Numerical Evaluation and Validation of Loop Integrals

A numerical loop-integral result is credible only when its mathematical definition, uncertainty estimate, analytic sheet, and independent checks are all explicit. A program’s successful exit status is evidence about execution, not about the physical correctness of the supplied family, normalization, or continuation.

Required background. One-Loop Integral Families and Analytic Functions supplies exact and parameter representations that can serve as benchmarks.

Helpful background. Differential Equations for Master Integrals supplies an independent transport method and clarifies why boundary points and paths must accompany numerical values.

Define the numerical target

Before evaluation, record the exact object:

$$I(\epsilon)=\sum_{k=k_{\min}}^{k_{\max}} c_k\epsilon^k+O(\epsilon^{k_{\max}+1}).$$

The record must include denominators and powers, numerator, loop measure, $d$, $\mu$, conventional gamma and $4\pi$ factors, masses and invariants with units, momentum orientation, complex infinitesimals or contour deformation, and the requested coefficients $c_k$. For complex values, quote uncertainties for real and imaginary parts, including their correlation if it matters downstream.

Dimensional analysis supplies the first cheap check. Rescaling all masses, external momenta, and the scale $\mu$ together by $\lambda$ must multiply an integral of mass dimension $\Delta$ by exactly $\lambda^\Delta$. If $\mu$ is held fixed while only physical scales are rescaled, the predictable logarithmic shifts must instead be included coefficient by coefficient.

Choose a representation suited to the point

No representation is uniformly best.

• In a Euclidean region, Feynman parameters often give a smooth real-domain integral.
• Sector decomposition factorizes endpoint singularities and produces numerical integrals for Laurent coefficients; the algorithm and its numerical-period interpretation are described in Weinzierl 2022, §10.1, pp. 342–346; §10.3, pp. 350–351.
• Differential equations efficiently transport a high-quality boundary value across many points, provided singular points and continuation paths are controlled.
• Mellin–Barnes or dispersive representations can turn threshold behavior into explicit contour or spectral data.
• Direct loop-momentum integration may be useful after analytic energy integration or a justified contour deformation, but naive real-axis quadrature is unstable at pinches.

Above threshold, a parameter contour deformation must preserve endpoints, avoid singular hypersurfaces, and have the sign required by $-i0$. Varying the deformation size within a safe window should not change the result beyond the quoted uncertainty.

For a parameter polynomial $\mathcal F(\mathbf x)$ on the simplex, a common local deformation is

$$z_i=x_i-i\lambda x_i(1-x_i) \frac{\partial\mathcal F}{\partial x_i}, \qquad \lambda>0.$$

To first order,

$$\operatorname{Im}\mathcal F(\mathbf z) =-\lambda\sum_i x_i(1-x_i) \left(\frac{\partial\mathcal F}{\partial x_i}\right)^2 +O(\lambda^2),$$

which has the causal sign away from stationary points. It cannot cross a Landau pinch, where $\mathcal F=0$ and all allowed derivatives vanish together; there the boundary value must be approached from its prescribed side or a different representation used.

A bubble benchmark with a known imaginary part

Define the dimensionless finite function

$$F(s)=-\int_0^1\mathrm d x\, \log\!\left[1-\frac{s}{m^2}x(1-x)-i0\right].$$

For $0<s<4m^2$, the logarithm never meets its cut, so $\operatorname{Im}F=0$. For $s>4m^2$, the negative interval lies between

$$x_\pm=\frac{1\pm\beta}{2}, \qquad \beta=\sqrt{1-\frac{4m^2}{s}}>0.$$

Because $\log(-a-i0)=\log a-i\pi$ for $a>0$,

$$\operatorname{Im}F(s)=\pi(x_+-x_-)=\pi\beta.$$

This gives a sharp sign and threshold test for a numerical implementation. At $s=5m^2$,

$$\operatorname{Im}F=\frac{\pi}{\sqrt5} =1.404962946208\ldots .$$

The real part can be checked against direct one-dimensional quadrature below threshold. Above threshold, split the real integral at $x_-$ and $x_+$ or use a controlled contour deformation before comparing with analytic continuation. Close to $s=4m^2$, compare a threshold expansion and absolute error rather than demanding uniform relative error from a representation with cancellation.

Independent validation, not repeated execution

Running the same code with more samples mainly probes its internal convergence model. Stronger tests change the source of possible error:

1. compare two representations, such as parameters and differential equations;
2. compare with a closed one-loop result or a simpler boundary point;
3. verify symmetry, scaling, complex conjugation, and differential equations;
4. test UV, soft, large-mass, or threshold limits against analytic expansions;
5. vary precision, integration partition, and contour-deformation parameters;
6. substitute reduced masters back into unreduced identities at sample points.

For polylogarithmic functions, continuation and numerical evaluation require transformations into convergent regions and explicit branch control; examples are given in Weinzierl 2022, §§15.1–15.2, pp. 521–529.

Reporting a result

A useful numerical statement has the form

At the stated kinematic point and normalization, method A gives $c_0=a\pm\delta_a+i(b\pm\delta_b)$; method B gives a compatible value. The differential-equation residual, symmetry test, and threshold-sign check are below specified tolerances.

The uncertainty should distinguish stochastic or quadrature error from systematic effects such as truncation in $\epsilon$, asymptotic expansion, interpolation, or finite precision. Agreement to digits smaller than either method’s systematic uncertainty is not meaningful.

Common pitfalls

Evaluating exactly on a singular surface. A physical boundary value is a limit from a specified side. Replace “at $s=s_0$” by a controlled limiting procedure when $s_0$ is a threshold or pseudothreshold.

Comparing different normalizations. Factors of $i$, $(4\pi)^\epsilon$, $e^{-\gamma_E\epsilon}$, and $\mu^{2\epsilon}$ shift Laurent coefficients. Convert conventions analytically before comparing numbers.

Quoting only relative error near zero. When cancellations make the true value small, absolute error and component-wise residuals are more informative.

Exercises

1. At $s=5m^2$, the benchmark has $\beta=1/\sqrt5$ and $\operatorname{Im}F=\pi/\sqrt5>0$ for the displayed $-i0$ prescription.
2. Name two independent checks for a differential-equation evaluation. A direct parameter integral at one point and substitution into the differential equation probe different errors.

Where a validated value is used

• Landau Equations and Physical Singularities identifies candidate singular surfaces that a numerical contour and continuation path must respect.
• The observable-level validation matrix combines independent loop checks with real-radiation, scale, scheme, and assembly tests.

References

• Weinzierl, Stefan. Feynman Integrals. Cham: Springer, 2022. doi:10.1007/978-3-030-99558-4. Open PDF.