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II. THEORETICAL FRAMEWORK

The next-to-leading order (NLO) effective Hamiltonian for s dνν¯ reads [13]: → GF α ∗ l ∗ eff = [V VcdX + V VtdX(xt)](¯sd)V −A(¯νlνl)V −A + h.c., (5) H √ 2π sin2 θ cs NL ts 2 W l=Xe,µ,τ

l where X(xt) and XNL are relevant for the top and the charm contribution, respectively. Their explicit expressions 2 2 can be found in Ref. [13]. Here, xt = mt /mW . To leading order in αs, the function X(xt) relevant for the top contribution reads [14, 15] α X(x)= X (x)+ s X (x), 0 4π 1 x 2+ x 3x 6 X (x)= + − ln x , 0 8 −1 x (1 x)2  − − 23x +5x2 4x3 x 11x2 + x3 + x4 8x +4x2 + x3 x4 X (x)= − + − ln x + − ln2 x 1 − 3(1 x)2 (1 x)3 2(1 x)3 − − − 4x x3 ∂X (x) − L (1 x)+8x 0 ln x , (6) − (1 x)2 2 − ∂x µ − where x = µ2/M 2 with µ = (m ) and µ W O t x ln t L2(1 x)= dt . (7) − Z 1 t 1 − l The function XNL corresponds to X(xt) in the charm sector. It results from the renormalization group (RG) calcu- lation in next-to-leading-order logarithmic approximation (NLLA) and is given as follows:

Xl = C 4B(1/2), (8) NL NL − NL (1/2) 0 where CNL and BNL correspond to the Z -penguin and the box-type contribution, respectively, given as [16]

24 x(mc) 25 48 24 696 4π 15212 −1 CNL = Kc K+ + K− K33 + (1 Kc ) 32  7 11 − 77  αs(µ) 1875 −  µ2 1176244 2302 3529184 + 1 ln 2 (16K+ 8K−) K+ K− + K33  − mc  − − 13125 − 6875 48125 56248 81448 4563698 + K K K− + K ,  4375 + − 6875 144375 33

24 (1/2) x(mc) 25 4π 15212 −1 BNL = Kc 3(1 K2) + (1 Kc ) 4  −  αs(µ) 1875 −  µ2 r ln r 305 15212 15581 ln + K + KK , (9) − m2 − 1 r − 12 625 2 7500 2 c − where r = m2/m2(µ), µ = (m ) and l c O c αs(MW ) αs(µ) 6 − 12 − 1 K = ,Kc = ,K+ = K 25 ,K− = K 25 ,K33 = K2 = K 25 . (10) αs(µ) αs(mc) In the following, we consider the transitions between the baryon octet (Ξ, Σ, Λ and N) and the transitions from the baryon decuplet to the octet Ω− Ξ−. The transition matrix elements of→ the vector and axial-vector currents between the baryon octets can be 2 2 parametrized in terms of six form factors f1,2,3(q ) and g1,2,3(q ): qν q B′ (P ′,S′ ) dγ¯ (1 γ )s B (P,S ) =u ¯(P ′,S′ ) γ f (q2)+ iσ f (q2)+ µ f (q2) u(P,S ) h 8 z | µ − 5 | 8 z i z  µ 1 µν M 2 M 3  z qν q u¯(P ′,S′ ) γ g (q2)+ iσ g (q2)+ µ g (q2) γ u(P,S ), (11) − z  µ 1 µν M 2 M 3  5 z 3

TABLE I: The form factors for B → B′ transition f1(0), f2(0) and g1(0) [17], where the experimental anomalous magnetic moments are κp = 1.793 ± 0.087 and κn = −1.913 ± 0.069 [18], with the two coupling constants F = 0.463 ± 0.008 and D = 0.804 ± 0.008 [18]. Here, g1/f1 is positive for the neutron decay, and all other signs are fixed using this sign convention.

+ 0 0 0 B → B′ Λ → n Σ → p Ξ → Λ Ξ → Σ Ξ− → Σ− 3 3 1 f1(0) −q 2 −1 q 2 - √2 1 3 3 1 f2(0) −q 2 κp −(κp + 2κn) q 2 (κp + κn) − √2 (κp − κn) κp − κn 3 3 1 g1(0) −q 2 (F + D/3) −(F − D) q 2 (F − D/3) − √2 (F + D) F + D

′ ′ where q = P P , and M denotes the mass of the parent baryon octet B8. The form factors for B8 B8 transition 2 −2 → fi(q ) and gi(q ) can be expressed by the following formulas [17]:

2 2 2 2 fm = aFm(q )+ bDm(q ), gm = aFm+3(q )+ bDm+3(q ), (m =1, 2, 3), (12)

2 2 2 where Fi(q ) and Di(q ) with i = 1, 2, , 6 are different functions of q for each of the six form factors. Some remarks are necessary [17]: ··· The constants a and b in Eq. (12) are SU(3) Clebsch-Gordan coefficients that appear when an octet operator is • sandwiched between octet states. For q2 = 0, the form factor f (0) is equal to the electric charge of the baryon, therefore F (0) = 1 and D (0) = 0. • 1 1 1 The weak f2(0) form factor can be computed using the anomalous magnetic moments of proton and neutron • (κ and κ ) in the exact SU(3) symmetry. Here F (0) = κ + 1 κ and D (0) = 3 κ . p n 2 p 2 n 2 − 2 n g (0) is a linear combination of two parameters, F and D. • 1 − 0 Since gn→p = F (q2)+ D (q2) = 0 and gΞ →Ξ = D (q2) F (q2) = 0, we could get F (q2) = D (q2) = 0. • 2 5 5 2 5 − 5 5 5 Therefore, all the pseudo-tensor form factors g2 vanish in all decays up to symmetry-breaking effects.

In the s dνν¯ decay, the f3 and g3 terms are proportional to the neutrino mass and thus can be neglected for • the decays→ considered in this work. Since the invariant mass squared of lepton pairs in the hyperon decays is relatively small, it is expected that the 2 q distribution in the form factors has small impact on the decay widths. We list the expressions for f1, f2 and g1 at q2 = 0 in TAB. I. Hence, Eq. (11) can be rewritten as:

qν B′ (P ′,S′ ) dγ¯ (1 γ )s B (P,S ) =u ¯(P ′,S′ ) γ f (q2)+ iσ f (q2) γ g (q2)γ u(P,S ). (13) h 8 z | µ − 5 | 8 z i z  µ 1 µν M 2 − µ 1 5 z

The helicity amplitudes of hadronic part are defined as

V ′ ′ ′ µ ∗ H ′ B (P , λ ) dγ¯ s B (P, λ) ǫ (λ ), (14) λ ,λV ≡ h 8 | | 8 i V µ V A ′ ′ ′ µ ∗ H ′ B (P , λ ) dγ¯ γ s B (P, λ) ǫ (λ ). (15) λ ,λV ≡ h 8 | 5 | 8 i V µ V (′) Here λ denotes the helicity of the parent (daughter) baryon in the initial (final) state, and λV is the helicity of the V,A ′ virtual intermediate vector particle. It can be shown that the helicity amplitudes Hλ ,λV have the following simple forms [19]:

2 V Q− ′ q A Q+ ′ H 1 0 = i (M + M )f1 f2 , H 1 0 = i (M M )g1, 2 , − p q2  − M  2 , − p q2 − p ′ p V M + M A H 1 1 = i 2Q− f1 + f2 , H 1 1 = i 2Q+g1. (16) 2 , − M  2 , − p p 4

′ 2 2 ′ In the above, Q± = (M M ) q , and M (M ) is the parent (daughter) baryon mass in the initial (final) state. The amplitudes for the negative± helicity− are obtained from the relations,

V V A A H ′ = H ′ , H ′ = H ′ . (17) −λ ,−λV λ ,λV −λ ,−λV − λ ,λV The complete helicity amplitudes are obtained by

V A H ′ = H ′ H ′ . (18) λ ,λV λ ,λV − λ ,λV

Due to the lack of experimental data for the M1 and E2 transitions from the baryon decuplet to the octet, the vector transition matrix element for Ω− Ξ− can not be determined. In this work we follow Ref. [18]. and consider only the axial-vector current matrix element→ [18, 20, 21]:

− ′ ′ − ′ ′ A 2 A 2 Ξ (P ,S ) dγ¯ γ s Ω (P,S ) =u ¯ − (P ,S ) C (q )g + C (q )q q h z | µ 5 | z i Ξ z 5 µν 6 µ ν A 2  α A 2 ′ ν + C (q )γ + C (q )p (q g q g ) u − (P,S ). (19) 3 4 α µν − ν αµ Ω z   ν − 3 Here, uΩ− (P,Sz) represents the Rarita-Schwinger spinor that describes the baryon decuplet Ω with spin 2 . In A 2 A 2 Ref. [22] it is shown that C3 (q ) and C4 (q ) are proportional to the mass difference of the initial and final baryons, A 2 A 2 A 2 2 A 2 2 and thus are suppressed. In the chiral limit, C5 (q ) and C6 (q ) are related by C6 (q )= MN C5 (q )/q [20]. In our A − − − 0 calculations, we use C5 (0) = 1.653 0.006 for Ω Ξ , which is the same as Ω Ξ in the SU(3) limit [18]. The helicity amplitude can be expressed± as: → →

A − ′ ′ − ∗µ H ′ = Ξ (P , λ ) dγ¯ γ s Ω (P, λ) ǫ (λ ) (20) λ ,λV h | µ 5 | i V V ′ ′ A 2 A 2 ν ∗µ − =u ¯Ξ (P , λ ) C5 (q )gµν + C6 (q )qµqν uΩ− (P, λ)ǫV (λV ). (21)   (′) A ′ Here, λ and λV have the same definition as in Eqs.(14)-(15). It can be shown that the helicity amplitudes Hλ ,λV have the following simple forms [19]:

A A 2Q+ EV A 2 A A Q+ A 2 A A A 2 H 1 0 = H− 1 0 = i C5 (q ), H 1 1 = H− 1 −1 = i C5 (q ), H 1 −1 = H− 1 1 = i Q+C5 (q ). (22) 2 , 2 , r 3 2 2 , 2 , r 3 2 , 2 , q p p The differential decay width for B B′νν¯ is given as: → dΓ dΓ dΓ = L + T . (23) dq2 dq2 dq2

2 2 Here, dΓL/dq and dΓT /dq are the longitudinal and transverse parts of the decay width, and their explicit expressions are given by

2 ′ dΓL q p 2 2 = N ( H 1 0 + H− 1 0 ), (24) dq2 12(2π)3M 2 | 2 , | | 2 , | 2 ′ dΓT q p 2 2 2 2 = N ( H 1 1 + H− 1 −1 + H 1 −1 + H− 1 1 ). (25) dq2 12(2π)3M 2 | 2 , | | 2 , | | 2 , | | 2 , |

′ ′ In Eqs. (24) and (25), p = Q+Q−/2M is the magnitude of the momentum of B in the rest frame of B, and N =2N1(0) + N1(mτ ) with p

2 GF α ∗ l ∗ N1(ml)= V VcsX (ml)+ V VtsX(xt) . (26) √2 2π sin2 Θ cd NL td W

Note that we have neglected the electron and muon masses. One can then obtain the decay width

(M−M ′)2 2 dΓ Γ= dq 2 . (27) Z0 dq 5

TABLE II: The input parameters of this work.

The masses and lifetimes of baryons in the initial and final states[23]

mp = 938.2720813MeV mΣ+ = 1189.37MeV mΞ0 = 1314.86MeV

mn= 939.5654133MeV mΣ− = 1197.45MeV mΞ− = 1321.71MeV

mΛ = 1115.683MeV mΣ0 = 1192.642MeV mΩ− = 1672.45MeV 10 10 10 τΞ0 = 2.90 × 10− s τΞ− = 1.639 × 10− s τΩ− = 0.821 × 10− s 10 10 τΛ = 2.632 × 10− s τΣ+ = 0.8018 × 10− s Physical constants and CKM Parameters[23, 24] 5 2 2 GF = 1.16637387 × 10− GeV− sin θW = 0.23122 αs(mZ )= 0.1182 α ≡ α(mZ )= 1/128

mτ =1776.86MeV mc = 1.275GeV mt = 173.0GeV mW =80.379 GeV mZ= 91.1876GeV A = 0.836 λ = 0.22453 ρ¯ = 0.122 η¯ = 0.355

III. NUMERICAL RESULTS AND DISCUSSION

A. Calculations in SM

With the inputs parameters given in TAB. II and the formulae from the last section, the LO and NLO results for µc = 1 GeV,µt = 100 GeV and µc = 3 GeV,µt = 300 GeV are listed in TAB. III. From the results in TAB. III one can see that: The branching fractions of s dνν¯ rare hyperon decays range from 10−14 to 10−11. • → For µ = 1GeV,µ = 100GeV, the NLO results are smaller than the LO ones by about 30%, while for µ = • c t c 3GeV,µt = 300GeV the NLO results are larger than the LO ones by about 10%.

The LO results vary by about 50% from µc = 1GeV,µt = 100GeV to µc = 3GeV,µt = 300GeV, while the NLO • ones vary by about 30%. As expected, the NLO results depend less on the mass scales. The branching ratio of Ω− Ξ−νν¯ is the largest among the 6 channels. It is of the same order as for K+ π+νν¯ • and K π0νν¯. → → L → At present, there is a small number of experimental studies, and thus most experimental constraints are less severe. The prospects for rare and forbidden hyperon decays at BESIII were analyzed in a recent publication Ref.[12]. We quote the experimental sensitivity for all decay modes in TAB. III. Unfortunately, one can see that the current BESIII experiment will not be able to probe these hyperon decays. We hope this may be improved at future experimental facilities like the Super Tau-Charm Factory.

B. Form factors Uncertainties

Note that due to the Ademollo-Gatto theorem [25], the form factor f1(0) does not receive any SU(3) symmetry breaking correction. However, f2(0) can be computed using the anomalous magnetic moments of proton and neutron (κp and κn) in the exact SU(3) symmetry. The experimental data for κp and κn already include the SU(3) symmetry breaking effects [18]:

κ (m0) =1.363 0.069, κ (m0) = 1.416 0.049, (28) p O s ± n O s − ± κ (m0)+ (m1) =1.793 0.087, κ  (m0)+ (m1) = 1.913 0.069. (29) p O s O s ± n O s O s − ±     The uncertainties from κp and κn in the effect of SU(3) symmetry breaking is approximately 25%. We calculated the effect of κ and κ on the branching ratio of Σ+ pνν¯ in the case of NLO with the energy scale µ = 1GeV and p n → c µt = 100GeV such that:

(Σ+ pνν¯) (m0) = (4.86 0.04) 10−13, (Σ+ pνν¯) (m0)+ (m1) = (5.01 0.08) 10−13. (30) B → O s ± × B → O s O s ± ×     6

TABLE III: The LO, NLO, NLO+SUSY and NLO+M331 results for the branching ratio of rare hyperon decays for µc = 1 GeV, µt = 100 GeV and µc = 3 GeV, µt = 300 GeV.

+ 0 0 0 Branching ratio B(Λ → nνν¯) B(Σ → pνν¯) B(Ξ → Λνν¯) B(Ξ → Σ νν¯) B(Ξ− → Σ−νν¯) B(Ω− → Ξ−νν¯) 12 13 12 13 13 11 LO 2.85 × 10− 6.88 × 10− 1.06 × 10− 1.77 × 10− 2.17 × 10− 1.78 × 10− µc = 1 GeV 12 13 13 13 13 11 NLO 1.98 × 10− 5.01 × 10− 7.35 × 10− 1.24 × 10− 1.52 × 10− 1.93 × 10− 12 12 12 13 13 11 NLO+SUSY (Set.I) 8.14 × 10− 2.06 × 10− 3.02 × 10− 5.08 × 10− 6.23 × 10− 7.94 × 10− 12 13 12 13 13 11 µt = 100 GeV NLO+SUSY (Set.II) 3.78 × 10− 9.55 × 10− 1.40 × 10− 2.36 × 10− 2.89 × 10− 3.69 × 10− 11 12 12 13 13 10 NLO+M331 1.24 × 10− 3.13 × 10− 4.59 × 10− 7.71 × 10− 9.45 × 10− 1.20 × 10− 12 13 13 14 14 11 LO 1.10 × 10− 2.65 × 10− 4.10 × 10− 6.83 × 10− 8.37 × 10− 1.07 × 10− µc = 3 GeV 12 13 13 14 14 11 NLO 1.20 × 10− 3.04 × 10− 4.46 × 10− 7.50 × 10− 9.19 × 10− 1.17 × 10− 12 12 12 13 13 11 NLO+SUSY (Set.I) 5.85 × 10− 1.48 × 10− 2.17 × 10− 3.65 × 10− 4.47 × 10− 5.71 × 10− 12 13 13 13 13 11 µt = 300 GeV NLO+SUSY (Set.II) 2.35 × 10− 5.94 × 10− 8.72 × 10− 1.47 × 10− 1.80 × 10− 2.29 × 10− 11 12 12 13 13 11 NLO+M331 1.02 × 10− 2.58 × 10− 3.80 × 10− 6.37 × 10− 7.81 × 10− 9.95 × 10− 7 7 7 7 5 BESIII sensitivity [12] 3 × 10− 4 × 10− 8 × 10− 9 × 10− −− 2.6 × 10−

Next, we consider the uncertainty of the branching ratio of Σ+ pνν¯ and Λ nνν¯ in the case of NLO with → → the energy scale µc = 1GeV and µt = 100GeV. This uncertainty comes from the parameters F =0.463 0.008 and D =0.804 0.008 [18] in the form factor g (0): ± ± 1 (Σ+ pνν¯)=(5.01 0.12) 10−13, (Λ nνν¯)=(2.03 0.05) 10−12. (31) B → ± × B → ± × For the decay Ω− Ξ−νν¯, CA(0) = 1.653 0.006 in the SU(3) symmetry, while CA(0) = 1.612 0.007 in the SU(3) → 5 ± 5 ± symmetry breaking [18]. In the case of NLO with the energy scale µc = 1 GeV and µt = 100 GeV the branching ratio (Ω− Ξ−νν¯) is then calculated as: B → (Ω− Ξ−νν¯)(sy)=(1.84 0.01) 10−11, (Ω− Ξ−νν¯)(br)=(1.93 0.01) 10−11. (32) B → ± × B → ± × As an illustration of the effects of q2 distribution in the form factors, we attempt to use the following parametrization for all form factors:

2 F (0) F (q )= q2 , (33) 1 2 − m with m representing the initial hyperon mass. For example, for the NLO case of µc = 1 GeV and µt = 100 GeV, we obtain: (Λ nνν¯)(F (0)) = 1.98 10−12, (Λ nνν¯)(F (q2))=2.03 10−12, B → × B → × (Σ+ pνν¯)(F (0)) = 5.05 10−13, (Σ+ pνν¯)(F (q2))=5.16 10−13. (34) B → × B → × We find that the differences between the two cases are small, about a few percent. When all the above errors in the form factors are included, we find that the final branching ratios for most decay modes have an uncertainty of about 5% to 10%.

C. Contribution from MSSM

The effective Hamiltonian for s dνν¯ in the generalized supersymmetry (SUSY) extension of SM is given in Eq. → (5), with X(xt) replaced by [26]

Xnew = X(xt)+ XH (xtH )+ Cχ + CN . (35) 2 2 Here, xtH = mt /mH± , and XH (xtH ) corresponds to the charged Higgs contribution. Cχ and CN denote the chargino and neutralino contributions

0 LL U LR U LR∗ U Cχ = Xχ + Xχ RsLdL + Xχ RsLtR + Xχ RtRdL , D CN = XN RsLdL , 7

U U TABLE IV: Upper limits for the R parameters. Notice that the phase of RsLtR and RtRdL is unconstrained.

quantity upper limit m D d˜L RsLdL (−112 − 55i) 500GeV U mu˜L RsLdL (−112 − 54i) 500GeV 3 U mu˜L iφ RsLtR Min{231  500GeV  , 43}× e , 0 <φ< 2π 2 U mu˜L iφ RtRdL 37  500GeV  × e , 0 <φ< 2π

i where Xχ and XN depend on the SUSY masses, and respectively on the chargino and neutralino mixing angles. The explicit expressions for XH (x), Cχ and CN can be found in Ref. [26]. The R parameters are defined in terms of mass U insertions, and their upper limits are listed in TAB. IV [26]. It should be mentioned that the phase φ of RsLtR and U RtRdL is a free parameter which ranges from 0 to 2π. We set φ = 0 as a central result. The parameters in TAB. V are adopted for detailed calculations [27]. The assumption M1 0.5M2 was made [28]. With the above parameters, the branching ratios of hyperon decays are listed in TAB. III,≈ and are significantly

TABLE V: Parameters and their ranges used in the paper[27]. All mass parameters are in GeV. parameters[27] the meaning of parameters[27] the range of parameters[27] Set.I[27] Set.II[27]

β The angle of unitarity triangle −180◦ ≤ β ≤ 180◦ tan β = 2 tan β = 20

MA CP-odd Higgs boson mass 150 ≤ MA ≤ 400 333 260 M2 SU(2) gaugino mass; we use M1 GUT-related to M2 50 ≤ M2 ≤ 800 181 750 µ Supersymmetric Higgs mixing parameter −400 ≤ µ ≤ 400 −375 −344

Msl Common flavour diagonal slepton mass parameter 95 ≤ Msl ≤ 1000 105 884

Msq Common mass parameter for the first two generations of squarks 240 ≤ Msq ≤ 1000 308 608

Mt˜L Squark mass parameter for the right stop 50 ≤ Mt˜R ≤ 1000 279 338 enhanced compared with the SM results. Taking as examples the decays Λ nνν¯ and Ω− Ξ−νν¯ with the energy → → scale µc = 1 GeV and µt = 100 GeV, we obtain:

NLO : (Λ nνν¯)=1.98 10−12, (Ω− Ξ−νν¯)=1.93 10−11, (36) B → × B → × Set.I: (Λ nνν¯)=8.14 10−12, (Ω− Ξ−νν¯)=7.94 10−11, (37) B → × B → × Set.II: (Λ nνν¯)=3.78 10−12, (Ω− Ξ−νν¯)=3.69 10−11. (38) B → × B → × Comparing the results of NLO+SUSY (Set. I) and (Set. II) with the ones of NLO, we see that all the branching ratios are roughly enhanced by a factor of 4 and 2, respectively. However, none of these results can be probed at the ongoing experimental facilities, like BESIII experiment [12].

D. Contribution from minimal 331 model

The so-called minimal 331 model is an extension of SM at the TeV scale, where the weak gauge group of SM SU(2)L ′ is extented to SU(3)L. In this model, a new neutral Z gauge boson can give very important additional contributions, for it can transmit FCNC at the tree level. In TAB. III, we denote this model as M331. More details of this model can be found in Ref. [29]. The minimal 331 model leads to a new term in the effective Hamiltonian [30]:

′ ˜ ∗ ˜ 2 Z GF V32V31 MZ 2 eff = cos θW (¯sd)V −A(¯νlνl)V −A + h.c., (39) H √ 3 M ′ l=Xe,µ,τ 2  Z  8

′ ˜ ∗ ˜ 2 −6 ˜ ∗ ˜ 2 −8 with MZ = 1 TeV, Re[(V32V31) ]=9.2 10 and Im[(V32V31) ]=4.8 10 [30]. The other parameters are the × × SM same as the SM inputs [23, 24]. The function X(xt) in Eq. (5) can be redefined as X(xt)= X (xt) + ∆X with

2 2 ˜ ∗ ˜ 2 sin θW cos θW 2π V32V31 MZ ∆X = ∗ . (40) α 3 VtsVtd MZ′ 

With the modified function X(xt) and considering the NLO contribution, the branching ratios of rare hyperon decays in the minimal 331 model can be calculated, as shown in TAB. III. The NLO+M331 predictions are much larger than the NLO results in SM, and are two and four times larger than the results of NLO+SUSY (Set. I) and NLO+SUSY (Set. II), respectively.

IV. CONCLUSIONS

FCNC processes offer important tools to test SM, and to search for possible new physics. The two decays K+ + 0 → π νν¯ and KL π νν¯ have been widely studied, while the corresponding baryon sector has not been explored. In this work, we studied→ the s dνν¯ rare hyperon decays. We adopted the leading order approximations for the form factors for small q2, and derived→ expressions for the decay width. We applied the decay width formula to both SM and new physics contributions. Different energy scales were considered. The branching ratios in SM range from 10−14 −11 + + 0 to 10 , and the largest is of the same order as for the decays K π νν¯ and KL π νν¯. When all the errors in the form factors are included, we found that the final branching ratios→ for most decay→ modes have an uncertainty of about 5% to 10%. After taking into account the contribution from MSSM, the branching ratios are enhanced by a factor of 2 4. The branching ratios of hyperon decays in the minimal 331 Model are seven times larger than the SM results. ∼

Acknowlegement

The authors are grateful to Profs. Hai-Bo Li and Wei Wang for useful discussions. This work is supported in part by National Natural Science Foundation of China under Grant No.11575110, 11735010, 11911530088, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100, and by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education.

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