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Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification

Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom

arXiv:1304.8050v4hep-thhep-phmath.QA

TL;DR

The paper examines whether relaxing the first-order condition can extend the noncommutative-geometric Standard Model framework. It classifies the resulting spaces and finds a Pati-Salam-type model, with Higgs content that depends on the initial Dirac operator.

  • Problem

    The Standard Model’s three gauge couplings fail to unify at a high scale by a few percent, motivating study of physics beyond it.

    Source: §I · p. 3

  • Method

    The authors classify product noncommutative spaces without imposing the first-order condition on the Dirac operator.

    Source: §I · p. 4

  • Results

    The classification yields almost uniquely a Pati-Salam SU(2)R × SU(2)L × SU(4) model.

    Source: §I · p. 4

  • Takeaways & Limitations

    The model’s Higgs fields can include quadratic composites or additional fundamental fields, depending on the initial Dirac operator.

    Source: §I · p. 4

Abstract

from arXiv · show

The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_{R}x SU(2)_{L}x SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,1,4)+(1,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.

I. INTRODUCTION

The paper asks whether relaxing the first-order condition can expand the noncommutative-geometric model beyond the Standard Model. It classifies the resulting space and presents a Pati-Salam model with Higgs fields whose composition depends on the initial Dirac operator.

I. INTRODUCTION

  • The first-order condition restricts inner fluctuations to be linear and is central to the Standard Model’s unique selection in this setting.Without it, fluctuations can acquire a quadratic part expressed in terms of the linear part.Source: §I · p. 3
  • Dropping the first-order condition leads almost uniquely to the algebra HR ⊕ HL ⊕ M4(C) and a Pati-Salam SU(2)R × SU(2)L × SU(4) model.The model is built on a product of a continuous four-dimensional space and a finite space.Source: §I · p. 4
  • For certain initial Dirac-operator configurations, some Higgs fields are fundamental while others are quadratic products of fundamental fields; for generic initial Dirac operators, all Higgs fields are fundamental.Source: §I · p. 4
  • The additional Higgs fields provide spontaneous breaking of the Pati-Salam symmetry to U(1)em × SU(3)c.The paper also reports that the model truncates correctly to the Standard Model.Source: §I · p. 4

II. FIRST-ORDER CONDITION AND INNER FLUCTUATIONS

The first-order condition makes the usual linear inner fluctuations behave covariantly under gauge transformations. Without that condition, the fluctuations require a quadratic contribution and their gauge transformation is nonlinear.

II. FIRST-ORDER CONDITION AND INNER FLUCTUATIONS

  • The usual gauge-transformation argument for a fluctuated Dirac operator works only when the first-order condition is satisfied.The stated condition is [JuJ−1, A] = 0 for the gauge potential A.Source: §II · pp. 4–5
  • Without the first-order condition, the inner fluctuation includes a quadratic term A(2) in addition to the linear term A(1).A(2) vanishes when the first-order condition holds.Source: §II · p. 5
  • When the first-order condition fails, the quadratic contribution transforms nonlinearly using the gauge-transformed linear term.The authors interpret A(2) as the nonlinear correction to the inner fluctuations.Source: §II · p. 6
  • The semigroup structure ensures that fluctuating an already fluctuated Dirac operator still gives an inner fluctuation.The same algebraic rules apply to ordinary and real spectral triples.Source: §II · pp. 6–7

III. CLASSIFICATION OF FINITE GEOMETRIES WITHOUT FIRST ORDER

The classification considers finite noncommutative geometries without imposing the first-order condition on the Dirac operator. It yields a Pati-Salam-type gauge symmetry with 16 fundamental fermions, while the Higgs content depends on the initial Dirac operator’s condition.

III. CLASSIFICATION OF FINITE GEOMETRIES WITHOUT FIRST ORDER

  • The geometric framework had already reproduced the Standard Model’s fermion count, gauge symmetries, fermionic representations, and Higgs fields in this setting.The authors describe the resulting spectral action as giving the Standard Model coupled to gravity at a high-energy scale.Source: CONDITION · p. 8
  • The first-order condition had restricted HR ⊕ HL ⊕ M4(C) to the subalgebra C ⊕ H ⊕ M3(C).Removing that restriction motivates examining the more general algebra and its physical significance.Source: CONDITION · p. 8
  • Without imposing the first-order condition, the resulting gauge symmetry is almost uniquely Pati-Salam-type, and the number of fundamental fermions remains 16.SU(4) is the color group with lepton number as the fourth color.Source: CONDITION · pp. 8–9
  • With that initial order-one condition, Higgs fields in A(2) are quadratic composites of those in A(1); otherwise additional Higgs fields are fundamental.The additional fundamental representations are (3R, 1L, 10), (1R, 1L, 6), and (2R, 2L, 1 + 15).Source: CONDITION · p. 9
  • When the initial Dirac operator satisfies the order-one condition, the fundamental Higgs fields transform as (2R, 2L, 1), (2R, 1L, 4), and (1R, 1L, 1 + 15) with respect to SU(2)R × SU(2)L × SU(4).Source: CONDITION · p. 9

IV. SUMMARY OF TENSOR NOTATION

The section introduces tensor notation as a more practical way to present computations that would otherwise use large matrix representations. The notation also summarizes how the Dirac operator’s couplings distinguish Dirac masses from Majorana masses.

IV. SUMMARY OF TENSOR NOTATION

  • Tensorial notation offers a more practical alternative to handling the model as a 384 × 384 matrix representation.The authors say the tensorial computations are done by hand and can also be checked with algebraic manipulation programs.Source: §IV · p. 9
  • Non-zero mixing between primed and unprimed indices is possible only when a and b are restricted to C ⊕ H ⊕ M3(C).The generation-space matrices k*νR are assumed to be 3 × 3.Source: §IV · p. 11
  • The Yukawa-coupling structure gives Dirac masses to all fermions but Majorana masses only to right-handed neutrinos.The paper describes this as the unique possibility consistent with the stated setup.Source: §IV · p. 12

V. DIRAC OPERATOR AND INNER FLUCTUATIONS ON HR ⊕HL ⊕M4 (C)

Inner fluctuations in the SU(2)R × SU(2)L × SU(4) model include a nonlinear quadratic connection component. Depending on the initial Dirac operator’s order-one condition, additional Higgs fields are composite or independent.

V. DIRAC OPERATOR AND INNER FLUCTUATIONS ON HR ⊕HL ⊕M4 (C)

  • A(2) transforms nonlinearly and includes terms quadratic in the gauge-transformed linear connection A(1).The quadratic component is computed using the gauge-transformed A(1).Source: §V · p. 13
  • The stated field content includes traceless SU(4) generators and gauge fields for SU(2)R, SU(2)L, and SU(4).The passages also assign a (1R, 1L, 1 + 15) representation to a field and state that eφ is not independent.Source: §V · p. 14
  • With the subalgebra order-one condition, A(2) contains composite Higgs fields formed from products of fundamental fields.For generic initial Dirac operators, the field described by equation 24 is not defined by that relation, and additional fields occur in specified SU(2)R × SU(2)L × SU(4) representations.Source: §V · p. 15
  • The authors describe the Higgs representations as minimal and state that the additional Higgs fields in the connections may be independent or depend quadratically on fundamental fields.The text identifies whether these fields are independent or composite as depending on the order-one condition.Source: §V · p. 15

VI. THE SPECTRAL ACTION FOR THE SU (2)R × SU (2)L × SU (4) MODEL

The spectral action yields a bosonic SU(2)R × SU(2)L × SU(4) Pati-Salam model with a specified Higgs structure. It includes gauge-coupling unification and is expected to break spontaneously to the Standard Model symmetry at high energies.

VI. THE SPECTRAL ACTION FOR THE SU (2)R × SU (2)L × SU (4) MODEL

  • The bosonic action is derived from the Dirac operator and its inner fluctuations using the spectral action and heat-kernel coefficients.The action is expanded using a cutoff scale and an even, positive function.Source: §VI · pp. 15–16
  • The Higgs-field dependence of mass terms changes when the initial Dirac operator satisfies the order-one condition: specified fields become dependent on fundamental Higgs fields.In that case the mass terms can be expressed in terms of the fundamental Higgs field.Source: §VI · p. 17
  • The bosonic spectral action contains cosmological and Einstein-Hilbert terms, a Weyl-tensor-square term, gauge and Higgs kinetic terms, and Higgs mass and quartic terms.The authors characterize the result as a Pati-Salam model with a fixed Higgs structure.Source: §VI · p. 18
  • The action gives gauge-coupling unification and is expected to break spontaneously at high energies to the Standard Model symmetry U(1) × SU(2) × SU(3).The text proposes running the coupling relation with renormalization-group equations as a test against measured low-energy values.Source: §VI · p. 18

VII. TRUNCATION TO THE STANDARD MODEL

The section describes how the Pati-Salam model is truncated to the Standard Model. It identifies the Higgs-field substitutions and reports that the resulting bosonic action agrees with the cited Standard Model result.

VII. TRUNCATION TO THE STANDARD MODEL

  • The Higgs field φb.a in (2R, 2L, 1) is truncated to the Higgs doublet H, yielding the Standard Model bosonic action.The authors state that the resulting action agrees with reference.Source: §VII · pp. 18, 20, 21
  • The Higgs field Δ in (2R, 1, 4) is truncated to a real singlet scalar field.The text identifies an SU(4) gauge-field generator with B − L.Source: §VII · p. 19
  • The resulting Dirac-operator components are identified with the operators acting on the right-handed neutrino and right-handed electron.The text says analogous substitutions yield the expected results for the remaining fermions.Source: §VII · p. 19

VIII. THE POTENTIAL AND SYMMETRY BREAKING

The scalar potential is examined for symmetry-breaking vacua, including the Standard Model vacuum and alternative minima. The analysis also connects these vacua to gauge-boson masses and the emergence of the first-order condition.

VIII. THE POTENTIAL AND SYMMETRY BREAKING

  • The Higgs fields considered in the potential are composite rather than fundamental, and their potential is expressed in terms of fundamental fields.The discussion includes the quartic terms in that expression.Source: §VIII · p. 21
  • The potential has eighth-order field dependence, making it difficult to determine all its vacua; the authors identify full vacuum and unification analyses as future work.They expand around the starting vacuum, and also flag top-quark and Higgs masses for further analysis.Source: §VIII · pp. 22, 26
  • A Δ-field vacuum breaks SU(2)R × SU(2)L × SU(4) to U(1) × SU(2)L × SU(3)c.The Δ-fields transform as (2R, 1L, 4), and the vacuum is invariant only under the stated subgroup.Source: §VIII · p. 22
  • Unimodularity on U(A) induces unimodularity on the subalgebra, yielding the correct hypercharges for the fermions.Source: §VIII · p. 22
  • The φ-field potential can further break U(1) × SU(2)L × SU(3)c to U(1)em × SU(3)c, while its right-hand plot suggests a minimum other than the Standard Model vacuum.This potential is considered after the Δ- and Σ-fields acquire vevs.Source: §VIII · p. 22
  • The gauge-boson masses follow the breaking pattern: Bµ remains massless, heavy vectors acquire masses of order w², and the Standard Model WµL and Z have electroweak-scale masses.The W± mass receives a small correction of order gw; the Z′ has a mass of order the weak scale gv.Source: §VIII · pp. 24, 26
  • At low energies, one Higgs-field vacuum yields a Dirac operator satisfying the first-order condition, making that condition a spontaneously broken phase rather than an external imposition.The paper also identifies the first-order condition as a vacuum solution of the spectral action.Source: §VIII · pp. 22, 26

IX. APPENDIX: DETAILED CALCULATIONS FOR THE PRACTITIONER

The appendix details calculations for the product-space Dirac operator and spectral action, deriving the model’s gauge fields and Higgs content. It also distinguishes composite Higgs fields from additional fields that can be fundamental.

IX. APPENDIX: DETAILED CALCULATIONS FOR THE PRACTITIONER

  • The appendix identifies composite Higgs fields and states that spontaneous breaking yields Dirac and Majorana masses.The fundamental fields are described as similar in form to the fermion bilinear.Source: §IX · p. 31
  • The fundamental Higgs fields transform in (2R, 2L, 1), (2R, 1L, 4), and (1R, 1L, 1 + 15) representations.The last representation is stated as 1 + 15.Source: §IX · p. 32
  • For generic initial Dirac operators, additional fields can be independent fundamentals in (2R, 2L, 1 + 15), (3R, 1L, 10), and (1R, 1L, 6).Source: §IX · pp. 32–33
  • Gauge-field calculations identify SU(2)R and SU(2)L gauge fields alongside U(4) fields, with SU(4) generators.Source: §IX · pp. 32–33
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