{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0",
   "metadata": {},
   "source": [
    "# 3HDM with S₃ flavor symmetry: forcing the vacuum, not tuning it\n",
    "\n",
    "A generic three-Higgs-doublet potential has dozens of independent quartic\n",
    "couplings — far too many to be predictive. One standard way to cut that\n",
    "down is to impose a **discrete flavor symmetry** on the scalar sector: pick\n",
    "a finite group, assign the doublets to its representations, and keep only\n",
    "the quartic invariants that group theory allows.\n",
    "\n",
    "This notebook builds the simplest non-abelian example: $S_3$, the\n",
    "permutation group on 3 objects, with $(H_1, H_2)$ forming its 2-dimensional\n",
    "irrep and a third doublet $H_S$ sitting in the trivial singlet. It follows\n",
    "`examples/thdm_s3.py` step by step, showing two things no earlier feynlag\n",
    "tutorial needed:\n",
    "\n",
    "1. **Finite discrete-symmetry invariance.** Every gauge check in prior\n",
    "   tutorials linearizes the transformation at $\\alpha=0$ (a continuous Lie\n",
    "   group). $S_3$ has no such infinitesimal neighborhood of the identity —\n",
    "   `feynlag` instead applies the group's *exact, finite* substitution and\n",
    "   checks the Lagrangian term-by-term for literal equality.\n",
    "2. **A vacuum that isn't free to choose.** With three VEVs ($v_1, v_2,\n",
    "   v_S$) but an $S_3$-covariant potential that only has *two* independent\n",
    "   mass parameters ($\\mu_0^2,\\mu_1^2$) to absorb tadpole conditions, the\n",
    "   third condition becomes a real constraint. Solving it doesn't tune a\n",
    "   free ratio — it **forces** $v_1/v_2$ to a fixed value.\n",
    "\n",
    "Along the way we also see the $S_3$ Clebsch–Gordan decomposition\n",
    "$2\\otimes2 = 1\\oplus1'\\oplus2$ that the potential is built from, the\n",
    "resulting CP-even, pseudoscalar and charged mass matrices, and — following\n",
    "the literature model this potential is drawn from — a geometric rotation\n",
    "ansatz that diagonalizes them into seven physical scalar masses."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1",
   "metadata": {},
   "source": [
    "**Assumed background:** the Higgs mechanism / spontaneous gauge symmetry\n",
    "breaking, Goldstone's theorem, and enough Lie-group representation theory\n",
    "to know what a tensor-product (Clebsch–Gordan) decomposition is — the kind\n",
    "of material a first QFT course plus a group-theory-for-physicists chapter\n",
    "covers. If tensor products/irreps are new, `SUN_Groups_Tutorial.ipynb`\n",
    "builds that machinery up from scratch first (for continuous groups; $S_3$\n",
    "here is discrete, but the tensor-product idea is the same). The\n",
    "library-specific machinery this notebook exercises (`S3`, `Model`,\n",
    "`Rotation`, ...) is documented in `docs/manual/declaration.md` and\n",
    "`docs/manual/ssb.md`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2",
   "metadata": {},
   "source": [
    "**Notation used throughout this notebook:**\n",
    "\n",
    "| Symbol | Meaning |\n",
    "|---|---|\n",
    "| $H_1,H_2,H_S$ | the three $SU(2)_L\\times U(1)_Y$ Higgs doublets ($H_1,H_2$ in the $S_3$ **2**, $H_S$ in the **1**) |\n",
    "| $v_1,v_2,v_S$ | their vacuum expectation values |\n",
    "| $\\lambda_1,\\ldots,\\lambda_8$ | the eight $S_3$-invariant quartic couplings (§2 table) |\n",
    "| $\\mu_0^2,\\mu_1^2$ | the singlet/doublet-sector mass² parameters |\n",
    "| $M_S^2,\\,M_A^2,\\,M_C^2$ | the CP-even, pseudoscalar (CP-odd) and charged $3\\times3$ mass-squared matrices |\n",
    "| $R(\\varphi,\\theta)$ | the geometric (coupling-independent) rotation fixed by the vacuum direction alone |\n",
    "| $\\alpha$ | the one genuinely dynamical mixing angle, needed only for the CP-even sector |\n",
    "| $h_0,H_1,H_2$ | the three physical CP-even scalars |\n",
    "| $A_1,A_2$ | the two physical pseudoscalars (the third pseudoscalar direction is the $Z$'s Goldstone) |\n",
    "| $H_1^\\pm,H_2^\\pm$ | the two physical charged scalars (the third charged direction is the $W^\\pm$'s Goldstone) |\n",
    "\n",
    "Keep this table in view — the notebook builds every one of these from\n",
    "scratch, in this order."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3",
   "metadata": {},
   "source": [
    "## 1. Symmetries, parameters, fields\n",
    "\n",
    "$S_3$ has three irreducible representations:\n",
    "\n",
    "- $\\mathbf{1}$ — trivial: every group element acts as $+1$.\n",
    "- $\\mathbf{1'}$ — the sign representation: the 3-cycle acts as $+1$, the\n",
    "  transposition as $-1$.\n",
    "- $\\mathbf{2}$ — a genuine 2-dimensional (real, orthogonal) doublet:\n",
    "  `feynlag`'s `S3` represents the 3-cycle generator as a $2\\pi/3$ rotation\n",
    "  matrix and the transposition generator as a reflection\n",
    "  $\\mathrm{diag}(1,-1)$.\n",
    "\n",
    "`s3.assign(irrep, *fields)` below registers which fields sit in which\n",
    "irrep and builds, from those generator matrices, an explicit finite\n",
    "substitution map (one linear combination of components per group\n",
    "generator) — this is what `check_discrete_invariance` later applies\n",
    "*exactly*, in contrast to the $O(\\alpha)$ linearized check every gauge\n",
    "symmetry in this library uses.\n",
    "\n",
    "We also declare the usual electroweak gauge sector ($SU(2)_L\\times U(1)_Y$)\n",
    "alongside $S_3$ — the potential must satisfy *both* symmetries at once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "from IPython.display import display\n",
    "sp.init_printing()\n",
    "\n",
    "from feynlag import (\n",
    "    ExternalParameter, InternalParameter, Lagrangian, Model, S3, SU2,\n",
    "    Scalar, U1, check_discrete_invariance, dag,\n",
    ")\n",
    "\n",
    "gw = ExternalParameter(\"g_w\", 0.6535, positive=True)\n",
    "g1 = ExternalParameter(\"g_1\", 0.3580, positive=True)\n",
    "SU2L, U1Y = SU2(\"SU2L\", coupling=gw), U1(\"U1Y\", coupling=g1)\n",
    "s3 = S3()\n",
    "\n",
    "v1 = ExternalParameter(\"v_1\", 200.0, positive=True, unit_dim=1)\n",
    "v2 = ExternalParameter(\"v_2\", 115.0, positive=True, unit_dim=1)\n",
    "vS = ExternalParameter(\"v_S\", 80.0, positive=True, unit_dim=1)\n",
    "lams = {k: ExternalParameter(f\"lambda_{k}\", 0.05 * k) for k in range(1, 9)}\n",
    "mu0sq = InternalParameter(\"mu0sq\", unit_dim=2)\n",
    "mu1sq = InternalParameter(\"mu1sq\", unit_dim=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5",
   "metadata": {},
   "source": [
    "$v_1,v_2,v_S=200,115,80$ GeV aren't arbitrary: they're chosen so that\n",
    "$\\sqrt{v_1^2+v_2^2+v_S^2}\\approx246$ GeV, the single familiar SM Higgs VEV\n",
    "— [GomezBock21]'s Eq. (8) constraint. Three doublets, one combined scale."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6",
   "metadata": {},
   "source": [
    "Before wiring up real fields, let's see the $2\\otimes2$\n",
    "Clebsch–Gordan decomposition on abstract symbols — `S3.doublet_product`\n",
    "returns the three channels the tensor product of two doublets splits\n",
    "into."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'1': a1*b1 + a2*b2, '1p': a1*b2 - a2*b1, '2': (a1*b1 - a2*b2, -a1*b2 - a2*b1)}"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1, a2, b1, b2 = sp.symbols(\"a1 a2 b1 b2\")\n",
    "cg = s3.doublet_product((a1, a2), (b1, b2))\n",
    "cg"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8",
   "metadata": {},
   "source": [
    "So $2\\otimes2 = 1\\oplus1'\\oplus2$: a singlet ($a_1b_1+a_2b_2$), a\n",
    "pseudo-singlet ($a_1b_2-a_2b_1$), and — since a doublet's tensor square\n",
    "must contain another doublet — a second copy of the **2** itself. We'll\n",
    "reuse these exact three combinations below, applied not to raw field\n",
    "components but to the gauge-invariant bilinears $H_i^\\dagger H_j$.\n",
    "\n",
    "## 2. Building the $S_3$-covariant potential from Clebsch–Gordan invariants\n",
    "\n",
    "$H_1, H_2, H_S$ are three ordinary $SU(2)_L\\times U(1)_Y$ Higgs doublets.\n",
    "`s3.assign(\"2\", H1, H2)` puts $(H_1,H_2)$ in the doublet irrep;\n",
    "`s3.assign(\"1\", HS)` puts $H_S$ in the trivial singlet."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle H_{S} = (H_{Sp}, H_{S0})$"
      ],
      "text/plain": [
       "Scalar('H_S')"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def doublet(name):\n",
    "    return Scalar(name, reps={SU2L: 2, U1Y: sp.Rational(1, 2)},\n",
    "                  component_names=[f\"{name}p\", f\"{name}0\"])\n",
    "\n",
    "H1, H2, HS = doublet(\"H_1\"), doublet(\"H_2\"), doublet(\"H_S\")\n",
    "s3.assign(\"2\", H1, H2)\n",
    "s3.assign(\"1\", HS)\n",
    "\n",
    "H1.expand_vev({H1.components[1]: v1})\n",
    "H2.expand_vev({H2.components[1]: v2})\n",
    "HS.expand_vev({HS.components[1]: vS})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10",
   "metadata": {},
   "source": [
    "`bra(a, b)` builds the $SU(2)_L\\times U(1)_Y$-invariant bilinear\n",
    "$a^\\dagger b$. The doublet-index CG formulas from above apply directly to\n",
    "these bilinears, with $(a_1,a_2)=(H_1^\\dagger,H_2^\\dagger)$ on the bra side\n",
    "and $(b_1,b_2)=(H_1,H_2)$ on the ket side: $a_1b_1\\to H_1^\\dagger H_1$,\n",
    "$a_1b_2\\to H_1^\\dagger H_2$, and so on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "11",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle H_{10} \\overline{H_{10}} + H_{1p} \\overline{H_{1p}} + H_{20} \\overline{H_{20}} + H_{2p} \\overline{H_{2p}}$"
      ],
      "text/plain": [
       "    ___       ___       ___       ___\n",
       "H₁₀⋅H₁₀ + H₁ₚ⋅H₁ₚ + H₂₀⋅H₂₀ + H₂ₚ⋅H₂ₚ"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - H_{10} \\overline{H_{20}} - H_{1p} \\overline{H_{2p}} + H_{20} \\overline{H_{10}} + H_{2p} \\overline{H_{1p}}$"
      ],
      "text/plain": [
       "      ___       ___       ___       ___\n",
       "- H₁₀⋅H₂₀ - H₁ₚ⋅H₂ₚ + H₂₀⋅H₁₀ + H₂ₚ⋅H₁ₚ"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( H_{10} \\overline{H_{10}} + H_{1p} \\overline{H_{1p}} - H_{20} \\overline{H_{20}} - H_{2p} \\overline{H_{2p}}, \\  - H_{10} \\overline{H_{20}} - H_{1p} \\overline{H_{2p}} - H_{20} \\overline{H_{10}} - H_{2p} \\overline{H_{1p}}\\right)$"
      ],
      "text/plain": [
       "⎛    ___       ___       ___       ___        ___       ___       ___       ___⎞\n",
       "⎝H₁₀⋅H₁₀ + H₁ₚ⋅H₁ₚ - H₂₀⋅H₂₀ - H₂ₚ⋅H₂ₚ, - H₁₀⋅H₂₀ - H₁ₚ⋅H₂ₚ - H₂₀⋅H₁₀ - H₂ₚ⋅H₁ₚ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def bra(a, b):\n",
    "    return (dag(a) * b.mat)[0]\n",
    "\n",
    "x11, x22 = bra(H1, H1), bra(H2, H2)\n",
    "x12, x21 = bra(H1, H2), bra(H2, H1)\n",
    "s1, s2 = bra(HS, H1), bra(HS, H2)\n",
    "sss = bra(HS, HS)\n",
    "\n",
    "inv1 = x11 + x22          # the '1' channel:  a1*b1 + a2*b2\n",
    "inv1p = x12 - x21         # the '1\\'' channel: a1*b2 - a2*b1\n",
    "d2 = (x11 - x22, -(x12 + x21))   # the '2' channel\n",
    "\n",
    "display(inv1)\n",
    "display(inv1p)\n",
    "display(d2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12",
   "metadata": {},
   "source": [
    "$H_S$ enters the potential through its own doublet of bilinears,\n",
    "$(s_1,s_2)=(H_S^\\dagger H_1,\\,H_S^\\dagger H_2)$, which transforms exactly\n",
    "like $(H_1,H_2)$ under $S_3$. Contracting this doublet with itself, or\n",
    "with the $(H_1,H_2)$ doublet channel `d2`, gives two more singlet\n",
    "invariants — both need an explicit $+\\text{h.c.}$ since neither\n",
    "combination is manifestly real on its own."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vFqO6luSCKMgnV7KgIExuufKWNlpkrOBowdpbxpLtlrAEnn/DwbNw0rCnDZIx0KCyKGKJM0tYiqR1KrRkryUsnegtqrFkryUsRdI6Flqy2QoW6Vwh7g4GWc+xtQZYbIC3C0sBFCe3IyYsKzjIydHJku2WsIR+4O1CvsqhddVQOgZCO9q9Jc4sYdHyKalnyV5LWCQcamUt2WsJi5ZPaT1LNlvBIp0rpJyfblCDE9GfLTUxabnoE7If4Zg/d08l6vtvqqBXnhUcveyR6LFkuyUsGw6DP3+EfA6iWmoeAzVFtXJLnFnCUuOtR7kley1h6cFtTYcley1hqfHWq9ySzZawgF/pXCHuEgZZT7D93lgzvOLhcxB1Nnkhj6944OsdVs9jIZ+T2EvfBic+ptVrHSDz7X326Q/s/4bzF/48tVPhiBVBPwNG6mF7TLfYiPMXlC22CXFRz+iksh12VPtAAVyFJW6ntR/iOg3H7Esm+nZLkoyBFn0lGTVnvg/5QD/fBXb2y17kScYQMaqweBw9x3OJr55lKnsJwNuc5V4BUoWlhXsFliOqDLMXnBzi9ySp5gcKLEdwzzaG8a8wQIWF7dT4V2CRzhXiJkQrWdDuVrJg6BKEbFoM5G1vJ36LOstf9eCYr4z4FZt7xQPO+cuuP7B3b+rG/hNsP2ML+jbNqHE4PdDLQJCrcUsgh2MuX/6K/fKfajiW4gr6aZ/jymW8+3CTPvQuXLwrcq/ECM+zRdlnh0P64KyVtgwtFqcdPDT1QxuUlVT4drLKLJwwAM3WAU6Wv8bGfWtKfhFBZRVnwED/DD51hgPlGl9VYQGO3uPZcdqZZ6dz86Gyt8b9po3WUxUWKC9y39p4Sm4w/0PsPdLva36gxOK6YjD3bGMI/yk/ashTYanx39BuSiR73d8Ko33VvH4DRbfY3OS/VZo454V+G0DFYiSP78+izjh9hbwfsYW6/ObBvKfYGLDx/B+hAuWwMchiIJRaZdPiCE28gt4Pwwn3OCfun+I8HEtxuerQkwqi2EYIKlR/++KxaW2v9cHG9KZTLZagvLUfgnzr/okXbB1AxTGAfmP52cpRK5iNnIozYOA4oY/Sh1JJ46sqLGi85ksaLKfOPKc4UtnbwH2qrVqeCksD97V2s+WD+R9lr8bXVFga/ECDxfXHYO7Zhspm1KuNdYdf+KHC0sC/EIYTf+IrVecKtK+a1xlkNSU04KJPCK9uBYbKKCdxTCGQuj+7/yS45CsiUI+3Dx9j2wZTnNioc5W/EwfUucSHouk8/BuWOMUrWyJcsZJRxzttz/YB8ZIP7BhIcIUxJK4kfoMyTvCrtBNL0FXtBwpKsaEK/YmJPmQmdeLszB7oFfvqTixZX9JgOTNoQMZOe4uIoPvIsZPlniClWIqGdSrcyX3WXug92u+zjGiwZJV1LhjFP2FC95G+X2RGisUrGz5XMMhiEBMaKhkRbt2lgijWywZhMH4bzHDg8BeI/GYegrNt24wsQ5QZl6lxREoYTH2Htrn8R3u4yvY9tjigy3GSwxWpH3aotp32bVDFfRBs5Y8ZGET/nVxgY2DMvNSb+9VYoC+kaj8AgwZb8JvqtxMPpHUMBNzafQ/OUm0HjrZlJV9VY6n4knQ8bzGPOlfbWwKk9E81lgr3wQ8k47hkXq+yUfZqfE2NpUJG4H4rVhqDW9lR52qbB/ibGkuJHOU4pErpXFGCkSx7hNw32J4lS9eZ7jYFjHm7zl7OHHkozwVhThDlHBjUVbv9QuNTjrsbBzDwdh1vF/IZKE6wDLh4YWpJOVwtdffK7LadABJ9QJv4i1D2zU8oD8HmLc75bTGVdmNBOy39oMFGzPybKOJvSa1joEVXSWY3ZyXlibKSr3bBkvClBAyXVcKSq9Mzv4u9CUAa/+yCJcG9BkvCpO5Zo+zNAS35WhcsuYYT+SUsCfEhWV1s7uRvXbAkWNL6vnSuSDRdzuJKFoMm906h7aSEcwY5r1iOzU22yGMw8gZ791wR9lwNYplbycI5V0L4PJUrx/GSkEc53rf+GMdhAgz7Rc4fsE032UOWx11wUBfb9u3zGSwGFbSJuONUxRULjzomXujuYjsxQt9ZHyAv8MxAOebhKc4XHnpioS5s1X5oxUbbosQAvhjsR7I8zI6BjZz4tCdnhcaXPtrI0HdCwHzqjQX6znwJ7TVh2eAcctrb3hRItHH42CGOFPetWFJ29M4DFvre0OsW9Df5Wm8sGa6asGTqds/ubTP0nY115F3E91NktWJJ1JXOFQkV5awbgOND5nSQZ9hWE5PPL/7iDTK87VRNkGMnvcA+rHjx/ITz8OJSnm9Xydw5ZIhvNw7oYSD4GNv2AXc6y9b2Ki7UGZ562U6g0JXtA28IfWB5Lg3HX2DjCpdLvbBAT3M/hLaxL2KL5HhIWQb/TQl4smOgSUFBqBdnhSZOaKPJV3tiga6sL/l2suO5ZEvPsp72NuAq+mdPLCXuPc4ilgZbdoscZa9vp+hrPbHkiEEbTWMwV793fk+boSs71j3uor/1xNLAUxFLoj7lm+eKRP1qFleymDiRMvhZBRos6JF8J9EQBllcHWFicBYm9G9wzIk3BFU8jm9b4bRLYqDGZcVtkMXVtVSw2BsXg0Vuh6daH/hyBqAOnz9nX9Vu62psEfWDBBtkaQNxL8FhI8ChY6ARwx6x3r6axeL7w8J4zmI8skDin3tx1bg/EsteW1rq1+yFjsP8vgGvJSwNcOsiNf4t+ZsUC+S1c8WWOM6Zbt7cFvD8g7u7uxMa46T0GvvVaw1SFTR50PsX6tGgVUJ+/E4qBjpMfDkoH7RmQJYFTkFpgj5ODNy+9nWDfj6TtdxWifUifziuuL1Rx7Cj2Aco569EyMsPHsOQPqButCXqBwk2L/sp9sWVT2/jsoP80DGwNKQ8AD5+m2T/8AsIsf4H2/JuORyfIHOIr6Kdoi8diYVtjU417lF+5Ngpcn8kltG8Uz/sKdrrZY7yezNj8AjuW/g/0t/QVpF/KRYvL54rpNy7IMuTyeeSfkDD21Ueqc4pf4UMoN8Z+PAZqbC6aMYKCTbI8g8/OXCSQXPJKNSZY6BE0CxLMiDxz6SCjpmWsHQ0a6oyyoAlf5Nigbx6rpB0x6NI+Escv4zO5+HDYoArJG+MmtyEDYOGKwrZVckG2+YYaCBpipwx0OSfZ7XGZFjCMsbCqdUSA5b8rRlLh7miuQ+WIAuN8tYZXzwZll6blUzB62bA9zmXYrkCdHZb95LWtWKDHPHzr2x4G02VUHeOARVzD7dSq38ewZAlLEfYO9u4LAOW/E2CBbK75woJ88vtwlAJALgawF9KDHkIPrQz95OBngzAX3m7s8tzfHMM9OyZqWsyMBmYDNhhoOdc0WLVWZDVUmnKTAYmA5OBycBkYDIwGZgMlBlYbheWxWbpZGAyMBmYDEwGJgOTgcmAhIEZZEnYmrKTgcnAZGAyMBmYDEwGGhm42cpZfx4F+O62mC91DizLe74uheFS7VrqB88B/+A7vP9sFy3QY/K5REucA8uD8n1L3NO5HxL/lrh/SLyHi6gl/j2mbtf6YOPI/eqZLJD5GRrjL8y6TFYjgU/dk4GYAfgsfzHC1ze4v22KyyTHcwxI2Jqyk4HJwGTgWAZ6XeuPQr0EWQDOn+7zre8j/kblKHtmOw+YAfguV6D4x9Oq1zjMMfCAnWeaPhmYDFwNA3uv9UcaGj+Txbdd82fwM00GrpIBDLzvAfxrHyxpbJhjQMParDMZmAxMBg5koMO1/jC07pksAOatlmfYF2+1oPxfkHuOjbcVmfguLf4Vi/ufOOy5kkAdfPMqV8ZcOfZfoowveuyeLGLqbmSk0Kq9hnDxi8IrbNL/Lry6MWCI88hDxx1astcSlnGMv9NsyV5LWN4xNO7Imr2G8Kiu9eN6Kq3Z3S4EaQTb/L91kOfD52+xT95aRD7/1PNP7Pknw4cki5hGGm7V3kvjQvsM7ul/H+K4ObCH7NWOgUtzPtLPU7ot2WsJS4qr3nmW7LWEpTfPKX3W7L00HrSvutanuB2ZF24XcgXq55aGYNhTL5d8IzzKuSJA45PlLW1IZSxiStkAnJ9gI9e7klV7LeACBgZWb7H9U0jyVY4BC5zXeAbGLn7PdizZawlLrQ96lFuy1xKWHtzWdFiz1wIeYNBe62t0i8qBo3h9exSR9b9GzV94uVxQxluFTLny+9K+nxYxpSxk8Mltb7JqrxVc9OXm24VXPgascF7y6V5+zzYs2WsJS4n/XmWW7LWEpRe/JT3W7LWCR3StLxG8o6x4fXsExQyK+F+Ft42NuCAK8rmVqjC55cobmxGJWcQkMkAobNVeK7h+A5/PBJwS97WOASucC+jeJWrJXktYdpHaWNmSvZawNNK3S8yavVbwSK/1uzpBU5lB1nNsrQEW2+DtwlIAxclNMmFR595kEdNem0r1rdprBRdvF/JVDvyG0ZKueQxY4byF5x4yluy1hKUHtzUdluy1hKXGW49ya/ZawSO91vfoC5GOG0hzIvqzpRYmLRe9QvYjHPPn7qlEff9NFYzIs4hphJ1Bp1V7jeEK/vwReOMgrKWrHAPGOK9xvLvckr2WsOwmtkGBJXstYWmgbreINXuN4ZFe63f3h1QBg6wn2H5vrBhe8fA5iD6bvJDHVzzw9Q7Z57F8B1HPH77NW+w5yf0SdGL/rS+jzN9w/sKfp3ZqTNDLdvmANN+tdPZLSSGOFLYReWp7AxjYxWC52AdBVrBX46r1gwBDEKVPMdG3W9IhY6Bmp8LfhnGuwNLC816ZYfYqgFnCooAvrqKy1/v8S98av/QwrV7po/A1S1juLRr7OcxeJWxLeKTXeqXJ+mqilSw041ayMCjOAiwPIZCfvJ2IegzAuAq2BE045rLjr9i7/0LDnr/y+gN799Zu7Pnk/s/Ygm7f1LJTYYI+tuvqYs9ga5VQLsWx1Eddvg4g6F7yceAmfZSn/raIr8QIz7PFdbbHKnuDErRR7YMgK9yrcAFPsR+EGIJ4+HYTzmt79n9rnSF2ggeNv1nCcoINI/2efTjE3ppzZMotYXEQwT/9+DW2s+tZxgZmJ78wJ+RV9kLPt8C1XO+8j/yKfPd6H5wf5veDsDiqrpF7B1z3McQXdFCar9tK9e+qed91tr/LdUfFef0GIrfYnNCmYuqUE2IygPLCBMBggTpT6RXKPowLcE75n6I8rmL9I5yjjCtcDLIYnKVW3FSYoIuBIttm0JFKUhyLDuhcLipLJg58W7RD9bcvXpfK3ghHSx9E4s2HKlzgotYPzQAiwSf+uDVwGj4GGuzU+NsozjVYTrBxpN+zS0fZG7lO86ElLA40+Kcfn63IN1tUFlTZC5VfAdeP2MK8Qd9i3lNsHPsaX7OExbEGW66Re4dd8TGKfwWUJXZpvdZr2nB10Meq6xuDrKaEBkIEl7wViHISzxQG0/3Z+pMPI3OA8e9P4uRWtpDP5eTH2LbBFB2Y+lf5kO+BCWrXSYpjXXvcWSd7i30Q0KMtfsPkBZvfOkPiauI3KOPFcUmdcC364gMJjqgefYiJftMtjbITekV+T4MsYelGcEHRKHs9l82+7uWHXHc0WAqUdSvayT0npuTrgaD3aL/viqUbwQVFo7hnk9At8ntfZ4/vZ/nX4kG9Idd64umVHkERA5cAtKQ33K7LBVFF8r1iBlPfoXPvsHF1ip18wj4ETzkcjFKfUHaTemDaqHSnUhwpHSPyethb6wP2R7A//JcfVxMZGH+DLfWDhx64zvhS4Ag6gq+0frs5cgwEjPE+8B3n8Tjn9ywbwjn0arAQz+g0xF6lj1nCMpp36lfby+sGttsIJCda/vqcX9Q0vmYJS2TWsMMh9ir9nkZawyO91g/rqJxiBllvsD3LCUT57raaHxxR9nLoyEd5Lgg7oYy3yXi7kM8ecWJjwJWatFG0SiQyNSB3Y1q1Uj/J4ajX7COx297GPqCd/IXoU2w/oU4Ignmx5LfPbdqNa6vQn0txBDXEyL+Jii/uoSy1P2wMpBov5JX8bftNKQUAABCCSURBVBTnOTglLLk6PfNH2avxMUtYenKc09XFXozHp2iAumq3NEu+ZglLjq+e+aPs1fg97bKGR3qt79k3TbpuIOW+UWAA8DbSalJiHspfYePeTa7IY1D0BvvwYDofdmWZW8lCPm8ncpVq9dwRzp1+7NkGn8Hi5E1d8e3HVfsoC4ntu4meenDcBVNQnthXcSTqDMnqaS91YQvBR64PTpAJXDNwjvuHF0nHDXXheGg/tODIkM6LeDbYT9Q5ZAwk2g1ZTf52BOcA1IQlAB+5P8LeVh+zhGUk50F3b3uhj3MEn7/6GMfBx8I+NBv2j3EQvtidLGEJAEfuj7AXbVSv8cFGa3gCLr+XXus31cef3oBA3gqisz/DtpqYfH7xF2+Q4fJvMUGGARgHTvyAO+uwo5c2IRdeYsoByYkvTu68F6ZY8fa4Bce2zqjzXvZCT1MfbOygTyy/BMXxF9i4wnXqhYu6GlIWR6Yu5Rn8NyXYMnwMlICg/arfs/4RnLdiKdnTq+wIeyOsRR+zhCXCPOywp73Qxev5C+zD3Q6en3B+uN/vxTKM8EgxMHI+3j3vUmXJXt9k0e+9DlN4PO6wE13rQ6Uj91zJYuLEyQGwBDzM7JjoMFye3AZZ/GazDdL43A8DghBU8Ti+ZYXTQ9IIHHRWbpdIkj4Ig/MxgDq8frByJau21N/VNt9uMw7IU5Y4XTAoADN6DNSgjPC3Wpu58t5Y6EPOj3INXjJf6mMjsVrC0sNObw+/8DDI4rhk+hpb+PLW29dcA6kPS1hS+Hrn1ez15c3X1r34euOBPu21fq8p2/rF69sHd3d3J4Cl87/GfvV6ha0m7Tn0cpBx4+BiChdcPpO1LAvfF7nom8EXE19GyvepcICGOszfnaCP36aIh0Ec7f8PtuX9XDg+QWY4DrZzRIIt0j7gjxLIzw8e30X6AbhFOLz8p9gXvwluOYf86DFgxt9gqxks234YcV6zF+UiH9uD0RKWPXa01oW9f0GWk+EqId+9F5GZOD7kOmsJy4qMQSc1e1F+mN/7fi76ghSPlxdf6wfRnVXrgixPAJ+P+gHAt6tN2cqz4P1lAH7AoIzPb4VvnBcxVooD8vzDUA68s+C9ZgDqzDFQI+k9LJf62EgKLGEZaefUfXkGrPmaFA/k1df6I9l/FDX2JY5fRufz8GEzwBW+NwYoaMaBQcdvZsnV0UY75hhoJOo9E2v2sQPstoTlAHNnExdkwJqvNePpcK0/jPYlyAJo3o7jiybD0u1hIGZDthjwPsBbSlwROlvqPwqtBAdkiZd/EcLbvqqEunMMqJi73koSHxttpSUso22d+i/LgDVfk+CB7O5r/ZHsL7cLQ6MwgKsB/NXHqIfgQ1NzPxnoxgD8lbc3uzy7N8dAt26ZiiYDk4HJQFcGel7ruwLLKDsLsjJyM3syMBmYDEwGJgOTgcnAZEDAwHK7UFBnik4GJgOTgcnAZGAyMBmYDFQYmEFWhaBZPBmYDEwGJgOTgcnAZEDDwM22kvXnUYDvbov5UufAsrzr5VIYLtWupX7wHPCPaMN72HbRAj3v5XOJlvoMWB7U2LHEPQfHQ+LfEvcPifdwEbbEv8fUba4INpb2q2eyQMZnEOYvyrpMVqWGZ9lkoCcD8Fn+4oSvb3B/3aHVPceAlrlZbzIwGZgM2Geg11zRaukSZKFh/lSfb30/9G9TWoFOuclAjQH4Lleg+CfYqtc4zDFQY3iWTwYmA5OB62dg71whYSAOsn5GxR/R+Pc5BSj7F8qeY+OKFxNf88C3gru/MPHAuZLAl4oxaHPl2H+JMr6DqHuyiKm7kZFCq/ZawQUcfAvwx9iL/Q11qmMg6grVoRWeCN4SFhWZwkqW7LWERUijStySvZawqMhUVLJksxUswKGeKyRd4IIsNMZbLb9i3/TfhZDjc1FvsU+ueiGf/1H0J/b8v7tDkkVMIw23au+lcaF990UAe+l/F4rGwN6+vTRPMX5LWGJco44t2WsJyyi+Y72W7LWEJeZo5LElmy+NBe2r5gpp/4RfF/L/6bIrWLFSAAv/pJ58WSnKOVmFVay46rBji5hSxgLnJ9h4S2tXsmqvEVz048+AhT4oSc1jQKI0JWuEJwfNEpYUV73zLNlrCUtvnlP6LNlrCUuKqxF5lmw2gkU7V4i6JwRZnPh5q6QlfeGFcvK8VciUK78v7ftpEVPKQk780sk/pceqvRfHhcF7C8LeYvtnirhCnmQMFNQ0FV2cpwilJSwRrGGHluy1hGUY4ZFiS/ZawhJRNPTQks0Xx7JjrhB10qMoovxfY00XRKFeciULOsJtmlx5YzMiMYuYRAYIha3aawUXfTn4YZVaxRio6qwIWOGJMC1hqdDWpdiSvZawdCG3osSSvZawVGjrVmzJZitYRHOFpie4kkVj+V+FXAFoSbxdWAqgnqFcoq+lzZqMRUw1zHvKrdprBRcfaKQftibpGGjVm5OzwhPxWcKS46tnviV7LWHpyXFOlyV7LWHJ8dU735LNVrBI5wpxn9ygxnNsTQEWAjEXfUL+Ixz/mGmNt8P+mynrnm0RU3cjI4VW7TWGi7cL+SoHbi2+3TwGoq5QHVriyRIWFZnCSpbstYRFSKNK3JK9lrCoyFRUsmSzJSygUjpXiNlnkMWg6M/GmuFFj5+DKIJbJeTxaX2+3iH7PJYnmHr+8JU5CRLDL0En9t/6Msr8Decv/Hlqp8IEnWzzpVfIh/WZVq+aEOK41zD+U2VvDAt2MVgu9kEs33iswtXSD43tx2LBn9mvZ34aC/pjyRhIVBdlqXhiC54rPmv2NY7PftmLPMm4oUoVFo/jQY2dGvckU5iGcS/EcZT4MHuP8nsSVfMDBZar519hgMoXWvhXYJHOFeImGGQ9wfZ7Y023kgVHyk1cgbzk7UTUYwDGVbAlaMIxlw35+gj3NxvY8wHkP7B3L5TEnr/I+xlb0L2FqsX0LXQub7bH8XfEgc29dgLnUhwLLq/L4Voy7w/I9QnlS7tROV+J0fIckdZe1xTaqPZBhElyqMVV7AcJgEiWgTuT4/v+sPhZHAPgjEHYa2zct6bkFxFUVvEEDBwnwafOcKBc468qLMBR7DMllhPq9eQ51U8qe4GryH2qoYY8FZYa9w3tZkUG8z/EXmA+zO9rfqDE4vpjMPdsYwj/WWcqF6iw1PgvN5ktvfUlrXNFVlGu4AYFvLCFaC4nF/J5sUkGUF6A5DFYCMBDvbB/hbLVu7hwTvmfggD2/Db+j3COMq5wMchicJYKBrWYvoI+vnw12MN2mfcUG4NIKY4A+YT6qSCK+SHAUb2R3DegtTfga+mDICvZa3HV+kGCIci2+nOQL44B9Bv9+WzlKFQW7lU8AQN9kmOFPpRKGn9VYUHjtT7TYDl15jnFkcreBu5TbdXyVFgauK+1my0fzP8oezW+psLS4AcaLK4/BnPPNlQ2o15trDv8wg8Vlgb+hTCcuHSuELfBIIsTyJNaTRjook/IJW8FopzEMYWg5f5s/clnZNhp23dyuZUt5PP2Die8bTBFjNS/yof8HkwMhJK/qJTigJ5D0k57A8ZiHwQhtMVviAwsuLoXElcTv0EZJ/wl7cSV7Qc2IMGxAHrnz60DqGkMRPpVhzt5yrYJvaJxQ0U7sWT7TIMla1jHgp32FpFAd/NYoaKdWLLce90iLEXDOhWOshd6j/b7LCMaLFllnQtG8U+Y0C3yt51YisxIsXhlIfZpnSuKGFKFDLJaU7hdlwuiigGPb4TBFP/El7fmqMf9jQ/OQ/DEACuVSEAgIy5XY0Kb20CPFy/+KpKrBSFgjNvicQ7HVm7UudreCFCtD06wP/QDf9zAoPrvyCM3DEqZt32TvxoXdJb6QYoD0FwK9W5DhpG9mqcK/mDvVqzkr2oslT57n8fOlt8TuAjct44V6hjFvQbLmU0DMkbZq/E1NZYKL4H7rVhpDG5lR52rba6M9WDzIb5fIkc5Dqky2HBb0r+n7BEqM8AJDZV0udsUMOZtRsh1JMpzQdgJZbxNxtuFfPaI7TLgYgfVEgOsFMbdmNgwMHCwUlftllAOB9UckXbb29gHtJO/ECUvP6FOCILpiPz2uE27cVFhoh+kOAIu1mPiBa4ltY6BFl0lmS48lRrYlJGH1LihWBcsiT7bQFhOS1gWoYEHXexN4NP4aBcsCe41WBImdc8aZW8OaMnXumDJNZzIL2FJiA/J6mJzJ3/rgiXBktb3WY+pda64lxZ8ciXrDbbwS6FVVZDKC/QrbNy7yRV5DIreYB8eTOeqFMvcShbyufLBZ6hWzx3h3P2cHntO1HwGi5N3iIBx6hLLUontu4meenDcBRMbgj5i5730+E+FqzhY94jU017qwnbLDdhzfXBCeeCagTP7M6SnOHDcQGZ4P7TgCMA2e/ZpsHNTlDzNjoGktCCzN0+Zppv8tTcW6HswYyfD+wkcVMcK60LO8njJmafOP8JegLuI32dIacKSqds9+wj+0cZFfD9FViuWRF3pXJFQUc5ikMWVKU6+bgKOxZFHxyn+4g0yvM1WTJBhAMaLDCf2OLGTlpUvyIWXmNLw7YqZO++FiSCgi+28wD6swvGc+VUclDsiAcuhfbCx6RnOl1+C4vgLbFzhIkddcHld2X5gOVIWx33x2SdXJBe/Ois9z8iOgXNRWU5PnnIto40mf+2JBbqyfebbYXlyDOfs6J3f094GbEUf7YmlxL3HWcTSYMtukaPs9e0Ufa0nlhwxaKNpDObq987vaTN0Zce6x130t55YGngqYknUl84VCRXlrBsQwF/vccIkOMnEVNa8LmWgxmW5bZDFFaRtkPYN8hiUhaCKx/EtK5zuT95xuArHIIsrNEzEEoKKETjIM7dLJEkfnDw/jwHU4fXn5Kl2S1Vkm9eb7QclDvoydTYltHHEGGjCskNohL8m4dT6DJUOw5IEeHCm0kdVKGvcH4lFZYCwUs1eqLPka5awCJlOi9f4t+RvSiyiuSLNUjn3g7u7uxPAcULi7ZUQYJRrCUu9frbxta8aAg0+k+WWHGOVyGPwxcSXkfIhawZCoQ7zdyfo+wtKGESsEvLd+7qYiePhOFaNDzyBLeRf0gf81Qj76wcP6yL9ANwiHJBnn7JvP8Rxs894foaNAc+hegd8/DbJ/uCXDga7vB2/vE8OxyfIHOKvaOehjZ0i9+BD5KPsK22qcX8kFq0Nkno1e6kLMkf5fdEPjsQi4XCPbI1/lB/p+0X+pVggr5orpHyGIIsX7ddodPUOK6myKf/+MABfGBp4tzIlxQF5DvpPsS/e5t62D/k5BrakzPMmBqQ+2qRUKWQJi9KEWe2KGLDkb1IskFfNFdLuecQKaIy35njL5DOpgin/3jLAFRM+EH7pJMXB1VjxiuwcA5fu5qtuX+qjI421hGWknVO3DQYs+ZsUi2qukNLugixf6UvsX0oVTPn3jwEEHFx+59IsV4TObqkeZbEUB+T5zSR5C7oR8xwDjURNsXsGpD46kjdLWEbaOXXbYMCSv0mxQH7vXNHcCe52YZBGw1zJeo69eCUg6Jj7ycAlGIDPMihkgOV+KarFMMeAlrlZbzIwGZgM2Geg11zRamm8knVC4/z132/Yc9ltpsnANTHALwai57BSxs0xkGJl5k0GJgOTgfeGgS5zRSsb/wefNyJr2x27kAAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left(H_{10} \\overline{H_{S0}} + H_{1p} \\overline{H_{Sp}}\\right) \\left(H_{10} \\overline{H_{10}} + H_{1p} \\overline{H_{1p}} - H_{20} \\overline{H_{20}} - H_{2p} \\overline{H_{2p}}\\right) + \\left(H_{20} \\overline{H_{S0}} + H_{2p} \\overline{H_{Sp}}\\right) \\left(- H_{10} \\overline{H_{20}} - H_{1p} \\overline{H_{2p}} - H_{20} \\overline{H_{10}} - H_{2p} \\overline{H_{1p}}\\right) + \\left(H_{S0} \\overline{H_{10}} + H_{Sp} \\overline{H_{1p}}\\right) \\left(H_{10} \\overline{H_{10}} + H_{1p} \\overline{H_{1p}} - H_{20} \\overline{H_{20}} - H_{2p} \\overline{H_{2p}}\\right) + \\left(H_{S0} \\overline{H_{20}} + H_{Sp} \\overline{H_{2p}}\\right) \\left(- H_{10} \\overline{H_{20}} - H_{1p} \\overline{H_{2p}} - H_{20} \\overline{H_{10}} - H_{2p} \\overline{H_{1p}}\\right)$"
      ],
      "text/plain": [
       "⎛    ____       ____⎞ ⎛    ___       ___       ___       ___⎞   ⎛    ____      ↪\n",
       "⎝H₁₀⋅H_S0 + H₁ₚ⋅H_Sp⎠⋅⎝H₁₀⋅H₁₀ + H₁ₚ⋅H₁ₚ - H₂₀⋅H₂₀ - H₂ₚ⋅H₂ₚ⎠ + ⎝H₂₀⋅H_S0 + H₂ ↪\n",
       "\n",
       "↪   ____⎞ ⎛      ___       ___       ___       ___⎞   ⎛     ___        ___⎞ ⎛  ↪\n",
       "↪ ₚ⋅H_Sp⎠⋅⎝- H₁₀⋅H₂₀ - H₁ₚ⋅H₂ₚ - H₂₀⋅H₁₀ - H₂ₚ⋅H₁ₚ⎠ + ⎝H_S0⋅H₁₀ + H_Sp⋅H₁ₚ⎠⋅⎝H ↪\n",
       "\n",
       "↪    ___       ___       ___       ___⎞   ⎛     ___        ___⎞ ⎛      ___     ↪\n",
       "↪ ₁₀⋅H₁₀ + H₁ₚ⋅H₁ₚ - H₂₀⋅H₂₀ - H₂ₚ⋅H₂ₚ⎠ + ⎝H_S0⋅H₂₀ + H_Sp⋅H₂ₚ⎠⋅⎝- H₁₀⋅H₂₀ - H ↪\n",
       "\n",
       "↪    ___       ___       ___⎞\n",
       "↪ ₁ₚ⋅H₂ₚ - H₂₀⋅H₁₀ - H₂ₚ⋅H₁ₚ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left(H_{10} \\overline{H_{S0}} + H_{1p} \\overline{H_{Sp}}\\right)^{2} + \\left(H_{20} \\overline{H_{S0}} + H_{2p} \\overline{H_{Sp}}\\right)^{2} + \\left(H_{S0} \\overline{H_{10}} + H_{Sp} \\overline{H_{1p}}\\right)^{2} + \\left(H_{S0} \\overline{H_{20}} + H_{Sp} \\overline{H_{2p}}\\right)^{2}$"
      ],
      "text/plain": [
       "                     2                        2                        2       ↪\n",
       "⎛    ____       ____⎞    ⎛    ____       ____⎞    ⎛     ___        ___⎞    ⎛   ↪\n",
       "⎝H₁₀⋅H_S0 + H₁ₚ⋅H_Sp⎠  + ⎝H₂₀⋅H_S0 + H₂ₚ⋅H_Sp⎠  + ⎝H_S0⋅H₁₀ + H_Sp⋅H₁ₚ⎠  + ⎝H_ ↪\n",
       "\n",
       "↪                   2\n",
       "↪    ___        ___⎞ \n",
       "↪ S0⋅H₂₀ + H_Sp⋅H₂ₚ⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lam4_term = s1 * d2[0] + s2 * d2[1]\n",
    "lam4_term += sp.conjugate(lam4_term)     # (HS†H)_2 ⊗ (H†H)_2 → 1, + h.c.\n",
    "\n",
    "lam7_term = s1**2 + s2**2\n",
    "lam7_term += sp.conjugate(lam7_term)     # (HS†H)_2 ⊗ (HS†H)_2 → 1, + h.c.\n",
    "\n",
    "display(lam4_term)\n",
    "display(lam7_term)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14",
   "metadata": {},
   "source": [
    "All 8 quartic couplings are now S₃-invariant combinations built from\n",
    "`inv1`, `inv1p`, `d2`, `lam4_term`, `lam7_term`, `sss`:\n",
    "\n",
    "| Coupling | CG channel | Term | Role |\n",
    "|---|---|---|---|\n",
    "| $\\lambda_1$ | $1\\otimes1$ | $(H_1^\\dagger H_1+H_2^\\dagger H_2)^2$ | ordinary doublet-sector quartic |\n",
    "| $\\lambda_2$ | $1'\\otimes1'$ | $(H_1^\\dagger H_2-H_2^\\dagger H_1)^2$ | pseudo-singlet quartic |\n",
    "| $\\lambda_3$ | $2\\otimes2\\to1$ | $d_2\\cdot d_2$ | doublet-channel self-contraction |\n",
    "| $\\lambda_4$ | $(H_S^\\dagger H)_2\\otimes(H^\\dagger H)_2\\to1$, + h.c. | $s\\cdot d_2$ + h.c. | singlet–doublet mixed quartic |\n",
    "| $\\lambda_5$ | $1\\otimes1$ | $(H_S^\\dagger H_S)(H_1^\\dagger H_1+H_2^\\dagger H_2)$ | portal-like singlet×doublet-norm coupling |\n",
    "| $\\lambda_6$ | $2\\otimes\\bar2\\to1$ | $s_1(H_1^\\dagger H_S)+s_2(H_2^\\dagger H_S)$ | $\\|s_1\\|^2+\\|s_2\\|^2$-type cross term |\n",
    "| $\\lambda_7$ | $(H_S^\\dagger H)_2\\otimes(H_S^\\dagger H)_2\\to1$, + h.c. | $s_1^2+s_2^2$ + h.c. | CP-sensitive; needs + h.c. to be Hermitian |\n",
    "| $\\lambda_8$ | $1\\otimes1$ | $(H_S^\\dagger H_S)^2$ | pure singlet quartic |\n",
    "\n",
    "plus a mass term for each $S_3$ singlet sector, $\\mu_1^2(H_1^\\dagger\n",
    "H_1+H_2^\\dagger H_2)$ and $\\mu_0^2 H_S^\\dagger H_S$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "15",
   "metadata": {},
   "outputs": [],
   "source": [
    "l = {k: p.s for k, p in lams.items()}\n",
    "V = (mu1sq.s * inv1 + mu0sq.s * sss\n",
    "     + l[1] * inv1**2 + l[2] * inv1p**2\n",
    "     + l[3] * (d2[0]**2 + d2[1]**2)\n",
    "     + l[4] * lam4_term\n",
    "     + l[5] * sss * inv1\n",
    "     + l[6] * (s1 * bra(H1, HS) + s2 * bra(H2, HS))\n",
    "     + l[7] * lam7_term\n",
    "     + l[8] * sss**2)\n",
    "\n",
    "L = Lagrangian().add(-V, sector=\"potential\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16",
   "metadata": {},
   "source": [
    "## 3. Assemble the model and check invariance\n",
    "\n",
    "Everything above was built from gauge-invariant bilinears and $S_3$\n",
    "Clebsch–Gordan combinations by hand — but nothing has actually *verified*\n",
    "gauge or $S_3$ invariance yet. `Model.check_invariance()` checks every\n",
    "term against both `gauge_groups` and `discrete_groups` at once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "17",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "invariant: True\n"
     ]
    }
   ],
   "source": [
    "model = Model(\"3HDM-S3\", gauge_groups=[SU2L, U1Y], discrete_groups=[s3],\n",
    "              fields=[H1, H2, HS],\n",
    "              parameters=[gw, g1, v1, v2, vS, mu0sq, mu1sq,\n",
    "                          *lams.values()],\n",
    "              lagrangian=L)\n",
    "\n",
    "report = model.check_invariance()\n",
    "print(\"invariant:\", report.ok)\n",
    "if not report.ok:\n",
    "    print(report.failures)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18",
   "metadata": {},
   "source": [
    "In plain terms: every coupling written into $V$ is simultaneously\n",
    "*allowed* — there's no hidden tension between the gauge and $S_3$\n",
    "symmetries that would force some $\\lambda_k$ to vanish."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19",
   "metadata": {},
   "source": [
    "What if a term *weren't* $S_3$-invariant? Take, for instance,\n",
    "$(H_S^\\dagger H_1)(H_1^\\dagger H_1) + \\text{h.c.}$ — it singles out $H_1$\n",
    "without pairing it with the matching $H_2$ combination the doublet\n",
    "structure requires. `check_discrete_invariance` catches this directly,\n",
    "without needing a full `Model`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "20",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "forbidden term is S3-invariant? False\n"
     ]
    }
   ],
   "source": [
    "bad = (dag(HS) * H1.mat)[0] * (dag(H1) * H1.mat)[0]\n",
    "bad = bad + sp.conjugate(bad)\n",
    "\n",
    "ok, _ = check_discrete_invariance(bad, s3)\n",
    "print(\"forbidden term is S3-invariant?\", ok)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21",
   "metadata": {},
   "source": [
    "## 4. Tadpoles and the forced $\\sqrt3$ vacuum alignment\n",
    "\n",
    "There are three VEVs ($v_1,v_2,v_S$) but the $S_3$-covariant potential\n",
    "only has **two** independent mass parameters ($\\mu_0^2,\\mu_1^2$) free to\n",
    "absorb tadpole conditions. Solving the $v_2$ and $v_S$ conditions for\n",
    "$\\mu_0^2,\\mu_1^2$ and substituting into the $v_1$ condition therefore\n",
    "leaves a genuine constraint on $v_1,v_2$ alone — not an identity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "22",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "residual v1-tadpole condition:\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{3 \\lambda_{4} v_{S} \\left(3 v_{1}^{2} - v_{2}^{2}\\right)}{2}$"
      ],
      "text/plain": [
       "         ⎛    2     2⎞\n",
       "3⋅λ₄⋅v_S⋅⎝3⋅v₁  - v₂ ⎠\n",
       "──────────────────────\n",
       "          2           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tadpoles = model.tadpoles()\n",
    "t1, t2, tS = tadpoles[v1.s], tadpoles[v2.s], tadpoles[vS.s]\n",
    "\n",
    "sol = sp.solve([sp.Eq(t2, 0), sp.Eq(tS, 0)], [mu0sq.s, mu1sq.s], dict=True)[0]\n",
    "residual = sp.factor(sp.expand(t1.subs(sol)))\n",
    "print(\"residual v1-tadpole condition:\")\n",
    "display(residual)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23",
   "metadata": {},
   "source": [
    "The residual factors as $\\propto v_S(3v_1^2 - v_2^2)$: since\n",
    "$v_S\\neq0$, the only way to satisfy all three tadpole conditions\n",
    "simultaneously is $v_1^2 = v_2^2/3$ — a fixed ratio, not a free parameter.\n",
    "(`docs/manual/ssb.md` frames this as a structural consequence of the CG\n",
    "structure, not a coincidence of the numeric benchmark point; the\n",
    "`declaration.md` chapter notes that `feynlag`'s real-orthogonal $S_3$\n",
    "basis swaps which doublet component the literature's $v_1=\\sqrt3\\,v_2$\n",
    "convention refers to — the ratio *squared* is the basis-independent\n",
    "statement.)\n",
    "\n",
    "This exact relation was independently derived in the literature model this\n",
    "potential is drawn from: **[GomezBock21]** M. Gómez-Bock, M. Mondragón, A.\n",
    "Pérez-Martínez, *\"Scalar and gauge sectors in the 3-Higgs Doublet Model\n",
    "under the S₃-symmetry\"*, Eur. Phys. J. C **81**, 942 (2021),\n",
    "[arXiv:2102.02800](https://arxiv.org/abs/2102.02800),\n",
    "[doi:10.1140/epjc/s10052-021-09731-3](https://doi.org/10.1140/epjc/s10052-021-09731-3)\n",
    "— their Eq. (13) tadpole solution is exactly $v_1^2=3v_2^2$ (their\n",
    "$v_1,v_2$; the basis swap above is what turns this into our\n",
    "$v_1^2=v_2^2/3$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "24",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v1 solutions: [sqrt(3)*v_2/3]\n",
      "  (v1/v2)^2 = 1/3\n"
     ]
    }
   ],
   "source": [
    "v1_solutions = sp.solve(sp.Eq(residual, 0), v1.s)\n",
    "print(\"v1 solutions:\", v1_solutions)\n",
    "for s_v1 in v1_solutions:\n",
    "    ratio_sq = sp.simplify((s_v1 / v2.s) ** 2)\n",
    "    print(\"  (v1/v2)^2 =\", ratio_sq)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25",
   "metadata": {},
   "source": [
    "In plain terms: naively, three VEVs minus three tadpole conditions\n",
    "should still leave one free ratio to fit experimentally — instead the\n",
    "$S_3$ structure removes it. The model has one fewer free continuous\n",
    "parameter than the field content alone would suggest."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26",
   "metadata": {},
   "source": [
    "## 5. CP-even mass matrix on the aligned vacuum\n",
    "\n",
    "Imposing the alignment and re-solving the two mass parameters\n",
    "consistently, the $v_1$ tadpole condition is now automatically satisfied\n",
    "— confirming the alignment is exactly what's needed, not an\n",
    "approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "27",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v1 tadpole after alignment + resolving mu0sq, mu1sq: 0\n"
     ]
    }
   ],
   "source": [
    "align = {v1.s: v2.s / sp.sqrt(3)}\n",
    "\n",
    "sol2 = sp.solve([sp.Eq(tadpoles[v2.s].subs(align), 0),\n",
    "                 sp.Eq(tadpoles[vS.s].subs(align), 0)],\n",
    "                [mu0sq.s, mu1sq.s], dict=True)[0]\n",
    "\n",
    "print(\"v1 tadpole after alignment + resolving mu0sq, mu1sq:\",\n",
    "      sp.simplify(tadpoles[v1.s].subs(align).subs(sol2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28",
   "metadata": {},
   "source": [
    "`Model.mass_matrix` takes the real CP-even fluctuations\n",
    "(`expand_vev`'s auto-generated `{name}0_r` symbols) and returns\n",
    "$\\partial^2V/\\partial\\phi_i\\partial\\phi_j$ evaluated on the vacuum, with\n",
    "any cached tadpole solutions applied automatically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "29",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{2 v_{2} \\left(\\lambda_{1} v_{2} + \\lambda_{3} v_{2} + 3 \\sqrt{3} \\lambda_{4} v_{S}\\right)}{3} & \\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{1} v_{2} + 2 \\sqrt{3} \\lambda_{3} v_{2} - 9 \\lambda_{4} v_{S}\\right)}{3} & \\frac{v_{2} \\left(- 3 \\lambda_{4} v_{2} + \\sqrt{3} \\lambda_{5} v_{S} + \\sqrt{3} \\lambda_{6} v_{S} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{3}\\\\\\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{1} v_{2} + 2 \\sqrt{3} \\lambda_{3} v_{2} - 9 \\lambda_{4} v_{S}\\right)}{3} & 2 v_{2}^{2} \\left(\\lambda_{1} + \\lambda_{3}\\right) & v_{2} \\left(- \\sqrt{3} \\lambda_{4} v_{2} + \\lambda_{5} v_{S} + \\lambda_{6} v_{S} + 2 \\lambda_{7} v_{S}\\right)\\\\\\frac{v_{2} \\left(- 3 \\lambda_{4} v_{2} + \\sqrt{3} \\lambda_{5} v_{S} + \\sqrt{3} \\lambda_{6} v_{S} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{3} & v_{2} \\left(- \\sqrt{3} \\lambda_{4} v_{2} + \\lambda_{5} v_{S} + \\lambda_{6} v_{S} + 2 \\lambda_{7} v_{S}\\right) & \\frac{2 \\left(2 \\sqrt{3} \\lambda_{4} v_{2}^{3} + 9 \\lambda_{8} v_{S}^{3}\\right)}{9 v_{S}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡        2⋅v₂⋅(λ₁⋅v₂ + λ₃⋅v₂ + 3⋅√3⋅λ₄⋅v_S)             v₂⋅(2⋅√3⋅λ₁⋅v₂ + 2⋅√3⋅ ↪\n",
       "⎢        ──────────────────────────────────             ────────────────────── ↪\n",
       "⎢                        3                                                 3   ↪\n",
       "⎢                                                                              ↪\n",
       "⎢      v₂⋅(2⋅√3⋅λ₁⋅v₂ + 2⋅√3⋅λ₃⋅v₂ - 9⋅λ₄⋅v_S)                          2      ↪\n",
       "⎢      ───────────────────────────────────────                      2⋅v₂ ⋅(λ₁  ↪\n",
       "⎢                         3                                                    ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢v₂⋅(-3⋅λ₄⋅v₂ + √3⋅λ₅⋅v_S + √3⋅λ₆⋅v_S + 2⋅√3⋅λ₇⋅v_S)                           ↪\n",
       "⎢───────────────────────────────────────────────────  v₂⋅(-√3⋅λ₄⋅v₂ + λ₅⋅v_S + ↪\n",
       "⎣                         3                                                    ↪\n",
       "\n",
       "↪ λ₃⋅v₂ - 9⋅λ₄⋅v_S)    v₂⋅(-3⋅λ₄⋅v₂ + √3⋅λ₅⋅v_S + √3⋅λ₆⋅v_S + 2⋅√3⋅λ₇⋅v_S)⎤\n",
       "↪ ─────────────────    ───────────────────────────────────────────────────⎥\n",
       "↪                                               3                         ⎥\n",
       "↪                                                                         ⎥\n",
       "↪                                                                         ⎥\n",
       "↪ + λ₃)                    v₂⋅(-√3⋅λ₄⋅v₂ + λ₅⋅v_S + λ₆⋅v_S + 2⋅λ₇⋅v_S)    ⎥\n",
       "↪                                                                         ⎥\n",
       "↪                                                                         ⎥\n",
       "↪                                    ⎛          3           3⎞            ⎥\n",
       "↪                                  2⋅⎝2⋅√3⋅λ₄⋅v₂  + 9⋅λ₈⋅v_S ⎠            ⎥\n",
       "↪  λ₆⋅v_S + 2⋅λ₇⋅v_S)              ───────────────────────────            ⎥\n",
       "↪                                             9⋅v_S                       ⎦"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = [sp.Symbol(f\"{n}0_r\", real=True) for n in (\"H_1\", \"H_2\", \"H_S\")]\n",
    "M = model.mass_matrix(h).subs(sol2).subs(align)\n",
    "M = M.applyfunc(lambda e: sp.simplify(sp.expand(e)))\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "30",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "symmetric: True\n",
      "nonzero diagonal: True\n"
     ]
    }
   ],
   "source": [
    "print(\"symmetric:\", sp.simplify(M - M.T) == sp.zeros(3, 3))\n",
    "print(\"nonzero diagonal:\", M[0, 0] != 0 and M[1, 1] != 0 and M[2, 2] != 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31",
   "metadata": {},
   "source": [
    "## 6. The other two scalar sectors: pseudoscalar and charged mass matrices\n",
    "\n",
    "`thdm_s3.py`'s potential `V` was written from full doublet bilinears\n",
    "(`bra`, `dag(a)*b.mat`) — it already implicitly contains the charged\n",
    "($H_1^\\pm,H_2^\\pm,H_S^\\pm$) and imaginary neutral (pseudoscalar) pieces of\n",
    "every field, even though only the real neutral fluctuations (`_r`\n",
    "symbols) have been differentiated so far. No new Lagrangian terms are\n",
    "needed to get the other two sectors, just two more second-derivative\n",
    "matrices:\n",
    "\n",
    "- **Pseudoscalar** ($M_A^2$): the same `Model.mass_matrix` used for the\n",
    "  CP-even sector, applied to `expand_vev`'s imaginary (`_i`) fluctuation\n",
    "  symbols instead of the real ones.\n",
    "- **Charged** ($M_C^2$): `Model.mass_matrix(..., charged=True)`, which\n",
    "  differentiates $\\partial^2V/\\partial\\bar\\phi_i\\partial\\phi_j$ via a\n",
    "  Dummy-conjugate trick (SymPy can't differentiate w.r.t. `conjugate(φ)`\n",
    "  directly) — this one takes the **raw** charged component symbols\n",
    "  (`H1.components[0]`, etc.), not fluctuation symbols, since a charged\n",
    "  field never gets a VEV and so never had `_r`/`_i` symbols generated for\n",
    "  it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "32",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M_A symmetric: True\n",
      "M_C symmetric: True\n"
     ]
    },
    {
     "data": {
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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 2 \\lambda_{2} v_{2}^{2} - 2 \\lambda_{3} v_{2}^{2} + \\frac{4 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - 2 \\lambda_{7} v_{S}^{2} & \\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{2} v_{2} + 2 \\sqrt{3} \\lambda_{3} v_{2} - 3 \\lambda_{4} v_{S}\\right)}{3} & \\frac{v_{2} \\left(- \\lambda_{4} v_{2} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{3}\\\\\\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{2} v_{2} + 2 \\sqrt{3} \\lambda_{3} v_{2} - 3 \\lambda_{4} v_{S}\\right)}{3} & - \\frac{2 \\lambda_{2} v_{2}^{2}}{3} - \\frac{2 \\lambda_{3} v_{2}^{2}}{3} + \\frac{2 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - 2 \\lambda_{7} v_{S}^{2} & \\frac{v_{2} \\left(- \\sqrt{3} \\lambda_{4} v_{2} + 6 \\lambda_{7} v_{S}\\right)}{3}\\\\\\frac{v_{2} \\left(- \\lambda_{4} v_{2} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{3} & \\frac{v_{2} \\left(- \\sqrt{3} \\lambda_{4} v_{2} + 6 \\lambda_{7} v_{S}\\right)}{3} & \\frac{4 v_{2}^{2} \\left(\\sqrt{3} \\lambda_{4} v_{2} - 6 \\lambda_{7} v_{S}\\right)}{9 v_{S}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡         2          2   4⋅√3⋅λ₄⋅v₂⋅v_S           2       v₂⋅(2⋅√3⋅λ₂⋅v₂ + 2⋅√ ↪\n",
       "⎢- 2⋅λ₂⋅v₂  - 2⋅λ₃⋅v₂  + ────────────── - 2⋅λ₇⋅v_S        ──────────────────── ↪\n",
       "⎢                              3                                             3 ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                             2          2     ↪\n",
       "⎢     v₂⋅(2⋅√3⋅λ₂⋅v₂ + 2⋅√3⋅λ₃⋅v₂ - 3⋅λ₄⋅v_S)          2⋅λ₂⋅v₂    2⋅λ₃⋅v₂    2 ↪\n",
       "⎢     ───────────────────────────────────────        - ──────── - ──────── + ─ ↪\n",
       "⎢                        3                                3          3         ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢            v₂⋅(-λ₄⋅v₂ + 2⋅√3⋅λ₇⋅v_S)                           v₂⋅(-√3⋅λ₄⋅v₂ ↪\n",
       "⎢            ─────────────────────────                           ───────────── ↪\n",
       "⎣                        3                                                   3 ↪\n",
       "\n",
       "↪ 3⋅λ₃⋅v₂ - 3⋅λ₄⋅v_S)         v₂⋅(-λ₄⋅v₂ + 2⋅√3⋅λ₇⋅v_S) ⎤\n",
       "↪ ───────────────────         ───────────────────────── ⎥\n",
       "↪                                         3             ⎥\n",
       "↪                                                       ⎥\n",
       "↪                                                       ⎥\n",
       "↪ ⋅√3⋅λ₄⋅v₂⋅v_S           2   v₂⋅(-√3⋅λ₄⋅v₂ + 6⋅λ₇⋅v_S) ⎥\n",
       "↪ ───────────── - 2⋅λ₇⋅v_S    ───────────────────────── ⎥\n",
       "↪      3                                  3             ⎥\n",
       "↪                                                       ⎥\n",
       "↪                                2                      ⎥\n",
       "↪  + 6⋅λ₇⋅v_S)               4⋅v₂ ⋅(√3⋅λ₄⋅v₂ - 6⋅λ₇⋅v_S)⎥\n",
       "↪ ────────────               ───────────────────────────⎥\n",
       "↪                                       9⋅v_S           ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 2 \\lambda_{3} v_{2}^{2} + \\frac{4 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - \\frac{\\lambda_{6} v_{S}^{2}}{2} - \\lambda_{7} v_{S}^{2} & \\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{3} v_{2} - 3 \\lambda_{4} v_{S}\\right)}{3} & \\frac{v_{2} \\left(- 2 \\lambda_{4} v_{2} + \\sqrt{3} \\lambda_{6} v_{S} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{6}\\\\\\frac{v_{2} \\left(2 \\sqrt{3} \\lambda_{3} v_{2} - 3 \\lambda_{4} v_{S}\\right)}{3} & - \\frac{2 \\lambda_{3} v_{2}^{2}}{3} + \\frac{2 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - \\frac{\\lambda_{6} v_{S}^{2}}{2} - \\lambda_{7} v_{S}^{2} & \\frac{v_{2} \\left(- 2 \\sqrt{3} \\lambda_{4} v_{2} + 3 \\lambda_{6} v_{S} + 6 \\lambda_{7} v_{S}\\right)}{6}\\\\\\frac{v_{2} \\left(- 2 \\lambda_{4} v_{2} + \\sqrt{3} \\lambda_{6} v_{S} + 2 \\sqrt{3} \\lambda_{7} v_{S}\\right)}{6} & \\frac{v_{2} \\left(- 2 \\sqrt{3} \\lambda_{4} v_{2} + 3 \\lambda_{6} v_{S} + 6 \\lambda_{7} v_{S}\\right)}{6} & \\frac{2 v_{2}^{2} \\left(2 \\sqrt{3} \\lambda_{4} v_{2} - 3 v_{S} \\left(\\lambda_{6} + 2 \\lambda_{7}\\right)\\right)}{9 v_{S}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                    2                                         ↪\n",
       "⎢         2   4⋅√3⋅λ₄⋅v₂⋅v_S   λ₆⋅v_S          2            v₂⋅(2⋅√3⋅λ₃⋅v₂ - 3 ↪\n",
       "⎢- 2⋅λ₃⋅v₂  + ────────────── - ─────── - λ₇⋅v_S             ────────────────── ↪\n",
       "⎢                   3             2                                     3      ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                          2                   ↪\n",
       "⎢          v₂⋅(2⋅√3⋅λ₃⋅v₂ - 3⋅λ₄⋅v_S)               2⋅λ₃⋅v₂    2⋅√3⋅λ₄⋅v₂⋅v_S  ↪\n",
       "⎢          ──────────────────────────             - ──────── + ──────────────  ↪\n",
       "⎢                      3                               3             3         ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢    v₂⋅(-2⋅λ₄⋅v₂ + √3⋅λ₆⋅v_S + 2⋅√3⋅λ₇⋅v_S)          v₂⋅(-2⋅√3⋅λ₄⋅v₂ + 3⋅λ₆⋅v ↪\n",
       "⎢    ───────────────────────────────────────          ──────────────────────── ↪\n",
       "⎣                       6                                               6      ↪\n",
       "\n",
       "↪                                                             ⎤\n",
       "↪ ⋅λ₄⋅v_S)             v₂⋅(-2⋅λ₄⋅v₂ + √3⋅λ₆⋅v_S + 2⋅√3⋅λ₇⋅v_S)⎥\n",
       "↪ ────────             ───────────────────────────────────────⎥\n",
       "↪                                         6                   ⎥\n",
       "↪                                                             ⎥\n",
       "↪         2                                                   ⎥\n",
       "↪   λ₆⋅v_S          2  v₂⋅(-2⋅√3⋅λ₄⋅v₂ + 3⋅λ₆⋅v_S + 6⋅λ₇⋅v_S) ⎥\n",
       "↪ - ─────── - λ₇⋅v_S   ────────────────────────────────────── ⎥\n",
       "↪      2                                 6                    ⎥\n",
       "↪                                                             ⎥\n",
       "↪                          2                                  ⎥\n",
       "↪ _S + 6⋅λ₇⋅v_S)       2⋅v₂ ⋅(2⋅√3⋅λ₄⋅v₂ - 3⋅v_S⋅(λ₆ + 2⋅λ₇)) ⎥\n",
       "↪ ──────────────       ────────────────────────────────────── ⎥\n",
       "↪                                      9⋅v_S                  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hi = [sp.Symbol(f\"{n}0_i\", real=True) for n in (\"H_1\", \"H_2\", \"H_S\")]\n",
    "M_A = model.mass_matrix(hi).subs(sol2).subs(align)\n",
    "M_A = M_A.applyfunc(lambda e: sp.simplify(sp.expand(e)))\n",
    "\n",
    "hc = [H1.components[0], H2.components[0], HS.components[0]]\n",
    "M_C = model.mass_matrix(hc, charged=True).subs(sol2).subs(align)\n",
    "M_C = M_C.applyfunc(lambda e: sp.simplify(sp.expand(e)))\n",
    "\n",
    "print(\"M_A symmetric:\", sp.simplify(M_A - M_A.T) == sp.zeros(3, 3))\n",
    "print(\"M_C symmetric:\", sp.simplify(M_C - M_C.T) == sp.zeros(3, 3))\n",
    "display(M_A)\n",
    "display(M_C)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33",
   "metadata": {},
   "source": [
    "## 7. The Gómez-Bock–Mondragón–Pérez-Martínez rotation ansatz\n",
    "\n",
    "**[GomezBock21]** (full citation above) diagonalizes all three sectors with\n",
    "a rotation built purely from the geometry of the vacuum. Writing\n",
    "$v_1=v\\cos\\varphi\\sin\\theta$, $v_2=v\\sin\\varphi\\sin\\theta$, $v_S=v\\cos\\theta$\n",
    "(their Eq. 21) — i.e. $\\varphi,\\theta$ are just the *angles* the VEV vector\n",
    "makes in $(v_1,v_2,v_S)$-space — their Eq. 25–29 rotation is\n",
    "\n",
    "$$R(\\varphi,\\theta)=\\begin{pmatrix}\\sin\\theta\\cos\\varphi&-\\sin\\varphi&-\\cos\\theta\\cos\\varphi\\\\\\sin\\theta\\sin\\varphi&\\cos\\varphi&-\\cos\\theta\\sin\\varphi\\\\\\cos\\theta&0&\\sin\\theta\\end{pmatrix}.$$\n",
    "\n",
    "Nothing here depends on the potential's couplings — it's fixed the moment\n",
    "the vacuum direction is known. We build it directly from `feynlag`'s own\n",
    "$v_1,v_2,v_S$ (no basis translation needed — tested below against the\n",
    "swapped assignment too, which does *not* diagonalize; this direct one\n",
    "does)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "34",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{v_{2}}{\\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}} & - \\frac{\\sqrt{3}}{2} & - \\frac{\\sqrt{3} v_{S}}{2 \\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}}\\\\\\frac{\\sqrt{3} v_{2}}{\\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}} & \\frac{1}{2} & - \\frac{3 v_{S}}{2 \\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}}\\\\\\frac{\\sqrt{3} v_{S}}{\\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}} & 0 & \\frac{2 v_{2}}{\\sqrt{4 v_{2}^{2} + 3 v_{S}^{2}}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡        v₂           -√3         -√3⋅v_S        ⎤\n",
       "⎢───────────────────  ────  ─────────────────────⎥\n",
       "⎢   ________________   2         ________________⎥\n",
       "⎢  ╱     2        2             ╱     2        2 ⎥\n",
       "⎢╲╱  4⋅v₂  + 3⋅v_S          2⋅╲╱  4⋅v₂  + 3⋅v_S  ⎥\n",
       "⎢                                                ⎥\n",
       "⎢       √3⋅v₂                      -3⋅v_S        ⎥\n",
       "⎢───────────────────  1/2   ─────────────────────⎥\n",
       "⎢   ________________             ________________⎥\n",
       "⎢  ╱     2        2             ╱     2        2 ⎥\n",
       "⎢╲╱  4⋅v₂  + 3⋅v_S          2⋅╲╱  4⋅v₂  + 3⋅v_S  ⎥\n",
       "⎢                                                ⎥\n",
       "⎢      √3⋅v_S                       2⋅v₂         ⎥\n",
       "⎢───────────────────   0     ─────────────────── ⎥\n",
       "⎢   ________________            ________________ ⎥\n",
       "⎢  ╱     2        2            ╱     2        2  ⎥\n",
       "⎣╲╱  4⋅v₂  + 3⋅v_S           ╲╱  4⋅v₂  + 3⋅v_S   ⎦"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v12 = sp.sqrt(v1.s**2 + v2.s**2)\n",
    "vtot = sp.sqrt(v1.s**2 + v2.s**2 + vS.s**2)\n",
    "cphi, sphi = v1.s / v12, v2.s / v12\n",
    "cth, sth = vS.s / vtot, v12 / vtot\n",
    "\n",
    "R = sp.Matrix([\n",
    "    [sth * cphi, -sphi, -cth * cphi],\n",
    "    [sth * sphi,  cphi, -cth * sphi],\n",
    "    [cth,          0,    sth],\n",
    "])\n",
    "R = sp.simplify(R.subs(align))\n",
    "R"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35",
   "metadata": {},
   "source": [
    "**Checking the expressions are right** (not assuming): apply\n",
    "$R^TM_AR$ and $R^TM_CR$ and look at which entries are *identically* zero,\n",
    "for arbitrary $\\lambda_1,\\ldots,\\lambda_8$ — no numbers substituted yet."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "36",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "D_A = R^T M_A R zero pattern:\n",
      "   [True, True, True]\n",
      "   [True, False, True]\n",
      "   [True, True, False]\n",
      "D_C = R^T M_C R zero pattern:\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   [True, True, True]\n",
      "   [True, False, True]\n",
      "   [True, True, False]\n",
      "\n",
      "Goldstone eigenvalues (should be exactly 0): 0 0\n",
      "\n",
      "pseudoscalar physical mass^2 formulas:\n"
     ]
    },
    {
     "data": {
      "image/png": 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DLD5PHUxCx13oWmALN35FxFeqHnSM4Vq65liuY6Z8ntYxTyfpZe+Hjty5tn+GmQL0KD4MF8rTRA9XEXAsq7B0F+nz1N2U9NOh3gV2+EJVCpe0bQ4d0bj5UAvM7dSGgGPZhtOpc/k8nXoGOm6/+0NHCWXs1MNHxxX2jz9NZCjHciJwCxfzeVoY8BU193/IL+HRbb6DPQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle - \\frac{8 \\lambda_{2} v_{2}^{2}}{3} - \\frac{8 \\lambda_{3} v_{2}^{2}}{3} + \\frac{5 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - 2 \\lambda_{7} v_{S}^{2}$"
      ],
      "text/plain": [
       "         2          2                             \n",
       "  8⋅λ₂⋅v₂    8⋅λ₃⋅v₂    5⋅√3⋅λ₄⋅v₂⋅v_S           2\n",
       "- ──────── - ──────── + ────────────── - 2⋅λ₇⋅v_S \n",
       "     3          3             3                   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\sqrt{3} \\lambda_{4} v_{2}^{3}}{9 v_{S}} + \\frac{\\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - \\frac{8 \\lambda_{7} v_{2}^{2}}{3} - 2 \\lambda_{7} v_{S}^{2}$"
      ],
      "text/plain": [
       "          3                         2            \n",
       "4⋅√3⋅λ₄⋅v₂    √3⋅λ₄⋅v₂⋅v_S   8⋅λ₇⋅v₂            2\n",
       "─────────── + ──────────── - ──────── - 2⋅λ₇⋅v_S \n",
       "   9⋅v_S           3            3                "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "charged physical mass^2 formulas:\n"
     ]
    },
    {
     "data": {
      "image/png": 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zcda/Fp5QAk1XJxT+jqalG6zYnR8s2lFNS86QwOJAlj5pgNj/e+p8vCGjvy3LkSXQdHVkgWc0J53gcsNNEI50ZRRrWWZKoOrTBRoE9l9J0+6ZuwAXQqMKJNB0VYESMliQnjhqd90ANkNYhbJUDbLqI7/K4MB0e/tZSOEHrKbp6oDCLVG15hGuNo4uhW8a6x6LdnVHuErIM7eO2kGWIyf8giI9evLEd5CjKY3qkEDTVR166OVCc4iXx08Vpi+6AN53vYVaZBEJ1H66IB0QGw0SBgWWLR+FSMG3iFBaJbMk0HQ1S2yHL6R5gqXKy2J+jZieLx/78MnhmVtBC9W/+NKgwO8atje6Z8BU+4GYFYyZwS42XQ2K5qQJ0os7DzvABCcMHg+ktegCEvg/lUFXr+y2MvEAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle - \\frac{8 \\lambda_{3} v_{2}^{2}}{3} + \\frac{5 \\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - \\frac{\\lambda_{6} v_{S}^{2}}{2} - \\lambda_{7} v_{S}^{2}$"
      ],
      "text/plain": [
       "         2                          2          \n",
       "  8⋅λ₃⋅v₂    5⋅√3⋅λ₄⋅v₂⋅v_S   λ₆⋅v_S          2\n",
       "- ──────── + ────────────── - ─────── - λ₇⋅v_S \n",
       "     3             3             2             "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\sqrt{3} \\lambda_{4} v_{2}^{3}}{9 v_{S}} + \\frac{\\sqrt{3} \\lambda_{4} v_{2} v_{S}}{3} - \\frac{2 \\lambda_{6} v_{2}^{2}}{3} - \\frac{\\lambda_{6} v_{S}^{2}}{2} - \\frac{4 \\lambda_{7} v_{2}^{2}}{3} - \\lambda_{7} v_{S}^{2}$"
      ],
      "text/plain": [
       "          3                         2         2          2          \n",
       "4⋅√3⋅λ₄⋅v₂    √3⋅λ₄⋅v₂⋅v_S   2⋅λ₆⋅v₂    λ₆⋅v_S    4⋅λ₇⋅v₂          2\n",
       "─────────── + ──────────── - ──────── - ─────── - ──────── - λ₇⋅v_S \n",
       "   9⋅v_S           3            3          2         3              "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def zero_pattern(D):\n",
    "    return [[bool(sp.simplify(D[i, j]) == 0) for j in range(3)] for i in range(3)]\n",
    "\n",
    "D_A = sp.simplify(R.T * M_A * R)\n",
    "D_C = sp.simplify(R.T * M_C * R)\n",
    "\n",
    "print(\"D_A = R^T M_A R zero pattern:\")\n",
    "for row in zero_pattern(D_A):\n",
    "    print(\"  \", row)\n",
    "print(\"D_C = R^T M_C R zero pattern:\")\n",
    "for row in zero_pattern(D_C):\n",
    "    print(\"  \", row)\n",
    "\n",
    "print(\"\\nGoldstone eigenvalues (should be exactly 0):\", D_A[0, 0], D_C[0, 0])\n",
    "print(\"\\npseudoscalar physical mass^2 formulas:\")\n",
    "display(D_A[1, 1])\n",
    "display(D_A[2, 2])\n",
    "print(\"charged physical mass^2 formulas:\")\n",
    "display(D_C[1, 1])\n",
    "display(D_C[2, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37",
   "metadata": {},
   "source": [
    "$R$ **exactly, fully diagonalizes both sectors** — for any couplings,\n",
    "not just the benchmark point — with one eigenvalue identically zero (the\n",
    "Goldstone boson eaten by $Z$ and $W^\\pm$ respectively) and two compact\n",
    "closed-form mass² formulas. This matches [GomezBock21]'s Eq. 30–33: the\n",
    "pseudoscalar and charged sectors need *no* mixing angle beyond this one\n",
    "geometric rotation, because it's fixed entirely by the Goldstone theorem\n",
    "— gauge symmetry alone, nothing dynamical about the potential.\n",
    "\n",
    "Does the same trick fully diagonalize the CP-even sector `M` from §5?\n",
    "\n",
    "In plain terms: two of the twelve real scalar degrees of freedom the three doublets started with are *exactly* massless before any dynamics is switched on at all — pure kinematics from the Goldstone theorem, later eaten as the longitudinal modes of $Z$ and $W^\\pm$, not a coincidence of these particular $\\lambda_k$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "38",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "D_S = R^T M R zero pattern:\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   [False, True, False]\n",
      "   [True, False, True]\n",
      "   [False, True, False]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{2 \\left(16 \\lambda_{1} v_{2}^{4} + 16 \\lambda_{3} v_{2}^{4} - 16 \\sqrt{3} \\lambda_{4} v_{2}^{3} v_{S} + 12 \\lambda_{5} v_{2}^{2} v_{S}^{2} + 12 \\lambda_{6} v_{2}^{2} v_{S}^{2} + 24 \\lambda_{7} v_{2}^{2} v_{S}^{2} + 9 \\lambda_{8} v_{S}^{4}\\right)}{3 \\left(4 v_{2}^{2} + 3 v_{S}^{2}\\right)} & 0 & \\frac{2 v_{2} \\left(- 8 \\sqrt{3} \\lambda_{1} v_{2}^{2} v_{S} - 8 \\sqrt{3} \\lambda_{3} v_{2}^{2} v_{S} - 8 \\lambda_{4} v_{2}^{3} + 18 \\lambda_{4} v_{2} v_{S}^{2} + 4 \\sqrt{3} \\lambda_{5} v_{2}^{2} v_{S} - 3 \\sqrt{3} \\lambda_{5} v_{S}^{3} + 4 \\sqrt{3} \\lambda_{6} v_{2}^{2} v_{S} - 3 \\sqrt{3} \\lambda_{6} v_{S}^{3} + 8 \\sqrt{3} \\lambda_{7} v_{2}^{2} v_{S} - 6 \\sqrt{3} \\lambda_{7} v_{S}^{3} + 6 \\sqrt{3} \\lambda_{8} v_{S}^{3}\\right)}{3 \\left(4 v_{2}^{2} + 3 v_{S}^{2}\\right)}\\\\0 & 3 \\sqrt{3} \\lambda_{4} v_{2} v_{S} & 0\\\\\\frac{2 v_{2} \\left(- 8 \\sqrt{3} \\lambda_{1} v_{2}^{2} v_{S} - 8 \\sqrt{3} \\lambda_{3} v_{2}^{2} v_{S} - 8 \\lambda_{4} v_{2}^{3} + 18 \\lambda_{4} v_{2} v_{S}^{2} + 4 \\sqrt{3} \\lambda_{5} v_{2}^{2} v_{S} - 3 \\sqrt{3} \\lambda_{5} v_{S}^{3} + 4 \\sqrt{3} \\lambda_{6} v_{2}^{2} v_{S} - 3 \\sqrt{3} \\lambda_{6} v_{S}^{3} + 8 \\sqrt{3} \\lambda_{7} v_{2}^{2} v_{S} - 6 \\sqrt{3} \\lambda_{7} v_{S}^{3} + 6 \\sqrt{3} \\lambda_{8} v_{S}^{3}\\right)}{3 \\left(4 v_{2}^{2} + 3 v_{S}^{2}\\right)} & 0 & \\frac{v_{2} \\left(72 \\lambda_{1} v_{2} v_{S}^{3} + 72 \\lambda_{3} v_{2} v_{S}^{3} + 16 \\sqrt{3} \\lambda_{4} v_{2}^{4} + 72 \\sqrt{3} \\lambda_{4} v_{2}^{2} v_{S}^{2} - 27 \\sqrt{3} \\lambda_{4} v_{S}^{4} - 72 \\lambda_{5} v_{2} v_{S}^{3} - 72 \\lambda_{6} v_{2} v_{S}^{3} - 144 \\lambda_{7} v_{2} v_{S}^{3} + 72 \\lambda_{8} v_{2} v_{S}^{3}\\right)}{9 v_{S} \\left(4 v_{2}^{2} + 3 v_{S}^{2}\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                        ⎛        4           4              3 ↪\n",
       "⎢                                      2⋅⎝16⋅λ₁⋅v₂  + 16⋅λ₃⋅v₂  - 16⋅√3⋅λ₄⋅v₂  ↪\n",
       "⎢                                      ─────────────────────────────────────── ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢                                                                              ↪\n",
       "⎢     ⎛            2                 2              3               2          ↪\n",
       "⎢2⋅v₂⋅⎝- 8⋅√3⋅λ₁⋅v₂ ⋅v_S - 8⋅√3⋅λ₃⋅v₂ ⋅v_S - 8⋅λ₄⋅v₂  + 18⋅λ₄⋅v₂⋅v_S  + 4⋅√3⋅λ ↪\n",
       "⎢───────────────────────────────────────────────────────────────────────────── ↪\n",
       "⎢                                                                              ↪\n",
       "⎣                                                                              ↪\n",
       "\n",
       "↪                2    2           2    2           2    2           4⎞         ↪\n",
       "↪ ⋅v_S + 12⋅λ₅⋅v₂ ⋅v_S  + 12⋅λ₆⋅v₂ ⋅v_S  + 24⋅λ₇⋅v₂ ⋅v_S  + 9⋅λ₈⋅v_S ⎠         ↪\n",
       "↪ ────────────────────────────────────────────────────────────────────         ↪\n",
       "↪        ⎛    2        2⎞                                                      ↪\n",
       "↪      3⋅⎝4⋅v₂  + 3⋅v_S ⎠                                                      ↪\n",
       "↪                                                                              ↪\n",
       "↪               0                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪     2                  3             2                  3             2      ↪\n",
       "↪ ₅⋅v₂ ⋅v_S - 3⋅√3⋅λ₅⋅v_S  + 4⋅√3⋅λ₆⋅v₂ ⋅v_S - 3⋅√3⋅λ₆⋅v_S  + 8⋅√3⋅λ₇⋅v₂ ⋅v_S  ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪        ⎛    2        2⎞                                                      ↪\n",
       "↪      3⋅⎝4⋅v₂  + 3⋅v_S ⎠                                                      ↪\n",
       "\n",
       "↪                                                      ⎛            2          ↪\n",
       "↪                                                 2⋅v₂⋅⎝- 8⋅√3⋅λ₁⋅v₂ ⋅v_S - 8⋅ ↪\n",
       "↪                                       0         ──────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                 3⋅√3⋅λ₄⋅v₂⋅v_S                               ↪\n",
       "↪                                                                              ↪\n",
       "↪              3              3⎞                                     ⎛         ↪\n",
       "↪ - 6⋅√3⋅λ₇⋅v_S  + 6⋅√3⋅λ₈⋅v_S ⎠                                  v₂⋅⎝72⋅λ₁⋅v₂ ↪\n",
       "↪ ──────────────────────────────        0                         ──────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪         2              3               2             2                  3    ↪\n",
       "↪ √3⋅λ₃⋅v₂ ⋅v_S - 8⋅λ₄⋅v₂  + 18⋅λ₄⋅v₂⋅v_S  + 4⋅√3⋅λ₅⋅v₂ ⋅v_S - 3⋅√3⋅λ₅⋅v_S  +  ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                         ⎛    2        2⎞     ↪\n",
       "↪                                                       3⋅⎝4⋅v₂  + 3⋅v_S ⎠     ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                0             ↪\n",
       "↪                                                                              ↪\n",
       "↪     3               3              4              2    2               4     ↪\n",
       "↪ ⋅v_S  + 72⋅λ₃⋅v₂⋅v_S  + 16⋅√3⋅λ₄⋅v₂  + 72⋅√3⋅λ₄⋅v₂ ⋅v_S  - 27⋅√3⋅λ₄⋅v_S  - 7 ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                           ⎛    2        2⎞   ↪\n",
       "↪                                                     9⋅v_S⋅⎝4⋅v₂  + 3⋅v_S ⎠   ↪\n",
       "\n",
       "↪           2                  3             2                  3              ↪\n",
       "↪ 4⋅√3⋅λ₆⋅v₂ ⋅v_S - 3⋅√3⋅λ₆⋅v_S  + 8⋅√3⋅λ₇⋅v₂ ⋅v_S - 6⋅√3⋅λ₇⋅v_S  + 6⋅√3⋅λ₈⋅v_ ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "↪            3               3                3               3⎞               ↪\n",
       "↪ 2⋅λ₅⋅v₂⋅v_S  - 72⋅λ₆⋅v₂⋅v_S  - 144⋅λ₇⋅v₂⋅v_S  + 72⋅λ₈⋅v₂⋅v_S ⎠               ↪\n",
       "↪ ──────────────────────────────────────────────────────────────               ↪\n",
       "↪                                                                              ↪\n",
       "↪                                                                              ↪\n",
       "\n",
       "↪  3⎞⎤\n",
       "↪ S ⎠⎥\n",
       "↪ ───⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎥\n",
       "↪    ⎦"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_S = sp.simplify(R.T * M * R)\n",
    "print(\"D_S = R^T M R zero pattern:\")\n",
    "for row in zero_pattern(D_S):\n",
    "    print(\"  \", row)\n",
    "D_S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39",
   "metadata": {},
   "source": [
    "Only **one** state decouples this time (row/column 1 — the analogue of\n",
    "[GomezBock21]'s $h_0$, with the compact closed form\n",
    "$m_{h_0}^2=3\\sqrt3\\,\\lambda_4v_2v_S$, independent of every other coupling\n",
    "including $\\lambda_8$). The remaining $(0,2)$ block is *not* diagonal — a\n",
    "genuine, coupling-dependent mixing survives. This is the qualitative\n",
    "difference [GomezBock21] report: the CP-even sector needs a **second**,\n",
    "dynamical mixing angle (their $\\alpha$, Eq. 36) that the Goldstone-protected\n",
    "pseudoscalar/charged sectors don't."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40",
   "metadata": {},
   "source": [
    "## 8. Finishing the CP-even sector: the leftover $2\\times2$ block\n",
    "\n",
    "This is exactly the shape `feynlag.vacuum.diagonalize.solve_mixing_angle_2x2`\n",
    "/ `diagonalize_orthogonal_2x2` already handles — the same tool\n",
    "`examples/thdm.py` uses to diagonalize the plain 2HDM's $2\\times2$ CP-even\n",
    "matrix, reused here on the block left over after the geometric rotation\n",
    "above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "41",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tan(2 theta) is a closed form of length 707 characters (long, but a genuine analytic formula — unlike the raw cubic-root diagonalization of the full 3x3, this 2x2 reduction stays closed-form)\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\text{True}$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from feynlag import solve_mixing_angle_2x2, diagonalize_orthogonal_2x2\n",
    "\n",
    "block = sp.Matrix([[D_S[0, 0], D_S[0, 2]], [D_S[2, 0], D_S[2, 2]]])\n",
    "theta_expr, tan2theta = solve_mixing_angle_2x2(block)\n",
    "print(\"tan(2 theta) is a closed form of length\", len(str(tan2theta)),\n",
    "      \"characters (long, but a genuine analytic formula — unlike the raw\",\n",
    "      \"cubic-root diagonalization of the full 3x3, this 2x2 reduction stays\",\n",
    "      \"closed-form)\")\n",
    "\n",
    "h_a, h_b, H_1, H_2 = sp.symbols(\"h_a h_b H_1 H_2\", real=True)\n",
    "rot = diagonalize_orthogonal_2x2(block, [h_a, h_b], [H_1, H_2])\n",
    "rot.angle_relation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42",
   "metadata": {},
   "source": [
    "707 characters is unreadable as one blob — but it's a sum of terms each\n",
    "proportional to a single $\\lambda_k$. Factoring groups them, which at\n",
    "least makes the *structure* (which couplings enter, and how) legible even\n",
    "though the full expression stays long."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle - \\frac{12 v_{2} v_{S} \\left(8 \\sqrt{3} \\lambda_{1} v_{2}^{2} v_{S} + 8 \\sqrt{3} \\lambda_{3} v_{2}^{2} v_{S} + 8 \\lambda_{4} v_{2}^{3} - 18 \\lambda_{4} v_{2} v_{S}^{2} - 4 \\sqrt{3} \\lambda_{5} v_{2}^{2} v_{S} + 3 \\sqrt{3} \\lambda_{5} v_{S}^{3} - 4 \\sqrt{3} \\lambda_{6} v_{2}^{2} v_{S} + 3 \\sqrt{3} \\lambda_{6} v_{S}^{3} - 8 \\sqrt{3} \\lambda_{7} v_{2}^{2} v_{S} + 6 \\sqrt{3} \\lambda_{7} v_{S}^{3} - 6 \\sqrt{3} \\lambda_{8} v_{S}^{3}\\right)}{96 \\lambda_{1} v_{2}^{4} v_{S} - 72 \\lambda_{1} v_{2}^{2} v_{S}^{3} + 96 \\lambda_{3} v_{2}^{4} v_{S} - 72 \\lambda_{3} v_{2}^{2} v_{S}^{3} - 16 \\sqrt{3} \\lambda_{4} v_{2}^{5} - 168 \\sqrt{3} \\lambda_{4} v_{2}^{3} v_{S}^{2} + 27 \\sqrt{3} \\lambda_{4} v_{2} v_{S}^{4} + 144 \\lambda_{5} v_{2}^{2} v_{S}^{3} + 144 \\lambda_{6} v_{2}^{2} v_{S}^{3} + 288 \\lambda_{7} v_{2}^{2} v_{S}^{3} - 72 \\lambda_{8} v_{2}^{2} v_{S}^{3} + 54 \\lambda_{8} v_{S}^{5}}$"
      ],
      "text/plain": [
       "                  ⎛          2                 2              3                ↪\n",
       "       -12⋅v₂⋅v_S⋅⎝8⋅√3⋅λ₁⋅v₂ ⋅v_S + 8⋅√3⋅λ₃⋅v₂ ⋅v_S + 8⋅λ₄⋅v₂  - 18⋅λ₄⋅v₂⋅v_S ↪\n",
       "────────────────────────────────────────────────────────────────────────────── ↪\n",
       "        4               2    3           4               2    3              5 ↪\n",
       "96⋅λ₁⋅v₂ ⋅v_S - 72⋅λ₁⋅v₂ ⋅v_S  + 96⋅λ₃⋅v₂ ⋅v_S - 72⋅λ₃⋅v₂ ⋅v_S  - 16⋅√3⋅λ₄⋅v₂  ↪\n",
       "\n",
       "↪ 2             2                  3             2                  3          ↪\n",
       "↪   - 4⋅√3⋅λ₅⋅v₂ ⋅v_S + 3⋅√3⋅λ₅⋅v_S  - 4⋅√3⋅λ₆⋅v₂ ⋅v_S + 3⋅√3⋅λ₆⋅v_S  - 8⋅√3⋅λ ↪\n",
       "↪ ──────────────────────────────────────────────────────────────────────────── ↪\n",
       "↪                3    2                  4            2    3            2    3 ↪\n",
       "↪  - 168⋅√3⋅λ₄⋅v₂ ⋅v_S  + 27⋅√3⋅λ₄⋅v₂⋅v_S  + 144⋅λ₅⋅v₂ ⋅v_S  + 144⋅λ₆⋅v₂ ⋅v_S  ↪\n",
       "\n",
       "↪     2                  3              3⎞        \n",
       "↪ ₇⋅v₂ ⋅v_S + 6⋅√3⋅λ₇⋅v_S  - 6⋅√3⋅λ₈⋅v_S ⎠        \n",
       "↪ ────────────────────────────────────────────────\n",
       "↪             2    3           2    3            5\n",
       "↪  + 288⋅λ₇⋅v₂ ⋅v_S  - 72⋅λ₈⋅v₂ ⋅v_S  + 54⋅λ₈⋅v_S "
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tan2theta.factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44",
   "metadata": {},
   "source": [
    "## 9. Physical spectrum at a benchmark point\n",
    "\n",
    "Substituting numeric values (each parameter's own declared\n",
    "`ExternalParameter.value`, as in previous benchmark checks) into the\n",
    "*original* λ = 0.05·k demonstration point shows the same lesson as\n",
    "before, now for all three sectors at once — solving the tadpoles only\n",
    "guarantees a *stationary* point, and the mass matrices are the\n",
    "second-derivative stability test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "45",
   "metadata": {},
   "outputs": [],
   "source": [
    "lam_syms = [lams[k].s for k in range(1, 9)]\n",
    "spectrum_exprs = [D_S[1, 1], D_S[0, 0], D_S[0, 2], D_S[2, 2],\n",
    "                  D_A[1, 1], D_A[2, 2], D_C[1, 1], D_C[2, 2]]\n",
    "spectrum_fn = sp.lambdify(lam_syms + [v2.s, vS.s], spectrum_exprs, \"math\")\n",
    "\n",
    "def block_eigs(a, b, c):\n",
    "    # real symmetric 2x2: discriminant (a-c)^2 + 4b^2 is never negative\n",
    "    tr, det = a + c, a * c - b * b\n",
    "    root = (tr**2 - 4 * det) ** 0.5\n",
    "    return (tr + root) / 2, (tr - root) / 2\n",
    "\n",
    "def spectrum(lam_values, v2_val, vS_val):\n",
    "    h0, a, b, c, A1, A2, Hp1, Hp2 = spectrum_fn(*lam_values, v2_val, vS_val)\n",
    "    H1m, H2m = block_eigs(a, b, c)\n",
    "    return dict(h0=h0, H1=H1m, H2=H2m, A1=A1, A2=A2, Hpm1=Hp1, Hpm2=Hp2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46",
   "metadata": {},
   "source": [
    "`spectrum(...)` turns any set of couplings into all seven physical\n",
    "mass² values at once, reusing the closed forms just derived. Now evaluate\n",
    "it at the same \"democratic\" $\\lambda_k=0.05k$ point used throughout the\n",
    "notebook so far."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "47",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "naive benchmark mass^2 (GeV^2): {'h0': 9560.92, 'H1': 14214.63, 'H2': -2301.34, 'A1': -7985.04, 'A2': -12834.08, 'Hpm1': -3178.38, 'Hpm2': -8027.42}\n",
      "negative directions: ['H2', 'A1', 'A2', 'Hpm1', 'Hpm2']\n"
     ]
    }
   ],
   "source": [
    "naive_lams = [0.05 * k for k in range(1, 9)]\n",
    "naive = spectrum(naive_lams, v2.value, vS.value)\n",
    "print(\"naive benchmark mass^2 (GeV^2):\", {k: round(v, 2) for k, v in naive.items()})\n",
    "print(\"negative directions:\", [k for k, v in naive.items() if v < 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48",
   "metadata": {},
   "source": [
    "In plain terms: a negative mass² here doesn't mean an imaginary-mass\n",
    "particle — it flags a direction in which the potential curves *downward*\n",
    "at this point. This benchmark sits on a saddle/hilltop of $V$, not in a\n",
    "valley, in five of the seven directions."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49",
   "metadata": {},
   "source": [
    "Five of the seven directions are tachyonic at this naive, \"democratic\"\n",
    "benchmark. [GomezBock21] face exactly this problem at scale — their\n",
    "Section 3.4 scans $\\mathcal O(10^{11})$ random points over the eight\n",
    "self-couplings and keeps only the ones passing unitarity and stability\n",
    "constraints before ever reporting a mass. We do the same thing here, just\n",
    "at notebook scale: a small random scan over $\\lambda_1,\\ldots,\\lambda_8$\n",
    "(same $v_2,v_S$) for the first point where all seven mass² come out\n",
    "positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "50",
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "\n",
    "rng = random.Random(7)\n",
    "stable = None\n",
    "for trial in range(500_000):\n",
    "    lam_values = [rng.uniform(0.0, 1.5) for _ in range(8)]\n",
    "    spec = spectrum(lam_values, v2.value, vS.value)\n",
    "    if all(m2 > 0 for m2 in spec.values()):\n",
    "        stable = (lam_values, spec)\n",
    "        break"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51",
   "metadata": {},
   "source": [
    "The scan above just stops at the first point where all seven mass²\n",
    "come out positive — a stand-in, at notebook scale, for [GomezBock21]'s\n",
    "$\\mathcal O(10^{11})$-point stability/unitarity scan. Read off what it\n",
    "found."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "52",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stable point found at trial 272 — lambda_k: {1: 0.955, 2: 0.063, 3: 0.617, 4: 1.181, 5: 0.46, 6: 1.036, 7: 0.006, 8: 0.457}\n",
      "\n",
      "mass^2 (GeV^2): {'h0': 56478.91, 'H1': 46526.5, 'H2': 13218.39, 'A1': 7299.66, 'A2': 23283.43, 'Hpm1': 6260.27, 'Hpm2': 10974.67}\n",
      "\n",
      "physical masses (GeV): {'h0': 237.65, 'H1': 215.7, 'H2': 114.97, 'A1': 85.44, 'A2': 152.59, 'Hpm1': 79.12, 'Hpm2': 104.76}\n"
     ]
    }
   ],
   "source": [
    "lam_values, spec = stable\n",
    "print(\"stable point found at trial\", trial, \"— lambda_k:\",\n",
    "      {k + 1: round(lam_values[k], 3) for k in range(8)})\n",
    "print(\"\\nmass^2 (GeV^2):\", {k: round(v, 2) for k, v in spec.items()})\n",
    "masses = {k: v ** 0.5 for k, v in spec.items()}\n",
    "print(\"\\nphysical masses (GeV):\", {k: round(v, 2) for k, v in masses.items()})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53",
   "metadata": {},
   "source": [
    "Seven positive physical masses — three CP-even ($h_0,H_1,H_2$), two\n",
    "pseudoscalar ($A_1,A_2$), two charged ($H_1^\\pm,H_2^\\pm$) — plus the two\n",
    "Goldstone bosons eaten by $Z$ and $W^\\pm$, exactly [GomezBock21]'s \"nine\n",
    "physical Higgs bosons, one of which corresponds to the Standard Model\n",
    "one.\" As a last cross-check, the CP-even three masses above should agree\n",
    "with brute-force `M.eigenvals()` at the same point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "54",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "brute-force M.eigenvals():    [13218.3871, 46526.5033, 56478.9149]\n",
      "rotation-ansatz h0,H1,H2:     [13218.3871, 46526.5033, 56478.9149]\n"
     ]
    }
   ],
   "source": [
    "trial_vals = dict(zip(lam_syms, lam_values))\n",
    "trial_vals[v2.s] = v2.value\n",
    "trial_vals[vS.s] = vS.value\n",
    "\n",
    "Mn = sp.N(M.subs(trial_vals))\n",
    "brute_force = sorted(complex(e).real for e in Mn.eigenvals().keys())\n",
    "via_rotation = sorted([spec[\"h0\"], spec[\"H1\"], spec[\"H2\"]])\n",
    "print(\"brute-force M.eigenvals():   \", [round(x, 4) for x in brute_force])\n",
    "print(\"rotation-ansatz h0,H1,H2:    \", [round(x, 4) for x in via_rotation])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55",
   "metadata": {},
   "source": [
    "## 10. Recap\n",
    "\n",
    "| Step | Tool | Result |\n",
    "|---|---|---|\n",
    "| Declare $S_3$ irreps + gauge groups | `S3`, `SU2`, `U1`, `s3.assign` | $(H_1,H_2)$ in the **2**, $H_S$ in the **1** |\n",
    "| Build the potential from CG invariants | `S3.doublet_product`, `bra` | 8 $S_3$-covariant quartics, matching the abstract $1\\oplus1'\\oplus2$ split |\n",
    "| Check both symmetries at once | `Model.check_invariance` | gauge **and** finite $S_3$ invariance both pass |\n",
    "| Catch a broken term | `check_discrete_invariance` | a term missing its $H_2$ partner is rejected |\n",
    "| Solve the tadpoles | `Model.tadpoles`, `sp.solve` | 3 VEVs, 2 free mass parameters → $v_1^2=v_2^2/3$ forced, matching [GomezBock21] |\n",
    "| CP-even mass matrix | `Model.mass_matrix` | symmetric $3\\times3$ matrix on the aligned vacuum |\n",
    "| Pseudoscalar + charged mass matrices | `Model.mass_matrix(..., charged=True)` | two more $3\\times3$ (Hermitian) matrices, no new Lagrangian terms needed |\n",
    "| Geometric rotation ansatz | hand-built $R(\\varphi,\\theta)$ | **exactly** diagonalizes pseudoscalar + charged sectors, symbolically, for any couplings |\n",
    "| Leftover CP-even $2\\times2$ block | `solve_mixing_angle_2x2`, `diagonalize_orthogonal_2x2` | analytic $\\tan2\\alpha$, reusing the same tool `thdm.py` uses for the plain 2HDM |\n",
    "| Benchmark spectrum | numeric scan over $\\lambda_1,\\ldots,\\lambda_8$ | 7 positive physical masses, matching [GomezBock21]'s \"nine physical Higgs bosons\" (7 + 2 Goldstones) |\n",
    "\n",
    "The headline results: a discrete flavor symmetry doesn't just forbid\n",
    "individual couplings — an over-constrained potential can force the vacuum\n",
    "itself onto a specific, symmetry-preserving direction, derived here\n",
    "directly from the symbolic tadpole system rather than assumed. And once\n",
    "the vacuum is fixed, the Goldstone theorem alone (not any dynamical\n",
    "coupling) is enough to fully diagonalize two of the three scalar sectors —\n",
    "only the CP-even sector needs a genuine, coupling-dependent mixing angle,\n",
    "and only a real (not merely stationary) vacuum, checked by scanning for\n",
    "positive mass², gives physical particles instead of tachyons.\n",
    "\n",
    "**Where to go next:**\n",
    "\n",
    "- `SUN_Groups_Tutorial.ipynb` — a deeper dive into representation theory,\n",
    "  for continuous gauge groups.\n",
    "- `docs/manual/declaration.md` — how `S3`'s real-orthogonal generators are\n",
    "  chosen, and `S3.doublet_product`.\n",
    "- `docs/manual/ssb.md` — the general tadpole-solving machinery and this\n",
    "  exact worked case.\n",
    "- `docs/manual/invariance.md` — the finite vs. infinitesimal invariance\n",
    "  check, and why the fermion bar-leg transform differs between $S_3$ and\n",
    "  $Z_N$.\n",
    "- **[GomezBock21]** M. Gómez-Bock, M. Mondragón, A. Pérez-Martínez,\n",
    "  *\"Scalar and gauge sectors in the 3-Higgs Doublet Model under the\n",
    "  S₃-symmetry\"*, Eur. Phys. J. C **81**, 942 (2021),\n",
    "  [arXiv:2102.02800](https://arxiv.org/abs/2102.02800),\n",
    "  [doi:10.1140/epjc/s10052-021-09731-3](https://doi.org/10.1140/epjc/s10052-021-09731-3)\n",
    "  — the literature model this whole notebook follows."
   ]
  }
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