Ultraviolet Complete Technicolor and Higgs Physics at LHC

We construct a Technicolor model which provides masses for the electroweak gauge bosons and for all the Standard Model matter fields. Starting from an ultraviolet complete supersymmetric technicolor, we propose a scenario where all elementary scalars, gauginos, and higgsinos are decoupled at an energy scale substantially higher than the electroweak scale, therefore avoiding the little hierarchy problem of the minimal supersymmetric standard model. The resulting low energy theory has an SU(3) global symmetry whose breaking to SO(3) leads to electroweak symmetry breaking. We study in detail the phenomenology of this theory and demonstrate that it reproduces the present LHC data at the same level of precision as the Standard Model itself.


Introduction
The data collected at the LHC experiments during the 7 and 8 TeV runs, with the epochal discovery of the Higgs boson [1,2] and the measurement of its couplings [3,4], seem to have provided the experimental verification of the Standard Model (SM) in its entirety below a TeV. Because of this, new physics coupled with the Electroweak (EW) SM currents must have a typical scale of the order of a TeV or higher. One possibility is that the new physics scale is much above the terascale, and then naturality as a model building paradigm should be reinterpreted [5,6]. If on the other hand naturality is realized in nature, then the new physics scale should be near the terascale and the present LHC data would provide hints of a new spectrum awaiting discovery in the future runs at the LHC. In this paper we investigate a model framework falling into the latter category.
In Technicolor (TC) theories [7,8] the new physics scale is naturally of the order of a TeV. The EW symmetry is broken by a new strong interaction which generates a fermion condensate, in a way analogous to QCD. The absence of light elementary scalars in TC automatically solves the SM fine-tuning problem. The mass of the lightest composite scalar is usually expected to be of O(TeV), well above the measured 126 GeV value. A light scalar can arise as consequence of approximate global symmetries, chiral symmetry [9][10][11] or scale invariance [12][13][14][15][16]. However, only recently it has been realized that also with simple QCD-like TC dynamics the scalar particle can become light because of loop corrections originating from extended sectors, which are always required in TC models to account for the generation of fermion masses [17][18][19].
The observed mass pattern of matter fields is generated in TC by coupling the technifermions with the SM fermions either via heavy gauge bosons of an Extended Technicolor (ETC) sector [20][21][22][23] or through scalar fields in bosonic technicolor (BTC) [24][25][26][27][28]. In case the scalar mediators are elementary, one can control fine-tuning at scales above the elementary scalar masses by introducing supersymmetry (SUSY) [29,30]. Combining SUSY with TC is therefore appealing because of two general features • The fundamental Higgs fields do not participate in electroweak symmetry breaking, but serve as natural messengers between the symmetry breaking sector and SM matter fields. • The strong TC dynamics responsible for the EW symmetry breaking alleviates the little hierarchy problem of supersymmetric scenarios.
Recently supersymmetric TC models based on N = 4 super Yang-Mills were considered in [31,32] where the low energy effective theory at scales below the TC scale, TC , was taken to be Minimal Walking Technicolor (MWT) [14,15,33]. In this paper we study a simpler but still, as we shall show, phenomenologically viable possibility within the UV complete theory of [31,32]: we assume a larger portion of the supersymmetric spectrum to be heavy which still allows the resulting low energy effective theory, featuring an SU(3) global symmetry in the TC sector, to break correctly EW symmetry. Due to its global symmetry and the near conformal TC coupling we call this SU(3) Walking Technicolor (3WT).
The paper is structured as follow: In Section 2 we review the particle content and renormalizable Lagrangian of MSCT, which represents the elementary description of 3WT. Next we integrate out the heavy mass eigenstates of MSCT and derive the effective Lagrangian at scales below the SUSY breaking scale, m SUSY , in Section 3. In the subsequent Section 4 we write the effective Lagrangian at scales below TC , where the TC interaction becomes strong and is assumed to bind technifermions and technigluons within composite states. We continue in Sec- Table 1 Non-MSSM superfield content of MSCT. Here Adj and denote the adjoint and fundamental representations, respectively. None of the fields above is charged under SU(3) c .
tion 5 by working out the mass eigenstates, light Higgs couplings, and EW precision parameters of the model. Then, in Section 6, we test the 3WT viability by scanning the parameter space for data points satisfying the direct search limits on new particles and then performing a goodness of fit analysis of Higgs physics data at LHC with the viable scanned data points. The result of this analysis, which is that 3WT fits the current experimental data at a goodness level comparable to that of the SM, is the main result of the paper. Finally, we offer our conclusions in Section 7.

An UV complete technicolor model
The UV complete supersymmetric theory which provides our starting point is the same which has been introduced in [31,32] and called Minimal Supersymmetric Conformal Technicolor (MSCT). The gauge symmetry group of MSCT extends the SM one by the TC gauge group, SU(2) TC . The TC vector superfields can be rearranged with the chiral superfields, containing the technifermions which transform under the adjoint representation of SU(2) TC , in N = 4 superfields. The MSCT Lagrangian can hence be expressed in compact form as that of the Minimal Supersymmetric Standard Model (MSSM) extended by an N = 4 Super Yang Mills (4SYM) sector, which contains the TC sector. We furthermore add to the superpotential a fourth lepton superfamily sector, in which the fermion components are needed to cancel the Witten topological anomaly generated by the odd number of left-handed technifermions. The non-MSSM superfields introduced in MSCT and their quantum numbers are summarized in Table 1. Analogously the MSCT superpotential can be expressed in compact form in terms of the MSSM superpotential and its extension where P MSSM is the MSSM superpotential, and P TC is expressed by with H u (H d ) denoting the Y = +1/2 (−1/2) Higgs superfield. The dot (·) indicates a contraction between the SU(2) L doublets with the antisymmetric two-index Levi-Civita tensor .
To the potential obtained from Eq. (1) we add the soft SUSY breaking terms of the MSSM as well as those corresponding to P TC , with the latter expressed by: where we write a hat on top of the scalar component of the chiral supermultiplets. The model defined by Eqs. (1), (2) and (3) constitutes the fundamental description of the theory we study in this paper. The relevant scales of the model are the SUSY breaking scale, m SUSY , and the EW scale that we identify with the low-energy strongly coupled regime of the TC theory TC ∼ 4πv w , which for v w = 246 GeV implies TC ∼ 3 TeV. We will assume here that the two scales satisfy With this ordering the EW symmetry is broken dynamically. Furthermore, we assume the mass spectrum to feature roughly the following hierarchy: 1) All SUSY breaking masses as well as the μ parameter are of O(m SUSY ), therefore all the superpartners as well as the elementary Higgs scalars have masses of the same order. 2) All the lightest composite states acquire masses, which are at most of the order of TC .
In the following section we proceed to derive the effective Lagrangian describing the physics below the SUSY breaking scale by integrating out the heavy states.

Mesoscopic Lagrangian
After we integrate out all particles with mass greater than m SUSY , the only technifermions left are the fermionic components of the EW doublet superfield L and EW singlet 3 in Table 1: To derive the effective Lagrangian, valid between the scales TC and m SUSY , one first writes down the Higgs Yukawa sector in MSCT: where i = 1, 2, 3 is the flavor index and it is summed over. The matrices Y u , Y d , and Y l are diagonal, and the CKM matrix V is contained in the definitions of the vectors Given that the potential of the MSSM Higgs fields is Similarly, the last term originates in an analogous way from the potential and the Yukawa sector of 4SYM. In Eq. (9) we have defined where σ 2 is the second Pauli matrix, and the indices i and j denote SU(3) flavor; the first letters of the alphabet are reserved for the adjoint SU(2) technicolor indices, while the Greek indices label the spin component, and the TC indices, running from 1 to 3, are written explicitly only in the last term. We suppress summed spin indices as long as it can be done consistently. Finally, we have defined In the rest of this paper we use abbreviations s θ ≡ sin θ , c θ ≡ cos θ and t θ ≡ tan θ . The four-fermion interaction terms in Eq. (9) are relevant because they eventually give mass to the SM fermions once the technifermions condense. The first line in Eq. (9) derives from decoupling the Higgs scalars, and breaks the global SU(3) symmetry, while the last term in Eq. (9), stemming from the 4SYM sector, respects the global SU(3) symmetry of the pure TC sector.
At energy scales below TC the TC interaction becomes strong, and physical states charged under TC get bound in composite states with zero TC charge. In the next section therefore we derive the effective Lagrangian involving such states.

Effective Lagrangian at the electroweak scale
Similarly to QCD, in TC a tower of composite states is predicted to arise at low energies. At scales below TC the new physics degrees of freedom are the composite states associated with the strong TC interaction, and the form of the effective Lagrangian is constrained to satisfy the approximate global symmetries of the fundamental Lagrangian. In the following we derive the effective Lagrangian introducing first the composite scalars and then the composite vectors.

Technicolor scalar sector
The composite scalar matrix field M, singlet under SU(2) TC , has minimal particle content given by the techniquark bilinears: The field M transforms under the full SU(3) group according to The effective linearly transforming SU(3) invariant Lagrangian reads: Table 2 Transformation properties of the component fields of the matrix M under SU(2) L × U(1) Y . The complex scalars are grouped, based on their transformation properties under SU(2) L , into one triplet, one doublet, and one singlet.
where the covariant derivative is given by In the above equation λ a are the Gell-Mann matrices, t ξ = tan ξ with ξ the EW mixing angle, W μ and B μ are the SM EW gauge fields, and The most general SU(3) preserving effective potential, including operators up to dimension four, is 2 which breaks SU(3) spontaneously to SO(3) for positive m 2 , as we show explicitly in Appendix A. The TC gauge interaction is actually invariant under U(3) ≡ SU(3) × U(1) A , rather than just SU(3). However the U(1) A axial symmetry is anomalous, and is therefore broken at the quantum level. This corresponds to the det M term in Eq. (17). The components of the matrix M ∼ η T η can be described in terms of the transformation properties of the composite states under SU(2) L × U(1) Y . This notation is introduced in Table 2.
Using this notation, the matrix M is written in terms of complex scalars as This notation is suitable to study the vacuum, since the flavor extension sector breaks the global symmetry of the potential from SU(3) down to the EW gauge group SU(2) × U(1). Next, we discuss how to consistently introduce also the composite vector fields.

Vector sector
A minimal set of composite vector fields transforming homogeneously under SU(3) can be written, in terms of Gell-Mann matrices λ a , as which transform under SU(3) according to The elementary particle content of A μ is expressed by the equivalence where the components of in SU(3) space are Dirac spinors, with the superscript c on the last entry denoting the charge conjugation. The vector and axial-vector charge eigenstates and their elementary particle content are given in Appendix B.
The effective Lagrangian including composite vector fields, A μ , has already been derived in [33] for a theory with SU(4) global symmetry in the TC sector by applying the hidden local symmetry principle [34,35]. Those results can be straightforwardly used for SU(3) symmetric TC by defining the corresponding vector field and the field strength tensor: with G μ defined in Eq. (15). The vector field C μ has the same transformation law as A μ : The kinetic and mass terms for the vector fields can then be written as while the scalar-vector field interaction terms up to dimension four operators read with constants r i ∼ O (1). A few remarks are in order: First, higher dimensional operators are suppressed by powers of TC , and are therefore subleading. Second, terms proportional to y U , which explicitly break SU(3) global symmetry, are small compared to those proportional to g 2 TC and can therefore be neglected at leading order. Third, to simplify the phenomenological analysis of 3WT, presented in the next section, we neglect also a covariant derivative coupling term (see [33] for its precise definition). 3 Finally, in the next subsection, we determine the effective Lagrangian terms of the flavor extension of 3WT below scale TC and then summarize the 3WT full Lagrangian.

Flavor extension sector
The four-fermion theory, Eq. (9), is given just below the SUSY breaking scale and the techniquark condensate needs to be evolved down to the EW scale. This is achieved by multiplying the techniquark Yukawa coupling y U , renormalized at the SUSY breaking scale, with the dimensionless factor written under the assumption that the anomalous dimension γ of the techniquark mass operator is constant. Note that in the following we neglect the contribution of the last term in Eq. (9) because that term respects the global SU(3) symmetry, and therefore its effects should already be parametrized by the quartic couplings in the TC effective Lagrangian, Eq. (17). The masses of the SM fermions and the fourth family leptons arise from the terms on the first line of Eq. (9): more specifically those masses are generated by the following four-fermion operator with upon condensation of the techniquarks. Under SU(3) global symmetry the spurion K transforms as K → u * Ku † . The four-techniquark term on the other hand is where α and β are spin indices. For this term to be invariant under SU(3), the spurion K must transform as K ij kl → u im u jn u * ko u * lp K mnop , with u ∈ SU(3). To estimate the effects of renormalization, we simply assume factorization, leading to a multiplicative factor of ω 2 .
At the lowest order in the spurions, the SU(3) breaking effective Lagrangian, obtained from Eqs. (28) and (29) is: where we introduced factors of TC to define the dimensionless coefficients c i , which parametrize the couplings of the effective Lagrangian in terms of those of the underlying theory. We estimate these coefficients using dimensional analysis [36][37][38] and find To summarize, the 3WT effective Lagrangian is given by where L k contains all the covariant kinetic terms for the composite scalar and vector fields given respectively in Eqs. (14) and (24). The terms included in L M−V are given by Eq. (25), while the Yukawa interactions and the potential are defined by with TC potential V M given by Eq. (17). Having at hand the full Lagrangian for the model, we proceed to work out its phenomenological consequences in the next section.

Phenomenology
In this section we work out the mass eigenstates arising at low energy in 3WT, the couplings of the light composite Higgs boson to SM mass eigenstates, and the EW precision parameters S and T .

EW symmetry breaking & mass eigenstates
The most general ground state which breaks the EW gauge group down to the electromagnetic U(1) Q can be parametrized by the following form of the vacuum expectation value (vev) of the matrix field M Minimizing the scalar potential given in Eqs. (17) and (30) leads to the following relations between the parameters: A sufficient condition for the potential to be bounded from below is λ, λ > 0. In the limit v σ = √ 2v we obtain m = 0, consistently with the pure TC result in Appendix A. We notice that the ground state changes because of the four-technifermion interaction in Eq. (29). An analogous change of ground state occurs in ETC theories where some of the chiral symmetries of the pure TC theory are broken by extended gauge interactions. In our model setup the effective four fermion interactions at low energy arise from attractive Yukawa couplings which is different from the usual ETC scenario where the underlying gauge interactions can be either repulsive or attractive.
The mass spectrum of 3WT, corresponding to the vev in Eq. (34), includes two neutral scalars, h 0 and H 0 , as well as one neutral pseudoscalar, 0 , one singly charged and two doubly charged scalars, H ± , h ±± , and H ±± , respectively, with the corresponding squared mass matrices given in Appendix C. We introduce for later use the mixing angle ϕ defined by Assuming that only the third generation SM fermions have non-negligible Yukawa couplings, it follows from Eq. (30) that the masses of the upper component u and the lower component d of a generic SM fermion EW doublet are given, respectively, by Using the previous equation, the fact that both members of the fourth fermion doublet, E and N , have to be heavier than about 100 GeV, and requiring the Yukawa couplings to be perturbative, we deduce that θ cannot be close to either 0 or π/2. To simplify the study of 3WT from here on we take m s = m SUSY , which from Eq. (11) is equivalent to imposing: From the fact that neither t θ nor its inverse are large, it follows that our choice does not introduce any large hierarchy between b and μ 2 , and both can be taken of O(m 2 s ). This is consistent with our assumption that the higgsinos are heavy and decoupled from low energy phenomenology.
Finally, the composite axial-vector and vector resonances mix with the SM gauge bosons, while the doubly charged baryon technivector does not. The resulting physical massive vector with the corresponding squared mass matrices given in Appendix C. The masses of W ± and Z in the limit of negligible mixing ( = 0) read: where t ρ = √ 2v /v σ , and the EW scale is given by In the next subsection we give a more detailed discussion on the Higgs mass.

Higgs mass
The mass of a light Higgs in TC models deserves a little introduction. In [17] it has been shown that m 2 h 0 receives a large negative corrections at one loop from the top quark and the W ± and Z bosons. In [18,19] these considerations have been extended to fully dynamical settings of electroweak symmetry breaking and generation of fermion masses. Because of these corrections the tree level Higgs mass can be significantly larger than 126 GeV: adapting the formulas given in [17] to the 3WT case, while neglecting in first approximation any mixing between the two neutral scalar states, we can write where (m 2 h 0 ) tree is the scalar mass due to pure strong dynamics, neglecting the scalar coupling with the EW gauge currents and SM matter. Furthermore, κ is an O(1) renormalization coefficient, while a f and a π are rescaling coefficients of the SM Higgs linear coupling to fermions and quadratic coupling to gauge bosons, respectively. In writing the formula above we set the dimension of the TC representation to 3 and the number of technidoublets to 0.75, as it is the case for 3WT. To illustrate the idea we evaluate the formula above by assuming couplings and renormalization coefficient to be SM like (a F = a π = κ = 1), and m E = m N = m t . With these choices we find a light Higgs, (m 2 h 0 ) tree = 740 GeV. This value is not far from the naive TC estimate for 3WT, obtained by scaling up the mass of the f 0 (500) QCD resonance [39] (m f 0 = 400-550 GeV): The required suppression of the tree level Higgs mass, around 50%, might realistically come from near-conformal dynamics. This should be indeed the case for 3WT, which has 1.5 adjoint Dirac flavors and is outside but close to the conformal window [40]. As this discussion implies, there will be some amount of fine tuning involved in this scenario. However, the same applies to practically any BSM scenario which explains the relatively small mass of the observed Higgs boson. In our case the amount of fine tuning can be quantified in a simple way as The concrete values are on the level of few percents. For example, for m h 0 naive 1 TeV, FT 1.6%, whereas for m h 0 naive ∼ 740 GeV, FT 2.9%. We observe that the couplings involved in the one loop correction to m h 0 are SM-like in size, and this ensures that higher order corrections are indeed perturbative and do not spoil the partial cancellation with the effective TC tree level contribution.

Coupling coefficients
In our model the linear Higgs coupling to charged vector bosons can be written in compact form as with the non-zero terms of the matrix given by Here ϕ is the mixing angle of the neutral scalars, Eq. (37), and To simplify the analysis we select the slice of parameter space where the axial vector coupling to the light Higgs is zero ( 1,3 = 3,3 = 0). Moreover, we require the mixing mass term of the axial-vector to arise only from mixing ((M 2 W ) 1,3 = − m 2 A / √ 2), which is taken to be small. Together these conditions are satisfied by imposing With the help of the two equations above, the charged vector boson squared mass matrix in Eq. (C.6) can be written in compact form as [41]: with The mass eigenvalues can be expanded in x and , which in TC are both expected to be small: where we kept contributions up to O(x n 4−n ), with n = 0, . . . , 4. Generally the linear couplings of the light Higgs are conveniently expressed in terms of coupling coefficients defined by The corresponding result for fermions reads simply Among the neutral vector resonances only Z is relevant for the LHC observables we include in this study. As we discuss in the next section, the numerical values of a Z that we find are very close to a W , and for all practical purposes they can therefore be taken equal to each other. 4

Oblique corrections
The precision EW parameters [42,43] can be calculated directly from the vector-boson sector of the effective Lagrangian, Eqs. (24) and (25), by integrating out the heavy charged and neutral states and then using the formulas provided by [44,45]. 5 At tree level and linear order in the mixing parameter we find: For the T parameter to be consistent with the experiments, the vev component v clearly has to be small in comparison to the EW scale. We note that the S parameter is zero up to corrections of order 4 while the T parameter obtains further contributions of order 2 which we neglect since 1. The intrinsic TC contribution is usually calculated from the one loop perturbative diagrams of the technifermions, which are assumed to be massive because of dynamical symmetry breaking. The dynamical mass divided by m Z is usually taken infinite, as this gives a meaningful result with no unknown parameters. These contributions are denoted by S naive and T naive . For our underlying technicolor theory, Eq. (5), the dynamical masses are such that while the up-techniquark, U , gains only a Dirac mass, m U , the down-techniquark, D L , acquires also a Majorana mass, m L . Oblique corrections for this general case have been calculated in [46] in terms of integral functions, which we use to derive the explicit formulas given in Appendix D. In the pure 3WT limit the Dirac and Majorana masses are equal to each other, 6 m L = m U , and we find The dependence of T on the renormalization scale NP is due to the Majorana mass, and it should be matched onto a renormalizable term in the underlying theory. Indeed in the full theory, Eqs. (1), (2), (3), the contribution to T of the technifermion D R cancels the one in Eq. (56) when its mass is the same as that of U . Phenomenological viability therefore would require the two masses to be close in value, and we implement this requirement in the current analysis by assuming m D ∼ NP and NP to be close to m U . With this assumption the one-loop contribution T naive is negligible when compared to the tree-level contribution, T tree , and therefore we approximate the naive parameters as given by: Another independent contribution to S and T in 3WT comes from the fourth family leptons N and E. These have been already evaluated in [47]: for m E > m N , the S N,E contribution is actually negative and can offset the positive S naive . We can therefore summarize the non-negligible contributions to the S and T parameters, with the assumptions described above, as: where T tree , S naive , and the Higgs contributions are given in Eqs. (55) and (57). The contribution of the heavy leptons to the S parameter is indeed critical to partially cancel the naive TC contribution (∼0.24) and match the experimental result (0.04 ± 0.09): the required cancellation imposes a moderate constraint on the ratio m E /m N [15,46].
In the next section we study the viability of 3WT by performing a goodness of fit analysis based on the recent LHC and Tevatron data on Higgs physics, as well as the experimental values of the S and T parameters.

Experimental validation
To test the model's viability, we start by performing a numerical scan over the parameter space and collecting data points that satisfy the direct search lower bounds on the new physics mass spectrum and the EW precisions tests. We also require perturbativity of all couplings excluding g TC , and stability of the potential at large values of the scalar fields. The unknown parameters derived from strong dynamics are estimated using dimensional analysis [36][37][38]. Finally, we fix the anomalous dimension to γ = 1.5, and require the supersymmetry breaking scale to be larger than 5 TeV. Putting all this together, the free parameters in our scan acquire values in the domain defined by the following relations: The remaining model parameters, λ , θ , x and v , are determined in terms of the ones above by using, respectively, Eqs. (C.1), (38), (41), and (51) together with the observed masses of the Higgs boson, top quark, W boson, and the EW vev. The last relation above ensures that the physical W is mostly made of the SU(2) L gauge field. We scan the parameter space defined above and collect 1000 data points, each satisfying the following constraints, as well as the experimental limits on S and T [39]. We also check that the collected data points satisfy the ATLAS lower limit on the mass of a sequential W boson [48], equal to 2.55 TeV, after an appropriate rescaling of the limit which takes into account the non-SM value of the W coupling to fermions [41].
Having at hand a sufficiently large collection of viable points in the 3WT parameter space, we perform a goodness of fit analysis by using the observed Higgs decay rates to γ γ , ZZ, W W , τ τ , bb, γ γ J J at ATLAS [49-53] and CMS [54-57], and to γ γ , W W , and bb at Tevatron [58], as well as the S and T experimental values [39], for a total of 19 observables. The LHC and Tevatron results are expressed in terms of the signal strengths, defined aŝ where is the efficiency associated with the given final state in an exclusive search, while for inclusive searches one simply has σ tot = σ pp→h 0 (X) , the h 0 production total cross section.
The combined signal strengths from ATLAS, CMS, 7 and Tevatron are given in Table 3, while the signal strengths and efficiencies 8 for dijet associated γ γ production at ATLAS and CMS are listed in Table 4.
Finally, the observed values for the S and T parameters are [39] S = 0.04 ± 0.09 , T = 0.07 ± 0.08 , r(S, T ) = 88% , with the last quantity defining the correlation of the two parameters. For a detailed description of the present fit we refer the reader to [41], where the same statistical analysis has been performed for a different model. Given that no new physics has been detected, only the contributions of new charged particles at one loop to h→γ γ become relevant when comparing the 3WT predictions to the data in Tables 3, 4. More explicitly one has [59] h→γ γ = with i summed over all the charged particles, N i is the number of colors, e i the charge in electron units, and F i a function of the mass m i and the coupling coefficient defined in [41]. In the limit of new particles being much heavier than the light Higgs, one finds with the coupling coefficients defined by Eq. (52). We can therefore mimic the contribution of the charged non-SM particles in 3WT to the observables in Tables 3, 4 by including only the new contribution of a heavy singly charged vector boson with coupling coefficient a V determined by where the factors of 4 account for the double charge of the corresponding states. Moreover, to simplify the analysis we redefine consistently with [41] where the numerical deviations from the first approximate equality above turn out to be negligible for the collected data points compared to the uncertainties on the coupling coefficients. At each collected data point we determine the numerical values of a f , a V , and a V by Eqs. (53), (65) and (66), while we calculate numerically the coupling coefficients of the charged scalars. In Fig. 1 we plot the viable data points on the (a V , a f ) (left panel) and (a V , a V ) (right panel) planes, while in Fig. 2 we plot the data points on the (a V , a f ) plane, together with the 68% (green), 90% (blue), and 95% (yellow) confidence level (CL) regions. In both plots the missing parameter is fixed to the optimal value marked with a blue star. It is clear from Fig. 1, left In calculating χ 2 min /d.o.f. in the above equations we assumed the model to allow three free parameters (a f , S, T ), since a V is strongly correlated with a f near χ 2 min and a S is basically constant. The contribution of the new charged vector bosons, and especially that of the vector baryon , to where the number of degrees of freedom (d.o.f.) has decreased by one, since a V is a free parameter. It is interesting to notice that the optimal value of a V above is equal to the average a V , calculated over the 1000 data points, while the average values of a f and a V are, respectively, 0.98 and 1.03, which are very close to the corresponding optimal values given above. This shows that strong dynamics, which we used to determine the scanned range of values of the free parameters, generates rather naturally the coupling strengths favored by LHC data, at least once the direct constraints on the mass spectrum and the EW precision parameters are satisfied. The 3WT result in Eq. (68) should be compared to the SM one: While the SM fit is less satisfactory than the 3WT one, it clearly shows that the SM is still perfectly viable in light of present collider data. It is instructive to notice that the fit performed with completely free coupling coefficients, therefore not motivated by any specific underlying theory, produces a worse fit than the 3WT: This is because the underlying strong dynamics introduces a large correlation between a f and a V , hence increasing the number of d.o.f. by one, while achieving a χ 2 min very close to the corresponding result obtained with free coupling coefficients (Fig. 2).

Conclusions
In this paper we derived the low energy effective theory of a supersymmetric model with a new strong interacting sector and tested its viability at the LHC. We started from MSSM extended by a strong interacting N = 4 Super Yang Mills (4SYM) sector as well as by a fourth lepton superfamily. By integrating out all the elementary scalars (as well as gauginos and higgsinos), which we assume to be very heavy, we obtained a Technicolor (TC) sector extended by four-fermion interactions between the (4SYM) TC fermions and the SM ones. Due to these interactions, the TC fermion condense gives mass to the EW gauge bosons and to the SM fermion as well. The advantage of this setup is twofold: Supersymmetry naturalizes the scalars which allow ETC-type generation of fermion masses, while the strong sector disentangles the SUSY breaking scale from the electroweak scale and solves the little hierarchy problem.
Given that at low energy the strong interacting states form bound states, we constructed the effective Lagrangian at the EW scale expressed in terms of composite scalar and vector fields, in addition to the SM fields. Because the TC potential features an SU(3) global symmetry and the TC coupling is near conformal, we called this model SU(3) walking technicolor (3WT). To test the viability of the model, we worked out the Higgs couplings to the fermion and vector mass eigenstates, as well as the S and T EW parameters. We then scanned the model parameter space for data points featuring a viable mass spectrum, ensuring that the couplings remain perturbative at large scales. By performing a goodness of fit analysis using Higgs physics data from LHC as well as the experimental values of the EW precision parameters, we showed that 3WT fits the experimental data with a level of goodness comparable to that of the SM. Remarkably, the role played by heavy composite vector resonances turned out to be critical, as their contribution to the diphoton decay of the light Higgs is absolutely necessary to bring the corresponding 3WT prediction within the experimental constraints. These composite vector resonances, having mass of O(TeV), should in principle be observable at LHC.
To conclude, we highlight that SU(3) Walking Technicolor is an UV complete model, which, by avoiding any scalars at the EW scale, in principle solves fine tuning problem. This model, moreover, is favored by Higgs physics and EW precision data at a level comparable to that of the SM. Using the generators above, it is straightforward to obtain the vector and axial-vector charge eigenstates and their elementary particle content from Eqs. (19), (21). First note that the charge operator is Q = S 3 . We find first the linear combinations of the generators corresponding to charge eigenvalues 0, ±1 and ±2. Then we project the corresponding vector states, e.g. v 0 μ = 2Tr(S 3 A μ ), and obtain: (B.10) The particle contents given above reproduce the corresponding results in [33] if one applies there the substitution D R → D c L .

Appendix C. Squared mass matrices
For the neutral scalar and pseudoscalar states, the charged and doubly charged states, the squared mass matrices are, respectively in the σ 0 , 0 basis, in the σ 0 , 0 basis, 2v v σ λ + λ 4v 2 λ + λ , (C.3) in the ± , σ ± basis, in the ±± , δ ±± basis. We define, besides in Eq. (22), the following dimensionless parameters: The contributions of the corresponding heavy neutrinos mass eigenstates and of the heavy electron E to the S and T parameters have been derived in terms of integral functions in [46]. From those, we derived the corresponding explicit results: 1 + 2c 4 ζ 1 + log ν 2 1 − 2 log ν 2 E + 2s 4 ζ 1 + log ν 2