Standard Model operator sectors through dimension six

The Standard Model provides a demanding test of operator enumeration: its fields carry several simultaneous representations, repeated fermions have nontrivial flavor symmetry, and gauge invariance permits interactions that violate baryon or lepton number. A useful calculation should expose those distinctions in the result.

The operator enumerator includes one- and three-generation Standard Model inventories, with every conjugate declared explicitly. This page derives their benchmark totals without derivatives or gauge-field strengths. The results are exact dimensions of complex invariant-polynomial spaces. They are a specified part of the Standard Model effective field theory, rather than a complete SMEFT basis.

The inventory and its conventions

Use an all-left convention for the five fermion species. Thus $u^c,d^c,e^c$ are left Weyl charge-conjugate fields; they have the charges opposite to the physical right-handed fermions. Write $Q_Y=6Y$ for the integer-normalized hypercharge, to distinguish it from the quark doublet $Q$.

Field	Lorentz type	$(SU(3)_c,SU(2)_w)_{Q_Y}$	$3B$	$L$
$Q$	Left Weyl	$(\mathbf3,\mathbf2)_1$	$1$	$0$
$u^c$	Left Weyl	$(\overline{\mathbf3},\mathbf1)_{-4}$	$-1$	$0$
$d^c$	Left Weyl	$(\overline{\mathbf3},\mathbf1)_2$	$-1$	$0$
$L$	Left Weyl	$(\mathbf1,\mathbf2)_{-3}$	$0$	$1$
$e^c$	Left Weyl	$(\mathbf1,\mathbf1)_6$	$0$	$-1$
$H$	Scalar	$(\mathbf1,\mathbf2)_3$	$0$	$0$

Each fermion has $n_g$ flavors; there is one Higgs doublet. Add the conjugate of every row, giving twelve species in total. Conjugating reverses the abelian charges, dualizes color, and exchanges left and right Weyl types. The weak doublet is pseudoreal, so its dual has the same Dynkin label. These assignments agree with Grzadkowski and collaborators, Table 1 after translating their right-handed fields into the all-left convention.

The presets choose the standard $\mathbb Z_6$ quotient. Every declared multiplet descends to that group. For this inventory, switching to its covering product or the order-two or order-three diagonal quotient leaves the local invariant counts unchanged: the allowed representations factor through the quotient, and invariance under the cover is equivalent to invariance under its image. The global groups still differ in other questions, as explained in the global-group guide.

The complete polynomial census within this scope

The following table includes both members of each non-self-conjugate pair and excludes the identity. The counts are recomputed by the published inventory, rather than stored as answers in the algorithm.

Canonical dimension	Nonzero monomial sectors	Complex dimension, $n_g=1$	Complex dimension, $n_g=3$
$1$	$0$	$0$	$0$
$2$	$1$	$1$	$1$
$3$	$0$	$0$	$0$
$4$	$7$	$7$	$55$
$5$	$2$	$2$	$12$
$6$	$36$	$45$	$2806$
Total through six	46	55	2874

A sector means a field occupation, not one contraction. The same 46 occupations survive for one and three generations, but their flavor spaces have different dimensions. In particular, enlarging the number of flavors can increase the number of contractions within an existing field structure.

Dimensions two and four

The only quadratic scalar sector is $H H^\dagger$, with invariant $H^\dagger H$. There is no gauge-invariant fermion mass bilinear in this inventory: no sterile neutrino has been included, and the chiral charged fermions cannot acquire a mass before a Higgs insertion.

At dimension four there is one Higgs quartic and three Yukawa structures,

$$Qu^cH,\qquad Qd^cH^\dagger,\qquad Le^cH^\dagger,$$

together with their conjugates. Each Yukawa structure has $n_g^2$ complex contractions, because its two flavor labels belong to distinct species. Therefore

$$C_4=1+6n_g^2.$$

The result is $7$ for one generation and $55$ for three. Kinetic terms contain derivatives or field strengths and are outside this table. Thus this is the full dimension-four polynomial answer for the chosen letters, not the complete renormalizable Lagrangian.

Dimension five

The two sectors are $L^2H^2$ and its conjugate. The flavor tensor in either sector is symmetric, so

$$C_5=2\frac{n_g(n_g+1)}2=n_g(n_g+1).$$

This gives $2$ and $12$, respectively. One often describes the Weinberg operator as the unique dimension-five field structure, with its Hermitian conjugate understood. The enumerator uses a different explicit convention: the two conjugate multidegrees are separate rows. The Weinberg guide derives the symmetric tensor. Compare also Fonseca, Section 1 for the distinction between a field structure and its flavor coefficients.

Resolve dimension six into physical sectors

There are three types of field content at this dimension: six Higgs fields, two fermions with three Higgs fields, and four fermions. The exact result splits as follows.

Polynomial class	$n_g=1$	$n_g=3$
$(H^\dagger H)^3$	$1$	$1$
Yukawa structures multiplied by $H^\dagger H$, including conjugates	$6$	$54$
Four-fermion sectors preserving $B$ and $L$	$30$	$2205$
Four-fermion sectors violating $B$, including conjugates	$8$	$546$
Total	45	2806

There is a useful independent published check of the largest entry. Summing the CP-even and CP-odd four-fermion columns in Alonso, Jenkins, Manohar, and Trott, Table 2 and Appendix A gives

$$C_{\psi^4,\,B=0} =\frac{107n_g^4+2n_g^3+11n_g^2}{4}.$$

It evaluates to $30$ and $2205$. The four-fermion classes contain no derivatives or field strengths, so their algebraic count can be compared directly with the corresponding part of this computation. The connection to their real Hermitian coefficient count is explained below.

Baryon-violating structures

Choose the $\Delta B=\Delta L=1$ orientation of each pair. In the all-left convention used here, these structures and their multiplicities are

Field occupation	Complex multiplicity for common $n_g$	At $n_g=3$
$Q L (u^c)^\dagger(d^c)^\dagger$	$n_g^4$	$81$
$Q^2(u^c)^\dagger(e^c)^\dagger$	$n_g^3(n_g+1)/2$	$54$
$Q^3L$	$n_g^2(2n_g^2+1)/3$	$57$
$((u^c)^\dagger)^2(d^c)^\dagger(e^c)^\dagger$	$n_g^4$	$81$

The total is $273$ at three generations. Including the conjugate sectors doubles it to $546$. The flavor representations and their dimensions are tabulated in Alonso, Chang, Jenkins, Manohar, and Shotwell, equation (1) and Table I. Their $u,d,e$ denote right-handed fermions, which explains the daggers in the table above.

The $Q^3L$ multiplicity is a particularly sensitive check: three identical quarks correlate spin, weak, color, and flavor permutations. The four-fermion guide derives the count $57$ and its reduction from distinguishable slots. A charge-neutral field list alone would not determine that number.

Combining the four baryon-violating structures gives

$$C_{\psi^4,\,B\ne0} =\frac{19n_g^4+3n_g^3+2n_g^2}{3},$$

where conjugates are included. Adding this to $1+6n_g^2+C_{\psi^4,B=0}$ reproduces the dimension-six subtotal in the census.

Impose baryon and lepton selection rules

Use the integer charges $3B$ and $L$ from the inventory table and reverse them for conjugate fields. Two continuous rules with target zero impose $\Delta B=\Delta L=0$. In the three-generation preset the result becomes

Dimension	Nonzero sectors	Complex invariant dimension
$2$	$1$	$1$
$4$	$7$	$55$
$5$	$0$	$0$
$6$	$28$	$2260$
Total	36	2316

The Weinberg pair disappears because it has $\Delta L=\pm2$. The eight baryon-violating occupations disappear because they have $\Delta B=\Delta L=\pm1$. The other sectors remain unchanged. Selecting the continuous rule $3(B-L)=0$ instead removes the Weinberg pair but retains the baryon-violating dimension-six sectors. Preserving $B-L$ therefore does not by itself forbid these proton-decay operators.

As a discrete example, lepton parity $(-1)^L$ allows the Weinberg pair, since its charge is even, while excluding the displayed baryon-violating sectors with odd $L$. An exact lepton $U(1)$ is a stronger assumption than its parity subgroup.

For a nonzero target, the interface selects a charge sector rather than imposing conservation. Choosing target $3$ for $3B$ and target $1$ for $L$ isolates the $273$ complex directions above at dimension six. A Hermitian interaction would also involve their conjugate sector.

A general consistency check is

$$d\equiv\frac{\Delta B-\Delta L}{2}\pmod2, \qquad \frac{\Delta B-\Delta L}{2}\in\mathbb Z,$$

for Standard Model Lorentz- and hypercharge-invariant operators. Kobach, equations (15)–(18) derives this relation; it also permits any number of the specified sterile-neutrino fields. The Weinberg sector has odd $d$ and $\Delta B-\Delta L=\mp2$, whereas the displayed baryon-violating sectors have even $d$ and $\Delta B-\Delta L=0$. This is a necessary check. It does not replace the nonzero-contraction test or determine a multiplicity, and it is not asserted for arbitrary new field assignments.

Global charge filters are algebraic assumptions on the operators. They do not check whether a proposed global symmetry is exact quantum mechanically, whether it is anomalous, or whether a chosen gauge theory is consistent.

Compare counts with matching conventions

There are three distinct objects to keep track of:

• A field structure records occupations and may package many flavor contractions.
• A complex polynomial space includes every independent contraction and, in these inventories, both conjugate sectors.
• A Hermitian interaction space identifies coefficients through Hermitian conjugation; a further flavor-basis quotient can reduce the number of physical parameters.

If a finite complex space $V$ is closed under an antilinear involution $\dagger$, its Hermitian fixed space $V_{\mathbb R}$ has

$$\dim_{\mathbb R}V_{\mathbb R}=\dim_{\mathbb C}V.$$

Indeed, every $v\in V$ decomposes uniquely as

$$v=\frac{v+v^\dagger}{2} +i\frac{v-v^\dagger}{2i},$$

with both displayed components Hermitian. This explains why a conjugation-closed polynomial count can numerically equal a real Hermitian coefficient count. It does not mean that the program has imposed Hermiticity or chosen a physical flavor basis. A charge-selected sector that is not closed under conjugation does not satisfy the premise by itself.

The familiar full dimension-six SMEFT totals include derivative and field-strength classes and specify their treatment of redundancies and conjugation. This page keeps a smaller alphabet and algebraic equivalence. Comparing the total $2806$ directly with a complete SMEFT total would mix different questions. The class-by-class checks above establish the appropriate comparison.

Open an inventory in the enumerator, choose dimension six alone, and inspect a row in the fixed-monomial workbench. Save the inventory to preserve the global group, charge normalization, flavor multiplicities, and selection rules together with the result. The method reference states the completeness and replay contracts.