{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.actions.one-loop-chiral-supergraph-d-algebra-flow",
  "title": "One chiral loop: topology, D-algebra, and destination",
  "artifact_class": "original monochrome exact supergraph-reduction flow with a three-destination locality classifier",
  "quantitative": true,
  "generated_by": "figures-src/supersymmetry-duality/one-loop-chiral-supergraph-d-algebra-flow.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-24",
  "owner_page": {
    "id": "qft.topic.susy-actions.supergraphs-quantum-effective-actions",
    "file": "src/content/docs/supersymmetry-duality/actions-supercurrents-effective-theory/supergraphs-quantum-effective-actions.md",
    "route": "/supersymmetry-duality/actions-supercurrents-effective-theory/supergraphs-quantum-effective-actions/"
  },
  "canonical_anchor": "one-loop-chiral-supergraph-d-algebra-flow",
  "public_assets": {
    "svg": "/figures/supersymmetry-duality/one-loop-chiral-supergraph-d-algebra-flow.svg",
    "structured_json": "/figures/supersymmetry-duality/one-loop-chiral-supergraph-d-algebra-flow.json"
  },
  "reader_question": "How do the Lorentzian phases, cubic-vertex symmetry factor, chiral projectors, Grassmann-delta collapse, and DRED scalar bubble combine in one massless Wess-Zumino two-point supergraph, and which pieces belong to a local counterterm, a nonlocal 1PI form factor, or a Wilsonian shell action?",
  "takeaway": "One chiral and one antichiral cubic vertex with two mixed lines reduce through an exact factor-sixteen projector identity to a full-superspace Phi-dagger Phi D-term; its ultraviolet pole is local, its massless 1PI logarithm is nonlocal, and a Wilsonian shell yields a separate local derivative expansion below the shell.",
  "alt_text": "A vertical black-and-gray supergraph flow. A chiral vertex iy and an antichiral vertex iy-star are connected by two mixed internal lines i G-hat of momenta q and q plus p, with external Phi and Phi-dagger and symmetry factor one half. The two negative quarter vertex projectors multiply to bar-D-squared D-squared over sixteen; acting on the Grassmann delta produces sixteen and leaves one d-four-theta integral of Phi-dagger Phi. A DRED panel keeps the D-algebra four-dimensional while continuing the loop momentum to d equals four minus two epsilon. The result distinguishes i Gamma from Gamma and then branches to a momentum-independent local ultraviolet counterterm, a logarithmic nonlocal massless 1PI form factor, and a shell-local Wilsonian derivative expansion for external momenta below the infrared edge of the shell.",
  "caption_semantics": "Exact convention-controlled reduction of the massless one-loop Wess-Zumino chiral two-point supergraph with interaction y Phi cubed over three factorial. The chiral and antichiral vertex phases iy and iy-star, two Lorentzian mixed-line phases i, and graph symmetry factor one half combine to plus absolute-y-squared over two. The chiral projectors give bar-D-one-squared D-two-squared over sixteen, whose action on the second Grassmann delta supplies the compensating factor sixteen and leaves the full-superspace Phi-dagger Phi operator. In DRED, the spinor algebra remains four-dimensional while the loop momentum is d equals four minus two epsilon. The momentum-independent pole is a local Kähler counterterm; the massless logarithm is a nonlocal 1PI form factor; and integrating a finite momentum shell gives a distinct local derivative expansion when the external momentum is below that shell. The diagram is exact for the printed normalization and schematic in layout, not to scale.",
  "conventions": {
    "spacetime": "four-dimensional Lorentzian spacetime",
    "metric": "eta=diag(+1,-1,-1,-1)",
    "supersymmetry": "rigid N=1",
    "path_integral_weight": "exp(i S)",
    "interaction": "S_int = integral d^6z (y/3!) Phi^3 + integral d^6 bar z (y*/3!) Phi_dagger^3",
    "vertex_phases": {
      "chiral": "i y",
      "antichiral": "i y*"
    },
    "reduced_mixed_line": "G_hat(k;theta_1,theta_2)=delta^4(theta_1-theta_2)/(k^2+i0)",
    "full_lorentzian_mixed_line": "i G_hat",
    "derivative_algebra": "{D_alpha,bar D_dot_alpha}=-2 i sigma^mu_alpha_dot_alpha partial_mu",
    "derivative_order": "the chiral vertex contributes -bar D_1^2/4 and the antichiral vertex contributes -D_2^2/4; after integration by parts the product is bar D_1^2 D_2^2/16 on one internal delta",
    "grassmann_delta": "delta^4_12=(theta_1-theta_2)^2(bar theta_1-bar theta_2)^2 in the page's Berezin orientation",
    "regulator": "dimensional reduction: D_alpha, bar D_dot_alpha, theta, and bar theta remain four-dimensional while loop momentum is continued to d=4-2 epsilon",
    "subtraction_scheme": "modified dimensional reduction, DRbar",
    "external_kinematics": "Phi carries momentum p into the chiral vertex and Phi_dagger carries -p into the antichiral vertex; internal denominators carry q and q+p"
  },
  "fixture": {
    "model": "massless cubic Wess-Zumino model",
    "interaction_normalization": "y Phi^3/3! plus hermitian conjugate",
    "external_fields": [
      {
        "field": "Phi",
        "momentum": "p",
        "endpoint": "chiral vertex"
      },
      {
        "field": "Phi_dagger",
        "momentum": "-p",
        "endpoint": "antichiral vertex"
      }
    ],
    "vertices": [
      {
        "chirality": "chiral",
        "phase": "i y",
        "measure": "d^6 z_1"
      },
      {
        "chirality": "antichiral",
        "phase": "i y*",
        "measure": "d^6 bar z_2"
      }
    ],
    "internal_lines": [
      {
        "type": "mixed",
        "phase": "i",
        "momentum": "q",
        "denominator": "q^2+i0"
      },
      {
        "type": "mixed",
        "phase": "i",
        "momentum": "q+p",
        "denominator": "(q+p)^2+i0"
      }
    ],
    "symmetry_factor_derivation": {
      "chiral_external_choices": 3,
      "antichiral_external_choices": 3,
      "internal_pairings": 2,
      "chiral_vertex_factorial": 6,
      "antichiral_vertex_factorial": 6,
      "cross_term_expansion_factor": 1,
      "result": 0.5,
      "explanation": "The two orderings of the chiral-antichiral cross term cancel the 1/2! expansion factor; 3 times 3 external choices and 2! internal bijections divided by (3!)^2 leave 1/2."
    },
    "phase_ledger": [
      {
        "source": "chiral vertex",
        "factor": "i y",
        "scalar_i_power": 1
      },
      {
        "source": "antichiral vertex",
        "factor": "i y*",
        "scalar_i_power": 1
      },
      {
        "source": "q mixed line",
        "factor": "i G_hat(q)",
        "scalar_i_power": 1
      },
      {
        "source": "q+p mixed line",
        "factor": "i G_hat(q+p)",
        "scalar_i_power": 1
      }
    ],
    "phase_product": "(i y)(i y*)[i G_hat(q)][i G_hat(q+p)]=+|y|^2 G_hat(q)G_hat(q+p)",
    "before_d_algebra": "(+|y|^2/2) integral d^6z_1 d^6barz_2 Phi(1) Phi_dagger(2) [G_hat(1,2)]^2",
    "projector_reduction": {
      "chiral_projector": "-bar D_1^2/4",
      "antichiral_projector": "-D_2^2/4",
      "product": "+bar D_1^2 D_2^2/16",
      "kernel": "delta^4_12 [(bar D_1^2 D_2^2/16) delta^4_12]=delta^4_12",
      "factor_sixteen": "bar D_1^2 D_2^2 delta^4_12=16 under the two-delta superspace integral in the declared orientation",
      "surviving_measure": "integral d^4 theta",
      "surviving_operator": "Phi_dagger(-p,theta) Phi(p,theta)",
      "external_spinor_derivatives": "none"
    },
    "dred_bubble": {
      "loop_dimension": "d=4-2 epsilon",
      "spinor_dimension": 4,
      "integral": "I(p^2)=mu^(2epsilon) integral d^(4-2epsilon)q/(2pi)^(4-2epsilon) 1/[(q^2+i0)((q+p)^2+i0)]",
      "value": "I(p^2)=i/(16 pi^2)[1/bar_epsilon-log((-p^2-i0)/mu^2)+2+O(epsilon)]",
      "bar_epsilon": "1/bar_epsilon=1/epsilon-gamma_E+log(4pi)",
      "finite_constant_origin": "-integral_0^1 dx log[x(1-x)]=2"
    },
    "effective_action": {
      "i_gamma": "i Gamma_2^(1)=(|y|^2/2) integral d^4p/(2pi)^4 d^4theta Phi_dagger(-p)Phi(p) I(p^2)",
      "gamma": "Gamma_2^(1)=|y|^2/(32pi^2) integral d^4p/(2pi)^4 d^4theta Phi_dagger(-p)Phi(p)[1/bar_epsilon-log((-p^2-i0)/mu^2)+2]",
      "coefficient": "|y|^2/(32 pi^2)",
      "operator_class": "real full-superspace Kähler or wavefunction D-term"
    },
    "destinations": [
      {
        "id": "uv_local",
        "status": "local",
        "term": "1/bar_epsilon",
        "functional": "counterterm in the renormalized action",
        "operator": "delta Z integral d^8z Phi_dagger Phi",
        "reason": "momentum independent and polynomial"
      },
      {
        "id": "onepi_nonlocal",
        "status": "nonlocal",
        "term": "-log((-p^2-i0)/mu^2)",
        "functional": "massless one-particle-irreducible effective action",
        "operator": "Phi_dagger Phi with a logarithmic momentum form factor",
        "reason": "nonanalytic at p^2=0"
      },
      {
        "id": "wilsonian_shell_local",
        "status": "local derivative expansion under a scale hierarchy",
        "term": "finite shell integral",
        "functional": "Wilsonian action after Wick rotation and integration over Lambda_IR<|q_E|<Lambda_UV",
        "operator": "local superspace derivative series",
        "reason": "analytic expansion in |p_E|/Lambda_IR when |p_E|<<Lambda_IR"
      }
    ],
    "component_round_trip": "Expanding the surviving D-term multiplies scalar and fermion kinetic terms by the same wavefunction factor; unequal factors would signal a normalization or D-algebra error."
  },
  "inference_boundaries": [
    "The full four-dimensional D-algebra is combined with d-dimensional loop momentum specifically as dimensional reduction, not ordinary dimensional regularization.",
    "DRED makes the displayed superspace bookkeeping manifest but is not by itself an all-orders proof of supersymmetry or gauge invariance.",
    "The ultraviolet pole and massless 1PI logarithm are parts of one regulated 1PI calculation; the Wilsonian shell contribution is defined by a different integration domain and must not be obtained by simply relabelling the full 1PI logarithm.",
    "The logarithmic nonlocal D-term is not a local superpotential correction.",
    "The massless result assumes nonexceptional external momentum before the infrared limit; contact terms and zero-mode subtleties require separate distributional control."
  ],
  "accessibility_encoding": {
    "color_independence": "The figure uses only black, white, and gray; direct status labels, equations, solid reduction arrows, and dashed destination arrows encode every relationship without color.",
    "explicit_light_canvas": true,
    "structured_equivalent": "This JSON records both vertices, both internal lines, every Lorentzian i phase, the symmetry-factor derivation, projector order, factor sixteen, momentum routing, regulator split, i-Gamma and Gamma equations, and all three destination tests.",
    "narrow_width_strategy": "The visual uses one vertical spine and full-width destination boxes; the structured JSON preserves all details when the figure is magnified or read nonvisually."
  },
  "scientific_references": [
    {
      "citation": "Marcus T. Grisaru, Warren Siegel, and Martin Rocek, Improved Methods for Supergraphs, Nuclear Physics B 159 (1979) 429-450",
      "doi": "https://doi.org/10.1016/0550-3213(79)90344-4",
      "locator": "pp. 429-450",
      "use": "supergraph projector placement, D-algebra, and loop organization"
    },
    {
      "citation": "S. James Gates Jr., Marcus T. Grisaru, Martin Rocek, and Warren Siegel, Superspace, or One Thousand and One Lessons in Supersymmetry, Frontiers in Physics 58 (1983; corrected open edition 2001)",
      "arxiv": "https://arxiv.org/abs/hep-th/0108200",
      "locator": "chapters 6-7",
      "use": "chiral supergraph rules, Grassmann delta functions, and D-algebra"
    },
    {
      "citation": "Dominik Stockinger, Regularization by Dimensional Reduction: Consistency, Quantum Action Principle, and Supersymmetry, JHEP 03 (2005) 076",
      "doi": "https://doi.org/10.1088/1126-6708/2005/03/076",
      "arxiv": "https://arxiv.org/abs/hep-ph/0503129",
      "locator": "sections 2-4",
      "use": "consistent formulation and limitations of dimensional reduction"
    },
    {
      "citation": "Steven Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press (2000)",
      "doi": "https://doi.org/10.1017/CBO9781139644198",
      "locator": "chapter 30, printed pp. 307-316",
      "use": "supergraph normalization and one-loop effective-action organization"
    }
  ]
}