Heavy-quark mass relation from a standard-model boson operator representation in terms of fermions

The standard model[1] (SM) is pivotal in obtaining information about elementary particles, yet theoretical puzzles remain as the connection between the electroweak, Higgs and Yukawa sectors, and the origin of the electroweak breaking. New connections may lead to understanding of SM dimension and dimensionless variables that are fixed phenomenologically today.

The SM’s properties provide hints to address these puzzles. Quantum numbers of SM spin-1/2 and vector particles are related, as these belong to the fundamental and the adjoint representations, respectively, of the gauge and Lorentz symmetry groups; to implement such symmetries, the vector bosons and the Higgs scalar have charge and spin of specific pairs of SM fermions.

One productive approach to obtain new information on the SM relies on these patterns. Ref. [2] rewrites the SM Lagrangian equivalently in terms of a matrix basis, for all particles (scalar, vector, spin 1/2). It derives a mass relation (classically) for the quark masses, under a weak SM assumption of a common scalar operator between the electroweak sectors, using a discrete equivalent basis induced by a SM extension; spin and gauge degrees of freedom are separated, and the bosons’ degrees of freedom are formally composite of the fermions. This suggests the possibility of a SM spin-1/2 basis and associated quantum numbers, under the weak assumption of truncation that can be tested.

On the other hand, SM extensions with a bottom-up approach, generalizing SM aspects, have led to restrictions on these parameters, beyond the insight of an encompassing theory. For example, grand-unified theories[3] assume a common group for the interactions, requiring a unique coupling constant at the unification scale, constraining the SM couplings.

The discrete composite SM quantum numbers suggest such a description of the elementary particles, and hint at generalization directions. Compositeness is a central tenet behind many physical systems, in which features are explained in terms of simpler elements. It means a system’s configurations are divided into two classes, elementary and composite, where the latter’s degrees of freedom are constructed in terms of the former’s, and observables manifest these relations. In the case of the quark model[4], hadron masses are explained in terms of constituent-quark masses. In superconductivity, Cooper pairs[5], produced by a residual phonon interaction between two electrons, conform the relevant free-streaming variable.

Thus, in the BCS theory of superconductivity[6] the Cooper-pair energy gap depends on interactions. In an application of this theory to quantum field theory and elementary particles[7], a four-fermion interaction produces fermion and composite-boson masses, linking their values.

Based on this type of interaction, applying dynamic mass generation, models were researched[8] that produce a Higgs-scalar condensate[9], composed of top and antitop quarks. These models comprise, among others, an extension to vector particles[10], applying interaction resummation[11], with consideration of a chiral condensate[12].

In finding the appropriate description of a composite system, the focus is on effective degrees of freedom, concentrating the relevant information. In quantum field theory, fundamental variables as coupling constants and masses run, depending on the energy scale, as renormalization equations take into account interactions.

Once these properties are integrated, a composite configuration has a simple description in terms of the elementary ones. Compositeness may be present at a fundamental level or partially, depending on whether it involves all or selected number of the relevant degrees of freedom. SM compositeness features may prove useful to address the SM problem that the fermion sector remains disconnected from the boson elements, and so the masses that they generate, which arise from independent Lagrangian terms.

In this paper, we derive a mass relation for the heavy quarks in terms of electroweak parameters, and a hierarchy constraint, relying on SM compositeness, using the common mass-generating Higgs component, in a quantization framework. Concentrating on SM heavy particles, boson degrees of freedom can be expressed in terms of the fermions’, considering spin and isospin. From the electroweak and Yukawa Lagrangian components, the vector Z and W masses are reproduced. The material is organized thus: in Section II, mass-generating operators are extracted from the electroweak Lagrangian. Section III relates the paper scheme to composite top quark anti quark Higgs condensate models. In Section IV, such operators are written in a second-quantized fermion basis; these are used to calculate particle masses and derive the quark hierarchy constraint, in Section V. Conclusions are drawn in Section VI.

The SM assumes a Higgs doublet

parameterized in terms of $\mbox{\boldmath$\theta$\unboldmath}(x$) phase fields, with Pauli-matrix generators $\tau$, and the vacuum expectation value (vev) ${\bf v}=\frac{v}{\sqrt{2}}\left(\begin{array}[]{r}0\\ 1\end{array}\right)$; through the Higgs mechanism, an expansion around the scalar potential minimum produces $h(x)$ as the remaining only physical field, which minimizes the potential. In the unitary gauge, the $\mbox{\boldmath$\theta$\unboldmath}(x$) are absorbed into the vectors’ longitudinal components, and the exponential dependence is eliminated.

Next, we write the scalar-vector (SV) Lagrangian

where ${W^{i}}_{\mu}(x)$, $i=1,2,3,$ are weak vector components, ${B}_{\mu}(x)$ is the hypercharge vector term, and $g$, $g^{\prime}$ are the respective coupling constants. In the Higgs mechanism, ${\bf H}(x)$ acquires a vev $\langle{\bf H}(x)\rangle={\bf v}$, producing the SM mass-generating constant component, contained in ${\cal L}_{SV}$:

which provides masses to the vectors. The fields’ quadratic form is resolved in

where the weak isospin generators are $\frac{1}{2}{\mbox{\boldmath$\tau$\unboldmath}}={\mbox{\boldmath${\it I}$\unboldmath}}$, with $I^{\pm}=I^{1}\pm iI^{2}$, and $Y$ is the hypercharge operator. This form is expressed in terms of the charged ${W^{\pm}}_{\mu}(x)=\frac{1}{\sqrt{2}}[W^{1}_{\mu}(x)\mp iW^{2}_{\mu}(x)]$, and neutral components that decouple in the above quadratic form:

where the photon field $A_{\mu}(x)$ remains massless: ${g}g^{\prime}(I^{3}+\frac{1}{2}Y)A_{\mu}(x){\bf v}=0,$ whereas the obtained masses for the ${\rm Z}$ and ${\rm W}^{\pm}$ particles are

We focus on the heavy-quark chiral-(t,b) field components with left-handed doublet

and right-handed singlets $\psi_{tR}(x)=P_{R}\psi_{t}(x)$, $\psi_{bR}(x)=P_{R}\psi_{b}(x)$, with the projectors $P_{R}=\frac{1}{2}(1+\gamma_{5}),$ $P_{L}=\frac{1}{2}(1-\gamma_{5}).$ The scalar-fermion (Yukawa) interaction component is

where $\chi_{t}$, $\chi_{b}$ are Yukawa constants, and

The extracted mass-generating term from ${\cal L}_{SF}$, after the Higgs mechanism, is

As suggested by the Lagrangian components in Eqs. 9, 10, 16, the fields may be written in terms of a fermion basis, also indicated by the fermion-vector interaction term. Recalling the SM quantum-number compositeness property, from gauge invariance, bosons are written in terms of fermions, as some vector-field zero components constitute conserved terms. We concentrate on relevant discrete degrees of freedom: spin and isospin, in their chiral components, contributing to the masses, as implied by Eqs. 9, 10, 16. The vector field has the form[13] $A_{\mu}=g_{\mu\nu}A^{\nu}=\frac{1}{4}{\rm tr}\gamma_{\mu}\gamma_{\nu}A^{\nu}$ in a spinor basis. The SM Lagrangian is reformulated in a spin-isospin basis, in a similar procedure to the spin-extended model[2]. The vector-scalar component leading to the vectors’ masses is, from Eq. II,

where the trace here is only over $4\times 4$ gamma-matrix indices, and the equality uses

The inclusion of the $P_{R}$, $P_{L}$ projections shows the freedom in the representation choice, where the independence of symmetry-operator spaces (spin, isospin) is manifest, and the tensor product of spaces applies for its generators. Eqs. 18, 19 signal a new basis that separates Lorentz and gauge degrees of freedom, satisfying the Coleman-Mandula theorem[14], similarly but independently of a proposed SM extension, in Ref. [2]. With hindsight, the scalar doublet ${\bf H}(x)$, is written in the spin basis (using $\gamma_{0}$) while a full quantum version in second quantization is provided in Section IV.

Such a formulation introduces an equivalent basis with creation and annihilation operators incorporating spin and scalar (weak isospin, flavor) components as relevant degrees of freedom, and their vev. Next, we consider SM limits and composite extensions for which the expansion in this paper is relevant.

To face SM puzzles, as the origin of the electroweak breaking or the connection of the electroweak and Yukawa sectors, relevant limiting-environments are described for this work; in turn, its calculation leading to the SM vector masses complements these methods.

We use the 8-element massive basis for the t quark (and the b), conformed of a tensor product of the spaces isospin, momentum, spin, $q_{ks}=a_{q}a_{k}a_{s}$. We concentrate on the top and bottom quarks $q=t,b$, the up and down spin polarization, $i=\uparrow,\downarrow$, as these constitute the most massive fermions, with masses of the order of the SM massive bosons. As we deal with a QCD-scalar SM sector, any quark color is understood. The momentum (or space) degrees of freedom are factored out; for definiteness, among the new components, we choose the lowest-energy fermion operator, the massive mode, with ${\bf k}\rightarrow 0,$ as any mode is representative, given the Lorentz and gauge invariance. Such objects satisfy the anticommutation relations

with Kronecker deltas understood for defined indices. Quarks $q_{i}^{\dagger}$ have the same quantum numbers as antiquark operators $\bar{q}_{i}$, with antiquarks given by $\bar{q}_{i}^{\dagger}$. Action on the vacuum is $q_{i}|0\rangle=0$, $\bar{q}_{i}|0\rangle=0$. The normalization is set for the massive states, $\langle 0|q_{i}q_{i}^{\dagger}|0\rangle=\langle 0|\bar{q}_{i}\bar{q}_{i}^{\dagger}|0\rangle=1$.

Given the chirality’s admixture of positive and negative massive frequencies, these operators can be broken into their right and left components and are represented¹¹1Strictly speaking, the massive ${\bf k\rightarrow 0}$ quark component is obtained from the off-shell massless operators; their ${\bf k}$, ${-\bf k}$ redundancy is eliminated in such a limit. For example, $t_{R\uparrow}^{\dagger}({\bf k})$ corresponds to $\bar{t}_{L\downarrow}({-\bf k})$. by

and the antiquark annihilation operator, the orthogonal combination

These relations imply for the quiral $L,R$ quark-basis components, using Eq. 32, the anticommutation relations $\{q_{Qi}^{\dagger},q^{\prime}_{Q^{\prime}j}\}=\delta_{ij}\delta_{QQ^{\prime}}\delta_{qq^{\prime}}$, and $Q,Q^{\prime}=R,L$, and for anticommutator of creation-creation and annihilation-annihilation operators, zero.

On Table 1 we describe these fermions written in terms of right-handed ($R$) and left-handed components ($L$), with their quantum numbers whose operators are presented next. The relevant SM bosons and conserved quantities are bilinear components that can be written in this basis; we assume they are formulated after the Higgs mechanism. One-body operators are constructed from matrix elements in a generic basis $|i\rangle$, and associated operators $a_{i}$, as $Op\rightarrow\sum_{ij}\langle i|Op|j\rangle a_{i}^{\dagger}a_{j}$. We write the operators that define these states and also the SM vertices; some of these operators constitute symmetry generators obtained from conserved charges, and we list relevant ones:

For the scalar isospin and hypercharge, the same separation can be made, where the sequence in Eq. LABEL:SVmass contains the adjoint-representation fields $W_{\mu}^{i}$, and the field $B_{\mu}$. Starting with the latter’s associated hypercharge operator, we include a flavor space for the t, b quark pair:

where $P_{t_{L}}=\sum_{i}|t_{L}i\rangle\langle t_{L}i|$ a neutral component that sums over the spin elements generated by the $\gamma_{\mu}$ in Eq. LABEL:SVmass. In its second quantized form,

Other operators are given in such a form. The SU(2)_L generators $I^{i}$:

satisfying $[I^{3},I^{+}]=I^{+},$ $[I^{3},I^{-}]=-I^{-},$ $[I^{+},I^{-}]=2I^{3}$, $[Y,I^{\pm,3}]=0;$ the baryon number

and spin component along $\hat{z}$

satisfying $[S_{z},B]=0,$ $[S_{z},Y]=0,$ $[B,Y]=0$, $[S_{z},I^{\pm,3}]=0,$ $[Y,I^{\pm,3}]=0,$ $[B,I^{\pm,3}]=0$.

Composite states may also be constructed that generalize Cooper pairs in superconductivity. In particular, bispinor operators describe bosons, except for combinations of the form $q_{iL}^{\dagger}q_{iR}^{\dagger}$ or $q_{iL}q_{iR},$ which cannot form a state, as each term in the pair requires the same creation operator, so that the non-vanishing operator acting on $|0\rangle$ is squared, based on the massive-state normalization of $q_{i}^{\dagger}|0\rangle$, $\bar{q}_{i}^{\dagger}|0\rangle$.

Similarly, the neutral vector

has quantum numbers as in Table 1. $W^{3}_{z}$ is derived from terms of the form $\tau^{3}W_{\mu}^{3}(x)$ in Eq. LABEL:SVmass, shares with $I^{3}$ weak isospin quantum numbers, and has also spin 1 under the Lorentz group. Regarding the freedom in defining boson operators as $I^{i}$ and $W_{z}^{3}$, we note we choose them so as to reproduce the fermion coupling, and in particular, the chirality property. Other components can be constructed, e. g., by applying step operators; these are presented in the Appendix.

The Higgs field is constructed so that gauge invariance is satisfied, with quantum numbers common for the t and b field components:

with isospin-hypercharge $i^{3}=-1/2$, $y=1$, also manifest in Eq. 15. A similarity transformation can be used to obtain the component:

with $i^{3}=1/2$, $y=-1$.

The SM assumes conventionally a classical procedure for the vev; it assumes action on fields is through a multiplicative constant. Here we extend such an approach in that relevant degrees of freedom are considered in the expectation value, departing from the same field arrangement. The main purpose of this calculation is to write SM vertices in this basis, and to relate the vector-boson masses in the scalar-vector (SV) vertex, and fermion masses in the scalar-fermion (SF) Yukawa term. Vector operators are constructed so as to match vertices, fermion chirality and coupling.

Thus, the electroweak vertex operator in Eq. LABEL:SVmass that defines the mass-component Hamiltonian squared contains the scalar

where $\chi_{t},$ $\chi_{b}$ are parameters, and

are associated to $H_{ot}$, $H_{ob}$ in Eqs. 43, 44, respectively. The freedom choice of $\chi_{t},$ $\chi_{b}$ reflects an extended parameter space in the scalar-vector term implying a symmetry is present, akin to custodial symmetry[2].

Using the second-quantized fermion basis, we calculate the SM-vector masses. From Eqs. 33, 34, we find the conditions

(see Appendix).

This paper deals with the low-energy regime in which the scalar Higgs field acquires a vev, generating mass upon itself and the other fields. Boson fields are written in a composite description, with a chiral fundamental-representation fermion basis producing vectors in the adjoint one, and reproducing the SM⁴⁴4By construction, the Higgs particle remains in the weak doublet representation.. Their associated SM quantum numbers support such connections.

Effectively, we focus on the scalar-vector and scalar-fermion vertex mass components, providing the vector and fermion masses. Within such vertices, the field’s spatial component is factored out, so we concentrate on discrete spin-isospin degrees of freedom, relevant in mass generation. A common mass-giving scalar element appears in each vertex that identifies the scalar field in the two expressions. The second-quantized SM W and Z masses in Eqs. 50, 53, respectively, are reproduced under the normalization condition 51, which restricts quark masses, with Eq. 59 as particular case, from shared quantum numbers of scalar generators with $\chi_{b}$, $\chi_{t}$ parameters; under a hierarchy condition, $m_{t}\sim v/\sqrt{2}$ is reproduced. The vev normalization condition in Eq. 55 implies Eq. 51; demanding the classical normalization in Eq. 63, one derives general hierarchy condition for $m_{t}$ and $m_{b}$, as shown in Fig. 1. Based on current values of $m_{t}=172-176$ GeV, extracted kinetically or through pole methods[22], this relation is satisfied with a .6% accuracy.

In comparison, Ref. [2] constructs an equivalent spin basis, departing from a SM extension, within a classical description; it also digresses on applying the same scheme to the lower-mass quarks, although they are irrelevant at $v$ scales. Consistency with the W, Z masses imposes a normalization for Ref. [2] for the arbitrary Yukawa parameters $\chi_{b}$, $\chi_{t}$ with the freedom of choice within a symmetry.