Probing the mechanism of EWSB with a rho parameter defined in terms of Higgs couplings

A definition of the rho parameter based on the Higgs couplings with the gauge bosons, rho_h= g_{hWW}}/c^2_W g_{hZZ}, is proposed as a new probe into the origin of the mechanism of electroweak symmetry breaking. While $\rho_h=1$ holds in the SM, deviations from one are predicted in models with extended Higgs sector. We derive a general expresion of rho_h for a model with arbitrary Higgs multiplets, and discuss its size within the context of specific models with Higgs triplets, including the ``Little Higgs'' models recently proposed. We find the even for Higgs models that incorporate the custodial symmetry to make rho=1, one could have rho_h\neq 1, which could be tested at the level of a few percent, with the precision Higgs meassurements expected at the NLC.


Introduction
The mechanism of electroweak symmetry breaking (EWSB), which is triggered spontaneously through a Higgs doublet in the minimal standard model (SM), has remained without direct experimental verification so far. Precision measurements of electroweak observables constrain the Higgs mass below about 200 GeV at 95% CL [1,2,3] within the standard model. Thus, it is expected that a Higgs particle could be discovered at the Run 2 of the Tevatron, provided sufficient luminosity is achieved [4]. But it is intriguing to notice that the EW observables prefer a SM like Higgs with mass below 114.1 GeV [1,3], which is the present lower limit from LEP 2. The data indicate that the Higgs boson should have already been discovered [1], and the fact that it was not, could be taken as a hint of new physics, which could be related with the freedom to choose the Higgs sector [3]. Extensions of the Higgs sector have been proposed for a while [5], and in particular models with Higgs triplets (real or complex) have been considered well motivated, partly because such representations arise in the context of left-right symmetric models [6], or are associated with low-energy mechanisms aimed to generate neutrino masses [7], as an alternative to the usual see-saw mechanism. More recently, Higgs triplets with O (TeV) masses, have been predicted in connection with the so-called "Little Higgs" models [8], which attempt to explain the required lightness of the Higgs as being associated with a global symmetry.
Models with Higgs triplets can violate the custodial symmetry SU(2) c of the Higgs-Gauge sectors. This symmetry protects the relation between the gauge boson masses and the weak mixing angle, which can be conveniently parameterized through Veltman's rho parameter [9], i.e. ρ = m 2 W /m 2 Z c 2 W , which is equal to one at tree-level; loop corrections to this parameter could be very important, as it was exemplified by the prediction of a top quark heavier than originally expected. However, when one considers models with Higgs triplets, with their neutral component acquiring a v.e.v. that contributes to EWSB, then the ρ parameter could deviate from one even at tree-level. Several Higgs triplets, with ad hoc quantum numbers, are required in order to preserve the custodial symmetry [10].
A simple analysis of the SM Lagrangian reveals that the Gauge boson masses and their Higgs couplings originate from the terms: After SSB, one can write the neutral component in terms of the SM Higgs boson (h 0 ) and the Goldstone boson (G 0 Z ), i.e. Φ 0 = (v + h 0 + iG 0 Z )/ √ 2, and the gauge bosons (W ± and Z 0 ) acquire the masses: m 2 W = g 2 v 2 /4 and m 2 Z = g ′ 2 v 2 /4, respectively, with g ′ = g/c W . In this case it happens that the same source of EWSB that contributes to the Gauge boson masses, induces the Higgs-Gauge couplings, which in turn are given by: g hW W = g 2 v/2, g hZZ = g ′ 2 v/2, and therefore one can define the parameters ρ and ρ h , which satisfy: We could also express this result by saying that both the neutral Higgs and Goldstone boson have the same couplings to the gauge bosons in the SM. Thus, if the SM is the correct theory of EWSB, a measurement of the Higgs-Gauge couplings should give ρ h = 1. However, small deviation from one for ρ H can be expected to appear because of radiative effects, while the experimental value of ρ h will deviate from one because of the systematic and statistical errors.
Furthermore, when one considers physics beyond the SM aimed to explain EWSB, it is conceivable that the Goldstone bosons could have a different origin from other neutral scalar of the model, as it could happen in composite scenarios. Alternatively, even if both the Higgs and Goldstone bosons have a common origin, their Higgs-Gauge boson couplings could have different values, either because of mixing factors or because of renormalization effects. In all these cases, one would have ρ = ρ H . Given the possibility that an scalar particle could be detected in the near future, it will be important to verify whether this particle is indeed a type of Higgs boson, and the parameter ρ H could play a major role in this regard. This will be illustrated in the next sections with several examples.
The organization of this letter goes as follows: In section 2, we shall present a general expression for ρ H for a Higgs multiplet of arbitrary isospin T and hypercharge Y ; its size is discussed in detail within the context of a minimal extension of the SM that includes one doublet and a real (Y = 0) Higgs triplet; one of our main result is the argument that ρ ≃ 1 does not implies ρ H = 1. We shall also evaluate a similar parameter, but in terms of the Higgs decay widths, which would be closer to the output from future high-precision experiments for the Higgs boson. We then discuss, in section 2.2, a model with extended Higgs sectors, which do respects the custodial symmetry, i.e. ρ = 1, but the Higgs particles do not necessarily satisfy ρ H i = 1. Then, in section 4 we shall discuss the above parameter, for the Higgs sector that arises within the context of the "Little Higgs" model. Finally, we shall present our conclusions in section 5.
2 Higgs multiplets and the Rho parameter

A general expression for ρ H
Let us consider a model with an arbitrary Higgs sector, consisting of a number of Higgs multiplets Φ K of isospin T K and hypercharge Y K . From the expression for the kinetic terms, written in terms of the covariant derivative, one obtains the gauge boson masses, which satisfy the following expression for the rho parameter, where v K denotes the v.e.v. of the neutral component of the Higgs multiplet, while c K = 1/2 (1) for real (complex) representations. It is well known that Higgs representations for which T K (T K +1) = 3 4 Y 2 K , satisfy ρ = 1, regardless of their v.e.v.'s. Examples of this case are: (T, Y ) = (1/2, 1), (3,4), . . .. Alternatively, one could choose ad hoc v.e.v.'s for models with several types of Higgs multiplets, such as triplets, to have ρ = 1.
On the other hand, when one writes down the Gauge boson coupling with the neutral Higgs components Φ 0 K , which are weak eigenstates, ρ H satisfies a similar relation, namely: Thus, whatever choice makes ρ = 1 for the Higgs multiplet Φ K , it will also make ρ φ 0 K = 1. However, when one has several multiplets, one needs to consider the Higgs mass eigenstates instead, which are indeed the ones that could be detected and probed at future colliders. Thus, we have to consider the rotations that diagonaliazes the real parts of the neutral components, such that the Higgs mass eigenstates H i are related to the weak eigenstates Re φ 0 K as: Re φ 0 K = U Ki H i . Then, the rho parameter for the Higgs bosons H i is given by: From this important relation, we can discuss several consequences: (1) For models that contains several Higgs multiplets of the same type (say doublets), for which (2) On the other hand for a model that includes doublets and some other multiplet (say triplets), for which ρ ≃ 1 is satisfied with a hierarchy of v.e.v.'s, i.e. v K ≪ v D , then one has that ρ H i could be significantly different from one (as will be shown next). (as in the model to be discussed in section 3), then because of the factors U Ki , it turns out that in general ρ = 1 does not necessarily imply ρ H = 1, and this could provide an important test of the type of Higgs multiplet that participates in EWSB.

A model with one doublet and one real triplet
We shall evaluate now the size of ρ H for an extension of the SM, where the Higgs sector includes one real (Y = 0) Higgs triplet, Ξ = (ξ + , ξ 0 , ξ − ), in addition to the usual SM Higgs doublet Φ. The Higgs potential of the model is written as [11]: where the last term involves the linear form, namely: and τ is the vector of Pauli's spin matrices.
After constructing the mass matrices, and performing its diagonalization, we arrive to the following mass eigenstates: where h d = Re Φ 0 and h tr = Re ξ 0 ; while the mixing angle, α, is defined by: here v D = h d and v T 0 = h tr .
The Higgs-Gauge Lagrangian is given by: where and From this Lagrangian we can identify the masses of the Gauge bosons: , and the couplings hW W , hZZ, HW W and HZZ, where h (H) corresponds to the lighter SM-like (heavier) neutral Higgs mass eigenstate: Therefore in this model we have: Fig. 1, as a function of α (we are plotting the square values, just to get positive defined quantities, as future colliders will not know about signs for Higgs couplings). For ∆ρ ρ , which depends on the parameters of the model, we take the maximum value allowed by data [12], i.e. ∆ρ ρ ≃ 1%. We can appreciate that ρ h can deviate significantly from one for α → π/2, while ρ H can show large deviating from the SM prediction for α → 0. Thus clearly ρ h = ρ = ρ H .
Given the estimated precision expected for the measurements of the Higgs couplings at NLC, in particular for the ratios of Higgs-Gauge couplings which were analized in [13], it happends that ρ H i could be measured with a precision of order 2 %, which will allow to constrain considerably the parameter α in this Higgs triplet model. On the other hand, one could also use the prediction for the Higgs decays into gauge bosons as a possible test of violations of the custodial symmetry. For the real decays one has: Then the ratio R Γ h = Γ(h → W W )/2Γ(h → ZZ), is given by: while the ratio R Γ H = Γ(H → W W )/2Γ(H → ZZ), is given by: This ratios are plotted in Fig. 2 as a function of the Higgs mass, for two fixed values of α (0.04 and π/4), which represent two typical cases of small and large mixing, respectively. In this plot, we have included the decays into one real and one virtual gauge boson, (h, H) → V V * , for the appropriate range of Higgs masses.

An extended model with custodial symmetry
The Higgs sector can be extended to include extra Higgs multiplets in a manner that respects the custodial symmetry. A minimal model with Higgs triplets that gives ρ = 1 was discussed in reference [10], and studied in further detail in [14]. This model includes a real (Y=0) triplet, Ξ = (ξ + , ξ 0 , ξ − ), and a complex (Y=2) Higgs triplet, χ = (χ ++ , χ + , χ 0 ), in addition to the SM Higgs doublet, Φ = (φ + , φ 0 ). The v.e.v. of the neutral components can be choossen such that the gauge boson masses are given by: where the coefficients f H 0 i and g H 0 i are shown in Table 3. From this table we conclude that ρ H 0 1 = ρ H ′0 1 = 1, while ρ H 0 5 = 1/2. Thus, using our definition of the rho parameter, one can clearly distinguish a Higgs state of the type H 0 5 , which transforms non-trivially under the custodial symmetry, from the states H 0 1 and H ′ 0 1 , which are singlets under SU(2) c . However, it should be said that these states are not yet mass eigenstates.
While H 0 1 and H ′ 0 1 predict ρ H i = 1, their couplings with gauge bosons deviate from the SM prediction. Thus, in order to probe this sector of the model, one could compare the decay widths Γ(H 0 i → ZZ), or Γ(H 0 i → W W ), and using the expected precision on the Higgs measurement, determine that range of Table 1 Coefficients f H 0 i and g H 0 i for the Higgs-Gauge boson couplings.
parameters that could be excluded.
On the other hand, in terms of mass eigenstates the Higgs-Gauge boson couplings induce a ρ H i parameter, whose expression is given by: Thus, as anticipated in section 2, the choice v T 2 = v T 0 = v T , which makes ρ = 1, does not imply that ρ H i = 1. In fact, to get ρ H i = 1, for Higgs states that transforms as singlets under the custodial symmetry one would need all the Higgs interactions, including the ones appearing in the Higgs potential to respect the symmetry SU(2) c .

Higgs triplets from the Little Higgs models
A new approach was recently proposed to address the naturalness problem of the Higgs sector, dubbed the "little Higgs models", where the Higgs mass is protected from acquiring quadratic divergences by being promoted as a pseudo-Goldstone boson of a global symmetry [8]. The SM Higgs acquires mass via symmetry breaking at the EW scale (v). While the global symmetry is broken at high-energy scale Λ s . The important new feature of these models is that the Higgs remains light thanks to the global symmetry, which includes new fields that cancel the quadratic divergences. Furthermore, these extra Higgs fields exist as Goldstone boson multiplets from the global symmetry breaking.
A minimal model, called the "littlest Higgs", is based on a global symmetry SU(5) which is broken into SO(5) at the scale Λ s = 4πf , while the locally gauged subgroup is [SU(2) × U(1)] 2 , which in turn breaks into the EW gauge symmetry of the SM. This leaves 14 Goldstone bosons, including a real singlet and a real triplet, which become the longitudinal modes of the heavy gauge bosons, as well as a complex doublet and complex triplet, which acquire masses radiatively, of order v and f , respectively. Thus, the "littlest Higgs model", predicts the existence of several states with O (TeV) masses, which give place to violations of the custodial symmetry [15].
On the other hand, for the light Higgs state h, the model predicts the following Higgs-Gauge couplings, For the purpose of comparison with ρ, we expand ρ h in terms of r f and r t , which gives: and now: A ′ = (c 2 − s 2 ) 2 + 5 2 (c ′ 2 − s ′ 2 ) 2 , and B ′ = −6 √ 2s 0 . Therefore, since A = A ′ , B = B ′ one clearly has: ρ = ρ H . Thus, a measurement of the Higgs couplings at NLC will provide an independent test of the underlying symmetry of the Higgs sector.

Conclusions and discussion
In this letter we proposed a definition of the rho parameter based on the Higgs couplings with the gauge bosons, namely, , as a possible test of the custodial symmetry. We discuss the size of such violation in the context of general models with Higgs triplets, including the "Little Higgs" model recently proposed. We find that even for Higgs models that incorporate the custodial symmetry, to make ρ = 1, the Higgs couplings allow ρ H = 1. Furthermore, in models where ρ ≃ 1 we also obtain that the Higgs bosons could aquire values of ρ H significantly different from one. We find that ρ H could be tested at the level of few percent, given the expected Higgs tests that may be achieved at the planned next linear collider (NLC), where we will be entering into the era of precision measurements for the Higgs sector. Violations of the custodial symmetry could also be tested through the ratio of decay widths, R Γ = Γ(h→W W ) 2Γ(h→ZZ) , with similar precision.
In summary, given the possibility that an scalar particle could be detected in the near future, it will be important to verify whether this particle is indeed a type of Higgs boson, and the parameter ρ H could play a major role in this regard. This parameter meassures the transformation properties of the Higgs bosons under the custodial symmetry.