# Loop functions

We used for the calculation of the one-loop self energies and the one-loop corrections to the tadpoles in ${\overline{\text{DR}}}$-scheme the scalar functions defined in . The basic integrals are

\begin{aligned} A_0(m) &=& 16\pi^2Q^{4-n}\int{\frac{d^nq}{ i\,(2\pi)^n}}{\frac{1}{ q^2-m^2+i\varepsilon}} \thickspace ,\\ B_0(p, m_1, m_2) &=& 16\pi^2Q^{4-n}\int{\frac{d^nq}{ i\,(2\pi)^n}} {\frac{1}{\biggl[q^2-m^2_1+i\varepsilon\biggr]\biggl[ (q-p)^2-m_2^2+i\varepsilon\biggr]}} \thickspace , \label{B0 def}\end{aligned}

with the renormalization scale $Q$. The integrals are regularized by integrating in $n=4-2\epsilon$ dimensions. The result for $A_0$ is

$A_0(m)\ =\ m^2\left({\frac{1}{\hat\epsilon}} + 1 - \ln{\frac{m^2}{Q^2}}\right)~,\label{A}$

where $1/\hat\epsilon =1/\epsilon-\gamma_E+\ln 4\pi$. The function $B_0$ has the analytic expression

$B_0(p, m_1, m_2) \ =\ {\frac{1}{\hat\epsilon}} - \ln\left(\frac{p^2}{Q^2}\right) - f_B(x_+) - f_B(x_-)~,$

with

$x_{\pm}\ =\ \frac{s \pm \sqrt{s^2 - 4p^2(m_1^2-i\varepsilon)}}{2p^2}~, \qquad f_B(x) \ =\ \ln(1-x) - x\ln(1-x^{-1})-1~,$

and $s=p^2-m_2^2+m_1^2$. All the other, necessary functions can be expressed by $A_0$ and $B_0$. For instance,

$B_1(p, m_1,m_2) \ =\ {\frac{1}{2p^2}}\biggl[ A_0(m_2) - A_0(m_1) + (p^2 +m_1^2 -m_2^2) B_0(p, m_1, m_2)\biggr]~,$

and

\begin{aligned} B_{22}(p, m_1,m_2) &=& \frac{1}{6}\ \Bigg\{\, \frac{1}{2}\biggl(A_0(m_1)+A_0(m_2)\biggr) +\left(m_1^2+m_2^2-\frac{1}{2}p^2\right)B_0(p,m_1,m_2)\nonumber \\ &&+ \frac{m_2^2-m_1^2}{2p^2}\ \biggl[\,A_0(m_2)-A_0(m_1)-(m_2^2-m_1^2) B_0(p,m_1,m_2)\,\biggr] \nonumber\\ && + m_1^2 + m_2^2 -\frac{1}{3}p^2\,\Bigg\}~.\end{aligned}

Furthermore, for the vector boson self-energies it is useful to define

\begin{aligned} F_0(p,m_1,m_2) =& A_0(m_1)-2A_0(m_2)- (2p^2+2m^2_1-m^2_2)B_0(p,m_1,m_2) \ , \\ G_0(p,m_1,m_2) =& (p^2-m_1^2-m_2^2)B_0(p,m_1,m_2)-A_0(m_1)-A_0(m_2)\ ,\\ H_0 (p,m_1,m_2) =& 4B_{22}(p,m_1,m_2) + G(p,m_1,m_2)\ ,\\ \tilde{B}_{22}(p,m_1,m_2) =& B_{22}(p,m_1,m_2) - \frac{1}{4}A_0(m_1) - \frac{1}{4}A_0(m_2)\end{aligned}