From fields to allowed interactions

A field inventory is the starting point of a Lagrangian. It specifies which variables exist and how they transform. To find the interactions, one must then answer two different questions: which combinations of fields can occur, and how many independent invariant contractions does each combination admit? A list of neutral monomials answers only the first question.

The operator enumerator performs both steps for the scalar and Weyl fields supported by the local-operator workbench. It lists the nonzero Lorentz and gauge invariant sectors up to a selected canonical dimension, with optional additional abelian charge constraints. The present calculation concerns complex polynomials without derivatives or field strengths in four-dimensional Lorentzian spacetime.

1. Declare complete fields

For each species $\Phi_i$, specify its Lorentz type, color and weak representations, integer abelian charge, and number of flavors. These data define a complete component space

$$V_i=L_i\otimes C_i\otimes W_i\otimes F_i, \qquad F_i\simeq\mathbb C^{f_i}.$$

The flavor factor records independent components. It is not an additional group under which the interaction must be invariant. For example, three copies of a quark multiplet permit a general flavor tensor unless an extra symmetry is explicitly imposed.

The gauge group is the specified product $SU(3)\times SU(2)\times U(1)$ or one of its standard diagonal quotients. Choose that global group before enumerating. Every field must be a representation of it, even if a particular output row does not use that field. The global-form guide explains why a representation of the Lie algebra need not descend to the chosen group.

Conjugates are explicit entries. The conjugate of a left Weyl field is right Weyl, its color representation is dual, and all abelian charges change sign. A scalar and its conjugate are separate polynomial variables. Adding both does not yet impose a reality condition on their coefficients.

This convention has a useful consequence: a saved inventory records exactly which polynomial algebra was counted. Omitting a conjugate restricts that algebra. It does not ask the program to infer that the omitted variables should be restored later.

2. Turn the dimension cutoff into a finite search

Let $n_i\geq0$ be the number of occurrences of $\Phi_i$. The occupation vector $\mathbf n=(n_1,\ldots,n_s)$ labels a monomial sector, before its indices have been contracted. In four dimensions,

$$d(\mathbf n)=\sum_{i\in\mathrm{scalar}}n_i +\frac32\sum_{i\in\mathrm{Weyl}}n_i.$$

For a maximum dimension $D$, only vectors with $d(\mathbf n)\leq D$ need to be considered. The inequality is finite because every field has positive canonical dimension. Integer arithmetic suffices if the program works with twice the dimension:

$$2d(\mathbf n)=2N_s+3N_f\leq2D.$$

The current limit $D\leq6$ also implies at most six field insertions. There is therefore no additional loss from the workbench’s six-insertion bound within this domain.

A monomial such as $Q u^c H$ represents all its allowed spin, gauge, and flavor contractions. It is one sector even when it contains nine independent complex operators. Conversely, a charge-neutral sector can contain no operator at all. The result table reports sector counts and invariant dimensions separately.

The identity $1$ is the empty occupation vector. Its invariant space has dimension one, and its canonical dimension is zero. The enumerator excludes it by default and provides an explicit option to include it. Its coefficient would be a constant term in a Lagrangian; whether that constant is dynamically relevant depends, for example, on whether gravity is included in the physical problem.

3. Reject impossible occupations, then impose statistics

A candidate gauge singlet must have total integer $U(1)$ charge zero:

$$\sum_i n_i Q_i=0.$$

Color triality, weak center charge, and Lorentz spinor parity provide other quick necessary tests. With only scalars and Weyl fields and no derivatives, a Lorentz scalar must contain an even number of left Weyl fields and an even number of right Weyl fields. Each spin factor must be able to combine to integer spin zero.

These tests save work; they do not establish that a nonzero contraction exists. The decisive space is

$$\mathcal P_{\mathbf n} =\bigotimes_{i\in\mathrm{scalar}}\operatorname{Sym}^{n_i}V_i \otimes \bigotimes_{j\in\mathrm{Weyl}}\Lambda^{n_j}V_j, \qquad c_{\mathbf n}=\dim_{\mathbb C} \left(\mathcal P_{\mathbf n}^{\mathrm{Lorentz}\times G}\right).$$

Symmetric and exterior powers act on the whole field representation, including flavor. Only distinct species are tensored without an exchange projection. Character projection onto the singlet then gives $c_{\mathbf n}$ exactly. The invariant-series formulation of this construction is described by Marinissen, Rahn, and Waalewijn, Section II A, equations (2)–(4).

For a simple example, a single gauge-singlet left Weyl species has two spinor components. Its mass bilinear is nonzero, but its fourth exterior power vanishes. With two flavors there are four components, and the fourth exterior power can supply a Lorentz scalar. Treating the four occurrences as distinguishable spinor slots would miss that difference. The statistics guide develops the component-space viewpoint.

4. Add a global selection rule deliberately

An extra abelian rule assigns an integer charge $q_i$ to each field. To preserve a continuous $U(1)$, select

$$\sum_i n_iq_i=0.$$

To preserve a cyclic symmetry $\mathbb Z_m$, select

$$\sum_i n_iq_i\equiv0\pmod m.$$

The enumerator also permits a nonzero target charge. This selects a charged sector; it does not make its operators invariant under that extra symmetry. Such a sector is useful for studying symmetry violation or a possible spurion whose charge compensates the operator. The program does not add that spurion or assign its dynamics.

A continuous normalization and a cyclic normalization must be handled differently. Multiplying every continuous charge and its target by the same nonzero integer preserves the equality being tested. Multiplying cyclic charges by a number that is not invertible modulo $m$ can change which operators survive. The charges and modulus are consequently part of the saved specification.

Consider a neutral complex scalar $S$ and its explicit conjugate $S^\dagger$. At dimension at most four, without an extra symmetry, there are

$$\sum_{k=1}^{4}(k+1)=14$$

nonconstant monomial sectors $S^a(S^\dagger)^b$, each with one invariant. Give $S$ charge $+1$ and $S^\dagger$ charge $-1$ under a continuous global $U(1)$. Only $S^\dagger S$ and $(S^\dagger S)^2$ remain. Give both odd parity under $\mathbb Z_2$ instead: every even total degree is allowed, yielding three quadratic and five quartic sectors. The field inventory is unchanged; the physical assumption about symmetry is different.

5. Read a result as a graded coefficient space

The enumerator returns one row for every occupation vector with $c_{\mathbf n}>0$. The dimension subtotal is

$$C_d=\sum_{d(\mathbf n)=d}c_{\mathbf n}.$$

The sum is valid because different field occupations span distinct graded subspaces of the polynomial algebra. Multiplying fields in another order does not create a second row, and choosing another contraction does not create another occupation vector.

A familiar Higgs-only example makes the grading visible. Declare one $H\sim(\mathbf1,\mathbf2)_3$ and its conjugate. Abelian neutrality requires equal powers. Since $\operatorname{Sym}^n\mathbf2$ is irreducible of spin $n/2$, the product with its dual has one singlet. Through dimension six the nonconstant answer is

$$H^\dagger H,\qquad (H^\dagger H)^2,\qquad (H^\dagger H)^3.$$

There are three sectors and three complex invariant directions. With additional Higgs flavors, the symmetric powers involve the combined weak and flavor space; the same argument with a single doublet no longer applies.

For an inventory closed under conjugation, non-self-conjugate sectors occur in pairs of equal dimension. A Hermitian Lagrangian relates their coefficients. For a pair $\mathcal O,\mathcal O^\dagger$, the expression

$$C\mathcal O+C^*\mathcal O^\dagger$$

contains one complex coefficient. The table instead records the complex polynomial spaces of both sectors. The relation between those counts and real parameters requires the conjugation map; further removal of flavor-basis redundancies is another operation. The Standard Model examples illustrate these distinctions numerically.

A practical research workflow

1. Start from a published inventory or import a saved physical setup. Check the meaning of every conjugated field and every charge normalization.
2. Choose the global gauge group, a dimension interval, and any additional charge sector. Use a zero target when the extra symmetry is meant to be preserved.
3. Enumerate, then inspect dimension subtotals and the individual nonzero sectors. An unexpected absence often points to chirality, charge, or identical-field statistics.
4. Open a row in the fixed-monomial workbench to inspect its exact calculation. The transfer retains the entire field inventory and the selected quotient.
5. Save the inventory and result together with their versions. Loading recomputes the data, so a stored answer is reproducible evidence rather than an instruction to trust a cached number.

This workflow identifies allowed local polynomial interactions and their algebraic multiplicities. Derivatives and field strengths require additional field representations; equations of motion and integration by parts require additional equivalences. They are essential to a complete EFT basis and are outside this enumeration contract. See the method reference for the completeness statement within the supported domain.