ϵ′/ϵ anomaly and neutron EDM in SU(2) L × SU(2) R × U(1)B−L model with charge symmetry

Journal of High Energy Physics, May 2018

Abstract The Standard Model prediction for ϵ′/ϵ 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 × SU(2) R × 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 × U(1)B−L gauge group and charge symmetry. We show that \( {M_W}_{{}_R}<58 \) TeV is required to account for the ϵ′/ϵ anomaly in this model. Next, we make a prediction for the neutron EDM in the same model and study a correlation between ϵ′/ϵ and the neutron EDM. We confirm that the model can solve the ϵ′/ϵ anomaly without conflicting the current bound on the neutron EDM, and further reveal that almost all parameter regions in which the ϵ′/ϵ anomaly is explained will be covered by future neutron EDM searches, which leads us to anticipate the discovery of the neutron EDM.

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ϵ′/ϵ anomaly and neutron EDM in SU(2) L × SU(2) R × U(1)B−L model with charge symmetry

JHE symmetry Naoyuki Haba 0 Hiroyuki Umeeda 0 Toshifumi Yamada 0 0 Graduate School of Science and Engineering, Shimane University The Standard Model prediction for 0= based on recent lattice QCD results exhibits a tension with the experimental data. We solve this tension through WR+ gauge boson exchange in the SU(2)L Beyond Standard Model; CP violation - SU(2)L HJEP05(218) SU(2)R U(1)B L SU(2)R 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 U(1)B L gauge group and charge symmetry. We show that MWR < 58 TeV is required to account for the 0= anomaly in this model. Next, we make a prediction for the neutron EDM in the same model and study a correlation between 0= and the neutron EDM. We con rm that the model can solve the 0= anomaly without con icting the current bound on the neutron EDM, and further reveal that almost all parameter regions in which the 0= anomaly is explained will be covered by future neutron EDM searches, which leads us to anticipate the discovery of the neutron EDM. 1 Introduction 2 3 0= 3.1 3.2 3.3 4.1 4.2 Wilson coe cients for S = 1 operators Hadronic matrix elements Numerical analysis of 0= Wilson coe cients for operators contributing to the neutron EDM Hadronic matrix elements 4.2.1 4.2.2 calculation of the hadronic matrix elements with lattice QCD [1, 3, 4] enables us to evaluate the K ! 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 [5{7] by 2.1 , and other groups [9, 10] have also obtained predictions for 0= that show a discrepancy of 2:9 and 2:8 , respectively. More importantly, the lattice result corroborates the calculation with dual QCD approach [11, 12], which has derived a theoretical upper bound on 0= that is violated by the experimental data and has thus claimed anomaly in this observable. (However, ref. [13] presents a di erent calculation that claims the absence of the anomaly.) Some authors have tackled this 0= anomaly in new physics scenarios, such as a general right-handed current [14], the Littlest { 1 { Higgs model with T-parity [15], supersymmetry [16{18], non-standard interaction with Z0 and/or Z [19, 20], vector-like quarks [21] and SU(3)c SU(3)L U(1)X gauge group [22]. The SU(2)L U(1)B L gauge extension of the SM is a well-motivated framework for addressing the 0= puzzle, because the avor mixing matrix for right-handed quarks automatically introduces new CP-violating phases, and WR+ gauge boson exchange 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. [14] has shown that a general SU(2)L ancy. However, a major theoretical motivation for the SU(2)L by adding either the left-right parity [24] or the `charge symmetry' [27].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 0= in the SU(2)L symmetric matrices, which restricts the quark mixing matrix associated with WR+ to be the complex conjugate of the SM Cabibbo-Kobayashi-Maskawa (CKM) matrix multiplied by a new CP phase factor for each quark avor. Given the above restriction, one can evaluate 0= only in terms of two new CP phases, the mass of WR+ and the ratio of two vacuum expectation values (VEVs) of the bifundamental scalar, which leads to a speci c prediction for the model parameters. Our analysis on 0= proceeds as follows. By integrating out WR, WL and the top quark, we obtain the Wilson coe cients for S = 1 operators that contribute to K ! 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. [30, 31]. The hadronic matrix elements for current-current operators are seized from the lattice results [3, 4]. We nd that among new physics operators, (su)L(ud)R and (su)R(ud)R (each with two ways of color contraction) both dominantly contribute to 0= . Their contributions are of the same order because the Wilson coe cients of the (su)L(ud)R operators are suppressed by the hierarchy of two bifundamental scalar VEVs v1=v2 = tan , which is about mb=mt if there is no ne-tuning in accommodating the top and bottom quark Yukawa couplings, whereas this suppression does not enter into the Wilson coe cients of the (su)R(ud)R operators. On the other hand, the lattice computation has con rmed that the hadronic matrix elements for the former operators are enhanced compared to the latter. Thus, these operators possibly equally contribute to 0= . This result is in contrast to the study of ref. [14], which has concentrated solely on the (su)L(ud)R operators. 1The `charge symmetry' is inspired by D-parity [28] in the SO(10) grand uni cation theory. However, the model we consider cannot be embedded in the SO(10) theory, since we assume the charge symmetry breaking scale to be below O(100) TeV. { 2 { Once the 0= anomaly is explained in the SU(2)L 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 signi cant contributions from fourquark operators in SU(2)L U(1)B L models [14, 34{38],2 allowing us to discuss future detectability of the neutron EDM in relation to the 0= anomaly. Our analysis on the neutron EDM starts by integrating out WR, WL and top quark to obtain the Wilson coe cients for CP-violating operators. The leading order expression for the anomalous dimension matrix is found in refs. [41, 42, 44, 45]. Regarding the hadronic matrix elements of CP-violating operators, we reveal that the pion VEV h by four-quark operators [33] gives the leading contribution to the neutron EDM, which 0i induced is enhanced by the quark mass ratio ms=(mu + md) in comparison to the rest. This enhancement is understood as follows: since WR+-WL+ mixing gives rise to CP-odd and isospin-odd interactions, the pion VEV h 0i, which is isospin-odd, can arise without the mu, and thus can be directly proportional to 1=(mu + md). The pion VEV induces a CP-violating coupling for neutron n, baryon, and kaon K+ without 0= is shown at the end of the section. In section 4, we give the Wilson coe cients for CPviolating operators contributing to the neutron EDM. Special care is taken in evaluating meson condensates and their impact on the neutron EDM. The nal result is a prediction for the neutron EDM in light of the 0= anomaly. Section 5 is devoted to summary and discussions. 2 U(1)B L model with charge symmetry We consider SU(3)C SU(2)L SU(2)R U(1)B L gauge theory with charge symmetry. QiL QcRi L i L LcRi L R with La and Ra 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 elds; and SU(2)L $ SU(2)R; QL $ QcRi; i LL $ LcRi; i $ T ; L $ R: The part of the Lagrangian describing SU(2)L gauge groups, respectively, and Yq and Y~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. (2.2) leads to the following tree-level relations: gL = gR; (Yq)ij = (Yq)ji; (Y~q)ij = (Y~q)ji: The SU(2)R triplet scalar R develops a VEV, vR, to break SUR(2) U(1)B L ! U(1)Y , and the bi-fundamental scalar takes a VEV con guration, h i = p 1 2 v sin 0 0 v cos e i ! ; to break SU(2)L U(1)Y ! U(1)em, where is the spontaneous CP phase. The VEV of L is hereafter neglected, as it is severely constrained from -parameter. The resultant mass matrices for WLa, WRa and X gauge bosons read gLgR sin(2 )e i v2=4! gR2(vR2 + v2=4) WL+! WR+ gLgR v2=2 2gLgX v 2 R 0 1 0WL31 2gX2 v 2 R 2gLgX vR2CA B@WR3 AC : X (2.2) QcRi (2.3) (2.4) (2.5) (2.6) { 4 { The mass matrix for the charged gauge bosons is diagonalized as v and gL = gR, we have an important relation for , which indicates that when we assume tan couplings are naturally derived, the WL-WR mixing angle the factor 2mb=mt 0:05. ' mb=mt so that the top and bottom Yukawa is smaller than M W2 =M W2 0 by The quark mass matrices are given by3 { 5 { (2.7) (2.8) (2.10) (2.11) which we diagonalize as Mu = VuyLdiag(mu; mc; mt)VuR and Md = VdyLdiag(md; ms; mb)VdR, with VuL; VuR; VdL; VdR being unitary matrices. However, since Yq and Y~q are complex symmetric matrices, so are Mu and Md, and one can most generally write 0ei u 0 0 0 ei b Hence, the SM CKM matrix, VL = VuLVdyL, and the corresponding avor mixing matrix for right-handed quarks, VR = VuRVdyR, are related as 3U Ri s(U Rci) , DRi s(DRci) . 0ei u 0ei u 0 0 0 ei t 0 0 0 e i d 0 0 e i s 0 0 e i s 0 0 0 0 e i b 0 0 e i b 1 C : A 1 C A 1 2 2 1 2 = p U i W + + p U i W 0+ Eventually, the part of the Lagrangian eq. (2.3) describing avor-changing W; W 0 interactions is recast, in the unitary gauge, into the form, In this paper, we adopt the following convention for the quark phases and u; c; t; rst, we rede ne the phases of ve quarks to render the CKM matrix in the 0 s12s23 c12c13 c12s23s13ei c12c23s13e i s12c13 c12c23 c12s23 s12s23s13ei s12c23s13ei Next, we rede ne c; t; d; s; b to set c; t; d; s; b, and . where operators O's are de ned in appendix A. We determine the Wilson coe cients as follows: we approximate gR = gL by ignoring di erence in RG evolutions of gL and gR at scales below MW 0. Also, for each Wilson coe cient, if multiple terms have an identical phase, we only consider the one in the leading order of M W2 =M W2 0 or sin . By integrating u = 0: (2.13) (2.14) (3.1) X (VL)is(VL )id ei( s d) 1 2 F1(yi); C30 = C50 = 13 C40; (3.4) X (VL)is(VL )id ei( s d) 2 3 E1d(yi); C2 = (VL )us(VL)ud cos2 ; C2c = (VL )cs(VL)cd cos2 ; out W 0, one obtains the following leading-order matching conditions at a scale (note our convention with u = 0): (3.2) (3.3) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) By further integrating out W and the top quark, one gains the following leading-order matching conditions at a scale MW (note our convention with u = 0): C2RL = (VL)us(VL)ud ei( s ) sin cos ; C2LR = (VL )us(VL )ud ei( d+ ) sin cos ; C2RcL = (VL)cs(VL)cd ei( c+ s ) sin cos ; C2LcR = (VL )cs(VL )cd ei( c d+ ) sin cos ; C4 = C6 = C7 = C9 = Cg = X (VL )is(VL)id 2 F1(xi); X (VL )is(VL)id 3 E1d(xi); 1 2 cos2 (VL )is(VL)id F2(xi) + sin cos cos2 (VL )is(VL)id E2d(xi) + sin cos mi (VL)is(VL)id ei( i+ s )E3d(xi) ; C3 = C5 = 1 mi (VL)is(VL)id ei( i+ s )F3(xi) ; i=u;c;t C0 = X md (VL )is(VL)id F2(xi)+sin cos mi (VL )is(VL )id ei( i d+ )F3(xi) ; cos2 md (VL )is(VL)id E2d(xi)+sin cos mi (VL )is(VL )id ei( i d+ )E3d(xi) ; ms ms where loop functions F1; F2; F3 and E1d; E2d; E3d are de ned in appendix B. We are aware that the dipole operators receive two contributions with di erent phases when W 0 is integrated out and when W is. The two are expressed as Cg; C ; Cg0; C0 and Cg; C ; Cg0; C0 , respectively. (fCjRLg) and (fCjLRg), we assume that their initial conditions at scale by eqs. (3.10){(3.15) and solve the RG equations from We take into account RG evolutions of the Wilson coe cients at order O( s). The fact that four sets of operators, (fOig), (fOi0g) (i = 1; 2; : : : ; 10; 1c; 2c), (fORLg), (fOjLRg) j (j = 1; 2; 1c; 2c), do not mix with each other facilitates the computation. For (fCig), = MW to the scale for which = MW are given ref. [31], and those for (fCiRLg) and (Cg; C ) are in ref. [30]. the lattice results are reported. For (fCi0g), we assume that their initial conditions at = MW 0 are provided by eqs. (3.2){(3.9) and solve the RG equations from = MW 0 to the scale of lattice results. Finally, we compute RG evolutions of the coe cients of the dipole operators (Cg; C ) and (Cg0; C0 ), which receive contributions from (fCi; CjRLg) and (fCi0; CjLRg), respectively. The O( s) RG equations for (fCig) and (fCiRLg) are found in 3.2 Hadronic matrix elements We employ the lattice calculations of hadronic matrix elements h( 1; 2; : : : ; 10 for I = 0; 2 reported by RBC/UKQCD in refs. [3, 4]. )I jOijK0i for i = Since lattice calculations for the matrix elements of O1LR and O2LR are missing, we estimate them from the RBC/UKQCD results using isospin symmetry. In the limit of exact isospin symmetry, we nd, for I = 3=2 amplitudes, h ( )I=2jO7jK0i = h( )I=2j(sd)L uu 1 2 dd 1 2 ss R jK0i = = = 3 4 3 4 3 2 h h h ( ( ( )I=2j(sd)L(uu dd)RjK0i )I=2jp2(su)Lp2(ud)RjK0i )I=2jO2LRjK0i; where we have discarded I = 1=2 part when obtaining the second line, and when deriving the third line, we have inserted Clebsch-Gordan coe cients for constructing the I = 3=2 { 8 { (3.18) (3.19) (3.20) (3.21) (3.22) = h( p = ( 3 4 3 )I=0j 2 3 2 where in the rst line, we have separated (uu)R into second line, we have inserted Clebsch-Gordan coe cients for constructing the hold between the matrix elements for O1LR and O8; O6. The hadronic matrix elements for the chromo-dipole operators Og; Og0 are extracted from the calculation based on dual QCD approach [46]. Note that the above calculation is corroborated by the fact that it is consistent with a lattice calculation of the K- hadronic matrix element [47], which is related to the Kone by chiral perturbation theory. 3.3 Numerical analysis of 0= The de nition for the decay amplitudes of K0 ! is A0ei 0 = h( )I=0j H S=1 jK0i ; A2ei 2 = h( )I=2j H S=1 jK0i ; where 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 0 Re = Re i!ei( 2 0) ! p 2 ImA2 ReA2 ImA0 ReA0 ; where ! = ReA2=ReA0 is a suppression factor due to the I = 1=2 rule. For the strong phases, we use the values of refs. [3, 4], 2 = 23:8 5:0 degree and 0 = 11:6 2:8 degree. For the real parts of the decay amplitudes, we employ the experimental data [8], ReA2 = 1:479 10 8 GeV and ReA0 = 33:20 10 8 GeV, which leads to ! = 4:454 10 2. In our analysis, we separate the SM and new physics contributions as { 9 { (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) (3.29) Re NP 0 = Re 0 SM + Re 0 NP : For the SM part, we quote the calculation in the literature Re( 0= )SM = (1:38 6:90) 10 4 [4]. It is the new physics part, Re 0 = Re i!ei( 2 0) ! p 2 ImA2NP ReA2 ImA0NP ReA0 ; that we compute in this paper. In doing so, we approximate cos2 = 1 in the Wilson coe cients eqs. (3.10){(3.19), so that the SM contribution is separated from the new physics one at the operator level. 4 ] 40 0 1 -20 10 40 4 ] 20 0 -40 α-ψd=π/3, α-ψs=π/4 α-ψd=π/2, α-ψs=π/3 α-ψd=π, α-ψs=-π/4 20 30 40 50 60 data given by PDG [8], while model predictions with speci c choice of phases are shown by the lines. 20 40 60 PDG [8], while each red dot corresponds to the model prediction with a randomly generated set of ( d, s). In the analysis, we x the ratio of the bifundamental scalar VEVs at its natural value as tan = mb=mt. We have found numerically that for MW 0 > 1 TeV, the chromodipole contribution to Re( 0= ) does not exceed O(10 4) and is hence safely neglected.4 Consequently, only two combinations of new CP phases, mass determine the new physics contribution. d and s, and the W 0 First, we choose speci c values for the new CP phases in the calculation of 0= to illustrate the model prediction. In gure 1, the prediction for Re( 0= ) is presented with speci c choices of d and Next, we randomly vary s . d and parameters. In gure 2, we show the region of Re( 0= ) obtained by varying d and s. One observes that MW 0 < 58 TeV is necessary for 1 explanation of the anomaly. 4When we use a calculation based on the chiral quark model in ref. [48] to evaluate the hadronic matrix elements of the chromo-dipole operators, we are again lead to the result that the chromo-dipole contribution to Re( 0= ) is below O(10 4) for MW 0 > 1 TeV. s in the range [0; 2 ], since they are free We have con rmed that among the terms of S = 1 Hamiltonian eq. (3.1), Pi=1;2 Ci0Oi0 and Pj=1;2(CjRLORL + CjLROjLR) are the leading sources of the new physics contribution. j Wilson coe cients for operators contributing to the neutron EDM In the e ective QCD QED theory in which W; W 0 and the top quark are integrated out, the part of the CP-violating Hamiltonian that contributes to the neutron EDM is parametrized as HJEP05(218) HnEDM = p GF 2 X X where operators O's are de ned in appendix C. We determine the Wilson coe cients C's as follows: again, for each coe cient, if multiple terms have an identical phase, we exclusively consider the one in the leading order of M W2 =M W2 0 or sin . By integrating out W and the top quark, one obtains the following leading-order matching conditions at C2cd = 4 sin cos Im h(VL)cd(VL)cd ei( c+ d )i ; where loop functions F1; F2; F3 and E1d; E2d; E3d; E3u are de ned in appendix B. In eq. (4.10) (which corresponds to the Weinberg operator [49]), we present the dominant part proportional to mt. Terms obtained by integrating out W 0 possess the same phases as eqs. (4.2){(4.10) and are simply suppressed by M W2 =M W2 0 compared to eqs. (4.2){(4.10). They are therefore neglected in our analysis. e e 4 2 sin cos 4 2 sin cos 4 2 sin cos 4gs2 sin cos 4gs2 sin cos 4gs2 sin cos X X X i=d;s;b mu i=d;s;b mu i=u;c;t md X X i=d;s;b mu i=d;s;b mu X i=u;c;t md mt mb 4gs3 C3 ' (16 2)2 sin cos with xi mi Im h(VL)id(VL)id ei( i+ d )i E3d(xi); (d ! s); (4.6) mi Im h(VL)ui(VL)ui ei( i )i F3(xi); mi Im h(VL)ui(VL)ui ei( c+ i )i F3(xi); mi Im h(VL)id(VL)id ei( i+ d )i F3(xi); (d ! s); (4.9) F3(xt) Im h(VL)tb(VL)tb ei( t+ b )i ; (4.4) (4.5) (4.7) (4.8) (4.10) The RG equations at order O( s) for the Wilson coe cients are obtainable in refs. [41, 42, 44, 45]. We assume that the initial conditions at tions are given by eqs. (4.2){(4.10), and solve the equations from At the 1 GeV scale, we evaluate the hadronic matrix elements. = MW for the RG equa= MW to = 1 GeV. U(1)B L model with charge symmetry, the Wilson coe cients for the four-quark operators O1q0q = (q0q0)(qi 5q) and O2q0q = (q0 q0 )(q i 5q ) (q 6= q0; q; q0 = u; d; s) are particularly large. Therefore, we scrutinize how these operators contribute to the neutron EDM. Operators O1q0q contribute in the following three ways: HJEP05(218) The rst one is through meson condensation [33]; O1q0q operators give rise to tadpole terms for pseudoscalar mesons and induce their VEVs. These VEVs generate CPviolating 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 O1q0q with baryons and mesons, hBM jO1q0qjBi (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 O1q0q with neutrons and photon. On the other hand, O2q0q 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 h 0i, which belongs to the rst category, is enhanced by the factor ms=(mu + md) compared to the latter two. We therefore investigate how O1q0q operators bring about meson condensation, thereby contributing to the neutron EDM. We are aware that if Peccei-Quinn mechanism [50] exists, it a ects the meson condensation and also induces an e ective non-zero term due to incomplete cancellation between the genuine term and the axion VEV. Alternatively, it is logically possible to assume = 0 without Peccei-Quinn mechanism, by considering an unknown mechanism or through ne-tuning, in which case we do not need to take into account the e ect of PecceiQuinn mechanism or that of non-zero . In this paper, we consider both cases where (i) one has = 0 without Peccei-Quinn mechanism, and (ii) Peccei-Quinn mechanism is at work. We start from the case with = 0 without Peccei-Quinn mechanism. The meson condensation contribution is evaluated by the following steps: (L $ R); X q6=q0 with Cijkl LRLR = Cijkl RLLR First, we implement P C1q0qO1q0q part of the Hamiltonian eq. (4.1) into the meson chiral Lagrangian. To this end, we rewrite X q6=q0;q;q0=u;d;s C1q0qO1q0q = iCiLjRklLR (qiLqjR)(qkLqlR) + iCiRjkLlLR (qiRqjL)(qkLqlR) It then becomes clear that the theory would be invariant (except for U(1)A anomaly) if From the above transformation property and the parity invariance of QCD, the meson chiral Lagrangian at order O(p2) plus the leading CP-violating terms is found to be (remind that tr h(D U )yD U + (U + U y)i + tr hU D U yi tr hU yD U i F 2 0 F 2 12 = 0 has been assumed) Lmesons = h + a0 tr log U GF + p F 2 4 X 2 i;j;k;l=u;d;s where Cijkl LRLR; Cijkl as U ! RU Ly, and log U yi2 n iCiLjRklLR c1[U ]ji[U ]lk +iCiRjkLlLR c3[U y]ji[U ]lk c3[U ]ji[U y]lk o ; U = exp 2i p6F0 0 I3 + ; 2i F I3 diag(1; 1; 1); = 2B0 diag(mu; md; ms): (4.11) (4.12) (4.13) (4.15) c1[U y]ji[U y]lk + c2[U ]li[U ]jk c2[U y]li[U y]jk 2 p1 K 2 + 1 p 2 2 p1 K0 2 p1 K+ 1 1 RLLR have been de ned in eq. (4.12). Here, U is a nonlinear representation of the nine Nambu-Goldstone bosons that transforms under U(3)L U(3)R rotations L R includes the quark mass term, which are given by [U ]ij denotes the (i; j) component of matrix U . F is the pion decay constant in the chiral limit and F0 is the decay constant for 0, which we approximate as F0 ' F . B0 satis es B0 ' m2 =(mu + md). The term with log U represents instanton e ects, whose expression is exact in the large Nc limit [51], and a0 satis es 48a0=F02 ' m2 + m20 2m2K . c1, c2 and c3 are unknown low energy constants (LECs), which can be estimated by nave dimensional analysis [53] as c1 c2 c3 (2). The CP-violating part of the Lagrangian eq. (4.13) 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 HJEP05(218) mesons 0 , 8 and 0 : V ( 0; 8; 0) = F 2B0 p The above potential is minimized with non-zero meson VEVs, h as we are concerned with vertices with one meson, the physical modes of 0 0i, h 8i and h 0i. Insofar , 8 and 0 elds can be approximated as 0 phys ' 0 h i 0 ; 8phys ' 8 h 8i; 0phys ' 0 h 0i: In the SU(2)L U(1)B L model with charge symmetry, there hold relations C1ud ' C1du and jC1udj jC1sqj; jC1qsj (q = u; d). When (C1ud + C1du) and C1sq; C1qs are neglected accordingly, one nds, for small VEVs, 0 h i ' p2 (C1ud C1du) B0F 2 B0F 2mumdms + 8a0(mumd + mdms + msmu) ; c3 B0F 2(mu + md)ms + 8a0(mu + md + 4ms) ' p2 (C1ud C1du) p ' p2 (C1ud C1du) p c3 3B0F 2 (md p3B20cF3 2 (md B0F 2ms + 24a0 B0F 2ms mu) B0F 2mumdms + 8a0(mumd +mdms +msmu) mu) B0F 2mumdms +8a0(mumd +mdms +msmu) (4.17) (4.18) (4.19) ; : Note that h 8i and h 0i are proportional to md mu. This is because these VEVs are isospin singlets and hence must be constructed from the product of isospin-odd coe cient C1ud C1du and isospin-odd mass term md mu. In contrast, h because this VEV is isospin-violating. Since 20a0 B0F 2ms holds empirically, we nd 0i does not contain md mu from eq. (4.19) that h 0i is much larger than h 8i and h 0i by the factor ms=(md mu). (3). Meson condensation breaks CP symmetry (and U(3)L U(3)R symmetry) and in duces CP-violating interactions for baryons and mesons. To study these interactions, we write the baryon chiral Lagrangian at order O(p2) as (terms irrelevant in the current dis D 2 tr B 5f ; Bg F 2 MBBB tr B 2 5 where B represents baryons and L; R include mesons as i r ) R + i r ) R i l ) L; i l ) L; U = R Ly; R = Ly: + 0 1 p 2 is a covariant derivative for baryons, is a combination of meson elds, and + contains quark masses, which are de ned as + 2B0 Ly diag(mu; md; ms) R + 2B0 Ry diag(mu; md; ms) L: MB is the baryon mass in the chiral limit. We insert meson VEVs h baryon chiral Lagrangian eq. (4.20) and extract CP-violating interaction terms involving 0i, h 8i, h 0i into the neutron n. We thus obtain Lbaryons gnn nn p0hys + gnn8 nn 8phys + gnn0 nn 0phys + gnp (pn + + np ) + gn 0K0 ( 0nK0 + n 0K0) + gn K+ ( +nK + n K+) + gn K ( nK0 + n K0); (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) (4.27) gnn = gnn8 = gn 0K0 = g n K+ = 4 3 B0 F 4 p 2ms) + bF (md + 2ms)g hF00i ; (4.29) gnn0 = " p 2ms) + bF (md + 2ms)g hF8i 3 fb0(mu + md + ms) + bD(md + ms) + bF (md gnp = F B0 (bD + bF ) " p2(md mu) 0 h i F p p 3 (mu + md) ms)g hF00i ; F h 8i + p2 h 0i F0 (3md + ms) 0 h i + F 1 where the coupling constants are given by F B0 4 f b0(mu + md) (bD + bF )mdg F + p fb0(md mu) + (bD + bF )mdg 4 3 p fb0(md mu) + (bD + bF )mdg F 0 h i F 0 h i ; gn K = B0 (bD + 3bF ) p (3md + ms) 0 h i + F 1 6 (md 5ms) h 8i F Note in particular that h 0i enters into the expression for gn K+ eq. (4.33) without the mu, which is allowed because the coupling gn K+ violates isospin. It follows K+ is enhanced by the factor ms=(mu + md), as it contains a term msh i We compare the above meson-VEV-induced CP-violating couplings with those arising from direct hadronic matrix elements of O1q0q and O2q0q. The latter are estimated by nave dimensional analysis [53] as5 (B and M represent any baryon and meson, respectively) gBBM jdirect GF p X X where eq. (4.19) and the nave dimensional analysis on c3 eq. (4.16) are in use. Noting that (bD bF )(4 F ) 1 holds numerically, we observe that the meson VEV contribution eq. (4.36) dominates over the direct hadronic matrix element one eq. (4.35) by the factor ms=(mu + md). 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. (4.27), a CP-conserving baryon-meson axial-vector coupling and a photon coupling. We refer to the loop calculation of ref. [58] performed with infrared regularization [60, 61], from which the neutron EDM, dn, is obtained as (4.35) (4.36) HJEP05(218) dnjloop = 8 2F e g n p 2 K+ (D 2 p gnp (D + F ) F ) 1 1 1 1 log log m2K2 + m2 m 2mN mK 2mN (m mN ) mK : (4.37) Here, the divergent part 1= 1= E + log(4 ) and the scale stem from dimensional regularization in 4 2 dimension with mass parameter . In fact, the baryon chiral Lagrangian contains a LEC which cancels the above divergence and whose nite part contributes to the neutron EDM. The impact of the nite part of the LEC is assessed by nave dimensional analysis [53] as dnjLEC GF p X X 2 i=1;2 q;q0 jCiq0qj e (4 )2 : 4 F (4.38) On the other hand, from eqs. (4.19) and (4.33) and the estimate on c3 eq. (4.16), the nite part of the loop contribution eq. (4.37) is estimated to be dnjloop e 8 2F e 8 2F g n p 2 GF p2 (C1ud F ) C1du)(bD 2ms 1, we nd that the loop contribution eq. (4.39) dominates over the LEC one eq. (4.38) by the factor ms=(mu + md). It is thus justi able 5There are also studies in which the direct hadronic matrix elements are estimated with vacuum saturation approximation [54{57] and with hadron models [35, 36]. to estimate dn by simply extracting the nite part of the loop contribution. We further set = mN , since mN is a natural cuto scale, and arrive at dn 8 2F e g n 2 p log log m2 m2N + m 2mN mm2K2N + mK 2mN (m mN ) mK Next, we study the case with Peccei-Quinn mechanism. We incorporate the axion eld, a, into the meson Lagrangian eq. (4.13) by performing U(3)A chiral rotations to remove the gluon theta term and transform the quark elds as where u; d; s include the axion eld a as dL ! e i d=2dL; dR ! ei d=2dR; (4.41) u = d = s = mumd + mdms + msmu mumd + mdms + msmu mdms msmu mumd mumd + mdms + msmu a fa a fa a fa + + + ; ; ; with fa denoting the axion decay constant and being the genuine theta term. (With the above choice of u; d ; s, the axion does not mix with 0 or 8.) As a result, the axion eld is associated with the quark masses and the coe cients C1q0q, and can thus be implemented in the meson chiral Lagrangian through these terms. Accordingly, the meson potential eq. (4.17) is modi ed to the potential of 0 , 8 , 0 and axion a, V ( 0; 8; 0; a) = F 2B0 2c1 (C1ud + C1du) sin (4.42) where it should be reminded that u; d ; s are functions of a. The minimization condition for eq. (4.42) yields meson VEVs h 0i, h 8i, h 0i and an axion VEV hai. When only the term (C1ud C1du) is non-zero, these VEVs are given by 0 h i h 8i F F a h i + fa GF GF GF ' p2 (C1ud ' p2 (C1ud ' p2 (C1ud C1du) Bp03Fc32 (md C1du) B20cF3 2 (md c3 C1du) B0F 2 mumd + mdms + msmu ; mu + md + 4ms 1 mu) mumd + mdms + msmu ; h 0i F0 ' 0; 1 mu) mumd : (4.43) The VEVs of 0 and 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 genuine term and the leftover induces an e ective term; we estimate its contribution by employing the result of ref. [62] as (4.44) (4.45) (4.46) (4.47) (4.48) (4.49) dnjind = (2:7 1:2) a fa nal result is the sum of the meson VEV contribution estimated analogously to eq. (4.40), plus eq. (4.44). 4.2.2 Other CP-violating operators O1q, O2q and O3 The contributions of the dipole operators in eq. (C.1) and the Weinberg operator in eq. (C.2) to the neutron EDM can be obtained with the QCD sum rule. The former is calculated in ref. [63] while the latter is in ref. [64], resulting in the following relations: dnjquark = 0:47dd dnjqPuQark = 0:47dd dnjWeinberg = GpF egsC3 2 0:12du + e(0:18dcd 0:18dcu 0:008dcs); 0:12du + e(0:35dcd + 0:17dcu); (10 30) MeV; where r.h.s. must be evaluated at 1 GeV. In eqs. (4.45), (4.46), dq and dcq(q = u; d; s), so-called quark EDM and quark chromo-EDM, are de ned as, dq( ) = c dq( ) = GF GF 2 p e eqC1q( )mq( ); p2 C2q( )mq( ): Equations (4.45) and (4.46) represent the quark EDM contirbutions without and with Peccei-Quinn mechanism, respectively. For the case without Peccei-Quinn mechanism, we have taken Numerical analysis of neutron EDM versus 0= For numerical analysis of dn, we employ the following values: the chiral-limit pion decay constant F is obtained from a lattice calculation as F = 86:8 MeV [65]. D; F have been measured to be D = 0:804 and F = 0:463. For bD; bF , we quote the result of refs. [66, 67] with a NLO calculation in Lorentz covariant baryon chiral perturbation theory with decuplet contirbutions, which reads bD = 0:161 GeV 1 and bF = Since the same calculation formalism, combined with experimental data 0:502 GeV 1 . N ' 59(7) MeV, predicts a small value of the strange quark contribution to the nucleon mass s [66], we infer that these values of bD; bF are most robust. For the quark masses, we adopt lattice results in ref. [68], mud(2 GeV) = 3:373 MeV and ms(2 GeV) = 92:0 MeV, and further evaluate QCD ve-loop RG evolutions to obtain the masses at 1 GeV in M S scheme, which are used in our analysis. Also, we exploit an estimate mu=md = 0:46 [68]. The main source of uncertainty in our analysis is the unknown LEC c3 in the meson chiral Lagrangian eq. (4.13). The other unknown LEC c1 is ine ective, because the Wilson coe cients satisfy jC1ud C1uj jC1ud + C1uj; jC1sqj; jCqsj. Our calculations of loop-induced dn eq. (4.40) and axion-induced dn eq. (4.40) are hence proportional to c3 and subject to O(1) uncertainty originating from its nave dimensional analysis eq. (4.16). The fact that our results depend only on one LEC c3 is good news, because it excludes the possibility of accidental cancellation between contributions with di erent LECs. Another source of uncertainty is the renormalization scale in the loop calculation eq. (4.37), but this is subdominant compared to the uncertainty of c3. In the analysis, the ratio of the bifundamental scalar VEVs is again xed as tan = mb=mt. The values of the new CP phases c; t; d ; s ; b ; are randomly generated. We nd that the contribution of the Weinberg operator is suppressed by roughly 10 7 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 0= in gure 3. One observes that the contribution of the four-quark operators is dominant over that of the quark EDMs. As stated previously, an e ective term is induced in the presence of Peccei-Quinn mechanism. In gure 4, we additionally show the numerical prediction based on eq. (4.44). One nds that the induced gives subleading contribution to the neutron EDM. Next, the correlated prediction for jdnj and Re( 0= ) is presented in gures 5 and 6 in the cases without and with Peccei-Quinn mechanism, respectively. Here, small contributions from the quark EDMs are neglected. The cases with and without Peccei-Quinn mechanism yield almost identical results because the induced has a subdominant e ect, as seen in gure 4. We observe that MW 0 = 20 TeV and 50 TeV can be consistent with the data on Re( 0= ) at 1 level, whereas the case with MW 0 = 70 TeV cannot explain it. However, the case with MW 0 = 20 TeV has already been excluded by the current bound on the neutron EDM, and only MW 0 = 50 TeV can be compatible with the neutron EDM bound and the data on Re( 0= ). Figures 5 and 6 further inform us that almost all parameter points that account for the Re( 0= ) data will be covered by future neutron EDM searches [71]. Therefore, unless the tree-level 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. 1[10-2 0 10-6 104 ] m 102 c 6 2 e 0 1 100 [ n|10-2 d | 10-4 Only the contributions of four-quark operators and quark-level EDMs including both quark EDM and chormo-EDM are shown. A dashed line represents the current bound on the neutron EDM [69], while a dashed dotted line stands for the future bound [71]. 10 20 30 50 60 70 40 MW' [TeV] in the presence of Peccei-Quinn mechanism. A gray dashed line represents the current bound of EDM [69] while a black dashed dotted line stands for the future bound [71]. 5 Summary and discussions We have addressed the 0= 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, 0= can be evaluated only in terms of two new CP phases d and s, the mass of W 0 gauge boson (mostly composed of WR), and the bifundamental scalar VEV ratio tan . By xing tan at its natural value mb=mt, and by randomly varying d and s, we have shown that MW 0 < 58 TeV must be satis ed to account for the experimental value of 0= at 1 level. 10 100 current bound 10-2 future bound MW' =20TeV MW' =50TeV MW' =70TeV 104 ] 10 100 current bound 10-2 future bound MW' =20TeV MW' =50TeV MW' =70TeV 1σ range 1σ range -20 -10 0 10 20 Re(ϵ'/ϵ) [10-4] = 0 without Peccei-Quinn mechanism. A gray dashed line and a black dashed dotted line represent the current [69] and the future [71] bounds on the neutron EDM, while a cyan band stands for the 1 range of the direct CP violation in K ! decay obtained from PDG [8]. HJEP05(218) -20 -10 0 Re(ϵ'/ϵ) [10-4] 10 20 Next, we have made a prediction for the neutron EDM dn when the SU(2)L U(1)B L model with charge symmetry solves the 0= anomaly. We have investigated the contribution of meson condensates induced by four-quark operators, and revealed that the 0 VEV dominantly contributes to the neutron EDM, whose impact is enhanced by ms=(mu + md) compared to other contributions. This enhancement is attributable to the isospin violating coupling of W 0 gauge boson, which allows the 0 VEV to arise without mu. Additionally, we have found that the induced term in the presence of Peccei-Quinn mechanism yields only a subleading e ect on dn. On the basis of the above observations, we have shown that the 0= anomaly can be explained without con icting the current experimental bound on dn, and that the parameter space where the 0= data are accounted for will be almost entirely covered by future experiments [71]. We comment on the constraint from Re( ) on the model. Since W 0 gauge boson F = 2 processes only at loop levels, for MW 0 > 20 TeV, its contribution to Re( ) is safely below the experimental bound [73]. 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 MW 0 if there is no ne-tuning in the scalar potential, these particles may lead to a tension with the data on Re( ) [73] (constraint from Re( ) on general left-right models is found in ref. [74], and that on the model with left-right parity is in ref. [75]) (for early studies on the Re( ) constraint, see, e.g., ref. [76]). Acknowledgments The authors would like to thank Monika Blanke, Andrzej Buras, Antonio Pich and Amarjit Soni for valuable comments. This work is partially supported by Scienti c Grants by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 24540272, 26247038, 15H01037, 16H00871, and 16H02189). A Operators of e 2 q=u;d;s 8 2 mss X (s d )L(q q )L; X (s d )L(q q )R; X (s d )Leq(q q )R; X (s d )Leq(q q )L; F PLd; (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) ed = es = corresponding operators. where (qq0)L q (1 5)q0 and (qq0)R q (1 + 5)q0, ; are color indices, and color summation is taken in each quark bilinear unless ; are displayed. eu = 2=3 and 1=3. The operators Oi0, OjLR are obtained by interchanging L $ R in the The loop functions in the main text are de ned as follows: F1(x) = F2(x) = F3(x) = E1d(x) = E2d(x) = E3d(x) = E3u(x) = 12(1 x(2 + 5x 4(1 4 + x + x2 2(1 25x2 36(1 6(1 3(1 x)2 + (1 19x3 x)3 + 5x 8x2) 12(1 2(1 18(1 x2(2 2(1 x)4 log x; 3x) x)3 log x; 3x) x)3 log x: 4 9 2 3 x( 18 + 11x + x2) x2( 15 + 16x 4x2) 6(1 x)4 log x + log x + ; 2 3 x2(6 + 2x 5x2) 4 9 log x + log x + ; CP-violating operators that contribute to the neutron EDM O1q = O3 = 2 eq mqq 1 f abc 6 O4q = qq qi 5q; O1q0q = q0q0 qi 5q; O3q0q = q0 q0 q i 5q; i 5q F G a G b Ga ; O2q = 2 gs mqq i 5T aq Ga ; O5q = q q q i 5q; O2q0q = q0 q0 q i 5q ; O4q0q = q0 q0 q i 5q ; (C.1) (C.2) (C.3) (C.4) (C.5) where q0; q = u; d; s and q0 6= q. each quark bilinear unless ; are displayed. are color indices, and color summation is taken in Open Access. 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Naoyuki Haba, Hiroyuki Umeeda, Toshifumi Yamada. ϵ′/ϵ anomaly and neutron EDM in SU(2) L × SU(2) R × U(1)B−L model with charge symmetry, Journal of High Energy Physics, 2018, 52, DOI: 10.1007/JHEP05(2018)052