[Paper Review] Unlocking the Standard Model. I. 1 generation of quarks. Symmetries

B. Machet|arXiv (Cornell University)|Jun 11, 2012

Particle physics theoretical and experimental studies2 references3 citations

TL;DR

This paper proposes a minimal two-Higgs-doublet extension of the Glashow-Salam-Weinberg model for one quark generation, where the two Higgs doublets are parity-transformed copies of each other and isomorphic to the Standard Model Higgs. The chiral group $U(2)_L \times U(2)_R$ is spontaneously broken to $U(1) \times U(1)_{\text{em}}$, with the diagonal $U(1)$ linked to parity, and the Higgs fields are shown to be in one-to-one correspondence with bilinear quark-antiquark operators, unifying dynamical symmetry breaking with a two-Higgs-doublet structure.

ABSTRACT

A very specific two-Higgs-doublet extension of the Glashow-Salam-Weinberg model for one generation of quarks is advocated for, in which the two doublets are parity transformed of each other and both isomorphic to the Higgs doublet of the Standard Model. The chiral group U(2)_L X U(2)_R gets broken down to U(1) X U(1)_{em}. In there, the first diagonal U(1) is directly connected to parity through the U(1)_LX U(1)_R algebra. Both chiral and weak symmetry breaking can be accounted for, together with their relevant degrees of freedom. The two Higgs doublets are demonstrated to be in one-to-one correspondence with bilinear quark operators.

Motivation & Objective

• To resolve the inability of the standard Glashow-Salam-Weinberg model to simultaneously account for chiral and weak symmetry breaking.
• To propose a minimal extension of the Standard Model using two Higgs doublets that restores parity to a fundamental role.
• To establish a one-to-one correspondence between the two Higgs doublets and bilinear quark-antiquark operators, linking dynamical symmetry breaking to the Higgs mechanism.
• To demonstrate that the $U(2)_L \times U(2)_R$ chiral symmetry breaking pattern naturally leads to both Goldstone bosons and a physical Higgs boson, with the diagonal $U(1)$ connected to parity.

Proposed method

• Introduce a two-Higgs-doublet model where the two doublets are parity-transformed versions of each other and both isomorphic to the Standard Model Higgs doublet.
• Use the $U(2)_L \times U(2)_R$ chiral group as the global symmetry, with $SU(2)_L$ embedded as the weak gauge group.
• Define the Higgs doublets $H$ and $K$ such that their transformation laws under $T^i_L$ and $T^i_R$ match those of the Pauli matrices, leading to the key transformation rule (14).
• Establish a correspondence between the Higgs components and bilinear quark operators via the isomorphism $\mathfrak{K} \sim \bar{d}\gamma_5 u$, $\mathfrak{H} \sim \bar{d}u$, with vacuum expectation values $\langle \bar{u}u + \bar{d}d \rangle = \mu^3$, $\langle \bar{u}u - \bar{d}d \rangle = \nu^3$.
• Show that the diagonal $U(1)$ subgroup of $U(2)_L \times U(2)_R$ is tied to parity through the algebraic structure of $I_L$ and $I_R$, and that $U(1)_{\text{em}}$ is the electromagnetic symmetry.
• Derive the transformation rules for the Higgs components in terms of the $h^i$ fields (eq. 14), and extend them to the composite scalar and pseudoscalar operators via the $\mathfrak{K}$ and $\mathfrak{H}$ multiplets (eqs. 68 and 78).

Experimental results

Research questions

• RQ1 Can a two-Higgs-doublet model restore parity as a fundamental symmetry while preserving the structure of the Standard Model's Higgs mechanism?
• RQ2 How can both chiral and weak symmetry breaking be simultaneously accounted for within a minimal extension of the Glashow-Salam-Weinberg model?
• RQ3 Is there a one-to-one correspondence between the eight components of two Higgs doublets and the eight bilinear quark-antiquark operators in a one-generation quark system?
• RQ4 What is the role of the diagonal $U(1)$ subgroup of $U(2)_L \times U(2)_R$ in connecting parity to the electroweak symmetry breaking pattern?
• RQ5 How do the vacuum expectation values of quark bilinears $\langle \bar{u}u + \bar{d}d \rangle$ and $\langle \bar{u}u - \bar{d}d \rangle$ serve as catalysts for symmetry breaking?

Key findings

• The two Higgs doublets $H$ and $K$ are shown to be in one-to-one correspondence with the bilinear quark operators $\bar{d}\gamma_5 u$ and $\bar{d}u$, respectively, via the isomorphism established in equations (68) and (78).
• The vacuum expectation value $\langle \bar{u}u + \bar{d}d \rangle = \mu^3$ is identified as the scale for the first Higgs doublet, while $\langle \bar{u}u - \bar{d}d \rangle = \nu^3$ sets the scale for the second, with $\mu^3$ and $\nu^3$ playing distinct roles in symmetry breaking.
• The diagonal $U(1)$ subgroup of $U(2)_L \times U(2)_R$ is directly linked to parity through the algebraic action of $I_L$ and $I_R$, as shown in equation (61), making parity a fundamental symmetry in the model.
• The eight components of the two Higgs doublets are fully accounted for by the eight independent scalar and pseudoscalar bilinear quark operators, with the Higgs bosons $h^0$ and $s^0$ corresponding to $\bar{u}u + \bar{d}d$ and $\bar{u}u - \bar{d}d$, respectively, as per equation (82).
• The breaking of $U(2)_L \times U(2)_R$ down to $U(1) \times U(1)_{\text{em}}$ produces six Goldstone bosons: three from $SU(2)_L \times SU(2)_R$ breaking and three from the chiral symmetry breaking, all traced to the $U(1)_L \times U(1)_R$ structure.
• The model successfully unifies two-Higgs-doublet phenomenology with dynamical symmetry breaking, providing a natural framework where the Higgs fields emerge from quark condensates without introducing new fundamental scalars.