5. Spontaneous Symmetry Breaking¶

Physics statement¶

A scalar potential invariant under the gauge/discrete symmetries can still have a minimum away from the origin. Expanding the fields around that minimum — the vacuum expectation value (VEV) — rather than around zero is what turns gauge bosons and fermions massive and produces the physical Higgs/Goldstone spectrum. feynlag splits this into two concerns: the Vacuum object (the VEV shift itself) and the tadpole system (the condition that the shift point is actually a stationary point of the potential).

The VEV shift¶

Scalar.expand_vev({component: vev}) (2. Declaration: Parameters, Fields, Groups) registers, per scalar component, the CONVENTIONS.md-pinned expansion

\[ \phi^0 \;\to\; \frac{v + h + i\,a}{\sqrt2} \qquad \text{(complex component)}, \qquad \phi \;\to\; v + \phi \qquad \text{(real component, no $1/\sqrt2$)}, \]

introducing two fresh real symbols phi_r, phi_i per complex component (the CP-even and CP-odd fluctuations). Vacuum (vacuum/ewsb.py:18) aggregates every scalar’s shift_map into one substitution dict and provides two operations built on it:

Vacuum.shift(expr) — expand around the vacuum, keeping fluctuations (xreplace the shift map, then sp.expand). This is what turns the weak Lagrangian into the physical-basis Lagrangian’s raw material.
Vacuum.at_vacuum(expr) — evaluate strictly at the vacuum point: shift, then set every fluctuation to zero, and every scalar component that never received a VEV (e.g. a charged component, which cannot get one by charge conservation) to zero as well — both the component and its conjugate, so the potential really is evaluated at the true minimum candidate, not just the neutral-VEV slice of it.

Tadpoles: algorithm¶

The tadpole conditions are $\{\text{vev}: \partial V/\partial\text{vev} \vert_{\rm vacuum} = 0\}$ — the requirement that the vacuum point is a stationary point. extract_tadpoles (vacuum/tadpoles.py:13) computes this directly: V0 = vacuum.at_vacuum(potential), then sp.diff(V0, vev) for every registered VEV symbol (raising if a VEV isn’t a plain Symbol — a composite VEV expression can’t be differentiated against meaningfully).

Derivation: why $\partial V/\partial\text{vev}$ is the tadpole coefficient¶

The tadpole condition is usually stated as “the coefficient of the linear-in-fluctuation term vanishes” — i.e. the coefficient of $h$ (not $v$) in $V(v+h+\ldots)$. extract_tadpoles instead differentiates $V$ evaluated at the vacuum with respect to $v$ itself. These give the same answer by the chain rule: since $v$ and $h$ enter the shift additively ($\phi \to (v+h+ia)/\sqrt2$), $\partial/\partial v$ acting on any function of $(v+h)$ and $\partial/\partial h$ acting on the same function are literally the same differential operator up to which symbol survives the subsequent $h\to0$ evaluation:

\[ \left.\frac{\partial V}{\partial v}\right|_{h=0} \;=\; \left.\frac{\partial V}{\partial h}\right|_{h=0} \]

because both compute $V'(v+h)|_{h=0}$ for the same one-variable function $V'$. Differentiating with respect to the VEV symbol directly — rather than first extracting a linear-order Taylor coefficient in the fluctuation — is therefore not an approximation or a different quantity; it is the identical tadpole condition, computed the cheaper way (one sp.diff call instead of a full Taylor expansion and coefficient extraction).

Solving¶

solve_tadpoles(potential, vacuum, for_params) (vacuum/tadpoles.py:31) turns the tadpole dict into sp.Eq(..., 0) equations and calls sp.solve for the requested parameters. It raises rather than guessing whenever sp.solve returns zero solutions (over-constrained/inconsistent system) or more than one branch (under-determined choice of sign/root — the caller must pick a branch manually and use InternalParameter.define directly). On success, every InternalParameter among for_params gets its expr defined with the solution — completing the declaration made in 2. Declaration: Parameters, Fields, Groups (an InternalParameter starts life with expr=None and is only usable once some pipeline stage, most often this one, fills it in).

Model.solve_tadpoles (lagrangian.py:175) wraps this, remembers the solution in self._tadpole_solutions (applied automatically by mass_matrix and physical_lagrangian, 1. The Pipeline), and invalidates the Model cache.

Worked case: the over-constrained 3HDM+S₃ system¶

examples/thdm_s3.py is deliberately built so the tadpole system has more independent conditions than free VEV ratios — the S₃-symmetric potential, when its tadpole equations are solved simultaneously, does not admit an arbitrary vacuum: it forces a specific alignment, $v_1 = \sqrt3\,v_2$ in the literature basis (components swap in feynlag’s real-orthogonal S₃ irrep basis — see 2. Declaration: Parameters, Fields, Groups) — exactly the tadpole solution of the S₃-3HDM model this example follows [GomezBock21]. This is not a numerical accident of the benchmark point chosen; it is a structural consequence of the S₃ Clebsch–Gordan structure of the potential terms (S3.doublet_product) forcing the minimum to sit on a symmetry-preserving submanifold of VEV space. solve_tadpoles reproduces this alignment directly from the symbolic system, without it being hand-imposed.

Validation¶

tests/test_scalar_pipeline_sm.py::test_tadpole_solution — the SM Higgs case, $\mu^2 = \lambda v^2$.
tests/test_scalar_pipeline_thdm.py::test_tadpole_solutions_match_literature — 2HDM tadpoles cross-checked against the standard Gunion–Haber form.
tests/test_thdm_s3.py::test_tadpole_alignment_sqrt3 — the forced $\sqrt3$ vacuum alignment described above.

Minimal snippet¶

model.check_invariance(raise_on_failure=True)
solution = model.solve_tadpoles([mu2])   # {mu2.symbol: lam.symbol * v.symbol**2}

References¶

[GomezBock21] M. Gómez-Bock, M. Mondragón, A. Pérez-Martínez, “Scalar and gauge sectors in the 3-Higgs Doublet Model under the S₃-symmetry”, Eur. Phys. J. C 81, 942 (2021), arXiv:2102.02800, doi:10.1140/epjc/s10052-021-09731-3 — the literature S₃-3HDM model examples/thdm_s3.py and examples/THDM_S3_Tutorial.ipynb follow; their Eq. (13) tadpole solution is exactly the $v_1=\sqrt3v_2$ alignment above, and their Eq. (25)–(29) geometric rotation ansatz is the one the tutorial notebook builds and verifies against feynlag’s own pseudoscalar/charged/CP-even mass matrices.