Standard Model setup: fields, global group, and operators

A reproducible Standard Model symmetry calculation starts with an explicit field inventory. This walkthrough uses the implemented four-dimensional scalar/Weyl polynomial problem through canonical dimension six, with no derivatives or gauge-field strengths. It connects the global gauge group, all-left fermion convention, field statistics, selected sectors and saved files.

Open the one-generation inventory or the three-generation inventory. Both contain twelve species: the six rows below and an explicit conjugate of each. They use one Higgs doublet and no right-handed neutrino. For the same inputs outside the browser, download one-generation JSON or three-generation JSON.

1. Identify the fields before counting

Use metric signature $(+---)$, left/right Weyl labels $(\tfrac12,0)$ and $(0,\tfrac12)$, and $Q_Y=6Y$ for integer hypercharge. The quark field $Q$ is a different object from the charge $Q_Y$. Electric charge is $T_3+Y$.

ID	Field	Lorentz type	Color labels $(p,q)$	Weak label $n$	$Q_Y$	Flavors
Q	$Q$	Left Weyl	$(1,0)$	1	1	$n_g$
uc	$u^c$	Left Weyl	$(0,1)$	0	$-4$	$n_g$
dc	$d^c$	Left Weyl	$(0,1)$	0	2	$n_g$
L	$L$	Left Weyl	$(0,0)$	1	$-3$	$n_g$
ec	$e^c$	Left Weyl	$(0,0)$	0	6	$n_g$
H	$H$	Scalar	$(0,0)$	1	3	1

Here $u^c,d^c,e^c$ are left-handed charge-conjugates of the usual right-handed fields. This explains their antitriplet and reversed-charge assignments; compare Grzadkowski et al., Table 1 and §2, p.2. The conjugate IDs are Qdag, ucdag, dcdag, Ldag, ecdag, Hdag. Conjugation exchanges $(p,q)$, reverses $Q_Y$, exchanges Weyl chirality and preserves the number of flavors. The weak doublet’s dual is equivalent to the same $SU(2)$ irrep. Thus ucdag is right-handed; it is not another all-left row.

The algebraic count keeps conjugate variables explicit and independent over $\mathbb C$. It imposes Bose/Fermi statistics on the full Lorentz × gauge × flavor representation, but does not impose Hermiticity or quotient by flavor rotations. Read matter conventions and the field-statistics lesson for the rationale.

2. Check the chosen global group

The preset selects

$$G_6=\frac{SU(3)\times SU(2)\times U(1)}{\langle\gamma\rangle}, \qquad\gamma=(\omega I_3,-I_2,e^{i\pi/3}),\quad\omega=e^{2\pi i/3}.$$

For a field labeled $((p,q),n,Q_Y)$, the action of $\gamma$ is

$$\omega^{p+2q}(-1)^n e^{i\pi Q_Y/3} =\exp\!\left[\frac{2\pi i}{6}\bigl(2(p+2q)+3n+Q_Y\bigr)\right].$$

The six rows give integer exponents $6,0,6,0,6,6$, respectively, before reduction modulo six. Every row therefore descends; its conjugate has the inverse phase and descends too. The order-$d$ subgroup generated by $\gamma^{6/d}$ only requires the numerator to vanish modulo $d$, for $d=1,2,3,6$. All four choices admit this field inventory.

This reproduces the embedded center in Tong, §3, p.7. It does not select the experimentally realized global group or determine its line operators. Follow the Standard Model global-group calculation for the exact kernel and additional choices. The SU(5) block map provides a separate route to this subgroup; its primitive charge is $X=-Q_Y$.

Switching this inventory between compatible quotients leaves local invariant counts unchanged: each field action factors through the quotient, so the invariant subspace under the covering group is exactly the invariant subspace under its image. Nonlocal and bundle questions can still change.

3. Resolve the Weinberg sector from the inventory

In the enumeration table, select the $L^2H^2$ row and open its fixed-monomial calculation. The transferred setup retains all twelve species and gives zero powers to the other ten. The smaller dedicated three-flavor Weinberg preset contains only $L,H$ and returns the same count, but records a smaller field list.

The checks are immediate: $2(-3)+2(3)=0$ for hypercharge and $2(3/2)+2=5$ for canonical dimension. With spinor indices $\alpha,\beta$, weak indices $i,j,k,l$, and flavor indices $a,b$, a representative is

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

The fermion-exchange sign cancels the spinor-epsilon sign, and the commuting Higgs factors leave a symmetric flavor tensor. Hence this row has $n_g(n_g+1)/2$ complex directions: one or six for the two inventories. Its conjugate is another row, giving dimension-five totals two and twelve. The Weinberg article supplies the detailed contraction and source comparison. Adding an exact lepton-number selection rule removes the row; gauge invariance alone does not.

4. Resolve the QQQL sector

The $Q^3L$ row has zero hypercharge, canonical dimension six, baryon number one and lepton number one. Three identical quarks require their simultaneous spin, color, weak and flavor permutations to be projected together. Counting three distinguishable quark slots would overcount.

The surviving three-quark flavor tensors consist of one symmetric, one mixed, and one antisymmetric Young symmetry. With the additional lepton flavor index, Alonso et al., equation (10) and Table I, pp.3–4 give the dimensions that sum to

$$\begin{aligned} C_{QQQL} &=n_g\left[\frac{n_g(n_g+1)(n_g+2)}6 +\frac{n_g(n_g^2-1)}3 +\frac{n_g(n_g-1)(n_g-2)}6\right]\\ &=\frac{n_g^2(2n_g^2+1)}3. \end{aligned}$$

For one generation only the symmetric contribution survives and the count is one. For three generations the contributions are $30+24+3=57$. Recompute the dedicated QQQL preset and compare with the inventory row; the four-fermion derivation explains the underlying permutation constraint. Its Hermitian-conjugate sector is counted separately. A $B-L$ filter allows it, while separate conservation of $B$ and $L$ forbids it.

5. Compare a complete bounded census

The inventory totals count all scalar/Weyl monomials in the selected range, including distinct conjugate sectors and excluding the identity:

Canonical dimension	One generation	Three generations	Three generations, conserving B and L
2	1	1	1
4	7	55	55
5	2	12	0
6	45	2806	2260
Total	55	2874	2316

There are no nonzero dimension-one or dimension-three sectors in these inventories. Dimension four checks as one Higgs quartic plus three Yukawa structures and their conjugates: $1+6n_g^2$. At dimension six, separate $1+6n_g^2$ Higgs/Yukawa terms, baryon-preserving four-fermion terms, and baryon-violating four-fermion terms including conjugates. The latter two give

$$\frac{107n_g^4+2n_g^3+11n_g^2}{4},\qquad \frac{19n_g^4+3n_g^3+2n_g^2}{3}.$$

The first follows by adding the CP-even and CP-odd four-fermion columns of Alonso–Jenkins–Manohar–Trott, Table 2 and Appendix A, pp.31–32. The second sums the four non-sterile operator classes in the Table I cited above and their conjugates. At $n_g=3$ they are $2205$ and $546$, giving $1+54+2205+546=2806$. See the sector benchmark article for the full derivation and matching-convention comparison.

The B/L-conserving preset uses integer $3B$ and $L$, with reversed values on conjugates. It removes the twelve Weinberg directions and 546 baryon-violating directions, leaving $2874-12-546=2316$. This is an algebraic selection on operators, not a proof that these continuous symmetries are exact quantum symmetries.

6. Preserve and modify the calculation

Save the inventory or calculation envelope and reload it in the enumerator to recompute. A saved inventory includes the range, all species, selected quotient and extra charge filters. Opening a row in the local-operator tool preserves the species and quotient but records a fixed monomial; it does not carry the inventory’s B/L rules as new symmetries. Retain the inventory when that provenance matters.

Changing flavors, adding a sterile field, removing conjugates, selecting a nonzero charge sector, or changing the dimension range asks a different question. Recalculate after edits; stale results cannot be saved as current answers. An unsupported field or exceeded resource bound produces an explicit refusal rather than a partial census.

The result is a dimension of a complex invariant space. It is not a generated normalized contraction basis, a set of physical parameters after flavor redefinitions, or a full SMEFT basis. Derivatives, field strengths, integration by parts and equations of motion require additional machinery. Return to the setup library for smaller diagnostic examples or reproducibility for citation and replay details.