \nonumber \\&\quad -5 m_{q^\prime }^4-7 m_{q^\prime }^2 m_W^2+6 m_W^4\Big ] \mathbf {B}_0(0;m_{q^\prime },m_{q^\prime })-\Big [m_{q}^4+m_{q}^2 \left( 4 m_{q^\prime }^2-9 m_W^2\right) \nonumber \\&\quad -5 \left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \Big ] \mathbf {B}_0\left( m_{q}^2;m_{q^\prime },m_W\right) +\Big [12 m_{q}^2 m_W^2 \left( m_{q}^2+m_{q^\prime }^2\right) \nonumber \\&\quad -m_W^4 \left( 17 m_{q}^2+9 m_{q^\prime }^2\right) -\left( m_{q}^2-3 m_{q^\prime }^2\right) \left( m_{q}^2-m_{q^\prime }^2\right) ^2+6 m_W^6\Big ]\nonumber \\&\quad \mathbf {C}_0\left( m_{q}^2,m_{q}^2,0;m_{q^\prime },m_W,m_{q^\prime }\right) \nonumber \\&\quad -2 \left( m_{q}^2+m_{q^\prime }^2-m_W^2\right) \left( m_{q}^2+m_{q^\prime }^2+2 m_W^2\right) \Big \}, \end{aligned}$$

(52)

$$\begin{aligned} \mathcal {F}_q^{h}\left( q^2\right)&=\frac{1}{\eta ^4(\Vert q\Vert ,m_q)} \Big \{\eta ^2(\Vert q\Vert ,m_q)\left( m_h^2\mathbf {B}_0(0;m_h,m_h)-m_q^2 \mathbf {B}_0(0;m_q,m_q)\right) \nonumber \\&\quad +\left( 4m_q^2\eta ^2(\Vert q\Vert ,m_q)+m_h^2(10 m_q^2- q^2)\right) \mathbf {B}_0\left( m_q^2;m_q,m_h\right) \nonumber \\&\quad -3m_q^2\left( \eta ^2(\Vert q\Vert ,m_q)-2 m_h^2\right) \mathbf {B}_0\left( q^2;m_q,m_q\right) \nonumber \\&\quad -6m_h^2 m_q^2\left( \eta ^2(\Vert q\Vert ,m_q)+m_h^2\right) \mathbf {C}_0\left( m_q^2,m_q^2,q^2;m_q,m_h,m_q\right) \nonumber \\&\quad +(m_h^2-2m_q^2)\eta ^2(\Vert q\Vert ,m_q)\Big \}, \end{aligned}$$

(53)

and

$$\begin{aligned} \mathcal {F}_q^{h}(0)&=-\frac{1}{8m_q^2}\Big \{\left( 3 m_h^2-8 m_q^2\right) \mathbf {B}_0(0;m_q,m_q)+\left( 8 m_q^2-5 m_h^2\right) \mathbf {B}_0\left( m_q^2;m_q,m_h\right) \nonumber \\&\quad +2 m_h^2 \mathbf {B}_0(0;m_h,m_h)+3 m_h^2 \left( m_h^2-4 m_q^2\right) \mathbf {C}_0\left( m_q^2,m_q^2,0;m_q,m_h,m_q\right) \nonumber \\&\quad +2 (m_h^2-2m_q^2)\Big \}. \end{aligned}$$

(54)

1.3 Closed form results

We also present the explicit solutions for the two-point scalar functions in terms of closed form functions. Below $C_0\left( m_i^2,m_j^2,q^2;m_k,m_l,m_n\right)$ stands for a three-point Passarino-Veltman scalar function.

$$\begin{aligned} {\mathcal {F}}_q^{\mathrm{QCD}_1}\left( q^2\right)&=-\frac{m_q^2}{\Vert q\Vert \eta (\Vert q\Vert ,m_q)}\log \left( \frac{\Vert q\Vert \eta (\Vert q\Vert ,m_q)+2 m_q^2-q^2}{2 m_q^2}\right) , \end{aligned}$$

(55)

$$\begin{aligned} {\mathcal F}_q^{\mathrm{QCD}_2}\left( q^2\right)&=-\frac{m_q^2}{\eta ^4(\Vert q\Vert ,m_q)} \Bigg \{q^2 \Bigg [6 m_q^2 C_0\left( m_q^2,m_q^2,q^2;0,m_q,0\right) +\log \left( \frac{m_q^2}{q^2}\right) +2+i\pi \Bigg ]\nonumber \\&\quad +8 m_q^2 \Bigg [\log \left( \frac{m_q^2}{q^2}\right) -1+i\pi \Bigg ]\Bigg \}, \end{aligned}$$

(56)

$$\begin{aligned} {\mathcal F}_q^{\mathrm{QCD}_2}(0)&= \frac{1}{2} \Bigg \{\frac{1}{\epsilon }+\log \left( \frac{\mu ^2}{m_q^2}\right) +3\Bigg \}, \end{aligned}$$

(57)

$$\begin{aligned} {\mathcal F}_q^{A}\left( q^2\right)&=-\frac{m_q^2}{\Vert q\Vert \eta (\Vert q\Vert ,m_q)}\log \left( \frac{\Vert q\Vert \eta (\Vert q\Vert ,m_q)+2 m_q^2-q^2}{2 m_q^2}\right) , \end{aligned}$$

(58)

$$\begin{aligned} {\mathcal {A}}_q^{Z}\left( q^2\right)&=\frac{m_Z^2}{2 m_q^2\eta ^4(\Vert q\Vert ,m_q)} \Bigg \{ \Bigg (20 m_Z \left( 2 m_q^2-m_Z^2\right) \nonumber \\&\quad +\frac{2 m_Z \left( m_Z^2-8 m_q^2\right) q^2 }{m_q^2}\Bigg ) \eta (m_Z,m_q) \log \left( \frac{m_Z+\eta (m_Z,m_q)}{2 m_q}\right) \nonumber \\&\quad + 2\left( \frac{ \left( 9 m_Z^2-2 m_q^2\right) }{m_Z^2}+\frac{ \left( 8 m_q^4-24 m_Z^2 m_q^2+6 m_Z^4\right) }{m_Z^2 q^2} \right) m_q^2\Vert q\Vert \eta (\Vert q\Vert ,m_q)\nonumber \\&\quad \log \left( \frac{2 m_q^2-q^2+\Vert q\Vert \eta (\Vert q\Vert ,m_q)}{2 m_q^2}\right) \nonumber \\&\quad +\left( 2 \left( 8 m_q^4+14 m_Z^2 m_q^2-5 m_Z^4\right) +\frac{\left( -4 m_q^4-10 m_Z^2 m_q^2+m_Z^4\right) q^2}{m_q^2}\right) \nonumber \\&\quad \log \left( \frac{m_q^2}{m_Z^2}\right) + \Big (8 \left( q^2\right) ^2 +12 \left( 8 m_q^4-6 m_Z^2 m_q^2+m_Z^4\right) \nonumber \\&\quad +8 \left( 3 m_Z^2-7 m_q^2\right) q^2\Big ) m_q^2C_0\left( m_q^2,m_q^2,q^2;m_q,m_Z,m_q\right) \nonumber \\&\quad +2 \left( 2 m_q^2+m_Z^2\right) \eta ^2(\Vert q\Vert ,m_q) \Bigg \}, \end{aligned}$$

(59)

$$\begin{aligned} {\mathcal {V}}_q^{Z}\left( q^2\right)&=\frac{m_Z^2}{2 m_q^2\eta ^4(\Vert q\Vert ,m_q)}\Bigg \{ 2m_Z^2\Big ( \left( 3 m_Z^2-8 m_q^2\right) +2 q^2\Big ) m_q^2C_0\nonumber \\&\quad \left( m_q^2,m_q^2,q^2;m_q,m_Z,m_q\right) \nonumber \\&\quad +\Bigg (\frac{4 \left( 3 m_Z^2-2 m_q^2\right) }{q^2}+2\Bigg ) m_q^2\Vert q\Vert \eta ^2(\Vert q\Vert ,m_q) \log \nonumber \\&\quad \left( \frac{2 m_q^2-q^2+\Vert q\Vert \eta (\Vert q\Vert ,m_q)}{2 m_q^2}\right) \nonumber \\&\quad +2 m_Z^2 \eta (\Vert q\Vert ,m_q)+\Bigg (2 m_Z^2 \left( 8 m_q^2-5 m_Z^2\right) +\frac{m_Z^2 \eta ^2(m_Z,m_q)q^2 }{m_q^2}\Bigg ) \nonumber \\&\quad \log \left( \frac{m_q^2}{m_Z^2}\right) +\Bigg (4 m_Z \left( 4 m_q^2-5 m_Z^2\right) +\frac{2 m_Z \left( m_Z^2-2 m_q^2\right) q^2 }{m_q^2}\Bigg ) \eta (m_Z,m_q) \log \nonumber \\&\quad \left( \frac{m_Z+\eta (m_Z,m_q)}{2 m_q}\right) \Bigg \}, \end{aligned}$$

(60)

$$\begin{aligned} {\mathcal {A}}_q^{Z}(0)&=\frac{m_Z^2}{2 m_q^6}\Bigg \{-\frac{m_q^2 \left( m_q^2-2 m_Z^2\right) \left( 2 m_q^2-m_Z^2\right) }{m_Z^2}+\left( -2 m_q^4+4 m_q^2 m_Z^2-m_Z^4\right) \nonumber \\&\quad \log \left( \frac{m_q^2}{m_Z^2}\right) \nonumber \\&\quad -\frac{2\left( 8 m_q^4-6 m_q^2 m_Z^2+m_Z^4\right) m_Z}{\eta (m_Z,m_q)} \log \left( \frac{m_Z+\eta (m_Z,m_q)}{2 m_q }\right) \Bigg \}, \end{aligned}$$

(61)

$$\begin{aligned} {\mathcal {V}}_q^{Z}(0)&=\frac{m_Z^2}{2 m_q^6}\Bigg \{m_q^2 \left( m_q^2-2 m_Z^2\right) -m_Z^2 \left( m_Z^2-2 m_q^2\right) \log \left( \frac{m_q^2}{m_Z^2}\right) \nonumber \\&\quad -\frac{2 \left( 2 m_q^4-4 m_q^2 m_Z^2+m_Z^4\right) }{\eta (m_Z,m_q)} \log \left( \frac{\eta (m_Z,m_q)+m_Z}{2 m_q}\right) \Bigg \}, \end{aligned}$$

(62)

$$\begin{aligned} {\mathcal {F}}_{qq'}^{W}\left( q^2\right)&= \frac{1}{2 m_q^2\eta ^4(\Vert q\Vert ,m_q)} \Bigg \{-4 m_q^2 \Big (m_q^6-m_q^4 \left( 5 m_{q^\prime }^2+12 m_W^2\right) \nonumber \\&\quad +m_q^2 \left( 7 m_{q^\prime }^4-12 m_{q^\prime }^2 m_W^2+17 m_W^4\right) \nonumber \\&\quad -q^2 \left( m_q^4-2 m_q^2 \left( m_{q^\prime }^2+4 m_W^2\right) +m_{q^\prime }^4-6 m_{q^\prime }^2 m_W^2+8 m_W^4\right) \nonumber \\&\quad -3 \left( m_{q^\prime }^6-3 m_{q^\prime }^2 m_W^4+2 m_W^6\right) \nonumber \\&\quad -2 m_W^2 \left( q^2\right) ^2\Big ) C_0\left( m_q^2,m_q^2,q^2;m_{q^\prime },m_W,m_{q^\prime }\right) \nonumber \\&\quad +2 \eta ^2(\Vert q\Vert ,m_q) \big (m_q^4+3 m_q^2 m_W^2-m_{q^\prime }^4-m_{q^\prime }^2 m_W^2+2 m_W^4\big )\nonumber \\&\quad +\frac{2 m_q^2 }{q^2}\Vert q\Vert \eta (\Vert q\Vert ,m_q') \Big (2 \left( m_q^4+m_q^2\left( 6 m_{q^\prime }^2-11 m_W^2\right) \right. \nonumber \\&\quad -\left. 3\left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \right) \nonumber \\&\quad +q^2 \big (-3 m_{q^\prime }^2+m_q^2+10 m_W^2\big )\Big )\log \left( \frac{\Vert q\Vert \eta (\Vert q\Vert ,m_q')+2 m_{q^\prime }^2-q^2}{2 m_{q^\prime }^2}\right) \nonumber \\&\quad -\frac{2 }{m_q^2}\sqrt{((m_q - m_q')^2 - m_W^2) ((m_q + m_q')^2 - m_W^2)} \Big (q^2 \big (m_q^4\nonumber \\&\quad +m_q^2 \left( 9 m_W^2-2m_{q^\prime }^2\right) +m_{q^\prime }^4+m_{q^\prime }^2 m_W^2\nonumber \\&\quad -2 m_W^4\big )+2 m_q^2 \left( m_q^4+m_q^2 \left( 4 m_{q^\prime }^2-9 m_W^2\right) -5 \left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \right) \Big )\nonumber \\&\quad \times \log \left( \frac{\sqrt{((m_q - m_q')^2 - m_W^2) ((m_q + m_q')^2 - m_W^2)}+m_{q^\prime }^2-m_q^2+m_W^2}{2 m_{q^\prime } m_W}\right) \nonumber \\&\quad -\frac{1}{m_q^2} \Big (q^2 \big (m_q^6-3 m_q^4 \left( m_{q^\prime }^2-4 m_W^2\right) +m_q^2 \left( 3 m_{q^\prime }^4-8 m_{q^\prime }^2 m_W^2+11 m_W^4\right) \nonumber \\&\quad -m_{q^\prime }^6+3 m_{q^\prime }^2 m_W^4-2 m_W^6\big )+2 m_q^2 \big (m_q^6+3 m_q^4 \left( m_{q^\prime }^2-4 m_W^2\right) \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\quad +m_q^2 \left( -9 m_{q^\prime }^4+4 m_{q^\prime }^2 m_W^2-7 m_W^4\right) \nonumber \\&\quad +5 \left( m_{q^\prime }^6-3 m_{q^\prime }^2 m_W^4+2 m_W^6\right) \big )\Big )\log \left( \frac{m_{q^\prime }^2}{m_W^2}\right) \Bigg \}, \end{aligned}$$

(63)

$$\begin{aligned} {\mathcal {F}}_{qq'}^{W}(0)&=\frac{1}{2m_q^6} \left\{ m_q^2 \Big (-m_q^4-m_q^2 \left( 3 m_{q^\prime }^2-4 m_W^2\right) +2 \left( m_{q^\prime }^4+m_{q^\prime }^2 m_W^2-2 m_W^4\right) \Big )\right. \nonumber \\&\quad -\left( m_q^4 m_{q^\prime }^2+m_q^2 \left( -2 m_{q^\prime }^4+2 m_{q^\prime }^2 m_W^2-3 m_W^4\right) +m_{q^\prime }^6-3 m_{q^\prime }^2 m_W^4+2 m_W^6\right) \nonumber \\&\quad \log \left( \frac{m_{q^\prime }^2}{m_W^2}\right) \nonumber \\&\quad -\frac{2}{\sqrt{((m_q - m_q')^2 - m_W^2) ((m_q + m_q')^2 - m_W^2)}} \Big (m_q^6 m_{q^\prime }^2\nonumber \\&\quad +m_q^4 \left( -3 m_{q^\prime }^4+m_{q^\prime }^2 m_W^2+3 m_W^4\right) \nonumber \\&\quad +m_q^2 \left( 3 m_{q^\prime }^6-2 m_{q^\prime }^4 m_W^2+4 m_{q^\prime }^2 m_W^4-5 m_W^6\right) \nonumber \\&\quad -\left( m_{q^\prime }^2-m_W^2\right) ^3 \left( m_{q^\prime }^2+2 m_W^2\right) \Big ) \nonumber \\&\quad \times \left. \log \left( \frac{\sqrt{((m_q - m_q')^2 - m_W^2) ((m_q + m_q')^2 - m_W^2)}+m_{q^\prime }^2-m_q^2+m_W^2}{2 m_{q^\prime } m_W}\right) \right\} , \end{aligned}$$

(64)

$$\begin{aligned} {\mathcal {F}}_q^{h}\left( q^2\right)&=-\frac{1}{2m_q^2 \eta ^4(\Vert q\Vert ,m_q)} \Bigg \{12 m_q^4 m_h^2 \left( \eta (\Vert q\Vert ,m_q)+m_h^2\right) \nonumber \\&\quad C_0\left( m_q^2,m_q^2,q^2;m_q,m_h,m_q\right) \nonumber \\&\quad +2 m_q^2 m_h^2 \eta (\Vert q\Vert ,m_q)+\frac{6 m_q^4}{q^2} \Vert q\Vert \eta (\Vert q\Vert ,m_q) \left( \eta ^2(\Vert q\Vert ,m_q)\right. \nonumber \\&\quad +\left. 2m_h^2\right) \log \left( \frac{\Vert q\Vert \eta (\Vert q\Vert ,m_q)+2 m_q^2-q^2}{2 m_q^2}\right) \nonumber \\&\quad +m_h^2 \left( 24 m_q^4+q^2 \left( m_h^2-6 m_q^2\right) -10 m_q^2 m_h^2\right) \log \left( \frac{m_q^2}{m_h^2}\right) \nonumber \\&\quad -2m_h\eta (m_h,m_q) \left( -16 m_q^4-q^2 \eta (m_h,m_q)+10 m_q^2 m_h^2\right) \nonumber \\&\quad \log \left( \frac{\eta (m_h,m_q)+m_h}{2 m_q }\right) \Bigg \}, \end{aligned}$$

(65)

and

$$\begin{aligned} {\mathcal {F}}_q^{h}(0)&=-\frac{1}{2m_q^4} \Bigg \{m_q^2(3 m_q^2-2 m_h^2)+2 m_h \left( m_q^2-m_h^2\right) \eta (m_h,m_q) \log \left( \frac{\eta (m_h,m_q)+m_h}{2 m_q}\right) \nonumber \\&\quad +m_h^2\left( 3 m_q^2-m_h^2\right) \log \left( \frac{m_q^2}{m_h^2}\right) \Bigg \}. \end{aligned}$$

(66)