Feynman parameter combination of denominators

For positive Euclidean quantities A and B, integrate the derivative of the reciprocal of $xA+(1-x)B$ between zero and one. The endpoints immediately give the identity for $1/(AB)$. When A equals B, the parameter integral is simply constant, so the apparent difference denominator in an antiderivative is removable. The positivity assumption makes the elementary derivation uncomplicated. Minkowski denominators instead use a shared causal prescription and analytic continuation, rather than an arbitrary real-axis integral through poles.

For higher powers or more propagators, the general identity introduces positive parameters constrained to sum to one. Gamma-function factors reflect powers and the simplex volume. They are not graph symmetry factors: one comes from an integration identity and the other from Wick counting. Keeping these origins separate prevents multiplying by a spurious factorial. The same identity can be derived through Schwinger parameters, where each inverse power is a Laplace integral and an overall scale is separated from normalized ratios.

Use Minkowski signature $(+---)$ and equal-mass denominators $D_1=k^2-m^2+i0$ and $D_2=(k+p)^2-m^2+i0$. Weight the first by x and the second by one minus x. Completing the square shifts the loop variable to $\ell=k+(1-x)p$ and leaves the mass combination $\Delta=m^2-x(1-x)p^2$. The sign of the invariant follows from the original metric and routing; replacing it by a plus without Wick rotation changes the threshold structure.

For a spacelike invariant $p^2=-Q^2$, the mass combination becomes $m^2+x(1-x)Q^2$, positive for a positive mass and real Euclidean external momentum. This is a convenient domain in which to integrate first. Timelike amplitudes are then continued with the causal prescription. The maximum of the parameter product is one quarter, which later identifies the equal-mass threshold. A negative intermediate mass combination above threshold is not an instruction to take an absolute value inside a logarithm.

A convergent integral over all momentum space is invariant under a change of integration variable. For a divergent integral, that statement must be interpreted inside the chosen regulator. Dimensional regularization is translation invariant and supports the shifts used in standard loop reductions. A hard cutoff centered at one momentum origin is not unchanged by simply shifting the domain. Boundary terms can matter, especially in superficially divergent tensor expressions and in checks of gauge identities.

A useful workflow first records the propagator prescription and regulator, then combines denominators, shifts the loop, integrates it, and finally studies parameter endpoints. Ultraviolet poles may arise from the momentum integral while infrared or threshold behavior can appear through the remaining parameters. The quadratic lab can inspect the shape of the parameter product after recentering, but it does not implement a complex logarithm, a causal contour deformation, or a general Feynman-parameter integral.