The Light Composite Higgs in Strong Extended Technicolor

This paper extends an earlier one describing the Higgs boson $H$ as a light composite scalar in a strong extended technicolor model of electroweak symmetry breaking. The Higgs mass $M_H$ is made much smaller than $\Lambda_{ETC}$ by tuning the ETC coupling very close to the critical value for electroweak symmetry breaking. The technicolor interaction, neglected in the earlier paper, is considered here. Its weakness relative to extended technicolor is essential to understanding the lightness of $H$ compared to the low-lying spin-one technihadrons. Technicolor cannot be completely ignored, but implementing technigluon exchange together with strong extended technicolor appears difficult. We propose a solution that turns out to leave the results of the earlier paper essentially unchanged. An argument is then presented that masses of the spin-one technifermion bound states, $\rho_H$ and $a_H$, are much larger than $M_H$ and, plausibly, controlled by technicolor. Assuming $M_{\rho_H}$ and $M_{a_H}$ are in the TeV-energy region, we identify $\rho_H$ and $a_H$ with the diboson excesses observed near $2\,{\rm TeV}$ by ATLAS and CMS in LHC Run 1 data, and we discuss their phenomenology for Runs 2 and 3.

partners of the top quark and weak bosons to cancel their quadratically divergent contributions to its mass. Rather, this divergence is removed by the condition that m t and m T are much less than Λ. Thus, there is no need to fine-tune partners' masses and couplings to explain why they haven't been seen in LHC experiments.
In Ref. [16] TC dynamics were not included. But, TC cannot be ignored. First, there must be an unbroken TC subgroup of ETC. If all its symmetries were spontaneously broken, ETC would be infrared free at energies below the ETC boson masses. It is unclear whether such a theory can be free in the ultraviolet [20]. Second, at the scale Λ T C < ∼ 1 TeV Λ ET C = Λ, the TC gauge coupling α T C becomes strong enough that it can break EW symmetry all by itself. So, this is a situation with two very different but nonetheless important energy scales. ETC is the dominant force in driving EWSB and making the Higgs boson light. But what sets the mass scale for the technihadrons, the bound states of technifermions? Are they bound by TC alone or, like the Higgs boson, by ETC, or by some cooperative combination? A major of purpose of this paper is to include the effects of TC on EWSB and to estimate the mass scale of the spin-one technihadrons. Because of their potential experimental importance, we need to whether they are much heavier than H and, if so, whether they are within reach of the LHC experiments.
That TC necessarily plays a minor role compared to ETC in EW symmetry breaking was not emphasized in Ref. [16], even though it was one of the two main approximations of that paper. The relative contributions that TC and ETC make in binding the spin-one technihadrons and generating their masses, and to the requirement that the Higgs boson is much lighter than they, is what brings this issue to the fore.
We now review the main results of Ref. [16]: the fermions and their ETC interaction, the fine-tuned gap equations for the fermions' masses, the Higgs mass, and the principal results of EWSB. Then we preview the rest of this paper.
The model of Ref. [16] involved the third-generation quarks and a single doublet of technifermions transforming under (SU (2)⊗U (1)) EW , ordinary color SU (3) C and technicolor SU (N T C ) as follows: Here, d T C denotes the d T C -dimensional TC representation of the technifermions, not necessarily the fundamental representation of dimension N T C . Light quarks and leptons and other technifermions were not dealt with, but they are not difficult to include. The ETC interaction inducing EWSB at energies below Λ was taken to be The SU (2) EW and color-SU (3) C and SU (N T C ) indices, i and a, b and α, β are summed over. This interaction is obtained by Fierzing ETC contact terms of left times right-handed currents. The color and TC indices appearing here do not correspond to exchange of massless color and TC gluons. The couplings G 1,2,3 are nominally positive and of O(1/Λ 2 ). 4 In the neglect of EW interactions, the model has an (SU (2) L ⊗ U (1) R ) q ⊗ (SU (2) L ⊗ U (1) R ) T flavor symmetry that is explicitly broken to SU (2) L ⊗ U (1) by the G 2 -term. If L ET C generates both t and U masses and G 2 = 0, this flavor symmetry is spontaneously broken to U (1) and just three Goldstone bosons appear. In fact, G 2 must not equal zero; if it were, there would be an extra triplet of Goldstone bosons. They acquire only very small EW masses [21] and, so, are excluded experimentally. With G 2 = 0, this model has exactly one Higgs boson. Its vev is v = 246 GeV, setting the scale for m t,U ; see Eqs. (9,11) below. The low-energy theory, below the technihadron masses, is just the standard model (plus suppressed higher-dimension operators), with SM couplings of the Higgs to fermions and gauge bosons -a feature in accord with all measurements so far. 5 The TC interaction was neglected in Ref. [16], and calculations were carried out in the Nambu-Jona-Lasinio (NJL) approximation of large N T C and N C . The gap equations for the hard masses m t and m U , assumed much less than Λ and renormalized at the scale Λ, are The independence of N C and d T C imply that (just multiply Eq. (3) by m U and Eq. (4) by m t ) Then, Eqs. (3)(4)(5) yield the condition: It was shown in Ref. [16] that m t and m U are comparable and, so, the three G i are comparable as well.
Requiring m t , m U Λ is this model's only fine tuning. Once Eq. (6) is enforced in the fermion-antifermion scattering amplitudes in the spin-zero channels, all other sensitivity to the cutoff Λ is logarithmic. The mass parameters m t , m U , M W , M H and Λ are not independent. In the large-N approximation, their magnitude is set by requiring Eq. (11) below, and the Higgs mass M H is then determined by m t , m U and N C , d T C .
The fermion-antifermion scattering amplitudes (involving t and/or U ) have a pole in the 0 + channel at squared c.m. energy This M H is the Higgs boson mass at scale Λ. A good approximation to the solution of Eq. (7) is Thus, M H is indeed of order m t , m U and all these masses are much less than Λ because the ETC couplings have been tuned to be very close to the critical point at which EWSB first occurs.
The fermion-antifermion scattering amplitudes in the charged and neutral pseudoscalar channels have Goldstone poles at p 2 = 0. These poles appear in the W and Z propagators, g −2 2 D W (p) and (g 2 1 + g 2 2 ) −1 D Z (p) with residues and The EW mass scale is introduced by setting The ρ-parameter, is just a few percent greater than one. Table 1 contains numerical results for the model obtained from a simple scheme described in Ref. [16].  Table 1: The fermion masses, Higgs boson mass, ρ-parameter, SU (2) ⊗ U (1)) EW couplings and the W , Z-pole masses calculated for ETC scales Λ = 20 and 500 TeV. The top mass is an input determined by renormalizing from its value of 173 GeV. The Higgs boson's vev v = √ 2m t /Γ t is determined as a check on the calculation of thett scattering amplitude in the scalar channel, where Γ 2 t (Λ)/2 is the residue of the Higgs pole. The calculation scheme used is described in Ref. [16].
In Sec. 2 we discuss the difficulty of adding an interaction involving dynamical TC-gluon exchange to the ETC contact interaction in Eq. (2) and propose an approximation that surmounts the problem for fermion-antifermion scattering in the spin-zero channel. The approximation is inspired by analyses of the effect of TC on the Schwinger-Dyson equation for the technifermion dynamical mass function, Σ(p) [23,24]. In Sec. 3 we take up the matter of estimating the masses of the lightest spin-one vector and axial vector bound states, analogs of ρ, ω and a 1 . We shall refer to them as ρ H , ω H and a H to emphasize their relation to the composite Higgs boson H. In this strong-ETC model, it is not obvious a priori whether their masses are of order Λ ET C , Λ T C or something else, though that is a question of obvious phenomenological importance. We present a calculation that suggests they are of O(Λ T C ). As in any strong interaction theory, a more precise estimate is technically difficult. Assuming they are within reach of LHC Run 2, their LHC phenomenology is discussed in Sec. 4. 6 There we review our recent proposal [25] that ρ H and a H are the source of the apparent diboson (V V and V H, where V = W, Z) resonances near 2 TeV observed by ATLAS and CMS in their Run 1 data [26,27,28,29,30,31] and we propose tests of of our hypothesis for Run 2.
There has been much previous work using the NJL mechanism [32,33] to describe the Higgs boson, including especially Refs. [22,34,35,36]. Topcolor led to the top-seesaw models of Dobrescu and Hill [37] and Chivukula, et al. [38] and, more recently, Refs. [39,40]. 7 Bar-Shalom and collaborators proposed a "hybrid model" with a dynamical Higgs-like scalar plus an elementary scalar to describe H [41,42]. They used an NJL Lagrangian with fourth generation quarks interacting via a topcolor interaction with scale Λ ∼ 1 TeV to generate the dynamical scalar. Apart from the use of the NJL bubble approximation, these models do not resemble ours, and the use of fourth generation quarks is reminiscent of the top-seesaw mechanism. Di Chiara, et al., proposed a model of H based on TC and ETC [43,44], using an ETC Lagrangian similar to Eq. (2). Their model bears no further resemblance to ours. They assume that ETC plays no role in EWSB. But, through a sequence of calculations, they argue that ETC lowers their Higgs boson's mass from O(1 TeV) to 125 GeV. Finally, the authors of Ref. [45] proposed an interesting variation on the Higgs boson as a PGB of the familiar SO(5) → SO(4) model. They used strong ETC-like contact interactions to drive this symmetry breakdown, and constructed a UV completion of this model.

Adding TC to Strong ETC
In Ref. [16], the Higgs and Goldstone bosons were seen as poles in the fermion-antifermion scattering amplitudes calculated in the large-N , weak-TC limit. Figure 1 shows the first few terms intt →tt in the J P = 0 + channel. The four-fermion vertices are the appropriate terms in From Eq. (5), G 2 2 = G 1 G 3 , and this makes the scattering amplitudes geometric sums. The Higgs pole-mass condition Eq. (7) follows once Eq. (6) is imposed to eliminate the Λ 2divergence in the 0 + scattering amplitude.
In the large-N approximation, the inclusion of TC-gluon exchange between the technifermions is accomplished by using the kernel K 0 + in Fig. 2. The TC-gluon term of this kernel is the familiar ladder approximation. The difficulty with it is how to deal with the momentum carried by the TC-gluon and, worse, whether the sum is a geometric series for which something like Eq. (6) eliminates the Λ 2 -divergence.
The only situation we know that the ETC+TC kernel in Fig. 2 has been used successfully is the studies of the dynamical mass function Σ(p 2 ) in the technifermion propagator S −1 (p) = Figure 2: The kernel for scattering ofŪ U →tt andŪ U →Ū U including ETC contact terms and one-TC-gluon exchange. [23,24]. Remembering that the ETC boson mass Λ is a physical cutoff of momentum integrals whose integrands are strongly damped above Λ, a good approximation to the Schwinger-Dyson gap equation for Σ(p 2 ) is (for zero bare mass and Euclidean momentum p < ∼ Λ) We consider a simplified model with just G 3 contributing to Σ. Then λ = G 3 d T C Λ 2 /8π 2 ; α T C is the running TC gauge coupling; α c is the critical value of α T C for spontaneous chiral symmetry breaking in a pure-technicolor theory [46]; its value in the ladder approximation is π/3C 2 (d T C ); finally, M 2 = max(k 2 , p 2 ). In a pure ETC theory, Σ(0) = 0 for λ < 1, there is a (presumed) second-order phase transition at λ = 1, and Σ(0) rises rapidly to O(Λ) just above the transition. In a pure asymptotically-free TC theory, α T C reaches α c at a scale Λ c , Σ(0) Σ(Λ c ) = O(Λ c ), and Σ(p 2 ) falls off approximately as Λ 3 c /p 2 when α T C becomes weak [47]. Ref. [23] studied Eq. (14) for constant α T C . For α T C < α c , the behavior of Σ(0) was as in a pure-ETC theory except that, for α T C < α c , the phase transition occurred at Takeuchi studied the gap equation for a running α T C governed by the one-loop beta function β(α T C ) = −b 1 α 2 T C , with b 1 > 0 [24]. So long as Λ c Λ (as we expect), he found that Σ(0) = O(Λ c ) for λ < λ α T C . 8 Here, α T C α T C (Λ). At this critical value of λ, there is a smooth but rapid transition up to Σ(0) = Λ/few. The transition is more abrupt for small α T C (Λ)/α c so that λ α T C ∼ 1. The reason that α T C (Λ) is the controlling coupling for λ α T C is that, for this β-function and Λ c Λ, α T C α T C (Λ) α c and it is slowly running for most of the momentum range in the gap equation integral. We have verified Takeuchi's results for a more realistic walking-TC β-function, one with an infrared fixed point [48]. We also studied the momentum dependence of Σ(p 2 ). For λ < λ α T C (Λ) , we found that Σ is small and falls off approximately as 1/p 2 for Λ c < ∼ p < ∼ Λ, as for a pure-TC dynamical mass. At the critical λ, Σ(p) rises rapidly to O(Λ/10) and then remains nearly constant in p, as for a hard mass. This is an important result for us. In the weak dynamical-TC case needed for a light composite Higgs with M 2 H M 2 ρ H , this behavior of Σ is nearly what we get in the complete neglect of TC: it is much smaller than Λ below λ α T C and rises abruptly above, almost to O(Λ).
(Had α T C (Λ) been large, the transition from small to large Σ would have been gradual and there could be no large separation between H and ρ H masses.) The critical λ α T C is smaller than one because, to a good approximation, TC produces an interaction in the spin-zero channels of the same form and sign as the G 3 -term in L ET C . Thus, a smaller value of G 3 , i.e., λ, is needed to trigger the phase transition. The critical value of the sum of the two interaction strengths is still fixed by a condition like Eq. (6); i.e., the effective G 3 in L ET C is essentially unchanged. Since L ET C is the interaction determining the Higgs and Goldstone poles and their couplings to fermions in the large-N limit, the results reviewed in Sec. 1 are also unchanged.
To see this in detail, we use the fact that the TC coupling involved in the EW phase transition is approximately α T C (Λ). The relevant TC interaction then involves exchange of a technigluon with Euclidean momentum transfer ≈ −Λ 2 , where T = (U, D) is the technifermion doublet, t A are the TC generators in the representation d T C , and other indices are suppressed. A factor of 3/4 has been introduced into L T C to compensate for using Landau instead of Feynman gauge. 9 Use Then, in the large-N T C limit, L T C Fierz-transforms into Adding this to the G 3 term in Eq. (13), the effective λ is 9 Although this term is isospin-symmetric, its strength is is not sufficient to produce m D = 0.
For the critical value λ ef f = 1, λ = 1−α T C (Λ)/4α c . This is less than 20% higher than λ α T C (Λ) for α T C (Λ)/α c < 0.5, which is the range that Takeuchi considered. Thus, our approximation for L T C captures well the main effect of adding TC to ETC in the gap equation and spin-zero scattering amplitudes.

Masses of the Spin-One Technihadrons
As we stressed at the outset, the challenge for a TC-based composite Higgs model is to explain convincingly why H is much lighter than the lowest-lying spin-one technihadrons. In our model, there is the additional matter that there are two scales, Λ T C and Λ ET C = Λ. Which of these controls M ρ H ? If it is just Λ, are these masses of that order or, as for the Higgs, very much lighter? In this section we present an argument suggesting they are at least as heavy as Λ T C and therefore well above the Higgs mass. We assume that M ρ H ,... are due entirely to ETC and find that this results in unphysical or implausible masses for these states.
We start by considering a simplified model with the doublet T = (U, D) as the only fermions. Its SU (2) L ⊗ U (1) R invariant ETC interaction is This interaction produces nonzero m U , but not m D , if While L T can generate a light Higgs boson and three Goldstone bosons, it has the wrong chiral structure to generate masses for the spin-one technihadrons in the large-N T C limit. Therefore, we expand it to include terms capable of this. We assume that ETC generates V V and AA contact interactions which add to L T . For simplicity, we can take them to be V -A symmetric and flavor-U (2) invariant without affecting our argument: where τ a are Pauli matrices acting in the (U, D)-flavor space. The parameter δ allows freedom in the choice of the ETC coupling of the V V and AA terms. We write (with a unit ρ H coupling to the U (2) current) where p µ µ (p) = 0. Then, to leading order in N T C , the technivector masses are given by the poles in the ρ A → ρ B amplitude where In this model, with m D = 0, the fermion mass matrix is The momentum integral (25) is cutoff at Λ, just as the ones in the spin-zero channels were, giving The p µ p ν terms in these integrals do not contribute to T AB . Then, in the leading-log approximation, the ρ H -ω H mixing term is negligible and the poles in T AB are at Using the gap Eq. (21), the poles in T 11,22 and T 33,00 are atp 2 satisfyinḡ This is unphysical unless 0 < δ < 1 for ρ ± H and 0 < δ < 1−m 2 U /Λ 2 ln(Λ 2 /m 2 U ) for ρ 0 H and ω H . For δ at its upper limit,p 2 0, much less than Λ 2 T C . We believe this is unreasonable because no symmetry is responsible for such light masses. Our calculations break down beyond the Λ-cutoff, so M ρ H > ∼ Λ is an unreliable result. Over a large part of the physical range of δ, We cannot exclude this, but it seems implausible for a mass generated solely by strong ETC. A more believable result is that TC generates the ρ H and ω H masses and that they are of order Λ T C , the scale at which α T C becomes large and TC interactions confine. In these latter two cases, the ρ H , ω H masses are significantly larger than the Higgs mass, and that is a necessary condition for the viability of this type of model.
Let us extend this argument to the full G 1 -G 2 -G 3 model. There are two obvious possibilities for the V V + AA terms: we could add just the δG 3 interaction as we did in Eq. (22) or we could add similar terms with the appropriate coefficient, − 1 4 δG i , to all three interactions. The gap-equation condition is now given by Eq. (6). In the first case, the poles are always at , which we believe is implausible. The second case is similar to the pure-G 3 model discussed above. Finally, similar results and conclusions hold for ETC-generated masses of the axial vectors a H ; they are either unreasonably small or much larger than Λ T C but much smaller than Λ. The conclusion we draw is that an ETC origin of the technivector masses is implausible and, therefore, that they arise from the confined TC interactions and are of O(1 TeV).

Phenomenology of ρ H and a H
Preliminary remarks about the phenomenology of the model's technifermion bound states were made in Ref. [16]. They included, in particular, the expectations that: (1) the most accessible low-lying states, in addition to the Higgs H and longitudinal weak bosons W L and Z L (which really are bound by the ETC interaction, Eq. (2)), are the spin-one, techniisospin one and zero ρ and ω-like composites; (2) their masses are 1/2-2 TeV and they are produced at the LHC via the Drell-Yan process; (3) their principal decay modes would be to W + L W − L , W ± L , Z L or W + L W − L Z L and to W L H, Z L H. In this section we refine -and correctthese expectations, presenting some specific predictions of production and decay rates. We concentrate on the I = 1 vectors and axial vectors, ρ H and a H , which have simple two-body decay modes and Drell-Yan-size production rates.
The ρ H and a H may be described by a hidden local symmetry (HLS) Lagrangian [49] as the gauge bosons of an SU (2) L × SU (2) R chiral isospin symmetry. The vectorial nature and parity invariance of TC interactions suggest equal gauge couplings g L = g R ≡ g ρ H [50,51,52,53]. This coupling is analogous to g ρππ and is expected to be large, say, g ρ H 3-5. Furthermore, because the ETC interaction is tuned to be close to the EW phase transition, ρ H and a H are nearly parity-doubled isotriplets with M ρ H ∼ = M a H . The dimensionthree and four interactions of ρ H , a H with EW gauge bosons respect this parity up to EW corrections of O(g, g ). Furthermore, the parity makes the ρ H -a H contribution to the S-parameter [54,55,56,57,58] small [50,51]. The principal ρ H , a H decay modes are to lighter states withT T content, namely, the longitudinally-polarized V L = W L , Z L and the Higgs boson H. 10 The two-body decays allowed by parity and isospin are The fact that V L , H also contain third generation quarks may deplete somewhat the ρ H , a H couplings to them. This does not alter the major decay modes in Eqs. (34,35) nor affect their production rates at the LHC. In the absence of significant depletion, the relevant ρ H , a H couplings are induced by their O(gM 2 ρ H /g ρ H ) mixing with the EW gauge bosons [51,25]; they are    Table 2. No K-factor has been applied to the cross sections; from Ref. [25].
We proposed in Ref. [25] that these excesses are due to production of the ρ H and a H modes in Eqs. (34,35). The decay rates and cross sections for M ρ H = 1.8-2.0 TeV, M a H = 1.05M ρ H and g ρ H = 1.9 TeV/2v = 3.862 (i.e., M ρ H 1 2 g ρ H (4v)) are given in Tables 2 and 3 Our proposal leads to several predictions and recommendations for Run 2 data analyses: 1) The only V V diboson resonances come from ρ H production. Isospin invariance implies equal decay rates to W ± Z and W + W − , but no ZZ-signal. 13 It is therefore desirable that the separation of p T ∼ 1 TeV, nonleptonically-decaying W and Z-bosons be sharpened, and the overlap between nonleptonic W W , W Z and ZZ selections be minimized. Failing that, semileptonic and all-leptonic V V events can determine the content of the diboson resonances. This may be feasible in Run 2 with its planned luminosity of 300 fb −1 .
2) The V H resonances in our model are due to a H , not ρ H , production, but they are expected to be nearly degenerate with the V V resonances. They may be distinguished by looking for forward jets. At √ s = 13 TeV, about 1/3 of ρ H → V V production is due to VBF, which is accompanied by forward jets with a rapidity gap. Because a H → V V is so strongly suppressed, production of a H → V H is almost entirely due to DY, with no forward jets.
3) It is difficult to explain DY+VBF rates at 8 TeV of more than a few fb unless rather extreme parameter choices are made for DY amplitudes [25]. Therefore, estimates for Run 1 diboson signal rates of > ∼ 10 fb −1 in Ref.
[59] must be due to an up-fluctuation.

Summary and Plans
In this paper we developed further our strong-ETC model of electroweak symmetry breaking. We stressed that weak-TC -meaning a minimal role for TC in EWSB -is a necessary ingredient of our model if it is to explain the large mass gap between the Higgs boson H(125) and technihadrons. Our two main purposes were to include the effect of weak TC on EWSB and to establish that the model provides a plausible explanation for the lightness of H relative to the technihadrons ρ H and a H . For the first, we used Takeuchi's analysis [24] to show that weak TC modifies only slightly the analysis of Ref. [16] in which TC was ignored altogether. Specifically, in the J P = 0 ± channels of fermion-fermion scattering, the effective TC interaction has the same form and sign as the corresponding ETC interaction, thus having the effect of requiring only a slightly smaller ETC coupling to trigger EWSB with a light Higgs and three Goldstone bosons (H, V L ). Our argument that M ρ H M H was an indirect one: We showed that an ETC interaction designed to generate a ρ H pole in technifermion scattering amplitudes leads to values of M ρ H which are implausibly small or much greater than Λ T C but much less than Λ ET C . We have not found an argument more direct than this. Nor did we address the question of why the Λ T C = O(1 TeV) if TC has little to do with EWSB. Of course, we expect that it is if the diboson excesses near 2 TeV turn out to be confirmed in LHC Run 2. But that's not an answer. If the dibosons are not confirmed, the question is moot. That would be unfortunate for our model, because it does not appear to have another readily accessible "smoking-gun" prediction.
We provided in Sec. 4 our expectations for the ρ H and a H widths and production cross sections (with details in Ref. [25]) and tests that can help distinguish our model for the diboson resonances from others that have been proposed (see Ref. [59]). If they are confirmed, there will be plenty for the experimentalists to do to reveal the nature and the interactions responsible for the diboson resonances.
For our model, the main task remaining is to carry out a renormalization group analysis for the Higgs and heavy fermion masses. If this analysis can produce results in agreement with experiment, particularly M H below m t , it will give strong support to our approach to understanding the Higgs as a light composite state. [60] CMS Collaboration, "Search for massive resonances decaying into pairs of boosted W and Z bosons at √ s = 13 TeV,".
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