Yukawa couplings and four-fermion operators

A Yukawa interaction with two distinct Weyl species gives a relatively direct singlet count. Four-fermion operators introduce additional Lorentz contractions, and repeated species correlate those contractions with gauge and flavor symmetry. The two examples below show why a model-building calculation should begin with complete fields rather than dimensions of gauge representations alone.

The local-operator workbench reproduces these examples using exact characters of the full Lorentz, gauge, and flavor spaces. Its answer counts independent complex invariant polynomials at a specified field content. The method reference states the algebraic equivalence and the limits of that interpretation.

Fix an all-left Weyl convention

Use integer abelian charge $Q=6Y$, where electric charge is $Q_{\mathrm{em}}=T_3+Y$. An all-left convention represents a physical right-handed fermion by its left-handed charge-conjugate field. Thus $u^c$ and $d^c$ are color antifundamentals, and all their gauge charges are conjugated relative to the corresponding right-handed fields.

Species	Lorentz type	Gauge representation $(SU(3)_c,SU(2)_w)_Q$
$Q$	Left Weyl	$(\mathbf3,\mathbf2)_1$
$u^c$	Left Weyl	$(\overline{\mathbf3},\mathbf1)_{-4}$
$d^c$	Left Weyl	$(\overline{\mathbf3},\mathbf1)_2$
$L$	Left Weyl	$(\mathbf1,\mathbf2)_{-3}$
$e^c$	Left Weyl	$(\mathbf1,\mathbf1)_6$
$H$	Scalar	$(\mathbf1,\mathbf2)_3$
$H^\dagger$	Scalar	$(\mathbf1,\overline{\mathbf2})_{-3}$

For $SU(2)$, $\overline{\mathbf2}$ is equivalent to $\mathbf2$, so both Higgs rows use weak Dynkin label one. The lower weak index of $H^\dagger$ still usefully records its dual realization. The last row denotes the conjugate scalar variables, which are declared explicitly when the monomial requires them. Compare Grzadkowski–Iskrzyński–Misiak–Rosiek, Table 1 and equation (2.1), whose right-handed notation is related by this conjugation.

All these fields descend to the standard $\mathbb Z_6$ quotient and its order-two and order-three subgroups. The global-group guide specifies that embedding and the associated integer charge convention.

The up-type Yukawa matrix

Let $A$ be a color index, $i,j$ weak indices, $\alpha,\beta$ left spinor indices, and $p,r$ flavor indices. With $\epsilon_{12}=+1$, an up-type Yukawa operator is

$$\mathcal Y^u_{pr} =\epsilon_{\alpha\beta}\epsilon_{ij} Q_p^{\alpha Ai}u^c_{rA}{}^\beta H^j.$$

Its charge is $1-4+3=0$, and its canonical dimension is $3/2+3/2+1=4$. Each representation factor supplies one singlet:

$$\mathbf3_c\otimes\overline{\mathbf3}_c\supset\mathbf1_c, \qquad \mathbf2_w\otimes\mathbf2_w\supset\mathbf1_w, \qquad \mathbf2_\ell\otimes\mathbf2_\ell\supset\mathbf1_\ell.$$

Because $Q$ and $u^c$ are distinct species, no symmetry identifies their flavor labels. For one Higgs doublet and flavor multiplicities $f_Q,f_u$, there are

$$N_{Qu^cH}=f_Qf_u$$

independent complex operators. Three quark-doublet and three up-singlet flavors give nine entries of a general complex Yukawa matrix. The dimension of that coefficient space precedes any choice of flavor basis and any reduction to masses and mixing parameters.

The down-type and charged-lepton contractions use the conjugate Higgs:

$$\begin{aligned} \mathcal Y^d_{pr} &=\epsilon_{\alpha\beta} Q_p^{\alpha Ai}d^c_{rA}{}^\beta(H^\dagger)_i,\\ \mathcal Y^e_{pr} &=\epsilon_{\alpha\beta} L_p^{\alpha i}e^c_r{}^\beta(H^\dagger)_i. \end{aligned}$$

Their charges are $1+2-3=0$ and $-3+6-3=0$, respectively. Replacing $H^\dagger$ by $H$ in either expression fails the abelian test. Leaving out the Higgs also leaves an uncontracted weak doublet. These elementary failures are useful checks of a field declaration before a larger calculation.

A physical right-handed field entered as right Weyl with its original charges is not the same input as the left Weyl field $u^c$. Translating a four-component Lagrangian requires translating both the Lorentz type and the gauge representation of every conjugated field. The two-component reference of Dreiner–Haber–Martin, Sections 2 and 4 gives a systematic convention for doing so.

Four left spinors: three pairings, two invariants

For four distinguishable left Weyl spinor slots, the number of Lorentz scalars is two:

$$\mathbf2_\ell\otimes\mathbf2_\ell =\mathbf1_\ell\oplus\mathbf3_\ell, \qquad \dim\operatorname{Hom}_{SU(2)_\ell} (\mathbf1,\mathbf2_\ell^{\otimes4})=2.$$

The singlet comes once from $\mathbf1\otimes\mathbf1$ and once from $\mathbf3\otimes\mathbf3$. The three epsilon pairings obey the two-dimensional Schouten identity

$$\epsilon_{\alpha\beta}\epsilon_{\gamma\delta} -\epsilon_{\alpha\gamma}\epsilon_{\beta\delta} +\epsilon_{\alpha\delta}\epsilon_{\beta\gamma}=0.$$

This identity can be multiplied by four Grassmann fields in a fixed order. If the fields are subsequently reordered into bilinears, their exchange signs must be included. A convenient bilinear form of the resulting Fierz relation is given in Dreiner–Haber–Martin, equations (2.28) and (2.66). Counting visually different pairings as independent misses this relation even before repeated-field statistics are considered.

For a particularly transparent repeated-field example, take a gauge-singlet left Weyl species $N$ with $f$ flavors and power four. The relevant space is $\Lambda^4(\mathbb C^2_\ell\otimes\mathbb C^f)$. Its Lorentz singlet has flavor Young shape $(2,2)$, whose hook-length dimension is

$$N_{N^4}=\frac{f^2(f^2-1)}{12}.$$

The counts for $f=1,2,3$ are $0,1,6$. One flavor has only two Grassmann components, so every quartic monomial vanishes. Four distinguishable slots would instead have two Lorentz contractions even at one flavor. This example is discussed independently in Fonseca, Section 1, footnote 1.

The baryon-violating monomial QQQL

For three quark doublets and one lepton doublet,

$$Q^3L:\qquad Q_{\mathrm{total}}=3(1)-3=0, \qquad \Delta=4\left(\frac32\right)=6.$$

A representative contraction is

$$\mathcal Q_{prst} =\epsilon_{ABC}\epsilon_{il}\epsilon_{jk} \epsilon_{\alpha\beta}\epsilon_{\gamma\delta} Q_p^{\alpha Ai}Q_r^{\beta Bj}Q_s^{\gamma Ck}L_t^{\delta l}.$$

Flavor permutations of this expression are subject to linear relations. Treating all $f_Q^3f_L$ flavor entries as unrelated coefficients of every possible spin and weak contraction would overcount. The standard $qqq\ell$ operator and its permutation identities appear in Alonso–Chang–Jenkins–Manohar–Shotwell, equations (1), (10), and (13).

Here is a direct count. The three colors form a singlet through $\epsilon_{ABC}$, which transforms by the sign representation of permutations of the three quarks. Total Fermi antisymmetry then requires the remaining spin, weak, and quark-flavor tensor to be symmetric under the simultaneous permutation group $S_3$.

To couple to the lepton spin doublet, the three quark spin doublets must have total spin $1/2$. The multiplicity space of this doublet is the two-dimensional standard representation $[2,1]$ of $S_3$. The same statement holds for the three weak doublets, which must couple to a weak doublet before contracting with $L$. Their combined permutation representation is

$$[2,1]\otimes[2,1] =[3]\oplus[2,1]\oplus[1,1,1].$$

Consequently the surviving quark-flavor multiplicity space is

$$\operatorname{Sym}^3F_Q \oplus S_{(2,1)}F_Q \oplus\Lambda^3F_Q.$$

The symbol $S_{(2,1)}$ denotes the Schur functor of Young shape $(2,1)$. Its occurrence describes how the operator tensors transform under changes of flavor basis; it does not impose a flavor symmetry on the Lagrangian. Each of the three permutation types occurs once. At $f_Q=f$, their dimensions add to

$$\begin{aligned} &\frac{f(f+1)(f+2)}6 +\frac{f(f^2-1)}3 +\frac{f(f-1)(f-2)}6\\ &\hspace{2em}=\frac{f(2f^2+1)}3. \end{aligned}$$

The independent lepton-flavor index supplies a further factor $f_L$. Thus

$$\boxed{N_{Q^3L}=\frac{f_Lf_Q(2f_Q^2+1)}3.}$$

For equal quark and lepton flavor multiplicities,

Flavors $f_Q=f_L$	Symmetric quark-flavor part	Mixed part	Antisymmetric part	Total
$1$	$1$	$0$	$0$	$1$
$2$	$8$	$4$	$0$	$12$
$3$	$30$	$24$	$3$	$57$

Each column includes the lepton multiplicity. For three quark flavors the quark-flavor pieces have dimensions $10,8,1$, and multiplying their sum by three lepton flavors gives $57$. These contributions agree with Alonso and collaborators, Table I and the permutation-based treatment in Fonseca, Section 3, equation (49).

Before statistics, the distinguishable-slot count is $4f_Q^3f_L$: one color singlet, two weak singlets, two spin singlets, and the flavor choices. It is $324$ at three flavors. The reduction to $57$ is an algebraic effect of Fermi statistics together with contraction identities. It is not a numerical approximation or a choice to discard particular flavor interactions.

Interpret the answer as a local coefficient space

With the conventional assignments $B(Q)=1/3$, $L(L)=1$, and the other relevant charges zero, $Q^3L$ has $\Delta B=\Delta L=1$. An additional exact baryon or lepton symmetry would forbid it. The present workbench projects only the declared gauge and Lorentz groups; these global selection rules must be imposed as additional physical assumptions.

The number $57$ concerns the $Q^3L$ complex polynomial sector. Its Hermitian conjugate belongs to the conjugate sector. A Hermitian interaction has the form $C_I\mathcal O_I+C_I^*\mathcal O_I^\dagger$, so those conjugate coefficients are not additional independent complex parameters. Nor has the count been quotiented by all possible flavor-basis transformations.

Select the Yukawa or $QQQL$ preset in the workbench and compare the statistics-aware result with the distinguishable-slot count. The exported physical setups make the fields, powers, flavors, and global quotient reproducible. The examples above provide explicit contractions and derivations for these cases; the generic engine counts invariants without producing a contraction basis for arbitrary inputs. Derivative operators, equations of motion, integration by parts, and a complete SMEFT basis remain separate tasks.