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A finite-order SU(2) scalar operator contract

Physical scope

Formal finite-order scalar operator counting for an SU(2) gauge representation and its complex-algebraic invariant model; no complete basis, matching coefficient, truncation error, continuum construction, or empirical claim.

Example

Assume countable choice ${A C}_{ω}$ , meaning that every countable family of nonempty sets has a choice function, for the supplied gauge datum. Use $M = R^{1, 3}$ with $η = diag (- 1, 1, 1, 1)$ and $c = ℏ = 1$ . Let $S U (2) = {u \in G L_{2} (C) : u^{†} u = I, \det u = 1}$ . Use the trivial right principal bundle $P = M \times S U (2)$ with projection $π (x, g) = x$ and right action $(x, g) h = (x, g h)$ . Admit exactly $Σ = {[P]}$ , where $[P]$ is the minimum-rank set code for bundles isomorphic over $M$ to $P$ in the non-Abelian gauge-data definition; thus $[P] \in Σ$ . Its product connection has horizontal spaces $H_{(x, g)} = T_{x} M \times {0_{g}} \subset T_{x} M \times T_{g} S U (2)$ , complementary to the vertical spaces ${0_{x}} \times T_{g} S U (2)$ and invariant under the right action. The associated $s u (2)$ -valued connection form $ω$ has kernel $H$ ; for the global section $s (x) = (x, e)$ its potential is $A = s^{*} ω = 0$ , hence $A_{μ} = 0$ . Identify the associated standard and dual bundles with their product bundles using this chosen global section; the maps $H$ and $\tilde{H}$ are their components in this fixed trivialization. Gauge transformations are smooth maps $u : M \to S U (2)$ acting by $H (x) \mapsto u (x) H (x)$ and $\tilde{H} (x) \mapsto \tilde{H} (x) u (x)^{- 1}$ . Take one even Lorentz-scalar complex doublet $H \in C^{\infty} (M, C^{2})$ in the standard unitary representation. Before imposing the real-action condition, treat an independent dual field $\tilde{H} \in C^{\infty} (M, (C^{2})^{*})$ as a smooth field variable. In the complex-algebraic polynomial model, use $G_{a l g} = S O (η, C) \times S L (2, C)$ , with $S O (η, C) = {L \in G L_{4} (C) : L^{T} η L = η, \det L = 1}$ and $S L (2, C) = {g \in G L_{2} (C) : \det g = 1}$ . The orthogonal factor acts trivially on these Lorentz-scalar fields, and the special-linear factor acts by the standard and dual representations. Here $S O^{+} (1, 3) = {L \in G L_{4} (R) : L^{T} η L = η, \det L = 1, L^{0}_{0} \geq 1}$ , and the physical Lorentz and gauge groups map to $G_{a l g}$ by $(L, u) \mapsto (L, u)$ , viewing $L$ as a complex matrix and $u \in S U (2)$ as a matrix in $S L (2, C)$ ; the standard and dual actions restrict to the specified physical actions. Impose $\tilde{H} = H^{†}$ for the real action. Take no flavor symmetry, field weight one, derivative cutoff $r = 0$ , maximum field degree $q = 6$ , retained engineering dimension $D_{\max} = 6$ , loop order $L_{\max} = 0$ , and a positive scalar coefficient scale $Λ_{o p}$ of mass dimension one as assigned by the physical EFT operator contract. For this illustration the declared relation subspace is zero. The quotient uses $G_{a l g}$ as a formal algebraic selection only; no equality with the full invariant subspace for the specified physical symmetry group is asserted. Then $O_{2} = \tilde{H} H$ , $O_{4} = (\tilde{H} H)^{2}$ , and $O_{6} = (\tilde{H} H)^{3}$ are gauge-invariant scalar polynomials of engineering dimensions $2$ , $4$ , and $6$ . With real dimensionless $C_{6}$ , the term $C_{6} O_{6} / Λ_{o p}^{2}$ has Lagrangian density dimension $4$ and belongs to this dimension-six truncation. For $a, b \in {1, 2}$ , define $ε_{12} = 1$ , $ε_{21} = - 1$ , and $ε_{11} = ε_{22} = 0$ ; then the antisymmetric polynomial $ε_{a b} H^{a} H^{b}$ vanishes because the even field components commute. This is a finite scalar-sector illustration, not a complete $S U (2)$ EFT basis or a claim that these coefficients describe nature.

Example references

C. P. Burgess, Introduction to Effective Field Theory, §§3.1–3.3

David B. Kaplan, Five lectures on effective field theory, §1.2 and §§2.4.1–2.4.2

Brian Conrad, Reductive Group Schemes, Definition 1.1.11 and Example 1.1.12

Physical EFT operator data and power-counting contract

Countable-choice assumption inherited by the non-Abelian gauge datum

Compact non-Abelian gauge group, bundle, and unitary representation

Finite-jet local operator algebra and engineering dimension

Declared finite-order operator quotient

Facts & Assumptions

Given: The finite scalar field, the standard $S U (2)$ representation, the dual algebraic representation, the physical conjugation condition $\tilde{H} = H^{†}$ , and the fixed truncation data in the Example.

[A1]

Adopted model data: the local gauge action is $H (x) \mapsto u (x) H (x)$ for smooth $u : M \to S U (2)$ on the trivial bundle, and the action is restricted to its real slice $\tilde{H} = H^{†}$ . This is a formal model choice, not an observed physical sector.

[F1]

Physical EFT operator data and power-counting contract: the contract fixes field representations, engineering weights, finite jet and field-degree cutoffs, symmetry compatibility, and a finite canonical-dimension and loop truncation; it does not identify the operator scale with regulator scales.

[F2]

Non-Abelian gauge data and bundle sectors: the physical group is the actual compact noncommutative group $S U (2)$ , and its standard representation is unitary; the algebraic polynomial model records the standard and dual representations separately.

[F3]

Finite-jet local operators and engineering dimension: local operators are weight-homogeneous polynomials in a finite field-jet representation, with even generators commuting and odd generators anticommuting; a dimensionless coefficient multiplies $O_{D} / Λ_{o p}^{D - 4}$ in four dimensions.

[F4]

Finite-order invariant operator-basis quotient: a finite-order basis quotient is by its explicitly declared invariant relation subspace and asserts no omitted-relation completeness.

[F5]

For the nondegenerate quadratic form $η$ on $C^{4}$ , $S O (η, C)$ is connected semisimple because $4 > 3$ ; $S L (2, C)$ is also connected semisimple. These are reductive over $C$ under the complex algebraic-group definition stated in the local-operator contract (Conrad, Definition 1.1.11 and Example 1.1.12).

[A2]

In natural units, each scalar component of $H$ and $\tilde{H}$ has engineering weight one, $Λ_{o p}$ has weight one, and the selected terms are evaluated without derivatives.

[A3]

${A C}_{ω}$ is countable choice inherited through the supplied non-Abelian gauge-data construction; no full Axiom of Choice is assumed. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Verification

Given: The facts and model data above.

1.1givenA1A3F2F3algebra

For every $u (x) \in S U (2)$ , unitarity gives $u (x)^{†} u (x) = I$ . Therefore $(u H)^{†} (u H) = H^{†} u^{†} u H = H^{†} H$ pointwise. Equivalently, for $g \in S L (2, C)$ the dual transformation is $\tilde{H} \mapsto \tilde{H} g^{- 1}$ while $H \mapsto g H$ , so $\tilde{H} H$ is invariant under the algebraic action as well. Since the expressions contain no derivatives, allowing $u$ to depend on $x$ introduces no derivative-of- $u$ term. Each power $(\tilde{H} H)^{n}$ is thus a local gauge-invariant scalar polynomial.

1.2givenF3algebra

The two doublet components are even commuting variables, so $ε_{a b} H^{a} H^{b} = H^{1} H^{2} - H^{2} H^{1} = 0$ . This is an algebraic identity, not an additional nonzero invariant or an assumption about an operator basis.

1.3givenA2F1F3algebra

Each factor $\tilde{H} H$ has engineering weight $1 + 1 = 2$ . Hence $O_{2}, O_{4}, O_{6}$ have weights $2, 4, 6$ , respectively. Since $Λ_{o p}$ has weight one, $O_{6} / Λ_{o p}^{2}$ has weight $6 - 2 = 4$ , so its real coefficient $C_{6}$ is dimensionless and the term lies within $D_{\max} = 6$ , $q = 6$ , and $L_{\max} = 0$ . No coefficient value or omitted-order bound follows from this counting.

2.1givenF4step 1.1step 1.2step 1.3∎

The relation space in this example is not used to eliminate or identify further operators; the displayed singlet is an allowed representative and the antisymmetric polynomial is zero before quotienting. Thus the calculation establishes membership and dimensions only, not a complete basis.

A finite-order SU(2) scalar operator contract

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources