{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.holomorphy.spurion-anomaly-exactness-flow",
  "title": "From spurions to an exact Wilsonian term",
  "artifact_class": "logical scientific flow diagram",
  "quantitative_plot": false,
  "generated_by": "figures-src/supersymmetry-duality/spurion-anomaly-exactness-flow.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-24",
  "owner_pages": [
    {
      "id": "qft.topic.susy-holomorphy.holomorphic-couplings-background-superfields",
      "route": "/supersymmetry-duality/holomorphy-anomalies-exact-constraints/holomorphic-couplings-background-superfields/",
      "file": "src/content/docs/supersymmetry-duality/holomorphy-anomalies-exact-constraints/holomorphic-couplings-background-superfields.md"
    },
    {
      "id": "qft.topic.susy-holomorphy.r-symmetry-anomalies-holomorphic-scale",
      "route": "/supersymmetry-duality/holomorphy-anomalies-exact-constraints/r-symmetry-anomalies-holomorphic-scale/",
      "file": "src/content/docs/supersymmetry-duality/holomorphy-anomalies-exact-constraints/r-symmetry-anomalies-holomorphic-scale.md"
    },
    {
      "id": "qft.topic.susy-holomorphy.nonperturbative-superpotentials",
      "route": "/supersymmetry-duality/holomorphy-anomalies-exact-constraints/nonperturbative-superpotentials/",
      "file": "src/content/docs/supersymmetry-duality/holomorphy-anomalies-exact-constraints/nonperturbative-superpotentials.md"
    },
    {
      "id": "qft.topic.susy-holomorphy.quantum-chiral-rings-konishi-anomalies",
      "route": "/supersymmetry-duality/holomorphy-anomalies-exact-constraints/quantum-chiral-rings-konishi-anomalies/",
      "file": "src/content/docs/supersymmetry-duality/holomorphy-anomalies-exact-constraints/quantum-chiral-rings-konishi-anomalies.md"
    }
  ],
  "assets": {
    "editable_source": "figures-src/supersymmetry-duality/spurion-anomaly-exactness-flow.tex",
    "rendered_svg": "/figures/supersymmetry-duality/spurion-anomaly-exactness-flow.svg",
    "semantic_json": "/figures/supersymmetry-duality/spurion-anomaly-exactness-flow.json"
  },
  "reader_question": "Which steps merely constrain a holomorphic Wilsonian term, and which independent dynamical inputs establish its generation, normalization, and branches?",
  "takeaway": "Holomorphy, dimensions, and anomalous spurion charges fix the ADS functional form on the nonsingular SQCD meson patch, but a controlled instanton plus holomorphic decoupling are independent inputs needed to establish a nonzero coefficient and propagate its normalization.",
  "alt_text": "A portrait black-and-gray flow separates formal constraints from dynamical inputs. In SU(Nc) SQCD with zero less than Nf less than Nc and nonzero determinant M, dimensions and axial and R charges allow a candidate proportional to the one over Nc minus Nf power of Lambda_h to the 3Nc minus Nf divided by determinant M. A dashed gate marks that symmetry alone cannot establish a nonzero coefficient or branch. A controlled Nf equals Nc minus one instanton and the threshold relation Lambda_low to the 3Nc minus Nf plus one equals m Lambda_high to the 3Nc minus Nf yield the ADS coefficient Nc minus Nf. A dotted exit lists neutral invariants, extra zero modes, singular meson patches, massless 1PI limits, and missing phase or branch data as reasons to weaken the conclusion.",
  "caption_semantics": "Formal constraints and independent dynamical inputs for the Affleck-Dine-Seiberg superpotential in four-dimensional N=1 SU(Nc) SQCD. On the local Wilsonian meson patch with 0<Nf<Nc, det M nonzero, a supersymmetric regulator, the stated holomorphic-scale convention, and a chosen fractional-power branch, dimensions and anomalous axial and anomaly-free R charges allow the displayed candidate but do not prove generation or normalization. A controlled instanton at Nf=Nc-1 fixes the seed coefficient; holomorphic decoupling with the corrected low-energy exponent 3Nc-Nf+1 propagates it to C=Nc-Nf and checks the Nc pure-SYM branches after full-rank mass deformation. The diagram is exact for the printed equations and implication statuses and schematic, not to scale.",
  "conventions": {
    "spacetime": "four-dimensional rigid N=1 supersymmetry",
    "gauge_theory": "SU(Nc) SQCD with Nf fundamental-antifundamental pairs and 0<Nf<Nc",
    "dynkin_index": "T(fundamental)=1/2",
    "object": "local Wilsonian F-term at scale mu>0 in holomorphic variables with a supersymmetric regulator",
    "meson": "M^i_j=Q^i tilde Q_j, restricted to det M nonzero",
    "axial_assignment": "q_A(Q)=q_A(tilde Q)=1",
    "anomaly_free_r_assignment": "R(Q)=R(tilde Q)=1-Nc/Nf and R(Lambda_h^(3Nc-Nf))=0",
    "holomorphic_scale": "Lambda_h^b0=mu^b0 exp(2 pi i tau_h), b0=3Nc-Nf, q_A(Lambda_h^b0)=2Nf",
    "branch": "one branch of the fractional power is chosen; after full-rank mass deformation pure SU(Nc) SYM has Nc branches",
    "scale_matching": "Lambda_(Nf-1)^(3Nc-Nf+1)=m Lambda_(Nf)^(3Nc-Nf)"
  },
  "formal_charge_and_dimension_table": [
    {
      "quantity": "det M",
      "dimension": "2 Nf",
      "axial_charge": "2 Nf",
      "r_charge": "2(Nf-Nc)"
    },
    {
      "quantity": "Lambda_h^(3Nc-Nf)",
      "dimension": "3Nc-Nf",
      "axial_charge": "2Nf",
      "r_charge": "0"
    },
    {
      "quantity": "Lambda_h^(3Nc-Nf)/det M",
      "dimension": "3(Nc-Nf)",
      "axial_charge": "0",
      "r_charge": "2(Nc-Nf)"
    },
    {
      "quantity": "W_candidate",
      "dimension": "3",
      "axial_charge": "0",
      "r_charge": "2"
    }
  ],
  "implication_legend": [
    {
      "line_style": "solid",
      "status": "formal implication within the declared domain"
    },
    {
      "line_style": "long dashed",
      "status": "independent dynamical input"
    },
    {
      "line_style": "short dotted",
      "status": "failure or underdetermination exit"
    }
  ],
  "nodes_in_reading_order": [
    {
      "order": 1,
      "label": "Declare the object and domain",
      "content": "A local Wilsonian F-term at mu>0 in holomorphic variables; for the SQCD example 0<Nf<Nc, det M is nonzero, and a fractional-power branch is chosen."
    },
    {
      "order": 2,
      "label": "Apply formal constraints",
      "content": "Holomorphy, dimensions, anomalous axial charge, and anomaly-free R-charge leave the displayed monomial as the unique candidate on this patch when no neutral dimensionless holomorphic invariant exists."
    },
    {
      "order": 3,
      "label": "Allowed candidate",
      "equation": "W_candidate=C_(Nc,Nf)[Lambda_h^(3Nc-Nf)/det M]^(1/(Nc-Nf))",
      "fixed": [
        "functional dependence on Lambda_h and det M on the declared patch",
        "mass dimension three",
        "axial charge zero",
        "R-charge two"
      ],
      "open": [
        "whether C is nonzero",
        "the normalization of C",
        "the physical fractional-power branch"
      ]
    },
    {
      "order": 4,
      "label": "Formal-constraints gate",
      "content": "Formal constraints stop before generation, normalization, and physical branch selection; dynamics must enter."
    },
    {
      "order": 5,
      "label": "Controlled instanton",
      "equation": "2Nc-2Nf=2 for Nf=Nc-1",
      "content": "Complete Higgsing regulates the instanton calculation, gauge-Yukawa lifting leaves two fermion modes, and the seed coefficient is one in the stated convention."
    },
    {
      "order": 6,
      "label": "Holomorphic decoupling",
      "equation": "Lambda_(Nf-1)^(3Nc-Nf+1)=m Lambda_(Nf)^(3Nc-Nf)",
      "content": "The low-energy exponent is one larger than the high-energy exponent. Heavy-field F-term elimination propagates the seed to C=Nc-Nf, while full-rank mass deformation recovers Nc pure-SYM branches."
    },
    {
      "order": 7,
      "label": "Exact result",
      "equation": "W_ADS=(Nc-Nf)[Lambda_h^(3Nc-Nf)/det M]^(1/(Nc-Nf))",
      "hypotheses": "local Wilsonian F-term, declared scale convention, det M nonzero, chosen branch, controlled generation input, and verified holomorphic decoupling"
    },
    {
      "order": 8,
      "label": "Failure exits",
      "content": [
        {
          "condition": "a neutral dimensionless holomorphic invariant z exists",
          "conclusion": "a prefactor times arbitrary F(z), not a unique answer"
        },
        {
          "condition": "extra fermion zero modes remain unlifted",
          "conclusion": "a correlator, higher F-term, or zero may result rather than this superpotential"
        },
        {
          "condition": "det M=0",
          "conclusion": "the chosen meson coordinate patch fails"
        },
        {
          "condition": "the massless 1PI limit replaces a local Wilsonian action",
          "conclusion": "infrared nonlocality invalidates the local-term inference"
        },
        {
          "condition": "phase, branch, or dynamical input is absent",
          "conclusion": "the candidate is allowed but not established"
        }
      ]
    }
  ],
  "exact_result": "W_ADS=(Nc-Nf)[Lambda_h^(3Nc-Nf)/det M]^(1/(Nc-Nf))",
  "independent_dynamical_inputs": [
    {
      "id": "controlled_instanton",
      "relation": "2Nc-2Nf=2 at Nf=Nc-1",
      "conclusion": "the one-instanton sector generates the term and fixes C_(Nc,Nc-1)=1 in the declared scale convention"
    },
    {
      "id": "holomorphic_decoupling",
      "relation": "Lambda_(Nf-1)^(3Nc-Nf+1)=m Lambda_(Nf)^(3Nc-Nf)",
      "conclusion": "heavy-field F-term elimination gives C_(Nc,Nf)=Nc-Nf recursively"
    },
    {
      "id": "pure_sym_branch_check",
      "relation": "Lambda_SYM^(3Nc)=Lambda_h^(3Nc-Nf) det m",
      "conclusion": "full-rank masses recover the Nc pure-SYM branches"
    }
  ],
  "underdetermination_counterexample": {
    "model": "one-field Wess-Zumino spurion example",
    "assignments": [
      {
        "quantity": "Phi",
        "dimension": 1,
        "axial": 1,
        "r_charge": 1
      },
      {
        "quantity": "m",
        "dimension": 1,
        "axial": -2,
        "r_charge": 0
      },
      {
        "quantity": "y",
        "dimension": 0,
        "axial": -3,
        "r_charge": -1
      }
    ],
    "invariant": "z=y Phi/m",
    "invariant_dimension": 0,
    "invariant_axial_charge": 0,
    "invariant_r_charge": 0,
    "consequence": "when such a neutral dimensionless holomorphic invariant exists, formal constraints leave an arbitrary holomorphic function F(z)"
  },
  "failure_exits": [
    {
      "condition": "a neutral dimensionless holomorphic invariant z exists",
      "conclusion": "a prefactor times arbitrary F(z), not a unique answer"
    },
    {
      "condition": "extra fermion zero modes remain unlifted",
      "conclusion": "a correlator, higher F-term, or zero may result rather than this superpotential"
    },
    {
      "condition": "det M=0",
      "conclusion": "the chosen meson coordinate patch fails"
    },
    {
      "condition": "the massless 1PI limit replaces a local Wilsonian action",
      "conclusion": "infrared nonlocality invalidates the local-term inference"
    },
    {
      "condition": "phase, branch, or dynamical input is absent",
      "conclusion": "the candidate is allowed but not established"
    }
  ],
  "inference_boundaries": [
    "A spurionic symmetry is a bookkeeping constraint after a coupling is promoted to a background chiral field; it is not an independently conserved microscopic symmetry.",
    "Symmetry and holomorphy can permit a candidate without proving that its coefficient is nonzero.",
    "The absence of a neutral dimensionless invariant is patch- and field-content-dependent; when one exists, arbitrary F(z) dependence remains.",
    "The zero-mode count is necessary but does not replace a controlled measure, lifting, phase, and normalization calculation.",
    "The ADS expression is singular at det M=0, where the meson coordinate patch and low-energy description require reconsideration.",
    "The exact statement concerns a local Wilsonian F-term at nonzero separation scale, not an infrared-nonlocal massless 1PI functional.",
    "Fractional powers require branch data; full-rank mass deformation checks the Nc pure-SYM branches but does not erase this requirement."
  ],
  "long_semantic_equivalent": "Read from top to bottom. First declare a local Wilsonian F-term at a nonzero scale and the exact field-content, regulator, coordinate-patch, and branch assumptions. For SU(Nc) SQCD with 0<Nf<Nc, M=Q tilde Q and det M nonzero, b0=3Nc-Nf. With qA(Q)=qA(tilde Q)=1, det M and Lambda_h^b0 both have axial charge 2Nf. The anomaly-free R assignment gives R(det M)=2(Nf-Nc) and R(Lambda_h^b0)=0. Consequently Lambda_h^b0/det M has dimension 3(Nc-Nf), axial charge zero, and R-charge 2(Nc-Nf). Raising it to 1/(Nc-Nf) gives a dimension-three, R-charge-two candidate. These are formal implications, shown by solid arrows. They fix the functional form on this patch if no neutral dimensionless holomorphic invariant exists, but leave generation, normalization, and physical branch open. The dashed gate therefore says dynamics must enter. A controlled instanton at Nf=Nc-1, where a generic Higgs expectation value completely breaks the gauge group and lifting leaves 2Nc-2Nf=2 fermion modes, establishes a nonzero term and coefficient one in the declared convention. Holomorphic decoupling supplies a second independent input: after giving one flavor mass m, Lambda_low^(3Nc-Nf+1)=m Lambda_high^(3Nc-Nf). The plus one is required because the low-energy theory has Nf-1 flavors. Heavy-field F-term elimination propagates the coefficient to Nc-Nf. Giving all flavors full-rank masses yields Lambda_SYM^(3Nc)=Lambda_h^(3Nc-Nf) det m and checks the Nc pure-SYM branches. The resulting ADS superpotential is (Nc-Nf)[Lambda_h^(3Nc-Nf)/det M]^(1/(Nc-Nf)), with every listed hypothesis attached. The dotted failure exit weakens this conclusion whenever a neutral invariant allows arbitrary F(z), extra zero modes remain, det M vanishes, the massless 1PI rather than local Wilsonian functional is requested, or a dynamical phase, normalization, or branch input is missing.",
  "accessibility_encoding": {
    "reading_order": "top to bottom",
    "color_independence": "black and gray only; solid, long-dashed, and short-dotted lines plus explicit status words distinguish implication classes without color",
    "structured_equivalent": "this JSON preserves the charge table, every node, arrow status, equation, hypothesis, failure exit, and inference boundary",
    "responsive_strategy": "portrait layout uses direct labels at large source type; at narrow width the exact structured equivalent preserves equations and relations without relying on raster detail"
  },
  "scientific_checks": {
    "benchmark": {
      "Nc": 3,
      "Nf": 2,
      "k": 1
    },
    "b0_high": 7,
    "b0_low_after_decoupling": 8,
    "threshold_exponent_increment": 1,
    "threshold_dimension_lhs": 8,
    "threshold_dimension_rhs": 8,
    "det_M_dimension": 4,
    "ratio_dimension": 3,
    "candidate_dimension": 3,
    "ratio_axial_charge": 0,
    "candidate_axial_charge": 0,
    "det_M_r_charge": -2,
    "ratio_r_charge": 2,
    "candidate_r_charge": 2,
    "instanton_modes_after_lifting": 2,
    "high_scale_axial_plus_mass": 2,
    "low_scale_axial": 2,
    "propagated_coefficient": 1,
    "pure_sym_branch_count": 3,
    "neutral_invariant_z": {
      "dimension": 0,
      "axial_charge": 0,
      "r_charge": 0
    }
  },
  "arrow_source_map": [
    {
      "edge": "declared Wilsonian object to formal constraints",
      "status": "formal",
      "sources": [
        "Seiberg 1993, printed pp. 470-475"
      ]
    },
    {
      "edge": "formal charge table to ADS candidate",
      "status": "formal",
      "sources": [
        "Intriligator and Seiberg 1996, section 4.1, printed pp. 12-15",
        "Konishi and Shizuya 1985, printed pp. 111-134"
      ]
    },
    {
      "edge": "candidate to constraints-stop gate",
      "status": "inference boundary",
      "sources": [
        "Seiberg 1993, printed pp. 470-475"
      ]
    },
    {
      "edge": "gate to controlled instanton",
      "status": "independent dynamical input",
      "sources": [
        "Affleck, Dine, and Seiberg 1984, printed pp. 493-534"
      ]
    },
    {
      "edge": "controlled instanton to holomorphic decoupling",
      "status": "independent dynamical input",
      "sources": [
        "Intriligator, Leigh, and Seiberg 1994, sections 2-3.2, printed pp. 1094-1101"
      ]
    },
    {
      "edge": "holomorphic decoupling to exact result",
      "status": "independent dynamical input",
      "sources": [
        "Intriligator, Leigh, and Seiberg 1994, sections 2-3.2, printed pp. 1094-1101",
        "Intriligator and Seiberg 1996, section 4.1, printed pp. 12-15"
      ]
    },
    {
      "edge": "exact result to failure exits",
      "status": "failure and scope boundary",
      "sources": [
        "Seiberg 1993, printed pp. 470-475",
        "Cachazo, Douglas, Seiberg, and Witten 2002, section 3.1, equations (3.14)-(3.15), printed pp. 28-29"
      ]
    }
  ],
  "primary_sources": [
    {
      "citation": "Nathan Seiberg, Naturalness versus Supersymmetric Non-renormalization Theorems, Physics Letters B 318 (1993) 469-475",
      "doi": "https://doi.org/10.1016/0370-2693(93)91541-T",
      "locator": "printed pp. 470-475",
      "use": "background chiral couplings, holomorphy, anomalous spurion transformations, and the boundary between selection rules and dynamics"
    },
    {
      "citation": "Ian Affleck, Michael Dine, and Nathan Seiberg, Dynamical Supersymmetry Breaking in Supersymmetric QCD, Nuclear Physics B 241 (1984) 493-534",
      "doi": "https://doi.org/10.1016/0550-3213(84)90058-0",
      "locator": "printed pp. 493-534",
      "use": "controlled one-instanton generation and normalization of the ADS seed case"
    },
    {
      "citation": "Kenneth Intriligator, Robert G. Leigh, and Nathan Seiberg, Exact Superpotentials in Four Dimensions, Physical Review D 50 (1994) 1092-1104",
      "doi": "https://doi.org/10.1103/PhysRevD.50.1092",
      "arxiv": "https://arxiv.org/abs/hep-th/9403198",
      "locator": "sections 2-3.2, printed pp. 1094-1101",
      "use": "holomorphic decoupling, integrating fields in and out, coefficient recursion, and branch control"
    },
    {
      "citation": "Kenneth Intriligator and Nathan Seiberg, Lectures on Supersymmetric Gauge Theories and Electric-Magnetic Duality, Nuclear Physics B Proceedings Supplements 45BC (1996) 1-28",
      "doi": "https://doi.org/10.1016/0920-5632(95)00626-5",
      "arxiv": "https://arxiv.org/abs/hep-th/9509066",
      "locator": "section 4.1, printed pp. 12-15",
      "use": "SQCD charge table, ADS superpotential, scale matching, decoupling, and pure-SYM vacua"
    },
    {
      "citation": "Kenichi Konishi and Ken-ichi Shizuya, Functional-Integral Approach to Chiral Anomalies in Supersymmetric Gauge Theories, Il Nuovo Cimento A 90 (1985) 111-134",
      "doi": "https://doi.org/10.1007/BF02724227",
      "locator": "printed pp. 111-134",
      "use": "anomalous chiral Ward identities underlying spurionic charge constraints"
    },
    {
      "citation": "Freddy Cachazo, Michael R. Douglas, Nathan Seiberg, and Edward Witten, Chiral Rings and Anomalies in Supersymmetric Gauge Theory, JHEP 12 (2002) 071",
      "doi": "https://doi.org/10.1088/1126-6708/2002/12/071",
      "arxiv": "https://arxiv.org/abs/hep-th/0211170",
      "locator": "section 3.1, equations (3.14)-(3.15), printed pp. 28-29",
      "use": "generalized Konishi constraints and the distinction between Ward identities and theory-specific dynamical closure"
    }
  ]
}