Top-quark flavor-changing $tqZ$ couplings and rare $\Delta F=1$ processes

We model-independently study the impacts of anomalous $tqZ$ couplings ($q=u,c$), which lead to the $t\to q Z$ 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. Using the ATLAS's branching ratio (BR) upper bound of $BR(t\to uZ)<1.7\times 10^{-4}$, the influence of the anomalous $tuZ$-coupling on the rare decays can be found as follows: (a) The contribution to the Kaon direct CP violation can be up to $Re(\epsilon'/\epsilon) \lesssim 0.8 \times 10^{-3}$; (b) $BR(K^+\to \pi^+ \nu \bar \nu) \lesssim 12 \times 10^{-11}$ and $BR(K_L \to \pi^0 \nu\bar \nu)\lesssim 7.9 \times 10^{-11}$; (c) the BR for $K_S \to \mu^+ \mu^-$ including the long-distance effect can be enhanced by $11\%$ with respect to the standard model result, and (d) $BR(B_d \to \mu^+ \mu^-) \lesssim 1.97 \times 10^{-10}$. In addition, although $Re(\epsilon'/\epsilon)$ cannot be synchronously enhanced with $BR(K_L\to \pi^0 \nu \bar\nu)$ and $BR(K_S\to \mu^+ \mu^-)$ in the same region of the CP-violating phase, the values of $Re(\epsilon'/\epsilon)$, $BR(K^+ \to \pi^+ \nu \bar\nu)$, and $BR(B_d \to \mu^+ \mu^-)$ can be simultaneously increased.


I. INTRODUCTION
Top-quark flavor changing neutral currents (FCNCs) are extremely suppressed in the standard model (SM) due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [1]. The branching ratios (BRs) for the t → q(g, γ, Z, h) decays with q = u, c in the SM are of order of 10 −12 − 10 −17 [2,3], and these results are far below the detection limits of LHC, where the expected sensitivity in the high luminosity (HL) LHC for an integrated luminosity of 3000 fb −1 at √ s = 14 TeV is in the range 10 −5 − 10 −4 [4,5]. Thus, the top-quark flavor-changing processes can serve as good candidates for investigating the new physics effects. Extensions of the SM, which can reach the HL-LHC sensitivity, can be found in [6][7][8][9][10].
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 → qZ as [11]: Based on the current upper bounds, we model-independently study the implications of anomalous tqZ couplings in the low energy flavor physics. It is found that the tqZ couplings through the Z-penguin diagram can significantly affect the rare decays in K and B systems, such as ǫ ′ /ǫ, 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 → qZ decays, especially the t → uZ decay.
From a phenomenological perspective, the importance of investigating the influence of these rare decays are stated as follows: The inconsistency in ǫ ′ /ǫ between theoretical calculations and experimental data was recently found based on two analyses: (i) The RBC-UKQCD collaboration obtained the lattice QCD result with [12,13]: where the numbers in brackets denote the errors. (ii) Using a large N c dual QCD [14][15][16][17][18], the authors in [19,20] obtained: Both results show that the theoretical calculations exhibit an over 2σ deviation from the experimental data of Re(ǫ ′ /ǫ) exp = (16.6±2.3)×10 −4 , measured by NA48 [21] and KTeV [22,23]. 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 tuZ-coupling can be ǫ ′ /ǫ 0.8 × 10 −8 when the bound of BR(t → uZ) < 1.7 × 10 −4 is satisfied.
In addition, the result combined CMS and LHCb was reported as BR(B d → µ + µ − ) = (3.9 +1.6 −1.4 ) × 10 −10 [72]. It can be seen that the measured sensitivity is close to the SM result. We find that using the current upper limit of BR(t → uZ), the BR(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 Sec. II, we introduce the effective interactions for t → qZ and derive the relationship between the tqZ-coupling and BR(t → qZ). The Zpenguin FCNC processes induced via the anomalous tqZ couplings are given in Sec. III. The influence on ǫ ′ /ǫ is shown in the same section. The tqZ-coupling contribution to the other rare K and B decays is shown in Sec. IV. A summary is given in Sec. V.

II. ANOMALOUS tqZ COUPLINGS AND THEIR CONSTRAINTS
We write the anomalous tqZ interactions as [2]: 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 effects. 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 χ q = |ζ χ q |e −iθ χ q with χ = L, R.
Using the interactions in Eq. (6), we can calculate the BR for t → qZ decay. Since our purpose is to examine whether the anomalous tqZ-coupling can give sizable contributions to the rare K and B decays when the current upper bound of BR(t → qZ) is satisfied, we express the parameters ζ L,R q as a function of BR(t → qZ) to be: For the numerical analysis, the relevant input values are shown in Table I. Using the numer- 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.
In this section, we discuss the tqZ-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 processes in this study are the decays, such as K → ππ, K → πνν, and K S (B d ) → ℓ + ℓ − . It is found that the contributions to K L → πℓ + ℓ − and B → πℓ + ℓ − are not significant; therefore, we do not discuss the decays in this work. Based on the tqZ 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 tuZ-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 [20]: where ω = ReA 2 /ReA 0 ≈ 1/22.35 denotes the ∆I = 1/2 rule, and |ǫ 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, ǫ ′ /ǫ also strongly depends on the Wilson coefficients at the µ = m c scale. It is known that the main new physics contributions to ǫ ′ /ǫ are from the Q (′) 6 and Q (′) 8 operators [25,73]. Although these operators are not generated through the tqZ 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 [74]. 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(ǫ ′ /ǫ) provided in [20], the O 8 contribution can be written as: where r 2 = ωG F /(2|ǫ K |) ≈ 1.17 × 10 −4 GeV −2 , B  [75], and the matrix element of Q 8 2 is defined as: 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 ǫ ′ /ǫ, we use BR(t → qZ) ≡ Min(BR(t → cZ), BR(t → uZ)) instead of BR(t → u(c)Z) as the upper limit. The contours for Re(ǫ ′ /ǫ) Z P ( in units of 10 −3 ) as a function of BR(t → qZ) 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 BR(t → qZ). It can be seen that the Kaon direct CP violation arisen from the anomalous tuZ-coupling can reach 0.8 × 10 −3 , and the contribution from tcZ-coupling is only a minor effect. When the limit of t → qZ approaches BR(t → qZ) ∼ 0.5 × 10 −4 , the induced ǫ ′ /ǫ can be as large as Re(ǫ ′ /ǫ) Z P ∼ 0.4 × 10 −3 . is defined as the minimal one between BR(t → uZ) and BR(t → cZ). The horizontal dashed line (red) is the current upper limit of BR(t → qZ).

MERICAL 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)Zcoupling contributions to the b → sℓl (ℓ = ν, ℓ − ) processes can deviate from the SM result being less than 7% in terms of amplitude. However, the influence of the tuZ coupling on d → sℓl and b → dℓl 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 [37], we write the effective Hamiltonian for d i → d j ℓl induced by the tuZ coupling as: where we have ignored the small contributions from the tcZ-coupling; 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 [25]: where λ c = V * cs V cd , ∆ EM = −0.003; P c (X) = 0.404 ± 0.024 denotes the charm-quark contribution [76,77]; the values of κ L,+ 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 [25]. 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 [37]: with C SM 10 ≈ −4.21. Including the LD effect [66,67], the BR for K S → µ + µ − can be estimated using BR(K S → µ + µ − ) LD+SD ≈ 4.99 LD × 10 −12 + BR(K S → µ + µ − ) SD [68]. 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 V * td V tb C SM 10 + C Z 10 . Note that the associated Wilson coefficient in whereas it is C Z * 10 in the K decays. After formulating the BRs for the investigated processes, we now numerically analyze the tuZ-coupling effect on these decays. Since the involved parameter is the complex ζ L u = |ζ L u |e −iθ L u , we take BR(t → uZ) instead of |ζ L u |. Thus, we show BR(K L → π 0 νν) (in units of 10 −11 ) as a function of BR(t → uZ) and θ L u in Fig. 3(a), 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 BR(t → uZ). It can be clearly seen that BR(K L → π 0 νν) can be enhanced to 7 × 10 −11 in θ L u > 0 when BR(t → uZ) < 1.7 × 10 −4 is satisfied. Moreover, the result of BR(K L → π 0 νν) ≈ 5.3 × 10 −11 can be achieved when BR(t → uZ) = 0.5 × 10 −4 and θ u L = 2.1 are used. Similarly, the influence of ζ L u on BR(K + → π + νν) is shown in Fig. 3(b). Since BR(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 BR(t → uZ) = 0.5 × 10 −4 and θ u L = 2.1, the BR(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 BR(K → πνν) cannot constrain BR(t → uZ), 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 of K S → γγ → µ + µ − , predominantly contributes to the BR(K S → µ + µ − ). Thus, if the new physics contribution is much smaller than the LD effect, the new physics may not be so significant. In order to show the tuZ-coupling effect, we plot the contours for BR(K S → µ + µ − ) LD+SD ( in units of 10 −12 ) in Fig. 3(c). From the result, it can be clearly seen that BR(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 BR(t → uZ) = 0.5×10 −4 and θ L u = 2.1. 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 tuZ-coupling. We show the contours of BR(B d → µ + µ − ) ( in units of 10 −10 ) as a function of BR(t → uZ) and θ L u in Fig. 3(d). It can be seen that the maximum of the allowed BR(B d → µ + µ − ) can reach 1.97×10 −10 , which is a factor of 1.8 larger than the SM result of BR(B d → µ + µ − ) SM ≈ 1.06 × 10 −10 . Using BR(t → uZ) = 0.5 × 10 −4 and θ L u = 2.1, the enhancement factor to BR(B d → µ + µ − ) SM becomes 1.38. Since the maximum of BR(B d → µ + µ − ) has been close to the ATLAS upper bound of 2.1 × 10 −10 , the constraint from the rare B decay measured in the LHC could further constrain the allowed range of θ L u V. SUMMARY We studied the impacts of the anomalous tqZ couplings in the low energy physics, especially the tuZ coupling. It was found that the anomalous coupling can have significant contributions to ǫ ′ /ǫ, BR(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 ǫ ′ /ǫ, their designed experiment sensitivities are good enough to test the SM. It was found that using the sensitivity of BR(t → uZ) ∼ 5 × 10 −5 designed in HL-LHC, the resulted BR(K → πνν) and BR(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(ǫ ′ /ǫ), BR(K L → π 0 νν), and BR(K S → µ + µ − ) in the same region of the CP violating phase, where the positive Re(ǫ ′ /ǫ) requires θ L u < 0, but the large BR(K L → π 0 νν) and BR(K S → µ + µ − ) shown in each plot. The long-distance effect has been included in the K S → µ + µ − decay.
have to rely on θ L u > 0. Since BR(K + → π + νν) and BR(B d → µ + µ − ) involve both real and imaginary parts of Wilson coefficients, their BRs are not sensitive to the sign of θ L u . Hence, Re(ǫ ′ /ǫ), BR(K + → π + νν) and BR(B d → µ + µ − ) can be enhanced at the same time.