SecDec: Automated Feynman Integral Evaluation

• SecDec is a public program suite that automates the numerical evaluation of dimensionally regulated, multi-loop Feynman integrals using sector decomposition and contour deformation.
• It systematically factorizes ultraviolet and infrared singularities through iterative decomposition, subtraction, and Laurent expansion, producing finite numerical coefficient functions.
• SecDec has been applied in NNLO collider integrals, MSSM Higgs mass corrections, and soft-collinear effective theory, highlighting its versatility in high-energy physics computations.

SecDec is a public program family for the automated numerical evaluation of dimensionally regulated parameter integrals, especially multi-loop, multi-scale Feynman integrals. Its defining workflow is to factorize ultraviolet and infrared singularities by sector decomposition, perform subtraction and Laurent expansion in the regulator $\epsilon$, and numerically integrate the resulting finite coefficient functions. For non-Euclidean kinematics it supplements this with contour deformation in Feynman-parameter space, so that threshold singularities dictated by the Feynman $+i\delta$ prescription can be treated directly. In this sense, SecDec occupies the interface between symbolic singularity extraction and numerical integration, with applications ranging from NNLO collider integrals to phase-space and other endpoint-singular parametric problems (Borowka, 2015, Jahn, 2018).

1. Historical development and scope

The earliest public description presented SecDec as a general program for the numerical evaluation of multi-dimensional polynomial parameter integrals in which singularities regulated by dimensional regularisation are extracted using iterated sector decomposition. In that formulation, its main target was multi-loop Feynman integrals in Euclidean space, but the same machinery also applied to other parametric integrals, including phase-space integrals (Carter et al., 2010).

Version 2.0 removed the Euclidean restriction for loop integrals. The central advance was the automated deformation of the integration contour into the complex plane, allowing the numerical evaluation of multi-loop integrals with no restriction on the kinematics in the case of loop integrals. The same release also emphasized more general parameter integrals and phase-space applications, and introduced topology-based graph input for scalar loop integrals (Borowka et al., 2012).

Version 2.1 broadened the range further by allowing user-defined functions in the loop part and by adding direct support for contracted tensor integrals, with no limitation on the rank. This was motivated in part by difficult massive non-planar two-loop box integrals, where analytic preprocessing could yield non-canonical but numerically superior parametric representations (Borowka et al., 2013).

SecDec-3.0 was presented as a major upgrade over version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability. The version-3 papers stress guaranteed-termination geometric strategies, support for zero and negative propagator powers, support for linear propagators, improved kinematic scans and cluster workflows, and easier interfacing to reduction codes and databases (Borowka et al., 2015, Borowka, 2015).

A later successor, pySecDec, reorganized the algebraic part in Python, generated optimized C++ libraries, exposed a modular interface, and supported multiple regulators. This changed the package from a mainly standalone integral evaluator into a toolbox that could also be embedded into user-defined amplitude codes (Borowka et al., 2017, Jahn, 2018).

2. Parametric formulation and classes of integrals

For loop integrals, SecDec starts from the standard momentum-space representation. A scalar $L$-loop integral in $D$ dimensions with $N$ propagators and propagator powers $\nu_j$ is written as

$$G=\int\prod\limits_{l=1}^{L} d^D\kappa_l\; \frac{1} {\prod\limits_{j=1}^{N} P_{j}^{\nu_j}(\{k\},\{p\},m_j^2)}\;, \qquad d^D\kappa_l=(i\pi^{D/2})^{-1}d^D k_l\;, \qquad P_j(\{k\},\{p\},m_j^2)=q_j^2-m_j^2+i\delta\;.$$

After Feynman parametrization and loop-momentum integration, it takes the Symanzik form

$G = \frac{ (-1)^{N_{\nu} }{ \prod_{j=1}^{N} \Gamma(\nu_j) } \, \Gamma( N_{\nu}-LD/2 ) \int\limits_{0}^{\infty} \!\prod \limits_{j=1}^{N} dx_j\,x_j^{\nu_j-1}\, \delta(1-\sum_{l=1}^N x_l) \frac{\cal U}^{N_{\nu}-(L+1) D/2}(\vec{x})} {\cal F}^{N_\nu-L D/2}(\vec{x},s_{ij})}\;,$

with $N_\nu=\sum_j \nu_j$. The first Symanzik polynomial ${\cal U}$ contains sub-UV singularities, while $+i\delta$0 contains IR singularities and the kinematic dependence (Borowka, 2015).

This standard loop setup is not the only input class. In the userdefined setup, SecDec accepts more general parameter integrals of the form

$+i\delta$1

where $+i\delta$2 may contain products of polynomials with direct or exponential dependence on $+i\delta$3, negative exponents are allowed for $+i\delta$4 and $+i\delta$5, and powers $+i\delta$6 are also allowed. In the more general general setup, the integrand may contain an arbitrary number of products, each possibly with a negative exponent, and later versions also admit $+i\delta$7-dependent dummy functions that do not themselves contain non-factorized poles (Borowka, 2015).

Within this framework, SecDec is able to treat scalar loop integrals, tensor integrals of rank $+i\delta$8, inverse-propagator numerators, non-integer propagator powers, linear propagators, dimensionally regulated phase-space integrals, and certain hypergeometric or other parametric representations, provided the singularity structure is compatible with the decomposition machinery (Borowka et al., 2015, Schlenk et al., 2016).

3. Sector decomposition, pole extraction, and contour deformation

The central algorithmic idea is sector decomposition. In loop integrals, SecDec first performs a primary sector decomposition to eliminate the constraint $+i\delta$9 without generating endpoint singularities at $L$0. It then applies iterated sector decomposition so that overlapping singularities are disentangled and each sector integral is rewritten with singular behavior explicit as monomials in the integration variables. Conceptually, the resulting integrands have the form

$L$1

with $L$2 finite as $L$3 (Borowka et al., 2014).

Once the singular factors have been exposed, SecDec constructs subtraction terms at the endpoints and extracts the poles in the dimensional regulator. The output is a Laurent expansion,

$L$4

or, equivalently, a Laurent series whose coefficients are finite parameter integrals over the unit hypercube and can be evaluated numerically (Borowka et al., 2014, Borowka et al., 2015).

For physical kinematics, the algebraic extraction of UV and IR poles is not sufficient, because $L$5 can vanish inside the integration region. SecDec therefore deforms the Feynman-parameter contour into the complex plane. The deformation ansatz used in the version-2 papers is

$L$6

To first order in $L$7,

$L$8

so the deformation generates the required negative imaginary part consistent with the Feynman prescription (Borowka et al., 2012, Borowka et al., 2012).

A practical feature emphasized repeatedly is the separation between the algebraic and numerical stages. Up to the subtraction and expansion stage, kinematic invariants may remain symbolic; numerical values can be inserted only at the integration stage, which is advantageous for scans over many phase-space points. Conversely, if only one or two especially difficult kinematic points are needed, early numerical substitution can simplify the sector functions and improve convergence (Borowka, 2015).

4. Program organization, strategies, and later architectural changes

The main user-facing organization in SecDec 3 distinguishes the loop, userdefined, and general setups. The loop setup targets ordinary Feynman integrals specified by propagators and optional numerators; the userdefined setup accepts near-Feynman-parametric inputs built from $L$9, $D$0, and a numerator factor; the general setup is for arbitrary products of functions with endpoint singularities that can still be disentangled by decomposition. Version 3 also restructured the input specifically to facilitate interfaces to reduction programs and databases such as FIRE, Reduze, LiteRed, and Loopedia (Borowka, 2015).

SecDec-3.0 placed unusual emphasis on decomposition strategy. The older strategy X is heuristic and often efficient, but not guaranteed to terminate. Version 3 introduced the deterministic geometric strategies G1 and G2, based on convex geometry and the Cheng–Wu theorem. The G2 strategy is presented as the fastest implemented method and usually yields the smallest number of sectors. In its geometric formulation, one computes the Newton polytope of the relevant polynomial structure and maps each sector back to the unit hypercube through facet data, replacing recursive heuristic blow-ups by a deterministic construction (Borowka et al., 2015, Schlenk et al., 2016).

Other version-3 extensions included support for zero and negative propagator powers, which naturally represent inverse propagators and hence tensor numerators; support for loop integrals with linear propagators; inclusion of CQuad and Mathematica’s NIntegrate alongside the updated Cuba library; and improved cluster and batch-system workflows. Support for complex masses was added from version 3.0.8 and implemented compatibly with contour deformation, with the complex-mass scheme contributing a negative imaginary part to $D$1 through

$D$2

and hence

$D$3

This is explicitly of the correct sign for positive Feynman parameters and widths (Schlenk et al., 2016).

pySecDec retained the mathematical core but changed the architecture decisively. The algebraic part was rewritten as Python modules; generated C++ code was optimized with FORM; equivalent sectors could be identified algorithmically; the output became a reusable C++ library; and multiple regulators were supported natively. This modularization made it possible to use only selected parts of the pipeline, such as Feynman parametrization or library generation, rather than a monolithic end-to-end run (Borowka et al., 2017, Jahn, 2018).

5. Applications in perturbative quantum field theory

A recurrent application domain is massive two-loop collider phenomenology. In the non-planar two-loop four-point integrals relevant to NNLO $D$4 production, SecDec was used on the ggtt1 and ggtt2 topologies, including physical-region threshold behavior. For the more difficult light-fermion topology ggtt2, a combination of analytic subloop integration, new transformations, and the version-2.1 user-defined-function interface was used to reduce spurious singular behavior before numerical treatment (Borowka et al., 2013).

Another major application is the momentum-dependent $D$5 correction to neutral Higgs-boson masses in the MSSM. In that setting, FeynArts generates the diagrams, FormCalc and TwoCalc perform trace evaluation and tensor reduction, and SecDec evaluates the analytically unknown two-loop propagator integrals with several masses and nonzero external momentum. The hard integrals fall into four two-loop two-point topologies and occur in 34 different mass configurations with up to four different mass scales; the corresponding self-energy shifts were interfaced back into FeynHiggs (Borowka et al., 2014, Borowka et al., 2014).

SecDec has also served as a numerical backend in full two-loop amplitude calculations. In Higgs boson pair production in gluon fusion at NLO with full top-quark mass dependence, the virtual amplitude was generated with GoSam, reduced as far as possible with Reduze, and the resulting two-loop integrals were evaluated numerically using SecDec-3.0. The calculation involved 8 integral families with 9 propagators each, leading after partial reduction to 145 planar master integrals, 70 non-planar integrals, and 112 integrals that differ by a crossing; in the hardest non-planar sectors, tensor integrals with contracted loop momenta in the numerator were evaluated directly by SecDec (Borowka et al., 2016).

In soft-collinear effective theory, SecDec has been used in a different mode. For dijet soft functions, the singularity structure was first isolated analytically into universal Laplace-space kernels, after which SecDec’s general mode handled sector decomposition, pole extraction, and numerical integration while the observable-dependent measurement function remained symbolic until the final stage. This allowed numerical two-loop results for several SCET-1 soft functions, including the $D$6-parameter and angularities (Bell et al., 2015).

The program has also been integrated into nonstandard workflows outside conventional multiloop amplitude reduction. In one MSR-field-theory application, FeynArts was adapted to generate 1PI diagrams, Mathematica code was used to integrate over loop frequencies and reduce the resulting expressions to scalar Euclidean Feynman integrals, and SecDec then supplied the dimensional regularization and $D$7-expansion of the remaining scalar integrals (Pereira, 2022).

6. Methodological position, strengths, and limitations

A persistent misconception is that SecDec is only a Euclidean scalar-master-integral code. The literature describes a broader object: already by version 2 it handled arbitrary loop kinematics via contour deformation; version 2.1 supported contracted tensor integrals and user-defined loop-like functions; version 3 added inverse propagators, linear propagators, and later complex masses; pySecDec generalized the framework further to multiple regulators and reusable integral libraries (Borowka et al., 2012, Borowka et al., 2013, Jahn, 2018).

A second misconception is that SecDec is a universally optimal black box. The papers are explicit about trade-offs. Sector decomposition can generate many sectors and large expressions; runtime and memory can become large for complicated integrals; convergence may slow down near thresholds or in the presence of strong cancellations; and heuristic decomposition strategies can fail by infinite recursion, which is why deterministic strategies such as G1 and G2 were introduced (Borowka et al., 2015).

Its methodological role is therefore complementary to analytic methods rather than a replacement for them. The MSSM Higgs application states explicitly that SecDec is not a fully dedicated two-loop self-energy solver like TSIL; it is a general-purpose decomposition-and-integration framework, and this can make it computationally heavier than methods based on differential equations when those are available (Borowka et al., 2014). Similarly, the SCET soft-function application highlights the tradeoff between universality plus automation and the loss of exact analytic expressions, together with potential slow convergence or cancellation-induced loss of precision in angular-dependent cases (Bell et al., 2015).

Within those limits, SecDec’s distinctive niche is clear. It is most useful when integrals are multi-loop, multi-scale, threshold-sensitive, and analytically unavailable or cumbersome; when one needs direct access to physical kinematics rather than Euclidean continuations; and when a numerical backend must be integrated into a larger reduction, renormalization, or amplitude-evaluation chain. In that regime, SecDec functions as an automated bridge from singular parametric representations to finite numerical coefficient functions, and that role has remained stable across the evolution from the original package to SecDec 3 and pySecDec (Borowka, 2015, Borowka et al., 2017).