Electroweak Chiral Lagrangian for a Hypercharge-universal Topcolor Model

Electroweak chiral Lagrangian for a hypercharge-universal topcolor model is investigated. We find that the assignments of universal hypercharge improve the results obtained previously from K.Lane's prototype natural TC2 model by allowing a larger Z' mass resulting in a very small T parameter and the S parameter is still around the order of +1

Topcolor-assisted technicolor (TC2) is a class of new physics models which combines technicolor and topcolor together to realize the electroweak symmetry breaking dynamically.
In these theories, a technicolor condensate provides the masses to the weak vector bosons and an extended technicolor (ETC) sector gives masses to the light quarks and leptons, and a bottom-quark-sized mass to the top. The majority of the top-quark mass is due to the formation of a top-quark condensate through the dynamics of an extended color gauge sector. The typical gauge group of the TC2 models is , in which the extended color and hypercharge groups SU (3) spontaneously break to their diagonal subgroup SU(3) C ⊗ U(1) Y at a few TeVs and the remaining electroweak groups SU(2) L ⊗ U(1) Y spontaneously break to their electromagnetic subgroup U(1) em at the electroweak scale due to a combination of a top-quark condensate and a technifermion condensate. In the original TC2 model [1,2], the extended hyper- In this paper, we are mainly interested in the effects from flavor-universal hypercharge sector, to reduce the computations and to be convenient for comparison with flavor-nonuniversal hypercharge model, we base our calculations on the K.Lane's prototype natural TC2 model [2] discussed in Ref.
[5], but change its hypercharge assignments to that given in Ref. [3]. The gauge charges are shown as Table I.
In later numerical computations, technicolor group representation will be taken to be The action of the symmetry breaking sector is with L techniquark , L breaking and L 4T being the same as those in Ref.
[5] and the modified techinquark Lagrangian with flavor-universal hypercharge is Rotating hypercharge gauge fields B 1µ and B 2µ as The techinquark Lagrangian (3) is then reduced to where all three doublets techniquarks are arranged in one by six matrix ψ = As done in Ref.
[5], the EWCL for present model is where U(x) is a dimensionless unitary unimodular matrix field in EWCL, and Dµ(U) denotes the normalized functional integration measure on U. The normalization factor N [W a µ , B µ ] is determined through the requirement that when the technicolor interactions are switched The following computation procedure is exactly the same as those given in Ref.
[5], in which we integrated out the technigluons, the techniquarks and the colorons. We abbreviate the detailed process and only write down the resulted action, with where M 0 is the bare mass of Z ′ boson from spontaneously breaking of SU (3)  Further in which W a ξµ and B ξµ are rotated electroweak gauge fields given in Eq. (26) and (27) in Ref.
[5] which absorb Goldstone field U into the definition of gauge fields.
We can further decompose (10) into . We find that the Z ′ independent part S Z ′ [U, W a µ , B µ , 0] is just the same as that given in Ref.
With similar procedure of Ref.
[5] to integrate out the Z ′ field, we find that S eff [U, W a µ , B µ ] defined in (8) has exactly the standard structure of EWCL given by Ref.
[6], from which we can read out coefficients up to order of p 4 as follows, where L 1D i for i = 1, 3, 9, 10 are EWCL coefficients for one doublet technicolor model dis- The features of these results which are the same as those in K.Lane's model are: 1. The contributions to the p 4 order coefficients are divided into two parts: the three doublets technicolor model contribution and the Z ′ contribution.
2. All corrections from the Z ′ particle are at least proportional to β 1 which vanish if the mixing disappear by θ ′ = 0.
[6] is always positive in present model.
Since α 1 and α 2 depend on γ which from (21) further rely on an extra parameter K. We can combine (26),(15) and (16) together to fix K, Once K is fixed, with the help of (25), we can determine the ratio of infrared cutoff κ and ultraviolet cutoff Λ, in Fig.1, we draw the κ/Λ as functions of T and M Z ′ , we find natural criteria Λ > κ offers stringent constraints on the allowed region for T and M Z ′ that present theory prefers small T parameter. The upper bound for Z ′ mass increases as the value of T decrease, for example, upper bound is below 1TeV for T being order of 10 −3 and 2-3TeV for T being order of 10 −5 . In Fig.2, we draw Z ′ mass as a function of T parameter and the gray region is the forbidden zone where κ ≥ Λ. Not like K.Lane's model discussed in Ref.
[5] where we have the upper bound of Z ′ mass 400GeV, now this upper bound is pushed higher as long as we have a very small T parameter. Considering that Ref. [3] already gives lower bound of M Z ′ = 2.08TeV, from Fig.2 we find it corresponds to T < 7.09 × 10 −5 .
With this constraints on M Z ′ , in Fig.3 we further draw the S parameter in terms of T and M Z ′ . From this graph, we find that the S parameter in the region of T < 7.09 × 10 −5 and M Z ′ > 2TeV is still at order of +1 which implies present model is still not fully matching with the experiment data. Compared to previous result for K.Lane's natural TC2 model with nonuniversal hypercharge assignments, we find that the value of the S parameter does decrease due to the universal hypercharge. For example, S ≈ 1.1 at T = 10 −2 for K.Lane's model, while S ≈ 0 at T = 10 −2 for present model, this is compatible with result obtained in Ref. [3], but for more smaller T parameter, S increases and finally for M Z ′ at 2-3TeV, S is still at order of +1. Finally for completion of our discussion, we depict all nonzero coefficients α i . Fig.4 is the graph for α 1 and α 2 , Fig.5 is for α 3 , α 4 and α 7 , Fig.6 is for α 5 , α 6 , α 9 and α 8 , Fig.7 is for α 10 . In all these diagrams, we find that the curves are not sensitive to M Z ′ when M Z ′ > 1 − 2TeV, therefore we do not label the M Z ′ on the graph. For   Fig.5, Fig.6 and Fig.7, the T axis starts from 10 −3 instead of 10 −6 , since below T = 10 −3 , all curves approach to zero.
To summarize, we apply the formulation developed in Ref.