Fields and interactions · research workbench III

Local operators & field statistics

From field content to independent interactions. Keep Lorentz indices, gauge charges, flavor, and statistics in the same calculation.

Scope: 4D Lorentzian scalar and Weyl fields. Fixed monomial, no derivatives, canonical dimension ≤ 6. Read the precise counting contract.

Start from a worked example

L²H² has six complex invariants at canonical dimension five: a symmetric 3 × 3 lepton-flavor tensor, with one Higgs doublet.

4D LorentzianComplex local polynomialsNo derivativesView current result

Setup nameGlobal gauge group

The quotient and charge convention

G_d=\frac{SU(3)\times SU(2)\times U(1)}{\Gamma_d},\qquad \Gamma_d=\langle\gamma^{6/d}\rangle

\gamma=(e^{2\pi i/3}I_3,-I_2,e^{2\pi i/6}),\qquad d\in\{1,2,3,6\}

The circle has period 2π. Every declared field must descend to this group, including fields with zero occurrences in the chosen monomial. A quotient changes the admissible fields; for fields that descend, it does not change their local invariant tensors.

01 Field species L

Display nameA label; the stable ID stays with saved data.Lorentz representationColor SU(3)Weak SU(2)Integer charge QQ = 6Y in the Standard Model examples.Flavors1–3; ungauged component multiplicity.Occurrences in the monomial0–6; zero keeps this field in the setup.

\Phi_{1}

labels this species in the result.

02 Field species H

\Phi_{2}

labels this species in the result.

Exact arithmetic · inputs stay in your browser

Exact local invariant countFields descend

6independent complex invariants

Canonical dimension $5$ 2 active species · 4 field insertions

\dim_{\mathbb C}\operatorname{Inv}\!\left[\Lambda^{2}R_{1}\otimes \operatorname{Sym}^{2}R_{2}\right]=6

Selected monomial

\Phi_{1}^{2}\,\Phi_{2}^{2}

Net charge

0

This is a dimension over ℂ for one fixed field monomial. Flavor components are counted individually. Algebraic index identities are included; a real Lagrangian, its Hermitian conjugates, and equivalences under flavor-basis changes require further interpretation.

Worked contraction · Weinberg operator · three flavors

\mathcal O_{ab}=\epsilon_{\alpha\beta}\epsilon_{ij}\epsilon_{kl}L_a^{\alpha i}L_b^{\beta k}H^jH^l,\qquad \mathcal O_{ab}=\mathcal O_{ba}

Here α, β are left-handed spinor indices, i, j, k, l are weak indices, and a, b are lepton flavors. The Higgs has one flavor. Anticommutation and the spinor contraction make the flavor tensor symmetric, leaving n(n+1)/2 independent complex coefficients. The conjugate monomial is a separate calculation.

Derivation and sources

What the statistics change

Distinguishable tensor slots18

Bose/Fermi projection6

The first number treats repeated field occurrences as independent tensor slots. The second imposes symmetric powers for scalars and exterior powers for Weyl fields on their complete spin × gauge × flavor space. Only the second is the local polynomial count.

Selection rules and zeroes

U(1) charge neutrality
A fixed multidegree is neutral exactly when the sum of its integer U(1) charges is zero.
Condition satisfied · both tensor spaces
SU(3) center
The total triality Σ n_i(p_i+2q_i) must vanish modulo 3.
Condition satisfied · both tensor spaces
Weak SU(2) center
The sum of weak Dynkin labels, including field occupations, must be even.
Condition satisfied · both tensor spaces
Left Lorentz spin
Without derivatives, an even number of left Weyl factors is necessary for a Lorentz scalar.
Condition satisfied · both tensor spaces
Right Lorentz spin
Without derivatives, an even number of right Weyl factors is necessary for a Lorentz scalar.
Condition satisfied · both tensor spaces
L: Grassmann capacity
An exterior power above the full Lorentz × gauge × flavor component dimension vanishes. This condition does not constrain labeled tensor slots.
Condition satisfied · statistics-projected space

Passing these necessary conditions does not guarantee an invariant. The exact character projection determines the final count.

Field inventory and component dimensions

The exact field inventory used by this calculation
Species	Lorentz (jℓ, jᵣ)	SU(3) labels	Weak label	Q	Flavors	Power
$\Phi_{1}$ · L	$(\tfrac12,0)$	$(0,0)$	1	-3	3	2
$\Phi_{2}$ · H	$(0,0)$	$(0,0)$	1	3	1	2

Component dimensions before projecting onto invariants
Species	Lorentz	Gauge	Flavor	Full field	Graded power
L · power 2	$2$	$2$	$3$	$12$	$66$
H · power 2	$1$	$2$	$1$	$2$	$3$

Full graded component space: $198$ . Before imposing statistics: $576$ . These dimensions are generally much larger than the invariant subspace.

Global descent for every field

Every declared field is checked against the selected quotient
Field species	Generator phase (mod 1)	Global descent
L	$0$	Descends
H	$0$	Descends

For Γ₁ there is no quotient constraint. For the other choices, each displayed phase must be zero modulo one. This checks representation descent, without asserting anomaly cancellation.

✓ dimension identities✓ nonnegative count✓ graded not larger than distinguishable✓ weyl invariance

Inspect the global gauge group

Send every declared multiplet, its integer charge, and this exact quotient generator to the product-group workbench.

Check global structure

Saved setups preserve field species and assumptions. Saved calculations also preserve exact results, sources, and versions. Loading either file validates the inputs and recomputes them locally. A conjugate copy is a separate polynomial generator; Hermiticity is not imposed automatically.

Method & interpretation

Impose statistics on the whole field.

R_i=L_i\otimes C_i\otimes W_i\otimes\mathbb C^{n_i}

For each species, form the symmetric or exterior power of its full Lorentz, color, weak, and flavor representation. Multiply the exact graded characters, enforce vanishing U(1) charge, and extract the singlet with a Weyl-character projection. Integer multiplicities are computed exactly.

The finite-dimensional Lorentz spin algebra is complexified to two copies of $\mathfrak{sl}_2$ . Compact SU(2) character projection computes its algebraic invariants. The setup retains four-dimensional Lorentzian spacetime; these are finite-dimensional field representations, not Wigner particle representations.

Included

A fixed field monomial, scalar and Weyl fields, explicit gauge representations, ungauged flavors, Bose/Fermi symmetry, algebraic index identities, and global descent.

Beyond this release

Derivatives, gauge-field strengths, equations of motion, integration by parts, Hermiticity, flavor-basis quotients, anomaly checks, and a generated basis of explicit contractions.

Exact counting method · Fields and statistics · Setup schema and replay

Bounds, primary sources, and versions

At most 16 species, 3 flavors per species, 6 field insertions, and canonical dimension 6. Color labels p+q≤2, weak label 0–2, integer charges −24…24. Exact completion or an explicit resource-limit error. A calculation either completes exactly or reports that its resource guard was reached. No truncated answer is presented as a count. Engine limits are recorded in the method manifest.

Efficient Hilbert Series for Effective Theories§II A, Eqs. (2)–(4): symmetric/exterior powers and Haar projection without derivatives.
Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox§4.1, Eq. (4.2): 57 QQQL operators for three generations; Appendix A: characters, plethystic exponentials and Haar measure.
Enumerating the operators of an effective field theory§3, Eq. (49) and Table 2: QQQL flavor count n_L n_Q (2 n_Q² + 1)/3; Eq. (51): n²(n²−1)/12 four-sterile-Weyl invariants.
Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry§2 and §3.2: two-component Lorentz spinors, spinor contractions and symmetric fermion mass matrices.
Higher dimension operators in the SM EFT§2: operator Hilbert series and the additional treatment required for derivatives, equations of motion and integration by parts.
Steven Weinberg, Baryon- and Lepton-Nonconserving ProcessesPhys. Rev. Lett. 43, 1566–1570 (1979), Eq. (20): the dimension-five lepton-number-violating interaction

Algorithm: local-operators-1.0.0
Convention: four-d-local-polynomial-v1
Application: 6.0.0

Source checking and automated mathematical verification are recorded separately from independent human expert review, which remains pending.

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