Matter representations and charge conventions

A matter multiplet needs more information than a dimension such as “a doublet” or “a ten.” Specify its global gauge group, representation labels, abelian charges and periods, Lorentz representation, statistics, and flavor multiplicity. Conjugating a field changes several of these entries together.

This page uses four-dimensional Lorentzian spin spacetime, compact internal gauge groups, and finite-dimensional complex field representations in characteristic zero. Left Weyl fields have Lorentz label $(1/2,0)$ and right Weyl fields $(0,1/2)$ for the complexified Lorentz algebra. These are field-component representations, not a classification of unitary one-particle states.

Field-space lesson · Gauge-theory workflow · Convention profiles

Write a complete field declaration

For a field $\Phi$ with $f$ flavors, its component space has the form

$$V_\Phi=V_{\mathrm{Lorentz}}\otimes R_G\otimes F, \qquad F\cong\mathbb C^f.$$

For a product gauge group, $R_G$ is itself a tensor product. Flavor is a multiplicity space here; choosing $f=3$ does not impose a new $U(3)$ invariance on the interaction coefficients.

As an example, a quark doublet with three generations has declaration

$$Q:\quad (1/2,0)_{\mathrm{Lorentz}}\otimes (\mathbf3,\mathbf2)_{q=1}\otimes\mathbb C^3_{\mathrm{flavor}}, \qquad q=6Y.$$

It has $2\cdot3\cdot2\cdot3=36$ complex Grassmann components before field equations. This is not a count of on-shell particle polarizations. Color and weak indices are components of each multiplet, not independent species to be entered as extra workbench rows.

The group must be stated too. For $[SU(3)\times SU(2)\times U(1)]/\Gamma$, every field must have trivial action of the specified embedded $\Gamma$. A neutral composite does not repair an individual field that fails this descent test. The global-group guide fixes the standard diagonal subgroups.

Dual, conjugate, and charge-conjugate are related operations

For a representation $\rho$ on $V$, the dual representation acts on linear functionals:

$$\rho^*(g)=\rho(g^{-1})^{\mathsf T}.$$

The complex-conjugate representation acts on $\overline V$ by $\overline{\rho(g)}$. After choosing an invariant positive Hermitian form for a compact group, its matrices are unitary and these two matrices coincide. This identifies the conjugate and dual representation categories; it does not mean that a representation is equivalent to its own dual. For example, $\mathbf3$ and $\overline{\mathbf3}$ of $SU(3)$ remain inequivalent.

For a $U(1)$ character $e^{i\theta}\mapsto e^{iq\theta}$, conjugation sends $q$ to $-q$. It also exchanges left and right Lorentz spinors. In two-component notation, an undotted left field $\psi_\alpha$ has a dotted Hermitian conjugate $\psi^\dagger_{\dot\alpha}$ in the conjugate gauge representation. Dreiner–Haber–Martin, §§2 and 5, fix these conventions and distinguish field names from particle names.

A right-handed field $\psi_R$ may instead be represented by its left-handed charge conjugate $\psi_R^c$. This operation changes $(0,1/2)$ to $(1/2,0)$, replaces $R$ by $R^*$, and reverses all abelian charges. It is a change of variables, not an assumption that charge conjugation is a symmetry of the theory. Nor does it identify an independently specified left field $\psi_L$ with $\psi_R^c$.

For a vectorlike Dirac fermion, $\psi_L$ and $\psi_R$ have the same gauge representation $R$ in the usual chiral notation. In all-left notation, the two independent fields instead carry $R$ and $R^*$. A Dirac mass pairs these two fields. Calling both all-left entries “$R$” would give the wrong charge and tensor-product tests when $R$ is complex.

Translate one Standard Model generation

Use color before weak isospin and define

$$q=6Y\in\mathbb Z,\qquad \theta\sim\theta+2\pi, \qquad Q_{\mathrm{em}}=T_3+Y.$$

The familiar right-handed singlets translate as follows. Subscripts in the representation columns are integer $q$, not $Y$.

Usual right field	Gauge representation	Independent all-left field	Gauge representation
$u_R$	$(\mathbf3,\mathbf1)_4$	$u^c$	$(\overline{\mathbf3},\mathbf1)_{-4}$
$d_R$	$(\mathbf3,\mathbf1)_{-2}$	$d^c$	$(\overline{\mathbf3},\mathbf1)_2$
$e_R$	$(\mathbf1,\mathbf1)_{-6}$	$e^c$	$(\mathbf1,\mathbf1)_6$

The remaining fermions are already left-handed:

$$Q=(\mathbf3,\mathbf2)_1, \qquad L=(\mathbf1,\mathbf2)_{-3}.$$

The Higgs $H=(\mathbf1,\mathbf2)_3$ is a complex scalar. These values follow from Grzadkowski et al., Table 1 and §2 by conjugating the right-handed entries and multiplying $Y$ by six. No right-handed neutrino is included in this inventory.

A quick sign check uses electric charges. For $Q$, the two values $T_3=\pm1/2$ give $2/3,-1/3$; for $L$, they give $0,-1$. The left fields $u^c,d^c,e^c$ have charges $-2/3,+1/3,+1$. They are the conjugates of the corresponding right-field charges, as required.

The full gauge representation of $H$ is complex even though the weak doublet alone is pseudoreal: conjugation changes its nonzero $q$ from $3$ to $-3$. The matrix $\epsilon=i\sigma_2$ identifies a conjugate weak doublet with a doublet, giving

$$\widetilde H=\epsilon H^*,\qquad \widetilde H:(\mathbf1,\mathbf2)_{-3}.$$

It does not remove the hypercharge sign. The definition of $\widetilde H$ with $\epsilon_{12}=+1$ is also in Grzadkowski et al., §2.

A rescaling of charges must include the circle convention. At the level of a local covariant derivative, $g_Y Y=(g_Y/6)q$. At the compact-group level, replacing all $q$ by $kq$ while keeping $\theta$ of period $2\pi$ changes the action and can add a kernel. See Tong, Line Operators in the Standard Model, §3 and footnote 2, for the integer normalization and its global role.

Check a Yukawa interaction in both notations

Let a bar on a four-component fermion denote the Dirac adjoint. With the usual right-handed fields, the Yukawa interaction pairs may be written using the following structures; the all-left column chooses their conjugate orientation. Coupling coefficients must be conjugated correspondingly if the orientation is changed.

Right-field notation, plus Hermitian conjugate	All-left monomial, plus Hermitian conjugate	All-left $q$ sum
$\overline Q_L\widetilde H u_R$	$Q u^c H$	$1-4+3=0$
$\overline Q_L H d_R$	$Q d^c H^\dagger$	$1+2-3=0$
$\overline L_L H e_R$	$L e^c H^\dagger$	$-3+6-3=0$

The first column is the form used in Grzadkowski et al., equation (2.1). Each all-left row has two undotted Weyl fields, hence a Lorentz scalar contraction with $\epsilon_{\alpha\beta}$. The weak pair contracts with $\epsilon$ or the fundamental-dual pairing, and the quark rows use the color fundamental-dual pairing. The gauge-theory guide writes the up-type indices explicitly.

For $n_g$ copies of each fermion species, each Yukawa tensor has $n_g^2$ complex entries before flavor-basis changes. The two fermions in each row are different species, so no exchange symmetry relates their two flavor indices. A wrong hypercharge sign would exclude a valid interaction; a missing conjugation could also replace the color singlet channel by $\mathbf3\otimes\mathbf3$, which has no singlet.

Gauge reality does not choose the Lorentz field

A complex irrep of a compact group is of real type if it admits an equivariant antilinear map $J$ with $J^2=+1$, and of quaternionic or pseudoreal type if the corresponding map has $J^2=-1$. A complex-type irrep is inequivalent to its dual. These are properties of the gauge representation, as in the reality lesson.

For example, the $SU(3)$ adjoint is of real type. One can place a real scalar, a complex scalar, or a left Weyl fermion in that gauge representation. They have different Lorentz representations, statistics, and possible reality conditions. A complex adjoint scalar has eight complex components; imposing the invariant real structure gives eight real scalar components. A left Weyl adjoint has sixteen complex Grassmann components before field equations and admits a gauge-invariant Majorana mass. The gauge label “adjoint” alone chooses none of these field types.

The mass test illustrates why statistics matters. For left Weyl fields $\psi_a^{\alpha i}$, with gauge index $i$ and flavor index $a$, use an invariant bilinear $B_{ij}$ and write

$$\mathcal M_{ab}=\epsilon_{\alpha\beta}B_{ij} \psi_a^{\alpha i}\psi_b^{\beta j}.$$

Exchanging $a,b$ supplies a Grassmann minus sign and a Lorentz-epsilon minus sign. Therefore $\mathcal M_{ba}=+\mathcal M_{ab}$ when $B$ is symmetric, but $\mathcal M_{ba}=-\mathcal M_{ab}$ when $B$ is antisymmetric. The basic anticommuting-spinor identity is Dreiner–Haber–Martin, equation (2.60). Consequently:

Gauge irrep and neutral remaining charges	Flavor symmetry of the mass bilinear	Dimension for $f$ flavors
Real type, symmetric $B$	Symmetric	$f(f+1)/2$
Pseudoreal type, antisymmetric $B$	Antisymmetric	$f(f-1)/2$

A single neutral weak doublet has no such mass, while two independent doublets can pair. In contrast, a single Weyl field in a real gauge representation can have a Majorana mass. Dreiner–Haber–Martin, §3.2, equations (3.2.35)–(3.2.40), develop the antisymmetric mass matrix for pseudoreal multiplets. These algebraic statements do not certify anomaly cancellation of the proposed spectrum. They also require all additional charges to permit the bilinear.

For a commuting scalar doublet $h$, $\epsilon_{ij}h^ih^j=0$ for a different reason: its component product is symmetric. The field-space lesson develops both cases and explains why statistics must be imposed on the combined Lorentz, gauge, and flavor space.

Transfer the declaration into a calculation

In the local-operator workbench, choose the global gauge group and integer charge normalization first. Enter each independent field with its Lorentz type, gauge labels, flavor count, and power in the monomial. A conjugate letter used in a polynomial needs the conjugate gauge labels, reversed abelian charges, and opposite Weyl chirality. The conjugate row represents that letter; it does not introduce a new physical particle species by itself.

For example, conjugating $Q$ gives a right Weyl letter with color $(0,1)$, weak label $1$, and $q=-1$, retaining its flavor count. The field $u^c$ is instead an independent left Weyl species with color $(0,1)$, weak label $0$, and $q=-4$. The two cannot be interchanged merely because both are antitriplets.

Use the Standard Model setup to retain the full inventory and compare the operator sectors. The implemented calculation counts complex, derivative-free scalar/Weyl polynomial invariants through canonical dimension six. It does not impose Hermiticity, quotient coupling space by flavor changes, or supply a complete EFT basis. Save the complete declaration and convention profile along with the result so a reader can reconstruct the actual question.