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We study the impacts of anomalous tqZ couplings (q=u,c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=u,c$$\end{document}), which lead to the t→qZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow q Z$$\end{document} decays, on low energy flavor physics. It is found that the tuZ-coupling effect can significantly affect the rare K and B decays, whereas the tcZ-coupling effect is small. 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Using the data collected with an integrated luminosity of 36.1 fb −1 at √ s = 13 TeV, ATLAS reported the current strictest upper limits on the BRs for t → q Z as [18]: Based on the current upper bounds, we model-independently study the implications of anomalous tq Z couplings in the low energy flavor physics. It is found that the tq Z couplings through the Z -penguin diagram can significantly affect the rare decays in K and B systems, such as / , K → πνν, K S → μ + μ − , and B d → μ + μ − . Since the gluon and photon in the top-FCNC decays are on-shell, the contributions from the dipole-operator transition currents are small. In this study we thus focus on the t → q Z decays, especially the t → u Z decay. From a phenomenological perspective, the importance of investigating the influence of these rare decays are stated as follows: The inconsistency in / between theoretical calculations and experimental data was recently found based on two analyses: (i) The RBC-UKQCD collaboration obtained the lattice QCD result with [19,20]: where the numbers in brackets denote the errors. (ii) Using a large N c dual QCD [21][22][23][24][25], the authors in [26,27] obtained: Re( / ) SM = (1.9 ± 4.5) × 10 −4 .
Note that the authors in [28] could obtain Re( / ) = (15 ± 7) × 10 −4 when the short-distance (SD) and long-distance (LD) effects are considered. Both RBC-UKQCD and DQCD results show that the theoretical calculations exhibit an over 2σ deviation from the experimental data of Re( / ) exp = (16.6 ± 2.3) × 10 −4 , measured by NA48 [29] and KTeV [30,31]. Based on the results, various extensions of the SM proposed to resolve the anomaly can be found in . We find that the direct Kaon CP violation arisen from the tu Z-coupling can be / 0.8 × 10 −3 when the bound of B R(t → u Z) < 1.7 × 10 −4 is satisfied.
It has been found that the tu Z-coupling-induced Zpenguin can significantly enhance the B d → μ + μ − decay, where the SM prediction is given by B R(B d → μ + μ − ) = (1.06 ± 0.09) × 10 −10 [78]. From the data, which combine the full Run I data with the results of 26.3 fb −1 at √ s = 13 TeV, ATLAS reported the upper limit as B R(B d → μ + μ − ) < 2.1 × 10 −10 [79]. In addition, the result combined CMS and LHCb was reported as B R(B d → μ + μ − ) = (3.9 +1.6 −1.4 )×10 −10 [80], and LHCb recently obtained the upper limit of B R(B d → μ + μ − ) < 3.4 × 10 −10 [81]. It can be seen that the measured sensitivity is close to the SM result. We find that using the current upper limit of B R(t → u Z), the B R(B d → μ + μ − ) can be enhanced up to 1.97 × 10 −10 , which is close to the ATLAS upper bound.
The paper is organized as follows: In Sect. 2, we introduce the effective interactions for t → q Z and derive the relationship between the tq Z-coupling and B R(t → q Z). The Z -penguin FCNC processes induced via the anomalous tq Z couplings are given in Sect. 3. The influence on / is shown in the same section. The tq Z-coupling contribution to the other rare K and B decays is shown in Sect. 4. A summary is given in Sect. 5.

Anomalous tq Z couplings and their constraints
Based on the prescription in [2], we write the anomalous tq Z interactions as: where g is the SU (2) L gauge coupling; c W = cos θ W and θ W is the Weinberg angle; P L(R) = (1 ∓ γ 5 )/2, and ζ L(R) q denote the dimensionless effective couplings and represent the new physics effects. In this study, we mainly concentrate the impacts of the tq Z couplings on the low energy flavor physics, in which the rare K and B decays are induced through the penguin diagram. The rare D-meson processes, such as D-D mixing and D → ¯ , can be induced through the box diagrams; however, the processes in D system can always be suppressed by taking one of the involved anomalous couplings, e.g. tcZ, to be small. Thus, in the following analysis, we focus on the study in the rare K and B decays. In order to study the influence on the Kaon CP violation, we take ζ L ,R q as complex parameters, and the new CP violating phases are defined as ζ The top anomalous couplings in Eq. (6) can basically arise from the dimension-six operators in the SM effective field theory (EFT), where the theory with new physics effects obeys the SU (2) L × U (1) Y gauge symmetry. For clarity, we show the detailed analysis for the left-handed quark couplings in Appendix. It can be found that the couplings in Eq. (6), which are generated from the SM-EFT, are not completely excluded by the low-energy flavor physics when the most general couplings are applied. The case with the strict constraints can be found in [10]. In addition to the SM-EFT [82][83][84], the top anomalous tq Z couplings can be induced from the lower dimensional operators in the extension of the SM, such as SU (2) singlet vector-like up-type quark model [8], extra dimensions [9], and generic two-Higgs-doublet model [16]. Hence, in this study, we take ζ χ q are the free parameters and investigate the implications of the sizable ζ χ q effects without exploring their producing mechanism.
Using the interactions in Eq. (6), we can calculate the BR for t → q Z decay. Since our purpose is to examine whether the anomalous tq Z-coupling can give sizable contributions to the rare K and B decays when the current upper bound of B R(t → q Z) is satisfied, we express the parameters ζ L ,R q as a function of B R(t → q Z) to be: For the numerical analysis, the relevant input values are shown in Table 1. Using the numerical inputs, we obtain 3) × 10 −4 measured by ATLAS are applied, the upper limits on |ζ L u(c) | 2 + |ζ R u(c) | 2 can be respectively obtained as: Since the current measured results of the t → (u, c)Z decays are close each other, the bounds on ζ χ u and ζ χ c are very similar. We note that BR cannot determine the CP phase; therefore, θ χ u and θ χ c are free parameters.

Anomalous tq Z effects on /
In this section, we discuss the tq Z-coupling contribution to the Kaon direct CP violation. The associated Feynman diagram is shown in Fig. 1, where q = u, c; q and q are down type quarks, and f denotes any possible fermions. That is, the involved rare K and B decay processes in this study are such as K → ππ, K → πνν, and K S ( Sketched Feynman diagram for q → q ff induced by the tq Z coupling, where q and q denote the down-type quarks; q = u, c, and f can be any possible fermions that the contributions to K L → π + − and B → π + − are not significant; therefore, we do not discuss the decays in this work. Based on the tq Z couplings shown in Eq. (6), the effective Hamiltonian induced by the Z -penguin diagram for the K → ππ decays at μ = m W can be derived as: where λ t = V * ts V td ; the operators Q 3,7,9 are the same as the SM operators and are defined as: with e q being the electric charge of q -quark, and the effective Wilson coefficients are expressed as: with α = e 2 /4π , x t = m 2 t /m 2 W , and s W = sin θ W . The penguin-loop integral function is given as: Since W -boson can only couple to the left-handed quarks, the right-handed couplings ζ R u,c in the diagram have to appear with m u(c) and m t , in which the mass factors are from the mass insertion in the quark propagators inside the loop. When we drop the small factors m c,u /m W , the effective Hamiltonian for K → ππ only depends on ζ L u,c . Since |V ud /V td | is larger than |V cs /V ts | by a factor of 4.67, the dominant contribution to the S = 1 processes is from the first term of η Z defined in Eq. (11). In addition, V ud is larger than |V cd | by a factor of 1/λ ∼ 4.44; therefore, the main contribution in the first term of η Z comes from the V ud ζ L * u /V td effect. That is, the anomalous tu Z-coupling is the main effect in our study.
Using the isospin amplitudes, the Kaon direct CP violating parameter from new physics can be estimated using [27]: where ω = Re A 2 /Re A 0 ≈ 1/22.35 denotes the I = 1/2 rule, and | K | ≈ 2.228 × 10 −3 is the Kaon indirect CP violating parameter. It can be seen that in addition to the hadronic matrix element ratios, / also strongly depends on the Wilson coefficients at the μ = m c scale. It is known that the main new physics contributions to / are from the Q ( ) 6 and Q ( ) 8 operators [33,85]. Although these operators are not generated through the tq Z couplings at μ = m W in our case, they can be induced via the QCD radiative corrections. The Wilson coefficients at the μ = m c scale can be obtained using the renormalization group (RG) evolution [86]. Thus, the induced effective Wilson coefficients for Q 6,8 operators at μ = m c can be obtained as: It can be seen that y Z 6 (m c ) is much smaller than y Z 8 (m c ); that is, we can simply consider the Q 8 operator contribution.
According to the K → ππ matrix elements and the formulation of Re( / ) provided in [27], the O 8 contribution can be written as: where Although the Q 8 operator can contribute to the isospin I = 0 state of ππ, because its effect is a factor of 15 smaller than the isospin I = 2 state, we thus neglect its contribution. Since the t → (u, c)Z decays have not yet been observed, in order to simplify their correlation to / , we use B R(t → q Z) ≡ Min(B R(t → cZ), B R(t → u Z)) instead of B R(t → u(c)Z ) as the upper limit. The contours for Re( / ) Z P ( in units of 10 −3 ) as a function of B R(t → q Z) and θ L u are shown in Fig. 2, where the solid and dashed lines denote the results with θ L c = −θ L u and ζ L c = 0, respectively, and the horizontal dashed line is the current upper limit of B R(t → q Z). It can be seen that the Kaon direct CP violation arisen from the anomalous tu Z-coupling can reach 0.8 × 10 −3 , and the contribution from tcZ-coupling is only a minor effect. When the limit of t → q Z approaches B R(t → q Z) ∼ 0.5 × 10 −4 , the induced / can be as large as Re( / ) Z P ∼ 0.4 × 10 −3 .

Z-penguin induced (semi)-leptonic K and B decays and numerical analysis
The same Feynman diagram as that in Fig. 1 can be also applied to the rare leptonic and semi-leptonic K (B) decays when f is a neutrino or a charged lepton. Because |V us /V ts | |V cs /V ts | ∼ |V us /V td | |V ud /V td |, it can be found that the anomalous tu(c)Z -coupling contributions to the b → s ¯ ( = ν, − ) processes can deviate from the SM result being less than 7% in terms of amplitude. However, the influence of the tu Z coupling on d → s ¯ and b → d ¯ can be over 20% at the amplitude level. Accordingly, in the following analysis, we concentrate the study on the rare decays, such as K → πνν, K S → μ + μ − , and B d → μ + μ − , in which the channels are sensitive to the new physics effects and are theoretically clean.
According to the formulations in [45], we write the effective Hamiltonian for d i → d j ¯ induced by the tu Z coupling as: where we have ignored the small contributions from the tcZcoupling; d i → d j could be the s → d or b → d transition, and the effective Wilson coefficients are given as: Because −1 + 4s 2 W ≈ −0.08, the C Z 9 effect can indeed be neglected.
Based on the interactions in Eq. (17), the BRs for the K L → π 0 νν and K + → π + νν decays can be formulated as [33]: where λ c = V * cs V cd , E M = −0.003; P c (X ) = 0.404 ± 0.024 denotes the charm-quark contribution [88,89]; the values of κ L and κ + are respectively given as κ L = (2.231 ± 0.013) × 10 −10 and κ + = (5.173 ± 0.025) × 10 −11 , and X eff is defined as: with X SM L = 1.481 ± 0.009 [33]. Since K L → π 0 νν is a CP violating process, its BR only depends on the imaginary part of X eff . Another important CP violating process in K decay is K S → μ + μ − , where its BR from the SD contribution can be expressed as [45]: with C SM 10 ≈ −4.21. Including the LD effect [74,75], the BR for K S → μ + μ − can be estimated using B R(K S → μ + μ − ) LD+SD ≈ 4.99 LD × 10 −12 + B R(K S → μ + μ − ) SD [76]. Moreover, it is found that the effective interactions in Eq. (17) can significantly affect the B d → μ + μ − decay, where its BR can be derived as: Because B d → μ + μ − is not a pure CP violating process, the BR involves both the real and imaginary part of After formulating the BRs for the investigated processes, we now numerically analyze the tu Z-coupling effect on these decays. Since the involved parameter is the complex ζ L u = |ζ L u |e −iθ L u , we take B R(t → u Z) instead of |ζ L u |. Thus, we show B R(K L → π 0 νν) (in units of 10 −11 ) as a function of B R(t → u Z) and θ L u in Fig. 3a, where the CP phase is taken in the range of θ L u = [−π, π]; the SM result is shown in the plot, and the horizontal line denotes the current upper limit of B R(t → u Z). It can be clearly seen that B R(K L → π 0 νν) can be enhanced to 7 × 10 −11 in θ L u > 0 when B R(t → u Z) < 1.7×10 −4 is satisfied. Moreover, the result of B R(K L → π 0 νν) ≈ 5.3 × 10 −11 can be achieved when B R(t → u Z) = 0.5×10 −4 and θ u L = 2.1 are used. Similarly, the influence of ζ L u on B R(K + → π + νν) is shown in Fig. 3b. Since B R(K + → π + νν) involves the real and imaginary parts of X eff , unlike the K L → π 0 νν decay, its BR cannot be enhanced manyfold due to the dominance of the real part. Nevertheless, the BR of K + → π + νν can be maximally enhanced by 38%; even, with B R(t → u Z) = 0.5 × 10 −4 and θ u L = 2.1, the B R(K + → π + νν) can still exhibit an increase of 15%. It can be also found that in addition to |ζ L u |, the BRs of K → πνν are also sensitive to the θ L u CP-phase. Although the observed B R(K → πνν) cannot constrain B R(t → u Z), the allowed range of θ L u can be further limited.
For the K S → μ + μ − decay, in addition to the SD effect, the LD effect, which arises from the absorptive part (a) (b) (c) (d) Fig. 3 Contours of the branching ratio as a function of B R(t → u Z) and θ L u for a K L → π 0 νν, b K + → π + νν, c K S → μ + μ − , and d B d → μ + μ − , where the corresponding SM result is also shown in each plot. The long-distance effect has been included in the K S → μ + μ − decay of K S → γ γ → μ + μ − , predominantly contributes to the B R(K S → μ + μ − ). Thus, if the new physics contribution is much smaller than the LD effect, the influence on B R(K S → μ + μ − ) LD+SD = B R(K S → μ + μ − ) LD + B R(K S → μ + μ − ) SD from new physics may not be so significant. In order to show the tu Z-coupling effect, we plot the contours for B R(K S → μ + μ − ) LD+SD ( in units of 10 −12 ) in Fig. 3c. From the result, it can be clearly seen that B R(K S → μ + μ − ) LD+SD can be at most enhanced by 11% with respect to the SM result, whereas the BR can be enhanced only ∼ 4.3% when B R(t → u Z) = 0.5 × 10 −4 and θ L u = 2.1 are used. We note that the same new physics effect also contributes to K L → μ + μ − . Since the SD contribution to K L → μ + μ − is smaller than the SM SD effect by one order of magnitude, we skip to show the case for the K L → μ + μ − decay.
As discussed earlier that the tcZ-coupling contribution to the B s → μ + μ − process is small; however, similar to the case in K + → π + νν decay, the BR of B d → μ + μ − can be significantly enhanced through the anomalous tu Z-coupling. We show the contours of B R(B d → μ + μ − ) ( in units of 10 −10 ) as a function of B R(t → u Z) and θ L u in Fig. 3d. It can be seen that the maximum of the allowed B R(B d → μ + μ − ) can reach 1.97 × 10 −10 , which is a factor of 1. We studied the impacts of the anomalous tq Z couplings in the low energy physics, especially the tu Z coupling. It was found that the anomalous coupling can have significant contribu-tions to / , B R(K → πνν), K S → μ + μ − , and B d → μ + μ − . Although these decays have not yet been observed in experiments, with the exception of / , their designed experiment sensitivities are good enough to test the SM. It was found that using the sensitivity of B R(t → u Z) ∼ 5 × 10 −5 designed in HL-LHC, the resulted B R(K → πνν) and B R(B d → μ + μ − ) can be examined by the NA62, KOTO, KELVER, and LHC experiments.
According to our study, it was found that we cannot simultaneously enhance Re( / ), B R(K L → π 0 νν), and B R(K S → μ + μ − ) in the same region of the CP violating phase, where the positive Re( / ) requires θ L u < 0, but the large B R(K L → π 0 νν) and B R(

Appendix A: Anomalous gauge couplings from the SM-EFT
If we take the SM as an effective theory at the electroweak scale, the new physics effects should appear in terms of higher dimensional operators when the heavy fields above electroweak scale are integrated out. Thus, the effective Lagrangian with respect to the SM gauge symmetry can be generally expressed as [82][83][84]: where L k are the associated Wilson coefficients. The top flavor-changing anomalous couplings can be generated from the dimension-6 operators, where based on the notations in [84], the relevant operators in our study can be written as [84]: where ϕ denotes the SM Higgs doublet, Q T L = (U L , D L ) is left-handed quark doublet, D μ ϕ is the covariant derivative acting on ϕ, τ I are the Pauli matrices; ϕ = iτ 2 ϕ * , ϕ † ← D μ ϕ = (D μ ϕ) † ϕ, and The flavor indices are suppressed; therefore, the Wilson coefficients {C i } are 3 × 3 matrices. Since the top anomalous gauge couplings in this study are mainly related to the lefthanded couplings, in the following discussions, we focus on the couplings to the left-handed quarks. After electroweak symmetry breaking, the relevant Z and W gauge couplings to the quark weak eigenstates in Eq. (A2) can be formulated as: where ϕ = v/ √ 2 is the vacuum expectation value (VEV) of ϕ. It can be seen that the Z couplings to the down-type quarks can be removed if we assume C (1) Under such circumstance, the FCNCs at the tree level could only occur in the up-type quarks. In order to use the physical quark states to express Eq. (A4), we introduce the unitary matrices U u,d L ,R to diagonalize the quark mass matrices. Thus, defining C q L = V u L C q L V u † L , Eq. (A4) can be written as: where ξ q L = 2C q L +2C † q L , and V = V u L V d † L is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It can be seen that the anomalous gauge couplings in the neutral current interactions are strongly correlated with those in the charged-current interactions.
It is known that the CKM matrix has a hierarchical structure, such as V 11 (22,33) ∼ 1, |V 12(21) | ∼ λ, |V 23(32) | ∼ λ 2 , and |(V 13(31) | ∼ λ 3 , where λ ≈ 0.22 is the Wolfenstein parameter [90]. Since each CKM matrix element is measured well, it is necessary to examine if the sizable t → q Z FCNCs are excluded by the experimental measurements, which are dictated by the charged current interactions. Thus, in the following analysis, we concentrate on the modifications of V ub , V ts , and V td . First, we consider (ξ q L V ) ub for the b → u transition effect and decompose it as: where the V ub,cb terms in the second line are dropped due to V ub,cb V tb . In order to obtain a small effect in the b → u transition, we have to require v 2 (ξ q L ) ut /( 2 ) to be much less than 0.02, which is the current upper limit shown in Eq. (8). Similarly, the (ξ q L V CKM ) ts(td) factors can be expressed in terms of λ as: (ξ q L V ) ts ≈ λ(ξ q L ) tu + (ξ q L ) tc − λ 2 (ξ q L ) tt , (ξ q L V ) td ≈ (ξ q L ) tu − λ(ξ q L ) tc + λ 3 (ξ q L ) tt . (A7) If we take (ξ q L ) tu ∼ λ(ξ q L ) tc − λ 3 (ξ q L ) tt , i.e., the (ξ q L V ) td effect is suppressed, (ξ q L V ) ts can be rewritten as: (ξ q L V ) ts ≈ (1 + λ 2 ) (ξ q L ) tc − λ 2 (ξ q L ) tt ∼ (ξ q L ) tu λ . (A8) Because (ξ q L ) tu,tc,tt are taken as the free parameters, we have the degrees of freedom to obtain v 2 |(ξ q L V ) ts |/(2 2 ) < |V ts | ∼ λ 2 without (ξ q L ) tu(tc) 1. Using the result, we can obtain |ζ L u | = v 2 |(ξ q L ) tu |/ 2 < 0.021, where the upper limit is consistent with that shown in Eq. (8). Hence, although v 2 (ξ q L ) ut / 2 in a general SM-EFT is bounded by the measured CKM matrix elements, ζ L u = v 2 (ξ q L ) tu / 2 could be a free parameter and ζ L u < 0.021 is still allowed.