A custodial symmetry for Zbb

We show that a subgroup of the custodial symmetry O(3) that protects delta rho from radiative corrections can also protect the Zbb coupling. This allows one to build models of electroweak symmetry breaking, such as Higgsless, Little Higgs or 5D composite Higgs models, that are safe from corrections to Z->bb. We show that when this symmetry protects Zbb it cannot simultaneously protect Ztt and Wtb. Therefore one can expect to measure sizable deviations from the SM predictions of these couplings at future collider experiments. We also show under what circumstances Zb_R b_R can receive corrections in the right direction to explain the anomaly in the LEP/SLD forward-backward asymmetry A^b_{FB}.


Introduction
One of the most elegant solutions to the hierarchy problem is to consider that the Higgs boson, the scalar field responsible for electroweak symmetry breaking (EWSB), is not a fundamental particle. This approach is clearly inspired by QCD, where scalar and pseudoscalar states appear as composites of the strong dynamics. In recent years there has been a revival of interest in this approach. The important new ingredient has been calculability, achieved by using either the idea of "collective breaking" [1] or extra dimensions.
As in the old technicolor [2] or composite Higgs models [3], the main challenge of these new scenarios is to pass successfully all the electroweak precision tests (EWPT). This is a non-trivial task, since in these theories deviations from the Standard Model (SM) predictions usually arise at the tree level due to mixing effects between SM fields and the heavy states of the new sector. One of the main difficulties is to avoid large deviations in the Zb LbL coupling, whose measured value is in agreement with the SM prediction at the 0.25% level. This is difficult to overcome, since in these models the top, being heavy, couples strongly to the new sector. Since b L is in the same weak doublet as t L , it usually suffers from large modifications to its couplings.
In this article we will show that the custodial symmetry O(3), advocated long ago to protect ∆ρ [4], can also protect Zbb. In particular we will see that the Zb LbL coupling can be safe from corrections and at the same time the SU(2) L -related couplings Zt LtL and W t LbL can receive sizable modifications. As an example, we will present the explicit calculations of these effects in a 5D scenario of EWSB. The custodial symmetry can also be used to protect the coupling of the b R to the Z. However, the LEP and SLD experimental measurements of the forward-backward asymmetry A b F B suggest that the coupling Zb RbR might deviate from its SM value. We will then study the possibility of having large effects in Zb RbR of the right magnitude and sign as suggested by the experimental data.
Our analysis can be useful for any scenario of EWSB that contains a new sector beyond the SM (BSM) invariant under the global custodial symmetry. This sector is defined to include the Higgs field as well. Examples are the strongly interacting sector of technicolor models, the extra fields added in Little Higgs theories to avoid quadratic divergences, or the bulk of a warped extra dimension present in some Higgsless [5] and composite Higgs [6,7] models. 2 The coupling Zψψ and the custodial symmetry We will consider BSM sectors with the following global symmetry breaking pattern [4]: gether with a parity defined as the interchange L ↔ R (P LR ). As we will see below, this discrete symmetry plays an important role to protect the coupling of the Z to fermions from non-zero corrections. The BSM sector also has to respect an SU(3) c ⊗U(1) X symmetry corresponding to the SM color group and an extra U(1) needed to fit the hypercharges of the SM fields (Y = T 3 R + X). As usual [4], we will parametrize the symmetry breaking in Eq. (1) by a 2 × 2 unitary matrix field U transforming as a (2, 2) 0 under SU(2) L ⊗SU(2) R ⊗U(1) X , whose VEV is given by U = 1l.
Since the BSM sector is invariant under O(4), we can rotate to a basis in which each BSM field (or operator), O BSM , has a definite left and right isospin quantum number, T L,R , and its 3rd component, T 3 L,R . We will assume that every SM field Φ is coupled to a single BSM field (or operator): . This assumption is always fulfilled in the BSM models that we are interested in. It guarantees that we can univocally assign to each SM field definite quantum numbers T L,R , T 3 LR , corresponding to those of the operator O BSM to which it couples. Notice that this does not imply that the SM fields are in complete representations of SU(2) L ⊗ SU(2) R , as it is known not to be the case.
Let us consider the implications of the custodial symmetry O(3)=SU(2) V ⊗ P LR on the coupling Zψψ, where ψ denotes a generic SM fermion. At zero momentum, this coupling is given by where Q 3 L and Q are respectively the 3rd-component SU(2) L charge and the electric charge of ψ. Since the electric charge Q is conserved, possible modifications to the coupling Zψψ can only arise from corrections to Q 3 L . Before EWSB we have Q 3 L = T 3 L , but this is not guaranteed anymore after EWSB. We will be interested only in non-universal corrections induced by the BSM fields, and we will treat the SM W 3 L field as an external classical source which probes the left charge Q 3 L . This is consistent since corrections induced through the renormalization of the Z kinetic term are universal.
We found two subgroups of the custodial symmetry SU(2) V ⊗ P LR that can protect Q 3 L . The first one is the subgroup U(1) L ⊗U(1) R ⊗ P LR that it is broken by U down to U(1) V ⊗ P LR .
Although P LR is a symmetry of the BSM sector, it is not, in general, respected by the coupling of ψ to the BSM sector. For P LR to be a symmetry also of L int =ψO ψ + h.c., we must demand that ψ is an eigenstate of P LR . This implies for the field ψ. If this is the case, the non-universal corrections to the charge Q 3 L of ψ are zero. The proof goes as follows. By U(1) V invariance, we have that Q 3 V = Q 3 L + Q 3 R is conserved, and therefore it cannot receive corrections: On the other hand, by P LR invariance we have that the shift in Q 3 L must be equal to the shift in Q 3 R : Eq. (4) and Eq. (5) imply that δQ 3 L = 0. This proves that SM fermions that fulfill the condition (3) have their coupling to the Z protected by the subgroup U(1) V ⊗ P LR of the custodial symmetry.
The second example of a symmetry that can protect Q 3 L is that of the discrete transformation We will denote this symmetry by P C . Its action on 2-component spinors is given by P C = iσ 1 , while SO(3) vectors transform with P C = Diag(1, −1, −1). According to our rule then, the SM W 3 L can be assigned an odd parity under P C : For ψ to be an eigenstate of this symmetry, it must have If this is the case, we have that δQ 3 L = 0 at any order. Indeed, if ψ is an eigenstate of P C , then ψγ µ ψ is even under P C and it cannot couple to W 3 L that is odd. Thus, the coupling of the Z to SM fermions that fulfill Eq. (6) is protected by the subgroup P C of the custodial symmetry.
It is important to notice that the symmetries discussed above can only protect the coupling of the Z to fermions at zero momentum. However, momentum dependent corrections to Zψψ are parametrically suppressed in strongly coupled BSM sectors. For example, in the case of Zb LbL a naive estimate gives δg/g ∼ is the degree of mixing between t L (t R ) and BSM states (0 ≤ ξ L,R ≤ 1), and g BSM is the coupling among the BSM particles. Therefore, δg/g can be sufficiently small for g BSM ≫ λ t (and ξ R not too small).

Corrections to Zb LbL in custodial invariant models
The symmetry argument given in the previous section shows how to build Higgsless or composite Higgs models in which Zbb does not receive corrections from the BSM sector. Let us start with the Zb LbL coupling. In these models it has been commonly assumed that b L transforms as a (2, 1) 1/6 representation of the SU(2) L ⊗SU(2) R ⊗U(1) X group. In that case, b L has the quantum numbers T L = 1/2, T R = 0, T 3 L = −1/2 and T 3 R = 0, which fulfill neither the condition (3) nor (6). As a consequence, Zb LbL is not protected by the custodial symmetry. Condition (3), however, suggests us a better assignment for the b L quantum numbers: implies that t L , being in the same SU(2) L doublet as b L , has to have the following assignments: (3) is not satisfied for t L and there will be corrections to the Zt LtL coupling. Similarly, the custodial symmetry does not protect W t LbL (see below), and one can have large modifications in this coupling as well, without affecting Zb LbL .
At present, the couplings of the top to the gauge bosons are not accurately measured. Future accelerators, however, will improve the measurements of these couplings and will be able to test this scenario.

Operator analysis
We give here an operator analysis for the couplings of q L = (t L , b L ) to the Z and the W based on the custodial symmetry. For the assignment of Eq. (7), we must embed b L in a 4 2/3 of O(4)⊗U(1) X , or, equivalently, under SU(2) L ⊗SU(2) R ⊗ U(1) X . In addition to the SM doublet, this representation contains an extra SU(2) L doublet q ′ L that, not corresponding to any SM field, will play the role of a nondynamical spectator. We find two single-trace dimension-4 operators that can contribute to the Z couplings: where Q L = σ α Q α L is a 2 × 2 matrix field, 1 V µ = (iD µ U )U † ,V µ = (iD µ U ) † U , and the covariant derivative is defined as D µ U = ∂ µ U + igσ a W a µ U/2 − ig ′ B µ U σ 3 /2. By imposing P LR , under which U → U † , V µ ↔V µ and Q L → σ α † Q α L , we obtain c 1 = c 2 . There is also a double-trace operator that can contribute to the Z coupling to q L : To obtain the contributions to Zb LbL , Zt LtL and W t LbL we plug into Eqs. (9) and (10), where σ ± = (σ 1 ± iσ 2 )/2 and σ 0 = (1l + σ 3 )/2. This gives As expected from the symmetry argument, the contributions to Zb LbL vanish after imposing invariance under P LR (c 1 = c 2 ), while the contributions to the couplings of the top quark are different from zero. 1 We use the basis σ α = (1l, iσ1, iσ2, iσ3) where σa, a = 1, 2, 3, are the Pauli matrices.
The embedding of t R in a multiplet of SU(2) L ⊗SU(2) R ⊗ U(1) X is determined by the top mass operatorq L U t R . There are two possible invariant operators: implying respectively the two following embeddings for t R : which correspond respectively to a 1 2/3 and a 6 2/3 multiplet of O(4)⊗U(1) X . In both cases t R has T 3 L = T 3 R = 0, fulfilling the condition (6). Therefore, its coupling to the Z is protected by the P C symmetry. 2 We can also perform an operator analysis for the Z coupling to t R . For the case Using U R = σ 3 t R + ... we find that, as expected, the contribution to Zt RtR vanishes.
In we can embed the top in a 10 2/3 : In the composite Higgs model of Ref.

Explicit calculations in 5D models of EWSB
In this section we focus on 5D composite Higgs models realized in AdS 5 space-time [6,7], and compute the correction to Zψψ induced by the first Kaluza-Klein (KK) mode. In these theories the EWSB scale is given by v = ǫf π , where f π is the analog of the pion decay constant and ǫ is a model-dependent parameter bounded to be 0 < ǫ ≤ 1. The experimental constraint from the Peskin-Takeuchi S parameter generically requires ǫ 0.5. Our result for Zψψ will also apply to the class of Higgsless models in AdS 5 [5] after setting ǫ = 1.
Let us denote with c the fermion 5D bulk mass in units of the AdS curvature. We will assume −1/2 < c < 1/2, since for |c| > 1/2 the fermion zero modes are quite decoupled from the 5D bulk and non-universal corrections to Zψψ from the exchange of KK modes are exponentially suppressed (this is the case for the first and second generation fermions). There are two types of diagrams contributing to Zψψ, one involving the exchange of gauge KKs, the other involving fermionic KKs.
The contribution from the tower of SU(2) L ⊗SU(2) R gauge KKs is, at the tree level and at zero momentum: where δg(g/ cos θ W )ψγ µ ψZ µ gives the non-universal correction to the SM vertex. 3 Effects from the fermion KKs are of the form where θ KK is the mixing angle between the KK and ψ. This mixing occurs after EWSB and it is of order sin θ KK ∼ ǫ 1/2 − c. 4 Although the sum in Eq. (19) is over all the KK tower, a good approximation is obtained by considering only the lowest mode.
The coefficients of the operators in Eqs. (9) and (10) then read: Here θ (1,1) KK is the mixing angle between t L and the KK in the (1, 1) 2/3 representation, and θ

The coupling Zb RbR
The small ratio m b /m t can be naturally explained in the class of models under consideration by assuming that the SM b R couples weakly to the BSM sector. The shift in the coupling of b R to the Z due to the BSM sector, δg Rb , will then be small. This is the case, for example, when q L ∈ (2, 2) 2/3 and both b R and t R couple to the same BSM operator transforming as a (1, 3) 2/3 ⊕ (3, 1) 2/3 , case (b) of Eq. (13).
It is however interesting to consider the possibility that b R couples more strongly to the BSM sector, since a positive shift δg Rb ∼ +0.02 would explain the 3σ anomaly in the forward-backward asymmetry A b FB measured by the LEP and SLD experiments (see [10]).
Here and in the following, θ (r,s) KK denotes the mixing angle between b R and the KK state with electric charge −1/3 in a (r, s) representation of SU(2) L ⊗SU(2) R (if the representation (r, s) contains more than one state with electric charge −1/3, then θ (r,s) KK will refer to the KK with T 3 L = −1/2). Thus, one can obtain a positive δg Rb from the mixing of b R with the KKs in the (2, 4) 2/3 , as needed to explain the A b F B anomaly. A different possibility is that the SM q L itself couples to two different BSM operators: the first responsible for generating the top mass, the second for generating the bottom mass. 6 The coupling to this latter operator will in general violate the custodial symmetry subgroup protecting g Lb , but it is natural to assume that its coefficient is small, in order to reproduce the small ratio m b /m t .
The resulting δg Lb will also be small, allowing at the same time a large coupling of b R to the BSM 5 A larger and negative shift, δg Rb ∼ −0.17 would also explain the data [11], but to obtain such a large shift would require a very light spectrum of new particles. We do not consider here this possibility. 6 An explicit realization of this scenario in the context of a 5D composite Higgs model will be given in [9].
sector. There are many choices for embedding b R in SU(2) L ⊗SU(2) R ⊗U(1) X , giving δg Rb of either sign. The simplest choice is which can be embedded in a 5 of SO (5). In this case the BSM operator coupled to q L responsible for the bottom mass has to transform as a (2, 2) −1/3 . Since however T 3 L ,R = 0 for b R , the P C symmetry argument of section 2 implies δg Rb = 0 for both gauge and fermionic contributions.
Another possible choice is b R ∈ (1, 2) 1/6 , which can be embedded into a 4 of SO (5). In this case the BSM operator coupled to q L can transform as either a (2, 1) 1/6 or a (2, 3) 1/6 . At order ǫ, b R can mix with KKs in (2, 1) 1/6 and (2, 3) 1/6 . We find Thus, one has δg Rb > 0 from mixing with KKs in the (2, 3) 1/6 , as needed to explain the A b F B anomaly. A few other examples with 1 Higgs insertion are indicated in Table 1.

Conclusions
In models where the electroweak symmetry breaking is induced by a new (strongly interacting) sector coupled to the SM fields, it is crucial for the new sector to respect a custodial symmetry in order to prevent large corrections to ∆ρ. We have shown that the custodial symmetry O(3) can also protect the Zb LbL coupling from corrections. This suggests that the custodial invariance might be a key ingredient to build natural models of electroweak symmetry breaking with a relatively light spectrum of new fermions, as required by naturalness arguments. A way to test this scenario is to look for modifications in the couplings Ztt, W tb, which cannot be protected at the same time by the custodial symmetry and can receive potentially large shifts. Finally, we investigated the possibility of a modification of the Zb RbR coupling, showing that a positive shift, as required to explain the anomaly in the LEP/SLD forward-backward asymmetry A b F B , is possible for certain choices of the b R custodial quantum numbers.