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

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 ,

with the renormalization scale Q. The integrals are regularized by integrating in n = 4 − 2ϵ dimensions. The result for A₀ is

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

where 1/ϵ̂ = 1/ϵ − γ_E + ln 4π. The function B₀ 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 = p² − m₂² + m₁². All the other, necessary functions can be expressed by A₀ and B₀. 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

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\}~.

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

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)

See also