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---
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title: Loop functions
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permalink: /Loop_functions/
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---
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[Category:Calculations](/Category:Calculations "wikilink") 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
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$\\begin{aligned}
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A_0(m) &=& 16\\pi^2Q^{4-n}\\int{\\frac{d^nq}{ i\\,(2\\pi)^n}}{\\frac{1}{
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q^2-m^2+i\\varepsilon}} \\thickspace ,\\\\
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B_0(p, m_1, m_2) &=& 16\\pi^2Q^{4-n}\\int{\\frac{d^nq}{ i\\,(2\\pi)^n}}
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{\\frac{1}{\\biggl\[q^2-m^2_1+i\\varepsilon\\biggr\]\\biggl\[
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(q-p)^2-m_2^2+i\\varepsilon\\biggr\]}} \\thickspace ,
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\\label{B0 def}\\end{aligned}$
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with the renormalization scale *Q*. The integrals are regularized by integrating in *n* = 4 − 2*ϵ* dimensions. The result for *A*<sub>0</sub> is
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$A_0(m)\\ =\\ m^2\\left({\\frac{1}{\\hat\\epsilon}} + 1 -
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\\ln{\\frac{m^2}{Q^2}}\\right)~,\\label{A}$
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where 1/*ϵ̂* = 1/*ϵ* − *γ*<sub>*E*</sub> + ln 4*π*. The function *B*<sub>0</sub> has the analytic expression
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$B_0(p, m_1, m_2) \\ =\\ {\\frac{1}{\\hat\\epsilon}} -
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\\ln\\left(\\frac{p^2}{Q^2}\\right) - f_B(x_+) - f_B(x_-)~,$
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with
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$x_{\\pm}\\ =\\ \\frac{s \\pm \\sqrt{s^2 - 4p^2(m_1^2-i\\varepsilon)}}{2p^2}~,
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\\qquad f_B(x) \\ =\\ \\ln(1-x) - x\\ln(1-x^{-1})-1~,$
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and *s* = *p*<sup>2</sup> − *m*<sub>2</sub><sup>2</sup> + *m*<sub>1</sub><sup>2</sup>. All the other, necessary functions can be expressed by *A*<sub>0</sub> and *B*<sub>0</sub>. For instance,
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$B_1(p, m_1,m_2) \\ =\\ -{\\frac{1}{2p^2}}\\biggl\[ A_0(m_2) - A_0(m_1) + (p^2
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+m_1^2 -m_2^2) B_0(p, m_1, m_2)\\biggr\]~,$
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and
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$\\begin{aligned}
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B_{22}(p, m_1,m_2) &=& \\frac{1}{6}\\ \\Bigg\\{\\,
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\\frac{1}{2}\\biggl(A_0(m_1)+A_0(m_2)\\biggr)
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+\\left(m_1^2+m_2^2-\\frac{1}{2}p^2\\right)B_0(p,m_1,m_2)\\nonumber \\\\ &&+
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\\frac{m_2^2-m_1^2}{2p^2}\\ \\biggl\[\\,A_0(m_2)-A_0(m_1)-(m_2^2-m_1^2)
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B_0(p,m_1,m_2)\\,\\biggr\] \\nonumber\\\\ && + m_1^2 + m_2^2
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-\\frac{1}{3}p^2\\,\\Bigg\\}~.\\end{aligned}$
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Furthermore, for the vector boson self-energies it is useful to define
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$\\begin{aligned}
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F_0(p,m_1,m_2) =& A_0(m_1)-2A_0(m_2)- (2p^2+2m^2_1-m^2_2)B_0(p,m_1,m_2)
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\\ , \\\\
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G_0(p,m_1,m_2) =&
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(p^2-m_1^2-m_2^2)B_0(p,m_1,m_2)-A_0(m_1)-A_0(m_2)\\ ,\\\\
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H_0 (p,m_1,m_2) =& 4B_{22}(p,m_1,m_2) + G(p,m_1,m_2)\\ ,\\\\
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\\tilde{B}_{22}(p,m_1,m_2) =& B_{22}(p,m_1,m_2) - \\frac{1}{4}A_0(m_1) -
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\\frac{1}{4}A_0(m_2)\\end{aligned}$
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See also
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-------- |
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\ No newline at end of file |