Chapter

22 Multi-Higgs-Doublet Models

https://doi.org/10.1093/oso/9780198503996.003.0022

Pages

239–259

Published:

July 1999

Abstract

The scalar sector of the standard model (SM) consists of only one doublet with weak hypercharge Y = 1/2. Most extensions of the SM include an enlargement of the Higgs sector. There are many theoretical motivations to enlarge the scalar sector of the standard electroweak theory, even if one only considers extensions of that theory based on the standard SU(3)c^®SU(2)^®U(l) gauge group. Among the specially important theoretical motivations, one may include:because of the need to give masses to both the up-type and the downtype quarks, on the other hand in order to eliminate the gauge anomalies generated by the fermionic supersymmetric partners of the scalars. Spontaneous CP violation-if one wishes to have CP as a good symmetry of the Lagrangian, only broken by the vacuum, then an extension of the Higgs sector is required. This will be explained in detail in the next chapter, where specific examples are presented.

22.1 Introduction

The scalar sector of the standard model (SM) consists of only one doublet with weak hypercharge $Y = 1 / 2$ ⁠. Most extensions of the SM include an enlargement of the Higgs sector. There are many theoretical motivations to enlarge the scalar sector of the standard electroweak theory, even if one only considers extensions of that theory based on the standard $SU {(3)}_{c} \otimes SU (2) \otimes U (1)$ gauge group. Among the specially important theoretical motivations, one may include:

Supersymmetry—in a supersymmetric extension of the SM (for a review, see e.g. Nilles 1984) a minimum of two Higgs doublets, with weak hypercharges $Y = 1 / 2$ and $Y = - 1 / 2$ ⁠, are necessary.⁵⁶ This is done, on the one hand because of the need to give masses to both the up-type and the down-type quarks, on the other hand in order to eliminate the gauge anomalies generated by the fermionic supersymmetric partners of the scalars.

Spontaneous CP violation—if one wishes to have CP as a good symmetry of the Lagrangian, only broken by the vacuum, then an extension of the Higgs sector is required. This will be explained in detail in the next chapter, where specific examples are presented.

Strong CP problem—most of the proposed solutions for this problem (see Chapter 27), and in particular the Peccei–Quinn solution in any of its variations, require an enlargement of the Higgs sector.

Baryogenesis—one of the exciting features of the electroweak gauge theories is the fact that they have all the necessary ingredients (Sakharov 1967)—namely baryon-number violation, C and CP violation, and departure from thermal equilibrium—to generate a net baryon asymmetry in the early Universe. However, it is by now clear that the SM cannot provide the observed baryon asymmetry, for various reasons which include

the fact that the electroweak phase transition is not strongly first order (Anderson and Hall 1992; Buchmüller et al. 1994; Kajantie et al. 1996), and as a result any baryon asymmetry generated during the transition would be subsequently washed out by unsuppressed B-violating processes in the broken phase;

CP-violating effects arising through the Kobayashi–Maskawa mechanism in the three-generation SM are too small (Gavela et al. 1994; Huet and Sather 1995).

Therefore, the need of having extra sources of CP violation which could lead to a successful baryogenesis is an important motivation to consider physics beyond the SM. As will be seen in this chapter and in the following one, the enlargement of the Higgs sector is one of the simplest ways of having new sources of CP violation beyond the Kobayashi–Maskawa mechanism.

No fundamental scalars have yet been experimentally observed, and therefore at present one only has experimental bounds on the masses and coupling constants of the scalar sector. The experimental search—see e.g. Gunion et al. (1990)—for Higgs particles is one of the most important tasks of particle physics.

An important constraint on the enlargement of the Higgs sector arises from the experimentally well-established relationship $m_{W} = c_{w} m_{Z}$ ⁠. This equality holds at classical level if the scalar fields which get a vacuum expectation value (VEV) are either singlets of $SU (2) \otimes U (1)$ —whose VEVs contribute neither to $m_{W}$ nor to $m_{Z}$ —or the neutral components of doublets of SU(2). Almost any other neutral scalar getting a VEV will make $m_{W} \neq c_{w} m_{Z}$ at tree level.⁵⁷ Hence their VEVs must be sufficiently small. Thus, from all types of scalar multiplets that we may think of adding to the SM, two are outstanding: SU(2) doublets with $Y = \pm 1 / 2$ ⁠, and SU(2) singlets with $Y = 0$ ⁠. Both types of multiplets have the advantage of having relatively few components; in the case of doublets, there is the added advantage that they may have Yukawa couplings to the usual fermions, allowing some interesting effects to arise.

In this chapter we dwell on multi-Higgs-doublet models (MHDMs). These models have gauge group $SU (2) \otimes U (1)$ and the usual fermion content: $n_{g}$ families of left-handed doublets $Q_{L}$ and $L_{L}$ and of right-handed singlets $p_{R}, n_{R}$ ⁠, and $l_{R}$ ⁠. The scalar sector of the model consists of $n_{d} > 1$ doublets $ϕ_{a} (a = 1, 2, \dots, n_{d})$ with $Y = 1 / 2$ ⁠. Thus,

ϕ_{a} = (\begin{array}{c} φ_{a}^{+} \\ φ_{a}^{0} \end{array}) .

(22.1)

Then,

{\tilde{ϕ}}_{a} \equiv i τ_{2} ϕ_{a}^{†^{T}} = (\begin{array}{c} φ_{a}^{0^{†}} \\ - φ_{a}^{-} \end{array})

(22.2)

are doublets of SU(2) with $Y = - 1 / 2$ ⁠.

In the next four sections we study the general features of the MHDMs. We give special attention to the two-Higgs-doublet model (THDM), which is the object of § 22.3 and 22.5. We start the study of CP violation in multi-Higgs-doublet models in § 22.6.

22.2 General multi-Higgs-doublet model

The Yukawa Lagrangian reads

ℒ_{Y} = - \bar{Q_{L}} (Γ_{a} ϕ_{a} n_{R} + Δ_{a} {\tilde{ϕ}}_{a} p_{R}) - \bar{L_{L}} Π_{a} ϕ_{a} l_{R} + H . c . .

(22.3)

The matrices $Γ_{a}, Δ_{a}$ ⁠, and $Π_{a}$ have dimension $n_{g} \times n_{g}$ ⁠. A sum over a from 1 to $n_{d}$ is implicit in eqn (22.3). The scalar potential is

V = Y_{ab} ϕ_{a}^{†} ϕ_{b} + Z_{abcd} (ϕ_{a}^{†} ϕ_{b}) (ϕ_{c}^{†} ϕ_{d}) .

(22.4)

Once again, sums from 1 to $n_{d}$ over the indices a, b, c, and d are implicit. The coefficients $Y_{ab}$ have dimensions of mass squared; the coefficients $Z_{abcd}$ are dimensionless. We assume

Z_{abcd} = Z_{cdab}

(22.5)

without loss of generality. Hermiticity of V implies

\begin{array}{l} Y_{ab} = Y_{ba}^{*}, \\ Z_{abcd} = Z_{badc}^{*} . \end{array}

(22.6)

Hence, there are $n_{d}^{2}$ independent real parameters in the quadratic couplings $Y_{ab}$ ⁠, while the quartic couplings $Z_{abcd}$ are parametrized by $n_{d}^{2} (n_{d}^{2} + 1) / 2$ real quantities.

We assume that the vacuum preserves a U(1) gauge symmetry corresponding to electromagnetism. Thus,

〈 0 ∣ ϕ_{a} ∣ 0 〉 = (\begin{array}{c} 0 \\ v_{a} e^{i θ_{a}} \end{array}),

(22.7)

with the $v_{a}$ real and non-negative. Without loss of generality, we may use a U(1) gauge transformation to make the VEV of $φ_{1}^{0}$ real and positive, just as in the SM. We shall assume this from now on. Thus, $θ_{1} = 0$ ⁠.

We write

\begin{array}{l} ϕ_{a} = e^{i θ_{a}} (\begin{matrix} φ_{a}^{+} \\ v_{a} + (ρ_{a} + i η_{a}) / \sqrt{2} \end{matrix}), \\ {\tilde{ϕ}}_{a} = e^{- i θ_{a}} (\begin{matrix} v_{a} + (ρ_{a} - i η_{a}) / \sqrt{2} \\ - φ_{a}^{-} \end{matrix}) . \end{array}

(22.8)

We define the $n_{d} \times n_{d}$ Hermitian matrix $V_{ab}$ as

V_{ab} \equiv v_{a} v_{b} e^{i (θ_{a} - θ_{b})} .

(22.9)

V_{ab} = V_{ba}^{*} .

(22.10)

Let us define

v = \sqrt{v_{1}^{2} + v_{2}^{2} + \dots + v_{n_{d}}^{2}} > 0 .

(22.11)

As $m_{W}^{2}$ and $m_{Z}^{2}$ receive additive contributions from $v_{1}^{2}, v_{2}^{2}, \dots, v_{n_{d}}^{2}$ ⁠, we find that v is related to $m_{W}$ and $m_{Z}$ by the same eqns (11.18) as in the SM.

22.3 The two-Higgs-doublet model

The simplest example of a MHDM is the THDM, in which only two scalar doublets, $ϕ_{1}$ and $ϕ_{2}$ ⁠, are introduced. The most general renormalizable scalar potential invariant under $SU (2) \otimes U (1)$ then is

\begin{array}{l} V = m_{1} ϕ_{1}^{†} ϕ_{1} + m_{2} ϕ_{2}^{†} ϕ_{2} + m_{3} (e^{i δ_{3}} ϕ_{1}^{†} ϕ_{2} + e^{- i δ_{3}} ϕ_{2}^{†} ϕ_{1}) \\ + a_{1} {(ϕ_{1}^{†} ϕ_{1})}^{2} + a_{2} {(ϕ_{2}^{†} ϕ_{2})}^{2} + a_{3} (ϕ_{1}^{†} ϕ_{1}) (ϕ_{2}^{†} ϕ_{2}) + a_{4} (ϕ_{1}^{†} ϕ_{2}) (ϕ_{2}^{†} ϕ_{1}) \\ + a_{5} [e^{i δ_{5}} {(ϕ_{1}^{†} ϕ_{2})}^{2} + e^{- i δ_{5}} {(ϕ_{2}^{†} ϕ_{1})}^{2}] + a_{6} (ϕ_{1}^{†} ϕ_{1}) (e^{i δ_{6}} ϕ_{1}^{†} ϕ_{2} + e^{- i δ_{6}} ϕ_{2}^{†} ϕ_{1}) \\ + a_{7} (ϕ_{2}^{†} ϕ_{2}) (e^{i δ_{7}} ϕ_{1}^{†} ϕ_{2} + e^{- i δ_{7}} ϕ_{2}^{†} ϕ_{1}) . \end{array}

(22.12)

The coupling constants $m_{i}$ (with i from 1 to 3) and $a_{j}$ (with j from 1 to 7) are real; all phases have been explicitly displayed and, as a matter of fact, $m_{3}, a_{5}, a_{6},$ and $a_{7}$ may be taken to be non-negative without loss of generality.

In the language of the previous section, the tensors $Y_{ab}$ and $Z_{abcd}$ are given by

\begin{array}{l} Y_{11} = m_{1}, Y_{12} = m_{3} e^{i δ_{3}}, \\ Y_{21} = m_{3} e^{- i δ_{3}}, Y_{22} = m_{2}; \end{array}

(22.13)

\begin{array}{l} \begin{array}{l} Z_{1111} & = & a_{1}, \\ Z_{1122} = Z_{2211} & = & a_{3} / 2, \\ Z_{1212} & = & a_{5} e^{i δ_{5}}, \\ Z_{1112} = Z_{1211} & = & a_{6} e^{i δ_{6}} / 2, \\ Z_{2212} = Z_{1222} & = & a_{7} e^{i δ_{7}} / 2, \end{array} & \begin{array}{l} Z_{2222} & = & a_{2}, \\ Z_{1221} = Z_{2112} & = & a_{4} / 2, \\ Z_{2121} & = & a_{5} e^{- i δ_{5}}, \\ Z_{1121} = Z_{2111} & = & a_{6} e^{- i δ_{6}} / 2, \\ Z_{2221} = Z_{2122} & = & a_{7} e^{- i δ_{7}} / 2. \end{array} \end{array}

(22.14)

The VEVs are given by

〈 0 ∣ ϕ_{1} ∣ 0 〉 = (\begin{array}{c} 0 \\ v_{1} \end{array}), 〈 0 ∣ ϕ_{2} ∣ 0 〉 = (\begin{array}{c} 0 \\ v_{2} e^{iθ} \end{array}),

(22.15)

with $v_{1}$ and $v_{2}$ real and positive by definition. The expectation value of the potential in the vacuum is

\begin{array}{l} V_{0} \equiv 〈 0 | V | 0 〉 = m_{1} v_{1}^{2} + m_{2} v_{2}^{2} + 2 m_{3} v_{1} v_{2} \cos (δ_{3} + θ) \\ + a_{1} v_{1}^{4} + a_{2} v_{2}^{4} + (a_{3} + a_{4}) v_{1}^{2} v_{2}^{2} + 2 a_{5} v_{1}^{2} v_{2}^{2} \cos (δ_{5} + 2 θ) \\ + 2 a_{6} v_{1}^{3} v_{2} \cos (δ_{6} + θ) + 2 a_{7} v_{1} v_{2}^{3} \cos (δ_{7} + θ) . \end{array}

(22.16)

The stability of the vacuum requires that

\begin{array}{l} 0 = \frac{- 1}{2 v_{1} v_{2}} \frac{\partial V_{0}}{\partial θ} \\ \begin{array}{l} = m_{3} sin (δ_{3} + θ) + 2 a_{5} v_{1} v_{2} sin (δ_{5} + 2 θ) \\ + a_{6} v_{1}^{2} sin (δ_{6} + θ) + a_{7} v_{2}^{2} sin (δ_{7} + θ) . \end{array} \end{array}

(22.17)

The quark Yukawa Lagrangian is

ℒ_{Y}^{(q)} = - \bar{Q_{L}} [(Γ_{1} ϕ_{1} + Γ_{2} ϕ_{2}) n_{R} + (Δ_{1} {\tilde{ϕ}}_{1} + Δ_{2} {\tilde{ϕ}}_{2}) p_{R}] + H . c . .

(22.18)

The quark mass matrices are

\begin{array}{l} M_{p} = v_{1} Δ_{1} + v_{2} Δ_{2} e^{- iθ}, \\ M_{n} = v_{1} Γ_{1} + v_{2} Γ_{2} e^{iθ} . \end{array}

(22.19)

They are bi-diagonalized in the usual way, see eqns (12.14) and (12.15), and the CKM matrix is in eqn (12.19).

22.4 The Higgs basis

In any MHDM there is an advantageous basis for the scalar doublets, which we shall refer to as ‘the Higgs basis’. We use for the doublets in the Higgs basis the notation $H_{1}, H_{2}, \dots, H_{n_{d}}$ ⁠. The Higgs basis is defined in the following way: the doublet $H_{1}$ has VEV v; all other doublets have zero VEV. The defining property of the Higgs basis is that only $H_{1}$ has a VEV, which is real and positive.

The Higgs basis is obtained by means of a unitary transformation—a weak-basis transformation—of the original scalar doublets $ϕ_{1}, ϕ_{2}, \dots, ϕ_{n_{d}}$ ⁠, which mixes them without altering the gauge-kinetic Lagrangian. However, the Higgs basis is not completely well defined, because one has the freedom to redefine the doublets with vanishing VEV by means of an $(n_{d} - 1) \times (n_{d} - 1)$ unitary transformation.

The Higgs basis is useful because, when using it, the Goldstone bosons are isolated as components of $H_{1}$ ⁠. Thus,

\begin{array}{l} H_{1} = (\begin{array}{c} φ^{+} \\ v + (H + iχ) / \sqrt{2} \end{array}), \\ {\tilde{H}}_{1} = (\begin{array}{c} v + (H - iχ) / \sqrt{2} \\ - φ^{-} \end{array}), \end{array}

(22.20)

just as in eqns (11.15) and (11.16). The fields $φ^{\pm}$ and $χ$ are the Goldstone bosons. Contrary to what happens in the SM, the Hermitian neutral field H is not, in general, an eigenstate of mass, rather it mixes with the neutral components of $H_{2}, H_{3}, \dots, H_{n_{d}}$ ⁠.

22.5 The Higgs basis in the THDM

In the THDM, with the VEVs given by eqn (22.15), one reaches the Higgs basis by performing the following unitary transformation of the scalar multiplets:

(\begin{array}{c} H_{1} \\ H_{2} \end{array}) = \frac{1}{v} (\begin{array}{c} v_{1} & v_{2} \\ v_{2} & - v_{1} \end{array}) (\begin{array}{c} ϕ_{1} \\ e^{- iθ} ϕ_{2} \end{array}) .

(22.21)

We write

\begin{array}{l} H_{2} = (\begin{array}{c} C^{+} \\ (N + iA) / \sqrt{2} \end{array}), \\ {\tilde{H}}_{2} = (\begin{array}{c} (N - iA) / \sqrt{2} \\ - C^{-} \end{array}) . \end{array}

(22.22)

The Hermitian fields N and A mix with H to form the three physical neutral scalars $S_{1}, S_{2},$ and $S_{3},$ as we shall soon see.

Thus,

\begin{matrix} (\begin{array}{l} φ^{+} \\ C^{+} \end{array}) = \frac{1}{v} (\begin{matrix} v_{1} & v_{2} \\ v_{2} & - v_{1} \end{matrix}) (\begin{array}{l} φ_{1}^{+} \\ φ_{2}^{+} \end{array}), \\ (\begin{matrix} H \\ N \end{matrix}) = \frac{1}{v} (\begin{matrix} v_{1} & v_{2} \\ v_{2} & - v_{1} \end{matrix}) (\begin{array}{l} ρ_{1} \\ ρ_{2} \end{array}), \\ (\begin{matrix} χ \\ A \end{matrix}) = \frac{1}{v} (\begin{matrix} v_{1} & v_{2} \\ v_{2} & - v_{1} \end{matrix}) (\begin{array}{l} η_{1} \\ η_{2} \end{array}) . \end{matrix}

(22.23)

22.5.1 The potential

The scalar potential in the Higgs basis is

\begin{array}{l} V = μ_{1} H_{1}^{†} H_{1} + μ_{2} H_{2}^{†} H_{2} + (μ_{3} H_{1}^{†} H_{2} + H .c .) + λ_{1} {(H_{1}^{†} H_{1})}^{2} + λ_{2} {(H_{2}^{†} H_{2})}^{2} \\ + λ_{3} (H_{1}^{†} H_{1}) (H_{2}^{†} H_{2}) + λ_{4} (H_{1}^{†} H_{2}) (H_{2}^{†} H_{1}) \\ + [(λ_{5} H_{1}^{†} H_{2} + λ_{6} H_{1}^{†} H_{1} + λ_{7} H_{2}^{†} H_{2}) (H_{1}^{†} H_{2}) + H .c .] . \end{array}

(22.24)

The $μ_{i}$ (with i from 1 to 3) have dimensions of mass squared, while the $λ_{j}$ (with j from 1 to 7) are dimensionless. All coupling constants are real but for $μ_{3}, λ_{5}, λ_{6},$ and $λ_{7}$ ⁠, which are not in general real.

The vacuum state is assumed to be a stability point of the potential; therefore, the terms of V linear in the fields H, N, and A vanish. This yields

\begin{array}{l} μ_{1} = - 2 λ_{1} v^{2}, \\ μ_{3} = - λ_{6} v^{2} . \end{array}

(22.25)

We shall use eqns (22.25) to trade $μ_{1}$ and $μ_{3}$ by $λ_{1}$ and $λ_{6}$ ⁠, respectively.

One expands the potential V in terms of the fields, after making the substitutions in eqns (22.25). The terms quadratic in the fields are the mass terms of the scalars:

V = - λ_{1} v^{4} + m_{C}^{2} C^{+} C^{-} + \frac{1}{2} (H N A) ℳ (\begin{array}{c} H \\ N \\ A \end{array}) + cubic and quartic terms,

(22.26)

where $m_{C}^{2} = μ_{2} + v^{2} λ_{3}$ is the squared mass of the charged scalars $C^{\pm}$ ⁠, and

M = (\begin{matrix} 4 v^{2} λ_{1} & 2 v^{2} Re λ_{6} & - 2 v^{2} Im λ_{6} \\ 2 v^{2} Re λ_{6} & m_{C}^{2} + (λ_{4} + 2 Re λ_{5}) v^{2} & - 2 v^{2} Im λ_{5} \\ - 2 v^{2} Im λ_{6} & - 2 v^{2} Im λ_{5} & m_{C}^{2} + (λ_{4} - 2 Re λ_{5}) v^{2} \end{matrix}) .

(22.27)

The symmetric matrix $ℳ$ is diagonalized by an orthogonal matrix T. The diagonalization yields the masses $m_{1}, m_{2},$ and $m_{3}$ of the physical neutral scalars of the THDM. Thus,

ℳ = T diag (m_{1}^{2}, m_{2}^{2}, m_{3}^{2}) T^{T}

(22.28)

and

(\begin{array}{c} H \\ N \\ A \end{array}) = T (\begin{array}{l} S_{1} \\ S_{2} \\ S_{3} \end{array}) .

(22.29)

Without loss of generality, we may choose the determinant of T to be 1.

The cubic and quartic terms in eqn (22.26) give rise to scalar self-interactions. For instance, there is a cubic interaction between the neutral and the charged scalars:

V = \dots + (c_{1} H + c_{2} N + c_{3} A) C^{+} C^{-} = \dots + \sum_{k = 1}^{3} f_{k} S_{k} C^{+} C^{-},

(22.30)

where $f_{k} \equiv T_{1 k} c_{1} + T_{2 k} c_{2} + T_{3 k} c_{3}$ for k = 1, 2, and 3, and

(c_{1} c_{2} c_{3}) = \sqrt{2} v (λ_{3} Re λ_{7} − Im λ_{7}) .

(22.31)

22.5.2 The Yukawa interactions

The Yukawa interactions with the quarks read

ℒ_{Y}^{(q)} = - \frac{\bar{Q_{L}}}{v} [(M_{n} H_{1} + Y_{n} H_{2}) n_{R} + (M_{p} {\tilde{H}}_{1} + Y_{p} {\tilde{H}}_{2}) p_{R}] + H . c .,

(22.32)

where the matrices $Y_{n}$ and $Y_{p}$ are in principle arbitrary and unrelated to the mass matrices $M_{n}$ and $M_{p}$ ⁠. Namely,

\begin{array}{l} Y_{p} \equiv v_{2} Δ_{1} - v_{1} Δ_{2} e^{- iθ}, \\ Y_{n} \equiv v_{2} Γ_{1} - v_{1} Γ_{2} e^{iθ} . \end{array}

(22.33)

We define

\begin{array}{l} Y_{u} \equiv U_{L}^{p †} Y_{p} U_{R}^{p}, \\ Y_{d} \equiv U_{L}^{n †} Y_{n} U_{R}^{n} . \end{array}

(22.34)

While $M_{u}$ and $M_{d}$ are diagonal, real and positive by definition, $Y_{u}$ and $Y_{d}$ in general are arbitrary $n_{g} \times n_{g}$ matrices. Then,

\begin{array}{l} L_{Y}^{​ (q)} = - (1 + \frac{H}{\sqrt{2} v}) (\bar{d} M_{d} d + \bar{u} M_{u} u) + \frac{i χ}{\sqrt{2} v} (\bar{u} M_{u} γ_{5} u - \bar{d} M_{d} γ_{5} d) \\ + \frac{φ^{†}}{v} (\bar{u_{R}} M_{u} V d_{L} - \bar{u_{L}} V M_{d} d_{R}) + \frac{φ^{-}}{v} (\bar{d_{L}} V^{†} M_{u} u_{R} - \bar{d_{R}} M_{d} V^{†} u_{L}) \\ \begin{array}{l} - \frac{N}{\sqrt{2} v} [\bar{d} (Y_{d} γ_{R} + Y_{d}^{†} γ_{L}) d + \bar{u} (Y_{u} γ_{R} + Y_{u}^{†} γ_{L}) u] \\ - \frac{i A}{\sqrt{2} v} [\bar{d} (Y_{d} γ_{R} - Y_{d}^{†} γ_{L}) d - \bar{u} (Y_{u} γ_{R} - Y_{u}^{†} γ_{L}) u] \\ + \frac{C^{+}}{v} (\bar{u_{R}} Y_{u}^{†} V d_{L} - \bar{u_{L}} V Y_{d} d_{R}) + \frac{C^{-}}{v} (\bar{d_{L}} V^{†} Y_{u} u_{R} - \bar{d_{R}} Y_{d}^{†} V^{†} u_{L}) . \end{array} \end{array}

(22.35)

The first two lines of eqn (22.35) display the same interactions as in the SM, cf. eqns (12.23)–(12.26). The last three lines of eqn (22.35) include the Yukawa interactions of N, A, and $C^{\pm}$ ⁠, which depend on the non-diagonal, arbitrary matrices $Y_{u}$ and $Y_{d}$ ⁠. This is one of the most important features of multi-Higgs-doublet models: in general, there are flavour-changing neutral Yukawa interactions (FCNYI), mediated by neutral scalars. We shall come back to this important question in § 22.10.

22.6 CP transformation

In this chapter, our starting point when defining the CP transformation is the usual one: we require the gauge-kinetic terms of the Lagrangian to be CP-invariant, and this requirement fixes the most general CP transformation allowed. In particular, the pattern of spontaneous symmetry breaking—of the VEVs—influences the gauge interactions of the various fields, notably, those involving the scalar fields, and therefore the explicit values of the VEVs must be taken into account in the definition of the CP transformation—see eqn (22.37) below.⁵⁸

Thus, from the requirement of CP invariance of the gauge interactions of the fermions, one finds that they transform as in eqns (14.2).⁵⁹ We write $φ_{a}^{0} = v_{a} e^{i θ_{a}} + H_{a}^{0}$ ⁠, where the $H_{a}^{0}$ are quantum fields, while the VEVs are c-numbers constant over space–time. From the requirement of CP invariance of the gauge interactions of the scalars, one finds that they transform as

\begin{array}{l} (CP) φ_{a}^{+} (t, \vec{r}) {(CP)}^{†} = U_{ab}^{CP} φ_{b}^{-} (t, - \vec{r}), \\ (CP) H_{a}^{0} (t, \vec{r}) {(CP)}^{†} = U_{ab}^{CP} H_{b}^{0 †} (t, - \vec{r}), \end{array}

(22.36)

where the $n_{d} \times n_{d}$ unitary matrix $U^{CP}$ must be chosen such that

v_{a} e^{i θ_{a}} = U_{ab}^{CP} v_{b} e^{- i θ_{b}} .

(22.37)

Equations (22.36) and (22.37) may be put together as

(CP) ϕ_{a} (t, \vec{r}) {(CP)}^{†} = U_{ab}^{CP} ϕ_{b}^{†^{T}} (t, - \vec{r}) .

(22.38)

In the Higgs basis, eqn (22.37) constrains $U^{CP}$ to be of the form

U^{CP} = (\begin{array}{c} 1 & 0_{1 \times (n_{d} - 1)} \\ 0_{(n_{d} - 1) \times 1} & K^{CP} \end{array}),

(22.39)

with $K^{CP}$ an arbitrary $(n_{d} - 1) \times (n_{d} - 1)$ unitary matrix. Thus, the fields $φ^{\pm},$ H, and $χ$ transform under CP in exactly the same way as in the SM—see eqns (13.8), (13.11), and (13.12), respectively.

22.7 CP violation in the scalar potential: simple examples

22.7.1 THDM with a discrete symmetry

Consider the THDM with a discrete symmetry under which $ϕ_{2} \to - ϕ_{2}$ ⁠. The scalar potential in eqn (22.12) then has

m_{3} = a_{6} = a_{7} = 0 .

(22.40)

Then, there is only one $θ$ -dependent term in the vacuum potential of eqn (22.16). As $a_{5}$ is positive by definition, the minimum is attained when

cos (δ_{5} + 2 θ) = - 1 .

(22.41)

The matrix $U^{CP}$ is fixed by eqn (22.37): $U^{CP} = diag (1, e^{2 iθ})$ ⁠. Therefore, from eqn (22.38),

(CP) {(ϕ_{1}^{†} ϕ_{2})}^{2} {(CP)}^{†} = e^{4 iθ} {(ϕ_{2}^{†} ϕ_{1})}^{2} .

(22.42)

CP-invariance of the $a_{5}$ -term of the potential then requires

e^{i (δ_{5} + 4 θ)} = e^{- i δ_{5}} .

(22.43)

But, eqn (22.41) implies eqn (22.43). One concludes that CP is conserved in this simple model—as long as no Yukawa couplings are introduced, at least.

Notice that this happens in spite of the potential not being real and in spite of the existence of a complex phase between the VEVs of the two doublets. This implies that neither condition by itself alone, or even both of them together, leads to CP violation. The crucial point is that the vacuum phase $θ$ is determined by only one term in the potential: there is only one $θ$ -dependent term in $V_{0}$ ⁠. Such a situation, in which there is only one term in $V_{0}$ for each (relative) phase in the vacuum, usually leads to CP invariance.

22.7.2 Softly broken discrete symmetry

The situation changes when one allows the discrete symmetry $ϕ_{2} \to - ϕ_{2}$ to be softly broken. A symmetry is said to be broken softly when all terms which break it have dimension lower than four. In this specific case, allowing for soft breaking of the symmetry will lead to the presence of only one extra quadratic term in the potential, the one with coefficient $m_{3}$ in eqn (22.12). CP-invariance would now require

\begin{array}{l} e^{2 i (δ_{5} + 2 θ)} = 1, \\ e^{2 i (δ_{3} + θ)} = 1 . \end{array}

(22.44)

However, $θ$ is now determined by the stability condition in eqn (22.17), with $a_{6} = a_{7} = 0$ ⁠, which will in general yield $θ$ satisfying neither of the two eqns (22.44). Therefore, CP is violated. The point is that $V_{0}$ now contains two clashing terms depending on the sole vacuum phase $θ$ ⁠; under such conditions, one may in general expect CP violation to occur.

This two-Higgs-doublet model, in which the reflection symmetry $ϕ_{2} \to - ϕ_{2}$ is softly broken by the quadratic terms proportional to $m_{3}$ ⁠, has been used as a toy model for CP violation in the scalar sector (Branco and Rebelo 1985; Weinberg 1990).

22.7.3 Weinberg model

In the three-Higgs-doublet model of Weinberg (1976) there are two distinct symmetries: the first one transforms $ϕ_{2} \to - ϕ_{2}$ and leaves all other fields unchanged; the second one transforms $ϕ_{3} \to - ϕ_{3}$ and leaves all other fields unchanged. These two symmetries are assumed not to be softly broken—though they end up being spontaneously broken by the non-vanishing VEVs of $ϕ_{2}$ and $ϕ_{3}$ ⁠. The scalar potential is

\begin{array}{l} V = m_{1} ϕ_{1}^{†} ϕ_{1} + m_{2} ϕ_{2}^{†} ϕ_{2} + m_{3} ϕ_{3}^{†} ϕ_{3} + a_{1} {(ϕ_{1}^{†} ϕ_{1})}^{2} + a_{2} {(ϕ_{2}^{†} ϕ_{2})}^{2} + a_{3} {(ϕ_{3}^{†} ϕ_{3})}^{2} \\ + b_{1} (ϕ_{2}^{†} ϕ_{2}) (ϕ_{3}^{†} ϕ_{3}) + b_{2} (ϕ_{3}^{†} ϕ_{3}) (ϕ_{1}^{†} ϕ_{1}) + b_{3} (ϕ_{1}^{†} ϕ_{1}) (ϕ_{2}^{†} ϕ_{2}) \\ + c_{1} (ϕ_{2}^{†} ϕ_{3}) (ϕ_{3}^{†} ϕ_{2}) + c_{2} (ϕ_{3}^{†} ϕ_{1}) (ϕ_{1}^{†} ϕ_{3}) + c_{3} (ϕ_{1}^{†} ϕ_{2}) (ϕ_{2}^{†} ϕ_{1}) \\ + d_{1} [e^{i ε_{1}} {(ϕ_{2}^{†} ϕ_{3})}^{2} + e^{- i ε_{1}} {(ϕ_{3}^{†} ϕ_{2})}^{2}] + d_{2} [e^{i ε_{2}} {(ϕ_{3}^{†} ϕ_{1})}^{2} + e^{- i ε_{2}} {(ϕ_{1}^{†} ϕ_{3})}^{2}] \\ + d_{3} [e^{i ε_{3}} {(ϕ_{1}^{†} ϕ_{2})}^{2} + e^{- i ε_{3}} {(ϕ_{2}^{†} ϕ_{1})}^{2}], \end{array}

(22.45)

with real and positive $d_{1}, d_{2}$ ⁠, and $d_{3}$ ⁠. The vacuum potential is

\begin{array}{l} V_{0} = m_{1} v_{1}^{2} + m_{2} v_{2}^{2} + m_{3} v_{3}^{2} + a_{1} v_{1}^{4} + a_{2} v_{2}^{4} + a_{3} v_{3}^{4} \\ + (b_{1} + c_{1}) v_{2}^{2} v_{3}^{2} + (b_{2} + c_{2}) v_{3}^{2} v_{1}^{2} + (b_{3} + c_{3}) v_{1}^{2} v_{2}^{2} \\ + 2 d_{1} v_{2}^{2} v_{3}^{2} \cos (2 θ_{2} - 2 θ_{3} - ε_{1}) + 2 d_{2} v_{3}^{2} v_{1}^{2} \cos (2 θ_{3} - ε_{2}) \\ + 2 d_{3} v_{1}^{2} v_{2}^{2} \cos (2 θ_{2} + ε_{3}) . \end{array}

(22.46)

In the vacuum there are two gauge-invariant relative phases, $θ_{2}$ and $θ_{3}$ ⁠, but in $V_{0}$ there are three terms which depend on them. The fact that there are less phases than phase-dependent terms in $V_{0}$ leads, once again, to CP violation in the self-interactions of the scalars.

22.8 General treatment of CP violation

CP violation is associated with the presence of irremovable phases in the Lagrangian of the theory. However, a weak-basis transformation of the fields—which includes the rephasing of the fields as a particular case—can bring new phases in and out of the Lagrangian. The spurious phases thus generated or eliminated have no bearing on CP violation. Therefore, it is important to find quantities which characterize CP violation in a given theory and which do not depend on the weak basis chosen to write the Lagrangian. We have encountered this problem in Chapter 14, where we have derived the weak-basis (WB) invariant $tr {[H_{p}, H_{n}]}^{3}$ for the three-generation SM using a general method (Bernabéu et al. 1986a) to construct CP-violating WB invariants. This method has been applied to some extensions of the SM, such as models with vector-like quarks (Branco and Lavoura 1986), models with Majorana neutrinos (Branco et al. 1986), and left-right-symmetric models (Branco and Rebelo 1985). In all these applications, the CP-violating WB invariants were related to clashes between the CP-transformation properties required by the gauge interactions, on the one hand, and the fermion mass terms, on the other hand. Botella and Silva (1995) have extended the method to the Higgs sector. The method involves the construction of tensors of increasing complexity, whose indices lie in the various family spaces—the families of identical multiplets which may be mixed by weak-basis transformations (WBT). By taking traces over all those indices one obtains weak-basis-invariant quantities, thus removing all the spurious phases created by WBT. Any remaining imaginary part constitutes a hallmark of CP violation. This method for constructing weak-basis-invariant quantities is quite general: it works for any gauge group, and for any (basic or effective) Lagrangian. The method can also be extended to provide weak-basis-invariant measures for the breaking of other discrete symmetries, like R-parity in supersymmetric theories (Davidson and Ellis 1997).

22.8.1 Weak-basis transformations

Weak-basis transformations of the fermion fields are identical to the ones in the SM, see eqns (14.10). In an MHDM there are several identical scalar multiplets; therefore, we may perform a WBT of the scalar fields too:

ϕ_{a}^{'} = U_{ab} ϕ_{b},

(22.47)

where U is an $n_{d} \times n_{d}$ unitary matrix, so that the gauge-kinetic Lagrangian of the scalars does not get changed. The VEVs transform in the same way as the doublets, therefore

V_{cd}^{'} = U_{ca} U_{db}^{*} V_{ab},

(22.48)

where the matrix $V_{ab}$ was defined in eqn (22.9). The couplings in the scalar potential transform as

\begin{array}{l} Y_{cd}^{'} = U_{ca} U_{db}^{*} Y_{ab}, \\ Z_{efgh}^{'} = U_{ea} U_{fb}^{*} U_{gc} U_{hd}^{*} Z_{abcd} . \end{array}

(22.49)

In a simultaneous WBT of the scalar doublets and of the fermion multiplets—eqns (22.47) and (14.10)—the Yukawa-coupling matrices transform as

\begin{array}{c} Γ_{a}^{'} = U_{ab}^{*} W_{L}^{†} Γ_{b} W_{R}^{n}, \\ Δ_{a}^{'} = U_{ab} W_{L}^{†} Δ_{b} W_{R}^{p} . \end{array}

(22.50)

22.8.2 Weak-basis invariants

Meaningful physical quantities must be invariant under a WBT. In order to construct such quantities, one first considers the matrices

\begin{array}{l} H_{Γ ab} \equiv Γ_{b} Γ_{a}^{†}, \\ H_{Δ ab} \equiv Δ_{a} Δ_{b}^{†} . \end{array}

(22.51)

From eqns (22.50), one has

\begin{array}{l} H_{Γ cd}^{'} = U_{ca} U_{db}^{*} (W_{L}^{†} H_{Γ ab} W_{L}), \\ H_{Δ cd}^{'} = U_{ca} U_{db}^{*} (W_{L}^{†} H_{Δ ab} W_{L}) . \end{array}

(22.52)

Taking traces over the fermionic indices one obtains quantities which are invariant under a WBT of the fermion fields, but are tensors under a WBT of the scalar fields. Taking traces over the indices a, b, c, … too, one finally obtains the weak-basis invariants (Botella and Silva 1995). Simple examples are $V_{ab} Y_{ba}, V_{ab} tr H_{Γ ba}$ ⁠, and $V_{ab} Y_{bc} tr H_{Δ ca}$ ⁠. For instance, since the mass matrix of the down-type quarks is $M_{n} = v_{a} Γ_{a} e^{i θ_{a}}$ ⁠,

V_{ab} tr H_{Γ ba} = tr (M_{n} M_{n}^{†}) = tr M_{d}^{2} = m_{d}^{2} + m_{s}^{2} + m_{b}^{2} + \dots .

(22.53)

22.8.3 CP violation

The CP-invariance conditions for the Yukawa-coupling matrices $Γ_{a}$ and $Δ_{a}$ and for the couplings of the scalar potential $Y_{ab}$ and $Z_{abcd}$ contain several CP-transformation matrix elements and are not very transparent. However, when considering weak-basis-invariant combinations of the couplings and of the VEVs, one obtains the simple result that all weak-basis-invariant quantities must be real in order for CP symmetry to hold.⁶⁰

If, for definiteness, one wants to study the CP-invariance conditions for the scalar potential only, under the given pattern of spontaneous symmetry breaking, one must take into account the tensors $V_{ab}, Y_{ab}$ ⁠, and $Z_{abcd}$ ⁠. With these three tensors one may construct various weak-basis-invariant quantities. Some of these are real, whether CP is conserved or not, because of eqns (22.5), (22.6), and (22.10); others are not necessarily real. The simplest non-real invariants are (Lavoura and Silva 1994; Botella and Silva 1995)

\begin{array}{l} I_{1} \equiv Y_{ab} Z_{bccd} V_{da}, \\ I_{2} \equiv V_{ab} Y_{bc} V_{de} Y_{ef} Z_{cafd} . \end{array}

(22.54)

If either $I_{1}$ or $I_{2}$ is not real, then the scalar potential together with the vacuum structure violates CP.

If one also considers the Yukawa interactions, extra tensors come into play, for instance $tr H_{Γ ab}$ ⁠. If one only considers this extra tensor, one may construct two more weak-basis invariants which may, in principle, be non-real:

\begin{array}{l} I_{3} \equiv V_{ab} Y_{bc} tr H_{Γ ca}, \\ I_{4} \equiv V_{ab} tr H_{Γ bc} V_{de} tr H_{Γ ef} Z_{cafd} . \end{array}

(22.55)

If either $I_{3}$ or $I_{4}$ is not real, there is CP violation in the clash between the Yukawa interactions and the scalar sector of the model.

22.9 CP violation in the two-Higgs-doublet model

In this section we apply the general methods of the previous section to the study of CP violation in the THDM. Our aim is to identify sources of CP violation in the THDM which are not present in the SM.

The tensor $V_{ab}$ has a very simple form in the Higgs basis: $V_{11} = v^{2}$ and all other $V_{ab} = 0$ ⁠. This simplifies considerably the computation of the weak-basis-invariants and, therefore, the analysis of CP violation becomes much simpler in the Higgs basis. In this section we shall use the Higgs basis throughout.

22.9.1 $I_{1}$ and $I_{2}$

Computing the invariants $I_{1}$ and $I_{2}$ one finds

\begin{array}{l} I_{1} = v^{2} Y_{1 b} Z_{b c c 1} \\ = - v^{4} (2 λ_{1} Z_{1 c c 1} + λ_{6} Z_{2 c c 1}) \\ = - v^{4} (2 λ_{1}^{2} + λ_{1} λ_{4} + {|λ_{6}|}^{2} / 2 + λ_{6} λ_{7}^{*} / 2), \end{array}

(22.56)

\begin{array}{l} I_{2} = v^{4} Y_{1 c} Y_{1 f} Z_{c 1 f 1} \\ = - v^{6} Y_{1 f} (2 λ_{1} Z_{11 f 1} + λ_{6} Z_{21 f 1}) \\ = v^{8} [2 λ_{1} (2 λ_{1} Z_{1111} + λ_{6} Z_{2111}) + λ_{6} (2 λ_{1} Z_{1121} + λ_{6} Z_{2121})] \\ = v^{8} (4 λ_{1}^{3} + 2 λ_{1} {|λ_{6}|}^{2} + λ_{6}^{2} λ_{5}^{*}) . \end{array}

(22.57)

Thus, $Im (λ_{6} λ_{7}^{*}) \propto Im I_{1}$ and $Im (λ_{6}^{2} λ_{5}^{*}) \propto Im I_{2}$ are CP-violating quantities.

Using eqns (22.27) and (22.31), one may reproduce these quantities. One easily finds

\begin{array}{r} ℳ_{12} c_{3} - ℳ_{13} c_{2} = 2 \sqrt{2} v^{3} Im (λ_{6} λ_{7}^{*}), \\ ℳ_{12} ℳ_{13} (ℳ_{22} - ℳ_{33}) + ℳ_{23} (ℳ_{13}^{2} - ℳ_{12}^{2}) = - 8 v^{6} Im (λ_{6}^{2} λ_{5}^{*}) . \end{array}

(22.58)

Using eqn (22.28) and the choice det $T = 1$ ⁠, one obtains

\begin{array}{c} ℳ_{12} c_{3} - ℳ_{13} c_{2} = f_{1} T_{12} T_{13} (m_{3}^{2} - m_{2}^{2}) + f_{2} T_{11} T_{13} (m_{1}^{2} - m_{3}^{2}) \\ + f_{3} T_{11} T_{12} (m_{2}^{2} - m_{1}^{2}), \end{array}

(22.59)

\begin{array}{l} ℳ_{12} ℳ_{13} (ℳ_{22} - ℳ_{33}) \\ + ℳ_{23} (ℳ_{13}^{2} - ℳ_{12}^{2}) = (m_{1}^{2} - m_{2}^{2}) (m_{1}^{2} - m_{3}^{2}) (m_{2}^{2} - m_{3}^{2}) T_{11} T_{12} T_{13} . \end{array}

(22.60)

The fact that the quantity in eqn (22.60) violates CP was first noticed by Méndez and Pomarol (1991); the proof of that fact was later given by Lavoura and Silva (1994), who also discovered the quantity in eqn (22.59). Equation (22.60) means that the mixing of the neutral scalars violates CP if the masses of the three physical scalars are all different and if, moreover, all three matrix elements of the first row of T are different from zero. Notice the similarity of eqn (22.60) with the expression for the CP-violating invariant of the SM, cf. eqn (14.24).

22.9.2 $I_{3}$ and $I_{4}$

Comparing the Yukawa Lagrangians in eqns (22.3) and (22.32), one concludes that, in the Higgs basis,

\begin{array}{l} v^{2} tr H_{Γ 11} = tr (M_{d}^{2}), \\ v^{2} tr H_{Γ 22} = tr (Y_{d} Y_{d}^{†}), \\ v^{2} tr H_{Γ 12} = tr (M_{d} Y_{d}), \\ v^{2} tr H_{Γ 21} = tr (M_{d} Y_{d}^{†}) . \end{array}

(22.61)

One then finds, after a tedious yet straightforward computation,

Im I_{3} = \sum_{i = d, s, b, \dots} \frac{m_{i}}{2} \sum_{k = 1}^{3} m_{k}^{2} T_{1 k} [T_{2 k} Im {(Y_{d})}_{ii} + T_{3 k} Re {(Y_{d})}_{ii}] .

(22.62)

Similarly,

\begin{matrix} Im I_{4} - \frac{tr (M_{d}^{2})}{v^{2}} Im I_{3} = \sum_{i, j = d, s, b, \dots} \frac{m_{i} m_{j}}{2 v^{2}} \sum_{k = 1}^{3} m_{k}^{2} [T_{2 k} Im {(Y_{d})}_{i i} + T_{3 k} Re {(Y_{d})}_{i i}] \\ \times [T_{2 k} Re {(Y_{d})}_{j j} - T_{3 k} Im {(Y_{d})}_{j j}] . \end{matrix}

(22.63)

22.9.3 Feynman rules and CP violation: $I_{1}$ and $I_{2}$

We call a scalar field $S (t, \vec{r})$ ‘CP-even’ when

(CP) S (t, \vec{r}) {(CP)}^{†} = S (t, - \vec{r});

(22.64)

on the other hand, S is ‘CP-odd’ when

(CP) S (t, \vec{r}) {(CP)}^{†} = - S (t, - \vec{r}) .

(22.65)

From eqn (13.11) one knows that H is CP-even. The CP transformation of N and A is determined by eqns (22.38) and (22.39):

(CP) [N (t, \vec{r}) + iA (t, \vec{r})] {(CP)}^{†} = e^{iϑ} [N (t, - \vec{r}) - iA (t, - \vec{r})] .

(22.66)

Therefore, $N cos (ϑ / 2) + A sin (ϑ / 2)$ is CP-even, while $N sin (ϑ / 2) - A cos (ϑ / 2)$ is CP-odd. (The phase $ϑ$ is arbitrary, and therefore the exact determination of which linear combination of N and A is CP-odd is meaningless. However, the reasoning is not affected by this.) If CP were conserved, the CP-even scalars would not mix with the CP-odd scalar in the mass matrix $ℳ$ ⁠. Then, out of the three physical neutral scalars, two would be CP-even and one would be CP-odd.

The Feynman rules for the vertices of one neutral scalar with two Z bosons, and of two distinct neutral scalars with one Z boson, are given in Fig. 22.1. One sees that the vertex $S_{k} Z_{α} Z_{β}$ is proportional to $T_{1 k} g_{αβ}$ ⁠. Under CP, $Z_{α} \to - Z^{α}$ ⁠. As $g_{αβ} = g^{αβ}$ ⁠, it follows that this vertex only exists if $S_{k}$ is CP-even. If $T_{11}, T_{12},$ and $T_{13}$ are all non-zero, all three neutral scalars $S_{1}, S_{2}$ ⁠, and $S_{3}$ couple to $Z_{α} Z_{β}$ in this way, and therefore all of them are CP-even. However, we have seen in the previous paragraph that, when CP is conserved, one out of the three neutral scalars must be CP-odd. We thus conclude that $T_{11} T_{12} T_{13} \neq 0$ implies CP violation. This coincides with what we deduced from eqn (22.60).

Fig. 22.1.

Feynman rules for the $Z_{α} Z_{β} S_{k}$ and $Z_{α} S_{k} S_{l}$ vertices in the THDM. (In the latter case, k must be different from l.) Notice the similarity with the vertices $Z_{α} Z_{β} H$ and $Z_{α} χH$ ⁠, respectively, of the SM, in Fig. 11.2.

Open in new tab Download slide

The same may be seen in yet another way. Suppose that both $S_{1}$ and $S_{2}$ couple to $Z_{α} Z_{β}$ ⁠, with vertices proportional to $g_{αβ}$ ⁠. Then, both $S_{1}$ and $S_{2}$ must be CP-even. Now suppose that there also is a vertex $S_{1} S_{2} Z_{α}$ which, according to Fig. 22.1, is proportional to $T_{13} {(p_{2} - p_{1})}_{α}$ ⁠. Under CP, $Z_{α} \to - Z^{α}$ ⁠, but ${(p_{2} - p_{1})}_{α} \to {(p_{2} - p_{1})}^{α}$ ⁠. It follows that either $S_{1}$ is CP-even and $S_{2}$ is CP-odd, or vice versa. This contradicts the above conclusion that both $S_{1}$ and $S_{2}$ are CP-even. Thus, if $T_{11} T_{12} T_{13} \neq 0$ all three above-mentioned Feynman vertices exist and there is CP violation.

We may thus construct three simple cases in which the co-existence of three different Feynman vertices displays CP violation:

\begin{array}{l} S_{1} S_{2} Z_{α} \propto {(p_{1} - p_{2})}_{α} \\ S_{1} S_{3} Z_{α} \propto {(p_{1} - p_{3})}_{α} \\ S_{2} S_{3} Z_{α} \propto {(p_{2} - p_{3})}_{α} \end{array}} \Rightarrow CP violation,

(22.67)

\begin{array}{l} S_{1} S_{2} Z_{α} \propto {(p_{1} - p_{2})}_{α} \\ S_{1} Z_{α} Z_{β} \propto g_{αβ} \\ S_{2} Z_{α} Z_{β} \propto g_{αβ} \end{array}} \Rightarrow CP violation,

(22.68)

\begin{array}{l} S_{1} Z_{α} Z_{β} \propto g_{αβ} \\ S_{2} Z_{α} Z_{β} \propto g_{αβ} \\ S_{3} Z_{α} Z_{β} \propto g_{αβ} \end{array}} \Rightarrow CP violation .

(22.69)

One should notice however that, while eqns (22.67) and (22.68) are valid in any model, eqn (22.69) holds only in the THDM, because in the THDM there are only three neutral scalars and one of them must be CP-odd. In the context of, say, a three-Higgs-doublet model, the existence of three scalars with gauge interactions as in eqn (22.69) would not imply CP violation.

One may interpret eqn (22.59) along similar lines. The interaction $f_{k} S_{k} C^{+} C^{-}$ in eqn (22.30) implies that $S_{k}$ is CP-even. Similarly, the interaction $T_{1 k} S_{k} Z^{μ} Z_{μ}$ implies that $S_{k}$ is CP-even. Hence, if for instance the product $f_{1} T_{12} T_{13} \neq 0$ ⁠, this means that $S_{1}, S_{2}$ ⁠, and $S_{3}$ all are CP-even, which is impossible in the THDM. For this reason, the quantity in eqn (22.59) signals CP violation in the THDM.

22.9.4 Feynman rules and CP violation: $I_{3}$ and $I_{4}$

In order to understand the results in eqns (22.62) and (22.63), let us study in more detail the Yukawa interactions of the neutral scalars with the down-type quarks. From eqn (22.35), they read

ℒ_{Y}^{(q)} = \dots - \sum_{k = 1}^{3} \frac{S_{k}}{\sqrt{2} v} \bar{d} [T_{1 k} M_{d} + (T_{2 k} + {iT}_{3 k}) Y_{d} γ_{R} + (T_{2 k} - {iT}_{3 k}) Y_{d}^{†} γ_{L}] d .

(22.70)

Let us consider only the diagonal interactions, in which the incoming and outgoing quarks have the same flavour. They are

ℒ_{Y}^{(q)} = \dots - \sum_{k = 1}^{3} \frac{S_{k}}{\sqrt{2} v} \sum_{i} {\bar{d}}_{i} (a_{ki} + i γ_{5} b_{ki}) d_{i},

(22.71)

where

\begin{array}{l} a_{ki} \equiv T_{1 k} m_{i} + T_{2 k} Re {(Y_{d})}_{ii} - T_{3 k} Im {(Y_{d})}_{ii}, \\ b_{ki} \equiv T_{2 k} Im {(Y_{d})}_{ii} + T_{3 k} Re {(Y_{d})}_{ii} . \end{array}

(22.72)

Now, a glance at eqns (3.74) and (3.75) tells us that, in order for the interactions in eqn (22.71) to be CP-invariant, $S_{k}$ must be CP-even when $a_{ki} \neq 0$ ⁠, but it must be CP-odd when $b_{ki} \neq 0$ ⁠. Clearly, if $a_{ki}$ and $b_{kj}$ are simultaneously non-zero for any two down-type quarks $d_{i}$ and $d_{j}$ —even when $d_{i} \neq d_{j}$ —then $S_{k}$ does not have a definite CP-parity, and CP is violated. This is precisely what is reflected in eqns (22.62) and (22.63).

CP violation in the Yukawa couplings of the neutral scalars leads to effects like the generation of electric-dipole moments—in the coupling of a fermion with the photon— or weak electric-dipole moments—in the coupling of a fermion with the Z—at one-loop level. Weak electric-dipole moments have a particularly rich variety of contributing diagrams; some examples are presented in Fig. 22.2.

Fig. 22.2.

Some diagrams which may contribute to the weak electric-dipole moment in the coupling of a fermion f to the Z boson. In some vertices we have explicitly written down the form that that vertex might assume in order for the diagram to violate CP.

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22.10 Flavour-changing neutral Yukawa interactions

This and the next section, on the problem of flavour-changing neutral currents, may be skipped by the reader.

We have seen in § 22.5.2 that the matrices $Y_{u}$ and $Y_{d}$ defined by eqns (22.34) are not in general diagonal, and therefore the neutral scalars have flavour-changing neutral Yukawa interactions (FCNYI) with quarks of identical electric charge. Let us introduce the Hermitian matrices $X_{k}$ and $X_{k}^{'}$ ⁠, defined by⁶¹

\begin{array}{l} X_{k} \equiv \frac{(T_{2 k} + {iT}_{3 k}) Y_{d} + (T_{2 k} - {iT}_{3 k}) Y_{d}^{†}}{2 \sqrt{2} v}, \\ X_{k}^{'} \equiv \frac{- i (T_{2 k} + {iT}_{3 k}) Y_{d} + i (T_{2 k} - {iT}_{3 k}) Y_{d}^{†}}{2 \sqrt{2} v} . \end{array}

(22.73)

We may then rewrite eqn (22.70) in the form

ℒ_{Y}^{(q)} = \dots - \sum_{k = 1}^{3} S_{k} \bar{d} (T_{1 k} \frac{M_{d}}{\sqrt{2} v} + X_{k} + i X_{k}^{'} γ_{5}) d .

(22.74)

There is for instance a Yukawa interaction connecting the s quark with the d quark,

ℒ_{Y}^{(q)} = \dots - \sum_{k = 1}^{3} S_{k} {\bar{s} [{(X_{k})}_{21} + i {(X_{k}^{'})}_{21} γ_{5}] d + \bar{d} [{(X_{k})}_{21}^{*} + i {(X_{k}^{'})}_{21}^{*} γ_{5}] s} .

(22.75)

This interaction leads to a contribution $M_{21}^{S}$ to the off-diagonal matrix element $M_{21}$ in the neutral-kaon system. That contribution reads—see Appendix B—

\begin{array}{l} M_{21}^{S} = e^{i (ξ_{K} + ξ_{d} - ξ_{s})} \sum_{k = 1}^{3} \frac{f_{K}^{2} m_{K}}{24 m_{k}^{2}} {{[{({X^{'}}_{k})}_{21}]}^{2} [1 - \frac{11 m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}] \\ + {[{(X_{k})}_{21}]}^{2} [1 - \frac{m_{K}^{2}}{{(m_{s} + m_{d})}^{2}}]}, \end{array}

(22.76)

where $m_{K} \approx 498 MeV, f_{K} \approx 160 MeV$ ⁠, and $m_{s} + m_{d} \approx 180 MeV$ ⁠.

We know that the mass difference between $K_{L}$ and $K_{S}$ is $2 ∣ M_{21} ∣ \approx 3.49 \times 10^{- 12} MeV$ ⁠. This suggests that the masses $m_{k}$ should be rather high. Indeed, let us assume that $M_{21}^{S}$ is the largest contribution to $M_{21}$ ⁠, so that $∣ M_{21} ∣ \approx ∣ M_{21}^{S} ∣$ ⁠. We would then have

\sum_{k = 1}^{3} \frac{1}{m_{k}^{2}} {83.20 {[{(X_{k}^{'})}_{21}]}^{2} + 6.65 {[{(X_{k})}_{21}]}^{2}} = 3.29 \times 10^{- 6} {TeV}^{- 2} .

(22.77)

The complex numbers ${(X_{k})}_{21}$ and ${(X_{k}^{'})}_{21}$ are in principle arbitrary. However, we may reasonably guess that $∣ {(Y_{d})}_{21} ∣$ and $∣ {(Y_{d})}_{12} ∣$ should be of order $\sqrt{m_{s} m_{d}} \approx 40 MeV$ ⁠. If this is so, and as $v = 174 GeV$ ⁠, we have $∣ {(X_{k}^{'})}_{21} ∣, ∣ {(X_{k})}_{21} ∣ \sim 10^{- 4}$ ⁠.

Let us further assume that in the sum in the left-hand side of eqn (22.77) there are no large cancellations. It is then reasonable to estimate that, for each value of k, $6 \times 10^{- 7} / m_{k}^{2} \sim 3 \times 10^{- 6} {TeV}^{- 2},$ i.e.,

m_{k} ≳ 0.5 TeV .

(22.78)

This is a rather high value for the masses of the Higgs scalars. Of course, the derivation of eqn (22.78) involved various ad hoc assumptions Still, it is clear that we are confronted with a potential problem for the THDM: unless the neutral scalars have masses of order 1 TeV, their contribution to the mass difference of the neutral kaons may be too large.

The Yukawa interactions of the neutral scalars are a potential source of CP violation, too. The contribution $M_{21}^{S}$ to $M_{21}$ might generate, not only a large mass difference $Δ m$ ⁠, but also a large CP-violating parameter $ϵ$ ⁠. In principle, the constraints on the masses of the scalars from consideration of the contribution of $M_{21}^{S}$ to $ϵ$ will be stronger than the ones from the contribution of $M_{21}^{S}$ to $Δ m$ ⁠. However, there are natural ways of suppressing the imaginary parts in the FCNYI (Branco and Rebelo 1985), and thus their contribution to $ϵ$ ⁠. We therefore stick to the bound in eqn (22.78), which is somewhat more difficult to avoid.

The FCNYI are a general problem of multi-Higgs-doublet models. Those models in general have neutral scalar particles whose Yukawa couplings are not flavour-diagonal. Then, in order to satisfy experimental constraints arising from $K^{0} - \bar{K^{0}}, B^{0} - \bar{B^{0}}$ ⁠, and $D^{0} - \bar{D^{0}}$ mixing, as well as from some rare decays, either one has to find a natural mechanism to suppress the non-diagonal couplings, or the masses of the neutral scalars have to be rather high, in the TeV range.

22.11 Mechanisms for natural suppression of the FCNYI

22.11.1 Natural flavour conservation

In order to solve the problem of FCNYI, the concept of natural flavour conservation (NFC) was developed. With NFC, one avoids the FCNYI by imposing some extra symmetry on the Lagrangian of the MHDM; the extra symmetry should constrain the Yukawa interactions of the neutral scalars in such a way that they turn out diagonal. Glashow and Weinberg (1977) and Paschos (1977) have shown that the only way to achieve NFC is to ensure that only one Higgs doublet has Yukawa interactions with quarks of a given charge—see also Ecker et al. (1988).

Consider the Yukawa interactions in eqn (22.3). FCNYI arise because not all $Γ_{a}$ can be simultaneously bi-diagonalized, i.e., diagonalized by the same two unitary matrices. When one bi-diagonalizes the particular linear combination of the $Γ_{a}$ which constitutes $M_{n}$ ⁠, the down-type-quark mass matrix, one is not bidiagonalizing other linear combinations of the $Γ_{a}$ ⁠, orthogonal to $M_{n}$ ⁠. A simple solution to this problem is the following: all $Γ_{a}$ ⁠, except one of them, should be identically zero; the same thing happening in the up-quark sector, where all $Δ_{a}$ except one should vanish.

This leads to two possibilities: either the matrices $Γ_{a}$ and $Δ_{b}$ which do not vanish correspond to different Higgs doublets $(a \neq b)$ ⁠, or they correspond to the same Higgs doublet $(a = b)$ ⁠. The first situation—which in the context of two-Higgs-doublet models is sometimes called ‘model 1’, while the second choice is called ‘model 2’—is more interesting from the theoretical point of view, in particular because it automatically arises in a supersymmetric theory. Let us study it in more detail.

We start from eqn (22.3). We assume that

\begin{array}{l} Γ_{2} = Δ_{1} = 0, \\ Γ_{a} = Δ_{a} = 0 for a > 2 . \end{array}

(22.79)

(We leave open the possibility that there are more than two doublets; we just assume that the extra doublets, if they exist, do not have Yukawa couplings to the quarks.) Equation (22.79) may be enforced by two discrete symmetries:

\begin{array}{l} D_{1} : ϕ_{a} \to - ϕ_{a} for a > 2, \\ D_{2} : ϕ_{2} \to - ϕ_{2} and p_{R} \to - p_{R}, \end{array}

(22.80)

cf. § 22.7.3. The Yukawa Lagrangian in eqn (22.18), under the condition in eqn. (22.79), is

\begin{array}{l} L_{Y}^{(q)} = - (1 + \frac{ρ_{1}}{\sqrt{2} v_{1}}) \bar{d} M_{d} d - (1 + \frac{ρ_{2}}{\sqrt{2} v_{2}}) \bar{u} M_{u} u \\ - \frac{i η_{1}}{\sqrt{2} v_{1}} \bar{d} M_{d} γ_{5} d + \frac{i η_{2}}{\sqrt{2} v_{2}} \bar{u} M_{u} γ_{5} u \\ + (\frac{φ_{2}^{+}}{v_{2}} \bar{u_{R}} M_{u} V d_{L} - \frac{φ_{1}^{+}}{v_{1}} \bar{u_{L}} V M_{d} d_{R} + H .c .), \end{array}

(22.81)

cf. eqns (12.23)–(12.26). All neutral Yukawa interactions are flavour-diagonal and proportional to the mass of the quark. FCNYI do not arise because only one Yukawa-coupling matrix must be bi-diagonalized in each quark sector.

22.11.2 Non-vanishing but naturally small FCNYI

In this section we shall consider the possibility of having non-vanishing but naturally suppressed FCNYI. By natural suppression we mean that whatever mechanism is responsible for the suppression, it should result from either an exact or a softly broken symmetry of the Lagrangian. This naturalness requirement is essential in order to guarantee that the suppression mechanism is stable under radiative corrections.

A possible suppression mechanism could arise if the flavour-changing couplings of the neutral scalars were entirely fixed by quark masses and elements of the CKM matrix V. Since some of these matrix elements are experimentally known to be very small, one could then have a suppression. For definiteness, let us consider the flavour-changing neutral coupling vertex connecting two down-type quarks $d_{i}$ and $d_{j}$ with a scalar. Let us assume that the corresponding Yukawa coupling $Y_{ij}$ depends only on quark masses and on matrix elements of V. Of course, $Y_{ij}$ will have to be invariant under rephasing of the fields $d_{i}$ and $d_{j}$ ⁠. This restricts the functional dependence of $Y_{ij}$ on the CKM-matrix elements. The simplest dependence which conforms to the constraint of rephasing invariance is $V_{αi}^{*} V_{αj}$ ⁠, where $u_{α}$ denotes any of the up-type quarks. If one considers the specific case $d_{i} = d$ and $d_{j} = s$ ⁠, and if $u_{α}$ turns out to be the top quark, then one has a very strong suppression factor (Joshipura and Rindani 1991) ${∣ V_{ts} V_{td} ∣}^{2} \propto λ^{10} \sim 10^{- 8}$ in the neutral-scalar contribution to the $∣ Δ S ∣ = 2$ effective Hamiltonian.

The important question is whether it is possible to have such a functional dependence as a result of an exact or softly broken symmetry of the full Lagrangian. It has been shown (Branco et al. 1996) that this is indeed possible through the introduction of a symmetry $Z_{n}$ ⁠. There is some freedom in the choice of n, and the quark $u_{α}$ may be any of the up-type quarks, depending on the specific transformation properties of the quark fields under $Z_{n}$ ⁠. Within this class of models the Higgs bosons may often be relatively light, with masses $\sim 100 GeV$ ⁠. The important point that we want to emphasize is that the constraint of NFC may be too restrictive and other interesting scenarios are possible, with non-vanishing but naturally suppressed FCNYI. The interest in this possibility has been revived by the suggestion (Antaramian et al. 1992; Hall and Weinberg 1993) that some suppression factors could result from approximate family symmetries.

22.12 Main conclusions

We next collect the main conclusions of this chapter on what has to do with CP violation.

•

CP violation in the self-interactions of the scalars arises when the number of gauge-invariant phases between the vacuum expectation values is smaller than the number of terms in the scalar potential which feel those phases.

•

In the two-Higgs-doublet model, there is CP violation in the mixing of the neutral scalar fields if and only if

All neutral scalar fields $S_{k}$ have an interaction with two Z bosons of the form $S_{k} Z_{μ} Z^{μ}$ ⁠;

All pairs of neutral scalar fields have an interaction with a Z boson of the form $Z_{μ} (S_{k} \partial^{μ} S_{l} - S_{l} \partial^{μ} S_{k})$ ⁠.

For any pair of neutral scalar fields $S_{k}$ and $S_{l}$ ⁠, the three interactions $Z_{μ} (S_{k} \partial^{μ} S_{l} - S_{l} \partial^{μ} S_{k}), Z_{μ} Z^{μ} S_{k}$ ⁠, and $Z_{μ} Z^{μ} S_{l}$ ⁠, are simultaneously present.

All neutral scalar fields $S_{k}$ have an interaction with a charged scalar $C^{\pm}$ of the form $S_{k} C^{+} C^{-}$ ⁠.

•

CP is violated if any neutral scalar S has an interaction with a quark q of the form $S \bar{q} (a + ib γ_{5}) q$ ⁠, with the real numbers a and b simultaneously non-zero.

Notes

⁵⁶

In a non-supersymmetric theory, a scalar doublet with $Y = 1 / 2$ is equivalent to a scalar doublet with $Y = - 1 / 2$ ⁠, cf. eqns (11.15) and (11.16). In a supersymmetric theory this is not true any more, because each scalar multiplet belongs to a chiral supermultiplet which also includes a fermion multiplet of definite chirality. Indeed, the C-conjugate of a left-handed fermion is a right-handed antifermion, and not a left-handed antifermion.

⁵⁷

In general (Tsao 1980), Higgs multiplets with weak isospin T and weak hypercharge Y lead to the relation $m_{W} = c_{w} m_{Z}$ provided $T (T + 1) = 3 Y^{2}$ ⁠. Solutions to this equation apart from $T = Y = 0$ and $T = Y = 1 / 2$ are usually not considered, because they correspond to large scalar multiplets which cannot have Yukawa couplings to the known fermions.

⁵⁸

In the next chapter, which will be dedicated to spontaneous CP violation, the starting point will be different, in that we shall postulate the invariance of the Lagrangian, before spontaneous symmetry breaking, under a certain CP transformation, fixed a priori, and require that, after spontaneous symmetry breaking, there is no CP transformation under which the Lagrangian is invariant.

⁵⁹

We shall set the phase $ξ_{W}$ to zero. This leads to a simplification in some equations, without thereby losing generality.

⁶⁰

We are excluding the artificial procedure in which one would be introducing by hand some phases in the definition of otherwise real WB invariants.

⁶¹

The Yukawa interactions of the scalars with the up-type quarks in general also display FCNYI. However, the strongest experimental bounds on this type of interaction arise in the down-type-quark sector.

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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/9780198503996.003.0022

This is an open access publication, available online and distributed under the terms of a Creative Commons Attribution-Non Commercial-No Derivatives 4.0 International licence (CC BY-NC-ND 4.0), a copy of which is available at https://creativecommons.org/licenses/by-nc-nd/4.0/. Subject to this license, all rights are reserved.

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Contents

22 Multi-Higgs-Doublet Models

Abstract

22.1 Introduction

22.2 General multi-Higgs-doublet model

22.3 The two-Higgs-doublet model

22.4 The Higgs basis

22.5 The Higgs basis in the THDM

22.5.1 The potential

22.5.2 The Yukawa interactions

22.6 CP transformation

22.7 CP violation in the scalar potential: simple examples

22.7.1 THDM with a discrete symmetry

22.7.2 Softly broken discrete symmetry

22.7.3 Weinberg model

22.8 General treatment of CP violation

22.8.1 Weak-basis transformations

22.8.2 Weak-basis invariants

22.8.3 CP violation

22.9 CP violation in the two-Higgs-doublet model

22.9.1 $I_{1}$ and $I_{2}$

22.9.2 $I_{3}$ and $I_{4}$

22.9.3 Feynman rules and CP violation: $I_{1}$ and $I_{2}$

22.9.4 Feynman rules and CP violation: $I_{3}$ and $I_{4}$

22.10 Flavour-changing neutral Yukawa interactions

22.11 Mechanisms for natural suppression of the FCNYI

22.11.1 Natural flavour conservation

22.11.2 Non-vanishing but naturally small FCNYI

22.12 Main conclusions

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Contents

22 Multi-Higgs-Doublet Models

Abstract

22.1 Introduction

22.2 General multi-Higgs-doublet model

22.3 The two-Higgs-doublet model

22.4 The Higgs basis

22.5 The Higgs basis in the THDM

22.5.1 The potential

22.5.2 The Yukawa interactions

22.6 CP transformation

22.7 CP violation in the scalar potential: simple examples

22.7.1 THDM with a discrete symmetry

22.7.2 Softly broken discrete symmetry

22.7.3 Weinberg model

22.8 General treatment of CP violation

22.8.1 Weak-basis transformations

22.8.2 Weak-basis invariants

22.8.3 CP violation

22.9 CP violation in the two-Higgs-doublet model

22.9.1 I1 and I2

22.9.2 I3 and I4

22.9.3 Feynman rules and CP violation: I1 and I2

22.9.4 Feynman rules and CP violation: I3 and I4

22.10 Flavour-changing neutral Yukawa interactions

22.11 Mechanisms for natural suppression of the FCNYI

22.11.1 Natural flavour conservation

22.11.2 Non-vanishing but naturally small FCNYI

22.12 Main conclusions

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22.9.1 $I_{1}$ and $I_{2}$

22.9.2 $I_{3}$ and $I_{4}$

22.9.3 Feynman rules and CP violation: $I_{1}$ and $I_{2}$

22.9.4 Feynman rules and CP violation: $I_{3}$ and $I_{4}$