aa r X i v : . [ phy s i c s . g e n - ph ] J u l Particle generations in R | dust gravity Robert N. C. Pfeifer ∗ Dunedin, Otago, New Zealand (Dated: July 12, 2020)The R | dust gravity model contains analogues to the particle spectrum and interactions of theStandard Model and gravity, but with only four tunable parameters. As the structure of this model ishighly constrained, predictive relationships between its counterparts to the constants of the StandardModel 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 , m e , and m µ , with no free fittingparameters. They are shown to be 1776 . /c , 80 . /c , 91 . /c ,and 125 . /c respectively, all within 0 . σ or better of the corresponding observed values of1776 . /c , 80 . /c , 91 . /c , and 125 . /c . This resultsuggests the existence of a unifying relationship between lepton generations and the electroweakmass scale, which is proposed to arise from preon interactions mediated by the strong nuclear force. CONTENTS
I. Introduction 1II. Boson Mass Interactions 2A. W mass 21. Boson loops 32. QL photon and scalar interactions 53. Universality of loop corrections 6B. Z mass 71. Boson loops 72. QL photon and scalar interactions 8C. Weak mixing angle 9D. Gluon masses 9E. Scalar boson mass 101. Vector boson loops 102. Scalar boson loops 11F. Neutral boson gravitation 11III. Lepton Mass Interaction 12A. Leading order 121. Action on colour sector 122. Mass from photon and gluon componentsof QL 143. Mass from scalar component of QL 154. Gluon and scalar field mass deficits 17B. Foreground Loop Corrections 171. 1-loop EM corrections 172. 1-loop gluon corrections 183. 1-loop weak force corrections 204. 1-loop scalar corrections 215. 2-loop EM corrections 21C. Corrections to the lepton mass angle 211. Origin of corrections 212. Preamble 213. First-order correction to K ℓ from the tauchannel 22 ∗ [email protected]
4. Second-order correction to K ℓ from thetau channel 245. Dilaton corrections to ∆ e ( m e i ) 256. Corrections to [ K e ( θ e )] from the muonand electron channels 25IV. Relationships from R | dust gravity 27A. Mass relationships 27B. Minimum requirements for particlegenerations from preon substructure 28V. Conclusion 29A. Gell-Mann matrices 29References 29 I. INTRODUCTION
Introduced in Ref. 1, R | dust gravity is a modelcomprising a free dust field on the manifold R | . Whenthis dust field is in a highly disordered and hence (from acoarse-grained perspective) a highly homogeneous state,it admits a description in the low-energy limit where soli-ton waves in the dust field behave as interacting quasi-particles on a R , submanifold of R | . Although thegeometry of R | limits the order of wavefunctions andtherefore prevents the dust field from being normalisableon the R , submanifold, the number of dust particlesparticipating in each quasiparticle is of O(10 ), allow-ing the effective particles of the low-energy limit to ap-proximate normalisable wavefunctions to order O( x ).Fermionic preons condense into leptons, quarks, and ascalar boson, and choices of gauge cause these fields toadmit approximate interpretation as normalised parti-cle wavefunctions over a (1,3)-disc submanifold of R , taken to correspond to the observable universe. Furtherchoices of gauge collapse an emergent SU(9) symmetryto U(1) ⊗ SU(2) ⊗ SU(3) C while mapping the (1,3)-discto a region of curved Riemannian space-time denoted ˜ M .Breaking of the weak equivalence principle is anticipated,but only in limited domains. (1) This paper examines the lepton mass interactions of R | dust gravity, in which composite leptons acquiremass through coupling to the high-entropy backgrounddust field. On ˜ M , this dust field may be representedas a quantum liquid (QL) described in terms of its non-vanishing expectation values. In the presence of a com-posite lepton, in the notation of Ref. 1 the familiar non-vanishing bosonic components are h [ A µ ( x ) A µ ( y )] QL i = − f (2) ( x − y ) h E (2)QL i (1) h [ c ˜ cµ ( x ) c ˜ cµ ( y )] QL i (cid:12)(cid:12) ˜ c ∈{ ,..., } = − f (2) ( x − y ) h E (2)QL i (2)where f (2) ( x ) is a Gaussian satisfying Z d x h E (2)QL i f (2) ( x − y ) = 1 ∀ y. (3)These arise from a more general expression h (cid:2) ∂σ∂ (cid:3) ˙ mmµ ϕ ( x ) (cid:2) ∂σ∂ (cid:3) ˙ nnµ ϕ ( y ) i (4)= − Tr ( e ˙ mm e ˙ nn ) f (2) ( x − y ) h E (2)QL i which is written in terms of the underlying dust field ϕ .The leptons are made up of three differently-colouredpreons, genericallyΨ agα ( x ) ∝ (cid:0) ε αβ ε γδ − ε αγ ε βδ + ε αδ ε βγ (cid:1) (5) × C gc c c ψ ′ ac β ( x ) ψ ′ ac γ ( x ) ψ ′ ac δ ( x )where g is the generation index, and the coefficients C gc c c are constrained by requiring that particle Ψ aα beboth colourless and an eigenstate of the mass-generatinginteraction with the QL. The preon expectation valuessatisfy h [ ψ ′ ˙ m ( x ) ψ ′ ˙ n ( y ) ψ ′ m ( x ) ψ ′ ( y ) n ] QL i = 12 (cid:3) ( ˙ m, ˙ n, m, n ) f (1) ( x − y ) h E (1)QL i (6) Z d x h E (1)QL i f (1) ( x − y ) = 1 ∀ y (7)where m (or ˙ m ) and n (or ˙ n ) range from 1 to 9 andenumerate pairs of index values a (or ˙ a ) ∈ { , , } , c (or ˙ c ) ∈ { r, g, b } . The symbol (cid:3) ( ˙ m, ˙ n, m, n ) is definedto equal 1 iff h [ ψ ′ ˙ m ( x ) ψ ′ ˙ n ( y ) ψ ′ m ( x ) ψ ′ ( y ) n ] QL i (8) For massive bosons, higher-generation leptons, and for normalfermionic matter in regimes with an extremely low two-photonscalar potential—comparable to deep interstellar, perhaps inter-galactic space. is an absolute square, and 0 otherwise. Interestingly, thescalar boson is a composite particle made up of preonsand satisfies h [H ′ ( x )H ′∗ ( y )] QL i = 92 f (1) ( x − y ) h E (1)QL i . (9)In the QL, the field A µ corresponds to the photon, and c ˜ cµ corresponds to gluons associated with the Gell-Mannbasis of SU(3) C given in Eq. (A3). The QL introduces apreferred rest frame, but this is largely undetectable atenergy scales small compared to E (2)QL , which is compara-ble to the Planck energy.Finally, there is also a dilaton field. This is introducedin Sec. III E 1 a of Ref. 1, and is denoted ∆ . The gradientof the dilaton field, ∆ µ = ∂ µ ∆ (10)acts as a vector boson having no charge with respect toeither SU(3) A or SU(3) C . As its QL expectation valuehas not been zeroed by gauge, by maximisation of en-tropy in the QL this boson also satisfies a relationship[ ∆ µ ( x ) ∆ µ ( y )] QL = − f (2) ( x − y ) h E (2)QL i . (11)By Eqs. (57–64) of Ref. 1, boson ∆ µ is analogous tovector boson ϕ µ [1, Eq. (63)], and therefore couples toforeground fields with the same bare interaction strengthas the gluons, which are derived from the related bosons ϕ zµ | z ∈{ ,..., } of SU(9) [1, Sec. III D 1]. The dilaton gradi-ent field may be treated as a ninth gluon, associated withthe trivial representation of su(3) C . In what follows, thefamily of gluons will be taken to include the vector dila-ton gradient, giving a combined symmetry groupSU(3) C ⊕ ∼ = GL(3 , R ) C . (12)When referring to uncertainty in results, experimentaluncertainties will be denoted σ exp , and uncertainties inthe theoretical calculation will be denoted σ th .In this paper, it is generally assumed that any paricleunder study is at rest or near-rest with respect to theisotropy frame of the QL. It is worth noting that, as perSec. IV E of Ref. 1, a particle may be considered close toat rest if its boost parameter satisfies β < − . (13) II. BOSON MASS INTERACTIONSA. W mass In Sec. III F 3 of Ref. 1, a first-order expression for the W boson mass was obtained in terms of two of the freeparameters of R | dust gravity, f and E (1)QL : m W = 18 f h E (1)QL i [1 + O ( α )] . (14)To obtain the high-precision numerical results presentedin the present paper, it is necessary to evaluate somehigher-order corrections to this expression. FIG. 1. (i) Leading-order contribution to the W bosonmass. Loop diagrams may (ii)-(iv) modify the W boson mass-squared interaction, or (v) merely involve it. The StandardModel counterpart to diagram (v) is shown in diagram (vi).
1. Boson loops
The leading contribution to the mass of the W bo-son arises from interactions between the W boson andthe preon components of the QL. This interaction is cor-rected by numerous boson loops. When evaluating thesecorrections, it is important to distinguish between loopswhich modify the mass-squared interaction, and thosewhich merely involve the mass-squared interaction, asper the examples in Fig. 1.A diagram is said to “merely involve” the mass-squaredinteraction if the leading-order interaction of Fig. 1(i) canbe replaced by a simple mass vertex m W W † W and thefigure in question then obtained from expansion of thepropagator in terms of the simple mass vertex and higher-order loop corrections of the Standard Model. Wherethese same diagrams are present both in the StandardModel and in R | dust gravity, they may be discountedas they do not contribute to calculation of the valueof m W on the mass vertex. (In MS renormalisation,the mass vertex value of m W corresponds to the phys-ically observable mass.) An example of such a diagramis Fig. 1(v), with Standard Model counterpart shown inFig. 1(vi).Characteristic of the diagrams which do actually mod-ify the mass-squared vertex, and thus impact the valueof m W , is that the loop boson couples to the fermions which take on QL values in the leading-order diagram ofFig. 1(i). These loops, in turn, may then either (ii-iii) acton a single preon triplet, or (iv) span from that tripletto another part of the diagram. However, once such aloop is introduced, the preons which are evaluated us-ing the QL mean field values no longer all arise fromthe same vertex. For these preons to remain correlatedwith their counterparts on the lower vertex, such that thediagram’s contribution to m W does not vanish, the QLsources and sinks must continue to be within O( L QL ) ofone another, and the loop correction must not introduceany correlations with particles outside the local region(both spatial and temporal) within which the QL is self-correlated. This region has dimensions of order L QL , andis termed the autocorrelation region.Now recognise that in Figs. 1(ii)-(iii) the loop spansbetween two components of a composite fermion. Theseare necessarily separated, on average, by a distance ofO( L Ψ ) = O[ L (2)QL ], which is both the length scale asso-ciated with fermion mass-squared interactions (as theseare quadratic in the QL fields), and the maximum sep-aration of the preon components of a fermion triplet. Ifthe W boson is at rest in the isotropy frame of the QL, aloop boson propagating between two of these preons willtraverse a distance O[ L (2)QL ] in a time equal to or greaterthan O[ c − L (2)QL ], depending on whether there are massinteractions along its course. For zero mass interactionsa trajectory along the light cone is on-shell, and attractsno mass ratio factor f (cid:16) m f /m b (cid:17) (for masses of a fermion f and boson b ) as the propagating foreground boson ismassless in this context. Where mass interactions takeplace during boson propagation, these slow the propaga-tion of the boson across the distance of O[ L (2)QL ]. For anynon-negligible mass, the time to traverse this distancethen exceeds O( c − L (2)QL ) causing the ends of the loopcorrection to no longer both lie within the autocorrela-tion region for the W mass interaction. Thus only thelight cone trajectory need be considered, and the parti-cles participating in the loop are effectively massless.Note that this effect (in which particles may appearmassless) is only seen in contexts where an observableproperty is dependent on nonvanishing QL correlatorsdistributed across spatially disparate vertices, and thusunder normal circumstances the zero-mass channel is not apparent as massless propagation of any particle across adistance large compared with L QL is extremely improb-able due to the large number of candidate backgroundQL interactions. Even over a distance of O( L QL ) thereare at least O(10 ) such candidate interactions, and themassless channel is only observed when evaluation of theQL preon correlators systematically eliminates the con-tributions involving massive particles in the loop.Next, consider Fig. 1(iv) which connects the QL pre-ons to the external W boson line. This process maytake place at arbitrary length scales as connection to the W boson may take place anywhere along its previous orsubsequent world line. There is no reason for the mostheavily-weighted contributions to be those over lengthscales of O( L QL ) or less, and thus the loop will in gen-eral encircle multiple interactions between the W bosonand the QL in the manner of Fig. 1(i). In the absenceof any mechanism eliminating interactions over largerlength scales, the contributions arising from this diagramsum over all length scales and the long-range couplingsintroduced by the loop boson thus eliminate sufficientcorrelations between the QL preon fields to cause thecontribution of this diagram to m W to vanish.Having thus established (i) that it is only necessaryto consider boson loops spanning from one preon to an-other within a single composite fermion, and (ii) thatall bosons in these loops are effectively massless over thelength scales involved, these loop corrections may be eval-uated as follows: a. Gluon loops: It is convenient to work in the e ij basis of gl(3 , R ). A gluon may arise from one preon andbe absorbed by another, as in Fig. 1(ii), or be emittedand absorbed by the same preon as in Fig. 1(iii). Notethat the choice of preon propagator to evaluate in theQL mean-field regime in Fig. 1(iii) prevents this diagrambeing reduced to a foreground self-energy diagram andrequires its inclusion in the calculation of m W .There are three choices of source preon and threechoices of sink preon, for a total of nine gluon loop correc-tion diagrams. By GL(3 , R ) symmetry, all of these makeequal contributions to m W . It is convenient to work inthe e ij basis of gl(3 , R ), evaluate the correction associ-ated with a gluon which is off-diagonal in colour, andmultiply by nine.For such a diagram, the vertices are each associatedwith a factor of f , lasing in the presence of the QL yieldsa factor of h N QL i + 1, and the definition of a fermion in-troduces a factor of 1 /
3. As the boson is off-diagonal, theloop correction attracts a structural factor of 10 / W boson correc-tion to the electromagnetic anomaly [2], though found toadd to, rather than subtract from, the original vertex.A factor of six arises from the admissible permutationsof preon colour, but this results in a double-counting onintegrating over preon position (corresponding to inter-change of both position and colour), reducing this factorto three. The Gaussian integral yields 1 / (2 π ). The netcorrection associated with this diagram is therefore ofweight f ( h N QL i + 1) · · · · π = 60 α · π (cid:2) (cid:0) α (cid:1)(cid:3) (15)relative to Fig. 1(i), and the sum over all nine gluon cor-rections is of weight 60 α π . (16)Note that it is not necessary to evaluate a similar set ofcorrections for the lower vertex: The decision to replacethe lower preons in Fig. 1(ii) with the QL mean field FIG. 2. Underlying diagram which gives rise to Fig. 1(ii) onexpansion using mean field theory in the presence of a preonQL. value is a mathematical one, not a physical one. The basediagram is shown in Fig. 2. If, in evaluating this diagram,the preons above the loop boson were approximated bythe QL mean field regime, then the loop boson wouldyield a correction to the lower vertex, for the same overallresult. This yields two equivalent ways to evaluate thesame diagram in the presence of the QL to lowest order inthe mean field theory expansion, rather than two differentphysical processes. b. Photon, W , and Z boson loops: For the pho-ton, one can proceed as with the gluons, enumeratingall the single-preon interactions, or alternatively one canconsider a photon loop correction to the fermion as awhole. In this latter approach, consider the interac-tion between W boson and leptons, and recognise thata fermion/photon vertex sums over interactions with allthree preons and thus automatically enumerates the ninefigures described in Sec. II A 1 a. Appropriate selectionof preons to evaluate as arising from the QL is assumed.The resulting correction is then immediately seen to be α/ (2 π ).For the W /quark vertex, it is helpful to look atphoton/preon interactions and compare these with the W /lepton vertex. There are numerous ways to count theinteractions; one of the simplest is as follows: First, foreach vertex recognise that a single-preon photon loop isconstructed on the subspace of charged preons. The po-sitions and colours of these preons are immaterial. Thesubspace is equivalent for the two diagrams, and thereforeso is the contribution. Next, consider two-preon photonloops, which occur on the up quark limb for quark ver-tices, and the electron limb for the lepton vertices. In thiscase, focus on the subspace corresponding to the partic-ipating fermion. For the lepton there are three choicesof which pair of preons participates, giving three differ-ent locations for the photon loop (three possibilities onone preon configuration). For the up quark there arethree choices of preon configuration, with the unchargedpreon being in a different location for each, again givingthree different locations for the photon loop (one for eachof the three configurations). Again there is equivalency. FIG. 3. (i) Scalar boson loop on a fermion propagator.(ii) Crossing of scalar boson constituents.
The electromagnetic factor for the W /quark vertex isconsequently equivalent to that for the W /lepton vertex.For W boson loop corrections a similar approach maybe adopted, and it follows immediately from the whole-fermion perspective that since the W boson is species-changing, no valid W loop corrections exist.For Z boson loop corrections the corresponding inter-action weight is − f Z α π (17)where sin θ W = 1 − m W m Z (18) f Z = 13 h(cid:0) θ W − (cid:1) − i = 13 (cid:18) − m W m Z + 16 m W m Z (cid:19) (19)and the overall sign reflects that the customary defini-tion of f Z is negative, while the Z and photon terms areadditive. By the same argument as in the gluon sector,the Z boson in the loop is effectively massless over thislength scale and thus there is no factor of m f /m Z forsome fermion f . c. Scalar boson loop: A scalar boson loop may inter-act with a fermion as shown in Fig. 3(i)-(ii). As in Ref. 1,Sec. III F 4, since the vertex separation is ≤ L Ψ the scalarboson constituents may either connect to the vertices col-lectively as a single composite particle [Fig. 3(i)] or withcrossing as shown in Fig. 3(ii). To evaluate Fig. 3(i),recognise that spatially exchanging two preons as in di-agram (iii) reveals equivalency to a loop correction aris-ing from a composite vector boson, which is foregroundby construction, massless by the same argument as inSec. II A 1 a, and may be replaced by the equivalent fun-damental vector boson using Eqs. (4) and (6) to elimi-nate the factor of (cid:2) E (2)QL / E (1)QL (cid:3) which accompanies termsin H ′ H ′∗ . Following this reduction, the resulting masslessforeground boson loop yields a factor of α/ (2 π ). This is FIG. 4. (i) Coupling between W boson and QL photon field.(ii) Coupling between W boson and QL scalar field. For thescalar boson there is no need to separately consider crossedand uncrossed configurations, as all such symmetry factorsare incorporated into the mean-squared QL field value. multiplied by three for the choice of which preon to in-vert in Fig. 3(i), and by [1 + 1 / (2 π )] for diagram (ii) bythe same arguments as presented in Sec. III F 4 of Ref. 1.The net weight of the scalar boson correction is therefore3 α π (cid:18) π (cid:19) . (20) d. Net effect of all boson loops: The net effect of theboson loop corrections is therefore to amend the W bosonmass equation to m W = 18 f h E (1)QL i (cid:18)
64 + 32 π − f Z (cid:19) α π (21)+ O " E (2)QL E (1)QL + O (cid:0) α (cid:1)! f Z = 13 (cid:18) − m W m Z + 16 m W m Z (cid:19) (22)where the next-most-relevant corrections are those due tothe coupling of the W boson to the QL photon and scalarfields, and the second-order electromagnetic corrections.
2. QL photon and scalar interactions
Another potentially relevant correction is the W /QLphoton coupling, Fig. 4(i). At tree level this readily eval-uates to f h E (2)QL i (23)where a symmetry factor of 2 arises from the presence oftwo identical photon operators on the interaction vertex.This gives a net expression m W = 18 f h E (1)QL i ( (cid:18)
64 + 32 π − f Z (cid:19) α π + 118 " E (2)QL E (1)QL + O (cid:0) α (cid:1)) . (24)Inserting the leading-order expressions for E (1)QL (14) and E (2)QL [1, Eq. (192)] into this correction yields m W = 18 f h E (1)QL i ( (cid:18)
64 + 32 π − f Z (cid:19) α π (25)+ m e h k ( e )1 i m W [1 + O ( α )] + O (cid:0) α (cid:1)) where evaluation of E (2)QL in terms of electron parametershas been chosen as the electron mass is known to thehighest precision of the lepton masses. Parameter k (f) g isdefined in Sec. III A 1, and as in Ref. 1 it is an eigenvalueof the preon colour mixing matrix. However, when pro-ceeding beyond the leading order calculations of Ref. 1this parameter is seen to run with energy scale. It there-fore carries labels f and g where f indicates the family offermions for which k is calculated, and g is the particlegeneration; k (f) g is then eigenvalue g of the colour mixingmatrix K f which is associated with the fermion familycontaining species f, evaluated at the energy scale m f g where f g denotes generation g of the family containingf. As a matter of notation, f will generally be taken asthe lightest member of the family, and thus (for example) k ( e )2 represents the second eigenvalue of K e , evaluated atenergy scale m µ c .Energy scale E (2)QL is independent of the particle speciesused to compute m g / (cid:2) k (f) g (cid:3) . Using the calculations ofthe present paper it is only possible to obtain the approx-imate value m g h k (f) g i ≈ . × eV /c , (26)but this suffices to determine that (cid:2) E (2)QL / E (1)QL (cid:3) lies be-tween α/ (4 π ) and α / (4 π ) in magnitude. Neglecting thecorrection factor [1 + O( α )] on the mass ratio in Eq. (25)introduces errors of O (cid:8) α (cid:2) E (2)QL / E (1)QL (cid:3) (cid:9) which are there-fore small compared with O( α ) and may be ignored.The composite QL scalar boson field yields a similarcontribution, but: • Its internal Einstein sum means it receives contri-butions from nine times as many sectors of the QL,for an additional factor of nine. • Its internal two-component structure allows forcrossed and uncrossed connection of sources andsinks, for an additional factor of two. • However, the QL field operators on the interactionvertex are no longer interchangeable, for a factor ofa half. • The vertex factor is f , not f / • The Lagrangian term carries an additional factorof − (cid:2) E (2)QL / E (1)QL (cid:3) by construction. • The mean field value of [H ′ H ′∗ ] QL is − (cid:2) E (1)QL (cid:3) / (cid:2) E (2)QL (cid:3) for [ AA ] QL .Incorporating the QL scalar boson contribution to m W therefore yields m W = 18 f h E (1)QL i ( (cid:18)
64 + 32 π − f Z (cid:19) α π + 19 m e h k ( e )1 i m W + O (cid:0) α (cid:1)) . (27)In terms of standard error, the most significant ef-fect arising from the next order of corrections beyondthose considered is a contribution to m Z at approxi-mately 10 − σ exp , while the largest relative contributionis to m H ′ and m Z , to approximate order of parts in 10 .(Note that all attempts to estimate higher-order errorsin this paper are approximate, and are determined usingthe expressions presented in Sec. IV A.)
3. Universality of loop corrections
Up to now, the photon in Fig. 4 has been treated asa fundamental particle. However, in principle any occur-rence of a fundamental boson may be re-expressed as apair of preons using the identity ϕ ˙ a ˙ cacµ = ∂ ˙ a ˙ c σ µ ∂ ac ϕ ≈ f ∂ ˙ a ˙ c ϕσ µ ∂ ac ϕ = f ψ ′ ˙ a ˙ c σ µ ψ ′ ac (28)derived from Eqs. (43) and (80–81) of Ref. 1. Such pairsare bound by the colour interaction with a characteris-tic separation of O( L Ψ ), and thus for energies E ≪ E Ψ they appear collocated and at energies small comparedto E Ψ this reparameterisation is redundant. However, thelength scale L Ψ is comparable to the length scales L (1)QL and L (2)QL associated with mass vertices. The QL fields arecharacterised by energy scales E ( k )QL = hc/ L ( k )QL and thusare capable of discriminating the two constituents whenenergy is carried in the two-preon mode rather than thefundamental boson mode. Over the length scales associ-ated with mass vertices, this reparameterisation thereforecannot be ignored.First, consider the preon-mediated mass vertex ofFig. 1(i) and recognise that during evaluation, four ofthe preon lines are integrated over and eliminated. Theremaining two preon lines form a composite vector bo-son consisting of one preon and one antipreon, and theleading-order expression for W mass comes from evalu-ating this in terms of E (1)QL using the fermion sector mean-field theory expression (6). However, it is also valid totransform the composite vector boson vertices into funda-mental boson vertices and obtain a boson loop diagramsimilar to Fig. 4(i) which gives a coupling to the bo-son sector expressed in terms of E (2)QL . For the W bosonthis contribution vanishes, however, as the resulting QLbosons would be W or W † bosons, and the QL W ( † ) fieldsvanish by gauge except along foreground boson lines [1,Sec. III E 1].Now consider the diagrams of Fig. 1(ii)-(iii) which con-tain loop corrections. The preons persisting after inte-gration no longer necessarily arise from the same ver-tices, but this is unimportant as they are still within L Ψ of one another and can therefore once again be con-sidered to make up a composite vector boson, and thusbe mapped to fundamental vector bosons, to again yielda coupling to the boson sector of the QL expressed interms of E (2)QL (and again these vanish as the bosons are W bosons). However, now recognise that the inverse pro-cess may be applied to any boson loop diagram such asFig. 4(i), mapping it into a composite vector boson loopand then introducing additional arbitrarily selected pre-ons from the QL. Such a reconstructed six-preon-line di-agram then admits loop corrections in the manner de-scribed above. This mapping therefore identifies a set ofnontrivial higher-order correction to the photon loop dia-gram of Fig. 4(i) which can only be obtained by mappingthe interacting photons into preon constituents and re-cruiting additional preons from the QL to make up preonvertices. (2) Further recalling that the scalar boson loop inSec. II A 1 c was evaluated by a mathematical mappingvia a composite boson loop to a fundamental boson loop,the converse mapping once again reveals a route backto preon triplet vertices for this diagram, and impartsequivalent corrections to the scalar boson loop diagramof Fig. 4(ii).The net outcome is that every term in Eq. (21) hasa counterpart on the boson loops of Fig. 4, and the W boson mass may then concisely be written m W = 18 f h E (1)QL i " (cid:18)
64 + 32 π − f Z (cid:19) α π + O (cid:0) α (cid:1) × ( m e h k ( e )1 i m W [1 + O ( α )] ) . (29)The O( α ) correction to the term in the second bracketsis smaller than the O (cid:0) α (cid:1) term in the first brackets, so Note that this recruitment (i) is always possible due to the homo-geneity of the QL and the extremely large number of QL particleswithin autocorrelation length L (1)QL , and (ii) is obligatory as theenergy scale for a boson at rest in the isotropy frame of the QLmandates that it interacts with triplets. this abbreviates to m W = 18 f h E (1)QL i " (cid:18)
64 + 32 π − f Z (cid:19) α π × ( m e h k ( e )1 i m W ) (cid:2) (cid:0) α (cid:1)(cid:3) . (30) B. Z mass Higher order corrections to the Z boson mass are alsorequired, and their calculation is similar to that for the W boson.
1. Boson loops a. Gluon loops:
The calculation is analogous to thatperformed for the W boson. However, introduction ofan off-diagonal gluon coupling eliminates the end-to-endsymmetry of the Z boson mass-squared interaction re-sulting in the loss of a symmetry factor of 2. A similardivision by 2 for diagonal gluon couplings is necessaryby GL(3 , R ) symmetry, and occurs because replacing thepreon line in the loop as in Fig. 1(iii) eliminates inter-changeability of the two gluon/preon vertices. The netgluon contribution is therefore30 α π . (31) b. Photon, W , and Z boson loops: The Z boson isuncharged, so attracts no net correction from the photon.Once again, since the W boson is species-changing, the Z boson mass diagram also attracts no intra-fermion loopcorrection involving the W boson.For Z loops, consider the relative contributions of thedifferent fermions which contribute to the leading orderdiagram [analogous to Fig. 1(i)]. At tree level the Z boson couples only to electrons, neutrinos, and the downquark family. The relative weights of these contributionsto the leading order expression for m Z are given in thefirst numerical column of Table I. The coefficients arisingfrom the vertices of the loop correction are given in thesecond numerical column. The net weight of the Z bosoncorrection to each channel is the product of these, given inthe third numerical column. The total correction arisingfrom Z boson loops is seen to be512 α π . (32) c. Scalar boson loop: The scalar boson loop calcu-lation is identical to that for the W boson, yielding acorrection of 3 α π (cid:18) π (cid:19) . (33) Species Weight Vertex factors Net loop weight e • L
18 16 148 e • R
18 16 148 ν • e
12 23 13 d • L
18 16 148 d • R
18 16 148
Total: 1
TABLE I. List of channels contributing to the leading order Z boson mass diagram, their relative weights, the vertex coeffi-cients arising when a Z loop correction is introduced, and theweights of the net contributions of these loop corrections, ex-pressed as multipliers applied to α/ (2 π ). The net correctionarising from Z boson loops is seen to be { / α/ (2 π )]. d. Net effect of all boson loops: The net effect of theboson loop corrections is therefore to amend the Z bosonmass equation to m Z = 24 f h E (1)QL i (cid:18) π (cid:19) α π (34)+ O " E (2)QL E (1)QL + O (cid:0) α (cid:1)! where the next-most-relevant corrections are those dueto the coupling of the Z boson to the QL scalar field,and the second-order electromagnetic corrections.
2. QL photon and scalar interactions a. Direct coupling:
As the Z boson is uncharged, itacquires no mass through direct coupling to the QL pho-ton field. However, it still interacts with the QL scalarboson field. Following a similar calculation to Sec. II A 2yields m Z = 24 f h E (1)QL i " (cid:18) π (cid:19) α π × ( m e h k ( e )1 i m W ) (cid:2) (cid:0) α (cid:1)(cid:3) . (35) b. Universality coupling: Rather surprisingly, how-ever, there does exist a mechanism whereby the Z bosonmay acquire mass from the QL photon field. Consideragain the mechanism behind the universal applicability ofboson mass-squared vertex loop corrections in Sec. II A 3.When the Z boson is interacting with the preons of theQL, this attracts the obvious leading-order term associ-ated with Fig. 1(i), but when these preons are charged,the residual pair after integration over one set of spatialco-ordinates may then be mapped onto fundamental vec-tor bosons. For charged fermions this mapping results in a superposition of the photon, Z boson, and dilatongradient [which may, in this context, be recruited as athird diagonal boson on SU(3) A as it carries trivial rep-resentation on both the SU(3) A and SU(3) C sectors].Recognising that the off-diagonal fundamental bosonsdo not contribute to particle mass, only diagonal com-posite vector bosons need be considered. On-diagonal,for any given composite vector boson with representa-tion matrix e ii a basis of fundamental bosons may bechosen consisting of the photon, Z boson, and a bosonderived from the dilaton gradient which has a vanishingentry at e ii . Consequently, any mass arising from thissector may always be attributed to the QL photon field.First, consider the lepton channels. When the pre-ons are electron-type preons ( a ∈ { , } ) there are threechoices of charged preon and three choices of charged an-tipreon, and freedom to choose which preons to integrateout gives nine ways to make a composite vector bosonwhose a -charges indicate it relates to the photon. Whenthe preons are neutrino-type preons ( a = 3), these haveno overlap with the photon and so can be ignored. Thelepton sector thus offers a total of 18 channels (nine from e L and nine from e R ).Next, consider the quark channels. Again, only diag-onal contributions from electron-type preons are nonva-nishing and thus each down quark contributes only onechannel. The up quark does not couple to the Z boson.For these twenty channels, now determine the weightof each channel when compared with the W W † AA vertexof Fig. 4(i): • Each channel only constructs half of a photon, ei-ther the e or the e term, for a factor of . (Theoverall scaling of the representation matrix is ad-dressed in the next item, in terms of the resultingvertex factor.) • Vertex and symmetry factors for two couplings ofthe Z boson to e L , e R , d L , or d R yield a factor of g = α , compared with g = 2 α for the W bosonand the photon. • Given a preon and an antipreon comprising fouroperators eeee all within a single autocorrelationregion, the pairing of e with e to yield a bosonoperator is arbitrary, introducing another factor oftwo in the mapping from composite to fundamentalvector bosons.Summing over all twenty channels, the result is a factorof 5 relative to Fig. 4(i), increasing the Z boson mass to m Z = 24 f h E (1)QL i " (cid:18) π (cid:19) α π × ( m e h k ( e )1 i m W ) (cid:2) (cid:0) α (cid:1)(cid:3) . (36)It should be noted that no counterpart to the universal-ity coupling applies for the W boson as its QL preons donot appear in the correct combinations to be assembledpairwise into components of the composite counterpartto the photon. Also note that when evaluating univer-sality couplings, a choice must be made to work on eitherthe SU(3) A or the SU(3) C sector. As the Z boson is asso-ciated with a nontrivial representation on SU(3) A but atrivial representation on SU(3) C , it is necessary to workin the SU(3) A sector. Construction of colour-agnosticcomposite bosons then implicitly spans all valid preonpairs and hence colour choices, making it unnecessary toindependently consider the gluon sector.This completes calculation of Z boson mass to the levelof precision employed in this paper. In terms of standarderror, the most significant effect arising from the next or-der of corrections beyond those considered is a contribu-tion to m Z at approximately 10 − σ exp , while the largestrelative contribution is to m Z , to approximate order ofparts in 10 . C. Weak mixing angle
If the weak mixing angle is defined in terms of W and Z boson mass, the above results for m W and m Z implya weak mixing anglesin θ W = 1 − m W m Z (37)= 1 − (cid:2) (cid:0)
64 + π − f Z (cid:1) α π (cid:3) (cid:26) m e h k ( e )1 i m W (cid:27) (cid:2) (cid:0) + π (cid:1) α π (cid:3) (cid:26) m e h k ( e )1 i m W (cid:27) × (cid:2) (cid:0) α (cid:1)(cid:3) (38)where f Z in turn depends on sin θ W (19) and it is neces-sary to solve for consistency. It is worth noting that thecorrections described above for the W and Z boson massdiagrams also apply to foreground fermion/weak bosoninteraction vertices. However, note that for the Z boson,the magnitude of the corrections show some variation be-tween different species, with (for example) the electronsattracting EM loop corrections which are not present forthe neutrino. The value obtained for the weak mixingangle will consequently depend on the different weight-ings given to the various species which may be involved,and thus on the details of individual experiments to mea-sure this parameter. For this reason, the present paperconcentrates on particle mass rather than seeking to re-produce experimentally determined values of sin θ W . D. Gluon masses
By the unbroken GL(3 , R ) symmetry of the colour sec-tor, in the e ij basis all gluons have identical mass, and itsuffices to calculate the mass of one off-diagonal gluon. FIG. 5. As the gluon interaction with the QL gluon fieldneed not conserve colour on a per-interaction basis, fore-ground gluons may interact with a pair of gluons having non-complementary charges.
Evaluation of the gluon mass is therefore similar to eval-uation of W boson mass, and indeed evaluation of theleading order diagram and preon-to-preon gluon loop cor-rections proceed equivalently. Where the W and gluonmass calculations diverge is in the contribution from in-teractions with the QL boson fields. For the W bosonthis contribution arose from the QL photon and scalarboson fields. For the gluons, there are couplings to theQL gluon and scalar boson fields.To evaluate the gluon contribution to theO (cid:8)(cid:2) E (2)QL / E (1)QL (cid:3) (cid:9) term, recognise that the preserva-tion of colour cycle invariance across the entirety of R | dust gravity guarantees that all gluons always ap-pear in the context of a superposition of all nine possiblespecies. Interactions with the QL need not thereforeconserve colour charge on an individual gluon on aterm-by-term basis provided colour charge is collectivelyconserved across the superposition. (Individual gluoncolour will, however, be preserved on average over lengthor time scales sufficiently large compared with L QL asthe QL has net trivial colour.) For mass interactions,the consequence of this is that rather than interactingwith a single looped boson as per Fig. 1(i), a gluon caninteract with a pair of different gluons from the QL asper Fig. 5. To evaluate the QL gluon contribution, note: • This interaction has coefficient f , compared with f / • A diagram in which the two QL gluons have dif-ferent field operators on the vertex, ϕ c ˙ c ϕ c ˙ c , re-ceives a factor of relative to the photon term dueto loss of vertex symmetry, but a factor of two asthese fields may be pulled down from the action Z in either order. A diagram in which the two QLgluons have the same field operator attracts neitherof these factors. • Regardless of how the vertex operators are pulledfrom Z , each QL boson independently ranges overall nine possible species for a factor of 81.The net contribution to gluon mass from the QL gluonfield is therefore 81 times larger than the contribution to W boson mass from the QL photon field. To the same0order as used in Eq. (29) above, the QL gluon mass istherefore given by m c = 18 f h E (1)QL i " (cid:18)
64 + 32 π − f Z (cid:19) α π + O (cid:0) α (cid:1) × ( m e h k ( e )1 i m W [1 + O ( α )] ) . (39)As noted in Ref. 1, this is a bare mass and will be cor-rected by self-interaction terms at energy scales smallcompared with the strong nuclear force.With regards to the universality coupling, the gluonshave nontrivial representation on SU(3) C and thus thiscoupling must also be evaluated in the SU(3) C sector.However, all gluons already couple to all QL boson mem-bers of this sector, and thus there are no missing cou-plings to be recovered using this technique. Any attemptto do so would result in double counting, so no furtherterms are acquired.In terms of standard error, and assuming O( α ) termsare equivalent for W bosons and gluons, the most sig-nificant effect arising from the next order of correctionsbeyond those considered is a contribution to m Z at ap-proximately 10 − σ exp , while the largest relative contri-bution is to m W , m Z , and m H ′ , and is of approximateorder of parts in 10 . E. Scalar boson mass
As with the W and Z bosons and the gluon, somehigher-order corrections to gluon mass are computedhere. However, for purposes of this paper it suffices toevaluate only the O( α ) loop terms and ignore the cou-pling with the QL scalar boson field as this is not relevantto the numerical results presented at the level of precisionemployed in this paper.For convenience, as in Ref. 1 the subleading term inthe scalar boson mass interaction [Fig. 6(ii), l.h.s.] willbe rewritten as a correction to the leading term [Fig. 6(i)],associated with a factor of (2 π ) − .
1. Vector boson loops
To evaluate the gluon loop corrections, recognise thatthe four preons and two antipreons (or vice versa) of theQL in Fig. 6(i) and the r.h.s. of Fig. 6(ii) may be groupedinto a QL composite scalar boson and two QL compositevector bosons, with the latter then being mapped intotrue vector bosons and a numerical scaling factor. Thepreons may be grouped together in any valid combina-tion which yields these three effective QL particles, withthe four preons contributing a factor of (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) and thetwo antipreons contributing (cid:0) (cid:1)(cid:0) (cid:1) for a net factor of 24.However, two of the preons are the same colour, dividing FIG. 6. (i) Leading diagram contributing to scalar bosonmass. (ii) First correction.FIG. 7. First-order electromagnetic correction to lepton mag-netic moment. the number of colour-distinct combinations by two (andby construction from fermion channels, these preons alsocarry the same a -charge—see Ref. 1, Sec. III F 4), and thetwo composite vector bosons are functionally equivalent,again reducing the number of distinguishable arrange-ments by a factor of two, for an overall net factor of 6from colour arrangements.Given a particular arrangement of participating pre-ons, it is convenient to work with composite scalar andvector bosons so that each composite particle then ad-mits a gluon loop correction between its two compo-nents. However, the loop factor arising from the compos-ite scalar boson is of the opposite sign to those arisingfrom the composite vector bosons as the antipreon is re-placed by a second preon, giving a factor of −
1. Otherthan that, all calculations proceed equivalently.For a gluon which is diagonal in the e ij basis, thecalculation is directly analogous to the electromagneticloop correction to magnetic moment (Fig. 7), save thatthe vertex factors are f not f /
2. The contributionsfrom the off-diagonal gluons are necessarily the same byGL(3 , R ) symmetry.Summing these terms therefore yields a net weight forthe gluon loop corrections of12 α π . (40)There are no contributions from electroweak boson loops1 FIG. 8. Scalar boson corrections to the species appearingwithin the scalar boson mass vertex: (i) When a scalar bo-son loop correction is added to the scalar boson propagator,this replaces one preon in the original propagator with itsantiparticle, resulting in a sum over composite vector bosons.The resulting diagram is then identical to that obtained whenvector boson loop corrections are applied to the scalar bosonpropagator. These diagrams have also already been counted.(ii) The scalar boson loop correction to a composite vector bo-son propagator yields an internal loop comprising two scalarbosons. Swapping the sides of this internal loop reveals equiv-alency to a sum over composite vector boson corrections to thecomposite vector boson propagator, and these can be rewrit-ten as fundamental vector boson corrections to the compositevector boson propagator. These diagrams have also alreadybeen counted. as the H ′ boson carries no net electroweak charges, sothis is the total contribution from vector boson loops.
2. Scalar boson loops
Now consider scalar boson loop corrections to thescalar boson mass term. Once again, these act on theeffective composite vector and scalar bosons connectingthe mass interaction vertices. However, recalling thata scalar boson interaction vertex always maps one con-nected preon into its antiparticle, this always results in achange of particle species within the loop. The interac-tion between the looping species and the QL preon fieldsis inserted explicitly in one arm of the loop, but is alsoimplicitly present in the other arm of the loop as theother particle is also taken to be massive. Thus the ex-plicit QL preon interaction may be freely moved from onearm to the other. Therefore, first consider the diagramsinto which these interaction vertices will be inserted: • The scalar boson loop correction to the scalar bo-son propagator has an internal loop comprising onescalar boson and one composite vector boson. Thecomposite vector boson may be mapped into a fun-damental vector boson and a numerical factor us-ing Eqs. (80–81) of Ref. 1. The factor cancels withthat arising from the construction of H ′ H ′∗ . Free-dom to insert the explicit QL coupling in either limb of the loop yields that the scalar boson cor-rection to the scalar boson propagator is exactlyequivalent to a sum over vector boson loop correc-tions to the scalar boson propagator. They mustnot be counted twice. • The scalar boson loop correction to the compositevector boson propagator has an internal loop com-prising two scalar bosons. As in Sec. II A 1 c thismay be evaluated by rearranging the internal preonlines. However, on doing so, it is immediately seento be equivalent to a vector boson loop correctionto the composite vector boson field and thus alsohas already been counted.For a propagating scalar boson, there are therefore noindependent scalar boson loop corrections at O( α ), andthe net expression for scalar boson mass incorporatingall boson loop corrections is m ′ = 40 f h E (1)QL i (cid:18) π (cid:19) × απ + O " E (2)QL E (1)QL + O (cid:0) α (cid:1)! . (41)In terms of standard error, the most significant effectarising from the next order of corrections beyond thoseconsidered is a contribution to m H ′ at approximately10 − σ exp , while the largest relative contribution is to m H ′ , to approximate order of parts in 10 . F. Neutral boson gravitation
One final note regarding the universality coupling de-scribed in Secs. II B 2 b and II B 2 b. Although the Z bo-son is uncharged, through this process it acquires a meansof coupling to the photon pair field and thus engaging inthe gravity-equivalent interaction of Ref. 1, Sec. III G 3.The foreground gluons (including the vector dilatongradient) couple to the colour charges on both the QLpreon and gluon fields, and application of the universal-ity coupling allows the QL preon fields to be reduced tocomposite vector bosons in 1:1 correspondence with thegluons, so the composite to fundamental boson dualitydoes not reveal any new couplings. However, the inclu-sion of all possible colour couplings implies that a subsetof the composite vector boson fields may also be rewrit-ten to correspond to the two-photon coupling (carryingthe photon a -charge, and being summed over all pos-sible neutral colour combinations). As this coupling isconstructed from the same composite vector boson fieldsalready accounted for, it makes no additional contribu-tion to inertial mass. However, as it is a coupling to thephoton pair field it does grant gravitational mass to thenine gluon fields. Most notably, this interaction impartsgravitational mass to the neutral gluon, or vector dila-ton gradient (which is an axion-like particle). This istherefore a potential dark matter candidate.2 FIG. 9. The fundamental interaction giving rise to leptonmass: The triplet of preons scatters twice off the bosoniccomponent of the background quantum liquid. Either (i) thesame, or (ii) different preons may be involved on each occa-sion, with the upper and lower vertices independently eachconnecting to any of the three preons. This results in a totalof nine diagrams, three having the form of diagram (i) andsix having the form of diagram (ii). These nine diagrams arethen summed.
III. LEPTON MASS INTERACTIONA. Leading order
The fundamental interaction giving rise to lepton massis a double scattering of the preon triplet off the vectorboson component of the QL (Fig. 9). Both of these dia-grams may be considered mean field theory expansions ofa loop correction to the fermion propagator in the pres-ence of the QL, but it is more convenient to refer to theseas “leading order” diagrams, and to count the numberof (foreground) loop corrections to these leading orderdiagrams, e.g. 1-loop corrections to the leading order di-agram, etc. Henceforth such corrections will be termedsimply “1-loop corrections”.Figure 9 is constructed by generating a pair of inter-action vertices between the lepton and both the SU(9)-valued gauge boson field (representation ) and the dila-ton gradient field (trivial representation, ). These rep-resentations are then factorised according toSU(9) ⊕ ∼ = [SU(3) A ⊕ A ] ⊗ [SU(3) C ⊕ C ] + = ( + ) × ( + ) (42)where each instance of represents the appropriate triv-ial representation. Finally, each SU(3)-valued boson fieldis expanded in terms of its components, and terms arediscarded where the mean magnitude-squared of the QLcomponent of the boson field vanishes by gauge. Theresulting interactions may each involve any of the threepreons making up the lepton. As per Eqs. (1–2) and (11),the bosons may be • the photon, associated with the 1-dimensional rep- resentation of SU(3) C whose generator is λ = 1 √ , (43) • gluons, associated with the 8-dimensional represen-tation { λ i | i ∈ { , . . . , } } of SU(3) C generated bythe rescaled Gell-Mann matrices (A3), or • the vector dilaton, also associated with λ .By Eq. (3), the separation of the two boson source/sinksis of order L QL .To evaluate the contribution of the QL fields to lep-ton mass, it is helpful to separate the consequences ofthese boson interactions into two parts. First, there isthe action of the representation matrices of SU(3) C onthe preon fields, and second, there is the numerical massterm arising from the mean square value of the QL bosonfield. As noted in the introduction, the family of gluonswill be taken to include the vector dilaton gradient, giv-ing a combined symmetry groupSU(3) C ⊕ ∼ = GL(3 , R ) C . (12)It is then helpful to note that this 9-membered familyadmits representation both in terms of the Gell-Mannand identity matrices (basis { λ i | i ∈ { , . . . , } } ) and interms of the elementary matrices { e ij | i ∈ { , , } , j ∈{ , , } } .
1. Action on colour sector
To begin with the action of the generators ofGL(3 , R ) C , note that over the course of a propagatorof length L ≫ L QL , a lepton will engage in a near-arbitrarily large number of interactions with the back-ground QL fields. Each interaction will apply a gl(3 , R )representation matrix from { λ i | i ∈ , . . . , } dependingon the boson species with which the lepton interacts. Inthe absence of foreground W or Z bosons, the QL is madeup entirely of photons and gluons.Given a preon of colour c , this may have nonvanish-ing interaction with the photon or any of three gluons inthe elementary basis e ij . For example, if c = r then ad-missible gluons are c rr , c gr , and c br . Heuristically, theiraction on the colour space may be represented as c rr | r i → | r i c gr | r i → | g i c br | r i → | b i (44)where all associated numerical factors have been ignoredfor illustrative purposes.More generally, the family of gluons acts on a vectorof preon colours as indicated by c rr c rg c rb c gr c gg c gb c br c bg c bb | r i| g i| b i . (45)3It is worth noting that there is no emergence of colour su-perselection for individual particles in R | dust gravity,as the GL(3 , R ) C symmetry is unbroken, so there existsa freedom of basis on the colour sector correspondingto an arbitrary global transformation in SU(3) C . Anycoloured fundamental or composite particle may be putinto an arbitrary superposition of colours using such atransformation, though relative colour charges of differ-ent particles remain unchanged, as does the magnitudeof the overall colour charge of a composite particle.Recognise now that Fig. 9 contains contributions totwo mass vertices. Although their contributions tofermion mass are nonvanishing only when they appearpairwise, as shown in Fig. 9, there is no requirement forthis pair to be consecutive. It suffices that each vertex bepaired with a conjugate vertex separated by distance andtime no greater than L QL (which is within a few orders ofmagnitude of Planck length) in the isotropy frame of theQL. Indeed, these vertices are connected by a foregroundfermion propagator which in general also undergoes fur-ther interactions with the QL, represented by using amassive propagator for this fermion and requiring consis-tency with the outcome of the mass vertex calculation.In general a foreground fermion exhibiting a net propaga-tion over distance or time of O( L QL ) in the isotropy frameof the QL will scatter back and forth multiple times inthis process, interacting with O ( h N QL i ) ∼ particlesfrom the QL, and consequently the number of unpairedvertices is negligible. Furthermore, where unpaired ver-tices do exist, their net effect vanishes on average overlength or time scales larger than L QL . It is thereforereasonable to assume during evaluation that each vertexbelongs to a pair.Counting vertices arising from Fig. 9, for every photonvertex there is also on average one vertex for each of thenine gluons. Across macroscopic length scales, deviationsfrom this average relative frequency will be negligible.Consider now the specific case of interactions betweena propagating lepton and the bosons of the QL. Eachpreon may interact either with a photon or with any ofthe nine gluons, and the action of the photon on the spaceof preon colours is trivial, so it is convenient to ignore thephoton for now and reintroduce it later.As already noted, paired interactions with the QLgluon field conserve the net colour-neutrality of a leptonicpreon triplet. However, overlapping and intercalation ofmultiple interaction pairs implies that this property onlyholds on average, as any colour measurement will inter-rupt a finite number of mass interactions and thus sum-mation to yield no net colour charge on the preon tripletcannot be assured. It is desirable that any measurementof lepton colour should be null, not just the average,and thus a local change of co-ordinates on SU(3) C mustbe performed on a co-ordinate patch encompassing thenon-interacting preons such that changes in their colourstrack those of the interacting preon. This change of co-ordinates is not part of the choice of gauge on SU(3) C ,and thus is in principle associated with construction of some synthetic boson interactions where it intersects withparticle worldlines. As described in Sec. III F 2 of [1],the vertex factors associated with these interactions arisefrom the representation matrices of SU(3) C given as λ i in Appendix A. By construction these bosons are con-strained to have no effect beyond the colour shifts associ-ated with the boundary of the patch, and to leading orderthis effect is parameterless. In the leading order diagramthese bosons consequently have no degrees of freedom,carry no momentum, and thus may be integrated out toyield a factor of 1. Consequently they are not drawn.Only the factors arising from the representation matricespersist, acting on the colour vector of an individual preonas the matrix K ℓ = A A † A † AA A † , A = ± ± i2 . (46)The sign on i is free to be chosen by convention, while theoverall sign on A is fixed by noting that cyclic permuta-tion of colours, which is in ( K ℓ ) , is required to leave thesign of an eigenstate of K ℓ unchanged. The eigenvaluesof K ℓ must therefore be non-negative, setting A = − ± i2 . (47)Choosing a sign for i, the mixing matrix K ℓ may then bewritten K ℓ ( θ ℓ ) = e i θℓ √ e − i θℓ √ e − i θℓ √ e i θℓ √ e i θℓ √ e − i θℓ √ θ ℓ = − π . (48)As noted in Ref. 1, Sec. III F 2, this matrix bears a strongresemblance to Koide’s K matrix for leptons [3]. Theminus sign on Koide’s off-diagonal component S ( θ f ) hasbeen absorbed into the phase θ ℓ , and the free parameters a f , b f , and θ f are fixed by the geometry of the model, inkeeping with the predictive capacity of R | dust gravity.Recognising that on average all QL gluons act identi-cally and with equal frequency, it is convenient to collectthese together into a single gl(3 , R )-valued gluon asso-ciated with two applications of matrix K ℓ to the non-interacting preons, as shown in Fig. 10.Now recognise that since the gluons of R | dust grav-ity are massive, and since the fundamental mass interac-tions take the form of loop diagrams with respect to thelepton propagator (Fig. 9), these will be suppressed by afactor of O( m ℓ /m c ) relative to the photon contribution.With each gluon interacting, on average, once for everyphoton interaction, it is convenient to write the directcontributions of the gluon terms to particle mass as cor-rections to the larger photon term and to associate copiesof the matrix K ℓ with the photon vertices in a mannerequivalent to that shown in Fig. 10.Next, consider that the preons on which matrix K ℓ areacting are just two of three preons in a colour-neutral4 FIG. 10. When a composite fermion interacts with the glu-ons from the QL, represented as a single gl(3 , R )-valued boson,the colour mixing process represented by matrix K ℓ acts onall preons not coupling to the boson at any given vertex. Di-agrams (i) and (ii) correspond to Figs. 9(i)-(ii) respectively.Once again these are just two representative diagrams froma family of nine, as the upper and lower vertices may inde-pendently each be connected to any of the three preons. Thisresults in a total of nine diagrams, three having the form ofdiagram (i) and six having the form of diagram (ii). triplet. For leptons, all three a -charges are identical andthus over macroscopic scales, where chance fluctuationsbecome negligible, matrix K ℓ will act identically on eachmember of the triplet. This symmetry is convenient, asit allows the study of an individual preon prior to thereconstruction of the triplet as a whole.As per Eq. (5), the preons making up observable lep-tons are now eigenstates of this matrix K ℓ , correspondingto the eigenvectors v = 1 √ (49) v = 1 √ e π i3 e − π i3 − (50) v = 1 √ e π i3 e − π i3 (51)which are independent of θ ℓ and have eigenvalues { k ( ℓ ) i | i ∈ { , , } } given by k ( ℓ ) n = 1 + √ (cid:18) θ ℓ − π ( n − (cid:19) . (52)To reconstruct the lepton as a whole, recognise thatfor three preons at { x i | i ∈ { , , } } , with correspondingcolours c i , and with the preon at x having a well-definedcolor, say c = r , colour neutrality and colour cycle in-variance imply that a choice c = g , c = b is equal up toa sign to the alternative choice c = b , c = g (as this ex-change corresponds to spatial exchange of two fermions),and thus it suffices to consider only one such colour as-signment (say c = g , c = b ) along with spatial permu-tations. Putting preon 1 into a superposition of colour states then corresponds to a superposition of cyclic spa-tial rearrangements of the members of the triplet, withcolours c = r , c = g , and c = b corresponding tocolour assignments with respect to spatial co-ordinate x of rgb , gbr , and brg respectively. It follows that for lep-tons, colour superselection need not be violated, and in-stead the different spatial configurations of colours onthe preon triplet are eigenvectors of a matrix K (3) ℓ witheigenvalues identical to those of K ℓ . It appears unlikelythat a similar recovery of colour superselection can beachieved for the quarks.Having established through colour cycle invariancethat the matrix K ℓ acts identically on all constituentsof a lepton, and through Fig. 10 that two copies of K ℓ act per QL photon interaction, it follows that the effectof matrix K ℓ is to contribute a factor of [ k ( ℓ ) i ] to themass of a lepton of generation i . It might seem problem-atic that for θ ℓ = − π/ k ( ℓ )1 = 0, but it will be seen inSec. III C that θ ℓ acquires corrections from higher-orderdiagrams, resulting in k i > ∀ i , so k ( ℓ )1 may be assumedreal and positive, and this concern may be disregarded.
2. Mass from photon and gluon components of QL
The zeroth-order electromagnetic term is readily eval-uated by making a mean-field substitution (1) for[ A µ ( x ) A µ ( y )] QL . For a charged lepton ℓ i of generation i , this initial approximation may be written m ℓ i = f h k ( ℓ ) i i h E (2)QL i . (53)Note that there is a symmetry factor of two correspond-ing to exchange of the two QL interactions. This may beunderstood in either of two related ways:1. For diagrams such as Fig. 9 which separate into twoseparate terms on performing the mean field expan-sion of the QL, one approach is to recognise thateach term corresponds to a mass vertex and has itsexternal legs truncated independently. These twovertices are then indistinguishable.2. Alternatively, for any diagram, including oneswhich do not separate, recognise that the mass-squared is always applied in the context of an un-truncated fermion propagator, say from x to y . Inthis context, all fermion connections to the inter-action vertices are again untruncated. (Optionally,the full expression for propagation from x to y isthen used to infer an equivalent mass term, and thediagram may then be replaced by one in which thismass term is inserted into the propagator twice.)Applying either form of this approach to Fig. 9,the diagram for propagation between two points isagain seen to attract a symmetry factor of two.Now consider interactions between a foregroundfermion and the QL gluon fields. As with the photon,5these interactions take the form of loop diagrams evalu-ated in the mean-field regime for the QL, and as noted inRef. 1, the fermion may transiently surrender momentumto or borrow momentum from the QL. However, in con-trast with the photon loop evaluated to obtain Eq. (53),the gluon field is massive, and when a foreground particletransfers momentum to a gluon field, this results in aneffective foreground excitation of that gluon field. Fore-ground excitations acquire mass, and thus both limbs ofthe loop must be considered massive. However, the sep-aration of the two vertices is of O( L QL ), which is muchshorter than the strong interaction scale, so the gluonexhibits only its bare mass (39). This gives rise to aloop-associated factor of m ℓ /m c . This factor vanishesfor the photon, as it is massless, but not for interactionswith the QL gluon field.In both cases, evaluation of momentum flux aroundthe loop may be taken to yield a factor of α π f (cid:18) m ℓ m b (cid:19) , b ∈ { A, c } (54)where f ( n ) → n → ∞ . For the photon the factorof α/ (2 π ) is absorbed into the QL mean-field term bydefinition, and factor f ( · ) reduces to 1. For gluons, de-pendence on the same energy scale E (2)QL indicates that anidentical factor of α/ (2 π ) is absorbed into the mean fieldterm, while the mass dependence of f ( · ) reveals that thegluon terms are suppressed by a factor of m ℓ /m c relativeto the photon term.Working in the e ij basis, consider first an off-diagonalgluon. When this gluon is emitted, it changes the colourof the emitting preon. For example, the colours of thepreon triplet may change from rgb to rgg . However, amass vertex must by definition leave the fermion on whichit acts unchanged, and thus to generate a mass term thegluon must be absorbed by the same preon as it wasemitted, restoring the original state ( rgb ). Absorptionby a different preon is a valid physical process, but in-stead comprises part of the binding interaction betweenthe preons. This is in contrast to the photon, where emis-sion and absorption may be on different preons withoutaffecting either the a -charge or the c -charge configura-tion.Overall, expressed relative to the photon interaction,factors for an off-diagonal gluon arise as follows: • A given gluon may only be emitted by a preon ofspecific colour, for a relative factor of . • However, any of the three preons may be thiscolour, for a relative factor of three. • The photon may be absorbed by any of the threepreons, whereas each gluon may be absorbed onlyby the preon which emitted it, for a relative factorof . • Vertices are associated with factors of f , not f / √ • Vertices are no longer interchangeable, for a relativefactor of . • Interactions between a fermion and an off-diagonalboson attract a structural factor of . (For a com-parable calculation see the W correction to the EMvertex [2], in which the arising factor of com-prises a factor of 2 from the change in vertex factorand a structural factor of .) • As previously noted, there is a mass factor of m ℓ /m c .By GL(3 , R ) symmetry, each of the nine gluons will yieldthe same magnitude contribution to fermion mass, for afurther factor of nine. Taking both photon and gluonterms into account, the leading-order expression for pho-ton and gluon contribution to lepton mass is thereforegiven by m ℓ i = f h k ( ℓ ) i i h E (2)QL i q ℓ e + 5 m ℓ i m c ! (55)where q ℓ is the charge of lepton ℓ i .As an aside, for the fermions there is no equivalent tothe bosonic universality coupling explored in Sec. II B 2 b.For the Z boson, this coupling arises as the basic Z mass diagram [equivalent to Fig. 1(i)] intrinsically incor-porates six QL preon lines, and two co-ordinates to inte-grate over, permitting reduction to two preon lines whenone of these integrals is performed. In contrast, the ba-sic fermion mass diagram (Fig. 10) contains no intrinsicmechanism for adding extra preon lines. Although extrapreons may be recruited from the QL, consistent nor-malisation [1, Sec. III F 1] requires that integrating overthe additional co-ordinate thus introduced will inevitablyeliminate them again.
3. Mass from scalar component of QL
Next to be considered is the interaction between thecomposite lepton and the QL composite scalar bosonfield. Again it is desirable to write this term as a cor-rection to the photon term.To achieve this, recognise that the scalar boson is madeup of a sum over nine components. Likewise, when theSU(3) A sector is supplemented by the vector dilaton gra-dient this results in a symmetry group GL(3 , R ) A whoseLie algebra gl(3 , R ) A also has nine elements. The in-teraction vertex for each component of the scalar bosonmay be obtained by transforming that of a boson fromgl(3 , R ) A associated with basis element e ij . There arenine such bosons, which include the W and W † bosons.Note that the boson gauges [1, (127–131)] do not impactthe fermion sector, and the scalar boson is constructedfrom two fermionic preons, so all nine terms must betaken into account regardless of whether the associated6boson is eliminated by gauge, and since the symmetry ofthe fermion sector is unbroken, all nine terms are equal.Note that although the numerical factors associatedwith the scalar boson correction is obtained by trans-forming the equivalent vector boson corrections, this isnot double counting because the scalar boson involvedis a foreground particle, and therefore massive, with amass shell distinct from that of the numerically conve-nient bosons. This is in contrast with Sec. II E 2, whereall involved particles are effectively massless and thus anumerical identification corresponds with a physical iden-tification.By the internal GL(3 , R ) symmetry of the scalar bo-son, all terms contribute equally to the fermion/scalarboson interaction. Considering the off-diagonal term cor-responding to W , the following factors arise: • Relative to the photon, the W boson interactionacquires a structural factor of . • Relative to the photon, the W boson coupling isstronger by a factor of two. • Compared with the photon, there is a loss of vertexinterchangeability for a factor of . (Recall thatexternal legs are not truncated so vertices in Fig. 9are not distinguishable.) • However, this is offset by a factor of two corre-sponding to the different ways to attach the ex-ternal source and sink to the now-distinguishablevertices. • Transforming a
W W † pair into a term in theH ′ H ′∗ pair [in principle performed using Eqs. (80–81) of Ref. 1 and integration by parts] yields oneof nine terms in H ′ H ′∗ , so introduces a factor of − (cid:2) E (2)QL / E (1)QL (cid:3) from Eqs. (4) and (6). • The scalar boson contributes an internal symmetryfactor of two relating to crossing or not crossing itsinternal preon lines. • There is no factor of [1 + 1 / (2 π )] for this interac-tion, as there is only one relevant type of vertexinteraction involving the scalar boson field and thefermion field. This may be contrasted with Fig. 6,where there are two relevant types of vertex [seenin diagrams (i) and (ii) respectively] with each hav-ing different couplings to the preonic constituentsof the scalar boson. • Related to the above, twisting the inner preonline to yield an additional loop particle as shownin Fig. 12 changes the scalar boson loop correc-tion into a composite vector boson loop correction.However, in contrast with the vector boson loops inSec. II A 1 (and by similar argument the compositevector bosons appearing in Sec. II E) this loop is notconstrained to vanish at length scales greater than
FIG. 11. Leading-order contribution to fermion mass fromthe QL scalar field.FIG. 12. Twisting the inner preon line to yield an additionalloop particle changes the scalar boson loop correction into acomposite vector boson loop correction. As the loop is fore-ground, with dominant contribution from massive trajectoriestraversing length scales large compared with L (2)QL , it must bere-expressed in terms of fundamental vector bosons. L (1)QL . As observed in Sec. III D 3 of Ref. 1, incor-porating both fundamental and composite vectorbosons into the particle spectrum of the low-energy( L ≫ L QL ) regime is redundant and therefore thecomposite vector bosons must be re-expressed asfundamental vector bosons. The resulting diagraminvolves two boson loop corrections, and is not ascalar loop term at all. Similarly, twisting twice re-stores a scalar boson, but once again yields a redun-dant representation of vector boson loops, whichare higher-order terms in the correction series.There are nine bosons in the e ij basis for a final factor of9. Using lowest-order expressions for E (1)QL (14) and E (2)QL [1, Eq. (192)] yields a weight relative to the photon termof 240 m ℓ i m ′ m ℓ h k ( ℓ )1 i m W [1 + O ( α )] (56)for a total lepton mass m ℓ i = f h k ( ℓ ) i i h E (2)QL i × q ℓ e + 5 m ℓ i m c + 240 m ℓ i m ℓ h k ( ℓ )1 i m ′ m W . (57)7The most significant contribution from the higher-ordercorrection to Eq. (56) is to m Z at O(10 − σ exp ) (and inrelative terms, to m W , m Z , and m H ′ to order of parts in10 ) so this term may safely be dropped.
4. Gluon and scalar field mass deficits
Conservation of energy/momentum implies that therest mass imparted to the fermion must be compensatedby a reduction in the zeroth component of 4-momentumof some of the QL fields. Likelihood of contribution fromany given QL sector will be governed by availability ofzeroth-component energy within that sector, i.e. the restmass of the associated species, and the strength of cou-pling to that sector. It therefore follows that this borrow-ing of rest mass occurs with equal likelihood from eachof the nine gluon channels of the QL, with much lowerlikelihood from the scalar boson channel (due to a muchweaker coupling), and not at all from the photon chan-nel (due to zero rest mass). For a first approximation,consider only the gluon channel. Borrowing a mass of m ∗ from a QL gluon field takes place at the first of theexisting gluon/fermion interaction vertices of Fig. 9, andcorresponds to deletion of a gluon of mass m ∗ from theQL. This hole then propagates as a quasiparticle, and isfilled by the conjugate interaction at the second vertex.More generally, with multiple overlapping pairs of QLgluon field interactions occurring along a fermion prop-agator, there is a consistent propagating hole in the QLgluon sector corresponding to an energy deficit of m ∗ c ,and individual vertices may cause transient fluctuationsand may change which specific gluon fields (with respectto some arbitrary choice of colour basis) are involved inpropagating this hole, but in general it may be in any ofthe nine gluon channels at any time. This hole is in addi-tion to the effect discussed in Sec. III A 2 where fermionsmay surrender momentum to or borrow momentum fromthe QL, and thus gives an additional correction factor notyet discussed.This hole propagates as a quasiparticle accompanyingthe lepton. Any time that a fermion interacts with theQL gluon sector this hole is necessarily also present, andmay occupy any of nine channels.For simplicity, further consider the case when the holeoccupies an off-diagonal channel. It would be convenientto write the effect of this hole as a correction to the massof gluon, m c −→ m c − km ∗ (58)for some factor k . Recognising that the co-propagatinghole’s interaction with the fermion is trivial (it is only re-quired to be present), with any local energy/momentumtransfer to or from the QL being mediated by thefermion/gluon coupling of Fig. 9(i), the hole’s interac-tions attract no structural vertex factor and thus wherethe direct gluon interactions of Fig. 9(i) generate a factorof 5 / (3 m c ), the hole will contribute only − m ∗ . Further, the presence of the hole breaks the time reversal symme-try of a portion of the QL and this gives rise to a symme-try factor of relative to the original gluon interaction inwhich the QL was assumed time-reversal-invariant. Fi-nally, there are nine gluon channels, each of which yieldsan equivalent correction by GL(3 , R ) symmetry. The netoutcome is to correct the gluon mass to an effective massof ( m ∗ c ) = m c (cid:18) − m ∗ m c (cid:19) . (59)Note that m ∗ c is a function of m ∗ , but for a lepton ℓ i whichis on-shell and at (or close to) rest in the isotropy frameof the QL this admits the replacement m ∗ → m ℓ i andcan then be conveniently left implicit. Also note that thegluon mass deficit effect is a whole-field effect, acting onboth the foreground and QL gluon fields. The correctedgluon mass m ∗ c should be used anywhere a particle inter-acts with a gluon field in the presence of a lepton. Thelepton mass equation is amended to m ℓ i = f h k ( ℓ ) i i h E (2)QL i × q ℓ e + 5 m ℓ i ( m ∗ c ) + 240 m ℓ i m ℓ h k ( ℓ )1 i m ′ m W . (60)Similarly, the foreground lepton may also borrow itsmass from the scalar boson field. However, coupling be-tween leptons and scalar bosons is weaker than that be-tween leptons and gluons, and this correction falls belowthe threshold of relevance, giving rise primarily to cor-rections to m Z at O(10 − σ exp ) (or in relative terms, to m τ , m Z , m W , and m H ′ to order of parts in 10 ). B. Foreground Loop Corrections
Now consider the effects of foreground loop correctionson the leading-order diagrams of Figs. 9 and 11. Notethat since momentum is continually redistributed amongthe constituent preons by means of gluon-mediated in-teractions even over length scale L Ψ , a boson need notstart and finish its trajectory on the same preon in orderto be considered a loop correction to an emission vertex.Further note that the massive nature of the loop bosondoes not disrupt the QL correlators in these diagrams,as these are brought together through the use of spinoridentities at the vertices making these diagrams more ro-bust against interference from outside the autocorrelationregion than the boson mass diagrams of Sec. II.
1. 1-loop EM corrections
The O( α ) EM loop corrections to the lepton mass in-teraction are shown in Fig. 13. These should be compared8 FIG. 13. One-foreground-loop EM corrections to (i) vectorboson and (ii)-(iii) scalar boson lepton mass interactions. Thebroad oval interaction vertices indicate that the boson mayinteract with any of the three preons, and all configurationsshould be summed over.FIG. 14. Standard Model one-foreground-loop EM correc-tions to the electron mass vertex. The example shown is forthe left-helicity electron Weyl spinor; equivalent diagrams forthe right-helicity spinor exchange L and R . with their Standard Model counterparts in Fig. 14.As in Sec. II A 1, the only corrections which need tobe incorporated into the electron mass vertex are thosewhich do not also appear in the Standard Model. How-ever, comparing the diagrams of Fig. 13(i) with theircounterparts in Fig. 14 reveal these to be directly equiv-alent. The diagrams of Fig. 13(i) are therefore accounted for in the usual Standard Model corrections, making nocontribution to m ℓ i .Next, consider the scalar boson loops of Fig. 13(ii). Inthese diagrams the intermediate lepton has one preon re-versed, and the first two diagrams therefore yield factorsof α/ (6 π ) rather than α/ (2 π ). In conjunction with nega-tion of the Standard Model corrections, they thereforeyield a net correction to the QL scalar term with weight − αq ℓ πe . (61)The third diagram is consistent with the Standard Modelequivalent and so once again does not contribute to m ℓ i .Finally, there are two more diagrams from the scalarboson sector to consider. Moving a single scalar bosonvertex onto the photon loop is prohibited as the dia-gram as a whole does not then leave the preon tripletunchanged (one preon gets replaced by an antipreon), butmoving both vertices onto the loop is admissible. Thisresults in the diagrams shown in Fig. 13(iii), of whichonly the second is anticipated to be non-vanishing by thearguments of Ref. 1, Sec. III F 3. This loop contributes tothe mass of the loop photon (which vanishes by gauge),but also affects the fermion propagator to which it con-tributes a correction with weight α/ (2 π ) to the scalar bo-son term, combining with the above factor (61) to yielda correction of (cid:18) − αq ℓ πe (cid:19) . (62)The net expression for lepton mass is thus m ℓ i = f h k ( ℓ ) i i h E (2)QL i (63) × ( q ℓ e + 5 m ℓ i ( m ∗ c ) + 240 m ℓ i m ℓ h k ( ℓ )1 i m ′ m W (cid:18) − αq ℓ πe (cid:19)) but the final factor of (cid:18) − αq ℓ πe (cid:19) (64)is on the same scale as the O( α ) term in Eq. (56) andso may be dropped, as its most significant contributionis likewise only to m Z at O(10 − σ exp ) (and in relativeterms, to m W , m Z , and m H ′ to order of parts in 10 ).
2. 1-loop gluon corrections
The next loop corrections to consider are those arisingfrom gluon loops, analogous to the diagrams of Fig. 13.These gluons are foreground particles capable of mak-ing excursions outside of the local autocorrelation zone,and hence are massive, but the length scale involved isof order L Ψ so self-energy terms may be neglected, andthe gluon loops exhibit at most only the bare gluon mass9 FIG. 15. (i)-(ii) Gluon loop equivalents to the first diagram ofFig. 13(i). When the fermion couples to the QL photon field,only diagram (i) may be constructed as the gluon does notcarry an electromagnetic charge. When the fermion couplesto the QL gluon field, both diagram (i) and diagram (ii) maybe constructed. Conservation of colour charge indicates thaton summing the coupling of the QL gluon to the fermion andthe QL gluon to the loop gluon, this is equal to the couplingof the QL gluon to the fermion in the original leading-orderdiagram (iii). Similarly, diagram (iv) is the counterpart tothe third diagram of Fig. 13(i). This diagram is especiallyinteresting as it contributes both to the fermion mass and tothe mass of the loop gluon. Note that the two QL gluons mustcouple with the loop gluon at a single vertex, as discussed inRef. 1, Sec. III F 3. (including the gluon deficit correction). Only loop cor-rections to the QL photon and QL gluon interactionsneed be considered in the present paper; corrections tothe QL scalar boson interaction contribute to m Z atO(10 − σ exp ) (in relative terms, to m W , m Z , and m H ′ at parts in 10 ) and are therefore ignored.Consider the gluon loop counterparts to Fig. 13(i).Where the fermion interacts with the QL photon field,these corrections take on the form of Fig. 15(i). For in-teractions between the fermion and the QL gluon fieldsthey may take on the form of either Fig. 15(i) or (ii), andconservation of colour charge indicates that when thesecouplings are summed, this is equal to the coupling ofthe QL gluon to the fermion in the original leading-orderdiagram [Fig. 15(iii)]. Further, when the loop correc-tion is evaluated, this collapses to a numerical multiplieron the original QL interaction vertex and the custom-ary application of spinor identities [1, (170–171)] yieldsa non-vanishing contribution to m ℓ i . It then suffices toconsider an example where the coupling of the QL gluonand loop gluon vanishes, making it equivalent to the QLphoton case, evaluate the correction to the interaction vertex, and extrapolate this across all gluon colour com-binations by GL(3 , R ) symmetry.Figure 15(iv) shows a further diagram which may beconstructed using gluon loop corrections. This is a coun-terpart to the gluon version of the third diagram ofFig. 13(i), and also contributes to the mass of the loopgluon.Note that in contrast to the photon terms, where thethird diagram of Fig. 13(i) was accounted for in the Stan-dard Model, for preon/gluon interactions all loop correc-tions must be evaluated as there are no correspondingStandard Model terms.To evaluate these corrections begin with Fig. 15(i),which is the gluon loop counterpart to Fig. 13(i). Startwith the fermion coupling to the QL photon field, and aspecific choice of off-diagonal gluon. Compare with theequivalent EM loop figure and note the following changes: • The vertex factors increase from f / f , for arelative factor of two. • The boson is off-diagonal, for a structural factor of . • The photon source may be any of three chargedpreons, but the gluon may only be emitted by apreon of appropriate colour. However, any of thethree preons may be the preon of that colour, for anet factor of one. • Emission of the off-diagonal gluon changes thecolour of the emitting preon. There are then twopreons of that colour which are capable of absorb-ing the loop gluon, and both of the resulting dia-grams count as loop corrections due to the implicitexchange (not shown) of further gluons as a bindinginteraction sharing momentum between all mem-bers of the preon triplet. The choice of absorbingpreons gives a factor of two. • The loop gluon is massive, and is in the presenceof a foreground fermion so experiences a gluon fieldmass deficit giving a loop factor of m ℓ i / ( m ∗ c ) . • These factors multiply the equivalent photon loopfactor, which is α/ (2 π ).The per-gluon correction weight from this diagram istherefore 10 α π m ℓ i ( m ∗ c ) . (65)This calculation is repeated for the gluon counterpartsto the other two figures of Fig. 13(i), and each of thesefigures may involve any of nine gluons, for a total weightof 90 απ m ℓ i ( m ∗ c ) . (66)0 FIG. 16. (i)-(iii) W boson loop corrections to the QL vectorboson interactions (QL photon or QL gluon). For QL photonterms, ignore the orientation arrows on the QL boson lines.(iv) Loop involving a W boson and a neutrino, appearing inthe proper self-energy corrections of the Standard Model. Correction of the QL gluon interactions proceeds equiv-alently, and as noted above the correction to the scalarboson may be ignored at current precision, for a net lep-ton mass so far of m ℓ i = f h k ( ℓ ) i i h E (2)QL i (67) × ( q ℓ e " αm ℓ i π ( m ∗ c ) + 5 m ℓ i ( m ∗ c ) " αm ℓ i π ( m ∗ c ) + 240 m ℓ i m ℓ h k ( ℓ )1 i m ′ m W ) .
3. 1-loop weak force corrections
In the interest of brevity, this Section specialises to thecharged leptons only. a. QL photon interaction:
When the fermion inter-acts with the QL photon field, W boson loops may correctthis interaction as shown in Fig. 16. As with the gluonloops in Sec. III B 2, the boson loops may be collapsedto numerical factors on the vertices of the correspondingleading-order diagram, and the QL field terms contractedusing spinor/sigma matrix identities.Some caution is required with sign—first consider dia-gram (iii). Evaluating this diagram as a tensor networkin the manner of Ref. 4–6, it is readily seen to be an ab-solute square and therefore yields a contribution to m ℓ i which is additive to the leading order term. Diagrams (i)and (ii) contain as a subdiagram a correction to the EM emission process which is usually associated with the op-posite sign to direct emission by the fermion; however,compared with diagram (iii) they have also acquired anadditional fermion propagator segment which offsets this,so they are also additive. Relative to the leading-orderdiagram, Figs. 16(i)-(ii) acquire factors • (10 / m ℓ i /m W ), being the usual factor for a W loop correction to an EM vertex, • / / m ℓ i /m W ) only.Finally, consider the Standard Model counterpart: Thecharged lepton Proper Self-Energy (PSE) terms in theStandard Model give rise to the W /neutrino loop shownin Fig. 16(iv). This gives rise to a term analogous toFig. 16(iii), but has no counterpart to the symmetry fac-tor associated with the pair of identical photon operatorson the QL interaction vertex. Its associated factor istherefore (5 / m ℓ i /m W ), which must be deducted. Thenet weight of the W boson loop corrections is thus5 αm ℓ i πm W . (68)Next considering the Z boson loops, these take formdirectly analogous to Fig. 13, and by arguments similarto the above are also seen to all be additive. Notingthat the Z boson couples with opposite sign to e L and e R , the Z boson equivalents to the first two diagrams ofFig. 13(i) are therefore of opposite sign to their StandardModel counterparts and hence attract a factor of two, tofirst offset the Standard Model-derived term in the PSE,and then implement the actual correction. The thirddiagram of Fig. 13(i), on the other hand, has the samesign as its Standard Model counterpart and thus yieldsno net contribution to m ℓ i . The net outcome from Z boson loops is a term with weight − f Z αm ℓ i πm W (69)where f Z is as defined in Eq. (19). Overall, the weight ofthe weak boson corrections to the QL photon interactionis seen to be (5 − f Z ) αm ℓ i πm W . (70)Note that the use of f Z , which is dependent on the weakboson masses, implicitly incorporates the loop correc-tions to W and Z boson vertex interaction strengths.1This value is based on the mean Z vertex interactionstrength, evaluated as a weighted average across allspecies which interact with the Z boson, but this approxi-mation does not have a significant effect on the numericalresults obtained, being responsible for corrections to m Z at O(10 − σ exp ) (in relative terms, parts in 10 on m W , m Z , and m H ′ ). b. QL gluon interactions: This term is readilyobtained, as the calculation proceeds similarly toSec. III B 3 a, and it is found to be right at the thresholdof relevance. It is larger by a factor of 5 than the largestterms which have not been retained, and corrects m Z at0 . σ exp .As usual, start with the fermion interaction with a sin-gle off-diagonal QL gluon channel and then use GL(3 , R )symmetry to multiply by nine for the contributions ofthe other eight gluons. For the gluon channels, the QLgluons continue to couple to the fermion and not thelooping W boson, so when evaluating the equivalents ofFigs. 16(i)-(ii) there is no factor of for lost symmetry.The calculation is otherwise equivalent, for net W and Z corrections of (25 − f Z ) αm ℓ i πm W . (71) c. QL scalar boson interaction: These terms fall be-low the threshold of reliable evaluation with current nu-merical methods employed, with the largest correctionappearing to be approximately 10 − σ exp on m τ , or partsin 10 .The cumulative expression for charged lepton mass in-corporating weak boson corrections is therefore m ℓ i = f h k ( ℓ ) i i h E (2)QL i (72) × ( " αm ℓ i π ( m ∗ c ) + (5 − f Z ) αm ℓ i πm W + 5 m ℓ i ( m ∗ c ) " αm ℓ i π ( m ∗ c ) + (25 − f Z ) αm ℓ i πm W + 240 m ℓ i m ℓ h k ( ℓ )1 i m ′ m W ) .
4. 1-loop scalar corrections
These terms fall below the threshold of relevance, lead-ing to corrections to m Z at O(10 − σ exp ), or in relativeterms, to m W , m Z , and m H ′ at approximate order ofparts in 10 .
5. 2-loop EM corrections
These terms fall below the threshold of relevance, lead-ing to corrections to m Z at O(10 − σ exp ), or in relative terms, to m τ at approximate order of parts in 10 . C. Corrections to the lepton mass angle
1. Origin of corrections
There remains one more substantial correction to eval-uate. The leading-order contributions to the lepton massvertices and the preon colour mixing matrix were evalu-ated in Sec. III A. Loop corrections to the mass verticeswere evaluated in Sec. III B. It remains now to determinehow these loop corrections impact the preon colour mix-ing matrix K ℓ .When this matrix was constructed in Sec. III A 1 therewas an implicit assumption that all eigenvectors expe-rience equal couplings to the QL fields. The eigenval-ues of matrix K ℓ then imparted different masses to thethree eigenvectors, corresponding to the three membersof a given fermion family. However, from Eq. (72) itis apparent that the higher-order corrections to the QLcoupling themselves depend on particle mass, leading todifferential augmentation of the three eigenvectors (masschannels). As these eigenvectors correspond to differ-ent vectors in colour space, imparting different relativephases to the gluon couplings between the preon triplets,a change in the relative QL couplings for the differenteigenvectors will impact the colour mixing process. Con-veniently, it proves possible to represent this modificationby a correction to the value of θ ℓ .Recognising that the fermion masses vary between dif-ferent particle families, it is necessary to specify the fam-ily for which the corrected value of θ ℓ is being evaluated.This is done by replacing ℓ with a member of the particlefamily in question, e.g. e , µ , or τ for the electron family.
2. Preamble
To obtain a corrected expression for the electron massangle at some energy scale E , first recall that the eigen-vectors { v i | i ∈ { , , } } of K ℓ (49–51) are independentof the value of θ ℓ . For the electron family, matrix K ℓ therefore always admits the decomposition K ℓ = X i =1 k ( ℓ ) i v i v † i (73)2where v v † = 13 (74) v v † = 13 e π i3 e − π i3 e − π i3 e π i3 e π i3 e − π i3 (75) v v † = 13 e − π i3 e π i3 e π i3 e − π i3 e − π i3 e π i3 . (76)Then note that the imaginary components of the off-diagonal matrix elements arise from components 2 and 3only (75–76), with an increase in component 2 (cor-responding, for charged leptons, to the muon) makinga positive-signed contribution to the imaginary compo-nent in [ K ℓ ] , [ K ℓ ] , and [ K ℓ ] . Examining the vec-tors on the complex plane associated with exp( θ e ) =exp( − π i /
4) and a small correction proportional toexp(2 π i / θ ℓ (making it more nega-tive). Similarly, an isolated increase in component 3 (thetau) decreases θ ℓ . Finally, an isolated increase in compo-nent 1 (the electron) affects only the real component ofan entry such as [ K ℓ ] and thus when [ K ℓ ] is complex,once again affects the value of θ ℓ . When θ ℓ is no longerfixed to be − π/
4, it is labelled by the relevant particlefamily, e.g. θ e , and it becomes necessary to specify K ℓ as K ℓ ( θ ℓ ) for some family ℓ .Next, recognise that it is convenient to write the massinteraction as a leading-order term derived from the QLphoton field (and associated with colour mixing ma-trix K ℓ as derived Sec. III A 1) plus corrections. Forthe charged lepton masses, this corresponds to rewritingEq. (72) as m e i = f h k ( e ) i i h E (2)QL i [1 + ∆ e ( m e i )]= h m (0) e i i [1 + ∆ e ( m e i )] (77)∆ e ( m e i ) = 90 αm e i π ( m ∗ c ) + (5 − f Z ) αm e i πm W (78)+ 5 m e i ( m ∗ c ) (cid:20) αm e i π ( m ∗ c ) + (25 − f Z ) αm e i πm W (cid:21) + 240 m e i m e h k ( e )1 i m ′ m W where { e , e , e } = { e, µ, τ } and the subscript e on ∆ e indicates the family { e, µ, τ } rather than the electron inparticular. As can be readily seen from Eq. (78), theamplitudes of these corrections to particle mass are de-pendent upon the squared lepton masses m e i , with the largest corrections being those associated with the tau.The charged lepton mixing matrix K e ( θ e ) is implicitlydependent on the same corrections, as changes to the par-ticle mass ratios affect colour mixing, and hence the co-ordinate transformation required to restore colour neu-trality. Allowing the corrections to augment mass andalso to adjust K e ( θ e ) is not double-counting, as the masscorrection induces the corresponding adjustment in theco-ordinate transformation implemented by K e ( θ e ). Theco-ordinate transformation does, however, have an effecton the magnitude of the mass correction term, and thiseffect must be compensated for—see Sec. III C 5.
3. First-order correction to K ℓ from the tau channel To understand how these corrections to lepton massesaffect the matrices K ℓ , consider the leading order term (cid:2) m (0) e i (cid:3) which arises from the set of nine diagrams de-scribed in the caption of Fig. 10, and may be thought ofas the action of an operator (cid:2) ˆ m (0) (cid:3) on the preon tripletcomprising a sum over nine terms. The action of thisoperator on the colour space of the preon triplet breakscolour neutrality, but this is then corrected by applica-tion term-by-term of four matrices (or operators) K ℓ ,heuristically K ℓ (with implicit action of each operatoron the appropriate Hilbert space as per the appropriatediagram of Fig. 10), such that K ℓ (cid:2) ˆ m (0) (cid:3) as a wholeleaves the colour of the preon triplet unchanged up to acycle r → g → b → r .As seen in Eq. (77), higher-order corrections enhancethe action of (cid:2) ˆ m (0) (cid:3) on eigenstate e i by a factor of[1 + ∆ e ( m e i )]. If colour neutrality is to be conserved,then for colour-changing diagrams [including those which can mix colours under some global SU(3) C transforma-tion, such as interaction with the diagonal gluons of the representation of SU(3) C ] there must be an equivalentenhancement of the matrices K ℓ , K ℓ → K e ( θ e ) (79)[where dependence on θ e ( { m e i | i ∈ { , , } } ) is as-sumed but will subsequently be shown] such that theaction of [ K e ( θ e )] on e i is equivalent to the action of K ℓ [1 + ∆ e ( m e i )].As the largest such correction arises from the mostmassive particle, begin by considering e , the tau. Firstnote that introduction of a mass dependency for the ma-trices K e ( θ e ) implies that the bosons associated withthe colour-neutrality-preserving co-ordinate transforma-tion are no longer parameterless, and may therefore carrymomentum in the higher-order diagrams. They musttherefore be represented explicitly as per Fig. 17. Byconstruction these bosons only interact at the bound-ary of the transformed co-ordinate patch, and are con-sequently both foreground and massless. Let the dia-grams of Fig. 17(i)-(iii), having two co-ordinate trans-formation bosons, be termed “first order”, and for now3 FIG. 17. When the matrices K ℓ are replaced by K e ( θ e ), whichdepends on energy scale, this induces a pair of gauge bosonswhich may in principle connect the four matrices K e ( θ e ) inany of three ways [shown for Fig. 10(i) in diagrams (i)-(iii)]. Indiagram (i), the bosons each connect to two matrices K e ( θ e )on the same preon and thus enact a co-ordinate transforma-tion followed by its inverse. In diagram (ii), where the bosonsare crossed, there may be a permutation of the preon colourlabels but overall colour neutrality is nevertheless conserved.Diagram (iii) contrasts with the other two in that the bosontrajectories are orthogonal to the preon trajectories. For aworldsheet such as that shown (labelled W ), which is equidis-tant between the boson vertices and isochronous in the restframe of the particle, the total number of oriented crossingsfor each of the bosons in diagram (iii) is zero regardless of thetrajectory followed. This implies that these bosons do notdescribe a co-ordinate transformation extant at this world-sheet, and thus this diagram does not contribute to K e ( θ e ).Diagram (iv) is an example second-order diagram having fourbosons arising from the co-ordinate transformation and span-ning from the lower to the upper collection of matrices K e ( θ e ).This diagram represents the consecutive application of two(possibly different) transformations in SU(3) C to each preon. ignore diagrams having four co-ordinate transformationbosons (“second order”).Regarding the two co-ordinate transformation bosonsof Fig. 17(i)-(ii), their vertex factors are incorporatedinto the leading order matrix K ℓ . Integration over theirdegrees of freedom then yields a factor per diagram of(4 π ) − [not (2 π ) − , as they are not self-dual and thustheir loop is less symmetric than that for the photon].Note that Fig. 17(ii) counts as a loop diagram as there areimplicit gluon-mediated couplings between the preonicconstituents of the lepton, which are massless over thelength-scale involved [O( L QL )] and complete closure ofthe loop.Recognising that the leading term calculation does not perform any net colour mixing beyond the unmodified,constant factors of K ℓ , the resulting factor of 2 · (4 π ) − does not appear in the leading term calculation. Its ef-fect is seen only on the diagrams in ∆ e ( m τ ), where itrepresents variations in colour mixing relative to K ℓ , andmultiplies any correction to K ℓ by a factor of 2 · (4 π ) − .Next, recognise that the components of each matrix K e ( θ e ) which perform rotations on SU(3) C are composedfrom a weighted superposition of some set of representa-tion matrices λ ′ i (possibly, but not necessarily the ma-trices λ i of Appendix A), of which there are eight. (Aninth basis element is associated with the trivial repre-sentation and acts to preserve norm, but is strictly di-agonal, and independent of θ ℓ , so is not yet of interesthere.) An appropriate choice of basis permits every off-diagonal element in K e ( θ e ) to be associated with a singlebasis element { for example, in K ℓ , entry [ K ℓ ] is associ-ated with ( λ + λ ) / √ } . A change to θ e may correspondto a change in choice of representation matrices [e.g. to λ cos β + λ sin β for some angle β ] but this ability toconstruct a single basis element corresponding to a par-ticular entry in K e ( θ e ) persists for any reasonably smallperturbation around θ e = − π/ K e ( θ e ) acting on it, and recognise that ifthe portion of K e ( θ e ) which performs colour transforma-tions is decomposed into these eight components, then ifa given component λ ′ i acts on the preon in the lower posi-tion, then its conjugate must act in the upper position forthe foreground preon colour to be left invariant overall. (3) The action of the pair of matrices K e ( θ e ) on this bosonconsequently admits decomposition into eight channels,enumerated by the family of orthogonal representationmatrices λ ′ i appearing at the lower matrix K e ( θ e ).Next, consider the remaining two matrices K e ( θ e )when these are not on the same preon. In this situationthe representation matrices present will be conjugate upto a colour cycle r → g → b → r or its inverse. Asthe only colour-changing bosons in the QL are gluons,and SU(3) C symmetry is preserved on the gluon sectorand in the definition of the charged lepton, evaluation ofany diagram where the entries from K e ( θ e ) are offset by acolour cycle is necessarily equivalent to evaluation of a di-agram where they are not offset. It is therefore acceptable Note that if one diagram family as per Fig. 10 can be made tohave no net effect on colour, then all paired QL bosons have nonet effect on colour. Further, as unpaired bosons (as well as beingexceedingly rare) have on average no net effect on colour, thesemay also be grouped into clusters having zero net colour effectoverall, and an analogous co-ordinate transformation performedsuch that their effect on colour vanishes instant-by-instant aswell as on average, with the associated bosons vanishing overdistances large compared with L QL . Thus no error is made inrequiring that colour is left unchanged on a per-diagram basis.Also note that once the mechanism for corrections to θ e is fullyelucidated, the co-ordinate transformations arising from the un-paired bosons yield no systematic effect on θ e and thus theireffect may be ignored over macroscopic scales. K e ( θ e ) under the assumption thatall matrices K e ( θ e ) appear in conjugate pairs. In addi-tion, in all diagrams all representation matrices are mul-tiplied by their conjugate (whether independently or inconjunction with the representation matrices associatedwith the bosonic vertices) and thus the terms arising fromall 64 channels contributing to the actions of the matri-ces K e ( θ e ) on the preon triplet are additive on the same(net trivial) charge sector, and therefore summed. BySU(3) C symmetry, each channel will contribute equallyto the overall correction to the leading order diagram.Now consider the single channel associated with aspecific off-diagonal entry in at least one lower matrix K e ( θ e ). For definiteness, let this be [ K e ( θ e )] . As thereare, overall, 64 contributing channels, each channel mustproduce a net correction of ∆ e ( m τ ) /
64. However, thesynthetic bosons multiply all correction diagrams by afactor of 2 · (4 π ) − and thus [ K e ( θ e )] and [ K e ( θ e )] must satisfy[ K e ( θ e )] [ K e ( θ e )] = [ K ℓ ] [ K ℓ ] (cid:20) π ) ·
64 ∆ e ( m τ ) (cid:21) = [ K ℓ ] [ K ℓ ] (cid:20) π e ( m τ ) (cid:21) (80) ⇒ [ K e ( θ e )] [ K e ( θ e )] = [ K ℓ ] [ K ℓ ] r π e ( m τ ) . (81)Noting that [ K ℓ ] = [ K ℓ ] † , it is convenient to write δ e ( n ) = r π n − δ e ( n )] = [1 + i p δ e ( n )][1 − i p δ e ( n )] (83)and assign the corresponding off-diagonal entries of K e ( θ e ) to be[ K e ( θ e )] = [ K ℓ ] [1 ± i p δ e [∆ e ( m τ )]][ K e ( θ e )] = [ K ℓ ] [1 ∓ i p δ e [∆ e ( m τ )]] , (84)preserving the Hermeticity of K e ( θ e ).By the factor of ± i on the correction term, this correc-tion is orthogonal to the leading-order value of [ K ℓ ] .As this orthogonality is independent of the value of θ e , this implies that for a non-infinitesimal correction, p δ e [∆ e ( m τ )] is the length of an arc. Since [ K ℓ ] is ofunit length, the corresponding correction to θ ℓ is θ ℓ → θ ℓ ± p δ e [∆ e ( m τ )] + O ( { δ e [∆ e ( m τ )] } ) . (85)In the vicinity of θ ℓ = − π/
4, this may be rewritten θ ℓ → θ ℓ ( ∓ p δ e [∆ e ( m τ )]3 π ) . (86) As per the discussion under Eqs. (74–76), the tau correc-tion is known to decrease the magnitude of θ ℓ , giving[ K e ( θ e )] = e i θ e √ K ℓ ] n p δ e [∆ e ( m τ )] o (87) θ e = − π ( − p δ e [∆ e ( m τ )]3 π ) (88)where the value of θ ℓ has been written explicitly as − π .
4. Second-order correction to K ℓ from the tau channel The first-order correction to K ℓ is applied to all di-agrams which modify the colour mixing process, i.e. alldiagrams contributing to ∆ e . For these diagrams, thiscorrection is equivalent by construction to enacting thetransformation K ℓ → K ℓ [1 + ∆ e ( m τ )] , (89)resulting in a relative increase in these diagrams’ contri-bution to particle mass equivalent to∆ e ( m τ ) → ∆ e ( m τ )[1 + ∆ e ( m τ )] . (90)Any species-dependent increase in mass affects colourmixing in precisely the way described above for correc-tion ∆ e ( m τ ), regardless of whether this increase is di-agrammatic (as in the first-order component) or gauge-dependent and derived from a change to θ e (as here).This correction therefore attracts a further smaller cor-rection to K ℓ , enhancing the first-order correction calcu-lated above. While this series may be continued indefi-nitely, it is convenient to truncate at second order for aprecision of O[∆ e ( m τ )].To implement the corresponding adjustment to K e ( θ e ),let the O[∆ e ( m τ )] term be compensated by the second-order diagrams of which Fig. 17(iv) is a prototype. Inthese diagrams, a first-order correction [Fig. 17(i)-(ii)] issupplemented by a further, smaller correction.For Figs. 17(i)-(ii) there is a symmetry factor of twocorresponding to crossing or not crossing the bosons. ForFig. 17(iv), • the inner pair may be crossed (factor of two), • the outer pair may be crossed (factor of two), • the inner and outer bosons on the left may be ex-changed (factor of two), and • the inner and outer bosons on the right may beexchanged (factor of two).The factor of 2 · (4 π ) − associated with the set of diagramsFigs. 17(i)-(ii) becomes a factor of 16 · (4 π ) − for the set5of diagrams derived from Fig. 17(iv). The expression for[ K e ( θ e )] [ K e ( θ e )] then becomes[ K e ( θ e )] [ K e ( θ e )] = [ K ℓ ] [ K ℓ ] (cid:20) π ) ·
64 ∆ e ( m τ ) + (4 π ) · (cid:21) ∆ e ( m τ )= [ K ℓ ] [ K ℓ ] (cid:26) π e ( m τ ) (cid:20) π
32 ∆ e ( m τ ) (cid:21)(cid:27) (91)for a redefinition of δ e ( n ) in Eq. (88) to δ e ( n ) = s π n (cid:18) π n (cid:19) − . (92)The next order term extends δ e ( n ) to δ e ( n ) = s π n (cid:20) π n (cid:18) π n (cid:19)(cid:21) − m Z at O(10 − σ exp ), or in relative terms, to m W , m Z ,and m H ′ in order of parts in 10 .
5. Dilaton corrections to ∆ e ( m e i ) a. Running of m e i In Sec. III E 1 a of Ref. 1, thedilaton parameter was gauged by requiring normalisa-tion of probabilities for all foreground particles. However,probability is a property normalised on integrating overa three-dimensional worldsheet in R , , whereas the dila-ton coefficient may vary in time as well as space. Withprobability amplitudes depending only on x i | i ∈ { , , } but with the dilaton coefficient being a function of x µ ,the dilaton gauge is therefore underconstrained. Thischoice of gauge may therefore be extended to normaliseone further parameter, provided that parameter is en-ergy/timelike and its rescaling is independent of the prob-ability parameters being normalised over R .Now recognise that in the above, the lepton mass m e i has been written as a function of ∆ e and θ e , which arein turn a function of both the lepton mass itself and alsothe energy scale of the gluon deficit. For a particle whichis both on-shell and at rest, these two energy scales areidentical, but more generally, the gluon mass deficit willrun with energy scale. This running leads to a running ofelectron mass which has no counterpart in the StandardModel. Therefore complete the gauging of the dilatonparameter by enforcing that everywhere, after rescaling, h m e i O( L QL ) takes on its rest value regardless of the energyof the electron. The implications of this gauge will beseen in Sec. IV A. b. Rescaling arising from colour mixing The abovetransformations have maintained colour neutrality, but at the expense of introducing an additional scaling fac-tor of [1+∆ e ( m τ )] on all diagrams which affect the colourmixing process [i.e. all diagrams in ∆ e ( m τ )]. This rescal-ing is unphysical, being the result of a forced (non-gauge)change in local co-ordinate system, and may not in gen-eral be assumed to conserve the normalisation of proba-bility and electron mass generated by the choice of dila-ton gauge [1, Sec. III E 1 a; Sec. III C 5 a above]. This theninduces a change in the dilaton gauge which rescales theaffected sector to eliminate this factor.The effect of this change of gauge on particle mass isto rescale the modified correction factor∆ e ( m τ )[1 + ∆ e ( m τ )] → ∆ e ( m τ )[1 + ∆ e ( m τ )][1 + ∆ e ( m τ )] − = ∆ e ( m τ ) (94)recovering a particle mass of m τ = h m (0) e i [1 + ∆ e ( m τ )] . (95)This correction factor of [1 + ∆ e ( m τ )] − arises from thedilaton field and thus is insensitive to colour charge. Incontrast, the calculation of θ e is performed on a sin-gle channel of SU(3) C , and thus on a single channel ofGL(3 , R ). Recognising that the correction ∆ e ( m τ ) toparticle mass arises from a symmetric sum over all ninechannels of GL(3 , R ), it consequently follows that the cal-culation of θ e “sees” only one ninth of the total correctionarising from the regauging of the dilaton field.Further recognise that to maintain colour neutralitythis correction factor must be partitioned equally be-tween the mass interaction and the compensatory matri-ces [ K e ( θ e )] . The result is to rescale ∆ e ( m τ ) in Eq. (88)according to ∆ e ( m τ ) → r [∆ e ( m τ )], θ e = − π ( − p δ e { r [∆ e ( m τ )] } π + O h ( δ e { r [∆ e ( m τ )] } ) i) (88b) r ( n ) = n · r n ) − − , (96)again to O[∆ e ( m τ )].
6. Corrections to [ K e ( θ e )] from the muon and electronchannels Now recognise that a propagating preon is a massiveparticle and undergoes multiple scatterings off the QLfields. These scatterings may impart energy to or takeenergy from the preon, and are generally ignored as theiraverage contributions vanish over length scales large com-pared with L QL . Bringing these into consideration, itis necessary to integrate over all possible scatterings of6the preon off the QL fields, and for a foreground preonwith a rest energy small compared with E QL , the net ef-fect is to integrate near-homogeneously over all energiesfrom −E QL to + E QL [effectively homogeneously at ener-gies small compared with E QL , corresponding to boostssmall compared with 1 − as per Eq. (13)]. On-shellcontributions appear at the three eigenvalues of K e ( θ e ),and thus even for a foreground lepton in a definite coloursector eigenstate, all three channels contribute to the cor-rection to θ e . Over length scales of O( L QL ), in expres-sions involving the matrices K e ( θ e ), particle propagationis summed over the three generation channels. In expres-sions which do not involve these matrices, such as evalua-tion of boson masses, fermion symbols are generationless.Evaluation of the parameter θ e inevitably involves themixing matrices K e ( θ e ), and thus corrections to θ e fromall channels of propagation must be considered regardlessof the generation of the foreground particle which maybe measured in the low-energy limit.Now that there are multiple energy scales involved,it is advisable to be more careful with the gluon massdeficit. This also permits generalisation to particles fur-ther from rest in the isotropy frame of the QL, and isuseful preparation for Sec. IV A, in which off-shell exci-tations are considered. Note that the mean energy scaleof the gluon deficit is determined by the mean additionalenergy acquired by the foreground species on account ofits nonzero rest mass, i.e.[ k ] fg − q [ k ] + [ k ] + [ k ] , (97)and that this is not in general the same as the instan-taneous energy of the propagating particle. The gluondeficit undergoes its own energy fluctuations around thisvalue as a result of energy exchange between the deficitand the QL, but these are not necessarily simultaneouswith the fluctuations of the representative foregroundparticle, and in the absence of any generations mech-anism for bosons, may be separately averaged to zero.Further, when a particle is off-shell, or undergoes a boostrelative to the QL, this affects the energy scale of thegluon deficit via Eq. (97) without affecting particle restmass save through the dependency of rest mass on thegluon deficit as per Eq. (78). Recognising the indepen-dence of these energy scales, therefore write∆ e ( m e i , E ) = 90 αm e i π [ m ∗ c ( E )] + (5 − f Z ) αm e i πm W + 5 m e i [ m ∗ c ( E )] " αm e i π [ m ∗ c ( E )] + (25 − f Z ) αm e i πm W + 240 m e i m e h k ( e )1 i m ′ m W (98)[ m ∗ c ( E )] = m c (cid:18) − E m c c (cid:19) , (59b)where E is the energy scale used in calculation of thegluon mass deficit correction. Previous occurrences of ∆ e ( m τ ) in Secs. III C 3–III C 5 are thus noted to implic-itly be ∆ e ( m τ , m τ c ).To evaluate the correction arising from the e or muonchannel, recognise from Eqs. (75–76) that the effect ofthis correction is in opposition to the tau correction, andincreases the magnitude of θ e . As its scale is seen fromEq. (78) to be small compared with the tau correction,and from the construction of Eq. (84) it is seen to actin direct opposition to the tau correction, its effects areconveniently represented at energy scale E by subtractingthe muon correction from the tau correction at the levelof the interaction diagrams. This yields θ e ( E ) = − π − p δ e { r [∆ e ( m τ , E ) − ∆ e ( m µ , E )] } π ! (99)where θ e ( E ) is the effective value of θ e experienced by aparticle whose mass is E c − , and henceforth all instancesof θ e are acknowledged to depend, either explicitly orimplicitly, on a particle’s energy scale E .Finally, the e or electron channel is seen from Eq. (74)to contribute purely to the real part of [ K e ( θ e )] . Forthe imaginary part, all positive contributions to the imag-inary portion of [ K e ( θ e )] arise from the muon channeland all negative contrinutions from the tau channel, en-abling the relatively simple form of Eq. (99). In contrast,while the electron channel contributes to the real partof [ K e ( θ e )] the bulk of the real part (and, indeed, for θ ℓ = − π/ m e → m e [1+∆ e ( m e , E )], resulting in some rescaling (1 + ε ) of k ( e )1 . Asthe real and imaginary components of exp(i θ ℓ ) are bothnegative, the increase in the real component of [ v v † ] associated with this correction will decrease the magni-tude of θ ℓ . However, the electron mass is one of the fixedinput parameters to the model, and consequently thisrescaling (and its associated effect on θ ℓ ) must be offsetby a decrease in magnitude of the associated energy scale E (2)QL { and thus of (cid:2) m (0) e (cid:3) } corresponding to multiplica-tion by (1 + ε ) − . If this value is pulled out as an inde-pendent factor, in the electron mass diagram it cancelswith the (1 + ε ) arising from the electron mass rescaling,leaving m e and k unchanged (and eliminating any needto evaluate colour effects from this sector). However, forthe muon and the tau it may be seen as a rescaling of k ( e )2 or k ( e )3 respectively by a factor of (1 + ε ) − . To eluci-date the effect of this scaling factor on θ ℓ , restore colourneutrality by likewise holding the muon and tau massesfixed, corresponding to enacting a transformation k ( e ) i → k ( e ) i (1 + ε ) | i ∈ { , } . (100)The leading colour effect arising from this transforma-tion is equivalent to a rescaling of k ( e )3 (which generatesthe leading correction to θ ℓ ) by (1 + ε ) and is therefore7associated with an increase in the magnitude of θ ℓ . Therescaling of the muon term is by the same factor. Thismultiplier is independent of the existing muon and taucorrections, having its origins on the real rather than theimaginary portion of [ K e ( θ e )] , and therefore must beevaluated as a separate correction to θ ℓ = − π/ θ e ( E ) = − π − p δ e { r [∆ e ( m τ , E ) − ∆ e ( m µ , E )] } π ! × p δ e { r [∆ e ( m e , E )] } π ! . (101) IV. RELATIONSHIPS FROM R | DUSTGRAVITYA. Mass relationships
To obtain testable predictions from R | dust grav-ity, it is necessary to eliminate the model parameters f , E (1)QL , and E (2)QL from the mass expressions derivedabove. This is readily achieved by taking ratios of particlemasses. Recognising that the choice of gauge articulatedin Sec. III C 5 fixes the value of m e so this does not runwith energy scale, the value of m τ ( E τ ) /m e ( E τ ) evaluatedat energy scale E τ = m τ c then readily admits interpre-tation as the ratio of the rest masses of the electron andthe tau, and similarly for m µ ( E µ ) /m e ( E µ ) and the restmass of the muon.In conjunction with the boson mass expressions devel-oped in Sec. II, and noting the freedom to substitute m e h k ( e )1 ( E e ) i −→ m µ h k ( e )2 ( E µ ) i (102)or m e h k ( e )1 ( E e ) i −→ m τ h k ( e )3 ( E τ ) i (103)as convenient, as discussed in Sec. II A 2, this yields therelationships (at the present level of precision) m τ m e = h k ( e )3 ( E τ ) i ( αm τ π [ m ∗ c ( E τ )] + (5 − f Z ) αm τ πm W + m τ [ m ∗ c ( E τ )] h αm τ π [ m ∗ c ( E τ )] + (25 − f Z ) αm τ πm W i + m τ h k ( e )3 ( E τ ) i m ′ m W )h k ( e )1 ( E τ ) i ( αm e π [ m ∗ c ( E τ )] + (5 − f Z ) αm e πm W + m e [ m ∗ c ( E τ )] h αm e π [ m ∗ c ( E τ )] + (25 − f Z ) αm e πm W i + m e m τ h k ( e )3 ( E τ ) i m ′ m W ) (104) m µ m e = h k ( e )2 ( E µ ) i ( αm µ π [ m ∗ c ( E µ )] + (5 − f Z ) αm µ πm W + m µ [ m ∗ c ( E µ )] h αm µ π [ m ∗ c ( E µ )] + (25 − f Z ) αm µ πm W i + m µ h k ( e )2 ( E µ ) i m ′ m W )h k ( e )1 ( E µ ) i ( αm e π [ m ∗ c ( E µ )] + (5 − f Z ) αm e πm W + m e [ m ∗ c ( E µ )] h αm e π [ m ∗ c ( E µ )] + (25 − f Z ) αm e πm W i + m e m µ h k ( e )2 ( E µ ) i m ′ m W ) (105) m W m Z = 3 (cid:2) (cid:0)
64 + π − f Z (cid:1) α π (cid:3) ( m µ h k ( e )2 ( E µ ) i m W ) (cid:2) (cid:0) + π (cid:1) α π (cid:3) ( m µ h k ( e )2 ( E µ ) i m W ) (106) m m W = 20 (cid:0) π (cid:1) (cid:0) απ (cid:1) (cid:2) (cid:0)
64 + π − f Z (cid:1) α π (cid:3) ( m µ h k ( e )2 ( E µ ) i m W ) (107) m c m W = 1 + m µ h k ( e )2 ( E µ ) i m W m µ h k ( e )2 ( E µ ) i m W (108)where k ( ℓ ) n ( E ) = 1 + √ (cid:20) θ ℓ ( E ) − π ( n − (cid:21) (52b) f Z = 13 (cid:18) − m W m Z + 16 m W m Z (cid:19) (19b)8[ m ∗ c ( E )] = m c (cid:18) − E m c c (cid:19) (59b) E ℓ = m ℓ c (109) and θ e ( E ) = − π − p δ e { r [∆ e ( m τ , E ) − ∆ e ( m µ , E )] } π ! p δ e { r [∆ e ( m e , E )] } π ! (101)∆ e ( m e i , E ) = 90 αm e i π [ m ∗ c ( E )] + (5 − f Z ) αm e i πm W + 5 m e i [ m ∗ c ( E )] " αm e i π [ m ∗ c ( E )] + (25 − f Z ) αm e i πm W + 240 m e i m µ h k ( e )2 ( E µ ) i m ′ m W (98) r ( n ) = n · r n ) − −
19 (96) δ e ( n ) = s π n (cid:20) π n (cid:21) − . (92)The lower precision on m H ′ compared with the otherweak bosons is acceptable as the main sensitivity to thesehigher-order mass corrections is in the lepton mass ratiosvia θ e , in which m H ′ plays only a very small part. Theterms provided suffice to calculate m ′ to current exper-imental precision.Taking m e , m µ , and α as input parameters [7, 8], m e = 0 . /c (110) m µ = 105 . /c (111) α = 7 . × − , (112)these relationships may be solved numerically to yieldthe results given in Table II. Parameter Calculated value Observed value Discrepancy(GeV /c ) (GeV /c ) m τ . . . σ exp m W . . . σ exp m Z . . . σ exp m H ′ . . . σ exp m c . R | dustgravity. Quantity m c is the bare gluon mass. The displayeduncertainties in calculated masses arise from estimates ofhigher-order terms. The leading corrections due to uncer-tainty in m e , m µ , and α are smaller, being of order 10 − σ exp ,or parts in 10 on all masses. The gluon mass shown is thebare mass at scales below E Ψ , though with the caveat that itfrequently becomes massless in interactions above this scale,as seen in the calculations above. B. Minimum requirements for particle generationsfrom preon substructure
These results strongly suggest that particle generationsarise from a preonic substructure for leptons. It wouldappear that the minimum requirements for direct importof this structure from R | dust gravity into a modifica-tion of the Standard Model while retaining its predictivecapacity are as follows: • A tripartite preon structure for leptons bound bya GL(3 , R )-symmetric force sector [which need notnecessarily correspond to the strong force plus anadditional (axion?) companion, as here]. • A bipartite preon structure for the Higgs/scalar bo-son. • A duality between preon/antipreon pairs and theelectroweak and preon-binding bosons. • A multi-species quantum liquid in place of thepresent nonvanishing vacuum expectation value ofthe Higgs boson.However, these features come with costs, including: • Importing the quantum liquid relegates the scalarboson to a bystander having only a small role toplay in generating particle mass. The Higgs mecha-nism might still be responsible generating the quan-tum liquid, but will require some modification to beapplied in this context. • If the preon binding bosons are the gluons of thestrong force, there must exist an axion (this maybe useful as a dark matter candidate). • If the preon binding bosons are not the gluons ofthe strong force, further work is required to de-termine whether the inhomogeneous preon tripletscorrespond to quarks and why they carry colourcharge.9Alternatively the current role for the Higgs boson maybe retained, but only at the cost of introducing numer-ous fine tuning parameters. Unless an alternative butequivalent mechanism for fixing these parameters is con-structed, this comes at the cost of predictive power.Although R | dust gravity incorporates a colour-based mechanism for particle generations with substan-tial predictive power in the electroweak sector, it is pre-mature to identify this with electrostrong unificationwithout at least some evidence of predictive capacity inthe strong nuclear sector. It is, however, clearly a candi-date for a unifying mechanism, and an obvious test wouldbe calculation of the quark masses.Multiple other avenues for further testing of R | dustgravity exist, both theoretical and experimental, and per-haps one of the most accessible relates to the weak mix-ing angle. In Eq. (37), the mass ratio used to computesin θ W effectively takes a weighted average over all in-teractions of the Z boson, whereas in Sec. II C it is notedthat differences in higher-order corrections which dependon fermion charge could result in differing electroweakcouplings for different species, and thus different mea-sured values of sin θ W . There do appear to be somediscrepancies in the values of sin θ W measured using dif-ferent species [9, p. 172], and it would be fruitful to assesswhether this can be explained using R | dust gravity. V. CONCLUSION R | dust gravity is a model having many elementsin common both with the Standard Model of particlephysics and with the observable universe. As it has onlyfour tunable parameters, which must be set by referenceto physical constants, R | dust gravity may readily betested against observation.In the present paper, relationships in the electroweaksector of R | dust gravity were used to predict the val-ues of several fundamental particle masses in the Stan-dard Model, with results in excellent agreement with ob-servation. These results arise from the particle genera-tion mechanism used in R | dust gravity. Appendix A: Gell-Mann matrices
When working with the group SU(2), a well-knownbasis for the tangent Lie Algebra su(2) is given by the Pauli matrices σ i , rescaled by a factor of 1 / √ { τ i | i ∈ { , , } } τ i = σ i √ . (A1)A similar basis for su(3) is provided by rescaling the Gell-Mann matrices C a , { λ i | i ∈ { , . . . , } } λ i = C i √ C = C = − i 0i 0 00 0 0 C = − C = C = − i0 0 0i 0 0 C = C = − i0 i 0 C = √
00 0 − . (A3)Like the rescaled Pauli matrices τ i , the re-scaled Gell-Mann matrices satisfyTr [( λ i ) ] = 1 ∀ i ∈ { , . . . , } . (A4)In conjunction with a multiple of the identity matrix λ = 1 √ , (A5)the matrices λ i also yield a basis for gl(3 , R ).An alternative basis for gl(3 , R ) is given by the ele-mentary matrices e ij , consisting of a matrix with 1 inposition ( i, j ) and zero elsewhere. [1] R. N. C. Pfeifer, arXiv:0805.3819v14 (2020).[2] M. E. Peskin and D. V. Schroeder, “An Introduction toQuantum Field Theory,” (Westview Press, USA, 1995) p.773.[3] Y. Koide, in Proc. 30th Int. Conf. High-energy Phys., Os-aka, Japan , edited by C. S. Lim and T. Yamanaka (WorldScientific, Singapore, 2001) arXiv:hep-ph/0005137v1. [4] R. Penrose, in
Combinatorial Mathematics and its Appli-cations , edited by D. J. A. Welsh (Academic Press, 1971)pp. 221–244.[5] R. N. C. Pfeifer, P. Corboz, O. Buerschaper, M. Aguado,M. Troyer, and G. Vidal, Phys. Rev. B , 115126 (2010).[6] R. N. C. Pfeifer, Simulation of Anyons Using SymmetricTensor Network Algorithms , Ph.D. thesis, The University of Queensland (2011), arXiv:1202.1522v2.[7] P. A. Zyla et al. , to be published in Prog. Theor. Exp.Phys. , 083C01, (2020), http://pdg.lbl.gov/.[8] E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor,“The 2018 CODATA Recommended Values of the Funda-mental Physical Constants,” (2018), (Web Version 8.1). Database developed by J. Baker, M. Douma, and S. Ko-tochigova. Available at http://physics.nist.gov/constants,National Institute of Standards and Technology, Gaithers-burg, MD 20899.[9] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98