Chapter

26 The Left-Right-Symmetric Model

https://doi.org/10.1093/oso/9780198503996.003.0026

Pages

310–328

Published:

July 1999

Abstract

The main motivation for considering a left-right-symmetric model (LRSM) (Pati and Salam 1974; Mohapatra and Pati 1975; Senjanovic and Mohapatra 1975) is having an extension of the standard model in which parity is a spontaneously broken symmetry. This means that the Lagrangian is symmetric under a parity transformation and this symmetry is only broken by the vacuum. In the SM the left-handed fermions are in doublets and the right-handed fermions are singlets of an SU(2) gauge group. This arrangement is not parity-symmetric, because parity must interchange left-handed and right-handed fermions. The simplest way of having a left-right-symmetric extension of the SM is through the introduction of a second SU(2) gauge group which transforms the right-handed fermions as doublets and the left-handed fermions as singlets. The gauge group of the LRSM is thus SU(2)L®SU(2)R®U(l). When one considers grand unified theories, the gauge group of the LRSM can be elegantly interpreted as a subgroup of S0(10). The lepton fields are in doublets.

26.1 Overview of the model

26.1.1 Introduction

The main motivation for considering a left–right-symmetric model (LRSM) (Pati and Salam 1974; Mohapatra and Pati 1975; Senjanović and Mohapatra 1975) is having an extension of the standard model in which parity is a spontaneously broken symmetry. This means that the Lagrangian is symmetric under a parity transformation and this symmetry is only broken by the vacuum. In the SM the left-handed fermions are in doublets and the right-handed fermions are singlets of an SU(2) gauge group. This arrangement is not parity-symmetric, because parity must interchange left-handed and right-handed fermions. The simplest way of having a left–right-symmetric extension of the SM is through the introduction of a second SU(2) gauge group which transforms the right-handed fermions as doublets and the left-handed fermions as singlets. The gauge group of the LRSM is thus $SU {(2)}_{L} \otimes SU {(2)}_{R} \otimes U (1)$ ⁠. When one considers grand unified theories, the gauge group of the LRSM can be elegantly interpreted as a subgroup of SO(10). The lepton fields are in doublets

L_{L} = (\begin{matrix} v_{L} \\ l_{L} \end{matrix}); L_{R} = (\begin{matrix} v_{R} \\ l_{R} \end{matrix})

(26.1)

of $SU {(2)}_{L}$ and $SU {(2)}_{R}$ ⁠, respectively. The quark fields are in doublets

Q_{L} = (\begin{matrix} p_{L} \\ n_{L} \end{matrix}); Q_{R} = (\begin{matrix} p_{R} \\ n_{R} \end{matrix})

(26.2)

of $SU {(2)}_{L}$ and $SU {(2)}_{R}$ ⁠, respectively. Parity interchanges fermionic and bosonic multiplets of $SU {(2)}_{L}$ with analogous multiplets of $SU {(2)}_{R}$ ⁠; for each multiplet of $SU {(2)}_{L}$ there is a similar multiplet of $SU {(2)}_{R}$ ⁠. Therefore, under parity the fermion fields transform as

\begin{array}{l} L_{L} \leftrightarrow L_{R}, \\ Q_{L} \leftrightarrow Q_{R}, \end{array}

(26.3)

while the gauge bosons $W_{Lk}$ and $W_{Rk}$ (k from 1 to 3) associated with the gauge groups $SU {(2)}_{L}$ and $SU {(2)}_{R}$ ⁠, respectively, transform as

W_{Lk} \leftrightarrow W_{Rk} .

(26.4)

Parity invariance of the Lagrangian constrains the gauge coupling constants $g_{L}$ of $SU {(2)}_{L}$ and $g_{R}$ of $SU {(2)}_{R}$ to be equal. So one has $g_{L} = g_{R} = g$ ⁠. The covariant derivative is

\partial - ig \sum_{k = 1}^{3} (W_{Lk} T_{Lk} + W_{Rk} T_{Rk}) - i g^{'} BY,

(26.5)

where $g^{'}$ is the U(1) coupling constant and Y denotes the (weak) hypercharge, which in the LRSM takes values different from the SM ones. The $T_{Lk}$ and $T_{Rk}$ are the generators of $SU {(2)}_{L}$ and of $SU {(2)}_{R}$ ⁠, respectively.

The formula for the electric charge should be a left–right-symmetric extension of the SM expression $Q = T_{3} + Y$ ⁠. An attractive feature of the LRSM is that, if one makes the obvious extension $Q = T_{L 3} + T_{R 3} + Y$ ⁠, then the hypercharge Y acquires a simple physical meaning. As the lepton doublets in eqn (26.1) have hypercharge $- 1 / 2$ while the quark doublets in eqn (26.2) have hypercharge 1/6, one concludes that a general formula for the hypercharge is $Y = (B - L) / 2$ ⁠, where B and L denote the baryon number and the lepton number, respectively. Therefore in the LRSM one has

Q = T_{L 3} + T_{R 3} + \frac{B - L}{2} .

(26.6)

In view of eqn (26.6), the U(1) factor of the LRSM gauge group is often denoted $U {(1)}_{B - L}$ ⁠.

26.1.2 Gauge couplings

We proceed in a fashion similar to what was done in the treatment of the SM in Chapter 11. We define

\begin{array}{l} T_{L \pm} \equiv \frac{T_{L 1} \pm {iT}_{L 2}}{\sqrt{2}}, & T_{R \pm} \equiv \frac{T_{R 1} \pm {iT}_{R 2}}{\sqrt{2}}, \\ W_{L}^{\pm} \equiv \frac{W_{L 1} \mp {iW}_{L 2}}{\sqrt{2}}, & W_{R}^{\pm} \equiv \frac{W_{R 1} \mp {iW}_{R 2}}{\sqrt{2}} . \end{array}

(26.7)

Then,

\sum_{k = 1}^{2} (W_{Lk} T_{Lk} + W_{Rk} T_{Rk}) = W_{L}^{+} T_{L +} + W_{L}^{-} T_{L -} + W_{R}^{+} T_{R +} + W_{R}^{-} T_{R -} .

(26.8)

The gauge bosons $W_{L}^{\pm}$ and $W_{R}^{\pm}$ in general mix, i.e., they are not eigenstates of mass.

In the neutral sector, instead of g and $g^{'}$ it is convenient to use the angle $θ_{w}$ and the electric-charge unit e, defined by

\begin{array}{l} s_{w} \equiv sin θ_{w} = - \frac{g^{'}}{\sqrt{g^{2} + 2 {g^{'}}^{2}}}, \\ c_{w} \equiv cos θ_{w} = \sqrt{\frac{g^{2} + {g^{'}}^{2}}{g^{2} + 2 {g^{'}}^{2}},} \\ e = {gs}_{w} . \end{array}

(26.9)

It is useful to introduce the neutral gauge bosons A, $X_{1},$ and $X_{2},$ defined by the orthogonal transformation

(\begin{array}{c} W_{L 3} \\ W_{R 3} \\ B \end{array}) = (\begin{array}{c} - s_{w} & c_{w} & 0 \\ - s_{w} & - s_{w}^{2} / c_{w} & \sqrt{c_{w}^{2} - s_{w}^{2}} / c_{w} \\ \sqrt{c_{w}^{2} - s_{w}^{2}} & s_{w} \sqrt{c_{w}^{2} - s_{w}^{2}} / c_{w} & s_{w} / c_{w} \end{array}) (\begin{array}{c} A \\ X_{1} \\ X_{2} \end{array}),

(26.10)

where A is to be identified with the photon, and we choose the sign of $\sqrt{c_{w}^{2} - s_{w}^{2}}$ such that it equals $g / \sqrt{g^{2} + 2 {g^{'}}^{2}}$ ⁠. The neutral gauge couplings are given by

\begin{array}{l} g (W_{L 3} T_{L 3} + W_{R 3} T_{R 3}) + g^{'} B Y = - e A Q + \frac{g}{c_{w}} X_{1} (T_{L 3} - Q s_{w}^{2}) \\ + \frac{g \sqrt{c_{w}^{2} - s_{w}^{2}}}{c_{w}} X_{2} (T_{R 3} - Y \frac{s_{w}^{2}}{c_{w}^{2} - s_{w}^{2}}) . \end{array}

(26.11)

With the above definitions, $X_{1}$ interacts like the Z of the SM. However, $X_{1}$ is not in general an eigenstate of mass, it mixes with $X_{2}$ ⁠.

26.1.3 Scalar multiplets

The Higgs sector has several important functions to perform:

It should lead to an appropriate spontaneous breaking of the $SU {(2)}_{L} \otimes SU {(2)}_{R} \otimes U (1)$ gauge group. In view of the left-handed character of the observed charged-current interaction, the breaking of the LRSM gauge group should occur in two steps (Senjanović 1979): at a first stage—at high energy—the breaking should be to $SU {(2)}_{L} \otimes U {(1)}_{Y}$ ⁠, where Y denotes the hypercharge of the SM. Parity invariance is broken at this stage. The non-observation of right-handed charged currents at low energies requires that the mass $m_{W_{R}}$ of $W_{R}^{\pm}$ be substantially larger than $m_{W_{L}} .$ ⁶⁸ At a second stage—at a lower energy—the SM gauge group should be broken to the U(1) of electromagnetism.

It should give quarks and charged leptons a mass, while at the same time giving either zero or naturally small masses to the neutrinos.

Let us first consider the requirement of fermion masses. A general $SU {(2)}_{L}$ transformation is represented in the doublet representation by the $2 \times 2$ unitary matrix $U_{L}$ ⁠. Similarly, an $SU {(2)}_{R}$ transformation is represented in the doublet representation by the $2 \times 2$ unitary matrix $U_{R}$ ⁠. The quark doublets transform as $Q_{L} \to U_{L} Q_{L}$ and $Q_{R} \to U_{R} Q_{R}$ ⁠. Therefore, a scalar multiplet $ϕ$ which gives mass to the quarks via a Yukawa coupling $\bar{Q_{L}} ϕ Q_{R}$ must be a $2 \times 2$ matrix of fields transforming as $ϕ \to U_{L} ϕ U_{R}^{†}$ ⁠. Moreover, as $Q_{L}$ and $Q_{R}$ have the same hypercharge, the hypercharge of $ϕ$ must be zero. Then,

ϕ = (\begin{array}{c} φ_{1}^{0^{†}} & φ_{2}^{+} \\ - φ_{1}^{-} & φ_{2}^{0} \end{array}),

(26.12)

where we have already displayed the electric charge of each component field.

The multiplet

\tilde{ϕ} \equiv τ_{2} ϕ^{†^{T}} τ_{2} = (\begin{array}{c} {φ_{2}^{0}}^{†} & φ_{1}^{+} \\ - φ_{2}^{-} & φ_{1}^{0} \end{array})

(26.13)

transforms under a gauge transformation in the same way as $ϕ$ ⁠. Notice that

(\begin{array}{c} φ_{1}^{+} \\ φ_{1}^{0} \end{array}) and (\begin{array}{c} φ_{2}^{+} \\ φ_{2}^{0} \end{array})

(26.14)

are doublets of $SU {(2)}_{L}$ ⁠; from this point of view, the LRSM is like a two-Higgs-doublet model with an extra SU(2) symmetry. Indeed,

(\begin{array}{c} φ_{1}^{+} \\ - φ_{2}^{0^{†}} \end{array}) and (\begin{array}{c} φ_{2}^{+} \\ - φ_{1}^{0^{†}} \end{array})

(26.15)

are doublets of $SU {(2)}_{R}$ ⁠.

The VEV of $φ_{1}^{0}$ is $k_{1}$ and the VEV of $φ_{2}^{0}$ is $k_{2}$ ⁠. Both $k_{1}$ and $k_{2}$ are in general complex. We assume that the scalar potential is such that the other components of $ϕ$ do not acquire a VEV. The Yukawa couplings of $ϕ$ and $\tilde{ϕ}$ will generate Dirac masses for all fermions, including neutrinos. However, $ϕ$ is not sufficient, other Higgs multiplets have to be introduced. Both $φ_{1}^{0}$ and $φ_{2}^{0}$ have $T_{L 3} = - 1 / 2, T_{R 3} = 1 / 2$ ⁠, and $Y = 0$ ⁠. Therefore, when they acquire a VEV they keep unbroken the two U(1) groups generated by $T_{L 3} + T_{R 3}$ and by Y. Thus, $k_{1}$ and $k_{2}$ keep two neutral gauge bosons massless, instead of giving mass to every gauge boson but the photon. Furthermore, $k_{1}$ and $k_{2}$ cannot perform the spontaneous breaking of parity symmetry. This is because both $φ_{1}^{0}$ and $φ_{2}^{0}$ are components of doublets of $SU {(2)}_{L}$ and $SU {(2)}_{R}$ ⁠, and they cannot distinguish between the two gauge groups. We must introduce some extra Higgs multiplets which distinguish between the two SU(2) groups. In the first versions of the LRSM, this was achieved by introducing, apart from $ϕ$ ⁠, multiplets $χ_{L}$ and $χ_{R}$ transforming as doublets of $SU {(2)}_{L}$ and of $SU {(2)}_{R}$ ⁠, respectively, while being singlets of the other SU(2) gauge group, and having $B - L = 1$ ⁠. At present, a more attractive choice is introducing a triplet of $SU {(2)}_{L}$ which is a singlet of $SU {(2)}_{R}$ ⁠, together with a triplet of $SU {(2)}_{R}$ which is a singlet of $SU {(2)}_{L}$ ⁠. Both triplets are chosen to have $B - L = 2$ ⁠. The advantage of using these triplets is that their Yukawa couplings can generate $∣ Δ L ∣ = 2$ Majorana masses, thus leading to naturally small neutrino masses through the seesaw mechanism (see § 25.3). We may write a triplet $Δ^{'}$ of SU(2) as

Δ^{'} = (\begin{array}{c} Δ_{1} \\ Δ_{2} \\ Δ_{3} \end{array}) .

(26.16)

An infinitesimal SU(2) transformation of $Δ^{'}$ reads

Δ^{'} \to [1 + i (θ_{+} T_{+}^{(3)} + θ_{+}^{*} T_{-}^{(3)} + θ_{3} T_{3}^{(3)})] Δ^{'} .

(26.17)

The infinitesimal parameter $θ_{3}$ is real but the infinitesimal parameter $θ_{+}$ is complex. It is convenient to write the triplet in the form of a $2 \times 2$ traceless matrix

Δ \equiv \sqrt{2} (Δ_{3} T_{-}^{(2)} + Δ_{2} T_{3}^{(2)} - Δ_{1} T_{+}^{(2)}) = (\begin{array}{c} Δ_{2} / \sqrt{2} & - Δ_{1} \\ Δ_{3} & - Δ_{2} / \sqrt{2} \end{array}) .

(26.18)

By using the commutation algebra of the SU(2) generators, we find that the SU(2) transformation in eqn (26.17) may be written

Δ \to [1 + i (θ_{+} T_{+}^{(2)} + θ_{+}^{*} T_{-}^{(2)} + θ_{3} T_{3}^{(2)})] Δ [1 - i (θ_{+} T_{+}^{(2)} + θ_{+}^{*} T_{-}^{(2)} + θ_{3} T_{3}^{(2)})] .

(26.19)

Generalizing to a non-infinitesimal SU(2) transformation, represented in the doublet representation by the unitary matrix U, this means that $Δ \to U Δ U^{†}$ ⁠.

Thus, the triplets are $2 \times 2$ traceless matrices

Δ_{L} = (\begin{array}{c} Δ_{L}^{+} / \sqrt{2} & - Δ_{L}^{+ +} \\ Δ_{L}^{0} & - Δ_{L}^{+} / \sqrt{2} \end{array}) and Δ_{R} = (\begin{array}{c} Δ_{R}^{+} / \sqrt{2} & - Δ_{R}^{+ +} \\ Δ_{R}^{0} & - Δ_{R}^{+} / \sqrt{2} \end{array}),

(26.20)

which transform as $Δ_{L} \to U_{L} Δ_{L} U_{L}^{†}$ and $Δ_{R} \to U_{R} Δ_{L} U_{R}^{†}$ ⁠, and have $B - L = 2$ ⁠. This form of writing triplets is particularly convenient since eventually we want to build gauge singlets out of tensor products of triplets and doublets of $SU {(2)}_{L}$ and $SU {(2)}_{R}$ ⁠. The VEV of $Δ_{L}^{0}$ is $F_{L}$ and the VEV of $Δ_{R}^{0}$ is $F_{R}$ ⁠. We assume that all other Higgs fields have vanishing VEV, so that the U(1) of electromagnetism remains unbroken.

26.1.4 Gauge-boson masses

The covariant derivative of $φ_{1}^{+}$ is

\begin{array}{l} \partial φ_{1}^{+} - i [g (\frac{W_{L}^{+}}{\sqrt{2}} φ_{1}^{0} - \frac{W_{R}^{+}}{\sqrt{2}} φ_{2}^{0^{†}}) - eA φ_{1}^{+} \\ + \frac{g (c_{w}^{2} - s_{w}^{2})}{2 c_{w}} X_{1} φ_{1}^{+} + \frac{g \sqrt{c_{w}^{2} - s_{w}^{2}}}{2 c_{w}} X_{2} φ_{1}^{+}] . \end{array}

(26.21)

Substituting the fields by their VEVs and taking the squared modulus, we obtain

{∣ \frac{g}{\sqrt{2}} (W_{L}^{+} k_{1} - W_{R}^{+} k_{2}^{*}) ∣}^{2} .

(26.22)

Analogously, from the covariant derivatives of $φ_{2}^{+}, φ_{1}^{0}$ ⁠, and $φ_{2}^{0}$ ⁠, we obtain

\begin{array}{c} {∣ \frac{g}{\sqrt{2}} (W_{L}^{+} k_{2} - W_{R}^{+} k_{1}^{*}) ∣}^{2}, \\ {∣ \frac{{gk}_{1}}{2 c_{w}} (- X_{1} + \sqrt{c_{w}^{2} - s_{w}^{2}} X_{2}) ∣}^{2}, \\ {∣ \frac{{gk}_{2}}{2 c_{w}} (- X_{1} + \sqrt{c_{w}^{2} - s_{w}^{2}} X_{2}) ∣}^{2}, \end{array}

(26.23)

respectively. The covariant derivatives of $Δ_{L}^{+ +}$ and of $Δ_{R}^{+ +}$ do not yield any mass term for the gauge bosons, but the covariant derivatives of $Δ_{L}^{+}, Δ_{R}^{+}, Δ_{L}^{0}$ ⁠, and $Δ_{R}^{0}$ ⁠, yield

\begin{array}{c} {∣ {gW}_{L}^{+} F_{L} ∣}^{2}, \\ {∣ {gW}_{R}^{+} F_{R} ∣}^{2}, \\ {∣ \frac{g}{c_{w}} (X_{1} + \frac{s_{w}^{2}}{\sqrt{c_{w}^{2} - s_{w}^{2}}} X_{2}) F_{L} ∣}^{2}, \\ {∣ \frac{{gc}_{w}}{\sqrt{c_{w}^{2} - s_{w}^{2}}} X_{2} F_{R} ∣}^{2}, \end{array}

(26.24)

respectively. Putting everything together, we obtain the mass terms for the charged gauge bosons,

g^{2} (W_{L}^{-} W_{R}^{-}) (\begin{array}{c} \frac{{∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}}{2} + {∣ F_{L} ∣}^{2} & - k_{1}^{*} k_{2}^{*} \\ - k_{1} k_{2} & \frac{{∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}}{2} + {∣ F_{R} ∣}^{2} \end{array}) (\begin{array}{c} W_{L}^{+} \\ W_{R}^{+} \end{array}),

(26.25)

and for the neutral gauge bosons,

\frac{g^{2}}{c_{w}^{2}} (X_{1} \sqrt{c_{w}^{2} - s_{w}^{2}} X_{2}) ℳ (\begin{array}{c} X_{1} \\ \sqrt{c_{w}^{2} - s_{w}^{2}} X_{2} \end{array}),

(26.26)

where

ℳ = (\begin{matrix} \frac{{| k_{1} |}^{2} + {| k_{2} |}^{2}}{4} + {| F_{L} |}^{2} & - \frac{{| k_{1} |}^{2} + {| k_{2} |}^{2}}{4} + \frac{s_{w}^{2}}{c_{w}^{2} - s_{w}^{2}} {| F_{L} |}^{2} \\ - \frac{{| k_{1} |}^{2} + {| k_{2} |}^{2}}{4} + \frac{s_{w}^{2}}{c_{w}^{2} - s_{w}^{2}} {| F_{L} |}^{2} & \frac{{| k_{1} |}^{2} + {| k_{2} |}^{2}}{4} + \frac{s_{w}^{4} {| F_{L} |}^{2} + c_{w}^{4} {| F_{R} |}^{2}}{{(c_{w}^{2} - s_{w}^{2})}^{2}} \end{matrix}) .

(26.27)

In eqn (26.25) we see that $W_{L}^{\pm}$ and $W_{R}^{\pm}$ mix if $k_{1}$ and $k_{2}$ are simultaneously non-zero.⁶⁹The product $k_{1} k_{2}$ has $T_{L 3} = - 1$ and $T_{R 3} = + 1$ ⁠. We may therefore without loss of generality choose a gauge in which $k_{1} k_{2}$ is real, rendering the $W_{L}^{\pm} - W_{R}^{\pm}$ mixing real. The physical charged gauge bosons are the eigenstates of the mass matrix in eqn (26.25) and can be written

(\begin{array}{l} W_{1} \\ W_{2} \end{array}) = (\begin{array}{c} cos ζ & - sin ζ \\ sin ζ & cos ζ \end{array}) (\begin{array}{l} W_{L} \\ W_{R} \end{array}) .

(26.28)

It will be shown in § 26.2 that there is a region of parameters of the Higgs potential which leads to a minimum with $∣ F_{R} ∣ \neq ∣ F_{L} ∣$ ⁠. In this case, parity is spontaneously broken. The crucial assumption of the LRSM is that $∣ F_{R} ∣$ is much larger than $∣ k_{1} ∣, ∣ k_{2} ∣$ ⁠, and $∣ F_{L} ∣$ ⁠. In this case, the mixing angle $ζ$ and the masses of $W_{1}$ and $W_{2}$ are approximately given by

\begin{array}{l} ζ \approx \frac{k_{1} k_{2}}{{∣ F_{R} ∣}^{2}}, \\ m_{W_{1}}^{2} \approx g^{2} ({∣ F_{L} ∣}^{2} + \frac{{∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}}{2}), \\ m_{W_{2}}^{2} \approx g^{2} ({∣ F_{R} ∣}^{2} + \frac{{∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}}{2}) . \end{array}

(26.29)

Since $ζ ≪ 1, W_{1}$ and $W_{2}$ coincide, to a good approximation, with $W_{L}$ and $W_{R}$ ⁠, respectively. Similarly, $X_{1}$ and $X_{2}$ are approximate eigenstates of mass, with squared masses

\begin{array}{l} m_{X_{1}}^{2} \approx \frac{g^{2}}{c_{w}^{2}} (\frac{{∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}}{2} + 2 {∣ F_{L} ∣}^{2}), \\ m_{X_{2}}^{2} \approx \frac{g^{2}}{c_{w}^{2}} [\frac{({∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}) (c_{w}^{2} - s_{w}^{2})}{2} + 2 \frac{s_{w}^{4} {∣ F_{L} ∣}^{2} + c_{w}^{4} {∣ F_{R} ∣}^{2}}{c_{w}^{2} - s_{w}^{2}}] . \end{array}

(26.30)

Both $W_{R}^{\pm}$ and $X_{2}$ are much heavier than $W_{L}^{\pm}$ and $X_{1}$ ⁠, and therefore the interactions mediated by the former gauge bosons are suppressed when compared to the ones mediated by the latter. In particular, the charged gauge interactions of the right-handed fermions are much weaker than those among the left-handed fermions. The gauge boson $W_{L}^{\pm}$ is identified with the $W^{\pm}$ of the SM, and $X_{1}$ is identified with the Z of the SM.

We would like the mass of $X_{1}$ to be approximately equal to the mass of $W_{L}^{\pm}$ divided by $c_{w}$ ⁠, because this is an experimental fact. From eqns (26.29) and (26.30) we see that, in order to obtain this, we must assume ${∣ F_{L} ∣}^{2}$ to be much smaller than ${∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2}$ ⁠. Indeed, the SM relationship $m_{W} = c_{w} m_{Z}$ is a consequence of the fact that the breaking of $SU {(2)}_{L}$ is effected by doublets. The VEV of a triplet of $SU {(2)}_{L}$ must be very small compared to the VEV of at least one of the doublets.

26.2 Spontaneous symmetry breaking

This section contains an analysis of the scalar potential of the LRSM, and of the conditions under which spontaneous P and CP breaking may be obtained. Some readers may prefer to skip all but the first subsection.

26.2.1 The scalar potential

Under a gauge transformation,

\begin{array}{c} ϕ \to U_{L} ϕ U_{R}^{†}, \tilde{ϕ} \to U_{L} \tilde{ϕ} U_{R}^{†}, \\ ϕ^{†} \to U_{R} ϕ^{†} U_{L}^{†}, {\tilde{ϕ}}^{†} \to U_{R} {\tilde{ϕ}}^{†} U_{L}^{†} . \end{array}

(26.31)

Parity interchanges $U_{L}$ and $U_{R}$ ⁠. Therefore, $ϕ$ should transform under parity into some unitary combination of $ϕ^{†}$ and ${\tilde{ϕ}}^{†}$ ⁠. We shall assume that $ϕ$ is transformed into $ϕ^{†}$ ⁠. It can be shown (Ecker et al. 1981a,b) that this is the only choice leading to realistic quark masses and mixings. Thus, we assume that the Higgs multiplets transform under parity in the following way:

ϕ \overset{P}{\to} ϕ^{†}, Δ_{L} \overset{P}{\leftrightarrow} Δ_{R} .

(26.32)

After elimination of all redundant terms, the scalar potential may then be written

\begin{array}{l} V = μ_{1} tr (ϕ^{†} ϕ) + μ_{2} tr ({\tilde{ϕ}}^{†} ϕ + ϕ^{†} \tilde{ϕ}) \\ + λ_{1} {[tr (ϕ^{†} ϕ)]}^{2} + λ_{2} tr ({\tilde{ϕ}}^{†} ϕ {\tilde{ϕ}}^{†} ϕ + ϕ^{†} \tilde{ϕ} ϕ^{†} \tilde{ϕ}) \\ + λ_{3} tr (ϕ^{†} ϕ) tr ({\tilde{ϕ}}^{†} ϕ + ϕ^{†} \tilde{ϕ}) + λ_{4} tr ({\tilde{ϕ}}^{†} ϕ) tr (ϕ^{†} \tilde{ϕ}) \\ + μ_{3} tr (Δ_{L}^{†} Δ_{L} + Δ_{R}^{†} Δ_{R}) + λ_{5} tr (Δ_{L}^{†} Δ_{L} Δ_{L}^{†} Δ_{L} + Δ_{R}^{†} Δ_{R} Δ_{R}^{†} Δ_{R}) \\ + λ_{6} tr (Δ_{L}^{†} Δ_{L}^{†} Δ_{L} Δ_{L} + Δ_{R}^{†} Δ_{R}^{†} Δ_{R} Δ_{R}) + λ_{7} tr (Δ_{L}^{†} Δ_{L}) tr (Δ_{R}^{†} Δ_{R}) \\ + λ_{8} [tr (Δ_{L} Δ_{L}) tr (Δ_{R}^{†} Δ_{R}^{†}) + tr (Δ_{R} Δ_{R}) tr (Δ_{L}^{†} Δ_{L}^{†})] \\ + λ_{9} tr (Δ_{L}^{†} Δ_{L} ϕ ϕ^{†} + Δ_{R}^{†} Δ_{R} ϕ^{†} ϕ) + λ_{10} tr (Δ_{L}^{†} Δ_{L} \tilde{ϕ} {\tilde{ϕ}}^{†} + Δ_{R}^{†} Δ_{R} {\tilde{ϕ}}^{†} \tilde{ϕ}) \\ + λ_{11} tr (Δ_{L}^{†} Δ_{L} ϕ {\tilde{ϕ}}^{†} + Δ_{R}^{†} Δ_{R} ϕ^{†} \tilde{ϕ}) + λ_{11}^{*} tr (Δ_{L}^{†} Δ_{L} \tilde{ϕ} ϕ^{†} + Δ_{R}^{†} Δ_{R} {\tilde{ϕ}}^{†} ϕ) \\ + λ_{12} tr (Δ_{L}^{†} ϕ Δ_{R} ϕ^{†} + Δ_{R}^{†} ϕ^{†} Δ_{L} ϕ) + λ_{13} tr (Δ_{L}^{†} \tilde{ϕ} Δ_{R} ϕ^{†} + Δ_{R}^{†} {\tilde{ϕ}}^{†} Δ_{L} ϕ) \\ + λ_{14} tr (Δ_{L}^{†} ϕ Δ_{R} {\tilde{ϕ}}^{†} + Δ_{R}^{†} ϕ^{†} Δ_{L} \tilde{ϕ}) . \end{array}

(26.33)

We have used the letter $μ$ for couplings with dimension of mass squared, and the letter $λ$ for dimensionless couplings. All couplings except maybe $λ_{11}$ are real because of Hermiticity together with parity.

Let us consider the vacuum expectation value of the potential. The terms in the potential with coefficients $λ_{6}$ and $λ_{8}$ have zero VEV. We introduce the notation

\begin{array}{c} ∣ k_{1} ∣ = K cos s, \\ ∣ k_{2} ∣ = K sin s, \\ ∣ F_{R} ∣ = F cos r, \\ ∣ F_{L} ∣ = F sin r, \end{array}

(26.34)

where K and F are two positive quantities with mass dimension, while s and r are two angles of the first quadrant. Also, we denote the two gauge-invariant vacuum phases

\begin{array}{l} δ \equiv arg (k_{1} k_{2} F_{L}^{*} F_{R}), \\ α \equiv arg (k_{1} k_{2}^{*}) . \end{array}

(26.35)

One then obtains

\begin{array}{l} 〈 0 | V | 0 〉 = μ_{1} K^{2} + λ_{1} K^{4} + μ_{3} F^{2} + λ_{5} F^{4} + \frac{λ_{9} + λ_{10}}{2} F^{2} K^{2} \\ + 2 μ_{2} K^{2} \sin 2 s \cos α + λ_{2} K^{4} \sin^{2} 2 s \cos 2 α \\ + 2 λ_{3} K^{4} \sin 2 s \cos α + λ_{4} K^{4} \sin^{2} 2 s \\ + \frac{λ_{7} - 2 λ_{5}}{4} F^{4} \sin^{2} 2 r + \frac{λ_{9} - λ_{10}}{2} F^{2} K^{2} \cos 2 s \\ + F^{2} K^{2} \sin 2 s (Re λ_{11} \cos α - Im λ_{11} \cos 2 r \sin α) \\ + F^{2} K^{2} \sin 2 r [λ_{12} \sin s \cos s \cos δ \\ + λ_{13} \cos^{2} s \cos (δ + α) + λ_{14} \sin^{2} s \cos (δ - α)] . \end{array}

(26.36)

26.2.2 Spontaneous breaking of P

In order to analyse whether parity can be spontaneously broken, one has to examine in detail the r-dependence of the Higgs potential. We shall assume that the parameters of the scalar potential are chosen so that F is much larger than K. In order to obtain this, one clearly has to make some fine-tuning of the couplings in the potential. We shall assume, though, that no other fine-tuning of parameters beyond this one is done, i.e., that no coupling, and no combination of couplings, is assumed to be $\sim K^{2} / F^{2}$ in order to obtain spontaneous breaking of either parity or CP.

As $F ≫ K$ ⁠, the r-dependent part of the potential is dominated by the term $(λ_{7} - 2 λ_{5}) F^{4} {sin}^{2} r {cos}^{2} r$ ⁠. If $λ_{7} - 2 λ_{5} < 0$ ⁠, the minimum of this term occurs for

sin r = cos r = \frac{1}{\sqrt{2}} \Leftrightarrow ∣ F_{R} ∣ = ∣ F_{L} ∣ .

(26.37)

For $λ_{7} - 2 λ_{5} > 0$ ⁠, one has two possible minima:

\begin{array}{l} sin r = 0 \Leftrightarrow F_{L} = 0, \\ sin r = 1 \Leftrightarrow F_{R} = 0 . \end{array}

(26.38)

The extremum of eqn (26.37) is parity-conserving, while those of eqns (26.38) lead to spontaneous parity violation. Obviously, we are interested in the minimum corresponding to $F_{L} = 0$ ⁠, at this level of approximation.

We next consider the subleading r-dependent terms in the potential. There are two such terms, one proportional to $F^{2} K^{2} cos 2 r$ and another proportional to $F^{2} K^{2} sin 2 r$ ⁠. These subleading terms pull the minimum of the potential to $sin 2 r \sim K^{2} / F^{2}$ ⁠, thus leading to

F_{L} \sim \frac{K^{2}}{F} \approx \frac{K^{2}}{F_{R}} .

(26.39)

One has thus a minimum characterized by $F_{R}^{2} ≫ {∣ k_{1} ∣}^{2} + {∣ k_{2} ∣}^{2} ≫ F_{L}^{2}$ ⁠, ensuring that $m_{W} \approx c_{w} m_{Z}$ ⁠, as desired. The vacuum state is related by parity symmetry to another possible vacuum with the same energy density, characterized by $∣ F_{R} / F_{L} ∣ \sim K^{2} / F^{2}$ ⁠. We assume that it is the first vacuum which is realized in nature, or at least in that part of the Universe in which we live. Then, the lowest-energy charged-current weak interaction is among the left-handed fermions and not among the right-handed ones, as experimentally observed.

26.2.3 Spontaneous CP breaking

We next investigate whether CP can be spontaneously broken in the minimal LRSM. In the LRSM it is natural to assume CP to be a spontaneously broken symmetry, since parity is spontaneously broken too. We impose CP at the Lagrangian level, assuming the trivial CP transformation:

ϕ \overset{CP}{\to} ϕ^{†^{T}}, Δ_{L} \overset{CP}{\to} Δ_{L}^{†^{T}}, Δ_{R} \overset{CP}{\to} Δ_{R}^{†^{T}} .

(26.40)

CP invariance then constrains $λ_{11}$ to be real. It is convenient to write the VEV of the potential as

\begin{array}{l} 〈 0 | V | 0 〉 = μ_{1} K^{2} + λ_{1} K^{4} + μ_{3} F^{2} + λ_{5} F^{4} + \frac{λ_{9} + λ_{10}}{2} F^{2} K^{2} \\ + λ_{4} K^{4} \sin^{2} 2 s + \frac{λ_{9} - λ_{10}}{2} F^{2} K^{2} \cos 2 s \\ + (m_{3} K^{2} + l_{6} K^{4} + l_{7} F^{2} K^{2}) \sin 2 s \cos α + l_{8} K^{4} \sin^{2} 2 s \cos 2 α \\ + V_{r}, \end{array}

(26.41)

with

\begin{array}{l} V_{r} = l_{9} F^{4} {sin}^{2} 2 r + F^{2} K^{2} V_{δ} sin 2 r, \\ V_{δ} = l_{10} sin 2 s cos δ + l_{11} {sin}^{2} s cos (δ - α) + l_{12} {cos}^{2} s cos (δ + α) . \end{array}

(26.42)

We have changed the notation in the following way: $2 μ_{2} \to m_{3}, 2 λ_{3} \to l_{6}$ ⁠, $Re λ_{11} \to l_{7}, λ_{2} \to l_{8}, λ_{7} - 2 λ_{5} \to 4 l_{9}, λ_{12} \to 2 l_{10}, λ_{14} \to l_{11}$ ⁠, and $λ_{13} \to l_{12}$ ⁠. All the $δ$ -dependent terms are in $V_{δ}$ ⁠, and all the r-dependent terms are in $V_{r}$ ⁠.

Consider the minimization of $V_{r}$ as a function of r. We choose $l_{9} > 0$ as in the previous subsection. We may always obtain $V_{δ} < 0$ ⁠, if necessary by transforming $δ \to π + δ$ ⁠. With $V_{δ} < 0$ ⁠, the minimum occurs at the extremum

sin 2 r = - \frac{V_{δ} K^{2}}{2 l_{9} F^{2}} ≪ 1 .

(26.43)

This gives $V_{r} = - V_{δ}^{2} K^{4} / (4 l_{9})$ ⁠.

We proceed with the minimization relative to $δ$ ⁠. Defining

Δ \equiv {[l_{10} sin 2 s + (l_{11} {sin}^{2} s + l_{12} {cos}^{2} s) cos α]}^{2} + {(l_{11} {sin}^{2} s - l_{12} {cos}^{2} s)}^{2} {sin}^{2} α,

(26.44)

the minimum is given by

\begin{array}{l} \sin δ = \frac{(l_{11} \sin^{2} s - l_{12} \cos^{2} s) \sin α}{\sqrt{Δ}}, \\ \cos δ = \frac{l_{10} \sin 2 s + (l_{11} \sin^{2} s + l_{12} \cos^{2} s) \cos α}{\sqrt{Δ}}, \end{array}

(26.45)

which leads to $V_{δ} = \sqrt{Δ}$ at the minimum. As we want $V_{δ} < 0$ ⁠, we choose the negative sign for the square root.

Under these conditions, the extremum condition for $α$ is

\begin{array}{l} 0 = - \frac{\partial 〈 0 | V | 0 〉}{\partial α} = K^{2} \sin 2 s \sin α [- \frac{l_{10}}{2 l_{9}} (l_{11} \sin^{2} s + l_{12} \cos^{2} s) K^{2} \\ + m_{3} + l_{6} K^{2} + l_{7} F^{2} \\ + (4 l_{8} - \frac{l_{11} l_{12}}{2 l_{9}}) K^{2} \sin 2 s \cos α] . \end{array}

(26.46)

The trivial solution of eqn (26.46) is $sin α = 0$ ⁠, which implies $sin δ = 0$ ⁠, see eqns (26.45), and is therefore a CP-conserving solution. If we want to have spontaneous CP breaking we must choose the non-trivial solution of eqn (26.46),

cos α = \frac{2 l_{9} (m_{3} + l_{6} K^{2} + l_{7} F^{2}) - l_{10} (l_{11} {sin}^{2} s + l_{12} {cos}^{2} s) K^{2}}{(l_{11} l_{12} - 8 l_{8} l_{9}) K^{2} sin 2 s} .

(26.47)

However, because $∣ cos α ∣$ cannot exceed 1, this solution only exists if $l_{7} \sim K^{2} / F^{2}$ ⁠, or else if some cancellation occurs in the numerator such that it ends up being of order $K^{2}$ instead of being $\sim F^{2}$ ⁠, as one should a priori expect. We conclude that having spontaneous CP violation in the minimal LRSM requires fine-tuning.

Our derivation was based on the assumption that under CP the scalar fields transform as in eqn (26.40). We might try and define CP symmetry in a different way. However, it can be shown that no spontaneous-CP-breaking solution ever exists in the minimal LRSM unless some unnatural fine-tuning is assumed, in the sense that some quantity which should be of order $F^{2}$ is assumed to be of order $K^{2}$ instead. Thus, spontaneous CP violation is unnatural in the minimal LRSM (Branco and Lavoura 1985).

There are however extensions of the LRSM, with an enlarged scalar sector, in which spontaneous CP breaking is possible without any contrived fine-tuning. The simplest extensions of the minimal LRSM which can lead to natural spontaneous CP breaking are:

Add a real scalar singlet $η$ which transforms under P and under CP in the following way (Chang et al. 1984):

η \overset{P}{\to} - η, η \overset{CP}{\to} - η,

(26.48)

while the fields $ϕ, Δ_{L}$ ⁠, and $Δ_{R}$ still have the transformation properties in eqns (26.32) and (26.40). In this case there are new terms in the Higgs potential, in particular $iη tr ({\tilde{ϕ}}^{†} ϕ - ϕ^{†} \tilde{ϕ})$ and $η^{2} tr ({\tilde{ϕ}}^{†} ϕ + ϕ^{†} \tilde{ϕ})$ ⁠, which make it possible to have spontaneous CP violation without fine-tuning of couplings.

Add another scalar multiplet $ϕ^{'}$ transforming under the gauge group in the same way as $ϕ$ ⁠. In this case, there are many more phase-dependent terms in the Higgs potential and, again, a CP-breaking vacuum can be obtained without fine-tuning.

26.3 Quark masses and mixing matrices

26.3.1 Mass matrices

The Yukawa couplings of the quarks are given by

ℒ_{Y}^{(q)} = \bar{Q_{L}} (ϕ Γ_{1} + \tilde{ϕ} Γ_{2}) Q_{R} + \bar{Q_{R}} (ϕ^{†} Γ_{1}^{†} + {\tilde{ϕ}}^{†} Γ_{2}^{†}) Q_{L},

(26.49)

where $Γ_{1}$ and $Γ_{2}$ are $n_{g} \times n_{g}$ matrices in generation space. We shall assume that $ϕ$ transforms under parity as in eqn (26.32). As we have mentioned, this is the only choice leading to a realistic quark spectrum. Taking into account that parity interchanges $Q_{La}$ and $Q_{Ra}$ —where $a = 1, 2, \dots, n_{g}$ is a generation index—it is clear that parity invariance constrains the Yukawa-coupling matrices to be Hermitian, i.e., $Γ_{1} = Γ_{1}^{†}$ and $Γ_{2} = Γ_{2}^{†}$ ⁠.

The mass matrices for the up-type and down-type quarks, defined by the mass terms $\bar{p_{L}} M_{p} p_{R}$ and $\bar{n_{L}} M_{n} n_{R}$ ⁠, are, respectively,

\begin{array}{c} M_{p} = k_{1}^{*} Γ_{1} + k_{2}^{*} Γ_{2} \\ M_{n} = k_{2} Γ_{1} + k_{1} Γ_{2} . \end{array}

(26.50)

These are not the most general complex matrices, still they do not lead to any constraints on observable quantities like the quark masses.

26.3.2 Mixing matrices and CP-violating phases

The quark mass matrices are bi-diagonalized in the usual way, see eqns (12.14) and (12.15). The charged-current Lagrangian can be written in terms of the quark mass eigenstates as

\frac{g}{\sqrt{2}} \bar{u} γ^{μ} (W_{Lμ}^{+} V_{L} \frac{1 - γ_{5}}{2} + W_{Rμ}^{+} V_{R} \frac{1 + γ_{5}}{2}) d + H . c .,

(26.51)

where

\begin{array}{c} V_{L} \equiv U_{L}^{p †} U_{L}^{n}, \\ V_{R} \equiv U_{R}^{p †} U_{R}^{n} \end{array}

(26.52)

are the charged-current mixing matrices. They are $n_{g} \times n_{g}$ unitary matrices. Just as in the SM, the neutral-current Lagrangian does not change its form when expressed in terms of the quark mass eigenstates $u_{α}$ and $d_{k}$ ⁠. This is due to the fact that all quark fields of a given charge and helicity have the same $T_{L 3}, T_{R 3},$ and Y. Hence, the neutral gauge interaction is not a source of CP violation in the LRSM.

In the charged-current Lagrangian we should take into account the fact that $W_{L}^{\pm}$ and $W_{R}^{\pm}$ are not the eigenstates of propagation, because they mix. However, as we have stressed before, the $W_{L}^{\pm} - W_{R}^{\pm}$ mixing may be made real by a gauge choice, and therefore it is not a source of CP violation.

In general, the two mixing matrices $V_{L}$ and $V_{R}$ contain a total of $n_{g} (n_{g} + 1)$ phases. However, we may rephase the quark fields,

\begin{array}{c} u_{Lα}^{'} = exp (i θ_{Lα}) u_{Lα}, \\ u_{Rα}^{'} = exp (i θ_{Rα}) u_{Rα}, \\ d_{Lk}^{'} = exp (i χ_{Lk}) d_{Lk}, \\ d_{Rk}^{'} = exp (i χ_{Rk}) d_{Rk} . \end{array}

(26.53)

Then,

\begin{array}{c} {(V_{L}^{'})}_{αk} = exp [i (θ_{Lα} - χ_{Lk})] {(V_{L})}_{αk}, \\ {(V_{R}^{'})}_{αk} = exp [i (θ_{Rα} - χ_{Rk})] {(V_{R})}_{αk}, \\ m_{α}^{'} = exp [i (θ_{Lα} - θ_{Rα})] m_{α}, \\ m_{k}^{'} = exp [i (χ_{Lk} - χ_{Rk})] m_{k} . \end{array}

(26.54)

We see that, if we require the mass $m_{α}$ of the up-type quark $α$ to remain real and positive, we must impose $θ_{Lα} = θ_{Rα}$ ⁠. Similarly, we must set $χ_{Lk} = χ_{Rk}$ in order that the mass $m_{k}$ of the down-type quark k remains real and positive. This means that the rephasings of the left-handed and right-handed quark fields must be identical, and as a consequence we can only eliminate $2 n_{g} - 1$ phases from $V_{L}$ and/or from $V_{R}$ by means of rephasings. The total number of meaningful phases is thus

N_{phase}^{LR} = n_{g} (n_{g} + 1) - (2 n_{g} - 1) = n_{g}^{2} - n_{g} + 1 .

(26.55)

Therefore, even in the one-generation case there is one CP-violating phase remaining. We shall soon see the origin of this phase.

In general, the total number of CP-violating phases in the LRSM is given in eqn (26.55). However, the distribution of these phases by the mixing matrices $V_{L}$ and $V_{R}$ has some arbitrariness, since it depends on which phases we choose to eliminate. A convenient choice consists in using the rephasing freedom to eliminate the maximum number of phases from $V_{L} .$ With this choice the number of CP-violating phases in $V_{L}$ and in $V_{R}$ will be given by, respectively,

\begin{array}{l} N_{L} = \frac{(n_{g} - 1) (n_{g} - 2)}{2}, \\ N_{R} = \frac{n_{g} (n_{g} + 1)}{2} . \end{array}

(26.56)

For two generations one has $N_{L} = 0$ and $N_{R} = 3$ ⁠, and in this case all CP violation arises from the right-handed charged currents.⁷⁰

The origin of the extra CP-violating phases in the LRSM lies in the fact that we now have two mixing matrices, $V_{L}$ and $V_{R}$ ⁠, with the same rephasing properties. This means that $arg [{(V_{L})}_{αk} {(V_{R})}_{αk}^{*}]$ ⁠, for any up-type quark $α$ and for any down-type quark k, is a meaningful phase; the sine of any of these phases is CP-violating. In practical calculations these phases always appear multiplied by the masses of the corresponding quarks. Indeed, from eqns (26.54) we see that the rephasing-invariant quantities are $m_{α}^{*} m_{k} {(V_{L})}_{αk} {(V_{R})}_{αk}^{*}$ ⁠. Of course, in practice one takes $m_{α}$ and $m_{k}$ to be real and positive.

We now see the reason why in the LRSM there is one CP-violating phase even in the case $n_{g} = 1$ ⁠. Indeed, if there were only the up and the down quarks, then the quantity ${(V_{L})}_{ud} {(V_{R})}_{ud}^{*}$ might be complex and cause CP violation. In a practical calculation such a quantity might arise in the interference of two diagrams, one of them with a V – A interaction and the other one with a V + A interaction; or in a single diagram with both interactions, like for instance a diagram for the electric dipole moment of a quark mediated by a $W_{L}^{\pm}$ mixing with a $W_{R}^{\pm}$ (Fig. 26.1).

fig. 26.1.

Diagram generating an electric dipole moment for the down quark in the LRSM with one generation of quarks. (a) The diagram is drawn in the basis of the unphysical gauge bosons $W_{L}^{\pm}$ and $W_{R}^{\pm}$ ⁠. (b) The diagram is drawn in the basis of the physical gauge bosons $W_{1}^{\pm}$ and $W_{2}^{\pm}$ ⁠.

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26.3.3 Manifest and pseudo-manifest left–right symmetry

In the above analysis, we have assumed that the quark mass matrices are arbitrary, i.e., that they are neither Hermitian nor symmetric. This leads to $V_{L}$ and $V_{R}$ being completely independent, with different mixing angles and phases in the left-handed and right-handed charged currents. This case is often referred to as having ‘non-manifest left–right symmetry’ and it can be realized if $Γ_{1}$ and $Γ_{2}$ are complex and $α \neq 0$ ⁠. Next we shall consider two particular scenarios:

CP is explicitly broken, but $α = 0$ ⁠. This means that we may choose $k_{1}$ and $k_{2}$ simultaneously real which, in view of the fact that the Yukawa-coupling matrices are Hermitian, leads to $M_{p} = M_{p}^{†}$ and $M_{n} = M_{n}^{†}$ ⁠. This case is called ‘manifest left–right symmetry’.

Spontaneously broken CP $(α \neq 0)$ ⁠. Let us assume that CP is a good symmetry of the Lagrangian, with the scalars transforming as in eqn (26.40). CP invariance then constrains $Γ_{1}$ and $Γ_{2}$ to be real, while P invariance enforces Hermiticity. The mass matrices $M_{p}$ and $M_{n}$ are then symmetric. However, they will in general be complex, since $α \neq 0$ ⁠. This case is often referred to as displaying ‘pseudo-manifest left–right symmetry’.

One should emphasize that the scenario of manifest left–right symmetry is quite contrived. Indeed, once CP is not assumed to be a symmetry of the Lagrangian, there are in general terms in the vacuum potential dependent on $sin α$ and other terms dependent on $cos α$ (see eqn 26.36), and it is difficult to see how $α$ can turn out to vanish.

In the case of manifest left–right symmetry $M_{p}$ and $M_{n}$ are Hermitian. They are diagonalized by unitary transformations, i.e., $U_{L}^{p} = U_{R}^{p} X^{p}$ and $U_{L}^{n} = U_{R}^{n} X^{n}$ ⁠, where $X^{p}$ and $X^{n}$ are diagonal orthogonal matrices, needed to render the quark masses non-negative. As a result $V_{L} = X^{p} V_{R} X^{n}$ ⁠, i.e., the mixing matrices appearing in the left-handed and right-handed charged currents are essentially the same. In this case, the number of CP-violating phases coincides with the corresponding number in the SM.

In the case of pseudo-manifest left–right symmetry the quark mass matrices are symmetric. From the first eqn (12.15) one obtains

U_{L}^{p †} M_{p} M_{p}^{†} U_{L}^{p} = U_{R}^{p †} M_{p}^{†} M_{p} U_{R}^{p} = M_{u}^{2} .

(26.57)

Taking into account that $M_{p}$ is symmetric, one derives from eqn (26.57)

U_{L}^{p T} M_{p}^{*} M_{p} U_{L}^{p *} = U_{R}^{p †} M_{p}^{*} M_{p} U_{R}^{p} = M_{u}^{2} .

(26.58)

From eqn (26.58) one concludes that $U_{L}^{p *} = U_{R}^{p} K^{p}$ ⁠, where $K^{p}$ is a diagonal unitary matrix. Similarly, for the down sector one has $U_{L}^{n *} = U_{R}^{n} K^{n}$ ⁠. This leads to

V_{R} = K^{p} V_{L} K^{n *} .

(26.59)

Taking into account that $K^{p}$ and $K^{n}$ contain $2 n_{g} - 1$ meaningful phases, one concludes that the total number of phases in this case is $n_{g} (n_{g} + 1) / 2$ ⁠. If $n_{g} = 1$ or $n_{g} = 2$ this is the same as the general case in eqn (26.55): one phase for one generation, three phases for two generations. For three generations, however, the general case admits seven CP-violating phases, while the case of pseudo-manifest left–right symmetry only has six phases.

Another way to look at this case consists in using the rephasing freedom of the quark fields to transform

\begin{array}{l} V_{R}^{'} = {(K^{p})}^{- 1 / 2} V_{R} {(K^{n})}^{1 / 2}, \\ V_{L}^{'} = {(K^{p})}^{- 1 / 2} V_{L} {(K^{n})}^{1 / 2} . \end{array}

(26.60)

One then obtains $V_{R}^{'} = {V^{'}}_{L}^{*}$ ⁠.

Thus, in the case of manifest left–right symmetry the two mixing matrices may be chosen to be equal; in the case of pseudo-manifest left–right symmetry, they may be chosen to be the complex-conjugate of each other.

26.4 Weak-basis invariants and CP violation

In this section we study the CP properties of the LRSM through the method of weak-basis invariants, which was derived and applied to the SM in Chapter 14.

26.4.1 Conditions for CP invariance

Let us consider the Lagrangian of the LRSM, assuming that the original gauge symmetry has already been broken into the U(1) of electromagnetism. The most general CP transformation of the quark fields which leaves invariant the gauge interactions is

\begin{array}{l} (C) p_{L} (t, \vec{r}) {(C)}^{†} = K_{L} γ^{0} C {\bar{p_{L}}}^{T} (t, - \vec{r}), \\ (C) n_{L} (t, \vec{r}) {(C)}^{†} = K_{L} γ^{0} C {\bar{n_{L}}}^{T} (t, - \vec{r}), \\ (C) p_{R} (t, \vec{r}) {(C)}^{†} = K_{R} γ^{0} C {\bar{p_{R}}}^{T} (t, - \vec{r}), \\ (C) n_{R} (t, \vec{r}) {(C)}^{†} = K_{R} γ^{0} C {\bar{n_{R}}}^{T} (t, - \vec{r}), \end{array}

(26.61)

where $K_{L}$ and $K_{R}$ are $n_{g} \times n_{g}$ unitary matrices acting in family space. Note that, due to the presence of the right-handed charged current, $p_{R}$ and $n_{R}$ must transform in the same way under CP; this is the crucial difference between the CP properties of the LRSM and of the SM.

One readily concludes that, in order for CP invariance to hold, the quark mass matrices must satisfy

\begin{array}{l} K_{L}^{†} M_{p} K_{R} = M_{p}^{*}, \\ K_{L}^{†} M_{n} K_{R} = M_{n}^{*} . \end{array}

(26.62)

One may thus state

Theorem 26.1
The gauge interactions of the LRSM, together with the quark mass terms, are CP-invariant if and only if the quark mass matrices $M_{p}$ and $M_{n}$ are such that unitary matrices $K_{L}$ and $K_{R}$ exist which satisfy eqns (26.62).

26.4.2 Weak-basis transformations

In the LRSM, a weak-basis transformation is more restricted than in the SM:

\begin{array}{c} Q_{L} = W_{L} Q_{L}^{'}, \\ Q_{R} = W_{R} Q_{R}^{'} . \end{array}

(26.63)

The right-handed quarks $p_{R}$ and $n_{R}$ now transform in the same way, because of the presence of the right-handed charged current. The quark mass matrices transform as

\begin{array}{c} M_{p}^{'} = W_{L}^{†} M_{p} W_{R}, \\ M_{n}^{'} = W_{L}^{†} M_{n} W_{R} . \end{array}

(26.64)

Following a line analogous to the one in § 14.3, it is easy to show that Theorem 26.1 is weak-basis independent.

26.4.3 CP restrictions in a special weak basis

In order to understand the restrictions that eqns (26.62) imply on $M_{p}$ and $M_{n}$ ⁠, it is useful to work in the weak basis in which $M_{p} = M_{u}$ is diagonal and real. Such a weak basis always exists in the LRSM, as was also the case in the SM. Assuming the up-type-quark masses to be non-degenerate, one then finds

K_{L} = K_{R} = diag [exp (i δ_{j})],

(26.65)

and

arg {(M_{n})}_{jk} = \frac{δ_{j} - δ_{k}}{2} (mod π) .

(26.66)

Thus, CP invariance constrains $M_{n}$ to have cyclic phases in the weak basis where $M_{p}$ is diagonal and real.

The matrix $M_{n}$ is in general complex, having $n_{g}^{2}$ independent phases. The requirement that it has cyclic phases corresponds to $n_{g}^{2} - (n_{g} - 1)$ independent restrictions. As expected, this number coincides with the number $N_{phase}^{LR}$ of independent CP-violating phases appearing in the left-handed and right-handed charged currents, in the mass-eigenstate basis.

It is useful to compare the LRSM with the SM. In the SM, CP invariance requires the Hermitian matrix $M_{n} M_{n}^{†}$ to have cyclic phases in the weak basis in which $M_{p} M_{p}^{†}$ is diagonal. In the LRSM, CP constrains an arbitrary complex matrix $M_{n}$ to have cyclic phases in the weak basis in which $M_{p}$ is diagonal. Thus, CP invariance is a much stronger requirement on the LRSM than on the SM. One understands in a simple way why in the LRSM it is possible to have CP violation for one or two generations, while in the SM this is not possible. Indeed, $1 \times 1$ and $2 \times 2$ Hermitian matrices automatically have cyclic phases, while a $1 \times 1$ or a $2 \times 2$ general matrix does not necessarily have phases with that property.

26.4.4 Weak-basis invariants

In this subsection we shall construct necessary conditions for CP invariance, expressed in terms of weak-basis invariants (Branco and Rebelo 1985). From eqns (26.62) one obtains

\begin{array}{c} K_{L}^{†} H_{p} K_{L} = H_{p}^{T}, \\ K_{L}^{†} H_{n} K_{L} = H_{n}^{T}, \end{array}

(26.67)

where $H_{p} \equiv M_{p} M_{p}^{†}$ and $H_{n} \equiv M_{n} M_{n}^{†}$ ⁠. It follows that

tr {[H_{p}, H_{n}]}^{r} = 0 for r odd .

(26.68)

These are the weak-basis-invariant conditions for CP invariance which were already encountered in the SM. The LRSM has, however, other WB invariants which must also vanish in order for CP to be conserved. Indeed, one also obtains from eqns (26.62) that

\begin{array}{c} K_{R}^{†} H_{p}^{'} K_{R} = {H_{p}^{'}}^{T}, \\ K_{R}^{†} H_{n}^{'} K_{R} = {H_{n}^{'}}^{T}, \end{array}

(26.69)

with $H_{p}^{'} \equiv M_{p}^{†} M_{p}$ and $H_{n}^{'} \equiv M_{n}^{†} M_{n}$ ⁠. It follows that

tr {[H_{p}^{'}, H_{n}^{'}]}^{r} = 0 for r odd .

(26.70)

The matrices $H_{p}^{'}$ and $H_{n}^{'}$ are diagonalized by unitary transformations of the right-handed quark fields. Therefore, the conditions of eqn (26.70), when written in terms of quark masses and mixing angles, have the same form as those of eqn (26.68), with the mixing angles of $V_{L}$ substituted by those of $V_{R}$ ⁠.

Obviously, the lowest value of r for which the conditions of eqn (26.68) and eqn (26.70) are non-trivial is 3. Therefore, those invariants have mass dimensions of at least twelve. In the LRSM there are other WB invariants, relevant for the study of CP violation, which have no counterpart in the SM and have a much lower mass dimension. Indeed, from eqns (26.62) one obtains

K_{L}^{†} M_{p} M_{n}^{†} K_{L} = M_{p}^{*} M_{n}^{T} .

(26.71)

Taking the trace, one obtains

Im tr (M_{p} M_{n}^{†}) = 0 .

(26.72)

This is a necessary condition for CP invariance of the LRSM for any number of generations. The remarkable feature of eqn (26.72) is that it is non-trivial even for one generation. Indeed, eqn (26.72) reads

\sum_{α} \sum_{k} m_{α} m_{k} Im [{(V_{R})}_{αk} {(V_{L})}_{αk}^{*}] = 0 .

(26.73)

In particular, for one quark generation, one has

Im [{(V_{R})}_{ud} {(V_{L})}_{ud}^{*}] = 0 .

(26.74)

This result agrees with our considerations in § 26.3.2.

26.5 Phenomenological implications

The phenomenological implications of left–right-symmetric models depend to a large extent on whether the left–right symmetry is manifest or pseudo-manifest. Considering the latter case, with spontaneously broken CP, a detailed phenomenological analysis was carried out by Ecker and Grimus (1985), who extended earlier work by Chang (1983a,b) and by Branco et al. (1983). Imposing the requirement that the absolute value of the new contribution to $Δ m$ in the $K^{0} - \bar{K^{0}}$ system should not exceed the experimental value, they obtained the bound $m_{W_{R}} > 2.5 TeV$ ⁠, which improved the earlier bound derived by Beall et al. (1982). It has also been pointed out by Ecker and Grimus (1985) that the ratio $ϵ^{'} / ϵ$ in the LRSM can have either sign, and is correlated with the value of the neutron electric dipole moment. The possibility of having a small value of $ϵ^{'} / ϵ$ was, at the time of their calculation, very interesting, since the prediction of the SM was thought to be $ϵ^{'} / ϵ \geq 2 \times 10^{- 3}$ ⁠. At present, and taking into account the high value of the top-quark mass, values $\sim 10^{- 4}$ for $∣ ϵ^{'} / ϵ ∣$ can be obtained within the SM, and therefore that ratio is no longer a very useful parameter to distinguish between the SM and the LRSM.

26.6 Main conclusions

•

Due to the presence of a charged current with the right-handed quarks, there are new sources of CP violation in the left–right-symmetric model. These arise from the possibility that the corresponding matrix elements for the left-handed-quarks mixing matrix and for the right-handed-quarks mixing matrix might have different phases.

•

In the LRSM, CP violation may be present even in the case of only one family.

•

In the LRSM, there are weak-basis invariants relevant for CP violation of much lower mass dimension than in the SM.

It should also be pointed out that the scalar sector of the LRSM is quite complicated, and may be the source of various effects, including CP violation.

Notes

⁶⁸

Explicit lower bounds on $m_{W_{R}}$ vary between 240 GeV and 1600 GeV, depending on the experimental technique and on the specific variant of the LRSM that one is considering. The Particle Data Group (1996, p. 231) recommends $m_{W_{R}}$ > 406 GeV.

⁶⁹

It is difficult to have $k_{2} = 0$ when $k_{1} \neq 0$ (or vice versa), because the scalar potential contains terms linear in $k_{2}$ when $k_{1}, F_{L}$ ⁠, and $F_{R}$ do not vanish. Those terms draw $k_{2}$ away from $k_{2} = 0$ ⁠. See eqn (26.36) below.

⁷⁰

The fact that even for two generations one can obtain CP violation in the LRSM provided some of the original motivation to introduce this model. One should keep in mind that at the time when the model was proposed, only two incomplete generations were known, since the charm quark had not yet been discovered.

When there are only two generations, eqns (26.56) imply that we may set $V_{L}$ real. All CP violation then arises from the right-handed gauge interactions and, as such, it is expected to be suppressed by ${(m_{W_{L}} / m_{W_{R}})}^{2}$ ⁠. This suppression of CP-violating effects (Chang 1983a,b) was another attractive feature of the LRSM; however, it does not hold for three or more generations (Branco et al. 1983; Mohapatra 1985).

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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/9780198503996.003.0026

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

26 The Left-Right-Symmetric Model

Abstract

26.1 Overview of the model

26.1.1 Introduction

26.1.2 Gauge couplings

26.1.3 Scalar multiplets

26.1.4 Gauge-boson masses

26.2 Spontaneous symmetry breaking

26.2.1 The scalar potential

26.2.2 Spontaneous breaking of P

26.2.3 Spontaneous CP breaking

26.3 Quark masses and mixing matrices

26.3.1 Mass matrices

26.3.2 Mixing matrices and CP-violating phases

26.3.3 Manifest and pseudo-manifest left–right symmetry

26.4 Weak-basis invariants and CP violation

26.4.1 Conditions for CP invariance

26.4.2 Weak-basis transformations

26.4.3 CP restrictions in a special weak basis

26.4.4 Weak-basis invariants

26.5 Phenomenological implications

26.6 Main conclusions

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Contents

26 The Left-Right-Symmetric Model

Abstract

26.1 Overview of the model

26.1.1 Introduction

26.1.2 Gauge couplings

26.1.3 Scalar multiplets

26.1.4 Gauge-boson masses

26.2 Spontaneous symmetry breaking

26.2.1 The scalar potential

26.2.2 Spontaneous breaking of P

26.2.3 Spontaneous CP breaking

26.3 Quark masses and mixing matrices

26.3.1 Mass matrices

26.3.2 Mixing matrices and CP-violating phases

26.3.3 Manifest and pseudo-manifest left–right symmetry

26.4 Weak-basis invariants and CP violation

26.4.1 Conditions for CP invariance

26.4.2 Weak-basis transformations

26.4.3 CP restrictions in a special weak basis

26.4.4 Weak-basis invariants

26.5 Phenomenological implications

26.6 Main conclusions

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