Feynman and Schwinger Parameters

Parameterization converts a product of propagators into one denominator raised to a higher power. Schwinger parameters make Gaussian loop integration and graph homogeneity transparent; Feynman parameters quotient out their common scale and leave an integral over a simplex. Neither transformation discards the causal boundary value.

Required background. Anatomy of a Loop Integral supplies the denominators, powers, routing, and $i0$ data that are being combined.

From reciprocal powers to a simplex

For $\operatorname{Re}A>0$ and $\operatorname{Re}\nu>0$, the Euclidean Schwinger identity is

$$\frac{1}{A^\nu}=\frac{1}{\Gamma(\nu)} \int_0^\infty \mathrm d\alpha\,\alpha^{\nu-1}e^{-\alpha A}.$$

In a Lorentzian integral it is used after the Feynman $i0$ has selected a Wick rotation, or with the corresponding oscillatory representation and its convergence factor. Dropping that prescription before rotating loses the sheet information.

For two denominator factors, introduce $\alpha_1=\rho x$ and $\alpha_2=\rho(1-x)$. The Jacobian is $\rho$, and the $\rho$ integral gives

$$\frac{1}{A^aB^b} =\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^1\mathrm d x\, \frac{x^{a-1}(1-x)^{b-1}} {[xA+(1-x)B]^{a+b}}.$$

For $n$ factors this becomes an integral over $x_j\ge0$ with $\sum_jx_j=1$. The gamma-function normalization is fixed by the Dirichlet integral; checking the special case $A=B$ immediately recovers $A^{-(a+b)}$.

The Schwinger and Feynman constructions, including the projective nature of the latter, are derived in Weinzierl 2022, §§2.5.2–2.5.3, pp. 43–55.

Completing the square in the bubble

Apply the two-denominator identity to

$$D_1=\ell^2-m_1^2+i0, \qquad D_2=(\ell+p)^2-m_2^2+i0.$$

With weight $x$ on $D_2$, their combination is

$$(1-x)D_1+xD_2 =(\ell+xp)^2-\Delta(x)+i0,$$

where

$$\Delta(x)=(1-x)m_1^2+xm_2^2-x(1-x)p^2.$$

The shift $k=\ell+xp$ is valid in a translation-invariant regulator. Thus

$$B(p^2)=\mu^{2\epsilon}\int_0^1\mathrm d x \int\frac{\mathrm d^d k}{(2\pi)^d} \frac{1}{[k^2-\Delta(x)+i0]^2}.$$

This representation makes the denominator geometry visible. In the Euclidean region $p^2<0$ with positive masses, $\Delta(x)>0$ throughout the interval. As $p^2$ is continued, a zero can enter the integration domain. At the normal threshold the quadratic has a double zero, and beyond it the logarithm generated by the loop integration samples $\Delta-i0$ on the lower side of its cut.

The double-zero conditions $\Delta(x_*)=\Delta'(x_*)=0$ are solved, for positive masses and $x_*\in(0,1)$, by

$$x_* = \frac{m_1}{m_1+m_2}, \qquad p^2=(m_1+m_2)^2.$$

The algebraic companion $(m_1-m_2)^2$ belongs to a different parameter or sheet configuration and is not the normal positive-parameter threshold. The domain condition is what prevents a root of the discriminant from being mistaken for an accessible physical pinch.

The same bubble calculation and its massless specialization appear in Weinzierl 2022, §2.5.3, pp. 46–52.

Graph polynomials at many loops

After Gaussian integration of an $L$-loop scalar graph with $N=\sum_j\nu_j$, the projective form has the schematic structure

$$I\propto \Gamma\!\left(N-\frac{Ld}{2}\right) \int_{x_j\ge0}\!\Omega\, \frac{\mathcal U^{N-(L+1)d/2}} {(\mathcal F-i0\,\mathcal U)^{N-Ld/2}} \prod_j x_j^{\nu_j-1}.$$

$\Omega$ denotes the projective parameter measure in the simplex gauge $\sum_jx_j=1$. $\mathcal U$ is homogeneous of degree $L$ and records spanning-tree information; $\mathcal F$ is homogeneous of degree $L+1$ and contains masses and external invariants. In this gauge the momentum-space denominator $k^2-\mathcal F/\mathcal U+i0$ produces $\mathcal F-i0\,\mathcal U$. In a fully homogeneous projective expression the infinitesimal is instead written $-i0\,\mathcal U\sum_jx_j$; some references absorb it into the definition of $\mathcal F$. Overall prefactors depend on the chosen measure normalization, but the homogeneities and boundary value do not. Parameter boundaries $x_j=0$ correspond to contracted lines and are natural places to inspect UV or IR subregions. Interior zeros of $\mathcal F$ can participate in threshold pinches.

A complementary derivation and concise statement of these polynomials is given in Abreu, Britto, and Duhr 2022, §1.2, pp. 5–8.

Common pitfalls

Shifting before regulating. Completing the square is a change of variables in a defined regulated integral. For a cutoff that is not translation invariant, the shifted integration region contributes a surface term.

Reading every zero as a physical singularity. A zero of the parameter polynomial is only part of the pinch analysis. The domain, derivative conditions, $i0$ prescription, and possible numerator cancellation still matter.

Removing the common scale twice. Either retain all positive Schwinger parameters or fix one projective condition such as $\sum x_j=1$. Mixing the two without the correct Jacobian changes the normalization.

Exercises

1. Set $a=b=1$ and $A=B$ in the Feynman identity. The parameter integral is one and the result is $1/A^2$.
2. For $m_1=m_2=m$, minimize $\Delta(x)=m^2-x(1-x)p^2$. Its first zero occurs at $x=1/2$ and $p^2=4m^2$, the two-particle threshold.

Where parameter geometry leads

• Dimensional Regularization as an Amplitude Tool analytically continues the resulting Gaussian and parameter integrals in $d$.
• Landau Equations and Physical Singularities adds the stationarity and sheet tests needed to turn polynomial zeros into candidate pinches.

References

• Abreu, Samuel, Ruth Britto, and Claude Duhr. “The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical Structures in Feynman Integrals.” Journal of Physics A: Mathematical and Theoretical 55 (2022): 443004. doi:10.1088/1751-8121/ac87de.
• Weinzierl, Stefan. Feynman Integrals. Cham: Springer, 2022. doi:10.1007/978-3-030-99558-4. Open PDF.