Exact local-operator counting

The local-operator workbench computes the number of complex Lorentz-and-gauge invariant polynomials in a specified monomial of scalar and Weyl fields. Identical bosons are symmetrized and identical fermions antisymmetrized before singlets are extracted. The calculation returns an exact nonnegative integer, or an explicit validation or resource error.

Its convention is four-d-local-polynomial-v1; its counting algorithm is local-operators-1.0.0. This reference states what that integer means and how it is obtained. The field-space guide explains the physics behind the construction.

The mathematical question

For each field species $a$, let

$$R_a=V_{j_{\ell,a},j_{r,a}}\otimes V_{(p_a,q_a)}\otimes V_{n_a}\otimes\mathbb C_{Q_a} \otimes\mathbb C^{f_a}.$$

Here $(p_a,q_a)$ is the $SU(3)$ highest weight, $n_a=2j_{w,a}$ is the $SU(2)$ weak Dynkin label, and $f_a$ is an ungauged flavor multiplicity. The one-dimensional factor $\mathbb C_{Q_a}$ carries integer $U(1)$ charge $Q_a$, in a circle of period $2\pi$. For selected powers $m_a$, define

$$\mathcal A_{\boldsymbol m} =\bigotimes_{a\ \mathrm{bosonic}}\operatorname{Sym}^{m_a}R_a \otimes \bigotimes_{a\ \mathrm{fermionic}}\Lambda^{m_a}R_a.$$

The answer is

$$N_{\boldsymbol m} =\dim_{\mathbb C}\operatorname{Hom}_{K} \bigl(\mathbf1,\mathcal A_{\boldsymbol m}\bigr), \qquad K=SU(2)_\ell\times SU(2)_r\times SU(3)_c \times SU(2)_w\times U(1).$$

The two compact spin factors compute invariants of the complexified Lorentz algebra. The selected spacetime is still four-dimensional and Lorentzian. No parity projection is included. The construction of the derivative-free invariant algebra by symmetric/exterior powers and Haar projection is described in Marinissen–Rahn–Waalewijn, Section II A, equations (2)–(4).

The tensor product between different species has a fixed species ordering. Anticommuting two distinct fermion fields fixes a sign convention; it does not impose an additional antisymmetrization between their separately specified representation spaces. A repeated field must therefore be represented by one row and its power, with flavor degeneracy included in that row.

Validate the global group first

The physical setup specifies one of

$$G_d=(SU(3)\times SU(2)\times U(1))/\Gamma_d, \qquad d=1,2,3,6.$$

These are the standard embedded diagonal subgroups: writing

$$z=(e^{2\pi i/3}I_3,-I_2,e^{i\pi/3}), \qquad \Gamma_d=\langle z^{6/d}\rangle,$$

a field descends precisely when

$$2(p_a+2q_a)+3n_a+Q_a\equiv0\pmod d.$$

This follows by multiplying the three central characters on the given generator. The product-center reference develops the general embedded-quotient calculation. Every declared field is checked, including rows with power zero. An inadmissible field blocks the physical calculation; a neutral composite does not make the individual field a representation of the selected group.

If all fields descend, the invariant subspace under $G_d$ equals the invariant subspace under its covering product, because that product maps surjectively onto $G_d$. Thus changing among compatible quotients leaves this local count unchanged. It can still change line operators, bundles, and other global questions; the local count does not identify the full gauge theory.

Construct exact weight characters

Use five integral weight coordinates: two for the $SU(3)$ fundamental weights, one for weak $SU(2)$, and one for each spin factor. An $SU(2)$ irrep with Dynkin label $n$ has character

$$\chi_n(x)=x^n+x^{n-2}+\cdots+x^{-n}.$$

The left Weyl spin character is $u+u^{-1}$; the right Weyl character is $v+v^{-1}$. A scalar contributes one. Multiplying the relevant spin and gauge characters and then multiplying all weight multiplicities by $f_a$ gives the full one-field character.

If a weight $w$ has multiplicity $b_w$, its contributions to the symmetric and exterior generating functions are

$$\begin{aligned} \sum_{m\ge0}t^m\chi_{\operatorname{Sym}^mR}(z) &=\prod_w(1-tz^w)^{-b_w},\\ \sum_{m\ge0}t^m\chi_{\Lambda^mR}(z) &=\prod_w(1+tz^w)^{b_w}. \end{aligned}$$

For a single weight, the coefficient of $t^kz^{kw}$ is respectively $\binom{b_w+k-1}{k}$ or $\binom{b_w}{k}$. The binomial coefficients account for distinct field components with the same weight, including flavor copies. Omitting those multiplicities would change the operator count.

The implementation evaluates these finite coefficients through Newton identities. With $h_0=e_0=1$, it uses

$$\begin{aligned} m h_m(z)&=\sum_{k=1}^{m}\chi_R(z^k)h_{m-k}(z),\\ m e_m(z)&=\sum_{k=1}^{m}(-1)^{k-1}\chi_R(z^k)e_{m-k}(z), \end{aligned}$$

where $h_m=\chi_{\operatorname{Sym}^mR}$ and $e_m=\chi_{\Lambda^mR}$. The operation $z\mapsto z^k$ multiplies every weight by $k$ and leaves its multiplicity intact. It does not replace a flavor multiplicity $f$ by $f^k$. Each division by $m$ must be exact. The engine combines equal weights and finally tensors the requested species characters.

The equivalent exponential formulas are the bosonic and fermionic plethystic exponentials. See Henning–Lu–Melia–Murayama, equations (2.3)–(2.6). The present calculation extracts one finite multidegree; it does not claim to produce a complete rational Hilbert series or generators and relations for the entire invariant algebra.

Project to singlets, not merely zero weight

A zero-weight vector need not be invariant. The adjoint of $SU(2)$ has a zero weight but no singlet. Weyl projection removes precisely these false candidates.

For a Weyl-invariant finite character $\chi$, let $\Phi_+$ be the positive roots of the semisimple factors. The singlet multiplicity is

$$N=\operatorname{CT}_z\left[ \chi(z)\prod_{\alpha\in\Phi_+}(1-z^{-\alpha}) \right].$$

Here $\operatorname{CT}$ extracts the coefficient of weight zero. There is no Weyl-group prefactor in this positive-root formula. To verify the normalization, apply the Weyl character formula to an irreducible character:

$$\chi_\lambda(z)\prod_{\alpha\in\Phi_+}(1-z^{-\alpha}) =\sum_{w\in W}(-1)^{\ell(w)} z^{w(\lambda+\rho)-\rho}.$$

For dominant $\lambda$, a zero exponent occurs only for $\lambda=0$ and the identity Weyl element. The constant term is therefore one on the trivial irrep and zero on every other irrep. Linearity proves the formula for an arbitrary finite character. Equivalently one can use the product over all roots with a prefactor $1/|W|$, as in the normalized Haar integrals of Henning–Lu–Melia–Murayama, equations (3.10)–(3.12).

In the chosen coordinates, the $SU(3)$ positive roots are $(2,-1)$, $(-1,2)$, and $(1,1)$; each $SU(2)$ factor has positive root $2$. For example,

$$\operatorname{CT}_x\bigl[(x^2+1+x^{-2})(1-x^{-2})\bigr]=0.$$

For $U(1)$, every term at the selected multidegree has the same charge $\sum_a m_aQ_a$. A nonzero sum gives zero invariants immediately. There is no abelian Weyl denominator.

Checks and the comparison with distinguishable slots

During character expansion, the computation checks the one-field dimensions and the dimensions of the symmetric/exterior powers. If $r=\dim R$, those dimensions are

$$\dim\operatorname{Sym}^mR=\binom{r+m-1}{m}, \qquad \dim\Lambda^mR=\binom rm.$$

Evaluating a character at the identity means summing its weight multiplicities; this must reproduce the corresponding dimension. A fermion power larger than its component-space dimension vanishes. Final singlet counts must be nonnegative integers.

The workbench also calculates a distinguishable-slot count, by replacing each power with $R_a^{\otimes m_a}$ and performing the same Lorentz and gauge projection. This is a comparison before repeated-field statistics, not an alternative operator answer. In characteristic zero the required exchange sector is a direct summand of the full tensor product, so its invariant dimension cannot exceed the distinguishable-slot count.

Two useful comparisons are

$$\begin{array}{c|c|c} \text{Monomial} & \text{Distinguishable slots} & \text{Local operators}\\ \hline L^2H^2 & 2f_L^2 & f_L(f_L+1)/2\\ Q^3L & 4f_Q^3f_L & f_Lf_Q(2f_Q^2+1)/3 \end{array}$$

The first has one Higgs doublet. The second uses the Standard Model quark and lepton assignments. Their reductions are derived in the Weinberg and four-fermion examples.

Published computational domain

Input	Supported range
Spacetime	Four-dimensional Lorentzian
Field kinds	Scalar, left Weyl, right Weyl
Gauge structure	One $SU(3)$, one $SU(2)$, one integer-normalized $U(1)$; standard $\Gamma_d$ above
$SU(3)$ labels	$p,q\ge0$, $p+q\le2$: $\mathbf1,\mathbf3,\overline{\mathbf3},\mathbf6,\overline{\mathbf6},\mathbf8$
Weak label	$n=0,1,2$
Abelian charge	Integer $Q$ from $-24$ through $24$
Flavors	$1,2,3$ per species; ungauged multiplicity
Field rows	At most $16$
Monomial	Each power $0$–$6$, total power at most $6$
Canonical dimension	At most $6$, with scalar dimension $1$ and Weyl dimension $3/2$
Derivatives	Exactly zero
Intermediate character	At most $120{,}000$ distinct weights
Arithmetic work	At most $20{,}000{,}000$ coefficient multiplication pairs per request

The representation and degree bounds describe admissible inputs; they do not promise that every such input fits the resource bounds. Exceeding a resource bound terminates the calculation with an explicit error and returns no partial count. Changing a bound is an implementation decision, not an approximation to the invariant space.

Character multiplicities and dimensions use exact integer arithmetic (BigInt); large public integers are serialized as decimal strings. There is no floating-point quadrature or external algebra service. Input setups and worked examples are available in the physical-setup download; imported results are recomputed from the field data rather than trusted as cached answers.

What equivalence has been imposed?

Algebraic identities are already included: the representation space is the actual symmetric/exterior algebra, so alternative contractions related by Schouten identities, Fierz identities, or finite-dimensional tensor relations do not create extra dimensions. The algorithm counts this space without constructing a general explicit basis of contractions.

The equivalence is algebraic. It does not identify operators by integration by parts, equations of motion, field redefinitions, Hermitian conjugation, or a change of flavor basis. There are no derivative or field-strength letters, no anomaly-cancellation calculation, and no imposed baryon, lepton, discrete, or additional flavor symmetry. An allowed invariant is a candidate interaction, not a certificate of consistency for a quantum theory.

Derivatives and equation-of-motion relations require a further construction; see Henning–Lu–Melia–Murayama, Section 2 for the distinction between polynomial expressions and an EFT basis. In particular, this derivative-free result must not be presented as a complete Standard Model EFT basis through dimension six. Its value is a precisely specified and independently checkable answer to one local field-polynomial question.