Representation and Spurion Constraints on Operator Bases

Before integration by parts, equations of motion, or Fierz identities can reduce an operator list, symmetry must decide which local tensors are candidates at all. Lorentz and gauge invariance require singlets in explicit representation products; exact global symmetries impose selection rules; controlled symmetry breaking is encoded by spurions with declared transformation laws. These filters produce a checkable candidate space, but they do not yet settle flavor multiplicities, Hermitian completion, CP conventions, or redundancy.

Required background. From Operator Lists to Independent Bases defines the ambient invariant space. Multiplets, Invariants, and Selection Rules supplies the representation-theory logic, while Gauge Fields, Redundancy, and Observable Content distinguishes exact gauge redundancy from global symmetry. Helpful background. Compact Lie Groups, Roots, Weights, and Weyl Structure supplies weights and characters for non-Abelian projections.

Symmetry filters define the candidate space

Fix a multiset of fields and derivatives. If a bosonic species $\Phi_a$ appears $n_a$ times in representation $R_a$ and a fermionic species $\psi_b$ appears $m_b$ times in representation $S_b$, the relevant representation is contained in

$$\mathcal R_{\mathrm{raw}} =\bigotimes_a\operatorname{Sym}^{n_a}R_a \otimes \bigotimes_b\wedge^{m_b}S_b,$$

with additional permutation structure when fields carry distinguishable flavor labels. The number of invariant contractions is

$$n_{\mathrm{inv}} =\dim\operatorname{Hom}_{G_{\mathrm L}\times G_{\mathrm g}} (\mathbf1,\mathcal R_{\mathrm{raw}}),$$

where $G_{\mathrm L}$ is the Lorentz group and $G_{\mathrm g}$ the gauge group. A nonzero multiplicity says that singlet contractions exist; multiplicity greater than one means that every independent invariant tensor must be constructed explicitly.

In four dimensions it is often convenient to use the complexified Lorentz labels $(j_L,j_R)$:

Building block	Lorentz representation
Scalar $\phi$	$(0,0)$
Left-handed Weyl field $\psi_\alpha$	$(1/2,0)$
Conjugate Weyl field $\psi^\dagger_{\dot\alpha}$	$(0,1/2)$
Covariant derivative $D_{\alpha\dot\alpha}$	$(1/2,1/2)$
Self-dual and anti-self-dual field strengths	$(1,0)$ and $(0,1)$

Spinor indices are contracted with the invariant antisymmetric tensors, vector indices with the metric or equivalent spinor tensors, and gauge indices with invariant intertwiners. The Bose or Fermi projection is imposed before the singlet multiplicity is interpreted. Otherwise a contraction can be counted even though it vanishes for identical fields.

For a compact group, character orthogonality implements the singlet projection:

$$n_{\mathrm{inv}} =\int_G d\mu(g)\,\chi_{\mathcal R_{\mathrm{raw}}}(g).$$

Plethystic exponentials package all symmetric or antisymmetric field products, while the Haar integral extracts the trivial representation. This gives a powerful independent count, but not the index contractions themselves; that distinction is explicit in Lehman and Martin 2015, §§ 2–3, preprint pp. 3–8, Open PDF. Derivatives, IBP, and EOM require further quotient data, as developed in Henning et al. 2016, §§ 1–2, preprint pp. 1–7, Open PDF.

Continuous and discrete charge tests

For Abelian global factors, the representation test becomes charge addition. A monomial containing $n_a$ fields of charge $q_a$ and $r_A$ spurions of charge $s_A$ is invariant under $U(1)$ only if

$$\sum_a n_aq_a+\sum_A r_As_A=0.$$

For a unitary $\mathbb Z_N$ symmetry, the corresponding condition is

$$\sum_a n_aq_a+\sum_A r_As_A \equiv0\pmod N.$$

Conjugate fields carry conjugate representations and opposite Abelian charges. A derivative is neutral under an internal symmetry unless a background construction assigns it additional structure. Antiunitary CP is not merely another additive charge: phases, complex conjugation, field maps, and coefficient reality conditions must be declared, which is the subject of the next page.

Gauge invariance is stricter than a phenomenological global selection rule. A Wilson coefficient cannot simply carry an uncompensated gauge index. A “gauge spurion” must be a legitimate background field, compensator, or vacuum value in a gauge-covariant formulation; otherwise it represents explicit violation of the gauge redundancy and the candidate is inadmissible.

Spurions record controlled breaking

Suppose a coupling $Y$ breaks a global group $G_F$. Promote it temporarily to a nondynamical tensor transforming in a representation $R_Y$ such that every term is formally $G_F$ invariant. Count operators at a fixed number of $Y$ insertions, construct every contraction, and only then freeze $Y$ to its physical background value.

This procedure separates two statements:

• the formal covariance of the operator-plus-spurion tensor under $G_F$;
• the actual unbroken subgroup that leaves the chosen background $\langle Y\rangle$ invariant.

Freezing the spurion does not restore an exact symmetry. It records the direction and order of explicit breaking. D’Ambrosio, Giudice, Isidori, and Strumia use this logic by assigning the Yukawa matrices definite flavor representations and then fixing their background values in D’Ambrosio et al. 2002, § 2, preprint pp. 3–5, Open PDF.

A complete spurion declaration includes its group representation, Abelian and discrete charges, canonical or EFT weight, complex-conjugation rule, maximum insertion order, and fixed background. If several spurions have the same labels, their identities and relative power counting must remain distinct.

Counting is only the first stage

The symmetry projection supplies a dimension or a candidate multiplicity. A usable basis still needs explicit representatives, conventions, renormalization closure, and a coefficient translation.

An operator-basis result is a five-stage package. Counting fixes $n=\dim\mathcal Q$; construction supplies explicit contractions and relations; normalization fixes ordering, conjugation, and phases; closure enlarges the $d$-dimensional renormalization space when EOM or evanescent operators are required; and translation applies $O'=BO$ with the dual coefficient map $C'=B^{-T}C$. The lower row states the acceptance evidence at each stage. A count alone is neither an explicit basis nor proof of RG closure. The diagram is schematic and not to scale.

For each bounded field multiset, retain a record like this before proceeding:

Filter	Required input	Output
Canonical and EFT order	Field dimensions, derivative count, spurion weights	Allowed graded multisets
Lorentz	$(j_L,j_R)$ labels and statistics	Independent scalar contractions
Gauge	Representations, index orientation, invariant tensors	Gauge-singlet multiplicity and contractions
Exact global symmetry	Non-Abelian representations and Abelian charges	Allowed exact-symmetry sectors
Discrete symmetry	$\mathbb Z_N$ charges and unitary action	Congruence-allowed sectors
Controlled breaking	Spurion representations, insertion order, and background	Formally covariant symmetry-breaking structures

Only after this list is explicit should flavor permutations, Hermitian pairing, CP, IBP, EOM, and dimension-specific identities reduce or reorganize it.

First application: the dimension-five lepton operator

Use two left-handed lepton doublets $L_p^{i\alpha}$ and two Higgs doublets $H^j$. Here $i,j$ are weak $SU(2)_L$ indices, $\alpha$ is a left-handed spinor index, and $p$ is left open as a flavor label. Their relevant data are

Field	Lorentz	$SU(2)_L$	Hypercharge $Y$	Lepton number	Dimension
$L_p^{i\alpha}$	$(1/2,0)$	$\mathbf2$	$-1/2$	$+1$	$3/2$
$H^i$	$(0,0)$	$\mathbf2$	$+1/2$	$0$	$1$

The field content $L_pL_rHH$ has dimension five and total hypercharge zero. Contract each weak doublet pair with $\epsilon_{ij}$ and the two left-handed spinors with $\epsilon_{\alpha\beta}$:

$$O_5^{pr} =\epsilon_{\alpha\beta} \bigl(\epsilon_{ij}L_p^{i\alpha}H^j\bigr) \bigl(\epsilon_{kl}L_r^{k\beta}H^l\bigr).$$

This is a Lorentz and gauge singlet. Alternative pairings of the four weak doublets reduce to the same structure after the $SU(2)$ epsilon identity and field statistics, so the singlet multiplicity is one for fixed ordered flavor labels. The operator carries lepton number $+2$. An exact continuous $U(1)_L$ forbids it, whereas lepton parity $\mathbb Z_2$ permits it because $2\equiv0\pmod2$.

Introduce a coefficient spurion $\kappa_{pr}$ with lepton number $-2$. Then the oriented interaction

$$\frac{1}{\Lambda}\kappa_{pr}O_5^{pr}$$

is formally invariant. Freezing $\kappa$ records lepton-number violation by two units; it does not make $U(1)_L$ exact. The symmetry of the flavor tensor, its conjugate completion, and the number of independent real parameters are deliberately left to Flavor, Hermiticity, and CP Bookkeeping.

This is the unique dimension-five Standard Model field-content class first identified in Weinberg 1979, pp. 1566–1570. The calculation above supplies the representation and charge evidence rather than using that classification as a substitute for it.

Common pitfalls

Adding charges but ignoring tensor multiplicity. Charge neutrality is sufficient only for Abelian one-dimensional representations. Non-Abelian products can contain zero, one, or several singlets.

Counting before imposing field statistics. A formal invariant tensor can vanish or become related to another after identical bosons are symmetrized or identical fermions antisymmetrized.

Calling a frozen spurion an exact symmetry. The background value breaks the formal group to its stabilizer. State the insertion order and the actual residual symmetry.

Using a Wilson coefficient to hide gauge noninvariance. Ordinary coefficients are gauge singlets. Any object carrying gauge indices needs a genuine gauge-covariant origin.

Stopping at the Hilbert-series coefficient. A multiplicity is a cross-check on dimension. Construct the index contractions and later prove redundancy reduction and RG closure.

Exercises

For which nontrivial $\mathbb Z_N$ lepton-number subgroups is an operator with $\Delta L=2$ invariant?

Solution

The condition is $2\equiv0\pmod N$, so $N$ must divide $2$. The only nontrivial cyclic choice is $\mathbb Z_2$. In particular, a $\mathbb Z_4$ lepton-number symmetry forbids the operator unless a spurion supplies charge $2$ modulo $4$.

Show that the $U(1)$ Haar projection selects charge-neutral monomials.

Solution

A charge-$Q$ monomial has character $e^{iQ\theta}$. Therefore

$$\int_0^{2\pi}\frac{d\theta}{2\pi}e^{iQ\theta} =\delta_{Q,0}.$$

Applying the integral to a generating function deletes every nonzero-charge term and retains precisely the invariant sector.

References

• D’Ambrosio, G., G. F. Giudice, G. Isidori, and A. Strumia. “Minimal Flavour Violation: An Effective Field Theory Approach.” Nuclear Physics B 645, no. 1–2 (2002): 155–187. DOI; Open PDF
• Henning, Brian, Xiaochuan Lu, Tom Melia, and Hitoshi Murayama. “Hilbert Series and Operator Bases with Derivatives in Effective Field Theories.” Communications in Mathematical Physics 347, no. 2 (2016): 363–388. DOI; Open PDF
• Lehman, Landon, and Adam Martin. “Hilbert Series for Constructing Lagrangians: Expanding the Phenomenologist’s Toolbox.” Physical Review D 91, no. 10 (2015): 105014. DOI; Open PDF
• Weinberg, Steven. “Baryon- and Lepton-Nonconserving Processes.” Physical Review Letters 43, no. 21 (1979): 1566–1570. DOI