{ "artifact_id": "qft.artifact.supersymmetry-duality.n1-gauge.sequential-confinement-theory-card-flow", "title": "Sequential confinement: update the neighboring theory card", "artifact_class": "schematic scientific theory-card flow", "reader_question": "When one node of a two-node N=1 bifundamental quiver becomes strongly coupled first, which exact variables, constraints, neighboring-node data, branch information, and failure boundaries must be carried into the next effective description?", "takeaway": "For |Lambda_1| much greater than |Lambda_2|, node 1 has the exact N_f=N quantum-modified description, but node 2 remains dynamical: its composite representations, one-loop coefficient, gauge-R anomaly, inherited one-form symmetry, branch stabilizers, and mass exits must all be recomputed before any second strong-dynamics claim.", "alt_text": "A black-and-gray three-stage theory-card flow starts from simply connected SU(N)_1 times SU(N)_2 with bifundamentals X and X-tilde, b_1=b_2=2N, and diagonal Z_N one-form symmetry. Under the controlled hierarchy |Lambda_1| much greater than |Lambda_2|, node 1 is replaced by M, B_1, and B-tilde_1 obeying det M minus B_1 B-tilde_1 equals Lambda_1 to the 2N. The downstream node-2 card contains Phi=M/Lambda_1 in adjoint plus singlet and singlet baryons, with b_2,IR=2N, SU(N)_2 squared R anomaly zero, and the inherited one-form symmetry represented as the remaining center. Labeled exits distinguish mesonic and baryonic strata, a large common-mass threshold to two pure-SYM effective nodes, and a crossed-out comparable-scale inference that would incorrectly impose two isolated-node constraints.", "caption_semantics": "Sequential confinement for four-dimensional rigid N=1 simply connected SU(N)_1 times SU(N)_2 with N at least three, X in (N,anti-N), X-tilde in (anti-N,N), and zero tree superpotential. In the hierarchy |Lambda_1| much greater than |Lambda_2|, node 1 has N flavors and is replaced by M=X-tilde X, B_1=det X, and B-tilde_1=det X-tilde subject to det M-B_1 B-tilde_1=Lambda_1^(2N). Phi=M/Lambda_1 transforms under node 2 as adjoint plus singlet, the baryons are singlets, b_2 remains 2N, and the mixed SU(N)_2 squared R anomaly is N-N=0 for scalar R(Phi)=0. The ultraviolet diagonal Z_N one-form symmetry becomes the center one-form symmetry of the remaining node. Mesonic and baryonic strata have different stabilizers. A common mass gives Lambda_(a,pure)^(3N)=m^N Lambda_a^(2N) for both nodes; separate pure-node centers are emergent only in the decoupled effective theory. When the two strong scales are comparable, two isolated-node constraints are not a valid simultaneous description. Layout is schematic and not to scale; equations are exact only under their displayed hypotheses.", "exactness_status": "displayed beta coefficients, anomaly, center kernel, quantum constraint, representation decomposition, and mass thresholds are exact in their declared theory and hierarchy; layout is schematic", "conventions": { "spacetime": "four-dimensional Lorentzian spacetime with eta=diag(+1,-1,-1,-1)", "supersymmetry": "rigid N=1", "rank_domain": "integer N>=3", "gauge_group": "simply connected SU(N)_1 x SU(N)_2; quotienting by the diagonal center defines a different genuine-line lattice", "matter": "X:(N,anti-N), X_tilde:(anti-N,N), W_tree=0; B(X)=+1 and B(X_tilde)=-1", "dynkin_indices": "T(fundamental)=T(anti-fundamental)=1/2 and T(adjoint)=N", "beta_convention": "b=3 T(adjoint)-sum_i T(r_i), with the other gauge index counted as flavor multiplicity", "holomorphic_scales": "Lambda_a^(b_a)=mu^(b_a) exp(2 pi i tau_a(mu)), independently for a=1,2", "r_symmetry": "scalar R(X)=R(X_tilde)=0; chiral fermions have R=-1 and each gaugino has R=+1", "baryon_global_form": "the faithful ordinary baryon symmetry is U(1)_B/Z_N in the covering-space normalization B(X)=1", "hierarchy": "|Lambda_1|>>|Lambda_2|, so node 2 is weakly gauged at mu of order |Lambda_1|", "composite_normalization": "M=X_tilde X has engineering dimension two and Phi=M/Lambda_1 has engineering dimension one; this rescaling does not determine a canonical Kahler metric", "d_flatness": "a physical adjoint representative satisfies [M,M_dagger]=0; complex diagonalizability alone does not establish the unitary-gauge stabilizer", "mass_deformation": "W_mass=m Tr(X_tilde X), with |m|>>|Lambda_1|,|Lambda_2| for the clean two-node pure-SYM threshold exit" }, "ultraviolet_fields": [ { "id": "X", "representation_node_1": "fundamental", "representation_node_2": "anti-fundamental", "multiplicity_at_each_node": "N from the other gauge index", "scalar_R": 0, "fermion_R": -1, "baryon_charge": 1 }, { "id": "X_tilde", "representation_node_1": "anti-fundamental", "representation_node_2": "fundamental", "multiplicity_at_each_node": "N from the other gauge index", "scalar_R": 0, "fermion_R": -1, "baryon_charge": -1 } ], "composite_fields": [ { "id": "Phi", "definition": "Phi=M/Lambda_1 with M=X_tilde X", "representation_node_2": "adjoint plus singlet", "scalar_R": 0, "fermion_R": -1, "baryon_charge": 0, "engineering_dimension": 1, "normalization_boundary": "engineering normalization only; Kahler metric unknown" }, { "id": "B_1", "definition": "det X", "representation_node_2": "singlet", "scalar_R": 0, "baryon_charge": "N", "engineering_dimension": "N" }, { "id": "B_tilde_1", "definition": "det X_tilde", "representation_node_2": "singlet", "scalar_R": 0, "baryon_charge": "-N", "engineering_dimension": "N" } ], "stage_table": [ { "stage": "ultraviolet", "active_variables_under_node_2": "X and X_tilde, giving N flavor pairs", "exact_relation_or_check": "b_1=b_2=2N", "one_form_symmetry": "diagonal Z_N^(1)", "domain_of_control": "microscopic simply connected product-group theory" }, { "stage": "at mu of order Lambda_1", "active_variables_under_node_2": "M=X_tilde X, B_1=det X, B_tilde_1=det X_tilde", "exact_relation_or_check": "det M-B_1 B_tilde_1=Lambda_1^(2N), with every term of dimension 2N", "one_form_symmetry": "the same diagonal Z_N^(1)", "domain_of_control": "|Lambda_1|>>|Lambda_2|" }, { "stage": "below Lambda_1", "active_variables_under_node_2": "Phi=M/Lambda_1 in adjoint plus singlet; B_1 and B_tilde_1 are singlets", "exact_relation_or_check": "b_2,IR=2N and A[SU(N)_2^2 U(1)_R]=0", "one_form_symmetry": "center of the remaining node 2", "domain_of_control": "composite Kahler metric is not fixed; branch stabilizer requires a D-flat representative" }, { "stage": "large common-mass exit", "active_variables_under_node_2": "two pure-SYM effective nodes below |m|", "exact_relation_or_check": "Lambda_(1,pure)^(3N)=m^N Lambda_1^(2N) and Lambda_(2,pure)^(3N)=m^N Lambda_2^(2N)", "one_form_symmetry": "exact microscopic diagonal subgroup; separate centers emerge in the decoupled effective theory", "domain_of_control": "|m|>>|Lambda_1|,|Lambda_2|" } ], "branches": [ { "id": "mesonic_generic", "condition": "B_1=B_tilde_1=0, det M=Lambda_1^(2N), [M,M_dagger]=0, and distinct eigenvalues", "node_2_stabilizer": "maximal torus U(1)^(N-1)" }, { "id": "mesonic_symmetric", "condition": "B_1=B_tilde_1=0 and M proportional to the identity with determinant Lambda_1^(2N)", "node_2_stabilizer": "SU(N)_2" }, { "id": "baryonic_representative", "condition": "M=0, B_1=Lambda_1^N, B_tilde_1=-Lambda_1^N", "constraint_check": "det M-B_1 B_tilde_1=Lambda_1^(2N)", "node_2_stabilizer": "SU(N)_2 for this M=0 representative" } ], "flow_relations": [ { "from": "ultraviolet", "to": "node_1_quantum_modified", "label": "|Lambda_1|>>|Lambda_2|", "status": "controlled hierarchy" }, { "from": "node_1_quantum_modified", "to": "downstream_node_2_card", "label": "retain weakly gauged node-2 indices and recompute", "status": "required theory-card update" }, { "from": "downstream_node_2_card", "to": "mesonic_and_baryonic_strata", "label": "solve the quantum constraint and node-2 D terms", "status": "branch-dependent" }, { "from": "microscopic_theory", "to": "two_pure_SYM_effective_nodes", "label": "large common mass", "status": "separate relevant-deformation exit" }, { "from": "comparable_scales", "to": "two_independent_single_node_constraints", "label": "rejected", "status": "failure boundary" } ], "failure_exit": { "condition": "|Lambda_1| is of the same order as |Lambda_2|", "rejected_inference": "impose the two hierarchy-limit N_f=N constraints as independent relations on independent flavor fields", "reason": "the two nodes share X and X_tilde, so the limiting composite coordinates overlap", "retained_data": [ "the complex ratio Lambda_1^(2N)/Lambda_2^(2N)", "common gauge-invariant coordinates and branch identifications", "singular loci where additional states become light" ], "status": "failure boundary, not a claim that the complete comparable-scale theory is inconsistent" }, "inference_boundaries": [ "The single-node quantum-modified constraint is used only while the neighboring gauge coupling is parametrically weak at the first strong scale.", "Phi=M/Lambda_1 is an engineering normalization; no canonical composite kinetic term or physical pole mass follows without the Kahler metric.", "A complexified-gauge diagonal form does not by itself prove a physical Higgs pattern; the displayed mesonic stabilizers require [M,M_dagger]=0.", "Only the diagonal electric one-form symmetry is exact in the finite-mass microscopic bifundamental theory; separate pure-node centers belong to the decoupled effective description.", "The SU(2)^r ring result is source context for combined-scale singularities, not an exact solution of this N>=3 two-node model." ], "accessibility_encoding": { "explicit_light_canvas": true, "color_independence": "Black outlines, white and gray fills, solid controlled arrows, dashed hypothesis and failure arrows, a dotted deformation arrow, a multiplication sign, and direct status words carry every distinction without color.", "structured_equivalent": "This JSON preserves every stage-table field, representation, coefficient, anomaly, one-form statement, branch condition, threshold, failure mode, exactness qualification, and source locator shown or summarized by the figure.", "narrow_width_strategy": "At narrow width the three bold stage headings and arrow grammar preserve the overview; the adjacent page table and this structured record are the authoritative equation-level equivalent and support magnification without raster loss." }, "scientific_references": [ { "authors": "Nathan Seiberg", "title": "Exact Results on the Space of Vacua of Four Dimensional SUSY Gauge Theories", "journal": "Physical Review D", "volume": "49", "pages": "6857-6863", "year": 1994, "doi": "10.1103/PhysRevD.49.6857", "arxiv": "hep-th/9402044", "url": "https://arxiv.org/abs/hep-th/9402044", "locator": "sections 3-4", "use": "N_f=N_c quantum-modified moduli constraint and holomorphic decoupling" }, { "authors": "Spencer Chang and Howard Georgi", "title": "Quantum Modified Mooses", "journal": "Nuclear Physics B", "volume": "672", "pages": "101-122", "year": 2003, "doi": "10.1016/j.nuclphysb.2003.09.006", "arxiv": "hep-th/0209038", "url": "https://arxiv.org/abs/hep-th/0209038", "locator": "sections 3-8", "use": "local quantum-modified relations, scale dependence, splitting relations, and controlled moose limits" }, { "authors": "Erich Poppitz, Yael Shadmi, and Sandip P. Trivedi", "title": "Duality and Exact Results in Product Group Theories", "journal": "Nuclear Physics B", "volume": "480", "pages": "125-169", "year": 1996, "doi": "10.1016/S0550-3213(96)00464-6", "arxiv": "hep-th/9605113", "url": "https://arxiv.org/abs/hep-th/9605113", "locator": "sections 2-4", "use": "successive simple-group descriptions and renormalization-flow qualifications in product groups" }, { "authors": "Girma Hailu", "title": "N=1 Supersymmetric SU(2)^r Moose Theories", "journal": "Physical Review D", "volume": "67", "article": "085023", "year": 2003, "doi": "10.1103/PhysRevD.67.085023", "arxiv": "hep-th/0209266", "url": "https://arxiv.org/abs/hep-th/0209266", "locator": "sections 6-7", "use": "analogue showing that ring-moose singular loci depend on all node scales; not an exact solution of the displayed N>=3 model" } ] }