Chapter

24 Models with Vector-Like Quarks

https://doi.org/10.1093/oso/9780198503996.003.0024

Pages

276–292

Published:

July 1999

Abstract

There are many extensions of the standard model in which fermions with nonstandard SU(2)©U(l) quantum numbers-’exotic fermions’-naturally occur.The 27 contains, apart from the fifteen usual chiral fields in the first line of eqn (24.1)-the doublet of quarks, the up antiquark, the down antiquark, the doublet of leptons, and the positron-, a vector-like isosinglet quark of charge In this chapter we concentrate our attention on vector-like isosinglet quarks. Our interest in extensions of the SM with those exotic fermions is justified for the following reasons: They provide a framework for having a naturally small violation of the 3 x 3 unitarity of the CKM matrix. This leads to non-vanishing, but naturally suppressed, flavour-changing neutral currents (FCNC). The presence of FCNC opens up many interesting possibilities for rare Kand B decays, as well as for CP asymmetries in neutral-B decays.

24.1 Motivation

There are many extensions of the standard model in which fermions with non-standard $SU (2) \otimes U (1)$ quantum numbers—‘exotic fermions’—naturally occur.⁶² For example, in the grand unified theory based on the Lie algebra $E_{6}$ ⁠, each family of left-handed fermions is in the 27 representation of that algebra. If we consider the $SU {(3)}_{c} \otimes SU (2) \otimes U (1)$ subgroup of $E_{6}$ ⁠, that representation has the following branching rule:

\begin{array}{l} \underline{27} = (3, 2)_{1 / 6} + {(\bar{3}, 1)}_{- 2 / 3} + {(\bar{3}, 1)}_{1 / 3} + {(1, 2)}_{- 1 / 2} + {(1, 1)}_{1} \\ + {(3, 2)}_{- 1 / 3} + {(\bar{3}, 1)}_{1 / 3} + {(1, 2)}_{1 / 2} + {(1, 2)}_{- 1 / 2} + {(1, 1)}_{0} + {(1, 1)}_{0} . \end{array}

(24.1)

The 27 contains, apart from the fifteen usual chiral fields in the first line of eqn (24.1)—the doublet of quarks, the up antiquark, the down antiquark, the doublet of leptons, and the positron—, a vector-like isosinglet quark of charge $- 1 / 3$ ⁠, a vector-like isodoublet of leptons, and two isosinglet neutrinos. Here, we denote by ‘vector-like fermions’ those fermions whose left-handed and right-handed components transform in the same way under $SU {(3)}_{c} \otimes SU (2) \otimes U (1)$ ⁠.

In this chapter we concentrate our attention on vector-like isosinglet quarks. Our interest in extensions of the SM with those exotic fermions is justified for the following reasons:

They provide a framework for having a naturally small violation of the $3 \times 3$ unitarity of the CKM matrix. This leads to non-vanishing, but naturally suppressed, flavour-changing neutral currents (FCNC). The presence of FCNC opens up many interesting possibilities for rare K and B decays, as well as for CP asymmetries in neutral-B decays.

Adding isosinglet quarks to the SM leads to new sources of CP violation. In particular, isosinglet quarks enable one to achieve spontaneous CP violation (SCPV) using a very simple set of scalar fields, with only one non-Hermitian $SU (2) \otimes U (1)$ -singlet scalar added to the usual doublet—see § 24.7.

Extensions of the SM with isosinglet quarks may solve the strong CP problem through an implementation of the mechanism of Barr (1984) and Nelson (1984)—see § 27.7.1.

Models with vector-like quarks have been considered extensively in the literature. Early references include the papers by Aguila and Cortés (1985), Fishbane et al. (1985, 1986), Barger et al. (1986), Branco and Lavoura (1986), and Enqvist et al. (1986). A revived interest in these models was generated by the study of the impact of heavy isosinglet quarks on the CP-violating asymmetries in neutral-B decays, both in the limit where Z-exchange gives the dominant contribution to $B_{d}^{0} - \bar{B_{d}^{0}}$ mixing (Nir and Silverman 1990) and in the general case where both Z-exchange and the standard box-diagram contributions are taken into account (Branco et al. 1993). Recent analyses of the experimental constraints on these models may be found in the papers by Silverman (1996) and by Barenboim and Botella (1998).

24.2 The model

24.2.1 Quark spectrum

Our extension of the SM has $n_{g}$ doublets of left-handed quarks,

(\begin{array}{l} p_{Lo} \\ n_{Lo} \end{array}), with o = 1, 2, \dots, n_{g},

(24.2)

together with the following singlet quark fields: $n_{p}$ charge 2/3 left-handed $P_{L}, n_{n}$ charge –1/3 left-handed $N_{L}, n_{g} + n_{p}$ charge 2/3 right-handed $P_{R}$ ⁠, and $n_{g} + n_{n}$ charge –1/3 right-handed $N_{R} :$

\begin{array}{l} P_{L r}, with r = 1, 2, \dots, n_{p}, \\ N_{L s}, with s = 1, 2, \dots, n_{n}, \\ P_{R α}, with α = 1, 2, \dots, n_{g} + n_{p}, \\ N_{R k}, with k = 1, 2, \dots, n_{g} + n_{n} . \end{array}

(24.3)

Therefore, the electromagnetic current is

\begin{array}{l} J_{em}^{μ} \equiv \frac{2}{3} (\bar{p_{L o}} γ^{μ} p_{L o} + \bar{P_{L r}} γ^{μ} P_{L r} + \bar{P_{R α}} γ^{μ} P_{R α}) \\ - \frac{1}{3} (\bar{n_{L o}} γ^{μ} n_{L o} + \bar{N_{L s}} γ^{μ} N_{L s} + \bar{N_{R k}} γ^{μ} N_{R k}), \end{array}

(24.4)

and the electroweak gauge interactions are

\begin{array}{c} ℒ_{gauge} = - {eA}_{μ} J_{em}^{μ} + \frac{g}{\sqrt{2}} (W_{μ}^{+} \bar{p_{Lo}} γ^{μ} n_{Lo} + W_{μ}^{-} \bar{n_{Lo}} γ^{μ} p_{Lo}) \\ + \frac{g}{2 c_{w}} Z_{μ} (\bar{p_{Lo}} γ^{μ} p_{Lo} - \bar{n_{Lo}} γ^{μ} n_{Lo} - 2 s_{w}^{2} J_{em}^{μ}) . \end{array}

(24.5)

The quark mass terms may in general be written

\begin{array}{c} ℒ_{M} = - \bar{p_{Lo}} {(m_{p})}_{oα} P_{Rα} - \bar{n_{Lo}} {(m_{n})}_{ok} N_{Rk} \\ - \bar{P_{Lr}} {(M_{p})}_{rα} P_{Rα} - \bar{N_{Ls}} {(M_{n})}_{sk} N_{Rk} + H . c . . \end{array}

(24.6)

We shall use the letter x to denote either p or n in equations which hold for both x = p and x = n. The $m_{x}$ are $n_{g} \times (n_{g} + n_{x})$ matrices. They contain the usual $∣ Δ T ∣ = 1 / 2$ mass terms (T is the weak isospin), arising from the VEV of one or more Higgs doublets. Their mass scale should be $m \sim v .$ The $M_{x}$ are $n_{x} \times (n_{g} + n_{x})$ matrices. They contain $∣ Δ T ∣ = 0$ mass terms. Since these are $SU (2) \otimes U (1)$ -invariant, they may be present in the Lagrangian prior to the spontaneous breaking of the gauge symmetry. Not being protected by that symmetry, their mass scale M may be significantly larger than v.

We may denote $ℳ_{p}$ and $ℳ_{n}$ the full mass matrices for the up-type and down-type quarks, respectively:

ℳ_{x} = (\begin{array}{c} m_{x} \\ M_{x} \end{array}) .

(24.7)

These are $(n_{g} + n_{x}) \times (n_{g} + n_{x})$ matrices.

24.2.2 Mixing matrices

We denote the quark mass eigenstates by $u_{α}$ and $d_{k}$ ⁠, where $α$ and k run over the ranges in eqns (24.3). The unitary matrices $W_{p}$ and $W_{n}$ relate the left-handed-quark weak and mass eigenstates:

\begin{array}{c} (\begin{array}{l} p_{L} \\ P_{L} \end{array}) = W_{p} u_{L}, \\ (\begin{array}{l} n_{L} \\ N_{L} \end{array}) = W_{n} d_{L} . \end{array}

(24.8)

The $W_{x}$ are $(n_{g} + n_{x}) \times (n_{g} + n_{x})$ matrices. It is useful to write them as

W_{x} = (\begin{array}{c} A_{x} \\ B_{x} \end{array}),

(24.9)

where the $A_{x}$ are rectangular matrices consisting of the first $n_{g}$ rows of $W_{x}$ ⁠, while the $B_{x}$ consist of the last $n_{x}$ rows of $W_{x}$ ⁠. The unitarity of $W_{x}$ implies, on the one hand,

A_{x}^{†} A_{x} + B_{x}^{†} B_{x} = 1_{n_{g} + n_{x}},

(24.10)

and, on the other hand,

\begin{array}{l} A_{x} A_{x}^{†} = 1_{n_{g}}, \\ B_{x} B_{x}^{†} = 1_{n_{x}}, \\ A_{x} B_{x}^{†} = 0_{n_{g} \times n_{x}} . \end{array}

(24.11)

The $W_{x}$ must be chosen such as to diagonalize the Hermitian mass matrices $ℳ_{x} ℳ_{x}^{†}$ ⁠:

ℳ_{x} ℳ_{x}^{†} W_{x} = W_{x} D_{x}^{2} .

(24.12)

The real diagonal matrices $D_{p}$ and $D_{n}$ contain the masses of the up-type and down-type quarks, respectively.

The electromagnetic current in eqn (24.4) may be written in terms of the mass eigenstates as

J_{em}^{μ} = \frac{2}{3} \bar{u_{α}} γ^{μ} u_{α} - \frac{1}{3} \bar{d_{k}} γ^{μ} d_{k} .

(24.13)

The gauge interactions in eqn (24.5) are

\begin{matrix} L_{gauge} = - e A_{μ} J_{em}^{μ} + \frac{g}{\sqrt{2}} (W_{μ}^{+} \bar{u_{L α}} γ^{μ} V_{α k} d_{L k} + W_{μ}^{-} \bar{d_{L k}} γ^{μ} V_{α k}^{*} u_{L α}) \\ + \frac{g}{2 c_{w}} Z_{μ} [\bar{u_{L α}} γ^{μ} {(Z_{p})}_{α β} u_{L β} - \bar{d_{L k}} γ^{μ} {(Z_{n})}_{k j} d_{L j} - 2 s_{w}^{2} J_{em}^{μ}], \end{matrix}

(24.14)

where the mixing matrices are

\begin{array}{c} V = A_{p}^{†} A_{n} \\ Z_{x} = A_{x}^{†} A_{x} . \end{array}

(24.15)

The (generalized) CKM matrix V is an $(n_{g} + n_{p}) \times (n_{g} + n_{n})$ rectangular matrix. The CKM matrix in this model is not unitary as in the SM. The mixing matrices for the neutral currents $Z_{x}$ are Hermitian $(n_{g} + n_{x}) \times (n_{g} + n_{x})$ matrices. In general, they are not diagonal, i.e., flavour-changing neutral currents are present.

24.2.3 Non-unitarity of V and its relation to the FCNC

The generalized CKM matrix is not, in general, unitary. Indeed, if $n_{p} \neq n_{n}$ it is not even a square matrix. However, from the unitarity of $W_{p}$ and $W_{n}$ it follows (Branco et al. 1993) that there is an $(n_{g} + n_{p} + n_{n}) \times (n_{g} + n_{p} + n_{n})$ matrix

Y \equiv (\begin{array}{c} V & B_{p}^{†} \\ B_{n} & 0 \end{array}) = (\begin{array}{c} A_{p}^{†} A_{n} & B_{p}^{†} \\ B_{n} & 0 \end{array})

(24.16)

which is unitary. This may easily be checked using eqns (24.10) and (24.11).

The deviations of V from unitarity are closely related to the presence of FCNC. Indeed, because of the first eqn (24.11),

\begin{array}{l} Z_{p} = A_{p}^{†} A_{p} = A_{p}^{†} A_{n} A_{n}^{†} A_{p} = {VV}^{†}, \\ Z_{n} = A_{n}^{†} A_{n} = A_{n}^{†} A_{p} A_{p}^{†} A_{n} = V^{†} V . \end{array}

(24.17)

Thus, if V was unitary then $Z_{p}$ and $Z_{n}$ would be the unit matrix, and FCNC would be absent. From the first eqn (24.11) it also follows that

Z_{x}^{2} = Z_{x} .

(24.18)

Equations (24.17) may be elegantly derived without ever referring to the diagonalization of the quark mass matrices (Bamert et al. 1996). From eqns (11.12) and (11.3), we know that the gauge interactions in the $SU (2) \otimes U (1)$ gauge theory are given by

- eAQ + g (W^{+} T_{+} + W^{-} T_{-}) + \frac{g}{c_{w}} Z ([T_{+}, T_{-}] - {Qs}_{w}^{2}) .

(24.19)

In the vector space spanned by the $u_{Lα}$ and the $d_{Lk}$ ⁠, the matrices $T_{+}$ and $T_{-}$ are given by

T_{+} = \frac{1}{\sqrt{2}} (\begin{array}{l} 0 & V \\ 0 & 0 \end{array}), T_{-} = \frac{1}{\sqrt{2}} (\begin{array}{c} 0 & 0 \\ V^{†} & 0 \end{array}),

(24.20)

cf. eqn (24.14). Indeed, eqns (24.20) may be looked upon as the definition of the CKM matrix. From eqn (24.19), the matrix which determines that part of the neutral current which is not proportional to $J_{em}^{μ}$ is

\frac{1}{2} (\begin{array}{c} Z_{p} & 0 \\ 0 & - Z_{n} \end{array}) \equiv [T_{+}, T_{-}] = \frac{1}{2} (\begin{array}{c} {VV}^{†} & 0 \\ 0 & - V^{†} V \end{array}) .

(24.21)

This result is very general: it holds in the $SU (2) \otimes U (1)$ gauge theory for any standard or exotic $Q = 2 / 3$ and $Q = - 1 / 3$ quarks. It is valid if one adds to the SM not only vector-like isosinglet quarks, but also vector-like isodoublet quarks and mirror quarks (Lavoura and Silva 1993b).

24.3 Natural suppression of the FCNC

It is known experimentally that the FCNC are very much suppressed, although the detailed bounds depend on the flavours involved—see for example Lavoura and Silva (1993a) and Silverman (1996). Therefore, in order for the class of models considered here to be plausible, they must have a mechanism for natural suppression of the FCNC. In order to show that such a mechanism exists, we have to explicitly diagonalize the quark mass matrices.

In the discussion that follows, it will be convenient to have $ℳ_{p}$ and $ℳ_{n}$ in a special form, which we can obtain by using the freedom that one has to make weak-basis transformations (WBT). These are transformations of the quark fields which leave $ℒ_{gauge}$ in eqn (24.5) invariant. It is easy to convince oneself that, by making a WBT, one can bring both $ℳ_{p}$ and $ℳ_{n}$ to the form

ℳ_{x} = (\begin{array}{c} G_{x} & J_{x} \\ 0 & {\hat{M}}_{x} \end{array}),

(24.22)

where the ${\hat{M}}_{x}$ are $n_{x} \times n_{x}$ matrices, diagonal and real by definition. The matrices $G_{x}$ and $J_{x}$ are complex and have dimensions $n_{g} \times n_{g}$ and $n_{g} \times n_{x}$ ⁠, respectively. There is still some freedom left to make a WBT such that either $G_{p}$ or $G_{n}$ is made diagonal and real; we shall take $G_{p}$ to be diagonal and real.

Let us write

W_{x} = (\begin{array}{c} A_{x} \\ B_{x} \end{array}) = (\begin{array}{c} K_{x} & R_{x} \\ S_{x} & T_{x} \end{array}) .

(24.23)

The matrices $K_{x}$ and $T_{x}$ have dimensions $n_{g} \times n_{g}$ and $n_{x} \times n_{x}$ ⁠, respectively. The matrix $R_{x}$ is $n_{g} \times n_{x}$ ⁠, while $S_{x}$ is $n_{x} \times n_{g}$ ⁠. Equation (24.10) now reads

\begin{array}{l} K_{x}^{†} K_{x} + S_{x}^{†} S_{x} = 1_{n_{g}}, \\ R_{x}^{†} R_{x} + T_{x}^{†} T_{x} = 1_{n_{x}}, \\ K_{x}^{†} R_{x} + S_{x}^{†} T_{x} = 0_{n_{g} \times n_{x}} . \end{array}

(24.24)

We shall also write the matrices $D_{p}$ and $D_{n}$ as

D_{x} = (\begin{array}{c} {\bar{m}}_{x} & 0 \\ 0 & {\bar{M}}_{x} \end{array}) .

(24.25)

Equation (24.12) reads

\begin{array}{r} (G_{x} G_{x}^{†} + J_{x} J_{x}^{†}) K_{x} + J_{x} {\hat{M}}_{x} S_{x} = K_{x} {\bar{m}}_{x}^{2}, \\ (G_{x} G_{x}^{†} + J_{x} J_{x}^{†}) R_{x} + J_{x} {\hat{M}}_{x} T_{x} = R_{x} {\bar{M}}_{x}^{2}, \\ {\hat{M}}_{x} J_{x}^{†} K_{x} + {\hat{M}}_{x}^{2} S_{x} = S_{x} {\bar{m}}_{x}^{2}, \\ {\hat{M}}_{x} J_{x}^{†} R_{x} + {\hat{M}}_{x}^{2} T_{x} = T_{x} {\bar{M}}_{x}^{2} . \end{array}

(24.26)

All the above equations are exact, no approximations have been done yet. We now assume that $G_{x}$ and $J_{x}$ are $\sim m$ ⁠, while ${\hat{M}}_{x} \sim M$ ⁠, with $M ≫ m$ ⁠. One obtains the following solution of eqns (24.24) and (24.26) to leading order in m/M:

\begin{array}{l} T_{x} = 1_{n_{x}}, \\ R_{x} = J_{x} {\hat{M}}_{x}^{- 1}, \\ S_{x} = - {\hat{M}}_{x}^{- 1} J_{x}^{†} K_{x}, \end{array}

(24.27)

together with ${\bar{M}}_{x}^{2} = {\hat{M}}_{x}^{2}$ ⁠; while $K_{x}$ is the unitary matrix which diagonalizes $G_{x} G_{x}^{†}$ ⁠:

K_{x}^{†} G_{x} G_{x}^{†} K_{x} = {\bar{m}}_{x}^{2} .

(24.28)

In particular, since we have chosen to work in the weak basis where $G_{p}$ is diagonal, $K_{p} = 1_{n_{g}}$ to leading order.

The quark mixing matrix entering the charged current may now be computed and one obtains

V = (\begin{array}{c} K_{n} & J_{n} {\hat{M}}_{n}^{- 1} \\ {\hat{M}}_{p}^{- 1} J_{p}^{†} K_{n} & {\hat{M}}_{p}^{- 1} J_{p}^{†} J_{n} {\hat{M}}_{n}^{- 1} \end{array}) .

(24.29)

It is clear from eqn (24.29) that the mixing of standard quarks with isosinglet quarks is suppressed by m/M. Also, the matrix $K_{n}$ ⁠, which gives the interactions among the usual quarks, is unitary up to terms $\sim m^{2} / M^{2}$ ⁠.

The mixing matrices for the weak neutral current are

\begin{array}{l} Z_{n} = (\begin{matrix} 1 - K_{n}^{†} J_{n} {\hat{M}}_{n}^{- 2} J_{n}^{†} K_{n} & K_{n}^{†} J_{n} {\hat{M}}_{n}^{- 1} \\ {\hat{M}}_{n}^{- 1} J_{n}^{†} K_{n} & {\hat{M}}_{n}^{- 1} J_{n}^{†} J_{n} {\hat{M}}_{n}^{- 1} \end{matrix}), \\ Z_{p} = (\begin{matrix} 1 - J_{p} {\hat{M}}_{p}^{- 2} J_{p}^{†} & J_{p} {\hat{M}}_{p}^{- 1} \\ {\hat{M}}_{p}^{- 1} J_{p}^{†} & {\hat{M}}_{p}^{- 1} J_{p}^{†} J_{p} {\hat{M}}_{p}^{- 1} \end{matrix}) . \end{array}

(24.30)

The $n_{g} \times n_{g}$ upper-left submatrix, which governs the neutral-current interaction among the usual quarks, is the unit matrix up to a correction of the form $K_{x}^{†} J_{x} {\hat{M}}_{x}^{- 2} J_{x}^{†} K_{x}$ ⁠. Thus, flavour-changing neutral currents do arise, but they are naturally suppressed by a factor $m^{2} / M^{2}$ ⁠, because $J_{x} \sim m$ while ${\hat{M}}_{x} \sim M$ ⁠.

24.4 CP-violating phases

When the SM is extended through the addition of vector-like quarks, the CKM matrix is no longer unitary. Counting the independent CP-violating phases in V becomes complicated, especially when there are both $Q = - 1 / 3$ and $Q = 2 / 3$ vector-like quarks. We shall return to this question later; at this stage, we only want to count the number of phases in V when that matrix is evaluated in the approximation leading to eqn (24.29). Since in this approximation $K_{n}$ is unitary, the number of physical phases in it is just the same as in the SM, i.e., $(n_{g} - 1) (n_{g} - 2) / 2$ ⁠. In the upper-right block of V, we have the matrix $J_{n}$ which, being an $n_{g} \times n_{n}$ complex matrix, would in general contain $n_{g} n_{n}$ phases. However, $n_{n}$ phases may be eliminated by rephasing the last $n_{n}$ fields $d_{L}$ ⁠. Analogously, $J_{p}$ ⁠, in the lower-left block of V, has $(n_{g} - 1) n_{p}$ phases. Therefore, one has altogether

\frac{(n_{g} - 1) (n_{g} - 2)}{2} + (n_{g} - 1) (n_{p} + n_{n})

(24.31)

physical phases in V. We shall later show that this is indeed the correct number of CP-violating phases.

24.5 The invariant approach

24.5.1 CP-invariance conditions

The most general CP transformation of the quark fields leaving eqn (24.5) invariant is⁶³

\begin{array}{l} (C P) p_{L} {(C P)}^{†} = U_{L} γ^{0} C {\bar{p_{L}}}^{T}, \\ (C P) n_{L} {(C P)}^{†} = U_{L} γ^{0} C {\bar{n_{L}}}^{T}, \\ (C P) P_{R} {(C P)}^{†} = W_{R}^{p} γ^{0} C {\bar{P_{R}}}^{T}, \\ (C P) N_{R} {(C P)}^{†} = W_{R}^{n} γ^{0} C {\bar{N_{R}}}^{T}, \\ (C P) P_{L} {(C P)}^{†} = W_{L}^{p} γ^{0} C {\bar{P_{L}}}^{T}, \\ (C P) N_{L} {(C P)}^{†} = W_{L}^{n} γ^{0} C {\bar{N_{L}}}^{T} . \end{array}

(24.32)

We have suppressed the space-time variables for the sake of clarity. The matrix $U_{L}$ is $n_{g} \times n_{g}$ unitary, and the matrices $W_{R}^{x}$ and $W_{L}^{x}$ are $(n_{g} + n_{x}) \times (n_{g} + n_{x})$ and $n_{x} \times n_{x}$ unitary, respectively.

Requiring CP invariance of the mass terms in eqn (24.6) leads to the following conditions:

\begin{array}{r} U_{L}^{†} m_{p} W_{R}^{p} = m_{p}^{*}, \\ U_{L}^{†} m_{n} W_{R}^{n} = m_{n}^{*}, \\ W_{L}^{p †} M_{p} W_{R}^{p} = M_{p}^{*}, \\ W_{L}^{n †} M_{n} W_{R}^{n} = M_{n}^{*} . \end{array}

(24.33)

The existence of unitary matrices $U_{L}, W_{R}^{p}, W_{R}^{n}, W_{L}^{p}$ ⁠, and $W_{L}^{n}$ which satisfy eqns (24.33) is necessary and sufficient for CP invariance of $ℒ_{gauge} + ℒ_{M}$ ⁠. The fulfilment of this condition in any particular weak basis is equivalent to its fulfilment in any other weak basis, as one easily checks.

24.5.2 Analysis in a specific weak basis

In order to find the number of independent CP restrictions, we shall again use the weak basis in which the mass matrices have the form in eqn (24.22). We remind the reader that ${\hat{M}}_{p}, {\hat{M}}_{n}$ ⁠, and $G_{p}$ are diagonal and real by definition. In order to eliminate exceptional cases, of null measure in parameter space, we assume that the diagonal matrix elements of each of these three matrices are all different—the matrices are non-degenerate. Under these conditions, we easily find that eqns (24.33) imply

\begin{array}{c} U_{L} = diag (e^{i α_{o}}), \\ W_{L}^{p} = diag (e^{i β_{r}}), \\ W_{L}^{n} = diag (e^{i γ_{s}}), \end{array}

(24.34)

while the matrices $W_{R}^{p}$ and $W_{R}^{n}$ must take the block form

\begin{array}{c} W_{R}^{p} = (\begin{array}{c} U_{L} & 0 \\ 0 & W_{L}^{p} \end{array}), \\ W_{R}^{n} = (\begin{array}{c} X_{L} & 0 \\ 0 & W_{L}^{n} \end{array}), \end{array}

(24.35)

with $X_{L}$ an arbitrary $n_{g} \times n_{g}$ unitary matrix. The CP-invariance conditions of eqns (24.33) get reduced to

\begin{array}{r} diag (e^{- i α_{o}}) G_{n} X_{L} = G_{n}^{*}, \\ diag (e^{- i α_{o}}) J_{n} diag (e^{i γ_{s}}) = J_{n}^{*}, \\ diag (e^{- i α_{o}}) J_{p} diag (e^{i β_{r}}) = J_{p}^{*} . \end{array}

(24.36)

As $X_{L}$ may be chosen at will, the first eqn (24.36) is in fact equivalent to the simpler condition (see the argument in § 14.2)

diag (e^{- i α_{o}}) (G_{n} G_{n}^{†}) diag (e^{i α_{o}}) = {(G_{n} G_{n}^{†})}^{*} .

(24.37)

Thus, CP invariance imposes restrictions on the complex phases in the matrices $J_{p}, J_{n}$ ⁠, and $G_{n} G_{n}^{†} .$ Remember that the $J_{x}$ are complex matrices of dimension $n_{g} \times n_{x}$ ⁠, while $G_{n} G_{n}^{†}$ is an $n_{g} \times n_{g}$ Hermitian matrix. The number of independent phases in these three matrices is

n_{g} (n_{p} + n_{n}) + \frac{n_{g} (n_{g} - 1)}{2} .

(24.38)

CP invariance constrains these phases to be equal to differences of $n_{g}$ phases $α_{o}, n_{p}$ phases $β_{r}$ ⁠, and $n_{n}$ phases $γ_{s}$ ⁠. The number of independent CP restrictions therefore is

n_{g} (n_{p} + n_{n}) + \frac{n_{g} (n_{g} - 1)}{2} - (n_{g} + n_{p} + n_{n} - 1) = (n_{g} - 1) (\frac{n_{g}}{2} - 1 + n_{p} + n_{n}) .

(24.39)

This coincides with the number of CP-violating phases in the CKM matrix, given by eqn (24.31).

24.5.3 Invariant CP restrictions

From the first two eqns (24.33) one readily derives the following necessary conditions for CP invariance:

tr {[h_{p}^{a}, h_{n}^{b}]}^{f} = 0,

(24.40)

where a and b are integers, f is an odd integer, and $h_{x} \equiv m_{x} m_{x}^{†}$ ⁠. The conditions in eqn (24.40) are entirely analogous to those obtained in the SM, cf. eqn (14.21). In models with vector-like quarks, there are necessary conditions for CP invariance of a different type, for example

\begin{array}{l} Im tr (h_{p} h_{n} H_{n}) = 0, \\ Im tr (H_{p} h_{p} h_{n}) = 0, \end{array}

(24.41)

where $H_{x} \equiv m_{x} M_{x}^{†} M_{x} m_{x}^{†}$ ⁠.

The question of finding a set of necessary and sufficient conditions for CP invariance, expressed in terms of weak-basis invariants, is in general very complicated. For the case of a minimal extension in which only one vector-like isosinglet down-type quark is added to the three-generation SM—i.e., $n_{g} = 3, n_{n} = 1$ ⁠, and $n_{p} = 0$ —it has been shown (Aguila et al. 1998) that

\begin{array}{r} Im tr (h_{p}^{2} h_{n} h_{p} h_{n}^{2}) = 0, \\ Im tr (h_{p} h_{n} H_{n}) = 0, \\ Im tr (h_{p}^{2} h_{n} H_{n}) = 0, \\ Im tr (h_{p} h_{n}^{2} H_{n}) = 0, \\ Im tr (h_{p}^{2} h_{n}^{2} H_{n}) = 0, \\ Im tr [h_{p}^{2} (h_{p} h_{n} - h_{n} h_{p}) H_{n}] = 0, \\ Im tr [h_{p}^{2} (h_{p} h_{n}^{2} - h_{n}^{2} h_{p}) H_{n}] = 0, \end{array}

(24.42)

are necessary and sufficient conditions for CP conservation.

These invariant conditions are especially useful for analysing the limit where some of the quark masses are effectively degenerate. This is the case in very-high-energy collisions, where the natural asymptotic states no longer are hadronic states but rather quark jets. At high energies, it should be very difficult, if not impossible, to identify the flavour of the quark jets. In the extreme chiral limit $m_{u} = m_{c} = m_{d} = m_{s} = 0$ there is no CP violation in the SM, because there are identical-charge quarks which are degenerate. However, in the model with one down-type vector-like quark, there is CP violation even in that limit. Then, the strength of CP violation is controlled by the second invariant in eqns (24.42), all other invariants being proportional to that one (Aguila et al. 1998).

24.6 Parametrization of the CKM matrix

With the addition of vector-like quarks to the SM, parametrizing the CKM matrix V becomes rather involved, essentially due to the fact that it no longer is a unitary matrix. Still, in some cases one may use the fact that V is a submatrix of the larger unitary matrix Y in eqn (24.16). Various approaches to this problem have been proposed, including:

parametrization through Euler angles and phases (Branco and Lavoura 1986);

parametrization through the moduli of matrix elements and the arguments of quartets (Branco et al. 1993);

Wolfenstein-type parametrization (Lavoura and Silva 1993a).

Here we shall only describe the first of these approaches, which is interesting because it yields a different way of counting the number of CP-violating phases in V.

24.6.1 Parametrization with Euler angles and phases

This type of parametrization is possible when there are vector-like quarks of only one electric charge, either $Q = - 1 / 3$ or $Q = 2 / 3$ ⁠. In this case V consists of the first $n_{g}$ rows of a unitary matrix which, in principle, does not have any zero matrix elements. This is no longer true when both $n_{p}$ and $n_{n}$ are non-zero; then, the unitary matrix Y defined in eqn (24.16) has a zero submatrix, and its parametrization through Euler angles and phases is awkward.

For definiteness, let us assume that there are $n_{n}$ down-type vector-like quarks, but no up-type vector-like quarks, i.e., that $n_{p} = 0$ ⁠. In this case, one may choose, without loss of generality, a weak basis in which the up-type-quark mass matrix is diagonal and real non-negative. Then, V consists of the first $n_{g}$ rows of the $(n_{g} + n_{n}) \times (n_{g} + n_{n})$ unitary matrix $W_{n}$ which diagonalizes $ℳ_{n} ℳ_{n}^{†}$ ⁠. The task then consists of finding a parametrization of $W_{n}$ with only $(n_{g} - 1) (n_{g} / 2 - 1 + n_{n})$ phases in its first $n_{g}$ rows, according to eqns (24.31) and (24.39).

In order to construct such a parametrization, let us introduce the orthogonal matrices

O_{ij} \equiv (\begin{array}{c} 1 \\ ⋱ \\ cos θ_{ij} & \dots & - sin θ_{ij} \\ ⋮ & ⋮ \\ sin θ_{ij} & \dots & cos θ_{ij} \\ ⋱ \\ 1 \end{array}) \begin{array}{c} \leftarrow i \\ \leftarrow j \end{array}

(24.43)

and the rephasing matrices

I_{j} (δ_{k}) \equiv (\begin{array}{r} 1 \\ ⋱ \\ 1 \\ e^{i δ_{k}} \\ 1 \\ ⋱ \\ 1 \end{array}) \leftarrow j .

(24.44)

We may write $W_{n}$ as a product of ‘complex rotations’ $Ω_{ij}$ ⁠:

W_{n} = \prod_{i = N - 1}^{1} \prod_{j = i + 1}^{N} Ω_{ij},

(24.45)

where $N = n_{g} + n_{n}$ ⁠, and the unitary matrices $Ω_{ij}$ mix the $i^{th}$ and $j^{th}$ rows and columns, and contain one rotation angle and three phases:

Ω_{ij} = I_{i} (δ_{a}) I_{j} (δ_{b}) O_{ij} I_{i} (δ_{c}) .

(24.46)

Most of the rephasings $I_{j} (δ_{k})$ in the product in eqn (24.45) may be exchanged in position in such a way that they are brought out of the product of rotations $O_{ij}$ ⁠, and then they become equivalent to rephasings of the quark fields. If we omit writing down those trivial rephasings, it can be shown (Anselm et al. 1985) that the matrix $W_{n}$ may be written as

\begin{array}{l} W_{n} = O_{(N - 1) N} I_{N} O_{(N - 2) (N - 1)} O_{(N - 2) N} I_{N - 1} I_{N} \\ \times O_{(N - 3) (N - 2)} O_{(N - 3) (N - 1)} O_{(N - 3) N} I_{N - 2} I_{N - 1} I_{N} \dots O_{23} \dots O_{2 N} \\ \times I_{3} \dots I_{N} O_{12} \dots O_{1 N} . \end{array}

(24.47)

Explicit computation of the $N \times N$ unitary matrix in eqn (24.47) yields that it has $(n_{g} - 1) (N - n_{g} / 2 - 1)$ phases in its upper $n_{g}$ rows. This coincides with eqn (24.31), as $n_{n} = N - n_{g}$ ⁠.

Let us consider the example of $E_{6}$ with three families in the 27 representation. In each 27 there is one standard quark doublet together with one vector-like isosinglet down-type quark; therefore, $n_{g} = n_{n} = 3$ and $n_{p} = 0$ ⁠. The quark mixing matrix V consists of the first three rows of the $6 \times 6$ unitary matrix $W_{n}$ ⁠, in the weak basis in which the up-type-quark mass matrix is diagonal. We parametrize

\begin{array}{l} W_{n} = O_{56} I_{6} (δ_{10}) O_{45} O_{46} I_{5} (δ_{9}) I_{6} (δ_{8}) O_{34} O_{35} O_{36} I_{4} (δ_{7}) I_{5} (δ_{6}) I_{6} (δ_{5}) \\ \times O_{23} O_{24} O_{25} O_{26} I_{3} (δ_{4}) I_{4} (δ_{3}) I_{5} (δ_{2}) I_{6} (δ_{1}) O_{12} O_{13} O_{14} O_{15} O_{16} . \end{array}

(24.48)

The distribution of phases among the rows of this matrix turns out to be

(\begin{matrix} no phases \\ δ_{1}, δ_{2}, δ_{3}, δ_{4} \\ δ_{1}, δ_{2}, \dots, δ_{7} \\ δ_{1}, δ_{2}, \dots, δ_{9} \\ δ_{1}, δ_{2}, \dots, δ_{10} \\ δ_{1}, δ_{2}, \dots, δ_{10} \end{matrix}),

(24.49)

i.e., the first row of V is real, the second row depends on four phases, the third row depends on seven phases, and so on. This coincides with the formula $(n_{g} - 1) (N - n_{g} / 2 - 1)$ for $N = 6$ ⁠.

24.7 Spontaneous CP violation: a simple model

Models with vector-like quarks provide one of the simplest scenarios for spontaneous CP violation. We shall illustrate this feature by considering a minimal model proposed by Bento et al. (1991). This consists of the SM supplemented with only one charge $- 1 / 3$ vector-like quark and one non-Hermitian scalar singlet S. For later use, we shall also introduce a discrete symmetry under which $S, N_{L}$ ⁠, and $N_{R 4}$ change sign. This discrete symmmetry will not, however, play any role in the discussion of SCPV; it will only be important in the discussion of the strong CP problem in § 27.7.1

The scalar potential and the pattern of vacuum expectation values (VEVs) for this model have been discussed in § 23.6. One obtains the VEVs for $ϕ$ and S in eqns (23.38), with a non-trivial phase $α$ ⁠. However, we have emphasized in § 23.6 that, in the context of the SM with standard quarks only, the non-trivial vacuum phase does not lead to spontaneous CP breaking. In the extension of the SM with vector-like quarks the situation is different, because the presence of extra interactions means that there is less freedom in the choice of a CP transformation and, in particular, eqns (23.41) are no longer a valid CP transformation. This can readily be seen by examining the most general quark Yukawa couplings and the mass term consistent with the $SU (2) \otimes U (1)$ gauge symmetry and with the discrete symmetry:

ℒ_{Y} = - \bar{Q_{L}} Γ ϕ n_{R} - \bar{Q_{L}} Δ \tilde{ϕ} p_{R} - μ \bar{N_{L}} N_{R} - \bar{N_{L}} (FS + F^{'} S^{†}) n_{R} + H . c . .

(24.50)

Here, we have reserved the notation $N_{R}$ only for $N_{R 4}$ ⁠, the singlet quark which changes sign under the discrete symmetry, while for the other $P_{R}$ and $N_{R}$ fields we have returned to the SM notation and denoted them $p_{R}$ and $n_{R}$ ⁠, respectively. We assume the CP transformation in eqn (23.36), together with

\begin{array}{l} (C P) p_{R o} {(C P)}^{†} = γ^{0} C {\bar{p_{R o}}}^{T}, \\ (C P) n_{R o} {(C P)}^{†} = γ^{0} C {\bar{n_{R o}}}^{T}, \\ (C P) N_{R} {(C P)}^{†} = γ^{0} C {\bar{N_{R}}}^{T}, \\ (C P) \bar{p_{L o}} {(C P)}^{†} = - p_{L o}^{T} C^{- 1} γ^{0}, \\ (C P) \bar{n_{L o}} {(C P)}^{†} = - n_{L o}^{T} C^{- 1} γ^{0}, \\ (C P) \bar{N_{L}} {(C P)}^{†} = - N_{L}^{T} C^{- 1} γ^{0} . \end{array}

(24.51)

Hence, the $1 \times 3$ matrices of Yukawa couplings F and $F^{'}$ are real, just as the $3 \times 3$ matrices $Γ$ and $Δ$ ⁠, and the $Δ T = 0$ mass $μ$ ⁠. On the other hand, the transformation in eqn (23.41) would be a symmetry of the Lagrangian only if $F^{'}$ were equal to $F^{*}$ ⁠. We thus see that the presence of extra Yukawa interactions ensures that exactly the same vacuum structure now leads to CP violation, while in § 23.6 it did not.

Since the model has no charge 2/3 vector-like quarks, we may choose a weak basis in which the up-type-quark mass matrix is diagonal and real. The down-type-quark mass matrix is

ℳ_{n} = (\begin{array}{c} v Γ & 0 \\ V ℱ & μ \end{array}),

(24.52)

where

ℱ \equiv e^{iα} F + e^{- iα} F^{'} .

(24.53)

(Notice that $ℱ$ would be real if $F^{'}$ were equal to $F^{*}$ ⁠.) The matrix in eqn (24.52) is not in the standard form of eqn (24.22), but we can bring it to that form by means of a unitary transformation of the right-handed quark fields. Let us define

M \equiv \sqrt{μ^{2} + V^{2} {ℱℱ}^{†}} .

(24.54)

We may then construct the $4 \times 4$ unitary matrix

U = (\begin{array}{c} X & V ℱ^{†} / M \\ Y & μ / M \end{array}),

(24.55)

where X is a $3 \times 3$ matrix and Y is a $1 \times 3$ matrix. The unitarity of U implies, in particular,

{XX}^{†} = 1_{3} - \frac{V^{2}}{M^{2}} ℱ^{†} ℱ

(24.56)

and

V ℱ X + μY = 0 .

(24.57)

We then effect the weak-basis transformation

ℳ_{n} \to ℳ_{n} U = (\begin{array}{c} v Γ X & (vV / M) Γ ℱ^{†} \\ 0 & M \end{array}),

(24.58)

where we have used eqns (24.54) and (24.57). We thus obtain a matrix of the form in eqn (24.22). One may now use the results of § 24.3, provided that one assumes that $M ≫ v$ and $M^{2} ≫ vV \cdot$ ⁶⁴ The CKM matrix is a $3 \times 4$ matrix; its block connecting the standard quarks is a $3 \times 3$ matrix K which, in a first approximation, is the unitary matrix which diagonalizes

(v Γ X) {(v Γ X)}^{†} = v^{2} {ΓΓ}^{T} - \frac{v^{2} V^{2}}{M^{2}} Γ ℱ^{†} ℱ Γ^{T},

(24.59)

where we have used eqn (24.56). Because of the presence of the complex matrix $ℱ^{†} ℱ$ in eqn (24.59), it is clear that, if V and M are of the same order of magnitude—or, equivalently, if V and $μ$ are of the same order of magnitude—then the matrix K will in general contain a CP-violating phase and, moreover, this phase will not be suppressed by small mass ratios (Bento et al. 1991).

It is worth recalling the main features of the model that we have described. The scalar sector consists of the standard doublet and a non-Hermitian singlet S. In the fermion sector, only one charge $- 1 / 3$ isosinglet quark N is added. One imposes CP invariance of the Lagrangian. CP is spontaneously broken through the phase of $〈 0 ∣ S ∣ 0 〉$ ⁠. The crucial role played by the Yukawa couplings connecting N with the standard quarks should be emphasized. On the one hand, those couplings render the phase of $〈 0 ∣ S ∣ 0 〉$ genuinely CP-violating, by restricting the allowed CP transformations of the Lagrangian; on the other hand, it is through these couplings that that phase leads to the complexity of the $3 \times 3$ block of the CKM matrix connecting the standard quarks.

It is important to stress that this model may also be important because, if $μ$ and V are made higher than the cosmological-inflation scale, then domain walls, which are a problem which typically plagues models of SCPV, will be absent. Spontaneous CP breaking can occur at such a high energy scale that domain walls disappear during inflation, while CP violation in the CKM matrix remains unsuppressed.

24.8 Phenomenological implications

As we have emphasized, one of the salient features of models with vector-like quarks is the fact that the $3 \times 3$ CKM matrix connecting the standard quarks is no longer unitary and, as a result, there are FCNC coupling to the Z.

24.8.1 Experimental bounds on $Z_{kj}$

We shall first present some experimental constraints on FCNC in the down-type-quark sector. The relevant Lagrangian is given by

ℒ_{gauge} = \dots - \frac{g}{2 c_{w}} Z_{μ} \bar{d_{Lk}} γ^{μ} Z_{kj} d_{Lj} .

(24.60)

The flavour-changing parameters $Z_{kj}$ are closely related to the deviation of the CKM matrix V from unitarity, through the relations

\begin{array}{l} V_{u d}^{*} V_{u s} + V_{c d}^{*} V_{c s} + V_{t d}^{*} V_{t s} = Z_{d s}, \\ V_{u b}^{*} V_{u d} + V_{c b}^{*} V_{c d} + V_{t b}^{*} V_{t d} = Z_{b d}, \\ V_{u b}^{*} V_{u s} + V_{c b}^{*} V_{c s} + V_{t b}^{*} V_{t s} = Z_{b s} . \end{array}

(24.61)

The strongest constraint on $Z_{ds}$ stems from the experimental upper bound

BR (K^{+} \to π^{+} ν \bar{ν}) < 5.2 \times 10^{- 9} .

(24.62)

As in Appendix D, we compare the process $K^{+} \to π^{+} ν \bar{ν}$ with the charged-current process $K^{+} \to π^{0} e^{+} ν_{e}$ ⁠, which has

BR (K^{+} \to π^{0} e^{+} ν_{e}) = 0.482 \pm 0.006,

(24.63)

and using flavour-SU(3) symmetry, one obtains (Lavoura and Silva 1993b)

\frac{BR (K^{+} \to π^{+} ν \bar{ν})}{BR (K^{+} \to π^{0} e^{+} ν_{e})} = \frac{3 {∣ Z_{ds} ∣}^{2}}{2 {∣ V_{us} ∣}^{2}},

(24.64)

where the factor 3 corresponds to the sum over the three neutrino flavours. From eqns (24.62)–(24.64) one obtains

∣ Z_{ds} ∣ < 5.9 \times 10^{- 5} .

(24.65)

There are other limits, on $∣ Re (Z_{ds}^{2}) ∣$ and on $∣ Im (Z_{ds}^{2}) ∣$ ⁠, arising from $Δ m_{K}$ and from $ϵ_{K}$ ⁠, but eqn (24.65) is the stringest bound on $Z_{ds}$ ⁠.

The best limit on $∣ Z_{db} ∣$ and on $∣ Z_{sb} ∣$ is derived from the experimental bound (Particle Data Group 1996)

\frac{BR (B \to μ^{+} μ^{-} X)}{BR (B \to {μν}_{μ} X)} < \frac{5 \times 10^{- 5}}{(10.3 \pm 0.5) \times 10^{- 2}} .

(24.66)

The FCNC contributes to the decay $B \to μ^{+} μ^{-} X$ ⁠, where X is an arbitrary set of particles. A straightforward calculation gives (Parada 1996)

\frac{BR (B \to μ^{+} μ^{-} X)}{BR (B \to μ ν_{μ} X)} = \frac{[{(\frac{1}{2} - \sin^{2} θ_{w})}^{2} + \sin^{4} θ_{w}] ({|Z_{d b}|}^{2} + {|Z_{s b}|}^{2})}{{|V_{u b}|}^{2} + {|V_{c b}|}^{2} f (m_{c}^{2} / m_{b}^{2})},

(24.67)

where the function f has been given in eqn (15.18). From eqns (24.66) and (24.67) one obtains

{∣ Z_{db} ∣}^{2} + {∣ Z_{sb} ∣}^{2} < 5 \times 10^{- 6} .

(24.68)

24.8.2 Implications for CP asymmetries

The new contribution to $B^{0} - \bar{B^{0}}$ mixing arising from the Z-mediated $∣ Δ B ∣ = 2$ tree-level diagram can have a significant impact on CP asymmetries in $B^{0}$ decays. Let us write the complete matrix element $M_{12}$ of $B_{d}^{0} - \bar{B_{d}^{0}}$ mixing as

M_{12} = M_{12}^{SM} {({re}^{- iθ})}^{2},

(24.69)

where $M_{12}^{SM}$ is the standard-model box-diagram contribution, while (Barenboim et al. 1998)

{({re}^{iθ})}^{2} = 1 + \frac{4 π s_{w}^{2}}{α S_{0} (x_{t})} {(\frac{Z_{bd}}{V_{td} V_{tb}^{*}})}^{2} - \frac{R_{0} (x_{t})}{S_{0} (x_{t})} \frac{Z_{bd}}{V_{td} V_{tb}^{*}},

(24.70)

where $x_{t} = m_{t}^{2} / m_{W}^{2}$ ⁠, the Inami–Lim (1981) function $S_{0}$ has been defined in eqn (B.16), and

R_{0} (x) = \frac{x}{1 - x} (4 - x + \frac{3 x ln x}{1 - x})

(24.71)

is another Inami–Lim function. The first term in eqn (24.70) corresponds to the box diagram; the second term results from the Z-exchange tree-level diagram; the third term arises from a one-loop Z-vertex correction, which is relevant for small values of $Z_{bd}$ (Barenboim and Botella 1998). Analogous expressions hold in the case of $B_{s}^{0} - \bar{B_{s}^{0}}$ mixing. From eqn (24.68) it can be readily verified (Branco et al. 1993) that in the case of $B_{d}^{0} - \bar{B_{d}^{0}}$ mixing the Z contribution can be the dominant one, while in the case of $B_{s}^{0} - \bar{B_{s}^{0}}$ mixing it can at most compete with the usual box diagram.

The study of CP asymmetries should provide a very sensitive probe of FCNC. The measurement of those asymmetries with the expected experimental uncertainties may detect FCNC effects (Branco et al. 1993; Barenboim et al. 1998) even for rather small values of $Z_{bd}$ ⁠, at the level

∣ \frac{Z_{bd}}{V_{td} V_{tb}^{*}} ∣ \sim 10^{- 2} .

(24.72)

24.8.3 Baryogenesis

It has been shown (Branco et al. 1998) that, in a model with an extra singlet complex scalar, the first-order electroweak phase transition can be strong enough to avoid the baryon-asymmetry washout by sphalerons. The crucial point is the fact that in this model there is CP violation even in the extreme chiral limit where $m_{d} = m_{s} = m_{u} = m_{c} = 0$ ⁠. As a result, and contrary to what happens in the SM, there is no strong suppression of the baryon asymmetry by the ratio of light-quark masses over the critical temperature $T_{c}$ ⁠. The dominant contribution to the baryon asymmetry was estimated to be (Branco et al. 1998)

\frac{n_{B}}{s} \sim - 10^{- 3} v_{w} \frac{Im tr (h_{p} h_{n} H_{n})}{T_{c}^{8}},

(24.73)

where $v_{w}$ is the bubble-wall velocity. The appearance of the weak-basis (WB) invariant in the numerator of eqn (24.73) was to be expected, since it is the invariant which controls the strength of CP violation in the extreme chiral limit. That WB invariant can be expressed in terms of quark masses and mixing angles, and one obtains in leading order

\frac{n_{B}}{s} \sim - 10^{- 3} v_{w} \frac{m_{t}^{2} m_{b}^{2} m_{D}^{4}}{T_{c}^{8}} Im (V_{tb} V_{4 D} V_{tD}^{*} V_{4 b}^{*}),

(24.74)

when there is only one vector-like $Q = - 1 / 3$ quark D, with mass $m_{D}$ ⁠. If $m_{D} = 200 GeV$ and $∣ Im (V_{tb} V_{4 D} V_{tD}^{*} V_{4 b}^{*}) ∣ = 10^{- 4}$ one obtains the right order of magnitude for $n_{B} / s$ ⁠.

24.9 Main conclusions

•

Models with vector-like quarks have new sources of CP violation and provide a well-defined framework to study deviations from unitarity of the $3 \times 3$ CKM matrix, as well as flavour-changing neutral currents mediated by the Z. Those models have a rich phenomenology, which may be tested through the search for rare K and B decays, and through the study of CP asymmetries at B factories.

•

With the addition of at least one vector-like quark to the SM, spontaneous CP violation can be achieved with a very simple Higgs system, consisting of only one non-Hermitian scalar, singlet under $SU (2) \otimes U (1)$ ⁠. CP violation originates in the phase of the VEV of the singlet scalar. Through the mixing of the vector-like quarks with standard quarks, this phase generates an unsuppressed CP-violating phase in the $3 \times 3$ block of the CKM matrix connecting the usual quarks.

We are now in a position to add another item to the summary of models in § 23.7:

ONE HIGGS DOUBLET AND ONE SCALAR SINGLET

(CP SYMMETRY IMPOSED ON THE LAGRANGIAN)

•

No extra fermions: The vacuum contains one irremovable phase, but that phase does not lead to spontaneous CP breaking. The scalar sector does not lead to any new sources of CP violation.

•

Isosinglet quarks: The physical phase in the VEV of the singlet non-Hermitian scalar leads to spontaneous CP violation—it generates a complex CKM matrix. This is a minimal realization of the Nelson–Barr mechanism.

Notes

⁶²

Generic properties of models with exotic quarks have been studied, for example, by Langacker and London (1988), Joshipura (1992), Lavoura and Silva (1993a, b), and Nardi et al. (1995).

⁶³

We have eliminated the inconsequential phase $ξ_{W}$ from the CP transformation of $W^{\pm}$ in eqn (13.9).

⁶⁴

The assumption is indeed that the vacuum expectation value V and the $ΔT = 0$ mass $μ$ are of the same order of magnitude, which is much larger than v. Then, both X and Y have matrix elements $\sim 1$ ⁠.

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CP Violation. G. C. Branco, L. Lavoura, and J. P. Silva, Oxford University Press. © G. C. Branco, L. Lavoura, and J. P. Silva 2024. DOI: 10.1093/oso/9780198503996.003.0024

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Contents

24 Models with Vector-Like Quarks

Abstract

24.1 Motivation

24.2 The model

24.2.1 Quark spectrum

24.2.2 Mixing matrices

24.2.3 Non-unitarity of V and its relation to the FCNC

24.3 Natural suppression of the FCNC

24.4 CP-violating phases

24.5 The invariant approach

24.5.1 CP-invariance conditions

24.5.2 Analysis in a specific weak basis

24.5.3 Invariant CP restrictions

24.6 Parametrization of the CKM matrix

24.6.1 Parametrization with Euler angles and phases

24.7 Spontaneous CP violation: a simple model

24.8 Phenomenological implications

24.8.1 Experimental bounds on $Z_{kj}$

24.8.2 Implications for CP asymmetries

24.8.3 Baryogenesis

24.9 Main conclusions

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Contents

24 Models with Vector-Like Quarks

Abstract

24.1 Motivation

24.2 The model

24.2.1 Quark spectrum

24.2.2 Mixing matrices

24.2.3 Non-unitarity of V and its relation to the FCNC

24.3 Natural suppression of the FCNC

24.4 CP-violating phases

24.5 The invariant approach

24.5.1 CP-invariance conditions

24.5.2 Analysis in a specific weak basis

24.5.3 Invariant CP restrictions

24.6 Parametrization of the CKM matrix

24.6.1 Parametrization with Euler angles and phases

24.7 Spontaneous CP violation: a simple model

24.8 Phenomenological implications

24.8.1 Experimental bounds on Zkj

24.8.2 Implications for CP asymmetries

24.8.3 Baryogenesis

24.9 Main conclusions

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24.8.1 Experimental bounds on $Z_{kj}$