form reminiscent of the cutting rules, i.e. as a sum of phase-space integrals of higher-point amplitudes. To make this statement more precise, we shall now explicitly show that the cutting rules indeed exactly reproduce the above results.

According to the cutting rules, the integral $I_2(\mu)$ of eq. (\[eq:abc2\]) can be directly written in the form $$\begin{aligned}
 I_2(\mu)&\equiv& I_2^\text{0-cut}+ I_2^\text{1-cut}(\mu)+I_2^\text{2-cut}(\mu)\,.
  \end{aligned}$$ Here, the first term reads $$\begin{aligned}
I_2^\text{0-cut}&=& I_2(0)\:=\!\!\!\!
        \begin{gathered}
            \begin{fmfgraph*}(60,60)
                \fmfleft{in2}
                \fmfright{out2}
                \fmf{phantom,tension=5}{in2,vin2}
                \fmf{phantom,tension=5}{vout2,out2}
                \fmf{dashes,left,tension=0.4}{vin2,vout2,vin2}
                \fmf{dots,width=2}{vin2,vout2}
            \end{fmfgraph*}
        \end{gathered}\!\!\!\!,\end{aligned}$$ where we have denoted a $\mu=0$ massive propagator by a dashed line. This clearly agrees with eq. (\[eq:mot1\]). For the one-cut piece, on the other hand, we obtain $$\begin{aligned}
I_2^\text{1-cut}(\mu)&=& -\int_p \frac{\theta(\mu-E_p)}{2E_p} \Bigg[\int_Q \frac{1}{q_0^2+E_q^2}\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Bigg]_{p_0\to iE_p} \nonumber \\
&-&\int_q \frac{\theta(\mu-E_q)}{2E_q} \Bigg[\int_P \frac{1}{p_0^2+E_p^2}\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Bigg]_{q_0\to iE_q} \nonumber 
\\
&\equiv&
    -\:2\intop_{p}\frac{\theta(\mu - E_{p})}{2 E_{p}}\;
    \begin{gathered}
            \begin{fmfgraph*}(60,60)
                \fmfleft{in3}
                \fmfright{out3}
                \fmf{dashes,tension=5}{in3,vin3}
                \fmf{dashes,tension=5}{vout3,out3}
                \fmf{dashes,left,tension=0.4}{vin3,vout3}
                \fmf{dots,width=2}{vin3,vout3}
            \end{fmfgraph*}
        \end{gathered}\,
        \Bigg{|}_{p_{0}\rightarrow iE_{p}}\,, \label{sunset1}\end{aligned}$$ where we have used the symmetry of the two terms in the first form of the result. This can be identified with the second and third terms of eq. (\[I2int\]), using eq. (\[eq:I21\]). Finally, the two-cut part of the graph takes the form $$\begin{aligned}
I_2^\text{2-cut}(\mu)&=& \int_p \frac{\theta(\mu-E_p)}{2E_p}\int_q \frac{\theta(\mu-E_q)}{2E_q} \Bigg[\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Bigg]_{p_0\to iE_p,\,q_0\to iE_q}\;\;\nonumber
\\
&\equiv&
    \intop_{p}\frac{\theta(\mu - E_{p}) }{2 E_{p}}
            \intop_{q}\frac{\theta(\mu - 