The Flavor of the Composite Twin Higgs

The assumption of anarchic quark flavor puts serious stress on composite Higgs models: flavor bounds imply a tuning of a few per-mille (at best) in the Higgs potential. Composite twin Higgs (CTH) models significantly reduce this tension by opening up a new region of parameter space, obtained by raising the coupling among the composites close to the strong coupling limit $g_* \sim 4\pi$, thereby raising the scale of composites to around 10 TeV. This does not lead to large tuning in the Higgs potential since the leading quantum corrections are canceled by the twin partners (rather than the composites). We survey the leading flavor bounds on the CTH, which correspond to tree-level $\Delta F=2$ four-Fermi operators from Kaluza-Klein (KK) Z exchange in the kaon system and 1-loop corrections from KK fermions to the electric dipole moment of the neutron. We provide a parametric estimate for these bounds and also perform a numeric scan of the parameter space using the complete calculation for both quantities. The results confirm our expectation that CTH models accommodate anarchic flavor significantly better than regular composite Higgs (CH) models. Our conclusions apply both to the identical and fraternal twin cases.


I. INTRODUCTION
The search for traditional top partners responsible for canceling the leading contribution to the Higgs boson mass has so far come up empty handed, leading to bounds of order 700 GeV for stops and vectorlike top partners [1,2]. These bounds are starting to put significant pressure on supersymmetric and composite Higgs models. An alternative approach to solving the hierarchy problem is to consider models with top partners that are not charged under ordinary QCD, but rather under a mirror QCD group, related by a discrete symmetry to the Standard Model (SM) [3][4][5][6][7][8][9][10][11][12][13][14]. In this case the direct bounds on the top partners disappear, and natural models with small fine tuning are still viable. The best known example of such models is the Twin Higgs (TH) [3], in which the entire SM gauge group is doubled, and the twin sector is related by a softly broken Z 2 symmetry to the visible SM. In these models the one-loop quadratic divergences are automatically canceled, and the hierarchy problem is postponed until the cutoff scale of the theory of order 5-10 TeV, where a UV completion of the model becomes necessary. In particular, theories of flavor, whose scale is generally well above the multi-TeV, can only be incorporated into the TH once its UV completion is specified.
The simplest UV completion of the TH is by extending it to a composite Higgs (CH) model [8][9][10], with the scale of composite resonances as high as O (10) TeV. This could either be a warped extra dimensional (ED) model [8] or a 4D composite Higgs model with partial compositeness for the fermions [9,10]. In this paper we will be using the warped extra dimensional language, though every result can be restated in terms of the corresponding 4D CH model.
The main feature of this UV completion is the appearance of additional gauge and fermion partners (the KK modes in the extra dimensional language). However, unlike traditional CH models [15][16][17][18][19][20][21][22][23][24][25], these KK gauge and fermion partners are not the states responsible for the cancellation of the 1-loop divergences of the Higgs mass -that role is played by the twin partners: the twin top, twin W and Z, etc. The KK modes are simply there to UV complete the theory. Within this framework it is now possible to examine the question of the flavor hierarchy and flavor constraints on TH models. Of particular interest is the fact that warped ED models actually provide a natural framework for explaining the origin of the observed flavor hierarchies [26][27][28][29]. The appearance of small Yukawa couplings in this scenario is due to exponentially small overlaps of extra dimensional wave functions. This RSflavor mechanism also incorporates a natural suppression of flavor changing neutral currents (FCNC) called the RS-GIM mechanism: the same wave function suppressions appearing in the Yukawa couplings will also suppress the flavor changing operators. While RS-flavor is a very intriguing possibility, a detailed examination of the flavor constraints shows that there is a significant tension left between the KK scale needed for natural electroweak symmetry breaking (EWSB) and that needed to evade flavor constraints. See refs. [27][28][29][30][31][32][33][34] for the discussion of ∆F = 2 bounds and refs. [29,[44][45][46][47][48][49][50][51] for the leading dipole bounds in warped models. Generically in CH models, M KK < 1 TeV is required for a fully natural Higgs mass, while the flavor bounds require M KK > 10 − 20 TeV. To this end, flavor symmetries have been introduced to reconcile flavor bounds with naturalness [35][36][37][38][39][40][41].
The purpose of this paper is to reexamine these flavor bounds in the context of CTH models. While the form of the expressions for the flavor bounds in CTH models are practically identical to those in warped ED, the KK modes no longer play the role of the top partners, allowing us to explore a different region of parameter space. In particular, in CH models the tuning of the Higgs mass grows with (g * f ) 2 , where g * is the interaction strength of the KK modes, and f is the global symmetry breaking scale. Thus, one cannot raise g * without increasing the fine-tuning in the Higgs sector. In contrast, the fine tuning in the CTH model grows as f 2 and is insensitive to g * (since the cancellation is achieved by the twin partners). As we will show in this paper, the tension in the CTH model between flavor constraints and Higgs sector tuning is significantly reduced for larger values of g * . When g * ∼ 2π (4π), we obtain a scenario with f ∼ 3 (2) TeV, where all flavor constraints are obeyed and the tuning is at a percent level. One interesting consequence of raising g * is that the leading contribution to ∆F = 2 FCNC is no longer from KK gluon exchange, but rather from KK Z exchange. In this work we are agnostic to the description of the light mirror quarks and our analysis is relevant whether the low energy theory is fraternal [11] or identical [3] twin Higgs. We focus on flavor constraints from the quark sector, and leave the lepton sector for a future study.
The paper is organized as follows: in Sec. II we provide a brief overview of our main results; in Sec. III we define the CTH model and calculate the Higgs potential and the tuning. In Sec. IV we go over the calculation of the flavor bounds and in Sec. V we give the details of the numerical scan. Our conclusions are contained in Sec. VI. A series of appendices detail the construction of fermions in the CTH model (App. A), the fraternal CTH model (App. B), the evaluation of the Higgs potential (App. C), the Z 2 breaking via hypercharge (App. D), the tuning in CH models (App. E), and the loop functions relevant for the dipole calculations (App. F).

II. OVERVIEW OF FLAVOR BOUNDS
Before presenting the detailed calculations of the flavor bounds in the CTH model, we present a summary of the expected results based on simple estimates. We then emphasize the improvement of the fine tuning in the Higgs sector in the CTH model vs. standard composite Higgs models, in the presence of flavor bounds. The details of the CTH model will be reviewed in Sec. III, and will be used in the full evaluation of the flavor bounds in The key parameters of the model are the global symmetry breaking scale f , and the (dimensionless) interaction strength g * of the composite states (KK modes). The KK mass is related to these parameters via There is an additional coupling g s * which measures the interaction strength of the KK gluon.
Since the models under consideration are 5D warped theories with the Higgs arising as a scalar component of the 5D gauge field, there is no actual Yukawa coupling parameter Y in these models. Instead the Yukawa couplings arise from the 5D gauge interactions, and will also be proportional to some dimensionless boundary localized mixing parameters denoted bym u,d . The full meaning of these parameters can be obtained by the expression of the standard model masses where the functions f Q,−u,−d are the standard RS zero mode wave functions evaluated at the IR brane. The usual 5D RS Yukawa couplings can thus be identified as Y u,d = g * m u,d 2 . Note that there is also kinetic mixing in these 5D models among the matter fields, which gives additional contributions to the expressions above. However it does not play a role in the simple estimate and we ignore it for now. An estimate for the C 4 K and C 5 K coefficients of the ∆F = 2 color-singlet L-R 4-Fermi operator relevant for the kaon sector are given by [28]: with C 4 K mediated by the KK gluon and C 5 K by both the KK gluon and KK Z. The KK Z contribution to C 5 K dominates at higher values of g * . Note that the basic structure of this expression does not depend on whether one considers a CTH model or a standard CH model in warped space. Clearly there are various O(1) factors showing up due to the enlarged group structure. These will be included in the full scan of Sec. V, and we ignore them for now.
When this bound is saturated we havẽ In ordinary composite Higgs models the tuning is given by where a is an O(1) constant that depends on the particular CH model (for example, a = 0.35 in the model of [28] and [43], see Appendix E). In other words, the flavor bounds constrain the tuning directly, to the sub per-mille level. This estimate might be somewhat pessimistic for the composite Higgs models, since it assumes a universal scale for both the composite vector bosons and the composite fermions. Relaxing this condition, the estimates of Eq. (3) become where g V * f and g F * f are the scales of the vector and the fermion excitations. Of these two scales, it is only the scale of the fermion excitation that dominates in the tuning. The limit is then implying in this case a tuning of a few per-mille at best. In comparison, the tuning in CTH models is only linked to f and not g * : For larger values of g * , Eq. (10) is satisfied with a smaller f , and consequently, smaller tuning. In this regime the dominant ∆F = 2 bound comes from the KK Z mediated C 5 K . Of course in this extreme case of g * ∼ 4π the 5D description of the CTH model itself becomes strongly coupled. For a more realistic value of g * ∼ 2π, f ∼ 3 TeV and the tuning is at the percent level. To this extent, the idea of anarchic flavor can be revived in the framework of composite twin Higgs. These considerations apply both to identical and fraternal [11] versions of the CTH.

III. THE COMPOSITE TWIN HIGGS
In this section we define our benchmark composite twin Higgs model, whose flavor structure will be studied in detail in the upcoming sections of this paper. This model is realized in 5D anti-de Sitter (AdS) space with gauge-Higgs unification (GHU). We parametrize the 5D AdS metric in the standard form: where R is the AdS curvature and R ≤ z ≤ R is the coordinate of the extra dimension. The UV brane is located at z = R and the IR brane at z = R , with the hierarchy R /R ∼ 10 16 between the weak and Planck scales.
The gauge symmetry in the bulk is where the SO(8) contains the SM electroweak SU(2) L as well as the SU(2) R needed for custodial symmetry, and their twin partners The bulk symmetry is broken on the UV and IR branes to by imposing Dirichlet (-) boundary conditions for the gauge bosons corresponding to broken generators. The UV brane boundary conditions ensure that the surviving low-energy gauge symmetry is SM×SM twin , while the IR breaking SO(8)→SO (7) ensures the emergence of the pseudo Nambu-Goldstone boson (pNGB) Higgs necessary for the twin Higgs mechanism.
The hypercharge is as usual Y = X + T 3 R , and the mirror hypercharge is defined analogously as Y m = X m +T 3m R . The seven broken generators in the coset SO(8)/SO(7) are denoted T i8 , i = 1, . . . , 7. The A 5 i components of the gauge bosons corresponding to the these generators get IR Neumann (+) boundary conditions. Of these only A 5 1,...,4 have also UV Neumann boundary conditions -the four zero modes corresponding to the pNGB Higgs doublet, in the 4 of (SU (2) L × SU (2) R ) SM . A Coleman-Weinberg potential for these zero modes arises through loops of SM and mirror gauge bosons and fermions, resulting in the SM electroweak symmetry breaking.
The A 5 1,...,4 enter the equations of motion (EOM) of the gauge and fermion fields through the Wilson line between the two branes: where , while the scale f and coupling g * (introduced in Sec. II) are formally defined by f can be thought of as the vacuum expectation value (VEV) corresponding to the SO(8)/SO (7) breaking, and g * is the dimensionless bulk gauge coupling of SO(8) characterizing the interaction strength of the KK modes. The KK scale is defined as: the latter relation was already used in Sec. II. Using gauge transformations we can always bring Ω into the form: A. The Quark Sector The SM and mirror quarks in our model are embedded in bulk multiplets. Specifically, singlet: Here (+/−) denote the UV/IR boundary conditions, and the subscripts are the represen- The alternative fraternal CTH realization of the fermion sector (where only the third generation fermions have zero modes in the mirror sector) is detailed in App. B. Note that the leading flavor constraints will be identical in both cases.
Under the IR SO(7) symmetry, the bulk multiplets decompose as For each component we only show the chirality that has (+) boundary conditions (b.c.) on the IR brane. In the mirror sector, due to the Z 2 on the IR brane we have: The IR symmetry allows for the following mass terms on the IR brane: The m u,d are dimensionless IR mass parameter matrices. Note, that the IR Lagrangian has to be invariant under the Z 2 on the IR brane and so the SM and mirror multiplets share the same IR masses. These masses are generically 3 × 3 anarchic matrices, unless a flavor symmetry is postulated on th IR brane. With this choice of representations and boundary conditions, the lightest states in q L , u R , d R are zero modes prior to EWSB. The effect of the IR masses is to rotate the zero modes among the bulk multiplets, and as a result some zero modes will appear in more than one bulk multiplet. Specifically the left handed q L lives in Ψ 8 and in Ψ 28 , the right-handed u R lives in Ψ 1 and in Ψ 8 and the right-handed d R lives only in Ψ 28 . This results in kinetic mixing for each zero mode in the "bulk basis" -the basis in which the bulk masses are diagonal. This kinetic mixing can be parameterized by three Hermitian matrices K q , K u , K d so that the kinetic term isΨK / DΨ [28]: where f c is the standard RS flavor function These fermions get mass due to the VEV of A 5 . This VEV enters the bulk EOM through the covariant derivative of each multiplet in the bulk Lagrangian. To find the masses in the fermion KK tower in the presence of the A 5 VEV, we use an "auxiliary" fermion field for each multiplet: This auxiliary field has the Wilson lines rotated away from its EOM and thus satisfies the same bulk EOM as Ψ in the A 5 → 0 case, and Ω defined in Eq. (24), however the b.c. of the auxiliary fields will now contain Ω. This rotation now mixes the different zero modes appearing in the same multiplet (due to the IR localized masses) generating mass terms for the fermions after EWSB. Using the same basis as in Eq. (32), the mass terms are: B. The top sector and the Higgs potential To find the Higgs potential, we find the profile of the top and mirror top multiplets by solving the bulk EOM and imposing the b.c. The solution is expressed in the form of the spectral function ρ(p 2 ) = det(−1 + m 2 p 2 ), whose zeroes in p are exactly the (VEV dependent) masses forming the full the KK tower: and The contribution of the top/mirror top sector to the Coleman-Weinberg potential of the Higgs is then given by As explained in [28] (and also in App. C), we can approximate the contribution V ef f by: For all our results we use the full calculation for these terms that appears in App. C. Here we give the NDA estimates: These are the standard loop generated terms in the composite Higgs potential, where α is the logarithmically divergent term and α 2 is the quadratically divergent term.
In the Higgs potential of Eq. (40), the sin h f terms are generated by the top and the cos h f are generated by the mirror top. The quadratically divergent contribution sums up to a constant piece in the potential independent of the Higgs field. The only remaining piece in the potential is: Due to the Z 2 invariance in the top sector, the Higgs potential in Eq. (43) has a minimum for To obtain a realistic v f , an additional (at this point unspecified) Z 2 breaking contribution must be present which we parametrize as For any value of f we can now calculate α and β that produce the right Higgs mass and VEV: where for convenience we define α 0 as the value of α and β for f → ∞ In Fig. 1 we plot α/α 0 as a function of g * for a variety of input parameters in the 5D model. We find that the points in the 5D model naturally populate the various values of α α 0 for large enough values of g * . For α < α 0 there is no solution that gives the right EWSB, and similarly for β > α 0 .
If the Z 2 is exact in the gauge sector, then the contribution of gauge boson loops to the Higgs potential is negligible for our purposes: In App. D we give an example of a Z 2 breaking contribution to the Higgs potential that comes from a difference in the bulk gauge couplings of U (1) X and U (1) m X breaking the Z 2 , such that for every value of f there is a g m X < g X that gives the right β. The precise origin of the Z 2 breaking has no bearing on the flavor observables therefore we remain agnostic to it. The tuning is calculated using the Barbieri-Giudice measure: where p i are all the parameters of the 5D theory. The tuning for different points in the parameter space of the model with the Z 2 contribution of App. D is plotted in Fig. 2. The red line is a quadratic fit to the tuning given by: We note that the tuning only depends on f and not on g * which is essentially a free parameter. This is the major difference compared to the CH, where the tuning ∆ CH ∼ g 2 * , and thus g * can be raised only at the expense of increased tuning.

IV. ANARCHIC FLAVOR BOUNDS
The main purpose of this paper is to establish the basic flavor bounds on the CTH with anarchic flavor. To this end we focus on the flavor physics in the quark sector, while the lepton sector is left for a future study.
The flavor constraints on CH have been studied extensively in [27-35, 37, 38, 40-42], providing a limit on M KK for anarchic flavor. In CH models, the tuning in the CH is proportional to M 2 KK , and so any bound on M KK is also a lower bound on the tuning. 1 The strongest constraint in the quark sector M KK > ∼ 20 TeV arises from K , implying a per-mille level tuning at the least. We note that this bound may be lower in the case when the masses of the vector excitations are parametrically lower than the mass of the KK tops that enters the Higgs potential.
In the CTH, the flavor bounds are parametrically the same as in the CH. The Higgs potential, however, is only logarithmically dependent on M KK , and so g * is essentially a free parameter. As we will see, the lower bound on the tuning in the CTH implied by the flavor bounds will get weaker for larger values of g * as it is raised toward the strong coupling limit We start with the flavor structure of the CTH, working in the "bulk basis" -the basis in 1 The tuning in CH depends on the representations in the top sector [24]. In App. E we calculate the tuning in the model with two adjoints and a fundamental of SO(5), which has "double tuning" according to the definitions of [24].
which the bulk masses are diagonal. The anarchic mass matrices are given by: where we have simply extended Eq. (35) to all three generations. The kinetic terms are: HereM ij u/d are the anarchic 3 × 3 IR mass matrices, and F c is a diagonal matrix whose diagonal elements are the RS flavor functions f c (see Eq. (33)): To find the physical quark masses we must first diagonalize the kinetic terms by rotating the quarks with the Hermitian matrices H q and H u (H d is already the identity matrix due the structure of the b.c. ): The rotation to the mass basis is then via the usual unitary matrices U L , U R , D L , D R . The resulting diagonal mass matrices are: As a result, all of the quark couplings are rotated as well, yielding the source of flavor violations.
We now present the relevant constraints for the ∆F = 2 and the dipole operators.
A. ∆F = 2 ∆F = 2 processes are mediated mainly by KK gluons and KK Z bosons (see Fig. 3). The KK gluon couplings in the mass basis are: where g x Ψ (V ) is a diagonal matrix that gives the wavefunction overlap of the vector boson V with the x component of the Ψ multiplet: where where a x Ψ (Z) and a x Ψ (B ) are functions of the quantum numbers of the x component in Ψ given by: With these couplings we can now calculate the ∆F = 2 constraints on g * and f . We focus on the kaon system, specifically on the ∆F = 2 operators with the strongest experimental constraints: The KK gluon contribution to C 4 K , C 5 K can be calculated: While the KKZ contribution is: In the numerical scan we calculate C 4 k using the full expressions, and the estimate for the bound is given by:

B. Dipole Operators
The second type of constraints arise from loop induced dipole operators 1 . We follow here the method of [48]. The main contribution to these constraints is the KK fermion loop (see Fig. 4). For the dipole calculation, we work in the approximation where we keep only one KK level for the fermions, and only the zero modes for the gauge bosons. With this approximation the fermion fields are: L,R In this notation, the zero mode multiplets include only the components with (+, +) b.c. , and the rest of the components are set to zero; for instance, the only non-zero component in Ψ Q8 L is Q L . The mass terms are: where we have assumed that all the components in the first KK-level of each multiplet have a common mass, denoted by m Q , m D and m U for Ψ KK8 , Ψ KK28 and Ψ KK1 , respectively. We note that there is a slight difference in the KK masses of states with (+, +) and (−, +) b.c.
In our approximation we neglect this difference, and set m Q = m U = m D = g * f . Rotating to the mass basis, we find the interactions with the gauge bosons of the form where q is a zero mode quark and Ψ is a first KK mode quark. All the dipole operators can be calculated from the couplings V qΨ X . The strongest bound is the electric dipole moment of the neutron, where the parton level contributions are: The coefficients c,c can be calculated [48]: Here, L X ,L X are the loop functions defined in [48], and are given in Appendix F. Our calculation gives to the leading order in f 8 , f 28 , f 1 and v: where Y d = g * 2m d . The experimental bounds are estimated to be [48] f √ c > 3.11 TeV , f √c > 3.79 TeV.
We note that this estimation is based on QCD sum rules, evaluating both contributions to the neutron EDM individually, and may underestimate the theoretical uncertainty. We can now establish the previously quoted estimate of the resulting bound: wherem d is an average IR mass. and only get weaker with higher g * . Here we give an example of t → cZ. For high g * , The main contribution is t L → c L Z, as the right handed chirality is protected by the custodial symmetry that protects Z → b L b L . The P LR symmetry [19,20] is satisfied for right handed up quarks T 3 L = T 3 R = 0, and for left handed down quarks T 3 L = T 3 R = − 1 2 . Thus, all non universal contribution to Z coupling to these states has to come from mixing with the other chirality. The branching ratio for left handed tops can be estimated: For g * → 4π, this branching fraction is far below the LHC reach even for f below 1 TeV.
These constraints are subdominant and we do not include them in our numerical scan. We note here that similar contributions to Z m mediated FCNC processes are absent due to the exact P m LR symmetry for all the SM quarks (T 3m L = T 3m R = 0).

V. NUMERICAL SCAN
In this section we establish the validity of our estimates by performing a numerical scan of the flavor bounds. The full expressions for the magnitudes of the flavor violating processes together with the previously obtained simple estimates are: for the ∆F = 2 operators, and for the coefficients of the operators contributing to the neutron EDM.
To test our estimates, we generated sets of bulk and IR masses that reproduce the flavor texture of the SM and give the right EWSB. We then calculated the above expressions with these sets of parameters and compared to the estimates. We assumed that the IR masses form a 3 × 3 matrix, with eigenvalues randomly distributed and a random unitary rotation from the eigenbasis. The comparison with the estimate is performed by taking them d in the estimates as the mean of the probability distribution of the eigenvalues of the 3 × 3 matrix.
The procedure is then the following. We start by generating 7000 sets of 5D parameters in the top sector that give a realistic Higgs potential, as explained in App. C. These parameters are defined by the bulk masses of the third generation quarks c q 3 , c u 3 , c d 3 , the third generation IR boundary mass parameterm u 3 and g * , R, R , f . For each set of parameters we then follow the following steps • Fix the remaining bulk masses such that the physical quark masses and CKM angles will be naively reproduced.
• Choose random complex boundary mass matricesm u ,m d , with one of the eigenvalues ofm u beingm u 3 . The rest of the eigenvalues are chosen uniformly between 1 3m u 3 , 5 3m u 3 . These are the anarchic IR mass matrices.
• Calculate the full mass matrix and perform a χ 2 -fit for the six mass parameters, the three CKM angles, and the Jarlskog invariant. If the fit is reasonable (χ 2 < ξ) keep these parameters, otherwise throw it away.
• Using this procedure we generate 100 sets of flavor parameters for each 5D parameter point that reproduces correct EWSB. For each set we calculate magnitudes of all the relevant flavor violation processes.

Appendix B: Fraternal Boundary Conditions
In the model above the low energy 4D theory contains light mirror fermions. However, it is trivial to eliminate them and the mirror photon from the spectrum by assigning Dirichlet b.c. to them on the UV brane -and explicitly breaking Z 2 in the light sector. In this case the b.c. for the light mirror quark multiplets read: The mirror leptons (except for the mirror tau and mirror tau neutrino) can be eliminated from the spectrum in a similar manner. In this way our composite framework can also UV complete the fraternal twin Higgs of [11].
where α 2 , α 4 , n t are given by . This dependence on f, g * and m h is similar to the ones estimated in [24] for similar doubly-tuned CH models.