quantum field theory - Loop integral using Feynman's trick

I am trying to show for the one-loop integral with three propagators with different internal masses $m_1$, $m_2$, $m_3$, and all off-shell external momenta $p_1$, $p_2$, $p_3$ the following formula appearing in 't Hooft(1979), Bardin (1999), Denner (2007): (unfortunate metric $-,+,+,+$)

$$\int d^d q\frac{1}{(q^2+m_1^2)((q+p1)^2+m_2^2)((q+p_1+p_2)^2+m_3^2)}$$ $$=i\pi^2\int_0^1dx\int_0^xdy\frac{1}{ax^2+by^2+cxy+dx+ey+f}$$

where $a$, $b$, $c$, ... are coefficients depending on the momenta in the following way:

$a=-p_2^2$,

$b=-p_1^2$,

$c=-2p_1.p_2$,

$d=m_2^2-m_3^2+p_2^2$,

$e=m_1^2-m_2^2+p_1^2+2(p_1.p_2)$,

$f=m_3^2-i\epsilon$.

I don't really care about factors in fromt like $i\pi^2$. My simple problem is: I am totally unable to reproduce coefficients $d$, $e$ and $f$. The problem is, when I integrate over the third Feynman parameter, $m_3$ appears in all three coefficients $d$, $e$ and $f$. How do I squeeze the denominators to reproduce this formula?

Answer

Define the LHS of the equation above:

$$I=\int d^d q\frac{1}{(q^2+m_1^2)((q+p_1)^2+m_2^2)((q+p_1+p_2)^2+m_3^2)}$$

The first step is to squeeze the denominators using Feynman's trick:

$$I=\int_0^1 dx\,dy\,dz\,\delta(1-x-y-z)\int d^d q\frac{2}{[y(q^2+m_1^2)+z((q+p_1)^2+m_2^2)+x((q+p_1+p_2)^2+m_3^2)]^3}$$

The square in $q^2$ may be completed in the denominator by expanding:

$$[\text{denom}]=q^2+2q.(z p_1+x(p_1+p_2))+y m_1^2+z (p_1^2+m_2^2)+x(m_3^2+(p_1+p_2)^2)$$ $$=q^2+2q.Q+A^2\,$$

where $Q^\mu=z p_1^\mu+x(p_1+p_2)^\mu$ and $A^2=y m_1^2+z (p_1^2+m_2^2)+x(m_3^2+(p_1+p_2)^2)$, and by shifting the momentum, $q^\mu=(k-Q)^\mu$ as a change of integration variables. Upon performing the $k$ integral, we are left with integrals over Feynman parameters (because this integral has three propagators, it is UV finite):

$$I=i\pi^2\int_0^1 dx\,dy\,dz\,\delta(1-x-y-z)\frac{1}{[-Q^2+A^2]}$$

Now integrate over $z$ with the help of the Dirac delta:

$$I=i\pi^2\int_0^1 dx\int_0^{1-x}dy \frac{1}{[-Q^2+A^2]_{z\rightarrow1-y-z}}$$

To arrive at the RHS of the OP's equation(which is the part I forgot to do), we make a final change of variables: $x=1-x'$:

So that the denominator reads $ax^2+by^2+cxy+dx+ey+f$, with the coefficients $a,b,c,\ldots$ exactly defined in the question of OP. Note the change in the range of integration in $y$.

$$I=i\pi^2\int_0^1dx\int_0^xdy\frac{1}{ax^2+by^2+cxy+dx+ey+f}$$