{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.effective-actions.wilsonian-1pi-domain-map",
  "title": "Wilsonian–1PI domain map: ask about locality before exactness",
  "artifact_class": "original monochrome Wilsonian-versus-1PI locality and infrared decision map",
  "quantitative_status": "the coupling relation, massive-kernel coefficients, loop orders, coupling structure, and domain predicates are exact under the stated conventions; layout and arrow lengths are schematic and not to scale",
  "generated_by": "figures-src/supersymmetry-duality/wilsonian-1pi-domain-map.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-24",
  "registry_lifecycle": {
    "current_status": "read from the governed registry when provenance is generated",
    "recommended_materialized_status": "prototype",
    "note": "Materialization does not itself promote the governed record, edit its public route, integrate a page, or establish release acceptance."
  },
  "owner_page": {
    "id": "qft.topic.susy-holomorphy.nonrenormalization-wilsonian-1pi",
    "file": "src/content/docs/supersymmetry-duality/holomorphy-anomalies-exact-constraints/nonrenormalization-wilsonian-1pi.md",
    "route": "/supersymmetry-duality/holomorphy-anomalies-exact-constraints/nonrenormalization-wilsonian-1pi/"
  },
  "canonical_anchor": "wilsonian-1pi-domain-map",
  "public_assets": {
    "svg": "/figures/supersymmetry-duality/wilsonian-1pi-domain-map.svg",
    "structured_json": "/figures/supersymmetry-duality/wilsonian-1pi-domain-map.json"
  },
  "reader_question": "When does supersymmetric perturbative superpotential nonrenormalization constrain a local Wilsonian action, and when can gapped or massless 1PI infrared behavior require a different conclusion?",
  "takeaway": "A Wilsonian action at positive cutoff is shell-local for external momenta below that cutoff, whereas a full 1PI action is locally expandable only after a mass-gap test; in massless cubic Wess–Zumino theory the one-loop correction is a nonlocal Kähler D-term and the first chiral holomorphic anomaly occurs at two loops, never as a local Wilsonian superpotential correction.",
  "alt_text": "A portrait monochrome decision map begins by asking which modes were integrated out. One branch identifies a Wilsonian action at positive cutoff as local for external momentum well below the cutoff, states perturbative local-superpotential nonrenormalization, permits Kähler and wavefunction running, and gives Phi-c equals square-root Z times Phi and y-c equals y-h divided by Z to the three-halves. A parallel branch identifies the full 1PI action and asks whether the full spectrum is gapped. The gapped branch gives an analytic low-momentum expansion of a representative massive kernel. The massless branch warns that logarithms and inverse boxes are nonanalytic and can produce local-looking chiral integrals outside the Wilsonian theorem. A final massless cubic Wess–Zumino audit labels the one-loop correction as a nonlocal D-term with no chiral anomaly and the first nonzero chiral 1PI term as a two-loop y-h cubed times y-h-dagger squared Phi cubed term. The closing rule requires the functional, cutoff, regulator, scheme, momentum domain, mass gap, and loop order before transferring an exactness claim.",
  "caption_semantics": "Domain classifier for four-dimensional rigid N=1 supersymmetric effective actions. For a Wilsonian cutoff mu greater than zero and external momentum much smaller than mu, the shell contribution has a local derivative expansion; the local Wilsonian superpotential has no perturbative loop correction under a supersymmetry-preserving regulator, while Kähler data and canonical couplings run. With K containing Z Phi-dagger Phi, Phi-c equals Z to the one-half Phi and y-c equals y-h over Z to the three-halves for the displayed cubic coupling. The full 1PI functional instead requires an infrared test: a complete mass gap permits a local expansion below threshold, while massless propagation produces nonanalytic logarithms or inverse boxes. In massless cubic Wess–Zumino theory the one-loop 1PI effect is a nonlocal Kähler D-term and no chiral holomorphic anomaly occurs at one loop; the first nonzero chiral 1PI term occurs at two loops and is proportional to y-h cubed times y-h-dagger squared Phi cubed. It is an infrared 1PI effect, not a local Wilsonian superpotential correction. Layout is schematic; formulas and status labels are exact only in their declared domains.",
  "conventions": {
    "spacetime": "four-dimensional Lorentzian spacetime with eta=diag(+1,-1,-1,-1); the massive threshold check is Wick-rotated to Euclidean p_E",
    "supersymmetry": "rigid N=1",
    "wilsonian_definition": "S_mu integrates high-momentum modes above a positive Wilsonian cutoff mu while retaining lower modes",
    "wilsonian_locality_domain": "external invariants are nonexceptional and |p| is much smaller than mu, so shell kernels admit a local derivative expansion",
    "one_pi_definition": "Gamma integrates all quantum modes and therefore includes infrared propagation",
    "one_pi_locality_domain": "a local low-momentum expansion requires a complete mass gap and external momentum below the first threshold",
    "kinetic_normalization": "K_mu contains Z(mu) Phi_dagger Phi with positive real Z; Phi_c=Z(mu)^(1/2) Phi",
    "cubic_superpotential": "W=y_h Phi^3/3! in holomorphic normalization; y_c(mu)=y_h/Z(mu)^(3/2) after canonical normalization",
    "nonrenormalization_scope": "the no-loop-correction statement applies to the local perturbative Wilsonian superpotential with a supersymmetry-preserving regulator; it does not exclude Kähler, wavefunction, threshold, nonperturbative, or infrared 1PI effects",
    "regulator": "supersymmetry-preserving regulator and local counterterms; a finite subtraction scheme must be fixed before comparing local coefficients",
    "massless_limit": "take nonexceptional external momenta before any p->0 limit; log(-p^2/mu^2) and 1/Box signal failure of a Taylor expansion at zero momentum",
    "spurion_charge_check": "under Phi charge +1 and y_h charge -3, y_h^3(y_h_dagger)^2 Phi^3 is neutral; with R(Phi)=2/3 and R(y_h)=0 its chiral integrand has R charge 2"
  },
  "domains": [
    {
      "id": "wilsonian_local",
      "functional": "S_mu",
      "integrated_modes": "momenta above mu, with mu>0",
      "infrared_spectrum": "light fields may remain below mu",
      "momentum_domain": "|p|<<mu",
      "locality": "local derivative expansion of shell contributions",
      "superpotential_status": "no perturbative loop correction to the local Wilsonian superpotential under the declared assumptions",
      "allowed_quantum_effects": [
        "Kahler and wavefunction corrections",
        "running canonical couplings",
        "threshold matching",
        "separately justified nonperturbative superpotentials"
      ]
    },
    {
      "id": "one_pi_gapped",
      "functional": "Gamma",
      "integrated_modes": "all quantum modes",
      "infrared_spectrum": "complete mass gap m_gap>0",
      "momentum_domain": "|p|<<m_gap and below the first threshold",
      "locality": "local low-momentum derivative expansion",
      "superpotential_status": "compare with Wilsonian data only after matching fields, thresholds, regulator, and finite scheme",
      "allowed_quantum_effects": [
        "local threshold coefficients",
        "Kahler and wavefunction corrections",
        "nonperturbative terms consistent with the gapped low-energy theory"
      ]
    },
    {
      "id": "one_pi_massless",
      "functional": "Gamma",
      "integrated_modes": "all quantum modes",
      "infrared_spectrum": "one or more massless propagating modes",
      "momentum_domain": "nonexceptional external momentum first; p->0 is an infrared limit",
      "locality": "not guaranteed; logarithmic and inverse-Box kernels are nonanalytic",
      "superpotential_status": "a local-looking chiral 1PI term can arise from an infrared nonlocal D-term and is not constrained as a local Wilsonian vertex",
      "allowed_quantum_effects": [
        "nonlocal Kahler form factors",
        "infrared logarithms",
        "holomorphic anomalies at their actual perturbative order"
      ]
    }
  ],
  "representative_massive_case": {
    "model": "one representative equal-mass Euclidean bubble kernel",
    "definition": "B_m(p_E^2)=integral_0^1 dx log[(m^2+x(1-x)p_E^2)/mu^2]",
    "mass": 2,
    "subtraction_scale": 1,
    "external_p_squared": 0.04,
    "series": "log(m^2/mu^2)+p_E^2/(6m^2)-p_E^4/(60m^4)+O(p_E^6/m^6)",
    "analytic_radius_note": "the expansion is used only below the first physical threshold; the displayed Euclidean fixture has p_E^2/m^2=0.01"
  },
  "representative_massless_case": {
    "model": "massless cubic Wess-Zumino theory",
    "action": "integral d^8z Phi_dagger Phi + [integral d^6z y_h Phi^3/3! + h.c.]",
    "mass": 0,
    "one_loop": {
      "loop_order": 1,
      "operator_class": "nonlocal 1PI Kahler or wavefunction D-term",
      "schematic_term": "abs(y_h)^2 integral d^4theta Phi_dagger log(-Box/mu^2) Phi",
      "chiral_holomorphic_anomaly": false
    },
    "two_loop": {
      "loop_order": 2,
      "operator_class": "infrared chiral 1PI effective-potential term",
      "schematic_term": "C_2 integral d^2theta y_h^3(y_h_dagger)^2 Phi^3+h.c., with C_2 nonzero",
      "chiral_holomorphic_anomaly": true,
      "wilsonian_local_superpotential_correction": false,
      "coefficient_boundary": "the visual prints only proportionality because the numerical coefficient depends on the superpotential and loop normalization conventions"
    }
  },
  "semantic_table": [
    {
      "statement": "shell locality",
      "wilsonian": "yes for mu>0 and |p|<<mu",
      "one_pi": "not a general property",
      "perturbative": "yes under the scale hierarchy",
      "nonperturbative": "definition remains meaningful, but extra sectors require separate control",
      "infrared_sensitive": "no integrated zero-momentum modes in the shell statement"
    },
    {
      "statement": "local superpotential nonrenormalization",
      "wilsonian": "no perturbative loop correction under the declared assumptions",
      "one_pi": "not transferable without a separate infrared and locality proof",
      "perturbative": "protected only for the local Wilsonian F-term",
      "nonperturbative": "not generically protected; instanton or strong-dynamics terms are analyzed separately",
      "infrared_sensitive": "yes when restated as a claim about massless 1PI data"
    },
    {
      "statement": "Kahler and wavefunction data",
      "wilsonian": "may run with mu",
      "one_pi": "may contain momentum-dependent form factors",
      "perturbative": "allowed",
      "nonperturbative": "allowed",
      "infrared_sensitive": "massless 1PI logarithms are infrared sensitive"
    },
    {
      "statement": "canonical cubic coupling",
      "wilsonian": "y_c(mu)=y_h/Z(mu)^(3/2)",
      "one_pi": "a canonically normalized parameter is not by itself a full physical amplitude",
      "perturbative": "runs through Z even when y_h is perturbatively holomorphic",
      "nonperturbative": "requires the separately determined local effective data",
      "infrared_sensitive": "matching to an on-shell or momentum-subtraction observable requires 1PI data"
    },
    {
      "statement": "gapped 1PI derivative expansion",
      "wilsonian": "separate cutoff construction",
      "one_pi": "local for |p|<<m_gap below threshold",
      "perturbative": "analytic massive kernels furnish the expansion",
      "nonperturbative": "requires the exact spectrum to remain fully gapped",
      "infrared_sensitive": "controlled by m_gap>0"
    },
    {
      "statement": "massless cubic Wess-Zumino at one loop",
      "wilsonian": "no local superpotential correction",
      "one_pi": "nonlocal Kahler D-term; no chiral anomaly at one loop",
      "perturbative": "one loop",
      "nonperturbative": "not used",
      "infrared_sensitive": "yes; logarithmic at p^2=0"
    },
    {
      "statement": "massless cubic Wess-Zumino chiral anomaly",
      "wilsonian": "not a local Wilsonian superpotential correction",
      "one_pi": "first nonzero chiral term occurs at two loops and is proportional to y_h^3(y_h_dagger)^2 Phi^3",
      "perturbative": "two loops, not one loop",
      "nonperturbative": "not needed for this counterexample",
      "infrared_sensitive": "yes; enabled by massless nonlocality"
    }
  ],
  "inference_boundaries": [
    "Locality is asserted for a scale hierarchy, not for arbitrary external momenta or exceptional kinematics.",
    "A mass gap must cover the full spectrum relevant to the selected vacuum; a mass parameter in one multiplet alone is not sufficient.",
    "The perturbative Wilsonian theorem does not exclude nonperturbative superpotentials, Kahler corrections, wavefunction running, or threshold matching.",
    "The gapped 1PI branch establishes a derivative expansion below threshold but does not identify its finite coefficients with Wilsonian coefficients without matching.",
    "The massless Wess-Zumino counterexample is a 1PI infrared statement: its first chiral anomaly is at two loops, while the one-loop correction is a D-term.",
    "The four-dimensional cubic Wess-Zumino model is used perturbatively as an effective field theory; no nonperturbative UV completion is asserted."
  ],
  "accessibility_encoding": {
    "explicit_light_canvas": true,
    "color_independence": "Black outlines, white and gray fills, direct YES/NO and loop-order labels, equations, and solid arrows carry every distinction without color.",
    "structured_equivalent": "This JSON contains every functional, cutoff, momentum domain, mass-gap predicate, locality status, exactness qualification, allowed correction, coupling relation, massive and massless case, loop order, inference boundary, and source locator.",
    "narrow_width_strategy": "The portrait flow preserves reading order when scaled; the structured semantic table remains the authoritative equivalent when direct labels become small."
  },
  "scientific_references": [
    {
      "authors": "Nathan Seiberg",
      "title": "Naturalness Versus Supersymmetric Non-renormalization Theorems",
      "journal": "Physics Letters B",
      "volume": "318",
      "pages": "469-475",
      "year": 1993,
      "doi": "10.1016/0370-2693(93)91541-T",
      "arxiv": "hep-ph/9309335",
      "url": "https://arxiv.org/abs/hep-ph/9309335",
      "locator": "section 3, arXiv PDF pp. 5-7; section 4.1, Eqs. (4.1)-(4.3), arXiv PDF p. 7",
      "use": "Wilsonian versus 1PI infrared scope and the massless Wess-Zumino holomorphic-anomaly coupling structure"
    },
    {
      "authors": "Marcus T. Grisaru, Warren Siegel, and Martin Rocek",
      "title": "Improved Methods for Supergraphs",
      "journal": "Nuclear Physics B",
      "volume": "159",
      "pages": "429-450",
      "year": 1979,
      "doi": "10.1016/0550-3213(79)90344-4",
      "url": "https://doi.org/10.1016/0550-3213(79)90344-4",
      "locator": "pp. 429-450",
      "use": "perturbative supergraph nonrenormalization and D-term organization"
    },
    {
      "authors": "I. L. Buchbinder, S. M. Kuzenko, A. Yu. Petrov, and J. V. Yarevskaya",
      "title": "Superfield Effective Potential",
      "year": 1995,
      "arxiv": "hep-th/9501047",
      "url": "https://arxiv.org/abs/hep-th/9501047",
      "locator": "section 4, Eqs. (34)-(43), arXiv PDF pp. 8-9",
      "use": "explicit first nonzero two-loop chiral effective-potential term in the massless Wess-Zumino model"
    },
    {
      "authors": "I. Jack, D. R. T. Jones, and P. West",
      "title": "Not the No-Renormalisation Theorem?",
      "journal": "Physics Letters B",
      "volume": "258",
      "pages": "382-385",
      "year": 1991,
      "doi": "10.1016/0370-2693(91)91103-3",
      "url": "https://doi.org/10.1016/0370-2693(91)91103-3",
      "locator": "pp. 382-385",
      "use": "original massless two-loop apparent superpotential-renormalization result interpreted as an infrared 1PI effect"
    },
    {
      "authors": "Steven Weinberg",
      "title": "The Quantum Theory of Fields, Volume III: Supersymmetry",
      "publisher": "Cambridge University Press",
      "year": 2000,
      "doi": "10.1017/CBO9781139644198",
      "local_path": "sources/Weinberg - 2000 - The Quantum Theory of Fields Volume 3 Supersymmetry.pdf",
      "locator": "section 27.6, printed pp. 148-153; section 30.3, printed pp. 313-316",
      "use": "local textbook cross-check of Wilsonian nonrenormalization and supergraph effective-action scope"
    }
  ]
}