Survey of vector-like fermion extensions of the Standard Model and their phenomenological implications

With the renewed interest in vector-like fermion extensions of the Standard Model, we present here a study of multiple vector-like theories and their phenomenological implications. Our focus is mostly on minimal flavor conserving theories that couple the vector-like fermions to the SM gauge fields and mix only weakly with SM fermions so as to avoid flavor problems. We present calculations for precision electroweak and vector-like state decays, which are needed to investigate compatibility with currently known data. We investigate the impact of vector-like fermions on Higgs boson production and decay, including loop contributions, in a wide variety of vector-like extensions and their parameter spaces.


Vector-like-fermion Models
We add to the SM a vector-like pair of SU (2) doublets and some number of vector-like pair of SU (2) singlets. We assume all these fields to be charged under U (1) Y . Consider the vector-like doublets as two Weyl spinors χ α and χ c α , that transform as conjugates with respect to SU (2) L and U (1) Y , namely χ = (2, Y χ ), χ c = (2, −Y χ ). Although we are considering only one such doublet pair, one can add any number of such pairs and our statements below can be extended to include this case. The theory can equivalently be written in terms of a Dirac fermion where we follow the usual Lorentz index conventions α,α. Thus the Dirac spinor X transforms the same way as χ under SU (2) L and U (1) Y . A vector-like mass term can always be added Let us consider a vector-like pair of Weyl spinors χ = (2, Y χ ), χ c = (2, −Y χ ). Expanding the SU (2) structure we can write where X , X 1 and X 2 are Dirac fermions. The W 3 and B interactions to new vector-like fermions have the structure with the W 3 also coupling to X R . The vector-like nature is exhibited by the L, R chiralities having the same T 3 and Y couplings. Furthermore, SU (2) L invariance requires Y i L = Y i R = Y for the SU (2) L component fields. The W 1 interactions to new fermions are given by At this level, the X 1 and X 2 are degenerate owing to the SU (2) L symmetry. By introducing Yukawa couplings to the SM Higgs, this degeneracy can be broken. In order to write down Yukawa couplings with the SM Higgs and the X , we can introduce two SU (2) L singlet vector-like fermions ξ = (1, Y χ + 1/2) and Υ = (1, Y χ − 1/2), written as Dirac fermions.
In this case we can write the Yukawa couplings where the "dot" represents the antisymmetric product. EWSB will then mix these new fermions and will split the X 1 and X 2 masses. The sign of λ is not physical, since in the change λ ξ,Υ → −λ ξ,Υ , the sign can be absorbed away by a redefinition ξ → −ξ and Υ → −Υ without affecting anything else.
The only gauge interactions that the vector-like singlets have is the hypercharge B µ interactions given by

5
Next we comment on possible mixing terms for the SM hypercharge assignments Y χ = 1/6 or Y χ = −1/2. With only the X added without the Υ and ξ, for the SM Y χ assignment Y χ = 1/6, we can write the additional terms L ⊃ −M qχq X − λ uX · H * u R − λ dX Hd R + h.c. , (6) where q is the SM quark doublet, and u R , d R are SM SU (2) singlets. Alternately, if Y χ = −1/2, then we can write the additional terms where is the SM lepton doublet, and ν R , e R are SM SU(2) singlets. After EWSB the Yukawa couplings will split the X 1 and X 2 masses, and will also mix the new vector-like fermions with either the u L and d L in the first case, or with ν L and e L in the latter case. In addition to the X , with the ξ, Υ also added, for Y χ = 1/6, we can also write the terms Alternately, if Y χ = −1/2, then we can also write the terms Above, the M and λ are 3 × 3 Hermitian matrices but the generation indices have been suppressed. If lepton-number is not a good symmetry, and if ξ is a gauge singlet, one can in addition write a Majorana mass term 1 where ξ c is the charge conjugated field of the 4-component spinor ξ, and similarly for ν c R . With the mixing M and λ terms added, there is no longer freedom to rotate away the sign of the λ, λ , and therefore the sign becomes physical. Since in this work, to be safe from flavor constraints, we take the M SM −V L M V L , and λ 1, these physical effects will be suppressed, and we therefore do not investigate it further. 2 We will comment further about these possibilities later.
X c equivalence: Since we are dealing with a vector-like theory, it should not matter whether we write the theory in terms of X with hypercharge Y χ or in terms of X c ≡ −iγ 2 X * with hypercharge −Y χ . To show this, let us rewrite the theory in terms of X c . To relate conjugated and unconjugated terms, we use the relationsΨ c X c =X Ψ,Ψ c γ µ X c = −X γ µ Ψ. Using these identities we can write the Yukawa term either as λ ΥX HΥ or as λ ΥῩ c H T X c . The mass terms, their diagonalization, mass eigenvalues are all identical whether written in terms of X or X c . The gauge interactions on the other hand are opposite in sign, which we understand to mean that the conjugated fields have opposite "charge".
In the theoretical structure just described, we consider in the next few subsections various models.

The 11 Model
For SM-like hypercharge assignments, these vector-like SU (2) singlets can mix with the SM SU (2) singlets and will alter Higgs, electroweak, and flavor observables. If such mixing is sizable, the constraints from flavor observables can be rather strong, due to which we will not consider this possibility here. Ref. [49] discusses the Higgs phenomenology of vector-like leptons with mixing to SM singlets.
For non-SM-like hypercharge assignments with vector-like pairs of SU(2) singlet fermions added, no new (renormalizable) interactions involving the Higgs field can be written down. Thus, there is no modification to Higgs observables, to electroweak precision, or to flavor observables. This will therefore be uninteresting in the present context.

The 22 Model
For SM Y χ assignments, as mentioned earlier, Yukawa couplings can be written down between the χ and the SM right-handed singlets. We do not analyze in much detail the situation when sizable Yukawa couplings with SM SU(2) singlets are allowed as constraints on this (for example, coming from shifts in Zbb coupling, FCNC, etc.) are quite severe. This is discussed for instance in Ref. [36], where it is argued that after imposing constraints, the deviation from the SM case of the Higgs observables are at most a few percent.
For non-SM-like hypercharge assignments, with only a vector-like pair of SU (2)-doublets addedin with no singlets, no new (renormalizable) interactions can be written down involving the Higgs field. Such an extension will therefore not lead to any new contributions to Higgs boson phenomenology that are significant. So this will be uninteresting in the present context, and we will move to discussing in the following subsections, models with one or more singlets added.

The 22(11) 1 Model
To understand the nature of this model, we consider in turn the model with only one singlet field present at a time in addition to the X , namely, first with only the Υ present and then with only the ξ.

Υ Model
In addition to the SU (2) L multiplet X , lets consider only one vector-like SU (2) singlet pair in the theory, namely the Υ (without the ξ) with only the λ Υ term present in Eq. (4). This, as we will see, is sufficient to split the X 1 and X 2 masses. Since Y H = 1/2 as usual, for Consistent with all the SM gauge symmetries we can write a vector-like mass term as shown in After EWSB by H = (0 v) T / √ 2, we have the mass matrix where m ≡ λ Υ v/ √ 2, which we will assume to be real for simplicity. The above mass matrix is diagonalized by an O(2) rotation where {X i } denote the mass eigenstates, and, c V ≡ cos θ V , s V ≡ sin θ V , with the mixing angle given by The mass eigenvalues M i are which can also be written as To have non-negative M 2 one requires M χ M Υ − m 2 ≥ 0. In Fig. 1 we show the mass eigenvalues M i as a function of M Υ taking M χ = 1000 GeV and λ Υ = 1. Henceforth we drop the primes on the mass eigenstate fields for notational ease. Taking the EM charges of the mass eigenstates X i as Q i , we have, Table 1 the Q i for various choices of Y χ in the Υ model, and Q 4 should be ignored for this model. The W 1 µ , W 3 µ and B µ interaction terms in Eqs. (2), (3) and (5) in the fermion mass basis become 8 Writing this in the A, Z basis we have The Higgs interactions are got by replacing v → v(1 + h/v) in Eq. (12). In the mass basis this gives We commented below Eqs. (4) and (9) that the sign of the Yukawa couplings can be absorbed away by redefining the fields if the Yukawa terms mixing the X and SM fermions are taken negligibly small. They are therefore not physical in the limit of the mixing Yukawa couplings not present. For later use we note in this model that under Therefore, off-diagonal (in fermion fields) couplings of the h and Z µ change sign (i.e. the hX 3 X 2 and ZX 3 X 2 couplings), while all the diagonal couplings do not change sign.

ξ Model
In addition to the SU (2) L multiplet X , lets consider only one vector-like SU (2) singlet pair in the theory, namely the ξ (without the Υ) with only the λ ξ term present in Eq. (4).
After EWSB by H = (0 v) T / √ 2, we have the mass matrix wherem ≡ λ ξ v/ √ 2, which we will assume to be real for simplicity. The above mass matrix is diagonalized by an O(2) rotation where {X i } denote the mass eigenstates, and, c V ≡ cos θ V , s V ≡ sin θ V , with the mixing angle given by The mass eigenvalues M i are which can also be written as To have non-negative M 1 one requires M χ M ξ −m 2 ≥ 0. Henceforth we drop the primes on the mass eigenstate fields for notational ease. Taking the EM charges of the mass eigenstates X i as Q i , we have, Q 2 = Y χ − 1/2, Q 1 = Q 4 = Y χ + 1/2. We show in Table 1 the Q i for various choices of Y χ in the ξ model, and Q 3 should be ignored for this model. The W 1 µ , W 3 µ and B µ interaction terms in Eqs. (2), (3) and (5) in the fermion mass basis become The Higgs interactions are got by replacing v → v(1 + h/v) in Eq. (12). In the mass basis this gives

Alternate Yukawa coupling
Instead of the Yukawa coupling of the Υ shown in Eq. (4) we can alternately write where we use the fact that iσ 2 X * transforms the same way as X under SU (2) and have defined X c to mean this conjugation in gauge space and also conjugation in spinor space. In this case we wherem ≡λ Υ v/ √ 2. This mass matrix is identical to that in the theory with the ξ model shown in Eq. (20), but now written for (X c 1 , X c 2 ) instead of (X 1 , X 2 ). So the mass eigenvalues are the same as in the ξ model. Furthermore, due to the conjugation, all the gauge couplings are with opposite sign to what is in the model with only the ξ.
After EWSB, Eq. (4) will generate mass mixing terms where m ≡ λ Υ v/ √ 2 andm ≡ λ ξ v/ √ 2, both of which we will assume to be real for simplicity. The above mass matrix is diagonalized by two O(2) rotations where {X i } denote the mass eigenstates, and, c V ≡ cos θ V , s V ≡ sin θ V , with the mixing angle given by The mass eigenvalues M i are which can also be written as To have non-negative M 1 and M 2 one requires M χ M Υ − m 2 ≥ 0 and M χ M ξ −m 2 ≥ 0 respectively. Henceforth we drop the primes on the mass eigenstate fields for notational ease. Taking the EM charges of the mass eigenstates X i as Q i , we have, Table 1 the Q i for various choices of Y χ . The W 1 µ , W 3 µ and B µ interaction terms in Eqs. (2), (3) and (5) in the fermion mass basis become The Higgs interactions are got by replacing v → v(1 + h/v) in Eq. (12), which in the fermion mass basis is

Some vector-like Extensions of the SM
We add to the SM some number of vector-like colored (quark) and uncolored (lepton) doublets and singlets corresponding to the 22(11) 1 or 22(11) 2 framework outlined in the previous subsection. For notational ease, we refer to the quark SU(2) doublet X Q simply as Q, quark SU(2) singlets ξ U as U and Υ D as D, and, the lepton SU(2) doublet X L as L, SU(2) singlets Υ E as E and ξ N as N . From the context it should be clear that we are referring to the new vector-like fermions and not the SM ones. These vector-like quarks and leptons have EM charges same as the corresponding SM quarks and leptons only for the specific hypercharge Y χ assignments of Y Q = 1/6 and Y L = −1/2 respectively. We give below the nomenclature of the models with the addition of various vector-like SU(2) doublet and singlet quarks and leptons to the SM: • Minimal vector-like Quark: The 22(11) 1 model in the SU (3) c fundamental representation, with an SU (2) doublet quark X Q and one SU (2) singlet quark; with only the Υ D will be the MVQD 1 , while with only the ξ U will be the MVQU 1 . n copies of these will be called MVQD n and MVQU n .
• vector-like quark (VQ): The 22(11) 2 model in the SU (3) c fundamental representation, with an SU (2) doublet quark X Q and two SU (2) singlet quarks Υ D and ξ U . n copies of these will be called VQ n .
• Minimal vector-like lepton: The 22(11) 1 model which is an SU (3) c singlet, with an SU (2) doublet lepton X L and one SU (2) singlet lepton; with only the Υ E will be the MVLE 1 , while with only the ξ N will be the MVLN 1 . n copies of these will be called MVLE n and MVLN n .
• vector-like lepton (VL): The 22(11) 2 model which is an SU (3) c singlet, with an SU (2) doublet lepton X L and two SU (2) singlet leptons Υ E and ξ N . n copies of these will be called VL n . • The vector-like extension of the SM (VSM): V Q + V L forms the 1-generation VSM (VSM 1 ). n copies of these will form the n-generation VSM, called here VSM n .
For some Y Q and Y L assignments and in some region of parameter-space it is possible that the lightest vector-like particle is uncolored and EM neutral. Such a state, if stable, can be a possible dark matter candidate. For example, in the M V LE model with Y L = 1/2 there are two EM neutral uncolored states, and if stable, the lighter of these can be a dark matter candidate. Alternately, if the lightest vector-like particle is colored and stable, is it possible that a color-singlet bound-state of this (either with another vector-like particle or with an SM quark) could be the dark matter? A detailed investigation of this possibility is beyond the scope of this work, but such a strongly interacting dark matter (SIMP) candidate appears to be heavily disfavored as discussed for example in Refs. [50,51,52,53,54,55].
The addition of vector-like fermions changes the running of the Higgs quartic coupling and the vacuum can become unstable at some energy scale Λ. We will treat the theory we have written down as an effective theory valid below the scale Λ. Refs. [31,34] derive the vacuum stability constraint with vector-like leptons only, in a model similar to our V L n . Ref. [34] finds, for example for λ = 1, that Λ ≈ 2.5 TeV for the lightest charged lepton mass of 100 GeV. For vector-like lepton masses of 200, 450 GeV (for the two mass eigenstates), Ref. [31] finds Λ ∼ 10 TeV. Since their main motivation is obtaining a large µ γγ ≈ (1.5, 2), they need large Yukawa couplings for large vector-like masses, and therefore one would expect stronger constraints in their case than in ours. Ref. [56] analyzes the vacuum stability issue when new physics contribution changes the sign of the ggh and γγh effective vertices (while keeping the magnitude the same as in the SM). Ref. [57] works out the vacuum stability constraints in the case when the vector-like fermion mass is tied to the scale of EWSB. Ref. [58] considers the possibility of a strongly first-order electroweak phase transition driven by heavy vector-like fermions. Working out the precise value of Λ in our case will be taken up in future work.
We focus here on SM-like SU (3), SU (2) L and U (1) Y representations, and a more general analysis with non-SM-like representations will be postponed to future work. Our focus in this work will be the modifications of precision electroweak observables, and, LHC Higgs production and decay, due to the presence of additional vector-like fermions in SM-like representations.

Precision Electroweak Observables
One of the great successes of the SM is the near perfect agreement between the indirect constraints on the Higgs mass from precision electroweak observables [59,60,61] and the mass of the observed state at the LHC. The precision electroweak constraints have been investigated in the context of a chiral fourth generation elsewhere (see for example Ref. [62] and references therein). In this section, we present the oblique corrections [63,64,65] written in terms of the S, T, U parameters [63] due to vector-like fermions. The S, T, U parameters are given by Vector-like quark contributions to S, T and U have been presented earlier in Ref. [66]. Here we derive these including contributions from vector-like quarks and leptons, and in the following sections make contact with properties of the recently observed Higgs boson.

S, T , U Constraints
The allowed regions in the S and T plane is shown for instance in Refs. [67,68]. We take the reference Higgs mass and top mass values as m h = 125.5 GeV [69,70] and m t = 173.2 GeV [71, 72]. We include the ∆S, ∆T and ∆U contributions due to vector-like fermions and ascertain whether they lie within the 68 % or 95 % allowed regions. The U parameter best-fit value is 0.08 ± 0.11 [67]. Since contributions to U are typically quite small, it does not impose any nontrivial constraints on models. It is therefore acceptable to set U to zero and obtain constraints from S and T ; we adopt this method here.
New vector-like fermions X i will generate additional contributions to S, T and U due to the new W 1 , W 3 and B interactions going into the gauge 2-point functions Π shown in Fig. 2. The detailed computation of the Π with vector-like fermions is given in Appendix A. There we define Π LL with a chirality projector P L inserted in the first vertex and P L in the second vertex, and similarly for Π RR , Π RL and Π LR , and, Π ≡ Π LL + Π LR , and also, We similarly define Π . In terms of these Πs computed in Appendix A, we present next the S, T , and U due to vector-like fermions in the 22(11) 1 and 22(11) 2 models of Sections 2.3 and 2.4 respectively.

S, T , U in the 22(11) 1 model
We work out in turn the S, T , U , first for the model of Sec. 2.3.1 with only the Υ present, and then for the model of Sec. 2.3.2 with only the ξ present.

Υ Model
With only the Υ present, in the 22(11) 1 vector-like model described in Sec. 2.3.1, from the interactions in Eqs. (2), (3) and (5) we compute the gauge boson 2-point functions of Fig. 2 and then S, T and U . We compute Π 3Y (0) and then the S parameter from Eq. (36) as where d 3 is the dimension of the SU (3) representation of the vector-like fermion (for example, for a fundamental of SU (3), d 3 = N c = 3). The divergent 2/ pieces (along with the −γ + log 4π pieces) cancel leaving a finite S. We compute next the T -parameter, for which we compute Π 11 (0) and Π 33 (0) as T is then given by Eq. (37). Again, the divergent 2/ pieces (along with the −γ + log 4π pieces) cancel leaving a finite T . U is computed using Eq. (38) for which the Π 11 (0) and Π 33 (0) are got from equations identical to Eqs. (40) and (41) except for replacing Π → Π .
From the X c equivalence noted in Sec. 2, we infer that the vector-like fermion contributions to S and T parameters is the same whether written in terms of the X or X c . This is because the 2-point functions shown in Fig. 2 have two gauge interaction vertices although each with opposite sign compared between X and X c formulations, and the loop functions have the same value since the mass eigenvalues are exactly the same in the two formulations. This is not to say, however, that Y χ and −Y χ should necessarily give the same S, and we do indeed find that S can change sign with the sign of Y χ . This is because isospin offers a reference, i.e., in X , a given isospin component has hypercharge Y χ , while in X c it has hypercharge −Y χ . Furthermore, the Yukawa interaction terms in Eq. (4) couples the vector-like sector to a chiral SM sector, in particular to the Higgs with a specific hypercharge assignment, namely Y H = +1/2. For the two cases with Y χ and −Y χ , the U (1) Y invariance of this Yukawa coupling implies different Υ (or ξ) hypercharges and consequently different electromagnetic charges, and observables can be sensitive to this difference, as for instance the S-parameter.
It is interesting that vector-like fermions can easily give a negative S-parameter. This is unlike in the case of chiral fermions where one finds that the S-parameter is usually positive leading to a tight constraint on dynamical EWSB BSM sectors [63]. Some examples of theories that give a negative S-parameter are also discussed in Refs. [73,74].

ξ Model
Let us turn to the ξ model described in Sec The T -parameter is the same in both the models of Sec. 2.3.1 and Sec. 2.3.2. We have checked that our numerical results are consistent with these expectations.

Υ Model with alternate Yukawa coupling
For the Υ model with the alternate Yukawa coupling of Eq. (27), we noted in Sec. 2.3.3 that the mass matrices are identical to that of the ξ model. Also, the gauge couplings are of opposite sign, but since the vector-like fermion contributions to the gauge 2-point functions have two such couplings, this will not affect the S and T parameters. The last change is the different hypercharge assignment Y Υ = −Y χ −1/2. The S and T parameters for the Υ coupled as in L Yuk Alt will in general be different when compared to the Υ model or ξ model. For Y χ = −1/2, however, we haveỸ Υ = Y χ , and we therefore have In the 22(11) 2 vector-like model described in Sec. 2.4, from the interactions in Eqs. (2), (3) and (5) we can compute the gauge boson 2-point functions of Fig. 2. We find where the Π and Π are as given in Eqs. (A.74) and (A.75) respectively. The S, T and U are then given by Eqs. (36), (37) and (38) respectively.

Shift in the Zbb coupling
For Y χ = 1/6, −1/6 there are vector-like states with EM charge −1/3, and if they also have color, these states can mix with the SM b-quark after EWSB. This mixing will imply a shift in the Zbb coupling which is constrained by LEP. The measurements R b = 0.21629 ± 0.00066 and Γ hadrons = 1.7444 ± 0.0020 GeV at LEP [67], imply constraints on the Zbb coupling. A Yukawa coupling between the SM b and the vector-like b induces a mixing, which if it's only in the L sector, i.e. between b L ↔ b L only, 3 the Zb LbL coupling shifts to −1/2(c 2 L ) + (1/3)s 2 W where c L ≡ cos θ L and the mixing angle is given by tan(2θ L ) = (λ/ √ 2)v/M b . This mixing is constrained by the above data and will imply the limit M b /λ ≥ 3509 GeV at 1 σ ; and M b /λ ≥ 2481 GeV at 2 σ .
This limit can be evaded by ensuring that the Yukawa couplings between the SM b and b is not too large, as dictated by Eq. (43). We will assume that this is the case in the rest of our analysis. This limit will be evaded completely in the model where a custodial symmetry protects the Zbb coupling (see Ref. [75]). Also, this limit will not be relevant for Y χ assignments that do not result in a charge −1/3 vector-like state.

Mass and gauge eigenstates
In order to consider the constraints from colliders, we look at the decays of the vector-like matter.
In particular, we are interested in decays via the Higgs and the electroweak gauge bosons. We write the following Lagrangian for the mass sector of the model. We denote the VSM quarks as T R and B R , where these are the singlet states under SU(2) that are referred to in previous sections as ξ U and Υ D . The states denoted as TR, BR are the left-handed projections of the vector-like singlets. We assume that we are dealing with only one new generation of VSM quarks and leptons, i.e. the VSM 1 in order to get an idea of the constraints. For the sake of simplicity, we do not consider the effects of mixing with the doublet under SU(2), previously denoted as X Q , as it will not substantially alter the results. Aside from the assumption that off-diagonal couplings are small relative to the diagonal entries, we do not make any a priori assumptions about their sizes relative to each other, so the mixing between T and the 2nd and 3rd generations can in principle be of similar sizes. Motivated by simplicity and data, we do not consider mixing with the first generation, although this could be implemented, given that the off-diagonal couplings are small. We can write the mass terms as where m ij is used to denote a mass obtained due to the Higgs VEV in the Yukawa interactions of Sec. 2, and µ i , which appeared in Sections 2 and 3 as M ξ , is a vector-like mass. Because this produces a non-diagonal mass matrix, in order to perform decay calculations we must diagonalise to the mass-eigenvalue basis. We write the up-type mass matrix as follows: which will allow us to use a perturbative approach in m of f diag /m diag to find the diagonalising matrices. The resulting left-handed diagonalization matrix,V u L , is given by Using the relation V u we can look at the mass terms such that the mass eigenstates are given by and The same procedure is followed for the down-type quarks, yielding the same results as above, only with c ↔ s, t ↔ b, T ↔ B.

Higgs sector
Since the mass matrix we wrote above includes vector-like mass terms that do not arise via interaction with the Higgs, there is a misalignment between the mass matrix and the Yukawa interaction matrix. The Lagrangian for the Yukawa interactions with the Higgs is given by which can be written in matrix form as It is clear that applying the same rotation matrices that diagonalised to the mass eigenstate basis will not diagonalise the Yukawa interaction matrix, giving rise to non-zero off-diagonal couplings via the Higgs. We can write the Lagrangian in terms of the mass eigenstates as which is not diagonal. The α ij coefficients are listed in Appendix B. They give the size of the mixing through the Higgs between generations in the mass eigenstate basis. To make our expressions more clear, we define quantities a u,d i where i = 1, ..., 6 in the Appendix, where a u,d 1,2,3 correspond to the off-diagonal entries in the up (down)-type left diagonalization matrix, and a u,d 4,5,6 correspond to the right diagonalization matrix.

Electroweak Sector
Due to the addition of the new family of quarks, we must modify the CKM matrix accordingly. The matrix is defined as with the full 4 × 4 matrices approximated as where we use the coefficients a u,d i defined in Appendix B, and explained above, such that the new CKM matrix is approximately given by Because we did not include the first generation earlier, we have approximated V us , V ub , V cd , V td = 0, but doing a full 4 × 4 analysis would generate these properly as required.
Flavour changing neutral currents (FCNCs) may arise through couplings to the Z boson at tree level. The quark neutral current is given by The right handed current does not change with the addition of the T R state, as it carries the same charge as the other up-type quarks. However, the addition of the TR singlet to the left handed states will result in flavour-changing neutral currents, as it does not carry the same isospin as the other left handed quarks. This can be seen by writing down the current in matrix form as This allows it to be seen that there will be a FCNC induced due to the appearance of the − 1 2 in the matrix (henceforth denoted as δg Zqq added to the diagonal matrix, which accounts for the isospin of the TR being 0). It is then quite clear that the matrix entries for our FCNC will be given by β = g Zqq + V † L δg Zqq V L , and is found to be: where the coefficients β ij are defined in Appendix B, and give the sizes of the modified interactions between the Z boson and the SM and vector-like quark. Thus the interaction Lagrangian can be written as We have shown here the analysis for the up-type sector, and a similar analysis follows for the down-type sector.

Computation of decay widths
Having introduced these small couplings between the 3rd generation SM quarks and the VL quarks, we now want to check that there is prompt decay via W to b, or H to t, while also ensuring minimal effects on the gluon fusion rate and the diphoton production rate. The Lagrangian term governing the decay of the T to a W boson and b quark is given by L = g This generates the decay partial width where we can see that this will depend on |V T b | 2 . We denote the fermion mass eigenvalues as M t and M T . For the decay of the T to a Z and a SM top, the Lagrangian term is L = g cos θ W Z µ TRγ µ β 32 t L . This generates the decay partial width We can see that this depend on β 2 32 . For a T decaying to a SM top and a Higgs, the Lagrangian term is L = −α tT ht L T R . This leads to the decay partial width So this depends on α 2 tT .

Experimental Limits from VLQuark searches
In this section we review the current experimental limits from ATLAS and CMS on searches for vector-like quarks, taking the example of a top-like quark. The decays of such a vector-like quark have a dependence on the off-diagonal Yukawa couplings of the T to the t, as well as the other offdiagonal couplings (see definition of α tT in Appendix B). The decay of T to tH will provide a bound on (λ 2 tT · µ T ), as the decay width is Γ(T → Ht) ∼ α 2 tT · M T /(32π 2 ), and to leading order in M t /M T , we have α tT ∼ λ tT and M T ∼ µ T , such that Γ(T → Ht) ∼ λ 2 tT · µ T /(32π 2 ). This approximate expression only applies for M T M t (see Eq. (63)), and λ ij 1 for i = j. The condition λ ij 1 comes from the expression for M T as a function of λ ij that can be seen from Eq. (B.79), recalling that m ij are the masses obtained due to the Higgs VEV in the Yukawa interactions of Sec. 2 such that m ij = λ ij v/ √ 2. Depending on the decay width, the decay can either be prompt or displaced. We define a prompt decay to be one where cτ < 100µm, such that it would be smaller than cτ for the D 0 meson [67], which requires (λ 2 tT · µ T ) 1.6 × 10 −10 GeV. Based on 19.6 fb −1 , CMS studied the properties of a possible t-like VL quark with decays into bW, tZ and tH. They find limits of between 687 and 782 GeV [76]. The search at ATLAS used 14.3 fb −1 of data, with possible t-like decays into bW and tH. They propose limits of between 640 and 790 GeV [77].
If the VL quark is long-lived such that cτ 3m, it will decay outside the detector, which requires (λ 2 tT · µ T ) 6 × 10 −15 GeV. Heavy charged long-lived states such as these usually have low, but not zero, energy loss in the detector and may charge oscillate. Limits on long-lived VL quarks can be derived from the interpretation of ATLAS and CMS searches [78]. Using CMS results [79] one can exclude m Q < 800 GeV for cτ > 3m, assuming behaviour similar to that of long-lived heavy scalar top quarks. A caveat is that these limits assume pair-production of the VL quarks, with kinematics independent of the lifetime of the particle [78]. The MoEDAL collaboration discusses [80] in more detail the limits on production cross-section-dependent limits for various decay lifetimes of long-lived VL quarks.
The current FCNC experimental limits also place an upper bound on the off-diagonal Yukawa couplings. In particular, we compute the decay width of t → cX, where X = Z, H and use that as a means of limiting the mixing between the VL generation and the SM quarks. The current limit on FCNC top decays to a charm and Higgs are given by |α tc | 2 + |α ct | 2 < 0.14 [81]. The 20 current limit on top decays into up-type quarks and Z is given by BR(t → Zq) < 0.05 [82], with eventual limits expected to reach BR(t → Zq) < 10 −5 [83]. We simplify our analysis by setting all the off-diagonal Yukawa couplings equal to each other, λ ij ≡ δ. We find that δ 0.24 is required to avoid the constraint on |α tc | 2 + |α ct | 2 for µ T 600 GeV. This bound comfortably avoids the current bounds set by the decay of the top into Z and an up-type quark.

Experimental Limits from VLLepton searches
We do not compute limits on vector-like leptons here, so we summarise briefly here the results of Ref. [49]. In the case of singlet vector-like leptons, the constraints are not very tight, with M l 100 GeV if it decays into an electron or muon, and a lower limit less than 100 GeV if it decays into a tau. In the case of doublet vector-like leptons, the constraints are more strict, with M l 450 GeV if it decays into an electron or muon, and M l 300 GeV if it decays into a tau. These lower bounds motivate our choice of lightest mass eigenvalues for the vector-like fermions in Table 2.

Higgs Production and Decay Processes
In the SM, the hgg and hγγ couplings occur at loop level, the leading order being at one-loop. If vector-like fermions couple to the Higgs boson, they could potentially contribute in these loops and affect these amplitudes. Colored vector-like fermions that couple to the Higgs boson contribute to the gg → h loop amplitude, and electromagnetically charged ones to the h → γγ loop amplitude. Here we present these contributions. We use leading order expressions since we always deal with ratios of these quantities to corresponding SM ones. These ratios are quite insensitive to higher order corrections; for example, Ref. [84] finds that due to vector-like quarks in a composite-Higgs model, the ratios are changed by at most a few percent. Ref. [85] computes the fermion contribution to the ggh effective coupling using the low-energy Higgs theorem. They show the typical size of modification of this vertex in a few models of new physics. They argue that QCD corrections should largely cancel in the ratio Γ gg /Γ SM gg where the numerator includes the contribution of heavy new physics.
The h → gg partial width at leading order (1-loop) in the SM is given for example in Ref. [86].vectorlike fermions that carry color also contribute to the h → gg decay rate while uncolored ones do not. We generalize this result to include the contributions due to vector-like fermions in the fundamental representation of SU (3) (i.e. vector-like quarks), denoted as χ, by rescaling with the Higgs coupling shown below in parenthesis (...) as where τ Q = m 2 h /4M 2 Q , Q now includes the SM and the new vector-like quarks, κ hQQ are the Higgs couplings to the quarks, i.e. the usual Yukawa couplings for the SM quarks and the couplings shown in Eqs. (19), (26) and (35) for the new vector-like quarks X . The function A 1/2 can be found in Ref. [86]. The rescaling is as shown above since the SM hgg and hγγ fermion contributions are proportional to the fermion Yukawa coupling. Also, such a rescaling of the SM (chiral) result is applicable in the vector-like case since a vector-like Weyl fermion pair counts as a Dirac fermion, similar to the SM situation, and the γ and the g interactions are vector-like with respect to the corresponding unbroken gauge-group and do not see the distinction between chiral and vector-like fermions. For our numerical analysis, we take α s at the scale M Z . The gg → h production can be obtained from the h → gg decay rate. At leading order this is (see for example, Ref. [87]) Another Higgs production channel is via vector-boson fusion. Although the signal cross-section is lower than Higgs production via gluon fusion, the presence of the forward tagging jets help suppress background, making this mode also promising. We will separate the Higgs production in these categories by including ggh and V BF superscripts respectively. For brevity, we denote Γ h→XX as Γ XX , etc.
The "signal strength" for the h → XX decay mode is defined as Processes that have contributions at the tree-level are modified at loop-level by a relatively small amount, and therefore the contributions of heavy vector-like fermions to such processes can be neglected. For instance, Γ ZZ ≈ Γ SM ZZ , σ V BF ≈ σ V BF SM , etc. Furthermore, with heavy vector-like fermions present, the total Higgs decay width remains dominated by tree-level decays as it is in the SM, and to a very good approximation, Γ tot ≈ Γ SM tot . Thus, in the case of heavy vector-like fermion extensions of the SM, these imply the approximate relations

Numerical Results for Electroweak and Higgs Boson Observables
Here we explore the possible impact of vector-like fermions on precision electroweak observables and LHC Higgs observables. We begin with some general comments. Whenever µ ZZ deviates from 1, it is entirely due to the vector-like quark contributions to the ggh vertex as the hZZ vertex is not shifted from its tree-level value. Thus when there are no vector-like quarks present µ ZZ = 1, which for instance is the case with only vector-like leptons present. In the latter case, µ γγ may be shifted. µ ZZ , µ W W and µ bb are all equal to each other since all of them measure the ggh vertex.
In the following subsections we present some illustrative examples and show the deviations in µ γγ and µ ZZ we find in the various models presented. In all the models we consider, the vector-like quark contribution adds constructively with the SM top contribution and increases the magnitude of the ggh vertex when compared to the SM; as a consequence, µ ZZ,W W,bb ≥ 1 always. The hγγ coupling receives a contribution from the W ± as well which is larger and of opposite sign to the SM quark/lepton contribution. The vector-like fermion contribution being of the same sign as the SM quark/lepton contribution therefore decreases the magnitude of the hγγ coupling when compared to the SM; as a consequence, µ V BF γγ ≤ 1 always. µ ggh γγ being the product of the above two (see Eq. (68)) can be either bigger than or lesser than 1. We argued below Eqs. (4) and (9) that the sign of the Yukawa couplings are not physical in the limit of negligibly small Yukawa couplings mixing SM fermions with vector-like fermions, as we take here. We demonstrate this here explicitly by taking the Υ model as an example. For this, we use the argument for how various quantities change under λ Υ → −λ Υ we gave below Eq. (19). It is easy to see that in the vector-like fermion triangle-loop diagrams for hγγ and hgg couplings, there are either zero or two couplings that change sign in a particular diagram, thus making the overall sign unchanged. Also, the precision electroweak observables, S, T, U , given in terms of Π 11 , Π 33 , Π 3Y (and their derivatives with respect to p 2 ) all depend on s 2 V , and thus remain unchanged under λ Υ → −λ Υ . Although we explicitly demonstrate here for the Υ model, this is true for all of the models we have discussed in Sec. 2 due to the general arguments given below Eqs. (4) and (9). We therefore restrict ourselves to positive Yukawa couplings in the following, without loss of generality.

MVLE 1 vector-like Leptons Model
In Fig. 4 we show S, T , U as a function of M L for MVLE 1 with M E = M L and λ E = 1. Note that T and U are independent of Y L . Furthermore, U 0.02 for M L,E > 250 GeV, and will not place any nontrivial constraints; we therefore ignore U in the rest of our analysis.
For Y L = 1/2 (−1/2), the χ 2 , χ 3 (χ 1 ) mass eigenstates are EM neutral (and color singlet) as can be seen from Table 1. These assignments of Y L are therefore interesting for dark matter if the lightest χ is stable and in regions of parameter space where it is EM neutral. In the Y L = 1/2 (−1/2) Υ (ξ) model case, the EM charged state χ 1 (χ 2 ) does not couple to the Higgs as seen from Eq. (19), (26) and thus µ γγ will be 1. Since there are no new colored fermions, the gg → h production is unchanged, and µ ZZ,W W,bb etc. will also be 1.
We show in Fig. 5 the constraint from S and T parameters and µ ggh γγ . S and T are within the 68 % CL ellipse in the light gray region, between 68 % and 95 % ellipse in the medium gray region, and excluded worse than 95 % CL in the dark gray region. We also show the (100, 250, 450) GeV contours (with boxed numbers) of the the minimum mass eigenvalue, i.e. min(m 1 , m 2 , m 3 ). It is possible that in some specially constructed models such light masses may still be allowed; for a discussion of the limits on vector-like leptons, see for example Ref. [49]. To the left of the black solid line there is a mass eigenvalue < 250 GeV. Therefore the safest region in most models from both S and T parameters and from direct collider constraints is the light gray region to the right of all black lines. With only vector-like leptons added, the hgg coupling is unaltered and the deviations of interest are all in the hγγ coupling. Thus, µ ggh γγ is altered as shown, whereas µ ggh ZZ,bb = 1 and Γ gg = Γ SM gg and are not shown.

MVQD 1 vector-like Quarks Model
In Fig. 6 we show S, T , U as a function of M Q for MVQD 1 . Note that T is always positive and independent of Y Q . U 0.003 for M Q,D > 500 GeV, and will not place any nontrivial constraints; we therefore ignore U in the rest of our analysis. In Fig. 7 we show the the constraint from S and T parameters as shaded regions for Y Q = 1/6, λ D = 0.5, 1, 2, and for Y Q = −1/6, λ D = 1. We also show the (500, 700) GeV contours (with boxed numbers) of the the minimum mass eigenvalue, i.e. min(m 1 , m 2 , m 3 ). In Fig. 8 we show µ ggh γγ , µ ggh ZZ and µ ggh γγ /µ ggh ZZ for Y Q = 1/6 and −1/6 for λ D = 1. Keeping in mind the direct LHC limits on a vector-like b and t we ensure that all mass eigenvalues are > 500 GeV and show them as the colored region. The deviation in µ ggh ZZ is entirely due to a change in the ggh vertex since the hZZ vertex is unchanged, and this change is the same for either sign of Y Q . In general, in gg → h → γγ, i.e. for µ ggh γγ , both production and decay vertices are shifted; the ggh vertex is shifted due to the presence of new colored vector-like fermions, and the hγγ vertex is shifted since these states carry EM charge as well. Interestingly, as can be seen from Eq. (19), the hγγ coupling shift is small in this model since X 1 does not couple to the h, while the h couplings to X 2 and X 3 are opposite sign and will cancel up to mass differences. In all cases, the µ asymptote to 1 as the vector-like fermion contributions decouple.

VSM 1 vector-like Standard Model
Keeping in mind the direct collider limits, we restrict to the parameter-space with all vector-like quark mass eigenvalue ≥ 500 GeV and lepton mass eigenvalues ≥ 250 GeV. In Fig. 9 we show the signal strength in the VSM 1 with all the vector-like quark and lepton masses equal to the value shown in the X-axis, i.e. M {Q,U,D,L,E,N } = M V L , with Y Q = 1/6 and Y L = −1/2, and all the Yukawa couplings λ = 1. All these points satisfy the S and T constraints at or better than 2 σ level. The color of the dots denote the lightest mass eigenvalue; the red, blue and green dots respectively stand for light, medium and heavy mass categories given in Table 2  λ set equal (λ {U,D,E,N } = λ V L ), for Y Q = 1/6 and Y L = −1/2. All these points satisfy the S and T constraints at or better than 2 σ level. The color of the dots denote the lightest mass eigenvalue; the red, blue and green dots respectively stand for light, medium and heavy mass categories given in Table 2.
Next, we present some more results where we perform the scan in a more unconstrained fashion. We show in Fig. 11 the signal strength µ ggh γγ , in Fig. 12 the signal strength µ V BF γγ , in Fig. 13 the signal strength µ ZZ and in Fig. 14    The three categories for the lightest mass eigenvalue of the vector-like quark and lepton. "Light" is with either the M q or the M l as shown, "Medium" is with each in the interval shown, and "Heavy" is with both above the values shown. the red (dark gray), blue and green (light gray) dots respectively stand for light, medium and heavy mass categories given in Table 2. All these points satisfy the S and T constraints at or better than 2 σ level. The color (or shade of gray, if viewing in gray-scale) of the dots denote the lightest mass eigenvalue; the red (dark gray), blue and green (light gray) dots respectively stand for light, medium and heavy mass categories given in Table 2.
the lightest mass eigenvalue; the red, blue and green dots respectively stand for light, medium and heavy mass categories given in Table 2. These results indicate the sizes of the deviations one can  Table 2.
expect in the models that we have considered. In the next subsection, we ask to what degree such deviations are allowed by the present LHC Higgs data and the constraints on the model parameter space.

Fit to the LHC Higgs data
The ATLAS and CMS collaborations have extracted the effective hgg and hγγ couplings from a combined fit to the various observed diboson Higgs decay modes. We give this result in Table 3, where we quote the ATLAS result [69] directly as given, while we have translated the CMS result [70] on the 95 % C.L intervals of κ g = [0.63, 1.05] and κ γ = [0.59, 1.30] to the 1 σ values shown. We Table 3: κ g and κ γ values from ATLAS [69] and CMS [70].
Coupling ATLAS CMS κ g 1.04 ± 0.14 0.83 ± 0.11 κ γ 1.2 ± 0.15 0.97 ± 0.18 for Y Q = 1/6 and Y L = −1/2. All the points satisfy the S and T constraints at or better than 2 σ level. The color (or shade of gray, if viewing in gray-scale) of the dots denote the lightest mass eigenvalue; the red (dark gray), blue and green (light gray) dots respectively stand for light, medium and heavy mass categories given in Table 2. perform a χ 2 fit of the SM plus vector-like fermions model to these data. We treat the ATLAS and CMS channels shown in the Table as independent, implying four degrees of freedom (dof ). We neglect correlations between κ g and κ γ , which is a reasonable approximation.
We compute the χ 2 function where the κ Th i = {κ g , κ γ } for the SM plus vector-like fermion models discussed in Sec. 2, and compared with the respective four κ Exp i ATLAS and CMS experimental values shown in Table 3. The κ i are given by As is standard (see for example, Ref. [89]), from the χ 2 value for that model with vector-like fermions and the above LHC data (with dof = 4), we compute F χ 2 (which is twice the "p-value"), the fraction of times a worse fit is obtained, which is the integral of the tail of the χ 2 distribution from that χ 2 value up to infinity. Roughly speaking, the regions of parameter space where F χ 2 < 0.05 can be for Y Q = 1/6 and Y L = −1/2. All the points satisfy the S and T constraints at or better than 2 σ level. The color (or shade of gray, if viewing in gray-scale) of the dots denote the lightest mass eigenvalue; the red (dark gray), blue and green (light gray) dots respectively stand for light, medium and heavy mass categories given in Table 2. taken to be excluded at about 2 σ Gaussian equivalent, and regions where F χ 2 < 0.01 excluded at about 2.6 σ.
For the SM alone, without the addition of vector-like fermions, we have κ g,γ = 1 by definition, and from the data in Table 3 we obtain χ 2 /dof = 1.07 which yields F χ 2 = 0.37, an acceptable fit to the data. Next, we present χ 2 and F χ 2 for the SM plus vector-like fermions with the goal of identifying regions of vector-like fermion parameter space that have values of F χ 2 bigger than about 0.05, which can be taken as the allowed regions for that model, given the present data.
In Fig. 15 we show the χ 2 /dof and F χ 2 for the VL 1 model with Y L = −1/2, λ L,N = 1, M E = M N , i.e., the vector-like singlet masses taken equal. The parameter space shown in color is with all lepton mass eigenvalues ≥ 250 GeV, and satisfy the S and T constraints at or better than 2 σ.
In Fig. 16 we show the χ 2 /dof and F χ 2 for the VQ 1 model with Y Q = 1/6, λ U,D = 1, M U = M D , i.e., the vector-like singlet masses taken equal. The parameter space shown in color is with all quark mass eigenvalues ≥ 500 GeV, and satisfy the S and T constraints at or better than 2 σ.
In Fig. 17 we show the χ 2 /dof and F χ 2 for the VSM 1 model with Y Q = 1/6, Y L = −1/2, λ U,D ≡ λ q = 1, λ E,N ≡ λ = 1, M Q,U,D ≡ M q = 1000 GeV, M L,E,N ≡ M = 500 GeV, i.e. all the vector-like quark masses are taken equal, and all the vector-like lepton masses are taken equal (but not necessarily the same as the quark masses). The plots show the dependence on the and λ E = λ N ≡ λ L , for Y Q = 1/6 and Y L = −1/2. All the points satisfy the S and T constraints at or better than 2 σ level. The color (or shade of gray, if viewing in gray-scale) of the dots denote the lightest mass eigenvalue; the red (dark gray), blue and green (light gray) dots respectively stand for light, medium and heavy mass categories given in Table 2. two parameters that are varied with the other parameters fixed at the above mentioned values. The colored regions are with all quark mass eigenvalues ≥ 500 GeV, all lepton mass eigenvalues ≥ 250 GeV, and satisfy the S and T constraints at or better than 2 σ. For given masses, the χ 2 approaches the SM value as λ decreases; for example, for λ = 0.5, M q = 1000 GeV, M = 500 GeV we have χ 2 /dof = 1.26, F χ 2 = 0.28. Figure 16: For the VQ 1 model, for Y Q = 1/6, λ U = 1, λ D = 1, M U = M D . The colored regions are with all quark mass eigenvalues ≥ 500 GeV, and satisfy the S and T constraints at or better than 2 σ. In all the plots, χ 2 values decrease and F values increase as we go toward larger masses, or lower λ.

Conclusions
In this paper we have surveyed numerous vector-like fermion extensions of the SM. Our purpose has been to illuminate the structure of the theories, detail the precision electroweak implications and constraints, investigate LHC direct limits, and investigate the implications of vector-like fermions for Higgs boson production and decay.
The phenomenological implications of vector-like theories depend crucially on the details of the precise theory in question. Precision electroweak constraints depend on masses and couplings to vector bosons, flavor constraints depend on their flavor mixings with SM fermions, direct detection constraints depend on mass hierarchies and the size of the couplings and mixings to SM states that enable prompt or non-prompt decays into various model-specific final state ratios. In addition, the Higgs boson observables depend on overall mass scales and the ratios of the strongly interacting vector-like masses to electroweak interacting states. Our aim was to demonstrate all of these features through examples and extensive computations.
One of the recent motivations for considering vector-like states added to the SM was to account for possible deviations in the µ γγ = σ(h)B(h → γγ) rate at the LHC. As discussed in a Sec. 6.4, early indications from the LHC Higgs studies suggested an enhanced rate to two photons. Presently the central values of the ATLAS data are higher than the SM expected rates, and the central values of the CMS data are lower than the SM expected rates, making a combined total more consistent with the SM than originally thought. Nevertheless, it is one of the few observables that is sensitive to new physics in loops, and the uncertainties in the experimental measurements and QCD uncertainties [90] give plenty of room for large effects (tens of percent) from new physics. In our investigations we have shown that the addition of vector-like fermions with vector-like masses less than a few TeV, and with chiral couplings to the Higgs boson, leads to tens of percent shifts in Higgs production (gg → h → XY ) and decay (h → γγ) observables. In time, measurements of Higgs observables and direct searches for vector-like states will go far to confirm or exclude this possibility near the weak scale. Figure 17: For the VSM 1 model, for Y Q = 1/6, Y L = −1/2, λ q = 1, λ = 1, M q = 1000 GeV and M = 500 GeV. The colored regions are with all quark mass ≥ 500 GeV, all lepton mass eigenvalues ≥ 250 GeV, and satisfy the S and T constraints at or better than 2 σ. In all the plots, χ 2 values decrease and F values increase as we go toward larger masses, or lower λ.

B Explicit expressions for the mixing coefficients
This appendix contains expressions for the various coefficients that we defined in order to simplify the formulae in Sec. 4.

Mixing through the Higgs boson
We give here the explicit expressions for the mixing coefficients from section 5 between the new vector-like generation and the SM quarks through the Higgs boson.