Viability of minimal left-right models with discrete symmetries
VViability of minimal left-right models with discrete symmetries
Wouter Dekens, Dani¨el Boer
Van Swinderen Institute, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The Netherlands
Abstract
We provide a systematic study of minimal left-right models that are invariant under P , C , and/or CP transformations. Due to the high amount of symmetry such models are quitepredictive in the amount and pattern of CP violation they can produce or accommodateat lower energies. Using current experimental constraints some of the models can alreadybe excluded. For this purpose we provide an overview of the experimental constraints onthe different left-right symmetric models, considering bounds from colliders, meson-mixingand low-energy observables, such as beta decay and electric dipole moments. The featuresof the various Yukawa and Higgs sectors are discussed in detail. In particular, we give theHiggs potentials for each case, discuss the possible vacua and investigate the amount of fine-tuning present in these potentials. It turns out that all left-right models with P , C , and/or CP symmetry have a high degree of fine-tuning, unless supplemented with mechanisms tosuppress certain parameters. The models that are symmetric under both P and C are not inaccordance with present observations, whereas the models with either P , C , or CP symmetrycan not be excluded by data yet. To further constrain and discriminate between the modelsmeasurements of B -meson observables at LHCb and B -factories will be especially important,while measurements of the EDMs of light nuclei in particular could provide complementarytests of the LRMs. a r X i v : . [ h e p - ph ] N ov Introduction
Left-right models (LRMs) have been studied extensively as possible physics beyond the SM(BSM) [1–5]. LRMs extend the standard model (SM) gauge-group to SU (3) c × SU (2) L × SU (2) R × U (1) B − L and possess several attractive features. They offer an interpretation of the U (1) generator in terms of baryon and lepton number and naturally allow for neutrino massesthrough the see-saw mechanism. Furthermore, the gauge group of the LRM can appear in grandunified theories (GUTs), such as SO (10) and E , as an intermediate step [6], while avoiding the SU (5) group which has problems with proton decay. But perhaps their most appealing feature isthe possibility of having a symmetry between left- and right-handed particles at high energies, aso-called LR symmetry. LR models employing such a symmetry, LR symmetric models (LRSMs),restore parity ( P ) and/or charge conjugation ( C ) invariance at high energies, thereby explainingthe P - and/or C -violating nature of the SM as a low-energy effect.From a theoretical standpoint, the most attractive LRSMs might be those which exhibit both P and C symmetries, and thereby CP symmetry, at high energies. Such models in principle canexplain the observed CP violation as resulting from spontaneous CP violation rather than fromexplicit CP violation as in the SM. However, the LR symmetries of such “ C + P ” models stronglyconstrain the left and right CKM matrices, dictating the amount and pattern of CP and flavorviolation. For the so-called minimal LRSMs, which are most commonly considered and whichhave a minimally extended Higgs sector, these model constraints turn out to be incompatiblewith measurements of Kaon and B -meson mixing, as will be discussed. Therefore, minimalLRSMs require explicit P or C violation. It is the goal of this paper to assess the viabilityof these options, of which many aspects have already been discussed in the literature before.Nevertheless, it seems useful to collect the available results, combine and supplement them, andarrive at clear conclusions about which models are ruled out by current experimental constraintsand which models require an unacceptably large amount of fine-tuning. Apart from the LRSMswith “ C + P ”, P or C symmetry, we also will consider a LRM that is CP symmetric, but notnecessarily P and C symmetric. Since this option does not correspond to a LR symmetry, it isnot a left-right symmetric model.For all these models we consider the quark and Higgs sectors, review the relations betweenthe left- and right-handed CKM matrices, consider the possible vacua and calculate a measureof the fine-tuning in the Higgs potential in each case. Furthermore, we give an overview ofthe relevant experimental constraints on the different LRMs, considering bounds from directsearches at the LHC, from B -meson-mixing measurements at LHCb and B -factories, from Kaonmixing, and low-energy observables, such as beta decay and electric dipole moments (EDMs).As said, in the “ C + P ” models current constraints are sufficiently strong to exclude them, butfor the other options future measurements, in particular on CP violation by LHCb will be ableto limit the options further considerably and may also be able to differentiate between the C -symmetric and P -symmetric LRMs. Measurements on EDMs for the neutron, but also for theproton, other light nuclei and the electron would offer additional tests of LRMs. Currently theLR scale as given by the mass of the right-handed W boson, commonly referred to as W (cid:48) boson,is required to be at least 2 TeV by direct searches and in the case of P or C -symmetric LRSMs3 TeV by indirect Kaon and B -meson constraints [7]. In the coming decade this bound couldextend to 8 TeV or higher. As this scale gets pushed upwards, the already considerable if nothuge fine-tuning required in the models will increase further and the models become increasinglyless likely scenarios. These bounds and perhaps the fine-tuning may be weakened though by1onsidering non-minimal [8, 9] and/or less symmetric models [10–12]. We will not include suchmodels here, not for lack of theoretical motivation, but simply in order to limit the scope.The large amount of fine tuning in the LRMs models considered here is due to the fact thatthe Higgs potential necessarily relates the electroweak scale to the LR scale. As the LR scalehas been pushed into the TeV range there is a hierarchy between the scales which requires thetuning of some of the parameters in the potential. In fact, unless some of the parameters arechosen to be zero (or exactly related) the fine-tuning becomes extreme. Although it is not clearwhat amount of fine-tuning should be considered acceptable, it does affect the attractivenessof the LRSMs. We have introduced a measure of fine-tuning often employed in studies ofsupersymmetric extensions of the SM, in order to quantify the amount of fine-tuning. This mayexpedite the discussion about the viability of such models and hopefully stimulate the searchfor new mechanisms to mitigate the fine-tuning problem.The outline of this paper is as follows. First we introduce the general minimal LR model insection 2 and experimental bounds on CP violation in section 3, while discussing the specificLRSMs in subsequent sections. We first discuss the “ C + P ” LRSMs in detail in section 4, which,although they turn out not to be viable, have many features in common with the LRSMs witha single LR symmetry to be discussed in section 5. We present a summary and conclusions insection 6. In this section we will discuss minimal left-right models and highlight some of their features.We will start with the basic ingredients of the model, namely, its field content.
The gauge group of left-right (LR) models is given by SU (2) L × SU (2) R × U (1) B − L [1–5]. Asin the standard model (SM) the left-handed fermions form doublets under SU (2) L . New, withrespect to the SM, is that the right-handed fermions now form doublets under the added gaugegroup, SU (2) R . In order to build these doublets right-handed neutrinos have to be introduced.In short, the fermions are assigned to representations of the above gauge group as follows, Q L = (cid:18) u L d L (cid:19) ∈ (2 , , / , Q R = (cid:18) u R d R (cid:19) ∈ (1 , , / ,L L = (cid:18) ν L l L (cid:19) ∈ (2 , , − , L R = (cid:18) ν R l R (cid:19) ∈ (1 , , − . (1)With the fermions in the above representations a scalar, φ ∈ (2 , ∗ , L,R assigned to (3 , ,
2) and(1 , , φ = (cid:18) φ φ +1 φ − φ (cid:19) , ∆ L,R = (cid:32) δ + L,R / √ δ ++ L,R δ L,R − δ + L,R / √ (cid:33) . (2)2e will refer to LR models with such a Higgs sector as minimal LRMs. Symmetry breaking isrealized through the vacuum expectation values (vevs) of the scalar fields, (cid:104) φ (cid:105) = (cid:112) / (cid:18) κ κ (cid:48) e iα (cid:19) , (cid:104) ∆ L (cid:105) = (cid:112) / (cid:18) v L e iθ L (cid:19) , (cid:104) ∆ R (cid:105) = (cid:112) / (cid:18) v R (cid:19) , (3)where all parameters are real after gauge transformations have been used to eliminate two of thepossible phases [5]. In the first step of symmetry breaking the vev of the right-handed triplet, v R ,breaks the SU (2) L × SU (2) R × U (1) B − L group down to SU (2) L × U (1) Y . This vev also definesthe high scale of the model, and gives the main contribution to the masses of the additional gaugebosons, W ± R and Z R belonging to SU (2) R . At the electroweak scale the vevs of the bidoublet, κ and κ (cid:48) e iα , then break SU (2) L × U (1) Y to U (1) EM . In turn, these vevs dictate the masses of the W ± L and Z L bosons, which belong to SU (2) L , while α is the parameter indicating spontaneous CP violation. This implies these vevs are of the electroweak scale, indeed, we have κ + = v (cid:39)
246 GeV , (4)where κ ± ≡ κ ± κ (cid:48) . Finally, while the Dirac masses of the fermions are generated by the vevs of φ , the vevs of the triplets generate Majorana masses for the neutrinos. Thus, v L contributes tothe light neutrino masses and so is not expected to exceed this scale by much, i.e., v L (cid:46) O (1 eV).A (much) less stringent upper bound without theoretical prejudice can be derived from the ρ parameter; ρ ≡ M W M Z c W defined such that it is 1 to all orders in the SM. Since v L breaks custodialsymmetry it contributes to ρ −
1, from a global fit of ρ [14] one can then deduce, v L (cid:46) >
10 TeV in LRMs [7, 17], as they give rise to stringently constrained flavor-changingneutral currents, see section 4.1.2. The scalars arising from the triplet fields are not as wellconstrained and can still be relatively light while keeping the flavor-changing scalars heavy [18].In fact, the doubly charged scalars can still have masses ∼
450 GeV, while in the future, at √ s = 14 TeV with 300 fb − , the LHC is expected to probe masses up to 600 GeV [18, 19]. One of the main motivations for LRMs is the possibility of explaining the broken symmetrybetween left and right in the SM as a low-energy phenomenon. In LR models it is possible torestore this symmetry at high energies, which is then spontaneously broken at lower energies bythe vevs of the scalar fields. There are two possible transformations which qualify as symmetries3etween left and right P : Q L ←→ Q R , φ ←→ φ † , ∆ L,R ←→ ∆ R,L ,C : Q L ←→ ( Q R ) c , φ ←→ φ T , ∆ L,R ←→ ∆ ∗ R,L , (5)where the superscript c indicates charge conjugation. A LR model with such a P or C symmetryis called left-right symmetric. Note that the combination of the two symmetries in Eq. (5), CP , does not interchange left- and right-handed fields and so is not a LR symmetry. BothLR symmetries require the SU (2) L,R gauge couplings to be equal, g L = g R , at the LR scale,although a difference between the two could be induced when they are evolved down to theelectroweak scale. Two more specific (albeit not necessarily minimal) models, often discussed inthe literature, are the manifest and pseudomanifest LR models. The former refers to a LR modelwith P -symmetric Yukawa couplings and the additional assumption of a vanishing spontaneousphase, α = 0 [10, 21]. A pseudomanifest LR model on the other hand assumes C - and P -symmetric Yukawa couplings [10, 22]. In the past the case of a P -symmetric LR model wasmainly studied [1, 2, 13], however, recently there has been renewed interest in the C -symmetriccase as well [7, 20]. In either case these symmetries impose important restrictions, as we will seelater. Perhaps the most characteristic way in which LR models affect observables is through the right-handed charged-current interaction of the W ± R boson. For the quarks it is given by (in thequark-mass basis) L CC = g L √ U L γ µ V L D L W + Lµ + g R √ U R γ µ V R D R W + Rµ + h.c. , (6)where V L and V R are the SM CKM matrix and its right-handed equivalent. However, the gaugefields W ± L,R are not quite mass eigenstates. The two charged gauge-bosons mix because both ofthem couple to the bidoublet φ which is charged under both SU (2) groups. The mass terms forthe charged gauge-bosons are given by L W mass = ( W − Lµ W − Rµ ) (cid:32) g L ( κ + κ (cid:48) + 2 v L ) − g L g R κκ (cid:48) e − iα − g L g R κκ (cid:48) e iα g R ( κ + κ (cid:48) + 2 v R ) (cid:33) (cid:18) W + µL W + µR (cid:19) , (7)where g L,R are the coupling constants of SU (2) L,R . These gauge couplings will be equal in boththe P - and C -symmetric case. The gauge eigenstates are related to the mass eigenstates asfollows (cid:18) W + µL W + µR (cid:19) = (cid:18) cos ζ − sin ζe − iα sin ζe iα cos ζ (cid:19) (cid:18) W + µ W + µ (cid:19) , tan ζ (cid:39) g L g R κκ (cid:48) v R , (8) There are two other possible transformations on the φ fields which would qualify as a left-right symmetry,namely, φ → ˜ φ † and φ → ˜ φ T , where ˜ φ = τ φ ∗ τ . However, as observed in [20] these lead to unrealistic quarkmass matrices, M u = M † d in the former case and Tr( M u M † u ) = Tr( M d M † d ) in the latter. To be general these transformations should also include the possibility of changing the flavors of the quarks.However, when a single LR symmetry applies we can always choose a basis such that the transformations are asin Eq. (5). When both LR symmetries apply flavor rotations can play a role as we will see in section 4. W ± , refer to the mass eigenstates of the charged gauge-bosons. The masses themselvesare approximately given by M (cid:39) g L κ , M (cid:39) g R v R . (9)Direct searches at the LHC set a lower limit of 2 TeV (95% CL) on the mass of the right-handed W ± R from the W + R → t ¯ b channel [23]. More stringent limits have been obtained inleptonic decays which rely on certain assumptions about right-handed neutrinos. These limitsextend to M ≥ . − M N . Although these bounds on M depend on M N the two masses become correlated in someLRSMs after applying constraints from low-energy precision experiments in muon decay, therebyconsiderably reducing the allowed region in parameter space [27]. The collider bounds from bothtypes of channels assume the right-handed couplings to be the same as the left-handed couplings,e.g. for the W + R → t ¯ b channel g L | V tbL | = g R | V tbR | . Thus, the strength of the above bounds is inpart determined by whether or not the model is LR symmetric. If we do not assume any LRsymmetry these bounds can be weakened and even be evaded in some cases. In turn, the masses for the quarks are generated by the interactions of the bidoublet with thequarks. The most general form of the Yukawa interactions respecting the gauge symmetries inthe weak basis is, − L Y = ¯ Q L (cid:0) Γ φ + ˜Γ ˜ φ (cid:1) Q R + h.c. , (10)where Γ and ˜Γ are complex 3 × φ ≡ τ φ ∗ τ . After the Higgs fields acquire theirvevs this leads to the following mass matrices for the quarks M u = (cid:112) / κ Γ + κ (cid:48) e − iα ˜Γ) , M d = (cid:112) / κ (cid:48) e iα Γ + κ ˜Γ) . (11)The implications of the possible LR symmetries on the Yukawa sector are the following P : Γ = Γ † , ˜Γ = ˜Γ † , (12) C : Γ = Γ T , ˜Γ = ˜Γ T . (13)For the P -symmetric case this means that if α were zero, as in the manifest LR symmetricmodel, the mass matrices would be hermitian as well. In this limit there is a relation betweenthe left- and right-handed CKM matrices, namely, V R = S u V L S d , (14)where S u,d are diagonal matrices of signs. In general α (cid:54) = 0 and the above relation will notbe satisfied. Nonetheless, in the P -symmetric case, in order to reproduce the observed quarkmasses, the combination κ (cid:48) /κ sin α should be small [20]. Thus, the quark mass matrices will benearly hermitian, implying that Eq. (14) is approximately correct and the right and left mixingangles should be nearly equal. This was already shown numerically in Refs. [20, 28] and wasrecently confirmed by an explicit solution of V R [29].5n the C -symmetric case the mass matrices will be symmetric which implies the followingrelation between the two CKM matrices [30] V R = K u V ∗ L K d , (15)where K u = diag( θ u , θ c , θ t ) and K d = diag( θ d , θ s , θ b ) are diagonal matrices of phases, of whichone combination can be set to zero, while the rest remains unconstrained. This relation holdsirrespective of the value of α . As a result the mixing angles in both matrices will be equal.Which relation between the left- and right-handed CKM matrices applies has implications forthe bounds that can be set on these models. We will come back to this issue when discussingthe P - and C -symmetric LR models in more detail. The final part of the Lagrangian to be discussed is the Higgs potential. As we intend to discussthe potential for the LR symmetries, P and C as well as the CP -symmetric case, we will givethe potentials for these three cases.The potential invariant under the gauge group and the P symmetry is given by [5] V PH = − µ Tr( φ † φ ) − µ (cid:2) Tr( ˜ φ † φ ) + Tr( φ † ˜ φ ) (cid:3) − µ (cid:2) Tr(∆ L ∆ † L ) + Tr(∆ R ∆ † R ) (cid:3) + λ (cid:2) Tr( φ † φ ) (cid:3) + λ (cid:0)(cid:2) Tr( ˜ φ † φ ) (cid:3) + (cid:2) Tr( φ † ˜ φ ) (cid:3) (cid:1) + λ Tr( ˜ φ † φ ) Tr( φ † ˜ φ ) + λ Tr( φ † φ ) (cid:2) Tr( ˜ φ † φ ) + Tr( φ † ˜ φ ) (cid:3) + ρ (cid:0)(cid:2) Tr(∆ L ∆ † L ) (cid:3) + (cid:2) Tr(∆ R ∆ † R ) (cid:3) (cid:1) + ρ (cid:2) Tr(∆ L ∆ L )Tr(∆ † L ∆ † L ) + Tr(∆ R ∆ R )Tr(∆ † R ∆ † R ) (cid:3) + ρ Tr(∆ L ∆ † L )Tr(∆ R ∆ † R )+ ρ (cid:2) Tr(∆ L ∆ L )Tr(∆ † R ∆ † R ) + Tr(∆ R ∆ R )Tr(∆ † L ∆ † L ) (cid:3) + α Tr( φ † φ ) (cid:2) Tr(∆ L ∆ † L ) + Tr(∆ R ∆ † R ) (cid:3) + α (cid:0) e iδ (cid:2) Tr( ˜ φ † φ )Tr(∆ R ∆ † R ) + Tr( φ † ˜ φ )Tr(∆ L ∆ † L ) (cid:3) + h.c. (cid:1) + α (cid:2) Tr( φφ † ∆ L ∆ † L ) + Tr( φ † φ ∆ R ∆ † R ) (cid:3) + β (cid:2) Tr( φ ∆ R φ † ∆ † L ) + Tr( φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) Tr( ˜ φ ∆ R φ † ∆ † L ) + Tr( ˜ φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) Tr( φ ∆ R ˜ φ † ∆ † L ) + Tr( φ † ∆ L ˜ φ ∆ † R ) (cid:3) , (16)while the potential in the case of an unbroken C symmetry at high energies is given by, V CH = − µ Tr( φ † φ ) − µ (cid:2) e iδ µ Tr( ˜ φφ † ) + h.c. (cid:3) − µ (cid:2) Tr(∆ L ∆ † L ) + Tr(∆ R ∆ † R ) (cid:3) + λ (cid:2) Tr( φ † φ ) (cid:3) + λ (cid:0) e iδ λ (cid:2) Tr( ˜ φφ † ) (cid:3) + h.c. (cid:1) + λ Tr( ˜ φ † φ ) Tr( φ † ˜ φ ) + λ Tr( φ † φ ) (cid:2) e iδ λ Tr( ˜ φφ † ) + h.c. (cid:3) + ρ (cid:0)(cid:2) Tr(∆ L ∆ † L ) (cid:3) + (cid:2) Tr(∆ R ∆ † R ) (cid:3) (cid:1) + ρ (cid:2) Tr(∆ L ∆ L )Tr(∆ † L ∆ † L ) + Tr(∆ R ∆ R )Tr(∆ † R ∆ † R ) (cid:3) + ρ Tr(∆ L ∆ † L )Tr(∆ R ∆ † R )+ ρ (cid:2) e − iδ ρ Tr(∆ L ∆ L )Tr(∆ † R ∆ † R ) + e iδ ρ Tr(∆ R ∆ R )Tr(∆ † L ∆ † L ) (cid:3) + α Tr( φ † φ ) (cid:2) Tr(∆ L ∆ † L ) + Tr(∆ R ∆ † R ) (cid:3) + α (cid:2) e iδ α Tr( ˜ φ † φ ) + h.c. (cid:3)(cid:2) Tr(∆ L ∆ † L ) + Tr(∆ R ∆ † R ) (cid:3) + α (cid:2) Tr( φφ † ∆ L ∆ † L ) + Tr( φ † φ ∆ R ∆ † R ) (cid:3) + β (cid:2) e iδ β Tr( φ ∆ R φ † ∆ † L ) + e − iδ β Tr( φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) e iδ β Tr( ˜ φ ∆ R φ † ∆ † L ) + e − iδ β Tr( ˜ φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) e iδ β Tr( φ ∆ R ˜ φ † ∆ † L ) + e − iδ β Tr( φ † ∆ L ˜ φ ∆ † R ) (cid:3) . (17)6inally, the CP -symmetric, but not necessarily C - or P -symmetric, potential is given by, V CPH = − µ Tr( φ † φ ) − µ (cid:2) Tr( ˜ φ † φ ) + Tr( φ † ˜ φ ) (cid:3) − µ L Tr(∆ L ∆ † L ) − µ R Tr(∆ R ∆ † R )+ λ (cid:2) Tr( φ † φ ) (cid:3) + λ (cid:0)(cid:2) Tr( ˜ φ † φ ) (cid:3) + (cid:2) Tr( φ † ˜ φ ) (cid:3) (cid:1) + λ Tr( ˜ φ † φ ) Tr( φ † ˜ φ ) + λ Tr( φ † φ ) (cid:2) Tr( ˜ φ † φ ) + Tr( φ † ˜ φ ) (cid:3) + ρ L (cid:2) Tr(∆ L ∆ † L ) (cid:3) + ρ R (cid:2) Tr(∆ R ∆ † R ) (cid:3) + ρ L Tr(∆ L ∆ L )Tr(∆ † L ∆ † L ) + ρ R Tr(∆ R ∆ R )Tr(∆ † R ∆ † R )+ ρ Tr(∆ L ∆ † L )Tr(∆ R ∆ † R )+ ρ (cid:2) Tr(∆ L ∆ L )Tr(∆ † R ∆ † R ) + Tr(∆ R ∆ R )Tr(∆ † L ∆ † L ) (cid:3) + α L Tr( φ † φ )Tr(∆ L ∆ † L ) + α R Tr( φ † φ )Tr(∆ R ∆ † R )+ (cid:2) Tr( ˜ φ † φ ) + Tr( φ † ˜ φ ) (cid:3)(cid:2) α L Tr(∆ L ∆ † L ) + α R Tr(∆ R ∆ † R ) (cid:3) + α L Tr( φφ † ∆ L ∆ † L ) + α R Tr( φ † φ ∆ R ∆ † R ) + β (cid:2) Tr( φ ∆ R φ † ∆ † L ) + Tr( φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) Tr( ˜ φ ∆ R φ † ∆ † L ) + Tr( ˜ φ † ∆ L φ ∆ † R ) (cid:3) + β (cid:2) Tr( φ ∆ R ˜ φ † ∆ † L ) + Tr( φ † ∆ L ˜ φ ∆ † R ) (cid:3) . (18)In all cases all parameters are real, the P -symmetric potential contains 18 parameters while inthe C - and CP -symmetric cases there are 25 and 23 parameters, respectively. This implies thatthe P -symmetry is the most constraining when it comes to the Higgs potential. However, as wewill see later these potentials are all closely related.In the upcoming discussion about the amount of fine-tuning in these potentials we imposethe condition that the dimensionless parameters are in the perturbative regime, i.e. take onvalues of order 1. As will be discussed, in some cases the amount of fine-tuning can be greatlyreduced by setting some parameters to zero. In the literature this has been done for the β i parameters, without further justification, e.g. [13]. This in turn requires v L = 0. The choice β i = 0 is in principle unstable under renormalization unless enforced by a symmetry. However,in Refs. [5, 31] it was argued that such symmetries do not allow for Majorana masses for theneutrinos, it may thus not be a viable option.We note that for avoiding or strongly reducing the fine-tuning it is not needed to set β i = 0.As we will demonstrate below, the same reduction in the amount of fine-tuning can be achievedby arranging v L = 0, which does however require relations among some of the β i parameters [31].Another option is to introduce a mechanism that yields small β i . In Ref. [16] a softly brokenhorizontal U (1) symmetry was introduced to enforce β i of order v L /v R , with v L ∼ . C P violation
In this section we will discuss a number of experimental constraints on CP violation in LRMs,namely those from Kaon mixing and decays, B d,s − B d,s mixing, electric dipole moments (EDMs)and neutron β decay. We will discuss the impact of these bounds in specific LRSMs in moredetail in the subsequent sections. 7 b cFigure 1: Figures a and b show some of the LR contributions to meson mixing. The wavyand dashed lines represent W ± L,R -bosons and flavor-changing Higgs bosons, respectively. Theexternal fermions lines are the quarks in the mesons. Figure c shows the dominant diagramcontributing to the neutron EDM and CP violation in neutron β decay (assuming ¯ θ = 0). Thefermion lines now represent up and down quarks while one line represents e and ν in case of β decay and the dot denotes W L - W R mixing. The well-known indirect and direct CP -violating parameters in the Kaon sector, ε and ε (cid:48) , arecurrently determined to be [14] | ε | = (2 . ± . · − , (19)R e ( ε (cid:48) /ε ) = (1 . ± . · − . (20)Depending on the particular realization of the LR symmetry, these parameters can lead to strongconstraints on the phases in the matrix K u,d relating the left and right CKM matrices. For therelevant expressions we refer to Refs. [7, 32–34].In LRMs there are additional contributions to ε compared to the SM from box diagramsinvolving W ± R bosons and tree-level diagrams involving flavor-changing Higgs bosons [35]. Thisis analogous to the case of B meson mixing which will be discussed more explicitly next. B d,s − B d,s mixing The B d,s − B d,s mixing is described by the off-diagonal matrix element M q = (cid:104) B q |H| B q (cid:105) / M B q .In the SM M q is determined by box diagrams involving W ± L bosons. The magnitude of M q isrelated to the mass difference ∆ M B q between the mesons while its phase signifies CP violation,∆ M B q = 2 | M q | , φ q = Arg M q . (21)In LRMs there are additional contributions from box diagrams involving W ± R bosons and tree-level diagrams involving flavor-changing Higgs bosons. Separating the SM and LRM contribu-tions, M = M SM12 + M LR12 the new contributions can be parametrized by the following quantities, M q = M SM12 (1 + h q ) , h q ≡ M LR12 M SM12 , h q = | h q | e iσ q . (22)Thus, the magnitude of 1 + h q can be constrained by the mass differences, while CP violationin B q mixing as measured by φ d,s is sensitive to its phase. The LR contribution to these anglesis given by φ LR q = Arg(1 + | h q | e iσ q ) , σ q (cid:39) Arg (cid:18) − V tbR V tq ∗ R V tbL V tq ∗ L (cid:19) . (23)8he expressions for h q can be found in Refs. [7,32,33]. Clearly, this contribution depends on thephases present in the CKM matrices. This in turn depends on the choice of LR symmetry. Whenthe phases in V R are free they can be tuned so as to avoid the bounds from the CP -violatingobservables, φ d,s . However, in the more constrained LRSMs these bounds will be important.These phases appear in asymmetries of B d,s decays, currently the averages of experimentalmeasurements give the following values [36] − A mix CP ( B d → f ) = sin φ d = 0 . ± .
02 (68% CL) , (24) A mix CP ( B s → f (cid:48) ) = sin φ s = 0 . ± .
07 (68% CL) , (25)where f = ( J/ψ K S , J/ψ K L , . . . ) and f (cid:48) = ( J/ψ φ, J/ψ f (980) , . . . ) are all final states involving¯ cc ¯ sd and ¯ cc ¯ ss valence quarks, respectively. Currently the value for sin φ d is still compatible withits SM prediction, sin φ SM d ∼ .
83 [37], within (theoretical) errors, but is approaching a 3 σ level deviation [38]. The SM prediction for sin φ SM s ∼ .
036 is also still consistent with theexperimental value [37]. The precision of these measurements is expected to improve of course.In the long run, the error in the φ d ( φ s ) measurements should decrease by roughly a factor 3(10), while the determination of the mass differences is not expected to improve significantly.This assumes 50 fb − LHCb and 50 ab − Belle II data, which may be achieved by the mid 2020’sat the earliest [39].Implications of these measurements in terms of bounds for the specific C + P , P , C , and CP symmetric LRSMs will be discussed in sections 4.1.1, 5.1,5.2 and 5.3, respectively. As mentioned before, LR models introduce a number of additional CP -violating sources. Atlow energies these will generally contribute to electric dipole moments (EDMs). In the leptonsector this leads to a nonzero electron EDM while CP -violating interactions in the quark sectorEDMs can induce the EDMs of the neutron, proton and light nuclei. In the following we discussthe resulting bounds. Hadronic EDMs receive contributions from the CP -violating phases in the CKM matrices and α as well as the QCD-theta term, ¯ θ . In a general LRM the latter is a free parameter. Wewill first discuss the case where ¯ θ = 0, simply assuming this has been achieved through theimplementation of a Peccei-Quinn mechanism or in some other way. Note, however, that thisis not always the case, in fact, there is an interesting scenario in which ¯ θ becomes calculable,leading to strong constraints on the LR scale [40].When ¯ θ = 0 hadronic EDMs are dominated by a single interaction which appears at tree levelwhile other contributions only appear at the loop level [41, 42]. At the scale of ∼ L LR = − i v g R g L sin ζ Im (cid:16) e iα V ud ∗ L V udR (cid:17) (cid:20) η (cid:0) u R γ µ d R d L γ µ u L − d R γ µ u R u L γ µ d L (cid:1) (26)+ η (cid:0) ¯ u R γ µ t a d R ¯ d L γ µ t a u L − ¯ d R γ µ t a u R ¯ u L γ µ t a d L (cid:1) (cid:21) , t a are the SU (3) c generators and η = 1 . η = 1 . W ± L − W ± R bosons, see Fig. 1c. Nonperturbativetechniques are required to determine the contribution of this operator to the neutron EDM.Using naive dimensional analysis (NDA) [44, 45] and the upper limit on the neutron EDM, d n ≤ . · − e cm [46] one finds (cid:12)(cid:12) g R g L sin ζ Im (cid:0) V ud ∗ L V udR e iα (cid:1)(cid:12)(cid:12) ≤ · − , (27)with a considerable theoretical uncertainty. This is about a factor 40 weaker than the upperbound found in Ref. [13], however, a recent analysis [47] using chiral perturbation theory ( χ PT)indicates that the constraint obtained there may have been overestimated. A stronger bound onthe same combination may be derived from the limit on the mercury EDM [48], however, largenuclear uncertainties now play a role. In fact, taking the estimated uncertainties [48] at facevalue, the contribution of the operator in Eq. (26) to the mercury EDM is consistent with zero.It is also interesting to consider, in LR models, the EDMs of the proton, deuteron and helion( He), for which there are plans for measurements in storage rings [49–52]. Using NDA estimates,one would expect the proton and neutron EDMs to be of similar size while the deuteron EDMis enhanced by about one order of magnitude [53]. This implies the deuteron EDM is a moresensitive probe of LRMs than the neutron and proton EDMs.Furthermore, although the lack of knowledge of the nonperturbative physics does not allowfor a prediction of the absolute size of these EDMs, it is possible to relate them. This is due tothe fact that the dominant contributions come from a single operator whose chiral symmetryproperties imply these EDMs are not independent. An LRM in which the operator of Eq. (26)is indeed dominant would predict [54] , d He = (0 . ± . d D + (0 . ± . d n − (0 . ± . d p , (28)where the errors are mainly due to nuclear uncertainties. Thus the measurements of the EDMsof light nuclei would provide a test of LR models.The above no longer applies when ¯ θ (cid:54) = 0. When ¯ θ is a free parameter it becomes hard to saysomething in general about hadronic EDMs. A (partial) cancellation between the ¯ θ contributionand that of the operator in Eq. (26) can weaken or even evade the bound of Eq. (27). However,the P and CP symmetries in principle forbid θ , in which case ¯ θ is induced by the CP -violatingphases in the quark mass matrices. ¯ θ then becomes calculable in terms of r sin α and Yukawacouplings [40]. For the P -symmetric case this means that instead of the operator in Eq. (26)¯ θ gives the dominant contribution to the nEDM by far. The result is a very strong bound on r sin α (cid:46) m b m t · − which in turn implies (through ε (cid:48) ) a strong bound on M (cid:38)
20 TeV [40].In some scenarios this bound on α also has consequences for the leptonic Yukawa couplingsas they contribute to α at loop level. Assuming the Dirac Yukawa couplings for the quarks andleptons are similar the leptonic Dirac phase should naturally be (cid:46) − , which would suppress CP violation in neutrino oscillations beyond the reach of upcoming experiments [55].However, it is possible to suppress ¯ θ and thereby the contribution to the nEDM, for example,by implementation of the Peccei-Quinn mechanism. The strong bounds on α and M then nolonger apply, the bounds that can be derived instead will be discussed in section 5.1. It should be noted however that there exists a caveat in the form of a three-pion interaction induced byEq. (26) which can also contribute to the tri-nucleon EDMs, possibly spoiling such a relation, see Ref. [54] fordetails. θ = 0 case, from the comparison of Eqs. (27) and (23) it is clear that theneutron EDM and the phases φ d,s probe different combinations of CP -violating phases. The B -mixing observables, φ d,s , depend on the phases in V L,R while the neutron EDM is also sensitiveto α . Furthermore, the neutron EDM only receives contributions from W L - W R mixing (Fig. 1c)and thus depends on ζ . The best model independent bound on ζ allows a maximal value of0 .
02 [56]. Barring cancellations between α and the phases in the right and left CKM matrices, ζ ∼ .
01 will require sin α < − , but if ζ is smaller, sin α is of course allowed to be larger.Instead, sin φ d,s receive contributions from box-diagrams and flavor-changing Higgs exchange(diagrams a and b of Fig. 1), resulting in an M and M H dependence. The CP violation inKaon mixing ε is similar to φ d,s concerning the dependence on the model parameters while ε (cid:48) isalso sensitive to α . This emphasizes the importance of the different precision measurements of CP violation in order to probe all aspects of CP violation in LR models.Thus, the flavor-diagonal CP violation in the neutron EDM would seem to be complemen-tary to the flavor-changing CP violation appearing in meson mixing. Nonetheless, as we willdiscuss in upcoming sections, there are some scenarios in which these observables are no longerindependent. Such correlations then lead to strong bounds on the right-handed scale. Measurements of the electron EDM (eEDM) have recently improved considerably and also leadto a strong bound, at present d e ≤ . · − e cm [58] . However, the eEDM is sensitive toother parameters than the hadronic EDMs, in this case the phases of the neutrino mixing matrixenter. Thus, the eEDM and hadronic EDMs are complementary observables. Furthermore, inprinciple the coupling of the right-handed bosons in the lepton sector may differ from those inthe quark sector. Upon demanding anomaly cancellation one can relate the couplings from onesector to the other, but this could be altered by an as yet undiscovered fourth generation.Another difference is that for the eEDM there are no contributions from a leptonic equivalentof the four-quark operators in Eq. (26). This means that there are no tree-level contributionsand the eEDM is generated at loop level. The generated eEDM is given by [59, 60] d e (cid:39) − e π M W g R g L sin ζ Im (cid:0) e − iα ( M ν D ) ee (cid:1) , (29)where ( M ν D ) ee is the ee element of the neutrino Dirac-mass-matrix. It is in general not possible tocompare the electron EDM to the neutron EDM as the two involve different phases. Nonetheless,we can still try to estimate their relative sizes. Taking the different phases to be of the sameorder and assuming | ( M ν D ) ee | (cid:39) m e one finds d e /d n ∼ − (again assuming ¯ θ = 0) [54]. To be precise this bound holds for the combination Re (cid:2) tan ζe iα g R V udR g L V udL (cid:3) ≤ .
02. Limits of this order ofmagnitude were already derived in Ref. [57] using hyperon decays, while more stringent limits O (10 − ) can bederived if one is willing to make certain assumptions about the CKM matrices [56]. What is probed in these experiments is actually a combination of the eEDM and semi-leptonic four-fermioninteractions. The semi-leptonic interactions originate from tree-level diagrams which involve non-SM Higgs fieldsand thereby small Yukawa couplings. This, together with the fact that these Higgs fields should be heavy, oforder >
10 TeV for LRSMs, see section 4.1.2, implies that d e will generally dominate. .4 Neutron β decay Bounds on LR models can also be obtained from neutron β decay (n β d). In fact, Ref. [61]even claims that it already provides evidence against the manifest LRSM [21]. Although notstatistically significant, their result does show that especially the neutrino-neutron spin asym-metry α ν is very sensitive to the mass M . Here α ν = 2[ N ( θ ν < π/ − N ( θ ν > π/ / [ N ( θ ν <π/
2) + N ( θ ν > π/ θ ν is the angle between the neutrino direction and the polarizationdirection of the neutron. The analysis of [21] assumes that the right-handed current couplesequally to leptons and quarks, which is an implicit assumption on the existence and mass ofthe right-handed neutrinos, namely that the decay to right-handed neutrinos is kinematicallyallowed. This requires that they should be light ( m ν R ≤ m N ), which is not in accordance witha see-saw mechanism such as in the minimal LR models discussed here.Neutron β decay is sensitive to the CP-violating phase of the mixing between the W and W bosons in a similar but not completely identical way as the neutron EDM. Due to hadronicuncertainties which are hard to improve upon, the best current bound from n β d is not as strongas that of the neutron EDM. Assuming ¯ θ = 0 and assuming heavy right-handed neutrinos, thebest bound from n β d is Im (cid:18) tan ζe iα g R g L V udR V udL (cid:19) = (1 . ± . · − (68% CL) , (30)which for LRSMs and small mixing angles translates into κκ (cid:48) v R sin( α + θ u + θ d ) = (1 . ± . · − . (31)This bound is obtained from Ref. [56] using updated experimental results [62, 63] and a latticedetermination of g A /g V = 1 . θ = 0) and is much stronger, see Eq. (27), thus for LRSMs this boundis superseded by the nEDM constraint. However, for more general LRMs a comparison of thetwo observables would be sensitive to a deviation from | V udL | = | V udR | . Another way to detectdeviations from the LR symmetric case is proposed in Ref. [65] which shows that a study using b -tags at the LHC would be sensitive to deviations of | V tbR | from | V tbL | .Having discussed the LR model and parts of its Lagrangian rather generally, we will nowdiscuss in more detail the models which have an unbroken discrete symmetry, P , C , and/or CP at high energies. Starting with the most symmetric option, we will discuss in the next sectionthe LR models that are both P and CP symmetric. P - and C P -symmetric LR models
A LR model with both discrete symmetries has the appealing feature that P , C , and CP violation are explained as low-energy phenomena. However, there is no unique LR theory withboth a C and a P symmetry as there are several ways of implementing both LR symmetries This bound is altered if we assume the right-handed neutrinos to be light. Again, this is not what one wouldexpect in a LRM using the scalar triplets of Eq. (2), however, this can be achieved in a LRM using doubletsinstead [4].
12n Eq. (5). This is due to the fact that the P and C transformations need not be aligned inflavor-space [66]. In what follows we will briefly discuss all possible ways of implementing boththe P and C symmetries of Eq. (5), for a more detailed discussion see Refs. [32, 67].Whenever the C and P symmetries are not aligned the transformation rules for P or C willnot have the simple form of Eq. (5). We will select the basis in which the P transformationdoes have the form of Eq. (5), while the C transformation may be different. The implicationsin Eq. (13) for the P symmetry are then unchanged so that Γ and ˜Γ will be hermitian matrices.Following Ref. [66] we next demand invariance under CP symmetry.The most general CP transformation can be written as follows [66] Q L,R → U L,R Q cL,R , Φ → H Φ ∗ , ∆ L,R → e iφ L,R ∆ ∗ L,R , (32)where U L,R are unitary 3 × ≡ ( φ, ˜ φ ) T , and H is a unitary 2 × φ and˜ φ are not independent fields this implies a relation between the elements of H . Taking this andthe unitarity of H into account there are two possible forms of H , H = (cid:18) ± ± (cid:19) , H = (cid:18) e iϕ e − iϕ (cid:19) , (33)where ϕ is a real number. These possibilities for H and U L,R give rise to a number of possible CP transformations, which in principle lead to different models. To simplify the discussion, andwithout loss of generality, we work in the basis where Γ is diagonal. The possible transformationrules and the consequences of the resulting models are summarized in Fig. 2. For a more detaileddiscussion see Refs. [32,67]. In short, only the option H = H with e iϕ = 1 remains as a possible CP transformation for the Φ fields, while the other possibilities are unable to reproduce the quarkmasses or their mixing. Thus, there are two CP transformations which cannot be excluded onthe basis of yielding unrealistic quark masses, namely [66] CP : H = ± , U L = , U R = ± ,CP : H = ± iσ , U L = , U R = ∓ i . (34)We now discuss these two possibilities in more detail. CP -symmetric LR models The CP case is the more widely studied possibility, see for instance Refs. [17,32,66,68]. At firstsight, this model is able to produce phenomenologically viable mass and mixing angles for thequarks. However, as we will discuss, the model is unable to produce the observed CP violationdue to the CP symmetry which constrains the Yukawa interactions as follows,Γ = Γ T = Γ ∗ , ˜Γ = ˜Γ T = ˜Γ ∗ . (35)The symmetric Yukawa couplings imply V R = K u V ∗ L K d [30] for the CKM matrices, as in Eq. (15).The hermiticity of the Yukawa couplings implies additional conditions which allows all phasesin the CKM matrices to be solved in terms of known mixing angles, quark masses, and theparameters α and r ≡ κ (cid:48) /κ [32, 69]. There are seven such phases to be solved which can13 -symmetric Yukawaswith a CP symmetry Tr( M u M † u ) = Tr( M d M † d )Γ is diagonal.Degenerate?Γ = 0 or Γ = 0? Γ = diag( γ , (cid:15)γ , γ )with (cid:15) = ± m u m c ≤ m s m d m t m b ≤ ( m u m c m s m d ) / U L = e iϕ U R = , e iϕ = 1 γ = 0 or γ = 0? CP : ˜Γ = ˜Γ T CP : ˜Γ = − ˜Γ T H = H H = H YesNo (cid:15) = +1 (cid:15) = − γ = 0 γ = 0 e iϕ = ± e iϕ = ± i Γ = 0Γ = 0 Figure 2: Flowchart for a P - and CP -symmetric Yukawa sector, depicting the possible choicesfor the CP transformation, Eq. (32), and their consequences. We work in a basis where Γ isdiagonal. See Refs. [32, 67] for a detailed discussion.be parametrized by the usual SM phase in the left-handed CKM matrix and six additionalphases in the right-handed CKM matrix. This solution allows for the prediction of CP -violatingobservables in terms of the combination r sin α . However, in order to reproduce the quark massesthis combination has to be small [20], thus the above solution only exists for relatively smallamounts of CP violation, namely [17, 68] r sin α − r (cid:46) m b m t . (36)14he SM CKM phase δ can be expressed in terms of this combination as well, the above boundthen requires this phase to be rather small, δ < .
25 [17]. In contrast, the SM CKM fit requires δ to be rather large, δ (cid:39) . CP LR model cannot reproduce the CP violation of the SM in the decoupling limit, v R → ∞ . This limit can therefore be excluded [17].The above observations have further implications. Since the decoupling limit does not re-produce the SM there is not necessarily a range in parameter space where the CP LR modelreproduces the SM. In fact, from recent measurements of the CP violation in B − ¯ B mixing thepredictions of the model can be shown to be too small. After taking into account constraintsfrom CP -violating parameters in the Kaon sector, ε and ε (cid:48) , the model predicts (for any value of M and M H ) [68, 70], | sin φ d | < . , sin φ s < − . . (37)Clearly, this is incompatible with the measured value in Eq. (24). Thus, although the modelcan reproduce the observed quark masses and mixing angles, it is untenable when discussing CP -violating observables. As the Yukawa couplings of the pseudomanifest LRM coincide withthat of the P - and CP -symmetric LRM under discussion here, it follows that the minimal pseudomanifest LRM can be excluded in the same way. Further problems arise when considering the Higgs sector of the CP model. Although the aboveconsiderations may be considered to rule out the model we will nevertheless review some of theseproblems as they are exemplary of what will be encountered in other versions of LR symmetricmodels. Our discussion will largely follow that of Refs. [5, 31].In this case the Higgs potential takes the form of Eq. (16) with the additional constraint fromthe CP transformation that there is no explicit CP violation: δ = 0 . (38)Note that if the phases φ L,R are introduced in the CP transformation of the ∆ L,R fields, Eq. (32),additional constraints, β i = ρ = 0, are acquired. However, as the potential Eq. (16) is ( B − L )-symmetric no such constraints appear when e i ( φ L − φ R ) = 1. Such a Higgs potential has beenwidely studied in the literature [5, 16, 31, 71, 72]. From the requirement that the potentialis minimized one can obtain the following expressions for the dimensionful parameters of thepotential [31], µ v R = α − κ (cid:48) κ − α + κ v R λ + 2 κκ (cid:48) v R λ cos α,µ v R = α κκ (cid:48) κ − cos α α + κ v R λ + κκ (cid:48) v R cos α ( λ + 2 λ cos 2 α ) ,µ v R = ρ + κ v R α + 2 κκ (cid:48) cos αv R α + κ (cid:48) v R α , (39)where we neglected terms of order v L /v R and v L /v R . Substituting the exact expressions for µ i
15n the three remaining minimum equations we obtain2 ρ − ρ = β κκ (cid:48) cos( α − θ L ) + β κ cos θ L + β κ (cid:48) cos(2 α − θ L ) v R v L , (40)0 = κκ (cid:48) (cid:2) α (1 + v L v R ) + κ − v R (4 λ − λ ) (cid:3) sin α + v L v R (cid:2) β (cid:0) κ sin θ L + κ (cid:48) sin( θ L − α ) (cid:1) + 2( β + β ) κκ (cid:48) sin( θ L − α ) (cid:3) , (41)0 = β κκ (cid:48) sin( α − θ L ) − β κ sin θ L + β κ (cid:48) sin(2 α − θ L ) . (42)These equations were obtained by minimizing with respect to v L , α and θ L , respectively. FromEq. (41) we approximately have (for v R (cid:29) κ + (cid:29) v L ) α sin α = O ( β i v L /v R ) . (43)As we require the hierarchy v R (cid:29) v L , this implies that α sin α must be small in order for β to be in the perturbative regime. As two extreme cases we could have a small spontaneousphase, i.e. α = O ( v L /v R ), while α can be of order one, or α is small with a sizeable α , thus, α = O ( v L /v R ) with α = O (1). It turns out that in both extremes, as well as in all intermediatecases, some of the additional Higgs fields become too light, such that their effects should havebeen detected experimentally already [5, 31]. In one extreme, α = O ( v L /v R ) and α = O (1),Eqs. (40) and (42) imply 2 ρ − ρ = O ( κ /v R ), which in turn implies that the left-handedtriplet fields become light, O ( κ + ). Explicit calculation shows that they are even lighter, namelyof order O ( v L ). As these fields couple to the electroweak gauge-bosons such light fields shouldalready have been discovered at LEP-I [31].For the other extreme, whenever α is small, O ( κ /v R ), there are problems with flavour-changing neutral-currents (FCNCs). In the minimal LR model these FCNCs are generated bythe neutral scalars of the bidoublet. FCNCs are stringently constrained by Kaon- and B -mixing,in fact, the mass of such a scalar should be of the order of >
10 TeV [7,17] in LRSMs. An analysisof the masses of the physical Higgs fields [31] shows that whenever α is small, O ( κ /v R ), theHiggs fields with the FCNC couplings remain light and the FCNC bounds cannot be evaded.The remaining scenarios interpolate between the two extremes. Here one finds three lightneutral states, which are now mixtures of the neutral triplet field, δ L , and the flavor-changingneutral Higgs field [31].Thus, whenever sin α (cid:54) = 0 the potential Eq. (16) can not reproduce the SM Higgs spectrum.The addition of extra scalar fields could solve this, simultaneously allowing for a SM-like Higgsspectrum and spontaneous CP violation in the CP potential [17, 73]. This option will not bediscussed any further here. Another possibility is to have a potential without spontaneous CP violation, α = θ L = 0. In this case all the non-SM scalars can obtain a large mass and decouple,thereby allowing for a SM-like Higgs spectrum. Even in this case, however, there is a price topay in the form of fine-tuning as we will discuss next. The fact that there should be a hierarchy between the vevs, v R (cid:29) κ, κ (cid:48) (cid:29) v L , induces fine-tuning in the potential. This occurs because the minimum equations relate the different scalesto one another. For the parameters of the potential to be in the perturbative range this requires16 certain amount of fine-tuning. Similar to the Higgs potential itself it is useful to review thefine-tuning in the CP invariant potential, as it will turn out to be exemplary of the cases wewill study in the following sections.One of the dominant sources of fine-tuning appears in Eq. (40), schematically we have,2 ρ − ρ ∼ κκ (cid:48) v L v R β i , (44)which has been called the vev see-saw relation [5]. If we insist on the desired hierarchy, one maymake the following estimates for the vevs, v R ∼
10 TeV, κ, κ (cid:48) ∼
100 GeV and v L ∼ ∼ . Thus, in order for the ρ parameters to beof order one, the β i parameters should cancel to a precision of ∼ − , implying a very fine-tunedpotential. There are two ways to avoid this fine-tuning. One can either accept a very high scalefor the LR model, v R ∼ GeV, making the additional gauge fields, Z R and W ± R unobservable,or eliminate the vev see-saw relation. The latter option can be achieved by setting v L and β i tozero. In this case Eq. (40) vanishes and is no longer a source for fine-tuning. This option wasconcluded to be the only viable option leading to observable effects in Ref. [5]. However, evenin the case there is still a considerable amount of fine-tuning. Looking, for instance, at the thirdequation in Eq. (39) we see that the ρ and µ terms should cancel to O ( κ v R ) in order for α to be of order one. Similarly, from the first equation in Eq. (39) the α , and µ terms shouldcancel to O ( κ v R ) in order for λ i to be of order one. Combining the two, we see that cancellationsto a precision of order O ( κ v R ), i.e. O (10 − ) for the above selected values, are needed. Some ofthis fine-tuning may be avoided if we set some of the parameters to be small by hand or byintroducing an additional mechanism [16].As we will discuss later this type of fine-tuning tends to occur in more general scenarios aswell. Before moving on to these LR models, however, we will first review the second P - and CP -symmetric case. CP -symmetric LR models This case has received somewhat less attention than the CP possibility [66, 73]. Although theYukawa sector is distinct from the CP case, which is relevant for the amount of CP violationallowed, we will see that the Higgs potential is very similar. As in the previous case, the Yukawa interactions are constrained by the P and CP trans-formations. For the Yukawa interactions the P and CP transformations have the followingimplications Γ = Γ T = Γ ∗ , ˜Γ = − ˜Γ T = − ˜Γ ∗ . (45)The fact that ˜Γ is antisymmetric means that in this case the mass matrices will not be symmetric.Thus, there is, in general, no simple relation between the left and right-handed CKM matrices.However, the Higgs potential is clearly more constrained in this case.17 .2.2 The Higgs potential The CP invariant Higgs potential is that of Eq. (16) with the additional constraints, µ = λ = 0 , δ α = ± π/ ,β (1 − e i ( φ L − φ R ) ) = 0 , β , (1 + e i ( φ L − φ R ) ) = 0 , ρ (1 − e i ( φ L − φ R ) ) = 0 . (46)Thus, we have β = 0 and/or β , = 0 (and possibly ρ = 0) depending on φ L − φ R .As was already noted in Ref. [73], barring fine-tuning, there will be very little CP violationin this case. Neglecting subleading terms in the potential, one finds α = ± π/
2, which in thiscase is a CP conserving minimum, as, after an SU (2) L,R gauge-transformation, the vevs of thebidoublet can then be written as (cid:104) φ (cid:105) = e ± iπ/ (cid:18) κ κ (cid:48) (cid:19) , (47)which is invariant under the CP transformation.The feature that without fine-tuning the spontaneous CP violation will be small is reminiscentof the CP case. In fact, we observe that the CP -symmetric potential is very similar to a specialcase of the CP -symmetric potential. This can be seen by use of a field redefinition. In case wetake e i ( φ L − φ R ) = 1, and thereby β , = 0, we can apply the following redefinition φ → e ∓ iπ/ φ, (48)the resulting potential is then, after an SU (2) L,R transformation, nearly equal to the CP casewith the following identifications, µ CP = λ CP = β CP , = 0 , λ CP = − λ CP ,α CP = α CP ± π/ , θ CP L = θ CP L ± π/ . (49)The only remaining difference comes from the α terms involving Tr(∆ L ∆ † L ). Clearly, theseterms are suppressed with respect to their right-handed equivalent, due to the hierarchy v R (cid:29) κ, κ (cid:48) (cid:29) v L , meaning that to good approximation the two potentials are equivalent. Thus, in thiscase the minimum equations correspond to those of Eqs. (39) and (40)-(42) to O ( κ /v R ), withthe identifications of Eq. (49). A similar redefinition can be made for the case e i ( φ L − φ R ) = − CP potential is, to good approximation, equal to a special case of the CP invari-ant potential. This also implies that the conclusions about the CP -symmetric potential carryover. The case with spontaneous CP violation implies a non-SM-like Higgs spectrum whereasthe case without spontaneous CP violation has no CP violation at all. Therefore, we concludethat also the CP -symmetric case is not viable.In the following sections we will study minimal LR models with fewer discrete symmetries.We will see that although there may be important differences between the Yukawa sectors ofthese models, their Higgs potentials will tend to be very similar, like for the CP and CP cases.18 P - or C -symmetric LR models P -symmetric LR models P -symmetric LRMs have been studied quite extensively in the literature, e.g. [1, 2, 7, 13, 16]. Inthis case there is an approximate relation between the mixing matrices, V L (cid:39) K u V R K d , (50)where K u,d are diagonal matrices of phases. As was already mentioned, this is due to the factthat the combination r sin α should be small in order to be able to reproduce the small ratio m b /m t [20]. In fact, the same bound (36) as in the CP case applies here too. In order to satisfythis bound and simultaneously the experimental constraints on CP violation, one can arrive atconstraints on the phases in K u,d and on M . In other words, even though CP violation canarise in this type of model, the pattern of CP violation in Kaon and B-meson mixing and thenEDM may not be reproducible unless M has some minimum value. Although this has beendiscussed before in the literature [7, 13, 20], we will briefly summarize this point.At the moment we do not know the value of r , but if it is small ( r (cid:28) m b /m t ) one can usethe analytical expressions for the phases in K u,d derived in terms of r sin α [13], which haverecently been generalized to general r values [29]. For M in the TeV range, the constraint onindirect CP -violation in Kaon mixing, (cid:15) , then drives a combination of the phases in K u,d toa nonzero value, | θ d − θ s | (cid:39) .
17. As in this case all the phases are functions of r sin α , thisrequires a nonzero value for this combination. Both the neutron EDM as well as ε (cid:48) then set astrong bound on ζ sin α . These observables can then only be reconciled with ε for large valuesof M (cid:38)
10 TeV [7, 13].On the other hand, if r is large ( r (cid:38) m b /m t ) the phases in K u,d can be sizable and tuned tosatisfy the CP violation constraints. This leads to θ c − θ t (cid:39) π/ ε and θ d − θ s (cid:39) π from ε (cid:48) . In addition, for the CP -violation in B meson mixing (see Eq. (23)) we have σ d (cid:39) π + θ b − θ d , σ s (cid:39) π + θ b − θ s , (51)which leads to a correlation between φ d and φ s , on which we will comment further below (57).These constraints together with the Kaon and B -meson mass-differences, ∆ M K and ∆ M B d,s ,require M (cid:38) r probably allows toput an even more stringent bound, as in this case all phases in K u,d are known expressions of r sin α , like in the CP case .In the limit of vanishing α the quark sector of the P -symmetric LRM coincides with that ofthe minimal manifest LRM. In this case ε sets a strong bound on the W ± R mass of M (cid:38) P -symmetric LRMs are more stringentthan the direct limits on M and will be even more so in the future thanks to LHCb, B -factoriesand improvements in lattice determinations of the relevant matrix elements. The increase ofexperimental sensitivity discussed in section 3, which will take at least another 10 years torealize, is expected to push the lower bound on M to roughly 8 TeV [7], which will likely allowconfrontation of the P -symmetric LRMs with data, although in this case there is no upper limiton M , as the decoupling limit has not been excluded, in contrast to the CP case. Note that in the CP scenario also the SM phase δ can be expressed in terms of r sin α . The ranges taken for the parameters of the potential when generating random points in pa-rameter space. ρ ρ λ , ρ , α , λ , , , α , ρ , β , , α, δ , θ L v L (eV) κ (GeV) v R (GeV)[0 ,
5] [2 ρ ,
10] [0 ,
10] [ − ,
10] [0 , π ] [0 ,
10] [0 , , · ] The Higgs potential in this case is that of Eq. (16). As pointed out in Ref. [31] it can be mappedonto the CP case to good approximation by a field redefinition, similar to that of Eq. (48), φ → φe − iϕ/ , ϕ = Arg( α v R / e iδ − µ ) . (52)After an SU ( L ) L -gauge transformation this gives α CP → α P + ϕ, θ CP L → θ PL + ϕ. (53)The remaining differences between the two potentials then are terms subleading in v R , O ( κ /v R ).Thus, the minimum equations of Eqs. (39)–(41) with the above replacement apply here too,to O ( κ /v R ). The remaining minimum equations result from subleading terms and are notobtained from their CP equivalents after applying the identifications Eq. (53). Instead theysimply equal the corresponding CP equations, Eqs. (40) and (42). This near-equivalence meansthe conclusions of Ref. [31] about CP models discussed in section 4.1 should apply here too.Thus, again it will not be possible to obtain a SM-like Higgs spectrum for arbitrary values of α .However, as mentioned in section 4.1 for the CP case there is the possibility of a SM-like Higgsspectrum in the limit v R → ∞ for a specific value of α which now occurs very close to α = ϕ .Note that this is now an acceptable possibility as it still allows for spontaneous CP violation andwe already allowed explicit CP violation in the Yukawa couplings. Thus, in the P -symmetricLR model it is possible to have a SM-like spectrum in the decoupling limit in combination withspontaneous CP violation. However, the amount of spontaneous CP violation then entirelyresults from the explicit CP violation present in the Higgs potential, as ϕ = 0 when δ = 0. Ina sense this means that the CP violation is put in by hand and can be as large as allowed bythe value of r . Nevertheless, as discussed in the previous section, the pattern of CP violationin Kaon and B -meson mixing and the nEDM also put stringent constraints on the model, inparticular on M . Moreover, there is the issue of fine-tuning. The fact that the minimum equations relate several very different scales, namely, v R (cid:29) κ + (cid:29) v L to one another means that some of the parameters in the potential will tend to be fine-tuned.This is especially true for cases where v L (cid:54) = 0 [5] as was already noted in section 4.1. In thiscase one can obtain the vev see-saw relation of Eq. (44). This requires some parameters to befine-tuned to a precision of order O ( v L /v R ). However, as we will show, more fine-tuning may berequired, similar to that discussed in section 4.1, due to the remaining minimum equations. Infact, solving the minimum equations for µ , , β , , α and ρ we see that the leading terms in ρ are proportional to v R /v L . This implies that if ρ is to be of order one, i.e. in the perturbativeregime, these terms should cancel to a precision of v L /v R .20igure 3: The figure shows the fine-tuning measure ∆ Max as a function of v R in TeV for a P -symmetric V H . The blue points are randomly generated points satisfying the minimum equationsand the ranges in Table 1. Here the red line is chosen such that 0 .
1% of the points are foundbelow it. It is parametrized by 6 · − v R /v L taking an average value for v L of 10 eV. v L = β i = 0 v L = 0 , β i (cid:54) = 0Figure 4: Similar plots to that of Fig. 3. The plot on the left shows the fine-tuning in the casewhere β i and v L are set to zero, while the figure on the right does the same in the case whereonly v L is set to zero. The red lines are again chosen such that 0 .
1% of the points are foundbelow it. It is parametrized by 3 · − v R /κ and 2 · − v R /κ in the left and right plots,respectively.In order to study the matter quantitatively we use the minimum equations to solve for asmany parameters, which we will denote by p i , as there are equations. Subsequently we study thedependence of these p i on the remaining parameters, p j . More specifically, we adopt the followingquantity as a measure for the fine-tuning in p i typically used for supersymmetric extensions ofthe SM [74, 75], ∆ i = Max j (cid:12)(cid:12)(cid:12)(cid:12) d ln p i d ln p j (cid:12)(cid:12)(cid:12)(cid:12) . (54)Our procedure is as follows, we first generate random O (1) values for nearly all parametersin the potential while obtaining values for the remainder through the minimum equations. Theallowed ranges for the parameters are shown in Table 1. As the dimensionless parameters shouldremain in the perturbative range we conservatively constrain their values to lie in the interval[ − , λ and ρ and to havepositive masses-squared values of the Higgs fields, like for ρ , ρ − ρ , and α [16]. However, theimposed constraints are actually not sufficient to keep all mass-squared values positive, but thiswill not affect the conclusions. In addition, we have not imposed any experimental constraintson these masses. Thus, only a subset of the generated points will be phenomenologically viable.We further take values for the vevs which adhere to their naive expectations. We make noassumptions for the µ i parameters, instead we calculate their values through the minimumequations.We then solve the minimum equations for as many parameters as there are equations. Therandom points in parameter space are then used to calculate the fine-tuning measures for thesolved parameters p i . The results are shown in Fig. 3 where we plot the maximum value of∆ Max ≡ Max i ∆ i against v R .Clearly, the degree of fine-tuning can be significantly larger than one might expect from thesee-saw relation of Eq. (44) alone and be enhanced through the coupled minimum equations. Ascan be seen from the plot in Fig. 4 and was noted in Ref. [5] the fine-tuning may be considerablydecreased by setting v L = β i = 0, as was done in e.g. [13]. In this case, however, still a fine-tuning of order ∆ = O ( v R /κ ) (cid:38)
100 remains. Since setting β i = 0 may lack justification [5,31],we observe that setting only v L to zero leads to the same reduction in the amount of fine-tuning.In this case the vev see-saw relation vanishes and instead we obtain two relations for the β i parameters, namely, β = − β κ (cid:48) κ cos α, β = κ (cid:48) κ β . (55)It remains to be seen whether these relations can be justified or not. Therefore, setting v L andpossibly β i to zero is a simple way to greatly reduce the fine-tuning in the Higgs potential, butin both cases one may wonder whether this is justified. C -symmetric LR models C -symmetric LRMs have been less investigated in the literature so far, although recently therehas been renewed interest in Refs. [20] and [7]. In this case the mixing matrices are related byEq. (15), V R = K u V ∗ L K d , (56)with K u,d diagonal matrices of phases. The mass matrices are now less constrained than in the P -symmetric case; the Yukawa couplings are symmetric as opposed to being hermitian. One ofthe consequences is that the combination r sin α is no longer required to be small in order toreproduce the quark masses, as opposed to the P -symmetric case. Furthermore, now the phasesin K u,d are free parameters of the model. These can be tuned in order to evade the constraintsfrom CP -violating observables. The ε constraint can be evaded when | θ d − θ s | (cid:39) nπ ( n = 0 , ε (cid:48) constraint can be satisfied when θ d − θ s (cid:39) π or r is small. From the relation inEq. (15) we have for the phases σ d,s (see Eq. (23)) σ d = π + θ b − θ d + 2 φ, σ s = π + θ b − θ s , (57)where φ = Arg (cid:0) V tb ∗ L V tdL (cid:1) . Note that σ d and σ s are again correlated in an M and M H dependentway through the ε ( (cid:48) ) constraints. To be specific, for the quantities which determine the B d,s Max as a function of v R in TeV for a C -symmetric V H . With respect to the P -symmetric potential the C -symmetric potential hasseven additional phases, for which we choose the range [0 , π ]. The red line is parametrized by30 · − v R /v L taking an average value for v L of 10 eV.observables one has1 + h d (cid:39) − | h d | e i ( θ b − θ d +2 φ ) , h s (cid:39) − ( − n | h s | e i ( θ b − θ d ) , (58)while in the P -symmetric case, again taking into account ε ( (cid:48) ) constraints, one has1 + h d (cid:39) − | h d | e i ( θ b − θ d ) , h s (cid:39) | h s | e i ( θ b − θ d ) . (59)From the B d,s -mixing observables (∆ M B d,s and φ d,s ) it should in principle be possible to seewhich of the above patterns, Eq. (58) or (59), fits the data better (especially since | h d | / | h s | isconstant to good approximation [17, 20]). In other words, if a sign of an LRSM is found in B d,s -mixing these observables are in principle also sensitive to the difference between the P - and C -symmetric options.After the ε ( (cid:48) ) constraints have been used to fix the phases, the constraints from ∆ M K andfrom B d,s mixing can again be used to put a strong limit on the W ± R mass. In this case the B d,s meson limits are competitive with the bound from Kaon mixing [7] which is similar to the P -symmetric case, i.e. M (cid:38) The Higgs potential in this case is that of Eq. (17). The form of this potential is quite similarto that of the P -symmetric potential, Eq. (16). In this case, however, phases appear in the µ , λ , , ρ and β i terms which were absent in the P -symmetric potential. Nonetheless, to goodapproximation the potential of Eq. (17) can again be mapped onto the CP -symmetric case bya field redefinition, φ → e − iϕ (cid:48) / φ, ϕ (cid:48) = Arg (cid:0) α v R e iδ α − µ e − iδ µ (cid:1) . (60)After an SU (2) L -gauge transformation this then implies the following identifications, α CP = α C + ϕ, θ CP L = θ CL + ϕ. (61)23 L = β i = 0 v L = 0 , β i (cid:54) = 0Figure 6: Similar plots to that of Fig. 5. The plot on the left shows the fine-tuning in the casewhere β i and v L are set to zero, while the figure on the right does the same in the case whereonly v L is set to zero. The red lines are again chosen such that 0 .
1% of the points are foundbelow it. It is parametrized by 2 · − v R /κ and 1 · − v R /κ in the left and right plots,respectively.This redefinition removes the phases from the µ and α terms, but does not remove the phasesrelated to the λ , , ρ , and β i terms. However, these terms are subleading, i.e. of order O ( κ /v R )and smaller. At first sight the ρ term may be an exception to this, however this term onlycontributes an O ( v L v R ) term to the masses of doubly charged scalars and does not appear inthe minimum equations. Thus, up to terms subleading in v R , after the redefinition Eq. (60) the C -symmetric potential is equal to the CP -symmetric potential. Indeed four of the minimumequations are again those of the CP case, Eq. (39) and (41), to O ( κ /v R ), with the aboveidentifications. The remaining two, do not follow this rule as they emerge from subleadingterms in the potential. Instead they are given by,2 ρ − ρ = β κκ (cid:48) cos( δ β + α − θ L ) + β κ cos( δ β − θ L ) + β κ (cid:48) cos( δ β + 2 α − θ L ) v R v L , β κκ (cid:48) sin( δ β + α − θ L ) + β κ sin( δ β − θ L ) + β κ (cid:48) sin( δ β + 2 α − θ L ) (62)Nevertheless, up to small corrections the conclusions of P - and CP -symmetric case, discussedin section 4.1, should apply once more.The conclusions for the C -symmetric potential are very much like those for the P -symmetriccase. Again it will not be possible to obtain a SM-like Higgs spectrum for arbitrary values of α . Nonetheless, there is a possibility of a SM-like Higgs spectrum in the decoupling limit for aspecific value of the spontaneous phase which in this case occurs for α = ϕ (cid:48) . Thus, much likethe P -symmetric potential, it is possible to have a SM-like spectrum in the decoupling limit incombination with spontaneous CP violation, however, the size of the spontaneous phase is againentirely dictated by the explicit CP violation present in the potential, through Eq. (60). Not surprisingly, the fine-tuning measures in the C - and P -symmetric potentials are rathersimilar. If we do not eliminate the vev see-saw relation, Eq. (44), the potential must again bevery fine-tuned, as can be seen in Fig. 5, which can be compared with Fig. 3 of the P -symmetriccase. 24igure 7: The figure shows the fine-tuning measure ∆ Max as a function of v R in TeV for a CP -symmetric V H . The red line is parametrized by 1 · − v R /κ v L taking an average valuefor v L of 10 eV.One can again choose to eliminate the see-saw relation in order to reduce the amount of fine-tuning substantially. As before, this can be achieved by setting v L = 0 and possibly β i = 0, seeFig. 6. In both cases still a considerable amount of fine-tuning remains, ∆ = O ( v R /κ ) (cid:38) v L = 0 only, one obtains again two relations for the β i parameters, which howeverdiffer from those of Eq. (55), β = rβ sin( δ β − δ β − α )sin( δ β − δ β + α ) , β = − rβ . (63) CP -symmetric LR models In a CP -symmetric LR model the CKM matrices are in principle unrelated to one another,the same is true for the gauge couplings, generally, g R (cid:54) = g L . Nevertheless, in a similar fashionto the P and CP cases [20] one may again derive the upper bound of Eq. (36). Despite thissimilarity, here it is possible to tune the right-handed gauge coupling and CKM elements inorder to weaken the constraints from direct searches and B and K mixing. In fact, the studyof general LRMs (without the constraints on V R of the C and P cases) of Ref. [76] shows thatfor M in the 2 − V R , much like that of the SM CKMmatrix, is allowed, but also regions with large off-diagonal ( V cbR and V ubR ) elements are possible. The Higgs potential in this case is that of Eq. (18). This potential contains 5 more parametersthan the P -symmetric potential. Note, however, that if we were to neglect Tr(∆ L ∆ † L ) comparedto Tr(∆ R ∆ † R ), we would obtain a potential very similar to the P -symmetric case. Indeed, five ofthe minimum equations are, up to terms of O ( v L /v R ) given by those of the CP -symmetric case,Eqs. (39), (41) and (42), with the translations µ → µ R , ρ → ρ R and α i → α iR . The finalminimum equation, the vev see-saw relation, is also given by the corresponding CP equation,Eq. (40), to O ( κ /v R ), where µ L /v R now plays the role of ρ . Schematically,2 µ L v R − ρ ∼ κ v L v R β i . (64)25 L = β i = 0 v L = 0 , β i (cid:54) = 0Figure 8: Similar plots to that of Fig. 7. The plot on the left shows the fine-tuning in the casewhere β i and v L are set to zero, while the figure on the right does the same in the case whereonly v L is set to zero. The red lines are again chosen such that 0 .
1% of the points are foundbelow it. It is parametrized by 2 · − v R /κ and 5 · − v R /κ in the left and right plots,respectively.Thus, unless the β i terms cancel to good precision, the natural value for µ L is of the orderof O ( v R v L κ ). The µ L parameter thereby is the main difference between this and the CP case. Since, if µ L = µ R we could have identified it with µ of the CP case. Note, however,that µ L only appears in the mass terms for the left-handed triplet fields. Furthermore, the twomechanisms which led to small masses of additional Higgs fields in the CP case (see section 4.1)are still in place here. The equivalent of Eq. (41) now implies α R sin α to be small. Choosing α R small again implies small flavor-changing Higgs masses. On the other hand, a small valueof α implies, through Eqs. (42) and (40), small 2 µ L − v R ρ which dictates the masses of theleft-handed triplet fields.Thus, the condition that α R sin α be small and the lower bound on the flavor-changing Higgsmasses force α to be small. As this is the only source of CP violation in the quark sector, onemight expect the model to predict hardly any CP violation. This would lead one to doubt theviability of the model. However, the lower bounds on the CP -violating Higgs mass in the C and P cases assumed the relation between the CKM matrices these symmetries imply. As there isno such relation in this case the bounds can be weakened. For example, the analysis of Ref. [76]shows that points in parameters space for values of M H as low as M H ∼ . α = O ( v L /v R ). However, even smaller values of M H mightbe achieved at the price of additional fine-tuning [76], which would allow for larger α . The parameters multiplying the left-handed triplet terms do become important when discussingthe fine-tuning in this potential. These terms contribute terms to the minimum equations whichare smaller than those encountered in the P - and C -symmetric cases. This means these equationsnow relate high scales to even smaller scales, indicating more fine-tuning.We again go through the same procedure as in the P - and C - symmetric cases, generatingrandom points in parameter space and calculating the measure of fine-tuning. The results areshown in Figs. 7 and 8. Clearly, when v L (cid:54) = 0 and β i (cid:54) = 0 the fine-tuning measure reaches newheights, due to the vev see-saw relation and the newly added Tr(∆ L ∆ † L ) terms. However, when26e choose to eliminate the vev see-saw relation we obtain fine-tuning measures comparableto the C - and P -symmetric cases. This may have been expected as when v L = 0 the termswhich differ from the P case do not contribute to the minimum equations. We then obtainthe minimum equations of the P -symmetric potential (with δ = 0), with the translations, µ → µ R , ρ → ρ R and α i → α iR . Thus, as far as the fine-tuning is concerned, when v L = 0the CP -symmetric potential simplifies to a special case of the P -symmetric potential. As such,the relations between the β i parameters one then obtains is that of the P -symmetric case, Eq.(55). The most symmetric minimal LR models, the ones invariant under P , C , and CP , turn outnot to be viable. There are just two possible implementations of both P and C that are ableto produce the observed quark masses, yielding the CP and CP models. The models differin the relation among the left and right CKM matrices. In the case of the CP model thisrelation puts constraints on CP -violating observables from Kaon and B -meson mixing that areincompatible with measurements, in particular, the bounds on the B -mixing angle φ d . As theYukawa couplings of the minimal pseudomanifest LRM coincide with those of the CP model,it follows that it is also excluded. The CP model cannot be excluded in the same way, as thereis in general no simple relation between left and right CKM matrices. In this case the Higgspotential is more constraining. Here it was shown that the Higgs potentials of the CP model is,to good approximation, equal to a special case of the CP invariant potential. For that potentialit was shown in Ref. [31] that whenever there is spontaneous CP violation, α (cid:54) = 0, the potentialcan not reproduce the SM Higgs spectrum and since the model has no explicit CP violation,the case without spontaneous CP violation has no CP violation at all. Upon field redefinitions,these conclusions carry over to the CP model, which can therefore be considered excluded. Inaddition, both models generally require a large amount of fine-tuning. The minimum equationsgenerally relate very different scales, κ + , v R and v L , which implies that some of the parameterswill have to be fine-tuned. The most extreme tuning results from the so-called vev see-sawrelation Eq. (44) [5], which implies a huge amount of fine-tuning.Less symmetric possibilities are the P -, C - and CP -symmetric LRMs, the first two are LRsymmetric, while the last possibility, CP , does not relate left- and right-handed fields. Themost widely studied case is the P -symmetric LRM. This type of LRSM is most constrained inthe limit that the ratio of vevs r ≡ κ (cid:48) /κ is small (cid:46) m b /m t . Based on an analytical solutionfor the phases in the CKM matrices, which allows for strong constraints from CP -violatingobservables Kaon and B -meson mixing and the neutron EDM, a lower bound M (cid:38)
10 TeV wasobtained [13]. Outside this regime this solution for the phases does not exist and the boundis weakened, M (cid:38) P -symmetric LRMs aremore stringent than the direct limits on M for left-right symmetric models, which currently are2 TeV if one does not wish to make assumptions about right-handed neutrinos. The increaseof experimental sensitivity in the coming 10 years is expected to push the lower indirect boundon M to roughly 8 TeV, thereby exploring a considerable part of the still available parameterspace of the P -symmetric LRMs. The minimal manifest LRSM, the case of vanishing α , is moreconstrained, here the current bound on the W ± R mass is M (cid:38)
20 TeV [40].27he Higgs potential of the P -symmetric LRM has been widely studied in the literature[5, 31, 71, 72]. This potential is very similar to that of the CP -symmetric LRM [31], whichimplies that a SM-like Higgs spectrum is only possible for a specific value of the spontaneousphase. In this case, however, the spontaneous phase can be nonzero, although it is now entirelydictated by the explicit CP violation present in the potential. In a sense this means that the CP -violating phase α is put in by hand and can be as large as allowed by the value of r and theconstraints from Kaon and B -meson mixing and the nEDM.Finally, there is the issue of fine-tuning resulting from the minimum equations. Again themost extreme tuning results from the vev see-saw relation, which implies a huge amount offine-tuning. It is possible to avoid this by setting v L and β i to zero by hand as was done inRef. [13]. As we have demonstrated, the same reduction in the amount of fine-tuning can beachieved with v L = 0 only, in which case there are two relations among the β i parameters [31].The minimum amount of fine-tuning, assuming O (1) parameters, is then found to be ∆ ∼ O (1)implies a change in another parameter by a factor O (100). This may be acceptable for a theorywith such widely varying scales.Accepting such a minimum amount of fine-tuning, the P -symmetric LRM with some con-straints on its β i parameters is still viable, but is expected to be much further constrained in thecoming decade. As the lower bound on M and hence on v R increases, the amount of fine-tuningwill also increase.The C -symmetric LRM has been studied less in the literature. In this case the phases inthe CKM matrices are free parameters, which can be tuned to avoid constraints from CP -violating observables. Nonetheless, a lower-bound similar to the P -symmetric case can be set, M (cid:38) C -symmetric and P -symmetric LRMs the Kaon and B observables are the mostsensitive probes of the mass of the W ± R boson and the most promising way to explore theparameter space of these models. This is due to the fact that these observables are determinedby the left- and right-handed CKM elements which, in the P / C -symmetric case, are constrainedby the LR symmetries. Since the relation between the left and right CKM matrices is differentfor the C - and the P -symmetric LRMs, these two types of models predict a different pattern in B d,s -mixing. Thus, if signs of an LRM are observed in this sector one should in principle be ableto tell the difference between C - and P -symmetric LR models using B d,s -mixing observables.EDMs constrain the spontaneous CP -violating phase α and the W L − W R mixing angle ζ andare thereby complementary to the meson-mixing observables. At present the neutron EDMsets a strong limit on these parameters, although the deuteron EDM is expected to be a moresensitive probe by about an order of magnitude. Additionally, in an LR model one expectscertain relations to hold between the EDMs of light nuclei, measurements of which would allowfor another type of test of LRMs.Although new phases arise in the Higgs sector of the C -symmetric LRM, the Higgs potentialis again very similar to that of the CP -symmetric LRM. A SM-like Higgs spectrum is onlypossible for a specific value of the spontaneous phase, which is again entirely dictated by theexplicit CP violation present in the potential. Perhaps unsurprisingly, the amount of fine-tuningrequired in the potential is similar to the P -symmetric case. By setting v L = 0 (and possibly β i = 0) the vev see-saw relation can be eliminated and the fine-tuning dramatically reduced.Nevertheless, in this case a minimum fine-tuning of ∆ ∼
100 is required as well.28inally, the CP -symmetric model, which need not be P and C symmetric separately, isconsiderably different from the above cases. This is due to the fact that CP is not a LRsymmetry. Although, like the P case, there is a bound on r sin α , the CKM matrices andcoupling constants are now generally unrelated to one another ( g L (cid:54) = g R ). This means thereis more freedom in this model, such that both direct and indirect bounds are expected to beweakened, see, for instance, Ref. [76]. The relation between the EDMs of light nuclei is unaffectedby this and would still allow for a test of the model. As most features of the Higgs sector againresemble the CP case, a nonzero spontaneous phase again requires light Higgs fields. However,due to the additional freedom, the bounds on these fields may be weaker in this case. On theother hand, the potential generally requires more fine-tuning than the CP potential, exceptwhen considering the v L = 0 cases, in which the fine-tuning is similar to the P -symmetric case.Recapitulating, LRSMs are possibly the most attractive of the possible LRMs, but also themost constrained. The LR scale of these models is currently constrained to be in the TeV range, M (cid:38) CP -symmetric LRMs seem less constrained, allowing for more freedomin the right-handed CKM matrix. In the Higgs sector the potentials of the LRSMs all turnout to be quite similar. The CP -symmetric case allows some more freedom in the masses ofthe left-handed triplet fields, but is otherwise similar as well. The Higgs potentials of all threemodels require a considerable amount of fine-tuning, which poses the biggest challenge to theirviability. Acknowledgments
We would like to thank Wilco den Dunnen, Robert Fleischer, Jean-Marie Fr`ere and Jordy deVries for useful discussions.