{ "schema_version": 1, "artifact_id": "qft.artifact.supersymmetry-duality.breaking.order-parameters-goldstino-flow", "title": "From auxiliary order parameters to the Goldstino", "artifact_class": "original monochrome hypothesis-gated supersymmetry-breaking and Goldstino decision flow", "quantitative_status": "all displayed formulas, branch locations, vacuum energies, auxiliary directions, and mass spectra are exact at tree level under the declared canonical conventions; layout and arrow lengths are schematic and not to scale", "generated_by": "figures-src/supersymmetry-duality/order-parameters-goldstino-flow.mjs", "source_revision": 2, "generated_on": "2026-08-24", "registry_lifecycle": { "current_status": "read from the governed registry when provenance is generated", "accepted_drafting_statuses": [ "planned", "prototype" ], "recommended_materialized_status": "prototype", "note": "Materialization does not promote the registry, assign its public route, edit either owner page, or establish release acceptance." }, "owner_pages": [ { "role": "canonical_embed_owner", "id": "qft.topic.susy-holomorphy.f-d-breaking-goldstino", "file": "src/content/docs/supersymmetry-duality/susy-breaking-controlled-deformations/f-d-breaking-goldstino.md", "route": "/supersymmetry-duality/susy-breaking-controlled-deformations/f-d-breaking-goldstino/" }, { "role": "contextual_link_owner", "id": "qft.topic.susy-breaking.oraifeartaigh-fayet-iliopoulos-models", "file": "src/content/docs/supersymmetry-duality/susy-breaking-controlled-deformations/oraifeartaigh-fayet-iliopoulos-models.md", "route": "/supersymmetry-duality/susy-breaking-controlled-deformations/oraifeartaigh-fayet-iliopoulos-models/" } ], "canonical_anchor": "order-parameters-goldstino-flow", "canonical_contextual_target": "/supersymmetry-duality/susy-breaking-controlled-deformations/f-d-breaking-goldstino/#order-parameters-goldstino-flow", "public_assets": { "svg": "/figures/supersymmetry-duality/order-parameters-goldstino-flow.svg", "structured_json": "/figures/supersymmetry-duality/order-parameters-goldstino-flow.json" }, "reader_question": "When does nonzero F- or D-term evidence suffice for spontaneous supersymmetry breaking, and how is the massless Goldstino identified?", "takeaway": "Only after a candidate passes the exact-rigid-SUSY, stationary, stable, normalizable-vacuum and conserved-charge gate does the positive-energy order parameter imply breaking; stationarity and gauge invariance then produce the fermion-mass null vector and the supercurrent Goldstino pole.", "alt_text": "A monochrome portrait decision flow first gates a candidate on exact rigid four-dimensional N=1 supersymmetry, stationarity, physical stability, normalizability, and a conserved supercharge without unaccounted boundary or central terms. A no branch exits with status unknown. A yes branch defines D equal to g times the moment map plus xi and f-br squared equal to the vacuum value of F-star F plus D squared over two. Under the gate, nonzero F or D is necessary and sufficient for spontaneous breaking. Stationarity and gauge invariance make the mixed chiral-gaugino mass matrix annihilate the vector with entries F and i D over square root two, identifying the normalized Goldstino. Its inhomogeneous supersymmetry shift becomes a massless pole in the supercurrent matrix element. Four exact classical evidence cards then compare the O'Raifeartaigh branches y at most one and y greater than one, and the Fayet-Iliopoulos branches g xi below and above m squared, with their vacua, energies, auxiliary fields, spectra, gauge status, and boundary cautions.", "caption_semantics": "Hypothesis-gated order-parameter and Goldstino flow in a positive-metric, exactly supersymmetric rigid four-dimensional N=1 theory. At a stationary, physically stable and normalizable Lorentz-invariant vacuum with a well-defined conserved supercharge, f-br squared equals the vacuum value of F-star F plus D squared over two, using D equals g times phi-dagger T phi plus xi. Then f-br greater than zero is equivalent to spontaneous supersymmetry breaking. Stationarity plus gauge invariance makes the mixed chiral-gaugino mass matrix annihilate the vector (F, iD over square root two), and the same normalized direction has an inhomogeneous supersymmetry shift and a massless supercurrent pole. Failure to establish the gate leaves the breaking conclusion unknown. The O'Raifeartaigh and Fayet-Iliopoulos cards are exact tree-level model evidence in their declared domains; the layout and arrow lengths are schematic and not to scale.", "conventions": { "spacetime": "four-dimensional Minkowski spacetime with eta=diag(+1,-1,-1,-1)", "supersymmetry": "exact rigid N=1 supersymmetry unless a failure exit says otherwise", "kinetic_metrics": "canonical positive chiral Kahler metric and canonical positive gauge kinetic metric for the displayed formulas", "auxiliary_lagrangian": "L_aux=F_i^*F^i+W_iF^i+W_i^*F_i^*+D^aD^a/2-D^a P^a", "auxiliary_solutions": "F_i=-W_i^* and D^a=P^a=g_a phi^dagger T^a phi+xi^a", "scalar_potential": "V=F_i^*F^i+D^aD^a/2", "moment_map_sign": "the site convention is D=P=g mu+xi; no private D=-g(mu+xi) convention is used", "generator_normalization": "Hermitian generators T^a; the Abelian evidence model has charges q(phi_+)=+1 and q(phi_-)=-1", "fermion_mass_lagrangian": "L_mass=-(1/2) chi^T M_F chi+h.c. for chi=(psi^i,lambda^a)", "goldstino_phase": "G=(F_i^* psi^i-i D^a lambda^a/sqrt(2))/f_br up to one physically irrelevant overall phase", "current_normalization": "delta G=sqrt(2) f_br epsilon+... and <0|S^mu_alpha|G(p)>=i sqrt(2) f_br (sigma^mu bar u)_alpha" }, "decision_tree": { "hypothesis_gate": { "required": [ "exact rigid supersymmetry", "Lorentz-invariant candidate vacuum", "stationarity modulo gauge directions", "nonnegative physical Hessian, with flat directions identified", "normalizable vacuum state; a local or metastable configuration remains explicitly qualified and does not pass the full ground-state gate", "well-defined conserved supercharge with no unaccounted boundary, central, or gauge obstruction" ], "no_exit": "status unknown for the full theorem: a generic point, tachyon or saddle, runaway, merely local or metastable state, explicit breaking, boundary/BPS sector, or supergravity does not support the displayed rigid-ground-state inference" }, "order_parameter": { "formulas": [ "F_i=-W_i^*", "D^a=P^a=g_a phi^dagger T^a phi+xi^a", "f_br^2=V_vac=F_i^*F^i+(1/2)D^aD^a" ], "status": "necessary local diagnostic before the gate; necessary and sufficient only after the gate" }, "algebra_theorem": { "formula": "{Q_alpha,Qdot_beta^dagger}=2 sigma^mu_{alpha dot_beta} P_mu", "conclusion": "for a gated Lorentz-invariant vacuum, f_br>0 iff Q_alpha|Omega> is nonzero", "status": "theorem: necessary and sufficient under the declared hypotheses" }, "mass_null_vector": { "definitions": [ "A_i^a=(phi^dagger T^a)_i", "0=-W_ij F^j+g_a D^a A_i^a", "M_F=[[W_ij,i sqrt(2) g_a A_i^a],[i sqrt(2) g_b A_j^b,0]]", "M_F (F^j,iD^a/sqrt(2))^T=0" ], "gauge_identity": "A_i^a F^i=0", "goldstino": "G=[F_i^* psi^i-iD^a lambda^a/sqrt(2)]/f_br, up to an overall phase", "status": "theorem consequence of stationarity and gauge invariance" }, "current_pole": { "formulas": [ "delta_epsilon G=sqrt(2) f_br epsilon+...", "<0|S^mu_alpha(0)|G(p)>=i sqrt(2) f_br (sigma^mu bar u(p))_alpha" ], "status": "Goldstone theorem in exact rigid supersymmetry" } }, "logical_statuses": [ { "id": "vacuum_hypothesis_gate", "labels": [ "hypothesis" ], "visual_label": "HYPOTHESIS GATE", "statement": "Exact rigid supersymmetry, Lorentz invariance, stationarity, physical stability, normalizability, and a well-defined conserved charge are predicates, not consequences of a nonzero auxiliary field at an arbitrary point." }, { "id": "local_auxiliary_diagnostic", "labels": [ "necessary", "diagnostic" ], "visual_label": "NECESSARY LOCAL DIAGNOSTIC", "statement": "At a gated vacuum of the displayed positive-metric rigid EFT, unbroken supersymmetry requires every F_i and D^a to vanish." }, { "id": "positive_energy_equivalence", "labels": [ "theorem", "necessary", "sufficient" ], "visual_label": "THEOREM — SUFFICIENT AND NECESSARY UNDER THE GATE", "statement": "At a gated vacuum, f_br^2=V_vac>0 if and only if Q_alpha|Omega> is nonzero; equivalently, at least one F_i or D^a is nonzero." }, { "id": "goldstino_mass_null_vector", "labels": [ "theorem" ], "visual_label": "THEOREM CONSEQUENCE", "statement": "Stationarity and gauge invariance imply M_F (F,iD/sqrt(2))^T=0, so the normalized conjugate coefficient vector defines a massless Goldstino." }, { "id": "supercurrent_pole", "labels": [ "theorem" ], "visual_label": "GOLDSTONE / SUPERCURRENT POLE", "statement": "The inhomogeneous Goldstino shift fixes the massless one-particle pole residue of the conserved supercurrent in exact rigid supersymmetry." }, { "id": "failed_gate", "labels": [ "unknown" ], "visual_label": "FAILURE EXIT — STATUS UNKNOWN", "statement": "When a gate predicate is absent or false, nonzero F or D alone does not establish the spontaneous-breaking theorem." }, { "id": "worked_models", "labels": [ "model_evidence" ], "visual_label": "MODEL EVIDENCE", "statement": "The displayed O'Raifeartaigh and FI branches verify exact classical stationary points, global energies, stability boundaries, auxiliary directions, and mass sum rules; they do not replace the hypothesis gate." } ], "model_evidence": { "oraifeartaigh": { "id": "oraifeartaigh_three_chiral", "status": "model_evidence", "assumptions": "canonical Kahler potential and real positive f,h,m", "superpotential": "W=fX+(h/2)X phi_1^2+m phi_1 phi_2", "parameter": "y=hf/m^2", "reduced_global_potential": "after minimizing F_1 with phi_2=-(hX/m)phi_1 and choosing arg(phi_1^2)=pi, V(rho)=f^2+(m^2-hf)rho+(h^2/4)rho^2 for rho=|phi_1|^2>=0", "branches": [ { "id": "y_at_most_one", "domain": "01", "definitions": [ "Delta=hf-m^2>0", "r^2=2Delta/h^2" ], "stationary_set": "phi_1=+/- i r, phi_2=-(hX/m)phi_1, with X arbitrary", "vacuum_energy": "V_min=f^2-Delta^2/h^2=2fm^2/h-m^4/h^2", "auxiliaries": { "F_X": "-m^2/h", "F_1": "0", "F_2": "-m phi_1^*" }, "goldstino": "G=(F_X^* psi_X+F_2^* psi_2)/sqrt(V_min), up to phase", "spectrum_at_X_zero": { "real_scalar_mass_squared": "{0 x2, (2hf-m^2) x2, 2(hf-m^2) x1, 2hf x1}", "Weyl_fermion_mass_squared": "{0 x1, (2hf-m^2) x2}", "supertrace_mass_squared": 0 }, "stability": "all non-pseudomodulus scalar eigenvalues are positive for y>1", "global_status": "global classical pseudomoduli space; the reduced convex quadratic in rho has its minimum at rho=2Delta/h^2 and has no lower runaway" } ], "quantum_boundary": "the noncompact tree-level X direction must be lifted and its normalizable quantum vacuum established before applying the full vacuum theorem" }, "fayet_iliopoulos": { "id": "abelian_fayet_iliopoulos_pair", "status": "model_evidence", "assumptions": "rigid anomaly-free U(1), q(phi_+)=+1, q(phi_-)=-1, W=m phi_+ phi_-, and real positive g,xi,m", "potential": "V=m^2(x+z)+(1/2)[g(x-z)+xi]^2, x=|phi_+|^2, z=|phi_-|^2", "auxiliary": "D=g(x-z)+xi", "global_reduction": "for q=x-z and s=x+z>=|q|, V>=m^2|q|+[gq+xi]^2/2; q>=0 is minimized at q=0, while q=-z<=0 yields the two displayed branches", "branches": [ { "id": "unhiggsed", "domain": "gxim^2", "stationary_set": "phi_+=0 and |phi_-|^2=v^2=xi/g-m^2/g^2, modulo the U(1) phase", "vacuum_energy": "V_min=m^2xi/g-m^4/(2g^2)", "auxiliaries": { "D": "m^2/g", "F_plus": "-m phi_-^*", "F_minus": "0" }, "goldstino": "G=[F_plus^* psi_+-iD lambda/sqrt(2)]/sqrt(V_min), up to phase", "gauge_status": "U(1) Higgsed", "spectrum": { "vector_mass_squared": "m_A^2=2g^2v^2", "radial_scalar_mass_squared": "2g^2v^2", "phi_plus_complex_scalar_mass_squared": "2m^2", "massive_Weyl_fermion_mass_squared": "m^2+2g^2v^2 x2", "goldstino_mass_squared": 0, "supertrace_mass_squared": 0 }, "stability": "strictly stable modulo the eaten gauge orbit for gxi>m^2" } ], "boundary": "at gxi=m^2 the origin and Higgsed description meet; one charged complex scalar, equivalently two real components, is massless at quadratic order and stabilized by the positive D-term quartic", "consistency_boundary": "the charges +1 and -1 cancel the cubic and mixed gravitational U(1) anomalies and make W gauge invariant; a constant FI term has additional current-multiplet and supergravity consistency conditions outside this rigid model" } }, "failure_exits": [ "generic field point without stationarity", "tachyon, saddle, or unremoved gauge artifact in the Hessian", "runaway or absence of a normalizable vacuum state", "explicit supersymmetry breaking and therefore a nonconserved supercurrent", "boundary flux, central charge, or BPS sector outside the P_mu-only vacuum algebra", "local supersymmetry, where the Goldstino is eaten by the gravitino", "unresolved gauge anomaly or FI current-multiplet consistency condition" ], "structured_checks": [ { "id": "site_D_sign", "kind": "convention", "assertion": "D=g phi^dagger T phi+xi and V_D=D^2/2" }, { "id": "stationary_mass_null_vector", "kind": "exact_algebra", "assertion": "the O'Raifeartaigh y>1 and FI Higgsed fixtures make their displayed fermion mass matrices annihilate the auxiliary vector" }, { "id": "oraifeartaigh_global_minima", "kind": "convex_reduction", "assertion": "the rho>=0 reduced potential selects rho=0 for y<=1 and rho=2(hf-m^2)/h^2 for y>1" }, { "id": "oraifeartaigh_spectra_and_supertrace", "kind": "mass_spectrum", "assertion": "both X=0 branch spectra are non-tachyonic in domain and have STr M^2=0" }, { "id": "fi_global_minima", "kind": "constrained_minimization", "assertion": "the x,z>=0 potential selects the origin below gxi=m^2 and the Higgsed z=v^2 branch above it" }, { "id": "fi_spectra_and_supertrace", "kind": "mass_spectrum", "assertion": "both FI branch spectra are non-tachyonic in domain and have STr M^2=0" }, { "id": "goldstino_shift_normalization", "kind": "normalization", "assertion": "the coefficient vector normalized by f_br=sqrt(|F|^2+D^2/2) has delta G=sqrt(2)f_br epsilon" }, { "id": "logical_status_coverage", "kind": "semantic", "assertion": "theorem, necessary, sufficient, diagnostic, model_evidence, and unknown statuses are explicit" } ], "inference_boundaries": [ "Nonzero F or D at an arbitrary field point is only local data, not a spontaneous-breaking theorem.", "The equivalence between positive vacuum energy and broken supersymmetry uses the exact rigid algebra, a Lorentz-invariant vacuum, positive kinetic metrics, and well-defined charges.", "The displayed mass-null vector additionally uses stationarity and gauge invariance; it is not an independent assumption.", "The current matrix element asserts a massless pole only in exact rigid supersymmetry; explicit breaking or supergravity changes the conclusion.", "A local minimum or a classically flat noncompact set is not automatically a normalizable global quantum vacuum.", "The two model cards are exact classical evidence, not a claim that every O'Raifeartaigh or FI deformation has the same branch structure.", "The constant FI model is treated as a rigid anomaly-free field theory; coupling it to supergravity requires extra analysis." ], "accessibility_encoding": { "explicit_light_canvas": true, "color_independence": "Black outlines, white and gray fills, solid arrows, YES/NO branches, direct status words, equations, and branch inequalities carry every distinction without color.", "nested_text_equivalent": "The decision_tree object preserves the gate, no-exit, order parameter, algebra theorem, mass-null-vector consequence, and current pole in reading order.", "structured_equivalent": "This JSON preserves every formula, hypothesis, status, model branch, stationary set, vacuum energy, auxiliary direction, spectrum, failure exit, inference boundary, structured check, and source locator.", "narrow_width_strategy": "The portrait flow retains top-to-bottom order when scaled; this semantic JSON is the authoritative equivalent when dense equations require magnification." }, "scientific_references": [ { "authors": "Abdus Salam and J. A. Strathdee", "title": "On Goldstone Fermions", "journal": "Physics Letters B", "volume": "49", "pages": "465-467", "year": 1974, "doi": "10.1016/0370-2693(74)90637-6", "url": "https://doi.org/10.1016/0370-2693(74)90637-6", "locator": "pp. 465-467", "use": "Goldstone-fermion theorem and massless spin-one-half mode" }, { "authors": "Lochlainn O'Raifeartaigh", "title": "Spontaneous Symmetry Breaking for Chiral Scalar Superfields", "journal": "Nuclear Physics B", "volume": "96", "pages": "331-352", "year": 1975, "doi": "10.1016/0550-3213(75)90585-4", "url": "https://doi.org/10.1016/0550-3213(75)90585-4", "locator": "pp. 331-352", "use": "original chiral-model mechanism and tree-level mass formula" }, { "authors": "Pierre Fayet and John Iliopoulos", "title": "Spontaneously Broken Supergauge Symmetries and Goldstone Spinors", "journal": "Physics Letters B", "volume": "51", "pages": "461-464", "year": 1974, "doi": "10.1016/0370-2693(74)90310-4", "url": "https://doi.org/10.1016/0370-2693(74)90310-4", "locator": "pp. 461-464", "use": "original Abelian FI breaking model and Higgsed branch" }, { "authors": "Steven Weinberg", "title": "The Quantum Theory of Fields, Volume III: Supersymmetry", "publisher": "Cambridge University Press", "year": 2000, "doi": "10.1017/CBO9781139644198", "url": "https://doi.org/10.1017/CBO9781139644198", "local_path": "sources/Weinberg - 2000 - The Quantum Theory of Fields Volume 3 Supersymmetry.pdf", "locator": "section 27.2, printed pp. 122-127; section 27.4, printed pp. 134-135; section 27.5, printed pp. 144-148; sections 29.1-29.2, printed pp. 248-264", "use": "component gauge conventions, positive-energy criterion, and Goldstino direction" }, { "authors": "Zohar Komargodski and Nathan Seiberg", "title": "Comments on the Fayet-Iliopoulos Term in Field Theory and Supergravity", "year": 2009, "arxiv": "0904.1159", "url": "https://arxiv.org/abs/0904.1159", "locator": "sections 2-3", "use": "gauge-current and supergravity boundary of a constant FI term" } ] }