Accidental Symmetries and Their Violations

An accidental symmetry is not imposed when the theory is defined; it appears because the stated fields and gauge symmetries admit no renormalizable operator that violates it. In the minimal Standard Model, baryon and lepton selection rules are accidental at dimension four, but electroweak anomalies and higher-dimensional operators show why neither should be promoted to an unconditional exact law.

Required background. Use the quantum-consistency distinction from Standard Model anomaly cancellation and the operator expansion from degrees of freedom, symmetries, and local operators.

Helpful background. Accidental symmetries and emergent selection rules explains the associated naturalness logic without treating it as an ultraviolet proof.

Symmetries before and after the Yukawa couplings

With all Yukawa matrices set to zero, the five fermion species of three generations have the large kinetic-term flavor symmetry

$$U(3)_Q\times U(3)_u\times U(3)_d\times U(3)_L\times U(3)_e.$$

The Yukawa matrices act as spurions,

$$Y_u\mapsto V_QY_uV_u^\dagger, \qquad Y_d\mapsto V_QY_dV_d^\dagger, \qquad Y_e\mapsto V_LY_eV_e^\dagger,$$

and generic nonzero values break most of this group. In the renormalizable model with no neutrino mass, the surviving continuous particle-number transformations may be represented by baryon number $B$ and three individual lepton numbers $L_e,L_\mu,L_\tau$. That statement is classical and operator-level; anomaly and nonperturbative qualifications come next.

The spurion form is operational. A proposed flavor observable must be invariant when fields and Yukawa spurions are transformed together. If it changes, the expression contains an undeclared flavor basis choice or a missing spurion insertion.

Four distinct symmetry tests

The word “conserved” is ambiguous unless the level is stated. For each candidate current $J^\mu_X$, apply four tests:

Level	Question	Standard Model example
Classical action	Does every dimension-$\leq4$ monomial remain invariant?	$B$ and each $L_\alpha$ in the massless-neutrino model
Perturbative quantum theory	Is the current free of relevant triangle anomalies?	$B+L$ has an electroweak anomaly
Nonperturbative theory	Do topological sectors change the charge?	electroweak instanton/sphaleron transitions violate $B+L$
Effective theory	What is the lowest gauge-invariant higher-dimensional violation?	the dimension-five Weinberg operator violates $L$ by two units

For $N_g$ generations, an electroweak topological transition obeys

$$\Delta B=\Delta L=N_g\,\Delta N_{\mathrm{CS}}, \qquad \Delta(B-L)=0,$$

where $L=L_e+L_\mu+L_\tau$. Thus the anomalous direction is $B+L$, whereas $B-L$ survives the Standard Model electroweak anomaly. The underlying fermion zero-mode mechanism was established in ’t Hooft 1976, pp. 8–11. This selection rule does not provide a present-day transition rate; that requires a thermal or semiclassical calculation in a specified regime.

Lowest higher-dimensional violations

The unique dimension-five operator built only from Standard Model fields is, up to flavor indices and normalization,

$$\mathcal L_5 =\frac{C_5^{\alpha\beta}}{\Lambda} \left(\overline{L_\alpha^c}\,\widetilde H^*\right) \left(\widetilde H^\dagger L_\beta\right)+\text{h.c.}$$

After electroweak symmetry breaking it generates a Majorana mass matrix and violates total lepton number by two units. Its operator-level role and scale suppression were identified in Weinberg 1979, pp. 1566–1570. A neutrino mass generated this way also destroys the separate $L_e,L_\mu,L_\tau$ symmetries when the mass matrix is flavor off-diagonal.

Baryon-number violation first appears in Standard Model EFT at dimension six. Schematically,

$$\mathcal O_{qqql}\sim\epsilon_{abc} (q^aq^b)(q^c\ell), \qquad \mathcal O_{duue}\sim\epsilon_{abc} (d^au^b)(u^ce),$$

with Lorentz, weak, color, and flavor contractions understood. These operators can mediate nucleon decay; their mere existence does not predict a lifetime because the Wilson coefficients, matching scale, running, and hadronic matrix elements remain inputs. A complete nonredundant dimension-six classification is given in Grzadkowski et al. 2010, §§2–3.

Exact, anomalous, approximate, and accidental are different labels

The most reliable classification attaches a reason and a failure mode:

Label	Why it holds	How it can fail
Exact gauge invariance	defining redundancy plus anomaly cancellation	inconsistent field content or regulator
Accidental at dimension four	no allowed renormalizable violating operator	higher-dimensional operator
Anomalous global symmetry	classical current exists but the measure is not invariant	topological gauge configurations
Approximate symmetry	selected couplings or masses are small	restore the omitted spurions
Emergent low-energy rule	infrared degrees of freedom and expansion suppress violation	leave the controlled regime

Custodial symmetry illustrates the approximate case: it becomes transparent in limits involving hypercharge and Yukawa spurions, but it is not an exact symmetry of the full Standard Model with physical couplings. Proton stability illustrates the accidental case: dimension-four invariance is real, yet it is neither an all-orders theorem about the UV theory nor a statement that baryon-violating EFT coefficients vanish.

Naturalness statements with a ceiling

If setting a parameter $\epsilon$ to zero increases a symmetry, radiative corrections to $\epsilon$ are proportional to symmetry-breaking spurions. This is the technical-naturalness statement. It says that a hierarchy can be stable under quantum corrections inside the declared EFT; it does not explain why the UV matching condition selected a particular small value.

For example, $C_5=0$ restores lepton number in the minimal low-energy EFT, so a small $C_5$ is radiatively stable against interactions that respect that symmetry. But once another lepton-number-violating spurion is present, operator mixing can regenerate $C_5$. The input must therefore list all breaking spurions, not just the coefficient being discussed.

Selection-rule test

For any claimed symmetry:

1. list the fields, gauge group, and maximum operator dimension;
2. transform every allowed operator, not only the kinetic terms;
3. compute local and relevant global anomalies of the current;
4. identify nonperturbative sectors and their charge violation;
5. find the lowest higher-dimensional violating operator;
6. state whether small breaking is assumed, derived by matching, or measured.

A claim passes only if its adjective matches the strongest completed test. The output then feeds the running and vacuum-criterion page when scale evolution matters, or SMEFT/HEFT observables when the violating operators enter measurements.

References

• Grzadkowski, Bohdan, Michał Iskrzyński, Mikołaj Misiak, and Janusz Rosiek. “Dimension-Six Terms in the Standard Model Lagrangian.” Journal of High Energy Physics 2010, no. 10 (2010): 085, §§2–3. DOI.
• ’t Hooft, Gerard. “Symmetry Breaking through Bell–Jackiw Anomalies.” Physical Review Letters 37, no. 1 (1976): 8–11. DOI.
• Weinberg, Steven. “Baryon- and Lepton-Nonconserving Processes.” Physical Review Letters 43, no. 21 (1979): 1566–1570. DOI.