# Chiral pair density wave as a precursor of the pseudogap in kagomé
superconductors

Narayan Mohanta
Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667,
India

###### Abstract

Motivated by scanning tunneling microscopy experiments on $A$V3Sb5 ($A$ =
Cs, Rb, K) that revealed periodic real-space modulation of electronic states
at low energies, I show using model calculations that a triple-Q chiral pair
density wave (CPDW) is generated in the superconducting state by a charge
order of $2a\\!\times\\!2a$ superlattice periodicity, intertwined with a time-
reversal symmetry breaking orbital loop current. In the presence of such a
charge order and orbital loop current, the superconducting critical field is
enhanced beyond the Chandrasekhar-Clogston limit. The CPDW correlation
survives even when the long-range superconducting phase coherence is
diminished by a magnetic field or temperature, stabilizing an exotic granular
superconducting state above and in the vicinity of the superconducting
transition. The presented results suggest that the CPDW can be regarded as the
origin of the pseudogap observed near the superconducting transition.

Understanding electronic properties arising from coexisting superconductivity
and various density-wave orders has remained as a central problem in condensed
matter physics. It has captivated the physics community for decades in the
context of high-temperature cuprate superconductors; the recently-synthesized
kagomé metals $A$V3Sb5 ($A$ = Cs, Rb, K) have revived the interest Ortiz
_et al._ (2019). The V atoms in these compounds form a kagomé lattice, and the
Fermi level is predominantly populated by V $3d$ orbitals. The electronic band
structure exhibits Dirac points and nearly-flat less-dispersive bands Ortiz
_et al._ (2020); Shi  _et al._ (2022). Strong correlation of the nearly-flat
bands, topological effects from the Dirac fermions, van Hove singularities,
and frustration effects in the kagomé geometry are favorable conditions for
instabilities towards long-range many-body order to set in. Superconductivity
with a gap-to-$T_{c}$ ratio $2\Delta_{0}/k_{B}T_{c}\\!\approx\\!5$ was found
below $T_{c}\\!\approx\\!2.5$ K Ortiz  _et al._ (2020). A chiral charge order
was found to appear below $T_{\small{\rm co}}\\!\approx\\!94$ K with broken
time-reversal symmetry (TRS) but without the trace of any long-range magnetic
order, indicating the presence of an intertwined orbital loop current Jiang
_et al._ (2021); Mielke  _et al._ (2022); Khasanov  _et al._ (2022). The
absence of acoustic phonon anomaly at the charge-order wave vector rules out
the Pierls instability related to the Fermi surface nesting and phonon
softening as a possible mechanism, and implies that extended Coulomb
interactions at a van Hove filling may be responsible for it Li  _et al._
(2021); Denner  _et al._ (2021); Christensen  _et al._ (2021). A pressure-
driven transition from fully-gapped to partially-gapped superconductivity and
the coexistence of the superconductivity with the charge order over a large
parameter regime suggest unconventional pairing in these compounds Guguchia
_et al._ (2023). Alternative scenarios include non-chiral, anisotropic
$s$-wave superconductivity, supported by recent experimental findings at
different pressures Roppongi  _et al._ (2023).

A suppressed electronic density of states at the Fermi level, known as the
‘pseudogap’, posed an enigmatic problem in the high-temperature cuprate
superconductors. A similar pseudogap with a V-shaped density of states was
observed in scanning tunneling microscopy experiments on $A$V3Sb5 and
subsequent theoretical analysis, with periodic modulations of both charge
density and Cooper pair density of $2a\\!\times\\!2a$ superlattice periodicity
($a$ being the lattice constant) Zhao  _et al._ (2021); Chen  _et al._ (2021);
Xu  _et al._ (2021); Jiang  _et al._ (2023). The findings are usually
indicative of a nodal pairing symmetry or ungapped sections of the Fermi
surface. The concomitant periodic modulations of both superfluid and normal
fluid raised a series of questions including the origin of the pseudogap found
in the tunneling spectra.

Here, I focus on the observed variation of the density of states in the
pseudogap near the superconducting transition and show that a chiral density
wave of $s$-wave Cooper pairs can account for it. The chiral pair density wave
(CPDW) is generated in the superconducting state by the TRS breaking charge
order and it persists above the superconducting transition without long-range
superconducting phase coherence. This CPDW state can be described by a pairing
gap
$\Delta(\mathbf{r})\\!=\\!\sum_{a}\Delta_{a}e^{i(\mathbf{Q}_{a}\cdot\mathbf{r}+\varphi_{a})}$
at a lattice site position $\mathbf{r}$, $\Delta_{a}$ and $\varphi_{a}$ being
the magnitude and the relative phase of the pairing amplitude along three
characteristic momentum $\mathbf{Q}_{a}$ ($a\\!=\\!1,2,3$), set by the charge
order periodicity. The presented theoretical arguments are based on calculated
density of states $\rho(E)$ and Fourier transformed local density of states
$\rho({\mathbf{Q}_{p}},E)$ at a CPDW wave vector ${\mathbf{Q}_{p}}$. In the
vicinity and above the critical field $B_{c}$ or critical temperature $T_{c}$
for the superconducting transition, determined by a vanishing superfluid
density $n_{s}$, $\rho({\mathbf{Q}_{p}},E)$ reveals a particle-hole symmetric
density of states around zero energy, thereby ruling out the charge-ordered
electronic states as a possible origin of the pseudogap. Remarkably, the
critical value of the magnetic field, perpendicular to the kagomé plane, is
found to be enhanced beyond the usual Chandrasekhar-Clogston limit in the
presence of the orbital loop current, implying that a highly-provoking quantum
state prevails above the superconducting transition.

Two types of charge order were reported—star of David and tri-hexagonal or
inverse star of David patterns, both of $2a\\!\times\\!2a$ periodicity Kato
_et al._ (2022); Gupta  _et al._ (2022); Tan  _et al._ (2021). A chiral flux
phase, compatible with the symmetry of the kagomé lattice and broken TRS, was
shown to be energetically favorable Feng  _et al._ (2021). The tri-hexagonal
charge order which has been observed prominently in most compounds, as shown
in Fig. 1, is considered in the theoretical model and results presented below.
Such an unusual charge ordered state is also supported by a number of
interesting phenomena such as anomalous Hall effect and Nernst effect Yang
_et al._ (2020); Gan  _et al._ (2021); Yu  _et al._ (2021a).

Figure 1: Charge order configuration with intertwined orbital loop current on
the kagomé lattice, analogous to the tri-hexagonal pattern observed in
experiments. The yellow and cyan colors represent the modulation in the
chemical potential $\mu_{\rm co}$, while the arrows represent the loop current
propagation direction and the associated flux $\varphi$.

Model and observables.—To model the superconducting state and the
experimentally-observed pseudogap, the minimal tight-binding Hamiltonian at
the mean spin-singlet pairing field on the kagomé lattice is expressed as

$\displaystyle{\cal H}\\!=$ $\displaystyle\\!-t\sum_{\langle
ij\rangle,\sigma}(c_{i\sigma}^{\dagger}c_{j\sigma}+{\rm
H.c.})-\sum_{i,\sigma}(\mu_{0}+\xi_{i}\mu_{\rm
co})c_{i\sigma}^{\dagger}c_{i\sigma}$ (1)
$\displaystyle-\sum_{i}(\Delta_{i}c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}+{\rm
H.c.})-{i}t_{\rm lc}\sum_{\langle
ij\rangle,\sigma}(c_{i\sigma}^{\dagger}c_{j\sigma}-{\rm H.c.}),$

where $t$ is the nearest-neighbor hopping energy, $\mu_{0}$ is the global
chemical potential, $\mu_{\rm co}$ is the charge order amplitude, $\xi_{i}$ is
a local variable ($\pm 1$) that generates the tri-hexagonal charge order
pattern, shown in Fig. 1, $\Delta_{i}$ is the local spin-singlet pairing gap,
and the complex nearest-neighbor hopping ${i}t_{\rm lc}$ incorporates the TRS
breaking orbital loop current. The Hamiltonian is diagonalized by using the
standard unitary transformation of the fermionic fields
$c_{i\sigma}\\!=\\!\sum_{n}u_{ni}^{\sigma}\gamma_{n}+v_{ni}^{\sigma*}\gamma_{n}^{\dagger}$,
where $\gamma_{n}$ is an annihilation operator acting on the $n^{\rm th}$
eigenstate, and $u_{ni}^{\sigma}$ ($v_{ni}^{\sigma}$) is the corresponding
quasiparticle (quasihole) amplitude at site $i$ and spin $\sigma$. The
eigenstates are obtained by solving the Bogoliubov-de Gennes equations
$\sum_{j}{\cal H}_{ij}\psi_{nj}\\!=\\!E_{n}\psi_{ni}$, subject to the self-
consistent gap equation sup

$\displaystyle\Delta({\bf r}_{i})=\frac{{\cal
U}}{2}\sum_{n}\big{[}u_{ni}^{\uparrow}v_{ni}^{\downarrow*}-u_{ni}^{\downarrow}v_{ni}^{\uparrow*}\big{]}\tanh\Big{(}\frac{E_{n}}{2k_{B}T}\Big{)},$
(2)

where
$\psi_{ni}\\!=\\![u_{ni}^{\uparrow},u_{ni}^{\downarrow},v_{ni}^{\uparrow},v_{ni}^{\downarrow}]^{T}$,
${\cal U}$ is the pair-wise attractive potential and $T$ is the temperature.
Throughout the presented results, $\mu_{0}$ is kept at zero which places the
Fermi level close to one of the van Hove singularities, $t\\!=\\!1$ and ${\cal
U}\\!=\\!2$. The relevant energy scale is the maximum pairing gap magnitude,
which was found experimentally to be $\Delta\\!\approx\\!0.52$ meV Chen  _et
al._ (2021), is taken here to be the unit for all energies in what follows.

To keep track of the superconducting transition, the global superconducting
phase rigidity, determined by the superfluid density, is calculated from the
effective Drude weight, given by Scalapino  _et al._ (1992); sup

$\displaystyle n_{s}=\frac{D_{s}}{\pi
e^{2}}=-\langle\kappa\rangle+\Lambda({\bf Q}\rightarrow 0,i\omega\rightarrow
0),$ (3)

where the first term on the right hand side is the diamagnetic response, with
the local kinetic energy expressed in terms of the Bogoliubov quasiparticle
weights as

$\displaystyle\kappa_{i}=$ $\displaystyle-t\sum_{\langle
j\rangle,n,\sigma}\big{[}u_{ni}^{\sigma}u_{nj}^{\sigma*}+{\rm
c.c.}\big{]}f(E_{n})$
$\displaystyle+\big{[}v_{ni}^{\sigma}v_{nj}^{\sigma*}+{\rm
c.c.}\big{]}(1-f(E_{n})).$ (4)

The second term represents the paramagnetic response, obtained by the
transverse current-current correlation function

$\displaystyle\Lambda({\bf Q}\\!\rightarrow\\!0,i\omega\\!\rightarrow\\!0)\\!$
$\displaystyle=\\!\frac{1}{N}\\!\sum_{i,j,n_{1},n_{2}}^{\sigma,\sigma^{\prime}}\\!{\cal
A}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}\\!\big{[}{\cal
A}_{n_{1}n_{2}}^{j\sigma\sigma^{\prime}*}\\!+\\!{\cal
B}_{n_{1}n_{2}}^{j\sigma\sigma^{\prime}}\big{]}$
$\displaystyle\times\frac{f(E_{n1})-f(E_{n2})}{E_{n1}-E_{n2}},$ (5)

where $N$ is the total number of lattice sites and

$\displaystyle{\cal
A}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}=2\big{[}u_{n_{1}j}^{\sigma^{\prime}*}u_{n_{2}i}^{\sigma}-u_{n_{1}i}^{\sigma*}u_{n_{2}j}^{\sigma^{\prime}}\big{]},$
$\displaystyle{\cal
B}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}=2\big{[}v_{n_{1}j}^{\sigma^{\prime}*}v_{n_{2}i}^{\sigma}-v_{n_{1}i}^{\sigma*}v_{n_{2}j}^{\sigma^{\prime}}\big{]}.$
(6)

The local density of states, an observable that can be compared with the
scanning tunneling microscopy data, is calculated via

$\displaystyle\rho({\mathbf{r}_{i},E})\\!=\\!\sum_{n}\big{[}|u_{ni}^{\sigma}|^{2}\delta(E\\!-\\!E_{n})\\!+\\!|v_{ni}^{\sigma}|^{2}\delta(E\\!+\\!E_{n})\big{]}.$
(7)

The total density of states $\rho(E)$ is obtained by summing over all lattice
sites, and the Fourier transformed local density of states at a momentum
$\mathbf{Q}$ is obtained using

$\displaystyle\rho({\mathbf{Q},E})\\!=\\!\frac{1}{N}\sum_{i}\cos\\!\big{(}\mathbf{Q}\cdot\mathbf{r}_{i}\big{)}\rho({\mathbf{r}_{i},E}).$
(8)

The local and non-local modulations of the density of states are useful to
analyze the presence of the particle-hole symmetry and hence, to differentiate
the contributions from the normal fluid and the superfluid, as will be evident
from the numerical results presented below.

The CPDW state.—The density wave of $s$-wave bosons is envisaged from the
pairing gap $\Delta({\bf r}_{i})\\!=\\!\Delta_{m}e^{i\theta_{i}}$, both real
and imaginary parts of which reveal $2a\\!\times\\!2a$ periodic modulation
(real part is shown on the color scale in Fig. 2(a)). The phase angle
$\theta_{i}$ also shows a periodic structure (arrows in Fig. 2(a)). This
intriguing CPDW state is confirmed further by the Fourier transform of the
pair-pair correlation function

$\displaystyle
C({\mathbf{Q}})\\!=\\!\frac{1}{N}\sum_{i,j}\langle\Delta(\mathbf{r}_{i})\Delta(\mathbf{r}_{j})\rangle
e^{-i\mathbf{Q}\cdot\mathbf{r}_{ij}},$ (9)

which shows three characteristic momenta (see Fig. 2(b)), given by ($\pm\pi$,
$\pi/\sqrt{3}$) and (0, 2$\pi/\sqrt{3}$). It is confirmed that the CPDW state
is generated in the superconducting state due to the interplay between
$s$-wave onsite pairing, charge order and the TRS breaking orbital loop
current. Three types of such triple-$\mathbf{Q}$ correlation, at different
momenta, have been observed in the experiments Zhao  _et al._ (2021); Chen
_et al._ (2021), implying that the density waves are cascaded between normal
fluid and superfluid i.e. the CPDW can also induce subsequent charge orders.

Figure 2: (a) Profile of the pairing gap $\Delta({\bf
r}_{i})\\!=\\!\Delta_{m}e^{i\theta_{i}}$ solution on the considered $10a\times
10a$ lattice with periodic boundary conditions—the color scale shows the real
part; the arrows show the phase $\theta_{i}$. (b) Fourier transform of the
pair-pair correlation function $C({\mathbf{Q}})$ obtained on a $20a\times 20a$
lattice with periodic boundary conditions, showing the three characteristic
momenta (six-peak structure), indicative of the CPDW state. The hexagon
plotted with dashed lines depicts the Brillouin zone. Parameters used are
$\mu_{\rm co}=0.5$ and $t_{\rm lc}=1$. Figure 3: (a), (b) Variation of the
average gap magnitude $|\Delta|$ (without and with the charge order and the
loop current) and superfluid density $n_{s}$ with magnetic field $B_{z}$ and
temperature $T$. (c), (d) Density of states $\rho(E)$ for different $B_{z}$
and $T$ near the critical values $B_{c}$ and $T_{c}$, determined by vanishing
$n_{s}$. Insets in (c), (d) show the density of states at zero energy
$\rho(0)$ as a function of $B_{z}$ and $T$, respectively. The results were
obtained on a $20a\times 20a$ lattice with periodic boundary conditions. All
other parameters are the same as in Fig. 2. A constant offset has been added
to the vertical axis for each curve in (c) and (d) for clarity.

Pseudogap.—The coexistence of charge order and superconductivity in AV3Sb5
over a large parameter regime raised the natural question whether there is a
cooperation between the two commonly-known competing orders Yu  _et al._
(2021b). The present analysis shows that the superconducting gap is suppressed
in the presence of the charge order and the orbital loop current at zero
temperature and zero magnetic field. However, the average pairing gap
$|\Delta|$ vanishes at a magnetic field and a temperature, larger than the
critical values $B_{c}$ and $T_{c}$, determined by a vanishing superfluid
density $n_{s}$ (Fig. 3(a)-(b)). The magnetic field of amplitude $B_{z}$ was
incorporated by the Hamiltonian ${\cal
H}_{{}_{Z}}\\!=\\!-\mu_{{}_{B}}B_{z}\sum_{i,\sigma,\sigma^{\prime}}{\bm{\sigma}}_{\sigma\sigma^{\prime}}^{z}c_{i\sigma}^{\dagger}c_{i\sigma^{\prime}}$
which describes the Zeeman exchange coupling. Remarkably, the critical field
at which $|\Delta|$ drops to zero is enhanced by more than 20$\%$ above
$B_{c}$ in the presence of the charge order and the loop current.

Figure 4: Profile of the absolute value of the pairing gap $|\Delta({\bf
r}_{i})|$, self-consistently obtained on a $8a\times 8a$ lattice with periodic
boundary conditions, at different magnetic fields across the superconducting
transition: (a) $B_{z}\\!=\\!0$, (b) $B_{z}\\!=\\!1.1B_{c}$, and (c)
$B_{z}\\!=\\!1.5B_{c}$. In the vicinity of the superconducting transition (the
case of plot (b)), the pairing gap is locally suppressed in the charge-ordered
clusters, creating a granular phase having superconducting patches separated
by non-superconducting regions. All other parameters are the same as in Fig.
2. Figure 5: Fourier transformed pair-pair correlation function
$C({\mathbf{Q}})$ at (a) magnetic field $B_{z}\\!=\\!1.1B_{c}$, temperature
$T\\!=\\!0$, and (b) $B_{z}\\!=\\!0$, $T\\!=\\!1.03T_{c}$, indicating the
presence of the CPDW correlation above the superconducting transition. (c),
(d) Fourier-transformed local density of states $\rho({\mathbf{Q}_{p},E})$ at
a characteristic CPDW momentum $\mathbf{Q}_{p}$, shown in (a), at different
values of $B$ and $T$ across the superconducting transition, revealing
particle-hole symmetric quasiparticle states with a non-zero weight at
$E\\!=\\!0$. The results were obtained on a $20a\times 20a$ lattice with
periodic boundary conditions. All other parameters are the same as in Fig. 2.
A constant offset has been added to the vertical axis for each curve in (c)
and (d) for clarity.

It is known that a conventional spin-singlet superconductor, with a gap around
the Fermi level, has a vanishing paramagnetic susceptibility at $T\\!=\\!0$,
and hence it cannot lower its Free energy indefinitely by spin-polarizing the
quasiparticle states in the presence of a Zeeman magnetic field. Consequently,
when the Zeeman energy gain is comparable to the superconducting condensation
energy, given by
$\mu_{{}_{B}}B_{c0}\\!=\\!\Delta_{0}/\sqrt{2}\\!\approx\\!0.7\Delta_{0}$,
known as the Chandrasekhar-Clogston limit Chandrasekhar (1962); Clogston
(1962), there is a transition to the normal state. An exception to this
stringent condition occurs in the case of Fulde-Ferrell-Larkin-Ovchinikov type
finite-momentum condensates Fulde and Ferrell (1964); Larkin and Ovchinnikov
(1964) and in thin superconducting films with a large spin-orbit coupling
Bruno and Schwartz (1973). The enhancement in the critical field for vanishing
$|\Delta|$ in the kagomé lattice with a charge order and orbital loop current
indicates the formation of an unconventional state above and in the vicinity
of the superconducting transition. However, it can be attributed to the non-
zero density of states $\rho(E)$ within the superconducting gap (Fig. 3(c)) in
the pseudogap state. At the critical field $B_{c}$ for vanishing $n_{s}$,
$|\Delta|$ exhibits a dip while $\rho(E\\!=\\!0)$ shows a zero weight (inset
in (Fig. 3(c))), attesting the appearance of an unconventional state
immediately above $B_{c}$. The enhancement of the critical magnetic field and
the appearance of the pseudogap in the CPDW state suggest that there is a
correlation among these phenomena. The temperature driven transition to the
normal state also reveals a similar pseudogap (Fig. 3(d)), though the
variation of $\rho(E\\!=\\!0)$ with $T$ is rather monotonic. Moreover, the
V-shaped density of states and the multiple coherence peaks around the gap
show similarities with those observed in the tunneling spectra Chen  _et al._
(2021); Zhao  _et al._ (2021); Xu  _et al._ (2021).

CPDW correlation above superconducting transition.—To gain insights into the
origin of the pseudogap, the profile of the absolute value of the pairing gap
at different fields across the superconducting transition was looked at in
Fig. 4(a),(b),(c). It is found that above and in the vicinity of the critical
magnetic field, the pairing gap is locally suppressed in charge-ordered
clusters, stabilizing an emergent granular superconducting state in which the
pairing gap survives locally even though the long-range superconducting phase
coherence vanishes. The magnetization profile also reveals periodic
modulation, and the interplay of magnetization and superconducting gap
prevails in the entire granular superconducting phase sup . The fourier-
transformed pair-pair correlation function $C({\mathbf{Q}})$ and the fourier-
transformed local density of states $\rho({\mathbf{Q}_{p},E})$ at
$\mathbf{Q}_{p}$, one of the three characteristic momenta for the CPDW, were
also investigated (Fig. 5(a)-(d)) across the superconducting transition.
Interestingly, the CPDW correlation survives above the critical magnetic field
$B_{c}$ and critical temperature $T_{c}$. The observable
$\rho({\mathbf{Q}_{p},E})$ is particle-hole symmetric i.e. it is symmetric
when $E\\!\rightarrow\\!-E$, above and in the vicinity of $B_{c}$ and $T_{c}$,
ruling out other possible mechanisms of the pseudogap such as charge order of
electronic states or modulation due to electron scattering from a periodic
potential. From these findings, it can be argued that the modulations of the
density of states in the pseudogap, observed in the experiments, are a
consequence of the CPDW of $s$-wave Cooper pairs without a global phase
coherence.

Discussion and conclusion.—The superconducting state can be influenced by
multiple properties of the compounds such as the TRS breaking loop current,
the rotational symmetry-breaking nematic order, fermi surface nesting, Coulomb
interactions and sublattice interference Xiang  _et al._ (2021); Jin  _et al._
(2022); Nie  _et al._ (2022); Zhou and Wang (2022); Wu  _et al._ (2021).
Despite the complex nature of the pairing mechanism, there are growing
experimental evidences in support of spin-singlet $s$-wave pairing such as the
absence of a nodal state while transitioning from an anisotropic full-gap
state to an isotropic full-gap state driven by impurity concentration Roppongi
_et al._ (2023), and the appearance of a prominent Hebel-Slichter coherence
peak immediately below $T_{c}$ Mu  _et al._ (2021). The proposed CPDW of
$s$-wave Cooper pairs, therefore, provides a natural explanation for many
paradoxical experimental observations, including the pseudogap in the
tunneling spectra.

To summarize, it is shown that a CPDW state of $2a\\!\times\\!2a$ periodicity
emerges spontaneously in the kagomé lattice due to the interplay of onsite
spin-singlet superconductivity with a charge order of the same periodicity and
an orbital loop current. The CPDW correlation survives beyond the
superconducting transition in granular regions without global superconducting
phase coherence, producing a V-shaped particle-hole symmetric density of
states and pseudogap, which are otherwise indicative of a nodal unconventional
pairing symmetry.

Acknowledgements.—The work was supported by Science and Engineering Research
Board, India (Research Grant No. SRG/2023/001188). Numerical calculations were
performed at the computing resources of PARAM Ganga at the Indian Institute of
Technology Roorkee, provided by National Supercomputing Mission, implemented
by C-DAC, and supported by the Ministry of Electronics and Information
Technology and Department of Science and Technology, Government of India.

## References

  * Ortiz  _et al._ (2019) B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. Mangum, I. W. H. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, E. Ertekin, T. M. McQueen, and E. S. Toberer, “New kagomé prototype materials: discovery of KV3Sb5, RbV3Sb5, and CsV3Sb5,” Phys. Rev. Mater. 3, 094407 (2019).
  * Ortiz  _et al._ (2020) B. R. Ortiz, S. M. L. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He, and S. D. Wilson, “CsV3Sb5: A ${{\cal Z}}_{2}$ topological kagomé metal with a superconducting ground state,” Phys. Rev. Lett. 125, 247002 (2020).
  * Shi  _et al._ (2022) M. Shi, F. Yu, Y. Yang, F. Meng, B. Lei, Y. Luo, Z. Sun, J. He, R. Wang, Z. Jiang, Z. Liu, D. Shen, T. Wu, Z. Wang, Z. Xiang, J. Ying, and X. Chen, “A new class of bilayer kagomé lattice compounds with Dirac nodal lines and pressure-induced superconductivity,” Nat. Commun. 13, 2773 (2022).
  * Jiang  _et al._ (2021) Y.-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, G. Xu, Z. Guguchia, J. He, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, Z. Wang, R. Thomale, T. Neupert, S. D. Wilson, and M. Z. Hasan, “Unconventional chiral charge order in kagomé superconductor KV3Sb5,” Nat. Mater. 20, 1353 (2021).
  * Mielke  _et al._ (2022) C. Mielke, D. Das, J.-X. Yin, H. Liu, R. Gupta, Y.-X. Jiang, M. Medarde, X. Wu, H. C. Lei, J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y. Shi, R. Khasanov, M. Z. Hasan, H. Luetkens, and Z. Guguchia, “Time-reversal symmetry-breaking charge order in a kagomé superconductor,” Nature 602, 245 (2022).
  * Khasanov  _et al._ (2022) R. Khasanov, D. Das, R. Gupta, C. Mielke, M. Elender, Q. Yin, Z. Tu, C. Gong, H. Lei, E. T. Ritz, R. M. Fernandes, T. Birol, Z. Guguchia, and H. Luetkens, “Time-reversal symmetry broken by charge order in CsV3Sb5,” Phys. Rev. Res. 4, 023244 (2022).
  * Li  _et al._ (2021) H. Li, T. T. Zhang, T. Yilmaz, Y. Y. Pai, C. E. Marvinney, A. Said, Q. W. Yin, C. S. Gong, Z. J. Tu, E. Vescovo, C. S. Nelson, R. G. Moore, S. Murakami, H. C. Lei, H. N. Lee, B. J. Lawrie, and H. Miao, “Observation of unconventional charge density wave without acoustic phonon anomaly in kagomé superconductors AV3Sb5 (A=Rb, Cs),” Phys. Rev. X 11, 031050 (2021).
  * Denner  _et al._ (2021) M. M. Denner, R. Thomale, and T. Neupert, “Analysis of charge order in the kagomé metal AV3Sb5 (A=K, Rb, Cs),” Phys. Rev. Lett. 127, 217601 (2021).
  * Christensen  _et al._ (2021) M. H. Christensen, T. Birol, B. M. Andersen, and R. M. Fernandes, “Theory of the charge density wave in AV3Sb5 kagomé metals,” Phys. Rev. B 104, 214513 (2021).
  * Guguchia  _et al._ (2023) Z. Guguchia, C. Mielke, D. Das, R. Gupta, J.-X. Yin, H. Liu, Q. Yin, M. H. Christensen, Z. Tu, C. Gong, N. Shumiya, M. S. Hossain, T. Gamsakhurdashvili, M. Elender, P. Dai, A. Amato, Y. Shi, H. C. Lei, R. M. Fernandes, M. Z. Hasan, H. Luetkens, and R. Khasanov, “Tunable unconventional kagomé superconductivity in charge ordered RbV3Sb5 and KV3Sb5,” Nat. Commun. 14, 153 (2023).
  * Roppongi  _et al._ (2023) M. Roppongi, K. Ishihara, Y. Tanaka, K. Ogawa, K. Okada, S. Liu, K. Mukasa, Y. Mizukami, Y. Uwatoko, R. Grasset, M. Konczykowski, B. R. Ortiz, S. D. Wilson, K. Hashimoto, and T. Shibauchi, “Bulk evidence of anisotropic $s$-wave pairing with no sign change in the kagomé superconductor CsV3Sb5,” Nat. Commun. 14, 667 (2023).
  * Zhao  _et al._ (2021) H. Zhao, H. Li, B. R. Ortiz, S. M. L. Teicher, T. Park, M. Ye, Z. Wang, L. Balents, S. D. Wilson, and I. Zeljkovic, “Cascade of correlated electron states in the kagomé superconductor CsV3Sb5,” Nature 599, 216 (2021).
  * Chen  _et al._ (2021) H. Chen, H. Yang, B. Hu, Z. Zhao, J. Yuan, Y. Xing, G. Qian, Z. Huang, G. Li, Y. Ye, S. Ma, S. Ni, H. Zhang, Q. Yin, C. Gong, Z. Tu, H. Lei, H. Tan, S. Zhou, C. Shen, X. Dong, B. Yan, Z. Wang, and H.-J. Gao, “Roton pair density wave in a strong-coupling kagomé superconductor,” Nature 599, 222 (2021).
  * Xu  _et al._ (2021) H.-S. Xu, Y.-J. Yan, R. Yin, W. Xia, S. Fang, Z. Chen, Y. Li, W. Yang, Y. Guo, and D.-L. Feng, “Multiband superconductivity with sign-preserving order parameter in kagomé superconductor CsV3Sb5,” Phys. Rev. Lett. 127, 187004 (2021).
  * Jiang  _et al._ (2023) H.-M. Jiang, M.-X. Liu, and S.-L. Yu, “Impact of the orbital current order on the superconducting properties of the kagome superconductors,” Phys. Rev. B 107, 064506 (2023).
  * Kato  _et al._ (2022) T. Kato, Y. Li, T. Kawakami, M. Liu, K. Nakayama, Z. Wang, A. Moriya, K. Tanaka, T. Takahashi, Yu. Yao, and T. Sato, “Three-dimensional energy gap and origin of charge-density wave in kagomé superconductor KV3Sb5,” Commun. Mater. 3, 30 (2022).
  * Gupta  _et al._ (2022) R. Gupta, D. Das, C. Mielke, E. T. Ritz, F. Hotz, Q. Yin, Z. Tu, C. Gong, H. Lei, T. Birol, R. M. Fernandes, Z. Guguchia, H. Luetkens, and R. Khasanov, “Two types of charge order with distinct interplay with superconductivity in the kagomé material CsV3Sb5,” Commun. Phys. 5, 232 (2022).
  * Tan  _et al._ (2021) H. Tan, Y. Liu, Z. Wang, and B. Yan, “Charge density waves and electronic properties of superconducting kagomé metals,” Phys. Rev. Lett. 127, 046401 (2021).
  * Feng  _et al._ (2021) X. Feng, K. Jiang, Z. Wang, and J. Hu, “Chiral flux phase in the kagomé superconductor AV3Sb5,” Sci. Bull. 66, 1384 (2021).
  * Yang  _et al._ (2020) S.-Y. Yang, Y. Wang, B. R. Ortiz, D. Liu, J. Gayles, E. Derunova, R. Gonzalez-Hernandez, L. Šmejkal, Y. Chen, S. S. P. Parkin, S. D. Wilson, E. S. Toberer, T. McQueen, and M. N. Ali, “Giant, unconventional anomalous Hall effect in the metallic frustrated magnet candidate, KV3Sb5,” Sci. Adv. 6, eabb6003 (2020).
  * Gan  _et al._ (2021) Y. Gan, W. Xia, L. Zhang, K. Yang, X. Mi, A. Wang, Y. Chai, Y. Guo, X. Zhou, and M. He, “Magneto-Seebeck effect and ambipolar Nernst effect in the CsV3Sb5 superconductor,” Phys. Rev. B 104, L180508 (2021).
  * Yu  _et al._ (2021a) F. H. Yu, T. Wu, Z. Y. Wang, B. Lei, W. Z. Zhuo, J. J. Ying, and X. H. Chen, “Concurrence of anomalous Hall effect and charge density wave in a superconducting topological kagomé metal,” Phys. Rev. B 104, L041103 (2021a).
  * (23) See Supplemental Materials for more information.
  * Scalapino  _et al._ (1992) D. J. Scalapino, S. R. White, and S. C. Zhang, “Superfluid density and the Drude weight of the Hubbard model,” Phys. Rev. Lett. 68, 2830 (1992).
  * Yu  _et al._ (2021b) F. H. Yu, D. H. Ma, W. Z. Zhuo, S. Q. Liu, X. K. Wen, B. Lei, J. J. Ying, and X. H. Chen, “Unusual competition of superconductivity and charge-density-wave state in a compressed topological kagomé metal,” Nat. Commun. 12, 3645 (2021b).
  * Chandrasekhar (1962) B. S. Chandrasekhar, “A note on the maximum critical field of high-field superconductors,” Appl. Phys. Lett. 1, 7 (1962).
  * Clogston (1962) A. M. Clogston, “Upper limit for the critical field in hard superconductors,” Phys. Rev. Lett. 9, 266 (1962).
  * Fulde and Ferrell (1964) P. Fulde and R. A. Ferrell, “Superconductivity in a strong spin-exchange field,” Phys. Rev. 135, A550 (1964).
  * Larkin and Ovchinnikov (1964) A. I. Larkin and Y. N. Ovchinnikov, “Nonuniform state of superconductors,” Zh. Eksp. Teor. Fiz 47, 1136 (1964).
  * Bruno and Schwartz (1973) R. C. Bruno and B. B. Schwartz, “Magnetic field splitting of the density of states of thin superconductors,” Phys. Rev. B 8, 3161 (1973).
  * Xiang  _et al._ (2021) Y. Xiang, Q. Li, Y. Li, W. Xie, H. Yang, Z. Wang, Y. Yao, and H.-H. Wen, “Twofold symmetry of $c$-axis resistivity in topological kagomé superconductor CsV3Sb5 with in-plane rotating magnetic field,” Nat. Commun. 12, 6727 (2021).
  * Jin  _et al._ (2022) J.-T. Jin, K. Jiang, H. Yao, and Y. Zhou, “Interplay between pair density wave and a nested Fermi surface,” Phys. Rev. Lett. 129, 167001 (2022).
  * Nie  _et al._ (2022) L. Nie, K. Sun, W. Ma, D. Song, L. Zheng, Z. Liang, P. Wu, F. Yu, J. Li, M. Shan, D. Zhao, S. Li, B. Kang, Z. Wu, Y. Zhou, K. Liu, Z. Xiang, J. Ying, Z. Wang, T. Wu, and X. Chen, “Charge-density-wave-driven electronic nematicity in a kagomé superconductor,” Nature 604, 59 (2022).
  * Zhou and Wang (2022) S. Zhou and Z. Wang, “Chern Fermi pocket, topological pair density wave, and charge-4e and charge-6e superconductivity in kagomé superconductors,” Nat. Commun. 13, 7288 (2022).
  * Wu  _et al._ (2021) X. Wu, T. Schwemmer, T. Müller, A. Consiglio, G. Sangiovanni, D. Di Sante, Y. Iqbal, W. Hanke, A. P. Schnyder, M. M. Denner, M. H. Fischer, T. Neupert, and R. Thomale, “Nature of unconventional pairing in the kagomé superconductors AV3Sb5 (A=K, Rb, Cs),” Phys. Rev. Lett. 127, 177001 (2021).
  * Mu  _et al._ (2021) C. Mu, Q. Yin, Z. Tu, C. Gong, H. Lei, Z. Li, and J. Luo, “S-wave superconductivity in kagomé metal CsV3Sb5 revealed by 121/123 Sb NQR and 51V NMR measurements,” Chinese Phys. Lett. 38, 077402 (2021).

Supplemental Materials for "Chiral pair density wave as a precursor of the
pseudogap in kagomé superconductors"

1\. Robustness of the CPDW state
The CPDW state, discussed in the main text, appears also for smaller values of
the attractive pair-wise interaction strength ${\cal U}$, as shown in Fig.
S1(a), (b), (c). The discrepancy between the pairing amplitude and the
superfluid density exists at smaller values of ${\cal U}$, as shown in Fig.
S1(d). Similarly, different sets of the charge-order strength $\mu_{\rm co}$
and loop current parameter $t_{\rm lc}$ also give rise to the CPDW state and
the pseudogap behavior near the superconducting transition. Fig. S2 shows the
fourier transformed pair-pair correlation function $C({\mathbf{Q}})$ for two
sets of $\mu_{\rm co}$ and $t_{\rm lc}$, showing the C3-symmetric CPDW
correlations.

Figure S1: (a)-(c) Fourier transformed pair-pair correlation function
$C({\mathbf{Q}})$ at different values of ${\cal U}$: (a) ${\cal
U}\\!=\\!1.5t$, (b) ${\cal U}\\!=\\!1.0t$, and (c) ${\cal U}\\!=\\!0.5t$. (d)
Normalized pairing gap magnitude $|\Delta|/\Delta_{0}$ and the superfluid
density $n_{s}$ as a function of the magnetic field $B_{z}$. The results were
obtained on a $20a\times 20a$ lattice with periodic boundary conditions. All
other parameters are the same as in Fig. 1 of the main text. Figure S2:
Fourier transformed pair-pair correlation function $C({\mathbf{Q}})$ at
different values of the charge-order strength $\mu_{\rm co}$ and loop current
parameter $t_{\rm lc}$: (a) $\mu_{\rm co}\\!=\\!0.2t$, $t_{\rm
lc}\\!=\\!0.5t$, and (b) $\mu_{\rm co}\\!=\\!0.1t$, $t_{\rm lc}\\!=\\!0.25t$.
The results were obtained on a $20a\times 20a$ lattice with periodic boundary
conditions. All other parameters are the same as in Fig. 1 of the main
manuscript. Figure S3: Profile of the absolute value of the pairing gap (top
panel) and $z$ component of magnetization (bottom panel) at
$B\\!=\\!1.1B_{c}$. All other parameters are the same as in Fig.4 of the main
text.

2\. Self-consistent gap equation
The superconducting order parameter, the local pairing gap $\Delta({\bf
r}_{i})\\!=\\!-{\cal U}\langle c_{i\uparrow}c_{i\downarrow}\rangle$, can be
re-written using the Bogoliubov-Valatin transformation
$c_{i\sigma}\\!=\\!\sum_{n}u_{ni}^{\sigma}\gamma_{n}+v_{ni}^{\sigma*}\gamma_{n}^{\dagger}$,
where $\gamma_{n}$ is fermionic annihilation operator acting on the $n^{\rm
th}$ eigenstate and $u_{ni}^{\sigma}$ ($v_{ni}^{\sigma}$) is the corresponding
quasiparticle (quasihole) amplitude at site $i$ and spin $\sigma$, as

$\displaystyle\Delta({\bf r}_{i})=-{\cal
U}\sum_{n}\big{[}u_{ni}^{\uparrow}v_{ni}^{\downarrow*}(1-f(E_{n}))+u_{ni}^{\downarrow}v_{ni}^{\uparrow*}f(E_{n})\big{]}$
(E1)

where $f(E_{n})$ is the fermi function corresponding to the $n^{\rm th}$
eigen-energy $E_{n}$. Using the relation
$f(x)=\frac{1}{2}-\frac{1}{2}\tanh{(\frac{x}{2})}$, the above equation can be
expressed as

$\displaystyle\Delta({\bf r}_{i})$ $\displaystyle=\frac{{\cal
U}}{2}\sum_{n}\big{[}u_{ni}^{\uparrow}v_{ni}^{\downarrow*}-u_{ni}^{\downarrow}v_{ni}^{\uparrow*}\big{]}\tanh\Big{(}\frac{E_{n}}{2k_{B}T}\Big{)},$
(E2)

where $k_{B}$ is the Boltzmann constant and $T$ is the temperature. In the
above equation, the energy-independent terms were dropped because the
contribution from those terms will vanish. The above equation is the self-
consistent gap equation used in the calculations. The self-consistency
iteration is performed until convergence is achieved at each lattice sites.

3\. Calculation of superfluid density
The superfluid density is given by the effective Drude weight, derived earlier
in Ref. [Scalapino  _et al._ , 1992]

$\displaystyle n_{s}=\frac{D_{s}}{\pi
e^{2}}=-\langle\kappa\rangle+\Lambda({\bf Q}\rightarrow 0,i\omega\rightarrow
0),$ (E3)

where the first term on the right hand side is the diamagnetic response and
the second term is the paramagnetic response. The diamagnetic term, associated
with the local kinetic energy, can be expressed in terms of the Bogoliubov-de
Gennes (BdG) quasiparticle weights as

$\displaystyle\kappa_{i}=$ $\displaystyle-t\sum_{\langle
j\rangle,n,\sigma}\big{[}u_{ni}^{\sigma}u_{nj}^{\sigma*}+c.c.\big{]}f(E_{n})$
$\displaystyle+\big{[}v_{ni}^{\sigma}v_{nj}^{\sigma*}+c.c.\big{]}(1-f(E_{n})).$
(E4)

The paramagnetic response is given by the transverse current-current
correlation function

$\displaystyle\Lambda({\bf Q},\omega_{n})\\!$
$\displaystyle=\\!\frac{1}{N}\int_{0}^{1/T}e^{i\omega_{n}\tau}\langle
j_{x}^{p}({\bf Q},\tau)j_{x}^{p}(-{\bf Q},0)\rangle~{}d\tau,$ (E5)

with $\omega_{n}\\!=\\!2\pi nT$ and the paramagnetic current $j_{x}^{p}({\bf
Q})$ is given by

$\displaystyle j_{x}^{p}({\bf
Q})=it\sum_{i,\sigma}(c_{i+\hat{x},\sigma}^{\dagger}c_{i,\sigma}-c_{i,\sigma}^{\dagger}c_{i+\hat{x},\sigma})e^{-{\bf
Q}\cdot{\bf r}_{i}}.$ (E6)

The uniform (i.e. Q=0) paramagnetic response can be obtained Scalapino  _et
al._ (1992), in terms of the BdG quasiparticle weights as

$\displaystyle\Lambda({\bf Q}\\!\rightarrow\\!0,i\omega\\!\rightarrow\\!0)\\!$
$\displaystyle=\\!\frac{1}{N}\\!\sum_{i,j,n_{1},n_{2}}^{\sigma,\sigma^{\prime}}\\!{\cal
A}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}\\!\big{[}{\cal
A}_{n_{1}n_{2}}^{j\sigma\sigma^{\prime}*}\\!+\\!{\cal
B}_{n_{1}n_{2}}^{j\sigma\sigma^{\prime}}\big{]}$
$\displaystyle\times\frac{f(E_{n1})-f(E_{n2})}{E_{n1}-E_{n2}},$ (E7)

where $N$ is the total number of lattice sites and

$\displaystyle{\cal
A}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}=2\big{[}u_{n_{1}j}^{\sigma^{\prime}*}u_{n_{2}i}^{\sigma}-u_{n_{1}i}^{\sigma*}u_{n_{2}j}^{\sigma^{\prime}}\big{]},$
$\displaystyle{\cal
B}_{n_{1}n_{2}}^{i\sigma\sigma^{\prime}}=2\big{[}v_{n_{1}j}^{\sigma^{\prime}*}v_{n_{2}i}^{\sigma}-v_{n_{1}i}^{\sigma*}v_{n_{2}j}^{\sigma^{\prime}}\big{]}.$
(E8)

Figure S4: Density of states of a kagome lattice with periodic boundary
conditions for different lattice sizes (different columns) and different
values of the loop current amplitude $t_{lc}$ (different rows), without
chemical potential modulation and at a magnetic field $B\\!=\\!2t$.

4\. Magnetization in the emergent granular state
Near the critical magnetic field $B_{c}$ for the superconducting transition
(as dictated by the vanishing superfluid density), the system goes through an
emergent granular phase in which the superconducting gap magnitude is
suppressed in the charge-ordered clusters, as shown in Fig. S3(a) (and Fig.4
in the main text). The magnetization profile, obtained via
$m_{zi}=c_{i\uparrow}^{\dagger}c_{i\uparrow}-c_{i\downarrow}^{\dagger}c_{i\downarrow}$,
shown in Fig. S3(b), also reveals a periodic modulation. The magnetization is
enhanced slightly in the regions in which the pairing amplitude is decreased.
This interplay of magnetization and superconducting order parameter exists in
the entire granular superconducting phase, without any long-range
superconducting phase coherence.

5\. Effect of loop current in the density of states
To explore the role of the loop current in formation of the pseudogap near the
superconducting transition and to understand the finite size effect, in Fig.
S4 the density of states has been presented for loop current magnitude
$t_{lc}$ and different lattice sizes. The results show that, despite the
little noise in the density of state originating due to finite size effect, it
is evident that quasiparticle states build up within the superconducting gap
due to the loop current.

## References

  * Scalapino  _et al._ (1992) D. J. Scalapino, S. R. White, and S. C. Zhang, “Superfluid density and the Drude weight of the Hubbard model,” Phys. Rev. Lett. 68, 2830 (1992).


[1]$V_{#1}^{*}$

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

Andreas Crivellin<EMAIL_ADDRESS>Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  Matthew Kirk<EMAIL_ADDRESS>Departament de Física Quàntica i Astrofísica (FQA), Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona (UB), Spain  Teppei Kitahara
<EMAIL_ADDRESS>Institute for Advanced Research, Nagoya University,
Nagoya 464–8601, Japan  Kobayashi-Maskawa Institute for the Origin of
Particles and the Universe, Nagoya University, Nagoya 464–8602, Japan  Theory
Center, IPNS, High Energy Accelerator Research Organization (KEK), Tsukuba
305–0801, Japan  Federico Mescia<EMAIL_ADDRESS>Departament de Física Quàntica
i Astrofísica (FQA), Institut de Ciències del Cosmos (ICCUB), Universitat de
Barcelona (UB), Spain

###### Abstract

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.

††preprint: PSI-PR-22-09, ZU-TH-12/22, KEK-TH-2411

##  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 $4^{\rm th}$ 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]).

VLFs are not only interesting from the theoretical perspective, but also from
the phenomenological point of view as they could be involved in an explanation
of $b\to s\ell^{+}\ell^{-}$ data [15, 16, 17, 18, 19], the tension in
$(g-2)_{\mu}$ [20, 21, 22, 23, 24, 25, 26, 27, 28, 26, 29, 30, 31, 32, 33, 34,
35] or account for the Cabibbo angle anomaly [36, 37, 38, 39, 40, 41, 42, 43,
44, 45]. Furthermore, vector-like quarks (VLQs) can lead to tree-level effects
in $Z$-$t$-$c$ and $h$-$t$-$c$ couplings after electroweak (EW) symmetry
breaking, and therefore generate sizeable effects in the related flavour-
changing neutral current (FCNC) decays of the top quark [46, 47, 48, 49, 43,
44, 45].

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 down-quark FCNCs at tree-
level, such that the former can be sizable. However, even these VLQs affect
e.g. the $W$ mass 111 The contribution of VLQs to the $W$ mass, via the
oblique $S$ 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.  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 $t\to Z(h)c$ 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
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$) of VLQs, given in Table 1,
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
terms222Note that mass terms such as $m_{i}^{U}\bar{U}{u_{i}}$ 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). are

$\mathcal{L}=\sum_{F}\bar{F}\left(i\not{D}-M_{F}\right)F\,,$ (II.1)

where $F=\\{U,\,D,\,Q_{1},\,Q_{5},\,Q_{7},\,T_{1},\,T_{2}\\}$ and

$D_{\mu}=\partial_{\mu}+ig_{1}Y_{F}B_{\mu}+ig_{2}S^{I}W^{I}_{\mu}+ig_{s}T^{A}G^{A}_{\mu}\,.$
(II.2)

Here $T^{A}=\frac{1}{2}\lambda^{A}$ and $(S^{I})_{jk}$ are $0$,
$\frac{1}{2}(\tau^{I})_{jk}$, and $-i\epsilon_{Ijk}$ for the $SU(2)_{L}$
singlet, doublet, and triplet representations, respectively, and $\lambda^{A}$
and $\tau^{I}$ are the Gell-Mann and the Pauli matrices. The (generalized)
Yukawa couplings are encoded in the Lagrangian

$\displaystyle{{\cal L}}=$ $\displaystyle{{\cal L}^{H}_{qq}}+{{\cal
L}^{H}_{q\text{VLQ}}}+{{\cal L}^{H}_{\text{VLQ}\text{VLQ}}}\,,$ (II.3)

where the first term contains the SM Yukawa couplings

$\displaystyle\begin{aligned} -{{\cal
L}^{H}_{qq}}=&\,Y^{u}_{ij}\bar{q}_{i}\tilde{H}u_{j}+Y^{d}_{ij}\bar{q}_{i}Hd_{j}+\text{h.c.}\,,\end{aligned}$
(II.4)

the second term the Higgs interactions with vector-like and SM quarks

$\displaystyle-{{\cal L}^{H}_{q\text{VLQ}}}$
$\displaystyle=\xi_{i}^{U}{{\bar{U}}}{{\tilde{H}}^{\dagger}}{q_{i}}+\xi_{i}^{D}{{\bar{D}}}{H^{\dagger}}{q_{i}}+\xi_{i}^{u_{1}}{{\bar{Q}}_{1}}\tilde{H}{u_{i}}$
$\displaystyle+\xi_{i}^{d_{1}}{{\bar{Q}}_{1}}H{d_{i}}+\xi_{i}^{{Q_{5}}}{{\bar{Q}}_{5}}\tilde{H}{d_{i}}+\xi_{i}^{{Q_{7}}}{{\bar{Q}}_{7}}H{u_{i}}$
(II.5)
$\displaystyle+\frac{1}{2}\xi_{i}^{{T_{1}}}{H^{\dagger}}\tau\cdot{{\bar{T}}_{1}}{q_{i}}+\frac{1}{2}\xi_{i}^{{T_{2}}}{{\tilde{H}}^{\dagger}}\tau\cdot{{\bar{T}}_{2}}{q_{i}}+\mathrm{h.c.}\,,$

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
$\tau\cdot\bar{T}=\sum_{I}\tau^{I}\bar{T}^{I}$.

| $u$ | $d$ | $q$ | $H$ | $U$ | $D$ | $Q_{1}$ | $Q_{5}$ | $Q_{7}$ | $T_{1}$ | $T_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | 3 | 3 | 3 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3
$SU(2)_{L}$ | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3
$U(1)_{Y}$ | $\nicefrac{{2}}{{3}}$ |  $-$$\nicefrac{{1}}{{3}}$ | $\nicefrac{{1}}{{6}}$ | $\nicefrac{{1}}{{2}}$ | $\nicefrac{{2}}{{3}}$ |  $-$$\nicefrac{{1}}{{3}}$ | $\nicefrac{{1}}{{6}}$ |  $-$$\nicefrac{{5}}{{6}}$ | $\nicefrac{{7}}{{6}}$ |  $-$$\nicefrac{{1}}{{3}}$ | $\nicefrac{{2}}{{3}}$

Table 1: 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.

###  II.1 SMEFT and Matching

We write the SMEFT Lagrangian as

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\rm SM}+\sum_{i}C_{i}Q_{i}\,,$ (II.6)

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

$\displaystyle Q_{Hq}^{(1)}\,,Q_{Hq}^{(3)}\,,Q_{Hu}\,,Q_{Hd}\,,Q_{Hud}\,,$
(II.7)

and the four-quark operators generating $\Delta F=2$ processes read

$\displaystyle
Q^{(1)}_{qq}\,,Q^{(3)}_{qq}\,,Q_{uu}\,,Q_{dd}\,,Q^{(1)}_{qu}\,,Q^{(1)}_{qd}\,,Q^{(8)}_{qu}\,,Q^{(8)}_{qd}\,,$
(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
$Q_{HWB}$, with their contributions approximately given by

$\delta M_{W}\approx-v^{2}(29\,C_{HD}+64\,C_{HWB}+\cdots)\,\mathrm{GeV}\,,$
(II.9)

where $v\simeq 246$ GeV and ($\cdots$) indicates SMEFT operators not relevant
in our scenario with VLQs.333 Note that the SMEFT effects in the $W$ mass are
known fully at leading order [53, 54], but only partially at next-to-leading
order (NLO) [55], since in that work flavour universality of the SMEFT
coefficients is assumed. However we have checked that, after making some
conservative assumptions about the flavour dependence, the NLO effects are
small. An example diagram for the $W$ mass correction is shown on the left in
Fig. 1.

Figure 1:  Examples of Feynman diagrams showing the $U$ contributions to the
operator $Q_{HD}$, affecting the $W$-boson mass (left), and $Q_{qq}^{(1,3)}$,
affecting $B_{s}\\!-\\!\,\overline{\\!B}{}_{s}\,$ mixing (right).

The tree-level matching of the operators generating modified $Z$-quark
couplings is given by

$\displaystyle C_{Hq}^{(1)ij}+C_{Hq}^{(3)ij}$
$\displaystyle=-\frac{{\xi_{i}^{D*}\xi_{j}^{D}}}{{2M_{{D}}^{2}}}-\frac{{\xi_{i}^{{T_{1}}*}\xi_{j}^{{T_{1}}}}}{{8M_{{T_{1}}}^{2}}}+\frac{{\xi_{i}^{{T_{2}}*}\xi_{j}^{{T_{2}}}}}{{4M_{{T_{2}}}^{2}}}\,,$
$\displaystyle C_{Hq}^{(1)ij}-C_{Hq}^{(3)ij}$
$\displaystyle=\frac{{\xi_{i}^{U*}\xi_{j}^{U}}}{{2M_{{U}}^{2}}}-\frac{{\xi_{i}^{{T_{1}}*}\xi_{j}^{{T_{1}}}}}{{4M_{{T_{1}}}^{2}}}+\frac{{\xi_{i}^{{T_{2}}*}\xi_{j}^{{T_{2}}}}}{{8M_{{T_{2}}}^{2}}}\,,$
$\displaystyle C_{Hu}^{ij}$
$\displaystyle=-\frac{{\xi_{i}^{u_{1}*}\xi_{j}^{u_{1}}}}{{2M_{{Q_{1}}}^{2}}}+\frac{{\xi_{i}^{{Q_{7}}*}\xi_{j}^{{Q_{7}}}}}{{2M_{{Q_{7}}}^{2}}}\,,$
(II.10) $\displaystyle C_{Hd}^{ij}$
$\displaystyle=\frac{{\xi_{i}^{d_{1}*}\xi_{j}^{d_{1}}}}{{2M_{{Q_{1}}}^{2}}}-\frac{{\xi_{i}^{{Q_{5}}*}\xi_{j}^{{Q_{5}}}}}{{2M_{{Q_{5}}}^{2}}}\,,$

for $Z$-$d^{i}_{L}$-$d^{j}_{L}$, $Z$-$u^{i}_{L}$-$u^{j}_{L}$,
$Z$-$u^{i}_{R}$-$u^{j}_{R}$, and $Z$-$d^{i}_{R}$-$d^{j}_{R}$ respectively.
Modified $W$ couplings to left-handed quarks arise from $C_{Hq}^{(3)}$ alone,
while right-handed modifications do not appear in our scenario, due to our
(later) choice to set $\xi^{d_{1}}$ 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 $\xi^{u_{1}}$ and $Q_{7}$ (shown in
bold in Table 1) lead to effects in $t\to cZ$ while avoiding tree-level FCNCs
in the down sector. An approximate formula for this branching ratio is

$\\!\\!\\!\\!\text{Br}(t\to
cZ)\approx\frac{v^{4}}{2}\left\\{\left[C_{Hq}^{(1)23}\\!-\\!C_{Hq}^{(3)23}\right]^{2}+\left[C_{Hu}^{23}\right]^{2}\right\\}\,.$
(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.

##  III Phenomenological analysis

The current 95% CL upper bounds for $t\to cZ$ and $t\to ch$, based on the full
LHC Run 2 data set, are [57, 58, 59, 60]

${\rm Br}(t\to cZ)<1.3\times 10^{-4}\,,\quad{\rm Br}(t\to ch)<9.9\times
10^{-4}\,.$ (III.1)

While this already constrains some beyond the SM scenarios, at the high-
luminosity (HL-)LHC [61, 62], FCC-hh [63], ILC [64], or the FCC-ee [65], one
can expect to be sensitive to $t\to cZ$ branching ratios on the order of
$10^{-5}$ to $10^{-6}$ [66, 64]. For $t\to ch$, see Ref. [67] and references
therein, sensitivities on the order of $10^{-4}$ and $10^{-5}$ for the HL-LHC
[68] and FCC-hh [67, 69, 70] are estimated, respectively. A summary of the
future prospects for these FCNC top decays is given in Table 2.

| $\mathrm{Br}(t\to cZ)\times 10^{5}$ | $\mathrm{Br}(t\to ch)\times 10^{5}$
---|---|---
Current LHC | $13$ [59] | $99$ [60]
($$13\text{\,}\mathrm{TeV}$,$139\text{\,}{\mathrm{fb}}^{-1}$$)
HL-LHC |  $3.13$ [66] (0%) |  $15$ [68]
($$14\text{\,}\mathrm{TeV}$,$3\text{\,}{\mathrm{ab}}^{-1}$$) |  $6.65$ [66] (10%) |
HE-LHC |  $0.522$ [66] (0%) |  $7.7$ [67] (0%)
($$27\text{\,}\mathrm{TeV}$,$15\text{\,}{\mathrm{ab}}^{-1}$$) |  $3.84$ [66] (10%) |  $8.5$ [67] (10%)
FCC-hh |  | $7.7$ [71]
($$100\text{\,}\mathrm{TeV}$,$3\text{\,}{\mathrm{ab}}^{-1}$$) |
FCC-hh |  |  $2.39$ [70] (5%)
($$100\text{\,}\mathrm{TeV}$,$10\text{\,}{\mathrm{ab}}^{-1}$$) |  |  $9.68$ [69] (10%)
FCC-hh |  $0.0887$ [66] (0%) |  $0.96$ [67] (0%)
($$100\text{\,}\mathrm{TeV}$,$30\text{\,}{\mathrm{ab}}^{-1}$$) |  $3.54$ [66] (10%) |  $3.0$ [67] (10%)
|  |  $4.3$ [71]
ILC | $9.1$ [64] |
($$250\text{\,}\mathrm{GeV}$,$2\text{\,}{\mathrm{ab}}^{-1}$$) |
ILC | $2.9$ [64] |
($$1\text{\,}\mathrm{TeV}$,$8\text{\,}{\mathrm{ab}}^{-1}$$) |
FCC-ee | $2.8$ [65] |
($$350\text{\,}\mathrm{GeV}$,$10\text{\,}{\mathrm{ab}}^{-1}$$) |
Table 2: Summary of current limits and future sensitivities for $t\to Zc$ and
$t\to hc$. The values in brackets are the assumed systematic uncertainties on
the underlying experimental measurements at the future colliders (if
provided).

For the numerical analysis we use the software package smelli [72, 73] (based
on flavio [74] and wilson [75]), with $\\{\alpha,M_{Z},G_{F}\\}$ 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 $Y^{u}\approx
V^{\dagger}\cdot\text{diag}(0,0,y_{t})$, 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
$\Gamma(K^{+}\to\mu^{+}\nu)/\Gamma(\pi^{+}\to\mu^{+}\nu)$, $\mathrm{Br}(B\to
X_{c}e^{+}\nu)$, $\mathrm{Br}(B^{+}\to\tau^{+}\nu)$, and $\Delta M_{d}/\Delta
M_{s}$ as observables (see supplementary material for details).

Concerning the EW fit, the long standing tension in the $W$ mass, previously
with a significance of $\approx$$1.8\text{\,}\sigma$ [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
$4.7\text{\,}\mathrm{MeV}$ systematic uncertainty, and give a new world
average of

$\displaystyle M_{W}^{\rm exp}=80413.3\pm 8.0\,\text{MeV}\,.$ (III.2)

This value is $5.5\text{\,}\sigma$ higher than the SM prediction $M_{W}^{\rm
SM}=80358.7\pm 6.0$ MeV [78].

Concerning $B$ physics, even though the hints for lepton flavour universality
(LFU) violation in $b\to s\ell^{+}\ell^{-}$ 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 $B_{s}\\!-\\!\,\overline{\\!B}{}_{s}\,$ mixing (we use
inputs from Ref. [91] for the SM prediction).

In all our analyses, we set the masses of the VLQs to
$2\text{\,}\mathrm{TeV}$. This is consistent the published model-independent
bounds for third generation VLQs of
$M_{\text{VLQ}}>$1.31\text{\,}\mathrm{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 non-existent. Let us now consider the three
cases of $U$, $Q_{1}$ and $Q_{7}$ numerically:

Figure 2:  Preferred regions in the $\xi_{2}$-$\xi_{3}$ plane for the three
representations of VLQ that generate $t\to cZ$ 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
$\mathrm{Br}(t\to cZ)\times 10^{5}$. The region preferred by all data (the
global fit region with using the new experimental average in Eq. (III.2)) is
shown at the $1\text{\,}\sigma$ and $2\text{\,}\sigma$ level, while the others
regions correspond to $1\text{\,}\sigma$. 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 $t\to cZ$ 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 $t\to cZ$.

$U$: In addition to the modified $Z$-$t$-$c$ coupling, this VLQ also generates
relevant effects in $b\to s\ell^{+}\ell^{-}$ transitions via a $Z$ penguin,
resulting in an $C_{9}\approx-C_{10}/4$ pattern. In fact, mainly due to the
measurements of $P_{5}^{\prime}$ [95] and $B_{s}\to\phi\mu^{+}\mu^{-}$ [96,
97] there is a preference for a non-zero contribution with such a structure.
The bounds from $B_{s}\\!-\\!\,\overline{\\!B}{}_{s}\,$ 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 $b\to s\gamma$ suffers from a cancellation, but here between
terms generated by the matching on the SMEFT and integrating out the $W$ at
the weak scale ($b\to s\gamma$ is included within the $b\to s\ell^{+}\ell^{-}$
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 $\xi^{U}_{2}$ is constrained by the total $Z$ width. These
finding are summarised in Fig. 2 (top-left) where one can see that
$\mathrm{Br}(t\to cZ)$ can be of the order of $2\times 10^{-6}$, which could
be probed by FCC-hh.

$Q_{1}$ with $\xi^{u_{1}}$: The VLQ $Q_{1}$ with the couplings $\xi^{u_{1}}$
is found to be a very promising candidate for sizable rates of $t\to cZ$,
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$(t\to cZ)$ up to $1\text{\times}{10}^{-5}$ 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 $\mathrm{Br}(t\to ch)$ can be at most
$3\text{\times}{10}^{-6}$, 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 $b\to s\ell^{+}\ell^{-}$ processes is generated,
but in order for its size to be relevant, quite large couplings are required.
Furthermore, for small third generation couplings ($\xi^{Q_{7}}_{3}<1$) 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 $(\xi^{Q_{7}}_{3})^{4}$ the other involving
$(\xi^{Q_{7}}_{3})^{2}y_{t}^{2}$. Note that in the regime of such large
couplings, small tensions with Higgs data arise in the $h\to
ZZ,\,WW,\,\gamma\gamma$ partial widths, with tensions of $1.8$, $1.5$, and
$1.2\text{\,}\sigma$, respectively. Concerning $\mathrm{Br}(t\to cZ)$, again
an enhancement of the branching ratio up to $1\text{\times}{10}^{-5}$ is
possible, which could be probed by the HE-LHC, FCC-hh, FCC-ee, or ILC. Given
the large couplings allowed by data, $\mathrm{Br}(t\to ch)$ can be enhanced up
to $3\text{\times}{10}^{-5}$, 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 $t\to cZ$ 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 $t\to cZ$ of the order of
$1\text{\times}{10}^{-6}$, $1\text{\times}{10}^{-5}$ and
$1\text{\times}{10}^{-4}$, for $U$, $Q_{1}$ and $Q_{7}$, respectively.
Therefore, the parameter space of $Q_{7}$ is already constrained by LHC limits
on $t\to cZ$, 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 $t\to cZ$, showing that these observables
are correlated. Furthermore, $U$ and $Q_{7}$ lead to LFU effects in $b\to
s\ell^{+}\ell^{-}$ which cannot explain $R(K^{(*)})$ but affect observables
like $P_{5}^{\prime}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ and, in combination
with LFU violating effects, can further improve the description of data. In
conclusion, $t\to cZ$ 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.

###### Acknowledgements.

A. C. gratefully acknowledges financial support by the Swiss National Science
Foundation (PP00P_2176884). T. K. is supported by the Grant-in-Aid for Early-
Career Scientists (No. 19K14706) and by the JSPS Core-to-Core Program (Grant
No. JPJSCCA20200002) from the Ministry of Education, Culture, Sports, Science,
and Technology (MEXT), Japan. M. K. and F. M. acknowledge financial support
from the State Agency for Research of the Spanish Ministry of Science and
Innovation through the “Unit of Excellence María de Maeztu 2020-2023” award to
the Institute of Cosmos Sciences (CEX2019-000918-M) and from
PID2019-105614GB-C21 and 2017-SGR-929 grants.

## References

  * [1] O. Eberhardt, A. Lenz, A. Menzel, U. Nierste, and M. Wiebusch, “Status of the fourth fermion generation before ICHEP2012: Higgs data and electroweak precision observables,” Phys. Rev. D 86 (2012) 074014 [arXiv:1207.0438].
  * [2] O. Eberhardt, et al., “Impact of a Higgs boson at a mass of 126 GeV on the standard model with three and four fermion generations,” Phys. Rev. Lett. 109 (2012) 241802 [arXiv:1209.1101].
  * [3] J. L. Hewett and T. G. Rizzo, “Low-Energy Phenomenology of Superstring Inspired E(6) Models,” Phys. Rept. 183 (1989) 193.
  * [4] P. Langacker, “Grand Unified Theories and Proton Decay,” Phys. Rept. 72 (1981) 185.
  * [5] F. del Aguila and M. J. Bowick, “The Possibility of New Fermions With $\Delta$ I = 0 Mass,” Nucl. Phys. B 224 (1983) 107.
  * [6] I. Antoniadis, “A Possible new dimension at a few TeV,” Phys. Lett. B 246 (1990) 377–384.
  * [7] N. Arkani-Hamed, S. Dimopoulos, and J. March-Russell, “Stabilization of submillimeter dimensions: The New guise of the hierarchy problem,” Phys. Rev. D 63 (2001) 064020 [hep-th/9809124].
  * [8] N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, “The Littlest Higgs,” JHEP 07 (2002) 034 [hep-ph/0206021].
  * [9] T. Han, H. E. Logan, B. McElrath, and L.-T. Wang, “Phenomenology of the little Higgs model,” Phys. Rev. D 67 (2003) 095004 [hep-ph/0301040].
  * [10] B. A. Dobrescu and C. T. Hill, “Electroweak symmetry breaking via top condensation seesaw,” Phys. Rev. Lett. 81 (1998) 2634–2637 [hep-ph/9712319].
  * [11] R. S. Chivukula, B. A. Dobrescu, H. Georgi, and C. T. Hill, “Top Quark Seesaw Theory of Electroweak Symmetry Breaking,” Phys. Rev. D 59 (1999) 075003 [hep-ph/9809470].
  * [12] H.-J. He, C. T. Hill, and T. M. P. Tait, “Top Quark Seesaw, Vacuum Structure and Electroweak Precision Constraints,” Phys. Rev. D 65 (2002) 055006 [hep-ph/0108041].
  * [13] C. T. Hill and E. H. Simmons, “Strong Dynamics and Electroweak Symmetry Breaking,” Phys. Rept. 381 (2003) 235–402 [hep-ph/0203079]. [Erratum: Phys.Rept. 390, 553–554 (2004)].
  * [14] C. Anastasiou, E. Furlan, and J. Santiago, “Realistic Composite Higgs Models,” Phys. Rev. D 79 (2009) 075003 [arXiv:0901.2117].
  * [15] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, “Quark flavor transitions in $L_{\mu}-L_{\tau}$ models,” Phys. Rev. D 89 (2014) 095033 [arXiv:1403.1269].
  * [16] B. Gripaios, M. Nardecchia, and S. A. Renner, “Linear flavour violation and anomalies in B physics,” JHEP 06 (2016) 083 [arXiv:1509.05020].
  * [17] P. Arnan, L. Hofer, F. Mescia, and A. Crivellin, “Loop effects of heavy new scalars and fermions in $b\to s\mu^{+}\mu^{-}$,” JHEP 04 (2017) 043 [arXiv:1608.07832].
  * [18] P. Arnan, A. Crivellin, M. Fedele, and F. Mescia, “Generic Loop Effects of New Scalars and Fermions in $b\to s\ell^{+}\ell^{-}$, $(g-2)_{\mu}$ and a Vector-like $4^{\rm th}$ Generation,” JHEP 06 (2019) 118 [arXiv:1904.05890].
  * [19] A. Crivellin, C. A. Manzari, M. Alguero, and J. Matias, “Combined Explanation of the Z→bb¯ Forward-Backward Asymmetry, the Cabibbo Angle Anomaly, and $\tau$→$\mu$$\nu$$\nu$ and b→s$\ell$+$\ell$\- Data,” Phys. Rev. Lett. 127 (2021) 011801 [arXiv:2010.14504].
  * [20] A. Czarnecki and W. J. Marciano, “The Muon anomalous magnetic moment: A Harbinger for ’new physics’,” Phys. Rev. D 64 (2001) 013014 [hep-ph/0102122].
  * [21] K. Kannike, M. Raidal, D. M. Straub, and A. Strumia, “Anthropic solution to the magnetic muon anomaly: the charged see-saw,” JHEP 02 (2012) 106 [arXiv:1111.2551]. [Erratum: JHEP 10, 136 (2012)].
  * [22] R. Dermisek and A. Raval, “Explanation of the Muon g-2 Anomaly with Vectorlike Leptons and its Implications for Higgs Decays,” Phys. Rev. D 88 (2013) 013017 [arXiv:1305.3522].
  * [23] A. Freitas, J. Lykken, S. Kell, and S. Westhoff, “Testing the Muon g-2 Anomaly at the LHC,” JHEP 05 (2014) 145 [arXiv:1402.7065]. [Erratum: JHEP 09, 155 (2014)].
  * [24] G. Bélanger, C. Delaunay, and S. Westhoff, “A Dark Matter Relic From Muon Anomalies,” Phys. Rev. D 92 (2015) 055021 [arXiv:1507.06660].
  * [25] A. Aboubrahim, T. Ibrahim, and P. Nath, “Leptonic $g-2$ moments, CP phases and the Higgs boson mass constraint,” Phys. Rev. D 94 (2016) 015032 [arXiv:1606.08336].
  * [26] K. Kowalska and E. M. Sessolo, “Expectations for the muon g-2 in simplified models with dark matter,” JHEP 09 (2017) 112 [arXiv:1707.00753].
  * [27] S. Raby and A. Trautner, “Vectorlike chiral fourth family to explain muon anomalies,” Phys. Rev. D 97 (2018) 095006 [arXiv:1712.09360].
  * [28] A. Choudhury, L. Darmé, L. Roszkowski, E. M. Sessolo, and S. Trojanowski, “Muon g $-$ 2 and related phenomenology in constrained vector-like extensions of the MSSM,” JHEP 05 (2017) 072 [arXiv:1701.08778].
  * [29] L. Calibbi, R. Ziegler, and J. Zupan, “Minimal models for dark matter and the muon g$-$2 anomaly,” JHEP 07 (2018) 046 [arXiv:1804.00009].
  * [30] R. Capdevilla, D. Curtin, Y. Kahn, and G. Krnjaic, “Discovering the physics of $(g-2)_{\mu}$ at future muon colliders,” Phys. Rev. D 103 (2021) 075028 [arXiv:2006.16277].
  * [31] R. Capdevilla, D. Curtin, Y. Kahn, and G. Krnjaic, “No-lose theorem for discovering the new physics of (g-2)$\mu$ at muon colliders,” Phys. Rev. D 105 (2022) 015028 [arXiv:2101.10334].
  * [32] A. Crivellin and M. Hoferichter, “Consequences of chirally enhanced explanations of $(g-2)_{\mu}$ for $h\to\mu\mu$ and $Z\to\mu\mu$,” JHEP 07 (2021) 135 [arXiv:2104.03202].
  * [33] L. Calibbi, X. Marcano, and J. Roy, “Z lepton flavour violation as a probe for new physics at future $e^{+}e^{-}$ colliders,” Eur. Phys. J. C 81 (2021) 1054 [arXiv:2107.10273].
  * [34] G. Arcadi, L. Calibbi, M. Fedele, and F. Mescia, “Systematic approach to B-physics anomalies and t-channel dark matter,” Phys. Rev. D 104 (2021) 115012 [arXiv:2103.09835].
  * [35] P. Paradisi, O. Sumensari, and A. Valenti, “The high-energy frontier of the muon g-2.” arXiv:2203.06103.
  * [36] B. Belfatto, R. Beradze, and Z. Berezhiani, “The CKM unitarity problem: A trace of new physics at the TeV scale?” Eur. Phys. J. C 80 (2020) 149 [arXiv:1906.02714].
  * [37] Y. Grossman, E. Passemar, and S. Schacht, “On the Statistical Treatment of the Cabibbo Angle Anomaly,” JHEP 07 (2020) 068 [arXiv:1911.07821].
  * [38] C.-Y. Seng, X. Feng, M. Gorchtein, and L.-C. Jin, “Joint lattice QCD–dispersion theory analysis confirms the quark-mixing top-row unitarity deficit,” Phys. Rev. D 101 (2020) 111301 [arXiv:2003.11264].
  * [39] A. M. Coutinho, A. Crivellin, and C. A. Manzari, “Global Fit to Modified Neutrino Couplings and the Cabibbo-Angle Anomaly,” Phys. Rev. Lett. 125 (2020) 071802 [arXiv:1912.08823].
  * [40] A. Crivellin and M. Hoferichter, “$\beta$ Decays as Sensitive Probes of Lepton Flavor Universality,” Phys. Rev. Lett. 125 (2020) 111801 [arXiv:2002.07184].
  * [41] M. Endo and S. Mishima, “Muon $g-2$ and CKM unitarity in extra lepton models,” JHEP 08 (2020) 004 [arXiv:2005.03933].
  * [42] M. Kirk, “Cabibbo anomaly versus electroweak precision tests: An exploration of extensions of the Standard Model,” Phys. Rev. D 103 (2021) 035004 [arXiv:2008.03261].
  * [43] B. Belfatto and Z. Berezhiani, “Are the CKM anomalies induced by vector-like quarks? Limits from flavor changing and Standard Model precision tests,” JHEP 10 (2021) 079 [arXiv:2103.05549].
  * [44] G. C. Branco, J. T. Penedo, P. M. F. Pereira, M. N. Rebelo, and J. I. Silva-Marcos, “Addressing the CKM unitarity problem with a vector-like up quark,” JHEP 07 (2021) 099 [arXiv:2103.13409].
  * [45] S. Balaji, “Asymmetry in flavour changing electromagnetic transitions of vector-like quarks,” JHEP 05 (2022) 015 [arXiv:2110.05473].
  * [46] E. Nardi, “Top - charm flavor changing contributions to the effective b s Z vertex,” Phys. Lett. B 365 (1996) 327–333 [hep-ph/9509233].
  * [47] G. Cacciapaglia, et al., “Heavy Vector-like Top Partners at the LHC and flavour constraints,” JHEP 03 (2012) 070 [arXiv:1108.6329].
  * [48] F. J. Botella, G. C. Branco, and M. Nebot, “The Hunt for New Physics in the Flavour Sector with up vector-like quarks,” JHEP 12 (2012) 040 [arXiv:1207.4440].
  * [49] Y. Okada and L. Panizzi, “LHC signatures of vector-like quarks,” Adv. High Energy Phys. 2013 (2013) 364936 [arXiv:1207.5607].
  * [50] C.-Y. Chen, S. Dawson, and E. Furlan, “Vectorlike fermions and Higgs effective field theory revisited,” Phys. Rev. D 96 (2017) 015006 [arXiv:1703.06134].
  * [51] CDF Collaboration, “High-precision measurement of the $W$ boson mass with the CDF II detector,” Science 376 (2022) 170–176.
  * [52] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian,” JHEP 10 (2010) 085 [arXiv:1008.4884].
  * [53] L. Berthier and M. Trott, “Towards consistent Electroweak Precision Data constraints in the SMEFT,” JHEP 05 (2015) 024 [arXiv:1502.02570].
  * [54] M. Bjørn and M. Trott, “Interpreting $W$ mass measurements in the SMEFT,” Phys. Lett. B 762 (2016) 426–431 [arXiv:1606.06502].
  * [55] S. Dawson and P. P. Giardino, “Electroweak and QCD corrections to $Z$ and $W$ pole observables in the standard model EFT,” Phys. Rev. D 101 (2020) 013001 [arXiv:1909.02000].
  * [56] A. Carmona, A. Lazopoulos, P. Olgoso, and J. Santiago, “Matchmakereft: automated tree-level and one-loop matching,” SciPost Phys. 12 (2022) 198 [arXiv:2112.10787].
  * [57] ATLAS Collaboration, “Search for flavour-changing neutral current top-quark decays $t\to qZ$ in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP 07 (2018) 176 [arXiv:1803.09923].
  * [58] ATLAS Collaboration, “Search for top-quark decays $t\to Hq$ with 36 fb-1 of $pp$ collision data at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP 05 (2019) 123 [arXiv:1812.11568].
  * [59] ATLAS Collaboration, “Search for flavor-changing neutral-current couplings between the top quark and the Z boson with LHC Run2 proton-proton collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector.”. http://cds.cern.ch/record/2781174.
  * [60] ATLAS Collaboration, “Searches of top FCNC interactions with the ATLAS detector,” 13 Mar. 2022.  https://cds.cern.ch/record/2804177. Moriond EW 2022.
  * [61] N. A. Graf, M. E. Peskin, and J. L. Rosner, eds., “Physics at a High-Luminosity LHC with ATLAS.” arXiv:1307.7292.
  * [62] ATLAS Collaboration, “Sensitivity of searches for the flavour-changing neutral current decay $t\rightarrow qZ$ using the upgraded ATLAS experiment at the High Luminosity LHC.”. http://cds.cern.ch/record/2653389.
  * [63] FCC Collaboration, “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 (2019) 755–1107.
  * [64] ILC International Development Team Collaboration, “The International Linear Collider: Report to Snowmass 2021.” arXiv:2203.07622.
  * [65] H. Khanpour, S. Khatibi, M. Khatiri Yanehsari, and M. Mohammadi Najafabadi, “Single top quark production as a probe of anomalous $tq\gamma$ and $tqZ$ couplings at the FCC-ee,” Phys. Lett. B 775 (2017) 25–31 [arXiv:1408.2090].
  * [66] Y.-B. Liu and S. Moretti, “Probing tqZ anomalous couplings in the trilepton signal at the HL-LHC, HE-LHC and FCC-hh,” Chin. Phys. C 45 (2021) 043110 [arXiv:2010.05148].
  * [67] Y.-B. Liu and S. Moretti, “Probing the top-Higgs boson FCNC couplings via the $h\to\gamma\gamma$ channel at the HE-LHC and FCC-hh,” Phys. Rev. D 101 (2020) 075029 [arXiv:2002.05311].
  * [68] ATLAS Collaboration, “Sensitivity of ATLAS at HL-LHC to flavour changing neutral currents in top quark decays t → cH, with H → $\gamma$$\gamma$.”.
  * [69] H. Khanpour, “Probing top quark FCNC couplings in the triple-top signal at the high energy LHC and future circular collider,” Nucl. Phys. B 958 (2020) 115141 [arXiv:1909.03998].
  * [70] A. Papaefstathiou and G. Tetlalmatzi-Xolocotzi, “Rare top quark decays at a 100 TeV proton–proton collider: $t\rightarrow bWZ$ and $t\rightarrow hc$,” Eur. Phys. J. C 78 (2018) 214 [arXiv:1712.06332].
  * [71] FCC study Group Collaboration, “Prospect for top quark FCNC searches at the FCC-hh,” J. Phys. Conf. Ser. 1390 (2019) 012044 [arXiv:1812.00902].
  * [72] J. Aebischer, J. Kumar, P. Stangl, and D. M. Straub, “A Global Likelihood for Precision Constraints and Flavour Anomalies,” Eur. Phys. J. C 79 (2019) 509 [arXiv:1810.07698].
  * [73] P. Stangl, “smelli – the SMEFT Likelihood,” PoS TOOLS2020 (2021) 035 [arXiv:2012.12211].
  * [74] D. M. Straub, “flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond.” arXiv:1810.08132.
  * [75] J. Aebischer, J. Kumar, and D. M. Straub, “Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale,” Eur. Phys. J. C 78 (2018) 1026 [arXiv:1804.05033].
  * [76] S. Descotes-Genon, A. Falkowski, M. Fedele, M. González-Alonso, and J. Virto, “The CKM parameters in the SMEFT,” JHEP 05 (2019) 172 [arXiv:1812.08163].
  * [77] Particle Data Group Collaboration, “Review of Particle Physics,” PTEP 2020 (2020) 083C01.
  * [78] M. Awramik, M. Czakon, A. Freitas, and G. Weiglein, “Precise prediction for the W boson mass in the standard model,” Phys. Rev. D 69 (2004) 053006 [hep-ph/0311148].
  * [79] J. de Blas, et al., “Global analysis of electroweak data in the Standard Model.” arXiv:2112.07274.
  * [80] J. de Blas, M. Pierini, L. Reina, and L. Silvestrini, “Impact of the recent measurements of the top-quark and W-boson masses on electroweak precision fits.” arXiv:2204.04204.
  * [81] ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, Tevatron Electroweak Working Group, SLD Electroweak, Heavy Flavour Groups Collaboration, “Precision Electroweak Measurements and Constraints on the Standard Model.” arXiv:1012.2367.
  * [82] ATLAS Collaboration, “Measurement of the $W$-boson mass in pp collisions at $\sqrt{s}=7$ TeV with the ATLAS detector,” Eur. Phys. J. C 78 (2018) 110 [arXiv:1701.07240]. [Erratum: Eur.Phys.J.C 78, 898 (2018)].
  * [83] LHCb Collaboration, “Measurement of the W boson mass,” JHEP 01 (2022) 036 [arXiv:2109.01113].
  * [84] L.-S. Geng, et al., “Towards the discovery of new physics with lepton-universality ratios of $b\to s\ell\ell$ decays,” Phys. Rev. D 96 (2017) 093006 [arXiv:1704.05446].
  * [85] A. Crivellin, C. Greub, D. Müller, and F. Saturnino, “Importance of Loop Effects in Explaining the Accumulated Evidence for New Physics in B Decays with a Vector Leptoquark,” Phys. Rev. Lett. 122 (2019) 011805 [arXiv:1807.02068].
  * [86] M. Algueró, B. Capdevila, S. Descotes-Genon, P. Masjuan, and J. Matias, “Are we overlooking lepton flavour universal new physics in $b\to s\ell\ell$ ?” Phys. Rev. D 99 (2019) 075017 [arXiv:1809.08447].
  * [87] M. Algueró, et al., “Emerging patterns of New Physics with and without Lepton Flavour Universal contributions,” Eur. Phys. J. C 79 (2019) 714 [arXiv:1903.09578]. [Addendum: Eur.Phys.J.C 80, 511 (2020)].
  * [88] W. Altmannshofer and P. Stangl, “New physics in rare B decays after Moriond 2021,” Eur. Phys. J. C 81 (2021) 952 [arXiv:2103.13370].
  * [89] M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias, and M. Novoa-Brunet, “$b\rightarrow s\ell^{+}\ell^{-}$ global fits after $R_{K_{S}}$ and $R_{K^{*+}}$,” Eur. Phys. J. C 82 (2022) 326 [arXiv:2104.08921].
  * [90] D. London and J. Matias, “$B$ Flavour Anomalies: 2021 Theoretical Status Report.” arXiv:2110.13270.
  * [91] L. Di Luzio, M. Kirk, A. Lenz, and T. Rauh, “$\Delta M_{s}$ theory precision confronts flavour anomalies,” JHEP 12 (2019) 009 [arXiv:1909.11087].
  * [92] ATLAS Collaboration, “Combination of the searches for pair-produced vector-like partners of the third-generation quarks at $\sqrt{s}=$ 13 TeV with the ATLAS detector,” Phys. Rev. Lett. 121 (2018) 211801 [arXiv:1808.02343].
  * [93] ATLAS Collaboration, “Search for pair-production of vector-like quarks in $pp$ collision events at $\sqrt{s}=13$ TeV with at least one leptonically-decaying $Z$ boson and a third-generation quark with the ATLAS detector.”. http://cds.cern.ch/record/2773300.
  * [94] ATLAS Collaboration, “Search for single production of vector-like $T$ quarks decaying to $Ht$ or $Zt$ in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector.”. http://cds.cern.ch/record/2779174.
  * [95] LHCb Collaboration, “Measurement of $CP$-Averaged Observables in the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ Decay,” Phys. Rev. Lett. 125 (2020) 011802 [arXiv:2003.04831].
  * [96] LHCb Collaboration, “Angular analysis of the rare decay ${B}_{s}^{0}$→ $\phi$$\mu$+$\mu$-,” JHEP 11 (2021) 043 [arXiv:2107.13428].
  * [97] LHCb Collaboration, “Branching Fraction Measurements of the Rare $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B^{0}_{s}\rightarrow f_{2}^{\prime}(1525)\mu^{+}\mu^{-}$\- Decays,” Phys. Rev. Lett. 127 (2021) 151801 [arXiv:2105.14007].
  * [98] J. J. Heckman, “Extra $W$-Boson Mass from a D3-Brane.” arXiv:2204.05302.
  * [99] J. de Blas, J. C. Criado, M. Perez-Victoria, and J. Santiago, “Effective description of general extensions of the Standard Model: the complete tree-level dictionary,” JHEP 03 (2018) 109 [arXiv:1711.10391].

Supplemental Material

##  1 SMEFT operators

Here we give the definitions of the SMEFT operators relevant for flavour and
electroweak precision observables according to Ref. [52]. $p$, $r$, $s$ and
$t$ are flavour indices, while color as well as $SU(2)_{L}$ indices are
contracted within the bi-linears and $T^{A}$ stands for the generators of
$SU(3)_{C}$.

#### Modified gauge boson couplings:

The operators generating modified gauge boson couplings to quarks after EW
symmetry breaking are

$\displaystyle Q_{Hq}^{(1)ij}$
$\displaystyle=(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\bar{q}_{i}\gamma^{\mu}q_{j})\,,$
$\displaystyle Q_{Hq}^{(3)ij}$
$\displaystyle=(H^{\dagger}i\overleftrightarrow{D_{\mu}^{I}}H)(\bar{q}_{i}\tau^{I}\gamma^{\mu}q_{j})\,,$
(S.1.1) $\displaystyle Q_{Hu}^{ij}$
$\displaystyle=(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\bar{u}_{i}\gamma^{\mu}u_{j})\,,$
$\displaystyle Q_{Hd}^{ij}$
$\displaystyle=(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\bar{d}_{i}\gamma^{\mu}d_{j})\,,$
(S.1.2) $\displaystyle Q_{Hud}^{ij}$
$\displaystyle=i(\tilde{H}^{\dagger}D_{\mu}H)(\bar{u}_{i}\gamma^{\mu}d_{j})\,,$
(S.1.3)

with the covariant derivative given in Eq. (II.2) and

$\overleftrightarrow{D_{\mu}}=(D_{\mu}-\overset{\leftarrow}{D}_{\mu})\,,\qquad\overleftrightarrow{D_{\mu}^{I}}=(\tau^{I}D_{\mu}-\overset{\leftarrow}{D}_{\mu}\tau^{I})\,.$
(S.1.4)

It is useful to write explicitly the modifications of the
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
and
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
couplings (after EW symmetry breaking) as a function of the SMEFT
coefficients:

$\displaystyle\delta\mathcal{L}_{W,Z}=$
$\displaystyle-v^{2}\frac{g}{\sqrt{2}}W^{+}_{\mu}\,\bar{u}_{i}\gamma^{\mu}\left(\left[V\cdot
C^{(3)}_{Hq}\right]_{ij}P_{L}+\frac{1}{2}\left[C_{Hud}\right]_{ij}P_{R}\right)d_{j}+\mathrm{h.c.}$
(S.1.5)
$\displaystyle-v^{2}\frac{g}{2c_{W}}Z_{\mu}\,\bar{u}_{i}\gamma^{\mu}\left(\left[V\cdot\left\\{C^{(3)}_{Hq}-C^{(1)}_{Hq}\right\\}\cdot
V^{\dagger}\right]_{ij}P_{L}-\left[C_{Hu}\right]_{ij}P_{R}\right)u_{j}$
(S.1.6)
$\displaystyle-v^{2}\frac{g}{2c_{W}}Z_{\mu}\,\bar{d}_{i}\gamma^{\mu}\left(\left[C^{(3)}_{Hq}+C^{(1)}_{Hq}\right]_{ij}P_{L}+\left[C_{Hd}\right]_{ij}P_{R}\right)d_{j}\,,$
(S.1.7)

where $V$ is the CKM matrix and $v\simeq 246$ GeV.

####  $\Delta F=2$ processes:

The operators giving rise to, e.g., $B_{s}\\!-\\!\,\overline{\\!B}{}_{s}\,$
mixing, read

$\displaystyle\left[Q^{(1)}_{qq}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}q_{r})(\bar{q}_{r}\gamma_{\mu}q_{t})\,,$
$\displaystyle\left[Q^{(3)}_{qq}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}\tau^{I}q_{r})(\bar{q}_{r}\gamma_{\mu}\tau^{I}q_{t})\,,$
(S.1.8) $\displaystyle\left[Q_{uu}\right]_{prst}$
$\displaystyle=(\bar{u}_{p}\gamma^{\mu}u_{r})(\bar{u}_{s}\gamma_{\mu}u_{t})\,,$
$\displaystyle\left[Q_{dd}\right]_{prst}$
$\displaystyle=(\bar{d}_{p}\gamma^{\mu}d_{r})(\bar{d}_{s}\gamma_{\mu}d_{t})\,,$
(S.1.9) $\displaystyle\left[Q^{(1)}_{qu}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}q_{r})(\bar{u}_{s}\gamma_{\mu}u_{t})\,,$
$\displaystyle\left[Q^{(1)}_{qd}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}q_{r})(\bar{d}_{s}\gamma_{\mu}d_{t})\,,$
(S.1.10) $\displaystyle\left[Q^{(8)}_{qu}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}T^{A}q_{r})(\bar{u}_{s}\gamma_{\mu}T^{A}u_{t})\,,$
$\displaystyle\left[Q^{(8)}_{qd}\right]_{prst}$
$\displaystyle=(\bar{q}_{p}\gamma^{\mu}T^{A}q_{r})(\bar{d}_{s}\gamma_{\mu}T^{A}d_{t})\,.$
(S.1.11)

#### Down-quark magnetic dipoles:

The operators generating $b\to s\gamma$ after EW breaking are

$Q_{dW}^{ij}=(\bar{q}_{i}\sigma^{\mu\nu}d_{j})\tau^{I}HW^{I}_{\mu\nu}\,,\qquad
Q_{dB}^{ij}=(\bar{q}_{i}\sigma^{\mu\nu}d_{j})HB_{\mu\nu}\,.$ (S.1.12)

#### Modified Higgs couplings:

Here we have

$Q_{uH}^{ij}=(H^{\dagger}H)(\bar{q}_{i}u_{j}\tilde{H})\,,\qquad
Q_{dH}^{ij}=(H^{\dagger}H)(\bar{q}_{i}d_{j}{H})\,.$ (S.1.13)

#### The $W$ mass:

The prediction for the $W$-boson mass is affected already at tree-level by

$Q_{HD}=(H^{\dagger}D_{\mu}H)^{*}(H^{\dagger}D^{\mu}H)\,,\qquad
Q_{HWB}=(H^{\dagger}\tau^{I}H)W^{I}_{\mu\nu}B^{\mu\nu}\,.$ (S.1.14)

##  2 CKM treatment

Since we modify $W$-quark couplings at tree-level, the determination of the
“correct” CKM elements is non-trivial. In smelli, the method of Ref. [76] is
implemented which fixes the four inputs needed to determine the CKM using four
observables, fully taking into account both the SM and new physics
contributions, at each point in parameter space. These four observables are
$\Gamma(K^{+}\to\mu^{+}\nu)/\Gamma(\pi^{+}\to\mu^{+}\nu)$, $\mathrm{Br}(B\to
X_{c}e^{+}\nu)$, $\mathrm{Br}(B^{+}\to\tau^{+}\nu)$, and $\Delta M_{d}/\Delta
M_{s}$. Once the CKM matrix is fixed, it is then used as the input to all the
other theory predictions, including the “SM” part. Note however that this
means there is essentially a scheme dependence to which observables show
discrepancies with data (since the four used to fix the CKM must agree with
experiment, by construction), while only the global $\Delta\chi^{2}$ is
physical.

##  3 VLQ to SMEFT matching coefficients

The part of the Lagrangian detailing the interaction between the SM Higgs and
two VLQs is given by

$\displaystyle-\mathcal{L}^{H}_{\mathrm{VLQVLQ}}=$
$\displaystyle\lambda^{L,UQ}{{\bar{U}}}{{\tilde{H}}^{\dagger}}P_{L}{Q_{1}}+\lambda^{L,DQ}{{\bar{D}}}{H^{\dagger}}P_{L}{Q_{1}}+\lambda^{L,{Q_{5}}D}{{\bar{Q}}_{5,\alpha}}\tilde{H}P_{L}{D_{\beta}}+\lambda^{L,{Q_{7}}U}{{\bar{Q}}_{7}}HP_{L}{U}$
(S.3.1)
$\displaystyle+\frac{1}{2}\lambda^{L,{T_{1}}Q}{H^{\dagger}}\tau\cdot{{\bar{T}}_{1}}P_{L}{Q_{1}}+\frac{1}{2}\lambda^{L,{T_{2}}Q}{{\tilde{H}}^{\dagger}}\tau\cdot{{\bar{T}}_{2}}P_{L}{Q_{1}}+\frac{1}{2}\lambda^{L,{T_{1}}{Q_{5}}}{\tilde{H}^{\dagger}}\tau\cdot{{\bar{T}}_{1}}P_{L}{Q_{5}}$
$\displaystyle+\frac{1}{2}\lambda^{L,{T_{2}}{Q_{7}}}{{H}^{\dagger}}\tau\cdot{{\bar{T}}_{2}}P_{L}{Q_{7}}+(L\leftrightarrow
R)+\mathrm{h.c.}\,.$

Note that, generalizing Ref. [99], the interaction between two VLQs and the SM
Higgs can be different for the left-handed and right-handed components.

We give the coefficients relevant for our phenomenological study in the main
text. At tree-level, in addition to those given in the main text (Eq.
(II.10)), , we also find for the Wilson coefficients of the three Higgs
operator

$\displaystyle C_{uH}^{ij}$
$\displaystyle=\frac{Y^{u}_{kj}\xi^{U}_{k}\xi^{U*}_{i}}{2M_{U}^{2}}+\frac{Y^{u}_{ik}\xi^{u_{1}}_{j}\xi^{u_{1}*}_{k}}{2M_{Q_{1}}^{2}}+\frac{Y^{u}_{ik}\xi^{Q_{7}}_{j}\xi^{Q_{7}*}_{k}}{2M_{Q_{7}}^{2}}\,.$
(S.3.2)

At one-loop, we only present here the one-loop matching terms proportional to
either the new physics coupling $\xi$, or the SM up-type Yukawa $Y^{u}$, thus
neglecting the down-type and lepton Yukawas as well as the gauge couplings
$g_{1},g_{2}$. One exception are the electroweak-dipole operators, where we
include $Y^{d}g_{1,2}\xi^{2}$ terms since the SM dipole operators are already
suppressed by the same factor. Another exception is the $W$ mass operator
$Q_{HWB}$ where we also include $g_{1}g_{2}\xi^{2}$ coefficients, since this
operator can potentially have large contribution (see Eq. (II.9)). In
principle we also consider $\alpha_{s}$ 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 $Q_{Hq}^{(1,3)}$ operators to
contribute to the renormalization of the $Q_{qq}^{(1,3)}$ operators, and
similarly for the other $\Delta F=2$ operators. We perform the calculation by
matching off-shell amplitudes calculated in the full VLQ theory and the SMEFT,
and allowing for a $\overline{\rm MS}$ counter-term contribution on the SMEFT
side. A potential IR divergence is regulated using the Higgs doublet mass
$\mu_{H}$, which is set to zero at the end of the calculation.

#### One-loop matching for $\Delta F=2$ processes:

Figure S.1: Typical box diagrams giving rise to the $\Delta F=2$ matching
coefficients, where the external fermions $\psi$ 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.

The one-loop matchings onto the $\Delta F=2$ operators are obtained from the
diagrams in Fig. S.1. Among the three representations ($U$, $Q_{1}$ with
$\xi^{u_{1}}$, and $Q_{7}$), only $U$ contributes to
$B_{s}\\!-\\!\,\overline{\\!B}{}_{s}\,$ mixing. The one-loop matching
condition for $U$ is

$\displaystyle\left[C^{(1)}_{qq}\right]_{ijij}$
$\displaystyle=-\frac{\xi^{U\ast}_{i}\xi^{U}_{j}\xi^{U\ast}_{i}\xi^{U}_{j}}{256\pi^{2}M_{U}^{2}}+\frac{\xi^{U*}_{i}\xi^{U}_{j}(Y^{u}Y^{u\dagger})_{ij}}{128\pi^{2}}\widetilde{F}(M_{U})\,,$
(S.3.3) $\displaystyle\left[C^{(3)}_{qq}\right]_{ijij}$
$\displaystyle=-\frac{\xi^{U\ast}_{i}\xi^{U}_{j}\xi^{U\ast}_{i}\xi^{U}_{j}}{256\pi^{2}M_{U}^{2}}+\frac{\xi^{U*}_{i}\xi^{U}_{j}(Y^{u}Y^{u\dagger})_{ij}}{128\pi^{2}}\widetilde{F}(M_{U})\,,$
(S.3.4)

where the IR-finite loop function is

$\widetilde{F}(m)=\frac{1}{m^{2}}\left(\frac{3}{2}+\ln\frac{\mu^{2}}{m^{2}}\right)\,,$
(S.3.5)

and the renormalization scale $\mu$ should be $\mathcal{O}(M_{\rm VLQ})$.

Figure S.2: Typical diagrams giving rise to the dipole matching coefficients,
where the internal fermions $\psi$ can be a SM quark or a 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,

$\displaystyle\left[C_{dB}\right]_{ij}$
$\displaystyle=\frac{7g_{1}Y^{d}_{kj}\xi^{U*}_{i}\xi^{U}_{k}}{1152\pi^{2}M_{U}^{2}}\,,\qquad\left[C_{dW}\right]_{ij}=-\frac{5g_{2}Y^{d}_{kj}\xi^{U*}_{i}\xi^{U}_{k}}{384\pi^{2}M_{U}^{2}}\,.$
(S.3.6)

Tree |  $U$ |  $D$ |  $Q_{1}$ |  $Q_{5}$ |  $Q_{7}$ |  $T_{1}$ |  $T_{2}$
---|---|---|---|---|---|---|---
$Z$-$u_{L}^{i}$-$u_{L}^{j}$ |  ✓ |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
$Z$-$u_{R}^{i}$-$u_{R}^{j}$ |  $\times$ |  $\times$ |  ✓ |  $\times$ |  ✓ |  $\times$ |  $\times$
$Z$-$d_{L}^{i}$-$d_{L}^{j}$ |  $\times$ |  ✓ |  $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓
$Z$-$d_{R}^{i}$-$d_{R}^{j}$ |  $\times$ |  $\times$ |  ✓ |  ✓ |  $\times$ |  $\times$ |  $\times$
Table S.1: Overview on modified $Z$-quark couplings (in broken $SU(2)_{L}$) at
tree level in the VLQ models.

#### 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 $C_{Hq}^{(1)}-C_{Hq}^{(3)}$, $C_{Hu}$ and
$C_{Hq}^{(1)}+C_{Hq}^{(3)}$, $C_{Hd}$, 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
$b\to s\ell\ell$, we also calculate the one-loop matching for the
$U,Q_{1},Q_{7}$ cases where they are not already present at tree level.

$\displaystyle\left[C_{Hq}^{(1)}\right]_{ij}=$
$\displaystyle{\frac{\xi_{j}^{U}\xi_{k}^{U*}Y_{il}^{u}Y_{lk}^{u\dagger}}{32\pi^{2}M_{U}^{2}}\left(1+\ln\frac{\mu^{2}}{M_{U}^{2}}\right)}{+\frac{\xi_{k}^{U}\xi_{i}^{*U}Y_{kl}^{u}Y_{lj}^{u\dagger}}{32\pi^{2}M_{U}^{2}}\left(1+\ln\frac{\mu^{2}}{M_{U}^{2}}\right)}{-\frac{\xi_{j}^{U}\xi_{k}^{U}\xi_{i}^{*U}\xi_{k}^{*U}}{256\pi^{2}M_{U}^{2}}\left(17+14\log\frac{\mu^{2}}{M_{U}^{2}}\right)}$
$\displaystyle-\frac{Y^{u}_{il}Y^{u\dagger}_{kj}\xi^{u_{1}}_{k}\xi^{u_{1}*}_{l}}{384\pi^{2}M_{Q_{1}}^{2}}\left(1+6\ln\frac{\mu^{2}}{M_{Q_{1}}^{2}}\right){-\frac{Y^{u}_{il}Y^{u\dagger}_{kj}\xi^{Q_{7}}_{k}\xi^{Q_{7}*}_{l}}{64\pi^{2}M_{Q_{7}}^{2}}\left(-13+6\ln\frac{\mu^{2}}{M_{Q_{7}}^{2}}\right)}\,,$
(S.3.7) $\displaystyle\left[C_{Hq}^{(3)}\right]_{ij}=$
$\displaystyle\frac{\xi_{j}^{U}\xi_{k}^{U}\xi_{i}^{U*}\xi_{k}^{U*}}{256\pi^{2}M_{U}^{2}}\left(9+14\ln\frac{\mu^{2}}{M_{U}^{2}}\right)-\frac{\xi^{u_{1}}_{k}\xi^{u_{1}*}_{l}Y^{u}_{il}Y^{u\dagger}_{kj}}{96\pi^{2}M_{Q_{1}}^{2}}-\frac{5\xi^{Q_{7}}_{k}\xi^{Q_{7}*}_{l}Y^{u}_{il}Y^{u\dagger}_{kj}}{192\pi^{2}M_{Q_{7}}^{2}}\,.$
(S.3.8)

#### One-loop matching for the $W$ mass:

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

The one-loop matching onto the operators $Q_{HD}$ and $Q_{HWB}$, 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

$\displaystyle C_{HD}=$
$\displaystyle-\frac{3\xi^{U}_{i}\xi^{U}_{j}\xi^{U\ast}_{i}\xi^{U\ast}_{j}}{32\pi^{2}M_{U}^{2}}+\frac{3\xi^{U}_{i}(Y^{u}Y^{u\dagger})_{ij}\xi^{U*}_{j}}{16\pi^{2}}F_{1}(M_{U})$
$\displaystyle-\frac{\xi^{u_{1}}_{i}\xi^{u_{1}}_{j}\xi^{u_{1}\ast}_{i}\xi^{u_{1}\ast}_{j}}{8\pi^{2}M_{Q_{1}}^{2}}+\frac{3\xi^{u_{1}}_{i}(Y^{u\dagger}Y^{u})_{ij}\xi^{u_{1}*}_{j}}{8\pi^{2}}F_{2}(M_{Q_{1}})-\frac{\xi^{Q_{7}}_{i}\xi^{Q_{7}}_{j}\xi^{{Q_{7}}\ast}_{i}\xi^{{Q_{7}}\ast}_{j}}{8\pi^{2}M_{Q_{7}}^{2}}-\frac{3\xi^{Q_{7}}_{i}(Y^{u\dagger}Y^{u})_{ij}\xi^{Q_{7}*}_{j}}{8\pi^{2}}F_{2}(M_{Q_{7}})\,,$
(S.3.9) $\displaystyle C_{HWB}$
$\displaystyle=-\frac{g_{1}g_{2}\xi^{U}_{i}\xi^{U\ast}_{i}}{64\pi^{2}M_{U}^{2}}-\frac{g_{1}g_{2}\xi^{u_{1}}_{i}\xi^{u_{1}\ast}_{i}}{96\pi^{2}M_{Q_{1}}^{2}}+\frac{g_{1}g_{2}\xi^{Q_{7}}_{i}\xi^{{Q_{7}}\ast}_{i}}{96\pi^{2}M_{Q_{7}}^{2}}\,,$
(S.3.10)

with

$F_{1}(m)=\frac{1}{m^{2}}\left(\frac{1}{2}+\ln\frac{\mu^{2}}{m^{2}}\right)\,,\quad
F_{2}(m)=\frac{1}{m^{2}}\left(\frac{3}{2}+\ln\frac{\mu^{2}}{m^{2}}\right)\,.$
(S.3.11)

##  4 $t\to cZ$ and $t\to ch$

For $t\to cZ$ [66], we take the branching ratio to the $Z$ boson to be:

$\mathrm{Br}(t\to cZ)\approx 0.47\left|\lambda_{tcZ}\right|^{2}\,,$ (S.4.1)

with the couplings defined by the Lagrangian terms

$\mathcal{L}=\frac{g}{2c_{W}}\lambda_{tcZ}Z_{\mu}\bar{c}\gamma^{\mu}(\lambda_{L}P_{L}+\lambda_{R}P_{R})t+\mathrm{h.c.}\,,$
(S.4.2)

with the normalisation $\lambda_{L}^{2}+\lambda_{R}^{2}=1$. In Ref. [58], they
give the simple formula for the FCNC Higgs branching ratio:

$\mathrm{Br}(t\to ch)\approx
0.27\left|\lambda_{t_{L}c_{R}h}^{2}+\lambda_{c_{L}t_{R}h}^{2}\right|\,,$
(S.4.3)

where the couplings are defined by the Lagrangian terms

$\mathcal{L}=\lambda_{t_{L}c_{R}h}\bar{t}P_{R}c\,h+\lambda_{c_{L}t_{R}h}\bar{c}P_{R}t\,h+\mathrm{h.c.}\,.$
(S.4.4)

Comparing to the SMEFT Lagrangian, we see the correspondence:

$\displaystyle\lambda_{tcZ}\lambda_{L}$
$\displaystyle=v^{2}\left[V\cdot\left(C_{Hq}^{(3)}-C_{Hq}^{(1)}\right)V^{\dagger}\right]_{23}\,,\quad\lambda_{tcZ}\lambda_{R}=v^{2}\left[C_{Hu}\right]_{23}\,,$
(S.4.5) $\displaystyle\lambda_{t_{L}c_{R}h}$
$\displaystyle=-\frac{v^{2}}{\sqrt{2}}\left[C_{uH}\right]_{32}\,,\quad\lambda_{c_{L}t_{R}h}=-\frac{v^{2}}{\sqrt{2}}\left[C_{uH}\right]_{23}\,.$
(S.4.6)

For the estimation of the $t\to cZ,h$ branching ratios, at the level of
precision we are considering, we can take the CKM matrix to be the unit
matrix.