Heavy quarks within the electroweak multiplet

Standard-model fields and their associated electroweak Lagrangian are equivalently expressed in a shared spin basis. The scalar-vector terms are written with scalar-operator components acting on quark-doublet elements, and shown to be parametrization-invariant. Such terms, and the t- and b-quark Yukawa terms are linked by the identification of the common mass-generating Higgs operating upon the other fields, after acquiring a vacuum expectation value $v$. Thus, the customary vector masses are related to the fermions', fixing the t-quark mass $m_t$ with the relation $m^2_t+m^2_b=v^2/2$ either for maximal hierarchy, or given the b-quark mass $m_b$, implying $m_t \simeq 173.9$ GeV, for $v=246$ GeV. A sum rule is derived for all quark masses that generalizes this restriction. An interpretation follows that electroweak bosons and heavy quarks belong in a multiplet.


Introduction
The standard model (SM) describes elementary-particle features and their interactions, which is praiseworthy, given its relatively limited required input, consisting of specific gauge and flavor symmetries, representations, and parameters, yet aspects remain within the model whose origin and connection to other tenets is absent, and that need to be addressed.
Thus, among its successes, the SM predicts mass values for the W and Z bosons [1] that carry the short-range electroweak interaction, in terms of electroweak parameters, through the Higgs mechanism [2,3]. However, one salient SM problem is that the fermion sector and its masses remain arbitrary, as they arise from Lagrangian terms, independent from the boson elements.
The electroweak sector hints it may provide this link, given that the W and Z vectors have universal couplings to SM fermions, and the Higgs field collectively gives mass to fermions and bosons. In addition, the similar order of magnitude of the measured masses [4] of the W, Z, the recently discovered scalar excitation, associated with the Higgs [5], and the top quark (with the bottom quark's the next highest), suggests connections among them, and thus, a common energy scale. Furthermore, fermions occupy the spin-1/2 and fundamental representations of the Lorentz and scalar groups, respectively, as vector bosons belong to the adjoint representation of each group, 1,2 which implies bosons can be constructed in terms of fermions, suggesting composite structures and/or a common origin.
The above motivates looking for a formalism that takes account of discrete degrees of freedom in a single basis, including group representation properties, such as the fermion-boson fundamental-adjoint duality for the Lorentz-scalar representations, and 1 As the Higgs occupies the SU L (2) fundamental representation. 2 For the Abelian hypercharge group U(1) Y , gauge invariance ensures boson-fermion quantumnumber additivity.
that describes the combined action of operators on fields.
A previously proposed SM extension [6], based on a shared extended spin space, with a matrix formalism, satisfies these requirements, as it replicates SM fields with their features, and matrix multiplication accounts for operator action on fields. This space contains a (3+1)-dimensional [d] subspace and one beyond 3+1, linked, respectively, to Lorentz and scalar degrees of freedom [7]. At each dimension, a finite number of Lorentz-invariant partitions are generated with specific symmetries and representations, reproducing particular SM features, where the cases with dimension 5+1 [8], 7+1 [9], and 9+1 [10] were studied.
In this connection, it is worth recalling that a basis or representation choice can be useful, even essential, in the description of a system and its dynamics. It may reveal otherwise-hidden connections between its components, and provide a simpler framework to understand physical properties. Such a basis may describe effective degrees of freedom [11] accounting for collective interactions, allowing for a simpler near freeparticle description, in a first approximation. For example, nucleon and associated boson interactive configurations give a tractable account of nuclear-motion modes [12].
Within condensed matter and low-temperature superconducting systems, a residual attractive interaction related to phonons couple electrons into Cooper pairs [13], which propagate freely, and lead to frictionless currents. In an application of this theory to quantum field theory and elementary particles [14], a four-fermion interaction produces fermion and composite-boson masses, linking their values. The quark model [15] conceives mesons and baryons in terms of constituent (dressed) quarks.
Leaving aside the more speculative nature of the spin SM extension, but complementarily to it, in this paper, we use it as a basis to derive SM connections, and the fields' mass values in particular: SM heavy-fermion (F ), vector (V ), and scalar (S) fields are equivalently expressed in terms of the obtained common basis [6] for both Lorentz and electroweak degrees of freedom, in turn, recasting their Lagrangian com-ponents L = L F V + L SV + L SF ; the identification of the scalar operator within the L SF and L SV vertices links univocally its defining (mass) parameters. Indeed, such universal electroweakly-invariant terms lead, under the Higgs mechanism, to a scalar whose lowest-energy condensate state pervades space, and generates particle masses through its vacuum expectation value v. Within the spin basis, this mechanism is similarly represented; as these fields shape elements on a matrix space, with a single associated scalar operator acting upon the others, their mass-generation property relates their coefficients.
Next, as we give the paper's organization, we sketch the argument in more detail. Section 2 reviews the applied spin-extended space for symmetry generators and states. The paper focuses then on the (7+1)-d case that can describe the electroweak sector, and a quark doublet. For all sectors, L F V , L SV , L SF , the conventional and spin-space Lagrangian are equivalent, which is shown term-by-term in Appendices 1,2. Section 3 chooses one among two vector bases within L F V , where vectors with chiral properties are adequate. Section 4 writes L SV equivalently with combinations of the scalars and their conjugates, with universal couplings to vectors, shown explicitly in Appendix 2; similarly for the spin-base representation, in which these two scalars induce a projection to flavor-doublet components (as t,b quarks). Schematically, given the spin-space basis element B f for a field f (x), we write L SV in terms of B S containing these two scalar components, obtaining the vector mass squared In Section 5, we show that the fermion masses within L SF can be written [B S , B F ], where B F contains two terms with appropriate Yukawa coefficients. Within the spin-basis formalism, we derive that B S , B S have the same operator structure; given their mass-giving nature, the identification of these operators and their coefficients translates a v-normalization restriction on B S to B S , implying a relation for the t and b quark masses. Section 6 shows a procedure exists that generalizes consistently this relation to all quarks in terms of a sum rule for their masses, taking advantage of the chiral projection properties of the scalar field in the spin basis. In Section 7, we draw conclusions. We work in the classical framework afforded by the Lagrangian, and at tree-level, but also rely on a quantum-mechanical interpretation.
2 Symmetry generators and states in spin-extended space In the following, we introduce the spin basis and its main features, where more information may be found in previous treatments [7]- [10]. Mainly, it describes SM discrete degrees of freedom in a single scheme, namely, for the Lorentz and scalar groups, and for both symmetry generators and state representations, using a common matrix space:
The gamma matrices have Hermiticity properties 3 Following standard practice, the label 4 is omitted.

Operators and symmetry transformations
The Lorentz generators and transformations acting on spinors have standard expressions in the 4-d Clifford algebra C 4 , namely, with the (3 + 1)-d gamma matrices γ µ transforming as vectors, while the remaining N − 4 gamma matrices γ a , a = 5, . . . , N , and their products commuting with σ µν , so they are indeed Lorentz scalars identified with generators of continuous symmetries, either gauge or global. Together with the 4-d pseudoscalar the scalars are accommodated in the unitary symmetry set where 1 stands for the N -d identity matrix.
A projector operator P, obtained from elements of S N −4 , within a limited number partitions, is chosen to fit as closely the SM. The combined operator that acts on both the Lorentz generators J µν = i (x µ ∂ ν − x ν ∂ µ )+ 1 2 σ µν and the S N −4 symmetry-operator space is likewise projected Lorentz transformations are thus and scalar transformations have the form with I a ∈ S N −4 . Symmetry generators within this space are described schematically in Fig. 1 in Ref. [9].
The inner product of two fields is defined according to a matrix space Under a unitary transformation, Ψ → U ΨU † , given the ket-bra matrix structure [7], with the bras interpreted as conjugate states. Thus, a Hermitian operator Op within this space characterizes a state Ψ with the eigenvalue rule for real λ. This definition is consistent with the action of a derivative operator on a Hilbert space: The direct product trΨ † b Ψ a is also consistent associativity-wise with the operator rule, as tr[Op,

Field Representation
Fields are usually assumed to exist on a Cartesian basis; for example, a vector field has components A µ (x) = g µ ν A ν (x); alternatively, in the spin basis, it is expressed as indices now specify the vector character.) More generally, a physical field with scalar quantum numbers is associated with elements of C N , classified by operators from C 4 ⊗ S N −4 , so it has the structure (elements of 3+1 space ) × ( elements of S N −4 ) .
Fig. 2 in Ref. [9] shows the corresponding Lorentz states: scalars, vectors, fermions, and anti-symmetric tensors, arranged in the same matrix space. Next, we provide more details on the first three (physical) fields.

Fermion field
When Ψ is a spin-1/2 particle, it may be seen schematically conformed as Ψ ∼ |ψ 1 |a 1 F 1 F 2 |, with the ket carrying spin-1/2 and gauge-group fundamental representation ψ i , a i quantum numbers, respectively, and both the bra and ket carrying flavor group F i .
More specifically, a fermion can have the form where Γ F a is an element of S N −4 , and L α represents a spin polarization component, e.g., L 1 = (γ 1 + iγ 2 ). The operator P F is a projection operator, e. g., P F = L 5 , where implying with P c F = 1 − P F , so that Lorentz and gauge generators act trivially on its rhs when evaluating commutators as in Eq. (11), since P c F P F = (1 − P F )P F = 0.
Thus, for U accounting for the Lorentz representation in Eq. 8 and the scalar transformation in Eq. 9, Ψ transforms, unlike vector and scalar fields, as This leads to fermions transforming as the fundamental representation of both the Lorentz and gauge groups.

Vector field
We may view vectors constructed as Ψ ∼ |ψ 1 |a 1 a 2 | ψ 2 |, with the bra-ket configuration producing Lorentz vector and gauge group adjoint configurations, given the vector and scalar γ µ , µ = 0, . . . , 3 and γ a , a = 5, . . . , N , respective transformation properties. Thus a vector field has form where γ 0 γ µ ∈ C 4 and I a ∈ S N −4 is a generator of a given unitary group.

Scalar field
Ψ ∼ |ψ 1 |a 1 a 2 | ψ 2 |, with the bra-ket configuration producing Lorentz vector and gauge group fundamental configurations. In this case, a ket contains right-handed and a bra left-handed spin-1/2 components (or vice versa), reproducing the mass term and Higgs quantum numbers.
with Γ S a an element of S N −4 .

Lagrangian formulation
Interactive Lagrangians [7] can be given in terms of vector, scalar and fermion fields conforming to the general structure of operator action as in Eq. 11 and the inner product in Eq. 10. For example, a gauge-invariant fermion-vector Lagrangian is given where Ψ is a fermion field as in Eq. (13), g is the coupling constant, M is an appropriate mass operator, and N f contains the normalization. In the next subsection, we address the spin model in 7+1 d in connection with the SM, and whose basis states will allow to write L F V , L SV , L SF in the next Sections.

(7+1)-dimensional model
We next make a brief description of resulting states in a (7+1)-dimensional spin space under a useful partition for the SM description, sketching the way to obtain it, and providing graphic description.
Since h is conformed of all simultaneously diagonalizable operators, it is convenient to recast this basis in terms of the projection operators which run along the diagonal in the matrix space ( Fig. 1).
the U(1) hypercharge generator and I 3 within the SU(2) weak isospin generators The charge operator is defined in the standard way by the Gell-Mann-Nishijima There are also flavor operators, forming the groups SU(2) f , SU(2)f , U(1) f , and U(1)f , and given by respectively for SU(2) f and SU (2)f , and for U(1) f , and U(1)f . The operators f 3 ,f 3 , f 0 andf 0 belong to h. In Fig. 1 the matrix space is represented schematically. The diagonal operators classify the states (off-diagonal) acting from the left for states in the same row, and from the right for states in the same column, which is consistent with matrix multiplication.
We also define a combination of diagonal flavor operators that further classifies states, given byF Baryon number zero, Higgs-like scalars

States
States contain scalars, fermions and vectors. Only the first two are considered in this Section. The matrix space admits two Higgs doublets φ 1 and φ 2 ( Table 1 and Fig. 2). They satisfy φ 1 =γ 5 φ 2 . Their connection to Hermitian and SU(2) conjugates is clarified in Section 4.3.
Non-Higgs scalars can also be constructed that contribute to the diagonalization of massive states. Ref. [9] provides further information on their nature and their application to obtain fermion properties.
The massless-fermion states satisfy the general structure of Eq. 12, and have massless quark quantum numbers, when classified by baryon number, isospin, and hypercharge. The matrix space admits four generations of quarks of different flavor ( Fig. 2), arranged in four SU(2) L doublets and eight right-handed singlets, shown in Tables 2, 3, respectively. After electroweak symmetry breaking, the Higgs generates a mass operator used in Section 5 to obtain fermion mass states.

Fermion Yukawa elements
Bilinear fermion terms can be constructed that produce scalar elements transforming quarks into their different combinations. We use the (7+1)-d space represented in Fig. 2, with particular and general properties that can be distinguished.
There are two matrix configurations: is contained in the Dirac projector with (α, β)-spin components and (positive or negative)-energy; the three P F αβ i are the same up to a phase; Q are U -or D-type fermions obtained from Tables 2, 3, defining F , the R, L case taken as an example, φ 0 1 defined in Table 1. The i, j imply we choose a 3-generation (arbitrary) projection to reproduce the SM; we also note that Q α RiQ β Lj = 0 for i = j.
On the other hand, defines the Yukawa basis (full flavor transition matrix) to be used in Section 6, for the complete scalar-fermion SM Lagrangian component. One can check that Y U ij , Y D ij are the same (up to phases), so they are commonly labelled Y F ij . The set R, L, α is arbitrary and other choices will reproduce (up to phases) the nine Y F ij terms. Indeed, although the (7+1)-d basis can accommodate four generations, the projection operator for, say, flavors 1,2,3 induces the 3-generation subset with 9 elements, As the set is closed under matrix multiplication, the 4th generation is discarded (see Section 6.) The resulting projection operators may be understood from the products of a fermion with matrix structure |spin flavor| and an hermitian conjugate one, resulting in the form |spin spin| for Eq. 32, and, inverting the order, |flavor flavor| in Eq. 34.

Fermion-vector Lagrangian: chiral basis in spin space
Concentrating on the heaviest fermions, the SM two-quark 4 electroweak interaction where the spin-1/2 fields consist of respectively; each term contains two polarizations as, e. g., ; ψ α qh (x) are wave functions 6 for quarks q = t, b, with spin components α = 1, 2, and chirality h = L, R; W a µ (x), a = 1, 2, 3, and B µ (x), are associated gauge-group weak and hypercharge vector bosons, with coupling constants, g, g , respectively; τ a are the Pauli matrices representing the SU(2) L generators.
The (7+1)-d space allows for a description of quark fields with hypercharges 1/3, 4/3, −2/3, respectively, and spinor components chosen in Table 4, given explicitly in Tables 2, 3; the quantum numbers λ are obtained from the operator structure [Op, Ψ] = λΨ for the weak component The SM Lagrangian L F V in Eq. 36 can be equivalently written 7 in this basis: as derived in Ref. [7], and examined in Ref. [17] while gauge and Lorentz symmetries can be checked with the above transformation rule, or given the equivalence to the traditional formulation. A projection operator P f that connects the two expressions [17] can be omitted by finding phases for Ψ, which translates into finding an adequate γ µ basis. The trace coefficient is usually 1, as the field normalization factor accounts for reducible representations. A complete proof of the equivalence is given in Appendix 1.
The W-fermion vertex in L F V , Eq. 38, contains the matrix element F |W i oµ |F , where the W contribution describes the SU(2) L inherently chiral action on fermion states |F , |F , as it carries the projection L 5 = 1 2 (1 −γ 5 ), predicted by the spin basis [9]; it is thus the natural 7 The commutator is omitted as the operator acts trivially on one side.
(a) hypercharge 1/3 left-handed doublet In the SM, the Higgs particle is present [1] in the SU(2) L ×U(1) Y gauge-invariant interacting Lagrangian-density component with , composed of two charged (upper), and two neutral (lower) fields.

which uses an antiunitary transformation
C expressing charge-conjugation invariance (in addition to the CP symmetry in the electroweak sector, and approximate SU(2) L ×SU(2) R symmetry [18]; a Hilbert space is assumed;) this is also a consequence of the SU(2) property that the conjugate representation is obtained from a similarity transformation, which ensures independence of the doublet choice. Appendix 2 shows that results. This generalizes the expression [19,20] for L SV in terms ofH 1 With the U(1) overall phase, a three-parameter subspace of the norm-conserving constraint |χ t | 2 +|χ b | 2 = 1 is generated. We associate this isometry with the L SV invariance under together with the combination Cτ 3 .
Further extension can be made for the scalars in the spin basis by attaching thẽ γ 5 operator. Using the projection operators in Eq. 14, L SV in Eq. 42 is generalized with the substitutions thus including spin degrees of freedom, leading to a combined spinor-electroweak de- scription. An intermediate expression that connects to the spin basis, and ultimately to Yukawa components, is obtained with the trace also over spin degrees of freedom, the second equality using hermitian conjugates, R 5 L 5 = L 5 R 5 = 0, and trace properties which lead to only two identical non-trivial terms. These forms will prove useful in comparing with Yukawa terms below.

L SV in (7+1)-d spin space
In the spin basis, the four-scalar doublet structure above is reproduced. Indeed, it emerges naturally in the (7+1)-d spin basis, with the Higgs potential not altered under different definitions (chiral ones or not.) Table 1 presents two of these scalar elements (with two additional as their conjugates.) Together with coordinate dependence, they and whose quantum numbers associate them to the Higgs doublet. These are unique within the (7+1)-d space [9]. Although new scalar fields are introduced in principle, here we concentrate on the SM-equivalent projections. Given the SM Higgs conjugate representationH(x) the scalar components are interpreted through the assignments (see Table 1), This leads to the equivalent expressions where we introduced H af (x) = aφ 1 (x) + f φ 2 (x), and the subindex sym means only symmetric γ µ γ ν components are taken, to avoid the Pauli components; and the ± index means the commutator and the anticommutator should be used for the temporal and spatial γ µ components, respectively. The equality for L SV implies that it accommodates SM parity-conserving scalar representations.
The complex parameters a, f , are constrained by the normalization rule |a| 2 +|f | 2 = 1.
These properties for L SV are shown explicitly in Appendix 2.

L SV mass components in conventional and (7+1)-d spin space
The spin representation can be connected with that ofH χtχ b (x) with the expression with φ i defined in Eq. 99, and this parameterization applies the unitary transforma- Under the Higgs mechanism, the SM scalars acquire where the normalized Higgs operator H n is defined, with the same 0, + component conventions as for the φ i , implying, as The vector-Higgs vertex in L SV determines the vector-boson masses, and within the spin basis, the trace is taken consistently with H n . Thus, the mass component, extracted from Eq. 50, taking for F the W, Z field terms, and for H ab its vacuum expectation value in Eq. 54, is produced. For the neutral massive vector boson, one derives the normalized Z µ (x) = (−gW 3 µ (x) + g B µ (x))/ g 2 + g 2 , and massless photon A µ (x) = (g W 3 µ (x) + gB µ (x))/ g 2 + g 2 , giving, e. g., the 0-component Similarly, for L SW m , the W i oµ basis in Eq. 39 emerges, and defines the masses of the charged boson fields W ± µ (x) = 1 ). Thus, the charged-vector boson component where m t and m b are the top and bottom masses, respectively, and the fermion fields Ψ are defined in Eq. 37. We note that the Higgs scalar components have the correct chiral action over fermions: under the projection operators in Eq. 14 L 5 , and R 5 , e. g., Eq. 60, the underlying mass operator is H m ( Examples of quark massive basis states are summarized on Table 5 (see Tables 2-4), for both u and d-type quarks, with their quantum numbers. Only one polarization and one flavor are shown, as a more thorough treatment of the fermion-flavor states are given elsewhere [9].
This results in, e. g., where H h m = H m + H † m , and T c1 M , B c1 M correspond to negative-energy solution states (and similarly for opposite spin components) and Eq. 62 justifies the m t and m b mass interpretation.
Under the assumption of a single mass-producing field operator, we match a reparameterized H n in Eq. 58 that gives the Z mass, to the fermion-mass term H m , in Eq. 62, resulting in √ 2H n =H m ; a multiplet structure is suggested. In other words, the operator identification derives from their mass eigenvalues, expressed schematically as | Z| √ 2H n |Z | 2 = m 2 Z and t|H m + H † m |t = m t , and the proportionality constant is derived accordingly. In this association, the simple real-field Z µ (x) nature justifies its use (similarly for each W i µ (x)), as opposed to the complex W ± µ (x). Similarly, Eq. 50 is chosen over Eq. 51, as the latter adds the Higgs conjugate representation, unlike the SM. Thus, the vacuum expectation value reproduces the parameterization in Eq. 54, and identifies χ t , χ b as Yukawa parameters: The same argument can be made using the second scalar form in Eq. 51, as it also leads to Eq. 58. This results in Using Eq. 55, we obtain the relation for the t, b quark masses The commutator arrangement in Eq. 57 is used in the above comparison; as it is set on the demand of a normalized scalar, the argument strengthens on the use of

Extended quark-mass relation
We place the heavy-quark mass relation in Eq. 65 in the larger SM context, and argue for a plausible generalization for all quarks, based on it. For these purposes, we first derive some SM field properties using the spin basis, assuming they can be also derived within the conventional SM basis, given their equivalent application. Needless to say, we demand consistency with the SM, and with experiment. At the Section's end we identify some underlaying general assumptions.
Thus, we concentrate on the SM three-generation subset of the (7+1)-d model [9], as can be effected by the Yukawa operators in Eq. 34. Eq. 65 uses that the same single-scalar operator acts on the fermions and the vector bosons: such an operator is reproduced in the SV and SF terms, as the SV term admits a basis expression that applies the associated C-symmetries in Section 4. This connection implies the equivalent expression that can be read from the Appendices, which shows separation of quark i =u-and d-type L SV i components, depending on scalars, and no mixing among them. We focus on the mass-generating scalar elements corresponding to the neutral H 0 t , H 0 b , from Eqs. 53, 54, and their hermitian conjugates. As mass relations are considered, we assume fields after the Higgs mechanism is applied.
In particular, a connection emerges between the normalized bilinear Higgs term that gives masses to the vector bosons, as the Z mass in Eq. 58, and the fermions.
where H m is defined in Eq. 61, T α R , T α L , B α R , B α L , are quarks at rest, defined in Table  4, Eq. 67 can also be understood from the substitutions in trH m † H m with terms extracted from L SF in Eq. 60, using the trace permutation property. The identity in each substitution provides the link to the t, b Yukawa constants for the q = tb doublet, t, b singlet cases. The arguments leading to the mass relation in Eq.
65 imply y U qt = χ t , y D bq = χ b , as given in Eq. 63, namely, a diagonal mass basis is assumed.
Since one can pick any fermion generation on Tables 2, 3, the interpretation of the χ t , χ b coefficients as Yukawa constants within the SV term leads to a generalization to other families and non-diagonal Yukawa elements. We now consider the extension of L SF in Eqs. 60 and 64 with a fermion expansion that uses all Yukawa coefficients, where the Yukawa operators Y F from Eq. 34 are necessary to connect the u-and d-type quark fields defined in Eq. 37, and y U qi , y D qj are Yukawa coefficients, with the up, down, charm and strange quarks, also included, relabelling singlets i = u, c, t, j = d, s, b, and doublets q = ud, cs, tb.
The allowed Yukawa terms, diagonal and mixed, can be included using all combinations of a 3-generation set of normalized fermions on Table 2, where a projection operator as in Eq. 35 is applied. We evaluate the trace of bilinear q L † terms with L SV components, extending Eq. 67, producing which may be also obtained by the substitution of the associated scalar coefficients in bilinear neutral Higgs terms trH m The correspondence of L SF in Eqs. 60 and 64 to L SF T in Eq. 69 induces the sum of square mass-matrix elements in Eq. 70, which is equal (given the property trM † M = trM † M , M a matrix, M its diagonal form) to the sum over the square masses, A generalization with such a sum is induced, similar to relation Eq. 65, with the Higgs normalization condition, Eq. 55. Since Eq. 70 maintains the same structure as Eq. 67, following the generalization of L SF to L SF T , Implicitly, we used the SV -fermion symmetry, namely, no fermion preference. With today's uncertainties in the quark-mass values, this relation is phenomenologically consistent with Eq. 65, as the same maximal hierarchy or quark b-mass input argument follows, and the rest of the quarks have comparably negligible masses. As this relation is independent of the mass diagonalization matrix, it is also of the CKM matrix [22].
This implies that no operator will connect the initial fermions outside the 3 generations. So is the case for the 3-generation extension of L F V in Eq. 38, requiring a sum over the (electroweak) flavors. We conclude the 3-generation spin-basis projection consistently describes the SM.
By construction, Eqs. 68, 71 imply masses represent O(m q /m t ) corrections. This is also the order of the Hamiltonian needed to obtain the other fermion masses. More assumptions are necessary to get further information on masses, and CKM matrix elements. For example, hierarchy arguments on the masses' order of magnitude difference were derived [9] that explain how the associated W,Z,t,b, large scale mostly cancels for the other fermions at the vertical level (within a doublet) and horizontal level (between families). This leads to a consistent description in which such mechanisms coexist with the Higgs-generated one. While we produce above further consistency arguments for their parameters, more stringent constraints from the (7+1)-d will be tested elsewhere. Other arguments leading to hierarchy exist as textures [23].
We conclude Yukawa coefficients, contained in rest fermions as a device, connect to bilinear scalar combinations containing mass-generating Higgs terms in L SV , keeping the Lorentz or gauge structure of SV unmodified, and ultimately consistently with the SM. We show above L SF T in Eq. 69 induces a generalized sum rule for the square quark masses in Eq. 73. The latter is a plausible extension of Eq. 65, based on a subset of L SV terms, after the Higgs mechanism. The same type of argument can be made for leptons, but given their smaller masses, their influence will be lesser, while similar conditions as in Eq. 67 will also lead to PNRS matrix [24] independence.

Conclusions and Outlook
In summary, the formalism used places fields on a basis that simultaneously contains SM bosons and fermions. SV and SF terms are linked through the mass rendering of the scalar operator within them, using the electroweak SV vertex independence of its components acting on different fermion-doublet elements, implicitly expected, but which we now expose. Supporting a SM prediction of a unique scalar, input from the normalized scalar-vector vertex, and the mass-parameter interpretation in the SF vertex, relates v and m t , cf Eq. 65, the main result in the paper. The same relation can be argued by considering the scalar operator's matrix rank, or assuming normalized Yukawa components. Based on chiral properties, the same Higgs-operator rule, and a correspondence between fermion-boson inner products and Yukawa terms, a plausible extended sum rule for the fermion square masses is proposed, given in Eq. 73. Both relations are consistent with the SM, given today's particle-mass uncertainties. We conclude the spin basis is a useful platform to obtain, within the SM, the quark-mass electroweak relations.
The central argument input can be also read when V terms in L F V , attached with the projector L 5 in Eq. 14, are carried into the intermediate L SV chiral version in Eq. 46 and, after the 1/2 factor cancellation in its mass component, relate to F terms in the Yukawa L SF . The spin-basis gives it further support as it classifies discrete degrees and produces SM features. Thus, the matrix space restricts representations, in turn, exhausting the space; electroweak V fields belong to the adjoint, and S, F fields to the fundamental representations. Additionally, the chiral property in the F V electroweak term, associated to V , translates naturally to the SV interaction components. Normalized fields define the Lagrangian terms, setting the trace coefficient, and the stage for the L SV , L SF comparison. In the spin-basis context, the S field's chiral property is nominal, but consistent, as L SV contains the L 5 projector from V , and within L SF , S acts on chiral fermion components.
The scalar operator acting on vectors and fermions links their matrix elements, connecting parameters. The particles' simultaneous participation in mass generation through the Higgs mechanism and related SM vertices, with assigned representations, implies a description with common dynamics, and at a given energy scale, already at the classical level, and suggests fields belong in a multiplet, supporting a commonorigin unification assumption [7].
It follows that the arguments provide a geometric approach to address problems as the electroweak-symmetry breaking origin. The formalism facilitates the fields' composite description, as boson degrees of freedom may be written in terms of two fermions'. Expansions in such fields may be useful, independently of whether compositeness is physical or only a device.
Naturalness is hinted at in the φ 1 , φ 2 associated single scale, which produces a hierarchy effect [7]. Thus, while this symmetry-breaking effect applies for heavy-quark masses, it could be valid also horizontally between generations in accordance with the fermions' low masses. While here we considered the top-quark mass, the other fermions, besides the b-quark, may be included in this scheme, namely, considering bilinear fermion components for scalar particles, but they will have little influence on this result, as their SF interaction is proportional to their masses.
As the spin basis connects the vector and quark sectors, constraints may be derived for SM extensions as supersymmetry [25], composite models that require dynamical symmetry breaking [14] as technicolor [26] or, in an extension of such models, top and bottom quarks [27] that conform condensate-producing massive particles.
Besides the fields' spin representation connecting the scalar operator in two vertices, it highlights chiral components of particles and interactions that maintain their SM equivalence. Indeed, we showed two such valid chiral and non-chiral scalar bases for the SV Lagrangian. This freedom could be clarified in other vertices, as with a SM extension with additional scalar degrees of freedom, whereas in this paper, we considered only their SM projection.
then, the (5 + 1)-d representation and finally, the (7 + 1)-d representation The commuting property of the Lorentz and scalar symmetry operators implies that they can be represented as a tensor product. To compare with the spin-space basis, we write the conventional-basis generators as tensor products, choosing the (7+1)d space to represent them; thus the spin-1/2 and SU(2) L terms, expressed by the 4×4 Clifford basis, and Pauli matrices, respectively, generalize to, e.g., τ 3 1 s2×2 ∼ I 3 (actually, it exchanges γ 1 and γ 3 .) and |L1T is represented, after the unitary transformation U † D , by (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, i, 0). Next, we write all the conventional-basis states in this basis, and their association to the spin-extended basis states, with corresponding quantum numbers (notation used in Table 4 where the spin-basis states are shown in extenso in Tables 2 and 3. For the fermion wave functions ψ α qh (x), we use polar coordinates, where the conventional and spin terms contain, respectively, ψ α qh (x) exp [ip α qh (x)] ↔ ψ α qh (x) exp [ic α qh (x)], for quarks q = t, b, with spin components α = 1, 2, and chirality h = L, R. The magnitude part can be shown to be the same for both cases, as can be derived by comparing, e. g., the mass term. The vectors W a µ (x), B µ (x) are real fields.
The phases appear in each term in both bases. For example, for the conventional basis and for the two polarizations t L ( within the left-handed hyper- , with U D applied to transform back from the Dirac representation, and we used the terms in Eqs. (4) and (5) in this Appendix; for the spin basis, Ψ tL (x) = ψ 1 The Lagrangians' identity is shown, by checking that the same terms are reproduced in both bases, and finding independent constant phases that connect the two representations. In the following, we present the fermion-vector L F V Lagrangian components: interactive (weak and hypercharge), kinetic; also the fermion-scalar (Yukawa) L SF Lagrangian. The subtitle contains the two-basis Lagrangian expressions in a concise notation, and then one component is given in an expanded form; the equations that link the phases in the two representations are written as they derive from the terms.
Comparing the corresponding expression in the spin basis, we derive the following phase relations (which retroactively provide such an expression). (86) for arbitrary real constants p W 1 , p W 2 , p W 3 , p W 4 , p Z1 , p Z2 , requiring the identities Using the fields' integrability property (belonging to Hilbert space), integration by parts has been applied to make the derivative substitution i∂ µ → i 1 The representation of scalars in the conventional and spin bases uses the association, e. g., Hγ 0 4×4 → H t ; the conventional phases, written explicitly in Appendix 2, are set to fit the spin basis, as both operators act equally on fermions, and we applied the gamma-matrix representation freedom of choice.

Higgs invariance
For the scalar components, we also use expressions in polar coordinates, and in which the phase is written explicitly, to see its workings. Thus, for the conventional basis, where p 1 b , p 1 t , p 0 b , p 0 t are charged and neutral phases, respectively, andH χt,χ b (x) = (χ t H(x), χ bH (x)) is a 4 × 4 matrix, χ t , χ b , can be assumed real and their dependence in all terms is through the factor χ 2 t + χ 2 b , so their explicit form constitutes a likewise demonstration for L SV .
For the spin basis, we use a generalized expression for the scalar term with conjugated terms weighted by a multiplicative parameter λ, to keep track of terms, and with a normalization that makes L SV λ-independent: where φ + 1,2 , φ 0 1,2 are defined in Table 1, and φ 1 t , φ 1 b , φ 0 t , φ 0 b are charged and neutral phases, respectively, and those with λ correspond to the hermitian-conjugate function (see Eqs. (50), (51) in the paper.) Given the chiral nature of the scalar components, they do not mix with their hermitian-conjugate components.
Each of the L SV terms is indeed proportional to the combination χ 2 b + χ 2 t , which manifests the t-b symmetry of this component, as the phases that connect the two representations were obtained.
We thus completed the demonstration of the SM Lagrangian terms' equivalence in two bases; we conclude the spin-space representation reproduces the same properties of SM generators.