CP 3 - Origins Particle Physics & Origin of Mass Preprint typeset in JHEP style - HYPER VERSION arXiv:1009.1624v2 [hep-ph] 15 Jun 2011 Minimal Supersymmetric Conformal Technicolor: The Perturbative Regime Matti Antola∗ Department of Physics and Helsinki Institute of Physics, P.O.Box 64, FI-000140, University of Helsinki, Finland Stefano Di Chiara† CP3 -Origins, Campusvej 55, DK-5230 Odense M, Denmark Francesco Sannino‡ CP3 -Origins, Campusvej 55, DK-5230 Odense M, Denmark Kimmo Tuominen § Department of Physics, P.O.Box 35, FI-40014, University of Jyväskylä, Finland and Helsinki Institute of Physics, P.O.Box 64, FI-000140, University of Helsinki, Finland Abstract: We investigate the perturbative regime of the Minimal Supersymmetric Conformal Technicolor and show that it allows for a stable vacuum correctly breaking the electroweak symmetry. We find that the particle spectrum is richer than the MSSM one since it features several new particles stemming out from the new N = 4 sector of the theory. The parameter space of the new theory is reduced imposing naturalness of the couplings and soft supersymmetry breaking masses, perturbativity of the model at the EW scale as well as phenomenological constraints. By studying the RGEs at two loops we find that the Yukawa couplings of the heavy fermionic states flow to a common fixed point at a scale of a few TeVs. Our preliminary results on the spectrum of the theory suggest that the Tevatron and the LHC can rule out a significant portion of the parameter space of this model. Preprint: CP3 -Origins-2010-37 ∗ matti.antola@helsinki.fi dichiara@cp3.sdu.dk ‡ sannino@cp3.sdu.dk § kimmo.i.tuominen@jyu.fi † Contents 1. Introduction 1 2. The Model 3 3. Vacua and Stability Conditions 5 4. Mass Spectrum 4.1 Gauge Bosons 4.2 Fermions 4.3 Scalars 4.3.1 Tree-Level 4.3.2 One-Loop 6 7 7 8 8 10 5. Phenomenological Viability 11 6. Conclusions and Outlook 14 A. MSCT Lagrangian 16 B. Scalar Squared Mass Matrices 18 C. Renormalization Group Equations 20 1. Introduction The earliest models of technicolor [1, 2] have problems with the electroweak (EW) precision data [3, 4, 5, 6]. Technicolor models must be extended in order to give mass to the standard model (SM) fermions [7, 8]. In these extensions one, typically, expects potentially large flavor changing neutral current (FCNC) processes. Using near conformal dynamics alleviates the FCNC problem [9, 10, 11]. For over a decade it was hoped that such a near conformal dynamics could strongly reduce the tension with precision data even if one had a large number of technidoublets gauged under the electroweak (EW) symmetry. However very recently it has been shown [12, 13, 14] that to be phenomenologically viable the near conformal models should contain the most minimal number of flavors gauged under the EW symmetry. The simplest models of this type which are shown to pass the precision tests, or have the smallest deviation from the precision data, while still providing a (near) conformal behavior were put forward recently in [15, 16, 17, 18, 19, 20, 21, 22]. Among these, the Minimal Walking Technicolor (MWT) features the most economical particle content. 1 In MWT the gauge group is SU (2)T C × SU (3)C × SU (2)L × U (1)Y and the field content of the technicolor sector is constituted by two flavors of techni-fermions and one techni-gluon all in the adjoint representation of SU (2)T C . The model features also a pair of Dirac leptons, whose left-handed components are assembled in a weak doublet, necessary to cancel the Witten anomaly [23] arising when gauging the new technifermions with respect to the weak interactions. The model requires, however, still additional ingredients in order to give mass to the standard model (SM) fermions. For example, one may postulate the existence of an Extended Technicolor (ETC) sector, traditionally featuring new gauge interactions linking the SM fermions to the techniquarks, which can generate mass terms for the SM fermions (as well as for the techni-mesons and -baryons) via a new dynamical mechanism. Interesting developments recently appeared in the literature [24, 25, 26, 27, 28, 29]. Nonperturbative chiral gauge theories dynamics is expected to play a relevant role in models of ETC since it allows, at least in principle, the self breaking of the gauge symmetry. Recent progress on the phase diagrams of these theories has appeared in [30]. Another alternative is to reintroduce new bosons (bosonic technicolor) [31, 32, 33, 34, 35] able to give masses to the SM fermions using standard Yukawa interactions. Eventhough these models are phenomenologically viable, they suffer from a SM-like fine tuning and are therefore unnatural. Supersymmetric technicolor has been considered [36, 37] as a way to naturalize bosonic technicolor. Another possibility would be to imagine the new scalars also to be composite of some new strong dynamics. In [38] we made the observation that the techni-fermions and techni-gluons of the Minimal Walking Technicolor fit perfectly in an N = 4 supermultiplet, provided that we also include three scalar superpartners. In fact the SU (4) global symmetry of MWT is nothing but the well known SU (4)R R symmetry of the N = 4 Super Yang Mills (4SYM) theory. This is the global quantum symmetry that does not commute with the supersymmetry transformations. Supersymmetrizing MWT in this way leads to an approximate N = 4 supersymmetry of the technicolor sector that is broken only by EW gauge and Yukawa interactions. Due to approximate N = 4 invariance the beta function of the technicolor gauge coupling is zero at one loop, i.e. the associated technicolor model is approximately conformal. We called this model Minimal Supersymmetric Conformal Technicolor (MSCT). This model can also be viewed as the first extension of the SM featuring maximal supersymmetry in four dimensions when neglecting the EW gauging of the R-symmetry. MSCT constitutes an interesting theoretical as well as phenomenological model to explore since it naturally allows to investigate different regimes according to how strongly coupled the maximally supersymmetric Yang-Mills theory is taken to be. In this phenomenological work we analyze the situation in which such a theory is weakly coupled at the EW scale. To determine the spectrum of the theory we first analyze the ground state and afterwards compute the masses. We will show that at tree level all the states are massive except for the lightest CP-even and -odd Higgses that will acquire mass at one loop. We find that the physical spectrum is phenomenologically viable and can be investigated for possible discovery at the Large Hadron Collider (LHC) and the Tevatron. 2 We also analyze the running of the gauge and Yukawa couplings at two loops. The numerical solutions exhibit behavior in which some of the Yukawa couplings flow to a common fixed point, while the remaining ones go to zero with increasing energy. The technicolor coupling, whose one loop beta function is zero, at two loops instead decreases slowly with increasing energy. The paper is organized as follows: We recall the model details in Section 2 and in 3 we impose the minimization conditions on the scalar potential and derive the additional conditions for the stability of a non-trivial vacuum. In Section 4 we present the mass spectrum of MSCT in the perturbative regime (pMSCT), including the one-loop correction for the physical massless states, and finally in Section 5 we study the viability of pMSCT based on these results. 2. The Model The fermionic particle content of the MWT is given explicitly as ! ! a U N a QaL = , URa , DR , a = 1, 2, 3 ; LL = , E Da NR , E R , (2.1) L L where U and D are techni-fermions in the adjoint representation of SU (2)T C , whose lefthanded components form a doublet under SU (2)L , and the chiral leptons required to cancel the Witten anomaly are denoted by N and E. The following generic hypercharge assignment is free from gauge anomalies:   y y+1 y−1 Y (QL ) = , Y (UR , DR ) = , , 2 2 2   y −3y + 1 −3y − 1 Y (NR , ER ) = , . (2.2) Y (LL ) = − 3 , 2 2 2 The global symmetry of the technicolor theory, per se, is SU (4) which breaks explicitly to SU (2)L × U (1)Y by the natural choice of the electroweak embedding [15, 17]. The vacuum choice is stable against the SM quantum corrections [39]. a is a singlet under EW symmetry To build MSCT we set y = 1 in Eqs. (2.2) so that D̄R and can play the role of the techni-gaugino.1 We define the N = 4 supermultiplet in terms N = 1 superfields, whose scalar and fermionic components are expressed by:      
˜ , Ū ŨL , UL ∈ Φ1 , D̃L , DL ∈ Φ2 , Ū G, D̄R ∈ V, (2.3) R R ∈ Φ3 , where we used a tilde to label the scalar superpartner of each fermion. We indicated with Φi , i = 1, 2, 3 the three chiral superfields of 4SYM and with V the vector superfield. The superfields associated with the remaining MWT fermions N and E are given by:         ˜ , N̄ ˜ , Ē ÑL , NL ∈ Λ1 , ẼL , EL ∈ Λ2 , N̄ ∈ N, Ē (2.4) R R R R ∈ E. The quantum numbers of the superfields in Eqs.(2.3,2.4) and of those labeled by H and H ′ , which contain each a Higgs scalar weak doublet, are given in Table 1. 1 In Section 5 we briefly comment on the alternative choice of y = −1. 3 Superfield SU(2)T C SU(3)c SU(2)L U(1)Y Φ1,2 Adj 1  1/2 Φ3 Adj 1 1 -1 V Adj 1 1 0 Λ1,2 1 1  -3/2 N 1 1 1 1 E 1 1 1 2 H 1 1  1/2 H′ 1 1  -1/2 Table 1: MSCT N = 1 superfields The renormalizable lepton and baryon number2 conserving superpotential for the MSCT is P = PM SSM + PT C , (2.5) where PM SSM is the minimal supersymmetric standard model (MSSM) superpotential, and gT C PT C = − √ ǫijk ǫabc Φai Φbj Φck + yU ǫij Φai Hj Φa3 + yN ǫij Λi Hj N + yE ǫij Λi Hj′ E + yR EΦa3 Φa3 . 3 2 (2.6) a a In the last equation Φi = Qi , i = 1, 2, with a the technicolor index. Gauge invariance alone does not ensure the Yukawa coupling of the first term to be equal to gT C , however, setting it to this value amounts to the N = 4 limit. We have also investigated in Appendix C the independent running of a more general Yukawa coupling and shown that it tends towards the gT C value at low energies. This result further justifies our choice to set it equal to the technicolor gauge coupling itself. The Lagrangian of the MSCT is L = LM SSM + LT C , (2.7) where the supertechnicolor Lagrangian LT C , by following the notation of Wess and Bagger [40], can be written in the form:
1 LT C = Tr W α Wα |θθ + W̄α̇ W̄ α̇ |θ̄θ̄ + Φ†f exp (2gX VX ) Φf |θθθ̄θ̄ + (PT C |θθ + h.c.) , (2.8) 2 In the last equation Wα = − 1 D̄D̄ exp (−2gV ) Dα exp (2gV ) , V = V a TAa , (TAa )bc = −if abc , 4g (2.9) and Φf = Q, Φ3 , Λ, N, E; 2 X = T C, L, Y . (2.10) We assume all the superfields in Table 1 to have both lepton and baryon numbers equal to zero. 4 The product gX VX is assumed to include the gauge charge of the superfield on which it acts. The charge is Y for U (1)Y , and 1 (0) for a multiplet (singlet) of a generic group SU (N ). The technicolor vector superfield VT C is identified with V defined in Eq.(2.3). The remaining vector superfields are those already defined in the MSSM [41] while the superpotential PT C is given in Eq.(2.6). We have written the Lagrangian LT C in terms of elementary fields in Appendix A. The full MSSM Lagrangian LM SSM can be found in [41] and references therein. 3. Vacua and Stability Conditions The gauge group breaking of pMSCT (excluding the color group) follows the pattern SU (2)T C × SU (2)L × U (1)Y → U (1)EM × U (1)T C , were the first U (1) on the right determines the conservation of the ordinary electromagnetic (EM) charge. Even though the breaking involves also the TC group, besides the EW one, we will still refer to it simply as EWSB. Because of the symmetry of the Lagrangian, we are free to choose the vacuum of the techni-Higgs scalar to be aligned in the third direction of the SU (2)T C gauge space. We define the vacuum expectation values (vev)s of the scalar fields neutral under the residual symmetry by D E v D E D E vH vH TC D̃L3 = √ , H̃0 = sβ √ , H̃0′ = cβ √ , (3.1) 2 2 2 where all the vevs are chosen to be real, sβ = sin β, and cβ = cos β.3 The vacuum expectation values of the remaining fields is chosen to be zero so that U (1)EM is conserved. The scalar potential is obtained from the D, F , and sof t terms of the Lagrangian given in Appendix A, and by the corresponding MSSM scalar potential. The potential is:  

2 |D̃L3 |2 + m2u + |µ|2 |H̃0 |2 + m2d + |µ|2 |H̃0′ |2 − bH̃0 H̃0′ + c.c. Vin = MQ 2
 1 2 + (3.2) gL + gY2 |D̃L3 |2 − |H̃0′ |2 + |H̃0 |2 . 8 The terms depending on the phase of the different fields are the b term and its conjugate. As in the MSSM invariance under the U (1)Y symmetry together with the fact that H̃ and H̃ ′ have opposite hypercharges allow to redefine their vevs and b parameter to be real. The quartic terms in this potential cancel when |D̃L3 |2 = |H̃0′ |2 − |H̃0 |2 . To make the potential bound from below we impose the Hessian of Vin to be semi-definite positive along this D flat plane, which gives the conditions: 2 m2u + |µ|2 − MQ

2 m2d + |µ|2 + MQ > b2 , 2|µ|2 + m2u + m2d > 0 . (3.3) At the minimum Vin satisfies the equations ∂D̃3 Vin |φ=<φ> = 0, L ∂H̃0 Vin |φ=<φ> = 0, 3 ∂H̃ ′ Vin |φ=<φ> = 0 . 0 (3.4) Notice that, consistently with the rest of the paper, we indicated the scalar component of each Higgs weak doublet superfield with a tilde. 5 These equations can be solved with respect to the vevs and used to express the soft SUSY breaking parameters according to:

1 2 2 , (3.5) MQ = − gL2 + gY2 vT2 C − c2β vH 8

1 2 − |µ|2 + b t−1 (3.6) m2u = − gL2 + gY2 vT2 C − c2β vH β , 8

1 2 2 − |µ|2 + b tβ , (3.7) m2d = gL + gY2 vT2 C − c2β vH 8 where tβ = tan β. We impose the trivial vacuum to be unstable, both on the |H̃0′ |, |H̃0 | plane and the |D̃L3 | direction so that both vT C , vH > 0, by requiring the corresponding Hessian of Vin evaluated at the origin of the moduli space to have (at least) one negative eigenvalue. This translates to the conditions

2 < 0, m2u + |µ|2 m2d + |µ|2 < b2 . (3.8) MQ Finally, requiring the potential to be stable at the vacuum point determines the extra conditions (3.9) m2A1 > 0 , m2h0 > 0 , 1 where mh01 , mA0 are defined in Eqs.(4.16,4.17,4.4). Without loss of generality one can choose 0 < β < π/2. After plugging Eqs.(3.5,3.6,3.7) in (3.3,3.8,3.9) all these conditions are satisfied for

t2β 2 π 2 2 (3.10) gL + gY2 vT2 C − c2β vH , c2β vH < vT2 C , 0 < β < , 0<b< 16 4 or b > 0, π/4 6 β < π/2 . (3.11) We will investigate in the following the region of parameter space defined by the conditions (3.11) since, as it will become clear in in Section 5, this is the one which is phenomenologically appealing. 4. Mass Spectrum The superpotential in Eq.(2.6) and the soft SUSY-breaking terms in Eq.(A.13) conserve both lepton L and baryon B numbers. From Eqs.(A.5,A.6) one can see that the terms generated by the superpotential respect the same conservation laws. This is true also for the remaining contributions to the Lagrangian given in Eqs.(A.2,A.3,A.4). Moreover, after EWSB the MSCT Lagrangian is still invariant under the residual U (1)EM × U (1)T C .4 We can therefore write the gauge boson, fermion, and scalar (squared) mass matrices in block diagonal form in the basis of EM- and TC-charges and L and B numbers. The mass matrices of all the SM fermions and their superpartners assume the same form, in terms of the Higgs vevs, as of those obtained in the MSSM and can be found for example in [41]. The EW gauginos, Higgs scalar doublets and their superpartners mix with the N = 4 technicolor sector. Finally the fields NL , N̄R , and their scalar superpartners will not mix at tree level with other SM fields with EM charge QEM = 1 (where we defined QEM = TL3 +Y ) by constuction. 4 We can neglect SU (3)C , since none of the fields presented here carries color charge. 6 4.1 Gauge Bosons 4.1 Gauge Bosons After EWSB has occurred some techni-gluons and EW gauge bosons acquire mass. The corresponding sector of the MSCT Lagrangian can be written as a function of the mass eigenstates as: −µ + −Lg-mass = gT2 C vT2 C G+ µG where
+ −µ gL2 + gY2 2
gL2 2 2 2 Wµ W + Zµ Z µ (4.1) vT C + vH vT C + vH 2 4

1 gY 1 G± G1µ ∓ i G2µ , Wµ± = √ Wµ1 ∓ i Wµ2 , Zµ = cw Wµ3 − sw B , tw = . (4.2) µ = √ gL 2 2 The ± exponent of the techni-gluon refers to the U (1)T C charge, while the ± exponent on the EW gauge bosons refer to the usual EM charge. The remaining, massless states are the techni-photon and the EW photon: Gµ = G3µ , Aµ = sw Wµ3 + cw B . (4.3) The phenomenological constraints on a new U (1) massless gauge boson were studied in [42] and found to be phenomenologically viable. The tree-level masses of G, W and Z can be read off from Eq.(4.1): q mW gL 2 , mG = gT C vT C , mW = vT2 C + vH mZ = . (4.4) 2 cw From these masses and the eigenstates in Eq.(4.2) it is immediate to evaluate the EW oblique parameters at tree level by using the formulas in [43]: we find S = T = U = 0 at tree level. 4.2 Fermions The fermion mass terms are:
T
1 0
T + T M c χ− Mtc χ− χ M n χ0 + χ+ tc + χ tc 2
T −−
T −+ + mtc-c χ++ χtc + m̄tc-c χ+− χtc + mcc χ++ χ−− + c.c. , tc tc −Lf -mass = where5 0 χ = + χ = χ±+ tc =  3 H2 , H1′ , W̃3 , B̃, DL3 , D̄R  W̃1 − i W̃2 3 √ H1 , , UL , N̄R 2 UL1 ∓ i UL2 √ , 2 χ±− tc = , ! χ± tc − , = χ = ŪR1 ∓ i ŪR2 √ , 2 5  1 ∓ i D̄ 2 DL1 ∓ i DL2 D̄R R √ √ , 2 2 H2′ , W̃1 + i W̃2 3 √ , ŪL , NL 2 χ++ = ĒR , χ−− = EL , (4.5)  ! , , (4.6) Notice that a tc subscript here and in the following indicates that the leftmost superscript ± refers to the techni-charge under U (1)T C 7 4.3 Scalars and, at tree-level,   0 −2µ isβ gL vH −isβ gY vH 0 0  −2µ 0 −icβ gL vH icβ gY vH 0 0     1  isβ gL vH −icβ gL vH 2MW̃ 0 igL vT C 0   Mn =   , 2  −isβ gY vH icβ gY vH 0 2MB̃ −igY vT C 0     0 0 igL vT C −igY vT C 0 0  0 0 0 0 0 2MD Mtc =  0 −igT C vT C igT C vT C MD ! , (4.7) (4.8) √  2µ −isβ gL vH −yU vT C 0 √  1  2MW̃ 0 0  −icβ gL vH  Mc = √   ,  0 −igL vT C yU sβ vH 0 2 0 0 0 yN s β v H (4.9) yE c β v H yU s β v H √ , mcc = √ . (4.10) 2 2 In the previous equations with the labels n, tc, c, tc-c, cc, we referred to, respectively, neutralinos, techni-neutralinos, charginos, techni-charginos, and doubly-charged chargino. Furthermore the barred fields indicate Hermitian conjugation while a tilde indicates the fermion superpartner of the corresponding gauge boson. MW̃ and MB̃ correspond to the wino and the bino soft masses, respectively. It is important for the phenomenological bounds on MSCT to notice that at tree-level, from the last equation, mtc-c = −igT C vT C + mt = yt tβ mcc , yE (4.11) where the subscript t here refers to the top quark. 3 The squared masses are obtained diagonalizing Mp M†p , p = n, tc, c. We note that D̄R has become the gaugino of the residual U (1)T C with mass MD . For illustration we provide the explicit form of the techni-neutralino masses obtained diagonalizing the seesaw-like matrix in Eq.(4.8): r 2 MD MD + gT2 C vT2 C ∓ . (4.12) mtc0,1 = 4 2 4.3 Scalars 4.3.1 Tree-Level The complete potential is given by V = VT C + VM SSM , VT C = −LD − LF − Lsof t −  1 a a D̄R + c.c. MD D̄R 2  , (4.13) where VM SSM can be found in [41], while LD , LF , and Lsof t , are given in Appendix A. As for the SM fermions also the scalar superpartners do not mix, at the tree-level, with the 8 4.3 Scalars N = 4 techni-scalars or heavy scalar leptons. Therefore their mass spectrum assumes the same form as in the MSSM. The Higgs scalar fields, H̃ and H̃ ′ , on the other hand, mix with the techni-scalars. The squared mass matrices of the CP-even and -odd EM neutral Higgs scalars are given by, respectively,  


2 2 2 + 4bt−1 −c 2 gL2 + gY2 s2β vH gL2 + gY2 sβ vH vT C β gL + gY sβ vH − 4b β


1  2 − 4b c2 g 2 + g 2 v 2 + 4bt 2 2 M2h =  −cβ gL2 + gY2 sβ vH β −cβ gL + gY vH vT C  , L Y H β


4 gL2 + gY2 sβ vH vT C −cβ gL2 + gY2 vH vT C gL2 + gY2 vT2 C  
∂2V 3 ′ h , (4.14) H̃ , H̃ , D̃ , φ = ℜ M2h ij = 2 1 L ∂φhi ∂φhj φ=hφi and   bt−1 b 0 β
∂2V   M2A =  b btβ 0  , M2A ij = A ∂φA i ∂φj 0 0 0   φA = ℑ H̃2 , H̃1′ , D̃L3 (4.15) , φ=hφi From Eqs.(4.14,4.15) the squared masses of the CP-even and -odd Higgs scalars are m2h0 0 = m2A0 = 0, m2h0 1,2 1 = 2 m2A1 m2Z + ∓ r m2A1 − m2Z 2 + 4m2A1 m2B s22β ! 2b , s2β (4.16) , m2A1 = where we have defined the quantity: m2B = gY2 + gL2 2 vH 4 (4.17) which does not correspond to any particle. In the limit vT C = 0, however, mB = mZ and one recovers the MSSM results for the masses of the CP-even Higgs scalars. The massless eigenstates h00 , πZ (the longitudinal degree of freedom of the Z boson), and A0 , are expressed by h00 =Nh (sβ vT C , cβ vT C , c2β vH ) · φh , πZ =NZ (sβ vH , −cβ vH , vT C ) · φA , A A0 =NA (sβ vT C , −cβ vT C , −vH ) · φ , 2 , Nh−2 = vT2 C + c22β vH (4.18) 2 , NZ−2 = vT2 C + vH (4.19) NA−2 (4.20) = vT2 C + 2 vH , with φh,A defined respectively in Eqs. (4.14,4.15). The masslessness of h00 and A0 will not survive at the one-loop level. The remaining scalar squared mass matrices are given in Appendix B. By using these results and those given in Eqs.(4.4,4.7,4.14,4.15), and taking into account the multiplicities of each mass matrix, we can calculate the supertrace of the tree level squared mass matrices, defined by X 1 STrM2 = (−1)2j (2j + 1) TrM2j , j = 0, , 1 , (4.21) 2 j 9 4.3 Scalars where Mj are the complete squared mass matrices of scalars, femions, and gauge bosons. We obtain:   2 2 2 2 + ME2 + 6MQ + 3MU2 , + 2m2d + 2m2u + 2ML2 + MN − 3MD STrM2 = 2 −MB̃2 − 3MW̃ (4.22) where the numerical factors in front of the SUSY breaking squared mass parameters reflect the degrees of freedom of the corresponding fields. The equation above shows that the SUSY invariant contributions to the squared mass matrices cancel out, as they should. 4.3.2 One-Loop We calculate the one-loop contributions to the CP-even and -odd neutral (both under U (1)EM and U (1)T C ) scalars. We expect the lightest eigenstates, h00 and A0 , that are accidentally massless at tree level, to receive non-zero contributions to their masses from the one-loop effective potential. The one loop potential is [44]:     1 M2 (φ) 3 4 2 2 ∆V1 = (4.23) − STr M (φ) ln + 2M (φ) µ r , 64π 2 µ2r 2 where M2 (φ) are field-dependent mass matrices not evaluated at their vevs, defined by: M2 (φ)
ij = ∂2V , ∂φi ∂φj (4.24) and µr is the renormalization scale. The last term in Eq.(4.23) renormalizes the one-loop contributions to the scalar masses to zero when µ2r = M2 (hφi).6 The last term in Eq.(4.23) gives a very small contribution to ∆V1 since only the SUSY breaking terms (generally small to avoid a large fine tuning) do not cancel in the supertrace, and therefore we neglect it. To minimize the correction from higher order contributions to V, we take µr equal to the mass of the heaviest particle among the eigenstates presented in Sections 4.1, 4.2, and the last subsection. The one-loop mass matrix correction, ∆M2a , for any real field a with n components can be extracted from ∆V1 by numerically evaluating the derivatives of the mass eigenvalues with respect to the fields evaluated on the vevs [45], where ∂ 2 ∆V1 (a) + ∆Mij2 , (4.25) ∂ai ∂aj a=hai  2 2  X 1 ∂m2 ∂m2 X 1 m2k m2k 2 ∂ mk k k = ln 2 m ln 2 − 1 , + 32π 2 ∂ai ∂aj µr a=hai 32π 2 k ∂ai ∂aj µr a=hai (∆M2a )ij = ∂ 2 ∆V1 (a) ∂ai ∂aj a=hai k k δij ∂∆V1 (φh ) ∆Mij2 = − h φi ∂φhi =− X k φh =h φh i 2 1 2 δij ∂mk m 32π 2 k φhi ∂φhi  m2 ln 2k − 1 µr 6  . (4.26) φh =h φh i In case there is more than one field, one should use different scales µr for each contribution to the supertrace to get an exactly vanishing one-loop correction to the mass. 10 The second term in Eq.(4.25) takes into account the shift in the minimization conditions (see [45]), and m2k is the set of mass eigenvalues of the field dependent mass matrix M2 (φ). Notice that ∆Mij2 has to be included in the expression of (∆M2a )ij only when ai are the CP-even or -odd Higgses, since ∆Mij2 gives the shift of the soft mass parameters of the scalar fields that develop a non-zero vev. The Goldstone bosons do not contribute to ∆M2a . In this first estimate we compute ∆M2 for the neutral Higgses neglecting the contributions from top-stop mass splitting. We consider the fields given in Table 1, plus the W and B bosons and their superpartners. In this way the supertrace receives contributions only from the soft mass terms. We therefore consider our results for the one-loop masses of the CP-even and -odd Higgses an estimate of the values that can be obtained when taking into account the full MSCT spectrum. It is seen that except for the ordinary EM neutral Goldstone boson which can be interpreted as the longitudinal component of the Z boson no other neutral scalar is massless. The mass of the lightest physical states, h00 and A0 , has a strong dependence on the size of the Yukawa couplings in the superpotential, Eq.(2.6). A random scan of the parameter space, with the constraint that the SUSY breaking scale, given in Eqs.(A.13,3.2), is around the TeV region and with π/4 < β < π/2, gives:7 : mh00 = 10.6 ± 5.5 GeV , g T C = yU = yN = yE = yR = 1 , mh00 = 125 ± 54 GeV , g T C = y U = y N = yE = yR = π . (4.27) From the scan we read off the central value for the mass of the lightest Higgs and the associated standard deviation. The latter represents the spread in the distributions of the values of the parameters. We have also tried to reach a larger value of the masses by optimizing the search around the maximum value of the initial sample of parameters and obtain in this case mmax = 30.5 GeV and mmax = 276 GeV for the same choice of Yukawas h00 h00 above. The mass of A0 for the parameter values that maximize mh00 is mA0 = 8 (27) GeV for gT C = ... = 1 (π). It is interesting to notice also that mA0 = 0 for aT C = 0: consequentially in the following we take the soft parameter aT C to be rather large (though still within the TeV region). In the following section we impose the experimental bounds on the mass spectrum to determine its phenomenological viability and use the renormalization group equations to determine the perturbative range of our results. 5. Phenomenological Viability The lower bounds on the mass of the lightest neutralino and chargino are [46]: mχ00 > 46 GeV , mχ± > 94 GeV . 0 (5.1) These limits refer to the MSSM, but are rather general, since they are extracted mostly from the Z decay to neutralino-antineutralino pair the former, and from photo-production 7 We have checked that the potential is actually at a minimum and have considered two sample values of the Yukawa couplings. 11 of a chargino-antichargino pair at LEPII the latter. We can therefore assume these limits to hold also for the MSCT. Because of their generality and independence from the coupling strength (as long as it is not negligible), we use the lower bound on the chargino mass also for the mass of the doubly-charged chargino E. The presence of the term proportional to yR in the superpotential, Eq.(2.6) allows it to decay into singly charged ordinary particles. Therefore it escapes cosmological constraints 3 is an EW singlet fermion and therefore on charged stable particles. The techni-gaugino D̄R is a right-handed neutrino, which can be very light. Because of this, and the fact that the mass of the lightest techineutralino, Eq.(4.12), is a monotonically decreasing function of 3 , we assume M to be small with respect to the g the mass of D̄R D T C vT C energy scale. Other useful limits on the parameters are obtained by using the fact that the smallest eigenvalue of a semi-positive definite square matrix is smaller or equal to any eigenvalue of the principal submatrices. From the absolute square of the neutralino mass matrix, Eq.(4.7), we get MB̃2 > (46 GeV)2 −
gY2 2 vH + vT2 C = (13.5 GeV)2 , µ > 46 GeV , 4 46 GeV vTC > 2 q = 124 GeV , vH < 213 GeV , gL2 + gY2 (5.2) where we used, from Eq.(4.4), q 2 + v 2 = 246 GeV. vH TC (5.3) From the absolute square of the chargino mass matrix and the doubly-charged chargino mass, Eqs.(4.9,4.10), we get yE c β v H 1 2 2 √ > 94 GeV . = (63.5 GeV)2 , > (94 GeV)2 − c2β gL2 vH MW̃ 2 2 (5.4) From Eq.(4.11), with mt = 173 GeV, and the bounds (5.2,5.4), it follows that v u 1 173 u . yt > t 213 1 − 2942 2 (5.5) 2 yE 213 This last bound is plotted in Figure 1, where the shaded area shows the values of yt and yE excluded by the experiment: it is evident from the plot in Figure 1 that either yt or yE is constrained to be larger than about 1.3.8 To further study the phenomenological viability of the pMSCT spectrum we now analyze the evolution of the couplings using the two-loop renormalization group equations (RGE) given in Appendix C. In this calculation we assume a generic Yukawa coupling 8 Had we chosen the hypercharge parameter y=-1 rather than 1, the constraints in Eqs.(5.2,5.4,5.5) would be the same with yE and yN interchanged. Although a more detailed study would be necessary, we expect that the choice y = −1 produces the same general results and conclusions that we present in this paper for y=1. 12 2.0 1.8 yt 1.6 1.4 1.2 1.0 0.8 0.8 1.0 1.2 1.4 1.6 1.8 2.0 yE Figure 1: Shaded area shows experimentally excluded values of the Yukawa couplings yt and yE . yT C in place of gT C in Eq.(2.6). We find that typical behavior for the phenomenologically favored large Yukawa couplings is that most of them flow to an ultraviolet fixed point, while the remaining ones, such as yT C , flow toward zero. The fixed point behavior begins rather quickly, as the largest couplings reach their fixed point value y⋆ ≃ 6 at around 2 TeV. At two-loops the technicolor gauge coupling gT C decreases as a function of increasing scale, but the evolution is very slow in comparison to the Yukawa couplings. We also find that at two loops gT C = yT C is an infrared fixed point, in agreement with the findings in [47]. We scanned the parameter space of the model for Yukawa couplings that delay the onset of the fixed point, while satisfying the neutralino and chargino mass limits, Eq. (5.1), and maximizing the mass of the CP even Higgs scalar9 . The dimensionful soft SUSY breaking parameters were taken to be less than 1 TeV. In Figure 2 are plotted gT C , yT C , yU , yt , yN , yE as a function of the renormalization scale M : the couplings are normalized for M = mZ to yN = 1.8, gT C = yT C = yU = yt = 2.3, yE = 2.4. For such values of the Yukawa couplings we can achieve a phenomenologically viable spectrum: max mmax χ0 = 47 GeV , mχ± = 96 GeV , mh00 = 95 GeV , mA0 = 32 GeV . 0 (5.6) 0 While the running of the couplings described above maintains and improves perturbativity in gT C for increasing energy, the Yukawa couplings yU , yN , yt , responsible for the mass of the heavy upper components of weak doublets, increase and flow close to an ultraviolet fixed point at around 2 TeV. This behavior is general, since it is caused by the large anomalous dimension of the up-type Higgs. This, in turn, results from its coupling to the technicolor sector. 9 Mass limits on the scalars from studies of the MSSM are not applicable to this model. Both the CP even and odd scalars result from a mixing of the MSSM Higgses with the techni-Higgs, thereby reducing couplings compared to the MSSM. A more detailed study would be necessary to determine the actual phenomenologically viable parameter space. 13 12 yU yN 10 yt Coupling strength yTC 8 yE gTC 6 4 2 0 100 104 106 108 1010 M HGeVL Figure 2: Plot of gT C , yT C , yU , yt , yN , yE as a function of the renormalization scale M : the couplings are normalized for M = mZ to yN = 1.8, gT C = yT C = yU = yt = 2.3, yE = 2.4. One may expect important non-perturbative contributions from the largest Yukawa couplings. At the same time, pMSCT should present clear signatures in processes in which the cross section depends mainly on any of the yU , yN , yt couplings. 6. Conclusions and Outlook We have investigated the perturbative regime of MSCT, called pMSCT. In pMSCT the technicolor gauge coupling is taken to be large but still perturbative. In this regime the SUSY breaking scale is constrained by naturalness requirements to be at the TeV scale. First we showed that the model allows for a stable vacuum, in which the EW symmetry is broken by expectation values of the MSSM Higgses and the left-handed technisquark doublet. We then reduced the parameter space by imposing naturalness of the couplings and masses, one loop vacuum stability, perturbativity at the EW scale, and experimental constraints. The technisquark vev contributes to the masses of the EW gauge bosons but not to the masses of quarks and leptons. Therefore in pMSCT all Yukawa couplings must be larger than in the (MS)SM. By running the two loop renormalization group equations of the dimensionless couplings, for initial conditions compatible with phenomenological viability, we found that the up-type Yukawa couplings flow to an ultraviolet fixed point at about 2 TeV. The remaining Yukawa couplings go to zero with increasing energy, while the gauge coupling gT C decreases very slowly. Our preliminary conclusion is that if we require perturbative behavior of the Yukawa sector at TeV scales, the Tevatron and the LHC can rule out a significant portion of the parameter space of this model. 14 There are many possible interesting signatures of pMSCT in collider experiments. Compared to the MSSM, pMSCT features several new light states, such as the doubly charged particles. Therefore a possible signature would be a doubly charged particle together with a light chargino and/or neutralino. Also, since the Yukawa couplings are larger than in the SM, we expect O(100%) increase of several production cross sections, such as the Higgs scalar (h00 ) via the gluon-gluon fusion process. Another characteristic of the model is that due to the presence of one extra Higgs-type particle coming from the supertechnicolor sector, the spectrum features scalars and pseudoscalars lighter than in the MSSM. In the future we plan to explore the processes relevant for collider experiments, as well as dark matter phenomenology, which will be substantially different than in the MSSM. Since our model features a new N = 4 supertechnicolor sector at the EW scale, collider experiments have the possibility to explore string theory directly. This is because the new scalars coming from this sector can be directly identified with the extra six space coordinates of ten dimensional supergravity. This link is even more clear when considering the present supertechnicolor sector in the nonperturbative regime which can be investigated using AdS/CFT techniques. Acknowledgments We would like to thank Matti Järvinen for useful discussions, and R. Sekhar Chivukula for valuable comments. 15 A. MSCT Lagrangian The Lagrangian of a supersymmetric theory can, in general, be defined by L = Lkin + Lg−Y uk + LD + LF + LP −Y uk + Lsof t , (A.1) where the labels refer to the kinetic terms, the Yukawa ones given by gauge and superpotential interactions, the D and F scalar interaction terms, and the soft SUSY breaking ones. All these terms can be expressed in function of the elementary fields of the theory with the help of the following equations: 1 a a µ a a − iλ̄aj σ̄ µ Dµ λaj − Dµ φa† Lkin = − Fjµνa Fjµν i Dµ φi − iχ̄i σ̄ Dµ χi , 4   X √ Lg−Y uk = i 2gj φ†i Tja χi λaj − λ̄aj χ̄i Tja φi , (A.2) (A.3) j LD = − 1 X 2  † a 2 , g j φ i Tj φ i 2 (A.4) j ∂P 2 , ∂φai  2  1 ∂ P a b =− χ χ + h.c. , 2 ∂φai ∂φbl i l LF = − LP −Y uk (A.5) (A.6) where i, l run over all the scalar field labels, while j runs over all the gauge group labels, and a, b are the corresponding gauge group indices. Furthermore, we normalize the generators in the usual way, by taking the index T (F ) = 21 , where TrTRa TRb = T (R)δ ab , with R here referring to the representation (F =fundamental). The SUSY breaking soft terms, moreover, are obtained by re-writing the superpotential in function of the scalar fields alone, and by adding to it its Hermitian conjugate and the mass terms for the gauginos and the scalar fields. We refer to [41] and references therein for the explicit form of LM SSM in terms of the elementary fields of the MSSM, and focus here only on LT C . The kinetic terms are trivial and therefore we do not write them here. The gauge Yukawa terms are given by   √ ˜ c abc ˜ b Dc D̄a − Da D̄b D̃c + Ũ b Ū c D̄a − Da U b Ū ˜ b U c D̄a − Da Ū b Ũ c + D̄ Lg−Y uk = 2gT C Ū R R R ǫ R R R R L L L L R R L L L L R  gL  ˜ i j k k ˜k i j ˜i j k ˜k i j + i √ Q̄ L QL W̃ − W̄ Q̄L Q̃L + L̄L LL W̃ − W̄ L̄L L̃L σij 2  X  √ ˜ χ̄ χ̃ , χ = U a , Da , Ū a , N , E , N̄ , Ē , ˜ χ B̃ − B̄ + i 2g Y χ̄ (A.7) p Y p p p p p L L R L L R R p where W̃ k and B̃ are respectively the wino and the bino, σ k the Pauli matrices, i, j = 1, 2; k, a, b, c = 1, 2, 3; and the hypercharge Yp is given for each field χp in Table 1. 16 The D terms are given by LD = −  1  1 2 , gT C DTa C DTa C + gL2 DLk DLk + gY2 DY DY + gL2 DLk DLk + gY2 DY DY 2 2 M SSM (A.8) where k     ˜ c , Dk = σij Q̄ ˜ b D̃c + Ũ b Ū ˜ b Ũ c + D̄ ˜ i a Q̃j a + L̄ ˜ i L̃j + Dk DTa C = −iǫabc Ū L R R L L L L L,M SSM L L L L 2 X ˜p χ̃p + DY,M SSM . DY = Yp χ̄ (A.9) p k In these equations the DL,M SSM and DY,M SSM auxiliary fields are assumed to be expressed in function of the MSSM elementary fields [41]. The rest of the scalar interaction terms10 is given by     b ˜c 2 ˜ b D̃c ˜ b Ũ c + D̄ ˜b c ˜c ˜ b + D̃b D̄ ˜b ˜ b b 2 − Ũ b Ū Ū LF = −gT C ŨLb Ū L L L L L L + D̃L D̄L + ŪR ŨR L L L + ŪR ŨR     h i a ˜a ˜ ˜ ˜ ˜a ˜a ˜ D̄ ˜ c − y 2 H̃ D̃a − H̃ Ũ a H̄ + ŨRb Ū 1 L − H̄2 ŪL + ŨR ŪR H̃1 H̄1 + H̃2 H̄2 1 L 2 L U R  h i   2 ˜ H̄ ˜ − Ē ˜ H̄ ˜ ˜ a D̃b ˜ a Ũ b + D̄ ˜ b Ū N̄ Ñ H̃ − Ẽ H̃ − y + ŨRa Ū L 1 L 2 L 1 L 2 N L L L L R    i h ′ ′ 2 ˜ H̃ H̄ ˜ + H̃ H̄ ˜ + Ñ N̄ ˜ + Ẽ Ē ˜ ˜ H̄ ˜ ′ − Ē ˜ H̄ ˜′ − Ẽ H̃ H̃ + ÑR N̄ Ñ N̄ − y 1 1 2 2 R L L L L L 2 L 1 L 2 L 1 E   i  2 ′ ˜′ ˜a a ˜ ˜b ˜ b Ū ˜ ˜ ˜′ ˜ H̃ ′ H̄ ŨRa ŨRa Ū − yR + ẼR Ē R R R + 4ŪR ŨR ĒR ẼR 1 1 + H̃2 H̄2 + ÑL N̄L + ẼL ĒL n√  i   h c ˜ c ˜ a ˜b ˜ D̄ ˜a ˜ ˜a + 2yU gT C ǫabc ŨLb D̃Lc H̄ 1 L − H̄2 ŪL + ŨR ŪR ŨL H̄1 + D̃L H̄2     ˜ Ū ˜ a Ẽ − y y Ñ Ē ˜ a Ñ + D̄ ˜ ˜ ′ ˜ ′ − yU yN ŨRa N̄ R N E R R H̄1 H̃1 + H̄2 H̃2 L L L L h i    ˜ a 2√2g ǫabc Ū ˜ b D̄ ˜c ˜ ˜ ˜ ˜′ ˜a ˜ ˜a ˜ ˜a ˜ ˜′ + yR Ū TC R L L ĒR + 2yU ĒR D̄L H̄1 − ŪL H̄2 + yE ŪR ĒL H̄1 − N̄L H̄2 + h.c.} + Lmix , (A.10) with Lmix defined in function of the F auxiliary fields associated with the MSSM two Higgs super-doublets:  X ˜ , Lmix = − Fφp ,T C Fφ†p ,M SSM + h.c. , φp = H1′ , H2′ , H1 , H1 , FH1′ ,T C = −yE ẼL Ē R φp a ˜a a ˜a ˜ , F ˜ ˜ FH2′ ,T C = yE ÑL Ē R H1 ,T C = −yU D̃L ŪR − yN ẼL N̄R , FH2 ,T C = yU ŨL ŪR + yN ÑL N̄R . (A.11) The corresponding MSSM auxiliary fields F can be found in [41] and references therein. Also, in the Eqs.(A.10,A.11) we used H̃ and H̃ ′ to indicate the scalar Higgs doublets, for consistency with the rest of the notation where the tilde identifies the scalar component of a chiral superfield or the fermionic component of a vector superfield. The remaining 10 We consider the constants in the superpotential to be real to avoid the contribution of CP violating terms. 17 Yukawa interaction terms are determined by the superpotential, and can be expressed as √  h  ˜a ˜ c + U a D̃b Ū c + Ũ a Db Ū c + y (H Da − H U a ) Ū 2gT C ǫabc ULa DLb Ū 1 L 2 L U R L L R L L R R h  i    ˜ + H̃1 DLa − H̃2 ULa ŪRa + H1 D̃La − H2 ŨLa ŪRa + yN (H1 EL − H2 NL ) N̄ R     i h
˜ + H1 ẼL − H2 ÑL N̄R + H̃1 EL − H̃2 NL N̄R + yE H1′ EL − H2′ NL Ē R       i ¯ a Ē ˜ + Ũ + H1′ ẼL − H2′ ÑL ĒR + H̃1′ EL − H̃2′ NL ĒR − yR ŪRa ŪRa Ē R R R LP −Y uk = + h.c.. (A.12) The soft SUSY breaking terms, finally, can be written straightforwardly starting from the superpotential in Eq.(2.6), to which we add the techni-gaugino and scalar mass terms as well:    h  ˜ ˜ c + a H̃ D̃a − H̃ Ũ a Ū ˜a + a H̃ Ẽ − H̃ Ñ N̄ Lsof t = − aT C ǫabc ŨLa D̃Lb Ū 1 2 1 2 N L L R U R R L L    1 2 ˜a a a a ˜ + a Ū ˜a ˜a ˜ + aE H̃1′ ẼL − H̃2′ ÑL Ē R R R ŪR ĒR + MD D̄R D̄R + c.c. − MQ Q̄L Q̃L 2 ˜ L̃ − M 2 N̄ ˜ Ñ − M 2 Ē ˜ Ẽ . ˜ a Ũ a − M 2 L̄ (A.13) − M 2 Ū U R R L L L R N R R E R B. Scalar Squared Mass Matrices The techni-Higgs squared mass matrix is M2tc-h 1 = 2 φtc-h = ℜ gT2 C vT2 C −gT2 C vT2 C −gT2 C vT2 C gT2 C vT2 C D̃L1 − √ iD̃L2 2 , D̃L1 + √ ! iD̃L2 2 , M2tc-h !
ij = ∂2V ∂φtc-h ∂φtc-h i j , mhT C = gT C vT C . , φ=hφi (B.1) The massless eigenstate in the last matrix is the longitudinal degree of freedom of the techni-photon G in Eq.(4.3): 1 πT C = √ (1, 1) · φtc-h . 2 (B.2) The charged-Higgs squared mass matrix is M2h± = M2hc 0 0 M2hl 18 ! , (B.3) M2hc M2hc M2hc M2hc M2hc M2hc M2hl M2hl M2hl








11 = 13 = 22 = 24 = 34 = 44 = 11 = 12 = 22 = M2h±


1 1 2 2 sβ − vT2 C gL2 − 2yU2 , M2hc 12 = b + cβ gL2 vH 4bctβ + c2β gL2 vH 4 4

1 aU v T C , vH sβ vT C gL2 − 2yU2 , M2hc 14 = − √ 4 2

1 1 2 2 sβ + vT2 C , M2hc 23 = cβ gL2 vH vT C , btβ + gL2 vH 4 4

1 2 µvT C yU 2 2 − √ , Mhc 33 = vH c2β gL + 2s2β yU2 , 4 2 1 √ vH (aU sβ − µcβ yU ) , 2


1 2 2 2 2 − vT2 C + 2yU2 vH sβ + vT2 C + 4MU2 , gY c2β vH 4

1 1 2 2 2 2 2 ML2 + s2β vH , yN + − vTC gL + 3gY2 c2β vH 2 8 1 √ vH (aN sβ − µcβ yN ) , 2

1 2 2 1 2 2 2 + vH , (B.4) yN − c2β gY2 + yN + gY2 vTC MN 4 4
ij = ∂2V ± ± ∂φhi ∂φhj φ=hφi   ± ˜ 3 , Ñ , N̄ ˜ , . , φh = ℜ H̃1 , H̃2′ , ŨL3 , Ū L R R (B.5) The massless eigenstate in the Hermitian matrix M2hc , Eq(B.4), is the longitudinal degree of freedom of the W gauge boson: ± −2 2 πW = NW (sβ vH , −cβ vH , vT C ) · φh , NW = vT2 C + vH . (B.6) The remaining eigenvalues of M2hc and those of M2hl are all non-zero: they have rather lengthy and not particularly instructive expressions, and therefore we do not write them here. The techni-charged Higgs squared mass matrix is M2tc-h± M2d M2d

M2o 11 22
ij φtc-h ± = M2d −M2o M2o M2d ! , (B.7)

1 1 1 1 2 2 2 c2β gL2 vH + s2β yU2 vH − gL − 4gT2 C vT2 C , M2d 12 = √ vH (aU sβ − µcβ yU ) 4 2 4 2

1 1 1 2 2 2 yU + c2β vH gY2 − yU2 , 4gT2 C − gY2 vT2 C + vH = MU2 + 4 4 4
1 ∂2V = √ aT C vV C ǫij , M2tc-h± ij = , ± ± ∂φtc-h ∂φtc-h 2 i j φ=hφi ! ! ˜ 1 + iŪ ˜2 [ ˜ 1 + iŪ ˜2 ŨL1 − iŨL2 Ū ŨL1 − iŨL2 Ū R R R R √ √ √ √ . (B.8) =ℜ , ℑ , 2 2 2 2 = 19 The doubly charged-Higgs squared mass matrix is M2h2± M2h2±

11 12

1 1 2 2 2 2 2 − vTC , (B.9) = ML2 + c2β vH yE − gL − 3gY2 c2β vH 2 8  
1 1 1 1 2 2 2 2 = √ vH (µsβ yE − aE cβ ) , M2h2± 22 = gY2 + ME2 + c2β vH vTC − c2β vH yE 2 2 2 2 M2h2±
ij = M2h2± ∂2V 2± 2± ∂φhi ∂φhj , φh φ=hφi 2±   ˜ , . = ℜ ẼL , Ē R (B.10) M2tc-h± The eigenvalues of and are all non-zero: they have rather lengthy and not particularly instructive expressions, and therefore we do not write them here. C. Renormalization Group Equations In the following we write the two loop beta functions [48] of the gauge couplings. Notice that while the one loop beta function of gT C is zero the running of the coupling at two loops is non-trivial. 1 (1) 1 dga = βa + βa(2) ; g1 = gY , g2 = gL , g3 = gC , g4 = gT C ; t = log (E/mZ ) ; 2 dt 16π (16π 2 )2 (C.1) (1) 3 β1 = 15g1 , (C.2) (2) g1−3 β1 = − 42 2 26 2 108 2 54 78 2 1297 2 81 2 88 2 108 2 yN − y t − yTC − yU2 − yE + g + g + g + g , (C.3) 5 5 5 5 5 25 1 5 2 5 3 5 4 (1) β2 = 3g23 , (C.4) (2) 2 2 2 + 39g23 + − 6yU2 − 2yE − 6yt2 − 12yTC g2−3 β2 = −2yN 27 2 g + 24g32 + 12g42 , 5 1 (1) β3 = −3g33 , (2) g3−3 β3 = −4yt2 + 14g32 + (C.5) (C.6) 11 2 g + 9g22 , 5 1 (1) β4 = 0, (C.7) (C.8) 36 2 g + 12g22 . (C.9) 5 1 In the following we write the beta functions at two loops of the Yukawa couplings appearing in the superpotential Eq.(2.6) and of that of the top quark. Notice that we substituted gT C in the superpotential with yT C , since their respective beta functions are indeed different, and assumed yR = 0, as we did in the rest of the paper. All the beta functions below are divided by the respective Yukawa coupling. (2) 2 g4−3 β4 = −48yTC − 16yU2 + 48g42 + yp−1 dyp 1 ′(1) 1 = βp + βp′(2) ; p = T C, U, N, E, t ; 2 dt 16π (16π 2 )2 ′(1) βT C = − 9g12 2 − 3g22 − 12g42 + 12yTC + 4yU2 , 5 20 (C.10) (C.11) REFERENCES ′(2) 36 2 2 6 1431g14 9 2 2 72 2 2 27g24 2 2 + 48g42 yTC + g12 yU2 + 6g22 yU2 + g1 yTC + 12g22 yTC + g2 g1 + g4 g1 + + 48g44 5 5 50 5 5 2 2 4 2 2 yU2 − 96yTC − 18yU4 , (C.12) + 24g22 g42 − 4yN yU − 12yt2 yU2 − 48yTC βT C = ′(1) βU ′(2) βU =− 9g12 2 2 + 3yt2 + 8yTC + 6yU2 , − 3g22 − 8g42 + yN 5 18 2 2 12 18 4 2 2 2 + 32g42 yTC + g12 yU2 + 6g22 yU2 + 24g42 yU2 + 12g22 yTC g y + g 2 y 2 + 16g32 yt2 + g12 yTC 5 1 N 5 1 t 5 5 4 4 27g2 9 1431g1 2 2 4 2 2 + yN − 9yt2 yU2 − 9yt4 − yE yU − 3yN + g22 g12 + 12g42 g12 + + 32g44 + 12g22 g42 − 3yN 50 5 2 4 2 (C.14) − 22yU4 , − 56yTC yU2 − 64yTC = ′(1) βN = − ′(2) βN 21g12 2 2 , − 3g22 + 4yN + 3yt2 + 3yU2 + yE 5 39g12 2 2 − 3g22 + yN + 4yE , (C.17) 5 7371g14 27g24 6 4 2 2 2 4 2 2 2 2 2 2 , + yN −10yE −3yE +6g22 yE yU −3yN +6g12 yE yt −3yN +9g22 g12 + −3yN = − g12 yN 5 50 2 (C.18) 2 16g 13g12 ′(1) 2 3 − 3g22 − + yN + 6yt2 + 3yU2 , (C.19) βt = − 15 3 ′(1) ′(2) ′(2) βt (C.15) 4 18 12 3591g14 27g24 2 2 2 + g12 yt2 + 16g32 yt2 + g12 yU2 + 24g42 yU2 + g12 yE + 6g22 yN = 6g12 yN + + 9g22 g12 + 5 5 5 50 2 2 2 2 2 4 2 2 2 4 − 9yN yt − 9yN yU − 10yN − 3yE yN − 9yt4 − 24yTC yU2 − 9yU4 − 3yE , (C.16) βE βE (C.13) =− 18 2 2 6 18 6019g14 136 2 2 27g24 g1 yN + g12 yt2 + 6g22 yt2 + 16g32 yt2 + g12 yU2 + 24g42 yU2 + + g22 g12 + g g + 5 5 5 450 45 3 1 2 16g34 2 2 4 2 2 2 + 8g22 g32 − 3yN yt − 3yN − yE yN − 9yt2 yU2 − 22yt4 − 24yTC yU2 − 9yU4 . 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