Contributions from SUSY-FCNC couplings to the interpretation of the HyperCP events for the decay Σ^+ \to p μ^+ μ^-
aa r X i v : . [ h e p - ph ] D ec Contributions from SUSY-FCNC couplings to the interpretationof the HyperCP events for the decay Σ + → pµ + µ − Gao Xiangdong, ∗ Chong Sheng Li, † Zhao Li, ‡ and Hao Zhang § Department of Physics, Peking University, Beijing 100871, China (Dated: November 21, 2018)
Abstract
The observation of three events for the decay Σ + → pµ + µ − with a dimuon invariant mass of214 . ± . A in the NMSSM can be used to explain the HyperCPevents without contradicting all the existing constraints from the measurements of the kaon decays,and the constraints from the K − ¯ K mixing are automatically satisfied once the constraints fromkaon decays are satisfied. PACS numbers: 14.80.Cp, 12.60.Jv, 14.20.Jn ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] . INTRODUCTION Recently, there are a great deal of interest [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] in the interpretationof the observed three events for the decay Σ + → pµ + µ − with a dimuon invariant mass of214 . ± . . +6 . − . ± . × − [11]. It has beenargued that in the framework of the Standard Model(SM) it is possible to account for thetotal branching ratio when the long-distance contributions are properly included, but all thethree events are around 214MeV cannot be explained [11, 12]. If no new evidence supportthe SM explanations in the future experiments with more events, it is most likely to interpretthe three events with the existence of a new particle, X, beyond the SM. However, a newparticle explanation for the HyperCP events seems too radical because there are not earlierexperiments which observe such a 214MeV new particle. If this new light particle doesindeed exists and contributes to the hyperon decay, it may also contribute to the kaon andB-meson decays. So, the fact that lots of experiments at this low-energy region did notobserve the new 214MeV particle means that strong constraints has been imposed on thenew particle explanation for the HyperCP events.The authors of Ref. [8] proposed a argument to explain the HyperCP events for thehyperon decay with the new particle X without contradicting with the constraints on the Xfrom the low-energy experiments. As shown in Ref. [8], in addition to the flavor-changingtwo-quark contributions, there are also four-quark contributions arising from the combinedeffects of the usual SM | ∆ S | = 1 operators and the flavor-conserving couplings of X, whichare comparable with the two-quark ones and cancel sufficiently to lead to suppressed rarekaon decays rates while combining the above two kinds of contributions yields Σ → pµ + µ − rates within the required bounds.Based on the analysis in Ref. [8], the authors of Ref. [7] pointed out that a light pseu-doscalar Higgs particle A in the next-to-minimal supersymmetric standard model (NMSSM)[13, 14, 15] can be identified with X. In fact, the mass of the light pseudoscalar Higgs par-ticle A in the NMSSM can be as small as 214MeV in the large-tan β limit. Under someassumptions, it has been shown in Ref. [7] that there are regions in the parameter space2here A can satisfy the following constraints: B ( K ± → π ± A ) < ∼ . × − , B ( K s → π A ) < ∼ . × − , B ( B → X s A ) < ∼ . × − , (1)which are obtained in Ref. [8] from the measurements of the kaon and B-meson decays[16, 17, 18, 19, 20], and simultaneously explain the HyperCP events.However, the author of Ref. [7] only considered contributions from the SUSY chargedcurrent, i.e.,contributions arising from the exchanges of chargino and squark, and do notinclude contributions arising from SUSY-flavor-changing neutral currents(FCNC). It is wellknown that the SUSY-FCNC couplings can yield important(and, sometimes, even dominate)contributions to low-energy flavor physics, so further investigation on the possibility of theSUSY-FCNC mediating HyperCP events is needed. In this paper, we show that the SUSY-FCNC effects also can explain the HyperCP events and satisfy all the constraints in Eq.(1).We adopt the mass insertion method[21, 22, 23, 24] to parameterize the flavor-changingeffects and calculate SUSY-FCNC contributions to branching ratio of Σ → pA and rare kaondecays. This method introduce the super-CKM basis for the quark and squark states. Thecouplings of quarks and squarks to the neutral gauginos are flavor diagonal, while the flavor-changing SUSY effects are exhibited in the off-diagonal terms of the squark mass matrixdenoted by (∆ qij ) IJ , where I, J = L, R and i, j = 1 , , q = u, d denote the type of quark. The squark propagator is then expandedas a series of ( δ qij ) IJ = (∆ qij ) IJ / ˜ m , where ˜ m is an average squark mass. Using the massinsertion method, we can perform calculations of the SUSY-FCNC contributions to Σ → pA and rare kaon decays. Since the relevant ( δ qij ) IJ does not involve in the B-mesons decay, we donot consider the constraints from B-mesons decay. It is well known[23, 24, 25, 26, 27, 28, 29]that the parameters ( δ d ) IJ used in our calculations also yield important contributions to the K − ¯ K mixing, however, our calculations will show that the measurements of the K L − K S mass difference and the indirect CP violation observable ǫ K do not lead to more stringentconstraints than ones from kaon decays.We organize our paper as follows. In Sec. II we give a brief summary of the NMSSM.In Sec. III we calculate the two-quark flavor-changing contributions to the Σ and kaondecays arising from the SUSY-FCNC effects mediated by neutralino and gluino. In Sec. IV3e combine our two-quark contributions with four-quark contributions in Ref.[8] to give anumerical results and discussion. Feynman rules and analytical expressions for the four-quark contributions are collected in the Appendix A and B, respectively. II. NMSSM
In order to make our paper self-contained, we start with a brief description of the NMSSMand the relevant couplings considered in our paper. The superpotential of the NMSSM isgiven by[13, 14, 15] W = QY u H u U + QY d H d D + LY e H d E + λH d H u N − kN , (2)where H u and H d are the SU (2) doublet with the hypercharge 1/2 and -1/2 and are re-sponsible for the up- and down-type quark mass, respectively. The ratio of the vacuumexpectation values(VEVs) of the H u and H d is defined as tan β , which are just like thosein the the minimal supersymmetric standard model(MSSM). Compared with the MSSM,there is one more gauge-singlet Higgs Field N with the hypercharge 0 and VEV x in theNMSSM. After breaking of the supersymmetry, there are seven physical Higgs bosons inthe NMSSM, including two charged Higgs bosons, three neutral scalar and two pseudoscalarHiggs bosons.The Higgs potential of NMSSM is[30] V Higgs = V soft + V F + V D , (3)where V soft = m H d | H d | + m H u | H u | − ( λA λ H d H u N − kA k N + H . c . ) ,V F = | λ | ( | H d | + | H u | ) | N | + | λH d H u − kN | ,V D = g + g ′ | H d | − | H u | ) + g | H † u H d | . (4)The above Higgs potential has a global U (1) R symmetry in the limit of vanishing parameters A k , A λ → U (1) R symmetry is broken slightly, the lighter pseudoscalar A has a natural small mass: m A = 3 kxA λ + O (cid:18) β (cid:19) , (5)4n the large tan β limit, m A can be as low as ∼ A to the up- and down-type quarks and tothe leptons are given by [7] L A qq = − ( l u m u ¯ uγ u + l d m d ¯ dγ d ) iA v (6)and L A ℓ = ig ℓ m ℓ v ¯ ℓγ ℓA , (7)respectively, where l d = − g ℓ = v √ x (cid:18) A λ − kxA λ + kx (cid:19) , l u = l d tan β . (8)Note that l u can be neglected in the large-tan β limit. The four-quark contributions can bededuced from the interactions in Eq. (6) combined with the operators due to W exchangebetween quarks [7].It has been shown in Ref. [7] that an A of mass 214.3MeV decay dominantly to muon-antimuon pair, and B ( A → µ + µ − ) ∼ | g ℓ | < ∼ . . (9)Moreover, the neutralino sector of the NMSSM is different from that in the MSSM. Thereare five neutralinos in the NMSSM, and the Lagrangian for the mass term of the neutralinoscan be written as[15] L m χ = −
12 ( ψ ) T Y ψ + H.c., (10)where Y is the symmetric neutralino mixing matrix ,its expression can be found in Ref. [15].The masses of physical neutralinos can be obtained by diagonalizing Y by a unitary 5 × N m χ i δ ij = N ∗ im Y mn N jn . (11) III. THE SUSY-FCNC EFFECTS
The full hadronic amplitudes for the kaon decays in Eq. (1) and Σ → pA all can bewritten in the following form M full = M q + M q , (12)5 s ˜ χ i , ˜ gs dA s d ˜ χ i , ˜ gs dA ˜ d J s ˜ χ j ˜ d J ˜ s I ( e ) ( d )( c )( a ) ( b ) ˜ d J ˜ s I ˜ s I A ˜ χ i , ˜ g ˜ χ i , ˜ g s I ˜ d J ˜ s I d ˜ s I ˜ d J d J d ˜ s I ˜ χ i ˜ d J A FIG. 1: Feynman diagrams for | ∆ S | = 1 transitions, with I,J = L,R, and i,j = 1...5 where M q arises from the SUSY-FCNC interactions at the quark level, and M q arises fromthe four-quark interactions. Using the method shown in Ref. [8], we calculate the four-quarkcontributions, and their expressions and numerical results are given in Appendix B. In thissection, we mainly concentrate on the hadronic amplitudes M q from two-quark contributionand give our analytical results. In general, there are two kinds of FCNC contributions arisingfrom neutralino and gluino exchange, respectively. The relevant Feynman diagrams for the s → d transitions are shown in Fig. 1. Note that there are five kinds of neutralinos in theloops in the NMSSM. Calculating these Feynman diagrams, one can obtain the two-quarkFCNC Lagrangian for s → dA L A sd = iC L ¯ d − γ sA + iC R ¯ d γ sA + H.c. , (13)6ith C L ( R ) = C ˜ χ L ( R ) + C ˜ gL ( R ) , (14)where C ˜ χ L ( R ) and C ˜ gL ( R ) denote contributions from neutralino and gluino exchange, respec-tively. They are given by C ˜ χ L = α π X i,j =1 (cid:8) ( δ d ) LL (cid:2) I ij m d L i R ∗ j − I i m d L i R ∗ i + I i m d ( m s L i R ∗ i − L i R ∗ i ) (cid:3) +( δ d ) LR (cid:2) I ij m d m s L i R ∗ j − I i m d m s L i R ∗ i − I i m d m s ( L i R ∗ i + L i R ∗ i ) (cid:3) +( δ d ) RL (cid:2) I ij L i R ∗ j − I i L i R ∗ i + I i ( m d L i R ∗ i + m s L i R ∗ i ) (cid:3) +( δ d ) RR (cid:2) I ij m s L i R ∗ j − I i m s L i R ∗ i + I i m s ( m d L i R ∗ i − L i R ∗ i ) (cid:3)(cid:9) ,C ˜ χ R = α π X i,j =1 (cid:8) ( δ d ) LL (cid:2) I ij m s R i L ∗ j + I i m s R i L ∗ i + I i m s ( R i L ∗ i − m d R i L ∗ i ) (cid:3) +( δ d ) LR (cid:2) I ij R i L ∗ j + I i R i L ∗ i − I i ( m d R i L ∗ i + m s R i L ∗ i ) (cid:3) +( δ d ) RL (cid:2) I ij m d m s R i L ∗ j + I i m d m s R i L ∗ i + I i m d m s ( R i L ∗ i + R i L ∗ i ) (cid:3) +( δ d ) RR (cid:2) I ij m d R i L ∗ j + I i m d R i L ∗ i + I i m d ( R i L ∗ i − m s R i L ∗ i ) (cid:3)(cid:9) ,C ˜ gL = α s π C F m ˜ g [2 V Add C ( y ˜ g )( δ d ) RL + V A ˜ d ˜ d m d D ( y ˜ g )( m d ( δ d ) LL + m s ( δ dsd ) RR )] ,C ˜ gR = α s π C F m ˜ g [2 V Add C ( y ˜ g )( δ d ) LR − V A ˜ d ˜ d m d D ( y ˜ g )( m d ( δ d ) RR + m s ( δ dsd ) LL )] , (15)with V Add = g m W cos β U P ,V A ˜ d ˜ d = g m W cos β [ λ ( v U P + xU P ) − A D U P ] , (16) L i = 2 N ∗ i, √ c W ,L i = N ∗ i, √ m W s W cos β ,R i = L ∗ i ,R i = s W N i, − c W N i, √ c W s W , (17) I ij = D ( y i , y j ) R R ′′ ij + m ˜ χ i m ˜ χ j m d D ( y i , y j ) R L ′′ ij ,I ij = D ( y i , y j ) R L ′′ ij + m ˜ χ i m ˜ χ j m d D ( y i , y j ) R R ′′ ij ,I i = 2 V Add m ˜ χ i C ( y i ) , i = V A ˜ d ˜ d m ˜ χ i m d D ( y i ) , (18) y ˜ g = m g m d , y i = m χ i m d , i = 1 ... , (19)where U P is used to diagonalize the pseudoscalar Higgs mass matrices, the matrix N ij , i, j =1 · · · × c W = cos θ W , where θ W is the Weinbergangle as usual, and R ′′ R ij = 12 (cid:2) ( U P cos β + U P sin β ) × (cid:18) gc W ( N i N ∗ j + N j N ∗ i ) − √ λ ( N i N ∗ j + N j N ∗ i ) (cid:19) +( U P sin β − U P cos β ) (cid:18) gc W ( N i N ∗ j + N j N ∗ i ) + √ λ ( N i N ∗ j + N j N ∗ i ) (cid:19)(cid:21) −√ kU P ( N i N ∗ j + N j N ∗ i ) ,R ′′ R ij = − R ′′ L ∗ ij . (20)And the loop functions C ( x ), D ( x ) and D ( x, y ) are defined as C ( x ) = x − − x log( x )(1 − x ) ,D ( x ) = 1 − x + 2 x log( x )2(1 − x ) ,D ( x, y ) = − − x )(1 − y ) − x log( x )(1 − x ) ( x − y ) − y log( y )(1 − y ) ( y − x ) . (21)The quark level effective Lagrangian in Eq.(13) can be mapped onto the chiral Lagrangianat the leading order[8] L A = b D h ¯ B { h A , B }i + b F h ¯ B [ h A , B ] i + b h h A ih ¯ BB i + 12 f π B h h A i + H.c ., (22)where B = √ Σ + √ Λ Σ + p Σ − − √ Σ + √ Λ n Ξ − Ξ − √ Λ represents the baryon fields, f π = 92 . h· · · i ≡ Tr( · · · ) inflavor- SU (3). And h A = − i ( C R ξ † hξ † + C L ξhξ ) A (23)8here h = T + iT = is used to specify the s → d transition, ξ = e iπ/f π , Σ = ξξ = e iπ/f π , (24)and π = 1 √ √ π + √ η π + K + π − − √ π + √ η K K − ¯ K − √ η is the pion octet.The two-quark amplitude M q can be deduced from L A plus the usual chiral Lagrangian L s for the strong interactions of hadrons, which is expressed as[32, 33, 34, 35] L s = i h ¯ Bγ µ D µ B i − m h ¯ BB i + D h ¯ Bγ µ γ { A µ , B }i + F h ¯ Bγ µ γ [ A µ , B ] i + b D h ¯ B { M + , B }i + b F h ¯ B [ M + , B ] i + b h M + ih ¯ BB i + 14 f π h ∂ µ Σ † ∂ µ Σ i + 12 f π B h M + i , (25)with D µ B = ∂ µ B + [ V µ , B ] ,A µ = i ξ∂ µ ξ † − ξ † ∂ µ ξ ) , M + = ξ † M ξ † + ξM † ξ,V µ = 12 ( ξ∂ µ ξ † + ξ † ∂ µ ξ ) , where M = diag( ˆ m, ˆ m, m s ) is the quark mass matrix in the m u = m d = ˆ m limit.Using the mass relations m Σ − m p = 2( b D − b F )( m s − ˆ m ), m K − m π = B ( m s − ˆ m ) and m K = B ( m s + ˆ m ) , the amplitudes for the different decay modes can be written as [8] M q ( K + → π + A ) = −√ M q ( K → π A )= i (cid:18) C L + C R (cid:19) B , (26) M q (Σ + → pA ) = i (cid:18) C L + C R (cid:19) B ( m Σ − m p ) m K − m π ¯ p Σ + + i ( D − F ) (cid:18) C L − C R (cid:19) B ( m Σ + m p ) m K − m A ¯ pγ Σ + , (27)where B = 2031MeV[4]. 9 V. NUMERICAL RESULTS AND DISCUSSION
In this section we present our numerical results. The Higgs sector of NMSSM is describeby the six independent parameters λ, k, A λ , A k , tan β, µ, where µ = − λx . For convenience, we will take m A and the coupling of down-type quarks to A , l d instead of k and λ . We follow Ref.[7] to set l d = 0 . − λx = 150GeV and tan β = 30. A k and A λ are set as 0.001 and 0.002, respectively. The mass of A is set as 214 . ∼ C ˜ χ L,R arelarger than C ˜ gLR in most regions of parameter space. This is due to the effects of α s .Our numerical results are shown in Figs.3 - 7. We first assume that ( δ ) IJ are real.The allowed regions in parameter space are shown in Figs.3 - 5, where the grey areas arethe allowed regions for A to explain the HyperCP events. When the constraints obtainedfrom the kaon decays are considered, the allowed parameter space are greatly reduced tothe dark regions. From Figs.3 - 5, the constraints on the combinations of the parameterscan be obtained as following:( δ d ) LL ( RR ) ( δ d ) LR ( RL ) ≤ . × − (without kaon bounds) , . × − (with kaon bounds) , ( δ d ) LL ( δ d ) RR ≤ . × − (without kaon bounds) , . × − (with kaon bounds) , ( δ d ) LR ( δ d ) RL ≤ . × − (without kaon bounds) , . × − (with kaon bounds) . (28)It has been widely studied in the literature that the SUSY-FCNC effects has great impacton the K − ¯ K mixing if the relevant ( δ d ) IJ are complex. So we further investigate thepossible constraints on ( δ d ) IJ from the K L − K S mass difference ∆ m K and indirect CPviolation parameter ǫ K . However, the constraints shown in Eq.(28) are roughly several orderssmaller than those given in the literatures involving the SUSY-FCNC mediated K − ¯ K δ d ) LL and ( δ d ) LR ( RL ) are around O (10 − ∼ − )and O (10 − ∼ − ), respectively. This fact indicates that the constraints from the K − ¯ K mixing may be automatically satisfied once the constraints from Σ → pA and the rare kaondecays in Eq.(1) are satisfied. Our numerical results do indeed confirm that the constraintsfrom the K − ¯ K mixing do not lead to more stringent constraints than those ones givenfrom the kaon decays in Eq.(1) .Figs.6 and 7 show the constraints on the complex ( δ d ) IJ from Σ → pA and rare kaondecays in Eq.(1). And the corresponding constraints on the combinations of parameters aregiven by Re( δ d ) LR ( RL ) Im( δ d ) LR ( RL ) ≤ . × − (without kaon bounds) , . × − (with kaon bounds) , Re( δ d ) LL ( RR ) Im( δ d ) LL ( RR ) ≤ . × − (without kaon bounds) , . × − (with kaon bounds) . (29)From Figs.6 and 7, it can be seen that the grey areas are the allowed regions of the SUSY-FCNC parameters for A to explain the HyperCP events and the grey regions are greatlyreduced to the dark ones when the constraints from rare kaon decays are considered. Evenso, there are still regions in the SUSY-FCNC parameter space where A in the NMSSM canbe used to explain the HyperCP events without contradicting with the constraints from therare kaon decays and the K − ¯ K mixing.In conclusion, we have calculated the two-quark contributions to the decay Σ + → pµ + µ − arising from the transition s → dA via the SUSY-FCNC couplings. Combining the two-quark contributions with the four-quark contributions, we show that there are regions inthe SUSY-FCNC parameter space where the A in the NMSSM can be identified with anew particle of mass 214.3MeV, X, which can be used to explain the HyperCP events, whilesatisfying all the constraints from the measurements of the rare kaon decays. And once theconstraints from the kaon decays are satisfied, the constraints from the K − ¯ K mixing areautomatically satisfied. 11 cknowledgments This work was supported in part by the National Natural Science Foundation of China,under Grant No. 10421503, No. 10575001 and No. 10635030, and the Key Grant Project ofChinese Ministry of Education under Grant No. 305001.
Appendix A
We give the Feynman rules used in our calculations in Fig.2, where V A ˜ d ˜ d , V Add , L ( R ) i and R L ( R ) ′′ ij are defined in Eq. (16), Eq. (17) and Eq. (20), respectively. And δ IJ = I = J, I = J. Appendix B
We collect expressions of the four-quark amplitudes for the different decay modes in thisappendix. A detailed description can be found in Ref. [8], we cite their results here. Aspointed in Ref. [7], the couplings of A to the up-type quarks tends to zero at the limit oflarge tan β , so we neglect terms that are proportional to l u .The four-quark contributions for the kaon decays are as follow: M q ( K + → π + A ) = i l d γ v (cid:26) − m π m K + m π − m A ) c θ − √ m K − m π ) s θ ] × (4 m K − m π ) c θ + √ m K − ˜ m ) s θ m η − m A ) + h (2 m K + m π − m A ) s θ + √ m K − m π ) c θ i (4 m K − m π ) s θ − √ m K − ˜ m ) c θ m η ′ − m A ) ) , (30) M q ( K → πA ) = i l d γ v ( − (2 m K − m π − m A ) m π √ m A − m π )+[(2 m K + m π − m A ) c θ − √ m K − m π ) s θ ] × (4 m K − m π ) c θ + √ m K − ˜ m ) s θ √ m A − m η ) + h (2 m K + m π − m A ) s θ + √ m K − m π ) c θ i (4 m K − m π ) s θ − √ m K − ˜ m ) c θ √ m A − m η ′ ) ) , (31)12 χ i ˜ d I d ˜ d I ˜ d J A A d d m d V A ˜ d ˜ d ( δ IL δ JR − δ IR δ JL )˜ d I ˜ d J ˜ d I − i (∆ d ) IJ = − i ( δ d ) IJ m d − ie [( L i δ IR + m d L i δ IL ) P L − m d V Add γ + ( m d R i δ IR + R i δ IL ) P R i √ g s ( δ IR P L − δ IL P R ) T Aba ˜ g ˜ d aI d b R L ′′ ij P L + R R ′′ ij P R A ˜ χ i ˜ χ j FIG. 2: Feynman rules used in our paper. where γ = − . × − , s θ and c θ are short for sin θ and cos θ , θ = − . ◦ . And, thefour-quark contributions to the Σ → pA process are M q (Σ + → pA ) = i ¯ p ( A pA − B pA γ )Σ + , (32)with A pA = l d f π v A pπ ( m π m A − m π + (4 m K − m π ) c θ + √ m K − ˜ m ) c θ s θ m η − m A + (4 m K − m π ) s θ − √ m K − ˜ m ) c θ s θ m η ′ − m A ) (33)13nd B pA = l d f π v B pπ ( m π m A − m π + (4 m K − m π ) c θ + √ m K − ˜ m ) c θ s θ m η − m A + (4 m K − m π ) s θ − √ m K − ˜ m ) c θ s θ m η ′ − m A ) , (34)where A pπ = − . × − , B pπ = 26 . × − .Numerically, the above amplitudes are M q (Σ → pA ) = i ¯ p ( − . × − l d f π v − (5 . × − ) l d f π v γ )Σ + , M q ( K + → π + A ) = − i . × − l d m K v , M q ( K → πA ) = i . × − l d m K v . (35) [1] D. S. Gorbunov and V. A. Rubakov, Phys. Rev. D73 , 035002 (2006), hep-ph/0509147.[2] C. Q. Geng and Y. K. Hsiao, Phys. Lett.
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B375 , 561 (1992).[35] E. E. Jenkins and A. V. Manohar (1991), talk presented at the Workshop on Effective FieldTheories of the Standard Model, Dobogoko, Hungary, Aug 1991. -5 · ( RR ) d12 d = ( LL ) d12 d (-1.5 -1 -0.5 0 0.5 1 1.5 2 ) - · ( R L ) d d = ( L R ) d d ( -1-0.500.511.5 FIG. 3: The allowed values of ( δ d ) LR = ( δ d ) RL as a function of ( δ d ) LL = ( δ d ) RR . The greyarea is the regions where A can explain the HyperCP events, when the constraints from rare kaondecays are considered, the allowed regions are greatly reduced to the dark ones. ) -3 · ( RR ) d12 d (-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ) - · ( LL ) d d ( -0.6-0.5-0.4-0.3-0.2-0.100.10.2 FIG. 4: The allowed values of ( δ d ) LL as a function of ( δ d ) RR , The grey area is the regions where A can explain the HyperCP events, when the constraints from rare kaon decays are considered,the allowed regions are greatly reduced to the dark ones. -7 · ( LR ) d12 d (-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) - · ( R L ) d d ( -0.8-0.6-0.4-0.2-00.20.40.6 FIG. 5: The allowed values of ( δ d ) RL as a function of ( δ d ) LR , The grey area is the regions where A can explain the HyperCP events, when the constraints from rare kaon decays are considered,the allowed regions are greatly reduced to the dark ones. ) -8 · ) ( RL ) d12 d = ( LR ) d12 d Re((-0.5 0 0.5 1 1.5 ) - · ) ( R L ) d d = ( L R ) d d I m (( -1-0.500.51 FIG. 6: The allowed values of Im { ( δ d ) LR = ( δ d ) RL } as a function of Re { ( δ d ) LR = ( δ d ) RL } . Thegrey regions denotes the survival regions for explaining the HyperCP events alone; In the darkregions, the A can explain the HyperCP events and simultaneously satisfy the bounds originatekaon decays and the K − ¯ K mixing. -6 · ) ( RR ) d12 d = ( LL ) d12 d Re((-2 -1.5 -1 -0.5 0 0.5 ) - · ) ( RR ) d d = ( LL ) d d I m (( -1.5-1-0.500.511.5 FIG. 7: The allowed values of Im { ( δ d ) LL = ( δ d ) RR } as a function of Re { ( δ d ) LL = ( δ d ) RR } . Thegrey regions denotes the survival regions for explaining the HyperCP events alone; In the darkregions, the A can explain the HyperCP events and simultaneously satisfy the bounds originatekaon decays and the K − ¯ K mixing.mixing.