$\epsilon'/\epsilon$ Anomaly and Neutron EDM in $SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ model with Charge Symmetry

The Standard Model prediction for $\epsilon'/\epsilon$ based on recent lattice QCD results exhibits a tension with the experimental data. We solve this tension through $W_R^+$ gauge boson exchange in the $SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ model with `charge symmetry', whose theoretical motivation is to attribute the chiral structure of the Standard Model to the spontaneous breaking of $SU(2)_R\times U(1)_{B-L}$ gauge group and charge symmetry. We show that $M_{W_R}<58$ TeV is required to account for the $\epsilon'/\epsilon$ anomaly in this model. Next, we make a prediction for the neutron EDM in the same model and study a correlation between $\epsilon'/\epsilon$ and the neutron EDM. We confirm that the model can solve the $\epsilon'/\epsilon$ anomaly without conflicting the current bound on the neutron EDM, and further reveal that almost all parameter regions in which the $\epsilon'/\epsilon$ anomaly is explained will be covered by future neutron EDM searches, which leads us to anticipate the discovery of the neutron EDM.


Introduction
The direct CP violation in K → ππ decay parametrized by the parameter is sensitive to physics beyond the Standard Model (SM) due to the suppressed SM contribution. Recent calculation of the hadronic matrix elements with lattice QCD [1,2,3] enables us to evaluate the K → ππ decay amplitude without relying on any hadron model. On the basis of the above calculation, the same collaboration has reported that the SM prediction is separated from the experimental value [4,5,6] by 2.1σ, and other groups [8,9] have also obtained predictions for / that show a discrepancy of 2.9σ and 2.8σ, respectively. More importantly, the lattice result corroborates the calculation with dual QCD approach [10,11], which has derived a theoretical upper bound on / that is violated by the experimental data and has thus claimed anomaly in this observable. (However, Ref. [12] presents a different calculation that claims the absence of the anomaly.) Some authors have tackled this / anomaly in new physics scenarios, such as a general right-handed current [13], the Littlest Higgs model with T-parity [14], supersymmetry [15,16,17], non-standard interaction with Z and/or Z [18,19], vector-like quarks [20] and [21]. The SU (2) L × SU (2) R × U (1) B−L gauge extension of the SM is a well-motivated framework for addressing the / puzzle, because the flavor mixing matrix for right-handed quarks automatically introduces new CP-violating phases, and W + R gauge boson exchange contributes to ∆F = 1 processes at tree level while it contributes to ∆F = 2 processes at loop levels so that other experimental constraints, in particular the constraint from Re( ), are readily evaded.
Previously, Ref. [13] has shown that a general SU (2) L × SU (2) R × U (1) B−L model with an arbitrary right-handed quark mixing can solve the / discrepancy. However, a major theoretical motivation for the SU (2) L × SU (2) R × U (1) B−L model lies in its capability of explaining the origin of the chiral nature of the SM, which is achieved by adding either the left-right parity [22] or the 'charge symmetry' [23] 1 . The left-right parity requires invariance of the theory under the Lorentzian parity transformation plus the exchange of SU (2) L and SU (2) R gauge groups, while the charge symmetry requires invariance under the charge conjugation plus the exchange of SU (2) L and SU (2) R , both of which endow the model with a symmetric structure for the left and right-handed fermions at high energies.
In this paper, we study / in the SU (2) L × SU (2) R × U (1) B−L model with charge symmetry. As a consequence of the charge symmetry, the Yukawa matrices are complex symmetric matrices, which restricts the quark mixing matrix associated with W + R to be the complex con-jugate of the SM Cabibbo-Kobayashi-Maskawa (CKM) matrix multiplied by a new CP phase factor for each quark flavor. Given the above restriction, one can evaluate / only in terms of two new CP phases, the mass of W + R and the ratio of two vacuum expectation values (VEVs) of the bifundamental scalar, which leads to a specific prediction for the model parameters.
Our analysis on / proceeds as follows. By integrating out W R , W L and the top quark, we obtain the Wilson coefficients for ∆S = 1 operators that contribute to K → ππ decay.
The anomalous dimension matrix is divided into the same two 18 × 18 pieces for 36 operators, for which leading order expressions are obtainable from Refs. [25,26]. The hadronic matrix elements for current-current operators are seized from the lattice results [2,3]. We find that among new physics operators, (su) L (ūd) R and (su) R (ūd) R (each with two ways of color contraction) both dominantly contribute to / . Their contributions are of the same order because the Wilson coefficients of the (su) L (ūd) R operators are suppressed by the hierarchy of two bifundamental scalar VEVs v 1 /v 2 = tan β, which is about m b /m t if there is no fine-tuning in accommodating the top and bottom quark Yukawa couplings, whereas this suppression does not enter into the Wilson coefficients of the (su) R (ūd) R operators. On the other hand, the lattice computation has confirmed that the hadronic matrix elements for the former operators are enhanced compared to the latter. Thus, these operators possibly equally contribute to / . This result is in contrast to the study of Ref. [13], which has concentrated solely on the (su) L (ūd) R operators.
Once the / anomaly is explained in the SU (2) L × SU (2) R × U (1) B−L model with charge symmetry, correlated predictions for other CP violating observables are of interest. In particular, the neutron electric dipole moment (EDM), an observable sensitive to CP violation in the presence of CPT invariance, receives significant contributions from four-quark operators in SU (2) L × SU (2) R × U (1) B−L models [28,29,30,31,32,13] 2 , allowing us to discuss future detectability of the neutron EDM in relation to the / anomaly.
Our analysis on the neutron EDM starts by integrating out W R , W L and top quark to obtain the Wilson coefficients for CP-violating operators. The leading order expression for the anomalous dimension matrix is found in Refs. [35,36,37,38]. Regarding the hadronic matrix elements of CP-violating operators, we reveal that the pion VEV π 0 induced by four-quark operators [27] gives the leading contribution to the neutron EDM, which is enhanced by the quark mass ratio m s /(m u + m d ) in comparison to the rest. This enhancement is understood as follows: Since W + R -W + L mixing gives rise to CP-odd and isospin-odd interactions, the pion VEV π 0 , which is isospin-odd, can arise without the factor of m d − m u , and thus can be directly proportional to 1/(m u + m d ). The pion VEV induces a CP-violating coupling for neutron n, Σ − baryon, and kaon K + without the factor of m d − m u because thenΣ − K + vertex is not isospin-even. Consequently, the CP-violating coupling for n, Σ − , K + can appear with the factor of m s /(m u + m d ). This coupling contributes to the neutron EDM at the leading chiral order through charged baryon-meson loops. Considering the above-mentioned importance of the pion VEV, we in this paper investigate meson condensation, the resultant CP-violating baryon-meson couplings, and their contributions to the neutron EDM through baryon-meson loops, using chiral perturbation theory. This paper is organized as follows: In Sec. 2, we review the SU We consider SU (3) C × SU (2) L × SU (2) R × U (1) B−L gauge theory with charge symmetry. The field content is in Table 1. Hereafter, the fields are expressed in a way that they transform Table 1: Field content and charge assignments. i labels the three generations.
with θ a L and θ a R being gauge parameters for SU (2) L and SU (2) R , respectively. We demand the theory to be invariant under the following 'charge symmetry' transformation: charge conjugation of all gauge fields, The part of the Lagrangian describing SU (2) L ×SU (2) R ×U (1) B−L and Yukawa interactions of quarks is given by where g L , g R and g X are the gauge coupling constants for SU (2) L , SU (2) R and U (1) B−L gauge groups, respectively, and Y q andỸ q are the quark Yukawa couplings. s denotes the antisymmetric tensor for Lorentz spinors and g denotes that for the fundamental representation of SU (2) L or SU (2) R . Invariance under the charge symmetry transformation Eq. (2) leads to the following tree-level relations: The SU (2) R triplet scalar ∆ R develops a VEV, v R , to break SU R (2) × U (1) B−L → U (1) Y , and the bi-fundamental scalar Φ takes a VEV configuration, to break SU (2) L × U (1) Y → U (1) em , where α is the spontaneous CP phase. The VEV of ∆ L is hereafter neglected, as it is severely constrained from ρ-parameter. The resultant mass matrices for W a L , W a R and X gauge bosons read The mass matrix for the charged gauge bosons is diagonalized as For v R v and g L = g R , we have an important relation for ζ, which indicates that when we assume tan β m b /m t so that the top and bottom Yukawa couplings are naturally derived, the W L -W R mixing angle ζ is smaller than M 2 W /M 2 W by the factor 2m b /m t ∼ 0.05.
The quark mass matrices are given by 3 which we diagonalize as Hence, the SM CKM matrix, V L = V uL V † dL , and the corresponding flavor mixing matrix for right-handed quarks, V R = V uR V † dR , are related as Eventually, the part of the Lagrangian Eq. (3) describing flavor-changing W, W interactions is recast, in the unitary gauge, into the form, (12) where U i and D i denote the Dirac fields of the up and down-type quarks, respectively.
In this paper, we adopt the following convention for the quark phases and φ u , φ c , φ t , ψ d , ψ s , ψ b : First, we redefine the phases of five quarks to render the CKM matrix in the standard form, Next, we redefine φ c , φ t , ψ d , ψ s , ψ b to set Phase convention fixed in this way, all sources of CP violation are parametrized by Im

Wilson Coefficients for ∆S = 1 Operators
We match the SU In the effective theory, the ∆S = 1 Hamiltonian is parametrized as where operators O's are defined in Appendix A. We determine the Wilson coefficients as follows: We approximate g R = g L by ignoring difference in RG evolutions of g L and g R at scales below M W . Also, for each Wilson coefficient, if multiple terms have an identical phase, we only consider the one in the leading order of M 2 W /M 2 W or sin ζ. By integrating out W , one obtains the following leading-order matching conditions at a scale µ ∼ M W (note our convention with φ u = 0): By further integrating out W and the top quark, one gains the following leading-order matching conditions at a scale µ ∼ M W (note our convention with φ u = 0): where loop functions F 1 , F 2 , F 3 and E 1d , E 2d , E 3d are defined in Appendix B. We are aware that the dipole operators receive two contributions with different phases when W is integrated out and when W is. The two are expressed as δC g , δC γ , δC g , δC γ and ∆C g , ∆C γ , ∆C g , ∆C γ , respectively.
We take into account RG evolutions of the Wilson coefficients at order O(α s ). The fact that four sets of operators, , we assume that their initial conditions at scale µ = M W are given by Eqs. (24)(25)(26)(27)(28)(29) and solve the RG equations from µ = M W to the scale for which the lattice results are reported. For ({C i }), we assume that their initial conditions at µ = M W are provided by Eqs. (16)(17)(18)(19)(20)(21)(22)(23) and solve the RG equations from µ = M W to the scale of lattice results. Finally, we compute RG evolutions of the coefficients of the dipole operators (C g , C γ ) and (C g , C γ ), which receive contributions from [26], and those for ({C RL i }) and (C g , C γ ) are in Ref. [25].

Hadronic Matrix Elements
We employ the lattice calculations of hadronic matrix elements (ππ) I |O i |K 0 for i = 1, 2, ..., 10 for I = 0, 2 reported by RBC/UKQCD in Refs. [2,3]. Since lattice calculations for the matrix elements of O LR 1 and O LR 2 are missing, we estimate them from the RBC/UKQCD results using isospin symmetry. In the limit of exact isospin symmetry, we find, for ∆I = 3/2 amplitudes, where we have discarded ∆I = 1/2 part when obtaining the second line, and when deriving the third line, we have inserted Clebsch-Gordan coefficients for constructing the ∆I = 3/2 operator from a ∆I = 1/2 one and a ∆I = 1 one. For ∆I = 1/2 amplitudes, we find where in the first line, we have separated (ūu) R into ∆I = 0 and ∆I = 1 parts, and in the second line, we have inserted Clebsch-Gordan coefficients for constructing the ∆I = 1/2 operator from a ∆I = 1/2 one and a ∆I = 0 one or a ∆I = 1 one. The same relations hold between the matrix elements for O LR The hadronic matrix elements for the chromo-dipole operators O g , O g are extracted from the calculation based on dual QCD approach [39]. Note that the above calculation is corroborated by the fact that it is consistent with a lattice calculation of the K-π hadronic matrix element [40], which is related to the K-ππ one by chiral perturbation theory.

Numerical Analysis of /
The definition for the decay amplitudes of K 0 → ππ is where δ 0,2 represent the strong phases. In terms of the above amplitudes, one writes the direct CP violation parameter divided by the indirect one as where ω = ReA 2 /ReA 0 is a suppression factor due to the ∆I = 1/2 rule. For the strong phases, we use the values of Refs. [3,2], δ 2 = 23.8 ± 5.0 degree and δ 0 = −11.6 ± 2.8 degree. For the real parts of the decay amplitudes, we employ the experimental data [7], ReA 2 = 1.479 × 10 −8 GeV and ReA 0 = 33.20 × 10 −8 GeV, which leads to ω = 4.454 × 10 −2 . In our analysis, we separate the SM and new physics contributions as For the SM part, we quote the calculation in the literature Re( / ) SM = (1.38 ± 6.90) × 10 −4 [3]. It is the new physics part, that we compute in this paper. In doing so, we approximate cos 2 ζ = 1 in the Wilson coefficients Eqs. (24)(25)(26)(27)(28)(29)(30)(31)(32)(33), so that the SM contribution is separated from the new physics one at the operator level.
In the analysis, we fix the ratio of the bifundamental scalar VEVs at its natural value  Next, we randomly vary α−ψ d and α−ψ s in the range [0, 2π], since they are free parameters.
In Fig. 2, we show the region of Re( / ) obtained by varying α − ψ d and α − ψ s . One observes that M W < 58 TeV is necessary for 1σ explanation of the anomaly. Re(ϵ'/ϵ) [10 -4 ] Figure 2: Numerical result of Re( / ). The blue band represents the experimental data given by PDG [7], while each red dot corresponds to the model prediction with a randomly generated set of (α − ψ d , α − ψ s ).
We have confirmed that among the terms of ∆S = 1 Hamiltonian Eq. (15) where operators O's are defined in Appendix C. We determine the Wilson coefficients C's as follows: Again, for each coefficient, if multiple terms have an identical phase, we exclusively consider the one in the leading order of M 2 W /M 2 W or sin ζ. By integrating out W and the top quark, one obtains the following leading-order matching conditions at µ ∼ M W (note our convention with φ u = 0): where loop functions F 1 , F 2 , F 3 and E 1d , E 2d , E 3d , E 3u are defined in Appendix B. In Eq. (53) (which corresponds to the Weinberg operator [42]), we present the dominant part proportional to m t . Terms obtained by integrating out W possess the same phases as Eqs. (45)(46)(47)(48)(49)(50)(51)(52)(53) and are simply suppressed by M 2 W /M 2 W compared to Eqs. (45)(46)(47)(48)(49)(50)(51)(52)(53). They are therefore neglected in our analysis.

Hadronic Matrix Elements
In the SU (2) L ×SU (2) R ×U (1) B−L model with charge symmetry, the Wilson coefficients for the four-quark operators O 1q q = (q q )(qiγ 5 q) and O 2q q = (q α q β )(q β iγ 5 q α ) (q = q ; q, q = u, d, s) are particularly large. Therefore, we scrutinize how these operators contribute to the neutron EDM. Operators O 1q q contribute in the following three ways: • The first one is through meson condensation [27]; O 1q q operators give rise to tadpole terms for pseudoscalar mesons and induce their VEVs. These VEVs generate CP-violating interactions for baryons and mesons, which contribute to the neutron EDM through baryon-meson loop diagrams.
• The second one is through hadronic matrix elements of O 1q q with baryons and mesons, BM |O 1q q |B (B denotes a baryon and M a meson), which contribute to the neutron EDM through baryon-meson loop diagrams.
• The third one is directly through the hadronic matrix element of O 1q q with neutrons and photon.
On the other hand, O 2q q operators do not yield meson condensation, but do contribute to the neutron EDM in the latter two ways. Later, it will be shown that the contribution from the pion VEV π 0 , which belongs to the first category, is enhanced by the factor m s /(m u + m d ) compared to the latter two. We therefore investigate how O 1q q operators bring about meson condensation, thereby contributing to the neutron EDM.
We are aware that if Peccei-Quinn mechanism [43] exists, it affects the meson condensation and also induces an effective non-zeroθ term due to incomplete cancellation between the genuineθ term and the axion VEV. Alternatively, it is logically possible to assumeθ = 0 without Peccei-Quinn mechanism, by considering an unknown mechanism or through fine-tuning, in which case we do not need to take into account the effect of Peccei-Quinn mechanism or that of non-zeroθ. In this paper, we consider both cases where (i) one hasθ = 0 without Peccei-Quinn mechanism, and (ii) Peccei-Quinn mechanism is at work.
We start from the case withθ = 0 without Peccei-Quinn mechanism. The meson condensation contribution is evaluated by the following steps: (1) First, we implement C 1q q O 1q q part of the Hamiltonian Eq. (44) into the meson chiral Lagrangian. To this end, we rewrite q =q ;q,q =u,d,s It then becomes clear that the theory would be invariant (except for From the above transformation property and the parity invariance of QCD, the meson chiral Lagrangian at order O(p 2 ) plus the leading CP-violating terms is found to be (remind that θ = 0 has been assumed) where C LRLR ijkl , C RLLR ijkl have been defined in Eq. (55). Here, U is a nonlinear representation of the nine Nambu-Goldstone bosons that transforms under U (3) L × U (3) R rotations L × R as U → RU L † , and χ includes the quark mass term, which are given by (1, 1, 1), [U ] ij denotes the (i, j) component of matrix U . F π is the pion decay constant in the chiral limit and F 0 is the decay constant for η 0 , which we approximate as F 0 F π . B 0 satisfies B 0 m 2 π /(m u + m d ). The term with log U represents instanton effects, whose expression is exact in the large N c limit [44], and a 0 satisfies 48a 0 /F 2 0 m 2 η + m 2 η − 2m 2 K . c 1 , c 2 and c 3 are unknown low energy constants (LECs), which can be estimated by naïve dimensional analysis [45] as (2) The CP-violating part of the Lagrangian Eq. (56) contains tadpole terms for mesons, which lead to non-zero meson VEVs. Assuming that electric charge and strangeness are not broken spontaneously, we obtain the following potential for neutral mesons π 0 , η 8 and η 0 : The above potential is minimized with non-zero meson VEVs, π 0 , η 8 and η 0 . Insofar as we are concerned with vertices with one meson, the physical modes of π 0 , η 8 and η 0 fields can be approximated as In the SU (2) L × SU (2) R × U (1) B−L model with charge symmetry, there hold relations C 1ud −C 1du and |C 1ud | |C 1sq |, |C 1qs | (q = u, d). When (C 1ud + C 1du ) and C 1sq , C 1qs are neglected accordingly, one finds, for small VEVs, Note that η 8 and η 0 are proportional to m d − m u . This is because these VEVs are isospin singlets and hence must be constructed from the product of isospin-odd coefficient C 1ud − C 1du and isospin-odd mass term m d − m u . In contrast, π 0 does not contain m d − m u because this VEV is isospin-violating. Since 20a 0 ∼ B 0 F 2 π m s holds empirically, we find from Eq. (62) that π 0 is much larger than η 8 and η 0 by the factor m s /(m d − m u ).
where B represents baryons and ξ L , ξ R include mesons as Γ µ is a covariant derivative for baryons, ξ µ is a combination of meson fields, and χ + contains quark masses, which are defined as M B is the baryon mass in the chiral limit. We insert meson VEVs π 0 , η 8 , η 0 into the baryon chiral Lagrangian Eq. (63) and extract CP-violating interaction terms involving neutron n. We thus obtain L baryons ⊃ḡ nnπn n π 0 phys +ḡ nn8n n η 8phys +ḡ nn0n n η 0phys +ḡ npπ (pnπ + +npπ − ) where the coupling constants are given bȳ Note in particular that π 0 enters into the expression forḡ nΣ − K + Eq. (76) without the factor of m d − m u , which is allowed because the couplingḡ nΣ − K + violates isospin. It follows thatḡ nΣ − K + is enhanced by the factor m s /(m u + m d ), as it contains a term m s π 0 .
We compare the above meson-VEV-induced CP-violating couplings with those arising from direct hadronic matrix elements of O 1q q and O 2q q . The latter are estimated by naïve dimen-sional analysis [45] as 5 (B and M represent any baryon and meson, respectively) On the other hand,ḡ nΣ − K + Eq. (76), for example, is estimated to bē where Eq. (62) and the naïve dimensional analysis on c 3 Eq. (59) are in use. Noting that This fact allows us to neglect the latter contribution in the rest of the analysis.
(4) The neutron EDM receives contributions from baryon-meson loop diagrams involving a CP-violating coupling of Eqs. (70), a CP-conserving baryon-meson axial-vector coupling and a photon coupling. We refer to the loop calculation of Ref. [50] performed with infrared regularization [51,52], from which the neutron EDM, d n , is obtained as Here, the divergent part 1/¯ ≡ 1/ − γ E + log(4π) and the scale µ stem from dimensional regularization in 4−2 dimension with mass parameter µ. In fact, the baryon chiral Lagrangian contains a LEC which cancels the above divergence and whose finite part contributes to the neutron EDM. The impact of the finite part of the LEC is assessed by naïve dimensional analysis [45] as On the other hand, from Eqs. (62) Since (b D −b F )(4πF π ) ∼ 1 and D−F ∼ 1, we find that the loop contribution Eq. (82) dominates over the LEC one Eq. (81) by the factor m s /(m u + m d ). It is thus justifiable to estimate d n by simply extracting the finite part of the loop contribution. We further set µ = m N , since m N is a natural cutoff scale, and arrive at Next, we study the case with Peccei-Quinn mechanism. We incorporate the axion field, a, into the meson Lagrangian Eq. (56) by performing U (3) A chiral rotations to remove the gluon theta term and transform the quark fields as where α u , α d , α s include the axion field a as with f a denoting the axion decay constant andθ being the genuine theta term. (With the above choice of α u , α d , α s , the axion does not mix with π 0 or η 8 .) As a result, the axion field is associated with the quark masses and the coefficients C 1q q , and can thus be implemented in the meson chiral Lagrangian through these terms. Accordingly, the meson potential Eq. (60) is modified to the potential of π 0 , η 8 , η 0 and axion a, where it should be reminded that α u , α d , α s are functions of a. The minimization condition for Eq. (85) yields meson VEVs π 0 , η 8 , η 0 and an axion VEV a . When only the term (C 1ud − C 1du ) is non-zero, these VEVs are given by The VEVs of π 0 and η 8 remain of the same order as the case without Peccei-Quinn mechanism, and hence they contribute to the neutron EDM in an analogous way. The axion VEV no longer cancels the genuineθ term and the leftover induces an effectiveθ term; we estimate its contribution by employing the result of Ref. [53] as The final result is the sum of the meson VEV contribution estimated analogously to Eq. (83), plus Eq. (87).

Other CP-violating operators
The contributions of the dipole operators in Eq. (102) and the Weinberg operator in Eq. (103) to the neutron EDM can be obtained with the QCD sum rule. The former is calculated in Ref. [54] while the latter is in Ref. [55], resulting in the following relations: where r.h.s. must be evaluated at 1 GeV. In Eqs. (88, 89), d q and d c q (q = u, d, s), so-called quark EDM and quark chromo-EDM, are defined as, Equations (88) and (89) represent the quark EDM contirbutions without and with Peccei-Quinn mechanism, respectively. For the case without Peccei-Quinn mechanism, we have takenθ = 0.

Numerical Analysis of Neutron EDM versus /
For numerical analysis of d n , we employ the following values: The chiral-limit pion decay constant F π is obtained from a lattice calculation as F π = 86.8 MeV [56]. In the analysis, the ratio of the bifundamental scalar VEVs is again fixed as tan β = m b /m t . The values of the new CP phases φ c , φ t , ψ d , ψ s , ψ b , α are randomly generated. We find that the contribution of the Weinberg operator is suppressed by roughly 10 −7 − 10 −9 compared with that of the four-quark operators, and thus we neglect it in the analysis.
First, we show the numerical result for the neutron EDM without the constraint from / in Fig. (3). One observes that the contribution of the four-quark operators is dominant over that of the quark EDMs.  As stated previously, an effectiveθ term is induced in the presence of Peccei-Quinn mechanism. In Fig. 4, we additionally show the numerical prediction based on Eq. (87). One finds that the inducedθ gives subleading contribution to the neutron EDM. inform us that almost all parameter points that account for the Re( / ) data will be covered by future neutron EDM searches [61]. Therefore, unless the tree-levelθ in the case without Peccei-Quinn mechanism miraculously cancels the contribution of the model, we anticipate the discovery of the neutron EDM in the near future.

Summary and Discussions
We have addressed the / anomaly in the SU (2) L × SU (2) R × U (1) B−L gauge extension of the SM with charge symmetry. Since the charge symmetry gives strong restrictions on the mixing matrix for right-handed quarks, / can be evaluated only in terms of two new CP phases α−ψ d and α − ψ s , the mass of W gauge boson (mostly composed of W R ), and the bifundamental scalar VEV ratio tan β. By fixing tan β at its natural value m b /m t , and by randomly varying α − ψ d and α − ψ s , we have shown that M W < 58 TeV must be satisfied to account for the experimental value of / at 1 σ level.
Next, we have made a prediction for the neutron EDM d n when the SU (2) L × SU (2) R × U (1) B−L model with charge symmetry solves the / anomaly. We have investigated the contribution of meson condensates induced by four-quark operators, and revealed that the π 0 VEV dominantly contributes to the neutron EDM, whose impact is enhanced by m s /(m u + m d ) compared to other contributions. This enhancement is attributable to the isospin violating coupling of W gauge boson, which allows the π 0 VEV to arise without the factor of m d − m u .
Additionally, we have found that the inducedθ term in the presence of Peccei-Quinn mechanism yields only a subleading effect on d n . On the basis of the above observations, we have shown that the / anomaly can be explained without conflicting the current experimental bound on d n , and that the parameter space where the / data are accounted for will be almost entirely covered by future experiments [61]. We comment on the constraint from Re( ) on the model. Since W gauge boson contributes to ∆F = 2 processes only at loop levels, for M W > 20 TeV, its contribution to Re( ) is safely below the experimental bound [62]. However, the heavy neutral scalar particles coming from the bifundamental scalar induce ∆F = 2 processes at tree level. Since their mass is of the same order as or below M W if there is no fine-tuning in the scalar potential, these particles may lead to a tension with the data on Re( ) [62] (constraint from Re( ) on general left-right models is found in Ref. [63], and that on the model with left-right parity is in Ref. [64]) (for early studies on the Re( ) constraint, see, e.g., Ref. [65]).