New Vector-Like Fermions and Flavor Physics

We study renormalizable extensions of the standard model that contain vector-like fermions in a (single) complex representation of the standard model gauge group. There are 11 models where the vector-like fermions Yukawa couple to the standard model fermions via the Higgs field. These models do not introduce additional fine- tunings. They can lead to, and are constrained by, a number of different flavor- changing processes involving leptons and quarks, as well as direct searches. An interesting feature of the models with strongly interacting vector-like fermions is that constraints from neutral meson mixings (apart from CP violation in neutral kaon mixing) are not sensitive to higher scales than other flavor-changing neutral-current processes. We identify order 1/(4 pi M)^2 (where M is the vector-like fermion mass) one-loop contributions to the coefficients of the four-quark operators for meson mixing, that are not suppressed by standard model quark masses and/or mixing angles.


Introduction
The driving force behind many of the extensions of the standard model (SM) has been the hierarchy puzzle. However, there may be reasons that the fine tuning of quadratically large contributions to the Higgs mass from very high momentum scales is acceptable (e.g., environmental reasons). This motivates the consideration of simple extensions of the standard model that do not address the hierarchy puzzle and do not introduce any additional fine tunings of parameters.
The SM provides no understanding why there are three generations of chiral fermions with masses that require weak symmetry breaking or why they only come in such simple representations of the gauge group. Vector-like fermions can be much heavier than the SM fermions, since their masses do not require weak symmetry breaking. For some choices of quantum numbers, such vector-like fermions can Yukawa couple to SM fermions. Given our lack of understanding of the number of generations and their quantum numbers, it seems worthwhile exploring the possibility that vector-like fermions exist. Of course, their masses may all be much larger than the weak scale, but it is not unreasonable that one such vector-like representation has a mass light enough that it can influence the next generation -1 -

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of flavor physics experiments. This motivates a study of extensions of the standard model with additional fermions that reside in a single vector-like representation of the SM gauge group with a mass M . (For some earlier studies of such models, see, e.g., refs. [1][2][3][4][5][6][7][8][9][10]. ) We focus on vector-like fermions that can have Yukawa couplings involving the Higgs field and the SM fermions. This allows them to influence flavor physics. There are several such models. These models have been studied in the literature previously, but here we consider all such models and compare the patterns of deviations from the SM they would give rise to. These models predict a very specific pattern for the contributions of beyond the standard model (BSM) physics to ∆F = 1 flavor-changing neutral-current (FCNC) processes compared to ∆F = 2 neutral meson mixings. For large masses M the dominant order 1/(4πM ) 2 contribution to the coefficients of the four-quark operators responsible for neutral meson mixing arise at one loop, and are not suppressed by SM quark masses and/or weak mixing angles. We compute this contribution to meson mixing in the 7 models that involve strongly interacting vector-like fermions. In addition to considering ∆F = 1 and ∆F = 2 flavor-changing neutral-current processes, we also briefly discuss violations of lepton universality and unitarity of the CKM matrix that arises from the corrections to the W boson couplings in these models.
The light neutrino masses play no role in our analysis so we take the SM to contain massless left-handed neutrinos. We are most interested in experiments that have reach in vector-like fermion mass of more than ∼ 10 TeV when the Yukawa coupling constants of the vector-like fermions to the standard model fermions are around unity. For such masses, effects originating from the non-unitarity of the 3 × 3 CKM matrix can be neglected.
If the new fermion is in a real representation it can have a Majorana mass term which does not require a helicity partner in the spectrum. In that case, unless the Majorana mass is very large or the Yukawa coupling of the new neutral fermion to the SM neutrinos is very small, a SM neutrino gets an unacceptably large mass. Hence we do not consider models where the additional fermions come in a real representation of the gauge group. 1 ATLAS and CMS have searched for strongly interacting vector-like fermions contained in several of the models discussed below [11][12][13][14][15]. The bounds from the Run 1 data constrain the masses of vector-like quarks typically at about the M > 800 GeV level. These constraints come from the pair production of the vector-like fermions via their gauge couplings, and are (essentially) independent of the Yukawa couplings, which are the focus of this paper.
In section 2 we describe the models, section 3 explores the constraints from measurements both in the lepton and quark sectors, and section 4 contains our conclusions.

The models
There are 11 renormalizable models with vector-like fermions in complex representations of the standard model gauge group, where the vector-like fermions have renormalizable Yukawa couplings to the SM fermions through the Higgs doublet.
Model III has vector-like fermions, , with same gauge quantum numbers as the SM left-handed doublets, (1, 2, −1/2). They Yukawa couple to the SM right-handed charged lepton fields e i R . Model IV contains vector-like fermions, l, with (1, 2, −3/2) gauge quantum numbers, and they also Yukawa couple to e i R . Models V and VI contain vector-like fermions D and U with the same quantum numbers as the SM right-handed down and up-type quarks, (3, 1, −1/3) and (3, 1, 2/3). They Yukawa couple to the SM left-handed quark doublets Q i L . Models VII and VIII are similar with vector-like SU(2) L triplet fermions T d and T u with gauge quantum numbers (3, 3, −1/3) and (3, 1, 2/3) respectively.
There are three more models where the vector-like fermions Yukawa couple to the SM right-handed up-type, u i R and down-type d i R quarks. In Model IX the vector-like fermions q ud have the same quantum numbers as the SM left-handed quark doublets, (3, 2, 1/6). In this case the new vector-like quark doublet couples to both u i R and d i R . Model X has a vector-like quark doublet q u with quantum numbers (3, 2, 7/6) that Yukawa couple through the Higgs doublet to the right-handed up-type quarks. Model XI has a vector-like quark doublet q d with quantum numbers (3, 2, −5/6) that Yukawa couples to the right-handed down-type quarks.

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Here the SU(2) triplets T e , T u and T d are represented by the two-by-two matrices where the subscripts denote the charges of the fermions. It is important to note that there are no new one-loop contributions to the ordinary lepton and quark mass matrices of the form where v 174 GeV is the Higgs vacuum expectation value. This is due to the approximate chiral symmetry, e.g., e i R → e iα e i R in Model I, which is only broken by the SM Yukawa couplings. This prevents such contributions to the light fermion mass matrix. So the λ i couplings can be order unity without the need for a cancellation in the SM fermion mass matrix.
The BSM effects caused by the vector-like fermions vanish as their mass M → ∞. We are interested in very large M , greater than 10 TeV, and it is flavor-changing neutralcurrent processes that are sensitive to such BSM physics. In the SM, flavor-changing neutral currents are suppressed by small weak mixing angles and small SM quark or lepton masses. Furthermore, they do not occur at tree level. Since the neutrino masses are very small, charged lepton flavor violation in the SM is negligibly small. In the quark sector, despite their suppression, flavor-changing neutral-current processes that change flavor by one (∆F = 1, e.g., K + → π + νν) and by two (∆F = 2, e.g., K 0 −K 0 mixing) have been observed.
In the limit where the SM Yukawa couplings vanish, the SM has a U(3 flavor symmetry. This symmetry forbids flavor-changing neutral currents. However in Models I-XI, even when the SM quark and lepton Yukawas are zero, the terms in the Lagrange density proportional to the Yukawa couplings λ i break that symmetry. That pattern of flavor symmetry breaking can be characterized using the spurion method. For example in Model V the BSM terms in the Lagrange density involving ij λ j , then the U(3) Q symmetry is restored. This means that in Model V one-loop BSM physics associated with the high mass scale M generates meson mixing through a term in the effective Lagrangian of the form, 16) and similarly for the other models. These corrections are not suppressed by small quark masses and/or mixing angles, and we have not found expressions for them in the literature. (We take the BSM Yukawa couplings λ i to be of order unity.) They are computed for Models V-XI in section 3.2.4. Corrections to meson mixing of order λ i λ * j /(4πM ) 2 that are also suppressed by weak mixing angles and/or quark masses were considered for Model V in ref. [16].

New interactions with gauge bosons
The new BSM contribution to the Z coupling and to the W arise at tree level, after integrating out the heavy fermion(s). They can be obtained ether by calculating Feynman diagrams or by diagonalizing the 4 × 4 fermion mass matrices as was discussed in ref. [17].
Here we explicitly show how to obtain the BSM contributions to the Z couplings for Model I. (The same method can be applied to the other models.) After electroweak symmetry breaking, the mass matrix in charged lepton sector becomes, where roman capital indices A and B go over {0, 1, 2, 3},ê is defined byê ≡ (E, e 1 , e 2 , e 3 ) T , and Here we assumed, without loss of generality, that the charged lepton fields e i L (i.e., the lower components of the doublets L i L ) are eigenstates of the charged lepton mass matrix in the SM (i.e., in the λ i → 0 limit). This matrix is diagonalized by the 4 × 4 unitary transformations Vê L,R ,ê L,R = Vê L,Rê L,R , (2.19) where the prime denotes a mass eigenstate field. So Up to corrections suppressed by (v/M ) 2 Consequently the masses of the charged leptons in the SM part of the Lagrangian are approximately equal to the charged lepton masses, and the vector mass parameter M is approximately the heavy vector-like lepton mass. The Z boson coupling is, where the ellipses denote terms not containing the matrix Vê L . Here g Z = g 2 1 + g 2 2 and g 1,2 are the gauge couplings of U(1) Y and SU(2) L , respectively. There is some ambiguity in how the terms are organized since Vê L is unitary. We have written the Z couplings involving Vê L so that the order (v/M ) 2 corrections to the SM Z couplings to the charged leptons can be read off using the order (v/M ) terms in Vê L that we have explicitly calculated. From this it follows (removing the hats and primes) that the BSM couplings of the Z boson to the light mass eigenstate charged leptons are, with m Z being the Z boson mass. For simplicity, hereafter we remove the primes used to specify the mass eigenstate fields.
In the same way, the new Z couplings to the quarks and charged leptons are obtained for the other models.
Note that in Models III and IV the BSM tree-level Z couplings differ only by an overall sign. Similarly for hadronic models, Model VI : L Model IX : L Model XI : L So far, we have worked in the mass eigenstate basis most convenient for expressing the Z couplings. For example, in Model V we have worked in a basis where in the λ i → 0 limit the down-type quarks in the Lagrangian are mass eigenstates, while in Model VI we have worked in the basis where the down-type quarks in the Lagrangian are mass eigenstate. In -6 -

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Model IX the couplings λ (d) correspond to the down quark mass eigenstate basis and the couplings λ (u) are in the up-quark mass eigenstate basis.
In case of vector-like triplets, i.e., Models VII and VIII, it is not possible to choose a basis where the CKM matrix, V , is absent from the Z couplings. 2 Choosing the up-type quarks in the Lagrange density to be mass eigenstates (in the λ i → 0 limit) implies for Model VII, Similarly, in Model VIII the result is We can transform these Z couplings to the basis where the down-type quarks are mass eigenstates in the λ i → 0 limit by redefining the couplings λ i → k λ k V † ki . In this basis eqs. (2.32) and (2.33) become and Model VIII : L There are corrections to the W boson couplings as well. We write the couplings of the W -bosons to the left-handed light lepton and quark mass eigenstates in terms of 3 × 3 matrices X and Y as, 3 In the SM limit, X is the inverse of the PMNS matrix and Y is the CKM matrix. Because of the mixing with the vector-like leptons in the extensions of the SM discussed in this paper, X and Y are no longer unitary matrices. It is straightforward to express X and Y in terms of the components of the 4 × 4 diagonalization matrices V L,R in the various models.
In this paper we focus on violations of lepton universality and violations of unitarity of the CKM matrix. For that purpose we compute the quantities.

38)
2 More precisely, the unitary matrix V is the CKM matrix in the λi → 0 limit. 3 There are also right-handed currents in some of the models. at quadratic order in (v 2 /M 2 ), neglecting terms of higher order. We find in the models with vector-like leptons that In Model II we have worked in a basis where the charged leptonsê j are mass eigenstates in the SM part of the charged lepton mass matrix and so In the models with vector-like quarks (neglecting the off diagonal elements of the CKM matrix)

Experimental constraints
For leptonic-extension models, i.e., Models I-IV, the constraints from µ → e conversion, µ → 3e and τ → 3e are important. They are induced by the tree-level Z couplings, which we derived in the previous section. On the other hand, for hadronic-extension models, i.e., Models V-XI, meson mixing, such as K 0 -K 0 and D 0 -D 0 , are induced at one-loop level. In addition, ∆F = 1 FCNC processes, such as K → πνν, K → µ + µ − , B s → + − , etc., are induced at tree level. We will derive effective Hamiltonians which are relevant for these processes and discuss the current experimental bounds, as well as future prospects. (For some recent studies of constraints on some of these models, see, e.g., refs. [18][19][20].) For upper bounds on lepton flavor violating processes we quote bounds at 90% CL, as do most experiments, whereas for other measurements we quote the 1σ limits that are approximately 84% CL as one-sided bounds for Gaussian distributions.

Leptonic models
The flavor violating tree-level couplings in eqs. (2.23)-(2.26) give BSM tree-level contributions to the flavor-changing neural-current processes; for example, in the muon sector, to the µ → 3e rate and muon conversion to an electron in the presence of a nucleus. The amplitudes for these processes are both proportional to the same combination of parameters λ 1 λ * 2 /M 2 . The flavor diagonal terms in eqs. to large values of M as the flavor-changing charged lepton neutral currents are. Note that in these models the radiative decay µ → eγ does not arise at tree level, only at one loop.
Let us first discuss µ → e conversion in the presence of a nucleus. It is among the most powerful probes of charged lepton flavor violation beyond the standard model. The µ − conversion rate to an e − in the presence of a nucleus N is usually quoted as a branching ratio normalized to the SM weak interaction, µ − N → ν µ N capture rate. At present the most stringent bound is Br(µ → e conv. in Au) < 7 × 10 −13 at the 90% C.L. [21]. Future experiments will have a dramatically improved sensitivity to a branching ratio Br(µ → e conv. in Al) ∼ 10 −16 [22,23].
The reach, in mass scale for new physics, of the next generation charged lepton flavor violation (CLFV) experiments that search for µ to e conversion on Al and for the radiative decay µ → eγ was discussed in ref. [24]. They assume an effective Lagrangian of the form, and present the reach of these experiments in the κ−Λ plane. For κ 1 the limit on Λ from conversion on gold is close to 1 × 10 3 TeV while future planned experiments for conversion on aluminum are sensitive to Λ 7.2 × 10 3 TeV. In Models I-IV the weak radiative decay proceeds at the one-loop level and so indeed κ 1. Taking into account left-handed and right-handed quark currents which couple to Z boson, the relations between Λ and the vector-like lepton mass M in Models a = I, II, III, and IV are µ→e = 1 and Z, A are atomic number, mass number of a nucleus, respectively. To obtain this expression we used the vector part in the quark current, neglecting the axial current. For Models III and IV, the right-handed lepton current occurs but it gives the same contribution to the rate as the left-handed one. On gold and aluminum eq. (3.2) implies that, We obtain from figure 2 of ref.
Competitive constraints to the above processes also arise from the upper bounds on the τ → eπ, τ → µπ, τ → eρ, and τ → µρ branching ratios. The vector-like fermions generate Of the current 90% CL experimental limits [29][30][31] the strongest bounds arise from Br(τ → eπ) < 2. These imply that the strongest expected bounds in Model I will be M/ |λ 1 λ 3 | > 19 TeV from τ → eπ, and M/ |λ 2 λ 3 | > 21 TeV from τ → µρ. The expected sensitivities of the other channels are only slightly weaker, thus we can have high confidence in the experimental reach, but not in which channel will give the best bounds. For generic BSM Yukawa couplings, λ, the reach for new physics is much greater for experiments that search for charged lepton flavor violation than those that seek violation of lepton universality. However, it is possible for non-generic λ's that the violations of universality are more important. We close this section by briefly commenting on the implications of the powerful constraint on e-µ charged current universality coming from pion decay π + → e +ν (γ), µ +ν (γ). The latest experimental result gives g e /g µ = 0.9996 ± 0.0012 [32] where g e,µ are the charged-current couplings of e and µ. 4 In Models I-IV, g e /g µ = R 1 /R 2 , which implies, for example, in Model I that, It should be possible to improve this bound by a factor of two in the future. Tau decay, on the other hand, gives constraints on other charged-current couplings, such as g τ /g µ = 1.0011 ± 0.0015 [31], which yields, e.g., in Model I,

Hadronic models
To constrain Models V-XI, FCNC processes in the quark sector are most important. (For a recent study of Z-mediated FCNC effects, see, e.g., ref. [35].) We focus on leptonic and semileptonic decays and neutral meson mixing. In these models, constraints from nonleptonic decays are weaker.

Meson decays involving a νν pair
For kaon decays involving a neutrino-antineutrino pair, such as K + → π + νν, the effective Hamiltonian is where in the second line of eq. (3.14) the subscripts V, A on the coefficients refer to the fact that these coefficients are for the vector and axial quark currents and c ν, Only the vector part of the quark current contributes to this process. In the SM c ν = 0 and at NNLO, |c , with a few percent uncertainty [36][37][38][39][40]. The hadronic models modify c ν,V additively. Obviously in this case it is convenient to work in the basis where the down-type quarks are mass eigenstates. 5 4 The constraint from Kaon decay [33,34] is weaker. 5 Going forward, for different processes we work in whatever basis makes the theoretical expressions simplest. This means that the λ parameters are not always the same, but are linearly related through the CKM matrix.
where for the FCNC down-type BSM contributions Note that for Model IX it is the λ (d) 's that occur in this expression. Models VI and X do not contribute to this process through tree-level Z exchange. The uncertainty of the SM prediction, Br (SM) (K + → π + νν) = (7.8 ± 0.8) × 10 −11 , is dominated by that of c Zd |λ 1 λ 2 | < 1.7 × 10 2 TeV. For the similar 3rd-2nd generation transition mediated by b → sνν, the SM prediction for the coefficient of the operator obtained from eq. (3.14) by d → s ands →b replacements is |c (SM) ν,V | 1/(9.8 TeV) 2 [42]. This process has not been observed yet, and the best current bound, Br(B → Kνν) < 1.6 × 10 −5 , is about 4 times the SM prediction [43]. This yields for new physics aligned with the SM contribution, M (a) > η Zd |λ 2 λ 3 | < 11 TeV). For B → πνν, Belle II expects to reach a sensitivity at the 1 × 10 −5 level [28], which will provide weaker bounds than B → π + − in the models considered in this paper. 6 Despite the stated uncertainty, the probability that all 7 observed events were due to background was quoted as 0.001 [41]. 7 At the SM level, B → Kνν is expected to get large backgrounds from B → K * νν [28]. While Br(B → Kνν) depends only on C+ν , Br(B → K * νν) also depends on C−ν , slightly complicating the analysis.

Meson decays to an + − pair
The effective Hamiltonian differs from eq. (3.14) in that the coupling to the + − pair can either be left-or right-handed. We write the effective Hamiltonian as where up to normalization the conventional choice of operator basis is with obvious replacements for b → d or s → d decays. We use the notation c i and c i to emphasize that these are dimensionful couplings, containing all terms multiplying the four-fermion operators Q i in eq. (3.20). For B s → µ + µ − the SM gives |c and the η Zd coefficients are the same as in eq. (3.17). The LHCb-CMS combination of their measurements, Br(B s → µ + µ − ) = (2.8 +0.7 −0.6 ) × 10 −9 , is quoted as the SM prediction times 0.76 +0. 20 −0.18 [44]. The 1σ range of the measured B s → µ + µ − rate is slightly outside the SM and corresponds to the region, 0.96 > |1 + c  A future 10% measurement [45,46] would increase this sensitivity to ∼ 40 TeV. Again we note that eq. Zd |λ 3 λ 1 | 19 TeV. In the HL-LHC era, this measurement will reach an uncertainty around 20% of the SM prediction [45,46], increasing the mass reach to above 60 TeV.
The extraction of the short-distance part of the measured rate Br(K L → µ + µ − ) = (6.84 ± 0.11) × 10 −9 [47] is subject to considerable uncertainties. The estimate Br(K L → µ + µ − ) SD ≤ 2.5 × 10 −9 [48] is about 3 times the SM short-distance rate which follows from |c SM 10,A | = 1/(1.8 × 10 2 TeV) 2 . If the BSM component is aligned with the SM, we find Unlike the bound from K + → π + νν, the prospect of improving this is not good, and the uncertainties are greater.

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Recently LHCb established a strong bound Br(D 0 → µ + µ − ) < 6.2 × 10 −9 at 90% CL [49], which is well above the SM level. In Models VI-X, It is possible that in the HL-LHC era the experimental bound will improve by a factor of ∼ 20 [51].
The inclusive decay rate Br(B → X s + − ) = (5.0 ± 0.6) × 10 −6 [31] depends in the models we consider on |c 9 | 2 + |c 9 | 2 and |c 10 | 2 + |c 10 | 2 . In Models IX and XI, there is no interference between the SM and the new physics contributions, so the constraints are weak. We follow ref. [53], which studied the rates in the low-and high-q 2 regions and found that for the SM value of c 9 the constraint on c 10 in Models V, VII, and VIII is 0.88 < 1 + c
For FCNC top decays, CMS has set the best bound so far, Br(t → qZ) < 5 × 10 −4 [55] at 95% CL (where q = c, u, corresponding to i = 2, 1 below, respectively). Comparing to the dominant t → bW rate, in models VI-X, For Models VII and VIII we used again the basis in eqs. (2.32) and (2.33). We find Zu |λ 3 λ i | > 0.96 TeV. The HL-LHC is expected to reach sensitivity at the 10 −5 level [56], which will improve this bound to about 2.3 TeV. However, the direct (and λ-independent) searches are comparably sensitive, and are expected to remain to be so.

Neutral meson mixing
Since the new fermions interact with the Higgs field and the ordinary light quarks via Yukawa couplings, meson mixing, such as K 0 -K 0 , D 0 -D 0 , or B 0 -B 0 , is induced. The effective Hamiltonian for these processes contains dimension-six four-quark operators with coefficients of mass dimension −2. At tree level, through Z exchange, the coefficients are of the form ∼ (λ i λ * j ) 2 v 2 /M 4 . However, coefficients of order (λ i λ * j ) 2 /(4πM ) 2 are generated at one loop that are not CKM and/or quark-mass suppressed. For large M , these one-loop matching contributions are more important than tree-level Z exchange. Furthermore, they are independent of the Higgs vacuum expectation value, v, and arise from short distances ∼ 1/M . They can be calculated in the symmetric phase and come from box diagrams with virtual scalars and the heavy vector-like fermions in the loop; see figure 3. The resulting effective Lagrangians are,  Figure 3. Diagrams contributing to K and D mixing in Model V.
The operators above are renormalized with the subtraction point at the scale M . They can be related through a QCD correction factor ξ to scale invariant operators whose matrix elements can be evaluated using lattice QCD. So we write whereB P does not depend on subtraction point. For K mixing, f K = 156 MeV and B K 0.76. In that case the QCD correction factor, ξ, is given in the leading logarithmic approximation [57] by The scale µ 0 does not affect any physical results. For 10 TeV < M < 100 TeV, ξ 0.5, and this value is approximately the same for mixing in the K, D, and B d,s systems. We use the lattice QCD averages from ref. [50], andB D = B MS D (3 GeV)/[α s (3 GeV)] 6/25 1.0 from a recent calculation of B MS D (3 GeV) [58]. In the case of kaon mixing, we demand the new physics contribution to the real part of M 12 to be less than 40% of the experimental value of ∆m K [59]. We take the 1σ uncertainty on the determination of K in the SM to be 25% (reading off the bound h K < 0.35 from ref. [60]). For all the (hadronic) models, except Model X, this results in the constraint (Note that the 90% CL constraint from K is only slightly weaker, replacing 6.7 × 10 2 TeV by 6.3 × 10 2 TeV.) In the future, the K constraint is expected to improve by about √ 2.4 [60], replacing 6.7 × 10 2 TeV by 1.0 × 10 3 TeV in eq. (3.37). Improvement in the real part is contingent upon lattice QCD calculations of the long-distance contributions to ∆m K ; reaching x% precision would replace 42 TeV in eq. (3.37) by 260 TeV/ √ x. In B d,s meson mixing, the new physics contribution is conventionally parametrized as M d,s 12 = (M d,s 12 ) SM × 1+h d,s e 2iσ d,s . Until recently, the bounds on real (2σ = 0, mod π, that is MFV-like) and imaginary (2σ = π/2, mod π) new physics contributions have been quite different [61,62]. This is no longer the case [60], and since we are most interested in physics reach, we simply quote the limits on the absolute values of the new physics contribution. In B d,s mixing, h d < 0.3 and h s < 0.2 [60] yield in all models except Model X, using (XI) mix = 1, and for Model IX it is the λ (d) 's that occur in the constraints. In the next decade these limits will improve to h d < 0.05 and h s < 0.04 [60], which will replace 25 TeV by 61 TeV and 6.4 TeV by 14 TeV in eq. (3.38).
The mixing of D mesons is probably dominated by long-distance physics. Thus, we can only require that the new physics contribution does not exceed the measurement, i.e., ∆m D /Γ D < 0.006 [31]. (The significance of ∆m D = 0 is less than 2σ, so we use the upper bound of the 1σ region. We do not distinguish between imaginary and real contributions to M 12 relative to the SM; the bounds on the mass scale may differ by a factor ∼ 2, depending on the value of ∆m D .) In contrast to above, the λ (u) couplings occur in the constraint for Model IX. Furthermore, in this case we choose the up-type quarks to be the SM mass eigenstate fields, so the factors of the CKM matrix move to the terms with the down-type quarks. Then D mixing implies for all models except Model XI, where now η (X) mix = 1. Note that for Models V-VIII the constraint from D mixing is slightly stronger than from the real part of K mixing. However, the constraints are actually a little different since the λ's in eqs. (3.38) and in (3.39) are not the same. They are linearly related through the CKM matrix. The future evolution of this bound is uncertain. While Belle II expects to measure ∆m D /Γ D with an uncertainty of 0.001 [28], the central value will matter for the bounds on new physics, and therefore we do not assume that this bound will improve.

Unitarity of the CKM matrix
The CKM matrix is not unitary in the extensions of the SM we are considering. The strongest constraints on violations of unitarity come form the first row and first column of the CKM matrix [47] S 11 = 0.9999 ± 0.0006 and T 11 = 1.000 ± 0.004 . (3.40) In the models with vector-like quarks both of these constraints involve the same combination of couplings and we find There are 11 renormalizable models that add to the SM vector-like fermions in a single (complex) representation of the gauge group that can Yukawa couple to the SM fermions through the Higgs field. These BSM fermions can have a mass M that is much greater than the weak scale, since they have a mass term even in the absence of weak symmetry breaking. However, unlike BSM scalars, such fermions are technically natural. These models are a class of very simple extensions of the SM that do not worsen the SM hierarchy puzzle. The masses of these vector-like fermions can take any value, up to the ultraviolet cutoff (∼ M Pl ), and so there is no particular reason that they should be in a region that can be probed experimentally. However, there are many such models and it is not unreasonable that one of them has vector-like fermions with masses in the experimentally testable range.
We considered the experimental constraints from flavor physics on the mass of these vector-like fermions to get a feel for the mass reach that the present experiments have for this class of models. We are primarily interested in very heavy vector-like fermions, say, with masses greater than 10 TeV. Hence it is flavor-changing neutral-current processes that provide the most important constraints. However, we also discussed violations of lepton universality and CKM matrix unitarity.
An important feature of these models is that for large M the BSM contribution to meson mixing is either suppressed by a loop factor (∼ 1/16π 2 ) or a factor of m 2 Z /M 2 compared to processes that change flavor by one unit and are dominated by tree level through flavor-changing Z exchange. This implies that, except for the case of the kaon CP violation parameter K , the constraints on M from meson mixing are not overwhelmingly strong. We computed the order 1/(4πM ) 2 one-loop contribution to the coefficients of the four-quark operators responsible for meson mixing that is not suppressed by SM quark masses or weak mixing angles.
We are interested in a rough assessment of the experimental reach. The strongest current bounds on the vector-like fermion masses and couplings in each of the 11 models studied in this paper are summarized in table 1. We display bounds which are above or will get near the 10 TeV level in the near future. Muon to electron conversion in a nucleus, µ → 3e, the kaon CP -violating parameter K , and K L → µ + µ − are sensitive to vector-like fermion masses 100 TeV (for Yukawa couplings with magnitude unity). However, it is important to remember that the couplings λ i may have a flavor structure that suppresses these contributions relative to those involving the third generation.
We summarize the expected future sensitivity in table 2, where we only display sensitivities near or above 10 TeV. The Mu2e constraint will be improved dramatically in the next generation of experiments [22,23]. The measured K + → π + νν branching ratio corresponds to a mass reach around 70 TeV, which will increase substantially as the next generation experiments reach an uncertainty at about 10% of the SM rate, especially since the current central value is above the SM prediction (which has very small theoretical uncertainty). The improvement in the ∆m K bound is entirely dependent on lattice QCD calculations, as discussed after eq. (3.37). The B d,s and D mixing sensitivities will be improved by Belle II and LHCb. These experiments, and CMS and ATLAS, will also probe - FCNC B d,s , D, and τ decays much better than current bounds. The future sensitivities in table 2 correspond to estimated 50/ab Belle II [28] and 50/fb LHCb [45] sensitivities and CMS/ATLAS reach in rare decays on the same time scale. Compared to table 1, a greater number of the best bounds will come from purely leptonic rather than semileptonic decays. The sensitivities in table 2 may be realized in ∼ 10 years.

A Diagonalizing matrix
Here we summarize the 4 × 4 diagonalizing matrices in the 11 models. Terms of order (v/M ) 2 and higher are not explicitly displayed.
Model I : Model III : Model IX : Model XI : . (A.14) Finally, there are tree-level mass splittings among the heavy fermions, except for the SU(2) L singlet models, due to electroweak symmetry breaking. The results are

B Numerical inputs and Wilson coefficients
We collect in this appendix expressions used to derive bounds on vector-like fermions from flavor-changing neutral-current processes, to facilitate easier comparison with and reproduction of our numerical results. Our goal in this paper is to study the sensitivities of many processes, so leading or next-to leading order results suffice. (For most processes the state of the art is one or two orders higher.) In many cases we ignore the SM uncertainties, when we know that they are subdominant effects.

JHEP10(2015)027
We adopt for the numerical values of the coupling constants α(m b ) = 1/133, sin 2 θ W = 0.23, g Z = 0.73. For the top quark mass we use m t (m t ) = 165 GeV, obtained from the one-loop relation from the m t = 173 GeV (presumed) pole mass, extracted from fits to tt production at the Tevatron and the LHC.
The B → Kµ + µ − rate is given by [42] dΓ(B → Kµ + µ − ) dq 2 = G 2 F α 2 m 3 B 1536π 5 |V tb V ts | 2 [λ K (q 2 )] 3/2 f 2 + |C eff 9 (q 2 )| 2 + |C 10 + C 10 | 2 where λ K (q 2 ) is a phase space factor, f + and f T are q 2 -dependent form factors. The B → πµ + µ − rate is obtained with obvious replacements. In the heavy quark limit f + /f T = 1 + O(Λ QCD /m b ) [64], and model calculations are consistent with a mild q 2 dependence of this ratio. Hence, at the desired level of precision, the form factors can be pulled out of the {. . .} expression in eq. (B.5), obtaining simple approximations for the effect of new physics via C 10 and C 10 . (This is impossible for B → K * µ + µ − , as C 7 enters with a 1/q 2 dependence in that case.) In the numerical analysis, we use C 7 = −0.33, C 10 = −4.2, and a mean value |C eff 9 (q 2 )| = 4.4. These estimates can be refined as the measurements improve. 8 In our definition of c (SM) ν all lepton flavors (labeled by i) are included by using parameter Pc(X), which is defined in [36,37]. Then c (SM) ν corresponds to the decay amplitude.