Probing anomalous $t\bar t Z$ interactions with rare meson decays

Anomalous couplings of the $Z$ boson to top quarks are only marginally constrained by direct searches and are still sensitive to new particle dynamics at the TeV scale. Employing an effective field theory approach we consider the dimension-six operators which generate deviations from the standard-model vector and axial-vector interactions. We show that rare $B$ and $K$ meson decays together with electroweak precision observables provide strong constraints on these couplings. We also consider constraints from t-channel single-top production.


Introduction
In the case of no direct observation of new particles at the Large Hadron Collider (LHC), precision measurements of the Standard Model (SM) interactions provide an alternative route to discover New Physics (NP). The measurement of the ttZ production cross section at the LHC [1,2] offers a direct test of anomalous ttZ couplings. In a recent publication, Schulze and Röntsch presented the calculation and analysis of ttZ production at next-toleading order in QCD [3]. They used current and future-projected LHC data to constrain anomalous vector and axial-vector couplings of the Z boson to top quarks.
In this work we study constraints on anomalous ttZ couplings arising from electroweak precision observables (EWPO), rare B and K meson decays, and t-channel single-top production. The ttZ couplings contribute to the rate of rare meson decays and to EWPO via radiative corrections. Thus, in the absence of contributions to these observables from other sources, rare meson decays and EWPO offer strong bounds on anomalous ttZ couplings. In fact, we show that the indirect constraints exceed the direct bounds, even after the projected high-luminosity upgrade of the LHC [3].
We focus on the rare decays B s → µ + µ − , K + → π + νν and K L → π 0 νν. These decays proceed via Z-penguin diagrams 1 which, in the SM, are dominated by the top-quark loop. This makes these processes especially sensitive to anomalous ttZ couplings. For this reason, the current observation of the B s → µ + µ − rate already competes with constraints from EWPO and the expected future improvements in the measurements of rare meson decay rates are especially encouraging as they may provide the first evidence for anomalous ttZ interactions.
The use of rare decays to obtain bounds on anomalous Z couplings has already been discussed in Refs. [4,5]. The authors derived constraints on bbZ couplings by relating them to flavor-changing couplings within the minimal flavor violation (MFV) framework. In this work, we consider constraints arising from operator mixing, induced by electroweak loops, within an effective field theory (EFT) approach. This strategy has already been proposed in Refs. [6,7] for flavor-changing neutral currents (FCNCs) involving the top quark, as well as for constraining the anomalous W tb couplings [8,9].
The remainder of the paper is structured as follows: In Sec. 2 we define the appropriate EFT approach to study possible deviations in ttZ couplings. In Sec. 3 we revisit the main constraints from EWPO and t-channel single-top production, and present our analysis of rare meson decays. In Sec. 4 we discuss the numerical results and conclude.

Effective field theory
Our starting point is the SM augmented by gauge-invariant dimension-six operators constructed out of the SM fields. This may be thought of as the effective field theory after integrating out NP at some scale Λ significantly larger than the electroweak scale. We write the effective Lagrangian as We are interested in the subset of operators that induce modifications of the vector and axial-vector ttZ couplings at tree-level. They will have the main impact on direct searches for anomalous ttZ couplings. If we assume no correlations between Wilson coefficients, a complete set of such operators is given by (cf. [10,11]): These operators contain the Higgs doublet φ, the left-handed third-generation quark doublet Q L,3 , and the right-handed top quark t R . Moreover, σ a are the Pauli matrices and D µ is the SM gauge-covariant derivative and we defined so that the operators are manifestly Hermitian. Therefore, all Wilson coefficients considered in this work are real.
In order to include in Eq. (2.2) all operators that induce ttZ at tree-level, we have chosen the basis in which the up-type quark Yukawa matrix is diagonal. Accordingly, we have The fields t L(R) , d L,j are mass eigenstates and V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. A generic NP model can generate FCNC transitions in the up-quark sector; to take these effects into account we have to consider additional operators involving the first and second quark generation. However, in the restricted, but phenomenologically well-motivated, framework of MFV [12] these operators are suppressed with respect to those in Eq. (2.2) by elements of the CKM matrix. Thus, the resulting bounds on ttZ couplings are negligible. Moreover, in the limit that only the top-quark Yukawa is non-vanishing in the MFV spurion counting, such additional operators are absent. For simplicity, we will use this approximation in the following, but we will comment on the effect of keeping a large bottom-quark Yukawa in Sec. 4. This discussion will cover MFV models with down-type quark alignment, i.e. models in which, at tree-level, only up-type quark FCNCs are generated by NP. In our analysis, only the operators in Eq. (2.2) receive non-zero initial conditions at the scale Λ. However, electroweak corrections and corrections involving the SM top-quark Yukawa coupling y t will induce mixing into the following operators relevant for our analysis: where i = 1, 2 and j = 1, 2, 3. We can safely neglect the remaining Yukawa couplings in the mixing calculation. The operators in Eq. (2.5) will lead to additional bounds on the anomalous ttZ couplings.
To study the effects of these operators on rare decays we need to know their Wilson coefficients at the electroweak scale where we match to the five-flavor effective theory. The running of the Wilson coefficients is given by the renormalization group equations (RGE). The matching is most conveniently performed in the broken phase where the Higgs field obtains its vacuum-expectation value, v. The relevant part of the effective Lagrangian in the broken phase reads Using the conventions of Ref. [13], the flavor-diagonal neutral-current part of the Lagrangian is given by This Lagrangian induces a shift of the SM Z-boson couplings to third-generation quarks. We follow the conventions in Ref. [14] and parametrize this shift by (2.8) The operators Q (1) φq,33 and Q φu,33 involve only neutral quark currents, whereas Q φq,33 will necessarily induce deviations also in the effective W couplings. These charged-current contributions include The FCNC contributions relevant for us involve only down-type quarks and are given by Note that Eq. (2.10) also receives contributions from the operators in Eq. (2.5), but these contributions cancel via the unitarity of the CKM matrix (see Sec. 3.3) and we do not include them here. Finally, we remark that the operators in Eq. (2.2) do not induce modifications of couplings involving the Higgs boson. Constraints from flavor physics and EWPO suggest the absence of both tree-level downtype FCNC transitions and modifications of the bbZ couplings if NP enters at the TeV scale. Thus, we take C φq,33 = 0 at the scale Λ. This relation can be realized in models with additional vector-like quarks [15]. Moreover, these models do not generate four-fermion operators at tree level. Under these assumptions, rare meson decays and the effective bbZ couplings are sensitive to anomalous ttZ couplings generated by the operators in Eq. (2.2).

t-channel single top-quark production
The operator Q (3) φq,33 induces a tree-level correction to the W tb coupling given by Eq. (2.9). Both the ATLAS and the CMS collaborations have recently measured the t-channel single top-quark production cross section [16,17], which constitutes a direct measurement of the W tb coupling. In particular, they report the measured cross-section normalized with respect to the SM expectation, σ t-ch (t)/σ theo t-ch (t) ≡ R σ . The measured values reported by the two collaborations are where we have combined the experimental and theoretical errors in quadrature. In our analysis this ratio is given by Combining the measurements, we find

Electroweak precision observables
We saw in Sec. 2 that it is phenomenologically reasonable to assume C with y t , g 1 , and g 2 the SM top-Yukawa coupling and the electroweak gauge couplings in the conventions of Ref. [13], respectively. The terms proportional to the top-quark Yukawa coupling have been presented in Ref. [18] and those proportional to the gauge couplings in Ref. [19]. However, our result for the latter differs from the one of Ref. [19] in the last term of the first line in Eq. (3.4). We have calculated the mixing in a general R ξ gauge and find a gauge-independent result. While we agree with Ref. [19] on the terms arising from penguin-type insertions of the operators, we find an additional contribution for the mixing of Q φq,33 into itself. It arises from the diagrams shown in Figure 1. An additional non-trivial check of our result can be found in the next section 2 .
Solving the RGE in Eqs. (3.4), we determine the Wilson coefficients at the electroweak scale: (3.5) We used the leading-log approximation to relate the Wilson coefficient at the scale Λ to the electroweak scale µ W . Using Eq. (2.8), we find Here, we defined x t ≡ m 2 t /M 2 W and used the relation C φq,33 (Λ) + C φq,33 (Λ) = 0. The above corrections lead to a shift in the parameter b , defined in Ref. [20], given by The authors of Ref. [3] have compared their direct constraints on anomalous ttZ couplings with the indirect constraint derived from δ b . They use the expression for δ b given in Ref. [21] while using the same effective operators as in our work. However, the calculation in Ref. [21] has been carried out in a different framework, namely, the nonlinearly realized electroweak chiral Lagrangian [22]. Their result therefore does not agree with ours, and the expression for δ b in Ref. [3] should be replaced by the one in Eq. (3.6).
Quantum corrections also induce the mixing of the operators in Eq. (2.2) into the last two operators, Q φW B and Q φD , in Eq. (2.5). They are tightly constrained by EWPO, since they lead to the universal oblique S and T parameters [23][24][25], respectively. The mixing of the operators in Eq. (2.2) into Q φW B and Q φD is given by [18,19] φq,33 − C φu,33 . (3.7) Accordingly, we obtain the expressions for the S and T parameters (cf. Ref. [26]) (3.8) We checked that the T parameter obtained via the above RGE analysis agrees with a direct computation of the vacuum-polarization diagrams. The T parameter is directly related to the quantity 1 , defined in Ref. [27], via 1 ≡ αT . The term proportional to x t in Eq. (3.8) can be deduced from the corresponding result for δ 1 in Ref. [28]; we agree with that result. However, our result disagrees with that quoted in Ref. [3], where the contributions from the modified charged current have not been included.

Rare meson decays
We now advocate the use of rare meson decays to constrain anomalous ttZ couplings. Recall that in this work we assume C φq,33 = 0, and thus the absence of tree-level FCNC transitions at the scale Λ. Operator mixing reintroduces these transitions at lower scales. We calculate the running of the Wilson coefficients from Λ to the electroweak scale, where we match onto the five-flavor effective theory. We then compute the modifications of the rare meson decay rates, which allows us to bound the Wilson coefficients at the scale Λ. We focus here on the processes B (s,d) → µ + µ − and K → πνν. The reason is that all these decays are dominated by the Z-penguin within the SM and are thus the best candidates to constrain anomalous ttZ couplings.
The part of the RGE relevant for the rare meson decays is the same as in Eq. (3.4), with the addition As a non-trivial check of our calculation we reproduce the logarithmic part of the loop functions f (j) νν given in Ref. [9]. We solve the RGE and find the following expressions for the Wilson coefficients at the electroweak scale: (3.10) Note that in this equation we used the unitarity of the CKM matrix. The rate of the rare decay B s → µ + µ − was measured by the CMS [29] and LHCb [30] collaborations at the LHC and it was found to be consistent with the SM expectations [31]. The experiments are expected to reach the sensitivity of observing the SM B d → µ + µ − rate in the next LHC run. Within the SM and the restricted set of operators we consider, a single operator Q A ≡ (bγ µ γ 5 q)(μγ 5 µ) with q = d, s mediates the decays to a very good approximation. The rates depend on a single hadronic quantity, the decay constant f Bq of the meson. The average time-integrated branching ratios then read [31]: with M Bq the mass of the B meson, r qµ = 2m µ /M Bq , β qµ = 1 − r 2 qµ , N = V * tb V tq G 2 F M 2 W /π 2 and Γ q H the total width of the heavy mass eigenstate in B q -B q mixing. In general, the timeintegrated rates depend on the details of B q -B q mixing [32], but Eq. (3.11) holds to a very good approximation for both the B s and the B d system within the SM. C A (µ b ) is the Wilson coefficient of Q A . It incorporates effects from heavy NP to the SM contribution and is evolved to the scale of the B-meson, µ b ∼ m b . The SM part includes perturbative next-to-leading-order (NLO) QCD [33][34][35] as well as recently calculated next-to-next-toleading (NNLO) QCD [36] and NLO electroweak [37] corrections, which we all include in our analysis. However, at leading order, we have that is the SM loop function and δY NP is the additional contribution from NP.
Similarly, the branching ratio for K + → π + νν can be written as [38][39][40] Br K + → π + νν(γ) where λ ≡ |V us | and λ i ≡ V * is V id . The leading contribution is contained in the function X t which comprises the top-quark loops. It is known to NLO in QCD and electroweak interactions [33,35,39,41]. The parameter P c describes the short-distance contribution of the charm quark and has been calculated including NNLO QCD [42][43][44] and NLO electroweak corrections [45]. Other long-distance contributions are contained in the parameter δP c,u [40]. The hadronic matrix element of the low-energy operator is parameterized by κ + . This parameter is extracted from K 3 data and known precisely, including long-distance QED radiative corrections (∆ EM ), and NLO and partially NNLO corrections in chiral perturbation theory [46,47].
The branching ratio of the CP -violating neutral mode involves only the top-quark contribution and can be written as 3 (3.13) Again, the hadronic matrix element can be extracted from the K l3 decays, parametrized here by κ L [46]. As in the B q → µ + µ − decays, we include all NP effects as additional contributions to the SM top-quark function X t = X SM t + δX NP . At the order we consider, all NP effects originate from the modifications of the ttZ coupling. Thus, these are the same for B q → µ + µ − and K → πνν decays where x t = m 2 t /M 2 W and we used again the relation C φq,33 = −C (1) φq,33 at the scale Λ. We conclude this section by comparing our work to existing results in the literature. Rare meson decays have been used to constrain FCNC Zqt couplings, where q = u, c, in Ref. [6,7], and to constrain anomalous W tb couplings in Ref. [9]. The corrections to the Wilson coefficients presented in the above publications amount to one-loop threshold corrections at the electroweak scale. These corrections are scheme dependent [49,50]; the scheme dependence would cancel only when performing the two-loop running of the Wilson coefficients in the effective theory above the electroweak scale. On the other hand, the logarithmic dependence of the corrections on the scale Λ is scheme independent (since the leading-order anomalous dimensions are scheme independent). By effectively choosing a questionably low matching scale, of the order of the W -boson mass, these terms are rendered numerically insignificant in the above articles.

Numerics and discussion
In this section we present the constraints on anomalous ttZ couplings derived from the observables discussed above. We show the individual constraints and perform a combined fit using current experimental data. In addition, we show the impact of future precision measurements of the rare B and K decay branching ratios.
Our main results are summarized in Fig. 2. In the left panel we show the individual 68% CL regions resulting from the measurement of the T parameter, δg b L , Br(K + → π + νν), and Br(B s → µ + µ − ). In addition, we show the region compatible with the measurements in Table 1 at 68% and 95% CL.
We find that the branching ratio of B s → µ + µ − and the T parameter currently lead to the most stringent constraints. In particular, the combination of the two leads to a strong bound on both Wilson coefficients C (1) φq,33 log(µ W /Λ) v 2 /Λ 2 and C φu,33 log(µ W /Λ) v 2 /Λ 2 , of the order of a few percent. We note that t-channel single-top production leads to the bound −0.032 < v 2 C (1) φq,33 /Λ 2 < 0.044 (cf. Eq. (3.3)). This is weaker than the indirect bounds and leaves C φu,33 completely unconstrained.
In the future, we expect improvements in the measurement of Br(B s → µ + µ − ), with a final uncertainty of ∼ 5% [52]. In addition, various experiments plan to measure the branching ratios of the rare K decays with high precision. The NA62 experiment at CERN aims at a final precision of ∼ 10% for the charged mode, which could be improved to ∼ 3% at an experiment at Fermilab [53]. The KOTO experiment aims at a similar precision for the neutral mode. On the other hand, the bounds from EWPO are mainly obtained from fits to LEP data and we do not expect any significant improvement within the next few years. In the right panel of Fig. 2 we show our future projections. As an illustration we assume a branching ratio measurement of all three rare decay modes with the SM central values and a precision of 5%. We keep the current constraints from the EWPO, but note that these bounds could be improved at future e + e − colliders [54].
The indirect constraints on the anomalous ttZ couplings are much stronger than the constraints from direct searches, i.e. from tt + Z production, even after a high-luminosity upgrade of the LHC. For instance, the authors of Ref. [3] give the bounds −0.04 < v 2 /Λ 2 C (1) φq,33 < 0.19 and −0.13 < v 2 /Λ 2 C φu,33 < 0.32, assuming 3000 fb −1 of data. However, one has to keep in mind that indirect constraints rely on a set of assumptions. In this work we assumed i) only C Assumption i) is compatible only with NP models with non-trivial flavor structure, Current measurements Future projections while assumption ii) can be motivated by explicit models [15]. A simple way to deviate from assumption iii) is to consider models with a large enhancement of the bottom-quark Yukawa coupling; a generic example is a two Higgs-doublet model with large tan β. The large bottom-Yukawa coupling will induce flavor off-diagonal versions of the operators in Eq. (2.2) and Eq. (2.5). These off-diagonal operators lead to additional contributions to FCNC top decays and D 0 −D 0 mixing. In order to relate these observables to ttZ couplings, we assume MFV. Thus the resulting constraints on anomalous ttZ couplings are suppressed by CKM-matrix elements. As an illustrative example consider an extreme case where the bottom-Yukawa coupling is much larger than the top-Yukawa coupling. In this case, we have C φq,23 ∼ λ 2 C φq,33 etc., where λ ≡ |V us | ≈ 0.22 is the Wolfenstein parameter. Then D 0 − D 0 mixing is suppressed by λ 10 ≈ 10 −7 and thus completely negligible. Also, the branching ratio for t → cZ is φq,33 − C Using the present bound Br(t → cZ) < 0.05% given by the CMS collaboration [55] we see that the resulting bounds are not competitive with bounds from EWPO and rare B/K decays. Note that the off-diagonal operators will also lead to additional contributions to rare B/K decays and anomalous bbZ couplings. The generalization of our assumption ii) can be used to eliminate such contribution from these off-diagonal operators [15].
More generally, all rare decays in the down sector which are mediated by a Z penguin can be used to obtain bounds on anomalous ttZ couplings with our method. Suitable decays which will be measured in the future include B d → µ + µ − [56] and B → Kνν [57]. It would be interesting to allow for complex Wilson coefficients of the operators in Eq. (2.2) and study their effect on CP violation in rare meson decays.
To conclude, in this work we studied the effects of dimension-six operators, generating anomalous vector and axial-vector ttZ couplings at tree-level, on precision observables. In particular, we advocate the use of rare K and B meson decays to obtain strong constraints on ttZ couplings.