Large $t\to cZ$ as a Sign of Vector-Like Quarks in Light of the $W$ Mass

The rare flavour changing top quark decay $t\to cZ$ is a clear sign of new physics and experimentally very interesting due to the huge number of top quarks produced at the LHC. However, there are few (viable) models which can generate a sizable branching ratio for $t\to cZ$ -- in fact vector-like quarks seem to be the only realistic option. In this paper, we investigate all three representations (under the Standard Model gauge group) of vector-like quarks ($U$, $Q_1$ and $Q_7$) that can generate a sizable branching ratio for $t\to cZ$ without violating bounds from $B$ physics. Importantly, these are exactly the three vector-like quarks which can lead to a sizable positive shift in the prediction for $W$ mass, via the couplings to the top quark also needed for a sizable Br($t\to cZ$). Calculating and using the one-loop matching of vector-like quarks on the Standard Model Effective Field Theory, we find that Br($t\to cZ$) can be of the order of $10^{-6}$, $10^{-5}$ and $10^{-4}$ for $U$, $Q_1$ and $Q_7$, respectively and that in all three cases the large $W$ mass measurement can be accommodated.


I. INTRODUCTION
The Standard Model (SM) of particle physics contains three generations of chiral fermions, i.e.Dirac fields whose left and right-handed components transform differently under its gauge group.While a combination of LHC searches and flavour observables excludes a chiral 4 th generation [1,2], vector-like fermions (VLFs) can be added consistently to the SM without generating gauge anomalies.In fact, VLFs appear in many extensions of the SM such as grand unified theories [3][4][5], composite models or models with extra dimensions [6,7] and little Higgs models [8,9] (including the option of top condensation [10][11][12][13][14]).
There are three VLQs (U , Q 1 and Q 7 ) that generate a Z-t-c (and h-t-c) coupling but do not give rise to downquark FCNCs at tree-level, such that the former can be sizable.However, even these VLQs affect e.g. the W mass 1 and B decays at the loop-level.Therefore, it is important to calculate and include these effects in a phenomenological analysis in order to assess the possible size of t → Z(h)c and to evaluate if one can account for the recent measurement of the W mass by the CDF collaboration [51], which suggests that M W is larger than the expected within the SM.

II. SETUP AND MATCHING CALCULATION
There are seven possible representations (under the SM gauge group SU (3) C × SU (2) L × U (1) Y ) of VLQs, given in Table I, defining them as heavy fermions which are triplets of SU (3) C and that can mix with the SM quarks after EW symmetry breaking, i.e. fermions which can have couplings to the SM Higgs and a SM quark.The kinetic and mass terms 2 are where Here T A = 1 2 λ A and (S I ) jk are 0, 1 2 (τ I ) jk , and −i Ijk for the SU (2) L singlet, doublet, and triplet representations, respectively, and λ A and τ I are the Gell-Mann and the and T parameters, has previously been calculated at fixed order in Ref. [50], where they studied the contribution to electroweak observables and Higgs decays only. 2 Note that mass terms such as m U i Ū u i can always be removed by a field redefinition, such that the kinetic terms and the mass terms take the diagonal form shown in Eq. (II.1).
Table I.Representations of the Higgs, the SM quarks and of the VLQs under the SM gauge group.The three representations in bold are the ones relevant for our analysis as they generate flavour-changing top decays at tree level but down-quark FCNCs first appear at one-loop level.
Pauli matrices.The (generalized) Yukawa couplings are encoded in the Lagrangian where the first term contains the SM Yukawa couplings the second term the Higgs interactions with vector-like and SM quarks and the last term defines the Higgs interactions with two VLQs (given in the supplementary material as they are not relevant for our analysis).Here i, j = {1, 2, 3} are flavour indices and τ • T = I τ I T I .

A. SMEFT and Matching
We write the SMEFT Lagrangian as such that the Wilson coefficients have dimensions of inverse mass squared.Using the Warsaw basis [52], the operators generating modified gauge-boson couplings to quarks are Hq , Q (3) and the four-quark operators generating ∆F = 2 processes read qd , Q (8)  qu , Q (8) qd , (II.8) The explicit definitions of all these operators can be found in Ref. [52] and in the supplementary material.The dipole operators, responsible for radiative down-type quark decays after EW symmetry breaking, are Q dW and Q dB .In addition, we have the operator involving three Higgs fields, Q uH , that generates modifications of the Higgs-up-quark coupling, including possibly flavour changing ones, after EW symmetry breaking.Finally we also need two bosonic operators that lead to a modification to the W mass, Q HD and Q HW B , with their contributions approximately given by where v 246 GeV and (• • • ) indicates SMEFT operators not relevant in our scenario with VLQs. 3 An example diagram for the W mass correction is shown on the left in Fig. 1.
The tree-level matching of the operators generating modified Z-quark couplings is given by Hq alone, while right-handed modifications do not appear in our scenario, due to our (later) choice to set ξ d 1 to zero which removes all contributions to the C Hud coefficient.From these equations, we can see that only the representations U , Q 1 with coupling ξ u 1 and Q 7 (shown in bold in Table I) lead to effects in t → cZ while avoiding tree-level FCNCs in the down sector.An approximate formula for this branching ratio is (1)23 Hq −C (II.11)We calculated the one-loop matching on the SMEFT for these VLQs for the operators relevant for B physics, the W mass and EW precision observables (EWPOs) using MatchMakerEFT [56] and compared the results to our own calculation, finding perfect agreement.Details of our calculation and explicit expressions for the relevant Wilson coefficients are given in the supplementary material.
For the numerical analysis we use the software package smelli [72,73] (based on flavio [74] and wilson [75]), with {α, M Z , G F } constituting the input scheme.Furthermore, we work in the down-basis such that Cabibbo-Kobayashi-Maskawa (CKM) elements appear in transitions involving left-handed up-type quarks after EW symmetry breaking, meaning that Y d is diagonal in unbroken SU (2) L while Y u ≈ V † • diag(0, 0, y t ), with V being the CKM matrix.Note that in our setup the determination of CKM elements is already modified at tree-level.The resulting effects are consistently accounted for in smelli using the method described in Ref. [76], but choosing Γ(K , and ∆M d /∆M s as observables (see supplementary material for details).
Concerning the EW fit, the long standing tension in the W mass, previously with a significance of ≈1.8 σ [77][78][79], was recently increased by the measurement of the CDF collaboration [51].In [80], they have made a naive combination of the existing measurements (Tevatron [51], LEP [81], ATLAS [82] and LHCb [83]), assuming a common 4.7 MeV systematic uncertainty, and give a new world average of This value is 5.5 σ higher than the SM prediction M SM W = 80358.7 ± 6.0 MeV [78].Concerning B physics, even though the hints for lepton flavour universality (LFU) violation in b → s + − data cannot be explained by our LFU effects, an additional LFU part [84][85][86][87][88][89][90], generated by Z-b-s penguins, can further increase the agreement with data.In addition, box diagrams, like the one shown on the right in Fig. 1 also generate effect in B s − B s mixing (we use inputs from Ref. [91] for the SM prediction).
In all our analyses, we set the masses of the VLQs to 2 TeV.This is consistent the published model-independent bounds for third generation VLQs of M VLQ > 1.31 TeV limits from ATLAS [92] and recent conference reports [93, 94] which give slightly stronger limits.We also checked single VLQ production, which is model-dependent, and found the bounds for our scenarios to be weaker or nonexistent.Let us now consider the three cases of U , Q 1 and Q 7 numerically: U : In addition to the modified Z-t-c coupling, this VLQ also generates relevant effects in b → s + − transitions via a Z penguin, resulting in an C 9 ≈ −C 10 /4 pattern.In fact, mainly due to the measurements of P 5 [95] and B s → φµ + µ − [96, 97] there is a preference for a nonzero contribution with such a structure.The bounds from B s − B s mixing turn out to be weakened due to a partial (accidental) cancellation between the one-loop matching and the renormalization group equation (RGE) effect.Similarly, the contribution to b → sγ suffers from a cancellation, but here between terms generated by the The region preferred by all data (the global fit region with using the new experimental average in Eq. (III.2)) is shown at the 1 σ and 2 σ level, while the others regions correspond to 1 σ.We also show in the preferred region from the EW fit without the inclusion of the new M W result from CDF (red, dashed-dotted), where it can be seen that a large t → cZ branching ratio is also possible in this scenario.Note that in the plot for Q 7 the hatched regions on the top-left and top-right are already excluded by the current LHC limits on t → cZ.matching on the SMEFT and integrating out the W at the weak scale (b → sγ is included within the b → s + − region in Fig. 2).Concerning EWPOs, a shift in M W is dominantly generated by top-loop effects within the SMEFT (left diagram in Fig. 1), bringing theory and experiment into total agreement.Meanwhile, the second generation coupling ξ U 2 is constrained by the total Z width.These finding are summarised in Fig. 2 (top-left) where one can see that Br(t → cZ) can be of the order of 2 × 10 −6 , which could be probed by FCC-hh.
Q 1 with ξ u 1 : The VLQ Q 1 with the couplings ξ u 1 is found to be a very promising candidate for sizable rates of t → cZ, since it has small effects in B physics as it generates at tree-level only right-handed corrections to Z-up-quark couplings.At the same time, we can get an improvement concerning the agreement between theory and experiment in M W through the direct 1-loop contribution to C HD for large couplings is induced through top loops in the SMEFT (thus favouring the third generation coupling), while large couplings to charm quarks are ruled out by the total Z width, as shown in Fig. 2 (top-right).From there we see that an enhancement of Br(t → cZ) up to 1 × 10 −5 is possible, which could already be probed by the HE-LHC (albeit in an optimistic scenario with zero systematic errors).Note, however, that even in this quite unconstrained scenario Br(t → ch) can be at most 3 × 10 −6 , which is still a factor of three smaller than the reach of even the most optimistic FCC-hh scenario.Q 7 : In case of the VLQ Q 7 (see Fig. 2 (bottom-left)), the preferred sign for the contribution in b → s + − processes is generated, but in order for its size to be relevant, quite large couplings are required.Furthermore, for small third generation couplings (ξ 3 < 1) an effect with the wrong sign arises in M W , while for large couplings the sign reverses, which can be traced back to two different contribution, one proportional to (ξ 3 ) 4 the other involving (ξ Note that in the regime of such large couplings, small tensions with Higgs data arise in the h → ZZ, W W, γγ partial widths, with tensions of 1.8, 1.5, and 1.2 σ, respectively.Concerning Br(t → cZ), again an enhancement of the branching ratio up to 1 × 10 −5 is possible, which could be probed by the HE-LHC, FCChh, FCC-ee, or ILC.Given the large couplings allowed by data, Br(t → ch) can be enhanced up to 3 × 10 −5 , therefore potentially visible at the FCC-hh if the systematic uncertainties are well controlled.

IV. CONCLUSIONS
In this paper we examined the possibility of obtaining a sizable branching ratio for t → cZ within models containing VLQs.This is only feasible for representations which solely change Z couplings to the up-type quarks at tree-level while not not generating down-type FCNCs at this perturbative order, i.e.U , Q 1 and Q 7 .However, at the loop-level, B physics and electroweak observables are still affected.We therefore calculated the one-loop matching of these VLQs onto the SMEFT operators relevant for flavour and electroweak precision observables.
Using these results, we found in our phenomenological analysis that one can generate a sizable branching ratio for t → cZ of the order of 1 × 10 −6 , 1 × 10 −5 and 1 × 10 −4 , for U , Q 1 and Q 7 , respectively.Therefore, the parameter space of Q 7 is already constrained by LHC limits on t → cZ, while Q 1 and U can be tested by the HL-LHC and the FCC-hh respectively.Importantly, these three VLQ representations are also the ones which lead to a relevant and positive shift in the W mass and can thus explain the larger value of M W , compared to the SM prediction, obtained recently by the CDF collaboration.In fact, accounting for a larger M W requires sizable couplings to top quarks (see also Ref. [98]) which are also important for measurable effects in t → cZ, showing that these observables are correlated.Furthermore, U and Q 7 lead to LFU effects in b → s + − which cannot explain R(K ( * ) ) but affect observables like P 5 and B s → φµ + µ − and, in combination with LFU violating effects, can further improve the description of data.In conclusion, t → cZ is an unambiguous signal of VLQs and sizable branching ratios of it, within the range of the HL-LHC, are motivated by the recent CDF measurement of the W mass. consider α s terms but these only arise in a) the four-quark operators with a Kronecker delta, which therefore cannot induce a change of flavour, b) totally bosonic operators which are not interesting for our purposes, and c) gluon dipoles at one-loop, which are too small to generate an observable effect.
The main technicality associated with matching is to correctly account of the one-loop renormalization of the SMEFT, specifically the terms proportional to the SM quark Yukawas, as these allow the Q (1,3) Hq operators to contribute to the renormalization of the Q (1,3)   qq operators, and similarly for the other ∆F = 2 operators.We perform the calculation by matching off-shell amplitudes calculated in the full VLQ theory and the SMEFT, and allowing for a MS counter-term contribution on the SMEFT side.A potential IR divergence is regulated using the Higgs doublet mass µ H , which is set to zero at the end of the calculation.
One-loop matching for ∆F = 2 processes: The one-loop matchings onto the ∆F = 2 operators are obtained from the diagrams in Fig. S.1.Among the three representations (U , Q 1 with ξ u 1 , and Q 7 ), only U contributes to B s −B s mixing.The one-loop matching condition for U is C (1) where the IR-finite loop function is and the renormalization scale µ should be O(M VLQ ).
One-loop matching for down-quark magnetic dipoles: Some partial work was done in Ref. [32] for the dipole operators, but note that we find our VLQ interactions are more general than those considered in that work, and additional diagrams contribute to the one-loop matching onto the dipole operators C dB and C dW .Some typical diagrams are shown in Fig. S.2.Among the three representations, only U produces the one-loop matching condition,  One-loop matching for modified gauge boson couplings: While the tree-level SMEFT operators, generated by the VLQs, can affect W and Z couplings, the low-energy Z coupling to up-and down-type quarks specifically depends on C Hq , C Hd , respectively (see Eq. (II.10)).As we can see that this leads to some of the Z quark couplings remaining SM-like for certain VLQs (as summarised in Table S.1).Since these interactions are constrained by EWPO, and also contribute to many interesting processes such as b → s , we also calculate the one-loop matching for the U, Q 1 , Q 7 cases where they are not already present at tree level.

Figure 1 .
Figure 1.Examples of Feynman diagrams showing the U contributions to the operator Q HD , affecting the W -boson mass (left), and Q (1,3) qq , affecting B s −B s mixing (right).

Figure 2 .
Figure 2. Preferred regions in the ξ 2 -ξ 3 plane for the three representations of VLQ that generate t → cZ at tree-level but give rise to down-quark FCNCs only at the loop level: U (top-left), Q 1 (top-right), and Q 7 (bottom-left).The contour lines show the predicted size of Br(t → cZ) × 105 .The region preferred by all data (the global fit region with using the new experimental average in Eq. (III.2)) is shown at the 1 σ and 2 σ level, while the others regions correspond to 1 σ.We also show in the preferred region from the EW fit without the inclusion of the new M W result from CDF (red, dashed-dotted), where it can be seen that a large t → cZ branching ratio is also possible in this scenario.Note that in the plot for Q 7 the hatched regions on the top-left and top-right are already excluded by the current LHC limits on t → cZ.

Figure S. 1 .Figure S. 2 .
Figure S.1.Typical box diagrams giving rise to the ∆F = 2 matching coefficients, where the external fermions ψ can be any of the SM quarks {q L , u R , d R }, and in the right diagram the internal q represents any SM quark and gives rise to the Yukawa corrections.

Figure S. 3 .
Figure S.3.Typical box diagrams giving rise to the W mass, where the internal fermions q represents any SM quark.

Q 7 .
(S.3.8)One-loop matching for the W mass: The one-loop matching onto the operators Q HD and Q HW B , which modify the W -boson mass prediction, are obtained from the diagrams in Fig. S.3.The matching conditions for the three representation VLQs are

Table S .
1. Overview on modified Z-quark couplings (in broken SU (2) L ) at tree level in the VLQ models.