Spectral Pati-Salam model from noncommutative geometry: vacuum structure, quantum stability of the inner-fluctuation connection, and quantitative phenomenology

The noncommutative-geometric construction of Chamseddine, Connes and van Suijlekom (JHEP 2013) derives, from the spectral action on an almost-commutative space with finite algebra $\mathbb{H}_R\oplus\mathbb{H}_L\oplus M_4(\mathbb{C})$ used without imposing the first order condition, a Pati-Salam model with gauge group $SU(2)_R\times SU(2)_L\times SU(4)$, a rigid spectrally determined scalar sector (fundamental Higgs fields plus composite or additional fundamental scalars depending on the initial Dirac operator), and tree-level coupling unification $g_R=g_L=g$ at the cutoff scale. The paper itself left the complete analysis of this model open. The authors' 2015 follow-up established one-loop gauge-coupling unification for all three scalar-content scenarios; the surviving open core is: (i) global minimization of the spectral scalar potential (effective order-eight dependence, fixed coefficients) and classification of vacua and symmetry-breaking chains — including whether a viable Standard-Model-breaking vacuum exists at all, which an unpublished 2019 preprint contests; (ii) whether the connection structure — the semigroup of inner perturbations and the composite versus fundamental character of the $A_{(2)}$ scalars — is preserved under quantization, for which no quantum field theory formalism faithful to the noncommutative structure yet exists; (iii) complete quantitative phenomenology: gauge-boson mass matrices and $Z$–$Z'$, $W_L$–$W_R$ mixing with experimental constraints, renormalization-group analysis beyond one loop with threshold effects, and predictions for the top-quark mass, the Higgs mass and the singlet $\sigma$; and (iv) systematic classification of admissible initial finite Dirac operators delimiting the composite-Higgs and fundamental-Higgs regimes.

Background

In noncommutative geometry (NCG), a physical space is described by a spectral triple $(\mathcal{A},\mathcal{H},D)$: an involutive algebra $\mathcal{A}$ of generalized coordinate functions represented on a Hilbert space $\mathcal{H}$, together with a self-adjoint Dirac operator $D$ whose spectrum encodes the metric. An almost-commutative manifold is the product of an ordinary four-dimensional Riemannian spin manifold $M$ with a finite noncommutative space, meaning $\mathcal{A}=C^\infty(M)\otimes A_F$ for a finite-dimensional *-algebra $A_F$; the Hilbert space carries the fermions of the model, one copy per generation. The spectral action principle of Chamseddine and Connes postulates that the bosonic action is

$$S=\mathrm{Tr}\,f(D_A/\Lambda),$$

where $D_A$ is the fluctuated Dirac operator and $f$ a positive cutoff function at scale $\Lambda$; its heat-kernel expansion (an asymptotic expansion in powers of $\Lambda$ with coefficients given by curvature invariants of $D_A$) produces Einstein gravity together with Yang-Mills and scalar terms whose coefficients are fixed by the moments $F_n=\int f(u)u^n\,du$ of the cutoff function.

Gauge bosons and Higgs fields both arise as inner fluctuations of the Dirac operator: invariance under inner automorphisms of $\mathcal{A}$ requires adding to $D$ a one-form $A$ and its conjugate $JAJ^{-1}$ (with $J$ the charge-conjugation operator implementing the real structure), and, once the first order condition is dropped, also a quadratic piece $A_{(2)}$. The first order condition — the requirement $[[D,a],Jb^*J^{-1}]=0$ for all algebra elements $a,b$ — was the key restriction that, in the earlier construction, singled out the finite algebra $\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ and essentially uniquely yielded the Standard Model (SM) with its fermion representations, hypercharges and Higgs field. Chamseddine, Connes and van Suijlekom first showed (arXiv:1304.7583) that without the first order condition the inner fluctuations form a semigroup extending the unitary group, and then applied the classification of such product spaces to the finite algebra $\mathbb{H}_R\oplus\mathbb{H}_L\oplus M_4(\mathbb{C})$ (arXiv:1304.8050, the source paper; here $\mathbb{H}$ is the quaternion algebra, the two copies $\mathbb{H}_R$ and $\mathbb{H}_L$ acting on right- and left-handed fermion doublets, and $M_4(\mathbb{C})$ the algebra mixing the four colors). The result is a Pati-Salam-type model — gauge group $SU(2)_R\times SU(2)_L\times SU(4)$, where $SU(4)$ unifies quarks and leptons with the lepton number as the fourth color — with, per generation, sixteen Weyl fermions in the $(2_R,1_L,4)\oplus(1_R,2_L,4)$ representation, fundamental Higgs fields in the $(2_R,2_L,1)$, $(2_R,1_L,4)$ and $(1_R,1_L,1+15)$ representations, and additional scalars in $A_{(2)}$ that are composites (quadratic in the fundamental Higgs fields) when the initial Dirac operator satisfies the first order condition relative to the SM subalgebra, or additional fundamental fields in the $(3_R,1_L,10)$, $(1_R,1_L,6)$, $(2_R,2_L,1+15)$ and possibly $(1,2_L,4)$ representations for a generic initial Dirac operator.

The spectral action then dictates the dynamics. Normalizing the gauge kinetic terms forces the tree-level unification $g_R=g_L=g$ at the cutoff scale. The scalar potential is rigid: quartic in the fields with coefficients fixed by the Yukawa matrices (the matrices of Higgs–fermion couplings) and the moments $F_n$ — no free parameters beyond the fermion sector — and of effective order eight in the fields once the composite scalars are expressed through the fundamental ones. Around one candidate vacuum, with the fields $\phi$, $\Sigma$ and $\Delta$ acquiring vacuum expectation values (vevs) $v$, $u$, $w$, the authors verified by a Hessian computation that the SM vacuum is a local minimum, that the symmetry can break directly to $U(1)_{em}\times SU(3)_c$, that the $\Delta$-vev $w$ must supply the heavy Majorana masses of the right-handed neutrinos, placing it at the see-saw scale — at least of order $10^{11}$ GeV (the observed small neutrino masses then arise from heavy right-handed neutrinos via the see-saw mechanism) — and that the first order condition re-emerges as a vacuum property rather than an external imposition. The gauge-boson mass pattern is $W_R$ and $Z'$ heavy, of order $gw$, with the SM $W$ and $Z$ at the electroweak scale. The authors state in conclusion that it remains to minimize the potential to determine all possible minima, to check whether the model allows unification of the coupling constants, to determine the top-quark and Higgs masses, and that the $Z$–$Z'$ and $W_L$–$W_R$ mixing suppressions must be seen explicitly; in the body they flag 'the interesting question... whether the structure of the connection is preserved at the quantum level'.

Problem Statement

Complete the analysis of the spectral Pati-Salam model derived by Chamseddine, Connes and van Suijlekom (JHEP 2013) — the 'careful analysis' the authors deferred to future work — so as to determine the model's dynamics and phenomenological viability. The problem has four jointly supporting components (a single scientific objective: deciding whether the model is a viable unified extension of the Standard Model and what it predicts):

1. Vacuum structure. Determine the vacua of the spectral scalar potential — a fixed polynomial with effective order-eight dependence on the fields once composites are expanded, with coefficients fixed by the Yukawa matrices and the cutoff-function moments — for both the composite and the fundamental scalar content; classify the symmetry-breaking chains $SU(2)_R\times SU(2)_L\times SU(4)\to\cdots\to U(1)_{em}\times SU(3)_c$ and their scales; identify the global minimum for physically relevant parameter ranges; and settle whether a viable Standard-Model-breaking vacuum exists at all (an unpublished 2019 preprint contests this against the source paper's local analysis). The question whether a viable Standard-Model-breaking vacuum exists is independent of the classification of the vacua and of the identification of the global minimum over the full parameter ranges; a verdict either way on the existence question neither presupposes nor supplies the classification or the global minimum.

2. Quantum stability of the connection. Determine whether, and under what conditions, the structure of the connection is preserved at the quantum level: does quantization (functional integration over the inner perturbations) preserve the semigroup structure of the inner fluctuations and the composite — or independently fundamental — character of the $A_{(2)}$ scalars, do radiative corrections induce independent scalar degrees of freedom, or is the question not yet well posed because no quantization framework faithful to the noncommutative structure has been established?

3. Quantitative phenomenology. Compute the explicit mass matrices and mixing angles of the Standard-Model and heavy gauge bosons, including the suppression of $Z$–$Z'$ and $W_L$–$W_R$ mixing, with their experimental constraints; extend the renormalization-group analysis beyond the one-loop level, including threshold effects at the intermediate and unification scales (one-loop unification itself is established and need not be redone); and derive the model's quantitative predictions for the top-quark mass, the Higgs mass and the singlet $\sigma$, confronting them with experiment.

4. Classification of initial Dirac operators. Classify systematically the admissible initial finite Dirac operators (self-adjoint, satisfying the reality axioms and unimodularity — the trace condition on the inner fluctuations that restricts the emergent gauge group to its special-unitary part and fixes the fermion hypercharges — with or without the first order condition relative to the Standard-Model subalgebra) and delimit the parameter space separating the composite-Higgs and fundamental-Higgs regimes.

Here a vacuum counts as 'viable' when the symmetry-breaking chain reaches $U(1)_{em}\times SU(3)_c$ with the Standard-Model $W$ and $Z$ at the electroweak scale, $W_R$ and $Z'$ heavy at the order of the $\Delta$-vev, that vev $w$ at or above the see-saw scale ($w\gtrsim10^{11}$ GeV, so that the observed light-neutrino masses follow), and the potential stable — a minimum — about it; 'physically relevant parameter ranges' means the ranges of the model's free inputs, the Yukawa matrices and the cutoff-function moments $F_n$ under the normalization and positivity conditions of the spectral action, rather than an externally imposed grid.

References

1. Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom, 'Beyond the spectral standard model: emergence of Pati-Salam unification', Journal of High Energy Physics 11 (2013) 132, DOI 10.1007/jhep11(2013)132; arXiv:1304.8050, https://arxiv.org/abs/1304.8050 (source paper)
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