Spontaneous CP Violation and the Strong CP Problem in Left-Right Symmetric Theories

We study spontaneous CP violation as a solution to the strong CP problem in left-right symmetric theories. The discrete CP symmetry is broken by a complex vacuum expectation value of a right-handed Higgs doublet. Heavy vector-like down-type quarks mix with the Standard Model quarks introducing the known CP violation. A non-vanishing QCD vacuum angle is generated at two-loop level. The implementation in the ultraviolet complete theory of trinification at low scales is discussed. We further comment on the phenomenology and future testability of the model.


I. INTRODUCTION
The Standard Model of particle physics has been a success story for the last four decades. However, there still remain numerical puzzles such as the hierarchy problem, the proton stability, or the flavor structure of the Standard Model. In this article, we want to address one of these puzzles: the strong CP problem.
The non-Abelian SU (3) C gauge symmetry of the Standard Model has a nontrivial vacuum structure which allows for explicit CP violation via the QCD vacuum angle θ QCD [1, 2] L ⊃ g 2 s θ QCD 32π 2 Tr G µνG µν , where g s is the QCD gauge coupling and G µν is the gluon field strength tensor withG µν = (1/2) µναβ G αβ .
Measurements of the neutron electric dipole moment [3] and of the electric dipole moment of mercury [4,5] have set an experimental upper limit on the QCD vacuum angle The CP violation in QCD is therefore strikingly small and the strong CP problem arises: Why is the CP violation in the weak interaction so strong compared to the CP violation in the strong interaction?
In principle, there are three known solutions to the strong CP problem.
• A zero up-quark mass at tree-level. The measured up-quark mass is then only generated via nonperturbative QCD processes. The theory then exhibits an axial symmetry which renders θ QCD nonphysical [6,7]. arXiv:1808.02075v2 [hep-ph] 19 Jun 2019 • Add a new scalar field with a nontrivial charge with respect to an anomalous global U (1) P Q symmetry. After spontaneously breaking U (1) P Q the pseudo-Nambu-Goldstone boson of U (1) P Q , the QCD axion, dynamically forces θ QCD to be small. This solution was first discussed in Ref. [8,9] and the U (1) P Q symmetry is thus referred to as Peccei-Quinn symmetry.
• The CP symmetry is an exact symmetry of nature which is broken spontaneously by a complex vacuum expectation value [10][11][12][13]. All the CP violation in the Standard Model is then connected to the complex phase of a vacuum expectation value.
Today, vanishing quark masses at tree-level are in conflict with state-of-the-art lattice calculations [14].
A zero up-quark mass at tree-level as solution to the strong CP problem is thus mostly discarded. The theoretically preferred solution is the QCD axion. Axionlike particles received more and more attention in recent years [15,16] -especially as there is a significant experimental effort to test the axion hypothesis [17,18]. Nevertheless, we should not forget that the Peccei-Quinn symmetry also has theoretical short comings such as the stabilization with respect to Planck scale physics [19][20][21].
In this paper, we evaluate the theoretical attractiveness of spontaneous CP violation as solution to the strong CP problem. We thereby focus on the Nelson-Barr mechanism. The Nelson-Barr mechanism was first introduced and phenomenologically discussed in Ref. [10][11][12][13], see Ref. [22] for a recent review. The mechanism requires additional heavy vectorlike quarks which mix with the Standard Model quarks via a complex coupling. As the determinant of the quark mass matrix has to be real at tree-level in order to allow for a vanishing QCD vacuum angle θ QCD at leading order, the mixing has to be asymmetric. A small nonvanishing vacuum angle θ QCD can then be produced only at loop level. A first bottom-up toy model discussing the Nelson-Barr mechanism had to require additional discrete symmetries to suppress θ QCD [23,24]. This is a problem which generally arises in the Nelson-Barr mechanism [22]. We therefore consider left-right symmetric theories [25][26][27][28] because we only observe left-handed currents at low energies and thus it is plausible that such theories provide the preferred framework for such an asymmetric symmetry breaking and quark mixing.
The paper is organized as follows. In Sec. II, we introduce the minimal realization of the Nelson-Barr mechanism in left-right symmetric theories. We discuss the embedding of this minimal theory in the UV complete theory of trinification in Sec. III. Section IV is then dedicated to the phenomenology of spontaneous CP violation and the possibility to definitively probe these theories in future experiments.
Finally, we conclude in Sec. V.

II. NELSON-BARR MECHANISM IN LEFT-RIGHT SYMMETRY
The Nelson-Barr mechanism realizes the idea of spontaneous CP violation as a solution to the strong CP problem. Thereby, new vectorlike quarks are added to the Standard Model of particle physics which mix with the known quarks. The mixing takes place via a new scalar degree of freedom which acquires a complex vacuum expectation value and hence spontaneously breaks the CP symmetry. The complex phase of the vacuum expectation value then allows for a connection of the CP violation in the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the QCD vacuum angle.
An especially appealing scenario for spontaneous CP violation are left-right symmetric theories. It was first realized [29,30] that parity is sufficient to solve the strong CP problem in left-right symmetric theories without a scalar bidoublet. The Standard Model fermion masses are then generated by nonrenormalizable dimension five operators. A recent discussion of the dimension five mass terms in the context of Froggatt-Nielsen theories can be found in Ref. [31]. A solution to the strong CP problem without an axion and without a scalar bidoublet but with renormalizable mass terms in the context of left-right symmetric theories was first studied in Ref. [32]. By not including a scalar bidoublet in the theory, heavy vectorlike quarks and leptons had to be introduced in order to give mass to the Standard Model fermions via left-and right-handed Higgs doublets.
A scalar bidoublet is the preferred mechanism to generate Standard Model fermion masses in left-right symmetric theories. However, it was argued that without supersymmetry a scalar bidoublet in a left-right symmetric theory cannot have a complex vacuum expectation value [33]. Further nonsupersymmetric studies of spontaneous CP violation in left-right symmetric theories can be found in Refs. [34][35][36][37][38][39]. In this paper, we want to address the question: What is the left-right symmetric theory with the minimal number of propagating degrees of freedom which can spontaneously break CP and solve the strong CP problem?
Minimal left-right symmetric gauge theories are based on the gauge symmetry group Without supersymmetry and with only new vectorlike down-type quarks to realize the Nelson-Barr mechanism, the minimal fermionic particle content is given by q L ∼ (3, 2, 1, 1/3) , q R ∼ (3, 1, 2, 1/3) , L ∼ (1, 2, 1, −1) , R ∼ (1, 1, 2, −1) , ever, in a unified picture we would expect three generations of additional down-type quarks D. Furthermore, we add two types of vectorlike fermions per generation less than in Ref. [32]. The scalar sector we consider is given by with i, j ∈ {1, 2}. Compared to Ref. [32] only a scalar bidoublet was added. The interactions are thus given by withφ = σ 2 φ * σ 2 and implicit summation over i and j. The discrete spacetime symmetries P , C, and CP are conserved for real Yukawa couplings and the additional condition Note that in Ref. [40] it was demonstrated that at least two right-handed Higgs doublets are required to spontaneously violate CP . The two left-handed Higgs doublets do not acquire a vacuum expectation value and are only included to have a left-right symmetric particle content. The scalar potential takes the form with i, j, k, l ∈ {1, 2}, φ 1 ≡ φ, and φ 2 ≡φ. We had to assume a discrete Z 3 symmetry to forbid linear terms and couplings introducing a complex electroweak symmetry breaking vacuum expectation value. CP conservation only allows for real couplings in the scalar potential. The relevant interactions to generate a complex vacuum expectation value are those with differing spurion charges [40].
Assigning H R,1 and H R,2 the spurion charges we find that the relevant terms in the scalar potential to spontaneously break CP are given by with spurion charges As a proof of principle that such a scalar potential can generate spontaneous CP violation and realize the Nelson-Barr mechanism, we assume that all couplings mixing the bidoublet φ with the Higgs doublets H L,i and H R,j are negligible, To further simplify the discussion, we take the limit such that the mixing of the four Higgs doublets is controlled by only four couplings A minimum of the scalar potential is then given by with The vacuum expectation values (16) then are a valid minimum for v 2 in the limit of λ (1) have to be satisfied. The potential is therefore bounded and the extremal solutions belong to a minimum of the potential. This completes our proof of principle and demonstrates that spontaneous CP violation with complex vacuum expectation values in the right-handed Higgs sector and vanishing vacuum expectation values in the left-handed Higgs sector in left-right symmetric theories can be realized.
The scalar bidoublet φ gives mass to the Standard Model fermions. The lepton and up-type quark mass matrices are given by with implicit flavor indices. We will work in the basis where m u is diagonal. Note that neutrinos are Dirac fermions in this minimal configuration. Majorana neutrinos could be accounted for by introducing a singly charged scalar field [41]. The down-type quark mass matrix is given by in the down-type quark basis (d, D) and with where flavor indices are again implicit. We can then define a biunitary transformation where we parametrize the transformation as with C the known 3 × 3 CKM matrix with the abbreviations s ij = sin(θ ij ) and c ij = cos(θ ij ) and the measured values θ 12 = (13.04 ± 0.05) • , θ 13 = (0.201 ± 0.011) • , θ 23 = (2.38 ± 0.06) • , and δ = (1.20 ± 0.08)rad [42]. Using the relation [24] CM 2 we can establish a direct connection between the measured CP violation in the CKM matrix and the CP violating phase α. A direct correspondence for example is achieved in the limit where we parametrized the full flavor matrices as and Due to the asymmetric mixing in the down-type quark mass matrix, we find at tree-level Thus, as CP is an exact symmetry of the Lagrangian the QCD vacuum angle θ QCD vanishes at tree-level. This changes at the quantum level. One-loop processes as shown for example in Fig. 1 contribute to m d with the result that θ QCD is no longer vanishing. As these corrections are not only loop suppressed but also Yukawa suppressed, the resulting θ QCD is compatible with current constraints ∆θ QCD 10 −11 y 1/2 0.01 The theory is thus also testable with future electric dipole moment measurements of the neutron [43][44][45]. The first higher dimensional operator contributing to m D is given by which results in a mass correction assuming order one couplings. If we further assume v R ≡ v R1 v R2 , we find for a Planck scale suppressed operator the upper bound [22] v R 10 8 GeV .
The upper bound on the scale of spontaneous CP violation is therefore close to the lower bound on new flavor violating physics from meson mixing [46] Λ F 10 7 GeV, which hints toward future detection possibilities. Moreover, recent low scale ultraviolet completions of the Standard Model could be realized at energy scales as low as 10 8 GeV [47,48].

III. NELSON-BARR MECHANISM IN TRINIFICATION
The introduction of vectorlike down-type quarks D and additional scalar doublets is ad hoc from a bottom-up perspective. However, this particle content arises automatically in trinification [49][50][51][52][53][54], a gauge theory based on which is one possible ultraviolet completion of left-right symmetric theories. The Nelson-Barr mechanism in E 6 grand unified theories was recently discussed in Ref. [55]. The fermion content per generation in trinification is given by where the additional down-type quarks are embedded in Q. To reproduce the Standard Model mass spectrum, at least two scalar bitriplets are required with i ∈ {1, 2}. The appearance of at least two left-and right-handed Higgs doublets thus follows from the necessity to reproduce the Standard Model. The interactions take the form Trinification thus motivates the minimal particle content needed in left-right symmetric theories to solve the strong CP problem via spontaneous CP violation. Unfortunately, this minimal scalar sector requires tuning of the scalar potential to solve the strong CP problem via spontaneously breaking CP . To illustrate the tuning, we choose the simplified potential In order to have a vacuum structure of the form with v L1 = 0 and v L2 = 0, which is a necessary condition to solve the strong CP problem via spontaneous CP violation, we have to require In order to avoid such tuning, we have to introduce a scalar bisextet such that the dimension five and six operators can generate a complex left-right symmetry breaking vacuum expectation value without also introducing a complex SU (2) L violating vacuum expectation value.
However, the bisextet Σ also has the advantage of enabling a type-II seesaw mechanism in the context of trinification. In the minimal theory of trinification, Majorana neutrino masses are generated at the one-loop level [50,56,57]. As the neutral component of the SU (2) R triplet embedded in the bisextet Σ develops a vacuum expectation value which breaks SU (2) R , the right-handed neutrinos receive a Majorana mass term via the interaction The active neutrino mass is then suppressed by the ratio of electroweak vacuum expectation values to leftright symmetry breaking vacuum expectation values.

IV. PHENOMENOLOGY OF THE NELSON-BARR MECHANISM
The clearest prediction of the Nelson-Barr mechanism are new vectorlike quarks. Current LHC constraints from Atlas and CMS on vectorlike quarks can be found in Ref. [58][59][60][61][62][63][64] and give an approximate lower bound Constraints on theories with vectorlike quarks and additional colored fields such as for example trinification were studied in Ref. [65]. Also the possibility to search for vectorlike quarks with a broad width at the LHC was recently discussed in Ref. [66]. Assuming M D v R , we expect that a large range of the allowed parameter space can be tested at future hadron colliders. For projected limits on vectorlike quarks at a 100 TeV proton-proton collider see Ref. [67].
The lower limits from right-handed current searches give the lower limit on the mass of W R [68-71] where the mass is related to the scale of left-right symmetry breaking by the relation where g R is the gauge coupling constant of SU (2) R . Thus, for order one coupling constants the lower bound on v R is in the TeV range. We can therefore conclude that the left-right symmetry breaking scale v R in a theory with spontaneous CP violation is bounded from above and below Note that for left-right symmetric theories with complex electroweak symmetry breaking vacuum expectation values stronger constraints on M W R from the kaon sector apply [72].
Baryogenesis requires C and CP violation. A possibility to explain the baryon asymmetry in the Universe with spontaneous CP violation is electroweak baryogenesis. The prospects of a strong first order phase transition at the electroweak scale in the minimal left-right symmetric theory with spontaneous CP violation will be discussed in future work. Note that electroweak baryogenesis is testable at future gravitational wave observatories.
Measuring the reheating temperature of the Universe T RH or the Hubble rate during inflation H inf to be above ∼ 10 8 GeV would immediately rule out this minimal theory of spontaneous CP violation. By spontaneously breaking the discrete P and CP symmetry at the scale v R , we introduce domain walls into the thermal history of the Universe. As no such topological defects have been observed, we have to require that the left-right symmetry was never in the unbroken phase after inflation. Low-scale standard single field inflation predicts the tensor-to-scalar ratio r to be [73] r 1.5 · 10 −13 H inf 10 8 GeV The upper bound on the Hubble rate during inflation thus translates to an upper bound on the tensor-to-scalar ration Therefore, if a sizeable tensor-to-scalar ratio r is measured, the presented model of spontaneous CP violation is ruled out.
The minimal Nelson-Barr mechanism in left-right symmetric theories presented in Sec. II predicts no CP violation in the leptonic sector. This prediction is currently in tension with global fits to current neutrino data [74][75][76]. Future neutrino experiments such as DUNE [77] and Hyper-Kamiokande [78] will therefore be able to conclusively probe the minimal Nelson-Barr mechanism in left-right symmetric theories.
There are two possibilities to account for CP violation in the leptonic sector at the renormalizable level.
The interactions lead to a mixing of the right-handed neutrinos ν R and the Majorana fermion N L . The complex vacuum expectation value H R,1 therefore enters the neutrino mass matrix. The gauge singlet N L is a potential dark matter candidate and can generate the baryon asymmetry in the early Universe via oscillations [79].
• We can include new charged vectorlike leptons which mix via the interaction with the Standard Model leptons. The complex phase which spontaneously breaks CP is thus introduced to the charged lepton mass matrix.
In both scenarios the CP violation in the quark sector and in the leptonic sector have a common origin.
New vectorlike leptons up to masses of 100 TeV can be probed with future experiments which measure the electric dipole moment of the electron, the rate of µ → 3e, and the rate of µ → eγ [80]. Additionally, vectorlike leptons are a striking signature at future lepton colliders, projected limits can be found in Ref. [81].
We thus see the complementarity of future neutrino experiments, low energy flavor experiments, and lepton and hadron colliders.

V. CONCLUSIONS
In this paper, we reevaluated the possibility of solving the strong CP problem by spontaneous CP As a consequence, the theory predicts that the reheating temperature after inflation and the Hubble rate during inflation have to be below ∼ 10 8 GeV. We can therefore derive an upper bound on the tensor-to-scalar ratio r 10 −13 such that this theory is testable.
The particle content of the minimal bottom-up approach to spontaneous CP violation in left-right symmetric theories is well-motivated in the context of trinification. Moreover, trinification is an ultraviolet completion of the Standard Model which can be realized at scales as low as ∼ 10 8 GeV. Note that a scalar bisextet Σ is needed to break CP spontaneously in trinification. However, the scalar bisextet also opens up the possibility of a type II seesaw mechanism in trinification.
The minimal model predicts no leptonic CP violation. In the near future, current and up-coming neutrino experiments can therefore falsify the minimal model. To account for leptonic CP violation new fermions have to be introduced. We presented the two options, including a gauge singlet fermion or additional vectorlike charged leptons, to incorporate the CP violating phase in the leptonic sector at the renormalizable level.
We can thus conclude that spontaneous CP violation is still a theoretically motivated solution to the strong CP problem. The minimal model of spontaneous CP violation in left-right symmetric theories is highly predictive. Determining the Hubble rate during inflation and measuring the CP phase in the neutrino mixing matrix can both potentially rule out the presented model.

VI. ACKNOWLEDGMENT
We thank Pavel Fileviez Pérez, Jisuke Kubo, Ravi Kuchimanchi, Manfred Lindner, and Kai Schmitz for very useful discussions and comments on the manuscript.