Sector Decomposition: Methods and Applications

• Sector decomposition is a method that partitions complex integrals or constrained systems into simpler sectors to explicitly isolate singularities and enable efficient analytic and numerical evaluations.
• It underpins diverse applications, from isolating UV/IR divergences in multi-loop Feynman integrals to exact decompositions in portfolio optimization and large-scale energy systems.
• Advanced algorithmic variants—including iterative and geometric approaches—preserve the original structure, enhance convergence, and facilitate hybrid analytic-numeric computations.

Sector decomposition denotes a family of decomposition procedures that partition a structured object into sectors so that singular, constrained, or weakly coupled behavior becomes explicit and tractable. In perturbative quantum field theory, it is a constructive way to expose and factorize ultraviolet and infrared singularities of multi-loop Feynman integrals in dimensional regularization, extract the Laurent expansion in the regulator $\varepsilon$, and evaluate finite coefficients numerically (Tentyukov et al., 2010). In other arXiv literatures, the same expression refers to exact decompositions of constrained portfolio simplices in reinforcement learning (Winkel et al., 2024), sectoral and spatial decompositions of multi-sector energy-system optimization models (Parolin et al., 11 Apr 2025), and orthogonal decompositions of completely positive maps that underlie localized sector theory in algebraic quantum field theory (Ojima et al., 2015).

1. Terminological scope and common structural idea

Across these literatures, sector decomposition is not a single algorithm but a recurring structural motif: a complicated global object is rewritten as a sum, product, or direct integral of simpler sectorwise objects, together with deterministic rules that preserve the original problem. In the Feynman-integral setting, the sectors are subregions of parameter space on which singular behavior factorizes into monomials (Carter et al., 2010). In constrained portfolio allocation, the sectors are padded standard simplices attached to asset groups and recombined through a weighted Minkowski sum (Winkel et al., 2024). In large linear energy-system models, the relevant structure is block-angular: sector blocks are loosely coupled by a small set of shared constraints or variables, so decomposition separates sector subproblems from a coordinating master problem (Hadidi et al., 29 Jul 2025). In AQFT, the sectors arise from orthogonal decompositions of local states viewed as completely positive maps (Ojima et al., 2015).

The common algebraic purpose is preservation rather than approximation. The QFT algorithms isolate the same Laurent coefficients that were implicit in the original dimensionally regularized integral (Tentyukov et al., 2010). The portfolio construction proves exact equivalence between the constrained simplex and its decomposed representation (Winkel et al., 2024). The AQFT framework identifies local sectors through direct-integral decompositions of Stinespring representations (Ojima et al., 2015). In the energy-optimization literature, the decomposition is exact at the model level when the master and subproblem coupling is formulated correctly, although practical performance depends strongly on coupling structure and cut quality (Hadidi et al., 29 Jul 2025).

2. Multi-loop Feynman integrals: parametric form and singularity factorization

In perturbative QFT, sector decomposition begins after Schwinger or Feynman parametrization. A generic scalar $L$-loop integral can be written in parameter space as

$$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$

where $U$ and $F$ are graph polynomials and $\gamma,\beta$ are functions of $\varepsilon$ (Kaneko et al., 2010). In the formulation used by SecDec, loop integrals reduce to integrals involving the first and second Symanzik polynomials $\mathcal{U}$ and $\mathcal{F}$, while more general polynomial parameter integrals can be treated directly in the form $$\int_0^1 dx_1\cdots dx_N \prod_i P_i(\vec x,\{\alpha\})^{\nu_i}$$ with $L$0 (Carter et al., 2010).

The singular structure is concentrated at parameter-space boundaries. UV and IR divergences appear when subsets of Feynman parameters vanish so that $L$1 or $L$2 also vanish, producing overlapping endpoint singularities. The core algorithm therefore divides the integration region into sectors and applies variable changes so that, in each final sector, the singular behavior is encoded in explicit monomials such as $L$3, while the remaining polynomial factor is finite and non-zero at the origin (Tentyukov et al., 2010). After iterated decomposition, one obtains sector integrals of the schematic form

$L$4

with $L$5 finite as any subset of $L$6 (Tentyukov et al., 2010).

For loop integrals, a standard preliminary step is primary sector decomposition: the simplex constrained by $L$7 is partitioned into sectors indexed by the largest Feynman parameter, and each sector is mapped to a unit hypercube (Tentyukov et al., 2010). SecDec then recursively chooses minimal sets of variables whose simultaneous vanishing forces at least one polynomial factor to vanish, splits the corresponding cube into ordered subsectors, and rescales variables until every dangerous factor has a non-zero constant term after monomial extraction (Carter et al., 2010). Pole extraction proceeds by Taylor subtraction or equivalent plus-distribution manipulations, yielding a Laurent series

$L$8

whose coefficients $L$9 are finite parameter integrals (Carter et al., 2010).

3. Algorithmic variants and software implementations

The iterative strategy associated with Binoth–Heinrich, Bogner–Weinzierl, and Smirnov–Tentyukov is the standard computational template: primary sectors, recursive subsectoring, monomial factorization, analytic pole isolation, and numerical integration of the finite coefficients (Tentyukov et al., 2010). FIESTA implements this workflow in Mathematica and C, uses the Cuba library for multidimensional integration, and is parallelizable on modern multicore computers and even on multiple computers (Tentyukov et al., 2010). SecDec provides a related general framework in Mathematica, Perl, and Fortran 77; it accepts both loop integrals and general polynomial parameter integrals, supports Euclidean multi-loop integrals, tensor numerators, $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$0-dependent propagator powers, and phase-space integrals, and produces Fortran routines for each Laurent coefficient after subtraction and expansion (Carter et al., 2010).

A distinct line of development replaces iterative factorization by computational geometry. The non-iterative method based on convex geometry maps monomials of a polynomial to power vectors, defines dominance cones

$$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$1

and triangulates the non-empty cones into simplicial cones whose monomial coordinate changes give the desired factorization (Kaneko et al., 2010). This construction is deterministic, eliminates the possibility of infinite decomposition loops, and in the reported test implementation often yields fewer final sectors than standard iterative strategies; for the triple box, for example, the geometric method produced 6568 final sectors versus 10155 for strategy X (Kaneko et al., 2010).

A further simplification appears in specialized self-energy topologies. In the 2-loop scalar self-energy treatment, diagram-adapted variable changes bring the first Symanzik polynomial to the common form

$$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$2

after which six primary sectors and an explicit factor $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$3 isolate the UV behavior (Kato, 24 Jan 2026). For the depictive 3-loop two-point functions, a tailored reparametrization yields the “complete” four-variable form

$$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$4

and a three-step decomposition factors $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$5 as $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$6 with a universal residual function $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$7 (Doncker et al., 2024). These structured constructions retain the central purpose of sector decomposition while sharply reducing combinatorial complexity.

4. High-order perturbative applications

The practical value of sector decomposition in QFT is established by multi-loop benchmarks. FIESTA was used to evaluate four-loop massless propagator master integrals numerically, including the most complicated topologies among the 28 independent four-loop massless propagator-like masters (Tentyukov et al., 2010). For coefficients already known analytically, the numerical results agreed to 3–4 significant digits, and higher $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$8 coefficients not available analytically at the time were obtained for future higher-loop applications (Tentyukov et al., 2010).

The method also enters precision Higgs phenomenology. In the three-loop computation of the momentum-dependent Higgs self-energy at order $$G = \int_{x_i\ge 0} d^N x\;\delta\!\left(1-\sum_{j=1}^N x_j\right)\,\Bigl(\prod_j x_j^{\nu_j-1}\Bigr)\,U(x)^\gamma\,F(x)^\beta,$$9, IBP reduction first produced an inconvenient master basis with spurious $U$0 denominators and kinematic singularities. A “good” basis of 294 master integrals was then constructed so that the coefficients of the masters were free of poles in $U$1, and the integrals were evaluated numerically with FIESTA 5.0 (R. et al., 2022). The calculation was performed below threshold, so ComplexMode=False was sufficient; the default precision was about 6 digits for each master integral, and cancellations reduced the final precision of the Higgs-mass shift to about 4 digits. The reported correction was about $U$2 MeV at $U$3 GeV (R. et al., 2022).

The depictive 3-loop study illustrates a complementary mode of use: the coefficients of the ultraviolet divergent part are determined analytically, while the coefficients of the finite part are computed numerically, with explicit energy dependence and threshold behavior (Doncker et al., 2024). The 2-loop self-energy treatment likewise separates the divergent and finite parts into integrals that can be renormalized analytically and evaluated numerically, with the UV singularity concentrated in one sector variable after decomposition (Kato, 24 Jan 2026). Together these examples show that sector decomposition functions both as a fully numerical back end for otherwise inaccessible master integrals and as an analytic-numeric hybrid method for extracting pole structure in closed form.

5. Exact simplex and sector decomposition in constrained reinforcement learning

A different usage of sector decomposition appears in constrained portfolio optimization. The action space of an unconstrained allocation problem is the simplex

$U$4

but sector-based rules such as $U$5 and $U$6 cut out a convex polytope inside that simplex (Winkel et al., 2024). The paper recasts this constrained action set as a weighted Minkowski sum of padded standard simplices,

$U$7

with $U$8, $U$9, $F$0, and $F$1 (Winkel et al., 2024). The construction is exact rather than approximate: every feasible constrained allocation has a preimage in the decomposed space, and the weighted sum reproduces exactly the original H-representation of the polytope (Winkel et al., 2024).

This exactness underlies CAOSD, “Constrained Allocation Optimization with Simplex Decomposition.” The surrogate action space is a product of four unconstrained simplexes, sampled auto-regressively from Dirichlet distributions and mapped surjectively to the original constrained portfolio (Winkel et al., 2024). Because the mapping preserves optimality, standard PPO can be applied in the surrogate space without penalty terms or Lagrange multipliers, while every action seen by the environment is feasible by construction (Winkel et al., 2024).

On Nasdaq-100 monthly data from 2010–2020 with 2021 backtesting, and across 100 randomly generated two-constraint experiments, CAOSD outperformed the CRL baselines RCPO, IPO, and P3O. The average annualized return in simulation was $F$2 for CAOSD, compared with $F$3 for P3O, $F$4 for RCPO, and $F$5 for IPO; in backtesting the corresponding figures were $F$6, $F$7, $F$8, and $F$9, with the random baseline at $\gamma,\beta$0 in simulation and $\gamma,\beta$1 in backtesting (Winkel et al., 2024). The paper interprets this as a hierarchical sector decomposition of the portfolio decision: joint-sector allocation, sector-specific top-ups, and a residual global allocation.

6. Sectoral decomposition in large-scale energy-system optimization

In multi-sector energy-system models, the reviewed literature does not always use “sector decomposition” as a separate term, but it explicitly identifies the block structures and decomposition methods that make sectorwise parallelization possible (Hadidi et al., 29 Jul 2025). The starting point is a large linear or mixed-integer linear program with a constraint matrix that can often be permuted into block-diagonal, horizontally bordered block-diagonal, vertically bordered block-diagonal, arrowhead, or staircase form. Sector decomposition in this context means treating each sector block—electricity, heat, gas, hydrogen, transport, industry, or sector-region-time combinations—as a subproblem and handling the relatively small set of coupling constraints or shared variables in a master problem or through prices (Hadidi et al., 29 Jul 2025).

The appropriate decomposition method depends on the coupling. Constraint-coupled structures align with Dantzig–Wolfe or Lagrangian decomposition; variable-coupled structures align with Benders decomposition or variable splitting; arrowhead structures combine both; and staircase structures support time-stage decomposition (Hadidi et al., 29 Jul 2025). The review emphasizes that no single approach is universally superior and that the lack of standardized benchmark suites complicates comparison (Hadidi et al., 29 Jul 2025). It also surveys generic software frameworks, including GCG, BaPCod, Coluna, DIP, Plasmo.jl + PlasmoAlgorithms, StructJuMP, StochasticPrograms.jl, mpi-sppy, DSP, SMS++, and urbs-DecEnSys (Hadidi et al., 29 Jul 2025).

A concrete sectoral formulation is given for multi-sector capacity expansion models. The extensive-form LP contains investment variables $\gamma,\beta$2 and operational variables $\gamma,\beta$3 indexed by subperiod $\gamma,\beta$4, zone $\gamma,\beta$5, and sector $\gamma,\beta$6. Sectoral decomposition introduces export budgets

$\gamma,\beta$7

treats them as master variables, and solves separate subproblems for each $\gamma,\beta$8 pair subject to these budgets and sector-specific emission budgets (Parolin et al., 11 Apr 2025). The resulting temporal + sectoral Benders algorithm achieved runtime reductions within 15%–70% relative to existing decomposition algorithms on continental United States case studies, while temporal + sectoral BD specifically produced 20%–70% runtime reduction against temporal BD across the tested configurations (Parolin et al., 11 Apr 2025). The paper also reports a limitation of the budget-based formulation: hydrogen storage capacity is severely under-estimated, motivating a two-stage correction procedure (Parolin et al., 11 Apr 2025).

7. Orthogonal decomposition of local states in algebraic quantum field theory

In AQFT, sector decomposition refers neither to parameter-space sectors nor to block-angular optimization, but to the decomposition of localized completely positive maps. A local state on a net of observables is defined as a unital CP map $\gamma,\beta$9 with region $\varepsilon$0 such that $\varepsilon$1 for $\varepsilon$2 in the algebra of the causal complement of $\varepsilon$3, and $\varepsilon$4 for $\varepsilon$5 and some state $\varepsilon$6 (Ojima et al., 2015). This formalizes localized state preparation on a background representation rather than a global state on the full algebra.

The representation-theoretic content is extracted through the Stinespring representation of the CP map. The paper develops an orthogonal decomposition theory of CP maps: two CP maps $\varepsilon$7 and $\varepsilon$8 are mutually orthogonal if they have no non-trivial common CP submap, equivalently if the Stinespring representation of $\varepsilon$9 splits by a projection into the Stinespring representations of $\mathcal{U}$0 and $\mathcal{U}$1 (Ojima et al., 2015). On this basis it defines CP-measure spaces and orthogonal CP-measure spaces, and proves a Tomita-type theorem for CP maps: orthogonal CP-measure spaces with barycenter $\mathcal{U}$2 are categorically isomorphic to abelian von Neumann subalgebras of the $\mathcal{U}$3-relative commutant $\mathcal{U}$4 (Ojima et al., 2015).

This decomposition feeds directly into localized sector theory. Local factor states are local states whose full center is inherited from the center of the reference representation, and local sectors are quasi-equivalence classes of such local factor states (Ojima et al., 2015). The paper further shows that, under an appropriate condition on the Stinespring representation of a local state in the vacuum background, the DHR selection criterion follows automatically, so that the induced representation is equivalent to a localized endomorphism supported in a bounded region (Ojima et al., 2015). In this setting, sector decomposition becomes a direct-integral decomposition of localized preparations into mutually disjoint superselection sectors.

Sector decomposition therefore names a class of exact or structure-preserving decompositions whose mathematical forms differ sharply across fields but whose objective is stable: isolate the part of a problem that obstructs direct treatment, encode it in simpler sectorwise components, and recover the original object without losing the information needed for analysis, optimization, or representation theory.