{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n1-gauge.quantum-moduli-superpotentials-confinement",
  "artifact_type": "semantic figure record",
  "title": "One mass-deformation staircase through exact SQCD",
  "evidence_cutoff": "2026-08-24",
  "reader_question": "How do exact SU(Nc) SQCD superpotentials and constraints transform under one-flavor mass deformations from Nf=Nc+1 to pure SYM?",
  "takeaway": "A fixed holomorphic normalization makes the s-confining superpotential, quantum-modified constraint, ADS branches, and Nc pure-SYM vacua one decoupling chain; dimensions, charges, anomalies, and scale exponents independently test every step.",
  "conventions": {
    "theory": "four-dimensional N=1 simply connected SU(Nc) SQCD",
    "composites": "M^i_j=Q^i Qtilde_j; baryons contain Nc quarks",
    "baryon_number": "B(Q)=+1, B(Qtilde)=-1, hence B(B)=Nc and B(Btilde)=-Nc",
    "anomaly_free_R": "R(Q)=R(Qtilde)=1-Nc/Nf",
    "holomorphic_scale": "Lambda_Nf denotes the scale of the theory with Nf full flavors in one fixed finite scheme",
    "ADS_branch_scope": "fractional powers are local holomorphic branches; the local branch count is Nc-Nf"
  },
  "operator_charge_formulas": [
    {
      "operator": "M",
      "engineering_dimension": "2",
      "R": "2*(1-Nc/Nf)",
      "baryon_number": "0"
    },
    {
      "operator": "B",
      "engineering_dimension": "Nc",
      "R": "Nc*(1-Nc/Nf)",
      "baryon_number": "+Nc"
    },
    {
      "operator": "Btilde",
      "engineering_dimension": "Nc",
      "R": "Nc*(1-Nc/Nf)",
      "baryon_number": "-Nc"
    }
  ],
  "regimes": [
    {
      "domain": "Nf=Nc+1",
      "variables": "M, B, Btilde",
      "exact_record": "W=(B M Btilde-det M)/Lambda^(2*Nc-1)",
      "checks": "both numerator terms have dimension 2*Nc+2, R=2, and baryon number zero; division leaves dimension three"
    },
    {
      "domain": "Nf=Nc",
      "variables": "M, B, Btilde",
      "exact_record": "det M-B Btilde=Lambda^(2*Nc)",
      "checks": "all terms have dimension 2*Nc, R=0, and baryon number zero"
    },
    {
      "domain": "0<Nf<Nc",
      "variables": "M on a generic full-rank meson patch",
      "exact_record": "W=(Nc-Nf)*(Lambda^(3*Nc-Nf)/det M)^(1/(Nc-Nf))",
      "checks": "dimension three, R=2, baryon number zero, and Nc-Nf local branches"
    },
    {
      "domain": "Nf=0",
      "variables": "glueball/gaugino chiral operator",
      "exact_record": "Nc chiral branches after index and holomorphic decoupling",
      "checks": "the branch count is protected; a nonzero condensate coefficient requires dynamical normalization"
    }
  ],
  "one_flavor_thresholds": [
    "Lambda_Nc^(2*Nc)=m*Lambda_(Nc+1)^(2*Nc-1)",
    "Lambda_(Nc-1)^(2*Nc+1)=m*Lambda_Nc^(2*Nc)",
    "Lambda_(Nf-1)^(3*Nc-Nf+1)=m_f*Lambda_Nf^(3*Nc-Nf)"
  ],
  "anomaly_fixture": {
    "regime": "Nf=Nc+1",
    "anomaly": "SU(Nf)_L^3",
    "ultraviolet": "Nc quark colors give Nc",
    "infrared": "M gives +Nf and B gives -1, so Nf-1=Nc",
    "note": "B is the SU(Nf)_L antifundamental; Btilde is neutral under SU(Nf)_L"
  },
  "SU2_enhanced_flavor_fixtures": [
    {
      "Nc": 2,
      "Nf": 2,
      "flavor": "SU(4)",
      "exact_record": "Pf V=Lambda^4"
    },
    {
      "Nc": 2,
      "Nf": 3,
      "flavor": "SU(6)",
      "exact_record": "W=-Pf V/Lambda^3",
      "sign_scope": "the displayed sign follows the chosen composite and scale phase convention"
    }
  ],
  "integer_rank_fixtures": [
    {
      "Nc": 2,
      "quantum_modified": {
        "Nf": 2,
        "constraint_scale_exponent": 4,
        "M": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": 0
        },
        "B": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": 2
        },
        "Btilde": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": -2
        }
      },
      "s_confining": {
        "Nf": 3,
        "scale_denominator_exponent": 3,
        "R_Q": "1/3",
        "R_M": "2/3",
        "R_B": "2/3",
        "numerator_engineering_dimension": 6,
        "superpotential_engineering_dimension": 3,
        "superpotential_R": 2,
        "SU_Nf_L_cubic_anomaly": {
          "ultraviolet": 2,
          "infrared": "3-1=2"
        }
      },
      "thresholds": {
        "from_Nf_Nc_plus_1_to_Nc": {
          "ultraviolet_exponent": 3,
          "infrared_exponent": 4
        },
        "from_Nf_Nc_to_Nc_minus_1": {
          "ultraviolet_exponent": 4,
          "infrared_exponent": 5
        }
      },
      "branch_counts": {
        "pure_SYM": 2,
        "ADS_by_Nf": [
          {
            "Nf": 1,
            "local_branches": 1
          }
        ]
      }
    },
    {
      "Nc": 3,
      "quantum_modified": {
        "Nf": 3,
        "constraint_scale_exponent": 6,
        "M": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": 0
        },
        "B": {
          "engineering_dimension": 3,
          "R": "0",
          "baryon_number": 3
        },
        "Btilde": {
          "engineering_dimension": 3,
          "R": "0",
          "baryon_number": -3
        }
      },
      "s_confining": {
        "Nf": 4,
        "scale_denominator_exponent": 5,
        "R_Q": "1/4",
        "R_M": "2/4",
        "R_B": "3/4",
        "numerator_engineering_dimension": 8,
        "superpotential_engineering_dimension": 3,
        "superpotential_R": 2,
        "SU_Nf_L_cubic_anomaly": {
          "ultraviolet": 3,
          "infrared": "4-1=3"
        }
      },
      "thresholds": {
        "from_Nf_Nc_plus_1_to_Nc": {
          "ultraviolet_exponent": 5,
          "infrared_exponent": 6
        },
        "from_Nf_Nc_to_Nc_minus_1": {
          "ultraviolet_exponent": 6,
          "infrared_exponent": 7
        }
      },
      "branch_counts": {
        "pure_SYM": 3,
        "ADS_by_Nf": [
          {
            "Nf": 1,
            "local_branches": 2
          },
          {
            "Nf": 2,
            "local_branches": 1
          }
        ]
      }
    },
    {
      "Nc": 4,
      "quantum_modified": {
        "Nf": 4,
        "constraint_scale_exponent": 8,
        "M": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": 0
        },
        "B": {
          "engineering_dimension": 4,
          "R": "0",
          "baryon_number": 4
        },
        "Btilde": {
          "engineering_dimension": 4,
          "R": "0",
          "baryon_number": -4
        }
      },
      "s_confining": {
        "Nf": 5,
        "scale_denominator_exponent": 7,
        "R_Q": "1/5",
        "R_M": "2/5",
        "R_B": "4/5",
        "numerator_engineering_dimension": 10,
        "superpotential_engineering_dimension": 3,
        "superpotential_R": 2,
        "SU_Nf_L_cubic_anomaly": {
          "ultraviolet": 4,
          "infrared": "5-1=4"
        }
      },
      "thresholds": {
        "from_Nf_Nc_plus_1_to_Nc": {
          "ultraviolet_exponent": 7,
          "infrared_exponent": 8
        },
        "from_Nf_Nc_to_Nc_minus_1": {
          "ultraviolet_exponent": 8,
          "infrared_exponent": 9
        }
      },
      "branch_counts": {
        "pure_SYM": 4,
        "ADS_by_Nf": [
          {
            "Nf": 1,
            "local_branches": 3
          },
          {
            "Nf": 2,
            "local_branches": 2
          },
          {
            "Nf": 3,
            "local_branches": 1
          }
        ]
      }
    },
    {
      "Nc": 5,
      "quantum_modified": {
        "Nf": 5,
        "constraint_scale_exponent": 10,
        "M": {
          "engineering_dimension": 2,
          "R": "0",
          "baryon_number": 0
        },
        "B": {
          "engineering_dimension": 5,
          "R": "0",
          "baryon_number": 5
        },
        "Btilde": {
          "engineering_dimension": 5,
          "R": "0",
          "baryon_number": -5
        }
      },
      "s_confining": {
        "Nf": 6,
        "scale_denominator_exponent": 9,
        "R_Q": "1/6",
        "R_M": "2/6",
        "R_B": "5/6",
        "numerator_engineering_dimension": 12,
        "superpotential_engineering_dimension": 3,
        "superpotential_R": 2,
        "SU_Nf_L_cubic_anomaly": {
          "ultraviolet": 5,
          "infrared": "6-1=5"
        }
      },
      "thresholds": {
        "from_Nf_Nc_plus_1_to_Nc": {
          "ultraviolet_exponent": 9,
          "infrared_exponent": 10
        },
        "from_Nf_Nc_to_Nc_minus_1": {
          "ultraviolet_exponent": 10,
          "infrared_exponent": 11
        }
      },
      "branch_counts": {
        "pure_SYM": 5,
        "ADS_by_Nf": [
          {
            "Nf": 1,
            "local_branches": 4
          },
          {
            "Nf": 2,
            "local_branches": 3
          },
          {
            "Nf": 3,
            "local_branches": 2
          },
          {
            "Nf": 4,
            "local_branches": 1
          }
        ]
      }
    }
  ],
  "limitations": [
    "The figure records exact Wilsonian chiral data, not the Kähler metric, mass spectrum, string tension, or existence of every proposed wall.",
    "The ADS expression is local away from det M=0 and its fractional power must not be treated as one global single-valued function.",
    "SU(2) uses antisymmetric mesons V and Pfaffians because pseudoreality enhances the flavor symmetry."
  ],
  "primary_sources": [
    {
      "citation": "Seiberg 1994, Exact results on the space of vacua of four-dimensional SUSY gauge theories",
      "url": "https://arxiv.org/abs/hep-th/9402044"
    },
    {
      "citation": "Intriligator and Seiberg 1996, Lectures on supersymmetric gauge theories and electric-magnetic duality",
      "url": "https://arxiv.org/abs/hep-th/9509066"
    },
    {
      "citation": "Veneziano and Yankielowicz 1982, An effective Lagrangian for the pure N=1 supersymmetric Yang-Mills theory",
      "url": "https://doi.org/10.1016/0370-2693(82)90623-6"
    }
  ],
  "governance_snapshot": {
    "registry_status": "prototype",
    "registry_public_route": null,
    "integration_state": "complete"
  }
}