arXiv:2008.05893v1 [physics.gen-ph] 11 Jul 2020 I.Lpo asInteraction Mass III. ∗ I oo asInteractions Mass II. .Introduction I. [email protected] .Gunmasses D. .Laigorder Leading A. A. .Cretost h etnms angle mass lepton the to Corrections C. angle mixing Weak C. B. .Frgon opCorrections Loop Foreground B. .Saa oo mass boson Scalar E. .Nurlbsngravitation boson Neutral F. Z .Frtodrcreto to correction First-order 3. Preamble 2. corrections of Origin 1. corrections EM 2-loop 5. corrections scalar 1-loop corrections 4. force weak 1-loop 3. corrections gluon 1-loop 2. corrections EM 1-loop 1. deficits mass field scalar QL and of Gluon component 4. scalar from Mass 3. components gluon and from Mass sector 2. colour on Action 1. loops boson Scalar 2. loops boson Vector 1. interactions scalar and photon QL 2. loops Boson 1. corrections loop of interactions Universality scalar 3. and photon QL 2. loops Boson 1. W mass fQL of channel mass assae hc spooe oaiefo ro interactio l from between arise to relationship proposed unifying is which a scale, of mass existence the suggests 1776 n 125 and aaees hyaesont e1776 be to shown are They parameters. oos n ig-iesaa oo r acltda func as calculated are boson scalar Higgs-like a and , oe a eotie.I hsppr h oe ausfrthe for values model the paper, this In obtained. be may its Model between param relationships tunable predictive four constrained, only highly with but gravity, and Model Standard The . 61)MeV 86(12) R . 61 GeV 16(1) 0 CONTENTS | 18 utgaiymdlcnan nlge otepril spect the to analogues contains model gravity dust /c /c 2 atcegnrtosin generations Particle 80 , 2 epciey l ihn0 within all respectively, . 7(2 GeV 379(12) K ℓ 4 rmtetau the from uei,Oao e Zealand New Otago, Dunedin, oetN .Pfeifer C. N. Robert /c Dtd uy1,2020) 12, July (Dated: 2 91 , . 6()MeV 867(1) . 22 21 21 21 21 21 20 18 17 17 17 15 14 12 12 12 11 11 10 10 862)GeV 1876(21) 3 2 2 1 9 9 8 7 7 6 5 . 5 σ rbte ftecrepnigosre ausof values observed corresponding the of better or opiigafe utfil ntemanifold the on field dust free a comprising o ae nteds edbhv sitrcigquasi- interacting as behave a field on dust soli- the where limit in low-energy waves the ton in description a state, a (from admits homogeneous hence it and highly disordered a highly perspective) a coarse-grained in is field dust this hrfr rvnsteds edfo en normalisable being the from on field dust the prevents therefore emtyof geometry n h ffcieprilso h o-nrylmtt ap- to limit O( order low-energy to the wavefunctions normalisable of O(10 proximate particles of effective is the ing each in participating l aeucin vra(,)ds umnfl of submanifold (1,3)-disc to parti- a fields over normalised these wavefunctions as cause cle interpretation gauge a approximate of and choices admit , and , boson, into scalar condense Fermionic hie fgueclas neegn U9 symmetry SU(9) emergent an U(1) Further collapse to universe. gauge observable of the choices to correspond to taken oargo fcre imninsaetm denoted space-time Riemannian curved of region a to V eainhp from Relationships IV. .Gl-anmatrices Gell-Mann A. Conclusion V. /c R nrdcdi Ref. in Introduced oneprst h osat fteStandard the of constants the to counterparts 2 0 References .Ms relationships Mass A. .Mnmmrqieet o particle for requirements Minimum B. 80 , | in of tions 18 /c smdae ytesrn ula force. nuclear strong the by mediated ns po eeain n h electroweak the and generations epton .Cretost [ to Corrections ∆ 6. to corrections 5. to correction Second-order 4. eeain rmpensubstructure preon from generations R ⊗ ∗ 2 . utgravity dust tr.A h tutr fti oe is model this of structure the As eters. 763 GeV 3786(3) 1 n 125 and , , SU(2) a channel n lcrnchannels and 3 α R umnfl,tenme fds particles dust of number the submanifold, asso h a,the tau, the of masses , 0 R m | 18 .INTRODUCTION I. 1 ⊗ e , and , 3 . u n neatoso the of interactions and rum 01)GeV 10(14) iisteodro aeucin and wavefunctions of order the limits SU(3) umnfl of submanifold /c 1 m 2 , R 91 , µ C 0 R ihn refitting free no with , K | 18 0 hl apn h (1,3)-disc the mapping while . | e 874 GeV 1877(4) 18 ( /c utgravity dust θ e 2 utgaiyi model a is gravity dust )] hsresult This . 4 W e rmtemuon the from R ( m 0 K and | 18 e ℓ i 4 /c ) lhuhthe Although . rmthe from Z 2 , R 0 36 | 18 ,allow- ), When . x 10 R 36 M 1 29 29 29 28 27 27 25 25 24 ˜ , ). 3 . 2

Breaking of the weak equivalence principle is anticipated, is an absolute square, and 0 otherwise. Interestingly, the but only in limited domains.(1) scalar boson is a composite particle made up of preons This paper examines the lepton mass interactions of and satisfies R0 18 2 | dust gravity, in which composite leptons acquire 9 (1) (1) [H′(x)H′∗(y)] = f (x y) . (9) mass through coupling to the high-entropy background h QLi 2 − EQL dust field. On M˜ , this dust field may be represented In the QL, the field A corresponds to theh photon,i and as a quantum liquid (QL) described in terms of its non- µ cc˜ corresponds to associated with the Gell-Mann vanishing expectation values. In the presence of a com- µ basis of SU(3) given in Eq. (A3). The QL introduces a posite lepton, in the notation of Ref. 1 the familiar non- C preferred rest frame, but this is largely undetectable at vanishing bosonic components are (2) energy scales small compared to QL, which is compara- 2 E µ (2) (2) ble to the Planck energy. [A (x)Aµ(y)] = f (x y) (1) h QLi − − EQL Finally, there is also a dilaton field. This is introduced h i2 in Sec. IIIE1a of Ref. 1, and is denoted ∆. The gradient [ccµ˜ (x)cc˜ (y)] = f (2)(x y) (2) (2) µ QL c˜ 1,...,8 QL of the dilaton field, h i ∈{ } − − E h i ∆ = ∂ ∆ (10) where f (2)(x) is a Gaussian satisfying µ µ acts as a having no charge with respect to 4 (2) either SU(3) or SU(3) . As its QL expectation value d4x f (2)(x y)=1 y. (3) A C EQL − ∀ has not been zeroed by gauge, by maximisation of en- Z h i tropy in the QL this boson also satisfies a relationship These arise from a more general expression 2 µ f (2) (2) mm˙ nnµ˙ [∆ (x)∆µ(y)]QL = (x y) QL . (11) ∂σ∂ ϕ(x) ∂σ∂ ϕ(y) (4) − − E h µ i h i 2 By Eqs. (57–64) of Ref. 1, boson ∆µ is analogous to    mm˙ nn˙ (2) (2) vector boson ϕ81 [1, Eq. (63)], and therefore couples to = Tr (e e )f (x y) QL µ − − E foreground fields with the same bare interaction strength h i which is written in terms of the underlying dust field ϕ. as the gluons, which are derived from the related bosons z The leptons are made up of three differently-coloured ϕµ z 1,...,80 of SU(9) [1, Sec. IIID1]. The dilaton gradi- | ∈{ } preons, generically ent field may be treated as a ninth gluon, associated with the trivial representation of su(3)C. In what follows, the Ψagα(x) εαβεγδ εαγ εβδ + εαδεβγ (5) family of gluons will be taken to include the vector dila- ∝ − g ac1 ac ac3 ton gradient, giving a combined symmetry group ψ′ (x )ψ′ 2 (x )ψ′ (x ) c1c2c3 β 1 γ 2  δ 3 ×C R SU(3)C 1 ∼= GL(3, )C. (12) where g is the generation index, and the coefficients ⊕ g aα When referring to uncertainty in results, experimental c1c2c3 are constrained by requiring that particle Ψ be bothC colourless and an eigenstate of the mass-generating uncertainties will be denoted σexp, and uncertainties in interaction with the QL. The preon expectation values the theoretical calculation will be denoted σth. satisfy In this paper, it is generally assumed that any paricle under study is at rest or near-rest with respect to the m˙ n˙ m n isotropy frame of the QL. It is worth noting that, as per [ψ′ (x) ψ′ (y) ψ′ (x) ψ′(y) ] h QLi Sec. IVE of Ref. 1, a particle may be considered close to 2 (6) 1  (1) (1) at rest if its boost parameter satisfies = (m, ˙ n,m,n˙ ) f (x y) QL 2 − E 32 4 h i β < 1 10 . (13) d4x (1) f (1)(x y)=1 y (7) − EQL − ∀ Z h i II. BOSON MASS INTERACTIONS where m (orm ˙ ) and n (orn ˙ ) range from 1 to 9 and enumerate pairs of index values a (ora ˙ ) 1, 2, 3 , c (orc ˙) r,g,b . The symbol (m, ˙ n,m,n˙ ∈) is { defined} A. W mass to equal∈ 1{ iff } In Sec. IIIF3 of Ref. 1, a first-order expression for the m˙ n˙ m n [ψ′ (x) ψ′ (y) ψ′ (x) ψ′(y) ]QL (8) W boson mass was obtained in terms of two of the free 0 18 (1) h i parameters of R | dust gravity, f and : EQL 2 2 2 (1) mW = 18f QL [1 + O (α)] . (14) 1 E For massive bosons, higher-generation leptons, and for normal h i fermionic in regimes with an extremely low two-photon To obtain the high-precision numerical results presented scalar potential—comparable to deep interstellar, perhaps inter- in the present paper, it is necessary to evaluate some galactic space. higher-order corrections to this expression. 3

which take on QL values in the leading-order diagram of Fig. 1(i). These loops, in turn, may then either (ii-iii) act on a single preon triplet, or (iv) span from that triplet to another part of the diagram. However, once such a loop is introduced, the preons which are evaluated us- ing the QL mean field values no longer all arise from the same vertex. For these preons to remain correlated with their counterparts on the lower vertex, such that the 2 diagram’s contribution to mW does not vanish, the QL sources and sinks must continue to be within O( QL) of one another, and the loop correction must not introduceL any correlations with particles outside the local region (both spatial and temporal) within which the QL is self- correlated. This region has dimensions of order QL, and is termed the autocorrelation region. L Now recognise that in Figs. 1(ii)-(iii) the loop spans between two components of a composite . These are necessarily separated, on average, by a distance of O( ) = O[ (2) ], which is both the length scale asso- LΨ LQL ciated with fermion mass-squared interactions (as these are quadratic in the QL fields), and the maximum sep- aration of the preon components of a fermion triplet. If the W boson is at rest in the isotropy frame of the QL, a loop boson propagating between two of these preons will traverse a distance O[ (2) ] in a time equal to or greater FIG. 1. (i) Leading-order contribution to the W boson LQL 1 (2) mass. Loop diagrams may (ii)-(iv) modify the W boson mass- than O[c− QL], depending on whether there are mass squared interaction, or (v) merely involve it. The Standard interactionsL along its course. For zero mass interactions Model counterpart to diagram (v) is shown in diagram (vi). a trajectory along the light cone is on-shell, and attracts 2 2 no mass ratio factor f mf /mb (for masses of a fermion 1. Boson loops f and boson b) as the propagating foreground boson is massless in this context. Where mass interactions take place during boson propagation, these slow the propaga- The leading contribution to the mass of the W bo- (2) tion of the boson across the distance of O[ ]. For any son arises from interactions between the W boson and QL non-negligible mass, the time to traverseL this distance the preon components of the QL. This interaction is cor- 1 (2) then exceeds O(c− ) causing the ends of the loop rected by numerous boson loops. When evaluating these LQL corrections, it is important to distinguish between loops correction to no longer both lie within the autocorrela- which modify the mass-squared interaction, and those tion region for the W mass interaction. Thus only the which merely involve the mass-squared interaction, as light cone trajectory need be considered, and the parti- per the examples in Fig. 1. cles participating in the loop are effectively massless. A diagram is said to “merely involve” the mass-squared Note that this effect (in which particles may appear interaction if the leading-order interaction of Fig. 1(i) can massless) is only seen in contexts where an observable 2 property is dependent on nonvanishing QL correlators be replaced by a simple mass vertex mW W †W and the figure in question then obtained from expansion of the distributed across spatially disparate vertices, and thus propagator in terms of the simple mass vertex and higher- under normal circumstances the zero-mass channel is not order loop corrections of the . Where apparent as massless propagation of any particle across a these same diagrams are present both in the Standard distance large compared with QL is extremely improb- 0 18 L Model and in R | dust gravity, they may be discounted able due to the large number of candidate background as they do not contribute to calculation of the value QL interactions. Even over a distance of O( QL) there 2 are at least O(1036) such candidate interactions,L and the of mW on the mass vertex. (In MS renormalisation, 2 massless channel is only observed when evaluation of the the mass vertex value of mW corresponds to the phys- ically observable mass.) An example of such a diagram QL preon correlators systematically eliminates the con- is Fig. 1(v), with Standard Model counterpart shown in tributions involving massive particles in the loop. Fig. 1(vi). Next, consider Fig. 1(iv) which connects the QL pre- Characteristic of the diagrams which do actually mod- ons to the external W boson line. This process may ify the mass-squared vertex, and thus impact the value take place at arbitrary length scales as connection to the 2 of mW , is that the loop boson couples to the W boson may take place anywhere along its previous or 4 subsequent world line. There is no reason for the most heavily-weighted contributions to be those over length scales of O( QL) or less, and thus the loop will in gen- eral encircleL multiple interactions between the W boson and the QL in the manner of Fig. 1(i). In the absence of any mechanism eliminating interactions over larger length scales, the contributions arising from this diagram sum over all length scales and the long-range couplings introduced by the loop boson thus eliminate sufficient correlations between the QL preon fields to cause the 2 contribution of this diagram to mW to vanish. Having thus established (i) that it is only necessary to consider boson loops spanning from one preon to an- FIG. 2. Underlying diagram which gives rise to Fig. 1(ii) on other within a single composite fermion, and (ii) that expansion using mean field theory in the presence of a preon all bosons in these loops are effectively massless over the QL. length scales involved, these loop corrections may be eval- uated as follows: a. Gluon loops: It is convenient to work in the eij R value is a mathematical one, not a physical one. The base basis of gl(3, ). A gluon may arise from one preon and diagram is shown in Fig. 2. If, in evaluating this diagram, be absorbed by another, as in Fig. 1(ii), or be emitted the preons above the loop boson were approximated by and absorbed by the same preon as in Fig. 1(iii). Note the QL mean field regime, then the loop boson would that the choice of preon propagator to evaluate in the yield a correction to the lower vertex, for the same overall QL mean-field regime in Fig. 1(iii) prevents this diagram result. This yields two equivalent ways to evaluate the being reduced to a foreground self-energy diagram and 2 same diagram in the presence of the QL to lowest order in requires its inclusion in the calculation of mW . the mean field theory expansion, rather than two different There are three choices of source preon and three physical processes. choices of sink preon, for a total of nine gluon loop correc- b. Photon, W , and Z boson loops: For the pho- tion diagrams. By GL(3, R) symmetry, all of these make ton, one can proceed as with the gluons, enumerating equal contributions to m2 . It is convenient to work in W all the single-preon interactions, or alternatively one can the e basis of gl(3, R), evaluate the correction associ- ij consider a photon loop correction to the fermion as a ated with a gluon which is off-diagonal in colour, and whole. In this latter approach, consider the interac- multiply by nine. tion between W boson and leptons, and recognise that For such a diagram, the vertices are each associated a fermion/photon vertex sums over interactions with all with a factor of f, lasing in the presence of the QL yields three preons and thus automatically enumerates the nine a factor of N + 1, and the definition of a fermion in- QL figures described in Sec. IIA1a. Appropriate selection troduces a factorh i of 1/3. As the boson is off-diagonal, the of preons to evaluate as arising from the QL is assumed. loop correction attracts a structural factor of 10/3 anal- The resulting correction is then immediately seen to be ogous to that seen when calculating the W boson correc- α/(2π). tion to the electromagnetic anomaly [2], though found to For the W / vertex, it is helpful to look at add to, rather than subtract from, the original vertex. photon/preon interactions and compare these with the A factor of six arises from the admissible permutations of preon colour, but this results in a double-counting on W /lepton vertex. There are numerous ways to count the interactions; one of the simplest is as follows: First, for integrating over preon position (corresponding to inter- change of both position and colour), reducing this factor each vertex recognise that a single-preon photon loop is constructed on the subspace of charged preons. The po- to three. The Gaussian integral yields 1/(2π). The net correction associated with this diagram is therefore of sitions and colours of these preons are immaterial. The subspace is equivalent for the two diagrams, and therefore weight so is the contribution. Next, consider two-preon photon 1 10 1 60α loops, which occur on the limb for quark ver- f 2( N + 1) 3 = 1+O α2 (15) h QLi · 3 · 3 · · 2π 9 2π tices, and the electron limb for the lepton vertices. In this ·   case, focus on the subspace corresponding to the partic- relative to Fig. 1(i), and the sum over all nine gluon cor- ipating fermion. For the lepton there are three choices rections is of weight of which pair of preons participates, giving three differ- 60α ent locations for the photon loop (three possibilities on . (16) one preon configuration). For the up quark there are 2π three choices of preon configuration, with the uncharged Note that it is not necessary to evaluate a similar set of preon being in a different location for each, again giving corrections for the lower vertex: The decision to replace three different locations for the photon loop (one for each the lower preons in Fig. 1(ii) with the QL mean field of the three configurations). Again there is equivalency. 5

FIG. 4. (i) Coupling between W boson and QL photon field. (ii) Coupling between W boson and QL scalar field. For the scalar boson there is no need to separately consider crossed FIG. 3. (i) Scalar boson loop on a fermion propagator. and uncrossed configurations, as all such symmetry factors (ii) Crossing of scalar boson constituents. are incorporated into the mean-squared QL field value.

multiplied by three for the choice of which preon to in- The electromagnetic factor for the W /quark vertex is vert in Fig. 3(i), and by [1 + 1/(2π)] for diagram (ii) by consequently equivalent to that for the W /lepton vertex. the same arguments as presented in Sec. IIIF4 of Ref. 1. For W boson loop corrections a similar approach may The net weight of the scalar boson correction is therefore be adopted, and it follows immediately from the whole- 3α 1 fermion perspective that since the W boson is species- 1+ . (20) changing, no valid W loop corrections exist. 2π 2π   For Z boson loop corrections the corresponding inter- d. Net effect of all boson loops: The net effect of the action weight is boson loop corrections is therefore to amend the W boson f α mass equation to Z (17) − 2π 2 2 2 (1) 3 α mW = 18f QL 1+ 64 + fZ (21) where E 2π − 2π h i   2 (2) 2 2 mW QL 2 sin θW =1 2 (18) +O E +O α − mZ  (1)  " QL # ! 1 2 2  E   fZ = 4 sin θW 1 5 3 − − 2 4 h 2 4i (19) 1 mW mW 1 mW  mW f = 4 24 + 16 (22) = 4 24 + 16 Z 3 m2 m4 3 − m2 m4 − Z Z  Z Z    where the next-most-relevant corrections are those due to and the overall sign reflects that the customary defini- the coupling of the W boson to the QL photon and scalar tion of fZ is negative, while the Z and photon terms are fields, and the second-order electromagnetic corrections. additive. By the same argument as in the gluon sector, the Z boson in the loop is effectively massless over this 2 2 length scale and thus there is no factor of mf /mZ for 2. QL photon and scalar interactions some fermion f. c. Scalar boson loop: A scalar boson loop may inter- Another potentially relevant correction is the W /QL act with a fermion as shown in Fig. 3(i)-(ii). As in Ref. 1, photon coupling, Fig. 4(i). At tree level this readily eval- Sec. IIIF4, since the vertex separation is the scalar ≤ LΨ uates to boson constituents may either connect to the vertices col- 2 lectively as a single composite particle [Fig. 3(i)] or with 2 (2) f QL (23) crossing as shown in Fig. 3(ii). To evaluate Fig. 3(i), E h i recognise that spatially exchanging two preons as in di- where a symmetry factor of 2 arises from the presence of agram (iii) reveals equivalency to a loop correction aris- two identical photon operators on the interaction vertex. ing from a composite vector boson, which is foreground This gives a net expression by construction, massless by the same argument as in 2 2 2 (1) 3 α Sec. IIA1a, and may be replaced by the equivalent fun- m = 18f 1+ 64+ fZ W EQL 2π − 2π damental vector boson using Eqs. (4) and (6) to elimi- (   (2) (1) 2 h i 2 (24) nate the factor of QL/ QL which accompanies terms (2) E E 1 QL in H′H′∗. Following this reduction, the resulting massless + E +O α2 .   18 (1) foreground boson loop yields a factor of α/(2π). This is " QL # ) E  6

(1) Inserting the leading-order expressions for QL (14) and The Lagrangian term carries an additional factor (2) E • (2) (1) 2 [1, Eq. (192)] into this correction yields of 2 QL/ QL by construction. EQL − E E   (1) 2 2 The mean field value of [H′H′∗]QL is /2, 2 2 (1) 3 α • − EQL mW = 18f QL 1+ 64+ fZ (25) (2) E ( 2π − 2π compared with QL for [AA]QL.   h i   E m2 Incorporating the QL scalar boson contribution to m2 + e [1 + O (α)]+O α2 W 4 therefore yields k(e) m2 ) 1 W  h i 2 3 α (2) 2 2 (1) where evaluation of in terms of electron parameters mW = 18f QL 1+ 64+ fZ EQL E ( 2π − 2π has been chosen as the electron mass is known to the h i   (f) 2 (27) highest precision of the lepton masses. Parameter kg is 19me 2 defined in Sec. IIIA1, and as in Ref. 1 it is an eigenvalue + 4 +O α . k(e) m2 ) of the preon colour mixing matrix. However, when pro- 1 W  ceeding beyond the leading order calculations of Ref. 1 h i this parameter is seen to run with energy scale. It there- In terms of standard error, the most significant ef- fore carries labels f and g where f indicates the family of fect arising from the next order of corrections beyond fermions for which k is calculated, and g is the particle those considered is a contribution to mZ at approxi- (f) 2 generation; kg is then eigenvalue g of the colour mixing mately 10− σexp, while the largest relative contribution ′ 7 matrix Kf which is associated with the fermion family is to mH and mZ , to approximate order of parts in 10 . (Note that all attempts to estimate higher-order errors containing species f, evaluated at the energy scale mfg where fg denotes generation g of the family containing in this paper are approximate, and are determined using f. As a matter of notation, f will generally be taken as the expressions presented in Sec. IV A.) the lightest member of the family, and thus (for example) (e) k2 represents the second eigenvalue of Ke, evaluated at 2 3. Universality of loop corrections energy scale mµc . Energy scale (2) is independent of the particle species EQL 2 (f) 4 Up to now, the photon in Fig. 4 has been treated as used to compute m / kg . Using the calculations of fg a fundamental particle. However, in principle any occur- the present paper it is only possible to obtain the approx-   rence of a fundamental boson may be re-expressed as a imate value pair of preons using the identity 2 mf a˙ c˙ a˙ c˙ a˙ c˙ g 16 2 4 a˙ cac˙ ac ac ′ ac 4 9.85 10 eV /c , (26) ϕµ = ∂ σµ∂ ϕ f∂ ϕσµ∂ ϕ = fψ σµψ′ (f) ≈ × ≈ kg (28) derived from Eqs. (43) and (80–81) of Ref. 1. Such pairs h i 2 are bound by the colour interaction with a characteris- but this suffices to determine that (2)/ (1) lies be- EQL EQL tic separation of O( Ψ), and thus for energies Ψ tween α/(4π) and α2/(4π2) in magnitude. Neglecting the L E ≪E   they appear collocated and at energies small compared correction factor [1+O(α)] on the mass ratio in Eq. (25) to Ψ this reparameterisation is redundant. However, the (2) (1) 2 E introduces errors of O α / which are there- length scale is comparable to the length scales (1) EQL EQL Ψ QL fore small compared with O(α2) and may be ignored. (2) L L    and QL associated with mass vertices. The QL fields are The composite QL scalar boson field yields a similar L (k) (k) contribution, but: characterised by energy scales QL = hc/ QL and thus are capable of discriminating theE two constituentsL when Its internal Einstein sum means it receives contri- energy is carried in the two-preon mode rather than the • butions from nine times as many sectors of the QL, fundamental boson mode. Over the length scales associ- for an additional factor of nine. ated with mass vertices, this reparameterisation therefore cannot be ignored. Its internal two-component structure allows for • First, consider the preon-mediated mass vertex of crossed and uncrossed connection of sources and Fig. 1(i) and recognise that during evaluation, four of sinks, for an additional factor of two. the preon lines are integrated over and eliminated. The remaining two preon lines form a composite vector bo- However, the QL field operators on the interaction son consisting of one preon and one antipreon, and the • vertex are no longer interchangeable, for a factor of leading-order expression for W mass comes from evalu- a half. ating this in terms of (1) using the fermion sector mean- EQL The vertex factor is f 2, not f 2/2. field theory expression (6). However, it is also valid to • 7 transform the composite vector boson vertices into funda- this abbreviates to mental boson vertices and obtain a boson loop diagram 2 similar to Fig. 4(i) which gives a coupling to the bo- 2 2 (1) 3 α (2) mW = 18f QL 1+ 64+ fZ son sector expressed in terms of . For the W boson E " 2π − 2π # EQL h i   this contribution vanishes, however, as the resulting QL 2 (30) ( ) 19m bosons would be W or W † bosons, and the QL W † fields e 2 1+ 4 1+O α . × (e) vanish by gauge except along foreground boson lines [1, ( k m2 ) Sec. IIIE1]. 1 W   h i Now consider the diagrams of Fig. 1(ii)-(iii) which con- tain loop corrections. The preons persisting after inte- B. Z mass gration no longer necessarily arise from the same ver- tices, but this is unimportant as they are still within Higher order corrections to the Z boson mass are also Ψ of one another and can therefore once again be con- Lsidered to make up a composite vector boson, and thus required, and their calculation is similar to that for the be mapped to fundamental vector bosons, to again yield W boson. a coupling to the boson sector of the QL expressed in terms of (2) (and again these vanish as the bosons are EQL 1. Boson loops W bosons). However, now recognise that the inverse pro- cess may be applied to any boson loop diagram such as a. Gluon loops: The calculation is analogous to that Fig. 4(i), mapping it into a composite vector boson loop performed for the W boson. However, introduction of and then introducing additional arbitrarily selected pre- an off-diagonal gluon coupling eliminates the end-to-end ons from the QL. Such a reconstructed six-preon-line di- symmetry of the Z boson mass-squared interaction re- agram then admits loop corrections in the manner de- sulting in the loss of a symmetry factor of 2. A similar scribed above. This mapping therefore identifies a set of division by 2 for diagonal gluon couplings is necessary nontrivial higher-order correction to the photon loop dia- by GL(3, R) symmetry, and occurs because replacing the gram of Fig. 4(i) which can only be obtained by mapping preon line in the loop as in Fig. 1(iii) eliminates inter- the interacting into preon constituents and re- changeability of the two gluon/preon vertices. The net cruiting additional preons from the QL to make up preon gluon contribution is therefore vertices.(2) Further recalling that the scalar boson loop in 30α . (31) Sec. IIA1c was evaluated by a mathematical mapping 2π via a composite boson loop to a fundamental boson loop, the converse mapping once again reveals a route back b. Photon, W , and Z boson loops: The Z boson is to preon triplet vertices for this diagram, and imparts uncharged, so attracts no net correction from the photon. equivalent corrections to the scalar boson loop diagram Once again, since the W boson is species-changing, the of Fig. 4(ii). Z boson mass diagram also attracts no intra-fermion loop The net outcome is that every term in Eq. (21) has correction involving the W boson. a counterpart on the boson loops of Fig. 4, and the W For Z loops, consider the relative contributions of the boson mass may then concisely be written different fermions which contribute to the leading order diagram [analogous to Fig. 1(i)]. At tree level the Z boson couples only to , , and the down 2 2 2 (1) 3 α 2 quark family. The relative weights of these contributions m = 18f 1+ 64+ fZ +O α W EQL 2π − 2π to the leading order expression for m2 are given in the "   # Z h i  first numerical column of Table I. The coefficients arising 2 19me from the vertices of the loop correction are given in the 1+ 4 [1 + O (α)] . (29) × (e) ( k m2 ) second numerical column. The net weight of the Z boson 1 W correction to each channel is the product of these, given in h i the third numerical column. The total correction arising The O(α) correction to the term in the second brackets from Z boson loops is seen to be is smaller than the O α2 term in the first brackets, so 5 α . (32)  12 2π c. Scalar boson loop: The scalar boson loop calcu- 2 Note that this recruitment (i) is always possible due to the homo- lation is identical to that for the W boson, yielding a geneity of the QL and the extremely large number of QL particles (1) correction of within autocorrelation length LQL, and (ii) is obligatory as the energy scale for a boson at rest in the isotropy frame of the QL 3α 1 1+ . (33) mandates that it interacts with triplets. 2π 2π   8

Species Weight Vertex factors Net loop weight in a superposition of the photon, Z boson, and dilaton • 1 1 1 eL 8 6 48 gradient [which may, in this context, be recruited as a • 1 1 1 eR 8 6 48 third diagonal boson on SU(3)A as it carries trivial rep- • 1 2 1 resentation on both the SU(3)A and SU(3)C sectors]. νe 2 3 3 • 1 1 1 Recognising that the off-diagonal fundamental bosons dL 8 6 48 • d 1 1 1 do not contribute to particle mass, only diagonal com- R 8 6 48 posite vector bosons need be considered. On-diagonal, Total: 1 5 12 for any given composite vector boson with representa- tion matrix eii a basis of fundamental bosons may be TABLE I. List of channels contributing to the leading order Z chosen consisting of the photon, Z boson, and a boson boson mass diagram, their relative weights, the vertex coeffi- derived from the dilaton gradient which has a vanishing cients arising when a Z loop correction is introduced, and the entry at eii. Consequently, any mass arising from this weights of the net contributions of these loop corrections, ex- sector may always be attributed to the QL photon field. pressed as multipliers applied to α/(2π). The net correction First, consider the lepton channels. When the pre- arising from Z boson loops is seen to be {1+(5/12)[α/(2π)]. ons are electron-type preons (a 1, 2 ) there are three choices of charged preon and three∈{ choices} of charged an- tipreon, and freedom to choose which preons to integrate d. Net effect of all boson loops: The net effect of the out gives nine ways to make a composite vector boson boson loop corrections is therefore to amend the Z boson whose a-charges indicate it relates to the photon. When mass equation to the preons are -type preons (a = 3), these have no overlap with the photon and so can be ignored. The 2 401 3 α m2 = 24f 2 (1) 1+ + (34) lepton sector thus offers a total of 18 channels (nine from Z EQL 12 2π 2π e and nine from e ).   L R h i Next, consider the quark channels. Again, only diag- (2) 2 onal contributions from electron-type preons are nonva- +O EQL +O α2  (1)  nishing and thus each contributes only one " QL # !  E   channel. The up quark does not couple to the Z boson. For these twenty channels, now determine the weight where the next-most-relevant corrections are those due of each channel when compared with the W W AA vertex to the coupling of the Z boson to the QL scalar field, † of Fig. 4(i): and the second-order electromagnetic corrections. Each channel only constructs half of a photon, ei- • 1 ther the e11 or the e22 term, for a factor of 2 . (The 2. QL photon and scalar interactions overall scaling of the representation matrix is ad- dressed in the next item, in terms of the resulting a. Direct coupling: As the Z boson is uncharged, it vertex factor.) acquires no mass through direct coupling to the QL pho- Vertex and symmetry factors for two couplings of ton field. However, it still interacts with the QL scalar • the Z boson to eL, eR, dL, or dR yield a factor of boson field. Following a similar calculation to Sec. IIA2 2 2 g = α , compared with g = 2α for the W boson yields 8 2 2 and the photon. 2 401 3 α Given a preon and an antipreon comprising four m2 = 24f 2 (1) 1+ + Z QL • operators eeee all within a single autocorrelation E " 12 2π 2π # h i   region, the pairing of e with e to yield a boson 2 (35) 18me 2 operator is arbitrary, introducing another factor of 1+ 4 1+O α . × (e) two in the mapping from composite to fundamental ( k m2 ) 1 W   vector bosons. h i b. Universality coupling: Rather surprisingly, how- Summing over all twenty channels, the result is a factor ever, there does exist a mechanism whereby the Z boson of 5 relative to Fig. 4(i), increasing the Z boson mass to may acquire mass from the QL photon field. Consider 2 401 3 α again the mechanism behind the universal applicability of 2 2 (1) mZ = 24f QL 1+ + boson mass-squared vertex loop corrections in Sec. IIA3. E " 12 2π 2π # h i   When the Z boson is interacting with the preons of the (36) 23m2 QL, this attracts the obvious leading-order term associ- e 2 1+ 4 1+O α . ated with Fig. 1(i), but when these preons are charged, ×( (e) 2 ) k1 mW the residual pair after integration over one set of spatial   h i co-ordinates may then be mapped onto fundamental vec- It should be noted that no counterpart to the universal- tor bosons. For charged fermions this mapping results ity coupling applies for the W boson as its QL preons do 9 not appear in the correct combinations to be assembled pairwise into components of the composite counterpart to the photon. Also note that when evaluating univer- sality couplings, a choice must be made to work on either the 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 a trivial representation on SU(3)C , it is necessary to work in the SU(3)A sector. Construction of colour-agnostic composite bosons then implicitly spans all valid preon pairs and hence colour choices, making it unnecessary to FIG. 5. As the gluon interaction with the QL gluon field independently consider the gluon sector. need not conserve colour on a per-interaction basis, fore- This completes calculation of Z boson mass to the level ground gluons may interact with a pair of gluons having non- of precision employed in this paper. In terms of standard complementary charges. error, the most significant effect arising from the next or- der of corrections beyond those considered is a contribu- 2 tion to mZ at approximately 10− σexp, while the largest Evaluation of the gluon mass is therefore similar to eval- relative contribution is to mZ , to approximate order of uation of W boson mass, and indeed evaluation of the parts in 107. leading order diagram and preon-to-preon gluon loop cor- rections proceed equivalently. Where the W and gluon mass calculations diverge is in the contribution from in- C. Weak mixing angle teractions with the QL boson fields. For the W boson this contribution arose from the QL photon and scalar boson fields. For the gluons, there are couplings to the If the weak mixing angle is defined in terms of W and QL gluon and scalar boson fields. Z boson mass, the above results for m2 and m2 imply W Z To evaluate the gluon contribution to the a weak mixing angle 2 O (2)/ (1) term, recognise that the preserva- EQL EQL m2 tion of colour cycle invariance across the entirety of 2 W 018  sin θW =1 (37) R dust gravity guarantees that all gluons always ap- − m2 | Z pear in the context of a superposition of all nine possible 2 3 α 19me species. Interactions with the QL need not therefore 3 1+ 64 + fZ 1+ 4 2π 2π (e) 2 − k1 m conserve colour charge on an individual gluon on a =1  h i W     2 term-by-term basis provided colour charge is collectively − 401 3 α 23me 4 1+ + 1+ 4 conserved across the superposition. (Individual gluon 12 2π 2π k(e) m2  h 1 i W  colour will, however, be preserved on average over length    1+O α2 (38) or time scales sufficiently large compared with as × QL the QL has net trivial colour.) For mass interactions,L   2 where fZ in turn depends on sin θW (19) and it is neces- the consequence of this is that rather than interacting sary to solve for consistency. It is worth noting that the with a single looped boson as per Fig. 1(i), a gluon can corrections described above for the W and Z boson mass interact with a pair of different gluons from the QL as diagrams also apply to foreground fermion/weak boson per Fig. 5. To evaluate the QL gluon contribution, note: interaction vertices. However, note that for the Z boson, This interaction has coefficient f 2, compared with the magnitude of the corrections show some variation be- • f 2/2 for the photon. tween different species, with (for example) the electrons attracting EM loop corrections which are not present for A diagram in which the two QL gluons have dif- the neutrino. The value obtained for the weak mixing • ferent field operators on the vertex, ϕc1c˙1 ϕc2c˙2 , re- angle will consequently depend on the different weight- 1 ceives a factor of 2 relative to the photon term due ings given to the various species which may be involved, to loss of vertex symmetry, but a factor of two as and thus on the details of individual experiments to mea- these fields may be pulled down from the action sure this parameter. For this reason, the present paper Z in either order. A diagram in which the two QL concentrates on particle mass rather than seeking to re- gluons have the same field operator attracts neither 2 produce experimentally determined values of sin θW . of these factors. Regardless of how the vertex operators are pulled • D. Gluon masses from Z, each QL boson independently ranges over all nine possible species for a factor of 81. By the unbroken GL(3, R) symmetry of the colour sec- The net contribution to gluon mass from the QL gluon tor, in the eij basis all gluons have identical mass, and it field is therefore 81 times larger than the contribution to suffices to calculate the mass of one off-diagonal gluon. W boson mass from the QL photon field. To the same 10 order as used in Eq. (29) above, the QL gluon mass is therefore given by

2 2 2 (1) 3 α 2 m = 18f 1+ 64 + fZ +O α c EQL 2π − 2π "   # h i  2 99me 1+ 4 [1 + O (α)] . (39) × ( (e) 2 ) k1 mW h i As noted in Ref. 1, this is a bare mass and will be cor- rected by self-interaction terms at energy scales small compared with the strong nuclear force. With regards to the universality coupling, the gluons FIG. 6. (i) Leading diagram contributing to scalar boson have nontrivial representation on SU(3)C and thus this mass. (ii) First correction. coupling 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 attempt to do so would result in double counting, so no further terms are acquired. In terms of standard error, and assuming O(α2) terms are equivalent for W bosons and gluons, the most sig- nificant effect arising from the next order of corrections beyond those considered is a contribution to mZ at ap- 1 proximately 10− σexp, while the largest relative contri- bution is to mW , mZ , and mH′ , and is of approximate 7 FIG. 7. First-order electromagnetic correction to lepton mag- order of parts in 10 . netic moment.

E. Scalar boson mass the number of colour-distinct combinations by two (and by construction from fermion channels, these preons also As with the and the gluon, some carry the same a-charge—see Ref. 1, Sec. IIIF4), and the higher-order corrections to gluon mass are computed two composite vector bosons are functionally equivalent, here. However, for purposes of this paper it suffices to again reducing the number of distinguishable arrange- evaluate only the O(α) loop terms and ignore the cou- ments by a factor of two, for an overall net factor of 6 pling with the QL scalar boson field as this is not relevant from colour arrangements. to the numerical results presented at the level of precision Given a particular arrangement of participating pre- employed in this paper. ons, it is convenient to work with composite scalar and For convenience, as in Ref. 1 the subleading term in vector bosons so that each composite particle then ad- the scalar boson mass interaction [Fig. 6(ii), l.h.s.] will mits a gluon loop correction between its two compo- be rewritten as a correction to the leading term [Fig. 6(i)], nents. However, the loop factor arising from the compos- 1 associated with a factor of (2π)− . ite scalar boson is of the opposite sign to those arising from the composite vector bosons as the antipreon is re- placed by a second preon, giving a factor of 1. Other − 1. Vector boson loops than that, all calculations proceed equivalently. For a gluon which is diagonal in the eij basis, the To evaluate the gluon loop corrections, recognise that calculation is directly analogous to the electromagnetic loop correction to magnetic moment (Fig. 7), save that the four preons and two antipreons (or vice versa) of the 2 2 QL in Fig. 6(i) and the r.h.s. of Fig. 6(ii) may be grouped the vertex factors are f not f /2. The contributions into a QL composite scalar boson and two QL composite from the off-diagonal gluons are necessarily the same by R vector bosons, with the latter then being mapped into GL(3, ) symmetry. true vector bosons and a numerical scaling factor. The Summing these terms therefore yields a net weight for preons may be grouped together in any valid combina- the gluon loop corrections of tion which yields these three effective QL particles, with 12α the four preons contributing a factor of 4 2 1 and the . (40) 2 1 1 2π two antipreons contributing 2 1 for a net factor of 24. 1 1    However, two of the preons are the same colour, dividing There are no contributions from electroweak boson loops   11

limb of the loop yields that the scalar boson cor- rection to the scalar boson propagator is exactly equivalent to a sum over vector boson loop correc- tions to the scalar boson propagator. They must not be counted twice. The scalar boson loop correction to the composite • vector boson propagator has an internal loop com- prising two scalar bosons. As in Sec. IIA1c this may be evaluated by rearranging the internal preon lines. However, on doing so, it is immediately seen to be equivalent to a vector boson loop correction to the composite vector boson field and thus also FIG. 8. Scalar boson corrections to the species appearing has already been counted. within the scalar boson mass vertex: (i) When a scalar bo- son loop correction is added to the scalar boson propagator, For a propagating scalar boson, there are therefore no this replaces one preon in the original propagator with its independent scalar boson loop corrections at O(α), and , resulting in a sum over composite vector bosons. the net expression for scalar boson mass incorporating The resulting diagram is then identical to that obtained when all boson loop corrections is vector boson loop corrections are applied to the scalar boson 2 2 2 (1) 1 propagator. These diagrams have also already been counted. mH′ =40f QL 1+ (ii) The scalar boson loop correction to a composite vector bo- E 2π h i   son propagator yields an internal loop comprising two scalar (2) 2 (41) 6α bosons. Swapping the sides of this internal loop reveals equiv- 1+ +O EQL +O α2 . alency to a sum over composite vector boson corrections to the × π  (1)  " QL # ! composite vector boson propagator, and these can be rewrit-  E   ten as fundamental vector boson corrections to the composite In terms of standard error, the most significant effect vector boson propagator. These diagrams have also already arising from the next order of corrections beyond those been counted. considered is a contribution to mH′ at approximately 1 10− σexp, while the largest relative contribution is to 5 mH′ , to approximate order of parts in 10 . as the H′ boson carries no net electroweak charges, so this is the total contribution from vector boson loops. F. Neutral boson gravitation

2. Scalar boson loops One final note regarding the universality coupling de- scribed in Secs. IIB2b and IIB2b. Although the Z bo- Now consider scalar boson loop corrections to the son is uncharged, through this process it acquires a means scalar boson mass term. Once again, these act on the of coupling to the photon pair field and thus engaging in effective composite vector and scalar bosons connecting the gravity-equivalent interaction of Ref. 1, Sec. IIIG3. the mass interaction vertices. However, recalling that The foreground gluons (including the vector dilaton a scalar boson interaction vertex always maps one con- gradient) couple to the colour charges on both the QL nected preon into its antiparticle, this always results in a preon and gluon fields, and application of the universal- change of particle species within the loop. The interac- ity coupling allows the QL preon fields to be reduced to tion between the looping species and the QL preon fields composite vector bosons in 1:1 correspondence with the is inserted explicitly in one arm of the loop, but is also gluons, so the composite to fundamental boson duality implicitly present in the other arm of the loop as the does not reveal any new couplings. However, the inclu- other particle is also taken to be massive. Thus the ex- sion of all possible colour couplings implies that a subset plicit QL preon interaction may be freely moved from one of the composite vector boson fields may also be rewrit- arm to the other. Therefore, first consider the diagrams ten to correspond to the two-photon coupling (carrying into which these interaction vertices will be inserted: the photon a-charge, and being summed over all pos- sible neutral colour combinations). As this coupling is The scalar boson loop correction to the scalar bo- constructed from the same composite vector boson fields • son propagator has an internal loop comprising one already accounted for, it makes no additional contribu- scalar boson and one composite vector boson. The tion to inertial mass. However, as it is a coupling to the composite vector boson may be mapped into a fun- photon pair field it does grant gravitational mass to the damental vector boson and a numerical factor us- nine gluon fields. Most notably, this interaction imparts ing Eqs. (80–81) of Ref. 1. The factor cancels with gravitational mass to the neutral gluon, or vector dila- that arising from the construction of H′H′∗. Free- ton gradient (which is an -like particle). This is dom to insert the explicit QL coupling in either therefore a potential dark matter candidate. 12

resentation of SU(3)C whose generator is

1 0 0 1 λ9 = 0 1 0 , (43) √3   0 0 1     gluons, associated with the 8-dimensional represen- • tation {λi i 1,..., 8 } of SU(3)C generated by the rescaled| ∈ Gell-Mann { } matrices (A3), or

the vector dilaton, also associated with λ9. FIG. 9. The fundamental interaction giving rise to lepton • mass: The triplet of preons scatters twice off the bosonic By Eq. (3), the separation of the two boson source/sinks component of the background quantum liquid. Either (i) the is of order QL. same, or (ii) different preons may be involved on each occa- To evaluateL the contribution of the QL fields to lep- sion, with the upper and lower vertices independently each ton mass, it is helpful to separate the consequences of connecting to any of the three preons. This results in a total these boson interactions into two parts. First, there is of nine diagrams, three having the form of diagram (i) and the action of the representation matrices of SU(3)C on six having the form of diagram (ii). These nine diagrams are the preon fields, and second, there is the numerical mass then summed. term arising from the mean square value of the QL boson field. As noted in the introduction, the family of gluons will be taken to include the vector dilaton gradient, giv- III. LEPTON MASS INTERACTION ing a combined symmetry group

SU(3)C 1 = GL(3, R)C. (12) A. Leading order ⊕ ∼ It is then helpful to note that this 9-membered family admits representation both in terms of the Gell-Mann The fundamental interaction giving rise to lepton mass and identity matrices (basis {λi i 1,..., 9 }) and in is a double scattering of the preon triplet off the vector | ∈ { } terms of the elementary matrices {eij i 1, 2, 3 , j boson component of the QL (Fig. 9). Both of these dia- 1, 2, 3 }. | ∈ { } ∈ grams may be considered mean field theory expansions of { } a loop correction to the fermion propagator in the pres- ence of the QL, but it is more convenient to refer to these 1. Action on colour sector as “leading order” diagrams, and to count the number of (foreground) loop corrections to these leading order To begin with the action of the generators of diagrams, e.g. 1-loop corrections to the leading order di- GL(3, R) , note that over the course of a propagator agram, etc. Henceforth such corrections will be termed C of length , a lepton will engage in a near- simply “1-loop corrections”. QL arbitrarilyL large ≫ Lnumber of interactions with the back- Figure 9 is constructed by generating a pair of inter- ground QL fields. Each interaction will apply a gl(3, R) action vertices between the lepton and both the SU(9)- representation matrix from λi i 1,..., 9 depending valued field (representation 80) and the dila- on the boson species with which{ | the∈ lepton interacts.} In ton gradient field (trivial representation, 1). These rep- the absence of foreground W or Z bosons, the QL is made resentations are then factorised according to up entirely of photons and gluons. Given a preon of colour c1, this may have nonvanish-

SU(9) 1 = [SU(3)A 1A] [SU(3)C 1C ] ing interaction with the photon or any of three gluons in ⊕ ∼ ⊕ ⊗ ⊕ (42) the elementary basis e . For example, if c = r then ad- 80 + 1 = (8 + 1) (8 + 1) ij 1 × missible gluons are crr, cgr, and cbr. Heuristically, their action on the colour space may be represented as where each instance of 1 represents the appropriate triv- crr r r cgr r g cbr r b (44) ial representation. Finally, each SU(3)-valued boson field | i → | i | i → | i | i → | i is expanded in terms of its components, and terms are where all associated numerical factors have been ignored discarded where the mean magnitude-squared of the QL for illustrative purposes. component of the boson field vanishes by gauge. The More generally, the family of gluons acts on a vector resulting interactions may each involve any of the three of preon colours as indicated by preons making up the lepton. As per Eqs. (1–2) and (11), the bosons may be crr crg crb r | i cgr cgg cgb g . (45)    | i  the photon, associated with the 1-dimensional rep- cbr cbg cbb b •    | i      13

It is worth noting that there is no emergence of colour su- some synthetic boson interactions where it intersects with 0 18 perselection for individual particles in R | dust gravity, particle worldlines. As described in Sec. IIIF2 of [1], as the GL(3, R)C symmetry is unbroken, so there exists the vertex factors associated with these interactions arise a freedom of basis on the colour sector corresponding from the representation matrices of SU(3)C given as λi to an arbitrary global transformation in SU(3)C . Any in Appendix A. By construction these bosons are con- coloured fundamental or composite particle may be put strained to have no effect beyond the colour shifts associ- into an arbitrary superposition of colours using such a ated with the boundary of the patch, and to leading order transformation, though relative colour charges of differ- this effect is parameterless. In the leading order diagram ent particles remain unchanged, as does the magnitude these bosons consequently have no degrees of freedom, of the overall colour charge of a composite particle. carry no momentum, and thus may be integrated out to Recognise now that Fig. 9 contains contributions to yield a factor of 1. Consequently they are not drawn. two mass vertices. Although their contributions to Only the factors arising from the representation matrices fermion mass are nonvanishing only when they appear persist, acting on the colour vector of an individual preon pairwise, as shown in Fig. 9, there is no requirement for as the matrix this pair to be consecutive. It suffices that each vertex be 1 A A paired with a conjugate vertex separated by distance and † 1 i Kℓ = , A = ± . (46) time no greater than QL (which is within a few orders of  A† 1 A  L ± 2 magnitude of Planck length) in the isotropy frame of the A A† 1   QL. Indeed, these vertices are connected by a foreground   fermion propagator which in general also undergoes fur- The sign on i is free to be chosen by convention, while the ther interactions with the QL, represented by using a overall sign on A is fixed by noting that cyclic permuta- 3 massive propagator for this fermion and requiring consis- tion of colours, which is in (Kℓ) , is required to leave the tency with the outcome of the mass vertex calculation. sign of an eigenstate of Kℓ unchanged. The eigenvalues In general a foreground fermion exhibiting a net propaga- of Kℓ must therefore be non-negative, setting tion over distance or time of O( QL) in the isotropy frame 1 i of the QL will scatter back andL forth multiple times in A = ± . (47) − 2 this process, interacting with O ( N ) 1036 particles h QLi ∼ from the QL, and consequently the number of unpaired Choosing a sign for i, the mixing matrix Kℓ may then be vertices is negligible. Furthermore, where unpaired ver- written tices do exist, their net effect vanishes on average over iθ −iθ 1 e ℓ e ℓ length or time scales larger than QL. It is therefore √2 √2 −iθ iθ 3π reasonable to assume during evaluationL that each vertex K (θ )= e ℓ 1 e ℓ θ = . (48) ℓ ℓ  √2 √2  ℓ − − 4 belongs to a pair. eiθℓ e iθℓ  1  Counting vertices arising from Fig. 9, for every photon  √2 √2    vertex there is also on average one vertex for each of the As noted in Ref. 1, Sec. IIIF2, this matrix bears a strong nine gluons. Across macroscopic length scales, deviations resemblance to Koide’s K matrix for leptons [3]. The from this average relative frequency will be negligible. minus sign on Koide’s off-diagonal component S(θf ) has Consider now the specific case of interactions between been absorbed into the phase θℓ, and the free parameters a propagating lepton and the bosons of the QL. Each af , bf , and θf are fixed by the geometry of the model, in 0 18 preon may interact either with a photon or with any of keeping with the predictive capacity of R | dust gravity. the nine gluons, and the action of the photon on the space Recognising that on average all QL gluons act identi- of preon colours is trivial, so it is convenient to ignore the cally and with equal frequency, it is convenient to collect photon for now and reintroduce it later. these together into a single gl(3, R)-valued gluon asso- As already noted, paired interactions with the QL ciated with two applications of matrix Kℓ to the non- gluon field conserve the net colour-neutrality of a leptonic interacting preons, as shown in Fig. 10. 0 18 preon triplet. However, overlapping and intercalation of Now recognise that since the gluons of R | dust grav- multiple interaction pairs implies that this property only ity are massive, and since the fundamental mass interac- holds on average, as any colour measurement will inter- tions take the form of loop diagrams with respect to the rupt a finite number of mass interactions and thus sum- lepton propagator (Fig. 9), these will be suppressed by a 2 2 mation to yield no net colour charge on the preon triplet factor of O(mℓ /mc) relative to the photon contribution. cannot be assured. It is desirable that any measurement With each gluon interacting, on average, once for every of lepton colour should be null, not just the average, photon interaction, it is convenient to write the direct and thus a local change of co-ordinates on SU(3)C must contributions of the gluon terms to particle mass as cor- be performed on a co-ordinate patch encompassing the rections to the larger photon term and to associate copies non-interacting preons such that changes in their colours of the matrix Kℓ with the photon vertices in a manner track those of the interacting preon. This change of co- equivalent to that shown in Fig. 10. ordinates is not part of the choice of gauge on SU(3)C , Next, consider that the preons on which matrix Kℓ are and thus is in principle associated with construction of acting are just two of three preons in a colour-neutral 14

states then corresponds to a superposition of cyclic spa- tial rearrangements of the members of the triplet, with colours c1 = r, c1 = g, and c1 = b corresponding to colour 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 on (3) the preon triplet are eigenvectors of a matrix Kℓ with eigenvalues identical to those of Kℓ. It appears unlikely that a similar recovery of colour superselection can be achieved for the quarks. FIG. 10. When a composite fermion interacts with the glu- Having established through colour cycle invariance ons from the QL, represented as a single gl(3, R)-valued boson, that the matrix Kℓ acts identically on all constituents the colour mixing process represented by matrix Kℓ acts on all preons not coupling to the boson at any given vertex. Di- of a lepton, and through Fig. 10 that two copies of Kℓ agrams (i) and (ii) correspond to Figs. 9(i)-(ii) respectively. act per QL photon interaction, it follows that the effect (ℓ) 2 Once again these are just two representative diagrams from of matrix Kℓ is to contribute a factor of [ki ] to the a family of nine, as the upper and lower vertices may inde- mass of a lepton of generation i. It might seem problem- pendently each be connected to any of the three preons. This (ℓ) atic that for θℓ = 3π/4, k = 0, but it will be seen in results in a total of nine diagrams, three having the form of − 1 Sec. IIIC that θℓ acquires corrections from higher-order diagram (i) and six having the form of diagram (ii). (ℓ) diagrams, resulting in ki > 0 i, so k1 may be assumed real and positive, and this concern∀ may be disregarded. triplet. For leptons, all three a-charges are identical and thus over macroscopic scales, where chance fluctuations become negligible, matrix Kℓ will act identically on each 2. Mass from photon and gluon components of QL member of the triplet. This symmetry is convenient, as it allows the study of an individual preon prior to the The zeroth-order electromagnetic term is readily eval- reconstruction of the triplet as a whole. uated by making a mean-field substitution (1) for µ As per Eq. (5), the preons making up observable lep- [A (x)Aµ(y)]QL. For a charged lepton ℓi of generation tons are now eigenstates of this matrix Kℓ, corresponding i, this initial approximation may be written to the eigenvectors 4 2 m2 = f 2 k(ℓ) (2) . (53) 1 ℓi i QL 1 E v1 = 1 (49) h i h i √3   Note that there is a symmetry factor of two correspond- 1 ing to exchange of the two QL interactions. This may be    πi understood in either of two related ways: e 3 1 πi v2 = e− 3 (50) 1. For diagrams such as Fig. 9 which separate into two √3   1 separate terms on performing the mean field expan-  −  sion of the QL, one approach is to recognise that  2πi  e 3 each term corresponds to a mass vertex and has its 1 2πi v3 = e− 3 (51) external legs truncated independently. These two √3   1 vertices are then indistinguishable.     2. Alternatively, for any diagram, including ones which are independent of θℓ and have eigenvalues {k(ℓ) i 1, 2, 3 } given by which do not separate, recognise that the mass- i | ∈{ } squared is always applied in the context of an un- (ℓ) 2π(n 1) truncated fermion propagator, say from x to y. In k =1+ √2cos θℓ − . (52) n − 3 this context, all fermion connections to the inter-   action vertices are again untruncated. (Optionally, To reconstruct the lepton as a whole, recognise that the full expression for propagation from x to y is for three preons at {x i 1, 2, 3 }, with corresponding i then used to infer an equivalent mass term, and the colours c , and with the| preon∈{ at x} having a well-defined i 1 diagram may then be replaced by one in which this color, say c = r, colour neutrality and colour cycle in- 1 mass term is inserted into the propagator twice.) variance imply that a choice c = g, c = b is equal up to 2 3 Applying either form of this approach to Fig. 9, a sign to the alternative choice c = b, c = g (as this ex- 2 3 the diagram for propagation between two points is change corresponds to spatial exchange of two fermions), again seen to attract a symmetry factor of two. and thus it suffices to consider only one such colour as- signment (say c2 = g, c3 = b) along with spatial permu- Now consider interactions between a foreground tations. Putting preon 1 into a superposition of colour fermion and the QL gluon fields. As with the photon, 15 these interactions take the form of loop diagrams evalu- Vertices are no longer interchangeable, for a relative • 1 ated in the mean-field regime for the QL, and as noted in factor of 2 . Ref. 1, the fermion may transiently surrender momentum to or borrow momentum from the QL. However, in con- Interactions between a fermion and an off-diagonal • 5 trast with the photon loop evaluated to obtain Eq. (53), boson attract a structural factor of 3 . (For a com- the gluon field is massive, and when a foreground particle parable calculation see the W correction to the EM 10 transfers momentum to a gluon field, this results in an vertex [2], in which the arising factor of 3 com- effective foreground excitation of that gluon field. Fore- prises a factor of 2 from the change in vertex factor 5 ground excitations acquire mass, and thus both limbs of and a structural factor of 3 .) the loop must be considered massive. However, the sep- As previously noted, there is a mass factor of aration of the two vertices is of O( QL), which is much L • 2 2 shorter than the strong interaction scale, so the gluon mℓ /mc. exhibits only its bare mass (39). This gives rise to a 2 2 By GL(3, R) symmetry, each of the nine gluons will yield loop-associated factor of mℓ /mc. This factor vanishes for the photon, as it is massless, but not for interactions the same magnitude contribution to fermion mass, for a with the QL gluon field. further factor of nine. Taking both photon and gluon In both cases, evaluation of momentum flux around terms into account, the leading-order expression for pho- the loop may be taken to yield a factor of ton and gluon contribution to lepton mass is therefore given by 2 α mℓ f 2 , b A, c (54) 4 2 2 2 2π m (ℓ) (2) q 5mℓi b ∈{ } m2 = f 2 k ℓ + (55)   ℓi i QL 2 2 E e mc ! where f (n) 1 as n . For the photon the factor h i h i of α/(2π) is→ absorbed→ into ∞ the QL mean-field term by where q is the charge of lepton ℓ . definition, and factor f ( ) reduces to 1. For gluons, de- ℓ i · As an aside, for the fermions there is no equivalent to pendence on the same energy scale (2) indicates that an EQL the bosonic universality coupling explored in Sec. IIB2b. identical factor of α/(2π) is absorbed into the mean field For the Z boson, this coupling arises as the basic Z term, while the mass dependence of f ( ) reveals that the mass diagram [equivalent to Fig. 1(i)] intrinsically incor- · 2 2 gluon terms are suppressed by a factor of mℓ /mc relative porates six QL preon lines, and two co-ordinates to inte- to the photon term. grate over, permitting reduction to two preon lines when Working in the eij basis, consider first an off-diagonal one of these integrals is performed. In contrast, the ba- gluon. When this gluon is emitted, it changes the colour sic fermion mass diagram (Fig. 10) contains no intrinsic of the emitting preon. For example, the colours of the mechanism for adding extra preon lines. Although extra preon triplet may change from rgb to rgg. However, a preons may be recruited from the QL, consistent nor- mass vertex must by definition leave the fermion on which malisation [1, Sec. IIIF1] requires that integrating over it acts unchanged, and thus to generate a mass term the the additional co-ordinate thus introduced will inevitably gluon must be absorbed by the same preon as it was eliminate them again. emitted, restoring the original state (rgb). Absorption by a different preon is a valid physical process, but in- stead comprises part of the binding interaction between 3. Mass from scalar component of QL the preons. This is in contrast to the photon, where emis- sion and absorption may be on different preons without affecting either the a-charge or the c-charge configura- Next to be considered is the interaction between the tion. composite lepton and the QL composite scalar boson Overall, expressed relative to the photon interaction, field. Again it is desirable to write this term as a cor- factors for an off-diagonal gluon arise as follows: rection to the photon term. To achieve this, recognise that the scalar boson is made A given gluon may only be emitted by a preon of up of a sum over nine components. Likewise, when the • 1 specific colour, for a relative factor of 3 . SU(3)A sector is supplemented by the vector dilaton gra- dient this results in a symmetry group GL(3, R) whose However, any of the three preons may be this A Lie algebra gl(3, R) also has nine elements. The in- • colour, for a relative factor of three. A teraction vertex for each component of the scalar boson The photon may be absorbed by any of the three may be obtained by transforming that of a boson from • R preons, whereas each gluon may be absorbed only gl(3, )A associated with basis element eij . There are by the preon which emitted it, for a relative factor nine such bosons, which include the W and W † bosons. 1 Note that the boson gauges [1, (127–131)] do not impact of 3 . the fermion sector, and the scalar boson is constructed Vertices are associated with factors of f, not f/√2, from two fermionic preons, so all nine terms must be • for a relative factor of two. taken into account regardless of whether the associated 16 boson is eliminated by gauge, and since the symmetry of the fermion sector is unbroken, all nine terms are equal. Note that although the numerical factors associated with the scalar boson correction is obtained by trans- forming the equivalent vector boson corrections, this is not double counting because the scalar boson involved is a foreground particle, and therefore massive, with a mass shell distinct from that of the numerically conve- nient bosons. This is in contrast with Sec. IIE2, where all involved particles are effectively massless and thus a numerical identification corresponds with a physical iden- FIG. 11. Leading-order contribution to fermion mass from tification. the QL scalar field. By the internal GL(3, R) symmetry of the scalar bo- son, all terms contribute equally to the fermion/scalar boson interaction. Considering the off-diagonal term cor- responding to W , the following factors arise:

Relative to the photon, the W boson interaction • 5 acquires a structural factor of 3 . Relative to the photon, the W boson coupling is • stronger by a factor of two.

Compared with the photon, there is a loss of vertex • 1 FIG. 12. Twisting the inner preon line to yield an additional interchangeability for a factor of 2 . (Recall that loop particle changes the scalar boson loop correction into a external legs are not truncated so vertices in Fig. 9 composite vector boson loop correction. As the loop is fore- are not distinguishable.) ground, with dominant contribution from massive trajectories (2) traversing length scales large compared with LQL, it must be However, this is offset by a factor of two corre- re-expressed in terms of fundamental vector bosons. • sponding to the different ways to attach the ex- ternal source and sink to the now-distinguishable vertices. (1) QL. As observed in Sec. IIID3 of Ref. 1, incor- poratingL both fundamental and composite vector Transforming a W W pair into a term in the † bosons into the particle spectrum of the low-energy • H H pair [in principle performed using Eqs. (80– ′ ′∗ ( ) regime is redundant and therefore the 81) of Ref. 1 and integration by parts] yields one QL compositeL ≫ L vector bosons must be re-expressed as of nine terms in H′H′∗, so introduces a factor of 2 fundamental vector bosons. The resulting diagram 2 (2)/ (1) from Eqs. (4) and (6). − 9 EQL EQL involves two boson loop corrections, and is not a scalar loop term at all. Similarly, twisting twice re- The scalar boson contributes an internal symmetry stores a scalar boson, but once again yields a redun- • factor of two relating to crossing or not crossing its dant representation of vector boson loops, which internal preon lines. are higher-order terms in the correction series. There is no factor of [1+1/(2π)] for this interac- There are nine bosons in the e basis for a final factor of • tion, as there is only one relevant type of vertex ij 9. Using lowest-order expressions for (1) (14) and (2) interaction involving the scalar boson field and the EQL EQL fermion field. This may be contrasted with Fig. 6, [1, Eq. (192)] yields a weight relative to the photon term where there are two relevant types of vertex [seen of in diagrams (i) and (ii) respectively] with each hav- m2 m2 ing different couplings to the preonic constituents ℓi ℓ1 240 2 4 [1 + O (α)] (56) of the scalar boson. mH′ (ℓ) 2 k1 mW Related to the above, twisting the inner preon h i for a total lepton mass • line to yield an additional loop particle as shown in Fig. 12 changes the scalar boson loop correc- 4 2 m2 =f 2 k(ℓ) (2) tion into a composite vector boson loop correction. ℓi i EQL However, in contrast with the vector boson loops in h i h i Sec. IIA1 (and by similar argument the composite 2 5m2 240m2 m2 (57) qℓ ℓi ℓi ℓ1 vector bosons appearing in Sec. IIE) this loop is not  2 + 2 + 4  . × e mc (ℓ) 2 2 constrained to vanish at length scales greater than  k1 mH′ mW  h i   17

The most significant contribution from the higher-order the presence of the hole breaks the time reversal symme- 2 correction to Eq. (56) is to mZ at O(10− σexp) (and in try of a portion of the QL and this gives rise to a symme- ′ 1 relative terms, to mW , mZ , and mH to order of parts in try factor of 2 relative to the original gluon interaction in 107) so this term may safely be dropped. which the QL was assumed time-reversal-invariant. Fi- nally, there are nine gluon channels, each of which yields an equivalent correction by GL(3, R) symmetry. The net 4. Gluon and scalar field mass deficits outcome is to correct the gluon mass to an effective mass of

Conservation of energy/momentum implies that the 2 2 2 27 m rest mass imparted to the fermion must be compensated (m∗) = m 1 ∗ . (59) c c − 10 m2 by a reduction in the zeroth component of 4-momentum  c  of some of the QL fields. Likelihood of contribution from any given QL sector will be governed by availability of Note that mc∗ is a function of m , but for a lepton ℓi which is on-shell and at (or close to)∗ rest in the isotropy frame zeroth-component energy within that sector, i.e. the rest 2 2 of the QL this admits the replacement m mℓ and mass of the associated species, and the strength of cou- ∗ → i pling to that sector. It therefore follows that this borrow- can then be conveniently left implicit. Also note that the ing of rest mass occurs with equal likelihood from each gluon mass deficit effect is a whole-field effect, acting on of the nine gluon channels of the QL, with much lower both the foreground and QL gluon fields. The corrected likelihood from the scalar boson channel (due to a much gluon mass mc∗ should be used anywhere a particle inter- weaker coupling), and not at all from the photon chan- acts with a gluon field in the presence of a lepton. The nel (due to zero rest mass). For a first approximation, lepton mass equation is amended to consider only the gluon channel. Borrowing a mass of 4 2 2 2 2 (ℓ) (2) m from a QL gluon field takes place at the first of the mℓi =f ki QL ∗ E existing gluon/fermion interaction vertices of Fig. 9, and h i h i corresponds to deletion of a gluon of mass m from the 2 5m2 240m2 m2 (60) ∗ qℓ ℓi ℓi ℓ1 QL. This hole then propagates as a quasiparticle, and is  2 + 2 + 4  . × e (mc∗) (ℓ) 2 2 filled by the conjugate interaction at the second vertex.  k1 mH′ mW  More generally, with multiple overlapping pairs of QL h i gluon field interactions occurring along a fermion prop- Similarly, the foreground lepton may also borrow its agator, there is a consistent propagating hole in the QL mass from the scalar boson field. However, coupling be- gluon sector corresponding to an energy deficit of m c2, tween leptons and scalar bosons is weaker than that be- ∗ and individual vertices may cause transient fluctuations tween leptons and gluons, and this correction falls below and may change which specific gluon fields (with respect the threshold of relevance, giving rise primarily to cor- 2 to some arbitrary choice of colour basis) are involved in rections to mZ at O(10− σexp) (or in relative terms, to 7 propagating this hole, but in general it may be in any of mτ , mZ , mW , and mH′ to order of parts in 10 ). the nine gluon channels at any time. This hole is in addi- tion to the effect discussed in Sec. IIIA2 where fermions may surrender momentum to or borrow momentum from B. Foreground Loop Corrections the QL, and thus gives an additional correction factor not yet discussed. Now consider the effects of foreground loop corrections This hole propagates as a quasiparticle accompanying on the leading-order diagrams of Figs. 9 and 11. Note the lepton. Any time that a fermion interacts with the that since momentum is continually redistributed among QL gluon sector this hole is necessarily also present, and the constituent preons by means of gluon-mediated in- may occupy any of nine channels. teractions even over length scale Ψ, a boson need not For simplicity, further consider the case when the hole start and finish its trajectory on theL same preon in order occupies an off-diagonal channel. It would be convenient to be considered a loop correction to an emission vertex. to write the effect of this hole as a correction to the mass Further note that the massive nature of the loop boson of gluon, does not disrupt the QL correlators in these diagrams, 2 2 2 as these are brought together through the use of spinor mc mc km (58) −→ − ∗ identities at the vertices making these diagrams more ro- for some factor k. Recognising that the co-propagating bust against interference from outside the autocorrelation hole’s interaction with the fermion is trivial (it is only re- region than the boson mass diagrams of Sec. II. quired to be present), with any local energy/momentum transfer to or from the QL being mediated by the fermion/gluon coupling of Fig. 9(i), the hole’s interac- 1. 1-loop EM corrections tions attract no structural vertex factor and thus where the direct gluon interactions of Fig. 9(i) generate a factor The O(α) EM loop corrections to the lepton mass in- 2 2 of 5/(3mc), the hole will contribute only m . Further, teraction are shown in Fig. 13. These should be compared − ∗ 18

for in the usual Standard Model corrections, making no

contribution to mℓi . Next, consider the scalar boson loops of Fig. 13(ii). In these diagrams the intermediate lepton has one preon re- versed, and the first two diagrams therefore yield factors of α/(6π) rather than α/(2π). In conjunction with nega- tion of the Standard Model corrections, they therefore yield a net correction to the QL scalar term with weight

2αq2 ℓ . (61) − 3πe2 The third diagram is consistent with the Standard Model

equivalent and so once again does not contribute to mℓi . Finally, there are two more diagrams from the scalar boson sector to consider. Moving a single scalar boson vertex onto the photon loop is prohibited as the dia- gram as a whole does not then leave the preon triplet unchanged (one preon gets replaced by an antipreon), but moving both vertices onto the loop is admissible. This results in the diagrams shown in Fig. 13(iii), of which only the second is anticipated to be non-vanishing by the arguments of Ref. 1, Sec. IIIF3. This loop contributes to the 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/(2π) to the scalar bo- son term, combining with the above factor (61) to yield a correction of αq2 1 ℓ . (62) − 6πe2 FIG. 13. One-foreground-loop EM corrections to (i) vector   boson and (ii)-(iii) scalar boson lepton mass interactions. The The net expression for lepton mass is thus broad oval interaction vertices indicate that the boson may 4 2 interact with any of the three preons, and all configurations m2 = f 2 k(ℓ) (2) (63) should be summed over. ℓi i EQL h2 i5mh 2 i 240m2 m2 2 qℓ ℓi ℓi ℓ1 αqℓ 2 + 2 + 4 1 2 ×(e (mc∗) (ℓ) 2 2 − 6πe ) k1 mH′ mW   h i but the final factor of αq2 1 ℓ (64) − 6πe2   is on the same scale as the O(α) term in Eq. (56) and so may be dropped, as its most significant contribution 3 is likewise only to mZ at O(10− σexp) (and in relative FIG. 14. Standard Model one-foreground-loop EM correc- 8 tions to the electron mass vertex. The example shown is for terms, to mW , mZ , and mH′ to order of parts in 10 ). the left-helicity electron Weyl spinor; equivalent diagrams for the right-helicity spinor exchange L and R. 2. 1-loop gluon corrections with their Standard Model counterparts in Fig. 14. The next loop corrections to consider are those arising As in Sec. IIA1, the only corrections which need to from gluon loops, analogous to the diagrams of Fig. 13. be incorporated into the electron mass vertex are those These gluons are foreground particles capable of mak- which do not also appear in the Standard Model. How- ing excursions outside of the local autocorrelation zone, ever, comparing the diagrams of Fig. 13(i) with their and hence are massive, but the length scale involved is counterparts in Fig. 14 reveal these to be directly equiv- of order Ψ so self-energy terms may be neglected, and alent. The diagrams of Fig. 13(i) are therefore accounted the gluonL loops exhibit at most only the bare gluon mass 19

vertex, and extrapolate this across all gluon colour com- binations by GL(3, R) symmetry. Figure 15(iv) shows a further diagram which may be constructed using gluon loop corrections. This is a coun- terpart to the gluon version of the third diagram of Fig. 13(i), and also contributes to the mass of the loop gluon. Note that in contrast to the photon terms, where the third 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 corresponding Standard Model terms. To evaluate these corrections begin with Fig. 15(i), which is the gluon loop counterpart to Fig. 13(i). Start with the fermion coupling to the QL photon field, and a specific choice of off-diagonal gluon. Compare with the equivalent EM loop figure and note the following changes:

The vertex factors increase from f 2/2 to f 2, for a • relative factor of two. FIG. 15. (i)-(ii) Gluon loop equivalents to the first diagram of Fig. 13(i). When the fermion couples to the QL photon field, The boson is off-diagonal, for a structural factor of • 5 only diagram (i) may be constructed as the gluon does not 3 . carry an electromagnetic charge. When the fermion couples The photon source may be any of three charged to the QL gluon field, both diagram (i) and diagram (ii) may • be constructed. Conservation of colour charge indicates that preons, but the gluon may only be emitted by a on summing the coupling of the QL gluon to the fermion and preon of appropriate colour. However, any of the the QL gluon to the loop gluon, this is equal to the coupling three preons may be the preon of that colour, for a of the QL gluon to the fermion in the original leading-order net factor of one. diagram (iii). Similarly, diagram (iv) is the counterpart to the third diagram of Fig. 13(i). This diagram is especially Emission of the off-diagonal gluon changes the interesting as it contributes both to the fermion mass and to • colour of the emitting preon. There are then two the mass of the loop gluon. Note that the two QL gluons must preons of that colour which are capable of absorb- couple with the loop gluon at a single vertex, as discussed in ing the loop gluon, and both of the resulting dia- Ref. 1, Sec. III F 3. grams count as loop corrections due to the implicit exchange (not shown) of further gluons as a binding interaction sharing momentum between all mem- (including the gluon deficit correction). Only loop cor- bers of the preon triplet. The choice of absorbing rections to the QL photon and QL gluon interactions preons gives a factor of two. need be considered in the present paper; corrections to the QL scalar boson interaction contribute to mZ at The loop gluon is massive, and is in the presence 3 • O(10− σexp) (in relative terms, to mW , mZ , and mH′ of a foreground fermion so experiences a gluon field 8 2 2 at parts in 10 ) and are therefore ignored. mass deficit giving a loop factor of mℓi /(mc∗) . Consider the gluon loop counterparts to Fig. 13(i). These factors multiply the equivalent photon loop Where the fermion interacts with the QL photon field, • factor, which is α/(2π). these corrections take on the form of Fig. 15(i). For in- teractions between the fermion and the QL gluon fields The per-gluon correction weight from this diagram is they may take on the form of either Fig. 15(i) or (ii), and therefore conservation of colour charge indicates that when these couplings are summed, this is equal to the coupling of 2 10α mℓi the QL gluon to the fermion in the original leading-order 2 . (65) 3π (mc∗) diagram [Fig. 15(iii)]. Further, when the loop correc- tion is evaluated, this collapses to a numerical multiplier This calculation is repeated for the gluon counterparts on the original QL interaction vertex and the custom- to the other two figures of Fig. 13(i), and each of these ary application of spinor identities [1, (170–171)] yields figures may involve any of nine gluons, for a total weight 2 a non-vanishing contribution to mℓi . It then suffices to of consider an example where the coupling of the QL gluon 2 and loop gluon vanishes, making it equivalent to the QL 90α mℓi 2 . (66) photon case, evaluate the correction to the interaction π (mc∗) 20

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 an additional fermion propagator segment which offsets this, so they are also additive. Relative to the leading-order diagram, Figs. 16(i)-(ii) acquire factors 2 2 (10/3)(mℓi /mW ), being the usual factor for a W • loop correction to an EM vertex, 1/2 due to loss of exchange symmetry for the two • QL photon/composite fermion interactions. (Even with a composite fermion, an effective vertex is de- fined on summing over all possible preon/photon pairings and when identical, these effective vertices may still be exchanged for a symmetry factor of two.) Diagram (iii) retains the exchange symmetry, this time on the two copies of the photon operator at the vertex, 2 2 so attracts the factor of (10/3)(mℓi /mW ) only. FIG. 16. (i)-(iii) W boson loop corrections to the QL vector Finally, consider the Standard Model counterpart: The boson interactions (QL photon or QL gluon). For QL photon charged lepton Proper Self-Energy (PSE) terms in the terms, ignore the orientation arrows on the QL boson lines. Standard Model give rise to the W /neutrino loop shown (iv) Loop involving a W boson and a neutrino, appearing in in Fig. 16(iv). This gives rise to a term analogous to the proper self-energy corrections of the Standard Model. Fig. 16(iii), but has no counterpart to the symmetry fac- tor associated with the pair of identical photon operators on the QL interaction vertex. Its associated factor is Correction of the QL gluon interactions proceeds equiv- 2 2 therefore (5/3)(mℓi /mW ), which must be deducted. The alently, and as noted above the correction to the scalar net weight of the W boson loop corrections is thus boson may be ignored at current precision, for a net lep- 2 ton mass so far of 5αmℓi 2 . (68) 4 2 2πmW 2 2 (ℓ) (2) mℓ = f ki (67) i EQL Next considering the Z boson loops, these take form h 2 i h 90iαm2 5m2 90αm2 directly analogous to Fig. 13, and by arguments similar qℓ ℓi ℓi ℓi 2 1+ 2 + 2 1+ 2 to the above are also seen to all be additive. Noting ×( e " π(mc∗) # (mc∗) " π(mc∗) # that the Z boson couples with opposite sign to eL and 240m2 m2 ℓi ℓ1 eR, the Z boson equivalents to the first two diagrams of + 4 . Fig. 13(i) are therefore of opposite sign to their Standard (ℓ) 2 2 ) k1 mH′ mW Model counterparts and hence attract a factor of two, to h i first offset the Standard Model-derived term in the PSE, and then implement the actual correction. The third 3. 1-loop weak force corrections diagram of Fig. 13(i), on the other hand, has the same sign as its Standard Model counterpart and thus yields 2 In the interest of brevity, this Section specialises to the no net contribution to mℓi . The net outcome from Z charged leptons only. boson loops is a term with weight a. QL photon interaction: When the fermion inter- 2 4fZαm acts with the QL photon field, W boson loops may correct ℓi 2 (69) this interaction as shown in Fig. 16. As with the gluon − 2πmW loops in Sec. IIIB2, the boson loops may be collapsed where f is as defined in Eq. (19). Overall, the weight of to numerical factors on the vertices of the corresponding Z the weak boson corrections to the QL photon interaction leading-order diagram, and the QL field terms contracted is seen to be using spinor/sigma matrix identities. 2 Some caution is required with sign—first consider dia- (5 4fZ)αm − ℓi . (70) gram (iii). Evaluating this diagram as a tensor network 2πm2 in the manner of Ref. 4–6, it is readily seen to be an ab- W 2 solute square and therefore yields a contribution to mℓi Note that the use of fZ , which is dependent on the weak which is additive to the leading order term. Diagrams (i) boson masses, implicitly incorporates the loop correc- and (ii) contain as a subdiagram a correction to the EM tions to W and Z boson vertex interaction strengths. 21

8 This value is based on the mean Z vertex interaction terms, to mτ at approximate order of parts in 10 . strength, evaluated as a weighted average across all species which interact with the Z boson, but this approxi- mation does not have a significant effect on the numerical results obtained, being responsible for corrections to mZ 2 8 at O(10− σexp) (in relative terms, parts in 10 on mW , mZ , and mH′ ). C. Corrections to the lepton mass angle b. QL gluon interactions: This term is readily obtained, as the calculation proceeds similarly to Sec. IIIB3a, and it is found to be right at the threshold 1. Origin of corrections of relevance. It is larger by a factor of 5 than the largest terms which have not been retained, and corrects mZ at There remains one more substantial correction to eval- 0.18 σexp. As usual, start with the fermion interaction with a sin- uate. The leading-order contributions to the lepton mass gle off-diagonal QL gluon channel and then use GL(3, R) vertices and the preon colour mixing matrix were evalu- symmetry to multiply by nine for the contributions of ated in Sec. IIIA. Loop corrections to the mass vertices the other eight gluons. For the gluon channels, the QL were evaluated in Sec. IIIB. It remains now to determine gluons continue to couple to the fermion and not the how these loop corrections impact the preon colour mix- looping W boson, so when evaluating the equivalents of ing matrix Kℓ. 1 Figs. 16(i)-(ii) there is no factor of 2 for lost symmetry. When this matrix was constructed in Sec. IIIA1 there The calculation is otherwise equivalent, for net W and Z was an implicit assumption that all eigenvectors expe- corrections of rience equal couplings to the QL fields. The eigenval- 2 ues of matrix Kℓ then imparted different masses to the (25 12fZ)αmℓ − i . (71) three eigenvectors, corresponding to the three members 6πm2 W of a given fermion family. However, from Eq. (72) it c. QL scalar boson interaction: These terms fall be- is apparent that the higher-order corrections to the QL low the threshold of reliable evaluation with current nu- coupling themselves depend on particle mass, leading to merical methods employed, with the largest correction differential augmentation of the three eigenvectors (mass 11 appearing to be approximately 10− σexp on mτ , or parts channels). As these eigenvectors correspond to differ- in 1016. ent vectors in colour space, imparting different relative The cumulative expression for charged lepton mass in- phases to the gluon couplings between the preon triplets, corporating weak boson corrections is therefore a change in the relative QL couplings for the different eigenvectors will impact the colour mixing process. Con- 4 2 2 2 (ℓ) (2) veniently, it proves possible to represent this modification mℓi = f ki QL (72) E by a correction to the value of θℓ. h i h 2i 2 90αm (5 4fZ )αm ℓi − ℓi Recognising that the fermion masses vary between dif- 1+ 2 + 2 ×( " π(mc∗) 2πmW # ferent particle families, it is necessary to specify the fam- 2 2 2 ily for which the corrected value of θℓ is being evaluated. 5m 90αm (25 12fZ)αm ℓi ℓi − ℓi This is done by replacing ℓ with a member of the particle + 2 1+ 2 + 2 (mc∗) " π(mc∗) 6πmW # family in question, e.g. e, µ, or τ for the electron family. 2 2 240mℓi mℓ1 + 4 . (ℓ) 2 2 ) k1 mH′ mW h i

4. 1-loop scalar corrections 2. Preamble

These terms fall below the threshold of relevance, lead- To obtain a corrected expression for the electron mass ing to corrections to m at O(10 2 σ ), or in relative Z − exp angle at some energy scale , first recall that the eigen- terms, to mW , mZ , and mH′ at approximate order of E vectors {vi i 1, 2, 3 } of Kℓ (49–51) are independent parts in 107. | ∈ { } of the value of θℓ. For the electron family, matrix Kℓ therefore always admits the decomposition 5. 2-loop EM corrections

3 (ℓ) These terms fall below the threshold of relevance, lead- Kℓ = k viv† (73) 3 i i ing to corrections to mZ at O(10− σexp), or in relative i=1 X 22 where largest corrections being those associated with the tau. The charged lepton mixing matrix Ke(θe) is implicitly 1 1 1 1 dependent on the same corrections, as changes to the par- v1v† = 1 1 1 (74) ticle mass ratios affect colour mixing, and hence the co- 1 3   1 1 1 ordinate transformation required to restore colour neu-   trality. Allowing the corrections to augment mass and  2πi 2πi 1 e 3 e− 3 also to adjust Ke(θe) is not double-counting, as the mass 1 2πi 2πi v2v2† =  e− 3 1 e 3  (75) correction induces the corresponding adjustment in the 3 2πi 2πi e 3 e− 3 1 co-ordinate transformation implemented by Ke(θe). The   co-ordinate transformation does, however, have an effect  −2πi 2πi  1 e 3 e 3 on the magnitude of the mass correction term, and this 1 2πi 2πi v3v3† =  e 3 1 e− 3  . (76) effect must be compensated for—see Sec. IIIC5. 3 2πi 2πi e− 3 e 3 1     4 Then note that the imaginary components of the off- 3. First-order correction to Kℓ from the tau channel diagonal matrix elements arise from components 2 and 3 only (75–76), with an increase in component 2 (cor- To understand how these corrections to lepton masses responding, for charged leptons, to the ) making affect the matrices Kℓ, consider the leading order term a positive-signed contribution to the imaginary compo- (0) 2 mei which arises from the set of nine diagrams de- nent in [Kℓ]12,[Kℓ]23, and [Kℓ]31. Examining the vec- scribed in the caption of Fig. 10, and may be thought of   2 tors on the complex plane associated with exp(θe) = as the action of an operator mˆ (0) on the preon triplet exp( 3πi/4) and a small correction proportional to − comprising a sum over nine terms. The action of this exp(2πi/3), it is seen that the action of this correction operator on the colour space of the preon triplet breaks is to increase the magnitude of θℓ (making it more nega- colour neutrality, but this is then corrected by applica- tive). Similarly, an isolated increase in component 3 (the tion term-by-term of four matrices (or operators) Kℓ, tau) decreases θℓ. Finally, an isolated increase in compo- 4 heuristically Kℓ (with implicit action of each operator nent 1 (the electron) affects only the real component of on the appropriate Hilbert space as per the appropriate an entry such as [Kℓ]12 and thus when [Kℓ]12 is complex, 4 (0) 2 once again affects the value of θ . When θ is no longer diagram of Fig. 10), such that Kℓ mˆ as a whole ℓ ℓ leaves the colour of the preon triplet unchanged up to a fixed to be 3π/4, it is labelled by the relevant particle   family, e.g. θ−, and it becomes necessary to specify K as cycle r g b r. e ℓ As seen→ in→ Eq.→ (77), higher-order corrections enhance Kℓ(θℓ) for some family ℓ. (0) 2 Next, recognise that it is convenient to write the mass the action of mˆ on eigenstate ei by a factor of interaction as a leading-order term derived from the QL [1 + ∆e(mei )]. If colour neutrality is to be conserved,   photon field (and associated with colour mixing ma- then for colour-changing diagrams [including those which trix Kℓ as derived Sec. IIIA1) plus corrections. For can mix colours under some global SU(3)C transforma- the charged lepton masses, this corresponds to rewriting tion, such as interaction with the diagonal gluons of the Eq. (72) as 8 representation of SU(3)C ] there must be an equivalent enhancement of the matrices Kℓ, 4 2 m2 = f 2 k(e) (2) [1 + ∆ (m )] ei i QL e ei Kℓ Ke(θe) (79) E → 2 (77) h(0) i h i [where dependence on θe({me i 1, 2, 3 }) is as- = mei [1 + ∆e(mei )] i sumed but will subsequently be| shown]∈ { such} that the h i 4 action of [Ke(θe)] on ei is equivalent to the action of 2 2 4 90αm (5 4fZ )αm K [1 + ∆e(me )]. ei − ei ℓ i ∆e(mei )= 2 + 2 (78) As the largest such correction arises from the most π(mc∗) 2πmW 2 2 2 massive particle, begin by considering e3, the tau. First 5m 90αm (25 12fZ)αm + ei 1+ ei + − ei note that introduction of a mass dependency for the ma- (m )2 π(m )2 6πm2 c∗  c∗ W  trices Ke(θe) implies that the bosons associated with 2 2 the colour-neutrality-preserving co-ordinate transforma- 240mei me1 + 4 tion are no longer parameterless, and may therefore carry (e) 2 2 k1 mH′ mW momentum in the higher-order diagrams. They must h i therefore be represented explicitly as per Fig. 17. By where e1,e2,e3 = e,µ,τ and the subscript e on ∆e construction these bosons only interact at the bound- indicates{ the family} {e,µ,τ } rather than the electron in ary of the transformed co-ordinate patch, and are con- particular. As can be{ readily} seen from Eq. (78), the sequently both foreground and massless. Let the dia- amplitudes of these corrections to particle mass are de- grams of Fig. 17(i)-(iii), having two co-ordinate trans- 2 pendent upon the squared lepton masses mei , with the formation bosons, be termed “first order”, and for now 23

perform any net colour mixing beyond the unmodified, 4 2 constant factors of Kℓ , the resulting factor of 2 (4π)− does not appear in the leading term calculation.· Its ef- fect is seen only on the diagrams in ∆e(mτ ), where it 4 represents variations in colour mixing relative to Kℓ , and 4 2 multiplies any correction to Kℓ by a factor of 2 (4π)− . Next, recognise that the components of each· matrix Ke(θe) which perform rotations on SU(3)C are composed from a weighted superposition of some set of representa- tion matrices λi′ (possibly, but not necessarily the ma- trices λi of Appendix A), of which there are eight. (A ninth basis element is associated with the trivial repre- sentation and acts to preserve norm, but is strictly di- agonal, and independent of θℓ, so is not yet of interest here.) An appropriate choice of basis permits every off- diagonal element in Ke(θe) to be associated with a single basis element for example, in Kℓ, entry [Kℓ] is associ- { 12 ated with (λ1 +λ2)/√2 . A change to θe may correspond to a change in choice of} representation matrices [e.g. to λ1 cos β + λ2 sin β for some angle β] but this ability to FIG. 17. When the matrices Kℓ are replaced by Ke(θe), which construct a single basis element corresponding to a par- depends on energy scale, this induces a pair of gauge bosons ticular entry in Ke(θe) persists for any reasonably small which may in principle connect the four matrices Ke(θe) in perturbation around θe = 3π/4. any of three ways [shown for Fig. 10(i) in diagrams (i)-(iii)]. In Now consider a preon in− Fig. 10(i) or (ii) which has diagram (i), the bosons each connect to two matrices Ke(θe) two matrices Ke(θe) acting on it, and recognise that if on the same preon and thus enact a co-ordinate transforma- the portion of K (θ ) which performs colour transforma- tion followed by its inverse. In diagram (ii), where the bosons e e are crossed, there may be a permutation of the preon colour tions is decomposed into these eight components, then if labels but overall colour neutrality is nevertheless conserved. a given component λi′ acts on the preon in the lower posi- Diagram (iii) contrasts with the other two in that the boson tion, then its conjugate must act in the upper position for trajectories are orthogonal to the preon trajectories. For a the foreground preon colour to be left invariant overall.(3) worldsheet such as that shown (labelled W), which is equidis- The action of the pair of matrices Ke(θe) on this boson tant between the boson vertices and isochronous in the rest consequently admits decomposition into eight channels, frame of the particle, the total number of oriented crossings enumerated by the family of orthogonal representation for each of the bosons in diagram (iii) is zero regardless of the matrices λi′ appearing at the lower matrix Ke(θe). trajectory followed. This implies that these bosons do not Next, consider the remaining two matrices Ke(θe) describe a co-ordinate transformation extant at this world- when these are not on the same preon. In this situation sheet, and thus this diagram does not contribute to K (θ ). e e the representation matrices present will be conjugate up Diagram (iv) is an example second-order diagram having four to a colour cycle r g b r or its inverse. As bosons arising from the co-ordinate transformation and span- → → → ning from the lower to the upper collection of matrices Ke(θe). the only colour-changing bosons in the QL are gluons, This diagram represents the consecutive application of two and SU(3)C symmetry is preserved on the gluon sector (possibly different) transformations in SU(3)C to each preon. and in the definition of the charged lepton, evaluation of any diagram where the entries from Ke(θe) are offset by a colour cycle is necessarily equivalent to evaluation of a di- agram where they are not offset. It is therefore acceptable ignore diagrams having four co-ordinate transformation bosons (“second order”). Regarding the two co-ordinate transformation bosons 3 Note that if one diagram family as per Fig. 10 can be made to of Fig. 17(i)-(ii), their vertex factors are incorporated have no net effect on colour, then all paired QL bosons have no into the leading order matrix Kℓ. Integration over their net effect on colour. Further, as unpaired bosons (as well as being degrees of freedom then yields a factor per diagram of exceedingly rare) have on average no net effect on colour, these 2 2 may also be grouped into clusters having zero net colour effect (4π)− [not (2π)− , as they are not self-dual and thus overall, and an analogous co-ordinate transformation performed their loop is less symmetric than that for the photon]. such that their effect on colour vanishes instant-by-instant as Note that Fig. 17(ii) counts as a loop diagram as there are well as on average, with the associated bosons vanishing over implicit gluon-mediated couplings between the preonic distances large compared with LQL. Thus no error is made in constituents of the lepton, which are massless over the requiring that colour is left unchanged on a per-diagram basis. Also note that once the mechanism for corrections to θ is fully length-scale involved [O( )] and complete closure of e LQL elucidated, the co-ordinate transformations arising from the un- the loop. paired bosons yield no systematic effect on θe and thus their Recognising that the leading term calculation does not effect may be ignored over macroscopic scales. 24 to evaluate entries in Ke(θe) under the assumption that As per the discussion under Eqs. (74–76), the tau correc- all matrices Ke(θe) appear in conjugate pairs. In addi- tion is known to decrease the magnitude of θℓ, giving tion, in all diagrams all representation matrices are mul- tiplied by their conjugate (whether independently or in eiθe [Ke(θe)] = = [Kℓ] 1 + i δe[∆e(mτ )] (87) conjunction with the representation matrices associated 12 √2 12 with the bosonic vertices) and thus the terms arising from n p o all 64 channels contributing to the actions of the matri- ces Ke(θe) on the preon triplet are additive on the same 3π 4 δe[∆e(mτ )] (net trivial) charge sector, and therefore summed. By θe = 1 (88) − 4 ( − 3π ) SU(3)C symmetry, each channel will contribute equally p to the overall correction to the leading order diagram. 3π where the value of θℓ has been written explicitly as . Now consider the single channel associated with a − 4 specific off-diagonal entry in at least one lower matrix Ke(θe). For definiteness, let this be [Ke(θe)] . As there 12 4 are, overall, 64 contributing channels, each channel must 4. Second-order correction to Kℓ from the tau channel produce a net correction of ∆e(mτ )/64. However, the 4 synthetic bosons multiply all correction diagrams by a The first-order correction to Kℓ is applied to all di- 2 factor of 2 (4π)− and thus [Ke(θe)] and [Ke(θe)] agrams which modify the colour mixing process, i.e. all · 12 21 must satisfy diagrams contributing to ∆e. For these diagrams, this correction is equivalent by construction to enacting the (4π)2 [K (θ )]2 [K (θ )]2 = [K ]2 [K ]2 1+ ∆ (m ) transformation e e 12 e e 21 ℓ 12 ℓ 21 2 64 e τ  ·  4 4 π2 Kℓ Kℓ [1 + ∆e(mτ )] , (89) = [K ]2 [K ]2 1+ ∆ (m ) → ℓ 12 ℓ 21 8 e τ   resulting in a relative increase in these diagrams’ contri- (80) bution to particle mass equivalent to

∆e(mτ ) ∆e(mτ )[1 + ∆e(mτ )]. (90) π2 → [Ke(θe)]12 [Ke(θe)]21 = [Kℓ]12 [Kℓ]21 1+ ∆e(mτ ). ⇒ r 8 Any species-dependent increase in mass affects colour (81) mixing in precisely the way described above for correc- tion ∆e(mτ ), regardless of whether this increase is di- Noting that [Kℓ]12 = [Kℓ]21† , it is convenient to write agrammatic (as in the first-order component) or gauge- dependent and derived from a change to θe (as here). π2n This correction therefore attracts a further smaller cor- δe(n)= 1+ 1 (82) rection to K , enhancing the first-order correction calcu- r 8 − ℓ lated above. While this series may be continued indefi- [1 + δe(n)] = [1 + i δe(n)][1 i δe(n)] (83) − nitely, it is convenient to truncate at second order for a 2 and assign the correspondingp off-diagonalp entries of precision of O[∆e(mτ )]. To implement the corresponding adjustment to K (θ ), Ke(θe) to be e e 2 let the O[∆e(mτ )] term be compensated by the second- order diagrams of which Fig. 17(iv) is a prototype. In [Ke(θe)] = [Kℓ] [1 i δe[∆e(mτ )]] 12 12 ± (84) these diagrams, a first-order correction [Fig. 17(i)-(ii)] is [Ke(θe)] = [Kℓ] [1 ipδe[∆e(mτ )]], 21 21 ∓ supplemented by a further, smaller correction. p For Figs. 17(i)-(ii) there is a symmetry factor of two preserving the Hermeticity of Ke(θe). corresponding to crossing or not crossing the bosons. For By the factor of i on the correction term, this correc- ± Fig. 17(iv), tion is orthogonal to the leading-order value of [Kℓ]12. As this orthogonality is independent of the value of the inner pair may be crossed (factor of two), θe, this implies that for a non-infinitesimal correction, • δ [∆ (m )] is the length of an arc. Since [K ] is of the outer pair may be crossed (factor of two), e e τ ℓ 12 • unit length, the corresponding correction to θℓ is p the inner and outer bosons on the left may be ex- 3 • changed (factor of two), and θℓ θℓ δe[∆e(mτ )]+O( δe[∆e(mτ )] 2 ). (85) → ± { } p the inner and outer bosons on the right may be In the vicinity of θℓ = 3π/4, this may be rewritten − • exchanged (factor of two).

4 δe[∆e(mτ )] 2 θℓ θℓ 1 . (86) The factor of 2 (4π)− associated with the set of diagrams · 4 → ( ∓ 3π ) Figs. 17(i)-(ii) becomes a factor of 16 (4π)− for the set p · 25 of diagrams derived from Fig. 17(iv). The expression for at the expense of introducing an additional scaling fac- 2 2 tor of [1+∆ (m )] on all diagrams which affect the colour [Ke(θe)]12 [Ke(θe)]21 then becomes e τ mixing process [i.e. all diagrams in ∆e(mτ )]. This rescal- 2 2 ing is unphysical, being the result of a forced (non-gauge) [Ke(θe)]12 [Ke(θe)]21 change in local co-ordinate system, and may not in gen- (4π)2 (4π)4 = [K ]2 [K ]2 1+ ∆ (m )+ ∆2(m ) eral be assumed to conserve the normalisation of proba- ℓ 12 ℓ 21 2 64 e τ 16 642 e τ  · ·  bility and electron mass generated by the choice of dila- π2 π2 ton gauge [1, Sec. IIIE1a; Sec. IIIC5a above]. This then = [K ]2 [K ]2 1+ ∆ (m ) 1+ ∆ (m ) ℓ 12 ℓ 21 8 e τ 32 e τ induces a change in the dilaton gauge which rescales the    (91) affected sector to eliminate this factor. The effect of this change of gauge on particle mass is for a redefinition of δe(n) in Eq. (88) to to rescale the modified correction factor 1 ∆e(mτ )[1 + ∆e(mτ )] ∆e(mτ )[1 + ∆e(mτ )][1 + ∆e(mτ )]− π2n π2n → δe(n)= 1+ 1+ 1. (92) = ∆e(mτ ) (94) 8 32 − s   recovering a particle mass of The next order term extends δe(n) to 2 2 (0) mτ = me [1 + ∆e(mτ )] . (95) π2n π2n π2n δe(n)= 1+ 1+ 1+ 1 (93) h i 8 32 72 − 1 s    This correction factor of [1 + ∆e(mτ )]− arises from the dilaton field and thus is insensitive to colour charge. In but is not used in this paper as its effects fall below the contrast, the calculation of θe is performed on a sin- threshold of relevance. Its most significant correction is gle channel of SU(3)C, and thus on a single channel of 2 to mZ at O(10− σexp), or in relative terms, to mW , mZ , GL(3, R). Recognising that the correction ∆e(mτ ) to 7 and mH′ in order of parts in 10 . particle mass arises from a symmetric sum over all nine channels of GL(3, R), it consequently follows that the cal- culation of θe “sees” only one ninth of the total correction

5. Dilaton corrections to ∆e(mei ) arising from the regauging of the dilaton field. Further recognise that to maintain colour neutrality this correction factor must be partitioned equally be- a. Running of mei In Sec. IIIE1a of Ref. 1, the tween the mass interaction and the compensatory matri- dilaton parameter was gauged by requiring normalisa- 4 tion of probabilities for all foreground particles. However, ces [Ke(θe)] . The result is to rescale ∆e(mτ ) in Eq. (88) according to ∆e(mτ ) r[∆e(mτ )], probability is a property normalised on integrating over → a three-dimensional worldsheet in R1,3, whereas the dila- ton coefficient may vary in time as well as space. With 3π 4 δe r[∆e(mτ )] θ = 1 { } probability amplitudes depending only on xi i 1, 2, 3 e − 4 ( − p 3π but with the dilaton coefficient being a function| ∈{ of xµ}, (88b) the dilaton gauge is therefore underconstrained. This 3 +O (δe r[∆e(mτ )] ) 2 choice of gauge may therefore be extended to normalise { } ) one further parameter, provided that parameter is en- h i ergy/timelike and its rescaling is independent of the prob- R3 1 ability parameters being normalised over . (1 + n)− 1 Now recognise that in the above, the lepton mass m r(n)= n 1+ − , (96) ei · r 9 has been written as a function of ∆e and θe, which are 2 in turn a function of both the lepton mass itself and also again to O[∆e(mτ )]. the energy scale of the gluon deficit. For a particle which is both on-shell and at rest, these two energy scales are 4 identical, but more generally, the gluon mass deficit will 6. Corrections to [Ke(θe)] from the muon and electron run with energy scale. This running leads to a running of channels electron mass which has no counterpart in the Standard Model. Therefore complete the gauging of the dilaton Now recognise that a propagating preon is a massive parameter by enforcing that everywhere, after rescaling, particle and undergoes multiple scatterings off the QL 2 me O( QL) takes on its rest value regardless of the energy fields. These scatterings may impart energy to or take ofh thei L electron. The implications of this gauge will be energy from the preon, and are generally ignored as their seen in Sec. IV A. average contributions vanish over length scales large com- b. Rescaling arising from colour mixing The above pared with QL. Bringing these into consideration, it transformations have maintained colour neutrality, but is necessaryL to integrate over all possible scatterings of 26 the preon off the QL fields, and for a foreground preon ∆e(mτ ) in Secs. IIIC3–IIIC5 are thus noted to implic- 2 with a rest energy small compared with , the net ef- itly be ∆e(mτ ,mτ c ). EQL fect is to integrate near-homogeneously over all energies To evaluate the correction arising from the e2 or muon from to + [effectively homogeneously at ener- channel, recognise from Eqs. (75–76) that the effect of −EQL EQL gies small compared with QL, corresponding to boosts this correction is in opposition to the tau correction, and small compared with 1 10E32 as per Eq. (13)]. On-shell increases the magnitude of θ . As its scale is seen from − e contributions appear at the three eigenvalues of Ke(θe), Eq. (78) to be small compared with the tau correction, and thus even for a foreground lepton in a definite colour and from the construction of Eq. (84) it is seen to act sector eigenstate, all three channels contribute to the cor- in direct opposition to the tau correction, its effects are rection to θe. Over length scales of O( QL), in expres- conveniently represented at energy scale by subtracting L E sions involving the matrices Ke(θe), particle propagation the muon correction from the tau correction at the level is summed over the three generation channels. In expres- of the interaction diagrams. This yields sions which do not involve these matrices, such as evalua- tion of boson masses, fermion symbols are generationless. 3π 4 δe r[∆e(mτ , ) ∆e(mµ, )] θ ( )= 1 { E − E } Evaluation of the parameter θe inevitably involves the e E − 4 − p 3π ! mixing matrices Ke(θe), and thus corrections to θe from all channels of propagation must be considered regardless (99) of the generation of the foreground particle which may where θ ( ) is the effective value of θ experienced by a be measured in the low-energy limit. e e particle whoseE mass is c 2, and henceforth all instances Now that there are multiple energy scales involved, − of θ are acknowledgedE to depend, either explicitly or it is advisable to be more careful with the gluon mass e implicitly, on a particle’s energy scale . deficit. This also permits generalisation to particles fur- Finally, the e or electron channel is seenE from Eq. (74) ther from rest in the isotropy frame of the QL, and is 1 to contribute purely to the real part of [K (θ )] . For useful preparation for Sec. IV A, in which off-shell exci- e e 12 the imaginary part, all positive contributions to the imag- tations are considered. Note that the mean energy scale inary portion of [K (θ )] arise from the muon channel of the gluon deficit is determined by the mean additional e e 12 and all negative contrinutions from the tau channel, en- energy acquired by the foreground species on account of abling the relatively simple form of Eq. (99). In contrast, its nonzero rest mass, i.e. while the electron channel contributes to the real part 2 2 2 of [Ke(θe)] the bulk of the real part (and, indeed, for [k0]fg [k1]fg + [k2]fg + [k3]fg, (97) 12 − θℓ = 3π/4 the entirety of the real part) arises instead and that this is notq in general the same as the instan- from− the muon and tau channels. 2 2 taneous energy of the propagating particle. The gluon Let the electron mass undergo a rescaling me me[1+ → (e) deficit undergoes its own energy fluctuations around this ∆e(me, )], resulting in some rescaling (1 + ε) of k . As E 1 value as a result of energy exchange between the deficit the real and imaginary components of exp(iθℓ) are both and the QL, but these are not necessarily simultaneous negative, the increase in the real component of [v1v1†]12 with the fluctuations of the representative foreground associated with this correction will decrease the magni- particle, and in the absence of any generations mech- tude of θℓ. However, the electron mass is one of the fixed anism for bosons, may be separately averaged to zero. input parameters to the model, and consequently this Further, when a particle is off-shell, or undergoes a boost rescaling (and its associated effect on θℓ) must be offset relative to the QL, this affects the energy scale of the by a decrease in magnitude of the associated energy scale gluon deficit via Eq. (97) without affecting particle rest (2) (0) 2 and thus of me corresponding to multiplica- mass save through the dependency of rest mass on the EQL { } tion by (1 + ε) 1. If this value is pulled out as an inde- gluon deficit as per Eq. (78). Recognising the indepen- −   dence of these energy scales, therefore write pendent factor, in the electron mass diagram it cancels with the (1 + ε) arising from the electron mass rescaling, 90αm2 (5 4f )αm2 ei Z ei leaving me and k1 unchanged (and eliminating any need ∆e(mei , )= 2 + − 2 E π [m ( )] 2πmW to evaluate colour effects from this sector). However, for c∗ E 2 2 2 the muon and the tau it may be seen as a rescaling of 5m 90αm (25 12fZ)αm ei ei ei k(e) or k(e) respectively by a factor of (1+ε) 1. To eluci- + 2 1+ 2 + − 2 2 3 − [mc∗( )] " π [mc∗( )] 6πmW # E E date the effect of this scaling factor on θℓ, restore colour 2 2 neutrality by likewise holding the muon and tau masses 240mei me1 + 4 (98) fixed, corresponding to enacting a transformation (e) 2 2 k1 mH′ mW (e) (e) h i 2 ki ki (1 + ε) i 2, 3 . (100) 2 2 27 → | ∈{ } [m∗( )] = m 1 E , (59b) c E c − 10 m2c4  c  The leading colour effect arising from this transforma- where is the energy scale used in calculation of the tion is equivalent to a rescaling of k(e) (which generates E 3 gluon mass deficit correction. Previous occurrences of the leading correction to θℓ) by (1 + ε) and is therefore 27

(1) (2) associated with an increase in the magnitude of θℓ. The f, QL, and QL from the mass expressions derived rescaling of the muon term is by the same factor. This above.E This is readilyE achieved by taking ratios of particle multiplier is independent of the existing muon and tau masses. Recognising that the choice of gauge articulated 2 corrections, having its origins on the real rather than the in Sec. IIIC5 fixes the value of me so this does not run 2 2 imaginary portion of [Ke(θe)]12, and therefore must be with energy scale, the value of mτ ( τ )/me( τ ) evaluated 2 E E evaluated as a separate correction to θℓ = 3π/4 rather at energy scale τ = mτ c then readily admits interpre- than being conflated into a single term (as− was possible tation as the ratioE of the rest masses of the electron and 2 2 for the muon and tau channels by their linearity on the the tau, and similarly for mµ( µ)/me( µ) and the rest imaginary component). This correction acts on both the mass of the muon. E E muon and the tau channel to yield In conjunction with the boson mass expressions devel- oped in Sec. II, and noting the freedom to substitute 3π 4 δe r[∆e(mτ , ) ∆e(mµ, )] θe( )= 1 { E − E } E − 4 − 3π ! p 2 2 me mµ 4 δe r[∆e(me, )] 4 4 (102) 1+ { E } . (e) −→ (e) k ( e) k ( µ) × p 3π ! 1 E 2 E (101) h i h i or

IV. RELATIONSHIPS FROM R0|18 DUST 2 2 GRAVITY me mτ 4 4 (103) (e) −→ (e) k1 ( e) k3 ( τ ) A. Mass relationships E E h i h i 0 18 To obtain testable predictions from R | dust grav- as convenient, as discussed in Sec. IIA2, this yields the ity, it is necessary to eliminate the model parameters relationships (at the present level of precision)

4 2 2 2 2 2 4 (e) 90αmτ (5 4fZ )αmτ 5mτ 90αmτ (25 12fZ )αmτ 240mτ k ( τ ) 1+ ∗ 2 + − 2 + ∗ 2 1+ ∗ 2 + − 2 + 4 3 π[m ( τ )] 2πm [m ( τ )] π[m ( τ )] 6πm (e) 2 2 2 c W c c W k ( ) m ′ m m E ( E E E h 3 τ i H W ) τ h i h i E 2 = me 4 2 2 2 2 2 2 2 (e) 90αme (5 4fZ )αme 5me 90αme (25 12fZ )αme 240memτ k ( τ ) 1+ ∗ 2 + − 2 + ∗ 2 1+ ∗ 2 + − 2 + 4 1 π[m ( τ )] 2πm [m ( τ )] π[m ( τ )] 6πm (e) 2 2 c W c c W k ( ) m ′ m E ( E E E h 3 τ i H W ) h i h i E (104)

4 2 2 2 2 2 4 (e) 90αmµ (5 4fZ )αmµ 5mµ 90αmµ (25 12fZ )αmµ 240mµ k ( µ) 1+ ∗ 2 + − 2 + ∗ 2 1+ ∗ 2 + − 2 + 4 2 π[m ( µ)] 2πm [m ( µ)] π[m ( µ)] 6πm (e) 2 2 2 c W c c W k ( ) m ′ m m E ( E E E h 2 µ i H W ) µ h i h i E 2 = me 4 2 2 2 2 2 2 2 (e) 90αme (5 4fZ )αme 5me 90αme (25 12fZ )αme 240memµ k ( µ) 1+ ∗ 2 + − 2 + ∗ 2 1+ ∗ 2 + − 2 + 4 1 π[m ( µ)] 2πm [m ( µ)] π[m ( µ)] 6πm (e) 2 2 c W c c W k ( ) m ′ m E ( E E E h 2 µ i H W ) h i h i E (105)

2 2 99mµ 3 α 19mµ 1+ 4 2 (e) 2 3 1+ 64 + π fZ π 1+ (e) 4 k ( µ) m 2 2 2 k ( ) m2 mc h 2 E i W m − h 2 µ i W W ( E ) 2 = m2 (108) = m 19 µ 2    W 1+ 4 mZ 23m2 k(e) m2 401 3 α µ h 2 ( µ)i W 4 1+ + 1+ 4 E 12 2π 2π (e) 2 k ( µ) m ( h 2 E i W )    (106) where 2 1 6α m 20 1+ π 1+ π H = 2 (ℓ) 2π(n 1) 2 kn ( )=1+ √2cos θℓ( ) − (52b) mW 2 3 α  19mµ E E − 3 9 1+ 64 + fZ 1+ 4   2π 2π (e) 2 − k ( µ) m 2 4 ( h 2 E i W ) 1 mW mW fZ = 4 24 + 16 (19b)    (107) 3 − m2 m4  Z Z  28

2 2 2 27 and [m∗( )] = m 1 E (59b) c E c − 10 m2c4  c  2 ℓ = mℓc (109) E

3π 4 δe r[∆e(mτ , ) ∆e(mµ, )] 4 δe r[∆e(me, )] θe( )= 1 { E − E } 1+ { E } (101) E − 4 − p 3π ! p 3π !

2 2 2 2 2 2 2 90αmei (5 4fZ )αmei 5mei 90αmei (25 12fZ)αmei 240mei mµ ∆e(mei , )= 2 + − 2 + 2 1+ 2 + − 2 + 4 E π [m ( )] 2πmW [mc∗( )] π [m ( )] 6πmW (e) 2 2 c∗ E " c∗ # k ( µ) m ′ m E E 2 E H W h i (98)

1 (1 + n)− 1 B. Minimum requirements for particle generations r(n)= n 1+ − (96) from preon substructure · r 9 π2n π2n δe(n)= 1+ 1+ 1. (92) These results strongly suggest that particle generations 8 32 − s   arise from a preonic substructure for leptons. It would appear that the minimum requirements for direct import 0 18 The lower precision on mH′ compared with the other of this structure from R | dust gravity into a modifica- weak bosons is acceptable as the main sensitivity to these tion of the Standard Model while retaining its predictive higher-order mass corrections is in the lepton mass ratios capacity are as follows: via θe, in which mH′ plays only a very small part. The 2 terms provided suffice to calculate m ′ to current exper- A tripartite preon structure for leptons bound by H • R imental precision. a GL(3, )-symmetric force sector [which need not necessarily correspond to the strong force plus an Taking m , m , and α as input parameters [7, 8], e µ additional (axion?) companion, as here].

2 me =0.5109989461(31) MeV/c (110) A bipartite preon structure for the Higgs/scalar bo- • 2 son. mµ = 105.6583745(24) MeV/c (111) 3 α =7.2973525693(11) 10− , (112) A duality between preon/antipreon pairs and the × • electroweak and preon-binding bosons. these relationships may be solved numerically to yield A multi-species quantum liquid in place of the the results given in Table II. • present nonvanishing vacuum expectation value of the .

Parameter Calculated value Observed value Discrepancy However, these features come with costs, including: (GeV/c2) (GeV/c2) Importing the quantum liquid relegates the scalar m 1.776867(1) 1.77686(12) 0.06 σ τ exp • boson to a bystander having only a small role to mW 80.3786(3) 80.379(12) 0.03 σexp play in generating particle mass. The Higgs mecha- mZ 91.1877(4) 91.1876(21) 0.05 σexp nism might still be responsible generating the quan- mH′ 125.16(1) 125.10(14) 0.45 σexp tum liquid, but will require some modification to be mc 80.4276(4) – – applied in this context.

R0|18 If the preon binding bosons are the gluons of the TABLE II. Calculated values of particle masses in dust • gravity. Quantity mc is the bare gluon mass. The displayed strong force, there must exist an axion (this may uncertainties in calculated masses arise from estimates of be useful as a dark matter candidate). higher-order terms. The leading corrections due to uncer- −3 If the preon binding bosons are not the gluons of tainty in me, mµ, and α are smaller, being of order 10 σexp, • or parts in 107 on all masses. The gluon mass shown is the the strong force, further work is required to de- bare mass at scales below EΨ, though with the caveat that it termine whether the inhomogeneous preon triplets frequently becomes massless in interactions above this scale, correspond to quarks and why they carry colour as seen in the calculations above. charge. 29

Alternatively the current role for the Higgs boson may Pauli matrices σi, rescaled by a factor of 1/√2: be retained, but only at the cost of introducing numer- ous fine tuning parameters. Unless an alternative but σi equivalent mechanism for fixing these parameters is con- {τi i 1, 2, 3 } τi = . (A1) | ∈{ } √2 structed, this comes at the cost of predictive power. 0 18 Although R | dust gravity incorporates a colour- based mechanism for particle generations with substan- A similar basis for su(3) is provided by rescaling the Gell- tial predictive power in the electroweak sector, it is pre- Mann matrices Ca, mature to identify this with electrostrong unification without at least some evidence of predictive capacity in Ci the strong nuclear sector. It is, however, clearly a candi- {λi i 1,..., 8 } λi = (A2) | ∈{ } √2 date for a unifying mechanism, and an obvious test would be calculation of the quark masses. where 0 18 Multiple other avenues for further testing of R | dust gravity exist, both theoretical and experimental, and per- 0 1 0 0 i 0 haps one of the most accessible relates to the weak mix- − C1 = 1 0 0 C2 = i 0 0 ing angle. In Eq. (37), the mass ratio used to compute     2 0 0 0 0 0 0 sin θW effectively takes a weighted average over all in-     teractions of the Z boson, whereas in Sec. IIC it is noted 1 0 0  0 0 1  that differences in higher-order corrections which depend C3 = 0 1 0 C4 = 0 0 0  −    on fermion charge could result in differing electroweak 0 0 0 1 0 0 couplings for different species, and thus different mea-     (A3) 2  0 0 i   0 0 0  sured values of sin θW . There do appear to be some − 2 C5 = 00 0 C6 = 0 0 1 discrepancies in the values of sin θW measured using dif-     ferent species [9, p. 172], and it would be fruitful to assess i0 0 0 1 0 0 18     whether this can be explained using R | dust gravity.    1  00 0 2 0 0 2 1 C7 = 0 0 i C8 = 0 0 .  −  √3  2  0i 0 0 0 1 V. CONCLUSION    −      0 18 R | dust gravity is a model having many elements Like the rescaled Pauli matrices τi, the re-scaled Gell- in common both with the Standard Model of particle Mann matrices satisfy physics and with the observable universe. As it has only four tunable parameters, which must be set by reference 2 R0 18 Tr [(λi) ]=1 i 1,..., 8 . (A4) to physical constants, | dust gravity may readily be ∀ ∈{ } tested against observation. In the present paper, relationships in the electroweak In conjunction with a multiple of the identity matrix 0 18 sector of R | dust gravity were used to predict the val- ues of several fundamental particle masses in the Stan- 1 0 0 dard Model, with results in excellent agreement with ob- 1 servation. These results arise from the particle genera- λ9 =  0 1 0  , (A5) R0 18 √3 tion mechanism used in | dust gravity. 0 0 1    

Appendix A: Gell-Mann matrices the matrices λi also yield a basis for gl(3, R). An alternative basis for gl(3, R) is given by the ele- When working with the group SU(2), a well-known mentary matrices eij , consisting of a matrix with 1 in basis for the tangent Lie Algebra su(2) is given by the position (i, j) and zero elsewhere.

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