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[Physics.Gen-Ph] 11 Jul 2020 I.Lpo Asinteraction Mass Lepton III

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[Physics.Gen-Ph] 11 Jul 2020 I.Lpo Asinteraction Mass Lepton III Particle generations in R0|18 dust gravity Robert N. C. Pfeifer∗ Dunedin, Otago, New Zealand (Dated: July 12, 2020) The R0|18 dust gravity model contains analogues to the particle spectrum and interactions of the Standard Model and gravity, but with only four tunable parameters. As the structure of this model is highly constrained, predictive relationships between its counterparts to the constants of the Standard Model may be obtained. In this paper, the model values for the masses of the tau, the W and Z bosons, and a Higgs-like scalar boson are calculated as functions of α, me, and mµ, with no free fitting parameters. They are shown to be 1776.867(1) MeV/c2, 80.3786(3) GeV/c2, 91.1877(4) GeV/c2, and 125.16(1) GeV/c2 respectively, all within 0.5 σ or better of the corresponding observed values of 1776.86(12) MeV/c2, 80.379(12) GeV/c2, 91.1876(21) GeV/c2, and 125.10(14) GeV/c2. This result suggests the existence of a unifying relationship between lepton generations and the electroweak mass scale, which is proposed to arise from preon interactions mediated by the strong nuclear force. 4 CONTENTS 4. Second-order correction to Kℓ from the tau channel 24 I. Introduction 1 5. Dilaton corrections to ∆e(mei ) 25 4 6. Corrections to [Ke(θe)] from the muon II. Boson Mass Interactions 2 and electron channels 25 A. W mass 2 0 18 1. Boson loops 3 IV. Relationships from R | dust gravity 27 2. QL photon and scalar interactions 5 A. Mass relationships 27 3. Universality of loop corrections 6 B. Minimum requirements for particle B. Z mass 7 generations from preon substructure 28 1. Boson loops 7 2. QL photon and scalar interactions 8 V. Conclusion 29 C. Weak mixing angle 9 D. Gluon masses 9 A. Gell-Mann matrices 29 E. Scalar boson mass 10 1. Vector boson loops 10 References 29 2. Scalar boson loops 11 F. Neutral boson gravitation 11 I. INTRODUCTION III. Lepton Mass Interaction 12 A. Leading order 12 Introduced in Ref. 1, R0 18 dust gravity is a model 1. Action on colour sector 12 | comprising a free dust field on the manifold R0 18. When 2. Mass from photon and gluon components | this dust field is in a highly disordered and hence (from a of QL 14 coarse-grained perspective) a highly homogeneous state, 3. Mass from scalar component of QL 15 it admits a description in the low-energy limit where soli- 4. Gluon and scalar field mass deficits 17 ton waves in the dust field behave as interacting quasi- B. Foreground Loop Corrections 17 1,3 0 18 particles on a R submanifold of R | . Although the 1. 1-loop EM corrections 17 0 18 geometry of R | limits the order of wavefunctions and arXiv:2008.05893v1 [physics.gen-ph] 11 Jul 2020 2. 1-loop gluon corrections 18 therefore prevents the dust field from being normalisable 3. 1-loop weak force corrections 20 on the R1,3 submanifold, the number of dust particles 4. 1-loop scalar corrections 21 participating in each quasiparticle is of O(1036), allow- 5. 2-loop EM corrections 21 ing the effective particles of the low-energy limit to ap- 36 C. Corrections to the lepton mass angle 21 proximate normalisable wavefunctions to order O(x10 ). 1. Origin of corrections 21 Fermionic preons condense into leptons, quarks, and a 2. Preamble 21 scalar boson, and choices of gauge cause these fields to 4 3. First-order correction to Kℓ from the tau admit approximate interpretation as normalised parti- channel 22 cle wavefunctions over a (1,3)-disc submanifold of R1,3 taken to correspond to the observable universe. Further choices of gauge collapse an emergent SU(9) symmetry to U(1) SU(2) SU(3)C while mapping the (1,3)-disc ∗ ⊗ ⊗ [email protected] to a region of curved Riemannian space-time denoted M˜ . 2 Breaking of the weak equivalence principle is anticipated, is an absolute square, and 0 otherwise. Interestingly, the but only in limited domains.(1) scalar boson is a composite particle made up of preons This paper examines the lepton mass interactions of and satisfies R0 18 2 | dust gravity, in which composite leptons acquire 9 (1) (1) [H′(x)H′∗(y)] = f (x y) . (9) mass through coupling to the high-entropy background h QLi 2 − EQL dust field. On M˜ , this dust field may be represented In the QL, the field A corresponds to theh photon,i and as a quantum liquid (QL) described in terms of its non- µ cc˜ corresponds to gluons associated with the Gell-Mann vanishing expectation values. In the presence of a com- µ basis of SU(3) given in Eq. (A3). The QL introduces a posite lepton, in the notation of Ref. 1 the familiar non- C preferred rest frame, but this is largely undetectable at vanishing bosonic components are (2) energy scales small compared to QL, which is compara- 2 E µ (2) (2) ble to the Planck energy. [A (x)Aµ(y)] = f (x y) (1) h QLi − − EQL Finally, there is also a dilaton field. This is introduced h i2 in Sec. IIIE1a of Ref. 1, and is denoted ∆. The gradient [ccµ˜ (x)cc˜ (y)] = f (2)(x y) (2) (2) µ QL c˜ 1,...,8 QL of the dilaton field, h i ∈{ } − − E h i ∆ = ∂ ∆ (10) where f (2)(x) is a Gaussian satisfying µ µ acts as a vector boson having no charge with respect to 4 (2) either SU(3) or SU(3) . As its QL expectation value d4x f (2)(x y)=1 y. (3) A C EQL − ∀ has not been zeroed by gauge, by maximisation of en- Z h i tropy in the QL this boson also satisfies a relationship These arise from a more general expression 2 µ f (2) (2) mm˙ nnµ˙ [∆ (x)∆µ(y)]QL = (x y) QL . (11) ∂σ∂ ϕ(x) ∂σ∂ ϕ(y) (4) − − E h µ i h i 2 By Eqs. (57–64) of Ref. 1, boson ∆µ is analogous to mm˙ nn˙ (2) (2) vector boson ϕ81 [1, Eq. (63)], and therefore couples to = Tr (e e )f (x y) QL µ − − E foreground fields with the same bare interaction strength h i which is written in terms of the underlying dust field ϕ. as the gluons, which are derived from the related bosons z The leptons are made up of three differently-coloured ϕµ z 1,...,80 of SU(9) [1, Sec. IIID1]. The dilaton gradi- | ∈{ } preons, generically ent field may be treated as a ninth gluon, associated with the trivial representation of su(3)C. In what follows, the Ψagα(x) εαβεγδ εαγ εβδ + εαδεβγ (5) family of gluons will be taken to include the vector dila- ∝ − g ac1 ac ac3 ton gradient, giving a combined symmetry group ψ′ (x )ψ′ 2 (x )ψ′ (x ) c1c2c3 β 1 γ 2 δ 3 ×C R SU(3)C 1 ∼= GL(3, )C. (12) where g is the generation index, and the coefficients ⊕ g aα When referring to uncertainty in results, experimental c1c2c3 are constrained by requiring that particle Ψ be bothC colourless and an eigenstate of the mass-generating uncertainties will be denoted σexp, and uncertainties in interaction with the QL. The preon expectation values the theoretical calculation will be denoted σth. satisfy In this paper, it is generally assumed that any paricle under study is at rest or near-rest with respect to the m˙ n˙ m n isotropy frame of the QL. It is worth noting that, as per [ψ′ (x) ψ′ (y) ψ′ (x) ψ′(y) ] h QLi Sec. IVE of Ref. 1, a particle may be considered close to 2 (6) 1 (1) (1) at rest if its boost parameter satisfies = (m, ˙ n,m,n˙ ) f (x y) QL 2 − E 32 4 h i β < 1 10 . (13) d4x (1) f (1)(x y)=1 y (7) − EQL − ∀ Z h i II. BOSON MASS INTERACTIONS where m (orm ˙ ) and n (orn ˙ ) range from 1 to 9 and enumerate pairs of index values a (ora ˙ ) 1, 2, 3 , c (orc ˙) r,g,b . The symbol (m, ˙ n,m,n˙ ∈) is { defined} A. W mass to equal∈ 1{ iff } In Sec. IIIF3 of Ref. 1, a first-order expression for the m˙ n˙ m n [ψ′ (x) ψ′ (y) ψ′ (x) ψ′(y) ]QL (8) W boson mass was obtained in terms of two of the free 0 18 (1) h i parameters of R | dust gravity, f and : EQL 2 2 2 (1) mW = 18f QL [1 + O (α)] . (14) 1 E For massive bosons, higher-generation leptons, and for normal h i fermionic matter in regimes with an extremely low two-photon To obtain the high-precision numerical results presented scalar potential—comparable to deep interstellar, perhaps inter- in the present paper, it is necessary to evaluate some galactic space. higher-order corrections to this expression. 3 which take on QL values in the leading-order diagram of Fig. 1(i). These loops, in turn, may then either (ii-iii) act on a single preon triplet, or (iv) span from that triplet to another part of the diagram. However, once such a loop is introduced, the preons which are evaluated us- ing the QL mean field values no longer all arise from the same vertex. For these preons to remain correlated with their counterparts on the lower vertex, such that the 2 diagram’s contribution to mW does not vanish, the QL sources and sinks must continue to be within O( QL) of one another, and the loop correction must not introduceL any correlations with particles outside the local region (both spatial and temporal) within which the QL is self- correlated.
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