pyCHM

Open, pure-numpy/scipy Composite Higgs Model calculator — the radiatively-generated Coleman–Weinberg Higgs potential, the electroweak vacuum, the spectrum, and four fine-tuning measures (Barbieri–Giudice, HOT, the information measure I = ½ log det(I+F), and the prior→posterior KL tuning), with two evaluation routes for the one-loop potential.

No private dependencies. The physics is the published two-site M4DCHM (SO(5)→SO(4)) of Murnane, The Landscape of Composite Higgs Models (arXiv:2606.18364).

There is no other public composite-Higgs potential-and-fine-tuning calculator — the analogue of SoftSUSY/FeynHiggs for SUSY. pyCHM fills that gap.

📖 Documentation: murnanedaniel.github.io/pyCHM — tutorial, physics, derivations, and the auto-generated API reference.

Install

pip install -e ".[test]"      # runtime + tests
pip install -e ".[docs]"      # to build the docs site (mkdocs build --strict)

Use

import pychm
m = pychm.Model('5-5-5')

# validated benchmark (xi ~ 0.070, m_t ~ 0.163 TeV, Delta_BG ~ 127). masses in TeV;
# the Y sector is stored as (mY, Y) with the singlet-Y mass mSY = mY + Y.
point = dict(
    mU=2.82899, mUt=1.5395, mD=2.30783, mDt=0.969391,
    mYu=0.00754655, Yu=7.95104, mYd=0.03112, Yd=0.60624,
    Delta_uL=1.12956, Delta_uR=1.11262, Delta_dL=0.5056, Delta_dR=0.106068,
    f=0.871937, f1=1.50568, fX=2.99808, g=0.67095, gp=0.358138, grho=5.44665, gX=2.96355)

m.spectrum(point)   # -> {xi, f, mt, mb, mh, mW, mZ, mtop_partner}  or None (no EWSB)
m.tuning(point)     # -> {BG, HOT, I, KL, J}

# is the point still allowed? map the spectrum onto current LHC/EWPT/SMEFT bounds
import pychm
print(pychm.constraints.report(m.spectrum(point)))   # pass/fail table with arXiv citations

The two routes

Both evaluate the same one-loop CW potential — ½ Tr log(p² + M²(s_h)) — by diagonalising the mass matrices M(s_h); they differ only in the per-eigenvalue kernel:

eigenvalue (default, the precision route): closed-form CW, K(m²) = m⁴ log m². Exact, manifestly finite. This is the route used for the spectrum and fine-tuning.
momentum: the divergence-subtracted Euclidean momentum integral, K(m²) = 4∫₀^∞ pₑ³ [log(pₑ²+m²) − log(pₑ²+1) − (m²−1)/(pₑ²+1) + (m²−1)²/2(pₑ²+1)²] dpₑ. The two subtractions cancel the quadratic and log UV divergences at the integrand level; the leftover polynomial in m² drops out of V(s_h)−V(0) because the supertraces Str 1, m², m⁴ are all s_h-independent.

The two routes agree on the potential curve V(s_h) to ~1% of its depth (CI enforces it on random points, tests/test_routes_equivalence.py). Caveat: a tuned electroweak vacuum is a near-cancellation, which amplifies the momentum route's residual quadrature error into a several-percent error on xi (and a large error on m_h''). That is exactly why fine-tuning is computed with the closed-form eigenvalue route — and is the central lesson of the project.

Validation

On the benchmark point above, pyCHM reproduces an independent mass-eigenvalue engine (the one behind the published global fits of arXiv:2101.00428; not redistributed here) with no shared code:

quantity	pyCHM	independent engine
`xi` (vacuum)	0.07006	0.07005
`Delta_BG`	126.9	126.7
HOT (`	J	`)
`I` (nats)	5.27	5.27

i.e. 0.03% on xi and <0.5% on Delta_BG. The fine-tuning is differentiated in the fundamental Lagrangian-mass basis {mU, mUt, mY, mSY, mD, mDt, …, Δ}, which is the basis the BG number is defined over.

14-14-10

The 14-14-10 representation (q_L and t_R partners in the symmetric 14 of SO(5), b_R in the 10) is wired into the same eigenvalue-route pipeline:

m = pychm.Model('14-14-10')
point = dict(
    mQ=3.965100934, mU=2.397190093, mD=1.482462237,
    mYu=0.1478991938, Yu=0.4989142322, Ytu=2.538912347, Yd=0.5293788277,
    Delta_q=2.850102384, Delta_u=1.915943759, Delta_d=0.2222079908,
    f=1.4396717743254574, f1=1.8934618718580186, fX=2.2076564878795057,
    g=0.6709494105248374, gp=0.3581380846874656, grho=5.525258191589697, gX=4.696070132753916)
m.spectrum(point)   # xi ~ 0.0268, m_t ~ 0.128 TeV
m.tuning(point)     # Delta_BG ~ 16.2

On this benchmark pyCHM reproduces the independent engine to 0.0002% on sh, <0.01% on m_t, 0.06% on m_h and ~0.1% on Delta_BG (tests/test_mchm14.py). The BG measure is differentiated in the 14-14-10 Lagrangian-mass basis {mQ, mU, mD, mYu, mSYu, mSYtu, Yd, Δq, Δu, Δd}, with the two singlet-Y combinations mSYu = mYu + Yu/2 and mSYtu = mYu + 4(Yu+Ytu)/5 held independent of mYu.

14-1-10

The 14-1-10 representation (q_L partner in the symmetric 14 of SO(5), t_R partner an SO(4) singlet, b_R in the 10) is the simpler cousin: with a singlet t_R partner the up Y-sector collapses to a single (mU, Yu) — no mYu/Ytu — so the fundamental massive basis is just {mQ, mU, mD, Yu, Yd, Δq, Δu, Δd}, each mapping directly onto a mass-matrix entry.

m = pychm.Model('14-1-10')
point = dict(
    mQ=1.5774575432442343, mU=0.053295037861913734, mD=3.9639579843374463,
    Yu=3.213522953043455, Yd=1.5033041681268544,
    Delta_q=0.8229454568382461, Delta_u=3.843401100178204, Delta_d=0.19968869968780683,
    f=1.1329813572955627, f1=1.5049448725076554, fX=1.4519742141888066,
    g=0.6709494105248374, gp=0.3581380846874656, grho=3.3254933316652062, gX=9.7650801141742978)
m.spectrum(point)   # xi ~ 0.0485, m_t ~ 0.154 TeV
m.tuning(point)     # Delta_BG ~ 7.3 (argmax Delta_q)

On this benchmark pyCHM reproduces the independent engine to 0.035% on sh, 0.034% on m_t, <0.01% on m_h (matched scheme at the oracle f) and 0.038% on Delta_BG (tests/test_mchm14_1_10.py).

6-6-6 (NMCHM, SO(6)/SO(5))

The Next-to-Minimal model promotes the coset to SO(6)/SO(5): the five pNGBs are the Higgs doublet plus a real SO(5) singlet s, and the quark partners sit in the 6 (the NM4DCHM6). The full SO(6) representation tower — the 6, the adjoint 15, the symmetric-traceless 20', the self-dual 10/10bar and the Weyl spinors 4/4bar — is built from group theory in pychm.groups.so6 (closed-form Goldstone via Rodrigues, lifts via the same tensor / Clifford machinery as SO(5)), and validated against the thesis in closed form (the Goldstone Φ eq. 474, the broken generators eq. 632, the 6 = 4+1+1 embedding eq. 633) and by the SO(6)→SO(5) branchings.

m = pychm.Model('6-6-6')
m.spectrum(point)   # same xi/m_t/m_h as 5-5-5 at <s>=0 (inherits the pypngb anchor)
m.tuning(point)     # same Delta_BG
from pychm import nmchm6
nmchm6.singlet_mass2(point, thv, f=f)   # the new SO(5)-singlet pNGB mass (massless in the MCHM)

At <s>=0 the SO(6) mass matrices reduce to the validated 5-5-5 entry-for-entry (so EWSB and the anchor carry over), and the model adds the genuine NMCHM observable: a finite, calculable singlet pNGB mass from the fermion loop (tests/test_so6.py, test_so6_spinors.py, test_nmchm6.py; see docs/THESIS_VALIDATION.md for the derived-vs-input-vs-anchored boundary).

5-5-su5 / 15-15-su5 (SU(5)/SO(5), littlest Higgs)

The SU(5)/SO(5) coset (Ferretti real-rep / littlest Higgs) carries 14 pNGBs — a Higgs doublet, a complex triplet, and a singlet (14 → (3,3)+(2,2)+(1,1) under custodial SO(4)). It is built on the same engine via the type-AI Cartan split (groups.lie.cartan_split), with the SU(N) symmetric-15 rep available through the no-trace-removal tensor builder (tensor_basis(..., group='SU')). Two runnable variants: partners in the fundamental 5 ('5-5-su5') and in the symmetric 15 ('15-15-su5').

m = pychm.Model('5-5-su5')
s = m.spectrum(point)          # EWSB fires; m_t, m_h match the MCHM Higgs sector
from pychm import su5so5
su5so5.singlet_mass2(point, thv, f=s['f'])   # the NEW observables: the singlet pNGB mass
su5so5.triplet_mass2(point, thv, f=s['f'])   # and the complex-triplet pNGB mass

Beyond EWSB/m_t/m_h, the model predicts the triplet and singlet pNGB masses (curvatures of the multi-field CW potential), the genuinely new observables relative to the MCHM/NMCHM (tests/test_su5so5.py).

The coset landscape

pychm.groups.landscape turns the abstraction around: it enumerates cosets G/H on the engine, branches each under a custodial SO(4), and flags those with a (2,2) Higgs — reproducing a slice of the 642-model classification of Chala & Fonseca (arXiv:2309.10635) from pyCHM's own Coset:

>>> from pychm.groups import landscape; landscape.print_scan(max_ngb=14)
SO(5)/SO(4)   4  YES  (2,2)                    # MCHM
SO(6)/SO(5)   5  YES  (1,1) + (2,2)            # NMCHM
SO(7)/SO(6)   6  YES  2x(1,1) + (2,2)
SU(5)/SO(5)  14  YES  (1,1) + (2,2) + (3,3)    # littlest Higgs

See docs/COSET_LANDSCAPE.md for the survey: the custodial-(2,2) + symmetric-space organizing principle, the systematic tables, and the completeness status.

Toward a generic spectrum generator (`groups/so5.py`)

The three models above hand-code their fermion mass matrices. Their only representation-specific content is the Higgs (Goldstone) dressing — the s_h factors cos(h/f), sin(h/f), cos²(h/2f) for the 5; (3+5cos2h/f)/8, √5 sin(2h/f)/4 for the 14 — which are just the matrix elements of the Goldstone matrix U(h) in the chosen SO(5) irrep. pychm.groups.so5 builds U_R(h) for the 5, 10 and 14 from group theory, so this dressing follows from one construction for any partner representation:

import numpy as np
from pychm.groups import so5
ES = so5.embedding('14', 'singlet')
so5.overlap('14', ES, ES, h_over_f)        # == (3 + 5 cos(2 h/f)) / 8, exactly

tests/test_ccwz.py checks these reproduce the hand-coded factors (the 14 singlet overlap to machine precision). Everything downstream — the Coleman–Weinberg potential, the vacuum, the spectrum and the four tuning measures — is already representation-agnostic and dispatches on a model= string.

The assembler (assemble.py) closes the loop. Given a declarative spec — the partner representation, the elementary embeddings, and the composite states (masses + SO(5) content) — it emits mass_U/mass_D with the Higgs dressing supplied by groups.so5, for any representation. It is validated to reproduce all three hand-coded models — 5-5-5, 14-1-10 and 14-14-10 — entry-for-entry to machine precision, exercising the 5, 10 and 14 of SO(5). Each assembled model is registered (pychm.Model('5-5-5-assembled'), '14-1-10-assembled', '14-14-10-assembled') and reproduces the full validated spectrum and tuning end-to-end, to the precision the tuned vacuum permits:

import pychm
pychm.Model('14-14-10-assembled').spectrum(point)  # 19x19 up sector from 14/10 embeddings + groups.so5

So a composite-Higgs model is now specifiable purely by group-theoretic data: choose the partner representations, write down where the elementary fermions embed, and groups.so5 + assemble build the Higgs-dependent mass matrices — no per-model transcription. tests/test_assemble.py is the regression harness (every model checked entry-for-entry and end-to-end against its hand-coded oracle).

Symbolic derivation from scratch, and arbitrary representations (`pychm.groups`)

The dressing factors are not only computed numerically — they are derived in closed form. The symbolic engine builds the Goldstone matrix U_R(θ) (θ = h/f) with sympy and reduces each overlap to a closed trigonometric expression, so the benchmark factors follow from group theory with no curve-fitting:

from pychm.groups import core, derive
ES = so5.embedding_sym('14', 'singlet')
so5.overlap_sym('14', ES, ES)                 # -> (5*cos(2*θ) + 3)/8, derived symbolically

The previous assembler reverse-fit its composite states by least squares over s_h samples; that is gone. Every benchmark dressing is now proven to be a genuine Goldstone matrix element ⟨c|U_R|E⟩ by an exact symbolic solve (groups.derive.solve_composite), and the assembled models lambdify these closed forms to numpy callables (no sympy on the hot path).

The construction extends to any compatible representation:

Arbitrary tensor irreps — groups.tensors builds an orthonormal basis of any rank-k symmetric-traceless / antisymmetric SO(5) irrep and lifts U to it; groups.so5.U_rep(('sym', 3), s_h) is the 30, etc.
SO(4) decomposition — groups.decompose finds the (j_L, j_R) sub-multiplets via the two SU(2) Casimirs (5 = (2,2)+(1,1); 10 = (2,2)+(3,1)+(1,3); 14 = (3,3)+(2,2)+(1,1)), with a compatible(rep, jL, jR) predicate for placing the elementary fermions.
Spinorial reps — groups.spinors gives the SO(5)≅Sp(4) gamma matrices and the Goldstone matrix in the 4 (the MCHM4 partner, 4 = (2,1)+(1,2)) and the 16.
NMCHM SO(6)/SO(5) — groups.so6 / groups.so6_spinors give the SO(6) coset, the closed-form 5-pNGB Goldstone, and the full irrep tower 6 / 15 / 20' / 10 / 4 with their SO(6)→SO(5)→SO(4) branchings; nmchm6 is the worked NM4DCHM6 model.

New representations have no hand-coded oracle, so they are validated by internal consistency (unitarity, the representation homomorphism, the SO(4) branching) in tests/test_tensors.py, tests/test_spinors.py, tests/test_so6.py and tests/test_so6_spinors.py.

Validation against the thesis

tests/test_thesis_equations.py cross-checks pyCHM against the published equations of Murnane, The Landscape of Composite Higgs Models (arXiv:2606.18364) in closed form: the Goldstone matrix and SO(5) generators, the SO(4) bases and branchings, the App. A7 form-factor building blocks (verbatim), the per-representation Higgs dressing, the Coleman–Weinberg kernel and pole mass, the vacuum / Higgs-mass / gauge relations, and the Barbieri–Giudice tuning. The 14-rep prefactors are derived as exact SO(4) Clebsch weights (groups.decompose. channel_weights_sym); the overall 4/5 normalization is itself derived from the 14-singlet embedding (|S₄₄|²=4/5), so the prefactors are group theory end to end, not a thesis read-off.

The first-principles derivations behind every "derived" claim are written up in docs/DERIVATIONS.md (and docs/DERIVATIONS.pdf), each step tied to the function and the test that closes it.

Scope is stated precisely in docs/THESIS_VALIDATION.md: the Goldstone dressing is derived from group theory, the model structure is thesis input, and the validated equation classes are the ones the library implements. The NMCHM SO(6)/SO(5) representations (Ch.7/8) are now implemented and validated in closed form (Goldstone Φ eq. 474, broken generators eq. 632, the 6 embedding eq. 633, the full rep tower and its branchings); higher-order tuning, Bayesian evidence, large-N and the scanning machinery remain out of scope (not implemented). The thesis has no per-point numeric tables: the 14 models are anchored to the independent pypngb engine to <0.1%, and the NM4DCHM6 inherits that anchor by reducing to the 5-5-5 at <s>=0. This is a precise validation of the implemented physics, not a sweep of all 218 pages.

Status

	state
5-5-5 fermion + gauge mass matrices, both routes	✅
electroweak vacuum, spectrum (`xi, f, mt, mb, mh, mW, mZ`)	✅
four fine-tuning measures (BG/HOT/I/KL), validated to <0.5%	✅
CI route-equivalence on random points	✅
14-14-10 representation (eigenvalue route), validated to <0.1%	✅
14-1-10 representation (eigenvalue route), validated to <0.1%	✅
generic CCWZ Goldstone dressing (`groups/so5.py`, reps 5/10/14), rep factors validated	✅
generic mass-matrix assembler (`assemble.py`), reproduces all 3 models (reps 5/10/14)	✅
symbolic CCWZ engine (`groups/`): dressing derived in closed form, curve-fitting removed	✅
arbitrary tensor irreps + SO(4) decomposition (5/10/14/30/…); spinor reps 4, 16	✅
NMCHM SO(6)/SO(5): reps 6/15/20'/10/4, closed-form Goldstone + branchings; NM4DCHM6 model + singlet pNGB	✅
group-agnostic `Coset(G,H)` engine: SO(N), SU(N), Sp(N) generators, one Goldstone, one branching	✅
SU(4)/Sp(4) coset cross-checked against SO(6)/SO(5); SU(5)/SO(5) littlest-Higgs models (5 + 15) + triplet/singlet pNGB masses	✅
coset-landscape enumerator (`groups.landscape`): reproduces a slice of the 642-model classification (arXiv:2309.10635)	✅
CI: all models + cosets + the landscape (146 tests, 3.9/3.11/3.12)	✅

Licence

MIT.

Name		Name	Last commit message	Last commit date
Latest commit History 43 Commits
.github/workflows		.github/workflows
docs		docs
examples		examples
site		site
src/pychm		src/pychm
tests		tests
validation		validation
.gitignore		.gitignore
CLAUDE.md		CLAUDE.md
LICENSE		LICENSE
README.md		README.md
mkdocs.yml		mkdocs.yml
pyproject.toml		pyproject.toml

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pyCHM

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The two routes

Validation

14-14-10

14-1-10

6-6-6 (NMCHM, SO(6)/SO(5))

5-5-su5 / 15-15-su5 (SU(5)/SO(5), littlest Higgs)

The coset landscape

Toward a generic spectrum generator (`groups/so5.py`)

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Validation against the thesis

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pyCHM

Install

Use

The two routes

Validation

14-14-10

14-1-10

6-6-6 (NMCHM, SO(6)/SO(5))

5-5-su5 / 15-15-su5 (SU(5)/SO(5), littlest Higgs)

The coset landscape

Toward a generic spectrum generator (groups/so5.py)

Symbolic derivation from scratch, and arbitrary representations (pychm.groups)

Validation against the thesis

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Toward a generic spectrum generator (`groups/so5.py`)

Symbolic derivation from scratch, and arbitrary representations (`pychm.groups`)