Ultraviolet complete technicolor and Higgs physics at LHC

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Ultraviolet complete technicolor and Higgs physics at LHC Antola, Matti; Di Chiara, Stefano; Tuominen, Kimmo Antola, M., Di Chiara, S., & Tuominen, K. (2015). Ultraviolet complete technicolor and Higgs physics at LHC. Nuclear Physics B, 899, 55-77. https://doi.org/10.1016/j.nuclphysb.2015.07.012 2015 Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 899 (2015) 55–77 www.elsevier.com/locate/nuclphysb Ultraviolet complete technicolor and Higgs physics at LHC Matti Antola a,1, Stefano Di Chiara b,a,∗, Kimmo Tuominen c,a aHelsinki Institute of Physics, P.O. Box 64, FI-000140, Univ. of Helsinki, Finland bDepartment of Physics, P.O. Box 35, FI-40014, Univ. of Jyväskylä, Finland cDepartment of Physics, P.O. Box 64, FI-000140, Univ. of Helsinki, Finland Received 23 February 2015; received in revised form 27 June 2015; accepted 11 July 2015 Available online 23 July 2015 Editor: Hong-Jian He Abstract We consider a supersymmetric model with a new strong interacting sector. The model is built on a strongly interacting N=4 Super Yang Mills sector, broken explicitly to N=1 supersymmetry by embedding within the Minimal Supersymmetric Standard Model (MSSM). Due to cancellation of global and gauge anomalies, the model additionally features a fourth lepton superfamily. We propose a scenario where all elementary scalars, gauging and higgsinos are decoupled at an energy scale substantially higher than the electroweak (EW) scale, thereby avoiding the little hierarchy problem of MSSM. We construct a low energy effective model, where EW symmetry breaking and viable mass spectrum are produced dynamically. To test further the viability of the model, we work out the Higgs couplings as well as the EW precision parameters and then perform a goodness of fit analysis using LHC and EW precision data. The model fits the given experimental data at a level comparable to that of the Standard Model. ©2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. *Corresponding author. E-mail addresses: [email protected] (M. Antola), [email protected] (S. Di Chiara), [email protected] (K. Tuominen). 1Currently at Eniram, Helsinki, Finland. http://dx.doi.org/10.1016/j.nuclphysb.2015.07.012 0550-3213/©2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 56 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 1. Introduction The data collected at the LHC experiments during the 7 and 8TeV 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 [9–11] or scale invariance [12–16]. 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 [17–19]. 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 [20–23] or through scalar fields in bosonic technicolor (BTC) [24–28]. 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=4super 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 2we 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, mSUSY, in Section 3. In the subsequent Section 4we 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- M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 57 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. Superfield SU(2)TC SU(2)LU(1)Y LAdj 1/2 3Adj 1 −1 VAdj 1 0 L1−3/2 N111 E112 tion 5by 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. 2. 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 P=PMSSM +PTC,(1) where PMSSM is the MSSM superpotential, and PTC is expressed by PTC =−gTC √2abca L·b Lc 3+yUa L·Hua 3+yNL·HuN+yEL·HdE +yREa 3a 3,(2) with Hu(Hd) denoting the Y=+1/2 (−1/2)Higgs superfield. The dot (·) indicates a contraction between the SU(2)Ldoublets 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 PTC, with the latter expressed by: 58 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 LTC soft =−aTCabc ˆ Ua Lˆ Db Lˆ U∗c R+aUˆ Qa L·ˆ Huˆ U∗a R+aNˆ L·ˆ Huˆ N∗ R +aEˆ L·ˆ Hdˆ E∗ R+aRˆ U∗a Rˆ U∗a Rˆ E∗ R+1 2MDD†a RD†a R+c.c.−M2 Qˆ Q†a Lˆ Qa L −M2 Uˆ U∗a Rˆ Ua R−M2 Lˆ L† Lˆ LL−M2 Nˆ N∗ Rˆ NR−M2 Eˆ E∗ Rˆ ER,(3) 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, mSUSY, and the EW scale that we identify with the low-energy strongly coupled regime of the TC theory TC ∼4πvw, which for vw=246 GeV implies TC ∼3TeV. We will assume here that the two scales satisfy mSUSY TC.(4) 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 μparameter are of O(mSUSY), 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. 3. Mesoscopic Lagrangian After we integrate out all particles with mass greater than mSUSY, the only technifermions left are the fermionic components of the EW doublet superfield Land EW singlet 3in Table 1: Qa L=Ua L Da L,U a R,a=1,2,3.(5) To derive the effective Lagrangian, valid between the scales TC and mSUSY, one first writes down the Higgs Yukawa sector in MSCT: −LMSCT Yukawa =ˆ Hu·Fu+ˆ Hd·Fd+h.c. , Fu=qi LuYi uu†i R+yUQLU† R+yNLLN† R, Fd=qi LdYi dd†i R+li LYi le†i R+yELLE† R,(6) where i=1, 2, 3is the flavor index and it is summed over. The matrices Yu, Yd, and Ylare diagonal, and the CKM matrix Vis contained in the definitions of the vectors qTi Lu =(ui L,Vij dj L)and qTi Ld =(V †ij uj L,di L). (7) Given that the potential of the MSSM Higgs fields is VMSSM =m2 SUSY +|μ|2|ˆ Hu|2+m2 SUSY +|μ|2|ˆ Hd|2−bˆ Huˆ Hd+h.c.+... (8) by solving the equation of motion in terms of the Higgs mass eigenstates and plugging the solutions back into Eq. (6), leads us to the first line of the following dimension six interaction terms for the fermions in the intermediate scale (or mesoscopic) effective Lagrangian M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 59 L4-fermion =c2 θ m2 sF† uFu+F† dFd−cθsθ m2 s (Fu·Fd+h.c.) +g2 TC m2 SUSY abccdeηαa iηb jαη†d i˙ βη†˙ βe j.(9) Similarly, the last term originates in an analogous way from the potential and the Yukawa sector of 4SYM. In Eq. (9) we have defined ηT α=ULα,D Lα,−iσ2 αβ U†β R,(10) where σ2is the second Pauli matrix, and the indices iand jdenote 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 m2 s=μ2+m2 SUSY(μ2+m2 SUSY)2−b2 μ2+m2 SUSY2+b2,tan θ=b μ2+m2 SUSY .(11) In the rest of this paper we use abbreviations sθ≡sin θ, cθ≡cos θand tθ≡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. 4. 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. 4.1. Technicolor scalar sector The composite scalar matrix field M, singlet under SU(2)TC, has minimal particle content given by the techniquark bilinears: Mij ∼ηα iηβ jεαβ =ηiηj,with i, j =1...3.(12) The field Mtransforms under the full SU(3) group according to M→uMuT,with u∈SU(3). (13) The effective linearly transforming SU(3)invariant Lagrangian reads: 60 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 Table 2 Transformation properties of the component fields of the matrix Munder 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. Field SU(2)LU(1)Y ∼QLQL 1 σ∼QLU† R−1 2 δ−− ∼U† RU† R1−2 LM=1 2Tr DμM†DμM−VM,(14) where the covariant derivative is given by DμM=∂μM−igLGμM+MGT μ, with Gμ=˜ Wa μ λa 2+tξBμYM,a=1,2,3.(15) In the above equation λaare the Gell-Mann matrices, tξ=tan ξwith ξthe EW mixing angle, ˜ Wμand Bμare the SM EW gauge fields, and YM=diag 1 2,1 2,−1.(16) The most general SU(3)preserving effective potential, including operators up to dimension four, is2 VM=−m2 2Tr M†M+λ 4Tr M†M2+λTr M†MM†M−2mdet M+det M†, (17) which breaks SU(3)spontaneously to SO(3)for positive m2, 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)Aaxial symmetry is anomalous, and is therefore broken at the quantum level. This corresponds to the detMterm in Eq. (17). The components of the matrix M∼ηTηcan 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 Mis written in terms of complex scalars as M=⎛ ⎝ √2++ +σ0 +√20σ− σ0σ−√2δ−− ⎞ ⎠.(18) 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. 2In principle the higher dimensional operators can play a role and should be systematically included. For an initial investigation and qualitative account of the various constraints, we truncate the effective theory at the level of dimension four operators. This provides a quantitative baseline for possibly more refined analyses in the future. M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 61 4.2. Vector sector A minimal set of composite vector fields transforming homogeneously under SU(3)can be written, in terms of Gell-Mann matrices λa, as Aμ=Aa μ λa 2,(19) which transform under SU(3) according to Aμ→uAμu†,with u∈SU(3). (20) The elementary particle content of Aμis expressed by the equivalence Aμj i∼ηα iσμ α˙ βη†˙ βj −1 3δj iηα kσμ α˙ βη†˙ βk =jγμi−1 3δj ikγμk, =¯ UL,¯ DL,¯ Uc R,(21) where the components of in SU(3)space are Dirac spinors, with the superscript con 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, Aμ, 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: Cμ=Aμ−G μ,=gL gTC ,F μν =∂μAν−∂νAμ−igTC Aμ,A ν,(22) with Gμdefined in Eq. (15). The vector field Cμhas the same transformation law as Aμ: Cμ→uCμu†,with u∈SU(3). (23) The kinetic and mass terms for the vector fields can then be written as LV=−1 2Tr ˜ Wμν ˜ Wμν−1 4BμνBμν −1 2Tr FμνFμν+m2 ATr CμCμ,(24) while the scalar-vector field interaction terms up to dimension four operators read LM–V=g2 TCr1Tr CμCμMM†+g2 TCr2Tr CμMCμT M† −g2 TC r3 4Tr CμCμTr MM†,(25) with constants ri∼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 yU, which explicitly break SU(3)global symmetry, are small compared to those proportional to g2 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).3Finally, 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. 3Neglecting this term is simply a restriction on the parameter space: this term could be included in more thorough future analyses. 62 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 4.3. 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 yU, renormalized at the SUSY breaking scale, with the dimensionless factor ω=ULU† RmSUSY ULU† RTC =mSUSY TC γ ,(26) written under the assumption that the anomalous dimension γof 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 ηTKη , (27) with Kij =yUcθω m2 sδikcθq†k LuY∗ uuR+y∗ NL†k LNR −iksθqk LdYdd† R+lk LYle† R+yELk LE† Rδ3j, i, j =1,...,3;k=1,2;3k≡0,(28) upon condensation of the techniquarks. Under SU(3)global symmetry the spurion Ktransforms as K→u∗Ku†. The four-techniquark term on the other hand is y2 Uc2 θ m2 s ω2(QLU† R)(Q† LUR)=K ij kl ηα iηjαη† kβ η†β l, K ij kl =y2 Uc2 θ m2 s ω2(δik1+δik2)δjl3,(29) where αand βare spin indices. For this term to be invariant under SU(3), the spurion Kmust transform as K ij kl →uimujnu∗ kou∗ lp K mnop, with u ∈SU(3). To estimate the effects of renormalization, we simply assume factorization, leading to a multiplicative factor of ω2. At the lowest order in the spurions, the SU(3)breaking effective Lagrangian, obtained from Eqs. (28) and (29) is: LF=c12 TCTr [MK]+c24 TCK ij kl Mij M∗ kl +h.c. ,(30) where we introduced factors of TC to define the dimensionless coefficients ci, which parametrize the couplings of the effective Lagrangian in terms of those of the underlying theory. We estimate these coefficients using dimensional analysis [36–38] and find c1=Oϒ−1,c 2=Oϒ−2,ϒ≡TC vw .(31) M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 69 Table 3 Combined signal strengths from LHC and Tevatron experiments. ij ATLAS CMS Tevatron ZZ 1.50 ±0.40 0.91 ±0.27 γγ 1.65 ±0.32 1.11 ±0.31 6.20 ±3.30 WW 1.01 ±0.31 0.76 ±0.21 0.89 ±0.89 ττ 0.70 ±0.70 1.10 ±0.40 bb −0.40 ±1.10 1.30 ±0.70 1.54 ±0.77 Table 4 Signal strengths and efficiencies for Higgs decay to γγ associated to a dijet at LHC. ATLAS 7 TeV ATLAS 8 TeV CMS 7 TeV CMS 8 TeV γγJJ 2.7±1.92.8±1.62.9±1.90.3±1.3 pp →h22.5% 45.0% 26.8% 46.8% pp →qqh 76.7% 54.1% 72.5% 51.1% pp →t¯ th 0.6% 0.8% 0.6% 1.7% pp →Vh 0.1% 0.1% 0% 0.5% where is the efficiency associated with the given final state in an exclusive search, while for inclusive searches one simply has σtot =σpp→h0(X), the h0production total cross section. The combined signal strengths from ATLAS, CMS,7and Tevatron are given in Table 3, while the signal strengths and efficiencies8for dijet associated γγ production at ATLAS and CMS are listed in Table 4. Finally, the observed values for the Sand Tparameters are [39] S=0.04 ±0.09 ,T=0.07 ±0.08 ,r(S,T)=88% ,(62) 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→γγ become relevant when comparing the 3WT predictions to the data in Tables 3, 4. More explicitly one has [59] h→γγ =α2 em3 h 256π3v2 w i Nie2 iFi 2 ,(63) with isummed over all the charged particles, Niis the number of colors, eithe charge in electron units, and Fia function of the mass miand the coupling coefficient defined in [41]. In the limit of new particles being much heavier than the light Higgs, one finds FWi=7aWi,F E=FN=−af 4 3,F Si=−aSi 1 3,(64) 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, 4by including only the new contribution of a heavy singly charged vector boson with coupling coefficient aVdetermined by 7We use the mass cut based result for CMS result on the Higgs to diphoton decay. 8We chose to include only the loose categories from the ATLAS and CMS dataset at 8 TeV. 70 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 Fig. 1. Viable data points in the (aV, af)(left panel) and (aV, aV)(right panel) planes, together with the 68% (green), 90% (blue), and 95% (yellow) CL region. The blue star in each plot marks the optimal coupling coefficients on the respective planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) aV≡1 7(FW+FW +4F)−aS 21 , aS≡−3(16FE+4FN+FH±+4Fh±± +4FH±±),(65) where the factors of 4 account for the double charge of the corresponding states. Moreover, to simplify the analysis we redefine consistently with [41] aZ≈aW≡aV,(66) 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 af, aV, and aV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 (aV, af)(left panel) and (aV, aV)(right panel) planes, while in Fig. 2 we plot the data points on the (aV, af)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 panel, that the Wand Zcouplings are enhanced, compared to their SM values, while the SM fermion couplings are suppressed. This result for the 3WT couplings enhances the Higgs decay to diphotons. However, the contribution of the new charged fermions and scalars, expressed by Eq. (65), is large and interferes destructively with the Wcontribution to the same process. As a consequence the data point minimizing χ2in the (af, aV, aS)space, obtained from a 3WT particle spectrum without the composite vector resonances at low energy, is ruled out: aV=1.00 ,a f=1.00 ,a S=20.5,S=0.04 ,T=0.07 ; χ2 min/d.o.f. =3.42 ,P χ2>χ2 min=0.0004 % ,d.o.f. =16 .(67) In calculating χ2 min/d.o.f. in the above equations we assumed the model to allow three free parameters (af, S, T), since aVis strongly correlated with afnear χ2 min and aSis basically constant. The contribution of the new charged vector bosons, and especially that of the vector baryon , to M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 71 Fig. 2. Viable data points in the (aV, af)plane passing through the point with optimal coupling coefficients in the (aV, af, aV)space, marked by a blue star, together with the 68% (green), 90% (blue), and 95% (yellow) CL region. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) the Higgs decay into diphoton is large, and offsets entirely the negative contribution of E,N, and charged scalars in Eq. (65). Among the 1000 viable data points, the one producing the minimum value for χ2is: aV=1.01 ,a f=0.99 ,a V=0.21 ,S=0.04 ,T=0.07 ; χ2 min/d.o.f. =0.83 ,P χ2>χ2 min=65 % ,d.o.f. =15 ,(68) where the number of degrees of freedom (d.o.f.) has decreased by one, since aVis a free parameter. It is interesting to notice that the optimal value of aVabove is equal to the average aV, calculated over the 1000 data points, while the average values of afand aVare, 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: χ2 min/d.o.f. =0.89 ,P χ2>χ2 min=60% ,d.o.f. =19 .(69) 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: aV=0.97+0.10 −0.11 ,a f=1.02+0.25 −0.32 ,a V=0.21+0.16 −0.18 , χ2 min/d.o.f. =0.85 ,P χ2>χ2 min=62% ,d.o.f. =14 .(70) This is because the underlying strong dynamics introduces a large correlation between afand aV, hence increasing the number of d.o.f. by one, while achieving a χ2 min very close to the corresponding result obtained with free coupling coefficients (Fig. 2). 72 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 7. 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 Sand TEW 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. Acknowledgements We thank R. Foadi for providing the code to evaluate the EW oblique corrections and for discussions. This work was financially supported by the Academy of Finland project 267842. Appendix A. EW symmetry breaking in global SU(3)invariant technicolor The SU(3)symmetry of the microscopic TC Lagrangian is spontaneously broken to the maximal diagonal subgroup, SO(3). The symmetry breaking pattern leaves us with five broken generators with associated Goldstone bosons. Such a breaking is driven by the condensate ηα iηβ jαβ Eij =2U† RUL+DLDL,(A.1) where the indices i, j=1, ..., 3 denote the components of the triplet of η, and the Greek indices indicate the ordinary spin. The matrix Eis a 3 ×3matrix defined as M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 73 E=⎛ ⎝ 001 010 100 ⎞ ⎠.(A.2) The above condensate is invariant under an SO(3)symmetry. It is convenient to separate the eight generators of SU(3) into the three that leave the vacuum invariant, Sa, and the remaining five that do not, Xa. Then the Sagenerators of the SO(3) subgroup satisfy the relation SaE+ESaT=0,with a=1,...,3,(A.3) so that uEuT=E, for u ∈SO(3). An explicit realization of the generators is shown in Appendix B. The scalar and pseudoscalar degrees of freedom, necessary to model the Goldstone bosons and spontaneous symmetry breaking, consist of a composite Higgs and its pseudoscalar partner, as well as five pseudoscalar Goldstone bosons and their scalar partners. These can be assembled in the matrix M=σ+i √3I3+√2(ia+ a)XaE, (A.4) which transforms under the full SU(3) group according to M→uMuT,with u ∈SU(3). (A.5) The Xa’s, a=1, ..., 5are the generators of the SU(3) group which do not leave the vacuum expectation value (VEV) of Minvariant M= v √3E. (A.6) Appendix B. SU(3)generators The generators Siof SO(3)satisfy SiE+ESiT =0. The other generators of SU(3)are written as Xi. The generators are normalized as Tr[SiSj]=δij /2Tr[XiXj]=δij /2Tr[XiSj]=0(B.1) and given in terms of the Gell-Mann matrices λiby S1=1 2√2λ1−λ6(B.2) S2=1 2√2λ2−λ7(B.3) S3=1 4λ3+√3λ8(B.4) X1=1 2√2λ1+λ6(B.5) X2=1 2√2λ2+λ7(B.6) X3=1 4√3λ3−λ8(B.7) 74 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 X4=1 2λ4(B.8) X5=1 2λ5(B.9) 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 =S3. 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. v0 μ= 2Tr(S3Aμ), and obtain: v0 μ≡A3 μ+√3A8 μ 2∼¯ ULγμUL+¯ URγμUR, v+ μ≡A1 μ−A6 μ 2−iA2 μ−A7 μ 2∼¯ DLγμUL+¯ Dc LγμUR, v− μ≡A1 μ−A6 μ 2+iA2 μ−A7 μ 2∼¯ ULγμDL+¯ URγμDc L, a0 μ≡√3A3 μ−A8 μ 2∼¯ ULγμUL−¯ URγμUR−2¯ DLγμDL, a+ μ≡A1 μ+A6 μ 2−iA2 μ+A7 μ 2∼¯ DLγμUL−¯ Dc LγμUR, a− μ≡A1 μ+A6 μ 2+iA2 μ+A7 μ 2∼¯ ULγμDL−¯ URγμDc L, ++ μ≡A4 μ−iA5 μ √2∼¯ Uc RγμUL, −− μ≡A4 μ+iA5 μ √2∼¯ ULγμUc R.(B.10) The particle contents given above reproduce the corresponding results in [33] if one applies there the substitution DR→Dc 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 M2 ¯ h0=2v2 σλ+2λ2vvσλ−2λ 2vvσλ−2λ2v2 σλ +v2 λ+4λ,(C.1) in the  σ0,  0basis, M2 ¯π0=8v2 λ 4vvσλ 4vvσλ 2v2 σλ ,(C.2) in the  σ0,  0basis, M2 ¯ h±=2v2 σλ +λ2√2vvσλ +λ 2√2vvσλ +λ4v2 λ +λ,(C.3) in the ±, σ±basis, M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 75 M2 ¯ h±± =2v2 σλ −4v2 −v2 σλ4v2 λ +2v2 σλ 4v2 λ +2v2 σλ2v2 σλ −4v2 −v2 σλ,(C.4) in the ±±, δ±± basis. We define, besides in Eq. (22), the following dimensionless parameters: x=gLvw 2mA ,t ρ=√2v vσ ,z i=gTCvw 2mA2 ri,i=1,2,3.(C.5) Then the non-zero terms of the charged vector boson squared mass matrix (which by definition is symmetric) are M2¯ W1,1=m2 Ax2+21+z1−z3 21+c2 ρ, M2¯ W2,2=m2 A1+z1+z2s2ρ−z3 21+c2 ρ,M2¯ W1,2=−  √2M2¯ W2,2, M2¯ W3,3=m2 A1+z1−z2s2ρ−z3 21+c2 ρ,M2¯ W1,3=−  √2M2¯ W3,3, (C.6) in the ˜ W± μ, V± μ, A± μbasis, with furthermore the squared mass of the doubly charged vector boson given by m2 =m2 A1+2c2 ρ(z1+z2)−z3 21+c2 ρ.(C.7) Finally, the non-zero terms of the neutral vector boson squared mass matrix in the ˜ W3 μ, Bμ, V3 μ, A3 μbasis are M2 ¯ Z1,1=m2 Ax21+s2 ρ+21+z1−1 21+c2 ρz3+z2s2 ρ, M2 ¯ Z1,2=−m2 Ax21+s2 ρ+2z1−c2 ρ(2z1−z2)+z2tξ, M2 ¯ Z2,2=m2 Ax21+s2 ρ+23+z1+c2 ρ(4z1−5z2)+z2−3 21+c2 ρz3t2 ξ, M2 ¯ Z3,3=m2 A1+2c2 ρ(z1−z2)−z3 21+c2 ρ, M2 ¯ Z1,3=− 2M2 ¯ Z3,3,M2 ¯ Z2,3=−3 2t ξM2 ¯ Z3,3, M2 ¯ Z4,4=m2 A1+z1−z2s2ρ−z3 21+c2 ρ, M2 ¯ Z1,4=−√3 2M2 ¯ Z4,4,M2 ¯ Z2,4=√3 2t ξM2 ¯ Z4,4.(C.8) Appendix D. Sand Tparameters for general neutrino mass matrix The most general mass terms for a pair of rightand left-handed neutrinos is defined by L⊃−mE¯ EREL−1 2nT LMnL+h.c.,M=MLmD mDMR,n L=(NL,¯ NR)T(D.1) 76 M. Antola et al. / Nuclear Physics B 899 (2015) 55–77 with eigenvalues λ1,2=1 2(ML+MR)±(ML−MR)2+4m2 D.(D.2) The contributions of the corresponding heavy neutrinos mass eigenstates and of the heavy electron Eto the Sand Tparameters have been derived in terms of integral functions in [46]. From those, we derived the corresponding explicit results: S=1 12π1+2c4 ζ1+log ν2 1−2logν2 E+2s4 ζ1+log ν2 2 +s2 2ζ 36π 91−log ν2 1ν4 1ν2 2−91−log ν2 2ν2 1ν4 2−1−3logν2 1ν6 1+1−3logν2 2ν6 2 ν2 1−ν2 23 −(−1)βs2 2ζ 8π ν1ν2ν4 1−2ν2 1ν2 2log ν2 1 ν2 2−ν4 2 ν2 1−ν2 23,(D.3) T=2 NP 64πc2 ξs2 ξm2 Z16c4 ζν2 1log ν2 1+16s4 ζν2 2log ν2 2+8ν2 Elog ν2 E −s2 2ζ1−2logν2 1ν4 1−1−2logν2 2ν4 2 ν2 1−ν2 2 +4(−1)βs2 2ζ1−log ν2 1ν3 1ν2−1−log ν2 2ν1ν3 2 ν2 1−ν2 2 +4c2 ζ1−2logν2 1ν4 1−1−2logν2 Eν4 E ν2 1−ν2 E +4s2 ζ1−2logν2 2ν4 2−1−2logν2 Eν4 E ν2 2−ν2 E,(D.4) where NP is the given renormalization scale, ξis the EW mixing angle, and ν1=λ1 NP ,ν 2=λ2 NP ,ν E=mE NP ,t 2ζ=2mD MR−ML , β=1 2⎡ ⎢ ⎣1+⎛ ⎝λ1 |λ1| ∗λ2 |λ2|⎞ ⎠ 2⎤ ⎥ ⎦.(D.5) In the limit MR→∞, and ML=mE≡mU, one recovers the results in Eqs. (56). References [1] G. Aad, et al., ATLAS Collaboration, Phys. Lett. B 716 (2012) 1, arXiv:1207.7214. [2] S. Chatrchyan, et al., CMS Collaboration, Phys. Lett. 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